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\begin{document}
\begin{frontmatter}
\title{\textbf{\LARGE{On the number of hyperelliptic limit cycles of Li\'{e}nard systems}}}
\author{\textbf{Xinjie Qian }}
\cortext[]{Corresponding author.}
\ead{[email protected]}
\author{ \textbf{Jiazhong Yang}}
\ead{[email protected]}
\address{School of Mathematical Science, Peking University, 100871 Beijing, P. R. China}
\begin{abstract}
In this paper, we study the maximum number, denoted by $H(m,n)$, of hyperelliptic limit cycles
of the Li\'{e}nard systems
$$\dot x=y, \qquad \dot y=-f_m(x)y-g_n(x),$$
where, respectively, $f_m(x)$ and $g_n(x)$ are real polynomials of degree $m$ and $n$, $g_n(0)=0$.
The main results of the paper are as follows: We obtain the upper bound and lower bound of $H(m,n)$ in all the cases with $n\neq 2m+1$. When $n=2m+1$, we derive the lower bound of $H(m,n)$. Furthermore, these upper bound can be reached in some cases.
\end{abstract}
\begin{keyword}
Hyperelliptic Limit Cycles \sep Li\'{e}nard Systems \sep Configuration
\end{keyword}
\end{frontmatter}
\section {Introduction}
Consider the following Li\'{e}nard differential system
\begin{equation}\label{1}
\dot x=y, \quad\quad\dot y=-f_m(x)y-g_n(x),
\end{equation}
where $f_m(x)$ and $ g_n(x)$ are polynomials of degrees $m$ and $n$, respectively, with the following explicit expressions
$$f_m(x)=\sum_{i=0}^ma_ix^i, \qquad \ g_n(x)=\sum_{i=1}^nb_ix^i, \quad a_mb_n\not=0.$$
We shall call this system a {\it Li\'{e}nard system of type $(m,n)$},
or simply a Li\'{e}nard system if no confusion arises.
This paper is primarily devoted to a study
of the maximum number $H(m,n)$ of hyperelliptic limit cycles of the Li\'enard system in terms of $m$ and $n$.
Here we adopt the conventional definition of a limit cycle. Namely,
by {\it a limit cycle} of a polynomial system we mean that it is
an isolated closed orbit of the system.
It is called an {\it algebraic limit cycle} if it is a limit cycle and is contained in
an invariant algebraic curve \{$(x,y)\mid F(x,y)=0$\}.
In particular, if $F(x,y)$ takes the form
$F(x,y)=(y+P(x))^2-Q(x),$
where $P$ and $Q$ are polynomials,
then we call the invariant curve
hyperelliptic.
Correspondingly, a limit cycle is called a hyperelliptic limit cycle if it is contained in a hyperelliptic curve.
The investigation of limit cycles of the Li\'enard system has been one
of the most interesting topics for decades (see \cite{JL},\cite{FD}).
In the most general setting, however, it is a very hard subject
and the problem of existence is quite
elusive. Therefore certain assumptions are reasonably imposed, and special categories are
technically restricted. Among them, the algebraic and hyperelliptic versions of the problem have caught
particular attention of the study. A brief survey of the situation is as follows.
Odani \cite{Odani} in 1995 proved that if $n\leq m$ and
$f_mg_n(f_m/g_n)'\not\equiv0$,
then any Li\'{e}nard system of $(m,n)$-type has no invariant algebraic curves.
Therefore in this case, it is impossible to have
any hyperelliptic limit cycles.
Chavarriga et al. \cite{JIJH}, Zoladek \cite{Z}, and Makoto Hayashi \cite {MH} proved that
any Li\'{e}nard systems of the types
$(0,n)$, $(1,n)$, $(2, 4)$ and $(m,m +1)$
have no algebraic limit cycles, hence there are no hyperelliptic limit cycles.
In 2008, Llibre and Zhang \cite{LZ} proved that no Li\'enard
system of $(3,5)$-type has hyperelliptic limit cycles.
On the other hand, in the same paper \cite{LZ}, they found
that in the following cases there are Li\'{e}nard systems of $(m,n)$-type which
can possess hyperelliptic limit cycles:
\begin{itemize}
\item [(i)] $(m, n)$-type, for $m\geq 2$ and $n\geq 2m+1$;
\item[(ii)] $(m,2m)$-type for $m\geq 3$;
\item[(iii)] $(m, 2m-1)$-type for $m \geq 4$;
\item[(iv)] $(m, 2m-2)$-type for $m\geq 4$.
\end{itemize}
An individual type $(5,7)$
of the Li\'enard system is discussed in \cite{YZ},
where Yu and Zhang clarified that
there exist Li\'{e}nard systems of $(5,7)$-type
which have hyperelliptic limit cycles.
A recent paper \cite{Liu} is conclusive, where the authors considered the remaining types of the systems and proved that, in all these cases, there always exist Li\'enard systems
of $(m,n)$-type which have hyperelliptic limit cycles.
Thus the problem of the existence of hyperelliptic limit cycles
for all the possible types of the Li\'enard systems is
completely answered.
Collecting all the known results mentioned above and arranging them into Fig.1,
we can provide a visual
way to exhibit the distribution of the hyperelliptic limit cycles. Namely, in the $(m,n)$-plane,
there is a clearly-cut boundary
dividing all the types of the Li\'enard systems into two regions:
Systems falling in region $1$ can
never have any hyperelliptic limit cycle which means $H(m,n)=0$, and
in the other region, for each pair of
$(m,n)$, there always exists such a Li\'enard system
which admits at least one hyperelliptic limit cycle, thus $H(m,n)\geq1$.
Systems falling on the boundary are also unambiguously specified.
\begin{figure}\label{fig}
\end{figure}
The present paper grows from a very casual observation.
If one looks at Figure and takes region $1$ as land and region $2$ as sea, and if
we walk from the land to the sea,
we are in fact traveling from a region where systems have no hyperelliptic limit cycle
to a region where such limit cycles start to appear.
A very natural question like this can pop up:
when we walk from the land to the sea,
does the water become deeper and deeper? In other words,
does the maximum number of hyperelliptic limit cycles
increase as we walk into the sea further and further?
Such curiosity leads us to explore this problem and
to see if there is any algebraic mechanism behind this.
The investigation turns out to be quite interesting:
While only those Li\'enard systems falling in the sea
can have hyperelliptic
limit cycles, we prove that
those systems in ``deeper" water indeed can have larger $H(m,n)$.
A detailed classification is summarized in the following theorem.
Notice that we also consider the configuration of these limit cycles,
another one of very important aspects of the subject.
{\bf \large{Main Theorem}}: {\it
Consider Li\'enard systems of the type $(m,n)$ where $m\geq 2$, the maximum number of hyperelliptic limit cycles admits the following estimations:
$$ H(m,n)\geq\left\{
\begin{array}{lcl}
n-m-1 & & {m+2 \leq n \leq [\frac{4m+2}{3}]}\\
{[\frac{n-1}{4}]} & & {[\frac{4m+2}{3}]+1\leq n\leq 2m, m\geq 4}\\
{[\frac{m}{2}]} & & {n\geq 2m+1}
\end{array} \right.
$$
and
$$ H(m,n)\leq\left\{
\begin{array}{lcl}
{[\frac{n+1}{4}]} & & {m+2 \leq n \leq 2m-2, m\geq 4}\\
{[\frac{n-1}{4}]} & & {n=2m-1, or~n=2m , m\geq 4}\\
{[\frac{m}{2}]} & & {n>2m+1}
\end{array} \right. $$
In all the cases with $ n\neq {2m+1}$ and $H(m,n)>1$,
the hyperelliptic limit cycles of the system can only have non-nested configuration.
}
{\bf \large{ Remark: }}{\it It immediately follows from the main theorem that
\begin{itemize}
\item [\emph{(i)}] When $1+[\frac{4m+2}{3}]\leq n\leq 2m-2$, if $n-1\equiv0$ \emph{(mod $4$)} or $n-1\equiv1$ \emph{(mod $4$)}, then $H(m,n)=[\frac{n-1}{4}]$;
\item[\emph{(ii)}] If $n=2m-1$ or $n=2m$, $m\geq4$, then $H(m,n)=[\frac{n-1}{4}]$;
\item[\emph{(iii)}] If $ n>{2m+1}$ , then $H(m,n)=[\frac{m}{2}]$.
\end{itemize}
}
The paper is organized as follows: In section 2, we shall introduce
some preliminaries including definitions, notation
and basic methods. In section 3 ,4 and 5, we present a detailed proof of the results.
\section{Preliminaries}
In this section, we shall collect some related properties of Li\'enard
systems and introduce a complete discrimination system for polynomials. For the proof of these results, we refer the reader to
(\emph{\emph{\cite{ Liu, LZ, YZ, Z, MH, YL, YLJZ}}})for details.
\subsection{Hyperelliptic limit cycles of Li\'enard systems}
Recall that the Li\'enard system takes the form
$$\dot x=y, \quad\quad\dot y=-f_m(x)y-g_n(x).$$
Assume that the system has a hyperelliptic invariant curve
\begin{equation}\label{2}
F(x, y)= (y + P(x))^2-Q(x)=0.
\end{equation}
The following properties hold, whose proof is standard and hence omitted.
\begin{Lemma}
There exists $K(x, y)\in \R[x, y]$ such that
$$y\frac{\partial F}{\partial x}-(f_m(x)y+g_n(x))\frac{\partial F}{\partial
y}=K(x,y)F.$$
\end{Lemma}
\begin{Lemma} If relation \emph{(\ref{2})} holds, then
the degree of polynomial $P(x)$ is $m+1$, and the polynomials $f_m$ and $g_n$
can be expressed in terms of $P$ and $Q$ as follows.
\begin{equation}\label{3}
f_m=P'+\frac{PQ'}{2Q}, \quad
g_n=\frac{Q'(P^2-Q)}{2Q}.\end{equation}
\end{Lemma}
Since any
singular point of system (\ref{1}) must be located on the $x$-axis, thus
when a hyperelliptic curve $F(x, y)=0$ contains a limit cycle of system (\ref{1}),
the limit cycle should intersect the $x$-axis at two different points, denoted
by $(s_1, 0)$ and $(s_2, 0)$. The following properties hold.
\begin{Lemma}
\emph{(i)} $s_1$ and $s_2$ are real simple roots of $Q (x)$.\emph{ (ii)} Any root of $Q(x)$ must be a root of
$P(x)$.
\end{Lemma}
\begin{Lemma}\label{lemma2}
If $s_1$ and $s_2$ are simple roots of $Q (x)$ and
$Q(x)>0$ in $(s_1, s_2)$, then the hyperelliptic curve \emph{(\ref{2})}
contains a closed curve in the strip $s_1\leq x\leq s_2$.
\end{Lemma}
Now one step further: assume that
(i) the hyperelliptic curve $F(x,y)=0$ contains a
closed curve $C$ in the strip $s_1\leq x\leq s_2$, where $s_1$ and $s_2$
are simple roots of $Q (x)$,
(ii) this closed curve $C$ surrounds only
one singularity $(\alpha, 0)$ of system (\ref{1}),
(iii) the
singularity $(\alpha, 0)$ is a focus or node.
Then this closed
curve $C$ is a limit cycle.
We have the following criteria to recognize the type of the singular point.
\begin{Lemma}
If $g_n(\alpha)=0, g_n'(\alpha)>0$ and $f_m(\alpha)\not=0$, then
$(\alpha, 0)$ is a focus or a node of system \emph{(\ref{1})} .
\end{Lemma}
Combining all the known result, we give the following lemma which is very useful in determining if an algebraic curve
is a hyperelliptic limit cycle of the Lienard system.
\begin{Lemma} \label{lemma4}
An algebraic curve \emph{(\ref{2})} in the strip $x\in [s_1,s_2]$
is a hyperelliptic limit cycle if the following sufficient conditions
are met.
\emph{(i)} $f_m$ and $g_n$ satisfy (\ref{3}),
\emph{(ii)} All the roots of $Q(x)=0$ are real and $s_1,s_2$ are simple root
and $Q(x)>0$ for $x\in (s_1,s_2)$.
\emph{(iii)} $P^2(x)-Q(x)<0$ for
$x\in (s_1,s_2)$.
\emph{(iv) } If $\alpha \in (s_1,s_2)$ such that $Q'(\alpha)=0$, then $f_m(\alpha)\neq 0$.
\end{Lemma}
Proof of Lemma: Condition (i) means that $F=0$ is the invariant curve
of the system, and all the roots of $Q(x)$ are the roots of $P(x)$.
From (iii) we know that the curve $F(x,y)=0$ in the strip
bounded by $x=s_1$ and $x=s_2$ intersects the $x$ axis only at these two
endpoints. Condition (ii) means that $Q'(s_i)\neq 0$. It follows that the curve in the strip
has no singular points and is closed. From (ii)
we also know that $Q'(x)$ has only one real root $\alpha$ for $[s_1,s_2]$.
Again, from (iii) we see that $g_n(x)$ has a unique real root $\alpha$.
Therefore the system has only one singular point
inside the closed orbit formed by $F=0$ when restricted to the strip.
We can even see that this singular point is either a focus type or
a node. In fact, $g_n'(\alpha)=Q''(\alpha)\cdot \frac{P^2(\alpha)-Q(\alpha)}{2Q(\alpha)}
+Q'(\alpha)(\frac{P^2-Q}{2Q})'(\alpha)$.
Notice that the second term vanishes, and since $\alpha$ is the maximal
value point of $Q$, therefore $Q''(\alpha)<0$. It follows that
$g'(\alpha)>0$. Condition (iv) says that
$f_m(\alpha)\neq 0$, therefore $(\alpha,0)$ is
a focus or a node. Therefore the closed orbit
is hyperelliptic limit cycle of the system.
\subsection{Algorithm for root classification}
Given a polynomial
$$f(x)=a_0x^n+a_1x^{n-1}+\cdots+a_n,$$
we write the derivative of $f(x)$ as
$$f'(x)=na_0x^{n-1}+(n-1)a_1x^{n-2}+\cdots+a_{n-1}.$$
For the n-degree polynomial $f(x)$, $\alpha_1, \alpha_2, \cdots, \alpha_n$ denote all the roots of it. Let $s_p=\sum\limits_{j=1}^n\alpha_j^p$, $p=0,1,2,\cdots,n$, $S_k=|s_{i+j}|,i,j=0,1,\cdots,k-1,$ that is,
\begin{equation}
S_k=\left|
\begin{array}{cccc}
s_0 & s_1 & \cdots & s_{k-1} \\
s_1 & s_2 & \cdots & s_{k} \\
\cdots & \cdots & \cdots & \cdots \\
s_{k-1} & s_{k} & \cdots & s_{2k-2}\\
\end{array}
\right|.
\end{equation}
\vskip0.2cm
\begin{Definition}
\emph{(discrimination matrix)} The Sylvester matrix of $f(x)$ and $f'(x)$, denoted by \emph{Discr(f)}
$$
\left(
\begin{array}{cccccccc}
a_0 & a_1 & a_2 & \cdots & a_n & 0 & \cdots & 0 \\
0 & na_0 & (n-1)a_1& \cdots & a_{n-1} & 0 & \cdots & 0 \\
0 & a_0 & a_1 & \cdots & a_{n-1} & a_n & \cdots & 0 \\
0 & 0 & na_0 & \cdots & 2a_{n-2} & a_{n-1} & \cdots & 0 \\
\cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\
\cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\
0 & 0 & 0 & \cdots & a_0 & a_1 & \cdots & a_n\\
0& 0 & 0 & \cdots & 0 & na_0 & \cdots & a_{n-1}\\
\end{array}
\right)
$$
is called the discrimination matrix of $f(x)$.
\end{Definition}
\vskip0.2cm
\begin{Definition}
\emph{(discriminant sequence)} Denoted by $D_k$, the determinant of the submatrix of \emph{Discr(f)}, formed by the first 2k rows and the first $2k$ columns, for $k=1,\cdots,n$. We call the n-tuple \emph{$(D_1,D_2,\cdots,D_n)$}
the discriminant sequence of polynomial $f(x)$.
\end{Definition}
\vskip0.2cm
\begin{Definition}
\emph{(sign list)}
we call the list
$$\emph{$[ sign(D_1), sign(D_2), \cdots, sign(D_n)]$}$$
the sign list of the discrimination sequence \emph{$(D_1,D_2,\cdots,D_n)$}.
\end{Definition}
\vskip0.2cm
\begin{Definition}
\emph{(revised sign list)} Given a sign list $[s_1,s_2,\cdots,s_n]$, we construct
a new list $[\varepsilon_1,\varepsilon_2,\cdots,\varepsilon_n]$ as follows:
\begin{itemize}
\item If $[s_1,s_2,\cdots,s_n]$ is a section of given list, where
$s_i\neq0, s_{i+1}=\cdots=s_{i+j-1}=0, s_{i+j}\neq0$, then we replace the subsection $[s_{i+1},s_{i+2},\cdots,s_{i+j-1}]$
by $[-s_i,-s_i,s_i,s_i,-s_i,-s_i,s_i,s_i,-s_i,\cdots]$.
i.e. let
$\varepsilon_{i+r}=(-1)^{[{\frac{r+1}{2}}]}s_i$, for $r=1,2,\cdots{j-1}.$
\item Otherwise, let $\varepsilon_k=s_k$, there are no changes for other terms.
\end{itemize}
\end{Definition}
\vskip0.2cm
From \emph{\cite{YLJZ}}, we already know the following lemma.
\vskip0.35cm
\begin{Lemma} \label{lemma5}
For $k=1,2,\cdots,n,$ we have $D_k=S_k.$
\end{Lemma}
\vskip0.35cm \begin{Lemma} \label{lemma6}
Given a polynomial \emph{$f(x)=a_0x^n+a_1x^{n-1}+\cdots+a_n$} with real coefficients, if the number of the sign changes of the revised sign list of
$$\{D_1(f),D_2(f),\cdots,D_n(f)\}$$
is $v$, then the number of the pairs of distinct conjugate imaginary roots of \emph{$f(x)$ } equals $v$.
Furthermore, if the number of non-vanishing members of the revised sign list is $l$, then the number of the distinct real roots of \emph{$f(x)$} equals $l-2v$.
\end{Lemma}
\section{The Proof of the Results about Lower Bounds}
According to all the possible pairs $(m,n)$ where $m\geq 2$, we divide the proof into the following cases.
\begin{itemize}
\item[] (i) $ m+2\leq n\leq [\frac{4m+2}{3}]$;
\item[] (ii) $[\frac{4m+2}{3}]+1\leq n\leq 2m$ and $(m,n)$ is not in $\{(3,5),(2,4)\}$;
\item[] (iii)$n\geq 2m+1$.
\end{itemize}
\subsection{Case (i)}
When $ m+2\leq n\leq [\frac{4m+2}{3}]$, it suffices to construct
a Li\'enard system of type $(m,n)$ which can have $n-m-1$ hyperelliptic limit
cycles on invariant curve ({\ref 2}).
Suppose $n$ is odd. Now let $t=\frac{4m-3n+3}{2}$. By Corollary 3.1 in
\cite{Liu}, there exist a positive constant $c$ and a polynomial
\begin{displaymath}
Q_1(x)=(x-x_0)(x-1)\prod^{t}_{i=1}(x-x_i)^2\prod^{n-m-2}_{i=1}(x-y_i)^2,
\end{displaymath}
such that
\begin{displaymath}
P_1(x)=Q_1(x)+c=\prod^{t}_{i=1}(x-z_i)^2\prod^{n-m-1}_{i=1}(x-a_i)(x-b_i),
\end{displaymath}
where $x_0<z_1<x_1<z_2<...<z_t<x_t$ and $x_t<a_1<b_1<y_1<a_2<b_2<y_2<...<a_{n-m-1}<b_{n-m-1}<1$. We set
$$G(x)=(x-x_0)^2(x-1)^2\prod^{t}_{i=1}(x-x_i)^2\prod^{n-m-2}_{i=1}(x-y_i)^2\prod^{n-m-1}_{i=1}(x-a_i)(x-b_i),$$
\begin{displaymath}
\begin{aligned}
P(x)&=\sqrt{G(x)P_1(x)}\\
&=(x-x_0)(x-1)\prod^{t}_{i=1}(x-x_i)(x-z_i)\prod^{n-m-2}_{i=1}(x-y_i)\prod^{n-m-1}_{i=1}(x-a_i)(x-b_i),
\end{aligned}
\end{displaymath}
\begin{displaymath}
\begin{aligned}
Q(x)&=G(x)Q_1(x)\\
&=(x-x_0)^3(x-1)^3\prod^{t}_{i=1}(x-x_i)^4\prod^{n-m-2}_{i=1}(x-y_i)^4\prod^{n-m-1}_{i=1}(x-a_i)(x-b_i).
\end{aligned}
\end{displaymath}
then
$$
f_{m}(x)=P'(x)+\frac{P(x)Q'(x)}{2Q(x)},\quad g_{n}(x)=\frac{Q'(x)(P^2(x)-Q(x))}{2Q(x)}
$$
are polynomials of degree $m$ and $n$ respectively.
We claim, for each $i$, $i=1,2,...,n-m-1$, when $x\in[a_i,b_i]$, the closed curve given by ({\ref 2})
is a hyperelliptic limit cycle of the system.
1. In fact, it is easy to see that the
condition(i), (ii), (iii) of Lemma \ref{lemma4} is satisfied.
2. Let us verify condition(iv) by contradiction. Assume $Q'(x)$ and $f_{m}(x)$ have a common root $\alpha$ in $(a_i,b_i)$, then $P'(\alpha)=0$. With $G(\alpha)\neq0$, then $G'(\alpha)=P_1'(\alpha)=Q_1'(\alpha)=0$, and $(\frac{G(x)}{P_1(x)Q_1(x)})'\Big| _{x=\alpha}=0$.
\begin{displaymath}
\left(\frac{G(x)}{P_1(x)Q_1(x)}\right)'=\frac{(x-x_0)(x-1)}{\prod_{i=1}^{t}(x-z_i)^2}\left(\frac{1}{x-x_0}+\frac{1}{x-1}-2\sum_{i=1}^{t}\frac{1}{x-z_i}\right),
\end{displaymath}
we have $(G/(P_1Q_1))'(\alpha)>0$
, this leads to a contradiction.
By Lemma \ref{lemma4}, we can prove the system has
$n-m-1$ hyperelliptic limit cycles.
Suppose $n$ is even, let $t=(4m-3n+2)/2$. By Corollary 3.2 in
\cite{Liu}, there exist a positive constant $c$ and a polynomial
\begin{displaymath}
Q_1(x)=(x-1)\prod^{t}_{i=1}(x-x_i)^2\prod^{n-m-1}_{i=1}(x-y_i)^2,
\end{displaymath}
such that
\begin{displaymath}
P_1(x)=Q_1(x)+c=(x-x_0)\prod^{t}_{i=1}(x-z_i)^2\prod^{n-m-1}_{i=1}(x-a_i)(x-b_i),
\end{displaymath}
where $x_0<x_1<z_1<x_2<...<x_t<z_t$ and $z_t<y_1<a_1<b_1<y_2<a_2<b_2<...<y_{n-m-1}<a_{n-m-1}<b_{n-m-1}<1$. We set
$$G(x)=(x-x_0)(x-1)^2\prod^{t}_{i=1}(x-x_i)^2\prod^{n-m-1}_{i=1}(x-y_i)^2\prod^{n-m-1}_{i=1}(x-a_i)(x-b_i),$$
\begin{displaymath}
\begin{aligned}
P(x)&=\sqrt{G(x)P_1(x)}\\
&=(x-x_0)(x-1)\prod^{t}_{i=1}(x-x_i)(x-z_i)\prod^{n-m-1}_{i=1}(x-y_i)\prod^{n-m-1}_{i=1}(x-a_i)(x-b_i),
\end{aligned}
\end{displaymath}
\begin{displaymath}
\begin{aligned}
Q(x)&=G(x)Q_1(x)\\
&=(x-x_0)(x-1)^3\prod^{t}_{i=1}(x-x_i)^4\prod^{n-m-1}_{i=1}(x-y_i)^4\prod^{n-m-1}_{i=1}(x-a_i)(x-b_i).
\end{aligned}
\end{displaymath}
then
$$
f_{m}(x)=P'(x)+\frac{P(x)Q'(x)}{2Q(x)},\quad g_{n}(x)=\frac{Q'(x)(P^2(x)-Q(x))}{2Q(x)}
$$
are polynomials of degree $m$ and $n$ respectively.
We claim, for each $i$, $i=1,2,...,n-m-1$, when $x\in[a_i,b_i]$, the closed curve given by ({\ref 2})
is a hyperelliptic limit cycle of the system.
1. In fact, it is easy to see that the
condition(i), (ii), (iii) of Lemma \ref{lemma4} is satisfied.
2. Let us verify condition(iv) by contradiction.
Assume $Q'(x)$ and $f_{m}(x)$ have a common root $\alpha$ in $(a_i,b_i)$. Analogous the argument
above, we can get $\alpha$ is a root of $(G/(P_1Q_1))'$, while
\begin{displaymath}
\left(\frac{G(x)}{P_1(x)Q_1(x)}\right)'=\frac{(x-1)}{\prod_{i=1}^{t}(x-z_i)^2}\left(\frac{1}{x-1}-2\sum_{i=1}^{t}\frac{1}{x-z_i}\right),
\end{displaymath}
for each $i$, we can observe $x_0<z_i<\alpha<1$, thus $(G/P_1Q_1)'(\alpha)>0$ ,
this leads to a contradiction.
By Lemma \ref{lemma4}, we can prove the system has $n-m-1$ hyperelliptic limit cycles.
\subsection{Case(ii)}
Now we come to case \emph{$(ii)$}, when $[\frac{4m+2}{3}]+1\leq n\leq 2m-1$,
we shall construct
a Li\'enard system ({\ref1}) that can have $[\frac{n-1}{4}]$ hyperelliptic limit
cycles on invariant curve ({\ref 2}), from which we can infer that $H(m,n)\geq [\frac{n-1}{4}]$.
In the proof of case $(i)$, we perturbed the polynomial with a constant to transform
repeated roots into single roots, but this perturbation doesn't work in case $(ii)$. To prove case $(ii)$,
firstly we divide the case $(ii)$ into the following cases:
\begin{itemize}
\item[] (ii-i) $n-1\equiv0$ (mod $4$);
\item[] (ii-ii) $n-1\equiv1$ (mod $4$);
\item[] (ii-iii) $n-1\equiv2$ or $n-1\equiv3$ (mod $4$);
\end{itemize}
{\bf Case (ii-i)}: $n-1\equiv0$ (mod $4$)
\begin{Lemma}\label{lemma7}
For $h,l\in N$, define the polynomial
\begin{displaymath}
Q_1(x)=(x-s)x^{2h+1}\prod_{i=1}^l(x-i)^2,
\end{displaymath}
where $s>l+1$, then there exists a polynomial $c(x)$ of degree $2h$ which is positive in $[0,s]$ and such that
\begin{displaymath}
Q_1(x)+c(x)=(x-y_{l+1})\prod_{i=1}^l(x-y_i)(x-z_i)\prod_{i=1}^{2h+1}(x-x_i)
\end{displaymath}
where $0<x_1<x_2<\cdots<x_{2h+1}<y_1,$ $ y_1<1<z_1<y_2<\cdots<z_l<y_{l+1}<s$.
\end{Lemma}
{\bf Proof:}
We prove this lemma by mathematical induction. For
$h=0$, let $c(x)$ be a positive constant $\epsilon$. It easily follows
that the proposition for $h=0$ holds, if $\epsilon$ is sufficiently
small. Assume the proposition holds for $h=k$, it must been shown
that the proposition holds for $h=k+1$.
Decompose $Q_1(x)$
into two fractions $x^2$ and $Q_1^*(x)$, then $0$ is a repeated
root of degree of $2k+1$ of $Q_1^*(x)$. Using the induction
hypothesis, there exists a polynomial $c^*(x)$ of degree $2k$
which is positive in $[0,s]$, (we can choose $c^*(x)$ which satisfied the maximum absolute value of its coefficients is sufficient small) and such that
\begin{eqnarray}
Q_1(x)+x^2c^*(x)&=&x^2(Q_1^*(x)+c^*(x)) \nonumber\\
&=&x^2(x-y_{l+1}')
\prod_{i=1}^l(x-y_i')(x-z_i')\prod_{i=3}^{2k+3}(x-x_i'), \nonumber
\end{eqnarray}
where $0<x_3'<\cdots<x_{2k+3}'<y_1'$, $y_1'<1<z_1'<y_2'<\cdots<z_l'<y_{l+1}'<s$.
Choose a sufficiently small $d$ which satisfied $d>0$ and $xc^*(x)-d$ has only one root $\alpha<<1$ in $[0,s]$. For the maximum absolute value of coefficients of $c^*(x)$ is sufficient small, the local maximum of $Q_1(x)+x^2c^*(x)$ in
$(0,x_3')$ is the least maximum among all the maxima of
$Q_1(x)+x^2c^*(x)$ in $[0,s]$. Perturbing $Q_1(x)+x^2c^*(x)$ with $-dx$,
we get a polynomial
\begin{displaymath}
Q_1(x)+x^2c^*(x)-dx= x(x-y_{l+1}'')
\prod_{i=1}^l(x-y_i'')(x-z_i'')\prod_{i=2}^{2k+3}(x-x_i''),
\end{displaymath}
where $0<x_2''<x_3''<\cdots<x_{2k+3}''<y_1''$, $y_1''<z_1''<y_2''<\cdots<z_l''<y_{l+1}''$.
Since $\alpha<<1$ is the only root of $xc^*(x)-d$ in $[0,s]$, we have $Q_1(s)+s^2c^*(s)-ds>0$ and $Q_1(i)+i^2c^*(i)-di>0, 1\leq i\leq l$ , then $y_{l+1}''<s$ and $y_i''<i<z_i'', 1\leq i\leq l$. When $0<x<\alpha$, we have $x^2c^*(x)-dx<0$, while $x_2''$ is the root of $Q_1(x)+x^2c^*(x)-dx$, for $Q_1( x_2'')<0$, then $\alpha< x_2''$.
Assume $\gamma$ is minimum point of $x^2c^*(x)-dx$ in $[0,s]$, then $0<\gamma<\alpha<x_2''$. Choose $b>0$ satisfied $\gamma^2c^*(\gamma)-d\gamma+b>0$, $Q_1(\gamma)+\gamma^2c^*(\gamma)-d\gamma+b<0$. (The existence of $b$ relies on $Q_1(\gamma)<0$.) Now we start to proof all roots of $Q_1(x)+x^2c^*(x)-dx+b$ are real. Assume $\beta$ is minimum point of $Q_1(x)+x^2c^*(x)-dx+b$ in $[0,x_2'']$, we obtain $Q_1(\beta)+\beta^2c^*(\beta)-d\beta+b\leq Q_1(\gamma)+\gamma^2c^*(\gamma)-d\gamma+b<0$. For $d$ is sufficiently small, the local minimum $Q_1(\beta)+\beta^2c^*(\beta)-d\beta+b$ in
$(0,x_2'')$ is the largest minimum among all the minima of
$Q_1(x)+x^2c^*(x)-dx$ in $[0,s]$,
we know that
all roots of $Q_1(x)+x^2c^*(x)-dx+b$ are real.
Perturbing $Q_1(x)+x^2c^*(x)-dx$ with $b$, we get a polynomial
\begin{displaymath}
Q_1(x)+x^2c^*(x)-dx+b=(x-y_{l+1})\prod_{i=1}^l(x-y_i)(x-z_i)\prod_{i=1}^{2k+3}(x-x_i).
\end{displaymath}
where $0<x_1<x_2<\cdots<x_{2k+3}<y_1$, $y_1<y_1''<z_1''<z_1<y_2<\cdots<z_l<y_{l+1}<y_{l+1}''$. For $y_{l+1}''<s$ and $y_i''<i<z_i'', 1\leq i\leq l$, we have $0<x_1<x_2<\cdots<x_{2k+3}<y_1$,$y_1<1<z_1<y_2<\cdots<z_l<y_{l+1}<s$. On the other hand, we know the degree of $c(x)=x^2c^*(x)-dx+b$ is $2k+2$, and $c(x)\geq \gamma^2c^*(\gamma)-d\gamma+b>0$ in $[0,s]$. This completes the proof of lemma.
\vskip 0.4cm
Denote $\frac{n-1}{4}=t$, We set
\begin{displaymath}
Q_1(x)=(x-2m+2t)x^{6t-2m+1}\prod_{i=1}^{m-2t-1}(x-i)^2,
\end{displaymath}
by Lemma \ref{lemma7}, we can perturb $Q_1(x)$ with a polynomial $c(x)$ of degree $6t-2m$ which is positive in $[0,s]$, then
\begin{displaymath}
P_1(x)=Q_1(x)+c(x)= \prod_{i=1}^{t}(x-a_i)(x-b_i),
\end{displaymath}
where
$0<a_1<b_1<\cdots<a_{3t-m+1}$, $a_{3t-m+1}<b_{3t-m+1}<1<a_{3t-m+2}<\cdots <m-2t+1<a_t<b_t<2m-2t$. We define
$$G(x)=\prod_{i=1}^t(x-a_i)(x-b_i)\prod_{i=0}^{m-2t-1}(x-i)^2(x-2m+2t)^2.$$
\begin{displaymath}
P(x)=\sqrt{G(x)P_1(x)},\qquad Q(x)=G(x)Q_1(x),
\end{displaymath}
then
$$P(x)=(x-2m+2t)\prod_{i=1}^{t}(x-a_i)(x-b_i)\prod_{i=0}^{m-2t-1}(x-i),$$
\begin{displaymath}
Q(x)=(x-2m+2t)^3x^{6t-2m+3}\prod_{i=1}^{t}(x-a_i)(x-b_i)\prod_{i=1}^{m-2t-1}(x-i)^4,
\end{displaymath}
and
$$f_{m}(x)=P'(x)+\frac{P(x)Q'(x)}{2Q(x)},\quad
g_{n}(x)=\frac{Q'(x)(P^2(x)-Q(x))}{2Q(x)}$$
are polynomials of degree $m$ and $n$ respectively.
We claim, for each $i$, $i=1,2,...,t$, when $x\in[a_i,b_i]$,
the closed curve given by ({\ref 2}) is a hyperelliptic limit cycle of the system.
1. In fact, it is easy to see that the condition(i), (ii), (iii) of Lemma \ref{lemma4}
is satisfied.
2. Let us verify condition(iv) by contradiction. Assume $Q'(x)$
and $f_{m}(x)$ have a common root $\alpha$ in $(a_i,b_i)$.
Suppose $6t-2m=0$, then $n=\frac{4m+3}{3}$, which is possible when $\frac{4m+3}{3}=[\frac{4m+2}{3}]+1$.
For $G(\alpha)\neq0$, we get $\alpha$
is the common root of $P_1'$, $Q_1'$, $G'$ and $(G/(P_1Q_1))'$, then
\begin{displaymath}
\left(\frac{G}{P_1Q_1}\right)'=[x^2(x-2m+2t)/x^{6t-2m+1}]'=2x-2m+2t,
\end{displaymath}
thus $\alpha=m-t$, but it is impossible for $Q_1'(m-t)<0$ and this leads
to a contradiction.
By Lemma \ref{lemma4}, we can prove the system has $t$
hyperelliptic limit cycles.
On the other hands, $6t-2m>0$, for $6t-2m$ is even, then $6t-2m\geq 2$. With $f_{m}(\alpha)=Q'(\alpha)=0$, then $P'(\alpha)=0$. For $G(\alpha)\neq0$, and $\frac {P_1}{Q_1}=\frac {P^2}{Q}$, we get $(\frac {Q}{P})'\Big|_{x=\alpha}=(\frac {P_1}{Q_1})'\Big|_{x=\alpha}=0$. Since
\begin{displaymath}
\frac{Q(x)}{P(x)}=x^{6t-2m+2}(x-2m+2t)^2\prod_{i=1}^{m-2t-1}(x-i)^3,
\end{displaymath}
we know $\alpha$ is irrelevant of $c(x)$.
Differentiating $P_1/Q_1$, we have
\begin{eqnarray}\label{5}
\left(\frac{P_1(x)}{Q_1(x)}\right)'=\frac{c'(x)Q_1(x)-Q_1'(x)c(x)}{Q_1^2(x)}
\end{eqnarray}
For $Q_1(\alpha)\neq0$,
we have \begin{eqnarray}\label{6}\frac{c'(\alpha)}{c(\alpha)}=\frac{Q_1'(\alpha)}{Q_1(\alpha)}.\end{eqnarray}
With the degree of $c(x)$ is more than 2 and the right side of (\ref{6}) is irrelevant of $c(x)$ , we can change the polynomial coefficients of $c(x)$ to make the left hand side of (\ref{6}) doesn't equal the right hand side, such that the root of (\ref{5})
in $(a_j, b_j)$ is different to the root of equation $(Q/P)'$. Therefore, such $\alpha$ doesn't exist and
this verifies condition(iv).
By Lemma \ref{lemma4}, we prove the system has $t$
hyperelliptic limit cycles. This completes the proof of the case $n-1\equiv0$ (mod $4$).
\vskip 0.4cm
\noindent{\bf Case (ii-ii)}: $n-1\equiv1$ (mod $4$)
\vskip 0.2cm
\noindent For the proof of Lemma \ref{lemma8} is similar to Lemma \ref{lemma7}, we omit it.
\begin{Lemma}\label{lemma8}
For $h\in N^+, l\in N$, define the polynomial
\begin{displaymath}
Q_1(x)=(x-s_1)(x-s_2)x^{2h}\prod_{i=1}^l(x-i)^2,
\end{displaymath}
where $ s_1<-1, s_2>l+1$, then there exists a polynomial $c(x)$ of degree $2h-1$ which is positive in $[s_1,s_2]$ and such that
\begin{displaymath}
Q_1(x)+c(x)=(x-z_{-1})(x-y_{l+1})\prod_{i=1}^l(x-y_i)(x-z_i)\prod_{i=1}^{2h}(x-x_i)
\end{displaymath}
where $s_1<z_{-1}<x_1<0<x_2<x_3<\cdots<x_{2h}<y_1$, $y_1<1<z_1<y_2<\cdots<z_l<y_{l+1}<s_2$.
\end{Lemma}
Denote $\frac{n-2}{4}=t$.
We set
\begin{displaymath}
Q_1(x)=(x+2)(x-s)x^{6t-2m+2}\prod_{i=1}^{m-2t-2}(x-i)^2,
\end{displaymath}where $s>>m-2t-2$.
Since $6t-2m+2\geq2$, by lemma \ref{lemma8}, we can perturb $Q_1(x)$ with a polynomial $c(x)$ of degree $6t-2m+1$ which is positive in $[-2,s]$, then
\begin{displaymath}
P_1(x)=Q_1(x)+c(x)= \prod_{i=1}^{t}(x-a_i)(x-b_i),
\end{displaymath}
where
$-2<a_1<b_1<0<a_2<\cdots<a_{3t-m+2}<b_{3t-m+2}$, $b_{3t-m+2}<1<a_{3t-m+3}<b_{3t-m+3}<2<a_{3t-m+4}<\cdots <m-2t-2<a_t<b_t<s$. Define
\begin{displaymath}
G(x)=\prod_{i=1}^t(x-a_i)(x-b_i)\prod_{i=0}^{m-2t-2}(x-i)^2(x+2)^2(x-s)^2,
\end{displaymath}
\begin{displaymath}
P(x)=\sqrt{G(x)P_1(x)},\qquad Q(x)=G(x)Q_1(x),
\end{displaymath}
we have
\begin{displaymath}
P(x)=(x+2)(x-s)\prod_{i=1}^{t}(x-a_i)(x-b_i)\prod_{i=0}^{m-2t-2}(x-i),
\end{displaymath}
\begin{displaymath}
Q(x)=(x+2)^3(x-s)^3x^{6t-2m+4}\prod_{i=1}^{t}(x-a_i)(x-b_i)\prod_{i=1}^{m-2t-2}(x-i)^4.
\end{displaymath}
then
$$f_{m}(x)=P'(x)+\frac{P(x)Q'(x)}{2Q(x)},\quad
g_{n}(x)=\frac{Q'(x)(P^2(x)-Q(x))}{2Q(x)}$$
are polynomials of degree $m$ and $n$ respectively.
We claim, for each $i$, $i=1,2,...,t$, when $x\in[a_i,b_i]$,
the closed curve given by ({\ref 2}) is a hyperelliptic limit cycle of the system.
1. In fact, it is easy to see that the condition(i), (ii), (iii) of Lemma \ref{lemma4}
is satisfied.
2. Let us verify condition(iv) by contradiction. Assume $Q'(x)$
and $f_{m}(x)$ have a common root $\alpha$ in $(a_i,b_i)$, then $P'(\alpha)=0$.
Note that $\frac {P_1}{Q_1}=\frac {P^2}{Q}$, and $G(\alpha)\neq0$, we get $(\frac {Q}{P})'\Big|_{x=\alpha}=(\frac {P_1}{Q_1})'\Big|_{x=\alpha}=0$. Since
\begin{displaymath}
\frac{Q(x)}{P(x)}=x^{6t-2m+3}(x+2)^2(x-s)^2\prod_{i=1}^{m-2t-2}(x-i)^3,
\end{displaymath}
we get $\alpha$ is irrelevant of $c(x)$ immediately.
Differentiating $P_1/Q_1$, then
\begin{eqnarray}\label{7}
\left(\frac{P_1(x)}{Q_1(x)}\right)'=\frac{c'(x)Q_1(x)-Q_1'(x)c(x)}{Q_1^2(x)},
\end{eqnarray}
for $Q_1(\alpha)\neq0$,
we have \begin{eqnarray}\label{8}\frac{c'(\alpha)}{c(\alpha)}=\frac{Q_1'(\alpha)}{Q_1(\alpha)}.\end{eqnarray}
While $6t-2m+1>0$, the degree of $c(x)$ is more than 1 and the right side of (\ref{8}) is irrelevant of $c(x)$, we can change the polynomial coefficients of $c(x)$ to make the left hand side of (\ref{8}) doesn't equal the right hand side, such that the root of (\ref{7})
in $(a_j, b_j)$ is different to the root of equation $(Q/P)'$.
Therefore, such $\alpha$ doesn't exist and
this verifies condition(iv).
By Lemma \ref{lemma4}, we can prove the system has $t$
hyperelliptic limit cycles. Since $t=\frac{n-2}{4}=[\frac{n-1}{4}]$, we complete the proof.
\vskip 0.4cm
\noindent{\bf Case (ii-iii)}: $n-1\equiv2$ or $n-1\equiv3$(mod $4$)
\begin{Lemma}\label{lemma9}
If a Li\'enard system of $(m,n)$-type has $t$ hyperelliptic limit cycles on
invariant curve $(y + P(x))^2 - Q(x)=0$ and for each limit cycle the conditions
of Lemma \ref{lemma4} are met, then there exists Li\'enard system of $(m+1,n+2)$-type
with at least $t$ hyperelliptic limit cycles.
\end{Lemma}
{\bf Proof.} It suffices that, based on the system in the assumption, we construct
a new Li\'enard system of $(m+1,n+2)$-type in the form of
\begin{displaymath}
\dot { x } = y, \qquad \dot { y } = -f_{m+1}(x)y - g_{n+2}(x)
\end{displaymath}
with the same number of hyperelliptic limit cycles. We take $\tilde P_s(x)$
and $\tilde Q_s(x)$ in the form
\begin{displaymath}
\tilde P_s(x)=P(x)(x-s),\quad \tilde Q_s(x)=Q(x)(x-s)^2.
\end{displaymath}
Changing $P(x)$ and $Q(x)$ in equation (\ref{3}) to $\tilde P_s(x)$ and
$\tilde Q_s(x)$ respectively, we get
\begin{displaymath}
f_{m+1}(x)=\tilde P_s'(x)+\frac{\tilde P_s(x)\tilde Q_s'(x)}{2\tilde Q_s(x)},\quad g_{n+2}(x)=\frac{\tilde Q_s'(x)(\tilde P_s^2(x)-\tilde Q_s(x))}{2\tilde Q_s(x)}.
\end{displaymath}
Note they are polynomials of $m+1$, $n+2$ degree respectively.
Consider a hyperelliptic limit cycle of the original system on the
invariant curve (\ref{2}) that intersect with $x$-axis on $a_1$ and $b_1$. We claim there exists a sufficient large $s_1$, which satisfied the closed curve with $x\in [a_1,b_1]$ on invariant curve $(y +\tilde P_{s_1}(x))^2-\tilde Q_{s_1}(x)=0$ is a hyperelliptic
limit cycle of the new system.
We observe that the condition(i),
(ii) and (iii) of Lemma \ref{lemma4} are trivially verified when $s$ is larger
than all the roots of $Q(x)$. Then we just have to consider
condition(iv). Differentiating $\tilde Q_s(x)$, we get
\begin{displaymath}
\tilde Q_s'(x)=(x-s)^2(Q'(x)+\frac{2}{x-s}Q(x)).
\end{displaymath}
It follows that, $\tilde \alpha_s \rightarrow \alpha$ as $s\rightarrow\infty$,
where $\alpha$ and $\tilde \alpha_s $ denote the root of $Q'(x)$ and
$\tilde Q_s'(x)$ in $(a_1,b_1)$ respectively.
Differentiating $\tilde P_s(x)$, we get
\begin{displaymath}
\tilde P_s'(x)=(x-s)(P'(x)+\frac{1}{x-s}P(x)).
\end{displaymath}
Hence, $\tilde P_s'(\tilde \alpha_s)/(\tilde \alpha_s-s)\rightarrow P'(\tilde \alpha_s)\rightarrow P'(\alpha)$
as $s\rightarrow\infty$. Furthermore, $P'(\alpha)\neq 0$ which follows from the assumption that
$f_m'(\alpha)\neq 0$. Thus, we can find a sufficient large $s_1$ satisfied
$\tilde P_{s_1}'(\tilde \alpha_{s_1})\neq 0$
to make $f_{m+1}(\tilde \alpha_{s_1})\neq 0$ . By Lemma \ref{lemma4}, we can prove the system $$\dot { x } = y,\quad \dot { y } = -f_{m+1}(x)y - g_{n+2}(x)$$ has at least $t$ hyperelliptic limit cycles ,where \begin{displaymath}
f_{m+1}(x)=\tilde P_{s_1}'(x)+\frac{\tilde P_{s_1}(x)\tilde Q_{s_1}'(x)}{2\tilde Q_{s_1}(x)},\quad g_{n+2}(x)=\frac{\tilde Q_{s_1}'(x)(\tilde P_{s_1}^2(x)-\tilde Q_{s_1}(x))}{2\tilde Q_{s_1}(x)},
\end{displaymath}this completes the proof of the lemma.
\vskip0.2cm
{\bf Proof of the case $n-1\equiv2$ (mod $4$)\emph{}}:
Suppose $(m-1,n-2)$ is still in case $(ii)$, then $(m-1,n-2)$ is in the case (ii-i) , use the above argument, we have a Li\'enard system of $(m-1,n-2)$-type that has $[\frac{n-3}{4}]$
hyperelliptic limit cycles. Since $n-1\equiv2$ (mod $4$), we have $[\frac{n-3}{4}]=[\frac{n-1}{4}]$. By the argument of Lemma \ref{lemma9} , we can construct a new Li\'enard system of
$(m,n)$-type with $[\frac{n-1}{4}]$ hyperelliptic limit cycle based on the system of $(m-1,n-2)$-type.
On the other hand, $(m-1,n-2)$ is in case $(i)$, then $[\frac {4m+4}{3}]=[\frac {4m+5}{3}]=n$, but $n-1\equiv2$ (mod $4$), which yields a contradiction. Therefore $(m-1,n-2)$ can only in case $(ii)$, this completes the proof.
\vskip0.2cm
{\bf Proof of the case $n-1\equiv3$ (mod $4$))}:
Suppose $(m-1,n-2)$ is still in case $(ii)$, then $(m-1,n-2)$ is in the case (ii-ii), use the above argument, we have a Li\'enard system of $(m-1,n-2)$-type that has $[\frac{n-3}{4}]$
hyperelliptic limit cycles. Since $n-1\equiv3$ (mod $4$), we have $[\frac{n-3}{4}]=[\frac{n-1}{4}]$. By the argument of Lemma \ref{lemma9} , we can construct a new Li\'enard system of
$(m,n)$-type with at least $[\frac{n-1}{4}]$ hyperelliptic limit cycles based on the system of $(m-1,n-2)$-type, thus $H(m,n)\geq[\frac{n-1}{4}]$.
On the other hand, $(m-1,n-2)$ is in case $(i)$, we can construct a Li\'enard system of $(m-1,n-2)$ type that has $n-m-2$
hyperelliptic limit cycles. For $(m-1,n-2)$ is in case $(i)$, and $n-1\equiv3$ (mod $4$), we have $n-m-2=[\frac{n-1}{4}]$. This completes the proof of the case $n-1\equiv3$ (mod $4$).
\vskip0.3cm
When $n=2m$,
we define
$$P(x)=\prod\limits_{i=1}^m(x-i)(x+s)\quad Q(x)=\prod\limits_{i=1}^m(x-i)(x+s)^{m+2},$$
where $s>>m$ is sufficiently large. If $m$ is odd, for each $i=1,2,\cdots\frac{m-1}{2}$, when $x\in[2i-1,2i]$, the closed curve
given by (\ref{2}) is a hyperelliptic limit cycle of the system, therefore $H(m,2m)\geq\frac{m-1}{2}=[\frac{2m-1}{4}]$.
On the other hand, $m$ is even, for each $i=1,2,\cdots\frac{m-2}{2}$, when $x\in[2i,2i+1]$, the closed curve
given by (\ref{2}) is a hyperelliptic limit cycle of the system, therefore $H(m,2m)\geq\frac{m-2}{2}=[\frac{2m-1}{4}]$.
\subsection{Case(iii)}
We set
$$P(x)=\prod_{i=1}^m(x-i)(x+s),$$
$$Q(x)=-s\prod_{i=1}^m(x-i)(x+s)^{n-m+1},$$ where $s>>m$ is sufficiently large, we take $f_{m}(x)$ and $g_{n}(x)$ in system ({\ref 1}) in the form of equation ({\ref 3}).
It is easy to see $f_m(x)$ and $g_n(x)$ are polynomials of degree $m$ and $n$ respectively.
Suppose $m$ is even. We claim, for each $i=1,2,...\frac{m}{2}$, when $x\in[2i-1,2i]$, the closed curve
given by (\ref{2}) is a hyperelliptic limit cycle of the system.
1. In fact, it is easy to see
that the condition(i), (ii), (iii) of Lemma \ref{lemma4} is satisfied.
2. Let us verify condition(iv)
by contradiction. Assume $Q'(x)$ and $f_{m}(x)$ have a common root $\alpha$ in $(2i-1,2i)$, then $P'(\alpha)=0$. With
\begin{displaymath}
R'(x)=(\frac{Q(x)}{P(x)})'=-s(n-m)(x+s)^{n-m-1},
\end{displaymath}
we would have $R'(\alpha)=0$, but $R'(x)$ only have one root $-s$, this leads to a contradiction.
By Lemma \ref{lemma4}, we can prove the system
has $\frac{m}{2}$ hyperelliptic limit cycles.
Suppose $m$ is odd. In an analogous way, when $x\in[2i,2i+1]$, we can prove the closed curve
given by (\ref{2}) is a hyperelliptic limit cycle of the system for each
$i=1,2,\cdots\frac{m-1}{2}$.
Therefore, we obtain $H(m,n)\geq \left[\frac{m}{2}\right]$, when $m\geq2$ and $n\geq2m+1$.
\section{Configuration Of Hyperelliptic Limit Cycles}
\begin{Lemma}\label{lemma10}
If an $(m,n)$-Lienard system \emph{(\ref{1})}
has a hyperelliptic curve
$$(y+P(x))^2-Q(x)=0,$$
where $n\neq2m+1$,
then the system only has this one hyperelliptic curve.
\end{Lemma}
{\bf Proof.} From equation (\ref{3}), we have
\begin{equation}\label{9}
2Q(x)f_m(x)=2Q(x)P'(x)+P(x)Q'(x),
\end{equation}
and
\begin{equation}\label{10}
2Q(x)g_n(x)=Q'(x)(P^2(x)-Q(x)).
\end{equation}
Therefore, we know that the degree of $P(x)$ is $m+1$, while the degree
of $P^2(x)-Q(x)$ is $n+1$.
Let $f_m(x)$ and $g_n(x)$ take the form
\begin{displaymath}
f_m(x)=\sum_{i=0}^{m}a_i x^i,\qquad g_n(x)=\sum_{i=0}^{n}b_i x^i.
\end{displaymath}
If $n>2m+1$, the degree of $P^2(x)-Q(x)$ equals the degree of $Q(x)$.
Let us denote $P(x)$ and $Q(x)$ by
\begin{displaymath}
P(x)=\sum_{i=0}^{m+1}p_i x^i,\qquad Q(x)=\sum_{i=0}^{n+1}q_i x^i.
\end{displaymath}
then the coefficients of the highest degree terms of the each side of equations (\ref{9}) and (\ref{10}) are:
\begin{displaymath}
2a_m q_{n+1}=(2m+n+3)p_{m+1}q_{n+1},
\quad 2q_{n+1}b_n=-(n+1)q_{n+1}^2.
\end{displaymath}
Thus
$p_{m+1}=\frac{2a_m}{2m+n+3}$ and $q_{n+1}=\frac{-2b_n}{n+1}$ are uniquely
determined.
Comparing the coefficients of the second highest degree terms of the polynomials on each side of equations
(\ref{9}) and (\ref{10}) respectively, we have
\begin{displaymath}
2a_m q_n+2a_{m-1}q_{n+1}=(2m+2+n)p_{m+1}q_n+(2m+n+1) p_m q_{n+1},
\end{displaymath}
\begin{displaymath}
2b_n q_n+ 2b_{n-1}q_{n+1}=-(2n+1) q_n q_{n+1}+(n+1)q_{n+1}c,
\end{displaymath}
where $c=0$ or $c=p_{m+1}^2$. For the coefficients of $p_m$ and $q_n$ in the linear equations mentioned above which derive from comparing the coefficients of the second highest degree terms of the polynomials on each side of equations
(\ref{9}) and (\ref{10}) are $(2m+n+1)q_{n+1}$ and $nq_{n+1}$ respectively, we have the values of $p_m$ and $q_n$ are uniquely defined.
More generally, by comparing the coefficients of $x^{n+i}$ and $x^{2n+i-m}$ of the equation (\ref{9}) and (\ref{10}) respectively, we can get the values of
$p_i$ and $q_{n+i-m}$ are uniquely defined, where $i=0,1,\cdots,{m-1}$. We also can derive the value of $q_j$ is uniquely defined, where $j=1,2,\cdots,{n-m-1}$.
For the value of $q_0$, We compare the coefficients of $x^{2m+1}$ and $x^{n+2m+2}$ of the equation (\ref{10}) respectively, we have
\begin{equation}\label{11}
\begin{aligned}
2q_{2m+1} b_0+\cdots+ 2q_0 b_{2m+1}=\left({(2m+2)q_{2m+2}(p_0^2-q_0)+\cdots+}\right. \\
\left.{ q_1(2 p_m p_{m+1}-q_{2m+2})}\right) \\
\end{aligned}
\end{equation}
\begin{equation}\label{12}
\begin{aligned}
2b_{2m+1} q_{n+1}+\cdots+ 2b_{n}q_{2m+2}=\left({(n+1)q_{n+1}(p_{m+1}^2-q_{2m+2})+\cdots+}\right. \\
\left.{(-2m-2)q_{2m+2}q_{n+1}}\right) \\
\end{aligned}
\end{equation}
the coefficient of $q_{0}$ in the linear equation (\ref{11}) is $2b_{2m+1}+(2m+2)q_{2m+2}$, while $2b_{2m+1}+(2m+2)q_{2m+2}=(n+1)p_{m+1}^2\neq 0$ which derive from the equation (\ref{12}). Therefore the value of $q_{0}$ is uniquely defined. Finally the polynomial $P(x)$ and $Q(x)$ are determined, we complete the proof of the lemma in the case $n>2m+1$.
If $n<2m+1$, the degree of $P^2(x)-Q(x)$ is smaller than the degree of $P^2(x)$.
Thus, the coefficients of some higher terms of
$P^2(x)$ and $Q(x)$ are same, namely,
\begin{equation}\label{13}
(P^2)^{(n+i)}(x)=(Q)^{(n+i)}(x), \quad 2\leq i\leq 2m+2-n.
\end{equation}
Let us separate $P(x)$ and $Q(x)$ in the form
\begin{displaymath}
P(x)=\sum_{i=0}^{m+1}p_i x^i,\qquad Q(x)=\sum_{i=0}^{2m+2}q_i x^i.
\end{displaymath}
Comparing the coefficients of the highest degree terms of polynomials
on each side of equation (\ref{9}) and (\ref{13}), we have
\begin{displaymath}
2a_m q_{2m+2}=(4m+4)p_{m+1}q_{2m+2}, \qquad p_{m+1}^2=q_{2m+2}.
\end{displaymath}
Thus $p_{m+1}=\frac{a_m}{2(m+1)}$ are uniquely defined.
More generally, by comparing the coefficients of $x^{3m+2-i}$ and $x^{2m+2-i}$ of the equation (\ref{9}) and (\ref{13}) respectively, we have $ p_{m+1-i}=\frac{a_{m-i}}{2(m+1-i)}$ and the valve of $ q_{2m+2-i}$ is uniquely defined, where $0\leq i\leq {2m-n}$.
Then we compare the coefficients of $x^{n+m+1}$ and $x^{n+2m+2}$ of the equation (\ref{9}) and (\ref{10}), we have $ p_{n-m}=\frac{a_{n-m-1}}{2(n-m)}+\frac{(n-2m-1)b_n}{2(n-m)a_m}$, and the valve of $ q_{n+1}$ is uniquely defined. Repeating the above process, we derive that $ p_{n-m-2}£¬\cdots£¬ p_1$ and $q_{n-1}£¬\cdots£¬q_{m+2}$ are uniquely defined.
For the value of $ p_0$, we can compare the coefficient of $x^n$ of equation (\ref{9}), then we have a linear equation for $ p_0$ while the coefficient of $ p_0$ can only be $-b_n$ or $\frac{n-4m-3}{m+1}b_n$, therefore the value of $ p_0$ is uniquely defined, then the values of $ q_{m+1},q_m,\cdots,q_0$ which are depend on the value of $ p_0$ are uniquely defined. Finally the polynomial $P(x)$ and $Q(x)$ are determined, we complete the proof of the lemma.
We know from the above discuss, if an $(m,n)$-Lienard system (\ref{1}), where $n\neq2m+1$,
has a hyperelliptic curve
$(y+P(x))^2-Q(x)=0,$
then the system can only has this hyperelliptic curve. Thus, there are at most two points in the hyperelliptic limit cycles of the system when we fix the value of $x$ which means no hyperelliptic limit cycle can contained other hyperelliptic limit cycle. Therefore, the hyperelliptic limit cycles only have non-nested configuration. (see Fig.2)
\begin{figure}\label{fig}
\end{figure}
\section{The Proof of the Results about Upper Bounds}
By the argument of Lemma \ref{lemma10}, we know a system (\ref{1}) in the case $n\neq2m+1$
has a hyperelliptic curve $(y+P(x))^2-Q(x)=0$,
then the system can only has this hyperelliptic curve. Take the polynomial $P$, $Q$ of the hyperelliptic curve in the form
$$P(x)=\prod_{i=1}^a(x-x_i)^{\alpha_i+1}\prod_{j=1}^b(x-y_j)^{\beta_j+1}\prod_{l=1}^c(x-z_l)^{\gamma_l+1}$$
$$Q(x)=\prod_{i=1}^a(x-x_i)\prod_{j=1}^b(x-y_j)^{\omega_j+2},$$
where $a,b,c,\alpha_i,\beta_j,\gamma_l,\omega_j\geq0$, $\alpha=\sum\limits_{i=1}^a{\alpha_i}$, $\beta=\sum\limits_{j=1}^b{\beta_j}$, $\gamma=\sum\limits_{l=1}^c{\gamma_l}$, and $\omega=\sum\limits_{j=1}^b{\omega_j}$. We set $x_i\neq y_j\neq z_l$, $x_1\neq x_2\neq\cdots\neq x_a$, $y_1\neq y_2\neq\cdots\neq y_b$ and $ z_1\neq z_2\neq\cdots\neq z_c$.
If $2\beta_i>\omega_i$, $i=1,2,\cdots,b$, then we replace $\beta_i$ and $\omega_i$ with $\beta_i^-$ and $\omega_i^-$ respectively, and use $b^-$ denotes the number of i which satisfied $2\beta_i>\omega_i$.
Otherwise, we replace $\beta_i$ and $\omega_i$ with $\beta_i^+$ and $\omega_i^+$, and use $b^+$ denotes the number of i which satisfied $2\beta_i\leq\omega_i$, then $b=b^++b^-$, $\omega=\omega_i^++\omega_i^-$, $\beta=\beta_i^++\beta_i^-$.
For proving the result of upper bounds, firstly, we discuss the case $m+2\leq n\leq2m-2$.
If
$F(x, y)=(y + P(x))^2-Q(x)=0$ is an invariant algebraic curve of system $(1)$, it is necessary that $P(x)$ has degree $m+1$, and $P^2(x)-Q(x)$ has degree $n+1$, thus
$a+b+c+\alpha+\beta+\gamma=m+1,$
$a+2b+\omega=2m+2$,
and $ln(P^2/Q)=O(x^{n-2m-1})$ , which implies that
\begin{eqnarray}\label{14}
\sum\limits_{i=1}^a(2\alpha_i+1)x_i^j+\sum\limits_{i=1}^{b^-}(2\beta_i^--\omega_i^-)y_i^j+\sum\limits_{i=1}^c(2\gamma_i+2)z_i^j
=\sum\limits_{i=1}^{b^+}(\omega_i^+-2\beta_i^+)y_i^j,
\end{eqnarray}
where $ j=1,2,\cdots,2m-n$. Assume
\begin{equation}
f(x)=\prod\limits_{i=1}^a(x-x_i)^{2\alpha_i+1}\prod\limits_{j=1}^{b^-}(x-y_j)^{2\beta_j^--\omega_j^-}\prod\limits_{l=1}^c(x-z_l)^{2\gamma_l+2},
\end{equation}
\begin{equation}
g(x)=\prod\limits_{j=1}^{b^+}(x-y_j)^{\omega_j^+-2\beta_j^+}.
\end{equation}
We use $k$ denotes the number of the distinct roots of \emph{$f(x)$ }, $t$ denotes the number of the distinct roots of \emph{$g(x)$}, $s$ denotes the degree of $f(x)$, $\tau$ denotes the degree of $f(x)-g(x)$, and $t_0$ denotes the number of the distinct real roots of $Q(x)$. It is easy to see $s=2\alpha+a+2\beta^--\omega^-+2\gamma+2c$, $\tau=n+1-a-2b-2\beta^+-\omega^-$, and $t=b^+$, $k=a+b^-+c$.
From \emph{\cite{YLJZ}}, we know the discrimination sequence \emph{$(D_1,D_2,\cdots,D_n)$} of \emph{$f(x)$ } satisfied $D_k\neq0, D_{k+1}=D_{k+2}=\cdots=D_s=0$. When $n$ is even, if $t\geq
\frac{s-\tau+1}{2}$, then $t_0\leq k \leq m+1-b^+\leq\frac{n}{2}$. Otherwise, $t\leq
\frac{s-\tau-1}{2}$, for the n-degree polynomial $f(x)$, we have
$$S_l=\left|
\begin{array}{cccc}
n & \sum\limits_{i=1}^{b^+}(\omega_i^+-2\beta_i^+)y_i & \cdots & \sum\limits_{i=1}^{b^+}(\omega_i^+-2\beta_i^+)y_i^{l-1} \\
\sum\limits_{i=1}^{b^+}(\omega_i^+-2\beta_i^+)y_i & \sum\limits_{i=1}^{b^+}(\omega_i^+-2\beta_i^+)y_i^2 & \cdots & \sum\limits_{i=1}^{b^+}(\omega_i^+-2\beta_i^+)y_i^{l} \\
\cdots & \cdots & \cdots & \cdots \\
\sum\limits_{i=1}^{b^+}(\omega_i^+-2\beta_i^+)y_i^{l-1} & \sum\limits_{i=1}^{b^+}(\omega_i^+-2\beta_i^+)y_i^{l} & \cdots & \sum\limits_{i=1}^{b^+}(\omega_i^+-2\beta_i^+)y_i^{2l-2}\\
\end{array}
\right|,$$ where $0\leq l\leq \frac{s-\tau+1}{2}$.
From Lemma \ref{lemma5}, we have $D_{t+1}=D_{t+2}=\cdots=D_{\frac{s-\tau+1}{2}}=0$ . If $k\leq\frac{s-\tau+1}{2}$, then $t_0\leq k \leq m+1-\frac{n}{2}<\frac{n}{2}$. Otherwise, from Lemma \ref{lemma6}, we have
$t_0\leq k-([{\frac{\frac{s-\tau+1}{2}-t}{4}}]\times2+[\frac{\frac{s-\tau+1}{2}-t-[{\frac{\frac{s-\tau+1}{2}-t}{4}}]\times4+1}{2}])\times2\leq k-\frac{s}{2}+\frac{\tau}{2}-\frac{1}{2}+t\leq a+b+c+\frac{n+1}{2}-(a+b+c)-\frac{1}{2}\leq\frac{n}{2}$.
When $n$ is odd, if $t\geq
\frac{s-\tau}{2}$, then $t_0\leq k\leq m+1-b^+\leq\frac{n+1}{2}$. Otherwise, $t\leq
\frac{s-\tau-2}{2}$, we have $D_{t+1}=D_{t+2}=\cdots=D_{\frac{s-\tau}{2}}=0$. If $k\leq\frac{s-\tau}{2}$, then $t_0\leq k <\frac{n+1}{2}$. Otherwise, from Lemma \ref{lemma6}, we have
$t_0\leq k-([{\frac{\frac{s-\tau}{2}-t}{4}}]\times2+[\frac{\frac{s-\tau}{2}-t-[{\frac{\frac{s-\tau}{2}-t}{4}}]\times4+1}{2}])\times2\leq k-\frac{s}{2}+\frac{\tau}{2}+t\leq a+b+c+\frac{n+1}{2}-(a+b+c)\leq\frac{n+1}{2}$.
Since the system (\ref{1}) can have at most one hyperelliptic limit curves, and the hyperelliptic limit cycle should intersect the $x$-axis at two different points $x_1, x_2,$ where $x_1, x_2,$ are simple root of $Q(x)$, we have $H(m,n)\leq \frac{t_0}{2}$. This completes the proof of case $m+2\leq n\leq2m-2$.
When $n=2m-1$, if $m$ is odd, we know from the preliminaries, any root of $Q(x)$ must be a root of $P(x)$, while the degree of $P(x)$ is $m+1$, $Q(x)$ can have at most $m$ simple roots, then $H(m,n)\leq\frac{m-1}{2}=\left[\frac{n-1}{4}\right]$. For $m$ is even, if $Q(x)$ have $m$ simple roots, then there are at most $\frac{m-2}{2}$ intervals which satisfied $Q(x)>0$ in the interval. Otherwise $Q(x)$ can have at most $m-1$ simple roots, then $H(m,n)\leq\frac{m-2}{2}=\left[\frac{n-1}{4}\right]$. When $n=2m$, the proof is similar to the case $n=2m-1$, so we omit it.
Recall that we want to prove
$H(m,n)\leq \left[\frac{m}{2}\right]$ when $ n>{2m+1}$.
Since the system (\ref{1}) can have at most one hyperelliptic limit curves, and $Q(x)$ can have not more than $m$ simple roots when $n>2m+1$, we obtain the upper bound of $H(m,n)$ is
$\left[\frac{m}{2}\right]$ .
\section*{References}
\end{document}
|
\begin{document}
\author[Ban, Bertrand, Jaber Chehayeb, Salha, Tabbara]{Seok Ban, Florian Bertrand, Amir Jaber Chehayeb, Adam Salha, Walid Tabbara}
\title{On the higher order Kobayashi pseudometric}
\begin{abstract}
We study the higher order Kobayashi pseudometric introduced by Yu. We first obtain estimates of this pseudometric in a special pseudoconvex domain in $\mathbb C^3$. We then study the structure of the higher order extremal discs and their connection with the standard extremal discs for the Kobayashi metric.
\end{abstract}
\thanks{Research of the second author was supported by the Center for Advanced Mathematical Sciences}
\maketitle
\section*{Introduction}
In his thesis and related papers \cite{yu,yu1,yu2} Yu introduced a higher order version of the Kobayashi pseudometric for the purpose of measuring precisely the type invariants of boundaries of pseudoconvex domains. This invariant pseudometric shares many of the fundamental properties of the standard Kobayashi pseudometric; in particular, in the complex plane,
the higher order Kobayashi pseudometric and the Kobayashi pseudometric coincide on any domain (see for instance \cite{ja-pf}). The pseudometric introduced by Yu was later on studied by many different authors \cite{ki-hw-ki-le,ki,ni1,ni2,ja-pf}. In this paper, we focus on two aspects of this pseudometric. We first consider a particular domain given by Yu \cite{yu} and obtain new estimates of the higher order Kobayashi metric (Theorem \ref{theoest}); as an application of our method, we obtain the exact value of the Kobayashi metric in certain cases (Proposition \ref{propexact}). This answers partially two questions addressed by Jarnicki and Pflug (problems 3.7 and 3.8 p.149 \cite{ja-pf}). Then, inspired by the works of Poletski \cite{po}, Edigarian \cite{ed}, Jarnicki and Pflug \cite{ja-pf}, and the paper \cite{be-de-jo}, we study the higher order pseudometric by means of the corresponding extremal discs and focus on their connection with the usual extremal discs (Theorem \ref{theopext}). We conclude the paper with an appendix in which we prove higher order Schwarz type lemmas.
\section{Preliminaries}
We denote by $\mathbb D=\{\zeta \in \mathbb C \ | \ |\zeta|<1\}$ the unit disc in $\mathbb C$ and by $\mathbb D_r=\{\zeta \in \mathbb C \ | \ |\zeta|<r\}$ the disc centered at $0$ and radius $r>0$.
\subsection{Invariant pseudometrics and the Kobayashi pseudometric of higher order}
Let $\Omega \subset \mathbb C^n$ be a domain. Following \cite{ko,ro}, the {\it Kobayashi pseudometric} of the domain $\Omega$ at $p\in \Omega$ and
$v \in T_p\Omega=\mathbb C^n$ is defined by
$$K_{\Omega}\left(p,v\right)=\inf
\left\{\frac{1}{r}>0 \ \big| \ \exists \ f: \mathbb D \to \Omega \ \mbox{holomorphic}, f\left(0\right)=p, f'(0)=rv\right\}.$$
We also recall the {\it Carath\'eodory pseudometric} $C_\Omega$ at $p \in \Omega$ and $v \in \mathbb C^n$ defined by
$$C_\Omega(p,v) = \text{sup}\{\left|d_pg (v)\right| \ | \ g: \Omega \rightarrow \mathbb D \ \mbox{holomorphic}, g(p) = 0\},$$
where $d_pg (v)$ is the differential at $p$ of $g$ in the direction $v$.
Note that in the case of the unit disc $\mathbb D \subset \mathbb C$, it follows from Schwarz Lemma that both $K_\mathbb D$ and $C_\mathbb D$ coincide with the Poincar\'e metric, that is, for $\zeta \in \mathbb D$ and $v \in \mathbb C$
$$K_\mathbb D(\zeta,v) = C_{\mathbb D}(p,v)=\cfrac{|v|}{1 - |\zeta|^2}.$$
Kobayashi pseudometrics of higher order were introduced by different authors \cite{ve,yu}. In this paper, we focus on the one defined by Yu in \cite{yu,yu1,yu2}. More precisely, for a positive integer $k>0$, the {\it $k^{th}$ order Kobayashi pseudometric} is defined by
$$K^k_{\Omega}\left(p,v\right)=\inf
\left\{\frac{1}{r}>0 \ | \ \exists \ f: \mathbb D \to \Omega \ \mbox{holomorphic}, f\left(0\right)=p, \nu(f)\geq k, f^{(k)}(0)=k!rv\right\},$$
where $p\in \Omega$, $v \in \mathbb C^n$, and $\nu(f)$ is the vanishing order of $f$ defined as the degree of the first nonzero term in the power expansion of $f$.
Before stating some fundamental properties of the higher order Kobayashi metric, we first establish the following higher order Schwarz Lemma on the unit disc.
\begin{lem}[Higher order Schwarz Lemma]\label{lemhigh}
Let $f:\mathbb D \rightarrow \mathbb D$ be a holomorphic function satisfying $f^{(\ell)}(0) = 0$ for all $\ell=0,\ldots,k-1$. Then we have for all $\zeta \in \mathbb D$
$$|f(\zeta)| \leq |\zeta|^{k}$$
and
$$|f^{k}(0)|\leq k!$$
Moreover, if $|f(\zeta)| = |\zeta|^k$ for some $\zeta \neq 0$ or $|f^{(k)}(0)| = k!$, then $f(\zeta) = e^{i\theta}\zeta^k$ for some $\theta \in \mathbb{R}$.
\end{lem}
Note that the proof of this lemma is essentially contained in the proof of Proposition 2.2 in \cite{yu}. We give a slightly different proof.
\begin{proof}
We proceed by induction. For $k = 1$, this is the classical Schwarz Lemma. Assume that the $k^{th}$ order Schwarz Lemma holds. Let $f:\mathbb D \rightarrow \mathbb D $ be a holomorphic function with $f^{(\ell)}(0) = 0$ for $\ell=0, \dots k$. Define the disc $g(\zeta)=f(\zeta)/\zeta^k$. According to the $k^{th}$-order Schwarz Lemma,
we have $g: \mathbb D \to \mathbb D$. Since $g(0)=0$, the classical Schwarz Lemma applied to $g$ leads to $|f(\zeta)| \leq |\zeta|^{k+1}$ for all $\zeta \in \mathbb D$ and
$$|g'(0)|=\frac{|f^{k+1}(0)|}{(k+1)!}\leq 1.$$
In case of equality, the result follows once again by induction.
\end{proof}
We now summarize important basic properties of the higher order Kobayashi pseudometric which have been established by many authors \cite{yu,ja-pf}.
\begin{prop}[\cite{yu,ja-pf}]\label{propprop}
Let $\Omega,\Omega' \subset \mathbb C^n$ be two domains. Let $k>0$ be a positive integer.
\begin{enumerate}[i.]
\item Let $F:\Omega \to \Omega'$ be a holomorphic map. Then for any $p\in \Omega$ and $v \in T_p\Omega$, we have
$$K^k_{\Omega'}(F(p),d_pF(v))\leq K^k_{\Omega}(p,v).$$
In particular, the higher order Kobayashi pseudometric is invariant under biholomorphisms.
\item In case $n=1$, the pseudometrics $K^k_{\Omega}$ and $K_{\Omega}$ coincide.
\item We have $K^{mk}_\Omega \leq K^{k}_\Omega$ for any $m \in \; \mathbb N$.
\item For any $p\in \Omega$ and $v \in T_p\Omega$, we have
$$C_{\Omega}(p,v) \leq K^k_{\Omega}(p,v) \leq K_{\Omega}(p,v).$$
\end{enumerate}
\end{prop}
For completeness, we include the proof.
\begin{proof} {\phantom{m}}
We first prove {\rm i}.
Let $g: \mathbb D \rightarrow \Omega$ be a holomorphic disc satisfying $g(0) = p$ and
$g^{(k)}(0) = k!r v$ for some $r>0$. Consider the composition $F\circ g: \mathbb D \rightarrow \Omega'$ and note that $(F\circ g)(0) = F(p)$ and
$$(F\circ g)^{(k)}(0) = d_pF(g^{(k)}(0)) = k!r d_pF(v).$$ It follows directly that $K_{\Omega'}^k(F(p),d_pF(v)) \leq K_\Omega^k(p,v)$.
The proof of {\rm ii} follows from Proposition 3.8.8 in \cite{ja-pf} and the uniformization theorem. The fact that $K^k_{\mathbb D}$ and $K_{\mathbb D}$ coincide follows directly from the higher order Schwarz Lemma \ref{lemhigh} since by invariance of both metric, it is enough to show that
$K^k_{\mathbb D}(0,v)=K_{\mathbb D}(0,v)=|v|$. Their equality on the punctured disc follows, for instance, from Lemma \ref{lempdhigh}.
For {\rm iii}, we follow exactly \cite{ja-pf}. Let $p \in \Omega$ and $v \in T_p\Omega$ and let $f:\mathbb D \to \Omega$ be a holomorphic disc with $f\left(0\right)=p$, $\nu(f)\geq k$, and $f^{(k)}(0)=k!rv$. We define $g(\zeta)=f(\zeta^m)$. The holomorphic disc $g:\mathbb D\to \mathbb D$ satifsies
$g\left(0\right)=0$,
$\nu(g)\geq km$, and
$$g^{(km)}(0)=(mk)!\frac{f^{(k)}(0)}{k!}=(mk)!rv.$$ Thus $K^{mk}_\Omega (p,v) \leq K^{k}_\Omega(p,v)$.
Finally, we prove {\rm iv}. The inequality $K^k_{\Omega}(p,v) \leq K_{\Omega}(p,v)$ follows immediately from {\rm iii}.
For the inequality concerning the Carath\'eodory metric, we consider $f: \mathbb D \to \Omega$ and $ g:\Omega \to \mathbb D$ both holomorphic and satisfying
$f\left(0\right)=p, \nu(f)\geq k, f^{(k)}(0)=k!rv, \mbox{and } g(p)=0$. The composition $g\circ f:\mathbb D\to\mathbb D$ is such that $g\circ f(0)=0$ and, since $\nu(f)\geq k$, we have
$(g\circ f)^{(l)}(0)=0$ for $l<k$. We then apply the higher order Schwarz Lemma \ref{lemhigh} to obtain $|(g\circ f)^{(k)}(0)| \leq k!$, and since
$$(g\circ f)^{(k)}(0)= d_{f(0)}g(f^{(k)}(0))= d_{p}g(k!rv),$$
we get $|d_{p}g(v)|\leq \frac{1}{r}.$
This shows {\rm iv} and ends the proof
\end{proof}
\begin{rem} The following higher order Kobayashi metric was introduced by Venturini in \cite{ve}
$$F^k_{\Omega}\left(p,\xi\right):=\inf
\left\{\frac{1}{r}>0 \ | \ f: \mathbb D \rightarrow \Omega
\mbox{ holomorphic}, f\left(0\right)=p, f^{\ell}(0)=r^\ell\xi_\ell, 1 \leq \ell \leq k\right\}$$
for any $p\in M$ and any $k$-jet $\xi=(\xi_1,\ldots,\xi_k)\in J^k_p(\Omega)=\mathbb C^{kn}$ at $p$.
The relation between these two higher order Kobayashi pseudometrics is given by
$$k!\left(F^k_{\Omega}\left(p,(0,\cdots,0,v)\right)\right)^k=K^k_{\Omega}\left(p,v\right).$$
\end{rem}
\subsection{Higher order extremal and stationary discs}
Following Lempert \cite{le}, we define higher order extremal discs as follows:
\begin{defi} Let $k>0$ be a positive integer and let $\Omega \subset \mathbb C^n$ be a domain. A holomorphic disc $f: \mathbb D \rightarrow \Omega$ is a {\it $k$-extremal disc for the pair $(p,v) \in \Omega \times T_p\Omega$} if $f(0)=p$, $\nu(f)\geq k$, $f^{(k)}(0)=k!\lambda v$
with $\lambda>0$ and if
$g: \mathbb D \rightarrow \Omega$ is holomorphic and such that $g(0)=p$, $\nu(g)\geq k$, $g^{(k)}(0)=k!\mu v$ with $\mu>0$, then $\mu\leq \lambda$. In case $k=1$, we will simply say that $f$ is an {\it extremal disc}.
We denote by $X_{\Omega}^k(p,v)$ the set of $k$-extremal discs for the pair $(p,v)$, and we set $X_{\Omega}^1(p,v)=X_{\Omega}(p,v).$
\end{defi}
Note that in case the domain $\Omega$ is bounded, then by Montel's theorem, for any pair
$(p,v) \in \Omega \times T_p\Omega$ there exists a corresponding $k$-extremal disc.
The following result is a direct consequence of Propostion \ref{propprop} $i.$
\begin{lem}\label{leminv}
Let $\Omega \subset \mathbb C^n$ be a domain and let $F$ be an automorphism of $\Omega$. Then for any $p \in D, v \in T_p\Omega$ we have
$$X_\Omega^k(F(p),d_pF(v)) = F_* X_\Omega^k(p,v).$$
\end{lem}
Let $\Omega = \{\rho<0\}\subset \mathbb C^n$ be a smooth domain, where $\rho$ is a defining function. We set
$\displaystyle \partial \rho = \left(\frac{\partial \rho}{\partial z_1},\ldots,\frac{\partial \rho}{\partial z_n}\right)$. Let $k$ be
a positive integer. Following \cite{be-de}, we define
\begin{defi}
A bounded holomorphic map $f:\mathbb Delta\to \mathbb C^n$ is a \emph{$k$-stationary disc attached to $b\Omega$ in the $L^{\infty}$ sense} if $f(b\mathbb Delta)
\subset b\Omega$ a.e. and if there exists a $L^{\infty}$ function $c:b\mathbb Delta\to \mathbb R^+$ such that the function $ \zeta\mapsto \zeta^k
c(\zeta)\partial \rho(f(\zeta))\in \mathbb C^n$ defined on $b\mathbb Delta$ extends holomorphically to $\mathbb Delta$.
\end{defi}
These discs were introduced in \cite{be-de} and generalize the notion of stationary discs introduced by Lempert \cite{le}. They are particularly well adapted to study Levi degenerate hypersurfaces.
\section{An example of Yu}
In this section, we are interested in the behavior of the higher order Kobayashi pseudometric in a particular domain. Following Yu \cite{yu}, we consider the domain
$\Omega =\{z=(z_1, z_2, z_3) \in \mathbb C^3 \ | \ \rho(z,\overline z)<0\} \subset \mathbb C^3$, where
$$ \rho(z,\overline z) = \mathbb{R}e e z_3 + \big|z_1^2 - z_2^3\big|^2.$$
The defining function $\rho$ is plurisubharmonic, and thus, the domain $\Omega$ is pseudoconvex. This example is the first example of a domain in which the higher order Kobayashi metric does not coincide with the usual Kobayashi metric (see Proposition \ref{propyu} below).
\subsection{Estimates of the higher order Kobayashi pseudometric}
Our goal is to study the higher order Kobayashi pseudometric for the pair
$$(p,v) = \left((0,0,-1), (0,1,0)\right) \in \Omega \times \mathbb C^3.$$
The following proposition was proved in \cite{yu}.
\begin{prop}[\cite{yu}]\label{propyu} We have
$$K_\Omega\left((0,0,-1), (0,1,0)\right) = 1,$$
and for any positive integer $k>0$
$$K^{2k}_\Omega\left((0,0,-1), (0,1,0)\right) = 0.$$
\end{prop}
As pointed out in the book \cite{ja-pf}, it is relevant to find good estimates of $K^k_\Omega\left((0,0,-1), (0,1,0)\right)$ for $k\geq3$ odd. We will now focus on this question. A first important observation is:
\begin{lem}
For any positive integer $k >0$, we have
$$K^{2k+1}_\Omega\left((0,0,-1), (0,1,0)\right) \leq K^{2k- 1}_\Omega\left((0,0,-1), (0,1,0)\right).$$
\end{lem}
\begin{proof}
Recall that $(p,v) = \left((0,0,-1), (0,1,0)\right)$. Let $ f = (f_1, f_2, f_3) : \mathbb D \to \Omega$ be a holomorphic disc satisfying
\begin{equation*}
\begin{cases}
f(0) = p\\
f^{(\ell)}(0) = 0, \ \ell = 1,\ldots, 2n-2\\
f^{(2n-1)}(0) = (2n-1)!rv\\
\end{cases}
\end{equation*}
for some $r>0$. Consider the holomorphic disc $g: \mathbb D \to \mathbb C^3$ defined by
$$g(\zeta) = (\zeta^{3}f_1(\zeta), \zeta^2f_2(\zeta), f_3(\zeta))$$
for $\zeta \in \mathbb D.$ We first note that $g: \mathbb D \to \Omega$ since
\begin{eqnarray*}
\rho\left(g(\zeta),\overline{g(\zeta)}\right) &= & \mathbb{R}e e f_3(\zeta) + \big |\zeta^6f_1^2(\zeta) - \zeta^6{f_2}^3(\zeta)\big|^2 \\
& = & \mathbb{R}e e f_3(\zeta) + \big|\zeta\big|^{12}\big| {f_1}^2(\zeta) - {f_2}^3(\zeta)\big|^2 \\
& < & \rho\left(f(\zeta),\overline{f(\zeta)}\right)\\
\end{eqnarray*}
which is negative since $f:\mathbb D \to \Omega$. Moreover we have $g(0) = p$, $ g^{(\ell)}(0) = 0$ for any $\ell =1,\ldots, 2n$, and $g^{(2n+1)}(0) = (2n+1)!rv.$
This proves the lemma.
\end{proof}
Accordingly, we will then focus on estimating $K^{3}_\Omega\left((0,0,-1), (0,1,0)\right)$. We notice the basic estimate
\begin{equation}\label{eqbasic}
K^{3}\left((0,0,-1), (0,1,0)\right) \leq 1
\end{equation}
which simply follows from the previous lemma and Proposition \ref{propyu}.
Our main result is
\begin{theo}\label{theoest}
The $3^{rd}$ order Kobayashi pseudometric satisfies
\begin{equation}\label{eqK3}
K^3_\Omega\left((0,0,-1), (0,1,0)\right)\leq \left(\cfrac{8\pi}{1-e^{-2\pi}}\right)^{-1/3} \approx 0.3412.
\end{equation}
\end{theo}
\begin{proof}
The idea is to find relevant holomorphic discs contained in $\Omega$. Instead of simply giving the explicit expression of the disc leading to the estimate (\ref{eqK3}), we explain its
construction.
We seek a holomorphic disc $f = (f_1, f_2, f_3): \mathbb D \to \Omega$ satisfying
\begin{equation}\label{eqcond}
\begin{cases}
f(0) = (0,0,-1) \\
f^{(\ell)}(0) = 0, \ \ell = 1, 2\\
\displaystyle f^{(3)}(0)= 6r (0,1,0)\\
\end{cases}
\end{equation}
with $r>0$. We will fix $f_3 \equiv -1$. Such a disc can be written as
$$f(\zeta)= \left(f_1(\zeta), f_2(\zeta), -1\right) = \left(\zeta ^4 h_1(\zeta), \zeta^3 h_2(\zeta), -1\right)$$
for some holomorphic functions $h_1, h_2: \mathbb D \to \mathbb C$. We assume that $h_1(0) = 1$ and we
write
$$h_1(\zeta) = 1 + \zeta\varphi(\zeta)$$
for some holomorphic function $\varphi$. The problem is then to find two holomorphic functions $\varphi$ and $h_2$ with $h_2(0) = r >1$ with the condition that the corresponding disc $f$ is contained in $\Omega$. Using the defining function $\rho$, we
need to ensure
$$-1+ \big|\zeta^8(1 + \zeta\varphi(\zeta))^2 - \zeta^9 h_2^3(\zeta)\big|^2 = -1+ \big|\zeta\big|^{16} \big|(1 + \zeta\varphi(\zeta))^2 - \zeta h_2(\zeta)^3\big|^2 <0,$$
that is,
$$ \big|\zeta\big|^{16} \big|1 + 2\zeta\varphi(\zeta) + \zeta^2\varphi(\zeta)^2 - \zeta h_2(\zeta)^3\big|^2 < 1$$
This leads us to find holomorphic functions $\varphi$ and $h_2$ satisfying the equation
\begin{equation}\label{eqkey}
\varphi(\zeta) (2+ \zeta\varphi(\zeta)) = h_2(\zeta)^3
\end{equation}
We then construct $\varphi$ so that both $\varphi(\zeta)$ and $2+ \zeta\varphi(\zeta)$ have no zero in the unit disc $\mathbb D$. Consider
\begin{equation}\label{eqphi}
\varphi(\zeta)=\frac{e^\zeta-1}{\zeta}.
\end{equation}
This holomorphic function has no zeros on $\mathbb D$ and the same holds for
$$2+ \zeta\varphi(\zeta)=1+e^\zeta.$$
We may then take for $h_2$ a cubic root of $\varphi(\zeta) (2+ \zeta\varphi(\zeta))$. It follows that the holomorphic disc
$$\left(\zeta^4\left(1+\zeta \frac{e^\zeta -1}{\zeta}\right),\zeta^3h_2(\zeta),-1\right)=\left(\zeta^4e^\zeta,\zeta^3h_2(\zeta),-1\right)$$
is contained in $\Omega$, satisfies the conditions (\ref{eqcond}), and since $h_2(0)=2^{1/3}$ we obtain the estimate
$$K^3_\Omega\left((0,0,-1), (0,1,0)\right)\leq \frac{1}{2^{1/3}}$$
which refines our basic estimate (\ref{eqbasic}). Following this approach, we now modify the above function $\varphi$ in (\ref{eqphi}) to sharpen this estimate. Thus we consider
$$\varphi(z) = \alpha \cfrac{e^{\beta \zeta} -1}{\zeta},$$
with $\alpha,\beta>0$.
We have $\varphi(0) = \alpha \beta$, and $\varphi(\zeta) = 0$ if and only if $\beta\zeta \in 2\pi i\mathbb{Z}\setminus\{0\}$. Therefore, to ensure that $\varphi$ has no zeros in $\mathbb D$, we need to have
\begin{equation}\label{eqbeta}
\beta \leq 2 \pi.
\end{equation}
We turn to the function
$$2+ \zeta\varphi(\zeta)=2-\alpha+\alpha e^{\beta \zeta}$$
which vanishes when $\displaystyle e^{\beta \zeta}=\frac{\alpha-2}{\alpha}$. To make sure that $2+ \zeta\varphi(\zeta)$ has no zeros in $\mathbb D$, we need
\begin{equation}\label{eqalpha}
\log\left(\cfrac{\alpha-2}{\alpha}\right) \leq -\beta.
\end{equation}
Moreover, using Equation (\ref{eqkey}), we have
$$r^3=h_2^3(0)=2\alpha \beta.$$
The value $r$ is the largest when equality occurs in (\ref{eqbeta}) and (\ref{eqalpha}), namely when $\beta=2\pi$ and $\alpha=\cfrac{2}{1 - e^{-2\pi}}$. This construction leads us to consider the holomorphic disc
$$\displaystyle f(z) = \left(\zeta^4\left(1 +\frac{2(e^{2\pi \zeta} -1)}{1 - e^{-2\pi}}\right), \zeta^3h_2(\zeta), -1\right),$$
where $h_2$ is a well defined holomorphic disc defined by Equation (\ref{eqkey}). The constructed disc is contained in $\Omega$, satisfies the conditions (\ref{eqcond}), and since
$h_2(0)=\left(\cfrac{8\pi}{1-e^{-2\pi}}\right)^{1/3}$ we obtain the desired estimate
\begin{equation*}
K^3_\Omega\left((0,0,-1), (0,1,0)\right)\leq \left(\cfrac{8\pi}{1-e^{-2\pi}}\right)^{-1/3}.
\end{equation*}
\end{proof}
\begin{rem}
Note that instead of $f_1(\zeta) = \zeta^4h_1(\zeta)$, one could have considered a disc of the form $f_1(\zeta) = \zeta^nh_1(\zeta)$ with $n >4$. However, in case $n>4$, the condition to ensure that the disc
$\left(\zeta^nh_1(\zeta),\zeta^3h_2(\zeta),-1\right)$ is contained in the domain $\Omega$ becomes
$$\left| f_1^2 - f_2^3\right|^2 = |\zeta|^{18}\left|\zeta^{2n-9}h_1^2(\zeta) - h_2^3(\zeta)\right|^2 < 1.$$
A classical subharmonicity argument (see e.g. p.105 \cite{yu}) implies $|h_2(0)|=r \leq 1$, which does not improve the estimate we have obtained in Theorem \ref{theoest}.
\end{rem}
\subsection{On the standard Kobayashi metric}
As an interesting application of the method developed in the proof of Theorem \ref{theoest}, we are able to obtain the exact value of the standard Kobayashi metric in the domain $\Omega$ in certain new cases.
Consider the point $z_t=(0,0,-t)$ with $0<t<1$, and the vector $X=(a,b,0)$ with $|a|^2+|b|^2=1$. In case $a \neq 0$, the following estimate was obtained by Yu in \cite{yu}:
\begin{equation}\label{eqeq}
|a|t^{-\frac{1}{4}} \leq K_{\Omega}(z_t,X)\leq t^{-\frac{1}{4}}.
\end{equation}
As pointed out by Jarnicki and Pflug in p.145 \cite{ja-pf}, it would be interesting to know the exact value of $K_{\Omega}(z_t,X)$. We are able to answer this question in some cases.
\begin{prop}\label{propexact}
Assume that $\displaystyle \frac{|b|^3}{|a|^3}\leq \frac{2}{\sqrt{t}}\min\left\{2\pi,\log \left(1+2t^{\frac{1}{4}}\right)\right\}$. Then we have
$$ K_{\Omega}(z_t,X)= |a|t^{-\frac{1}{4}}.$$
\end{prop}
Essentially, this proposition states that for any point $z_t=(0,0,-t) \in \Omega$ with $0<t<1$ on the normal line through the origin, there exists a region of directions for which we know the exact value of the Kobayashi metric.
\begin{proof}
We follow the strategy of the proof of Theorem \ref{theoest}.
Consider the holomorphic disc $f=(f_1,f_2,-t)$ of the form
$$f(\zeta)=\left(\zeta \left(\frac{a}{|a|t^{-\frac{1}{4}}}+\zeta \varphi(\zeta)\right),\zeta h_2(\zeta) -t\right),$$
where
$$\displaystyle \varphi(\zeta)=\frac{e^{\frac{b^3\sqrt{t}}{2a|a|^2}\zeta}-1}{\zeta},$$
and where $h_2$ is such that
\begin{equation}\label{eqkey2}
\varphi(\zeta)\left(2\frac{a}{|a| t^{-\frac{1}{4}}}+\zeta \varphi(\zeta)\right) = h_2^3(\zeta)
\end{equation}
We first show that $h_2$ is a well defined holomorphic disc. We note that $\varphi$ has no zero in the unit disc. We need to show that the same occurs for
$$2\frac{a}{|a|t^{-\frac{1}{4}}}+\zeta \varphi=2\frac{a}{|a|t^{-\frac{1}{4}}}+e^{\frac{b^3\sqrt{t}}{2a|a|^2}\zeta}-1.$$ Note that on $\partial \mathbb D$, we have
$$\left|e^{\frac{b^3\sqrt{t}}{2a|a|^2}\zeta}-1\right|=\left|\sum_{k\geq1} \left(\frac{b^3\sqrt{t}}{2a|a|^2}\right)^k\frac{\zeta^k}{k!}\right |\leq \sum_{k\geq1} \frac{1}{k!}\left(\frac{|b|^3\sqrt{t}}{2|a|^3}\right)=e^{\frac{|b|^3\sqrt{t}}{2|a|^3}}-1.$$
The latter is less than $2t^{\frac{1}{4}}$ since $\frac{|b|^3\sqrt{t}}{|a|^3}\leq \log \left(1+2t^{\frac{1}{4}}\right)$. By Rouch\'e's theorem, it follows that $2\frac{a}{|a|t^{-\frac{1}{4}}}+\zeta \varphi$ has no zeros in the unit disc. Thus the function $h_2$, and so the disc $f$, are well defined.
Moreover, by construction we have $f(\mathbb D) \subset \Omega$. Indeed, using (\ref{eqkey2}) we have
\begin{eqnarray*}
-t+ \big|f_1^2(\zeta)-f_2^3(\zeta)\big|^2 &=& -t+ |\zeta|^4\left|\left(\frac{a}{|a|t^{-\frac{1}{4}}}+\zeta \varphi(\zeta)\right)^2-\zeta h_2^3(\zeta)\right|^2\\
\\
&=& -t+ |\zeta|^4\left|\left(\frac{a}{|a|t^{-\frac{1}{4}}}\right)^2\right|^2= -t+ |\zeta|^4t<0.\\
\end{eqnarray*}
Finally, we have $f(0)=z_t$ and $f'(0)=\frac{1}{ |a|t^{-\frac{1}{4}}}X$. Combined with (\ref{eqeq}), this proves the proposition.
\end{proof}
\section{On higher order extremal discs}
In this section, we are interested in the structure of the set of $k$-extremal discs. For $a\in \mathbb D$, we denote by $B_a:\mathbb D\to \mathbb D$ the Blaschke function $B_a(\zeta)=\cfrac{\zeta-a}{1-\overline{a}\zeta}$.
\subsection{General results}
We first start with the following proposition. Recall that for a domain $\Omega \subset \mathbb C^n$, the set of $k$-extremal discs for the pair $(p,v)$ is denoted by $X_{\Omega}^k(p,v)$.
\begin{lem}\label{lemext}
Let $\Omega \subset \mathbb C^n$ be a bounded domain. Let $k>0$ be a positive integer and $p\in \Omega$ and $v \in T_p\Omega$.
Then $K^k_{\Omega}(p,v) = K_{\Omega}(p,v)$ if and only if $\{f(\zeta^k) \ | \ f \in X_{\Omega}(p,v)\} \subset X_{\Omega}^k(p,v)$.
\end{lem}
\begin{proof}
Let $p \in \Omega$ and $v \in T_p\Omega$.
Assume first that $K^k_{\Omega}(p,v) = K_{\Omega}(p,v)$. Let $f\in X_{\Omega}(p,v)$ and set $h(\zeta)=f(\zeta^k)$. We note that $h(0)=p$, $\nu(h)\geq k$,
$$h^{(k)}(0)=k!f^{(k)}(0)=\frac{k!}{K_{\Omega}(p,v)}v=\frac{k!}{K^k_{\Omega}(p,v)}v$$
which shows directly that $h \in X_{\Omega}^k(p,v)$.
Now assume that $\{f(\zeta^k) \ | \ f \in X_{\Omega}(p,v)\} \subset X_{\Omega}^k(p,v)$. Let $f$ be an extremal disc for the pair $(p,v)$ and define the disc $h(\zeta)=f(\zeta^k)$. As before we have
$h(0)=p$, $\nu(h)\geq k$, and
$$h^{(k)}(0)=\frac{k!}{K_{\Omega}(p,v)}v,$$
and since $h\in X_{\Omega}^k(p,v)$, we have $h^{(k)}(0)=\frac{k!}{K^k_{\Omega}(p,v)}v$, leading to $K^k_{\Omega}(p,v) = K_{\Omega}(p,v)$.
\end{proof}
In general, the equality does not hold, even if $K^k_{\Omega}(p,v) = K_{\Omega}(p,v)$.
\begin{ex}
Consider the bidisc $\mathbb D\times \mathbb D \subset \mathbb C^2$. Note that
$$K^2_{\mathbb D\times \mathbb D}((0,0),(1,0))= K_{\mathbb D\times \mathbb D}((0,0),(1,0))$$
by the product property of the higher order Kobayashi pseudometric (see e.g. Proposition 3.8.7 in \cite{ja-pf}) and by the fact that $K^2_{\mathbb D} \equiv K_{\mathbb D}$.
According to Schwarz Lemma, we have
$$X_{\mathbb D\times \mathbb D}((0,0),(1,0)) = \{ (\zeta, \zeta^2\psi(\zeta)) \ | \ \psi: \mathbb D \to \mathbb D \text{ holomorphic}\}.$$
The holomorphic disc $\zeta \mapsto (\zeta^2 , \zeta^3) \in X_{\Omega}^2((0,0),(1,0))$ and is not of the form $f(\zeta^2)$ for some extremal disc $f \in X_{\Omega}((0,0),(1,0)).$
\end{ex}
It is interesting to note that in case $\Omega$ is the unit disc $\mathbb D$ or the punctured disc $\mathbb D\setminus\{0\}$, we have
\begin{prop}
Let $\Omega=\mathbb D$ or $\mathbb D\setminus\{0\}$. Then for any $(p,v) \in \Omega \times T_p\Omega$, we have
$$\{f(\zeta^k) \ | \ f \in X_{\Omega}(p,v)\} = X_{\Omega}^k(p,v).$$
\end{prop}
\begin{proof}
According to Proposition \ref{propprop} and Lemma \ref{lemext} we only need to show that $X_{\Omega}^k(p,v) \subset \{f(\zeta^k) \ | \ f \in X_{\Omega}(p,v)\}$. This follows from the equality cases established in the corresponding higher order Schwarz lemmas (Lemma \ref{lemhigh} and Lemma \ref{lempdhigh}) and from the invariance of extremal discs (Lemma \ref{leminv}).
\end{proof}
\subsection{The case of complex ellipsoids in $\mathbb C ^2$}
The Kobayashi metric on convex and nonconvex complex ellipsoids has been studied by many authors \cite{po,bl-fa-kl-kr-ma-pa,ja-pf-ze,pf-zw,ed}. It is interesting to note that in the papers \cite{po,ja-pf-ze,pf-zw,ed}, the metric is studied by means of extremal discs.
In this section we focus on the {\it nonconvex complex ellipsoid}
$$\mathcal{E}(1,m) = \{ (z_1,z_2) \in \mathbb C^2 \; \big| \; |z_1|^{2} + |z_2|^{2m} < 1\}$$
where $m \in (0,1/2)$.
We first note that $\mathcal{E}(1,m)$ is a bounded balanced pseudoconvex domain, that is if $z\in\mathcal{E}(1,m)$ and $\lambda\in\bar{\mathbb D}$ then $\lambda z\in\mathcal{E}(1,m)$. Accordingly, the higher order Kobayashi metric coincide with the Kobayashi metric at the origin $(0,0)$ (see e.g. Theorem 2.4 in \cite{ki}).
\begin{theo}\label{theopext}
Let $(a,b) \in \mathcal{E}(1,m)$ and $v \in \mathbb C^2$. Let $f \in X_{\mathcal{E}(1,m)}\left((a,b),v\right)$. Then the map $f(\zeta^k)$ is $k$-stationary in the $L^{\infty}$ sense.
\end{theo}
\begin{proof}
Following \cite{pf-zw} and \cite{ed}, we consider two kind of maps (see Proposition 2 \cite{pf-zw} and Theorem 2 \cite{ed}).
The first sort $\varphi=(\varphi_1,\varphi_2): \mathbb D \to \mathcal{E}(1,m)$ of is of the form
\begin{equation}\label{eqkind1}
\varphi(\zeta)=\left(a_1B_{\alpha_1}^{r_1}(\zeta)\cdot \frac{1-\overline{\alpha_1}\zeta}{1-\overline{\alpha_0}\zeta},
a_2 B_{\alpha_2}^{r_2}(\zeta)\cdot \left(\frac{1-\overline{\alpha_2}\zeta}{1-\overline{\alpha_0}\zeta}\right)^{\frac{1}{2m}}\right)
\end{equation}
with
$$
\begin{cases}
a_1,a_2 \in \mathbb C^{*}, \alpha_0 \in \mathbb D, \alpha_1,\alpha_2 \in \overline{\mathbb D}\\
\\
r_j \in \{0,1\}, j=1,2 \mbox{ and if} \ r_j=1 \mbox{ then}\ \alpha_j \in \mathbb D\\
\\
\alpha_0=|a_1|^2\alpha_1+|a_2|^{2m}\alpha_2\\
\\
1+|\alpha_0|^2=|a_1|^2(1+|\alpha_1|^2)+|a_2|^{2m}(1+|\alpha_2|^2)
\end{cases}
$$
The second kind of maps $\psi=(\psi_1,\psi_2): \mathbb D \to \mathcal{E}(1,m)$ is of the form
\begin{equation}\label{eqkind2}
\psi(\zeta)=\left(a_1\prod_{\ell=1}^kB_{\alpha_{\ell 1}}^{r_{\ell 1}}(\zeta)\cdot \frac{1-\overline{\alpha_{\ell 1}}\zeta}{1-\overline{\alpha_{\ell 0}}\zeta},
a_2 \prod_{\ell=1}^k B_{\alpha_{\ell2}}^{r_{\ell2}}(\zeta)\cdot \left(\frac{1-\overline{\alpha_{\ell2}}\zeta}{1-\overline{\alpha_{0\ell}}\zeta}\right)^{\frac{1}{2m}}\right)
\end{equation}
with
$$
\begin{cases}
a_1,a_2 \in \mathbb C\setminus\{0\}, \alpha_{\ell j} \in \overline{\mathbb D}, \ell=1,\ldots,k, j=0,1,2\\
\\
r_{\ell j} \in \{0,1\}, \ell=1,\ldots,k, j=1,2, \mbox{ and if} \ r_{\ell j}=1 \mbox{ then} \ \alpha_{\ell j} \in \mathbb D\\
\\
\displaystyle |a_1|^2\prod_{\ell=1}^k(\zeta-\alpha_{\ell 1})(1-\overline{\alpha_{\ell 1}}\zeta)+|a_2|^{2m}\prod_{\ell=1}^k(\zeta-\alpha_{\ell2})(1-\overline{\alpha_{\ell2}}\zeta)
=\prod_{\ell=1}^k(\zeta-\alpha_{\ell 0})(1-\overline{\alpha_{\ell 0}}\zeta)
\\
\end{cases}
$$
Consider now an extremal disc $\varphi=(\varphi_1,\varphi_2) \in X_{\mathcal{E}(1,m)}\left((a,b),v\right)$.
According to Pflug and Zwonek \cite{pf-zw}, the map $\varphi$ is necessarily of the form (\ref{eqkind1}). We wish to show that $\varphi(\zeta^k)$ is of the kind (\ref{eqkind2}).
For this purpose, for $j=0,1,2$, we consider the $k^{th}$-roots of $\alpha_j$ and denote them by $\alpha_{1j},\alpha_{2j}\ldots,\alpha_{kj}$. We also set $r_{\ell j}=r_j$ for all $\ell=1,\ldots,k$ and $ j=1,2$.
Define
$$I=\displaystyle |a_1|^2\prod_{\ell=1}^k(\zeta-\alpha_{\ell 1})(1-\overline{\alpha_{\ell 1}}\zeta)+|a_2|^{2m}\prod_{\ell=1}^k(\zeta-\alpha_{\ell2})(1-\overline{\alpha_{\ell2}}\zeta)
$$
We compute, using $\zeta^k-\alpha_j=\prod_{\ell=1}^k(\zeta-\alpha_{kj})$ and $1-\overline{\alpha_j}\zeta^k=\prod_{\ell=1}^k(1-\overline{\alpha_{kj}}\zeta^2)$,
\begin{eqnarray*}
I&=&\displaystyle |a_1|^2(\zeta^k-\alpha_1)(1-\overline{\alpha_{1}}\zeta^k)+|a_2|^{2m}(\zeta^k-\alpha_2)(1-\overline{\alpha_{2}}\zeta^k)\\
\\
& =& -\left(|a_1|^2\overline{\alpha_{1}}+|a_2|^{2m}\overline{\alpha_{2}}\right)\zeta^{2k}+ \left(|a_1|^2(1+|\alpha_1|^2)+|a_2|^{2m}(1+|\alpha_2|^2)\right)\zeta^k -\left(|a_1|^2\alpha_{1}+|a_2|^{2m}\alpha_{2}\right)\\
\\
& =& -\overline{\alpha_0}\zeta^{2k}+ \left(1+|\alpha_0|^2\right)\zeta^k -\alpha_0=(\zeta^k-\alpha_0)(1-\overline{\alpha_0}\zeta^k)=\prod_{\ell=1}^k(\zeta-\alpha_{\ell 0})(1-\overline{\alpha_{\ell 0}}\zeta).\\
\end{eqnarray*}
and
\begin{eqnarray*}
\varphi(\zeta^k)&=&\left(a_1B_{\alpha_1}^{r_1}(\zeta^k)\cdot \frac{1-\overline{\alpha_1}\zeta^k}{1-\overline{\alpha_0}\zeta^k},
a_2 B_{\alpha_2}^{r_2}(\zeta^k)\cdot \left(\frac{1-\overline{\alpha_2}\zeta^k}{1-\overline{\alpha_0}\zeta^k}\right)^{\frac{1}{2m}}\right)\\
\\
& =&\left(a_1\prod_{\ell=1}^kB_{\alpha_{\ell 1}}^{r_{\ell 1}}(\zeta)\cdot \frac{1-\overline{\alpha_{\ell 1}}\zeta}{1-\overline{\alpha_{\ell 0}}\zeta},
a_2 \prod_{\ell=1}^k B_{\alpha_{\ell2}}^{r_{\ell2}}(\zeta)\cdot \left(\frac{1-\overline{\alpha_{\ell2}}\zeta}{1-\overline{\alpha_{0\ell}}\zeta}\right)^{\frac{1}{2m}}\right).\\
\end{eqnarray*}
Therefore the map $\varphi(\zeta^k)$ is exactly of the form (\ref{eqkind2}). Following a variational approach due to Poletsky \cite{po},
Edigarian \cite{ed} showed that maps of the form (\ref{eqkind2}) are solutions of a certain Euler-Lagrange equation
(see Problem ($\mathcal{P}$) of $m$-type p.84 \cite{ed}). Together with Remark 11.4.4 in \cite{ja-pf}, it follows that such maps $k$-stationary in the $L^{\infty}$ (see also \cite{be-de-jo}).
This shows that $\varphi(\zeta^k)$ is then $k$-stationary in the $L^{\infty}$.
\end{proof}
\begin{rem}
We could have focused on maps of the form (\ref{eqkind1}) and (\ref{eqkind2}) centered at $(0,b)$. Indeed, we know (see e.g. \cite{pf-zw}) that for $a \in \mathbb D$ and $\theta \in \mathbb{R}$, the map $F_{a,\theta}$ defined by
$$F_{a,\theta}(z_1,z_2)= \left(\cfrac{z_1 - a}{1 - a\bar z_1} , \cfrac{e^{i \theta}(1 - |a|^2)^{\frac{1}{2m}} z_2}{(1 - a\bar z_1)^{\frac{1}{m}}}\right)$$
is an automorphism of $\mathcal{E}(1,m)$ which maps any point $(a,c) \in \mathcal{E}(1,m)$ to a point $(0,b) \in \mathcal{E}(1,m)$ with $b \in [0,1)$. Moreover, by Lemma \ref{leminv}, we have $$ (F_{a, \theta})_* X^k_{\mathcal{E}(1,m)}\left((a,c), d_{(0,0)}F_{a, \theta}(v)\right) =X^k_{\mathcal{E}(1,m)} \big((0,b), v\big).$$
Nevertheless, we decided to keep the more general form of such maps since the simplification of notation is barely noticeable.
\end{rem}
\begin{rem}
In case the complex ellipsoid $\mathcal{E}(1,m)$ is convex, that is, when $m>\frac{1}{2}$. We know from Lempert theory \cite{le} that the Kobayashi metric and the Carath\'eodory metric
coincide. Therefore by Proposition \ref{propprop}, the higher order Kobayashi metrics and the standard Kobayashi metric coincide; this fact was already observed by many authors
\cite{ja-pf,yu}. According to Lemma \ref{lemext}, we have
$$\left\{f(\zeta^k) \ | \ f \in X_{\mathcal{E}(1,m)}\left((a,b),v\right)\right\} \subset X^k_{\mathcal{E}(1,m)}\left((a,b),v\right).$$
It follows that if $\varphi$ is a map of the form (\ref{eqkind1}), then $\varphi(\zeta^k)$ is a $k$-extremal and, thus, must be of the form (\ref{eqkind2}). Nevertheless, without a direct computation, the dependance of the parameters $\alpha_{\ell j}$ on the parameters $\alpha_j$ is only implicit.
\end{rem}
\section{Appendix: on higher order Schwarz lemmas}
In this appendix, we establish higher order versions of Schwarz type lemmas in the vein of Lemma \ref{lemhigh}.
\begin{lem}\label{lemsphigh}
Let $f: \mathbb D \rightarrow \mathbb D$ be a holomorphic function and $\zeta \in \mathbb D$ with $f^{(\ell)}(\zeta) = 0$ for all $ \ell=1,\ldots,k-1$. We have for all $w \in \mathbb D$
\begin{equation}\label{eqsphigh1}
\left|\cfrac{f(\zeta)-f(w)}{1 - \overline{f(\zeta)}f(w)}\right| \leq \bigg|\cfrac{\zeta-w}{1-\mathbb Bar{\zeta}w}\bigg|^k
\end{equation}
and
\begin{equation}\label{eqsphigh2}
\left|f^{(k)}(\zeta)\right| \leq k! \cfrac{1-|f(\zeta)|^2}{(1-|\zeta|^2)^k}.
\end{equation}
\noindent
Moreover, in case of equality (with $w\neq \zeta$) then $f$ is of the form
$f(w) = \cfrac{e^{i\theta}(B_{\zeta}(w))^k + f(\zeta)}{1 + e^{i\theta}\overline{f(\zeta)}(B_{\zeta}(w))^k}.$
\end{lem}
\begin{proof}
Let $\zeta,w \in \mathbb D$. Set $a=f(\zeta) $ and consider the function $g= B_a\circ f\circ B_{-\zeta}:\mathbb D \to \mathbb D$. We have $g(0) = 0$ and, for $\ell=1,\ldots,k-1$
$$g^{(\ell)}(0) = (B_a \circ f\circ B_{-\zeta})^{(\ell)}(0) = 0$$
since $(f\circ B_{-\zeta})^{(\ell)}(0) = 0$. Moreover, a straightforward computation also leads to
$$g^{(k)}(0) = B_a'((f\circ B_{-\zeta})(0))\cdot (f\circ B_{-\zeta})^{(k)}(0) = B_a'(a)\cdot f^{(k)}(\zeta) \cdot (B_{-\zeta}'(0))^k.$$
By applying Lemma \ref{lemhigh} with $g$, we obtain for $\tilde{w}\in \mathbb D$
$$|g(\tilde{w})| = \bigg|\cfrac{f(B_{-\zeta}(\tilde{w})) - a}{1 - \overline{a}f(B_{-\zeta_1}(\tilde{w}))}\bigg| \leq |\tilde{w}|^k,$$
which, for $\tilde{w}= B_{\zeta}(w)$, gives (\ref{eqsphigh1}), and
$$\left|g^{(k)}(0)\right| = \left|f^{(k)}(\zeta)\right| \cdot \left|B_a'(a)\right| \cdot \left|(B_{-\zeta}'(0))\right|^k \leq k!$$
which implies (\ref{eqsphigh2}).
Finally, if equality holds, then by Lemma \ref{lemhigh} we have $g(\zeta) = e^{i\theta}\zeta^{k}$ for some $\theta \in \mathbb{R}$, and thus
$$f(w) = B_{-a}\circ g \circ B_{\zeta}(w) = \cfrac{e^{i\theta}(B_{\zeta}(w))^k + f(\zeta)}{1 + e^{i\theta}\overline{f(\zeta)}(B_{\zeta}(w))^k}.$$ \end{proof}
We also establish a higher order Schwarz Lemma in the case of the punctured disc.
\begin{lem}\label{lempdhigh}
Let $f: \mathbb D \rightarrow \mathbb D\setminus\{0\}$ be a holomorphic function such that $f^{(\ell)}(0) = 0$ for all $\ell=1,\ldots,k-1$. Then
\begin{equation}\label{eqpdhigh}
\left|f^{(k)}(0)\right| \leq -2k!|f(0)|\log|f(0)|.
\end{equation}
Moreover in case of equality then $f$ is of the form
\begin{equation*}
f(\zeta) = e^{i\alpha} e^{ \log|f(0)|\cfrac{1+e^{i\theta}\zeta^k}{1 - e^{i\theta}\zeta^k}}
\end{equation*}
for some $\theta \in \mathbb{R}$ and where $f(0)=|f(0)|e^{i\alpha}$.
\end{lem}
\begin{proof} We write $f$ as an exponential $f=e^g$ for some holomorphic function
$$g: \mathbb D \to \left\{ \zeta \in \mathbb C \ | \ \mathbb{R}e e\zeta<0\right\}.$$
It follows that $\varphi \circ (-ig)$, where $\varphi:\mathbb{H} \to \mathbb D$ is the Cayley transform $\varphi(\zeta)=\frac{\zeta-i}{\zeta+i}$, is a self map of the unit disc. For $\ell=1,\ldots,k-1$ we have $g^{(\ell)}(0)$ and thus, after a direct computation we obtain
$$(\varphi \circ (-ig))^{(\ell)}(0) = 0$$
and
$$(\varphi \circ (-ig))^{(k)}(0) = \varphi'(-ig(0)) g^{(k)}(0) = \frac{\varphi'(-ig(0)) f^{(k)}(0)}{f(0)}.$$
We now apply Lemma \ref{lemsphigh} to $\varphi \circ (-ig)$ and get
$$\bigg|\cfrac{(\varphi \circ (-ig))^{(k)}(0)}{k!}\bigg| = \frac{1}{k!} \bigg|\cfrac{2}{(g(0) - 1)^2}\, \cdot \, \cfrac{f^{(k)}(0)}{ f(0)}\bigg| \leq 1 - |(\varphi\circ (-ig))(0)|^2 = 1 - \left| \cfrac{g(0) +1 }{g(0) -1}\right|^2.$$
This implies
$$\frac{2}{k!}\left|\cfrac{f^{(k)}(0)}{ f(0)}\right| \leq |1 - g(0)|^2 - |1 + g(0)|^2 = 2(-g(0) - \overline{g(0)}) = -4\mathbb{R}e e g(0) = -4\log|f(0)|$$
and proves (\ref{eqpdhigh}).
We now write $f(0) = |f(0)|e^{i\alpha}$ with $0 \leq \alpha \leq 2\pi$. Then, we can renormalize the holomorphic function g such that $ f = e^{i\alpha + g}$ where $g(0) = \log|f(0)| \in \mathbb{R}$.
In case of equality in (\ref{eqpdhigh}), then by Lemma \ref{lemsphigh} we have
$$\varphi \circ (-ig) = B_{-\varphi(-ig(0))} \circ (e^{i\theta}\zeta^k)$$
and so
$$ g(\zeta) =- \cfrac{B_{-\varphi(-ig(0))}\circ (e^{i\theta}\zeta^k) + 1}{1 - B_{-\varphi(-ig(0))} \circ (e^{i\theta}\zeta^k)} =
g(0)\cfrac{e^{1+i\theta}\zeta^k}{1 - e^{i\theta}\zeta^k}$$
leading to (\ref{eqpdhigh}).
\end{proof}
\noindent {\it Acknowledgments.} This work was done in the framework of the Summer Research Camp in Mathematics designed by the Department of Mathematics at the American University of Beirut (AUB) and that benefitted from a generous support from the Center for Advanced Mathematical Sciences at AUB.
{\small
\noindent Seok Ban, Florian Bertrand, Amir Jaber Chehayeb, Adam Salha, Walid Tabbara\\
Department of Mathematics\\\
American University of Beirut, Beirut, Lebanon\\{\sl E-mail addresses}: [email protected], [email protected], [email protected], [email protected], [email protected]\\
}
\end{document}
|
\begin{document}
\title[Separated bump conditions]
{On separated bump conditions for Calder\'on-Zygmund operators}
\author[A. K. Lerner]{Andrei K. Lerner}
\address[A. K. Lerner]{Department of Mathematics,
Bar-Ilan University, 5290002 Ramat Gan, Israel}
\email{[email protected]}
\thanks{The author was supported by ISF grant No. 447/16 and ERC Starting Grant No. 713927.}
\begin{abstract}
We improve bump conditions for the two-weight boundedness of Calder\'on-Zygmund operators introduced recently by R. Rahm and S. Spencer \cite{RS}.
\end{abstract}
\keywords{Calder\'on-Zygmund operators, sparse operators, bump conditions.}
\subjclass[2010]{42B20, 42B25}
\maketitle
\section{Introduction}
Let $T$ be a Calder\'on-Zygmund operator on ${\mathbb R}^n$. In this note we are concerned with separated bump conditions on a couple of weights $(w,\si)$ for which
\begin{equation}\label{twocz}
\|T(f\si)\|_{L^p(w)}\lesssim \|f\|_{L^p(\si)}\quad(1<p<\infty).
\end{equation}
Given a cube $Q\subset {\mathbb R}^n$, denote $f_Q=\frac{1}{|Q|}\int_Qf$. It is well known that the standard $A_p$ condition,
$$
[w,\si]_{A_p}=\sup_Qw_Q(\si_Q)^{p-1}<\infty,
$$
is not sufficient for (\ref{twocz}). In fact, the $A_p$ condition is not sufficient even for the maximal operator instead of $T$ \cite{M}.
A number of works were devoted to finding slightly stronger conditions that are sufficient for (\ref{twocz}). Among such conditions one can distinguish joint and separated bump conditions.
By a joint bump condition we generally mean a condition of the form
$$\sup_QB_1[w;Q](B_2[\si;Q])^{p-1}<\infty,$$
where $B_1[w;Q]$ and $B_2[\si;Q]$ are referred to as bumps, that is, expressions slightly larger than $w_Q$ and $\si_Q$, respectively.
For joint bump conditions, see, e.g., \cite{L,Li,NRTV}.
By a separated bump condition one means a more delicate and weaker condition of the form
$$\sup_QB_1[w;Q](\si_Q)^{p-1}<\infty\quad\text{and}\quad \sup_Qw_Q(B_2[\si;Q])^{p-1}<\infty.$$
There are several different ways of ``bumping", see \cite{CRV,La,Li} for the Orlicz bumps and \cite{LS,RS,TV} for the so-called entropy bumps; we recall them below, in Section 5.
In a recent work by R. Rahm and S. Spencer \cite{RS}, yet another bumps were introduced. Assume that $\f_p$ is a function that is decreasing on $(0,1)$ and increasing on $(1,\infty)$
and such that $\int_0^{\infty}\frac{1}{\f_p(t)^{1/p}}\frac{dt}{t}<\infty$. It was shown in \cite{RS} that if
$$\sup_Qw_Q(\si_Q)^{p-1}\f_p(\si_Q)<\infty\quad\text{and}\quad \sup_Qw_Q(\si_Q)^{p-1}\f_{p'}^{p-1}(w_Q)<\infty,$$
then (\ref{twocz}) holds.
In this note we improve the integrability assumptions on $\f_p$ in the above result. In particular, we will show that for $t\ge 1$, the assumption $\int_1^{\infty}\frac{1}{\f_p(t)^{1/p}}\frac{dt}{t}<\infty$
can be improved to $\int_1^{\infty}\frac{1}{\f_p(t)}\frac{dt}{t}<\infty$. For $0<t<1$ our condition looks a bit technical but it shows that, for example, one can take
$\f_p(t)=\log\big(e+\frac{1}{t}\big)\log\log^{p+\e}\big(e^{e}+\frac{1}{t}\big).$
As in the previous works on this topic, our proof is based on the sparse domination.
The paper is organized as follows. In Section 2 we recall the standard scheme of reducing the initial problem to analysis of testing conditions.
Section 3 contains some, mostly known, auxiliary statements. The main result is contained in Section~4. In Section 5 we give a brief overview of known bump conditions.
Section 6 contains some further remarks and complements.
Throughout the paper we use the notation $A\lesssim B$ if $A\le CB$ with some independent constant $C$. We write $A\simeq B$ if $A\lesssim B$ and $B\lesssim A$.
\section{Standard reductions}
As in most of the previous works dealing with bump conditions, we will make use of the following tools.
\begin{list}{\labelitemi}{\leftmargin=1em}
\item Reducing to sparse operators. Recall that the sparse operator $A_{\mathcal S}$ is defined by
$$A_{\mathcal S}f=\sum_{Q\in {\mathcal S}}f_Q\chi_Q,$$
where ${\mathcal S}$ is a sparse family of dyadic cubes, which means that there exist disjoint subsets $E_Q\subset Q\in {\mathcal S}$ such that $|E_Q|\simeq |Q|$.
Since a Calder\'on-Zygmund operator~$T$ is pointwise bounded by at most $3^n$ sparse operators $A_{\mathcal S}$ (see, e.g., \cite{LO} for a very short proof of this fact), the problem is reduced
to finding sufficient conditions for $A_{\mathcal S}$ instead of $T$ in (\ref{twocz}).
\item In turn, the two-weight boundedness for $A_{\mathcal S}$ is characterized by testing conditions. Denote by $[w,\si]_p$ the best possible constant such that
for every $R\in {\mathcal S}$,
\begin{equation}\label{ws}
\Big\|\sum_{Q\in {\mathcal S}, Q\subseteq R}\si_Q\chi_Q\Big\|_{L^p(w)}\le [w,\si]_p\si(R)^{1/p}.
\end{equation}
Then (see \cite{H,LSU,T})
$$\|A_{{\mathcal S}}(\cdot\si)\|_{L^p(\si)\to L^p(w)}\simeq [w,\si]_p+[\si,w]_{p'}.$$
Thus, separated bump conditions typically appear as conditions on $(w,\si)$ for which $[w,\si]_p$ and $[\si,w]_{p'}$ are finite. By symmetry, it suffices to analyze $[w,\si]_p$.
\item As it was shown in \cite{HL, Li}, a very efficient way to handle the left-hand side of (\ref{ws}) is based on the following facts established respectively in \cite{COV} and \cite{H}:
for every dyadic lattice ${\mathscr D}$ and any non-negative sequence $\{a_Q\}_{Q\in {\mathscr D}}$,
\begin{equation}\label{cov}
\Big\|\sum_{Q\in {\mathscr D}}a_Q\chi_Q\Big\|_{L^p(w)}\simeq \left(\sum_{Q\in {\mathscr D}}a_Q\Big(\frac{1}{w(Q)}\sum_{Q'\in {\mathscr D}, Q'\subseteq Q}a_{Q'}w(Q')\Big)^{p-1}w(Q)\right)^{1/p},
\end{equation}
and for every sparse family ${\mathcal S}$ and $0<s<1$,
\begin{equation}\label{h}
\sum_{Q\in {\mathcal S}, Q\subseteq R}(w_Q)^{s}|Q|\lesssim (w_R)^{s}|R|.
\end{equation}
\item It was proved in \cite{H} that for every sparse family of dyadic cubes ${\mathcal S}$,
\begin{equation}\label{hyt}
\int_R\Big(\sum_{Q\in {\mathcal S}, Q\subseteq R}\si_Q\chi_Q\Big)^pw\lesssim (\sup_{Q\in{\mathcal S}}w_Q\si_Q^{p-1})\sum_{Q\in {\mathcal S},Q\subseteq R}\si(Q)
\end{equation}
(observe that this can be shown by combining (\ref{cov}) and (\ref{h})).
\end{list}
\section{Auxiliary propositions}
The following result is based on the same ideas as in \cite{La,Li}.
\begin{prop}\label{mp} Given a weight $\si$ and a cube $Q$, define $\la_Q(\si)$ such that $\la_Q(\si)\ge 1$ and for every sparse family ${\mathcal S}$,
\begin{equation}\label{conds}
\sum_{Q\in {\mathcal S}, Q\subseteq R}\la_Q(\si)^{-1}\si(Q)\lesssim \si(R).
\end{equation}
Then
$$[w,\si]_p\lesssim \sup_{Q\in {\mathcal S}}(w_Q)^{1/p}(\si_Q)^{1/p'}\la_Q(\si)^{1/p}\f^{1/p'}(\la_Q(\si)),$$
where $\f$ is an increasing function such that $\int_1^{\infty}\frac{1}{\f(t)}\frac{dt}{t}<\infty$.
\end{prop}
\begin{proof} For $k\ge 0$ define
\begin{equation}\label{fk}
{\mathcal F}_k=\{Q\in {\mathcal S}: 2^k\le \la_Q(\si)\le 2^{k+1}\}.
\end{equation}
Then, setting $M=\sup_{Q\in {\mathcal S}}w_Q(\si_Q)^{p-1}\la_Q(\si)\f^{p-1}(\la_Q(\si))$, by (\ref{hyt}) we obtain
\begin{eqnarray}
&&\Big\|\sum_{Q\in {\mathcal S}, Q\subseteq R}\si_Q\chi_Q\Big\|_{L^p(w)}\le \sum_{k=0}^{\infty}\Big\|\sum_{Q\in {\mathcal F}_k, Q\subseteq R}\si_Q\chi_Q\Big\|_{L^p(w)}\nonumber\\
&&\lesssim \sum_{k=0}^{\infty}\Big((\sup_{Q\in{\mathcal F}_k}w_Q\si_Q^{p-1})\sum_{Q\in {\mathcal F}_k,Q\subseteq R}\si(Q)\Big)^{1/p}\nonumber\\
&&\lesssim M^{1/p}\sum_{k=0}^{\infty}\frac{1}{\f(2^k)^{1/p'}}\Big(\sum_{Q\in {\mathcal F}_k,Q\subseteq R}\la_Q(\si)^{-1}\si(Q)\Big)^{1/p}\label{subs}\\
&&\lesssim M^{1/p}\Big(\sum_{k=0}^{\infty}\frac{1}{\f(2^k)}\Big)^{1/p'}\Big(\sum_{Q\in {\mathcal S}, Q\subseteq R}\la_Q(\si)^{-1}\si(Q)\Big)^{1/p}\lesssim M^{1/p}\si(R)^{1/p},\nonumber
\end{eqnarray}
which completes the proof.
\end{proof}
The following proposition is contained in \cite{RS}. We give its proof for the sake of completeness.
\begin{prop}\label{rs} Let ${\mathcal S}$ be a sparse family of dyadic cubes. For $k\in {\mathbb Z}$ define
$${\mathcal F}_k=\{Q\in {\mathcal S}: 2^k<\si_Q\le 2^{k+1}\}.$$
Then
$$\sum_{Q\in {\mathcal F}_k,Q\subseteq R}\si(Q)\lesssim \si(R).$$
\end{prop}
\begin{proof} Let $\{Q_j\}$ be the maximal cubes of $\{Q\in {\mathcal F}_k, Q\subseteq R\}$. Then, by maximality, they are pairwise disjoint, and also, by sparseness,
\begin{eqnarray*}
\sum_{Q\in {\mathcal F}_k,Q\subseteq R}\si(Q)&=&\sum_j\sum_{Q\in {\mathcal F}_k,Q\subseteq Q_j}\si(Q)\le 2^{k+1}\sum_j\sum_{Q\in {\mathcal F}_k,Q\subseteq Q_j}|Q|\\
&\lesssim& 2^k\sum_j|Q_j|\lesssim \sum_j\si(Q_j)\lesssim \si(R),
\end{eqnarray*}
which completes the proof.
\end{proof}
\begin{prop}\label{psi} Let $\psi$ be a function that is decreasing on $(0,1)$ and increasing on $(1,\infty)$, and such that $\int_0^{\infty}\frac{1}{\psi(t)}\frac{dt}{t}<\infty$.
Then
$$\sum_{Q\in {\mathcal S},Q\subseteq R}\frac{1}{\psi(\si_Q)}\si(Q)\lesssim \si(R).$$
\end{prop}
\begin{proof} Setting ${\mathcal F}_k$ as in the previous proposition, we obtain
$$
\sum_{Q\in {\mathcal S},Q\subseteq R}\frac{1}{\psi(\si_Q)}\si(Q)=\sum_{k\in {\mathbb Z}}\sum_{Q\in {\mathcal F}_k,Q\subseteq R}\frac{1}{\psi(\si_Q)}\si(Q)\lesssim
\Big(\sum_{k\in {\mathbb Z}}\frac{1}{\psi(2^k)}\Big)\si(R),
$$
and we are done.
\end{proof}
\section{Main result}
\begin{theorem}\label{mr}
Assume that $\psi$ is a function that is decreasing on $(0,1)$ and increasing on $(1,\infty)$, and such that $\int_0^{\infty}\frac{1}{\psi(t)}\frac{dt}{t}<\infty$. Assume also that
$\psi(t)\lesssim e^{\sqrt t}$ for $t\ge 1$. Next, let $\f$ be an increasing function on $(1,\infty)$ such that $\int_1^{\infty}\frac{1}{\f(t)}\frac{dt}{t}<\infty$. Define
$$
\nu_p(t) = \begin{cases} \psi(t)\f^{p-1}(\psi(t)), &0<t<1\\
\psi(t),& t\ge 1, \end{cases}
$$
and set
$$[w,\si]_{\nu_p}=\sup_{Q}w_Q(\si_Q)^{p-1}\nu_p(\si_Q).$$
Then
$$\|T(\cdot\si)\|_{L^p(\si)\to L^p(w)}\lesssim [w,\si]_{\nu_p}^{1/p}+[\si,w]_{\nu_{p'}}^{1/p'}.$$
\end{theorem}
\begin{remark}\label{r1} A typical example of $\nu_p$ on $(0,1)$ can be obtained by setting
$$\psi(t)=\log\Big(e+\frac{1}{t}\Big)\log\log^{1+\e}\Big(e^e+\frac{1}{t}\Big)\quad(0<t<1)$$
and
$$\f(t)=\log(e+t)\log\log^{1+\e}(e^e+t)\quad(1<t<\infty).$$
Then, for $0<t<1$,
$$
\nu_p(t)\simeq
\log\Big(e+\frac{1}{t}\Big)\log\log^{p+\e}\Big(e^e+\frac{1}{t}\Big)\log\log\log^{(p-1)(1+\e)}\Big(e^{e^e}+\frac{1}{t}\Big).
$$
\end{remark}
\begin{remark}\label{r2} In the cases of interest the growth of $\psi$ for $t\ge 1$ is logarithmic (e.g., $\psi(t)=\log^{1+\e}(e+t)$
or $\psi(t)=\log(e+t)\log\log^{1+\e}(e^e+t)$, etc.), and hence the assumption
$\psi(t)\lesssim e^{\sqrt t}$ for $t\ge 1$ holds trivially.
\end{remark}
\begin{proof}[Proof of Theorem \ref{mr}] As we have discussed in Section 2, it suffices to estimate $[w,\si]_p$ in (\ref{ws}), that is, our goal is to show that
\begin{equation}\label{goal}
\Big\|\sum_{Q\in {\mathcal S}, Q\subseteq R}\si_Q\chi_Q\Big\|_{L^p(w)}\lesssim [w,\si]_{\nu_p}^{1/p}\si(R)^{1/p}.
\end{equation}
Set
$${\mathcal S}_1=\{Q\in {\mathcal S}:\si_Q<1\}\quad\text{and}\quad {\mathcal S}_2=\{Q\in {\mathcal S}:\si_Q\ge 1\}.$$
An immediate combination of Propositions \ref{mp} and \ref{psi} yields
\begin{eqnarray*}
\Big\|\sum_{Q\in {\mathcal S}_1, Q\subseteq R}\si_Q\chi_Q\Big\|_{L^p(w)}
&\lesssim& \Big(\sup_{Q\in {\mathcal S}_1}(w_Q)^{1/p}(\si_Q)^{1/p'}\psi(\si_Q)^{1/p}\f^{1/p'}(\psi(\si_Q))\Big)\si(R)^{1/p}\\
&\lesssim& [w,\si]_{\nu_p}^{1/p}\si(R)^{1/p}.
\end{eqnarray*}
Therefore, in order to prove (\ref{goal}), it remains to show that
\begin{equation}\label{rem}
\Big\|\sum_{Q\in {\mathcal S}_2, Q\subseteq R}\si_Q\chi_Q\Big\|_{L^p(w)}^p\lesssim [w,\si]_{\nu_p}\si(R).
\end{equation}
At this point we apply (\ref{cov}), which says that
\begin{equation}\label{inn}
\Big\|\sum_{Q\in {\mathcal S}_2, Q\subseteq R}\si_Q\chi_Q\Big\|_{L^p(w)}^p\simeq \sum_{Q\in {\mathcal S}_2, Q\subseteq R}\si_Q\Big(\frac{1}{w(Q)}
\sum_{Q'\in {\mathcal S}_2, Q'\subseteq Q}\si_{Q'}w(Q')\Big)^{p-1}w(Q).
\end{equation}
Split the inner sum on the right-hand side of~(\ref{inn}) as follows:
\begin{equation}\label{nspl}
\sum_{Q'\in {\mathcal S}_2, Q'\subseteq Q}\si_{Q'}w(Q')=
\sum_{Q'\in {\mathcal S}_2, Q'\subseteq Q\atop \si_{Q'}\le \si_Q^{1/2}}\si_{Q'}w(Q')+
\sum_{Q'\in {\mathcal S}_2, Q'\subseteq Q\atop \si_{Q'}>\si_Q^{1/2}}\si_{Q'}w(Q').
\end{equation}
Suppose first that $p\ge 2$.
Consider the first sum on the right-hand side of (\ref{nspl}). Denote
$${\mathcal F}_0=\{Q'\in {\mathcal S}_2: w_{Q'}(\si_{Q'})^{p-1}\le [w,\si]_{A_p}\psi(\si_Q)^{-1}\}$$
and, for $k=1,\dots,N\simeq \log\big(\psi(\si_Q)\big)$,
$${\mathcal F}_k=\{Q'\in {\mathcal S}_2: 2^{k-1}[w,\si]_{A_p}\psi(\si_Q)^{-1}<w_{Q'}(\si_{Q'})^{p-1}\le 2^{k}[w,\si]_{A_p}\psi(\si_Q)^{-1}\},$$
where, abusing the notation, we set
$$[w,\si]_{A_p}=\sup_{Q\in {\mathcal S}_2}w_Q(\si_Q)^{p-1}.$$
Denote also
$$E_k=\{Q'\in {\mathcal F}_k: Q'\subseteq Q, \si_{Q'}\le \si_Q^{1/2}\}.$$
Then
\begin{eqnarray*}
\sum_{{Q'\in {\mathcal S}_2, Q'\subseteq Q}\atop \si_{Q'}\le \si_Q^{1/2}}\si_{Q'}w(Q')=\sum_{k=0}^N\sum_{Q'\in E_k}\si_{Q'}w(Q').\\
\end{eqnarray*}
By (\ref{hyt}),
\begin{eqnarray*}
\sum_{Q'\in E_0}\si_{Q'}w(Q')
&\le&\left(\frac{[w,\si]_{A_p}}{\psi(\si_Q)}\right)^{\frac{1}{p-1}}\sum_{{Q'\in {\mathcal S}_2, Q'\subseteq Q}}(w_{Q'})^{1-\frac{1}{p-1}}|Q'|\\
&\lesssim &\left(\frac{[w,\si]_{A_p}}{\psi(\si_Q)}\right)^{\frac{1}{p-1}}(w_{Q})^{1-\frac{1}{p-1}}|Q|.
\end{eqnarray*}
Fix $1\le k\le N$. Let $\{Q_j\}$ be the maximal cubes of $E_k$.
Then, by (\ref{hyt}),
\begin{eqnarray*}
\sum_{Q'\in E_k,Q'\subseteq Q_j}\si_{Q'}w(Q')
&\le&\left(\frac{2^{k}[w,\si]_{A_p}}{\psi(\si_Q)}\right)^{\frac{1}{p-1}}\sum_{Q'\in E_k,Q'\subseteq Q_j}(w_{Q'})^{1-\frac{1}{p-1}}|Q'|\\
&\lesssim& \left(\frac{2^{k}[w,\si]_{A_p}}{\psi(\si_Q)}\right)^{\frac{1}{p-1}}(w_{Q_j})^{1-\frac{1}{p-1}}|Q_j|\\
&\lesssim& \si_{Q_j}(w_{Q_j})^{\frac{1}{p-1}}(w_{Q_j})^{1-\frac{1}{p-1}}|Q_j|\lesssim \si_Q^{1/2}w(Q_j).
\end{eqnarray*}
Hence, since $\{Q_j\}$ are pairwise disjoint,
$$
\sum_{Q'\in E_k}\si_{Q'}w(Q')=\sum_j\sum_{Q'\in E_k,Q'\subseteq Q_j}\si_{Q'}w(Q')
\lesssim \si_Q^{1/2}w(Q).
$$
From this, and using also that, by our assumption, $\log\big(\psi(\si_Q)\big)\lesssim \si_Q^{1/2}$, we obtain
$$\sum_{k=1}^N\sum_{Q'\in E_k}\si_{Q'}w(Q')\lesssim \log\big(\psi(\si_Q)\big)\si_Q^{1/2}w(Q)\lesssim \si_Qw(Q).$$
Collecting the above estimates yields
$$\sum_{Q'\in {\mathcal S}_2, Q'\subseteq Q\atop \si_{Q'}\le \si_Q^{1/2}}\si_{Q'}w(Q')\lesssim
\left(\frac{[w,\si]_{A_p}}{\psi(\si_Q)}\right)^{\frac{1}{p-1}}(w_{Q})^{1-\frac{1}{p-1}}|Q|+\si_Qw(Q).$$
Therefore, by Proposition \ref{psi},
\begin{eqnarray}
&&\sum_{Q\in {\mathcal S}_2, Q\subseteq R}\si_Q\Big(\frac{1}{w(Q)}\sum_{Q'\in {\mathcal S}_2, Q'\subseteq Q\atop \si_{Q'}\le \si_Q^{1/2}}\si_{Q'}w(Q')\Big)^{p-1}w(Q)\label{part}\\
&&\lesssim [w,\si]_{A_p}\sum_{Q\in {\mathcal S}_2, Q\subseteq R}\frac{1}{\psi(\si_Q)}\si(Q)+\sum_{Q\in {\mathcal S}_2, Q\subseteq R}\si_Q^{p}w(Q)\nonumber\\
&&\lesssim [w,\si]_{\nu_p}\sum_{Q\in {\mathcal S}_2, Q\subseteq R}\frac{1}{\psi(\si_Q)}\si(Q)\lesssim [w,\si]_{\nu_p}\si(R).\nonumber
\end{eqnarray}
Consider the second sum on the right-hand side of (\ref{nspl}). Since $\psi$ is increasing on $(1,\infty)$,
\begin{eqnarray*}
\sum_{{Q'\in {\mathcal S}_2, Q'\subseteq Q}\atop \si_{Q'}>\si_Q^{1/2}}\si_{Q'}w(Q')
&\le& \psi(\si_Q^{1/2})^{-\frac{1}{p-1}}
\sum_{{Q'\in {\mathcal S}_2, Q'\subseteq Q}\atop \si_{Q'}>\si_Q^{1/2}}\si_{Q'}w(Q')\psi(\si_{Q'})^{\frac{1}{p-1}}\\
&\le& [w,\si]_{\nu_p}^{\frac{1}{p-1}}\psi(\si_Q^{1/2})^{-\frac{1}{p-1}}\sum_{{Q'\in {\mathcal S}_2, Q'\subseteq Q}}(w_{Q'})^{1-\frac{1}{p-1}}|Q'|\\
&\lesssim& [w,\si]_{\nu_p}^{\frac{1}{p-1}}\psi(\si_Q^{1/2})^{-\frac{1}{p-1}}(w_{Q})^{1-\frac{1}{p-1}}|Q|.\\
\end{eqnarray*}
Hence,
\begin{eqnarray*}
&&\sum_{Q\in {\mathcal S}_2, Q\subseteq R}\si_Q\Big(\frac{1}{w(Q)}\sum_{Q'\in {\mathcal S}_2, Q'\subseteq Q\atop \si_{Q'}>\si_Q^{1/2}}\si_{Q'}w(Q')\Big)^{p-1}w(Q)\\
&&\lesssim [w,\si]_{\nu_p}\sum_{Q\in {\mathcal S}_2, Q\subseteq R}\frac{1}{\psi(\si_Q^{1/2})}\si(Q)\lesssim [w,\si]_{\nu_p}\si(R),\\
\end{eqnarray*}
where we have used again Proposition \ref{psi} and that $\int_1^{\infty}\frac{1}{\psi(t^{1/2})}\frac{dt}{t}<\infty.$
This, along with (\ref{part}), completes the proof of (\ref{rem}) in the case $p\ge 2$.
In the case $1<p<2$ the proof is similar. Consider the first sum on the right-hand side of (\ref{nspl}).
Define the sets $E_k$ exactly as in the previous case. By (\ref{hyt}),
\begin{eqnarray*}
\sum_{Q'\in E_0}\si_{Q'}w(Q')
&\le&\frac{[w,\si]_{A_p}}{\psi(\si_Q)}\sum_{{Q'\in {\mathcal S}_2, Q'\subseteq Q}}\si_{Q'}^{2-p}|Q'|\\
&\lesssim & \frac{[w,\si]_{A_p}}{\psi(\si_Q)}\si_{Q}^{2-p}|Q|.
\end{eqnarray*}
Fix $1\le k\le N$. Let $\{Q_j\}$ be the maximal cubes of $E_k$.
Then
\begin{eqnarray*}
&&\sum_{Q'\in E_k,Q'\subseteq Q_j}\si_{Q'}w(Q')\le\frac{2^{k}[w,\si]_{A_p}}{\psi(\si_Q)}\sum_{Q'\in E_k,Q'\subseteq Q_j}\si_{Q'}^{2-p}|Q'|\\
&&\lesssim \frac{2^{k}[w,\si]_{A_p}}{\psi(\si_Q)}\si_{Q_j}^{2-p}|Q_j|\lesssim w_{Q_j}\si_{Q_j}^{p-1}\si_{Q_j}^{2-p}|Q_j|\lesssim \si_Q^{1/2}w(Q_j).
\end{eqnarray*}
Therefore,
$$
\sum_{Q'\in E_k}\si_{Q'}w(Q')=\sum_{j}\sum_{Q'\in E_k,Q'\subseteq Q_j}\si_{Q'}w(Q')\lesssim \si_Q^{1/2}w(Q),
$$
which implies
\begin{eqnarray*}
\sum_{{Q'\in {\mathcal S}_2, Q'\subseteq Q}\atop \si_{Q'}\le \si_Q^{1/2}}\si_{Q'}w(Q')&\lesssim& \frac{[w,\si]_{A_p}}{\psi(\si_Q)}\si_{Q}^{2-p}|Q|+\log\big(\psi(\si_Q)\big)\si_Q^{1/2}w(Q)\\
&\lesssim& \frac{[w,\si]_{A_p}}{\psi(\si_Q)}\si_{Q}^{2-p}|Q|+\si_Qw(Q).
\end{eqnarray*}
Hence, by Proposition \ref{psi},
\begin{eqnarray}
&&\sum_{Q\in {\mathcal S}_2, Q\subseteq R}\si_Q\Big(\frac{1}{w(Q)}\sum_{Q'\in {\mathcal S}_2, Q'\subseteq Q\atop \si_{Q'}\le \si_Q^{1/2}}\si_{Q'}w(Q')\Big)^{p-1}w(Q)\label{part2}\\
&&\lesssim [w,\si]_{A_p}^{p-1}\sum_{Q\in {\mathcal S}_2, Q\subseteq R}\frac{1}{\psi^{p-1}(\si_Q)}\big((\si_Q)^{p-1}w_Q\big)^{2-p}\si(Q)+\sum_{Q\in {\mathcal S}_2, Q\subseteq R}\si_Q^{p}w(Q)\nonumber\\
&&\lesssim [w,\si]_{\nu_p}\sum_{Q\in {\mathcal S}_2, Q\subseteq R}\frac{1}{\psi(\si_Q)}\si(Q)\lesssim [w,\si]_{\nu_p}\si(R).\nonumber
\end{eqnarray}
Further, arguing as above,
\begin{eqnarray*}
\sum_{{Q'\in {\mathcal S}_2, Q'\subseteq Q}\atop \si_{Q'}>\si_Q^{1/2}}\si_{Q'}w(Q')&\le& \psi(\si_Q^{1/2})^{-1}
\sum_{{Q'\in {\mathcal S}_2, Q'\subseteq Q}\atop \si_{Q'}>\si_Q^{1/2}}\si_{Q'}w(Q')\psi(\si_{Q'})\\
&\lesssim &\frac{[w,\si]_{\nu_p}}{\psi(\si_Q^{1/2})}\si_Q^{2-p}|Q|.
\end{eqnarray*}
Therefore,
\begin{eqnarray*}
&&\sum_{Q\in {\mathcal S}_2, Q\subseteq R}\si_Q\Big(\frac{1}{w(Q)}\sum_{Q'\in {\mathcal S}_2, Q'\subseteq Q\atop \si_{Q'}>\si_Q^{1/2}}\si_{Q'}w(Q')\Big)^{p-1}w(Q)\\
&&\lesssim [w,\si]_{\nu_p}^{p-1}\sum_{Q\in {\mathcal S}_2, Q\subseteq R}\frac{1}{\psi^{p-1}(\si_Q^{1/2})}\big((\si_Q)^{p-1}w_Q\big)^{2-p}\si(Q)\\
&&\lesssim [w,\si]_{\nu_p}\sum_{Q\in {\mathcal S}_2, Q\subseteq R}\frac{1}{\psi(\si_Q^{1/2})}\si(Q)\lesssim [w,\si]_{\nu_p}\si(R),
\end{eqnarray*}
which, along with (\ref{part2}), proves (\ref{rem}) in the case $1<p<2$. This completes the proof.
\end{proof}
\section{A brief survey of different bump conditions} The approach described in Proposition \ref{mp} is the key to different bump conditions, and we overview them briefly.
\subsection{Orlicz bumps}
Recall that for a Young function $A$, the normalized Luxemburg norm is defined by
$$
\|f\|_{A,Q}=\inf\Big\{\la>0:\frac{1}{|Q|}\int_QA(|f(y)|/\la)dy\le 1\Big\}.
$$
Define the maximal operator $M_A$ by
$$M_Af(x)=\sup_{Q\ni x}\|f\|_{A,Q}.$$
We say that a Young function $A$ satisfies the $B_p$ condition if $\int_1^{\infty}\frac{A(t)}{t^p}\frac{dt}{t}<\infty$.
Assume that $A\in B_p$, and set $\la_Q(\si)=\frac{\si_Q}{\|\si^{1/p}\|_{A,Q}^p}$. Then
\begin{eqnarray*}
&&\sum_{Q\in {\mathcal S},Q\subseteq R}\la_Q(\si)^{-1}\si(Q)=\sum_{Q\in {\mathcal S},Q\subseteq R}\|\si^{1/p}\|_{A,Q}^p|Q|\\
&&\lesssim \sum_{Q\in {\mathcal S},Q\subseteq R}\int_{E_Q}(M_A(\si^{1/p}\chi_R))^p \lesssim \int_{R}(M_A(\si^{1/p}\chi_R))^p \lesssim \si(R),
\end{eqnarray*}
where we have used that $M_A$ is bounded on $L^p$ for $A\in B_p$ \cite{P}.
Hence, by Proposition \ref{mp},
\begin{equation}\label{kli}
[w,\si]_p\lesssim \sup_Q(w_Q)^{1/p}\frac{\si_Q}{\|\si^{1/p}\|_{A,Q}}\f^{1/p'}\left(\frac{\si_Q}{\|\si^{1/p}\|_{A,Q}^p}\right).
\end{equation}
This result was obtained by K. Li \cite{Li}.
Let $\bar A$ denote the Young function complementary to $A$.
By generalized H\"older's inequality,
$$
\si_Q\le 2\|\si^{1/p}\|_{A,Q}\|\si^{1/p'}\|_{\bar A, Q}.
$$
From this and from (\ref{kli}),
\begin{equation}\label{la}
[w,\si]_p\lesssim \sup_Q(w_Q)^{1/p}\|\si^{1/p'}\|_{\bar A, Q}\f^{1/p'}\left(\frac{\|\si^{1/p'}\|_{\bar A,Q}^p}{(\si_Q)^{p-1}}\right).
\end{equation}
This result was obtained by M. Lacey \cite{La}.
\subsection{Entropy bumps}
Assume that instead of (\ref{conds}) we have
$$
\sum_{Q\in {\mathcal F}_k, Q\subseteq R}\si(Q)\lesssim 2^k\si(R),
$$
where the sets ${\mathcal F}_k$ are defined by (\ref{fk}). In this case, setting
$$M=\sup_Qw_Q(\si_Q)^{p-1}\la_Q(\si)\f^{p}(\la_Q(\si)),$$
instead of (\ref{subs}) we obtain
$$M^{1/p}\sum_{k=0}^{\infty}\frac{1}{\f(2^k)}\Big(\sum_{Q\in {\mathcal F}_k,Q\subseteq R}\la_Q(\si)^{-1}\si(Q)\Big)^{1/p}\lesssim M^{1/p}\si(R)^{1/p},$$
which implies
\begin{equation}\label{entb}
[w,\si]_p\lesssim \sup_Q(w_Q)^{1/p}(\si_Q)^{1/p'}\la_Q(\si)^{1/p}\f(\la_Q(\si)).
\end{equation}
Denote $\la_Q(\si)=\frac{\int_QM(\si\chi_Q)}{\si(Q)}$, and let us consider the sets ${\mathcal F}_k$ defined by (\ref{fk}).
Let $\{Q_j\}$ be the maximal cubes of $\{Q\in {\mathcal F}_k, Q\subseteq R\}$. Then
\begin{eqnarray*}
\sum_{Q\in {\mathcal F}_k, Q\subseteq R}\si(Q)=\sum_j\sum_{Q\in {\mathcal F}_k, Q\subseteq Q_j}\si(Q)\lesssim \sum_j\int_{Q_j}M(\si\chi_{Q_j})\lesssim
2^k\sum_j\si(Q_j)\lesssim 2^k\si(R).
\end{eqnarray*}
Therefore, by (\ref{entb}),
\begin{equation}\label{entropy}
[w,\si]_p\lesssim \sup_Q(w_Q)^{1/p}(\si_Q)^{1/p'}\left(\frac{\int_QM(\si\chi_Q)}{\si(Q)}\right)^{1/p}\f\left(\frac{\int_QM(\si\chi_Q)}{\si(Q)}\right),
\end{equation}
where $\int_1^{\infty}\frac{1}{\f(t)}\frac{dt}{t}<\infty$.
In the case $p=2$ this result was obtained by S. Treil and A.~Volberg \cite{TV} (who gave the name ``entropy bumps" to the bumps appearing in this expression),
and it was extended to any $p>1$ by M. Lacey and S. Spencer \cite{LS} (see also \cite{RS}).
\section{Remarks and complements}
\subsection{Comparison between different bump conditions}
Although we do not give concrete examples, it is not difficult to see that among bump conditions mentioned
above, there is no universally better condition.
For example, the entropy bump condition appearing in (\ref{entropy}) requires that $\si$ belongs locally to $L\log L$, while in (\ref{goal}) and (\ref{kli}) only local integrability of $\si$ is assumed.
The difference between (\ref{goal}) and (\ref{kli}) is expressed in the difference between
$$\la_Q(\si)=\psi(\si_Q)\quad\text{and}\quad \la_Q(\si)=\frac{\si_Q}{\|\si^{1/p}\|_{A,Q}^p}.$$
On the one hand, by homogeneity, $\psi(\si_Q)$ can not be estimated by $\frac{\si_Q}{\|\si^{1/p}\|_{A,Q}^p}$. On the other hand, for $A\in B_p$, it is easy to find a sequence $\si_j$ such that
$(\si_j)_Q=1$ and $\|\si_j^{1/p}\|_{A,Q}\to 0$ as $j\to \infty$ (it suffices to consider $\si=\frac{|Q|}{|E|}\chi_E$ for $E\subset Q$), and therefore, $\frac{\si_Q}{\|\si^{1/p}\|_{A,Q}^p}$ can not be estimated by
$\psi(\si_Q)$.
Concerning practical applications, the condition in Theorem \ref{mr} is the simplest as it basically requires only the computation of $w_Q$ and $\si_Q$. To check (\ref{kli}), one should estimate $\|\si^{1/p}\|_{A,Q}$
from below, which is a more difficult task.
\subsection{A new two-weight bound for the maximal operator}
Let $M$ denote the Hardy-Littlewood maximal operator. Using Sawyer's two-weight characterization for $M$ \cite{S}, it is easy to show that
$$\|M(\cdot\si)\|_{L^p(\si)\to L^p(w)}\lesssim \left(\sup_R\frac{1}{\si(R)}
\sum_{Q\in {\mathcal S}, Q\subseteq R}\si_Q^pw(Q)\right)^{1/p}.
$$
Let $\la_Q(\si)$ satisfy (\ref{conds}). Then
\begin{equation}\label{th}
\sum_{Q\in {\mathcal S}, Q\subseteq R}\si_Q^pw(Q)\lesssim \big(\sup_{Q}w_Q(\si_Q)^{p-1}\la_Q(\si)\big)\si(R).
\end{equation}
Therefore,
$$
\|M(\cdot\si)\|_{L^p(\si)\to L^p(w)}\lesssim \sup_{Q}w_Q^{1/p}\si_Q^{1/p'}\la_Q(\si)^{1/p}.
$$
Combining this with Proposition \ref{psi} yields
\begin{equation}\label{nb}
\|M(\cdot\si)\|_{L^p(\si)\to L^p(w)}\lesssim \sup_{Q}w_Q^{1/p}\si_Q^{1/p'}\psi(\si_Q)^{1/p}.
\end{equation}
This bound seems to be new.
\subsection{On the separated bump conjecture}
The separated bump conjecture (probably first formulated in \cite{CRV} in a slightly different form) asserts that if $A\in B_p$, then
\begin{equation}\label{sepcon}
[w,\si]_p\lesssim \sup_Q(w_Q)^{1/p}\|\si^{1/p'}\|_{\bar A,Q}.
\end{equation}
This conjecture is still open. In the particular case when $A(t)=\frac{t^p}{\log^{1+\e}(e+t)}$ it was confirmed in \cite{CRV,HP,La};
in general, (\ref{kli}) and (\ref{la}) represent the currently best known bounds towards this conjecture.
Informally speaking, the idea behind the separated bump conjecture is that a ``good" upper bound for $\|M(\cdot\si)\|_{L^p(\si)\to L^p(w)}$ should also be an upper bound for $[w,\si]_p$. Having this point of view in mind and taking into
account (\ref{nb}), one can also conjecture that
\begin{equation}\label{nc}
[w,\si]_p\lesssim \sup_{Q}w_Q^{1/p}\si_Q^{1/p'}\psi(\si_Q)^{1/p},
\end{equation}
where $\psi$ satisfies the assumptions of Proposition \ref{psi}. Theorem \ref{mr} shows that this conjecture holds on the set $\{Q:\si_Q>1\}$.
One can also ask a weaker question whether the finiteness of the right-hand side of (\ref{sepcon}) or (\ref{nc}) implies that $[w,\si]_p<\infty$.
At this point, we make an elementary observation that if $E=\{Q:w_Q(\si_Q)^{p-1}\ge 1\}$, then, by(\ref{hyt}),
\begin{eqnarray*}
\int_R\Big(\sum_{Q\in {\mathcal S}\cap E, Q\subseteq R}\si_Q\chi_Q\Big)^pw&\lesssim& (\sup_{Q\in{\mathcal S}\cap E}w_Q\si_Q^{p-1})\sum_{Q\in {\mathcal S}\cap E,Q\subseteq R}\si(Q)\\
&\lesssim& [w,\si]_{A_p}\sum_{Q\in {\mathcal S},Q\subseteq R}\si_Q^pw(Q).
\end{eqnarray*}
This along with (\ref{th}) shows that in order to get a counterexample (if exists) to such a weaker question, the principal role should be played by cubes $Q$ with $w_Q(\si_Q)^{p-1}<1$.
\end{document}
|
\begin{document}
\title{Harmonic and Monogenic Potentials in Low Dimensional Euclidean Half--Space}
\author{F.\ Brackx, H.\ De Bie, H.\ De Schepper}
\date{\small{Clifford Research Group, Department of Mathematical Analysis,\\ Faculty of Engineering and Architecture, Ghent University\\
Building S22, Galglaan 2, B-9000 Gent, Belgium\\}}
\maketitle
\begin{abstract}
\noindent In the framework of Clifford analysis, a chain of harmonic and monogenic potentials in the upper half of Euclidean space $\mathbb{R}^{m+1}$ was constructed recently, including a higher dimensional analogue of the logarithmic function in the complex plane. Their distributional limits at the boundary $\mathbb{R}^{m}$ were also determined. In this paper the potentials and their distributional boundary values are calculated in dimensions 3 and 4, dimensions for which the expressions in general dimension break down.
\end{abstract}
\maketitle
\section{Introduction}
\langlebel{intro}
Recently, see \cite{bdbds1, bdbds2}, a generalization to Euclidean upper half--space $\mathbb{R}^{m+1}_+$ was constructed of the logarithmic function $\ln{z}$ which is holomorphic in the upper half of the complex plane. This construction was carried out in the framework of Clifford analysis, where the functions under consideration take their values in the universal Clifford algebra $\mathbb{R}_{0,m+1}$ constructed over the Euclidean space $\mathbb{R}^{m+1}$ equipped with a quadratic form of signature $(0,m+1)$. The concept of a higher dimensional holomorphic function, mostly called {\em monogenic} function, is expressed by means of the generalized Cauchy--Riemann operator $D$, which is a combination of the derivative with respect to one of the real variables, say $x_0$, and the so--called Dirac operator $\partialux$ in the remaining real variables $(x_1, x_2, \ldots, x_m)$. This generalized Cauchy--Riemann operator $D$ and its Clifford algebra conjugate $\overline{D}$ linearize the Laplace operator, whence Clifford analysis may be seen as a refinement of harmonic analysis.\\[-2mm]
The starting point for constructing a higher dimensional monogenic logarithmic function, was the fundamental solution of the generalized Cauchy--Riemann operator $D$, also called Cauchy kernel, and its relation to the Poisson kernel and its harmonic conjugate in $\mathbb{R}^{m+1}_+$. We then proceeded by induction in two directions, {\em downstream} by differentiation and {\em upstream} by primitivation, yielding a doubly infinite chain of monogenic, and thus harmonic, potentials. This chain mimics the well--known sequence of holomorphic potentials in $\mathbb{C}_+$ (see e.g. \cite{slang}):
$$
\frac{1}{k!} z^k \left[ \ln z - ( 1 + \frac{1}{2} + \ldots + \frac{1}{k}) \right] \rightarrow \ldots \rightarrow z ( \ln z - 1) \rightarrow \ln z
\stackrel{\frac{d}{dz}}{\longrightarrow} \frac{1}{z} \rightarrow - \frac{1}{z^2} \rightarrow \ldots \rightarrow (-1)^{k-1} \frac{(k-1)!}{z^k}
$$
Identifying the boundary of upper half--space with $\mathbb{R}^m \cong \{(x_0,\underline{x}) \in \mathbb{R}^{m+1} : x_0 = 0\}$, the distributional limits for $x_0 \rightarrow 0+$ of those potentials were computed. They split up into two classes of distributions, which are linked by the Hilbert transform, one scalar--valued, the second one Clifford vector--valued. They belong to two out of the four families of Clifford distributions which were thoroughly studied in a series of papers, see \cite{fb1, fb2, distrib} and the references therein.\\
The general expressions for the potentials and their boundary values, as established in \cite{bdbds1, bdbds2}, break down for dimensions $3 \ (m=2)$ and $4 \ (m=3)$, which are particularly important with a view on applications. In these specific cases ad hoc calculations have to be carried out. This is the aim of the underlying paper, which is a valuable and useful complement to \cite{bdbds1, bdbds2} . To make the paper self--contained the basics of Clifford algebra and Clifford analysis are recalled.
\section{Basics of Clifford analysis}
\langlebel{basics}
Clifford analysis (see e.g. \cite{red, green}) is a function theory which offers a natural and elegant generalization to higher dimension of holomorphic functions in the complex plane and refines harmonic analysis. Let $(e_0, e_1,\ldots,e_m)$ be the canonical orthonormal basis of Euclidean space $\mathbb{R}^{m+1}$ equipped with a quadratic form of signature $(0,m+1)$. Then the non--commutative multiplication in the universal real Clifford algebra $\mathbb{R}_{0,m+1}$ is governed by the rule
$$
e_{\alpha} e_{\beta} + e_{\beta} e_{\alpha} = -2 \delta_{\alpha \beta}, \qquad \alpha,\beta = 0, 1,\ldots,m
$$
whence $\mathbb{R}_{0,m+1}$ is generated additively by the elements $e_A = e_{j_1} \ldots e_{j_h}$, where $A=\lbrace j_1,\ldots,j_h \rbrace \subset \lbrace 0,\ldots,m \rbrace$, with $0\leq j_1<j_2<\cdots < j_h \leq m$, and $e_{\emptyset}=1$.
For an account on Clifford algebra we refer to e.g. \cite{porteous}.\\[-2mm]
We identify the point $(x_0, x_1, \ldots, x_m) \in \mathbb{R}^{m+1}$ with the Clifford--vector variable
$$
x = x_0 e_0 + x_1 e_1 + \cdots x_m e_m = x_0 e_0 + \underline{x}
$$
and the point $(x_1, \ldots, x_m) \in \mathbb{R}^{m}$ with the Clifford--vector variable $\underline{x}$.
The introduction of spherical co--ordinates $\underline{x} = r \underline{\omega}$, $r = |\underline{x}|$, $\underline{\omega} \in S^{m-1}$, gives rise to the Clifford--vector valued locally integrable function $\underline{\omega}$, which is to be seen as the higher dimensional analogue of the {\em signum}--distribution on the real line.\\[-2mm]
At the heart of Clifford analysis lies the so--called Dirac operator
$$
\partial = \partialxn e_0 + \partial_{x_1} e_1 + \cdots \partial_{x_m} e_m = \partialxn e_0 + \partialux
$$
which squares to the negative Laplace operator: $\partial^2 = - \Delta_{m+1}$, while also $\partialux^2 = - \Delta_{m}$. The fundamental solution of the Dirac operator $\partial$ is given by (see \cite{red,green})
$$
E_{m+1} (x) = - \frac{1}{\sigma_{m+1}} \ \frac{x}{|x|^{m+1}}
$$
where $\sigma_{m+1} = \frac{2\partiali^{\frac{m+1}{2}}}{\Gamma(\frac{m+1}{2})}$ stands for the area of the unit sphere $S^{m}$ in $\mathbb{R}^{m+1}$. It thus holds
$$
\partial E_{m+1} (x) = \delta(x)
$$
with $\delta(x)$ the standard Dirac distribution in $\mathbb{R}^{m+1}$.\\
We also introduce the generalized Cauchy--Riemann operator
$$
D = \onehalf \overline{e_0} \partial = \onehalf (\partialxn + \overline{e_0} \partialux)
$$
which, together with its Clifford algebra conjugate $\overline{D} = \onehalf(\partialxn - \overline{e_0} \partialux)$, linearizes the Laplace operator: $D \overline{D} = \overline{D} D = \frac{1}{4} \Delta_{m+1}$. \\[-2mm]
A continuously differentiable function $F(x)$, defined in an open region $\Omega \subset \mathbb{R}^{m+1}$ and taking values in the Clifford algebra $\mathbb{R}_{0,m+1}$ (or subspaces thereof), is called (left--)monogenic if it satisfies in $\Omega$ the equation $D F = 0$, which is equivalent with $\partial F = 0$. \\
Singling out the basis vector $e_0$, we can decompose the real Clifford algebra $\mathbb{R}_{0,m+1}$ in terms of the Clifford algebra
$\mathbb{R}_{0,m}$ as $\mathbb{R}_{0,m+1} = \mathbb{R}_{0,m} \oplus \overline{e_0} \, \mathbb{R}_{0,m}$. Similarly we decompose the considered functions as
$$
F(x_0,\underline{x}) = F_1(x_0,\underline{x}) + \overline{e_0} \, F_2(x_0,\underline{x})
$$
where $F_1$ and $F_2$ take their values in the Clifford algebra $\mathbb{R}_{0,m}$; mimicking functions of a complex variable, we will call $F_1$ the {\em real} part and $F_2$ the {\em imaginary} part of the function $F$.\\[-2mm]
\section{Harmonic and monogenic potentials in $\mathbb{R}_+^{3}$}
\langlebel{potentials3}
In this section we calculate the harmonic and monogenic potentials in upper half--space $\mathbb{R}^{3}_+$, as defined in general in \cite{bdbds1, bdbds2}.
\subsection{The Cauchy kernel in $\mathbb{R}_+^{3}$}
The starting point is the Cauchy kernel of Clifford analysis, i.e. the fundamental solution of the generalized Cauchy--Riemann operator $D$, which is, up to a constant factor, the fundamental solution $E_3(x)$ of the Dirac operator $\partial$, and is given by
$$
C_{-1}(x_0,\underline{x}) = \frac{1}{\sigma_{3}} \, \frac{x \overline{e_0}}{|x|^{3}} = \frac{1}{4\partiali} \, \frac{x_0 - \overline{e_0} \underline{x}}{(x_0^2+r^2)^{3/2}}
$$
where we have put $|\underline{x}| = r$.
It may be decomposed in terms of the traditional Poisson kernel $P(x_0,\underline{x})$ and its conjugate $Q(x_0,\underline{x})$ in $\mathbb{R}^{3}_+$:
$$
C_{-1}(x_0,\underline{x}) = \frac{1}{2} A_{-1}(x_0,\underline{x}) + \frac{1}{2} \overline{e_0} \, B_{-1}(x_0,\underline{x})
$$
where for $x_0 >0$,
\begin{eqnarray*}
A_{-1}(x_0,\underline{x}) & = & P(x_0,\underline{x}) \ = \ \partialhantom{-} \frac{1}{2\partiali} \, \frac{x_0}{(x_0^2+r^2)^{3/2}} \langlebel{A-1}\\
B_{-1}(x_0,\underline{x}) & = & Q(x_0,\underline{x}) \ = \ - \frac{1}{2\partiali} \, \frac{\underline{x}}{(x_0^2+r^2)^{3/2}} \langlebel{B-1}
\end{eqnarray*}
Their distributional limits for $x_0 \rightarrow 0+$ are given by
\begin{eqnarray*}
a_{-1}(\underline{x}) & = & \lim_{x_0 \rightarrow 0+} A_{-1}(x_0,\underline{x}) \ = \ \delta(\underline{x})\\
b_{-1}(\underline{x}) & = & \lim_{x_0 \rightarrow 0+} B_{-1}(x_0,\underline{x}) \ = \ H(\underline{x}) \ = \ - \frac{1}{2\partiali} \, \mbox{Pv} \frac{\underline{x}}{r^3} \ = \ - \frac{1}{2\partiali} \, \mbox{Pv} \frac{\underline{\omega}}{r^2}
\end{eqnarray*}
with $\mbox{Pv}$ standing for the "principal value" distribution in $\mathbb{R}^2$. The distribution
$
H(\underline{x})
$
is the convolution kernel of the Hilbert transform $\mathcal{H}$ in $\mathbb{R}^2$ (see e.g.\ \cite{gilmur}). Note that $\mathcal{H}^2 = \mathbf{1}$ and that this Hilbert transform links both distributional boundary values, as expressed in the following property, which in fact holds regardless the dimension.
\begin{property} One has
$$
\begin{array}{rc}
(i) \quad \mathcal{H} \left [ a_{-1} \right ] = \mathcal{H} \left [ \delta \right ] \ = \ H \ast \delta \ = \ H \ = \ b_{-1} \\[2mm]
(ii) \quad \mathcal{H} \left [ b_{-1} \right ] = \mathcal{H} \left [ H \right ] \ = \ H \ast H \ = \ \delta \ = \ a_{-1}
\end{array}
$$
\end{property}
\noindent Note also that $a_{-1}(\underline{x}) = \delta(\underline{x}) = E_0$ can be seen as the fundamental solution of the identity operator $\partialux^0 = {\bf 1}$, while $b_{-1}(\underline{x}) = H(\underline{x}) = - \frac{1}{2\partiali} \, \mbox{Pv} \frac{\underline{\omega}}{r^2} = F_0$ is the fundamental solution of the Hilbert operator $^0\mathcal{H} = \mathcal{H}$ (see \cite{bdbds2}).
\subsection{The downstream potentials in $\mathbb{R}_+^{3}$}
The first in the sequence of so--called {\em downstream} potentials is the function $C_{-2}$ defined by
$$
\overline{D} C_{-1} = C_{-2} = \frac{1}{2} A_{-2} + \frac{1}{2} \overline{e_0} B_{-2}
$$
Clearly it is monogenic in $\mathbb{R}^{3}_+$, since $DC_{-2} = D \overline{D} C_{-1} = \frac{1}{4} \Delta_{m+1} C_{-1} = 0$.
The definition itself of $C_{-2}(x_0,\underline{x})$ implies that it shows the monogenic potential (or primitive) $C_{-1}(x_0,\underline{x})$ and the conjugate harmonic potentials $A_{-1}(x_0,\underline{x})$ and $\overline{e_0} B_{-1}(x_0,\underline{x})$. For the notion of higher dimensional {\em conjugate harmonicity} in the framework of Clifford analysis we refer the reader to \cite{fbrdfs}.\\
The harmonic component $A_{-2}(x_0,\underline{x})$ may be calculated as $\partialo A_{-1}$ or as $-\partialux B_{-1}$, and even the general expression is valid, leading to
$$
A_{-2}(x_0,\underline{x}) = \frac{1}{2\partiali} \, \frac{-2x_0^2 + r^2}{(x_0^2 + r^2)^{5/2}}
$$
Similarly the harmonic component $B_{-2}(x_0,\underline{x})$ may be calculated as $\partialo B_{-1}$ or as $-\partialux A_{-1}$, and also the general expression is valid, leading to
$$
B_{-2}(x_0,\underline{x}) = \frac{1}{2\partiali} \, \frac{3 x_0 \underline{x}}{(x_0^2 + r^2)^{5/2}}
$$
The distributional limits for $x_0 \rightarrow 0+$ of these harmonic potentials are given by
$$
\left \{ \begin{array}{rcl}
a_{-2}(\underline{x}) = \lim_{x_0 \rightarrow 0+} A_{-2}(x_0,\underline{x}) & = & \displaystyle\frac{1}{2\partiali} \, {\rm Fp} \displaystyle\frac{1}{r^3}\\[4mm]
b_{-2}(\underline{x}) = \lim_{x_0 \rightarrow 0+} B_{-2}(x_0,\underline{x}) & = & - \partialux \delta
\end{array} \right .
$$
where ${\rm Fp}$ stands for the "finite part" distribution on the real $r$--line.\\
Note that $a_{-2}(\underline{x}) = - F_{-1}$, with $F_{-1} = \partialux H$ the fundamental solution of the operator $^{-1}\mathcal{H}$, while
$b_{-2}(\underline{x}) = - E_{-1}$, with $E_{-1} = \partialux \delta$ the fundamental solution of the operator $\partialux^{-1}$ (see \cite{bdbds2}).\\
Proceeding in the same manner, the sequence of {\em downstream} monogenic potentials in $\mathbb{R}_+^{m+1}$ is defined by
$$
C_{-k-1} = \overline{D} C_{-k} = \overline{D}^2 C_{-k+1} = \ldots = \overline{D}^k C_{-1}, \qquad k=1,2,\ldots
$$
where each monogenic potential decomposes into two conjugate harmonic potentials:
$$
C_{-k-1} = \frac{1}{2} A_{-k-1} + \frac{1}{2} \overline{e_0} B_{-k-1}, \qquad k=1,2,\ldots
$$
with, for $k$ odd, say $k=2\ell-1$,
$$
\left \{ \begin{array}{rcl}
A_{-2\ell} & = & \partial_{x_0}^{2\ell-1} A_{-1} \ = \ - \partial_{x_0}^{2\ell-2} \partialux B_{-1} \ = \ \ldots \ = \ - \partialux^{2\ell-1} B_{-1} \\[2mm]
B_{-2\ell} & = & \partial_{x_0}^{2\ell-1} B_{-1} \ = \ - \partial_{x_0}^{2\ell-2} \partialux A_{-1} \ = \ \ldots \ = \ - \partialux^{2\ell-1} A_{-1}
\end{array} \right .
$$
while for $k$ even, say $k=2\ell$,
$$
\left \{ \begin{array}{rcl}
A_{-2\ell-1} & = & \partial_{x_0}^{2\ell} A_{-1} \ = \ - \partial_{x_0}^{2\ell-1} \partialux B_{-1} \ = \ \ldots \ = \ \partialux^{2\ell} A_{-1} \\[2mm]
B_{-2\ell-1} & = & \partial_{x_0}^{2\ell} B_{-1} \ = \ - \partial_{x_0}^{2\ell-1} \partialux A_{-1} \ = \ \ldots \ = \ \partialux^{2\ell} B_{-1}
\end{array} \right .
$$
The harmonic potentials $A_{-k}(x_0,\underline{x})$ are real--valued and given by
$$
A_{-k}(x_0,\underline{x}) = (-1)^{k+1} \, \frac{1}{2\partiali} \, k! \, \frac{1\cdot3\cdot\cdots(2k-1)}{(k+1)(k+2)\cdots(2k)} \, \frac{r^k}{(x_0^2 + r^2)^{k+\onehalf}} \, i^k \, C_k^{-k}(i \frac{x_0}{r})
$$
where $C_k^{\langlembda}$ stands for the Gegenbauer polynomial. We have explicitly calculated $A_{-k}$ for the $k$--values $1,2,3$ and $4$. We obtain\\
\noindent
for $k=1$
$$
A_{-1} = \frac{1}{2\partiali} \, \onehalf \, \frac{r}{(x_0^2 + r^2)^{3/2}} \, i \, C_1^{-1}(i \frac{x_0}{r}) = \frac{1}{2\partiali} \, \frac{x_0}{(x_0^2 + r^2)^{3/2}}
$$
for $k=2$
$$
A_{-2} = - \frac{1}{2\partiali} \, 2! \, \frac{1\cdot3}{3\cdot4} \, \frac{r^2}{(x_0^2 + r^2)^{5/2}} \, i^2 \, C_2^{-2}(i \frac{x_0}{r}) = \frac{1}{2\partiali} \, \frac{-2 x_0^2 + r^2}{(x_0^2 + r^2)^{5/2}}
$$
for $k=3$
$$
A_{-3} = \frac{1}{2\partiali} \, 3! \, \frac{1\cdot3\cdot5}{4\cdot5\cdot6} \, \frac{r^3}{(x_0^2 + r^2)^{7/2}} \, i^3 \, C_3^{-3}(i \frac{x_0}{r}) = \frac{1}{2\partiali} \, \frac{6 x_0^3 - 9 x_0 r^2}{(x_0^2 + r^2)^{7/2}}
$$
and for $k=4$
$$
A_{-4} = - \frac{1}{2\partiali} \, 4! \, \frac{1\cdot3\cdot5\cdot7}{5\cdot6\cdot7\cdot8} \, \frac{r^4}{(x_0^2 + r^2)^{9/2}} \, i^4 \, C_4^{-4}(i \frac{x_0}{r}) = \frac{1}{2\partiali} \, \frac{-24 x_0^4 + 72 x_0^2 r^2 - 9 r^4}{(x_0^2 + r^2)^{9/2}}
$$
The harmonic potentials $B_{-k}(x_0,\underline{x})$ are Clifford vector--valued and given by
$$
B_{-k}(x_0,\underline{x}) = (-1)^{k} \, \frac{1}{2\partiali} \, (k-1)! \, \frac{3\cdot5\cdots(2k-1)}{(k+2)(k+3)\cdots(2k)} \, \frac{r^{k-1} \underline{x}}{(x_0^2 + r^2)^{k+\onehalf}} \, i^{k-1} \, C_{k-1}^{-k}(i \frac{x_0}{r})
$$
We have explicitly calculated $B_{-k}$ for the $k$--values $1,2,3$ and $4$. We obtain\\
\noindent
for $k=1$
$$
B_{-1} = - \frac{1}{2\partiali} \, \frac{\underline{x}}{(x_0^2 + r^2)^{3/2}} \, C_0^{-1}(i \frac{x_0}{r}) = - \frac{1}{2\partiali} \, \frac{\underline{x}}{(x_0^2 + r^2)^{3/2}}
$$
for $k=2$
$$
B_{-2} = \frac{1}{2\partiali} \, \frac{3}{4} \, \frac{r \underline{x}}{(x_0^2 + r^2)^{5/2}} \, i \, C_1^{-2}(i \frac{x_0}{r}) = \frac{1}{2\partiali} \, \frac{3 x_0 \underline{x}}{(x_0^2 + r^2)^{5/2}}
$$
for $k=3$
$$
B_{-3} = - \frac{1}{2\partiali} \, 2! \, \frac{3\cdot5}{5\cdot6} \, \frac{r^2 \underline{x}}{(x_0^2 + r^2)^{7/2}} \, i^2 \, C_2^{-3}(i \frac{x_0}{r}) = - \frac{1}{2\partiali} \, \frac{(12 x_0^2 -3 r^2) \underline{x}}{(x_0^2 + r^2)^{7/2}}
$$
and for $k=4$
$$
B_{-4} = \frac{1}{2\partiali} \, 3! \, \frac{3\cdot5\cdot7}{6\cdot7\cdot8} \, \frac{r^3 \underline{x}}{(x_0^2 + r^2)^{9/2}} \, i^3 \, C_3^{-4}(i \frac{x_0}{r}) = \frac{1}{2\partiali} \, \frac{(60 x_0^3 - 45 x_0 r^2) \underline{x}}{(x_0^2 + r^2)^{9/2}}
$$
Their distributional limits for $x_0 \rightarrow 0+$ are given by
$$
\left \{ \begin{array}{rcl}
a_{-2\ell} & = & (- \partialux)^{2\ell-1} H = (-1)^{\ell-1} \, \frac{1}{2\partiali} \, (2\ell-1)!! \, (2\ell-1)!! \, {\rm Fp} \displaystyle\frac{1}{r^{2\ell+1}} \\[4mm]
b_{-2\ell} & = & (- \partialux)^{2\ell-1} \delta \end{array} \right .
$$
and
$$
\left \{ \begin{array}{rcl}
a_{-2\ell-1} & = & \partialux^{2\ell} \delta\\[2mm]
b_{-2\ell-1} & = & \partialux^{2\ell} H
= (-1)^{\ell-1} \, \frac{1}{2\partiali} \, (2\ell-1)!! \, (2\ell+1)!! \, {\rm Fp} \displaystyle\frac{1}{r^{2\ell+2}} \, \underline{\omega} \end{array} \right .
$$
They show the following properties, in fact valid regardless the dimension.
\begin{property}
\langlebel{lem2}
One has for $j,k=1,2,\ldots$
\begin{itemize}
\item[(i)] $a_{-k} \xrightarrow{\hspace*{1mm} -\partialux \hspace*{1mm}} b_{-k-1} \xrightarrow{\hspace*{1mm} -\partialux \hspace*{1mm}} a_{-k-2}$
\item[(ii)] $\mathcal{H} \left [ a_{-k} \right ] = b_{-k}$, $\mathcal{H} \left [ b_{-k} \right ] = a_{-k}$
\item[(iii)] $a_{-j} \ast a_{-k} = a_{-j-k+1}$ \\
$a_{-j} \ast b_{-k} = b_{-j} \ast a_{-k} = b_{-j-k+1}$ \\
$b_{-j} \ast b_{-k} = a_{-j-k+1}$.
\end{itemize}
\end{property}
\subsection{The upstream potentials in $\mathbb{R}_+^{3}$}
Let us have a look at the so--called {\em upstream} potentials. To start with the fundamental solution of the Laplace operator $\Delta_{3}$ in $\mathbb{R}^{3}$, sometimes called Green's function, and here denoted by $\frac{1}{2}A_0(x_0,\underline{x})$, is given by
$$
\frac{1}{2}A_0(x_0,\underline{x}) = - \frac{1}{4\partiali} \frac{1}{(x_0^2 + r^2)^\onehalf}
$$
Its conjugate harmonic in $\mathbb{R}^{3}_+$, in the sense of \cite{fbrdfs}, is
\begin{equation}
B_0(x_0,\underline{x}) = \frac{2}{\sigma_{3}} \, \frac{\underline{x}}{|\underline{x}|^2} \, \mathbb{F}_2 \left ( \frac{r}{x_0} \right )
\langlebel{B0}
\end{equation}
where
$$
\mathbb{F}_2(v) = \int_0^v \frac{\eta}{(1+\eta^2)^\frac{3}{2}} \, d\eta = \frac{v^2}{1+v^2+\sqrt{1+v^2}}
$$
leading to
$$
B_0(x_0,\underline{x}) = \frac{1}{2\partiali} \, \frac{\underline{x}}{ \sqrt{x_0^2+r^2}\left(x_0+\sqrt{x_0^2+r^2}\right)}
$$
It is verified that
$$
\partialo B_0(x_0,\underline{x}) = - \frac{1}{2\partiali} \, \frac{\underline{x}}{(x_0^2 + r^2)^{3/2}} = Q(x_0,\underline{x})
$$
and
$$
\partialux B_0(x_0,\underline{x}) = - \frac{1}{2\partiali} \, \frac{x_0}{(x_0^2 + r^2)^{3/2}} = - P(x_0,\underline{x})
$$
and also
$$
\lim_{x_0 \rightarrow 0+} \, B_0(x_0,\underline{x}) = \frac{1}{2\partiali} \, \frac{\underline{x}}{r^2} = b_0(\underline{x})
$$
Note that $b_0(\underline{x}) = \frac{1}{2\partiali} \, \frac{\underline{x}}{r^2} = - E_1$, with $E_1$ the fundamental solution of the Dirac operator $\partialux^1 = \partialux$.\\[2mm]
Green's function $A_0(x_0,\underline{x})$ itself shows the distributional limit
$$
\lim_{x_0 \rightarrow 0+} A_0(x_0,\underline{x}) = - \frac{1}{2\partiali} \, \frac{1}{r} = a_0(\underline{x})
$$
Note that $a_0(\underline{x}) = - F_1$, $F_1 = \frac{1}{2\partiali} \, \frac{1}{r}$ being the fundamental solution to the so--called Hilbert--Dirac operator $^1\mathcal{H} = (-\Delta_2)^\onehalf$ (see \cite{bdbds2, hidi}).\\
It is readily seen that $\overline{D} A_0 = \overline{D} \overline{e_0} B_0 = C_{-1}$. So $A_0(x_0,\underline{x})$ and $\overline{e_0} B_0(x_0,\underline{x})$ are conjugate harmonic potentials with respect to the operator $\overline{D}$, of the Cauchy kernel $C_{-1}(x_0,\underline{x})$ in $\mathbb{R}^{3}_+$. Putting $C_0(x_0,\underline{x}) = \frac{1}{2} A_0(x_0,\underline{x}) + \frac{1}{2} \overline{e_0} B_0 (x_0,\underline{x})$, it follows that also $\overline{D} C_0 (x_0,\underline{x}) = C_{-1}(x_0,\underline{x})$,
which implies that $C_0(x_0,\underline{x})$ is a monogenic potential (or monogenic primitive) of the Cauchy kernel $C_{-1}(x_0,\underline{x})$ in $\mathbb{R}^{3}_+$.
Their distributional boundary values are intimately related, as mentioned in the following property, valid in general.
\begin{property}
\langlebel{lemintiem}
One has
\begin{itemize}
\item[(i)] $-\partialux a_0 = b_{-1} = H$; \quad $-\partialux b_0 = a_{-1} = \delta$
\item[(ii)] $\mathcal{H} \left [a_0 \right ] = b_0$; \quad $\mathcal{H} \left [b_0 \right ] = a_0$
\end{itemize}
\end{property}
\begin{remark}
In the upper half of the complex plane the function $\ln(z)$ is a holomorphic potential (or primitive) of the Cauchy kernel $\frac{1}{z}$ and its real and imaginary components are the fundamental solution $\ln |z|$ of the Laplace operator, and its conjugate harmonic $i\, {\rm arg}(z)$ respectively. By similarity we could say that $C_0(x_0,\underline{x}) = \frac{1}{2} A_0(x_0,\underline{x}) + \frac{1}{2} \overline{e_0} B_0(x_0,\underline{x})$, being a monogenic potential of the Cauchy kernel $C_{-1}(x_0,\underline{x})$ and the sum of the fundamental solution $A_0(x_0,\underline{x})$ of the Laplace operator and its conjugate harmonic $\overline{e_0} B_0(x_0,\underline{x})$, is a {\em monogenic logarithmic function} in the upper half--space $\mathbb{R}^{3}_+$.
\end{remark}
The construction of the sequence of {\em upstream} harmonic and monogenic potentials in $\mathbb{R}^{3}_+$ is continued as follows.\\
The general expression for $A_1(x_0,\underline{x})$ established in \cite{bdbds1}, is not valid for $m=2$. By direct calculation we obtain
$$
A_1(x_0,\underline{x}) = - \frac{1}{2\partiali} \, \ln{\left(x_0+\sqrt{x_0^2 + r^2}\right)}
$$
and it is verified that $\partialo A_1 = A_0$, $-\partialux A_1 = B_0$ and $\lim_{x_0 \rightarrow 0+} \, A_1 = - \frac{1}{2\partiali} \ln{r} = a_1(\underline{x})$.
Note that $a_1(\underline{x}) = - \frac{1}{2\partiali} \ln{r} = E_2$ is the fundamental solution of the negative Laplace operator $\partialux^2 = - \Delta_2$.\\
For its conjugate harmonic in $\mathbb{R}^3_+$ we obtain
$$
B_1(x_0,\underline{x}) = \frac{1}{2\partiali} \, \frac{x_0 \underline{x}}{r^2} \, \mathbb{F}_2\left( \frac{r}{x_0}\right) - \frac{1}{2\partiali} \, \frac{\underline{x}}{(x_0^2 + r^2)^\onehalf} = \frac{1}{2\partiali} \, \frac{\underline{x}}{r^2} \left( x_0 - \sqrt{x_0^2 + r^2} \right)
$$
for which it is verified that $\partialo B_1 = B_0$, $-\partialux B_1 = A_0$ and $\lim_{x_0 \rightarrow 0+} \, B_1 = - \frac{1}{2\partiali} \frac{\underline{x}}{r} = b_1(\underline{x})$.\\[2mm]
Note that $b_1(\underline{x}) = - \frac{1}{2\partiali} \frac{\underline{x}}{r} = - \frac{1}{2\partiali} \underline{\omega} = F_2$ is the fundamental solution of the operator $^2\mathcal{H}$ (see \cite{bdbds2}).\\
It follows that $\overline{D} A_{-1} = \overline{D} \overline{e_0} B_{-1} = C_{0}$, whence $A_1(x_0,\underline{x})$ and $B_1(x_0,\underline{x})$ are conjugate harmonic potentials in $\mathbb{R}^{3}_+$ of the function $C_0(x_0,\underline{x})$ and
$$
C_1(x_0,\underline{x}) = \frac{1}{2} A_1(x_0,\underline{x}) + \frac{1}{2} \overline{e_0} B_1(x_0,\underline{x})
$$
is a monogenic potential in $\mathbb{R}^{3}_+$ of $C_0$.
The above mentioned distributional boundary values show the following properties, valid in general.
\begin{property}
\langlebel{lem54}
\rule{0mm}{0mm}
\begin{itemize}
\item[(i)] $- \partialux a_1 = b_0$, $-\partialux b_1 = a_0$
\item[(ii)] $\mathcal{H} \left [ a_1 \right ] = b_1$, $\mathcal{H} \left [ b_1 \right ] = a_1$
\end{itemize}
\end{property}
In the next step the general expressions for $A_2(x_0,\underline{x})$ and $B_2(x_0,\underline{x})$ are not valid. A direct computation yields
$$
A_2(x_0,\underline{x}) = \frac{1}{2\partiali} \, \left( \sqrt{x_0^2 + r^2} - x_0 \ln{\left(x_0 + \sqrt{x_0^2 + r^2}\right)} \right)
$$
and it is verified that $-\partialux A_2 = B_1$ and $\lim_{x_0 \rightarrow 0+} \, A_2 = \frac{1}{2\partiali} r = a_2(\underline{x})$.\\
Note that $a_2(\underline{x}) = - F_3$, with $F_3 = - \frac{1}{2\partiali} r$ the fundamental solution of the operator $^3\mathcal{H}$ (see \cite{bdbds2}).\\
For its conjugate harmonic in $\mathbb{R}^3_+$ we find
$$
B_2(x_0,\underline{x}) = \frac{\underline{x}}{4\partiali} \, \left( \onehalf - \frac{x_0}{x_0 + \sqrt{x_0^2 + r^2}} - \ln{\left( x_0 + \sqrt{x_0^2 + r^2} \right)} \right)
$$
for which it is verified that $-\partialux B_2 = A_1$ and $\lim_{x_0 \rightarrow 0+} \, B_2 = \frac{\underline{x}}{4\partiali} \, (- \ln r + \onehalf) = b_2(\underline{x})$.
Note that $b_2(\underline{x}) = - E_3$, with $E_3 = (\frac{1}{4\partiali} \ln{r} - \frac{1}{8\partiali})\underline{x}$ the fundamental solution of the operator $\partialux^3$.\\
It follows that
$$
C_2(x_0,\underline{x}) = \frac{1}{2} A_2(x_0,\underline{x}) + \frac{1}{2} \overline{e_0} B_2(x_0,\underline{x})
$$
is a monogenic potential in $\mathbb{R}^{3}_+$ of $C_1$. The distributional limits show the following properties, also valid in general.
\begin{property}
\langlebel{lem56}
\rule{0mm}{0mm}
\begin{itemize}
\item[(i)] $-\partialux a_2 = b_1$, $- \partialux b_2 = a_1$
\item[(ii)] $\mathcal{H} \left [ a_2 \right ] = b_2$, $\mathcal{H} \left [ b_2 \right ] = a_2$
\end{itemize}
\end{property}
Inspecting the above expressions for the harmonic potentials $A_1(x_0,\underline{x})$ and $A_2(x_0,\underline{x})$ we can put forward a general form for the potentials $A_j(x_0,\underline{x}), j=1,2,\ldots$
\begin{proposition}
For $j=1,2,\ldots$ one has
$$
2\partiali A_j(x_0,\underline{x}) = P_j(x_0,r^2) \, \ln{(x_0 + \sqrt{x_0^2+r^2})} + Q_j(x_0,r^2) \, \sqrt{x_0^2+r^2} + S_j(x_0,r^2)
$$
with
$$
P_{2k}(x_0,r^2) = p_{2k}^{2k-1} x_0^{2k-1} + p_{2k}^{2k-3} r^2 x_0^{2k-3} + \cdots + p_{2k}^{1} r^{2k-2}x_0
$$
$$
P_{2k+1}(x_0,r^2) = p_{2k+1}^{2k} x_0^{2k} + p_{2k+1}^{2k-2} r^2 x_0^{2k-2} + \cdots + p_{2k+1}^{0} r^{2k}
$$
and
$$
Q_{2k}(x_0,r^2) = q_{2k}^{2k-2} x_0^{2k-2} + q_{2k}^{2k-4} r^2 x_0^{2k-4} + \cdots + q_{2k}^{0} r^{2k-2}
$$
$$
Q_{2k+1}(x_0,r^2) = q_{2k+1}^{2k-1} x_0^{2k-1} + q_{2k+1}^{2k-3} r^2 x_0^{2k-3} + \cdots + q_{2k+1}^{1} r^{2k-2}x_0
$$
and
$$
S_{2k}(x_0,r^2) = s_{2k}^{2k-3} r^2 x_0^{2k-3} + s_{2k}^{2k-5} r^4 x_0^{2k-5} + \cdots + s_{2k}^{1} r^{2k-2} x_0
$$
$$
S_{2k+1}(x_0,r^2) = s_{2k+1}^{2k-2} r^2 x_0^{2k-2} + s_{2k+1}^{2k-4} r^4 x_0^{2k-4} + \cdots + s_{2k+1}^{0} r^{2k}
$$
all the coefficients $p_{2k}^j$, $p_{2k+1}^j$, $q_{2k}^j$, $q_{2k+1}^j$, $s_{2k}^j$ and $s_{2k+1}^j$ being real constants.
\end{proposition}
\partialf
The harmonic potentials $A_1(x_0,\underline{x})$ and $A_2(x_0,\underline{x})$ computed above, fit into this general form. Now we will show that it is possible to determine unambiguously all the coefficients in the expression of $A_j(x_0,\underline{x})$ in terms of the coefficients in $A_{j-1}(x_0,\underline{x})$. To that end we impose on $A_j(x_0,\underline{x})$ the following two conditions, in line with its definition:
$$
\partialo (2\partiali A_j(x_0,\underline{x})) = 2\partiali A_{j-1}(x_0,\underline{x})
$$
and
$$
\lim_{x_0 \rightarrow 0+} (2\partiali A_j(x_0,\underline{x})) = 2\partiali a_{j}(\underline{x})
$$
In the case where $j$ is even, say $j=2k$, this leads to the equations
\begin{equation}
\langlebel{evenP}
\partialo P_{2k} = P_{2k-1}
\end{equation}
\begin{equation}
\langlebel{evenS}
\partialo S_{2k} = S_{2k-1}
\end{equation}
\begin{equation}
\langlebel{evenQ}
P_{2k} + x_0 Q_{2k} + (x_0^2+r^2) \, \partialo Q_{2k} = (x_0^2+r^2) \, \partialo Q_{2k-1}
\end{equation}
\begin{equation}
\langlebel{evenlim}
q_{2k}^0 r^{2k-1} + s_{2k}^0 r^{2k-1} = 2\partiali a_{2k}(\underline{x})
\end{equation}
From (\ref{evenP}) all the $p_{2k}$--coefficients may be determined as
$$
p_{2k}^j = \frac{1}{j} \, p_{2k-1}^{j-1}
$$
while from (\ref{evenS}) all the $s_{2k}$--coefficients follow by the similar relation
$$
s_{2k}^j = \frac{1}{j} \, s_{2k-1}^{j-1}
$$
Then all the $q_{2k}$--coefficients follow recursively from (\ref{evenQ}), and equation (\ref{evenlim}) may be used as a check.
In the case where $j$ is odd, say $j=2k+1$, the similar equations read
\begin{equation}
\langlebel{oddP}
\partialo P_{2k+1} = P_{2k}
\end{equation}
\begin{equation}
\langlebel{oddS}
\partialo S_{2k+1} = S_{2k}
\end{equation}
\begin{equation}
\langlebel{oddQ}
P_{2k+1} + x_0 Q_{2k+1} + (x_0^2+r^2) \, \partialo Q_{2k+1} = (x_0^2+r^2) \, \partialo Q_{2k}
\end{equation}
and
\begin{equation}
\langlebel{oddlim}
p_{2k+1}^0 r^{2k} \ln{r}+ s_{2k+1}^0 r^{2k} = 2\partiali a_{2k+1}(\underline{x})
\end{equation}
From (\ref{oddP}) all the $p_{2k+1}$--coefficients, except $p_{2k+1}^0$, may be determined as
$$
p_{2k+1}^j = \frac{1}{j} \, p_{2k}^{j-1}
$$
while from (\ref{oddS}) all the $s_{2k+1}$--coefficients, except $s_{2k+1}^0$, follow by the similar relation
$$
s_{2k+1}^j = \frac{1}{j} \, s_{2k}^{j-1}
$$
The remaining coefficients $p_{2k+1}^0$ and $s_{2k+1}^0$ follow from (\ref{oddlim}).
All the $q_{2k+1}$--coefficients then follow recursively from (\ref{oddQ}).
~
{$\square$}\pagebreak[1]\par
\par
\begin{remark}
It is obvious that solving equations (\ref{evenlim}) and (\ref{oddlim}) requires the knowledge of the distributional boundary values $a_j(\underline{x}), j=1,2,\ldots$ There holds (see \cite{bdbds2})
$$
a_{2k}(\underline{x}) = (-1)^{k+1} \frac{1}{2\partiali} \frac{1}{((2k-1)!!)^2} \, r^{2k-1} \quad (k = 1,2,\ldots)
$$
and
$$
a_{2k+1}(\underline{x}) = (\alpha_{2k} \ln{r} + \beta_{2k}) \frac{\partiali^{k+1}}{k!} \, r^{2k} \quad (k = 0,1,2,\ldots)
$$
where the coefficients $\alpha_{2k}$ and $\beta_{2k}$ are defined recursively by
$$
\left \{ \begin{array}{rcl}
\alpha_{2k+2} & = & - \displaystyle\frac{1}{2\partiali} \displaystyle\frac{1}{2k+2} \, \alpha_{2k}\\[4mm]
\beta_{2k+2} & = & - \displaystyle\frac{1}{2\partiali} \displaystyle\frac{1}{2k+2} \, (\beta_{2k} - \displaystyle\frac{1}{k+1} \, \alpha_{2k})
\end{array} \right . \qquad k=0,1,2,\ldots
$$
with starting values $\alpha_{0} = - \displaystyle\frac{1}{2\partiali^2}$ and $\beta_{0} = 0$, leading to their closed form
$$
\alpha_{2k} = \displaystyle\frac{(-1)^{k+1}}{2^{2k+1} \partiali^{k+2} k!} \quad {\rm and } \quad
\beta_{2k} = \displaystyle\frac{(-1)^{k} H_{k}}{2^{2k+1} \partiali^{k+2} k!}
$$
with $H_k = \sum_{n=1}^k \, \frac{1}{n}$.
\end{remark}
Now by Proposition 3.1 all the upstream $A_j$--potentials may be calculated recursively. Using the shorthands
$$
LOG = \ln{(x_0+\sqrt{x_0^2+r^2})} \quad {\rm and} \quad SQRT = \sqrt{x_0^2+r^2}
$$
the outcome of our calculations is the following:
$$
2\partiali A_3(x_0,\underline{x}) = (-\onehalf x_0^2 + \frac{1}{4} r^2) LOG + \frac{3}{4} x_0 SQRT - \frac{1}{4} r^2
$$
$$
2\partiali A_4(x_0,\underline{x}) = (-\frac{1}{6} x_0^3 + \frac{1}{4} r^2 x_0) LOG + (\frac{11}{36} x_0^2 - \frac{1}{9} r^2) SQRT - \frac{1}{4} r^2 x_0
$$
$$
2\partiali A_5(x_0,\underline{x}) = (-\frac{1}{24} x_0^4 + \frac{1}{8} r^2 x_0^2 - \frac{1}{64} r^4) LOG + (\frac{25}{288} x_0^3 - \frac{55}{576} r^2 x_0) SQRT - \frac{1}{8} r^2 x_0^2 + \frac{3}{128} r^4
$$
$$
2\partiali A_6(x_0,\underline{x}) = (-\frac{1}{120} x_0^5 + \frac{1}{24} r^2 x_0^3 - \frac{1}{64} r^4 x_0) LOG + (\frac{137}{7200} x_0^4 - \frac{607}{14400} r^2 x_0^2 + \frac{1}{225} r^4) SQRT - \frac{1}{24} r^2 x_0^3 + \frac{3}{128} r^4 x_0
$$
Note that once an upstream $A_j$--potential is determined, the $B_{j-1}$--potential follows readily by
$$
(-\partialux)A_j(x_0,\underline{x}) = B_{j-1}(x_0,\underline{x})
$$
\section{Harmonic and monogenic potentials in $\mathbb{R}_+^{4}$}
\langlebel{potentials4}
In this section we calculate the harmonic and monogenic potentials in upper half--space $\mathbb{R}^{4}_+$. Quite naturally the structure of this section is completely similar to the foregoing one.
\subsection{The Cauchy kernel in $\mathbb{R}_+^{4}$}
The starting point is the Cauchy kernel, i.e. the fundamental solution of the generalized Cauchy--Riemann operator $D$:
$$
C_{-1}(x_0,\underline{x}) = \frac{1}{\sigma_{4}} \, \frac{x \overline{e_0}}{|x|^{4}} = \frac{1}{2\partiali^2} \, \frac{x_0 - \overline{e_0} \underline{x}}{(x_0^2 + r^2)o^{2}}
$$
where we have put now $|\underline{x}| = \rho$.
It may be decomposed in terms of the traditional Poisson kernels in $\mathbb{R}^{4}_+$:
$$
C_{-1}(x_0,\underline{x}) = \frac{1}{2} A_{-1}(x_0,\underline{x}) + \frac{1}{2} \overline{e_0} \, B_{-1}(x_0,\underline{x})
$$
where, also mentioning the usual notations, for $x_0 >0$,
\begin{eqnarray*}
A_{-1}(x_0,\underline{x}) & = & P(x_0,\underline{x}) \ = \ \partialhantom{-} \frac{1}{\partiali^2} \, \frac{x_0}{(x_0^2 + r^2)o^{2}} \langlebel{A-1}\\
B_{-1}(x_0,\underline{x}) & = & Q(x_0,\underline{x}) \ = \ - \frac{1}{\partiali^2} \, \frac{\underline{x}}{(x_0^2 + r^2)o^{2}} \langlebel{B-1}
\end{eqnarray*}
Their distributional limits for $x_0 \rightarrow 0+$ are given by
\begin{eqnarray*}
a_{-1}(\underline{x}) & = & \lim_{x_0 \rightarrow 0+} A_{-1}(x_0,\underline{x}) \ = \ \delta(\underline{x})\\
b_{-1}(\underline{x}) & = & \lim_{x_0 \rightarrow 0+} B_{-1}(x_0,\underline{x}) \ = \ H(\underline{x}) \ = \ - \frac{1}{\partiali^2} \, \mbox{Pv} \frac{\underline{x}}{\rho^4} \ = \ - \frac{1}{\partiali^2} \, \mbox{Pv} \frac{\underline{\omega}}{\rho^3}
\end{eqnarray*}
Note that, as in general, both distributional boundary values are linked by the Hilbert transform $\mathcal{H}$ in $\mathbb{R}^3$ with the above convolution kernel $H(\underline{x})$:
\begin{eqnarray*}
\mathcal{H} \left [ a_{-1} \right ] & = & \mathcal{H} \left [ \delta \right ] \ = \ H \ast \delta \ = \ H \ = \ b_{-1} \\
\mathcal{H} \left [ b_{-1} \right ] & = & \mathcal{H} \left [ H \right ] \ = \ H \ast H \ = \ \delta \ = \ a_{-1}
\end{eqnarray*}
Note also that $a_{-1}(\underline{x}) = \delta(\underline{x}) = E_0$ can be seen as the fundamental solution of the identity operator $\partialux^0 = {\bf 1}$, and that $b_{-1}(\underline{x}) = - \frac{1}{\partiali^2} \, \mbox{Pv} \frac{\underline{\omega}}{\rho^3} = H(\underline{x}) = F_0$ is the fundamental solution of the Hilbert operator $^0\mathcal{H} = \mathcal{H}$.
\subsection{The downstream potentials in $\mathbb{R}_+^{4}$}
The first in the sequence of the {\em downstream} potentials is the function $C_{-2}$ defined by
$$
\overline{D} C_{-1} = C_{-2} = \frac{1}{2} A_{-2} + \frac{1}{2} \overline{e_0} B_{-2}
$$
Clearly it is monogenic in $\mathbb{R}^{4}_+$, since $DC_{-2} = D \overline{D} C_{-1} = \frac{1}{4} \Delta_{m+1} C_{-1} = 0$.
The definition itself of $C_{-2}(x_0,\underline{x})$ implies that it shows the monogenic potential (or primitive) $C_{-1}(x_0,\underline{x})$ and the conjugate harmonic potentials $A_{-1}(x_0,\underline{x})$ and $\overline{e_0} B_{-1}(x_0,\underline{x})$.\\
The harmonic component $A_{-2}(x_0,\underline{x})$ may be calculated as $\partialo A_{-1}$ or as $-\partialux B_{-1}$, and even the general expression is valid, leading to
$$
A_{-2}(x_0,\underline{x}) = \frac{1}{\partiali^2} \, \frac{-3x_0^2 + \rho^2}{(x_0^2 + r^2)o^{3}}
$$
Similarly the harmonic component $B_{-2}(x_0,\underline{x})$ may be calculated as $\partialo B_{-1}$ or as $-\partialux A_{-1}$, and also the general expression is valid, leading to
$$
B_{-2}(x_0,\underline{x}) = \frac{4}{\partiali^2} \, \frac{ x_0 \underline{x}}{(x_0^2 + r^2)o^{3}}
$$
The distributional limits for $x_0 \rightarrow 0+$ of these harmonic potentials are given by
$$
\left \{ \begin{array}{rcl}
a_{-2}(\underline{x}) = \lim_{x_0 \rightarrow 0+} A_{-2}(x_0,\underline{x}) & = & \displaystyle\frac{1}{\partiali^2} \, {\rm Fp} \displaystyle\frac{1}{\rho^4}\\[4mm]
b_{-2}(\underline{x}) = \lim_{x_0 \rightarrow 0+} B_{-2}(x_0,\underline{x}) & = & - \partialux \delta
\end{array} \right .
$$
Note that $a_{-2}(\underline{x}) = - F_{-1}$, with $F_{-1} = \partialux H$ the fundamental solution of the operator $^{-1}\mathcal{H}$, while
$b_{-2}(\underline{x}) = - E_{-1}$, with $E_{-1} = \partialux \delta$ the fundamental solution of the operator $\partialux^{-1}$ (see \cite{bdbds2}).\\
The sequence of {\em downstream} monogenic potentials in $\mathbb{R}_+^{m+1}$ is defined, as in general dimension, by
$$
C_{-k-1} = \overline{D} C_{-k} = \overline{D}^2 C_{-k+1} = \ldots = \overline{D}^k C_{-1}, \qquad k=1,2,\ldots
$$
where each monogenic potential decomposes into two conjugate harmonic potentials:
$$
C_{-k-1} = \frac{1}{2} A_{-k-1} + \frac{1}{2} \overline{e_0} B_{-k-1}, \qquad k=1,2,\ldots
$$
The harmonic components $A_{-k}(x_0,\underline{x})$ are real--valued and given by
$$
A_{-k}(x_0,\underline{x}) = (-1)^{k+1} \, \frac{1}{\partiali^2} \, \frac{2^{k-1} (k!)^2}{(k+2)(k+3)\cdots(2k+1)} \, \frac{\rho^k}{(x_0^2 + r^2)o^{k+1}} \, i^k \, C_k^{-k-\onehalf}(i \frac{x_0}{\rho})
$$
We have explicitly calculated $A_{-k}$ for the $k$--values $1,2,3$ and $4$. We obtain\\
\noindent
for $k=1$
$$
A_{-1} = \frac{1}{\partiali^2} \, \frac{1}{3} \, \frac{\rho}{(x_0^2 + r^2)o^{2}} \, i \, C_1^{-3/2}(i \frac{x_0}{\rho}) = \frac{1}{\partiali^2} \, \frac{x_0}{(x_0^2 + r^2)o^{2}}
$$
for $k=2$
$$
A_{-2} = - \frac{2}{\partiali^2} \, \frac{2\cdot2}{4\cdot5} \, \frac{\rho^2}{(x_0^2 + r^2)o^{3}} \, i^2 \, C_2^{-5/2}(i \frac{x_0}{\rho}) = \frac{1}{\partiali^2} \, \frac{-3 x_0^2 + \rho^2}{(x_0^2 + r^2)o^{3}}
$$
for $k=3$
$$
A_{-3} = \frac{4}{\partiali^2} \, \frac{6\cdot6}{5\cdot6\cdot7} \, \frac{\rho^3}{(x_0^2 + r^2)o^{4}} \, i^3 \, C_3^{-7/2}(i \frac{x_0}{\rho}) = \frac{1}{\partiali^2} \, \frac{12 x_0^3 - 12 x_0 \rho^2}{(x_0^2 + r^2)o^{4}}
$$
and for $k=4$
$$
A_{-4} = - \frac{8}{\partiali^2} \, \frac{24\cdot24}{6\cdot7\cdot8\cdot9} \, \frac{\rho^4}{(x_0^2 + r^2)o^{5}} \, i^4 \, C_4^{-9/2}(i \frac{x_0}{\rho}) = \frac{1}{\partiali^2} \, \frac{-60 x_0^4 + 120 x_0^2 \rho^2 - 12 \rho^4}{(x_0^2 + r^2)o^{5}}
$$
The conjugate harmonic components $B_{-k}(x_0,\underline{x})$ are Clifford vector--valued and given by
$$
B_{-k}(x_0,\underline{x}) = (-1)^{k} \, \frac{1}{\partiali^2} \, \frac{2^{k-1} (k-1)! k!}{(k+3)(k+4)\cdots(2k+1)} \, \frac{\rho^{k-1} \underline{x}}{(x_0^2 + r^2)o^{k+1}} \, i^{k-1} \, C_{k-1}^{-k-\onehalf}(i \frac{x_0}{\rho})
$$
We have explicitly calculated $B_{-k}$ for the $k$--values $1,2,3$ and $4$. We obtain\\
\noindent
for $k=1$
$$
B_{-1} = - \frac{1}{\partiali^2} \, \frac{\underline{x}}{(x_0^2 + r^2)o^{2}} \, C_0^{-3/2}(i \frac{x_0}{\rho}) = - \frac{1}{\partiali^2} \, \frac{\underline{x}}{(x_0^2 + r^2)o^{2}}
$$
for $k=2$
$$
B_{-2} = \frac{2}{\partiali^2} \, \frac{1! 2!}{5} \, \frac{\rho \underline{x}}{(x_0^2 + r^2)^{3}} \, i \, C_1^{-5/2}(i \frac{x_0}{\rho}) = \frac{4}{\partiali^2} \, \frac{x_0 \underline{x}}{(x_0^2 + r^2)o^{3}}
$$
for $k=3$
$$
B_{-3} = - \frac{4}{\partiali^2} \, \frac{2!3!}{6\cdot7} \, \frac{\rho^2 \underline{x}}{(x_0^2 + r^2)^{4}} \, i^2 \, C_2^{-7/2}(i \frac{x_0}{\rho}) = \frac{4}{\partiali^2} \, \frac{(-5 x_0^2 + \rho^2) \underline{x}}{(x_0^2 + r^2)o^{4}}
$$
and for $k=4$
$$
B_{-4} = \frac{8}{\partiali^2} \, \frac{3!4!}{7\cdot8\cdot9} \, \frac{\rho^3 \underline{x}}{(x_0^2 + r^2)^{5}} \, i^3 \, C_3^{-9/2}(i \frac{x_0}{\rho}) = \frac{24}{\partiali^2} \, \frac{(5 x_0^3 - 3 x_0 \rho^2) \underline{x}}{(x_0^2 + r^2)o^{5}}
$$
Their distributional limits for $x_0 \rightarrow 0+$ are given by
$$
\left \{ \begin{array}{rcl}
a_{-2\ell} & = & (- \partialux)^{2\ell-1} H = (-1)^{\ell-1} 2^{\ell-1} (2\ell-1)!! \, \ell! \, \displaystyle\frac{1} {\partiali^2} \, {\rm Fp} \displaystyle\frac{1}{\rho^{2\ell+2}} \\[4mm]
b_{-2\ell} & = & (- \partialux)^{2\ell-1} \delta \end{array} \right .
$$
and
$$
\left \{ \begin{array}{rcl}
a_{-2\ell-1} & = & \partialux^{2\ell} \delta\\[2mm]
b_{-2\ell-1} & = & \partialux^{2\ell} H
= (-1)^{\ell-1} 2^{\ell} (2\ell-1)!! \, (\ell+1)! \, \displaystyle\frac{1}{\partiali^2} \, {\rm Fp} \displaystyle\frac{1}{\rho^{2\ell+3}} \, \underline{\omega} \end{array} \right .
$$
They show the by now traditional properties (see Property 2).
\subsection{The upstream potentials in $\mathbb{R}_+^{4}$}
For the so--called {\em upstream} potentials, we start with the fundamental solution of the Laplace operator $\Delta_{4}$ in $\mathbb{R}^{4}$, denoted by $\frac{1}{2}A_0(x_0,\underline{x})$, and given by
$$
\frac{1}{2}A_0(x_0,\underline{x}) = - \frac{1}{4\partiali^2} \frac{1}{x_0^2 + r^2o}
$$
Its conjugate harmonic in $\mathbb{R}^{4}_+$, in the sense of \cite{fbrdfs}, is
\begin{equation}
B_0(x_0,\underline{x}) = \frac{2}{\sigma_{4}} \, \frac{\underline{x}}{\rho^3} \, \mathbb{F}_3 \left ( \frac{\rho}{x_0} \right )
\langlebel{B0}
\end{equation}
where
$$
\mathbb{F}_3(v) = \int_0^v \frac{\eta^2}{(1+\eta^2)^2} \, d\eta = \onehalf \left( \arctan{v} - \frac{v}{1+v^2}\right)
$$
leading to
$$
B_0(x_0,\underline{x}) = \frac{1}{2\partiali^2} \, \frac{\underline{x}}{\rho^3} \, \left( \arctan{\frac{\rho}{x_0}} - \frac{x_0 \rho}{x_0^2 + r^2o} \right)
$$
It is verified that
$$
\partialo B_0(x_0,\underline{x}) = - \frac{1}{\partiali^2} \, \frac{\underline{x}}{(x_0^2 + r^2)o^{2}} = Q(x_0,\underline{x})
$$
and
$$
\partialux B_0(x_0,\underline{x}) = - \frac{1}{\partiali^2} \, \frac{x_0}{x_0^2 + r^2o} = - P(x_0,\underline{x})
$$
and also
$$
\lim_{x_0 \rightarrow 0+} \, B_0(x_0,\underline{x}) = \frac{1}{4\partiali} \, \frac{\underline{x}}{\rho^3} = b_0(\underline{x})
$$
Note that $b_0(\underline{x}) = \frac{1}{4\partiali} \, \frac{\underline{x}}{\rho^3} = \frac{1}{4\partiali} \, \frac{\underline{\omega}}{\rho^2} = -E_1$, with $E_1$ the fundamental solution of the Dirac operator $\partialux^1 = \partialux$.\\[2mm]
Green's function $A_0(x_0,\underline{x})$ itself shows the distributional limit
$$
\lim_{x_0 \rightarrow 0+} A_0(x_0,\underline{x}) = - \frac{1}{2\partiali^2} \, \frac{1}{\rho^2} = a_0(\underline{x})
$$
Note that $a_0(\underline{x}) = - F_1$, $F_1 = \frac{1}{2\partiali^2} \, \frac{1}{\rho^2}$ being the fundamental solution to the so--called Hilbert--Dirac operator $^1\mathcal{H} = (-\Delta_3)^\onehalf$ (see \cite{bdbds2, hidi}).\\
It is readily seen that $\overline{D} A_0 = \overline{D} \overline{e_0} B_0 = C_{-1}$. So $A_0(x_0,\underline{x})$ and $\overline{e_0} B_0(x_0,\underline{x})$ are conjugate harmonic potentials (or primitives), with respect to the operator $\overline{D}$, of the Cauchy kernel $C_{-1}(x_0,\underline{x})$ in $\mathbb{R}^{3}_+$. Putting $C_0(x_0,\underline{x}) = \frac{1}{2} A_0(x_0,\underline{x}) + \frac{1}{2} \overline{e_0} B_0 (x_0,\underline{x})$, it follows that also $\overline{D} C_0 (x_0,\underline{x}) = C_{-1}(x_0,\underline{x})$,
which implies that $C_0(x_0,\underline{x})$ is a monogenic potential (or primitive) of the Cauchy kernel $C_{-1}(x_0,\underline{x})$ in $\mathbb{R}^{3}_+$.
Their distributional boundary values are intimately related, as shown in the similar Property 3.
\begin{remark}
As in any dimension we could again say that $C_0(x_0,\underline{x}) = \frac{1}{2} A_0(x_0,\underline{x}) + \frac{1}{2} \overline{e_0} B_0(x_0,\underline{x})$, being a monogenic potential of the Cauchy kernel $C_{-1}(x_0,\underline{x})$ and the sum of the fundamental solution $A_0(x_0,\underline{x})$ of the Laplace operator and its conjugate harmonic $\overline{e_0} B_0(x_0,\underline{x})$, is a {\em monogenic logarithmic function} in the upper half--space $\mathbb{R}^{4}_+$.
\end{remark}
The construction of the sequence of {\em upstream} harmonic and monogenic potentials in $\mathbb{R}^{4}_+$ is continued as follows.\\
The general expression for $A_1(x_0,\underline{x})$, established in \cite{bdbds1}, remains valid for $m=3$:
$$
A_1(x_0,\underline{x}) = \frac{1}{2\partiali^2} \, \frac{1}{\rho} \, \arctan{\frac{\rho}{x_0}}
$$
and it is verified that $\partialo A_1 = A_0$, $-\partialux A_1 = B_0$ and $\lim_{x_0 \rightarrow 0+} \, A_1 = \frac{1}{4\partiali} \, \frac{1}{\rho} = a_1(\underline{x})$. Note that this distributional boundary value $a_1(\underline{x}) = E_2$ is the fundamental solution of the negative Laplace operator $\partialux^2 = - \Delta_3$.\\
For its conjugate harmonic in $\mathbb{R}^4_+$ we obtain by direct calculation
$$
B_1(x_0,\underline{x}) = \frac{1}{2\partiali^2} \, \frac{\underline{x}}{\rho^2} \, \left( \frac{x_0}{\rho} \arctan{\frac{\rho}{x_0}} - 1 \right)
$$
for which it is verified that $\partialo B_1 = B_0$, $-\partialux B_1 = A_0$ and $\lim_{x_0 \rightarrow 0+} \, B_1 = - \frac{1}{2\partiali^2} \frac{\underline{x}}{\rho^2} = b_1(\underline{x})$. Note that $b_1(\underline{x}) = - \frac{1}{2\partiali^2} \frac{\underline{\omega}}{\rho} = F_2$ is the fundamental solution of the operator $^2\mathcal{H}$ (see \cite{bdbds2}).\\[2mm]
It follows that $\overline{D} A_{-1} = \overline{D} \overline{e_0} B_{-1} = C_{0}$, whence $A_1(x_0,\underline{x})$ and $B_1(x_0,\underline{x})$ are conjugate harmonic potentials in $\mathbb{R}^{3}_+$ of the function $C_0(x_0,\underline{x})$ and
$$
C_1(x_0,\underline{x}) = \frac{1}{2} A_1(x_0,\underline{x}) + \frac{1}{2} \overline{e_0} B_1(x_0,\underline{x})
$$
is a monogenic potential in $\mathbb{R}^{3}_+$ of $C_0$.
The above mentioned distributional boundary values show properties similar to those of Property 4.\\
In the next step the general expressions for $A_2(x_0,\underline{x})$ is not valid. A direct computation yields
$$
A_2(x_0,\underline{x}) = \frac{1}{2\partiali^2} \, \left( \frac{x_0}{\rho} \arctan{\frac{\rho}{x_0}} + \ln{ \sqrt{x_0^2 + r^2o}} \right)
$$
and it is verified that $-\partialux A_2 = B_1$ and $\lim_{x_0 \rightarrow 0+} \, A_2 = \frac{1}{2\partiali^2} \ln{\rho} = a_2(\underline{x})$.
Note that $a_2(\underline{x}) = - F_3$, $F_3 = - \frac{1}{2\partiali^2} \ln{\rho}$ being the fundamental solution of the operator $^3\mathcal{H}$ (see \cite{bdbds2}).
For its conjugate harmonic in $\mathbb{R}^4_+$ we find
$$
B_2(x_0,\underline{x}) = \frac{1}{4\partiali^2} \, \left( \frac{\underline{x}}{\rho^3} (x_0^2 + r^2)o \arctan{\frac{\rho}{x_0}} - \frac{\underline{x} x_0}{\rho^2} \right)
$$
for which it is verified that $\partialo B_2 = B_1$, $-\partialux B_2 = A_1$ and $\lim_{x_0 \rightarrow 0+} \, B_2 = \frac{1}{8\partiali} \, \frac{\underline{x}}{\rho} = b_2(\underline{x})$. Note that $b_2(\underline{x}) = -E_3$, with $E_3$ the fundamental solution of $\partialux^3$.\\
It follows that
$$
C_2(x_0,\underline{x}) = \frac{1}{2} A_2(x_0,\underline{x}) + \frac{1}{2} \overline{e_0} B_2(x_0,\underline{x})
$$
is a monogenic potential in $\mathbb{R}^{3}_+$ of $C_1$. The distributional limits show the properties similar to those of Property 5.\\
Inspecting the above expressions for the harmonic potentials $A_1(x_0,\underline{x})$ and $A_2(x_0,\underline{x})$ we can put forward a general form for the potentials $A_j(x_0,\underline{x}), j=1,2,\ldots$
\begin{proposition}
For $j=1,2,\ldots$ one has
$$
2\partiali A_j(x_0,\underline{x}) = U_j(x_0,r^2) \, \frac{1}{r} \arctan{\frac{r}{x_0}} + V_j(x_0,r^2) \, \ln{\sqrt{x_0^2+r^2}} + W_j(x_0,r^2)
$$
with
$$
U_{2k}(x_0,r^2) = u_{2k}^{2k-1} x_0^{2k-1} + u_{2k}^{2k-3} r^2 x_0^{2k-3} + \cdots + u_{2k}^{1} r^{2k-2}x_0
$$
$$
U_{2k+1}(x_0,r^2) = u_{2k+1}^{2k} x_0^{2k} + u_{2k+1}^{2k-2} r^2 x_0^{2k-2} + \cdots + u_{2k+1}^{0} r^{2k}
$$
and
$$
V_{2k}(x_0,r^2) = v_{2k}^{2k-2} x_0^{2k-2} + v_{2k}^{2k-4} r^2 x_0^{2k-4} + \cdots + v_{2k}^{0} r^{2k-2}
$$
$$
V_{2k+1}(x_0,r^2) = v_{2k+1}^{2k-1} x_0^{2k-1} + v_{2k+1}^{2k-3} r^2 x_0^{2k-3} + \cdots + v_{2k+1}^{1} r^{2k-2}x_0
$$
and
$$
W_{2k}(x_0,r^2) = w_{2k}^{2k-2} x_0^{2k-2} + w_{2k}^{2k-4} r^2 x_0^{2k-4} + \cdots + w_{2k}^{0} r^{2k-2}
$$
$$
W_{2k+1}(x_0,r^2) = w_{2k+1}^{2k-1} x_0^{2k-1} + w_{2k+1}^{2k-3} r^2 x_0^{2k-3} + \cdots + w_{2k+1}^{1} r^{2k-2} x_0
$$
all the coefficients $u_{2k}^j$, $u_{2k+1}^j$, $v_{2k}^j$, $v_{2k+1}^j$, $w_{2k}^j$ and $w_{2k+1}^j$ being real constants.
\end{proposition}
\partialf
The proof is similar to that of Proposition 3.1. The harmonic potentials $A_1(x_0,\underline{x})$ and $A_2(x_0,\underline{x})$ computed above, fit into this general form. Now we will show that it is possible to determine unambiguously all the coefficients in the expression of $A_j(x_0,\underline{x})$ in terms of the coefficients in $A_{j-1}(x_0,\underline{x})$. To that end we impose on $A_j(x_0,\underline{x})$ the following two conditions, in line with its definition:
$$
\partialo (2\partiali A_j(x_0,\underline{x})) = 2\partiali A_{j-1}(x_0,\underline{x})
$$
and
$$
\lim_{x_0 \rightarrow 0+} (2\partiali A_j(x_0,\underline{x})) = 2\partiali a_{j}(\underline{x})
$$
In the case where $j$ is even, say $j=2k$, this leads to the equations
\begin{equation}
\langlebel{evenU}
\partialo U_{2k} = U_{2k-1}
\end{equation}
\begin{equation}
\langlebel{evenV}
\partialo V_{2k} = V_{2k-1}
\end{equation}
\begin{equation}
\langlebel{evenW}
- U_{2k} + x_0 V_{2k} + (x_0^2+r^2) \, \partialo W_{2k} = (x_0^2+r^2) \, W_{2k-1}
\end{equation}
\begin{equation}
\langlebel{evenlim3}
v_{2k}^0 r^{2k-2} \ln{r} + w_{2k}^0 r^{2k-2} = 2\partiali a_{2k}(\underline{x})
\end{equation}
From (\ref{evenU}) all the $u_{2k}$--coefficients may be determined as
$$
u_{2k}^j = \frac{1}{j} \, u_{2k-1}^{j-1}
$$
while from (\ref{evenV}) all the $v_{2k}$--coefficients, except $v_{2k}^0$, follow by the similar relation
$$
v_{2k}^j = \frac{1}{j} \, v_{2k-1}^{j-1}
$$
The coefficients $v_{2k}^0$ and $w_{2k}^0$ follow from (\ref{evenlim3}) and
all the other $w_{2k}$--coefficients follow recursively from the system of equations (\ref{evenW}). The last equation in this system can be used to check the value of $v_{2k}^0$.\\
In the case where $j$ is odd, say $j=2k+1$, the similar equations read
\begin{equation}
\langlebel{oddU}
\partialo U_{2k+1} = U_{2k}
\end{equation}
\begin{equation}
\langlebel{oddV}
\partialo V_{2k+1} = V_{2k}
\end{equation}
\begin{equation}
\langlebel{oddW}
- U_{2k+1} + x_0 V_{2k+1} + (x_0^2+r^2) \, \partialo W_{2k+1} = (x_0^2+r^2) \, W_{2k}
\end{equation}
and
\begin{equation}
\langlebel{oddlim3}
\frac{\partiali}{2} u_{2k+1}^0 r^{2k-1} = 2\partiali a_{2k+1}(\underline{x})
\end{equation}
From (\ref{oddU}) all the $u_{2k+1}$--coefficients, except $u_{2k+1}^0$, may be determined as
$$
u_{2k+1}^j = \frac{1}{j} \, u_{2k}^{j-1}
$$
The coefficient $u_{2k+1}^0$ follows from (\ref{oddlim3}).
From (\ref{oddV}) all the $v_{2k}$--coefficients follow by the similar relation
$$
v_{2k+1}^j = \frac{1}{j} \, v_{2k}^{j-1}
$$
All the $w_{2k+1}$--coefficients follow recursively from the system of equations (\ref{oddW}), and the last equation in this system can be used to check the value of $u_{2k+1}^0$.
~
{$\square$}\pagebreak[1]\par
\par
\begin{remark}
It is obvious that solving equations (\ref{evenlim3}) and (\ref{oddlim3}) requires the knowledge of the distributional boundary values $a_j(\underline{x}), j=1,2,\ldots$ There holds (see \cite{bdbds2})
$$
a_{2k+1}(\underline{x}) = (-1)^{k} \frac{1}{\partiali} \frac{1}{2^{k+2}} \frac{1}{k!} \frac{1}{(2k-1)!!} \, r^{2k-1} \quad (k = 0,1,2,\ldots)
$$
and
$$
a_{2k}(\underline{x}) = - (\alpha_{2k-2} \ln{r} + \beta_{2k-2}) \frac{2^k \partiali^{k}}{(2k-1)!!} \, r^{2k-2} \quad (k = 1,2,\ldots)
$$
where the coefficients $\alpha_{2k}$ and $\beta_{2k}$ are defined recursively by
$$
\left \{ \begin{array}{rcl}
\alpha_{2k+2} & = & - \displaystyle\frac{1}{2\partiali} \displaystyle\frac{1}{2k+2} \, \alpha_{2k}\\[4mm]
\beta_{2k+2} & = & - \displaystyle\frac{1}{2\partiali} \displaystyle\frac{1}{2k+2} \, (\beta_{2k} - \displaystyle\frac{4k+5}{(2k+2)(2k+3)} \, \alpha_{2k})
\end{array} \right . \qquad k=0,1,2,\ldots
$$
with starting values $\alpha_{0} = - \displaystyle\frac{1}{4\partiali^3}$ and $\beta_{0} = 0$, leading to their closed form
$$
\alpha_{2k} = \displaystyle\frac{(-1)^{k+1}}{2^{2k+2}\partiali^{k+3}k!} \quad {\rm and} \quad
\beta_{2k} = \displaystyle\frac{(-1)^{k} (H_{2k+1}-1)}{2^{2k+2}\partiali^{k+3}k!}
$$
\end{remark}
Now by Proposition 4.1 all the upstream $A_j$--potentials may be calculated recursively. Using the shorthands
$$
QUAT = \frac{1}{r} \arctan{\frac{r}{x_0}} \quad {\rm and} \quad LNSQ = \ln{\sqrt{x_0^2+r^2}}
$$
the outcome of our calculations is the following:
$$
2\partiali A_3(x_0,\underline{x}) = \frac{1}{2\partiali} (x_0^2 - r^2) QUAT + \frac{1}{\partiali} x_0 LNSQ - \frac{1}{2\partiali} x_0
$$
$$
2\partiali A_4(x_0,\underline{x}) = (\frac{1}{6\partiali} x_0^3 - \frac{1}{2\partiali} r^2 x_0) QUAT + (\frac{1}{2\partiali} x_0^2 - \frac{1}{6\partiali} r^2) LNSQ - \frac{5}{12\partiali} x_0^2 + \frac{5}{36\partiali} r^2
$$
$$
2\partiali A_5(x_0,\underline{x}) = (\frac{1}{24\partiali} x_0^4 - \frac{1}{4\partiali} r^2 x_0^2 + \frac{1}{24\partiali} r^4) QUAT + (\frac{1}{6\partiali} x_0^3 - \frac{1}{6\partiali} r^2 x_0) LNSQ - \frac{13}{72\partiali} x_0^3 + \frac{13}{72\partiali} r^2 x_0
$$
$$
2\partiali A_6(x_0,\underline{x}) = (\frac{1}{120\partiali} x_0^5 - \frac{1}{12\partiali} r^2 x_0^3 + \frac{1}{24\partiali} r^4 x_0) QUAT + (\frac{1}{24\partiali} x_0^4 - \frac{1}{12\partiali} r^2 x_0^2 + \frac{1}{120\partiali} r^4) LNSQ
$$
$$
- \frac{77}{1440\partiali} x_0^4 + \frac{77}{720\partiali} r^2 x_0^2 - \frac{77}{7200\partiali} r^4
$$
\vspace*{3mm}
Note that once an upstream $A_j$--potential is determined, the $B_{j-1}$--potential follows readily by
$$
(-\partialux)A_j(x_0,\underline{x}) = B_{j-1}(x_0,\underline{x})
$$
\section{Conclusion}
The problem of generalizing, within the framework of Clifford analysis, the holomorphic function $\ln{z}$ in the upper half of the complex plane to a monogenic logarithmic function in the upper half of Euclidean space $\mathbb{R}{m+1}_+$, led to a doubly infinite chain of monogenic and conjugate harmonic potentials or primitives with the Cauchy--kernel as a pivot element (see \cite{bdbds1}. Their distributional boundary limits in $\mathbb{R}^m$ turned out to be the fundamental solutions of positive and negative integer powers of the Dirac operator $\partialux$ and the Hilbert--Dirac operator, i. e. a convolution operator with kernel $\partialux H$, $H$ being the multidimensional Hilbert kernel (see \cite{hidi}). These operators are special cases of the operators $^\mu \partialux$ and $^\mu H$ defined for the complex parameter $\mu \in \mathbb{C}$ (see \cite{disturb, bdbds2}). Those results depend, quite naturally, on the dimension $m$ of the Euclidean space considered. There is the traditional phenomenon in Clifford analysis that the parity of the dimension has a substantial impact, one could even speak of an {\em even} and and {\em odd} world with diverging results. Next to that there is the problem that for specific dimensions $m=2$ and $m=3$, which are in fact the most important dimensions with a view on application, the general formulae are no longer valid. Both cases need fresh ad hoc calculations, and the obtained low dimensional results are reported on in the underlying paper.
\end{document}
|
\begin{document}
\title{Canonical description of 1D few-body systems with short range interaction}
\date{\today}
\author{Quirin Hummel}
\email{[email protected]}
\author{Juan Diego Urbina}
\author{Klaus Richter}
\affiliation{Institut f\"ur Theoretische Physik, Universit\"at Regensburg, D-93040
Regensburg, Germany}
\begin{abstract}
We address the fundamental interplay between indistinguishability and interactions when discreteness effects are neglected in systems with strictly fixed number of particles. For this end we supplement cluster expansions (many-body canonical techniques where quantum statistics is treated exactly) with short-time/large volume dynamical information where interparticle forces are described non-perturbatively. This approach, specially suitable for the few-body case where it overcomes the inappropriate use of virial expansions, can be consistently combined with scaling considerations, minimal ground-state information and strong coupling expansions in such a way that a single interaction event provides most of the thermodynamic and spectral properties of 1D systems with short range interactions. Our analytical results, in excellent agreement with numerical simulations, show a form of universal integrability of interaction effects for arbitrary confinements.
\end{abstract}
\pacs{}
\keywords{}
\maketitle
The description of the physical properties of systems with many (in general interacting) particles is one of the most intriguing and at the same time problematic subjects in modern physics.
As in most cases no exact solutions can be found one falls back either on full numerical simulations or on the problem of identifying simple, basic key features that build up the more complex systems as a whole and their emergent phenomena.
Progress in this direction can be achieved by the combination of quasiparticle, mean field and perturbative methods, with the physical picture corresponding to a system of particle-like excitations evolving under an effective external field and a weak residual interaction~\cite{negele2008,fetter2003}.
There are basically three reasons for the success of this approach in the past.
First, the previously unsurmountable difficulty in producing high excited states and the consequent focus on ground-state properties where the quasi-particle plus mean-field picture is valid.
Second, the natural interest on extreme regimes where a small parameter can be identified, thus justifying perturbation expansions.
Third, the macroscopically large number of particles typically involved, pushing the system into the limit where well developed grand-canonical methods could be used instead of the fundamental, but far less understood, canonical or microcanonical description.
The recent experimental realization of quantum systems made of few interacting, identical particles~\cite{greiner2002,serwane2011,preiss2015} and the consequent measurement of their spectral, thermodynamical and dynamical properties poses then a theoretical challenge: while for realistic few-body systems the very concept of mean-field is problematic, the fundamental issue is the lack of analytical tools to describe the interplay between indistinguishability and interaction within a strictly number-constraining formalism.
\begin{figure}
\caption{\label{fig:fig1}
\label{fig:fig1}
\end{figure}
\begin{figure*}
\caption{\label{fig:diagrams}
\label{fig:diagrams}
\end{figure*}
In this paper we fill this gap with a method that provides analytical results (given by sums over a {\it finite} set of diagrams) for spectral and thermodynamical properties of few-body systems where indistinguishability is treated exactly, interactions are treated non-perturbatively, and the total number of particles is strictly fixed.
At the heart of our approach lies the fact that the consistent use of short-time information responsible for the smooth properties of many-body spectra demands that interaction effects are universally given by cluster functions characteristic of quantum integrable models. We further show that this consistence must be applied order by order in the cluster expansion and therefore bounds any extra physical input like scaling considerations, condensation effects and fermionization. In this way most thermodynamic and spectral properties of interacting many-body systems, being quantum integrable or not, turned out to be analytical obtained in terms of a single interaction diagram.
The quality of our approach, the Quantum Cluster Expansion (QCE), is illustrated in \fref{fig:fig1} where we check our results against expensive numerical calculations of thermodynamic and spectral properties of a system made from few interacting bosonic atoms which are harmonically confined, a 1d many-body system of experimental relevance.
As is clearly seen in Fig.~\ref{fig:fig1}a, the failure of grand canonical approaches to describe the thermodynamics in this few-body system is {\it not} just a practical issue: already in the non-interacting limit, including more and more terms of the infinite virial expansions~\cite{gallavotti1999} does not reproduce the correct canonical description. On the contrary, the QCE (here used in its simplest form where condensation effects due to the discreteness of the ground state are not included) provides accurate results down to ultra low temperatures. As shown in Fig.~\ref{fig:fig1}b, when interactions are switched on the consistent combination of one single interaction cluster with scaling and strong coupling expansions provides analytical results for the many-body density of states in excellent agreement with numerical simulations.
The analytical description of few-body systems within the QCE requires two ingredients. First, all information about the discreteness of the many-body spectra is dropped, or included only to account for condensation effects. Like in text-book derivations of grand-canonical potentials for non-interacting systems, this is a standard assumption justified by the high level density of many body systems. Second, interactions will be included only at the pairwise level, but now in a way that is fully consistent with particle exchange symmetry and, importantly, with the short time expansion implicit in the smooth contribution to the spectrum.
The technical implementation of these assumptions begins with an exact, finite cluster expansion~\cite{ursell1927,kahn1938} of the quantum propagator $K^{(N)}$ for $N$ distinguishable but interacting particles which to first order reads
\begin{eqnarray} \label{eq:QCEK}
&&K^{(N)}({\bf q}^\mathrm{f},{\bf q}^\mathrm{i};t) = K^{(N)}_0({\bf q}^\mathrm{f}, {\bf q}^\mathrm{i};t) \nonumber\\
&&+ \sum_{k<l} K^{(N-2)}_0({\bf q}^\mathrm{f}_{\overline{kl}},{\bf q}^\mathrm{i}_{\overline{kl}};t) \Delta K^{(2)}({\bf q}^\mathrm{f}_{kl},{\bf q}^\mathrm{i}_{kl};t)
+ \ldots \,.
\end{eqnarray}
Here ${\bf q}^\mathrm{f}$ and ${\bf q}^\mathrm{i}$ are the final and initial coordinates, $\overline{kl}$ denotes the set of particle labels excluding $k,l$ and the subscript $0$ refers to non-interacting propagation related through $\Delta K^{(2)}$ with the full two-body propagator by
\begin{equation} \label{eq:K2}
K^{(2)} = K^{(2)}_0 + \Delta K^{(2)} \,.
\end{equation}
The canonical partition function $Z(\beta)={\rm Tr}K(t = - \mathrm{i} \hbar \beta)$ is then given by tracing in the properly (anti)symmetrized coordinate basis, while its inverse Laplace transform
\footnote{We prefer to use the bilateral form of the Laplace transform which gives the correct behavior at negative energies in form of Heaviside-step-functions [\textit{e.g.}\ $\Linv[\beta^{-1/2}](E)=(\pi E)^{-1/2} \theta(E)$].}
yields the many-body density of states $\varrho(E) = \Linv[Z(\beta)](E)$. Finally, following the approach of~\cite{hur14}, Weyl's method to obtain the smooth single-particle spectrum by replacing the exact quantum propagation for its short-time limit is here generalized to the many-body, interacting case. In the case of $N$ identical non-interacting particles of mass $m$, confined by the homogeneous potential $V(q)=w^{\mu}V(q/w)$ one gets ~\cite{hur14}
\begin{equation} \label{eq:Znonint}
Z^{(N)}_{0,\pm}(\beta) = \sum_{l=1}^N z_l \left( \frac{V_{\rm eff}}{\lambda_T^d} \right)^l \,, \quad z_l = (\pm 1)^{N-l} C_l^{(N,d)} / l! \,,
\end{equation}
where $\lambda_T = \sqrt{2\pi\hbar^2 \beta / m}$ is the thermal wavelength, the universal constants $C_l^{(N,d)}$ can be found in~\cite{hur14}, and plus~(minus) refer to bosons~(fermions), while the effective dimension $d=D+\frac{2}{\mu}D$ and effective volume
$V_{\rm eff} = (2 \hbar^2 / m e_{0})^{D / \mu} \int \mathrm{d}^D\!q \exp({-V({\bf q})/e_{0}})$ with $e_{0}$ an arbitrary energy unit, are given in terms of the physical dimension $D$ and degree of homogeneity $\mu$. The special case of zero external potential is included as $\mu \rightarrow \infty$, $d=D$, and with the available physical volume $V_{\rm eff} = V_D$.
An overview of the possible contributions to the QCE is given by the diagrams shown in Fig.~\ref{fig:diagrams}. The non-interacting part $K^{(N)}_0$ of the propagator factorizes into single-particle (SP) propagators~(see~\fref{fig:diagrams}a) and the contribution to $Z$ corresponding to a permutation $P \in S_N$ is a product of {\it cluster}-contributions, each involving a subset of the particles as large as the cycle-lengths in the cycle-decomposition of $P$.
Let $\mathfrak{N}_P$ denote the multiset with elements $n_i \in \mathbb{N}$ corresponding to the cycle lengths of a permutation $P \in S_N$.
Clearly $\sum_i n_i = N$ and we may use $\mathfrak{N}$ without subscript wherever the assignment is clear from context.
The contribution to the trace of the propagator from one cycle of length $n_i$ is the amplitude~(see~\fref{fig:diagrams}b)
\begin{eqnarray} \label{eq:Ani}
\mathcal{A}_{n_i}(t) &=& \int \mathrm{d}^D\!q_1 \ldots \mathrm{d}^D\!q_{n_i} \prod_{k=1}^{n_i} K^{(1)}_0({\bf q}_{k+1},{\bf q}_k; t) \nonumber \\ &=& \int \mathrm{d}^D\!q \, K^{(1)}_0({\bf q}, {\bf q};n_i t) \,,
\end{eqnarray}
where we use the semigroup property of the SP propagator, with the identification ${\bf q}_{n_i+1}:={\bf q}_1$.
Consistently with the short time propagation (as discussed in~\cite{hur14}), we will also use $K_0^{(1)}({\bf q},{\bf q},t) \simeq \mathrm{e}^{-\frac{i}{\hbar}V({\bf q})t}K_{\rm free}^{(1)}({\bf q},{\bf q},t)$ where $K_{\rm free}$ stands for unconfined propagation.
The full contribution to the non-interacting partition-function corresponding to a permutation is then~(see~\fref{fig:diagrams}c)
\begin{equation}
\mathcal{A}_\mathfrak{N}(-\mathrm{i} \hbar \beta) = \prod_{n \in \mathfrak{N}} \mathcal{A}_{n}(- \mathrm{i} \hbar \beta)
\end{equation}
while the partition function is
\begin{equation} \label{eq:Znonintgen}
Z_{0,\pm}^{(N)}(\beta) = \frac{1}{N!} \sum_{\mathfrak{N} \vdash N} (\pm 1)^{N-l} c^{(N)}_\mathfrak{N} \mathcal{A}_\mathfrak{N}(- \mathrm{i} \hbar \beta) \,.
\end{equation}
Here, the sum runs over all partitions $\mathfrak{N}$ of $N$ and
\begin{equation} \label{eq:c}
c^{(N)}_\mathfrak{N} := \frac{N!}{\prod_{n\in \mathfrak{N}} n \prod_n m_{\mathfrak{N}}(n)!}
\end{equation}
denotes the number of permutations of $N$ with a cycle-decomposition corresponding to $\mathfrak{N}$, where $m_{\mathfrak{N}}(n)$ is the multiplicity of $n$ in $\mathfrak{N}$.
Evaluation of~\eref{eq:Znonintgen} yields then the explicit result~\eref{eq:Znonint}.
At the pairwise level, corresponding to~\eref{eq:QCEK} the effect of interactions is calculated by choosing all possible pairs $\{k,l\}$ of particles and replacing the product $K_0^{(1)}(q_{P(k)},q_k;t) K_0^{(1)}(q_{P(l)},q_l;t)$ in $\mathcal{A}_\mathfrak{N}$ by the interaction term $\Delta K^{(2)}((q_{P(k)},q_{P(l)}),(q_k,q_l);t)$ defined in~\eref{eq:K2}~(see~\fref{fig:diagrams}d).
In the corresponding corrections to $\mathcal{A}_\mathfrak{N}$, the interaction can link two particles involved in either the same or in two different cycles of $P$, referred as {\it intra-cycle}- and {\it inter-cycle}-contributions respectively.
Basic combinatorics show that the joint contribution to $Z$ from all inter-cycle contributions is
\begin{equation} \label{eq:Zinter}
\begin{split}
Z_{\rm inter}^{(N)} = \left( \begin{matrix} N \\ 2 \end{matrix} \right) \frac{1}{N!} \sum_{n_1=1}^{N-1} \sum_{n_2=1}^{N-n_1} \sum_{\mathfrak{N} \vdash N-n_1-n_2} (\pm 1)^{N-l-2} \\
\times A_{(n_1,n_2)}^{\rm inter} A_{\mathfrak{N}} \, c_{\mathfrak{N}}^{(N-2)} \,,
\end{split}
\end{equation}
where $A_{(n_1,n_2)}^{\rm inter}$ is the amplitude of two cycles with $n_1$ and $n_2$ particles involving the interacting part $\Delta K^{(2)}$ for a pair of particles, one of which living in each of the two cycles (see~\fref{fig:diagrams}e). Note that in definition~\eref{eq:c} the cardinality of $\mathfrak{N}$ may differ from the upper index $N$, which is the case in~\eref{eq:Zinter} and that $c_{\mathfrak{N}}^{(N-2)}$ has the meaning of counting the number of permutations of $N$ with two distinct cycles of length $n_1$ and $n_2$ involving particle 1 and 2 respectively and the remaining $N-n_1-n_2$ being composed of cycles with lengths $\mathfrak{N}$. Finally, for the case $n_1+n_2=N$ we consistently define $A_{\{\}}:=1$ and $c_{\{\}}^{(N-2)}:=(N-2)!$.
For practical use, it is convenient to write~\eref{eq:Zinter} as
\begin{equation}
Z_{\rm inter}^{(N)} = \frac{1}{2} \sum_{n=2}^{N} (\pm 1)^n Z_{0,\pm}^{(N-n)} \sum_{n_1=1}^{n-1} A_{(n_1,n-n_1)}^{\rm inter} \,,
\end{equation}
which recursively generates $Z_{\rm inter}^{(N)}$ depending on non-interacting partition functions of all smaller particle numbers. Analogue considerations on the intra-cycle contributions $A_{(n_1,n_2)}^{\rm intra}$ shown in~\fref{fig:diagrams}f finally yield the (first order) QCE correction to the partition function
\begin{equation} \label{eq:DeltaZ}
\Delta Z^{(N)}_\pm = \sum_{n=2}^N (\pm 1)^n Z^{(N-n)}_{0,\pm} \sum_{n_1=1}^{n-1}
\frac{A_{(n_1,n-n_1)}^{\rm inter} \pm A_{(n_1,n-n_1)}^{\rm intra} }{2}
\end{equation}
and its extension for the case with multiple distinguishable species in~\cite{SM}.
At the level of general interactions, equation~(\ref{eq:DeltaZ}) is the main result of this paper. It results from the consistent use of short-time dynamical information, in the spirit of the celebrated Weyl expansion, to obtain thermodynamic and spectral properties that are not sensitive to the discreteness of the many-body spectrum. It is organized in a way such that all contributions coming from indistinguishability are included, while interaction takes place among one pair of particles at the time.
In the following we will show how the QCE can be used to further provide analytical results in situations where the explicit calculation of $A_{(n_1,n_2)}$ is possible. Remarkably, this is the case for the broad case of 1D systems with contact interactions that covers both the integrable Lieb-Liniger model~\cite{lieb1963} as well as the non-integrable and experimentally important case of homogeneous (in particular harmonic) confinement.
We choose units by setting $\hbar^2/2m = 1$ such that the thermal wavelength becomes $\lambda_T = \sqrt{4\pi \beta}$.
In these units, the Hamiltonian of the $N$ particle system with coordinates $x_i$ is
\begin{equation} \label{eq:Hdelta}
\hat{H} = \sum_{i=1}^N \left(-\frac{\partial^2}{\partial x_i^2} +V(x_{i})\right)+ \sqrt{8 \alpha} \sum_{i<j} \delta(x_i - x_j) \,,
\end{equation}
where $\alpha$ is an energy associated with the strength of the interaction.
\Q{Explain LPA.}
Using the explicit expression for the interacting part of the two-body propagator for this potential~\cite{manoukian1989}, $A_{(n_1,n_2)}^{\rm inter}$ is found to be given by~(\fref{fig:diagrams}e)
\begin{equation} \label{eq:Adelta}
\begin{split}
A_{(n_1,n_2)}^{\rm inter} = - \frac{V_{\rm eff}}{\lambda_T^d n^{\frac{d}{2}}} \frac{\sqrt{2\beta \alpha}}{4 \pi } \int_0^\infty \mathrm{d} r \int_{-\infty}^\infty \mathrm{d} z \int_0^\infty \mathrm{d} u \\
\times \exp \left[-\frac{1}{8} z^2 - \sqrt{\frac{\beta \alpha}{2}} u - \frac{1}{8}( | \bar{\nu} z + r | + |r| + u )^2 \right] \,,
\end{split}
\end{equation}
where $n$ and $\bar{\nu}$ are related to the numbers of particles involved in the process by
\begin{equation}
\bar{\nu} = \sqrt{(2 n_1 n_2 - n_1 - n_2)/n} \,, \quad n=n_1+n_2 \,.
\end{equation}
For $\delta$-interactions it turns out that $A_{(n_1,n_2)}^{\rm intra}=A_{(n_1,n_2)}^{\rm inter}=A_{(n_1,n_2)}$, thus confirming (due to~\eref{eq:DeltaZ}) their vanishing effect on spinless fermions.
Finally, the multiple integrals in~\eref{eq:Adelta} can be reduced by further manipulations to get
\begin{equation} \label{eq:Adelta2}
\begin{split}
A_{(n_1,n_2)} = \frac{V_{\rm eff}}{\lambda_T^d n^{\frac{d}{2}}} \left[ \frac{2}{\pi} \atan \bar{\nu} - 1 + \frac{2 \bar{\nu}^2}{\sqrt{\pi(1+\bar{\nu}^2)}} \sqrt{s} \right.\\
\left. - \frac{2}{\sqrt{\pi}} \bar{\nu} \sqrt{s} \mathrm{e}^s \erfc (\sqrt{s}) + \frac{2}{\sqrt{\pi}} (1 - 2 \bar{\nu}^2 s) F_{\bar{\nu}}(s) \right] \,,
\end{split}
\end{equation}
where we introduced the thermal interaction strength $s=\beta \alpha$.
The remaining integral is defined by
\begin{equation} \label{eq:F}
F_{\bar{\nu}}(s) = \mathrm{e}^{(1+\bar{\nu}^2)s} \int_0^\infty \mathrm{d} z \mathrm{e}^{-(z-\bar{\nu} \sqrt{s})^2} \erfc (\sqrt{s} + \bar{\nu} z) \,,
\end{equation}
and therefore setting $\bar{\nu} = 0$, $d=1$, and $V_{\rm eff}=L$ recovers the case without confinement involving only two particles $A_{(1,1)} = \frac{L}{\lambda_T} \frac{1}{\sqrt{2}} (-1 + \mathrm{e}^s \erfc (\sqrt{s}))$~\cite{ghur2015}.
By substituting $F_{\bar{\nu}}(s)$ into $A_{(n_1,n_2)}$ in Eq.~(\ref{eq:Adelta2}) and using the later into Eq.~(\ref{eq:DeltaZ}) we obtain an analytical expression for the canonical partition function (for specific applications the alternative form of~\eref{eq:F} given in~\cite{SM} improves accuracy in the numerical integration).
Up to this point, we have used the QCE in its simplest form where only one interaction event is taken into count, valid from vanishing to moderately high interaction strength $\alpha$. The quality of the results is, however, drastically extended to higher values of $\alpha$ using the same reduced information by means of a shifting method based on the following universal scaling of $A_{(n_1,n_2)}$ with the (effective) system size, temperature and interaction strength. The main contribution to the $n$-fold integrals in Eq.~(\ref{eq:Ani}) after pairwise replacement with the interacting parts $\Delta K^{(2)}$ comes from the region where all $n$ particles are close to each other, implying fast convergence and allowing us to extend all integrals over relative coordinates to infinity, whereas changes in the center-of-mass are only subject to the external potential, thus yielding the (effective) size $V_{\rm eff}$ of the system as prefactor.
From dimensional analysis the scaling behavior with $V_{\rm eff}/\lambda^{d}_T$ and $\beta \alpha$ then follows as is discussed to more detail in~\cite{SM}. Remarkably, the scaling with $1 / \sqrt{n}$ reminds of the scaling of a non-interacting cycle~\eref{eq:Ani} and fits to the physical picture that all particles that are connected by either symmetry-permutations and/or interaction form a single cluster whose internal wavefunction spreads with a velocity proportional to $n$.
Since the previous arguments hold also for other dimensions (as long as the interaction is ``short-ranged'' in the sense above and can be expressed in terms of a single energy-type parameter $\alpha$ representing its strength) we expect the universal form
\begin{equation} \label{eq:Ascaling}
A_{(n_1,n_2)}^{\rm inter/intra} = \frac{V_{\rm eff}}{\lambda_T^d} n^{-\frac{d}{2}} a_{(n_1,n_2)}^{\rm inter/intra}(\beta \alpha)
\end{equation}
with the internal part $a_{(n_1,n_2)}^{\rm inter/intra}(s)$ characteristic for the specific interaction and not depending on the external potential.
This scaling together with the scaling of the non-interacting contributions~\eref{eq:Znonint} used in~\eref{eq:DeltaZ} allow us to write the full partition function in first order QCE for $N$ bosons or fermions in a system of (effective) size $V_{\rm eff}$ in the form
\Q{write $\pm$ at $(\Delta)z$?}
\begin{equation} \label{eq:Z1}
Z_1^{(N)}(\beta) = \sum_{l=1}^{N} [ z_l + \Delta_1 z_l(\beta \alpha) ] \left( \frac{V_{\rm eff}}{\lambda_T^d} \right)^l \,.
\end{equation}
The interaction-related coefficients are
\Q{write $\pm$ at $(\Delta)z$?}
\begin{equation} \label{eq:Deltaz}
\Delta_1 z_l(s) = \sum_{n=2}^{N-l+1} (\pm 1)^n n^{-\frac{d}{2}} z_{l-1}^{(N-n)} \sum_{n_1=1}^{n-1} a_{(n_1,n-n_1)}(s) \,,
\end{equation}
where (defining $z_0^{(m)} := \delta_{m0}$) the general $z$ can be read off from~\eref{eq:Znonint} while $a_{(n_1,n_2)}=(a^{\rm inter}_{(n_1,n_2)} \pm a^{\rm intra}_{(n_1,n_2)})/2$ (given by~\eref{eq:Adelta2} and~\eref{eq:Ascaling} for $\delta$-interactions) must be evaluated for each particular interaction.
We will illustrate the validity of the QCE in general and of the scaling property~\eref{eq:Z1} in particular by comparing its thermodynamical and spectral consequences against numerical simulations.
The QCE mechanical equation of state~\cite{gallavotti1999}
\begin{equation} \label{eq:EOS}
P(V_{\rm eff},\beta,N,\alpha) = \frac{k_\mathrm{B} T}{V_{\rm eff}} \frac{\sum_{l=1}^N l [z_l + \Delta z_l(\beta \alpha)] \big( \frac{V_{\rm eff}}{\lambda_T^d} \big)^l}
{\sum_{l=1}^N [z_l + \Delta z_l(\beta \alpha)] \big( \frac{V_{\rm eff}}{\lambda_T^d} \big)^l} \,,
\end{equation}
gives a finite expression for the pressure $P$ in terms of $N$, contrary to virial expansions in the grand canonical treatment, that reproduces very well the exact numerical calculations.
In the same spirit, within QCE the many-body smooth density of states is found to be ($\hbar^2 / 2 m = 1$)
\Q{simplified to solely $f_l$, change SM.}
\begin{equation} \label{eq:rho}
\bar{\varrho}^{(N)}_\pm(E) = \sum_{l=1}^{N} \left[ \frac{z_l}{\Gamma\!\left(\frac{l d}{2}\right)} + f_l\!\left(\frac{E}{\alpha}\right) \right]
\frac{V_{\rm eff}^l E^{\frac{l d}{2}-1} \theta(E)}{(4\pi)^{\frac{l d}{2}}} \, ,
\end{equation}
where the second term between brackets corresponds to the interacting part and the functions $f_l$ can be expressed through elementary functions in the case of $\delta$-interaction~\cite{SM}.
Note that~\eref{eq:rho} shows that the effect of interactions gets suppressed either when the total energy $E \gg \alpha$ or $E \ll \alpha$ for interaction potentials that vanish for $\alpha \rightarrow 0$ or $\alpha \rightarrow \infty$, respectively.
\begin{figure}
\caption{\label{fig:fig4}
\label{fig:fig4}
\end{figure}
A special feature of 1D systems with contact interactions that is clearly seen in the numerical results is the apparent mapping linking the limits $\alpha\to 0$ with $\alpha\to \infty$.
This correspondence can be made precise using an exact boson-fermion duality valid for arbitrary $\alpha$~\cite{cheon1999}.
Thus, we can construct the QCE expansion around the strongly interacting regime as an effective spinless fermionic theory, providing again analytical expressions for the partition function~\cite{SM} that perfectly describe the numerical observations in the corresponding regime of large $\alpha$.
In particular, for infinitely strong repulsion, the system behaves as a gas of free fermions in the currently relevant aspects.
For harmonic confinement, fermionization is additionally reflected by the rigid shift $\Delta E_{\infty}$ between the DOS in the two limits.
This feature is incorporated by a suitable generalization/extension of our approach.
Motivated by the general scaling property~\eref{eq:rho} we propose the ansatz
\begin{equation}
\label{eq:Eshift}
\Delta E_{\alpha}=\chi(E/\alpha)\Delta E_{\infty}
\end{equation}
for the energy shift $\Delta E_\alpha$ for finite interaction strength.
In~\cite{SM} we show that the universal scaling $\chi(E/\alpha)$ is uniquely obtained from the first order QCE itself.
It interpolates between the different regimes for $\alpha$ without any fitting.
The shifting method provides again analytical results in good agreement with numerical calculations shown in~\fref{fig:fig1}b.
Besides the possibility of including consistently short-time dynamics into the analytical description given by the QCE when the later is supplemented with scaling considerations and fermionization, a final point is the description of condensation phenomena. Here, and similarly to the usual grand canonical approach, within QCE ultra low temperature effects require that the ground state is treated separately.
Within QCE, this can be achieved by a consistent method where minimal information about the lowest two MB states is combined with the QCE for non-zero temperatures by the ansatz
\begin{equation}
\label{eq:split}
Z(\beta) = \mathrm{e}^{-\beta E_0(V_\mathrm{eff})} + \mathrm{e}^{-\beta E_1(V_\mathrm{eff})} \sum_{l=0}^N w_l(\beta \alpha) \left( \frac{V_{\rm eff}}{\lambda_T^d} \right)^l \,,
\end{equation}
which can be analytically matched order by order for large $V_\mathrm{eff}$ with~\eref{eq:Z1} to determine the $w_l$ functions.
With this minimal modification, and using only one interaction event, the corresponding modification to~\eref{eq:EOS} shows again excellent agreement with numerical results for the Lieb-Liniger model covering a large regime of interactions and all system sizes~(see~\fref{fig:fig4}).
Moreover, numerical simulations require thousands of many-body energy levels to achieve convergence, and therefore are feasible only because the model at hand is quantum integrable.
Although the QCE exploits the universality of the smooth part of the many-body density of states, in the sense of its dependence with a very restricted set of universal functions together with few geometrical parameters, it can be used to study system specific effects. This is again illustrated in~\fref{fig:fig4} where the non-monotonicity of the pressure as a function of the system's length for three interacting bosons on a ring, a very peculiar consequence of the competition between interactions and bunching, is fully reproduced by our analytical formulas.
In conclusion, we have shown that the consistent use of short-time/large-volume dynamical information in the description of interacting 1d few-body systems leads to the emergence of robust features depending on a very restricted set of universal functions. In particular, most spectral and thermodynamical observable properties that are not sensitive to the discreteness of the spectrum are resembled by only two-body effects even for non-integrable models.
Our results show that the condition of integrability is too restrictive when one is not interested in the precise form of the many-body spectrum but instead on its smooth part and analytical results can be found for smooth observables for the, previously considered intractable, non-integrable cases.
We acknowledge financial support from the DPG through the FOR760, and illuminating discussions with Peter Schmelcher, Bruno Eckhart and Benjamin Geiger.
\cleardoublepage
\newcommand{\sectionQ}[1]{\section{}\vspace*{-10mm}\begin{center}{\bf #1}\end{center}}
\setcounter{section}{1}
\appendix
\sectionQ{Appendix A: Formal derivation of QCE in path-integral formulation}
This section is intended to give analytic support to~\Leref{eq:QCEK}.
To find the first correction to the $N$-body propagator within QCE we start with the exact path-integral representation of the distinguishable propagator
\begin{equation}
\begin{split}
\MoveEqLeft[6] K^{(N)}({\bf q}^\mathrm{f},{\bf q}^\mathrm{i};t) = \\
\int_{{\bf q}^\mathrm{i}}^{{\bf q}^\mathrm{f}} \mathcal{D}{\bf q}(s)
&\prod_{k=1}^N \exp \left[{\frac{\mathrm{i}}{\hbar}\int_0^t \frac{m}{2} [\dot{{\bf q}}_k(s)]^2 - V_{\rm ext}({\bf q}_k(s))} \mathrm{d} s \right] \\
\times &\prod_{k<l} \exp \left[{-\frac{\mathrm{i}}{\hbar} \int_0^t V_{\rm int}({\bf q}_k(s) - {\bf q}_l(s)) \mathrm{d} s} \right] \,.
\end{split}
\end{equation}
Analogous to the Mayer functions in the cluster expansion in classical statistical mechanics we define the \textit{Mayer functionals} $f_{kl}[{\bf q}(s)]$ by
\begin{equation}
1 + f_{kl}[{\bf q}(s)] := \exp \left[{-\frac{\mathrm{i}}{\hbar} \int_0^t V_{\rm int}({\bf q}_k(s) - {\bf q}_l(s)) \mathrm{d} s} \right] \,.
\end{equation}
The next step is to expand the product $\prod_{k<l}$ over pairs into a sum and order its terms by the number of Mayer functionals involved.
\begin{equation}
\prod_{k<l} (1 + f_{kl}[{\bf q}(s)]) = 1 + \sum_{k<l} f_{kl}[{\bf q}(s)] + \ldots
\end{equation}
In first order QCE we truncate all terms that involve more than one Mayer functional which physically corresponds to neglecting interaction effects that are affecting more than one pair of particles at a time.
Since fo indistinguishable particles all kinds of symmetry related cycle-structures are applied afterwards this will still give non-trivial interaction-induced contributions involving more than two particles.
The first summand involving no Mayer functional gives the non-interacting propagator whereas the next term is evaluated by factorizing the path-integral into independent factors and using the two-body identity
\begin{equation}
\begin{split}
&\int_{{\bf q}_{kl}^\mathrm{i}}^{{\bf q}_ {kl}^\mathrm{f}} \mathcal{D}{\bf q}_{kl}(s)
\prod_{j = k,l} \exp \left[{\frac{\mathrm{i}}{\hbar}\int_0^t \frac{m}{2} [\dot{{\bf q}}_j(s)]^2 - V_{\rm ext}({\bf q}_j(s))} \mathrm{d} s \right] \\
&\qquad\qquad\times \left( \exp \left[{-\frac{\mathrm{i}}{\hbar} \int_0^t V_{\rm int}({\bf q}_k(s) - {\bf q}_l(s)) \mathrm{d} s} \right] - 1 \right) \\
&= K^{(2)}({\bf q}_{kl}^\mathrm{f}, {\bf q}_{kl}^\mathrm{i};t) - K_0^{(1)}({\bf q}_k^\mathrm{f}, {\bf q}_k^\mathrm{i};t) K_0^{(1)}({\bf q}_l^\mathrm{f}, {\bf q}_l^\mathrm{i};t) \\
&= \Delta K^{(2)}({\bf q}_{kl}^\mathrm{f}, {\bf q}_{kl}^\mathrm{i};t) \,,
\end{split}
\end{equation}
where ${\bf q}_{kl} = ({\bf q}_k,{\bf q}_l)$, the subscript $0$ denotes propagation amplitudes of the corresponding non-interacting system and hence $\Delta K^{(2)}$ denotes the full interacting part of the two-body propagator $K^{(2)}$.
Together with the $N-2$ independent path-integrals for the remaining particles $j\neq k,l$, which lead to non-interacting single-particle propagators we obtain~\Leref{eq:QCEK}.
\sectionQ{Appendix B: Multiple species}
If multiple distinguishable species of particles are involved the full first order contribution to the overall partition function reads
\begin{equation} \label{sm:eq:DeltaZmultiS}
\begin{split}
\Delta Z^{(N_1,\ldots,N_s)} ={} &\sum_{i=1}^s \Delta Z_{\epsilon_i}^{(N_i)} \prod_{j \neq i} Z_{0,\epsilon_j}^{(N_j)} \\
&{}+ \sum_{i<j} \sum_{n_i=1}^{N_i} \sum_{n_j=1}^{N_j} \epsilon_i^{n_i-1} \epsilon_j^{n_j-1} A_{(n_i,n_j)}^{\rm inter} \\
&\quad\times Z_{0,\epsilon_i}^{(N_i-n_i)} Z_{0,\epsilon_j}^{(N_j-n_j)} \prod_{k \neq i,j} Z_{0,\epsilon_k}^{(N_k)} \,,
\end{split}
\end{equation}
where $N_1,\ldots,N_s$ are the numbers of particles in each of the $s$ species and $\epsilon_i = \pm$ reflects the exchange symmetry within species $i$.
The special case $s=2$ of~\eref{sm:eq:DeltaZmultiS} can be used for calculations on the Gaudin-Yang model.
In the given general form,~\eref{sm:eq:DeltaZmultiS} is valid for arbitrary short-ranged interactions addressable with the QCE approach using~\Leref{eq:DeltaZ}.
The interaction-related two-body information then finds its way into~\eref{sm:eq:DeltaZmultiS} through the diagrammatic calculation of $A_{(n_1,n_2)}^{\rm inter}$ and $A_{(n_1,n_2)}^{\rm intra}$ (see~\fref{fig:diagrams}e,f of the Letter) depending on the interacting part of the propagator of two particles living in free space and being subject to the specific interactions (see~\fref{fig:diagrams}d of the Letter).
In the case of $\delta$-interactions the given expression~\Leref{eq:Adelta} can be used in~\eref{sm:eq:DeltaZmultiS}.
Special care has to be taken if some of the particle species are allowed to differ in mass.
Then one has to relax the specific choice of units $\hbar^2 / (2m) =1$ because of ambiguity and take care of the correct masses $m_i$ in all calculations.
This is done by substituting the corresponding thermal de-Broglie wavelength $\lambda_{\rm T} \rightarrow \lambda_{\rm T}^i$ in all expressions involving only one species $i$ on the one hand.
We denote the modified quantities with a tilde and find the two trivial substitutions
\begin{align}
\tilde{Z}_{0,\epsilon_i} &{}= \left. Z_{0,\epsilon_i} \right|_{\lambda_{\rm T} \rightarrow \lambda_{\rm T}^{i}} \,, \nonumber \\
\Delta\tilde{Z}_{\epsilon_i} &{}= \left. \Delta Z_{\epsilon_i} \right|_{\lambda_{\rm T} \rightarrow \lambda_{\rm T}^{i}} \,,
\end{align}
with the corresponding thermal de-Broglie wavelength
\begin{equation}
\lambda_{\rm T}^i = \left( \frac{2 \pi \beta \hbar^2}{m_i} \right)^{\frac{1}{2}} \,.
\end{equation}
On the other hand, the inter-cycle contributions $A_{(n_i,n_j)}^{\rm inter}$ [see~\Leref{eq:Adelta}] between two different species $i$ and $j$ have to be altered by the prescription
\begin{equation}
\tilde{A}_{(n_i,n_j)}^{\rm inter} = \left( \frac{M_{ij}}{4 \mu_{ij}} \right)^{\frac{1}{2}} \left. A_{(n_i,n_j)}^{\rm inter}
\right|_{\substack{\lambda_{\rm T} \rightarrow \tilde{\lambda}_{\rm T}^{ij} \\ n \rightarrow \tilde{n}_{ij} \\ \bar{\nu} \rightarrow \tilde{\bar{\nu}}_{ij} }} \,,
\end{equation}
where the modified quantities
\begin{align}
\tilde{\lambda}_{\rm T}^{ij} &{}= \left( \frac{\pi \beta \hbar^2}{\mu_{ij}} \right)^{\frac{1}{2}} \,, \nonumber \\
\tilde{n}_{ij} &{}= \frac{2 m^{\rm tot}_{ij}}{M_{ij}} \,, \nonumber \\
\tilde{\bar{\nu}}_{ij} &{}= \sqrt{\frac{M_{ij}}{m^{\rm tot}_{ij}} n_i n_j - 1} \,,
\end{align}
are defined in terms of the reduced and total mass
\begin{align}
\mu_{ij} &{}= \frac{m_i m_j}{m_i + m_j} \,, \nonumber \\
M_{ij} &{}= m_i + m_j
\end{align}
of two representatives of the different species and the total cluster-mass
\begin{equation}
m^{\rm tot}_{ij} = n_i m_i + n_j m_j \,.
\end{equation}
Naturally, it is also possible to put different interaction-strengths $\alpha_{ij}$ between different species.
\sectionQ{Appendix C: Numerically stable representation of $F_{\bar{\nu}}(s)$}
The integral given in~\Leref{eq:F} is subject to numerical instability for large values of $s$.
In order to represent the function $F_{\bar{\nu}}(s)$ in a form where the numerical accuracy is not an essential issue, one can partially treat the integral analytically in a way that the remaining integral gives only small contributions also for large values of $s$.
To acchieve this we first recognize that $F_{\bar{\nu}}(s)$ can be written in terms of Owen's $T$-function
\begin{equation}
T(a,b) = \frac{1}{2\pi} \int_0^b \mathrm{d} x \frac{\mathrm{e}^{-\frac{1}{2} a^2 (1+x^2)}}{1+x^2} \,.
\end{equation}
The corresponding expression is
\begin{equation}
\begin{split}
F_{\bar{\nu}}(s) = \mathrm{e}^{(1+\bar{\nu}^2)s} &\left[ {\rm erf}(\bar{\nu} \sqrt{s}) - {\rm erf}(\sqrt{(1+\bar{\nu}^2)s}) \right. \\
&\left. {}+ 4 T(\bar{\nu} \sqrt{2s}, \bar{\nu}^{-1}) \right] \,,
\end{split}
\end{equation}
and by use of the general property
\begin{equation}
T(h,a) + T\!\left(a h, \frac{1}{a}\right) = \frac{1}{4} \left( 1 - {\rm erf}\!\left( \frac{h}{\sqrt{2}} \right) {\rm erf}\!\left( \frac{a h}{\sqrt{2}} \right) \right)
\end{equation}
it is equivalent to
\begin{equation} \label{eqSM:F2}
\begin{split}
F_{\bar{\nu}}(s) = \mathrm{e}^{(1+\bar{\nu}^2)s} &\left[ {\rm erfc}(\sqrt{(1+\bar{\nu}^2)s}) - {\rm erfc}(\bar{\nu} \sqrt{s}) {\rm erfc}(\sqrt{s}) \right. \\
&\left. {}+ {\rm erfc}(\sqrt{s}) - 4 T(\sqrt{2 s},\bar{\nu}) \right] \,.
\end{split}
\end{equation}
The terms in the first row of this equation are well behaved numerically, since the asymptotics $\mathrm{e}^{x^2} {\rm erfc}(x) = 1/(\sqrt{\pi} x) + \mathcal{O}(x^{-3})$ for $x\gg1$ are very well known.
The numerical problem now lies in cancellation effects between the two terms of the second row.
To overcome this, we split the Owen $T$ function
\begin{equation}
4 T(\sqrt{2 s}, \bar{\nu}) = \frac{2}{\pi} \int_0^\infty \mathrm{d} x \frac{\mathrm{e}^{-s (1+x^2)}}{1+x^2} -
\frac{2}{\pi} \int_{\bar{\nu}}^\infty \mathrm{d} x \frac{\mathrm{e}^{-s (1+x^2)}}{1+x^2} \,.
\end{equation}
The first term can be evaluated to
\begin{equation}
\frac{2}{\pi} \int_0^\infty \mathrm{d} x \frac{\mathrm{e}^{-s (1+x^2)}}{1+x^2} = {\rm erfc}(\sqrt{s}) \,,
\end{equation}
which gets obvious after derivation with respect to $s$, and therefore compensates exactly the term ${\rm erfc}(\sqrt{s})$ in~\eref{eqSM:F2}.
From the remaining integral a factor can be extracted to compensate for the exponential prefactor while keeping it still bounded.
In total one numerically well behaved form of the function $F$ is
\begin{equation}
\begin{split}
F_{\bar{\nu}}(s) ={} &\mathrm{e}^{(1+\bar{\nu}^2)s} \left[ {\rm erfc}(\sqrt{(1+\bar{\nu}^2)s}) - {\rm erfc}(\bar{\nu} \sqrt{s}) {\rm erfc}(\sqrt{s}) \right] \\
& {}+ \frac{2}{\pi} \int_{\bar{\nu}}^\infty \mathrm{d} x \frac{\mathrm{e}^{-s (x^2 - {\bar{\nu}}^2)}}{1+x^2} \,.
\end{split}
\end{equation}
\sectionQ{Appendix D: Calculation of QCE contributions}
For comparisons with exact or numerically calculated spectra it is more convenient to use the level counting function $\bar{\mathcal{N}}(E) = \int_{-\infty}^E \mathrm{d} E' \bar{\varrho}(E')$ rather than the DOS $\bar{\varrho}(E)$.
Therefore we will give the explicit expressions for the first order QCE-contributions to the coefficients of the former.
One may write
\begin{equation} \label{eq:Calc:N}
\bar{\mathcal{N}}(E) = \sum_{l=1}^N \left[ \frac{z_l}{\Gamma\left(\frac{l}{2} +1\right)} + g_l^{(N)}\!\left( \frac{E}{\alpha} \right) \right]
\frac{L^l E^{\frac{l}{2}} \theta(E)}{(4 \pi)^\frac{l}{2}} \,.
\end{equation}
This implies
\begin{equation} \label{sm:eq:ggeneral}
g_l^{(N)}(\epsilon) = \epsilon^{-\frac{l}{2}} \Linvs \left[ \Delta_1 z_l(s) s^{-\frac{l}{2}-1} \right](\epsilon) \,,
\end{equation}
where the functions $\Delta_1 z_l(s)$ are given by~\Leref{eq:Deltaz}.
The relation to the coefficients of the DOS~[\Leref{eq:rho}] is then given by
\begin{equation}
\epsilon^{1-\frac{l}{2}} f_l^{(N)}(\epsilon) = \frac{l}{2} g_l^{(N)}(\epsilon) + \epsilon \frac{\mathrm{d}}{\mathrm{d} \epsilon} g_l^{(N)}(\epsilon) \,.
\end{equation}
For the explicit calculation of~\eref{sm:eq:ggeneral} we split the function
\begin{equation}
a_{(n_1,n-n_1)}(s) = a_1(s) + a_2(s) + a_3(s) + a_4(s)
\end{equation}
into its four addends
\begin{equation} \label{sm:eq:a1234}
\begin{split}
a_1(s) &= \frac{2}{\pi} \atan{\bar{\nu}} - 1 + \frac{2 \bar{\nu}^2}{\sqrt{\pi (1+\bar{\nu}^2)}} \sqrt{s} \,,\\
a_2(s) &= - \frac{2}{\sqrt{\pi}} \bar{\nu} \sqrt{s} \mathrm{e}^s \erfc(\sqrt{s}) \,,\\
a_3(s) &= \frac{2}{\sqrt{\pi}} F_{\bar{\nu}}(s) \,,\\
a_4(s) &= - \frac{4}{\sqrt{\pi}} \bar{\nu}^2 s F_{\bar{\nu}}(s) = - 2 \bar{\nu}^2 s a_3(s) \,,
\end{split}
\end{equation}
where we have ommitted the dependence on $n_1$ and $n$ through $\bar{\nu} = \sqrt{{2n_1(n-n_1)}/{n}-1}$ to ease notation.
Together we have
\begin{equation} \label{sm:eq:gfromb}
g_l^{(N)}(\epsilon) = \sum_{n=2}^{N-l+1} \frac{1}{\sqrt{n}} z_{l-1}^{(N-n)} \sum_{n_1=1}^{n-1} \sum_{j=1}^{4} b_j^{(l)}(\epsilon)
\end{equation}
with
\begin{equation}
b_j^{(l)}(\epsilon) = \epsilon^{-\frac{l}{2}} \Linvs\left[ s^{-\frac{l}{2}-1} a_j(s) \right](\epsilon) \,.
\end{equation}
In the following explicit expressions for the four $b_j$ are calculated.
\subsection{Calculation of $b_1^{(l)}(\epsilon)$}
Applying standard rules of inverse Laplace transformation to powers of $s$ gives
\begin{equation} \label{sm:eq:Linva1}
\begin{split}
b_1^{(l)}(\epsilon) = &\left( \frac{2}{\pi} \atan{\bar{\nu}} -1 \right) \frac{\theta(\epsilon)}{\Gamma\left( \frac{l}{2}+1 \right)} \\
& {}+ \frac{2 \bar{\nu}^2}{\sqrt{\pi (1+\bar{\nu}^2)}} \frac{\theta(\epsilon)}{\Gamma\left(\frac{l}{2}+\frac{1}{2}\right) \sqrt{\epsilon} } \,.
\end{split}
\end{equation}
\subsection{Calculation of $b_2^{(l)}(\epsilon)$}
Following the recursive approach in~\cite{ghur2015} gives
\begin{equation}
\begin{split}
\MoveEqLeft b_2^{(l)}(\epsilon) = - \frac{2 \bar{\nu}}{\sqrt{\pi}}
\frac{ \left(1 + \frac{1}{\epsilon}\right)^{\frac{l}{2}-\frac{1}{2}} }{ \Gamma\left(\frac{l}{2}+\frac{1}{2}\right) \sqrt{\epsilon} } h_\lambda(\epsilon) \\
& {}+ \frac{2 \bar{\nu}}{\pi} \sum_{k=1}^{\lfloor \frac{l}{2} \rfloor} \frac{\Gamma\left( \frac{l}{2}-k+\frac{1}{2}\right)}
{ \Gamma\left( \frac{l}{2}-k+1\right) \Gamma\left( \frac{l}{2} + \frac{1}{2} \right) } \left( 1 + \frac{1}{\epsilon}\right)^{k-1} \frac{\theta(\epsilon)}{\epsilon} \,,
\end{split}
\end{equation}
with the definitions
\begin{equation} \label{sm:eq:h}
h_\lambda(\epsilon) = \begin{cases}
\frac{2}{\pi} \theta(\epsilon) \atan ( \sqrt{\epsilon} ) & : \quad \lambda = \frac{1}{2} \,,\\
\theta(\epsilon) & : \quad \lambda = 0 \,,
\end{cases}
\end{equation}
and
\begin{equation}
\lambda = \frac{1}{2} ( l\ {\rm mod}\ 2 ) = \begin{cases}
\frac{1}{2} & : \quad l\ {\rm odd} \,,\\
0 & : \quad l\ {\rm even} \,.\\
\end{cases}
\end{equation}
Here $\lfloor q \rfloor$ denotes the integer $n \leq q$ that is closest to $q$.
\\
\subsection{Calculation of $b_3^{(l)}(\epsilon)$}{}
First, we remove the exponential prefactor by defining
\begin{equation}
\tilde{F}_{\bar{\nu}}(s) := \mathrm{e}^{-(1+\bar{\nu}^2)s} F_{\bar{\nu}}(s) \,.
\end{equation}
The integral in $\tilde{F}_{\bar{\nu}}(s)$ can not be evaluated to elementary expressions directly.
In contrast to that its inverse Laplace transform can be related to the solvable derivative given by
\begin{equation} \label{sm:eq:Fprime}
\mathrm{e}^{(1+\bar{\nu}^2)s} \tilde{F}_{\bar{\nu}}^\prime(s) = \frac{\bar{\nu}}{2} s^{-\frac{1}{2}} \mathrm{e}^s \erfc(\sqrt{s}) - \frac{1}{2} \sqrt{1+\bar{\nu}^2} s^{-\frac{1}{2}} \,.
\end{equation}
Using this observation we calculate
\begin{align}
\Linvs\left[F_{\bar{\nu}}(s)\right](\epsilon) &= \Linvs\left[\tilde{F}_{\bar{\nu}}(s)\right](\epsilon + (1+\bar{\nu}^2)) \nonumber \\
&{}= -\frac{ \Linvs\left[\tilde{F}_{\bar{\nu}}^\prime(s)\right](\epsilon+(1+\bar{\nu}^2)) }{ \epsilon + (1+\bar{\nu}^2) } \nonumber \\
&{}= -\frac{ \Linvs\left[\mathrm{e}^{(1+\bar{\nu}^2)s}\tilde{F}_{\bar{\nu}}^\prime(s)\right](\epsilon) }{ \epsilon + (1+\bar{\nu}^2) } \nonumber \\
&{}= (\epsilon + (1+\bar{\nu}^2))^{-1} \nonumber \\
&\quad\times\left( \frac{\sqrt{1+\bar{\nu}^2}}{2\sqrt{\pi}} \frac{\theta(\epsilon)}{\sqrt{\epsilon}}
- \frac{\bar{\nu}}{2\sqrt{\pi}} \frac{\theta(\epsilon)}{\sqrt{1+\epsilon}} \right) \,.
\end{align}
From there we get
\begin{align} \label{sm:eq:InitInt}
\Linvs\left[s^{-1}F_{\bar{\nu}}(s)\right](\epsilon) &= \int_{-\infty}^{\epsilon} \mathrm{d} x \Linvs\left[F_{\bar{\nu}}(s)\right](x) \nonumber \\
&= \frac{\theta(\epsilon)}{\sqrt{\pi}} \left[ \atan\left( \sqrt{\frac{\epsilon}{1+\bar{\nu}^2}} \right) \right. \nonumber \\
& \qquad\quad\left. + \atan\left( \sqrt{\frac{\bar{\nu}^2}{1+\epsilon}} \right) - \atan \bar{\nu} \right] \,,
\end{align}
and
\begin{equation} \label{sm:eq:InitHalfInt}
\begin{split}
\MoveEqLeft \Linvs\left[s^{-\frac{1}{2}}F_{\bar{\nu}}(s)\right](\epsilon) \\
&=\int_{-\infty}^\infty \mathrm{d} x \Linvs\left[s^{-\frac{1}{2}}\right](\epsilon - x) \Linvs\left[F_{\bar{\nu}}(s)\right](x) \\
&\begin{split}
{}=\frac{\theta(\epsilon)}{2 \pi} \int_0^{\epsilon} \mathrm{d} x \frac{1}{\sqrt{\epsilon-x}} &\left[ \frac{\sqrt{1+\bar{\nu}^2}}{\sqrt{x}(x+(1+\bar{\nu}^2))} \right. \\
&\left. {} - \frac{\bar{\nu}}{\sqrt{1+x}(x+(1+\bar{\nu}^2))} \right]
\end{split} \\
&{}= \frac{\theta(\epsilon)}{\pi} (\epsilon + (1+\bar{\nu}^2))^{-\frac{1}{2}} \atan\left( \frac{1}{\bar{\nu}} \sqrt{1+\frac{1+\bar{\nu}^2}{\epsilon}} \right) \,.
\end{split}
\end{equation}
We calculate $\Linvs\left[s^{-n} \tilde{F}_{\bar{\nu}}(s)\right]$ for larger negative powers of $s$ using a recursive approach, where~\eref{sm:eq:InitInt} and~\eref{sm:eq:InitHalfInt} will serve as initial values.
We define
\begin{equation} \label{sm:eq:DefGn}
G_n(s) := \Gamma(n) s^{-n} \tilde{F}_{\bar{\nu}}(s) \,,
\end{equation}
where $n$ may be either integer or half-integer.
Taking the derivative of~\eref{sm:eq:DefGn} with respect to $s$ leads to
\begin{equation}
G_{n+1}(s) = - \frac{\partial}{\partial s} G_n(s) + \Gamma(n) s^{-n} \tilde{F}_{\bar{\nu}}^\prime(s) \,,
\end{equation}
which implies the recursion relation
\begin{equation} \label{sm:eq:RecRel}
\begin{split}
\Linvs\left[G_{n+1}(s)\right](\epsilon) ={} &\epsilon \Linvs\left[G_n(s)\right](\epsilon) \\
& {}+ \Gamma(n) \Linvs\left[s^{-n} \tilde{F}_{\bar{\nu}}^\prime(s)\right](\epsilon)
\end{split}
\end{equation}
for the inverse Laplace transformed objects, where the initial values $\Linvs\left[G_1(s)\right]$ or $\Linvs\left[G_{\frac{1}{2}}(s)\right]$ are given explicitely by~\eref{sm:eq:InitInt} and~\eref{sm:eq:InitHalfInt}.
The solution to~\eref{sm:eq:RecRel} is either given by
\begin{equation} \label{sm:eq:GsolInt}
\begin{split}
\Linvs\left[G_{n+1}(s)\right](\epsilon) ={} &\epsilon^n \Linvs\left[G_1(s)\right](\epsilon) \\
& {}+ \sum_{k=1}^n \epsilon^{n-k} \Gamma(k) \Linvs\left[s^{-k} \tilde{F}_{\bar{\nu}}^\prime(s)\right](\epsilon)
\end{split}
\end{equation}
for integer indexes or by
\begin{align} \label{sm:eq:GsolHalfInt}
&\Linvs\left[G_{n+\frac{1}{2}}(s)\right](\epsilon) = \epsilon^n \Linvs\left[G_\frac{1}{2}(s)\right](\epsilon) \nonumber\\
& {}\quad+ \sum_{k=0}^{n-1} \epsilon^{n-1-k} \Gamma\left(k+\frac{1}{2}\right) \Linvs\left[s^{-k-\frac{1}{2}} \tilde{F}_{\bar{\nu}}^\prime(s)\right](\epsilon)
\end{align}
for half-integer indexes.
In the given form, both solutions~\eref{sm:eq:GsolInt} and~\eref{sm:eq:GsolHalfInt} are valid for $n \in \mathbb{N}_0$.
After reintroducing the exponential prefactor,~\eref{sm:eq:GsolInt} and~\eref{sm:eq:GsolHalfInt} become
\begin{eqnarray}
\begin{split}
&\Gamma(n+1) \Linvs\left[s^{-n-1} F_{\bar{\nu}}(s)\right](\epsilon) \\
&{}= (\epsilon + (1+\bar{\nu}^2))^n \Linvs\left[s^{-1} F_{\bar{\nu}}(s)\right](\epsilon) \\
&\begin{split}
\quad {}+ \sum_{k=1}^n &(\epsilon+(1+\bar{\nu}^2))^{n-k} \Gamma(k) \\
&\times\Linvs\left[s^{-k} \mathrm{e}^{(1+\bar{\nu}^2)s} \tilde{F}_{\bar{\nu}}^\prime(s)\right](\epsilon) \,,
\end{split}
\end{split}
\\\nonumber
\end{eqnarray}
and
\begin{eqnarray}
\begin{split}
&\Gamma\!\left( n+\frac{1}{2}\right) \Linvs\left[s^{-n-\frac{1}{2}} F_{\bar{\nu}}(s)\right](\epsilon) \\
&\qquad\begin{split}
&{}= \sqrt{\pi} (\epsilon + (1+\bar{\nu}^2))^n \Linvs\left[s^{-\frac{1}{2}} F_{\bar{\nu}}(s)\right](\epsilon) \\
&\quad\begin{split}
{}+ \sum_{k=1}^n &(\epsilon+(1+\bar{\nu}^2))^{n-k} \Gamma\!\left(k-\frac{1}{2}\right) \\
&\times \Linvs\left[s^{-k+\frac{1}{2}} \mathrm{e}^{(1+\bar{\nu}^2)s} \tilde{F}_{\bar{\nu}}^\prime(s)\right](\epsilon) \,,
\end{split}
\end{split}
\end{split}
\\\nonumber
\end{eqnarray}
where $n \in \mathbb{N}_0$.
The remaining step is to calculate $\Linvs\left[s^{-n} \mathrm{e}^{(1+\bar{\nu}^2)s} \tilde{F}_{\bar{\nu}}^\prime(s)\right](\epsilon)$ for $n$ being either integer or half-integer.
Using~\eref{sm:eq:Fprime} leads to
\begin{equation}
\begin{split}
\MoveEqLeft[4] \Linvs\left[s^{-n} \mathrm{e}^{(1+\bar{\nu}^2)s} \tilde{F}_{\bar{\nu}}^\prime(s)\right](\epsilon) \\
{}={} &\frac{\bar{\nu}}{2} \Linvs\left[ s^{-n-1} \sqrt{s} \erfc(\sqrt{s}) \right](\epsilon) \\
&{}- \frac{1}{2} \sqrt{1+\bar{\nu}^2} \Linvs\left[ s^{-n-\frac{1}{2}} \right](\epsilon) \\
{}={} &- \frac{\sqrt{\pi}}{4} \epsilon^n b_2^{(2n)}(\epsilon)
- \frac{\sqrt{1+\bar{\nu}^2}}{2 \Gamma(n+\frac{1}{2})} \epsilon^{n-\frac{1}{2}} \theta(\epsilon) \,.
\end{split}
\end{equation}
For $l \geq -1$ we get
\begin{widetext}
\begin{equation}\label{sm:eq:b3}
\begin{split}
\MoveEqLeft[6] b_3^{(l)}(\epsilon) = \frac{ \left(1+\frac{1+\bar{\nu}^2}{\epsilon}\right)^{\frac{l}{2}} }{ \Gamma\!\left( \frac{l}{2} + 1 \right) }
\left[ t_\lambda(\epsilon) - \frac{1}{\sqrt{\pi}} \sum_{k=1}^{\lceil \frac{l}{2} \rceil}
\Gamma(k-\lambda) \left(1+\frac{1+\bar{\nu}^2}{\epsilon}\right)^{\lambda-k} \left( \frac{\sqrt{\pi}}{2} b_2^{(2(k-\lambda))}(\epsilon) + \frac{\sqrt{1+\bar{\nu}^2}}{\Gamma\!\left(k-\lambda+\frac{1}{2}\right)} \frac{\theta(\epsilon)}{\sqrt{\epsilon}} \right) \right] \,,
\end{split}
\end{equation}
\end{widetext}
where $\lceil q \rceil$ denotes the integer $n \geq q$ that is closest to $q$ and the function $t_\lambda$ is defined as
\begin{equation}
t_\lambda(\epsilon) = \begin{cases}
\frac{2}{\pi} \theta(\epsilon) \atan\left( \frac{1}{\bar{\nu}} \sqrt{1+\frac{1+\bar{\nu}^2}{\epsilon}} \right) & : \lambda = \frac{1}{2} \,, \\
\frac{2}{\pi} \theta(\epsilon) \left[ \atan\left( \sqrt{\frac{\epsilon}{1+\bar{\nu}^2}} \right) \right. & \\
\qquad \quad \left. {}+ \atan\left( \sqrt{\frac{\bar{\nu}^2}{1+\epsilon}} \right) - \atan \bar{\nu} \right] & : \lambda = 0 \,.\\
\end{cases}
\end{equation}
\subsection{Calculation of $b_4^{(l)}(\epsilon)$}
Since~\eref{sm:eq:b3} is not only valid for $l \in \mathbb{N}$ but also for the values $l=-1,0$ we can use the simple relation between $a_3$ and $a_4$~\eref{sm:eq:a1234} to get
\begin{equation}
b_4^{(l)}(\epsilon) = - 2 \bar{\nu}^2 \frac{1}{\epsilon} b_3^{(l-2)}(\epsilon)
\end{equation}
for all $l \in \mathbb{N}$.
\sectionQ{Appendix E: QCE in fermionization regime}
For arbitrary interaction strengths $\alpha$ a 1D bosonic system with $\delta$-interaction maps exactly to a spinless fermionic system with an effective attractive 0-range interaction potential~\cite{cheon1999} which will here simply be referred to as the anti-$\delta$-interaction.
In order to apply the first order QCE in the effective fermionic theory we need to derive the two-body propagator for the anti-$\delta$-interaction which can be completely achieved on an abstract level relating it back to the propagator in the $\delta$-interacting system.
First, for any two-body propagator $K$ we define the swapping operation denoted by $\bar{K}$ as
\begin{equation}
\begin{split}
&\bar{K}((q_1',q_2'),(q_1,q_2)) \\
&\;=\begin{cases}
K((q_1',q_2'),(q_1,q_2)), &\text{for } (q_1-q_2)(q_1'-q_2')>0 ,\\
-K((q_1',q_2'),(q_1,q_2)), &\text{for } (q_1-q_2)(q_1'-q_2')<0 \,,
\end{cases}
\end{split}
\end{equation}
which gives a relative sign inversion when the two particles have to cross each other along any classical path from $(q_1,q_2)$ to $(q_1',q_2')$.
Now consider the interacting propagator $K$ of two distinguishable particles subject to the $\delta$-interaction.
It is built from its symmetric part $K_+$ and its antisymmetric part $K_-$ \textit{w.r.t.}\ to particle exchange,
\begin{equation}
K = K_+ + K_- \,,
\end{equation}
where $K_+$($K_-$) is defined by all symmetric(antisymmetric) eigenfunctions $\psi_\pm(R,r)$ of the two-body system, where $R,r$ denote center-of-mass and relative coordinates, respectively.
The $\delta$-interaction only has an effect on the symmetric wavefunctions $\psi_+(R,r)$, whereas the antisymmetric ones are unaffected $\psi_-(R,r) = \psi_{0,-}(R,r)$, thus we write
\begin{align}
K_+ &= K_{0,+} + K_\alpha \,,\\
K_- &= K_{0,-} \,,
\end{align}
where $K_{0,\pm}$ denotes the (anti)symmetric part of the non-interacting propagator and $K_\alpha$ the modification to the symmetric part due to finite interaction.
For the anti-$\delta$-interaction (which will be denoted by a tilde) the opposite is the case and one has unaffected symmetric wavefunction $\tilde{\psi}_+(R,r) = \psi_{0,+}(R,r)$ whereas the antisymmetric wavefunctions $\tilde{\psi}_-(R,r)$ feel the interaction in form of a jump discontinuity at vanishing relative distance $r$ of the particles.
Because of the exact mapping, those antisymmetric wavefunctions are equivalent with the symmetric ones for the $\delta$-interaction with a conditional sign-inversion
\begin{equation}
\tilde{\psi}_-(R,r) = \mathrm{sign}(r) \psi_+(R,r) \,.
\end{equation}
This sign-inversion is then reflected in the propagator $\tilde{K}$ of two distinguishable particles being subject to the anti-$\delta$-interaction as
\begin{equation}
\begin{split}
\tilde{K} &= K_{0,+} + \bar{K}_+ \\
&= K_{0,+} + \bar{K}_{0,+} + \bar{K}_\alpha \,.
\end{split}
\end{equation}
For first order QCE calculations one needs then only the modification $\tilde{K}_\alpha$ of the porpagator due to anti-$\delta$-interaction, thus we write
\begin{equation}
\begin{split}
\tilde{K} &= K_0 + \tilde{K}_\alpha \\
&= K_{0,+} + K_{0,-} + \tilde{K}_\alpha \,,
\end{split}
\end{equation}
and obtain the final result
\begin{equation} \label{eq:Ferm:prop}
\tilde{K}_\alpha = \bar{K}_{0,+} + \bar{K}_\alpha - K_{0,-} \,.
\end{equation}
A simple test of this result can be done in the limit $\alpha\rightarrow\infty$ where the symmetric propagator for $\delta$-interaction becomes just the swapped version of the free antisymmetric propagator
\begin{equation}
K_{0,+} + K_\alpha \xrightarrow[\alpha\rightarrow\infty]{} \bar{K}_{0,-} \,,
\end{equation}
so that
\begin{equation}
\tilde{K}_\alpha \xrightarrow[\alpha\rightarrow\infty]{} 0 \,,
\end{equation}
which means the fermionic theory is non-interacting in this limit, which confirm the fermionization effect.
Using the relation~\eref{eq:Ferm:prop} in the calculation of the corresponding QCE diagrams involved in the cluster contribution $\tilde{A}_{(n_1,n-n_1)}(s)$ for the fermionic theory one gets then a replacement of the functions $a_{(n_1,n-n_1)} \mapsto \tilde{a}_{(n_1,n-n_1)}$ given by (see~\eref{sm:eq:a1234} for comparison)
\begin{equation} \label{sm:eq:a1234ferm}
\begin{split}
\tilde{a}_1(s) &= -\frac{2}{\pi} \frac{\bar{\nu}}{1+\bar{\nu}^2} - \frac{2 \bar{\nu}^2}{\sqrt{\pi (1+\bar{\nu}^2)}} \sqrt{s} \,,\\
\tilde{a}_2(s) &= \frac{2}{\sqrt{\pi}} \bar{\nu} \sqrt{s} \mathrm{e}^s \erfc(\sqrt{s}) = - a_2(s) \,,\\
\tilde{a}_3(s) &= \frac{2}{\sqrt{\pi}} F_{\bar{\nu}}(s) = a_3(s)\,,\\
\tilde{a}_4(s) &= \frac{4}{\sqrt{\pi}} \bar{\nu}^2 s F_{\bar{\nu}}(s) = - a_4(s) \,,
\end{split}
\end{equation}
and consequently
\begin{equation} \label{sm:eq:b1234ferm}
\begin{split}
\tilde{b}^{(l)}_1(\epsilon) &= -\frac{2}{\pi} \frac{\bar{\nu}}{1+\bar{\nu}^2} \frac{\theta(\epsilon)}{\Gamma(\frac{l}{2}+1)} - \frac{2 \bar{\nu}^2}{\sqrt{\pi (1+\bar{\nu}^2)}} \frac{\theta(\epsilon)}{\Gamma(\frac{l}{2}+\frac{1}{2}) \sqrt{\epsilon}} \,,\\
\tilde{b}^{(l)}_2(\epsilon) &= -b^{(l)}_2(\epsilon) \,,\\
\tilde{b}^{(l)}_3(\epsilon) &= b^{(l)}_3(\epsilon)\,,\\
\tilde{b}^{(l)}_4(\epsilon) &= -b^{(l)}_4(\epsilon) \,,
\end{split}
\end{equation}
which can then be used in~\eref{sm:eq:gfromb} and~\eref{eq:Calc:N} together with the non-interacting fermionic coefficients
\begin{equation}
\tilde{z}_l^{(n)} = (-1)^{n-l} z_l^{(n)}
\end{equation}
to get the corresponding counting functions for the fermionization regime.
\sectionQ{Appendix F: Universal Scaling}\label{sec:Scaling}
\subsection{Generic scaling of spatial potentials}
Suppose the system under observation with (in total) $D$ spatial degrees of freedom is described by a Hamiltonian
\begin{equation} \label{eq:SC:Ham}
\hat{H} = \hat{T} + V(\hat{\bf q}) \,,
\end{equation}
where $\hat{T}$ is the kinetic energy and $V$ is a spatial potential energy that can be an external potential as well as an interaction potential affecting different particle coordinates.
Suppose further that the potential $V$ scales with parameter $\alpha$ of unit energy that represents its strength.
Other dimensionless parameters $\boldsymbol{\lambda}$ might also be involved.
Moreover, two further constants $\hbar$ and a mass $m$ are allowed to be arguments of the potential.
In case that more than just one mass are entering the Hamiltonian (\textit{e.g.}\ different particle species or anisotropic mass) the dependence of $V$ on various masses can be substituted by a dependence on one reference-mass (then simply called $m$) and a number of dimensionless parameters $\boldsymbol{\lambda}$ representing the ratios between the actually participating masses and $m$.
In total, the generic assumption is that one can write
\begin{equation} \label{eq:SC:Vgen}
V({\bf q}) = V(\alpha,\hbar, m, \boldsymbol{\lambda}, {\bf q}) \,,
\end{equation}
with units $[\alpha]=[E]$, $[\lambda_j]=1$, $[q_i]=[x]$, and $[V({\bf q})]=[E]$.
Exceptions of~\eref{eq:SC:Vgen} are potentials that are homogeneous functions of ${\bf q}$ of degree $-2$, namely the (anisotropic) $\sim 1/q^2$ potential, Dirac-Delta potentials $\sim \delta^{(2)}\!\left(\sum_{ij} a_{ij} \left( \begin{smallmatrix} q_i \\ q_j \end{smallmatrix} \right) \right)$ involving two dimensions, linear combinations of the mentioned, and maybe other more exotic constructions.
The reason for this exception is that those potentials intrinsically are given by dimensionless couplings that cannot be transformed into energy-like couplings $\alpha$ by means of the available constants.
At the same time this means that such potentials yield scale-invariant Hamiltonians which need to be regularized to give them physical meaning.
To acchieve that usually the regularized forms are equipped with a physical parameter of the system that is to be modelled.
This parameter must not be dimensionless and is often given as a bound state energy or a scattering length.
Therefore also those exceptional cases are in their final regularized physically meaningful versions again admitting the form~\eref{eq:SC:Vgen}.
With the form~\eref{eq:SC:Vgen} one can write
\begin{equation}
V(\alpha, \hbar, m, \boldsymbol{\lambda}, {\bf q}) = \alpha \tilde{V}(\alpha, \hbar, m, \boldsymbol{\lambda}, {\bf q}) \,,
\end{equation}
where $\tilde{V}$ is dimensionless.
Therefore its functional dependence on all arguments must be in a way that the latter are combined to dimensionless quantities.
The unique way (up to a dimensionless factor) to do so is given by the scaled coordinates $(\sqrt{2m \alpha}/\hbar) {\bf q}$ which allows one to write
\begin{equation}
V(\alpha, \hbar, m, \boldsymbol{\lambda}, {\bf q}) = \alpha \bar{V}\!\left(\boldsymbol{\lambda}, \sqrt{\frac{2 m \alpha}{\hbar^2}} {\bf q} \right) \,,
\end{equation}
where $\bar{V}$ is again dimensionless.
The last step introduces a temperature $T$ and related inverse temperature $\beta = 1/ k_{\rm B} T$ which yields the thermal de-Broglie wavelength \begin{equation}
\lambda_{\rm T} = \left( \frac{m}{2 \pi \hbar^2 \beta} \right)^{-\frac{1}{2}}
\end{equation}
as a length-scale, which defines
\begin{equation} \label{eq:SC:x}
{\bf x} := \frac{1}{\lambda_{\rm T}} {\bf q}
\end{equation}
as dimensionless, scaled coordinates.
The final general scaling of the potential in terms of ${\bf x}$ is given by
\begin{equation}
V(\alpha, \hbar, m, \boldsymbol{\lambda}, {\bf q}) = \alpha U\!\left(\boldsymbol{\lambda}, \sqrt{\beta \alpha} {\bf x}\right) \,.
\end{equation}
\subsection{Generic scaling of the propagator}
In this section we consider the evolution of quantum states in the system given by~\eref{eq:SC:Ham} in imaginary time $t = - i \hbar \beta$.
The corresponding (non-unitary) evolution operator for a fixed {\it relaxation ``time''} $\beta$ is $\mathrm{e}^{-\beta \hat{H}}$.
Let $\left| {\bf q} \right\rangle$ denote the eigenstates of $\hat{\bf q}$ with eigenvalues ${\bf q}$ normalized as
\begin{equation}
\left\langle {\bf q}' | {\bf q} \right\rangle = \delta^{(D)}({\bf q}'-{\bf q}) \,,
\end{equation}
and let further denote $| \psi \rangle$ an arbitrary state of the system and $\psi({\bf q}) = \langle {\bf q} | \psi \rangle$ its wavefunction.
The action of the evolution operator is given by the action of its exponent which is (everything non-relativistic)
\begin{equation} \label{eq:SC:betaH}
\langle {\bf q} | \beta \hat{H} | \psi \rangle = \left[ - \beta \sum_i \frac{\hbar^2}{2 m_i} \nabla_{q,i}^2
+ \beta \alpha U\!\left( \boldsymbol{\lambda}, \sqrt{\beta \alpha} {\bf x} \right) \right] \psi({\bf q}) \,,
\end{equation}
where the scaled version of the potential energy is used (see last subsection).
Here, $\nabla_{q,i}^2$ are Laplacians with respect to some components of ${\bf q}$.
The different masses $m_i$ can be accounted for by defining a reference mass $m$ and additional dimensionless parameters $\lambda_i = m / m_i$ that might also be arguments to the potential $U$.
In addition we rewrite the Laplacians as derivatives $\nabla_{x,i}^2 = \lambda_{\rm T}^2 \nabla_{q,i}^2 $ with respect to the scaled coordinates $x$~\eref{eq:SC:x}.
Eq.~\eref{eq:SC:betaH} then reads
\begin{equation}
\begin{split}
&\langle \lambda_{\rm T} {\bf x} | \beta \hat{H} | \psi \rangle \\
&\qquad= \left[ - \frac{1}{4\pi} \sum_i \lambda_i \nabla_{x,i}^2
+ \beta \alpha U\!\left( \boldsymbol{\lambda}, \sqrt{\beta \alpha} {\bf x} \right) \right] \psi( \lambda_{\rm T} {\bf x}) \,.
\end{split}
\end{equation}
This essentially shows that the evolution operator only involves the scaled coordinates ${\bf x}$, some dimensionless parameters $\boldsymbol{\lambda}$ and the product $\beta \alpha$ of inverse temperature (or relaxation ``time'') and the potential coupling.
We write
\begin{equation} \label{eq:SC:evolution}
\mathrm{e}^{-\beta \hat{H}} = \mathrm{e}^{- \hat{h}(x,\boldsymbol{\lambda},\beta \alpha)} \,.
\end{equation}
In order to express the (imaginary) time evolution of any state in the scaled coordinates ${\bf x}$, we define the position eigenstates associated to the scaled coordinates~\eref{eq:SC:x} as
\begin{equation}
| {\bf x} \rangle = \lambda_{\rm T}^{\frac{D}{2}} | {\bf q} \rangle \,,
\end{equation}
which satisfy the normalization condition
\begin{equation} \label{eq:SC:xnorm}
\langle {\bf x}' | {\bf x} \rangle = \lambda_{\rm T}^D \langle {\bf q}' | {\bf q} \rangle = \lambda_{\rm T}^D \delta^{(D)}(\lambda_{\rm T} {\bf x}' - \lambda_{\rm T} {\bf x} ) = \delta^{(D)}({\bf x}' - {\bf x}) \,,
\end{equation}
only dependent on scaled coordinates ${\bf x}, {\bf x}^\prime$.
The evolution of any initial state $| \psi(0) \rangle$ to the corresponding final state $| \psi(\beta) \rangle$ in terms of the scaled positions is then given by
\begin{equation}
\langle {\bf x}^{\rm f} | \psi(\beta) \rangle = \int \mathrm{d}^D x^{\rm i}
\underbrace{\langle {\bf x}^{\rm f} | \mathrm{e}^{- \hat{h}(x, \boldsymbol{\lambda}, \beta \alpha)} | {\bf x}^{\rm i} \rangle}_{=: k({\bf x}^{\rm f}, {\bf x}^{\rm i}; \boldsymbol{\lambda}, \beta \alpha)}
\langle {\bf x}^{\rm i} | \psi(0) \rangle \,.
\end{equation}
The scaled evolution kernel $k$ only depends on ${\bf x}^{\rm f}, {\bf x}^{\rm i}, \boldsymbol{\lambda},$ and $\beta \alpha$ (but not on $\alpha$ or $\beta$ alone) and hence the scaling of the evolution kernel (or propagator) in real coordinates is given by
\begin{align}
K({\bf q}^{\rm f}, {\bf q}^{\rm i}; \beta) &= \langle {\bf q}^{\rm f} | \mathrm{e}^{-\beta \hat{H}} | {\bf q}^{\rm i} \rangle \nonumber \\
&= \lambda_{\rm T}^{-D} k(\lambda_{\rm T}^{-1} {\bf q}^{\rm i}, \lambda_{\rm T}^{-1} {\bf q}^{\rm f}; \boldsymbol{\lambda}, \beta \alpha) \,.
\end{align}
The sole dependence of $k$ on the scaled variables ${\bf x}i$, ${\bf x}f$, and $\beta \alpha$ is due to the scaling of the evolution operator~\eref{eq:SC:evolution} on the one hand and the normalization condition~\eref{eq:SC:xnorm} on the other hand.
The latter is here crucial, which can for example be seen when defining $k$ by its differential equation in ${\bf x}^{\rm i}$ and $\beta \alpha$ together with an {\it initial condition}.
One should distinguish two cases.
i) If the potential energy vanishes for $\alpha \rightarrow 0$, the initial condition can be taken at $\beta_0 = 0$ whereas
ii) in case that the potential energy vanishes for $\alpha \rightarrow \infty$, the reference point will be $\beta_0 \rightarrow \infty$.
Since the scaled kinetic part does not depend on $\beta$, both cases are then summarized by the initial condition
\begin{equation} \label{eq:SC:kIC}
\lim_{\beta \rightarrow \beta_0} k({\bf x}^{\rm f}, {\bf x}^{\rm i}; \boldsymbol{\lambda}, \beta \alpha)
= \prod_i \frac{1}{\sqrt{\lambda_i}} \exp \left[ -\frac{\pi}{\lambda_i} ({\bf x}_i^{\rm f} - {\bf x}_i^{\rm i})^2 \right] \,.
\end{equation}
The finiteness of the initial condition is thereby guaranteed by the proper normalization of $| {\bf x} \rangle$ states.
Note that the finite width of the gaussian in~\eref{eq:SC:kIC} is not contradictory to the point-like initial condition of the propagator in real coordinates
\begin{equation}
\lim_{\beta \rightarrow 0} \langle {\bf q}^{\rm f} | \mathrm{e}^{-\beta \hat{H}} | {\bf q}^{\rm i} \rangle = \delta^{(D)}({\bf q}^{\rm f} - {\bf q}^{\rm i}) \,,
\end{equation}
because the scaling ratio between ${\bf x}$ and ${\bf q}$ vanishes for $\beta \rightarrow 0$.
The specific form of the differential equation for $k({\bf x}f,{\bf x}i; \boldsymbol{\lambda}, \beta \alpha)$ is not important for this argument, rather it is the fact that it is an equation involving only ${\bf x}f, {\bf x}i,$ and $\beta \alpha$ which is crucial here.
Nevertheless for completeness the differential equation will be given in an abstract form here and in an explicit form in the next subsection.
To ease notation we drop the dependence on $\boldsymbol{\lambda}$, define the {\it thermal coupling} $s := \beta \alpha$ and write $\hat{h}(x, \boldsymbol{\lambda}, \beta \alpha) = \hat{h}(s)$.
The derivative \textit{w.r.t.}\ $s$ is given by
\begin{equation} \label{eq:SC:SEQkabstract}
\frac{\partial}{\partial s} k({\bf x}f, {\bf x}i; s) = \langle {\bf x}f | \left( \sum_{j=0}^\infty \frac{ [-\hat{h}(s), - \frac{\partial \hat{h}}{\partial s}]_j}{(j+1)!} \right)
\mathrm{e}^{-\hat{h}(s)} | {\bf x}i \rangle \,,
\end{equation}
with the multiple commutator defined as
\begin{equation}
[ \hat{A}, \hat{B} ]_j = [ \hat{A}, [ \hat{A}, \hat{B} ]_{j-1} ] \quad \mbox{with} \quad [ \hat{A}, \hat{B} ]_0 := \hat{B} \,.
\end{equation}
Since both $\hat{h}(s)$ as well as $\frac{\partial \hat{h}}{\partial s}$ are built from the operators $\hat{{\bf x}}$ and the corresponding conjugate variables $\hat{\bf k}$ with $[ \hat{k}_a , \hat{x}_b ] = - \mathrm{i} \delta_{ab}$, the term in brackets in~\eref{eq:SC:SEQkabstract} acts as a differential operator on ${\bf x}$ that depends on $s$.
Meaning one can write
\begin{equation}
\frac{\partial}{\partial s} k({\bf x}f, {\bf x}i; s) = \overset{\rightarrow}{\mathcal{D}_{x^{\rm f}}}(s) k({\bf x}f, {\bf x}i; s) \,,
\end{equation}
with some differential operator $\overset{\rightarrow}{\mathcal{D}_{x^{\rm f}}}(s)$ acting on ${\bf x}f$.
The differential equation can also be given in the more symmetric form
\begin{equation}
\left( \frac{\partial}{\partial s} - \frac{1}{2} \overset{\rightarrow}{\mathcal{D}_{x^{\rm f}}}(s)
- \frac{1}{2} \overset{\rightarrow}{\mathcal{D}^\ast_{x^{\rm i}}}(s) \right) k({\bf x}f, {\bf x}i; s) = 0 \,.
\end{equation}
\subsection{Alternative derivation using the Schr\"odinger equation}
In this subsection we give an alternative derivation of the scaling behaviour of the evolution kernel employing the Schr\"odinger equation
\begin{align} \label{eq:SC:SEQ}
&-\frac{\partial}{\partial \beta} K_\alpha({\bf q}f,{\bf q}i;\beta) \nonumber \\
&= \left[ - \frac{\hbar^2}{2m} \sum_i \lambda_i \nabla_{q^{\mathrm{f}},i}^2
+ \alpha U\!\left(\boldsymbol{\lambda},\sqrt{\beta \alpha} \frac{{\bf q}f}{\lambda_{\rm T}} \right) \right] K_\alpha({\bf q}f,{\bf q}i;\beta)
\end{align}
for the propagator $K_\alpha$ in real coordinates.
We switch now to scaled coordinates~\eref{eq:SC:x} and define the scaled kernel as
\begin{equation}
k_\alpha^{\rm sc}({\bf x}f,{\bf x}i;\beta \alpha) := \lambda_{\rm T}^D K_\alpha({\bf q}f,{\bf q}i;\beta) \,.
\end{equation}
The derivative \textit{w.r.t.}\ $\beta$ involves then also the scaled coordinates and the prefactor in the following way
\begin{widetext}
\begin{equation}
\frac{\partial}{\partial \beta} K_\alpha({\bf q}f, {\bf q}i; \beta) = \left[ \lambda_{\rm T}^{-D} \alpha \frac{\partial}{\partial s}
+ \lambda_{\rm T}^{-D} \frac{\partial {\bf x}f}{\partial \beta} \cdot \boldsymbol{\nabla}_{x^{\rm f}}
+ \lambda_{\rm T}^{-D} \frac{\partial {\bf x}i}{\partial \beta} \cdot \boldsymbol{\nabla}_{x^{\rm i}}
+ \frac{\partial \lambda_{\rm T}^{-D}}{\partial \beta} \right] k_\alpha^{\rm sc}({\bf x}f, {\bf x}i; s) \,,
\end{equation}
\end{widetext}
where we have again introduced the {\it thermal coupling} $s = \beta \alpha$.
Recognizing that
\begin{align}
\frac{\partial {\bf x}^{\rm i(f)}}{\partial \beta} &{}= -\frac{1}{2 \beta} {\bf x}^{\rm i(f)} \,, \\
\frac{\partial \lambda_{\rm T}^{-D}}{\partial \beta} &{}= - \frac{D}{2 \beta} \lambda_{\rm T}^{-D} \,, \\
\frac{\hbar^2}{2m} \nabla_{q^{\rm f},i}^2 &{}= \frac{1}{4 \pi \beta} \nabla_{x^{\rm f},i}^2 \,,
\end{align}
the Schr\"odinger equation~\eref{eq:SC:SEQ} for the scaled kernel becomes
\begin{widetext}
\begin{equation} \label{eq:SC:SEQsc}
\left[ -\frac{1}{4 \pi} \sum_i \lambda_i \nabla_{x^{\rm f},i}^2
- \frac{1}{2} {\bf x}i \cdot \boldsymbol{\nabla}_{x^{\rm i}}
- \frac{1}{2} {\bf x}f \cdot \boldsymbol{\nabla}_{x^{\rm f}}
+ s U\!\left(\boldsymbol{\lambda}, \sqrt{s} {{\bf x}} \right)
+ s \frac{\partial}{\partial s}
- \frac{D}{2}
\right] k_\alpha^{\rm sc}({\bf x}f, {\bf x}i; s) = 0 \,.
\end{equation}
\end{widetext}
Since the differential operator in~\eref{eq:SC:SEQsc} does not depend explicitely on $\alpha$ and the initial condition
\begin{equation}
\lim_{s \rightarrow s_0} k({\bf x}^{\rm f}, {\bf x}^{\rm i}; s)
= \prod_i \frac{1}{\sqrt{\lambda_i}} \exp \left[ -\frac{\pi}{\lambda_i} ({\bf x}_i^{\rm f} - {\bf x}_i^{\rm i})^2 \right] \,,
\end{equation}
where
\begin{equation}
s_0 =
\begin{cases}
0 & \mbox{if } \lim_{\alpha \rightarrow 0} V({\bf q}) = 0 \,, \\
\infty & \mbox{if } \lim_{\alpha \rightarrow \infty} V({\bf q}) = 0 \,,
\end{cases}
\end{equation}
is also independent of $\alpha$, the scaled evolution kernel is completely defined independently of $\alpha$, meaning there is no explicit dependence on $\alpha$.
Therefore one can ommit the subscript and write the scaling property for the propagator in real coordinates and imaginary time as
\begin{equation} \label{eq:SC:PropScaling}
K_\alpha({\bf q}^{\rm f}, {\bf q}^{\rm i}; \beta)
= \lambda_{\rm T}^{-D} k^{\rm sc}(\lambda_{\rm T}^{-1} {\bf q}^{\rm i}, \lambda_{\rm T}^{-1} {\bf q}^{\rm f}; \beta \alpha) \,.
\end{equation}
For simplicity we dropped the dependence on dimensionless parameters $\boldsymbol{\lambda}$ in $k^{\rm sc}$ which can always exist implicitely.
In the following section, the scaling property~\eref{eq:SC:PropScaling} will be used to derive universal scaling properties for QCE contributions.
\subsection{Universal scaling properties of QCE including external potentials}
Since~\eref{eq:SC:PropScaling} is a general property regardless of the dimension and explicit form of the potential $V$ it holds also for the propagator of systems of $N$ distinguishable particles where $V({\bf q}) = \sum_{ij} v_{ij}({\bf q}_i-{\bf q}_j)$ is an interaction potential relating different particles.
Remarkably this holds also if the interaction is applied only on a subset of particles.
Furthermore, also differences of two propagators where the interaction links different subsets of particles in the two cases are still subject to the general scaling with $\beta \alpha$.
This enables us to write Ursell operators of arbitrary order $n$ as
\begin{equation} \label{eq:SC:ursell}
U^{(n)}_\alpha ({\bf q}f, {\bf q}i; \beta) = \lambda_{\rm T}^{-n D} \tilde{u}^{(n)}({\bf x}f, {\bf x}i; s) \,.
\end{equation}
Moving to the indistinguishable case, an arbittrary cluster contribution involves a product of Ursell operators and the final configuarions are given as a permutation of the initial configuration ${\bf q}f = P({\bf q}i)$.
This means the integrand for an arbitrary cluster contribution of $n$ particles is given by a function
\begin{equation} \label{eq:SC:cluster}
K_\alpha^{(n)}(P({\bf q}),{\bf q};\beta) = \lambda_{\rm T}^{-n D} \tilde{k}^{(n)}(P({\bf x}), {\bf x}; s) \,,
\end{equation}
where $K$ stands now for a product of ursell operators~\eref{eq:SC:ursell} and $\tilde{k}$ for its scaled version.
The cluster contribution is then the amplitude
\begin{equation} \label{eq:SC:amplitude}
\int_\Omega \mathrm{d}^{D}q_1 \ldots \int_\Omega \mathrm{d}^{D}q_n K_\alpha^{(n)}(P({\bf q}),{\bf q};\beta) \,.
\end{equation}
Since we talk about a single cluster, there is only one invariant direction in ${\bf q}$-space of a so constructed integrand, which corresponds to the center-of-mass motion.
Otherwise the integral~\eref{eq:SC:amplitude} would be seperable into distinct cluster-contributions per definition.
If additionally a smooth external confinement potential $V_\mathrm{ext}({\bf q}) = \sum_i v_\mathrm{ext}({\bf q}_i)$ is applied, we address it by assuming it to be simultaniously constant for all $n$ involved particles.
This is consistent with the short-time philosophy of the QCE since for short times, the relevant spread of the cluster is small compared to the scale of variations in the external potential.
This assumption separates the amplitude~\eref{eq:SC:amplitude} into an \textit{internal part}
\begin{equation}
\begin{split}
&Z_\mathrm{int} = \int \mathrm{d}^{D}q_2 \ldots \int \mathrm{d}^{D}q_n \\
&\qquad \times K_{\alpha, \mathrm{int}}^{(n)}(P(({\bf 0},{\bf q}_2,\ldots,{\bf q}_n)),({\bf 0},{\bf q}_2,\ldots,{\bf q}_n);\beta) \,,
\end{split}
\end{equation}
that fixes one of the coordinates, extends the integration over the others to infinity and assumes zero external potential, and an external part
\begin{equation}
Z_\mathrm{ext} = \int \mathrm{d}^{D}q_1 K_{\mathrm{ext},n}^{(1)}({\bf q}_1,{\bf q}_1;\beta) \,,
\end{equation}
which corresponds to a single particle feeling the $n$-fold external potential.
For the external potential we may use the generic scaling property again, now introducing a parameter $\alpha_\mathrm{ext}$ to write
\begin{equation}
V_\mathrm{ext}({\bf q}_1) = \alpha_\mathrm{ext} U_\mathrm{ext}(\sqrt{\beta \alpha_\mathrm{ext}} {\bf x}_1) \,,
\end{equation}
and consequently
\begin{equation}
Z_\mathrm{ext} = \int \mathrm{d}^{D}x_1 \tilde{k}_{\mathrm{ext},n}^{(1)}({\bf x}_1, \beta \alpha_\mathrm{ext}) \,.
\end{equation}
The short-time effect of the external potential can thereby be considered as a local phase shift
\begin{equation}
K_{\mathrm{ext},n}^{(1)}({\bf q}_1,{\bf q}_1; \beta) \simeq \lambda_{\rm T}^{-D} \mathrm{e}^{-\beta n V_\mathrm{ext}({\bf q}_1)} \,,
\end{equation}
which leads to
\begin{equation}
Z_\mathrm{ext} = n^{D/2} \xi(\sqrt{n \beta \alpha_\mathrm{ext}}) \,,
\end{equation}
where
\begin{equation}
\xi(a) := \int \mathrm{d}^{D}y \, \mathrm{e}^{-a^2 U_\mathrm{ext}(a {\bf y})} \,.
\end{equation}
For homogeneously scaling external potentials this gives
\begin{equation}
Z_\mathrm{ext} = \left( n^{-d/2} \frac{V_\mathrm{eff}}{\lambda_{\rm T}^d} \right) n^{D/2} \,,
\end{equation}
with the definitions of effective dimension $d$ and effective volume $V_\mathrm{eff}$ given in the main text.
Using the scaling~\eref{eq:SC:cluster} for the internal part gives
\begin{equation}
\begin{split}
&Z_\mathrm{int}(\beta \alpha) = \int \mathrm{d}^{D}x_2 \ldots \mathrm{d}^{D}x_n \\
&\qquad \times \tilde{k}^{(n)}_\mathrm{int}(P(({\bf 0},{\bf x}_2,\ldots,{\bf x}_n)),({\bf 0},{\bf x}_2,\ldots,{\bf x}_n);s) \,,
\end{split}
\end{equation}
which is a function of only $s=\beta \alpha$ so that we are free to write
\begin{equation}
Z_\mathrm{int} = n^{D/2} a(\beta \alpha) \,,
\end{equation}
which defines the internal amplitude $a$ and where the prefactor is inspired by the free case and explicit QCE(1) calculations in the case of $\delta$-interaction but can be defined like that in any case.
In total, the contribution~\eref{eq:SC:amplitude} from a specific cluster in homogeneous external potentials has the form
\begin{equation}
A^{(\mathfrak{C})}_\alpha(\beta) = n^{-d/2} \frac{V_\mathrm{eff}}{\lambda_{\rm T}^d} a^{(\mathfrak{C})}(\beta \alpha) \,,
\end{equation}
where $\mathfrak{C}$ denotes a specific internal \textit{cluster-structure} that does not depend on any system parameters.
\sectionQ{Appendix G: Energy shifting method}
For the purpose of this section we write the non-interacting counting function as
\begin{equation} \label{eq:ES:N0}
\mathcal{N}_0(\tilde{E}) = c_N \tilde{E}^{N d/2} + c_{N-1} \tilde{E}^{(N-1)d/2} + \ldots \,,
\end{equation}
with the scaled total energy
\begin{equation}
\tilde{E} = E \left( \frac{\hbar^2}{2m} V_\mathrm{eff}^{-2/d} \right)^{-1} \,,
\end{equation}
which is a quantity depending on the (effective) system size.
The observation of full shifts $\Delta E_\infty$ between the limits $\alpha \rightarrow 0$ and $\alpha \rightarrow \infty$ is
\begin{equation}
\Delta \tilde{E}_\infty = \tilde{a} \mathcal{N}^{(2/d-1)/N} \,,
\end{equation}
with some constant $\tilde{a}$, so that the fully shifted counting function
\begin{equation}
\mathcal{N}_\infty(\tilde{E}) = \mathcal{N}_0(E-\Delta E_\infty)
\end{equation}
can be expanded in the large $\tilde{E}$ limit (which is at the same time a large $V_\mathrm{eff}$ limit) to
\begin{align}
&\mathcal{N}_\infty(\tilde{E}) \nonumber\\
&\quad= c_N \tilde{E}^{N \frac{d}{2}} \left( 1+ \frac{\Delta \tilde{E}_\infty}{\tilde{E}} \right)^{N \frac{d}{2}}
+ c_{N-1} \tilde{E}^{(N-1)\frac{d}{2}} + \ldots \nonumber \\
&\quad= c_N \tilde{E}^{N \frac{d}{2}} + \left( c_{N-1} + N \frac{d}{2} \tilde{a} c_N^{(\frac{2}{d}-1)/N} \right) \tilde{E}^{(N-1)\frac{d}{2}} + \ldots \,.
\end{align}
Thus the leading correction from the shift is of the same order in $\tilde{E}$ as the first sub-leading term in~\eref{eq:ES:N0}, so that it can be matched to the sub-leading term of the non-interacting fermionic counting function, which is simply a negative of the free bosonic term.
We identify
\begin{equation}
c_{N-1} + N \frac{d}{2} \tilde{a} c_N^{(\frac{2}{d}-1)/N} = - c_{N-1}
\end{equation}
to fix the constant $\tilde{a}$ for the full shift.
In the case of arbitrary interaction strength $\alpha$, the ansatz is
\begin{equation}
\Delta \tilde{E}_\alpha = \tilde{\chi}(E/\alpha) \mathcal{N}^{(2/d-1)/N} \,,
\end{equation}
where now one has to do a clear distinction between the variables $\tilde{E}$ (which scales with the volume) and $\frac{E}{\alpha}$ (which scales with the interaction strength).
Meaning, we are free to do an expansion for large $\tilde{E}$ while considering $\frac{E}{\alpha}$ as independent variable.
In other words one can do a large $V_\mathrm{eff}$ expansion to get
\begin{align}
&\mathcal{N}_\alpha(\tilde{E}) \nonumber\\
&\quad= c_N \tilde{E}^{N \frac{d}{2}} \left( 1+ \frac{\Delta \tilde{E}_\alpha}{\tilde{E}} \right)^{N \frac{d}{2}}
+ c_{N-1} \tilde{E}^{(N-1)\frac{d}{2}} + \ldots \nonumber \\
&\quad= c_N \tilde{E}^{N \frac{d}{2}} \nonumber \\
&\qquad+ \left( c_{N-1} + N \frac{d}{2} \tilde{\chi}(E/\alpha) c_N^{(\frac{2}{d}-1)/N} \right) \tilde{E}^{(N-1)\frac{d}{2}} + \ldots \,.
\end{align}
As also the first order QCE correction is in general of the order $\tilde{E}^{(N-1)\frac{d}{2}}$, the shifting fraction $\tilde{\chi}$ can be exactly matched to the later.
In the 1D case with $\delta$-interaction and without external potential this can be expressed in terms of~\eref{eq:Calc:N} as
\begin{equation}
\chi(E/\alpha) = - \frac{\Gamma(\frac{N+1}{2})}{2 z_{N-1}} g_{N-1}^{(N)}\left(\frac{E}{\alpha}\right) \,,
\end{equation}
where $\chi(E/\alpha) = \tilde{\chi}(E/\alpha) \tilde{a}^{-1}$ is now the unscaled energy shift fraction fulfilling
\begin{equation}
\Delta E_\alpha = \chi(E/\alpha) \Delta E_\infty \,.
\end{equation}
The corresponding expression for homogeneous external potentials is then related by the general scaling property (see appendix F).
\end{document}
|
\begin{document}
\title{A cospectral family of graphs for the normalized~Laplacian found by toggling}
\author{Steve Butler\thanks{Dept.\ of Mathematics, Iowa State University, Ames, IA 50011, USA\newline ({\tt \{butler,keheysse\}@iastate.edu})}~\thanks{Partially supported by an NSA Young Investigator Grant} \and Kristin Heysse\footnotemark[1]}
\maketitle
\begin{abstract}
We give a construction of a family of (weighted) graphs that are pairwise cospectral with respect to the normalized Laplacian matrix, or equivalently probability transition matrix. This construction can be used to form pairs of cospectral graphs with differing number of edges, including situations where one graph is a subgraph of the other. The method used to demonstrate cospectrality is by showing the characteristic polynomials are equal.
\end{abstract}
\section{Introduction}\label{sec:introduction}
Spectral graph theory studies the relationship between the structure of a graph and the eigenvalues of a particular matrix associated with that graph. There are several matrices that are commonly studied, each with merits and limitations. These limitations exist because graphs can be constructed which have the same spectrum with respect to the matrix and are fundamentally different in some structural aspect. Such graphs are called \emph{cospectral}.
There are many possible matrices to consider, and the matrix we consider in this paper is the normalized Laplacian (see \cite{BC,Chung}). The rows and columns of this matrix are indexed by the vertices, and for a simple graph the matrix is defined as follows:
\[
\mathcal{L}(i,j)=\left\{\begin{array}{c@{\qquad}l}
1 & \text{if $i=j$, and vertex $i$ is not isolated};\\[5pt]
{\displaystyle\frac{-1}{\sqrt{d_i d_j}}} & \text{if } i {\sim} j; \\[5pt]
0 & \text{otherwise;} \end{array} \right.
\]
where $d_i$ is the degree of vertex $i$.
In this paper we want to look at the more general setting of edge-weighted graphs, i.e., there is a symmetric, non-negative weight function, $w(i,j)$ on the edges. The degree of a vertex now corresponds to the sum of the weights of the incident edges, i.e., $d_i=\sum_{i{\sim}j}w(i,j)$. The normalized Laplacian for weighted graphs is defined in the following way:
\[
\mathcal{L}(i,j)=\left\{ \begin{array}{c@{\qquad}l}
1 & \text{if $i=j$, and vertex $i$ is not isolated};\\[5pt] {\displaystyle\frac{-w(i,j)}{\sqrt{d_i d_j}}} & \text{if }i {\sim} j;\\[5pt]
0 & \text{otherwise.} \end{array} \right.
\]
(A simple graph corresponds to the case where $w(i,j)\in\{0,1\}$ for all $i,j$.) We note that when the graph has no isolated vertices, $\mathcal{L}$ can be written as $\mathcal{L}=D^{-1/2}(D-A)D^{-1/2}$, where $A_{i,j}=w(i,j)$ and $D$ is the diagonal degree matrix. Finally, we point out that this matrix is connected with the probability transition matrix $D^{-1}A$ of a random walk. In particular, two graphs with no isolated vertices are cospectral for $\mathcal{L}$ if and only if they are cospectral for $D^{-1}A$.
There has been some interest in the construction of cospectral graphs for the normalized Laplacian. Cavers \cite{Cavers} showed that a restricted variation of Godsil-McKay switching (see \cite{GM}) preserves the spectrum, while Butler and Grout \cite{BG} showed that gluing in two different special bipartite graphs into some arbitrary graph resulted in a pair of cospectral graphs. In both cases, the operation preserved the number of edges in the graph.
On the other hand, it is possible for graphs with differing number of edges to be cospectral with respect to the normalized Laplacian. The classic example of this is complete bipartite graphs $K_{p,q}$ which have spectrum $\{0,1^{(p+q-2)},2\}$ (here the exponent is indicating multiplicity). For example, the (sparse) star $K_{1,2n-1}$ is cospectral with the (dense) regular graph $K_{n,n}$. Until recently, this was the \emph{only} known construction of cospectral graphs with differing number of edges. Butler and Grout \cite{BG} gave some examples of small graphs found by exhaustive computation that differ in the number of edges, including some where one graph was a subgraph of the other. Butler \cite{twins} expanded on this example to form an infinite family and showed how to construct many pairs of bipartite graphs which were cospectral.
In this paper we introduce a new construction of cospectral graphs for the normalized Laplacian which can differ in the number of edges. The basic idea is to form a ring of linked modules, and then a similar graph where we interchange the role of two of the modules (what we term ``toggling''). The resulting pair of graphs are cospectral with respect to the normalized Laplacian. An example of this construction is shown in Figure~\ref{firstexample}. Note that the left graph is a subgraph of the right graph.
\begin{figure}
\caption{A pair of cospectral graphs for $\mathcal{L}
\label{firstexample}
\end{figure}
In Section~\ref{Construction}, we give a formal description of this family, of toggling, and state the main result. In Section~\ref{sec:charpoly}, we show how to compute the characteristic polynomial of the normalized Laplacian by using decompositions. We then break the decompositions of a graph in our family into those which contain a ``long'' cycle (see Section~\ref{sec:long}) and those which do not (see Section~\ref{sec:short}), and in particular conclude the characteristic polynomials are equal so the graphs must be cospectral. In Section~\ref{sec:weighted} we show how to go from weighted graphs to simple graphs which are cospectral with respect to the normalized Laplacian.
\section{Construction}\label{Construction}
Our family of graphs are formed as a ring composed of three different types of (weighted) modules: the \emph{path} on four vertices, the \emph{cycle} on four vertices, and the \emph{edge} on two vertices, which we label as {\tt P}, {\tt C}, and {\tt E}, respectively. The modules are shown in Figure~\ref{modules} where we have marked the edge weights using a parameter $k$ where $k>0$ is for now arbitrary. Each module has special vertices marked with ``$+$'' and ``$-$'' which can be thought of as poles of a magnet to indicate how consecutive modules will connect. In particular, the ``$+$'' vertex on one module will connect with the ``$-$'' vertex on the next module. We will refer to these two special vertices as the \emph{signed} vertices.
\begin{figure}
\caption{The {\tt P}
\label{modules}
\end{figure}
A graph in our family is formed by connecting $\tau$ modules together in a cycle. In particular, such graph can be associated with a word using the letters {\tt P}, {\tt C} and {\tt E}. As an example, starting with the top module and reading clockwise, the two graphs shown in Figure~\ref{firstexample} (where $k=1$) have the words {\tt PPCCPPPC} and {\tt CCPPCCCP}. Note that given a graph in our family there are many possible words, i.e., we can choose any module to start and any possible direction. On the other hand, given a word, there is a unique graph.
\begin{definition}
Given a word $W=\ell_1\ell_2\ldots\ell_\tau$ where $\ell_i\in\{{\tt P},{\tt C},{\tt E}\}$ and $\tau\ge3$. Then $G(W)$ is the graph obtained by connecting the corresponding $\tau$ modules in cyclic order as indicated by the word where consecutive modules connect on the signed vertices, and where the final module will connect to the first module.
\end{definition}
We note that the two words we constructed for the graphs in Figure~\ref{firstexample} are related by interchanging the roles of ${\tt P}$ and ${\tt C}$. This will generalize as follows.
\begin{definition}
Given a cyclic word $W$ composed of the letters {\tt P}, {\tt C}, and {\tt E}. Then the \emph{toggling} of $W$ is $W^T$, the word formed by taking $W$ and replacing every {\tt P} by {\tt C} and every {\tt C} by {\tt P}. The occurrences of {\tt E} are unchanged.
\end{definition}
The motivation for the use of the word ``toggling'' is to notice that the difference between $G(W)$ and $G(W^T)$ is adding or removing the edge on a module which goes between the non-signed vertices. In essence, we are switching the states of these edges.
We can now state our main result.
\begin{theorem}\label{thm:main}
For a word $W$ of length at least three using the letters {\tt P}, {\tt C}, and {\tt E}, $G(W)$ and $G(W^T)$ are cospectral with respect to the normalized Laplacian.
\end{theorem}
We note that if $W$ does not contain the same number of occurrences of {\tt P} and {\tt C}, then the number of edges in $G(W)$ and $G(W^T)$ will differ and are clearly non-isomorphic. Among other things, we can construct cospectral simple graphs which differ by exactly $m$ edges by setting $k=1$ and using a word in $\tt{P}$ and $\tt{C}$ where there are $m$ more instances of $\tt{C}$ than of $\tt{P}$. There are also some special words $W$ so that $G(W)$ is a subgraph of $G(W^T)$. One example of this behavior is $W={\tt CC}\ldots{\tt C}$ and $W^T={\tt PP}\ldots{\tt P}$, though others exist (see Figure~\ref{firstexample}).
\section{Computing the characteristic polynomial}\label{sec:charpoly}
Our approach will involve showing the characteristic polynomials of $G(W)$ and $G(W^T)$ are equal. We start by determining how to compute the characteristic polynomial by the use of generalized cycle decompositions (see \cite{Brualdi}). For an $n \times n$ matrix $M=[m_{i,j}]$,
\[
\det(M)=\sum_{\sigma \in S_n} \sgn(\sigma)\underbrace{m_{1,\sigma(1)}m_{2,\sigma(2)}\cdots m_{n,\sigma(n)}}_{:=w_M(\sigma)}=\sum_{\sigma \in S_n} \sgn(\sigma) w_M(\sigma).
\]
Let $G_M$ denote the digraph which corresponds to $M$, meaning it has $i{\to}j$ if and only if $m_{i,j}\ne0$. We can consider a permutation $\sigma$ which contributes a nonzero term to $\det(M)$. The factors of $w_M(\sigma)$ correspond to $n$ edges such that each vertex has in-degree and out-degree equal to one, as each vertex will appear as the first and second index somewhere in $w_M(\sigma)$. Such a collection of edges is a \emph{generalized cycle decomposition} of $G_M$. There are three possible structures in a generalized cycle decomposition: loops (a directed edge that goes into and out of the same vertex), edges (pairs of directed edges $i{\to}j$ and $j{\to}i$), and longer directed cycles. More generally, if we think of loops and edges as cycles of length one and two, respectively, then a generalized cycle decomposition is a collection of disjoint cycles so that every vertex is in exactly one cycle.
In the case when the matrix $M$ is symmetric, many of these generalized cycle decompositions will contribute the same factor to the determinant. For example, changing the orientation on a long cycle gives a different decomposition but does not change $\sgn(\sigma)w_M(\sigma)$. With this in mind we consider decompositions.
\begin{definition}
Let $G$ be an undirected (weighted) graph. Then a \emph{decomposition}, $D$, is a subgraph consisting of disjoint edges and cycles.
\end{definition}
When $M$ is symmetric, we can treat $G_M$ as an undirected graph. Every generalized cycle decomposition now corresponds to a unique decomposition, $D$, by removing loops and dropping the orientation on the long cycles. Conversely, if we let $s=s(D)$ denote the number of cycles of length at least three in the decomposition $D$, then each decomposition corresponds to a collection of $2^s$ different generalized cycle decompositions. Namely, any vertex not in an edge or a cycle has a loop added, edges become cycles of length two, and each of the $s$ cycles of length at least $3$ have one of two possible orientations chosen.
If we let $e(D)$ count the number of cycles in the decomposition which have an even number of vertices (including edges), and $F(D)$ be the set of isolated edges in the decomposition $D$, then we have the following result.
\begin{proposition}\label{prop:charpoly}
Let $G$ be a weighted graph on $n$ vertices without loops or isolated vertices. Then the characteristic polynomial of the normalized Laplacian matrix is
\[
p(t)=\sum_{D}(-1)^{e(D)}2^{s(D)}(t-1)^{n-|V(D)|}\frac{ \prod_{\{i,j\}\in E(D)}w(i,j)\prod_{\{i,j\}\in F(D)}w(i,j)}{\prod_{i\in V(D)}d_i}
\]
where the sum runs over all decompositions $D$ of the graph $G$.
\end{proposition}
\begin{proof}
The characteristic polynomial with respect to the normalized Laplacian can be written as
\begin{align*}
p(t) &=\det(tI-\mathcal{L})\\
&=\det\big(tI-D^{-1/2}(D-A)D^{-1/2}\big)\\
&=\det\big(\underbrace{(t-1)I+D^{-1/2}AD^{-1/2}}_{=M}\big).
\end{align*}
The graph $G_M$ (ignoring loops) has the same edges and non-edges as $G$, and so we can use decompositions to compute the determinant.
In particular, every decomposition of $G$ will relate to $2^{s(D)}$ generalized cycle decompositions. For each such generalized cycle decomposition corresponding to a permutation $\sigma$, we have $\sgn(\sigma)=(-1)^{e(D)}$. We will have $n-|V(D)|$ loops which each contribute $(t-1)$. The non-loop edges $i{\to}j$ will contribute $w(i,j)/\sqrt{d_id_j}$. Now we recall that each vertex in a generalized cycle decomposition has one edge coming in and one edge going out, and therefore for each vertex $i$ in $V(D)$ we will have $\sqrt{d_i}$ occurring twice in the denominator giving us the $d_i$. Finally, for cycles of length three or greater we only use each edge once in the generalized cycle decomposition, but for cycles of length two we use the same edge for both directions and so we use the edge twice.
\end{proof}
\section{Decompositions of $G(W)$ with a long cycle}\label{sec:long}
Proposition~\ref{prop:charpoly} shows that we can determine the characteristic polynomial by looking at decompositions of the graph. In this section we will consider the collection of decompositions of a graph $G(W)$ which contain a long cycle, i.e., a cycle which passes through all of the signed vertices in $G(W)$. We denote the set of these decompositions as $L$.
\begin{lemma}\label{lem:long}
Let $W$ be a word of length $\tau$ with $\ell$ occurrences of {\tt P} and $m$ occurrences of {\tt C}. Then for $G(W)$ we have
\begin{multline*}
\sum_{D\in L}(-1)^{e(D)}2^{s(D)}(t-1)^{n-|V(D)|}\frac{ \prod_{\{i,j\}\in E(D)}w(i,j)\prod_{\{i,j\}\in F(D)}w(i,j)}{\prod_{i\in V(D)}d_i}\\
=\frac{(-1)^{\tau-1}(t-1)^{2(m+\ell)}}{2^{\tau-1}(k+1)^{m+\ell}}
\end{multline*}
\end{lemma}
\begin{proof}
Knowing we have a long cycle yields a lot of information about the decomposition $D$ in $G(W)$. In particular, for a module of type ${\tt P}$ or ${ \tt E}$, the decomposition will contain only the edge between the signed vertices. These are shown in Figure~\ref{xzlongdecomp}, where edge weights have been removed for clarity.
\begin{figure}
\caption{Forced decomposition for {\tt P}
\label{xzlongdecomp}
\end{figure}
For a module of type {\tt C} the situation is a more interesting as there are three different options for the decomposition. Namely, that the long cycle passes only through the signed vertices; the long cycle passes through the signed vertices and there is an edge between the unsigned vertices; the long cycle passes through all of the vertices. These three possibilities are shown in Figure~\ref{ylongdecomp}.
\begin{figure}
\caption{Three possible decompositions for {\tt C}
\label{ylongdecomp}
\end{figure}
Now suppose that among the $m$ modules of type {\tt C} that precisely $h$ of them are the configuration shown on the left in Figure~\ref{ylongdecomp}; $i$ of them are the configuration shown in the center in Figure~\ref{ylongdecomp}; and $j$ of them are the configuration shown on the right in Figure~\ref{ylongdecomp}. The choices of which {\tt C} modules behave in which way is arbitrary. Summing over all the possibilities gives the following.
\begin{multline*}
\sum_{D\in L}(-1)^{e(D)}2^{s(D)}(t-1)^{n-|V(D)|}\frac{ \prod_{\{i,j\}\in E(D)}w(i,j)\prod_{\{i,j\}\in F(D)}w(i,j)}{\prod_{i\in V(D)}d_i}\\
=\sum_{h+i+j=m}\frac{2(-1)^{\tau-1}(k+1)^{\tau-\ell-m}(t-1)^{2\ell}}{\big(2(k+1)\big)^\tau}\times\\
{m\choose h,i,j}\big((t-1)^2\big)^h\bigg({(-1)k^4\over\big(k(k+1)\big)^2}\bigg)^i\bigg({k^4\over\big(k(k+1)\big)^2}\bigg)^j
\end{multline*}
We have $2^{s(D)}=2$ because there is only one cycle of length greater than three, namely the long cycle which contains all the signed vertices. The $e(D)$ will count the number of $\tt{C}$ modules in the middle configuration and possibly the long cycle itself. Regardless of the number of $\tt{C}$ modules in the configuration on the right, the contribution from the long cycle to $(-1)^{e(D)}$ will be $(-1)^{\tau-1}$. Consider first the contributions of isolated vertices and edge weights of the $\tt{P}$ and $\tt{E}$ modules. The $(k+1)^{\tau-\ell-m}$ is the weight of the edges on the long cycle coming from the modules of type {\tt E}. The $(t-1)^{2\ell}$ accounts for the isolated vertices from the modules of type {\tt P}. Further, the $\big(2(k+1)\big)^\tau$ is the product of the degrees of the signed vertices (each such vertex has degree $2(k+1)$ as can be seen by noting that in the modules the signed vertices have degree $k+1$ and then we identify two such vertices).
It remains to account for the portions of the decomposotions formed on the $\tt{C}$ modules which are not the signed vertices. The ${m\choose h,i,j}=m!/(h!i!j!)$ is the multinomial coefficient for how many ways to choose the different module configurations for {\tt C}, and the final three factors are the contributions from each configuration formed by accounting for isolated vertices, edge weights, and the degrees of vertices in the decomposition (i.e., a pair of isolated vertices or the product of the edge weights over product of degrees). Notice that for the middle configuration, the contribution to $(-1)^{e(D)}$ has been appropriately grouped.
Now we can simplify by pulling out the terms which do not depend on the sum and cancelling. For the terms in the sum we can use the multinomial theorem to simplify. Continuing the above computation, we now have
\begin{align*}
&=\frac{(-1)^{\tau-1}(t-1)^{2\ell}}{2^{\tau-1}(k+1)^{\ell+m}}\bigg((t-1)^2-\frac{k^4}{\big(k(k+1)^2\big)^2}+\frac{k^4}{\big(k(k+1)^2\big)^2}\bigg)^m\\
&=\frac{(-1)^{\tau-1}(t-1)^{2\ell}}{2^{\tau-1}(k+1)^{\ell+m}}(t-1)^{2m}=\frac{(-1)^{\tau-1}(t-1)^{2(m+\ell)}}{2^{\tau-1}(k+1)^{m+\ell}}.\qedhere
\end{align*}
\end{proof}
The important thing to note is that the expression in Lemma~\ref{lem:long} will be the same for $W$ and $W^T$ because $m+\ell$ is invariant under toggling.
\section{Decompositions of $G(W)$ without a long cycle}\label{sec:short}
Any cycle in a decomposition with edges in consecutive modules would have to go through all of the modules to close up. In particular, if there is not a long cycle in our decomposition $D$, then the decomposition is composed of only edges and $C_4$'s which lie in individual modules.
We consider what decompositions can happen in a single module and how decompositions in consecutive modules interact. The first task is straightforward to carry out, and in Tables~\ref{table:P}, \ref{table:C}, and \ref{table:E} we show the possible \emph{local} decompositions for each module. To help facilitate the analysis we have grouped the local decompositions by which signed vertices (if any) are used.
\tableP
\tableC
\tableE
The next part is to understand the transitions between modules, i.e., how local decompositions interact. We have already grouped the local decompositions by which of the signed vertices are used. We now note that if signed vertices are used in by a local decompositon in one module, it influences which of the signed vertices are available for use in the next module. This is indicated by the following transition matrix with rows and columns indexed by subsets of the signed vertices:
\[
Q=\bordermatrix{
&~\emptyset~&~{+}~&~{-}~&{+}/{-}\cr
\emptyset&1&1&1&1\cr
{+}&1&1&1&1\cr
{-}&1&0&1&0\cr
{+}/{-}&1&0&1&0}.
\]
Using $Q$ we can now count the number of ways that we can have decompositions use the signed vertices in the modules for $G(W)$. This is done using the transfer matrix method (see \cite{GS}), and in particular is equal to the number of closed walks in the directed graph corresponding to $Q$ which have the same length as the length of the word.
We need to go one step further and for every module add the contribution of the local decomposition.
This final part is done by adding in diagonal weight matrices where the diagonal entries correspond to the contribution of the decomposition for that particular module. These contributions are found by $(-1)$ raised to the number of even cycles (i.e., edges or $C_4$'s) times the product of the edge weights used in the local decomposition (remembering for an edge to use that edge twice), divided by the product of the degrees of any vertex used in the decomposition. The only subtle part is handling the vertices which will not be a part of a decomposition in any module. What we do is assume at the beginning that \emph{every} vertex is isolated and contributes a $(t-1)$ then whenever a vertex becomes a part of the decomposition we divide by $(t-1)$ to correct (the choice of this approach is because signed vertices lie in two modules, hence while it might not be in the decomposition of one module it could be in the decomposition of the other). When there are several possible decompositions in a given case we add them together to form the entry for the weight matrix.
The contributions were previously listed in the tables and become the diagonal entries of the weight matrices. We therefore have the following weight matrices.
\begin{align*}
X_{\tt P}&=\left(\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & \frac{-k}{(t-1)^2(2k+2)} & 0 & 0 \\
0 & 0 & \frac{-k}{(t-1)^2(2k+2)} & 0 \\
0 & 0 & 0 & \frac{k^2-(t-1)^2}{(t-1)^4(2k+2)^2}
\end{array}\right)
\\
X_{\tt C}&=\left(\begin{array}{cccc}
1-\frac{k^2}{(t-1)^2(k+1)^2} & 0 & 0 & 0 \\
0 & \frac{-k}{2(t-1)^2(k+1)^2} & 0 & 0 \\
0 & 0 & \frac{-k}{2(t-1)^2(k+1)^2} & 0 \\
0 & 0 & 0 & \frac{-1}{4(t-1)^2(k+1)^2}
\end{array}\right)
\\
X_{\tt E}&=\left(\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & \frac{-1}{4(t-1)^2}
\end{array}\right)
\end{align*}
So for the graph $G(\ell_1\ell_2\cdots \ell_\tau)$, we have the following:
\begin{multline}\label{eq:short}
\sum_{D\notin L}(-1)^{e(D)}2^{s(D)}(t-1)^{n-|V(D)|}\frac{ \prod_{\{i,j\}\in E(D)}w(i,j)\prod_{\{i,j\}\in F(D)}w(i,j)}{\prod_{i\in V(D)}d_i}\\
=(t-1)^{|V(G(W))|}\trace(QX_{\ell_1}QX_{\ell_2}\cdots QX_{\ell_\tau}).
\end{multline}
We now focus on rewriting the trace expression in \eqref{eq:short}. To start we note that $Q=RSR^{-1}$ where
\[
R = \left(\begin{array}{cccc}
1&-1&1&1\\
1&-1&0&0\\
\frac12&1&0&-1\\
\frac12&1&-2&0
\end{array}\right),
\quad\text{and}\quad
S = \left(\begin{array}{cccc}
3&0&0&0\\
0&0&1&0\\
0&0&0&0\\
0&0&0&0
\end{array}\right).
\]
Combining this with $\trace(AB)=\trace(BA)$ we can conclude
\begin{multline*}
(t-1)^{|V(G(W))|}\trace(QX_{\ell_1}QX_{\ell_2}\cdots QX_{\ell_\tau})\\
=(t-1)^{|V(G(W))|}\trace(RSR^{-1}X_{\ell_1}RSR^{-1}X_{\ell_2}\cdots RSR^{-1}X_{\ell_\tau})\\
=(t-1)^{|V(G(W))|}\trace\big((SR^{-1}X_{\ell_1}R)(SR^{-1}X_{\ell_2}R)\cdots (SR^{-1}X_{\ell_\tau}R)\big).
\end{multline*}
Because $S$ has two rows of $0$'s this simplifies the matrices that we have to deal with. In particular we have
\begin{align*}
SR^{-1}X_{\tt P}R&=\left(\begin{array}{cc}Y_{\tt P}&Z_{\tt P}\\O&O\end{array}\right),\\
SR^{-1}X_{\tt C}R&=\left(\begin{array}{cc}Y_{\tt C}&Z_{\tt C}\\O&O\end{array}\right),\text{ and}\\
SR^{-1}X_{\tt E}R&=\left(\begin{array}{cc}Y_{\tt E}&Z_{\tt E}\\O&O\end{array}\right),
\end{align*}
where if we let $u:=t-1$ then
\begin{align*}
Y_{\tt P}&=\left(\begin{array}{cc}\frac{16k^2u^4 + 32ku^4 - 8k^2u^2 + 16u^4 - 8ku^2 + k^2 - u^2}{12(k+1)^2u^4}&\frac{-8k^2u^4 - 16ku^4 - 2k^2u^2 - 8u^4 - 2ku^2 + k^2 - u^2}{6(k+1)^2u^4}\\[5pt]
\frac{8k^2u^4 + 16ku^4 + 2k^2u^2 + 8u^4 + 2ku^2 - k^2 + u^2}{24(k+1)^2u^4}&\frac{16k^2u^4 + 32ku^4 - 8k^2u^2 + 16u^4 - 8ku^2 + k^2 - u^2}{12(k + 1)^2u^4
}\end{array}\right)\\
Y_{\tt C}&=\left(\begin{array}{cc}
\frac{16k^2u^2 + 32ku^2 - 16k^2 + 16u^2 - 8*k - 1}{12(k+1)^2u^2}& \frac{-8k^2u^2 - 16ku^2 + 8k^2 - 8u^2 - 2k - 1}{6(k+1)^2u^2}\\[5pt]
\frac{8k^2u^2 + 16ku^2 - 8k^2 + 8u^2 + 2k + 1}{24(k+1)^2u^2}&\frac{-4k^2u^2 - 8ku^2 + 4k^2 - 4u^2 - 4k + 1}{12(k+1)^2u^2}
\end{array}\right)\\
Y_{\tt E}&=\left(\begin{array}{cc}\frac{16u^2-1}{12u^2}&\frac{-8u^2-1}{6u^2}\\[5pt]\frac{8u^2+1}{24u^2}&\frac{-4u^2+1}{12u^2}\end{array}\right)
\end{align*}
Because we can carry out block matrix multiplication, we note that the resulting upper left block will be the product of the upper left blocks and that the resulting lower right block will be the all zeroes matrix. This allows us to conclude the following:
\begin{equation*}
(t-1)^{|V(G(W))|}\trace(QX_{\ell_1}QX_{\ell_2}\cdots QX_{\ell_\tau})
=(t-1)^{|V(G(W))|}\trace(Y_{\ell_1}Y_{\ell_2}\cdots Y_{\ell_\tau})
\end{equation*}
There is no convenient way to find a simple expression for these decompositions as we did for the long cycles. However, it suffices to show that the toggled words will produce equivalent results, which is what we now show.
\begin{lemma}\label{lem:short}
Let $W=\ell_1\ell_2\ldots \ell_\tau$ and $W^T=\gamma_1\gamma_2\ldots\gamma_\tau$. Then
\[
(t-1)^{|V(G(W))|}\trace(Y_{\ell_1}Y_{\ell_2}\cdots Y_{\ell_\tau}) =
(t-1)^{|V(G(W^T))|}\trace(Y_{\gamma_1}Y_{\gamma_2}\cdots Y_{\gamma_\tau}).
\]
\end{lemma}
\begin{proof}
Both sides are polynomials, and so it suffices to verify that the relationship holds for $t\ne0,1,2$ (i.e., if two polynomials agree at all but three points, they must agree everywhere). To show that they are equal, we will make use of the following special matrix,
\[
U=\left(\begin{array}{cc}
20u^2-2&-32u^2-4\\
8u^2+1&-20u^2+2
\end{array}\right).
\]
This matrix has the following special properties, which can be verified by carrying out matrix multiplication:
\begin{itemize}
\item $UY_{\tt P}=Y_{\tt C}U$.
\item $UY_{\tt C}=Y_{\tt P}U$.
\item $UY_{\tt E}=Y_{\tt E}U$.
\end{itemize}
These properties are key, in that they indicate we can pass $U$ through one of the $Y_{*}$ matrices but we need to change the matrix in the same way that we do in the toggling operation.
For $t\ne 0,1,2$ we have that $U$ is invertible and so by repeated application of the above properties we have
\begin{align*}
(t-1)^{|V(G(W))|}\trace(Y_{\ell_1}Y_{\ell_2}\cdots Y_{\ell_\tau})
&=(t-1)^{|V(G(W))|}\trace(UY_{\ell_1}Y_{\ell_2}\cdots Y_{\ell_\tau}U^{-1})\\
&=(t-1)^{|V(G(W))|}\trace(Y_{\gamma_1}UY_{\ell_2}\cdots Y_{\ell_\tau}U^{-1})\\
&=(t-1)^{|V(G(W))|}\trace(Y_{\gamma_1}Y_{\gamma_2}U\cdots Y_{\ell_\tau}U^{-1})\\
&=\cdots\\
&=(t-1)^{|V(G(W))|}\trace(Y_{\gamma_1}Y_{\gamma_2}\cdots U Y_{\ell_\tau}U^{-1})\\
&=(t-1)^{|V(G(W))|}\trace(Y_{\gamma_1}Y_{\gamma_2}\cdots Y_{\gamma_\tau}UU^{-1})\\
&=(t-1)^{|V(G(W))|}\trace(Y_{\gamma_1}Y_{\gamma_2}\cdots Y_{\gamma_\tau})\\
&=(t-1)^{|V(G(W^T))|}\trace(Y_{\gamma_1}Y_{\gamma_2}\cdots Y_{\gamma_\tau}),
\end{align*}
where in the last we use that toggling does not change the number of vertices in the graph.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:main}]
To show that the graphs $G(W)$ and $G(W^T)$ are cospectral we can show that they have the same characteristic polynomial. We use Proposition~\ref{prop:charpoly} and consider all the possible decompositions. Lemma~\ref{lem:long} shows that the sum over all the decompositions which contain a long cycle are equal while Lemma~\ref{lem:short} shows that the sum over all the decompositions which do not contain a long cycle are also equal. Thus the sum over all decompositions is equal, and the theorem is established.
\end{proof}
\section{Weighted Graphs to Simple Graphs}\label{sec:weighted}
We have considered graphs with edge weights in terms of a parameter $k$ as shown in Figure~\ref{modules}. By letting $k=1$ and restricting to {\tt P} and {\tt C} modules we will produce cospectral simple graphs.
Simple graphs can also be obtained by appropriately ``blowing up'' our graph. This works by replacing vertices by independent sets. An edge between $u$ and $v$ which has been replaced by $r$ and $s$ vertices respectively then becomes a complete bipartite graphs between the two independent sets with all edge weights $w(u,v)/rs$. (Note that $r$ and $s$ are generally chosen so that this new edge weight is $1$, i.e., so the new graph is a simple graph.) Similarly several consecutive {\tt E} edges with weight $k+1$ can become $k+1$ parallel paths. A discussion on how eigenvalues for the normalized Laplacian work for blowups can be found in \cite{twins}. In particular, it is known that the eigenvalues of the blowups are determined from the eigenvalues of the original graphs (which we have shown to be cospectral) and the remaining eigenvalues will come from the blowup procedure, which will be the same for both graphs.
As an example in Figure~\ref{blowup} we start with the cospectral graphs
$\tt{EEEPCC}$ and $\tt{EEECPP}$. This figure also contains the blowups which result by replacing the unsigned vertices in {\tt C} and {\tt P} modules with $k$ independent vertices (marked by putting $k$ inside the vertex and making the lines bold to represent complete bipartite graphs), the three consecutive {\tt E} edges become $k+1$ parallel paths of length three. In particular, the resulting blowups are simple graphs which are also cospectral.
\begin{figure}
\caption{The graphs $\tt{EEEPCC}
\label{blowup}
\end{figure}
There are other possibilities. For instance, from the definition of the normalized Laplacian we note that the matrix does not change if we scale all edge weights by a fixed amount. So we can first scale the edge weights and then perform a blowup. A partial example of this is shown in Figure~\ref{singleE} where we consider the graph corresponding to {\tt ECC}. By setting $k=1$ and then scaling all edge weights by $2$ we get a weighted graph which has as a blowup the graph shown on the right in Figure~\ref{singleE}. By a similar process we could also do the same for {\tt EPP} to construct a cospectral pair of simple graphs.
\begin{figure}
\caption{The graph $\tt{ECC}
\label{singleE}
\end{figure}
\section{Conclusion}
Many, if not most, approaches to establish cospectrality rely on showing that a small perturbation in the graph corresponds to a small, controllable perturbation in the eigenvectors and hence eigenvalues are preserved. This was \emph{not} the case in this construction, which is why we considered the characteristic polynomials. Also, while we show that the characteristic polynomials are equal, we never explicitly computed one. Instead, we showed that the method to determine these polynomials will produce the same answer for a pair of cospectral graphs. It would be interesting to find additional families where this can occur.
We have been able to establish a large family of cospectral graphs for the normalized Laplacian (and hence also probability transition matrix) which have unusual properties, including cospectral graphs with differing number of edges and graphs cospectral with subgraphs. There is a vast amount about the spectrum of the normalized Laplacian that is not well understood. We hope to see more of this area explored in future work.
\noindent\textbf{Acknowledgment.} The work on this paper was partially conducted while the authors were visiting the Institute for Mathematics and its Applications.
\end{document}
|
\begin{document}
\large
\centerline{\textbf{Curvilinear integral theorem for $G$-monogenic mappings}}
\centerline{\textbf{in the algebra of complex quaternion}}
\vskip 2mm
\centerline{Tetyana KUZMENKO}
\vskip 2mm
\small
\noindent{\textbf{Keywords:} quaternion algebra, $G$-monogenic mapping, curvilinear Cauchy integral theorem.}
\vskip 2mm
\noindent\textbf{Abstract.} For $G$-monogenic mappings taking values in the algebra of complex quaternion we prove a curvilinear analogue of the Cauchy integral theorem in the case where a curve of integration lies on the boundary of a domain.
\vskip 2mm
\noindent{\textbf{AMS 2010:} 30G35.}
\large
\vskip 5mm
\textbf{Introduction}
Let $\mathbb{H(C)}$ be the quaternion algebra over the field of complex numbers
$\mathbb{C}$, whose basis consists of the unit $1$ of the algebra and of the elements
$I,J,K$
satisfying the multiplication rules:
$$I^2=J^2=K^2=-1,$$
$$\,IJ=-JI=K,\quad JK=-KJ=I,\quad KI=-IK=J.$$
In the algebra $\mathbb{H(C)}$ there exists another basis $\{e_1,e_2,e_3,e_4\}$ such that multiplication table in a new
basis can be represented as (see, e. g., \cite{Cartan})
$$
\begin{tabular}{c||c|c|c|c|}
$\cdot$ & $e_1$ & $e_2$ & $e_3$ & $e_4$\\
\hline
\hline
$e_1$ & $e_1$ & $0$ & $e_3$ & $0$\\
\hline
$e_2$ & $0$ & $e_2$ & $0$ & $e_4$\\
\hline
$e_3$ & $0$ & $e_3$ & $0$ & $e_1$\\
\hline
$e_4$ & $e_4$ & $0$ & $e_2$ & $0$\\
\hline
\end{tabular}\,\,.
$$
The unit of the algebra can be decomposed as $1=e_1+e_2$.
Let us consider the vectors
$$i_1=e_1+e_2, \quad i_2=a_1e_1+a_2e_2, \quad i_3=b_1e_1+b_2e_2,$$
$a_k,b_k\in\mathbb{C},\,k=1,2$,
which are linearly independent over the field of real numbers $\mathbb{R}$.
It means that the equality $\alpha_1i_1+\alpha_2i_2+\alpha_3i_3=0$ for $\alpha_1,\alpha_2,\alpha_3\in\mathbb{R}$ holds if and only if
$\alpha_1=\alpha_2=\alpha_3=0$.
In the algebra $\mathbb{H(C)}$ we consider the linear span $E_3:=\{\zeta=xi_1+yi_2+zi_3:x,y,z\in\mathbb{R}\}$ generated by the
vectors $i_1,i_2,i_3$ over the field $\mathbb{R}$. A set $S\subset\mathbb{R}^3$ is associated with the set $S_\zeta:=
\{\zeta=xi_1+yi_2+zi_3:(x,y,z)\in S\}$ in $E_3$. We also note that a
topological property of a set $S_\zeta$ in $E_3$ understand as the same
topological property of the set $S$ in $\mathbb{R}^3$.
For example, we will say that a curve $\gamma_\zeta\subset E_3$ is homotopic to
a point if $\gamma\subset\mathbb{R}^3$ is homotopic to a point, etc.
We say (see \cite{Shpakivskiy-Kuzmenko}) that a continuous mapping $\Phi:\Omega_\zeta\rightarrow\mathbb{H(C)}$ \big(or $\widehat{\Phi}:\Omega_\zeta\rightarrow\mathbb{H(C)}$\big) is \emph{right-$G$-monogenic}
\big(or \emph{left-$G$-monogenic}\big) in a domain
$\Omega_\zeta\subset E_3$\,,
if $\Phi$ \big(or $\widehat{\Phi}$\big) is differentiable in the sense
of the G\^{a}teaux at every point of\, $\Omega_\zeta$\,, i.~e. for every $\zeta\in
\Omega_\zeta$ there exists an element $\Phi'(\zeta)\in\mathbb{H(C)}$ \big(or $\widehat{\Phi}'(\zeta)\in\mathbb{H(C)}$\big) such that
$$\lim\limits_{\varepsilon\rightarrow 0+0}\Big(\Phi(\zeta+\varepsilon h)-\Phi(\zeta)\Big)\varepsilon^{-1}= h\Phi'(\zeta)\quad\forall\,h\in E_3$$
$$\Biggr(\text{or }\,\, \lim\limits_{\varepsilon\rightarrow 0+0}
\left(\widehat{\Phi}(\zeta+\varepsilon h)-\widehat{\Phi}(\zeta)\right)
\varepsilon^{-1}= \widehat{\Phi}'(\zeta)h\quad\forall\,h\in E_3\Biggr).$$
The Cauchy integral theorems for holomorphic functions of the complex variable are
fundamental results of the classical complex analysis.
Analogues of these results are also important tools in the quaternionic analysis.
In the paper \cite{Shp-Kuzm-intteor} were established some analogues of classical
integral theorems of the theory of analytic functions of the complex variable: the surface and curvilinear Cauchy
integral theorems and the Cauchy integral formula. The Morera theorem was proved in the paper \cite{Shp-Kuz-umb}. Taylor's and Laurent's expansions of $G$-monogenic mappings are obtained in \cite{Kuzm-series}.
Namely, in the paper \cite{Shp-Kuzm-intteor} was proved a curvilinear analogue of the Cauchy integral theorem in the case where a curve of integration lies in a domain of $G$-monogeneity.
In the present paper we prove the curvilinear Cauchy integral theorem for $G$-monogenic mappings in the case where a curve of integration lies on the boundary of a domain of $G$-monogeneity.
\vskip 5mm
\textbf{The main result}
Let $\gamma$ be a Jordan rectifiable curve in $\mathbb{R}^3$. For a continuous
mapping $\Psi:\gamma_\zeta\rightarrow\mathbb{H(C)}$
of the form
\begin{equation}\label{Phi-form}
\Psi(\zeta)=\sum\limits_{k=1}^{4}\Big(U_k(x,y,z)+iV_k(x,y,z)\Big) e_k,
\end{equation}
where $(x,y,z)\in\gamma$ and $U_k:\gamma\rightarrow\mathbb{R}$,
$V_k:\gamma\rightarrow\mathbb{R}$,
we define integrals along a Jordan rectifiable curve $\gamma_\zeta$ by
the equalities
$$\int\limits_{\gamma_\zeta}{d\zeta\,\Psi(\zeta)}:=\sum\limits_{k=1}^{4}{e_k\int
\limits_{\gamma}U_k(x,y,z)dx}+\sum\limits_{k=1}^{4}{i_2e_k\int\limits_{\gamma}
U_k(x,y,z)dy}+$$
$$+\sum\limits_{k=1}^{4}{i_3e_k\int\limits_{\gamma}U_k(x,y,z)dz}+i\sum
\limits_{k=1}^{4}{e_k\int\limits_{\gamma}V_k(x,y,z)dx}+$$
$$+i\sum\limits_{k=1}^{4}{i_2e_k\int\limits_{\gamma}V_k(x,y,z)dy}+i\sum
\limits_{k=1}^{4}{i_3e_k\int\limits_{\gamma}V_k(x,y,z)dz}$$
\noindent and
$$\int\limits_{\gamma_\zeta}{\Psi(\zeta)\,d\zeta}:=\sum\limits_{k=1}^{4}
{e_k\int\limits_{\gamma}U_k(x,y,z)dx}+\sum\limits_{k=1}^{4}{e_ki_2\int
\limits_{\gamma}U_k(x,y,z)dy}+$$
$$+\sum\limits_{k=1}^{4}{e_ki_3\int\limits_{\gamma}U_k(x,y,z)dz}+i\sum
\limits_{k=1}^{4}{e_k\int\limits_{\gamma}V_k(x,y,z)dx}+$$
$$+i\sum\limits_{k=1}^{4}{e_ki_2\int\limits_{\gamma}V_k(x,y,z)dy}+i\sum
\limits_{k=1}^{4}{e_ki_3\int\limits_{\gamma}V_k(x,y,z)dz},$$
where $d\zeta:=dxi_1+dyi_2+dzi_3$.
In the paper \cite{Shp-Kuzm-intteor} for right-$G$-monogenic mappings was obtained the following analogue of the Cauchy integral theorem.
\vskip 2mm
\textbf{Theorem 1 \cite{Shp-Kuzm-intteor}.} \emph{Let $\Phi:\Omega_\zeta\rightarrow\mathbb{H(C)}$ be a
right-$G$-monogenic mapping in a domain $\Omega_\zeta$. Then for every
closed Jordan rectifiable curve $\gamma_\zeta$ homotopic to a point in
$\Omega_\zeta$\,, the following equality is true:
\begin{equation}\label{int=0}
\int\limits_{\gamma_\zeta}d\zeta\,\Phi(\zeta)=0.
\end{equation}}
\vskip 2mm
Below we establish sufficient conditions for the curve $\gamma_\zeta$\, lying on the boundary $\partial\Omega_\zeta$ of a domain $\Omega_\zeta$\, such that the equality (\ref{int=0}) holds. For this goal we apply a scheme of the paper \cite{Pl-Shp-dop} for $G$-monogenic mappings.
Let on a boundary $\partial\Omega_\zeta$ of the domain $\Omega_\zeta$ given closed Jordan rectifiable curve $\gamma_\zeta\equiv\gamma_\zeta(t),$ where $0\leq t\leq1,$ homotopic to an interior point $\zeta_0\in\Omega_\zeta$. It means that there exists the mapping $H(s,t)$ continuous on the square $[0,1]\times[0,1]$, such that $H(0,t)=\gamma_\zeta(t),\, H(1,t)\equiv\zeta_0$, and all curves $\gamma_\zeta^s\equiv\gamma_\zeta^s(t):=\{\zeta=H(s,t):0\leq t\leq1\}$ for $0<s<1$ are contained in the domain $\Omega_\zeta$.
Consider also the curves $\Gamma_\zeta^t\equiv\Gamma_\zeta^t(s):=\{\zeta=H(s,t):0\leq s\leq1\}$. Denote by $\Gamma[\zeta_1,\zeta_2]$ arc of Jordan oriented rectifiable curve, beginning at the point $\zeta_1$ and ending at the point $\zeta_2$, and denote by the mes a linear Lebesgue measure of a rectifiable curve.
As in the paper \cite{Shp-Kuz-umb}, for the element $\zeta=xi_1+yi_2+zi_3$ we define
the Euclidian norm
$$\|\zeta\|=\sqrt{x^2+y^2+z^2}.$$
Using the Theorem of equivalents of norms, for the element
$a:=\sum\limits_{k=1}^{4}(a_{1k}+ia_{2k})e_k$,\, $a_{1k},a_{2k}\in\mathbb{R}$, we have
the following inequalities
$$|a_{1k}+ia_{2k}|\leq\sqrt{\sum\limits_{k=1}^{4}\big(a_{1k}^2+a_{2k}^2\big)}\,\leq c \|a\|,$$
where $c$ is a positive constant does not dependent on $a$.
\vskip 2mm
\textbf{Theorem 2.} \emph{Suppose that $\Phi:\overline{\Omega}_\zeta\rightarrow\mathbb{H(C)}$ is a continuous mapping in the closure $\overline{\Omega}_\zeta$ of a domain $\Omega_\zeta$ and right-$G$-monogenic in $\Omega_\zeta$. Suppose also that $\gamma_\zeta\subset\partial\Omega_\zeta$ is a closed Jordan rectifiable curve homotopic to an interior point $\zeta_0\in\Omega_\zeta$\,, the curves of the family $\{\Gamma_\zeta^t:0\leq t\leq1\}$ are rectifiable and the set} $\{\text{mes}\,\gamma_\zeta^s:0\leq\ s\leq1\}$ \emph{is bounded, then the equality \em(\ref{int=0})\em\, is true.}
\vskip 2mm
\textbf{Proof.} Let $\varepsilon>0$. We fix the number $\rho\in\big(0,\frac{1}{2}\,\text{mes}\,\gamma_\zeta\big)$ such that for arbitrary $\zeta_1,\zeta_2\in\overline{\Omega}_\zeta$ from the condition $||\zeta_1-\zeta_2||<2\rho$ follows the inequality
\begin{equation}\label{teor-8-1}
||\Phi(\zeta_1)-\Phi(\zeta_2)||<\varepsilon.
\end{equation}
Since the mapping $H$ is uniformly continuous on the square $[0,1]\times[0,1]$, then there exists $\delta>0$ such that for all $s\in(0,\delta)$ and $t,t'\in[0,1]:|t-t'|<\delta$ the inequality $|H(0,t)-H(s,t')|<\rho$ is true.
Let numbers $0=t_0<t_1<\ldots<t_n<1$ such that for corresponding points $\zeta_{0,k}:=H(0,t_k)$ of the curve $\gamma_\zeta$\, the following relations are fulfilled
$$\text{mes}\,\gamma_\zeta[\zeta_{0,k},\zeta_{0,k+1}]=\rho \quad \text{for} \quad k=\overline{0,n-1},$$
$$\text{mes}\,\gamma_\zeta[\zeta_{0,n},\zeta_{0,0}]\leq\rho.$$
It is obvious that $2\leq n\leq\left[\frac{\text{mes}\,\gamma_\zeta}{\rho}\right]+1$.
Let us consider the points $\zeta_{s,k}:=H(s,t_k)$ of the curve $\gamma_\zeta^s$ and the curves
$$\Upsilon_{[k]}^s:=\gamma_\zeta[\zeta_{0,k},\zeta_{0,k+1}]\cup\Gamma_\zeta^{t_{k+1}}[\zeta_{0,k+1},\zeta_{s,k+1}]\cup\gamma_\zeta^s[\zeta_{s,k+1},\zeta_{s,k}]\cup\Gamma_\zeta^{t_k}[\zeta_{s,k},\zeta_{0,k}]$$
for $k=\overline{0,n},$ where $\zeta_{s,n+1}:=\zeta_{s,0}$ for $0\leq s\leq1$, setting that the orientation of curves $\Upsilon_{[k]}^s$ is induced by orientation of the curve $\gamma_\zeta.$
Let $s\in(0,\delta).$ Since for all $\zeta\in\Upsilon_{[k]}^s$ the inequality $||\zeta-\zeta_{0,k}||\leq2\rho$ is true, then by Theorem 2 \cite{Shp-Kuzm-intteor}, Lemma 4.1 \cite{Shp-Kuz-umb} and the inequality (\ref{teor-8-1}), we have
$$\Bigg\|\int\limits_{\gamma_\zeta}d\zeta\Phi(\zeta)\Bigg\|=\Bigg\|\sum_{k=0}^n\int\limits_{\Upsilon_{[k]}^s}d\zeta(\Phi(\zeta)-\Phi(\zeta_{0,k}))\Bigg\|\leq$$
$$\leq c\sum\limits_{k=0}^n\int\limits_{\Upsilon_{[k]}^s}||d\zeta||\,||\Phi(\zeta)-\Phi(\zeta_{0,k})||\leq c\varepsilon\sum\limits_{k=0}^n\text{mes}\,\Upsilon_{[k]}^s\leq$$
$$\leq c\varepsilon\Big(\text{mes}\,\gamma_\zeta+\text{mes}\,\gamma_\zeta^s+2(n+1)\max\limits_{k=\overline{0,n}}\,\text{mes}\,\Gamma_\zeta^{t_k}[\zeta_{s,k},\zeta_{0,k}]\Big)\leq$$
\begin{equation}\label{teor-8-2}
\leq M\varepsilon\Bigg(1+\frac{1}{\rho}\max\limits_{k=\overline{0,n}}\,\text{mes}\,\Gamma_\zeta^{t_k}[\zeta_{s,k},\zeta_{0,k}]\Bigg),
\end{equation}
and a constant $M$ does not depend on $\varepsilon$ and $\rho$.
Passing to the limit in the inequality (\ref{teor-8-2}) as $s\rightarrow0$, we have the inequality
$$\Bigg\|\int\limits_{\gamma_\zeta}d\zeta\Phi(\zeta)\Bigg\|\leq M\varepsilon.$$
Now passing to the limit in the last inequality as $\varepsilon\rightarrow0$, we obtain the equality (\ref{int=0}). The Theorem is proved.
\vskip 2mm
The similar statement is true for the left-$G$-monogenic mappings.
\vskip 2mm
\textbf{Theorem 3.} \emph{Suppose that $\widehat{\Phi}:\overline{\Omega}_\zeta\rightarrow\mathbb{H(C)}$ is a continuous mapping in the closure $\overline{\Omega}_\zeta$ of a domain $\Omega_\zeta$ and left-$G$-monogenic in $\Omega_\zeta$. Suppose also that $\gamma_\zeta\subset\partial\Omega_\zeta$ is a closed Jordan rectifiable curve homotopic to an interior point $\zeta_0\in\Omega_\zeta$\,, the curves of the family $\{\Gamma_\zeta^t:0\leq t\leq1\}$ are rectifiable and the set} $\{\text{mes}\,\gamma_\zeta^s:0\leq\ s\leq1\}$ \emph{is bounded, then the following equality is true:}
$$\int\limits_{\gamma_\zeta}\widehat{\Phi}(\zeta)d\zeta=0.$$
\vskip 2mm
\renewcommand{References}{References}
\end{document}
|
\begin{document}
\title{Nonuniqueness of semidirect decompositions for semidirect products with
directly decomposable factors and applications for dihedral groups}
\author{
Peteris\ Daugulis\thanks{Department of Mathematics, Daugavpils
University, Daugavpils, LV-5400, Latvia ([email protected]).
} }
\pagestyle{myheadings} \markboth{P.\ Daugulis}{Nonuniqueness of
semidirect decompositions and applications for dihedral groups}
\maketitle
\begin{abstract} Nonuniqueness of semidirect decompositions of groups is
an insufficiently studied question in contrast to direct
decompositions. We obtain some results about semidirect
decompositions for semidirect products with factors which are
nontrivial direct products. We deal with a special case of
semidirect product when the twisting homomorphism acts diagonally
on a direct product, as well as for the case when the extending
group is a direct product. We give applications of these results
in the case of generalized dihedral groups and classic dihedral
groups $D_{2n}$. For $D_{2n}$ we give a complete description of
semidirect decompositions and values of minimal permutation
degrees.
\end{abstract}
\begin{keywords}
semidirect product, direct product, diagonal action, generalized
dihedral group
\end{keywords}
\begin{AMS}
20E22, 20D40.
\end{AMS}
\section{Introduction}\
\subsection{Background} The aim of this article is
to study semidirect decompositions of groups both in general and
special cases.
By the well known Krull-Remak-Schmidt theorem the multiset of
isomorphism types of indecomposable direct factors for groups
satisfying ascending and descending chain conditions on normal
subgroups does not depend on the order of factors. Thus direct
decompositions of such groups, e.g. finite groups, may be
considered understood.
Few results of this type are known for semidirect and Zappa-Szep
decompositions. One can mention the Schur-Zassenhaus theorem as an
example.
We consider cases when the base group or the extending group is a
direct product. We present a general result which allows to
characterize some semidirect decompositions in the case when the
base group is a direct product and the twisting homomorphism acts
diagonally, Prop.\ref{4}.
Finally we obtain a
nonuniqueness result of semidirect decomposition in the case when
the extending group is a direct product, Prop.\ref{7}. We give
applications of some of these results in the case which is
relatively easy to understand - finite dihedral groups, both
classic and generalized.
We use traditional multiplicative notation for general groups and
additive notation for abelian groups. In this article the dihedral
group of order $m=2n$ is denoted by $D_{m}$ - $D_{m}=\langle a,x|
a^n=e, x^2=e, xax=a^{-1}\rangle$, $m=2n$. For any $m|n$ we
usually identify $\mathbb{Z}_{m}$ with the corresponding subgroup
of $\mathbb{Z}_{n}$. $Q_{m}$ denotes the dicyclic group of order
$m=4k$ - $Q_{m}=\langle a,x|a^{2k}=e, x^2=a^{k},
x^{-1}ax=a^{-1}\rangle$.
The cyclic group of order $m$ is denoted by $\mathbb{Z}_{m}$, in
additive notation we assume that $\mathbb{Z}_{m}=\langle
1\rangle$. In this article we identify elements of
$\mathbb{Z}_{m}$ and corresponding minimal nonnegative integers.
We use this identification for powers of group elements. For
example, if $r\in \mathbb{Z}_{3}$ and $r\equiv 2 (mod\ 3)$, then
$a^r=a^2$ for any group element $a$.
\subsection{Basic facts about semidirect products}\
We remind the reader that an external semidirect product of groups
$N$ (\sl base group\rm) and $H$ (\sl extending group\rm) is the
group $N\rtimes_{\varphi}H=(N\times H,\cdot)$ where the group
product is defined on the Cartesian product $N\times H$ using a
group homomorphism (\sl twisting homomorphism\rm) $\varphi\in
Hom(H, Aut(N))$ as follows: $(n_{1},h_{1})\cdot
(n_{2},h_{2})=(n_{1}\varphi(h_{1})(n_{2}),h_{1}h_{2})$. Sets
$\widetilde{N}=N\times \{e_{H}\}$ and
$\widetilde{H}=\{e_{N}\}\times H$ are subgroups in $N\times H$.
A group $G$ is an internal semidirect product of its subgroups $N$
and $H$ if $N$ is a normal subgroup, $G=NH$ and $N\cap H=\{e\}$.
If a group $G$ is finite then for $G$ to be an internal semidirect
product $NH$ is equivalent to 1) $N$ being normal in $G$, 2)
$|N|\cdot |H|=|G|$ and 3) $N\cap H=\{e\}$. In the internal case
the twisting homomorphism $H\rightarrow Aut(N)$ is given by the
map $h\mapsto (n\rightarrow hnh^{-1})$, for any $n\in N$, $h\in
H$.
Both expressions will be called semidirect decompositions of
$G$. If the twisting homomorphism is not discussed, we omit it and
use the notation $\rtimes$. We consider direct product to be a
special case of semidirect product with the twisting homomorphism
being trivial. For relevant treatment see \cite{R1}, \cite{R2}.
A nontrivial semidirect product may admit more than one semidirect
decomposition. Examples are abundant starting from groups of order
$8$.
\begin{example} Twisting homomorphisms are not given in these examples.
$D_{8}\simeq \mathbb{Z}_{4}\rtimes \mathbb{Z}_{2}\simeq
\mathbb{Z}^{2}_{2}\rtimes \mathbb{Z}_{2}$. $\Sigma_{4}\simeq
A_{4}\rtimes \mathbb{Z}_{2}\simeq \mathbb{Z}^{2}_{2}\rtimes
\Sigma_{3}$.
There are semidirect products such that $\mathbb{Z}_{3}\rtimes
Q_{8}\simeq Q_{24}$, but $Q_{8}\rtimes \mathbb{Z}_{3}\simeq
SL(2,\mathbb{F}_{3})$. On the other hand, there is a group
$G_{32}$ of order $32$, such that $G_{32}\simeq D_{8}\rtimes
\mathbb{Z}^{2}_{2}\simeq \mathbb{Z}^{2}_{2}\rtimes D_{8}$.
Finally, there is a group $G_{24}$ of order $24$ which can be
decomposed in $5$ different ways: $G_{24}\simeq
\mathbb{Z}_{3}\rtimes D_{8}\simeq \mathbb{Z}^{2}_{2}\rtimes
\mathbb{Z}_{3}\simeq D_{12}\rtimes \mathbb{Z}_{2}\simeq
(\mathbb{Z}_{3}\times \mathbb{Z}^{2}_{2})\rtimes
\mathbb{Z}_{2}\simeq Q_{12}\rtimes \mathbb{Z}_{2}$.
\end{example}
\section{Main results}
\subsection{Diagonal semidirect products}
\subsubsection{Automorphisms of direct products}\label{5}
We introduce a linear algebra style notation for direct products
of groups.
Let $G=G_{1}\times G_{2}$. Encode the element $(g_{1},g_{2})$ as a
column $\left[
\begin{array}{c}
g_{1} \\
\hline
g_{2} \\
\end{array}
\right] $. If $\varphi\in Aut(G)$, then $\varphi(
\left[
\begin{array}{c}
g_{1} \\
\hline
g_{2} \\
\end{array}
\right])=
\left[
\begin{array}{c}
\varphi_{1}(g_{1},g_{2}) \\
\hline
\varphi_{2}(g_{1},g_{2}) \\
\end{array}
\right] $. One can theck, that for all relevant parameter values
$\varphi_{i}$ satisfy the following properties:
\begin{enumerate}
\item $\varphi_{i}(ab,e)=\varphi_{i}(a,e)\varphi_{i}(b,e)$,
\item $\varphi_{i}(e,ab)=\varphi_{i}(e,a)\varphi_{i}(e,b)$,
\item
$\varphi_{i}(a,b)=\varphi_{i}(a,e)\varphi_{i}(e,b)=\varphi_{i}(e,b)\varphi_{i}(a,e)$,
\end{enumerate}
Define $\varphi_{11}(g_{1})=\varphi_{1}(g_{1},e)$,
$\varphi_{12}(g_{2})=\varphi_{1}(e,g_{2})$,
$\varphi_{21}(g_{1})=\varphi_{2}(g_{1},e)$,
$\varphi_{22}(g_{2})=\varphi_{2}(e,g_{2})$, for all $g_{i}\in
G_{i}$. All functions $\varphi_{ij}$ are group homomorphisms. Thus
$\varphi_{i}(g_{1},g_{2})=\varphi_{i}(g_{1},e)\varphi_{i}(e,g_{2})=\varphi_{i1}(g_{1})\varphi_{i2}(g_{2})$.
We can encode action of $\varphi$ as follows: $ \varphi(
\left[
\begin{array}{c}
g_{1} \\
\hline
g_{2} \\
\end{array}
\right])=
\left[
\begin{array}{c|c}
\varphi_{11}(g_{1}) & \varphi_{12}(g_{2}) \\
\hline
\varphi_{21}(g_{1}) & \varphi_{22}(g_{2}) \\
\end{array}
\right]
$. Thus an automorphism $\varphi\in Aut(G_{1}\times G_{2})$ is
determined by $4$ group homomorphisms $\varphi_{ij}:G_{j}\rightarrow
G_{i}$.
\begin{definition}
We call $\varphi\in Aut(G_{1}\times G_{2})$, $G_{1}\neq \{e\}$, $G_{2}\neq \{e\}$, a \sl diagonal
automorphism\rm\ if $\varphi_{12}$ and $\varphi_{21}$ are
trivial homomorphisms.
\end{definition}
\begin{definition}
We call $(G_{1}\times G_{2})\rtimes_{\varphi} H$ a \sl diagonal
semidirect product\rm\ if $\varphi(h)$ is a diagonal $G_{1}\times
G_{2}$-automorphism for any $h\in H$. Explicitly, there are group
homomorphisms $\varphi_{ii}(h):G_{i}\rightarrow G_{i}$ such that
$\varphi(h)(g_{1},g_{2})=(\varphi_{11}(h)(g_{1}),\varphi_{22}(h)(g_{2}))$.
\end{definition}
\begin{remark} $Ker(\varphi)=Ker(\varphi_{11})\cap Ker(\varphi_{22})$.
Note that $G_{i}$ may not be indecomposable as direct factors.
Described encodings and diagonal semidirect products can be
generalized to cases when the base groups splits into an arbitrary
finite number of direct factors. Similar encodings can be used
considering internal semidirect products.
\end{remark}
\subsubsection{Semidirect decompositions of diagonal semidirect products}\
We present a proposition showing nonuniqueness of semidirect
decomposition for diagonal semidirect products. Va\-gue\-ly
speaking, any direct factor of the base group which is invariant
with respect to the initial twisting homomorphism can be moved to
the extending group (nonnormal semidirect factor) to enlarge it.
The new twisting homomorphism is such that the moved direct factor
acts trivially on the remaining part of the base group.
\begin{proposition}\label{4} Let $N_{1},N_{2},H$ be groups. Let $G=(N_{1}\times N_{2})\rtimes_{\varphi}H$
be a diagonal semidirect product,
$\varphi(h)(g_{1},g_{2})=(\varphi_{11}(h)(g_{1}),\varphi_{22}(h)(g_{2}))$.
Then the following statements hold.
\begin{enumerate}
\item $G\simeq N_{1}\rtimes_{\Phi_{11}}
(N_{2}\rtimes_{\varphi_{22}}H)$, for some $\Phi_{11}\in
Hom(N_{2}\rtimes_{\varphi_{22}} H,Aut(N_{1}))$.
\item
$Ker(\Phi_{11})=\widetilde{N_{2}}\widetilde{Ker(\varphi_{11})}$.
\item If $\varphi_{11}(h)=id_{N_{1}}$, for any $h\in H$, i.e.
$\varphi(h)(
\left[
\begin{array}{c}
g_{1} \\
\hline
g_{2} \\
\end{array}
\right] )=
\left[
\begin{array}{c|c}
g_{1} & e \\
\hline
e & \varphi_{22}(h)(g_{2}) \\
\end{array}
\right] $, then $G\simeq N_{1}\times (N_{2}\rtimes_{\varphi_{22}}
H).$
\end{enumerate}
\end{proposition}
\begin{proof}
1. Consider $N_{1}\rtimes_{\Phi_{11}} (N_{2}\rtimes_{\varphi_{22}}
H)$ where $\Phi_{11}(n_{2},h)=\varphi_{11}(h)$. It is directly
checked that $\Phi_{11}\in Hom(N_{2}\rtimes H,Aut(N_{1}))$. We
will prove that
$$(N_{1}\times N_{2})\rtimes_{\varphi} H \simeq
N_{1}\rtimes_{\Phi_{11}} (N_{2}\rtimes_{\varphi_{22}} H).$$ Define
a bijective map $f:(N_{1}\times N_{2})\rtimes_{\varphi}
H\rightarrow N_{1}\rtimes_{\Phi_{11}} (N_{2}\rtimes_{\varphi_{22}}
H)$ by $f((n_{1},n_{2}),h)=(n_{1},(n_{2},h))$, for all $n_{i}\in
N_{i}$, $h\in H$. We prove that $f$ is a group homomorphism.
Let $a,a'\in N_{1}$, $b,b'\in N_{2}$, $h,h'\in H$. We have that
$((a,b),h)\cdot ((a',b'),h')=((a,b)\varphi(h)(a',b'),hh')=$
$=((a,b)(\varphi_{11}(h)(a'),\varphi_{22}(h)(b')),hh')=((a\varphi_{11}(h)(a'),b\varphi_{22}(h)(b')),hh').$
On the other hand,
$(a,(b,h))\cdot (a',(b',h'))=(a\Phi_{11}(b,h)(a'),(b,h)\cdot
(b',h'))=$
$=(a\varphi_{11}(h)(a'),(b\varphi_{22}(h)(b'),hh'))$. We see that
$f$ is a group isomorphism.
2. $Ker(\Phi_{11})=\{(n_{2},h)|h\in
Ker(\varphi_{11})\}=\widetilde{N_{2}}\widetilde{Ker(\varphi_{11})}$.
3. In notations given above, $\varphi_{11}(h)=id_{N_{1}}$ implies
$\Phi_{11}(n_{2},h)=id_{N_{1}}$, for any $n_{2}\in N_{2}$, $h\in
H$. Thus
it is the direct product.
\end{proof}
\begin{example}
Let $G=(\mathbb{Z}_{7}\times
\mathbb{Z}_{9})\rtimes_{\varphi}\mathbb{Z}_{3}$, where
$\varphi(1)(
\left[
\begin{array}{c}
g_{1} \\
\hline
g_{2} \\
\end{array}
\right] )=
\left[
\begin{array}{c|c}
g^{2}_{1}& e \\
\hline
e& g^{4}_{2} \\
\end{array}
\right]$. In additive notation this can be simplified as follows.
$\varphi(1)(
\left[
\begin{array}{c}
g_{1} \\
\hline
g_{2} \\
\end{array}
\right] )=
\left[
\begin{array}{c|c}
2g_{1}& 0 \\
\hline
0& 4g_{2} \\
\end{array}
\right]=
\left[
\begin{array}{c|c}
2 & 0 \\
\hline
0 & 4 \\
\end{array}
\right]
\left[
\begin{array}{c}
g_{1} \\
\hline
g_{2} \\
\end{array}
\right]
$. $G$ can be defined as the subgroup of $\Sigma_{16}$ generated
by three permutations:
\begin{enumerate}
\item[a)]
$(1,...,7)$ (generating $\mathbb{Z}_{7}$),
\item[b)] $(8,...,16)$ (generating
$\mathbb{Z}_{9}$) and
\item[c)]
$\underbrace{(1,2,4)(3,6,5)}_{\mathbb{Z}_{7}}\underbrace{(8,11,14)(9,15,12)}_{\mathbb{Z}_{9}}$
(generating action of $\mathbb{Z}_{3}$ on $\mathbb{Z}_{7}\times
\mathbb{Z}_{9}$).
\end{enumerate}
We have that $G\simeq \mathbb{Z}_{7}\rtimes
(\mathbb{Z}_{9}\rtimes_{4}
\mathbb{Z}_{3})\simeq \mathbb{Z}_{9}\rtimes (\mathbb{Z}_{7}\rtimes_{2} \mathbb{Z}_{3})$.
\end{example}
\subsection{Directly decomposable extending groups}\
We show that a direct factor of the extending group can be
transferred to the base group.
\begin{proposition} \label{7} Let $N,H_{1},H_{2}$ be groups Then $$N\rtimes_{\varphi} (H_{1}\times H_{2})\simeq
(N\rtimes_{\varphi_{1}} H_{1})\rtimes_{\varphi_{2}} H_{2},$$ where
$\varphi_{1}(h_{1})(n)=\varphi(h_{1},e_{H_{2}})(n)$ and
$\varphi_{2}(h_{2})(n,h_{1})=(\varphi(e_{H_{1}},h_{2})(n),h_{1})$,
for all $n\in N$, $h_{i}\in H_{i}$.
\end{proposition}
\begin{proof} It is checked that $\varphi_{i}$ are group homomorphisms.
We prove that the map $f:N\rtimes (H_{1}\times
H_{2})\longrightarrow (N\rtimes H_{1})\rtimes H_{2}$ given by
$f(n,(h_{1},h_{2}))=((n,h_{1}),h_{2})$ is a group homomorphism.
Let $n,n'\in N$, $h_{i},h'_{i}\in H_{i}$.
Consider the product $(n,(h_{1},h_{2}))\cdot (n',(h'_{1},h'_{2}))$
in $N\rtimes_{\varphi}(H_{1}\times H_{2})$:
$(n,(h_{1},h_{2}))\cdot
(n',(h'_{1},h'_{2}))=(n\varphi(h_{1},h_{2})(n'),(h_{1}h'_{1},h_{2}h'_{2}))$.
Consider the product $((n,h_{1}),h_{2})\cdot ((n',h'_{1}),h'_{2})$
in $(N\rtimes H_{1})\rtimes H_{2}$:
$ ((n,h_{1}),h_{2})\cdot ((n',h'_{1}),h'_{2})=
(((n,h_{1})\varphi_{2}(h_{2})(n',h'_{1})),h_{2}h'_{2})=$
$=(((n,h_{1})(\varphi(e,h_{2})(n'),h'_{1})),h_{2}h'_{2})=
((n\varphi_{1}(h_{1})(\varphi(e,h_{2})(n')),h_{1}h'_{1}),h_{2}h'_{2})=$
$=((n\varphi(h_{1},h_{2})(n'),h_{1}h'_{1}),h_{2}h'_{2})$.
We see that both products have equal corresponding components and
thus $f$ is a group isomorphism.
\end{proof}
\begin{example} Let $G=\mathbb{Z}_{7}\rtimes_{\varphi} (\mathbb{Z}_{2}\times
\mathbb{Z}_{3})$ where $\varphi(x,y)(1)\equiv (-1)^{x}2^{y} (mod\
7)$. $G$ can be defined as the subgroup of $\Sigma_{7}$ generated
by three permutations:
\begin{enumerate}
\item[a)]
$(1,...,7)$ (generating $\mathbb{Z}_{7}$),
\item[b)] $(1,6)(2,5)(3,4)$ (generating
action of $\mathbb{Z}_{2}$ on $\mathbb{Z}_{7}$) and
\item[c)]
$(1,2,4)(3,6,5)$ (generating
action of $\mathbb{Z}_{3}$ on $\mathbb{Z}_{7}$).
\end{enumerate}
Then $G\simeq D_{2\cdot 7}\rtimes \mathbb{Z}_{3}\simeq
(\mathbb{Z}_{7}\rtimes \mathbb{Z}_{3})\rtimes \mathbb{Z}_{2}$.
\end{example}
\subsection{Applications}\
\subsubsection{Generalized dihedral groups}\
We remind the reader that an external semidirect product
$D(A)=A\rtimes_{\varphi} \mathbb{Z}_{2}$ is called \sl generalized
dihedral group\rm\ provided 1) $A$ is abelian and 2) $\varphi(1)(g)=-g$ for any
$g\in A$, in additive notation. We can also denote $D(A)$ by
$A\rtimes_{-1}\mathbb{Z}_{2}$.
Using the well known classification of finite abelian groups we
can assume that $A=\bigoplus_{i=1}^{n}\mathbb{Z}_{m_{i}}$. We use
linear algebra style encoding - we encode $(g_{1},...,g_{n})\in A$
as a column vector $
\left[
\begin{array}{cc}
g_{1} \\
\hline
... \\
\hline
g_{n} \\
\end{array}
\right] $. Notations introduced in section \ref{5} are modified
for additive group notation. The action of the twisting
homomorphism is given by scalar or matrix multiplication:
$\varphi(1)(
\left[
\begin{array}{cc}
g_{1} \\
\hline
... \\
\hline
g_{n} \\
\end{array}
\right] )=
\left[
\begin{array}{c|c|c}
(-g_{1}) & 0 & 0 \\
\hline
0 & ... & 0 \\
\hline
0 & 0 & (-g_{n}) \\
\end{array}
\right]= -
\left[
\begin{array}{cc}
g_{1} \\
\hline
... \\
\hline
g_{n} \\
\end{array}
\right]=
(-\textbf{E}_{n})\cdot
\left[
\begin{array}{cc}
g_{1} \\
\hline
... \\
\hline
g_{n} \\
\end{array}
\right] $, where $\textbf{E}_{n}$ is the $n\times n$ identity
matrix.
\begin{remark} Generalized dihedral groups are diagonal semidirect
products with an injective twisting homomorphism.
\end{remark}
\begin{proposition} Let $A=\bigoplus_{i=1}^{n}\mathbb{Z}_{m_{i}}$, let $A=A_{1}\oplus
A_{2}$, where $A_{1}=\bigoplus_{i=1}^{n_{1}}\mathbb{Z}_{m_{i}}$,
$A_{2}=\bigoplus_{i=n_{1}+1}^{n}\mathbb{Z}_{m_{i}}$. Then
$$D(A)\simeq A_{1}\rtimes (A_{2}\rtimes_{-1}\mathbb{Z}_{2})=A_{1}\rtimes D(A_{2})\simeq A_{1}\rtimes D(A/A_{1}).$$
\end{proposition}
\begin{proof}
$D(A)=(A_{1}\oplus A_{2})\rtimes_{\varphi}\mathbb{Z}_{2}$, where
$\varphi(1)(g)=-g$, for any $g\in A$. Thus
$\varphi(g_{1},g_{2})=(-g_{1},-g_{2})$, for any $g_{i}\in G_{i}$.
It follows that $D(A)$ is a diagonal semidirect product with
respect to $A_{1}\oplus A_{2}$ decomposition. According to
Proposition \ref{4} we have that $D(A)\simeq
A_{1}\rtimes_{\Phi_{11}}(A_{2}\rtimes_{\varphi_{22}}
\mathbb{Z}_{2})=A_{1}\rtimes D(A_{2})$, where
$\Phi_{11}(g_{2},1)(g_{1})=\varphi_{11}(1)(g_{1})=-g_{1}$.
\end{proof}
\begin{example}
Let $G=D(\mathbb{Z}_{3}\oplus \mathbb{Z}_{5})$. $G$ can be defined
as a subgroup of $\Sigma_{8}$ generated by permutations $(1,2,3)$,
$(4,5,6,7,8)$ and $(1,2)(4,7)(5,6)$.
Then $G\simeq
\mathbb{Z}_{3}\rtimes D_{2\cdot 5}\simeq \mathbb{Z}_{5}\rtimes
D_{2\cdot 3}$.
\end{example}
\subsubsection{Dihedral groups}\
Classic dihedral groups are special cases of generalized dihedral
groups when the base group is a cyclic group. We give a complete
description of semidirect decompositions of $D_{2n}$ using both
Proposition \ref{4} and ad hoc computations.
We use a classical presentation of dihedral groups:
$$D_{2n}=\langle a,x| a^n=e, x^2=e, xax=a^{n-1}\rangle = \langle
a\rangle \cup \langle a\rangle x. $$
We note that $D_{2}\simeq \mathbb{Z}_{2}$ and $D_{4}\simeq
\mathbb{Z}_{2}\times \mathbb{Z}_{2}$, in all other cases $D_{2n}$
is nonabelian.
\paragraph{Subgroups}
Let $n\in \mathbb{N}$, $n\ge 3$, $d\in \mathbb{N}$, $d|n$,
$m=\frac{n}{d}$. It is known that $D_{2n}$ has the following
subgroups, see \cite{C}.
\begin{enumerate}
\item For each $m\in \mathbb{N}$ such that $m|n$ there is a
subgroup
$$A_{m}=\langle a^{\frac{n}{m}}\rangle = \langle a^{d} \rangle =
\{e,a^{d},a^{2d},...,a^{(m-1)d}\}\simeq \mathbb{Z}_{m}.$$
$A_{m}\trianglelefteq D_{2n}$ for all $m$. The number of such
subgroups is $d(n)$ (the number of natural $n$-divisors).
\item For each $m\in \mathbb{N}$ such that $m|n$ and each $r\in
\mathbb{Z}_{\frac{n}{m}}=\mathbb{Z}_{d}$ there is a subgroup
$$B_{2m,r}=\langle a^{\frac{n}{m}},a^{r}x\rangle = \langle
a^{d},a^{r}x\rangle =\langle A_{m}, A_{m}(a^rx)\rangle \simeq
D_{2m}.$$
Note that $r\in \mathbb{Z}_{\frac{n}{m}}$ is identified with an
integer as described in the introduction.
The number of such subgroups is $\sigma(n)$ (the sum of natural
$n$-divisors).
If $2|n$ then $B_{n,r}\trianglelefteq D_{2n}$. In all other
cases, if $1<m<n$ then $B_{2m,r}\not \trianglelefteq D_{2n}$.
\end{enumerate}
\paragraph{Classical decompositions} It known that
$D_{2n}\simeq \mathbb{Z}_{n} \rtimes_{\varphi} \mathbb{Z}_{2}$
where the twisting homomorphism is $\varphi(1)(g)=-g$. In internal
terms, $D_{2n}=A_{n}\rtimes B_{2,r}$, for all $r\in
\mathbb{Z}_{n}$. If $2|n$ and $4\not | n$, then $D_{2n}\simeq
D_{n}\times \mathbb{Z}_{2}$, or, in internal terms,
$D_{2n}=B_{n,r}\times A_{2}$. where $r\in \mathbb{Z}_{2}$. Again,
note, that second indices of $B$-type subgroups can be interpreted
as both integers and residues.
\paragraph{External semidirect decompositions of $D_{2n}$}Using Proposition \ref{4} we get an exaustive description of
external semidirect decompositions of $D_{2n}$.
\begin{proposition} \label{2}
\begin{enumerate}
\item $D_{2n}\simeq \mathbb{Z}_{m}\rtimes_{\varphi}
D_{\frac{2n}{m}}$, for any $m\in \mathbb{N}$, $m|n$, such that
$GCD(m,\frac{n}{m})=1$. $\varphi$ is defined as follows: if
$D_{\frac{2n}{m}}=\langle
a,x|a^{\frac{n}{m}}=e,x^2=e,xax=a^{-1}\rangle$ then
$\varphi(a)(1)=1$ and $\varphi(x)(1)=-1$.
\item $D_{2n}\simeq D_{n}\rtimes_{\varphi} \mathbb{Z}_{2}$, if
$n=2^{\alpha}q$, $\alpha\in \mathbb{N}$. $\varphi$ is defined as
follows: if $D_{n}=\langle
a,x|a^{\frac{n}{2}}=e,x^2=e,xax=a^{-1}\rangle$ then
$\varphi(1)(a)=a^{-1}$ and $\varphi(1)(x)=ax$.
\item If $2|n$ and $4\not| n$ then $$D_{2n}\simeq D_{n}\times
\mathbb{Z}_{2}.$$
\item There are no other nontrivial external semidirect
decompositions of $D_{2n}$ in the following sense. If
$D_{2n}\simeq X\rtimes Y$, $|X|>1$, $|Y|>1$, then there are two
poosibilities:
\begin{enumerate}
\item[a)] $X=\mathbb{Z}_{m}$ and $Y=D_{\frac{2n}{m}}$, where
$m|n$, $GCD(m,\frac{n}{m})=1$ or
\item[b)] $X=D_{n}$ and $Y=\mathbb{Z}_{2}$, if $2|n$.
\end{enumerate}
\end{enumerate}
\end{proposition}
\begin{proof} Statements 1.,2.,3. are proved by exhibiting a suitable
internal semidirect decomposition.
1. We use the primary decomposition theorem for cyclic groups: if
$n=\prod_{i=1}^{k}p^{\alpha_{i}}_{i}$, then $\mathbb{Z}_{n}\simeq
\bigoplus_{i=1}^{k}\mathbb{Z}_{p^{\alpha_{i}}_{i}}$. The statement
follows from Proposition \ref{4}. Note that $Ker(\varphi)=\langle
a \rangle$.
Alternatively, we prove the same statement using the information
about $D_{2n}$-subgroups. We show that if $GCD(m,\frac{n}{m})=1$
then $D_{2n}=A_{m}\rtimes B_{\frac{2n}{m},r}$.
We have that $A_{m}\trianglelefteq D_{2n}$ and $|A_{m}|\cdot
|B_{\frac{2n}{m},r}|=2n=|D_{2n}|$. $A_{m}\cap B_{\frac{2n}{m}}\le
\langle a^{\frac{n}{m}}\rangle$. Considering subgroups of $\langle
a^{\frac{n}{m}} \rangle$ it follows that $A_{m}\cap
B_{\frac{2n}{m}}=\{e\}$. Thus $D_{2n}=A_{m}\rtimes
B_{\frac{2n}{m},r}\simeq \mathbb{Z}_{m}\rtimes_{\varphi}
D_{\frac{2n}{m}}$. A direct computation shows that $\varphi$ is as
stated: $(a^{m})a^{d}(a^{-m})=a^{d}$,
$(a^{r}x)a^{d}(a^{r}x)=a^{-d}$.
Note that if $2|n$ and $4\not |n$ then $A_{2}\cap B_{n,r}=\{e\}$,
$r\in \mathbb{Z}_{2}$, hence $D_{2n}=A_{2}\times B_{n,r}\simeq
\mathbb{Z}_{2}\times D_{n}$. In this case there are no nontrivial
semidirect decompositions of type $\mathbb{Z}_{2}\rtimes D_{n}$.
2. This case is not covered by Proposition \ref{4}, we show
directly that $D_{2n}=B_{n,0}\rtimes B_{2,1}$.
If $2|n$, then $B_{n,0}\trianglelefteq D_{2n}$. $|B_{n,0}|\cdot
|B_{2,1}|=|D_{2n}|$. It can be checked that $B_{n,0}\cap
B_{2,1}=\{e\}$: $B_{n,0}=\langle a^2, x\rangle$, $B_{2,1}=\langle
ax \rangle$.
Thus $D_{2n}\simeq D_{n}\rtimes \mathbb{Z}_{2}$. A direct
computation shows that $\varphi$ is as stated: $(ax)a^2$ $(ax)=$
$a^{-2}$ (the generator $a^2$ gets inverted), $(ax)x(ax)=a^2x$
(the generator $x$ gets multiplied by the other generator $a^2$).
3. Using Proposition \ref{4} we see that
$D_{2n}=D(\mathbb{Z}_{n})=(\mathbb{Z}_{2}\oplus ...)\rtimes
\mathbb{Z}_{2}\simeq \mathbb{Z}_{2}\times
D(\mathbb{Z}_{n}/\mathbb{Z}_{2})\simeq \mathbb{Z}_{2}\times
D_{n}$.
It can also be proved using the list of subgroups. We remind that
$D_{2n}=B_{n,0}\times A_{2}\simeq D_{n}\times \mathbb{Z}_{2}$ for
the following reasons. Both subgroups are normal. $|B_{n,0}|\cdot
|A_{2}|=|D_{2n}|$. $B_{n,0}=\langle a^2, x\rangle$, $A_{2}=\langle
a^{\frac{n}{2}}\rangle$, $\frac{n}{2}$ is odd, therefore
$B_{n,0}\cap A_{2}=\{e\}$.
4. Consider all possible internal semidirect decompositions of
$D_{2n}$.
If $D_{2n}=X\rtimes Y$ then $X$ must be a normal subgroup of
$D_{2n}$ therefore $X$ must be $A_{m}$ or $B_{n,r}$ with $2|n$.
If $X=A_{m}$ then $Y$ must be $B_{m',r}$ ir order to generate
$D_{2n}$, with $m'=\frac{2n}{m}$. $A_{m}\cap
B_{\frac{2n}{m},r}=\{e\}$ iff $GCD(n,\frac{n}{m})=1$.
Let $X=B_{n,r}$ with $2|n$, $r\in \mathbb{Z}_{2}$. There are $n+1$
subgroups of $D_{2n}$ having order $2$: $B_{2,r}\ $, $r\in
\mathbb{Z}_{n}$ and $A_2=\langle a^{\frac{n}{2}}\rangle$. For any
$n$ such that $2|n$ this gives a semidirect decomposition of type
$D_{n}\rtimes \mathbb{Z}_{2}$. If $4\not |n$ then $A_{2}\cap
B_{n,r}=\{e\}$ which gives a direct decomposition $D_{n}\times
\mathbb{Z}_2$.
\end{proof}
\begin{remark} In terms of prime factorization the condition $GCD(m,\frac{n}{m})=1$ is
equivalent to the fact that $m$ and $\frac{n}{m}$ are products of
full prime powers of the prime factorization of $n$. Existence of
many members of this family also follows from Schur-Zassenhaus
theorem. If $m|n$ and $GCD(m,\frac{n}{m})=1$ then
$GCD(|A_{m}|,|D_{2n}/A_{m}|)=1$, $D_{2n}/A_{m}\simeq
D_{\frac{2n}{m}}$ and, hence $D_{2n}\simeq A_{m}\rtimes
D_{\frac{2n}{m}}$.
\end{remark}
\begin{remark} Note that there are at most $2$ external semidirect decompositions when $n$ is a
prime power:
\begin{enumerate}
\item if $n=p^{\alpha}$, $p$ an odd prime, then there is only one
(classical) external semidirect decomposition:
$D_{2p^{\alpha}}\simeq \mathbb{Z}_{p^{\alpha}}\rtimes
\mathbb{Z}_{2}$,
\item if $n=2^{\alpha}$, $\alpha\ge 3$, then there are two
external semidirect decompositions: $D_{2\cdot 2^{\alpha}}\simeq
\mathbb{Z}_{2^{\alpha}}\rtimes \mathbb{Z}_{2}\simeq
D_{2^{\alpha}}\rtimes \mathbb{Z}_{2}$.
\end{enumerate}
\end{remark}
\begin{remark}
The image of the twisting homomorphism in each case of a proper
semidirect product is isomorphic to $\mathbb{Z}_{2}$. If the
extending group is not $\mathbb{Z}_{2}$, then the twisting
homomorphism is not injective.
\end{remark}
\begin{example}
External semidirect decompositions of $D_{2\cdot 30}$:
$D_{60}\simeq \mathbb{Z}_{30}\rtimes \mathbb{Z}_{2}\simeq
\mathbb{Z}_{6}\rtimes D_{10}\simeq \mathbb{Z}_{10}\rtimes
D_{6}\simeq\mathbb{Z}_{15}\rtimes D_{4}\simeq\mathbb{Z}_{3}\rtimes
D_{20}\simeq \mathbb{Z}_{5}\rtimes D_{12}\simeq$
$\simeq D_{30} \rtimes \mathbb{Z}_{2}\simeq D_{30}\times
\mathbb{Z}_{2}$.
\end{example}
\paragraph{Internal semidirect decompositions of $D_{2n}$} We now describe all internal semidirect decompositions of
$D_{2n}$.
\begin{proposition} Let $n\in \mathbb{N}$.
\begin{enumerate}
\item If $m\in \mathbb{N}$, $m|n$, is such that
$GCD(m,\frac{n}{m})=1$, then $$D_{2n} = A_{m}\rtimes
B_{\frac{2n}{m},r},$$ for all $r\in \mathbb{Z}_{m}$.
\item If $n=2^{\alpha}q$, $\alpha\in \mathbb{N}$, then
$$D_{2n}=B_{n,0}\rtimes B_{2,r_{1}}=B_{n,1}\rtimes B_{2,r_{0}}$$
where $r_{i}\in \mathbb{Z}_{n}$, $r_{i}\equiv i(mod\ 2)$.
\item If $2|n$ and $4\not n$ then $D_{2n}=B_{n,0}\times A_{2}$ and
$D_{2n}=B_{n,1}\times A_{2}$.
\item There are no other internal semidirect decompositions of
$D_{2n}$.
\end{enumerate}
\end{proposition}
\begin{proof}
1. We look for internal semidirect decompositions of $D_{2n}$ in
form $A_{m}\rtimes B_{m',r}$. We must have $m'=\frac{2n}{m}$ and
$r\in \mathbb{Z}_{m}$. $A_{m}\cap B_{\frac{2n}{m},r}=\{e\}$ iff
$GCD(m,\frac{n}{m})=1$. Thus $D_{2n}=A_{m}\rtimes
B_{\frac{2n}{m},r}$ for all $m$ such that $GCD(m,\frac{n}{m})=1$
and all $r\in \mathbb{Z}_{m}$ are the only possible decompositions
of this kind.
2. We look for internal semidirect decompositions of $D_{2n}$ in
form $B_{m,r}\rtimes B_{m',r'}$. We must have
$B_{m,r}\trianglelefteq D_{2n}$ therefore $2|n$, $m=n$ and $r\in
\mathbb{Z}_{2}$, thus we have two possible decomposition series:
$B_{n,0}\rtimes B_{2,r'}$ and $B_{n,1}\rtimes B_{2,r''}$. To
ensure trivial intersections of semidirect factors we must have
$r'\equiv 1(mod\ 2)$ and $r''\equiv 0(mod\ 2)$.
3. If $2|n$ and $4\not n$ then $B_{n,0}\cap A_{2}=B_{n,1}\cap
A_{2}=\{e\}$ where all subgroups are normal.
4. It follows from the previous arguments.
\end{proof}
\paragraph{Permutation representations of dihedral groups}
Finally we find minimal degrees of faithful permutation
representations of $D_{2n}$. If $n$ is not a prime power then
these numbers are smaller than degrees of classical permutation
representations of dihedral groups. This is a consequence of
\ref{2} and Karpilovsky bounds for finite abelian groups \cite{K}.
Let $\mu(G)$ be the minimal faithful permutation representation
degree of $G$, i.e. the minimal $n\in \mathbb{N}$ such that there
is an injective group homomorphism $G\rightarrow \Sigma_{n}$. It
is known that for finite groups $G,H$ and a group homomorphism
$\varphi:H\rightarrow Aut(G)$ we have that $\mu(G\rtimes_{\varphi}
H)\le |G|+\mu(H)$. If, additionally, $\varphi$ is injective, then
$\mu(G\rtimes_{\varphi} H)\le |G|$.
\begin{proposition} Let $n=\prod_{i}p_{i}^{\alpha_{i}}$ be the prime
factorization of $n\in \mathbb{N}$. Then $\mu(D_{2n})=
\sum_{i}p_{i}^{\alpha_{i}}$.
\end{proposition}
\begin{proof} First we prove that
\begin{equation} \label{8} \mu(D_{2n})\le \sum_{i}p_{i}^{\alpha_{i}}.
\end{equation}
By statement 1. of \ref{2} we have that $D_{2n}\simeq
\mathbb{Z}_{p_{1}^{\alpha_{1}}}\rtimes D_{2n_{1}}$, where
$n_{1}=\frac{n}{p_{1}^{\alpha_{1}}}$. Thus $\mu(D_{2n})\le
p_{1}^{\alpha_{1}}+\mu(D_{2n_{1}})$. \ref{8} follows by induction
in $i$ using injectivity of the twisting homomorphism at the last
step.
To prove the opposite inequality and the statement, we note that
$D_{2n}\simeq \mathbb{Z}_{n}\rtimes \mathbb{Z}_{2}$, thus
$\mathbb{Z}_{n}\le D_{2n}$. It implies $\mu(\mathbb{Z}_{n})\le
\mu(D_{2n})$, therefore $\sum_{i}p_{i}^{\alpha_{i}}\le
\mu(D_{2n})$ by the Karpilovsky theorem for abelian groups
\cite{K}.
\end{proof}
\begin{example} $\min_{n\in \mathbb{N}}\{n: \mu(D_{2n})<n\}=6$: $\mu(D_{2\cdot 6})=5$, $D_{2\cdot 6}$ can be generated by $(1,2,3)$, $(1,2)$, $(4,5)$.
If, additionally, $D_{2n}$ is directly indecomposable, then the
minimum is $12$: $\mu(D_{2\cdot 12})=7$, $D_{2\cdot 12}$ can be
generated by $(1,2,3,4)$, $(5,6,7)$, $(1,3)(5,6)$.
\end{example}
\section{Conclusion} We have obtained results showing possibility of
various semidirect decompositions of a given semidirect product in
two cases: 1) if the original twisting homomorphism is diagonal
and the base group is directly decomposable and 2) if the
extending group is directly decomposable. These results may
stimulate further interest in looking for analogues of
Krull-Remak-Schmidt theorem type results for semidirect and
Zappa-Szep products.
We have presented semidirect decompositions of generalized
dihedral groups and classical dihedral groups as an application.
Apart from semidirect decompositions guarranteed by the general
proposition \ref{4}, for $D_{2n}$ there are additional
decompositions of external type $D_{n}\rtimes \mathbb{Z}_{2}$ if
$2|n$.
Semidirect decompositions of dihedral groups give the exact value
of $\mu(D_{2n})$.
\end{document}
|
\begin{document}
\title{$C^0$-limits of Legendrian Submanifolds}
\begin{abstract}
Laudenbach and Sikorav proved that closed, half-dimensional non-Lagrangian submanifolds of symplectic manifolds are immediately displaceable as long as there is no topological obstruction. From this they deduced that under certain assumptions the $C^0$-limit of a sequence of Lagrangian submanifolds is again Lagrangian, provided that the limit is smooth.
In this note we extend Laudenbach and Sikorav's ideas to contact manifolds. We prove correspondingly that certain non-Legendrian submanifolds of contact manifolds can be displaced immediately without creating short Reeb chords as long as there is no topological obstruction. From this it will follow that under certain assumptions the $C^0$-limit of a sequence of Legendrian submanifolds with uniformly bounded Reeb chords is again Legendrian, provided that the limit is smooth.
{\varepsilonilon}nd{abstract}
\section{Introduction}
The Lagrangian Arnold conjecture \cite{arn65} implies that a Lagrangian submanifold $L$ of a symplectic manifold $M$ always intersects its image under a Hamiltonian diffeomorphism. Furthermore, the number of intersection points should be bounded from below by the Betti number of $L$ if the intersection is transverse and by the cup-length of $L$ in the general case. For $C^1$-small Hamiltonian diffeomorphisms in a cotangent bundle the Arnold conjecture follows easily from Morse theory. Gromov proved in \cite{gro85} with the use of pseudo-holomorphic curves that it is impossible to displace a weakly exact Lagrangian in a geometrically bounded symplectic manifold by a Hamiltonian diffeomorphism. Of course, this cannot hold for arbitrary Lagrangians in arbitrary symplectic manifolds as the example of an embedded circle in $\mathbb{R}^2$ with its standard symplectic structure shows. But Polterovich \cite{pol93} showed under the assumptions that $L$ is rational and that $M$ is geometrically bounded that $L$ will always intersect its image under a Hamiltonian diffeomorphism $\psi$ as long as $\psi$ is sufficiently small in the Hofer norm. Floer \cite{flo88} introduced a homology theory for Lagrangian intersections in order to prove the Arnold conjecture in the case that $M$ is compact and $\pi_2(M,L) = 0$. Chekanov \cite{che98} used Floer's ideas to prove that the Arnold conjecture holds for all closed Lagrangians in geometrically bounded symplectic manifolds as long as the Hamiltonian diffeomorphism is sufficiently small in the Hofer norm.
Now, the question arises whether non-Lagrangian submanifolds can be rigid as well. To this end, Laudenbach and Sikorav proved in \cite{ls94} that half-dimensional closed non-Lagrangian submanifolds of symplectic manifolds are infinitesimally displaceable as long there is no topological obstruction. Here, infinitesimally displaceable means that there is a Hamiltonian vector field nowhere tangent to that submanifold.
Similarly to the symplectic case, there are also results about the rigidity of Legendrian submanifolds $L$ in a contact manifold $M$. For example, Rizell and Sullivan (\!\!\cite{rs16}, \cite{rs18}) proved that if the contact Hamiltonian $H$ generating a contactomorphism $\phi^H$ is ``sufficiently small", then there are short (compared to $H$) Reeb chords between $L$ and $\phi(L)$.
In this work, we extend Laudenbach and Sikorav's ideas to contact manifolds. We prove that under certain assumptions for a given $n$-dimensional non-Legendrian submanifold $L$ (where $\dim(M) = 2n+1$) there exists a contact vector field that is nowhere contained in the sum of the tangent space of $L$ and the span of the Reeb vector field along $L$.
Laudenbach and Sikorav \cite{ls94} noted that if a sequence $\{L_n\}_{n \in \mathbb{N}}$ of closed Lagrangian submanifolds of a geometrically bounded\footnote{They consider the cases $M = \mathbb{R}^{2n}$ and $\pi_2(M,L) = 0$ but their proof easily extends to general geometrically bounded symplectic manifolds, cf. Theorem \ref{c0-limit lagrange general} below.} symplectic manifold $C^0$-converges to an embedded submanifold $L$, then the displacement energies of the $L_n$ have to be uniformly bounded away from zero. But if $L$ has vanishing displacement energy, then the sequence of the displacement energies of the $L_i$ has to go to zero. From this they concluded that the limit has to be Lagrangian as well.
In a similar way, it will follow that the limit of a sequence of closed Legendrian submanifolds with uniformly bounded Reeb chords is again Legendrian (Theorem \ref{c0-limit legendre}).\\
\textit{Acknowledgements}: This work was carried out as part of the Master's program ``Theoretical and Mathematical Physics" at the Ludwig-Maximilans-University Munich, and it summarizes the results of my Master's thesis. I would first like to thank Thomas Vogel for supervising this work and for his numerous helpful remarks about this note. He always found the time to answer all of my questions. Furthermore, I am grateful to Yang Huang for many interesting and stimulating discussions. Also, I would like to thank Georgios Dimitroglou Rizell for explaining to me some of the results of his joint work with M. Sullivan.
\break
\section{Displacing non-Legendrian submanifolds}\label{sec:main results}
As mentioned in the introduction, closed Lagrangian submanifolds of many symplectic manifolds are rigid. Let us describe the following rather weak rigidity property. Let $(M^{2n},\omega)$ be a symplectic manifold and $L \subseteq M$ a closed Lagrangian submanifold. The restriction of any function $H:M \to \mathbb{R}$ to $L$ has a critical point $x \in L$ because $L$ is closed, i.e. $dH(x)|_{T_xL} = 0$. For the Hamiltonian vector field $X_H$ associated to $H$, defined by $i_{X_H} \omega = - dH$, this implies that $X_H(x) \in T_xL^{\perp_\omega} = T_xL$ since $L$ is Lagrangian. In other words, there exists no Hamiltonian vector field on $M$ that is nowhere tangent to $L$.
Now let $L^n$ be a closed non-Lagrangian submanifold of $M$ and we ask whether there exists a Hamiltonian vector field nowhere tangent to $L$. Of course, there might not exist any vector field that is nowhere tangent to $L$ as the self-intersection number of $L$ might be non-zero. But under the additional assumption that there is no such topological obstruction, Laudenbach and Sikorav proved the affirmative answer.
\begin{thm}\label{main thm symp}\hspace{-1mm}\normalfont{\cite{ls94}}\,
Let ($M^{2n},\omega$) be a symplectic manifold and $L$ a closed, connected submanifold of dimension n such that
(i) $L$ is $non$-$Lagrangian$, i.e. there exists a point $x \in L$ such that $T_xL$ is not a Lagrangian subspace of $T_xM$,
(ii) the normal bundle $\nu$ of $L \subseteq M$ has a nowhere vanishing section.
Then there exists a Hamiltonian vector field on $M$ that is nowhere tangent to $L$.\\
{\varepsilonilon}nd{thm}
\begin{rmk}\label{rmk:generalizations of symp non-rig}
Clearly, the generalization of Theorem \ref{main thm symp} to non-coisotropic submanifolds fails in general as such manifolds may contain closed Lagrangian submanifolds. However, Gürel \cite{gue08} noted that Theorem \ref{main thm symp} extends to nowhere coisotropic manifolds. Also, one can prove that even the parametric and a relative version of the h-principle for Hamiltonian vector fields that are nowhere tangent to $L$ holds.\\
{\varepsilonilon}nd{rmk}
Analogously to the Lagrangian case, Legendrians obey the following rigidity result. Let $(M, \xi = \ker \alpha)$ be a cooriented contact manifold and $L \subseteq M$ a closed Legendrian submanifold. Let $H: M \to \mathbb{R}$ be an arbitrary function. Then $H|_L$ has a critical point $x \in L$. From $dH(x)|_{T_xL} = 0$ it follows that $X_H(x) \in T_xL^{\perp_{d \alpha}} \oplus \langle R_\alpha(x) \rangle = T_xL \oplus \langle R_\alpha(x) \rangle$. Here, $X_H$ denotes the contact vector field associated to $H$ that is defined by
\begin{equation}
i_{X_H} d \alpha|_{\xi} = -dH|_\xi \quad \text{ and } \quad \alpha(X_H) = H,
{\varepsilonilon}nd{equation}
$(\cdot)^{\perp_{d \alpha}}$ denotes the $d \alpha|_\xi$ complement in $\xi$, and $R_\alpha$ denotes the Reeb vector field on $M$. This means that for a closed Legendrian submanifold there exists no contact vector field that is nowhere contained in $TL \oplus R_\alpha$.
We now also consider non-Legendrian submanifolds. Below, we will apply the proof of Theorem \ref{main thm symp} in \cite{ls94} in order to show that, as in the symplectic case, there exist contact vector fields nowhere tangent to $TL \oplus R_\alpha$ as long as there is no topological obstruction, at least for a generic non-Legendrian submanifold. Note that the flow of such a contact vector field displaces the non-Legendrian submanifold $L$ in such a way that there are no short Reeb chords between $L$ and its image under the flow.
\begin{thm}\label{main thm contact}
Let ($M^{2n+1}, \xi = \mathrm{ker}\,\alpha$) be a cooriented contact manifold. Denote its Reeb vector field by $R_\alpha$. Let $L \subseteq M$ be a closed, connected submanifold of dimension n such that
(i) $R_\alpha(x) \notin T_x L$ for all $x \in L$,
(ii) $L$ is $non$-$Legendrian$, i.e. there exists a point $x \in L$ with $T_xL \not\subseteq \xi_x$,
(iii) there exists a nowhere vanishing section of the normal bundle of the subvector bundle $TL \oplus \langle R_\alpha|_L \rangle \subseteq TM|_L$.
Then there exists a contact vector field $X$ such that $X(x) \notin T_xL \,\oplus \langle R_\alpha (x) \rangle$ for all $x \in L$.\\
{\varepsilonilon}nd{thm}
\begin{rmk}\label{rmk:conditions in cont non-rig are necessary}
For a generic $n$-dimensional submanifold $L \subseteq (M,\ker \alpha)$, $R_\alpha$ will be nowhere tangent to $L$. Hence, Theorem \ref{main thm contact} describes the generic case. With basically the same proof one can show that a similar statement also holds if we require that the Reeb vector field is everywhere tangent to $L$.\\
{\varepsilonilon}nd{rmk}
\begin{rmk}
Similarly to Gürel's result \cite{gue08} that was mentioned in Remark \ref{rmk:generalizations of symp non-rig}, Theorem \ref{main thm contact} also holds for submanifolds that have a dimension different from $n$ if one requires that $\left( \pi T_xL \right)^{\perp_{d \alpha}} \not\subseteq T_xL$ holds for all $x \in L$. Here, $\pi:TM = \xi \oplus \langle R_\alpha \rangle \to \xi$ denotes the projection onto the first factor. Also, the relative and a parametric h-principle hold in the setting of Theorem \ref{main thm contact} and in this case.\\
{\varepsilonilon}nd{rmk}
Laudenbach and Sikorav deduced Theorem \ref{main thm symp} from the following more general statement.
\begin{thm}\label{tech thm}\hspace{-1mm}\normalfont{\cite{ls94}}\,
Let $M$ be a manifold, $L$ a closed connected submanifold, and $E$ a subbundle of $TM|_L$ with ${\rm rk}(E) = \dim(L)$ such that
(i) $E\neq TL$, i.e. there exists a point $x \in L$ with $E_x\neq T_xL$,
(ii) there exists a nowhere vanishing section of $E$.\\
Then there exists a function $H$ on $M$ such that $dH|_{E_x}$ is non-zero for all $x \in L$.\\
{\varepsilonilon}nd{thm}
\noindent{\textit{Proof. }}of{Theorem~\ref{main thm contact}}
We will show how Theorem \ref{tech thm} implies Theorem \ref{main thm contact}.
The tangent bundle $TM$ of $M$ splits as $TM = \xi \oplus \langle R_\alpha \rangle$. As above, let $\pi$ denote the projection onto the first factor. In order to apply Theorem \ref{tech thm}, we define the vector bundle
\begin{equation}
E \coloneqq (\pi TL)^{\perp_{d \alpha}}
{\varepsilonilon}nd{equation}
on $L$. Since $R_\alpha$ is nowhere tangent to $L$, this indeed defines a vector bundle with ${\rm rk}(E) = \dim(L)$. Because $E \subseteq \xi$, it follows that $E = TL$ if and only if $L$ is Legendrian. Thus, condition $(ii)$ in Theorem \ref{main thm contact} is precisely condition $(i)$ in Theorem \ref{tech thm}.
It is convenient to consider a complex structure $J: \xi \to \xi$ on the contact distribution such that
\begin{equation}
g_J(v,w) \coloneqq d \alpha (v, Jw), \quad v,w \in \xi_x, x \in L,
{\varepsilonilon}nd{equation}
defines a metric on $\xi$. Such complex structures exist because $d \alpha|_\xi$ defines a symplectic structure on $\xi$ (cf. \cite{ms17}, Proposition 2.6.4). Then we can extend $g_J$ to a metric on $M$ in such a way that the Reeb vector field $R_\alpha$ is orthogonal to $\xi$.
By assumption, there exists a vector field $X$ that is orthogonal to $TL \oplus \langle R_\alpha \rangle$ at every point of $L$. Especially, $X$ is tangent to $\xi$ along $L$. Now it is easy to check that $J X$ defines a nowhere vanishing section of $E$.
Therefore, Theorem \ref{tech thm} implies that there exists a function $H: M \to \mathbb{R}$ such that $dH|_{E_x}$ is non-zero for all $x \in L$. For any $x \in L$, we have that
\begin{equation}
0 = dH|_{E_x} = dH|_{(\pi T_xL)^{\perp_{d \alpha}}} \quad \Leftrightarrow \quad X_H(x) \in T_xL \oplus \langle R_\alpha(x) \rangle.
{\varepsilonilon}nd{equation}
Hence, Theorem \ref{main thm contact} follows.
{\varepsilonilon}proof
\break
\section{$C^0$-limits of Legendrian submanifolds}\label{sec:applications}
Let $(M,\omega)$ be a symplectic manifold. Eliashberg \cite{eli87} proved that the group of symplectomorphisms of $M$ is $C^0$-closed as a subset of the group of diffeomorphisms of $M$. This theorem can be stated equivalently in terms of graphs of diffeomorphisms of $M$. For this, recall that a diffeomorphism of $M$ is a symplectomorphism if and only if its graph in $(M \times M, pr_1^* \omega - pr_2^* \omega)$ is Lagrangian. Then Eliashberg's result states that the $C^0$-limit of a sequence of smooth, Lagrangian graphs in $M \times M$ is again Lagrangian, provided that it is a smooth graph.
Now one can also consider the closure of the symplectomorphism group of $M$ inside the group of homeomorphisms of $M$. A homeomorphism that is a $C^0$-limit of symplectomorphisms is called a $C^0$-symplectomorphism. Humilière, Leclercq and Seyfaddini \cite{hls15} generalized Elishberg's Theorem: If a $C^0$-symplectomorphism maps a coisotropic submanifold to a smooth manifold, then the image will be coisotropic as well.
In these statements it is assumed that the $C^0$-limits of the Lagrangian (or coisotropic) submanifolds are induced by $C^0$-limits of symplectomorphisms. But Laudenbach and Sikorav showed that this assumption is not necessary in general.
\begin{thm}\label{c0-limit lagrange}\hspace{-1mm}\normalfont{\cite{ls94}}\,
Let $(M^{2n}, \omega)$ be a symplectic manifold and $L^n$ a closed manifold. Let $f_i: L \to M$ be a sequence of Lagrangian embeddings of $L$ into $M$ that $C^0$-converges to an embedding $f:L \to M$. If
(i) $(M, \omega)$ is geometrically bounded and $\pi_2(M, f(L)) = 0$, or
(ii) $(M, \omega) = (\mathbb{R}^{2n}, \omega_0)$,\\
then $f$ is a Lagrangian embedding.\\
{\varepsilonilon}nd{thm}
Recall the following definition.
\begin{defn}\label{geombdd} (cf. \cite{gro85}, \cite{al94}) Let $(M,\omega)$ be a symplectic manifold. It is \textit{geometrically bounded} if there exists an almost complex structure $J$ such that $g_J(\cdot,\cdot) \coloneqq \omega(\cdot,J \cdot)$ defines a complete Riemannian metric for which there exists an upper bound on the sectional curvature and a positive lower bound on the injectivity radius of $(M,g_J)$.\\
{\varepsilonilon}nd{defn}
We will show that Theorem \ref{c0-limit lagrange} even holds if we replace the conditions $(i)$ and $(ii)$ by the more general condition that $(M,\omega)$ is geometrically bounded.
\begin{thm}\label{c0-limit lagrange general}
Let $(M^{2n}, \omega)$ be a geometrically bounded symplectic manifold and $L^n$ a closed manifold. Let $f_i: L \to M$ be a sequence of Lagrangian embeddings of $L$ into $M$ that $C^0$-converges to an embedding $f:L \to M$. Then $f$ is a Lagrangian embedding.\\
{\varepsilonilon}nd{thm}
Now consider a cooriented contact manifold $(M, \ker \alpha)$. Correspondingly to Eliashberg's result, Müller and Spaeth \cite{ms14} showed that the group of contactomorphism of $M$ is $C^0$-closed as a subset of the group of diffeomorphisms. Again, we also obtain a statement about the graphs of contactomorphisms as follows. Consider the projections $pr_1, pr_2: E \coloneqq M \times M \times \mathbb{R} \to M$ onto the first and second factor, respectively. A section of $pr_1: (E, e^z pr_1^*\alpha - pr_2^*\alpha) \to M$ is Legendrian if and only if it is of the form $x \mapsto (x, \psi(x), g(x))$ for some contactomorphism $\psi:M \to M$. Here, $z$ denotes the coordinate on $\mathbb{R}$ and $g$ is the conformal factor of $\psi$ defined by $\psi^*\alpha = e^g \alpha$. If we now apply Müller and Spaeth's Theorem to a sequence of contactomorphisms for which their respective conformal factors converge uniformly (cf. also \cite{ms15}), then it follows that the $C^0$-limit of a sequence of smooth Legendrian sections of $pr_1: E \to M$ is again Legendrian as long as it is a smooth section.
Still under the assumption that the conformal factors converge uniformly, Rosen and Zhang \cite{rz18} proved a result analogous to the Humilière-Leclercq-Seyfaddini Theorem, namely, that smooth images of coisotropic submanifolds (i.e. $(TL \cap \xi)^{\perp_{d \alpha}} \subseteq TL$, cf. \cite{hua15}) under homeomorphisms that are $C^0$-limits of contactomorphisms are again coisotropic. Usher \cite{ush20} showed that the conclusion of this statement is still true if the conformal factors are only required to be uniformly bounded from below.
Now we want to examine the question under which conditions smooth $C^0$-limits of Legendrian submanifolds are again Legendrian, even if the limit in not induced by a $C^0$-limit of contactomorphisms. It is well-known that any n-dimensional submanifold of a contact manifold $(M^{2n+1}, \xi)$ can be $C^0$-approximated by Legendrian submanifolds as long as there is no topological obstruction (see \cite{em02}, 16.1.3), but we will show that under certain conditions such approximations must have short Reeb chords.
\begin{thm}\label{c0-limit legendre}
Let $(M^{2n+1}, \xi = \ker{\alpha)}$ be a cooriented contact manifold and $L^n$ a closed manifold. Let $f_i:L \to M$ be a sequence of Legendrian embeddings of $L$ into $M$ that $C^0$-converge to an embedding $f = f_\infty:L \to M$. Assume that there exists ${\varepsilonilon}ps > 0$ such that for all $i \in \mathbb{N}$ there are no Reeb chords of length less than ${\varepsilonilon}ps$ going from $f_i(L)$ to itself. If one of the following conditions is satisfied, then $f$ is a Legendrian embedding.
(a) The Reeb vector field is nowhere tangent to $f(L)$ and there exist real numbers $a,b \in \mathbb{R},~a < b$, and a geometrically bounded symplectic manifold $(N, \omega)$ such that there exists a symplectic embedding $i:(M \times [a,b], d(e^s \alpha)) \to (N,\omega)$.
(b) The Reeb vector field is nowhere tangent to $f(L)$ and $M$ is either compact or the contactization\footnote{In fact, we only have to require that the contact form on $M = P \times \mathbb{R}$ is equal to the standard contact form on $P \times \mathbb{R}$ outside of a compact set.}
$M = P \times \mathbb{R}$ of a Liouville manifold $P$.
(c) $M$ is the contactization $M = P \times \mathbb{R}$ of an exact, geometrically bounded symplectic manifold $P$.\\
{\varepsilonilon}nd{thm}
By a Liouville manifold $P$ we mean an open exact symplectic manifold that contains a compact domain $\overline{P} \subset P$ such that the Liouville vector field is transverse to $\partial \overline{P}$, and such that the Liouville flow $\Phi_t$ satisfies $P \setminus \overline{P} = \bigcup_{t>0} \Phi_t(\partial \overline{P})$.\\
\begin{rmk}
(1) Because the question whether the $C^0$-limit is Legendrian does not depend on the contact form, the theorem should be read as, ``If there exists a contact form such that there is a positive uniform lower bound on the length of the Reeb chords of the $f(L_i)$, then the limit is Legendrian".
(2) It is known that a Liouville manifold is always geometrically bounded. Therefore, $(c)$ immediately implies $(b)$ in the case that $M$ is the contactization of a Liouville manifold. We explicitly stated that part of $(b)$ nonetheless because the proofs of $(b)$ and $(c)$ rely on different results about Legendrian and non-Legendrian submanifolds.
(3) An embedding $i: M \times [a,b] \to N$ as in $(a)$ exists if $(M, \alpha)$ is a boundary component of a compact symplectic manifold with boundary of contact type.
(4) The fact that non-Legendrian submanifolds can be $C^0$-approximated by Legendrian submanifolds also shows that the closedness condition on $L$ in Theorem \ref{c0-limit lagrange general} cannot be removed. Indeed, a $C^0$-converging sequence of Legendrian submanifolds lifts in the symplectization to a $C^0$-converging (in the weak topology) sequence of cylindrical Lagrangian submanifolds and the limit of the latter sequence is Lagrangian if and only if the limit of the former sequence is Legendrian.\\
{\varepsilonilon}nd{rmk}
\begin{rmk}
Now let us consider $C^0$-approximations of paths in $\mathbb{R}^3$ with its standard contact structure $\xi = \ker(dz - y dx)$.
On the one hand, if the path is induced by the Reeb flow, then it is easy to see that it can be $C^0$-approximated by Legendrians that do not have any Reeb chords. For example, if $L$ is the interval
\begin{equation}
L \coloneqq I = \{(0,0,z) \in \mathbb{R}^3|\ z \in [0,1]\},
{\varepsilonilon}nd{equation}
then $L$ can be $C^0$-approximated by Legendrians, whose Lagrangian projection looks like a spiral (Figure \ref{fig:lagrangian projection of legendrian approximating interval}).
\begin{figure}[h]
\centering
\includegraphics[width=0.3\linewidth]{lagrangian_projection_of_legendrian_approximating_interval}
\caption{Lagrangian projection of a Legendrian submanifold that is $C^0$-approximating the interval.}\label{fig:lagrangian projection of legendrian approximating interval}
{\varepsilonilon}nd{figure}
On the other hand, if an embedded path $\gamma: [0,1] \to \mathbb{R}^3$ is not Legendrian and if the Reeb vector field is nowhere collinear to its velocity vector, then it cannot be $C^0$-approximated by a Legendrian path without Reeb chords. In order to see this, let\footnote{We assume that $\gamma$ is defined on the interval $[-1,2]$ instead of $[0,1]$ in order to make it easier to write down the argument below.} $\gamma: [-1,2] \to \mathbb{R}^3$ be such an embedded non-Legendrian path and let ${\varepsilonilon}ta: [-1,2] \to \mathbb{R}^3$ be a Legendrian embedding without Reeb chords that is ${\varepsilonilon}ps$-close to $\gamma$ for some ${\varepsilonilon}ps >0$. Let $\pi: \mathbb{R}^3 \to \mathbb{R}^2$ denote the Lagrangian projection. We write $\pi \gamma$ for $\pi \circ \gamma$ and $\pi {\varepsilonilon}ta$ for $\pi \circ {\varepsilonilon}ta$. $\pi \gamma: [-1,2] \to \mathbb{R}^2$ is an immersed path. Let $\widetilde{\gamma}: [-1,2] \to \mathbb{R}^3$ be the unique Legendrian lift of $\pi \gamma$ to $\mathbb{R}^3$ such that $\widetilde{\gamma}(0) = \gamma(0)$. Since $\gamma$ is not Legendrian, we can assume that, after possibly restricting to a subinterval of $[-1,2]$, $\pi \gamma$ is an embedding and that $\widetilde{\gamma}(1) \neq \gamma(1)$. Let $z$, $\widetilde{z}$ and $z_{\varepsilonilon}ta$ denote the $z$-coordinates of $\gamma$, $\widetilde{\gamma}$ and ${\varepsilonilon}ta$, respectively. Define $C \coloneqq \frac{1}{10}| z(1) - \widetilde{z}(1) | > 0$. Note that $\widetilde{z}$ (and, in fact, the $z$-coordinate of any Legendrian path) satisfies
\begin{equation}\label{eq:z coord of legendrian path}
\widetilde{z}(1) - \widetilde{z}(0) = \int_{\widetilde{\gamma}|_{[0,1]}} y dx.
{\varepsilonilon}nd{equation}
For any $\kappa > 0$, let $U_\kappa$ denote the closed $\kappa$-neighbourhood of $\pi \gamma([0,1])$. After possibly decreasing ${\varepsilonilon}ps$, we can assume that there exists a closed ball-shaped neighbourhood $V_{\varepsilonilon}ps$ of $\pi \gamma([0,1])$ that satisfies $U_{\varepsilonilon}ps \subseteq V_{\varepsilonilon}ps \subseteq U_{2{\varepsilonilon}ps}$. We define
\begin{align}
t_- \coloneqq \inf \{t \in [-1,0] | \pi \gamma (s) \in V_{\varepsilonilon}ps ~ \forall s \in [t,0] \}, \\
s_- \coloneqq \inf \{t \in [-1,0] | \pi {\varepsilonilon}ta (s) \in V_{\varepsilonilon}ps ~ \forall s \in [t,0] \},
{\varepsilonilon}nd{align}
and similarly we define
\begin{align}
t_+ \coloneqq \sup \{t \in [1,2] | \pi \gamma (s) \in V_{\varepsilonilon}ps ~ \forall s \in [1,t] \}, \\
s_+ \coloneqq \sup \{t \in [1,2] | \pi {\varepsilonilon}ta (s) \in V_{\varepsilonilon}ps ~ \forall s \in [1,t] \}.
{\varepsilonilon}nd{align}
Since $\gamma$ and ${\varepsilonilon}ta$ are embedded paths, it follows that $t_-, s_- \to 0$ and $t_+, s_+ \to 1$ as ${\varepsilonilon}ps \to 0$. Now choose ${\varepsilonilon}ps$ so small and $V_{\varepsilonilon}ps$ in such a way that the following conditions are satisfied:
(a) ${\varepsilonilon}ps < C$
(b) $t_-, s_- > -1, \quad t_+,s_+ < 2$,
(c) $\Vert \gamma(s_-) - \gamma(0) \Vert < C$ and $\Vert \gamma(s_+) - \gamma(1) \Vert < C$,
(d) $\Vert \widetilde{\gamma}(t_-) - \widetilde{\gamma}(0) \Vert < C$ and $\Vert \widetilde{\gamma}(t_+) - \widetilde{\gamma}(1 )\Vert < C$,
(e) For any four points $x^0_+,x^0_-,x^1_+,x^1_- \in \partial V_{\varepsilonilon}ps$ with $\Vert x^0_- - x^1_- \Vert < 11{\varepsilonilon}ps$ and $\Vert x^0_+ - x^1_+ \Vert < 11{\varepsilonilon}ps$ and for any two embedded paths $\sigma_0, \sigma_1: [0,1] \to V_{\varepsilonilon}ps$ with $\sigma_{i}(0)= x^{i}_-$ and $\sigma_{i}(1) = x^{i}_+$ for $i \in \{0,1\}$, we have that \begin{equation}
\Big| \int_{\sigma_0} y dx - \int_{\sigma_1} y dx \Big| < C
{\varepsilonilon}nd{equation}
(f) $\pi \gamma (\frac{-4 {\varepsilonilon}ps}{\Vert (\pi \gamma)'(0) \Vert}) \not\in U_{3 {\varepsilonilon}ps}$ and $\pi \gamma (1 + \frac{4 {\varepsilonilon}ps}{\Vert (\pi \gamma)'(0) \Vert}) \not\in U_{3 {\varepsilonilon}ps}$.
(g) For all $t \in \left[\frac{-4 {\varepsilonilon}ps}{\Vert (\pi \gamma)'(0) \Vert},0 \right]$ we have that $\Vert \pi \gamma(t) - \pi \gamma(0) \Vert < 5 {\varepsilonilon}ps$, and for all $t \in \left[1, 1 + \frac{4 {\varepsilonilon}ps}{\Vert (\pi \gamma)'(0) \Vert}\right]$ we have that $\Vert \pi \gamma(t) - \pi \gamma(1) \Vert < 5 {\varepsilonilon}ps$.\\
It is clear that (a)-(d) will be satisfied if ${\varepsilonilon}ps$ is sufficiently small.
To see that (e) can be satisfied, choose ${\varepsilonilon}ps$ so small and choose $V_{\varepsilonilon}ps$ in such a way that for any two points $y^0,y^1 \in \partial V_{\varepsilonilon}ps$ with $\left| y^0 -y^1 \right| < 11 {\varepsilonilon}ps$ there exists an embedded path $\chi:[0,1] \to V_{\varepsilonilon}ps$ with $\chi(0)=y^0$ and $\chi(1)=y^1$ such that $\left|\int_\chi ydx \right| < \frac{C}{10}$. Furthermore, we assume that ${\rm area} (V_{\varepsilonilon}ps) < \frac{C}{10}$. Let $x^0_+,x^0_-,x^1_+,x^1_- \in \partial V_{\varepsilonilon}ps$ be four points and $\sigma_0, \sigma_1: [0,1] \to V_{\varepsilonilon}ps$ be two paths as in (e).
By our assumptions, there exist two embedded paths $\chi_-$ and $\chi_+$ with $\chi_-(0) = x_-^0$, $\chi_-(1) = x_-^1$, $\chi_+(0) = x_+^0$ and $\chi_+(1) = x_+^1$ such that $\left|\int_{\chi_\pm} ydx \right| < \frac{C}{10}$. Now let $\lambda_-, \lambda_+,\lambda_0, \lambda_1: [0,1] \to \partial V_{\varepsilonilon}ps$ be four paths that are embeddings when restricted to $(0,1)$ such that $\lambda_-(0) = x_-^0$, $\lambda_-(1) = x_-^1$, $\lambda_+(0) = x_+^0$, $\lambda_+(1) = x_+^1$, $\lambda_0(0) = x_-^0$, $\lambda_0(1) = x_+^0$, $\lambda_1(0) = x_-^1$ and $\lambda_1(1) = x_+^1$. As $d(ydx) = - dx \wedge dy$, it follows from Stokes' Theorem that
\begin{equation}
\Big| \int_{\chi_-} y dx - \int_{\lambda_-} y dx \Big| \leq {\rm area} (V_{\varepsilonilon}ps)< \frac{C}{10}.
{\varepsilonilon}nd{equation}
Similarly, it follows that
\begin{equation}
\Big| \int_{\chi_+} y dx - \int_{\lambda_+} y dx \Big| < \frac{C}{10}, \quad \Big| \int_{\sigma_0} y dx - \int_{\lambda_0} y dx \Big| < \frac{C}{10}, \quad \Big| \int_{\sigma_1} y dx - \int_{\lambda_1} y dx \Big| < \frac{C}{10}.
{\varepsilonilon}nd{equation}
The absolute value of the winding number of the concatenation $\lambda_0 \ast \lambda_+ \ast \overline{\lambda_1} \ast \overline{\lambda_-}$ is at most four. Here, $\overline{(\cdot)}$ denotes the inversion of paths. Therefore, it follows again from Stokes' Theorem that
\begin{equation}
\Big| \int_{\lambda_0 \ast \lambda_+ \ast \overline{\lambda_1} \ast \overline{\lambda_-}} y dx \Big| \leq 4~ {\rm area}(V) < \frac{4}{10} C.
{\varepsilonilon}nd{equation}
Combining the above inequalities one easily concludes that
\begin{equation}
\Big| \int_{\sigma_0} y dx - \int_{\sigma_1} y dx \Big| < C.
{\varepsilonilon}nd{equation}
This proves (e).
By looking at the Taylor expansion of $\pi \gamma$ around $0$ and $1$, it can also be seen that (f) and (g) are satisfied if ${\varepsilonilon}ps$ is sufficiently small.
From now on assume that ${\varepsilonilon}ps$ and $V_{\varepsilonilon}ps$ are such that the conditions (a) - (g) are satisfied.
As ${\varepsilonilon}ta$ is ${\varepsilonilon}ps$-close to $\gamma$, (f) implies that $\pi {\varepsilonilon}ta (\frac{-4 {\varepsilonilon}ps}{\Vert (\pi \gamma)'(0) \Vert}) \not\in U_{2 {\varepsilonilon}ps}$ and $\pi {\varepsilonilon}ta (1 + \frac{4 {\varepsilonilon}ps}{\Vert (\pi \gamma)'(0) \Vert}) \not\in U_{2 {\varepsilonilon}ps}$. Since $V_{\varepsilonilon}ps \subseteq U_{2 {\varepsilonilon}ps}$, we can conclude from this observation together with (f) that $s_-, t_- > \frac{-4 {\varepsilonilon}ps}{\Vert (\pi \gamma)'(0) \Vert}$ and $s_+,t_+ < 1 + \frac{4 {\varepsilonilon}ps}{\Vert (\pi \gamma)'(0) \Vert}$. Using (g) and the fact that ${\varepsilonilon}ta$ is ${\varepsilonilon}ps$-close to $\gamma$ it follows that
\begin{equation}\label{eq:pi eta(s_-)-pi gamma(t_-)}
\Vert \pi {\varepsilonilon}ta (s_-) - \pi \gamma (t_-) \Vert = \Vert \big(\pi {\varepsilonilon}ta (s_-) - \pi \gamma (s_-)\big) + \big(\pi \gamma (s_-) - \pi \gamma(0)\big) + \big(\pi \gamma(0) - \pi \gamma (t_-)\big) \Vert < 11 {\varepsilonilon}ps
{\varepsilonilon}nd{equation}
and similarly also
\begin{equation}\label{eq:pi eta(s_+)-pi gamma(t_+)}
\Vert \pi {\varepsilonilon}ta (s_+) - \pi \gamma (t_+) \Vert < 11 {\varepsilonilon}ps.
{\varepsilonilon}nd{equation}
We can see that
\begin{equation}\label{eq:z_eta(0) - z_eta(s_-)}
\left| z_{\varepsilonilon}ta(0) - z_{\varepsilonilon}ta(s_-) \right| = \left| (z_{\varepsilonilon}ta(0) - z(0)) + (z(0) - z(s_-)) + (z(s_-) - z_{\varepsilonilon}ta(s_-)) \right| \leq 3C,
{\varepsilonilon}nd{equation}
where in the last inequality we used (a) and (c) together with the assumption that ${\varepsilonilon}ta$ is ${\varepsilonilon}ps$-close to $\gamma$. In the same way we also obtain
\begin{equation}\label{eq:z_eta(s_+) - z_eta(1)}
\left| z_{\varepsilonilon}ta(s_+) - z_{\varepsilonilon}ta(1) \right| \leq 3C.
{\varepsilonilon}nd{equation}
We can conclude that
\begin{equation}\label{eq:ztilde diff - z_eta diff}
\begin{gathered}
\left| (\widetilde{z}(1) - \widetilde{z}(0)) - (z_{\varepsilonilon}ta(1) - z_{\varepsilonilon}ta(0)) \right| \\
\overset{\text{(d)},(\ref{eq:z_eta(0) - z_eta(s_-)}), (\ref{eq:z_eta(s_+) - z_eta(1)})}{\leq} \left| (\widetilde{z}(t_+) - \widetilde{z}(t_-)) - (z_{\varepsilonilon}ta(s_+) - z_{\varepsilonilon}ta(s_-)) \right| + 8C \\
\overset{(\ref{eq:z coord of legendrian path})}{=} \Big| \int_{\pi \widetilde{\gamma}|_{[t_-,t_+]}} y dx - \int_{\pi {\varepsilonilon}ta|_{[s_-,s_+]}} y dx \Big| + 8C \overset{\text{(b)},(\ref{eq:pi eta(s_-)-pi gamma(t_-)}),(\ref{eq:pi eta(s_+)-pi gamma(t_+)}),\text{(e)}}{<} 9C,
{\varepsilonilon}nd{gathered}
{\varepsilonilon}nd{equation}
where in the last step we used the assumptions that $\pi \widetilde{\gamma} = \pi \gamma$ and $\pi {\varepsilonilon}ta$ are embeddings in order to apply (e) (recall that ${\varepsilonilon}ta$ does not have any Reeb chords).
Now,
\begin{equation}
\begin{gathered}
|z(1) - z(0) - (z_{\varepsilonilon}ta(1) - z_{\varepsilonilon}ta(0))| \\
\overset{z(0)=\widetilde{z}(0)}{\geq} |z(1) - \widetilde{z}(1)| - |\widetilde{z}(1) - \widetilde{z}(0) - (z_{\varepsilonilon}ta(1) - z_{\varepsilonilon}ta(0))| \overset{\text{def.\ C},~ (\ref{eq:ztilde diff - z_eta diff})}{>} C
{\varepsilonilon}nd{gathered}
{\varepsilonilon}nd{equation}
leads to a contradiction if ${\varepsilonilon}ps$ is small enough because ${\varepsilonilon}ta$ is ${\varepsilonilon}ps$-close to $\gamma$.
This shows that ${\varepsilonilon}ta$ must have Reeb chords if ${\varepsilonilon}ps$ is sufficiently small. It is also clear that these Reeb chords need to be short because ${\varepsilonilon}ta$ is contained in the ${\varepsilonilon}ps$-neighbourhood of $\gamma$ and the Reeb vector field is nowhere tangent to $\gamma$.
Using Darboux charts, it follows that this statement holds in any $3$-dimensional contact manifold. To the author's knowledge, it is an open question under which conditions it is possible or impossible to $C^0$-approximate open submanifolds $L^n \subseteq (M^{2n+1},\xi = \ker \alpha)$ by Legendrian submanifolds without short Reeb chords in the case $n > 1$.\\
{\varepsilonilon}nd{rmk}
\begin{rmk}
If we lower the dimension of $L$ and ask whether the $C^0$-limit of isotropic submanifolds are isotropic, then the answer is no since there is a $C^0$-dense h-principle for subcritical isotropic embeddings into symplectic and contact manifolds (\!\!\cite{em02}, Theorem 12.4.1).\\
{\varepsilonilon}nd{rmk}
Another open question is whether Theorem \ref{c0-limit lagrange general} fails if we do not require $M$ to be geometrically bounded, and, similarly, whether the assumptions in Theorem \ref{c0-limit legendre} on $M$ are necessary. Also, one might expect these theorems to hold even for non-compact $L$ if we require the embeddings to be fixed outside some compact subset.\\
\noindent{\textit{Proof. }}of{Theorem~\ref{c0-limit lagrange general}}
As we can apply the theorem to every connected component of $L$, we can assume that $L$ is connected.
Recall that for a compactly supported Hamiltonian symplectomorphism $\psi$ the Hofer norm (cf. \cite{hof90}) is defined as
\begin{equation}
\| \psi \| \coloneqq \underset{H} {\inf}\, \|H \|_{osc},
{\varepsilonilon}nd{equation}
where the infimum is taken over all time-dependent functions $H_t$ on $M$ whose associated Hamiltonian flow $\phi^H_t$ satisfies $\phi^H_1 = \psi$. Here, $\| H \|_{osc}$ denotes the oscillatory energy of $H$ which is defined as
\begin{equation}
\| H \|_{osc} \coloneqq \int_0^1 \left(\underset{x \in M}{\max}\, H(x,s) - \underset{x \in M}{\min}\, H(x,s)\right) ds.
{\varepsilonilon}nd{equation}
The Hofer norm is used to define the displacement energy $e(U)$ of a subset $U \subseteq M$ as
\begin{equation}
e(U) \coloneqq \inf \{\| \psi \| | \psi(U) \cap U = {\varepsilonilon}mptyset\}.
{\varepsilonilon}nd{equation}
Now assume that the conclusion of the theorem is false, i.e.\ there exists a sequence of Lagrangian embeddings $f_i:L \to M$ that $C^0$-converge to an embedding $f:L \to M$, but $f$ is not Lagrangian. Let $\iota :S^1 \to T^* S^1$ denote the zero-section. After possibly replacing $L$, $M$, $f_i$ and $f$ by $L \times S^1$, $M \times T^*S^1$, $f_i \times \iota$ and $f \times \iota$, respectively, we can assume that $f(L) \subseteq M$ admits a nowhere-vanishing section of its normal bundle. In order to simplify the notation, we will identify $f(L)$ with $L$ and write $L_i \coloneqq f_i(L)$. By Theorem \ref{main thm symp} there exists a Hamiltonian vector field nowhere tangent to $L$. Hence, for any ${\varepsilonilon}ps > 0$ there is a neighbourhood of $L \subseteq M$ that is displaced by this Hamiltonian isotopy from itself in a time less than ${\varepsilonilon}ps$ by compactness of $L$.
Since the $f_i$ converge uniformly towards $f$, we can find for any ${\varepsilonilon}ps > 0$ a number $N = N({\varepsilonilon}ps) \in \mathbb{N}$ such that $L_k$ is displaced form itself in a time less than ${\varepsilonilon}ps$ for all $k \geq N$. This implies that the displacement energy of the $L_i$ goes to zero as $i$ increases.
Chekanov proved in \cite{che98} that there is a lower bound on the displacement energy of a closed Lagrangian submanifold in a geometrically bounded symplectic manifold in terms of the minimal area of non-constant pseudoholomorphic spheres in $M$ and non-constant pseudoholomorphic discs in $M$ with boundary on $L$. Let $N \subseteq M$ be a compact tubular neighbourhood of $L$. If $L_i$ is sufficiently $C^0$-close to $L$, then $f_i$ and $f$ are homotopic as maps into $N$. For example, one can explicitly define such a homotopy by moving along the shortest geodesic connecting $f(x)$ and $f_i(x)$ for all $x \in L$. Since $f:L \to N$ is a homotopy equivalence, this implies that $f_i:L \to N$ is a homotopy equivalence as well if $i$ is sufficiently large. Without loss of generality we assume that this is the case for all $f_i$. A non-constant pseudoholomorphic curve with boundary on one of the $L_i$ has positive symplectic area. Hence, it defines a non-trivial class is $H_2(M,L;\mathbb{R}) \cong H_2(M,N;\mathbb{R})$.
According to Proposition $4.4.1$ in Chapter V in \cite{al94}, there exists a compact neighbourhood $V \subseteq M$ of $N$ such that every pseudoholomorphic curve whose image intersects $N$ lies completely in $V$. Let $U \subseteq M$ be a compact submanifold (possibly with boundary) that contains $V$. Then Lemma \ref{lem:lower bound of area of discs} below shows that the areas of non-constant pseudoholomorphic discs with boundary on one of the $L_i$ are bounded away from zero. Together with Chekanov's energy capacity inequality this implies that the displacement energies of the $L_i$ are uniformly bounded away from zero. This gives the desired contradiction. {\varepsilonilon}proof\\
\begin{lem}\label{lem:lower bound of area of discs}
Let $N$ be a compact submanifold of a compact manifold $U$ (possibly with boundary). Then there is a constant $C > 0$ such that any disc representing a non-trivial class in $H_2(U,N;\mathbb{R})$ has area larger than $C$.
{\varepsilonilon}nd{lem}
\noindent{\textit{Proof. }}of{Lemma}
The following proof is an adaptation of the proof of the corresponding lemma in \cite{ls94}.
By compactness, $H_2(U,N;\mathbb{Z})$ is finitely generated. Let $\{a_i\}_{i \in I}$ be a basis of the free quotient of $H_2(U,N;\mathbb{Z})$, where $I$ is a finite index set. Then the $a_i$ also form a real basis of $H_2(U,N;\mathbb{R})$. Denote by $\{\alpha_i\}_{i \in I}$ the basis of $H^2_{dR}(U,N;\mathbb{R})$ dual to the $a_i$. Then the homology class of any disc $D$ with boundary on $N$ can be written in the form $D = \sum_{i \in I} n_i a_i$, where $n_i \in \mathbb{Z}$ is the integral of $\alpha_i$ over $D$. Hence, we see that for all $i \in I$,
\begin{equation}
|n_i| = \Big| \int_D \alpha_i \Big| \leq \mathrm{area}(D) \Vert \alpha_i \Vert_{C^0},
{\varepsilonilon}nd{equation}
which implies that
\begin{equation}
\mathrm{area}(D) \geq \mathrm{max}_i \frac{|n_i|}{\Vert \alpha_i \Vert_{C^0}} \geq \mathrm{min}_i \frac{1}{\Vert \alpha_i \Vert_{C^0}},
{\varepsilonilon}nd{equation}
if the homology class of $D$ is non-zero (i.e. if not all of the $n_i$ vanish).
{\varepsilonilon}proof\\
\noindent{\textit{Proof. }}of{Theorem \ref{c0-limit legendre}} As before we can assume that $L$ is connected.
\textit{Part} $(a)$:
We will reduce Theorem \ref{c0-limit legendre} $(a)$ to Theorem \ref{c0-limit lagrange general} by using the following construction from \cite{moh01}.
First note that since $R_\alpha$ is nowhere tangent to $f_\infty(L)$, we can assume that, after possibly decreasing ${\varepsilonilon}ps > 0$ in the statement of the theorem, there is an ${\varepsilonilon}ps > 0$ such that
\begin{equation}
\begin{split}
L \times [0, {\varepsilonilon}ps] &\to M \\
(x,t) &\mapsto \left( \phi^\alpha_t \circ f_i \right)(x)
{\varepsilonilon}nd{split}
{\varepsilonilon}nd{equation}
is an embedding for all $i \in \mathbb{N} \cup \{\infty\}$, where $\phi^\alpha_t$ denotes the Reeb flow in $(M,\alpha)$. Let
\begin{equation}
(\gamma_1,\gamma_2):S^1 \to [0, {\varepsilonilon}ps] \times [a,b]
{\varepsilonilon}nd{equation}
be an embedded loop. Consider the embeddings
\begin{equation}
\begin{split}
F_i: L \times S^1 & \to (M \times [a,b], d(e^s \alpha)) \\
(x,t) & \mapsto ((\phi^\alpha_{\gamma_1(t)} \circ f_i)(x), \gamma_2(t)).
{\varepsilonilon}nd{split}
{\varepsilonilon}nd{equation}
It is clear that the $C^0$-convergence of the $f_i$ implies $C^0$-convergence of the $F_i$. Furthermore, a straightforward computation shows that $F_i$ is a Lagrangian embedding if and only if $f_i$ is a Legendrian embedding. Hence, we can apply Theorem \ref{c0-limit lagrange general} to conclude that $f_\infty$ is a Legendrian embedding.\\
The proofs of $(b)$ and $(c)$ are similar to the proof of Theorem \ref{c0-limit lagrange general}. We will use known rigidity results for Legendrian and non-rigidity results for non-Legendrian submanifolds to prove the statement. Again, we identify $L$ with $f(L)$ and write $L_i \coloneqq f_i(L)$.\\
\textit{Part} $(b)$: Assume that $L$ is not Legendrian. After possibly replacing $(M,\alpha)$, $L$, $f_i$ and $f$ by $(M\times T^*S^1,\alpha - p dq)$, $L \times S^1$, $f_i \times \iota$ and $f \times \iota$, respectively, we can assume that there exists a vector field that is nowhere (along $L$) contained in $TL \oplus \langle R_\alpha \rangle$. Here, $\iota:S^1 \to T^*S^1$ denotes the zero section.
Theorem \ref{main thm contact} implies that there exists a contact vector field $X$ that is nowhere contained in $TL \oplus \langleR_\alpha\rangle$. This implies that its flow $\phi_t \coloneqq \phi^X_t$ displaces $L$ for sufficiently small times such that there are no short (compared to the length of the Reeb chords of $L$) Reeb chords between $L$ and $\phi_t(L)$ for any $t>0$ that is sufficiently small. To be more precise, let $\sigma$ denote the minimal length of Reeb chords of $L$. Then, for any $\lambda > 0$ there exists a $\delta > 0$ such that there are no Reeb chords of length smaller than $\sigma - \lambda$ between $L$ and $\phi_t(L)$ for all $0 < t < \delta$. In this case, $\phi_t$ also displaces a neighbourhood of $L \subseteq M$ without short Reeb chords by compactness of $L$. Then for any sufficiently small $t > 0$, there exists an $N \in \mathbb{N}$ such that $\phi_t$ also displaces $L_i$ without short Reeb chords for all $i\geq N$. This shows that for any ${\varepsilonilon}ta > 0$ there is an $N \in \mathbb{N}$ and a function $H:M \to \mathbb{R}$ such that $\Vert H \Vert_{C^1} < {\varepsilonilon}ta$ and the contactomorphism associated to $H$ displaces $L_i$ without short Reeb chords for all $i \geq N$.
For any closed Legendrian submanifold $N \subseteq M$, let $\sigma(\alpha, N)$ denote the minimal length of Reeb chords $\gamma$ of $N$ and of closed Reeb orbits $\gamma$ in $M$ satisfying $[\gamma] = 0 \in \pi_1(M,N)$. Rizell and Sullivan proved that if the $C^1$-norm\footnote{In fact, they only required that the oscillatory energy of $H$ and the conformal factor of the contact flow associated to $H$ are sufficiently small.} of a generic function $H$ on $M$ is small compared to $\sigma(\alpha, N)$, there always exist short (compared to the $C^1$-norm of $H$) Reeb chords between $N$ and $\phi_1^H(N)$ (\!\!\cite{rs16}, Theorem 1.3) if $M$ satisfies the conditions in $(b)$. This gives the desired contradiction because, after possibly approximating $H$, we can assume that it is generic. \\
\textit{Part} $(c)$: For a compactly supported contactomorphism $\psi$ on $(M,\alpha)$ that is isotopic to the identity one can define
\begin{equation}
\| \psi \|_\alpha \coloneqq \underset{H}{\inf}\, \|H\|,
{\varepsilonilon}nd{equation}
where the infimum is taken over all time-dependent functions $H_t$ whose associated contact isotopy $\phi^H_t$ satisfies $\phi^H_1 = \psi$. Here, $\|H \|$ is defined by
\begin{equation}
\|H\| \coloneqq \int_0^1 \underset{x \in M}{\max}\, H(x,s) ds.
{\varepsilonilon}nd{equation}
Shelukhin \cite{she17} proved that this defines a (non-degenerate) norm on the group of compactly supported contactomorphisms isotopic to the identity.
Now assume that $L$ is not Legendrian. Note that for a generic contactomorphism $\phi$, $\phi(L)$ will not intersect $L$ since $\dim(L) = n$ and $\dim(M) = 2n+1$. Let $\phi$ be such a contactomorphism. Then there exists a lower bound $C > 0$ on the length of Reeb chords between $L$ and $\phi(L)$. Theorem 1.9 and Proposition 7.4 in \cite{rz18} together imply that there exist a sequence $\phi_n$ of contactomorphisms isotopic to the identity such that $\lim\limits_{n \to \infty} \Vert \phi_n \Vert_\alpha = 0$ and $\phi_n(L) = \phi(L)$ for all $n \in \mathbb{N}$. By compactness of $L$ we can find for any $n \in N$ and any ${\varepsilonilon}ta > 0$ a neighbourhood $U = U(n,{\varepsilonilon}ta)$ of $L \subseteq M$ such that there are no Reeb chords of length smaller than $C - {\varepsilonilon}ta$ between $U$ and $\phi_n(U)$. After possibly perturbing the $\phi_n$ and choosing a slightly larger ${\varepsilonilon}ta$, we can assume that the $\phi_n$ are generic and still have the above properties (except, of course, $\phi_n(L) = \phi(L)$).
Since the $L_i$ $C^0$-converge to $L$, we can find for any two positive numbers $\delta, {\varepsilonilon}ta > 0$ some numbers $n,K \in \mathbb{N}$ such that $L_i \subseteq U(n,{\varepsilonilon}ta)$ for all $i \geq K$ and $\Vert \phi_n \Vert_\alpha < \delta$. In particular, there are no Reeb chords of length smaller than $C - {\varepsilonilon}ta$ between $L_i$ and $\phi_n(L_i)$.
This is a contradiction to a result of Rizell and Sullivan \cite{rs18} that states that there have to exist short Reeb chords between $L_i$ and $\phi_n(L_i)$ in the above setting if $\Vert \phi_n \Vert_\alpha$ is sufficiently small.
{\varepsilonilon}proof\\
\begin{rmk}
In the proof of $(b)$ we only had to consider Reeb chords $\gamma$ that satisfy $[\gamma] = 0 \in \pi_1(M,f_i(L))$. One could seemingly strengthen the assumption in part $(b)$ of Theorem~\ref{c0-limit legendre} by only requiring that there exists a uniform lower bound on the length of the Reeb chords that satisfy this condition. But it is easy to see that, in fact, compactness of $L$ and the $C^0$-convergence of the $f_i$ imply that there cannot be a sequence of Reeb chords that are non-zero in $\pi_1(M,f_i(L))$ and whose length converges to zero. Indeed, for sufficiently large $i$, $f_i:L \to N$ is a homotopy equivalence between $L$ and a tubular neighbourhood $N$ of $f(L)$ and any sufficiently short Reeb chord of $f_i(L)$ is contained in $N$. Hence, such a Reeb chord is trivial in $\pi_1(M,N) \cong \pi_1(M,f_i(L))$.\\
{\varepsilonilon}nd{rmk}
{\varepsilonilon}nd{document}
|
\begin{document}
\title[Regularity for odd solutions to degenerate or singular problems]
{Liouville type theorems and regularity of solutions to degenerate or singular problems part II: odd solutions}
\author{Yannick Sire, Susanna Terracini and Stefano Vita}
\address[Y. Sire]{Department of Mathematics
\newline\indent
Johns Hopkins University
\newline\indent
3400 N. Charles Street, Baltimore, MD 21218, U.S.A.}
\email{[email protected]}
\address[S. Terracini]{Dipartimento di Matematica G. Peano
\newline\indent
Universit\`a degli Studi di Torino
\newline\indent
Via Carlo Alberto 10, 20123 Torino, Italy}
\email{[email protected]}
\address[S. Vita]{Dipartimento di Matematica
\newline\indent
Universit\`a degli Studi di Milano Bicocca
\newline\indent
Piazza dell'Ateneo Nuovo 1, 20126, Milano, Italy}
\email{[email protected]}
\date{\today}
\thanks{Work partially supported by the ERC Advanced Grant 2013 n.~339958 Complex Patterns for Strongly Interacting Dynamical Systems - COMPAT. Y.S. was partially supported by the Simons foundation.}
\keywords{Degenerate and singular elliptic equations; Liouville type Theorems; Blow-up; Fractional Laplacian; Fractional divergence form elliptic operator; Schauder estimates; Boundary Harnack; Fermi coordinates}
\subjclass[2010] {
35J70,
35J75,
35R11,
35B40,
35B44,
35B53,
}
\maketitle
\begin{center}{\it To Sandro, with friendship, admiration and much more}\end{center}
\begin{abstract}
We consider a class of equations in divergence form with a singular/degenerate weight \[
-\mathrm{div}(|y|^a A(x,y)\nabla u)=|y|^a f(x,y)+\textrm{div}(|y|^aF(x,y))\;.
\]
Under suitable regularity assumptions for the matrix $A$, the forcing term $f$ and the field $F$, we prove H\"older continuity of solutions which are odd in $y\in\mathbb{R}$, and possibly of their derivatives. In addition, we show stability of the $C^{0,\alpha}$ and $C^{1,\alpha}$ a priori bounds for approximating problems in the form
\[
-\mathrm{div}((\varepsilon^2+y^2)^{a/2} A(x,y)\nabla u)=(\varepsilon^2+y^2)^{a/2} f(x,y)+\textrm{div}((\varepsilon^2+y^2)^{a/2}F(x,y))
\]
as $\varepsilon\to 0$. Our method is based upon blow-up and appropriate Liouville type theorems.
\end{abstract}
\tableofcontents
\section{Introduction and main results}
Let $z=(x,y)\in\mathbb{R}^{n+1}$, with $x\in\mathbb{R}^{n}$ and $y\in\mathbb{R}$, $n\geq1$, $a\in\mathbb{R}$. Our aim is to study the boundary behaviour of solutions to a class of problems involving singular/degenerate operators in divergence form including
$$\mathcal L_au:=\mathrm{div}(|y|^aA(x,y)\nabla u),$$
and their regularizations. The boundary here coincides with $\Sigma:=\{y=0\}$ the \textsl{characteristic manifold}, where the weight becomes degenerate or singular, and this happens respectively when $a>0$ and $a<0$. Accordingly, this class of operators is called degenerate elliptic.
The first motivation for this work is to complete the study started in \cite{SirTerVit1} on local regularity for solutions to degenerate/singular problems including the following
\begin{equation}\label{La1}
-\mathrm{div}\left(|y|^a\nabla u\right)=|y|^af+\mathrm{div}\left(|y|^aF\right)\qquad\mathrm{in \ } B_1.
\end{equation}
In \cite{SirTerVit1}, we treated the regularity of even-in-$y$ solutions (corresponding to Neumann boundary conditions), including the case of variable coefficients. We provided local $\mathcal C^{0,\alpha}$ and $\mathcal C^{1,\alpha}$ estimates, which are uniform as the parameter $\varepsilon\to0^+$, for even solutions of regularized uniformly elliptic problems of the form
\begin{equation}\label{LarhoA}
-\mathrm{div}\left(\rho_\varepsilon^a(y)A(x,y)\nabla u_\varepsilon\right)=\rho_\varepsilon^a(y)f_\varepsilon+\mathrm{div}\left(\rho_\varepsilon^a(y)F_\varepsilon\right)\qquad\mathrm{in \ } B_1,
\end{equation}
where the regularized family of weights $\rho_\varepsilon^a$ is defined as:
\begin{equation}\label{rho}
\rho_\varepsilon^a(y):=\begin{cases}
(\varepsilon^2+y^2)^{a/2}\min\{\varepsilon^{-a},1\} &\mathrm{if \ }a\geq0,\\
(\varepsilon^2+y^2)^{a/2}\max\{\varepsilon^{-a},1\} &\mathrm{if \ }a\leq0.
\end{cases}
\end{equation}
A further motivation comes from a remarkable link between our operators and fractional powers of the Laplacian, from a Dirichlet-to-Neumann point of view, as highlighted in \cite{CafSil1}, when our weights belong to the $A_2$-class; i.e. $a\in(-1,1)$.
Goal of this paper is to deal with odd-in-$y$ solutions to \eqref{La1} (corresponding to Dirichlet boundary conditions), providing local regularity, when possible in the $\varepsilon$-stable sense, by proving uniform bounds for solutions to \eqref{LarhoA}.
Odd solutions make sense as energy solutions in the natural weighted Sobolev spaces whenever $a\in(-\infty,1)$ (in the sense of \S\ref{sect:sobolev}). At first, we notice that can not expect, for the odd solutions, the same estimates as for the even ones, where the regularity results from the combined effect of the ellipticity and the boundary condition. In fact, the function $y|y|^{-a}$ is $\mathcal L_a$-harmonic with finite energy when $a<1$ (in case of $A=\mathbb I$), and for $a\in(0,1)$ is no more than H\"older continuous. We will refer to this special solution as the \textsl{characteristic odd comparison solution}. Similar, yet smoother, characteristic odd comparison solutions exist for the full regularized family of $\varepsilon$-problems (in a rather general setting). Nonetheless, one major obstruction in the study of regularity is the fact that our weights need not to be locally integrable when $a\leq-1$, preventing the application of classical regularity theory such as that developed for degenerate weights of the $A_2$-Muckenhoupt class, starting from the seminal papers \cite{FabKenSer,FabJerKen1,FabJerKen2}. We shall adopt here a different perspective, exploiting suitably tailored Liouville type theorems as main tools. To this aim, a major hindrance is that the measure $|y|^a\mathrm{d}z$ is not absolutely continuous with respect to the Lebesgue measure. In order to overcome this difficulty, one can be guided by the following insight:
\begin{Proposition}\label{prop1}
Let $a\in(-\infty,1)$ and $u\in H^{1,a}(B_1)$ be an odd energy solution to \eqref{La1} in $B_1$ (for simplicity with $F=0$). Then for any $r<1$ the ratio $w=u/y|y|^{-a}\in H^{1,2-a}(B_r)$ and it is an even energy solution to
\begin{equation}\label{BHLaw}
-\mathrm{div}\left(|y|^{2-a}\nabla w\right)=|y|^{2-a}\bar f=|y|^{2-a}\frac{f}{y|y|^{-a}}\qquad\mathrm{in \ }B_r.
\end{equation}
\end{Proposition}
Proposition \ref{prop1} allows the application of the results for even solutions already proved in \cite{SirTerVit1}, providing regularity up to the multiplicative factor $y|y|^{-a}$. Thanks to this observation it is natural to shift the study of regularity for odd solutions to that of even solutions of the auxiliary problem above. A similar perspective has been adopted in \cite{ShaYer} for the obstacle problem in the same singular/degenerate setting.
As an example, by the Schauder estimates in \cite{SirTerVit1}, when the forcing $\bar f=:f/y|y|^{-a}$ in \eqref{BHLaw} is $C^{k,\alpha}$, then the ratio $w=u/y|y|^{-a}$ is locally $C^{k+2,\alpha}$. Thus, we understand that the correct way to face the regularity of odd solutions consists in seeking $\mathcal C^{0,\alpha}$ and $\mathcal C^{1,\alpha}$ bounds for the ratio between the solution and the characteristic odd one, depending on the regularity of the same ratio of the right hand side. This point of view corresponds to \textsl{(possibly higher order and/or non homogeneous) boundary Harnack principle} at $\Sigma$ in the sense of \cite{CafFabMorSal, DesSav, FabJerKen2, JavNeu, JerKen}. It is worthwhile noticing that, when $a\in(-\infty,1)$, then the exponent $2-a$ belongs to $(1,+\infty)$, placing equation \eqref{BHLaw} in the so called super degenerate case, again outside the land of $A_2$-Muckenhoupt weights theory, and which has been treated in \cite{SirTerVit1} when associated with Neumann boundary conditions. Furthermore, looking at the right hand side of \eqref{BHLaw}, we realize that the transition from the odd to the even case requires to pay a cost in terms of more stringent conditions on the forcing term $f$, in the sense that the ratio $\frac{f}{y|y|^{-a}}$ must possess some regularity (integrability at least); in other words, when $a<0$, it means that the forcing term is vanishing with a certain rate at $\Sigma$. In this regard, our results are connected with the recent paper \cite{AllLSha}, where a boundary Harnack principle with right hand side is established in the uniformly elliptic case.
As already pointed out, our results are not limited to the $A_2$-Muckenhoupt class of weights, which restricts $a$ in the interval $(-1,1)$. Nonetheless, we wish to state the following corollary, which joins the results contained in this paper with the Schauder theory for even solutions developed in \cite{SirTerVit1}, concerning full regularity for energy solutions of degenerate or singular problems when the weight is $A_2$-Muckenhoupt and $A=\mathbb I$.
\begin{Corollary}
Let $a\in(-1,1)$, $k\in\mathbb N\cup\{0\}$, $\alpha\in(0,1)$ and consider $u\in H^{1,a}(B_1)$ an energy solution to
\begin{equation*}
-\mathrm{div}\left(|y|^a\nabla u\right)=|y|^af\qquad\mathrm{in \ }B_1.
\end{equation*}
Let us consider the even and odd parts\footnote{Even and odd parts (in $y$) of a function are defined as usual as
$$f_e(x,y)=\frac{f(x,y)+f(x,-y)}{2},\qquad f_o(x,y)=\frac{f(x,y)-f(x,-y)}{2}.$$}
(with respect to $y$) of the forcing term $f$. Let
$$f=f_e+f_o=f_e+y|y|^{-a}\tilde f_e\qquad\mathrm{with \ }f_e,\tilde f_e\in C^{k,\alpha}(B_1).$$
Then
\begin{equation*}
u=u_e+u_o=u_e+y|y|^{-a}\tilde u_e,\qquad\mathrm{with \ }u_e,\tilde u_e\in C^{k+2,\alpha}_{\mathrm{loc}}(B_1).
\end{equation*}
\end{Corollary}
As a next step, we aim at deepening the $\varepsilon$-stability of these estimates with respect to the family of regularized weights \eqref{rho} (also including the variable coefficient case). In other words, we deal with odd-in-$y$ solutions to the family of equations \eqref{LarhoA}.
We will provide local uniform-in-$\varepsilon$ regularity estimates, enlightening their delicate link with curvature issues related with the matrix $A$. As we shall see, also the notion of characteristic solution must be suitably adjusted in order to deal with the variable coefficient cases. Finally, we will apply our results to a family of degenerate/singular equations naturally associated with the euclidean Laplacian expressed in Fermi coordinates in the neighbourhood of an embedded hypersurface.
Below we set the minimal assumptions on the matrix $A$ that we need throughout the paper:
\begin{Assumption}[HA]\label{(HA)}
The matrix $A=(a_{ij})$ is $(n+1,n+1)$-dimensional and symmetric $A=A^T$, has the following symmetry with respect to $\Sigma$:
we have
\begin{equation*}
A(x,y)=JA(x,-y)J,\qquad\qquad\mathrm{with}\qquad\qquad
J
=\left(
\begin{array}{c|c}
\mathbb I_n & 0 \\
\hline
0 & -1
\end{array}
\right).
\end{equation*}
Therefore, $A$ is continuous and satisfies the uniform ellipticity condition $\lambda_1|\xi |^2 \leq A(x,y)\xi\cdot\xi \leq \lambda_2|\xi |^2$, for all $\xi\in\brd{R}^{n+1}$, for every $(x,y)$ and some ellipticity constants $0<\lambda_1\leq\lambda_2$. Moreover, the characteristic manifold $\Sigma$ is assumed to be invariant with respect to $A$; that is, there exists a suitable scalar function $\mu$ such that there exists a positive constant such that
\begin{equation}\label{boundmu}
\frac{1}{C}\leq\mu(x,y)\leq C
\end{equation}
and with
$$A(x,0)\cdot e_{y}=\mu(x,0) e_y.$$
\end{Assumption}
Whenever the hypothesis on $A$ are not specified, we always imply Assumption (HA). From now on, through out the paper, whenever not otherwise specified, in order to simplify the notations, we will work with $A=\mathbb I$ every time this condition is not playing a role in the proofs. In the perspective of Proposition \ref{prop1}, but considering odd solutions for the family of regularized problems in \eqref{LarhoA}, it will be convenient to adopt the following notation on the matrix $A$.
\begin{Notation}[HA+]\label{(HA+)}
We can always write matrix $A$ as:
\begin{equation*}
A(x,y)=\mu(x,y)B(x,y),
\end{equation*}
with
\begin{equation*}
B(x,y)=
\left(
\begin{array}{c|c}
\tilde B(x,y) & T(x,y) \\
\hline
T(x,y) & 1
\end{array}
\right),
\end{equation*}
where $\tilde B$ is a $(n,n)$-dimensional matrix and $T:\brd{R}^{n+1}\to\brd{R}^n$ (we denote by $\tilde A=\mu\tilde B$). We remark here that under our hypothesis on the symmetries of coefficients; one has, for $y<0$
\begin{equation*}
A(x,y)=\mu(x,-y)\left(
\begin{array}{c|c}
\tilde B(x,-y) & -T(x,-y) \\
\hline
-T(x,-y) & 1
\end{array}
\right).
\end{equation*}
\end{Notation}
The structural assumption on the matrix $A$ is consistent with \cite{CafSti}. Moreover, it fits also with the metric induced by Fermi's coordinates, which allow to study phenomena of singularity or degeneration on a characteristic manifold $\Sigma$ which is a generic (regular enough) $n$-dimensional hypersurface embedded in $\brd{R}^{n+1}$.
Hence, the objective will be to consider the ratio $w_\varepsilon$ between odd solutions $u_\varepsilon$ to \eqref{LarhoA} and functions of the form
\begin{equation}\label{veps}
v^a_\varepsilon(x,y)=(1-a)\int_0^y\rho_\varepsilon^{-a}(s)\mu(x,s)^{-1}\mathrm{d}s,
\end{equation}
which play now the role of the \textsl{characteristic odd solution} for the regularized family of weights in the variable coefficients case. It is worthwhile stressing that the characteristic solutions $v^a_\varepsilon$ do not longer solve the homogenous problem, as a dependence on the curvature appears.
As said, we wish to obtain uniform local regularity estimates for $w_\varepsilon$ which will be even solutions to an auxiliary weighted problems having the following structure
\begin{equation}\label{simpleBH}
-\mathrm{div}\left(\rho_\varepsilon^a(v_\varepsilon^a)^2A\nabla w_\varepsilon\right)=\rho_\varepsilon^a(v_\varepsilon^a)^2f_\varepsilon+\mathrm{div}\left(\rho_\varepsilon^a(v_\varepsilon^a)^2F_\varepsilon\right)+\rho_\varepsilon^a(v_\varepsilon^a)^2b_\varepsilon\cdot\nabla w_\varepsilon.
\end{equation}
The new weights appearing in the auxiliary equation are equivalent, though not equal, (using \eqref{boundmu}) to
\begin{equation}
\omega_\varepsilon^a(y)
=\rho_\varepsilon^a(y)(1-a)^2(\chi_\varepsilon^a(y))^2
\end{equation}
where we have defined
\begin{equation}\label{eq:chi}
\chi_\varepsilon^a(y):=\int_0^y\rho_\varepsilon^{-a}(s)\mathrm{d}s\;.
\end{equation}
We remark that, as $a\in(-\infty,1)$, such a class of weights is always super degenerate; indeed, at $\Sigma$, they behave like
$$\omega_\varepsilon^a(y)\sim\begin{cases}
y^2 &\mathrm{if \ }\varepsilon>0\\
|y|^{2-a} &\mathrm{if \ }\varepsilon=0,
\end{cases}$$
with $2-a\in(1,+\infty)$.
Our first main result concerns in fact the even solutions to the auxiliary family of equations \eqref{simpleBH}. It essentially consists in extending (in a non trivial way) the analogous result already obtained in \cite{SirTerVit1} to the new family of weights $\rho_\varepsilon^a(v_\varepsilon^a)^2$.
\begin{Theorem}\label{theo:first}
Let $a\in(-\infty,1)$ and, as $\varepsilon\to0$, let $\{w_\varepsilon\}$ be a family of solutions in $B_1^+$ of \eqref{simpleBH} which are even-in-$y$; that is, satisfying the boundary condition
$$\rho_\varepsilon^a(v_\varepsilon^a)^2\partial_yw_\varepsilon=0 \qquad \mathrm{on \ }\partial^0B_1^+.$$
$1)$ Let $r\in(0,1)$, $\beta>1$, $p_1>\frac{n+3+(-a)^+}{2}$, $p_2,p_3>n+3+(-a)^+$, and $\alpha\in(0,2-\frac{n+3+(-a)^+}{p_1}]\cap(0,1-\frac{n+3+(-a)^+}{p_2}]\cap(0,1-\frac{n+3+(-a)^+}{p_3}]$. Let's moreover take $A$ with continuous coefficients and $\| b_\varepsilon\|_{L^{p_3}(B_1^+,\omega_\varepsilon^a(y)\mathrm{d}z)}\leq b$. There is a positive constant $c$ depending on $a$, $b$, $n$, $\beta$, $p_1$, $p_2$, $p_3$, $\alpha$ and $r$ only such that functions $w_\varepsilon$
satisfy
\begin{equation*}
\|w_\varepsilon\|_{C^{0,\alpha}(B_r^+)}\leq c\left(\|w_\varepsilon\|_{L^\beta(B_1^+,\omega_\varepsilon^a(y)\mathrm{d}z)}+ \| f_\varepsilon\|_{L^{p_1}(B_1^+,\omega_\varepsilon^a(y)\mathrm{d}z)}+\|F_\varepsilon\|_{L^{p_2}(B_1^+,\omega_\varepsilon^a(y)\mathrm{d}z)}\right).
\end{equation*}
$2)$ Let $r\in(0,1)$, $\beta>1$, $p_1,p_2>n+3+(-a)^+$, and $\alpha\in(0,1-\frac{n+3+(-a)^+}{p_1}]\cap(0,1-\frac{n+3+(-a)^+}{p_2}]$. Let $F_\varepsilon=(F^1_\varepsilon,...,F^{n+1}_\varepsilon)$ with the $y$-component vanishing on $\Sigma$: $F^{n+1}_\varepsilon(x,0)= F^y_\varepsilon(x,0)=0$ in $\partial^0B_1^+$.
Let's moreover take $A$ with $\alpha$-H\"older continuous coefficients and $\|b_\varepsilon\|_{L^{2p_2}(B_1^+,\omega_\varepsilon^a(y)\mathrm{d}z)}\leq b$. There is a positive constant $c$ depending on $a$, $b$, $n$, $\beta$, $p_1$, $p_2$, $\alpha$ and $r$ only such that functions $w_\varepsilon$ satisfy
\begin{equation*}
\|w_\varepsilon\|_{C^{1,\alpha}(B_r^+)}\leq c\left(\|w_\varepsilon\|_{L^\beta(B_1^+,\omega_\varepsilon^a(y)\mathrm{d}z)}+ \|f_\varepsilon\|_{L^{p_1}(B_1^+,\omega_\varepsilon^a(y)\mathrm{d}z)}+\|F_\varepsilon\|_{C^{0,\alpha}(B_1^+)}\right).
\end{equation*}
\end{Theorem}
We would like to remark here that local $C^{2,\alpha}$ uniform estimates (up to $\Sigma$) with respect to the regularization can not be proven (for a counterexample we refer to \cite[Remark 5.4]{SirTerVit1}).
When applying Theorem \ref{theo:first} to the quotient
\begin{equation}
w_\varepsilon=\dfrac{u}{v_\varepsilon^a}
\end{equation}
of a solution of \eqref{LarhoA} and the characteristic solution \eqref{veps}, we realise that the actual terms appearing in right hand side of \eqref{simpleBH} depend on the original forcings $f,F$ jointly with the parameters $\mu,T,B$ of the matrix $A$ written as in Notation (HA+). In particular, as shown in \eqref{BHepseq1}, we see the appearance of a drift term involving the $x$-derivatives of $\mu$ which, consequently, need to satisfy a $\mathcal C^{0,\alpha}$ condition. Our main result is Theorem \ref{holderBH}.
We give here below a simplified statement, suitable to be applied to the case of laplacians in Fermi coordinates treated in subsection \S\ref{subsect:fermi}.
\begin{Theorem}\label{holderBHsimple}
Let $a\in(-\infty,1)$, the matrix $A$ written as in Notation (HA+) with $T\equiv 0$. As $\varepsilon\to0$ let $\{u_\varepsilon\}$ be a family of solutions in $B_1^+$ of
\begin{equation*}\label{1oddBH}
\begin{cases}
-\mathrm{div}\left(\rho_\varepsilon^aA\nabla u_\varepsilon\right)=\rho_\varepsilon^af_\varepsilon+\mathrm{div}\left(\rho_\varepsilon^aF_\varepsilon\right) & \mathrm{in \ } B_1^+\\
u_\varepsilon=0 & \mathrm{on \ }\partial^0B_1^+.
\end{cases}
\end{equation*}
Let also $\{v_\varepsilon^a\}$ be the family of solutions defined in \eqref{veps} in $B_1^+$. Denote
$$w_\varepsilon=\frac{u_\varepsilon}{v_\varepsilon^a}\;.$$
$1)$ Assume $\mu$ be Lipschitz continuous, $r\in(0,1)$, $\beta>1$, $p_1>\frac{n+3+(-a)^+}{2}$, $p_2>n+3+(-a)^+$, and $\alpha\in(0,2-\frac{n+3+(-a)^+}{p_1}]\cap(0,1-\frac{n+3+(-a)^+}{p_2}]$. Let's moreover take $A$ with continuous coefficients. There is a positive constant $c$ depending on $a$, $n$, $\beta$, $p_1$, $p_2$, $\alpha$ and $r$ only such that the $w_\varepsilon$ satisfy
\begin{eqnarray*}
\|w_\varepsilon\|_{C^{0,\alpha}(B_r^+)} &\leq& c\left( \|w_\varepsilon\|_{L^\beta(B_1^+,\omega_\varepsilon^a(y)\mathrm{d}z)}+ \|{f_\varepsilon}/{v_\varepsilon^a}\|_{L^{p_1}(B_1^+,\omega_\varepsilon^a(y)\mathrm{d}z)} \right.\\
&& \left. + \|F_\varepsilon^y/(yv_\varepsilon^a)\|_{L^{p_1}(B_1^+,\omega_\varepsilon^a(y)\mathrm{d}z)} +\|F_\varepsilon/v_\varepsilon^a\|_{L^{p_2}(B_1^+,\omega_\varepsilon^a(y)\mathrm{d}z)}\right).
\end{eqnarray*}
$2)$ Assume $\mu\in \mathcal C^{1,\alpha}(B_1^+)$, and let $r\in(0,1)$, $\beta>1$, $p_1>n+3+(-a)^+$, and $\alpha\in(0,1-\frac{n+3+(-a)^+}{p_1}]$. Let $F_\varepsilon=(F^1_\varepsilon,...,F^{n+1}_\varepsilon)$ with the $\alpha$-H\"older continuous ratio between the $y$-component and $v_\varepsilon^a$ vanishing on $\Sigma$: $F^{n+1}_\varepsilon(x,0)/v_\varepsilon^a= F^y_\varepsilon(x,0)/v_\varepsilon^a=0$ in $\partial^0B_1^+$. Let's moreover take $A$ with $\alpha$-H\"older continuous coefficients. There is a positive constant $c$ depending on $a$, $n$, $\beta$, $p_1$, $\alpha$ and $r$ only such that
\begin{eqnarray*}
\|w_\varepsilon\|_{C^{1,\alpha}(B_r^+)} &\leq& c\left( \|w_\varepsilon\|_{L^\beta(B_1^+,\omega_\varepsilon^a(y)\mathrm{d}z)}+ \|{f_\varepsilon}/{v_\varepsilon^a}\|_{L^{p_1}(B_1^+,\omega_\varepsilon^a(y)\mathrm{d}z)} \right.\\
&& \left. + \|F_\varepsilon^y/(yv_\varepsilon^a)\|_{L^{p_1}(B_1^+,\omega_\varepsilon^a(y)\mathrm{d}z)} +\|F_\varepsilon/v_\varepsilon^a\|_{C^{0,\alpha}(B_1^+)}\right).
\end{eqnarray*}
\end{Theorem}
It is worthwhile noticing here that any energy odd solution to \eqref{LarhoA} for $\varepsilon=0$ (under suitable conditions on the matrix and the right hand side) can be approximated by a $\varepsilon$-sequence of solutions to \eqref{LarhoA} satisfying the hypothesis in our regularity results. The same happens for the auxiliary weighed problems solved by the even functions $w=u/y|y|^{-a}$. This is done in details in \cite[Section 2 and 6]{SirTerVit1}.
\begin{remark}
A special, yet fundamental, case is when take $A=\mathbb I$, so that $\mu\equiv 1$ and the family of fundamental comparison odd solutions $v_\varepsilon^a$'s are in fact the $\chi_\varepsilon^a$'s. Nevertheless, it has to be noticed that, in the presence of non trivial curvature, the ratio $v_\varepsilon^a/\chi_\varepsilon^a$ may not be uniformly (in $\varepsilon$) bounded in $\mathcal C^{1,\alpha}(B_1^+)$. Furthermore, in the variable coefficient case, the $\chi_\varepsilon^a$'s are not in the kernel of the corresponding operators, as a (possibly weird) right hand side appears.
\end{remark}
This Theorem finds a natural application to the study of the boundary behaviour of solutions of operators degenerate/singular at embedded manifolds, as shown by the following result.
\begin{Corollary}\label{holderBHFermi}
Let $\Sigma$ be an $n$-dimensional hypersurface embedded in $\brd{R}^{n+1}$, of class $\mathcal C^{3,\alpha}$ and let $d_\Sigma(X)$ denote the signed distance of $X$ to $\Sigma$. Let $a\in(-\infty,1)$, $R>0$ sufficiently small, and consider, as $\varepsilon\to0$, a family of solutions to
\begin{equation*}\label{1oddBHsigma}
\begin{cases}
-\mathrm{div}\left(\rho_\varepsilon^a\circ d_\Sigma\nabla u_\varepsilon\right)=\rho_\varepsilon^a\circ d_\Sigma f_\varepsilon+\mathrm{div}\left(\rho_\varepsilon^a\circ d_\Sigma F_\varepsilon\right) & \mathrm{in \ } B_R\cap\{d_\Sigma(X)>0\}\\
u_\varepsilon=0 & \mathrm{on \ } B_R\cap \Sigma.
\end{cases}
\end{equation*}
Let also $\{\chi_\varepsilon^a\}$ be the family of functions defined in \eqref{eq:chi} in $B_R$. Denote
$$w_\varepsilon=\frac{u_\varepsilon}{\chi_\varepsilon^a\circ d_\Sigma }\;,$$
$1)$ The same conclusion of point 1) of Theorem \ref{holderBHsimple} holds with $v_\varepsilon^a$ replaced by $\chi_\varepsilon^a$, $y$ by $d_\Sigma(X)$ and $e_{n+1}$ by the normal $\nu$ at $\Sigma$.\\
$2)$ The same conclusion of point 2) of Theorem \ref{holderBHsimple} holds in $C^{1,\alpha}(B_r\cap\{y\geq \sqrt{\varepsilon}\})$ where $c$ is independent of $\varepsilon$, and, again, $v_\varepsilon^a$ replaced by $\chi_\varepsilon^a$, $y$ by $d_\Sigma(X)$ and $e_{n+1}$ by the normal $\nu$ at $\Sigma$.\\
\end{Corollary}
\begin{remark}
In particular, letting $\varepsilon\to 0$ we find $C^{1,\alpha}(B_r^+)$ estimates in the degenerate/singular case, though not in the full $\varepsilon$-stable sense. The reason is the possible lack of uniform-in-$\varepsilon$ smoothness of the ratio $v_\varepsilon^a/\chi_\varepsilon^a$.
\end{remark}
\subsection{Proof of Corollary \ref{holderBHFermi}}\label{subsect:fermi}
As already mentioned, the structural assumption on the matrix $A$ done in Assumption (HA) with Notation (HA+) fits also with the metric induced by Fermi's coordinates around the characteristic manifold $\Sigma$ (see \cite{PacWei}). Let $\Sigma$ be an oriented regular enough hypersurface embedded in $\brd{R}^{n+1}$. We are concerned with operators associated with Dirichlet energies of the form
\begin{equation*}
\int_{\{d_\Sigma(X)>0\}}(\rho_\varepsilon^a\circ d_\Sigma)(X)|\nabla u|^2,
\end{equation*}
with $a\in\brd{R}$,
$X \in \brd{R}^{n+1}$ and $d_\Sigma(\cdot)$ the signed distance function to $\Sigma$. Let $g_e$ be the Euclidean metric on $\brd{R}^{n+1}$ and denote by $\nu$ the unit normal vector field on $\Sigma$. We define Fermi coordinates in a tubular neighborhood of $\Sigma$ as follows: let $z\in\Sigma$ and $y\in\brd{R}$, and define
\begin{equation*}
Z(z,y):=z+y\,\nu(z).
\end{equation*}
Nevertheless, $z$ and $y\,\nu(z)$ belong to $\brd{R}^{n+1}$. Given $y \in \brd{R}$, we define
$$\Sigma_y:=\{Z(z,y) \in \brd{R}^{n+1}\ : \ z\in\Sigma\}. $$
Following Lemma 6.1 in \cite{PacWei}, one has that the induced metric on $\Sigma_y$ is given by
\begin{equation*}
g^y= g^0-2yh^0+y^2h^0\otimes h^0,
\end{equation*}
where $g^0$ is the induced metric on $\Sigma$, $h^0$ is the second fundamental form on $\Sigma$ and $h^0\otimes h^0$ its square; namely we have
$$
h^0(t_1,t_2)=-g^0(\nabla_{t_1}\nu,t_2)
$$
for all $t_1,t_2$ on the tangent bundle of $\Sigma$. Notice in particular that, in local coordinates, the terms $g^0,h^0,h^0\otimes h^0$ depend only on $z$.
Therefore, invoking Lemma 6.3 in \cite{PacWei}, one finally has
$$
Z^*g_e=g^y+dy^2
$$
where $g^y$ is considered as a family of metrics on the tangent bundle of $\Sigma$, depending smoothly on $y$ in a neighborhood of $0$ in $\brd{R}$.
\begin{comment}
All in all, one obtains the following expression for the Euclidean laplacian in the Fermi coordinates
$$
\Delta_e=\Delta_{Z^*g_e}= \partial_y^2-H_y \partial_y +\Delta_{g^y}
$$
where $H_y$ is the mean curvature of $\Sigma_y$.
We now express the operator $\mathrm{div}\Big ( d_\Sigma(X)^a \,\nabla\,\,\Big )$ in those coordinates. For $y$ close enough to $0$, we have
$$
\mathrm{div}\Big ( d_\Sigma(X)^a \,\nabla u \Big)=|y|^a \Delta_e+\nabla_e |y|^a \cdot \nabla_e u.
$$
The well known formula holds for the gradient operator $\partial^i_e=g_e^{ij}\partial_j$ where we used Einstein convention. Furthermore invoking Lemma 3.1 in Wei's lecture notes, we have
$$
\Delta_e=\partial_y^2+\Delta_\Sigma-y|A|^2 \partial_y +y\Big [ a_{ij}(z,y)\partial_{ij}+b_i(z,y)\partial_i\Big ] +y^2\,\,c(y,z)\partial_y
$$
where the coefficients $a_{ij}, b_i,c$ are smooth and bounded and $|A|^2$ is the squared length of the second fundamental form on $\Sigma$. The term $\nabla_e |y|^a \cdot \nabla_e u$ simply gives $\partial_y |y|^a \, \partial_y u$, whereas the other one gives,
$$
|y|^a \Delta_eu=|y|^a \Big [ \partial_y^2u-y |A|^2\partial_yu + \Delta_\Sigma u \Big ] + y |y|^a \Big [ a_{ij}(z,y)\partial_{ij}u+b_i(z,y)\partial_i u\Big ] +y^2|y|^a\,\,c(y,z)\partial_yu.
$$
Therefore, grouping the terms in a convenient way, one obtains that the operator in Fermi coordinates is
$$
\mathrm{div}\Big ( d_\Sigma(X)^a \,\nabla u \Big)= |y|^a \Big [ \partial_y^2u + \Delta_\Sigma u \Big ]+\partial_y |y|^a \, \partial_y u + y|y|^a B(z,y) +y^2 |y|^a C(z,y)
$$
where $B,C$ are smooth and bounded in their arguments. Because of the formulas, $\mathrm{det}(g_{Z^*g_e})=\mathrm{det}(g^y)$ and $\mathrm{div}_e u=\frac{1}{\sqrt{\det g^y}}\partial^i \Big ( \sqrt{\det g^y} u_i\Big ) $, one obtains finally
\begin{equation}
\mathrm{div}\Big ( d_\Sigma(X)^a \,\nabla u \Big)=\mathrm{div} \Big ( {\color{red}|y|^a}A(z,y)\nabla u \Big ) + {\color{red}\sqrt{\det g^y}}y|y|^a B(z,y) +{\color{red}\sqrt{\det g^y}}y^2 |y|^a C(z,y)
\end{equation}
\end{comment}
In other words, we are obtaining a quadratic form for $v(z,y)=u(Z(z,y))$ of the form
\begin{equation*}
\int_{0}^{y_0} \rho_\varepsilon^a(y)\int_{\Sigma_y}\left(|\nabla_{g^y}v|^2+|\partial_yv|^2\right) \sqrt{\mathrm{det} g^y}.
\end{equation*}
Recall that the variation with respect to $y$ of of the volume form of the parallel hypersurfaces $\Sigma_y$ satisfy the equation:
\begin{equation}\label{eq:gy}
H_y=-\dfrac{1}{\sqrt{\mathrm{det} g^y}}\dfrac{d}{dy}\sqrt{\mathrm{det} g^y}.
\end{equation}
Hence, by considering a parametrization of $\Sigma$ of the form $z=\psi(x)$ with $x\in \brd{R}^n$, then one obtains for $w(x,y)=v(\psi(x),y)$
\begin{equation*}
\int \rho_\varepsilon^a(y)A\nabla w\cdot\nabla w,
\end{equation*}
where
\begin{equation*}
A(x,y)=
\left(
\begin{array}{c|c}
\tilde A (x,y) & 0 \\
\hline
0 & 1
\end{array}
\right)\cdot\sqrt{\det g^y}.
\end{equation*}
We remark that the matrix $A$ satisfies Assumption (HA), and can be expressed as in Notation (HA+) with $\mu(x,y)=\sqrt{\det g^y}$. As $\Sigma\in\mathcal C^{3,\alpha}$, we have $\mu\in\mathcal C^{1,\alpha}(B_{r_0}^+)$ for $r_0$ small enough. Hence we are in the position to apply Theorem \ref{holderBHsimple}. Next we have to compare the two families $v_\varepsilon^a$ and $\chi_\varepsilon^a$. At first, in order to prove point 1) we remark that Proposition \ref{c0alphaG} ensures uniform-in-$\varepsilon$ $\mathcal C^{0,\alpha}$ estimates for the ratio $v_\varepsilon^a/\chi_\varepsilon^a$.
Using \eqref{eq:gy}, we infer that $\partial_y\mu(\cdot,0)\in\mathcal C^{1,\alpha}(B_{r_0}^+)$ and finally, by virtue of Proposition \ref{c1alphaG}, we obtain that also the ratio $v_\varepsilon^a/\chi_\varepsilon^a$ satisfies the desired uniform bounds in $\mathcal C^{1,\alpha}(B_r\cap\{y\geq\sqrt{\varepsilon}\})$, for $r<r_0$.
\begin{comment}
\subsection{Ideas}
Study of odd solutions.\\\\
When $a\in(-1,1)$, then $|y|^a$ is still locally integrable, hence one can do the analysis as in EVEN obtaining $C^{0,\alpha}$ uniform estimates both for the case of $f$ and $\mathrm{div}F$. Moreover, if $a\in(-1,0]$ one can prove uniform estimates in $C^{1,\alpha}$ in the case of $f$.\\\\
When $a\leq-1$, then $|y|^a$ is not locally integrable anymore, and so one considers the suitable approximation of the ratio
$$\frac{u}{y|y|^{-a}},$$
given by
$$\frac{u_\varepsilon}{(1-a)\int_0^y(\varepsilon^2+t^2)^{-a/2}},$$
which are solutions for degenerate problem of order $b=2$; that is the weight is for $\varepsilon>0$
$$\rho_\varepsilon^a(y)\left((1-a)\int_0^y(\varepsilon^2+t^2)^{-a/2}\right)^2\sim y^2.$$
\end{comment}
\subsection*{\sl Notations.}
Below is the list of symbols we shall use throughout this paper.\\
\begin{tabular}{ll}
$\brd{R}^{n+1}_+=\brd{R}^n\times(0,+\infty)$ & $z=(x,y)$ with $x\in\brd{R}^n$, $y>0$ \\
$\Sigma=\{y=0\}$ & characteristic manifold \\
$B_r^+=B_r\cap\{y>0\}$ & half ball \\
$\partial^+B_r^+=S^n_+(r)=\partial B_r\cap\{y>0\}$ & upper boundary of the half ball \\
$\partial^0B_r^+=B_r\cap\{y=0\}$ & flat boundary of the half ball \\
$\rho_\varepsilon^a(y)=\left(\varepsilon^2+y^2\right)^{a/2}$ & regularized weight \\
$\omega_\varepsilon^a(y)=\rho_\varepsilon^a(y)\pi_\varepsilon^a(y)$ & regularized auxiliary weight \\
$\mathcal L_{\rho_\varepsilon^a}u=\mathrm{div}\left(\rho_\varepsilon^a(y)A(x,y)\nabla u\right)$ & regularized operator \\
$H^{1}(\Omega,\rho_\varepsilon^a(y)\mathrm{d}z)$ & weighted Sobolev space given by the completion of $C^\infty(\overline\Omega)$ \\
$H^{1}_0(\Omega,\rho_\varepsilon^a(y)\mathrm{d}z)$ & weighted Sobolev space given by the completion of $C^\infty_c(\Omega)$ \\
$\tilde H^{1}(\Omega,\rho_\varepsilon^a(y)\mathrm{d}z)$ & weighted Sobolev space given by the completion of $C^\infty_c(\overline\Omega\setminus\Sigma)$ \\
$H^{1,a}(\Omega)=H^{1}(\Omega,|y|^a\mathrm{d}z)$ & weighted Sobolev space for $\varepsilon=0$ \\
$\partial_y^au=|y|^a\partial_yu$ & "weighted" derivative \\
$y|y|^{-a}$ & characteristic odd solution \\
$v_\varepsilon^a$ & characteristic odd solution in presence of $A$ and $\varepsilon>0$\\
$a^+=\max\{a,0\}$ &\\
\end{tabular}
\section{Functional setting and preliminary results}\label{sect:sobolev}
In this section we collect the natural notions of Sobolev spaces, and their main properties, needed to work in our degenerate or singular context (for further details see \cite[Section 2]{SirTerVit1}). Let $\Omega\subset\brd{R}^{n+1}$ be non empty, open and bounded. Denoting by $C^\infty(\overline\Omega)$ the set of real functions $u$ defined on $\overline\Omega$ such that the derivatives $D^\alpha u$ can be continuously extended to $\overline\Omega$ for all multiindices $\alpha$, then for any $a\in\mathbb{R}$, $\varepsilon\geq0$ we define the weighted Sobolev space $H^1(\Omega,\rho_\varepsilon^a(y)\mathrm{d}z)$ as the closure of $C^\infty(\overline\Omega)$ with respect to the norm
$$\|u\|_{H^{1}(\Omega,\rho_\varepsilon^a(y)\mathrm{d}z)}=\left(\int_{\Omega}\rho_\varepsilon^au^2+\int_{\Omega}\rho_\varepsilon^a|\nabla u|^2\right)^{1/2}.$$
To simplify the notation we will denote
$$H^{1,a}(\Omega)=H^{1}(\Omega,|y|^a\mathrm{d}z)=H^{1}(\Omega,\rho_0^a(y)\mathrm{d}z).$$
In the same way, we define $H^{1}_0(\Omega,\rho_\varepsilon^a(y)\mathrm{d}z)$ as the closure of $C^\infty_c(\Omega)$ with respect to the norm
$$\|u\|_{H^{1}_0(\Omega,\rho_\varepsilon^a(y)\mathrm{d}z)}=\left(\int_{\Omega}\rho_\varepsilon^a|\nabla u|^2\right)^{1/2}.$$
We will denote by $\tilde H^1(\Omega,\rho_\varepsilon^a(y)\mathrm{d}z)$ the closure of $C^\infty_c(\overline\Omega\setminus\Sigma)$ with respect to the norm $\|\cdot\|_{H^1(\Omega,\rho_\varepsilon^a(y)\mathrm{d}z)}$. In particular, when $a<1$, there is a natural isometry (on balls $B$ centered in a point on $\Sigma$ of any radius)
$$T_\varepsilon^a: \tilde H^{1}(B,\rho_\varepsilon^a(y)\mathrm{d}z)\to\tilde H^{1}(B): u\mapsto v=\sqrt{\rho_\varepsilon^a} u,$$
where $\tilde H^{1}(B)$ is endowed with the equivalent norm with squared expression
\[
Q_\varepsilon(v)=\int_B|\nabla v|^2 +\left[ \left(\dfrac{\partial_y \rho_\varepsilon^a}{2\rho_\varepsilon^a}\right)^2+\partial_y\left(\dfrac{\partial_y \rho_\varepsilon^a}{2\rho_\varepsilon^a}\right)\right]v^2-\int_{ \partial B}\dfrac{\partial_y \rho_\varepsilon^a}{2\rho_\varepsilon^a}yv^2\;,
\]
(this is detailed in the appendix \ref{app:isometries}). We remark that both in the super singular and super degenerate cases, that is when $a\in(-\infty,-1]\cup[1,+\infty)$ and $\varepsilon=0$, when the weight is taken outside the $A_2$-Muckenhoupt class, one has
\begin{equation}\label{tildeH1=H1}
H^{1,a}(\Omega)=\tilde H^{1,a}(\Omega)\,.
\end{equation}
This happens for very opposite reasons: roughly speaking, when $a\leq-1$ then the singularity is so strong to force the function to annihiliate on $\Sigma$ (we will call this case the super singular case). Instead, when $a\geq1$, then the strong degeneracy leaves enough freedom to the function to allow it to be very irregular through $\Sigma$ (we will call this case the super degenerate case). In the latter case, $\Sigma$ has vanishing capacity with respect to the energy $\int |y|^a |\nabla u|^2$.
The Sobolev embedding theorems are stated in details in \cite{SirTerVit1} as inequalities which are uniform in $\varepsilon$. This point is fundamental in order to develop a local regularity theory which is stable with respect to the regularization parameter $\varepsilon$. Hence, following some results contained in \cite{Haj}, the critical Sobolev exponents do depend on how the weighted measures $\mathrm{d}\mu=\rho_\varepsilon^a(y)\mathrm{d}z$ scale on balls of small radius $r>0$: one can check that there exists $b,d>0$ independent from $\varepsilon\geq0$ (in the locally integrable case $a>-1$) such that for small radii
$$\mu(B_r(z))\geq br^d.$$
So, we can define the effective dimension
$$d=n+1+a^+=n^*(a),$$
and the Sobolev optimal exponent is
$$2^*(a)=\frac{2d}{d-2}=\frac{2(n+1+a^+)}{n+a^+-1}.$$
For details one can refer to Theorems 2.4 and 2.5 in \cite{SirTerVit1}.
In the very same way one can define weighted Sobolev spaces for the class of weights $\omega_\varepsilon^a$; that is, the spaces $H^1(\Omega,\omega_\varepsilon^a(y)\mathrm{d}z)=\tilde H^1(\Omega,\omega_\varepsilon^a(y)\mathrm{d}z)$ (the equality is due to the fact that $\omega_\varepsilon^a$ is always a super degenerate weight as $a<1$) and $H^1_0(\Omega,\omega_\varepsilon^a(y)\mathrm{d}z)$.
In this case one can check that there exist two positive constants $\overline b,\overline d>0$ independent on $\varepsilon\geq0$ such that $\mathrm{d}\overline\mu=\omega_\varepsilon^a(y)\mathrm{d}z$ has the following growth condition on small balls of radius $r>0$
$$\overline\mu(B_r(z))\geq \overline br^{\overline d},$$
and the effective dimension is given by
$\overline d=n+1+2+(-a)^+=n+3+(-a)^+=\overline n^*(a)$. Hence one can state the following
\begin{Theorem}\label{sobemb1}
Let $a\in(-\infty,1)$, $n\geq1$, $\varepsilon\geq0$ and $u\in C^1_c(\Omega)$. Then there exists a constant which does not depend on $\varepsilon\geq0$ such that
\begin{equation*}\label{soboBH}
\left(\int_{\Omega}\omega_\varepsilon^a|u|^{\overline2^*(a)}\right)^{2/\overline2^*(a)}\leq c(\overline d,\overline b,\Omega)\int_{\Omega}\omega_\varepsilon^a|\nabla u|^2,
\end{equation*}
where the optimal embedding exponent is
\begin{equation*}\label{2*a}
\overline2^*(a)=\frac{2\overline d}{\overline d-2}=\frac{2(n+3+(-a)^+)}{n+(-a)^++1}.
\end{equation*}
\end{Theorem}
\subsection{Energy solutions}
Throughout the paper, we are going to consider different elliptic equations depending on different families of weights. Nevertheless, we will deal with right hand sides having forcing terms, terms expressed by the divergence of a given field and drift terms (we will see that any other possible term that will appear can be translated in one of these). In order to give an unified definition of energy solutions to weighted problems, we will consider a generic measurable weight function $w$, and define an energy solution $u$ in $B_1$ to
\begin{equation}\label{divP}
-\mathrm{div}\left(w A\nabla u\right)=wf+\mathrm{div}\left(wF\right)+w\,b\cdot\nabla u\qquad\mathrm{in \ }B_1.
\end{equation}
We say that $u\in H^1(B_1,w\mathrm{d}z)$ is an energy solution to \eqref{divP} if
\begin{equation}\label{variationP}
\int_{B_1}wA(x,y)\nabla u\cdot\nabla\phi=\int_{B_1}wf\phi-\int_{B_1} wF\cdot\nabla\phi+\int_{B_1}w(b\cdot\nabla u)\phi,\qquad\forall\phi\in C^{\infty}_c(B_1)\cap H^1(B_1,w\mathrm{d}z),
\end{equation}
any time the terms in the right hand side give sense to the previous integrals. We remark that we are not assuming local integrability of the weight, and this is the reason why we must consider test functions in the suitable weighted Sobolev space.
Now, we recall the consequent definition of energy solutions in case the weight term is given by $\rho_\varepsilon^a(y)$, with $a\in\brd{R}$ and $\varepsilon\geq0$ (the following definition is contained in \cite{SirTerVit1}). Let us consider the following problem
\begin{equation}\label{La}
-\mathrm{div}\left(\rho_\varepsilon^a A\nabla u\right)=\rho_\varepsilon^af+\mathrm{div}\left(\rho_\varepsilon^aF\right)\qquad\mathrm{in \ }B_1.
\end{equation}
We say that $u\in H^{1}(B_1,\rho_\varepsilon^a(y)\mathrm{d}z)$ is an energy solution to \eqref{La} if
\begin{equation}\label{variationLa}
\int_{B_1}\rho_\varepsilon^aA(x,y)\nabla u\cdot\nabla\phi=\int_{B_1}\rho_\varepsilon^af\phi-\int_{B_1} \rho_\varepsilon^aF\cdot\nabla\phi,\qquad\forall\phi\in C^{\infty}_c(B_1)\cap H^{1}(B_1,\rho_\varepsilon^a(y)\mathrm{d}z).
\end{equation}
We remark that the condition in \eqref{variationLa} can be equivalently expressed testing with any $\phi\in C^\infty_c(B_1\setminus\Sigma)$ if $a\in(-\infty,-1]\cup[1,+\infty)$ and $\varepsilon=0$.
In order to give a sense to energy solutions to \eqref{La} we need the following minimal hypothesis on the right hand side.
\begin{Assumption}[H$f\rho_\varepsilon^a$]
Let $a\in(-1,+\infty)$. Then if $n\geq2$ or $n=1$ and $a^+>0$, the forcing term $f$ in \eqref{La} belongs to $L^p(B_1,\rho_\varepsilon^a(y)\mathrm{d}z)$ with $p\geq(2^*(a))'$ the conjugate exponent of $2^*(a)$; that is,
$$(2^*(a))'=\frac{2(n+1+a^+)}{n+a^++3}.$$
If $n=1$ and $a^+=0$ then $f\in L^p(B_1,\rho_\varepsilon^a(y)\mathrm{d}z)$ with $p>1$.
Let $a\in(-\infty,-1]$. Then if $n\geq2$, the condition on the forcing term is $(\rho_\varepsilon^a)^{1/2}f\in L^p(B_1)$ with $p\geq(2^*(a))'=(2^*)'$. If $n=1$, then any $p>1$ is allowed.
\end{Assumption}
\begin{Assumption}[H$F\rho_\varepsilon^a$]
Let $a\in(-1,+\infty)$. The condition on the field $F=(F^1,...,F^{n+1})$ in \eqref{La} is $F\in L^p(B_1,\rho_\varepsilon^a(y)\mathrm{d}z)$ with $p\geq2$. Let $a\in(-\infty,-1]$. Then the condition is $(\rho_\varepsilon^a)^{1/2}F\in L^p(B_1)$ with $p\geq2$.
\end{Assumption}
\subsection{Some preliminary results on the auxiliary equation}
\begin{comment}
\begin{Assumption}[H$\overline f_\varepsilon$]
Let $a\in(-\infty,1)$. Then the forcing term $\overline f$ in \eqref{LaBH} belongs to $L^p(B_1,\omega_\varepsilon^a(y)\mathrm{d}z)$ with $p\geq(\overline 2^*(a))'$ the conjugate exponent of $\overline 2^*(a)$; that is,
$$(\overline 2^*(a))'=\frac{2(n+3+(-a)^+)}{n+(-a)^++5}.$$
\end{Assumption}
\begin{Assumption}[H$\overline F_\varepsilon$]
Let $a\in(-\infty,1)$. Then the field term $\overline F$ in \eqref{LaBH} belongs to $L^p(B_1,\omega_\varepsilon^a(y)\mathrm{d}z)$ with $p\geq2$.
\end{Assumption}
We remark that under this minimal integrability assumption on the field $\overline F$, the forcing term
\begin{Assumption}[H$V_\varepsilon$]
Let $a\in(-\infty,1)$. Then the $0$-order term $V$ in \eqref{LaBH} belongs to $L^p(B_1,\omega_\varepsilon^a(y)\mathrm{d}z)$ with
$$p\geq\frac{\overline d}{2}=\frac{n+3+(-a)^+}{2}.$$
\end{Assumption}
\end{comment}
We are concerned with local regularity of energy odd solutions to \eqref{La} with $a\in(-\infty,1)$ and $\varepsilon\geq0$. Our analysis relies in the validity of suitable Liouville type theorems which hold true whenever $a>-1$; that is, when the weight $|y|^a$ is locally integrable. In order to ensure regularity results also in the super singular case $a\leq-1$, we will consider the ratio $w$ between the odd solution $u$ and the function $v^a_\varepsilon$ defined in \eqref{veps}
which is odd and satisfies
\begin{equation}\label{vepsa}
\mathrm{div}\left(\rho_\varepsilon^aA\cdot\nabla v_\varepsilon^a\right)=\mathrm{div}_x\left(\rho_\varepsilon^a\mu\tilde B\cdot \nabla_xv_\varepsilon^a\right)+\mathrm{div}_x(T) \qquad\mathrm{in \ } B_1,
\end{equation}
whenever the right hand side in the equation satisfies suitable integrability assumptions and the matrix $A$ is written as in Notation (HA+). As we have already remarked in the introduction, such a function $v^a_\varepsilon$ plays the role of the characteristic odd solution $y|y|^{-a}$ in presence of a matrix and of regularization.
The following Lemma is a formal computation
\begin{Lemma}\label{rhofrac}
Let $a\in\mathbb{R}$, $\varepsilon>0$ and let $u,v$ be solutions to
$$-\mathrm{div}\left(\rho_\varepsilon^aA\nabla u\right)=\rho_\varepsilon^af,\qquad-\mathrm{div}\left(\rho_\varepsilon^aA\nabla v\right)=\rho_\varepsilon^ag\qquad\mathrm{in \ } B_1,$$
with $v>0$ and $A$ satisfying Assumption (HA). Then the function $w=u/v$ is solution to
\begin{equation*}
-\mathrm{div}\left(\rho_\varepsilon^av^2A\nabla w\right)=\rho_\varepsilon^avf-\rho_\varepsilon^aug\qquad\mathrm{in \ } B_1.
\end{equation*}
\end{Lemma}
\proof
Let recall $\rho=\rho_\varepsilon^a$. Then
\begin{eqnarray*}
-\mathrm{div}\left(\rho v^2A\nabla w\right)&=&-\mathrm{div}\left(\rho v^2A\left(\frac{\nabla u}{v}-\frac{u\nabla v}{v^2}\right)\right)\nonumber\\
&=&-\mathrm{div}\left(\rho vA\nabla u-\rho uA\nabla v\right)\nonumber\\
&=&-v\mathrm{div}\left(\rho A\nabla u\right)-\rho\nabla v\cdot(A\nabla u)+u\mathrm{div}\left(\rho A\nabla v\right)+\rho\nabla u\cdot(A\nabla v)\nonumber\\
&=&-v\mathrm{div}\left(\rho A\nabla u\right)-\rho\nabla v\cdot(A\nabla u)+u\mathrm{div}\left(\rho A\nabla v\right)+\rho\nabla v\cdot(A^T\nabla u)\nonumber\\
&=&\rho vf-\rho ug.
\end{eqnarray*}
\endproof
The new class of weights appearing in the auxiliary equation for the ratio $w=u/v^a_\varepsilon$ is given by $\rho_\varepsilon^a(v_\varepsilon^a)^2$ and it will be equivalent (using \eqref{boundmu}) to
\begin{equation*}
\omega_\varepsilon^a(y)=\rho_\varepsilon^a(y)\pi_\varepsilon^a(y)=\rho_\varepsilon^a(y)(1-a)^2(\chi_\varepsilon^a(y))^2=\rho_\varepsilon^a(y)\left((1-a)\int_0^y\rho_\varepsilon^{-a}(s)\mathrm{d}s\right)^2.
\end{equation*}
We remark that, considering $a\in(-\infty,1)$, such a class of weights is always super degenerate; that is, at $\Sigma$
$$\omega_\varepsilon^a(y)\sim\begin{cases}
y^2 &\mathrm{if \ }\varepsilon>0\\
|y|^{2-a} &\mathrm{if \ }\varepsilon=0,
\end{cases}$$
with $2-a\in(1,+\infty)$.
Formal computations show that the auxiliary equation for $w$ (which corresponds to equation \eqref{BHLaw} in Proposition \ref{prop1} for $\varepsilon=0$ and $A=\mathbb I$) in $B_r$ for any $r<1$ is given by
\begin{equation}\label{BHepseq}
-\mathrm{div}\left(\rho_\varepsilon^a(v_\varepsilon^a)^2A\nabla w\right)=\rho_\varepsilon^a(v_\varepsilon^a)^2\left(\overline f +Vw-\frac{\overline F\cdot\nabla v_\varepsilon^a}{v_\varepsilon^a}\right)+\mathrm{div}\left(\rho_\varepsilon^a(v_\varepsilon^a)^2\overline F\right)\qquad\mathrm{in \ } B_r,
\end{equation}
with
$$\overline f:=\frac{f}{v_\varepsilon^a},\qquad\overline F:=\frac{F}{v_\varepsilon^a}$$
and
$$V:=\frac{\mathrm{div}_x\left(\mu\tilde B\cdot \nabla_xv_\varepsilon^a\right)}{v_\varepsilon^a}+\frac{\mathrm{div}_x(T)}{\rho_\varepsilon^av_\varepsilon^a}.$$
Actually we can rewrite the $0$-order term, obtaining that the auxiliary equation for $w$ in $B_r$ is given by
\begin{eqnarray}\label{BHepseq1}
-\mathrm{div}\left(\rho_\varepsilon^a(v_\varepsilon^a)^2A\nabla w\right)&=&\rho_\varepsilon^a(v_\varepsilon^a)^2\left(\overline f -\frac{\overline F\cdot\nabla v_\varepsilon^a}{v_\varepsilon^a}\right)+\mathrm{div}\left(\rho_\varepsilon^a(v_\varepsilon^a)^2\overline F\right)\nonumber\\
&&+\mathrm{div}_x\left(\rho_\varepsilon^a(v_\varepsilon^a)^2b^{\tilde A}w\right)-\rho_\varepsilon^a(v_\varepsilon^a)^2\left(b^{\tilde A}\cdot b^{\mathbb I}w+b^{\tilde A}\cdot\nabla_xw\right)\nonumber\\
&&+\mathrm{div}_x\left(\rho_\varepsilon^a(v_\varepsilon^a)^2\overline Tw\right)-\rho_\varepsilon^a(v_\varepsilon^a)^2\left(\overline T\cdot b^{\mathbb I}w+\overline T\cdot\nabla_xw\right),
\end{eqnarray}
where for a $(n,n)$-dimensional matrix $M$
\begin{equation*}
b^M=M\cdot\frac{\nabla_xv_\varepsilon^a}{v_\varepsilon^a},\qquad\mathrm{and}\qquad \overline T=\frac{T}{\rho_\varepsilon^av_\varepsilon^a}.
\end{equation*}
Thus we can write the equation the following form:
\begin{eqnarray}\label{BHepseq2}
-\mathrm{div}\left(\rho_\varepsilon^a(v_\varepsilon^a)^2A\nabla w\right)&=&\rho_\varepsilon^a(v_\varepsilon^a)^2f+\mathrm{div}\left(\rho_\varepsilon^a(v_\varepsilon^a)^2 F_1\right)\nonumber\\
&&+\mathrm{div}_x\left(\rho_\varepsilon^a(v_\varepsilon^a)^2F_2w\right)+\rho_\varepsilon^a(v_\varepsilon^a)^2Vw+\rho_\varepsilon^a(v_\varepsilon^a)^2b\cdot\nabla_xw.
\end{eqnarray}
We would like to prove that $w$ is an even energy solution to \eqref{BHepseq2} in $B_r$ in the sense that $w\in H^1(B_r,\omega_\varepsilon^a(y)\mathrm{d}z)$ and satisfies
\begin{eqnarray*}
\int_{B_r}\rho_\varepsilon^a(v_\varepsilon^a)^2A\nabla w\cdot\nabla\phi&=&\int_{B_r}\rho_\varepsilon^a(v_\varepsilon^a)^2f\phi-\int_{B_r}\rho_\varepsilon^a(v_\varepsilon^a)^2 F_1\cdot\nabla\phi\nonumber\\
&&-\int_{B_r}\rho_\varepsilon^a(v_\varepsilon^a)^2F_2w\cdot\nabla_x\phi\nonumber\\
&&+\int_{B_r}\rho_\varepsilon^a(v_\varepsilon^a)^2Vw\phi+\int_{B_r}\rho_\varepsilon^a(v_\varepsilon^a)^2(b\cdot\nabla_xw)\phi,
\end{eqnarray*}
for any $\phi\in C^{\infty}_c(B_r\setminus\Sigma)$ (as we have already remarked, super degeneracy allows us to take test functions compactly supported away from $\Sigma$). In order to give a sense to energy solutions to \eqref{BHepseq2} we need the following minimal hypothesis on the right hand side.
\begin{Assumption}[H$f\omega_\varepsilon^a$]
Let $a\in(-\infty,1)$. Then the forcing term $f$ in \eqref{BHepseq2} belongs to $L^p(B_1,\omega_\varepsilon^a(y)\mathrm{d}z)$ with $p\geq(\overline 2^*(a))'$ the conjugate exponent of $\overline 2^*(a)$; that is,
$$(\overline 2^*(a))'=\frac{2(n+3+(-a)^+)}{n+(-a)^++5}.$$
\end{Assumption}
\begin{Assumption}[H$F_1\omega_\varepsilon^a$]
Let $a\in(-\infty,1)$. Then the field term $F_1$ in \eqref{BHepseq2} belongs to $L^p(B_1,\omega_\varepsilon^a(y)\mathrm{d}z)$ with $p\geq2$.
\end{Assumption}
\begin{Assumption}[H$F_2\omega_\varepsilon^a$]
Let $a\in(-\infty,1)$. Then the field term $F_2$ in \eqref{BHepseq2} belongs to $L^p(B_1,\omega_\varepsilon^a(y)\mathrm{d}z)$ with $p\geq \overline d=n+3+(-a)^+$.
\end{Assumption}
\begin{Assumption}[H$V\omega_\varepsilon^a$]
Let $a\in(-\infty,1)$. Then the $0$-order term $V$ in \eqref{BHepseq2} belongs to $L^p(B_1,\omega_\varepsilon^a(y)\mathrm{d}z)$ with
$$p\geq\frac{\overline d}{2}=\frac{n+3+(-a)^+}{2}.$$
\end{Assumption}
\begin{Assumption}[H$b\omega_\varepsilon^a$]
Let $a\in(-\infty,1)$. Then the field $b$ the drift term in \eqref{BHepseq2} belongs to $L^p(B_1,\omega_\varepsilon^a(y)\mathrm{d}z)$ with $p\geq \overline d=n+3+(-a)^+$.
\end{Assumption}
We will need the following important result, which contains also Proposition \ref{prop1} when $\varepsilon=0$ and $A=\mathbb{I}$.
\begin{Proposition}\label{BoundaryHarnackeps}
Let $a\in(-\infty,1)$, $\varepsilon\geq0$ and let $u_\varepsilon\in H^1(B_1,\rho_\varepsilon^a(y)\mathrm{d}z)$ be an odd energy solution to \eqref{La} in $B_1$. Then, fixed $0<r<1$, the function $w_\varepsilon=u_\varepsilon/v_\varepsilon^a$ is an even energy solution in $H^1(B_r,\omega_\varepsilon^a(y)\mathrm{d}z)$ to \eqref{BHepseq1}, provided that the right hand side satisfies the suitable integrability assumptions stated above.
\end{Proposition}
\proof
First, we want to show that $w_\varepsilon\in H^1(B_r,\rho_\varepsilon^a(y)(v_\varepsilon^a)^2(x,y)\mathrm{d}z)$. We remark that since $\frac{1}{C}\leq\mu\leq C$ and since the weight is super degenerate, we have that at $\Sigma$
$$\rho_\varepsilon^a(v_\varepsilon^a)^2\sim\omega_\varepsilon^a\sim\begin{cases}
|y|^{2-a} & \mathrm{if \ }\varepsilon=0\\
|y|^2 & \mathrm{if \ }\varepsilon>0,
\end{cases}$$
with $2-a\in(1,+\infty)$, then the (H=W)-property does not necessarily hold (due to the lack of a Poincar\'e inequality, see \cite{SirTerVit1}). Nevertheless, we can argue as follows: let $\eta\in C^\infty_c(B_1)$ be a radial decreasing cut off function such that $0\leq\eta\leq1$ and $\eta\equiv1$ in $B_r$. Let also for $\delta>0$
$$f_\delta(y)=\begin{cases}
0 & \mathrm{in \ }B_1\cap\{|y|\leq\delta\} \\
\log\frac{y}{\delta} & \mathrm{in \ }B_1\cap\{\delta\leq|y|\leq\delta e\} \\
1 & \mathrm{in \ }B_1\cap\{\delta e\leq|y|\}.
\end{cases}$$
Let $\varphi_\delta=\eta f_\delta$, then $|\varphi_\delta|\leq1$ and $|\nabla\varphi_\delta|\leq c/y$ uniformly in $\delta>0$. We remark that one can replace $f_\delta$ with a function with the same properties which is $C^\infty(B_1)$. So,
\begin{equation}\label{L2delta}
\int_{B_1}\rho_\varepsilon^a(v_\varepsilon^a)^2|\varphi_\delta w_\varepsilon|^2\leq\int_{B_1}\rho_\varepsilon^au_\varepsilon^2\leq c.
\end{equation}
Obviously in $B_1\setminus\Sigma$ equation \eqref{BHepseq1} holds.
It is an easy consequence of Lemma \ref{rhofrac}, using that $v_\varepsilon^a$ is an odd energy solution to \eqref{vepsa} in $B_1$ and that $v_\varepsilon^a>0$ in $B_1\setminus\Sigma$. Then, testing the equation with $\varphi_\delta^2w_\varepsilon$, we obtain
\begin{eqnarray}\label{H1delta}
\int_{B_1}\rho_\varepsilon^a(v_\varepsilon^a)^2A\nabla(\varphi_\delta w_\varepsilon)\cdot\nabla(\varphi_\delta w_\varepsilon)&=&\int_{B_1}\rho_\varepsilon^a(v_\varepsilon^a)^2\left(\varphi_\delta^2A\nabla w_\varepsilon\cdot\nabla w_\varepsilon+2\varphi_\delta w_\varepsilon A\nabla w_\varepsilon\cdot\nabla\varphi_\delta+w_\varepsilon^2A\nabla\varphi_\delta\cdot\nabla\varphi_\delta\right)\nonumber\\
&=&\int_{B_1}(RHS)\varphi_\delta^2w_\varepsilon+\int_{B_1}\rho_\varepsilon^a(v_\varepsilon^a)^2w_\varepsilon^2A\nabla\varphi_\delta\cdot\nabla\varphi_\delta\nonumber\\
&\leq&\int_{B_1}(RHS)\varphi_\delta^2w_\varepsilon+c\int_{B_1}\frac{\rho_\varepsilon^a}{y^2}u_\varepsilon^2\leq c,
\end{eqnarray}
and this is true by the weighted Hardy inequality in \eqref{hard}, weighted Sobolev embeddings (Theorem \ref{sobemb1}) and Assumptions $(H f\omega_\varepsilon^a)$, $(H F_1\omega_\varepsilon^a)$, $(H F_2\omega_\varepsilon^a)$, $(H V\omega_\varepsilon^a)$ and $(H b\omega_\varepsilon^a)$ which give a bound on the term with $(RHS)$ of equation \eqref{BHepseq1}. We remark that, fixed $\delta>0$, the boundedness in norm $H^1(B_1,\rho_\varepsilon^a(y)(v_\varepsilon^a)^2(x,y)\mathrm{d}z)$ is enough to ensure that $\varphi_\delta w_\varepsilon$ belongs to the same space. In fact, they have compact support away from $\Sigma$, and hence these norms are equivalent to the usual $H^1$-norm. Since the bounds in \eqref{L2delta} and \eqref{H1delta} are uniform in $\delta>0$, this is enough to have weak convergence for the sequence $\varphi_\delta w_\varepsilon$ in $H^1(B_r,\rho_\varepsilon^a(y)(v_\varepsilon^a)^2(x,y)\mathrm{d}z)$ as $\delta\to0$ and of course the limit is $w_\varepsilon$ (it is almost everywhere pointwise limit).\\\\
\begin{comment}
{\color{blue}QUESTA PARTE SI PUO SEMPLIFICARE PERCHE SERVE TESTARE CON LE C INFINITO A SUPPORTO COMPATTO FUORI DA SIGMA. Now we want to show that $w_\varepsilon$ is an energy solution to \eqref{BHepseq} in $B_r$. To this end, for every $\varphi \in C^\infty_c(B_r)$ and $0<\delta<1$, let $\eta_\delta\in C^{\infty}(B_r)$ be a family of functions such that $0\leq\eta_\delta\leq 1$ and
$$
\eta_\delta(x,y) =
\begin{cases}
0 & \mathrm{in \ }B_r\cap \{\abs{y}\leq \delta\}, \\
1 &\mathrm{in \ }B_r\cap\{\abs{y}\geq 2\delta\},
\end{cases}
$$
with $\abs{\nabla \eta_\delta} \leq 1/\delta$.
Thus, by testing the equation with $\eta_\delta\varphi$ we obtain for every $\delta \in (0,1)$,
\begin{eqnarray}\label{integration}
\int_{B_r}{\rho_\varepsilon^a(v_\varepsilon^a)^2\left(\overline f+Vw_\varepsilon\right)\eta_\delta\varphi}&=&\int_{B_r}{\rho_\varepsilon^a(v_\varepsilon^a)^2A\nabla w_\varepsilon\cdot\nabla(\eta_\delta \varphi)}\nonumber\\
&=& \int_{B_r}{\rho_\varepsilon^a(v_\varepsilon^a)^2 \eta_\delta A\nabla w_\varepsilon\cdot\nabla \varphi}+ \int_{B_r}{\rho_\varepsilon^a(v_\varepsilon^a)^2\varphi A\nabla w_\varepsilon\cdot\nabla \eta_\delta}\;.
\end{eqnarray}
To conclude the proof, we observe that, by H\"older inequality, there holds
\begin{align*}
\int_{B_r}{\rho_\varepsilon^a(v_\varepsilon^a)^2 \varphi A\nabla w_\varepsilon\cdot \nabla \eta_\delta} &\leq c(A)\norm{\varphi}{L^\infty(B_r)}\left(\int_{B_r}{\rho_\varepsilon^a(v_\varepsilon^a)^2\abs{\nabla w_\varepsilon}^2}\right)^{1/2}\left( \int_{B_r}{\rho_\varepsilon^a(v_\varepsilon^a)^2\abs{\nabla \eta_\delta}^2}\right)^{1/2}\\
&\leq C\frac{1}{\delta}\left( \int_{\delta}^{2\delta}{\rho_\varepsilon^a(v_\varepsilon^a)^2\mathrm{d}y}\right)^{1/2}\leq C\begin{cases}\delta^{\frac{1-a}{2}} &\mathrm{if \ }\varepsilon=0\\ \delta^{\frac{1}{2}} &\mathrm{if \ }\varepsilon>0\end{cases},
\end{align*}
which implies, as $\delta \to 0$ in \eqref{integration}, via the Dominated Convergence Theorem, that
$$
\int_{B_r}{\rho_\varepsilon^a(v_\varepsilon^a)^2A\nabla w_\varepsilon\cdot\nabla \varphi} = \int_{B_r}{\rho_\varepsilon^a(v_\varepsilon^a)^2\left(\overline f+Vw_\varepsilon\right)\varphi} \quad \mbox{ for \ every \ }\varphi \in C^\infty_c(B_r)\;.
$$
}
\end{comment}
We have already remarked that in $B_r\setminus\Sigma$ equation \eqref{BHepseq1} holds. Then, one can conclude since the weighted Sobolev space $H^1(B_r,\rho_\varepsilon^a(y)(v_\varepsilon^a)^2(x,y)\mathrm{d}z)$ is super degenerate, and consequently test functions can be taken in $C^\infty_c(B_r\setminus\Sigma)$.
\endproof
\section{Liouville type theorems}\label{sec:liouville}
In this section we present two important results which will be our main tool in order to prove regularity local estimates which are uniform with respect to $\varepsilon\geq0$.
\begin{Theorem}\label{Liouvilleodd}
Let $a\in(-1,1)$, $\varepsilon\geq 0$ and $w$ be a solution to
\begin{equation*}
\begin{cases}
-\mathrm{div}(\rho_\varepsilon^a(y)\nabla w)=0 & \mathrm{in \ }\mathbb{R}^{n+1}_+\\
w(x,0)=0,
\end{cases}
\end{equation*}
and let us suppose that for some $\gamma\in[0,1-a)$, $C>0$ it holds
\begin{equation}\label{eq:wronggrowth0}
|w(z)|\leq C (1 + |z|^{\gamma})
\end{equation}
for every $z$. Then $w$ is identically zero.
\end{Theorem}
\proof
It is enough to prove the result only for $\varepsilon\in\{0,1\}$. In fact for any other value of $\varepsilon>0$ we can normalize the problem falling in the case $\varepsilon=1$.\\\\
$\bf{Case \ 1:}$ \ $\varepsilon=0$.\\
Let us consider $w\in H^{1,a}_{\mathrm{loc}}(\mathbb{R}^{n+1}_+)$ satisfying the conditions of the statement, that is, solution in the following sense
$$\int_{\mathbb{R}^{n+1}_+}y^a\nabla w\cdot\nabla\phi=0\qquad\forall\phi\in C^\infty_c(\mathbb{R}^{n+1}_+).$$
Let us define
$$E(r)=\frac{1}{r^{n+a-1}}\int_{B_r^+}y^{a}|\nabla w|^2,\qquad H(r)=\frac{1}{r^{n+a}}\int_{\partial^+ B_r^+}y^aw^2.$$
Note that, as the weight $y^a$ is locally integrable, \eqref{eq:wronggrowth0} implies
\begin{equation}\label{eq:wgh0}
H(r)\leq C(1+r^{2\gamma})\;, \forall r>0\;.
\end{equation}
Now, defining $w^r(x)=w(rx)$, we have
$$E(r)=\int_{B_1^+}y^a|\nabla w^r|^2\qquad\mathrm{and}\qquad H(r)=\int_{S^n_+}y^a(w^r)^2,$$
and hence
$$H'(r)=\frac{2}{r}E(r).$$
We are looking for the best constant in the following trace Poincar\'e inequality
\begin{equation}\label{tpti}
\int_{B_1^+}y^a|\nabla u|^2\geq\lambda(a)\int_{S^n_+}y^a u^2.
\end{equation}
Actually we are able to provide the best constant $\lambda(a)$ in \eqref{tpti}, since $u(x,y)=y^{1-a}$ is the unique function in $\tilde H^{1,a}(B_1^+)$ which solves
\begin{equation*}
\begin{cases}
-L_au=0 &\mathrm{in} \ B_1^+\\
u>0 &\mathrm{in} \ B_1^+\\
u(x,0)=0\\
\nabla u\cdot\nu=\lambda(a)u &\mathrm{in} \ S_+^{n},
\end{cases}
\end{equation*}
with $\lambda(a)=1-a$. However $\lambda(a)$ is the same of \eqref{lama}. Hence $H'(r)\geq \frac{2\lambda(a)}{r}H(r)$, and integrating, there we infer
\begin{equation*}
\frac{H(r)}{r^{2(1-a)}}\geq H(1),
\end{equation*}
we obtain that if $w$ is not trivial, its growth at infinity is at least $r^{1-a}$, in contradiction with \eqref{eq:wgh0} taking $r$ large.\\
$\bf{Case \ 2:}$ \ $\varepsilon=1$.\\
Let us define
$$E(r)=\frac{1}{r^{n+a-1}}\int_{B_r^+}(1+y^2)^{a/2}|\nabla w|^2,\qquad H(r)=\frac{1}{r^{n+a}}\int_{\partial^+ B_r^+}(1+y^2)^{a/2}w^2.$$
Note that, as $a>-1$, the $\rho_\varepsilon^{a}$'s are uniformly locally integrable and thus \ref{eq:wronggrowth0} implies again
\begin{equation}\label{eq:wgh05}
H(r)\leq C(1+r^{2\gamma})\;, \forall r>0\;, \text{with $\gamma<1-a$.}
\end{equation}
Hence,
\begin{equation}\label{H11}
H'(r)=\frac{2}{r}E(r)-\frac{a}{r^{n+a+1}}\int_{\partial^+ B_r^+}(1+y^2)^{a/2-1}w^2.
\end{equation}
Moreover, defining $w^r(x)=w(rx)$ one has
$$E(r)=\int_{B_1^+}\left(\frac{1}{r^2}+y^2\right)^{a/2}|\nabla w^r|^2\qquad\mathrm{and}\qquad H(r)=\int_{S^n_+}\left(\frac{1}{r^2}+y^2\right)^{a/2}(w^r)^2.$$
By Lemma \ref{A1} and Remark \ref{A2}, one can find for any radius $r>0$ the best constant $\lambda_r(a)$ such that
\begin{equation}\label{tptir}
\int_{B_1^+}\left(\frac{1}{r^2}+y^2\right)^{a/2}|\nabla u|^2\geq\lambda_r(a)\int_{S^n_+}\left(\frac{1}{r^2}+y^2\right)^{a/2} u^2.
\end{equation}
Defining $\rho_k(y)=\left(\frac{1}{r_k^2}+y^2\right)^{a/2}$ with $r_k\to+\infty$ as $k\to+\infty$, one can see
$$\lambda(a)=\min_{v\in\tilde H^1(B_1^+)\setminus\{0\}}\frac{Q_a(v)}{\int_{S^n_+}v^2}\qquad\mathrm{and}\qquad\lambda_k(a)=\min_{v\in\tilde H^1(B_1^+)\setminus\{0\}}\frac{Q_{\rho_k}(v)}{\int_{S^n_+}v^2}.$$
By Lemma \ref{A1}, $\lambda_k(a)\to\lambda(a)=1-a$ as $k\to+\infty$.\\\\
Now we want to prove that the correction term in \eqref{H11} is of lower order as $r\to+\infty$. By \eqref{tracehard}, we have that in $\tilde C^\infty_c(B_1^+)$
\begin{equation*}
\int_{B_1^+}\rho_r|\nabla u|^2\geq c_0\int_{\partial B_1^+}\frac{\rho_r}{y}u^2.
\end{equation*}
Hence
\begin{eqnarray*}
\left|\frac{a}{r^{n+a+1}}\int_{\partial^+ B_r^+}\left(1+y^2\right)^{a/2-1}w^2\right|&\leq&\frac{|a|}{r^{n+a+1}}\int_{\partial^+ B_r^+}\left(1+y^2\right)^{a/2-1/2}w^2\\
&=&\frac{|a|}{r^{2}}\int_{S^n_+}\left(\frac{1}{r^2}+y^2\right)^{a/2-1/2}(w^r)^2\\
&\leq&\frac{|a|}{r^{2}}\int_{S^n_+}\left(\frac{1}{r^2}+y^2\right)^{a/2}y^{-1}(w^r)^2\\
&\leq&\frac{|a|}{c_0r^{2}}\int_{B_1^+}\left(\frac{1}{r^2}+y^2\right)^{a/2}|\nabla w^r|^2\\
&=&\frac{|a|}{c_0r^{2}}E(r).
\end{eqnarray*}
Hence for $r$ large enough
\begin{equation*}
H'(r)\geq \frac{2\lambda_r(a)}{r}H(r),
\end{equation*}
and since $\lambda_r(a)\to\lambda(a)=1-a$, by integrating the above expression we deduce that, for all small $\delta$, there exists $r_0>0$ such that, for every $r>r_0$
\begin{equation*}
\frac{H(r)}{r^{2(1-a-\delta)}}\geq H(r_0),
\end{equation*}
which says that if $w$ is not trivial, its growth at infinity is at least $r^{1-a-\delta}$. Taking $\delta>0$ so small that $1-a-\delta>\gamma$ we find a contradiction with \eqref{eq:wgh05}.
\endproof
\begin{Theorem}\label{LiouvilleBH}
Let $a\in(-\infty,1)$, $\varepsilon\geq0$ and $w$ be a solution to
\begin{equation*}
\begin{cases}
-\mathrm{div}((\omega_\varepsilon^a(y))^{-1}\nabla w)=0 & \mathrm{in \ }\mathbb{R}^{n+1}_+\\
w=0 &\mathrm{in \ }\mathbb{R}^{n}\times\{0\},
\end{cases}
\end{equation*}
and let us suppose that for some $\gamma\in[0,1)$, $C>0$ it holds
\begin{equation}\label{eq:wronggrowthBH}
|w(z)|\leq C\omega_\varepsilon^a(y)(1 + |z|^\gamma)
\end{equation}
for every $z=(x,y)$. Then $w$ is identically zero.
\end{Theorem}
\proof
By a simple normalization argument, it is enough to prove the result only for $\varepsilon\in\{0,1\}$. We start with\\\\
$\bf{Case \ 1:}$ \ $\varepsilon=0$.\\
The case falls into the proof of $\bf{Case \ 1}$ in \cite[Theorem 3.4]{SirTerVit1} replacing $a\in(-\infty,1)$ with $(a-2)\in(-\infty,-1)$.\\\\
$\bf{Case \ 2:}$ \ $\varepsilon=1$.\\
Let us now define
$$E(r)=\frac{1}{r^{n+(a-2)-1}}\int_{B_r^+}(\omega_1^a(y))^{-1}|\nabla w|^2,\qquad\textrm{and}\qquad H(r)=\frac{1}{r^{n+(a-2)}}\int_{\partial^+ B_r^+}(\omega_1^a(y))^{-1}w^2.$$
Note that, defining $w^r(x)=w(rx)$ one has
$$E(r)=\int_{B_1^+}(\omega_{1/r}^a(y))^{-1}|\nabla w^r|^2,\qquad\textrm{and}\qquad H(r)=\int_{S^n_+}(\omega_{1/r}^a(y))^{-1}(w^r)^2.$$
First we remark that the growth condition \eqref{eq:wronggrowthBH} implies the following upper bound
\begin{equation}\label{eq:wgh}
H(r)\leq Cr^{-2(a-2)}(1+r^{2\gamma})\;, \forall r>0\;,
\end{equation}
(due to the local integrability of $y^{2-a}$) and heence
$$\int_{S^n_+}\omega_{1/r}^a(y)\leq c$$
uniformly in $r>0$. Therefore,
\begin{equation}\label{H111}
H'(r)=\frac{2}{r}E(r)+\int_{S^n_+}\frac{d}{dr}[(\omega_{1/r}^a(y))^{-1}](w^r)^2.
\end{equation}
By Lemma \ref{B1} and Lemma \ref{A2BH}, one can find for any radius $r>0$ the best constant $\mu_r(a)$ such that
\begin{equation}\label{tptir2}
\int_{B_1^+}(\omega_{1/r}^a(y))^{-1}|\nabla u|^2\geq\mu_r(a)\int_{S^n_+}(\omega_{1/r}^a(y))^{-1}|u|^2.
\end{equation}
Defining $(\omega^a_k(y))^{-1}=(\omega_{1/r_k}^a(y))^{-1}$ and $\mu_k=\mu_{r_k}$ with $r_k\to+\infty$ as $k\to+\infty$, by Lemma \ref{A2BH}, $\mu_k(a)\to\mu(a)=1-(a-2)=3-a$ as $k\to+\infty$.\\\\
Now we want to prove that the correction term in \eqref{H111} is of lower order as $r\to+\infty$. By \eqref{tracehardBH}, we have that\begin{equation*}
\int_{B_1^+}(\omega_{1/r}^a(y))^{-1}|\nabla u|^2\geq c_0\int_{S^n_+}\frac{(\omega_{1/r}^a(y))^{-1}}{y}u^2.
\end{equation*}
Using
\begin{eqnarray*}
\int_{S^n_+}\frac{d}{dr}[(\omega_{1/r}^a(y))^{-1}](w^r)^2&=&\frac{a}{r^3}\int_{S^n_+}\frac{1}{\left(\frac{1}{r^2}+y^2\right)}(\omega_{1/r}^a(y))^{-1}(w^r)^2\\
&&-\frac{2a}{r^3}\int_{S^n_+}\frac{\int_0^y\left(\frac{1}{r^2}+s^2\right)^{-a/2-1}}{\int_0^y\left(\frac{1}{r^2}+s^2\right)^{-a/2}}(\omega_{1/r}^a(y))^{-1}(w^r)^2,
\end{eqnarray*}
we can estimate the first term of the rest as follows
\begin{eqnarray*}
\left|\frac{a}{r^3}\int_{S^n_+}\frac{1}{\left(\frac{1}{r^2}+y^2\right)}(\omega_{1/r}^a(y))^{-1}(w^r)^2\right|&\leq&\frac{|a|}{r^2}\int_{S^n_+}\frac{1}{\left(\frac{1}{r^2}+y^2\right)^{1/2}}(\omega_{1/r}^a(y))^{-1}(w^r)^2\\
&\leq&\frac{|a|}{r^2}\int_{S^n_+}\frac{(\omega_{1/r}^a(y))^{-1}}{y}(w^r)^2\leq\frac{c}{r^{2}}E(r).
\end{eqnarray*}
Moreover, when $a\leq0$ the second term of the rest can be estimated as
\begin{eqnarray*}
\left|\frac{2a}{r^3}\int_{S^n_+}\frac{\int_0^y\left(\frac{1}{r^2}+s^2\right)^{-a/2-1}}{\int_0^y\left(\frac{1}{r^2}+s^2\right)^{-a/2}}(\omega_{1/r}^a(y))^{-1}(w^r)^2\right|&\leq&\frac{2|a|}{r^2}\int_{S^n_+}\frac{ry\int_0^{ry}\left(1+s^2\right)^{-a/2-1}}{\int_0^{ry}\left(1+s^2\right)^{-a/2}}\frac{(\omega_{1/r}^a(y))^{-1}}{y}(w^r)^2\\
&\leq&\frac{2|a|}{r^2}\int_{S^n_+}\frac{(\omega_{1/r}^a(y))^{-1}}{y}(w^r)^2\leq\frac{c}{r^{2}}E(r),
\end{eqnarray*}
and this is due to the fact that, calling $z=ry\in[0,+\infty)$, by the fact that
$$f(z)=\frac{z\int_0^{z}\left(1+s^2\right)^{-a/2-1}}{\int_0^{z}\left(1+s^2\right)^{-a/2}}$$
is continuous and such that $f(0)=0$ and
$$f(z)\sim_{z\to+\infty}\begin{cases}cz^a &\mathrm{if \ }a\in(-1,0]\\ \frac{\log z}{z} &\mathrm{if \ }a=-1\\ \frac{1}{z} &\mathrm{if \ }a<-1\end{cases}$$
and hence $f(z)\leq c$ in $[0,+\infty)$. Instead, when $a\in(0,1)$ the second term of the rest can be estimated as
\begin{eqnarray*}
\left|\frac{2a}{r^3}\int_{S^n_+}\frac{\int_0^y\left(\frac{1}{r^2}+s^2\right)^{-a/2-1}}{\int_0^y\left(\frac{1}{r^2}+s^2\right)^{-a/2}}(\omega_{1/r}^a(y))^{-1}(w^r)^2\right|&\leq&\frac{2|a|}{r^{2-a}}\int_{S^n_+}\frac{(ry)^{1-a}\int_0^{ry}\left(1+s^2\right)^{-a/2-1}}{\int_0^{ry}\left(1+s^2\right)^{-a/2}}\frac{(\omega_{1/r}^a(y))^{-1}}{y^{1-a}}(w^r)^2\\
&\leq&\frac{2|a|}{r^{2-a}}\int_{S^n_+}\frac{(\omega_{1/r}^a(y))^{-1}}{y^{1-a}}(w^r)^2\leq\frac{2|a|}{r^{2-a}}\int_{S^n_+}\frac{(\omega_{1/r}^a(y))^{-1}}{y}(w^r)^2\\
&\leq&\frac{c}{r^{2-a}}E(r),
\end{eqnarray*}
using the fact that $0\leq y\leq1$ and by the fact that
$$f(z)=\frac{z^{1-a}\int_0^{z}\left(1+s^2\right)^{-a/2-1}}{\int_0^{z}\left(1+s^2\right)^{-a/2}}$$
is continuous and such that $f(0)=0$ and
$$f(z)\sim_{z\to+\infty}c$$
and hence $f(z)\leq c$ in $[0,+\infty)$.
Hence for $r$ large enough
\begin{equation*}
H'(r)\geq \frac{2\mu_r(a)}{r}H(r),
\end{equation*}
and since $\mu_r(a)\to\mu(a)=1-(a-2)$, we can choose a small $\delta>0$ such that $1-(a-2)-\delta>\gamma-(a-2)$. Hence, by integrating the above expression we deduce that there exists $r_0>0$ such that, for every $r>r_0$, we have $\mu_r(a)>1-(a-2)-\delta>\gamma-(a-2)$ and
\begin{equation*}
\frac{H(r)}{r^{2(1-(a-2)-\delta)}}\geq H(r_0),
\end{equation*}
which is in contradiction with \eqref{eq:wgh} for $r$ large if $w$ is not trivial.
\endproof
\begin{Corollary}\label{LiouvilleBHbis}
Let $a\in(-\infty,1)$, $\varepsilon\geq0$ and $w$ be a solution to
\begin{equation*}
\begin{cases}
-\mathrm{div}(\omega_\varepsilon^a(y)\nabla w)=0 & \mathrm{in \ }\mathbb{R}^{n+1}_+\\
\omega_\varepsilon^a\partial_yw=0 &\mathrm{in \ }\mathbb{R}^{n}\times\{0\},
\end{cases}
\end{equation*}
and let us suppose that for some $\gamma\in[0,1)$, $C>0$ it holds
\begin{equation}\label{gro0BH}
|w(z)|\leq C(1 + |z|^\gamma)
\end{equation}
for every $z$. Then $w$ is constant.
\end{Corollary}
\proof
Again, it is enough to treat the cases $\varepsilon\in\{0,1\}$. Let us assume $\varepsilon=1$, the case $\varepsilon=0$ coincides with the case $\varepsilon=0$ in \cite[Corollary 3.5]{SirTerVit1}, by replacing in the proof $a\in(-1,+\infty)$ by $(2-a)\in(1,+\infty)$. Then we have (by an even reflection across $\Sigma$) an even solution $w$ to
\begin{equation*}
-\mathrm{div}\left(\omega_1^a(y)\nabla w\right)=0 \qquad\mathrm{in} \ \mathbb{R}^{n+1}.
\end{equation*}
Such a solution is $w\in H^{1,2}_{\mathrm{loc}}(\mathbb{R}^{n+1})=H^{1}_{\mathrm{loc}}(\mathbb{R}^{n+1},|y|^2\mathrm{d}z)$, with the growth condition \eqref{gro0BH}. Now we observe that, as $w$ is not constant with a sublinear growth at infinity, $v=\omega_1^a(y)\partial_y w$ can not be trivial, otherwise
$w$ would be globally harmonic and sublinear, in contradiction with the Liouville theorem in \cite{NorTavTerVer}. Hence, if $w$ is not constant, $v$ must be an odd and nontrivial solution to
\begin{equation*}
\begin{cases}
-\mathrm{div}\left((\omega_1^a(y))^{-1}\nabla v\right)=0 &\mathrm{in \ }\mathbb{R}^{n+1}_+\\
v=0 &\mathrm{in} \ \{y=0\}.
\end{cases}
\end{equation*}
By the arguments in the proof of Theorem \ref{LiouvilleBH}, we know that the weighted average of $v^2$ must satisfy a minimal growth rate as
\begin{equation*}\label{gro1}
H(r)=\frac{1}{r^{n+(a-2)}}\int_{\partial^+ B_r^+}(\omega_1^a(y))^{-1}v^2\geq cr^{2(1-(a-2)-\delta)}, \qquad 1-\delta>\gamma\;,
\end{equation*}
for $r\geq r_0$ depending on $\delta>0$ chosen. Therefore, by integrating, we obtain
\[
\int_{B_r^+}\omega_1^a(y)(\partial_y w)^2=\int_0^r\mathrm{d}t\int_{\partial^+ B_t^+}(\omega_1^a(y))^{-1}v^2\geq c r^{n-(a-2)+2-2\delta}\;.
\]
On the other hand, we have, by \eqref{gro0BH}
\begin{equation*}
\begin{split}
\int_{B_r^+}\omega_1^a(y) (\partial_y w)^2\leq \int_{B_r^+}\omega_1^a(y) |\nabla w|^2\\ \leq c \int_{B_{2r}^+}\omega_1^a(y) |w|^2\leq c(1+r^{n-(a-2)+2\gamma})
\end{split}
\end{equation*}
in contradiction with the previous inequality when $r$ is large, since $1-\delta>\gamma$.
\endproof
\section{Local uniform bounds in H\"older spaces for the auxiliary problem}
As a first step in our regularity theory for odd solutions, we state some results on local uniform estimates for solutions to \eqref{simpleBH}; that is,
\begin{equation*}
-\mathrm{div}\left(\rho_\varepsilon^a(v_\varepsilon^a)^2A\nabla u_\varepsilon\right)=\rho_\varepsilon^a(v_\varepsilon^a)^2f_\varepsilon+\mathrm{div}\left(\rho_\varepsilon^a(v_\varepsilon^a)^2F_\varepsilon\right)+\rho_\varepsilon^a(v_\varepsilon^a)^2b_\varepsilon\cdot\nabla u_\varepsilon\qquad\mathrm{in \ }B_1.
\end{equation*}
Using a Moser iteration argument (see also \cite[Section 8.4]{GilTru}), one can prove the following nowadays standard result.
\begin{Proposition}\label{Moser}
Let $a\in(-\infty,1)$ and $\varepsilon\geq0$. Let $u\in H^{1}(B_1,\omega_\varepsilon^a(y)\mathrm{d}z)$ be an energy solution to \eqref{simpleBH}. Let $\beta>1$,
$$p_1>\frac{\overline d}{2}=\frac{n+3+(-a)^+}{2}, \qquad p_2,p_3>\overline d.$$
Let moreover $\|b\|_{L^{p_3}(B_1,\omega_\varepsilon^a(y)\mathrm{d}z)}\leq b$.
Then, for any $0<r<1$ there exists a positive constant independent of $\varepsilon$ (depending on $n$, $a$, $p_1$, $p_2$, $p_3$, $\beta$, $b$, $r$ and $\alpha$) such that
$$\|u\|_{L^\infty(B_{r})}\leq c\left(\|u\|_{L^{\beta}(B_1,\omega_\varepsilon^a(y)\mathrm{d}z)}+\|f\|_{L^{p_1}(B_1,\omega_\varepsilon^a(y)\mathrm{d}z)}+\|F\|_{L^{p_2}(B_1,\omega_\varepsilon^a(y)\mathrm{d}z)}\right).$$
\end{Proposition}
\proof
The proof follows the same steps as in \cite[Proposition 2.17]{SirTerVit1}, but iterating the Sobolev embedding in Theorem \ref{sobemb1}.
\endproof
\subsection{Local uniform bounds in $C^{0,\alpha}$ spaces}
\begin{Theorem}\label{C0alphaBH}
Let $a\in(-\infty,1)$ and as $\varepsilon\to0$ let $\{u_\varepsilon\}$ be a family of solutions in $B_1^+$ of \eqref{simpleBH} satisfying boundary conditions (evenness)
$$
\rho_\varepsilon^a(v_\varepsilon^a)^2\partial_yu_\varepsilon=0 \qquad \mathrm{on \ }\partial^0 B_1^+.
$$
Let $r\in(0,1)$, $\beta>1$, $p_1>\frac{\overline d}{2}=\frac{n+3+(-a)^+}{2}$, $p_2,p_3>\overline d$, and $\alpha\in(0,1)\cap(0,2-\frac{n+3+(-a)^+}{p_1}]\cap(0,1-\frac{n+3+(-a)^+}{p_2}]\cap(0,1-\frac{n+3+(-a)^+}{p_3}]$. Let $\|b\|_{L^{p_3}(B_1,\omega_\varepsilon^a(y)\mathrm{d}z)}\leq b$. Let moreover $A$ satisfy assumption (HA) with continuous coefficients. There is a positive constant depending on $a$, $n$, $\beta$, $p_1$, $p_2$, $p_3$, $b$, $\alpha$ and $r$ only such that
$$\|u_\varepsilon\|_{C^{0,\alpha}(B_r^+)}\leq c\left(\|u_\varepsilon\|_{L^\beta(B_1^+,\omega_\varepsilon^a(y)\mathrm{d}z)}+ \|f_\varepsilon\|_{L^{p_1}(B_1^+,\omega_\varepsilon^a(y)\mathrm{d}z)}+\|F_\varepsilon\|_{L^{p_2}(B_1,\omega_\varepsilon^a(y)\mathrm{d}z)}\right).$$
\end{Theorem}
\proof
The proof follows the very same steps as in the proof of \cite[Theorem 4.1]{SirTerVit1}.
First, one has to remark that the suitable H\"older continuity for $\varepsilon\geq0$ fixed is given by the theory developed for even solutions to degenerate problems in \cite{SirTerVit1}. Then, one can argue by contradiction with the usual blow up argument considering two blow up sequences
\begin{equation*}
v_k(z)=\frac{(\eta u_k)(z_k+r_kz)-(\eta u_k)(z_k)}{L_kr_k^\alpha},\qquad w_k(z)=\frac{\eta(z_k)(u_k(z_k+r_kz)- u_k(z_k))}{L_kr_k^\alpha},
\end{equation*}
(with the same asymptotic behaviour on compact sets) defined in the rescaled domains $B(k)=\frac{B-z_k}{r_k}$ (where $B=B_{\frac{1+r}{2}}$ and $\{z_k\}$ is one of the two sequences of points where H\"older seminorms blow up), the first possessing some uniform H\"older continuity, and the second one satisfying suitable problems on rescaled domains which blow up. In order to complete the proof we prove some steps.\\\\
{\bf Step 1: blow-ups.} The first thing to do is to characterize the possible asymptotic behaviours of the weights $\rho_\varepsilon^a(v_\varepsilon^a)^2$ in the rescaled points: that is,
\begin{eqnarray*}
p_k(z)&:=&\rho_\varepsilon^a(y_k+r_ky)(v_\varepsilon^a(z_k+r_kz))^2\nonumber\\
&=&\left(\varepsilon_k^2+(y_k+r_ky)^2\right)^{a/2}\left(\int_0^{y_k+r_ky}\left(\varepsilon_k^2+s^2\right)^{-a/2}\mu(x_k+r_kx,s)^{-1}\mathrm{d}s\right)^2.
\end{eqnarray*}
To this end, let us define by $\Gamma_k=(\varepsilon_k,y_k,r_k)$ and denote $\nu_k=|\Gamma_k|$. The latter is a bounded sequence and, up to subsequences, has finite limit $\nu=|(0,y_\infty,0)|\geq0$ (where we have assumed $z_k\to z_\infty=(x_\infty,y_\infty)$). Taking possibly another subsequence, we may assume that the normalized sequence
$$\tilde\Gamma_k=\frac{\Gamma_k}{\nu_k}=(\tilde\varepsilon_k,\tilde y_k,\tilde r_k)=\left(\frac{\varepsilon_k}{\nu_k},\frac{y_k}{\nu_k},\frac{r_k}{\nu_k}\right)$$
has a limit
$$\tilde\Gamma_k\to\tilde\Gamma=(\tilde\varepsilon,\tilde y,\tilde r)\in S^2\;,$$
and moreover that
$$ \lim_{k\to+\infty}\frac{\tilde y_k}{\tilde r_k}=\tilde l\in[0,+\infty].$$
Thus we can consider $\tilde\Sigma=\lim\Sigma_k$; that is,
\begin{equation*}
\tilde\Sigma=\begin{cases}
\{(x,y)\in\mathbb{R}^{n+1} \ : \ y=-\tilde l \} &\mathrm{if}\; \tilde l <+\infty,\\
\emptyset &\mathrm{if}\; \tilde l =+\infty.
\end{cases}
\end{equation*}
After rescaling the independent variables, we find new weights having the form:
\begin{equation*}
p_k(z)=\nu_k^{2-a}\left(\tilde\varepsilon_k^2+\left(\tilde y_k+\tilde r_ky\right)^2\right)^{a/2}\left(\int_0^{\tilde y_k+\tilde r_ky}\left(\tilde\varepsilon_k^2+t^2\right)^{-a/2}\mu(x_k+r_kx,\nu_kt)^{-1}\mathrm{d}t\right)^2,
\end{equation*}
and, in order to study their asymptotics, we have to distinguish between different cases:\\\\
{\bf Case 1.} $\nu>0$. Then, $\tilde r=\tilde\varepsilon=0$ and $\tilde y=1$. Moreover, it is easy to see, using that $1/\mu$ is continuous, that $p_k(z)= c+o(1)$.\\\\
{\bf Case 2.} $\nu=0$ and $\tilde\varepsilon=0$ ($\tilde y\neq0\lor\tilde r\neq0$). Using the continuity of $1/\mu$, up to a vertical translation of $-\tilde l$, one obtains
$$p_k(z)=\nu_k^{2-a}\tilde p(y)(1+o(1))$$
where
$$\tilde p(y)=\begin{cases}
c &\mathrm{if \ }\tilde r=0,\\
c|y|^{2-a} &\mathrm{if \ }\tilde r\neq 0.
\end{cases}$$
{\bf Case 3.} $\nu=0$ and $\tilde\varepsilon\in(0,1)$. Using again the continuity of $1/\mu$, , up to a vertical translation of $-\tilde l$, we obtain
$$p_k(z)=\nu_k^{2-a}\tilde p(y)(1+o(1))$$
where
$$\tilde p(y)=\begin{cases}
c &\mathrm{if \ }\tilde r=0,\\
c \ \omega_1^a(y) &\mathrm{if \ }\tilde r\neq 0.
\end{cases}$$
in the second case up to a dilation of $\frac{\tilde\varepsilon}{\tilde r}$.\\\\
{\bf Case 4.} $\nu=0$ and $\tilde\varepsilon=1$ ($\tilde y=0\land\tilde r=0$). As usual, by the continuity of $1/\mu$, , up to a vertical translation of $-\tilde l$, one obtains, if $\tilde r_k=o(\tilde y_k)$
$$p_k(z)=\nu_k^{2-a}\tilde y_k^2c(1+o(1)),$$
and otherwise
$$p_k(z)=\nu_k^{2-a}\tilde r_k^2c|y|^2(1+o(1)).$$
\\\\
Let us define
$$h_k=\begin{cases}
\nu_k^{2-a} &\mathrm{in \ {\bf Cases \ 1,2,3}},\\
\nu_k^{2-a}\tilde y_k^2 &\mathrm{in \ {\bf Case \ 4}, \ and \ } \tilde r_k=o(\tilde y_k)\\
\nu_k^{2-a}\tilde r_k^2 &\mathrm{in \ {\bf Case \ 4}, \ otherwise,}
\end{cases}$$
and $\tilde p_k=\frac{p_k}{h_k}$. We have shown that, up to the suitable normalization, the rescaled weights $\tilde p_k$ do converge uniformly to $\tilde p$ on compact sets of $\brd{R}^{n+1}\setminus\tilde\Sigma$ (or the whole $\brd{R}^{n+1}$ whenever $\tilde\Sigma=\emptyset$). Note that this latter case is equivalent to the limiting weight $\tilde p$ be constant. \\\\
{\bf Step 2: the limiting equation and uniform-in-$k$ energy estimates.}
The equation for the rescaled variable $w_k$ becomes:
\begin{eqnarray}\label{eq:rescaled}
-\mathrm{div}(\tilde p_kA(z_k+r_k \cdot)\nabla w_k)(z)&=& \frac{\eta(z_k)}{L_k}r_k^{2-\alpha}\tilde p_k(z)f_k(z_k+r_kz)\nonumber\\
&&+\mathrm{div}\left(\frac{\eta(z_k)}{L_k}r_k^{1-\alpha} \tilde p_k(\cdot)F_k(z_k+r_k\cdot)\right)(z)\nonumber\\
&&+r_k\tilde p_k(z)b_k(z_k+r_kz)\cdot\nabla w_k(z).
\end{eqnarray}
By a Caccioppoli type inequality, easily obtained by multiplying \eqref{eq:rescaled} by $\overline\eta^2 w_k$, being $\overline\eta$ a cut-off function, taking into account that the $w_k$ are uniformly bounded and that
\begin{itemize}
\item the first term in the right hand side of \eqref{eq:rescaled} is bounded in $L^1_{loc}$;
\item the field $\frac{\eta(z_k)}{L_k}r_k^{1-\alpha} F_k(z_k+r_k\cdot)$ in the second term in the right hand side of \eqref{eq:rescaled} is bounded in $L^2_{loc}(\tilde p(z)\mathrm{d}z)$;
\item $r_kb_k(z_k+r_k\cdot)$ is bounded in $L^2_{loc}(\tilde p_k(z)\mathrm{d}z)$;
\end{itemize}
then we obtain uniform-in-$k$ energy bounds holding on compact subsets of $\brd{R}^{n+1}$:
\[
\forall R>0,\;\exists c>0,\;\forall k, \qquad
\int_{B_R}\tilde p_k A(z_k+r_k z) \nabla w_k\cdot \nabla w_k\leq c\;.
\]
The computations are very similar to the ones done in the following step.\\\\
{\bf Step 3: the right hand side vanishes as $k\to+\infty$.}
Next we wixh to check that the right hand sides in the rescaled equations vanish in $L^1_{\mathrm{loc}}(\brd{R}^{n+1}\setminus\tilde\Sigma)$ (or $L^1_{\mathrm{loc}}(\brd{R}^{n+1})$ whenever $\tilde\Sigma=\emptyset$), and that consequently the limit $w$ is an energy solution of
\begin{equation}\label{limittilde}
-\mathrm{div}\left(\tilde pA(z_\infty)\nabla w\right)=0\qquad \mathrm{in} \ \mathbb{R}^{n+1}\setminus\tilde\Sigma,
\end{equation}
even with respect to $\tilde\Sigma$ (when not empty). We use the continuity of the matrix $A$ in order to obtain a constant coefficients matrix in the limit equation \eqref{limittilde} together with the fact that $\tilde\Sigma$ is invariant with respect to the limit matrix to have evenness across the characteristic hyperplane. \\\\
Let us show that the right hand sides vanish in $L^1_{\mathrm{loc}}$ at least for one of the cases (the other cases are very similar), for instance when $h_k=\nu_k^{2-a}\tilde y_k^2$; that is, {\bf Case 4}, when $\tilde r_k=o(\tilde y_k)$. Indeed, let $\phi\in C^\infty_c(\mathbb{R}^{n+1})$: using the fact that for $k$ large enough $\mathrm{supp}(\phi)\subset B_R\subset B(k)$, using H\"older inequality, we have
\begin{eqnarray}
&&\left|\int_{B_R}\tilde p_k(z) f_{\varepsilon_k}(z_k+r_kz)\phi(z)\mathrm{d}z\right|\nonumber\\
&\leq&\|\phi\|_{L^\infty(B_R)}\left(\frac{1}{r_k^{n+1}h_k}\int_{B_{r_kR}(z_k)}\rho_{\varepsilon_k}^a(\zeta_{n+1})(v_{\varepsilon_k}^a(\zeta))^2|f_{\varepsilon_k}(\zeta)|^{p_1}\mathrm{d}\zeta\right)^{1/p_1}\left(\int_{B_{R}}\tilde p_k(z)\mathrm{d}z\right)^{1/p'_1}\nonumber\\
&\leq&cr_k^{-\frac{n+1}{p_1}}\nu_k^{-\frac{2-a}{p_1}}\tilde y_k^{-\frac{2}{p_1}},\nonumber
\end{eqnarray}
and hence the first term in the right hand side converges to zero since $\alpha\leq2-\frac{n+3+(-a)^+}{p_1}$, $\tilde r_k=\frac{r_k}{\nu_k}$, the fact that $0\leq r_k\leq \nu_k$ and having
$$\frac{\eta(z_k)}{L_k}r_k^{2-\alpha-\frac{n+3+(-a)^+}{p_1}}\left(\frac{r_k^{(-a)^+}}{\nu_k^{-a}}\right)^{1/p_1}\left(\frac{\tilde r_k}{\tilde y_k}\right)^{2/p_1}\to0.$$
With analogous computations one can check that also the second term in the right hand side vanishes.\\\\
Concerning the third term, one can estimate the integral as follows
\begin{eqnarray}\label{3p}
&&\\
&&r_k\left|\int_{B_R}\tilde p_k(z)b_k(z_k+r_kz)\cdot\nabla w_k(z)\phi(z)\right|\nonumber\\
&\leq& r_k\|\phi\|_{L^\infty(B_R)}\left(\int_{B_R}\tilde p_k|\nabla w_k|^2\right)^{\frac{1}{2}}\left(\int_{B_R}\tilde p_k\right)^{\frac{p_3-2}{2p_3}}\left(\frac{1}{h_kr_k^{n+1}}\int_{B_{r_kR}(z_k)} \rho_{\varepsilon_k}^a(\zeta_{n+1})(v_{\varepsilon_k}^a(\zeta))^2|b_{\varepsilon_k}(\zeta)|^{p_3}\right)^{\frac{1}{p_3}}\nonumber\\
&\leq&cb^{\frac{1}{p_3}}r_k^{1-\frac{n+3+(-a)^+}{p_3}}\left(\frac{r_k^{(-a)^+}}{\nu_k^{-a}}\right)^{1/p_3}\left(\frac{\tilde r_k}{\tilde y_k}\right)^{2/p_3}\left(\int_{B_R}\tilde p_k|\nabla w_k|^2\right)^{1/2}\nonumber\\
&=&t_k\left(\int_{B_R}\tilde p_k|\nabla w_k|^2\right)^{1/2}\nonumber.
\end{eqnarray}
The sequence $t_k$ converges to zero, having $p_3>n+3+(-a)^+$. Moreover, the full term vanishes using the uniform energy bound obtained in the previous step.\\\\
{\bf Step 4: the limit belongs to $H^1_{loc}(\brd{R}^{n+1},\tilde p\mathrm{d}z)$.} At this point, always up to subsequences, we know that the (pointwise) convergence to $w$ holds also in the weak $H^1_{loc}(\brd{R}^{n+1}\setminus\tilde\Sigma)$ topology. Now we wish to infer that the limit $w$ belongs to the space $H^1_{loc}(\brd{R}^{n+1},\tilde p \mathrm{d}z)$ as the closure of $C^\infty$ with respect to the weighted norm (as defined in \S\ref{sect:sobolev}). Let us start with the easiest case when $\tilde\Sigma=\emptyset$ and the limiting weight $\tilde p$ is constant. Moreover, we know tha $\tilde p_k$ converge uniformly to $\tilde p$ on compact sets. Thus the sequence $w_k$ converges weakly $H^1$ to $w$ on each compact subset. The convergence to The case when $\tilde \Sigma\neq \emptyset$ requires a more thorough analysis.
In order to ensure that $w\in H^1_{\mathrm{loc}}(\brd{R}^{n+1},\tilde p(y)\mathrm{d}z)$ also when $\tilde\Sigma\neq \emptyset$, one can argue as follows: using the fact that $\mu$ is continuous with $\frac{1}{C}<\mu<C$, then fixed a compact set of $\brd{R}^{n+1}$, we can find positive constants $c_k,C_k$ (which are uniformly bounded from above and below by two constants respectively $0<c_1<c_2<+\infty$) such that
$$c_k\tilde\omega_{\varepsilon_k}(y_k+r_ky)\leq \tilde p_k(x,y)\leq C_k\tilde \omega_{\varepsilon_k}(y_k+r_ky)\qquad\mathrm{and}\qquad\frac{C_k}{c_k}\to1.$$
where we have denoted
\begin{equation}\label{eq:tildeomega}
\tilde\omega_k=\frac{\omega_{\varepsilon_k}}{h_k}.
\end{equation}
Now, reabsorbing the weights as in \eqref{Qomega1+} and using the family of isometries given by
$$\overline T_k(w_k)=\left(\tilde\omega_{k}(y_k+r_ky)\right)^{1/2}w_k=W_k,$$
one obtains uniform boundedness of the $W_k$'s in $ H^1_{\mathrm{loc}}(\brd{R}^{n+1})$, and hence they weakly converge in the same space to $W$. Coming back with the inverse isometry associated with the limit weight
$$\overline T(w)=(\tilde p(y))^{1/2}w=W,$$
we obtain $w\in H^1_{\mathrm{loc}}(\brd{R}^{n+1},\tilde p(y)\mathrm{d}z)$.\\\\
{\bf{Step 5: end of the proof.}} Next we wish to show that $w$ solves the equation in \eqref{limittilde} also across the limiting characteristic hyperplane $\tilde \Sigma$. Indeed, using $w\in H^1_{\mathrm{loc}}(\brd{R}^{n+1},\tilde p(y)\mathrm{d}z)$ jointly with equation \eqref{limittilde} holding in $\brd{R}^{n+1}\setminus\Sigma$, using that $C^\infty_c(\overline B_R\setminus \Sigma)$ is actually dense in $H^1(B_R,\tilde p(y)\mathrm{d}z)$ (all the weights here, including the limit one, are super degenerate), as we have already remarked in \eqref{tildeH1=H1}.
Eventually, one can reach a contradiction by applying the suitable Liouville theorems in \cite{NorTavTerVer, SirTerVit1} and Corollary \ref{LiouvilleBHbis} for the case $\tilde p(y)=\omega_1^a(y)$.
\endproof
\subsection{Local uniform bounds in $C^{1,\alpha}$ spaces}
\begin{Theorem}\label{C1alphaBH}
Let $a\in(-\infty,1)$ and as $\varepsilon\to0$ let $\{u_\varepsilon\}$ be a family of solutions in $B_1^+$ of \eqref{simpleBH} satisfying boundary conditions (evenness)
$$
\rho_\varepsilon^a(v_\varepsilon^a)^2\partial_yu_\varepsilon=0 \qquad \mathrm{on \ }\partial^0 B_1^+.
$$
Let $r\in(0,1)$, $\beta>1$, $p_1,p_2>\overline d=n+3+(-a)^+$. Let $\|b\|_{L^{2p_2}(B_1,\omega_\varepsilon^a(y)\mathrm{d}z)}\leq b$. Let $F_\varepsilon=(F^1_\varepsilon,...,F^{n+1}_\varepsilon)$ with the $y$-component vanishing on $\Sigma$: $F^{n+1}_\varepsilon(x,0)= F^y_\varepsilon(x,0)=0$ in $\partial^0B_1^+$. Let moreover $A$ satisfy assumption (HA) with $\alpha$-H\"older continuous coefficients and $\alpha\in(0,1-\frac{n+3+(-a)^+}{p_1}]\cap(0,1-\frac{n+3+(-a)^+}{p_2}]$. There is a positive constant depending on $a$, $n$, $\beta$, $p_1$, $p_2$, $b$, $\alpha$ and $r$ only such that
$$\|u_\varepsilon\|_{C^{1,\alpha}(B_r^+)}\leq c\left(\|u_\varepsilon\|_{L^\beta(B_1^+,\omega_\varepsilon^a(y)\mathrm{d}z)}+ \|f_\varepsilon\|_{L^{p_1}(B_1^+,\omega_\varepsilon^a(y)\mathrm{d}z)}+\|F_\varepsilon\|_{C^{0,\alpha}(B_1^+)}\right).$$
\end{Theorem}
\proof
We wish to follow the same steps of proof of \cite[Theorems 5.1 and 5.2]{SirTerVit1}. Among others, we have to deal with an additional difficulty; that is, our weights here do depend on the full variable $z=(x,y)$ and not on $y$ only. For our purposes, we can take advantage of the fact that the ratio
\begin{equation}
\gamma_\varepsilon^a(x,y)=:\frac{v_\varepsilon^a(x,y)}{\chi_\varepsilon^a(y)}=\frac{\int_0^y\rho_\varepsilon^{-a}(s)\mu^{-1}(x,s)\mathrm{d}s}{\int_0^y\rho_\varepsilon^{-a}(s)\mathrm{d}s}
\end{equation}
is uniformly bounded in $C^{0,\alpha}$ with respect to $\varepsilon$ (just apply Lemma \ref{c0alphaG}, using the fact that $\mu^{-1}\in C^{0,\alpha}$ since the matrix $A$ possesses $\alpha$-H\"older continuous coefficients). Hence, one can rewrite our operator as
\begin{equation}
\textrm{div}(\rho_\varepsilon^a(y)(v_\varepsilon^a(x,y))^2A(x,y)\nabla u_\varepsilon)=\textrm{div}(\omega_\varepsilon^a(y)A_\varepsilon(x,y)\nabla u_\varepsilon),
\end{equation}
where, up to constants, the new family of matrices is defined as
\begin{equation}
A_\varepsilon(x,y)=(\gamma_\varepsilon^a(x,y))^2A(x,y),
\end{equation}
with coefficients which are uniformly bounded in $C^{0,\alpha}$ with respect to $\varepsilon$.\\\\
With these premises, we are now able to follow the construction made in \cite[Theorems 5.1 and 5.2]{SirTerVit1}. Just to give the idea, the contradiction argument uses two blow-up sequences
\begin{equation*}
v_k(z)=\frac{\eta(\hat{z}_k+r_kz)}{L_kr_k^{1+\alpha}}\left(u_k(\hat{z}_k+r_kz)-u_k(\hat{z}_k)\right),\qquad w_k(z)=\frac{\eta(\hat{z}_k)}{L_kr_k^{1+\alpha}}\left(u_k(\hat{z}_k+r_kz)-u_k(\hat{z}_k)\right),
\end{equation*}
for $z\in B(k):=\frac{B-\hat{z}_k}{r_k}$. Hence, one has to work with
$$\overline v_k(z)=v_k(z)-\nabla v_k(0)\cdot z,\qquad\overline w_k(z)=w_k(z)-\nabla w_k(0)\cdot z,$$
or
$$\overline v_k(z)=v_k(z)-\nabla_x v_k(0)\cdot x,\qquad\overline w_k(z)=w_k(z)-\nabla_x w_k(0)\cdot x,$$
respectively when $\frac{d(z_k,\Sigma)}{r_k}\to+\infty$ (in this case we choose $\hat{z}_k=z_k$), or $\frac{d(z_k,\Sigma)}{r_k}\leq c$ uniformly in $k$ (in this case we choose $\hat{z}_k=(x_k,0)$ to be the projection on $\Sigma$ of $z_k$, where $z_k=(x_k,y_k)$).\\\\
Hence, reasoning as in the previous Theorem \ref{C0alphaBH}, one can characterize all possible rescalings of the weights (in facts the possible scalings of weights $p_k$ and $\omega_k$ are the same), and prove that the limit $w$ is an energy entire solution to the suitable limiting problem.\\\\
We remark that in order to show that the limit equation has a constant coefficient matrix one has to reason as in \cite[Remark 5.3]{SirTerVit1}, using the $\alpha$-H\"older continuity of coefficients of the matrix (in this case we will invoke the uniform bounds with respect to $\varepsilon$ in $C^{0,\alpha}$ for the coefficients of $A_{\varepsilon_k}$).\\\\
Nevertheless, we need also to deal with drift terms in the rescaled equations, and we wish to show that they vanish once testing with the suitable test function $\phi$ supported in $B_R$. We assume here that $h_k=\nu_k^{2-a}$ (one of the possible cases). Hence, also in this case we use the fact that we know a priori that the sequence $\{u_k\}$ is uniformly locally bounded in $C^{0,\beta}$ spaces, for any choice of $\beta\in(0,1)$ (follows from Theorem \ref{C0alphaBH}). Reasoning as in \cite[Remark 5.3]{SirTerVit1}, this gives the following energy estimate
\begin{equation}
\int_{B_r} p_k(z)|\nabla u_k|^2\leq\frac{c}{r^{2(1-\beta)}}\int_{B_{2r}}p_k(z).
\end{equation}
Hence, we can estimate
\begin{comment}
Moreover, the same reasonings are needed also to make vanish the extra term
\begin{equation}
\mathrm{div}(p_k\nabla_x w_k(0)).
\end{equation}
\end{comment}
\begin{eqnarray*}
&&r_k\left\vert \int_{B_R}\tilde p_k(z)b_k(\hat z_k+r_kz)\cdot\nabla w_k(z)\phi(z)\right\vert\\
&\leq&\frac{r_k^{1-\alpha}\eta(\hat z_k)}{L_k}\|\phi\|_{L^\infty(B_R)}\int_{B_R}\tilde p_k(z)|b_k(\hat z_k+r_kz)|\cdot|\nabla u_k(\hat z_k+r_kz)|\\
&\leq&\frac{cr_k^{1-\alpha}}{L_k}\left(\int_{B_R}\tilde p_k(z)|b_k(\hat z_k+r_kz)|^2\right)^{1/2}\left(\int_{B_R}\tilde p_k(z)|\nabla u_k(\hat z_k+r_kz)|^2\right)^{1/2}\\
&\leq&\frac{cr_k^{1-\alpha}}{L_k}\left(\int_{B_R}\tilde p_k(z)\right)^{1/2p_2'}\left(\frac{1}{r_k^{n+1}h_k}\int_{B_{r_kR}(\hat z_k)} \rho_{\varepsilon_k}^a(\zeta_{n+1})(v_{\varepsilon_k}^a(\zeta))^2|b_k(\zeta)|^{2p_2}\right)^{1/2p_2}\\
&&\cdot\frac{c}{r_k^{1-\beta}}\left(\int_{2B_R}\tilde p_k(z)\right)^{1/2}\\
&\leq&\frac{c}{L_k}\left(\frac{r_k}{\nu_k}\right)^{1/p_2}\left(\frac{r_k^{(-a)^+}}{\nu_k^{-a}}\right)^{1/2p_2}r_k^{\frac{1}{2}\left(1-\alpha-\frac{n+3+(-a)^+}{p_2}\right)}r_k^{\beta-\frac{1+\alpha}{2}}\to0
\end{eqnarray*}
since $L_k\to+\infty$, $\nu_k\leq r_k$, $p_2>n+3+(-a)^+$, $\alpha\leq1-\frac{n+3+(-a)^+}{p_2}$ and choosing $\beta>\frac{1+\alpha}{2}$.
\endproof
\subsection{Local regularity for the auxiliary equation}
The following is the main result of the paper (we have already stated it in a simplified version in Theorem \ref{holderBHsimple} in the introduction). Let $a\in(-\infty,1)$, the matrix $A$ written as in Notation (HA+) and let $u_\varepsilon$ be an odd energy solution to \eqref{LarhoA} in $B_1^+$; that is,
\begin{equation}\label{1oddBH}
\begin{cases}
-\mathrm{div}\left(\rho_\varepsilon^aA\nabla u_\varepsilon\right)=\rho_\varepsilon^af_\varepsilon+\mathrm{div}\left(\rho_\varepsilon^aF_\varepsilon\right) & \mathrm{in \ } B_1^+\\
u_\varepsilon=0 & \mathrm{on \ }\partial^0B_1^+.
\end{cases}
\end{equation}
Let also $v_\varepsilon^a$ be defined as in \eqref{veps} in $B_1^+$. Then, we have already showed (in Proposition \ref{BoundaryHarnackeps}) that under suitable integrability assumptions on the terms in the right hand side and on coefficients of the matrix $A$, then functions
$$w_\varepsilon=\frac{u_\varepsilon}{v_\varepsilon^a}$$
are even energy solutions (for any $R<1$) to equations \eqref{BHepseq1}; that is,
\begin{eqnarray*}
-\mathrm{div}\left(\rho_\varepsilon^a(v_\varepsilon^a)^2A\nabla w_\varepsilon\right)&=&\rho_\varepsilon^a(v_\varepsilon^a)^2\left(\overline f_\varepsilon -\frac{\overline F_\varepsilon\cdot\nabla v_\varepsilon^a}{v_\varepsilon^a}\right)+\mathrm{div}\left(\rho_\varepsilon^a(v_\varepsilon^a)^2\overline F_\varepsilon\right)\nonumber\\
&&+\mathrm{div}_x\left(\rho_\varepsilon^a(v_\varepsilon^a)^2(b^{\tilde A}_\varepsilon+\overline T_\varepsilon) w_\varepsilon\right)-\rho_\varepsilon^a(v_\varepsilon^a)^2\left((b^{\tilde A}_\varepsilon+\overline T_\varepsilon)\cdot b^{\mathbb I}_\varepsilon w_\varepsilon+(b^{\tilde A}_\varepsilon+\overline T_\varepsilon)\cdot\nabla_xw_\varepsilon\right),
\end{eqnarray*}
with boundary condition
$$\rho_\varepsilon^a(v_\varepsilon^a)^2\partial_yw_\varepsilon=0 \qquad \mathrm{on \ }\partial^0B_R^+,$$
and where we denote by
\begin{equation*}
\overline f_\varepsilon=\frac{f_\varepsilon}{v_\varepsilon^a},\qquad \overline F_\varepsilon=\frac{F_\varepsilon}{v_\varepsilon^a},\qquad b^M_\varepsilon=M\cdot\frac{\nabla_xv_\varepsilon^a}{v_\varepsilon^a},\qquad\mathrm{and}\qquad \overline T_\varepsilon=\frac{T}{\rho_\varepsilon^av_\varepsilon^a},
\end{equation*}
($M$ is a general $(n,n)$-dimensional matrix).
\begin{Theorem}\label{holderBH}
Let $a\in(-\infty,1)$, the matrix $A$ written as in Notation (HA+) and as $\varepsilon\to0$ let $\{u_\varepsilon\}$ be a family of solutions in $B_1^+$ of \eqref{1oddBH}.\\\\
$1)$ Let $r\in(0,1)$, $\beta>1$, $p_1,p_2>\frac{n+3+(-a)^+}{2}$, $p_3,p_4>n+3+(-a)^+$, and $\alpha\in(0,2-\frac{n+3+(-a)^+}{p_1}]\cap(0,2-\frac{n+3+(-a)^+}{p_2}]\cap(0,1-\frac{n+3+(-a)^+}{p_3}]\cap(0,1-\frac{n+3+(-a)^+}{p_4}]$. Let also
$$\| (b^{\tilde A}_\varepsilon+\overline T_\varepsilon)\cdot b^{\mathbb I}_\varepsilon\|_{L^{p_2}(B_1^+,\omega_\varepsilon^a(y)\mathrm{d}z)}\leq b_1,\qquad \| b^{\tilde A}_\varepsilon+\overline T_\varepsilon\|_{L^{p_4}(B_1^+,\omega_\varepsilon^a(y)\mathrm{d}z)}\leq b_2.$$
Let us moreover take $A$ with continuous coefficients. There is a positive constant depending on $a$, $n$, $\beta$, $p_1$, $p_2$, $p_3$, $p_4$, $b_1$, $b_2$, $\alpha$ and $r$ only such that functions
$$w_\varepsilon=\frac{u_\varepsilon}{v_\varepsilon^a}$$
satisfy
\begin{eqnarray*}
\|w_\varepsilon\|_{C^{0,\alpha}(B_r^+)}&\leq& c\left(\|w_\varepsilon\|_{L^\beta(B_1^+,\omega_\varepsilon^a(y)\mathrm{d}z)}+ \left\|\overline f_\varepsilon-\frac{\overline F_\varepsilon\cdot \nabla v_\varepsilon^a}{v_\varepsilon^a}\right\|_{L^{p_1}(B_1^+,\omega_\varepsilon^a(y)\mathrm{d}z)} +\|\overline F_\varepsilon\|_{L^{p_3}(B_1^+,\omega_\varepsilon^a(y)\mathrm{d}z)}\right).
\end{eqnarray*}
$2)$ Let $r\in(0,1)$, $\beta>1$, $p_1,p_2>n+3+(-a)^+$, and $\alpha\in(0,1-\frac{n+3+(-a)^+}{p_1}]\cap(0,1-\frac{n+3+(-a)^+}{p_2}]$. Let $\overline F_\varepsilon=(\overline F^1_\varepsilon,...,\overline F^{n+1}_\varepsilon)$ with the $y$-component vanishing on $\Sigma$: $\overline F^{n+1}_\varepsilon(x,0)=\overline F^y_\varepsilon(x,0)=0$ in $\partial^0B_1^+$. Let also
$$\| (b^{\tilde A}_\varepsilon+\overline T_\varepsilon)\cdot b^{\mathbb I}_\varepsilon\|_{L^{p_2}(B_1^+,\omega_\varepsilon^a(y)\mathrm{d}z)}\leq b_1,\qquad \| b^{\tilde A}_\varepsilon+\overline T_\varepsilon\|_{C^{0,\alpha}(B_1^+)}\leq b_2.$$
Let's moreover take $A$ with $\alpha$-H\"older continuous coefficients. There is a positive constant depending on $a$, $n$, $\beta$, $p_1$, $p_2$, $b_1$, $b_2$, $\alpha$ and $r$ only such that functions
$$w_\varepsilon=\frac{u_\varepsilon}{v_\varepsilon^a}$$
satisfy
\begin{eqnarray*}
\|w_\varepsilon\|_{C^{1,\alpha}(B_r^+)}&\leq& c\left(\|w_\varepsilon\|_{L^\beta(B_1^+,\omega_\varepsilon^a(y)\mathrm{d}z)}+\left\|\overline f_\varepsilon-\frac{\overline F_\varepsilon\cdot \nabla v_\varepsilon^a}{v_\varepsilon^a}\right\|_{L^{p_1}(B_1^+,\omega_\varepsilon^a(y)\mathrm{d}z)}+\|\overline F_\varepsilon\|_{C^{0,\alpha}(B_1^+)}\right).
\end{eqnarray*}
\end{Theorem}
We remark that uniform estimates with respect to the regularization are optimal in $C^{1,\alpha}$-spaces (in \cite[Remark 5.4]{SirTerVit1} we provided a counterexample which show that $C^{2,\alpha}$ estimates could not be uniform up to $\Sigma$ as $\varepsilon\to0$).\\\\
In order to prove our main result we have the following useful preliminary result on equations of the form
\begin{equation}\label{BHepseq3}
-\mathrm{div}\left(\rho_\varepsilon^a(v_\varepsilon^a)^2A\nabla u_\varepsilon\right)=\rho_\varepsilon^a(v_\varepsilon^a)^2Vu_\varepsilon+\mathrm{div}\left(\rho_\varepsilon^a(v_\varepsilon^a)^2F u_\varepsilon\right)\qquad\mathrm{in \ } B_1.
\end{equation}
\begin{Lemma}
Let $a\in(-\infty,1)$ and $\varepsilon\geq0$. Let $u\in H^{1}(B_1,\omega_\varepsilon^a(y)\mathrm{d}z)$ be an energy solution to \eqref{BHepseq3}, where $V\in L^{p_1}(B_1,\omega_\varepsilon^a(y)\mathrm{d}z)$ with $p_1>\frac{\overline d}{2}=\frac{n+3+(-a)^+}{2}$ and $F\in L^{p_2}(B_1,\omega_\varepsilon^a(y)\mathrm{d}z)$ with $p_2>\overline d=n+3+(-a)^+$. Let
$$\|V\|_{L^{p_1}(B_1,\omega_\varepsilon^a(y)\mathrm{d}z)}\leq b_1,\qquad \|F\|_{L^{p_2}(B_1,\omega_\varepsilon^a(y)\mathrm{d}z)}\leq b_2.$$
$1)$ Then, for any $0<r<1$ and $\beta>1$ there exists a positive constant independent of $\varepsilon$ (depending on $n$, $a$, $r$, $\beta$, $p_1$, $p_2$, $b_1$, $b_2$), $m_1>\frac{\overline d}{2}$ and $m_2>\overline d$ such that
$$\|Vu\|_{L^{m_1}(B_{r},\omega_\varepsilon^a(y)\mathrm{d}z)}+\|Fu\|_{L^{m_2}(B_{r},\omega_\varepsilon^a(y)\mathrm{d}z)}\leq c\|u\|_{L^{\beta}(B_1,\omega_\varepsilon^a(y)\mathrm{d}z)}.$$
$2)$ If moreover $p_1>\overline d=n+3+(-a)^+$ and $F\in C^{0,\alpha}(B_1)$ for some $\alpha\in(0,1)$,
$$\|V\|_{L^{p_1}(B_1,\omega_\varepsilon^a(y)\mathrm{d}z)}\leq b_1,\qquad \|F\|_{C^{0,\alpha}(B_1)}\leq b_2,$$
then for any $0<r<1$ and $\beta>1$ there exists a positive constant independent of $\varepsilon$ (depending on $n$, $a$, $r$, $\beta$, $p_1$, $\alpha$, $b_1$, $b_2$), and $m_1>\overline d$ such that
$$\|Vu\|_{L^{m_1}(B_{r},\omega_\varepsilon^a(y)\mathrm{d}z)}+\|Fu\|_{C^{0,\alpha}(B_{r})}\leq c\|u\|_{L^{\beta}(B_1,\omega_\varepsilon^a(y)\mathrm{d}z)}.$$
\end{Lemma}
\proof
The proof is done applying Moser iterations on a finite number of small enough balls which cover $B_r$. The radius of such balls is chosen in order to ensure coercivity of the quadratic forms. Hence, using the fact that the weighted integrability of $V$ and $F$ is suitably large, by a finite number of Moser iterations one can promote the integrability of $u$ itself, up to guarantee that the products $Vu$ and $Fu$ have the desired integrability (this type of argument is classic, see for instance \cite[Section 8.4]{GilTru}). Since the number of iterations is finite, one can control uniformly the constants in the iterative process, proving point 1). At to point 2), thanks to point 1) we can apply Theorem \ref{C0alphaBH} in order to obtain that the solution is $C^{0,\alpha}$ with a bound which is independent from $\varepsilon$. Hence, we obtain the second inequality taking into account the H\"older continuity of $F$.
\endproof
A relevant consequence of this result is that, under suitable conditions on the $0$-order terms and divergence terms with the solution itself inside (the conditions stated in Theorem \ref{holderBH}), we can treat $Vw$ and $\mathrm{div}(\rho v^2Fw)$ respectively as a fixed forcing term and a divergence term with a given field. As a consequence, we obtain uniform local regularity estimates in Theorem \ref{holderBH} for solutions $w_\varepsilon$ to \eqref{BHepseq} by simply applying Theorems \ref{C0alphaBH} and \ref{C1alphaBH}.
\subsubsection{A criterion for local $C^{1,\alpha}$ estimates}
We would like to show an example of a set of hypothesis for which part $2)$ of our main Theorem \ref{holderBH} holds true; that is, local uniform $C^{1,\alpha}$ estimates for the ratio of odd solutions and the fundamental ones.\\\\
We remark that, as $a\to-\infty$, the decay of the data on $\Sigma$ becomes stronger and stronger.
\begin{Assumption}[$C^{1,\alpha}$]
Let $f_\varepsilon:=y^{\max\{1,1-a\}}g_\varepsilon$ with $g_\varepsilon$ uniformly bounded in $L^{p}(B_1^+,\omega_\varepsilon^a(y)\mathrm{d}z)$ as $\varepsilon\to0$ and
\begin{equation*}
p>n+3+(-a)^+.
\end{equation*}
Let $F_\varepsilon:=y^{\max\{1,1-a\}}G_\varepsilon$ with $G_\varepsilon$ uniformly bounded in $C^{0,\alpha}(B_1^+)$ as $\varepsilon\to0$ and
\begin{equation*}
\alpha>1-\frac{1+\min\{2,2-a\}}{n+3+(-a)^+}.
\end{equation*}
Nevertheless, the matrix $A$, which satisfies Assumption (HA+), must also satisfy some regularity assumptions: $A\in C^{0,\alpha}(B_1^+)$ with $\nabla_x\mu\in C^{0,\alpha}(B_1^+)$ and $T=y\tilde T$ with $\tilde T\in C^{0,\alpha}(B_1^+)$.
\end{Assumption}
\begin{comment}
\section{Local uniform bounds in H\"older spaces for odd solutions}
Eventually, in this section, we state the results on uniform local regularity estimates for odd solutions. As a corollary of Theorem \ref{holderBH}, we obtain local uniform estimates for odd solutions whenever $a<1$.
\begin{Corollary}
$1)$ Let $a\in(-\infty,1)$ and as $\varepsilon\to0$ let $\{u_\varepsilon\}$ be a family of solutions in $B_1^+$ of \eqref{1oddBH}. Under the hypotesis in Theorem \ref{holderBH} part $1)$, we obtain for $\alpha\in(0,1-a)\cap(0,2-\frac{n+3+(-a)^+}{p_1}]\cap(0,2-\frac{n+3+(-a)^+}{p_2}]\cap(0,1-\frac{n+3+(-a)^+}{p_3}]\cap(0,1-\frac{n+3+(-a)^+}{p_4}]$ that there is a positive constant independent from $\varepsilon$ (see the dependence in Theorem \ref{holderBH}) such that
$$\|u_\varepsilon\|_{C^{0,\alpha}(B_r^+)}\leq \|w_\varepsilon\|_{C^{0,\alpha}(B_r^+)}\|v_\varepsilon^a\|_{C^{0,\alpha}(B_r^+)}.$$
$2)$ Let $a\in(-\infty,0)$ and as $\varepsilon\to0$ let $\{u_\varepsilon\}$ be a family of solutions in $B_1^+$ of \eqref{1oddBH}. Under the hypotesis in Theorem \ref{holderBH} part $2)$, we obtain for $\alpha\in(0,-a)\cap(0,1-\frac{n+3+(-a)^+}{p_1}]\cap(0,1-\frac{n+3+(-a)^+}{p_2}]$ that there is a positive constant independent from $\varepsilon$ (see the dependence in Theorem \ref{holderBH}) such that
$$\|u_\varepsilon\|_{C^{1,\alpha}(B_r^+)}\leq \|w_\varepsilon\|_{C^{1,\alpha}(B_r^+)}\|v_\varepsilon^a\|_{C^{1,\alpha}(B_r^+)}.$$
\end{Corollary}
The $C^{1,\alpha}$ regularity in the case $a=0$ is a standard result.\\\\
\end{comment}
\subsection{Local uniform bounds in H\"older spaces for odd solutions in the $A_2$ case}
Moreover, when the weight is locally integrable; that is, $a\in(-1,1)$, we obtain local estimates for odd solutions working directly on the equation.
\begin{Theorem}\label{holderodd}
Let $a\in(-1,1)$ and as $\varepsilon\to0$ let $\{u_\varepsilon\}$ be a family of solutions in $B_1^+$ of either
\begin{equation}\label{1odd}
-\mathrm{div}\left(\rho_\varepsilon^aA\nabla u_\varepsilon\right)=\rho_\varepsilon^af_\varepsilon+\mathrm{div}\left(\rho_\varepsilon^aF_\varepsilon\right)
\end{equation}
satisfying the Dirichlet boundary condition
\[
u_\varepsilon=0\quad \mathrm{on \ }\partial^0B_1^+.
\]
Let $r\in(0,1)$, $\beta>1$, $p_1>\frac{n+1+a^+}{2}$, $p_2>n+1+a^+$ and $\alpha\in(0,1)\cap(0,1-a)\cap(0,2-\frac{n+1+a^+}{p_1}]\cap(0,1-\frac{n+1+a^+}{p_2}]$. Let moreover $A$ satisfy assumption (HA) with continuous coefficients. There are constants depending on $a$, $n$, $\beta$, $p_1$, $p_2$, $\alpha$ and $r$ only such that
$$\|u_\varepsilon\|_{C^{0,\alpha}(B_r^+)}\leq c\left(\|u_\varepsilon\|_{L^\beta(B_1^+,\rho_\varepsilon^a(y)\mathrm{d}z)}+ \|f_\varepsilon\|_{L^{p_1}(B_1^+,\rho_\varepsilon^a(y)\mathrm{d}z)} + \|F_\varepsilon\|_{L^{p_2}(B_1^+,\rho_\varepsilon^a(y)\mathrm{d}z)}\right).$$
\end{Theorem}
\proof
The proof is obtained by contradiction following the very same passages of \cite[Theorem 4.1]{SirTerVit1}, observing that in presence of the zero Dirichlet boundary condition at $\Sigma$ we obtain a contradiction by applying the Liouville Theorem \ref{Liouvilleodd}. The blow-up sequences invoked are centered in points $\hat z_k\in B^+=B_{\frac{1+r}{2}}\cap\{y\geq0\}$; that is,
\begin{equation*}
v_k(z)=\frac{(\eta u_k)(\hat z_k+r_kz)-(\eta u_k)(\hat z_k)}{L_kr_k^\alpha},\qquad w_k(z)=\frac{\eta(\hat z_k)(u_k(\hat z_k+r_kz)- u_k(\hat z_k))}{L_kr_k^\alpha},
\end{equation*}
with
$$z\in B(k):=\frac{B-\hat z_k}{r_k}.$$
Moreover, if $y_k/r_k\to+\infty$ (where $z_k=(x_k,y_k)$), then we choose $\hat z_k=z_k$, while if $y_k/r_k\leq c$ uniformly, then we choose $\hat z_k=(x_k,0)$. In this second case, we remark that $v_k$ and $w_k$ are antisymmetric with respect to $\{y=0\}$ so that the limit $w$ will be odd in $y$.
\endproof
\appendix
\section{Some special functions}
In this appendix we are going to state and prove some technical results which will allow us to compare, from the regularity point of view, the variable coefficient case with the constant one.
\begin{remark}\label{psia}
Let $a\in(-\infty,1)$, $\varepsilon\geq0$. Then the family of functions
\begin{equation}\label{psi}
\psi_\varepsilon^a(y):=\frac{y\rho_\varepsilon^{-a}(y)}{\int_0^y\rho_\varepsilon^{-a}(s)\mathrm{d}s}
\end{equation}
are monotone in $y$ and uniformly bounded in $L^{\infty}(B_1^+)$ by a constant which does not depend on $\varepsilon$. In fact, denoting $t=y/\varepsilon$, we have
\begin{equation*}
\psi_\varepsilon^a(y)=\psi_1^a\left(\frac{y}{\varepsilon}\right)=\psi_1^a\left(t\right)=\frac{t(1+t^2)^{-a/2}}{\int_0^t(1+s^2)^{-a/2}\mathrm{d}s}.
\end{equation*}
The latter function is continuous and monotone nondecreasing if $a<0$ and nonincreasing if $a\in(0,1)$. Since $\psi_1^a$ has limit $1$ as $t\to0$ and limit $1-a$ as $t\to+\infty$, then
$$\sup_{t>0}\psi_1^a(t)=\max\{1,1-a\}\qquad\mathrm{and}\qquad \inf_{t>0}\psi_1^a(t)=\min\{1,1-a\}.$$
Finally, note that the family $\psi_\varepsilon^a$ can not be equicontinuous, nor uniformly bounded in $C^{0,\alpha}(B_1^+)$, while it enjoys the following property:
\begin{equation}\label{eq:property}
\exists c>0\;:\;
\forall \varepsilon\in[0,\varepsilon_0)\;, \Vert \psi_\varepsilon^a\Vert_{\textrm{Lip}(B_1\cap\{y>\sqrt{\varepsilon}\})}<c\;,
\end{equation}
due to the fact that $\psi_1^a$ is bounded, has a finite limit as $t\to +\infty$ and its derivative vanishes as $1/t^2$.
\end{remark}
\begin{Lemma}\label{a2}
Let $a\in(-\infty,1)$, $\varepsilon\geq0$, $\alpha\in(0,1)$ and let $g(x,y,s)\in C^{0,\alpha}_{x,y}(B_1^+)$ uniformly in $s\in[0,y]$, such that $|g(x,y,s)|\leq c|y|^\alpha$ for $(x,y)\in B_1^+$ uniformly in $s\in[0,y]$. Then the family of functions
\begin{equation*}
\mathcal G_\varepsilon(x,y)=\frac{\int_0^y\rho_\varepsilon^{-a}(s)g(x,y,s)\mathrm{d}s}{\int_0^y\rho_\varepsilon^{-a}(s)\mathrm{d}s}
\end{equation*}
is uniformly bounded in $C^{0,\alpha}(B_1^+)$ by a constant which does not depend on $\varepsilon$.
\end{Lemma}
\proof
We remark that the proof follows some ideas of the proof in \cite[Lemma 7.5]{SirTerVit1}, where the case $\varepsilon=0$ is done. The uniform H\"older continuity in the $x$-variable is trivial. Hence, fixed $0<\delta<1$, let us consider the following two sets
$$I_1=\{(y_1,y_2)\ : \ 0\leq y_1\leq y_2<1, \ y_2-y_1\geq\delta y_2\}$$
and
$$I_2=\{(y_1,y_2)\ : \ 0\leq y_1\leq y_2<1, \ y_2-y_1<\delta y_2\}.$$
If we consider $(y_1,y_2)\in I_1$, using that for $i=1,2$, in the interval $(0,y_i)$ it holds $|g(x,y_i,s)|\leq cy_i^\alpha$ and thanks to the inequalities $(y_2-y_1)^\alpha\geq\delta^\alpha y_2^\alpha\geq\delta^\alpha y_i^\alpha$, then
\begin{eqnarray*}
\frac{|\mathcal G_\varepsilon(x,y_1)-\mathcal G_\varepsilon(x,y_2)|}{(y_2-y_1)^\alpha}&\leq&\frac{1}{(y_2-y_1)^\alpha}\sum_{i=1}^2|\mathcal G_\varepsilon(x,y_i)|\\
&\leq&\frac{c}{\delta^\alpha}\sum_{i=1}^2\frac{y_i^\alpha\int_0^{y_i}\rho_\varepsilon^{-a}(s)}{y_i^\alpha\int_0^{y_i}\rho_\varepsilon^{-a}(s)}=\frac{2c}{\delta^\alpha}.
\end{eqnarray*}
If we consider $(y_1,y_2)\in I_2$, then
\begin{eqnarray*}
\frac{|\mathcal G_\varepsilon(x,y_1)-\mathcal G_\varepsilon(x,y_2)|}{(y_2-y_1)^\alpha}&\leq&\frac{1}{(y_2-y_1)^\alpha}\frac{\int_{0}^{y_2}\rho_\varepsilon^{-a}(s)|g(x,y_2,s)-g(x,y_1,s)|}{\int_0^{y_2}\rho_\varepsilon^{-a}(s)}\\
&&+\frac{1}{(y_2-y_1)^\alpha}\frac{\int_{y_1}^{y_2}\rho_\varepsilon^{-a}(s)|g(x,y_1,s)|}{\int_0^{y_2}\rho_\varepsilon^{-a}(s)}\\
&&+\frac{1}{(y_2-y_1)^\alpha}\frac{\left(\int_{0}^{y_1}\rho_\varepsilon^{-a}(s)|g(x,y_1,s)|\right)\left(\int_{y_1}^{y_2}\rho_\varepsilon^{-a}(s)\right)}{\left(\int_{0}^{y_1}\rho_\varepsilon^{-a}(s)\right)\left(\int_{0}^{y_2}\rho_\varepsilon^{-a}(s)\right)}\\
&=&J_1+J_2+J_3.
\end{eqnarray*}
Hence, $J_1$ can be bounded using the fact that $|g(x,y_2,s)-g(x,y_1,s)|\leq c(y_2-y_1)^\alpha$. Working on $J_2$, there exists $y_1\leq\xi\leq y_2$ such that
\begin{eqnarray*}
J_2&\leq& c(y_2-y_1)^{1-\alpha}\frac{\rho_\varepsilon^{-a}(\xi)y_1^\alpha}{\int_0^{y_2}\rho_\varepsilon^{-a}(s)}\\
&\leq&c\left(\frac{y_2-y_1}{y_2}\right)^{1-\alpha}\frac{y_2\rho_\varepsilon^{-a}(\xi)}{\int_0^{y_2}\rho_\varepsilon^{-a}(s)}\\
&\leq&c\delta^{1-\alpha}\max\{1,(1-\delta)^{-a}\}\frac{y_2\rho_\varepsilon^{-a}(y_2)}{\int_0^{y_2}\rho_\varepsilon^{-a}(s)}
\end{eqnarray*}
using the fact that $y_2-y_1<\delta y_2$, the inequalities
\begin{equation*}
1-\delta<\frac{y_1}{y_2}\leq\frac{\xi}{y_2}\leq 1,
\end{equation*}
and the fact that $\rho_\varepsilon^{-a}(\xi)\leq \max\{1,(1-\delta)^{-a}\}\rho_\varepsilon^{-a}(y_2)$ (easy to check). Eventually, recalling $y_2/\varepsilon=t\in[0,+\infty)$, we have already remarked that the function defined in \eqref{psi} is bounded uniformly in $\varepsilon$
$$\frac{y_2\rho_\varepsilon^{-a}(y_2)}{\int_0^{y_2}\rho_\varepsilon^{-a}(s)}=\frac{t(1+t^2)^{-a/2}}{\int_0^{t}(1+s^2)^{-a/2}}=\psi(t)\leq\max\{1,1-a\}.$$
With analogous computations we can bound also $J_3$.
\endproof
\begin{Proposition}\label{c0alphaG}
Let $a\in(-\infty,1)$, $\varepsilon\geq0$, $\alpha\in(0,1)$ and let $\gamma\in C^{0,\alpha}(B_1^+)$. Then the family of functions
\begin{equation*}
\mathcal G_\varepsilon(x,y)=\frac{\int_0^y\rho_\varepsilon^{-a}(s)\left(\gamma(x,s)-\gamma(x,0)\right)\mathrm{d}s}{\int_0^y\rho_\varepsilon^{-a}(s)\mathrm{d}s}
\end{equation*}
is uniformly bounded in $C^{0,\alpha}(B_1^+)$ by a constant which does not depend on $\varepsilon$.
\end{Proposition}
\proof
Just notice that, since $g(x,y,s):=\gamma(x,s)-\gamma(x,0)$ for $s\leq y$, $g$ satisfies conditions of the previous Lemma \ref{a2}. Indeed is $\alpha$-H\"older continuous in $(x,y)$ uniformly in $s\leq y$ and
$$|g(x,y,s)|=|\gamma(x,s)-\gamma(x,0)|\leq c|s|^\alpha\leq c|y|^\alpha.$$
\begin{comment}
We remark that the proof follows some ideas of the proof in \cite[Lemma 7.5]{SirTerVit1}, where the case $\varepsilon=0$ is done. The uniform H\"older continuity in the $x$-variable is trivial. Hence, fixed $0<\delta<1$, let us consider the following two sets
$$I_1=\{(y_1,y_2)\ : \ 0\leq y_1\leq y_2<1, \ y_2-y_1\geq\delta y_2\}$$
and
$$I_2=\{(y_1,y_2)\ : \ 0\leq y_1\leq y_2<1, \ y_2-y_1<\delta y_2\}.$$
If we consider $(y_1,y_2)\in I_1$, using that for $i=1,2$, in the interval $(0,y_i)$ it holds $|g(x,s)-g(x,0)|\leq cs^\alpha\leq cy_i^\alpha$ and thanks to the inequalities $(y_2-y_1)^\alpha\geq\delta^\alpha y_2^\alpha\geq\delta^\alpha y_i^\alpha$, then
\begin{eqnarray*}
\frac{|\mathcal G_\varepsilon(x,y_1)-\mathcal G_\varepsilon(x,y_2)|}{(y_2-y_1)^\alpha}&\leq&\frac{1}{(y_2-y_1)^\alpha}\sum_{i=1}^2|\mathcal G_\varepsilon(x,y_i)|\\
&\leq&\frac{c}{\delta^\alpha}\sum_{i=1}^2\frac{\int_0^{y_i}\rho_\varepsilon^{-a}(s)s^\alpha}{y_i^\alpha\int_0^{y_i}\rho_\varepsilon^{-a}(s)}\leq\frac{2c}{\delta^\alpha}.
\end{eqnarray*}
If we consider $(y_1,y_2)\in I_2$, then
\begin{eqnarray*}
\frac{|\mathcal G_\varepsilon(x,y_1)-\mathcal G_\varepsilon(x,y_2)|}{(y_2-y_1)^\alpha}&\leq&\frac{1}{(y_2-y_1)^\alpha}\frac{c\int_{y_1}^{y_2}\rho_\varepsilon^{-a}(s)s^\alpha}{\int_0^{y_2}\rho_\varepsilon^{-a}(s)}+\frac{1}{(y_2-y_1)^\alpha}\frac{c\left(\int_{0}^{y_1}\rho_\varepsilon^{-a}(s)s^\alpha\right)\left(\int_{y_1}^{y_2}\rho_\varepsilon^{-a}(s)\right)}{\left(\int_{0}^{y_1}\rho_\varepsilon^{-a}(s)\right)\left(\int_{0}^{y_2}\rho_\varepsilon^{-a}(s)\right)}\\
&=&J_1+J_2.
\end{eqnarray*}
Working on $J_1$, there exists $y_1\leq\xi\leq y_2$ such that
\begin{eqnarray*}
J_1&\leq& c(y_2-y_1)^{1-\alpha}\frac{\rho_\varepsilon^{-a}(\xi)\xi^\alpha}{\int_0^{y_2}\rho_\varepsilon^{-a}(s)}\\
&\leq&c\delta^{1-\alpha}y_2^{1-\alpha}\frac{\rho_\varepsilon^{-a}(\xi)\xi^\alpha}{\int_0^{y_2}\rho_\varepsilon^{-a}(s)}\\
&\leq&c\delta^{1-\alpha}\left(\frac{\xi}{y_2}\right)^\alpha\frac{y_2\rho_\varepsilon^{-a}(\xi)}{\int_0^{y_2}\rho_\varepsilon^{-a}(s)}\\
&\leq&c\delta^{1-\alpha}\frac{y_2\rho_\varepsilon^{-a}(y_2)}{\int_0^{y_2}\rho_\varepsilon^{-a}(s)}
\end{eqnarray*}
using the fact that $y_2-y_1<\delta y_2$, the inequalities
\begin{equation*}
1-\delta<\frac{y_1}{y_2}\leq\frac{\xi}{y_2}\leq 1,
\end{equation*}
and the fact that $\rho_\varepsilon^{-a}(\xi)\leq \max\{1,(1-\delta)^{-a}\}\rho_\varepsilon^{-a}(y_2)$ (easy to check). Eventually, recalling $y_2/\varepsilon=t\in[0,+\infty)$, we have already remarked that the function defined in \eqref{psi} is bounded
$$\frac{y_2\rho_\varepsilon^{-a}(y_2)}{\int_0^{y_2}\rho_\varepsilon^{-a}(s)}=\frac{t(1+t^2)^{-a/2}}{\int_0^{t}(1+s^2)^{-a/2}}=\psi(t)\leq\max\{1,1-a\}.$$
Eventually, also for the second term
$$J_2\leq c\left(\frac{y_2-y_1}{y_2}\right)^{1-\alpha}\frac{y_2\rho_\varepsilon^{-a}(\xi)}{\int_0^{y_2}\rho_\varepsilon^{-a}(s)}\leq c\delta^{1-\alpha}\max\{1,1-a\}.$$
\end{comment}
\endproof
\begin{Proposition}\label{c1alphaG}
Let $a\in(-\infty,1)$, $\varepsilon\geq0$, $\alpha\in(0,1)$ and let $\gamma\in C^{1,\alpha}(B_1^+)$ with $\partial_y\gamma(x,0)\in C^{1,\alpha}(B_1^+)$. Consider the family of functions
\begin{equation*}
\mathcal G_\varepsilon(x,y)=\frac{\int_0^y\rho_\varepsilon^{-a}(s)\left(\gamma(x,s)-\gamma(x,0)\right)\mathrm{d}s}{\int_0^y\rho_\varepsilon^{-a}(s)\mathrm{d}s}\;.
\end{equation*}
Then there exists $c>0$ such that, for every $\varepsilon\in[0,\varepsilon_0]$,
$\mathcal G_\varepsilon$ is uniformly bounded in $C^{1,\alpha}(B_1\cap\{y\geq \sqrt{\varepsilon}\})$ by $c$.
\end{Proposition}
\proof
One can rewrite our function as
\begin{eqnarray*}
\mathcal G_\varepsilon(x,y)&=&\frac{\int_0^y\rho_\varepsilon^{-a}(s)\left(\gamma(x,s)-\gamma(x,0)-\partial_y\gamma(x,0)s\right)\mathrm{d}s}{\int_0^y\rho_\varepsilon^{-a}(s)\mathrm{d}s}+\partial_y\gamma(x,0)\frac{\int_0^y\rho_\varepsilon^{-a}(s)s\, \mathrm{d}s}{\int_0^y\rho_\varepsilon^{-a}(s)\mathrm{d}s}\\
&=& \frac{\int_0^y\rho_\varepsilon^{-a}(s)\left(\int_0^s(\partial_y\gamma(x,\tau)-\partial_y\gamma(x,0))\mathrm{d}\tau\right)\mathrm{d}s}{\int_0^y\rho_\varepsilon^{-a}(s)\mathrm{d}s}+\partial_y\gamma(x,0)\frac{\int_0^y\rho_\varepsilon^{-a}(s)s\, \mathrm{d}s}{\int_0^y\rho_\varepsilon^{-a}(s)\mathrm{d}s}\\
&=& \mathcal{H}_\varepsilon(x,y)+\partial_y\gamma(x,0)\frac{\int_0^y\rho_\varepsilon^{-a}(s)s\, \mathrm{d}s}{\int_0^y\rho_\varepsilon^{-a}(s)\mathrm{d}s}.
\end{eqnarray*}
First we show that the second term has the desired property uniformly in $\varepsilon$. At first we remark that $\partial_y\gamma(x,0)\in C^{1,\alpha}(B_1^+)$. Now consider that the family of functions
\begin{equation*}
\xi_\varepsilon^a(y):=\frac{\int_0^y\rho_\varepsilon^{-a}(s)s\, \mathrm{d}s}{\int_0^y\rho_\varepsilon^{-a}(s)\mathrm{d}s}
\end{equation*}
is uniformly bounded in $L^\infty(B_1^+)$. In fact, denoting $t=y/\varepsilon$,
\begin{equation*}
\xi_\varepsilon^a(y)=\varepsilon\xi_1^a(t)=\varepsilon\frac{\int_0^t(1+s^2)^{-a/2}s\, \mathrm{d}s}{\int_0^t(1+s^2)^{-a/2}\mathrm{d}s}=y \, \frac{\xi_1^a(t)}{t},
\end{equation*}
is bounded in $B_1^+$ (uniformly with respect to $\varepsilon\geq0$). In fact, the first factor $y$ is obviously bounded in $[0,1]$ and the second one is bounded for $t\in[0,+\infty)$. Now, let us consider the derivative in $y$,
\begin{equation*}
\partial_y\xi_\varepsilon^a(y)=(\xi_1^a)'(t)=\psi_1^a(t)\left(1-\frac{\int_0^t(1+s^2)^{-a/2}s\, \mathrm{d}s}{t\int_0^t(1+s^2)^{-a/2}\mathrm{d}s}\right)\;.
\end{equation*}
We claim that $\partial_y\xi_\varepsilon^a$ enjoys the property stated in \eqref{eq:property}, being the product of two functions, both bounded,
having a finite limit as $t\to +\infty$ and derivatives vanishing as $1/t^2$.
.
Eventually we consider $\mathcal H_\varepsilon$. Computing the gradient $\nabla_x\mathcal H_\varepsilon$, we obtain
\begin{equation*}
\nabla_x\mathcal H_\varepsilon(x,y)=\frac{\int_0^y\rho_\varepsilon^{-a}(s)\left(\tilde\gamma(x,s)-\tilde\gamma(x,0)\right)\mathrm{d}s}{\int_0^y\rho_\varepsilon^{-a}(s)\mathrm{d}s}
\end{equation*}
where
$$\tilde\gamma(x,s)=\nabla_x\gamma(x,s)-\nabla_x\partial_y\gamma(x,0)s\in C^{0,\alpha}(B_1^+),$$
and satisfies the assumptions in Proposition \ref{c0alphaG}.\\\\
It remains to consider the partial derivative in $y$ of $\mathcal H_\varepsilon$; that is,
\begin{eqnarray*}
\partial_y\mathcal H_\varepsilon(x,y)&=&\frac{y\rho_\varepsilon^{-a}(y)}{\int_0^y\rho_\varepsilon^{-a}(s)\mathrm{d}s}\cdot\dfrac{\int_0^y\rho_\varepsilon^{-a}(s)\left(\frac{1}{y}\int_s^y(\partial_y\gamma(x,\tau)-\partial_y\gamma(x,0))\mathrm{d}\tau\right)\mathrm{d}s}{\int_0^y\rho_\varepsilon^{-a}(s)\mathrm{d}s}\\
&=&\psi_\varepsilon^a(y)\cdot\mathcal I_\varepsilon(y).
\end{eqnarray*}
By Remark \ref{psia}, the family of functions $\psi_\varepsilon^a$ enjoy the desired propery \eqref{eq:property}. Now we wish to conclude that $\mathcal I_\varepsilon$ is uniformly bounded in $C^{0,\alpha}(B_1^+)$. To this aim, it is enough to prove that the function
$$g(x,y,s)=\frac{1}{y}\int_s^y(\partial_y\gamma(x,\tau)-\partial_y\gamma(x,0))\mathrm{d}\tau$$
satisfies conditions in Lemma \ref{a2}. Using the H\"older continuity of $\partial_y\gamma$, obviously
$$|g(x,y,s)|\leq\frac{1}{y}\int_s^y|\partial_y\gamma(x,\tau)-\partial_y\gamma(x,0)|\mathrm{d}\tau\leq\frac{c|y|^\alpha(y-s)}{y}\leq c|y|^\alpha.$$
The H\"older continuity of $g$ in the $x$-variable is trivial. Nevertheless, following the reasonings in the proof of Lemma \ref{a2}, fixed $0<\delta<1$, let us consider the following two sets
$$I_1=\{(y_1,y_2)\ : \ 0\leq y_1\leq y_2<1, \ y_2-y_1\geq\delta y_2\}$$
and
$$I_2=\{(y_1,y_2)\ : \ 0\leq y_1\leq y_2<1, \ y_2-y_1<\delta y_2\}.$$
If we consider $(y_1,y_2)\in I_1$, using that for $i=1,2$ it holds $|g(x,y_i,s)|\leq cy_i^\alpha$ and thanks to the inequalities $(y_2-y_1)^\alpha\geq\delta^\alpha y_2^\alpha\geq\delta^\alpha y_i^\alpha$, then
\begin{eqnarray*}
\frac{|g(x,y_1,s)-g(x,y_2,s)|}{(y_2-y_1)^\alpha}&\leq&\frac{1}{(y_2-y_1)^\alpha}\sum_{i=1}^2|g(x,y_i,s)|\\
&\leq&\frac{c}{\delta^\alpha}\sum_{i=1}^2\frac{y_i^\alpha}{y_i^\alpha}=\frac{2c}{\delta^\alpha}.
\end{eqnarray*}
If we consider $(y_1,y_2)\in I_2$, then, using the fact that $y_2-y_1<\delta y_2$
\begin{eqnarray*}
\frac{|g(x,y_1,s)-g(x,y_2,s)|}{(y_2-y_1)^\alpha}&\leq&\frac{1}{(y_2-y_1)^{\alpha}y_2}\int_{y_1}^{y_2}|\partial_y\gamma(x,\tau)-\partial_y\gamma(x,0)|\mathrm{d}\tau\\
&&+\frac{1}{(y_2-y_1)^{\alpha}}\left|\frac{1}{y_2}-\frac{1}{y_1}\right|\int_{s}^{y_1}|\partial_y\gamma(x,\tau)-\partial_y\gamma(x,0)|\mathrm{d}\tau\\
&\leq&c\delta^{1-\alpha}+c\frac{\delta^{1-\alpha}}{1-\delta}.
\end{eqnarray*}
\endproof
\begin{comment}
{\color{blue}\begin{Lemma}
Let $a\in(-\infty,1)$, $\varepsilon\geq0$, $\alpha\in(0,1)$ and let $g\in C^{1,\alpha}(B_1)$. Then the family of functions
\begin{equation*}
\mathcal G_\varepsilon(y)=\frac{\int_0^y\rho_\varepsilon^{-a}(s)\left(\int_0^s(g'(\tau)-g'(0))\mathrm{d}\tau\right)\mathrm{d}s}{\int_0^y\rho_\varepsilon^{-a}(s)\mathrm{d}s}
\end{equation*}
is uniformly bounded in $C^{1,\alpha}(B_1)$ by a constant which does not depend on $\varepsilon$.
\end{Lemma}
\proof
Let us compute the first derivative of $\mathcal G_\varepsilon$; that is,
\begin{eqnarray*}
\mathcal G_\varepsilon'(y)&=&\frac{y\rho_\varepsilon^{-a}(y)}{\int_0^y\rho_\varepsilon^{-a}(s)\mathrm{d}s}\cdot\dfrac{\int_0^y\rho_\varepsilon^{-a}(s)\left(\frac{1}{y}\int_s^y(g'(\tau)-g'(0))\mathrm{d}\tau\right)\mathrm{d}s}{\int_0^y\rho_\varepsilon^{-a}(s)\mathrm{d}s}\\
&=&\psi_\varepsilon^a(y)\cdot\mathcal H_\varepsilon(y).
\end{eqnarray*}
By Remark \ref{psia}, the family of functions $\psi_\varepsilon^a$ is uniformly bounded in $C^{0,\alpha}(B_1)$. Now we want to conclude that $\mathcal H_\varepsilon$ is uniformly bounded in $C^{0,\alpha}(B_1)$. Following the proof of Proposition \ref{c0alphaG}, fixed $0<\delta<1$, we can consider the following sets
$$I_1=\{(y_1,y_2)\ : \ 0\leq y_1\leq y_2<1, \ y_2-y_1\geq\delta y_2\}$$
and
$$I_2=\{(y_1,y_2)\ : \ 0\leq y_1\leq y_2<1, \ y_2-y_1<\delta y_2\}.$$
If we consider $(y_1,y_2)\in I_1$, using that for $i=1,2$, in the interval $(0,y_i)$ it holds $|g'(\tau)-g'(0)|\leq c\tau^\alpha\leq cy_i^\alpha$ and thanks to the inequalities $(y_2-y_1)^\alpha\geq\delta^\alpha y_2^\alpha\geq\delta^\alpha y_i^\alpha$, then
\begin{eqnarray*}
\frac{|\mathcal H_\varepsilon(y_1)-\mathcal H_\varepsilon(y_2)|}{(y_2-y_1)^\alpha}&\leq&\frac{1}{(y_2-y_1)^\alpha}\sum_{i=1}^2|\mathcal H_\varepsilon(y_i)|\\
&\leq&\frac{c}{\delta^\alpha}\sum_{i=1}^2\frac{\int_0^{y_i}\rho_\varepsilon^{-a}(s)(y_i-s)y_i^{\alpha-1}}{y_i^\alpha\int_0^{y_i}\rho_\varepsilon^{-a}(s)}\leq\frac{2c}{\delta^\alpha}.
\end{eqnarray*}
If we consider $(y_1,y_2)\in I_2$, then
\begin{eqnarray*}
\frac{|\mathcal H_\varepsilon(y_1)-\mathcal H_\varepsilon(y_2)|}{(y_2-y_1)^\alpha}&\leq&\frac{1}{(y_2-y_1)^\alpha}\dfrac{\int_0^{y_2}\rho_\varepsilon^{-a}(s)\left(\frac{1}{y_2}\int_{y_1}^{y_2}(g'(\tau)-g'(0))\mathrm{d}\tau\right)\mathrm{d}s}{\int_0^{y_2}\rho_\varepsilon^{-a}(s)\mathrm{d}s}\\
&&+\frac{1}{(y_2-y_1)^\alpha}\dfrac{\int_0^{y_2}\rho_\varepsilon^{-a}(s)\left(\left[\frac{1}{y_2}-\frac{1}{y_1}\right]\int_{s}^{y_1}(g'(\tau)-g'(0))\mathrm{d}\tau\right)\mathrm{d}s}{\int_0^{y_2}\rho_\varepsilon^{-a}(s)\mathrm{d}s}\\
&&+\frac{1}{(y_2-y_1)^\alpha}\dfrac{\int_{y_1}^{y_2}\rho_\varepsilon^{-a}(s)\left(\frac{1}{y_1}\int_{s}^{y_1}(g'(\tau)-g'(0))\mathrm{d}\tau\right)\mathrm{d}s}{\int_0^{y_2}\rho_\varepsilon^{-a}(s)\mathrm{d}s}\\
&&+\frac{1}{(y_2-y_1)^\alpha}\dfrac{\left(\int_0^{y_1}\rho_\varepsilon^{-a}(s)\left(\frac{1}{y_1}\int_{s}^{y_1}(g'(\tau)-g'(0))\mathrm{d}\tau\right)\mathrm{d}s\right)\left(\int_{y_1}^{y_2}\rho_\varepsilon^{-a}(s)\mathrm{d}s\right)}{\left(\int_0^{y_1}\rho_\varepsilon^{-a}(s)\mathrm{d}s\right)\left(\int_0^{y_2}\rho_\varepsilon^{-a}(s)\mathrm{d}s\right)}\\
&=&J_1+J_2+J_3+J_4.
\end{eqnarray*}
Easy calculations very similar to the ones in the proof of Proposition \ref{c0alphaG} show that there exists a constant uniform in $\varepsilon\geq0$ which bound the expression above. {\color{red}I CONTI LI HO FATTI CON CURA, TORNA TUTTO QUI.}
\endproof
}
\end{comment}
\begin{comment}
\begin{Proposition}\label{c1alphaG}
Let $a\in(-\infty,1)$, $\varepsilon\geq0$, $\alpha\in(0,1)$ and let $g\in C^{1,\alpha}(B_1)$ with $g_y(x,0)\in C^{1,\alpha}(B_1)$. Then the family of functions
\begin{equation*}
\mathcal G_\varepsilon(x,y)=\frac{\int_0^y\rho_\varepsilon^{-a}(s)\left(g(x,s)-g(x,0)\right)\mathrm{d}s}{\int_0^y\rho_\varepsilon^{-a}(s)\mathrm{d}s}
\end{equation*}
is uniformly bounded in $C^{1,\alpha}(B_1)$ by a constant which does not depend on $\varepsilon$.
\end{Proposition}
\proof
If we consider $\nabla_x \mathcal G_\varepsilon(x,y)$, then we end up with a function satisfying conditions in Proposition \ref{c0alphaG} and hence we can conclude the uniform H\"older continuity. Nevertheless, considering $\partial_y\mathcal G_\varepsilon(x,y)$, one can rewrite our function as
\begin{eqnarray*}
\mathcal G_\varepsilon(x,y)&=&\frac{\int_0^y\rho_\varepsilon^{-a}(s)\left(g(x,s)-g(x,0)-\partial_yg(x,0)s\right)\mathrm{d}s}{\int_0^y\rho_\varepsilon^{-a}(s)\mathrm{d}s}+\partial_yg(x,0)\frac{\int_0^y\rho_\varepsilon^{-a}(s)s\, \mathrm{d}s}{\int_0^y\rho_\varepsilon^{-a}(s)\mathrm{d}s}\\
&=& \frac{\int_0^y\rho_\varepsilon^{-a}(s)\left(\int_0^s(\partial_yg(x,\tau)-\partial_yg(x,0))\mathrm{d}\tau\right)\mathrm{d}s}{\int_0^y\rho_\varepsilon^{-a}(s)\mathrm{d}s}+\partial_yg(x,0)\frac{\int_0^y\rho_\varepsilon^{-a}(s)s\, \mathrm{d}s}{\int_0^y\rho_\varepsilon^{-a}(s)\mathrm{d}s}.
\end{eqnarray*}
The oscillations of $\partial_y\mathcal G_\varepsilon$ in the $x$-variables is easy to check. In order to control oscillations in the $y$ variable we observe that the first term satisfies the previous Lemma. For the second piece we recall that $\partial_yg(x,0)\in C^{1,\alpha}(B_1)$. Moreover, the family of functions
\begin{equation*}
\xi_\varepsilon^a(y):=\frac{\int_0^y\rho_\varepsilon^{-a}(s)s\, \mathrm{d}s}{\int_0^y\rho_\varepsilon^{-a}(s)\mathrm{d}s}
\end{equation*}
is uniformly bounded in $C^{1,\alpha}(B_1)$. In fact, denoting $t=y/\varepsilon$,
\begin{equation*}
\xi_\varepsilon^a(y)=\varepsilon\xi_1^a(t)=\varepsilon\frac{\int_0^t(1+s^2)^{-a/2}s\, \mathrm{d}s}{\int_0^t(1+s^2)^{-a/2}\mathrm{d}s}=y \, \frac{\xi_1^a(t)}{t},
\end{equation*}
is bounded in $B_1$ (uniformly with respect to $\varepsilon\geq0$). In fact, the first function $y$ is bounded in $[0,1]$ and the second one is bounded for $t\in[0,+\infty)$. Considering the derivative in $y$,
\begin{equation*}
\partial_y\xi_\varepsilon^a(y)=(\xi_1^a)'(t)=\psi_1^a(t)\left(1-\frac{\int_0^t(1+s^2)^{-a/2}s\, \mathrm{d}s}{t\int_0^t(1+s^2)^{-a/2}\mathrm{d}s}\right)
\end{equation*}
which is H\"older continuous in $[0,+\infty)$, since it is bounded and has bounded oscillations. In fact, we have already proved in Remark \ref{psia} that $\psi_1^a(t)$ is H\"older continuous in $[0,+\infty)$, and second function has the following asymptotic behaviour as $t\to0$
\begin{equation*}
1-\frac{\int_0^t(1+s^2)^{-a/2}s\, \mathrm{d}s}{t\int_0^t(1+s^2)^{-a/2}\mathrm{d}s}=\frac{1}{2}\left(1+\frac{a}{2}t^2+o(t^2)\right).
\end{equation*}
\endproof
\end{comment}
\section{Quadratic forms, stability and isometries}
In this appendix we are going to prove some useful inequalities, needed when working in weighted Sobolev spaces, specially whenever the weight does not belong to the $A_2$ class. These results will be the key of the validity of Liouville type theorems in Section \ref{sec:liouville}.
\subsection{Hardy type inequalities}
At first, we deal with the validity of Hardy (trace) type inequalities and their spectral stability. These results will be the key tools in order to establish a class of Liouville theorems contained in this section. Let $\mathbb{R}^{n+1}_+=\mathbb{R}^{n+1}\cap\{y>0\}$, $B_1^+=B_1\cap\{y>0\}$ and $S^n_+=S^{n}\cap\{y>0\}$. We define the space $\tilde H^{1}(B_1^+)$ as the closure of $C^\infty_c(\overline B_1^+\setminus\Sigma)$ with respect to the norm
$$\left(\int_{B_1^+}|\nabla v|^2\right)^{1/2}.$$
Then, we remark that the following trace Poincar\'e inequality holds
\begin{equation}\label{tracepoinc}
c\int_{S^{n}_+}v^2\leq\int_{B_1^+}|\nabla v|^2.
\end{equation}
We first state the following Hardy inequality.
\begin{Lemma}[Hardy inequality]\label{HARDY}
Let $v\in \tilde H^{1}(B_1^+)$. Then
\begin{equation}\label{Hardy}
\frac{1}{4}\int_{B_1^+}\frac{v^2}{y^2}\leq\int_{B_1^+}|\nabla v|^2.
\end{equation}
\end{Lemma}
\proof
The proof is an easy exercise based on the well known Hardy inequality on the half space
$$\frac{1}{4}\int_{\brd{R}^{n+1}_+}\frac{v^2}{y^2}\leq\int_{\brd{R}^{n+1}_+}|\nabla v|^2,$$
and using the Kelvin transform.
\endproof
Next, we will need a boundary version of the Hardy inequality
\begin{Lemma}[Boundary Hardy inequality]\label{BoundaryHardy}
There exists $c_0>0$ such that, for every $v\in\tilde H^1(B_1^+)$, there holds
\begin{equation}\label{tracehard}
c_0\int_{S^n_+}\frac{v^2}{y}\leq\int_{B_1^+}|\nabla v|^2.
\end{equation}
\end{Lemma}
\proof
By taking the harmonic replacement of $v$ on $B_1^+$, we may assume without loss of generality that
\( \Delta v=0\) in $B_1^+$. Now we consider the following inversion (stereographic projection) $\Phi:B_1^+\subset\mathbb{R}^{n+1}\to\mathbb{R}^{n+1}$ such that
$$\Phi:z=(x,y)=(x_1,...,x_{n},y)\mapsto \tilde z=(\tilde x,\tilde y)=(\tilde x_1,...,\tilde x_{n},\tilde y),$$
with
$$\Phi(z)=\frac{z+e_1}{|z+e_1|^2}-\frac{e_1}{2}\qquad\mathrm{and}\qquad \Phi^{-1}(\tilde z)=\frac{\tilde z+\frac{e_1}{2}}{|z+\frac{e_1}{2}|^2}-e_1.$$
This map is conformal and such that $\Phi(B_1^+)=\{\tilde x_1>0\}\cap\{\tilde y>0\}$ and $\Phi(S^n_+)=\{\tilde x_1=0\}\cap\{\tilde y>0\}$. Hence, the Kelvin transform
\begin{equation*}
w(\tilde z)=Kv(\tilde z):=\frac{1}{|\tilde z+\frac{e_1}{2}|^{n-1}}v(\Phi^{-1}(\tilde z))
\end{equation*}
is harmonic in $\{\tilde x_1>0\}\cap\{\tilde y>0\}$ and such that
$$\int_{B_1^+}|\nabla v|^2\mathrm{d}z=\int_{\{\tilde x_1>0\}\cap\{\tilde y>0\}}|\nabla w|^2\mathrm{d}\tilde z.$$
Using a fractional Hardy inequality (see \cite{BogDyd}) on the $n$-dimensional half space $\{\tilde x_1=0\}\cap\{\tilde y>0\}$, up to extending the function $w=0$ in $\{\tilde x_1=0\}\cap\{\tilde y<0\}$, we have
\begin{eqnarray*}
\int_{\{\tilde x_1>0\}\cap\{\tilde y>0\}}|\nabla w|^2\mathrm{d}\tilde z &\geq& c\iint_{(\{\tilde x_1=0\}\cap\{\tilde y>0\})^2}\frac{|w(\tilde \zeta_1)-w(\tilde \zeta_2)|^2}{|\tilde \zeta_1-\tilde \zeta_2|^{n+1}}\mathrm{d}\tilde \zeta_1\mathrm{d}\tilde \zeta_2\nonumber\\
&\geq& c\int_{\{\tilde x_1=0\}\cap\{\tilde y>0\}}\frac{w^2(\tilde z)}{\tilde y}\mathrm{d}\tilde z.
\end{eqnarray*}
Finally we compute
\begin{eqnarray*}
&&\int_{S^n_+}\frac{v^2(z)}{y}\mathrm{d}\sigma(z) \\
&=&\int_{\{\tilde x_1=0\}\cap\{\tilde y>0\}}\frac{w^2(\tilde z)}{\tilde y}\left|\tilde z+\frac{e_1}{2}\right|^{2(n-1)+2} \cdot|\Phi^{-1}_{\tilde x_2}(\tilde z)\wedge\Phi^{-1}_{\tilde x_3}(\tilde z)\wedge...\wedge\Phi^{-1}_{\tilde x_{n}}(\tilde z)\wedge\Phi^{-1}_{\tilde y}(\tilde z)|\mathrm{d}\tilde z\nonumber\\
&\leq&\int_{\{\tilde x_1=0\}\cap\{\tilde y>0\}}\frac{w^2(\tilde z)}{\tilde y}\mathrm{d}\tilde z.
\end{eqnarray*}
\endproof
\subsection{A stability result}
\begin{Lemma}\label{quad}
Let $\{Q_k\}_{k\in\mathbb{N}}$ be a family of quadratic forms $Q_k:\tilde H^1(B_1^+)\to[0,+\infty)$ defined by
\begin{equation*}
Q_k(v)=\int_{B_1^+}|\nabla v|^2+\int_{B_1^+}V_kv^2+\int_{S^n_+}W_kv^2.
\end{equation*}
Assume that the family $\{Q_k\}$ satisfies the following conditions:
\begin{itemize}
\item[i)] $|W_k|\leq c$ on $S^n_+$ and $|V_k|\leq\frac{c}{y^2}$ in $B_1^+$ uniformly on $k\in\mathbb{N}$;
\item[ii)] there exists a constant $C>0$ which does not depend on $k\in\mathbb{N}$ such that for any $v\in\tilde H^1(B_1^+)$
\begin{equation}\label{unifequiv}
\frac{1}{C}\|v\|_{\tilde H^1(B_1^+)}^2\leq Q_k(v)\leq C\|v\|_{\tilde H^1(B_1^+)}^2;
\end{equation}
\item[iii)] $V_k\to V$ in $B_1^+$ and $W_k\to W$ on $S^n_+$ pointwisely as $k\to+\infty$, where
$$Q(v)=\int_{B_1^+}|\nabla v|^2+\int_{B_1^+}Vv^2+\int_{S^n_+}Wv^2,$$
with $Q:\tilde H^1(B_1^+)\to[0,+\infty)$ satisfying $|W|\leq c$ on $S^n_+$, $|V|\leq\frac{c}{y^2}$ in $B_1^+$ and
$$\frac{1}{C}\|v\|_{\tilde H^1(B_1^+)}^2\leq Q(v)\leq C\|v\|_{\tilde H^1(B_1^+)}^2.$$
\end{itemize}
Let
$$\lambda_k=\min_{v\in\tilde H^1(B_1^+)\setminus\{0\}}\frac{Q_k(v)}{\int_{S^n_+}v^2},\qquad\lambda=\min_{v\in\tilde H^1(B_1^+)\setminus\{0\}}\frac{Q(v)}{\int_{S^n_+}v^2}.$$
Then, $\lambda_k\to\lambda$.
\end{Lemma}
\proof
Let $\{v_k\}\subset \tilde H^1(B_1^+)\setminus\{0\}$ be a sequence of minimizers for $\lambda_k$; that is, such that
$$\lambda_k=Q_{k}(v_k)=\int_{B_1^+}|\nabla v_k|^2+\int_{B_1^+}V_kv_k^2+\int_{S^n_+}W_kv_k^2,$$
and $\int_{S^n_+}v^2_k=1$. Since by compact embedding $\tilde H^1(B_1^+)\hookrightarrow L^2(S^n_+)$ the minimum
$$\min_{v\in\tilde H^1(B_1^+)\setminus\{0\}}\frac{\|v\|^2_{\tilde H^1(B_1^+)}}{\int_{S^n_+}v^2}=\frac{\|u\|^2_{\tilde H^1(B_1^+)}}{\int_{S^n_+}u^2}=\nu>0$$
is achieved by $u\in\tilde H^1(B_1^+)\setminus\{0\}$ and it is strictly positive by the trace Poincar\'e inequality, then there exists a positive constant $C$ independent from $k$ such that
$$\frac{\nu}{C}\leq\lambda_k\leq C\nu.$$
Moreover, we have that
$$\frac{1}{C}\|v_k\|^2_{\tilde H^1(B_1^+)}\leq\lambda_k\leq C\nu$$
and so there exists $\overline v\in\tilde H^1(B_1^+)$ such that $v_k\rightharpoonup\overline v$ in $\tilde H^1(B_1^+)$ and, up to passing to a subsequence, $v_k\to\overline v$ in $L^2(S^n_+)$. Moreover, the limit is non trivial by the condition $\int_{S^n_+}\overline v^2=1$.\\\\
We want to prove that the convergence is strong in $\tilde H^1(B_1^+)$. Testing the eigenvalue equation solved by $v_k$ with $v_k-\overline v$, we have
$$\int_{B_1^+}\nabla v_k\cdot\nabla(v_k-\overline v)+\int_{B_1^+}V_kv_k(v_k-\overline v)+\int_{S^n_+}W_kv_k(v_k-\overline v)=\lambda_k\int_{S^n_+}v_k(v_k-\overline v).$$
Using the fact that $|W_k|, |\lambda_k|\leq c$ uniformly in $k$, the strong convergence and the normalization in $L^2(S^n_+)$, by the H\"older inequality the terms over the half sphere $S^n_+$ go to 0 in the limit. So
\begin{equation}\label{testing11}
\int_{B_1^+}\nabla v_k\cdot\nabla(v_k-\overline v)+\int_{B_1^+}V_kv_k(v_k-\overline v)\to0.
\end{equation}
Hence,
\begin{eqnarray}\label{strong1}
Q_{k}(v_k-\overline v)&=&\int_{B_1^+}|\nabla (v_k-\overline v)|^2+\int_{B_1^+}V_k(v_k-\overline v)^2+\int_{S^n_+}W_k(v_k-\overline v)^2\nonumber\\
&=&\int_{B_1^+}\nabla v_k\cdot\nabla(v_k-\overline v)+\int_{B_1^+}V_kv_k(v_k-\overline v)-\int_{B_1^+}\nabla \overline v\cdot\nabla(v_k-\overline v)\nonumber\\
&&-\int_{B_1^+}V\overline v(v_k-\overline v)+\int_{B_1^+}(V-V_k)\overline v(v_k-\overline v)+\int_{S^n_+}W_k(v_k-\overline v)^2\to0.
\end{eqnarray}
In fact, the sum of the first two terms goes to 0 by \eqref{testing11}, the sum of the second two by weak convergence in $\tilde H^1(B_1^+)$. The third term is such that
\begin{eqnarray*}
\int_{B_1^+}(V-V_k)\overline v(v_k-\overline v)&\leq&\left(\int_{B_1^+}(V-V_k)\overline v^2\right)^{1/2}\left(\int_{B_1^+}(V-V_k)(v_k-\overline v)^2\right)^{1/2}\\
&\leq& c\left(\int_{B_1^+}(V-V_k)\overline v^2\right)^{1/2}\to0.
\end{eqnarray*}
We used that $V_k\to V$, the fact that $|V_k-V|\leq\frac{c}{y^2}$ and the Hardy inequality to ensure the dominated convergence theorem. Eventually the last term in \eqref{strong1} goes to 0 by the strong convergence in $L^2(S^n_+)$. Hence we obtain the strong convergence by \eqref{unifequiv}.\\\\
It is easy to see that $Q_{k}(v_k)\to Q(\overline v)$. This is enough to conclude because if we consider $\tilde v$ the normalized in $L^2(S^n_+)$ minimizer of $\lambda$, since it is competitor for the minimization of any $Q_{k}$, then
$$\lambda_k=Q_{k}(v_k)\leq Q_{k}(\tilde v),$$
and since $Q_{k}(v_k)\to Q(\overline v)$ and $Q_{k}(\tilde v)\to Q(\tilde v)$, then by $Q(\overline v)\leq Q(\tilde v)$, and by the minimality of $\overline v$, we finally obtain that $\overline v=\tilde v$ with $\lambda_k\to\lambda$.
\endproof
\subsection{Quadratic forms for the odd case}
Let $a\in(-\infty,1)$, $\varepsilon\geq0$ and consider a function $u\in C^\infty_c(\overline B_1^+\setminus\Sigma)$ and define $v=(\rho_\varepsilon^a)^{1/2}u\in C^\infty_c(\overline B_1^+\setminus\Sigma)$. Let us define the quadratic form
\begin{equation}\label{Qrho}
\int_{B_1^+}\rho_\varepsilon^au^2=Q_{\rho_\varepsilon^a}(v)=\int_{B_1^+}|\nabla v|^2+\int_{B_1^+}V_{\rho_\varepsilon^a}v^2+\int_{S^n_+}W_{\rho_\varepsilon^a}v^2,
\end{equation}
where
$$V_{\rho_\varepsilon^a}(y)=\frac{(\rho_\varepsilon^a)''}{2\rho_\varepsilon^a}-\left(\frac{(\rho_\varepsilon^a)'}{2\rho_\varepsilon^a}\right)^2=\frac{a[(a-2)y^2+2\varepsilon^2]}{4(\varepsilon^2+y^2)^2}$$
and
$$W_{\rho_\varepsilon^a}(y)=-\frac{(\rho_\varepsilon^a)'y}{2\rho_\varepsilon^a}=-\frac{ay^2}{2(\varepsilon^2+y^2)}.$$
Let
\begin{equation*}
Q_{a}(v)=\int_{B_1^+}|\nabla v|^2+\int_{B_1^+}V_{a}v^2+\int_{S^n_+}W_{a}v^2,
\end{equation*}
with $V_a(y)=\frac{a(a-2)}{4y^2}=V_{\rho_0^a}(y)$ and $W_{a}(y)=-\frac{a}{2}=W_{\rho_0^a}(y)$. Eventually consider a sequence $\varepsilon_k\to0$ as $k\to+\infty$ and define $\rho_k=\rho_{\varepsilon_k}^a$. Let us recall $Q_k=Q_{\rho_k}$ and $Q=Q_a$.
\begin{Lemma}\label{A1}
Under the previous hypothesis, the family $\{Q_{k}\}=\{Q_{\rho_{\varepsilon_k}}\}$ defined in \eqref{Qrho} and its limit $Q$ satisfy the conditions in Lemma \ref{quad}.
\end{Lemma}
\proof
Condition $i)$ is trivially satisfied. Moreover, combining $i)$, the trace Poincar\'e and the Hardy inequalities, we easily obtain the upper bound in $ii)$ for any $k\in\mathbb{N}$ with a constant independent on $\varepsilon_k$; that is,
$$Q_k(v)\leq c\|v\|_{\tilde H^1(B_1^+)}^2.$$
Let us consider $Q=Q_a$ and let us define $u=y^{-a/2}v\in C^\infty_c(\overline B_1^+\setminus\Sigma)$.
\begin{eqnarray}\label{Qa}
Q_a(v)&=&\int_{B_1^+}|\nabla v|^2+\left(\frac{a^2}{4}-\frac{a}{2}\right)\int_{B_1^+}\frac{v^2}{y^2}-\frac{a}{2}\int_{S^n_+}v^2\\\nonumber
&=&\int_{B_1^+}|\nabla v|^2+\left(\frac{a^2}{4}-\frac{a}{2}\right)\int_{B_1^+}\frac{v^2}{y^2}-\frac{a}{2}\int_{B_1^+}\mathrm{div}\left(\frac{v^2}{y}\vec{e_n}\right)=\int_{B_1^+}y^a|\nabla u|^2.
\end{eqnarray}
First of all we notice that if $a\leq 0$ the lower bound follows trivially. So we can suppose that $a\in(0,1)$. Since for $a\neq1$, $(\frac{a^2}{4}-\frac{a}{2})>-\frac{1}{4}$, hence by the Hardy inequality in \eqref{Hardy}, the quantity
$$G_a(v)=\int_{B_1^+}|\nabla v|^2+\left(\frac{a^2}{4}-\frac{a}{2}\right)\int_{B_1^+}\frac{v^2}{y^2}$$
defines an equivalent norm in $\tilde H^1(B_1^+)$. Hence by the compact embedding $\tilde H^1(B_1^+)\hookrightarrow L^2(S^n_+)$ we have that the minimum in
$$\xi(a)=\min_{v\in\tilde H^1(B_1^+)\setminus\{0\}}\frac{G_a(v)}{\int_{S^n_+}v^2}$$
is achieved. In fact, considering a minimizing sequence, we can take it such that $\int_{S^n_+}v_k^2=1$ and also such that $v_k\in C^\infty_c(\overline B_1^+\setminus\Sigma)$. So it is uniformly bounded in $\tilde H^1(B_1^+)$ and $v_k\rightharpoonup\overline v\in\tilde H^1(B_1^+)$ with $G_a(v_k)\to\xi(a)$. Moreover the convergence is strong in $L^2(S^n_+)$ by compact embedding. Since $\int_{S^n_+}v_k^2=1$, we also obtain convergence of the $\tilde H^1_0$-norms of the $v_k$ to that of the limit, yielding strong convergence in $\tilde H^1(B_1^+)$. In fact, by the lower semicontinuity of the norm
$$\xi(a)\leq\frac{G_a(\overline v)}{\int_{S^n_+}\overline v^2}\leq\liminf_{k\to+\infty}\frac{G_a(v_k)}{\int_{S^n_+}v_k^2}=\xi(a).$$
Obviously by the condition $\int_{S^n_+}\overline v^2=1$ the limit $\overline v$ is not trivial. This proves that $\overline v$ achieves the minimum. Moreover, defining
\begin{equation}\label{lama}
\lambda(a)=\min_{v\in\tilde H^1(B_1^+)\setminus\{0\}}\frac{Q_a(v)}{\int_{S^n_+}v^2}=\xi(a)-\frac{a}{2}\geq0,
\end{equation}
we want to prove that actually $\lambda(a)>0$. First of all, such a minimum is nonnegative since the minimizing sequence can be taken in $C^\infty_c(\overline B_1^+\setminus\Sigma)$ and so the equalities in \eqref{Qa} give this condition. By contradiction let $\lambda(a)=0$. Hence the minimizing sequence is such that $Q_a(v_k)\to0$. Defining $u_k=y^{-a/2}v_k$, one has $\int_{B_1^+}y^a|\nabla u_k|^2\to0$. Moreover, the strong convergence in $\tilde H^1(B_1^+)$ gives the almost everywhere convergence of $\nabla v_k\to\nabla\overline v$ which of course implies that $\nabla u_k\to \nabla(y^{-a/2}\overline v)$ almost everywhere in $B_1^+$. Hence, since $\nabla(y^{-a/2}\overline v)=0$ almost everywhere, $\overline v=cy^{a/2}$, but $\nabla\overline v$ does not belong to $L^2(B_1^+)$. This is a contradiction. So $\lambda(a)>0$. So we have the inequality
$$Q_a(v)\geq\lambda(a)\int_{S^n_+}v^2,$$
which says that
$$Q_a(v)\geq\frac{\lambda(a)}{\frac{a}{2}+\lambda(a)}\left(\int_{B_1^+}|\nabla v|^2+\left(\frac{a^2}{4}-\frac{a}{2}\right)\int_{B_1^+}\frac{v^2}{y^2}\right),$$
and by the equivalence of the norms we obtain the result for a constant which depends on $a$ and $\lambda(a)$. Eventually, we have proved that also $Q_a$ is an equivalent norm on $\tilde H^1(B_1^+)$.\\\\
In order to prove the lower bound for $Q_k$ which is uniform in $k$, it is enough to remark that if $a\geq0$, then $Q_k\geq Q_a$. If $a<0$, then one can check that
$$Q_k(v)\geq \int_{B_1^+}|\nabla v|^2-\int_{B_1^+}\frac{a}{4(a-4)}\frac{v^2}{y^2},$$
with $\frac{a}{4(a-4)}<\frac{1}{4}$ and hence by the Hardy inequality in \eqref{Hardy} we have also in this case an equivalent norm.
\endproof
Let us recall the definition of $\tilde H^1(B_1^+,\rho_\varepsilon^a(y)\mathrm{d}z)$ as the closure of $C^\infty_c(\overline B_1^+\setminus\Sigma)$ with respect to the norm
$$\int_{B_1^+}\rho_\varepsilon^a|\nabla u|^2.$$
\begin{Lemma}\label{A2}
Let $a\in(-\infty,1)$, $\varepsilon\geq0$ and $u\in\tilde H^1(B_1^+,\rho_\varepsilon^a(y)\mathrm{d}z)$. Then the following inequalities hold true for a positive constant $c$ independent of $\varepsilon\in[0,1]$
\begin{equation}\label{poin}
c\int_{B_1^+}\rho_\varepsilon^au^2\leq\int_{B_1^+}\rho_\varepsilon^a|\nabla u|^2,
\end{equation}
\begin{equation}\label{tracepoin}
c\int_{S^n_+}\rho^a_{\varepsilon}u^2\leq\int_{B_1^+}\rho_\varepsilon^a|\nabla u|^2,
\end{equation}
\begin{equation}\label{hard}
c\int_{B_1^+}\frac{\rho_\varepsilon^a}{y^2}u^2\leq\int_{B_1^+}\rho_\varepsilon^a|\nabla u|^2,
\end{equation}
\begin{equation}\label{tracehard}
c\int_{S^n_+}\frac{\rho_\varepsilon^a}{y}u^2\leq\int_{B_1^+}\rho_\varepsilon^a|\nabla u|^2,
\end{equation}
\begin{equation}\label{sob}
\left(\int_{B_1^+}(\rho_\varepsilon^a)^{2^*/2}|u|^{2^*}\right)^{2/2^*}\leq c\int_{B_1^+}\rho_\varepsilon^a|\nabla u|^2,
\end{equation}
which are respectively the Poincar\'e inequality, the trace Poincar\'e inequality, the Hardy inequality, the trace Hardy inequality and a Sobolev type inequality.
\end{Lemma}
\proof
The proof is performed for functions $u\in C^\infty_c(\overline B_1^+\setminus\Sigma)$ and then extending the inqualities to $u\in\tilde H^1(B_1^+,\rho_\varepsilon^a(y)\mathrm{d}z)$ by a density argument. By Lemma \ref{A1} there exists a positive constant uniform in $\varepsilon$ such that
\begin{equation}\label{Qr}
\int_{B_1^+}\rho_\varepsilon^a|\nabla u|^2=Q_{\rho_\varepsilon^a}((\rho_\varepsilon^a)^{1/2}u)\geq c\int_{B_1^+}|\nabla((\rho_\varepsilon^a)^{1/2}u)|^2,
\end{equation}
then all the inequalities are obtained by the validity of them in $\tilde H^1(B_1^+)$.
\endproof
\subsection{Quadratic forms for the auxiliary weights}
Consider now $a\in(-\infty,1)$ and define
$$\pi_\varepsilon^a(y)=\left((1-a)\int_0^y\rho_\varepsilon^{-a}(s)\mathrm{d}s\right)^2,$$
and
\begin{equation*}
\omega_\varepsilon^a(y)=\rho_\varepsilon^a(y)\pi_\varepsilon^a(y).
\end{equation*}
We observe that this weight is super degenerate; that is, at $\Sigma$
$$\omega_\varepsilon^a(y)\sim\begin{cases}
|y|^{2-a} & \mathrm{if \ }\varepsilon=0\\
|y|^2 & \mathrm{if \ }\varepsilon>0,
\end{cases}$$
with $2-a\in(1,+\infty).$
\begin{comment}
Let us consider the following equation
\begin{equation}
\begin{cases}
-\mathrm{div}(\omega\nabla w)=0 & \mathrm{in} \ \mathbb{R}^{n+1}_+,\\
\omega\partial_yw=0 & \mathrm{in} \ \mathbb{R}^{n}.
\end{cases}
\end{equation}
Let $u=\omega\partial_yw$. Then it solves
\begin{equation}\label{1}
\begin{cases}
-\mathrm{div}(\omega^{-1}\nabla u)=0 & \mathrm{in} \ \mathbb{R}^{n+1}_+,\\
u=0 & \mathrm{in} \ \mathbb{R}^{n}.
\end{cases}
\end{equation}
Let us call $v=\omega^{-1/2}u$.
\end{comment}
\subsubsection{Super singular weights $(\omega_\varepsilon^a)^{-1}$}
Let us consider $u\in C^\infty_c(\overline B_1^+\setminus\Sigma)$ and define $v=(\omega_\varepsilon^a)^{-1/2}u\in C^\infty_c(\overline B_1^+\setminus\Sigma)$. Then we consider the quadratic form
\begin{equation}\label{Qomega}
\int_{B_1^+}(\omega_\varepsilon^a)^{-1}|\nabla u|^2=Q_{\omega_\varepsilon^a}(v)=\int_{B_1^+}|\nabla v|^2+\int_{B_1^+}V_{\omega_\varepsilon^a} v^2+\int_{S^n_+}W_{\omega_\varepsilon^a} v^2,
\end{equation}
with
$$V_{\omega_\varepsilon^a}=\frac{1}{4}[(\log\omega_\varepsilon^a)']^2-\frac{1}{2}(\log\omega_\varepsilon^a)'',$$
and
$$W_{\omega_\varepsilon^a}=\frac{1}{2}(\log\omega_\varepsilon^a)'y.$$
Hence
$$V_{\omega_0^a}(y)=\frac{(2-a)(4-a)}{4y^2},\qquad\mathrm{and}\qquad W_{\omega_0^a}(y)=\frac{2-a}{2}.$$
Eventually consider a sequence $\varepsilon_k\to0$ as $k\to+\infty$ and define $\omega_k=\omega_{\varepsilon_k}^a$. Let us name $Q_k=Q_{\omega_k}$ and $Q=Q_{\omega_0^a}$.
In what follows it would be useful to consider for $t>0$, the continuous function defined in \eqref{psi}; that is,
\begin{equation*}
\psi(t)=\frac{t(1+t^2)^{-a/2}}{\int_0^{t}(1+s^2)^{-a/2}\mathrm{d}s},
\end{equation*}
which is monotone nondecreasing if $a<0$ and nonincreasing if $a\in(0,1)$. Since $\psi$ has limit $1$ as $t\to0$ and limit $1-a$ as $t\to+\infty$, then
$$\sup_{t>0}\psi(t)=\max\{1,1-a\}\qquad\mathrm{and}\qquad \inf_{t>0}\psi(t)=\min\{1,1-a\}.$$
Let us finally define for any $k\in\mathbb{N}$
\begin{equation}\label{Qomega1}
\tilde Q_k(v)=Q_{k}(v)+\left(-\frac{a}{2}\right)^+\int_{S^n_+}v^2.
\end{equation}
First we need the following technical result.
\begin{Lemma}\label{Phia}
Let us define for $a\in(-\infty,1)$ and $t\in[0,+\infty)$ the function
\begin{equation}\label{phia}
\Phi_a(t)=\left[\frac{\sqrt{2}t(1+t^2)^{-a/2}}{\int_0^t(1+s^2)^{-a/2}}+\frac{at^2}{\sqrt{2}(1+t^2)}\right]^2+\frac{at^2[(2-a)t^2-2]}{4(1+t^2)^2}.
\end{equation}
Hence there exists a positive constant $c_1(a)>-\frac{1}{4}$ such that
\begin{equation}
\inf_{t>0}\Phi_a(t)=c_1(a).
\end{equation}
\end{Lemma}
\proof
{\bf Step 1: $a\in(-3,1)$.}\\\\
Whenever $0\leq a<1$, there holds
$$\min_{t>0}f_a(t)=\min_{t>0}\frac{at^2[(2-a)t^2-2]}{4(1+t^2)^2}=f_a\left(\frac{1}{\sqrt{3-a}}\right)=\frac{a}{4(a-4)}>-\frac{1}{4}.$$
Moreover, if $a<0$,
$$\inf_{t>0}f_a(t)=\lim_{t\to+\infty}f_a(t)=\frac{a(2-a)}{4}.$$
Hence, whenever $1-\sqrt{2}<a<0$, then, the infimum remains strictly larger that $-1/4$.\\\\
Moreover, for $a<0$, then $f_a(t)\geq0$ in $\left[0,\sqrt{\frac{2}{2-a}} \ \right]$. From now on we will consider $a<0$ and $t>\sqrt{\frac{2}{2-a}} $. Now, let us compute the square in \eqref{phia}, and add $1/4$; that is
\begin{eqnarray*}
\Phi_a(t)+\frac{1}{4}&=&\frac{2t^2(1+t^2)^{-a}}{\left(\int_0^t(1+s^2)^{-a/2}\right)^2}+\frac{2at^3(1+t^2)^{-1-a/2}}{\int_0^t(1+s^2)^{-a/2}}+\frac{a^2t^4}{2(1+t^2)^2}+f_a(t)+\frac{1}{4}\\
&=&\frac{2t^3(1+t^2)^{-1-a/2}}{\left(\int_0^t(1+s^2)^{-a/2}\right)^2} \ \cdot g_a(t)+\frac{t^4(a^2+2a+1)+t^2(-2a+2)+1}{4(1+t^2)^2}\\
&=&I_a(t)+J_a(t),
\end{eqnarray*}
with
$$g_a(t)=\left(\frac{(1+t^2)^{1-a/2}}{t}+a\int_0^t(1+s^2)^{-a/2}\right).$$
It is easy to see that
$$\inf_{t>0}J_a(t)\begin{cases}>0 & \mathrm{if \ } a\neq-1\\ =0 & \mathrm{if \ }a=-1.\end{cases}$$
Nevertheless, since
$$g_a'(t)=\frac{(1+t^2)^{-a/2}}{t^2}(t^2-1),$$
then $g_a$ has its global minimum in $t=1$, and hence it is easy to see that
$$g_a(1)=2^{1-a/2}+a\int_0^1(1+s^2)^{-a/2}\geq 2^{1-a/2}+a\int_0^1(1+s)^{-a/2}=2^{1-a/2}\frac{2+a}{2-a}-\frac{2a}{2-a}>0,$$
surely if $a>-3$. Hence, when $a\in(-3,-1)\cup(-1,0)$, we have the result since $\inf_{t>0}I_a(t)\geq0$ and $\inf_{t>0}J_a(t)>0$. In the case $a=-1$ one can see that
$$\inf_{t>0}I_{-1}(t)=\min_{t>0}I_{-1}(t)>0,$$
using the explicit form
$$I_{-1}(t)=\frac{2t^3(1+t^2)^{-1-a/2}}{\frac{1}{4}\left(t\sqrt{t^2+1}+\log(\sqrt{t^2+1}+t)\right)^2}\left(\frac{(1+t^2)^{1-a/2}}{t}-\frac{1}{2}\left(t\sqrt{t^2+1}+\log(\sqrt{t^2+1}+t)\right)\right).$$
{\bf Step 2: $a\leq-3$.}\\\\
We can express
$$\Phi_a(t)+\frac{1}{4}=\frac{t^4}{(1+t^2)^2}\left(2\left(\frac{(1+t^2)^{-\frac{a}{2}+1}}{t\int_0^t(1+s^2)^{-\frac{a}{2}}}+\frac{a}{2}\right)^2+\frac{a(2-a)}{4}-\frac{a}{2t^2}+\frac{1}{4}\frac{(1+t^2)^2}{t^4}\right).$$
Hence
$$\tilde\Phi_a(t)=\frac{(1+t^2)^2}{t^4}\left(\Phi_a(t)+\frac{1}{4}\right),$$
and $\gamma_a(t)=\tilde\Phi_a(t/\sqrt{-a})-\frac{0.001}{4}\frac{(-a+t^2)^2}{t^4}$; that is,
\begin{equation}\label{gammaa}
\gamma_a(t)=2a^2\left(\frac{(1+\frac{t^2}{-a})^{-\frac{a}{2}+1}}{t\int_0^t(1+\frac{s^2}{-a})^{-\frac{a}{2}}}-\frac{1}{2}\right)^2+\frac{a(2-a)}{4}+\frac{a^2}{2t^2}+\frac{0.999}{4}\frac{(-a+t^2)^2}{t^4}.
\end{equation}
First need to highlight some fundamental properties of the functions
\begin{equation*}
w_a(t)=\frac{(1+\frac{t^2}{-a})^{-\frac{a}{2}+1}}{t\int_0^t(1+\frac{s^2}{-a})^{-\frac{a}{2}}}.
\end{equation*}
As $a\to-\infty$ one has the pointwise convergence $w_a(t)\to v(t)$ in $(0,+\infty)$ (which is however uniform on compact subsets) with
\begin{equation*}
v(t)=\frac{e^{\frac{t^2}{2}}}{t\int_0^te^{\frac{s^2}{2}}}.
\end{equation*}
We wish to prove the following\\\\
{\bf Claim:} $w_a/v\geq 1$ in $[0,+\infty)$. At first, elementary computations show that, in a neighbourhood of $t=0$, the expansion
$$w_a(t)=\frac{1}{t^2}+\frac{1}{2}+\frac{1}{-a}+o(1)\qquad\mathrm{and}\qquad v(t)=\frac{1}{t^2}+\frac{1}{2}+o(1),$$
holds, while in a neighbourhood of $t=+\infty$ we have
$$w_a(t)=\frac{1-a}{-a}+o(1)\qquad\mathrm{and}\qquad v(t)=1+o(1),$$
implying that $w_a/v>1$ near zero and at infinity. Thus, the claim is false if and only if there exists $t_0>0$ such that
\begin{equation}\label{syst0}
\begin{cases}
w_a(t_0)=v_a(t_0)\\
\left(\frac{w_a}{v}\right)'(t_0)\leq0.
\end{cases}
\end{equation}
Remark that, $w_a$ and $v$ solve respectively the following differential equations
\begin{equation*}
w_a'(t)=\frac{1}{t(1+\frac{t^2}{-a})}\left(\frac{1-a}{-a}t^2-1\right)w_a(t)-\frac{t}{1+\frac{t^2}{-a}}w_a^2(t)
\end{equation*}
and
\begin{equation*}
v'(t)=\frac{t^2-1}{t}v(t)-tv^2(t).
\end{equation*}
Using these equations we obtain
\begin{equation*}
\left(\frac{w_a}{v}\right)'=\frac{w_a}{v}\left(\frac{t}{-a+t^2}(2-t^2)-\frac{t}{1+\frac{t^2}{-a}}w_a+tv\right),
\end{equation*}
and \eqref{syst0} holds if and only if
\begin{equation*}
v(t_0)\leq 1-\frac{2}{t_0^2}.
\end{equation*}
Now we are going to show that, on the contrary,
\begin{equation}\label{v>z}
v(t)>z(t):=1-\frac{2}{t^2}.
\end{equation}
In $(0,\sqrt 2)$ we have $v>0$ and $z<0$. Moreove the inequality \eqref{v>z} can be checked numerically (with error estimate) on $[\sqrt 2,\sqrt 6]$, and is also valid in a neighbourhood of $t=+\infty$, by the exapnsion
$$v(t)=1-\frac{1}{t^2}+o\left(\frac{1}{t^2}\right)>z(t).$$
So, the function $v-z$ is positive near $0$ and at $+\infty$, and hence denying \eqref{v>z} yields the existence of $t_1\geq\sqrt 6$ such that
$$\begin{cases}
v(t_1)=z(t_1)\\
(v-z)'(t_1)\leq0.
\end{cases}$$
It is easy to see that at such a point $t_1$ one has $(v-z)'(t_1)>0$ if $t_1\geq\sqrt 6$ (using the fact that $v(t_1)=z(t_1)$).\\\\
Now we can turn back to \eqref{gammaa}, obtaining by convexity that
\begin{equation}\label{gammaaa}
\gamma_a(t)\geq2a^2\left(v(t)-\frac{1}{2}\right)^2+\frac{a(2-a)}{4}+\frac{a^2}{2t^2}+\frac{0.999}{4}\frac{(-a+t^2)^2}{t^4}.
\end{equation}
In order to complete the proof, we need to prove positivity of the right hand side. To this aim, we observe that the function $v$ changes monotonicity only once on $(0,+\infty)$ and its absolute minimum value $0,77836\pm 10^{-5}$ is larger than $1/2$. Moreover, as $v(5.1)=0.95774\pm10^{-5}$ and $v'(5.1)=0.001860\pm 10^{-5}>0$ we infer
positivity of the right hand side for all $t\in [5.1,+\infty)$, for all $a\leq -2.96767$. The remaining values $(a,t)$ lay in the compact rectangle $[-43.3272,-2.96767]\times[1,5.1]$ and can be easily dealt numerically with error controlled minimization.
\endproof
\begin{Lemma}\label{B1}
Under the previous hypothesis, the family $\{\tilde Q_{k}\}=\{\tilde Q_{\omega_{\varepsilon_k}}\}$ defined in \eqref{Qomega1} and its limit $\tilde Q$ satisfy the conditions in Lemma \ref{quad}.
\end{Lemma}
\proof
First, we want to prove property $i)$; that is, there exists a positive constant $c>0$ uniform in $\varepsilon\to0$ such that
$$|V_{\omega_\varepsilon^a}(y)|\leq\frac{c}{y^2}\qquad\mathrm{and}\qquad |W_{\omega_\varepsilon^a}(y)|\leq c.$$
We remark that there exists a positive constant $c>0$ uniform in $\varepsilon\to0$ such that
$$|(\log\rho_\varepsilon^a)'|=\left|\frac{(\rho_\varepsilon^a)'}{\rho_\varepsilon^a}\right|\leq |a|\frac{y}{\varepsilon^2+y^2}\leq\frac{c}{y}.$$
Moreover
$$|(\log\rho_\varepsilon^a)''|\leq \left|\frac{(\rho_\varepsilon^a)'}{\rho_\varepsilon^a}\right|\cdot\left|\frac{(\rho_\varepsilon^a)''}{(\rho_\varepsilon^a)'}\right|+\left|\frac{(\rho_\varepsilon^a)'}{\rho_\varepsilon^a}\right|^2\leq \frac{c}{y^2}.$$
It remains to prove the following uniform bounds
\begin{equation*}
\left|\frac{(\pi_\varepsilon^a)'}{\pi_\varepsilon^a}\right|\leq \frac{c}{y},\qquad\mathrm{and}\qquad\left|\frac{(\pi_\varepsilon^a)''}{(\pi_\varepsilon^a)'}\right|\leq \frac{c}{y}.
\end{equation*}
Then the result follows since we are considering the logarithm of a product by linearity of the derivative.
\begin{eqnarray*}
|(\log\pi_\varepsilon^a)'|=\left|\frac{(\pi_\varepsilon^a)'}{\pi_\varepsilon^a}\right|&=&2\frac{\rho_\varepsilon^{-a}(y)}{\int_0^y\rho_\varepsilon^{-a}(s)\mathrm{d}s}\\
&=&\frac{2}{y}\frac{\frac{y}{\varepsilon}(1+\left(\frac{y}{\varepsilon}\right)^2)^{-a/2}}{\int_0^{\frac{y}{\varepsilon}}(1+s^2)^{-a/2}\mathrm{d}s}\\
&\leq&\frac{2}{y}\sup_{t>0}\frac{t(1+t^2)^{-a/2}}{\int_0^{t}(1+s^2)^{-a/2}\mathrm{d}s}\leq \frac{2\max\{1,1-a\}}{y}.
\end{eqnarray*}
Moreover,
$$\left|\frac{(\pi_\varepsilon^a)''}{(\pi_\varepsilon^a)'}\right|\leq\frac{\rho_\varepsilon^{-a}(y)}{\int_0^y\rho_\varepsilon^{-a}(s)\mathrm{d}s}+|a|\frac{y}{\varepsilon^2+y^2}\leq\frac{\max\{1,1-a\}+|a|}{y}.$$
Eventually
$$|(\log\pi_\varepsilon^a)''|\leq \left|\frac{(\pi_\varepsilon^a)'}{\pi_\varepsilon^a}\right|\cdot\left|\frac{(\pi_\varepsilon^a)''}{(\pi_\varepsilon^a)'}\right|+\left|\frac{(\pi_\varepsilon^a)'}{\pi_\varepsilon^a}\right|^2\leq \frac{c}{y^2}.$$
Obviously, point $i)$ implies the uniform upper bound in \eqref{unifequiv} by trace Poincar\'e and the Hardy inequalities. In order to prove the uniform lower bound and eventually proving $ii)$, we only have to prove that there exists a positive constant $c_1>-\frac{1}{4}$ uniform in $\varepsilon\to0$ such that
$$V_{\omega_\varepsilon^a}(y)\geq\frac{c_1}{y^2}.$$
In fact,
$$W_{\omega_\varepsilon^a}(y)+\left(-\frac{a}{2}\right)^+\geq0.$$
Let $t=y/\varepsilon>0$. Then
$$V_{\omega_\varepsilon^a}(y)=\frac{\Phi_a(t)}{y^2},$$
with $\Phi_a$ as in definition \eqref{phia}. We can conclude by applying Lemma \ref{Phia}.\\\\
Eventually we remark that also condition $iii)$ holds true.
\endproof
Let us define $\tilde H^1(B_1^+,(\omega_\varepsilon^a(y))^{-1}\mathrm{d}z)$ as the closure of $C^\infty_c(\overline B_1^+)$ with respect to the norm
$$\int_{B_1^+}(\omega_\varepsilon^a)^{-1}|\nabla u|^2.$$
\begin{Lemma}\label{A2BH}
Let $a\in(-\infty,1)$ and $u\in\tilde H^1(B_1^+,(\omega_\varepsilon^a(y))^{-1}\mathrm{d}z)$. Then
the following inequalities hold true for a positive constant $c$ independent of $\varepsilon\in[0,1]$
\begin{equation}\label{poinBH}
c\int_{B_1^+}(\omega_\varepsilon^a)^{-1}u^2\leq\int_{B_1^+}(\omega_\varepsilon^a)^{-1}|\nabla u|^2,
\end{equation}
\begin{equation}\label{tracepoinBH}
c\int_{S^n_+}(\omega_\varepsilon^a)^{-1}u^2\leq\int_{B_1^+}(\omega_\varepsilon^a)^{-1}|\nabla u|^2,
\end{equation}
\begin{equation}\label{hardBH}
c\int_{B_1^+}\frac{(\omega_\varepsilon^a)^{-1}}{y^2}u^2\leq\int_{B_1^+}(\omega_\varepsilon^a)^{-1}|\nabla u|^2,
\end{equation}
\begin{equation}\label{tracehardBH}
c\int_{S^n_+}\frac{(\omega_\varepsilon^a)^{-1}}{y}u^2\leq\int_{B_1^+}(\omega_\varepsilon^a)^{-1}|\nabla u|^2,
\end{equation}
\begin{equation}\label{sobBH}
c\left(\int_{B_1^+}((\omega_\varepsilon^a)^{-1})^{2^*/2}|u|^{2^*}\right)^{2/2^*}\leq \int_{B_1^+}(\omega_\varepsilon^a)^{-1}|\nabla u|^2,
\end{equation}
which are respectively the Poincar\'e inequality, the trace Poincar\'e inequality, the Hardy inequality, the trace Hardy inequality and a Sobolev type inequality.
\end{Lemma}
\proof
First, we prove \eqref{tracepoinBH}. Thanks to Lemma \ref{B1} we can define for a sequence $\varepsilon_k\to0$
$$\tilde\mu_k=\min_{v\in\tilde H^1(B_1^+)\setminus\{0\}}\frac{\tilde Q_{k}(v)}{\int_{S^n_+}v^2}=\min_{v\in\tilde H^1(B_1^+)\setminus\{0\}}\frac{Q_{k}(v)}{\int_{S^n_+}v^2}+\left(-\frac{a}{2}\right)^+=\mu_k+\left(-\frac{a}{2}\right)^+,$$
and
$$\tilde\mu=\min_{v\in\tilde H^1(B_1^+)\setminus\{0\}}\frac{\tilde Q(v)}{\int_{S^n_+}v^2}=\min_{v\in\tilde H^1(B_1^+)\setminus\{0\}}\frac{Q(v)}{\int_{S^n_+}v^2}+\left(-\frac{a}{2}\right)^+=\mu+\left(-\frac{a}{2}\right)^+.$$
Actually, we are able to provide the value of $\mu$ since $u(x,y)=y^{1-(a-2)}$ is the unique function in $\tilde H^{1,a-2}(B_1^+)\setminus\{0\}$ which solves
\begin{equation*}
\begin{cases}
-L_{a-2}u=0 &\mathrm{in} \ B_1^+\\
u>0 &\mathrm{in} \ B_1^+\\
u(x,0)=0\\
\nabla u\cdot\nu=\mu u &\mathrm{in} \ S_+^{n},
\end{cases}
\end{equation*}
with $\mu=1-(a-2)=3-a$. Hence, by Lemma \ref{quad}, since $\tilde\mu_k\to\tilde\mu$, then $\mu_k\to\mu=3-a>0$ and one can find $\varepsilon_0>0$ such that for $0\leq\varepsilon_k\leq\varepsilon_{\overline k}=\varepsilon_0$ one has $\mu_k\geq\mu_{\overline k}>0$. Hence one has \eqref{tracepoinBH} with a constant $\mu_{\overline k}>0$ uniform in $0\leq\varepsilon\leq\varepsilon_0$.
For the other inequalities, the proof is done taking functions $u\in C^\infty_c(\overline B_1^+\setminus\Sigma)$ and then passing to functions $u\in\tilde H^1(B_1^+,(\omega_\varepsilon^a(y))^{-1}\mathrm{d}z)$ by density. By Lemma \ref{B1} there exists a positive constant uniform in $\varepsilon$ such that
\begin{equation}\label{QrBH}
\int_{B_1^+}(\omega_\varepsilon^a)^{-1}|\nabla u|^2+\left(-\frac{a}{2}\right)^+\int_{S^n_+}(\omega_\varepsilon^a)^{-1}u^2=\tilde Q_{\omega_\varepsilon^a}((\omega_\varepsilon^a)^{-1/2}u)\geq c\int_{B_1^+}|\nabla((\omega_\varepsilon^a)^{-1/2}u)|^2,
\end{equation}
then all the inequalities are obtained by the validity of them in $\tilde H^1(B_1^+)$ and using the trace Poincar\'e inequality \eqref{tracepoinBH}.
\endproof
\subsubsection{Super degenerate weights $\omega_\varepsilon^a$}
Let $a\in(-\infty,1)$ and let us consider $u\in C^\infty(B_1^+)$ and define $v=(\omega_\varepsilon^a)^{1/2}u\in C^\infty_c(\overline B_1^+\setminus\Sigma)$. Then we consider the quadratic form
\begin{equation}\label{Qomega1+}
\int_{B_1^+}\omega_\varepsilon^a\left(|\nabla u|^2+u^2\right)=\overline Q_{\omega_\varepsilon^a}(v)=\int_{B_1^+}\left(|\nabla v|^2+v^2\right)+\int_{B_1^+}\overline V_{\omega_\varepsilon^a} v^2+\int_{S^n_+}\overline W_{\omega_\varepsilon^a} v^2,
\end{equation}
with
$$\overline V_{\omega_\varepsilon^a}=\frac{1}{4}[(\log\omega_\varepsilon^a)']^2+\frac{1}{2}(\log\omega_\varepsilon^a)''=\frac{a}{4}\frac{(a-2)y^2+2\varepsilon^2}{(\varepsilon^2+y^2)^2},$$
and
$$\overline W_{\omega_\varepsilon^a}=-\frac{1}{2}(\log\omega_\varepsilon^a)'y.$$
We remark that $\overline V_{\omega_\varepsilon^a}=V_{\rho_\varepsilon^a}$ in \eqref{Qrho}. It is easy to check that the family of quadratic forms $\overline Q_{\omega_\varepsilon^a}$ are equivalent norms in $\tilde H^1(B_1^+)$ with constants which are uniform in $\varepsilon$; i.e. the following holds
\begin{Lemma}\label{B1+}
Under the previous hypothesis, the family $\{\overline Q_{k}\}=\{\overline Q_{\omega_{\varepsilon_k}}\}$ defined in \eqref{Qomega1+} and its limit $\overline Q$ satisfy the conditions in Lemma \ref{quad}.
\end{Lemma}
\subsection{Isometries}\label{app:isometries}
In this last section, we express a fundamental consequence of the previous estimate on uniform-in-$\varepsilon$ equivalence of norms. Indeed, for all exponents $a\neq 0$, the nature of the weighted Sobolev spaces changes drastically when switching between $\varepsilon>0$ and $\varepsilon=0$. For this reason, we need to embed them isometrically in the common space $\tilde H^1$ uniformly as $\varepsilon\to0$. To this aim, we can take advantage of some fundamental isometries between weighted spaces to $\tilde H^1$, which allow, by reabsorbing the weight, to obtain uniform estimates in a common space to any element in the approximating sequence.
Fixed $a\in(-\infty,1)$ and $\varepsilon\geq0$, then the map $$T^a_\varepsilon:\tilde H^1(B_1^+,\rho_\varepsilon^a(y)\mathrm{d}z)\to\tilde H^1(B_1^+)\;: u\mapsto v=T^a_\varepsilon(u)=(\rho_\varepsilon^a)^{1/2}u $$
is an isometry when we endow the space $\tilde H^1(B_1^+)$ with the squared norm $Q_{\rho_\varepsilon^a}$. Indeed we have:
$$\int_{B_1^+}\rho_\varepsilon^a|\nabla u|^2=Q_{\rho_\varepsilon^a}(v).$$
Is is worthwhile noticing that the family of quadratic forms $Q_{\rho_\varepsilon^a}$ is uniformly bounded (above and below) with respect to $\varepsilon\in[0,1]$.
Eventually, we remark that, similarily, fixed $a\in(-\infty,1)$ and $\varepsilon\geq0$, then the map
\begin{equation}\label{isomsd}
\overline T^a_\varepsilon: H^1(B_1^+,\omega_\varepsilon^a(y)\mathrm{d}z)\to\tilde H^1(B_1^+)\;: u\mapsto v=\overline T^a_\varepsilon(u)=(\omega_\varepsilon^a)^{1/2}u
\end{equation}
is also an isometry when the latter space is endowed with the squared norm $\overline Q_{\omega_\varepsilon^a}(v)$ as we have
$$\int_{B_1^+}\omega_\varepsilon^a\left(|\nabla u|^2+u^2\right)=\overline Q_{\omega_\varepsilon^a}(v).$$
Again, $\overline Q_{\omega_\varepsilon^a}$ is uniformly bounded (above and below) with respect to $\varepsilon\in[0,1]$. Once again, we remark that for these super degenerate weights Poincar\'e type inequalities do not hold true (see \cite{SirTerVit1}) and hence we can not consider only the weighted $L^2$-norm of the gradient in the equation above.
\begin{comment}
\section{Higher order boundary Harnack principles via Schauder estimates}
{\color{blue}Secondo me per fare $C^{2,\alpha}$ serve $A\in C^{2,\alpha}$ con (HA+) e $f=y^\beta g$ con $g\in C^{0,\alpha}$ e $\beta\geq 1-a$.}
\end{comment}
\end{document}
|
\begin{document}
\draft
\title{Generating entangled superpositions of macroscopically
distinguishable states \\ within a parametric oscillator}
\author{Francesco De Martini,$^{1,2}$ Mauro Fortunato,$^{2,3,}$\cite{mau}
Paolo Tombesi,$^{2,3}$ and David Vitali$^{2,3}$}
\address {$^1$Dipartimento di Fisica, Universit\`a ``La Sapienza''
I--00185 Roma, Italy \\
$^{2}$Istituto Nazionale di Fisica della Materia, Italy \\
$^3$~Dipartimento di Matematica e Fisica,
Universit\`a di Camerino, via Madonna delle Carceri I--62032
Camerino, Italy}
\date{\today}
\maketitle
\begin{abstract}
We suggest a variant of the recently proposed experiment for the generation
of a new kind of Schr\"odinger-cat states, using two coupled parametric
down-converter nonlinear crystals $\lbrack$F.~De~Martini, Phys. Rev. Lett. {\bf
81}, 2842 (1998)$\rbrack$. We study the parametric oscillator case and find that
an entangled Schr\"odinger-cat type state of two cavities, whose mirrors are
placed along the output beams of the nonlinear crystals, can be realized
under suitable conditions.
\end{abstract}
\pacs{PACS numbers: 03.65.Bz, 42.50.Dv}
\begin{multicols}{2}
\narrowtext
\section{Introduction}
\label{intro}
Schr\"odinger-cat states~\cite{kn:sc,kn:cat} are most important in the domain of
fundamental quantum mechanics, since the study of their progressive
decoherence~\cite{kn:zur,kn:prlha} would provide a better understanding of the
transition from the {\em quantum} to the {\em classical}
world~\cite{kn:tra}. However, due to their extreme sensitivity to the
decoherence caused by the interaction with the environment, such
linear superpositions of macroscopically distinguishable states are
difficult to produce and to observe~\cite{kn:zur,kn:prlha}. In the
last few years, a major effort in this field has led to the
experimental production and detection of {\em mesoscopic} superpositions
of distinct states, both in the context of the single-mode microwave
cavities~\cite{kn:prlha} and of the dynamics of the
center of mass motion of a trapped
ion~\cite{kn:win}. On the other hand, entanglement has been widely
recognized as one of the essential and most puzzling features of
quantum mechanics~\cite{kn:ent}, in that it allows the existence of
{\em quantum correlated} states of two noninteracting subsystems:
Entangled states play a crucial role in the so called Einstein,
Podolsky and Rosen (EPR) paradox~\cite{kn:epr}, and are essential in
the rapidly growing field of quantum information, as they allow the
feasibility of quantum state teleportation~\cite{kn:tel}, quantum
cryptography~\cite{kn:cryp}, and quantum computation~\cite{kn:qcomp}.
In two recent papers~\cite{kn:dem}, one of us has proposed an
original scheme for the generation of a new kind of {\em amplified}
Schr\"odinger-cat type states. It is based on the new concept of {\em
quantum injection} into an optical parametric amplifier (OPA) operating in
{\em entangled} configuration.
As a relevant variant and a natural extension of the above scheme,
in the present work we analyze the case of the quantum injection in an optical
parametric oscillator (OPO) in which two optical cavities are added to
the OPA scheme considered in Ref.~\cite{kn:dem}: refer to Fig.~\ref{fg:scheme}.
Since the presence of the cavities
leads to a large enhancement of the nonlinear (NL) parametric
interaction, the number of the photon couples which are expected to
be generated, in practical conditions, by the OPO scheme is far
larger than in the amplifier condition: In addition, the generation of
parametrically coupled quasi-coherent fields represents in this
context an appealing perspective.
The Schr\"odinger-cat state that has been put forward in
Ref.~\cite{kn:dem} and is being analyzed, in a more detailed fashion, in the
present paper, is a superposition of
two macroscopic states which are distinguished by their polarization.
It can be considered as a sort of amplified version of the
polarization-entangled
states which have been widely used in the last few years for the demonstration
of the violation of Bell's inequality~\cite{kn:kwiat,kn:bell}, of
teleportation~\cite{kn:tel}, and for the generation of
Greenberger-Horne-Zeilinger (GHZ) states~\cite{kn:ghz}.
The present paper is organized as follows:
In Sec.~\ref{spdc} we
briefly describe the process of type-II parametric down conversion,
with an emphasis on the kind of entangled states usually produced in
these experiments, and on the state we want to generate.
In Sec.~\ref{scheme} we outline the experimental apparatus needed for
the realization of our scheme.
We devote Sec.~\ref{evol} to the presentation of the dynamical time
evolution of the density matrix and of the Wigner function in our
system,
and Sec.~\ref{stability} to the discussion of the stability
conditions for our parametric oscillator.
In Sec.~\ref{choice} we set the initial conditions for the two coupled
nonlinear crystals and the two cavities,
whereas the way in which the
cat state is produced is discussed in detail in Sec.~\ref{genera}.
Sec.~\ref{detect} is devoted to the presentation of the three methods we
propose for detecting the Schr\"odinger-cat state: photodetection
(Sec.~\ref{photo}), measurement of the second-order quantum coherence
(Sec.~\ref{corr}), and Wigner function reconstruction (Sec.~\ref{wigner}).
We finally summarize and discuss our results in Sec.~\ref{conclu}. The
appendix is devoted to the development of the small interaction time
approximation.
\end{multicols}
\widetext
\begin{multicols}{2}
\section{Entanglement generating Parametric Down Conversion}
\label{spdc}
Let us first describe the kind of states commonly generated in the experiments
aimed at the violation of the Bell's inequalities.
In these experiments the NL crystal (typically beta-barium-borate:
BBO) is cut for Type II phase matching where the two down-converted
photons are emitted into two cones, one ``ordinary'' polarized
($o$), the other ``extraordinary'' polarized ($e$). When the angle between the
pump direction and the nonlinear crystal optical axis is sufficiently
large~\cite{kn:kwiat}, the two cones mutually intersect along two lines,
lying on opposite sides of the pump beam direction. These ones
identify the output modes of the parametric down conversion:
$\vec{k}_{j}$ ($j=1$, $2$).
Therefore the field belonging to the modes $\vec{k}_{j}$ can be
simultaneously $e$- and $o$-polarized. In typical conditions, the
output state of the emitted photon couple may be expressed
by~\cite{kn:tel,kn:dem,kn:shih}
\begin{equation}
|\psi \rangle =\frac{1}{\sqrt{2}}\left(|e_{1},o_{2}\rangle +e^{i\phi
}|o_{1}, e_{2}\rangle \right) \;.
\label{eq:pola}
\end{equation}
Since we have, for each couple, four degrees of freedom involved,
i.e. 2 states of orthogonal linear polarization $e$, $o$ for each
mode $\vec{k}_{j}$, we can rewrite state (\ref{eq:pola}) in the more
precise form
\begin{equation}
|\psi \rangle =\frac{1}{\sqrt{2}}\left(|1\rangle _{1e} |1\rangle _{2o}
|0\rangle _{2e} |0\rangle _{1o}+e^{i\phi }|1\rangle _{1o} |1\rangle _{2e}
|0\rangle _{1e} |0\rangle _{2o} \right) \;,
\label{eq:pola2}
\end{equation}
which will be used in the following.
The ``Schr\"odinger-cat state'' we want to generate is a sort of
amplification of this state, that is, it may be expressed in the form
\begin{eqnarray}
|\psi \rangle & = & \frac{1}{\sqrt{2}}\left(|\psi ^{N}\rangle _{1e}
|\psi ^{N}\rangle _{2o}
|0\rangle _{2e} |0\rangle _{1o} \right.
\nonumber \\
& & \left. +e^{i\phi}|\psi ^{N} \rangle _{1o}
|\psi ^{N} \rangle _{2e} |0\rangle _{1e} |0\rangle _{2o} \right) \;,
\label{eq:catpro}
\end{eqnarray}
where $|\psi ^{N}\rangle$ is a state with a large number of photons in
some sense, and the states $|0\rangle$ are to be interpreted here as
squeezed vacuum states. This kind of state is different from the
traditional Schr\"odinger cat states discussed in the quantum optics
literature \cite{kn:zur,kn:prlha}, where one has a {\it single} mode of the
electromagnetic field in a superposition of two macroscopic states with
different phases of the
field. The state (\ref{eq:catpro}) is a {\em nonlocal} superposition in which a
macroscopic optical field is ``localized'' simultaneously either in the
$e$- or in the $o$-polarized mode. In other words it is a state more
similar to nonlocal field states such as
\begin{equation}
|\psi \rangle =\frac{1}{\sqrt{2}}\left(|\alpha \rangle _{1}
|0\rangle _{2} + |0\rangle _{1} |\alpha \rangle _{2} \right) \;,
\label{eq:catloc}
\end{equation}
where the field can be simultaneously in one cavity or in another cavity
and whose generation is discussed in \cite{kn:haro}.
We shall present here an experimental scheme for the generation of a state
which is actually a mixed state, but nonetheless, has the same structure of
the state of Eq.~(\ref{eq:catpro}), that is, can be represented by the
density operator
\begin{eqnarray}
\rho & = & \frac{1}{2}\left(\rho (N)_{1e,2o} \otimes
\rho (0) _{1o,2e}+\rho (0)_{1e,2o} \otimes
\rho (N) _{1o,2e} \right.
\nonumber \\
& & \left. + \rho ({\rm INT})_{1e,2o} \otimes
\rho ({\rm INT'}) _{1o,2e}\right.
\nonumber \\
& & \left. +\rho ({\rm INT})_{1e,2o} ^{\dagger}\otimes
\rho ({\rm INT'})^{\dagger} _{1o,2e}\right)\;,
\label{eq:catpro2}
\end{eqnarray}
where $\rho (N)$ is a two-mode mixed state with a large number of
photons, $\rho (0)$ is a two-mode mixed state with a small number of
photons and $\rho ({\rm INT})$ and $\rho ({\rm INT'})$ are the
interference terms.
\section{The experimental scheme}
\label{scheme}
We shall consider an experimental arrangement, Fig.~\ref{fg:scheme},
based on the one proposed in Ref.~\cite{kn:dem} and similar to that
adopted in Refs.~\cite{kn:mandel} to show the realization of
inducing coherence, without induced emission. Two down-conversion
NL crystals, are arranged in such a way that the two corresponding idlers
beams are aligned along a common direction $\vec{k}_{2}$.
Moreover, both idler beams and the signal beam
of one NL crystal (with wave-vector $\vec{k}_{3}$) are placed within
couples of mirrors. This scheme can be thought of to realize the coupling
of two nondegenerate OPOs. The signal beam of the other crystal,
emitted along the direction $\vec{k}_{1}$ triggers the photodetector
${\rm D_{1}}$.
The directions $\vec{k}_{1}$, $\vec{k}_{2}$ and $\vec{k}_{3}$ are
selected to realize for both NL crystals the Type-II
phase matching described before. These beams are then associated with six
modes, with annihilation operator $a_{1o}$, $a_{1e}$, $a_{2o}$, $a_{2e}$,
$a_{3o}$ and $a_{3e}$. Note that the first two annihilation operators
refer to traveling-waves, while the last four refer to cavity
modes.
The dynamics of the system is determined by the nonlinear parametric
interaction at each crystal and by the damping terms associated
with losses and dissipation inside the cavities~\cite{kn:leg2}, as we
shall see in the next section.
\section{Time evolution for the density matrix and the Wigner function}
\label{evol}
The partial Hamiltonian operators describing the unitary dynamics
inside the crystals are given by~\cite{kn:parosc}
\begin{mathletters}
\label{eq:hnl}
\begin{eqnarray}
\hat{H}_{\rm NL1} & = & i\hbar\chi_{1}(\hat{a}_{1e}^{\dagger}
\hat{a}_{2o}^{\dagger}-\hat{a}_{1e}\hat{a}_{2o})
\nonumber \\
& & +i\hbar\chi_{1}
(\hat{a}_{1o}^{\dagger}\hat{a}_{2e}^{\dagger}-\hat{a}_{1o}\hat{a}_{2e})
\label{eq:hnl1}\;, \\
\hat{H}_{\rm NL2} & = & i\hbar\chi_{2}(\hat{a}_{2o}^{\dagger}
\hat{a}_{3e}^{\dagger}-\hat{a}_{2o}\hat{a}_{3e})
\nonumber \\
& & +i\hbar\chi_{2}
(\hat{a}_{2e}^{\dagger}\hat{a}_{3o}^{\dagger}-\hat{a}_{2e}\hat{a}_{3o})
\label{eq:hnl2}\;,
\end{eqnarray}
\end{mathletters}
where $\chi_{1}=\epsilon_{1}\chi^{(2)}$, $\chi_{2}=\epsilon_{2}\chi^{(2)}$,
$\chi^{(2)}$ is the second-order nonlinear susceptibility of the
crystals, and $\epsilon_{i}$ ($i=1,2$) is the pump intensity in
crystals 1 and 2, respectively, which is assumed to be ``classical''.
Due to the explicit presence of dissipation in this problem, one has to
write the master equation for the reduced density matrix of the
combined system which arises from the Hamiltonian terms
(\ref{eq:hnl1}) and (\ref{eq:hnl2}) and from the damping terms
\begin{equation}
{\cal L}_{i}\rho=\kappa_{i}(2\hat{a}_{i}\rho\hat{a}_{i}^{\dagger}
-\hat{a}_{i}^{\dagger}\hat{a}_{i}\rho
-\rho\hat{a}_{i}^{\dagger}\hat{a}_{i})\;,
\label{eq:damping}
\end{equation}
for $i=2e, 2o, 3e, 3o$. Since the damping constants $\kappa_{i}$ are
essentially connected to the transmittivity of the mirrors, it is quite
natural to assume $\kappa_{2e}=\kappa_{2o}=\kappa_{2}$ and
$\kappa_{3e}=\kappa_{3o}=\kappa_{3}$.
Upon writing the full master equation for the total density matrix
$\rho_{\rm T}$ of the (six-mode) system, it appears clear that the dynamics
of the six modes actually decouples into two independent dynamics for
two groups of three modes. In fact, one has
\begin{equation}
\dot{\rho}_{\rm T}= {\cal L}_{1e-2o-3e}\rho_{\rm T}
+{\cal L}_{1o-2e-3o}\rho_{\rm T}\;,
\label{eq:meq}
\end{equation}
where
\begin{eqnarray}
{\cal L}_{1e-2o-3e}\rho_{\rm T} & = &
-\frac{i}{\hbar}[\hat{H}_{1e-2o-3e},\rho_{\rm T}]
\label{eq:parmeq} \\
& + & \kappa_{2}(2\hat{a}_{2o}\rho_{\rm T}\hat{a}_{2o}^{\dagger}
-\hat{a}_{2o}^{\dagger}\hat{a}_{2o}\rho_{\rm T}
-\rho_{\rm T}\hat{a}_{2o}^{\dagger}\hat{a}_{2o})
\nonumber \\
& + & \kappa_{3}(2\hat{a}_{3e}\rho_{\rm T}\hat{a}_{3e}^{\dagger}
-\hat{a}_{3e}^{\dagger}\hat{a}_{3e}\rho_{\rm T}
-\rho_{\rm T}\hat{a}_{3e}^{\dagger}\hat{a}_{3e})\;,
\nonumber
\end{eqnarray}
and
\begin{eqnarray}
\hat{H}_{1e-2o-3e} & = & i\hbar\chi_{1}(\hat{a}_{1e}^{\dagger}
\hat{a}_{2o}^{\dagger}-\hat{a}_{1e}\hat{a}_{2o})
\nonumber \\
& & +i\hbar\chi_{2}
(\hat{a}_{2o}^{\dagger}\hat{a}_{3e}^{\dagger}-\hat{a}_{2o}\hat{a}_{3e})
\label{eq:hnl123}\;.
\end{eqnarray}
${\cal L}_{1o-2e-3o}$ is identical to ${\cal L}_{1e-2o-3e}$ up to the
substitution $e\to o$ and $o\to e$. As a consequence, the complete time
evolution will be of the form
\begin{equation}
\rho_{\rm T}(t)=e^{{\cal L}_{1e-2o-3e}t}e^{{\cal L}_{1o-2e-3o}t}
\rho_{\rm T}(0)\;.
\label{eq:timev}
\end{equation}
From Eq.~(\ref{eq:timev}) it is clear that if the initial condition is
factorized, namely, if
\begin{equation}
\rho_{\rm T}(0)=\rho_{1e-2o-3e}(0)\otimes \rho_{1o-2e-3o}(0)\;,
\label{eq:factin}
\end{equation}
the state will remain factorized at all times, unless specifically
designed conditional measurements~\cite{kn:cond} are performed on the system
(for example on the mode $\vec{k}_{1}$).
Due to the decoupling between the $1e-2o-3e$ and the $1o-2e-3o$
modes, we can simply restrict ourselves to the investigation of
the three-mode problem described by the master equation (\ref{eq:meq})
with (\ref{eq:parmeq}), and we shall drop the subscript $e$ and $o$
when not needed.
The Wigner function~\cite{kn:wig}
$W(x_{1},y_{1},x_{2},y_{2},x_{3},y_{3})=W(\alpha_{1},\alpha_{2},\alpha_{3})$,
with $\alpha_{i}=x_{i}+iy_{i}$ $(i=1,2,3)$, resulting from this density matrix
$\rho$ will then be a function of six real variables (or three complex
variables). Its time evolution, upon evaluating the commutator and the
damping terms and after some lengthy algebra, is described by the
six-dimensional Fokker-Planck equation
\begin{equation}
\frac{\partial}{\partial t} W(\vec{z},t) =
\gamma_{ij}\frac{\partial}{\partial z_{i}}
\left(z_{j}W(\vec{z},t)\right) + D_{ij}\frac{\partial}{\partial
z_{i}\partial z_{j}}W(\vec{z},t)\;,
\label{eq:6fp}
\end{equation}
where the vector $\vec{z}=(x_{1},y_{1},x_{2},y_{2},x_{3},y_{3})$, the
matrix $D={\rm diag}(0,0,\kappa_{2}/4,\kappa_{2}/4,\kappa_{3}/4,\kappa_{3}/4)$, and
\begin{equation}
\gamma=\left(\matrix{0&0&-\chi_{1}&0&0&0 \cr
0&0&0&\chi_{1}&0&0 \cr
-\chi_{1}&0&\kappa_{2}&0&-\chi_{2}&0 \cr
0&\chi_{1}&0&\kappa_{2}&0&\chi_{2} \cr
0&0&-\chi_{2}&0&\kappa_{3}&0 \cr
0&0&0&\chi_{2}&0&\kappa_{3}}\right)\;.
\label{eq:gamma6}
\end{equation}
The solution to Eq.~(\ref{eq:6fp}) can be written~\cite{kn:gar} as the
integral
\begin{equation}
W(\vec{z},t)=\int d^{4}z'\, W(\vec{z}\;',0) T(\vec{z},\vec{z}\;',t)\;,
\label{eq:solfp}
\end{equation}
where
\begin{mathletters}
\label{eq:where}
\begin{eqnarray}
T(\vec{z},\vec{z}\;',t) & = & \frac{1}{(2\pi)^{3}}
\frac{1}{\protect\sqrt{{\rm Det} \sigma (t)}}
\label{eq:tzzt}\\
& \times &
\exp\left[-\frac{1}{2}\langle\vec{z}-G(t)\vec{z}\;'|\sigma^{-1}(t)|\vec{z}
-G(t)\vec{z}\;'\rangle\right]\;,
\nonumber \\
G(t) & = & \exp\left(-\gamma t\right)\;,
\label{eq:gt}
\end{eqnarray}
and
\begin{equation}
\sigma(t)=2\int\limits_{0}^{t}d\tau\, G(\tau) D G^{t}(\tau)\;,
\label{eq:sigma}
\end{equation}
\end{mathletters}
$G^{t}$ being the transposed of the matrix $G$.
\section{Stability}
\label{stability}
The stability properties of the system are intimately connected to the
threshold of the overall OPO consisting of NL1 and NL2. Below threshold,
the system is stable and
reaches a stationary state, since all eigenvalues of $\gamma$ have
positive real parts. On the other hand, above threshold the system is
unstable and its energy exponentially increases, because some
eigenvalues of $\gamma$ have negative real part.
This result can be easily checked in the case in which the parametric
oscillator associated with NL2 is decoupled from NL1 ($\chi_{1}=0$):
in this case modes $\vec{k}_{2}$ and $\vec{k}_{3}$ decouple
from mode $\vec{k}_{1}$, and we end up with a four-dimensional
problem for the modes $\vec{k}_{2}$ and $\vec{k}_{3}$, described
by a Fokker-Planck equation of the same type as Eq.~(\ref{eq:6fp}),
but with
\begin{equation}
\gamma=\left(\matrix{\kappa_{2}&0&-\chi_{2}&0 \cr
0&\kappa_{2}&0&\chi_{2} \cr
-\chi_{2}&0&\kappa_{3}&0 \cr
0&\chi_{2}&0&\kappa_{3}}\right)\;,
\end{equation}
and $D={\rm diag}(\kappa_{2}/4,\kappa_{2}/4,\kappa_{3}/4,\kappa_{3}/4)$. In this case the
(doubly degenerate) eigenvalues of $\gamma$ are
\begin{equation}
\lambda_{\pm}=\frac{\kappa_{2}+\kappa_{3}}{2}\pm\sqrt{\left(\frac{\kappa_{2}-\kappa_{3}}{2}
\right)^{2}+\chi_{2}^{2}}\;,
\label{eq:eigenv}
\end{equation}
and the stability condition becomes
\begin{equation}
\chi_{2}^{2}\leq \kappa_{2}\kappa_{3}\;,
\label{eq:stability}
\end{equation}
which coincides with the customary threshold for the parametric
oscillator~\cite{kn:parosc}. However, if we turn on the first parametric
amplifier ($\chi_{1}\neq 0$) then the problem turns from 4-dimensional
to 6-dimensional, as we have seen: the eigenvalues of $\gamma$ change
and it is in principle possible to change the threshold, i.e.,
the stability condition.
As soon as $\chi_{1}\neq 0$, namely the first parametric amplifier is
present, the system becomes unstable, independently on the values
of $\chi_{2}$, $\kappa_{2}$, and $\kappa_{3}$. In fact, the eigenvalue equation
for $\gamma$ is
\begin{equation}
(\lambda^{3}-(\kappa_{2}+\kappa_{3})\lambda^{2}
+(\kappa_{2}\kappa_{3}-\chi_{1}^{2}-\chi_{2}^{2})\lambda
+\kappa_{3}\chi_{1}^{2})^{2} = 0\;.
\label{eq:eigenunst}
\end{equation}
As a consequence, we have three doubly degenerate eigenvalues
($\lambda_{1}$, $\lambda_{2}$, and $\lambda_{3}$). Since
$\lambda_{1}\lambda_{2}\lambda_{3}=-\kappa_{3}\chi_{1}^{2}$, at least one
of the $\lambda_{i}$ has a negative real part.
\section{Choice of the initial condition}
\label{choice}
We assume that at the beginning the first crystal is switched off (the
pump strength $\epsilon_{1}=0$). On the other hand, the second pump is
on ($\epsilon_{2}\neq 0$) and the second parametric oscillator is in
its equilibrium state below threshold.
We therefore have a factorized initial state
\begin{equation}
\rho_{\rm T}(0) = \rho_{1e-2o-3e}(0)\otimes\rho_{1o-2e-3o}(0)\;,
\label{eq:infact}
\end{equation}
where
\begin{eqnarray}
\rho_{1e-2o-3e}(0) & = & \rho_{1o-2e-3o}(0)=\rho_{1-2-3}(0)
\nonumber \\
& = & |0\rangle_{1}
{}_{1}\langle 0| \otimes \rho_{2-3}(0)\;,
\label{eq:rho123}
\end{eqnarray}
and $\rho_{2-3}(0)$ is the equilibrium state of the oscillator below
threshold. This can be easily determined upon considering the limits
\begin{equation}
\lim_{t \to \infty} G(t)=0\;, \,\,\,\,\,\,\, \lim_{t \to \infty} \sigma (t)
= \sigma(\infty)\;,
\label{eq:limit}
\end{equation}
and results in the following expression
\begin{eqnarray}
W(\vec{z}, t=\infty) & = & \int d^{4}z' W(\vec{z}\;',0) T(\vec{z},
\vec{z}\;', \infty)
\label{eq:weqbt} \\
& = & \frac{1}{(2\pi)^{2}}\frac{1}{\protect\sqrt{\det
\sigma(\infty)}}
\exp \left[ -\frac{1}{2}\vec{z}\sigma^{-1}(\infty) \vec{z}\right]\;.
\nonumber
\end{eqnarray}
The equilibrium state is thus a Gaussian state in which the modes $\vec{k}_{2}$
and $\vec{k}_{3}$ are correlated.
The initial state $\rho_{2-3}(0)$ is then given by the density matrix
corresponding to the Wigner function
\begin{eqnarray}
W^{2-3}_{\rm bt}(0) & = & \left(\frac{2}{\pi}\right)^{2}
\left(1-\frac{\chi_{2}^{2}}{k^{2}}\right)
\nonumber \\
& & \times \exp\left\{-2\left(x_{2}^{2}+y_{2}^{2}+x_{3}^{2}+y_{3}^{2}
-2\frac{\chi_{2}}{k}
x_{2}x_{3}\right.\right.
\nonumber \\
& & \left.\left. \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
+ 2\frac{\chi_{2}}{k} y_{2}y_{3}\right)\right\}\;,
\label{eq:w23bt}
\end{eqnarray}
where it is straightforward to realize that the modes 2 and 3 are
correlated. Moreover, it is not a pure state, because
\begin{eqnarray}
{\rm Tr}(\rho^{2}) & = & \pi \int dx_{2} dy_{2} dx_{3} dy_{3}
\left[W_{\rm bt}^{2-3} (0)\right]^{2}
\nonumber \\
& = & \left(1-\frac{\chi_{2}^{2}}{k^{2}}\right) < 1 \;,
\label{eq:trrhosq}
\end{eqnarray}
as expected.
The reduced density matrices of each mode are identical and coincide
with the thermal state
\begin{equation}
W_{\rm bt}^{\rm red}(0) = \frac{2}{\pi}
\left(1-\frac{\chi_{2}^{2}}{k^{2}}\right)
\exp\left\{-2|\alpha|^{2}\left(1-\frac{\chi_{2}^{2}}{k^{2}}\right)\right\}\;,
\label{eq:redtherm}
\end{equation}
with an initial mean number of photons given by
\begin{equation}
\bar{N}=\frac{\chi_{2}^{2}}{2(k^{2}-\chi_{2}^{2})}\;,
\label{eq:meannumb}
\end{equation}
which means that when the oscillator is initially sufficiently close
to threshold the initial mean number of photons in modes 2 and 3
within the cavities can be large.
\section{Generation of the cat state}
\label{genera}
At time $t=0$ the first pump is turned on ($\epsilon_{1}\neq 0$): Also
the first crystal begins to operate and the two groups of three modes
start their joint evolution, according to
\begin{eqnarray}
\rho_{\rm T}(t) & = & e^{{\cal L}_{1e-2o-3e}t}\rho_{1e-2o-3e}(0)
\nonumber \\
& & \otimes e^{{\cal L}_{1o-2e-3o}t}\rho_{1o-2e-3o}(0)\;,
\label{eq:evolrhot}
\end{eqnarray}
where the two factorized evolutions are identical because both the
operator ${\cal L}$ and the initial condition are identical in the
two cases. As a consequence, we end up with two identical
six-dimensional problems.
The solution of Eq.~(\ref{eq:evolrhot}) can be found as in
Sec.~\ref{evol} by using the Wigner functions
\begin{equation}
W_{123}(\vec{z}, t) = \int d^{6}z' W_{123}(\vec{z}\;',0)
T(\vec{z}, \vec{z}\;', t)\;,
\label{eq:wig123}
\end{equation}
where the initial Wigner function $W_{123}(\vec{z},0)$ corresponding
to the initial density matrix [Eq.~(\ref{eq:rho123})] is given by
\begin{mathletters}
\label{eq:wwww}
\begin{eqnarray}
W_{123}(\vec{z}, 0) & = & \left(\frac{2}{\pi}\right)^{3}
\left(1-\frac{\chi_{2}^{2}}{k^{2}}\right)
e^{-2(x_{1}^{2}+y_{1}^{2})}
\nonumber \\
& & \times \exp\Big\{-2\Big[x_{2}^{2}+y_{2}^{2}+x_{3}^{2}+y_{3}^{2}
\nonumber \\
& & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, -2\left.
\left.\frac{\chi_{2}}{k}\left(x_{3}x_{2}+y_{3}y_{2}
\right)\right]\right\}
\label{eq:w123in} \\
& = & \left(\frac{2}{\pi}\right)^{3}\sqrt{\det C}
\exp\left\{-2\langle\vec{z}|C|\vec{z}\rangle\right\}\;,
\label{eq:wcomp}
\end{eqnarray}
\end{mathletters}
with
\begin{equation}
C= \left(\matrix{
1 & 0 & 0 & 0 & 0 & 0 \cr
0 & 1 & 0 & 0 & 0 & 0 \cr
0 & 0 & 1 & 0 & -\frac{\chi_{2}}{k} & 0 \cr
0 & 0 & 0 & 1 & 0 & \frac{\chi_{2}}{k} \cr
0 & 0 & -\frac{\chi_{2}}{k} & 0 & 1 & 0 \cr
0 & 0 & 0 & \frac{\chi_{2}}{k} & 0 & 1 \cr
}\right)\;,
\label{eq:cmatrix}
\end{equation}
and
\begin{equation}
\det C = \left(1-\frac{\chi_{2}^{2}}{k^{2}}\right)^{2}\;.
\label{eq:detc}
\end{equation}
From Eq.~(\ref{eq:wig123}) one can immediately recognize that since
the initial state $W_{123}(\vec{z}, 0)$ is Gaussian and the propagator
$T(\vec{z}, \vec{z}\;', t)$ is also Gaussian, the Wigner function
$W_{123}(\vec{z}, t)$ of the evolved state must remain Gaussian at all
times.
Upon integrating over $d^{4}z'$, Eq.~(\ref{eq:wig123}) can be
rewritten as
\begin{equation}
W(\vec{z},t)=\frac{\protect\sqrt{\det B(t)}}{\pi^{3}}
\exp \left\{-\langle\vec{z}|B(t)|\vec{z}\rangle\right\}\;,
\label{eq:sol123}
\end{equation}
where
\begin{equation}
B(t)=\left[2\sigma(t)+\frac{G(t)C^{-1}G^{t}(t)}{2}\right]^{-1}\;,
\label{eq:bt}
\end{equation}
and $G(t)$ and $\sigma(t)$ are the six-dimensional matrices defined
in Eqs.~(\ref{eq:gamma6}), (\ref{eq:gt}), and (\ref{eq:sigma}).
This Gaussian evolution holds for a short time only. As a matter of
fact, one should distinguish between the mode along direction 1 and
those along directions 2 and 3: $a_{2}^{\dagger}$ and $a_{3}^{\dagger}$
denote creation of a photon in the {\em stationary-wave} modes within the
cavities, whereas $a_{1}^{\dagger}$ denotes the creation of a photon
in the {\em traveling-wave} mode along direction $\vec{k}_{1}$.
Therefore the interaction $H_{\rm NL1}=i\hbar\chi_{1}(a_{1}^{\dagger}
a_{2}^{\dagger}-a_{1}a_{2})$ exists only for the time period during
which this traveling wave mode 1 moves {\em within} the nonlinear
crystal. In order to prepare the desired state for the modes 2 and 3,
simultaneously taking full advantage of the degree of freedom
represented by the traveling-wave mode 1, we perform a {\em
conditional}~\cite{kn:cond} measurement on direction 1, thereby
conditioning the state of the four modes along directions 2 and 3
upon the detection of a photon along direction 1 polarized at $\pi/4$
with respect to the two output polarizations $e$ and $o$, which are
orthogonal to each other. In this way we also post-select (along
direction 2) the input state of the second crystal.
The projection operator associated to such a conditional measurement
is therefore given by
\begin{eqnarray}
\hat{P}_{\frac{\pi}{4}} & = & \frac{1}{2}\left\{|1\rangle_{1o}|0\rangle_{1e}
+ |0\rangle_{1o}|1\rangle_{1e}\right\}
\nonumber \\
& & \times \left\{{}_{1o}\langle 1|{}_{1e}\langle 0|
+ {}_{1o}\langle 0|{}_{1e}\langle 1|\right\}\;.
\label{eq:ppi4}
\end{eqnarray}
As a consequence of this measurement (whose success probability
amounts to 0.5) the state along direction 1 and directions 2 and 3
factorizes: The state along direction 1 is given by
\begin{equation}
|\psi\rangle_{1}=\frac{1}{\protect\sqrt{2}}\left\{|1\rangle_{1o}|0\rangle_{1e}
+ |0\rangle_{1o}|1\rangle_{1e}\right\}\;,
\label{eq:polph}
\end{equation}
which represents a photon polarized at $\pi/4$, whilst the
conditional state for directions 2 and 3 is represented by the
density matrix
\begin{eqnarray}
\rho^{c}_{2o-3e-2e-3o}(t) & \propto &
\left\{{}_{1o}\langle 1|{}_{1e}\langle 0|
+ {}_{1o}\langle 0|{}_{1e}\langle 1|\right\}
\rho_{1o-2e-3o}(t)
\nonumber \\
& & \otimes \rho_{1e-2o-3e}(t)
\left\{|1\rangle_{1o}|0\rangle_{1e}\right.
\nonumber \\
& & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
+ \left.|0\rangle_{1o}|1\rangle_{1e}\right\}\;,
\label{eq:rho2323}
\end{eqnarray}
which can be rewritten as
\begin{eqnarray}
\rho^{c}_{2o-3e-2e-3o}(t) & \propto &
\left[\rho^{(1)}_{2e-3o}\rho^{(0)}_{2o-3e}
+\rho^{(0)}_{2e-3o}\rho^{(1)}_{2o-3e}\right.
\nonumber \\
+ & & \left.\rho^{({\rm int})}_{2e-3o}\rho^{({\rm int})\,\dagger}_{2o-3e}
+\rho^{({\rm int})\,\dagger}_{2e-3o}\rho^{({\rm
int})}_{2o-3e}\right]\;,
\label{eq:rhocon}
\end{eqnarray}
where
\begin{mathletters}
\label{eq:rhoconiii}
\begin{eqnarray}
\rho^{(1)}_{2-3} & = & {}_{1}\langle 1|\rho_{1-2-3}(t)|1\rangle_{1}\;,
\label{eq:rhocon231} \\
\rho^{(0)}_{2-3} & = & {}_{1}\langle 0|\rho_{1-2-3}(t)|0\rangle_{1}\;,
\label{eq:rhocon230} \\
\rho^{({\rm int})}_{2-3} & = & {}_{1}\langle 1|\rho_{1-2-3}(t)|0\rangle_{1}\;.
\label{eq:rhocon23i}
\end{eqnarray}
\end{mathletters}
The state of Eq.~(\ref{eq:rhocon}) is of the same form of the desired
state, Eq.~(\ref{eq:catpro2}), and is a linear superposition of distinguishable
states, as long as $\rho^{(1)}_{2-3}$ is well distinguishable from
$\rho^{(0)}_{2-3}$.
It should also be emphasized at this stage that
the density matrix~(\ref{eq:rhocon}) directly corresponds to the
Wigner function, Eq.~(2) of Ref.~\cite{kn:dem}, obtained in the OPA
case. The similarity between the OPO and the OPA configurations is better
brought about in the limit of small interaction times (see the
appendix, where it is also shown that---in this limit---many
of our results are very similar to those obtained in the OPA
case~\cite{kn:dem}). Roughly speaking, one should recover the OPA
results from the OPO ones in the limit $\kappa \to \infty$, since this
condition means absence of cavity mirrors. However, this
correspondence does not hold exactly because the initial state in the
OPO case (the state present in the cavity at $t=0$, when the first
nonlinear crystal is switched on) is slightly different. This fact
explains the differences between the OPA and the OPO, which
result in a far larger effective number of photons in the latter case.
\section{Detection of the cat state}
\label{detect}
How can we probe the quantum state produced in this
parametric-oscillator entangled configuration, and prove that it
actually represents a Schr\"odinger-cat state? In order to do
this, one has to
independently show that i) the state is indeed made out of two
{\em macroscopically distinct components}, that ii) these two components
exhibit {\em quantum interference}, so that the state can be considered as a
true {\em linear superposition} rather than a {\em statistical
mixture}, and that iii) the ``separation'' between the two components
{\em scales with a macroscopic or mesoscopic parameter}, usually the number
of photons.
To achieve this goal, we propose three different and independent
methods---which can be used either alternatively or
simultaneously---as we shall explain in detail in the next three
subsections.
\subsection{Photodetection}
\label{photo}
Let us employ photon number measurements for the modes along
direction 2, thereby collecting the photon-number distributions
$P(n_{2o})$ and $P(n_{2e})$. We therefore consider the reduced
density matrix obtained by performing the trace on the state of
Eq.~(\ref{eq:rhocon}), that is,
\begin{eqnarray}
\rho_{2e} & = & {\rm Tr}_{2o-3e-3o}\left\{\rho_{2e-3o-2o-3e}(t)\right\}
\nonumber \\
& = & \frac{1}{2}\left\{{\rm Tr}_{3o}\left[\rho_{2e-3o}^{(1)}\right]
{\rm Tr}_{23}\left[\rho_{2-3}^{(0)}\right]\right.
\nonumber \\
& & + \left.{\rm Tr}_{3o}\left[\rho_{2e-3o}^{(0)}\right]
{\rm Tr}_{23}\left[\rho_{2-3}^{(1)}\right]\right\}
\left[P\left(\frac{\pi}{4}\right)\right]^{-1}\;,
\label{eq:rhored2e}
\end{eqnarray}
where $P(\pi/4)$ is the probability of finding one photon with
polarization at $\pi/4$, that is,
\begin{eqnarray}
P\left(\frac{\pi}{4}\right) & = & {\rm Tr}_{1o-1e-2e-3o-2o-3e}
\left[\hat{P}_{\frac{\pi}{4}}\rho_{1o-2e-3o}(t)\right.
\nonumber \\
& & \otimes \left.\rho_{1e-2o-3e}(t)\right]
\nonumber \\
& = & {\rm Tr}_{2-3}\left[\rho_{2-3}^{(1)}\right]
{\rm Tr}_{2-3}\left[\rho_{2-3}^{(0)}\right]\;,
\label{eq:probpi4}
\end{eqnarray}
represents the probability of the conditional measurement generating
the desired cat state. The interference terms in Eq.~(\ref{eq:rhocon})
obviously give no contribution to Eq.~(\ref{eq:rhored2e}), since
\begin{equation}
{\rm Tr}_{2-3}\left[\rho_{2-3}^{({\rm int})}\right]=
{\rm Tr}_{2-3}\left[{}_{1}\langle 1 | \rho_{1-2-3}(t) | 0\rangle_{1}\right]
=0\;.
\label{eq:trint}
\end{equation}
Combining Eqs.~(\ref{eq:rhored2e}) and (\ref{eq:probpi4}), one obtains
for the reduced state
\begin{equation}
\rho_{2e}=\frac{1}{2}\left[
\frac{{\rm Tr}_{3o}\rho_{2e-3o}^{(1)}}{{\rm Tr}_{2-3}\rho_{2-3}^{(1)}}
+\frac{{\rm Tr}_{3o}\rho_{2e-3o}^{(0)}}{{\rm Tr}_{2-3}\rho_{2-3}^{(0)}}
\right]\;,
\label{eq:2erhored}
\end{equation}
with an identical form for the reduced state $\rho_{2o}$. The reduced
density matrices $\rho_{3e}$ and $\rho_{3o}$ can be determined in a
similar way.
From Eq.~(\ref{eq:2erhored}) it is immediate to recognize that the
reduced state of the mode $2e$ is given by the sum of two density
matrices, conditioned upon the detection of {\em one} photon
and of {\em zero} photons in the mode $1o$ (or, more precisely,
{\em one} photon in the mode $1e$), respectively. Therefore the two
terms of the reduced density matrix can be experimentally obtained
by rotating the polarizer in front of the detector D$_{1}$ located along the
direction $\vec{k}_{1}$: When the polarizer is vertical (mode $1e$),
we have zero photons in the mode $1o$, and only the second term of
the sum in the right-hand side (rhs) of Eq.~(\ref{eq:2erhored}) is realized.
On the contrary, if the polarizer is set horizontally (mode $1o$), one
detects one photon in the mode $1o$, projecting the resulting
density matrix for the mode $2e$ onto the second term in the
sum~(\ref{eq:2erhored}). However, both terms are present when the
polarizer is set at 45$^{\circ}$. An experimentalist could then take
advantage of this property to test the presence of the two component
states: the distinction between the two states in the superposition
can be made via photon number measurements, yielding the probability
distribution $P(n_{2e})$. In fact, one has
\begin{equation}
P(n_{2e})=\frac{1}{2}\left(P_{\rm H}(n_{2e})+P_{\rm
V}(n_{2e})\right)\;,
\label{eq:pn2e}
\end{equation}
where $P_{\rm H}(n_{2e})$ [$P_{\rm V}(n_{2e})$] is the probability
distribution obtained when the polarized is set horizontally
(vertically). The results an experimentalist would obtain with a
simple photodetection in these two situations are shown in
Fig.~\ref{fg:photod}(a) and (b), together with the probability
distribution~(\ref{eq:pn2e}) one would obtain when the polarizer is
set at $45^{\circ}$ [Fig.~\ref{fg:photod}(c)].
In this way we have verified the existence of two distinct components
in the state~(\ref{eq:2erhored}). But how can we be sure that these
two components form a quantum superposition and not just a classical
mixture? To answer this question, one has to perform a measurement
able to distinguish the ``cat state''
\begin{eqnarray}
\rho^{\rm cat}_{2o-3e-2e-3o}(t) & = &
\left[2{\rm Tr}_{2-3}\rho_{2-3}^{(1)}{\rm
Tr}_{2-3}\rho_{2-3}^{(0)}\right]^{-1}
\nonumber \\
& \otimes & \left[\rho^{(1)}_{2e-3o}\rho^{(0)}_{2o-3e}
+ \rho^{(0)}_{2e-3o}\rho^{(1)}_{2o-3e}
\right.
\nonumber \\
& + & \left.\rho^{({\rm int})}_{2e-3o}\rho^{({\rm int})\,\dagger}_{2o-3e}
+ \rho^{({\rm int})\,\dagger}_{2e-3o}\rho^{({\rm
int})}_{2o-3e}\right]\;,
\label{eq:rhoconcomp}
\end{eqnarray}
from the corresponding statistical mixture
\begin{eqnarray}
\rho^{\rm mix}_{2o-3e-2e-3o}(t) & = &
\left[2{\rm Tr}_{2-3}\rho_{2-3}^{(1)}{\rm
Tr}_{2-3}\rho_{2-3}^{(0)}\right]^{-1}
\nonumber \\
& \times & \left[\rho^{(1)}_{2e-3o}\rho^{(0)}_{2o-3e}
+ \rho^{(0)}_{2e-3o}\rho^{(1)}_{2o-3e}
\right]\;,
\label{eq:rhostat}
\end{eqnarray}
which does not exhibit any interference.
In order to reach this goal, we perform an interference experiment,
involving the modes along direction $\vec{k}_{2}$ only, using a
detection system similar to the one proposed in Ref.~\cite{kn:dem},
as schematically described in Fig.~\ref{fg:scheme}. The measured
quantity is given by the photocounts at the detector D$_{\rm c}$, as
a function of the variable phase $\phi$. The annihilation operator $c$
corresponding to the mode traveling to the detector D$_{\rm c}$
can be written in terms of the annihilation operators of the modes
$2e$ and $2o$ as
\begin{equation}
c=\frac{1}{\protect\sqrt{2}}\left(a_{2o}+e^{i\phi}a_{2e}
\right)\;,
\label{eq:c}
\end{equation}
so that the operator number of photons for the mode $c$ will be given
by
\begin{equation}
c^{\dagger}c=\frac{1}{2}\left(
a_{2o}^{\dagger}a_{2o}
+a_{2e}^{\dagger}a_{2e}
+e^{i\phi}a_{2o}^{\dagger}a_{2e}
+e^{-i\phi}a_{2e}^{\dagger}a_{2o}\right)\;.
\label{eq:nc}
\end{equation}
In order to be able to distinguish between the superposition state and
the mixture, the expectation value
\begin{equation}
\langle c^{\dagger}c\rangle_{\rm cat} =
{\rm Tr}\left[c^{\dagger}c \rho_{2o-3e-2e-3o}^{\rm cat}(t)\right]
\label{eq:cdccond}
\end{equation}
has to be different from
\begin{equation}
\langle c^{\dagger}c\rangle_{\rm mix} =
{\rm Tr}\left[c^{\dagger}c \rho_{2o-3e-2e-3o}^{\rm mix}(t)\right]\;.
\label{eq:cdcmix}
\end{equation}
It is then clear that this interference experiment can answer our
question whenever the contributions of the off-diagonal terms
${\rm Tr}[c^{\dagger}c \rho_{2e-3o}^{\rm (int)}
\rho_{2o-3e}^{\rm (int)\, \dagger}]$
and its complex conjugate
${\rm Tr}[c^{\dagger}c \rho_{2e-3o}^{\rm (int)\, \dagger}
\rho_{2o-3e}^{\rm (int)}]$ are nonzero.
Let us start by evaluating the contribution of the diagonal terms,
namely, Eq.~(\ref{eq:cdcmix}). After explicit integration of the
corresponding Wigner function, it is easy to prove that the
phase-dependent terms [the third and the fourth term in
Eq.~(\ref{eq:nc})] vanish when one computes the expectation value,
Eq.~(\ref{eq:cdcmix}). Therefore the diagonal terms yield a phase
($\phi$)-independent contribution given by
\begin{mathletters}
\label{eq:cdc}
\begin{eqnarray}
\langle c^{\dagger}c\rangle_{\rm mix} & = &
\frac{1}{2}\left(\langle a_{2o}^{\dagger}a_{2o}\rangle_{\rm mix}
+ \langle a_{2e}^{\dagger}a_{2e}\rangle_{\rm mix}\right)
\nonumber \\
& = & \frac{1}{4}\left[\langle n_{2o}\rangle^{(1)}
+ \langle n_{2o}\rangle^{(0)}\right]
\nonumber \\
& & + \frac{1}{4}\left[\langle n_{2e}\rangle^{(1)}
+ \langle n_{2e}\rangle^{(0)}\right]
\label{eq:cdcexp} \\
& = & \frac{1}{2}\left[\langle n_{2}\rangle^{(1)}
+ \langle n_{2}\rangle^{(0)}\right]\;,
\label{eq:ncdiagexp}
\end{eqnarray}
\end{mathletters}
where
\begin{equation}
\langle n_{2}\rangle^{(i)} =
\frac{{\rm Tr}_{2-3}\left[\rho_{2-3}^{(i)} a_{2}^{\dagger}
a_{2}\right]}{{\rm Tr}_{2-3}\left[\rho_{2-3}^{(i)}\right]}\;,
\end{equation}
($i=0$, 1) is the mean photon number in one of the two diagonal states
in Eqs.~(\ref{eq:rhoconcomp}) and (\ref{eq:rhostat}). In the small
interaction-time limit, which is very well justified in the present case
(see appendix), $1\gg kt$, $\chi_{1}t$, $\chi_{2}t$,
we have
\begin{mathletters}
\label{eq:23rho}
\begin{eqnarray}
\rho_{2-3}^{(0)} & \simeq & \rho_{2-3}(0)\;,
\label{eq:23rho0} \\
\rho_{2-3}^{(1)} & \propto & a_{2}^{\dagger}\rho_{2-3}(0) a_{2}\;.
\label{eq:23rho1}
\end{eqnarray}
\end{mathletters}
As a consequence, the two expectation values in the rhs of
Eq.~(\ref{eq:ncdiagexp}) can be explicitly evaluated and are given by
\begin{mathletters}
\label{eq:n2}
\begin{eqnarray}
\langle n_{2} \rangle^{(0)} & = & \bar{N} =
\left[2\left(\protect\frac{k^{2}}{\chi_{2}^{2}} - 1 \right)\right]^{-1}\;,
\label{eq:n20} \\
\langle n_{2} \rangle ^{(1)} & = & 2\bar{N}+1\;,
\label{eq:n21}
\end{eqnarray}
\end{mathletters}
where $\bar{N}$ is the initial mean photon number in the cavity.
In conclusion, the diagonal contribution to
the expectation value in Eq.~(\ref{eq:cdccond}) amounts to
\begin{equation}
\langle c^{\dagger} c \rangle_{\rm mix} \simeq \frac{1 + 3\bar{N}}{2}\;,
\label{eq:mixcdc}
\end{equation}
which is indeed $\phi$-independent as expected.
We turn now our attention to the off-diagonal terms in
Eq.~(\ref{eq:rhoconcomp}), which are absent in Eq.~(\ref{eq:rhostat}).
First we note that the expectation values of the number operators
relative to the two polarizations in mode 2 computed on the off-diagonal
terms vanish, i.e.,
\begin{equation}
\langle a_{2o}^{\dagger}a_{2o}\rangle_{\rm o-d}=
\langle a_{2e}^{\dagger}a_{2o}\rangle_{\rm o-d}=0\;.
\label{eq:numopod}
\end{equation}
On the other hand, the third and the fourth term in the rhs of
Eq.~(\ref{eq:nc}) give to the expectation value on the off-diagonal
terms the contributions
\begin{mathletters}
\label{eq:phio}
\begin{eqnarray}
e^{i\phi}\langle a_{2o}^{\dagger} a_{2e}\rangle_{\rm o-d} & = &
\left\{2{\rm Tr}_{2-3}\left[\rho_{2-3}^{(1)}\right]
{\rm Tr}_{2-3}\left[\rho_{2-3}^{(0)}\right]\right\}^{-1}
\nonumber \\
& & \times \left(\langle a_{2e}\rangle^{\rm (int)}
\langle a_{2o}^{\dagger}\rangle^{{\rm (int)}\, \dagger}\right.
\nonumber \\
& & \,\,\,\,\,\,\,\,\, + \left.\langle a_{2e}\rangle^{{\rm (int)}\, \dagger}
\langle a_{2o}^{\dagger}\rangle^{\rm (int)}\right)\;,
\label{eq:phiod} \\
e^{-i\phi} \langle a_{2e}^{\dagger} a_{2o} \rangle_{\rm o-d} & = &
\left(e^{i\phi}\langle a_{2o}^{\dagger} a_{2e}\rangle_{\rm o-d}\right)^{*}\;,
\label{eq:phiodst}
\end{eqnarray}
\end{mathletters}
where
\begin{equation}
\langle a_{2}\rangle^{\rm (int)}={\rm Tr}_{2-3}\left[\rho_{2-3}^{\rm (int)}
a_{2}\right]\;.
\label{eq:a2int}
\end{equation}
These contributions are generally different from zero, and this
observation is sufficient to reach the conclusion that the proposed interference
experiment is able to distinguish the cat state from the corresponding
mixture.
We are able to evaluate these off-diagonal terms in the small
interaction-time limit developed in the appendix: At the lowest order in
$\chi_{1}t$, $\chi_{2}t$, and $kt$, we have
\begin{equation}
\rho_{2-3}^{\rm (int)}\simeq a_{2}^{\dagger} \rho_{2-3}(0)\;,
\label{eq:intrho23}
\end{equation}
and therefore, using Eq.~(\ref{eq:a2int}),
\begin{mathletters}
\label{eq:inta2}
\begin{eqnarray}
\langle a_{2e}\rangle^{\rm (int)} & = &
\chi_{1}t\left(\bar{N}+1\right)\;,
\label{eq:inta2e} \\
\langle a_{2e}\rangle^{{\rm (int)}\, \dagger} & = &
\chi_{1}t\langle a_{2e}^{2}\rangle = 0 \;,
\label{eq:inta2e+} \\
\langle a_{2o}^{\dagger}\rangle^{\rm (int)} & = &
\chi_{1}t \langle a_{2o}^{\dagger\, 2}\rangle = 0\;,
\label{eq:inta2o} \\
\langle a_{2o}^{\dagger}\rangle^{{\rm (int)}\, \dagger} & = &
\chi_{1}t\left(\bar{N}+1\right)\;.
\label{eq:inta2o+}
\end{eqnarray}
\end{mathletters}
On the other hand,
\begin{mathletters}
\label{eq:r23}
\begin{eqnarray}
\rho_{2-3}^{(1)} & \simeq &
\chi_{1}^{2}t^{2}a_{2}^{\dagger}\rho_{2-3}(0) a_{2}\;,
\label{eq:r123} \\
\rho_{2-3}^{(0)} & \simeq & \rho_{2-3}(0)\;,
\label{eq:r023}
\end{eqnarray}
\end{mathletters}
which yield, respectively,
\begin{mathletters}
\label{eq:tr23r23}
\begin{eqnarray}
{\rm Tr}_{2-3}\rho_{2-3}^{(1)} & = &
\chi_{1}^{2}t^{2}\left(\bar{N}+1\right)\;,
\label{eq:tr23r123} \\
{\rm Tr}_{2-3} \rho_{2-3}^{(0)} & \simeq & 1\;,
\label{eq:tr23r023}
\end{eqnarray}
\end{mathletters}
and, finally,
\begin{equation}
e^{i\phi}\langle a_{2o}^{\dagger} a_{2e}\rangle_{\rm o-d}
= \frac{\bar{N}+1}{2} e^{i\phi}\;.
\end{equation}
In conclusion, considering the off-diagonal contribution,
Eq.~(\ref{eq:cdccond}) can be rewritten as
\begin{equation}
\langle c^{\dagger}c\rangle_{\rm cat} =
\langle c^{\dagger}c\rangle_{\rm mix}
+ \frac{\bar{N}+1}{2} \cos\phi\;.
\label{eq:cdccondmix}
\end{equation}
It is then clear that the photocounts at the detector D$_{\rm c}$
exhibit interference fringes as a function of the variable phase
$\phi$, if and only if the state~(\ref{eq:2erhored}) is a true linear
superposition and not just a statistical mixture of the two
macroscopic components. The visibility of such interference fringes
is given by
\begin{equation}
V=\frac{1+\bar{N}}{1+3\bar{N}}
\label{eq:visi}
\end{equation}
and has therefore the lower bound $1/3$ for $\bar{N}\to \infty$.
\subsection{Correlation functions}
\label{corr}
Our aim in this subsection is to compute the first- and second-order
correlation functions relative to our output modes, in order to make
an independent test of the presence of quantum coherence in our system.
We keep in mind~\cite{kn:miwa} that a manifestation of quantum
coherence at second order is subpoissonian statistics, i.e.,
\begin{equation}
G^{(2)}(0) < \left[G^{(1)}(0)\right]^{2}\;,
\label{eq:subpo}
\end{equation}
where $G^{(1)}(0)$ and $G^{(2)}(0)$ are, respectively, the first-
and second-order correlation functions.
Let us consider the same experimental apparatus we have proposed
for the detection of interference (see Fig.~\ref{fg:scheme}). We
take now into account both output ports $c$ and $d$ of the polarizing
beam splitter, with annihilation operators
\begin{mathletters}
\label{eq:cd}
\begin{eqnarray}
c=\frac{1}{\protect\sqrt{2}} \left( a_{2o} +e^{i\phi} a_{2e}
\right)\;,
\label{eq:cc} \\
d=\frac{1}{\protect\sqrt{2}} \left( a_{2o} -e^{i\phi} a_{2e}
\right)\;,
\label{eq:dd}
\end{eqnarray}
\end{mathletters}
and evaluate the correlation functions
$\langle c^{\dagger}cc^{\dagger}c \rangle$,
$\langle d^{\dagger}dd^{\dagger}d \rangle$,
and $\langle c^{\dagger}cd^{\dagger}d \rangle$,
where $c^{\dagger}c$ is given by Eq.~(\ref{eq:nc}), and
\begin{equation}
d^{\dagger}d=\frac{1}{2}\left(
a_{2o}^{\dagger}a_{2o}
+a_{2e}^{\dagger}a_{2e}
-e^{i\phi}a_{2o}^{\dagger}a_{2e}
-e^{-i\phi}a_{2e}^{\dagger}a_{2o}\right)\;.
\label{eq:nd}
\end{equation}
We shall evaluate the functions $(c^{\dagger}c)^{2}$,
$(d^{\dagger}d)^{2}$, and $c^{\dagger}c d^{\dagger}d$ in the
small-time approximation limit (see appendix), in which
\begin{eqnarray}
\rho_{2e-3o-2o-3e} (t) & \propto &
\left(a_{2e}^{\dagger}+a_{2o}\dagger\right)
\rho_{2e-3o} (0) \rho_{2o-3e} (0)
\nonumber \\
& & \times \Big( a_{2e}+a_{2o} \Big)\;,
\label{eq:r23st}
\end{eqnarray}
where $\rho_{2o-3e} (0)$ is the Gaussian state described by the Wigner
function $W_{\rm bt}^{2-3} (0)$ of Eq.~(\ref{eq:w23bt}), for which
the Wigner function corresponding to the reduced density matrix of
mode $2$ alone is given by Eq.~(\ref{eq:redtherm}),
that represents a thermal state with a mean number of photons given by
$\bar{N}$ of Eq.~(\ref{eq:n20}).
Upon evaluating all the required expectation values, we obtain
\begin{mathletters}
\label{eq:corr}
\begin{eqnarray}
\langle \left(c^{\dagger}c\right)^{2}\rangle & = &
\frac{16\bar{N}^{2}+14\bar{N} + 2 + 2\cos\phi
\left(4\bar{N}^{2}+5\bar{N}+1\right)}{4}
\nonumber \\
& & \label{eq:corrc} \\
\langle \left(d^{\dagger}d\right)^{2}\rangle & = &
\frac{16\bar{N}^{2}+14\bar{N} + 2 - 2\cos\phi
\left(4\bar{N}^{2}+5\bar{N}+1\right)}{4}
\nonumber \\
& & \label{eq:corrd} \\
\langle c^{\dagger}cd^{\dagger}d\rangle & = &
\bar{N}\left(2\bar{N}+1\right)\;.
\label{eq:corrcd}
\end{eqnarray}
\end{mathletters}
From Eqs.~(\ref{eq:corr}) it is clear that the visibility of the
fringes in $\langle (c^{\dagger}c)^{2}\rangle$ and
$\langle (d^{\dagger}d)^{2}\rangle$ is given by
\begin{equation}
V = \frac{4\bar{N}^{2}+5\bar{N} + 1}{8\bar{N}^{2} + 7\bar{N} + 1}\;,
\label{eq:corrvis}
\end{equation}
and monotonically decreases from $V=1$ (for $\bar{N}=0$) to $V=1/2$
(for $\bar{N} \to \infty$).
Finally, considering the field at the output port $c$, the first-
and second-order correlation functions for mode $2$ can be written as
\begin{mathletters}
\label{eq:gg}
\begin{eqnarray}
G^{(1)}_{2}(0) & = & \langle c^{\dagger}c \rangle =
\frac{1+3\bar{N} + (1+\bar{N})\cos\phi}{2}\;,
\label{eq:gg1} \\
G^{(2)}_{2}(0) & = & \langle c^{\dagger}c^{\dagger}cc\rangle =
\langle \left(c^{\dagger}c\right)^{2}\rangle - \langle c^{\dagger}c
\rangle
\nonumber \\
& = & 2\bar{N}\left[1+2\bar{N}+\left(\bar{N}+1\right)\cos\phi\right]\;,
\label{eq:gg2}
\end{eqnarray}
\end{mathletters}
respectively. It should be noted that these results map into the
corresponding ones obtained in Ref.~\cite{kn:dem} for the OPA case
upon a redefinition of the phase angles.
By comparing $[G^{(1)}(0)]^{2}$ and $G^{(2)}(0)$ it is
possible to see that $G^{(2)}(0) < [G^{(1)} (0)]^{2}$ only at low
mean photon number, as it could have been easily expected.
The best situation is obtained when $\phi =0$, in which case
\begin{mathletters}
\label{eq:ggg}
\begin{eqnarray}
[G^{(1)}(0)]^{2} & = & \left(1+2\bar{N}\right)^{2}\;,
\label{eq:ggg1} \\
G^{(2)}(0) & = & 2\bar{N}\left(3\bar{N}+2\right)\;,
\label{eq:ggg2}
\end{eqnarray}
\end{mathletters}
and the condition for quantum coherence at second order is reached
when $\bar{N} < 1/\sqrt{2}$.
On the other hand, when $\phi=\pi$, $[G^{(1)}(0)]^{2}=\bar{N}^{2}$,
$G^{(2)}(0)=2\bar{N}^{2}$, and therefore $G^{(2)}(0)$ is always larger
than $[G^{(1)}(0)]^{2}$.
\subsection{Wigner function}
\label{wigner}
The aim of the present section is to provide a means to represent the
essential features of the Schr\"odinger-cat state, Eq.~(\ref{eq:rhocon}),
which ``lives'' in a 8-dimensional phase space, in the more customary
2-dimensional phase space, in order to make a comparison with the more
conventional cat states~\cite{kn:cat,kn:prlha}.
Let us start from Eq.~(\ref{eq:rhocon}) which we rewrite here for
convenience
\begin{eqnarray}
\rho^{c}_{2o-3e-2e-3o}(t) & \propto &
\left[\rho^{(1)}_{2e-3o}\rho^{(0)}_{2o-3e}
+\rho^{(0)}_{2e-3o}\rho^{(1)}_{2o-3e}\right.
\nonumber \\
+ & & \left.\rho^{({\rm int})}_{2e-3o}\rho^{({\rm int})\,\dagger}_{2o-3e}
+\rho^{({\rm int})\,\dagger}_{2e-3o}\rho^{({\rm
int})}_{2o-3e}\right]\;.
\label{eq:cat}
\end{eqnarray}
The Wigner function representation of the density matrix~(\ref{eq:cat})
would of course reflect its characteristic Schr\"odinger-cat properties.
However, in order to better understand the nature of this state, it
would be interesting and desirable to see whether it is possible to
find different optical modes in whose terms the state (and therefore
the Wigner function) may be rewritten in a simpler form. Our key idea is
then to look for linear combinations of mode operators (which can
easily be realized with linear elements: polarizers and beam-splitters)
such as to factorize the state~(\ref{eq:cat}) in smaller subspaces.
We first perform a transformation which changes the horizontally and
vertically polarized modes into the 45$^{\circ}$-polarized ones,
namely,
\begin{mathletters}
\label{eq:modetra}
\begin{eqnarray}
a_{+45,2} & = & \frac{a_{2e}+a_{2o}}{\protect\sqrt{2}}\;,
\;\;\;
a_{-45,2}=\frac{a_{2e}-a_{2o}}{\protect\sqrt{2}}\;,
\label{eq:amodetra} \\
a_{+45,3} & = & \frac{a_{3e}+a_{3o}}{\protect\sqrt{2}}\;,
\;\;\;
a_{-45,3}=\frac{a_{3e}-a_{3o}}{\protect\sqrt{2}}\;,
\label{eq:bmodetra}
\end{eqnarray}
\end{mathletters}
and the corresponding expressions for mode $\vec{k}_{1}$ and for
the creation operators. In terms
of these new operators, $H_{\rm NL1}$ and $H_{\rm NL2}$
[Eqs.~(\ref{eq:hnl1}) and (\ref{eq:hnl2})] can be rewritten as
\begin{mathletters}
\label{eq:hnlp}
\begin{eqnarray}
H_{\rm NL1} & = & i\hbar\chi_{1}\left({a}_{+45,2}^{\dagger}
{a}_{+45,1}^{\dagger}-{a}_{+45,2}{a}_{+45,1}\right)
\nonumber \\
& & -i\hbar\chi_{1}
\left({a}_{-45,2}^{\dagger}{a}_{-45,1}^{\dagger}
-{a}_{-45,1}{a}_{-45,2}\right)\;,
\label{eq:hnl1p} \\
{H}_{\rm NL2} & = & i\hbar\chi_{2}\left({a}_{+45,2}^{\dagger}
{a}_{+45,3}^{\dagger}-{a}_{+45,2}{a}_{+45,3}\right)
\nonumber \\
& & -i\hbar\chi_{2}
\left({a}_{-45,2}^{\dagger}{a}_{-45,3}^{\dagger}
-{a}_{-45,2}{a}_{-45,3}\right)
\label{eq:hnl2p}\;.
\end{eqnarray}
\end{mathletters}
We have already assumed [Sec.~\ref{evol}] that the cavity decay rates
$\kappa_{i}$ do not depend on the polarization. This in turn means that
$\kappa_{+45,2}=\kappa_{-45,2}=\kappa_{2}$ and
$\kappa_{+45,3}=\kappa_{-45,3}=\kappa_{3}$, and therefore we have that
for the $\pm 45^{\circ}$-polarized modes we have the same evolution equation as
that for the original modes (except for a minus sign).
Consequently, it is possible to repeat all the same arguments as before
[Secs.~\ref{evol} and \ref{genera}]. In particular, the modes
$a_{+45,1}$, $a_{+45,2}$, and $a_{+45,3}$ are decoupled from their
orthogonal counterparts $a_{-45,1}$, $a_{-45,2}$, and $a_{-45,3}$,
and the evolution equation may be rewritten as
\begin{equation}
\rho_{\rm T}(t)=e^{{\cal L}_{+45}t} e^{{\cal L}_{-45}t}
\rho_{+45,1;+45,2;+45,3;-45,1;-45,2;-45,3} (0)\;.
\label{eq:evolrt}
\end{equation}
In Eq.~(\ref{eq:evolrt}) the initial condition is given in the same
way by
\begin{equation}
\rho_{+45,1;+45,2;+45,3}(0) = |0\rangle_{+45,1}\langle 0|
\otimes \rho^{\rm bt}_{+45,2;+45,3} (0) \;,
\label{eq:in45}
\end{equation}
where $\rho^{\rm bt}_{+45,2;+45,3} (0)$ is the equilibrium state below
threshold of the parametric oscillator when NL1 is turned off, and the
same initial condition holds for the $-45^{\circ}$-polarized modes.
As a consequence, the same Gaussian evolution we have found in
Sec.~\ref{evol} holds. The only difference is that now the conditional
measurement is simply a projection onto the state $|1\rangle_{+45,1}$,
i.e., the one-photon state for the $a_{+45,1}$ mode, while the
$-45^{\circ}$-polarized modes remain decoupled from the orthogonal ones.
The cat state after the conditional detection of the $n=1$ photon for
the $+45,1$ mode is then written in the following way
\end{multicols}
\noindent\rule{0.5\textwidth}{0.4pt}\rule{0.4pt}{\baselineskip}
\widetext
\begin{eqnarray}
\rho^{\rm c} & \propto & {}_{-45,1}\langle 0|_{+45,1}\langle 1 |
\rho_{+45,1;+45,2;+45,3} (t)
\rho_{-45,1;-45,2;-45,3} |1\rangle_{+45,1}
|0\rangle_{-45,1} =
\rho_{+45,2;+45,3}^{(1)} \otimes \rho_{-45,2;-45,3}^{(0)} \;,
\label{eq:cr01}
\end{eqnarray}
\noindent
\rule{0.5\textwidth}{0.4pt}
\noindent\null
\rule{0.4pt}{\baselineskip}
\null
\begin{multicols}{2}
\noindent
where $\rho_{2-3}^{(0)}$ and $\rho_{2-3}^{(1)}$ are again given by the
expressions~(\ref{eq:rhocon231}) and (\ref{eq:rhocon230}).
It should be noted that, using these new $\pm 45^{\circ}$-polarized
modes, one gets a complete factorization of the $-45^{\circ}$-polarized
modes, which are {\em not affected} by the {\em quantum injection}
process induced by the conditional measurement. The
$-45^{\circ}$-polarized modes are not ``interesting'', in the sense
that all the ``cat'' properties of the state~(\ref{eq:cr01}) are
contained in $\rho_{+45,2;+45,3}^{(1)}$, and therefore we shall
neglect them from now on. We are then left with the state
$\rho_{+45,2;+45,3}^{(1)}$, which is an {\em entangled} state of the
modes $+45,2$ and $+45,3$.
As the second step of our procedure aimed at the further
simplification of the original 8-dimensional Wigner function, we
consider the transformation
\begin{equation}
d_{+} = \frac{a_{+45,2}+a_{+45,3}}{\protect\sqrt{2}}\;,
\;\;\;
d_{-} = \frac{a_{+45,2}-a_{+45,3}}{\protect\sqrt{2}}\;,
\label{eq:d+d-}
\end{equation}
which is suggested by the interaction term in Eq.~(\ref{eq:hnl2p}).
In terms of $d_{+}$ and $d_{-}$, Eq.~(\ref{eq:hnl2p}) becomes
\begin{equation}
H_{\rm NL2} = i\hbar\frac{\chi_{2}}{2}\left(d_{+}^{\dagger \, 2}
-d_{+}^{2}\right) -i\hbar\frac{\chi_{2}}{2}\left(d_{-}^{\dagger \, 2}
- d_{-}^{2}\right)\;,
\label{eq:hnl2d}
\end{equation}
and the two modes $d_{+}$ and $d_{-}$ are squeezed by the nonlinear
crystal. These modes can be experimentally realized outside the cavity
for example with two PBS and a 50\%--50\% BS, as schematically described in
Fig.~\ref{fg:schemwig}. The state of these two modes can be
represented by the Wigner function
\begin{equation}
W_{1} \left(
\frac{x_{d_{+}}+x_{d_{-}}}{\protect\sqrt{2}},
\frac{y_{d_{+}}+y_{d_{-}}}{\protect\sqrt{2}},
\frac{x_{d_{+}}-x_{d_{-}}}{\protect\sqrt{2}},
\frac{y_{d_{+}}-y_{d_{-}}}{\protect\sqrt{2}}
\right)\;,
\label{eq:w1d+d-}
\end{equation}
where [see Eqs.~(\ref{eq:sol123}) and (\ref{eq:bt})]
\end{multicols}
\noindent\rule{0.5\textwidth}{0.4pt}\rule{0.4pt}{\baselineskip}
\widetext
\begin{eqnarray}
W_{1} \left( x_{2},y_{2},x_{3},y_{3} \right)
& \propto & \int dx_{1} dy_{1} \, W_{n=1} \left( x_{1}, y_{1} \right)
\times \frac{\protect\sqrt{\det B}}{\pi^{2}} e^{-\langle \vec{z} |
B | \vec{z} \rangle}\;.
\label{eq:w12233}
\end{eqnarray}
What is the nature of this state? In order to answer this question,
we are naturally guided by two different approaches: i) the study
of the OPA case~\cite{kn:dem} and ii) the use of the small-time limit
$\chi_{1} t,\chi_{2} t,\kappa t \ll 1$ we have already considered in
Sec.~\ref{photo} and worked out in the appendix.
In the OPA case~\cite{kn:dem} the output state at time $t$ is given by
\begin{eqnarray}
|\psi(t)\rangle & = & \frac{1}{\protect\sqrt{2}}
e^{\chi_{2}t (a_{2e}^{\dagger}a_{3o}^{\dagger} - a_{2e} a_{3o})
+ \chi_{2}t(a_{2o}^{\dagger}a_{3e}^{\dagger} - a_{2o} a_{3e})}
\times \left(a_{2e}^{\dagger}+a_{2o}^{\dagger} \right) |0\rangle\;,
\label{eq:opapsi}
\end{eqnarray}
which can be rewritten in terms of the $\pm 45^{\circ}$-polarized
modes as
\begin{eqnarray}
|\psi (t) \rangle & = &
e^{-\chi_{2}t(a_{-45,2}^{\dagger}a_{-45,3}^{\dagger}-a_{-45,2}a_{-45,3})}
|0\rangle
\otimes e^{\chi_{2}t(a_{+45,2}^{\dagger}a_{+45,3}^{\dagger}
-a_{+45,2}a_{+45,3})} a_{+45,2}^{\dagger} |0\rangle=
\psi_{-45,2;-45,3}^{(0)} \psi_{+45,2;+45,3}^{(1)}\;.
\label{eq:opapsi45}
\end{eqnarray}
Neglecting the factorized state $\psi_{-45,2;-45,3}^{(0)}$, and using
the $d_{\pm}$ modes, we have
\begin{mathletters}
\label{eq:squent}
\begin{eqnarray}
\psi_{+45,2;+45,3}^{(1)} & = &
e^{\frac{\chi_{2}t}{2}(d_{+}^{\dagger\, 2}-d_{+}^{2})}
e^{-\frac{\chi_{2}t}{2}(d_{-}^{\dagger\, 2}-d_{-}^{2})}
\left(\frac{d_{+}^{\dagger}+d_{-}^{\dagger}}{\protect\sqrt{2}}\right)
|0\rangle
\label{eq:squenta} \\
& = &
\frac{1}{\protect\sqrt{2}}
\left( \left|\frac{\chi_{2}t}{2}, 1\right\rangle_{d_{+}}
\left|-\frac{\chi_{2}t}{2}, 0\right\rangle_{d_{-}}\right.
+ \left.\left|\frac{\chi_{2}t}{2}, 0\right\rangle_{d_{+}}
\left|-\frac{\chi_{2}t}{2}, 1\right\rangle_{d_{-}} \right)\;,
\label{eq:squentb}
\end{eqnarray}
\end{mathletters}
\noindent
\rule{0.5\textwidth}{0.4pt}
\noindent\null
\rule{0.4pt}{\baselineskip}
\null
\begin{multicols}{2}
\noindent
which is an entangled superposition of the {\em squeezed} one-photon
and vacuum states of the modes $d_{+}$ and $d_{-}$.
It is quite clear now that if we want to ``isolate'' one mode, say,
the $d_{+}$ mode, we need a second {\em conditional measurement} on
the mode $d_{-}$, e.g., a projection onto the state
\begin{equation}
|\varphi\rangle_{d_{-}} = \alpha |0\rangle_{d_{-}}
+ \beta |1\rangle_{d_{-}}\;.
\label{eq:prosup}
\end{equation}
Such a conditional measurement could be performed, for example, by sending a
two-level atom---resonant with the atomic transition---through the
cavity, and eventually post-selecting its internal state in a
corresponding superposition of its ground and excited states.
The conditional state, provided the measurement has given a successful
result, would then read as
\begin{equation}
|\psi^{\rm c}\rangle_{d_{+}} \propto \alpha^{\star} |\frac{\chi_{2}t}{2}
, 1\rangle_{d_{+}} + \frac{\beta^{\star}}{\cosh \chi_{2}t}
|\frac{\chi_{2}t}{2},0\rangle_{d_{+}}\;.
\label{eq:psicd+}
\end{equation}
We can reach a similar conclusion also by analyzing the OPO case using
the very well justified small-time approximation (see appendix) in the
limit $\chi_{1}t, \chi_{2}t, \kappa t \ll 1$, applied to the modes
$+45,1$, $+45,2$, and $+45,3$.
We have, at the lowest order in $\chi_{1}t$,
\begin{mathletters}
\label{eq:r12345}
\begin{eqnarray}
\rho_{+45,2;+45,3}^{(1)} & = & {}_{+45,1}\langle 1|\rho_{1-2-3}
|1\rangle_{+45,1}
\label{eq:r12345a} \\
& \propto & a_{+45,2}^{\dagger} \rho_{2-3}(0) a_{+45,2}\;,
\label{eq:r12345b}
\end{eqnarray}
\end{mathletters}
where the initial density matrix $\rho_{2-3}(0)$ is the state described
by the Wigner function~(\ref{eq:w23bt}).
If we now write Eq.~(\ref{eq:w23bt}) in terms of the new variables
corresponding to the modes $d_{+}$ and $d_{-}$, namely,
\begin{mathletters}
\label{eq:varchange}
\begin{eqnarray}
x_{d_{+}} & = & \frac{x_{2}+x_{3}}{\protect\sqrt{2}}\;, \;\;\;\;\;\;
x_{d_{-}} = \frac{x_{2}-x_{3}}{\protect\sqrt{2}}\;,
\label{eq:varchana} \\
y_{d_{+}} & = & \frac{y_{2}+y_{3}}{\protect\sqrt{2}}\;, \;\;\;\;\;\;
x_{d_{-}} = \frac{y_{2}-y_{3}}{\protect\sqrt{2}}\;,
\label{eq:varchanb}
\end{eqnarray}
\end{mathletters}
we obtain
\end{multicols}
\noindent\rule{0.5\textwidth}{0.4pt}\rule{0.4pt}{\baselineskip}
\widetext
\begin{eqnarray}
W_{\rm bt}^{d_{+}d_{-}} & = & W_{\rm bt}^{d_{+}} W_{\rm bt}^{d_{-}} =
\frac{2}{\pi} \sqrt{1-\frac{\chi_{2}^{2}}{\kappa^{2}}}
\exp\left[-2\left(1-\frac{\chi_{2}}{\kappa}\right) x_{d_{+}}^{2}\right.
\left.-2\left(1+\frac{\chi_{2}}{\kappa}\right) y_{d_{+}}^{2}\right]
\nonumber \\
& & \times
\frac{2}{\pi} \sqrt{1-\frac{\chi_{2}^{2}}{\kappa^{2}}}
\exp\left[-2\left(1+\frac{\chi_{2}}{\kappa}\right) x_{d_{-}}^{2}\right.
\left. -2\left(1+\frac{\chi_{2}}{\kappa}\right) y_{d_{-}}^{2}\right]\;.
\nonumber \\
\label{eq:wdb}
\end{eqnarray}
The initial states for the modes $d_{\pm}$ are generalized Gaussian
states~\cite{kn:gar}, of the kind
\begin{equation}
\rho_{\pm} \propto \exp \left( -nd_{\pm}^{\dagger}d_{\pm}
-\frac{1}{2} m^{\star}_{\pm} d_{\pm}^{2}-\frac{1}{2} m_{\pm}
d_{\pm}^{\dagger \, 2} \right)\;,
\label{eq:gengau}
\end{equation}
with
\begin{mathletters}
\label{eq:nmgau}
\begin{equation}
n=\frac{1}{\protect\sqrt{1-\frac{\chi_{2}^{2}}{\kappa^{2}}}} \log
\left(\frac{1+\protect\sqrt{1-\frac{\chi_{2}^{2}}{\kappa^{2}}}}
{1-\protect\sqrt{1-\frac{\chi_{2}^{2}}{\kappa^{2}}}} \right)\;,
\label{eq:ngau}
\end{equation}
and
\begin{equation}
m_{\pm} = \pm \frac{\chi_{2}}{\kappa} n\;.
\label{eq:mgau}
\end{equation}
\end{mathletters}
Since the initial state factorizes, we have
\begin{mathletters}
\label{eq:rfac}
\begin{eqnarray}
\rho_{d_{+}d_{-}}^{(1)} & \propto & a_{+45,2}^{\dagger} \rho_{d_{+}}(0)
\rho_{d_{-}} (0) a_{+45,2}
\propto (d_{+}^{\dagger}+d_{-}^{\dagger}) \rho_{d_{+}}(0)
\rho_{d_{-}} (0) (d_{+}+d_{-})
\label{eq:rfacb} \\
& = & \left(d_{+}^{\dagger} \rho_{d_{+}} (0) d_{+} \right) \otimes
\rho_{d_{-}} (0)
+ \rho_{d_{+}} (0) \otimes \left(d_{-}^{\dagger}
\rho_{d_{-}} (0) d_{-}^{\dagger} \right)
\nonumber \\
& &
+ \left(d_{+}^{\dagger} \rho_{d_{+}} (0) \right) \otimes
\left(\rho_{d_{-}} (0) d_{-} \right)
+ \left(\rho_{d_{+}} (0) d_{+}
\right) \otimes \left( d_{-}^{\dagger} \rho_{d_{-}} (0) \right)\;,
\label{eq:rfacc}
\end{eqnarray}
\end{mathletters}
which is a mixed state analogous to the pure state~(\ref{eq:squent})
obtained in the OPA case. Its Wigner function can be calculated from
Eq.~(\ref{eq:rfac}) and is given by
\begin{eqnarray}
W\left(x_{d_{+}}, y_{d_{+}}, x_{d_{-}}, y_{d_{-}}\right) & = &
\frac{4}{\pi^{2}} \frac{\left(1-\frac{\chi_{2}^{2}}{\kappa^{2}}
\right)^{2}}{2-\frac{\chi_{2}^{2}}{\kappa^{2}}}
\left[\left(x_{d_{+}}^{2}+y_{d_{-}}^{2}\right)
\left(2-\frac{\chi_{2}}{\kappa}\right)^{2} +
\left(y_{d_{+}}^{2}+x_{d_{-}}^{2}\right)
\left(2+\frac{\chi_{2}}{\kappa}\right)^{2} -2\right.
\nonumber \\
& & \;\; \left.+2\left(4-\frac{\chi_{2}^{2}}{\kappa^{2}}\right)
\left(x_{d_{+}}x_{d_{-}}+y_{d_{+}}y_{d_{-}}\right)\right]
e^{-2\left(1-\frac{\chi_{2}}{\kappa}\right)
\left(x_{d_{+}}^{2}+y_{d_{-}}^{2}\right)
-2\left(1+\frac{\chi_{2}}{\kappa}\right)
\left(x_{d_{-}}^{2}+y_{d_{+}}^{2}\right)}\;.
\label{eq:wigfour}
\end{eqnarray}
\noindent
\rule{0.5\textwidth}{0.4pt}
\noindent\null
\rule{0.4pt}{\baselineskip}
\null
\begin{multicols}{2}
\noindent
Two important features should be noted within the form of this Wigner
function: i) the interference term (the last term in the square
brackets) decreases when the number of photons in the initial state
increases. This behavior is governed by the factor
$4-\chi_{2}^{2}/\kappa^{2}$ and by the fact that [see
Eqs.~(\ref{eq:meannumb}) and (\ref{eq:n2})] $\bar{N} \to \infty$
when $\chi_{2}/\kappa \to 1$. ii) The Wigner function is negative
around the origin and its negativity scales to zero as the
initial mean photon number $\bar{N} \to \infty$. In fact,
\begin{equation}
W(0,0,0,0)=-\frac{8}{\pi^{2}}\frac{\left(1-\frac{\chi_{2}^{2}}{\kappa^{2}}
\right)^{2}}{2-\frac{\chi_{2}^{2}}{\kappa^{2}}} =
-\frac{4}{\pi^{2}}\frac{1}{(2\bar{N}+1)(\bar{N}+1)}\;.
\label{eq:wzero}
\end{equation}
We have already seen the same scaling behavior of quantum properties
with $\bar{N} \to \infty$ in the calculation of the second-order
correlation function $G^{(2)}$: This is one of the desired properties
of a Schr\"odinger-cat state, as we have emphasized at the beginning
of this section.
Again, Eq.~(\ref{eq:wigfour}) bears a remarkable similarity with the
corresponding result obtained in Ref.~\cite{kn:dem} for the OPA
configuration, in the limits $\kappa \to \infty$ and of small
interaction times. The main advantage of the OPO is given by the
larger effective number of photons per mode $N$
[see Eq.~(\ref{eq:meannumb})] with respect to the
$\sinh^{2}\chi t$ of the OPA~\cite{kn:dem}.
We have therefore learnt that in order to obtain a one-mode state
which embodies all the relevant features of the original four-mode
cat state one {\em has to perform} a conditional measurement on the
mode $d_{-}$. When this is successfully done, the final conditioned
state of the mode $d_{+}$ alone is described by the Wigner function
\begin{equation}
W(\delta_{+}) \propto \pi \int d^{2} \delta_{-}
W_{\alpha |0\rangle + \beta |1\rangle} (\delta_{-})
W(\delta_{+},\delta_{-})\;,
\label{eq:wigd+}
\end{equation}
where
\begin{eqnarray}
& & W(\delta_{+},\delta_{-}) =
\nonumber \\
& & W_{1} \left(
\frac{x_{d_{+}}+x_{d_{-}}}{\protect\sqrt{2}},
\frac{y_{d_{+}}+y_{d_{-}}}{\protect\sqrt{2}},
\frac{x_{d_{+}}-x_{d_{-}}}{\protect\sqrt{2}},
\frac{y_{d_{+}}-y_{d_{-}}}{\protect\sqrt{2}}
\right)\;,
\nonumber \\
\label{eq:wde+-}
\end{eqnarray}
is the Wigner function [see Eqs.~(\ref{eq:w1d+d-}) and
(\ref{eq:w12233})] of the state~(\ref{eq:rfac}), and
\begin{eqnarray}
W_{\alpha |0\rangle + \beta |1\rangle} (\delta_{-})
& = & \frac{2}{\pi}
\Big[1+4|\beta|^{2}\left(|\delta|^{2}-\frac{1}{2}\right)
\nonumber \\
& & +4{\rm Re} (\delta){\rm Re}(\alpha\beta^{\star})
+4{\rm Im}(\delta){\rm Im} (\alpha^{\star}\beta) \Big]
\nonumber \\
\label{eq:wab}
\end{eqnarray}
is the Wigner function of the state onto which the conditional
measurement projects the mode $d_{-}$ [Eq.~(\ref{eq:prosup})].
According to the small-time limit approximation (see appendix)
the explicit form of the Wigner function~(\ref{eq:wigd+}) can
be derived from Eqs.~(\ref{eq:rfac})--(\ref{eq:wab}) and,
after a lengthy calculation, reads as
\end{multicols}
\noindent\rule{0.5\textwidth}{0.4pt}\rule{0.4pt}{\baselineskip}
\widetext
\begin{eqnarray}
W(x_{d_{+}},y_{d_{+}}) & \propto &
\exp\left[-2\left(1-\frac{\chi_{2}}{\kappa}\right)x_{d_{+}}^{2}
-2\left(1+\frac{\chi_{2}}{\kappa}\right)y_{d_{+}}^{2}\right]
\Bigg\{\left(|\alpha|^{2}
+|\beta|^{2}\frac{\chi_{2}^{2}/\kappa^{2}}{4-\chi_{2}^{2}/\kappa^{2}}\right)
\left[x_{d_{+}}^{2}\left(2-\frac{\chi_{2}}{\kappa}\right)^{2}\right.
\nonumber \\
& & \left.+ y_{d_{+}}^{2}\left(2+\frac{\chi_{2}}{\kappa}\right)^{2} -1\right]
+ |\beta|^{2} + 2{\rm Re} \left(\alpha^{\star}\beta
\left[x_{d_{+}}\left(2-\frac{\chi_{2}}{\kappa}\right)\right.\right.
-iy_{d_{+}}
\left.\left.\left(2+\frac{\chi_{2}}{\kappa}\right)\right]\right)\Bigg\}\;,
\label{eq:wd}
\end{eqnarray}
\noindent
\rule{0.5\textwidth}{0.4pt}
\noindent\null
\rule{0.4pt}{\baselineskip}
\null
\begin{multicols}{2}
\noindent
which is in very good agreement with the numerically computed exact
one. As desired, the value of $W(x_{d_{+}},y_{d_{+}})$ at the origin
may also be negative (depending on the parameters $\alpha$ and $\beta$
specifying the conditional measurement), reflecting the quantum properties
of the original 4-dimensional Wigner function~(\ref{eq:wigfour}).
Explicitly, one has
\begin{equation}
W(x_{d_{+}}=0,y_{d_{+}}=0) = 2 |\beta|^{2}
\frac{2-\frac{\chi_{2}^{2}}{\kappa^{2}}}{4-\frac{\chi_{2}^{2}}{\kappa^{2}}}
- |\alpha|^{2}\;.
\label{eq:w00}
\end{equation}
These results are graphically shown in
Figs.~\ref{fg:wig1}--\ref{fg:margy}. In Figs.~\ref{fg:wig1} and
\ref{fg:wig2} we have plotted the Wigner function~(\ref{eq:wd}) for
two values of $N$. In both figures two different viewpoints have been
selected for the tridimensional plots, in order to display most
clearly the quantum superposition character of our cat-like state.
In particular, one should note that in both cases the Wigner function
is negative around the origin. However, a comparison between
Fig.~\ref{fg:wig1} and Fig.~\ref{fg:wig2} shows that even though the two
Gaussian peaks are better separated for a larger number of photons, the
negativity of the Wigner function tends to disappear as soon as the initial
number of photons increases, as expected. This behavior is further
confirmed by the inspection of the corresponding marginal distributions
of the Wigner function~(\ref{eq:wd}), shown in Figs.~\ref{fg:margx} and
\ref{fg:margy}: $P(x)=P(x_{d_{+}})$ displays a larger separation
between the peaks as the initial mean photon number $\langle n
\rangle = N$ increases. On the other hand, $P(y)=P(y_{d_{+}})$ displays
the interference between the two macroscopic components, which tends
to be washed out when the number of photons increases. In fact, for
$N=14.94$, the interference fringes have already disappeared.
\section{Discussion and Conclusions}
\label{conclu}
In this paper we have considered the generation of {\em entangled}
Schr\"odinger-cat states in an optical parametric oscillator, as a
relevant variant of the original proposal~\cite{kn:dem} which instead
had considered the amplifier case. In these works, the central point
(both conceptually and experimentally) is the
{\em quantum injection}~\cite{kn:dem} of the second nonlinear crystal
with the output of the first parametric medium. In the present paper,
we have computed the time evolution for the electromagnetic-field and
chosen the initial condition needed for the generation of the desired
cat state. Such a state, however, lives in a eight-dimensional
phase-space: therefore we have proposed three methods which are able
to prove that it is an actual Schr\"odinger-cat state: direct
photodetection, measurement of the correlation functions, and
measurement of the Wigner function. Our calculations show that the
state produced in this way has indeed two macroscopic (mesoscopic)
components---which are macroscopically (mesoscopically)
distinguishable---and that they are in a coherent superposition (and
not just in a statistical mixture), i.e. they display quantum
interference.
A comparison with the performance of the corresponding OPA
scheme~\cite{kn:dem} is in order here. First, the OPO has a larger
conversion efficiency due to the enhancement factor of the parametric
interaction, given by the presence of the cavities. This leads to a
larger number of photon couples with the same pump power. Second, our
Schr\"odinger-cat state is confined in the cavities, contrarily to
what happens in the OPA case, where it is a traveling wave.
However, the price one has to pay in order to have these advantages,
is given by the unavoidable cavity losses, that tend to destroy the
coherence of the state when the number $N$ of initial photons tends to
infinity. Such a phenomenon---decoherence~\cite{kn:zur,kn:prlha,kn:tra}---is
visualized by the
progressive disappearance of the interference fringes and of the
negativity of the Wigner function when $N$ increases. It is then clear
that one has to consider a trade-off condition between the enhancement
factor (a large $N$) and the losses (a low $\kappa$). This may lead to
a comparison between the performances of the OPO and the
OPA~\cite{kn:dem}: in particular, our OPO configuration is preferable
when the mean number of initial photons $N$ [see
Eq.~(\ref{eq:meannumb})] is larger than the corresponding
parameter ($\sinh^{2}\chi t$~\cite{kn:dem}) of the OPA.
In conclusion, we think that an experiment along the lines outlined
in this paper and in \cite{kn:dem}---which is realizable using
presently available technology---is a promising candidate for
producing entangled superpositions of macroscopically distinct quantum
states.
\acknowledgments
It is a pleasure for us to acknowledge interesting and stimulating
discussions with A.~Ekert and P.~Grangier.
This work has been partially supported by INFM (through the 1997
Advanced Research Project ``Cat''), by the European Union
in the framework of the TMR Network ``Microlasers and Cavity QED'',
and by MURST through ``Cofinanziamento 1997''.
\appendix
\section*{}
\label{app}
The fact that the time $t$ during which we have the interaction within
the first nonlinear crystal is very short is of fundamental importance,
and it allows an immediate description of the experiment. To bring out
this most clearly, we develop an approximate treatment, which is
however justified by the actual experimental values reported in
Ref.~\cite{kn:dem}.
The interaction time $t$, which is the time of flight of the photon
within the first nonlinear crystal NL1, is given by
\begin{equation}
t=\frac{L_{k}n}{c}\simeq 10^{-11}\,{\rm sec}\,
\label{eq:intt}
\end{equation}
where $L_{k}$ is the crystal length, $n$ its refraction index, and
$c$ is the speed of light in vacuum. On the other hand, for an average
pump power $P\simeq 300\, {\rm mW}$, the coupling strength is of the
order of $\chi_{2}\simeq 6\cdot 10^{8}\, {\rm Hz}$.
In order to obtain ``macroscopic'' states, one needs a quite large
initial mean number of photons in the parametric oscillator below
threshold. This fixes the damping rates $\kappa_{2}=\kappa_{3}=\kappa$ to be
slightly larger than $\chi_{2}$, since, from Eq.~(\ref{eq:meannumb}),
we have
\begin{equation}
\frac{\kappa^{2}}{\chi_{2}^{2}}=1+\frac{1}{2 \bar{N}}\;.
\label{eq:kchi}
\end{equation}
Therefore, we have $\kappa\simeq 6\cdot 10^{8}\, {\rm Hz}$, too. Since
the wavelength of the photon is $\lambda\simeq 7.3 \cdot 10^{-5}$ cm,
this amounts to having a {\em standard cavity}, with a quality factor
\begin{equation}
Q=\frac{2\pi c}{\lambda \kappa}\simeq 10^{5}\;.
\label{eq:qfactor}
\end{equation}
On the other hand, $\chi_{1}$ will be of the order of $\chi_{2}$.
In summary, we have
\begin{equation}
\chi_{2}t \simeq \chi_{1}t \simeq \kappa t\simeq 10^{-3}\;.
\label{eq:chit}
\end{equation}
From Eqs.~(\ref{eq:meq}--\ref{eq:hnl123}), (\ref{eq:infact}), and
(\ref{eq:rho123}), one has, for the time evolution of the combined
density matrix,
\begin{equation}
\rho_{123}(t)=e^{{\cal L}_{123}(t)}\rho_{23}(0)|0\rangle_{1}\langle
0|\;.
\label{eq:ellero}
\end{equation}
Since ${\cal L}_{123} \propto \kappa $, $\chi_{1}$, $\chi_{2}$, it is
appropriate to expand the exponential $e^{{\cal L}_{123}t}$ in power
series up to second order in $\kappa t$, $\chi_{1}t$, $\chi_{2}t$, yielding
\begin{equation}
e^{{\cal L}_{123}t} \simeq 1 + {\cal L}_{123}t +
\frac{1}{2} {\cal L}_{123}^{2}t^{2}\;,
\label{eq:expand}
\end{equation}
and
\begin{equation}
{\cal L}_{123}\rho = {\cal L}_{23}\rho +
\chi_{1}\left[a_{1}^{\dagger}a_{2}^{\dagger}-a_{1}a_{2}, \rho
\right]\;,
\label{eq:L123}
\end{equation}
where ${\cal L}_{23}$ is that part of the Liouvillian which only acts
on the modes $\vec{k}_{2}$ and $\vec{k}_{3}$, as given by
Eqs.~(\ref{eq:parmeq}) and (\ref{eq:hnl123}).
It is possible in this way to determine the conditional states
$\rho_{23}^{(0)}$, $\rho_{23}^{(1)}$, and the interference terms in
Eqs.~(\ref{eq:rhocon}--\ref{eq:rhocon23i}). We compute first
\begin{equation}
\rho_{2-3}^{(1)}\simeq\frac{a_{2}^{\dagger}\rho_{2-3}(0)a_{2}}{{\rm Tr}
(a_{2}^{\dagger}\rho_{2-3}(0)a_{2})}\;.
\label{eq:1rho1}
\end{equation}
The properties of this state are usually characterized by measuring
the photon-number distribution of the mode 2 along direction 2. We
have therefore to perform the trace over the mode 3 in
Eq.~(\ref{eq:1rho1}), obtaining
\begin{equation}
\rho_{2}^{(1)\, {\rm red}}={\rm Tr}\left(\rho_{2-3}^{(1)}\right)
=\frac{a_{2}^{\dagger}\left({\rm Tr}_{3}\rho_{2-3}(0)\right)a_{2}}{{\rm Tr}
(a_{2}^{\dagger}\rho_{2-3}(0)a_{2})}\;.
\label{eq:rho21red}
\end{equation}
We already know that ${\rm Tr}_{3}\rho_{2-3}(0)$ is a thermal state
with a mean number of photons given by $\bar{N}$ [see
Eqs.~(\ref{eq:meannumb}), (\ref{eq:kchi}), and (\ref{eq:n20})], i.e.,
\begin{equation}
{\rm Tr}_{3}\rho_{2-3}(0)=\sum_{n=0}^{\infty}
\left(\frac{\bar{N}}{1+\bar{N}}\right)^{n}|n\rangle\langle n|
\left(\frac{1}{1+\bar{N}}\right)\;,
\label{eq:thermal}
\end{equation}
[see Eq.~(\ref{eq:redtherm})] and consequently
\begin{eqnarray}
\rho_{2}^{(1)\, {\rm red}} & = & \sum_{n=0}^{\infty}P_{\rm H}(n)
|n\rangle\langle n|\;,
\label{eq:phnn} \\
P_{\rm H}(n) & = & n\left(\frac{\bar{N}}{1+\bar{N}}\right)^{n-1}
\frac{1}{(1+\bar{N})^{2}}\;,
\label{eq:phn}
\end{eqnarray}
which is a sort of {\em shifted} thermal state and is identical to the
state obtained in the case of the parametric amplifier~\cite{kn:dem} with
a mean number of photons given by Eq.~(\ref{eq:meannumb}).
On the other hand, we have, at the lowest order in $\chi_{1}t$,
\begin{mathletters}
\begin{eqnarray}
\rho_{2-3}^{(0)} & = & \langle 0|\rho_{1-2-3}(t) |0\rangle =
\left(1+{\cal L}_{23}t +\frac{{\cal L}_{23}^{2}t^{2}}{2}\right)
\rho_{2-3}(0)
\nonumber \\
& & - \frac{\chi_{1}^{2}t^{2}}{2}
\left(a_{2}a_{2}^{\dagger}\rho_{2-3}(0)
+\rho_{2-3}(0)a_{2}a_{2}^{\dagger}\right)
\label{eq:0rho0a} \\
& \simeq & \rho_{2-3}(0)\;,
\label{eq:0rho0b}
\end{eqnarray}
\end{mathletters}
and the state $\rho_{2}^{(0)\, {\rm red}}={\rm Tr}_{3}(\rho_{2-3}(0))$
conditioned upon the detection of no photons is essentially identical
to the initial usual thermal state.
\end{multicols}
\begin{multicols}{2}
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\end{multicols}
\begin{figure}
\caption{\widetext
Scheme of the experimental apparatus required for the
generation and detection of entangled superpositions of
macroscopically distinguishable states: The idler beam
($k_{2}
\label{fg:scheme}
\end{figure}
\begin{multicols}{2}
\begin{figure}
\caption{\narrowtext
Photon-number probability distributions for a photodetection
experiment on mode $2e$. $P_{\rm H}
\label{fg:photod}
\end{figure}
\begin{figure}
\caption{\narrowtext
Scheme of the experimental arrangement needed for the
measurement of the Wigner function (see text). $k_2$ and $k_3$
represent the cavity modes of Fig.~\protect\ref{fg:scheme}
\label{fg:schemwig}
\end{figure}
\begin{figure}
\caption{\narrowtext
Tridimensional plot of the Wigner function for the $d_{+}
\label{fg:wig1}
\end{figure}
\begin{figure}
\caption{\narrowtext
Tridimensional plot of the Wigner function for the $d_{+}
\label{fg:wig2}
\end{figure}
\begin{figure}
\caption{\narrowtext
Probability distributions $P(x)$ for the quadrature operator
$x_{d_{+}
\label{fg:margx}
\end{figure}
\begin{figure}
\caption{\narrowtext
Probability distributions $P(y)$ for the quadrature operator
$y_{d_{+}
\label{fg:margy}
\end{figure}
\end{multicols}
\end{document}
|
\begin{document}
\title{On Counting Perfect Matchings in General Graphs}
\begin{abstract}
Counting perfect matchings has played a central role in the theory of counting problems.
The permanent, corresponding to bipartite graphs, was shown to be \#P-complete to
compute exactly by Valiant (1979),
and a fully polynomial randomized approximation scheme (FPRAS) was presented
by Jerrum, Sinclair, and Vigoda (2004)
using a Markov chain Monte Carlo (MCMC) approach. However, it has
remained an open question whether there exists an FPRAS for counting
perfect matchings in general graphs. In fact, it was unresolved
whether the same Markov chain defined by JSV is rapidly mixing in general.
In this paper, we show that it is not. We prove torpid mixing for
any weighting scheme on hole patterns in the JSV chain.
As a first step toward overcoming this obstacle, we
introduce a new algorithm for counting matchings based on the
Gallai--Edmonds decomposition of a graph, and give an FPRAS for counting
matchings in graphs that are sufficiently close to bipartite. In
particular, we obtain a fixed-parameter tractable algorithm for counting
matchings in general graphs, parameterized by the greatest ``order'' of a
factor-critical subgraph.
\end{abstract}
\section{Introduction}\label{sec:intro}
Counting perfect matchings is a fundamental problem in the area of counting/sampling problems.
For an undirected graph $G=(V,E)$, let $\mathcal{P}$ denote the set of perfect matchings of $G$.
Can we compute (or estimate) $|\mathcal{P}|$ in time polynomial in $n=|V|$?
For which classes of graphs?
A polynomial-time algorithm for the corresponding decision and optimization
problems of determining if a given graph contains a
perfect matching or finding a matching of maximum size was presented
by Edmonds~\cite{Edmonds}.
For the counting problem, a classical algorithm of Kasteleyn~\cite{Kasteleyn}
gives a polynomial-time algorithm for exactly computing $|\mathcal{P}|$ for planar graphs.
For bipartite graphs, computing $|\mathcal{P}|$ is equivalent to computing the permanent of $n\times n$ $(0,1)$-matrices. Valiant~\cite{Valiant} proved that the $(0,1)$-Permanent is \#P-complete.
Subsequently attention turned to the Markov Chain Monte Carlo (MCMC) approach.
A Markov chain where the mixing time is polynomial in $n$ is said to be {\em rapidly mixing},
and one where the mixing time is exponential in $\Omega(n)$ is referred to as
{\em torpidly mixing}. A rapidly mixing chain yields an $\mathsf{FPRAS}$ (fully polynomial-time
randomized approximation scheme) for the corresponding counting problem of
estimating $|\mathcal{P}|$~\cite{JVV}.
For dense graphs, defined as those with minimum degree $>n/2$, Jerrum and Sinclair~\cite{JS}
proved rapid mixing of a Markov chain defined by Broder~\cite{Broder}, which yielded an $\mathsf{FPRAS}$ for
estimating $|\mathcal{P}|$. The Broder chain walks on the collection $\Omega=\mathcal{P}\cup\mathcal{N}$ of
perfect matchings $\mathcal{P}$
and near-perfect matchings $\mathcal{N}$; a near-perfect matching is a matching with
exactly 2 holes or unmatched vertices.
Jerrum and Sinclair~\cite{JS}, more generally, proved rapid mixing when
the number of perfect matchings is within a ${\sf poly}(n)$ factor of the
number of near-perfect matchings, i.e., $|\mathcal{P}|/|\mathcal{N}|\geq 1/{\sf poly}(n)$.
A simple example, referred to as a ``chain of boxes'' which is
illustrated in Figure~\ref{fig:chain-of-boxes},
shows that the Broder chain is torpidly mixing. This example was a useful
testbed for catalyzing new approaches to solving the general permanent problem.
Jerrum, Sinclair and Vigoda~\cite{JSV} presented a new Markov chain on $\Omega=\mathcal{P}\cup\mathcal{N}$
with a non-trivial weighting scheme on the matchings based on the holes (unmatched vertices).
They proved rapid mixing for any bipartite graph with the requisite weights used in the
Markov chain, and they presented a polynomial-time algorithm to learn these weights.
This yielded an $\mathsf{FPRAS}$ for estimating $|\mathcal{P}|$ for all bipartite graphs. That is the current
state of the art (at least for polynomial-time, or even sub-exponential-time algorithms).
Could the JSV-Markov chain be rapid mixing on non-bipartite graphs? Previously
there was no example for which torpid mixing was established, it was simply the case
that the proof in~\cite{JSV} fails. We present a relatively simple example where the
JSV-Markov chain fails for the weighting scheme considered in~\cite{JSV}. More
generally, the JSV-chain is torpidly mixing on our class of examples for any weighting
scheme based on the hole patterns, see Theorem~\ref{thm:jsv-fails} in
Section~\ref{sec:JSV-fails} for a formal statement following the precise definition
of the JSV-chain.
A natural approach for non-bipartite graphs is to consider Markov chains
that exploit odd cycles or blossoms in the manner of Edmonds' algorithm.
We observe that a Markov chain which considers \emph{all} blossoms for its
transitions is intractable since sampling all blossoms is NP-hard,
see~Theorem~\ref{thm:all-blossoms-hard}. On the other hand, a chain restricted
to minimum blossoms is not powerful enough to overcome our torpid mixing
examples. See Section~\ref{sec:blossoms-fail} for a discussion.
Finally we utilize the Gallai--Edmonds graph decomposition into
factor-critical graphs~\cite{Edmonds,GallaiA,GallaiB,Schrijver} to present
new algorithmic insights that may overcome the obstacles in our classes of
counter-examples. In Section~\ref{sec:EG}, we describe how the Gallai--Edmonds
decomposition can be used to efficiently estimate $|\mathcal{P}|$, the number of perfect
matchings, in graphs whose factor-critical subgraphs have bounded order
(Theorem~\ref{thm:fpt-alg}), as well as in the torpid mixing example
graphs (Theorem~\ref{thm:counterexample-alg}).
Although all graphs are explicitly defined in the text below, figures depicting
these graphs are deferred to the appendix,
\subsection{Markov Chains}
Consider an ergodic
Markov chain with transition matrix $P$ on a finite state space $\Omega$, and let
$\pi$ denote the unique stationary distribution. We will usually assume the
Markov chain is time reversible, i.e., that it satisfies the \textbf{detailed
balance condition} $\pi(x)P(x,y) = \pi(y)P(x,y)$ for all states $x, y \in
\Omega$.
For a pair of distributions $\mu$ and $\nu$ on $\Omega$ we denote
their total variation distance as $d_{\mathsf{TV}}(\mu,\nu) = \frac{1}{2}\sum_{x\in\Omega}
|\mu(x)-\nu(x)|$.
The
standard notion of \textbf{mixing time}
$T_{\mathrm{mix}}$ is the number of steps from
the worst starting state $X_0=i$ to reach total variation distance $\leq 1/4$ of
the stationary distribution $\pi$, i.e., we write
$T_{\mathrm{mix}} = \max_{i\in\Omega} \min\{t: d_{\mathsf{TV}}(P^t(i,\cdot),\pi)\leq 1/4\}$.
We use conductance to obtain lower bounds on the mixing time.
For a set $S\subset\Omega$ its \textbf{conductance} is defined as:
\[ \mathcal{P}hi(S) = \frac{\sum_{x\in S,y\notin S}\pi(x)P(x,y)}{\sum_{x\in S}\pi(x)}.
\]
Let $\mathcal{P}hi_* = \min_{S\subset\Omega:\pi(S)\leq 1/2} \mathcal{P}hi(S)$.
Then (see, e.g., \cite{Sinclair,LWP})
\begin{equation}
\label{eq:conductance}
T_{\mathrm{mix}} \geq \frac{1}{4\mathcal{P}hi_*}.
\end{equation}
\subsection{Factor-Critical Graphs}
A graph $G = (V,E)$ is \textbf{factor-critical} if for every vertex $v \in V$,
the graph induced on $V\setminus \{v\}$ has a perfect matching. (In particular,
$|V|$ is odd.)
Factor-critical graphs are characterized by their ``ear'' structure. The
\textbf{quotient} $G/H$ of a graph $G$ by a (not necessarily induced) subgraph
$H$ is derived from $G$ by deleting all edges in $H$ and contracting all
vertices in $H$ to a single vertex $v_H$ (possibly creating loops or
multi-edges). An \textbf{ear} of $G$ relative a subgraph $H$ of $G$ is simply a cycle in
$G/H$ containing the vertex $v_{H}$.
\begin{theorem}[Lov\'asz~\cite{Lovasz72}]
A graph $G$ is factor-critical if and only if there is a decomposition $G =
C_0 \cup \cdots \cup C_r$ such that $C_0$ is a single vertex, and $C_i$ is
an odd-length ear in $G$ relative to $\bigcup_{j < i} C_j$, for all $0 <
i \le r$.
Furthermore, if $G$ is factor critical, there exists such a decomposition
for every choice of vertex $C_0$, and the \emph{order} $r$ of the
decomposition is independent of all choices.
\end{theorem}
Since the number of ears in the ear decomposition of a factor-critical graph
depends only on the graph, and not on the choice made in the decomposition, we
say the \textbf{order} of the factor-critical graph $G$ is the number $r$ of
ears in any ear decomposition of $G$.
Factor-critical graphs play a central role in the Gallai--Edmonds structure
theorem for graphs. We state an abridged version of the theorem below.
Given a graph $G$, let $D(G)$ be the set of vertices that remain unmatched in
at least one maximum matching of $G$. Let $A(G)$ be the set of vertices not in
$D(G)$ but adjacent to at least one vertex of $D(G)$. And let $C(G)$ denote the
remaining vertices of $G$.
\begin{theorem}[Gallai--Edmonds Structure Theorem]
The connected components of $D(G)$ are factor-critical. Furthermore, every
maximum matching of $G$ induces a perfect matching on $C(G)$, a near-perfect
matching on each connected component of $D(G)$, and matches all vertices in
$A(G)$ with vertices from distinct connected components of $D(G)$.
\end{theorem}
\section{The Jerrum--Sinclair--Vigoda Chain}
\label{sec:JSV-fails}
We recall the definition of the original Markov chain proposed by
Broder~\cite{Broder}. The
state space of the chain is $\Omega = \ensuremath{\mathcal{P}} \cup \bigcup_{u,v} \ensuremath{\mathcal{N}}(u, v)$
where $\ensuremath{\mathcal{P}}$ is the collection of perfect matchings and
$\ensuremath{\mathcal{N}}(u,v)$ are near-perfect matchings with holes at $u$ and $v$
(i.e., vertices $u$ and $v$ are the only unmatched vertices).
The transition rule for a matching $M \in \Omega$ is as follows:
\begin{enumerate}
\item If $M \in \ensuremath{\mathcal{P}}$, randomly choose an edge $e \in M$ and transition to
$M\setminus \{e\}$.
\item If $M \in \ensuremath{\mathcal{N}}(u,v)$, randomly choose a vertex $x \in V$. If $x \in
\{u,v\}$ and $u$ is adjacent to $v$, transition to $M \cup \{(u,v)\}$. Otherwise,
let $y \in V$ be the vertex matched with $x$ in $M$, and randomly choose $w
\in \{u,v\}$. If $x$ is adjacent to $w$, transition to the matching $M \cup
\{(x,w)\}\setminus\{(x,y)\}$.
\end{enumerate}
The chain $\ensuremath{\mathfrak{X}_{\textrm{B}}}$ is symmetric, so its stationary distribution is uniform. In
particular, when $|\ensuremath{\mathcal{P}}|/|\Omega|$ is at least inverse-polynomial in $n$,
we can efficiently generate uniform samples from $\ensuremath{\mathcal{P}}$ via rejection sampling,
given access to samples from the stationary distribution of $\ensuremath{\mathfrak{X}_{\textrm{B}}}$.
In order to sample perfect matchings even when $|\Omega|/|\ensuremath{\mathcal{P}}|$ is
exponentially large, Jerrum, Sinclair, and Vigoda~\cite{JSV} introduce a new chain $\ensuremath{\mathfrak{X}_{\textrm{JSV}}}$
that changes the stationary distribution of $\ensuremath{\mathfrak{X}_{\textrm{B}}}$ by means of a Metropolis
filter. The new stationary distribution is uniform across \emph{hole patterns},
and then uniform within each hole pattern, i.e., for every $M \in \Omega$, the
stationary probability of $M$ is proportional to $1/|\ensuremath{\mathcal{N}}(u,v)|$ if $M \in
\ensuremath{\mathcal{N}}(u,v)$, and proportional to $1/|\ensuremath{\mathcal{P}}|$ if $M \in \ensuremath{\mathcal{P}}$.
We define $\ensuremath{\mathfrak{X}_{\textrm{JSV}}}$ in greater detail. For $M \in \Omega$, define the weight
function
\begin{equation}\label{eq:std-wt}
w(M) = \left\{\begin{matrix} \frac{1}{|\ensuremath{\mathcal{P}}|} & \textrm{if $M \in \ensuremath{\mathcal{P}}$} \\
\frac{1}{|\ensuremath{\mathcal{N}}(u,v)|} & \textrm{if $M \in
\ensuremath{\mathcal{N}}(u,v)$} \end{matrix}\right.
\end{equation}
\begin{definition}\label{def:jsv-chain}
The chain $\ensuremath{\mathfrak{X}_{\textrm{JSV}}}$ has the same state space as $\ensuremath{\mathfrak{X}_{\textrm{B}}}$. The transition rule
for a matching $M \in \Omega$ is as follows:
\begin{enumerate}
\item First, choose a matching $M' \in \Omega$ to which
$M$ may transition, according to the transition rule for $\ensuremath{\mathfrak{X}_{\textrm{B}}}$
\item With probability $\min\{1, w(M')/w(M)\}$, transition to $M'$.
Otherwise, stay at $M$.
\end{enumerate}
\end{definition}
In their paper, Jerrum, Sinclair, and Vigoda~\cite{JSV} in fact analyze a more general
version of the chain $\ensuremath{\mathfrak{X}_{\textrm{JSV}}}$ that allows for arbitrary edge weights in the graph.
They show that the chain is rapidly mixing
for bipartite graphs $G$. (They also study the separate problem of estimating
the weight function $w$, and give a ``simulating annealing'' algorithm that
allows the weight function $w$ to be estimated by gradually adjusting edge
weights to obtain an arbitrary bipartite graph $G$ from the complete bipartite
graph.) Their analysis of the mixing time uses a canonical paths argument that
crucially relies on the bipartite structure of the graph. However, it remained
an open question whether a different analysis of the same chain $\ensuremath{\mathfrak{X}_{\textrm{JSV}}}$, perhaps
using different canonical paths, might generalize to non-bipartite graphs. We
rule out this approach.
In fact, we rule out a more general family of Markov chains for sampling
perfect matchings. We say a Markov chain is ``of $\ensuremath{\mathfrak{X}_{\textrm{JSV}}}$ type'' if it has
the same state space as $\ensuremath{\mathfrak{X}_{\textrm{JSV}}}$, with transitions as defined in
Definition~\ref{def:jsv-chain}, for \emph{some} weight function $w(M)$ (not
necessarily the same as in Eq.~\eqref{eq:std-wt}) depending only the hole
pattern of the matching $M$.
\begin{theorem}\label{thm:jsv-fails}
There exists a graph $G$ on $n$ vertices such that for any Markov chain
$\ensuremath{\mathfrak{X}}$ of $\ensuremath{\mathfrak{X}_{\textrm{JSV}}}$ type on $G$, either the stationary probability of $\ensuremath{\mathcal{P}}$
is at most $\exp(-\Omega(n))$, or the mixing time of $\ensuremath{\mathfrak{X}}$ is at least
$\exp(\Omega(n))$.
\end{theorem}
The graph $G$ of Theorem~\ref{thm:jsv-fails} is constructed from several copies
of a smaller gadget $H$, which we now define.
\begin{definition}\label{def:chain-of-boxes}
The \textbf{chain of boxes gadget} $B_k$ of length $k$ is the graph on $4k$
vertices depicted in Figure~\ref{fig:chain-of-boxes}. To construct $B_k$,
we start with a path $P_{2k-1} = v_0, v_1, \ldots, v_{2k}$ of length
$2k-1$. Then, for every even edge $\{v_{2i}, v_{2i+1}\}$ on the path, we add two
additional vertices $a_i, b_i$, along with edges to form a path $v_{2i}, a_i, b_i,
v_{2i+1}$ of length $3$.
\end{definition}
\begin{figure}
\caption{The ``chain of boxes'' gadget $B_k$, which has $2^k$ perfect
matchings, but only a single matching in $\ensuremath{\mathcal{N}
\label{fig:chain-of-boxes}
\end{figure}
\begin{observation}\label{obs:chain-of-boxes}
The chain of boxes gadget $B_k$ has $2^k$ perfect matchings, but only one matching
in $\ensuremath{\mathcal{N}}(v_0, v_{2k+1})$.
\end{observation}
\begin{definition}\label{def:torpid-gadget}
The \textbf{torpid mixing gadget} $H_k$ is the graph depicted in
Figure~\ref{fig:gadget}. To construct $H$, first take a $C_{12}$ and label
two antipodal vertices as $a$ and $b$. Add an edge between $a$ and $b$, and
label the two vertices farthest from $a$ and $b$ as $u$ and $v$. Label the
neighbor of $u$ closest to $a$ as $w_1$, and the other neighbor of $u$ as
$w_2$. Label the neighbor of $v$ closest to $a$ as $z_1$ and the other
neighbor of $v$ as $z_2$. Finally, add four chain-of-boxes gadgets $B_k$,
identifying the vertices $v_0$ and $v_{2k}$ of the gadgets with $w_1$ and
$a$, with $a$ and $z_1$, with $w_2$ and $b$, and with $b$ and $z_2$,
respectively.
\end{definition}
Note that in Figures~\ref{fig:gadget} and~\ref{fig:gadget-alt}, one ``box'' from each
copy of $B_k$ in the torpid mixing gadget is left undrawn, for visual clarity.
\begin{figure}
\caption{The torpid mixing gadget $H_k$. The unique matching $M
\in \ensuremath{\mathcal{N}
\label{fig:gadget}
\end{figure}
\begin{figure}
\caption{A matching $M' \in \ensuremath{\mathcal{N}
\label{fig:gadget-alt}
\end{figure}
\begin{lemma}\label{lem:torpid-gadget}
The torpid mixing gadget $H = H_k$ has $16k + 4$ vertices and exactly $2$
perfect matchings. Furthermore, $|\ensuremath{\mathcal{N}}_H(u,v)| = 1$ and $\ensuremath{\mathcal{N}}_H(x_1, v) \ge
2^k$.
\end{lemma}
\begin{proof}
A matching $M \in \ensuremath{\mathcal{N}}_H(u,v)$ is depicted in Figure~\ref{fig:gadget}. We
argue that $M$ is the only matching in $\ensuremath{\mathcal{N}}(u,v)$. First note that $x_1$
must be matched with either $w_1$ or $a$. Either choice forces the matching
on the ``chain of boxes'' above $x_1$ remain identical to $M$. But then if
$x_1$ is matched with $a$, there are no vertices to which $w_1$ can be
matched. So $x_1$ must be matched with $w_1$, and the choice of edge for
$x_2$, $y_1$, and $y_2$ is forced symmetrically, giving the matching $M$.
Similarly, there are exactly two perfect matchings of $H$. Vertex $u$ is
matched with either $w_1$ or $w_2$, and either choice determines all other
edges. In particular, if $u$ is matched with $w_1$, then $x_1$ must be
matched with $a$, and $y_1$ with $z_1$, and so on along the entire
$12$-cycle containing $u$ and $v$. The edges on the four ``chains of
boxes'' are then also completely determined. The other case, when $u$ is
matched with $w_2$, is symmetric.
We now argue that $|\ensuremath{\mathcal{N}}_H(x_1, v)| \ge 2^k$.
Starting from the matching $M' \in \ensuremath{\mathcal{N}}_H(x_1, v)$ depicted in
Figure~\ref{fig:gadget-alt}, each of the $k$ copies of $C_4$ in the chain
of boxes above $x_1$ can be independently alternated, giving $2^k$ distinct
matchings in $\ensuremath{\mathcal{N}}_H(x_1, v)$. \qed
\end{proof}
The torpid mixing gadget already suffices on its own to show that the Markov
chain $\ensuremath{\mathfrak{X}}_{\ensuremath{\mathfrak{X}_{\textrm{JSV}}}}$ defined in~\cite{JSV} is torpidly mixing. In
particular, the conductance out of the set $\ensuremath{\mathcal{N}}_H(x_1, v) \subseteq \Omega(H)$
is $2^{-\Omega(k)}$. In order to prove the stronger claim of
Theorem~\ref{thm:jsv-fails}, that every Markov chain of $\ensuremath{\mathfrak{X}_{\textrm{JSV}}}$-type fails
to efficiently sample perfect matchings, we construct a slightly larger graph
from copies of the torpid mixing gadgets.
\begin{definition}\label{def:counterexample}
The \textbf{counterexample graph} $G_k$ is the graph depicted in
Figure~\ref{fig:counterexample}. It is defined
by replacing every third edge of the twelve-cycle $C_{12}$ with the gadget
$H_k$ defined in Figure~\ref{fig:gadget}. Specifically, let $\{u_i, v_i\}$
be the $3i$-th edge of $C_{12}$ for $i \in \{1, \ldots, 4\}$.
We delete each edge $\{u_i, v_i\}$ and replace it
with a copy of $H$, identifying the vertices $u$ and $v$ of $H$ with
vertices $u_i$ and $v_i$ of $C_{12}$. The resulting graph is $G_k$. Thus, of
the $12$ original vertices in $C_{12}$, $8$ of the corresponding vertices
in $G_k$ participate in a copy of the gadget $H$, and $4$ do not. These $4$
vertices of $G_k$ which do not participate in any copy of the gadget $H$ are
labeled $t_1, \ldots, t_4$ in cyclic order, and the copies of the gadget
$H$ are labeled $H_1, \ldots H_4$ in cyclic order, with $H_1$ coming
between $t_1$ and $t_2$, and so on. Thus, $t_1$ is adjacent to $u_1$ and
$v_4$, $t_i$ is adjacent to $u_i$ and $v_{i-1}$ for $i \in \{2,\ldots,
4\}$, and $H_i$ contains both $u_i$ and $v_i$.
\end{definition}
\begin{figure}
\caption{The ``counterexample graph'' $G_k$ on which $\ensuremath{\mathfrak{X}
\label{fig:counterexample}
\end{figure}
In particular, $G_k$ has $4|V(H)| + 4 = 64k + 8$ vertices.
The perfect and near-perfect matchings of $G_k$ are naturally divided into four
intersecting families. For $i \in \{1,\ldots, 4\}$ we define $S_i$
to be the collection of (perfect and near-perfect) matchings $M \in
\Omega(G_k)$ such that the restriction of $M$ to $H_i$ has two holes, at $u_i$ and
$v_i$, i.e., such that the vertices $u_i$ and $v_i$ either have holes in
$M$ or are matched outside of $H_i$.
\begin{lemma}\label{lem:counterexample-matchings}
The counterexample graph $G_k$ has exactly $8$ perfect matchings. Of these,
$4$ are in $S_1 \cap S_3 \setminus (S_2 \cup S_4)$ and $4$ are in $S_2 \cap
S_4 \setminus (S_1 \cup S_3)$.
\end{lemma}
\begin{proof}
The graph $G_k$ has exactly $8$ perfect matchings. To obtain a matching in
$S_1 \cap S_3 \setminus (S_2 \cup S_4)$, we may without loss of generality
start by matching the vertices in $H_1$ and $H_3$ according to a matching
in $\ensuremath{\mathcal{N}}_{H_1}(u_1, v_1)$ or $\ensuremath{\mathcal{N}}_{H_3}(u_3,
v_3)$, respectively. We must then match $t_1$ with $u_1$, $t_2$ with $v_1$,
$t_3$ with $u_3$, and $t_4$ with $v_3$. Finally, we must match the
remaining vertices according to a perfect matching on each of $H_2$ and
$H_4$. By Lemma~\ref{lem:torpid-gadget}, there are two perfect matchings on each
of $H_2$ and $H_4$, and a unique matching in each of $\ensuremath{\mathcal{N}}_{H_1}(u_1, v_1)$
and $\ensuremath{\mathcal{N}}_{H_3}(u_3, v_3)$, so indeed there are four matchings in $(S_1
\cap S_3) \setminus (S_2 \cup S_4)$. Similarly, there are exactly four
matchings in $(S_2 \cap S_4) \setminus (S_1 \cup S_3)$.
To see that there are no other perfect matchings, let $M$ be an arbitrary
perfect matching of $G_k$. Then $t_1$ is matched either with $u_1$ or
$v_4$. Suppose $t_1$ is matched with $u_1$. Then $v_4$ is matched within
$H_4$. Since $H_4$ has an even number of vertices, $u_4$ must also be
matched within $H_4$, and hence $M$ induces a perfect matching on $H_4$.
Continuing in a similar fashion, $M$ must also induce a perfect matching on
$H_2$. Then the restriction of $M$ to $H_1$ or $H_3$ has holes at $u_1$ and
$v_1$, and at $u_3$ and $v_3$, respectively, so $M \in S_1 \cap S_3
\setminus (S_2 \cup S_4)$. Symmetrically, if $t_1$ is matched with $v_4$
then $M \in S_2 \cap S_4 \setminus (S_1 \cup S_3)$. \qed
\end{proof}
In the proof below, we use the notation $\ensuremath{\mathcal{N}}(M)$ denote the collection of
matchings with the hole pattern as $M$. That is, $\ensuremath{\mathcal{N}}(M) = \ensuremath{\mathcal{P}}$ if $M \in
\ensuremath{\mathcal{P}}$, and $\ensuremath{\mathcal{N}}(M) = \ensuremath{\mathcal{N}}(u,v)$ if $M \in \ensuremath{\mathcal{N}}(u,v)$.
\begin{proof}[Proof of Theorem~\ref{thm:jsv-fails}]
Let $G_k$ be the counterexample graph of
Definition~\ref{def:counterexample}. We will show that the set $S_1 \cup
S_3 \subseteq \Omega(G_k)$ has poor conductance, unless the stationary
probability of $\ensuremath{\mathcal{P}}_{G_k}$ is small. We will write $A = S_1 \cup S_3$ and
$\overline{A} = \Omega(G_k)\setminus (S_1\cup S_3)$.
Let $M \in A$ and $M' \in \overline{A}$ be such that $P(M, M') > 0$. We
claim that neither $M$ nor $M'$ are perfect matchings. Assume without loss
of generality that $M \in S_1$. If $M \in S_1$ is a perfect matching, then
$M \in P_2$ and so $M \in S_3$. The only legal transitions from $M$ to
$\Omega \setminus S_1$ are those that introduce additional holes within
$H_1$, but none of these transitions to a matching outside of $S_3$. Hence,
$M$ cannot be perfect. But if $M'$ is perfect, then $M' \in P_1$, and so
$M'$ induces a perfect matching on $S_1$. But then the transition from $M$
to $M'$ must simultaneously affect $u_1$ and $v_1$, and no such transition
exists.
We denote by $\partial \overline{A}$ the set of matchings $M' \in
\overline{A}$ such that there exists a matching $M \in A$ with $P(M, M') >
0$. We claim that for every matching $M' \in \overline{A}$, we have
\begin{equation}\label{eq:small-boundary}
|\ensuremath{\mathcal{N}}(M') \cap \partial \overline{A}| \le 2^{k-1} |\ensuremath{\mathcal{N}}(M')|\,.
\end{equation}
Let $M' \in \partial\overline{A}$, and let $M \in A$ be such that
$P(M, M') > 0$. Suppose first that $M \in
S_1$. Label the vertices of $H_1$ as in Figure~\ref{fig:gadget},
identifying $u_1$ with $u$ and $v_1$ with $v$. Let $N$ be the matching on
$H = H_1$ induced by $M$, and let $N'$ be the matching on $H_1$ induced by
$M'$. We have $N \in \ensuremath{\mathcal{N}}_{H}(u_1, v_1)$. But by
Lemma~\ref{lem:torpid-gadget}, we have $|\ensuremath{\mathcal{N}}_H(u_1, v_1)| = 1$, i.e., $N$
is exactly the matching depicted in Figure~\ref{fig:gadget}. The only
transitions that remove the hole at $u$ are the two that shift the hole to
$x_1$ or $x_2$, and the only transitions that remove the hole at $v$ are
the two that shift the hole to $y_1$ or $y_2$. So, without loss of
generality, by the symmetry of $G_k$, we have $N' \in \ensuremath{\mathcal{N}}_H(x_1, v_1)$.
By Lemma~\ref{lem:torpid-gadget}, $|\ensuremath{\mathcal{N}}_H(x_1, v_1)| \ge 2^k$, but only
one matching in $\ensuremath{\mathcal{N}}_H(x_1, v_1)$ has a legal transition to $N$.
Therefore, if we replace the restriction of $M'$ to
$H_1$ with any other matching in $\ensuremath{\mathcal{N}}_H(x_1, v_1)$, we obtain another
matching $M'' \in \ensuremath{\mathcal{N}}(M')$, but $M''$ has no legal transition to any
matching in $\ensuremath{\mathcal{N}}(M)$. Hence, only a $2^{-k}$-fraction of $\ensuremath{\mathcal{N}}(M')$ has a
legal transition to $S_1$, and similarly only a $2^{-k}$-fraction of
$\ensuremath{\mathcal{N}}(M')$ has a legal transition to $S_3$. In particular, we have proved
Eq.~\eqref{eq:small-boundary}.
From Eq.~\eqref{eq:small-boundary}, it immediately follows that the
stationary probability of $\partial\overline{A}$ is
\begin{equation}\label{eq:small-boundary2}
\pi(\partial\overline{A}) = \sum_{M' \in \partial\overline{A}} \pi(M')
= \sum_{M' \in \overline{A}} \pi(M')\frac{|\ensuremath{\mathcal{N}}(M')\cap
\partial\overline{A}|}{|\ensuremath{\mathcal{N}}(M')|} = 2^{-k+1}\pi(\overline{A})
\end{equation}
We now compute
\begin{align*}
\sum_{\substack{M \in A, M' \in
\overline{A} \\ P(M, M') > 0}}\pi(M)P(M, M')
= \sum_{\substack{M \in A, M' \in
\overline{A} \\ P(M, M') > 0}}\pi(M')P(M', M)
&\le \pi(\partial(\overline{A})) \\ &< 2^{-k+1}\pi(\overline{A}),
\end{align*}
where we first use the detailed balance condition and then
Eq.~\eqref{eq:small-boundary2}.
Now by \eqref{eq:conductance} and the
definition of conductance, we have
\[
\frac{1}{4 \tau_{\ensuremath{\mathfrak{X}}}} < \mathcal{P}hi(A)
< 2^{-k}\frac{\pi(\overline{A})}{\pi(A)}\,.
\]
In particular, if $\tau_{\ensuremath{\mathfrak{X}}} < 2^{k/2 - 2}$, then $\pi(\overline{A}) >
2^{k/2+1}\pi(A)$. Suppose this is the case. By
Lemma~\ref{lem:counterexample-matchings}, half of the perfect matchings of
$G_k$ belong to $A$. In particular, $\pi(\ensuremath{\mathcal{P}}_{G_k}) \le 2\pi(A) <
2^{-k/2+2}$. Hence, either the stationary probability of $\ensuremath{\mathcal{P}}$ is at most
$2^{-k/2+2} = \exp(-\Omega(n))$, or the mixing time of $\ensuremath{\mathfrak{X}}$ is at least
$2^{k/2-2} = \exp(\Omega(n))$. \qed
\end{proof}
We remark that the earlier Markov chain studied by Broder~\cite{Broder} and Jerrum and
Sinclair~\cite{JS} is also torpidly mixing on the counterexample graph of
Definition~\ref{def:counterexample}, since the ratio of near-perfect matchings to
perfect matchings is exponential~\cite{JS}.
\section{Chains Based on Edmonds' Algorithm}
\label{sec:blossoms-fail}
Given that Edmonds' classical algorithm for \emph{finding} a perfect matching
in a bipartite graph requires the careful consideration of odd cycles in the
graph, it is reasonable to ask whether a Markov chain for counting perfect
matchings should also somehow track odd cycles. In this section, we briefly
outline some of the difficulties of such an approach.
A \emph{blossom} of length $k$ in a graph $G$ equipped with a matching $M$ is
simply an odd cycle of length $2k + 1$ in which $k$ of the edges belong to $M$.
Edmonds' algorithm finds augmenting paths in a graph by exploring the
alternating tree rooted at an unmatched vertex, and contracting blossoms to a
vertex as they are encountered. Given a blossom $B$ containing an unmatched
vertex $u$, there is an alternating path of even length to every vertex $v \in
B$. \emph{Rotating} $B$ to $v$ means shifting the hole at $u$ to $v$ by
alternating the $u$-$v$ path in $B$.
Adding rotation moves to a Markov chain in the style of $\ensuremath{\mathfrak{X}_{\textrm{JSV}}}$ is an attractive
possible solution to the obstacles presented in the previous section. Indeed,
if it were possible to rotate the $7$-cycle containing $u$ and $a$ in the graph in
Figure~\ref{fig:gadget}, it might be possible to completely avoid problematic
holes at $x_1$ or $x_2$.
The difficulty in introducing such an additional move the Markov chain $\ensuremath{\mathfrak{X}_{\textrm{JSV}}}$
is in defining the set of \emph{feasible} blossoms that may be rotated, along with
a probability distribution over such blossoms. In order to be useful, we must
be able to \emph{efficiently sample} from the feasible blossoms at a given
near-perfect matching $M$. Furthermore, the feasible blossoms must respect time
reversibility: if $B$ is feasible when the hole is at $u \in B$, then it must
also be feasible after rotating the hole to $v \in B$; reversibility of the Markov chain is needed
so that we understand its stationary distribution. Finally, the feasible
blossoms must be rich enough to avoid the obstacles outlined in the previous
section.
The set of ``minimum length'' blossoms at a given hole vertex $u$ satisfies the
first criterion of having an efficient sampling algorithm. But it is easy to
see that if only minimum length blossoms are feasible, then the obstacles
outlined in the previous section will still apply (simply by adding a $3$-cycle
at every vertex). Moreover, families blossoms characterized by minimality may
struggle to satisfy the second criterion of time-reversibility. In
Figure~\ref{fig:blossom}, there is a unique blossom containing $u$, but after
rotating the hole to $v$, it is no longer minimal.
\begin{figure}
\caption{After rotating the blossom so that the hole is moved from $u$ to
$v$, the blossom is no longer ``minimal''.}
\label{fig:blossom}
\end{figure}
On the other hand, the necessity of having an efficient sampling algorithm for
the feasible blossoms already rules out the simplest possibility, namely, the
uniform distribution over \emph{all} blossoms containing a given hole vertex
$u$. Indeed, if we could efficiently sample from the uniform distribution over
all blossoms containing a given vertex $u$, then by an entropy argument we
could find arbitrarily large odd cycles in the graph, which is NP-hard.
\begin{theorem}
\label{thm:all-blossoms-hard}
Let {\sc Sampling Blossoms} problem be defined as follows. The
input is an undirected graph $G$ and a near-perfect matching $M$
with holes at $w,r\in V(G)$. The output is a uniform sample from
the uniform distribution of blossoms containing $w$. Unless NP=RP
there is no randomized polynomial-time sampler for {\sc Sampling
Blossoms}.
\end{theorem}
\begin{proof}
We reduce from the problem of finding the longest $s$-$t$-path in
a directed graph $H$ (ND29 in~\cite{GJ}). We construct an instance
of {\sc Sampling Blossoms}, that is, $G$ and $M$ as follows. For
every $v\in V(H)$ we add two vertices $v_0,v_1$ into $V(G)$ and
also add $\{v_0,v_1\}$ into $M$. For every edge $(u,v)\in E(H)$ we
add edge $\{u_1,v_0\}$ into $E(G)$. Finally we add $w,r$ into
$V(H)$ and $\{w,s_0\},\{t_1,w\}$ into $E(H)$.
Note that there is one-to-one correspondence between blossoms that
contain $w$ in $G$ and $s$-$t$-paths in $H$. We now modify $G$ to
``encourage longer paths''. We replace each $\{v_0,v_1\}$ edge in
$G$ by a chain of boxes (with $\ell$ boxes) and replace
$\{v_0,v_1\}$ in $M$ by the unique perfect matching of the chain
of boxes. In the modified graph $G$ for every $s$-$t$-path $p$ in
$H$ there are now $2^{k \ell}$ blossoms that contain $w$ in $G$,
where $k$ is the number of vertices in $p$.
Taking $\ell = n^2$ a uniformly random blossom that contains $w$
in $G$ will with probability $1-o(1)$ correspond to a longest
$s$-$t$-path in $H$ (the number of $s$-$t$-paths is bounded by
$(n+1)^n = 2^{O(n\log n)}$ and hence the fraction of blossoms
corresponding to non-longest $s$-$t$-paths is $2^{O(n\log n)}
2^{-n^2} = o(1)$). \qed
\end{proof}
\section{A Recursive Algorithm}
\label{sec:EG}
We now explore a new recursive algorithm for counting matchings, based on the
Gallai--Edmonds decomposition. In the worst case, this algorithm may still
require exponential time. However, for graphs that have additional structural
properties, for example, those that are ``sufficiently close to bipartite'' in
a sense that will be made precise, our recursive algorithm runs in polynomial
time. In particular, it will run efficiently on examples similar to those used
to prove torpid mixing of Markov chains in the previous section.
We now state the algorithm. It requires as a subroutine an algorithm for
computing the permanent of the bipartite adjacency matrix of a bipartite graph
$G$ up to accuracy $\ensuremath{\varepsilon}$. We denote this subroutine by $\textsc{Permanent}(G,
\ensuremath{\varepsilon})$. The $\textsc{Permanent}$ subroutine requires time polynomial in $|V(G)|$
and $1/\ensuremath{\varepsilon}$ using the algorithm of Jerrum, Sinclair, and
Vigoda~\cite{JSV}, but we use it as a black-box.
\begin{algorithm}
\caption{Recursive algorithm for approximately counting the number of perfect matchings in
a graph}\label{alg:recursive}
\begin{algorithmic}[1]
\mathcal{P}rocedure{Recursive-Count}{$G,\ensuremath{\varepsilon}$}
\State If $V(G) = \emptyset$, return $1$.
\State Choose $u \in V(G)$.
\State Compute the Gallai--Edmonds decomposition of $G-u$.
\ForAll{$v \in D(G-u)$}
\State $H_v \gets$ the connected component of $G-u$ containing $v$
\State $m_v \gets$ \Call{Recursive-Count}{$H_v - v, \ensuremath{\varepsilon}/(2n)$}
\EndFor
\State $m_C \gets$ \Call{Recursive-Count}{$C(G-u), \ensuremath{\varepsilon}/3$}
\State Let $X = A(G-u) \cup \{u\}$, and let $Y$ be the set of connected
components in $D(G-u)$. Let $G'$ be the bipartite graph on $(X,Y)$
defined as follows: for every $x \in X$ and $H \in Y$, if $x$ has any
neighbors in $H$ in $G'$, add an edge $\{x, H\}$ in $G'$ with weight
\[
w(x, H) = \sum_{v \in N(x) \cap H} m_v\,.
\]
\State \textbf{return} $m_C * \textsc{Permanent}(G', \ensuremath{\varepsilon}/3)$
\EndProcedure
\end{algorithmic}
\end{algorithm}
We first argue the correctness of the algorithm.
\begin{theorem}\label{thm:main}
Algorithm~\ref{alg:recursive} computes the number of perfect matchings in $G$ to
within accuracy $\ensuremath{\varepsilon}$.
\end{theorem}
\begin{proof}
We show that the algorithm is correct for graphs on $n$ vertices, assuming
it is correct for all graphs on at most $n-1$ vertices.
We claim that permanent of the incidence matrix of $G'$ defined on line 10
equals the number of perfect matchings in $G$. Indeed, every perfect
matching $M$ of $G$ induces a maximum matching $M_u$ on $G-u$. By the
Gallai--Edmonds theorem, $M_u$ matches each element of $A(G')$ with a
vertex from a distinct component of $D(G')$, leaving one component of
$D(G')$ unmatched. Vertex $u$ must therefore be matched in $M$ with a
vertex from the remaining component of $D(G')$. Therefore, $M$ induces a
perfect matching $M'$ on $G'$. Now let $H_x \in Y$ be the vertex of $G'$
matched to $x$ for each $x \in X$. Then the number of distinct matchings of
$G$ inducing the same matching $M''$ on $G''$ is exactly
\[
\prod_{x \in X}\sum_{v \in N(x) \cap H_x} m_v = \prod_{x \in X} w(x,
H_x)
\]
which is the contribution of $M'$ to the permanent of $G'$. Similarly,
from an arbitrary matching $M'$ of $G'$, with $H_x$ defined as above, we
obtain $\prod_{x \in X} w(x, H_x)$ matchings of $G$, proving the claim.
Hence, it suffices to to compute the permanent of the incidence matrix of
$G'$ up to accuracy $\ensuremath{\varepsilon}$. We know the entries of the incidence matrix
up to accuracy $\ensuremath{\varepsilon}/(2n)$, and $(1 + \ensuremath{\varepsilon}/(2n))^{n/2} < 1 + \ensuremath{\varepsilon}/3$ for $\ensuremath{\varepsilon}$
sufficiently small. Therefore, it suffices to compute the permanent of our
approximation of the incidence matrix up to accuracy $\ensuremath{\varepsilon}/3$ to get
overall accuracy better than $\ensuremath{\varepsilon}$. \qed
\end{proof}
The running time of Algorithm~\ref{alg:recursive} is sensitive to the choice of
vertex $u$ on line 3. If $u$ can be chosen so that each component of
$D(G-u)$ is small, then the algorithm is an efficient divide-and-conquer
strategy. More generally, if $u$ can be chosen so that each component of
$D(G-u)$ is in some sense ``tractable'', then an efficient divide-and conquer
strategy results. In particular, since it is possible to exactly count the
number of perfect matchings in a factor-critical graph of bounded order in
polynomial time, we obtain an efficient algorithm for approximately counting
matchings in graphs whose factor-critical subgraphs have bounded order. This is
the sense in which Algorithm~\ref{alg:recursive} is efficient for graphs
``sufficiently close'' to bipartite.
\begin{theorem}\label{thm:fpt-alg}
Suppose every factor-critical subgraph of $G$ has order at most $k$. Then
the number of perfect matchings in $G$ can be counted to within accuracy
$\ensuremath{\varepsilon}$ in time $2^{O(k)} {\sf poly}(n,1/\ensuremath{\varepsilon})$.
\end{theorem}
The essential idea of the proof is to first observe that a factor-critical
graph can be shrunk to a graph with $O(k)$ edges having the same number of
perfect matchings after deleting any vertex. The number of perfect matchings
can then be counted by brute force in time $2^{O(k)}{\sf poly}(n)$. This procedure
replaces the recursive calls on line $6$ of the algorithm.
\begin{proof}
We first observe that if $H$ is a factor-critical graph of order $k$ with
$n$ vertices, then the number of perfect matchings in $H-v$ can be counted
exactly in time $2^{O(k)} {\sf poly}(n)$ for every vertex $v$. Writing $d_u$ for the degree of a
vertex $u$, we have
\begin{equation}\label{eq:fc-degree}
\sum_{u \in V(H)} (d_u - 2) = 2(k-1),
\end{equation}
since adding
one ear to a graph adds some number of vertices of degree $2$, and
increases the degree of two existing vertices by one each, or one vertex by
two. Fix $v \in H$, and suppose there is a vertex $u$ of degree $2$ in $H - v$,
with neighbors $w_1$ and
$w_2$. Let $H'$ denote the multigraph obtained from $H - v$ by contracting the edges from
$u$ to $w_1$ and $w_2$, so $H'$ has two fewer
vertices than $H$, and has a vertex $w$ with the same multiset of neighbors as
$w_1$ and $w_2$ (excluding $v$). Then there is a bijection between the perfect
matchings of $H'$ and of $H-v$; each perfect matching of $H'$ lifts to a
matching of $H-v$ with a hole at $u$ and exactly one of $w_1$ or $w_2$, and
each perfect matching of $H-v$ projects to a perfect matching of $H'$ by
ignoring the matched edge at $u$. Hence, we may contract away all degree-$2$
vertices of $H-v$, and obtain a graph with the same number of perfect matchings in which
every vertex (save at most two of degree $1$, the former neighbors of $v$)
has degree at least $3$. Then since the contraction does not change the sum in
Eq.~\eqref{eq:fc-degree}, we have
\[
3(|V(H')|-2) \le \sum_{u \in V(H')} d_u \le 2(k-1) + 2|V(H')|
\]
and hence $H'$ has $O(k)$ edges, and the perfect matchings of $H'$ can
be enumerated in time $2^{O(k)}$.
Now we modify Algorithm~\ref{alg:recursive} to run in time $2^{O(k)}
{\sf poly}(n, 1/\ensuremath{\varepsilon})$. First, we delete all edges not appearing in any perfect
matching, call \Call{Recursive-Count}{$G_i,\ensuremath{\varepsilon}/(2n)$} on each
connected component $G_i$, and multiply the results of all of these calls
to estimate the number of perfect matchings in $G$. We have $C(G_i-u) =
\emptyset$ for each such component $G_i$ and every vertex $u \in V(G_i)$,
since edges leaving $C(G_i - u)$ cannot appear in any matching of $G_i - u$.
Therefore, the recursive call on line 8 of the algorithm can be eliminated.
On line 6, instead of computing $m_v$ by a recursive
call, we instead use the procedure described above to compute it in time
$2^{O(k)}$. Hence, Algorithm~\ref{alg:recursive} requires $O(n)$ calls to a
procedure that takes time $2^{O(k)}$. The other lines of
Algorithm~\ref{alg:recursive} require only polynomial time in $n$ and
$1/\ensuremath{\varepsilon}$, so in all Algorithm~\ref{alg:recursive} requires time
$2^{O(k)}{\sf poly}(n, 1/\ensuremath{\varepsilon})$. \qed
\end{proof}
We note that Theorem~\ref{thm:fpt-alg} is proved by eliminating recursive calls in
the algorithm. Although the recursive calls of Algorithm~\ref{alg:recursive}
can be difficult to analyze, they can also be useful, as we now demonstrate by
showing that Algorithm~\ref{alg:recursive} runs as-is in polynomial time on the
counterexample graph of Definition~\ref{def:counterexample}, for appropriate choice of
the vertex $u$ on the line 3 of the algorithm.
\begin{theorem}\label{thm:counterexample-alg}
Algorithm~\ref{alg:recursive} runs in polynomial time on the counterexample
graph of Definition~\ref{def:counterexample}, for appropriate choice of the vertex
$u$ on the line 3 of the algorithm
\end{theorem}
\begin{proof}
After deleting the vertices $u$ and $v$ from the torpid mixing gadget $H$ in
Figure~\ref{fig:gadget}, no odd cycles remain the graph $H$. Let $U$ denote
the set of all four copies of the vertices $u$ and $v$ appearing in the
counterexample graph $G$, so $|U| = 8$. With every recursive call
\Call{Recursive-Count}{$G', \ensuremath{\varepsilon}'$}, if $U \cap V(G') \ne \emptyset$, we
choose $u \in U\cap V(G')$. Hence, after $8$ recursive calls, there are no
odd cycles remaining in $G'$, and each factor-critical subgraph is a single
vertex. When $U \cap V(G') = \emptyset$, we choose $u$ so that $A(G'-u) =
\Omega(n)$---for example taking $u$ at one end of a chain of boxes---so
that the overall recursive depth is $O(1)$. \qed
\end{proof}
\end{document}
|
\begin{document}
\title{Can ${\cal B}
\begin{abstract}
It is known that ${\cal B}({\widehat{\mathrm{env}}}ll^p)$ is not amenable for $p =1,2,\infty$, but whether or not ${\cal B}({\widehat{\mathrm{env}}}ll^p)$ is amenable for $p \in (1,\infty) \setminus \{ 2 \}$ is an open problem. We show that, if ${\cal B}({\widehat{\mathrm{env}}}ll^p)$ is amenable for $p \in (1,\infty)$, then so are ${\widehat{\mathrm{env}}}ll^\infty({\cal B}({\widehat{\mathrm{env}}}ll^p))$ and ${\widehat{\mathrm{env}}}ll^\infty({\cal K}({\widehat{\mathrm{env}}}ll^p))$. Moreover, if ${\widehat{\mathrm{env}}}ll^\infty({\cal K}({\widehat{\mathrm{env}}}ll^p))$ is amenable so is ${\widehat{\mathrm{env}}}ll^\infty({\mathbb M}hbb{I},{\cal K}(E))$ for any index set ${\mathbb M}hbb I$ and for any infinite-dimensional ${\cal L}^p$-space $E$; in particular, if ${\widehat{\mathrm{env}}}ll^\infty({\cal K}({\widehat{\mathrm{env}}}ll^p))$ is amenable for $p \in (1,\infty)$, then so is ${\widehat{\mathrm{env}}}ll^\infty({\cal K}({\widehat{\mathrm{env}}}ll^p {\mathrm{op}}lus {\widehat{\mathrm{env}}}ll^2))$. We show that ${\widehat{\mathrm{env}}}ll^\infty({\cal K}({\widehat{\mathrm{env}}}ll^p {\mathrm{op}}lus {\widehat{\mathrm{env}}}ll^2))$ is not amenable for $p =1,\infty$, but also that our methods fail us if $p \in (1,\infty)$. Finally, for $p \in (1,2)$ and a free ultrafilter $\cal U$ over ${{\mathbb M}hbb N}$, we exhibit a closed left ideal of $({\cal K}({\widehat{\mathrm{env}}}ll^p))_{\cal U}$ lacking a right approximate identity, but enjoying a certain, very weak complementation property.
{\widehat{\mathrm{env}}}nd{abstract}
\begin{keywords}
amenability; ${\cal L}^p$-spaces; maximal operator ideals; ultra-amenability.
{\widehat{\mathrm{env}}}nd{keywords}
\begin{classification}
Primary 47L10; Secondary 46B07, 46B08, 46B45, 46E30, 46H20, 47L20.
{\widehat{\mathrm{env}}}nd{classification}
\section*{Introduction}
In his seminal memoir \cite{Joh1}, B.\ E.\ Johnson initiated the theory of amenable Banach algebras. The choice of terminology is motivated by \cite[Theorem 2.5]{Joh1}: a locally compact group $G$ is amenable in the usual sense (see \cite{Pat}, for instance) if and only if its group algebra $L^1(G)$ is an amenable Banach algebra.
\par
Ever since \cite{Joh1} was published, there have been ongoing efforts to determine, for particular classes of Banach algebras, which algebras in them are the amenable ones. One spectacular result in this direction is the characterization of the amenable ${C^\ast}$-algebras: a ${C^\ast}$-algebra is amenable if and only if it is nuclear (this result, mostly credited to A.\ Connes and U.\ Haagerup, is the culmination of the efforts of many mathematicians; see \cite{LoA} or \cite{Tak} for self-contained accounts).
\par
One particular class of Banach algebras for which the problem of characterizing its amenable members is still wide open is the class of Banach algebras ${\cal B}(E)$, the algebras of all bounded linear operators on a Banach space $E$. From a philosophical point of view, this problem ought to be easy: amenability can often be thought of as a weak finiteness condition, and, for any infinite-dimensional Banach space $E$, the algebra ${\cal B}(E)$ should simply be too ``large'' to be amenable. Already Johnson asked in \cite{Joh1}:
\begin{itemize}
\item Is ${\cal B}(E)$ ever amenable for infinite-dimensional $E$? (\cite[10.4]{Joh1})
\item Is ${\cal B}({\mathfrak H})$ amenable for an infinite-dimensional Hilbert space ${\mathfrak H}$? (\cite[10.2]{Joh1})
{\widehat{\mathrm{env}}}nd{itemize}
\par
The Hilbert space case was settled relatively quickly: in \cite{Was}, S.\ Wassermann showed that a nuclear von Neumann algebra had to be subhomogeneous. In view of the equivalence of amenability and nuclearity for ${C^\ast}$-algebras, this means that ${\cal B}({\mathfrak H})$ can be amenable only if $\dim {\mathfrak H} < \infty$.
\par
Ever since, very little progress has been made in the general Banach space case. Until recently, it was not even known whether ${\cal B}({\widehat{\mathrm{env}}}ll^p)$ was amenable or not for any $p \in [1,\infty]$ other than $2$. This situation changed with C.\ J.\ Read's paper \cite{Rea}:
making ingenious use of random hypergraphs, Read showed that ${\cal B}({\widehat{\mathrm{env}}}ll^1)$ is not amenable. Moreover, he showed that, for any $p \in [1,\infty] \setminus \{ 2 \}$, the Banach algebra $\text{${\widehat{\mathrm{env}}}ll^\infty$-}\bigoplus_{n=1}^\infty {\cal B}({\widehat{\mathrm{env}}}ll^p_n)$ also fails to be amenable (the $p=2$ case already follows from Wassermann's result). Subsequently, G.\ Pisier simplified Read's proof by replacing the random hypergraphs of \cite{Rea} with expanders (\cite{Pis}). Eventually, N.\ Ozawa, simplified Pisier's argument even further and succeeded in giving a proof that simultaneously established the non-amenability of ${\cal B}({\widehat{\mathrm{env}}}ll^p)$ for $p=1,2,\infty$ and of $\text{${\widehat{\mathrm{env}}}ll^\infty$-}\bigoplus_{n=1}^\infty {\cal B}({\widehat{\mathrm{env}}}ll^p_n)$ for any $p \in [1,\infty]$ (\cite{Oza}); even though it is not explicitly stated in \cite{Oza}, the proof also works for ${\cal B}(c_0)$.
\par
In the present paper, we investigate what consequences the hypothetical amenability of ${\cal B}({\widehat{\mathrm{env}}}ll^p)$ for $p \in (1,\infty) \setminus \{ 2 \}$ would have.
\par
Our first result is that, if ${\cal B}({\widehat{\mathrm{env}}}ll^p)$ is amenable, then so is ${\widehat{\mathrm{env}}}ll^\infty({\cal B}({\widehat{\mathrm{env}}}ll^p))$. As the much ``smaller'' algebra $\text{${\widehat{\mathrm{env}}}ll^\infty$-}\bigoplus_{n=1}^\infty {\cal B}({\widehat{\mathrm{env}}}ll^p_n)$ is not amenable, this lends again support to the belief that ${\cal B}({\widehat{\mathrm{env}}}ll^p)$ is not amenable (even though, of course, this is a far cry from a proof).
\par
A straightforward consequence of the amenability of ${\widehat{\mathrm{env}}}ll^\infty({\cal B}({\widehat{\mathrm{env}}}ll^p))$ is that ${\widehat{\mathrm{env}}}ll^\infty({\cal K}({\widehat{\mathrm{env}}}ll^p))$ is amenable, too, and we shall devote most of this paper to exploring the consequences of the amenability of that particular Banach algebra and, more generally, of ${\widehat{\mathrm{env}}}ll^\infty({\cal K}(E))$ for particular Banach spaces $E$. (Incidentally, the question of whether ${\widehat{\mathrm{env}}}ll^\infty({\cal K}(E))$ is amenable for specific Banach spaces $E$ seems to have received almost no attention in the literature; the only references known to the authors are \cite{CSR} and \cite{LLW}, where the case $E = {\widehat{\mathrm{env}}}ll^2$ is settled in the negative.)
\par
First, we show that, due to the separability of ${\cal K}({\widehat{\mathrm{env}}}ll^p)$, the amenability of ${\widehat{\mathrm{env}}}ll^\infty({\cal K}({\widehat{\mathrm{env}}}ll^p))$ already entails the amenability of ${\widehat{\mathrm{env}}}ll^\infty({\mathbb M}hbb{I}, {\cal K}({\widehat{\mathrm{env}}}ll^p))$ for every index set ${\mathbb M}hbb I$ and thus of $({\cal K}({\widehat{\mathrm{env}}}ll^p))_{\cal U}$ for every ultrafilter $\cal U$ (so that ${\cal K}({\widehat{\mathrm{env}}}ll^p)$ is ultra-amenable in the terminology of \cite{Daw}).
\par
Next, we see that the amenability of ${\widehat{\mathrm{env}}}ll^\infty({\cal K}({\widehat{\mathrm{env}}}ll^p))$ forces ${\widehat{\mathrm{env}}}ll^\infty({\mathbb M}hbb{I}, {\cal K}(E))$ to be amenable for every index set ${\mathbb M}hbb I$ and every infinite-dimensional ${\cal L}^p$-space $E$ in the sense of \cite{LP}. In particular, if ${\widehat{\mathrm{env}}}ll^\infty({\cal K}({\widehat{\mathrm{env}}}ll^p))$ is amenable, the so is ${\widehat{\mathrm{env}}}ll^\infty({\cal K}({\widehat{\mathrm{env}}}ll^p {\mathrm{op}}lus {\widehat{\mathrm{env}}}ll^2))$, which is interesting because ${\cal B}({\widehat{\mathrm{env}}}ll^p {\mathrm{op}}lus {\widehat{\mathrm{env}}}ll^2)$ is known to be non-amenable.
\par
We then study the amenability of ${\widehat{\mathrm{env}}}ll^\infty({\cal K}(E {\mathrm{op}}lus F))$ for certain Banach spaces $E$ and $F$. Using the theory of operator ideals (see \cite{Pie}), we show that ${\widehat{\mathrm{env}}}ll^\infty({\cal K}(E {\mathrm{op}}lus {\widehat{\mathrm{env}}}ll^2))$ is not amenable for $E = c_0, {\widehat{\mathrm{env}}}ll^\infty$, and ${\widehat{\mathrm{env}}}ll^1$, but we also show that our methods fail to establish the non-amenability of ${\widehat{\mathrm{env}}}ll^\infty({\cal B}({\widehat{\mathrm{env}}}ll^p {\mathrm{op}}lus {\widehat{\mathrm{env}}}ll^2))$ for $p \in (1,\infty)$.
\par
Finally, we take a look at a particular left ideal of $({\cal K}({\widehat{\mathrm{env}}}ll^p))_{\cal U}$ for $p \in (1,2)$ and $\cal U$ a free ultrafilter over ${{\mathbb M}hbb N}$. We show that this ideal lacks a right approximate identity and, at the same time, enjoys a certain complementation property, which is unfortunately too weak to obtain a contradiction to the amenability of $({\cal K}({\widehat{\mathrm{env}}}ll^p))_{\cal U}$.
\subsubsection*{Acknowledgments}
This research was initiated while the first author was visiting the University of Alberta in the summer of 2007; the financial support and the kind hospitality are gratefully acknowledged. Both authors would like to thank Andreas Defant, Albrecht Pietsch, and Nicole Tomczak-Jaegermann for valuable help with operator ideals.
\section{Amenable Banach algebras}
The definition of an amenable Banach algebra given in \cite{Joh1} is in terms of certain derivations being inner. Throughout this paper, however, we shall not rely on that definition directly, but rather on a more intrinsic, but equivalent characterization, also due to Johnson (\cite{Joh2}).
\par
Let ${\mathfrak A}$ be a Banach algebra, and let $E$ and $F$ be a left and right Banach ${\mathfrak A}$-module, respectively. We use $\hat{\otimes}$ to denote the projective tensor product of Banach spaces. The Banach space $E \hat{\otimes} F$ becomes a Banach ${\mathfrak A}$-bimodule via
\[
a \cdot ( x \otimes y) := a \cdot x \otimes y \quad\text{and}\quad (x \otimes y) \cdot a : =x \otimes y \cdot a
\qquad (a \in {\mathfrak A}, \, x \in E, \, y \in F).
\]
In particular, ${\mathfrak A} \hat{\otimes} {\mathfrak A}$ is a Banach ${\mathfrak A}$-bimodule in a canonical manner. With respect to these module operatations, the diagonal map $\Delta \!: {\mathfrak A} \hat{\otimes} {\mathfrak A} \to {\mathfrak A}$ induced by multiplication, i.e., $\Delta(a \otimes b) = ab$ for $a,b \in {\mathfrak A}$, is a bimodule homomorphism; if we want to emphasize the algebra ${\mathfrak A}$, we sometimes write $\Delta_{\mathfrak A}$ for $\Delta$.
\begin{definition} \label{amdef}
Let ${\mathfrak A}$ be a Banach algebra. An {\widehat{\mathrm{env}}}mph{approximate diagonal} for ${\mathfrak A}$ is a bounded net $( \boldsymbol{d}_\alpha )_\alpha$ in ${\mathfrak A} \hat{\otimes} {\mathfrak A}$ such that
\begin{equation} \label{diag1}
a \cdot \boldsymbol{d}_\alpha - \boldsymbol{d}_\alpha \cdot a \to 0 \qquad (a \in {\mathfrak A})
{\widehat{\mathrm{env}}}nd{equation}
and
\begin{equation} \label{diag2}
a \Delta \boldsymbol{d}_\alpha \to a \qquad (a \in {\mathfrak A}).
{\widehat{\mathrm{env}}}nd{equation}
If ${\mathfrak A}$ has an approximate diagonal bounded, we say that ${\mathfrak A}$ is {\widehat{\mathrm{env}}}mph{amenable}.
{\widehat{\mathrm{env}}}nd{definition}
\begin{remarks}
\item If ${\mathfrak A}$ is amenable and has an identity $1_{\mathfrak A}$, then there is an approximate diagonal $( \boldsymbol{d}_\alpha )_{\alpha \in {\mathbb M}hbb A}$ for ${\mathfrak A}$ such that $\Delta \boldsymbol{d}_\alpha = 1_{\mathfrak A}$ for all $\alpha \in {\mathbb M}hbb A$.
\item If ${\mathfrak A}$ is amenable with an approximate diagonal bounded by $C \geq 1$, then ${\mathfrak A}$ is also called {\widehat{\mathrm{env}}}mph{$C$-amenable}. It is clear from ({{\mathrm{op}}eratorname{Re}}f{diag2}) that is doesn't make sense to speak of $C$-amenability for any $C \in (0,1)$.
{\widehat{\mathrm{env}}}nd{remarks}
\par
For modern accounts of the theory of amenable Banach algebras, see \cite{Dal} or \cite{LoA}.
\par
The hereditary properties of Banach algebraic amenability are well understood. For instance, quotients of amenable Banach algebras are again amenable (\cite[Corollary 2.3.2]{LoA}), and a closed ideal of an amenable Banach algebra is amenable if and only if it has a bounded approximate identity and if and only if it is weakly complemented (\cite[Theorem 2.3.7]{LoA}). Whether or not a particular subalgebra of an amenable Banach algebra is amenable is a much more delicate question and an elegant characterization is certainly out of reach. Nevertheless, some partial results exist such as the following (\cite[Theorem 6.2]{GJW}):
\begin{theorem} \label{bgn}
Let ${\mathfrak A}$ be a Banach algebra with a bounded approximate identity, let $P_1 \in {\cal M}({\mathfrak A})$ be a projection, and let $P_2 := {\mathrm{id}}_{\mathfrak A} - P_1$. Suppose further that $\Delta_{\mathfrak A}$ maps $P_2 {\mathfrak A} P_1 \hat{\otimes} P_1 {\mathfrak A} P_2$ onto $P_2 {\mathfrak A} P_2$. Then ${\mathfrak A}$ is amenable if and only if $P_1 {\mathfrak A} P_1$ is amenable.
{\widehat{\mathrm{env}}}nd{theorem}
\par
Here, ${\cal M}({\mathfrak A})$ stands for the {\widehat{\mathrm{env}}}mph{multiplier algebra} of ${\mathfrak A}$ (\cite[p.\ 60]{Dal}).
\par
The Banach algebra ${\mathfrak A}$ in Theorem {{\mathrm{op}}eratorname{Re}}f{bgn} has a matrix like structure thanks to the projections $P_1$ and $P_2$. We shall now prove a necessary condition for the non-amenability of such algebras:
\begin{proposition} \label{noamprop}
Let ${\mathfrak A}$ be a Banach algebra, let $P_1 \in {\cal M}({\mathfrak A})$ be an idempotent, and let $P_2 := {\mathrm{id}}_{\mathfrak A} - P_1$. Suppose that there is a closed ideal $I$ of ${\mathfrak A}$ such that $P_2 I P_1 = P_2 {\mathfrak A} P_1$, but $P_1 I P_2 \subsetneq P_1 {\mathfrak A} P_2$. Then ${\mathfrak A}$ is not amenable.
{\widehat{\mathrm{env}}}nd{proposition}
\begin{proof}
For $j,k \in \{1,2\}$, set
\[
{\mathfrak A}_{j,k} := P_j {\mathfrak A} P_k \qquad\text{and}\qquad I_{j,k} := P_j I P_k.
\]
It follows that
\[
{\mathfrak A} \cong \begin{bmatrix} {\mathfrak A}_{1,1} & {\mathfrak A}_{1,2} \\ {\mathfrak A}_{2,1} & {\mathfrak A}_{2,2} {\widehat{\mathrm{env}}}nd{bmatrix}
\qquad\text{and}\qquad
I \cong \begin{bmatrix} I_{1,1} & I_{1,2} \\ I_{2,1} & I_{2,2} {\widehat{\mathrm{env}}}nd{bmatrix}
\]
and thus
\[
{\mathfrak A} / I \cong \begin{bmatrix} {\mathfrak A}_{1,1} / I_{1,1} & {\mathfrak A}_{1,2} / I_{1,2} \\ {\mathfrak A}_{2,1} / I_{2,1} & {\mathfrak A}_{2,2} / I_{2,2} {\widehat{\mathrm{env}}}nd{bmatrix}
= \begin{bmatrix} {\mathfrak A}_{1,1} / I_{1,1} & {\mathfrak A}_{1,2} / I_{1,2} \\ 0 & {\mathfrak A}_{2,2} / I_{2,2} {\widehat{\mathrm{env}}}nd{bmatrix}.
\]
Consequently, $\left[ \begin{smallmatrix} 0 & {\mathfrak A}_{1,2} / I_{1,2} \\ 0 & 0 {\widehat{\mathrm{env}}}nd{smallmatrix} \right]$ is a non-zero, complemented, nilpotent ideal of ${\mathfrak A}/I$, which is impossible if ${\mathfrak A}/I$ is amenable.
{\widehat{\mathrm{env}}}nd{proof}
\begin{remark}
The idea behind Proposition {{\mathrm{op}}eratorname{Re}}f{noamprop} is implicitly already contained in \cite[Question 4]{Gro}, where it is attributed to G.\ A.\ Willis. It can be used to establish the non-amenability of ${\cal B}(E {\mathrm{op}}lus F)$ if ${\cal B}(F,E) = {\cal K}(F,E)$, but ${\cal B}(E,F) \subsetneq {\cal K}(E,F)$; this applies, for instance, to ${\cal B}({\widehat{\mathrm{env}}}ll^p {\mathrm{op}}lus {\widehat{\mathrm{env}}}ll^q)$ with $1 \leq p < q < \infty$ (\cite[Proposition 2.c.3]{LT}).
{\widehat{\mathrm{env}}}nd{remark}
\par
We conclude this section with the discussion of a stronger variant of amenability also introduced by Johnson (\cite{Joh3}).
\par
Given a Banach algebra ${\mathfrak A}$, let $\Sigma$ denote the flip map on ${\mathfrak A} \hat{\otimes} {\mathfrak A}$, i.e., $\Sigma(a \otimes b) = b \otimes a$ for $a,b \in {\mathfrak A}$. An element $\boldsymbol{a} \in {\mathfrak A} \otimes {\mathfrak A}$ is called {\widehat{\mathrm{env}}}mph{symmetric} if $\Sigma \boldsymbol{a} = \boldsymbol{a}$; somewhat abusing terminology, we will also call a net in ${\mathfrak A} \otimes {\mathfrak A}$ symmetric if it consists of symmetric elements of ${\mathfrak A} \hat{\otimes} {\mathfrak A}$.
\begin{definition}
A Banach algebra is called {\widehat{\mathrm{env}}}mph{symmetrically amenable} if it has a symmetric approximate diagonal.
{\widehat{\mathrm{env}}}nd{definition}
\begin{remark}
The group algebra $L^1(G)$ of a locally compact group $G$ is symmetrically amenable if and only if it is amenable (\cite[Theorem 4.1]{Joh3}) whereas the Cuntz algebras ${\cal O}_n$ for $n \in {{\mathbb M}hbb N}$, $n \geq 2$ (\cite{Cun}) are amenable, but not symmetrically amenable (\cite[p.\ 457]{Joh3}).
{\widehat{\mathrm{env}}}nd{remark}
\par
In view of how difficult it is, even for very well behaved Banach spaces $E$, to show that ${\cal B}(E)$ is not amenable, it is somewhat surprising to see how easily the correspoding question for symmetric amenability can be settled in the negative for a large class of Banach spaces:
\begin{proposition}
Let $E$ be a Banach space such that $E \cong E {\mathrm{op}}lus E$. Then ${\cal B}(E)$ is not symmetrically amenable.
{\widehat{\mathrm{env}}}nd{proposition}
\begin{proof}
Assume that ${\cal B}(E)$ is symmetrically amenable. By \cite[Corollary 2.5]{Joh3}, there is $\phi \in {\cal B}(E)^\ast$ such that
${{\mathrm{op}}eratorname{lan}}gle {\mathrm{id}}_E, \phi {{\mathrm{op}}eratorname{ran}}gle = 1$ and ${{\mathrm{op}}eratorname{lan}}gle ST, \phi {{\mathrm{op}}eratorname{ran}}gle = {{\mathrm{op}}eratorname{lan}}gle TS, \phi {{\mathrm{op}}eratorname{ran}}gle$ for $S,T \in {\cal B}(E)$.
\par
For $j=1,2$, let $P_j \!: E {\mathrm{op}}lus E \to E$ be the projection onto the $j$-th summand. Since $E \cong E {\mathrm{op}}lus E$, there are $U_j,V_j \in {\cal B}(E)$ with
\[
U_j V_j = {\mathrm{id}}_E \quad\text{and}\quad V_j U_j = P_j \qquad (j=1,2).
\]
It follows that
\[
1 = {{\mathrm{op}}eratorname{lan}}gle {\mathrm{id}}_E,\phi {{\mathrm{op}}eratorname{ran}}gle = {{\mathrm{op}}eratorname{lan}}gle P_1 + P_2 , \phi {{\mathrm{op}}eratorname{ran}}gle
= {{\mathrm{op}}eratorname{lan}}gle V_1 U_1, \phi {{\mathrm{op}}eratorname{ran}}gle + {{\mathrm{op}}eratorname{lan}}gle V_2 U_2, \phi {{\mathrm{op}}eratorname{ran}}gle
= {{\mathrm{op}}eratorname{lan}}gle U_1 V_1 , \phi {{\mathrm{op}}eratorname{ran}}gle + {{\mathrm{op}}eratorname{lan}}gle U_2 V_2 , \phi {{\mathrm{op}}eratorname{ran}}gle
= 2 {{\mathrm{op}}eratorname{lan}}gle {\mathrm{id}}_E,\phi {{\mathrm{op}}eratorname{ran}}gle = 2,
\]
which is nonsense.
{\widehat{\mathrm{env}}}nd{proof}
\section{Amenability of ${\cal B}({\widehat{\mathrm{env}}}ll^p)$ and ${\widehat{\mathrm{env}}}ll^\infty({\cal K}({\widehat{\mathrm{env}}}ll^p))$}
We begin with establishing some notation, part of which was already used in the introduction.
\par
Let ${\mathbb M}hbb I$ be any index set, and let $( E_i )_{i \in {\mathbb M}hbb I}$ be a family of Banach spaces; we write $\prod_{i \in {\mathbb M}hbb I} E_i$ for its Cartesian product. For $p \in [1,\infty)$, we set
\[
\text{${\widehat{\mathrm{env}}}ll^p$-}\bigoplus_{i \in {\mathbb M}hbb I} E_i
:= \left\{ (x_i)_{i \in {\mathbb M}hbb I} \in \prod_{i \in {\mathbb M}hbb I} E_i : \sum_{i \in {\mathbb M}hbb I} \| x_i \|^p < \infty \right\};
\]
it is a linear space which becomes a Banach space if equipped with the norm
\[
\| (x_i)_{i \in {\mathbb M}hbb I} \|_p := \left( \sum_{i \in {\mathbb M}hbb I} \| x_i \|^p \right)^\frac{1}{p}
\qquad \left( (x_i)_{i \in {\mathbb M}hbb I} \in \text{${\widehat{\mathrm{env}}}ll^p$-}\bigoplus_{i \in {\mathbb M}hbb I} E_i \right).
\]
Furthermore, we define
\[
\text{${\widehat{\mathrm{env}}}ll^\infty$-}\bigoplus_{i \in {\mathbb M}hbb I} E_i :=
\left\{ (x_i)_{i \in {\mathbb M}hbb I} \in \prod_{i \in {\mathbb M}hbb I} E_i : \sup_{i \in {\mathbb M}hbb I} \| x_i \| < \infty \right\};
\]
it, too, becomes a Banach space with the norm
\[
\| (x_i)_{i \in {\mathbb M}hbb I} \|_\infty := \sup_{i \in {\mathbb M}hbb I} \| x_i \| \qquad
\left( (x_i)_{i \in {\mathbb M}hbb I} \in \text{${\widehat{\mathrm{env}}}ll^\infty$-}\bigoplus_{i \in {\mathbb M}hbb I} E_i \right).
\]
We use $\text{$c_0$-}\bigoplus_{i \in {\mathbb M}hbb I} E_i$ to denote the closure of those $( x_i )_{i \in {\mathbb M}hbb I} \in \text{${\widehat{\mathrm{env}}}ll^\infty$-}\bigoplus_{i \in {\mathbb M}hbb I} E_i$ for which $x_i = 0$ for all but finitely many $i \in {\mathbb M}hbb I$. We note that, if $( {\mathfrak A}_i )_{i \in {\mathbb M}hbb I}$ is a family of Banach algebras, then $\text{${\widehat{\mathrm{env}}}ll^\infty$-}\bigoplus_{i \in {\mathbb M}hbb I} {\mathfrak A}_i$ is a Banach algebra (which contains $\text{$c_0$-}\bigoplus_{i \in {\mathbb M}hbb I} {\mathfrak A}_i$ as a closed ideal). If $E_i = E$ for all $i \in {\mathbb M}hbb I$, we simply write ${\widehat{\mathrm{env}}}ll^p({\mathbb M}hbb{I},E)$ or $c_0({\mathbb M}hbb{I},E)$ instead of $\text{${\widehat{\mathrm{env}}}ll^p$-}\bigoplus_{i \in {\mathbb M}hbb I} E$ and $\text{$c_0$-}\bigoplus_{i \in {\mathbb M}hbb I} E$, respectively. We apply the usual conventions: if $E_i = {{\mathbb M}hbb C}$ for all $i \in {\mathbb M}hbb I$ or ${\mathbb M}hbb{I} = {{\mathbb M}hbb N}$, we suppress the symbol for the space or the index set, respectively. For instance, if $p \in [1,\infty]$ and $E$ is any Banach space, then ${\widehat{\mathrm{env}}}ll^p(E)$ stands for ${\widehat{\mathrm{env}}}ll^p({{\mathbb M}hbb N},E)$, and if ${\mathbb M}hbb{I}$ is any index set, then $c_0({\mathbb M}hbb{I})$ means $c_0({\mathbb M}hbb{I},{{\mathbb M}hbb C})$. Also, we write ${\widehat{\mathrm{env}}}ll^p_n$ instead of ${\widehat{\mathrm{env}}}ll^p(\{ 1, \ldots, n \},{{\mathbb M}hbb C})$. Finally, for $i \in {\mathbb M}hbb I$, we let $\delta_i \!: {\mathbb M}hbb{I} \to {{\mathbb M}hbb C}$ denote the point mass at $i$; it is clear that $\delta_i \in {\widehat{\mathrm{env}}}ll^p({\mathbb M}hbb{I})$ for any $p \in [1,\infty]$.
\par
Given any Banach space $E$, we have isometric isomorphisms between ${\widehat{\mathrm{env}}}ll^p({\widehat{\mathrm{env}}}ll^p(E)) = {\widehat{\mathrm{env}}}ll^p({{\mathbb M}hbb N}^2,E)$ and ${\widehat{\mathrm{env}}}ll^p(E)$ for $p \in [1,\infty)$ and between $c_0(c_0(E)) = c_0({{\mathbb M}hbb N}^2,E)$ and $c_0(E)$ (simply due to the fact that ${{\mathbb M}hbb N}$ and ${{\mathbb M}hbb N}^2$ have the same cardinality). This simple observation lies at the heart of the proof of our first theorem:
\begin{theorem} \label{thm1}
Let $E$ be a Banach space. Then:
\begin{items}
\item for $p \in [1,\infty)$, the Banach algebra ${\cal B}({\widehat{\mathrm{env}}}ll^p(E))$ is amenable if and only if ${\widehat{\mathrm{env}}}ll^\infty({\cal B}({\widehat{\mathrm{env}}}ll^p(E)))$ is amenable;
\item ${\cal B}(c_0(E))$ is amenable if and only if ${\widehat{\mathrm{env}}}ll^\infty({\cal B}(c_0(E)))$ is amenable.
{\widehat{\mathrm{env}}}nd{items}
{\widehat{\mathrm{env}}}nd{theorem}
\begin{proof}
We only prove (i) ((ii) is proven analogously).
\par
Set ${\mathfrak A} := {\widehat{\mathrm{env}}}ll^\infty({\cal B}({\widehat{\mathrm{env}}}ll^p(E)))$. It is elementary that ${\cal B}({\widehat{\mathrm{env}}}ll^p(E))$ is amenable if ${\mathfrak A}$ is (\cite[Corollary 2.3.2]{LoA}).
\par
For the converse, suppose that ${\cal B}({\widehat{\mathrm{env}}}ll^p(E))$ is amenable, and let $( \boldsymbol{d}_\alpha )_{\alpha \in {\mathbb M}hbb A}$ be an approximate diagonal for it; we may suppose that $\Delta \boldsymbol{d}_\alpha = {\mathrm{id}}_{{\widehat{\mathrm{env}}}ll^p(E)}$ for all $\alpha \in {\mathbb M}hbb A$.
\par
First, observe that we can identify ${\mathfrak A}$ with the block diagonal matrices in ${\cal B}({\widehat{\mathrm{env}}}ll^p({\widehat{\mathrm{env}}}ll^p(E)))$. For $n \in {{\mathbb M}hbb N}$, let $P_n \!: {\widehat{\mathrm{env}}}ll^p({\widehat{\mathrm{env}}}ll^p(E)) \to {\widehat{\mathrm{env}}}ll^p(E)$ denote the projection onto the $n$-th coordinate. Define
\[
{\cal Q} \!: {\cal B}({\widehat{\mathrm{env}}}ll^p({\widehat{\mathrm{env}}}ll^p(E))) \to {\cal B}({\widehat{\mathrm{env}}}ll^p({\widehat{\mathrm{env}}}ll^p(E))), \quad T \mapsto \sum_{n=1}^\infty P_n T P_n,
\]
where the infinite series converges in the strong operator topology ${\mathbb M}hrm{SOT}$. Then $\cal Q$ is a projection onto ${\mathfrak A}$.
\par
Since we have ${\widehat{\mathrm{env}}}ll^p({\widehat{\mathrm{env}}}ll^p(E)) \cong {\widehat{\mathrm{env}}}ll^p(E)$, there are bounded sequences $(U_n)_{n=1}^\infty$ and $( V_n )_{n=1}^\infty$ in ${\cal B}({\widehat{\mathrm{env}}}ll^p({\widehat{\mathrm{env}}}ll^p(E)))$ such that
\[
U_n V_n = {\mathrm{id}}_{{\widehat{\mathrm{env}}}ll^p({\widehat{\mathrm{env}}}ll^p(E))} \quad\text{and}\quad V_n U_n = P_n \qquad (n \in {{\mathbb M}hbb N})
\]
and, consequently,
\begin{equation} \label{UV}
V_n = P_n V_n \quad\text{and}\quad U_n = U_n P_n \qquad (n \in {{\mathbb M}hbb N}).
{\widehat{\mathrm{env}}}nd{equation}
Define
\begin{align*}
{\cal Q}_L \!: {\cal B}({\widehat{\mathrm{env}}}ll^p({\widehat{\mathrm{env}}}ll^p(E))) \to {\cal B}({\widehat{\mathrm{env}}}ll^p({\widehat{\mathrm{env}}}ll^p(E))), & \quad T \mapsto \sum_{n=1}^\infty P_n T U_n \\
\intertext{and}
{\cal Q}_R \!: {\cal B}({\widehat{\mathrm{env}}}ll^p({\widehat{\mathrm{env}}}ll^p(E))) \to {\cal B}({\widehat{\mathrm{env}}}ll^p({\widehat{\mathrm{env}}}ll^p(E))), & \quad T \mapsto \sum_{n=1}^\infty V_n T P_n,
{\widehat{\mathrm{env}}}nd{align*}
where again the series are convergent in the strong operator topology. It is obvious that ${\cal Q}_L$ is a left and ${\cal Q}_R$ a right ${\mathfrak A}$-module homomorphism, and with ({{\mathrm{op}}eratorname{Re}}f{UV}) in mind, it is easy to see that both ${\cal Q}_L$ and ${\cal Q}_R$ attain their values in ${\mathfrak A}$.
\par
Let $S,T \in {\cal B}({\widehat{\mathrm{env}}}ll^p({\widehat{\mathrm{env}}}ll^p(E)))$. Since multiplication is jointly continuous on norm bounded subsets with respect to ${\mathbb M}hrm{SOT}$, we have
\[
\begin{split}
({\cal Q}_L S) ({\cal Q}_R T) & = \text{${\mathbb M}hrm{SOT}$-}\lim_{N\to\infty}
\left( \sum_{n=1}^N P_n S U_n \right)\left( \sum_{n=1}^N V_n T P_n \right) \\
& = \text{${\mathbb M}hrm{SOT}$-}\lim_{N\to\infty} \sum_{n,m=1}^N P_n S U_n V_m T P_m \\
& = \text{${\mathbb M}hrm{SOT}$-}\lim_{N\to\infty} \sum_{n=1}^N P_n S U_n V_n T P_n \\
& = \text{${\mathbb M}hrm{SOT}$-}\lim_{N\to\infty} \sum_{n=1}^N P_n ST P_n \\
& = {\cal Q}(ST).
{\widehat{\mathrm{env}}}nd{split}
\]
It follows that
\begin{equation} \label{Q}
\Delta \circ ({\cal Q}_L \otimes {\cal Q}_R) = {\cal Q} \circ \Delta.
{\widehat{\mathrm{env}}}nd{equation}
\par
Identifying ${\cal B}({\widehat{\mathrm{env}}}ll^p(E))$ and ${\cal B}({\widehat{\mathrm{env}}}ll^p({\widehat{\mathrm{env}}}ll^p(E))$, we claim that $( ({\cal Q}_L \otimes {\cal Q}_R) \boldsymbol{d}_\alpha )_{\alpha \in {\mathbb M}hbb A}$ is an approximate diagonal for ${\mathfrak A}$. Clearly, the net is bounded in ${\mathfrak A} \hat{\otimes} {\mathfrak A}$. Since ${\cal Q}_L \otimes {\cal Q}_R$ is an ${\mathfrak A}$-bimodule homomorphism, it follows that ({{\mathrm{op}}eratorname{Re}}f{diag1}) holds. Finally, since
\[
\Delta(({\cal Q}_L \otimes {\cal Q}_R) \boldsymbol{d}_\alpha) = {\cal Q}(\Delta \boldsymbol{d}_\alpha ) =
{\cal Q}\left({\mathrm{id}}_{{\widehat{\mathrm{env}}}ll^p({\widehat{\mathrm{env}}}ll^p(E))}\right) = {\mathrm{id}}_{{\widehat{\mathrm{env}}}ll^p({\widehat{\mathrm{env}}}ll^p(E))}
\]
by ({{\mathrm{op}}eratorname{Re}}f{Q}), condition ({{\mathrm{op}}eratorname{Re}}f{diag2}) holds as well.
{\widehat{\mathrm{env}}}nd{proof}
\par
Specializing to $E = {{\mathbb M}hbb C}$ yields:
\begin{corollary} \label{cor1}
Let $p \in (1,\infty)$ be such that ${\cal B}({\widehat{\mathrm{env}}}ll^p)$ is amenable. Then ${\widehat{\mathrm{env}}}ll^\infty({\cal B}({\widehat{\mathrm{env}}}ll^p))$ and ${\widehat{\mathrm{env}}}ll^\infty({\cal K}({\widehat{\mathrm{env}}}ll^p))$ are both amenable.
{\widehat{\mathrm{env}}}nd{corollary}
\begin{proof}
The claim for ${\widehat{\mathrm{env}}}ll^\infty({\cal B}({\widehat{\mathrm{env}}}ll^p))$ is an immediate consequence of Theorem {{\mathrm{op}}eratorname{Re}}f{thm1}. Since ${\cal K}({\widehat{\mathrm{env}}}ll^p)$ has a bounded approximate identity, so does ${\widehat{\mathrm{env}}}ll^\infty({\cal K}({\widehat{\mathrm{env}}}ll^p))$. Since ${\widehat{\mathrm{env}}}ll^\infty({\cal K}({\widehat{\mathrm{env}}}ll^p))$ is a closed ideal of the amenable Banach algebra ${\widehat{\mathrm{env}}}ll^\infty({\cal B}({\widehat{\mathrm{env}}}ll^p))$, it is amenable by \cite[Proposition 2.3.3]{LoA}.
{\widehat{\mathrm{env}}}nd{proof}
\begin{remarks}
\item The analogous statement of Corollary {{\mathrm{op}}eratorname{Re}}f{cor1} for ${\widehat{\mathrm{env}}}ll^1$ and $c_0$ is also true. Since, however, ${\cal B}({\widehat{\mathrm{env}}}ll^1)$ and ${\cal B}(c_0)$ are known to be not amenable by \cite{Oza}, it would be somewhat pointless to formulate it.
\item Even though ${\cal B}({\widehat{\mathrm{env}}}ll^1)$ and ${\cal B}(c_0)$ are not amenable, it seems to be unknown whether ${\widehat{\mathrm{env}}}ll^\infty({\cal K}({\widehat{\mathrm{env}}}ll^1))$ and ${\widehat{\mathrm{env}}}ll^\infty({\cal K}(c_0))$ are.
\item It is known that ${\widehat{\mathrm{env}}}ll^\infty({\cal K}({\widehat{\mathrm{env}}}ll^2))$ is not amenable (see \cite{LLW}), but proving it is at about the same level of difficulty as a proof for the non-amenability of ${\cal B}({\widehat{\mathrm{env}}}ll^2)$.
{\widehat{\mathrm{env}}}nd{remarks}
\par
We shall thus, from now on, focus on the (non-)amenability of ${\widehat{\mathrm{env}}}ll^\infty({\cal K}({\widehat{\mathrm{env}}}ll^p))$ instead of that of ${\cal B}({\widehat{\mathrm{env}}}ll^p)$.
\section{Ultra-amenability of ${\cal K}({\widehat{\mathrm{env}}}ll^p)$}
Let $E$ be a Banach space, let ${\mathbb M}hbb I$ be an index set, and let $\cal U$ be an ultrafilter over ${\mathbb M}hbb I$. We let
\[
{\cal N}_{\cal U} := \left\{ (x_i)_{i \in {\mathbb M}hbb I} : \lim_{i \in \cal U} \| x_i \| = 0 \right\}.
\]
It is immediate that ${\cal N}_{\cal U}$ is a closed subspace of ${\widehat{\mathrm{env}}}ll^\infty({\mathbb M}hbb{I},E)$. The quotient space ${\widehat{\mathrm{env}}}ll^\infty({\mathbb M}hbb{I},E)/ {\cal N}_{\cal U}$ is called the {\widehat{\mathrm{env}}}mph{ultrapower of $E$ with respect to $\cal U$}; we denote it by $(E)_{\cal U}$. Whenever $(x_i)_{i \in {\mathbb M}hbb I} \in {\widehat{\mathrm{env}}}ll^\infty({\mathbb M}hbb{I},E)$, we write $( x_i )_{\cal U}$ for its equivalence class in $\cal U$. For further material on ultrapowers, we refer to the survey article \cite{Hei} and the somewhat more detailed treatment in \cite{Sim}.
\par
If ${\mathfrak A}$ is a Banach algebra, then it is straightforward that $({\mathfrak A})_{\cal U}$ is again a Banach algebra. The following definition is due to the first author (\cite{Daw}):
\begin{definition} \label{uldef}
A Banach algebra ${\mathfrak A}$ is said to be {\widehat{\mathrm{env}}}mph{ultra-amenable} if $({\mathfrak A})_{\cal U}$ is amenable for every ultrafilter $\cal U$.
{\widehat{\mathrm{env}}}nd{definition}
\begin{remark}
Ultra-amenability implies amenability (\cite[Corollary 5.5]{Daw}), but is much stron\-ger: a ${C^\ast}$-algebra is ultra-amenable if and only if it is subhomogeneous (\cite[Theorem 5.7]{Daw}) and ${\widehat{\mathrm{env}}}ll^1(G)$, for a discrete group $G$, is ultra-amenable if and only if $G$ is finite (\cite[Theorem 5.11]{Daw}).
{\widehat{\mathrm{env}}}nd{remark}
\par
Suppose that ${\cal B}({\widehat{\mathrm{env}}}ll^p)$ is amenable for some $p \in (1,\infty)$. Then ${\widehat{\mathrm{env}}}ll^\infty({\cal K}({\widehat{\mathrm{env}}}ll^p))$ is amenable by Corollary {{\mathrm{op}}eratorname{Re}}f{cor1}, so that its quotient $({\cal K}({\widehat{\mathrm{env}}}ll^p))_{\cal U}$ is amenable for every ultrafilter $\cal U$ over ${{\mathbb M}hbb N}$. Alas, this does not allow us (yet) to say that ${\cal K}({\widehat{\mathrm{env}}}ll^p)$ is ultra-amenable because Definition {{\mathrm{op}}eratorname{Re}}f{uldef} requires us to consider ultrafilters over arbitrary index sets.
\par
Nevertheless, the amenability of ${\widehat{\mathrm{env}}}ll^\infty({\cal K}({\widehat{\mathrm{env}}}ll^p))$ allows us to conclude the ultra-amenability of ${\cal K}({\widehat{\mathrm{env}}}ll^p)$ by virtue of the following theorem:
\begin{theorem} \label{thm2}
The following are equivalent for a {\widehat{\mathrm{env}}}mph{separable} Banach algebra ${\mathfrak A}$:
\begin{items}
\item ${\widehat{\mathrm{env}}}ll^\infty({\mathfrak A})$ is amenable;
\item ${\widehat{\mathrm{env}}}ll^\infty({\mathbb M}hbb{I},{\mathfrak A})$ is amenable for every index set ${\mathbb M}hbb I$;
\item ${\mathfrak A}$ is ultra-amenable.
{\widehat{\mathrm{env}}}nd{items}
{\widehat{\mathrm{env}}}nd{theorem}
\par
For the proof, recall the following definitions from \cite{Daw}. Let ${\mathfrak A}$ be a Banach algebra, and let $n \in {{\mathbb M}hbb N}$. Then:
\begin{itemize}
\item let $S_n({\mathfrak A})$ denote the collection of all subsets of the unit sphere of ${\mathfrak A}$ of cardinality $n$;
\item for $C \geq 1$ and ${\widehat{\mathrm{env}}}psilon > 0$, let $D_n({\mathfrak A},C,{\widehat{\mathrm{env}}}psilon)$ consist of those $A \subset S_n({\mathfrak A})$ such that there is a sequence $( t_k )_{k=1}^\infty$ in $[0,\infty)$ with $\sum_{k=1}^\infty t_k \leq C$ with the property that, for each $S \in A$, there are sequences $( a_k )_{k=1}^\infty$ and $( b_k )_{k=1}^\infty$ in ${\mathfrak A}$ with $\| a_k \| \| b_k \| \leq t_k$ for $k \in {{\mathbb M}hbb N}$, so that
\[
\boldsymbol{d} := \sum_{k=1}^\infty a_k \otimes b_k \in {\mathfrak A} \hat{\otimes} {\mathfrak A},
\]
and
\[
\| a \cdot \boldsymbol{d} - \boldsymbol{d} \cdot a \| \leq {\widehat{\mathrm{env}}}psilon
\quad\text{and}\quad
\| \Delta_{\mathfrak A}(\boldsymbol{d}) a - a \| \leq {\widehat{\mathrm{env}}}psilon \qquad (a \in F).
\]
{\widehat{\mathrm{env}}}nd{itemize}
\begin{lemma} \label{ullem}
For a Banach algebra ${\mathfrak A}$ consider the following statements:
\begin{items}
\item there is $C \geq 1$ such that $S_n({\mathfrak A}) \in D_n({\mathfrak A},C,{\widehat{\mathrm{env}}}psilon)$ for each $n \in {{\mathbb M}hbb N}$ and ${\widehat{\mathrm{env}}}psilon > 0$;
\item ${\widehat{\mathrm{env}}}ll^\infty({\mathbb M}hbb{I},{\mathfrak A})$ is amenable for each index set ${\mathbb M}hbb I$;
\item ${\widehat{\mathrm{env}}}ll^\infty({\mathfrak A})$ is amenable.
{\widehat{\mathrm{env}}}nd{items}
Then
\[
{\widehat{\mathrm{env}}}mph{(i)} {{\mathrm{op}}eratorname{Im}}plies {\widehat{\mathrm{env}}}mph{(ii)} {{\mathrm{op}}eratorname{Im}}plies {\widehat{\mathrm{env}}}mph{(iii)},
\]
and {\widehat{\mathrm{env}}}mph{(iii)} implies {\widehat{\mathrm{env}}}mph{(i)} if ${\mathfrak A}$ is separable.
{\widehat{\mathrm{env}}}nd{lemma}
\begin{proof}
(i) $\Longrightarrow$ (ii) is routine in view of the definition of $D_n({\mathfrak A},C,{\widehat{\mathrm{env}}}psilon)$, and (ii) $\Longrightarrow$ (iii) is trivial.
\par
Suppose that (iii) holds and that ${\mathfrak A}$ is separable. Let $C \geq 1$ be such that ${\mathfrak A}$ is $C$-amenable, let $n \in {{\mathbb M}hbb N}$, and let ${\widehat{\mathrm{env}}}psilon > 0$.
\par
Define a metric $d$ on $S_n({\mathfrak A})$ by letting
\[
d(A,B) := \max_{a \in A} \min_{b \in B} \| a - b \| + \max_{b \in B} \min_{a \in A} \| a - b \|\qquad (A,B \in S_n({\mathfrak A})).
\]
The separability of ${\mathfrak A}$ implies that the metric space $(S_n({\mathfrak A}),d)$ is separable and so contains a dense, countable subset, say $\{ A_1, A_2, \ldots \}$. For each $k \in {{\mathbb M}hbb N}$, let $A_k = \left\{ a^{(k)}_1, \ldots, a^{(k)}_n \right\}$. For $j =1, \ldots, n$, set $a_j := \left( a^{(k)}_j \right)_{k=1}^\infty \in {\widehat{\mathrm{env}}}ll^\infty({\mathfrak A})$. Let $( b_k )_{k=1}^\infty$ and $( c_k )_{k=1}^\infty$ be sequences in ${\widehat{\mathrm{env}}}ll^\infty({\mathfrak A})$ with $\sum_{k=1}^\infty \| b_k \| \| c_k \| \leq C$ such that, for $\boldsymbol{d} := \sum_{k=1}^\infty b_k \otimes c_k \in {\widehat{\mathrm{env}}}ll^\infty({\mathfrak A}) \hat{\otimes} {\widehat{\mathrm{env}}}ll^\infty({\mathfrak A})$, we have
\begin{equation} \label{diag3}
\| a_j \cdot \boldsymbol{d} - \boldsymbol{d} \cdot a_j \| \leq \frac{{\widehat{\mathrm{env}}}psilon}{2}
{\widehat{\mathrm{env}}}nd{equation}
and
\begin{equation} \label{diag4}
\| \Delta_{{\widehat{\mathrm{env}}}ll^\infty({\mathfrak A})} ( \boldsymbol{d}) a_j - a_j \| \leq \frac{{\widehat{\mathrm{env}}}psilon}{2}
{\widehat{\mathrm{env}}}nd{equation}
for $j =1, \ldots, n$. (The existence of such $( b_k )_{k=1}^\infty$ and $( c_k )_{k=1}^\infty$ follows from the $C$-amenability of ${\widehat{\mathrm{env}}}ll^\infty({\mathfrak A})$.)
\par
For each $k \in {{\mathbb M}hbb N}$, let $b_k = \left( b^{(k)}_\nu \right)_{\nu=1}^\infty$ and $c_k = \left( c^{(k)}_\nu \right)_{\nu=1}^\infty$. Then ({{\mathrm{op}}eratorname{Re}}f{diag4}) yields that
\begin{equation} \label{diag5}
\sup_{\nu \in {{\mathbb M}hbb N}} \left\| \sum_{k=1}^\infty b^{(k)}_\nu c^{(k)}_\nu a_j^{(\nu)} - a_j^{(\nu)} \right\| \leq \frac{{\widehat{\mathrm{env}}}psilon}{2}
\qquad (j = 1, \ldots, n).
{\widehat{\mathrm{env}}}nd{equation}
For $\nu \in {{\mathbb M}hbb N}$, let $P_\nu \!: {\widehat{\mathrm{env}}}ll^\infty({\mathfrak A}) \to {\mathfrak A}$ be the projection onto the $\nu$-th coordinate, and note that $P_\nu \otimes P_\nu \!: {\widehat{\mathrm{env}}}ll^\infty({\mathfrak A}) \hat{\otimes} {\widehat{\mathrm{env}}}ll^\infty({\mathfrak A}) \to {\mathfrak A} \hat{\otimes} {\mathfrak A}$ is a contraction. From ({{\mathrm{op}}eratorname{Re}}f{diag3}), we conclude that
\begin{equation} \label{diag6}
\sup_{\nu \in {{\mathbb M}hbb N}} \left\| \sum_{k=1}^\infty a_j^{(\nu)} b^{(k)}_\nu \otimes c^{(k)}_\nu -
\sum_{k=1}^\infty b^{(k)}_\nu \otimes c^{(k)}_\nu a_j^{(\nu)} \right\| \leq \frac{{\widehat{\mathrm{env}}}psilon}{2}
\qquad (j = 1, \ldots, n).
{\widehat{\mathrm{env}}}nd{equation}
Finally, it is straightforward that
\[
\sum_{k=1}^\infty \sup_{\nu \in {{\mathbb M}hbb N}} \left\| b_\nu^{(k)} \right\|\left\| b_\nu^{(k)} \right\| \leq
\sum_{k=1}^\infty \| b_k \| \| c_k \| \leq C.
\]
\par
Let $A \in S_n({\mathfrak A})$ be arbitrary. Since $\{ A_1, A_2, \ldots \}$ is dense in $S_n({\mathfrak A})$, there is $\nu \in {{\mathbb M}hbb N}$ such that $d(A,A_\nu) \leq \frac{{\widehat{\mathrm{env}}}psilon}{4C}$. With $A_\nu = \left\{ a^{(\nu)}_1, \ldots, a^{(\nu)}_n \right\}$, this means that, for any $a \in A$, there is $j \in \{1, \ldots, n\}$ such that $\left\| a_j^{(\nu)} - a \right\| \leq \frac{{\widehat{\mathrm{env}}}psilon}{4C}$. From ({{\mathrm{op}}eratorname{Re}}f{diag6}), we infer that
\[
\begin{split}
\lefteqn{\left\| \sum_{k=1}^\infty a b^{(k)}_\nu \otimes c^{(k)}_\nu - \sum_{k=1}^\infty b^{(k)}_\nu \otimes c^{(k)}_\nu a \right\|}
& \\
& \leq 2 \left\| a - a_j^{(\nu)} \right\| \sum_{k=1}^\infty \left\| b_\nu^{(k)} \right\|\left\| b_\nu^{(k)} \right\| +
\left\| \sum_{k=1}^\infty a_j^{(\nu)} b^{(k)}_\nu \otimes c^{(k)}_\nu -
\sum_{k=1}^\infty b^{(k)}_\nu \otimes c^{(k)}_\nu a_j^{(\nu)} \right\| \\
& \leq \frac{{\widehat{\mathrm{env}}}psilon}{4C} 2C + \frac{{\widehat{\mathrm{env}}}psilon}{2} \\
& = {\widehat{\mathrm{env}}}psilon
{\widehat{\mathrm{env}}}nd{split}
\]
and from ({{\mathrm{op}}eratorname{Re}}f{diag5}) that
\[
\begin{split}
\lefteqn{\left\| \sum_{k=1}^\infty b^{(k)}_\nu c^{(k)}_\nu a - a \right\|} & \\
& \leq \left\| a - a_j^{(\nu)} \right\| +
\left\| a - a_j^{(\nu)} \right\|\sum_{k=1}^\infty \left\| b_\nu^{(k)} \right\|\left\| b_\nu^{(k)} \right\| +
\left\| \sum_{k=1}^\infty b^{(k)}_\nu c^{(k)}_\nu a_j^{(\nu)} - a_j^{(\nu)} \right\| \\
& \leq \frac{{\widehat{\mathrm{env}}}psilon}{4C} (1+C) + \frac{{\widehat{\mathrm{env}}}psilon}{2} \\
& = {\widehat{\mathrm{env}}}psilon.
{\widehat{\mathrm{env}}}nd{split}
\]
\par
All in all, we have established that $S_n({\mathfrak A}) \in D_n({\mathfrak A},C,{\widehat{\mathrm{env}}}psilon)$.
{\widehat{\mathrm{env}}}nd{proof}
\begin{proof}[Proof of Theorem {\widehat{\mathrm{env}}}mph{{{\mathrm{op}}eratorname{Re}}f{thm2}}]
By Lemma {{\mathrm{op}}eratorname{Re}}f{ullem}, (i) $\Longleftrightarrow$ (ii) holds, and (ii) $\Longrightarrow$ (iii) is trivial.
\par
We postpone the actual proof of (iii) $\Longrightarrow$ (i) for some preliminary considerations.
\par
Let $\cal S$ be the set of all sequences in $[0,\infty)$. For $( t_k )_{k=1}^\infty, ( s_k )_{k=1}^\infty \in \cal S$, define $( t_k )_{k=1}^\infty \ll ( s_k )_{k=1}^\infty$ if there are $( r_k )_{k=1}^\infty \in \cal S$, a bijection $\sigma \!: {{\mathbb M}hbb N} \to {{\mathbb M}hbb N}$, and $\nu,\mu \in {{\mathbb M}hbb N}$ such that
\begin{itemize}
\item $s_k = r_{\sigma(k)}$ for $k \in {{\mathbb M}hbb N}$,
\item $r_k = t_k$ for $k < \nu$,
\item $r_\nu + r_{\nu+1} + \cdots + r_{\nu+\mu-1} = t_\nu$, and
\item $r_{k+\mu-1} = t_k$ for $k > \nu$.
{\widehat{\mathrm{env}}}nd{itemize}
(Informally, one might want to say that $( r_k )_{k=1}^\infty$ is obtained from $( t_k )_{k=1}^\infty$ by splitting up one term and from $( s_k )_{k=1}^\infty$ through rearrangement.) It is clear from this definition that, whenever $( t_k )_{k=1}^\infty \ll ( s_k )_{k=1}^\infty$ and one of the sequences lies in ${\widehat{\mathrm{env}}}ll^1$, the so does the other and has the same norm in ${\widehat{\mathrm{env}}}ll^1$. We then define $( t_k )_{k=1}^\infty \preceq ( s_k )_{k=1}^\infty$ if there are $\left( r_k^{(1)} \right)_{k=1}^\infty, \ldots, \left( r_k^{(m)} \right)_{k=1}^\infty \in \cal S$ such that
\[
( t_k )_{k=1}^\infty \ll \left( r_k^{(1)} \right)_{k=1}^\infty \ll \cdots \ll
\left( r_k^{(m)} \right)_{k=1}^\infty \ll ( s_k )_{k=1}^\infty.
\]
Let ${\cal S}_0$ be the collection of all sequences in $\cal S$ that are eventually zero, and note that $({\cal S}_0, \preceq)$ is a directed set.
\par
Let $n \in {{\mathbb M}hbb N}$, let $F \in S_n({\mathfrak A})$, let ${\widehat{\mathrm{env}}}psilon > 0$, and let $( t_k )_{k=1}^\infty \in \cal {\widehat{\mathrm{env}}}ll^1 \cap S$. We call $F$ and $( t_k )_{k=1}^\infty$ {\widehat{\mathrm{env}}}mph{compatible} if there is $\boldsymbol{d} = \sum_{k=1}^\infty a_k \otimes b_k \in {\mathfrak A} \hat{\otimes} {\mathfrak A}$ with $\| a_k \| \| b_k \| \leq t_k$ for $k \in {{\mathbb M}hbb N}$ and
\[
\| a \cdot \boldsymbol{d} - \boldsymbol{d} \cdot a \| \leq {\widehat{\mathrm{env}}}psilon \quad\text{and}\quad
\| \Delta(\boldsymbol{d}) a - a \| \qquad (a \in F).
\]
Note that, if $( t_k )_{k=1}^\infty \preceq ( s_k )_{k=1}^\infty$ and $F$ and $( t_k )_{k=1}^\infty$ are compatible, then so are $F$ and $( s_k )_{k=1}^\infty$.
\par
Suppose now that ${\mathfrak A}$ is ultra-amenable, let $C \geq 1$ be as in \cite[Theorem 5.6]{Daw}, let $n \in {{\mathbb M}hbb N}$, and let ${\widehat{\mathrm{env}}}psilon > 0$. By \cite[Theorem 5.6]{Daw}, there is a partition of $S_n({\mathfrak A})$ into finitely many sets each of which has a compatible sequence in ${\widehat{\mathrm{env}}}ll^1 \cap \cal S$ with ${\widehat{\mathrm{env}}}ll^1$-norm at most $C$. We can suppose that each of these sequences lies in ${\cal S}_0$ and use the directedness of $({\cal S}_0, \preceq)$ to obtain one single sequence in ${\cal S}_0$---still with ${\widehat{\mathrm{env}}}ll^1$-norm at most $C$---that is compatible with all sets in the partition of $S_n({\mathfrak A})$. But then $S_n({\mathfrak A})$ itself and that sequence are compatible, which means that $S_n({\mathfrak A}) \in D_n({\mathfrak A},C,{\widehat{\mathrm{env}}}psilon)$. By Lemma {{\mathrm{op}}eratorname{Re}}f{ullem}, this implies the amenability of ${\widehat{\mathrm{env}}}ll^\infty({\mathfrak A})$.
{\widehat{\mathrm{env}}}nd{proof}
\begin{remark}
The proof of Theorem {{\mathrm{op}}eratorname{Re}}f{thm2} can be modified to yield the following generalization: a Banach algebra ${\mathfrak A}$ of density character $\kappa$ is ultra-amenable if and only if ${\widehat{\mathrm{env}}}ll^\infty({\mathbb M}hbb{I},{\mathfrak A})$ is amenable for any index set ${\mathbb M}hbb I$ and if and only if it is amenable for an index set ${\mathbb M}hbb I$ of cardinality $\kappa$. This closes a gap in the proof of \cite[Theorem 2.5]{LLW}, which claims the equivalence of Theorem {{\mathrm{op}}eratorname{Re}}f{thm2}(ii) and (iii) in the case of a ${C^\ast}$-algebra.
{\widehat{\mathrm{env}}}nd{remark}
\par
As ${\cal K}({\widehat{\mathrm{env}}}ll^p)$ for $p \in (1,\infty)$ and ${\cal K}(c_0)$ are separable, Theorem {{\mathrm{op}}eratorname{Re}}f{thm2} yields:
\begin{corollary} \label{cor2}
Let $E = {\widehat{\mathrm{env}}}ll^p$ with $p \in (1,\infty)$ or $E = c_0$. Then the following are equivalent:
\begin{items}
\item ${\widehat{\mathrm{env}}}ll^\infty({\cal K}(E))$ is amenable;
\item ${\widehat{\mathrm{env}}}ll^\infty({\mathbb M}hbb{I},{\cal K}(E))$ is amenable for every index set ${\mathbb M}hbb I$;
\item ${\cal K}(E)$ is ultra-amenable.
{\widehat{\mathrm{env}}}nd{items}
{\widehat{\mathrm{env}}}nd{corollary}
\section{Amenability of ${\widehat{\mathrm{env}}}ll^\infty({\cal K}(E))$ for ${\cal L}^p$-spaces}
Whether or not the Banach algebras of the form ${\widehat{\mathrm{env}}}ll^\infty({\cal K}(E))$ for a Banach space $E$ are amenable or not seems to have received very little attention in the literature so far. As Corollary {{\mathrm{op}}eratorname{Re}}f{cor1} shows, it is, for $E = {\widehat{\mathrm{env}}}ll^p$ with $p \in [1,\infty)$, intimately linked to the open problem of whether ${\cal B}({\widehat{\mathrm{env}}}ll^p)$ is amenable and thus certainly a question deserving further exploration.
\par
In this section, we show that the (possible) amenability of ${\widehat{\mathrm{env}}}ll^\infty({\cal K}({\widehat{\mathrm{env}}}ll^p))$ entails the amenability of ${\widehat{\mathrm{env}}}ll^\infty({\cal K}(E))$ for a large class of a Banach spaces $E$.
\par
For our first proposition, we denote by ${\cal F}(E,F)$ for two Banach spaces $E$ and $F$ the bounded finite rank operators from $E$ to $F$; as usual, we write ${\cal F}(E)$ as shorthand for ${\cal F}(E,E)$.
\begin{proposition} \label{facam}
Let $E$ and $F$ be Banach spaces such that $E^\ast$ and $F^\ast$ have the bounded approximation property. Suppose further that the following factorization property holds:
\begin{quote}
there is $C \geq 0$ such that, for each $T \in {\cal F}(F)$, there are $S \in {\cal F}(F,E)$ and $R \in {\cal F}(E,F)$ with $\| R \| \| S \| \leq C \| T \|$ and $RS = T$.
{\widehat{\mathrm{env}}}nd{quote}
Then, for any index set ${\mathbb M}hbb I$, the following are equivalent:
\begin{items}
\item ${\widehat{\mathrm{env}}}ll^\infty({\mathbb M}hbb{I},{\cal K}(E))$ is amenable;
\item ${\widehat{\mathrm{env}}}ll^\infty({\mathbb M}hbb{I},{\cal K}(E {\mathrm{op}}lus F))$ is amenable.
{\widehat{\mathrm{env}}}nd{items}
{\widehat{\mathrm{env}}}nd{proposition}
\begin{proof}
Let ${\mathbb M}hbb{I}$ be an index set, and let ${\mathfrak A} := {\widehat{\mathrm{env}}}ll^\infty({\mathbb M}hbb{I},{\cal K}(E {\mathrm{op}}lus F))$. We wish to apply Theorem {{\mathrm{op}}eratorname{Re}}f{bgn}.
\par
As both $E^\ast$ and $F^\ast$ have the bounded approximation property, $(E {\mathrm{op}}lus F)^\ast \cong E^\ast {\mathrm{op}}lus F^\ast$ has it, too. Consequently, ${\cal K}(E {\mathrm{op}}lus F)$ has a bounded approximate identity, and so does ${\mathfrak A}$.
\par
For $i \in {\mathbb M}hbb{I}$, let $P_{1,i} \!: E {\mathrm{op}}lus F \to E$ and $P_{2,i} \!: E {\mathrm{op}}lus F \to F$ be the canonical projection; for $j=1,2$, set $P_j := ( P_{j,i})_{i \in {\mathbb M}hbb I} \in {\widehat{\mathrm{env}}}ll^\infty({\mathbb M}hbb{I},{\cal B}(E {\mathrm{op}}lus F)) \subset {\cal M}({\widehat{\mathrm{env}}}ll^\infty({\mathbb M}hbb{I},{\cal K}(E {\mathrm{op}}lus F)))$. It follows that $P_1 {\mathfrak A} P_1 \cong {\widehat{\mathrm{env}}}ll^\infty({\mathbb M}hbb{I},{\cal K}(E))$ and $P_2 {\mathfrak A} P_2 \cong {\widehat{\mathrm{env}}}ll^\infty({\mathbb M}hbb{I},{\cal K}(F))$.
\par
The restriction of $\Delta_{\mathfrak A}$ to $P_2 {\mathfrak A} P_1\hat{\otimes} P_1 {\mathfrak A} P_2$ induces a quotient norm, say $| \cdot |$, on its range, which dominates the given norm $\| \cdot \|$. By our factorization hypothesis, the range of $\Delta_{\mathfrak A}( P_2 {\mathfrak A} P_1 \hat{\otimes} P_1 {\mathfrak A} P_2)$ contains ${\widehat{\mathrm{env}}}ll^\infty({\cal F}(F))$, and we have $| \cdot | \leq C \| \cdot \|$ on ${\widehat{\mathrm{env}}}ll^\infty({\cal F}(F))$. As $F^\ast$ has the approximation property, so does $F$, and, in particular, ${\cal F}(F)$ is dense in ${\cal K}(F)$, as is ${\widehat{\mathrm{env}}}ll^\infty({\cal F}(F))$ in ${\widehat{\mathrm{env}}}ll^\infty({\cal K}(F))$. Every element of ${\widehat{\mathrm{env}}}ll^\infty({\cal K}(F))$ is thus a limit---with respect to $\| \cdot \|$---of a sequence in ${\widehat{\mathrm{env}}}ll^\infty({\cal F}(F))$. This sequence is a Cauchy sequence with respect to $\| \cdot \|$ and thus with respect to $| \cdot |$; consequently, it converges---with respect to $| \cdot |$---to an element in $\Delta_{\mathfrak A}( P_2 {\mathfrak A} P_1 \hat{\otimes} P_1 {\mathfrak A} P_2)$. Since $\| \cdot \| \leq | \cdot |$, this limit with respect to $| \cdot |$ is the same limit as with respect to $\| \cdot \|$. So, $\Delta_{\mathfrak A}$ maps $P_2 {\mathfrak A} P_1 \hat{\otimes} P_1 {\mathfrak A} P_2$ onto $P_2 {\mathfrak A} P_2$.
\par
The hypotheses of Theorem {{\mathrm{op}}eratorname{Re}}f{bgn} are thus all satisfied, and the claim follows.
{\widehat{\mathrm{env}}}nd{proof}
\par
We shall now look at Banach spaces for which the factorization hypothesis of Proposition {{\mathrm{op}}eratorname{Re}}f{facam} is satisfied.
\par
Let $p \in [1,\infty]$ and $\lambda \geq 1$. A Banach space $E$ is called an {\widehat{\mathrm{env}}}mph{${\cal L}^p_\lambda$-space} if, for any finite-dimensional subspace $X$ of $E$, there are $n \in {{\mathbb M}hbb N}$ and an $n$-dimensional subspace $Y$ of $E$ containing $X$ such that $d(Y,{\widehat{\mathrm{env}}}ll^p_n) \leq \lambda$, where $d$ is the Banach--Mazur distance (\cite[Definition 3.1]{LP}). We call $E$ simply an {\widehat{\mathrm{env}}}mph{${\cal L}^p$-space} if it is an ${\cal L}^p_\lambda$-space for some $\lambda \geq 1$. All $L^p$-spaces, i.e., spaces of $p$-integrable functions on some measure space, are ${\cal L}^p$-spaces. The ${\cal L}^p$-spaces were introduced in \cite{LP} and studied further in \cite{LR}. We list some of their properties:
\begin{itemize}
\item Every ${\cal L}^p$-space is isomorphic to a subspace of an $L^p$-space (\cite[Theorem I(i)]{LR}). In particular, for $p \in (1,\infty)$, each ${\cal L}^p$-space is reflexive.
\item If $E$ is an ${\cal L}^p$-space, then $E^\ast$ is an ${\cal L}^{p'}$-space, where $p' \in [1,\infty]$ is conjugate to $p$, i.e., satisfies $\frac{1}{p} + \frac{1}{p'} = 1$ (\cite[Theorem III(a)]{LR}).
\item If $E$ is an ${\cal L}^p$-space, then there is a constant $\rho \geq 1$ such that, for each finite-dimensional subspace $X$ of $E$, there are $n \in {{\mathbb M}hbb N}$, an $n$-dimensional subspace $Y$ of $E$ containing $X$ with $d(Y,{\widehat{\mathrm{env}}}ll^p_n)$, and a projection $P$ onto $Y$ with $\| P \| \leq \rho$ (\cite[Theorem III(c)]{LR}). In particular, $E$ has the bounded approximation property.
{\widehat{\mathrm{env}}}nd{itemize}
\par
In \cite{GJW}, it is mentioned without proof before \cite[Theorem 6.4]{GJW} that, given any two infinite-dimensional ${\cal L}^p$-spaces $E$ and $F$, every operator in ${\cal F}(F)$ factors through $E$ with both factors being compact. Since, for our purpose, we need control over the norms of those factors, we give a refinement of this observation with a detailed proof:
\begin{lemma} \label{Lpfac}
Let $p \in [1,\infty]$, and let $E$ and $F$ be ${\cal L}^p$-spaces with $\dim E = \infty$. Then there is $C \geq 0$ such that, for each $T \in {\cal F}(F)$, there are $S \in {\cal F}(F,E)$ and $R \in {\cal F}(E,F)$ with $\| R \| \| S \| \leq C \| T \|$ and $RS = T$.
{\widehat{\mathrm{env}}}nd{lemma}
\begin{proof}
Let $T \in {\cal F}(F)$, and set $X := TF$. Let $\lambda \geq 1$ be such that $E$ is a ${\cal L}^p_\lambda$-space. Then there are $n \in {{\mathbb M}hbb N}$, a finite-dimensional subspace $Y$ of $F$ containing $X$, and a bijective linear map $\tau \!: Y \to {\widehat{\mathrm{env}}}ll^p_n$ such that $\| \tau \| \| \tau^{-1} \| \leq \lambda$.
\par
Let $\rho \geq 1$ be the constant for $E$ whose existence is guaranteed by \cite[Theorem III(c)]{LR}. Let $Z_0$ be an $n$-dimensional subspace of $E$. (Here, we require that $\dim E = \infty$.) Then there are $m \in {{\mathbb M}hbb N}$, an $m$-dimensional subspace $Z$ of $E$ containing $Z_0$, a bijective map $\sigma \!: Z \to {\widehat{\mathrm{env}}}ll^p_m$ with $\| \sigma \| \| \sigma^{-1} \| \leq \rho$, and a projection $P$ onto $Z$ with $\| P \| \leq \rho$; note that necessarily $m \geq n$. With $\iota \!: {\widehat{\mathrm{env}}}ll^p_n \to {\widehat{\mathrm{env}}}ll^p_m$ and $\pi \!: {\widehat{\mathrm{env}}}ll^p_m \to {\widehat{\mathrm{env}}}ll^p_n$ being the canonical embedding and projection, we see that $T= \tau^{-1} \pi \sigma \sigma^{-1} \iota \tau T$. Set $S := \sigma^{-1} \iota \tau T$ and $R := \tau^{-1} \pi \sigma P$. Then $S=RT$ holds, and we have
\[
\| R \| \| S \| \leq \| \tau^{-1} \| \| \pi \| \| \sigma \| \| P \| \| \sigma^{-1} \| \| \iota \| \| \tau \| \| T \|
= \| \tau \| \| \tau^{-1} \| \| \sigma \| \| \sigma^{-1} \| \| P \| \| T \| \leq \lambda \rho^2 \| T \|.
\]
Hence, $C := \lambda \rho^2$ has the desired property.
{\widehat{\mathrm{env}}}nd{proof}
\begin{remark}
In \cite{DF}, A.\ Defant and K.\ Floret introduced the class of ${\cal L}^p_g$-spaces, which contains all ${\cal L}^p$-spaces, but is somewhat better behaved. For $p=1,\infty$, the ${\cal L}^p_g$-spaces are just the ${\cal L}^p$-spaces whereas, for $p \in (1,\infty)$, a space is an ${\cal L}^p_g$-space if and only if it is an ${\cal L}^p$-space {\widehat{\mathrm{env}}}mph{or isomorphic to a Hilbert space} (\cite[23.3]{DF}). Therefore, Lemma {{\mathrm{op}}eratorname{Re}}f{Lpfac} does not hold true for ${\cal L}^p_g$-spaces if $p \neq 1,\infty$.
{\widehat{\mathrm{env}}}nd{remark}
\par
Together, Proposition {{\mathrm{op}}eratorname{Re}}f{facam} and Lemma {{\mathrm{op}}eratorname{Re}}f{Lpfac} yield the following dichotomy theorem:
\begin{theorem} \label{thm3}
Let $p \in [1,\infty]$, and let ${\mathbb M}hbb I$ be an index set. Then one of the following assertions is true:
\begin{items}
\item ${\widehat{\mathrm{env}}}ll^\infty({\mathbb M}hbb{I},{\cal K}(E))$ is amenable for every infinite-dimensional ${\cal L}^p$-space $E$;
\item ${\widehat{\mathrm{env}}}ll^\infty({\mathbb M}hbb{I},{\cal K}(E))$ is not amenable for any infinite-dimensional ${\cal L}^p$-space $E$.
{\widehat{\mathrm{env}}}nd{items}
{\widehat{\mathrm{env}}}nd{theorem}
\begin{proof}
Suppose that (ii) is false, i.e, there is an infinite-dimensional ${\cal L}^p$-space $E$ such that ${\widehat{\mathrm{env}}}ll^\infty({\mathbb M}hbb{I},{\cal K}(E))$ is amenable. Let $F$ be any infinite-dimensional ${\cal L}^p$-space. Then Lemma {{\mathrm{op}}eratorname{Re}}f{Lpfac} and Proposition {{\mathrm{op}}eratorname{Re}}f{facam} imply that ${\widehat{\mathrm{env}}}ll^\infty({\mathbb M}hbb{I},{\cal K}(E {\mathrm{op}}lus F))$ is amenable. Interchanging the r\^oles of $E$ and $F$ and invoking Lemma {{\mathrm{op}}eratorname{Re}}f{Lpfac} and Proposition {{\mathrm{op}}eratorname{Re}}f{facam} again, then yields the amenability of ${\widehat{\mathrm{env}}}ll^\infty({\mathbb M}hbb{I},{\cal K}(F))$. This proves that (i) is true.
{\widehat{\mathrm{env}}}nd{proof}
\par
Combining Theorems {{\mathrm{op}}eratorname{Re}}f{thm2} and {{\mathrm{op}}eratorname{Re}}f{thm3}, we obtain:
\begin{corollary} \label{cor3}
Let $p \in (1,\infty]$, let $E = {\widehat{\mathrm{env}}}ll^p$ if $p \in (1,\infty)$ and $E = c_0$ if $p = \infty$, and suppose that ${\widehat{\mathrm{env}}}ll^\infty({\cal K}(E))$ is amenable. Then ${\widehat{\mathrm{env}}}ll^\infty({\mathbb M}hbb{I},{\cal K}(F))$ is amenable for every index set ${\mathbb M}hbb I$ and every infinite-dimensional ${\cal L}^p$-space $F$. In particular, ${\cal K}(F)$ is ultra-amenable for every infinite-dimensional ${\cal L}^p$-space $F$.
{\widehat{\mathrm{env}}}nd{corollary}
\begin{remark}
Let $p \in (1,\infty) \setminus \{ 2 \}$. Then ${\widehat{\mathrm{env}}}ll^p({\widehat{\mathrm{env}}}ll^2)$ and ${\widehat{\mathrm{env}}}ll^p {\mathrm{op}}lus {\widehat{\mathrm{env}}}ll^2$ are not $L^p$-spaces, but still ${\cal L}^p$-spaces (\cite[Example 8.2]{LP}). Hence, if ${\widehat{\mathrm{env}}}ll^\infty({\cal K}({\widehat{\mathrm{env}}}ll^p))$ is amenable, then so are ${\widehat{\mathrm{env}}}ll^\infty({\cal K}({\widehat{\mathrm{env}}}ll^p({\widehat{\mathrm{env}}}ll^2)))$ and ${\widehat{\mathrm{env}}}ll^\infty({\cal K}({\widehat{\mathrm{env}}}ll^p {\mathrm{op}}lus {\widehat{\mathrm{env}}}ll^2))$. This is remarkable because ${\cal B}({\widehat{\mathrm{env}}}ll^p {\mathrm{op}}lus {\widehat{\mathrm{env}}}ll^2)$ is known to be not amenable (\cite[Question 4]{Gro}).
{\widehat{\mathrm{env}}}nd{remark}
\section{A non-amenability criterion for ${\widehat{\mathrm{env}}}ll^\infty({\cal K}(E {\mathrm{op}}lus F))$}
As we just observed, the amenability of ${\widehat{\mathrm{env}}}ll^\infty({\cal K}({\widehat{\mathrm{env}}}ll^p))$ implies the amenability of ${\widehat{\mathrm{env}}}ll^\infty({\cal K}({\widehat{\mathrm{env}}}ll^p {\mathrm{op}}lus {\widehat{\mathrm{env}}}ll^2))$. In this section, we shall thus explore the amenability of ${\widehat{\mathrm{env}}}ll^\infty({\cal K}(E {\mathrm{op}}lus F))$ for two Banach spaces $E$ and $F$.
\par
Recall that an {\widehat{\mathrm{env}}}mph{operator ideal} $\cal A$ is a rule that assigns to each pair $(E,F)$ of a Banach spaces a subspace ${\cal A}(E,F)$ of ${\cal B}(E,F)$ containing ${\cal F}(E,F)$ such that $RTS \in {\cal A}(X,Y)$ for any Banach spaces $X$ and $Y$, $T \in {\cal A}(E,F)$, $S \in {\cal B}(X,E)$, and $R \in {\cal B}(F,Y))$; if $E = F$, we convene again to simply write ${\cal A}(E)$. The seminal reference on operator ideals is \cite{Pie}. More recent treatments can be found in \cite{DF}, \cite{DJT}, or \cite{TJ}. We call $[{\cal A},\alpha]$---following \cite{DJT} in our notation---a {\widehat{\mathrm{env}}}mph{Banach operator ideal} if, for each pair $(E,F)$ of Banach spaces, there is a norm $\alpha$ on ${\cal A}(E,F)$ turning it into a Banach space such that
\[
\alpha(y \odot \phi) = \| y \| \| \phi \| \qquad (y \in F, \, \phi \in E^\ast),
\]
where $y \odot \phi \in {\cal F}(E,F)$ is the rank one operator corresponding to the elementary tensor $y \otimes \phi$, and
\[
\alpha(RTS) \leq \| R \| \alpha(T) \| S \| \qquad (T \in {\cal A}(E,F), \, S \in {\cal B}(X,E), \, R \in {\cal B}(F,Y))
\]
for any Banach spaces $X$ and $Y$. Given a Banach operator ideal $[{\cal A},\alpha]$, its {\widehat{\mathrm{env}}}mph{maximal hull} $[{\cal A}^{\max},\alpha^{\max}]$ is defined as follows. For two Banach spaces $E$ and $F$, let ${\mathbb M}hfrak{F}(E)$ denote the finite-dimensional subspaces of $E$, and let ${\mathbb M}hfrak{F}_c(F)$ stand for the closed subspaces of $F$ with finite co-dimension; for $X \in {\mathbb M}hfrak{F}(E)$ and $Y \in {\mathbb M}hfrak{F}_c(F)$, let $\iota_X \!: X \to E$ and $\pi_Y \!: F \to F/Y$ be the inclusion and quotient map, respectively. We define
\[
\alpha^{\max}(T) := \sup \{ \alpha(\pi_Y T \iota_X ): X \in {\mathbb M}hfrak{F}(E), \, Y \in {\mathbb M}hfrak{F}_c(F) \} \in [0,\infty]
\qquad (T \in {\cal B}(E,F))
\]
and
\[
{\cal A}^{\max}(E,F) := \left\{ T \in {\cal B}(E,F): \alpha^{\max}(T) < \infty \right\}.
\]
It is routinely checked that $[{\cal A}^{\max},\alpha^{\max}]$ is again a Banach operator ideal, and we call $[{\cal A},\alpha]$ {\widehat{\mathrm{env}}}mph{maximal} if $[{\cal A}^{\max},\alpha^{\max}] = [{\cal A},\alpha]$.
\par
An immediate consequence of \cite[17.5, Representation Theorem for Maximal Operator Ideals]{DF} is: if $[{\cal A},\alpha]$ is a maximal Banach operator ideal, then ${\cal A}(E,F^\ast)$ is a dual Banach space for any two Banach spaces $E$ and $F$ such that weak$^\ast$ topology of ${\cal A}(E,F^\ast)$ coincides with the weak$^\ast$ topology of ${\cal B}(E,F^\ast) = (E \hat{\otimes} F)^\ast$ on norm bounded subsets.
\par
In particular, we have (compare \cite[17.21, Proposition]{DF}):
\begin{lemma} \label{maxid}
Let $[{\cal A},\alpha]$ be a maximal Banach operator ideal, let $E$ and $F$ be Banach spaces, and let $( T_i )_{i \in {\mathbb M}hbb I}$ be a bounded net in ${\cal A}(E,F^\ast)$ that converges to $T \in {\cal B}(E,F^\ast)$ with respect to the weak$^\ast$ topology of ${\cal B}(E,F^\ast)$. Then $T$ lies in ${\cal A}(E,F^\ast)$.
{\widehat{\mathrm{env}}}nd{lemma}
\begin{proposition} \label{opidnonam}
Let $E$ and $F$ be Banach spaces, let ${\mathbb M}hbb{I}$ be an index set, and suppose that there are $( T_i )_{i \in {\mathbb M}hbb I} \in {\widehat{\mathrm{env}}}ll^\infty({\mathbb M}hbb{I},{\cal K}(F,E))$ and an ultrafilter $\cal U$ over ${\mathbb M}hbb I$ such that $\text{{\widehat{\mathrm{env}}}mph{weak}$^\ast$-}\lim_{i \in \cal U} T_i \notin {\cal K}(F,E^{\ast\ast})$, where the limit is with respect to the weak$^\ast$ topology of ${\cal B}(E,F^{\ast\ast})$. Suppose further that there is a maximal operator ideal $[{\cal A},\alpha]$ with the following properties:
\begin{alphitems}
\item ${\cal K}(E,F) \subset {\cal A}(E,F)$ such that the inclusion is continuous;
\item ${\cal A}(F,E^{\ast\ast}) \subset {\cal K}(F,E^{\ast\ast})$.
{\widehat{\mathrm{env}}}nd{alphitems}
Then ${\widehat{\mathrm{env}}}ll^\infty({\mathbb M}hbb{I}, {\cal K}(E {\mathrm{op}}lus F))$ is not amenable.
{\widehat{\mathrm{env}}}nd{proposition}
\begin{proof}
Let $P_1$ and $P_2$ be the projections in ${\widehat{\mathrm{env}}}ll^\infty({\mathbb M}hbb{I}, {\cal B}(E {\mathrm{op}}lus F))$ induced by the canonical projections onto $E$ and $F$, respectively (compare the proof of Proposition {{\mathrm{op}}eratorname{Re}}f{facam}). We wish to apply Proposition {{\mathrm{op}}eratorname{Re}}f{noamprop}.
\par
Letting
\[
I := \varcl{{\widehat{\mathrm{env}}}ll^\infty({\mathbb M}hbb{I},{\cal A}(E {\mathrm{op}}lus F)) \cap {\widehat{\mathrm{env}}}ll^\infty({\mathbb M}hbb{I},{\cal K}(E {\mathrm{op}}lus F))},
\]
with the closure taken in the norm topology of ${\widehat{\mathrm{env}}}ll^\infty({\mathbb M}hbb{I},{\cal K}(E {\mathrm{op}}lus F))$, defines a closed ideal of ${\mathfrak A} := {\widehat{\mathrm{env}}}ll^\infty({\mathbb M}hbb{I},{\cal K}(E {\mathrm{op}}lus F))$.
\par
From (a), it is obvious that $P_2 I P_1 = P_2 {\mathfrak A} P_1$.
\par
To see that $P_1 I P_2 \subsetneq P_1 {\mathfrak A} P_2$, let $( T_i )_{i \in {\mathbb M}hbb I} \in {\widehat{\mathrm{env}}}ll^\infty({\mathbb M}hbb{I},{\cal K}(F,E))$ and an ultrafilter $\cal U$ over ${\mathbb M}hbb I$ be as specified in the hypotheses. We claim that $(T_i )_{i \in {\mathbb M}hbb I} \in P_1 {\mathfrak A} P_2 \setminus P_1 I P_2$. Define
\[
Q_{\cal U} \!: {\widehat{\mathrm{env}}}ll^\infty({\mathbb M}hbb{I},{\cal K}(F,E)) \to {\cal B}(F,E), \quad
( S_i )_{i \in {\mathbb M}hbb I} \mapsto \text{weak$^\ast$-}\lim_{i \in \cal U} S_i.
\]
so that $T := Q_{\cal U}\left( (T_i )_{i \in {\mathbb M}hbb I} \right) \notin {\cal K}(F,E^{\ast\ast})$. Assume that $(T_i )_{i \in {\mathbb M}hbb I} \in P_1 I P_2$, and let ${\widehat{\mathrm{env}}}psilon >0$ be arbitrary. By the definition of $I$, there is thus $( R_i )_{i \in {\mathbb M}hbb I} \in
{\widehat{\mathrm{env}}}ll^\infty({\mathbb M}hbb{I},{\cal A}(F,E)) \cap {\widehat{\mathrm{env}}}ll^\infty({\mathbb M}hbb{I},{\cal K}(F,E))$ such that $\sup_{i \in {\mathbb M}hbb I} \| R_i - T_i \| < {\widehat{\mathrm{env}}}psilon$. Since $Q_{\cal U}$ is a contraction, this means that $\| R - T \| < {\widehat{\mathrm{env}}}psilon$, where $R := Q_{\cal U}\left( (R_i )_{i \in {\mathbb M}hbb I} \right)$. By Lemma {{\mathrm{op}}eratorname{Re}}f{maxid}, $R \in {\cal A}(F,E^{\ast\ast})$ holds, so that $R \in {\cal K}(F,E^{\ast\ast})$ by (b). Since ${\widehat{\mathrm{env}}}psilon > 0$ was arbitrary, this means that $T \in {\cal K}(F,E^{\ast\ast})$, which contradicts our hypotheses.
{\widehat{\mathrm{env}}}nd{proof}
\par
The hypotheses of Proposition {{\mathrm{op}}eratorname{Re}}f{opidnonam} appear to be technical and somewhat contrived, but as our next theorem shows, they do, in fact, occur naturally in certain situations:
\begin{theorem} \label{thm4}
The Banach algebra ${\widehat{\mathrm{env}}}ll^\infty({\cal K}(E {\mathrm{op}}lus {\widehat{\mathrm{env}}}ll^2))$ is not amenable for any of the following spaces $E$: $c_0$, ${\widehat{\mathrm{env}}}ll^\infty$, and ${\widehat{\mathrm{env}}}ll^1$.
{\widehat{\mathrm{env}}}nd{theorem}
\begin{proof}
We first consider the case $E = c_0$.
\par
For $n \in {{\mathbb M}hbb N}$, let $\pi_n \!: c_0 \to c_0$ denote the projection onto the first $n$ coordinates, and let $\iota \!: {\widehat{\mathrm{env}}}ll^2 \to c_0 \hookrightarrow {\widehat{\mathrm{env}}}ll^\infty$ be the natural inclusion. Then $( \pi_n \iota )_{n=1}^\infty \in {\widehat{\mathrm{env}}}ll^\infty({\cal K}({\widehat{\mathrm{env}}}ll^2,c_0))$, and $\text{weak$^\ast$-}\lim_{n \in \cal U} \pi_n \iota = \iota \notin {\cal K}({\widehat{\mathrm{env}}}ll^2,{\widehat{\mathrm{env}}}ll^\infty)$ for any free ultrafilter $\cal U$ over ${{\mathbb M}hbb N}$.
\par
Let $[\Pi_2,\pi_2]$ be the ideal of $2$-summing operators (see \cite[p.\ 31]{DJT}, for instance); then $[\Pi_2,\pi_2]$ is maximal (by \cite[6.16 Theorem]{DJT}). By \cite[3.7 Theorem]{DJT}, $\Pi_2(c_0,{\widehat{\mathrm{env}}}ll^2) = {\cal B}(c_0,{\widehat{\mathrm{env}}}ll^2)$ holds, so that Proposition {{\mathrm{op}}eratorname{Re}}f{opidnonam}(a) is satisfied. On the other hand, every $2$-summing operator is completely continuous (\cite[2.17 Theorem]{DJT}). Since ${\widehat{\mathrm{env}}}ll^2$ is reflexive, this means that $\Pi_2({\widehat{\mathrm{env}}}ll^2,{\widehat{\mathrm{env}}}ll^\infty) \subset {\cal K}({\widehat{\mathrm{env}}}ll^2,{\widehat{\mathrm{env}}}ll^\infty)$, so that Proposition {{\mathrm{op}}eratorname{Re}}f{opidnonam}(b) holds as well.
\par
The $E = {\widehat{\mathrm{env}}}ll^\infty$ case has an almost identical proof.
\par
Suppose now that $E = {\widehat{\mathrm{env}}}ll^1$. We shall apply Proposition {{\mathrm{op}}eratorname{Re}}f{opidnonam} to the Banach algebra ${\widehat{\mathrm{env}}}ll^\infty({\cal K}({\widehat{\mathrm{env}}}ll^2 {\mathrm{op}}lus {\widehat{\mathrm{env}}}ll^1 ))$, which is isomorphic to ${\widehat{\mathrm{env}}}ll^\infty({\cal K}({\widehat{\mathrm{env}}}ll^1 {\mathrm{op}}lus {\widehat{\mathrm{env}}}ll^2))$. With $\iota \!: {\widehat{\mathrm{env}}}ll^1 \to {\widehat{\mathrm{env}}}ll^2$ being the canonical inclusion and $\pi_n \!: {\widehat{\mathrm{env}}}ll^2 \to {\widehat{\mathrm{env}}}ll^2$ for $n \in {{\mathbb M}hbb N}$ denoting the projection onto the first $n$ coordinates, we have $\text{weak$^\ast$-}\lim_{n \in \cal U} \pi_n \iota = \iota \notin {\cal K}({\widehat{\mathrm{env}}}ll^1,{\widehat{\mathrm{env}}}ll^2)$ for any free ultrafilter $\cal U$ over ${{\mathbb M}hbb N}$---just as in the case $E = c_0$. Let Let $[\Pi_2^d,\pi_2^d]$ be the dual ideal of $[\Pi_2,\pi_2]$ (see \cite[p.\ 186]{DJT}), so that $[\Pi_2^d,\pi_2^d]$ is maximal by \cite[9.4 Corollary]{DJT}. Using the definition of $[\Pi_2^d,\pi_2^d]$ and arguing as in the $E= c_0$ case, we see that Proposition {{\mathrm{op}}eratorname{Re}}f{opidnonam}(a) and (b) are satisfied.
{\widehat{\mathrm{env}}}nd{proof}
\begin{remark}
Even though ${\widehat{\mathrm{env}}}ll^\infty({\cal K}({\widehat{\mathrm{env}}}ll^2))$ is known not to be amenable, there seems to be no way---by means of Theorem {{\mathrm{op}}eratorname{Re}}f{bgn}, for instance---to conclude directly from its non-amenability that the Banach algebras considered in Theorem {{\mathrm{op}}eratorname{Re}}f{thm4} are not amenable.
{\widehat{\mathrm{env}}}nd{remark}
\par
Having established the non-amenability of ${\widehat{\mathrm{env}}}ll^\infty({\cal K}({\widehat{\mathrm{env}}}ll^p {\mathrm{op}}lus {\widehat{\mathrm{env}}}ll^2))$ for $p =1,\infty$ with the help of Proposition {{\mathrm{op}}eratorname{Re}}f{opidnonam}, one might be tempted to try to extend this result to general $p \in [1,\infty] \setminus \{ 2 \}$ through the choice of a suitable maximal Banach operator ideal $[ {\cal A}, \alpha ]$. Alas, as we shall see now, this attempt is futile:
\begin{proposition} \label{lp2}
Let $p \in (1,\infty) \setminus \{ 2 \}$, let ${\mathfrak A} := {\widehat{\mathrm{env}}}ll^\infty({\cal K}({\widehat{\mathrm{env}}}ll^p {\mathrm{op}}lus {\widehat{\mathrm{env}}}ll^2))$, and let $P_1, P_2 \in {\widehat{\mathrm{env}}}ll^\infty({\cal B}({\widehat{\mathrm{env}}}ll^p {\mathrm{op}}lus {\widehat{\mathrm{env}}}ll^2))$ be the projections induced by the canonical projections onto ${\widehat{\mathrm{env}}}ll^p$ and ${\widehat{\mathrm{env}}}ll^2$, respectively. Then there is no closed ideal $I$ of ${\mathfrak A}$ with $P_2 I P_1 = P_2 {\mathfrak A} P_1$ and $P_1 I P_2 \subsetneq P_1 {\mathfrak A} P_2$.
{\widehat{\mathrm{env}}}nd{proposition}
\begin{proof}
We use the fact (\cite[p.\ 73]{LT}) that we have an isomorphism
\begin{equation} \label{iso}
{\widehat{\mathrm{env}}}ll^p \cong \text{${\widehat{\mathrm{env}}}ll^p$-}\bigoplus_{n=1}^\infty {\widehat{\mathrm{env}}}ll^2_n.
{\widehat{\mathrm{env}}}nd{equation}
For $n \in {{\mathbb M}hbb N}$, let $J_n \!: {\widehat{\mathrm{env}}}ll^2_n \to {\widehat{\mathrm{env}}}ll^p$ and $Q_n \!: {\widehat{\mathrm{env}}}ll^p \to {\widehat{\mathrm{env}}}ll^2_n$ the embedding of and projection onto the $n$-th summand in ({{\mathrm{op}}eratorname{Re}}f{iso}), respectively; note that those maps $J_n$ and $P_n$ are uniformly bounded. Furthermore, let $\iota_n \!: {\widehat{\mathrm{env}}}ll^2_n \to {\widehat{\mathrm{env}}}ll^2$ and $\pi_n \!: {\widehat{\mathrm{env}}}ll^2 \to {\widehat{\mathrm{env}}}ll^2_n$ be the canonical embedding and projection, respectively, for $n \in {{\mathbb M}hbb N}$.
\par
Assume that there is a closed ideal $I$ of ${\mathfrak A}$ with $P_2 I P_1 = P_2 {\mathfrak A} P_1$. Note that, since ${\mathfrak A}$ has a bounded approximate identity, $I$ is also a closed ideal of ${\widehat{\mathrm{env}}}ll^\infty({\cal B}({\widehat{\mathrm{env}}}ll^p {\mathrm{op}}lus {\widehat{\mathrm{env}}}ll^2))$, so that, in particular, $P_j I P_k \subset I$ for $j,k=1,2$.
\par
Let $( T_n )_{n=1}^\infty \in {\widehat{\mathrm{env}}}ll^\infty({\cal K}({\widehat{\mathrm{env}}}ll^2,{\widehat{\mathrm{env}}}ll^p))$, and let ${\widehat{\mathrm{env}}}psilon > 0$ be arbitrary. For each $n \in {{\mathbb M}hbb N}$, we can find $S_n \in {\cal F}({\widehat{\mathrm{env}}}ll^2,{\widehat{\mathrm{env}}}ll^p)$ with $\| T_n - S_n \| \leq {\widehat{\mathrm{env}}}psilon$ as well as $N_n \in {{\mathbb M}hbb N}$ such that $S_n \iota_{N_n} \pi_{N_n} = S_n$. Define $( U_n )_{n=1}^\infty \in {\widehat{\mathrm{env}}}ll^\infty({\cal K}({\widehat{\mathrm{env}}}ll^2,{\widehat{\mathrm{env}}}ll^p))$ and $( V_n )_{n=1}^\infty \in {\widehat{\mathrm{env}}}ll^\infty({\cal K}({\widehat{\mathrm{env}}}ll^p))$ by letting
\[
U_n := \iota_{N_n} Q_{N_n} \quad\text{and}\quad V_n := J_{N_n}\pi_{N_n} \qquad (n \in {{\mathbb M}hbb N}).
\]
so that
\[
\begin{bmatrix} 0 & ( S_n )_{n=1}^\infty \\ 0 & 0 {\widehat{\mathrm{env}}}nd{bmatrix}
= \begin{bmatrix} 0 & ( S_n )_{n=1}^\infty \\ 0 & 0 {\widehat{\mathrm{env}}}nd{bmatrix}
\begin{bmatrix} 0 & 0 \\ ( U_n)_{n=1}^\infty & 0 {\widehat{\mathrm{env}}}nd{bmatrix}
\begin{bmatrix} 0 & ( V_n )_{n=1}^\infty \\ 0 & 0 {\widehat{\mathrm{env}}}nd{bmatrix}.
\]
As $\left[ \begin{smallmatrix} 0 & 0 \\ ( U_n)_{n=1}^\infty & 0 {\widehat{\mathrm{env}}}nd{smallmatrix} \right] \in P_2 {\mathfrak A} P_1 = P_2 I P_1 \subset I$ and $I$ is an ideal, it follows that $\left[ \begin{smallmatrix} 0 & ( S_n )_{n=1}^\infty \\ 0 & 0 {\widehat{\mathrm{env}}}nd{smallmatrix} \right] \in I$ as well. Since ${\widehat{\mathrm{env}}}psilon > 0$ was arbitrary, this entails that $( T_n )_{n=1}^\infty \in P_1 I P_2$, and since $( T_n )_{n=1}^\infty \in {\widehat{\mathrm{env}}}ll^\infty({\cal K}({\widehat{\mathrm{env}}}ll^2,{\widehat{\mathrm{env}}}ll^p))$ was arbitrary, this means that $P_1 I P_2 = P_1 {\mathfrak A} P_2$.
{\widehat{\mathrm{env}}}nd{proof}
\begin{remarks}
\item Even though Proposition {{\mathrm{op}}eratorname{Re}}f{lp2} shows that the (still hypothetical) non-amenability of ${\widehat{\mathrm{env}}}ll^\infty({\cal K}({\widehat{\mathrm{env}}}ll^p {\mathrm{op}}lus {\widehat{\mathrm{env}}}ll^2))$ cannot be established in the same way as for ${\cal B}({\widehat{\mathrm{env}}}ll^p {\mathrm{op}}lus {\widehat{\mathrm{env}}}ll^2)$, it naturally leads to the question if, for sufficiently nice Banach spaces $E$, the amenability of ${\widehat{\mathrm{env}}}ll^\infty({\cal K}(E))$ forces ${\cal B}(E)$ to be amenable. Since ${\cal K}(E)^{\ast\ast} = {\cal B}(E)$ via trace duality for any reflexive Banach space with the approximation property, this question can, for such spaces, be put into a more general framework: If ${\widehat{\mathrm{env}}}ll^\infty({\mathfrak A})$ is amenable for some Banach algebra ${\mathfrak A}$, does this imply that ${\mathfrak A}^{\ast\ast}$, equipped with one of the Arens products (see \cite{Dal}), is amenable? Partial answers, which do not apply to the case where ${\mathfrak A} = {\cal K}({\widehat{\mathrm{env}}}ll^p {\mathrm{op}}lus {\widehat{\mathrm{env}}}ll^2)$, are given in \cite{CSR}.
\item In \cite[Theorem 1]{CSR}, the following is claimed to be a consequence of \cite{GI}: For a unital ${C^\ast}$-algebra ${\mathfrak A}$, there are an index set ${\mathbb M}hbb I$, which can be chosen as ${{\mathbb M}hbb N}$ if ${\mathfrak A}^\ast$ is separable, and a an algebra homomorphism from ${\widehat{\mathrm{env}}}ll^\infty({\mathbb M}hbb{I},{\mathfrak A})$ onto ${\mathfrak A}^{\ast\ast}$. An inspection of the proof of \cite[Theorem 1]{CSR} shows that the alleged algebra homomorphism is
\begin{equation} \label{bogus}
{\widehat{\mathrm{env}}}ll^\infty({\mathbb M}hbb{I},{\mathfrak A}) \to {\mathfrak A}^{\ast\ast}, \quad ( a_i )_{i \in {\mathbb M}hbb I} \mapsto \text{weak$^\ast$-}\lim_{i \in \cal U} a_i
{\widehat{\mathrm{env}}}nd{equation}
for a suitable ultrafilter $\cal U$ over ${\mathbb M}hbb I$. Let ${\mathfrak A}$ be the unitization of ${\cal K}({\widehat{\mathrm{env}}}ll^2)$, and let $\cal U$ be a free ultrafilter over ${{\mathbb M}hbb N}$. Then we have
\[
\text{weak$^\ast$-}\lim_{n \in \cal U} \delta_1 \odot \delta_n = \text{weak$^\ast$-}\lim_{n \in \cal U} \delta_n \odot \delta_1 = 0
\]
whereas
\[
\text{weak$^\ast$-}\lim_{n \in \cal U} (\delta_1 \odot \delta_n)(\delta_n \odot \delta_1) = \delta_1 \odot \delta_1 \neq 0,
\]
which means that ({{\mathrm{op}}eratorname{Re}}f{bogus}) cannot be multiplicative, contrary to what is claimed in \cite{CSR}.
{\widehat{\mathrm{env}}}nd{remarks}
\section{A left ideal in $({\cal K}({\widehat{\mathrm{env}}}ll^p))_{\cal U}$ without bounded right approximate identity}
If ${\mathfrak A}$ is an amenable Banach algebra, then a closed ideal of ${\mathfrak A}$ has a bounded approximate identity if and only if it is weakly complemented. By finding a closed, weakly complemented ideal of ${\mathfrak A}$ that lacks a bounded approximate identity, one can thus show that ${\mathfrak A}$ is not amenable, as was done in \cite{DGH} in the case of the measure algebra of a non-discrete, locally compact group. More generally, a closed left ideal of ${\mathfrak A}$ has a bounded right approximate identity if and only if it is weakly complemented (see \cite[Lemma 2.3.6]{LoA}, for instance).
\par
In this section, for $p \in (1,2)$ and a free ultrafilter $\cal U$ over ${{\mathbb M}hbb N}$, we shall exhibit a closed left ideal of $({\cal K}({\widehat{\mathrm{env}}}ll^p))_{\cal U}$ that lacks a right approximate identity (bounded or not) and present some, albeit circumstantial, evidence for it being weakly complemented.
\par
For $p \in (1,2)$, let $\iota \!: {\widehat{\mathrm{env}}}ll^p \to {\widehat{\mathrm{env}}}ll^2$ be the natural inclusion map, and note that the adjoint $\iota^\ast \!: {\widehat{\mathrm{env}}}ll^2 \to {\widehat{\mathrm{env}}}ll^{p'}$ is the canonical inclusion of ${\widehat{\mathrm{env}}}ll^2$ in ${\widehat{\mathrm{env}}}ll^{p'}$. Let $\cal U$ be a free ultrafilter over ${{\mathbb M}hbb N}$, and define
\begin{equation} \label{leftid}
L_2 := \varcl{\{ ( T_n \iota )_{\cal U} : ( T_n )_{\cal U} \in ({\cal B}({\widehat{\mathrm{env}}}ll^2,{\widehat{\mathrm{env}}}ll^p))_{\cal U} \}}^{({\cal B}({\widehat{\mathrm{env}}}ll^p))_{\cal U}}.
{\widehat{\mathrm{env}}}nd{equation}
Obviously, $L_2$ is a closed left ideal of $({\cal B}({\widehat{\mathrm{env}}}ll^p))_{\cal U}$. By Pitt's theorem (\cite[Proposition 2.c.3]{LT}), ${\cal B}({\widehat{\mathrm{env}}}ll^2,{\widehat{\mathrm{env}}}ll^p) = {\cal K}({\widehat{\mathrm{env}}}ll^2,{\widehat{\mathrm{env}}}ll^p)$ holds, so that $L_2$ is, in fact, a closed ideal of $({\cal K}({\widehat{\mathrm{env}}}ll^p))_{\cal U}$.
\par
Recall that, for ${\widehat{\mathrm{env}}}psilon > 0$, a $(1+{\widehat{\mathrm{env}}}psilon)$-isometry from a Banach space $E$ into a Banach space $F$, is a linear map $T \!: E \to F$ satisfying
\[
(1 -{\widehat{\mathrm{env}}}psilon)\| x \| \leq \| Tx \| \leq (1+{\widehat{\mathrm{env}}}psilon) \| x \| \qquad (x \in E).
\]
By \cite[19.1 Dvoretzky's Theorem]{DJT}, there is, for each infinite-dimensional Banach space $E$, for each $n \in {{\mathbb M}hbb N}$, and for each ${\widehat{\mathrm{env}}}psilon > 0$, a $(1+{\widehat{\mathrm{env}}}psilon)$-isometry from ${\widehat{\mathrm{env}}}ll^2_n$ into $E$. We shall use this theorem to obtain particular elements of $L_2$. For each $n \in {{\mathbb M}hbb N}$, let $\pi_n \!: {\widehat{\mathrm{env}}}ll^2 \to {\widehat{\mathrm{env}}}ll^2_n$ denote the canonical projection onto the first $n$ coordinates, and let, for each $n \in {{\mathbb M}hbb N}$, $\tau_n \!: {\widehat{\mathrm{env}}}ll^2_n \to {\widehat{\mathrm{env}}}ll^p$ be a $\left(1 + \frac{1}{n} \right)$-isometry, which exists by Dvoretzky's theorem. Then $(\tau_n \pi_n \iota )_{\cal U}$ lies in $L_2$.
\par
The following is our technical main result in this section:
\begin{lemma} \label{dvolem}
Let $p \in (1,2)$, let $\cal U$ be a free ultrafilter over ${{\mathbb M}hbb N}$, and let $L_2$ and $(\tau_n \pi_n \iota )_{\cal U}$ be defined as above. Then we have
\[
\| (\tau_n \pi_n \iota )_{\cal U} (T_n)_{\cal U} - (\tau_n \pi_n \iota )_{\cal U} \| \geq 1
\qquad ((T_n)_{\cal U} \in L_2).
\]
{\widehat{\mathrm{env}}}nd{lemma}
\begin{proof}
Assume towards a contradiction that there are $\theta \in [0,1)$ and $( T_n )_{\cal U} \in ({\cal B}({\widehat{\mathrm{env}}}ll^2,{\widehat{\mathrm{env}}}ll^p))_{\cal U}$ such that
\begin{equation} \label{dvoineq}
\theta > \| (\tau_n \pi_n \iota )_{\cal U} (T_n \iota)_{\cal U} - (\tau_n \pi_n \iota )_{\cal U} \|
= \lim_{n \in \cal U} \| \tau_n \pi_n \iota T_n \iota - \tau_n \pi_n \iota \|
= \lim_{n \in \cal U} \| \pi_n \iota T_n \iota - \pi_n \iota \|,
{\widehat{\mathrm{env}}}nd{equation}
where the last equality is due to the fact that $\tau_n$ is a $\left(1 + \frac{1}{n} \right)$-isometry for each $n \in {{\mathbb M}hbb N}$.
\par
Let $T := \text{weak-}\lim_{n \in \cal U} T_n \in {\cal K}({\widehat{\mathrm{env}}}ll^2,{\widehat{\mathrm{env}}}ll^p)$. (The limit exists by \cite[Propositon 1.45]{Hei} because ${\cal K}({\widehat{\mathrm{env}}}ll^2,{\widehat{\mathrm{env}}}ll^p)$ is reflexive, so that its closed unit ball is weakly compact.) Let $x \in {\widehat{\mathrm{env}}}ll^p$ and $\xi \in {\widehat{\mathrm{env}}}ll^2$. Then we have:
\[
\begin{split}
| {{\mathrm{op}}eratorname{lan}}gle \xi, (\iota T \iota - \iota)(x) {{\mathrm{op}}eratorname{ran}}gle |
& = | {{\mathrm{op}}eratorname{lan}}gle \iota^\ast(\xi), (T\iota)(x) {{\mathrm{op}}eratorname{ran}}gle - {{\mathrm{op}}eratorname{lan}}gle \xi, \iota(x) {{\mathrm{op}}eratorname{ran}}gle | \\
& = \lim_{n \in \cal U} | {{\mathrm{op}}eratorname{lan}}gle \iota^\ast(\xi), (T_n \iota)(x) {{\mathrm{op}}eratorname{ran}}gle - {{\mathrm{op}}eratorname{lan}}gle \xi, \iota(x) {{\mathrm{op}}eratorname{ran}}gle | \\
& = \lim_{n \in \cal U} | {{\mathrm{op}}eratorname{lan}}gle \xi, (\iota T_n \iota - \iota)(x) {{\mathrm{op}}eratorname{ran}}gle | \\
& = \lim_{n \in \cal U} | {{\mathrm{op}}eratorname{lan}}gle \pi_n^\ast(\xi), (\iota T_n \iota - \iota)(x) {{\mathrm{op}}eratorname{ran}}gle |,
\qquad\text{because $\lim_{n \to \infty} \| \pi_n^\ast(\xi) - \xi \|_2 = 0$}, \\
& = \lim_{n \in \cal U} | {{\mathrm{op}}eratorname{lan}}gle \xi, (\pi_n\iota T_n \iota - \pi_n\iota)(x) {{\mathrm{op}}eratorname{ran}}gle | \\
& \leq \lim_{n \in \cal U} \| \pi_n\iota T_n \iota - \pi_n\iota \| \| x \| \| \xi \| \\
& < \theta \| x \| \| \xi \|, \qquad\text{by ({{\mathrm{op}}eratorname{Re}}f{dvoineq})}.
{\widehat{\mathrm{env}}}nd{split}
\]
It follows that
\begin{equation} \label{dvoineq2}
\| \iota T \iota - \iota \| \leq \theta.
{\widehat{\mathrm{env}}}nd{equation}
\par
Since $T$ is compact, so is $\iota T \iota$. Hence, there is a strictly increasing sequence $( n_k)_{k=1}^\infty$ in ${{\mathbb M}hbb N}$ such that $((\iota T \iota)(\delta_{n_k}))_{k=1}^\infty$ is norm convergent in ${\widehat{\mathrm{env}}}ll^2$ with limit ${\widehat{\mathrm{env}}}ta$, say. It follows that
\begin{equation} \label{dvoeq}
\lim_{k \to \infty} {{\mathrm{op}}eratorname{lan}}gle \delta_{n_k}, (\iota T \iota)(\delta_{n_k}) {{\mathrm{op}}eratorname{ran}}gle =
\lim_{k \to \infty} {{\mathrm{op}}eratorname{lan}}gle \delta_{n_k},{\widehat{\mathrm{env}}}ta {{\mathrm{op}}eratorname{ran}}gle = 0.
{\widehat{\mathrm{env}}}nd{equation}
Together, ({{\mathrm{op}}eratorname{Re}}f{dvoineq2}) and ({{\mathrm{op}}eratorname{Re}}f{dvoeq}) yield
\begin{multline*}
1 = \lim_{k \to \infty} {{\mathrm{op}}eratorname{lan}}gle \delta_{n_k}, \iota(\delta_{n_k}) {{\mathrm{op}}eratorname{ran}}gle \\
= \lim_{k \to \infty} | {{\mathrm{op}}eratorname{lan}}gle \delta_{n_k}, (\iota T \iota)(\delta_{n_k}) {{\mathrm{op}}eratorname{ran}}gle -
{{\mathrm{op}}eratorname{lan}}gle \delta_{n_k}, \iota(\delta_{n_k}) {{\mathrm{op}}eratorname{ran}}gle |
= \lim_{k \to \infty} | {{\mathrm{op}}eratorname{lan}}gle \delta_{n_k}, (\iota T \iota - \iota)(\delta_{n_k}) {{\mathrm{op}}eratorname{ran}}gle | \leq \theta,
{\widehat{\mathrm{env}}}nd{multline*}
which is impossible because $\theta \in [0,1)$.
{\widehat{\mathrm{env}}}nd{proof}
\par
The following is now immediate:
\begin{proposition} \label{norai}
Let $p \in (1,2)$, let $\cal U$ be a free ultrafilter over ${{\mathbb M}hbb N}$, and let $L_2$ be the closed left ideal of $({\cal K}({\widehat{\mathrm{env}}}ll^p))_{\cal U}$ defined in {\widehat{\mathrm{env}}}mph{({{\mathrm{op}}eratorname{Re}}f{leftid})}. Then $L_2$ does not have a right approximate identity.
{\widehat{\mathrm{env}}}nd{proposition}
\begin{remark}
Both Lemma {{\mathrm{op}}eratorname{Re}}f{dvolem} and Proposition {{\mathrm{op}}eratorname{Re}}f{norai} remain true in the slightly more general situation where $\cal U$ is a countably incomplete ultrafilter over an arbitrary index set.
{\widehat{\mathrm{env}}}nd{remark}
\par
{\widehat{\mathrm{env}}}mph{If} we could establish that $L_2$ is weakly complemented, i.e., has a complemented annihilator in $({\cal K}({\widehat{\mathrm{env}}}ll^p))_{\cal U}^\ast$, then we know that $({\cal K}({\widehat{\mathrm{env}}}ll^p))_{\cal U}$---and thus, by Theorem {{\mathrm{op}}eratorname{Re}}f{thm1}, ${\cal B}({\widehat{\mathrm{env}}}ll^p)$---cannot be amenable. Unfortunately, such a proof eludes us, mostly due to the lack of a suitable description of $({\cal K}({\widehat{\mathrm{env}}}ll^p))_{\cal U}^\ast$. Nevertheless, we are able to show that the annihilator of $L_2$ in a certain closed subspace of $({\cal K}({\widehat{\mathrm{env}}}ll^p))_{\cal U}^\ast$ is indeed complemented.
\par
We achieve this as a by-product of a general complementation result for ultrapowers of vector valued ${\widehat{\mathrm{env}}}ll^p$-spaces.
\par
Given a set $S$ and an ultrafilter $\cal U$ over some index set ${\mathbb M}hbb{I}$, we use ${{\mathrm{op}}eratorname{lan}}gle S {{\mathrm{op}}eratorname{ran}}gle_{\cal U}$ for the corresponding {\widehat{\mathrm{env}}}mph{set theoretic ultrapower} (see \cite{Hei} for the definition). For $( s_i )_{i \in {\mathbb M}hbb I} \in S^{\mathbb M}hbb{I}$, we denote its image in ${{\mathrm{op}}eratorname{lan}}gle S {{\mathrm{op}}eratorname{ran}}gle_{\cal U}$ by ${{\mathrm{op}}eratorname{lan}}gle s_i {{\mathrm{op}}eratorname{ran}}gle_{\cal U}$.
\par
The following lemma relates, for $p \in [1,\infty)$, a Banach space $E$, and an ultrafilter $\cal U$ (over an arbitrary index set), the spaces ${\widehat{\mathrm{env}}}ll^p({{\mathrm{op}}eratorname{lan}}gle {{\mathbb M}hbb N} {{\mathrm{op}}eratorname{ran}}gle_{\cal U}, (E)_{\cal U})$ and $({\widehat{\mathrm{env}}}ll^p(E))_{\cal U}$. We identify the finitely supported functions in ${\widehat{\mathrm{env}}}ll^p({{\mathrm{op}}eratorname{lan}}gle {{\mathbb M}hbb N} {{\mathrm{op}}eratorname{ran}}gle_{\cal U}, (E)_{\cal U})$ with the algebraic tensor product $c_{00}({{\mathrm{op}}eratorname{lan}}gle {{\mathbb M}hbb N} {{\mathrm{op}}eratorname{ran}}gle_{\cal U}) \otimes (E)_{\cal U}$, where $c_{00}({{\mathrm{op}}eratorname{lan}}gle {{\mathbb M}hbb N} {{\mathrm{op}}eratorname{ran}}gle_{\cal U})$ are the finitely supported functions from ${{\mathrm{op}}eratorname{lan}}gle {{\mathbb M}hbb N} {{\mathrm{op}}eratorname{ran}}gle_{\cal U}$ into ${{\mathbb M}hbb C}$.
\begin{lemma} \label{subspace}
Let $p \in [1,\infty)$, let $E$ be a Banach space, and let $\cal U$ be an ultrafilter. Then there is a unique isometry $J_p \!: {\widehat{\mathrm{env}}}ll^p({{\mathrm{op}}eratorname{lan}}gle {{\mathbb M}hbb N} {{\mathrm{op}}eratorname{ran}}gle_{\cal U}, (E)_{\cal U}) \to ({\widehat{\mathrm{env}}}ll^p(E))_{\cal U}$ given by
\begin{equation} \label{embed}
J_p \left( \delta_{{{\mathrm{op}}eratorname{lan}}gle n_i {{\mathrm{op}}eratorname{ran}}gle_{\cal U}} \otimes ( x_i)_{\cal U} \right) = ( \delta_{n_i} \otimes x_i )_{\cal U}
\qquad \left(\delta_{{{\mathrm{op}}eratorname{lan}}gle n_i {{\mathrm{op}}eratorname{ran}}gle_{\cal U}} \in {{\mathrm{op}}eratorname{lan}}gle {{\mathbb M}hbb N} {{\mathrm{op}}eratorname{ran}}gle_{\cal U}, \, (x_i)_{\cal U} \in (E)_{\cal U} \right)
{\widehat{\mathrm{env}}}nd{equation}
{\widehat{\mathrm{env}}}nd{lemma}
\begin{proof}
It is routinely checked that ({{\mathrm{op}}eratorname{Re}}f{embed}) defines an isometry from $c_{00}({{\mathrm{op}}eratorname{lan}}gle {{\mathbb M}hbb N} {{\mathrm{op}}eratorname{ran}}gle_{\cal U}) \otimes (E)_{\cal U}$ into $({\widehat{\mathrm{env}}}ll^p(E))_{\cal U}$, which then extends to all of ${\widehat{\mathrm{env}}}ll^p({{\mathrm{op}}eratorname{lan}}gle {{\mathbb M}hbb N} {{\mathrm{op}}eratorname{ran}}gle_{\cal U}, (E)_{\cal U})$ by continuity.
{\widehat{\mathrm{env}}}nd{proof}
\par
Lemma {{\mathrm{op}}eratorname{Re}}f{subspace} enables us to canonically identify ${\widehat{\mathrm{env}}}ll^p( {{\mathrm{op}}eratorname{lan}}gle {{\mathbb M}hbb N} {{\mathrm{op}}eratorname{ran}}gle_{\cal U},(E)_{\cal U})$ with a closed subspace of $({\widehat{\mathrm{env}}}ll^p(E))_{\cal U}$.
\par
Given a Banach space $E$ and an ultrafilter $\cal U$, there is a canonical duality between $(E)_{\cal U}$ and $(E^\ast)_{\cal U}$, which induces an isometric embedding of $(E^\ast)_{\cal U}$ into $(E)_{\cal U}^\ast$; for countably incomplete $\cal U$, this embedding is an isomorphism if and only if $(E)_{\cal U}$ is reflexive (\cite[Proposition 7.1]{Hei}). Recall that $E$ is called {\widehat{\mathrm{env}}}mph{superreflexive} if every Banach space that can be finite represented in $E$ is reflexive; equivalently, $E$ is superreflexive if and only if $(E)_{\cal U}$ is reflexive for each ultrafilter $\cal U$ (\cite[Proposition 6.4]{Hei}). Also, if $E$ is superreflexive and $p \in (1,\infty)$, then ${\widehat{\mathrm{env}}}ll^p(E)$ is also superreflexive (\cite[Proposition 4]{Daw0}). All this guarantees that the map $\Pi_p$ in the following proposition is well defined.
\begin{proposition} \label{kernel}
Let $p \in (1,\infty)$, let $E$ be a superreflexive Banach space, let $\cal U$ be an ultrafilter, and let $J_p \!: \!: {\widehat{\mathrm{env}}}ll^p({{\mathrm{op}}eratorname{lan}}gle {{\mathbb M}hbb N} {{\mathrm{op}}eratorname{ran}}gle_{\cal U}, (E)_{\cal U}) \to ({\widehat{\mathrm{env}}}ll^p(E))_{\cal U}$ and $J_{p'} \!: \!: {\widehat{\mathrm{env}}}ll^{p'}({{\mathrm{op}}eratorname{lan}}gle {{\mathbb M}hbb N} {{\mathrm{op}}eratorname{ran}}gle_{\cal U}, (E^\ast)_{\cal U}) \to ({\widehat{\mathrm{env}}}ll^{p'}(E^\ast))_{\cal U}$ be as in Lemma {\widehat{\mathrm{env}}}mph{{{\mathrm{op}}eratorname{Re}}f{subspace}}. Then $\Pi_p := J_p J^\ast_{p'}$ is a norm one projection onto ${\widehat{\mathrm{env}}}ll^p( {{\mathrm{op}}eratorname{lan}}gle {{\mathbb M}hbb N} {{\mathrm{op}}eratorname{ran}}gle_{\cal U},(E)_{\cal U})$. Moreover, we have for any $q \in (p,\infty]$ that
\begin{equation} \label{kernelq}
\ker \Pi_p = \left\{ (x_i)_{\cal U} \in ({\widehat{\mathrm{env}}}ll^p(E))_{\cal U} : \lim_{i \in \cal U} \| x_i \|_{{\widehat{\mathrm{env}}}ll^q(E)} = 0 \right\}.
{\widehat{\mathrm{env}}}nd{equation}
{\widehat{\mathrm{env}}}nd{proposition}
\begin{proof}
It is easy to see that $\Pi_p$ is indeed a norm one projection onto ${\widehat{\mathrm{env}}}ll^p( {{\mathrm{op}}eratorname{lan}}gle {{\mathbb M}hbb N} {{\mathrm{op}}eratorname{ran}}gle_{\cal U},(E)_{\cal U})$.
\par
Let ${\mathbb M}hbb I$ be the index set over which $\cal U$ is defined. For $( n_i )_{i \in {\mathbb M}hbb I} \in {{\mathbb M}hbb N}^{\mathbb M}hbb{I}$, let $P_{n_i} \!: {\widehat{\mathrm{env}}}ll^p(E) \to E$ denote the projection onto the $n_i$-th coordinate. From the definition of $\Pi_p$, it is clear that $( x_i )_{\cal U} \in ({\widehat{\mathrm{env}}}ll^p(E))_{\cal U}$ belongs to $\ker \Pi_p$ if and only if $\lim_{i \in \cal U} \| P_{n_i} x_i \|_E = 0$ for any
$( n_i )_{i \in {\mathbb M}hbb I} \in {{\mathbb M}hbb N}^{\mathbb M}hbb{I}$. It follows that
\[
\ker \Pi_p \supset \left\{ (x_i)_{\cal U} \in ({\widehat{\mathrm{env}}}ll^p(E))_{\cal U} : \lim_{i \in \cal U} \| x_i \|_{{\widehat{\mathrm{env}}}ll^\infty(E)} = 0 \right\}
\]
For the converse inclusion, let $(x_i)_{\cal U} \in ({\widehat{\mathrm{env}}}ll^p(E))_{\cal U}$ be such that $\lim_{i \in \cal U} \| x_i \|_{{\widehat{\mathrm{env}}}ll^\infty(E)} =: \delta > 0$. Let $U \in \cal U$ be such that $\| x_i \|_{{\widehat{\mathrm{env}}}ll^\infty(E)} = \sup_{n \in {{\mathbb M}hbb N}} \| P_n x_i \| > \frac{\delta}{2}$ for each $i \in U$. For each $i \in U$, choose $n_i \in {{\mathbb M}hbb N}$ such that $\| P_{n_i} x_i \|_{{\widehat{\mathrm{env}}}ll^\infty(E)} > \frac{\delta}{2}$. It follows that $\lim_{i \in \cal U} \| P_{n_i} x_i \|_{{\widehat{\mathrm{env}}}ll^\infty(E)} \geq \frac{\delta}{2} > 0$, so that $(x_i)_{\cal U} \in ({\widehat{\mathrm{env}}}ll^p(E))_{\cal U} \notin \ker \Pi_p$. All in all, we have
\begin{equation} \label{kernelinfty}
\ker \Pi_p = \left\{ (x_i)_{\cal U} \in ({\widehat{\mathrm{env}}}ll^p(E))_{\cal U} : \lim_{i \in \cal U} \| x_i \|_{{\widehat{\mathrm{env}}}ll^\infty(E)} = 0 \right\}
{\widehat{\mathrm{env}}}nd{equation}
\par
Let $q \in (p,\infty)$. In view of ({{\mathrm{op}}eratorname{Re}}f{kernelinfty}), it is clear that
\[
\ker \Pi_p \supset \left\{ (x_i)_{\cal U} \in ({\widehat{\mathrm{env}}}ll^p(E))_{\cal U} : \lim_{i \in \cal U} \| x_i \|_{{\widehat{\mathrm{env}}}ll^q(E)} = 0 \right\}.
\]
For the converse inclusion, note that, for any $x = ( x_n )_{n=1}^\infty \in {\widehat{\mathrm{env}}}ll^p(E)$, we have
\[
\| x \|_{{\widehat{\mathrm{env}}}ll^q(E)}^q = \sum_{n=1}^\infty \| x_n \|^q =
\sum_{n=1}^\infty \| x_n \|^{q-p} \| x_n \|^p
\leq \| x \|_{{\widehat{\mathrm{env}}}ll^\infty(E)}^{q-p} \sum_{n=1}^\infty \| x_n \|^p
\leq \| x \|_{{\widehat{\mathrm{env}}}ll^\infty(E)}^{q-p} \| x \|^p_{{\widehat{\mathrm{env}}}ll^p(E)}.
\]
Consequently, if $( x_i )_{\cal U} \in \ker \Pi_p$, i.e., $\lim_{i \in \cal U} \| x_i \|_{{\widehat{\mathrm{env}}}ll^\infty(E)} = 0$, then
$\lim_{i \in \cal U} \| x_i \|_{{\widehat{\mathrm{env}}}ll^q(E)} = 0$ holds as well. This proves ({{\mathrm{op}}eratorname{Re}}f{kernelq}).
{\widehat{\mathrm{env}}}nd{proof}
\par
Let $p \in (1,2)$, and let $\cal U$ be a free ultrafilter over ${{\mathbb M}hbb N}$. We can canonically represent $({\cal B}({\widehat{\mathrm{env}}}ll^p))_{\cal U}$ on $({\widehat{\mathrm{env}}}ll^p)_{\cal U}$ by letting
\[
( T_n )_{\cal U} ( x_n )_{\cal U} = (T_n x_n)_{\cal U} \qquad ( ( T_n )_{\cal U} \in ({\cal B}({\widehat{\mathrm{env}}}ll^p))_{\cal U}, \,
( x_n )_{\cal U} \in ({\widehat{\mathrm{env}}}ll^p)_{\cal U}).
\]
Clearly, $( T_n )_{\cal U} ( x_n )_{\cal U} = 0$ holds for all $( T_n )_{\cal U} \in L_2$ if and only if $\lim_{n \in \cal U} \| x_n \|_2 = 0$, i.e., $( x_n )_{\cal U} \in \ker \Pi_p$ by Proposition {{\mathrm{op}}eratorname{Re}}f{kernel}.
\par
The Banach space ${\cal B}(({\widehat{\mathrm{env}}}ll^p)_{\cal U})$ has the canonical predual $({\widehat{\mathrm{env}}}ll^p)_{\cal U} \hat{\otimes} ({\widehat{\mathrm{env}}}ll^{p'})_{\cal U}$, which, by \cite[Proposition 4.7]{Daw}, embeds isometrically into $({\widehat{\mathrm{env}}}ll^p \hat{\otimes} {\widehat{\mathrm{env}}}ll^{p'})_{\cal U} = ({\cal K}({\widehat{\mathrm{env}}}ll^p)^\ast)_{\cal U}$ and thus into $({\cal K}({\widehat{\mathrm{env}}}ll^p))_{\cal U}^\ast$ (see \cite[p.\ 87]{Hei}). It therefore makes sense to speak of the annihilator of $L_2$ in $({\widehat{\mathrm{env}}}ll^p)_{\cal U} \hat{\otimes} ({\widehat{\mathrm{env}}}ll^{p'})_{\cal U}$.
\par
In view of the foregoing we have:
\begin{corollary} \label{complcor}
Let $p \in (1,2)$, and let $\cal U$ be a free ultrafilter. Then the annihilator of $L_2$ in $({\widehat{\mathrm{env}}}ll^p)_{\cal U} \hat{\otimes} ({\widehat{\mathrm{env}}}ll^{p'})_{\cal U}$ is its complemented subspace $\ker \Pi_p \hat{\otimes} ({\widehat{\mathrm{env}}}ll^{p'})_{\cal U}$, where $\Pi_p$ is the canonical projection from $({\widehat{\mathrm{env}}}ll^p)_{\cal U}$ onto ${\widehat{\mathrm{env}}}ll^p({{\mathrm{op}}eratorname{lan}}gle {{\mathbb M}hbb N} {{\mathrm{op}}eratorname{ran}}gle_{\cal U})$.
{\widehat{\mathrm{env}}}nd{corollary}
\begin{remark}
It would be interesting to know whether the annihilator of $L_2$ in $({\cal K}({\widehat{\mathrm{env}}}ll^p)^\ast)_{\cal U}$ is complemented: as $({\cal K}({\widehat{\mathrm{env}}}ll^p))_{\cal U}^\ast$ can be finitely represented in $({\cal K}({\widehat{\mathrm{env}}}ll^p)^\ast)_{\cal U}$ (\cite[Theorem 7.3]{Hei}), this would further support our belief that $L_2$ is weakly complemented.
{\widehat{\mathrm{env}}}nd{remark}
{{\mathrm{op}}eratorname{Re}}newcommand{1.2}{1.0}
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\begin{tabbing}
\textit{Second author's address}: \= Department of Mathematical and Statistical Sciences \kill
\textit{First author's address}: \> Department of Pure Mathematics \\
\> University of Leeds \\
\> Leeds, LS2 9JT \\
\> United Kingdom \\[
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\textit{E-mail}: \> \texttt{[email protected]} \\[
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\textit{Second author's address}: \> Department of Mathematical and Statistical Sciences \\
\> University of Alberta \\
\> Edmonton, Alberta \\
\> Canada T6G 2G1 \\[
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\textit{E-mail}: \> \texttt{[email protected]}
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\begin{document}
\title{On Axially Symmetric Incompressible
Magnetohydrodynamics in Three Dimensions}
\author{Zhen Lei
\footnote{School of Mathematical Sciences; LMNS and Shanghai Key
Laboratory for Contemporary Applied Mathematics, Fudan
University, Shanghai 200433, P. R. China. Email: [email protected]} }
\date{\today}
\maketitle
\begin{abstract}
In this short article, we prove the global regularity of axially
symmetric solutions to the systems of incompressible ideal magnetohydrodynamics
and resistive magnetohydrodynamics
in three dimensions in the csae that the magnetic fields are purely swirling and perpendicular
to the velocity fields.
\end{abstract}
Keywords: Magnetohydrodynamics, global regularity, axial symmetry.
\section{Introduction}
Magnetohydrodynamics (MHD) is to study the behavior of an
electrically-conducting fluids. Examples of such fluids include
plasmas, liquid metals, salt water, etc. The field of MHD was
initiated by Hannes Alfv${\rm \acute{e}}$n, for which he received
the Nobel Prize in Physics in 1970. However, the mathematical theory
on MHD is still very little known until today.
The fundamental concept behind MHD is that magnetic fields can
induce currents in a moving conductive fluid, which in turn creates
forces on the fluid and also changes the magnetic field itself. MHD
owes its peculiar interest and difficulty to this interaction
between the field and the fluid motion. The set of equations which
describe MHD are a combination of the Navier-Stokes equations of
fluid dynamics and Maxwell's equations of electromagnetism.
Our main result concerns the following incompressible three-dimensional ideal MHD:
\begin{equation}\label{iMHD}
\begin{cases}
\partial_t {\bf u} + ({\bf u} \cdot\nabla) {\bf u} + \nabla
p = \mu\Delta {\bf u} + \frac{1}{\mu_0}(\nabla\times {\bf B})\times {\bf B},\\[-4mm]\\
\partial_t{\bf B} = \nabla\times ({\bf u} \times {\bf B}),\\[-4mm]\\
\nabla\cdot {\bf u} =0,\quad \nabla\cdot {\bf B}=0,
\end{cases}
\end{equation}
where ${\bf B}$ denotes the magnetic field, ${\bf u}$ the bulk
plasma velocity and $p$ the plasma pressure. The magnetic constant $\mu_0$ and the fluid viscosity $\mu$ are both positive.
We will set all the constants to be 1 since they play no role in this paper. The ideal
MHD is used when the electrically-conducting fluid has so little resistivity that it can be treated as a
perfect conductor. This is the limit of infinite magnetic Reynolds
number. For applications of ideal MHD, see, for instance, \cite{CC}.
The following theorem shows that if the magnetic field is purely swirling
and is perpendicular to the velocity field, then the 3D
incompressible ideal MHD \eqref{iMHD} is globally well-posed in the axially symmetric case.
\begin{thm}\label{thm-iMHD}
Suppose that ${\bf u}_0$ and ${\bf B}_0$ are both axially symmetric
divergence-free vectors with $u_0^\theta = 0$ and $B^r_0 = B^z_0 =
0$. Moreover, we assume that $({\bf u}_0, {\bf B}_0) \in H^2$ and $\frac{B^\theta_0}{r} \in L^\infty$. Then there
exists a unique global solution $({\bf u}, {\bf B})$ for the ideal
MHD \eqref{iMHD} with the initial data $({\bf u}_0, {\bf B}_0)$
which satisfies
\begin{equation}\nonumber
\|{\bf u}(t, \cdot)\|_{H^2}^2 + \|{\bf B}(t, \cdot)\|_{H^2}^2 +
\int_0^t\|\nabla{\bf u}\|_{H^2}^2ds \lesssim
e^{e^{e^{t^{\frac{5}{4}}}}}.
\end{equation}
\end{thm}
The notations used here will be introduced in section 2. Note that the
Faraday's equation for ${\bf B}$ in \eqref{iMHD} is exactly the same
as the vorticity equation for the 3D incompressible Euler equations
(by identifying ${\bf B}$ and $\nabla \times {\bf u}$). This may
lead to an essential difficulties for the global well-posedness of
the ideal MHD \eqref{iMHD} in general case. Indeed, the global
regularity of \eqref{iMHD} is widely open in the
even two-dimensional case if the magnetic field is non-trivial. We
achieve Theorem \ref{iMHD} by exploring the underlying special
structures of the MHD system in axially symmetric case. The magnetic
stretching term ${\bf B}\cdot\nabla {\bf u}$ in Faraday's equation
can be absorbed into the convection term by dividing the equation by
$r$. This yields that $\Pi = \frac{B^\theta}{r}$ is only transported
by the velocity field ${\bf u}$. On the other hand, by dividing $r$
in the vorticity equation, one can absorb the vortex stretching term
into the convection term, leaving only one term involving $\Pi$ as a
forcing one in $\Omega$ equation. See section 2 and 3 for more details.
Similar result in Theorem \ref{iMHD} is of course expected to hold for the
following resistive MHD:
\begin{equation}\label{RMHD}
\begin{cases}
\partial_t {\bf u} + ({\bf u} \cdot\nabla) {\bf u} + \nabla
p = \Delta {\bf u} + (\nabla\times {\bf B})\times {\bf B},\\[-4mm]\\
\partial_t{\bf B} = \nu\Delta {\bf B} + \nabla\times ({\bf u} \times {\bf B}),\\[-4mm]\\
\nabla\cdot {\bf u} =0,\quad \nabla\cdot {\bf B}=0.
\end{cases}
\end{equation}
Again, we will set the resistivity constant $\nu > 0$ to be 1
since it plays no role here. We have the following theorem:
\begin{thm}\label{thm-RMHD}
Suppose that ${\bf u}_0$ and ${\bf B}_0$ are both axially symmetric
divergence-free vectors with $u_0^\theta = 0$ and $B^r_0 = B^z_0 =
0$. Moreover, we assume that $({\bf u}_0, {\bf B}_0) \in H^1$ and
$\frac{B^\theta_0}{r} \in L^\infty$. Then there exists a unique
global solution $({\bf u}, {\bf B})$ for the resistive MHD
\eqref{RMHD} with the initial data $({\bf u}_0, {\bf B}_0)$.
Moreover, $({\bf u}, {\bf B})$ is smooth in the sense that $({\bf
u}(t, \cdot), {\bf B}(t, \cdot)) \in H^s$ for any $s \geq 0$ and $t
> 0$.
\end{thm}
Our motivation of the above results is a novelty observation on the connections between
MHD and axially symmetric Navier-Stokes equations.
If we rewrite the 3D incompressible
axially symmetric Navier-Stokes equations as \eqref{NS-axi}, in terms of ${\bf u} =
u^r{\bf e}_r + u^z{\bf e}_z$ and ${\bf b} = u^\theta {\bf
e}_\theta$, then there is only a sign difference\footnote{In fact,
the pressure is also changed. But the pressure is not a troublesome
term for our purpose due to the divergence-free condition.} between
the Navier-Stokes equations \eqref{NS-axi} for $({\bf u}, {\bf b})$
and the resistive MHD \eqref{RMHD} for $({\bf u}, {\bf B})$ (see
Remark \ref{connection} in section 2 for details). However, this
difference of sign significantly changes the difficulties in solving
3D axially symmetric incompressible equations of MHD.
We remark that the perfect resistive case will be treated in a forthcoming paper \cite{LeiLi}. It is also
interesting to consider the case when $u^\theta = B^\theta = 0$.
Before ending the introduction, let us mention some important
results in the field of incompressible MHD. The local well-posedness
of the resistive MHD \eqref{RMHD} was established in
\cite{SermangeTemam} where the authors also proved the global
well-posedness in 2D case. A nontrivial blowup criterion for the
perfect resistive MHD was established in terms of only
$L^1_t({\rm BMO})$ norm of vorticity of the velocity field in
\cite{LeiZ-2}. Recently, Lin, Xu and Zhang \cite{LXZ} obtained the global well-posedness of classical
solutions for the 2D ideal MHD \eqref{iMHD} under the assumption
that the initial velocity field and the displacement of the magnetic
field from a non-zero constant is sufficiently small in appropriate
Sobolev spaces. Cao and Wu \cite{CWu} proved the global regularity
of 2D resistive MHD with partial viscosity and resistivity (see also \cite{CWY} and the references therein). We also
emphasis the partial regularity theory and Serrin type criterions in
\cite{HeXin1, HeXin2}, and various blowup criterions in \cite{CKS,
CMZ} (see also the reference therein).
The remaining of this paper is simply organized as follows: In
section 2 we will derive the axisymmetric MHD in cylindrical
coordinate. We will also make a comment on the difference between
the resistive MHD \eqref{RMHD} and the axially symmetric
Navier-Stokes equations and prove a maximum principle for $\Pi$. We
will prove Theorem \ref{thm-iMHD} in section 3. Then in section 4 we present the proof of Theorem \ref{RMHD}.
\section{Axially Symmetric MHD and A Maximum Principle}
In this section we will first derive the incompressible axially
symmetric MHD in cylindrical coordinate. Then we show that the
quantity $\Pi$ satisfies a maximum principle. We also present an
interesting connection between the axisymmetric MHD studied in
Theorem \ref{thm-RMHD} and the axisymmetric Navier-Stokes equations
with non-trivial swirl $u^\theta$ (see \eqref{mhd-axi} and
\eqref{NS-axi}).
Let us begin with some notations. A point in $\mathbb{R}^3$ is
denoted by ${\bf x} = (x_1, x_2, z)$. Let $r = \sqrt{x_1^2 + x_2^2}$
and
\begin{equation}\nonumber
{\bf e}_r = \begin{pmatrix}\frac{x_1}{r}\\ \frac{x_2}{r}\\ 0
\end{pmatrix},
\quad {\bf e}_\theta = \begin{pmatrix}- \frac{x_2}{r}\\ \frac{x_1}{r}\\
0
\end{pmatrix},
\quad {\bf e}_z = \begin{pmatrix}0\\ 0\\ 1
\end{pmatrix}
\end{equation}
be the three orthogonal unit vectors along the radial, the angular,
and the axial directions respectively. An axially symmetric solution
to the 3D incompressible MHD \eqref{RMHD} is a solution $({\bf u},
{\bf B}, p)$ which takes the following form
\begin{equation}\nonumber
\begin{cases}
{\bf u}(t, {\bf x}) = u^r(t, r, z){\bf e}_r + u^\theta(t, r, z){\bf
e}_\theta + u^z(t, r, z){\bf e}_z,\\[-4mm]\\
{\bf B}(t, {\bf x}) = B^r(t, r, z){\bf e}_r + B^\theta(t, r, z){\bf
e}_\theta + B^z(t, r, z){\bf e}_z,\\[-4mm]\\
p(t, {\bf x}) = p(t, r, z).
\end{cases}
\end{equation}
We will also write the vorticity field $\nabla\times {\bf u}$ in
cylindrical coordinate:
\begin{equation}\nonumber
\nabla\times {\bf u}(t, {\bf x}) = \omega^r(t, r, z){\bf e}_r +
\omega^\theta(t, r, z){\bf e}_\theta + \omega^z(t, r, z){\bf e}_z,
\end{equation}
where
\begin{equation}\nonumber
\omega^r = - \partial_zu^\theta,\quad \omega^\theta = \partial_zu^r
- \partial_ru^z,\quad \omega^z = \frac{1}{r}\partial_r(ru^\theta).
\end{equation}
Define
\begin{equation}\label{defn-1}
\Pi = \frac{B^\theta}{r},\quad \Omega =
\frac{\omega^\theta}{r},\quad \Gamma = ru^\theta.
\end{equation}
By expanding the Lorentz force term as
\begin{equation}\nonumber
(\nabla\times {\bf B}) \times {\bf B} = {\bf B}\cdot\nabla {\bf B} -
\nabla\frac{|{\bf B}|^2}{2},
\end{equation}
and then taking the inner product of ${\bf u}$ and ${\bf B}$
equations with ${\bf e_r}$, ${\bf e_\theta}$ and ${\bf e_z}$,
respectively, we can derive the resistive MHD in cylindrical
coordinate:
\begin{equation}\label{mhd-axi-g}
\begin{cases}
\partial_tu^r + u^r\partial_ru^r + u^z\partial_zu^r - \frac{(u^\theta)^2}{r}
+ \partial_rP\\
\quad\quad\quad = \big(\Delta - \frac{1}{r^2}\big)u^r
+ B^r\partial_rB^r + B^z\partial_zB^r - \frac{(B^\theta)^2}{r},\\[-4mm]\\
\partial_tu^\theta + u^r\partial_ru^\theta + u^z\partial_zu^\theta + \frac{u^ru^\theta}{r}\\
\quad\quad\quad = \big(\Delta - \frac{1}{r^2}\big)u^\theta
+ B^r\partial_rB^\theta + B^z\partial_zB^\theta + \frac{B^rB^\theta}{r},\\[-4mm]\\
\partial_tu^z + u^r\partial_ru^z + u^z\partial_zu^z + \partial_zP\\
\quad\quad\quad\quad\quad = \Delta u^z + B^r\partial_rB^z + B^z\partial_zB^z,\\[-4mm]\\
\partial_tB^r + u^r\partial_rB^r + u^z\partial_zB^r\\
\quad\quad\quad = \big(\Delta - \frac{1}{r^2}\big)B^r
+ B^r\partial_ru^r + B^z\partial_zu^r,\\[-4mm]\\
\partial_tB^\theta + u^r\partial_rB^\theta + u^z\partial_zB^\theta + \frac{B^ru^\theta}{r}\\
\quad\quad\quad = \big(\Delta - \frac{1}{r^2}\big)B^\theta
+ B^r\partial_ru^\theta + B^z\partial_zu^\theta + \frac{u^rB^\theta}{r},\\[-4mm]\\
\partial_tB^z + u^r\partial_rB^z + u^z\partial_zB^z\\
\quad\quad\quad = \Delta B^z + B^r\partial_ru^z + B^z\partial_zu^z,
\end{cases}
\end{equation}
where the pressure is given by
\begin{equation}\label{p}
P = p + \frac{|{\bf B}|^2}{2}.
\end{equation}
The incompressible
constraints are
\begin{equation}\label{incom-1}
\partial_ru^r + \frac{u^r }{r} + \partial_zu^z = 0,\quad
\partial_rB^r + \frac{B^r }{r} + \partial_zB^z =
0.
\end{equation}
The general axially symmetric resistive MHD is governed by
\eqref{mhd-axi-g} and \eqref{incom-1}. In this paper, we consider a
family of solutions with the form
\begin{equation}\label{axi-solu}
{\bf u}(t, {\bf x}) = u^r(t, r, z){\bf e}_r + u^z(t, r, z){\bf
e}_z,\quad {\bf B}(t, {\bf x}) = B^\theta(t, r, z){\bf e}_\theta.
\end{equation}
It is easy to check that $(u^\theta, B^r, B^z)$ can be zero for all
time if they are zero initially. In this case, $({\bf u}, {\bf B},
P)$ in \eqref{axi-solu} and \eqref{p} is governed by
\begin{equation}\label{mhd-axi}
\begin{cases}
\partial_tu^r + u^r\partial_ru^r + u^z\partial_zu^r
+ \partial_rP = \big(\Delta - \frac{1}{r^2}\big)u^r
- \frac{(B^\theta)^2}{r},\\[-4mm]\\
\partial_tu^z + u^r\partial_ru^z + u^z\partial_zu^z + \partial_zP = \Delta u^z,\\[-4mm]\\
\partial_tB^\theta + u^r\partial_rB^\theta + u^z\partial_zB^\theta
= \big(\Delta - \frac{1}{r^2}\big)B^\theta
+ \frac{u^rB^\theta}{r},
\end{cases}
\end{equation}
together with the incompressible constraint
\begin{equation}\label{incom}
\partial_ru^r + \frac{u^r }{r} + \partial_zu^z = 0.
\end{equation}
To avoid the explicit presence of pressure, we also need the
vorticity formula of \eqref{mhd-axi}:
\begin{equation}\label{vorticity-axi}
\begin{cases}
\partial_tB^\theta + u^r\partial_rB^\theta + u^z\partial_zB^\theta
= \big(\Delta - \frac{1}{r^2}\big)B^\theta
+ \frac{u^rB^\theta}{r},\\[-4mm]\\
\partial_t\omega^\theta + u^r\partial_r\omega^\theta + u^z\partial_z\omega^\theta
- \frac{u^r\omega^\theta}{r} = \big(\Delta - \frac{1}{r^2}\big)
\omega^\theta - \frac{\partial_z(B^\theta)^2}{r}.
\end{cases}
\end{equation}
\begin{remark}\label{connection}
It is well-known that the axially symmetric Navier-Stokes equations
(in the case of ${\bf B} \equiv 0$) are (see, for instance,
\cite{MB})
\begin{equation}\nonumber
\begin{cases}
\partial_tu^r + u^r\partial_ru^r + u^z\partial_zu^r
+ \partial_rp = \big(\Delta - \frac{1}{r^2}\big)u^r
+ \frac{(u^\theta)^2}{r},\\[-4mm]\\
\partial_tu^z + u^r\partial_ru^z + u^z\partial_zu^z + \partial_zp = \Delta u^z,\\[-4mm]\\
\partial_tu^\theta + u^r\partial_ru^\theta + u^z\partial_zu^\theta
= \big(\Delta - \frac{1}{r^2}\big)u^\theta
- \frac{u^ru^\theta}{r}.
\end{cases}
\end{equation}
If we denote ${\bf u} = u^r{\bf e}_r + u^z{\bf e}_z$ and ${\bf b} =
u^\theta {\bf e}_\theta$, we can rewrite the above axially symmetric
Navier-Stokes equations as
\begin{equation}\label{NS-axi}
\begin{cases}
\partial_t{\bf u} + {\bf u}\cdot\nabla {\bf u}
+ \nabla p = \Delta{\bf u}
- {\bf b}\cdot\nabla {\bf b},\\[-4mm]\\
\partial_t{\bf b} + {\bf u}\cdot\nabla{\bf b} = \Delta {\bf b} - {\bf b}\cdot\nabla{\bf u},\\[-4mm]\\
\nabla\cdot{\bf u} = \nabla\cdot{\bf b} = 0.
\end{cases}
\end{equation}
If we compare the MHD equations \eqref{RMHD} with the Navier-Stokes
equations \eqref{NS-axi}, we find that we can recover \eqref{RMHD}
from \eqref{NS-axi} by changing the sign of the terms ${\bf
b}\cdot\nabla {\bf b}$ and ${\bf b}\cdot\nabla {\bf u}$. The
significance of the tiny difference, especially, the sign of ${\bf
b}\cdot\nabla {\bf u}$, yields a much stronger \textit{a priori}
estimate in the MHD case.
\end{remark}
\begin{prop}[Maximum Principle]\label{prop-maxi}
Assume that $({\bf u}, {\bf B}, P)$ is a smooth bounded solution to
\eqref{mhd-axi} with or without resistivity. Then the quantity $\Pi$ satisfies the maximum principle
\begin{equation}\nonumber
\|\Pi(t, \cdot)\|_{L^\infty} \leq \|\Pi(0, \cdot)\|_{L^\infty},\quad
\forall\ t \geq 0.
\end{equation}
\end{prop}
\begin{proof}
In the case of zero resistivity, by dividing the
equation for $B^\theta$ by $r$, one has
\begin{equation}\label{Pi-eqn-1}
\partial_t\Pi + u^r\partial_r\Pi + u^z\partial_z\Pi
= 0,
\end{equation}
which gives the maximum principle for $\Pi$ in the case of zero
resistivity.
Similarly, in the resistive case, we have
\begin{equation}\label{Pi-eqn-2}
\partial_t\Pi + u^r\partial_r\Pi + u^z\partial_z\Pi
= (\partial_r^2 + \frac{3}{r}\partial_r + \partial_z^2)\Pi.
\end{equation}
Then the maximum principle follows by interpreting $(\partial_r^2 + \frac{3}{r}\partial_r + \partial_z^2)$
as a five-dimensional Laplacian operator.
\end{proof}
\section{Proof of Theorem \ref{thm-iMHD}}
In this section we prove Theorem \ref{thm-iMHD}. Throughout this
paper, we will use $A_1 \lesssim A_2$ to denote that $A_1 \leq
C_0A_2$ and $A_1 \simeq A_2$ to denote that $C_0^{-1}A_2 \leq A_1
\leq C_0A_2$ for a generic positive constant $C_0
> 1$ and two positive quantities $A_1$ and $A_2$.
\begin{proof}[Proof of Theorem \ref{thm-iMHD}]
Let us rewrite the vorticity equation in \eqref{vorticity-axi} in
terms of $\Omega$:
\begin{equation}\nonumber
\partial_t\Omega + u^r\partial_r\Omega + u^z\partial_z\Omega
= (\partial_r^2 + \frac{3}{r}\partial_r + \partial_z^2)\Omega -
\partial_z\Pi^2.
\end{equation}
By taking the $L^2$ inner product of the above equation with
$\Omega$ and preforming the standard energy estimate, one has
\begin{eqnarray}\nonumber
&&\frac{1}{2}\frac{d}{dt}\|\Omega\|_{L^2}^2 -
\int\Omega(\partial_r^2 + \frac{3}{r}\partial_r +
\partial_z^2)\Omega dx\\\nonumber
&&= - \frac{1}{2}\int (u^r\partial_r\Omega^2 + u^z\partial_z
\Omega^2)dx - \int\Omega\partial_z\Pi^2dx.
\end{eqnarray}
Using the incompressibility condition \eqref{incom} and the fact of
$dx = 2\pi rdrdz$, one has
\begin{eqnarray}\nonumber
\int (u^r\partial_r\Omega^2 + u^z\partial_z\Omega^2) dx = 0
\end{eqnarray}
and
\begin{eqnarray}\nonumber
- \int\Omega(\partial_r^2 + \frac{3}{r}\partial_r +
\partial_z^2)\Omega dx = \|\nabla\Omega\|_{L^2}^2 + 2\pi\int_{\mathbb{R}}|\Omega(t, 0, z)|^2dz.
\end{eqnarray}
By integration by part and interpolation, we have
\begin{eqnarray}\nonumber
\big|\int\Omega\partial_z\Pi^2dx\big| \leq
\|\Pi\|_{L^4}^2\|\partial_z\Omega\|_{L^2} \leq
\frac{1}{2}\|\Pi\|_{L^2}^2\|\Pi\|_{L^\infty}^2 +
\frac{1}{2}\|\partial_z\Omega\|_{L^2}^2.
\end{eqnarray}
Consequently, one has
\begin{eqnarray}\label{3-1}
\frac{d}{dt}\|\Omega\|_{L^2}^2 + \|\nabla\Omega \|_{L^2}^2 \leq
\|\Pi\|_{L^2}^2\|\Pi\|_{L^\infty}^2.
\end{eqnarray}
Similarly, using equation \eqref{Pi-eqn-2} and preforming the $L^2$
energy estimate, one has
\begin{eqnarray}\label{3-2}
\|\Pi(t, \cdot)\|_{L^2} \leq \|\Pi_0\|_{L^2},\quad \forall\ t \geq
0.
\end{eqnarray}
Consequently, by Proposition \ref{prop-maxi} and using \eqref{3-1},
\eqref{3-2}, we have
\begin{equation}\label{3-3}
\|\Omega(t, \cdot)\|_{L^2} \lesssim 1 + \sqrt{t},\quad
\int_0^t\|\nabla\Omega\|_{L^2}^2dt \lesssim 1 + t, \quad \forall\ t
\geq 0.
\end{equation}
Here we used that $\Omega_0 \in L^2$ which is due to the fact that
${\bf u}_0 \in H^2$ and
\begin{equation}\nonumber
\big|\nabla(\nabla\times {\bf u})\big|^2 = \big|({\bf e}_r\partial_r
+ \frac{1}{r}{\bf e}_\theta\partial_\theta + {\bf
e}_z\partial_z)\omega^\theta {\bf e}_\theta\big|^2 =
|\nabla\omega^\theta|^2 + |\Omega|^2.
\end{equation}
Similarly, one also has $\Pi_0 \in L^2$ since ${\bf B}_0 \in H^1$.
To proceed, we need a technical lemma regarding the property of a
Riesz operator on $\mathbb{R}^3$. We first recall the following
weighted Calderon-Zygmund inequality for a singular integral
operator with a weight function which is in the $\mathcal{A}_p$
class (see Stein \cite{Stein} pp. 194-217 for details). Let
$\mathcal{K}$ be a Riesz operator in $\mathbb{R}^n$ and $w(x)$ be a
weight in the $\mathcal{A}_p$ class (see page 194 of \cite{Stein}
for definition). One can extend the Calderon-Zygmund inequality for
the singular integral operator with the integral having weight
function $w(x)$. Specifically, for $1 < p < \infty$, there holds
\begin{equation}\nonumber
\|\mathcal{K}f\|_{L^p(\mathbb{R}^n)} \lesssim
\|f\|_{L^p(\mathbb{R}^n)},\quad \forall\ \ f \in L^p(\mathbb{R}^n).
\end{equation}
The following lemma plays an essential role in our global regularity
analysis.
\begin{lem}\label{C-Z}
There holds
\begin{equation}\nonumber
\int_0^T\|r^{-1}u^r(t, \cdot)\|_{L^\infty}dt \lesssim \sup_{0 \leq t
\leq T}\|\Omega(t,
\cdot)\|_{L^2}^{\frac{1}{2}}\int_0^T\|\nabla\Omega(t,
\cdot)\|_{L^2}^{\frac{1}{2}}dt.
\end{equation}
\end{lem}
\begin{rem}\nonumber
We pointed out that in \cite{HLL} the authors have established an
inequality $\|r^{-1}\partial_zu^r\|_{L^p} \lesssim \|\Omega\|_{L^p}$
for $1 < p < \infty$ for $\mathbb{R}^2 \times \mathbb{T}^1$, where
$\mathbb{T}^1$ is a one-dimensional torus.
\end{rem}
\begin{proof}
We follow the proof in \cite{HLL}. By the incompressible constraint
\eqref{incom}, we can introduce the angular stream function
$\psi^\theta$ such that
\begin{equation}\label{3-4}
- \big(\partial_r^2 + \frac{1}{r}\partial_r - \frac{1}{r^2} +
\partial_z^2\big)\psi^\theta = \omega^\theta,
\end{equation}
and
\begin{equation}\nonumber
u^r = - \partial_z\psi^\theta,\quad u^z =
\frac{1}{r}\partial_r(r\psi^\theta).
\end{equation}
We divide by $r$ in \eqref{3-4}, which gives that
\begin{equation}\label{3-5}
- \big(\partial_r^2 + \frac{3}{r}\partial_r +
\partial_z^2\big)\frac{\psi^\theta}{r} = \frac{\omega^\theta}{r}.
\end{equation}
Following \cite{HLL}, we interpret the Laplace operator in
\eqref{3-5} as a five-dimensional one. We formally write
$$y = (y_1, y_2, y_3, y_4, z),\quad r = \sqrt{y_1^2 + y_2^2 + y_3^2 + y_4^2},
\quad \Delta_y = \big(\partial_r^2 + \frac{3}{r}\partial_r +
\partial_z^2\big).$$
This way we have $\frac{\psi^\theta}{r} =
(-\Delta_y)^{-1}\frac{\omega^\theta}{r}.$ In the remaining part of
the proof of this lemma, we will use a subscript $y$ to denote the
derivatives with respect to $y$.
It is clear that
\begin{eqnarray}\nonumber
\nabla^2\frac{\psi^\theta}{r} &=& ({\bf e}_r\partial_r +
\frac{1}{r}{\bf e}_\theta\partial_\theta +
{\bf e}_z\partial_z)\big({\bf e}_r\partial_r\frac{\psi^\theta}{r}\big) +
\nabla\partial_z\frac{\psi^\theta}{r}\otimes {\bf e}_z\\\nonumber
&=& {\bf e}_r\otimes {\bf e}_r\partial_r^2\frac{\psi^\theta}{r} +
{\bf e}_\theta\otimes {\bf e}_\theta\frac{1}{r}\partial_r\frac{\psi^\theta}{r}
+ ({\bf e}_z\otimes {\bf e}_r + {\bf e}_r\otimes{\bf e}_z)\partial_{zr}^2\frac{\psi^\theta}{r} +
{\bf e}_z\otimes {\bf e}_z\partial_z^2\frac{\psi^\theta}{r}.
\end{eqnarray}
Consequently, one has
\begin{eqnarray}\label{S7-1}
\big|\nabla^2\frac{\psi^\theta}{r}\big|^2 \simeq
\big|\partial_r^2\frac{\psi^\theta}{r}\big|^2 +
\big|\frac{1}{r}\partial_r\frac{\psi^\theta}{r}\big|^2 +
\big|\partial_z^2\frac{\psi^\theta}{r}\big|^2 +
\big|\partial_{rz}^2\frac{\psi^\theta}{r}\big|^2.
\end{eqnarray}
On the other hand, one also has
\begin{eqnarray}\nonumber
\nabla_y^2\frac{\psi^\theta}{r} &=& (\widetilde{{\bf e}}_r\partial_r
+ \nabla_\theta + \widetilde{{\bf e}}_z\partial_z)\big(\widetilde{{\bf e}}_r
\partial_r\frac{\psi^\theta}{r}\big) +
\nabla_y\partial_z\frac{\psi^\theta}{r}\otimes \widetilde{{\bf e}}_z\\\nonumber
&=& \widetilde{{\bf e}}_r\otimes \widetilde{{\bf
e}}_r\partial_r^2\frac{\psi^\theta}{r} +
\nabla_\theta \widetilde{{\bf e}}_r\partial_r\frac{\psi^\theta}{r}
+ (\widetilde{{\bf e}}_z\otimes \widetilde{{\bf e}}_r + \widetilde{{\bf e}}_r
\otimes \widetilde{{\bf e}}_z)\partial_{zr}^2\frac{\psi^\theta}{r} +
\widetilde{{\bf e}}_z\otimes \widetilde{{\bf e}}_z\partial_z^2\frac{\psi^\theta}{r}\\\nonumber
&=& \widetilde{{\bf e}}_r\otimes \widetilde{{\bf e}}_r
\partial_r^2\frac{\psi^\theta}{r} + (I_0 - \widetilde{{\bf e}}_r\otimes
\widetilde{{\bf
e}}_r)\frac{1}{r}\partial_r\frac{\psi^\theta}{r}\\\nonumber
&& +\ (\widetilde{{\bf e}}_z\otimes \widetilde{{\bf e}}_r +
\widetilde{{\bf e}}_r
\otimes \widetilde{{\bf e}}_z) \partial_{zr}^2\frac{\psi^\theta}{r} +
\widetilde{{\bf e}}_z\otimes \widetilde{{\bf e}}_z\partial_z^2\frac{\psi^\theta}{r}.
\end{eqnarray}
where $I_0 = \begin{pmatrix}I_{4\times 4} & 0 \\ 0 & 0\end{pmatrix}$
and $\nabla_\theta$ is defined by
\begin{equation}\nonumber
\nabla_\theta = \nabla - \widetilde{{\bf e}}_r(\widetilde{{\bf
e}}_r\cdot\nabla_y) - \widetilde{{\bf e}}_z\partial_z,\quad
\widetilde{{\bf e}}_r = \frac{1}{r}\begin{pmatrix}y_1\\ y_2\\ y_3\\ y_4\\
0\end{pmatrix},
\widetilde{{\bf e}}_z = \begin{pmatrix}0\\ 0\\ 0\\ 0\\
1\end{pmatrix}.
\end{equation}
Clearly, $\widetilde{{\bf e}}_r\otimes \widetilde{{\bf e}}_r$, $I_0
- \widetilde{{\bf e}}_r\otimes \widetilde{{\bf e}}_r$,
$\widetilde{{\bf e}}_z\otimes \widetilde{{\bf e}}_r$,
$\widetilde{{\bf e}}_r\otimes \widetilde{{\bf e}}_z$ and
$\widetilde{{\bf e}}_z\otimes \widetilde{{\bf e}}_z$ are all
mutually orthogonal. Consequently, one also has
\begin{equation}\label{S7-2}
\big|\nabla_y^2\frac{\psi^\theta}{r}\big|^2 \simeq
\big|\partial_r^2\frac{\psi^\theta}{r}\big|^2 +
\big|\frac{1}{r}\partial_r\frac{\psi^\theta}{r}\big|^2 +
\big|\partial_z^2\frac{\psi^\theta}{r}\big|^2 +
\big|\partial_{rz}^2\frac{\psi^\theta}{r}\big|^2.
\end{equation}
By \eqref{S7-1} and \eqref{S7-2}, we have
\begin{eqnarray}\nonumber
\int\big|\nabla^2\frac{\psi^\theta}{r}\big|^pdx &\simeq&
\int_{-\infty}^\infty\int_0^\infty\Big(\big|\partial_r^2
\frac{\psi^\theta}{r}\big|^2 + \big|\frac{1}{r}\partial_r\frac{\psi^\theta
}{r}\big|^2 + \big|\partial_z^2\frac{\psi^\theta}{r}\big|^2 +
\big|\partial_{rz}^2\frac{\psi^\theta}{r}\big|^2\Big)^{\frac{p}{2}}rdrdz\\\nonumber
&=& \int_{-\infty}^\infty\int_0^\infty\Big(\big|\partial_r^2
\frac{\psi^\theta}{r}\big|^2 + \big|\frac{1}{r}\partial_r\frac{\psi^\theta
}{r}\big|^2 + \big|\partial_z^2\frac{\psi^\theta}{r}\big|^2 +
\big|\partial_{rz}^2\frac{\psi^\theta}{r}\big|^2\Big)^{\frac{p}{2}}w(r)r^3drdz\\\nonumber
&\simeq& \int_{-\infty}^\infty\int_0^\infty\big|\nabla_y^2
\frac{\psi^\theta}{r}\big|^pw(r)r^3drdz\\\nonumber
&\simeq& \int\big|\nabla_y^2(-\Delta_y)^{-1}
\frac{\omega^\theta}{r}\big|^pw(r)dy,
\end{eqnarray}
where $w(r)$ is a weight function $w(r) = \frac{1}{r^2}$.
Let $1 < p < \infty$. Using Lemma 2 in \cite{HLL} (see also a
general version in Lemma \ref{A-p} in Appendix of this paper), we
have
\begin{eqnarray}\nonumber
\int\big|\nabla_y^2(-\Delta_y)^{-1}
\frac{\omega^\theta}{r}\big|^pw(r)dy &\lesssim& \int\big|
\frac{\omega^\theta}{r}\big|^pw(r)dy\\\nonumber
&\simeq& \int\big|\frac{\omega^\theta}{r}\big|^pdx.
\end{eqnarray}
Consequently, one has
\begin{equation}\label{3-6}
\int\big|\nabla^2\frac{\psi^\theta}{r}\big|^pdx \lesssim
\int\big|\frac{\omega^\theta}{r}\big|^pdx.
\end{equation}
Repeating the above procedure, one also has
\begin{equation}\label{3-7}
\int\big|\nabla^2\frac{\partial_z\psi^\theta}{r}\big|^pdx \lesssim
\int\big|\frac{\partial_z\omega^\theta}{r}\big|^pdx.
\end{equation}
Taking $p = 2$ in \eqref{3-6} and \eqref{3-7} and using the
interpolation inequality $\|f\|_{L^\infty} \lesssim \|\nabla
f\|_{L^2}^{\frac{1}{2}}\|\nabla^2f\|_{L^2}^{\frac{1}{2}}$ in
$\mathbb{R}^3$, one has
\begin{eqnarray}\nonumber
\int_0^T\big\|r^{-1}u^r(t, \cdot)\big\|_{L^\infty}dt &=&
\int_0^T\big\|r^{-1}\partial_z\psi^\theta(t,
\cdot)\big\|_{L^\infty}dt\\\nonumber
&\lesssim& \int_0^T\big\|\nabla\partial_z(r^{-1}\psi^\theta(t,
\cdot))\big\|_{L^2}^{\frac{1}{2}}\big\|\nabla^2\partial_z(r^{-1}\psi^\theta(t,
\cdot))\big\|_{L^2}^{\frac{1}{2}}dt\\\nonumber
&\lesssim& \sup_{0 \leq t \leq T}\|\Omega(t,
\cdot))\|_{L^2}^{\frac{1}{2}}\int_0^T\|\partial_z\Omega(t,
\cdot)\|_{L^2}^{\frac{1}{2}}dt.
\end{eqnarray}
This finishes the proof of the lemma.
\end{proof}
Now we derive an $L^\infty$ estimate for $B^\theta$. Ignoring the
viscosity in the equation of $B^\theta$ in \eqref{mhd-axi}, one has
\begin{eqnarray}\nonumber
\|B^\theta(t, \cdot)\|_{L^\infty} &\leq& \|B^\theta_0\|_{L^\infty} +
\int_0^t\|B^\theta(s, \cdot)\|_{L^\infty}\big\|\frac{u^r}{r}\big\|_{L^\infty}ds.
\end{eqnarray}
By Gronwall's inequality and using \eqref{3-3} and Lemma \ref{C-Z},
we have
\begin{eqnarray}\label{9}
\|B^\theta(t, \cdot)\|_{L^\infty} \leq
\|B^\theta_0\|_{L^\infty}e^{\int_0^t\|r^{-1}u^r(s,
\cdot)\|_{L^\infty}ds} \lesssim e^{t^{\frac{5}{4}}}.
\end{eqnarray}
Let us coming back to \eqref{vorticity-axi} and estimate that
\begin{eqnarray}\nonumber
&&\frac{1}{2}\frac{d}{dt}\int|\omega^\theta|^2dx +
\int\big(|\nabla \omega^\theta|^2 +
\frac{|\omega^\theta|^2}{r^2}\big)dx\\\nonumber
&&\leq \big\|\frac{u^r}{r}\big\|_{L^\infty}\int(\omega^\theta)^2dx
+ \|\Pi\|_{L^\infty}\|B^\theta\|_{L^2}\|\partial_z\omega^\theta\|_{L^2}\\\nonumber
&&\leq \big\|\frac{u^r}{r}\big\|_{L^\infty}\int(\omega^\theta)^2dx
+ \frac{1}{2}\|\Pi\|_{L^\infty}^2\|B^\theta\|_{L^2}^2
+ \frac{1}{2}\|\partial_z\omega^\theta\|_{L^2}^2.
\end{eqnarray}
Recalling the following basic energy law
\begin{equation}\label{EnergyL}
\frac{1}{2}\frac{d}{dt}\big(\|{\bf u}\|_{L^2}^2 + \|{\bf
B}\|_{L^2}^2\big) + \int_0^t\big\|\nabla{\bf u}\|_{L^2}^2ds = 0,
\end{equation}
and using the \textit{a priori} estimate in \eqref{3-3} and Lemma
\ref{C-Z}, one has
\begin{equation}\label{10}
\|\nabla\times {\bf u}(t, \cdot)\|_{L^2} \lesssim
e^{t^{\frac{5}{4}}},\quad \int_0^t\|\nabla(\nabla\times {\bf
u})\|_{L^2}^2dt \lesssim e^{t^{\frac{5}{4}}}, \quad \forall\ t \geq
0.
\end{equation}
The next step is to bootstrap the regularity of ${\bf u}$ and ${\bf
B}$. We are going to show the $L^1([0, T], {\rm Lip}(\mathbb{R}^3))$
estimate of ${\bf u}$. We will make use of the structure of the
ideal MHD in \eqref{mhd-axi} to avoid some possible technical
complications. The key observation is that we can write the
vorticity equation as
\begin{equation}\nonumber
\partial_t(\nabla\times {\bf u}) + \nabla\times[(\nabla\times {\bf u})\times
{\bf u}] = \Delta(\nabla\times {\bf u})
- \partial_z(\Pi B^\theta e_\theta).
\end{equation}
Here by the maximum principle in Proposition \ref{prop-maxi} and
\eqref{9}, one has $\Pi B^\theta \in L^\infty([0, t],
L^\infty(\mathbb{R}^3))$. Moreover, we can apply \eqref{10} to
bootstrap the regularity of $(\nabla\times {\bf u})\times {\bf u}$.
Then we may apply the standard parabolic estimate to get the
$L^1([0, t], L^\infty(\mathbb{R}^3))$ estimate for
$\nabla\times {\bf u}$.
We first perform $L^4$ energy estimate for \eqref{vorticity-axi} and derive
that
\begin{eqnarray}\nonumber
&&\frac{1}{4}\frac{d}{dt}\int|\omega^\theta|^4dx +
\int\big(|\nabla |\omega^\theta|^2|^2 +
\frac{|\omega^\theta|^4}{r^2}\big)dx\\\nonumber
&&\leq \big\|\frac{u^r}{r}\big\|_{L^\infty}\int(\omega^\theta)^4dx
+ \|\Pi\|_{L^\infty}\|B^\theta\|_{L^\infty}\|\partial_z|\omega^\theta|^2\|_{L^2}
\|\omega^\theta\|_{L^2}\\\nonumber
&&\leq \big\|\frac{u^r}{r}\big\|_{L^\infty}\int(\omega^\theta)^4dx
+ \frac{1}{2}\|\Pi\|_{L^\infty}^2
\|B^\theta\|_{L^\infty}^2\|\omega^\theta\|_{L^2}^2
+ \frac{1}{2}\|\partial_z|\omega^\theta|\|_{L^2}^2.
\end{eqnarray}
Using the \textit{a priori} estimate in \eqref{3-3} and Lemma
\ref{C-Z}, one has
\begin{eqnarray}\nonumber
\||\omega^\theta|^2\|_{L^\infty([0, t], L^2(\mathbb{R}^3))}^2 +
\|\nabla|\omega^\theta|^2\|_{L^2([0, t], L^2(\mathbb{R}^3))}^2
\lesssim e^{t^{\frac{5}{4}}}.
\end{eqnarray}
By Sobolev imbedding inequality, one has
\begin{eqnarray}\nonumber
\|\omega^\theta\|_{L^\infty([0, t], L^4(\mathbb{R}^3))} +
\|\omega^\theta\|_{L^4([0, t], L^{12}(\mathbb{R}^3))} \lesssim
e^{t^{\frac{5}{4}}}.
\end{eqnarray}
On the other hand, by Sobolev imbedding, one also has
\begin{eqnarray}\nonumber
\|{\bf u}\|_{L^\infty([0, t], L^\infty(\mathbb{R}^3))} \lesssim
\|{\bf u}\|_{L^\infty([0, t], L^2(\mathbb{R}^3))} +
\|\omega^\theta\|_{L^\infty([0, t], L^4(\mathbb{R}^3))} \lesssim
e^{t^{\frac{5}{4}}}.
\end{eqnarray}
Hence, we have
\begin{equation}\nonumber
\|(\nabla\times {\bf u})\times {\bf u}\|_{L^4([0, t],
L^{12}(\mathbb{R}^3))} \lesssim e^{t^{\frac{5}{4}}}.
\end{equation}
Write
\begin{equation}\nonumber
\nabla\times {\bf u} = e^{t\Delta}\nabla\times {\bf u}_0 -
\int_0^te^{(t - s)\Delta}\big(\nabla\times[(\nabla\times {\bf
u})\times{\bf u}] + \partial_z(\Pi B^\theta e_\theta)\big)ds.
\end{equation}
A standard parabolic estimate gives that
\begin{equation}\nonumber
\|\nabla\nabla\times {\bf u}\|_{L^4([0, t], L^{12}(\mathbb{R}^3))}
\lesssim e^{t^{\frac{5}{4}}}.
\end{equation}
By Sobolev imbedding, we have
\begin{equation}\label{1-2}
\|\nabla {\bf u}\|_{L^4([0, t], L^\infty(\mathbb{R}^3))} \lesssim
e^{t^{\frac{5}{4}}}.
\end{equation}
Now let us derive the $L^1([0, T], {\rm Lip}(\mathbb{R}^3))$
estimate of ${\bf B}$. We first write
\begin{equation}\nonumber
\partial_t{\bf B} + {\bf u}\cdot\nabla{\bf B} = \frac{u^r}{r}{\bf
B}.
\end{equation}
Applying $\nabla$, one has
\begin{equation}\nonumber
\partial_t\nabla{\bf B} + {\bf u}\cdot\nabla\nabla{\bf B} = - \nabla{\bf u}\cdot\nabla{\bf B}
+ \frac{u^r}{r}\nabla{\bf B} + \nabla u^r\Pi {\bf e}_\theta +
(\nabla \frac{1}{r})u^r{\bf B}.
\end{equation}
Note that
\begin{equation}\nonumber
(\nabla \frac{1}{r})u^r{\bf B} = - \frac{u^r}{r}\Pi {\bf e}_r,
\end{equation}
one has
\begin{eqnarray}\nonumber
\|\nabla{\bf B}(t, \cdot)\|_{L^\infty} &\lesssim& \|\nabla{\bf
B}_0\|_{L^\infty} + \int_0^t\big(\|\nabla{\bf u}\|_{L^\infty} +
\big\|\frac{u^r}{r}\big\|_{L^\infty}\big)\|\nabla{\bf B}(s,
\cdot)\|_{L^\infty}ds\\\nonumber
&&\quad +\ \int_0^t\big(
\|\nabla{\bf u}\|_{L^\infty} + \big\|\frac{u^r}{r}
\big\|_{L^\infty}\big)\|\Pi(s, \cdot)\|_{L^\infty}ds.
\end{eqnarray}
We can use \eqref{3-3}, \eqref{1-2}, Lemma \ref{C-Z} and Gronwall's
inequality to estimate that
\begin{eqnarray}\label{1-3}
\|\nabla{\bf B}(t, \cdot)\|_{L^\infty} \lesssim
e^{e^{t^{\frac{5}{4}}}}.
\end{eqnarray}
The \textit{a priori} estimates \eqref{1-2} and \eqref{1-3} are
enough for the global regularity of the ideal MHD equations
\eqref{iMHD}. Indeed, applying the standard $H^2$ energy estimate,
one has
\begin{eqnarray}\nonumber
&&\frac{1}{2}\frac{d}{dt}\big(\|\nabla^2{\bf u}(t, \cdot)\|_{L^2}^2
+ \|\nabla^2{\bf B}(t, \cdot)\|_{L^2}^2\big) + \|\nabla^3{\bf u}
(t, \cdot)\|_{L^2}^2\\\nonumber
&&\lesssim \int\big(- \nabla^2{\bf u}\nabla^2({\bf u}\cdot\nabla
{\bf u}) + \nabla^2{\bf u}\nabla^2({\bf B}\cdot\nabla {\bf
B})\big) dx\\\nonumber
&&\quad +\ \int\big(- \nabla^2{\bf B}\nabla^2({\bf u}\cdot\nabla
{\bf B}) + \nabla^2{\bf B}\nabla^2({\bf B}\cdot\nabla {\bf
u})\big) dx\\\nonumber
&&\lesssim \frac{1}{2}\|\nabla^3{\bf u}(t, \cdot)\|_{L^2}^2
+ \|{\bf u}(t, \cdot)\|_{L^\infty}^2\|\nabla^2{\bf u}
(t, \cdot)\|_{L^2}^2 + \|{\bf B}(t, \cdot)\|_{L^\infty}^2\|\nabla^2{\bf B}
(t, \cdot)\|_{L^2}^2\\\nonumber
&&\quad +\ \big(\|\nabla{\bf u}(t, \cdot)\|_{L^\infty} +
\|\nabla{\bf B}(t, \cdot)\|_{L^\infty}\big)\big(\|\nabla^2{\bf
B}(t, \cdot)\|_{L^2}^2 + \|\nabla^2{\bf u}
(t, \cdot)\|_{L^2}^2\big).
\end{eqnarray}
Here we also used the Gagliardo-Nirenberg's inequality $\|\nabla
f\|_{L^4}^2 \lesssim \|f\|_{L^\infty}\|\nabla^2f\|_{L^2}$, the
integration by parts and $\int\nabla^2{\bf B}({\bf u}\cdot\nabla)
\nabla^2{\bf B}dx = 0$. Consequently, one has
\begin{equation}\nonumber
\begin{cases}
\|\nabla^2 {\bf u}(t, \cdot)\|_{L^2} + \|\nabla^2 {\bf B}(t,
\cdot)\|_{L^2} \lesssim e^{e^{e^{t^{\frac{5}{4}}}}},\\[-4mm]\\
\int_0^t\|\nabla^3{\bf u}\|_{L^2}^2dt \lesssim
e^{e^{e^{t^{\frac{5}{4}}}}},
\end{cases}
\quad \forall\ t \geq 0.
\end{equation}
We finished the proof of Theorem \ref{iMHD}.
\end{proof}
\begin{rem}
The proof of Lemma \ref{C-Z} can also be proved by using the following Biot-Savart law (see
\cite{ShirotaYanagisawa94}):
\begin{equation}\nonumber
|u^r(t, x)| \lesssim \int_{|y - x| \leq 4r}\frac{|\omega^\theta(t,
y)|}{|x - y|^2}dy + r\int_{|y - x| \geq 4r}\frac{|\omega^\theta(t,
y)|}{|x - y|^3}dy,
\end{equation}
which, by
Young's inequality, gives that
\begin{eqnarray}\nonumber
|u^r(t, r, z)| \lesssim r\frac{1}{|x|^2}\ast\Omega \leq
r\big\|\frac{1}{|x|^2}\big\|_{L^{\frac{3}{2},
\infty}}\|\Omega\|_{L^{3,1}} .
\end{eqnarray}
Here $L^{p, q}$ denotes the usual Lorentz norm. Then using the real interpolation
and Sobolev imbedding, one has
\begin{eqnarray}\label{8}
\big|\frac{u^r(t, r, z)}{r}\big| \lesssim \|\Omega\|_{L^{3, 1}} \leq
\|\Omega\|_{L^2}^{\frac{1}{2}}\|\Omega\|_{L^{\frac{1}{2}}}.
\end{eqnarray}
\end{rem}
\section{Proofs of Theorem \ref{thm-RMHD}}
In this section we prove Theorem \ref{thm-RMHD}.
\begin{proof}[Proof of Theorem \ref{thm-RMHD}]
Similarly as in obtaining \eqref{3-1}, one has
\begin{eqnarray}\nonumber
\frac{1}{2}\frac{d}{dt}\|\Omega\|_{L^2}^2 +
\|\nabla\Omega\|_{L^2}^2
&\lesssim& \big|\int\Omega\partial_z\Pi^2dx\big|\\\nonumber
&\leq& \|\Pi\|_{L^\infty}^{\frac{1}{3}}\|\Pi\|_{L^{\frac{10}{3}}}
^{\frac{5}{3}}\|\partial_z\Omega\|_{L^2}\\\nonumber &\lesssim&
\|\Pi\|_{L^\infty}^{\frac{1}{3}}\|\Pi\|_{L^2}^{\frac{2}{3}}
\|\nabla\Pi\|_{L^2}\|\partial_z\Omega\|_{L^2}.
\end{eqnarray}
Consequently, one has
\begin{eqnarray}\label{2}
\frac{d}{dt}\|\Omega\|_{L^2}^2 + \|\nabla\Omega \|_{L^2}^2 \lesssim
\|\Pi\|_{L^\infty}^{\frac{2}{3}}\|\Pi\|_{L^2}^{\frac{4}{3}}\|\nabla\Pi\|_{L^2}^2.
\end{eqnarray}
Applying a similar argument to $\Pi$ equation in \eqref{Pi-eqn-2},
one has
\begin{eqnarray}\label{3}
\frac{d}{dt}\|\Pi\|_{L^2}^2 + \|\nabla\Pi\|_{L^2}^2 \leq 0.
\end{eqnarray}
Clearly, the combination of \eqref{2}, \eqref{3} and the maximum
estimate in Proposition \ref{prop-maxi} gives the following
\textit{a priori} estimate
\begin{equation}\label{4}
\|\Pi(t, \cdot)\|_{L^2} + \|\Omega(t, \cdot)\|_{L^2} \lesssim 1\ \
(\forall\ t \geq 0),\quad \int_0^\infty\big(\|\nabla\Pi\|_{L^2}^2 +
\|\nabla\Omega\|_{L^2}^2\big)dt \lesssim 1.
\end{equation}
Now let us come back to the equation of $\omega^\theta$ in
\eqref{vorticity-axi}. Applying the standard energy estimate, one
has
\begin{eqnarray}\label{1-1}
&&\frac{1}{2}\frac{d}{dt}\int|\omega^\theta|^2dx +
\int\big(|\nabla \omega^\theta|^2 +
\frac{|\omega^\theta|^2}{r^2}\big)dx\\\nonumber
&&= \int\frac{u^r(\omega^\theta)^2}{r}dx - \int
\frac{\partial_z(B^\theta)^2}{r}\omega^\theta dx.
\end{eqnarray}
Using Sobolev imbedding theorem and interpolation, one has
\begin{eqnarray}\nonumber
\Big|\int\frac{u^r(\omega^\theta)^2}{r}dx\Big| &\leq&
\|u^r\|_{L^2}\|\Omega\|_{L^6}\|\omega^\theta\|_{L^3}\\\nonumber
&\lesssim& \|u^r\|_{L^2}\|\nabla\Omega\|_{L^2}\|\omega^\theta
\|_{L^2}^{\frac{1}{2}}\|\nabla\omega^\theta\|_{L^2}^{\frac{1}{2}}\\\nonumber
&\lesssim& \|u^r\|_{L^2}^2\|\nabla\Omega\|_{L^2}^2 + \|\omega^\theta
\|_{L^2}^2 + \frac{1}{4}\|\nabla\omega^\theta\|_{L^2}^2.
\end{eqnarray}
On the other hand, it is clear that one also has
\begin{eqnarray}\nonumber
\Big|\int\frac{\partial_z(B^\theta)^2}{r}\omega^\theta
dx\Big| &\leq& \|\Pi\|_{L^\infty}\|B^\theta\|_{L^2}
\|\partial_z\omega^\theta\|_{L^2}\\\nonumber
&\leq& \|\Pi\|_{L^\infty}^2\|B^\theta\|_{L^2}^2 +
\frac{1}{4}\|\partial_z\omega^\theta\|_{L^2}.
\end{eqnarray}
Using the \textit{a priori} estimate \eqref{4}, the basic energy law
\eqref{EnergyL} and Proposition \ref{prop-maxi}, we have
\begin{eqnarray}\label{5}
\|\nabla\times {\bf u}(t, \cdot)\|_{L^2} \lesssim 1\ \ (\forall\ t
\geq 0),\quad \int_0^\infty\|\nabla(\nabla\times{\bf
u})\|_{L^2}^2\big)dt \lesssim 1.
\end{eqnarray}
The \textit{a priori} estimate \eqref{5} is enough to get the global
regularity of the resistive MHD \eqref{RMHD}. Indeed, by using the
equation of ${\bf B}$, one can easily verifies that $\nabla{\bf B}$
also satisfies \eqref{5}. We have finished the proof of Theorem
\ref{thm-RMHD}.
\end{proof}
\section{Appendix}
In this appendix we first prove that $w(y) = r^\alpha$ is a
$\mathcal{A}_p$ for Riesz operator in $\mathbb{R}^5$ under $-4 <
\alpha < 4p\big(1 - \frac{1}{p}\big)$. The case of $\alpha = - 2$
has been studied in \cite{HLL}.
\begin{lem}[$\mathcal{A}_p$ Weight]\label{A-p}
Let $1 < p < \infty$ and $w(y) = r^\alpha$, $y \in \mathbb{R}^5$.
Then $w(x)$ is in $\mathcal{A}_p$ class if $-4 < \alpha < 4p\big(1 -
\frac{1}{p}\big)$.
\end{lem}
\begin{proof}
Recall that a real valued non-negative function $w(x)$ is said to be
in $\mathcal{A}_p( \mathbb{R}^n)$ class if it satisfies
\begin{equation}\nonumber
\sup_{B \subset \mathbb{R}^n
}\Big(\frac{1}{|B|}\int_Bw(x)dx\Big)\Big(\frac{1}{|B|}\int_Bw(x)^{-
\frac{q}{p}}dx\Big)^{\frac{p}{q}} < \infty.
\end{equation}
Here $p$ and $q$ are conjugate indices with $1 < p < \infty$.
For
any ball $B \subset \mathbb{R}^5$, denote $B = B(y_0, R)$. It is
easy to see that if $r_0 > 2R$, one has $r \simeq r_0$ for any $x
\in B$. Consequently, for any $\alpha \in \mathbb{R}$, one has
\begin{eqnarray}\nonumber
&&\Big(\frac{1}{|B|}\int_Bw(x)dx\Big)\Big(\frac{1}{|B|}\int_Bw(x)^{-
\frac{q}{p}}dx\Big)^{\frac{p}{q}}\\\nonumber
&&\lesssim \Big(\frac{1}{|B|}\int_Br_0^\alpha dx\Big)
\Big(\frac{1}{|B|}\int_Br_0^{- \frac{q}{p}
\alpha}dx\Big)^{\frac{p}{q}} \lesssim 1.
\end{eqnarray}
On the other hand, if $r_0 \leq 2R$, then for $\alpha + 3 > -1$ and
$- \frac{q\alpha}{p} + 3 > -1$, one has
\begin{eqnarray}\nonumber
&&\Big(\frac{1}{|B|}\int_Bw(x)dx\Big)\Big(\frac{1}{|B|}\int_Bw(x)^{-
\frac{q}{p}}dx\Big)^{\frac{p}{q}}\\\nonumber
&&\lesssim \Big(\frac{1}{R^5}\int_{r_0 - R}^{r_0 + R}dz\int_{0}^{
3R}r^{\alpha + 3}dr\Big)\Big(\frac{1}{R^5}\int_{r_0 - R}^{r_0 +
R}dz\int_{0}^{3R}r^{- \frac{q\alpha}{p} + 3}dr\Big)^{\frac{p}{q}}\\\nonumber
&&\lesssim R^\alpha R^{-\alpha} = 1.
\end{eqnarray}
Noting that the condition on $\alpha$ is $- 4 < \alpha < 4p\big(1 -
\frac{1}{p}\big)$, we in fact have completed the proof of the lemma.
\end{proof}
\end{document}
|
\begin{document}
\begin{abstract}
We investigate the (small) quantum cohomology ring of the moduli spaces
$\overline{\mathcal{M}}_{0,n}$ of stable $n$-pointed curves of genus $0$. In particular, we
determine an explicit presentation in the case $n=5$ and we outline a
computational approach to the case $n=6$.
\end{abstract}
\maketitle
\section{Introduction}
The (small) quantum cohomology ring of a smooth algebraic variety with
$\star$-product defined in terms of ($3$-point) Gromov-Witten invariants
is a formal deformation of the classical Chow ring in the sense that
the $\star$-product specializes to the cup-product when the formal
parameters are set to $0$.
The notion of quantum Chow ring has been recently extended also to
smooth orbifolds and its degree zero part is usually called the
stringy Chow ring.
In \cite{AGV:02} the stringy Chow ring of $\overline{\mathcal{M}}_{1,1}$
has been computed, while in \cite{Spencer:06} the case of $\mathcal{M}_2$
is handled and that of $\overline{\mathcal{M}}_{2}$ is announced.
Here instead we address the (small) quantum cohomology of the moduli
spaces $\overline{\mathcal{M}}_{0,n}$ of stable $n$-pointed curves of genus $0$. Even though
these spaces are smooth projective varieties with a quite explicit
description, nonetheless their geometry turns out to be rather involved:
indeed, just to quote a couple of astonishing facts, we mention that,
despite the serious efforts by many valuable mathematicians, their
ample (resp., effective) cone has been determined so far only for $n \le 7$
and $n \le 6$ resp. (see \cite{KMK:96} and \cite{HT:02} resp.).
In the present paper we provide an explicit presentation of the small
quantum cohomology ring in the case $n=5$ (see Corollary~\ref{n=5})
relying on previous work by G\"ottsche and Pandharipande (\cite{GP:98})
and we suggest a computational approach to the case $n=6$ (see
Remark~\ref{n=6}) inspired by Gathmann \cite{G:01} (see also \cite{BM:04}, where the small quantum cohomology of all del Pezzo surfaces is calculated, and \cite{B:04}, where a general theorem about semisimplicity conservation under blowing-up of points is proved).
The author is grateful to Gianfranco Casnati, Gianni Ciolli, and Barbara Fantechi for inspiring conversations and enlightening suggestions. Thanks are also due to
Yuri I. Manin and Dan Abramovich for kindly pointing
out references \cite{B:04}, \cite{BM:04}, and
\cite{BK:05}, respectively.
\section{Preliminaries}
\subsection{(Small) quantum cohomology}\label{quantum}
Let $X$ be a smooth complex projective variety, let $T_0 = 1 \in A^0(X),
T_1, \ldots,$ $T_m$ be a homogeneous basis of the graded vector space
$V := H^*(X, \mathbb{Q})$. Let $T^0 = \mathrm{point}, T^1, \ldots, T^m \in V$
be the (Poincar\'e) dual basis and let $E \subset H_2(X, \mathbb{Z})$ denote
the subset of effective curves.
The (small) quantum cohomology ring of $X$ is a $\star$-product structure
on $V \otimes R$, where $R$ is a formal power series ring and the quantum
product $\star$ reduces to the usual cup product $\cup$ when all formal
variables are set to zero. Namely, given classes $\alpha_1$ and $\alpha_2
\in H^*(X, \mathbb{Q})$, their quantum product is defined as follows:
$$
\alpha_1 \star \alpha_2 = \alpha_1 \cup \alpha_2 + \sum_{\beta \in E}
\sum_{i=0}^m I_\beta(\alpha_1, \alpha_2, T_i) T^i q^{\beta}
$$
where $I_\beta(\cdot,\cdot,\cdot)$ denotes the ($3$-point) Gromov-Witten
invariant relative to class $\beta$ (see for instance \cite{FP:97}, \S~7).
\subsection{Moduli spaces of genus zero (stable) curves}\label{moduli}
The moduli spaces $\overline{\mathcal{M}}_{0,n}$ of stable $n$-pointed curves of genus $0$
is a smooth projective variety of dimension $n-3$ which can be explicitely
obtained from $\mathbb{P}^{n-3}$ via the following construction due to
Kapranov. For every $n \ge 4$, let $X \subset \mathbb{P}^{n-3}$ be a set of $n-1$
points in linear general position, let $B^0 := \mathbb{P}^{n-3}$ and for $i \ge 0$
let $B^{i+1} \to B^i$ be the blow-up of $B^i$ along the proper transforms of
the $\binom{n-1}{i+1}$ $i$-planes through $i+1$ points. With the above
notation, we have $\overline{\mathcal{M}}_{0,n} \cong B^{n-4}$ (see for instance \cite{V:02}).
Let $P:= \{1,2, \ldots, n \}$ and for every $S \subset P$ with
$2 \le \vert S \vert \le n-2$ let $\Delta_{\{0,S\}}$ be the boundary
component of $\overline{\mathcal{M}}_{0,n}$ whose general element is the union of
two copies of $\mathbb{P}^1$, labelled respectively by $S$
and $P \setminus S$, meeting at one point. We denote by
$\delta_S$ the corresponding class in $\mathrm{Pic}(\overline{\mathcal{M}}_{0,n})$ and
we define inductively:
\begin{eqnarray*}
\mathscr B_4 &:=& \{ \delta_{\{2,3\}} \}\\
\mathscr B_i &:=& \mathscr B_{i-1} \cup \{ \delta_B: B \subseteq \{1, \ldots, i \},
i \notin B \supseteq \{i-1, i-2 \} \}\\
& &\cup \{\delta_{B^c \setminus \{ i \}}: \delta_B \in
\mathscr B_{i-1} \setminus \mathscr B_{i-2} \}.
\end{eqnarray*}
Then according to \cite{F:05}, Proposition~1, $\mathscr B_n$ is a basis of $\mathrm{Pic}(\overline{\mathcal{M}}_{0,n})$.
\section{The results}
\subsection{The case $n=5$}
In order to determine the quantum cohomology ring we first need to manage
the quantum product between two divisor classes.
\begin{Theorem}~\label{main}
Let $X$ be $\mathbb{P}^2$ blown up at $4$ points in linear general position.
Let $H, E_1, \ldots, E_4$ denote the strict transform of the hyperplane class
and the exceptional divisor classes respectively.
If $\Delta_1, \Delta_2 \in H^2(X, \mathbb{Q})$ then their quantum product can be expressed
as follows:
\begin{eqnarray*}
\Delta_1 \star \Delta_2 &=& \Delta_1 . \Delta_2
- \sum_{i=1}^4 \Delta_1.E_i
\Delta_2.E_i E_i q^{0, e_i} \\
& & + \sum_{1 \le i < j \le 4} \Delta_1.(H-E_i-E_j)
\Delta_2.(H-E_i-E_j)\\
& &+ (H+E_i+E_j)q^{1, e_i+e_j}\\
& &+ \sum \Delta_1.(H-\sum_{i=1}^4 \varepsilon_i E_i)
\Delta_2.(H-\sum_{i=1}^4 \varepsilon_iE_i) q^{1, \sum_{i=1}^4 \varepsilon_ie_i}\\
& & + \sum \Delta_1. (2H-\sum_{i=1}^4 \varepsilon_iE_i)
\Delta_2. (2H-\sum_{i=1}^4 \varepsilon_iE_i)
q^{2,\sum_{i=1}^4 \varepsilon_ie_i}
\end{eqnarray*}
where the sums run over $\varepsilon_i \in \{0,1 \}$ and if
$\beta = a H - \sum_{i=1}^4 b_i E_i$ we denote $q^{\beta}$ by
$q^{a,(b_1,b_2,b_3,b_4)}$ writing $e_i$ for the $i$-th vector of
the canonical basis of $\mathbb{R}^4$.
\end{Theorem}
\proof By \cite{BP:04}, Corollary~3.3, the effective cone of $X$ is
generated by the following divisors:
\begin{eqnarray*}
D_1 &=& H - E_1 - E_2 \\
D_2 &=& H - E_1 - E_3 \\
D_3 &=& H - E_1 - E_4 \\
D_4 &=& H - E_2 - E_3 \\
D_5 &=& H - E_2 - E_4 \\
D_6 &=& H - E_3 - E_4 \\
D_7 &=& E_1 \\
D_8 &=& E_2 \\
D_9 &=& E_3 \\
D_{10} &=& E_4 \\
\end{eqnarray*}
Hence if
$$
\beta = d H - \sum_{i=1}^4 a_i E_i
$$
is an effective curve, then we can write
$$
\beta = \sum_{i=1}^{10} c_i D_i
$$
with $c_i \ge 0$ for every $i$, in particular we have
$$
d = \sum_{i=1}^6 c_i
$$
By the divisor axiom,
$$
I_\beta(\Delta_1,\Delta_2,T_i) = (\Delta_1.\beta)(\Delta_2.\beta) I_\beta(T_i)
$$
therefore we need only to compute $1$-point Gromov-Witten invariants.
Since
$$
-K_X = \mathcal{O}_X(1) = 3H - E_1 - E_2 - E_3 - E_4
$$
we deduce that the expected dimension of the corresponding moduli space of stable
maps is
\begin{equation}\label{dimension}
\mathrm{vdim} \overline{\mathcal{M}}_{0,1}(X, \beta) = - K_X. \beta = \sum_{i=1}^{10} c_i
\end{equation}
We have to consider separately the three cases $T_i = 1$, $T_i = D$ a divisor
and $T_i = \mathrm{point}$. It is a general fact that $I_\beta(1)=0$ (see for
instance \cite{FP:97}, \S~7.(II)). Next, if $D$ is a divisor then
$I_\beta(D) \ne 0$ only if $\mathrm{vdim} \overline{\mathcal{M}}_{0,1}(X, \beta)=1$,
in particular we have $d \le 1$. If $d=0$, then $\beta$ is purely exceptional
and from \cite{G:01}, Lemma~2.3~(i), it follows that $I_\beta(D) \ne 0$ if
and only if $\beta = E_i$, $D = E_i$ and $I_{E_i}(E_i)=-1$. If instead $d=1$,
then $\beta = H - E_i - E_j$ and $I_\beta(D) \ne 0$ if and only if $D$ is either
$H$, or $E_i$, or $E_j$, and $I_\beta(D)=1$. Finally, if $T_i = \mathrm{point}$
then $I_\beta(\mathrm{point}) \ne 0$ only if
$\mathrm{vdim} \overline{\mathcal{M}}_{0,1}(X, \beta)=2$, in particular we have $d \le 2$.
If $d=0$, then $\beta$ is purely exceptional and $I_\beta(\mathrm{point}) = 0$
by \cite{G:01}, Lemma~2.3~(i). If $d=1$, we have $I_\beta(\mathrm{point}) = 1$
for $\beta = H - \sum_{i=1}^4 \varepsilon_i E_i$ with $\varepsilon_i \in \{0,1 \}$
and $I_\beta(\mathrm{point}) = 0$ otherwise by \cite{GP:98},
\S~5.2 and \S~3.(P4)--(P5). If $d=2$, we have $I_\beta(\mathrm{point}) = 1$
for $\beta = 2H - \sum_{i=1}^4 \varepsilon_i E_i$ with $\varepsilon_i \in \{0,1 \}$
and $I_\beta(\mathrm{point}) = 0$ otherwise by \cite{GP:98}, \S~5.2
and \S~3.(P4)--(P5). Hence our claim follows.
\qed
As a consequence, we can perform the computation we are interested in.
\begin{Corollary}\label{n=5}
The small quantum cohomology ring of $\overline{\mathcal{M}}_{0,5}$
admits the following explicit presentation:
$$
QH^*_s(\overline{\mathcal{M}}_{0,5}) = \frac{\mathbb{Q}[q^{0, \sum_{i=1}^4 \varepsilon_i e_i},
q^{1,\sum_{i=1}^4 \varepsilon_i e_i}, q^{2, \sum_{i=1}^4 \varepsilon_i e_i},
\delta_{2,3}, \delta_{3,4},\delta_{1,5},
\delta_{2,5},\delta_{1,4}]}{(f_i^*)_{i=1, \ldots, 5}}
$$
where $\varepsilon_i \in \{0,1\}$, $e_i$ denotes the $i$-th vector of
the canonical basis of $\mathbb{R}^4$ and
\begin{eqnarray*}
f_1^* &=& \delta_{2,3} \star \delta_{3,4}+ E_1 q^{0,(1,0,0,0)}-q^{1,(0,0,0,0)}
-q^{1,(0,0,1,0)}-q^{1,(1,1,0,1)}\\
& &-q^{1,(1,1,1,1)}-4q^{2,(0,0,0,0)}-q^{2,(1,0,0,0)}-2q^{2,(0,1,0,0)}-4q^{2,(0,0,1,0)}\\
& &-2q^{2,(0,0,0,1)}-q^{2,(1,0,1,0)}-2q^{2,(0,1,1,0)}-q^{2,(0,1,0,1)}\\
& &-2q^{2,(0,0,1,1)}-q^{2,(0,1,1,1)}\\
f_2^* &=& \delta_{2,3} \star \delta_{2,5}-(H+E_2+E_3)q^{1,(0,1,1,0)}-q^{1,(0,1,0,0)}
-q^{1,(0,1,1,0)}\\
& &+q^{1,(1,1,0,1)}+q^{1,(1,1,1,1)}-2q^{2,(0,1,0,0)}-q^{2,(1,1,0,0)}
-2q^{2,(0,1,1,0)}\\
& &-q^{2,(0,1,0,1)}-q^{2,(0,1,1,1)}
-q^{2,(1,1,1,0)}\\
f_3^* &=& \delta_{3,4} \star \delta_{1,4}+ E_2 q^{0,(0,1,0,0)}-q^{1,(0,0,0,0)}
-q^{1,(0,0,0,1)}-q^{1,(1,1,1,0)}\\
& &-q^{1,(1,1,1,1)}-4q^{2,(0,0,0,0)}-2q^{2,(1,0,0,0)}-q^{2,(0,1,0,0)}\\
& &-2q^{2,(0,0,1,0)}-4q^{2,(0,0,0,1)}-q^{2,(1,0,1,0)}-2q^{2,(1,0,0,1)}-q^{2,(0,1,0,1)}\\
& &-2q^{2,(0,0,1,1)}-q^{2,(1,0,1,1)}\\
f_4^* &=& \delta_{1,5} \star \delta_{2,5}-(H+E_1+E_2)q^{1,(1,1,0,0)}
-q^{1,(1,1,0,0)}-q^{1,(1,1,1,0)}\\
& &-q^{1,(1,1,0,1)}-q^{1,(1,1,1,1)}-q^{2,(1,1,0,0)}-q^{2,(1,1,1,0)}
-q^{2,(1,1,0,1)}\\
& &-q^{2,(1,1,1,1)}\\
f_5^* &=& \delta_{1,5} \star \delta_{1,4}-(H+E_1+E_4)q^{1,(1,0,0,1)}-q^{1,(1,0,0,0)}
-q^{1,(1,0,0,1)}\\
& &+q^{1,(1,1,1,0)}+q^{1,(1,1,1,1)}-2q^{2,(1,0,0,0)}-2q^{2,(1,0,0,1)}
-q^{2,(1,1,0,0)}\\
& &-q^{2,(1,0,1,0)}-q^{2,(1,1,0,1)}-q^{2,(1,0,1,1)}
\end{eqnarray*}
\end{Corollary}
\proof From Keel's results in \cite{K:92} we deduce the following presentation
of the classical Chow ring of $\overline{\mathcal{M}}_{0,5}$ in terms of the
basis $\mathscr B_5$ recalled in \S~\ref{moduli}:
$$
A^*(\overline{\mathcal{M}}_{0,5}) = \frac{\mathbb{Z}[\delta_{2,3}, \delta_{3,4},\delta_{1,5},
\delta_{2,5},\delta_{1,4}]}{(f_i)_{i=1, \ldots, 5}}
$$
where
\begin{eqnarray*}
f_1^* &=& \delta_{2,3}.\delta_{3,4} = 0 \\
f_2^* &=& \delta_{2,3}.\delta_{2,5} = 0 \\
f_3^* &=& \delta_{3,4}.\delta_{1,4} = 0 \\
f_4^* &=& \delta_{1,5}.\delta_{2,5} = 0 \\
f_5^* &=& \delta_{1,5}.\delta_{1,4} = 0
\end{eqnarray*}
According to Kapranov construction recalled in \S~\ref{moduli}, we can regard
$\overline{\mathcal{M}}_{0,5}$ as $\mathbb{P}^2$ blown up at $4$ points in linear
general position and obtain exactly as in \cite{V:02} the following identifications
(here we take $5$ to be the special point):
\begin{eqnarray*}
\delta_{2,3} &=& H - E_1 - E_4 \\
\delta_{3,4} &=& H - E_1 - E_2 \\
\delta_{1,5} &=& E_1 \\
\delta_{2,5} &=& E_2 \\
\delta_{1,4} &=& H - E_2 - E_3
\end{eqnarray*}
Notice moreover that by (\ref{dimension}) $I_\beta(\alpha_1,\alpha_2,T_i) \ne 0$
only when $\sum c_i (-K_X.D_i)$ is a fixed number
with both $c_i \ge 0$ and $-K_X.D_i > 0$ for every $i$, hence there are only
finitely many possible values for the exponents of the formal variables $q$ and
the quantum cohomology ring turns out to be a polynomial ring.
Hence our claim can be deduced from Theorem~\ref{main} by applying\cite{FP:97},
\S~10, Proposition~11 (see \cite{P:03}, Chapter~3, for analogous computations).
\qed
\subsection{The case $n=6$}
Here we have obtained only partial results.
\begin{Conjecture}\label{associativity}
Let $Y$ be the blow-up a smooth projective threefold $X$ along a curve $C$ such that
$g(C) \ge 1$ or $g(C)=0$ and $-K_X.C \ge 0$. Then the associativity equations of the
quantum product suffice to determine all (genus $0$) Gromov-Witten invariants of $Y$
in terms of those of $X$.
\end{Conjecture}
\begin{Remark}
In order to address Conjecture~\ref{associativity}, one
might wish to argue as in \cite{G:01}, proof of Theorem~2.1.
Indeed, if both $\beta$ and $T$ are non-exceptional classes,
then \cite{H:00}, Theorem~1.5, would even imply $I_\beta^Y(T) = I_\beta^X(T)$ (but see \cite{BK:05}, Remark~8, for a
pertinent counterexample to the statement in \cite{H:00}).
On the other hand, if $\beta$ is exceptional then
$\beta = F$ and the only eventually nonzero invariants
to be computed are $I_{dF}(d \varphi)$, where $F$ and $\varphi$ correspond to exceptional fibers. These invariants enumerate $d$-fold coverings of a fibre over a point in $C$,
hence they are zero for $d \ge 2$ (otherwise a curve should lie in two different fibers). If instead $d=1$ then $I_F(\varphi)=-1$ (see for instance \cite{C:05}, Lemma~2).
Unluckily, as far as we know, the analogue of \cite{G:01}, Algorithm~2.4, is still missing.
\end{Remark}
\begin{Remark}\label{n=6}
From Conjecture~\ref{associativity} it would follow that all Gromov-Witten invariants
of $\overline{\mathcal{M}}_{0,6}$ can be recursively computed. Indeed, as recalled in
\S~\ref{moduli}, $\overline{\mathcal{M}}_{0,6}$ can be identified with $\mathbb{P}^3$ blown up in $5$ points in linear general position and along the cords between pairs of points.
In order to check that $-K.C \ge 0$, let $C$ be the strict transform of the cord
$l_{ab}$ between points $a$ and $b$ and choose planes $\pi$, $\rho$ in
$\mathbb{P}^3$ such that $l_{ab} = \pi.\rho$. If $p: \tilde{X} \to X$
denotes the blow up of $l_{ab}$ we have
\begin{eqnarray*}
p^* \pi &=& \tilde{\pi} + E_a + E_b \\
p^* \rho &=& \tilde{\rho} + E_a + E_b \\
K_{\tilde{X}} &=& K_{\mathbb{P}^3} + 2 \sum_i E_i + 2 \sum_{i,j} E_{ij}
\end{eqnarray*}
(see \cite{H:77}, II., ex.~8.5) where $\tilde{\pi}$ and $\tilde{\rho}$ resp. are
the strict transforms of $\pi$ and $\rho$ resp., while $E_i$ and $E_{ij}$ are
the exceptional divisors corresponding to the points and the cords which have
been previously blown up. Hence
\begin{eqnarray*}
-K_{\tilde{X}}.C &=& (- K_{\mathbb{P}^3} - 2 \sum_i E_i - 2 \sum_{i,j} E_{ij}). \\
& & (p^* \pi - E_a - E_b) . (p^* \rho - E_a - E_b) \\
&=& - K_{\mathbb{P}^3}.\pi.\rho - 2 E_a^3 - 2 E_b^3 \\
&=& \mathcal{O}_{\mathbb{P}^3}(4).l_{ab}-2-2 = 0
\end{eqnarray*}
(recall that if $p: \tilde{X} \to X$ is the blow up of a smooth threefold along
a smooth curve with exceptional divisor $E$ then $E.p^*C=0$ for every curve
$C \subset X$, see for instance \cite{C:05}, Lemma~1).
\end{Remark}
\end{document}
|
\begin{document}
\author{N. Fabre\footnote{[email protected]}}
\affiliation{Telecom Paris, Institut Polytechnique de Paris, 19 Place Marguerite Perey, 91120 Palaiseau, France}
\begin{abstract}
We present a linear optical protocol for teleporting and correcting both temporal and frequency errors in two time-frequency qubit states. The first state is the frequency (or time-of-arrival) cat qubit, which is a single photon in a superposition of two frequencies (or time-of-arrival), while the second is the time-frequency Gottesman-Kitaev-Preskill (GKP) state, which is a single photon with a frequency comb structure. The proposed optical scheme could be valuable for reducing error rate in quantum communication protocols involving one of these qubits.
\end{abstract}
\title{Teleportation-based error correction protocol of time-frequency qubits states}
\maketitle
\section{Introduction}
Quantum information can be encoded in various degrees of freedom of single photons, which can be described by either discrete or continuous variables (CV). Frequency (or energy) and time-of-arrival are natural pairs of conjugate quantum continuous variables in the single photon subspace, along with the transverse position and momentum degrees of freedom \cite{fabre_generation_2020, fabre:tel-03191301, tasca_continuous_2011, fabre_time_2022}. Discretizing the frequency or time-of-arrival into temporal or frequency bins, or performing the mode decomposition of the the continuous variable distribution of the single photon, can be experimentally motivated due to the finite resolution of detection devices or specific requirements of a quantum protocol, such as in quantum metrology for super-resolution \cite{PhysRevX.6.031033}. We should stress that in any dimensional single-photon encoding, photon losses do not correspond to a logical error. The second way of encoding information is through a particle-number sensitive encoding, which can be used to define physical systems with either discrete or continuous variables. In this encoding, CV corresponds to the quadratures of the electromagnetic field, {\it{i.e}}, the amplitude and phase of the quantum field, in a particular mode. With the particle-sensitive encoding, photon loss corresponds to a logical error. Mathematically, the quadrature of an electromagnetic field in a given mode can be treated as the continuous degree of freedom of a single photon \cite{fabre_time_2022}, as long as an auxiliary discrete mode is occupied by only one single photon. \\
Error correction code for continuous variables encoding is defined by discretizing them. Three bosonic qubit codes have been studied, such as the cat-code \cite{cochrane_macroscopically_1999, PhysRevX.9.041053,Albert_2019}, Gottesman, Kitaev and Preskill (GKP) code \cite{gottesman_encoding_2001,fluhmann_encoding_2019,PhysRevA.99.032344,campagne-ibarcq_quantum_2020,https://doi.org/10.48550/arxiv.2205.09781,PhysRevA.103.032409,8482307} and the binomial code \cite{michael_new_2016,hu_quantum_2019}. Cat and GKP codes are candidates for achieving universal quantum computation, see \cite{PhysRevX.9.041053,PRXQuantum.3.010329} and \cite{PhysRevLett.123.200502,bourassa_blueprint_2021}. GKP codes could be employed for building quantum repeaters \cite{rozpedek_quantum_2021}, and for sensing application \cite{Duivenvoorden_2017,Terhal_2016}. The mathematical analogy between time-frequency and quadrature CV allows defining time-frequency qubits states, called time-frequency cat state \cite{fabre:tel-03191301, PhysRevA.102.023710} and time-frequency GKP state \cite{fabre_generation_2020,fabre:tel-03191301,https://doi.org/10.48550/arxiv.2301.03188}. Both of these codes are ways to discretize time-frequency continuous variables at the single photon level to define a qubit. CV or time-frequency CV codes possess an equivalent mathematical structure, they are common eigenvectors of non-commuting displacements operators \cite{fabre_generation_2020} and they are thus designed to be robust against small shift in one continuous variable (cat state) and the two canonically conjugated ones (GKP state). \\
In this paper, we start by reminding the mathematical structure of the time-frequency cat and GKP codes, and discuss the temporal and frequency errors from which they are designed to be robust against. We then analyse two different entanglement structures of time-frequency GKP state which can be generated experimentally, and can be interpreted as the entanglement of a noisy state of interest with a less noisy ancilla. However, these entanglement structures which lead to a natural error correction strategy for single photon encoding is difficult to realize experimentally with the current technology, since it requires to perform a frequency entanglement operation between two single photons. Therefore, the standard method for error correction for quadrature qubits cannot be applied straightforwardly. Inspired by the teleportation-based error correction protocol for quadrature variables \cite{PhysRevA.102.062411,PhysRevResearch.3.033118}, we develop a teleportation-based error correction protocol of frequency qubits states, using only linear optical elements, and allows correcting both time and frequency variables at once. The error correction is naturally performed because the ancilla EPR state which assists to the teleportation is less noisy in the temporal and frequency domains compared to the state of interest. Since the protocol requires the use of Bell's measurement, we also describe how to experimentally implement such a measurement for the two type of frequency qubits. The proposed protocol is intrinsically probabilistic but consist on the teleportation of non-orthogonal states \cite{sisodia_teleportation_2017}, instead of orthogonal ones \cite{PhysRevLett.70.1895, bouwmeester_high-fidelity_2000}. The non-orthogonality of the state reduces the efficiency of the teleportation protocol, and we mention that photon number resolving detectors can help increase the probability of success by reducing the number of rejected measurement events. The presented teleportation protocol is a new solution for correcting temporal broadening caused by dispersion effects affecting time-bin qubit states, thereby reducing the error rate of quantum communication protocols \cite{Zhong_2015,Jin:19,PhysRevApplied.14.014051}. \\
The paper is organized as follows. In Sec.~\ref{universal}, we provide a reminder of the definition of the time-frequency cat and GKP states and the reason of the experimental difficulty behind the natural error correction scheme of the time-frequency GKP states which requires frequency entanglement gates. In Sec.~\ref{concrete}, we explain the optical equivalent of the polarizing beam-splitter, a Mach-Zehnder interferometer, which allows separating spatially the two logical states of the time-frequency cat and GKP qubits. Such an interferometer is crucial for implementing Bell's measurement in the time-frequency degree of freedom. In Sec.~\ref{sectionteleportation}, we present a teleportation-based error correction protocol for the time-frequency GKP state, which makes use of the Bell's measurement. The protocol is probabilistic and can be achieved with current experimental devices. Finally, in Sec.~\ref{conclusion}, we summarize our results and present new perspectives.
\section{Time-frequency qubits states}\label{universal}
\subsection{Time-frequency cat state}
We will denote $\ket{\Omega}$ the vacuum state. A single photon state at frequency $\omega$ in the spatial port $a$ is denoted as $\ket{\omega}_{a}=\hat{a}^{\dagger}(\omega)\ket{\Omega}$. The frequency cat state as introduced in \cite{PhysRevA.102.023710}, is defined as the superposition of a single photon into two different frequency Gaussian distribution:
\begin{equation}\label{catstate}
\ket{\psi}=N_{\alpha\beta}(\alpha\ket{\omega_{1}}_{a}+\beta\ket{\omega_{2}}_{a})=N_{\alpha\beta}(\alpha \ket{0}_{a}+\beta \ket{1}_{a}),
\end{equation}
where $\ket{\omega_{1}}_{a}=\frac{1}{\sqrt{2\pi\sigma^{2}}} \int d\omega \text{exp}(-(\omega-\omega_{1})^{2}/2\sigma^{2}) \ket{\omega}_{a}$ and the normalization of the state is given by $1=N_{\alpha\beta}^{2}(\abs{\alpha}^{2}+\abs{\beta}^{2}+2\text{Re}(\alpha\beta^{*}) e^{-(\omega_{1}-\omega_{2})^{2}/2\sigma^{2}})$. This is a non-orthogonal qubit state as their overlap is ${}_a\bra{0}\ket{1}_{a}=\text{exp}(-(\omega_{1}-\omega_{2})^{2}/2\sigma^{2})$.
Experimental proposal for manipulating such a state was proposed in \cite{lukens_frequency-encoded_2017,lu_controlled-not_2019}, with pulse shapers and electro-optic modulators were used in cascade. The results showed an operation fidelity close to 100\% but the successive optical elements decrease drastically the probability of single photon detection. The wavefunction of the time cat state is defined as:
\begin{equation}\label{timebinqubit}
\ket{\psi}=N_{\alpha\beta}(\alpha \ket{t_{1}}_{a}+\beta \ket{t_{2}}_{a}).
\end{equation}
Note that for avoiding to have a normalization constant depending in the coefficients $\alpha,\beta$, and writing the wavefunction in an orthogonal basis, we can employ the Gram-Schmidt decomposition procedure. The normalized orthogonal basis $\ket{a}$ and $\ket{b}$ can be written as:
\begin{equation}
\ket{a}=\ket{0}, \ \ket{b}=N[\ket{1}-\bra{0}\ket{1}\ket{0}],
\end{equation}
where $N=1/\sqrt{1-r^{2}}$ where $r=\abs{\bra{0}\ket{1}}$. The GKP input state Eq.~(\ref{GKPinput}) can be written in the orthogonal basis as:
\begin{equation}
\ket{\psi}=(\alpha+\beta\bra{0}\ket{1})\ket{a}+\frac{\beta}{N}\ket{b},
\end{equation}
where we have now $\abs{(\alpha+\beta\bra{0}\ket{1})}^{2}+\abs{\frac{\beta}{N}}^{2}=1$.\\
The frequency entangled cat state, an EPR state could be written as:
\begin{align}\label{EPRfrequency}
\ket{\phi^{\pm}}=N_{\text{EPR}}(\ket{\omega_{1}\omega_{1}}_{ab}\pm\ket{\omega_{2}\omega_{2}}_{ab})\\
\ket{\psi^{\pm}}=N_{\text{EPR}}(\ket{\omega_{1}\omega_{2}}_{ab}\pm\ket{\omega_{2}\omega_{1}}_{ab})
\end{align}
where $N_{\text{EPR}}^{2}(2+2e^{-(\omega_{1}-\omega_{2})^{2}/2\sigma^{2}})=1$. When $\omega_{1}-\omega_{2} \gg \sigma$ we recover the normalization of an EPR state composed of orthogonal qubits $N_{\text{EPR}}=1/\sqrt{2}$. The frequency cat state can be produced by integrated optical waveguide \cite{https://doi.org/10.48550/arxiv.2207.10943}, and bulk system \cite{chen_hong-ou-mandel_2019}. The wavefunction of a temporal entangled EPR state Eq.~(\ref{EPRfrequency}) has the same mathematical structure, and such a quantum state can be produced by quantum dots for instance \cite{jayakumar_time-bin_2014}. This type of quantum state has potential applications in quantum communications \cite{kim_quantum_2022}.
\subsection{Time-frequency GKP state}
We define a frequency lattice of period $\overline{\omega}$. Centered on each of this interval, we define the ideal time-frequency GKP state as the following frequency comb at the single photon level:
\begin{equation}
\ket{\overline{\mu}_{\omega}}_{a}=\sum_{n\in\mathds{Z}} \ket{(2n+\mu)\overline{\omega}}_{a}
\end{equation}
where $\mu=0,1$ index the two logical states. Note that the equal weight superposition of the zero and the one logical time-frequency GKP states in the frequency domain are the zero and one in the temporal domain:
\begin{align}
\ket{\overline{+}_{\omega}}_{a}=\frac{1}{\sqrt{2}}(\ket{\overline{0}_{\omega}}_{a}+\ket{\overline{1}_{\omega}}_{a})=\ket{\overline{0}_{t}}_{a}\\
\ket{\overline{-}_{\omega}}_{a}=\frac{1}{\sqrt{2}}(\ket{\overline{0}_{\omega}}_{a}-\ket{\overline{1}_{\omega}}_{a})=\ket{\overline{1}_{t}}_{a}.
\end{align}
The periodicity of the state in the temporal domain: $\overline{\omega}=2\pi/\overline{\omega}$. Such a state is not physical since the state is an infinite sum of monochromatic state and will require an infinite energy to prepare it. The physical time-frequency GKP state can be built upon this ideal state by applying time and frequency noise, which are frequency and time displacement operations multiplied by Gaussian distribution which is detailed in \cite{fabre_generation_2020}. The wavefunction of the two logical states can be written as follows:
\begin{equation}
\ket{\mu_{\omega}}_{a}=N_{\mu}\sum_{n\in\mathds{Z}}\int d\omega G^{\kappa}(\omega)G^{\sigma}(\omega-(2n+\mu)\overline{\omega}) \ket{\omega}_{a}
\end{equation}
where $G$ are Gaussian functions representing the envelope of the comb of width $\kappa$ and the peaks of the comb of width $\sigma$. The frequency probability distribution of the grid state is represented in Fig.~\ref{GKP}. Alternatively for large comb $\overline{\omega}/\sigma \gg 1$ \cite{PhysRevA.94.022325}, we can write:
\begin{equation}
\ket{\mu_{\omega}}_{a}=N_{\mu}\sum_{n\in\mathds{Z}}c_{2n+\mu} \int d\omega G^{\sigma}(\omega-(2n+\mu)\overline{\omega}) \ket{\omega}_{a}
\end{equation}
with the envelope coefficients $c_{n}=\text{exp}(-(n\overline{\omega}/\kappa)^{2}/2)$. $N_{\mu}$ is the normalization constant found thanks to the relation $1=\abs{{}_a\bra{\mu_{\omega}}\ket{\mu_{\omega}}_{a}}^{2}=N_{\mu}^{2}\sum_{n\in\mathds{Z}} \abs{c_{2n+\mu}}^{2} \sqrt{\pi \sigma^{2}}$. \\
\begin{figure*}
\caption{\label{GKP}
\label{GKP}
\end{figure*}
In general, the physical GKP state can be in a superposition of the two logical states:
\begin{equation}\label{GKPinput}
\ket{\psi}=N_{\alpha\beta}(\alpha\ket{0_{\omega}}_{a}+\beta\ket{1_{\omega}}_{a}),
\end{equation}
where $N_{\alpha\beta}=(\abs{\alpha}^{2}+\abs{\beta}^{2}+2\text{Re}(\alpha^{*}\beta \bra{0}\ket{1})))^{-1/2}$. The two logical states are not orthogonal when a Gaussian wavepacket enters in the frequency bin of its neighbour. The overlap ${}_a\bra{0_{\omega}}\ket{1_{\omega}}_{a}$ is different than zero and is equal to:
\begin{equation}
{}_a\bra{0_{\omega}}\ket{1_{\omega}}_{a}=e^{-\overline{\omega}^{2}/4\sigma^{2}} \frac{\sum_{n}c_{2n}c^{*}_{2n+1}}{(\sqrt{\sum_{n} \abs{c_{2n}}^{2}\sum_{n} \abs{c_{2n+1}}^{2})} }.
\end{equation}
The full state is thus described by five importants parameters. The complex parameters $\alpha$ and $\beta$ where the quantum information is encoded, the frequency width $\sigma,\kappa$ and the periodicity of the state $\overline{\omega}$. If the state is not too much noisy, meaning that $ {}_a\bra{0_{\omega}}\ket{1_{\omega}}_{a}\sim 0$, then the normalisation condition of Eq.~(\ref{GKPinput}) is given by $\abs{\alpha}^{2}+\abs{\beta}^{2}=1$. Finally, in \cite{fabre_generation_2020,fabre_time_2022}, we define the time-of-arrival and frequency operators which do not commute and verify an Heisenberg algebra. This is mathematically equivalent to the non-commutativity of time-frequency displacement operators. This property leads to consider temporal and frequency bandwidth as quantum noise at the single photon level. \\
The GKP states which are defined as the sum of squeezed states in a given mode \cite{gottesman_encoding_2001,albert_performance_2018}, are designed to be robust against small shift in position and momentum, which can be caused by a Gaussian quantum channel, but they are also robust against photon losses \cite{albert_performance_2018}. On the other hand, time-frequency GKP states are designed to be robust against small shift in time and frequency. The major difference between GKP states and time-frequency GKP states is that photon losses do not result in errors for the latter. In general, temporal errors are the dominant source of errors, while frequency is considered a robust degree of freedom, as it is barely affected by linear physical processes. Additionally, GKP states can be used for fault-tolerant universal quantum computation \cite{bourassa_blueprint_2021}. Due to their mathematical similarity with time-frequency GKP states, it is expected that they would lead to the same mathematical result. However, the generation of non-Gaussian states using the degree of freedom of a single photon is relatively simple to implement experimentally. As a result, the experimental implementation of a time-frequency GKP state is considered straightforward. In contrast, creating entanglement gates between two single photons is a more challenging task. For particle-number sensitive encoding, non-Gaussian operations involving the quadrature degree of freedom can be difficult to implement, while two-mode Gaussian operations, such as with a beam splitter, are easier to perform.
\subsection{Sources of time-frequency noise}
In this section, we discuss the physical processes that lead to temporal-spectral broadening or distorsion. Coherent and incoherent errors will lead to either pure and mixed state respectively.\\
Temporal errors for both codes arise from temporal spreading of each wavepacket composing the state due to linear dispersion effect, described by a coherent model (see for instance \cite{hong_dispersive_2018} for an example in the single photon regime). After such second order dispersion effect, the temporal width of the Gaussian wavepacket becomes $\tau=\tau_{0}\sqrt{1+\tau_{c}^{4}/\tau_{0}^{4}}$ where $\tau_{0}$ is the initial width of the pulse and $\tau_{c}=\sqrt{\beta_{2}L}$, $L$ being the length of the dispersive medium and $\beta_{2}$ the dispersion coefficient. Such a dispersion process is described by an unitary operation, and it can in principle be undone by a reverse transformation. However, it requires the knowledge of the full characterization of the propagation channel. For the time-frequency GKP state, the dispersive effect not only lead to a temporal spreading of each wavepacket, but also lead to the formation of replica temporal images, called the temporal Talbot effect (see for instance \cite{maram_spectral_2013}). One has to consider specific length of the fiber or dispersive coefficient to recover the initial state, which will be also temporally broadened. Polarizing mode dispersion is one of incoherent temporal broadening \cite{antonelli_pulse_2005,poon_polarization_2008,gordon_pmd_2000}. Due to a coupling between the polarization and frequency degree of freedom, if the polarization is not measured, the single photon state becomes mixed in frequency \cite{chang-hua_polarization_nodate}. Thus, the error correction protocol that will be presented in Sec.~\ref{errorentangle} becomes particularly relevant because we can not cancel the error simply by a unitary operation. \\
Frequency noise that causes spectral broadening while the spectral distribution remains Gaussian, is not typically dominant at the single-photon level. Spectral broadening induced by the self-phase modulation effect \cite{matsuda_deterministic_2016} results from the accumulated phase $\phi_{NL}(t) = \frac{2\pi}{\lambda} n_{2} I(t)L$, where $n_{2}$ is the non-linear refractive index, $L$ is the length of the medium, and depends on the intensity $I(t)$ of the field, leading to a non-Gaussian spectral distribution. At the single-photon level, this non-linear process does not occur naturally. In \cite{fabre_generation_2020}, we also argue that frequency noise arises from frequency broadening caused by the generation of photon pairs itself, and we describe one method to correct such a noise. There are numerous processes that can distort the spectral distribution of single photons, such as distortions caused by frequency shifts induced by electro-optic modulators \cite{Kurzyna_2022,PhysRevLett.129.123605}, or the presence of a filter during single photon heralding.\\
An error correction protocol is implemented to mitigate the effects of potential errors from various sources. Its objective is to restore the Gaussian distribution of each frequency peak. This is because Gaussian probability distributions are better understood for setting confidence intervals and error thresholds, as discussed in \cite{fukui_high-threshold_2018}. Both qubits are sensitive to temporal and frequency broadening. While the time-frequency cat state is designed to be robust against errors over one variable, the time-frequency GKP state can correct errors along both orthogonal variables. It is not necessary to use the time-frequency GKP state if the main error is in the temporal domain. We now develop two error correction methods, one based on a direct frequency entanglement between the noisy state of interest and a less noisy ancilla Sec.~\ref{errorentangle}, the other method by using only linear optics and also less noisy ancilla Sec.~\ref{sectionteleportation}.
\subsection{Time-frequency entangled GKP state and error correction protocols}\label{errorentangle}
Error correction of continuous variables states can be done with a Steane error correction protocol, for the quadrature degree of freedom \cite{seshadreesan_coherent_2021} and for the time-frequency one \cite{fabre_generation_2020}. For the two encodings, the protocol consists of entangling the state of interest with one less noisy ancilla (a $\ket{+}$ logical state in one variable, time or frequency), with a beam splitter (resp. with frequency beam-splitter that will be explicated below), and performing one homodyne (resp. single photon frequency measurement) detection at the spatial output of the ancilla for correcting the error along one variable. The protocol is repeated to correct errors in the orthogonally (or canonically) conjugated variable. To achieve this, one must entangle the state of interest with a less noisy state using a $\ket{+}$ state in the canonically conjugated variable compared to the first step. Then, perform the entangling operation, project a measurement in the canonically conjugated variable (compared to step one), and finally conduct a conditional displacement operation. Important tools for quantifying the threshold of noise from which it is still possible to correct the GKP states, using such a Steane error correction protocol was done in \cite{fukui_high-threshold_2018}. Figures of merit for quantifying the probability of measuring the one logical state while this is the zero that should be obtained was done in \cite{PhysRevA.73.012325,fukui_high-threshold_2018}. We will refer to these figures of merit, when we will note one logical state is more (or less) noisy than the other. From now on, we develop two entanglement structures of time-frequency GKP state that can be generated in the laboratory, and how to perform the error correction in each case. \\
The first entanglement structure of time-frequency GKP state that we can studied is the one obtained by using a spontaneous parametric down conversion process (SPDC) from a non-linear crystal placed into an optical cavity \cite{fabre_generation_2020,maltese_generation_2020}. The corresponding wavefunction can be cast as:
\begin{equation}\label{firstGKP}
\ket{\psi}=\iint d\omega_{s} d\omega_{i} f_{+}(\omega_{+})f_{-}(\omega_{-})f(\omega_{s})f(\omega_{i}) \ket{\omega_{s},\omega_{i}}.
\end{equation}
The functions $f_{\pm}$ model respectively the energy conservation and the phase-matching of the SPDC process, and $f$ models the cavity function. The joint spectrum intensity is represented in Fig.~\ref{cestceci}(c). Instead of the traditional view where the wavefunction of two photons is written by applying a quadratic Hamiltonian on the vacuum state, it can also be expressed with a quantum circuit representation as shown in Fig.~\ref{cestceci}(a). This starts with two ideal separable GKP states (fictitious), which undergo initial frequency broadening, which can be interpreted as a frequency noise \cite{fabre_generation_2020}. Then, the state is entangled through the gate performed by the non-linear crystal:
\begin{equation}\label{FBS}
\hat{U}\ket{\omega_{s},\omega_{i}}=\ket{\frac{\omega_{s}+\omega_{i}}{\sqrt{2}},\frac{\omega_{s}-\omega_{i}}{\sqrt{2}}}
\end{equation}
that we called a frequency beam-splitter by analogy with the beam-splitter which acts mathematically similarly into the quadrature position-momentum degree of freedom \cite{doi:10.1080/09500340.2022.2073613,fabre_time_2022}. The state then undergoes a temporal broadening which can be interpreted as a temporal noise, and a final frequency beam-splitter operation is performed. The mathematical reason behind these four successive operations is that the envelop of the grid state are function of the collective variables $\omega_{\pm}$, while the cavity of the local variable $\omega_{s,i}$. The form of the frequency entanglement between the two single photon grid state generated by the non-linear process (see Eq.~(\ref{FBS})) allows for the reduction of the temporal broadening of one of the single photons by performing a temporally-resolved measurement of the other. The form of the pure wavefunction after the conditioned operation has been written in \cite{fabre_generation_2020}.
\begin{figure*}
\caption{\label{cestceci}
\label{cestceci}
\end{figure*}
After correcting the single photon state in the temporal domain, the next step is to correct the state in the frequency domain. For that, we must first entangle the single photon to correct with a less noisy ancilla single photon state using the entanglement gate Eq.~(\ref{FBS}). Since now we have two separable single photon states, we can not directly entangle them by using a non-linear crystal, which is non-efficient. Nevertheless, such a frequency entangled gate could be implemented with a quantum emitter embedded into a waveguide, which assists to the interaction between the two single photons \cite{le_jeannic_dynamical_2022,PhysRevLett.126.023603}. \\
The second entanglement structure of time-frequency GKP state that can be considered, is to start with two initially separable GKP states, with one being less noisy than the other. These states are then entangled using Eq.~(\ref{FBS}). The resulting wavefunction is:
\begin{equation}\label{secondGKP}
\ket{\psi'}=\iint d\omega_{s} d\omega_{i} f_{+}(\omega_{+})f_{-}(\omega_{-})f(\omega_{+})f(\omega_{-}) \ket{\omega_{s},\omega_{i}}.
\end{equation}
In this new spectral function (see Fig.~\ref{cestceci}(d)), the cavity function is now dependent on the collective variables, thus the periodicity of the grid state is not the same as in Eq.~(\ref{firstGKP}). The corresponding quantum circuit representation is pictured in Fig.~\ref{cestceci}(b). The error correction protocol is followed by a temporally-resolved measurement followed by a conditional displacement operation to correct only the temporal noise in the state of interest. A second entanglement operation is performed using a less noisy ancilla state, followed by a resolved-frequency measurement followed by a conditional displacement operation to correct the frequency noise. The difference in the joint spectral amplitude of the photon pairs results - Eq.~(\ref{firstGKP}) and Eq.~(\ref{secondGKP}) - in a different wave function when one of the photons undergoes a temporally-resolved (and frequency) measurement, and a conditional displacement operation. \\
\section{Spatial separation of the two logical time-frequency qubit states}\label{concrete}
In this section, we develop the equivalent of the polarizing beam-slitter operation for the time-frequency cat and GKP state, which is the crucial optical component for the teleportation-based error correction described in the next section. Such an optical element will be called the frequency qubit beam-splitter (FQBS) in what follows.
\subsection{Spatial separation of the two logical time-frequency cat states}\label{spatialtwocolors}
In this section, we explicit how to separate spatially the two Gaussian wavepackets with linear optics, using a Mach-Zehnder interferometer. The frequency cat state as described by Eq.~(\ref{catstate}) is introduced into a balanced beam-splitter and the associated wavefunction is:
\begin{equation}
\ket{\psi}=\frac{1}{2}(\ket{\omega_{1}}_{a}+\ket{\omega_{2}}_{a}+\ket{\omega_{1}}_{b}+\ket{\omega_{2}}_{b}).
\end{equation}
We assumed for simplicity that the two frequency state are well separated $\omega_{1}-\omega_{2} \gg \sigma$.
Then, a pulse shaper is placed at the spatial port $b$, described by the following unitary operation:
\begin{equation}
\hat{U}\ket{\omega_{1}}_{b}=\ket{\omega_{1}}_{b}, \ \hat{U}\ket{\omega_{2}}_{b}=e^{i\phi} \ket{\omega_{2}}_{b}.
\end{equation}
Such an operation can be implemented for instance by mapping the spectral to the spatial degree of freedom with a grating, a spatial light modulator at the focal length of two lenses, and then performing back the mapping from the spatial to spectral degree of freedom \cite{fabre:tel-03191301,PhysRevA.94.063842}. We assume that the frequency peaks are spaced enough so that the pulse shaper acts on each logical state independently. The two spatial paths are then recombined to another balanced beam-splitter. The output state has the final form:
{\small{\begin{equation}
\ket{\psi}=\frac{1}{\sqrt{2}} \ket{\omega_{1}}_{a}\ket{\Omega}_{b}+\frac{1}{2\sqrt{2}}((1+e^{i\phi}) \ket{\omega_{2}}_{a}\ket{\Omega}_{b}+(1-e^{i\phi}) \ket{\Omega}_{a}\ket{\omega_{2}}_{b}).
\end{equation}}}
If $\phi=\pi$, the two logical states are spatially separated $\ket{\psi}=\frac{1}{\sqrt{2}}(\ket{\omega_{1}}_{a}\ket{\Omega}_{b}+\ket{\Omega}_{a}\ket{\omega_{2}}_{b})$. This optical interferometer is the equivalent of the polarizing beam-splitter that separates the vertical and horizontal polarization of optical fields into two distinct spatial paths.
\subsection{Spatial separation of the two logical time-frequency GKP states}\label{separationGKP}
In this part, we introduce how to separate spatially the odd and the even frequencies with a Mach-Zehnder interferometer. Such a scheme was already proposed for manipulating large quadrature position-momentum continuous variables cluster states \citep{PhysRevA.94.032327}, and was described in \cite{fabre:tel-03191301,https://doi.org/10.48550/arxiv.2301.03188}.\\
We start with a time-frequency GKP state with a finite envelop but with infinitely narrow frequency width $\ket{\tilde{+}}=\frac{1}{\sqrt{2}}(\ket{\tilde{0}}_{a}+\ket{\tilde{1}}_{a}$), and is introduced into a balanced beam-splitter. The spatial output port of the beam-splitter is noted $a$ and $b$. A time-shift operation is performed in spatial port $b$, and then the two spatial ports are recombined into a balanced beam-splitter (see Fig.~\ref{scheme}). The final wave function can be written as:
{\small{\begin{equation}
\ket{\psi}=\frac{1}{2}\sum_{n\in\mathds{Z}}[c_{n}(e^{in\overline{\omega} t}-1)\ket{n\overline{\omega}}_{a}\ket{\Omega}_{b}-(e^{in\overline{\omega} t}+1)\ket{\Omega}_{a}\ket{n\overline{\omega}}_{b}].
\end{equation}}}
If we set $t=\pi/\overline{\omega}$, after the second beam-splitter the wavefunction is (see \cite{fabre:tel-03191301}):
\begin{equation}
\ket{\psi}=\frac{1}{\sqrt{2}}(-\ket{1}_{a}\ket{\Omega}_{b}+\ket{\Omega}_{a}\ket{0}_{b}).
\end{equation}
The odd and even frequency components are spatially separated, allowing for individual manipulation, such as correcting the phase accumulation of the one logical state. It is possible to achieve the same result in the temporal domain by shifting the frequency instead of the time, as shown in Fig.~\ref{scheme}.
\begin{figure*}
\caption{\label{scheme}
\label{scheme}
\end{figure*}
If each frequency peak is not infinitely narrow, then the output wavefunction after the Mach-Zehnder interferometer can be written as:
\begin{multline}\label{imperfectpfbs}
\ket{\psi}=\frac{1}{2}\sum_{n\in\mathds{Z}}c_{n}[(\int d\omega (e^{i\pi\omega/\overline{\omega} }-1) G^{\sigma}(\omega-n\overline{\omega}) \ket{\omega}_{a}\ket{\Omega}_{b}\\
-(\int d\omega (e^{i\pi\omega/\overline{\omega} }+1) G^{\sigma}(\omega-n\overline{\omega})\ket{\Omega}_{a} \ket{\omega}_{b}.
\end{multline}
The corresponding probability frequency distribution at spatial port $a$ and $b$ are
\begin{align}
P_{a}(\omega)=\frac{1}{4}\abs{\sum_{n\in\mathds{Z}}c_{n}(e^{i\pi\omega/\overline{\omega} }-1) G^{\sigma}(\omega-n\overline{\omega})}^{2},\\
P_{b}(\omega)=\frac{1}{4}\abs{\sum_{n\in\mathds{Z}}c_{n}(e^{i\pi\omega/\overline{\omega} }+1) G^{\sigma} (\omega-n\overline{\omega})}^{2}.
\end{align}
We represent in Fig.~\ref{PFBS11}(a) (b) the spectral distribution of two $\ket{+}$ states for $\sigma=0.1\overline{\omega}$ and $\sigma=0.2\overline{\omega}$ along with the output probability distribution after the Mach-Zehnder interferometer in Fig.~\ref{PFBS11}(c),(d). We observe that when $\sigma = 0.1\overline{\omega}$, it is valid to consider the zero and one logical states independently since they do not interfere due to their central frequencies being too far apart for overlap. However, the spatial separation is imperfect, the one logical state does not emerge from the correct spatial port. When $\sigma=0.2\overline{\omega}$ (see Fig.~\ref{PFBS11}), there is an interference term between the zero and one logical state because they now overlap significantly. The resulting state is outside the GKP subspace, it can be seen with the distorsion and because the probability at the center of the odd and even frequency bins is zero. The addition of a periodic frequency filter can enhance the projection into the GKP subspace and the choice of a frequency width would be crucial for rejecting the logical states emerging from the incorrect spatial port.\\
\begin{figure*}
\caption{\label{PFBS11}
\label{PFBS11}
\end{figure*}
In Appendix \ref{Twophoton}, we also investigate the separation of the odd and even components of the comb when the two-photon state is an input of the frequency qubit beam-splitter. This is relevant to the teleportation-based error correction protocol because the state to be teleported is combined with a single photon from an EPR pair during the Bell measurement. The spatial separation is, in that case, also completely effective when each Gaussian distribution approaches a Dirac distribution, and not otherwise. Explicitly, for the two photons states we obtain: $\ket{0\tilde{1}}_{aa}\rightarrow \ket{0_{G}}_{a}\ket{\tilde{1}_{G}}_{a'}+\ket{0_{E}}_{a'}\ket{\tilde{1}_{E}}_{a}$, whose the corresponding expressions in given in Appendix \ref{Twophoton}. The case to combine two different single photon states also plays a role in quantum communication scenarios, where the second qubit is from an attacker which tries to collect the information about the qubit carrying the information of interest. \\
\section{Teleportation-based error correction of time-frequency qubits states}\label{sectionteleportation}
In this section, we propose a protocol for correcting and teleporting frequency qubit states without relying on frequency entangling operations. This protocol is similar to the one used for teleporting polarization qubit states as described in \cite{lutkenhaus_bell_1999}, and is inspired from the GKP analog \cite{PhysRevA.102.062411,PhysRevResearch.3.033118}. Since the EPR state has a lower level of noise in both temporal and frequency variables, the protocol includes an additional component for correcting errors in the state being teleported.\\
The single photon state $\ket{\psi}$ to be teleported and corrected is described by the wavefunction $\ket{\psi}=N_{\alpha\beta}(\alpha \ket{0}_{a}+\beta\ket{1}_{a})$, where the two logical state are either the time-frequency cat or GKP qubits (see Eq.~(\ref{GKPinput})). The wavefunction of the entangled EPR time-frequency state in spatial port $b$ and $c$ which assists for the teleportation and the correction is:
\begin{equation}
\ket{\phi^{+}}_{bc}=N_{\text{EPR}}(\ket{\tilde{0}\tilde{1}}_{bc}+\ket{\tilde{1}\tilde{0}}_{bc})
\end{equation}
composed of logical states which are less noisy, indicated with the tilde notation, than the state to be teleported. Upon completion of the protocol, the single-photon state $\ket{\psi}$ will be localized to the spatial port $c$, and the correction is automatically performed since the EPR state is less noisy than the state of interest.\\
We now write the protocol for the error correction and teleportation of the time-frequency cat state, considering that the Bell's measurement perfectly separate the two logical states. This protocol allows correcting frequency errors affecting a frequency qubit states. The protocol is represented in Fig. \ref{teleportationscheme}, and we will now proceed to write the evolution of the wavefunction at each step of the protocol. The initial wavefunction, composed of the state to be corrected and teleported, and the EPR state, written in the Bell's basis is:
\begin{multline}
\ket{\psi}=\frac{N_{\alpha\beta}N_{\text{EPR}}}{2}(\ket{\omega_{1}\tilde{\omega}_{1}}+\ket{\omega_{2}\tilde{\omega}_{2}})(\alpha \ket{\tilde{\omega_{1}}}_{c}+\beta \ket{\tilde{\omega_{2}}}_{c})\\
+(\ket{\omega_{1}\tilde{\omega}_{1}}-\ket{\omega_{2}\tilde{\omega}_{2}})(\alpha \ket{\tilde{\omega_{1}}}_{c}-\beta \ket{\tilde{\omega_{2}}}_{c})\\
+(\ket{\omega_{1}\tilde{\omega}_{2}}+\ket{\omega_{2}\tilde{\omega}_{1}})(\beta \ket{\tilde{\omega_{1}}}_{c}+\alpha \ket{\tilde{\omega_{2}}}_{c})\\
+(-\ket{\omega_{1}\tilde{\omega}_{2}}+\ket{\omega_{2}\tilde{\omega}_{1}})(\beta \ket{\tilde{\omega_{1}}}_{c}-\alpha \ket{\tilde{\omega_{2}}}_{c}).
\end{multline}
The single photon state and one member of the EPR pair are then combined into a beam-splitter, followed by two parity-frequency beam-splitter are placed in spatial port $a'$ and $b'$. The first and second Bell's state are transformed as:
\begin{multline}
\frac{N_{\text{EPR}}}{2}[\ket{\omega_{1}\tilde{\omega}_{1}}_{a}-\ket{\omega_{1}}_{a}\ket{\tilde{\omega}_{1}}_{b}+\ket{\omega_{1}}_{b}\ket{\tilde{\omega}_{1}}_{a}-\ket{\omega_{1}\tilde{\omega}_{1}}_{b}\\
+\ket{\omega_{2}\tilde{\omega}_{2}}_{a}-\ket{\omega_{2}}_{a}\ket{\tilde{\omega}_{2}}_{b}+\ket{\omega_{2}}_{b}\ket{\tilde{\omega}_{2}}_{a}-\ket{\omega_{2}\tilde{\omega}_{2}}_{b}].
\end{multline}
The presence of a single photon in each port is a consequence of the distinguishability of the photons. While if the photons were indistinguishable, when the two logical states are orthogonal as it is the case for the polarization encoding, only bunching event will be measured. In order to suppress this coincidence events that leads to errors in the teleportation protocol, the utilisation of frequency filters with the same frequency width (envelope and peak) as the EPR state is a potential solution. Explicitly, after the filtering operation, the state becomes $\ket{\tilde{\omega}_{1}\tilde{\omega}_{1}}_{a}-\ket{\omega_{1}\tilde{\omega}_{1}}_{b}+\ket{\tilde{\omega}_{2}\tilde{\omega}_{2}}_{a}-\ket{\omega_{2}\tilde{\omega}_{2}}_{b}$, which leads only to two bunching events, that are ignored if single photon detectors are used. The same analysis can be employed for the second Bell's state.\\
For the third and fourth Bell's state, they are transformed as:
\begin{multline}
\frac{N_{\text{EPR}}}{2}[ \pm \ket{\omega_{1}\tilde{\omega}_{2}}_{aa'}\mp\ket{\omega_{1}\tilde{\omega}_{2}}_{ab'}\pm\ket{\omega_{1}\tilde{\omega}_{2}}_{ba'}\mp \ket{\omega_{1}\tilde{\omega}_{2}}_{bb'}\\
+\ket{\omega_{2}\tilde{\omega}_{1}}_{a'a}-\ket{\omega_{2}\tilde{\omega}_{1}}_{a'b}+\ket{\omega_{2}\tilde{\omega}_{1}}_{b'a}-\ket{\omega_{2}\tilde{\omega}_{1}}_{b'b}].
\end{multline}
The use of frequency filter is again imperative, since the measurement of coincidence once filtered of $a,a'$ (or $b,b'$) permits to ensure that the quantum state $N_{\alpha\beta}(\beta \ket{\tilde{\omega_{1}}}+\alpha \ket{\tilde{\omega_{2}}})$ has been teleported. In the same way, the measurement of the coincidence of $a,b'$ (or $b,a'$) allows to teleport the state $N_{\alpha\beta}(\beta \ket{\tilde{\omega_{1}}}-\alpha \ket{\tilde{\omega_{2}}})$. The receiver sends to spatial port $c$ which detectors have measured coincidences, and then a product of Pauli matrix gates must be applied to recover the initial state of interest. Pauli matrices for the time-frequency cat states are frequency and temporal shifts operations \cite{fabre_generation_2020}, which can be implemented by either a electro-optical modulator \cite{Kurzyna_2022,PhysRevLett.129.123605} and a delay line respectively. The full optical scheme of the teleportation-based error correction protocol is represented in Fig.~\ref{teleportationscheme}, along with a illustration of the effect of the error correction for qubit cat states.\\
The corresponding probability of each event is $P=\frac{1}{8(1+e^{-\Delta^{2}/2\sigma^{2}})}$, and the overall probability success of the teleportation is
\begin{equation}
P=\frac{1}{2(1+e^{-\Delta^{2}/2\sigma^{2}})},
\end{equation}
which is then lower than 50 \%, since only linear optics is used \cite{lutkenhaus_bell_1999}, and because the non-orthogonality of the encoding further decrease the probability of success. Note that the use of photon-number-resolving (PNR) detectors will able to not discard the bunching events coming from the first and second Bell's state. With the use of such a PNR detector, the overall probability of success of the teleportation protocol is:
\begin{equation}
P_{\text{PNR}}=\frac{3}{4(1+e^{-\Delta^{2}/2\sigma^{2}})}
\end{equation}
and thus allows to increase the probability of success of the protocol despite the non-orthogonality of the state. Experimentally, the choice of the PNR detector could be the one described in \cite{eaton_resolution_2023}. Reaching a $3/4$ probability of success has also been found by using non-linear process or ancilla entangled states \cite{PhysRevLett.113.140403,PhysRevA.59.116}.\\
We can formulate the previous protocol for time-frequency GKP states. If we employ a EPR state which is less noisy in the temporal and frequency domain, the teleported state is corrected in both temporal and frequency variables at once. It is in contrast with the error correction protocol based on frequency entanglement described in Sec.~\ref{errorentangle}, where we have to repeat twice the same protocol to correct both variables. In the Appendix \ref{correctionGKP}, we tackle the case of the teleportation error correction protocol when the spatial separation of the non-orthogonal qubit state is imperfect, discussing the special case of time-frequency GKP state and the Bell's measurement relying on the spatial separation of the two logical states described in Sec.~\ref{separationGKP}. The imperfect spatial separation affects both the efficiency and the fidelity of the teleportation protocol. The fidelity is always equal to one when the logical states are orthogonal, but this is not the case for non-orthogonal time-frequency qubit states. The use of frequency filters can eliminate detection events caused by imperfect spatial separation, but it comes at the cost of decreased brightness.\\
\begin{figure*}
\caption{\label{teleportationscheme}
\label{teleportationscheme}
\end{figure*}
\section{Conclusion}\label{conclusion}
In this paper, we have analyzed the teleportation-based error correction protocol for two types of frequency qubit states: time-frequency cat states and time-frequency GKP states. This optical scheme has the same goal as a quantum relay, reducing the error rate of wrong detection by decreasing the overlap of the two logical states composing the qubit. We have discussed the experimental realization of Bell's measurement for these two types of qubits. The advantage of discretizing a grid state into a qubit state by combining the even and odd peaks, rather than considering the state as a time-frequency qudit, is convenient because it simplifies the optical implementation of grid state manipulations. When the states are not infinitely frequency narrowed, the Bell's measurement leads to wrong detection, as the spatial separation of the two logical states into two spatial ports is imperfect. This can be corrected by using frequency filters, at the cost of losing single photon detection events. To tackle this issue, the use of frequency resolved detection and the fault-tolerance threshold defined in \cite{fukui_high-threshold_2018} could be valuable for avoiding the use of frequency filters. We have illustrate that our protocol can correct the errors of the qubit composed of two colors, but it could be also done for correcting the temporal error coming from broadening and dispersion, of a qubit composed of two Gaussian centered at two temporal bins (see Eq.~(\ref{timebinqubit})). In this context, it is important to study and evaluate the overall effectiveness of the teleportation protocol, as well as the final quality of the state being corrected, given the level of accuracy in separating the two logical states.
\section*{ACKNOWLEDGMENT}
N. Fabre acknowledges useful discussions with Filip Rozp\k{e}dek, Arne Keller and Pérola Milman for the completion of this manuscript.
\appendix
\section{Spatial separation of a two-photon state}\label{Twophoton}
We show in this section that a two-photon state as input can also be separated into the even and the odd components in two distinct spatial ports. We consider an initial separable two photon idea time-frequency GKP state as:
\begin{equation}
\ket{0\tilde{0}}_{aa}=\sum_{n,m\in\mathds{Z}^{2}} c_{2n}\tilde{c}_{2m} \hat{a}^{\dagger}(2n\overline{\omega})\hat{a}^{\dagger}(2m\overline{\omega})\ket{\Omega}.
\end{equation}
The tilde notation is here to indicate that the two states are not identical, one of them can be more noisy compared to the other. After the first beam-splitter and the time-displacement operator, the wave function of the two-photon state is:
\begin{multline}
\frac{1}{2}\sum_{n,m\in\mathds{Z}^{2}} c_{2n}\tilde{c}_{2m}(\hat{a}^{\dagger}_{\tau}(2n\overline{\omega})\hat{a}^{\dagger}_{\tau}(2m\overline{\omega})+\hat{b}^{\dagger}(2n\overline{\omega})\hat{a}^{\dagger}_{\tau}(2m\overline{\omega})\\+\hat{a}^{\dagger}_{\tau}(2n\overline{\omega})\hat{b}^{\dagger}(2m\overline{\omega})+\hat{b}^{\dagger}(2m\overline{\omega})\hat{b}^{\dagger}(2n\overline{\omega}))\ket{\Omega}.
\end{multline}
where $\hat{a}^{\dagger}_{\tau}(2n\overline{\omega})=e^{i2n\overline{\omega}\tau} \hat{a}^{\dagger}(2n\overline{\omega})$. The output wave function after the second beam-splitter is:
\begin{multline}
\frac{1}{4}\sum_{n,m\in\mathds{Z}^{2}} c_{2n}\tilde{c}_{2m} (\hat{a}^{\dagger}_{\tau}(2n\overline{\omega})+\hat{b}^{\dagger}_{\tau}(2n\overline{\omega}))(\hat{a}^{\dagger}_{\tau}(2m\overline{\omega})+\hat{b}^{\dagger}_{\tau}(2m\overline{\omega}))\\
+(\hat{a}^{\dagger}_{\tau}(2n\overline{\omega})-\hat{b}^{\dagger}_{\tau}(2n\overline{\omega}))(\hat{a}^{\dagger}_{\tau}(2m\overline{\omega})+\hat{b}^{\dagger}_{\tau}(2m\overline{\omega}))\\
+(\hat{a}^{\dagger}_{\tau}(2n\overline{\omega})+\hat{b}^{\dagger}_{\tau}(2n\overline{\omega}))(\hat{a}^{\dagger}(2m\overline{\omega})-\hat{b}^{\dagger}(2m\overline{\omega}))\\
+ (\hat{a}^{\dagger}(2n\overline{\omega})-\hat{b}^{\dagger}(2n\overline{\omega}))(\hat{a}^{\dagger}(2m\overline{\omega})-\hat{b}^{\dagger}(2m\overline{\omega}))\ket{\Omega}.
\end{multline}
We rearrange and post-select only the coincidence terms:
\begin{multline}
\frac{1}{4}\sum_{n,m\in\mathds{Z}^{2}} c_{2n}\tilde{c}_{2m}(\hat{a}^{\dagger}_{\tau}(2n\overline{\omega})+\hat{a}^{\dagger}(2n\overline{\omega}))\hat{b}_{\tau}^{\dagger}(2m\overline{\omega})\\
- (\hat{a}^{\dagger}_{\tau}(2n\overline{\omega})+\hat{a}^{\dagger}(2n\overline{\omega}))\hat{b}^{\dagger}(2m\overline{\omega})\\
+(\hat{a}^{\dagger}_{\tau}(2m\overline{\omega})+\hat{a}^{\dagger}(2m\overline{\omega}))\hat{b}_{\tau}^{\dagger}(2n\overline{\omega})\\
- (\hat{a}^{\dagger}_{\tau}(2m\overline{\omega})+\hat{a}^{\dagger}(2m\overline{\omega}))\hat{b}^{\dagger}(2n\overline{\omega})\ket{\Omega}.
\end{multline}
Let us first consider the ideal case, where the spectral distribution is a Dirac one $G_{2n}(\omega)=\delta(\omega-2n\overline{\omega})$. We point out that $(\hat{a}^{\dagger}_{\tau}(2n\overline{\omega})+\hat{a}^{\dagger}(2n\overline{\omega}))\ket{0}=(e^{2in\overline{\omega}\tau}+1)\ket{2n\overline{\omega}}$, with $\tau=\pi/\overline{\omega}$, we have $(\hat{a}^{\dagger}_{\tau}(2n\overline{\omega})+\hat{a}^{\dagger}(2n\overline{\omega}))\ket{0}=2\ket{2n\overline{\omega}}$.
We have also $(\hat{b}_{\tau}^{\dagger}(2m\overline{\omega})-\hat{b}^{\dagger}(2m\overline{\omega})\ket{2n\overline{\omega}}\ket{0}=0$. We can verify that the others terms are zero. It means that there is no coincidence event which is the desired outcome. \\
We now rearrange and post-select only the bunching terms:
\begin{multline}
\frac{1}{4}\sum_{n,m\in\mathds{Z}^{2}} c_{2n}\tilde{c}_{2m}(\hat{a}^{\dagger}_{\tau}(2n\overline{\omega})\hat{a}^{\dagger}_{\tau}(2m\overline{\omega})+\hat{a}^{\dagger}(2n\overline{\omega})\hat{a}^{\dagger}_{\tau}(2m\overline{\omega})\\
+\hat{a}^{\dagger}_{\tau}(2n\overline{\omega})\hat{a}^{\dagger}(2m\overline{\omega})+\hat{a}^{\dagger}(2n\overline{\omega})\hat{a}^{\dagger}(2m\overline{\omega})\\
+ \hat{b}^{\dagger}_{\tau}(2n\overline{\omega})\hat{b}^{\dagger}_{\tau}(2m\overline{\omega})-\hat{b}^{\dagger}(2n\overline{\omega})\hat{b}^{\dagger}_{\tau}(2m\overline{\omega})\\
+\hat{b}^{\dagger}_{\tau}(2n\overline{\omega})\hat{b}^{\dagger}(2m\overline{\omega})-\hat{b}^{\dagger}(2n\overline{\omega})\hat{b}^{\dagger}(2m\overline{\omega}))\ket{\Omega}.
\end{multline}
In the ideal case, the terms in the spatial port $a$ remains, and the ones in the spatial port $b$ interfere destructively.\\
For a time-frequency GKP state with a finite bandwidth, there is no longer a perfect destructive (or constructive) interference effect that separates the even and odd components of the comb perfectly. We analyze what happens to the state $\ket{0\tilde{1}}_{aa}$. Post-selecting on the coincidence we have:
\begin{equation}\label{attack}
\ket{0\tilde{1}}_{aa}\rightarrow \ket{0_{G}}_{a}\ket{\tilde{1}_{G}}_{a'}+\ket{0_{E}}_{a'}\ket{\tilde{1}_{E}}_{a}
\end{equation}
where we have defined:
\begin{align}
\ket{0_{G}}_{a}=\frac{N_{e}}{2}\sum_{n\in\mathds{Z}}c_{2n}\int d\omega (e^{i\omega\tau}+1)G_{2n}^{\sigma}(\omega) \ket{\omega}_{a}\\
\ket{\tilde{1}_{G}}_{a'}=\frac{N_{o}}{2}\sum_{n\in\mathds{Z}}c_{2n+1} \int d\omega (e^{i\omega\tau}-1)G_{2m+1}^{\tilde{\sigma}}(\omega) \ket{\omega}_{a'}\\
\ket{\tilde{1}_{E}}_{a}=\frac{N_{o}}{2}\sum_{n\in\mathds{Z}}c_{2n+1} \int d\omega (e^{i\omega\tau}+1)G_{2m+1}^{\tilde{\sigma}}(\omega) \ket{\omega}_{a}\\
\ket{0_{E}}_{a'}=\frac{N_{e}}{2}\sum_{n\in\mathds{Z}}c_{2n}\int d\omega (e^{i\omega\tau}-1)G_{2n}^{\sigma}(\omega) \ket{\omega}_{a'}.
\end{align}
The post-selection on coincidence results in only those events where both the even and odd components are in the correct (designated by $G$) and incorrect (designated by $E$) spatial ports. It should be noted that the output state is no longer in the GKP subspace and results in detection errors. These errors can be corrected through the use of frequency filters or by using frequency-resolved detection and setting a frequency threshold width to accept only certain events \cite{fukui_high-threshold_2018}. Additionally, the imperfect spatial separation described in Eq.~(\ref{attack}) can also be interpreted as an attack to extract some information of the quantum state of interest in a quantum communication protocol.
\section{Teleportation-based error correction protocol with physical time-frequency GKP state}\label{correctionGKP}
In the following, we will employ the GKP coherent picture \cite{seshadreesan_coherent_2021}. The GKP qubit defined by Eq.~(\ref{GKPinput}) is composed of an envelope of width $\kappa$ and each peak has a width of $\sigma$ (resp. $\kappa_{1},\sigma_{1}$), while the frequency widths of the EPR state will be noted $\tilde{\kappa}$ and $\tilde{\sigma}$.\\
As has been noted, when the time-frequency GKP state has a limited bandwidth, the frequency qubit beam-splitter is not able to perfectly separate the odd and even components of the comb. This results in some of the state leaking out of each spatial port, and not conforming to a GKP state. Furthermore, the state in the right port is also distorted. To address these two challenges, it is necessary to employ frequency filters that enable projection back into the GKP subspace and eliminate the undesirable components. Given that the state being teleported and the EPR state have different frequency widths, a frequency filter is placed prior to detection, which establishes the frequency bin and aligns with the reference EPR state. The projector modeling for the frequency filtering process has the following form:
\begin{align}
\hat{\Pi}_{a}=N^{2}_{e}(\sigma,\kappa)\sum_{n\in\mathds{Z}}\tilde{c}_{2n}\int G_{2n}^{\tilde{\sigma}}(\omega)\ket{\omega}\bra{\omega}\\
\hat{\Pi}_{a'}=N^{2}_{o}(\sigma,\kappa)\sum_{n\in\mathds{Z}}\tilde{c}_{2n+1}\int G_{2n+1}^{\tilde{\sigma}}(\omega)\ket{\omega}\bra{\omega}
\end{align}
The normalization constant is found by using $\hat{\Pi}^{2}=\mathds{I}$ and $\text{Tr}(\hat{\Pi}^{2})=1$, $N^{2}_{e}(\tilde{\sigma},\tilde{\kappa})=\sqrt{2\pi}/\tilde{\sigma} \sum_{n}\abs{\tilde{c}_{2n}}^{2}$. In the large comb approximation, we have $N_{e}(\tilde{\sigma},\tilde{\kappa})=N_{o}(\tilde{\sigma},\tilde{\kappa})$. \\
The coincidence destructive measurement is described by the positive operator value measurement:
\begin{equation}
\hat{\Pi}_{a,a'}=\hat{\Pi}_{a}\hat{\Pi}_{a'} \otimes \ket{1}\bra{1}.
\end{equation}
By assuming that the state is pure, the probability of coincidence in the spatial port $a,a'$, $P_{aa'}(0,1)=\text{Tr}(\hat{\Pi}_{a,a'}\ket{\psi}\bra{\psi}\hat{\Pi}_{a,a'}^{\dagger})$ is:
\begin{multline}
P_{aa'}(0,1)=\abs{\frac{1}{2\sqrt{2}}}^{2}\abs{a^{01}_{\sigma\tilde{\sigma}}+b^{01}_{\sigma\tilde{\sigma}}}^{2}\times \\
(\int d\omega (\abs{\alpha}^{2} \abs{\bra{\omega}\ket{\tilde{0}}}^{2}+\abs{\beta}^{2} \abs{\bra{\omega}\ket{\tilde{1}}}^{2}+2\text{Re}(\alpha^{*}\beta \bra{\omega}\ket{\tilde{0}}\bra{\omega}\ket{\tilde{1}}))
\end{multline}
where we have used $\abs{a^{01}_{\sigma\tilde{\sigma}}+b^{01}_{\sigma\tilde{\sigma}}}=\abs{a^{10}_{\sigma\tilde{\sigma}}+b^{10}_{\sigma\tilde{\sigma}}}$ which is shown afterward. For an EPR with a sufficiently narrow distribution, we assume that $\bra{\tilde{0}}\ket{\tilde{1}} =0$, $\int d\omega \abs{\bra{\omega}\ket{\tilde{0}}}^{2}=\int d\omega \abs{\bra{\omega}\ket{\tilde{1}}}^{2}=1$. These two conditions are important, since the probability then does not depend on $\alpha$ and $\beta$,
\begin{equation}
P_{aa'}(0,1)=\abs{\frac{1}{2\sqrt{2}}}^{2}\abs{a^{01}_{\sigma\tilde{\sigma}}+b^{01}_{\sigma\tilde{\sigma}}}^{2}
\end{equation}
since otherwise, information about the quantum state could be extracted during the measurement.\\
The wavefunction of the state after the detection is: $\ket{\psi}_{c}=\hat{\Pi}_{i,j}\ket{\psi}/\text{Tr}(\hat{\Pi}_{i,j}\ket{\psi}\bra{\psi}\hat{\Pi}_{i,j}^{\dagger})$, where $i,j=a,b; a',b'$. When coincidence is detected at the spatial port $a$ and $a'$, the post-selected state is:
\begin{multline}\label{postselectedstatewithout}
\ket{\psi}_{c}=\frac{1}{\abs{a^{01}_{\sigma\tilde{\sigma}}+b^{01}_{\sigma\tilde{\sigma}}}} (\alpha(a^{1,\tilde{\sigma}}_{0, \sigma}+b^{1,\sigma}_{0,\tilde{\sigma}}) \ket{\tilde{0}}_{c}\\+\beta(a^{1,\sigma}_{0, \tilde{\sigma}}
+b^{1,\tilde{\sigma}}_{0,\sigma}) \ket{\tilde{1}}_{c}).
\end{multline}
The expression is similar for the other coincidences events in the other spatial ports, and extra Pauli operations have to be performed. The coefficients $a^{10}_{\tilde{\sigma}\sigma}$ and $b^{10}_{\sigma \tilde{\sigma}}$ have the expression:
\begin{multline}
a^{10}_{\tilde{\sigma}\sigma}=\frac{1}{4} N_{e}(\sigma,\kappa)N_{o}(\tilde{\sigma},\tilde{\kappa})N_{e}(\tilde{\sigma},\tilde{\kappa})N_{o}(\tilde{\sigma},\tilde{\kappa})\\
\times \sum_{n,m,k,k'} c_{2n}\tilde{c}_{2m+1}\tilde{c}_{2k}\tilde{c}_{2k'+1}\int d\omega (e^{i\omega\tau}+1)G_{2n}^{\sigma}(\omega)G_{2k}^{\tilde{\sigma}}(\omega) \\
\times \int d\omega (e^{i\omega\tau}-1)G_{2m+1}^{\tilde{\sigma}}(\omega) )G_{2k'+1}^{\tilde{\sigma}}(\omega)
\end{multline}
\begin{multline}
b^{10}_{\sigma \tilde{\sigma}}=\frac{1}{4} N_{e}(\tilde{\sigma},\tilde{\kappa})N_{o}(\sigma,\kappa)N_{e}(\tilde{\sigma},\tilde{\kappa})N_{o}(\tilde{\sigma},\tilde{\kappa})\\
\times \sum_{n,m,k,k'} \tilde{c}_{2n}c_{2m+1}\tilde{c}_{2k}\tilde{c}_{2k'+1}\int d\omega (e^{i\omega\tau}+1)G_{2m+1}^{\sigma}(\omega)G_{2k}^{\tilde{\sigma}}(\omega)\\
\times \int d\omega (e^{i\omega\tau}-1)G_{2n}^{\tilde{\sigma}}(\omega) )G_{2k'+1}^{\tilde{\sigma}}(\omega).
\end{multline}
The $b$ coefficient contains odd (resp. even) terms in a spatial port where the frequency filters is centered at even (resp. odd) frequencies. In the ideal case, namely if both the EPR and the teleported state are ideal time-frequency GKP state we remind that $a^{01}_{\sigma\tilde{\sigma}}=2$ and $b^{01}_{\sigma\tilde{\sigma}}=0$. After evaluation of the integrals, and assuming that $2k=2n$, meaning that the temporal spreading only reach the next bin, we find that:
\begin{multline}\label{definitionaandb}
a^{10}_{\tilde{\sigma}\sigma}=\frac{\sqrt{\tilde{\sigma}\tilde{\sigma}}\sqrt{\sigma\tilde{\sigma}}}{4\sqrt{\tilde{\sigma}^{2}+\sigma^{2}}\sqrt{\tilde{\sigma}^{2}+\tilde{\sigma}^{2}}}(e^{-\frac{\pi^{2}\alpha^{2}}{2\overline{\omega}^{2}}}+1)(-e^{-\frac{\pi^{2}\tilde{\alpha}^{2}}{2\overline{\omega}^{2}}}-1)
\end{multline}
\begin{multline}
b^{10}_{\sigma \tilde{\sigma}}=\frac{\sqrt{\tilde{\sigma}\tilde{\sigma}}\sqrt{\sigma\tilde{\sigma}}}{4\sqrt{\tilde{\sigma}^{2}+\sigma^{2}}\sqrt{\tilde{\sigma}^{2}+\tilde{\sigma}^{2}}}\frac{\sum_{n,m} c_{2m+1}c_{2m}c_{2n}c_{2n+1}}{\sum_{n,m} \abs{c_{2m+1}}^{2}\abs{c_{2n}}^{2}}\\
\times (e^{-\frac{\pi^{2}\alpha^{2}}{2\overline{\omega}^{2}}} e^{-\frac{\overline{\omega}^{2}}{2(\tilde{\sigma}^{2}+\tilde{\sigma}^{2})}} e^{i\frac{\pi \tilde{\sigma}^{2}}{(\sigma^{2}+\tilde{\sigma}^{2})}}+1)\\
\times (e^{-\frac{\pi^{2}\alpha^{2}}{2\overline{\omega}^{2}}}e^{-\frac{\overline{\omega}^{2}}{2(\tilde{\sigma}^{2}+\tilde{\sigma}^{2})}} e^{i\frac{\pi \tilde{\sigma}^{2}}{(\sigma^{2}+\tilde{\sigma}^{2})}}-1)
\end{multline}
where we have defined that $\alpha^{2}=\sigma^{2}\tilde{\sigma}^{2}/(\sigma^{2}+\tilde{\sigma}^{2})$. While $a^{10}_{\tilde{\sigma}\sigma}$ is real, $b^{10}_{\sigma \tilde{\sigma}}$ is a complex quantity. From these expressions, we point out that $a^{1,\tilde{\sigma}}_{0, \sigma}=a^{0,\tilde{\sigma}}_{1, \sigma}$ and $b^{1,\sigma}_{0,\tilde{\sigma}}=b^{0,\sigma}_{1,\tilde{\sigma}}$. When the width of the frequency filter is $\tilde{\sigma}\rightarrow 0$, we have $a^{01}_{\sigma\tilde{\sigma}}=2$ and $b^{01}_{\sigma\tilde{\sigma}}=0$ which is as the ideal case and thus lead to a high fidelity of the state, but at the cost of losing many photons. As the imperfect spatial separation of the two logical states leads to a decreasing of the fidelity, it is a reminiscent fact coming that the state possesses continuous variables.\\
\end{document}
|
\begin{document}
\begin{abstract}
We show that
$$\sum_{\substack{p_n \in [x, 2x] \\ p_{n+1} - p_n \ge x^{1/2}}} (p_{n+1} - p_n) \ll x^{0.57+\epsilon}$$
and
$$\sum_{\substack{p_n \in [x, 2x] \\ p_{n+1} - p_n \ge x^{0.45}}} (p_{n+1} - p_n) \ll x^{0.63+\epsilon},$$
where $p_n$ is the $n$th prime number. The proof combines Heath-Brown's recent work with Harman's sieve, improving and extending his results. We give applications of the results to prime-representing functions, binary digits of primes and approximation of reals by multiplicative functions.
\end{abstract}
\title{On large differences between consecutive primes}
\section{Introduction}
\label{sec:intro}
A central problem in number theory is understanding the distribution of prime numbers. Notable work in this area include Baker-Harman-Pintz's~\cite{BHP} result on intervals of length $x^{0.525}$ containing prime numbers and Jia's work~\cite{jia} showing that almost all intervals of length $x^{1/20}$ contain primes, both being preceded by numerous weaker results on the problems.
Both of these results may be viewed as instances of the problem of bounding the number of intervals of length $x^c$ without primes. The case $c = 1/2$ is of special interest, as even under the Riemann hypothesis it is not known that intervals of length $\sqrt{x}$ necessarily contain primes.
The current best result for $c = 1/2$ is given in the recent work of Heath-Brown~\cite{HB-V}, where he shows that there are at most $X^{3/5 + \epsilon}$ intervals $[x, x + x^{1/2}]$ with $x \in \mathbb{Z} \cap [X, 2X]$ that do not contain primes. Heath-Brown's result relies on his mean square estimate (see~\cite[Proposition 1]{HB-V} or Proposition \ref{prop:HB_MVT} below) for the product of two Dirichlet polynomials, one of which is sparse. Heath-Brown's work improves the previous result of Matomäki~\cite{matomaki}, who obtained an exceptional set of size $X^{2/3}$. See \cite{wolke2}, \cite{HB1978}, \cite{HB1979} and \cite{peck2} for earlier results.
Heath-Brown's argument does not utilize Harman's sieve, in contrast to Matomäki's proof. One may strengthen the result by combining Heath-Brown's methods with Harman's sieve. We show the following.
\begin{theorem}
\label{thm:0.5}
Let $p_n$ denote the $n$th prime. We have
$$\sum_{\substack{p_n \in [x, 2x] \\ p_{n+1} - p_n \ge x^{1/2}}} (p_{n+1} - p_n) \ll x^{0.57+\epsilon}$$
for any fixed $\epsilon > 0$.
\end{theorem}
It is the best to view the bound as $x^{1/2 + 0.07 + \epsilon}$, so that the ``excess'' is $30\%$ smaller than in Heath-Brown's result $x^{1/2 + 0.1 + \epsilon}$.
We further demonstrate that Heath-Brown's method adapts to intervals shorter than $\sqrt{x}$.
\begin{theorem}
\label{thm:0.45}
Let $p_n$ denote the $n$th prime. We have
$$\sum_{\substack{p_n \in [x, 2x] \\ p_{n+1} - p_n \ge x^{0.45}}} (p_{n+1} - p_n) \ll x^{0.63+\epsilon}$$
for any fixed $\epsilon > 0$.
\end{theorem}
The Lindelöf hypothesis would imply a bound of $x^{1 - c + \epsilon}$ for intervals of length $x^c$~\cite{yu}. Hence the bound in Theorem~\ref{thm:0.45} should be viewed as $x^{(1 - 0.45) + 0.08 + \epsilon}$, the excess being of similar size as in Theorem~\ref{thm:0.5}.
For intervals shorter than $\sqrt{x}$, previously Peck~\cite{peck} has given a bound of $x^{1.25 - c + \epsilon}$ for intervals of length $x^c$ for any $1/4 < c \le 1/2$. Islam~\cite{islam} gives the bound $x^{2/3 + 5(1/2 - c)}$ for $c < 1/2$, improving on Peck's result for $c > 1/2 - 1/48$. Simultaneously to our work Stadlmann \cite{stadlmann} has given the bound $x^{1.23 - c + \epsilon}$ for $c > 0.23$ (also by using Heath-Brown's mean value theorem from \cite{HB-V}).
Theorem~\ref{thm:0.45} gives a substantial improvement on previous results, demonstrating the incredible strength of Heath-Brown's new methods: the excess in Theorem \ref{thm:0.45} is less than a third of the excess in Peck's result, and, with the exception of Heath-Brown's result~\cite{HB-V}, the bound in Theorem \ref{thm:0.45} for intervals of length $x^{0.45}$ is stronger than any previous bound for intervals of length $\sqrt{x}$.
Certainly, with more work one could obtain bounds for intervals of length $x^c$ with any $0.45 < c < 0.5$ (beating the bound $x^{0.63}$ one gets from Theorem \ref{thm:0.45}). One can also extend the results for even shorter intervals (see also \cite{stadlmann}). The $\epsilon$ term in the exponents could be dropped with a bit more work, and by more effort one could improve the exponents slightly.
The proofs of Theorems~\ref{thm:0.5} and~\ref{thm:0.45} actually show that there are few intervals of length $x^c$ which contain $o(x^c/\log x)$ primes. See Theorem~\ref{thm:many} for a precise formulation.
We present a couple of applications of the results (see Section \ref{sec:applications} for more detailed discussion). First, we show that there are prime-representing functions of the form $\lfloor A^{\alpha^n} \rfloor$ for any $\alpha \ge 20/11$.
\begin{theorem}
\label{thm:PRF}
Let $\alpha \ge 20/11 = 1.818\ldots$ be fixed. There exists $A > 1$ such that $\lfloor A^{\alpha^n} \rfloor$ is a prime for all $n \in \mathbb{Z}_+$.
\end{theorem}
Previous results on the problem include those of Mills~\cite{mills} (the first such result, giving $\alpha = 3$), Matomäki~\cite{matomaki-PRF} (allowing $\alpha \ge 2$) and Islam~\cite{islam} (with $\alpha \ge 1.946\ldots$).
Second, we show that there are infinitely many primes with very many (or very few) ones in their binary representation.
\begin{theorem}
\label{thm:bin}
Let $d \in \{0, 1\}$. There are infinitely many primes $p$ such that at least $74.2\%$ of the digits of the binary representation of $p$ are equal to $d$.
\end{theorem}
In \cite{naslund} it is noted that the bound $3/4 - 1/80 - \epsilon = 73.75\% - \epsilon$ for Theorem \ref{thm:bin} follows from the result of Baker, Harman and Pintz \cite{BHP} on primes in intervals of length $x^{1/2 + 1/40}$. To our knowledge this was the best previous bound on the problem. If there were primes in intervals of length $\sqrt{x}$, the same method would give the bound $75\% - \epsilon$.
Finally, we note an improvement on approximation of real numbers by multiplicative functions.
\begin{theorem}
\label{thm:approx}
Let $\epsilon > 0$ and $\alpha > 1$ be given. There are infinitely many integers $n$ such that
$$\left|\frac{\sigma(n)}{n} - \alpha\right| < n^{-0.55 + \epsilon},$$
where $\sigma(n)$ is the sum of divisors of $n$.
\end{theorem}
The previous best result is due to Harman \cite{harman-approx} with the bound $n^{-0.52}$.
\subsection{Overview of the method}
\label{sec:overview}
Our proof largely follows the one given by Heath-Brown in~\cite{HB-V}, with the modification that we in addition utilize Harman's sieve. We give an overview of the proof below. For convenience we mostly consider the case of intervals of length $x^{0.5}$.
First, we perform elementary manipulations, reducing to showing that for all but roughly $O(x^{0.07})$ integers $m \approx \sqrt{x}$ we have
\begin{align}
\label{eq:overview_1}
S(\mathcal{A}(m), 2\sqrt{x}) \ge \epsilon \frac{|\mathcal{A}(m)|}{|\mathcal{B}(m)|} S(\mathcal{B}(m), 2\sqrt{x}),
\end{align}
where $\mathcal{A}(m)$ is a certain interval roughly of length $\sqrt{X}$ associated to $m \in \mathbb{Z}$, $\mathcal{B}(m)$ is an interval of length $x^{1 - o(1)}$, and $S(\mathcal{C}, z)$ counts the number integers in $\mathcal{C}$ which have no prime factors smaller than $z$. This follows the usual approach to applying Harman's sieve, where the set of interest is compared to a larger set.
We then decompose the terms $S(\mathcal{A}(m), 2\sqrt{x})$ and $S(\mathcal{B}(m), 2\sqrt{x})$ in~\eqref{eq:overview_1} by the Buchstab identity, which states that
\begin{align*}
S(\mathcal{C}, z) = S(\mathcal{C}, z') - \sum_{z' \le p < z} S(\mathcal{C}_p, p),
\end{align*}
where $\mathcal{C}_n = \{k \in \mathbb{N} : kn \in \mathcal{C}\}$. This reduces the problem to obtaining asymptotics of the form
\begin{align}
\label{eq:overview_2}
\sum_{\substack{p_1, \ldots , p_n \\ p_i \in [x^{\alpha_i}, x^{\beta_i}] \\ p_n < \ldots < p_1}} S(\mathcal{A}_{p_1 \cdots p_n}(m), z) \approx \frac{|\mathcal{A}|}{|\mathcal{B}|}\sum_{\substack{p_1, \ldots , p_n \\ p_i \in [x^{\alpha_i}, x^{\beta_i}] \\ p_n < \ldots < p_1}} S(\mathcal{B}_{p_1 \cdots p_n}(m), z)
\end{align}
for all but $O(x^{0.07})$ exceptional values of $m$, where $z$ is either a function of $x$ or $z = p_n$.
We use a method of Heath-Brown~\cite[Proposition 2]{HB-V} to link asymptotics of the form~\eqref{eq:overview_2} with the problem of bounding the mean value of certain type of Dirichlet polynomials. More specifically, in order to show that~\eqref{eq:overview_2} holds for all but $O(x^{0.07})$ exceptional $m$, it suffices to show (roughly)
\begin{align}
\label{eq:overview_3}
\int_{|t| \in [T_0, T]} |F(it)M(it)| \d t = o\left(\frac{Rx}{\log x}\right),
\end{align}
where $R = x^{0.07}$, $T = \sqrt{x}$, $T_0 = (\log x)^A$, $F(s)$ is the Dirichlet polynomial of length $x$ whose coefficients correspond to the summands in~\eqref{eq:overview_2} and $M(s)$ is an arbitrary polynomial of the form
$$M(s) = \sum_{i = 1}^R \zeta_im_i^{-s}, \qquad |\zeta_i| = 1, m_i \approx \sqrt{x}.$$
Note that $F(s)$ factorizes as a product of at least $n$ polynomials corresponding to the sums over $p_i$ in~\eqref{eq:overview_2}.
To prove~\eqref{eq:overview_3}, we use a mean value theorem due to Heath-Brown~\cite[Proposition 1]{HB-V}. The mean value theorem applies to mean squares of the form
\begin{align}
\label{eq:overview_4}
\int_{-T}^{T} |Q(it)M(it)|^2 \d t.
\end{align}
Our strategy for showing~\eqref{eq:overview_3} is thus factorizing $F(s)$ as $F(s) = P(s)Q(s)$, applying the Cauchy-Schwarz inequality to obtain
\begin{align}
\label{eq:overview_5}
\int_{-T}^{T} |F(it)M(it)| \d t \le \sqrt{\int_{-T}^{T} |P(it)|^2 \d t}\sqrt{\int_{-T}^{T} |Q(it)M(it)|^2 \d t}
\end{align}
and bounding the latter integral by Heath-Brown's mean value theorem. The former mean square in~\eqref{eq:overview_5} is bounded by further factorizing $P(s)$ and using various pointwise bounds and large value theorems to the factors.
We note that this is a simplification of the actual proof. In practice we start by assuming $F$ factorizes as $F(s) = A(s)B(s)C(s)$ and decompose the integral over $t$ according to the sizes of $|A(it)|, |B(it)|, |C(it)|$. In each of the resulting cases $t \in \mathcal{T}$ we may, in addition to applying the Cauchy-Schwarz argument as above, simply bound
\begin{align}
\label{eq:overview_6}
\int_{\mathcal{T}} |F(it)M(it)| \d t \le R|\mathcal{T}| \max_{t \in \mathcal{T}} |A(it)B(it)C(it)|.
\end{align}
It suffices that at least one of these strategies yields a bound small enough to imply~\eqref{eq:overview_3}.
With these strategies, we are able to find a set such that if $F(s)$ factorizes as $F(s) = A(s)B(s)C(s)$ with the triplet $(A, B, C)$ of lengths lying in this set, then~\eqref{eq:overview_3} holds. In the case of intervals of length $\sqrt{x}$ the set of admissible $(A, B, C)$ is relatively simple, corresponding to a hexagon in a $(\log_x(A), \log_x(B))$ coordinate system. (Note that the length of $C(s)$ is essentially determined by the lengths of $A(s)$ and $B(s)$, as $F \approx x$.) For intervals of length $x^{0.45}$ the set is much more complicated and best described as a union of intersections of half-planes in the above coordinate system.
Recalling that $F(s)$ has at least $n$ factors corresponding to each of $p_i$ in~\eqref{eq:overview_2} and that there are possibly many ways of grouping the factors of $F(s)$ into three polynomials $A(s), B(s), C(s)$, we obtain numerous ranges of $n, \alpha_i, \beta_i$ for which the asymptotic~\eqref{eq:overview_2} holds.
However, we cannot establish an asymptotic of type~\eqref{eq:overview_2} for all $n, \alpha_i$ and $\beta_i$. Hence we apply Harman's sieve, discarding certain sums arising from the applications of the Buchstab identity, making sure that the resulting ``loss'' is less than $1 - \epsilon$. It would not be feasible to do this by hand and hence we perform this step with a computer calculation.
Of course, we cannot perform a check over all possible lengths of the factors of $F(s)$ (of which there are unboundedly many), and hence we have to manage with merely upper and lower bounds of the form $x^{\alpha} \le P \le x^{\beta}$ for the relevant polynomials $P(s)$. To overcome this issue, we perform an extensive casework, allowing us to reduce to cases where the differences $\beta - \alpha$ are small. In each case, we consider different ways of combining the factors of $F(s)$ to write $F(s) = A(s)B(s)C(s)$ and check whether our bounds on the lengths of polynomials are strong enough to imply that the resulting triplet $(A, B, C)$ necessarily lies in the set obtained before. We then sum the loss over those cases where we cannot find such a factorization $F(s) = A(s)B(s)C(s)$ and check that it indeed is less than one.
In the case of Theorem~\ref{thm:0.5} we obtain the result without using Heath-Brown's identity (except when establishing certain theoretical results). In contrast, for intervals of length $x^{0.45}$ we incorporate the Heath-Brown decomposition into our computer calculation. This is done simply by performing a casework on the lengths of the resulting polynomials.
The running times of the computations being roughly 15 minutes and 30 hours (for Theorems~\ref{thm:0.5} and~\ref{thm:0.45}, respectively) on a usual consumer laptop. Implementations in C++ are available with the arXiv version of the paper.
The organization of the paper is as follows.
We present notation and our choice of parameters in Section~\ref{sec:parameters}. An exposition of the key tools is given in Section~\ref{sec:heath-brown}.
We perform a reduction to~\eqref{eq:overview_1} in Section~\ref{sec:to_buchstab}.
We then link asymptotic formulas of the form~\eqref{eq:overview_2} to mean value bounds as in~\eqref{eq:overview_3} in Section~\ref{sec:dir}. This is a somewhat standard procedure based on tools such as dyadic decomposition, Perron's formula, the Heath-Brown decomposition and so on, though the implementation is technical. We use, in particular, Shiu's bound on the moments of the divisor function in short intervals to bound various error terms.
Having reduced the problem to Dirichlet polynomials, we lay out various tools (such as coefficient bounds, pointwise bounds and bounds for moments of zeta sums) in Section~\ref{sec:tools}. Using these and Huxley's large value theorem, admissible ranges of $(A, B, C)$ are obtained in Section~\ref{sec:ranges}.
We then discuss the application of Harman's sieve, starting with the case $c = 0.5$ in Section~\ref{sec:0.5}. We start with theoretical results and then present the computational procedure and its results. The procedure is adapted to the case $c = 0.45$ in Section~\ref{sec:0.45}.
Discussion and proofs of the applications (Theorems~\ref{thm:PRF},~\ref{thm:bin} and~\ref{thm:approx}) are given in Section~\ref{sec:applications}.
We remark that our research procedure relied heavily on numerical computations. There is, a priori, numerous ways one may bound the integral $\int |F(it)M(it)| \d t$ (such as via Cauchy-Schwarz's inequality as in~\eqref{eq:overview_5}, the $L^1$-type estimate~\eqref{eq:overview_6}, Hölder's inequality as in Lemma~\ref{lem:two_zetas}), numerous ways to bound $\mathcal{T}$ in~\eqref{eq:overview_6} via large value theorems (there are multiple polynomials one may apply the bound to, one may raise those polynomials to some power, one may apply large value theorems of Hal\'asz-Montgomery, Huxley or Jutila), numerous ways one may combine the factors of $F(s)$ to obtain a product $A(s)B(s)C(s)$ and so on (not to mention that initially we, of course, did not know how strong of a result one can prove in Theorems~\ref{thm:0.5} and~\ref{thm:0.45}). We wrote several programs to guide our intuitions and search through the vast search space, and many of the key results and their proofs (such as Proposition~\ref{prop:range_0.5} and~\ref{prop:range_0.45}) were found with the help of such computations. And while in many cases the final argument is, once identified, relatively simple, due to its sheer complexity the application of Harman's sieve relies on a computer calculation.
\subsection{Choice of parameters and notation}
\label{sec:parameters}
Throughout the paper the length of a Dirichlet polynomial $P(s)$ is denoted by the same letter $P$, and $P$ may also refer to the Dirichlet polynomial itself. We often denote the support of the coefficients of $P(s)$ by $[P, P']$ (or $(P, P']$ etc.).
The letters $p$ and $q$ denote prime numbers, $\epsilon$ denotes a small positive constant, not necessarily the same at each occurrence, and $x$ is a large parameter.
Let
$$c \in \{0.45, 0.5\},$$
with $c = 0.5$ in the case of Theorem~\ref{thm:0.5} and $c = 0.45$ in Theorem~\ref{thm:0.45}. We define the following parameters:
$$\delta_0 = \frac{x^{c-1}}{(\log \log x)^2},$$
$$\delta_1 = \exp(-\sqrt{\log x}),$$
$$z_1 = \exp(\log x / (\log \log x)^5),$$
$$z_2 = \exp(\log x / (\log \log x)^3),$$
$$\eta = \exp(-(\log \log \log x)^2),$$
$$S = \exp((\log \log x)^{17}),$$
$$T = x^{1 - c}S^2,$$
$$T_0 = \exp(\sqrt{\log x}/3),$$
$$L_{\zeta} = \begin{cases}x^{(1-c)/2} = x^{1/4} &\text{ if } c = 0.5, \\ x^{(1 - c)/2}\exp(\log x / \sqrt{\log \log x}) = x^{0.275}\exp(\log x / \sqrt{\log \log x}) &\text{ if } c = 0.45\end{cases}$$
and
\begin{align}
\label{eq:def-R}
R = \begin{cases} x^{0.07 + \nu} &\text{ if } c = 0.5, \\ x^{0.18 + \nu} &\text{ if } c = 0.45,\end{cases}
\end{align}
where $\nu > 0$ is an arbitrarily small but fixed constant.
Furthermore, we define
$$H' = x^c(\log \log x)^{-4},$$
and for an integer $m > 0$ we let
$$\mathcal{A}(m) = \{n \in \mathbb{Z}_+ : mH' < n \le mH'(1 + \delta_0)\}$$
and
$$\mathcal{B}(m) = \{n \in \mathbb{Z}_+ : mH' < n \le mH'(1 + \delta_1)\}.$$
We will always consider only those $m$ with $m \in [x/H', 3x/H']$. The variable $H'$ should not be confused with the letter $H$ used to refer to the length of a Dirichlet polynomial $H(s)$ introduced later in the proof.
For a set $\mathcal{C}$ of integers, we let $\mathcal{C}_d$ denote $\{n \in \mathbb{Z}_+ : dn \in \mathcal{C}\}$ and $S(\mathcal{C}, z)$ denote the number of integers in $\mathcal{C}$ which have no prime factors smaller than $z$. We will write $\mathcal{A}_d(m)$ and $\mathcal{B}_d(m)$ instead of (the technically correct) $\mathcal{A}(m)_d$ and $\mathcal{B}(m)_d$.
For the convenience of the reader, here is a brief account on the reasons and constraints behind the choices of parameters above.
We reduce the problem to considering primes in the intervals $\mathcal{A}(m)$ and comparing these intervals to the longer intervals $\mathcal{B}(m)$. The parameters $H'$, $\delta_0$ and $\delta_1$ are relevant for this step, being chosen so that $\mathcal{A}(m)$ is slightly shorter than $x^c$ and that $\mathcal{B}(m)$ is long enough that we have asymptotic formulas for the number of primes in $\mathcal{B}(m)$.
At the beginning of the proof we sieve out prime factors less than $z_1$. This is important for keeping the sizes of the coefficients of Dirichlet polynomials small, and is achieved if $z_1 = x^{f(x)}$ for $f(x)$ tending to zero fast enough. At certain places we use a simple sieve to replace $1_{p \mid n \implies p \ge z_1}$ with
$$\sum_{\substack{d \mid n \\ d < z_2 \\ p \mid d \implies p < z_1}} \mu(d),$$ the latter being occasionally more convenient to work with. This procedure requires $z_2$ to be somewhat larger than $z_1$ (namely $\log z_2 \ge (\log \log x)^{1+\epsilon} \log z_1)$. The pair $(z_1, z_2)$ chosen above satisfies these constraints.
It is convenient to discard polynomials whose length is too close to certain reals $s = s(x)$. We are able to discard lengths lying in $[sx^{-\eta}, sx^{\eta}]$ as long as $\eta^{-1}$ is larger than $(\log \log x)^A$ for some fixed (but large) $A$. Hence the choice of $\eta$ above.
The parameter $S$ encompasses many small losses and additional factors arising in the course of the proof, for example $\log$-powers arising from dyadic decompositions or upper bounds on $\tau(k)$ for integers $k$ which are $z_1$-rough. This imposes lower bounds on $S$ of the form $(\log x)^{O(1)}$ or $2^{\log x / \log z_1}$. A choice of the form $S = \exp((\log \log x)^C)$ for a large enough constant $C$ works. Any losses of powers of $S$ are insignificant, as the Vinogradov pointwise bound for Dirichlet polynomials wins $\exp((\log x)^{\alpha})$ with $\alpha > 0$.
The parameter $T$ corresponds to the length of the range of integration, chosen to be essentially $x^{1 - c}$. As noted above, powers of $S$ are insignificant and not worth too much attention.
As is common for Dirichlet polynomial methods, we handle the case $|t| \le T_0$ separately, as one obtains cancellations in sums such as $\sum p^{-it}$ only for large enough $|t|$. The specific value of $T_0$ is not too important.
The Heath-Brown decomposition essentially allows one to assume that any polynomials longer than a certain power of $x$ are ``zeta sums'', the benefit being that the fourth moment of the zeta function is known. This is useful when applied to zeta sums longer than $\sqrt{T} \approx x^{(1-c)/2}$, the savings being the larger the longer the zeta sums. The parameter $L_{\zeta}$ denotes the threshold starting from which we are interested in zeta sums. What we call zeta sums are not quite sums of the form $\sum 1/n^s$ (see Definition \ref{def:zeta_sum}), and for $c = 0.45$ our fourth moment estimate for the zeta sums is slightly lossy. Hence we leave a margin of $\exp(\log x / \sqrt{\log \log x})$, the constraints behind this term being that it is larger than $\max_{k \le x} \tau(k) = \exp(O(\log x / \log \log x))$ while being less than $x^{\epsilon}$.
The value of $R$ is such that $Rx^c$ corresponds to the bounds in Theorems~\ref{thm:0.5} and~\ref{thm:0.45}.
In the proof we will encounter many situations where a quantity $X$ is bounded by $Y$ up to losses of $S^{o(1)}$. We hence introduce the following notation:
\begin{align}
\label{eq:all}
X \lll Y \Leftrightarrow \text{ there exists } \epsilon > 0 \text{ such that } S^{\epsilon}X \ll Y.
\end{align}
\section{Key tools}
\label{sec:heath-brown}
The following two propositions form the core of the method employed in this work. The first one is~\cite[Proposition 2]{HB-V} formulated slightly more generally. We give a proof below.
\begin{proposition}
\label{prop:dir_to_arit}
Let $0 < c < 1$ be a fixed constant, let $x$ be large and define $H', S, T$ and $T_0$ as in Section~\ref{sec:parameters}. Let $0 < \epsilon \le 1$ be fixed.
Let $F(s) = \sum_{k} c_kk^{-s}$ for some $c_k \in \mathbb{C}$ supported on $k \in [x, 2x]$. Assume that there exists a constant $C \in \mathbb{Z}_+$ such that
$$|c_k| \ll (\log x)^C\tau(k)^C.$$
Assume there exists $R$ such that for any distinct integers $m_1, \ldots, m_R \in [x/H', 3x/H']$ and any complex numbers $\zeta_1, \ldots , \zeta_R$ of magnitude $1$ we have
\begin{align}
\label{eq:dir_to_arit}
\int_{T_0 \le |t| \le T} |F(it)M(it)| \d t \le \frac{Rx}{S^{\epsilon}},
\end{align}
where
$$M(s) = \sum_{i = 1}^R \zeta_i m_i^{-s}.$$
Then, for all but $O(R)$ integers $m \in [x/H', 3x/H']$ one has
\begin{align}
\label{eq:dir_to_arit_c}
\sum_{k \in \mathcal{A}(m)} c_k = \frac{\delta_0}{\delta_1}\sum_{k \in \mathcal{B}(m)} c_k + O\left(\frac{\delta_0 x}{S^{\epsilon}}\right).
\end{align}
\end{proposition}
Note that the left hand side of \eqref{eq:dir_to_arit_c} is bounded from above by $\delta_0 x(\log x)^{O(1)}$ (and is heuristically of this magnitude for many choices of $c_k$), so \eqref{eq:dir_to_arit_c} corresponds to an asymptotic formula for the average of $c_k$ in a short interval with savings of $S^{\epsilon}$ in the error term. Even though we have fixed the choices of $H', S, T$ and $T_0$ here, the result applies for a wider range of parameters. In this work we will be applying the result with $c \in \{0.45, 0.5\}$.
The second vital tool is Heath-Brown's mean value theorem~\cite[Theorem 4(iii)]{HB-MVT}.
\begin{proposition}
\label{prop:HB_MVT}
Let $T \ge 1$ and let $m_1, \ldots , m_R \in (0, T]$ be distinct integers. Let $\zeta_1, \ldots , \zeta_R$ be complex numbers of modulus $1$. Then, for any $N \in \mathbb{Z}_+$ and $q_1, \ldots, q_N \in \mathbb{C}$ we have
\begin{align*}
\int_0^T \left|\sum_{k = 1}^R \zeta_k m_k^{-it}\right|^2\left|\sum_{n \le N} q_nn^{-it}\right|^2 \d t \ll_{\epsilon} \left(N^2R^2 + (NT)^{\epsilon}(NRT + NR^{7/4}T^{3/4})\right)\max_n |q_n|^2
\end{align*}
for any $\epsilon > 0$.
\end{proposition}
For $R \le T^{1/3}$ the term $NR^{7/4}T^{3/4}$ is smaller than $NRT$ and may thus be dropped. This is the case for our choice of parameters.
Furthermore, at a couple of occasions we apply a bound on the moments of the divisor function on short intervals. This lemma follows from the more general result of Shiu~\cite{shiu}.
\begin{lemma}
\label{lem:shiu}
Let $\delta > 0$ and $N \in \mathbb{Z}_+$ be fixed. For any $X, Y, z \ge 2$ with $X^{\delta} \le Y \le X$ we have
\begin{align*}
\sum_{\substack{X < n \le X + Y \\ p \mid n \implies p \ge z}} \tau(n)^N \ll \frac{Y}{\log X} \left(\frac{\log X}{\log z}\right)^{2^N}.
\end{align*}
\end{lemma}
As our formulation of Proposition~\ref{prop:dir_to_arit} is more general than~\cite[Proposition 2]{HB-V}, we give a proof below (even though the proof is essentially the same as in~\cite{HB-V}).
\begin{proof}[Proof of Proposition~\ref{prop:dir_to_arit}]
Let $m \in [x/H', 3x/H']$. We start with an application of Perron's formula (see e.g. \cite[Lemma 1.1]{harman}), obtaining
\begin{align*}
\sum_{k \in \mathcal{A}(m)} c_k = \frac{1}{2\pi i}\int_{-iT}^{iT} F(s)\frac{(1 + \delta_0)^s - 1}{s}(H'm)^s ds + O(E),
\end{align*}
where the error $E$ is bounded by
\begin{align*}
E \ll \sum_{x \le k \le 2x} |c_k|\left(\frac{1}{\max(1, T|\log(mH'(1 + \delta_0)/k)|)} + \frac{1}{\max(1, T|\log(mH'/k)|)}\right).
\end{align*}
We first bound this error. For a parameter $J \ge x^{\epsilon}$, $k \in [x, 2x]$ and $m \in [x/H', 3x/H']$, the condition $J < |mH'(1 + \delta_0) - k| \le 2J$ implies
\begin{align*}
\frac{1}{T|\log (mH'(1 + \delta_0)/k)|} \ll \frac{x}{JT},
\end{align*}
and hence the corresponding terms contribute $ \ll xT^{-1}(\log x)^{O_C(1)}$ by Lemma~\ref{lem:shiu}. Summing over dyadic ranges of $J$ gives a contribution of $\ll xT^{-1}(\log x)^{O(1)}$. The case $J < |mH' - k| \le 2J$ is similar. Finally, for the case where $|mH'(1 + \delta_0) - k| < x^{\epsilon}$ or $|mH' - k| < x^{\epsilon}$, we bound the contribution by $2|c_k|$, obtaining an error of $\ll x^{\epsilon}(\log x)^{O(1)}$, again by Shiu's bound (Lemma \ref{lem:shiu}). Hence, the error is
$$E \ll \frac{x}{T}(\log x)^{O(1)} + x^{\epsilon}(\log x)^{O(1)} = O(\delta_0 x/S^{\epsilon}).$$
A similar analysis applies to $\mathcal{B}(m)$, leading to
\begin{align*}
\sum_{k \in \mathcal{B}(m)} c_k = \frac{1}{2\pi i}\int_{-iT}^{iT} F(s)\frac{(1 + \delta_1)^s - 1}{s}(H'm)^s ds + O\left(\frac{\delta_0 x}{S^{\epsilon}}\right).
\end{align*}
Writing $\mathbb{D}elta(n) = \mathbb{D}elta(n, m) = 1_{n \in \mathcal{A}(m)} - \frac{\delta_0}{\delta_1}1_{n \in \mathcal{B}(m)}$, it then follows that
\begin{align*}
\sum_{k \in \mathbb{Z}_+} c_k\mathbb{D}elta(k) = \frac{1}{2\pi i} \int_{-iT}^{iT} F(s)G(s)m^{s} ds + O\left(\frac{\delta_0 x}{S^{\epsilon}}\right),
\end{align*}
where
\begin{align*}
G(s) = \left(\frac{(1 + \delta_0)^s - 1}{s} - \frac{\delta_0}{\delta_1}\frac{(1 + \delta_1)^s - 1}{s}\right)(H')^s.
\end{align*}
For bounding the contribution of small values of $|s|$, we note that if $0 \le \mu \le 1$ and $t$ is real, we have
\begin{align*}
\frac{(1 + \mu)^{it} - 1}{it} = \mu + O(\mu^2(1 + |t|)),
\end{align*}
and hence $|G(it)| \ll \delta_0 \delta_1 (1 + |t|)$. Moreover, we have $|F(it)| \le \sum_{k} |c_k| \ll x(\log x)^{O(1)}$, and hence
\begin{align*}
\int_{-T_0}^{T_0} |F(it)G(it)| \d t \ll \delta_0\delta_1x T_0^2 (\log x)^{O(1)},
\end{align*}
which is $O(\delta_0 x/S^{\epsilon})$.
Hence, we are left with showing that
\begin{align}
\label{eq:prop2_int}
\left|\int_{\substack{T_0 \le |t| \le T}} F(it)G(it)m^{it} \d t\right| = O\left(\frac{\delta_0 x}{S^{\epsilon}}\right)
\end{align}
for all but $O(R)$ integers $m \in [x/H', 3x/H']$. Assume not, and let $m_1, \ldots , m_R$ be such that the integral in~\eqref{eq:prop2_int} is greater than $3\delta_0 x/S^{\epsilon}$ in absolute value. Choose complex coefficients $\zeta_j$ of absolute value $1$ such that
\begin{align*}
\overline{\zeta_j} \int_{T_0 \le |t| \le T} F(it)G(it)m_j^{it} \d t = \left|\int_{T_0 \le |t| \le T} F(it)G(it)m_j^{it} \d t \right|
\end{align*}
for $1 \le j \le R$. Let $M(s) = \sum_{i = 1}^R \zeta_i m_i^{-s}$. Now
\begin{align*}
\int_{T_0 \le |t| \le T} F(it)G(it)\overline{M(it)} \d t \ge 3R\frac{\delta_0 x}{S^{\epsilon}}.
\end{align*}
By
$$|G(it)| = \left|\int_{1}^{1 + \delta_0} v^{-it - 1} \d v - \delta_0 \delta_1^{-1} \int_{1}^{1 + \delta_1} v^{-it - 1} \d v\right| \le 2\delta_0,$$
we now have
\begin{align*}
\int_{T_0 \le |t| \le T} |F(it)M(it)| \d t \ge \frac{3}{2}\frac{Rx}{S^{\epsilon}},
\end{align*}
contrary to the assumption \eqref{eq:dir_to_arit}.
\end{proof}
\section{Reduction to Buchstab sums}
\label{sec:to_buchstab}
The purpose of this section is to reformulate Theorems~\ref{thm:0.5} and~\ref{thm:0.45} in terms of Buchstab sums. We start with the following lemma. Recall the notations $\mathcal{A}(m), \mathcal{B}(m)$ and $S(\mathcal{C}, z)$ from Section \ref{sec:parameters}.
\begin{lemma}
\label{lem:to_buch}
Let $0 < d < 1$ be a constant and let $c \in \{0.45, 0.5\}$ be given. Assume that the number of integers $m \in [x/H', 3x/H']$ with
\begin{align}
\label{eq:S-diff}
S(\mathcal{A}(m), 2\sqrt{x}) < d \frac{\delta_0}{\delta_1} S(\mathcal{B}(m), 2\sqrt{x})
\end{align}
is less than $kR$ for some constant $k > 0$. Then the measure of $y \in [x, 2x]$ such that
\begin{align}
\label{eq:pi-diff}
\pi(y(1 + \delta_0)) - \pi(y) \le (d - \epsilon)\frac{y\delta_0}{\log x}
\end{align}
is at most $kRx^c$. In particular, we then have
$$\sum_{\substack{p_n \in [x, 2x] \\ p_{n+1} - p_n \ge x^{c}}} (p_{n+1} - p_n) \ll Rx^c.$$
\end{lemma}
\begin{proof}
For the last claim, note that any prime gap $[p_n, p_{n+1}]$ with $p_{n+1} - p_n \ge x^c$ gives an interval $[p_n, (p_{n+1} + p_n)/2]$ of length $(p_{n+1} - p_n)/2$ of values of $y$ satisfying~\eqref{eq:pi-diff}. Hence the sum of lengths of such long prime gaps can be at most $2kRx^c = O(Rx^c)$.
Denote the set of $y$ satisfying \eqref{eq:pi-diff} by $\mathcal{I}(x)$. Assume that $\textup{Meas}(\mathcal{I}(x)) > kRx^c$. Denoting $R' = \ceil{kR}$, it follows that one may choose points $y_1, \ldots , y_{R'} \in \mathcal{I}(x)$ so that $|y_i - y_j| > H'$ for any $i \neq j$. Let $m_i = 1 + \floor{y_i/H'}$ for every $i = 1, \ldots, R$, so $m_i \in [x/H', 3x/H']$ and $m_i$ are pairwise distinct. We show that $m_i$ satisfy~\eqref{eq:S-diff}, resulting in a contradiction.
Note that, by the Brun-Titchmarsh theorem, we have, for $i = 1, \ldots , R$,
$$|\pi(m_iH') - \pi(y_i)| \ll \frac{H'}{\log x} = o\left(\frac{\delta_0 x}{\log x}\right).$$
Similarly $|\pi(m_iH'(1 + \delta_0)) - \pi(y_i(1 + \delta_0))| = o(\delta_0 x/\log x)$. It follows that
\begin{align}
\label{eq:small_A}
S(\mathcal{A}(m_i), 2\sqrt{x}) = \pi(m_iH'(1 + \delta_0)) - \pi(m_iH') \le \left(d - \frac{\epsilon}{2}\right) \frac{m_iH'\delta_0}{\log x}.
\end{align}
On the other hand, by the prime number theorem with Vinogradov's error term (see e.g. \cite[Corollary 8.30]{IK}), we have, for $i = 1, \ldots , R$,
\begin{align}
\label{eq:large_B}
S(\mathcal{B}(m_i), 2\sqrt{x}) = \int_{m_iH'}^{m_iH'(1 + \delta_1)} \frac{\d t}{\log t} + o(\delta_1x/\log x) \ge \left(1 - \frac{\epsilon}{2}\right)\frac{m_iH'\delta_1}{\log x}.
\end{align}
The equations \eqref{eq:small_A} and \eqref{eq:large_B} contradict the assumption \eqref{eq:S-diff}, from which the result follows.
\end{proof}
Hence, our task is to show that for all but $O(R)$ integers $m \in [x/H', 3x/H']$ we have
$$S(\mathcal{A}(m), 2\sqrt{x}) > d \frac{\delta_0}{\delta_1} S(\mathcal{B}(m), 2\sqrt{x})$$
for some (small) $d > 0$. As explained in Section~\ref{sec:overview}, we accomplish this by utilizing Buchstab's identity and Harman's sieve. Hence, in the next sections we present a method for obtaining asymptotic formulas of the form
\begin{align}
\label{eq:asy_form}
\sum_{\substack{p_1, \ldots, p_n \\ p_i \in [x^{\alpha_i}, x^{\beta_i}] \\ p_n < \ldots < p_1}} S(\mathcal{A}_{p_1 \cdots p_n}(m), z) = \frac{\delta_0}{\delta_1}\sum_{\substack{p_1, \ldots, p_n \\ p_i \in [x^{\alpha_i}, x^{\beta_i}] \\ p_n < \ldots < p_1}} S(\mathcal{B}_{p_1 \cdots p_n}(m), z) + o\left(\frac{\delta_0 x}{\log x}\right).
\end{align}
We note that in the course of establishing the assumption of Lemma~\ref{lem:to_buch} we do not only obtain Theorems~\ref{thm:0.5} and~\ref{thm:0.45}, but we in fact also get the following stronger result.
\begin{theorem}
\label{thm:many}
Fix $c \in \{0.45, 0.5\}$ and $\nu > 0$. There exist constants $C, d' > 0$ such that number of disjoint intervals $[n, n + n^c], n \in \mathbb{Z} \cap [x, 2x]$ with
\begin{align}
\label{eq:many}
\pi(n + n^c) - \pi(n) \le d' \frac{n^c}{\log x}
\end{align}
is less than $CR$, where $R = x^{0.07 + \nu}$ if $c = 0.5$ and $R = x^{0.18 + \nu}$ if $c = 0.45$.
\end{theorem}
\begin{proof}[Proof of Theorem~\ref{thm:many} assuming premise of Lemma~\ref{lem:to_buch}]
Consider an integer $n \in [x, 2x]$ such that $\pi(n + n^c) - \pi(n) \le d' \frac{n^c}{\log x}$. Write the interval $[n, n+n^c]$ as a disjoint union of $\ell = O((\log \log x)^2)$ half-open intervals $I$ of length $|I| \in [3x^c/(\log \log x)^2, 4x^c/(\log \log x)^2]$. At least $\ell/2$ of such $I$ must contain less than
$$10d'\frac{x^c}{(\log x)(\log \log x)^2}$$
primes. Each of these $\ell/2$ intervals $I$ results in a set $\mathcal{Y}$ of $y \in I \subset [x, 3x]$ satisfying
$$\pi(y(1 + \delta_0)) - \pi(y) < 10d' \frac{y\delta_0}{\log x}$$
with measure $\textup{Meas}(\mathcal{Y}) \gg x^c/(\log \log x)^2$.
Note that the intervals $I$ and thus the resulting sets $\mathcal{Y}$ obtained from different values of $n$ are pairwise disjoint. Hence, denoting by $N$ the number of $n \in [x, 2x]$ with $\pi(n + n^c) - \pi(n) \le d'\frac{n^c}{\log x}$, the measure of $y$ satisfying \eqref{eq:pi-diff} with $d = 11d'$ is $\gg N \ell x^c/(\log \log x)^2 \gg Nx^c$. Assuming that \eqref{eq:S-diff} holds for $d$ small enough, for $d'$ small enough Lemma \ref{lem:to_buch} then gives $N = O(R)$.
\end{proof}
\section{From Buchstab sums to Dirichlet polynomials}
\label{sec:dir}
In this section we reduce the problem of obtaining asymptotics of type~\eqref{eq:asy_form} to the problem of bounding mean values of Dirichlet polynomials.
Hence, we consider formulas of the form
\begin{align}
\label{eq:buch2}
\sum_{k \in \mathcal{A}(m)} c_k = \frac{\delta_0}{\delta_1} \sum_{k \in \mathcal{B}(m)} c_k + o\left(\frac{\delta_0 x}{\log x}\right),
\end{align}
where
\begin{align*}
c_k = \sum_{\substack{p_1, \ldots , p_n \in \mathbb{P} \\ p_i \in I_i \\ p_n < \ldots < p_1}} 1_{p_1 \cdots p_n \mid k}1_{p \mid k \implies p \ge z}
\end{align*}
and $I_i$ are intervals. Here and in what follows we will have $p_n \ge z$. The idea is to apply Proposition~\ref{prop:dir_to_arit} to reduce the problem to one on Dirichlet polynomials. However, a direct application of the proposition would not work, as the resulting Dirichlet polynomial $F(s)$ would not have certain desirable properties (such as factorizing as a product of shorter polynomials). Hence, we first have to ``clean up'' the sums before applying Proposition~\ref{prop:dir_to_arit}.
We first introduce some notation and preliminary tools, after which we perform the modifications on the sums.
We define
$$\xi(h) = 1_{p \mid h \implies p \ge z_1}$$
and
\begin{align}
\label{eq:def_xi_0}
\xi_0(h) = \begin{cases} \xi(h) & \text{ if } h < L_{\zeta} \\ \sum\limits_{\substack{d \mid h \\ d < z_2 \\ p \mid d \implies p < z_1}} \mu(d) & \text{ otherwise.}\end{cases}
\end{align}
For $N \in \mathbb{Z}_+$, denote
$$\mathbb{D}elta(N) = \mathbb{D}elta(N, m) = 1_{N \in \mathcal{A}(m)} - \frac{\delta_0}{\delta_1}1_{N \in \mathcal{B}(m)}.$$
For a set $\mathcal{S} = \mathcal{S}(x) \subset \mathbb{R}_+$ of reals, let
\begin{align*}
\mathbb{D}elta_{\mathcal{S}}(N, m) = \begin{cases} 0, \text{ if } d \in [sx^{-\eta}, sx^{\eta}] \text{ for some } d \mid N, s \in \mathcal{S} \\ \mathbb{D}elta(N, m) \text{ otherwise} \end{cases}
\end{align*}
and
\begin{align}
\label{def:delta_S'}
\mathbb{D}elta_{\mathcal{S}}'(N, m) = \begin{cases} 0, \text{ if } p_1p_2 \mid N \text{ for some } z_1 \le p_1, p_2 \le x^{c/2 - \epsilon} \text{ with } p_1/4 \le p_2 \le 4p_1 \\ \mathbb{D}elta_{\mathcal{S}}(N, m) \text{ otherwise}\end{cases}.
\end{align}
Hence $\mathbb{D}elta_{\mathcal{S}}'(N, m)$ removes those integers which have a divisor lying close to $s$ for some $s \in \mathcal{S}$ or which have two prime factors (of suitable size) which are almost equal in size.
In what follows we assume
\begin{align}
\label{eq:S_assume}
\sup_{s \in \mathcal{S}} s < x^{c - \epsilon} \quad \text{and} \quad |S| \ll (\log \log x)^5.
\end{align}
While the proofs in this section require no additional information on $\mathcal{S}$, we reveal that we will choose
\begin{align}
\label{eq:S_choice}
\mathcal{S} := \left(\{T^{2/n} \ | \ n \ge 4\} \cap [z_1, \infty)\right) \cup \{L_{\zeta}\},
\end{align}
so $\mathcal{S}$ will be of size $O(\log x / \log z_1) = O((\log \log x)^5)$ and $\sup_{s \in \mathcal{S}} s = \max(T^{1/2}, L_{\zeta}) < x^{c - \epsilon}$.
We first present some preliminary tools, after which we perform the modifications on the sums in~\eqref{eq:buch2}.
\subsection{Preliminary tools}
Many of the results and proofs of this section follow closely those given by Heath-Brown in~\cite{HB-V}, in particular Lemmas 3, 5, 6 and 8 there.
We first note that the contribution of integers divisible by $p^2$ for some $p \ge L_{\zeta}$ to our sums is negligble.
\begin{lemma}
\label{lem:p^2}
Let $D \in \mathbb{Z}_+$ be a constant. We have, for all but $O(x^{\epsilon})$ integers $m \in [x/H', 3x/H']$,
\begin{align}
\label{eq:p^2}
\sum_{p \ge L_{\zeta}} \sum_{\substack{N \in \mathcal{A}(m) \\ p^2 \mid N}} \tau(N)^D = o\left(\frac{\delta_0 x}{\log x}\right)
\end{align}
The corresponding result holds with $\mathcal{A}$ and $\delta_0$ replaced by $\mathcal{B}$ and $\delta_1$.
\end{lemma}
\begin{proof}
We first note that the contribution of $p > 2x^{1/2}$ to the sum in \eqref{eq:p^2} is zero, so we may assume $p \le 2x^{1/2}$.
Then note that $L_{\zeta}^2 > |\mathcal{A}(m)|$, so that for any fixed $m$ and $p \ge L_{\zeta}$ there is at most one $N \in \mathcal{A}(m)$ with $p^2 \mid N$. Hence, bounding $\tau(N)^D \le x^{\epsilon}$,
$$\sum_{L_{\zeta} \le p < \delta_0 x^{1 - 2\epsilon}} \sum_{\substack{N \in \mathcal{A}(m) \\ p^2 \mid N}} \tau(N)^D \ll \sum_{p \le \delta_0 x^{1 - 2\epsilon}} x^{\epsilon} = o\left(\frac{\delta_0 x}{\log x}\right),$$
so the contribution of $p < \delta_0 x^{1 - 2\epsilon}$ is negligble.
Finally note that as $m \in [x/H', 3x/H']$ varies, the intervals $\mathcal{A}(m)$ are disjoint and lie in $[x, 4x]$. Hence the total contribution of a single value $p$ to sums as in \eqref{eq:p^2} is $O(x/p^2)$, so
$$\sum_{m \in [x/H', 3x/H']} \sum_{\delta_0 x^{1-2\epsilon} < p \le 2x^{1/2}} \sum_{\substack{N \in \mathcal{A}(m) \\ p^2 \mid N}} 1 \ll \sum_{\delta_0 x^{1-2\epsilon} < p \le 2x^{1/2}} \frac{x}{p^2} \ll x^{1 - c + 3\epsilon}.$$
Thus the number of $m$ for which \eqref{eq:p^2} does not hold is bounded by $x^{1 - c + 4\epsilon}/(\delta_0 x) \ll x^{5\epsilon}$, which is the desired bound up to redefining $\epsilon$.
\end{proof}
The second result is used to remove integers whose some divisor lies inconveniently close to an element of $\mathcal{S}$.
\begin{lemma}
\label{lem:close}
Let $D \in \mathbb{Z}_+$ be a constant. Assume that $\mathcal{S} \subset \mathbb{R}_+$ is as in~\eqref{eq:S_assume}. We have, for any $m \in [x/H', 3x/H']$,
\begin{align*}
\sum_{s \in \mathcal{S}} \sum_{\substack{d \in [sx^{-\eta}, sx^{\eta}]}} \sum_{\substack{N \in \mathcal{A}(m) \\ d \mid N \\ p \mid N \implies p \ge z_1}} \tau(N)^D = o\left(\frac{\delta_0 x}{\log x}\right).
\end{align*}
The corresponding result holds with $\mathcal{A}$ and $\delta_0$ replaced by $\mathcal{B}$ and $\delta_1$.
\end{lemma}
\begin{proof}
We consider each $s \in S$ individually, and hence have to show
\begin{align}
\label{eq:close}
\sum_{d \in [sx^{-\eta}, sx^{\eta}]} \sum_{\substack{N \in \mathcal{A}(m) \\ d \mid N \\ p \mid N \implies p \ge z_1}} \tau(N)^D = o\left(\frac{\delta_0 x}{|S| \log x}\right).
\end{align}
Write $N = dN'$ in the inner sum and bound $\tau(N) \le \tau(d)\tau(N')$. Applying Lemma \ref{lem:shiu} to the resulting sum over $N'$ (which by \eqref{eq:S_assume} is longer than $x^{\epsilon}$) we obtain
\begin{align*}
\sum_{d \in [sx^{-\eta}, sx^{\eta}]} \sum_{\substack{N \in \mathcal{A}(m) \\ d \mid N \\ p \mid N \implies p \ge z_1}} \tau(N)^D &\ll \sum_{\substack{d \in [sx^{-\eta}, sx^{\eta}] \\ p \mid d \implies p \ge z_1}} \tau(d)^D \frac{\delta_0 x / d}{\log x} \left(\frac{\log x}{\log z_1}\right)^{2^D} \\
&\ll \frac{\delta_0 x}{\log x}(\log \log x)^{O(1)} \sum_{\substack{d \in [sx^{-\eta}, sx^{\eta}] \\ p \mid d \implies p \ge z_1}} \frac{\tau(d)^D}{d}.
\end{align*}
We then perform a dyadic decomposition over $d$. The contribution of the interval $[w, 2w]$ to the sum is, again by Lemma \ref{lem:shiu}, bounded by
$$\ll \frac{1}{w} \cdot \frac{w}{\log X} \left(\frac{\log X}{\log z_1}\right)^{2^D} = \frac{(\log \log x)^{O(1)}}{\log x}.$$
Sum over $O(\eta \log x)$ values of $w$. The left hand side of \eqref{eq:close} is hence bounded by
\begin{align*}
\eta \frac{\delta_0 x}{\log x}(\log \log x)^{O(1)},
\end{align*}
which is sufficient, as $\eta = \exp(-(\log \log \log x)^2)$ and $|S| = (\log \log x)^{O(1)}$.
The proof for $\mathcal{B}$ is similar.
\end{proof}
The next result is similar and used to remove integers which have two (not too large) prime factors close to each other.
\begin{lemma}
\label{lem:dyadic}
For any $m \in [x/H', 3x/H']$ and any $D \in \mathbb{Z}_+$ we have
\begin{align*}
\sum_{\substack{p_1, p_2 \in \mathbb{P} \\ z_1 \le p_1, p_2 < x^{c/2 - \epsilon} \\ p_1/4 \le p_2 \le 4p_1}} \sum_{\substack{N \in \mathcal{A}(m) \\ p_1p_2 \mid N \\ p \mid N \implies p \ge z_1}} \tau(N)^D = o\left(\frac{x \delta_0}{\log x}\right).
\end{align*}
The corresponding result holds with $\mathcal{A}$ and $\delta_0$ replaced by $\mathcal{B}$ and $\delta_1$.
\end{lemma}
\begin{proof}
Write $N = p_1p_2N'$ in the inner sum and bound $\tau(N)^D \ll \tau(N')^D$. Applying Lemma \ref{lem:shiu} to the resulting sum over $N'$ (which by $p_i \le x^{c/2 - \epsilon}$ is longer than $x^{\epsilon}$) we obtain
\begin{align*}
\sum_{\substack{p_1, p_2 \in \mathbb{P} \\ z_1 \le p_1, p_2 < x^{c/2 - \epsilon} \\ p_1/4 \le p_2 \le 4p_1}} \sum_{\substack{N \in \mathcal{A}(m) \\ p_1p_2 \mid N \\ p \mid N \implies p \ge z_1}} \tau(N)^D &\ll \sum_{\substack{p_1, p_2 \in \mathbb{P} \\ z_1 \le p_1, p_2 < x^{c/2 - \epsilon} \\ p_1/4 \le p_2 \le 4p_1}} \frac{\delta_0 x}{p_1p_2 \log x} \left(\frac{\log x}{\log z_1}\right)^{2^D} \\
&\ll \frac{\delta_0 x}{\log x}(\log \log x)^{O(1)} \sum_{\substack{p_1, p_2 \in \mathbb{P} \\ z_1 \le p_1, p_2 < x^{c/2 - \epsilon} \\ p_1/4 \le p_2 \le 4p_1}} \frac{1}{p_1p_2}.
\end{align*}
The sum over $p_1, p_2$ is bounded by
\begin{align*}
\ll \sum_{\substack{p_1 \in \mathbb{P} \\ z_1 \le p_1 < x^{c/2 - \epsilon}}} \frac{1}{p_1 \log p_1} \ll \frac{1}{\log z_1}.
\end{align*}
The result follows. The proof for $\mathcal{B}$ is similar.
\end{proof}
We then present Heath-Brown's identity (also known as the Heath-Brown decomposition).
\begin{lemma}
\label{lem:HB-dec}
Let $f : \mathbb{Z}_+ \to \mathbb{R}$ be an arbitrary function supported on $[1, 10x]$ and let $k \in \mathbb{Z}_+$ be fixed. Let $g(n) = \Lambda(n)1_{n \not\in \mathbb{P}}$. Assume that for any $N_1, \ldots N_{2k}$ and $N_1', \ldots , N_{2k}'$ and any $f_i \in \{1, \log, \mu, g\}$ satisfying $N_i > 5x^{1/k} \implies f_i \in \{1, \log, g\}$ we have
\begin{align*}
\sum_{\substack{n_1, \ldots , n_{2k} \\ N_i < n_i \le N_i'}} f_1(n_1) \cdots f_{2k}(n_{2k})f(n_1 \cdots n_{2k}) = o\left(\frac{\delta_0 x}{\log x}\right).
\end{align*}
Then
\begin{align*}
\sum_{p \le 10x} f(p) = o\left(\frac{\delta_0 x}{\log x}\right).
\end{align*}
\end{lemma}
There is of course nothing special with the bound $o(\delta_0 x/\log x)$. The function $g$ is an artifact arising from replacing $(\log p)1_{p \in \mathbb{P}}$ by $\Lambda(n)$. In practice when applying the Heath-Brown decomposition, the case where $f_i \in \{1, \log, \mu\}$ for all $i$ is the most difficult one.
\begin{proof}
First, in order to evaluate $\sum f(p)$, it suffices to evaluate $\sum (\log p)f(p)$. More precisely, by partial summation we have
\begin{align*}
\sum_{p \ge 2} f(p) &= \sum_{p \ge 2} \frac{1}{\log p}(\log p)f(p) \\
&= -\int_{2}^{\infty} \frac{-1}{t(\log t)^2}\sum_{2 \le p \le t} (\log p)f(p) \d t,
\end{align*}
and so it suffices to show
\begin{align}
\label{eq:HB-dec-proof}
\sum_{p \le t} (\log p)f(p) = o\left(\frac{\delta_0 x}{\log x}\right)
\end{align}
for any $t$.
For this we use Heath-Brown's identity (see e.g.~\cite[(13.37)]{IK})
\begin{align*}
\Lambda(n) = \sum_{1 \le j \le k} (-1)^{j-1} \binom{k}{j} \sum_{m_1, \ldots , m_j \le 5X^{1/k}} \mu(m_1) \cdots \mu(m_j) \sum_{m_1 \cdots m_j n_1 \cdots n_j = n} \log n_1, \quad n \le 10x
\end{align*}
which allows us to write
\begin{align*}
\sum_{n \in I} \Lambda(n)f(n),
\end{align*}
where $I$ is an interval, as $O(1)$ sums
\begin{align*}
\sum_{\substack{n_1, \ldots , n_{2k} \\ n_i \in [N_i, N_i'] \\ n_1 \cdots n_{2k} \in I}} f_1(n_1) \cdots f_{2k}(n_{2k})f(n_1 \cdots n_{2k})
\end{align*}
for $f_i \in \{1, \mu, \log\}$, where $N_i \ge 5x^{1/k}$ implies $f_i \in \{1, \log\}$. Note that by splitting the sums if necessary we may assume that $N_i' \ge 5x^{1/k}$ implies $N_i \ge 5x^{1/k}$.
Thus, we have
\begin{align*}
\sum_{p \le t} (\log p)f(p) = &\sum_{\substack{(N_i, N_i', f_i)}} \sum_{\substack{n_1, \ldots , n_{2k} \\ n_i \in [N_i, N_i'] \\ n_1 \cdots n_{2k} \le t}} f_1(n_1) \cdots f_{2k}(n_{2k})f(n_1 \cdots n_{2k}) \\
- &\sum_{e \ge 2} \sum_{p^e \le t} (\log p)f(p^e).
\end{align*}
We note that the last sum is of the same form as the others, as we may write
\begin{align*}
\sum_{e \ge 2} \sum_{p^e \le t} (\log p)f(p^e) =
\sum_{\substack{n_1 \in [1, t] \\ n_2, \ldots , n_{2k} \in [1, 1] \\ n_1 \cdots n_{2k} \le t}}f_1(n_1)f_2(n_2) \cdots f_{k}(n_{2k})f(n_1 \cdots n_{2k})
\end{align*}
with $f_1 = g, f_2 = \ldots = f_{2k} = 1$. The result follows.
\end{proof}
The next result is used to replace the indicator function $\xi(h) = 1_{p \mid h \implies p \ge z_1}$ with the more convenient function $\xi_0(h)$ defined in \eqref{eq:def_xi_0}.
\begin{lemma}
\label{lem:xi_to_xi0}
For any $m \in [x/H', 3x/H']$ and any $D \in \mathbb{Z}_+$ we have
\begin{align}
\label{eq:xi_to_xi0}
&\sum_{h \in \mathbb{Z}_+} \sum_{\substack{N \in \mathcal{A}(m) \\ h \mid N}} |\xi(h) - \xi_0(h)| \tau(N)^D(\log N)^D = o\left(\frac{\delta_0 x}{\log x}\right).
\end{align}
The corresponding result holds with $\mathcal{A}$ and $\delta_0$ replaced by $\mathcal{B}$ and $\delta_1$.
\end{lemma}
\begin{proof}
By~\cite[Lemma 7]{HB-V} we have
\begin{align*}
|\xi(h) - \xi_0(h)| \le \sum_{\substack{d \mid (h, \Pi_1) \\ z_2 \le d < z_1z_2}} 1,
\end{align*}
where $\Pi_1 = \prod_{p < z_1} p$. Hence the left hand side of~\eqref{eq:xi_to_xi0} is bounded by
\begin{align}
\label{eq:xi_proof_1}
(\log x)^{O(1)} \sum_{\substack{d \mid \Pi_1 \\ z_2 \le d < z_1z_2}} \sum_{\substack{N \in \mathcal{A}(m) \\ d \mid N \\ p \mid N \implies p \ge z_1}} \tau(N)^{D+1}.
\end{align}
Write $N = dN'$ in the inner sum and bound $\tau(N) \le \tau(d)\tau(N')$. As $z_1z_2 < x^{\epsilon}$, we may apply Lemma~\ref{lem:shiu} to bound \eqref{eq:xi_proof_1} by
\begin{align}
\label{eq:xi_proof_2}
\delta_0 x(\log x)^{O(1)} \sum_{\substack{d \mid \Pi_1 \\ z_2 \le d < z_1z_2}} \frac{\tau(d)^{D+1}}{d}.
\end{align}
We bound the sum over $\tau(d)^{D+1}/d$ by Rankin's trick as in the proof of~\cite[Lemma 6]{HB-V}. For a parameter $\theta > 0$, we have
\begin{align*}
\sum_{\substack{d \mid \Pi_1 \\ z_2 \le d < z_1z_2}} \frac{\tau(d)^{D+1}}{d} &\le z_2^{-\theta} \sum_{\substack{d \mid \Pi_1 \\ z_2 \le d < z_1z_2}} \frac{\tau(d)^{D+1}}{d^{1 - \theta}} \\
&\le z_2^{-\theta} \sum_{\substack{d = 1 \\ d \mid \Pi_1}}^{\infty} \frac{\tau(d)^{D+1}}{d^{1 - \theta}} \\
&= z_2^{-\theta} \prod_{p < z_1} \left(1 + \frac{2^{D+1}}{p^{1 - \theta}}\right) \\
&\le z_2^{-\theta} \exp\left(2^{D+1} \sum_{p < z_1} \frac{1}{p^{1 - \theta}}\right).
\end{align*}
Choosing $\theta = 1/\log z_1$ we have, for $x$ large enough
\begin{align*}
\sum_{p < z_1} \frac{1}{p^{1 - \theta}} \le 3 \log \log z_1.
\end{align*}
It follows that \eqref{eq:xi_proof_2} is bounded by
$$\delta_0 x (\log x)^{O(1)} \sum_{\substack{d \mid \Pi_1 \\ z_2 \le d < z_1z_2}} \frac{\tau(d)^{D+1}}{d} \ll \delta_0 x(\log x)^{O(1)}z_2^{-1/\log z_1}.$$
This is sufficient, as $\log z_2 = (\log \log x)^2 \log z_1$.
The proof for $\mathcal{B}$ is similar.
\end{proof}
Finally, we use the following lemma to truncate a certain sum at $T^{1 + \epsilon}$.
\begin{lemma}
\label{lem:long_h}
Let $m \in [x/H', 3x/H']$ be given and fix $\epsilon > 0$. We have, for any fixed $D \in \mathbb{Z}_+$,
\begin{align*}
\sum_{\substack{N' \in \mathbb{Z}_+}} \tau(N')^D(\log N')^D \left|\sum_{h > T^{1 + \epsilon}} \xi_0(h)\mathbb{D}elta(hN')\right| = o\left(\frac{\delta_0 x}{\log x}\right).
\end{align*}
\end{lemma}
\begin{proof}
Let $V = T^{1 + \epsilon}$. We have, for any $N' \in \mathbb{Z}_+$,
\begin{align*}
\sum_{h > V} \xi_0(h)\mathbb{D}elta(hN') &\ll \sum_{d < z_2} \left|\sum_{g > V/d} \mathbb{D}elta(gdN')\right|.
\end{align*}
If $VN' > mH'(1 + \delta_1)$, then the inner sum is empty. If $VN' \le mH'$, then
\begin{align*}
\sum_{g > V/d} \mathbb{D}elta(gdN') = \left(\frac{mH'\delta_0}{dN'} + O(1)\right) - \frac{\delta_0}{\delta_1}\left(\frac{mH'\delta_1}{dN'} + O(1)\right) = O(1).
\end{align*}
If $mH' < VN' \le mH'(1 + \delta_1)$, then
\begin{align*}
\sum_{g > V/d} |\mathbb{D}elta(gdN')| \le \frac{mH'\delta_0}{dN'} + O(1) + \frac{\delta_0}{\delta_1}\left(\frac{mH'\delta_1}{dN'} + O(1)\right) \ll \frac{mH'\delta_0}{dN'} + O(1).
\end{align*}
It follows that
\begin{align*}
&\sum_{\substack{N' \in \mathbb{Z}_+}} \tau(N')^D(\log N')^D \left|\sum_{h > T^{1 + \epsilon}} \xi_0(h)\mathbb{D}elta(hN)\right| \ll \\
&\sum_{N' \le x/T^{1+\epsilon/2}} z_2 \tau(N')^D(\log N')^D + \sum_{d < z_2} \sum_{mH'/V < N' \le mH'(1 + \delta_1)/V} \frac{mH'\delta_0}{dN'}\tau(N')^D(\log N')^D.
\end{align*}
The first sum gives a power saving bound over $\delta_0 x/\log x$. We bound the second sum as
\begin{align*}
&\frac{mH'\delta_0(\log x)^{D}}{mH'/V} \sum_{d < z_2} \frac{1}{d} \sum_{mH'/V < N' \le mH'(1 + \delta_1)/V} \tau(N')^D \ll \\
&V\delta_0(\log x)^D(\log z_2) \sum_{\substack{mH'/V < N' \le mH'(1 + \delta_1)/V}} \tau(N')^D,
\end{align*}
and apply Lemma~\ref{lem:shiu} to the sum over $N'$ to arrive at
\begin{align*}
V\delta_0 (\log x)^D (\log z_2) \frac{mH' \delta_1}{V}(\log x)^{O(1)} \ll \delta_0 x \delta_1 (\log x)^{O(1)},
\end{align*}
which is sufficient, as $\delta_1 \ll (\log x)^{-A}$ for any $A \ge 1$.
\end{proof}
\subsection{Modification of the sums}
We consider asymptotics of the form
\begin{align}
\label{eq:mod_1}
\sum_{\substack{p_1, \ldots , p_n \\ p_i \in I_i \\ p_n < \ldots < p_1}} S(\mathcal{A}_{p_1 \cdots p_n}(m), z) - \frac{\delta_0}{\delta_1} S(\mathcal{B}_{p_1 \cdots p_n}(m), z) = o\left(\frac{\delta_0 x}{\log x}\right)
\end{align}
and
\begin{align}
\label{eq:mod_2}
\sum_{\substack{p_1, \ldots , p_n \\ p_i \in I_i \\ p_n < \ldots < p_1}} S(\mathcal{A}_{p_1 \cdots p_n}(m), p_n) - \frac{\delta_0}{\delta_1}S(\mathcal{B}_{p_1 \cdots p_n}(m), p_n) = o\left(\frac{\delta_0 x}{\log x}\right)
\end{align}
for $m \in [x/H', 3x/H']$ and intervals $I_i \subset [z_1, 10\sqrt{x}]$. In~\eqref{eq:mod_1} we will assume $p_n \ge z$, that is, $I_i \subset [z, 10\sqrt{x}]$. We will always have $z \ge z_1$.
Our aim is to reduce the statements \eqref{eq:mod_1} and \eqref{eq:mod_2} to statements regarding mean values of Dirichlet polynomials via Proposition \ref{prop:dir_to_arit}. Before applying Proposition \ref{prop:dir_to_arit} we perform several modifications to the sums for the resulting Dirichlet polynomials to have certain desirable properties. For convenience we will mainly consider sums of the form~\eqref{eq:mod_1}, as the sum~\eqref{eq:mod_2} may be handled via similar methods (see Remark~\ref{rem:z=p_n}). We will perform the following modifications to~\eqref{eq:mod_1}.
\begin{itemize}
\item Handle integers divisible by $p^2$ for a large prime $p$.
\item Write the condition on $z$-roughness as sums over integers via Möbius inversion.
\item Discard cases where some product lies close to $s \in \mathcal{S}$ or where we have two prime factors close to each other (i.e. replace $\mathbb{D}elta$ with $\mathbb{D}elta_{\mathcal{S}}'$).
\item Apply Heath-Brown's identity to certain sums.
\item Replace occurrences of $\xi$ with $\xi_0$.
\item Restrict the size of a certain variable.
\item Decompose a certain sum as sums over primes.
\item Perform dyadic decomposition and remove cross conditions.
\end{itemize}
These steps are undertaken in Lemmas~\ref{lem:mod_qrh} to \ref{lem:mod_cross} below.
First, reduce to $m$ satisfying \eqref{eq:p^2} (and the similar conclusion for $\mathcal{B}$ and $\delta_1$). To this end, we let $\mathcal{M} \subset [x/H', 3x/H']$ denote the set of $m$ for which
$$\sum_{\substack{p_1, \ldots , p_n \in \mathbb{P} \\ p_i \in I_i \\ p_n < \ldots < p_1}} \sum_{\substack{\ell \in \mathbb{Z}_+ \\ \exists p \ge L_{\zeta} : p^2 \mid \ell}} \mathbb{D}elta(p_1 \cdots p_n\ell, m) = o\left(\frac{\delta_0 x}{\log x}\right).$$
By Lemma \ref{lem:p^2}, $\mathcal{M}$ contains all but $O(x^{\epsilon}) = O(R)$ values of $m \in [x/H', 3x/H']$.
We then write the condition on $z$-roughness in~\eqref{eq:mod_1} in a more convenient form.
\begin{lemma}
\label{lem:mod_qrh}
Let $n = O(1)$, $z \in [z_1, 10\sqrt{x}]$, intervals $I_1, \ldots , I_n \subset [z, 10\sqrt{x}]$ and $m \in \mathcal{M}$ be given. Assume that for any $0 \le n' \le 4$ we have
\begin{align*}
\sum_{\substack{p_1, \ldots , p_n \in \mathbb{P} \\ p_i \in I_i \\ p_n < \ldots < p_1}} \sum_{\substack{r \in \mathbb{Z}_+ \\ p \mid r \implies \\ z_1 < p < \min(z, L_{\zeta})}} \sum_{\substack{q_1, \ldots , q_{n'} \in \mathbb{P} \\ \min(z, L_{\zeta}) \le q_i < z \\ q_{n'} < \ldots < q_1}} \sum_{h \in \mathbb{Z}_+} \mu(r) \xi(h) \mathbb{D}elta(p_1 \cdots p_n r q_1 \cdots q_{n'} h, m) = o\left(\frac{\delta_0 x}{\log x}\right).
\end{align*}
Then~\eqref{eq:mod_1} holds.
\end{lemma}
\begin{proof}
Note that
\begin{align*}
& \sum_{\substack{p_1, \ldots , p_n \\ p_i \in I_i \\ p_n < \ldots < p_1}} S(\mathcal{A}_{p_1 \cdots p_n}(m), z) - \frac{\delta_0}{\delta_1} S(\mathcal{B}_{p_1 \cdots p_n}(m), z) \\
= & \sum_{\substack{p_1, \ldots , p_n \\ p_i \in I_i \\ p_n < \ldots < p_1}} \sum_{\substack{\ell \in \mathbb{Z}_+ \\ p \mid \ell \implies p \ge z}} \mathbb{D}elta(p_1 \cdots p_n \ell).
\end{align*}
As $m \in \mathcal{M}$, we may reduce to $\ell$ not divisible by $p^2$ for any $p \ge L_{\zeta}$. For such $\ell$ we then have
\begin{align*}
1_{p \mid \ell \implies p \ge z} &= \sum_{\substack{r' \in \mathbb{Z}_+ \\ p \mid r' \implies z_1 \le p < z}} \sum_{\substack{h \in \mathbb{Z}_+ \\ p \mid h \implies p \ge z_1}} \mu(r')1_{hr' = \ell} \\
&= \sum_{n' \le 4} \sum_{\substack{q_1, \ldots , q_{n'} \in \mathbb{P} \\ \min(z, L_{\zeta}) \le q_i < z \\ q_{n'} < \ldots < q_1}} \sum_{\substack{r \in \mathbb{Z}_+ \\ p \mid r \implies z_1 \le p < \min(z, L_{\zeta})}} \sum_{h \in \mathbb{Z}_+} (-1)^{n'}\xi(h)1_{q_1 \cdots q_{n'}hr = \ell},
\end{align*}
from which \eqref{eq:mod_1} follows.
\end{proof}
The next step is to discard cases where some product of the numbers is close to elements of $s \in \mathcal{S}$ or which have two not too large prime factors lying close to each other. In other words, we replace $\mathbb{D}elta$ with $\mathbb{D}elta_{\mathcal{S}}'$.
\begin{lemma}
\label{lem:mod_S}
Let $n, I_i, z$ and $m$ be as in Lemma~\ref{lem:mod_qrh}. Let $\mathcal{S}$ be as in \eqref{eq:S_choice}. Assume that for any $n' \le 4$ we have
\begin{align}
\label{eq:mod_S}
\sum_{\substack{p_1, \ldots , p_n \\ p_i \in I_i \\ p_n < \ldots < p_1}} \sum_{\substack{r \in \mathbb{Z}_+ \\ p \mid r \implies \\ z_1 \le p < \min(z, L_{\zeta})}} \sum_{\substack{q_1, \ldots , q_{n'} \in \mathbb{P} \\ \min(z, L_{\zeta}) \le q_i < z \\ q_{n'} < \ldots < q_1}} \sum_{h \in \mathbb{Z}_+} \mu(r) \xi(h) \mathbb{D}elta_{\mathcal{S}}'(p_1 \cdots p_n r q_1 \cdots q_{n'} h, m) = o\left(\frac{\delta_0 x}{\log x}\right).
\end{align}
Then~\eqref{eq:mod_1} holds.
\end{lemma}
\begin{proof}
We show the sum in Lemma~\ref{lem:mod_qrh} is $o(\delta_0 x/\log x)$ assuming \eqref{eq:mod_S}. First replace $\mathbb{D}elta$ with $\mathbb{D}elta_{\mathcal{S}}$, the difference being bounded in absolute value by
\begin{align*}
\sum_{s \in \mathcal{S}} \sum_{d \in [sx^{-\eta}, sx^{\eta}]} \sum_{\substack{N \in \mathcal{A}(m) \\ d \mid N \\ p \mid N \implies p \ge z_1}} \tau(N)^{O(1)} + \frac{\delta_0}{\delta_1} \sum_{s \in \mathcal{S}} \sum_{d \in [sx^{-\eta}, sx^{\eta}]} \sum_{\substack{N \in \mathcal{B}(m) \\ d \mid N \\ p \mid N \implies p \ge z_1}} \tau(N)^{O(1)},
\end{align*}
which by Lemma~\ref{lem:close} is $o(\delta_0 x/\log x)$. We then replace $\mathbb{D}elta_{\mathcal{S}}$ by $\mathbb{D}elta'_{\mathcal{S}}$, the error being similarly bounded by
\begin{align*}
\sum_{\substack{p_1, p_2 \in \mathbb{P} \\ z_1 \le p_1, p_2 < x^{c/2 - \epsilon} \\ p_1/4 \le p_2 \le 4p_1}} \sum_{\substack{N \in \mathcal{A}(m) \\ p_1p_2 \mid N \\ p \mid N \implies p \ge z_1}} \tau(N)^{O(1)} + \frac{\delta_0}{\delta_1} \sum_{\substack{p_1, p_2 \in \mathbb{P} \\ z_1 \le p_1, p_2 < x^{c/2 - \epsilon} \\ p_1/4 \le p_2 \le 4p_1}} \sum_{\substack{N \in \mathcal{B}(m) \\ p_1p_2 \mid N \\ p \mid N \implies p \ge z_1}} \tau(N)^{O(1)},
\end{align*}
which by Lemma~\ref{lem:dyadic} is small enough. The result follows.
\end{proof}
We then decompose the sums over $p_1, \ldots , p_n$ and $q_1, \ldots , q_{n'}$ by the Heath-Brown decomposition.
\begin{lemma}
\label{lem:mod_hbd}
Let $n = O(1)$, $z \in [z_1, 10\sqrt{x}]$, intervals $I_1, \ldots , I_n \subset [z, 10\sqrt{x}]$ and $m \in \mathcal{M}$ be given. Write $I_{i} = [\min(z, L_{\zeta}), z)$ for $i > n$. Assume that, for
\begin{enumerate}
\item any $n' \le 4$,
\item any $N_{i, j}$, where $1 \le i \le n+n', 1 \le j \le 8$, and any $N_{i, j}' > N_{i, j}$, and
\item any $f_{i, j}$, where $1 \le i \le n+n', 1 \le j \le 8$, with $f_{i, j} \in \{\xi, \xi \cdot \log, \xi \cdot g, \xi \cdot \mu\}$ and $N_{i, j} > L_{\zeta} \implies f_{i, j} \neq \xi \cdot \mu$,
\end{enumerate}
we have
\begin{align*}
\sum_{\substack{N_{i, j} < n_{i, j} \le N_{i, j}' \\ n_{i, 1} \cdots n_{i, 8} \in I_i}}^{\ast} \prod_{\substack{1 \le i \le n+n' \\ 1 \le j \le 8}} f_{i, j}(n_{i, j}) \sum_{\substack{r \in \mathbb{Z}_+ \\ p \mid r \implies z_1 \le p < \min(z, L_{\zeta})}} \sum_{h \in \mathbb{Z}_+} \mu(r) \xi(h) \mathbb{D}elta_{\mathcal{S}}'\left(rh \prod_{\substack{1 \le i \le n + n' \\ 1 \le j \le 8}} n_{i, j}, m\right) \\
= o\left(\frac{\delta_0 x}{\log x}\right),
\end{align*}
where the asterisk $\ast$ denotes that the sums is only over $n_{i, j}$ satisfying
$$\prod_{1 \le j \le 8} n_{i, j} < \prod_{1 \le j \le 8} n_{i-1, j} \qquad \text{for all } i \in \{2, 3, \ldots , n+n'\} \setminus \{n+1\}.$$
Then~\eqref{eq:mod_1} holds.
\end{lemma}
\begin{remark}
\label{rem:optional}
While in this lemma we have decomposed the sums over all of $p_1, \ldots , p_n$, $q_1, \ldots, q_{n'}$, we could choose to decompose the sums merely over a (possibly empty) subset of them. For clarity, we state the results here and below for the case where all the sums have been decomposed, understanding that we have this additional flexibility.
\end{remark}
\begin{proof}
We start from the sum in Lemma~\ref{lem:mod_S}. We apply Heath-Brown's decomposition (Lemma~\ref{lem:HB-dec}) with $k = 4$ to the sums over $p_1, \ldots , p_n, q_1, \ldots , q_{n'}$ one by one. For example, applying the decomposition to the sum over $p_1$ we take $f$ in Lemma~\ref{lem:HB-dec} to be
\begin{align*}
&f(\ell) = \\
&\xi(\ell)1_{\ell \in I_1} \sum_{\substack{p_2, \ldots , p_n \\ p_i \in I_i \\ p_n < \ldots < p_2 < \ell}} \sum_{\substack{r \\ p \mid r \implies \\ z_1 \le p < \min(z, L_{\zeta})}} \sum_{\substack{q_1, \ldots , q_{n'} \in \mathbb{P} \\ \min(z, L_{\zeta}) \le q_i < z \\ q_{n'} < \ldots < q_1}} \sum_{h \in \mathbb{Z}_+} \mu(r) \xi(h) \mathbb{D}elta'_{\mathcal{S}}(\ell p_2 \cdots p_n r q_1 \cdots q_{n'} h, m).
\end{align*}
Hence, the sum over $f(p_1)$ may be converted to a sum over $n_{1, 1}, \ldots, n_{1, 8}$ as in Lemma~\ref{lem:HB-dec}. Note that $\xi(\ell)$ is completely multiplicative.
Performing the decomposition for all $p_1, \ldots, p_n, q_1, \ldots , q_{n'}$ gives the result. Note that the cross conditions $p_{i} < p_{i-1}$ and $q_i < q_{i-1}$ transform to cross conditions of the form $n_{i, 1} \cdots n_{i, 8} < n_{i-1, 1} \cdots n_{i-1, 8}$.
Considering the implication $N_{i, j} > L_{\zeta} \implies f_{i, j} \neq \xi \cdot \mu$, note that for $c = 0.45$, we have simply relaxed the condition $N_i \ge 5x^{1/4}$ in Lemma \ref{lem:HB-dec} to $N_i \ge L_{\zeta}$. For $c = 0.5$, note that the the sum over $n_{i, j} \in [x^{1/4}, 5x^{1/4}]$ is empty as $x^{1/4} \in \mathcal{S}$, and we may thus assume $N_{i, j} > x^{1/4}$ implies $N_{i, j} > 5x^{1/4}$ and hence $f_{i, j} \neq \xi \cdot \mu$.
\end{proof}
Then we replace each occurrence of $\xi$ with $\xi_0$, as the latter will be more convenient to work with.
\begin{lemma}
\label{lem:mod_xi0}
Let $n = O(1)$, $z \in [z_1, 10\sqrt{x}]$, intervals $I_1, \ldots , I_n \subset [z, 10\sqrt{x}]$ and $m \in \mathcal{M}$ be given. Write $I_i = [\min(z, L_{\zeta}), z)$ for $i > n$. Assume that, for
\begin{enumerate}
\item any $n' \le 4$,
\item any $N_{i, j}, 1 \le i \le n+n', 1 \le j \le 8$ and $N_{i, j}' > N_{i, j}$, and
\item any $f_{i, j}, 1 \le i \le n+n', 1 \le j \le 8$ with $f_{i, j} \in \{\xi_0, \xi_0 \cdot \log, \xi_0 \cdot g, \xi_0 \cdot \mu\}$ and $N_{i, j} > L_{\zeta} \implies f_{i, j} \neq \xi_0 \cdot \mu$,
\end{enumerate}
we have
\begin{align*}
\sum_{\substack{N_{i, j} < n_{i, j} \le N_{i, j}' \\ n_{i, 1} \cdots n_{i, 8} \in I_i \\ \text{for } 1 \le i \le n+n', 1 \le j \le 8}}^{\ast} \prod_{\substack{1 \le i \le n+n' \\ 1 \le j \le 8}} f_{i, j}(n_{i, j}) \sum_{\substack{r \in \mathbb{Z}_+ \\ p \mid r \implies \\ z_1 \le p < \min(z, L_{\zeta})}} \sum_{h \in \mathbb{Z}_+} \mu(r) \xi_0(h) \mathbb{D}elta_{\mathcal{S}}'\left(rh \prod_{\substack{1 \le i \le n + n' \\ 1 \le j \le 8}} n_{i, j}, m\right) \\
= o\left(\frac{\delta_0 x}{\log x}\right),
\end{align*}
Then~\eqref{eq:mod_1} holds.
\end{lemma}
\begin{proof}
We start from a sum as in Lemma~\ref{lem:mod_hbd} and replace occurrences of $\xi$ with $\xi_0$ one by one. Note that $|\xi_0(k)| \le \tau(k)$, the sum is over $O(1)$ variables $n_{i, j}, r, h$, and any $N$ such that $\mathbb{D}elta_{\mathcal{S}}'(N, m)$ has non-zero coefficient in the sum is $z_1$-rough. As $\log, g, \mu$ are bounded by $\log$, it follows that every replacement of $\xi$ with $\xi_0$ induces an error bounded by
\begin{align*}
&\sum_{\substack{h' \in \mathbb{Z}_+}} |\xi(h') - \xi_0(h')| \sum_{\substack{N \in \mathcal{A}(m) \\ h' \mid N}} \tau(N)^{O(1)}(\log N)^{O(1)} \ + \\
\frac{\delta_0}{\delta_1} &\sum_{\substack{h' \in \mathbb{Z}_+}} |\xi(h') - \xi_0(h')| \sum_{\substack{N \in \mathcal{B}(m) \\ h' \mid N}} \tau(N)^{O(1)}(\log N)^{O(1)}.
\end{align*}
This error is small enough by Lemma~\ref{lem:xi_to_xi0}.
\end{proof}
Then we restrict the sum over $h$ to $h \le T^{1+\epsilon}$.
\begin{lemma}
\label{lem:mod_hlarge}
Let $n, I_i, z, m$ be as in Lemma \ref{lem:mod_hbd}. Assume that for any $n', N_{i, j}$ and $f_{i, j}$ as in Lemma~\ref{lem:mod_hbd} we have
\begin{align*}
\sum_{\substack{N_{i, j} < n_{i, j} \le N_{i, j}' \\ n_{i, 1} \cdots n_{i, 8} \in I_i \text{ for} \\ 1 \le i \le n+n', 1 \le j \le 8}}^{\ast} \prod_{\substack{1 \le i \le n+n' \\ 1 \le j \le 8}} f_{i, j}(n_{i, j}) \sum_{\substack{r \in \mathbb{Z}_+ \\ p \mid r \implies \\ z_1 \le p < \min(z, L_{\zeta})}} \sum_{h \le T^{1 + \epsilon}} \mu(r) \xi_0(h) \mathbb{D}elta'_{\mathcal{S}}\left(rh \prod_{\substack{1 \le i \le n + n' \\ 1 \le j \le 8}} n_{i, j}, m\right) \\
= o\left(\frac{\delta_0 x}{\log x}\right).
\end{align*}
Then~\eqref{eq:mod_1} holds.
\end{lemma}
\begin{proof}
We start from the sum in Lemma~\ref{lem:mod_xi0}. Let $V = T^{1+\epsilon}$. Consider separately the sums over $n_{i, j}, r$ and the sum over $h$. The sum over $n_{i, j}, r$ has $O(1)$ variables, the coefficients of each variable being bounded by $\tau \cdot \log$. The contribution of $h > V$ is thus bounded by
\begin{align*}
\sum_{N' \le 10x/V} \tau(N')^{O(1)}\log(N')^{O(1)} \left|\sum_{h > V} \xi_0(h)\mathbb{D}elta(N'h)\right|.
\end{align*}
This is small enough by Lemma~\ref{lem:long_h}.
\end{proof}
We then write the sum over $r$ in a more convenient form as multiple sums over primes.
\begin{lemma}
\label{lem:mod_ri}
Let $n = O(1)$, $z \in [z_1, 10\sqrt{x}]$, intervals $I_1, \ldots , I_n \subset [z, 10\sqrt{x}]$ and $m \in \mathcal{M}$ be given. Write $I_{i} = [\min(z, L_{\zeta}), z)$ for $i > n$. Assume that, for
\begin{enumerate}
\item any $n' \le 4$,
\item any $N_{i, j}, 1 \le i \le n+n', 1 \le j \le 8$ and $N_{i, j}' > N_{i, j}$,
\item any $f_{i, j}, 1 \le i \le n+n', 1 \le j \le 8$ with $f_{i, j} \in \{\xi_0, \xi_0 \cdot \log, \xi_0 \cdot g, \xi_0 \cdot \mu\}$ and $N_{i, j} > L_{\zeta} \implies f_{i, j} \neq \xi_0 \cdot \mu$,
\item any $t \in \mathbb{Z}_{\ge 0}$, and
\item any $R_i \in [z_1, \min(z, L_{\zeta}))$ and $R_i' \ge R_i$ where $1 \le i \le t$,
\end{enumerate}
we have
\begin{align*}
\sum_{\substack{N_{i, j} < n_{i, j} \le N_{i, j}' \\ n_{i, 1} \cdots n_{i, 8} \in I_i \\ \text{for } 1 \le i \le n+n', 1 \le j \le 8}}^{\ast} \prod_{\substack{1 \le i \le n+n' \\ 1 \le j \le 8}} f_{i, j}(n_{i, j}) \sum_{\substack{r_1, \ldots , r_t \in \mathbb{P} \\ r_i \in [R_i, R_i'] \\ r_t < \ldots < r_1}} \sum_{h \le T^{1 + \epsilon}} \xi_0(h) \mathbb{D}elta'_{\mathcal{S}}\left(h\prod_{\substack{1 \le i \le n + n' \\ 1 \le j \le 8}} n_{i, j} \prod_{1 \le i \le t} r_i, m\right) \\
= O\left(\frac{\delta_0 x}{(\log x)^2}\right).
\end{align*}
Then~\eqref{eq:mod_1} holds.
\end{lemma}
\begin{proof}
The result follows from the identity
\begin{align*}
\mu(r)1_{p \mid r \implies z_1 \le p < \min(z, L_{\zeta})} = \sum_{0 \le t \le \frac{\log 10x}{\log z_1}} \sum_{\substack{r_1, \ldots , r_t \in \mathbb{P} \\ r_t < \ldots < r_1 \\ r_1 < \min(z, L_{\zeta}) \\ r_t \ge z_1}} (-1)^t 1_{r_1 \cdots r_t = r}.
\end{align*}
\end{proof}
Finally, we perform a dyadic decomposition and remove the cross conditions.
\begin{lemma}
\label{lem:mod_cross}
Let $n = O(1)$, $z \in [z_1, 10\sqrt{x}]$, intervals $I_1, \ldots , I_n \subset [z, 10\sqrt{x}]$ and $m \in \mathcal{M}$ be given. Write $I_{i} = [\min(z, L_{\zeta}), z)$ for $i > n$. Assume that for
\begin{enumerate}
\item any $n' \le 4$,
\item any $J_1, \ldots , J_{n+n'} \le 8$,
\item any $N_{i, j} \ge z_1, 1 \le i \le n+n', 1 \le j \le J_i$ and $N_{i, j}' \in [N_{i, j}, 2N_{i, j}]$,
\item any $f_{i, j}, 1 \le i \le n+n', 1 \le j \le J_i$ with $f_{i, j} \in \{\xi_0, \xi_0 \cdot \log, \xi_0 \cdot g, \xi_0 \cdot \mu\}$ and $N_{i, j} > L_{\zeta} \implies f_{i, j} \neq \xi_0 \cdot \mu$,
\item any $t \in \mathbb{Z}_{\ge 0}$, and
\item any $R_i \in [z_1, \min(z, L_{\zeta})]$ and $R_i' \in [R_i, 2R_i]$ where $1 \le i \le t$
\end{enumerate}
such that the intervals $[R_i, R_i']$ are pairwise disjoint and
\begin{align}
\label{eq:prod_interval}
\prod_{1 \le j \le J_i} N_{i, j} \in \left[\frac{\inf(I_i)}{2^{8}}, \sup(I_i)\right]
\end{align}
we have
\begin{align*}
\sum_{\substack{N_{i, j} < n_{i, j} \le N_{i, j}' \\ \text{for } 1 \le i \le n+n', 1 \le j \le 8}} \prod_{\substack{1 \le i \le n+n' \\ 1 \le j \le J_i}} f_{i, j}(n_{i, j}) \sum_{\substack{r_1, \ldots , r_t \in \mathbb{P} \\ r_i \in [R_i, R_i']}} \sum_{h \le T^{1 + \epsilon}} \xi_0(h) \mathbb{D}elta'_{\mathcal{S}}\left(\prod_{\substack{1 \le i \le n + n' \\ 1 \le j \le 8}} n_{i, j} \prod_{1 \le i \le t} r_i h, m\right) \\
\lll \delta_0 x.
\end{align*}
Then~\eqref{eq:mod_1} holds.
\end{lemma}
Recall the notation $\lll$ from \eqref{eq:all}.
\begin{proof}
Note that the result is trivial if $t > \frac{\log 10x}{\log z_1}$, so assume $t = O((\log \log x)^5)$.
Consider the sum in Lemma~\ref{lem:mod_ri}. By a dyadic decomposition on the $O(1)$ sums over $n_{i, j}$ and $t$ sums over $r_i$, it suffices to obtain a bound of $S^{-\epsilon}\delta_0 x$ for sums of the form
\begin{align*}
\sum_{\substack{N_{i, j} < n_{i, j} \le N_{i, j}' \\ n_{i, 1} \cdots n_{i, J_i} \in I_i}}^{\ast} \prod_{\substack{1 \le i \le n+n' \\ 1 \le j \le J_i}} f_{i, j}(n_{i, j}) \sum_{\substack{r_1, \ldots , r_t \in \mathbb{P} \\ r_i \in [R_i, R_i'] \\ r_t < \ldots < r_1}} \sum_{h \le T^{1 + \epsilon}} \xi_0(h) \mathbb{D}elta'_{\mathcal{S}}\left(\prod n_{i, j} \prod r_i h, m\right)
\end{align*}
with $N_{i, j}' \le 2N_{i, j}$ and $R_i' \le 2R_i$. We may assume~\eqref{eq:prod_interval}, as otherwise the sum is empty and the result is trivial. Similarly, we may assume $R_{i+1} \le R_i'$.
By the definition \eqref{def:delta_S'} of $\mathbb{D}elta_{\mathcal{S}}'$ we may assume that for any $i$ with $R_i < x^{c/2 - \epsilon}$ the intervals $[R_i, R_i']$ and $[R_{i+1}, R_{i+1}']$ are disjoint, as otherwise the sum is empty. Hence we may assume $R_{i+1}' < R_i$ for any $i \ge 5$ (say). The cross conditions $r_{i+1} < r_i$ now follow automatically for $i \ge 5$.
We may remove the $O(1)$ cross conditions on $n_{i, j}$ and any possible $O(1)$ cross conditions on $r_i$ at $(\log x)^{O(1)} = S^{o(1)}$ by decomposing the sums into short intervals (cf. \cite[Section 3.2]{harman}). We may further assume $R_i, N_{i, j} \ge z_1$ as otherwise the corresponding sums are empty or over the set $\{1\}$. As a consequence, the number of variables $n_{i, j}$ may decrease -- hence the new parameters $J_1, \ldots , J_{n + n'}$.
\end{proof}
In terms of Dirichlet polynomials, one could describe our procedure as follows: we wish to establish the assumption of Proposition~\ref{prop:dir_to_arit} for a Dirichlet polynomial of the form
$$P_1(s) \cdots P_n(s)Q(s),$$
where $P_i(s) = \sum_{p \sim P_i} p^{-s}$ correspond to sums over $p_i$ (after a dyadic decomposition) in~\eqref{eq:mod_1} and $Q$ is some Dirichlet polynomial (itself equal to a certain product). We perform the Heath-Brown decomposition for $P_1, \ldots , P_n$, and similarly also decompose $Q$. Hence, we consider polynomials of the form
\begin{align}
\label{def:f}
F(s) = \prod_{\substack{1 \le i \le n+n' \\ 1 \le j \le J_i}} N_{i, j}(s) \prod_{1 \le i \le t} R_i(s) H(s).
\end{align}
We will apply Proposition~\ref{prop:dir_to_arit}. We note that each application of Proposition \eqref{eq:dir_to_arit} loses an exceptional set of $O(R')$ values of $m$, where $R'$ is the $R$-parameter in Proposition~\ref{prop:dir_to_arit}. As we will be applying the proposition more than $O(1)$ times, namely for $S^{o(1)}$ sums obtained by different choices of parameters in (1)--(6) in Lemma \ref{lem:mod_cross}, we have to take $R'$ slightly smaller than $R$ defined as in \eqref{eq:def-R}. The choice $R' = Rx^{-\nu/2}$ works. By redefining $\nu$ as $\nu/2$, we may talk about applying Proposition~\ref{prop:dir_to_arit} with the value of $R$ defined in \eqref{eq:def-R}.
Hence, our task is to show~\eqref{eq:dir_to_arit}. We have the following information on our polynomials (see Lemma \ref{lem:mod_cross}).
\begin{information}
\label{info}
The polynomials in \eqref{def:f} satisfy the following properties.
\begin{itemize}
\item For $i \le n$, the product of $N_{i, j}(s), 1 \le j \le J_i$ has length (approximately) equal to $P_i$. (For convenience, from now on we will write $\prod N_{i, j} = P_i$ or $\prod N_{i, j} \in [P_i, 2P_i]$ instead of the precise condition \eqref{eq:prod_interval}.)
\item All of $N_{i, j}(s), n+1 \le i \le n+n'$ have length lying in $[x^{1/4}, z]$ (and thus $n' = 0$ if $x^{1/4} > z$).
\item All of $R_i(s)$ have length bounded by $\min(x^{1/4}, z)$.
\item $H(s)$ is bounded in length by $T^{1+\epsilon}$.
\item Any polynomial is longer than $z_1$.
\item No product of the polynomials is close to $s, s \in \mathcal{S}$.
\item The coefficients of any polynomial are given by one of the functions $\xi_0, \xi_0 \cdot \log, \xi_0 \cdot g, \xi_0 \cdot \mu, 1_{\mathbb{P}}$.
\item The coefficients of any polynomial longer than $L_{\zeta}$ are given by $\xi_0, \xi_0 \cdot \log$ or $\xi_0 \cdot g$.
\item The coefficients of $F(s)$ are supported on the interval
$$[x/2^{\log x / \log z_1 - C}, x2^{\log x / \log z_1 + C}]$$
for some constant $C > 0$. Note that $2^{\log x / \log z_1} = S^{o(1)}$.
\end{itemize}
\end{information}
We remind that performing the Heath-Brown decomposition to any given polynomial is optional (Remark~\ref{rem:optional}). Our aim is to determine sufficient conditions for the lengths of our polynomials so that $F(s)$ satisfies the assumption of Proposition~\ref{prop:dir_to_arit}.
We highlight a particularly important class of Dirichlet polynomials. (Recall the definition of $\xi_0$ from \eqref{eq:def_xi_0} and $g$ from Lemma \ref{lem:HB-dec}.)
\begin{definition}
\label{def:zeta_sum}
A Dirichlet polynomial $P(s)$ is a \emph{zeta sum} if $P \ge L_{\zeta}$ and the coefficients of $P(s)$ are given by one of the functions $\xi_0, \xi_0 \cdot \log$ and $\xi_0 \cdot g$.
\end{definition}
Note that our definition is nonstandard, as zeta sums usually refer to Dirichlet polynomials whose coefficients are given by $1$ or $\log$. However, for our purposes coefficients $\xi_0, \xi_0 \cdot \log$ or $\xi_0 \cdot g$ work essentially as well as coefficients $1$ or $\log$ -- one only needs care when applying Proposition~\ref{prop:HB_MVT}, as the maximum of $\xi_0$ is quite large, in contrast to $1$ or $\log$.
In what follows we will choose $\mathcal{S}$ as in~\eqref{eq:S_choice}. We have included $L_{\zeta}$ in $\mathcal{S}$ mainly for convenience.
\begin{remark}
\label{rem:z=p_n}
We have above reduced showing~\eqref{eq:mod_1} to establishing the assumption of Lemma~\ref{lem:mod_cross}. The procedure adapts with slight modifications to~\eqref{eq:mod_2}. Namely, we replace $z$ with $\max(I_n)$ in the sums. When arriving at Lemma~\ref{lem:mod_ri}, one has the additional cross condition $r_1 < n_{n, 1} \cdots n_{n, 8}$, corresponding to the condition $p \mid r \implies p < p_n$. This cross condition is removed at Lemma~\ref{lem:mod_cross} similarly to the other cross conditions. We obtain that the polynomials $N_{i, j}, i \ge n+1$ and $R_i(s), i \le t$ are shorter than $P_n(s)$.
\end{remark}
\section{Tools for Dirichlet polynomials}
\label{sec:tools}
For $F(s)$ defined as in~\eqref{def:f}, we aim to show that
\begin{align*}
\int_{T_0 \le |t| \le T} |F(it)M(it)| \d t \lll Rx
\end{align*}
for any $M$ as in Proposition~\ref{prop:dir_to_arit}. (Whether or not we succeed depends on the lengths of the factors of $F(s)$ in a fashion we will describe in Section~\ref{sec:ranges}.)
For convenience we shall assume that $t > 0$, the case $t < 0$ being similar. We perform a dyadic decomposition over $t$, and consider $t \in [T_1, 2T_1]$ for some $T_1$ such that $[T_1, 2T_1] \subset [T_0, T]$. Hence our task is to show the following claim.
\begin{claim}
\label{claim}
We have
\begin{align}
\label{eq:target}
\int_{T_1}^{2T_1} |F(it)M(it)|\d t \lll Rx
\end{align}
for any $[T_1, 2T_1] \subset [T_0, T]$, $F(s)$ as in Information \ref{info} and $M(s)$ as in Proposition \ref{prop:dir_to_arit}.
\end{claim}
In this section we provide various tools that are employed in the next section, on the course establishing Claim \ref{claim} in certain easy cases. In the next five subsections we give five tools: Vinogradov-type pointwise bound, bound for the coefficients of the relevant Dirichlet polynomials, a fourth moment estimate for zeta sums, reduction to the case where the polynomials give power-saving bounds when $c = 0.5$, and handling the case with at least two zeta factors.
\subsection{Pointwise bound}
\label{sec:tools_pointwise}
\begin{lemma}
\label{lem:vinogradov}
Let $N \ge z_1$, $N' \le 2N$ and $f \in \{\xi_0, \xi_0 \cdot \log, \xi_0 \cdot g, \xi_0 \cdot \mu, 1_{\mathbb{P}}\}$ be given, with $f \neq \xi_0 \cdot \mu$ if $N' \ge L_{\zeta}$. We have
\begin{align*}
\left|\sum_{N < n \le N'} f(n)n^{-it}\right| \le N\exp(-(\log x)^{1/5})
\end{align*}
for all $|t| \in [T_0, T]$, assuming $x$ is large enough.
\end{lemma}
\begin{proof}
The proof is largely the same as in~\cite[Lemma 11]{HB-V}.
The result for $f = 1_{\mathbb{P}}$ is standard, following from Perron's formula and the Vinogradov-Korobov zero-free region for the zeta function. The result is immediate for $f = \xi_0 \cdot g$ by the sparsity of the support of $g$, noting that $\xi_0(p^e) \in \{0, 1\}$ for $e \ge 2$, $p$ prime.
For the case $f = \xi_0$, it suffices to obtain bounds for $N' < L_{\zeta}$ and $N \ge L_{\zeta}$ separately. In the former case, we have
\begin{align*}
\left|\sum_{N < n \le N'} f(n)n^{-it}\right| \le \sum_{\substack{m \le N' \\ P^+(m) \ge z_1}} \left|\sum_{\substack{N/m < p \le N'/m \\ p \ge P^+(m)}} p^{-it}\right|,
\end{align*}
where $P^+(m)$ denotes the largest prime factor of $m$. The inner sum is empty unless $N'/m \ge z_1$, in which case we have a bound of
$$\frac{N}{m}\exp(-2(\log x)^{1/5})$$
for the inner sum. Summing over $m$ results in a harmless log factor.
In the latter case $N \ge L_{\zeta}$ we have
\begin{align}
\label{eq:vinogradov_long}
\left|\sum_{N < n \le N'} f(n)n^{-it}\right| \le \sum_{d \le z_2} \left|\sum_{N/d < m \le N'/d} m^{-it}\right|.
\end{align}
For the inner sum we have the bound
$$\ll \sqrt{\frac{N}{d}}T_1^{1/6} + \frac{N}{d}T_1^{-1/6}$$
when $T_1 \le |t| \le 2T_1$ (see e.g.~\cite[Theorem 5.11]{titchmarsh}). By summing over $d$ we obtain that \eqref{eq:vinogradov_long} is
$$\ll z_2^{1/2}\sqrt{N}T_1^{1/6} + NT_1^{-1/6}\log z_2.$$
This is sufficient, as
$$z_2^{1/2}\sqrt{N}T_1^{1/6} \ll NT^{-1/12 + \epsilon} \ll Nx^{-\epsilon}$$
and
$$NT_1^{-1/6}\log z_2 \ll N\exp(-2(\log x)^{1/5})\log x$$
for $T_1 \in [T_0, T]$.
The case $f = \xi_0 \cdot \log$ follows similarly by partial summation. For $f = \xi_0 \cdot \mu$, by assumption we have $N' < L_{\zeta}$, and hence the same argument as above goes through.
\end{proof}
Note that since the factors of $F(s)$ have length at least $z_1$, Lemma~\ref{lem:vinogradov} applies to any (non-constant) factor of $F(s)$.
\subsection{Coefficient bound}
\label{sec:tools_coefficient}
\begin{lemma}
\label{lem:coefficients}
\begin{enumerate}[(i)]
\item There is a constant $C = O(1)$ such that the following holds: the coefficients $c_k$ of any product of the polynomials $N_{i, j}(s)$, $R_j(s)$ and $H(s)$ in~\eqref{def:f} are bounded in absolute value by $\tau(k)^C(\log k)^C$.
\item Let $c_k$ be the coefficients of any product of moments of those polynomials $N_{i, j}(s)$, $R_i(s)$ and $H(s)$ whose lengths do not exceed $L_{\zeta}$. Then, for any $k = x^{O(1)}$, we have $|c_k| = \exp(O((\log \log x)^{16})) = S^{o(1)}$.
\end{enumerate}
\end{lemma}
\begin{proof}
For the first claim, note that $R_i(s) = \sum_{r_i \in [R_i, R_i'] \cap \mathbb{P}} r_i^{-s}$ and $[R_i, R_i']$ are pairwise disjoint for $R_i < x^{c/2 - \epsilon}$, so any product of distinct $R_i(s), R_i < x^{c/2 - \epsilon}$ has coefficients lying in $\{0, 1\}$. As the coefficients of $N_{i, j}(s), H(s)$ and $R_i(s), R_i \ge x^{c/2 - \epsilon}$ are bounded by $\tau(k)^{O(1)}(\log k)^{O(1)}$ and there are only $O(1)$ such polynomials, the coefficients of the product have the same property.
For the second claim, under the assumption of the polynomials being shorter than $L_{\zeta}$ their coefficients $c_{k'}$ are bounded by $1_{p \mid k' \implies p \ge z_1}\log k'$. As we are only considering coefficients $c_k$ with $k = x^{O(1)}$, the number $\ell$ of polynomials in the product satisfies $\ell \ll \log x / \log z_1$. Now, given $k$ with $p \mid k \implies p \ge z_1$, the number of ways one can write $k$ as the product of $\ell$ integers is at most
$\tau(k)^{\ell}$. Noting that $\tau(k) = \exp(O(\log x / \log z_1))$, we obtain
$$|c_k| \ll (\log k)^{\ell}\tau(k)^{\ell} = \exp(O((\log \log x)^{16})) = S^{o(1)}.$$
\end{proof}
We note that while a bound of type $\tau(k)^C(\log k)^C$ for coefficients is good enough for most purposes, this bound is slightly problematic when applying Heath-Brown's mean value theorem (Proposition~\ref{prop:HB_MVT}). The reason is that Heath-Brown's result requires a bound on the maximum value of the coefficients (in contrast to many large value theorems which consider the mean square), and $\tau(k)$ has a small mean square but a large maximum (of type $\exp(\log k / \log \log k)$). Hence, we have to be slightly careful and distinguish between cases where $F(s)$ has or does not have zeta factors.
\subsection{Moment estimates}
\label{sec:tools_moment}
This subsection is devoted to obtaining fourth moment estimates for zeta sums. We first discard the case where there is a very long zeta factor.
\begin{lemma}
\label{lem:long_zeta}
Assume that $F(s)$ factorizes as $F(s) = P(s)Z(s)$, where $Z \ge \max(L_{\zeta}, T_1z_2)$ and $Z(s)$ is a polynomial whose coefficients are given by $\xi_0$ or $\xi_0 \cdot \log$. Then Claim~\ref{claim} holds.
\end{lemma}
Here and in what follows, when we say ``$F(s)$ factorizes as $A(s)B(s)$ with $X$'', we mean that one may arrange the factors on the right hand side of \eqref{def:f} as two products $A(s)$ and $B(s)$ so that $X$ holds.
\begin{proof}
This is similar to~\cite[Lemma 12]{HB-V}. The idea is to obtain a good pointwise bound for $F(s)$ via $Z(s)$ and to bound $M(s)$ trivially as $|M(it)| \le R$.
Assume first that the coefficients of $Z(s)$ are given by $1_{(Z, Z']}(n)\xi_0(n)$. Then
$$|Z(s)| = \left|\sum_{Z < n \le Z'} \xi_0(n)n^{-s}\right| \le \sum_{\substack{d < z_2}} \left|\sum_{\substack{Z/d < m \le Z'/m}} m^{-s}\right|.$$
It is well-known that (see e.g.~\cite[Theorem 4.11]{titchmarsh})
\begin{align*}
\sum_{N < n \le M} n^{-1/2 - it} \ll M^{1/2}/|t|
\end{align*}
uniformly for $M \ge N \ge |t|/2$. Hence, by partial summation,
\begin{align*}
\sum_{Z/d < m \le Z'/d} m^{-it} \ll \frac{Z}{dT_1}
\end{align*}
for $t \in [T_1, 2T_1]$, and thus
\begin{align}
\label{eq:Z_bound}
|Z(it)| \ll \frac{Z}{T_1}\log z_2.
\end{align}
Any polynomial in the factorization of $F(s)$ is shorter than $T^{1 + \epsilon}$ (see Information \ref{info}), and hence in particular $P(s)$ is non-constant. Thus the pointwise bound (Lemma~\ref{lem:vinogradov}) applies to $P(s)$, so we obtain
\begin{align*}
|F(it)| \ll S^{\epsilon}\frac{x\exp(-(\log x)^{1/5})}{T_1}\log z_2 \ll \frac{x}{S^{\epsilon}T_1},
\end{align*}
and hence
\begin{align*}
\int_{T_1}^{2T_1} |F(it)M(it)| \d t \ll RT_1 \frac{x}{S^{\epsilon}T_1},
\end{align*}
implying~\eqref{eq:target}.
The case where the coefficients of $Z(s)$ are given by $\xi_0 \cdot \log$ follows similarly using partial summation.
\end{proof}
\begin{lemma}
\label{lem:moments}
Let $M > 0$ and $t_1, \ldots , t_M \in [T_1, 2T_1]$ be such that $|t_i - t_j| \ge 1$ for $i \neq j$. Let $Q > x^{\epsilon}$ and $Q' \le 2Q$ be given, and let
$$Q(s) = \sum_{Q < q \le Q'} \frac{1}{q^s}.$$
Assume $T_1 \ge Q/2$. Then
\begin{align}
\label{eq:moment_sum}
\sum_{m = 1}^M |Q(it_m)|^4 \ll T_1Q^2(\log x)^8
\end{align}
and
\begin{align}
\label{eq:moment_int}
\int_{T_1}^{2T_1} |Q(it)|^4 \d t \ll T_1Q^2(\log x)^8.
\end{align}
\end{lemma}
\begin{proof}
The integral bound~\eqref{eq:moment_int} follows from the bound~\eqref{eq:moment_sum} on the sum by decomposing the integral over intervals of the form $[k, k+1], k \in \mathbb{Z}$, bounding the integrands by their maximums and bounding the contribution of odd and even $k$ separately via the bound~\eqref{eq:moment_sum}. Hence it suffices to establish \eqref{eq:moment_sum}.
By Perron's formula (see e.g. \cite[Lemma 1.1]{harman}) we have
\begin{align*}
Q(it) = \frac{1}{2\pi i} \int_{5/4 - iT_1/2}^{5/4 + iT_1/2} \zeta(s+it) \frac{2^s - 1}{s} Q^s ds + O(E(Q)) + O(E(Q')),
\end{align*}
where $E(q), q \in \{Q, Q'\}$ is bounded by
$$E(q) \ll \sum_{n = 1}^{\infty} \left(\frac{q}{n}\right)^{5/4}\min\left(1, \frac{1}{T|\log(q/n)|}\right).$$
The contribution of $n \ge 2q$ and $n \le q/2$ are bounded by $O(q^{5/4}/T)$. The contribution of $q/2 < n < 2q$ is bounded as in the proof of Proposition~\ref{prop:dir_to_arit}: given $J > 10$, the contribution of $J < |n - q| \le 2J$ is $O(q/T)$, and the contribution of $|n - q| \le 10$ is $O(1)$, from which
$$E(q) \ll \frac{q^{5/4}}{T}\log x + O(1) \ll Q^{1/2},$$
say.
Moving the line of integration into the line $\text{Re}(s) = 1/2$ produces an error of
$$\ll \max_{\frac{1}{2} \le \sigma \le \frac{5}{4}} \frac{Q^{\sigma} \cdot \left|\zeta\left(\sigma + \frac{iT_1}{2} + it\right)\right|}{T_1},$$
which by the convexity bound $|\zeta(\sigma + it')| \ll |t'|^{(1 - \sigma)/2 + \epsilon}$ for $0 \le \sigma \le 1$ (see e.g. \cite[Chapter 5.1]{titchmarsh}) is bounded by
$$T_1^{\epsilon}\left(\frac{Q^{5/4}}{T_1} + \frac{Q^{1/2}}{T_1^{3/4}}\right) = O(Q^{1/2}),$$
say.
Hence, we have
\begin{align*}
|Q(it)| \ll Q^{1/2} \int_{-T_1/2}^{T_1/2} \left|\zeta\left(\frac{1}{2}+i(\tau + t)\right)\right|\frac{1}{1 + |\tau|} d\tau + Q^{1/2}.
\end{align*}
By Hölder's inequality we thus have
\begin{align*}
\sum_{m = 1}^M |Q(it_m)|^4 \ll Q^2\left(T_1 + (\log T_1)^3 \sum_{m = 1}^M \int_{-T_1/2}^{T_1/2} \left|\zeta\left(\frac{1}{2} + i(\tau + t_m)\right)\right|^4 \frac{1}{1 + |\tau|} d\tau\right).
\end{align*}
We note that
\begin{align*}
\int_{-T_1/2}^{T_1/2} \left|\zeta\left(\frac{1}{2} + i(\tau + t_m)\right)\right|^4 \frac{1}{1 + |\tau|} d\tau &\ll \int_{t_m - T_1/2}^{t_m + T_1/2} \left|\zeta\left(\frac{1}{2} + i\tau\right)\right|^4 \frac{d\tau}{1 + |\tau - t_m|} \\
&\ll \int_{T_1/2}^{5T_1/2} \left|\zeta\left(\frac{1}{2} + i\tau\right)\right|^4 \frac{d\tau}{1 + |\tau - t_m|}
\end{align*}
and that for any $\tau \in \mathbb{R}$ we have
$$\sum_{m = 1}^M \frac{1}{1 + |\tau - t_m|} \ll \log T_1,$$
and thus we have
$$\sum_{m = 1}^M |Q(it_m)|^4 \ll Q^2T_1 + Q^2(\log T_1)^4 \int_{T_1/2}^{5T_1/2} \left|\zeta\left(\frac{1}{2} + i\tau\right)\right|^4 d\tau.$$
The fourth moment of the Riemann zeta function (see e.g. \cite[Chapter 7.6]{titchmarsh}) gives a bound of $O(T_1(\log T_1)^4)$ for the integral, giving the result.
\end{proof}
\begin{lemma}
\label{lem:moments_2}
Let $M > 0$ and $t_1, \ldots , t_M \in [T_1, 2T_1]$ be such that $|t_i - t_j| \ge 1$ for $i \neq j$. Let $Z(s) = \sum_{Z < n \le Z'} z_nn^{-s}$ be a polynomial satisfying either
\begin{enumerate}[(i)]
\item $L_{\zeta} \le Z < T_1z_2$ and the coefficients $z_n$ of $Z(s)$ are given by $\xi_0(n)$ or $\xi_0(n)\log(n)$, or
\item $L_{\zeta} \le Z$ and the coefficients $z_n$ of $Z(s)$ are given by $\xi_0(n)g(n)$.
\end{enumerate}
Then
\begin{align*}
\sum_{m = 1}^M |Z(it_m)|^4 \ll T_1Z^2z_2^4
\end{align*}
and
\begin{align*}
\int_{T_1}^{2T_1} |Z(it)|^4 \d t \ll T_1Z^2z_2^4.
\end{align*}
\end{lemma}
\begin{proof}
The second claim follows from the first as in the proof of Lemma~\ref{lem:moments}.
For (i), note that if the coefficients of $Z$ are given by $\xi_0$, we have
\begin{align*}
|Z(s)| = \left|\sum_{Z < n \le Z'} \xi_0(n)n^{-s} \right| \le \sum_{d < z_2} \left|\sum_{Z/d < m \le Z'/d} m^{-s}\right|,
\end{align*}
and by the power-mean inequality and Lemma~\ref{lem:moments} one thus obtains
\begin{align*}
\sum_{m = 1}^M |Z(it_m)|^4 &\le \sum_{m = 1}^M \left(\sum_{d < z_2} \left|\sum_{Z/d < m \le Z'/d} m^{-s}\right|\right)^4 \\
&\ll z_2^3 \sum_{m = 1}^M \sum_{d < z_2} \left|\sum_{Z/d < m \le Z'/d} m^{-s} \right|^4 \\
&\ll z_2^3 \sum_{d < z_2} T_1\left(\frac{Z}{d}\right)^2(\log x)^8 \\
&\ll T_1Z^2z_2^4.
\end{align*}
The case $\xi_0 \cdot \log$ is follows similarly by partial summation.
For (ii), note that $g$ is supported on proper powers of primes and $|\xi_0(p^e)| \le 1$ for prime powers $p^e$, from which the result immediately follows.
\end{proof}
In conclusion, from now on we may assume that the fourth moment bounds of Lemma~\ref{lem:moments_2} apply for zeta sums $Z(s)$: if the coefficients are given by $\xi_0$ or $\xi_0 \cdot \log$ and $Z \ge T_1z_2$, we are already done proving Claim~\ref{claim} by Lemma~\ref{lem:long_zeta}, and otherwise Lemma~\ref{lem:moments_2} applies.
We note that in the case $c = 0.5$ one may replace the $z_2^{O(1)}$ losses in Lemma~\ref{lem:moments_2} by $S^{\epsilon}$ losses.
\begin{lemma}
\label{lem:moments_3}
Assume $c = 0.5$. Let $M > 0, t_1, \ldots , t_M \in [T_1, 2T_1]$ and $Z(s)$ be given, where $|t_i - t_j| \ge 1$ for $i \neq j$, $x^{1/4} \le Z \le T_1z_2$ and the coefficients of $Z(s)$ are given by $\xi_0$, $\xi_0 \cdot \log$ or $\xi_0 \cdot g$. Then
\begin{align*}
\sum_{m = 1}^M |Z(it_m)|^4 \ll T_1Z^2S^{\epsilon}
\end{align*}
and
\begin{align*}
\int_{T_1}^{2T_1} |Z(it)|^4 \d t \ll T_1Z^2S^{\epsilon}.
\end{align*}
\end{lemma}
\begin{proof}
The second claim follows from the first and the case $\xi_0 \cdot g$ is trivial. For the case $\xi_0$, see~\cite[Lemma 13]{HB-V}. We note that our values of $S$ and $\eta$ are different from that of Heath-Brown, but the exact same proof works. The case $\xi_0 \cdot \log$ follows by partial summation.
\end{proof}
\subsection{Power-saving bounds}
\label{sec:tools_power}
We then note that Claim~\ref{claim} holds if at least one of our polynomials gives only little saving over the trivial bound, assuming $c = 0.5$.
\begin{lemma}
\label{lem:power_0.5}
Assume $c = 0.5$. Write $F(s) = Q_1(s) \cdots Q_k(s)$. Let $\mathcal{T}$ denote the set of $t \in [T_1, 2T_1]$ for which there exists at least one $1 \le i \le k$ with $|Q_i(it)| \ge Q_i^{4/5}$. Then
$$\int_{\mathcal{T}} |Q_1(s) \cdots Q_k(s)M(s)| \d t \lll Rx$$
for any $M(s)$ as in Proposition~\ref{prop:dir_to_arit}.
\end{lemma}
\begin{proof}
See~\cite[Section 9]{HB-V}. As with Lemma~\ref{lem:moments_3}, our values of $S$ and $\eta$ are different from those of Heath-Brown, but this changes nothing of improtance. Our value of $R$ is also different, but as the proof is based on the trivial bound $|M(s)| \le R$ and on large value theorems on the polynomials $Q_i(s)$, the proof of this lemma goes through for any value of $R$. Furthermore, while Heath-Brown's Dirichlet polynomials have coefficients $c_k = \xi_0(k)$, we also have the options $c_k = \xi_0(k)\log k$ and $c_k = \xi_0(k)g(k)$. However, the fourth moment estimate of Lemma~\ref{lem:moments_3} applies equally well in all of these cases.
\end{proof}
We note that we could establish a similar lemma when $c = 0.45$ but with $Q_i^{4/5}$ replaced by a larger threshold. However, we will take an approach which will not rely on pointwise bounds (other than Lemma~\ref{lem:vinogradov}).
\subsection{At least two zeta factors}
Recall the definition of zeta factor from Definition \ref{def:zeta_sum}. In this subsection we handle the case where $F(s)$ has at least two zeta factors.
\begin{lemma}
\label{lem:two_zetas}
Assume that $F(s)$ factorizes as $F(s) = P(s)Z_1(s)Z_2(s)$, where $Z_i$ are zeta sums with $Z_i > L_{\zeta}$. Then Claim~\ref{claim} holds.
\end{lemma}
Note that it may be the case that $P(s)$ has a zeta factor.
\begin{proof}
Assume first that $c = 0.45$. If $Z_i \ge T_1z_2$ for some $i \in \{1, 2\}$, we are done by Lemma~\ref{lem:long_zeta}, so assume not. We apply Hölder's inequality, the fourth moment estimate from Lemma~\ref{lem:moments_2} and Heath-Brown's mean value theorem from Proposition~\ref{prop:HB_MVT}. Noting that $RT > R^{7/4}T^{3/4}$ with out choice of parameters and that the coefficients $p_n$ of $P(s)$ satisfy $\max |p_n| \le \exp(\log x / (\log \log x)^{1 - \epsilon}) = x^{o(1)}$ by Lemma~\ref{lem:coefficients} and the classical bound $\tau(k) \ll \exp(O(\log x / \log \log x))$, we have
\begin{align}
\label{eq:two_zeta_0.45}
&\int_{T_1}^{2T_1} |F(it)M(it)| \d t \\
\ll &\left(\int_{0}^{T} |M(it)P(it)|^2 \d t\right)^{1/2}\left(\int_{T_1}^{2T_1} |Z_1(it)|^4 \d t\right)^{1/4}\left(\int_{T_1}^{2T_1} |Z_2(it)|^4 \d t\right)^{1/4} \nonumber \\
\ll &(P^2R^2 + PRTx^{\epsilon})^{1/2}\sqrt{T_1Z_1Z_2}z_2^2 \max |p_n| \nonumber \\
\ll &\frac{Rx\sqrt{T}\exp(\log x / (\log \log x)^{1-\epsilon})}{\sqrt{Z_1Z_2}} + R^{1/2}Tx^{1/2 + \epsilon} \nonumber,
\end{align}
which is acceptable.
If $c = 0.5$, we consider two cases depending on whether $P(s)$ has zeta factors or not. If $P(s)$ has no zeta factors (so its coefficients are bounded by $S^{o(1)}$ by Lemma \ref{lem:coefficients}), we proceed similarly as in \eqref{eq:two_zeta_0.45}, using the stronger fourth moment estimate from Lemma~\ref{lem:moments_3}. We obtain the bound
\begin{align*}
\int_{T_1}^{2T_1} |F(it)M(it)| \d t \ll (P^2R^2 + PRTx^{\epsilon})^{1/2}\sqrt{TZ_1Z_2}S^{o(1)}
\end{align*}
As we have disposed polynomials with length close to $L_{\zeta} \in \mathcal{S}$, we have $Z_i \ge L_{\zeta}x^{\eta}$. Note that $x^{\eta} \gg S$. It follows that $P \le x^{1/2 - \eta/2}$, and hence the above is bounded by
\begin{align*}
S^{o(1)}R\sqrt{TPZ_1Z_2} \cdot \sqrt{P} + x^{\epsilon}T \sqrt{RPZ_1Z_2} \ll
Rx^{1 - \eta/5} + x^{2\epsilon}\sqrt{R}T\sqrt{x},
\end{align*}
which is sufficient.
If $c = 0.5$ and $P(s)$ has a zeta factor $Z_3(s)$, we write $P(s) = Z_3(s)Q(s)$ and use Hölder's inequality to get
\begin{align*}
\int_{T_1}^{2T_1} |F(it)M(it)| \d t \ll I_1^{1/4}I_2^{1/4}I_3^{1/4} \left(\int_{T_1}^{2T_1} |Q(it)M(it)|^4 \d t\right)^{1/4},
\end{align*}
where by Lemma \ref{lem:moments_3}
\begin{align*}
I_i := \int_{T_1}^{2T_1} |Z_i(it)|^4 \d t \ll T_1Z_i^2S^{o(1)}.
\end{align*}
We further bound $|M(it)|^4 \le R^2|M(it)|^2$ and apply Proposition~\ref{prop:HB_MVT} to the polynomial $Q^2$. Note that $Q$ is shorter than $xS^{\epsilon}/Z_1Z_2Z_3 < x^{1/4 - \eta}$ and hence the coefficients of $Q^2$ are bounded by $S^{o(1)}$ by Lemma \ref{lem:coefficients}. We get
\begin{align*}
\int_{T_1}^{2T_1} |F(it)M(it)| \d t &\ll S^{o(1)}T_1^{3/4}\sqrt{RZ_1Z_2Z_3} \left(\int_{T_1}^{2T_1} |Q^2(it)M(it)|^2 \d t \right)^{1/4} \\
&\ll S^{o(1)}T^{3/4}\sqrt{RZ_1Z_2Z_3}\left(Q^4R^2 + x^{\epsilon}Q^2RT\right)^{1/4} \\
&\ll S^{o(1)}T^{3/4}R\sqrt{Qx} + x^{\epsilon}T\sqrt{x}R^{3/4}.
\end{align*}
The first term is small enough by $Q < x^{1/4 - \eta}$ and the second term clearly is small enough.
\end{proof}
\subsection{Writing \texorpdfstring{$F(s) = A(s)B(s)C(s)$}{F(s) = A(s)B(s)C(s)}}
We then take products of the factors of $F(s)$ in order to write $F(s) = A(s)B(s)C(s)$ for some $A(s), B(s), C(s)$. First note that the set of $t$ for which $\min(|A(it)|, |B(it)|, |C(it)|) \le x^{-1}$, say, has a negligible contribution to the integral in~\eqref{eq:target}. We may then partition the rest of $t \in [T_1, 2T_1]$ into $O((\log x)^{3+\epsilon}) = S^{o(1)}$ sets based on the values $\sigma_A, \sigma_B, \sigma_C$ satisfying $|A(it)| \sim A^{\sigma_A}$, $|B(it)| \sim B^{\sigma_B}$ and $|C(it)| \sim C^{\sigma_C}$. Note that we may assume $\sigma_A, \sigma_B, \sigma_C \le 1 - (\log x)^{-4/5}$ by Lemma~\ref{lem:vinogradov} (and $\sigma_A, \sigma_B, \sigma_C \le 4/5$ if $c = 0.5$ by Lemma~\ref{lem:power_0.5}). Given $\sigma_A, \sigma_B, \sigma_C$, we denote the set of such $t$ by $\mathcal{T}_{\sigma}$.
Hence, it suffices to show that
$$\int_{\mathcal{T}_{\sigma}} |A(it)B(it)C(it)M(it)| \d t \lll Rx.$$
We utilize two different strategies for bounding the integral. The first one is the simple bound
\begin{align*}
\int_{\mathcal{T}_{\sigma}} |A(it)B(it)C(it)M(it)| \d t \ll A^{\sigma_A}B^{\sigma_B}C^{\sigma_C}R|\mathcal{T}_{\sigma}|.
\end{align*}
This is sufficient if
\begin{align}
\label{eq:L1-cond}
|\mathcal{T}_{\sigma}| \lll A^{1 - \sigma_A}B^{1 - \sigma_B}C^{1-\sigma_C}.
\end{align}
The second strategy is to apply the Cauchy-Schwarz inequality and Proposition \ref{prop:HB_MVT} to get (recall that $RT > R^{7/4}T^{3/4}$ by our choice of parameters)
\begin{align*}
&\int_{\mathcal{T}_{\sigma}} |A(it)B(it)C(it)M(it)| \d t \\
\ll &\left(\int_{T_0}^T |B(it)M(it)|^2 \d t\right)^{1/2}\left(\int_{\mathcal{T}_{\sigma}} |A(it)C(it)|^2\right)^{1/2} \\
\ll &\left((\max |b_n|^2)\left(B^2R^2 + BRTx^{\epsilon}\right)\right)^{1/2}\left(|\mathcal{T}_{\sigma}|A^{2\sigma_A}C^{2\sigma_C}\right)^{1/2}.
\end{align*}
If $F(s)$ has no zeta factors, so that in particular $\max |b_n|^2 = S^{o(1)}$ by Lemma \ref{lem:coefficients}, Claim \ref{claim} reduces to showing that
\begin{align}
\label{eq:cond}
|\mathcal{T}_{\sigma}| \ll \min\left(S^{-\epsilon}A^{2-2\sigma_A}C^{2-2\sigma_C}, H'RA^{1-2\sigma_A}C^{1-2\sigma_C}\right)
\end{align}
holds for any choice of $\sigma_A, \sigma_B, \sigma_C \le 1 - (\log x)^{-4/5}$ (and $\sigma_A, \sigma_B, \sigma_C \le 4/5$ if $c = 0.5$). Note that we have dropped the $x^{\epsilon}$ loss in the second term of \eqref{eq:cond}, as one may decrease the value of $\nu$ in \eqref{eq:def-R} if necessary.
The case where $F(s)$ has a zeta factor is similar. By Lemma \ref{lem:two_zetas} we may assume there is only one zeta factor, which we choose to be $A(s)$, so that the coefficients of $B$ again satisfy $\max |b_n|^2 = S^{o(1)}$.
Our proofs for \eqref{eq:cond} rely on Huxley's large value theorem.
\begin{lemma}
\label{lem:huxley}
Let $P(s)$ be a Dirichlet polynomial, let $T$ and $\sigma \le 1$ be given and let $V$ denote the measure of $t \in [T, 2T]$ for which $|P(it)| \sim P^{\sigma}$. Write $G = \sum |c_p|^2$, where the sum is over the coefficients $c_p$ of $P(s)$. Then
\begin{align*}
V \lll \left(GP^{1 - 2\sigma} + T\min(GP^{-2\sigma}, G^3P^{1 - 6\sigma})\right).
\end{align*}
\end{lemma}
\begin{proof}
See~\cite[Theorem 9.7 and Corollary 9.9]{IK}.
\end{proof}
When we apply Lemma~\ref{lem:huxley}, $P(s)$ will always be a moment of products of factors of $F(s)$, where those factors are shorter than $L_{\zeta}$, with $P = x^{O(1)}$. Thus Lemma~\ref{lem:coefficients} gives $G \ll PS^{o(1)}$, and Lemma \ref{lem:huxley} implies
\begin{align}
\label{eq:huxley}
|\mathcal{T}_{\sigma}| \ll S^{o(1)}\left(P^{2 - 2\sigma} + T\min(P^{1 - 2\sigma}, P^{4 - 6\sigma})\right).
\end{align}
\section{Ranges of \texorpdfstring{$(A, B, C)$}{(A, B, C)}}
\label{sec:ranges}
In this section we determine certain cases where \eqref{eq:cond} and thus Claim \ref{claim} hold. We first consider the case $c = 0.5$.
\begin{proposition}[Ranges for $c = 0.5$]
\label{prop:range_0.5}
Let $c = 0.5$ and $R = x^{0.07 + \nu}$. Assume $F(s) = A(s)B(s)C(s)$, where $F(s), A(s), B(s)$ and $C(s)$ satisfy at least one of the following conditions:
\begin{itemize}
\item[(i)] $F(s)$ has no zeta factors, $A, B, C \ge z_1$ and $A, B \ge x^{0.43}$.
\item[(ii)] $F(s)$ has no zeta factors, $A, B, C \ge z_1$ and $B < x^{0.43}, AC^{3/5} \le x^{0.57}$ and $A \le x^{0.56}\min(x/C^8, 1)$.
\item[(iii)] $A(s)$ is a zeta sum, $A \ge L_{\zeta}$, $BS^3 \le x^{1/2}$ and $C \le x^{0.32}$.
\end{itemize}
Then Claim~\ref{claim} holds.
\end{proposition}
Note that in (iii) we allow $B(s)$ or $C(s)$ to be constant polynomials (though in the most difficult case $A = x^{1/4 + \epsilon}$ the conditions imply that $B(s)$ and $C(s)$ are non-constant).
The set of all $A \ge B \ge C$ with $ABC = x^{1 + o(1)}$ satisfying (i) or (ii) are illustrated in Figure~\ref{fig:kuva1} in black, the $x$-axis denoting the value of $\log A / \log x$ and the $y$-axis $\log B / \log x$. Note that the length of $C$ is then determined by $ABC = x^{1 + o(1)}$. We further mark the outlines of the five other symmetric cases in the figure, the axes of symmetry denoted by line segments. The triangle corresponds to the region $AB \le x^{1 + o(1)}, A, B \ge 1$.
\begin{figure}
\caption{Set of $(A, B, C)$ covered by Proposition~\ref{prop:range_0.5}
\label{fig:kuva1}
\end{figure}
\begin{proof}
We aim to show~\eqref{eq:cond}. By Lemma~\ref{lem:power_0.5} we may assume $\sigma_A, \sigma_B, \sigma_C \le 4/5$. Our proof is somewhat similar to the proof of \cite[Proposition 3]{HB-V}.
(i): We may assume $A \ge B$. Noting that $AC \le xS^{\epsilon}/B \le H'R$, it suffices to show
\begin{align}
\label{eq:proof_i}
|\mathcal{T}_{\sigma}| \lll A^{2-2\sigma_A}C^{2-2\sigma_C}.
\end{align}
By~\eqref{eq:huxley} we have
$$|\mathcal{T}_{\sigma}| \ll S^{o(1)}\left(A^{2 - 2\sigma_A} + T\min(A^{1 - 2\sigma_A}, A^{4 - 6\sigma_A})\right).$$
If the former term dominates, we are done, as $C \ge z_1$ and $\sigma_C \le 1 - (\log x)^{-4/5}$. Hence we may assume
\begin{align}
\label{eq:proof_iA}
|\mathcal{T}_{\sigma}| \ll S^{o(1)}T\min(A^{1-2\sigma_A}, A^{4-6\sigma_A}).
\end{align}
For any $w \ge 2$ such that $C^w = x^{O(1)}$ we have, by \eqref{eq:huxley},
\begin{align}
\label{eq:proof_iC}
|\mathcal{T}_{\sigma}| \ll S^{o(1)}\left(C^{2w - 2w\sigma_C} + TC^{w - 2w\sigma_C}\right).
\end{align}
(Note the implied constant does not depend on $w$.) We hence have, by taking weighted averages of \eqref{eq:proof_iA} and \eqref{eq:proof_iC},
\begin{align*}
|\mathcal{T}_{\sigma}| &\ll S^{o(1)}\left(TA^{1 - 2\sigma_A}\right)^{1 - 3/2w}\left(TA^{4 - 6\sigma_A}\right)^{1/2w}\left(C^{2w - 2w\sigma_C} + TC^{w - 2w\sigma_C}\right)^{1/w} \\
&\ll S^{o(1)}T^{1 - 1/w}A^{(2w+1)/2w - 2\sigma_A}\left(C^{2 - 2\sigma_C} + TC^{1 - 2\sigma_C}\right).
\end{align*}
Hence \eqref{eq:proof_i} follows if both
\begin{align}
\label{eq:proof_iTarget}
T^{(w-1)/w} \lll A^{(2w-1)/2w} \quad \text{ and } \quad T \lll A^{(2w-1)/2w}C
\end{align}
hold. The former condition may be written as $T^{(2w-2)/(2w-1)} \lll A$.
Let now $w$ be the integer such that $T^{2/(2w+1)} < C \le T^{2/(2w-1)}$. Clearly $C^w = x^{O(1)}$. As $T^{2/(2w+1)}, T^{2/(2w-1)} \in \mathcal{S}$, we then have $T^{2/(2w+1)}S^{10} < C < T^{2/(2w-1)}S^{-10}$, say. Now
\begin{align*}
A \ge \sqrt{F/C} > \sqrt{x/C}S^{-\epsilon} > T^{1 - 1/(2w-1)}S^{\epsilon}
\end{align*}
and
\begin{align*}
A^{(2w-1)/2w}C &\ge \sqrt{ABC}^{(2w-1)/(2w)}C^{\frac{1}{2} + \frac{1}{4w}} \ge \sqrt{x}^{(2w-1)/2w}C^{\frac{1}{2} + \frac{1}{4w}}S^{-\epsilon} \\
&> T^{(2w-1)/2w}C^{(2w+1)/4w}S^{-3} > TS^{\epsilon},
\end{align*}
implying \eqref{eq:proof_iTarget}.
(ii): We have $AC \ge S^{-\epsilon}x/B \ge H'RS^{-2\epsilon}$, so it suffices to show
\begin{align}
\label{eq:proof_ii}
|\mathcal{T}_{\sigma}| \lll H'RA^{1-2\sigma_A}C^{1-2\sigma_C}.
\end{align}
We have, by \eqref{eq:huxley},
\begin{align*}
|\mathcal{T}_{\sigma}| \ll S^{o(1)}\left(A^{2 - 2\sigma_A} + T\min(A^{1 - 2\sigma_A}, A^{4 - 6\sigma_A})\right).
\end{align*}
Consider first the case where $A^{2 - 2\sigma_A}$ dominates. We then have, by using the assumption on $AC^{3/5}$ and the fact $\sigma_C \le 4/5$,
\begin{align*}
|\mathcal{T}_{\sigma}| \ll S^{o(1)}A^{2 - 2\sigma_A} \ll S^{o(1)}A^{1 - 2\sigma_A}\frac{x^{0.57}S^{-\epsilon}}{C^{3/5}} \ll H'R A^{1 - 2\sigma_A}C^{1 - 2\sigma_C}S^{-\epsilon/2}.
\end{align*}
This suffices.
Assume then that $|\mathcal{T}_{\sigma}| \ll S^{o(1)}T\min(A^{1 - 2\sigma_A}, A^{4 - 6\sigma_A})$. Let $w = 4$. Using also \eqref{eq:huxley} to $C^w$ we obtain
\begin{align*}
|\mathcal{T}_{\sigma}| &\ll S^{o(1)}(TA^{1 - 2\sigma_A})^{1 - 3/2w}(TA^{4 - 6\sigma_A})^{1/2w}(C^{2w - 2w\sigma_C} + TC^{w - 2w\sigma_C})^{1/w} \\
&\ll S^{o(1)} T^{1 - 1/w}A^{1 + 1/2w - 2\sigma_A}C^{2 - 2\sigma_C} + TA^{1 + 1/2w - 2\sigma_A}C^{1 - 2\sigma_C} \\
&= S^{o(1)} T^{\frac{3}{4}}A^{\frac{9}{8} - 2\sigma_A}C^{2 - 2\sigma_C} + TA^{\frac{9}{8} - 2\sigma_A}C^{1 - 2\sigma_C}.
\end{align*}
This yields \eqref{eq:proof_ii} assuming
$$T^{\frac{3}{4}}A^{\frac{1}{8}}C \lll H'R \quad \text{and} \quad TA^{\frac{1}{8}} \lll H'R,$$
which hold under our assumptions.
(iii): If $B(s)C(s)$ has a zeta factor, we are done by Lemma~\ref{lem:two_zetas}. Assume this is not the case.
We have
\begin{align*}
|\mathcal{T}_{\sigma}| &\ll S^{o(1)}(TA^{2 - 4\sigma_A})^{1/2}(C^{4 - 4\sigma_C} + TC^{2 - 4\sigma_C})^{1/2} \\
&\ll S^{o(1)}\frac{T^{1/2}}{A}A^{2 - 2\sigma_A}C^{2-2\sigma_C} + S^{o(1)}\frac{T}{AC}A^{2 - 2\sigma_A}C^{2 - 2\sigma_C}.
\end{align*}
This implies \eqref{eq:cond}. Indeed, both of the terms above are dominated by $S^{-\epsilon}A^{2 - 2\sigma_A}C^{2 - 2\sigma_C}$, as $A \ge L_{\zeta}$ and $L_{\zeta} \in \mathcal{S}$ imply $A \ge L_{\zeta}x^{\eta} \ge T^{1/2}S^{2\epsilon}$ and $AC > TS^{2\epsilon}$ follows from $BS^3 \le x^{1/2}$. Both terms are also dominated by $H'RA^{1-2\sigma_A}C^{1-2\sigma_C}$, as $T^{1/2}C < H'RS^{-\epsilon}$ by assumption and $T < H'RS^{-\epsilon}$ by our choice of parameters.
\end{proof}
We then give the following ranges in the case $c = 0.45$.
\begin{proposition}[Ranges for $c = 0.45$]
\label{prop:range_0.45}
Let $c = 0.45$ and $R = x^{0.18 + \nu}$. Assume $F(s) = A(s)B(s)C(s)$, where $F(s), A(s), B(s)$ and $C(s)$ satisfy at least one of the following conditions:
\begin{itemize}
\item[(i)] $F(s)$ has no zeta factors, $A, B, C \ge z_1$, $AC \le H'R$, and for some $w \in \mathbb{Z}_+$ with $C^w = x^{O(1)}$ one has both $T^{2w}S^{w} \le A^{2w-1}C^{2w}$ and $T^{2w-2}S^{w} \le A^{2w-1}$.
\item[(ii)] $F(s)$ has no zeta factors, $A, B, C \ge z_1$, $AC > H'R$, and for some $w \in \mathbb{Z}_+$ with $C^w = x^{O(1)}$ one has $A^{1/2w}S \le H'RT^{-1}$ and $A^{1/2w}CS \le H'RT^{-1 + 1/w}$ and $B^{2w-1} \ge T^{2w-2}/R^{2w-3}$ and $B^{6w-1}C^{4w} \ge T^{6w}/R^{6w-3}$.
\item[(iii)] $A(s)$ is a zeta sum, $A \ge L_{\zeta}$, $BS^3 \le H'$ and $CS \le H'RT^{-1/2}$.
\end{itemize}
Then Claim~\ref{claim} holds.
\end{proposition}
Items (i) and (iii) are analogous to Proposition~\ref{prop:range_0.5}, with the proof of part (ii) requiring more work. For a given $(A, B, C)$, one should take $w$ so that $C^w$ is approximately $T$.
Figure~\ref{fig:kuva2} illustrates (an approximation of) the regions encompassed by (i) and (ii) (when $\nu \approx 0$). One sees that the regions are much more complicated than in the case $c = 0.5$. Furthermore, there are now six connected components instead of three. These matters make the task of finding suitable decompositions of $F(s)$ more difficult.
\begin{figure}
\caption{Set of $(A, B, C)$ covered by Proposition~\ref{prop:range_0.45}
\label{fig:kuva2}
\end{figure}
\begin{proof}
The proofs of (i) and (iii) follow from the proofs of the corresponding parts of Proposition~\ref{prop:range_0.5}. We are left with proving (ii).
Since $AC > H'R$, one of \eqref{eq:L1-cond} and \eqref{eq:cond} follows once we show
\begin{align}
\label{eq:proof_iiT}
|\mathcal{T}_{\sigma}| \lll \max\left(A^{1-\sigma_A}B^{1-\sigma_B}C^{1-\sigma_C}, H'RA^{1-2\sigma_A}C^{1-2\sigma_C}\right),
\end{align}
which then implies Claim~\ref{claim}.
We utilize Huxley's large value theorem \eqref{eq:huxley} to the polynomials $A, B$ and $C^w$, obtaining the bounds
\begin{align*}
|\mathcal{T}_{\sigma}| &\ll S^{o(1)}\left(A^{2-2\sigma_A} + T\min(A^{1-2\sigma_A}, A^{4-6\sigma_A})\right), \\
|\mathcal{T}_{\sigma}| &\ll S^{o(1)}\left(B^{2-2\sigma_B} + TB^{4-6\sigma_B}\right) \\
|\mathcal{T}_{\sigma}| &\ll S^{o(1)}\left(C^{2w - 2w\sigma_C} + T\min(C^{w - 2w\sigma_C}, C^{4w - 6w\sigma_C})\right).
\end{align*}
We will consider separate cases according to which terms in these bounds dominate. We use the shorthand $T\min(P^k)$ for $T\min(P^{k - 2k\sigma_P}, P^{4k - 6k\sigma_P})$.
First, assume that we have $|\mathcal{T}_{\sigma}| \ll S^{o(1)}T\min(A)$. This implies
\begin{align*}
|\mathcal{T}_{\sigma}| &\ll S^{o(1)}\left(TA^{1 - 2\sigma_A}\right)^{1 - 3/2w}\left(TA^{4-6\sigma_A}\right)^{1/2w}\left(C^{2w-2w\sigma_C} + TC^{w - 2w\sigma_C}\right)^{1/w} \\
&= S^{o(1)}T^{1 - 1/w}A^{1 + 1/2w - 2\sigma_A}\left(C^{2 - 2\sigma_C} + T^{1/w}C^{1 - 2\sigma_C}\right),
\end{align*}
and \eqref{eq:proof_iiT} follows if $A^{1/2w}CS^{\epsilon} \ll H'RT^{-1 + 1/w}$ and $A^{1/2w}S^{\epsilon} \ll H'RT^{-1}$.
Hence, we may from now on assume $|\mathcal{T}_{\sigma}| \ll S^{o(1)}A^{2 - 2\sigma_A}$. If $|\mathcal{T}_{\sigma}| \ll S^{o(1)}B^{2 - 2\sigma_B}$, then we have, by weighted averages and $\sigma_C \le 1 - (\log x)^{-4/5}$,
\begin{align*}
|\mathcal{T}_{\sigma}| \ll S^{o(1)}(A^{2-2\sigma_A})^{1/2}(B^{2-2\sigma_B})^{1/2} \ll S^{-\epsilon}A^{1-\sigma_A}B^{1-\sigma_B}C^{1-\sigma_C},
\end{align*}
implying \eqref{eq:proof_iiT}. Hence from now on we may also assume $|\mathcal{T}_{\sigma}| \ll S^{o(1)}TB^{4 - 6\sigma_B}$.
There are two cases to check, one where $|\mathcal{T}_{\sigma}| \ll S^{o(1)}C^{2w-2w\sigma_C}$ and one where $|\mathcal{T}_{\sigma}| \ll S^{o(1)}T\min(C^w)$.
Consider first the former case. To show \eqref{eq:proof_iiT} it suffices to show that the system
\begin{align*}
\begin{cases}
A^{2 - 2\sigma_A} &> A^{1 - \sigma_A}B^{1 - \sigma_B}C^{1 - \sigma_C}S^{-\epsilon} \\
A^{2 - 2\sigma_A} &> H'RA^{1 - 2\sigma_A}C^{1 - 2\sigma_C}S^{-\epsilon} \\
TB^{4 - 6\sigma_B} &> A^{1 - \sigma_A}B^{1 - \sigma_B}C^{1 - \sigma_C}S^{-\epsilon} \\
C^{2w - 2w\sigma_C} &> H'RA^{1 - 2\sigma_A}C^{1 - 2\sigma_C}S^{-\epsilon}
\end{cases}
\end{align*}
of inequalities has no solution in reals $\sigma_A, \sigma_B, \sigma_C$, when $\epsilon > 0$ is small enough. (Note that, after taking logarithms, this is a system of linear inequalities.) We first eliminate $\sigma_A$, by plugging the first inequality into the third and fourth. It follows that any solution to the above must also be a solution to the system
\begin{align*}
\begin{cases}
1 &> \frac{H'R}{AC}C^{2 - 2\sigma_C}S^{-\epsilon} \\
TB^{4 - 6\sigma_B} &> B^{2 - 2\sigma_B}C^{2 - 2\sigma_C}S^{-2\epsilon} \\
C^{2w - 2w\sigma_C} &> \frac{H'R}{AC}B^{2 - 2\sigma_B}C^{4 - 4\sigma_C}S^{-3\epsilon}.
\end{cases}
\end{align*}
We raise the first inequality to power $w - 5/2$, the second one to power $1/2$ and multiply all of the three inequalities together. We obtain
\begin{align*}
(TB^{4 - 6\sigma_B})^{1/2}C^{2w - 2w\sigma_C} > \left(\frac{H'R}{AC}\right)^{w - 3/2}B^{3 - 3\sigma_B}C^{2w(1 - \sigma_C)}S^{-(w+3/2)\epsilon}.
\end{align*}
Simplifying we obtain
$$T^{1/2}B > \left(\frac{H'R}{AC}\right)^{w - 3/2}S^{-(w+3/2)\epsilon}.$$
Note that $H'R/(AC) = S^{o(1)}H'R/(x/B) \ge SBR/T$. Hence no solutions exist if
$$T^{1/2}B \le \left(\frac{BR}{T}\right)^{w - 3/2},$$
which finally rearranges to
$$B \ge \frac{T^{(2w-2)/(2w-1)}}{R^{(2w-3)/(2w-1)}}.$$
Consider then the latter case where $|\mathcal{T}_{\sigma}| \ll S^{o(1)}T\min(C^w)$. We again reduce to a system of linear inequalities, namely
\begin{align*}
\begin{cases}
A^{2 - 2\sigma_A} &> A^{1 - \sigma_A}B^{1 - \sigma_B}C^{1 - \sigma_C}S^{-\epsilon} \\
A^{2 - 2\sigma_A} &> H'RA^{1 - 2\sigma_A}C^{1 - 2\sigma_C}S^{-\epsilon} \\
TB^{4 - 6\sigma_B} &> A^{1 - \sigma_A}B^{1 - \sigma_B}C^{1 - \sigma_C}S^{-\epsilon} \\
TC^{4w - 6w\sigma_C} &> H'RA^{1 - 2\sigma_A}C^{1 - 2\sigma_C}S^{-\epsilon}.
\end{cases}
\end{align*}
As before, we eliminate $\sigma_A$, and obtain
\begin{align*}
\begin{cases}
1 &> \frac{H'R}{AC}C^{2 - 2\sigma_C}S^{-\epsilon} \\
TB^{4 - 6\sigma_B} &> B^{2 - 2\sigma_B}C^{2 - 2\sigma_C}S^{-2\epsilon} \\
TC^{4w - 6w\sigma_C} &> \frac{H'R}{AC}B^{2 - 2\sigma_B}C^{4 - 4\sigma_C}S^{-3\epsilon}.
\end{cases}
\end{align*}
Similarly to before, by raising the first inequality to power $3w-5/2$, the second to $1/2$ and multiplying all of the resulting inequalities together on obtains, after applying $H'R/(AC) \ge SBR/T$ and rearrangement, that no solutions exist if
\begin{align*}
B^{6w-1}C^{4w} \ge \frac{T^{6w}}{R^{6w-3}}.
\end{align*}
\end{proof}
In the case $C$ is short (namely shorter than $x^{0.08-\epsilon}$), we give a simple approximation of the range of $(A, B, C)$ covered by (i) and (ii) in Proposition~\ref{prop:range_0.45}. The idea is that the regions in Figure~\ref{fig:kuva2} are well approximated as the region between two lines when one of the polynomials is short. This description is easier to work with when proving certain theoretical results in Section~\ref{sec:0.45}.
\begin{lemma}[Ranges for $c = 0.45$, simple approximation]
\label{lem:range_simple}
Let $c = 0.45$ and assume $R = x^{0.18 + \nu}$. Write $F(s) = A(s)B(s)C(S)$, and assume $F(s)$ has no zeta factors and that $(A, B, C)$ satisfies $A, B, C \ge z_1$, $C \le x^{0.08-\epsilon}$ and
$$x^{0.37+\epsilon}C^{-1/5} \le B \le x^{0.45-\epsilon}C^{-1/2}.$$
Then either (i) or (ii) of Proposition~\ref{prop:range_0.45} is satisfied.
\end{lemma}
The exponent $0.37$ comes from $T/R \approx x^{0.37}$ and $0.08$ comes from $H'R/T \approx x^{0.08}$.
\begin{proof}
We first consider the case $AC \le H'R$. We aim to find an integer $w \in \mathbb{Z}_+, w = O(1)$ so that (i) of Proposition~\ref{prop:range_0.45} is satisfied, i.e.
\begin{align}
\label{eq:proof_simple}
T^{2w}S^w \le A^{2w-1}C^{2w} \quad \text{and} \quad T^{2w-2}S^w \le A^{2w-1}.
\end{align}
Note that
\begin{align*}
A^{2w-1}C^{2w} = (AC)^{2w-1}C > (x^{0.55+\epsilon/2}C^{1/2})^{2w-1}C = x^{(0.55+\epsilon/2)(2w-1)}C^{(2w+1)/2}
\end{align*}
and
\begin{align*}
A^{2w-1} > x^{(0.55+\epsilon/2)(2w-1)}C^{-(2w-1)/2}.
\end{align*}
Recalling that $T = x^{0.55}S^2$, \eqref{eq:proof_simple} follows if $w$ satisfies
\begin{align*}
C^{(2w-1)/2} \le x^{0.55} \le C^{(2w+1)/2}.
\end{align*}
Such an integer $w$ clearly exists.
We then consider the case $AC > H'R$. We aim to find $w$ such that (ii) of Proposition~\ref{prop:range_0.45} is satisfied. We restrict our search to $w \ge 4$, in which case we have
\begin{align*}
A^{1/2w}S < S^2(x/BC)^{1/2w} \le (x^{1 - 0.37})^{1/8} < x^{0.08 - \epsilon} < H'R/T
\end{align*}
and
\begin{align*}
A^{1/2w}CS \le x^{1/2w}x^{0.08-\epsilon} < T^{1/w}\frac{H'R}{T}x^{-\epsilon},
\end{align*}
so the first two conditions of Proposition~\ref{prop:range_0.45}(ii) are satisfied. Hence, we are left with finding $w \ge 4$ such that
\begin{align*}
B^{2w-1} \ge T^{2w-2}/R^{2w-3} \quad \text{and} \quad B^{6w-1}C^{4w} \ge T^{6w}/R^{6w-3}.
\end{align*}
Writing $C = x^\alpha$ and using $B \ge x^{0.37 + \epsilon - \alpha/5}$ and $R \ge x^{0.18}$, the first inequality is satisfied if
\begin{align*}
(2w-1)(0.37 - \alpha/5) &> 0.37(2w-2) + 0.18 \Leftrightarrow \\
\frac{2\alpha}{5}w &< 0.37 - 0.18 + \frac{\alpha}{5}
\end{align*}
and the second one if
\begin{align*}
(6w-1)(0.37 - \alpha/5) + 4w\alpha &> 0.37 \cdot 6w + 0.54 \Leftrightarrow \\
\left(4 - \frac{6}{5}\right)\alpha w &> 0.37 + 0.54 - \frac{\alpha}{5}.
\end{align*}
It follows that $w$ satisfies both of these inequalities if
$$\frac{0.325}{\alpha} - \frac{1}{14} < w < \frac{0.475}{\alpha} + \frac{1}{2}$$
To show that there is an integer solution for $w$, it suffices to check that the difference between the upper and lower bounds is greater than one. This indeed is the case, as
$$\frac{0.15}{\alpha} + \frac{8}{14} \ge \frac{0.15}{0.08} + \frac{8}{14} > 1.$$
Finally, note that there exists a solution with $w \ge 4$, since
$$\frac{0.475}{\alpha} + \frac{1}{2} \ge \frac{0.475}{0.08} + \frac{1}{2} > 4.$$
\end{proof}
\section{Applying Harman's sieve: \texorpdfstring{$c = 0.5$}{c = 0.5)}}
\label{sec:0.5}
We have above established that we may obtain an asymptotic
\begin{align}
\label{eq:0.5_asy}
\sum_{\substack{p_1, \ldots , p_n \\ p_i \in I_i \\ p_n < \ldots < p_1}} S(\mathcal{A}_{p_1 \cdots p_n}(m), z) = \frac{\delta_0}{\delta_1}\sum_{\substack{p_1, \ldots , p_n \\ p_i \in I_i \\ p_n < \ldots < p_1}} S(\mathcal{B}_{p_1 \cdots p_n}(m), z) + o\left(\frac{\delta_0 x}{\log x}\right)
\end{align}
(for all but $O(R)$ values of $m$) under certain assumptions. Namely, we assume that the polynomial
$$F(s) = P_1(s) \cdots P_n(s)Q(s)H(s),$$
may be written as $F(s) = A(s)B(s)C(s)$, where $A, B, C$ satisfy the conditions of Proposition~\ref{prop:range_0.5}. Here $P_i(s)$ corresponds to the sum over $p_i$ and hence $P_i \in I_i$. If one wishes, one may apply the Heath-Brown decomposition to $P_i(s)$. Furthermore, $Q(s)$ is a product of polynomials shorter than $z$ and $H(s)$ is a polynomial of length bounded by $T^{1+\epsilon}$. In the case $z > L_{\zeta}$ one may also decompose $Q(s)$ by the Heath-Brown decomposition.
Our ultimate aim is to show that, for some constant $d > 0$, we have
$$S(\mathcal{A}(m), 2\sqrt{x}) \ge d\frac{\delta_0}{\delta_1} S(\mathcal{B}(m), 2\sqrt{x})$$
for all but $O(R)$ integers $m$ (recall Lemma \ref{lem:to_buch}). To this end, we utilize our asymptotics of form \eqref{eq:0.5_asy} together with Harman's sieve. Recall the basic idea of Harman's sieve: First, one uses the Buchstab identity to write $S(\mathcal{A}(m), 2\sqrt{x})$ as a linear combination of sums as in the left hand side of \eqref{eq:0.5_asy}. For example, one could write, by two applications of Buchstab's identity,
\begin{align}
\label{eq:harman_example}
S(\mathcal{A}(m), 2\sqrt{x}) &= S(\mathcal{A}(m), z) - \sum_{z \le p < 2\sqrt{x}} S(\mathcal{A}_p, p) \nonumber \\
&= S(\mathcal{A}(m), z) - \sum_{z \le p < 2\sqrt{x}} S(\mathcal{A}_p, z') + \sum_{\substack{z \le p < 2\sqrt{x} \\ z' \le q < p}} S(\mathcal{A}_{pq}, q).
\end{align}
(In practice we often apply Buchstab's identity four or six times.) For some of the sums one may apply asymptotics of the form \eqref{eq:0.5_asy}. Note that the right hand side of \eqref{eq:0.5_asy} is easy to evaluate, as $\mathcal{B}(m)$ is a long interval. For some of the sums one might not have an asymptotic as in \eqref{eq:0.5_asy}. When applying Harman's sieve, one arranges things so that such ``difficult'' sums have a positive sign (such as the first and third sum in \eqref{eq:harman_example}), so that they may be discarded and what remains is a lower bound for $S(\mathcal{A}(m), 2\sqrt{x})$. Of course, one has to be careful to not discard too many of the sums, so that the lower bound is strictly positive. The contribution of discarded terms is called \emph{loss}, normalized so that aim is to keep the loss strictly below $1$.
The problem has been thus reduced to a combinatorial task of finding ranges of $p_1, \ldots , p_n$ such that \eqref{eq:0.5_asy} holds, i.e. that for any choice of the polynomials $P_i(s)$, $Q(s)$ and $H(s)$ a suitable factorization $F(s) = A(s)B(s)C(s)$ may be found, and then applying Buchstab's identity suitably to deduce a lower bound for $S(\mathcal{A}(m), 2\sqrt{x})$.
In Section~\ref{sec:0.5_theoretical} we first present theoretical results covering certain situations where~\eqref{eq:0.5_asy} may be evaluated (mainly in the cases $n \le 2$). For the cases where we have several polynomials and the execution of Harman's sieve we employ a computational procedure presented in Section~\ref{sec:0.5_algorithm}.
\subsection{Theoretical results}
\label{sec:0.5_theoretical}
We start with the main lemma of this section.
\begin{lemma}
\label{lem:MN}
Let $0 \le n \ll 1$, $P_1, \ldots, P_n \ge z_1$ and $z \le x^{0.07 - \epsilon}$ be given. Assume that there is a subset $I \subset \{1, \ldots , n\}$ such that
$$M := \prod_{i \in I} P_i \quad \text{and} \quad N := \prod_{i \not\in I} P_i$$
satisfy $M < x^{1/2 - \epsilon}, Nz < x^{0.32 - \epsilon}$ and $MN < x^{3/4 - \epsilon}$. Then
\begin{align*}
\sum_{\substack{p_1, \ldots , p_n \\ z_1 < p_i \le P_i \\ p_n < \ldots < p_1}} S(\mathcal{A}_{p_1 \cdots p_n}(m), z) = \frac{\delta_0}{\delta_1} \sum_{\substack{p_1, \ldots, p_n \\ z_1 < p_i \le P_i \\ p_n < \ldots < p_1}} S(\mathcal{B}_{p_1 \cdots p_n}(m), z) + o\left(\frac{\delta_0 x}{\log x}\right).
\end{align*}
for all except $O(R)$ integers $m \in [x/H', 3x/H']$.
\end{lemma}
\begin{proof}
Write
$$F(s) = P_1(s) \cdots P_n(s)R_1(s) \cdots R_k(s)H(s)$$
as in \eqref{def:f}, and consider cases according to the length of $H(s)$.
Assume first that $H > L_{\zeta}$. We partition $P_i(s)$ and $R_i(s)$ as $(B', C')$ so that Proposition~\ref{prop:range_0.5}(iii) is satisfied with $(A, B, C) = (H, B', C')$. This is done via the following process: define $B_0 = M, C_0 = N$, and for each $1 \le i \le k$ define $(B_i, C_i)$ either by $(B_i, C_i) = (R_iB_{i-1}, C_{i-1})$ or by $(B_i, C_i) = (B_{i-1}, R_iC_{i-1})$. We claim that at each step we may define $(B_i, C_i)$ so that $B_i \le x^{1/2}S^{-3}$ and $C_i \le x^{0.32}$. By assumption this holds for $i = 0$. For $i \ge 1$, we cannot have both $B_{i-1}R_i > x^{1/2}S^{-3}$ and $C_{i-1}R_i > x^{0.32}$, as this would imply
$$HB_{i-1}C_{i-1}R_i \ge x^{1/4}B_{i-1}C_{i-1}R_i^2x^{-0.07+\epsilon} > x^{1/4 + 1/2 + 0.32 - 0.07 + \epsilon}S^{-3} > x^{1 + \epsilon/2},$$
a contradiction. Executing the process in this manner and choosing $(B', C') = (B_k, C_k)$ allows us to apply Proposition~\ref{prop:range_0.5}(iii).
We may then assume that $F(s)$ has no zeta factors. (Note that we did not decompose the polynomials $P_i(s)$, cf. Remark \ref{rem:optional}.) Note that necessarily $k \ge 1$, as $HMN < x^{1 - \epsilon}$.
Consider first the case $M \ge x^{0.43}$. Construct a pair $(A', B')$ by the following process: Begin with $A_0 = M, B_0 = NH$. At each step $1 \le i < k$, adjoin $R_i$ to the shorter of $A_{i-1}, B_{i-1}$. In the end we must have $B_{k-1} \ge x^{0.43}$. Indeed, this is by construction the case if $A_{k-1} \neq A_0$, and if $A_{k-1} = A_0$, we have $B_{k-1} \ge xS^{-\epsilon}/(MR_k) > x^{0.43+\epsilon/2}$. Furthermore, $A_{k-1} \ge A_0 = M \ge x^{0.43}$. Hence Proposition~\ref{prop:range_0.5}(i) is satisfied with $(A, B, C) = (A', B', R_k)$.
Consider then the case $M < x^{0.43}$. By $Nz < x^{0.32 - \epsilon}$ we have
$$MNHR_k < x^{0.43 + 0.32 + 1/4 - \epsilon/2} = x^{1 - \epsilon/2}$$
and hence $k \ge 2$. We adjoin $R_1$ to $M$, and in general keep adjoining $R_2, \ldots , R_{k-1}$ to $M$ as long as $M < x^{0.43}$. In the end we must have $x^{0.43} \le M \le x^{1/2 - \epsilon}$ as $NHR_k < x^{0.57 - \epsilon/2}$, and we may apply the process of the previous paragraph.
\end{proof}
We obtain the following lemma as an immediate consequence.
\begin{lemma}
\label{lem:SR}
We have
$$S(\mathcal{A}(m), x^{0.07 - \epsilon}) = \frac{\delta_0}{\delta_1}S(\mathcal{B}(m), x^{0.07 - \epsilon}) + o\left(\frac{\delta_0 x}{\log x}\right)$$
for all except $O(R)$ integers $m \in [x/H', 3x/H']$.
\end{lemma}
\begin{proof}
Apply Lemma~\ref{lem:MN} with $n = 0$.
\end{proof}
We now start the task of evaluating $S(\mathcal{A}(m), 2x^{1/2})$. We apply the Buchstab identity twice to obtain
\begin{align}
\label{eq:two_buchs}
S(\mathcal{A}(m), 2x^{1/2}) &= S(\mathcal{A}(m), x^{0.07 - \epsilon}) \nonumber\\
&- \sum_{x^{0.07 - \epsilon} \le p < 2x^{1/2}} S(\mathcal{A}_p(m), x^{0.07 - \epsilon}) \\
&+ \sum_{x^{0.07 - \epsilon} \le q < p < 2x^{1/2}} S(\mathcal{A}_{pq}(m), q).\nonumber
\end{align}
By Lemma~\ref{lem:SR}, we have an asymptotic for the first term (for all but $O(R)$ exceptional values of $m$).
Next, we dispose of the awkward case $p \approx x^{1/2}$.
\begin{lemma}
\label{lem:awkward}
We have
\begin{align*}
\sum_{x^{1/2 - \epsilon} \le p < 2x^{1/2}} S(\mathcal{A}_p(m), x^{0.07 - \epsilon}) = \frac{\delta_0}{\delta_1} \sum_{x^{1/2 - \epsilon} \le p < 2x^{1/2}} S(\mathcal{B}_p(m), x^{0.07 - \epsilon}) + o\left(\frac{\delta_0 x}{\log x}\right)
\end{align*}
for all except $O(R)$ integers $m \in [x/H', 3x/H']$.
\end{lemma}
\begin{proof}
We apply the Heath-Brown decomposition to the sum over $p$, and thus write
$$F(s) = \left(\prod_{i, j} N_{i, j}(s)\right)R_1(s) \cdots R_k(s)H(s),$$
where $N_{i, j}(s), R_i(s)$ and $H(s)$ are as in Information \ref{info}.
If $H < L_{\zeta}$ and no $N_{i, j}(s)$ is longer than $L_{\zeta}$, we apply Proposition~\ref{prop:range_0.5}(i) with
$$(A, B, C) = (P, R_1 \cdots R_{k-1}H, R_k).$$
If $H \ge L_{\zeta}$, by Lemma~\ref{lem:two_zetas} we are done if some $N_{i, j}(s)$ is longer than $L_{\zeta}$. Assume not. Combine any of $N_{i, j}(s)$ as long as their product is shorter than $L_{\zeta}$. In the end one has two or three polynomials, all shorter than $L_{\zeta}$ (otherwise their product would exceed $L_{\zeta}^2x^{\eta}$). In any case one can partition the polynomials into two sets, so that the product $B(s)$ of the first set satisfies $B < x^{1/2 - \epsilon}$ and the product $C(s)$ of the second set satisfies $C < x^{0.32 - \epsilon}$. The result now follows similarly as in the proof of Lemma~\ref{lem:MN} by adjoining polynomials $R_i(s)$ suitably one-by-one to $B(s)$ or $C(s)$.
If $H < L_{\zeta}$ and some $N_{i, j}(s)$ is longer than $L_{\zeta}$, we swap $H(s)$ and $N_{i, j}(s)$ and apply the argument of the previous paragraph.
\end{proof}
Lemmas~\ref{lem:MN} and~\ref{lem:awkward} together give an asymptotic for the second term on the right hand side of~\eqref{eq:two_buchs}. We further note that in the third term
$$\sum_{\substack{x^{0.07 - \epsilon} \le q < p < 2x^{1/2}}} S(\mathcal{A}_{pq}, q)$$
one may drop the region $p > x^{1/2 - \epsilon}$. The loss caused by this operation is $O(\epsilon)$ (see \eqref{eq:full_loss} below), which is negligible. Hence, our task is to show that the loss arising from the sum
$$\sum_{x^{0.07 - \epsilon} \le q < p < x^{1/2 - \epsilon}} S(\mathcal{A}_{pq}(m), q)$$
is bounded from above by $1 - c'$ for some constant $c' > 0$ independent of $\epsilon$.
Next, we show that there are certain regions with $q > x^{1/4}$ where $S(\mathcal{A}_{pq}, q)$ may be evaluated. Namely, let
\begin{align}
\label{eq:B_1}
B_1 &= \{(x, y) : x > y > \frac{1}{4}+\epsilon, x+y < 0.57-\epsilon, x + 0.7y > 0.43+\epsilon, x - 3y > -0.56+\epsilon\}, \\
B_2 &= \{(x, y) : (1-x-y, y) \in B_1\} \nonumber
\end{align}
and denote $p = x^{\alpha_p}, q = x^{\alpha_q}$. See Figure \ref{fig:kuva3} for an illustration of the regions where $(\alpha_p, \alpha_q) \in B_j$.
\begin{figure}
\caption{Sets $B_1$ and $B_2$.}
\label{fig:kuva3}
\end{figure}
\begin{lemma}
\label{lem:sum_B}
For $i \in \{1, 2\}$, we have
\begin{align*}
\sum_{(\alpha_p, \alpha_q) \in B_i} S(\mathcal{A}_{pq}(m), q) = \frac{\delta_0}{\delta_1} \sum_{(\alpha_p, \alpha_q) \in B_i} S(\mathcal{B}_{pq}(m), q) + o\left(\frac{\delta_0 x}{\log x}\right)
\end{align*}
for all except $O(R)$ integers $m \in [x/H', 3x/H']$.
\end{lemma}
\begin{proof}
The sums count products of three primes, essentially of size $x^{\alpha_p}, x^{\alpha_q}$ and $x^{1 - \alpha_p - \alpha_q}$. Hence, the cases $i = 1$ and $i = 2$ are analogous, and it suffices to consider $i = 1$.
Hence, consider
$$F(s) = P(s)Q(s)R_1(s) \cdots R_k(s)H(s),$$
where $R_i \le Q$ and the pair $(\log_x(P), \log_x(Q))$ lies in $B_1$. We note that in this proof we treat $R_1 \cdots R_kH$ as a single polynomial.
We apply the Heath-Brown decomposition to the polynomial $Q(s)$. Taking products of any two polynomials shorter than $L_{\zeta}$ and noting that $Q(s)$ is shorter than $x^{1/3+\epsilon}$, we may thus consider the case where $Q$ decomposes as the product of at most two polynomials, with a polynomial longer than $L_{\zeta}$ a zeta sum. Assume that we get two polynomials $Q_1, Q_2$ from the decomposition with $Q_1 \ge Q_2$, where possibly $Q_2 = 1$.
If $F(s)$ has at least two zeta factors, we are done by Lemma~\ref{lem:two_zetas}. Assume not.
If $Q_1 \ge L_{\zeta}$, we apply Proposition~\ref{prop:range_0.5}(iii) with
$$A = Q_1, B = R_1 \cdots R_kHQ_2, C = P.$$
Note that $AC \ge x^{1/4}P \ge x^{1/2+\epsilon}$ and hence $B < x^{1/2-\epsilon/2}$, and that $C = P < x^{0.32-\epsilon}$ by the condition $x+y < 0.57 - \epsilon$ in~\eqref{eq:B_1}.
Assume then that $Q_1 < L_{\zeta}$ (and hence $Q_2 > 1$). If $H(s)$ is a zeta sum, we swap $Q_1$ and $H$ and apply the argument above. Hence assume that there are no zeta factors.
We take
$$(A, B, C) = (\max(PQ_1, R_1 \cdots R_kH), \min(PQ_1, R_1 \cdots R_kH), Q_2),$$
and show that (i) or (ii) of Proposition~\ref{prop:range_0.5} holds.
We first note that by assumption $PQ < x^{0.57-\epsilon}$, and thus $R_1 \cdots R_kH > x^{0.43+\epsilon/2}$. Hence, we are done by Proposition \ref{prop:range_0.5}(i) if $PQ_1 > x^{0.43}$, and hence we may assume the contrary. In particular, $B = PQ_1$ and $A = R_1 \cdots R_kH$.
We apply Proposition \ref{prop:range_0.5}(ii). To do so, we have to check that the conditions
\begin{align*}
\frac{x}{PQ}Q_2^{3/5} < x^{0.57-\epsilon/2} \quad \text{ and } \quad \frac{x}{PQ} < x^{0.56-\epsilon/2}\min(1, x/Q_2^8)
\end{align*}
hold. The first one follows by
$$\frac{PQ}{Q_2^{3/5}} \ge PQ^{7/10} > x^{0.43+\epsilon}$$
and the second one follows from
$$PQ > x^{1/2} > x^{0.44+\epsilon/2} \quad \text{and} \quad \frac{PQ}{Q_2^8} \ge PQ^{-3} \ge x^{-0.56 + \epsilon}.$$
\end{proof}
\subsection{Computational procedure}
\label{sec:0.5_algorithm}
As the computations get very laborious to do by hand when the Buchstab identity is applied twice or even four times more, we will from now on rely on computer calculation for bounding the loss. Below we describe the algorithm used for the computation.
Consider the task of bounding the loss arising from
$$\sum_{\substack{(p_1, \ldots, p_n) \in I \\ p_n < \ldots < p_1}} S(\mathcal{A}_{p_1 \cdots p_n}, p_n)$$
for some product of intervals $I \subset [0, 1/2 - \epsilon]^n$. We first cover the set $I$ with a union of boxes
$$\mathcal{B} := [x^{\alpha_1}, x^{\beta_1}) \times \cdots \times [x^{\alpha_n}, x^{\beta_n})$$
for $0 \le \alpha_i < \beta_i \le 1/2 - \epsilon$, and consider the sum over a single box $\mathcal{B}$. We may assume $\beta_i \ge \alpha_{i+1}$, as otherwise the condition $p_{i+1} < p_i$ is not satisfied in $\mathcal{B}$, and that $\alpha_1 + \ldots + \alpha_{n-1} + 2\alpha_n \le 1 + \epsilon$, as otherwise the sum is empty. In practice we will choose the decompositions so that $\beta_i - \alpha_i$ are small (e.g. less than $1/100$) -- we specify the details in the end.
Next, we determine whether the Buchstab identity can be applied twice more. More precisely, the question is whether there exist parameters $z$ and $z_{p_{n+1}}$ so that, writing
\begin{align*}
\sum_{\substack{(p_1, \ldots, p_n) \in \mathcal{B} \\ p_n < \ldots < p_1}} S(\mathcal{A}_{p_1 \cdots p_n}, p_n) &= \sum_{\substack{(p_1, \ldots, p_n) \in \mathcal{B} \\ p_n < \ldots < p_1}} S(\mathcal{A}_{p_1 \cdots p_n}, z) \\
&- \sum_{\substack{(p_1, \ldots, p_n) \in \mathcal{B}, p_{n+1} \\ p_{n+1} < p_n < \ldots < p_1}} S(\mathcal{A}_{p_1 \cdots p_np_{n+1}}, z_{p_{n+1}}) \\
&+ \sum_{\substack{(p_1, \ldots, p_n) \in \mathcal{B}, p_{n+1}, p_{n+2} \\ p_{n+2} < p_{n+1} < p_n < \ldots < p_1}} S(\mathcal{A}_{p_1 \cdots p_n}, p_{n+2}),
\end{align*}
the first and second sums on the right hand side may be evaluated asymptotically. By Lemma~\ref{lem:MN}, one has an asymptotic for the first sum if $\beta_1 + \ldots + \beta_n < 3/4 - \epsilon$ and $\{1, \ldots , n\}$ may be partitioned into two sets $M, N$ such that
$$\sum_{i \in M} \beta_i < 1/2 - \epsilon \quad \text{and} \quad \sum_{i \in N} \beta_i < 0.32 - \epsilon.$$
Moreover, the better bound one has for the sum over $N$, the larger one may take $z$. Similarly, an asymptotic for the second sum is found if the sum $\beta_1 + \ldots + \beta_n + \beta_n$ is less than $3/4 - \epsilon$ and may be partitioned into two subsums smaller than $1/2 - \epsilon$ and $0.32 - \epsilon$, and better bounds for the latter subsum allow one to choose larger values of $z_{p_{n+1}}$.
We apply the Buchstab identity in this way until we can no more or until $n = 6$, after which the benefits from further applications of the identity would be negligible.
The question, then, is whether we have an asymptotic formula for
$$\sum_{\substack{(p_1, \ldots , p_n) \in \mathcal{B} \\ p_n < \ldots < p_1}} S(\mathcal{A}_{p_1 \cdots p_n}, p_n),$$
which corresponds to asking whether the polynomial
$$F(s) = P_1(s) \cdots P_n(s)R_1(s) \cdots R_k(s)H(s)$$
necessarily satisfies Claim~\ref{claim}. We do not utilize the Heath-Brown decomposition to $P_i(s)$ or $R_i(s)$ here. We have $R_i \le P_n$ and $P_i \in [x^{\alpha_i}, x^{\beta_i}]$ for all $i$.
If $H > L_{\zeta}$, we consider whether
$$P_1(s) \cdots P_n(s)R_1(s) \cdots R_k(s)$$
may be written as $B(s)C(s)$ with $B \le x^{1/2 - \epsilon}, C \le x^{0.32 - \epsilon}$, so that Proposition~\ref{prop:range_0.5}(iii) is satisfied. We note that
$$P_1 \le x^{\beta_1}, \ldots, P_n \le x^{\beta_n} \text{ and } R_1 \cdots R_k \le x^{1 - \alpha_1 - \ldots - \alpha_n - 1/4 + \epsilon},$$
and hence a suitable decomposition $(B, C)$ may be found (if one exists) by considering partitions of the multiset
$$\{\beta_1, \ldots , \beta_n, 3/4 - \alpha_1 - \ldots - \alpha_n\}$$
into two multisets and checking whether in any partition the two parts have sums less than $1/2 - \epsilon$ and $0.32 - \epsilon$.
If $H < L_{\zeta}$, there are no zeta factors, and we consider whether $F(s)$ may be written as $(A, B, C)$ so that Proposition~\ref{prop:range_0.5}(i) or (ii) is satisfied. We utilize two strategies.
The first strategy is a crude one, where we combine all of $R_1(s), \ldots , R_k(s)$ and $H(s)$ into one polynomial $Q(s)$, and go through all ways of writing $P_1(s) \cdots P_n(s)Q(s)$ as $A(s)B(s)C(s)$. The number of such ways is bounded by $3^{n+1}$.
The second strategy is slightly more careful, though it requires $\beta_1 + \ldots + \beta_n < 3/4$ so that $k \ge 1$. We perform a casework on the length of $R_1(s)$, combine all of $R_2(s) \cdots R_k(s)H(s)$ into one polynomial $Q(s)$, and check whether a suitable decomposition $A(s)B(s)C(s)$ for $P_1(s) \cdots P_n(s)R_1(s)Q(s)$ may be found for every possible length of $R_1(s)$. The benefit of this strategy is that we have more polynomials and in particular the short polynomial $R_1$ at our disposal.
In any case, we end up considering several decompositions $F(s) = A(s)B(s)C(s)$. Lower and upper bounds on the factors of $F(s)$ yield bounds on the lengths of $A(s), B(s), C(s)$ via the following (trivial) lemma. In the lemma and afterwards we denote lower and upper bounds on the length of $P(s)$ by $P_l$ and $P_u$ so that $P \in [x^{P_l}, x^{P_u}]$.
\begin{lemma}
\label{lem:operations}
\begin{itemize}
\item[(i)] Let $A(s)$ and $B(s)$ be Dirichlet polynomials. Then
$$AB \in [x^{A_l + B_l}, x^{A_u + B_u}].$$
\item[(ii)] Let $A(s), B(s)$ and $C(s)$ be Dirichlet polynomials with $C(s) = A(s)B(s)$. Then
$$B \in [x^{C_l - A_u}, x^{C_u - A_l}].$$
\end{itemize}
\end{lemma}
\begin{proof}
(i): Since $A \ge x^{A_l}$ and $B \ge x^{B_l}$, we have $AB \ge x^{A_l + B_l}$. The upper bound is proven similarly.
(ii): The upper bound follows from $x^{A_l}B \le AB = C \le x^{C_u}$, the lower bound being similar.
\end{proof}
Now, given lower and upper bounds on the lengths $A, B$ and $C$, Proposition~\ref{prop:range_0.5} applies assuming that
\begin{align}
\label{eq:loose_0.5}
&(\text{if } B_u > 0.43 - \epsilon, \text{ then } A_l > 0.43 + \epsilon) \text{ and} \nonumber \\
&(\text{if } B_l < 0.43 + \epsilon, \text{ then } A_u + 0.6C_u < 0.57 - \epsilon \text{ and } A_u < 0.56 + \min(0, 1 - 8C_u) - \epsilon).
\end{align}
In the case we do not have an asymptotic formula, the loss arising from discarding the sum is equal to (see~\cite{BH})
\begin{align}
\label{eq:full_loss}
\int_{x_1 = \alpha_1}^{\beta_1} \int_{x_2 = \alpha_2}^{\beta_2} \cdots \int_{x_n = \alpha_n}^{\beta_n} \omega\left(\frac{1 - x_1 - \ldots - x_n}{x_n}\right) 1_{x_n < \ldots < x_1} \frac{\d x_1 \cdots \d x_n}{x_1 \cdots x_{n-1}x_n^2},
\end{align}
where $\omega$ is the Buchstab function. Discarding the indicator function (which often has no effect, as the differences $\beta_i - \alpha_i$ are small and we have assumed $\beta_i \ge \alpha_{i+1}$) and bounding the integrand by its supremum, we obtain an upper bound of
\begin{align}
\label{eq:loss_bound}
\sup_{u \in J} \omega(u)\frac{1}{\alpha_n}\prod_{i = 1}^n \frac{\beta_i - \alpha_i}{\alpha_i},
\end{align}
where the supremum over $u$ is over the interval
$$J := \left[\frac{1 - \beta_1 - \ldots - \beta_n}{\beta_n}, \frac{1 - \alpha_1 - \ldots - \alpha_n}{\alpha_n}\right].$$
We apply the bounds
\begin{align*}
\omega(u) \le \begin{cases}
0, \qquad \qquad \qquad \ u < 1 \\
\frac{1}{u}, \qquad \qquad \ 1 \le u \le 2 \\
\frac{1 + \log(u-1)}{u}, \quad 2 \le u \le 3 \\
0.565, \qquad \quad 3 < u
\end{cases}
\end{align*}
to bound such supremums. (The first three items here are equalities.)
In practice, beginning from
$$\sum_{x^{0.07} < q < p \le x^{1/2 - \epsilon}} S(\mathcal{A}_{pq}, q),$$
we will decompose the sums over $p$ and $q$ into intervals of the form $[x^{\alpha_i}, x^{\beta_i}]$ with $\beta_i - \alpha_i = 1/3000$. In further applications of the Buchstab identity we will take $\beta_i - \alpha_i = 1/400$.
There are some additional implementation issues not discussed in detail here: In practice it suffices to consider only decompositions $F(s) = A(s)B(s)C(s)$ where $C(s)$ is equal to $P_n(s)$ or $R_1(s)$. Given a box $\mathcal{B} \subset \mathbb{R}^2$, we check whether $\mathcal{B}$ lies in the region $B_1 \cup B_2$ of Lemma~\ref{lem:sum_B} to handle the case $n = 2$. We take $\epsilon = 10^{-9}$ in various lemmas, and in general impose margins of $10^{-9}$ at various situations to avoid mistakes from rounding errors. The interested reader is invited to read the implementation.
The computation takes approximately fifteen minutes on a usual consumer laptop, giving an upper bound of $0.991 < 1$. As one would expect, most of the loss arises when $p$ is large (e.g. the case $p < x^{1/4}$ gives a loss of less than $0.03$). The program prints more detailed information during runtime.
\begin{remark}
\label{rem:easy}
There is an easier way (both computationally and conceptually) to obtain non-rigorous estimates for the loss. Instead of considering intervals of possible polynomial lengths, one takes a sample with the polynomial lengths being, for example, of the form $x^{k/n}, k \in \mathbb{Z}_+$ for some fixed $n$ (e.g. $n = 200$) to approximate the loss. Such a computation suggests that the value of $R$ could be somewhat improved from $x^{0.07}$, but not by much -- it seems to us that reaching the value $R = x^{0.06}$ would require new ideas.
\end{remark}
\section{Applying Harman's sieve: \texorpdfstring{$c = 0.45$}{c = 0.45}}
\label{sec:0.45}
We assume the reader has read Section~\ref{sec:0.5} before reading this section.
The case $c = 0.45$ is largely similar to the case $c = 0.5$. The central differences are that the results of Proposition~\ref{prop:range_0.45} are more complicated than those of Proposition~\ref{prop:range_0.5}, the resulting ranges of $(A, B, C)$ are more disconnected (see Figures~\ref{fig:kuva1} and~\ref{fig:kuva2}) and that we employ the Heath-Brown decomposition. Nevertheless, the modification is relatively straightforward.
We first give necessary theoretical results in Section~\ref{sec:theoretical}, after which we explain the computational procedure used in this case.
\subsection{Theoretical tools}
\label{sec:theoretical}
In this section we present tools which our computational procedure is based on. At many places we need results that rely only on lower and upper bounds on the length of relevant polynomials $P$. We denote these bounds by $P \in [x^{P_l}, x^{P_u}]$. These bounds behave well under multiplication and division, see Lemma~\ref{lem:operations}.
We extend Proposition~\ref{prop:range_0.45}(i) and (ii) to the case where we only have loose bounds on the lengths of polynomials (cf.~\eqref{eq:loose_0.5}). In what follows we write $h = 0.45, t = 0.55$ and $r = 0.18$.
\begin{lemma}
\label{lem:uncertain_typeII}
Let $F(s) = A(s)B(s)C(s)$ and let $\epsilon > 0$ be fixed. Assume $F(s)$ has no zeta factors and $A, B, C \ge z_1$. Denote by $\mathcal{C}_1$ the condition
\begin{align*}
& B_u \le t - r - \epsilon \quad \text{or for some } 1 \le w \le 20 \text{ we have} \\
& [2wt + \epsilon \le (2w-1)A_l + 2wC_l \text{ and } (2w-2)t + \epsilon \le (2w-1)A_l]
\end{align*}
and by $\mathcal{C}_2$ the condition
\begin{align*}
& B_l \ge t-r+\epsilon \quad \text{or for some } 1 \le w \le 20 \text{ we have} \\
& [A_u/2w \le h+r-t-\epsilon \text{ and } A_u/2w + C_u \le h+r-\left(1 - \frac{1}{w}\right)t-\epsilon \text{ and } \\
& (2w-1)B_l \ge (2w-2)t - (2w-3)r + \epsilon \text{ and } (6w-1)B_l + 4wC_l \ge 6wt - (6w-3)r + \epsilon].
\end{align*}
Assuming that both $\mathcal{C}_1$ and $\mathcal{C}_2$ hold, then Claim~\ref{claim} holds.
\end{lemma}
Note that even though we assume upper and lower bounds for $A, B, C$, we still have $F = xS^{o(1)}$ independent of those bounds and that the polynomials $A(s), B(s), C(s)$ are longer than $z_1$ (assuming they are non-constant).
\begin{proof}
This is a direct consequence of Proposition~\ref{prop:range_0.45}(i) and (ii). Note that the $\epsilon$-terms in Lemma~\ref{lem:uncertain_typeII} handle the $S^{O(1)}$-terms in Proposition~\ref{prop:range_0.45}. We have restricted to considering only $w \le 20$ for practical reasons (the exact threshold $20$ being somewhat arbitrary).
\end{proof}
We then give the corresponding result for Proposition~\ref{prop:range_0.45}(iii).
\begin{lemma}
\label{lem:uncertain_typeI}
Let $F(s) = A(s)B(s)C(s)$ and let $\epsilon > 0$ be fixed. Assume that $A_l > t/2 + \epsilon$ and that $A(s)$ is a zeta sum. If
$$B_u \le h-\epsilon \quad \text{and} \quad C_u \le h + r - t/2-\epsilon,$$
then Claim~\ref{claim} holds.
\end{lemma}
\begin{proof} Note that $A > L_{\zeta}$. The result follows from Proposition~\ref{prop:range_0.45}(iii).
\end{proof}
We next note that if $F(s)$ has a zeta factor, then polynomials shorter than $H'^2R/x$ do not cause us problems. With our choice of parameters we have $H'^2R/x = x^{0.08 + o(1)}$.
\begin{lemma}
\label{lem:ignore_short}
Let $A, B', C', P$ be Dirichlet polynomials with $AB'C'P \le xS^{o(1)}$ and $P \le H'^2Rx^{-1}$. Assume that $A > L_{\zeta}$ is a zeta sum and that
$$B'S^3 < H' \quad \text{and} \quad C'S < \frac{H'R}{\sqrt{T}}.$$
Defining $(B_1, C_1) = (B'P, C')$ and $(B_2, C_2) = (B', C'P)$, there is some $i \in \{1, 2\}$ such that
$$B_iS^3 < H' \quad \text{and} \quad C_iS < \frac{H'R}{\sqrt{T}}.$$
\end{lemma}
\begin{proof}
If not, then one has both
$$B'P \ge \frac{H'}{S^3} \quad \text{and} \quad C'P \ge \frac{H'R}{S\sqrt{T}},$$
so that
$$B'C'P \ge \frac{H'^2R}{\sqrt{T}S^4P} \ge \frac{x}{\sqrt{T}S^4},$$
which contradicts $A > L_{\zeta}$ and $AB'C'P \le xS^{o(1)}$.
\end{proof}
Hence, recalling Proposition~\ref{prop:range_0.45}(iii), in the presence of a zeta sum we may ignore polynomials $R_1(s), \ldots , R_k(s)$ assuming $z < x^{0.08 - \epsilon}$.
Our next result concerns the case where the polynomial
$$F(s) = P_1(s) \cdots P_n(s)R_1(s) \cdots R_k(s)H(s)$$
has many short factors $R_i < z$ with $z$ small. Heuristically, one should be able to find a suitable decomposition $F(s) = A(s)B(s)C(s)$ in this case, since having many short polynomials gives one plenty of options for adjusting the lengths $A, B, C$. The next result formalizes this intuition. The result is stronger the longer $R_1(s) \cdots R_k(s)$ is.
\begin{lemma}
\label{lem:MN-2}
Let $Q_1, \ldots , Q_n \ge z_1$ and $H \ge z_1$ be given. Let $\epsilon > 0$ be small and fixed and let $z \le x^{0.061}$. Assume that $Q_1 \cdots Q_nH < x/S$ and that at least one of the following conditions hold:
\begin{itemize}
\item[(i)] There is some subproduct $M(s)$ of $Q_1(s) \cdots Q_n(s)H(s)$ such that
$$M \in [x^{0.37 + \epsilon}, x^{0.45-\epsilon}z^{-1/2}] \cup [x^{0.55+\epsilon}, x^{0.63-\epsilon}z^{-4/5}].$$
\item[(ii)] We have $Q_1 \cdots Q_nH < x^{0.9-3\epsilon}/z$.
\item[(iii)] There is some subproduct $M(s)$ of $Q_1(s) \cdots Q_n(s)H(s)$ and some integer $K \ge 2$ such that
$$Z := \frac{x}{HQ_1 \cdots Q_n}$$
satisfies $Z > z^{K-1}S$ and
$$M \in [x^{0.37+\epsilon}Z^{-(5K-4)/5K}, x^{0.45-\epsilon}z^{-1/2}] \cup [x^{0.55+\epsilon}Z^{-(2K-1)/2K}, x^{0.63-\epsilon}z^{-4/5}].$$
\end{itemize}
Then, assuming
$$F(s) = Q_1(s) \cdots Q_n(s)H(s)R_1(s) \cdots R_k(s)$$
has no zeta factors, $F(s)$ satisfies Claim~\ref{claim} (for any $k \ge 1$, $R_i < z$).
\end{lemma}
As Lemma~\ref{lem:ignore_short} already essentially handles the case where $z$ is small and one has a zeta factor, restricting to the case where $F(s)$ has no zeta factors is not an issue. Note that the condition $Q_1 \cdots Q_nH < x/S$ implies $k \ge 1$. In (iii) the bound $Z > z^{K-1}S$ implies that $k \ge K$, so $K$ is a lower bound on the number of factors $R_i(s)$. In practice the polynomials $Q_i(s)$ correspond to the polynomials $P_i(s)$ or factors arising from applying the Heath-Brown decomposition to them.
\begin{proof}
We aim to write $F(s) = A(s)B(s)C(s)$ so that the conditions of Lemma~\ref{lem:range_simple} are satisfied. Note first that since $k \ge 1$, if a subproduct $M$ as in (i) may be found, we may simply take
$$(A, B, C) \in \left\lbrace\left(M, \frac{F}{MR_1}, R_1\right), \left(\frac{F}{MR_1}, M, R_1\right)\right\rbrace,$$
depending on which of the two intervals in (i) $M$ lies in.
Assume then that we are in the situation of (ii) or (iii). We have
$$Q_1 \cdots Q_n H = \frac{x}{Z},$$
where $Z$ is defined as in (iii), and hence
$$R_1 \cdots R_k \gg \frac{Z}{S^{1/2}}.$$
It follows that $k \ge K$. Note that we may assume $R_1 \ge \ldots \ge R_k$.
We choose $C = R_k$. It suffices to find a subproduct $P$ of $F/R_k$ such that $P$ has length
$$P \in [x^{0.37+\epsilon}R_k^{-1/5}, x^{0.45-\epsilon}R_k^{-1/2}] \cup [x^{0.55+\epsilon}R_k^{-1/2}, x^{0.63-\epsilon}R_k^{-4/5}] =: I_1 \cup I_2.$$
Indeed, if $P$ lies in the former interval, we take $B(s) = P(s)$ and $A(s) = F(s)/(P(s)C(s))$ in Lemma~\ref{lem:range_simple}. In the latter case one takes $B(s) = F(s)/(P(s)C(s))$ and $A(s) = P(s)$.
Let $L(s) = R_1(s) \cdots R_{k-1}(s)$. Since $C \le z \le x^{0.061}$, the lengths of the intervals $I_i$ are $x^{0.08-2\epsilon}C^{-3/10} > x^{0.0615} \ge z.$ Hence, it suffices to find a subset of $Q_1(s), \ldots , Q_n(s), H(s)$ whose product $Q(s)$ satisfies
\begin{align*}
Q \in [x^{0.37+\epsilon}R_k^{-1/5}L^{-1}, x^{0.45-\epsilon}R_k^{-1/2}] \cup [x^{0.55+\epsilon}R_k^{-1/2}L^{-1}, x^{0.63-\epsilon}R_k^{-4/5}] =: J_1 \cup J_2,
\end{align*}
as then one can construct a desired subproduct $P(s)$ of $F(s)/R_k(s)$ by adjoining factors of $L(s)$ to $Q(s)$ one by one until $Q$ lies in $I_1$ or $I_2$.
We note that if $L \ge x^{0.1 + 2\epsilon}$, then $J_1 \cup J_2$ is a single interval of length
$$\frac{x^{0.63-\epsilon}R_k^{-4/5}}{x^{0.37+\epsilon}R_k^{-1/5}L^{-1}} > x^{0.32}$$
It is easy to see that in this case a suitable subproduct $Q$ exists. This gives (ii).
Note that since $R_k(s)$ is the shortest of $R_1(s), \ldots , R_k(s)$, we have
$$R_k^{1/5}L = \frac{R_1 \cdots R_k}{R_k^{4/5}} \ge \frac{R_1 \cdots R_k}{(R_1 \cdots R_k)^{4/5k}} \ge Z^{(5k-4)/5k}$$
and similarly
$$R_k^{1/2}L \ge Z^{(2k-1)/2k}.$$
Hence the union $J_1 \cup J_2$ contains
$$[x^{0.37+\epsilon}Z^{-(5k-4)/5k}, x^{0.45-\epsilon}z^{-1/2}] \cup [x^{0.55+\epsilon}Z^{-(2k-1)/2k}, x^{0.63-\epsilon}z^{-4/5}].$$
As $k \ge K$ and the intervals are the longer the larger $k$ is, by the assumption of (iii) there is a subproduct of $Q_1(s) \cdots Q_n(s)H(s)$ lying in this set, implying the result.
\end{proof}
The next result is used when applying the Heath-Brown decomposition to a polynomial $P(s)$ to bound the lengths of the factors.
\begin{lemma}
\label{lem:HB-dec-lengths}
Let $L_{\zeta} \le P \le 10x^{1/2}$ be given, let $J = O(1)$ and let $N_1(s), \ldots , N_J(s)$ be such that $N_1 \cdots N_J = P$. Assuming that $N_i < L_{\zeta}$ for all $i$, one may partition $\{N_1(s), \ldots , N_J(s)\}$ into two sets $\mathcal{Q}_1, \mathcal{Q}_2$ such that the products $Q_i(s)$ of elements of $\mathcal{Q}_i$ satisfy
$$\max(Q_1, Q_2) \in [\sqrt{P}, \max(L_{\zeta}, P^{2/3})].$$
\end{lemma}
\begin{proof}
We first use a recursive algorithm for reducing the number of factors $N_i(s)$. As long as there exist $i \neq j$ such that $N_iN_j < L_{\zeta}$, replace $N_i(s)$ and $N_j(s)$ by their product $N_i(s)N_j(s)$, reducing the number of polynomials by one. In the end the number of polynomials must be two or three, as otherwise we would have $P < L_{\zeta}$ or $P > L_{\zeta}^2 > 10x^{1/2}$. If there remain two polynomials, we are done. If there remain three polynomials, combine the shortest two of them. The resulting polynomial has length not exceeding $P^{2/3}$.
\end{proof}
For the case $p_2 > L_{\zeta}$ we use the following lemma.
\begin{lemma}
\label{lem:p_2_large}
Let $x^{0.275+\epsilon} \le P_2 \le P_1 \le 10x^{1/2}$ be given with $P_1P_2^2 \le 10x$. Assume that $x/P_1P_2 \ge L_{\zeta}$ and $\min(P_1, x/P_1P_2) \le x^{0.355 - \epsilon}$. Then, assuming that the polynomial $F(s)$ obtained by applying the Heath-Brown decomposition to any polynomials longer than $L_{\zeta}$ has at least one zeta factor, $F(s)$ satisfies Claim~\ref{claim}.
\end{lemma}
\begin{proof}
If $F(s)$ has at least two zeta factors, we are done by Lemma~\ref{lem:two_zetas}. We let $Q(s)$ be the product of factors of $F(s)$ that are not factors resulting from the Heath-Brown decomposition applied to $P_1(s)$ or $P_2(s)$, so that $P_1P_2Q = xS^{o(1)}$. We may assume that $P_1 \le x^{0.355 - \epsilon}$, as the case $Q \le x^{0.355 - \epsilon}$ is symmetric.
If the zeta factor $Z(s)$ arises from decomposing $P_1(s)$, denote by $P_1'(s)$ the remaining polynomial of length $P_1/Z$. We apply Proposition \ref{prop:range_0.45}(iii) with
$$(A, B, C) = (Z, P_1'Q, P_2).$$
Note that $P_1'Q \le xS^{\epsilon}/ZP_2 \le xS^{\epsilon}/x^{0.55+\epsilon} < x^{0.45-\epsilon/2}$ and $P_2 \le x^{0.355 - \epsilon}$ by assumption.
If the zeta factor $Z(s)$ arises from decomposing $P_2(s)$, we similarly as above take $(A, B, C) = (Z, P_2'Q, P_1)$.
If the zeta factor $Z(s)$ is a factor of $Q(s)$, denote the product of the other factors of $Q(s)$ by $Q'(s)$, and take
$$(A, B, C) = (Z, P_2Q', P_1).$$
Now $P_2Q' \le xS^{\epsilon}/ZP_1 \le xS^{\epsilon}/x^{0.55+\epsilon} < x^{0.45-\epsilon/2}$ and $P_1 \le x^{0.355 - \epsilon}$ by assumption.
\end{proof}
For the case where $P_2 \ge x^{0.275+\epsilon}$ and there are no zeta factors we will employ a casework on the lengths of the polynomials arising from the Heath-Brown decomposition.
\subsection{Details of the procedure and results}
\label{sec:comp_results}
We employ Harman's sieve in a similar manner as in Section \ref{sec:0.5}. First, starting from the Buchstab sum $S(\mathcal{A}(m), 2x^{1/2})$, we apply Buchstab's identity twice to get
\begin{align*}
&S(\mathcal{A}(m), 2x^{1/2}) \\
= &S(\mathcal{A}(m), x^{0.06}) - \sum_{x^{0.06} \le p_1 < 2x^{1/2}} S(\mathcal{A}_{p_1}(m), x^{0.06}) + \sum_{x^{0.06} \le p_2 < p_1 \le 2x^{1/2}} S(\mathcal{A}_{p_1p_2}(m), p_2)
\end{align*}
(cf. \eqref{eq:two_buchs}). An asymptotic for the first term on the right hand side is obtained from Lemma~\ref{lem:MN-2}(ii) (if $H < L_{\zeta}$) and Lemma~\ref{lem:ignore_short} (if $H > L_{\zeta}$). We also have asymptotics for the second term: First, apply the Heath-Brown decomposition to $P_1(s)$. In case the resulting polynomial
$$F(s) = \prod_{1 \le i \le J} N_i(s)R_1(s) \cdots R_k(s)H(s)$$
has no zeta factors we may apply Lemma \ref{lem:MN-2}, and in the presence of a zeta factor one sees that the non-zeta factors of $N_1(s) \cdots N_J(s)H(s)$ may be partitioned into $B'(s)C'(s)$ such that $B'(s) < x^{0.45 - \epsilon}, C'(s) < x^{0.355 - \epsilon}$, from which the result follows via $k$ applications of Lemma \ref{lem:ignore_short} and Proposition \ref{prop:range_0.45}(iii).
Hence, the computation starts from
$$\sum_{\substack{x^{0.06} \le p_2 < p_1 < 2x^{1/2}}} S(\mathcal{A}_{p_1p_2}(m), p_2),$$
with the aim of showing that the Buchstab identity can be applied in such a manner that the resulting loss is less than one. As in Section \ref{sec:0.5_algorithm}, we split the sum into sums over $p_1 \in I_1, p_2 \in I_2$ for shorter intervals $I_i$.
We handle the case $p_2 > L_{\zeta}$ separately. In this case we apply the Heath-Brown decomposition to $P_1(s)$, $P_2(s)$ and any potential $R_i(s)$ longer than $L_{\zeta}$. We perform a casework on the lengths of the resulting factors, utilizing Lemma~\ref{lem:HB-dec-lengths} and using Lemma~\ref{lem:p_2_large} to discard the case with zeta factors, then considering ways of combining the factors to polynomials $A(s), B(s), C(s)$ and checking whether any satisfy Lemma~\ref{lem:uncertain_typeII}. An asymptotic is obtained if in all cases a suitable decomposition $F(s) = A(s)B(s)C(s)$ is found. (The loss arising from $p_2 > L_{\zeta}$ is roughly $0.09$ with our choice of parameters below.) From now on, assume that $p_2 < L_{\zeta}$.
We implement a procedure that determines if an asymptotic for
\begin{align*}
\sum_{\substack{p_1, \ldots , p_n \\ p_i \in I_i \\ p_n < \ldots < p_1}} S(\mathcal{A}_{p_1 \cdots p_n}(m), z)
\end{align*}
may be obtained, where $z$ is a function of $x$ or $z = p_n$. In practice $n \ge 2$ and $z$ is either at most $x^{0.06}$ or equal to $p_n$. Once again the problem is determining whether the polynomial
$$F(s) = P_1(s) \cdots P_n(s)R_1(s) \cdots R_k(s)H(s)$$
may be written as $A(s)B(s)C(s)$ so that Proposition \ref{prop:range_0.45} is satisfied, for any polynomials $R_i \le z$ and $H$. We may apply the Heath-Brown decomposition to $P_1(s)$ if we wish.
Our procedure is as follows (recall the notation $P_l, P_u, h, t$ and $r$ from the beginning of Section \ref{sec:theoretical}):
\begin{enumerate}
\item Check if the case $H > L_{\zeta}$ can be handled. (The answer is trivially positive if $(P_1)_l + \ldots + (P_n)_l > 1 - t/2 + \epsilon$. Assume otherwise.)
\begin{enumerate}[(i)]
\item Write $Q(s) = R_1(s) \cdots R_k(s)$. Consider whether $P_1(s) \cdots P_n(s)Q(s)$ may be partitioned as $B(s)C(s)$ as in Lemma~\ref{lem:uncertain_typeI}. By Lemma~\ref{lem:ignore_short}, the polynomial $Q(s)$ may be dropped if $z \le x^{0.06}$.
\item If this fails and $P_1 > L_{\zeta}$, apply the Heath-Brown decomposition to $P_1(s)$. Consider a casework on the lengths of the factors $Q_1(s), Q_2(s)$ of $P_1(s)$ (see Lemma~\ref{lem:HB-dec-lengths}), and in each case consider partitions $B(s)C(s)$ of the polynomial $Q_1(s)Q_2(s)P_2(s) \cdots P_n(s)Q(s)$ and check whether any satisfy Lemma~\ref{lem:uncertain_typeI}. Again, $Q(s)$ may be dropped if $z \le x^{0.06}$.
\end{enumerate}
\item Check if the case $H < L_{\zeta}$ can be handled without applying the Heath-Brown decomposition to $P_1(s)$.
\begin{enumerate}[(i)]
\item Write $Q(s) = H(s)R_1(s) \cdots R_k(s)$ and consider whether $P_1(s) \cdots P_n(s)Q(s)$ may be written as a product $A(s)B(s)C(s)$ satisfying Lemma~\ref{lem:uncertain_typeII}.
\item If not, and $P_1 \cdots P_n < x^{1 - t/2 - \epsilon}$ so that $k \ge 1$, consider cases depending on the size of $H(s)$. In each case, write $Q(s) = R_1(s) \cdots R_k(s)$ and consider partitions of $P_1(s) \cdots P_n(s)H(s)Q(s)$, again checking whether Lemma~\ref{lem:uncertain_typeII} applies. If $z \le x^{0.06}$, it also suffices if some condition of Lemma~\ref{lem:MN-2} is satisfied.
\end{enumerate}
\item Check if the case $H < L_{\zeta}$ can be handled by applying the Heath-Brown decomposition to $P_1(s)$ (assuming $P_1 > L_{\zeta}$).
\begin{enumerate}[(i)]
\item Check the case where $P_1(s)$ outputs a zeta sum. Let $P_1(s)$ output $Q_1(s), Q_2(s)$ with $Q_1 \ge L_{\zeta}$ a zeta sum (and $Q_2(s)$ possibly constant), and write $Q(s) = R_1(s) \cdots R_k(s)H(s)$. Consider decompositions of $Q_2(s)P_2(s) \cdots P_n(s)Q(s)$ as $B(s)C(s)$, and check whether any satisfy the conditions of Lemma~\ref{lem:uncertain_typeI}. The factors $R_1(s), \ldots , R_k(s)$ may be dropped if $z \le x^{0.06}$.
\item Check the case where $P_1(s)$ does not output a zeta sum. Perform a casework on the lengths of the factors $Q_1(s), Q_2(s)$ (see Lemma~\ref{lem:HB-dec-lengths}), write $Q(s) = R_1(s) \cdots R_k(s)H(s)$ and consider decomposition $A(s)B(s)C(s)$ of the polynomial $Q_1(s)Q_2(s)P_2(s) \cdots P_n(s)Q(s)$, checking whether any satisfy Lemma~\ref{lem:uncertain_typeII}.
\end{enumerate}
\end{enumerate}
If (1) fails, we return that an asymptotic cannot be established. Assuming (1) succeeds, we perform step (2), and only if it fails we perform step (3). Success of either (2) or (3) results in finding an asymptotic formula.
If no asymptotic formula is found, the loss is bounded as in~\eqref{eq:full_loss}
and~\eqref{eq:loss_bound}.
Values of $z$ for which we may apply the Buchstab identity twice more are determined by trial and error with the candidates $z = 0.06, 0.059, \ldots , 0.001$.
The intervals $I_i = [x^{\alpha_i}, x^{\beta_i}]$ are chosen so that $\beta_i - \alpha_i = 1/6000$ if $i \le 2$ and $\beta_i - \alpha_i = 1/450$ otherwise, and we take $\epsilon = 10^{-9}$ in various places. The computation gives the upper bound $0.996 < 1$ for the loss. As with $c = 0.5$, the program prints more detailed information on the contribution of different values of $p_1$ on the loss.
\begin{remark}
As in Remark~\ref{rem:easy}, one may approximate the loss by more straightforward means. Such approximations indicate that reaching $R = x^{0.17}$ would require new ideas.
\end{remark}
\section{Applications}
\label{sec:applications}
In this section we discuss the applications of Theorems~\ref{thm:PRF},~\ref{thm:bin} and~\ref{thm:approx} and show how the theorems follow from Theorem~\ref{thm:many}. We remark that likely one could obtain improvements to our results by proving variants of Theorem~\ref{thm:many} for different values of $c$.
We also note that Theorem \ref{thm:0.5} gives an improvement in a recent results of Kosyak, Moree, Sofos and Zhang \cite{KMSZ} on the maximum coefficients of cyclotomic polynomials. The author thanks Moree for pointing this out.
\subsection{Prime-representing functions}
\label{sec:PRF}
A folklore question in number theory is finding simple (non-trivial) functions that generate primes, i.e. functions $f : \mathbb{Z}_+ \to \mathbb{Z}_+$ such that $f(n)$ is a prime for all $n$. Mills~\cite{mills} famously showed that there exists a constant $A > 1$ such that $\lfloor A^{3^n} \rfloor$ is a prime for every $n \in \mathbb{Z}_+$. In short, the idea is to inductively construct a convergent sequence of constants $A_1, A_2, \ldots$ for which $\lfloor A_k^{3^n} \rfloor$ is a prime for any $n \le k$, and take $A = \lim_{k \to \infty} A_k$. The constant $3$ in the exponent arises from there being primes in intervals of the form $[x, x + x^{1 - 1/3}]$ for $x$ large enough, and stronger results on the length of such intervals allow one to reduce the constant $3$.
While we do not know whether there exist primes in intervals of length $x^{1 - 1/2}$, nevertheless Matomäki~\cite{matomaki-PRF} has shown that there exist constants $A > 1$ such that $\lfloor A^{2^n} \rfloor$ is a prime for any $n \ge 1$. The idea is to consider merely almost all intervals $[x, x + x^{1 - 1/2}]$ instead of all of them. With some modification the proof of Mills adapts to this case, assuming one has a strong enough bound for the set of exceptional $x$ for which $[x, x + x^{1 - 1/2}]$ has no or only few primes. Such a result is given by Matomäki in~\cite{matomaki}. Here again one may reduce the constant $2$ assuming one has analogous results for shorter intervals.
An improvement of the constant $2$ has been given by Islam in \cite{islam} by extending the result of Matomäki in~\cite{matomaki} to intervals slightly shorter than $\sqrt{x}$, reducing the constant to $\approx 1.946$.
The bound of Theorem~\ref{thm:0.45}, or Theorem~\ref{thm:many} to be precise, is strong enough that Matomäki's proof adapts to intervals of length $x^{0.45} = x^{1 - 1/(20/11)}$, leading to prime-representing functions of the form $\lfloor A^{(20/11)^n} \rfloor$. Numerically $20/11 \approx 1.8181\ldots$
\begin{proof}[Proof of Theorem~\ref{thm:PRF}]
Our proof follows those given by Mills~\cite{mills} and Matomäki~\cite[Corollary 4]{matomaki-PRF}.
Fix $\alpha \ge 20/11$ and $\epsilon > 0$ small enough. We inductively construct a sequence $p_0, p_1, \ldots$ of primes such that the interval
$$I_n = [p_n^\alpha, (p_n + 1)^\alpha - 1)$$
contains at least $\epsilon p_n^{\alpha-1}/\log p_n$ primes and
$$p_{n+1} \in I_n$$
for all $n \ge 0$. (Here we override the notation in Section \ref{sec:intro}, where $p_n$ denoted the $n$th prime.) Let $S_n = I_n \cap \mathbb{P}$.
Choose $p_0$ as a large prime, and assume we have already constructed $p_0, \ldots , p_n$ as above. We aim to construct $p_{n+1} \in S_n$ so that $[p_{n+1}^\alpha, (p_{n+1} + 1)^\alpha - 1)$ contains many primes. Note that for $p \in S_n$, the intervals
$$[p^\alpha, (p + 1)^\alpha - 1) \subset [p_n^{\alpha^2}, 2p_n^{\alpha^2}]$$
are disjoint and of length $(\alpha + o(1))p^{\alpha-1} = (\alpha + o(1))(p^\alpha)^{(\alpha-1)/\alpha} > (p^\alpha)^{0.45}$. By Theorem~\ref{thm:many} all but $O(p_n^{(0.18 + \epsilon)\alpha^2})$ primes $p \in S_n$ are such that $[p^\alpha, (p+1)^\alpha - 1)$ contains at least $\epsilon p^{\alpha-1}/\log p$ primes. By the induction hypothesis, $|S_n| \ge \epsilon p_n^{\alpha-1}/\log p_n$, which is much larger than $p_n^{(0.18 + \epsilon)\alpha^2}$. Hence, one may choose $p_{n+1}$ as desired.
Now, let
$$a_n = p_n^{\alpha^{-n}} \quad \text{and} \quad b_n = (p_n + 1)^{\alpha^{-n}}.$$
We trivially have $a_n < b_n$, by construction we have $a_{n+1} \ge a_n$ (as $a_{n+1} = p_{n+1}^{\alpha^{-(n+1)}} \ge (p_n^\alpha)^{\alpha^{-(n+1)}} = a_n$), and we have
$$b_{n+1} = (p_{n+1} + 1)^{\alpha^{-(n+1)}} < \left((p_n + 1)^{\alpha}\right)^{\alpha^{-(n+1)}} = (p_n + 1)^{\alpha^{-n}} = b_n.$$
It follows that $a_n$ is a bounded by $b_1$ and increasing. Furthermore, if one defines
$$A = \lim_{n \to \infty} p_n^{\alpha^{-n}},$$
we have, for all $n \ge 0$, $a_n \le A < b_n$ by above and thus $\lfloor A^{\alpha^n} \rfloor = p_n$.
\end{proof}
Similarly to~\cite{matomaki-PRF}, the proof could be generalized to prime-representing functions of the form $\lfloor A^{c_1 \cdots c_n} \rfloor$, where $c_i \ge 20/11$, and one sees that there are uncountably many admissible $A$ for any given $\alpha$ or $c_i$.
\subsection{Binary digits of primes}
In the last years there have been numerous results on primes with restricted digits. Mauduit and Rivat~\cite{mauduit-rivat} showed that the sum of digits function of prime numbers in a given base is equidistributed modulo $m$ for any fixed $m \in \mathbb{Z}_+$ (except in certain trivial cases). Bourgain~\cite{bourgain} has shown that one may prescribe a positive proportion of the binary digits of an integer at arbitrary places and find primes in the resulting set (assuming the final digit has not been set to $0$). Maynard~\cite{maynard} proved that, for any $d \in \{0, 1, \ldots , 9\}$, there are infinitely many primes without the digit $d$ in their decimal representation.
We consider the problem of finding primes with many digits $d$ in their binary representation for a given $d \in \{0, 1\}$. This is similar to the problem considered by Bourgain, differing in that we do not prescribe the places of the digit $d$ in the binary expansion. We note that the corresponding problem for smooth numbers was very recently studied by Hauck and Shparlinski \cite{hauck-shparlinski}.
We first give a useful lemma.
\begin{lemma}
\label{lem:thin_tail}
Fix $d \in \{0, 1\}$. For any $\epsilon > 0$ there exists a constant $c_{\epsilon} > 0$ such that the following holds: The number of integers $n \in [0, 2^k)$ whose binary expansion contains at most $(1/2 - \epsilon)k$ digits $d$ is $O((2^k)^{1 - c_{\epsilon}})$.
\end{lemma}
Note that while the lemma is stated for integers in the interval $[0, 2^k)$, the result may be applied to any $2^k$ consecutive integers, showing that most of those integers have approximately equal amounts of zeros and ones among their final $k$ binary digits.
\begin{proof}
The number of such $n$ is bounded by
$$\sum_{0 \le i \le (1/2 - \epsilon)k} {k \choose i}.$$
Via Striling's approximation one may show that for $i \le (1/2 - \epsilon)k$ we have ${k \choose i} \ll (2^k)^{1 - c_{\epsilon}}$ for some constant $c_{\epsilon} > 0$, from which the result follows.
\end{proof}
We then note that given $\epsilon > 0$ and $x = 2^k$ large enough (in terms of $\epsilon$), there are primes $p < x$ such that at least $(1/2 - \epsilon)k$ of the binary digits of $p$ are ones. Indeed, the number of integers $n < x$ having less than $k(1/2 - \epsilon)$ binary ones is $O(x^{1 - c_{\epsilon}})$ by Lemma \ref{lem:thin_tail} whereas the prime number theorem states that there are roughly $x/\log x$ primes less than $x$.
The $50\% - \epsilon$ bound may be improved by adapting the argument to short intervals. Let us sketch this argument: By~\cite{BHP}, intervals of length $x^{0.525}$ contain $\gg x^{0.525}/\log x$ primes. Consider then the interval
$$I := [2^k - 2^{k \cdot 0.525}, 2^k)$$
for $k$ large. The first $0.475k$ digits of any integer $n \in I$ are ones. Furthermore, by Lemma \ref{lem:thin_tail} there must be primes $p \in I$ such that out of the last $0.525k$ digits of $p$, at least a proportion of $50\% - \epsilon$ are ones. Hence the number of ones is at least
$$0.475k + \left(\frac{0.525}{2} - \epsilon\right)k = (0.7375 - \epsilon)k,$$
i.e. a proportion of $73.75\% - \epsilon$ of the digits are ones. A natural barrier for this method is $75\% - \epsilon$, which is what one would get if one could find primes in intervals of length $x^{1/2 + \epsilon}$.
One may improve the argument by considering merely almost all intervals. As in Section~\ref{sec:PRF}, this requires strong enough quantitative bounds on the size of the exceptional set.
\begin{proof}[Proof of Theorem~\ref{thm:bin}]
Let $k$ be a large enough integer divisible by $20$, let $x = 2^{k-1}$ and denote
$$I = [x, 2x) = [2^{k-1}, 2^k).$$
Any integer $n \in I$ has exactly $k$ binary digits.
Let $\epsilon > 0$ be small enough and let $t = 0.517 + \epsilon$. Let $n_1 < \ldots < n_m$ denote the integers $n \in I$ for which $2^{0.45k} \mid n$ and whose first $0.55k$ digits contain at least $\lfloor tk \rfloor$ instances of the digit $d$. (Recall that $20 \mid k$, so $0.45k$ and $0.55k$ are integers.) Hence
$$m \ge \binom{0.55k-1}{\lfloor tk \rfloor},$$
and from Stirling approximation we obtain
$$m \ge \left(\frac{0.55^{0.55}}{t^{t}(0.55 - t)^{0.55 - t}} - \epsilon\right)^k > 1.1329^k > 2^{(0.18 + \epsilon)k} > x^{0.18 + \epsilon}$$
for $\epsilon > 0$ small enough and $k$ large enough in terms of $\epsilon$. For each $n_i$, consider the interval
$$I_i = [n_i, n_i + 2^{0.45k}).$$
Note that $2^{0.45k} > n_i^{0.45}$ and that $I_i$ are pairwise disjoint. Since $m \ge x^{0.18 + \epsilon}$, by Theorem~\ref{thm:many} there exists $1 \le i \le m$ such that the interval $I_i$ contains $\gg 2^{0.45k}/k$ primes. Consider then the primes $p \in I_i$ for such $i$. By Lemma \ref{lem:thin_tail}, there exist primes $p \in I_i$ such that out of the last $0.45k$ digits of $p$ at least $(0.225 - \epsilon/2)k$ are equal to $d$. As the first $0.55k$ digits of $p$ have at least $tk = (0.517 + \epsilon)k$ digits equal to $d$, in total $p$ has at least
$$(0.517 + 0.225 + \epsilon/2)k = (0.742 + \epsilon/2)k$$
digits equal to $d$.
\end{proof}
We note that the under the Lindelöf hypothesis one may go beyond the $75\% - \epsilon$ barrier: It is known that the Lindelöf hypothesis implies that for any $c > 0$ the number of prime gaps $p_{n+1} - p_n$ longer than $x^c$ is at most $x^{1 - 2c + \epsilon}$ \cite{yu}. Applying this result with $c = 0.4$ in the above argument would allow one to replace $0.55$ by $0.6$ and $t = 0.517 + \epsilon$ by $0.542 + \epsilon$, resulting in a proportion of $76.3\%$ of the digit $d$.
\subsection{Approximation by multiplicative functions}
Harman~\cite{harman-approx} has considered the approximation of real numbers by multiplicative functions. More precisely, for a given real $\alpha > 1$, the aim is to find $n$ with
\begin{align}
\label{eq:harman-approx}
\left|\frac{\sigma(n)}{n} - \alpha\right|
\end{align}
as small as possible, where $\sigma(n) = \sum_{d \mid n} d$ is the sum-of-divisors function. Harman shows that there are infinitely many $n$ for which~\eqref{eq:harman-approx} is smaller than $n^{-0.52}$, improving on a result of Wolke~\cite{wolke}.
Harman~\cite[Theorem 2.2]{harman-approx} shows that if the number of disjoint intervals of length $x^c$ containing few primes up to $x$ is $o(x^{c/(2 - c)}/\log x)$, then~\eqref{eq:harman-approx} may be bounded by $n^{-(1 - c) + \epsilon}$ infinitely often. By a result of Peck~\cite{peck} this is true for $c \approx 0.471$, giving the bound $n^{-0.52}$ above. Theorem~\ref{thm:many} allows one to take $c = 0.45$, as our exponent $0.18 + \epsilon$ is smaller than $0.45/(2 - 0.45) \approx 0.29$. This implies Theorem~\ref{thm:approx}.
\end{document}
|
\begin{document}
\title[The cut loci on ellipsoids]{The cut
loci on ellipsoids and certain Liouville manifolds}
\author{Jin-ichi Itoh}
\address{Department of Mathematics, Faculty of Education,
Kumamoto University, Kumamoto 860-8555, Japan.}
\email{[email protected]}
\author{Kazuyoshi Kiyohara}
\address{Department of Mathematics, Faculty of Science, Okayama University, Okayama 700-8530, Japan.}
\email{[email protected]}
\subjclass[2000]{primary 53C22, secondary 53A05}
\begin{abstract}
We show that some riemannian manifolds
diffeomorphic to the sphere have the property that
the cut loci of general
points are smoothly embedded closed disks of codimension one.
Ellipsoids with distinct axes are typical examples of
such manifolds.
\end{abstract}
\maketitle
\section{Introduction}
On a complete riemannian manifold, any geodesic $\gamma(t)$
starting at a point $\gamma(0)=p$ has the property that
any segment $\{\gamma(t)\mid 0\le t\le T\}$ is minimal,
i.e., the length of the segment is equal to the distance
between the points $p$ and $\gamma(T)$, if $T>0$ is
small.
If the supremum $t_0$ of the set of such $T$ is
finite, then the point $\gamma(t_0)$ is called the {\it cut
point} of $p$ along the geodesic $\gamma(t)$ $(t\ge 0)$.
The {\it cut locus} of the point $p$ is then defined as the set of all cut
points of $p$ along the geodesics starting at $p$.
For the general properties of cut loci, we refer to
\cite{Kl}, \cite{Sa2}.
The study of cut locus was started at 1905 by H. Poincar\'{e} \cite{P}
in the case of convex surfaces, and there are several classical results,
for example, \cite{M}, \cite{We}, \cite{Wh}.
From its definition, the cut locus of a point $p$ on a compact manifold $M$
is homotopically equivalent to $M - \{p\}$,
but it can be very complicated, see \cite{GS}, \cite{I}.
The structure of cut locus was studied in
connection with
the singularity theory, see \cite{Bu}, \cite{Bu1}, \cite{Thom}.
Recently, a property of cut locus was used to solve Ambrose's problem on surfaces
\cite{Heb}, \cite{I}, and it was proved that the distance function to the cut locus has
Lipschitz continuity \cite{IT}, \cite{LN}.
Other applications of cut locus are found in \cite{DO}, \cite{LN} also.
It is well known that the cut locus of any point on the sphere of constant
curvature consists of a single point, and
it is also known that this property
characterizes the sphere of constant curvature (an
affirmatively solved case of the
Blaschke conjecture, see \cite{B}).
However, in most cases, to determine cut loci are quite difficult problems.
There are only a few cases where the cut loci are well
understood; for example, analytic surfaces \cite{M},
symmetric spaces and some homogeneous spaces \cite{H},
\cite{Sa-1}, \cite{Sa0}, \cite{Sa1}, \cite{Ta},
certain surfaces of revolution \cite{GMST}, \cite{ST2}, \cite{T1}, \cite{T2},
Alexandrov surfaces \cite{ShiT},
tri-axial ellipsoids and some Liouville
surfaces \cite{IK1}, \cite{IK2}, \cite{ST1}
(\cite{ST1} is an experimental work).
Especially in higher dimensional case
there are not many results without symmetric spaces and some singular spaces \cite{IV},
even if using computational approximations.
In the earlier paper \cite{IK1}, we proved that the cut
locus of a non-umbilic point on a tri-axial ellipsoid is a
segment of the curvature line containing the antipodal
point, inspired by an experimental work \cite{IS}. Also,
we gave the complete proof of Jacobi's last geometric statement (\cite{J}, \cite{JW}, see also
\cite{Si}, which contains historical remarks).
Furthermore, we have seen in \cite{IK2}
that there are many
surfaces possessing such simple cut loci.
Surfaces we considered in \cite{IK2} are
so-called Liouville surfaces, i.e., surfaces
whose geodesic flows possess
first integrals
which are fiberwise quadratic forms.
In such cases the geodesic equations are
explicitly solved by quadratures.
But, to determine cut loci we needed some
additional conditions, which is satisfied in
the case of ellipsoid.
In the present paper, we shall give a higher dimensional
version of the above-mentioned results. We shall consider
cut loci of points on certain Liouville manifolds
diffeomorphic to $n$-sphere, and prove that the cut locus
of any point is a smoothly embedded, closed
$(n-1)$-disk, if the point does not
belong to a certain submanifold of codimension two. We shall
also prove that the cut locus of a point on that submanifold
is a closed $(n-2)$-disk.
The $n$-dimensional ellipsoids with $n+1$ distinct axes
will be shown to possess such properties.
Here, ``Liouville manifold'' is a
higher dimensional version of Liouville surface, which
we shall explain in the next section.
Now, taking the ellipsoid $M: \sum_{i=0}^n u_i^2/a_i=1$ ($
0<a_n<\dots<a_0$) as an example, let us illustrate our results
in detail. Let $N_k$ and $J_k$ be the submanifolds of $M$ defined by
\begin{gather*}
N_k=\{u=(u_0,\dots,u_n)\in M\ |\ u_k=0\ \}\qquad(0\le k\le n)\\
J_k=\{u\in M\ |\ u_k=0,\quad \sum_{i\ne k}\frac{u_i^2}{a_i-a_k}=1\ \}\qquad (1\le k\le n-1)
\end{gather*}
Then: $N_k$ is totally geodesic, codimension 1; $J_k
\subset N_k$, $J_k$ is diffeomorphic to $S^{k-1}\times
S^{n-k-1}$; $\bigcup_kJ_k$ is the set of points where
some principal curvature with respect to the inclusion
$M\subset \mathbb R^{n+1}$ has multiplicity $\ge 2$;
denoting by $(\lambda_1,\dots,\lambda_n)$ the
elliptic coordinate system on $M$ such that
$a_k\le \lambda_k\le
a_{k-1}$ (see below), we have
\begin{equation*}
N_k=\{\lambda_k=a_k\quad\text{or}\quad\lambda_{k+1}=a_k\ \},
\qquad J_k=\{\lambda_k=\lambda_{k+1}=a_k\ \} .
\end{equation*}
Let us denote by $C(p)$ the cut locus of a point $p\in M$.
Let $(\lambda_1^0,\dots,\lambda_n^0)$ be the elliptic
coordinates of $p$.
Then:
\begin{itemize}
\item[(1)] If $p\not\in J_{n-1}$, then $C(p)$ is an
$(n-1)$-dimensional closed disk which is contained in
a submanifold (possibly with boundary) defined by
$\lambda_n=\lambda_n^0$. Also,
$C(p)$ contains the antipodal point of $p$ in its interior.
For each interior point $q$ of $C(p)$ there are exactly two
minimal geodesics joining $p$ and $q$; the tangent vectors
of those geodesics at $p$ are symmetric with respect to the
hyperplane $d\lambda_n=0$. For each boundary point $q$ of
$C(p)$, there is a unique minimal geodesic from $p$ to $q$,
along which $q$ is the first conjugate point of $p$ with
multiplicity one.
\item[(2)] If $p\in J_{n-1}$, then $C(p)$ is an
$(n-2)$-dimensional closed disk contained in $J_{n-1}$. It
is identical with the cut locus of $p$ in the
$(n-1)$-dimensional ellipsoid $N_{n-1}$. For each interior
point $q$ of $C(p)$ there is an $S^1$-family of minimal
geodesics joining $p$ and $q$; the tangent vectors
of those geodesics at $p$ form a cone whose orthogonal
projection to $T_pJ_{n-1}$ is one-dimensional. For each boundary point $q$ of
$C(p)$, there is a unique minimal geodesic from $p$ to $q$,
and along it $q$ is the first conjugate point of $p$; but the
multiplicity is two in this case.
\end{itemize}
Here, the elliptic coordinate system $(\lambda_1,\dots,\lambda_n)$ on $M$ $(\lambda_n\le \dots\le\lambda_1)$ is defined by the following identity in $\lambda$:
\begin{equation*}
\sum_{i=0}^n\frac{u_i^2}{a_i-\lambda}-1=\frac{\lambda\prod_{k=1}^n(\lambda_k-\lambda)}{\prod_i (a_i-\lambda)} .
\end{equation*}
For a fixed $u\in M$, $\lambda_k$ are determined by $n$
``confocal quadrics'' passing through $u$.
From $\lambda_k$'s, $u_i$ are explicitly described as:
\begin{equation*}
u_i^2=\frac{a_i\prod_{k=1}^n(\lambda_k-a_i)}{\prod_{j\ne i} (a_j-a_i)} .
\end{equation*}
The organization of the paper is as follows.
In \S2 we shall briefly explain Liouville manifolds in the form
what we need.
In \S 3 we shall illustrate how to solve geodesic
equations on a Liouville manifold. Since the geodesic flow is
completely integrable in this case, solutions are given by
integrating a system of closed $1$-forms. In this particular
case, a natural coordinate system provides ``separation
of variables''. This coordinate system is analogous to the
elliptic coordinate system on ellipsoids. In \S4 we shall give
an assumption under which the results on cut loci are
obtained. Some useful inequalities are proved there.
In \S5 basic properties of Jacobi fields and their
zeros are
investigated, which are crucial in the arguments
of the following sections. In \S6 we define a value $t_0(\eta)$ to each unit covector $\eta$, which will indicate
the cut point of the geodesic with initial covector $\eta$.
Then, we prove some preliminary
facts on the behavior of geodesics starting at
a fixed point. The main theorem,
Theorem \ref{thm:cut}, will be stated in \S7 and
proved in \S\S 7-9.
In the forthcoming paper, we shall clarify the
structures of conjugate loci of general points on
certain Liouville manifolds, which will be a higher
dimensional version of ``the last geometric statement
of Jacobi'' explained in \cite{IK1}, \cite{Si}.
\subsection*{Preliminary remarks and notations}
We shall consider geodesic equations in the hamiltonian formulation.
Let $M$ be a riemannian manifold and $g$ its riemannian
metric.
By $\flat:TM\to T^*M$ we denote the bundle isomorphism
determined by $g$ (Legendre transformation). We also use the
symbol $\sharp=\flat^{-1}$.
The canonical 1-form on $T^*M$ is denoted by $\alpha$.
For a canonical coordinate system $(x,\xi)$ on $T^*M$ ($
x$ being a coordinate system on $M$), $\alpha$ is expressed as
$\sum_i\xi_idx_i$. Then the 2-form $d\alpha$
represents the standard
symplectic structure on $T^*M$.
Let $E$ be the function on $T^*M$ defined by
\begin{equation*}
E(\lambda)=\frac12g(\sharp(\lambda),\sharp(\lambda))=
\frac12\sum_{i,j}g^{ij}(x)\xi_i\xi_j
\end{equation*}
We call it the (kinetic) energy function of $M$.
For a function $F, H$ on $T^*M$, we define a vector field
$X_F$ and the Poisson bracket $\{F,H\}$ by
\begin{equation*}
X_F=\sum_i\left(\frac{\partial F}{\partial \xi_i}\frac{\partial}
{\partial x_i}-\frac{\partial F}{\partial x_i}\frac{\partial}
{\partial \xi_i}\right)\, ,\qquad
\{F,H\}=X_FH\,.
\end{equation*}
Then $X_E$ generates the geodesic flow, i.e., the projection of each
integral curve of $X_E$ to $M$ is a geodesic of the riemannian manifold
$M$.
\section{Liouville manifolds}
By definition, Liouville manifold $(M,\mathcal F)$ is a pair
of an $n$-dimensional riemannian manifold $M$ and
an $n$-dimensional vector space
$\mathcal F$ of functions on $T^*M$ such that i) each $F\in \mathcal F$ is
fiberwise a quadratic polynomial; ii) those quadratic forms are
simultaneously normalizable on each fiber; iii) $\mathcal F$ is commutative with respect to the Poisson bracket; and, iv)
$\mathcal F$ contains the hamiltonian of the geodesic flow.
For the general theory of Liouville manifolds, we refer to \cite{Ki2}.
In this paper we only need a subclass of ``compact Liouville
manifolds of rank one and type (A)'', described in \cite{Ki2}.
So, in this section,
we shall briefly explain about it.
Each Liouville manifold treated here is constructed
from $n+1$ constants $a_0>\cdots > a_{n}>0$ and a positive $C^\infty$
function $A(\lambda)$ on the closed interval $a_n\le \lambda\le a_0$.
Let $\alpha_1,\dots,\alpha_n$ be positive numbers defined by
\begin{equation*}
\alpha_i=2\int_{a_i}^{a_{i-1}}\frac{A(\lambda)\ d\lambda}
{\sqrt{(-1)^i\prod_{j=0}^n (\lambda-a_j)}}\qquad
(i=1,\dots,n)
\end{equation*}
Define the function $f_i$ on the circle $\mathbb R/\alpha_i\mathbb Z=
\{x_i\}$ $(1\le i\le n)$
by the conditions:
\begin{gather}\label{eq:fdiff}
\left(\frac{df_i}{dx_i}\right)^2=\frac{(-1)^i4\prod_{j=0}
^n (f_i-a_j)}{A(f_i)^2}\\
f_i(0)=a_i,\ f_i(\frac{\alpha_i}4)=a_{i-1},\quad
f_i(-x_i)=f_i(x_i)=f_i(\frac{\alpha_i}2-x_i)\ .
\end{gather}
Then the range of $f_i$ is $[a_i,a_{i-1}]$.
Put
\begin{equation*}
R=\prod_{i=1}^n(\mathbb R/\alpha_i\mathbb Z)\ .
\end{equation*}
Let $\tau_i$ $(1\le i\le n-1)$ be the involutions
on the torus $R$ defined by
\begin{equation*}
\tau_i(x_1,\dots,x_n)=(x_1,\dots,x_{i-1},-x_i,
\frac{\alpha_{i+1} }2-x_{i+1},x_{i+2},\dots,x_n) \ ,
\end{equation*}
and let $G$ $(\simeq (\mathbb Z/2\mathbb Z)^{n-1})$ be the group of transformations generated by $\tau_1$, $\dots$,
$\tau_{n-1}$. Then it turns out that the quotient space
$M=R/G$ is homeomorphic to the $n$-sphere. Moreover,
let $p\in R$ be a ramification point of the branched covering
$R\to R/G$. Suppose $p$ is fixed by $\tau_{i_1}$,
$\dots$, $\tau_{i_k}$, and is not fixed by other $\tau_j$'s. Taking a suitable coordinate
system $(y_1,\dots,y_n)$ obtained from $(x)$ by exchanges $(x_i\leftrightarrow x_j)$ and translations $(x_i\to x_i+c)$,
it may be supposed that $p$ is represented by $y=0$ and
$\tau_{i_l}$ is given by
\begin{equation*}
(y_1,\dots,y_n)\mapsto (y_1,\dots,y_{2l-2},-y_{2l-1},
-y_{2l},y_{2l+1},\dots,y_n)\ .
\end{equation*}
Then we can define a differentiable structure on $M$ so
that
\begin{equation*}
(y_1^2-y_2^2,2y_1y_2,\dots,y_{2k-1}^2-y_{2k}^2,
2y_{2k-1}y_{2k},y_{2k+1},\dots,y_n)
\end{equation*}
is a smooth coordinate system around the image of $p$.
With this $M$ is diffeomorphic to
the standard $n$-sphere.
One can prove those facts by comparing the branched
covering $R\to R/G$ with the standard case; see
\cite[p.73]{Ki2}.
Now, put
\begin{equation*}
b_{ij}(x_i)=
\begin{cases}
(-1)^i\prod_{\substack{1\le k\le n-1\\ k\ne j}}(f_i(x_i)-a_k)\quad (1\le j\le n-1)\\
(-1)^{i+1}\prod_{k=1}^{n-1}(f_i(x_i)-a_k)\qquad (j=n)
\end{cases}\ ,
\end{equation*}
and define functions $F_1$, $\dots$, $F_n=2E$ on the cotangent
bundle by
\begin{equation}\label{integrals}
\sum_{j=1}^n b_{ij}(x_i)F_j=\xi_i^2\ ,
\end{equation}
where $\xi_i$ are the fiber coordinates with respect to
the base coordinates $(x_1,\dots,x_n)$. Although there
are points on $T^*R$ where $F_i$ are not well-defined,
it turns out that $F_i$ represent well-defined smooth
functions on $T^*M$. Computing the inverse matrix of $(b_{ij})$
explicitly, we have
\begin{align*}
2E=&\sum_i\frac{(-1)^{n-i} \xi^2_i}{\prod_{l\ne i}
(f_l-f_i)}\\
F_j=&\frac1{\prod_{\substack{1\le k\le n-1\\k\ne j}}(a_k-a_j)}
\sum_i\frac{(-1)^{n-i}\prod_{l\ne i}(f_l-a_j)}
{\prod_{l\ne i}(f_l-f_i)}\ \xi^2_i\qquad
(j\le n-1)\ .
\end{align*}
One can also see that $E$, restricted to each
cotangent space of $M$, is a positive definite quadratic
form. Therefore
\begin{equation}\label{eq:metric}
g=\sum_i(-1)^{n-i}\left(\prod_{l\ne i}(f_l-f_i)\right)\,dx_i^2
\end{equation}
is a well-defined riemannian metric on $M$, and $E$
is the hamiltonian of the associated geodesic flow.
We call $E$ the energy function of the riemannian manifold
$M$. From the formula (\ref{integrals}) one can easily see
that
\begin{equation*}
\{F_i,F_j\}=0\qquad (1\le i,j\le n)\ ,
\end{equation*}
where $\{,\}$ denotes the Poisson bracket (see
\cite[Prop.\,1.2.3]{Ki2}). In particular, the geodesic flow
is completely integrable in the sense of hamiltonian
mechanics.
As examples, if $A(\lambda)$ is a constant function, then
$M$ is the sphere of constant curvature. This case is
explained in detail in \cite[pp.71--74]{Ki2}. If $A(\lambda)
=\sqrt{\lambda}$, then $M$ is isometric to the ellipsoid
$\sum_{i=0}^n{u_i^2}/{a_i}=1$. In this case, the system
of functions
$(f_1(x_1),\dots,f_n(x_n))$
is nothing but the elliptic coordinate system (see Introduction), i.e., $f_i(x_i)=\lambda_i$.
One can easily check that the induced metric $\sum_i du_i^2$
coincides with the formula (\ref{eq:metric}) when $f_i$
satisfy the equations (\ref{eq:fdiff}) and
$A(\lambda)=\sqrt{\lambda}$.
Finally, let us define certain submanifolds of $M$ which are
analogous to those for the ellipsoid stated in
Introduction: Put
\begin{gather*}
N_k=\{x\in M\ |\ f_k(x_k)=a_k \quad\text{or}\quad
f_{k+1}(x_{k+1})=a_k\}\quad(0\le k\le n) ,\\
J_k=\{x\in M\ |\ f_k(x_k)=
f_{k+1}(x_{k+1})=a_k\}\quad (1\le k\le n-1) .
\end{gather*}
Then we have, putting $(F_k)_p=F_k\vert_{T_p^*M}$,
\begin{lemma}\label{lemma:subm}
\begin{itemize}
\item[(1)] $J_k=\{p\in M\ |\ (F_k)_p=0\}$.
\item[(2)] $N_k=\{p\in M\ |\ \text{\rm rank }(F_k)_p\le 1\}$
\quad $(1\le k\le n-1)$.
\item[(3)] $\bigcup_k J_k$ is identical with the branch
locus of the covering $R\to M=R/G$.
\item[(4)] $N_k$ is a totally geodesic submanifold of codimension one \ $(0\le k\le n)$.
\item[(5)] $J_k\subset N_k$, $J_k$ is diffeomorphic to
$S^{k-1}\times S^{n-k-1}$.
\end{itemize}
\end{lemma}
\begin{proof} For (1) and (2), see \cite[pp.52--56]{Ki2}.
(3) is obvious.
(4) follows from the fact that $N_k$ is the fixed point-set
of the involutive isometry $(x_1,\dots,x_n)\mapsto (x_1,\dots
,-x_k,\dots,x_n)$. (5) is easily seen by comparing the branched covering with the standard one, \cite[p.73]{Ki2}.
\end{proof}
\section{Geodesic equations}
The geodesic equations are generally written as
\begin{equation*}
\frac{dx_i}{dt}=\frac{\partial E}{\partial \xi_i},
\quad \frac{d\xi_i}{dt}=-\frac{\partial E}
{\partial x_i}.
\end{equation*}
But, since our geodesic flow is completely integrable, it is better to consider the equation
of geodesics with $F_j=c_j$ $(1\le j\le n-1)$
and $2E=1$. If $c=(c_1,\dots,c_{n-1},1)$ is a
regular value of the map
\begin{equation*}
\boldsymbol F=(F_1,\dots, F_{n-1},2E): T^*M\to \mathbb R^n\ ,
\end{equation*}
then its inverse image is a disjoint union of tori, and
the vector fields $X_{F_j}$, $X_E$ on it are
mutually commutative and linearly independent
everywhere. Here $X_f$ denotes the hamiltonian vector field
determined by a function $f$;
\begin{equation*}
X_f=\sum_i\left(\frac{\partial f}{\partial \xi_i}
\frac{\partial }{\partial x_i}-
\frac{\partial f}{\partial x_i}
\frac{\partial }{\partial \xi_i}\right)\ .
\end{equation*}
Let $\omega_j$ $(1\le j\le n)$ be
the dual $1$-forms of $\{\pi_*X_{F_j}\}$, where
$\pi:T^*M\to M$ is the bundle projection. Then, by
(\ref{integrals}) we have
\begin{equation*}
\omega_l=\sum_i\frac{b_{il}}{2\xi_i}\ dx_i\qquad
(1\le l\le n).
\end{equation*}
They are closed $1$-forms, and the geodesic
orbits are determined by
\begin{equation}\label{eq:geod0}
\omega_l=0 \qquad (1\le l\le n-1),
\end{equation}
and the length parameter $t$ on an orbit
is given by
\begin{equation}\label{eq:length}
dt=2\omega_n.
\end{equation}
Putting
\begin{equation*}
\Theta(\lambda)=\sum_{j=1}^{n-1}\left(
\prod_{\substack{1\le k\le n-1\\ k\ne j}}
(\lambda-a_k)\right)\,c_j-\prod_{k=1}^{n-1}(\lambda-a_k)\ ,
\end{equation*}
we have from (\ref{integrals})
\begin{equation*}
\xi_i=\epsilon_i\sqrt{\sum_j b_{ij}(x_i)c_j}
=\epsilon_i\sqrt{(-1)^i\Theta(f_i(x_i))}\qquad (1\le i\le n),
\end{equation*}
where $\epsilon_i=\text{sgn}\,\xi_i=
\text{sgn}\,\left(\frac{dx_i}{dt}\right)=\pm 1$.
If a covector $(x,\xi)$ with $F_1=c_1,\dots,
F_{n-1}=c_{n-1}$, $2E=1$ satisfies
$\xi_i\ne 0$
for any $1\le i\le n$, then we have
\begin{equation*}
(-1)^i \Theta(f_i(x_i))>0.
\end{equation*}
Therefore for such $c_1,\dots,c_{n-1}$, the equation
$\Theta(\lambda)=0$ has $n-1$ distinct real roots
$b_1>b_2>\dots>b_{n-1}$, and they satisfy
\begin{equation*}
f_1(x_1)>b_1>f_2(x_2)>b_2>\dots>f_{n-1}(x_{n-1})>b_{n-1}>f_n(x_n).
\end{equation*}
Thus we have the identity
\begin{equation*}
\Theta(\lambda)=-\prod_{l=1}^{n-1} (\lambda-b_l),
\end{equation*}
and $c_j$ are expressed by $b_l$'s as
\begin{equation}\label{b-c}
c_j=\frac{-\prod_{l=1}^{n-1}(a_j-b_l)}
{\prod_{\substack{1\le k\le n-1\\ k\ne j}}
(a_j-a_k)}
\qquad (1\le j\le n-1)\ .
\end{equation}
Conversely, let $b_1,\dots,b_{n-1}$ be any real numbers satisfying
\begin{equation}\label{brange}
a_{i+1}\le b_i\le a_{i-1}\ ,\qquad b_{i+1}\le b_i
\end{equation}
for any $i$, and define $c_1,\dots,c_{n-1}$ by (\ref{b-c}).
Then there is a covector $(x,\xi)$ with $F_1=c_1$, $\dots$,
$F_{n-1}=c_{n-1}$, $2E=1$. It can be verified that if
$b_1$, $\dots$, $b_{n-1}$ satisfy
\begin{equation}\label{b-cond}
a_{i+1}< b_i< a_{i-1}\ ,\quad b_i\ne a_i,\quad
b_{i+1}< b_i\qquad \text{for any }i
\end{equation}
then the corresponding $c=(c_1,\dots,c_{n-1},1)$ is a regular
value of $\boldsymbol F$.
To describe the behavior of the geodesics it is more convenient to
use the values $(b_1,\dots,b_{n-1})$ rather than using
$(c_1,\dots,c_{n-1})$ directly. So, we shall mainly use
$(b_1,\dots,b_{n-1})$ as the values of first integrals which determine
the Lagrange tori $\boldsymbol F^{-1}(c)$.
Also, we shall denote by $H_1,\dots,H_{n-1}$ the functions on
the unit cotangent bundle $U^*M$ whose values are $b_1,\dots,b_{n-1}$. Namely, $H_i$'s are determined by
\begin{gather*}
F_j(\mu)=\frac{-\prod_{l=1}^{n-1}(a_j-H_l(\mu))}
{\prod_{\substack{1\le k\le n-1\\ k\ne j}}
(a_j-a_k)}
\qquad (1\le j\le n-1),\\
H_1(\mu)\ge\dots\ge H_{n-1}(\mu),\qquad \mu\in U^*M.
\end{gather*}
The range of $H_i$ are given by (\ref{brange}).
Now, put
\begin{align*}
&a_i^+=\max\{a_i,b_i\}\quad (1\le i\le n-1),
\quad a_n^+=a_n\\
&a_i^-=\min\{a_i,b_i\}\quad (1\le i\le n-1),
\quad a_0^-=a_0\ .
\end{align*}
If $b_1,\dots,b_{n-1}$ satisfy the condition (\ref{b-cond}), then
the $\pi$-image of a connected component of
$\boldsymbol F^{-1}(c)$ (a Lagrange torus) is of
the form
\begin{equation*}
L_1\times\dots \times L_n\subset M\ ,
\end{equation*}
where each $L_i$ is a connected component of the inverse
image of $[a_i^+,a_{i-1}^-]$ by the map
\begin{equation*}
f_i:\mathbb R/\alpha_i\mathbb Z\to [a_i,a_{i-1}]\ .
\end{equation*}
(Observe that the ``generalized band'' $L_1\times\dots\times
L_n\subset R$ is injectively mapped to $M$
by the branched covering $R\to M$.)
Along a geodesic $(x_1(t),\dots,x_n(t))$, the coordinate function $x_i(t)$ oscillates on $L_i$ if $L_i$ is an interval,
or $x_i(t)$ moves monotonously if $L_i$ is the whole circle.
Also, the function $f_i(x_i(t))$ oscillates on the interval
$[a_i^+,a_{i-1}^-]$
After all, the equations of geodesic orbits
\begin{equation*}
\omega_l=0\qquad (1\le l\le n-1)
\end{equation*}
are described as
\begin{equation*}
\sum_{i=1}^n\frac{\epsilon_i(-1)^i\prod_{\substack{1\le k\le n-1\\ k\ne l}}(f_i(x_i)-a_k)\ dx_i}
{\sqrt{(-1)^{i-1}\prod_{k=1}^{n-1}(f_i(x_i)-b_k)}}
=0\qquad (1\le l\le n-1)\ .
\end{equation*}
Note that this system of equations is equivalent to
\begin{equation*}
\sum_{i=1}^n\frac{\epsilon_i(-1)^iG(f_i)\ dx_i}
{\sqrt{(-1)^{i-1}\prod_{k=1}^{n-1}(f_i-b_k)}}
=0
\end{equation*}
for any polynomial $G(\lambda)$ of degree $\le n-2$.
Since
\begin{equation*}
\left(\frac{df_i}{dx_i}\right)^2=\frac{(-1)^i4
\prod_{k=0}^n(f_i-a_k)}{A(f_i)^2},
\end{equation*}
those equations are also described as
\begin{equation}\label{eq:geod}
\sum_{i=1}^n\frac{\epsilon'_i(-1)^iG(f_i)A(f_i)\ df_i}
{\sqrt{-\prod_{k=1}^{n-1}(f_i-b_k)
\cdot\prod_{k=0}^n(f_i-a_k)}}
=0,
\end{equation}
where $\epsilon_i'=\text{sgn of }df_i(x_i(t))/dt$.
By (\ref{eq:geod}) we have
\begin{equation*}
\sum_{i=1}^n\int_s^{t}\frac{(-1)^iG(f_i)A(f_i)}
{\sqrt{-\prod_{k=1}^{n-1}(f_i-b_k)
\cdot\prod_{k=0}^n(f_i-a_k)}}\
\left|\frac{df_i(x_i(t))}{dt}\right|
\ dt=0
\end{equation*}
for any polynomial $G(\lambda)$ of degree $\le n-2$ and for
a fixed $s\in \mathbb R$.
By using the variables $\sigma_i$ defined by
\begin{equation*}
\sigma_{i}(t)=\int_0^t\left|\frac{df_i(x_i(t))}{dt}\right|
\ dt\ ,
\end{equation*}
this formula
is rewritten as
\begin{equation}\label{eq:geodsigma1}
\sum_{i=1}^n\int_{\sigma_i(s)}^{\sigma_i(t)}\frac{(-1)^iG(f_i)A(f_i)}
{\sqrt{-\prod_{k=1}^{n-1}(f_i-b_k)
\cdot\prod_{k=0}^n(f_i-a_k)}}\
\ d\sigma_i=0\ .
\end{equation}
Here, $f_i$ is regarded as a function of $\sigma_i$, i.e.,
putting $\phi_i(t)
=a_i+|t|$ for $|t|\le a_{i-1}-a_i$ and extending it to $\mathbb R$
as a periodic function with the period $2(a_{i-1}-a_i)$,
we have
\begin{equation*}
f_i=
\phi_i(\sigma_i+\epsilon_i(f_i(x_i(0))-a_i))\ ,
\end{equation*}
where $\epsilon_i=\pm 1$ is the sign of $df_i(x_i(t))/dt$ at
$t=0$.
Also, integrating $dt=\sum_i(b_{in}/\xi_i)dx_i$, we have
\begin{equation}\label{eq:geodlength}
\sum_{i=1}^n\int_{\sigma_i(s)}^{\sigma_i(t)}
\frac{(-1)^i\,\tilde G(f_i)A(f_i)}
{2\sqrt{-\prod_{k=1}^{n-1}(f_i-b_k)
\cdot\prod_{k=0}^n(f_i-a_k)}}\
d\sigma_i=t-s\ ,
\end{equation}
where $\tilde G(\lambda)$ is any monic polynomial in
$\lambda$ of degree $n-1$.
\section{A monotonicity condition for $A(\lambda)$}
We put the following conditions on the function
$A(\lambda)$:
\begin{equation}\label{cond2}
(-1)^{k-1}A^{(k)}(\lambda)>0\quad \text{on }
[a_n,a_0]\qquad (1\le k\le n-1)
\end{equation}
for $n\ge 3$, where $A^{(k)}$ denotes the $k$-th derivative of $A$.
For the case $n=\dim M=2$, we need (\ref{cond2}) for $1\le k\le 2$,
as described in our earlier paper \cite{IK2}.
A typical example satisfying the condition
(\ref{cond2}) is the ellipsoid,
in which case $A(\lambda)=\sqrt{\lambda}$.
Since the condition (\ref{cond2}) is $C^{n-1}$-open,
there are surely many $A(\lambda)$ satisfying it.
In the rest of this section, we shall prove some
inequalities which are obtained under the condition
(\ref{cond2}).
Put
\begin{equation*}
G_l(\lambda)=\prod_{\substack{1\le k\le n-1\\
k\ne l}}(\lambda-b_k)\qquad (1\le l\le n-1)\ .
\end{equation*}
\begin{prop}\label{prop:cond2}
If $A(\lambda)$ satisfies the condition {\rm(\ref{cond2})},
and if $b_1,\dots,b_{n-1}$ and $a_0,\dots,a_n$ are all
distinct, then the following inequalities hold:
\begin{enumerate}
\item \begin{equation*}
\sum_{i=1}^n\int_{a_i^+}^{a_{i-1}^-}
\frac{(-1)^{n-i+\# I}A(\lambda)\prod_{j\in I}(\lambda-b_j)
}
{\sqrt{-\prod_{k=1}^{n-1}(\lambda-b_k)
\cdot\prod_{k=0}^n(\lambda-a_k)}}\
d\lambda<0,
\end{equation*}
where $I$ is any (possibly empty)
subset of
$\{1,\dots, n-1\}$ such that $\# I\le n-2$;
\item \begin{equation*}
\frac{\partial }{\partial b_l}
\sum_{i=1}^n\int_{a_i^+}^{a_{i-1}^-}
\frac{(-1)^{i}G_l(\lambda)A(\lambda)\ d\lambda}
{\sqrt{-\prod_{k=1}^{n-1}(\lambda-b_k)
\cdot\prod_{k=0}^n(\lambda-a_k)}}
>0\ ,
\end{equation*}
where $1\le l\le n-1$.
\end{enumerate}
The inequality (1) is still valid if
$b_j$'s $(j\not\in I)$ are mutually distinct.
Precisely speaking, when a sequence of $b_j$'s with
$b_j$'s and $a_k$'s being all distinct converges to
some $b_j$'s which satisfy $b_k\ne b_l$ for any $k,l
\in J$, $k\ne l$, then the formula in (1) has a limit
and the limit is still negative.
\end{prop}
In the following two lemmas, we shall assume that
$b_1,\dots,b_{n-1}$ and $a_0,\dots,a_n$ are all distinct.
\begin{lemma}\label{lemma:flat}
\begin{equation*}
\sum_{i=1}^n\int_{a_i^+}^{a_{i-1}^-}
\frac{(-1)^{i}G(\lambda)\ d\lambda}
{\sqrt{-\prod_{k=1}^{n-1}(\lambda-b_k)
\cdot\prod_{k=0}^n(\lambda-a_k)}}
=0
\end{equation*}
for any polynomial $G(\lambda)$ of degree $\le n-2$.
\end{lemma}
\begin{proof}
Let $W=\{\lambda\}$ be the region $\mathbb C\cup\{\infty\}-\bigcup_{
i=1}^n [a_i^+,a_{i-1}^-]$. Then there are a
meromorphic function $\mu$ on $W$ such that
\begin{equation*}
\mu^2=-\prod_{k=1}^{n-1}(\lambda-b_k)
\cdot\prod_{k=0}^n(\lambda-a_k),
\end{equation*}
and the holomorphic
$1$-form $(G(\lambda)/\mu)d\lambda$ on $W$.
Taking the sum of contour integrals around the
intervals $[a_i^+,a_{i-1}^-]$, one obtains the
desired formula.
\end{proof}
\begin{lemma}\label{lemma:cond}
Let $J$ be any nonempty subset of $\{1,\dots,n-1\}$, and
let $B(\lambda)$ be the function defined by
\begin{equation}\label{eq:B}
\frac{A(\lambda)}{\prod_{k\in J}(\lambda-b_k)}=
\sum_{k\in J} \frac{e_k}{\lambda-b_k}+B(\lambda),
\quad e_k=\frac{A(b_k)}{\prod_{\substack{l\in J\\
l\ne k}}(b_k-b_l)}.
\end{equation}
Suppose $A(\lambda)$ satisfies the condition (\ref{cond2}).
Then $B(\lambda)$ satisfies
\begin{equation*}
(-1)^{\# J +m}B^{(m)}(\lambda)<0\quad \text{for}
\quad a_n\le \lambda\le a_0 \quad\text{and}
\quad 0\le m\le n-1-\# J.
\end{equation*}
\end{lemma}
\begin{proof}
We shall prove this by an induction on $\# J$.
When $J=\{k\}$, then
\begin{equation}\label{eq:prime}
B(\lambda)=\frac{A(\lambda)-A(b_k)}{\lambda-b_k}
=\int_0^1A'(t(\lambda-b_k)+b_k)dt,
\end{equation}
and we have $(-1)^{1+m}B^{(m)}(\lambda)<0$ by the
assumption on $A(\lambda)$.
Now suppose $\# J\ge 1$, $l\not\in J$ and let $J_1=J\cup\{l\}$.
Then
\begin{gather*}
\frac{A(\lambda)}{\prod_{k\in J_1}(\lambda-b_k)}=
\sum_{k\in J} \frac{e_k}{(\lambda-b_k)
(\lambda-b_l)}+\frac{B(\lambda)}{\lambda-b_l}\\
=\sum_{k\in J}\frac1{b_k-b_l}\left(\frac{e_k}
{\lambda-b_k}-\frac{e_k}{\lambda-b_l}\right)+
\frac{B(b_l)}{\lambda-b_l}+
\frac{B(\lambda)-B(b_l)}{\lambda-b_l}.
\end{gather*}
Let us denote the last term in the right-hand side
by $B_1(\lambda)$. Since it is written as
\begin{equation*}
\int_0^1B'(t(\lambda-b_l)+b_l)dt,
\end{equation*}
we have $(-1)^{\# J +1+m}B_1^{(m)}(\lambda)<0$
by the induction assumption.
\end{proof}
{\it Proof of Proposition \ref{prop:cond2}.}
First, suppose that $b_1,\dots,b_{n-1}$ and $a_0,\dots$, $a_n$ are all distinct.
Let $A(\lambda)$ be a positive function on
$[a_n,a_0]$ satisfying the condition (\ref{cond2}).
Let $I$ be as in Proposition \ref{prop:cond2} (1) and let $J$ be
its complement in $\{1,\dots,n-1\}$.
Define the function $B(\lambda)$ by the formula
(\ref{eq:B}). Then, by Lemmas \ref{lemma:cond} and
\ref{lemma:flat}
we have
\begin{equation}\label{eq:cond1}
\begin{gathered}
\sum_{i=1}^n\int_{a_i^+}^{a_{i-1}^-}
\frac{(-1)^{n-i+\# I}A(\lambda)\prod_{l\in I}(\lambda-b_l)
}
{\sqrt{-\prod_{k=1}^{n-1}(\lambda-b_k)
\cdot\prod_{k=0}^n(\lambda-a_k)}}\
d\lambda\\
=\sum_{i=1}^n\int_{a_i^+}^{a_{i-1}^-}
\frac{(-1)^{n-i+\# I}B(\lambda)\prod_{l=1}^{n-1}
(\lambda-b_l)}
{\sqrt{-\prod_{k=1}^{n-1}(\lambda-b_k)
\cdot\prod_{k=0}^n(\lambda-a_k)}}\
d\lambda\ .
\end{gathered}
\end{equation}
Since $(-1)^{i-1}\prod_{j=1}^{n-1}(\lambda-b_j)>0$
on $(a_i^+,a_{i-1}^-)$, and since
\begin{equation*}
(-1)^{n-1-\# I}B(\lambda)<0
\end{equation*}
by Lemma \ref{lemma:cond}, we have the inequality
(1) in this case.
Next, let us consider the limit case. The limit $b_j$'s are assumed that $b_k\ne b_l$ for any $k,l
\in J$, $k\ne l$. Note that the function $B(\lambda)$
is defined by the formula (\ref{eq:B}) and it only
depends on $A(\lambda)$ and $b_j$'s $(j\in J)$.
Since the limit $b_j$'s $(j\in J)$ are mutually
distinct, it follows that the function $B(\lambda)$
has a limit. Therefore the right-hand side of the
formula (\ref{eq:cond1}) has a finite limit and it is
still negative by the same reason as above.
To prove (2), we put
\begin{equation*}
\frac{A(\lambda)}{\lambda-b_l}=
\frac{A(b_l)}{\lambda-b_l}+B(\lambda, b_l).
\end{equation*}
Then the left-hand side of (2) is equal to
\begin{equation}\label{eq:cond}
\begin{gathered}
\frac{\partial }{\partial b_l}
\sum_{i=1}^n\int_{a_i^+}^{a_{i-1}^-}
\frac{(-1)^{i}B(\lambda, b_l)\prod_{j=1}^{n-1}(\lambda-b_j)}
{\sqrt{-\prod_{k=1}^{n-1}(\lambda-b_k)
\cdot\prod_{k=0}^n(\lambda-a_k)}}\
d\lambda\\
=\sum_{i=1}^n\int_{a_i^+}^{a_{i-1}^-}
\frac{(-1)^{i}\left(\frac{\partial }
{\partial b_l}B(\lambda, b_l)\right)
\prod_{j=1}^{n-1}(\lambda-b_j)}
{\sqrt{-\prod_{k=1}^{n-1}(\lambda-b_k)
\cdot\prod_{k=0}^n(\lambda-a_k)}}\
d\lambda\\
-\frac12\sum_{i=1}^n\int_{a_i^+}^{a_{i-1}^-}
\frac{(-1)^{i}B(\lambda, b_l)\prod_{\substack{
1\le j\le n-1\\j\ne l}}(\lambda-b_j)}
{\sqrt{-\prod_{k=1}^{n-1}(\lambda-b_k)
\cdot\prod_{k=0}^n(\lambda-a_k)}}\
d\lambda.
\end{gathered}
\end{equation}
The second line of the right-hand side is equal to
\begin{equation*}
-\frac12\sum_{i=1}^n\int_{a_i^+}^{a_{i-1}^-}
\frac{(-1)^{i}B_1(\lambda, b_l)\prod_{
1\le j\le n-1}(\lambda-b_j)}
{\sqrt{-\prod_{k=1}^{n-1}(\lambda-b_k)
\cdot\prod_{k=0}^n(\lambda-a_k)}},
\end{equation*}
where
\begin{equation*}
B_1(\lambda,b_l)=\frac{B(\lambda,b_l)-A'(b_l)}
{\lambda-b_l}=\frac{\partial}{\partial b_l}B(\lambda,b_l).
\end{equation*}
Since $B_1(\lambda,b_l)<0$, it follows that the right-hand
side of the formula (\ref{eq:cond}) is positive.
\qed
\section{Jacobi fields}\label{sec:jac}
In this section we shall consider Jacobi fields
along a geodesic which is not totally contained
in the submanifold $N_i$ for any $i$.
Let $\gamma(t)=(x_1(t),\dots,x_n(t))$ be such a
geodesic. In this case, the corresponding
values $b_i$ of the first integrals
$H_i$ satisfy
$b_i\ne a_{i+1}$ and $b_i\ne a_{i-1}$ for any $i$.
We shall consider the following
three cases separately: (i) $b_1,\dots,b_{n-1}$ and
$a_0,\dots,a_n$ are all distinct;
(ii) there are some $i$ such that $b_i=a_i$, but
other $b_j$'s are not equal to any $a_k$ nor
$b_k$; (iii) there are some $j$ such that
$b_j=b_{j-1}$, and there may be some $i$ such that
$b_i=a_i$, but there is no $l$ such that
$b_l=a_{l+1}$ or $b_l=a_{l-1}$.
First, let us consider the case where $b_1,
\dots,b_{n-1}$ and $a_0,\dots,a_n$ are all
distinct.
For each $i$, let $S_i\subset \mathbb R$ be the set of the time $s$
such that $f_i(x_i(s))=b_i$ $(b_i=a_i^+)$ or
$f_{i+1}(x_{i+1}(s))=b_i$ $(b_i=a_i^-)$.
Then $S_i$ are discrete subsets of $\mathbb R$. At each point $\gamma(s)$ where $s\not\in S_i$
for any $i$, the system of functions $(H_1,\dots,H_{n-1})$
can be used
as a coordinate system on the unit cotangent space
$U^*_{\gamma(s)}M$ around the covector $(x(s),\xi(s))=
\flat(\dot\gamma(s))$. Then, identifying $\partial/\partial
H_i\in T_{\flat(\dot\gamma(s))}(U^*_{\gamma(s)}M)$ with
a covector in $T^*_{\gamma(s)}M$ in a natural manner,
we put $\tilde V_i(s)=\sharp(\frac{\partial}{\partial H_i}
/|\frac{\partial}{\partial H_i}|)\in T_{\gamma(s)}M$ at
$\gamma(s)$. As is easily seen, the norm $|\partial/\partial
H_i|$ is equal to
\begin{equation*}
\frac12\sqrt{\frac{(-1)^{n-1}G_i(b_i)}{\prod_{m=1}^n(f_m(x_m)-b_i)}}\ .
\end{equation*}
At the point $\gamma(s)$ where $s\in S_i$, we put $\nu_i^2=f_i(x_i(s))-H_i$ if $b_i=a_i^+$ (resp.
$\nu_i^2=H_i-f_{i+1}(x_{i+1}(s))$ if $b_i=a_i^-$),
and use $\nu_i$ as a coordinate function on
$U^*_{\gamma(s)}M$
instead of $H_i$.
We choose the sign of $\nu_i$ so that it is equal
with the sign of $\xi_i$ (resp. $\xi_{i+1}$). Then we put $\tilde V_i(s)=
\sharp(\frac{\partial}
{\partial \nu_i}/|\frac{\partial}{\partial \nu_i}|)$
in this case.
It is easy to see that $\mathbb R\ni s\mapsto \tilde V_i(s)$ is
smooth up to the sign. Therefore we can take a smooth
vector field
$V_i(t)$ along the geodesic $\gamma(t)$ such that
$V_i(t)= \pm\tilde V_i(t)$ for any $t\in \mathbb R$.
We now define the Jacobi field $Y_{i, s}(t)$ along the geodesic
$\gamma(t)$ by the initial conditions $Y_{i,s}(s)=0$ and
$Y'_{i,s}(s)=V_i(s)$ for any $s\in \mathbb R$, where
$Y'_{i,s}(t)$ denotes the covariant derivative of
$Y_{i,s}(t)$ with respect to $\partial/\partial t$.
Let us denote by $\Omega(Y,Z)$ the symplectic inner product
of two Jacobi fields along $\gamma(t)$ which are
orthogonal to $\dot\gamma(t)$ for any $t$:
\begin{equation*}
\Omega(Y,Z)=g(Y(t), Z'(t))-g(Y'(t),Z(t))\ ,
\end{equation*}
which is constant in $t$.
Let $\mathcal Y_i$ be the vector space of Jacobi fields along
$\gamma(t)$ spanned by $\{Y_{i,s}(t)\ |\ s\in \mathbb R\}$.
\begin{prop}\label{prop:jf1}
Along the geodesic $\gamma(t)$ such that $b_1,\dots,b_{n-1}$
and $a_0, \dots, a_n$ are all distinct, the Jacobi fields
defined above have the following properties.
\begin{enumerate}
\item $Y_{i,s}(t)\in\mathbb R V_i(t)$ for any $i$ and $s,t\in\mathbb R$.
Also, $V_1(t),\dots,V_{n-1}(t)$, $\dot\gamma(t)$ are
mutually orthogonal for any $t\in\mathbb R$.
\item $\mathcal Y_i$ and $\mathcal Y_j$ $(i\ne j)$ are mutually orthogonal
with respect to the symplectic inner product $\Omega$,
i.e., $\Omega(Y_i,Y_j)=0$
for any $Y_i\in \mathcal Y_i$ and $Y_j\in\mathcal Y_j$.
\item Each $V_i(t)$ is parallel along the geodesic
$\gamma(t)$.
\item Each $\mathcal Y_i$ is two-dimanesional.
\item If $\gamma(s_1)$ and $\gamma(s_2)$ $(s_1<s_2)$
are mutually
conjugate along the geodesic $\gamma(t)$, then there is
$i$ and a nonzero Jacobi field $Y\in\mathcal Y_i$ such that
$Y(s_1)=Y(s_2)=0$.
\item $Y_{i,s_1}(s_2)\ne 0$ if $s_1\not\in S_i$,
$s_2\ne s_1$, and either $[s_1,s_2)\cap S_i=\emptyset$, $s_1<s_2$ or $(s_2,s_1]\cap S_i=\emptyset$, $s_2<s_1$.
\item The Jacobi field $Y_{i,s_1}(t)$ $(s_1\in S_i)$
vanishes at $t=s_2$ if and only if $s_2\in S_i$.
\end{enumerate}
\end{prop}
\begin{proof}
Let $\gamma(u,t)=(\dots,x_k(u,t),\dots)$ be a one-parameter
family of geodesics such that $x_k(0,t)=x_k(t)$ and
$(\partial/\partial u)|_{u=0}$ represents
the Jacobi field $Y_{i,s_1}(t)$. Suppose that $G=G_j$,
$i\ne j$, and $s=s_1$ and $t=s_2$ do not
belong to $S_i\cup S_j$ in the formula
(\ref{eq:geodsigma1}).
We then differentiate the
formula by $u$. Since
\begin{equation*}
\frac{\partial H_k}{\partial u}\big|_{u=0}\ne 0\quad (k=i)\ ;\quad
=0 \quad (k\ne i)\ ,
\end{equation*}
we have
\begin{equation}
\begin{gathered}
\sum_{l=1}^n\frac{\epsilon'_l(-1)^lG_j(f_l)A(f_l)}
{\sqrt{-\prod_{k=1}^{n-1}(f_l-b_k)
\cdot\prod_{k=0}^n(f_l-a_k)}}\,d(f_l(x_l))(Y_{i,s_1}(s_2))\\
-\frac1{2c}\sum_{l=1}^n\int_{\sigma_l(s_1)}^{\sigma_l(s_2)}\frac{(-1)^iG_{i,j}(f_l)A(f_l)}
{\sqrt{-\prod_{k=1}^{n-1}(f_l-b_k)
\cdot\prod_{k=0}^n(f_l-a_k)}}\
\ d\sigma_l=0\ ,
\end{gathered}
\end{equation}
where $c=\pm$ (the norm of $\partial/\partial H_i$ at
$\gamma(s_1)$) and $f_l=f_l(x_l(s_2))$ in the first line, and
$G_{i,j}(\lambda)=\prod_{k\ne i,j}(\lambda-b_k)$.
Observe that the second line in the above formula vanishes
by the formula (\ref{eq:geodsigma1}). Moreover, the
covector
\begin{equation*}
\frac14\sum_{l=1}^n\frac{\epsilon'_l(-1)^lG_j(f_l)A(f_l)}
{\sqrt{-\prod_{k=1}^{n-1}(f_l-b_k)
\cdot\prod_{k=0}^n(f_l-a_k)}}\
d(f_l(x_l))\bigg|_{f_l=f_l(x_l(s_2))}
\end{equation*}
is equal to the one which is represented by $\partial
/\partial H_j$ at $\gamma(s_2)$, which is a nonzero scalar
multiple of
$\flat(Y'_{j,s_2}(s_2))$. Thus we have
\begin{equation*}
\Omega(Y_{i,s_1},Y_{j,s_2})=
g(Y_{i,s_1}(s_2),Y'_{j,s_2}(s_2))=0\ ,
\end{equation*}
which is valid for any $s_1,s_2\in\mathbb R$ by continuity. In
particular, we have $g(Y_{i,s_1}(s_2),V_j(s_2))=0$ for any
$j\ne i$, and also $g(V_i(s_1),V_j(s_1))=0$ by differentiating it
at $s_2=s_1$. Thus we have (1) and (2).
(3) and (4) follow immediately from (1) and (2).
The assertion (5) is also obvious.
Next, we shall prove (6). First, we assume $s_1<s_2$ and
$s_2\not\in S_i$.
In the same way as above, we have
\begin{equation}\label{eq:jac4}
\begin{gathered}
\sum_{l=1}^n\frac{\epsilon'_l(-1)^lG_i(f_l)A(f_l)}
{\sqrt{-\prod_{k=1}^{n-1}(f_l-b_k)
\cdot\prod_{k=0}^n(f_l-a_k)}}\,
d(f_l(x_l))(Y_{i,s_1}(s_2))\\
+\frac1{2c}\sum_{l=1}^n\int_{\sigma_l(s_1)}^{\sigma_l(s_2)}\frac{(-1)^iG_{i}(f_l)A(f_l)}
{(f_l-b_i)\sqrt{-\prod_{k=1}^{n-1}(f_l-b_k)
\cdot\prod_{k=0}^n(f_l-a_k)}}\,
d\sigma_l=0\ .
\end{gathered}
\end{equation}
Note that, since $[s_1,s_2]\cap S_i=\emptyset$, $f_l-b_i$
never vanish on the interval $[\sigma_l(s_1),\sigma_l(s_2)]$.
The second line in the above formula being negative, we have
$g(Y_{i,s_1}(s_2),Y'_{i,s_2}(s_2))\ne 0$. Thus
$Y_{i,s_1}(s_2)\ne 0$.
Next, let us take $s_3\in S_i$ such that $s_1<s_3$ and $[s_1,s_3)\cap S_i=\emptyset$. As proved above,
\begin{gather*}
\left|\frac{\partial }{\partial H_i}\right|_{\gamma(s_1)}
\left|\frac{\partial }{\partial H_i}\right|_{\gamma(s_2)}
g(Y_{i,s_1}(s_2),Y'_{i,s_2}(s_2))=\\
-\frac18\sum_{l=1}^n\int_{\sigma_l(s_1)}^{\sigma_l(s_2)}\frac{(-1)^iG_{i}(f_l)A(f_l)}
{(f_l-b_i)\sqrt{-\prod_{k=1}^{n-1}(f_l-b_k)
\cdot\prod_{k=0}^n(f_l-a_k)}}\
\ d\sigma_l
\end{gather*}
for any $s_2$ such that $s_1<s_2<s_3$. Suppose $b_i=a_i^+$.
Since
\begin{equation*}
g(Y_{i,s_1}(s_2),Y'_{i,s_2}(s_2))=\Omega(Y_{s_1},Y_{s_2})
=-g(Y'_{i,s_1}(s_1),Y_{i,s_2}(s_1))\,,
\end{equation*}
multiplying both sides by $2|\nu_i|=2\sqrt{f_i(x_i(s_2))-b_i}$,
and taking a limit $s_2\to s_3$, we have
\begin{equation}\label{eq:singjac2}
-c'g(Y'_{i,s_1}(s_1), Y_{i,s_3}(s_1))=
\frac12\frac{(-1)^{i+1}G_{i}(b_i)A(b_i)}
{\sqrt{-\prod_{k\ne i}(b_i-b_k)
\cdot\prod_{k=0}^n(b_i-a_k)}}\ ,
\end{equation}
where $c'=|\partial/\partial H_i|_{\gamma(s_1)}
|\partial/\partial \nu_i|_{\gamma(s_3)}$.
Since the left-hand side of the above formula is equal to
\begin{equation*}
c'g(Y_{i,s_1}(s_3), Y'_{i,s_3}(s_3))\ ,
\end{equation*}
and since
the right-hand side does not vanish, we have
\begin{equation}\label{eq:singjac3}
Y_{i,s_1}(s_3)\ne 0\ ,\qquad
Y_{i,s_3}(s_1)\ne 0\ .
\end{equation}
The case where $s_2<s_1$
is similar. Therefore the assertion (6) follows.
Now, in the situation of (6), take $s_0\in S_i$ such that
$s_0<s_1$ and $(s_0,s_1]\cap S_i=\emptyset$. Then, again
multiplying both sides of the formula (\ref{eq:singjac2})
by $|\nu_i|=\sqrt{f_i(x_i(s_1))-b_i}$ and taking a limit
$s_1\to s_0$, we have
\begin{equation*}
g(Y_{i,s_0}(s_3),Y'_{i,s_3}(s_3))=0\ .
\end{equation*}
Thus it follows that $Y_{i,s_0}(s_3)=0$, and
combined with (\ref{eq:singjac3}) we have (7).
\end{proof}
The following corollary is immediate.
\begin{cor}\label{cor:conj}
Fix $t_0$ and let $t_0<t_1^i<t_2^i<\dots$ be the zeros of
the Jacobi field $Y_{i,t_0}(t)$ for $t\ge t_0$. Then:
\begin{enumerate}
\item If $t_0\in S_i$, then the set $\{t_k^i\}$
coincides with $\{t\in S_i\ |\ t>t_0\}$
\item If $t_0\not\in S_i$, then every $t_k^i\not\in S_i$,
and there is just one element of $S_i$ in the interval $(t_k^i,
t_{k+1}^i)$ for each $k$.
\item The set of conjugate points of $\gamma(t_0)$ along
$\gamma(t)$ $(t>t_0)$ is equal to $\{\gamma(t_k^i) \ |
\ k\ge1, 1\le i\le n-1\}$.
\end{enumerate}
\end{cor}
We shall prove one more result on the zeros of Jacobi fields
in this case, which needs
the assumption \eqref{cond2}.
\begin{prop}\label{prop:jacreg}
Fix $i$ and take $s_1$ and $s_2$ such that $s_1\not\in S_i$, $s_1<s_2$, and $\sigma_l(s_2)
-\sigma_l(s_1)\le 2(a_{l-1}^--a_l^+)$ for any $l$. Then $Y_{i,s_1}(s_2)
\ne 0$.
\end{prop}
\begin{proof}
Let $s_3\in S_i$ such that $s_1<s_3$ and $[s_1,s_3)\cap S_i=\emptyset$.
If $s_2\le s_3$, then the assertion follows from (5) of
the previous proposition. Now suppose $s_3<s_2$.
As above, we shall compute $g(Y_{i,s_1}(s_2),Y'_{i,s_2}(s_2))$. In this case, however,
the formula \eqref{eq:jac4} is invalid, because the integral
diverge at $t=s_3$. So, instead, we differentiate the
formula
\begin{equation}\label{eq:geod3}
\begin{aligned}
&-\sum_{l=1}^{n}\int_{\sigma_l(s_2)}^{
2(a_{l-1}^--a_l^+)+\sigma_l(s_1)}\frac{(-1)^lG_i(f_l)
A(f_l)\ d\sigma_l}
{\sqrt{-\prod_{k=1}^{n-1}(f_l-b_k)
\cdot\prod_{k=0}^n(f_l-a_k)}}\\
&+2\sum_{l=1}^n\int_{a_l^+}^{a_{l-1}^-}
\frac{(-1)^lG_i(\lambda)
A(\lambda)\ d\lambda}
{\sqrt{-\prod_{k=1}^{n-1}(\lambda-b_k)
\cdot\prod_{k=0}^n(\lambda-a_k)}}=0
\end{aligned}
\end{equation}
in terms of the deformation parameter defining $cY_{i,s_1}$,
$c$ being $\pm$ (the norm of $\partial/\partial H_i$ at
$\gamma(s_1)$):
\begin{equation}\label{eq:diff3}
\begin{aligned}
&\sum_{l=1}^n\frac{\epsilon'_l(-1)^lG_i(f_l)A(f_l)}
{\sqrt{-\prod_{k=1}^{n-1}(f_l-b_k)
\cdot\prod_{k=0}^n(f_l-a_k)}}\
d(f_l(x_l))(cY_{i,s_1}(s_2))\\
-&\frac1{2}\sum_{l=1}^{n}\int_{\sigma_l(s_2)}^{
2(a_{l-1}^--a_l^+)+\sigma_l(s_1)}\frac{(-1)^lG_i(f_l)
A(f_l)\ d\sigma_l}
{(f_l-b_i)\sqrt{-\prod_{k=1}^{n-1}(f_l-b_k)
\cdot\prod_{k=0}^n(f_l-a_k)}}\\
+&2\frac{\partial}{\partial b_i}
\sum_{l=1}^n\int_{a_l^+}^{a_{l-1}^-}
\frac{(-1)^lG_i(\lambda)
A(\lambda)\ d\lambda}
{\sqrt{-\prod_{k=1}^{n-1}(\lambda-b_k)
\cdot\prod_{k=0}^n(\lambda-a_k)}}=0\ ,
\end{aligned}
\end{equation}
Note that
$b_i$ is not contained in the range of $f_l$
while $\sigma_l$ moves in the interval $[\sigma_l(s_2),
2(a_{l-1}^--a_l^+)+\sigma_l(s_1)]$ $(l=i,i+1)$.
Since the second line of the formula \eqref{eq:diff3}
is positive or zero, and since the third line is positive
by Proposition \ref{prop:cond2} (2), it therefore follows
that $g(Y_{i,s_1}(s_2),Y'_{i,s_2}(s_2))\ne 0$.
\end{proof}
Next, we shall consider Jacobi fields along the geodesic
$\gamma(t)$ for which some $b_i$ is equal to $a_i$, but
other $b_j$'s are not equal to any $a_k$ nor
$b_k$.
For $i$ with
$b_i=a_i$, let $S_i$ be the set of $s\in\mathbb R$ where
$f_i(x_i(s))=b_i$. One can see from the formula
(\ref{eq:geodsigma1}) that $S_i$ is also the set of $s\in\mathbb R$
where $f_{i+1}(x_{i+1}(s))=b_i$, i.e., $s\in S_i$ if
and only if $\gamma(s)\in J_i$. For such $i$ and $s\in S_i$,
we define $\tilde Y_{i,s}(t)$ as the Jacobi field
$\pi_*(X_{F_i})$ along the geodesic $\gamma(t)$.
For $s\not\in S_i$, $Y_{i,s}(t)$ is defined as before.
Also, for $j$ with $b_j\ne a_j$, the set $S_j$
and the Jacobi fields $Y_{j,s}(t)$
are defined as before.
\begin{prop}
For a geodesic $\gamma(t)$ stated above, the statements in
Propositions \ref{prop:jf1}, \ref{prop:jacreg}
and Corollary \ref{cor:conj} equally hold.
\end{prop}
\begin{proof}
Only the parts related to the Jacobi field
$\tilde Y_{i,s}(t)=\pi_*(X_{F_i})$ would be nontrivial.
Suppose $b_i=a_i$ and $s_1\not\in S_j$, $s_2\in S_i$.
Considering the symplectic inner product of two Jacobi fields
$Y_{j,s_1}(t)$ and $\tilde Y_{i,s_2}(t)$, we have
\begin{gather*}
\Omega(Y_{j,s_1},\tilde Y_{i,s_2})=
c\,\omega\left(\frac{\partial}{\partial H_j},X_{F_i}
\right)_{\flat(\dot\gamma(s_1))}\\
= c\,\frac{\partial c_i}{\partial b_j}=
\frac{c\,\prod_{m\ne j}(a_i-b_m)}
{\prod_{\substack{1\le k\le n-1\\k\ne i}}(a_i-a_k)}\quad
\begin{cases}
=0\quad (j\ne i)\\
\ne 0\quad (j=i)
\end{cases}\ ,
\end{gather*}
where $\omega$ is the symplectic 2-form $\sum_kd\xi_k\wedge
dx_k$, $\partial/\partial H_j$ is the tangent vector
to $U^*_{\gamma(s_1)}M$ at $\flat(\dot\gamma(s_1))$ defined
as before, and $c=1/|\partial/\partial H_j|$.
The proposition follows from this formula.
\end{proof}
Next, we shall consider Jacobi fields along a
geodesic for which there are some $j$ such that
$b_j=b_{j-1}$ and there may be some $i$ such that
$b_i=a_i$, but there is no $l$ such that
$b_l=a_{l+1}$ or $b_l=a_{l-1}$.
In this case,
$f_j(x_j(t))(=b_j=b_{j-1})$
remains constant along the geodesic $\gamma(t)$. We put
this value $\lambda_j^0$ for convenience.
For each point $\gamma(s)$ on the geodesic,
we adopt $\mu_j,
\mu_{j-1}$ as the coordinate functions on the
unit cotangent space $U^*_{\gamma(s)}M$,
around the covector $\flat(\dot\gamma(s))$,
instead of $H_j, H_{j-1}$, defined by the
formula:
\begin{equation*}
\mu_{j-1}=H_{j-1}+H_j-2\lambda_j^0,\quad
\mu_j^2=4(H_{j-1}-\lambda_j^0)(\lambda_j^0-H_j)
\ .
\end{equation*}
We choose the sign of $\mu_j$ so that it is equal to that of
$\xi_j$. Let us denote by $Z_{j,s}(t)$,
$Z_{j-1,s}(t)$ the Jacobi fields along the
geodesic $\gamma(t)$ with the initial conditions
\begin{equation*}
Z_{k,s}(s)=0,\ Z_{k,s}'(s)=\sharp(\partial/
\partial \mu_k)/|\partial/\partial \mu_k|
\quad (k=j,j-1)\ .
\end{equation*}
Note that
\begin{equation*}
\left|\frac{\partial}{\partial \mu_{j-1}}\right|=
\left|\frac{\partial}{\partial \mu_{j}}\right|=
\frac12\sqrt{\frac{(-1)^nG_{j,j-1}(\lambda_j^0)}
{\prod_{m\ne j}(f_m-\lambda_j^0)}},\quad
\left\langle\frac{\partial}{\partial \mu_{j-1}},
\frac{\partial}{\partial \mu_{j}}\right\rangle=0
\end{equation*}
at each covector $\flat(\dot\gamma(s))$.
Define the real number $\theta_{s_1}(s_2)$ by
the formula
\begin{equation}\label{eq:theta}
\begin{gathered}
\sum_{\substack{1\le l\le n\\l\ne j}}\int_{\sigma_l(s_1)}^{\sigma_l(s_2)}\frac{(-1)^l
G_{j,j-1}(f_l)A(f_l)\ d\sigma_l}
{|f_l-\lambda_j^0|\sqrt{-\prod_{k\ne j,j-1}(f_l-b_k)
\cdot\prod_{k=0}^n(f_l-a_k)}}\\
+2\theta_{s_1}(s_2)\ \frac{(-1)^jG_{j,j-1}(\lambda_j^0)A(\lambda_j^0)}
{\sqrt{\prod_{k\ne j,j-1}(\lambda_j^0-b_k)
\prod_k(\lambda_j^0-a_k)}}\ =0\ .
\end{gathered}
\end{equation}
We then have the following proposition.
\begin{prop}
\begin{enumerate}
\item
$Z_{k,s_1}(s_2)=0$ for $k=j,j-1$ and any
$s_1,s_2$ such that $\theta_{s_1}(s_2)=\pi$.
\item $Z_{j,s_1}(s_2)$ and $Z_{j-1,s_1}(s_2)$
are linearly independent for any $s_1$ and $s_2$
such that $0<\theta_{s_1}(s_2)<\pi$.
\end{enumerate}
\end{prop}
\begin{proof}
We consider a one-parameter family of geodesics $t\to\gamma(u,t)$ such that $\gamma(0,t)=\gamma(t)$,
$\gamma(u,s_1)=\gamma(s_1)$, and the values $b_i$
of the first integrals $H_i$ for $\gamma(u,t)$ are the same as those
for $\gamma(t)$ except that $b_{j-1}(u)=
H_{j-1}(\flat(\dot\gamma(u,t)))=\lambda_j^0+u^2$.
Since $b_j=\lambda_j^0=f_j(x_j(u,s_1))$ for any $u$, it
follows that the Jacobi fields $Y_{j,s_1}(t)$ and
$Y_{j-1,s_1}(t)$ are defined along the geodesic
$\gamma(u,t)$ for $u\ne 0$. Observe that on the
unit cotangent space $U^*_{\gamma(s_1)}M$,
$(\partial/\partial \nu_j)/|\partial/\partial \nu_j|$ tends to
$\pm(\partial/\partial \mu_j)/|\partial/\partial \mu_j|$
and $(\partial/\partial H_{j-1})/|\partial/\partial H_{j-1}|$ tends to
$(\partial/\partial\mu_{j-1})/|\partial/
\partial\mu_{j-1}|$ as $u\to 0$. Thus the Jacobi
fields
$Y_{j,s_1}(t)$ and $Y_{j-1,s_1}(t)$ along
the geodesic $\gamma(u,t)$ converge to Jacobi fields
$Z_{j,s_1}(t)$ and $Z_{j-1,s_1}(t)$ up to the sign along the
geodesic $\gamma(t)$ as $u\to 0$.
Moreover, with this procedure of taking the limit, we claim
that the Jacobi fields $Y_{j,s_2}(t)$ and $Y_{j-1,s_2}(t)$
along the geodesic $\gamma(u,t)$ tend to
\begin{equation*}
\epsilon\left(\cos\theta Z_{j,s_2}(t)+
\sin\theta Z_{j-1,s_2}(t)\right)\
\text{and}\ \epsilon
\left(-\sin\theta Z_{j,s_2}(t)+
\cos\theta Z_{j-1,s_2}(t)\right)
\end{equation*}
respectively, where $\epsilon=\pm 1$ and
$\theta=\theta_{s_1}(s_2)$.
To see this, we begin with the formula before taking the
limit:
\begin{equation}
\sum_{i=1}^n\int_{\sigma_i(s_1)}^{\sigma_i(s_2)}\frac{(-1)^iG_{j,j-1}(f_i)A(f_i)}
{\sqrt{-\prod_{k=1}^{n-1}(f_i-b_k)
\cdot\prod_{k=0}^n(f_i-a_k)}}\
\ d\sigma_i=0\ .
\end{equation}
Define the function $\theta(u,t)$ by
\begin{gather*}
f_j(x_j(u,t))=b_j(\cos\theta(u,t))^2+b_{j-1}(u)
(\sin\theta(u,t))^2\ ,\\
\theta(u,s_1)=0,\qquad (\partial/\partial t)\theta\ge 0\ .
\end{gather*}
Then, taking the limit $u\to 0$,
we see that
\begin{equation*}
\int_{\sigma_j(s_1)}^{\sigma_j(s_2)}\frac{(-1)^jG_{j,j-1}(f_j)A(f_j)}
{\sqrt{-\prod_{k=1}^{n-1}(f_j-b_k)
\cdot\prod_{k=0}^n(f_j-a_k)}}\
\ d\sigma_j
\end{equation*}
tends to
\begin{equation*}
2\theta(0,s_2)\ \frac{(-1)^jG_{j,j-1}(\lambda_j^0)A(\lambda_j^0)}
{\sqrt{\prod_{k\ne j,j-1}(\lambda_j^0-b_k)
\prod_k(\lambda_j^0-a_k)}}\ .
\end{equation*}
Thus we have $\theta(0,t)=\theta_{s_1}(t)$ by (\ref{eq:theta}). The covector $\partial/\partial H_j$
at the point $\gamma(u,s_2)$ is equal to
\begin{equation*}
\frac14\sum_{i=1}^n\frac{\epsilon'_i(-1)^iG_j(f_i)A(f_i)
\ df_i}
{\sqrt{-\prod_{k=1}^{n-1}(f_i-b_k)
\cdot\prod_{k=0}^n(f_i-a_k)}}\ ,
\end{equation*}
which tends to, as $u\to 0$,
\begin{gather*}
\frac14\sum_{i\ne j}\frac{f_i-\lambda_j^0}
{|f_i-\lambda_j^0|}\frac{\epsilon'_i(-1)^i
G_{j,j-1}(f_i)A(f_i)\ df_i}
{\sqrt{-\prod_{k\ne j,j-1}(f_i-b_k)
\cdot\prod_{k=0}^n(f_i-a_k)}}\\
+\frac14\frac{(-1)^{j+1}\cot \theta\ G_{j,j-1}
(\lambda_j^0)A(\lambda_j^0)
\ df_j}
{\sqrt{\prod_{k\ne j,j-1}(\lambda_j^0-b_k)
\cdot\prod_{k=0}^n(\lambda_j^0-a_k)}}\ ,
\end{gather*}
where $\theta=\theta_{s_1}(s_2)$. Also,
$\partial/\partial H_{j-1}$ tends to
\begin{gather*}
\frac14\sum_{i\ne j}\frac{f_i-\lambda_j^0}
{|f_i-\lambda_j^0|}\frac{\epsilon'_i(-1)^i
G_{j,j-1}(f_i)A(f_i)\ df_i}
{\sqrt{-\prod_{k\ne j,j-1}(f_i-b_k)
\cdot\prod_{k=0}^n(f_i-a_k)}}\\
+\frac14\frac{(-1)^{j}\tan \theta\ G_{j,j-1}
(\lambda_j^0)A(\lambda_j^0)
\ df_j}
{\sqrt{\prod_{k\ne j,j-1}(\lambda_j^0-b_k)
\cdot\prod_{k=0}^n(\lambda_j^0-a_k)}}\ ,
\end{gather*}
As is easily seen, we have
\begin{gather*}
\flat(Z'_{j-1,s_2}(s_2))
=\frac{c}4\sum_{i\ne j}\frac{f_i-\lambda_j^0}
{|f_i-\lambda_j^0|}\frac{\epsilon'_i(-1)^i
G_{j,j-1}(f_i)A(f_i)\ df_i}
{\sqrt{-\prod_{k\ne j,j-1}(f_i-b_k)
\cdot\prod_{k=0}^n(f_i-a_k)}}\\
\flat(Z'_{j,s_2}(s_2))=\frac{c}4\frac{(-1)^{j+1}G_{j,j-1}
(\lambda_j^0)A(\lambda_j^0)
\ df_j}
{\sqrt{\prod_{k\ne j,j-1}(\lambda_j^0-b_k)
\cdot\prod_{k=0}^n(\lambda_j^0-a_k)}}\ ,
\end{gather*}
where $c=1/|\partial/\partial \mu_{j-1}|=
1/|\partial/\partial \mu_{j}|$ at $\gamma(s_2)$.
Therefore the claim follows.
From the formulas obtained above and
(\ref{eq:singjac2}), we thus have
\begin{equation}\label{eq:z1}
\begin{gathered}
g\left(Z_{j-1,s_1}(s_2),\ \cos\theta\ Z'_{j,s_2}(s_2)+
\sin\theta\ Z'_{j-1,s_2}(s_2)\right)=0\ ,\\
g\left(Z_{j,s_1}(s_2),\ -\sin\theta\ Z'_{j,s_2}(s_2)+
\cos\theta\ Z'_{j-1,s_2}(s_2)\right)=0\ ,\\
g\left(Z_{j,s_1}(s_2),\ \cos\theta\ Z'_{j,s_2}(s_2)+
\sin\theta\ Z'_{j-1,s_2}(s_2)\right)\\
=\frac{\sin\theta}{4cc'}\frac{(-1)^{j}G_{j,j-1}
(\lambda_j^0)A(\lambda_j^0)}
{\sqrt{-\prod_{k\ne j,j-1}(\lambda_j^0-b_k)
\cdot\prod_{k=0}^n(\lambda_j^0-a_k)}}\ ,
\end{gathered}
\end{equation}
where $c$ and $c'$ are the norms of $\partial/\partial \mu_j$ at $\gamma(s_1)$ and $\gamma(s_2)$
respectively.
In particular, we have:
\begin{align*}
&\cos\theta\ \Omega(Z_{j-1,s_1},Z_{j,s_2})
+\sin\theta\ \Omega(Z_{j-1,s_1},Z_{j-1,s_2})=0\\
&-\sin\theta\ \Omega(Z_{j,s_1},Z_{j,s_2})
+\cos\theta\ \Omega(Z_{j,s_1},Z_{j-1,s_2})=0\ ,
\end{align*}
where $\theta=\theta_{s_1}(s_2)$. As is easily seen, the above
formula is also valid when $s_2<s_1$, in which case
$\theta_{s_1}(s_2)=-\theta_{s_2}(s_1)<0$. Therefore,
exchanging $s_1$ and $s_2$ in the above formula, we have
\begin{equation}\label{eq:z2}
\begin{aligned}
&\Omega(Z_{j,s_1},Z_{j,s_2})=
\Omega(Z_{j-1,s_1},Z_{j-1,s_2})\\
& \Omega(Z_{j-1,s_1},Z_{j,s_2})
=- \Omega(Z_{j,s_1},Z_{j-1,s_2})\ .
\end{aligned}
\end{equation}
By (\ref{eq:z1}) and (\ref{eq:z2}) we also have
\begin{equation}\label{eq:z3}
\begin{aligned}
g\left(Z_{j-1,s_1}(s_2),\ -\sin\theta\ Z'_{j,s_2}(s_2)+
\cos\theta\ Z'_{j-1,s_2}(s_2)\right)\\
=\frac{\sin\theta}{4cc'}\frac{(-1)^{j}G_{j,j-1}
(\lambda_j^0)A(\lambda_j^0)}
{\sqrt{-\prod_{k\ne j,j-1}(\lambda_j^0-b_k)
\cdot\prod_{k=0}^n(\lambda_j^0-a_k)}}\ .
\end{aligned}
\end{equation}
Now the assertion (2) easily follows from (\ref{eq:z1}) and
(\ref{eq:z3}). Also, from those formulas we have
\begin{align*}
&g(Z_{j,s_1}(s_2),Z'_{j,s_2}(s_2))=
g(Z_{j,s_1}(s_2),Z'_{j-1,s_2}(s_2))=0\\
& g(Z_{j-1,s_1}(s_2),Z'_{j,s_2}(s_2))=
g(Z_{j-1,s_1}(s_2),Z'_{j-1,s_2}(s_2))=0\ ,
\end{align*}
provided $\theta_{s_1}(s_2)=\pi$. Since the Jacobi fields
$Z_{j,s}$, $Z_{j-1,s}$ belong to the limit of the vector
space $\mathcal Y_j+\mathcal Y_{j-1}$, and since it is
orthogonal to the limit of $\sum_{k\ne j,j-1}\mathcal Y_k$
with respect to the symplectic inner product $\Omega$,
it therefore follows that $Z_{j,s_1}(s_2)=Z_{j-1,s_1}(s_2)=0$. This finishes the proof of the proposition.
\end{proof}
\begin{remark}\label{remark:jac}
For $i$ with $b_i\ne b_{i-1}$ and $b_i\ne
b_{i+1}$, Propositions \ref{prop:jf1}, \ref{prop:jacreg}
and Corollary \ref{cor:conj} equally hold for
the Jacobi field $Y_{i,s}(t)$.
\end{remark}
\section{Geodesics starting at a one point}
In this and the subsequent sections we shall assume
that the condition (\ref{cond2}) are satisfied.
Let $p_0\in M$ be an arbitrary point. We may assume without
loss of generality that $p_0$ is
represented by $(x_1,\dots,x_n)=
(x_1^0,\dots,x_n^0)$, where $0\le x_i^0\le\alpha_i/4$ $(1\le i\le n)$.
Let $U^*_{p_0}M$ be the sphere of unit covectors at $p_0$.
We denote by
\begin{equation*}
t\mapsto\gamma(t,\eta)=
(x_1(t,\eta),\dots,x_n(t,\eta))
\end{equation*}
the geodesic with the initial covector
$\eta\in U^*_{p_0}M$ at $t=0$.
The function $x_i(t,\eta)$ is uniquely determined as a smooth
function when $b_i\ne a_i$ and $b_{i-1}\ne a_{i-1}$
for each $i$. In this case,
the geodesic does not meet $J_i\cup J_{i-1}$, a part of the branch locus.
If $b_i=a_i$, then the geodesic meets
$J_i$ and one gets more
than one representations for $x_i(t,\eta)$ and
$x_{i+1}(t,\eta)$ that are
continuous at the branch point and smooth elsewhere. Note
that $t\mapsto f_i(x_i(t,\eta))$ is uniquely determined in any
case.
As before, we put
\begin{equation*}
\sigma_i(t,\eta)=\int_0^t\left|
\frac{df_i(x_i(t,\eta))}{dt}\right|\,dt\ .
\end{equation*}
We shall assign a real number $t_0(\eta)>0$ to each
$\eta\in U^*_{p_0}M$. First we consider the case which is {\it not}
equal to any one of the following three cases: (i) the geodesic
$\gamma(t,\eta)$ is totally contained in the submanifold
$N_n$, i.e., $b_{n-1}= a_n$; (ii) $\gamma(t,\eta)$ is
totally contained in the submanifold $N_{n-1}$ and
$f_n(x_n^0)=a_{n-1}=b_{n-1}<f_{n-1}(x_{n-1}^0)$; and (iii)
$\gamma(t,\eta)$ is
totally contained in the submanifold $N_{n-1}$ and $p_0\in J_{n-1}$, in particular, $f_n(x_n^0)=a_{n-1}=b_{n-1}=f_{n-1}(x_{n-1}^0)$. Then,
define $t_0(\eta)$ by the formula
\begin{equation*}
\sigma_n(t_0(\eta),\eta)=2(a_{n-1}^--a_n^+)\ .
\end{equation*}
In the cases (i) and (ii) listed above, we define
$t_0(\eta)$ as follows: Let $Y(t)$
be the
Jacobi field along the geodesic $\gamma(t,\eta)$ such that
$Y(0)=0$ and $Y'(0)=(\partial/\partial x_n)/
|\partial/\partial x_n|$. Then $t=t_0(\eta)$
is the first positive time such that $Y(t)=0$.
In the case (iii) we define the Jacobi field
$Y(t)$ along the geodesic $\gamma(t,\eta)$ such that
$Y(0)=0$ and $Y'(0)$ is the unit normal vector to $N_{n-1}$. Then $t=t_0(\eta)$ is the first positive time such that
$Y(t)=0$. It is easily seen that $x_n(t_0(\eta),\eta)=
-x_n^0$, or $\frac{\alpha_n}2+x_n^0$ in any case.
It will be proved in Theorem \ref{thm:cut} that
the time $t=t_0(\eta)$ gives the cut point of
$p_0$ along the geodesic $\gamma(t,\eta)$. In particular,
it will become clear that $t_0(\eta)$ is a continuous function
of $\eta\in U^*_{p_0}M$ and $p_0\in M$. In this stage,
we shall only prove a partial result.
\begin{prop}\label{prop:conti}
For any $\eta\in U^*_{p_0}M$ and $p_0\in M$, there is a
sequence $\eta_k$ $(k=1,2,\dots)$ of unit covectors such
that the corresponding values $b_1,\dots,b_{n-1}$
of $H_1,\dots,$ $H_{n-1}$ at
$\eta_k$ and $a_0,\dots,a_n$ are all distinct for each $k$,
and
\begin{equation*}
\lim_{k\to\infty}\eta_k=\eta,\qquad \lim_{k\to\infty}
t_0(\eta_k)=t_0(\eta)\ .
\end{equation*}
\end{prop}
\begin{proof} At each covector $\eta$ which is not of the
cases (i), (ii), (iii), the function $t_0(\eta)$ is clearly
continuous, and we can find such $\{\eta_k\}$. For $\eta$
of the cases (i) or (ii) we note that $t_0(\eta)$ is equal to
the limit
$\lim_{s\to 0}t_0(\eta_s)$, where $\eta_s\in U^*_{p_0}$ is a
one-parameter family of covectors such that (i)
$b_{n-1}=a_n+s^2$, (ii) $b_{n-1}=a_{n-1}+s^2$, and other $b_j$'s are the same value as those
for $\eta=\eta_0$.
Now, for $\eta\in U^*_{p_0}M$ of the cases (ii), (iii),
we first choose $\{\tilde \eta_k\}\in U^*_{p_k}M$ such that
each $\tilde \eta_k$ is of the case (ii), $\tilde\eta_k\to\eta$ $(k\to\infty)$, and the values $b_1,\dots,b_{n-2}$
for each $\tilde\eta_k$ and $a_0,\dots,a_n$ are all distinct.
Then, for each $k$ we choose $\eta_k\in U^*_{p_k}M$ in the one-parameter family of covectors given above whose limit
is $\tilde\eta_k$ so that $\eta_k\to\eta$ as $k\to\infty$.
The case (i) is similar.
\end{proof}
For a while, we shall assume that $p_0\not\in J_{n-1}$.
Put
\begin{align*}
U_+=&\{\eta\in U^*_{p_0}M\ |\ \xi_n(\eta)>0\}\\
U_-=&\{\eta\in U^*_{p_0}M\ |\ \xi_n(\eta)<0\}\ .
\end{align*}
Note that they are well-defined hemispheres under the
assumption $p_0\not\in J_{n-1}$.
Let $\eta'\in U^*_{p_0}M$
be the reflection image of $\eta\in U^*_{p_0}M$
with respect to the hyperplane $H_n$ in
$T^*_{p_0}M$ defined by $\xi_n=0$, i.e.,
$\xi_n(\eta')=-\xi_n(\eta)$, $\xi_i(\eta')=
\xi_i(\eta)$ $(1\le i \le n-1)$.
\begin{prop}\label{prop:refl}
$\gamma(t_0(\eta'),\eta')=\gamma(t_0(\eta),\eta)$ for any $\eta\in U_+$.
\end{prop}
\begin{proof}
It is enough to show this for covectors $\eta$ such that
$b_i$'s and $a_j$'s are all distinct.
By (\ref{eq:geod}) we have
\begin{equation*}
\sum_{i=1}^n\int_0^{t_0(\eta)}\frac{(-1)^iG(f_i)A(f_i)}
{\sqrt{-\prod_{k=1}^{n-1}(f_i-b_k)
\cdot\prod_{k=0}^n(f_i-a_k)}}\
\left|\frac{df_i(x_i(t,\eta))}{dt}\right|
\ dt=0
\end{equation*}
for any polynomial $G(\lambda)$ of degree $\le n-2$.
By using the variables $\sigma_i$ given above, this formula
is rewritten as
\begin{equation}\label{eq:geodsigma}
\sum_{i=1}^n\int_0^{\sigma_i(t_0(\eta),\eta)}\frac{(-1)^iG(f_i)A(f_i)}
{\sqrt{-\prod_{k=1}^{n-1}(f_i-b_k)
\cdot\prod_{k=0}^n(f_i-a_k)}}\
\ d\sigma_i=0\ .
\end{equation}
Note that
\begin{equation}\label{eq:geodnth}
\begin{gathered}
\int_0^{\sigma_n(t_0(\eta),\eta)}\frac{(-1)^iG(f_n)A(f_n)}
{\sqrt{-\prod_{k=1}^{n-1}(f_n-b_k)
\cdot\prod_{k=0}^n(f_n-a_k)}}\
\ d\sigma_n\\
=2\int_{a_n^+}^{a_{n-1}^-}\frac{(-1)^iG(\lambda)A(\lambda)}
{\sqrt{-\prod_{k=1}^{n-1}(\lambda-b_k)
\cdot\prod_{k=0}^n(\lambda-a_k)}}
\ d\lambda\ .
\end{gathered}
\end{equation}
Since the values of each $b_i$ are the same for the two
covectors $\eta$ and $\eta'$, and since $\sigma_n(t_0(\eta),\eta)=2(a_{n-1}^--a_n^+)=\sigma_n(t_0(\eta'),\eta')$, we then have
\begin{equation}\label{eq:prime2}
\begin{aligned}
\sum_{i=1}^{n-1}\int_0^{\sigma_i(t_0(\eta),\eta)}\frac{(-1)^iG(f_i)A(f_i)}
{\sqrt{-\prod_{k=1}^{n-1}(f_i-b_k)
\cdot\prod_{k=0}^n(f_i-a_k)}}\
\ d\sigma_i\\
=\sum_{i=1}^{n-1}\int_0^{\sigma_i(t_0(\eta'),\eta')}\frac{(-1)^iG(f_i)A(f_i)}
{\sqrt{-\prod_{k=1}^{n-1}(f_i-b_k)
\cdot\prod_{k=0}^n(f_i-a_k)}}\
\ d\sigma_i
\end{aligned}
\end{equation}
Now, let $I$ be the set of $i\in\{1,\dots,n-1\}$ such that
\begin{equation*}
\sigma_i(t_0(\eta),\eta)>\sigma_i(t_0(\eta'),\eta')\ .
\end{equation*}
Then, as we shall prove in the next lemma, there is a
polynomial $G(\lambda)$ of degree $\le n-2$ such that
$(-1)^iG(\lambda)>0$ for $\lambda\in (a_i^+,a_{i-1}^-)$,
$i\in I$, and $(-1)^iG(\lambda)<0$ for $\lambda\in
(a_i^+, a_{i-1}^-)$, $i\not\in I$, if $I\ne \emptyset$.
With such
$G(\lambda)$, the formula (\ref{eq:prime2}) clearly yields
a contradiction. Therefore, $I=\emptyset$ and
\begin{equation*}
\sigma_i(t_0(\eta),\eta)=\sigma_i(t_0(\eta'),\eta')\ .
\end{equation*}
for every $1\le i\le n-1$. This indicates
\begin{equation*}
x_i(t_0(\eta),\eta)=x_i(t_0(\eta'),\eta')\ .
\end{equation*}
for any $1\le i\le n$, and therefore
$\gamma(t_0(\eta'),\eta')=\gamma(t_0(\eta),\eta)$\ .
\end{proof}
\begin{lemma}\label{lem:G}
Suppose $b_i$'s and $a_i$'s are all distinct. Let $I_1$ be a
subset of $\{1,\dots,n\}$ and let $I_2$ be its complement.
Assume both $I_1$ and $I_2$ are nonempty. Then there is a
polynomial $G(\lambda)$ of degree $\le n-2$ such that
\begin{equation*}
(-1)^iG(\lambda)
\begin{cases}
>0\quad \text{for } \lambda\in (a_i^+,a_{i-1}^-),\ i\in I_1\\
<0\quad \text{for } \lambda\in (a_i^+,a_{i-1}^-),\ i\in I_2
\end{cases}\ .
\end{equation*}
\end{lemma}
\begin{proof}
Assume $1\in I_1$. We put
\begin{equation*}
G(\lambda)=-\prod(\lambda-b_k)\ ,
\end{equation*}
where the product are taken over all such
$k\in\{1,\dots,n-1\}$ that both $k$ and $k+1$ belongs to $I_1$ or that both $k$ and $k+1$ belongs to $I_2$. Since both $I_1$
and $I_2$ are nonempty, it follows that $\deg G\le n-2$.
Also, it is clear that the signs of the function $G(\lambda)$
is different on the two intervals $(a_k^+,a_{k-1}^-)$ and
$(a_{k+1}^+,a_k^-)$ if and only if $\lambda-b_k$ is a factor
of $G(\lambda)$, i.e., $k$ and $k+1$ belong to the same
group. Since $-G(\lambda)>0$ on $(a_1^+,a_0^-)$,
it follows that this $G(\lambda)$ has the desired property.
In case $1\in I_2$, then $-G(\lambda)$ possesses the desired
property.
\end{proof}
\begin{prop}\label{prop:prime}
$t_0(\eta)=t_0(\eta')$ for any $\eta\in U^*_{p_0}M$.
\end{prop}
\begin{proof}
By (\ref{eq:length}) we have
\begin{equation}\label{eq:t0}
t_0(\eta)=\sum_{i=1}^n\int_0^{\sigma_i(t_0(\eta),\eta)}\frac{(-1)^{i+1}A(f_i)\prod_{k=1}^{n-1}(f_i-a_k)}
{2\sqrt{-\prod_{k=1}^{n-1}(f_i-b_k)
\cdot\prod_{k=0}^n(f_i-a_k)}}\
\ d\sigma_i
\end{equation}
Since $\sigma_i(t_0(\eta),\eta)=\sigma_i(t_0(\eta'),\eta')$
for any $i$ by Proposition \ref{prop:refl}, it therefore
follows that $t_0(\eta)=t_0(\eta')$.
\end{proof}
\begin{prop}\label{prop:sigma}
Suppose that the geodesic $\gamma(t,\eta)$ does not totally
contained in any $N_j$ for any $j$. Then,
$\sigma_i(t_0(\eta),\eta)< 2(a_{i-1}^--a_i^+)$ for
any $i\le n-1$ such that $b_i\ne b_{i-1}$.
\end{prop}
\begin{proof}
The assumption implies that there is no $i$
such that $b_i=a_{i+1}$ or $b_{i+1}=a_i$.
First, suppose that $b_1,\dots,b_{n-1}$ and $a_0,
\dots,a_n$ are all distinct. Let $I_1$ be the set of
$i\in\{1,\dots,n-1\}$ such that $\sigma_i(t_0(\eta),\eta)
\ge 2(a_{i-1}^--a_i^+)$. Assume that $I_1\ne\emptyset$.
Put $I_2=\{1,\dots,n\}-I_1$. Note that $n\in I_2$. For these
$I_1$ and $I_2$, let $G(\lambda)$ be the polynomial given in
the proof of Lemma \ref{lem:G}. Then we have
\begin{equation}\label{eq:ineq}
\begin{aligned}
&2\sum_{i=1}^n\int_{a_i^+}^{a_{i-1}^-}
\frac{(-1)^{i}G(\lambda)A(\lambda)\ d\lambda}
{\sqrt{-\prod_{k=1}^{n-1}(\lambda-b_k)
\cdot\prod_{k=0}^n(\lambda-a_k)}}\\
=&-\sum_{i\in I_1}\int_{2(a_{i-1}^--a_i^+)}^{\sigma_i(t_0(\eta),\eta)}\frac{(-1)^iG(f_i)A(f_i)}
{\sqrt{-\prod_{k=1}^{n-1}(f_i-b_k)
\cdot\prod_{k=0}^n(f_i-a_k)}}\
\ d\sigma_i\\
+&\sum_{i\in I_2-\{n\}}\int_{\sigma_i(t_0(\eta),\eta)}^{2(a_{i-1}^--a_i^+)}\frac{(-1)^iG(f_i)A(f_i)}
{\sqrt{-\prod_{k=1}^{n-1}(f_i-b_k)
\cdot\prod_{k=0}^n(f_i-a_k)}}\
\ d\sigma_i\ .
\end{aligned}
\end{equation}
Here, the polynomial $G(\lambda)$ is of the form
\begin{equation*}
G(\lambda)=
\begin{cases}
-\prod_{k\in K}(\lambda-b_k)\quad (\text{if }1\in I_1)\\
\prod_{k\in K}(\lambda-b_k)\quad (\text{if }1\in I_2)
\end{cases}\ ,
\end{equation*}
where $K$ is the subset of $\{1,\dots,n-1\}$ such that
$k\in K$ means $k$ and $k+1$ belong to the same
group, i.e., $k,k+1\in I_1$, or $k,k+1\in I_2$.
Therefore, $n-1-\#K$ is the number of such $k\in\{1,\dots,n-1\}$ that $k$ and
$k+1$ belong to the different groups. Since $n\in I_2$,
it follows that
\begin{equation*}
n-1-\#K\ \text{is }
\begin{cases}
\text{ odd\quad if }\ 1\in I_1\\
\text{ even\quad if }\ 1\in I_2.
\end{cases}
\end{equation*}
Therefore, by Proposition \ref{prop:cond2} (1)
it follows that the
first line in the formulas (\ref{eq:ineq}) is
positive, while the second and the third lines are
nonpositive, which is a contradiction. Thus $I_1$ must be
empty, and the proposition follows.
Next, we shall consider the case where $b_{j-1}=b_j$
for several $j$, but other $b_k$ and $a_k$ are all
distinct. In this case, we define the subset $I_1$
of $\{1,\dots,n-1\}$ as follows: For $k$ with $b_{k-1}\ne b_k$, $k\in I_1$ if and only if $\sigma_k
(t_0(\eta),\eta)\ge 2(a_{k-1}^--a_k^+)$; for $k$
with $b_{k-1}=b_k$, $k\in I_1$ if and only if $k-1
\in I_1$ or $k+1\in I_1$. Note that $b_{k-1}<b_{k-2}$
and $b_{k+1}<b_k$ if $b_k=b_{k-1}$.
Then, by the same way as above, we define the sets
$I_2$, $K$ and the polynomial $G(\lambda)$. Put
\begin{equation*}
J=\{j\ |\ b_j<b_{j-1}, 1\le j\le n-1\}\ .
\end{equation*}
Since $k-1\in K$ or $k\in K$ if $b_k=b_{k-1}$,
we then have, instead of \eqref{eq:ineq}, the
following formula:
\begin{equation}\label{eq:ineq2}
\begin{aligned}
&2\sum_{i\in J}\int_{a_i^+}^{a_{i-1}^-}
\frac{(-1)^{i}G(\lambda)A(\lambda)\ d\lambda}
{\sqrt{-\prod_{k=1}^{n-1}(\lambda-b_k)
\cdot\prod_{k=0}^n(\lambda-a_k)}}\\
=&-\sum_{i\in I_1\cap J}\int_{2(a_{i-1}^--a_i^+)}^{\sigma_i(t_0(\eta),\eta)}\frac{(-1)^iG(f_i)A(f_i)}
{\sqrt{-\prod_{k=1}^{n-1}(f_i-b_k)
\cdot\prod_{k=0}^n(f_i-a_k)}}\
\ d\sigma_i\\
+&\sum_{i\in I_2\cap J}\int_{\sigma_i(t_0(\eta),\eta)}^{2(a_{i-1}^--a_i^+)}\frac{(-1)^iG(f_i)A(f_i)}
{\sqrt{-\prod_{k=1}^{n-1}(f_i-b_k)
\cdot\prod_{k=0}^n(f_i-a_k)}}\
\ d\sigma_i\ .
\end{aligned}
\end{equation}
If $I_1\cap J\ne\emptyset$, then we have a
contradiction by the same reason as above.
Finally, let us further assume that $b_i=a_i$
for some $i$. In this case,
the times t such that $f_i(x_i(t,\eta))=a_i$ and those such
that $f_{i+1}(x_{i+1}(t,\eta))=a_i$ coincide. Therefore,
in each side of the formula \eqref{eq:ineq} or
\eqref{eq:ineq2}, the sum of the
integrals in $\sigma_i$ and $\sigma_{i+1}$ remains finite,
and the arguments above are also effective in this case.
\end{proof}
\begin{prop}\label{prop:ineqsing}
Suppose that the geodesic $\gamma(t,\eta)$ does not totally
contained in any $N_k$.
For a fixed $j$ with $b_j=b_{j-1}$, let $\theta_{s_1}(s_2)$
be the value defined in the formula
\eqref{eq:theta} in the previous section.
Then, $\theta_0(t_0(\eta))< \pi$
for such $j$.
\end{prop}
\begin{proof}
By \eqref{eq:theta} we have
\begin{equation*}
\begin{gathered}
\sum_{\substack{1\le l\le n\\l\ne j}}\int_{0}^{\sigma_l(s)}\frac{(-1)^l
G_{j,j-1}(f_l)A(f_l)\ d\sigma_l}
{|f_l-\lambda_j^0|\sqrt{-\prod_{k\ne j,j-1}(f_l-b_k)
\cdot\prod_{k=0}^n(f_l-a_k)}}\\
+2\theta_{0}(s)\ \frac{(-1)^jG_{j,j-1}(\lambda_j^0)A(\lambda_j^0)}
{\sqrt{\prod_{k\ne j,j-1}(\lambda_j^0-b_k)
\prod_k(\lambda_j^0-a_k)}}\ =0\ .
\end{gathered}
\end{equation*}
Also, taking a limit $a_j^+,a_{j-1}^-\to\lambda_j^0$ in Lemma \ref{lemma:flat}, we have
\begin{equation*}
\begin{gathered}
\sum_{\substack{1\le l\le n\\l\ne j}}\int_{0}^{2(a_{l-1}^--
a_l^+)}\frac{(-1)^l
G_{j,j-1}(f_l)A(\lambda_j^0)\ d\sigma_l}
{|f_l-\lambda_j^0|\sqrt{-\prod_{k\ne j,j-1}(f_l-b_k)
\cdot\prod_{k=0}^n(f_l-a_k)}}\\
+2\pi\ \frac{(-1)^jG_{j,j-1}(\lambda_j^0)A(\lambda_j^0)}
{\sqrt{\prod_{k\ne j,j-1}(\lambda_j^0-b_k)
\prod_k(\lambda_j^0-a_k)}}\ =0\ .
\end{gathered}
\end{equation*}
Therefore we obtain the following formula:
\begin{equation*}
\begin{gathered}
\sum_{\substack{1\le l\le n\\l\ne j}}\int_{\sigma_l(s)}^{2
(a_{l-1}^--a_l^+)}\frac{(-1)^l
G_{j,j-1}(f_l)A(f_l)\ d\sigma_l}
{|f_l-\lambda_j^0|\sqrt{-\prod_{k\ne j,j-1}(f_l-b_k)
\cdot\prod_{k=0}^n(f_l-a_k)}}\\
-\sum_{\substack{1\le l\le n\\l\ne j}}\int_{0}^{2
(a_{l-1}^--a_l^+)}\frac{(A(f_l)-A(\lambda_j^0))\,(-1)^l
G_{j,j-1}(f_l)\ d\sigma_l}
{|f_l-\lambda_j^0|\sqrt{-\prod_{k\ne j,j-1}(f_l-b_k)
\cdot\prod_{k=0}^n(f_l-a_k)}}\\
+2(\pi-\theta_{0}(s))\ \frac{(-1)^jG_{j,j-1}(\lambda_j^0)A(\lambda_j^0)}
{\sqrt{\prod_{k\ne j,j-1}(\lambda_j^0-b_k)
\prod_k(\lambda_j^0-a_k)}}\ =0\ .
\end{gathered}
\end{equation*}
We put $s=t_0(\eta)$.
The first line of this formula is nonpositive by the
previous proposition. Also, applying the $n-1$-dimensional
version of Proposition \ref{prop:cond2} (1) to the positive
function
\begin{equation*}
\left(A(\lambda)-A(\lambda_j^0)\right)/(\lambda-\lambda_j^0)
\ ,
\end{equation*}
the second line is negative. Since $(-1)^jG_{j,j-1}(\lambda_j^0)>0$,
it thus follows that $\theta_0(t_0(\eta))<
\pi$.
\end{proof}
As a consequence, we have the following proposition.
\begin{prop}\label{prop:noconj}
Suppose that the geodesic $\gamma(t,\eta)$ does not totally
contained in any $N_k$. Then:
\begin{enumerate}
\item There is no conjugate point of $p_0$ along the geodesic
$\gamma(t,\eta)$ in the interval $0<t<t_0(\eta)$.
\item $\gamma(t_0(\eta),\eta)$ is not a conjugate point of
$p_0$ along the geodesic $\gamma(t,\eta)$, unless $b_{n-1}(=H_{n-1}(\eta))=f_n(x_n^0)$.
\item If $b_{n-1}=f_n(x_n^0)$, then $\gamma(t_0(\eta),\eta)$
is a conjugate point of $p_0$ along the geodesic
$\gamma(t,\eta)$ with multiplicity one.
\end{enumerate}
\end{prop}
\begin{proof}
(1) and (2) follow from all results in \S4 and
Propositions \ref{prop:sigma} and
\ref{prop:ineqsing}. Now, let us prove (3).
Since $f_n(x_n^0)=b_{n-1}$, it follows
from Corollary \ref{cor:conj} (1) that
$Y_{n-1,0}(t_0(\eta))=0$. Hence $\gamma(t_0(\eta),\eta)$
is a conjugate point of $p_0$ along the geodesic
$\gamma(t,\eta)$. Now we show that $Y_{j,0}
(t_0(\eta))\ne 0$ (or, $Z_{j,0}(t_0(\eta))\ne 0$)
for any $j\le n-2$. First, suppose that $b_j\ne
b_{j-1}$ for any $j$. For $k\le n-2$ with
$b_k\ne f_k(x_k^0)$, $f_{k+1}(x_{k+1}^0)$ , we
have $Y_{k,0}(t_0(\eta))\ne 0$ by Propositions
\ref{prop:sigma} and \ref{prop:jacreg}. If
$b_k= f_k(x_k^0)$ or $f_{k+1}(x_{k+1}^0)$,
then again we have $Y_{k,0}(t_0(\eta))\ne 0$ by
Proposition \ref{prop:sigma} and Corollary
\ref{cor:conj} (1). In case $b_j=b_{j-1}$ for
some $j$, we also have $Z_{j,0}(t_0(\eta))\ne 0$
and $Z_{j-1,0}(t_0(\eta))\ne 0$ in the same way as above by Proposition \ref{prop:ineqsing}.
\end{proof}
\section{Cut locus (1)}
Let $p_0$ be a point as in \S5.
Let $N$ be the subset of $M$ represented by
$x_n=\frac{\alpha_n}2+x_n^0$ or $-x_n^0$,
which is a submanifold of $M$ diffeomorphic to
the $(n-1)$-sphere if $0\le x_n^0< \alpha_n/4$, and which is a
submanifold with boundary diffeomorphic to closed $(n-1)$-disk
if $x_n^0=\alpha_n/4$. Let $t_0(\eta)$ be the value defined
in the previous section.
\begin{thm}\label{thm:cut}
\begin{enumerate}
\item The cut point of $p_0$ along
the geodesic $\gamma(t,\eta)$ is given by $t=t_0(\eta)$
for any $p_0\in M$ and $\eta\in U^*_{p_0}M$.
\item Suppose $p_0\not\in J_{n-1}$. Then, the assignment
$\eta\mapsto \gamma(t_0(\eta),\eta)$ gives
a homeomorphism from $\overline{U_+}$ to its image
$C(p_0)$, the cut locus of $p_0$, and it gives $C^\infty$
embeddings of $U_+$ and $\partial \overline{U_+}$
respectively.
In particular, $C(p_0)$ is diffeomorphic to an $(n-1)$-closed
disk, and it is contained in (the interior of) $N$. Also,
for each $\eta\in \partial \overline{U_+}$, $\gamma(t_0(\eta),\eta)$ is the
first conjugate point of $p_0$ of multiplicity one along
the geodesic $t\mapsto\gamma(t,\eta)$ .
\item Suppose $p_0\in J_{n-1}$. Then the cut locus $C(p_0)$ coincides with the cut locus of
$p_0$ in the totally geodesic submanifold $N_{n-1}$, which is smoothly embedded $(n-2)$-disk
in $J_{n-1}$. For each interior
point $q$ of $C(p_0)$ there is an $S^1$-family of minimal
geodesics joining $p_0$ and $q$; the tangent vectors
of those geodesics at $p_0$ form a cone whose orthogonal
projection to $T_{p_0}J_{n-1}$ is one-dimensional. For each boundary point $q$ of
$C(p_0)$, there is a unique minimal geodesic from $p_0$ to $q$,
and along it $q$ is the first conjugate point of $p_0$ of
multiplicity two.
\end{enumerate}
\end{thm}
In this and the next two sections, we shall prove this theorem.
The proof will be divided into five cases: (I)
$p_0\not\in N_k$ for any $k$; (II) $0<x_n^0<\alpha_n/4$, but $p_0\in N_l$ for some $l$; (III) $x_n^0=0$; (IV) $x_n^0=\alpha_n/4$, and $p_0\not\in
J_{n-1}$; (V) $p_0\in J_{n-1}$.
In this section we shall consider the case (I) and prove (1) and (2)
of the theorem in this case. The proofs for the cases (II) $\sim$ (V) will be given in the next
two sections.
For each $\eta\in U_-$, let $t_-(\eta)$ be the
first positive time $t$ such that $x_n(t, \eta)
=-x_n^0$. Define the mapping $\Phi:U_{p_0}^*M\to N$ by
\begin{equation*}
\Phi(\eta)=\gamma(t_0(\eta),\eta)\quad (\eta\in \overline{
U^+});\qquad =\gamma(t_-(\eta),\eta)\quad (\eta\in \overline{
U_-})\ .
\end{equation*}
Then, $\Phi(\eta)\in N$ is the first point that the geodesic
$\gamma(t,\eta)$ meets $N$ for any $\eta$. We shall prove
that $\Phi$ is a homeomorphism. To do so, we need several
lemmas.
Take a point $p'_0$ in such
a way that $p'_0$ is represented as
$(x_1^0,\dots,x_{n-1}^0,\allowbreak x_n^1)$, where
$0\le x_n^1<x_n^0< \alpha_n/4$. Let $U_+'$ be the hemisphere of $U^*_{p'_0}M$ defined by
$\xi_n>0$. We define the mapping
$\psi: \overline{U_+}\to U'_+$
so that it preserves the values $b_i$ of $H_i$
$(1\le i\le n-1)$, i.e., by
$\psi(p_0; \xi_1,\dots,\xi_n)
=(p'_0; \tilde\xi_1,\dots,\tilde\xi_n)$, where
\begin{equation*}
\tilde\xi_i=\xi_i\quad (1\le i\le n-1),\qquad
\tilde\xi_n=\sqrt{(-1)^{n-1}\prod_{k=1}^{n-1}
(f_n(x_n^1)-b_k)}\ .
\end{equation*}
Note that $b_k$'s are functions of
$(p_0; \xi_1,\dots,\xi_n)\in \overline{U_+}$.
Since $b_{n-1}\ge f_n(x_n^0)>f_n(x_n^1)$,
the image $\psi(\overline{U_+})$ is contained in
the interior $U'_+$. Let $N'$ be the
submanifold of $M$ defined by
$x_n=-x_n^1$, and define the diffeomorphism
$\Psi: N\to N'$ by
\begin{equation*}
\Psi(x_1,\dots,x_{n-1},-x_n^0)=
(x_1,\dots,x_{n-1},-x_n^1).
\end{equation*}
We also define $\tilde\Phi:U_+'\to N'$
in the same way as $\Phi|_{\overline{U_+}}$.
\begin{lemma}\label{lemma:shift}
$\Psi(\Phi(\eta))=\tilde\Phi(\psi(\eta))$
for any $\eta\in\overline{U_+}$.
\end{lemma}
\begin{proof}
We write $\psi(\eta)=\tilde \eta$ for simplicity.
For the geodesics $\gamma(t,\eta)$ and $\gamma(t,\tilde\eta)$, we have the equality (\ref{eq:geodsigma}) and the similar one.
Taking the equality (\ref{eq:geodnth}) into
account, we have the similar formula as
(\ref{eq:prime2}):
\begin{gather*}
\sum_{i=1}^{n-1}\int_0^{\sigma_i(t_0(\eta),\eta)}\frac{(-1)^iG(f_i)A(f_i)}
{\sqrt{-\prod_{k=1}^{n-1}(f_i-b_k)
\cdot\prod_{k=0}^n(f_i-a_k)}}\
\ d\sigma_i\\
=\sum_{i=1}^{n-1}\int_0^{\sigma_i(t_0(\tilde\eta),\tilde\eta)}\frac{(-1)^iG(f_i)A(f_i)}
{\sqrt{-\prod_{k=1}^{n-1}(f_i-b_k)
\cdot\prod_{k=0}^n(f_i-a_k)}}\
\ d\sigma_i\ .
\end{gather*}
Therefore, in the same way as the proof of
Proposition \ref{prop:refl}, we have
$\sigma_i(t_0(\tilde\eta),\tilde\eta)=
\sigma_i(t_0(\eta),\eta)$ and hence
$x_i(t_0(\tilde\eta),\tilde\eta)=
x_i(t_0(\eta),\eta)$ for any $i\le n-1$.
Thus we have $\gamma(t_0(\tilde\eta),\tilde\eta)=
\Psi(\gamma(t_0(\eta),\eta))$. By the formula
(\ref{eq:t0}) we also have $t_0(\tilde\eta)=
t_0(\eta)$.
\end{proof}
By Proposition \ref{prop:noconj}, we know that
$\Phi|_{U_+}$ is a local diffeomorphism and so
is true for the initial point $p'_0$.
Therefore it follows from the above lemma that
$\Phi|_{\overline{U_+}}$ is a local
homeomorphism and $\Phi|_{\partial \overline{U_+}}$ is a local diffeomorphism. For the mapping $\Phi$ on
$\overline{U_-}$, we have the following
\begin{lemma}
$\Phi|_{\overline{U_-}}$ is a $C^1$ local
diffeomorphism.
\end{lemma}
\begin{proof}
By Proposition \ref{prop:noconj} and by the above
observation, we know that
$\Phi|_{U_-}$ and $\Phi|_{\partial\overline{U_-}}$ $(=\Phi|_{\partial\overline{U_+}})$ are $C^\infty$ immersions.
Let $\{\eta_s\}$
be a one-parameter family of unit covectors at
$p_0$ such that $\eta_s\in U_-$ $(s>0)$, $\eta_0
\in \partial\overline{U_-}$, and $\dot{\eta}_s
=\left(\partial/\partial \nu_{n-1}\right)/
|\partial/\partial \nu_{n-1}|$, where the variable
$\nu_{n-1}$ is the one defined in
\S\ref{sec:jac}.
We shall show that $\Phi|_{\overline{U_-}}$ is of class $C^1$ and a local diffeomorphism at $\eta_0$.
Differentiating the equality
\begin{equation*}
\sum_{l=1}^n\int_0^{\sigma_l(t_-(\eta_s),\eta_s)}\frac{(-1)^{l}G_{n-1}(f_l)A(f_l)\ d\sigma_l}
{\sqrt{-\prod_{k=1}^{n-1}(f_l-b_k)
\cdot\prod_{k=0}^n(f_l-a_k)}}=0
\end{equation*}
in $s$, one obtains
\begin{equation}\label{eq:t1}
\begin{gathered}
0=\beta(c_s\,Y_{n-1,0}(t_-(\eta_s))+\frac{\partial}{\partial s}
t_-(\eta_s)\cdot \dot\gamma(t_-(\eta_s),\eta_s))\\
-\nu_{n-1}\sum_{l=1}^n\int_{0}^{\sigma_l(t_-(\eta_s),\eta_s)}\frac{(-1)^iG_{n-1}(f_l)A(f_l)\ d\sigma_l}
{(f_l-b_{n-1})\sqrt{-\prod_{k=1}^{n-1}(f_l-b_k)
\cdot\prod_{k=0}^n(f_l-a_k)}}\ ,
\end{gathered}
\end{equation}
where $c_s=\pm |\partial/\partial \nu_{n-1}|$ at
$\eta_s$ and $\beta$ is the 1-form;
\begin{equation*}
\beta=\sum_{l=1}^{n-1}\frac{\epsilon'_l(-1)^lG_{n-1}(f_l(x_l))A(f_l(x_l))}
{\sqrt{-\prod_{k=1}^{n-1}(f_l(x_l)-b_k)
\cdot\prod_{k=0}^n(f_l(x_l)-a_k)}}\ d(f_l(x_l))\ .
\end{equation*}
Then, taking the limit $s\searrow 0$, we have
\begin{equation*}
\begin{gathered}
0=\frac{\partial}{\partial s}
t_-(\eta_s)\big|_{s=0}\ \beta(\dot\gamma(t_-(\eta_0),\eta_0))\\
+\ \frac{4\epsilon'_{n}(-1)^{n}G_{n-1}(b_{n-1})A(b_{n-1})}
{\sqrt{-\prod_{k\ne n-1}(b_{n-1}-b_k)
\cdot\prod_{k=0}^n(b_{n-1}-a_k)}}\ .
\end{gathered}
\end{equation*}
Noting that the covector $\flat(\dot\gamma(t_-(\eta_0),\eta_0))$
is equal to
\begin{equation*}
\frac12\sum_{l=1}^{n-1}\frac{\epsilon'_l(-1)^{l+1}A(f_l(x_l))
\prod_{k=1}^{n-1}(f_l(x_l)-b_k)}
{\sqrt{-\prod_{k=1}^{n-1}(f_l(x_l)-b_k)
\cdot\prod_{k=0}^n(f_l(x_l)-a_k)}}\ d(f_l(x_l))
\end{equation*}
at $\gamma(t_-(\eta_0),\eta_0)$, we see that
\begin{equation*}
\frac1{b_1-b_{n-1}}<-\beta(\dot\gamma(t_-(\eta_0),\eta_0))<\frac1{b_{n-2}-b_{n-1}}\ .
\end{equation*}
This indicates that $(\partial/\partial s)t_-(\eta_s)|_{s=0}$
is finite and nonzero.
Also, by similar formulas to \eqref{eq:t1}, the
derivatives of $\gamma(t_-(\eta),\eta)$
by the normalized $\partial/\partial H_j$ $(j\le
n-2)$ are of
the form $Y_{j,0}(t_-(\eta))
+c_\eta\dot\gamma(t_-(\eta)\eta)$ (or
$Z_{j,0}(t_-(\eta))+c_\eta\dot\gamma(t_-(\eta)\eta)$)
$\in T_{\gamma(t_-(\eta),\eta)}N$,
which are continuous in $\eta$ near the boundary
$\partial\overline{U_-}$.
Therefore the mapping $\Phi|_{\overline{U_-}}$ is of class $C_1$ and the lemma follows.
\end{proof}
The above lemma implies that $\Phi|_{\overline{U_-}}$ is a
local homeomorphism. Thus, combined with the above result,
we see that $\Phi: U_{p_0}^*M\to N$ is a local
homeomorphism. Since both $U_{p_0}^*M$ and $N$ are homeomorphic to the $(n-1)$-sphere, and since $n\ge 3$, it
therefore follows that $\Phi$ is really a homeomorphism.
We shall prove that the image
of the map $\overline{U_+}\ni\eta\mapsto \gamma(t_0(\eta),\eta)$ is just the cut locus of $p_0$. Let us temporarily denote this image by $\mathcal C$. Note that,
for any $\eta\in U^*_{p_0}M$, the cut point of $p_0$ along
the geodesic $\gamma(t,\eta)$ will appear at
$t\le t_0(\eta)$, because of Propositions \ref{prop:prime}
and \ref{prop:refl}.
In particular, putting
\begin{equation*}
V=\{t\eta\in T^*_{p_0}M\ |\ \eta\in U^*_{p_0}M,\
0\le t<t_0(\eta)\}\ ,
\end{equation*}
we have the following lemma. Put Exp$_{p_0}(t\eta)=\gamma(t,\eta)$.
\begin{lemma}\label{lem}
\begin{enumerate}
\item $\text{Exp}_{p_0}:\overline{V}\to M$ is surjective.
\item $\text{Exp}_{p_0}(V)\cap \mathcal C=\emptyset$.
\end{enumerate}
\end{lemma}
\begin{proof}
Let $q\in M$ be any point $(\ne p_0)$ and let
$\gamma(t,\eta)$
$(0\le t\le T)$ be a minimal geodesic joining $p_0$ and $q$
$(\eta\in U^*_{p_0}M)$. Since $T\le t_0(\eta)$, (1) follows.
Next, assume that there is some $\eta\in U^*_{p_0}M$ and
$0<T<t_0(\eta)$ such that $\gamma(T,\eta)\in \mathcal C$.
Then, $x_n(T,\eta)=-x_n^0$ or $\frac{\alpha_n}2+x_n^0$.
Note that, if $\eta\in \overline{U_+}$, then $t=t_0(\eta)$
is the first positive time when $x_n(T,\eta)=-x_n^0$ or
$\frac{\alpha_n}2+x_n^0$. Thus we have $\eta\in U_-$ and
$T=t_-(\eta)$. But, as we have proved in the previous lemma, $\gamma(T,\eta)
\not\in \mathcal C$ in this case, a contradiction. Thus
(2) follows.
\end{proof}
Fix $\eta\in U_{p_0}^*M$ and suppose that
the cut point of $p_0$ along the geodesic $\gamma(t,\eta)$
appear before $t=t_0(\eta)$, i.e., the geodesic segment
$\gamma(t,\eta)$ $(0\le t\le t_0(\eta))$ is no longer minimal.
Then there is another minimal geodesic $\gamma(t,\bar\eta)$
$(0\le t\le T)$
joining $p_0$ and $q=\gamma(t_0(\eta),\eta)$, $\bar\eta\in U^*_{p_0}M$.
Since the geodesic segment $\gamma(t,\bar\eta)$
$(0\le t\le T)$ is minimal, we have $T\le t_0(\bar\eta)$.
Also, since $\gamma(T,\bar\eta)=q\in\mathcal C$,
we have $T=t_0(\bar\eta)$ by Lemma~\ref{lem} (2).
Then, by the injectivity of $\Phi$ we have $\bar\eta=
\eta$ or $\eta'$. But this implies that the geodesic segment
$\gamma(t,\eta)$ $(0\le t\le t_0(\eta))$ is minimal, a
contradiction. Thus $t=t_0(\eta)$ gives the cut point of
$p_0$ along the geodesic $\gamma(t,\eta)$. This completes the
proof of (1) and (2) of the theorem
in the case where $0<x_i^0<\alpha_n/4$ for any $i$.
\section{Cut locus (2)}
In this section, we shall give a proof of Theorem \ref{thm:cut} for the case (II) described in the
previous section. The cases (III) $\sim$ (V) will be considered in the next section.
Note that the statement (1) of the theorem holds for
any $p_0$ and any $\eta\in U^*_{p_0}M$, which is a
consequence of the results in the previous section,
Proposition \ref{prop:conti}, and the continuous dependence
of cut points on the initial covectors. Thus we shall prove
(2) for the cases (II) $\sim$ (IV) and (3) for the case (V).
Now, let us consider the case (II); $0<x_n^0<\alpha_n/4$ and $p_0\in N_l$ for some
$l\le n-1$. As in the previous section, we shall show that
$\Phi:U_{p_0}^*M\to N$ is a homeomorphism.
\begin{prop}\label{prop:noconj2}
Suppose $p_0\in N_l$ and let $\eta\in U_{p_0}^*M$ be a
covector such that the geodesic $\gamma(t,\eta)$ is totally
contained in $N_l$. Let $Y_l(t)$ be a nonzero Jacobi field along
the geodesic $\gamma(t,\eta)$ such that $Y_l(0)=0$ and
$Y_l(t)$ is orthogonal to $N_l$ everywhere. Then,
$Y_l(t_0(\eta))\ne 0$.
\end{prop}
The proof will be given below. This proposition
together with Proposition \ref{prop:noconj}
applied to the intersection of the Liouville manifolds $N_l$ in which the geodesic is contained show that the
mapping $\Phi|_{U_+}$ and $\Phi|_{\partial\overline{U_+}}$ are immersions.
Then, in the same way as the previous section,
we see that $\Phi|_{\overline{U_+}}$ is a local
homeomorphism. On the other hand, since $t_0(\eta)$ represents the cut point, and since
$t_-(\eta)<t_0(\eta)$, the mapping $\Phi|_{U_-}$
is a $C^\infty$ embedding and $\Phi(U_-)\cap \Phi(\overline{U_+})=\emptyset$. Also $\Phi
(U^*_{p_0})=N$ by continuity. Therefore it follows
that $\Phi: U_{p_0}^*M\to N$ is a homeomorphism.
This indicates (2) of the theorem in this case.
In the rest of this section we shall prove
Proposition \ref{prop:noconj2}.
We may assume that
there is only one such $l$ that the geodesic is
totally contained in $N_l$.
According to the position of the geodesic
$\gamma(t,\eta)$, there are four different cases: (i) the
geodesic $\gamma(t,\eta)$ intersects $J_l$ transversally;
(ii) $\gamma(t,\eta)$ does not meet $J_l$; (iii)
$\gamma(t,\eta)$ is tangent to $J_l$, but not contained in
it; (iv) $\gamma(t,\eta)$ is contained in $J_l$.
First, let us consider the case (i), and first
assume $p_0\not\in J_l$. We may also assume
$f_{l+1}(x_{l+1}^0)<b_l=a_l=f_l(x_l^0)$; the case
where $f_{l+1}(x_{l+1}^0)=b_l=a_l<f_l(x_l^0)$ is
similar. Note that $f_l(x_l^0)<b_{l-1}$ in this case,
since the intersection of $\gamma(t,\eta)$
and $J_l$ is transversal in $N_l$. Then the
Jacobi field $Y_l(t)$ is given by the one-parameter family of geodesics $\{\gamma(t,\eta_s)
\}$, where $\eta_s\in U_{p_0}^*M$ satisfies
$\eta_0=\eta$ and $H_l(\eta_s)=b_l-s^2$,
$H_j(\eta_s)=b_j$ for $j\ne l$.
To show the proposition in this case,
we use a similar technique
as Lemma \ref{lemma:shift}, which is as follows.
Take a point $p'_0$ in such
a way that $p'_0$ is represented as
$(x_1^0,\dots,x_{l}^1,\dots,x_n^0)$, where
$0=x_l^0<x_l^1< \alpha_l/4$ and $f_l(x_l^1)<
b_{l-1}, a_{l-1}$. Let $U_{l-}'$ be the hemisphere of $U^*_{p'_0}M$ defined by
$\xi_l<0$ and so be $U_{l-}$ in $U^*_{p_0}M$.
Taking a sufficiently small neighborhood $W$ of
$\eta$ in $U_{p_0}^*M$, we define the mapping
$\psi: \overline{U_{l-}}\cap W\to U'_{l-}$
so that it preserves the values of $H_i$ $(1\le i\le n-1)$, i.e., by
$\psi(p_0; \xi_1,\dots,\xi_n)
=(p'_0; \tilde\xi_1,\dots,\tilde\xi_n)$, where
\begin{equation*}
\tilde\xi_i=\xi_i\quad (i\ne l),\qquad
\tilde\xi_l=\sqrt{(-1)^{l-1}\prod_{k\ne l}
(f_l(x_l^1)-H_k)}\ .
\end{equation*}
Note that $H_k$'s are functions of
$(p_0; \xi_1,\dots,\xi_n)\in \overline{U_{l-}}$.
Let $\tilde x_l^1$ be the value of $x_l(t,\psi(\eta_s))$ at the time when $\sigma_l(t,\psi(\eta_s))=2(a_{l-1}^--a_l^+)$, which is $-x_l^1$
or $x_l^1+\alpha_l/2$. Also, $\tilde x_l^0$
is similarly defined.
Let $N'$ be the
submanifold of $M$ defined by
$x_l=\tilde x_l^1$, and define the diffeomorphism
$\Psi: N'\to N_l$ by
\begin{equation*}
\Psi(x_1,\dots,,\tilde x_l^1,\dots,x_n)=
(x_1,\dots,\tilde x_l^0,\dots,x_n).
\end{equation*}
Then we have the following lemma. The proof being similar to that for Lemma \ref{lemma:shift},
we omit.
\begin{lemma}
$\Psi(\gamma(t_2(\psi(\eta_s)),\psi(\eta_s)))=
\gamma(t_2(\eta_s),\eta_s)$ for any $s>0$,
where $t_2(\eta_s)$
denotes the time when $\sigma_l
(t_2(\eta_s),\eta_s)=2(a_{l-1}^--a_l^+)$.
\end{lemma}
Since $t=t_2(\eta_s)$ is the first positive time
when the geodesic $\gamma(t,\eta_s)$ reach $N_l$
again, it follows that $t_2(\eta_0)=\lim_{s\to 0}
t_2(\eta_s)$ is the first positive time when
the Jacobi field $Y_l(t)$ vanishes.
Applying Proposition \ref{prop:noconj} to the
geodesic $\gamma(t,\psi(\eta_0))$, we have
$t_0(\psi(\eta_0))<t_2(\psi(\eta_0))$. Since
\begin{equation*}
\sigma_n(t_2(\psi(\eta_s)),\psi(\eta_s))=
\sigma_n(t_2(\eta_s),\eta_s),
\end{equation*}
we then have $\sigma_n(t_2(\eta_0),\eta_0)>2(a_{n-1}^--a_n^+)$, which implies
$t_0(\eta_0)<t_2(\eta_0)$, and hence $Y_l(t_0(\eta_0))\ne 0$.
Next, let us consider the case (i) with the condition $p_0\in J_l$.
Let $\eta_s\in U_{p_0}^*M$ be as above so that
the geodesic $\gamma(t,\eta_0)$ is transversal
to $J_l$ in $N_l$. Then the family of geodesics
$\{\gamma(t,\eta_s)\}_{s>0}$ coincides with the family
$\{\gamma(t,\zeta_r(\eta_{s_0}))\}$ for a fixed $s_0>0$, where $\{\zeta_r\}$ is the one-parameter group
of diffeomorphisms of $U^*M$ generated by
$X_{F_l}$. Thus, in this case, the first positive
time $t_2(\eta_0)$ when the Jacobi field $Y_l(t)$
vanishes has the property that
\begin{equation*}
\gamma(t_2(\eta_0),\eta_s)=\gamma(t_2(\eta_0),
\eta_0)\in J_l\ ,\quad \sigma_l(t_2(\eta_0),\eta_s)=2(a_{l-1}^--a_l^+)\ .
\end{equation*}
Now, let us consider $N_l$ as an $(n-1)$-dimensional Liouville manifold constructed from the constants $a_j$
$(j\ne l)$ and the function $A(\lambda)$. Then the
variables $f_l(x_l)$ and $f_{l+1}(x_{l+1})$ are connected
to a single variable whose range is $[a_{l+1},a_{l-1}]$,
and the total variation of this variable along the
geodesic $\gamma(t,\eta_0)$ $(0\le t\le t_2(\eta_0))$ is
equal to $2(a_{l-1}^--a_{l+1}^+)$. Hence by Proposition
\ref{prop:sigma} for the $(n-1)$-dimensional manifold $N_l$,
we have $t_0(\eta_0)<t_2(\eta_0)$, and thus $Y_l(t_0(\eta_0))
\ne 0$.
Next, we shall consider the case (ii); the geodesic
$\gamma(t,\eta)$ does not intersects $J_l$. There are
two cases: $a_l=f_l(x_l(t,\eta))=b_{l-1}$; $b_{l+1}=
f_{l+1}(x_{l+1}(t,\eta))=a_l$. The proofs for them are similar, so we may assume $a_l=b_{l-1}$. Note that
$b_l<a_l$ in this case, since $\gamma(t,\eta)$ does not meet
$J_l$.
The Jacobi field $Y_l(t)$ is given by the one-parameter family of geodesics $\{\gamma(t,\eta_s)
\}$, where $\eta_s\in U_{p_0}^*M$ satisfies
$\eta_0=\eta$ and $H_{l-1}(\eta_s)=a_l+s^2$,
$H_j(\eta_s)=b_j$ for $j\ne l-1$. Define $\theta_s(t)$ by the formula
\begin{equation*}
f_l(x_l(t,\eta_s))=a_l(\cos\theta_s(t))^2+H_{l-1}(\eta_s)
(\sin\theta_s(t))^2\ ,\quad \theta_s(0)=0
\end{equation*}
and put $\theta_0(t)=\lim_{s\to 0}\theta_s(t)$.
Let $t_2(\eta)$ be the time such that
$\theta_0(t_2(\eta))=\pi$. Then $t=t_2(\eta)$ is the first
positive time when $Y_l(t)=0$. We shall show that
$t_0(\eta)<t_2(\eta)$.
We have
\begin{gather*}
\sum_{i\ne l}\int_0^{\sigma_i(t_2(\eta),\eta)}\frac{(-1)^iG_{l,l-1}(f_i)A(f_i)\ d\sigma_i}
{|f_i-a_l|\sqrt{-\prod_{k\ne l-1}(f_i-b_k)
\cdot\prod_{k\ne l}(f_i-a_k)}}\\
+\frac{(-1)^l2\pi\ G_{l,l-1}(a_l)A(a_l)}
{\sqrt{-\prod_{k\ne l-1}(a_l-b_k)
\cdot\prod_{k\ne l}(a_l-a_k)}}
=0\ .
\end{gather*}
Also, a similar observation as in the proof of Lemma \ref{lemma:flat} indicates
\begin{gather*}
-2\sum_{i\ne l}\int_{a_i^+}^{a_{i-1}^-}
\frac{(-1)^{i}G_{l,l-1}(\lambda)A(a_l)\ d\lambda}
{|\lambda-a_l|\sqrt{-\prod_{k\ne l-1}(\lambda-b_k)
\cdot\prod_{k\ne l}(\lambda-a_k)}}\\
=\frac{(-1)^l2\pi\ G(a_l)A(a_l)}
{\sqrt{-\prod_{k\ne l-1}(a_l-b_k)
\cdot\prod_{k\ne l}(a_l-a_k)}}\ .
\end{gather*}
Thus we have the formula:
\begin{equation}\label{eq:theta7}
\begin{gathered}
2\sum_{i\ne l}\int_{a_i^+}^{a_{i-1}^-}
\frac{A(\lambda)-A(a_l)}{|\lambda-a_l|}
\frac{(-1)^{i}G_{l,l-1}(\lambda)\ d\lambda}
{\sqrt{-\prod_{k\ne l-1}(\lambda-b_k)
\cdot\prod_{k\ne l}(\lambda-a_k)}}\\
=\sum_{i\ne l}\int_{\sigma_i(t_2(\eta),\eta)}^{2(a_{i-1}^--a_i^+)}\frac{(-1)^iG_{l,l-1}(f_i)A(f_i)\ d\sigma_i}
{|f_i-a_l|\sqrt{-\prod_{k\ne l-1}(f_i-b_k)
\cdot\prod_{k\ne l}(f_i-a_k)}}\ .
\end{gathered}
\end{equation}
Take a sufficiently large constant $c>0$ and put
\begin{equation*}
B(\lambda)=c-\frac{A(\lambda)-A(a_l)}{\lambda-a_l}\ ,
\qquad [i]=i\ (i<l);\quad =i-1\ (i>l)\ .
\end{equation*}
Then, by Lemma \ref{lemma:flat} ($(n-1)$-dimensional
case), the left-hand side of the formula \eqref{eq:theta7}
is rewritten as
\begin{equation*}
2\sum_{[i]=1}^{n-1}\int_{a_i^+}^{a_{i-1}^-}
\frac{(-1)^{[i]+1}G_{l,l-1}(\lambda)B(\lambda)\ d\lambda}
{\sqrt{-\prod_{[k]=1}^{n-1}((\lambda-a_{k}^-)
(\lambda-a_{k}^+))}}\ .
\end{equation*}
Since $B(\lambda)$ satisfies the condition (\ref{cond2}), the
above value is positive by Proposition \ref{prop:cond2} (1)
($(n-1)$-dimensional case). If $t_0(\eta)\ge t_2(\eta)$,
then, applying
Proposition \ref{prop:sigma} to the Liouville manifold
$N_l$, we have $\sigma_i(t_2(\eta),\eta)\le 2(a_{i-1}^--a_i^+)$
for any $i\ne l$. This indicates that the right-hand side
of the
formula \eqref{eq:theta7} is nonpositive, a contradiction.
Therefore, it
follows that $t_0(\eta)<t_2(\eta)$, and $Y_l(t_0(\eta))\ne
0$.
Next, we shall consider the case (iii); $\gamma(t,\eta)$ is
tangent to $J_l$, but not contained in it. First, we assume
$p_0\not\in J_l$. In this case, it holds that either
$f_{l+1}(x_{l+1}^0)
<b_l=a_l=f_l(x_l^0)=b_{l-1}$ or $b_{l+1}=f_{l+1}(x_{l+1}^0)
=a_l=b_l<f_l(x_l^0)$.
Since the proofs are similar, we may assume
\begin{equation*}
f_{l+1}(x_{l+1}^0)
<b_l=a_l=f_l(x_l^0)=b_{l-1}\ .
\end{equation*}
Define a one-parameter family of unit covectors $\eta_s$
at $p_0$ such that $\eta_0=\eta$, $H_{l}(\eta_s)=a_l-s^2$,
and $H_j(\eta_s)=b_j$ for $j\ne l$. Then, the geodesics
$\gamma(t,\eta_s)$ $(s\ne 0)$ are still on $N_l$, but do
not meet $J_l$. Since the zeros of a
family of Jacobi fields are continuously depending on the
parameter, it follows that $\lim_{s\to 0}t_2(\eta_s)=
t_2(\eta)$ represents the first positive time $t$ such that
$Y_l(t)=0$.
Now, substitute $\eta=\eta_s$ in the formula \eqref{eq:theta7} and take a limit $s\to 0$. Then, if
$t_0(\eta)\ge t_2(\eta)$, one gets a similar contradiction
as above. Thus we have $t_0(\eta)<t_2(\eta)$, and $Y_l(t_0(\eta))\ne 0$ in this case.
Next, we assume that $p_0\in J_l$. Let $\eta_s\in U_{p_0}^*M$
$(\eta_0=\eta)$ be a one-parameter family of covectors such that the infinitesimal variation of the geodesics $\{\gamma(t,\eta_s)\}$ at $s=0$ is equal to $Y_l(t)$. Let $t_2(\eta_s)$ be the
first positive time such that $\gamma(t,\eta_s)\in N_l$.
Then, $t_2(\eta)=\lim_{s\to 0}t_2(\eta_s)$ is the first
positive time such that $Y_l(t)=0$. Also, by the same reason
as in the case (i), we have $\gamma(t_2(\eta_s),\eta_s)\in
J_l$ and so does for $s=0$. Hence we have $\sigma_{l+1}(t_2(
\eta),\eta)=2(a_l^--a_{l+1}^+)$, and thus $t_0(\eta)<t_2(
\eta)$ by Proposition \ref{prop:sigma}.
Finally, let us consider the case (iv); $\gamma(t,\eta)$ is
contained in $J_l$. In this case, we have
\begin{equation*}
b_{l+1}=f_{l+1}(x_{l+1}^0)
=b_l=a_l=f_l(x_l^0)=b_{l-1}\ .
\end{equation*}
Define the one-parameter family of the initial points $p_0(s)$ and the initial covectors $\eta_s\in U_{p_0(s)}^*M$
so that $H_{l+1}(\eta_s)=H_l(\eta_s)=b_l-s^2$ and $H_i(\eta_s)= b_i$ $(i\ne l,l+1)$. Then the formula \eqref{eq:theta7} is valid for $\eta_s$. Taking a limit $s\to 0$, we have:
\begin{gather*}
2\sum_{\substack{1\le [i]\le n-1\\ [i]\ne l}}
\int_{a_i^+}^{a_{i-1}^-}
\frac{(-1)^{[i]+1}G_{l,l-1}(\lambda)B(\lambda)\ d\lambda}
{\sqrt{-\prod_{[k]=1}^{n-1}((\lambda-a_{k}^-)
(\lambda-a_{k}^+))}}\\
=\sum_{i\ne l,l+1}\int_{\sigma_i(t_2(\eta),\eta)}^{2(a_{i-1}^--a_i^+)}\frac{(-1)^iG_{l,l-1}(f_i)A(f_i)\ d\sigma_i}
{|f_i-a_l|\sqrt{-\prod_{k\ne l-1}(f_i-b_k)
\cdot\prod_{k\ne l}(f_i-a_k)}}\ .
\end{gather*}
Since the left-hand side of the above formula is positive by
Proposition \ref{prop:cond2},
we have $t_0(\eta)<t_2(\eta)$ as before. This completes the
proof of Proposition \ref{prop:noconj2}.
\section{Cut locus (3)}
In this section, we shall give a proof of Theorem
\ref{thm:cut} (2) for the cases (III) and (IV), and (3) for
the case (V). First, we shall consider the case (III); $p_0
\in N_n$.
We use Lemma \ref{lemma:shift} in the case where $x_n^1=0$
and use it by exchanging $p_0$ and $p'_0$.
As a consequence, we see that the mapping
\begin{equation*}
(U_{p_0}^*M\supset)\ U_+\ni \eta\longmapsto\gamma(t_0(\eta),\eta)\in N_n
\end{equation*}
is a $C^\infty$ embedding. Therefore, to prove (2) in this
case it is enough to show that the mapping
\begin{equation}\label{eq:partial}
\partial\overline{U_+}\ni \eta\longmapsto\gamma(t_0(\eta),\eta)\in N_n
\end{equation}
is an embedding.
For $p_0\in N_{n}$ and $\eta\in U^*_{p_0}N_n$, let $\tilde
t_0(\eta)$ denotes the value which is defined in the same way
as
$t_0(\eta)$ for the $(n-1)$-dimensional Liouville manifold
$N_n$. (Note that $N_n$ is constructed from the constants
$0<a_{n-1}<\dots<a_0$ and the function $A(\lambda)$ as in
\S2.) As we have proved in (1), $t=\tilde t_0(\eta)$ gives the
cut point of $p_0$ along the geodesic $\gamma(t,\eta)$
in $N_n$. In particular, we have $t_0(\eta)\le
\tilde t_0(\eta)$. Therefore, the following proposition will indicate
that the mapping \eqref{eq:partial} is an embedding.
\begin{prop}\label{prop:sect7}
$t_0(\eta)<\tilde t_0(\eta)$ for any $p_0\in N_{n}$ and
$\eta\in U^*_{p_0}N_n$.
\end{prop}
\begin{proof}
We use the formula
\begin{gather*}
\sum_{i=1}^{n-1}\int_{a_i^+}^{a_{i-1}^-}\frac{
(-1)^{i+1}G_{n-1,n-2}(\lambda)B(\lambda)}{
\sqrt{-\prod_{k=1}^{n-2}(\lambda-b_k)\prod_{k=0}^{n-1}
(\lambda-a_k)}}\,d\lambda\\
=\sum_{i=1}^{n-1}\int_{\sigma_i(t_0(\eta),\eta)}^{
2(a_{i-1}^--a_i^+)}\frac{
(-1)^{i}G_{n-1,n-2}(f_i) A(f_i)}{
(f_i-a_n)
\sqrt{-\prod_{k=1}^{n-2}(f_i-b_k)\prod_{k=0}^{n-1}
(f_i-a_k)}}\,d\sigma_i\ ,
\end{gather*}
where
\begin{equation*}
B(\lambda)=c-\frac{A(\lambda)-A(a_n)}{\lambda-a_n}
\end{equation*}
and $c>0$ is a sufficiently large constant. As
before, the left-hand side of the above formula is
positive, whereas each integrand of the right-hand
side is negative for $i\le n-2$. Thus, if
$t_0(\eta)=\tilde t_0(\eta)$, then
\begin{equation*}
2(a_{n-2}^--a_{n-1}^+)=\sigma_{n-1}(\tilde t_0(\eta),
\eta)=\sigma_{n-1}(t_0(\eta),\eta),
\end{equation*}
and we have a contradiction. Therefore it follows
that $t_0(\eta)<\tilde t_0(\eta)$.
\end{proof}
Next, we shall consider the case (IV); $x_n^0=\alpha_n/4$
and $p_0\not\in J_{n-1}$. By the similar fact as
Lemma \ref{lemma:shift} and by
the proved cases, we see that the map $\eta\mapsto
\gamma(t_0(\eta),\eta)$ gives $C^\infty$ embeddings
$U_+\to N$ and $\partial \overline{U_+}\to N$,
where $N$ is the subset of $N_{n-1}$ such that
$x_n=-\alpha_n/4$. To see that
the cut locus $C(p_0)$, the union of the images of
those maps, is in the interior of $N$, it is enough to show
that $C(p_0)$ does not meet $J_{n-1}$, a connected component
of which is equal to the boundary of $N$. Assume that
$\gamma(t_0(\eta),\eta)\in J_{n-1}$ for some $\eta\in
\overline{U_+}$. By Lemma \ref{lemma:subm} we see that
$F_{n-1}(\eta)=0$. Since $p_0\not\in J_{n-1}$ and $p_0\in
N_{n-1}$, it thus follows that $\eta\in U^*_{p_0}N_{n-1}$,
i.e., $\eta\in\partial\overline{U_+}$. Now put
\begin{equation*}
\gamma(t)=\gamma(t_0(\eta)-t,\eta)
\end{equation*}
Then, $\gamma(t)$ is a geodesic starting at $\gamma(t_0(\eta),
\eta)\in J_{n-1}$ and its first conjugate point is $p_0=
\gamma(t_0(\eta))$. But, as we shall see just below, the first
conjugate point of any geodesic starting at a point in
$J_{n-1}$ also belongs to $J_{n-1}$, which is a contradiction.
Thus $C(p_0)$ is contained in the interior of $N$.
This finishes the proof of (2) of the theorem
in this case.
Finally we prove the statement (3) of the theorem
for the case (V); $p_0\in J_{n-1}$. Note that
$t=t_0(\eta)$
gives the cut point of $p_0$ along the geodesic
$\gamma(t,\eta)$ for any $\eta\in U^*_{p_0}M$.
We apply the results proved above
to the $(n-1)$-dimensional Liouville manifold $N_{n-1}$,
which is constructed from the constants $0<a_n<a_{n-2}<\dots
<a_0$ and the function $A(\lambda)$.
Noting the fact $J_{n-1}\cap J_{n-2}=\emptyset$,
we see that the cut locus $\tilde C(p_0)$
of $p_0$ in $N_{n-1}$ is an $(n-2)$-closed disk, and it is the
image of the map
\begin{equation*}
\overline{U_+}\cap T^*_{p_0}N_{n-1}\to J_{n-1},
\qquad \eta\mapsto \gamma(\bar t_0(\eta),\eta),
\end{equation*}
where $\bar t_0(\eta)$ is the value which is defined in the
same way as
$t_0(\eta)$ for the $(n-1)$-dimensional Liouville manifold
$N_{n-1}$. It has also been proved that the above map is
an embedding on the interior and on the boundary.
Let $\tilde \eta$ be a unit covector such that
$\tilde \eta\not\in T^*_{p_0}N_{n-1}$. Let $\{\zeta_s\}$
be the one-parameter transformation group of $T^*M$ generated
by $X_{F_{n-1}}$. Then $\tilde\eta_s=\zeta_s(\tilde \eta)\in U^*_{p_0}M$
whose orthogonal projection to $T^*_{p_0}J_{n-1}$
does not depend on $s$,
and $\tilde\eta_{\pm\infty}=\lim_{s\to\pm\infty}\tilde\eta_s\in
T^*_{p_0}N_{n-1}$. By the definition of $t_0(\tilde\eta_s)$
we have $\gamma(t_0(\tilde \eta_s),\tilde\eta_s)\in J_{n-1}$.
Therefore the Jacobi field $\pi_*X_{F_{n-1}}$ along the
geodesic $\gamma(t,\tilde\eta_s)$ also vanish at $t=
t_0(\tilde\eta_s)$. Thus we have
\begin{equation*}
\gamma(t_0(\tilde \eta_s),\tilde\eta_s)=
\gamma(t_0(\tilde\eta_{\pm\infty}),\tilde\eta_{\pm\infty}),\qquad
t_0(\tilde\eta_s)=t_0(\tilde\eta_{\pm\infty})
\end{equation*}
for any $s\in\mathbb R$. Since $t=t_0(\tilde\eta_s)$
gives the cut point of $p_0$ along the geodesic
$\gamma(t,\tilde\eta_s)$, and since $\tilde\eta_{+\infty}
\in U^*N_{n-1}$ and $\tilde\eta_{-\infty}\in U^*N_{n-1}$
are symmetric with respect to the hyperplane
$T^*_{p_0}J_{n-1}\subset T^*_{p_0}N_{n-1}$, it follows that
$\bar t_0(\eta_{\pm\infty})=t_0(\eta_{\pm\infty})$.
Thus we have proved that the cut locus $C(p_0)$ of $p_0$ in
$M$ coincides with $\tilde C(p_0)$ and that if $\eta_1$,
$\eta_2\in U^*_{p_0}M$
have the same $T^*_{p_0}J_{n-1}$-components, then $\gamma(
t_0(\eta_1),\eta_1)=\gamma(t_0(\eta_2),\eta_2)$. From these
it also follows that for $\eta\in U^*_{p_0}J_{n-1}$,
$t=t_0(\eta)$ gives the first conjugate
point of $p_0$ with multiplicity two along the geodesic
$\gamma(t,\eta)$. This finishes the proof of Theorem
\ref{thm:cut}.
\end{document}
|
\begin{document}
\title[A note on the Fourier coefficients of a Cohen-Eisenstein series]{\small {A note on the Fourier coefficients of a Cohen-Eisenstein series}
}
\author{SRILAKSHMI KRISHNAMOORTHY}
\date{\today}
\subjclass{}
{\mathrm {ad}}dress{Indian Institute of Technology Madras\\ Tamil Nadu, India.}
\email{[email protected], \ [email protected]}
\begin{abstract}
We prove a formula for the coefficients of a weight $3/2$ Cohen-Eisenstein series of square-free level $N$.
This formula generalizes a result of Gross and in particular, it proves a conjecture of Quattrini.
Let $l$ be an odd prime number. For any elliptic curve $E$ defined over ${\mathbb Q}$ of rank zero and square-free conductor $N$, if $l \mid |E({\mathbb Q})| $, under certain conditions on the Shafarevich-Tate group $\Sh_D$,
we show that $l$ divides $|\Sh_D|$ if and only if $l$ divides the class number $h(-D)$ of
${\mathbb Q}(\sqrt{-D}).$
\end{abstract}
\maketitle
Keywords: Half-integral weight modular forms; Fourier coefficients, Shafarevich-Tate group.\\
Mathematics Subject Classification: Primary 11F37; Secondary: 11F67, 11R52.
\section{{ Introduction }}
Let $E$ be an elliptic curve of prime conductor $N$ and analytic rank 0. Let $f$ be the new form of weight 2 of level $N$ on $\Gamma_0(N)$ associated to $E.$ Gross (Section 12, \cite{Gr87}) constructed
$$\mathcal{G}=\sum_{D} m_D q^D,$$ a weight 3/2 modular form interms of certain modular forms $g_i$, associated with $f.$
A special case of Waldspurger's formula (Proposition 13.5,
\cite{Gr87}) relates the product of the $L$-functions $$L(f,1)L(f \times \epsilon_{-D},1)$$ to $m^2_D,$ where $(\frac{-D}{N}) \mathrm{sgn}(W_N) \neq -1$ and $f \times \epsilon_{-D}$ is the
cusp form corresponding to the twist by $-D$ of $E,$ and
$W_N$ is the Atkin-Lehner involution. Bocherer and Schulze-Pillot generalized
Gross's construction (Section 3, \cite{BS90}) for square-free level $N.$
Quattrini collected many numerical examples (Section 3.7, \cite{Qu11}) of certain definite quaternion algebras ramified at exactly one prime and presented a conjecture (Conjecture \ref{main}) on the coefficients of the Cohen-Eisenstein series $$\mathcal{H} = \sum^{n}_{i=1} \frac{1}{w_i} g_i,$$ where
$g_i$ and $w_i$ are certain orders defined in Section \ref{prelims}.
We work with certain definite quaternion algebras ramified at finitely many primes $p_1,$ $p_2,$ ...$p_k,$ and we compute the coefficients of $\mathcal{H}$ for square-free level
in theorem \ref{mainthm}. As a consequence, we deduce the conjecture \ref{main} (Corollary \ref{main-conj}).
The estimation of the number of imaginary quadratic fields whose ideal class group has an element of order $l \geq 2$ and
the analogous questions for quadratic twists of elliptic curves has been the center of interest in many results.
For elliptic curves $E$ of prime conductors, using the theory of $p$-adic $L$-functions and Eisenstein quotients,
Mazur \cite{Ma79} showed that under certain conditions, the quadratic twist of $E$ by a primitive, odd quadratic
Dirichlet character $\chi$ has finite Mordell-Weil group of order not divisible by a prime $l$ if and only if
the quadratic field associated to $\chi$ has class number prime to $l.$
In \cite{Fr88}, Frey obtained the information about the elements of order $l$ in the Selmer group of $E_{D}$, the quadratic twist of $E$ by $-D$, by assuming
the elliptic curve $E$ over ${\mathbb Q}$ contains a ${\mathbb Q}$-rational torsion point of prime order $l$.
In \cite{Ja99}, James proved that 3 divides the order of the Selmer group of ${X_0(11)}_{D}$ if and only if 3 divides the class number $h(-D)$ under the similar assumption that
the elliptic curve $E$ contains a rational torsion point of order 3.
In \cite{Wo99}, Wong showed that there are infinitely many negative fundamental discriminants $-D$ such that the twist
$X_0(11)_{D}$ of the modular curve $X_0(11)$ has rank 0 over ${\mathbb Q}$ and an element of order 5 in its Shafarevich-Tate group.
Using the circle method and results of Frey, Kolyvagin, Ono \cite{Ono01} proved a result for the nontriviality of class groups of imaginary quadratic fields and results on the nontriviality
of the Shafarevich-Tate groups of certain elliptic curves. It is also known that for almost all primes $l$, there exist infinitely many
square-free integers $D$ such that $l$ $\nmid$ $|\Sh_D|$ (\cite{Ko99}).
We prove that (Theorem \ref{keyprop}) if $E$ is an elliptic curve with square-free conductor $N$ and $l$ is an odd prime dividing $|E({\mathbb Q})| $, under certain conditions on the Shafarevich-Tate group $\Sh_D$,
the proportion of $\Sh_D$ in the family, divisible by $l$, is the same as the proportion of class numbers $h(-D)$ divisible by $l$ in the family of
negative quadratic fields ${\mathbb Q}(\sqrt{-D})$ with the same Kronecker conditions.
To prove theorem \ref{mainthm}, we follow the strategy of Gross and we use Eichler's formula. The contents of this paper are as follows. In section \ref{prelims}, we discuss some preliminaries. In section \ref{optimal}, we compute the Fourier coefficients
of the modular forms $g_i$ in terms of $h(\mathcal{O}_{-D},R_i),$ the number of all optimal embeddings of the order of discrimiant $D$ into certain maximal orders $R_i$.
In section \ref{The order of the Shafarevich-Tate group}, we show that a certain odd prime divides the order of Shafarevich-Tate group of quadratic twists of elliptic curves
if and only if it divides the class number of the corresponding imaginary quadratic field.
In section \ref{Cohen-Eisenstein series}, we compute the coefficients of the Cohen-Eisenstein series and we deduce Conjecture \ref{main}.
\section{ {Preliminaries and statement of results}}\label{prelims}
Bocherer and Schulze-Pillot generalized
Gross's construction (Section 3, \cite{BS90}) for square-free level $N$ as follows. Let $B$ be a definite quaternion algebra ramified at primes $p_1, p_2,...,p_k$ and at $\infty$.
Let $N=p_1p_2...p_kM$ $(p_i \nmid M)$ be a square-free integer.
Let ${\mathbb O}O$ be an order of level $N$.
Let $I_1$,$I_2$,...,$I_n$ be a set of left ideals representing the distinct ideal classes of ${\mathbb O}O$, with $I_1 = {\mathbb O}O$.
Let $R_1$,$R_2$,...,$R_n$ be the respective right orders (of level $N$) of each ideal $I_i$.
For each $R_i$, let $L_i$ be the rank 3 lattice ${\mathbb Z} +2 R_i$. Denote the trace zero elements of $L_i$ by $S^{0}_i.$
For $b$ $\in$ $S^0_i,$ let ${\mathbb N}(b)$ be the norm of $b.$ Let $w_i$ be the order of the finite group $R_i^{*}/\pm 1$ for $i = 1$ to $n$.
Define $$g_i = \frac{1}{2} \sum_{b \in S^0_i} q^{{\mathbb N}(b)}.$$ The forms $g_i$ are in the Kohnen plus-space which is the space of modular forms $\sum a_n q^n$ of weight $3/2$ on $\Gamma_0(4N)$
whose Fourier coefficients $a_n$ are 0 if $-n \equiv 2, 3 \pmod 4.$
\subsection{Brandt matrices and Theta series}
Let $m$ be a positive integer. The Brandt matrix $B_m$ is defined by $B_m = (b_{ij}(m))_{n \times n},$ where $b_{ij}(m) = \frac{1}{e_j} |\{ \alpha \in I^{-1}_j I_i \ : \ N(\alpha) \frac{N(I_j)}{N(I_i)} = m\}|,$
where $e_j = | R^{*}_j |.$\\
{\mathcal H}space{0.25cm} The sum of any row in the matrix $B_m$ is given by $$b_m = \sum^{n}_{j=1} b_{ij}(m) = \sum_{d \mid m, (d, \frac{N}{M})=1 } d.$$
It is also the $m$-th coefficient of the zeta function $$\zeta_{\mathcal{O}} = \sum_{I} \frac{1}{{\mathbb{N}(I)}^{2s}} = \sum^{\infty}_{n=1}
\frac{b_n}{n^{2s}},$$ where the sum runs over all integral $\mathcal{O}$-left ideals $I.$
The vector $u=(1,1,...,1)$ is an eigenvector of the Brandt matrices, we have $B_m u^t = b_m u^t,$ for all positive integers $m.$
Fix $1 \leq i,j \leq n.$
These Bradnt matrices define a collection of theta series $$\theta_{ij}(\tau) = \frac{1}{e_j} \sum_{x \in I^{-1}_j I_i } q^{\frac{{\mathbb N}(x) {\mathbb N}(I_j)}{{\mathbb N}(I_i)} }= \sum^{\infty}_{m=0} b_{ij}(m) q^m$$
which are modular forms of weight 2 and level $N.$
The series $e_2(z) = \sum^{n}_{i=1} \frac{1}{2w_i} + \sum^{\infty}_{m=1} b_m q^m$ is an Eisenstein series of weight 2 and level $N.$\\
Let $f$ be a new form of square-free level $N=PM$ with $P=p_1p_2...p_k$ on $\Gamma_0(N)$ such that
\begin{equation}\label{sgn-eqn1}
k \ \mathrm{is} \ \mathrm{odd,} \
\mathrm{sgn}(W_p) = -1, \ \mathrm{if} \ p \ \mid P, \ \mathrm{ sgn}(W_q) = +1, \ \mathrm{if} \ q \ \mid M.
\end{equation}
Suppose the elliptic curve $E$ corresponding to $f$ has analytic rank 0 and $l$ is an odd prime dividing the order of the torsion group of $E,$ then it can be shown that (Proposition 3.2, \cite{Qu11})
\begin{equation}\label{eqn10}
f \equiv e_2 \pmod l.
\end{equation}
\subsection{Waldspurger's formula}
The Shimura correspondence \cite{Sh73} relates the modular forms of half integral weight $k+1/2$ with classical modular forms of even weight $2k.$ We will define the modular form
$\mathcal{G}$ of weight $3/2$ which corresponds to the new form $f$ satisfying (\ref{sgn-eqn1}).
Consider the quaternion algebra $B$ ramified exactly at $\infty$ and at the primes $p_i$ $\mid$ $N,$
where $\mathrm{sgn}(W_{p_i}) = -1.$ The Brandt matrices $B_m$ act on the the vector space
$V$ of formal linear combinations $\sum^n_{i=1} c_i I_i,$ $c_i \in \mathbb{C}.$ By Eichler's trace formula there is a one to
one correspondence between Hecke eigenforms of weight 2 and level $N$
and eigenvectors in
$V$
of all Brandt matrices (up to a constant multiple) (Section 2, \cite{Po09}). Hence the normalized new form $f \in S_2(N)$ corresponds to a one-dimensional eigenspace $\langle v= ( v_1, v_2,..., v_n ) \rangle,$
of the Brandt matrices $\{{B}_p\}$ (of level $N$ and prime degree $p$) in $B$, such that ${B}_p v^{t} = a_p v^t,$
where $a_p$ is the eigenvalue satisfying $T_p f = a_p f,$ for all $p.$ We can assume that $(\frac{v_1}{w_1}, \frac{v_2}{w_2},...,\frac{v_n}{w_n})$ is primitive and has integer coordinates. Then
$$\mathcal{G} = \sum^{n}_{i=1} \frac{v_i}{w_i} g_i = \sum_{D} m_D q^ D$$ is the weight $3/2$
modular form which corresponds to $f$ via the Shimura correspondence.\\
Let $P = p_1p_2...p_k.$ The modular form $\mathcal{G}$ is zero unless
$ \mathrm{sgn}(W_p) = \begin{cases}
-1, & \mbox{for } p \mid P \\
+1, & \mbox{for } p \mid M \\
\end{cases}$
If $-D$ is a fundamental discriminant such that $(\frac{-D}{p}) \mathrm{sgn}(W_p) \neq -1$ for every prime $p$
$\mid$ $\frac{N}{\mathrm{gcd}(N,D)},$ then the following
special case of Waldspurger's formula \cite{Wa81} holds (Section 3, \cite{BS90}).
\begin{equation}\label{eqn0}
\prod_{ p\mid \frac{N}{\mathrm{gcd}(N,D)} }( 1+ (\frac{-D}{p}) sgn (W_p) ) L(f,1)L(f \otimes \epsilon_{-D}, 1) =
\frac{ 2^{\omega(N)} (f, f) m^2_D } { \sqrt{D} \sum \frac{ v^2_i }{ w^2_i} }.
\end{equation}
\begin{defi}
The Cohen-Eisenstein series is the Eisenstein series of weight 3/2 corresponding to the eigenvector
$u=( 1, 1,..., 1 ),$ $\mathcal{H} := \sum^{n}_{i=1} \frac{1}{w_i} g_i.$
\end{defi}
\begin{remar}$(\mathrm{Multiplicity} \ \mathrm{one} \ \mathrm{modulo} \ l)$.\label{mul-remar}
When $N$ is prime, using the results of Mazur and Emerton \cite{Ma77}, \cite{Em02}, one can show that the Brandt matrices $\{B_p \}$ reduced modulo $l$ have a dimension one eigenspace for the eigenvalues $\sigma(p)_N$
(Theorem 3.6, \cite{Qu11}).
Since $u$ and $v$ are both eigenvectors for the Brandt matrices $\{ B_p \}$, we have $\lambda u \equiv v \pmod l$
for some $\lambda \in \mathbb{F}^{\times}_l$.
If $N$ is square-free, then it is not clear whether the eigenspace corresponding to $u = (1,1,...,1)$ is one dimensional modulo $l$.
\end{remar}
Let $D$ be a natural number and let ${\mathbb O}O_{-D}$ be the ring of integers in ${\mathbb Q}(\sqrt{-D})$. Let $h(-D)$ be the cardinality of the group $\mathrm{Pic}({\mathbb O}O_{-D})$, and
let $2u(-D)$ be the cardinality of the unit group ${\mathbb O}O^{*}_{-D}.$
When $N$ is a square-free number, Quattrini made the following conjecture by observations on known congruences among weight two modular forms and known congruences
among eigenvectors of Brandt matrices. The details can be found in Section 3.1 -- 3.5 of \cite{Qu11}.
\begin{conj}$(\mathrm{Conjecture}\ 3.7, \ $\cite{Qu11}$)$\label{main}Let $B$ be a definite quaternion algebra ramified at exactly one finite prime $p$ and let $N=pM$ $( \ p \nmid M \ )$
be a square-free integer. Let $ \mathcal{H} = \sum^{n}_{i=1} \frac{1}{w_i} g_i = \sum^{n}_{i =1} \frac{ 1}{2w_i} + \sum_{D > 0} \mathcal{H}(D) q^D.$
Let $D \in {\mathbb N}$ be such that $-D$ is a fundamental discriminant and $( \frac{-D}{p} ) \neq 1$, and
$( \frac{-D}{q} ) \neq -1$ for every prime $q$ $\mid$ $M$.
Then
$$\mathcal{H}(D) = \frac{2^{\omega(N)-1 - s(D)} h(-D)}{u(-D)},$$
\end{conj}
where $\omega(N)$ is the number of distinct primes that divide $N$ and $s(D)$
is the number of primes that divide $N$ and ramify in ${\mathbb Q}(\sqrt{-D}).$
If $M=1,$ then the above conjecture is the following result of Gross (Section 1, \cite{Gr87}).
\begin{prop}\label{gross-prop}
If $B$ is a definite quaternion algebra ramified only at a prime $N$ and $-D$ is a fundamental discriminant such that $( \frac{-D}{N} ) \neq 1,$ then the coefficients $\mathcal{H}(D)$ of the weight $3/2$ Eisenstein series
are given by $$\mathcal{H}(D) = \frac{( 1 - (\frac{-D}{N}) )}{2}\frac{ h(-D)}{u(-D)}.$$
\end{prop}
In Conjecture \ref{main} and in Proposition \ref{gross-prop}, Gross and Quattrini considered definite quaternion algebras ramified
at exactly one prime $p$ and at $\infty.$ We consider the generalized case, square-free level and definite quaternion algebras
ramified at finitely many primes $p_1,$ $p_2$,...$p_k$ and at $\infty$ (See Theorem \ref{mainthm}).
From Cremona's tables, the strong Weil curves of rank zero and
prime conductor with an odd torsion point, are listed by $E =
11A1$, $E=19A1$ and $E=37B1$. The first one has a 5-torsion point.
The other two curves have a 3-torsion point. For the $(-D)$
quadratic twists of $E$, $ | \Sh_{D} |$ is ${m^2_D}$, up to a
power of 2 and we also have $\lambda u \equiv v \pmod l,$ for some $\lambda \in \mathbb{F}^{\times}_l$
(remark \ref{mul-remar}). We state the following result of Quattrini (Proposition 3.8, \cite{Qu11}).
\begin{prop}
Let $E$ be the strong Weil curve of rank 0 and prime conductor $N$.
Consider the family $\{ E_D \}$ of negative quadratic twists of $E,$ for
$-D$ a fundamental discriminant and satisfying $(\frac{-D}{N}) =1.$ Suppose $E$ has a torsion point defined over ${\mathbb Q},$ of odd prime order $l.$ Then, $|\Sh_D|$ is divisible by $l,$ if and only if the class number $h(-D)$ of ${\mathbb Q}(\sqrt{-D})$ is divisible by
$l.$
\end{prop}
We generalize the above proposition to square-free level $N$ as follows.
Let $E$ be an elliptic curve of analytic rank zero and square-free conductor $N=PM$ with $P=p_1p_2...p_k.$
Let $f$ be the new form of level $N$ on $\Gamma_0(N)$ corresponding to $E$ satisfying
\begin{equation}\label{sgn-eqn}
\mathrm{Assume} \ k \ \mathrm{is} \ \mathrm{odd,} \
\mathrm{sgn}(W_p) = -1, \ \mathrm{if} \ p \ \mid P, \ \mathrm{ sgn}(W_q) = +1, \ \mathrm{if} \ q \ \mid M.
\end{equation}
Consider the family $\{ E_D \}$ of negative quadratic twists of $E$ satisfying the Kronecker condition
\begin{equation}\label{kronecker-eqn}
( \frac{-D}{p} ) \neq 1 \ \mathrm{ for} \ p \ \mid P, ( \frac{-D}{q} ) \neq -1, \ \mathrm{ for} \ q \ \mid M.
\end{equation}
We consider the definite quaternion algebra $B$ ramified exactly at all $p$ $\mid$ $P$ and at $\infty.$
We assume the following.
\begin{equation}\label{w-eqn}
\mathrm{If} \ P \ \mathrm{is} \ \mathrm{composite}, \ \mathrm{then} \ w_i \in \mathbb{F}^{\times}_l \ \mathrm{for} \ l = 3, 5 \ \mathrm{or} \ 7.
\end{equation}
The new form $f$ and the Eisenstein series $e_2$ of weight 2 correspond to the 3/2 weight forms
$\mathcal{G}$ and the Cohen-Eisenstein series $\mathcal{H}$ respectively, under the Shimura correspondence.
Let $v$ and $u$ be the eigenvectors of the Brandt matrices associated with the forms $f$ and $e_2$ respectively.
Suppose $\lambda u \equiv v \pmod l$
for some $\lambda \in \mathbb{F}^{\times}_l,$ then the congruence (\ref{eqn10}) in weight 2
can be lifted to a congruence in weight 3/2,
\begin{equation}\label{eqn11}
\lambda \mathcal{G} \equiv \mathcal{H} \pmod l.
\end{equation}
Thus we have the following result
\begin{thm}\label{keyprop}
Let $E$ be an elliptic curve of analytic rank zero and square-free conductor $N=PM.$ Let $f$ be the new form of level
$N$ corresponding to $E$ satisfying (\ref{sgn-eqn}).
Consider the family $\{ E_D \}$ of negative quadratic twists of $E$ satisfying the Kronecker condition
(\ref{kronecker-eqn}). Suppose $E$ has a torsion point defined over ${\mathbb Q},$ of odd prime order $l$ and that $|\Sh_{D}| =
{m^2_D}$ (upto a power of 2). Assume that $\lambda u \equiv v \pmod l,$ for some $\lambda \in \mathbb{F}^{\times}_l$ and
(\ref{w-eqn}) holds. Then, $|\Sh_D|$ is divisible by $l,$ if and only if the class number $h(-D)$ of ${\mathbb Q}(\sqrt{-D})$ is divisible by
$l.$
\end{thm}
\section{Optimal embeddings}\label{optimal}
We continue with the notation set out in the previous sections.
Let $K$ be a quadratic field over ${\mathbb Q}.$
Let $\phi$ be an embedding of $K$ into $B$.
The field $K$ is totally imaginary as $B$ is a definite quaternion algebra. Let ${\mathbb O}O_{-D}$ be an order of $K$ of discriminant $D$.
\begin{defi}
We say that $\phi$ is an optimal embedding of the order ${\mathbb O}O_{-D}$ into $R_i$ if $\phi$ is an embedding of $K$ into $B$ such that
$\phi({\mathbb O}O_{-D}) = \phi(K) \cap R_i$.
\end{defi}
Two optimal embeddings $i_1, i_2$ are equivalent if they are conjugate to each other by
an element in $R^{*}_i$. In other words, if there exists $x$ $\in$ $R^{*}_i$ such that $i_1(y) = x i_2(y) x^{-1}$
for all $y$ $\in$ $K$.
The Legendre symbol $( \frac{-D}{p} )$ is defined by
$( \frac{-D}{p} ):=$
$\begin{cases}
1, & \mbox{if } p \mbox{ splits in } K \\
0, & \mbox{if } p \mbox{ ramifies in } K \\
-1, & \mbox{if } p \mbox{ is inert in } K.
\end{cases}$
The Eichler symbol $\{ \frac{-D}{p} \}$ is defined by
$\{ \frac{-D}{p} \}:=$
$\begin{cases}
1, & \mbox{if} \ {p^2} \mid D \\
0, & \mbox{if} \ {p} \mid D, {p^2} \nmid {D}\\
( \frac{-D}{p} ), & \mbox{if } p \nmid D .
\end{cases}$
\\
\\
We prove a lemma and a proposition. We will use them in the proof of Theorem \ref{mainthm}.
\begin{lem}\label{for1} Let $h({\mathbb O}O_{-D}, R_i)$ be the number of equivalence classes of
optimal embeddings of the order of discriminant $D$ into $R_i$.
Then
$$ \sum^{n}_{i=1} h({\mathbb O}O_{-D}, R_i) = h(-D) \prod^{k}_{i=1} ( 1- \{ \frac{-D}{p_i} \} ) \prod_{q \mid M} ( 1+ \{ \frac{-D}{q} \} ).$$
\end{lem}
\begin{proof}
Let $\{{\mathfrak M}\}$ be a system of representatives of two-sided $R_i$ ideals modulo two-sided $R_i$ ideals of the form $R_i \xi $ where $\xi$ is an ${\mathbb O}O_{-D}$
ideal. Let $\{ {\mathfrak B} \}$ be a system of representatives of the ideal classes in ${\mathbb O}O_{-D}.$
Consider the set of all $({\mathfrak M}, {\mathfrak B} )$ such that\\
(1) The norm of ${\mathfrak M}$ is square-free and if $q$ is a prime divisor of the norm of ${\mathfrak M}$,
then either $q=p_i$ (for some $i =1$ to $k$) with $\{ \frac{-D}{p_i} \}=-1$ or $q$ is a prime divisor
of $M$ with $\{ \frac{-D}{q} \}=1$ and\\(2) ${\mathfrak B}$ is an integral ideal coprime to the conductor of ${\mathbb O}O_{-D}$.\\It is easy to observe that the
number of $ ({\mathfrak M}, {\mathfrak B})$ satisfying (1) and (2) is equal to
$$h(-D) \prod^{k}_{i=1} ( 1- \{ \frac{-D}{p_i} \} ) \prod_{q \mid M} ( 1+ \{ \frac{-D}{q} \} ).$$
There is a one-to-one correspondence between the set of all $({\mathfrak M}, {\mathfrak B})$ satisfying (1) and (2) and equivalence classes of optimal embeddings of the order of discriminant $-D$ into $R_i$.
For the proof of this correspondence, we refer to Section 3.2 of \cite{Sh65} (or) Satz 6,7 of \cite{Ei55}.
\end{proof}
We compute the Fourier coefficients of the modular forms $g_i,$ for $i=1$ to $k$ in the following proposition.
\begin{prop}\label{for2}
Let $g_i =\frac{1}{2} + \frac{1}{2} \sum_{ D > 0} a_i(D) q^{D}.$ Then
$a_i(D)$ is the number of elements $b$ $\in$ $R_i$ with $\mathrm{Tr}(b) =0$, $b \in {\mathbb Z} + 2 R_i$, ${\mathbb N}(b) = D.$
For $i = 1$ to $n$, we have $$a_i(D) = w_i \sum_{-D=df^2} \frac{h({\mathbb O}O_d, R_i)} {u(d)}, $$ where $u(d)=1$ unless $ d =-3, -4$ when $u(d) = 3,2$ respectively.
\end{prop}
\begin{proof}
Let $S$ be the set of elements $b$ $\in$ $R_i$ with $\mathrm{Tr}(b) =0$, $b \in {\mathbb Z} + 2 R_i$ and ${\mathbb N}(b) = D.$\\
For a negative integer $d$,
if $f: {\mathbb Q}(\sqrt{d}) {\mathcal H}ookrightarrow B$ is an embedding of an order ${\mathbb O}O_d$ into $R_i$, then\\
$b=f(\sqrt{d})$ is an element with trace 0 and norm $-d$. Since ${\mathbb O}O_{d} = {\mathbb Z} + {\mathbb Z} \frac{(-d + \sqrt{d})}{2},$ we have $b$ $\in$ $({\mathbb Z} + 2R_i)$.
Hence $b$ $\in$ $S^{0}_{i} = \{ x \in B | \mathrm{Tr}(x) =0 \} \cap ({\mathbb Z} + 2R_i).$\\ Conversely, if $b$ is an element in $S^{0}_i$ with norm $-d$ , then $f(\sqrt{d}) = b$ gives rise to an embedding
of the order ${\mathbb O}O_{d} = {\mathbb Z} + {\mathbb Z} \frac{ (-d + \sqrt{d})}{2}$ into $R_i$. The embedding $f(\sqrt{d})=b$ is optimal if and only if $b \notin f({\mathbb Z} + 2R_i)$ for some $f >1.$
Let $h^{*}({\mathbb O}O_{-D}, R_i)$ be the the number of optimal embeddings of ${\mathbb O}O_{-D}$ into $R_i$.
Using the above connection we proved that
$$a_{i}(D) = |S| = \sum_{-D=df^2} \{ b \in S, \frac{b}{f} \in S^{0}_{i}, \frac{b}{f} \notin n({\mathbb Z}+2R_i) \ \mathrm{for} \ n>1 \} = \sum_{-D=df^2} h^{*}({\mathbb O}O_d, R_i).$$
The group $\Gamma_i = R^{*}_i/\pm1$ acts on $S$. The $\Gamma_i$ orbits of $S$ correspond to equivalence classes of optimal embeddings.
Hence $$ | S/\Gamma_i | = \sum_{-D=df^2} h({\mathbb O}O_d, R_i).$$ The order of the stabilizer of an element $b \in S$ is 1 unless
the corresponding embedding extends to ${\mathbb Z}[\mu_6]$ or ${\mathbb Z}[\mu_4]$, when it is 3 or 2 respectively. Thus we have shown that
$$a_i(D) = w_i \sum_{-D=df^2} \frac{h({\mathbb O}O_d, R_i)} {u(d)}, $$ where $w_i = |\Gamma_i|.$
\end{proof}
Gross computed the traces of the Brandt matrices for prime level case\\(cf. Proposition 1.9, \cite{Gr87}).
It holds for square-free level, as we state in the following.
\begin{prop}
For all $m \geq 0,$
$$\mathrm{Tr}(B(m)) = \sum_{s \in \mathbb{Z}, s^2-4m \leq 0} \mathcal{H}(4m-s^2).$$
\end{prop}
\begin{proof} The diagonal entry of the brandt matrix $B(m)$ is $b_{ii}(m) = \frac{1}{e_i}|\{b, b \in R_i, {\mathbb N}(b) = m\} |.$\\
If $m =0,$ then $$\mathrm{Tr}(B(0)) = \frac{1}{24}
\prod^{k}_{i=1} ( {p_i} -1 ) \prod_{q \mid M} ( q+1 ) =
\sum^{n}_{i =1} \frac{ 1}{2w_i} = \mathcal{H}(0).$$ Let $A_i(s,m)$
be the set of elements $b$ $\in$ $R_i$ with $\mathrm{Tr}(b) =s$
and ${\mathbb N}(b) = m.$\\ This is a finite set. If $s^2-4m > 0$, then it
is an empty set. Hence
$$\mathrm{Tr}(B(m)) = \sum^n_{i=1} b_{ii}(m) = \sum^n_{i=1}
\sum_{s^2 \leq 4m} \frac{|A_i(s,m)|}{|R_i^{*}|} = \sum_{s^2 \leq
4m} ( \sum^n_{i=1} \frac{|A_i(s,m)|}{|R_i^{*}|} ).$$ If $s^2 =
4m$, then the inner sum $$\sum^n_{i=1}
\frac{|A_i(s,m)|}{|R_i^{*}|} = \sum^{n}_{i =1} \frac{ 1}{2w_i} =
\mathcal{H}(0).$$ Assume that $D =4m-s^2
>0.$
As in the
proof of Proposition \ref{for2}, we can show that
$$\frac{|A_i(s,m)|}{|R_i^{*}|} = \sum_{-D=df^2}
\frac{1}{2}\frac{h({\mathbb O}O_d, R_i)}{u(d)}.$$ By Lemma \ref{for1} and
Theorem \ref{mainthm}, $$\sum^n_{i=1} \frac{|A_i(s,m)|}{|R_i^{*}|}
= \sum^{n}_{i=1} \sum_{-D=df^2} \frac{1}{2}\frac{h({\mathbb O}O_d,
R_i)}{u(d)} = \mathcal{H}(4m-s^2).$$
\end{proof}
\section{The order of the Shafarevich-Tate group}\label{The order of the Shafarevich-Tate group}
Recall that we have equation (\ref{eqn0}) which relates the L-function of $f$ with the coefficients $m^2_D,$
$$\prod_{ p\mid \frac{N}{\mathrm{gcd}(N,D)} }( 1+ (\frac{-D}{p}) sgn (W_p) ) L(f,1)L(f \otimes \epsilon_{-D}, 1) =
\frac{ 2^{\omega(N)} (f, f ) m^2_D } { \sqrt{D} \sum \frac{ v^2_i }{ w^2_i} }.$$
If $E$ is the elliptic curve with conductor $N$ associated with $f \in S_2(\Gamma_0(N)),$
then we have $L(E,1)=L(f,1).$ Then the $L$-function $L(f \otimes \epsilon_{-D}, 1) = L(E_{D},1),$
where $E_{D}$ is the $-D$ quadratic twist of $E$ associated with $f \otimes \epsilon_{-D} \in S_2(\Gamma_0(ND^2)).$
Assume that the rank of $E$ is $0.$ The rank $0$ case of Birch and Swinnerton-Dyer Conjecture gives $$\frac{L(f \otimes \epsilon_{-D}, 1)}{\Omega_D}= \frac{L(E_{D},1)}{\Omega_D}= \frac{|\Sh_{D}| \prod c_{p,D}}{|\mathrm{Tor}(E_{D})|^2},$$
where $c_{p,D}$'s are the Tamagawa numbers and $\mathrm{Tor}(E_{D})$ is the torsion subgroup of $E_{D}({\mathbb Q}),$
${\Omega_D}$ is the real period of $E_D.$ Let $$C(D) = \frac{\prod_{ p\mid \frac{N}{\mathrm{gcd}(N,D)} }( 1+ (\frac{-D}{p}) sgn (W_p) ) } {2^{\omega(N)}}\frac{\Omega_D \prod c_{p,D} \sqrt{D} \frac{ v^2_i }{ w^2_i}L(f,1)}{(f, f ) |\mathrm{Tor}(E_{D})|^2 }.$$
Then $|\Sh_D| = \frac{m^2_D}{C(D)}.$
Math softwares can be used to compute the term $C(D).$
\subsection{Proof of Theorem \ref{keyprop}}
\begin{proof}We prove the theorem when $P$ is prime. One can conclude the theorem similarly when $P$ is composite.
If $l$ is an odd prime dividing the order of the group of torsion points of the elliptic curve $E$,
by Mazur's theorem, $l = 3, 5$ or $7.$ We know that $w_i \mid 12,$ the product $\prod^{n}_{i=1} w_i$ equals the exact
denominator of $\frac{N-1}{12}$ and $3$ divides the exact numerator of $\frac{N-1}{12}.$
Hence $w_i \in \mathbb{F}^{\times}_l$ for $l = 3, 5$ or $7.$
From $\lambda \mathcal{H} - \mathcal{G} = \sum^{n}_{i=1} \frac{(\lambda - v_i)}{w_i} g_i,$
it follows that the congruence $\lambda u \equiv v \pmod l,$ for some $\lambda \in \mathbb{F}^{\times}_l$ gives a congruence
$\lambda \mathcal{H} \equiv \mathcal{G} \pmod l.$
This yields a congruence on the coefficients $\lambda \mathcal{H}(D) \equiv m^2_D \pmod l.$ From Corollary \ref{main-cor}, we see that
$l$ divides $\mathcal{H}(D)$ if and only if $l$ divides $h(-D).$ We also have $|\Sh_{D}| =
{m^2_D}$ (up to a power of 2). Hence $|\Sh_D|$ is divisible by $l,$ if and only if the class number $h(-D)$ of ${\mathbb Q}(\sqrt{-D})$ is divisible by
$l.$
\end{proof}
By letting $k=1$ in the above Theorem, we deduce the following corollary.
\begin{cor}[Proposition 3.9, \cite{Qu11}] Let $E,$ $E_D,$ $l$ and $\Sh_D$ be as in Theorem \ref{keyprop}.
Assume that there is exactly one prime $p \mid N$ such that the sign of $ W_p = -1.$
Then, $|\Sh_D|$ is divisible by $l,$ if and only if the class number $h(-D)$ of ${\mathbb Q}(\sqrt{-D})$ is divisible by $l.$
\end{cor}
\section{Cohen-Eisenstein series}\label{Cohen-Eisenstein series}
\subsection{Examples}
We calculate the Fourier coeffients of the weight 3/2 Eisenstein series $\mathcal{H} = \sum^{n}_{i=1} \frac{1}{w_i} g_i$
and the class numbers of imaginary quadratic fields $K={\mathbb Q}(\sqrt{-D})$ for $d \leq 2000$ by using MAGMA.
Let $D > 0$ be a natural number and let ${\mathbb O}O_{-D}$ be the ring of integers in ${\mathbb Q}(\sqrt{-D})$. Let $h(-D)$ be the cardinality of the group $\mathrm{Pic}({\mathbb O}O_{-D})$, and
let $2u(-D)$ be the cardinality of the unit group ${\mathbb O}O^{*}_{-D}.$ A prime $l$ is inert, splits or ramifies in ${\mathbb O}O^{*}_{-D}$
if the Kronecker symbol $( \frac{-D}{l} )$ is -1, 1, 0 respectively.
We consider the strong Weil curves of rank zero with an odd torsion point from Cremona's table \cite{Cr97}.
\\
$\bullet N=66 = 2.3.11$
\\
We have elliptic curve $E = 66 \mathrm{C(I)} = [1,0,0,-45,81]$ of level $66$ with analytic rank zero and $|\mathrm{Tor}(E)| =10.$
We have $\mathrm{sgn} (W_2) = \mathrm{sgn} (W_3) = \mathrm{sgn} (W_{11}) = -1.$
We work in the quaternion algebra ramified at 2, 3, 11 and at $\infty.$
We calculate the Brandt matrices for an order of level 66. We have,
for $D \leq 2000$ such that $-D$ is a fundamental discriminant and $( \frac{-D}{2} ), ( \frac{-D}{3} )$ and $( \frac{-D}{11} ) \neq 1:$
\\
\\
$\mathcal{H}(D):=$
$\begin{cases}
\frac{2^{2} h(-D)}{u(-D)} , & \mbox{if none of the primes} \ 2,3, 11 \ \mbox{ramifies in} \ K \\
\frac{2^{1} h(-D)}{u(-D)}, & \mbox{if exactly one prime} \ p \mid 66 \ \mbox{ramifies in } K \\
\frac{h(-D)}{u(-D)}, & \mbox{if exactly two primes} \ p \mid 66 \ \mbox{ramify in } K\\
\frac{ h(-D)}{2 u(-D)}, & \mbox{if} \ 2, 3 \ \mbox{and} \ 11 \ \mbox{ramify in } K.
\end{cases}$
\\
\\
$\bullet N=210 = 2.3.5.7$
\\
We have elliptic curve $E = 210 \mathrm{A(A)} = [1,0,0,-41,-39]$ of level $210$ with analytic rank zero and $|\mathrm{Tor}(E)| = 6.$
We have $\mathrm{sgn} (W_2) = \mathrm{sgn} (W_3) = \mathrm{sgn} (W_{7}) = -1$ and $\mathrm{sgn} (W_5) = +1.$
We work in the quaternion algebra ramified at 2, 3, 7 and at $\infty.$
We calculate the Brandt matrices for an order of level 210. We have,
for $D \leq 2000$ such that $-D$ is a fundamental discriminant and $( \frac{-D}{2} ), ( \frac{-D}{3} ), ( \frac{-D}{7} ) \neq 1$ and
$( \frac{-D}{5} ) \neq -1:$
\\
\\
$\mathcal{H}(D):=$
$\begin{cases}
\frac{2^{3} h(-D)}{u(-D)} , & \mbox{if none of the primes} \ 2,3, 5, 7 \ \mbox{ramifies in} \ K \\
\frac{2^{2} h(-D)}{u(-D)} , & \mbox{if exactly one of the primes} \ p \mid 210 \ \mbox{ramifies in} K \\
\frac{2^{1} h(-D)}{u(-D)}, & \mbox{if exactly two of the primes} \ p \mid 210 \ \mbox{ramify in } K \\
\frac{h(-D)}{u(-D)}, & \mbox{if exactly three primes} \ p \mid 210 \ \mbox{ramify in } K\\
\frac{ h(-D)}{2 u(-D)}, & \mbox{if} \ 2, 3, 5 \ \mbox{and} \ 7 \ \mbox{ramify in } K.
\end{cases}$
\\
\\
We have also computed the Fourier coefficients $\mathcal{H}(D)$
for the rank 0 elliptic curves $E = 110A1(C) = [1,1,1,10,-45]$ with $\mathrm{Tor}(E) = 5,$
$E = 114A(A) = [1,0,0,-8,0]$ with $\mathrm{Tor}(E) = 6,$
$E = 130B(A) = [1,-1,1,-7,-1]$ with $\mathrm{Tor}(E) = 4,$
$E = 210B(A) = [1,0,1,-498,4228]$ with $\mathrm{Tor}(E) = 6$
and several other examples. Based on our numerical examples, we observed a generalization
of the conjecture \ref{main} which we prove in Corollary \ref{main-cor}.
\begin{thm}\label{mainthm}
Let $B$ be a definite quaternion algebra ramified at $p_1, p_2,...,p_k$. Let $N=p_1p_2...p_kM$ $( \ p_i \nmid M \ )$
be a square-free integer. Denote by $ \mathcal{H} = \sum^{n}_{i=1} \frac{1}{w_i} g_i = \sum^{n}_{i =1} \frac{ 1}{2w_i} +
\sum_{D > 0} \mathcal{H}(D) q^D.$ Then we have
$$ \mathcal{H}(D) =\frac{1}{2}\sum_{ -D=df^2 }
\Big[\frac{h(d)}{u(d)} \prod^{k}_{i=1} \Big( 1- \{ \frac{d}{p_i} \} \Big) \prod_{q \mid M} \Big( 1+ \{ \frac{d}{q} \} \Big) \Big].$$
\end{thm}
\begin{proof} Consider the weight $3/2$ Cohen-Eisenstein Series $\mathcal{H}$,
$$ \mathcal{H} = \sum^{n}_{i=1} \frac{1}{w_i} g_i = \frac{1}{2} \sum^{n}_{i=1} \frac{1}{w_i} + \sum_{D >0} \sum^{n}_{i=1} \frac{a_i(D)} {w_i} q^D.
$$
By Proposition \ref{for2}, we see that
\begin{equation}\label{eqn1}
\sum^{n}_{i=1} \sum_{D >0} \frac{a_i(D)}{w_i} q^D = \frac{1}{2} \sum_{ D >0} \sum_{ -D=df^2 }
\Big(\sum^{n}_{i=1} \frac{ h({\mathbb O}O_d, R_i)}{u(d)} \Big).
\end{equation}
By Lemma \ref{for1}, we have
\begin{equation}\label{eqn2}
\sum^{n}_{i=1} h({\mathbb O}O_d, R_i) =
h(d) \prod^{k}_{i=1} \Big( 1- \{ \frac{d}{p_i} \} \Big) \prod_{q \mid M} \Big( 1+ \{ \frac{d}{q} \} \Big).
\end{equation}
Substituting equations (\ref{eqn1}) and (\ref{eqn2}) in the Fourier expansion of $\mathcal{H},$ we get
$$\mathcal{H} = \sum^{n}_{i=1} \frac{1}{w_i} g_i = \frac{1}{2} \sum^{n}_{i=1} \frac{1}{w_i} + \frac{1}{2}
\sum_{ D>0} \sum_{ -D=df^2 }
\Big[\frac{h(d)}{u(d)} \prod^{k}_{i=1} \Big( 1- \{ \frac{d}{p_i} \} \Big) \prod_{q \mid M} \Big( 1+ \{ \frac{d}{q} \} \Big) \Big] q^D.$$
Hence
\begin{equation}\label{main-eqn}
\mathcal{H}(D) = \frac{1}{2}\sum_{ -D=df^2 }
\Big[\frac{h(d)}{u(d)} \prod^{k}_{i=1} \Big( 1- \{ \frac{d}{p_i} \} \Big) \prod_{q \mid M} \Big( 1+ \{ \frac{d}{q} \} \Big) \Big].
\end{equation}
This completes the proof.
\end{proof}
We deduce the following corollary which generalizes the result of Gross (Proposition \ref{gross-prop}) for square-free level $N.$
\begin{cor}\label{main-cor} Let $B$ and $\mathcal{H}$ be as in Theorem \ref{mainthm}.
If $-D$ is the fundamental discriminant, $\omega(N)$ is the number of distinct primes that divide $N$, $s(D)$
is the number of primes that divide $N$ and ramify in ${\mathbb Q}(\sqrt{-D}),$ and
$( \frac{-D}{p_i} ) \neq 1$, for every $i = 1$ to $k$, and
$( \frac{-D}{q} ) \neq -1$ for every prime $q$ $\mid$ $M$, then
$$\mathcal{H}(D) = \frac{2^{\omega(N)-1 - s(D)} h(-D)}{u(-D)}.$$
\end{cor}
\begin{proof} If $-D$ is the fundamental discriminant satisfying the Kronecker conditions, then from the equation (\ref{main-eqn}), we have
$$\mathcal{H}(D) = \prod^{k}_{i=1} \Big( 1- \{ \frac{-D}{p_i} \} \Big) \prod_{q \mid M} \Big( 1+ \{ \frac{-D}{q} \} \Big) \frac{1}{2}
\frac{h(-D)}{u(-D)}= \frac{2^{\omega(N)-1 - s(D)} h(-D)}{u(-D)}.$$
\end{proof}
\begin{cor}\label{main-conj}
Conjecture \ref{main} holds.
\end{cor}
\begin{proof}
Conjecture \ref{main} follows immediately from Corollary \ref{main-cor}
by letting $k=1.$
\end{proof}
\end{document}
|
\begin{document}
\title{Free-space continuous-variable quantum key distribution of unidimensional Gaussian modulation using polarized coherent-states in urban environment}
\author{Shi-yang Shen}
\author{Ming-wei Dai}
\author{Xue-tao Zheng}
\author{Qi-yao Sun}
\author{Bing Zhu}
\author{Guang-Can Guo}
\author{Zheng-Fu Han}
\affiliation{Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China}
\begin{abstract}
We use single homodyne detector to accomplish Continuous-Variable quantum key distribution(CV QKD) in a laboratory and urban environment free-space channel. This is based on Gaussian modulation with coherent-states in the polarization degree of freedom. We achieved a QKD distance at 460m, at the repetition rate of 10 kHz. We give the security of this protocol against collective attack in the asymptotic regime. The secure key rate is 0.152 kbps at the typical reconciliation efficiency of 0.95. The experiment setup of this scheme is simplified and the difficulty to realize has been remarkably reduced compared to traditional symmetric modulation ones, for example, GG02 protocol. The influence of security key rate brought by asymmetric modulation is small in a relative low channel loss condition in the free-space environment. This scheme is expected to be significance meaning to the future practically utilize.
\end{abstract}
\pacs{}
\keywords{}
\maketitle
\section{introduction}
\label{intro.}
Quantum key distribution allows the two authorized distant parties, Alice and Bob, to share a common key via a potential eavesdropped quantum channel. The first QKD protocol was been proposed in 1984 \cite{BB84}. While CV QKD protocols, especially ones with coherent-states light source, has been concerned recently \cite{GG02, reverse rec, reverse 2}, which utilize balanced homodyne detection technique, and light source is not at single photon level. Therefore, it has advantages of higher detection efficiency, thus higher secure key rate, and anti photon number attack. During a few years of development, CV QKD protocols and experiment implementations have been modified and simplified. First, the coherent-states protocols show substantial advantage against squeezed-states versions\cite{squeezed prot, squeezed, squeeze polar, squeeze free space} in the prepare of light source, and theoretical secure distance improves remarkably, which leads to the deep research of CV QKD theory and realizing variable experiment schemes. As far as the process of present experiments in the field of CV QKD, the symmetrically Gaussian modulated coherent-state(GMCS) protocols has been quite well studied\cite{GG02 experiment, free space exp, free space feasible, np exp, long distance, fiber exp} since the composable security analysis has been revealed \cite{composable}. Secondly, instead of Gaussian modulation, the discrete modulation reduces the complexity in the classical post-processing, in which case the signal-to-noise ratio is low because of the long distance propagation loss. Finally, the unidimensional CV QKD protocol proposed in 2015\cite{unidimension} further simplified apparatus in both preparing and detection sides since only one of the quadratures should be Gaussian modulated instead of both being modulated simultaneously. Experiment scheme in fiber channel has been accomplished in \cite{unidimension exp} and the security key rate at finite size scenario has been proved\cite{ud finite key}. Furthermore, composable security of unidimensional CV QKD has been revealed in \cite{ud composable}.
Besides, free-space channel is insensitivity to polarization compared to fiber channel, which results to be unchanged of the light polarization as propagating. Thus the polarization controller at receiver's side can be left out. In another words, the system needn't calibrate polarization direction frequently, reducing the calibrating time of non-key distribution, and thus increase the key rate. On the other hand, encoding with polarization avoids the non-synchronous disturbance of the phase. Therefore, the phase locking between local oscillator and signal is unnecessary as soon as the polarization has been aligned to the same direction, significantly simplifying the difficulty of system implementation. Security distance of CV QKD in free-space\cite{free space exp, free space feasible, squeeze free space} channel can reach dozens of kilometers, compatible with urban condition communication. It is expected to play an important role in the future practical applications.
This experiment uses unidimensional CV QKD scheme in the free-space channel, modulating the polarization quantum Stokes parameter with Gaussian distribution and gets the security key rate at real urban environment condition of 460m, which turned out to be little less performance but obviously simplified and more adaptable to experimental environment, compared to GG02 protocol at the same conditions.
The article is organized as follows, in section II and III, we explain the principle of coherent-state encoded in polarization degree and give the security key rate analysis against collective attack. In section IV, the experimental setup and environment will be revealed. Finally, in section V, we deal with the raw key data and give the security key rate.
\section{protocol description}
In section II, we describe the prepare and measure version of unidimensional protocol proposed in \cite{unidimension}, which correspond to the real experiment system implementation. In usual CV QKD protocols, both quadratures, X and P, must be modulated simultaneously. However, in this protocol, the situation is contrast. Only single quadrature, without loss of generality, denoted as X, will be modulated. Each coherent-state, is displaced by x in phase space, which obeys Gaussian distribution centered at 0 and has the variance of $V_{M}$. And the other quadrature has the variance of 1,normalized at the shot noise unit(SNU). Bob performs homodyne detective X quadrature in a certain time interval and sometimes monitor variance of P quadrature as a channel parameter. After Alice and Bob share a sufficient long sequence of real number raw key data, they estimate the channel parameters using a small random part of the data and perform reverse-reconciliation \cite{reverse rec}.
\label{Polarization encode principle}
For polarization encoding, quantum Stokes operators are treated as quadratures like phase encoding scheme, defined as follow \cite{stokes op}:
\begin{equation}
\begin{aligned}
\hat{S}_{0}&=\hat{a}^{\dagger}_{H}\hat{a}_{H}+\hat{a}^{\dagger}_{V}\hat{a}_{V},\\
\hat{S}_{1}&=\hat{a}^{\dagger}_{H}\hat{a}_{H}-\hat{a}^{\dagger}_{V}\hat{a}_{V},\\
\hat{S}_{2}&=\hat{a}^{\dagger}_{H}\hat{a}_{V}+\hat{a}^{\dagger}_{V}\hat{a}_{H},\\
\hat{S}_{3}&=i(\hat{a}^{\dagger}_{V}\hat{a}_{H}-\hat{a}^{\dagger}_{H}\hat{a}_{V}),\\
\end{aligned}
\end{equation}
where subscripts H and V label the creation and annihilation operators along horizontal and vertical polarization mode, respectively. These creation and annihilation operators obey the same commutation relations and Heisenberg uncertainty principle as the quadratures X and P in phase space except a constant coefficient:
\begin{equation}
[\hat{a}_{j},\hat{a}^{\dagger}_{k}]=\delta_{jk}, j,k=x,y
\end{equation}
and
\begin{equation}
[\hat{S}_{j},\hat{S}_{k}=2i\epsilon_{l}\hat{S}_{jkl}], for j,k,l=1,2,3
\end{equation}
While the variance of the latter three Stokes operators satisfy
\begin{equation}
\label{uncertainty}
Var[\hat{S}_{2}] Var[\hat{S}_{3}]\ge |\langle\hat{S}_{1}\rangle|^{2}
\end{equation}
In our work, we use polarization degree to encode information. The $S_{1}$ polarized (V mode) coherent-state light plays the role of local oscillator(LO). $S_{3}$ and $S_{2}$ polarized state generated by a electro-optical modulation(EOM) are two orthogonal quadratures. Since the intensity of the modulated light is far weaker (about 3 orders of magnitude lower) than the LO, the loss of LO is negligible, and the intensity of circle polarized light $|S_{1}|$ nearly remain unchanged. In other words, the right hand of equation (\ref{uncertainty}) is approximatively a constant. The output light is at strong vertical polarized mode with a superposition of a very weak circle mode. More explicitly, the shape of the polarization state in x-y space is an ellipse with a eccentricity of nearly unity, and its long and short axis are oriented to the direction of modulation intensity.
If the applied voltage of EOM is $U$, the phase difference between ordinary and extraordinary light is $\phi=\pi U/V_{\pi}$, where $V_{\pi}$ is the half wave voltage of EOM. Assume that the annihilation operator of input light is:
\begin{equation}
\hat{a}_{in}=\frac{a_{LO}}{\sqrt{2}}\left[
\begin{array}{ccc}
0\\
1\\
\end{array}\right]
\end{equation}
After EOM, without loss of generality, except for a phase factor, the output light is
\begin{equation}
\hat{a}_{in}=\frac{a_{LO}}{\sqrt{2}}\left[
\begin{array}{ccc}
0\\
e^{i \phi}\\
\end{array}\right]
\end{equation}
The light then passes through a quarter wave plate whose fast and slow axises is $45\deg$ to the axises of EOM, and the Jones Matrix of QWP is
\begin{equation}
J_{QWP}=\left[
\begin{array}{ccc}
\cos\frac{\pi}{4}&-\sin\frac{\pi}{4}\\
\sin\frac{\pi}{4}&\cos\frac{\pi}{4}\\
\end{array}\right]
\left[
\begin{array}{ccc}
1&0\\
0&i\\
\end{array}\right]
\left[
\begin{array}{ccc}
\cos\frac{\pi}{4}&\sin\frac{\pi}{4}\\
-\sin\frac{\pi}{4}&\cos\frac{\pi}{4}\\
\end{array}\right]
\end{equation}
Then the light is
\begin{equation}
\hat{a}=J\cdot \hat{a}_{out}=
\frac{a_{LO}}{2}\left[
\begin{array}{ccc}
1+i e^{i \phi}\\
1-i e^{i \phi}\\
\end{array}\right]
\end{equation}
After a 50:50 PBS and a balanced homodyne detector, the measured photon number difference is
\begin{equation}
n_{meas}=\hat{a}^{\dagger}_{H}\hat{a}_{H}-\hat{a}^{\dagger}_{V}\hat{a}_{V}=a^{2}_{LO}\sin{\frac{\pi U}{V_{\pi}}}
\end{equation}
When $U\ll V_{\pi}$,
\begin{equation}
\label{var}
n\approx a^{2}_{LO}\frac{\pi U}{V_{\pi}}
\end{equation}
Thus, if $U$ obeys Gaussian distribution, $U\sim N(0, \Sigma^{2})$, then $n\sim N(0,a^{4}_{LO}\frac{\pi^{2}\Sigma^{2}}{V^{2}_{\pi}})$.
The relation between applied voltage on EOM and modulation variance $V_{M}$ is
\begin{equation}
V_{M}=\frac{\pi^{2}\Sigma^{2}V^{2}_{LO}}{V^{2}_{\pi}N_{0}}
\end{equation}
according to Eq. \ref{var}, where $V_{LO}$ is the voltage of local oscillator in Alice's side measured by DAQ. However, the quadrature's variance is in SNU, the Stokes operator must be normalized as X and P. On one hand, the expectation value
\begin{equation}
\begin{aligned}
\langle\hat{S}_{3}\rangle&=\langle a_{H}a_{V}|i(\hat{a}^{\dagger}_{V}\hat{a}_{H}-\hat{a}^{\dagger}_{H}\hat{a}_{V})|a_{H}a_{V}\rangle\\
&=2a^{2}_{LO}\sin\phi\\
&=2n_{meas}\\
\end{aligned}
\end{equation}
On the other hand, note that the local oscillator is strong enough so that $\langle S_{1}\rangle\approx a^{2}_{LO}$, donated as $S_{1}$. By defined two new operators
\begin{equation}
\begin{aligned}
\hat{X}&=\frac{\hat{S}_{2}}{\sqrt{S_{1}}}\\
\hat{P}&=\frac{\hat{S}_{3}}{\sqrt{S_{1}}}\\
\end{aligned}
\end{equation}
$\hat{X}$ and $\hat{P}$ are evidently have the same form of commutation relations and Heisenberg uncertainty principle as usual defined quadrature X and P. To be distinguishable, we still use new symbols X and P instead of $S_{2}$ and $S_{3}$ below, unless either mentioned. Therefore, $X_{a}$ and $X_{b}$ are proportional to average photon numbers, thus proportional to the voltages measured by DAQs since the photo-diodes work on linear mode.
\begin{equation}
\begin{aligned}
X_{a}&=\frac{2n_{prep}}{\sqrt{n_{a}}}\\
X_{b}&=\frac{2n_{meas}}{\sqrt{n_{b}}}\\
\end{aligned}
\end{equation}
where $n_{a}=a^{2}_{LO}$ and $n_{b}=T \eta n_{a}$ are the average photon number of local oscillator of Alice and Bob respectively, and they can be monitored by power meter or wave oscilloscope in the unit as light intensity or voltage. And $T, \eta$ are overall transmittance and detection efficiency of homodyne.
\section{security analysis}
The security of CV QKD has been studied especially in recent years. \cite{gaussian extrem} give the extremality of Gaussian states, and as a consequence, the extremality of Gaussian attacks in \cite{gaussian attack extrem 1, gaussian attack extrem 2} against collective attack scheme in the asymptotic region. In 2010\cite{finite}, the finite-size effect analysis of CV QKD has been shown also by Leverrier. The security against general attacks in practical finite-size region has been proven in 2013 \cite{general attack}, which exploits symmetry in phase-space even with post-selection, as far as the modulation and post-election is symmetric in phase-space. Soon later, Leverrier\cite{composable} achieved the composable security for coherent CV QKD protocols against collective attacks, which established the security of coherent protocols against general attacks.
The security key rate for the unidimensional protocol is computed in \cite{unidimension, ud finite key} against collective attacks in asymptotic and finite size region. As already obviously known, the lower bound key rate is given by
\begin{equation}
K=I_{AB}-\chi_{BE}
\end{equation}
where
\begin{equation}
\chi_{BE}=S(E)-S(E|x_{B})
\end{equation}
is the Holevo information \cite{holevo, unidimension} between Bob and Eavesdropper in the scheme of reverse reconciliation. Since the eavesdropper holds the purification of the state $\rho_{ABE}$, the Von Neumann entropy can be expressed as
\begin{equation}
\begin{aligned}
S(E)&=S(AB),\\
S(E|x_{B})&=S(A|x_{B})\\
\end{aligned}
\end{equation}
which can be respectively calculated through the covariance matrix $\Gamma_{AB}$, and on the other hand, the conditioned Von Neumann entropy $S_{A|x_{B}}$ through the conditioned entropy $\Gamma_{A|x_{B}}$. More explicitly,
\begin{equation}
\chi_{BE}=G\left(\frac{\lambda_{1}-1}{2}\right)+G\left(\frac{\lambda_{2}-1}{2}\right)-G\left(\frac{\lambda_{c}-1}{2}\right)
\end{equation}
where the function $g(x)$ is defined as
\begin{equation}
G(x)=(x+1)\log(x+1)-x\log x
\end{equation}
and $\lambda_{1,2}$ are symplectic eigenvalues of $\Gamma_{AB}$ and $\lambda_{c}$ is symplectic eigenvalue of $\Gamma_{A|x_{B}}$.
In the description of entanglement-based scheme, the covariance matrix of a two-mode squeezed vacuum state is
\begin{equation}
\gamma=\left[
\begin{array}{ccc}
V\mathbb{I}_{2} & \sqrt{V^{2}-1}\mathbf{\sigma}_{z}\\
\sqrt{V^{2}-1}\mathbf{\sigma}_{z} & V\mathbb{I}_{2}\\
\end{array}\right]
\end{equation}
The variance V is equal to $\sqrt{V_{M}+1}$ in prepare-and-measure scheme with the modulation variance of $V_{M}$. According to \cite{unidimension}, the covariance for unidimensional protocol is built by a squeeze operation on one of its mode, for example, here $S_{1}$, with a squeezing parameter of $r=-\log\sqrt V$, then the covariance matrix reads:
\begin{equation}
\begin{split}
\gamma_{AB}=& S \gamma S^{T}\\
=&\left[
\begin{matrix}
\sqrt{V_{M}+1} & 0 & V_{M}\sqrt{V_{M}+1} & 0\\
0 & \sqrt{V_{M}+1} & 0 & -\frac{V_{M}}{\sqrt{V_{M}+1}}\\
V_{M}\sqrt{V_{M}+1} & 0 & V_{M}+1 & 0\\
0 & -\frac{V_{M}}{\sqrt{V_{M}+1}} & 0 & 1\\
\end{matrix}\right]
\end{split}
\end{equation}
where the squeezing operator S is
\begin{equation}
S=\left[
\begin{array}{ccc}
(V_{M}+1)^{1/4} & 0\\
0 & (V_{M}+1)^{-1/4}\\
\end{array}
\right]
\end{equation}
Now assume that the channel transmittance and noise in X(or equivalently, $S_{1}$) is $\eta_{x}, \epsilon_{x}$, respectively. After transmission through the noisy channel, the covariance becomes
\begin{equation}
\label{gammaAB1}
\gamma^{'}_{AB_{1}} =\left[
\begin{matrix}
\sqrt{V_{M}+1} & 0 & \sqrt{\eta_{x}V_{M}}({V_{M}+1})^{\frac{1}{4}} & 0\\
0 & \sqrt{V_{M}+1} & 0 & C_{p}\\
\sqrt{\eta_{x}V_{M}}({V_{M}+1})^{\frac{1}{4}} & 0 & 1+\eta_{x}(V_{M}+\epsilon_{x}) & 0\\
0 & C_{p} & 0 & V_{P1}\\
\end{matrix}\right]
\end{equation}
Since the P (or $S_{2}$) quadrature is unmodulated, its variance $V_{p}$ and the correlation between X (or $S_{1}$) and P (or $S_{2}$), $C_{p}$, remains unknown to all communication parties, including eavesdropper. $V_{p}$ is monitored at Bob's side. Considering the realized model of balanced homodyne detectors(BHD), it has a non-unity detection efficient $\eta_{e}$, and electronic noise $V_{e}$ in shot-noise unit. It is modeled as an ideal BHD, followed by a PBS of a transmittance efficient $\eta_{e}$. A thermal state of a noise variance of $1+\frac{V_{e}}{1-\eta_{e}}$ injects from one of the port of PBS. In this case, the covariance of Alice and Bob is \cite{unidimension exp}
\begin{equation}
\gamma^{'}_{AB} =\left[
\begin{matrix}
\sqrt{V_{M}+1} & 0 & \sqrt{\eta_{e}\eta_{x}V_{M}}({V_{M}+1})^{\frac{1}{4}} & 0\\
0 & \sqrt{V_{M}+1} & 0 & \sqrt{\eta_{e}}C_{p}\\
\sqrt{\eta_{e}\eta_{x}V_{M}}({V_{M}+1})^{\frac{1}{4}} & 0 & 1+\eta_{e}\eta_{x}(V_{M}+\epsilon_{x})+V_{e} & 0\\
0 & \sqrt{\eta_{e}}C_{p} & 0 & \eta_{e}V_{P1}+(1-\eta_{e})+V_{e}\\
\end{matrix}\right]
\end{equation}
Since the unknown parameters $C_{p}$ and $V_{p}$ must be physical, the Heisenberg Uncertainty principle gives their bound
\begin{equation}
\gamma_{AB}+i\Omega\ge 0
\end{equation}
where
\begin{equation}
\Omega=\left[
\begin{matrix}
0 & 1 & 0 & 0\\
-1 & 0 & 0 & 0\\
0 & 0 & 0 & 1\\
0 & 0 & -1 & 0\\
\end{matrix}\right]
\end{equation}
In the assumption of reverse reconciliation \cite{reverse rec}, Alice guesses the measurement of Bob's, so she holds the conditioned covariance matrix:
\begin{equation}
\gamma_{A|x_{B}}=\left[
\begin{matrix}
\frac{\sqrt{V_{M}+1}(1+\eta_{x}\epsilon_{x})}{1+\eta_{x}(V_{M}+\epsilon_{x})} & 0\\
0 & \sqrt{V_{M}+1}\\
\end{matrix}
\right]
\end{equation}
\section{experiment setup}
The experiment setup for the unidimensional CV QKD system in free-space is shown in \ref{setup}.
\begin{figure}
\caption{Free-space CV QKD experiment setup for unidimensional protocol. }
\label{setup}
\end{figure}
The laser(type) centered at 786nm, about 1 nm of full width at half maximum, is fiber pig-tailed and coupled to free-space in Gaussian mode. The output intensity is 15mW, and passes through two combinations of a PBS(Thorlabs PBS252, extinction ratio$\geq33dB$) and a half-wave plate whose axis is nearly perpendicular to the PBS axis. Since the axises of these two PBSs are aligned at the same direction, and the half-wave plate between them rotates the polarization of light nearly 90 degree, the output light intensity attenuates to $100\mu W$ level, equally the magnitude of orders about $1\times10^{14}$ photons per second. The quater-wave plate rotates the linear polarization light to a certain elliptic polarization, so as it becomes a circle polarization mode after the EOM since the ordinary and extraordinary axises are probably not at the same direction as PBS. Then, the polarization state is modulated in the EOM(Thorlabs, EO-AM C1, wavelength 600-900nm), whose modulation bandwidth is 100 MHz \cite{EOM datasheet}. Since the intensity of signal light is far weaker than local oscillator light (3 magnitude of orders), the voltage is controlled by computer, with Gaussian distributed random intensity whose variance is $\Sigma^{2}$ as mentioned in Section \ref{Polarization encode principle}, which is about 165 times to the shot-noise, at the modulation frequency of 10kHz, which is limited by the acquisition bandwidth of DAQ module(NI PCIe-6363, maximum acquisition rate is 2MHz). The width of modulation signal pulses is 10$\mu s$, which is much longer than the time difference of light distance between the two arms of BHD (about$10^{-11}s$).
The output signal with the local oscillator beam in spatial mode is focus within 2.1mm of $1/e^{2}$ diameter, and the full divergence angle of $4.5\times10^{-4}$, enter a Galileo beam expander(Thorlabs, GBE10-B, expansion is 10x) so that the transmitted full divergence is $4.5\times10^{-5}$. After propagated through a 460m free-space channel, the beam diameter at the receive aperture is about 4 cm.
At the receiver side, a 4-inch reflection mirror is to adjust beam direction. Two convex lenses whose diameter and focus are 4 inches, 20cm and 2 inches, 6cm reduce the beam diameter to about 1cm. HWP3 is used to calibrate the polarization direction such that the receiving local oscillator is polarized along vertical mode. QWP2 changes linear polarized light to circle so that the photon numbers of two ports of PBS corresponds to left and right circle polarized modes, respectively. Then, another EOM placed between QWP2 and PBS, acts as a basis switcher, to monitor the variance of $S_{2}$ quadrature. It is modulated randomly with the voltage of 0 or $V_{\pi}$. When applied voltage is 0, the measured phonon number is X quadrature, and when $V_{\pi}$, P quadrature. Two congruent convex lens focus the beam into photon diodes (Hamamatsu, S3883, photon-sensitivity 0.58A/W at 780nm, equal to detective efficient 0.872 for each individual diode \cite{hamamatsu datasheet}). Another DAQ module is used to acquisit the output of the difference voltage of two diodes for every pulse, at the sampling rate of 1 MHz. Thus, for each pulse it takes 100 samples, of which 10 for each signal pulse since the duty cycle is set to be 0.1. The average voltage of these 10 sample is raw key value for Bob. A consecutive 5 large pulse (10V) marks the start of each communication, in other words, the pulses followed the start pulses are as the distributed keys.
\section{experiment result}
First we record the beam spot behaviors caused by wandering and vibrating of the buildings. At receiver's side, a CCD camera beam profiler(Thorlabs, BC106N-VIS/M) at the lens focus records the profile and jitter of beam, as shown in Fig. \ref{profile}.
\label{profile}
\begin{figure}
\caption{Beam profiles recorded by CCD camera after two output ports of PBS at Bob's side at lens focus. Left: vertical polarized. Right: horizontal polarized. For convenience, the two beam profiles are seen in one figure. The horizontal and vertical axises are position of beam, in unit of $\mu m$. The diameter of two beams are not exactly the same because the CCD deviates from lens focus at the two directions. The profiles are Gaussian in both x and y direction but shaped as ellipse since beam is not vertical to lens.}
\end{figure}
The sensitivity area of photo-diodes is about 1.5mm, much larger than beam diameter so can collective all light intensity. However, the intensities are still fluctuate due to atmospheric turbulence.
The sender's side is placed at 9th floor of a building while receiver's side is at 16th floor of another building. Since the height of both buildings is high, their vibration is not negligible. The jitter of beam spot and trace of spot center are shown in Fig. \ref{jitter} and Fig. \ref{trace}, respectively.
\label{jitter}
\begin{figure}
\caption{The center of x and y as a function of time of beam at the transmission port of PBS. Measure time is 10 minutes. Red: peak position in y direction. Yellow: peak position in x direction. Blue: position of intensity center in y direction. Green: position of intensity center in x direction. The maximum jitter in vertical direction is about 100 $\mu m$(blue line), while 1000 $\mu m$(green line) in x direction.}
\end{figure}
The jitter in vertical direction is mainly caused by beam wandering while the buildings vibration in horizontal direction. The frequency of building vibration is much lower than beam wandering due to atmospheric turbulence.
\label{trace}
\begin{figure}
\caption{Positions of beam spot center, unit in $\mu m$. Jitter in horizontal direction is larger than vertical since the amplitude of building vibration is much larger than beam wandering.}
\end{figure}
Although beam spot always varies, it is at order of magnitude of $100~1000 \mu m$, and the beam diameter focused by lens is about 200$\mu m$ according to Fig.\ref{profile} and Fig.\ref{trace}, therefore the whole intensity cannot always impinges on the sensitivity area of photo-diodes, which is of 1.5mm diameter. However, since the homodyne detector subtract two intensities of transmission and reflection output of PBS, the jitter of differential intensity is suppressed.
Before modulate the signal, the shot-noise and electro noise must be measured by a DAQ. The intensity of lase output from EOM is $100 \mu W$, while $65 \mu W$ input to photo-diodes. When the laser is turned off, the variance of measured data is electro noise $V_{e}N_{0}$, in the unit of $V^{2}$. Then turn on the light, when the detection is balanced, the variance is $N_{0}(1+V_{e})$. Subtracting two variance gets the shot noise $N_{0}=15.4 mV^{2}$, and thus $V_{e}=0.0219$. Then $5\times10^{5}$ Gaussian distributed (pseudo) random variable, centered at 0, of variance of $1V^{2}$, are generated by computer software, and pulsed by the output of DAQ. Modulation variance $V_{M}=165$ when $\Sigma=1 V$.
However, smaller $V_{M}$ would be comparable to the leaked light from the local oscillator since the isolation ratio of a single PBS is only about 33 dB. The total transmittance measured with power meter is 0.65, including optical components reflecting loss and channel loss. A random chosen part about $1/5$ of data is used to estimate channel parameters $T$ and $\epsilon$ based on the equations:
\begin{equation}
\begin{aligned}
\tilde{T}&=(\frac{Cov(x,y)}{V_{M}})^{2}\\
\tilde{\epsilon}&=\frac{Var(y)-V_{e}-1}{\eta\tilde{T}}-V_{M}\\
\end{aligned}
\end{equation}
where x, y are the chosen string of data of Alice and Bob's. The excess noise $\tilde{\epsilon}=0.0375$ and $\tilde{T}=0.575$, and the latter is lower than that measured by power meter since the sensitivity area diameter of power meter is much larger than photo-diodes.
Another random $1/5$ part of data is used to monitor the variance of P quadrature, controlled by EOM2. When the applied voltage is $V_{\pi}=284 V$, the EOM2 acts as a half wave plate and rotate the polarization direction by $45\deg$ to measure $\pm45\deg$ modes. The accuracy of applied voltage is 0.1 V, $3.52\times10^{-4}$ of $V_{\pi}$, precise enough to suppress the modulated signal. Therefore, $V_{P1}$ of Eq. \ref{gammaAB1} is 1.00.
With all parameters achieved above, the security key rate can be evaluated. At distance of 460m atmospheric environment, secret key rate is 0.0254 bit per pulse at a typical reconciliation efficiency of 0.95, corresponding to 0.152 kbps, while secret key rate is 0.23 bit per pulse in laboratory environment at the same modulation voltage, electro noise but lower local oscillator intensity(30$\mu W$). The expected secret key rate of unidimensional versus GG02 protocols at different total loss with experiment measured parameter $V_{M}, \epsilon, V_{e} and V_{P1}$ is shown in Fig. \ref{comparison}.
\label{comparison}
\begin{figure}
\caption{Key rate at different channel transmittance. Solid line: GG02; dashed line: unidimensional protocol. Main parameters: $V_{M}
\end{figure}
At low channel loss, unidimensional performs close to GG02 protocol, but when transmittance is less than about 0.6, it is a magnitude of order lower than GG02.
In our experiment, the main factors to restrict secret key rate are the sample rate of DAQs, and the polarization fluctuation of laser. As mentioned above, sample rate of DAQs is up to 1 MHz, and modulation frequency is even lower, far less than the response bandwidth of EOM, 100 MHz. Besides, the memory and CPU of computer at Bob's side are not able to process too many key data (over $10^{7}$), the key rate considering finite-size region is expected to be further lower than the asymptotic limit\cite{ud finite key}. At last, the fiber pig-tailed laser coupled to free-space, may cause the polarization direction changing in the fiber and be unstable, lead to polarization noise and thus intensity fluctuation of $0.1\%$. The further improvement will be made to extend the secure distance.
\section{conclusions}
To conclude, we accomplished free-space unidimensional CV QKD experiment in a real urban environment through the atmospheric channel of 460m. In such a condition, the variance of the unmodulated quadrature, $S_{2}$ barely remains unchanged. With the correlation of two quadrature unknown, the pessimistic raw key rate against collective attacks reaches 0.0254 bit per pulse, at the modulation repetition of 10 kHz.
\end{document}
|
\mathbf{e}gin{document}
\title{A stabilized finite element method for advection-diffusion
equations on surfaces ootnotemark[1]}
\mathbf{e}gin{abstract}
A recently developed Eulerian finite element method is applied to solve
advection-diffusion equations posed on hypersurfaces. When transport processes on a surface
dominate over diffusion, finite element methods tend to be unstable unless the mesh
is sufficiently fine. The paper introduces a stabilized finite element formulation based on the SUPG technique.
An error analysis of the method is given.
Results of numerical experiments are presented that illustrate the performance of the stabilized method.
\end{abstract}
\mathbf{e}gin{keywords}surface PDE, finite element method, transport equations, advection-diffusion equation, SUPG stabilization
\end{keywords}
\mathbf{e}gin{AMS}
58J32, 65N12, 65N30, 76D45, 76T99
\end{AMS}
\partialgestyle{myheadings}
\thispagestyle{plain}
\markboth{M. A. OLSHANSKII, A.~REUSKEN, AND X.~XU }{A FEM FOR ADVECTION-DIFFUSION EQUATIONS ON SURFACES}
\section{Introduction}
Mathematical models involving partial differential
equations posed on hypersurfaces occur in many applications.
Often surface equations are coupled with other equations that are
formulated in a (fixed) domain which contains the surface.
This happens, for example, in common models of
multiphase fluids dynamics if one takes so-called surface active agents
into account \cite{GrossReuskenBook}. The surface transport of such surfactants is typically driven by convection and surface diffusion and the relative strength of these two is measured by the dimensionless surface Peclet number $Pe_s=\frac{UL}{D_s}$. Here $U$ and $L$ denote typical velocity and lenght scales, respectively, and $D_s$ is the surface diffusion coefficient. Typical surfactants have surface diffusion coefficients in the range $D_s \sim 10^{-3}-10^{-5}~ cm^2/s$ \cite{Agrawal1988}, leading to (very) large surface Peclet numbers in many applications. Hence, such applications result in advection-diffusion equations on the surface
with dominating advection terms. The surface may evolve in time and
be available only implicitly (for example, as a zero level of a level set function).
It is well known that finite element discretization methods for advection-diffusion problems need
an additional stabilization mechanism, unless the mesh size is sufficiently small to resolve
boundary and internal layers in the solution of the differential equation. For the planar case, this topic has been extensively studied
in the literature and a variety of stabilization methods has been developed, see, e.g., \cite{TobiskaBook}.
We are, however, not aware of any studies of stable finite element methods for
advection-diffusion equations posed on surfaces.
In the past decade the study of numerical methods for PDEs on surfaces has been a rapidly growing research area.
The development of finite element methods
for solving elliptic equations on surfaces can be traced back to the paper \cite{Dziuk88},
which considers a piecewise polygonal surface and uses a
finite element space on a triangulation of this discrete surface. This approach has been further analyzed and extended in several directions, see, e.g.,~\cite{Dziuk011} and the references therein.
Another approach has been introduced in \cite{Deckelnick07} and builds on the ideas of \cite{Bertalmio01}. The method in that paper applies to cases in which the surface is
given implicitly by some level set function and the key idea is to solve the partial differential
equation on a narrow band around the surface. Unfitted finite element spaces on this narrow band are
used for discretization. Another surface finite element method based on an outer (bulk) mesh has been introduced in \cite{Reusken08} and further studied in \cite{OlshanskiiReusken08,DemlowOlshanskii12}.
The main idea of this method is to use finite element spaces that are induced by
triangulations of an outer domain to discretize the partial differential
equation on the surface by considering \textit{traces} of the bulk finite element space on the surface, instead of
extending the PDE off the surface, as in \cite{Bertalmio01,Deckelnick07}.
The method is particularly suitable for problems in which
the surface is given implicitly by a level set or VOF function and in which there is a
coupling with a differential equation in a fixed outer domain.
If in such problems one uses finite element techniques for the discetization of
equations in the outer domain, this setting immediately results in an easy to implement discretization method for the surface equation. The approach does not require additional surface
elements.
In this paper we reconsider the volume mesh finite element method from \cite{Reusken08} and study a new aspect, that has not been studied in the literature so far, namely the stabilization of advection-dominated problems.
We restrict ourselves to the case of a stationary surface. To stabilize the discrete problem for the case
of large mesh Peclet numbers, we introduce a surface variant of the SUPG method.
For a class of stationary advection-diffusion equation an error analysis is presented. Although the convergence of the method is studied using a SUPG norm similar to the planar case~\cite{TobiskaBook}, the analysis is not standard and contains new ingredients:
Some new approximation properties for the traces of finite elements are needed and geometric errors
require special control. The main theoretical result is given in Theorem~\ref{Th1}. It yields an error estimate
in the SUPG norm which is almost robust in the sense that the dependence on the Peclet number is mild. This dependence is due to some insufficiently controlled geometric errors, as will be explained in section~\ref{sec_disc}.
The remainder of the paper is organized as follows. In
section~\ref{S_cd}, we recall equations for transport-diffusion processes on surfaces and present
the stabilized finite element method.
Section~\ref{sectanalysis} contains the theoretical results of
the paper concerning the approximation properties of the finite element space
and discretization error bounds for the finite element method. Finally, in section~\ref{sectexperiments}
results of numerical experiments are given for both stationary and time-dependent advection-dominated surface transport-diffusion equations, which show that the stabilization performs well and that numerical results are consistent with what is expected from the SUPG method in the planar case.
\section{Advection-diffusion equations on surfaces} \label{S_cd}
Let $\Omega$ be an open domain in $\mathbb{R}^3$ and $\Gamma$ be a connected $C^2$ compact
hyper-surface contained in $\Omega$.
For a sufficiently smooth function $g:\Omega\rightarrow \mathbb{R}$ the tangential derivative
at $\Gamma$ is defined by
\mathbf{e}gin{equation}
\nabla_{\Gamma} g=\nabla g-(\nabla g\cdot \mathbf{n}_{\Gamma})\mathbf{n}_{\Gamma},\lbl{e:2.1}
\end{equation}
where $\mathbf{n}_{\Gamma}$ denotes the unit normal to $\Gamma$. Denote by $\Delta_{\Gamma}$
the { Laplace-Beltrami operator} on $\Gamma$.
Let $\mathbf{w}:\Omega\rightarrow\mathbb{R}^3$ be a given divergence-free ($\operatorname{div}\mathbf{w}=0$) velocity field in $\Omega$. If the surface
$\Gamma$ evolves with a normal velocity of $\mathbf{w}\cdot\mathbf{n}_{\Gamma}$, then the conservation of a scalar quantity
$u$ with a diffusive flux on $\Gamma(t)$ leads to the surface PDE:
\mathbf{e}gin{equation}
\dot{u} + (\operatorname{div}_\Gamma\mathbf{w})u -\varepsilon\Delta_{\Gamma} u=0\qquad\text{on}~~\Gamma(t),
\label{transport}
\end{equation}
where $\dot{u}$ denotes the advective material derivative, $\varepsilon$ is the diffusion coefficient. In \cite{Dziuk07} the problem \eqref{transport} was shown to be well-posed in a suitable weak sense.
In this paper, we study a finite element method for an advection-dominated problem on a \textit{steady} surface. Therefore, we assume $\mathbf{w}\cdot\mathbf{n}_{\Gamma}=0$, i.e. the advection velocity is everywhere tangential to the surface. This and $\operatorname{div}\mathbf{w}=0$ implies $\operatorname{div}_\Gamma\mathbf{w}=0$, and the surface advection-diffusion equation takes the form:
\mathbf{e}gin{equation} u_t+ \mathbf{w}\cdot\nabla_{\Gamma} u -\varepsilon\Delta_{\Gamma} u=0
\qquad\text{on}~~\Gamma.
\lbl{e:2.2}
\end{equation}
Although the methodology and numerical examples of the paper are applied to the equations \eqref{e:2.2},
the error analysis will be presented for the stationary problem
\mathbf{e}gin{equation}
-\varepsilon\Delta_{\Gamma} u+\mathbf{w}\cdot\nabla_{\Gamma} u + c(\mathbf{x})u =f\qquad\text{on}~~\Gamma, \lbl{e:2.3}
\end{equation}
with $f\in L^2(\Gamma)$ and $c(\mathbf{x})\ge0$. To simplify the presentation we assume $c(\cdot)$ to be constant, i.e. $c(x)=c \geq 0$.
The analysis, however, also applies to non-constant $c$, cf. section~\ref{sec_disc}.
Note that \eqref{e:2.2} and \eqref{e:2.3} can be written in intrinsic surface quantities, since
$\mathbf{w}\cdot\nabla_{\Gamma} u = \mathbf{w}_{\Gamma}\cdot\nabla_{\Gamma} u$, with the tangential velocity $\mathbf{w}_{\Gamma}=\mathbf{w}-(\mathbf{w}\cdot\mathbf{n}_{\Gamma})\mathbf{n}_{\Gamma}$.
We assume $\mathbf{w}_{\Gamma}\in H^{1,\infty}(\Gamma)\cap L^{\infty}(\Gamma)$ and scale the equation so that
$\|\mathbf{w}_{\Gamma}\|_{L^\infty(\Gamma)}=1$ holds. Furthermore, since we are interested in the advection-dominated case we take $\varepsilon\in(0,1]$. Introduce the
bilinear form and the functional:
\mathbf{e}gin{equation*}
\mathbf{e}gin{aligned}
a(u,v)&:=\varepsilon\int_{\Gamma}\nabla_{\Gamma} u\cdot\nabla_{\Gamma} v \, \mathrm{d}\mathbf{s}+\int_{\Gamma}(\mathbf{w}\cdot\nabla_{\Gamma} u) v \, \mathrm{d}\mathbf{s}
+\int_{\Gamma}c \, uv\, \mathrm{d}\mathbf{s}, \\
f(v)&:=\int_{\Gamma}fv\, \mathrm{d}\mathbf{s}.
\end{aligned}
\end{equation*}
The weak formulation of \eqref{e:2.3} is as follows: Find $u\in V$ such that
\mathbf{e}gin{equation}
a(u,v)=f(v) \qquad \forall v\in V,\lbl{e:2.5}
\end{equation}
with
$$V= \left\{
\mathbf{e}gin{array}{ll}
\{v\in H^1(\Gamma)\ |\ \int_{\Gamma} v\, \mathrm{d}\mathbf{s}=0\} & \hbox{if } c= 0,\\
H^1(\Gamma) & \hbox{if } c>0.
\end{array}
\right.
$$
Due to the Lax-Milgram lemma, there exists a unique solution of \eqref{e:2.5}.
For the case $c=0$ the following Friedrich's inequality~\cite{Sobolev} holds:
\mathbf{e}gin{equation}
\|v\|_{L^2(\Gamma)}^2 \le C_F \|\nabla_\Gamma v\|_{L^2(\Gamma)}^2\quad
\forall~v\in V.
\lbl{FdrA}
\end{equation}
\subsection{The stabilized volume mesh FEM} \label{sectSUPG}
In this section, we recall the volume mesh FEM introduced in \cite{Reusken08} and describe its SUPG type stabilization.
Let $\{\mathcal{T}_h\}_{h>0}$ be a family of tetrahedral triangulations of
the domain $\Omega$. These triangulations
are assumed to be regular, consistent and stable. To simplify the presentation we assume that this family of triangulations is {quasi-uniform}. The latter assumption, however, is not essential for our analysis.
We assume that for each $\mathcal{T}_h$ a polygonal approximation of
$\Gamma$, denoted by $\Gamma_h$, is given:
$\Gamma_h$ is a $C^{0,1}$ surface without boundary and $\Gamma_h$ can
be partitioned in planar triangular segments.
It is important to note that $\Gamma_h$ is not a ``triangulation of $\Gamma$'' in the usual sense
(an $O(h^2)$ approximation of $\Gamma$, consisting of regular triangles). Instead,
we (only) assume that $\Gamma_h$ is \emph{consistent with the outer triangulation} $\mathcal{T}_h$
in the following sense. For any
tetrahedron $S_T\in\mathcal{T}_h$ such that $\mathrm{meas}_2(S_T\cap\Gamma_h)>0$, define
$T=S_T\cap\Gamma_h$. We assume that every $T\in\Gamma_h$ is a \textit{planar} segment
and thus it is either a triangle or a quadrilateral. Each quadrilateral segment can be divided into two triangles, so we may assume that every $T$ is a triangle. An illustration of such a triangulation is given in Figure~\ref{fig:tri}. The results shown in this figure are obtained by representing a sphere $\Gamma$ implicitly by its signed distance function, constructing the piecewise linear nodal interpolation of this distance function on a uniform tetrahedral triangulation $\mathcal{T}_h$ of $\Omega$ and then considering the zero level of this interpolant.
\mathbf{e}gin{figure}[ht!]
\mathbf{e}gin{center}
\centering
\includegraphics[width=0.45\textwidth]{Gamma_h0.eps}
\includegraphics[width=0.45\textwidth]{Gamma_h1.eps}
\caption{Approximate interface $\Gamma_h$ for a sphere, resulting from a coarse tetrahedral triangulation (left) and after one refinement (right).}
\label{fig:tri}
\end{center}
\end{figure}
Let $\mathcal{F}_h$ be the set of all triangular segments $T$, then $\Gamma_h$ can
be decomposed as
\mathbf{e}gin{equation} \label{defgammah}
\Gamma_h=\bigcup\limits_{T\in\mathcal{F}_h} T.
\end{equation}
Note that the triangulation $\mathcal{F}_h$ is {not} necessarily regular, i.e. elements from $T$ may have
very small internal angles and the size of neighboring triangles can vary strongly, cf.~Figure~\ref{fig:tri}.
In applications with level set functions (that represent $\Gamma$ implicitly), the approximation $\Gamma_h$ can be obtained as the zero level of a piecewise linear finite element approximation of the level set function on the tetrahedral triangulation $\mathcal{T}_h$.
The surface finite element space is \textit{the space of traces on $\Gamma_h$ of all piecewise linear continuous functions with respect to the outer triangulation $\mathcal{T}_h$}.
This can be formally defined as follows.
We define a subdomain that contains $\Gamma_h$:
\mathbf{e}gin{equation} \label{defomeg}
\omega_h= \bigcup_{T \in \mathcal{F}_h} S_T,
\end{equation}
an a corresponding volume mesh finite element space
\mathbf{e}gin{equation}
V_h:=\{v_h\in C(\omega_h)\ |\ v_h|_{S_T}\in P_1~~ \forall\ T \in\mathcal{F}_h\},\lbl{e:2.6}
\end{equation}
where $P_1$ is the space of polynomials of degree one. $V_h$ induces the following
space on $\Gamma_h$:
\mathbf{e}gin{equation}
V_h^{\Gamma}:=\{\psi_h\in H^1(\Gamma_h)\ |\ \exists ~ v_h\in V_h\ \text{such that }\ \psi_h=v_h|_{\Gamma_h}\}.
\lbl{e:fem-space}
\end{equation}
When $c=0$, we require that any function $v_h$ from $V_h^{\Gamma}$ satisfies $\int_{\Gamma_h}v_h \, \mathrm{d}\mathbf{s}=0$.
Given the surface finite element space $V_h^{\Gamma}$, the finite element
discretization of \eqref{e:2.5} is as follows: Find $u_h\in V_h^{\Gamma}$ such that
\mathbf{e}gin{equation}
\varepsilon\int_{\Gamma_h}\nabla_{\Gamma}h u\cdot\nabla_{\Gamma}h v\, \mathrm{d}\mathbf{s} + \int_{\Gamma_h}(\mathbf{w}^e\cdot\nabla_{\Gamma}h u) v\, \mathrm{d}\mathbf{s}
+ \int_{\Gamma_h}c \,uv\,\mathrm{d}\mathbf{s}=\int_{\Gamma_h}f^e v\, \mathrm{d}\mathbf{s} \lbl{plainFEM}
\end{equation}
for all $v_h\in V_h^{\Gamma}$. Here $\mathbf{w}^e$ and $f^e$ are
the extensions of $\mathbf{w}_{\Gamma}$ and $f$, respectively, along normals to $\Gamma$ (the precise definition is given in the next section). Similar to the plain Galerkin finite element for advection-diffusion equations, the method \eqref{plainFEM} is unstable unless the mesh is sufficiently fine such that the mesh Peclet number is less than one.
We introduce the following stabilized finite element method based on the standard SUPG approach, cf. \cite{TobiskaBook}:
Find $u_h\in V_h^{\Gamma}$ such that
\mathbf{e}gin{equation}
a_h(u_h,v_h)=f_h(v_h)\quad \forall~ v_h\in V_h^{\Gamma}, \lbl{e:2.8}
\end{equation}
with
\mathbf{e}gin{align}
a_h(u,v):=& \varepsilon\int_{\Gamma_h}\nabla_{\Gamma}h u\cdot\nabla_{\Gamma}h v \, \mathrm{d}\mathbf{s}
+ \int_{\Gamma_h}c \, uv\mathrm{d}\mathbf{s} \nonumber \\
&+ \frac12\left[\int_{\Gamma_h}(\mathbf{w}^e\cdot\nabla_{\Gamma}h u) v \, \mathrm{d}\mathbf{s} -\int_{\Gamma_h}(\mathbf{w}^e\cdot\nabla_{\Gamma}h v) u\, \mathrm{d}\mathbf{s} \right] \label{eqah}\\
&+\sum_{T\in\mathcal{F}_h}\delta_T\int_{T}(-\varepsilon\Delta_{\Gamma_h}u + \mathbf{w}^e\cdot\nabla_{\Gamma}h u + c\, u)\mathbf{w}^e\cdot\nabla_{\Gamma}h v\, \mathrm{d}\mathbf{s}, \nonumber \\
f_h(v):=&\int_{\Gamma_h}f^e v\mathrm{d}\mathbf{s} + \sum_{T\in\mathcal{F}_h}\delta_T\int_{T}f^e(\mathbf{w}^e\cdot\nabla_{\Gamma}h v)\, \mathrm{d}\mathbf{s}. \label{eqfh}
\end{align}
The stabilization parameter $\delta_T$ depends on $T \subset S_T$. The diameter of the tetrahedron $S_T$ is denoted by $h_{S_T}$. Let $\displaystyle \mathsf{Pe}_T:=\frac{h_{S_T} \|\mathbf{w}^e\|_{L^\infty(T)}}{2\varepsilon}$
be the cell Peclet number.
We take
\mathbf{e}gin{equation}
\widetilde{\delta_T}=
\left\{
\mathbf{e}gin{aligned}
&\frac{\delta_0 h_{S_T}}{\|\mathbf{w}^e\|_{L^\infty(T)}} &&\quad \hbox{ if } \mathsf{Pe}_T> 1,\\
&\frac{\delta_1 h^2_{S_T}}{\varepsilon} &&\quad \hbox{ if } \mathsf{Pe}_T\leq 1,
\end{aligned}
\right.\quad\text{and} \quad\delta_T=\min\{\widetilde{\delta_T},c^{-1}\}, \lbl{e:2.10}
\end{equation}
with some given positive constants $\delta_0,\delta_1\geq 0$.
Since $u_h \in V_h^{\Gamma}$ is linear on every $T$ we have
$\Delta_{\Gamma_h} u_h =0$ on $T$, and thus
$a_h(u_h,v_h)$ simplifies to
\mathbf{e}gin{multline} \label{eqqr}
a_h(u_h,v_h)=\varepsilon\int_{\Gamma_h}\nabla_{\Gamma}h u_h\cdot\nabla_{\Gamma}h v_h \, \mathrm{d}\mathbf{s}
+ \frac12\left[\int_{\Gamma_h}(\mathbf{w}^e\cdot\nabla_{\Gamma}h u_h) v_h -(\mathbf{w}^e\cdot\nabla_{\Gamma}h v_h) u_h \, \mathrm{d}\mathbf{s} \right] \\
+\int_{\Gamma_h}c \, u_h (v_h+\delta(\mathbf{x}) \mathbf{w}^e\cdot\nabla_{\Gamma}h v_h)\, \mathrm{d}\mathbf{s}+ \int_{\Gamma_h}\delta(\mathbf{x})(\mathbf{w}^e\cdot\nabla_{\Gamma}h u_h) (\mathbf{w}^e\cdot\nabla_{\Gamma}h v_h) \, \mathrm{d}\mathbf{s},
\end{multline}
where $\delta(\mathbf{x})=\delta_T$ for $\mathbf{x}\in T$.
\section{Error analysis} \label{sectanalysis}
The analysis in this section is organized as follows. First we collect some definitions and useful results in section~\ref{sectprelim}.
In section~\ref{sectcoercive}, we derive a coercivity result. In section~\ref{sectinter}, we present interpolation error bounds. In sections~\ref{sectcont} and \ref{sectconsistency}, continuity and consistency results are derived.
Combining these analysis we obtain the finite element error bound given in section~\ref{sectmain}.
In the error analysis we use the following mesh-dependent norm:
\mathbf{e}gin{equation} \label{defn}
\| u \|_{\ast}:=\left(\varepsilon\int_{\Gamma_h}|\nabla_{\Gamma}h u|^2\, \mathrm{d}\mathbf{s} + \int_{\Gamma_h}\delta(\mathbf{x})|\mathbf{w}^e\cdot\nabla_{\Gamma}h u|^2\, \mathrm{d}\mathbf{s} + \int_{\Gamma_h}c \,|u|^2\, \mathrm{d}\mathbf{s} \right)^{\frac{1}{2}}.
\end{equation}
Here and in the remainder $|\cdot |$ denotes the Euclidean norm for vectors and the corresponding spectral norm
for matrices.
\subsection{Preliminaries} \label{sectprelim}
For the hypersurface $\Gamma$, we define its $h$-neighborhood:
\mathbf{e}gin{equation}
U_h:=\{\mathbf{x}\in \mathbb{R}^3\ |\ \mathrm{dist}(\mathbf{x},\Gamma)<c_0 h\},\lbl{e:3.1}
\end{equation}
and assume that $c_0$ is sufficiently large such that $\omega_h\subset U_h$ and $h$ sufficiently small such that
\mathbf{e}gin{equation}
5c_0 h<\left(\mathrm{max}_{i=1,2} \|\kappa_i\|_{L^{\infty}(\Gamma)}\right)^{-1}\lbl{e:3.2}
\end{equation}
holds, with $\kappa_i$ being the principal curvatures of $\Gamma$. Here and in what follows $h$ denotes
the maximum diameter for tetrahedra of outer triangulation: $h=\max\limits_{S\in \omega_h} \text{diam}(S)$.
Let $d : U_h\rightarrow \mathbb{R}$ be the signed
distance function, $|d(\mathbf{x})|=\mathrm{dist}(\mathbf{x},\Gamma)$ for all $\mathbf{x}\in U_h$. Thus
$\Gamma$ is the zero level set of $d$. We assume $d<0$ in the interior of $\Gamma$ and $d>0$ in the exterior and define $\mathbf{n}(\mathbf{x}):=\nabla d(\mathbf{x})$ for all $\mathbf{x}\in U_h$. Hence, $\mathbf{n}=\mathbf{n}_{\Gamma}$ on $\Gamma$ and $|\mathbf{n}(\mathbf{x})|=1$ for all $\mathbf{x}\in U_h$. The Hessian of $d$ is denoted by
\mathbf{e}gin{equation}
\mathbf{H}(\mathbf{x}):= \nabla^2 d(\mathbf{x})\in \mathbb{R}^{3\times 3},\quad \mathbf{x}\in U_h.
\end{equation}
The eigenvalues of $\mathbf{H}(\mathbf{x})$ are denoted by $\kappa_1(\mathbf{x})$, $\kappa_2(\mathbf{x})$, and~$0$.
For $\mathbf{x}\in\Gamma$ the eigenvalues $\kappa_i, i=1,2,$ are the principal curvatures.
For each $\mathbf{x}\in U_h$, define the projection $\mathbf{p}: U_h\rightarrow\Gamma$ by
\mathbf{e}gin{equation}
\mathbf{p}(\mathbf{x})=\mathbf{x}-d(\mathbf{x})\mathbf{n}(\mathbf{x}).
\end{equation}
Due to the assumption \eqref{e:3.2}, the decomposition $\mathbf{x}=\mathbf{p}(\mathbf{x})+d(\mathbf{x})\mathbf{n}(\mathbf{x})$ is unique.
We will need the orthogonal projector
\mathbf{e}gin{equation*}
\mathbf{P}(\mathbf{x}):=\mathbf{I}-\mathbf{n}(\mathbf{x})\mathbf{n}(\mathbf{x})^T, \quad \hbox{for }\mathbf{x}\in U_h.
\end{equation*}
Note that $\mathbf{n}(\mathbf{x})= \mathbf{n}(\mathbf{p}(\mathbf{x}))$ and $\mathbf{P}(\mathbf{x})=\mathbf{P}(\mathbf{p}(\mathbf{x}))$ for $\mathbf{x} \in U_h$ holds.
The tangential derivative can be written as $\nabla_{\Gamma} g(\mathbf{x})=\mathbf{P}\nabla g(\mathbf{x})$ for $\mathbf{x}\in \Gamma$.
One can verify that for this projection and for the Hessian $\mathbf H$ the relation
$\mathbf H\mathbf{P}=\mathbf{P}\mathbf{H}=\mathbf{H}$ holds.
Similarly, define
\mathbf{e}gin{equation}
\mathbf{P}_h(\mathbf{x}):=\mathbf{I}-\mathbf{n}_{\Gamma_h}(\mathbf{x})\mathbf{n}_{\Gamma_h}(\mathbf{x})^T, \quad \hbox{for }\mathbf{x}\in \Gamma_h,~\mathbf{x} \hbox{ is not on an edge},
\end{equation}
where $\mathbf{n}_{\Gamma_h}$ is the unit (outward pointing) normal at $\mathbf{x}\in \Gamma_h$ (not on an edge). The tangential
derivative along $\Gamma_h$ is given by $\nabla_{\Gamma_h} g(\mathbf{x})=\mathbf{P}_h(\mathbf{x})\nabla g(\mathbf{x})$ (not on an edge).
\\
\mathbf{e}gin{Assumption} \label{ass1} \rm In this paper, we assume that for all $T\in \mathcal{F}_h$:
\mathbf{e}gin{align}
&\mathrm{ess\ sup}_{\mathbf{x}\in T}|d(\mathbf{x})| \le c_1 h_{S_T}^2, \lbl{e:3.9}\\
&\mathrm{ess\ sup}_{\mathbf{x}\in T}|\mathbf{n}(\mathbf{x})-\mathbf{n}_{\Gamma_h}(\mathbf{x})| \le c_2 h_{S_T},\lbl{e:3.10}
\end{align}
where $h_{S_T}$ denotes the diameter of the tetrahedron $S_T$ that contains $T$, i.e., $T=S_T \cap \Gamma_h$ and constants $c_1$, $c_2$ are independent of $h$, $T$.
\end{Assumption}
\ \\[1ex]
The assumptions \eqref{e:3.9} and \eqref{e:3.10} describe how accurate the piecewise planar approximation $\Gamma_h$ of $\Gamma$ is. If $\Gamma_h$ is constructed as the zero level of a piecewise linear interpolation of a level set function that characterizes $\Gamma$ (as in Fig.~\ref{fig:tri}) then these assumptions are fulfilled, cf. Sect. 7.3 in \cite{GrossReuskenBook}.
In the remainder, $A\lesssim B $ means $A\leq \tilde c B $ for some positive constant $\tilde c$ independent of $h$ and of the problem parameters $\varepsilon$ and $c$.
$A\simeq B$ means that both $A\lesssim B $ and $B\lesssim A $.
For $\mathbf{x}\in\Gamma_h$, define
\mathbf{e}gin{equation*}
\mu_h(\mathbf{x}) = (1-d(\mathbf{x})\kappa_1(\mathbf{x}))(1-d(\mathbf{x})\kappa_2(\mathbf{x}))\mathbf{n}^T(\mathbf{x})\mathbf{n}_h(\mathbf{x}).
\end{equation*}
The surface measures $\mathrm{d}\mathbf{s}$ and $\mathrm{d}\mathbf{s}_{h}$ on $\Gamma$ and $\Gamma_h$, respectively, are related by
\mathbf{e}gin{equation}
\mu_h(\mathbf{x})\mathrm{d}\mathbf{s}_h(\mathbf{x})=\mathrm{d}\mathbf{s}(\mathbf{p}(\mathbf{x})),\quad \mathbf{x}\in\Gamma_h. \lbl{e:3.16}
\end{equation}
The assumptions \eqref{e:3.9} and \eqref{e:3.10} imply that
\mathbf{e}gin{equation}
\mathrm{ess\ sup}_{\mathbf{x}\in\Gamma_h}(1-\mu_h)\lesssim h^2,\lbl{e:3.17}
\end{equation}
cf. (3.37) in \cite{Reusken08}.
The solution of \eqref{e:2.3} is defined on $\Gamma$,
while its finite element approximation $u_h \in V_h^\Gamma$ is defined on $\Gamma_h$.
We need a suitable extension of a function from $\Gamma$ to its neighborhood. For a function $v$ on $\Gamma$ we define
\mathbf{e}gin{equation}
v^e(\mathbf{x}):= v(\mathbf{p}(\mathbf{x})) \quad \hbox{for all } \mathbf{x}\in U_h.
\end{equation}
The following formula for this lifting function are known (cf. section 2.3 in \cite{Demlow06}):
\mathbf{e}gin{align}
\nabla u^e(\mathbf{x}) &= (\mathbf{I}-d(\mathbf{x})\mathbf{H})\nabla_{\Gamma} u(\mathbf{p}(\mathbf{x})) \quad \hbox{ a.e. on } U_h,\label{grad1}\\
\nabla_{\Gamma_h} u^e(\mathbf{x}) &= \mathbf{P}_h(\mathbf{x})(\mathbf{I}-d(\mathbf{x})\mathbf{H})\nabla_{\Gamma} u(\mathbf{p}(\mathbf{x})) \quad \hbox{ a.e. on } \Gamma_h,
\end{align}
with $\mathbf{H}=\mathbf{H}(\mathbf{x})$.
By direct computation one derives the relation
\mathbf{e}gin{multline}
\nabla^2 u^e(\mathbf{x})=(\mathbf{P} -d(\mathbf{x}) \mathbf{H})\nabla^2_{\Gamma} u(\mathbf{p}(\mathbf{x}))(\mathbf{P} -d(\mathbf{x}) \mathbf{H})-(n^T\nabla_{\Gamma} u(\mathbf{p}(\mathbf{x}))\mathbf{H}\\
-(\mathbf H\nabla_{\Gamma} u(\mathbf{p}(\mathbf{x}))) \mathbf{n}^T-\mathbf{n} (\mathbf H\nabla_{\Gamma} u(\mathbf{p}(\mathbf{x})))^T -d\nabla_{\Gamma} \mathbf H:\nabla_{\Gamma} u(\mathbf{p}(\mathbf{x})).
\lbl{e:3.14n}
\end{multline}
For sufficiently smooth $u$ and $|\mu| \leq 2$, using this relation one obtains the estimate
\mathbf{e}gin{equation}
|D^{\mu} u^e(\mathbf{x})|\lesssim \left(\sum_{|\mu|=2}|D_{\Gamma}^{\mu} u(\mathbf{p}(\mathbf{x}))| + |\nabla_{\Gamma} u(\mathbf{p}(\mathbf{x}))|\right)\quad \hbox{a.e. on } U_h,
\lbl{e:3.13}
\end{equation}
(cf. Lemma 3 in \cite{Dziuk88}). This further leads to (cf. Lemma 3.2 in \cite{Reusken08}):
\mathbf{e}gin{equation}
\|D^{\mu} u^e\|_{L^2(U_h)}\lesssim\sqrt{h}\|u\|_{H^2(\Gamma)}, \quad |\mu|\leq 2. \lbl{e:3.14}
\end{equation}
The next lemma is needed for the analysis in the following section.
\mathbf{e}gin{lemma}\label{lem_div} The following holds:
\[
\|\mathrm{div}_{\Gamma}h\mathbf{w}^e\|_{L^\infty(\Gamma_h)}\lesssim h \|\nabla_{\Gamma}\mathbf{w}\|_{L^\infty(\Gamma)}.
\]
\end{lemma}
\mathbf{e}gin{proof} We use the following representation for the tangential divergence:
\mathbf{e}gin{equation}\label{e:Marz}
\operatorname{div}_\Gamma\mathbf{w}(\mathbf{x})=\operatorname{tr}(\nabla_\Gamma\mathbf{w}(\mathbf{x}))= \operatorname{tr}(\mathbf{P}\nabla\mathbf{w}(\mathbf{x})).
\end{equation}
Take $\mathbf{x}\in\Gamma_h$, not lying on an edge. Using \eqref{grad1} we obtain
\mathbf{e}gin{align*}
& \mathrm{div}_{\Gamma}h\mathbf{w}^e(\mathbf{x}) \\ & =\operatorname{tr}(\mathbf{P}_h\nabla\mathbf{w}^e(\mathbf{x}))=
\operatorname{tr}\left(\mathbf{P}_h(\mathbf{I}-d(\mathbf{x})\mathbf{H})\nabla_{\Gamma} \mathbf{w}(\mathbf{p}(\mathbf{x}))\right)\\&=
\operatorname{tr}\left(\mathbf{P}\nabla_{\Gamma} \mathbf{w}(\mathbf{p}(\mathbf{x}))\right)
+\operatorname{tr}\left((\mathbf{P}_h-\mathbf{P})\nabla_{\Gamma} \mathbf{w}(\mathbf{p}(\mathbf{x}))\right)-
d(\mathbf{x})\operatorname{tr}\left(\mathbf{P}_h\mathbf{H}\nabla_{\Gamma} \mathbf{w}(\mathbf{p}(\mathbf{x}))\right).
\end{align*}
The first term vanishes due to
$
\operatorname{tr}\left(\mathbf{P}\nabla_{\Gamma} \mathbf{w}(\mathbf{p}(\mathbf{x}))\right)=
\operatorname{div}_\Gamma\mathbf{w}(\mathbf{p}(\mathbf{x}))=0$.
The second and the third term can be bounded using \eqref{e:3.9}, \eqref{e:3.10}:
\[
|\mathbf{P}_h-\mathbf{P}|\lesssim h,\quad |d(\mathbf{x})\mathbf{P}_h\mathbf{H}|\lesssim h^2.
\]
This proves the lemma.
\end{proof}
\subsection{Coercivity analysis} \label{sectcoercive}
In the next lemma we present a coercivity result. We use the norm introduced in \eqref{defn}.
\mathbf{e}gin{lemma} \label{thm0}
The following holds:
\mathbf{e}gin{equation}\lbl{coercivity}
a_h(v_h,v_h)\geq \frac{1}{2}\|v_h\|_{\ast}^2 \quad \text{for all}~~v_h \in V_h^\Gamma.
\end{equation}
\end{lemma}
\mathbf{e}gin{proof}
For any $v_h\in V_h^{\Gamma}$, we have
\mathbf{e}gin{equation}\label{aux2}
a_h(v_h,v_h)= \|v_h\|_{\ast}^2
+\int_{\Gamma_h}c\, \delta(\mathbf{x})v_h(\mathbf{w}^e\cdot\nabla_{\Gamma}h v_h)\, \mathrm{d}\mathbf{s}.
\end{equation}
The choice of $\delta_T$, cf. \eqref{e:2.10}, implies $c \, \delta(\mathbf{x}) \le1$. Hence the last term in \eqref{aux2} can be estimated as follows:
\mathbf{e}gin{align*}
& |\int_{\Gamma_h}c \, \delta(\mathbf{x})v_h(\mathbf{w}^e\cdot\nabla_{\Gamma}h v_h)\, \mathrm{d}\mathbf{s}|\\ & \le \frac12\int_{\Gamma_h} c \, v_h^2\, \mathrm{d}\mathbf{s} + \frac12\int_{\Gamma_h}c \, \delta(\mathbf{x})^2(\mathbf{w}^e\cdot\nabla_{\Gamma}h v_h)^2\, \mathrm{d}\mathbf{s}\le \frac12 \|v_h\|_{\ast}^2.
\end{align*}
This yields \eqref{coercivity}.
\end{proof}
\ \\
As a consequence of this result, we obtain the well-posedness of the discrete problem \eqref{e:2.8}.
\subsection{Interpolation error bounds} \label{sectinter}
Let $I_h:C(\bar{\omega}_h)\rightarrow V_h$ be the nodal interpolation operator.
For any $u\in H^2(\Gamma)$ the surface finite element function
$(I_h u^e)|_{\Gamma_h}\in V_h^{\Gamma}$ is an interpolant
of $u^e$ in $V_h^{\Gamma}$.
For any $u\in H^2(\Gamma)$ the following estimates hold~\cite{Reusken08}:
\mathbf{e}gin{align}
\|u^e-(I_h u^e)|_{\Gamma_h}\|_{L^2(\Gamma_h)}&\lesssim h^2\|u\|_{H^2(\Gamma)}, \lbl{e:3.18}\\
\|\nabla_{\Gamma}h u^e-\nabla_{\Gamma}h(I_h u^e)|_{\Gamma_h}\|_{L^2(\Gamma_h)}&\lesssim h\|u\|_{H^2(\Gamma)}.\lbl{e:3.19}
\end{align}
Using these results we easily obtain an interpolation error estimate in the $\|\cdot\|_{\ast}$- norm:
\mathbf{e}gin{lemma}
For any $u\in H^2(\Gamma)$ the following holds:
\mathbf{e}gin{equation} \lbl{e:3.20}
\| u^e-(I_h u^e)|_{\Gamma_h} \|_{\ast} \lesssim h(\varepsilon^{1/2}+h^{1/2}+c^{\frac12}h) \|u\|_{H^2(\Gamma)}.
\end{equation}
\end{lemma}
\mathbf{e}gin{proof} Define $\varphi:=u^e-(I_h u^e)|_{\Gamma_h} \in H^1(\Gamma_h)$.
Using the definition \eqref{e:2.10} of $\delta(\mathbf{x})$, we get
\mathbf{e}gin{equation}\label{aux3} \mathbf{e}gin{split}
\int_{\Gamma_h}\delta(\mathbf{x})|\mathbf{w}^e\cdot\nabla_{\Gamma}h\varphi|^2\mathrm{d}\mathbf{s} & =
\sum_{T\in\mathcal{F}_h}\int_{T}\delta_T|\mathbf{w}^e\cdot\nabla_{\Gamma}h\varphi|^2\, \mathrm{d}\mathbf{s} \\
& \lesssim h\|\nabla_{\Gamma}h\varphi\|_{L^2(\Gamma_h)}^2\lesssim h^3 \|u\|_{H^2(\Gamma)}^2 .
\end{split} \end{equation}
The remaining two terms in $\|u^e-(I_h u^e)\|_{\ast}$ are estimated in a straightforward way using \eqref{e:3.18} and \eqref{e:3.19}. This and \eqref{aux3} imply the inequality \eqref{e:3.20}.
\end{proof}
The next lemma estimates the interpolation error on the edges of the surface triangulation. In the remainder, $\mathcal{E}_h$ denotes the set of all edges in the interface triangulation $\mathcal{F}_h$.
\mathbf{e}gin{lemma}\lbl{prop:3.3}
For all $u\in H^2(\Gamma)$ the following holds:
\mathbf{e}gin{equation}
\left(\sum_{E\in\mathcal{E}_h}\int_{ E}(u^e-I_h u^e)|_{\Gamma_h}^2\, \mathrm{d}\mathbf{s}\right)^{1/2}\lesssim h^{3/2} \|u\|_{H^2(\Gamma)}.\lbl{e:3.24}
\end{equation}
\end{lemma}
\mathbf{e}gin{proof}
Define $\phi:= u^e-I_h u^e \in H^1(\omega_h)$.
Take $E\in\mathcal{E}_h$ and let $T \in \mathcal{F}_h$ be a corresponding planar segment of which $E$ is an edge. Let $W$ be a side of the tetrahedron $S_T$ such that $E \subset W$.
From Lemma 3 in \cite{Hansbo02} we have
\[
\|\phi\|_{L^2(E)}^2 \lesssim h^{-1} \|\phi\|_{L^2(W)}^2 + h \|\phi\|_{H^1(W)}^2.
\]
From the standard trace inequality
\[
\|w\|_{L^2(\partialrtial S_T)}^2 \lesssim h^{-1} \|w\|_{L^2(S_T)}^2 + h \|w\|_{H^1(S_T)}^2 \quad \text{for all}~~ w \in H^1(S_T),
\]
applied to $\phi$ and $\partialrtial_{x_i} \phi$, $i=1,2,3$, we obtain
\mathbf{e}gin{align*}
h^{-1} \|\phi\|_{L^2(W)}^2 & \lesssim h^{-2} \|\phi\|_{L^2(S_T)}^2 + \|\phi\|_{H^1(S_T)}^2, \\
h \|\phi\|_{H^1(W)}^2& \lesssim \|\phi\|_{H^1(S_T)}^2 + h^2 \|u^e\|_{H^2(S_T)}^2 .
\end{align*}
From standard error bounds for the nodal interpolation operator $I_h$ we get
\[
\|\phi\|_{L^2(E)}^2\lesssim h^{-2} \|\phi\|_{L^2(S_T)}^2+ \|\phi\|_{H^1(S_T)}^2 + h^2 \|u^e\|_{H^2(S_T)}^2 \lesssim h^2 \|u^e\|_{H^2(S_T)}^2.
\]
Summing over $E\in\mathcal{E}_h$ and using $\|u^e\|_{H^2(\omega_h)} \lesssim h^\frac12 \|u\|_{H^2(\Gamma)}$, cf. \eqref{e:3.14}, results in
\[
\sum_{E\in\mathcal{E}_h} \|\phi\|_{L^2(E)}^2 \lesssim h^2 \|u^e\|_{H^2(\omega_h)}^2 \lesssim h^3 \|u\|_{H^2(\Gamma)}^2,
\]
which completes the proof. \end{proof}
\subsection{Continuity estimates} \label{sectcont}
In this section we derive a continuity estimate for
the bilinear form $a_h(\cdot,\cdot)$. If one applies partial integration to the integrals that occur in $a_h(\cdot,\cdot)$ then \emph{jumps across the edges} $E \in \mathcal{E}_h$
occur. We start with a lemma that yields bounds for such jump terms.
Related to these jump terms we introduce the following notation.
For each $T\in\mathcal{F}_h$, denote by ${\mathbf{m}_h}|_{E}$ the outer normal to an edge $E$ in the plane which contains element $T$. Let $[\mathbf{m}_h]|_{E}=\mathbf{m}_h^+ +\mathbf{m}_h^-$ be the jump of the outer normals to the edge in two neighboring elements, c.f. Figure \ref{fig:3.1}.
\mathbf{e}gin{figure}[ht!]
\vspace*{-2mm}
\centering
\resizebox{!}{5.5cm}
{\includegraphics[ height=60mm]{normal.eps}}
\vspace*{-10mm}
\caption{ }
\lbl{fig:3.1}
\end{figure}
\mathbf{e}gin{lemma}\lbl{lem:jumpvel}
The following holds:
\mathbf{e}gin{equation} \label{jump_est}
|\mathbf{P}(\mathbf{x})[\mathbf{m}_h](\mathbf{x})|\lesssim h^2
\qquad a.e.\ \mathbf{x}\in E.
\end{equation}
\end{lemma}
\mathbf{e}gin{proof}
Let $E$ be the common side of two elements $T_1$ and $T_2$ in $\mathcal{F}_h$, and
$\mathbf{n}_h^+$, $\mathbf{n}_h^-$,
$\mathbf m _h^+$ and $\mathbf m_h^-$ are the unit normals as illustrated in Figure~\ref{fig:3.1}.
Denote by $\mathbf s_h$ the unit (constant) vector along the common side $E$, which can be represented as $\mathbf s_h=\mathbf{n}_h^{+}\times \mathbf m_h^{+}= \mathbf m_h^{-}\times\mathbf{n}_h^{-}$. The jump across $E$ is given by
\[
[\mathbf{m}_h]=\mathbf s_h\times(\mathbf{n}_h^{+}-\mathbf{n}_h^{-}).
\]
For each $\mathbf{x}\in E$ and $\mathbf{p}(\mathbf{x})\in\Gamma$, let $\mathbf{n}=\mathbf{n}(\mathbf{p}(\mathbf{x}))$ be the unit normal to $\Gamma$ at $\mathbf{p}(\mathbf{x})$ and $\mathbf{P}= \mathbf{P}(\mathbf{x})= \mathbf{I} - \mathbf{n} \mathbf{n}^T$ the corresponding orthogonal projection.
Using \eqref{e:3.10}, we get
\mathbf{e}gin{equation*}
|\mathbf{n}_h^--\mathbf{n}_h^+ |\le |\mathbf{n}_h^+-\mathbf{n}|+ |\mathbf{n}_h^--\mathbf{n}| \lesssim h_{S_{T_1}}+h_{S_{T_2}} \lesssim h.
\end{equation*}
Since $|\mathbf{n}_h^-|=|\mathbf{n}_h^+|=|\mathbf{n}|=1$, the above estimate implies
\[
\mathbf{n}_h^+-\mathbf{n}_h^- =c h^2 \mathbf{n} + \mathbf{e}_1,\quad \mathbf{e}_1\perp \mathbf{n},\quad |\mathbf{e}_1|\lesssim h.
\]
We also have
\[
\mathbf s_h=\mathbf{n}_h^{+}\times \mathbf m_h^{+}=(\mathbf{n}+(\mathbf{n}_h^{+}-\mathbf{n}))\times \mathbf m_h^{+}=
\mathbf{n}\times \mathbf m_h^{+}+\mathbf{e}_2,\quad |\mathbf{e}_2|\lesssim h.
\]
We use the decomposition
\[
\mathbf{P} [\mathbf{m}_h]= \mathbf{P} \left[(\mathbf{n}\times \mathbf m_h^{+}+\mathbf{e}_2) \times \left( c h^2 \mathbf{n} + \mathbf{e}_1\right)\right].
\]
Since $\mathbf{e}_1\perp \mathbf{n}$ we have $(\mathbf{n}\times \mathbf m_h^{+}) \times \mathbf{e}_1\partialrallel \mathbf{n}$ and thus $\mathbf{P} \big((\mathbf{n}\times \mathbf m_h^{+}) \times \mathbf{e}_1\big)=0$.
Therefore, we get
\mathbf{e}gin{equation}\label{aux20}
|\mathbf{P} [\mathbf{m}_h]|\lesssim h^2+|\mathbf{e}_1|\ |\mathbf{e}_2|\lesssim h^2,
\end{equation}
i.e., the result \eqref{jump_est} holds.
\end{proof}
\ \\[1ex]
In the analysis below, we need an inequality of the form $\|v_h\|_{L^2(\Gamma_h)} \lesssim \|v_h\|_\ast$ for all $v_h\in V_h^\Gamma$. This result can be obtained as follows. First we consider the case $c=0$. Then the functions $v_h \in V_h^\Gamma$ satisfy $\int_{\Gamma_h} v_h \, \mathrm{d}\mathbf{s} =0$. We assume that in $V_h^\Gamma$ a discrete analogon of the Friedrich's inequality \eqref{FdrA} holds uniformly with respect to $h$, i.e., there exists a constant $C_F$ independent of $h$ such that
\mathbf{e}gin{equation} \label{FrB}
\|v_h\|_{L^2(\Gamma_h)}^2 \leq C_F \|\nabla_{\Gamma_h} v_h\|_{L^2(\Gamma_h)}^2 \quad \text{for all}~~v_h \in V_h^\Gamma.
\end{equation}
Now we reduce the parameter domain $\varepsilon \in (0,1],~c>0$ as follows. For a given generic constant $c_0$ with $0< c_0 <1$, in the remainder we restrict to the parameter set
\mathbf{e}gin{equation} \label{pardomain}
\varepsilon \in (0,1],~~c \in \{0\} \cup [ c_0 \varepsilon, \infty).
\end{equation}
For $c>0$ we then have
\mathbf{e}gin{align*}
c_0 \|v_h\|_{L^2(\Gamma_h)}^2 & \leq \frac{2 c_0}{c_0+1} \frac{1}{c} \|c^\frac12 v_h\|_{L^2(\Gamma_h)}^2 \leq \frac{2}{c/c_0 + c} \|v_h\|_\ast^2 \leq \frac{2}{\varepsilon +c} \|v_h\|_\ast^2,
\end{align*}
and combing this with the result in \eqref{FrB} for the case $c=0$ we get
\mathbf{e}gin{equation} \label{basic}
\|v_h\|_{L^2(\Gamma_h)} \lesssim \frac{1}{\sqrt{\varepsilon + c}} \|v_h\|_\ast \quad \text{for all}~~v_h\in V_h^\Gamma
\end{equation}
and arbitrary $\varepsilon \in (0,1],~c \in \{0\} \cup [ c_0 \varepsilon, \infty)$.\\
In the proof of Theorem~\ref{thm2} below we need a bound for $\|v_h\|_{L^2(\mathcal{E}_h)}$ in terms of $\|v_h\|_\ast$. Such a result is derived with the help of the following lemma.
\mathbf{e}gin{lemma}\label{lem_edge_est}
Assume the outer tetrahedra mesh size satisfies $h\le h_0$, with some sufficiently small $h_0\simeq 1$,
depending only on the constant $c_2$ from \eqref{e:3.10}. The following holds:
\mathbf{e}gin{equation} \label{edge_est}
\sum_{E\in \mathcal{E}_h}\int_{E}v_h^2 \, \mathrm{d}\mathbf{s} \lesssim h^{-1} \|v_h\|^2_{L^2(\Gamma_h)}+
h \|\nabla_{\Gamma_h} v_h\|^2_{L^2(\Gamma_h)} \quad \text{for all}~~v_h \in V_h^\Gamma
\end{equation}
\end{lemma}
\mathbf{e}gin{proof}
Let $E\in \mathcal{E}_h$ be an edge of a triangle $T\in\mathcal{F}_h$ and $S_T\in\mathcal{T}_h$ is the corresponding tetrahedron of the outer triangulation. Consider the patch $\widetilde{\omega}(S_T)$ of all $S\in\mathcal{T}_h$
touching $S_T$. Denote $\omega(S_T)=\widetilde{\omega}(S_T)\cap\Gamma_h$. Let $v_h$ be an arbitrary fixed function
from $ V_h^\Gamma$.
We shall prove the bound
\mathbf{e}gin{equation}\label{local_est}
\int_{E}v_h^2 \, \mathrm{d}\mathbf{s} \lesssim h^{-1} \|v_h\|^2_{L^2(\omega(S_T))}+
h \|\nabla_{\Gamma_h} v_h\|^2_{L^2(\omega(S_T))}.
\end{equation}
Then summing over all $E\in \mathcal{E}_h$ and using that $\omega(S_T)$ consists of a uniformly bounded number of
tetrahedra (due to the regularity of the outer mesh), we obtain \eqref{edge_est}.
Let $\mathbb{P}$ be a plane containing $T$. We can define for sufficiently small $h$ an injective mapping $\phi:\omega(S_T)\to\mathbb{P}$ such that $|\nabla\phi|\lesssim1$ and $|\nabla(\phi^{-1})|\lesssim1$. For example, $\phi$ can be build by the orthogonal projection
on $\mathbb{P}$. Then $|\nabla\phi|\lesssim 1$ and $|\nabla(\phi^{-1})(\mathbf{x})|\lesssim (\sin\alpha)^{-1}$,
where $\alpha$ is the angle between $\mathbb{P}$ and $\mathbf{n}_h(\phi^{-1}(\mathbf{x}))$. Due to assumption \eqref{e:3.10} we have $1\lesssim \sin\alpha$ for sufficiently small $h$. If $\phi$ is the orthogonal projection on $\mathbb{P}$, then $\phi(E)=E$. Thus we get
\mathbf{e}gin{equation}\label{equiv}
\left\{
\mathbf{e}gin{split}
\int_{E}v_h^2\mathrm{d}\mathbf{s}&=\int_{\phi(E)}(v_h\circ\phi^{-1})^2\mathrm{d}\mathbf{s},\\
\|v_h\|_{L^2(\omega(S_T))}&\simeq\|v_h\circ\phi^{-1}\|_{L^2(\phi(\omega(S_T)))},\\
\|\nabla_{\Gamma_h} v_h\|_{L^2(\omega(S_T))}&\simeq\|\nabla_{\mathbb{P}}( v_h\circ\phi^{-1})\|_{L^2(\phi(\omega(S_T)))}.
\end{split}\right.
\end{equation}
Due to the shape regularity of $S\in\widetilde{\omega}(S_T)$ we have
\[h\lesssim\text{dist}(E,\partialrtial\,\widetilde{\omega}(S_T))\le\text{dist}(E,\partialrtial\,\omega(S_T)).\]
Hence, from $|\nabla\phi^{-1}|\lesssim1$ it follows that
$h\lesssim \text{dist}(\phi(E),\partialrtial\,\phi(\omega(S_T)))$. Thus, we may consider a rectangle $Q\subset\phi(\omega(S_T))$
such that $E=\phi(E)$ is a side of $Q$ and $|Q|\simeq h|E|$. By the standard trace theorem and scaling argument we
get
\[
\int_{\phi(E)}(v_h\circ\phi^{-1})^2\mathrm{d}\mathbf{s} \lesssim h^{-1}\|v_h\circ\phi^{-1}\|^2_{L^2(Q)}+ h\|\nabla_{\mathbb{P}} (v_h\circ\phi^{-1})\|^2_{L^2(Q)}.
\]
This together with \eqref{equiv} and $Q\subset\phi(\omega(S_T))$ implies \eqref{local_est}.
\end{proof}
An immediate consequence of the lemma and \eqref{basic} is the following corollary.
\mathbf{e}gin{corollary}\lbl{lem:3.2}
The following estimate holds:
\mathbf{e}gin{equation}\label{aux1}
h\sum_{E\in \mathcal{E}_h}\int_{E}v_h^2\, \mathrm{d}\mathbf{s} \lesssim \big( \frac{1}{\varepsilon +c} + \frac{h^2}{\varepsilon}\big) \|v_h\|_\ast^2 \quad \text{for all}~~v_h \in V_h^\Gamma.
\end{equation}
\end{corollary}
We are now in position to prove a continuity result for the surface finite element bilinear form.
\mathbf{e}gin{lemma} \label{thm2}
For any $u\in H^2(\Gamma)$ and $v_h\in V_h^{\Gamma}$, we have
\mathbf{e}gin{equation}
|a_h(u^e-(I_h u^e)|_{\Gamma_h},v_h)|\lesssim \left(\varepsilon^{1/2}+h^{1/2}+c^{\frac12}h+\frac{h^2}{\sqrt{\varepsilon+c}}+ \frac{h^3}{\sqrt{\varepsilon}}\right)h
\|u\|_{H^2(\Gamma)} \|v_h \|_{\ast}.
\end{equation}
\end{lemma}
\mathbf{e}gin{proof} Define $\phi=u^e-(I_h u^e)|_{{\Gamma_h}}$, then
\mathbf{e}gin{equation} \label{aux9}
\mathbf{e}gin{split}
a_h(\phi,v_h) & = \varepsilon\int_{\Gamma_h}\nabla_{\Gamma}h \phi\cdot\nabla_{\Gamma}h v_h \mathrm{d}\mathbf{s}\\
& + \int_{\Gamma_h} \frac12\big((\mathbf{w}^e\cdot\nabla_{\Gamma}h \phi) v_h-(\mathbf{w}^e\cdot\nabla_{\Gamma}h v_h) \phi\big) +c\,\phi v_h \,\mathrm{d}\mathbf{s} \\
& + \sum_{T\in\mathcal{F}_h} \delta_T \int_T \left(-\varepsilon\Delta_{\Gamma_h}\phi
+ \mathbf{w}^e\cdot\nabla_{\Gamma}h \phi + c\phi \right)\mathbf{w}^e\cdot\nabla_{\Gamma}h v_h \, \mathrm{d}\mathbf{s}.
\end{split}
\end{equation}
We estimate $a_h(\phi,v_h)$ term by term.
Due to \eqref{e:3.18}, \eqref{e:3.19}, we get for the first term on the righthand side of \eqref{aux9}:
\mathbf{e}gin{equation}\label{aux8}
\Big|\varepsilon\int_{\Gamma_h}\nabla_{\Gamma}h \phi\cdot\nabla_{\Gamma}h v_h \mathrm{d}\mathbf{s}\Big|\lesssim \varepsilon h\|u\|_{H^2(\Gamma)}\|\nabla_{\Gamma}h v_h\|_{L^2(\Gamma_h)}
\le \varepsilon^{\frac12}h\|u\|_{H^2(\Gamma)}\|v_h \|_{\ast}.
\end{equation}
To the second term on the righthand side of \eqref{aux9} we apply integration by parts:
\mathbf{e}gin{equation}\label{aux10}
\mathbf{e}gin{split} & \int_{\Gamma_h}\frac12\big((\mathbf{w}^e\cdot\nabla_{\Gamma}h \phi) v_h-(\mathbf{w}^e\cdot\nabla_{\Gamma}h v_h) \phi\big) + c \phi v_h \,\mathrm{d}\mathbf{s} \\
& =\int_{\Gamma_h}c \phi v_h\mathrm{d}\mathbf{s}-\int_{\Gamma_h}(\mathbf{w}^e\cdot\nabla_{\Gamma}h v_h) \phi\mathrm{d}\mathbf{s} \\
& + \frac12 \sum_{T\in\mathcal{F}_h}\int_{\partial T}(\mathbf{w}^e\cdot \mathbf{m}_h)\phi v_h\mathrm{d}\mathbf{s} -\frac12\int_{\Gamma_h}(\mathrm{div}_{\Gamma}h\mathbf{w}^e) \phi v_h\mathrm{d}\mathbf{s} \\
& =:I_1+I_2+I_3+I_4.
\end{split}
\end{equation}
The term $I_1$ can be estimated by
\mathbf{e}gin{equation*}
|I_1|\lesssim c^{\frac12}\|\phi \|_{L^2(\Gamma_h)}\|\sqrt{c}v_h\|_{L^2(\Gamma_h)} \lesssim h^2c^{\frac12}\|u \|_{H^2(\Gamma)}\|v_h\|_{\ast}.
\end{equation*}
To estimate $I_2$, we consider the advection-dominated case and the diffusion-dominated case separately. If $\mathsf{Pe}_T>1$, we have
\mathbf{e}gin{equation*}
\mathbf{e}gin{aligned}
\int_{T} (\mathbf{w}^e\cdot\nabla_{\Gamma}h v_h) \phi\mathrm{d}\mathbf{s}&\lesssim \delta_{T}^{-1/2}\|\phi\|_{L^2(T)}\left(\int_{T}\delta_T (\mathbf{w}^e\cdot\nabla_{\Gamma}h v_h)^2\mathrm{d}\mathbf{s}\right)^{1/2}
\\
&\lesssim { \max(h^{-1/2},c^{1/2})}\|\phi\|_{L^2(T)}\left(\int_{T}\delta_T (\mathbf{w}^e\cdot\nabla_{\Gamma}h v_h)^2\mathrm{d}\mathbf{s}\right)^{1/2},
\end{aligned}
\end{equation*}
and if $\mathsf{Pe}_T\leq 1$:
\mathbf{e}gin{equation*}
\mathbf{e}gin{aligned}
\int_{T} (\mathbf{w}^e\cdot\nabla_{\Gamma}h v_h) \phi\mathrm{d}\mathbf{s}&\lesssim \|\mathbf{w}^e\|_{L^{\infty}(T)} \|\nabla_{\Gamma}h v_h\|_{L^2(T)}\|\phi\|_{L^2(T)}
\\
&\lesssim \varepsilon^{1/2} h^{-1} \|\varepsilon^{1/2}\nabla_{\Gamma}h v_h\|_{L^2(T)}\|\phi\|_{L^2(T)} .
\end{aligned}
\end{equation*}
Summing over $T\in\mathcal{F}_h$ we obtain
\mathbf{e}gin{equation*}
|I_2|\lesssim (h^{-1/2}+\varepsilon^{1/2}h^{-1}) \|\phi\|_{L^2(\Gamma_h)} \|v_h\|_{\ast}\lesssim h(c^{1/2}h+h^{1/2}+\varepsilon^{1/2}) \|u\|_{H^2(\Gamma)} \|v_h\|_{\ast}.
\end{equation*}
The term $I_3$ is estimated using $\mathbf P \mathbf{w}^e=\mathbf{w}^e$, Lemmas~\ref{prop:3.3}, \ref{lem:jumpvel}, and Corollary~\ref{lem:3.2}:
\mathbf{e}gin{equation}\label{aux22}
\mathbf{e}gin{aligned}
|I_3| &\lesssim \Big|\sum_{E\in\mathcal{E}_h}\int_{E}( \mathbf{w}^e\cdot[\mathbf{m}_h])\phi v_h\mathrm{d}\mathbf{s}\Big| \\
&\lesssim \left(\sum_{E\in\mathcal{E}_h}\int_{E}|\phi|^2\mathrm{d}\mathbf{s}\right)^{1/2}
\left(\sum_{E\in\mathcal{E}_h}\int_{E} |\mathbf{P} [\mathbf{m}_h] |^2 v_h^2 \, \mathrm{d}\mathbf{s}\right)^{1/2}\\
&\lesssim h^{3}\|u\|_{H^2(\Gamma)}\left( h \sum_{E\in\mathcal{E}_h}\int_{E}v_h^2 \, \mathrm{d}\mathbf{s}\right)^{1/2} \lesssim \left( \frac{h^3}{\sqrt{\varepsilon +c}} +\frac{h^4}{\sqrt{\varepsilon}}\right) \|u\|_{H^2(\Gamma)}\|v_h\|_\ast.
\end{aligned}
\end{equation}
The term $I_4$ in \eqref{aux10} can be bounded due to Lemma~\ref{lem_div}, the interpolation
bounds and \eqref{basic}:
\[
\mathbf{e}gin{aligned}
|I_4|&\le \frac12\|\mathrm{div}_{\Gamma}h\mathbf{w}^e\|_{L^\infty(\Gamma_h)}\|\phi\|_{L^2(\Gamma_h)} \|v_h\|_{L^2(\Gamma_h)}\lesssim h^{3}\|u\|_{H^2(\Gamma)} \|v_h\|_{L^2(\Gamma_h)}\\&\le \frac{h^3}{\sqrt{\varepsilon+c}} \|u\|_{H^2(\Gamma)} \|v_h\|_{\ast}.
\end{aligned}
\]
Finally we treat the third term on the righthand side of \eqref{aux9}. Using $\delta_T \|\mathbf{w}^e\|_{L^\infty(T)} \lesssim h$, $\delta_T \varepsilon \lesssim 1$, $\delta_T c \leq 1$ and
the interpolation estimates \eqref{e:3.18} and \eqref{e:3.19} we obtain:
\mathbf{e}gin{equation}\label{aux11}
\mathbf{e}gin{aligned}
&\sum_{T\in\mathcal{F}_h} \delta_T \int_T (-\varepsilon\Delta_{\Gamma_h}\phi
+ \mathbf{w}^e\cdot\nabla_{\Gamma}h \phi + c \phi ) \mathbf{w}^e\cdot\nabla_{\Gamma}h v_h \, \mathrm{d}\mathbf{s} \\
\lesssim &\left(\sum_{T\in\mathcal{F}_h} \delta_{T}\big(\varepsilon^2 \|\Delta_{\Gamma_h}u^e\|^2_{L^2(T)}+\|\mathbf{w}^e\cdot\nabla_{\Gamma}h\phi\|^2_{L^2(T)}+c^2\|\phi\|^2_{L^2(T)}\big)\right)^{1/2}\|v_h\|_{\ast}\\
\lesssim & ~ (\varepsilon^{1/2}+h^{1/2}+c^{\frac12}h)h\|u\|_{H^2(\Gamma)}\|v_h\|_{\ast}.
\end{aligned}
\end{equation}
Combing the inequalities \eqref{aux8}--\eqref{aux11} proves the result of the lemma.
\end{proof}
\subsection{Consistency estimate} \label{sectconsistency}
The consistency error of the finite element method \eqref{e:2.8} is due to geometric errors resulting
from the approximation of $\Gamma$ by $\Gamma_h$. To estimate this geometric errors we need a few additional
definitions and results, which can be found in, for example, \cite{Demlow06}. For $\mathbf{x}\in \Gamma_h$ define $\tilde{\mathbf{P}}_{h}(\mathbf{x})= \mathbf{I}-\mathbf{n}_h(\mathbf{x})\mathbf{n}(\mathbf{x})^T/(\mathbf{n}_h(\mathbf{x})\cdot\mathbf{n}(\mathbf{x}))$.
One can represent the surface gradient of $u\in H^1(\Gamma)$ in terms of $\nabla_{\Gamma}h u^e$ as follows
\mathbf{e}gin{equation} \label{hhl}
\nabla_{\Gamma} u(\mathbf{p}(\mathbf{x}))=(\mathbf{I}-d(\mathbf{x})\mathbf{H}(\mathbf{x}))^{-1} \tilde{\mathbf{P}}_{h}(\mathbf{x}) \nabla_{\Gamma}h u^e(\mathbf{x})~~ \hbox{ a.e. }\mathbf{x}\in \Gamma_h.
\end{equation}
Due to \eqref{e:3.16}, we get
\mathbf{e}gin{equation}\label{aux13}
\int_{\Gamma}\nabla_{\Gamma} u\nabla_{\Gamma} v\, \mathrm{d}\mathbf{s}=\int_{\Gamma_h} \mathbf{A}_h\nabla_{\Gamma}h u^e\nabla_{\Gamma}h v^e \, \mathrm{d}\mathbf{s} \quad \hbox{for all } v\in H^1(\Gamma),
\end{equation}
with $\mathbf{A}_h(\mathbf{x})=\mu_h(\mathbf{x}) \tilde{\mathbf{P}}^T_h(\mathbf{x})(\mathbf{I}-d(\mathbf{x})\mathbf{H}(\mathbf{x}))^{-2}\tilde{\mathbf{P}}_h(\mathbf{x})$.
From $\mathbf{w} \cdot \mathbf{n} =0$ on $\Gamma$ and $\mathbf{w}^e(\mathbf{x})=\mathbf{w}(\mathbf{p}(\mathbf{x}))$, $\mathbf{n}(\mathbf{x}) =\mathbf{n}(\mathbf{p}(\mathbf{x}))$ it follows that $\mathbf{n}(\mathbf{x}) \cdot \mathbf{w}^e(\mathbf{x})=0$ and thus $\mathbf{w}(\mathbf{p}(\mathbf{x}))= \tilde \mathbf{P}_h(\mathbf{x})\mathbf{w}^e(\mathbf{x})$ holds. Using this, we get the relation
\mathbf{e}gin{equation}\label{aux14}
\int_{\Gamma}(\mathbf{w}\cdot \nabla_{\Gamma} u) v \, \mathrm{d}\mathbf{s} = \int_{\Gamma_h} (\mathbf{B}_h \mathbf{w}^e \cdot \nabla_{\Gamma}h u^e)v^e\, \mathrm{d}\mathbf{s},
\end{equation}
with $\mathbf{B}_h=\mu_h(\mathbf{x})\tilde{\mathbf{P}}_h^T(I-d\mathbf{H})^{-1}\tilde\mathbf{P}_h$.
In the proof we use the lifting procedure $\Gamma_h \to \Gamma$ given by
\mathbf{e}gin{equation} \label{deflift}
v_h^l(\mathbf{p}(\mathbf{x})): =v_h(\mathbf{x}) \quad \text{for}~~ \mathbf{x}\in\Gamma_h.
\end{equation}
It is easy to see that $v_h^l\in H^1(\Gamma)$.
The following lemma estimates the consistency error of the finite element method \eqref{e:2.8}.
\mathbf{e}gin{lemma} \label{thm3}
Let $u\in H^2(\Gamma)$ be the solution of \eqref{e:2.5}, then we have
\mathbf{e}gin{equation}
\sup_{v_h\in V_h^{\Gamma}}\frac{|f_h(v_h)-a_h(u^e,v_h)|}{\|v_h\|_{\ast}} \lesssim \big(h^{\frac12}+c^{\frac12}h+\frac{h}{\sqrt{c+\varepsilon}}\big)\,h(\|u\|_{H^2(\Gamma)}+\|f\|_{L^2(\Gamma)}).
\end{equation}
\end{lemma}
\mathbf{e}gin{proof}
The residual is decomposed as
\mathbf{e}gin{equation} \label{er}
f_h(v_h)-a_h(u^e,v_h)=f_h(v_h)-f(v_h^l)+a(u,v_h^l)-a_h(u^e,v_h).
\end{equation}
The following holds:
\mathbf{e}gin{align*}
f(v^l_h)=&\,\int_{\Gamma}fv_h^l\, \mathrm{d}\mathbf{s}= \int_{\Gamma_h}\mu_h f^e v_h\, \mathrm{d}\mathbf{s},\\
a(u,v^l_h)=&\,\varepsilon \int_{\Gamma}\nabla_{\Gamma} u \nabla_{\Gamma} v_h^l\, \mathrm{d}\mathbf{s}
+\int_{\Gamma}(\mathbf{w}\cdot\nabla_{\Gamma} u) v_h^l\mathrm{d}\mathbf{s}+\int_{\Gamma}cuv_h^l\, \mathrm{d}\mathbf{s}\\
=&\,\varepsilon \int_{\Gamma}\nabla_{\Gamma} u \nabla_{\Gamma} v_h^l\, \mathrm{d}\mathbf{s}
+\frac12\int_{\Gamma}(\mathbf{w}\cdot\nabla_{\Gamma} u) v_h^l-(\mathbf{w}\cdot\nabla_{\Gamma} v_h^l) u\,\mathrm{d}\mathbf{s} +\int_{\Gamma}cuv_h^l\, \mathrm{d}\mathbf{s}\\
=&\, \varepsilon \int_{\Gamma_h} \mathbf{A}_h\nabla_{\Gamma_h} u^e\nabla_{\Gamma_h} v_h\, \mathrm{d}\mathbf{s}+ \frac12\int_{\Gamma_h} (\mathbf{B}_h \mathbf{w}^e\cdot\nabla_{\Gamma}h u^e)v_h\, \mathrm{d}\mathbf{s}\\
&\,-\frac12\int_{\Gamma_h}(\mathbf{B}_h \mathbf{w}^e\cdot\nabla_{\Gamma}h v_h ) u^e\, \mathrm{d}\mathbf{s}+\int_{\Gamma_h}\mu_h c u^ev_h\, \mathrm{d}\mathbf{s}.
\end{align*}
Substituting these relations into \eqref{er} and using \eqref{eqah}, \eqref{eqfh} results in
\mathbf{e}gin{multline}\label{aux12}
f_h(v_h)-a_h(u^e,v_h)=
\int_{\Gamma_h}(1-\mu_h)f^ev_h\, \mathrm{d}\mathbf{s}+\varepsilon\int_{\Gamma_h}(\mathbf{A}_h-\mathbf{P}_h)\nabla_{\Gamma_h} u^e \cdot \nabla_{\Gamma_h}v_h\, \mathrm{d}\mathbf{s}\\
+\frac12\int_{\Gamma_h}((\mathbf{B}_h-\mathbf{P}_h)\mathbf{w}^e\cdot\nabla_{\Gamma}h u^e) v_h\, \mathrm{d}\mathbf{s}-\frac12\int_{\Gamma_h}((\mathbf{B}_h-\mathbf{P}_h)\mathbf{w}^e\cdot\nabla_{\Gamma}h v_h) u^e\, \mathrm{d}\mathbf{s} \\ + \int_{\Gamma_h}(\mu_h-1)cu^e v_h\, \mathrm{d}\mathbf{s}
+\sum_{T\in\mathcal{F}_h}\delta_T\int_T \big(f^e+\varepsilon\Delta_{\Gamma_h}u^e-\mathbf{w}^e\cdot\nabla_{\Gamma}h u^e-cu^e \big) \mathbf{w}^e\cdot\nabla_{\Gamma}h v_h \, \mathrm{d}\mathbf{s} \\
=:I_1+I_2+I_3+I_4+I_5+\varPi_1.
\end{multline}
We estimate the $I_i$ terms separately.
Applying \eqref{e:3.17} and \eqref{basic} we get
\mathbf{e}gin{align}
I_1&\lesssim h^2 \|f^e\|_{L^2(\Gamma_h)}\|v_h\|_{L^2(\Gamma_h)}\lesssim \frac{h^2}{\sqrt{c +\varepsilon }} \|f\|_{L^2(\Gamma)}\|v_h\|_{\ast},\label{aux15}\\
I_5&\lesssim h^2 c^{\frac12}\|u^e\|_{L^2(\Gamma_h)}\|\sqrt{c}v_h\|_{L^2(\Gamma_h)}\lesssim h^2c^{\frac12} \|u\|_{L^2(\Gamma)}\|v_h\|_{\ast}.
\end{align}
One can show, cf. (3.43) in \cite{Reusken08}, the bound
\mathbf{e}gin{equation*}
|\mathbf{P}_h-\mathbf{A}_h|=|\mathbf{P}_h(\mathbf I-\mathbf A_h)|\lesssim h^2.
\end{equation*}
Using this we obtain
\mathbf{e}gin{align}
& I_2\lesssim \varepsilon h^2 \|\nabla_{\Gamma}h u^e\|_{L^2(\Gamma_h)}\|\nabla_{\Gamma}h v_h\|_{L^2(\Gamma_h)} \lesssim \varepsilon^{1/2}h^2 \|u^e\|_{H^2(\Gamma)}\|v_h\|_{\ast}.
\end{align}
Since $(\mathbf I-d\mathbf{H})^{-1}=\mathbf I+O(h^2)$, we also estimate
\mathbf{e}gin{equation*}
|\mathbf B_h-\mathbf{P}_h|\lesssim h^2+|\mathbf{A}_h-\mathbf{P}_h| \lesssim h^2.
\end{equation*}
This yields
\mathbf{e}gin{equation}\label{aux16}
I_3\lesssim h^2 \|\nabla_{\Gamma}h u^e\|_{L^2(\Gamma_h)}\| v_h\|_{L^2(\Gamma_h)} \lesssim \frac{h^2}{\sqrt{c+\varepsilon }} \|u\|_{H^2(\Gamma)}\|v_h\|_{\ast}.
\end{equation}
To estimate $I_4$ we use the definition \eqref{e:2.10} of $\delta_T$.
If $\displaystyle \mathsf{Pe}_T \leq 1$, then
$\varepsilon^{-\frac12} \|\mathbf{w}^e\|_{L^\infty(T)} \leq \sqrt{2} \|\mathbf{w}^e\|_{L^\infty(T)}^{\frac12} h_{S_T}^{- \frac12}$ holds.
If $\displaystyle \mathsf{Pe}_T > 1$, then $\delta_T^{- \frac12} \leq {\max(c^{\frac12}, \delta_0^{-\frac12} \|\mathbf{w}^e\|_{L^\infty(T)}^{\frac12} h_{S_T}^{- \frac12})}$ holds.
Using the assumption that the outer triangulation is quasi-uniform we get $ h_{S_T}^{-1} \lesssim h^{-1}$. Thus, we obtain
\[
\min\{\varepsilon^{-\frac12}\|\mathbf{w}^e\|_{L^\infty(T)},\delta_T^{-\frac12}\}\lesssim { \max(c^{\frac12}, h^{-\frac12})\lesssim c^{\frac12}+ h^{-\frac12}},
\]
and
\mathbf{e}gin{align}\label{aux40}
I_4\lesssim& \max_{x\in\Gamma_h}|\mathbf B_h-\mathbf{P}_h| \sum_{T\in\mathcal{F}_h}(\varepsilon^{\frac12}\|\nabla_{\Gamma}h v_h\|_{L^2(T)} + \delta_T^{\frac12}\|\mathbf{w}^e_{\Gamma_h}\cdot\nabla_{\Gamma}h v_h\|_{L^2(T)})\notag\\
&~~\times\min\{\varepsilon^{-\frac12}\|\mathbf{w}^e\|_{L^\infty(T)},\delta_T^{-\frac12}\}\|u^e\|_{L^2(T)}\notag\\
\lesssim& ~ {h(c^{\frac12}h+ h^{\frac12}) } \|v_h\|_{\ast}\|u\|_{L^2(\Gamma)}.
\end{align}
Now we estimate $\varPi_1$. Using the equation $-\varepsilon \Delta_\Gamma u + \mathbf{w} \cdot \nabla_\Gamma u + c u=f$ on $\Gamma$ we get
\mathbf{e}gin{align}
\varPi_1&=\sum_{T\in\mathcal{F}_h} \delta_T \int_T \big( f\circ \mathbf{p} +\varepsilon\Delta_{\Gamma_h}u^e - \mathbf{w}^e\cdot\nabla_{\Gamma}h u^e -c u^e \big)\mathbf{w}^e\cdot\nabla_{\Gamma}h v_h\, \mathrm{d}\mathbf{s} \nonumber\\
&=\sum_{T\in\mathcal{F}_h}\delta_T \int_T \big(-\varepsilon(\Delta_{\Gamma} u) \circ \mathbf{p} +\varepsilon\Delta_{\Gamma_h}u^e \big) \mathbf{w}^e\cdot\nabla_{\Gamma}h v_h \, \mathrm{d}\mathbf{s} \nonumber\\
&\quad +\sum_{T\in\mathcal{F}_h}\delta_T \int_T \big((\mathbf{w}^e\cdot \nabla_{\Gamma} u) \circ \mathbf{p} -\mathbf{w}^e\cdot\nabla_{\Gamma}h u^e\big)\mathbf{w}^e\cdot\nabla_{\Gamma}h v_h \, \mathrm{d}\mathbf{s}\nonumber\\
& \quad +\sum_{T\in\mathcal{F}_h}\delta_T \int_T \big(cu \circ \mathbf{p} - c u^e\big)\mathbf{w}^e\cdot\nabla_{\Gamma}h v_h \, \mathrm{d}\mathbf{s} \nonumber\\
&=:\varPi_1^1+\varPi_1^2+\varPi_1^3. \label{aux17}
\end{align}
From $u^e = u\circ\mathbf{p} $ it follows that $\varPi_1^3=0 $ holds. For $\varPi_1^2$ we obtain, using \eqref{hhl} and $|\mu_h^{-1} \mathbf B_h-\mathbf{P}_h| \leq |\mu_h -1| |\mu_h|^{-1} |\mathbf B_h| + |\mathbf B_h - \mathbf P_h| \lesssim h^2$,
\mathbf{e}gin{align}
\varPi_1^2&=\sum_{T\in\mathcal{F}_h} \delta_T \int_T \big((\mu_h^{-1} \mathbf B_h-\mathbf{P}_h)\mathbf{w}^e\cdot\nabla_{\Gamma}h u^e\big)\mathbf{w}^e\cdot\nabla_{\Gamma}h v_h \, \mathrm{d}\mathbf{s} \nonumber\\
& \lesssim h^2 \left(\sum_{T\in\mathcal{F}_h}\delta_T\|\mathbf{w}^e\|_{L^\infty}^2 \| \nabla_{\Gamma}h u^e\|_{L^2(T)}^2\right)^{\frac{1}{2}}
\left(\sum_{T\in\mathcal{F}_h}\delta_T\|\mathbf{w}^e \cdot \nabla_{\Gamma}h v_h\|_{L^2(T)}^2\right)^{\frac{1}{2}}\nonumber\\
&\lesssim h^{\frac{5}{2}}\|u\|_{H^2(\Gamma)}\|v_h\|_{\ast}. \label{aaa}
\end{align}
Since $\delta_T\varepsilon\lesssim h^2$, we get
\mathbf{e}gin{align}\label{aux19}\notag
\varPi_2^1
& \lesssim \left(\sum_{T\in\mathcal{F}_h}\varepsilon^2\delta_T\| (\Delta_{\Gamma} u) \circ \mathbf{p}-\Delta_{\Gamma_h}u^e\|_{L^2(T)}^2\right)^{\frac{1}{2}}
\left(\sum_{T\in\mathcal{F}_h}\delta_T\|\mathbf{w}^e\cdot\nabla_{\Gamma}h v_h\|_{L^2(T)}^2\right)^{\frac{1}{2}}\\
&\lesssim h\varepsilon^{\frac12}\left(\sum_{T\in\mathcal{F}_h}\| (\Delta_{\Gamma} u) \circ \mathbf{p}-\Delta_{\Gamma_h}u^e\|_{L^2(T)}^2\right)^\frac{1}{2}\|v_h\|_{\ast}.
\end{align}
We finally consider the term between brackets in \eqref{aux19}.
Using the identity $\operatorname{div}_{\Gamma} \mathbf f = \mathrm{tr} (\nabla_{\Gamma} \mathbf f)$ and $\mathbf{n} \cdot \nabla u^e( \mathbf{p}(\mathbf{x}))=0$ we obtain for $\mathbf{x} \in T$, with $\nabla^2:= \nabla \nabla^T$.
\mathbf{e}gin{equation}\label{Delta1}
\Delta_{\Gamma} u(\mathbf{p}(\mathbf{x}))=
\operatorname{div}_{\Gamma} \nabla_\Gamma u(\mathbf{p}(\mathbf{x}))=\mathrm{tr}(\mathbf{P}\nabla \mathbf{P}\nabla u^e(\mathbf{p}(\mathbf{x})))=
\mathrm{tr}(\mathbf{P} \nabla^2 u^e(\mathbf{p}(\mathbf{x}))\mathbf{P}).
\end{equation}
From the same arguments it follows that
\mathbf{e}gin{equation}\label{Delta2}
\Delta_{\Gamma_h} u^e(\mathbf{x})=
\mathrm{tr}(\mathbf{P}_h\nabla \mathbf{P}_h \nabla u^e(\mathbf{x}))=
\mathrm{tr}(\mathbf{P}_h \nabla^2 u^e(\mathbf{x})\mathbf{P}_h)
\end{equation}
holds.
Using \eqref{e:3.14n} and $|d(\mathbf{x})| \lesssim h^2$, $|\mathbf{P}- \mathbf{P}_h| \lesssim h$, $|\mathbf{H}| \lesssim 1$, $| \nabla\mathbf{H}| \lesssim 1$ we obtain
\mathbf{e}gin{align*}
\mathbf{P}_h \nabla^2 u^e(\mathbf{x})\mathbf{P}_h & =\mathbf{P} \nabla^2 u^e(\mathbf{p}(\mathbf{x}))\mathbf{P} +\mathbf{R}, \\
|\mathbf{R}| & \lesssim h \big(|\nabla^2 u^e(\mathbf{p}(\mathbf{x})) | +|\nabla u^e(\mathbf{p}(\mathbf{x})) | \big).
\end{align*}
Thus, using \eqref{Delta1} and \eqref{Delta2}, we get
\mathbf{e}gin{align*}
| \Delta_{\Gamma} u (\mathbf{p}(\mathbf{x}))-\Delta_{\Gamma_h}u^e(\mathbf{x})| & \leq \big| \mathrm{tr} (\mathbf{P} \nabla^2 u^e(\mathbf{p}(\mathbf{x}))\mathbf{P}-\mathbf{P}_h \nabla^2
u^e(\mathbf{x})\mathbf{P}_h) \big| \\
& \lesssim h \big(|\nabla^2 u^e(\mathbf{p}(\mathbf{x})) | +|\nabla u^e(\mathbf{p}(\mathbf{x})) | \big),
\end{align*}
and combining this with \eqref{aux19} yields
\[
\varPi_2^1\lesssim h \varepsilon^{\frac12} \left(\sum_{T\in\mathcal{F}_h}\| (\Delta_{\Gamma} u) \circ \mathbf{p} -\Delta_{\Gamma_h}u^e\|_{L^2(T)}^2\right)^\frac{1}{2} \|v_h\|_\ast \lesssim\varepsilon^{\frac12} h^2 \|u\|_{H^2(\Gamma)} \|v_h\|_\ast.
\]
Combining this with the results \eqref{aux12}-\eqref{aux40} and \eqref{aaa} proves the lemma.
\end{proof}
\subsection{Main theorem} \label{sectmain}
Now we put together the results derived in the previous sections to prove the main result of the paper.
\mathbf{e}gin{theorem}\label{Th1}
Let Assumption~\ref{ass1} be satisfied. We consider problem parameters $\varepsilon$ and $c$ as in \eqref{pardomain}. Assume that the solution $u$ of \eqref{e:2.5} has regularity $u\in H^2(\Gamma)$. Let $u_h$ be the discrete solution of the SUPG finite element method \eqref{e:2.8}.
Then the following holds:
\mathbf{e}gin{equation}\label{errorEst}
\|u^e-u_h\|_{\ast}\lesssim h\big(h^{1/2}+\varepsilon^{1/2}+c^{\frac12}h+\frac{h}{\sqrt{\varepsilon+ c}}+ \frac{h^3}{\sqrt{\varepsilon}}\big) (\|u\|_{H^2(\Gamma)}+\|f\|_{L^2(\Gamma)}).
\end{equation}
\end{theorem}
\mathbf{e}gin{proof} The triangle inequality yields
\mathbf{e}gin{equation}
\|u^e-u_h\|_{\ast}\leq \|u^e-(I_h u^e)|_{\Gamma_h}\|_{\ast} + \|(I_h u^e)|_{\Gamma_h}- u_h \|_{\ast}.
\end{equation}
The second term in the upper bound can be estimated using Lemmas~\ref{thm0},~\ref{thm2},~\ref{thm3}:
\mathbf{e}gin{align*}
\frac12\|&(I_h u^e)|_{\Gamma_h}- u_h \|^2_\ast \le a_h((I_h u^e)|_{\Gamma_h}- u_h,(I_h u^e)|_{\Gamma_h}- u_h )\nonumber \\
&= a_h((I_h u^e)|_{\Gamma_h} -u^e, (I_h u^e)|_{\Gamma_h}- u_h)+a_h(u^e-u_h, (I_h u^e)|_{\Gamma_h}-u_h)\nonumber \\
&\lesssim h \big(h^{1/2}+\varepsilon^{1/2}+c^{\frac12}h +\frac{h^2}{\sqrt{\varepsilon+c}}+ \frac{h^3}{\sqrt{\varepsilon}}\big) \|u\|_{H^2(\Gamma)}\|(I_h u^e)|_{\Gamma_h}-u_h\|_{\ast}\nonumber \\
&\qquad + |a_{h}(u^e,(I_h u^e)|_{\Gamma_h}-u_h)-f_{h}((I_h u^e)|_{\Gamma_h}-u_h)|\nonumber \\
&\lesssim h \big(h^{1/2}+\varepsilon^{1/2}+c^{\frac12}h +\frac{h}{\sqrt{\varepsilon+c}}+ \frac{h^3}{\sqrt{\varepsilon}}\big)\big(\|u\|_{H^2(\Gamma)}+\|f\|_{L^2(\Gamma)}\big)\|(I_h u^e)|_{\Gamma_h}-u_h\|_{\ast}.\nonumber
\end{align*}
This results in
\mathbf{e}gin{equation}\label{aux24}
\|(I_h u^e)|_{\Gamma_h}- u_h \|_{\ast}\lesssim h \big(h^{1/2}+\varepsilon^{1/2}+c^{\frac12}h +\frac{h}{\sqrt{\varepsilon+c}}+ \frac{h^3}{\sqrt{\varepsilon}}\big)(\|u\|_{H^2(\Gamma)}+\|f\|_{L^2(\Gamma)}).
\end{equation}
The error estimate \eqref{errorEst} follows from \eqref{e:3.20} and \eqref{aux24}.\\
\end{proof}
\subsection{Further discussion}\label{sec_disc}
We comment on some aspects related to the main theorem. Concerning the analysis we note that the norm $\|\cdot\|_\ast$, which measures the error
on the left-hand side of \eqref{errorEst}, is the standard SUPG norm as found in standard analyses of
planar streamline-diffusion finite element methods. The analysis in this paper contains new ingredients compared to the planar case. To control the geometric errors (approximation of $\Gamma$ by $\Gamma_h$) we derived a consistency error bound in Lemma~\ref{thm3}. To derive a continuity result (Lemma~\ref{thm2}), as in the planar case, we apply partial integration to the term $\int_{\Gamma_h} (\mathbf{w}^e \cdot \nabla_{\Gamma_h} \phi) v_h \, \mathrm{d}\mathbf{s}$, cf. \eqref{aux10}. However, different from the planar case, this results in jumps across the edges $E \in \mathcal{E}_h$ which have to be controlled, cf.~\eqref{aux22}. For this the new results in the Lemmas~\ref{lem:jumpvel} and \ref{lem_edge_est} are derived. These jump terms across the edges cause the term $\frac{h^4}{\sqrt{\varepsilon}}$ in the error bound in \eqref{errorEst}.
Consider the error reduction factor $h^{3/2}+\varepsilon^{1/2}h+c^{\frac12}h^2+\frac{h^2}{\sqrt{\varepsilon+ c}}+ \frac{h^4}{\sqrt{\varepsilon}}$ on the right-hand side of \eqref{errorEst}. The first three terms of it are
typical for the error analysis of planar SUPG finite element methods for $P1$ elements. In the standard literature for the planar case, cf.~\cite{TobiskaBook}, one typically only considers the case $c >0$. Our analysis also applies to the case $c=0$, cf.~\eqref{pardomain}. Furthermore, the estimates are uniform w.r.t. the size of the parameter $c$. For a fixed $c>0$ and $\varepsilon \lesssim h$ the first four terms can be estimated by $\lesssim h^{3/2}$, a bound similar to the standard one for the planar case. The only ``suboptimal'' term is the last one, which is caused by (our analysis of) the jump terms. Note, however, that $\frac{h^4}{\sqrt{\varepsilon}} \lesssim h^{3/2}$ if $h^5 \lesssim \varepsilon$ holds, which is a very mild condition.
The norm $\|\cdot\|_\ast$ provides a robust control of streamline derivatives of the solution. Cross-wind oscillations, however, are not completely suppressed. It is well known that nonlinear stabilization methods can be used to get control over cross-wind derivatives as well. Extending such methods to surface PDEs is not within the scope of the present paper.
The error estimates in this paper are in terms of the maximum mesh size over tetrahedra in $\omega_h$, denoted by $h$.
In practice, the stabilization parameter $\delta_T$ is based on the \emph{local} Peclet number and the stabilization
is switched off or reduced in the regions, where the mesh is ``sufficiently fine''. To prove error estimates accounting for local smoothness of the solution $u$ and the local mesh size (as available for planar SUPG FE method), one needs local interpolation
properties of finite element spaces, instead of \eqref{e:3.18}, \eqref{e:3.19}. Since our finite element space is based on traces of piecewise linear functions, such local estimates are not immediately available. The extension of our analysis to this \emph{non}-quasi-uniform case is left for future research.
Finally, we remark on the case of a varying reaction term coefficient $c$. If the coefficient $c$ in the third term of \eqref{e:2.3} varies, the above analysis is valid with minor modifications. We briefly explain these modifications. The stabilization parameter $\delta_T$ should
be based on elementwise values $c_T=\max_{\mathbf{x}\in T} c(\mathbf{x})$.
For the well-possedness of \eqref{e:2.3}, it is sufficient to assume $c$ to be strictly positive on a subset of $\Gamma$ with positive measure:
\[
\mathcal{A}:=\operatorname{meas}\{\mathbf{x}\in\Gamma\,:\, c(\mathbf{x})\ge {c}_0\}>0,
\]
with some $c_0>0$.
If this is satisfied, the Friedrich's type inequality~\cite{Sobolev} (see, also Lemma~3.1 in \cite{Reusken08})
\mathbf{e}gin{equation*}
\|v\|_{L^2(\Gamma)}^2 \le C_F(\|\nabla_\Gamma v\|_{L^2(\Gamma)}^2+\|\sqrt{c} v\|_{L^2(\Gamma)}^2)\quad
\text{for all}~v\in V
\end{equation*}
holds, with a constant $C_F$ depending on $c_0$ and $\mathcal{A}$. Using this, all arguments in the analysis can be generalized to the case of a varying $c(\mathbf{x})$ with obvious modifications.
With $c_{\min}:=\text{ess\,inf}_{\mathbf{x}\in \Gamma} c(\mathbf{x})$
and $c_{\max}:=\text{ess\,sup}_{\mathbf{x}\in \Gamma} c(\mathbf{x})$, the final error estimate takes the form
\mathbf{e}gin{equation*}
\|u^e-u_h\|_{\ast}\lesssim h\big(h^{1/2}+\varepsilon^{1/2}+c^{\frac12}_{\max}h+\frac{h}{\sqrt{\varepsilon+ c_{\min}}}+ \frac{h^3}{\sqrt{\varepsilon}}\big) (\|u\|_{H^2(\Gamma)}+\|f\|_{L^2(\Gamma)}).
\end{equation*}
\section{Numerical experiments} \label{sectexperiments}
In this section we show results of a few numerical experiments which illustrate the performance of the method.
\mathbf{e}gin{example}
\label{example1} \rm The stationary problem \eqref{e:2.3} is solved on the unit sphere $\Gamma$, with
the velocity field
$$
\mathbf{w}(\mathbf{x})=(-x_2\sqrt{1-x_3^2},x_1\sqrt{1-x_3^2},0)^T,
$$
which is tangential to the sphere.
We set $\varepsilon=10^{-6}$, $c \equiv 1$ and consider the solution
$$
u(\mathbf{x})= \frac{x_1 x_2}{\pi}\mathrm{arctan}\left(\frac{x_3}{\sqrt{\varepsilon}}\right).
$$
Note that $u$ has a sharp internal layer along the
equator of the sphere. The corresponding right-hand side function $f$ is given
by
$$
f(\mathbf{x})=\frac{8\varepsilon^{3/2}(2+\varepsilon+2 x_3^2)x_1x_2x_3}{\pi(\varepsilon+4 x_3^2)^2}
+ \frac{6\varepsilon x_1x_2+\sqrt{x_1^2+x_2^2}(x_1^2-x_2^2)}{\pi}\mathrm{arctan}\left(\frac{x_3}{\sqrt{\varepsilon}}\right)+u.
$$
We consider the standard (unstabilized) finite element method in \eqref{plainFEM} and the stabilized method \eqref{e:2.8}.
A sequence of meshes was obtained by the gradual refinement of the outer triangulation. The induced \emph{surface} finite element spaces have dimensions $N=448,\, 1864,\, 7552,\, 30412$.
The resulting algebraic systems are solved by a direct sparse solver. Finite element errors are computed outside the layer: The variation of the quantities $err_{L^2}=\|u-u_h\|_{L^2(D)}$,
$err_{H^1}=\|u-u_h\|_{H^1(D)}$, and $err_{L^{inf}}=\|u-u_h\|_{L^{\infty}(D)}$,
with $D=\{\mathbf{x}\in\Gamma\ :\ |x_3|>0.3\}$, are shown in Figure~\ref{fig:err_ex1}. The results clearly show that the stabilized method performs much better than the standard one.
The results for the stabilized method indicate a $\mathcal{O}(h^2)$ convergence in the $L^2(D)$-norm and $L^{\infty}(D)$-norm. In the $H^1(D)$-norm
a first order convergence is observed.
Note that the analysis predicts (only) $\mathcal{O}(h^{\frac{3}{2}})$ convergence order in the (global) $L^2$-norm.
\mathbf{e}gin{figure}[ht!]
\mathbf{e}gin{center}
\includegraphics[width=\textwidth]{Ex1_errs1.eps}
\caption{Discretization errors for Example~\ref{example1}}\lbl{fig:err_ex1}
\end{center}
\end{figure}
Figure~\ref{fig:instable} shows the computed solutions with the two methods.
Since the layer is unresolved, the finite element method \eqref{plainFEM} produces globally oscillating solution. The stabilized method gives a much better approximation, although the layer
is slightly smeared, as is typical for the SUPG method.
\mathbf{e}gin{figure}[ht!]
\centering
\subfigure[]{
\includegraphics[width=2.2in]{solution1.eps}}
\subfigure[]{
\includegraphics[width=2.2in]{solution1-ns.eps}}
\caption{Example~\ref{example1}: solutions using the stabilized method and the standard method \eqref{plainFEM}.}
\lbl{fig:instable}
\end{figure}
\end{example}
\mathbf{e}gin{example}
\label{example2} \rm Now we consider the stationary problem \eqref{e:2.3} with $c \equiv 0$.
The problem is posed on the unit sphere $\Gamma$,
with this same velocity field $\mathbf{w}$ as in Example~\ref{example1}.
We set $\varepsilon=10^{-6}$, and consider the solution
$$
u(\mathbf{x})= \frac{x_1x_2}{\pi}\mathrm{arctan}\left(\frac{x_3}{\sqrt{\varepsilon}}\right).
$$
The corresponding right-hand side function $f$ is now given by
$$
f(\mathbf{x})=\frac{8\varepsilon^{3/2}(2+\varepsilon+2 x_3^2)x_1x_2x_3}{\pi(\varepsilon+4 x_3^2)^2}
+ \frac{6\varepsilon x_1x_2+\sqrt{x_1^2+x_2^2}(x_1^2-x_2^2)}{\pi}\mathrm{arctan}\left(\frac{x_3}{\sqrt{\varepsilon}}\right).
$$
Note that for $c = 0$ one looses explicit control of the $L_2$ norm in $\|\cdot\|_\ast$.
Thus we consider the streamline diffusion error :
$$
err_{SD}=\|\mathbf{w}_{\Gamma}\cdot\nabla_{\Gamma} (u-u_h)\|_{L^2(D)}.
$$
Results for this error quantity and for $err_{L^{inf}}=\|u-u_h\|_{L^{\infty}(D)}$ are shown in Figure~\ref{figg:err_ex2}.
We observe a $\mathcal{O}(h)$ behavior for the streamline diffusion error, which is consistent with
our theoretical analysis. The $L^{\infty}$-norm
of the error also shows a first order of convergence.
\mathbf{e}gin{figure}[ht!]
\mathbf{e}gin{center}
\includegraphics[width=\textwidth]{Ex2_errs2.eps}
\caption{Discretization errors for Example~\ref{example2}}\lbl{figg:err_ex2}
\end{center}
\end{figure}
\end{example}
\mathbf{e}gin{example} \label{example3} \rm
We show how this stabilization can be applied to a time dependent problem and illustrate its stabilizing effect. We consider a non-stationary problem \eqref{e:2.2} posed on the torus
\mathbf{e}gin{equation}\label{e:torus}
\Gamma=\{(x_1,x_2,x_3)\ |\ (\sqrt{x_1^2+x_2^2}-1)^2+x_3^2=\frac{1}{16}\}.
\end{equation}
We set $\varepsilon=10^{-6}$ and define the advection field
$$
\mathbf{w}(\mathbf{x})=\frac{1}{\sqrt{x_1^2+x_2^2}}(-x_2,x_1, 0)^T,
$$
which is divergence free and satisfies $\mathbf{w} \cdot \mathbf{n}_\Gamma =0$.
The initial condition is
$$
u_0(\mathbf{x}) = \frac{x_1x_2}{\pi}\mathrm{arctan}\left(\frac{x_3}{\sqrt{\varepsilon}}\right).
$$
The function $u_0$ possesses an internal layer, as shown in Figure~\ref{fig:rotate}(a).
The stabilized spatial semi-discretization of \eqref{e:2.2} reads: determine $u_h=u_h(t) \in V_h^\Gamma$ such that
\mathbf{e}gin{equation} \label{spatttie}
m(\partialrtial_t u_h, v_h) + \hat a_h(u_h,v_h)=0 \quad \text{for all}~~v_h \in V_h^\Gamma.
\end{equation}
with
\mathbf{e}gin{equation*}
\mathbf{e}gin{aligned}
m(\partialrtial_t u,v):=&\int_{\Gamma_h}\partialrtial_t u v \, \mathrm{d}\mathbf{s}+\sum_{T\in\mathcal{F}_h}\delta_{T}\int_{T} \partialrtial_t u (\mathbf{w}^e\cdot\nabla_{\Gamma_h}v)\, \mathrm{d}\mathbf{s},\\
\hat a_h(u,v):=& \varepsilon\int_{\Gamma_h}\nabla_{\Gamma_h} u\cdot\nabla_{\Gamma_h} v \, \mathrm{d}\mathbf{s} + \frac12\left[\int_{\Gamma_h}(\mathbf{w}^e\cdot\nabla_{\Gamma_h} u) v \, \mathrm{d}\mathbf{s} -\int_{\Gamma_h}(\mathbf{w}^e\cdot\nabla_{\Gamma_h} v) u \, \mathrm{d}\mathbf{s} \right]\\
&+\sum_{T\in\mathcal{F}_h}\delta_T\int_{T}(-\varepsilon\Delta_{\Gamma_h}u + \mathbf{w}^e\cdot\nabla_{\Gamma_h} u)\mathbf{w}^e\cdot\nabla_{\Gamma_h} v \, \mathrm{d}\mathbf{s}.
\end{aligned}
\end{equation*}
Note that $ \hat a_h(\cdot,\cdot)$ is the same as $a_h(\cdot,\cdot) $ in \eqref{eqah} with $c \equiv 0$.
The resulting system of ordinary differential equations is discretized in time by the Crank-Nicolson scheme.
For $\varepsilon=0$ the exact solution is the transport of $u_0(\mathbf{x})$
by a rotation around the $x_3$ axis. Thus the inner layer remains the same for all $t>0$. For $\varepsilon=10^{-6}$, the exact solution is
similar, unless $t$ is large enough for dissipation to play a noticeable role.
The space $V_h^\Gamma$ is constructed in the same way as in the previous examples. The spatial discretization has 5638 degrees of freedom.
The fully discrete problem is obtained by combining the SUPG method in \eqref{spatttie} and the Crank-Nicolson method with time step $\delta t=0.1$.
The evolution of the solution is illustrated in Figure~\ref{fig:rotate} demonstrating a smoothly `rotated' pattern.
\mathbf{e}gin{figure}[ht!]
\centering
\subfigure[]{
\includegraphics[width=2.1in]{solution000.eps}}
\subfigure[]{
\includegraphics[width=2.1in]{solution006.eps}}
\subfigure[]{
\includegraphics[width=2.1in]{solution012.eps}}
\subfigure[]{
\includegraphics[width=2.1in]{solution018.eps}}
\caption{Example~\ref{example3}: solutions for $t=0,0.6,1.2,1.8$ using the SUPG stabilized FEM.}\lbl{fig:rotate}
\end{figure}
We repeated this experiment with $\delta_T=0$ in the bilinear forms $m(\cdot,\cdot)$ and $\hat a_h(\cdot, \cdot)$ in \eqref{spatttie}, i.e. the method without stabilization.
As expected, we obtain (on the same grid) much less smooth discrete solutions (Figure~\ref{fig:rotate1}).
\mathbf{e}gin{figure}[ht!]
\centering
\subfigure[]{
\includegraphics[width=2.1in]{sfe_solution012.eps}}
\subfigure[]{
\includegraphics[width=2.1in]{sfe_solution018.eps}}
\caption{Example~\ref{example3}: solutions for $t=1.2,1.8$ using the standard FEM.}\lbl{fig:rotate1}
\end{figure}
\mathbf{e}gin{remark} \rm
With respect to mass conservation of the scheme we note the following. For $v_h\equiv 1$ in \eqref{spatttie} we get, with $M_h(t):=\int_{\Gamma_h} u_h(x,t) \, \mathrm{d}\mathbf{s}$,
\mathbf{e}gin{equation*}
\mathbf{e}gin{aligned}
|\frac{d}{dt} M_h(t)|&= |\int_{\Gamma_h} \frac12\mathbf{w}^e\cdot\nabla_{\Gamma_h} u_h\, \mathrm{d}\mathbf{s} |\\
& =|-\frac12\sum_{E\in\mathcal{E}_h}\int_{E}\mathbf{w}^e\cdot[\mathbf{m}] u_h\, \mathrm{d}\mathbf{s}+\frac12\int_{\Gamma_h} \mathrm{div}_{\Gamma}h\mathbf{w}^e u_h\, \mathrm{d}\mathbf{s}|
\\
&\lesssim h^2\sum_{E\in\mathcal{E}_h}\int_{E}|u_h|\, \mathrm{d}\mathbf{s}+Ch\int_{\Gamma_h}|u_h|\, \mathrm{d}\mathbf{s}.
\end{aligned}
\end{equation*}
Here we used estimates from Lemmas 3.1 and 3.5. Using Lemma 3.6 we get
\mathbf{e}gin{align*} \sum_{E\in\mathcal{E}_h}\int_{E}|u_h| \, \mathrm{d}\mathbf{s} & \lesssim
h^{- \frac12} \Big( \sum_{E\in\mathcal{E}_h}\int_{E} u_h^2\mathrm{d}\mathbf{s} \Big)^\frac12 \\ & \lesssim h^{-1}\big( \|u_h\|_{L^2(\Gamma_h)} + h \|\nabla_{\Gamma_h} u_h \|_{L^2(\Gamma_h)}).
\end{align*}
Assume that for the discrete solution we have a bound $\|u_h\|_{L^2(\Gamma_h)} + h \|\nabla_{\Gamma_h} u_h \|_{L^2(\Gamma_h)} \leq c$ with $c$ independent of $h$. Then we obtain $|\frac{d}{dt} M(t)| \lesssim h$ and thus
$|M_h(t)-M_h(0)| \leq c t h$, with a constant $c$ independent of $h$ and $t$.
Hence, with respect to mass conservation we
have an error that is (only) first order in $h$. Concerning mass conservation it would be better to use a discretization in which
in the discrete bilinear form in \eqref{eqah} one replaces
\mathbf{e}gin{equation} \label{convform}
\frac12\left[\int_{\Gamma_h}(\mathbf{w}^e\cdot\nabla_{\Gamma}h u) v \, \mathrm{d}\mathbf{s} -\int_{\Gamma_h}(\mathbf{w}^e\cdot\nabla_{\Gamma}h v) u\, \mathrm{d}\mathbf{s} \right]\quad\text{by}\quad -\int_{\Gamma_h}(\mathbf{w}^e\cdot\nabla_{\Gamma}h v) u\, \mathrm{d}\mathbf{s}.
\end{equation}
This method results in optimal mass conservation: $\frac{d}{dt} M_h(t)=0$.
It turns out, however, that (with our approach) it is more difficult to analyze. In particular, it is not clear how to derive a satisfactory coercivity bound.
In numerical experiments we observed that the behavior of the two methods (i.e, with the two variants given in \eqref{convform}) is very similar. In particular, the mass conservation error bound $|M(t)-M(0)| \leq c t h$ for the first method seems to be too pessimistic (in many cases). To illustrate this, we show results for the problem described above, but with initial condition
$$
u_0(\mathbf{x}) = 1+ \frac{1}{\pi}\mathrm{arctan}\left(\frac{x_3}{\sqrt{\varepsilon}}\right),\quad
\int_{\Gamma} u_0 \, \mathrm{d}\mathbf{s}=\pi^2\approx 9.8696.
$$
Figure~\ref{fig:mass} shows the quantity $M_h(t)$ for several mesh sizes $h$. For $t=0$ we have, due to interpolation of the initial condition $u_0$, a difference between $M_h(0)$ and $\int_{\Gamma} u_0 \mathrm{d}\mathbf{s}$ that is of order $h^2$. For $t >0$ we see, except for the very coarse mesh with $h=1/4$ a very good mass conservation.
\end{remark}
\mathbf{e}gin{figure}[ht!]
\mathbf{e}gin{center}
\includegraphics[width=0.7\textwidth]{mass.eps}
\caption{Total mass variation for Example~\ref{example3}}\lbl{fig:mass}
\end{center}
\end{figure}
\end{example}
\mathbf{e}gin{example}
\label{example4} \rm As a final illustration we show results for the non-stationary problem \eqref{e:2.2}, but now on a surface with a ``less regular'' shape. We take the surface given in \cite{Dziuk88}:
\mathbf{e}gin{equation}\label{e:Dziuk}
\Gamma=\{(x_1,x_2,x_3)\ |\ (x_1-x_3^2)^2+x_2^2+x_3^2=1\}.
\end{equation}
We set $\varepsilon=10^{-6}$ and define the advection field as the $\Gamma$-tangential part of
$
\tilde{\mathbf{w}}=(-1,0,0)^T.
$
This velocity field does not satisfy the divergence free condition.
The initial condition is taken as
$
u_0(\mathbf{x}) = 1.
$
We apply the same method as in Example~\ref{example3}. The mesh size is $1/8$ and the time step is $\delta t=0.1$.
Figure~\ref{fig:dziuk} shows the solution for several $t$ values. We observe that as time evolves mass is transported from the two poles on the right to the left pole,
just as expected. Our discretization yields for this strongly convection-dominated transport problem a qualitatively good discrete result, even with a (very) low grid resolution.
\mathbf{e}gin{figure}[ht!]
\centering
\subfigure[]{
\includegraphics[width=2.1in]{DziukShape.0000.eps}}
\subfigure[]{
\includegraphics[width=2.1in]{DziukShape.0005.eps}}
\subfigure[]{
\includegraphics[width=2.1in]{DziukShape.0010.eps}}
\subfigure[]{
\includegraphics[width=2.1in]{DziukShape.0020.eps}}
\caption{Example~\ref{example4}: solutions for $t=0,0.5,1.0,2.0$ using the SUPG stabilized FEM.}\lbl{fig:dziuk}
\end{figure}
\end{example}
\section*{Acknowledgments}
This work has been supported in part by the DFG through grant RE1461/4-1, by the Russian Foundation for Basic Research through grants 12-01-91330, 12-01-00283, and by NSFC project 11001260.
We thank J. Grande for his support with the implementation of the methods and C. Lehrenfeld for useful discussions.
\end{document}
|
\begin{document}
\title{Entanglement-assisted classical information capacity of the amplitude
damping channel}
\author{Xian-Ting Liang\thanks{
Email address: [email protected]} \\
Department of Physics and Institute of Modern Physics,\\
Ningbo University, Ningbo, Zhejiang 315211, China}
\maketitle
\begin{abstract}
In this paper, we calculate the entanglement-assisted classical information
capacity of amplitude damping channel and compare it with the particular
mutual information which is considered as the entanglement-assisted
classical information capacity of this channel in Ref. 6. It is shown that
the difference between them is very small. In addition, we point out that
using partial symmetry and concavity of mutual information derived from
dense coding scheme one can simplify the calculation of
entanglement-assisted classical information capacities for
non-unitary-covariant quantum noisy channels.
PACS number(s): 03.67. Hk, 03.65.Ta, 89.70.+c
Keywords: Entanglement; information capacity; amplitude damping channel
\end{abstract}
Entanglement-assisted classical information capacity of the quantum channel
describes the maximal rate, i.e. information sent per channel usage, when we
use dense coding scheme instead of simple encoding and decoding to transmit
the data through the channel $\varepsilon $. In the scheme, the sender, say
Alice, and receiver, say Bob, share a two-qubit entangled state prior to the
transmission. At first, Alice encodes information to be transmitted in an
entangled state by operating her holding qubit, and then she sends the qubit
through a quantum channel to Bob, finally Bob jointly measure two qubits
(one is sent from Alice and the other is held by him at the beginning of
this scheme) to decode the information. If no noise to be considered the
scheme called dense coding, and can transmit two bit classical information
by sending one qubit. However, in fact quantum noise is always exist. When
we consider the effect of noise, is this scheme still superior to the
traditional simple encoding and decoding scheme? If yes, then how superior
is the scheme to the traditional one? Up to now, these problems could only
be concretely answered by calculating the entanglement-assisted classical
information capacities for some concrete quantum noisy channels. So
developing the method of calculating the capacity is an interesting topic.
This problem was first investigated by Bennett, Shor, Smolin, and Thapliyal
(BSST) in \cite{Bennett et al01}, where the depolarizing and erasure
channels in $d$ dimensions were studied exactly. In Ref. \cite{Bennett et
al02} the same authors proposed a remarkable simple formula for calculating
the entanglement-assisted classical information capacity in terms of the
maximal mutual information between Alice and Bob, and the capacity of the
amplitude damping channel was also investigated. In this paper, we shall at
first recalculate the entanglement-assisted classical information capacity
of amplitude damping channel. Then we shall compare it with the particular
mutual information which is taken as the entanglement-assisted classical
information capacity of the amplitude damping channel in Ref. 6. In
addition, we shall summarize the method for calculating the
entanglement-assisted classical information capacity of this kind of
non-unitary-covariant channels. Let's review BSST theorem \cite{Bennett et
al02} \cite{Holevo01} first.
In the BSST theorem, the entanglement-assisted classical information
capacity $C\left( \varepsilon \right) $ of a quantum noisy channel $
\varepsilon :$ $B\left( \mathcal{H}\right) \rightarrow B\left( \mathcal{H}
\right) $ is given by
\begin{equation}
C\left( \varepsilon \right) =\sup_{\rho }I\left( \varepsilon ,\rho \right) ,
\label{e1}
\end{equation}
where
\begin{equation}
I\left( \varepsilon ,\rho \right) =S\left( \rho \right) +S\left( \varepsilon
\left( \rho \right) \right) -S\left( \varepsilon ,\rho \right) . \label{e2}
\end{equation}
Here, $S\left( \tau \right) $ denotes von Neumann entropy, $S\left(
\varepsilon ,\tau \right) $ denotes entropy exchange. Nielsen \emph{et al}.
in \cite{NandCbook} proposed a method for calculating the entropy exchange,
namely,
\begin{equation}
S\left( \varepsilon ,\rho \right) =S\left( \Omega \right) =-tr\left( \Omega
\log \left( \Omega \right) \right) , \label{e3}
\end{equation}
where $\Omega _{ij}=tr\left( E_{i}\rho E_{j}^{\dagger }\right) $, and $E_{i}$
denote Kraus operators of the channel $\varepsilon $. The proof of this
theorem was first given by \cite{Bennett et al02} and then improved by
Holevo in \cite{Holevo01}. However, the calculation of the
entanglement-assisted classical information capacity may still be a
difficult problem for some quantum noisy channels because in order to
maximize the mutual information $I\left( \varepsilon ,\rho \right) $, we
must choose the state $\rho $\ in Eq.(\ref{e1}) over all of the possible
states. Fortunately, $I\left( \varepsilon ,\rho \right) $ is a concave
function so if only we can prove the in question channel being a unitary
covariant channel the calculation become easy \cite{Liang01}. However, some
quantum noisy channels are not unitary covariant, so we cannot calculate
their capacities by simply replace $\rho $ with the maximally mixed state $
\mathbf{1}/d$ in Eq.(\ref{e2}), where $\mathbf{1}$ is the unitary matrix, $d$
is the dimension of the channel$.$ However, for the non-unitary covariant
channel the concavity and the partial symmetry of $I\left( \varepsilon ,\rho
\right) $ can still be used in the calculation.
In the quantum communication, the following problems are always encountered.
What are the dynamics of an atom which is spontaneously emitting a photon?
How does a spin system at high temperature approach equilibrium with its
environment? What is the state of a photon in an interferometer or cavity
when it is subject to scattering and attenuation? Each of these processes
has its own unique features, but the general behavior of all of them is well
characterized by a quantum operation known as amplitude damping. For the
qubit systems, the evolvement of the amplitude damping can be modeled by
amplitude damping channel. The amplitude damping channel is a non-unitary
covariant channel. In the following, we shall use the concavity and the
partial symmetry of $I\left( \varepsilon ,\rho \right) $ investigate its
entanglement-assisted classical information capacity. The Kraus operators of
amplitude damping channel are
\begin{equation}
E_{0}=\left(
\begin{array}{cc}
1 & 0 \\
0 & \sqrt{1-\eta }
\end{array}
\right) ,\qquad E_{1}=\left(
\begin{array}{cc}
0 & \sqrt{\eta } \\
0 & 0
\end{array}
\right) . \label{e4}
\end{equation}
Suppose the initial state $\rho \in \mathcal{H}=C^{2}$ is $\rho =\frac{1}{2}
\left( I+\vec{w}\cdot \vec{\sigma}\right) $, its eigenvalues are
\begin{equation}
\lambda _{1,2}=\frac{1}{2}\left( 1\pm \sqrt{w_{1}^{2}+w_{2}^{2}+w_{3}^{2}}
\right) . \label{e5}
\end{equation}
When the initial state pass through the amplitude damping channel, it will
become
\begin{equation}
\rho ^{\prime }=\varepsilon \left( \rho \right) =\frac{1}{2}\left(
\begin{array}{cc}
1+w_{3}+\eta \left( 1-w_{3}\right) & \sqrt{1-\eta }\left(
w_{1}-iw_{2}\right) \\
\sqrt{1-\eta }\left( w_{1}+iw_{2}\right) & \left( 1-\eta \right) \left(
1-w_{3}\right)
\end{array}
\right) . \label{e6}
\end{equation}
The eigenvalues of $\rho ^{\prime }$ are
\begin{equation}
\lambda _{1,2}^{\prime }=\frac{1}{2}\pm \sqrt{\left( 1-\eta \right)
^{2}w_{3}^{2}+\left( 1-\eta \right) \left( 2\eta
w_{3}+w_{1}^{2}+w_{2}^{2}\right) +\eta ^{2}}. \label{e7}
\end{equation}
By using the Kraus operators of amplitude damping channel and formula $
\Omega _{ij}=tr\left( E_{i}\rho E_{j}^{\dagger }\right) ,$ we obtain
\begin{equation}
\Omega =\frac{1}{2}\left(
\begin{array}{cc}
2-\eta +\eta w_{3} & \sqrt{\eta }\left( w_{1}-iw_{2}\right) \\
\sqrt{\eta }\left( w_{1}-iw_{2}\right) & \eta \left( 1-w_{3}\right)
\end{array}
\right) . \label{e8}
\end{equation}
The eigenvalues of $\Omega $ are
\begin{equation}
\chi _{1,2}=\frac{1}{2}\pm \sqrt{\left( 1-\eta \right) ^{2}+2\eta \left(
1-\eta \right) w_{3}+\eta w_{1}^{2}+\eta w_{2}^{2}+\eta ^{2}w_{3}^{2}}.
\label{e9}
\end{equation}
From Eq.(\ref{e2}) we can obtain the mutual information as
\begin{eqnarray}
I\left( \varepsilon ,\rho \right) &=&I\left( \eta ,\vec{w}\right) \notag
\\
&=&-\lambda _{1}\log _{2}\lambda _{1}-\lambda _{2}\log _{2}\lambda
_{2}-\lambda _{1}^{\prime }\log _{2}\lambda _{1}^{\prime } \notag \\
&&-\lambda _{2}^{\prime }\log _{2}\lambda _{2}^{\prime }+\chi _{1}\log
_{2}\chi _{1}+\chi _{2}\log _{2}\chi _{2}. \label{e10}
\end{eqnarray}
On one hand, from above results we have $I\left( \eta ,w_{1}\right) =I\left(
\eta ,-w_{1}\right) ,$ and $I\left( \eta ,w_{2}\right) =I\left( \eta
,-w_{2}\right) $, so we see $I\left( \eta ,\vec{w}\right) $ being
symmetrical on points $p\left( w_{1},w_{2}=0,w_{3}\right) $. On the other
hand, it was proven that $I\left( \eta ,\vec{w}\right) $ is a concave
function \cite{Keyl}, so the maximum of $I\left( \eta ,\vec{w}\right) $ must
be restricted on the points $p\left( w_{1},w_{2}=0,w_{3}\right) ;$ the
maximum of $I\left( \eta ,w\right) $ must be included in $I^{\prime }\left(
\varepsilon ,\vec{w}\right) :=I\left\vert _{w_{1},w_{2}=0}\right. \left(
\eta ,w_{3}\right) ,$ namely, $C\left( \eta \right) \subset I^{\prime
}\left( \eta ,\vec{w}\right) =I\left\vert _{w_{1},w_{2}=0}\right. \left(
\eta ,w_{3}\right) $. Further, we can calculate the capacities by taking a
series of $w_{3}^{\prime }$ in different $\eta .$ These $w_{3}^{\prime }$ in
different $\eta $ can be obtained as follows: first we take $I^{\prime
}\left( \eta ,\vec{w}\right) $ derivative with respect to $w_{3}$ as
\begin{equation}
\frac{dI^{\prime }\left( \eta ,\vec{w}\right) }{dw_{3}}=0, \label{e11}
\end{equation}
then we solve $w_{3}$ from Eq.(\ref{e11}) we can obtain a series of $w_{3}$,
which are $w_{3}^{\prime }.$ The numerical result of $w_{3}^{\prime }$ is
shown in Table 1 and their values as a function of $\eta $ are plotted in
Figure 1. The capacities $C\left( \eta \right) $, mutual information $
I\left( \eta ,w=0\right) $ and the difference of $C\left( \eta \right) $ and
$I\left( \eta ,w=0\right) $ are also given in the Table 1. We plot the $
I(\eta ,w=0)$ and $C\left( \eta \right) $ against $\eta $ in Fig. 2. It is
shown that the difference between $C\left( \eta \right) $ and $I(\eta ,w=0)$
is very small and we cannot distinguish them in the figure. In order to
compare them we plot the difference $C\left( \eta \right) -I(\eta ,w=0)$ in
Fig. 3.
\begin{eqnarray*}
&& \\
&& \\
&&Fig.1 \\
&& \\
&&
\end{eqnarray*}
\emph{Fig.1 }$w_{3}^{\prime }$\emph{\ versus }$\eta $ \emph{for the
amplitude damping channel, where }$w_{3}^{\prime }$\emph{\ make the mutual
information }$I(\eta ,w_{3})$\emph{\ be capacities }$C\left( \eta \right) $
\emph{.}
\begin{eqnarray*}
&& \\
&& \\
&&Fig.2 \\
&& \\
&&
\end{eqnarray*}
\emph{Fig.2 Capacity }$C\left( \eta \right) $\emph{\ and mutual information }
$I(\eta ,w=0)$\emph{\ versus }$\eta $\emph{\ for the amplitude damping
channel}$.$\emph{\ Their difference being very small; we cannot distinguish
them using this figure.}
\begin{eqnarray*}
&& \\
&& \\
&&Fig.3 \\
&& \\
&&
\end{eqnarray*}
\emph{Fig.3 Difference of capacity }$C\left( \eta \right) $\emph{\ and
mutual information }$I(\eta ,w=0)$\emph{\ versus }$\eta $\emph{\ for
amplitude damping channel.}
\begin{align*}
&
\begin{tabular}{|c|c|c|c|c|}
\hline
$\eta $ & $w_{3}^{\prime }$ & $C(\eta ,w_{3}^{\prime })$ & $I(\eta ,w_{3}=0)$
& $C-I$ \\ \hline
0.04 & .020707505 & 1.857993856 & 1.857404993 & .588863e-3 \\ \hline
0.08 & .029451443 & 1.754220384 & 1.753086250 & .1134134e-2 \\ \hline
0.12 & .034204349 & 1.663598636 & 1.662142602 & .1456034e-2 \\ \hline
0.16 & .036476402 & 1.580849799 & 1.579274705 & .1575094e-2 \\ \hline
0.20 & .036918238 & 1.503488311 & 1.501955000 & .1533311e-2 \\ \hline
0.24 & .035871433 & 1.430055143 & 1.428681156 & .1373987e-2 \\ \hline
0.28 & .033529523 & 1.359582064 & 1.358444378 & .1137686e-2 \\ \hline
0.32 & .030001598 & 1.291370839 & 1.290509150 & .861689e-3 \\ \hline
0.36 & .025341559 & 1.224884751 & 1.224304412 & .580339e-3 \\ \hline
0.40 & .019562439 & 1.159688417 & 1.159362804 & .325613e-3 \\ \hline
0.44 & .012643065 & 1.095410976 & 1.095283308 & .127668e-3 \\ \hline
0.48 & .004530009 & 1.031721423 & 1.031706094 & .15329e-4 \\ \hline
0.52 & -.0048640556 & .9683103674 & .9682939063 & .164611e-4 \\ \hline
0.56 & -.0156655130 & .9048748897 & .9047166920 & .1581977e-3 \\ \hline
0.60 & -.0280492412 & .8411041849 & .8406371958 & .4669891e-3 \\ \hline
0.64 & -.0422541602 & .7766639116 & .7756955885 & .9683231e-3 \\ \hline
0.68 & -.0586084818 & .7111767546 & .7094908497 & .16859049e-2 \\ \hline
0.72 & -.0775716652 & .6441954457 & .6415556220 & .26398237e-2 \\ \hline
0.76 & -.0998074512 & .5751615422 & .5713188441 & .38426981e-2 \\ \hline
0.80 & -.1263199222 & .5033365085 & .4980450000 & .52915085e-2 \\ \hline
0.84 & -.1587322020 & .4276745835 & .4207252951 & .69492884e-2 \\ \hline
0.98 & -.1999403638 & .3465572468 & .3378573979 & .86998489e-2 \\ \hline
0.92 & -.2560072406 & .2571288324 & .2469137502 & .102150822e-1 \\ \hline
0.96 & -.3442467036 & .1530143199 & .1425950071 & .104193128e-1 \\ \hline
\end{tabular}
\\
& \text{Table 1 The }w_{3}^{\prime }\text{, capacities }C\left( \eta \right)
\text{, mutual information }I\left( \eta ,w=0\right) \text{ and } \\
& \text{the difference of }C\left( \eta \right) \text{ and }I\left( \eta
,w=0\right) \text{ against parameter }\eta \text{ for amplitude } \\
& \text{damping channel.}
\end{align*}
In conclusion, on the one hand, by using the concavity and the partial
symmetry of $I\left( \rho ,\varepsilon \right) $ we investigate the
entanglement-assisted classical information capacity of the amplitude
damping channel. It is shown that the capacities $C\left( \eta \right) $ are
always a little bigger than the mutual information $I(\eta ,w=0),$ which
were taken as the entanglement-assisted classical information capacities of
the amplitude damping channel in Ref. \cite{Liang01}. From the results we
see the difference between $C\left( \eta \right) $ and $I(\eta ,w=0)$,
namely, $C\left( \eta \right) -I(\eta ,w=0),$ is very small for all of the
parameters $\eta .$ Hence, it is convenient and accurate to replace $C\left(
\eta \right) $ with $I(\eta ,w=0).$ On the other hand, we obtained some
insight into the calculation of entanglement-assisted classical information
capacity for non-unitary-covariant channels. We find that the concavity and
some symmetry of $I\left( \rho ,\varepsilon \right) $ for
non-unitary-covariant channels can help one simplify the calculations. In
particular, a unitary covariant channel corresponds to a entirety
symmetrical channel whose entanglement-assisted classical information
capacity can be calculated by simply replacing $\rho $ with the maximally
mixed state $\mathbf{1}/d$ in Eq.(\ref{e2}).
\begin{acknowledgement}
This work was supported by Youth Fund of Ningbo City of China.
\end{acknowledgement}
\end{acknowledgement}
\end{document}
|
\begin{document}
\title{Some Sufficient Conditions for Finding a Nesting of the Normalized Matching Posets of Rank 3}
\author{
Yu-Lun Chang
\thanks{Department of Applied Mathematics, National Chung Hsing University, Taichung 40227, Taiwan
{\tt Email:[email protected]} supported by MOST-105-2115-M-005-003-MY2.}
\and
Wei-Tian Li
\thanks{Department of Applied Mathematics, National Chung Hsing University, Taichung 40227, Taiwan
{\tt Email:[email protected]} supported by MOST-105-2115-M-005-003-MY2.}
}
\date{\small \today}
\maketitle
\begin{abstract}
Given a graded poset $P$, consider a chain decomposition $\mathcal{C}$ of $P$. If $|C_1|\le |C_2|$ implies that the set of the ranks of elements in $C_1$ is
a subset of the ranks of elements in $C_2$ for any chains $C_1,C_2\in \mathcal{C}$,
then we say $\mathcal{C}$ is a nested chain decomposition (or nesting, for short) of $P$,
and $P$ is said to be nested.
In 1970s, Griggs conjectured that every normalized matching rank-unimodal poset is nested.
This conjecture is proved to be true only for all posets of rank 2~\cite{W:05}, some posets of rank 3~\cite{HLS:09,ENSST:11},
and the very special cases for higher ranks.
For general cases, it is still widely open.
In this paper, we provide some sufficient conditions on the rank numbers of posets of rank 3 to satisfies the Griggs's conjecuture.
\end{abstract}
\section{Introduction}
We start with the necessary terminology of poset theory.
A {\em poset} $P=(P,\le)$ is a set $P$ equipped with a partially order relation $\le$.
Through out the paper, all posets are finite.
Let $P$ be a poset. We say a subposet $C$ of $P$ is a {\em chain} of length $\ell$
if $C=\{x_i\mid x_i< x_{i+1}\mbox{ for }0\le i \le \ell-1\}$.
A {\em chain decomposition} $\mathcal{C}$ of $P$ is a collection of disjoint chains of $P$ with $\cup_{C\in \mathcal{C}}C=P$.
We are looking for decompositions with as few number of chains as possible.
The most significant theorem in the literature was given by Dilworth~\cite{RPD:1950}. Here an {\em antichain} is a subposet
of $P$ such that neither $x\le y $ nor $y\le x$ holds for any $x\neq y$ in $Q$.
\begin{theorem}{\rm \cite{RPD:1950}}
For a poset $P$, the minimum number of chains in a chain decomposition is equal to the maximum size of an antichain of $P$.
\end{theorem}
In the following, we study the chain decompositions of a special class of posets, which includes example such as Boolean lattices, linear lattices, and divisor lattices, etc. A {\em graded poset} is a poset such that every maximal chain has the same length.
For a graded poset $P$, we define the rank function $r:P\longrightarrow \mathbb{N}$ such that
$r(x)=i$, if there exactly $i$ elements $y< x$ in a maximal chain.
Moreover, an element $x$ is of rank $i$ if $r(x)=i$ and the rank of $P$ is $\max_{x\in P}r(x)$.
The {\em $i$th level} of a graded poset $P$ is the collection of all elements of rank $i$, that is, $L_i=\{x\mid r(x)=i, x\in P\}$.
By the definitions, every level is an antichain.
Therefore, $\max_{i}|L_i|\le |\mathcal{C}|$ for every chain decomposition $\mathcal{C}$ of $P$.
For graded posets, we define a special chain decomposition:
\begin{definition}[nested chain decomposition]
Let $\mathcal{C}$ be a chain decomposition of a graded poset $P$.
For any chains $C_i,C_j\in \mathcal{C}$, if $|C_i|\le |C_j|$ implies
$\{r(x)\mid x\in C_i\}\subseteq \{r(x)\mid x\in C_j\}$, then $\mathcal{C}$ is called a
nested chain decomposition of $P$.
We say $P$ is nested, or it has a nesting, if such a decomposition of $P$ exists.
\end{definition}
Observe that a nesting $\mathcal{C}$ is a chain decomposition with minimum number of chains.
Because from the inclusion relation, $\cap_{C\in \mathcal{C}}\{r(x)\mid x\in C\}$ is not empty,
and there exists some $m$ and a level $L_m$
such that $m\in\cap_{C\in\mathcal{C}}\{r(x)\mid x\in C\}$.
Thus, every chain in $\mathcal{C}$ contains an element of rank $m$, and hence $|L_m|=|\mathcal{C}|$.
Since $\max_i|L_i|\le |\mathcal{C}|=|L_m|\le \max_i|L_i|$, we have $\max_i|L_i|\le |\mathcal{C}|$.
We refer the reader to see more properties of graded posets in~\cite{A:02,ENG:97}.
Anderson\cite{A:1967b} and Griggs\cite{JRG:1977} independently gave the same sufficient condition for the existence of a nesting in the graded posets.
Let $r_i$ denote the cardinality of level $i$. The {\em rank numbers} or {\em Whitney numbers} of a graded poset of rank $n$ is the
sequence $(r_0,r_1,\ldots, r_n)$. If $r_i=r_{n-i}$ for all $0\le i\le n$, then $P$ is said to be {\em rank-symmetric}.
Suppose there exists some $i$ such that $r_0\le\cdots \le r_i\ge \cdots \ge r_n$, then $P$ is {\em rank-unimodal}.
For any levels $L_i$ and $L_j$ of $P$, consider any subset $S$ of $L_i$ and denote the set
$\Gamma_j(S)=\{x\in L_j\mid x \le y\mbox { or }y\le x \mbox{ for some }y\in S\}$.
If the inequality
\[\frac{|S|}{r_i} \le \frac{|\Gamma_j(S)|}{r_j}\]
holds, then we say $P$ has {\em the normalized matching property} from $i$ to $j$.
By simple calculation, one can see if $P$ has the normalized matching property from $i$ to $j$,
then it also has the property from $j$ to $i$.
Moreover, if $P$ has the normalized matching property from $i$ to $j$ and $j$ to $k$ for $i<j<k$, then
it has the property from $i$ to $k$.
Once the normalized matching property holds between any two levels, then we say $P$ is a {\em normalized matching poset}.
\begin{theorem}{\rm \cite{A:1967b,JRG:1977}}\label{thm:scd}
A graded poset $P$ has a nesting if it is a normalized matching poset, and is rank-symmetric and rank-unimodal.
\end{theorem}
In fact, every chain in a nesting of a poset $P$ described in the theorem
contains elements of ranks $i,i+1,\ldots, n-i$ for some $i$, where $n$ is the rank of $P$.
Such a decomposition is also called a {\em symmetric chain decomposition} of $P$.
In addition to Theorem~\ref{thm:scd}, Griggs also posed the following conjecture~\cite{JRG:1977,JRG:1988,JRG:1995}:
\begin{conjecture}[\rm Griggs Nesting Conjecture]\label{gnc:un}
Every normalized matching rank-unimodal poset is nested.
\end{conjecture}
The conjecture turns out to be extremely difficult, although one can easily give an affirmative answer of graded posets of rank 1 using
the well-known Hall's Marriage Theorem~\cite{PH:1935}.
There are only a few graded posets of small ranks which are proven to satisfy the conjecture by Wang~\cite{W:05},
Hsu, Logan, and Shahriari.~\cite{HLS:09}, and
Escamilla, Nicolae, Salerno, Shahriari, and Tirrell~\cite{ENSST:11}, respectively.
In section 2, we introduce the early results on the graded posets of ranks 2 and 3,
and mention our theorems at the end of the section.
The main contribution of this paper is to give more sufficient conditions on the graded posets of rank 3 to satisfy the conjecture,
based on the ideas of proofs in the early papers.
The proofs of our theorems are presented in Section 3.
\section{Normalized Matching Posets of Rank 2 and 3}
Note that in Conjecture~$\ref{gnc:un}$, we only concern the normalized matching property and the conditions of the rank numbers.
The structure of the poset is irrelevant.
For convenience, we use the notation $NM(r_0,r_1,\ldots,r_n)$ to denote the collection
of all normalized matching posets of rank $n$ with rank numbers $(r_0,r_1,\ldots,r_n)$.
In 2005, Wang~\cite{W:05} dropped the rank-unimodal assumption and proved a stronger result.
\begin{theorem}{\rm \cite{W:05}}\label{thm:05}
Every poset $P\in NM(r_0,r_1,r_2)$ has a nesting.
\end{theorem}
For graded posets of rank 3, Shahriari with two research groups\cite{HLS:09,ENSST:11} developed some sufficient conditions on
the rank numbers $(r_0,r_1,r_2,r_3)$ to guarantee the existence of a nesting.
In~\cite{HLS:09}, the authors came up with a clever idea which can not only simplify the proof of Theorem~\ref{thm:05} but also reduce
the rank numbers to fewer cases that need to be considered for graded posets of rank 3.
Since we will use this idea in our proof, we introduce it below.
\begin{proposition}
Given a graded poset $P$, let $P'=P\cup \{x\}$ be obtained by adding a new element $x$ to the $i$th level of $P$ together
with the partial order relations $y< x$ (resp. $x<y$) if $y\in L_j$ and $j<i$ (resp. $j<i$).
If $P\in NM(r_0,r_1,\ldots, r_n)$, then $P'\in NM(r_0,\ldots,r_{i-1},r_{i}+1,r_{i+1},\ldots, r_n)$.
\end{proposition}
The proof of the proposition is straightforward, since if we pick a set $S$ in the $i$th level of $P'$,
either it contains $x$, then $|\Gamma_k(S)|/r_k=1$, or it does not contain $S$, then $|S|/(r_i+1)< |S|/r_i< |\Gamma_k(S)|/r_k$,
for all $k$. In~\cite{HLS:09}, such an element is called a {\em ghost}.
We demonstrate two instances of exploiting the ghosts to get a nesting.
Suppose $P$ is a poset in $NM(r_0,r_1,r_2)$ with $r_0<r_2<r_1$.
Then we add $r_2-r_0$ ghosts to the $0$th level to get a rank-symmetric poset $P'$.
By Theorem~\ref{thm:scd}, $P'$ has a nesting and each chain contains elements of ranks either $\{0,1,2\}$ or $\{2\}$.
After removing the ghosts from the chains of length 2,
we obtain a chain decomposition of $P$ such that each chain contains elements of ranks either $\{0,1,2\}$, or $\{1,2\}$, or $\{2\}$.
If the rank numbers satisfy $r_1<r_0<r_2$, then we add $r_0-r_1$ ghosts to the first level.
The new poset $P''$ restricted on the first and second levels is a poset of rank 1.
So we can partition it into chains of length either 0 or 1.
Meanwhile, the poset consisting of the 0th and first level of $P''$ can be partitioned
into chains of length of 1.
The chains of length 1 in two decompositions can be concatenated into chains of length 2.
Finally, we remove the ghosts to get of decomposition of $P$ with each chain containing elements of
rank either $\{0,1,2\}$, or $\{0,2\}$, or $\{2\}$.
Indeed, the arguments above are exact the ideas of Hsu et al. in~\cite{HLS:09},
used to reprove Theorem~\ref{thm:05} .
Using the ideas of the ghost elements, the induction, and the duality,
Hsu et al.~\cite{HLS:09} showed that to prove Conjecture~$\ref{gnc:un}$ for posets of rank 3, it suffices to verify that all posets $P\in (r_0,r_1,r_2,r_3)$ with $r_2>r_1>r_0=r_3$ are nested.
For example, if a poset $P\in (r_0,r_1,r_2,r_3)$ with $r_2>r_1>r_0>r_3$,
then we add $r_0-r_3$ ghosts to the third level of $P$ to get a new poset $P'\in (r_0,r_1,r_2,r_0)$.
Now suppose we already have a nesting of $P'$.
We then remove the ghosts in all the longest chains to get a nesting of $P$.
See~\cite{HLS:09} for the details of all the reduction methods.
With this assumption on the rank numbers, Hsu et al.~\cite{HLS:09}
and Escamilla et al.~\cite{ENSST:11} proved the following Theorem~\ref{thm:09} and Theorem~\ref{thm:11}, respectively.
\begin{theorem}{\rm \cite{HLS:09}}\label{thm:09}
Let $r_0, r_1$ and $r_2$ be positive integers with $r_0 <r_1 <r_2$. Assume that at least one of the following conditions are satisfied:
\begin{description}
\item{(a)} $r_{1} \ge r_{2}-\lceil{\frac{r_{2}}{r_{0}}}\rceil+1$;
\item{(b)} $r_{1}$ $=r_{0}+1$;
\item{(c)} $r_{2}> r_{0}r_{1}$;
\item{(d)} $r_{1}$ divides $r_{2}$.
\end{description}
Then every $P\in NM(r_0,r_1,r_2,r_0)$ is nested.
\end{theorem}
\begin{theorem}{\rm \cite{ENSST:11}}\label{thm:11}
Let $r_0, r_1$ and $r_2$ be positive integers with $r_0 <r_1 <r_2$. Assume that at least one of the following conditions are satisfied:
\begin{description}
\item{(a)} $r_{0}$ divides $r_{1}$, or
\item{(b)} $r_{0}+1$ divides $r_{1}$, or
\item{(c)} $f(i)\geq 0$ for all $1\leq i\leq r_{1}-r_{0}-1$, where the function $f$ is defined by
\[
f(i) = \left\lceil\frac{r_{0}(1+i)}{r_{2}-r_{0}}\right\rceil -
\left\lfloor\frac{r_{0}i}{r_{1}-r_{0}}\right\rfloor\mbox{; or}\]
\item{(d)} $r_{2}> r_{0}r_{1}-r_{0}\gcd(r_{1},r_{2})$.
\end{description}
Then every $P\in NM(r_0,r_1,r_2,r_0)$ is nested.
\end{theorem}
In~\cite{ENSST:11}, the authors also examined the posets of rank 3 with $r_2\le 13$.
Using Theorem~\ref{thm:11}, one can verify that if $r_0<r_1<r_2\le 13$, every poset
$P\in NM(r_0,r_1,r_2,r_0)$ satisfies Conjecture~\ref{gnc:un} except that
$(r_0,r_1,r_2,r_0)$ is equal to one of the six cases:
$(6,8,12,6)$, $(6,9,12,6)$, $(4,6,13,4)$, $(5,8,13,5)$, $(6,8,13,6)$, $(6,9,13,6)$.
We close Section 2 by stating our results.
For graded posets of rank 3,
we provide two more sufficient conditions on the rank numbers for the existence of a nesting:
\begin{theorem}\label{thm1}
Let $P\in NM(r_0,r_1,r_2,r_0)$.
If both $r_{1}$ and $r_{0}$ divide $r_{2}-1$, then $P$ is nested.
\end{theorem}
\begin{theorem}\label{thm2}
Let $P\in NM(r_0,r_1,r_2,r_0)$.
If $kr_{0}\le r_{1}\le k(r_{0}+1)$, then $P$ is nested.
\end{theorem}
Observe that by Theorem~\ref{thm1}, we see that every $P\in NM(4,6,13,4)$ has a nesting.
Unfortunately, other unsolved cases with $r_2\le 13$ mentioned in\cite{HLS:09} cannot be settled by our theorems.
Nevertheless, for $14\le r_2\le 15$, we can use Theorem~\ref{thm2} to show every $P\in NM(r_0,r_1,r_2,r_3)$ with
$(r_0,r_1,r_2,r_3)\in\{(4,9,14,4)$, $(3,7,15,3)$,$(4,9,15,4)$,$(5,11,15,5)\}$ is nested.
These are not covered by Theorem~\ref{thm:09} and~\ref{thm:11}.
\section{Proofs of the Main Theorems}
In this section, we give the proofs of Theorem~\ref{thm1} and Theorem~\ref{thm2} .
Let us begin with the proof of Theorem~\ref{thm1}.
\begin{proof1}
Pick a poset $P\in NM(r_0,r_1,r_2,r_0)$, where
$k_0r_0+1=r_2$ and $k_1r_1+1=r_2$ for some integers $k_1$ and $k_2$.
Let $P'$ be a poset obtained by removing an arbitrary element $x$ from $L_2$ .
To show that $P'$ is a normalized matching poset,
we only need to verify the inequality holds between $L_1$ and $L_2\setminus\{x\}$ as well as $L_2\setminus\{x\}$ and $L_3$.
First consider $L_1$ and $L_2\setminus\{x\}$.
By the symmetry, we only need to verify the normalized matching property from $L_1$ to $L_2\setminus \{x\}$.
For any $S\subseteq L_1$, since $P$ is a normalized matching poset, we have
\[
\frac{|S|}{r_1}\le \frac{|\Gamma_2(S)|}{r_2}=\frac{|\Gamma_2(S)|}{k_1r_1+1}.
\]
Equivalently,
\[{k_{1}|S|}+\frac{|S|}{r_1}\le |\Gamma_2(S)|.\]
When $S\neq\emptyset$, we have $k_1|S|+1\le |\Gamma_2(S)|$ since $|\Gamma_2(S)|$ is an integer.
Now, for $P'$, if the removed element $x$ is not in $\Gamma_2(S)$,
then
\[
\frac{|S|}{r_{1}}\le \frac{|\Gamma_2(S)|}{k_1r_1+1}\le \frac{|\Gamma_2(S)|}{k_1r_1}=\frac{|\Gamma_2(S)|}{r_{2}-1}=\frac{|\Gamma_2(S)|}{|L_2\setminus \{x\}|}.
\]
Otherwise, $x\in \Gamma_2(S)$ and then $S\neq\emptyset$.
We have
\[
\frac{|S|}{r_{1}}=\frac{k_{1}|S|}{k_{1}r_{1}}\le \frac{|\Gamma_2(S)|-1}{r_2-1}=\frac{|\Gamma_2(S)|-1}{|L_2\setminus \{x\}|}.
\]
The numerator $|\Gamma_2(S)|-1$ is just the number of elements $y\in L_2\setminus\{x\}$ satisfying $z<y$ for some $z\in S$.
So the normalized matching property holds between $L_1$ and $L_2\setminus\{x\}$.
Using a similar argument we can see that the normalized matching property also holds between $L_3$ and $L_2\setminus\{x\}$.
Now that $r_1$ divides $k_1r_1=r_2-1$,
so $P'$ has a nesting $\mathcal{C}$ by Theorem~\ref{thm:09} (d).
Finally, we view $\{x\}$ as a one-element chain and add it to $\mathcal{C}$ to get a nesting of $P$.
\end{proof1}
It is worth mentioning that the proof in Theorem~\ref{thm1} is similar to the next lemma in~\cite{HLS:09},
which is used to prove Theorem~\ref{thm:09} (b).
\begin{lemma}{\rm \cite{HLS:09}}\label{avoid}
Let $P\in NM(r_0,r_0+1)$. For any $x$ of rank 1 in $P$, there exists a chain partition of $P$ which consists of
$r_0$ chains of length 1 and another chain $\{x\}$ of length 0.
\end{lemma}
Before presenting the proof of Theorem~\ref{thm2}, we need more preparations.
In addition to adding the ghosts to a normalized matching poset,
there are some techniques to produce new normalized matching posets from the old ones.
We introduce two construction approaches.
\begin{definition}[$k$-clone]
Let $P$ be a graded poset and $L_i$ be a level of $P$. Then $L$ is said to be a $k$-clone of $L_i$ if $L=L_i\times\{1,\ldots,k\}$ and the partial
order relations of each $(y,i)$ and others elements in $L_{i-1}$ (resp. $L_{i+1}$) is $x<(y,i)$ (resp. $(y,i)<z$) if and only
if there exist some $x\in L_{i-1}$ and $y\in L_i$ (resp. $y\in L_j$ and $z\in L_{i+1}$). See Figure~\ref{fig1} as an illustration.
\end{definition}
\begin{figure}
\caption{A new poset obtained by replacing a 2-clone of the first level of $P$ to it.}
\label{fig1}
\end{figure}
\begin{definition}[$m$-bunch]
Let $P$ be a graded poset and $L_i$ be a level of $P$. Suppose $|L_i|=m\ell$ for some integers $m$ and $\ell$.
First partition $L_1$ into $m$ arbitrary subsets $A_{1}$,\ldots,$A_{m}$ of equal size $\ell$.
Then $L$ is an $m$-bunch of $L_i$ if $L=\{A_{1}$,\ldots,$A_{m}\}$ and the partial
order relations of each $A_j$ and others elements in $L_{i-1}$ (resp. $L_{i+1}$) is $x<A_j$ (resp. $A_j<z$)
if and only if there exist $x\in L_{i-1}$ and $y\in A_j$ (resp. $y\in A_j$ and $z\in L_{i+1}$).
See Figure~\ref{fig2} as an illustration.
\end{definition}
\begin{figure}
\caption{A new poset obtained by replacing a 2-bunch of the first level of $P$ to it.}
\label{fig2}
\end{figure}
The above operations on posets preserve the normalized matching property:
\begin{proposition}\label{prop:CB}
If $P$ is a normalized matching poset,
then the new poset obtained by replacing a $k$-clone or an $m$-bunch of some level
of $P$ to it is still a normalized matching poset.
\end{proposition}
The proof of this proposition was given by Hsu et al.~\cite{HLS:09} (clone),
and by Escamilla et al.~\cite{ENSST:11} (bunch), respectively.
Now we prove our second theorem.
\begin{proof2}
Consider $P\in NM(r_0,r_1,r_2,r_0)$ with $kr_0\le r_1\le k(r_0+1)$ for some integer $k$.
Note that the two ends of the inequality are in the statements of Theorem~\ref{thm:11} (a) and (b).
Thus, we may suppose $r_{1}=kr_{0}+t$ for some $1\le t\le k-1$.
Pick a poset $P\in NM(r_0,r_1,r_2,r_0)$.
We use the induction method to find the nestings of subposets induced by different levels of $P$.
Our goal is to combine the nestings properly to get a nesting of $P$.
First construct a poset $P_1$ of rank 2 induced by the top three levels of $P$
with a replacement of a $k$-clone of the highest level.
By Proposition~\ref{prop:CB}, $P_1\in NM(r_{1},r_{2},kr_{0})$,
and there exists a nesting $\mathcal{C}_1$ of $P_1$ by Theorem~\ref{thm:05}.
Observe that there are $kr_0$ chains of length 2 in $\mathcal{C}_1$ such that eahc of them contains an
element $(y,j)$ in the highest level of $P_1$.
Clearly, the bottom two levels $L_0$ and $L_1$ of $P$ induce a subposet of rank 1 and has a nesting.
However, we do not want a nesting of the above poset containing a chain of length 1 whose top element
is the bottom element of a chain of length 1 in $\mathcal{C}_1$.
This could lead to two chains of length 2 but the ranks of elements in one chain is $\{0,1,2\}$ and the other is $\{1,2,3\}$
when we combine the two nestings together.
To avoid this, we construct a poset $P_2$ of rank 1 as follows.
At the beginning, we add $k-t$ additional ghosts to $L_1$ of $P$ in advance.
Now this level contains $k(r_0+1)$ elements, and we will partition them into $r_0+1$ sets of size $k$.
Because $L_1$ is also the bottom level of $P_1$, for each $y_i\in L_3$ of $P$, there exist exactly $k$ elements in $L_1$ such that each of them lies in a chain,
containing $(y_i,j)$ for some $1\le j\le k$, of length 2 in $\mathcal{C}$.
In addition, there are $t=r_1-kr_0$ elements in $L_1$ which are not in any chain of length 2 in $\mathcal{C}$.
For $1\le i\le r_{0}$, let $A_{i}$ be the set consisting of every element in $L_1$, which lies in a chain in $\mathcal{C}_1$ containing the element $(y_i, j)$ for some $1\le j\le k$ .
Moreover, let $A_{r_{0}+1}$ be the set consisting of the $t$ remaining elements in $L_1$ and the $k-t$ ghosts.
We bunch all elements in $L_1$ and the ghosts into the above $r_0+1$ sets $A_1,\ldots, A_{r_0+1}$.
The poset induced by these $A_i$s and $L_0$ is $P_2$.
By Lemma $\ref{avoid}$, there is a chain partition $\mathcal{C}_2$ of $P_2$
with $r_0$ chains of length 1 and one chain of length 0
such that each chain of length 1 does not contain $A_{r_0+1}$ and each $x_i\in L_0$ is in a chain of length 1 in $\mathcal{C}_2$.
Assume the chains are $\{x_i,A_i\}$ for $1\le i\le r_0$.
It follows that for each $i$ there exists some element in $z\in A_i$ with $x_i<z$.
Fix some $i$. For those $k$ chains of length 2 in $\mathcal{C}_1$ containing $(y_i,j)$ for some $1\le j\le k$,
we extend one of them to length 3 by adding the element $x_i$
and delete the top elements of the remaining $k-1$ chains of length 2.
Repeating the operations for all $1\le i\le r_0$ gives us a nesting of $P$.
\end{proof2}
\end{document}
|
\begin{document}
\title{Field extensions and Galois descent\ for sheaves of vector spaces}
\begin{abstract}
We study extension of scalars for sheaves of vector spaces, assembling results that follow from well-known statements about vector spaces, but also developing some complements. In particular, we formulate Galois descent in this context, and we also discuss the case of derived categories and establish Galois descent for perverse sheaves. On the way, we prove interesting compatibilities between the six Grothendieck operations and extension of scalars, carefully distinguishing the cases of finite and infinite extensions, and we make use of some gluing techniques for $\mathbb{R}$-constructible and perverse sheaves.
\end{abstract}
\tableofcontents
\section{Introduction}
Given a field extension $L/K$, it is easy to produce an $L$-vector space $V$ out of a $K$-vector space $W$ by ``extending scalars'' (or ``change of rings''), i.e.\ setting $V\vcentcolon= L\otimes_K W$. The case of vector spaces is, of course, the most basic example, but similar constructions are performed in more advanced contexts: For instance, one can ``upgrade'' a scheme over $K$ to a scheme over $L$. While it is certainly interesting to study such an extension functor itself, one might also wonder if it is possible to describe more precisely its essential image and a possible way of reconstructing an object over $K$ from its associated object over $L$ and some extra data. The last question has nice answers mainly in the case where $L/K$ is a Galois extension, and the machinery behind its solution is commonly referred to as \emph{Galois descent}. In the case of vector spaces (where Galois descent is actually a special case of faithfully flat descent for modules), such questions have, for example, been studied in classical literature such as \cite{Winter}, \cite{Jacobson} \cite{Waterhouse}, \cite{Borel}. We also like to mention the nice surveys \cite{Conrad} and \cite{Jahnel}.
If $k$ is a field, $k$-vector spaces are nothing but sheaves (modules) over the constant sheaf $k$ on the one-point space, which is the same as the structure sheaf if we consider the one-point space as $\mathrm{Spec}\, k$. They therefore have two natural generalizations: One can think about $\mathcal{O}_X$-modules on more general varieties or schemes $X$ over $k$, or one can think about modules over the constant sheaf $k_X$ on more general topological spaces $X$. New questions arise in these contexts, since one can, for instance, ask the question of compatibility of extension of scalars with operations on sheaves (such as direct and inverse images, duality, etc.). The first viewpoint seems to be widely established (see e.g.\ \cite{Jahnel} for an overview and \cite[Tag 0CDQ]{Stacks}).
In this work, we are going to take the second viewpoint and study extension and descent questions in the case of sheaves of vector spaces and related categories. Most of the basic results from the theory of extension of scalars and Galois descent for vector spaces imply (more or less directly) similar results for sheaves of vector spaces, and indeed our main reference will be classical sheaf theory (see, for example, \cite{KS90}). However, the technical subtleties of certain statements are not always obvious, and it is difficult to keep the overview of the conditions that are required for every single statement: Some assertions hold for arbitrary field extensions and sheaves, others require finite or Galois extensions or certain constructibility assumptions on the sheaves. Although the main results in this direction are probably known to experts or follow from more general frameworks, a unique reference for the details in the concrete setting of sheaves of vector spaces does not seem to exist. These concepts play, however, a crucial role in theories like that of mixed Hodge modules, where extension of scalars is used for perverse sheaves.
The aim of this work is therefore twofold: Firstly, we like to collect and present the main definitions and statements about extension of scalars and Galois descent for sheaves of vector spaces, together with the arguments needed to deduce them from the well-known statements in linear algebra. Great parts are therefore meant to be rather expository and we do not claim originality for all of these statements, but we hope that this overview will serve as a useful reference.
Secondly, in the course of this detailed presentation, we establish some complementary technical results: We describe compatibilities between the six Grothendieck operations (and in particular the functor $\mathbb{R}Hom$) and extension of scalars in the (derived) category of sheaves of vector spaces. Moreover, we investigate Galois descent for complexes in the derived category of sheaves of vector spaces as well as for perverse sheaves. All these considerations involve in particular some interesting arguments using results on the structure of $\mathbb{R}$-constructible and perverse sheaves. Even for readers already familiar with the basic ideas, these statements may be an enlightening illustration of gluing techniques for constructible and perverse sheaves.
We originally got interested in this subject during the preparation of our joint article with Davide Barco, Marco Hien and Christian Sevenheck \cite{BHHS22}, where we studied certain differential equations from a topological viewpoint. The basic idea is the following: A Riemann--Hilbert correspondence is an equivalence between certain categories of differential systems and categories of topological objects (such as local systems and perverse sheaves, for example). These topological objects are a priori defined over the field of complex numbers, so one can ask under which conditions such an object ``descends'' to one over a subfield of $\mathbb{C}$, and it turned out that Galois descent serves as a useful technique there. Although this was our original motivation for studying Galois descent for sheaves of vector spaces, we will not make any reference to the concrete application in loc.~cit.\ here, nor to the more general framework of enhanced ind-sheaves we were studying in there. We rather consider this an independent, self-contained and accessible exposition of the subject, providing more details for a broader readership familiar with Galois and sheaf theory.
In our common work, we established some Galois descent results for sheaves (and more general objects), some of which we will reformulate and complement in this work. We will also investigate the case of complexes of sheaves.
We will mostly distinguish between results for finite Galois extensions (where we get the best results) and results for arbitrary (in particular infinite) field extensions. Let us note that the classical theory of Galois descent for vector spaces can also be adapted to the case of infinite Galois extensions, where again one gets better results than in the case of arbitrary infinite extensions. We will not discuss this case explicitly in this article, although it is certainly interesting to study this also in the context of sheaves.
\paragraph{Outline} After reviewing some basics of sheaf theory (mainly to set notation and terminology) in Section~\ref{sec:Sheaves}, we set up the concepts of $G$-structures, extension of scalars, $K$-structures and Galois descent in a quite abstract categorical framework in Section~\ref{sec:Cat}. On the one hand, this allows us to define these notions for multiple categories at the same time, on the other hand, it might also serve as a useful framework for studying them in different examples later. After each definition, we directly give the explicit description in our categories of interest, and the main immediate properties in the case of sheaves of vector spaces. We remark that several compatibilities between extension of scalars and the six Grothendieck operations remain open there (and are, in general, not true), cf.\ Lemma~\ref{lemma:compatExt}. We address them in Section~\ref{sec:Hom}, where we mainly study the compatibility of extension of scalars with the functor $\mathbb{R}Hom$. On the way, we will also discover compatibilities with the remaining functors. We distinguish the cases of finite and infinite field extensions since the constructions involved have different flavours. In the latter case, we will in particular work with $\mathbb{R}$-constructible sheaves and perform some interesting constructions using simplicial complexes.
In Section~\ref{sec:GaloisDescent}, we first describe explicitly Galois descent for sheaves and formulate it as an equivalence of categories, using in particular the results of the previous section to obtain full faithfulness. We then briefly discuss descent in derived categories of sheaves of vector spaces. We do not obtain an equivalence as in the abelian case, but we will give some background and explanations on the problems that arise. Finally, the last subsection is devoted to the study of Galois descent for perverse sheaves. We use a construction of A.\ Beilinson \cite{Bei} -- which we will briefly recall and study in the context of extension of scalars -- to realize perverse sheaves by ``gluing data'' and inductively reduce to the case of sheaves proved before.
\paragraph{Acknowledgements} I would like to thank Claude Sabbah for invaluable discussions, which in particular helped me to work out Galois descent for perverse sheaves using Beilinson's construction. I also thank Davide Barco, Marco Hien and Christian Sevenheck who, through our common work, inspired me to work out more details on this subject. Finally, I am particularly grateful to Takuro Mochizuki for answering my questions and providing some ideas during the preparation of \cite{BHHS22}, which helped me to better understand the theory. In particular, the proofs of Lemmas~\ref{lemma:RHomSimplices} and \ref{lemma:HomRcCpt} go back to his suggestions.
\section{A very short review of sheaf theory}\label{sec:Sheaves}
We assume the readers to be familiar with the theory of sheaves of vector spaces. We will recall here some basic facts and notations, and refer to the standard literature such as \cite{KS90} or \cite{Dimca} for details.
Although not strictly necessary in all places, we will assume all our topological spaces to be \emph{good}, i.e.\ Hausdorff, locally compact, second countable and of finite cohomological dimension. (This is particularly important for the construction of the functors $f_!$ and $f^!$.)
\paragraph{Presheaves and sheaves} Let $X$ be a topological space and let $\mathrm{Op}(X)$ be the category of open subsets of $X$, where $\mathrm{Hom}(U,V)$ has one element if $U\subseteq V$ and is empty otherwise. Let $k$ be a field.
A presheaf of $k$-vector spaces on $X$ is a functor $\mathrm{Op}(X)^\mathrm{op}\to \Vect{k}$. It is a sheaf if for every open $U\subseteq X$ qnd every open covering $U=\bigcup_{i\in I} U_i$ the natural sequence
$$0\to \mathcal{F}(U)\to \prod_{i\in I} \mathcal{F}(U_i) \rightrightarrows \prod_{i,j\in I} \mathcal{F}(U_i\cap U_j)$$
is exact. If $\mathcal{P}$ is a presheaf, its sheafification will be denoted by $\mathcal{P}^\#$. The sheafification functor is left adjoint to the natural inclusion of sheaves into presheaves.
We denote the category of sheaves of $k$-vector spaces on $X$ by $\Mod{k_X}$, and its bounded derived category by $\mathbb{D}bk{X}$. There are the six Grothendieck operations $\mathbb{R}R f_*$, $\mathbb{R}R f_!$, $f^{-1}$, $f^!$, $\otimes$ and $\mathbb{R}Hom$ (and the underived versions of all these functors except $f^!$). We will sometimes write $\otimes_{k_X}$ or $\mathbb{R}Hom_{k_X}$ if we want to emphasize the field, but it will usually be clear from the context.
\paragraph{(Locally) constant sheaves} If $V\in\Vect{k}$, we denote by $V_X\in\Mod{k_X}$ the constant sheaf with stalk $V$ on $X$. It is the sheafification of the constant presheaf $V^\mathrm{pre}_X$ defined by $V^\mathrm{pre}_X(U)=V$ for all $U\in \mathrm{Op}(X)$. If $p_X\colon X\to\{\mathrm{pt}\}$ is the map to the one-point space, we also have $V_X=p_X^{-1}V$ (note that sheaves of vector spaces on a one-point space are the same as vector spaces).
In particular, we have the constant sheaf $k_X$, and if $f\colon X\to Y$ is a morphism of topological spaces, then $f^{-1}k_Y\simeq k_X$.
If $\mathcal{F}\in\Mod{k_X}$ and $Z\subseteq X$ is a locally closed subset with inclusion $j\colon Z\hookrightarrow X$, we write $\mathcal{F}_Z\vcentcolon= j_!j^{-1}\mathcal{F}$. This is the restriction of $\mathcal{F}$ to $Z$, extended again to $X$ by zero outside of $Z$. We will also sometimes denote by $k_Z\in\Mod{k_X}$ the sheaf $j_!k_Z$ if there is no risk of confusion.
We moreover denote the duality functor by $\mathbb{D}D_X\vcentcolon= \mathbb{R}Hom(-,\omega_X)$, where $\omega_X=p_X^! k$ is the dualizing complex.
We call a sheaf $\mathcal{F}\in\Mod{k_X}$ \emph{locally constant} (or a \emph{local system}) if every point $x\in X$ has an open neighbourhood $U\subseteq X$ such that $\mathcal{F}|_U$ is isomorphic to a constant sheaf $V_X$ for some $V\in\Vect{k}$. It is \emph{locally constant of finite rank} if all the $V$ are finite-dimensional.
\paragraph{Constructibility and perversity} If $X$ is a real analytic manifold, $\mathcal{F}\in\Mod{k_X}$ is called \emph{$\mathbb{R}$-constructible} if there exists a locally finite covering $X=\bigcup_{\alpha\in A} X_\alpha$ by subanalytic subsets such that $\mathcal{F}|_{X_\alpha}$ is locally constant of finite rank for every $\alpha$. We denote by $\mathbb{D}bRck{X}$ the full subcategory of $\mathbb{D}bk{X}$ of complexes with $\mathbb{R}$-constructible cohomologies. We also note that this category is in fact equivalent to the bounded derived category of the category of $\mathbb{R}$-constructible sheaves.
If $X$ is a complex manifold, $\mathcal{F}\in\Mod{k_X}$ is called \emph{$\mathbb{C}$-constructible} if there exists a locally finite covering $X=\bigcup_{\alpha\in A} X_\alpha$ by $\mathbb{C}$-analytic subsets such that $\mathcal{F}|_{X_\alpha}$ is locally constant of finite rank for every $\alpha$. We denote by $\mathbb{D}bCck{X}$ the full subcategory of $\mathbb{D}bk{X}$ of complexes with $\mathbb{C}$-constructible cohomologies.
An object $\mathcal{F}^\bullet\in\mathbb{D}bCck{X}$ is called a \emph{perverse sheaf} if $\dim \mathop{\mathrm{supp}} \mathrm{H}^{-i}(\mathcal{F}^\bullet)\leq i$ for any $i\in \mathbb{Z}$ (we say that $\mathcal{F}^\bullet$ satisfies the \emph{support condition}) and $\dim \mathop{\mathrm{supp}} \mathrm{H}^{-i}(\mathbb{D}D_X\mathcal{F}^\bullet)\leq i$ for any $i\in \mathbb{Z}$ (i.e.\ $\mathbb{D}D_X\mathcal{F}^\bullet$ also satisfies the support condition). We denote by $\Pervk{X}$ the full subcategory of $\mathbb{D}bCck{X}$ consisting of perverse sheaves. We refer in particular to \cite{BBD} for the theory of perverse sheaves. The category $\Pervk{X}$ is an abelian category. Let us note that, in particular, a perverse sheaf has no nontrivial cohomologies in degrees less than $-\dim_\mathbb{C} X$. (This follows easily from \cite[p.\ 56]{BBD}, for example.)
\paragraph{A remark on operations and sheafification} Before entering the main topic of the article, let us remark the following elementary fact that we are going to use throughout the article:
If $\mathcal{P}$ is a presheaf and $\mathcal{F}$ is a sheaf, then the tensor product sheaf $\mathcal{P}^\#\otimes \mathcal{F}$ is isomorphic to the sheafification of the (naïve) presheaf tensor product $\mathcal{P}\overset{\mathrm{pre}}{\otimes}\mathcal{F}$. This is easy to show using tensor-hom adjunctions for presheaves and sheaves and the universal property of sheafification. It is, however, crucial that $\mathcal{F}$ is already a sheaf here. In particular, this means that if $L/K$ is a field extension (i.e.\ $L$ is a $K$-module) and $\mathcal{F}\in\Mod{K_X}$, then the sheaf $L_X\otimes\mathcal{F}$ is the sheafification of the presheaf $U\mapsto L\otimes_K \mathcal{F}(U)$.
Let us also note that sheafification does not commute with operations on sheaves in general: For example, it is not true in general that $f_*(\mathcal{P}^\#)\simeq (f_*^\mathrm{pre}\mathcal{P})^\#$ for a presheaf $\mathcal{P}$. Indeed, if this were true, we would not have examples as in Remark~\ref{rem:directImage}.
\section{Linear categories and field extensions}\label{sec:Cat}
The general philosophy of Galois descent is the following: Given a Galois extension $L/K$ with Galois group $G$ and an object $F$ over $L$, then the existence of a $G$-structure (i.e.\ a suitable collection of isomorphisms between $F$ and its Galois conjugates, often formulated as a suitable action of the Galois group on $F$) should guarantee the existence of a $K$-structure of $F$, i.e.\ an object over $K$ which is isomorphic to $F$ after extension of scalars. (Even more, a $G$-structure should in some sense determine a particular $K$-structure, since in the case where ``object'' means ``finite-dimensional vector space'', the pure existence of such a structure is not big news.)
We first set up a very general framework for the concepts of $G$- and $K$-structures, motivated by the notions set up in \cite{BHHS22}, but immediately describe these notions explicitly in our categories of interest. Recall that, given a field $k$, a $k$-linear category is a category whose hom spaces are $k$-vector spaces and composition of morphisms is $k$-linear. Note that if $L/K$ is a field extension, any $L$-linear category is automatically also $K$-linear. We will assume any functor between two linear categories to be linear (meaning that the induced map on hom spaces is linear).
\subsection{$G$-structures}
Let $L/K$ be a field extension and denote by $G=\mathrm{Aut}(L/K)$ the group of field automorphisms $g\colon L\to L$ such that $g|_K=\mathrm{id}_K$.
\begin{defi}\label{def:Gconj}
Let $\mathcal{C}(L)$ be an $L$-linear category. A \emph{$G$-conjugation} on $\mathcal{C}(L)$ is a collection of auto-equivalences
$$\gamma_g=\overline{(\bullet)}^g\colon \mathcal{C}(L) \overset{\sim}{\longrightarrow} \mathcal{C}(L)$$
(one for each $g\in G$) and natural isomorphisms $$I_{g,h}\colon \gamma_h\circ \gamma_g \overset{\sim}{\longrightarrow} \gamma_{gh}$$ for any $g,h\in G$ such that for any $g,h,k\in G$ the following diagram is commutative:
$$\begin{tikzcd}
& \gamma_{hk}\circ\gamma_g\arrow{dr}{I_{g,hk}} \\
\gamma_k\circ \gamma_h\circ\gamma_g\arrow{ur}{I_{h,k}\circ \gamma_g} \arrow{dr}[swap]{\gamma_k\circ I_{g,h}} & & \gamma_{ghk}\\
&\gamma_k\circ\gamma_{gh}\arrow{ur}[swap]{I_{gh,k}}
\end{tikzcd}$$
\end{defi}
Note that this implies in particular that $\gamma_{\mathrm{id}_L}\simeq \mathrm{id}_{\mathcal{C}(L)}$, where $\mathrm{id}_L\in G$ is the identity element of the $G$. In the categories we use, it will actually be equal to the identity functor. More precisely, all the $I_{g,h}$ will be equalities rather than isomorphisms and hence the $\gamma_g$ are actually automorphisms of $\mathcal{C}(L)$.
\begin{ex}\label{ex:GConj}
Here are the categories we are interested in in this article: The classical example is that of vector spaces, and it easily generalizes to presheaves and sheaves.
\begin{itemize}
\item[(a)] Let $\Vect{L}$ be the category of $L$-vector spaces. Then, for an object $V\in\Vect{L}$ and an element $g\in G$, the $L$-vector space $\overline{V}^g$ is defined as follows: As $K$-vector spaces (or sets), we set $\overline{V}^g=V$, and the action of $L$ on $\overline{V}^g$ is given by
$$\ell \cdot v \vcentcolon = g(\ell)v$$
for $\ell\in L$ and $v\in \overline{V}^g$, where the right-hand side is the given scalar multiplication on $V$.
Given a morphism $V\to W$ in $\Vect{L}$, then for any $g\in G$ the same set-theoretic map defines an $L$-linear morphism $\overline{f}^g\colon \overline{V}^g\to\overline{W}^g$.
Altogether, this gives a functor
$$\overline{(\bullet)}^g\colon \Vect{L}\to\Vect{L},$$
and it is easy to see that the functor $\overline{(\bullet)}^{g^{-1}}$ is a quasi-inverse, hence the above functor is indeed an auto-equivalence.
Moreover, given $g,h\in G$, the identification $\overline{\overline{V}^g}^h = \overline{V}^{gh}$ is immediate by the definition, as is compatibility of these identifications (that is, commutativity of the diagram in the above definition).
\item[(b)] Let $\mathcal{C}$ be a category and consider the category $\mathrm{Funct}(\mathcal{C},\Vect{L})$ of (covariant) functors from $\mathcal{C}$ to $\Vect{L}$. Then there is a $G$-conjugation on this category given as follows: Let $F\in\mathrm{Funct}(\mathcal{C},\Vect{L})$ and $g\in G$, then we define $\overline{F}^g$ by setting
$$\overline{F}^g(A)\vcentcolon= \overline{F(A)}^g$$
for any $A\in \mathcal{C}$. A morphism $A\to B$ in $\mathcal{C}$ is sent to the morphism $F(A)\to F(B)$, considered as a morphism $\overline{F(A)}^g\to\overline{F(B)}^g$, as remarked in (a). Thus, this clearly defines an element $\overline{F}^g\in\mathrm{Funct}(\mathcal{C},\Vect{L})$.
Given a morphism $F\to G$ in $\mathrm{Funct}(\mathcal{C},\Vect{L})$, it is equally easy to see that this induces a morphism $\overline{F}^g\to\overline{G}^g$ for any $g\in G$, and hence we obtain an auto-equivalence $\overline{(\bullet)}^g$ of $\mathrm{Funct}(\mathcal{C},\Vect{L})$ with the desired compatibilities.
\item[(c)] Consider the category $\Mod{L_X}$ of sheaves of $L$-vector spaces on a topological space $X$. It is a subcategory of $\mathrm{Funct}(\mathrm{Op}(X)^\mathrm{op},\Vect{L})$, where $\mathrm{Op}(X)$ is the category of open subsets of $X$ (with inclusions as morphisms). Hence, by (b), to a sheaf $\mathcal{F}\in\Mod{L_X}$ we can a priori associate a presheaf $\overline{\mathcal{F}}^g\in \mathrm{Funct}(\mathrm{Op}(X)^\mathrm{op},\Vect{L})$ for any $g\in G$. It is, however, clear that this presheaf is automatically a sheaf: The unique gluing condition required for sheaves is indeed independent of the action of $L$, it can be checked on the level of sheaves of $K$-vector spaces (or even sets), and on this level $\overline{\mathcal{F}}^g$ and $\mathcal{F}$ are the same object. We therefore get an auto-equivalence
$$\overline{(\bullet)}^g\colon \Mod{L_X}\overset{\sim}{\longrightarrow}\Mod{L_X}$$
satisfying the required compatibilities. Moreover, since this equivalence is exact, it also induces a functor
$$\overline{(\bullet)}^g\colon \mathbb{D}bL{X}\overset{\sim}{\longrightarrow} \mathbb{D}bL{X}$$
on the level of derived categories, equipping the latter with a $G$-conjugation.
\end{itemize}
\end{ex}
In the sequel, when we work in one of these categories, we will always use the $G$-conjugations described in Example~\ref{ex:GConj}.
Since we are mainly concerned with sheaves in this note, let us state the main properties of $G$-conjugation for sheaves of vector spaces.
\begin{lemma}[{cf.\ \cite[Lemma 2.1]{BHHS22}}]\label{lemma:compatConj}
Let $L/K$ be a field extension and $G\vcentcolon=\mathrm{Aut}(L/K)$. Let $g\in G$. Let $f\colon X\to Y$ be a continuous map between topological spaces.
\begin{itemize}
\item[(a)] Let $\mathcal{F}\in\mathbb{D}bL{X}$. Then $\overline{\mathbb{R}R f_* \mathcal{F}}^g\simeq \mathbb{R}R f_*\overline{\mathcal{F}}^g$ and $\overline{\mathbb{R}R f_! \mathcal{F}}^g\simeq \mathbb{R}R f_!\overline{\mathcal{F}}^g$.
\item[(b)] Let $\mathcal{G}\in\mathbb{D}bL{Y}$. Then $\overline{f^{-1} \mathcal{G}}^g\simeq f^{-1}\overline{\mathcal{G}}^g$ and $\overline{f^! \mathcal{G}}^g\simeq f^!\overline{\mathcal{G}}^g$.
\item[(c)] Let $\mathcal{F}_1,\mathcal{F}_2\in\mathbb{D}bL{X}$. Then $\overline{\mathbb{R}Hom(\mathcal{F}_1,\mathcal{F}_2)}^g\simeq \mathbb{R}Hom(\overline{\mathcal{F}_1}^g,\overline{\mathcal{F}_2}^g)$ and $\overline{\mathcal{F}_1\otimes\mathcal{F}_2}^g\simeq \overline{\mathcal{F}_1}^g\otimes \overline{\mathcal{F}_2}^g$.
\item[(c)] Let $\mathcal{F}\in\mathbb{D}bL{X}$. Then $\overline{\mathbb{D}D_X\mathcal{F}}^g\simeq \mathbb{D}D_X\overline{\mathcal{F}}^g$.
\item[(e)] If $\mathcal{F}\in\mathbb{D}bL{X}$ and $\mathrm{H}^i(\mathcal{F})$ is locally constant on $Z\subseteq X$, so is $\mathrm{H}^i(\overline{\mathcal{F}}^g)$.
In particular, if $X$ is a real analytic manifold and $\mathcal{F}\in\mathbb{D}bRcL{X}$ (resp.\ $X$ is a complex manifold and $\mathcal{F}\in\mathbb{D}bCcL{X}$), then $\overline{\mathcal{F}}^g\in\mathbb{D}bRcL{X}$ (resp.\ $\overline{\mathcal{F}}^g\in\mathbb{D}bCcL{X}$).
\item[(f)] If $X$ is a complex manifold and $\mathcal{F}\in\PervL{X}$, then $\overline{\mathcal{F}}^g\in\PervL{X}$.
\end{itemize}
\end{lemma}
\begin{proof}
This proof of (a)--(c) is taken from \cite[Lemma 2.1]{BHHS22}.
By the definitions of $G$-conjugation on sheaves and the direct image functor, we have $(\overline{f_*\mathcal{F}}^g)(U)=\overline{(f_*\mathcal{F})(U)}^g=\overline{\mathcal{F}(f^{-1}(U))}^g=\overline{\mathcal{F}}^g(f^{-1}(U))=f_*\overline{\mathcal{F}}^g(U)$, which shows $\overline{f_*\mathcal{F}}^g\simeq f_*\overline{\mathcal{F}}^g$. Moreover, we get $\overline{f_!\mathcal{F}}^g\simeq f_!\overline{\mathcal{F}}^g$, since the proper direct image sheaf is a subsheaf of the direct image sheaf, containing the sections on $U$ whose support is proper over $U$, and the notion of proper support does not depend on the action of $L$, so it is the same subsheaf in both cases.
f
For $L$-vector spaces $V$ and $W$, it is clear that we have an isomorphism of $L$-vector spaces
$$\overline{\mathrm{Hom}_L(V,W)}^g \simeq \mathrm{Hom}_L(\overline{V}^g,\overline{W}^g)$$
(sending a morphism on the left to the same set-theoretic map on the right).
Then for any open $U\subseteq X$, we have
$$\mathcal{H}om(\mathcal{F}_1,\mathcal{F}_2)(U)=\mathrm{Hom}(\mathcal{F}_1|_U,\mathcal{F}_2|_U)\subset \prod_{V\subset U \text{ open}} \mathrm{Hom}(\mathcal{F}_1(V),\mathcal{F}_2(V))$$
(namely the subset of families of morphisms compatible with restriction maps of $\mathcal{F}_1$ and $\mathcal{F}_2$).
This implies $\overline{\mathcal{H}om(\mathcal{F}_1,\mathcal{F}_2)}^g\simeq \mathcal{H}om(\overline{\mathcal{F}_1}^g,\overline{\mathcal{F}_2}^g)$ since conjugation is an equivalence and hence commutes with products.
Finally, (a) and the first part of (c) follow by deriving functors (noting that conjugation is exact), and the other statements follow by adjunction, using
$$\mathrm{Hom}_{\mathbb{D}bL{X}}(\overline{\mathcal{F}_1}^g,\mathcal{F}_2)\simeq \mathrm{Hom}_{\mathbb{D}bL{X}}(\mathcal{F}_1,\overline{\mathcal{F}_2}^{g^{-1}}).$$
For (d), note that
\begin{align*}
\mathbb{D}D_X\overline{\mathcal{F}}^g &= \mathbb{R}Hom(\overline{\mathcal{F}}^g,\omega_X) \simeq \mathbb{R}Hom(\overline{\mathcal{F}}^g,p_X^!L)\\
&\simeq \mathbb{R}Hom(\overline{\mathcal{F}}^g,p_X^!\overline{L}^g)\simeq \overline{\mathbb{R}Hom(\mathcal{F},p_X^!L)}^g \simeq \overline{\mathbb{D}D_X\mathcal{F}}^g.
\end{align*}
Here, $p_X\colon X\to\{\mathrm{pt}\}$ is the map to the one-point space, and $\omega_X\vcentcolon= p_X^!L$ is the dualizing complex. We have used an isomorphism of $L$-vector spaces $L\simeq \overline{L}^g$, which is given by $\ell\mapsto g(\ell)$, as well as (b) and (c).
We now prove (e). Let $U\subseteq Z$ be open such that $\mathrm{H}^i(\mathcal{F})|_U$ is a constant sheaf, i.e.\ $\mathrm{H}^i(\mathcal{F})|U\simeq V_U\simeq p_U^{-1}V$ for some $L$-vector space $V$ (where $p_U\colon U\to \{\mathrm{pt}\}$ denotes the map to the one-point space). Then clearly
$$\mathrm{H}^i(\overline{\mathcal{F}}^g)|_U\simeq \overline{\mathrm{H}^i(\mathcal{F}|_U)}^g\simeq \overline{p_U^{-1}V}^g\simeq p_U^{-1}\overline{V}^g$$
is a constant sheaf (with stalk $\overline{V}^g$), by exactness of conjugation and (b).
Finally, to show (f), note first that, by (e), the support of the cohomologies of an object $\mathcal{F}\in\mathbb{D}bCcL{X}$ does not change under conjugation. Therefore, if $\mathcal{F}$ satisfies the support condition, so does $\overline{\mathcal{F}}^g$. Moreover, by (d), the support condition for $\mathbb{D}D_X\mathcal{F}$ implies that for $\mathbb{D}D_X\overline{\mathcal{F}}^g$. Consequently, $\overline{\mathcal{F}}^g$ is perverse if $\mathcal{F}$ is.
\end{proof}
\begin{defi}\label{def:Gstructure}
Let $\mathcal{C}(L)$ be an $L$-linear category with $G$-conjugation. Let $F\in \mathcal{C}(L)$ be an object. A $G$-structure on $F$ is given by a family $(\varphi_g)_{g\in G}$ of isomorphisms $\varphi_g\colon F\overset{\sim}{\longrightarrow} \overline{F}^g$ such that for any $g,h\in G$ the following diagram is commutative:
$$\begin{tikzcd}
F \arrow[bend left]{rrrrdd}{\varphi_{gh}} \arrow{dd}{\varphi_h}\\ \\
\overline{F}^h\arrow{rr}{\gamma_h(\varphi_g)} && \overline{\overline{F}^g}^h\arrow{rr}{I_{g,h}} && \overline{F}^{gh}
\end{tikzcd}$$
\end{defi}
\begin{ex}\label{ex:Gstructures}
Consider again the category $\Vect{L}$ as in Example~\ref{ex:GConj}(a).
The trivial one-dimensional vector space $L\in \Vect{L}$ has a natural $G$-structure: 'Any element $g\in G$ defines an isomorphism of $L$-vector spaces $\varphi_g=g\colon L\overset{\sim}{\longrightarrow} \overline{L}^g$ given by $\ell\mapsto g(\ell)$ and satisfying the compatibilities from Definition~\ref{def:Gstructure}. (We will often just denote it by the letter $g$.) The same works, of course, for any $n$-dimensional vector space of the form $L^n$ for some $n\in\mathbb{Z}_{>0}$, applying $g$ componentwise.
Similarly, in the category $\Mod{L_X}$ of sheaves of $L$-vector spaces on a topological space, one gets a canonical $G$-structure on the constant sheaf $L_X$. Indeed, the constant sheaf $L_X$ is the sheafification of the constant presheaf $L^\mathrm{pre}_X$ defined by $L^\mathrm{pre}_X(U)=L$ for any open $U\subseteq X$. On $L^\mathrm{pre}_X$, we can define the $G$-structure just as in the example of the trivial vector space $L$ above. Then we conclude via Lemma~\ref{lemma:GstrSheafification} below.
\end{ex}
\begin{lemma}\label{lemma:GstrSheafification}
Assume that $\mathcal{P}\in\mathrm{Funct}(\mathrm{Op}(X)^\mathrm{op},\Vect{L})$ is a presheaf with a $G$-structure $(\phi_g)_{g\in G}$. Then its sheafification $\mathcal{F} \vcentcolon= \mathcal{P}^\#$ has an induced $G$-structure $(\varphi_g)_{g\in G}$.
\end{lemma}
\begin{proof}
For any $g\in G$, we are given
$$\phi_g\colon \mathcal{P}\overset{\sim}{\longrightarrow} \overline{\mathcal{P}}^g,$$
and this induces, by the universal property of sheafification, isomorphisms
$$\mathcal{P}^\#\overset{\sim}{\longrightarrow} (\overline{\mathcal{P}}^g)^\#.$$
It therefore remains to check that $(\overline{\mathcal{P}}^g)^\#\simeq (\overline{\mathcal{P}}^\#)^g$.
To see this, note that there is a natural homomorphism $\mathcal{P}\to\mathcal{P}^\#$ and hence a natural morphism $\overline{\mathcal{P}}^g\to \overline{(\mathcal{P}^\#)}^g$. By the universal property of sheafification, this induces a morphism $(\overline{\mathcal{P}}^g)^\#\to \overline{(\mathcal{P}^\#)}^g$. We can check on stalks that it is an isomorphism: $G$-conjugation commutes with taking stalks, and stalks of a presheaf and its associated sheaf are the same.
\end{proof}
The following is an easy corollary from Lemma~\ref{lemma:compatConj}. We leave it to the readers to verify that the necessary compatibilities are satisfied.
\begin{cor}
Let $f\colon X\to Y$ be a morphism of topological spaces. Let $\mathcal{F},\mathcal{F}_1,\mathcal{F}_2\in \mathbb{D}bL{X}$ and $\mathcal{G}\in\mathbb{D}bL{Y}$ be equipped with $G$-structures. Then $\mathbb{R}R f_*\mathcal{F}$, $\mathbb{R}R f_!\mathcal{F}$, $f^{-1}\mathcal{G}$, $f^!\mathcal{G}$, $\mathcal{F}_1\otimes\mathcal{F}_2$, $\mathbb{R}Hom(\mathcal{F}_1,\mathcal{F}_2)$, $\mathbb{D}D_X\mathcal{F}$ and $\mathrm{H}^i(\mathcal{F})$ for any $i\in\mathbb{Z}$ are equipped with an induced $G$-structure.
\end{cor}
We now introduce the following category associated to a category with $G$-conjugation. It will serve as the target category for Galois descent statements later.
\begin{defi}\label{def:catG}
Let $\mathcal{C}(L)$ be an $L$-linear category with $G$-conjugation. Then we define the category $\mathcal{C}(L)^G$ as follows:
\begin{itemize}
\item An object of $\mathcal{C}(L)^G$ is a pair $(F,(\varphi_g)_{g\in G})$, where $F\in \mathcal{C}(L)$ is an object and $(\varphi_g)_{g\in G}$ is a $G$-structure on $F$.
\item A morphism $(F,(\varphi_g)_{g\in G})\to (F',(\varphi'_g)_{g\in G})$ is a morphism $f\colon F\to\widetilde{F}$ in $\mathcal{C}(L)$ compatible with the $G$-structures, i.e.\ such that for any $g\in G$ the diagram
$$\begin{tikzcd}
F\arrow{r}{f}\arrow{d}{\simeq}[swap]{\varphi_g} & F'\arrow{d}{\varphi'_g}[swap]{\simeq}\\
\overline{F}^g \arrow{r}{\overline{f}^g} & \overline{F'}^g
\end{tikzcd}$$
commutes.
\end{itemize}
\end{defi}
\subsection{Extension of scalars}
As before, let $L/K$ be a field extension and set $G\vcentcolon=\mathrm{Aut}(L/K)$.
Extension of scalars (change of fields) is a well-known principle for vector spaces, sheaves or schemes, for example. In these examples, it is formed by taking tensor products or fibre products of the object over $K$ with an object determined by the field $L$. In the other direction, an object over $L$ can naturally be considered as an object over $K$. (Note that these processes are not inverse to each other, but adjoint.) In an abstract setting, one could define such an extension/restriction datum as follows.
\begin{defi}\label{def:ExtResScalars}
Let $\mathcal{C}(K)$ be a $K$-linear category and $\mathcal{C}(L)$ an $L$-linear category equipped with a $G$-conjugation $((\gamma_g)_{g\in G}, (I_{g,h})_{g,h\in G})$.
Consider a functor
$$\Phi_{L/K}\colon \mathcal{C}(K)\longrightarrow \mathcal{C}(L)$$
that factors as
$$\mathcal{C}(K) \xlongrightarrow{\Phi_{L/K}^G} \mathcal{C}(L)^G \longrightarrow \mathcal{C}(L),$$
where the second functor is just the one forgetting the $G$-structure. Consider furthermore a functor
$$\mathsf{for}_{L/K}\colon \mathcal{C}(L)\to\mathcal{C}(K)$$
together with natural isomorphisms of functors $J_g\colon \mathsf{for}_{L/K}\circ \gamma_g \overset{\sim}{\to}\mathsf{for}_{L/K}$ for any $g\in G$ such that for any $g,h\in G$ the diagram
$$\begin{tikzcd}
\mathsf{for}_{L/K}\circ \gamma_h\circ \gamma_g \arrow{rr}{\mathsf{for}_{L/K}\circ I_{g,h}} \arrow{d}{J_{h}\circ \gamma_g} && \mathsf{for}_{L/K}\circ \gamma_{gh}\arrow{d}{J_{gh}}\\
\mathsf{for}_{L/K}\circ \gamma_g \arrow{rr}{J_g} && \mathsf{for}_{L/K}
\end{tikzcd}$$
commutes.
If $\Phi_{L/K}$ is left adjoint to $\mathsf{for}_{L/K}$, i.e. there are natural isomorphisms
$$\mathrm{Hom}_{\mathcal{C}(K)}(Y,\mathsf{for}_{L/K}(X))\simeq \mathrm{Hom}_{\mathcal{C}(L)}(\Phi_{L/K}(Y),X)$$
for any $X\in\mathcal{C}(L)$, $Y\in\mathcal{C}(K)$, we call $\Phi_{L/K}$ a functor of \emph{extension of scalars} and $\mathsf{for}_{L/K}$ a functor of \emph{restriction of scalars}.
If, in such a situation, one has objects $F\in\mathcal{C}(L)$ and $A\in\mathcal{C}(K)$ such that $\Phi_{L/K}(A)\simeq F$, we say that $A$ is a \emph{$K$-structure} (or \emph{$K$-lattice}) of $F$.
For $F\in\mathcal{C}(L)$, we will often write $F^K\vcentcolon=\mathsf{for}_{L/K}(F)\in \mathcal{C}(K)$.
\end{defi}
Similar to what we saw above for the $I_{g,h}$ in Definition~\ref{def:Gconj}, the isomorphisms of functors $J_g$ will be equalities in our examples of interest.
\begin{ex}
The functor $\Vect{K}\to\Vect{L}$ defined by $W\mapsto L\otimes_K W$ is a functor of extension of scalars, and the functor $\Vect{L}\to\Vect{K}$ sending an $L$-vector space to itself (but only remembering the action of $K$) is a corresponding functor of restriction of scalars. Indeed, for $W\in\Vect{K}$, the object $L\otimes_K W$ carries a natural $G$-structure: It is clear that $\overline{L\otimes_K W}^g=\overline{L}^g\otimes_K W$, so the $G$-structure is just given by the natural one on $L$ (see Example~\ref{ex:Gstructures}). Moreover, it is clear that $V^K=(\overline{V}^g)^K$ for any $g\in G$, since $g|_K=\mathrm{id}_K$ and hence the action of $K$ is the same on each $\overline{V}^g$. The property of adjointness of these two functors is classical.
Similarly, a functor of extension of scalars for sheaves is given by $\Phi_{L/K}\colon \Mod{K_X}\to\Mod{L_X}, \mathcal{G}\mapsto L_X\otimes_{K_X} \mathcal{G}$ (with the obvious functor of restriction of scalars, which is defined as above for each space of sections $\mathcal{F}(U)$): The sheaf $L_X\otimes_{K_X} \mathcal{G}$ (a priori a tensor product of two $K_X$-modules) is the sheafification of the presheaf $U\mapsto L\otimes_K \mathcal{G}(U)$, and hence a sheaf of $L$-vector spaces. Again, the natural $G$-structure on $L_X\otimes_{K_X} \mathcal{G}$ is given by the natural $G$-structure on $L_X$ (see Example~\ref{ex:Gstructures}). As above, it is clear that $(\overline{\mathcal{F}}^g)^K=\mathcal{F}^K$ for any $g\in G$. The fact that these two functors are adjoint follows easily from the case of vector spaces, and follows from Lemma~\ref{lemma:ScalarExtAdj} below.
Noting that both $\Phi_{L_K}$ and $\mathsf{for}_{L/K}$ for sheaves of vector spaces are exact, we also obtain extension/restriction of scalars functors between derived categories of sheaves.
\end{ex}
When we work with these categories, we will always consider the functors of extension of scalars from the above example.
The following lemma describes the adjunction between extension and restriction of scalars in the case of sheaves of vector spaces. (This is rather classical, cf.\ e.g.\ \cite[Tag 0088]{Stacks}.)
\begin{lemma}\label{lemma:ScalarExtAdj}
Let $\mathcal{G}\in\mathbb{D}bK{X}$ and $\mathcal{F}\in\mathbb{D}bL{X}$. Then there are isomorphisms
$$\mathbb{R}Hom(\mathcal{G},\mathsf{for}(\mathcal{F})) \simeq \mathsf{for}_{L/K}\big(\mathbb{R}Hom(L_X\otimes_{K_X} \mathcal{G},\mathcal{F})\big)$$
in $\mathbb{D}bK{X}$ and
$$\mathrm{Hom}_{\mathbb{D}bK{X}}(\mathcal{G},\mathsf{for}(\mathcal{F})) \simeq \mathrm{Hom}_{\mathbb{D}bL{X}}(L_X\otimes_{K_X} \mathcal{G},\mathcal{F})$$
as $K$-vector spaces.
\end{lemma}
\begin{proof}
For vector spaces $W\in\Vect{K}$ and $V\in\Vect{L}$, it is well-known that
$$\mathrm{Hom}_K(W,\mathsf{for}_{L/K}(V))\simeq \mathrm{Hom}_L(L\otimes_K W, V)$$
(this is an isomorphism of $K$-vector spaces, i.e.\ there is a ``hidden'' functor $\mathsf{for}_{L/K}$ on the right-hand side),
and the morphism from left to right is given by $L$-linear continuation.
Let now $\mathcal{G}\in\Mod{K_X}$ and $\mathcal{F}\in\Mod{L_X}$. Then we get
$$\mathrm{Hom}_{\Mod{K_X}}(\mathcal{G},\mathsf{for}_{L/K}(\mathcal{F})) \simeq \mathrm{Hom}_{\Mod{L_X}}(L_X\otimes_{K_X} \mathcal{G},\mathcal{F})$$
as follows: An element of the left hom space is a compatible collection of $K$-linear morphisms $\mathcal{G}(U)\to\mathcal{F}(U)$ for any open $U\subseteq X$. By the statement for vector spaces, it is easy to see that this corresponds to a compatible collection of $L$-linear morphisms $L\otimes_K \mathcal{G}(U)\to \mathcal{F}(U)$, which represent a morphism of presheaves from the presheaf given by $U\mapsto L\otimes_K \mathcal{G}(U)$ to $\mathcal{F}$. By the universal property of sheafification (and since $\mathcal{F}$ is already a sheaf), this is equivalent to a morphism $L_X\otimes_{K_X} \mathcal{G}\to\mathcal{F}$.
Finally, we obtain an isomorphism
$$\mathcal{H}om(\mathcal{G},\mathsf{for}_{L/K}(\mathcal{F})) \simeq \mathsf{for}_{L/K}\big(\mathcal{H}om(L_X\otimes_{K_X} \mathcal{G},\mathcal{F})\big)$$
by defining it on sections over any open $U\subseteq X$, which means that we need a compatible collection of isomorphisms
$$\mathrm{Hom}_{\Mod{K_X}}(\mathcal{G}|_U,\mathsf{for}_{L/K}(\mathcal{F}|_U)) \simeq \mathrm{Hom}_{\Mod{L_X}}(L_U\otimes_{K_U} \mathcal{G}|_U,\mathcal{F}|_U)$$
for any such $U$, which is what we have constructed above.
The statements of the lemma now follows by deriving functors (noting that $\Phi_{L/K}$ and $\mathsf{for}_{L/K}$ are exact) and taking zeroth cohomology.
\end{proof}
Let us state the main compatibilities of extension of scalars for sheaves. (Parts (a) and (b) were also mentioned just before \cite[Lemma 2.4]{BHHS22}.)
\begin{lemma}\label{lemma:compatExt}
Let $L/K$ be a field extension. Let $X$ and $Y$ be topological spaces and let $f\colon X\to Y$ be a continuous map.
\begin{itemize}
\item[(a)] Let $\mathcal{G}\in \mathbb{D}bK{X}$. Then $\mathbb{R}R f_!(L_X\otimes_{K_X}\mathcal{G})\simeq L_Y\otimes_{K_Y} \mathbb{R}R f_!\mathcal{G}$.
\item[(b)] Let $\mathcal{H}\in \mathbb{D}bL{X}$. Then $f^{-1}(L_Y\otimes_{K_Y}\mathcal{H})\simeq L_X\otimes_{K_X} f^{-1}\mathcal{H}$.
\item[(c)] If $\mathcal{G}\in\mathbb{D}bK{X}$ and $\mathrm{H}^i(\mathcal{G})$ is locally constant on $Z\subseteq X$, then so is $\mathrm{H}^i(L_X\otimes_{K_X}\mathcal{G})$.
In particular, if $\mathcal{G}\in\mathbb{D}bRcK{X}$ (resp.\ $\mathcal{G}\in\mathbb{D}bCcK{X}$), then $L_X\otimes_{K_X} \mathcal{G}\in\mathbb{D}bRcL{X}$ (resp.\ $L_X\otimes_{K_X} \mathcal{G}\in\mathbb{D}bCcL{X}$).
\end{itemize}
\end{lemma}
\begin{proof}
These statements follow from basic properties of the tensor product: (a) follows from the projection formula, and (b) follows from the fact that tensor products commute with inverse images (see e.g.\ Proposition 2.6.6 and Proposition 2.6.5 of \cite{KS90}, respectively). Then (c) is easily deduced from (b) since constant sheaves are inverse images of vector spaces along the map to the one-point space.
\end{proof}
Note that this statement is ``less complete'', compared to Lemma~\ref{lemma:compatConj}. For example, it contains no statement about compatibility between extension of scalars and direct images $f_*$. In fact, such a compatibility is not true in general. We will come back to this point in Corollary~\ref{cor:directImage} and Remark~\ref{rem:directImage} below. On the other hand, the majority of Section~\ref{sec:Hom} is devoted to proving statements about the compatibility of extension of scalars with $\mathbb{R}Hom$. We will also give statements about the duality functor and the exceptional inverse image functor in Corollary~\ref{cor:dualityExt}.
\subsection{Galois descent}
Given a functor of extension of scalars as in the previous subsection immediately raises the question if there is a construction in the other direction: Given an object over $L$, can does it admit a $K$-structure? In general, this is certainly not the case, but the idea is that from a given $G$-structure on $L$ we might indeed be able to construct a $K$-structure. The context in which we can expect such a descent statement is mainly that a Galois extension $L/K$, since in this case the group $G=\mathrm{Aut}(L/K)$ (which is nothing but the Galois group) ``knows enough'' about the subfield $K$. This idea is made more precise by the concept of Galois descent.
\begin{defi}\label{def:GaloisDescent}
Let $L/K$ be a Galois extension, $\mathcal{C}(K)$ a $K$-linear category, $\mathcal{C}(L)$ an $L$-linear category with a $G$-conjugation, and let these categories be equipped with extension and restriction of scalars as in Definition~\ref{def:ExtResScalars}.
Then we say that \emph{Galois descent} is satisfied in this setting if $\Phi_{L/K}^G\colon \mathcal{C}(K)\to\mathcal{C}(L)^G$ is an equivalence of categories.
\end{defi}
Let now $L/K$ be a finite Galois extension with Galois group $G$. In this case, it is well-known that Galois descent is satisfied for vector spaces, see \cite{Conrad} for an exposition (as well as the references therein and in the introduction above). We briefly recall the construction: The functor
$$\Phi_{L/K}\colon \Vect{K}\longrightarrow \Vect{L},\qquad V\longmapsto L\otimes_K V$$
induces an equivalence between $K$-vector spaces and $L$-vector spaces equipped with a $G$-structure.
The quasi-inverse of $\Phi_{L/K}^G\colon \Vect{K}\to \Vect{L}^G$ can be explicitly described:
Let $V$ be an $L$-vector space equipped with a $G$-structure $(\varphi_g)_{g\in G}$. Recall that we write $V^K\vcentcolon= \mathsf{for}_{L/K}(V)$. Since $V^K=(\overline{V}^g)^K$ for any $g\in G$, the $L$-linear isomorphisms $\varphi_g\colon V\overset{\sim}{\to} \overline{V}^g$ can be interpreted as $K$-linear automorphisms $\varphi_g^K\vcentcolon= \mathsf{for}_{L/K}(\varphi_g)$ of $V^K$. Then one defines the space of invariants of all these automorphisms
\begin{align*}
V_K&\vcentcolon= \{v\in V^K\mid \varphi_g^K(v)=v\text{ for any $g\in G$}\} \\ &= \ker \left( \prod_{g\in G} (\varphi_g^K-\mathrm{id}_{V^K})\colon V^K \to \prod_{g\in G} V^K \right).
\end{align*}
This is a $K$-sub-vector space of $V$ and one can show that the natural morphism $L\otimes_K V_K\to V, \sum_i \ell_i\otimes v_i\mapsto \sum_i \ell_i v_i$ is an isomorphism.
The aim of the rest of this paper is to investigate the extension of scalars functor for sheaves of vector spaces, and to study Galois descent in this framework. We will first establish (not only in the case of Galois extensions) some statements about homomorphisms between extensions of scalars. Then, we will illustrate the problems that occur when we want to set up Galois descent in derived categories, and we will finally prove Galois descent for perverse sheaves.
\section{Homomorphisms between scalar extensions}\label{sec:Hom}
Let $L/K$ be a field extension. (We do not assume that $L/K$ is finite or Galois, if not explicitly stated.)
In this section, we want to describe spaces of morphisms $L_X\otimes_{K_X}\mathcal{G}_1\to L_X\otimes_{K_X} \mathcal{G}_2$ for $\mathcal{G}_1,\mathcal{G}_2\in\mathbb{D}bK{X}$ and relate them to spaces of morphisms $\mathcal{G}_1\to\mathcal{G}_2$.
As a motivation, consider the following well-known fact from linear algebra: Let $V,W\in\Vect{K}$. If $L/K$ is a finite field extension or $V$ and $W$ are finite-dimensional, then
\begin{equation}\label{eq:HomExtVS}
\mathrm{Hom}_L(L\otimes_K V,L\otimes_K W)\simeq L\otimes_K \mathrm{Hom}_K(V,W).
\end{equation}
This implies that in the case of finite field extensions the above functor of extension of scalars is faithful. In the case of arbitrary field extensions, it is still faithful if we restrict to finite-dimensional vector spaces.
We will develop analogous statements about homomorphism spaces in the context of sheaves. Similarly to the above, we impose some additional assumption if we do not require the field extension to be finite. For sheaves, we will replace the above finiteness assumption by $\mathbb{R}$-constructibility.
In the rest of this section, we will develop relations between homomorphisms of sheaves and their scalar extensions. First of all, we remark that there are always the following natural morphisms, and it is the main aim of the following two subsections to give conditions under which they are isomorphisms.
\begin{lemma}\label{lemma:natMorph}
Let $L/K$ be a field extension, $X$ a topological space and $\mathcal{F},\mathcal{G}\in\mathbb{D}bK{X}$. There are natural morphisms
$$L\otimes_{K} \mathrm{Hom}_{\mathbb{D}bK{X}}(\mathcal{F},\mathcal{G}) \to \mathrm{Hom}_{\mathbb{D}bL{X}}(L_X\otimes_{K_X}\mathcal{F},L_X\otimes_{K_X}\mathcal{G})$$
and
$$L_X\otimes_{K_X} \mathbb{R}Hom(\mathcal{F},\mathcal{G}) \to \mathbb{R}Hom(L_X\otimes_{K_X}\mathcal{F},L_X\otimes_{K_X}\mathcal{G}).$$
\end{lemma}
\begin{proof}
Since $\Phi_{L/K}=L_X\otimes_{K_X}(-)$ is a functor, we have a natural map
$$\mathrm{Hom}_{\mathbb{D}bK{X}}(\mathcal{F},\mathcal{G}) \to \mathrm{Hom}_{\mathbb{D}bL{X}}(L_X\otimes_{K_X} \mathcal{F}, L_X\otimes_{K_X} \mathcal{G}),$$
which can be extended $L$-linearly to a map
$$L\otimes_K \mathrm{Hom}_{\mathbb{D}bK{X}}(\mathcal{F},\mathcal{G}) \to \mathrm{Hom}_{\mathbb{D}bL{X}}(L_X\otimes_{K_X} \mathcal{F}, L_X\otimes_{K_X} \mathcal{G}).$$
For the second morphism in the assertion, consider first the case $\mathcal{F},\mathcal{G}\in\Mod{K_X}$.
In order to construct
$$L_X\otimes_{K_X} \mathcal{H}om(\mathcal{F},\mathcal{G}) \to \mathcal{H}om(L_X\otimes_{K_X}\mathcal{F},L_X\otimes_{K_X}\mathcal{G}),$$
we construct it on sections over any $U\subseteq X$.
The left-hand side is the sheaf associated to the presheaf $U\mapsto L\otimes_K \mathcal{H}om(\mathcal{F}|_U,\mathcal{G}|_U)$, and hence by the universal property of sheafification it suffices to construct for any such $U$ a morphism
$$L\otimes_K \mathcal{H}om(\mathcal{F}|_U,\mathcal{G}|_U)\to \mathrm{Hom}(L_U\otimes_{K_U} \mathcal{F}|_U, L_U\otimes_{K_U} \mathcal{G}|_U)$$
compatible with restrictions, which is what we did above.
The statement of the lemma follows now by deriving functors.
\end{proof}
Whenever we prove isomorphisms showing some commutation between $\mathbb{R}Hom$ or $\mathrm{Hom}$ and extension of scalars in the following, we will always implicitly mean that it is this natural morphism that induces the isomorphism.
We will study separately the cases of finite and general (in particular infinite) field extensions.
As a consequence, we get in particular the following property.
\begin{prop}
Let $L/K$ be a field extension and $X$ a topological space.
\begin{itemize}
\item[(a)] If $L/K$ is finite, the functor $\Phi_{L/K}$ is faithful on $\mathbb{D}bK{X}$.
\item[(b)] If $X$ is a compact real analytic manifold, then $\Phi_{L/K}$ is faithful on $\mathbb{D}bRcK{X}$.
\end{itemize}
\end{prop}
\begin{proof}
These statements will follow directly from Proposition~\ref{prop:morphLification} and Proposition~\ref{prop:morphLificationCpt}, respectively.
\end{proof}
\subsection{Finite field extensions}
In this subsection, we will study the case of finite field extensions, which will allow us to get results about homomorphisms between sheaves without any constructibility assumption.
First of all, sections of extensions of scalars along finite field extensions are particularly easily described, as the following lemma shows.
\begin{lemma}\label{lemma:sectionsScalarext}
If $L/K$ is finite and $\mathcal{G}\in\Mod{K_X}$, then
$$(L_X\otimes_{K_X}\mathcal{G})(U)=L\otimes_K \mathcal{G}(U).$$
\end{lemma}
\begin{proof}
We know that $L_X\otimes_{K_X}\mathcal{G}$ is the sheafification of $U\mapsto L\otimes_K \mathcal{G}(U)$. It therefore suffices to show that the presheaf $\mathcal{F}(U)\vcentcolon= L\otimes_K \mathcal{G}(U)$ is already a sheaf.
This follows directly since $L/K$ is finite and hence $L\otimes_K (-)$ commutes with arbitrary products:
If $U\subseteq X$ is open and $U=\bigcup_{i\in I} U_i$ is an open covering, then
$$0\to \mathcal{G}(U) \to \prod_{i\in I} \mathcal{G}(U_i) \rightrightarrows \prod_{i,j\in I} \mathcal{G}(U_i\cap U_j)$$
is exact since $\mathcal{G}$ is a sheaf.
Applying $L\otimes_K (-)$, we obtain an exact sequence
$$0\to \mathcal{F}(U) \to \prod_{i\in I} \mathcal{F}(U_i) \rightrightarrows \prod_{i,j\in I} \mathcal{F}(U_i\cap U_j),$$
proving that $\mathcal{F}$ is a sheaf, as claimed.
\end{proof}
We are now going to describe the homomorphisms between objects of the form $L_X\otimes_{K_X} \mathcal{F}$ more precisely and give a characterization of the image of morphisms in $\mathbb{D}bK{X}$ under the functor $\Phi_{L/K}$.
\begin{prop}\label{prop:morphLification}
Let $L/K$ be a finite field extension. Let $X$ be a topological space, and let $\mathcal{F},\mathcal{G}\in\mathbb{D}bK{X}$.
Then
$$\mathbb{R}Hom(L_X\otimes_{K_X}\mathcal{F}, L_X\otimes_{K_X}\mathcal{G}) \simeq L_X\otimes_{K_X} \mathbb{R}Hom(\mathcal{F}, \mathcal{G}) .$$
In particular, we have
\begin{equation}\label{eq:HomExtFin}
\mathrm{Hom}_{\mathbb{D}bL{X}}(L_X\otimes_{K_X}\mathcal{F}, L_X\otimes_{K_X}\mathcal{G}) \simeq L\otimes_{K}\mathrm{Hom}_{\mathbb{D}bK{X}}(\mathcal{F}, \mathcal{G}).
\end{equation}
If $L/K$ is finite Galois with Galois group $G$, then the subset of \eqref{eq:HomExtFin} consisting of morphisms $f$ fitting into the natural diagram
$$\begin{tikzcd}
L_X\otimes_{K_X}\mathcal{F} \arrow{r}{f} \arrow{d}{g\otimes\mathrm{id}_{\mathcal{F}}}[swap]{\simeq} & L_X\otimes_{K_X}\mathcal{G} \arrow{d}{g\otimes\mathrm{id}_\mathcal{G}}[swap]{\simeq}\\
\overline{L_X}^g\otimes_{K_X}\mathcal{F} \arrow{r}{\overline{f}^g} &\overline{L_X}^g\otimes_{K_X}\mathcal{G}
\end{tikzcd}$$
for any $g\in G$ is exactly the subset of morphisms of the form $1\otimes f_K$ for some $f_K\in\mathrm{Hom}_{\mathbb{D}bK{X}}(\mathcal{F},\mathcal{G})$.
\end{prop}
\begin{proof}
Let us first prove that for sheaves $\mathcal{F},\mathcal{G}\in\Mod{K_X}$, we have
$$\mathrm{Hom}_{L_X}(L_X\otimes_{K_X}\mathcal{F}, L_X\otimes_{K_X}\mathcal{G}) \simeq L\otimes_{K}\mathrm{Hom}_{K_X}(\mathcal{F}, \mathcal{G}).$$
An element in $\mathrm{Hom}_{K_X}(\mathcal{F}, \mathcal{G})$ is a family $(f_U)$ of morphisms $f_U\colon \mathcal{F}(U)\to\mathcal{G}(U)$ for any open $U\subseteq X$ such that for any inclusion $V\subset U$ we have a commutative diagram
$$\begin{tikzcd}
\mathcal{F}(U) \arrow{r}{f_U} \arrow{d} & \mathcal{G}(U) \arrow{d}\\
\mathcal{F}(V) \arrow{r}{f_V} & \mathcal{G}(V)
\end{tikzcd}$$
where the vertical arrows are the restriction morphisms of the sheaves.
In the same way, using to Lemma~\ref{lemma:sectionsScalarext}, an element in $\mathrm{Hom}_{L_X}(L_X\otimes_{K_X}\mathcal{F},L_X\otimes_{K_X}\mathcal{G})$ is a family $(\tilde{f}_U)$ of morphisms $\tilde{f}_U\colon L\otimes_K \mathcal{F}(U)\to L\otimes_K \mathcal{G}(U)$, compatible with restrictions. Choose a (finite) basis $\ell_1,\ldots,\ell_n$ of $L$ over $K$. Then, due to the isomorphism \eqref{eq:HomExtVS} for vector spaces, for any $U$ we can write $\tilde{f}_U = \sum_{i=1}^n \ell_i\otimes f_U^i$ for suitable $f_U^i\colon \mathcal{F}(U)\to\mathcal{G}(U)$. It is easy to check that for fixed $i$, the $f_U^i$ are still compatible with the restriction maps, and hence we can identify $(\tilde{f}_U)$ with an element $\sum_{i=1}^n \ell_i\otimes (f_U^i)\in L\otimes_K \mathrm{Hom}_{K_X}(\mathcal{F}, \mathcal{G})$.
Now, let us prove
$$\mathcal{H}om_{L_X}(L_X\otimes_{K_X}\mathcal{F}, L_X\otimes_{K_X}\mathcal{G}) \simeq L_X\otimes_{K_X}\mathcal{H}om_{K_X}(\mathcal{F}, \mathcal{G}).$$
Again in view of Lemma~\ref{lemma:sectionsScalarext}, applying sections on some $U\subseteq X$ to both sides of this isomorphism, what we need is an isomorphism
$$\mathrm{Hom}_{L_U}(L_U\otimes_{K_U}\mathcal{F}|_U, L_U\otimes_{K_U}\mathcal{G}|_U) \simeq L\otimes_{K}\mathrm{Hom}_{K_U}(\mathcal{F}|_U, \mathcal{G}|_U)$$
for any $U\subseteq X$, compatible with restrictions. This is exactly what we just proved.
Noting that $L_X\otimes_{K_X}(-)$ and $L\otimes_K(-)$ are exact, the first two statements of the proposition now follow by deriving functors and taking zeroth cohomology.
Now, let us prove the second part of the proposition: Let $\mathcal{F},\mathcal{G}\in\mathbb{D}bK{X}$. By what we proved above, an element of $\mathrm{Hom}_{\mathbb{D}bL{X}}(\mathcal{F}\otimes_{K_X}L_X, \mathcal{G}\otimes_{K_X}L_X)$ can therefore uniquely be written in the form $f=\sum_{k=1}^{d} \ell_k\otimes f_k$ for some $\ell_k\in L$ (understood as the map $L_X\to L_X$ given by multiplication with $\ell_k$) and some $f_k\in \mathrm{Hom}_{\mathbb{D}bK{X}}(\mathcal{F},\mathcal{G})$.
Assume that $f$ fits in a diagram as above for any $g\in G$. The vertical isomorphisms are induced by the natural $G$-structure on the constant sheaf $L_X$. Commutation of this diagram therefore means that $\sum_{k=1}^{d} g(\ell_k\cdot (-))\otimes f_k = \sum_{k=1}^{d} (g(\ell_k)\cdot g(-)) \otimes f_k$ (going right and then down in the diagram) and $\sum_{k=1}^{d} (\ell_k\cdot g(-)) \otimes f_k$ (going down and then right) coincide as morphisms in $\mathrm{Hom}_{\mathbb{D}bL{X}}(L_X\otimes_{K_X}\mathcal{F},\overline{L_X}^g\otimes_{K_X}\mathcal{G})$. Now there is an isomorphism
\begin{align*}
\mathrm{Hom}_{\mathbb{D}bL{X}}(L_X\otimes_{K_X}\mathcal{F},\overline{L_X}^g\otimes_{K_X}\mathcal{G})&\simeq \mathrm{Hom}_{\mathbb{D}bL{X}}(L_X\otimes_{K_X}\mathcal{F},L_X\otimes_{K_X}\mathcal{G})\\
&\simeq L\otimes_K \mathrm{Hom}_{\mathbb{D}bK{X}}(\mathcal{F},\mathcal{G})
\end{align*}
with the first isomorphism given by $\varphi\mapsto (g^{-1}\otimes \mathrm{id})\circ\varphi$, hence the above means that for any $g\in G$ we have $\sum_{k=1}^{d} \ell_k \otimes f_k= \sum_{k=1}^{d} g(\ell_k) \otimes f_k$ in $L\otimes_K \mathrm{Hom}_{\mathbb{D}bK{X}}(\mathcal{F},\mathcal{G})$. Then the result follows from the fact that the invariants of the Galois action on $L\otimes_K V$ (for a $K$-vector space $V$) are given by the $K$-subspace $1\otimes V$ (cf. e.g. \cite[Corollary 2.17]{Conrad}).
\end{proof}
Before continuing with the case of infinite extensions, let us establish a simple consequence of Lemma~\ref{lemma:sectionsScalarext}, extending the compatibility properties of Lemma~\ref{lemma:compatExt} in the case of a finite field extension.
\begin{cor}[of Lemma~\ref{lemma:sectionsScalarext}]\label{cor:directImage}
Let $L/K$ be a finite field extension. Let $f\colon X\to Y$ be a continuous map of topological spaces and $\mathcal{G}\in\mathbb{D}bK{X}$.
Then there is an isomorphism in $\mathbb{D}bL{X}$
$$\mathbb{R}R f_* (L_X\otimes_{K_X}\mathcal{G})\simeq L_Y\otimes_{K_Y} \mathbb{R}R f_* \mathcal{G}.$$
\end{cor}
\begin{proof}
Let us first treat the non-derived case. If $\mathcal{G}\in\Mod{K_X}$, then, by the definition of the direct image functor and Lemma~\ref{lemma:sectionsScalarext}, we have
$$\big(f_* (L_X\otimes_{K_X}\mathcal{G})\big)(U) = (L_X\otimes_{K_X}\mathcal{G})(f^{-1}(U)) \simeq L\otimes_{K}\mathcal{G}(f^{-1}(U))$$
and
$$\big(L_Y\otimes_{K_Y} (f_* \mathcal{G})\big)(U)\simeq L\otimes_K (f_* \mathcal{G})(U) = L\otimes_{K}\mathcal{G}(f^{-1}(U))$$
for any open $U\subseteq Y$, and thus $f_* (L_X\otimes_{K_X}\mathcal{G})\simeq L_Y\otimes_{K_Y}f_* \mathcal{G}$.
Since extension of scalars is exact, the derived statement follows.
\end{proof}
\begin{rem}\label{rem:directImage}
Let us remark that the assumption that $L/K$ is finite is crucial here, in contrast to the corresponding isomorphisms for the proper direct image $\mathbb{R}R f_!$ and the inverse image $f^{-1}$ (see Lemma~\ref{lemma:compatExt}), which are proved without this finiteness assumption, using standard properties of the tensor product.
Let us give a counterexample for the statement of Lemma~\ref{cor:directImage} in the case of an infinite field extension: Consider $X=\mathbb{R}_{> 0}=\;]0, \infty[\;\subset \mathbb{R}$, $Y=\mathbb{R}$ and $f\colon X\to Y$ the inclusion. Moreover, consider the field extension $\mathbb{Q}\subset \mathbb{C}$. Define the sheaf
$$\mathcal{G} \vcentcolon= \prod_{n\in \mathbb{Z}_{>0}} \mathbb{Q}_{\{\frac{1}{n}\}}.$$
It has stalk $\mathbb{Q}$ at any point $\frac{1}{n}$ and stalk $0$ otherwise. Therefore, we have
$$\mathbb{C}_X\otimes_{\mathbb{Q}_X}\mathcal{G}\simeq \prod_{n\in \mathbb{Z}_{>0}} \mathbb{C}_{\{\frac{1}{n}\}}.$$
On the other hand, if $U$ is a small neighbourhood of $0\in Y$ (automatically containing infinitely many points $\frac{1}{n}$), we have
$$(f_* \mathcal{G})(U) \simeq \prod_{n\geq N} \mathbb{Q}\simeq t^N\mathbb{Q}[\![t]\!]$$
for some $N$, and hence the stalk is
$$(f_* \mathcal{G})_0 \simeq \varinjlim_{N\to \infty} t^N\mathbb{Q}[\![t]\!].$$
Similarly,
$$\big(f_*(\mathbb{C}_X\otimes_{\mathbb{Q}_X}\mathcal{G})\big)_0 \simeq \varinjlim_{N\to \infty} t^N\mathbb{C}[\![t]\!],$$
and this is not isomorphic to $\mathbb{C}\otimes_\mathbb{Q}\varinjlim_{N\to \infty} t^N\mathbb{Q}[\![t]\!]\simeq \varinjlim_{N\to \infty} (\mathbb{C}\otimes_\mathbb{Q} t^N\mathbb{Q}[\![t]\!])$: An element of the first vector space is represented by a formal power series that might have infinitely many linearly independent (over $\mathbb{Q}$) coefficients, while an element of the second object needs to be represented by a finite sum of $\mathbb{Q}$-power series multiplied by a complex number, and two such series are equivalent only if they differ in a finite number of coefficients.
\end{rem}
\subsection{Infinite field extensions}
In this subsection, we will now study the case of infinite field extensions. We are mainly after a statement similar to (the first part of) Proposition~\ref{prop:morphLification}. (We certainly cannot expect an analogue of the second part of its statement outside the Galois case.)
Clearly, a statement like Lemma~\ref{lemma:sectionsScalarext} does not hold any more in this generality. To get the desired compatibility between hom spaces and extension of scalars, we will need to make some extra assumptions: Instead of general sheaves, we will work with $\mathbb{R}$-constructible sheaves (and hence on real analytic manifolds), since these can be modelled by constant sheaves on simplicial complexes, which gives the theory a combinatorial flavour. Moreover, for the final statement, we will assume that the manifold is compact, since this guarantees -- together with $\mathbb{R}$-constructibility -- that global sections are finite-dimensional, and we can thus get an analogue of Lemma~\ref{lemma:sectionsScalarext} at least for global sections.
We start with two lemmas about sheaves on simplices of a simplicial complex. We will not recall the theory of simplicial complexes here, and an intuitive understanding of simplicial complexes is probably enough to follow our arguments. We mainly use the terminology and notation of \cite[§8.1]{KS90}, where details can be found. We deviate from loc.~cit.\ in the following notation: For an abstract simplex $\sigma$, we denote by $Z_\sigma$ its geometric realization (this is the region not containing the lower-dimensional edges of the simplex, and it is denoted by $|\sigma|$ in \cite{KS90}).
\begin{lemma}\label{lemma:directImageCst}
Let $k$ be a field.
Let $\mathbf{S}$ be a simplicial complex with geometric realization $\mathcal{S}\vcentcolon=|\mathbf{S}|$, and let $\sigma$ be a simplex and write $Z\vcentcolon= Z_\sigma$. Consider the inclusion $j\colon Z\hookrightarrow |\mathbf{S}|$. Then
$$\mathbb{R}R j_* k_Z \simeq k_{\overline{Z}},$$
where the right-hand side denotes the constant sheaf on $\overline{Z}$, extended by zero on $\mathcal{S}\setminus\overline{Z}$.
\end{lemma}
\begin{proof}
We can decompose $j$ as $Z\overset{j_Z}{\hookrightarrow} \overline{Z}\overset{i_Z}{\hookrightarrow} |\mathbf{S}|$, where $j_Z$ is an open embedding, whereas $i_Z$ is a closed embedding. It follows that $\mathbb{R}R j_* k_Z\simeq {i_Z}_! \mathbb{R}R {j_Z}_* k_Z$ (since $i_Z$ is proper and proper direct images are exact). It therefore suffices to show that $\mathbb{R}R {j_Z}_* k_Z\simeq k_{\overline{Z}}$.
The constant sheaf $k_Z$ is the sheaf of locally constant $k$-valued functions on $Z$, and it is characterized by the fact that on connected open subsets of $Z$ its sections are $k$, and restriction maps $k_Z(U)\to k_Z(V)$ for connected open subsets $U,V\subseteq Z$ with $V\subseteq U$ are given by the identity. An analogous statement holds for the constant sheaf $k_{\overline{Z}}$.
By definition of the direct image, we have
$$({j_Z}_*k_Z)(U) = k_Z(U\cap Z)$$
for any open $U\in \overline{Z}$. It is easy to see that $U\cap Z$ is connected if $U\subseteq \overline{Z}$ is open and connected, and hence it follows that ${j_Z}_* k_Z = k_{\overline{Z}}$.
It remains to show that the higher direct images vanish. It is known that $\mathbb{R}R^k {j_Z}_* k_Z$ is the sheaf associated to the presheaf given by
$$U\mapsto \mathrm{H}^k(U\cap Z; k)$$
for any open $U\subseteq \overline{Z}$. For any contractible $U$, $U\cap Z$ is still contractible and hence $\mathrm{H}^k(U\cap Z; k)=0$ for $k\neq 0$. This shows the vanishing of $\mathbb{R}R^k {j_Z}_* k_Z$ since open contractible sets form a basis of the topology.
\end{proof}
\begin{lemma}\label{lemma:RHomSimplices}
Let $k$ be a field. Let $\mathbf{S}$ be a simplicial complex with geometric realization $\mathcal{S}\vcentcolon= |\mathbf{S}|$, and let $\sigma,\sigma'$ be two simplices with inclusions $\iota_\sigma\colon Z_\sigma\hookrightarrow\mathcal{S}$, $\iota_{\sigma'}\colon Z_{\sigma'}\hookrightarrow\mathcal{S}$. Then we have
$$\mathbb{R}Hom({\iota_\sigma}_! k_{Z_\sigma},{\iota_{\sigma'}}_! k_{Z_{\sigma'}})\simeq \begin{cases}
k_{\overline{Z_\sigma}}[\dim Z_\sigma-\dim Z_{\sigma'}] & \text{if $Z_\sigma\subseteq Z_{\sigma'}$}\\
0 &\text{if $Z_\sigma\cap \overline{Z_{\sigma'}}=\emptyset$}
\end{cases}$$
(Note that the first case covers in particular the case when $\sigma=\sigma'$. Note also that the two cases cover all possible situations by the definition of a simplicial complex.)
In particular, if $L/K$ is a field extension, there is an isomorphism
$$\mathbb{R}Hom(L_\mathcal{S}\otimes_{K_\mathcal{S}}{\iota_\sigma}_! K_{Z_\sigma},L_\mathcal{S}\otimes_{K_\mathcal{S}}{\iota_{\sigma'}}_! K_{Z_{\sigma'}})\simeq L_\mathcal{S}\otimes_{K_\mathcal{S}}\mathbb{R}Hom({\iota_\sigma}_! K_{Z_\sigma},{\iota_{\sigma'}}_! K_{Z_{\sigma'}}).$$
\end{lemma}
\begin{proof}
By adjunction (see e.g.\ \cite[Proposition 3.1.10]{KS90}), we have
\begin{align*}
\mathbb{R}Hom({\iota_\sigma}_! k_{Z_\sigma},{\iota_{\sigma'}}_! k_{Z_{\sigma'}}) \simeq \mathbb{R}R {\iota_\sigma}_* \mathbb{R}Hom(k_{Z_\sigma}, \iota_\sigma^! {\iota_{\sigma'}}_! k_{Z_{\sigma'}}).
\end{align*}
Now, note that
\begin{align*}
\iota_\sigma^! {\iota_{\sigma'}}_! k_{Z_{\sigma'}} &\simeq \mathbb{D}D_{Z_\sigma} \iota_\sigma^{-1} \mathbb{D}D_X {\iota_{\sigma'}}_! k_{Z_{\sigma'}}\\
&\simeq \mathbb{D}D_{Z_\sigma} \iota_\sigma^{-1} {\iota_{\sigma'}}_! \mathbb{D}D_{Z_{\sigma'}} k_{Z_{\sigma'}}\simeq \mathbb{D}D_{Z_\sigma} (\iota_\sigma^{-1} {\iota_{\sigma'}}_* k_{Z_{\sigma'}}[\dim Z_{\sigma'}]),
\end{align*}
where the last isomorphism follows from the fact that $Z_{\sigma'}$ is in particular an orientable differentiable manifold, so its dualizing complex is $\omega_{Z_{\sigma'}}\simeq k_{Z_{\sigma'}}[\dim Z_{\sigma'}]$ (see \cite[Proposition 3.3.6(iii)]{KS90}), and hence
$$\mathbb{D}D_{Z_{\sigma'}} k_{Z_{\sigma'}} = \mathbb{R}Hom(k_{Z_{\sigma'}},\omega_{Z_{\sigma'}})\simeq \mathbb{R}Hom(k_{Z_{\sigma'}},k_{Z_{\sigma'}})[\dim Z_{\sigma'}]\simeq k_{Z_{\sigma'}}[\dim Z_{\sigma'}].$$
If $Z_\sigma \cap \overline{Z_{\sigma'}}=\emptyset$, we have (using Lemma~\ref{lemma:directImageCst}) $\iota_\sigma^{-1} {\iota_{\sigma'}}_* k_{Z_{\sigma'}} \simeq \iota_\sigma^{-1} k_{\overline{Z_{\sigma'}}}\simeq 0$, which is easy to check on stalks. This proves the first part of the statement.
On the other hand, if $Z_\sigma \subset \overline{Z_{\sigma'}}$, we get $\iota_\sigma^{-1} {\iota_{\sigma'}}_* k_{Z_{\sigma'}} = \iota_\sigma^{-1} k_{\overline{Z_{\sigma'}}} = k_{Z_\sigma}$, and hence (with a similar argument for the dual as above)
\begin{align*}
\iota_\sigma^! {\iota_{\sigma'}}_! k_{Z_{\sigma'}} &\simeq \mathbb{D}D_{Z_\sigma} (\iota_\sigma^{-1} {\iota_{\sigma'}}_* k_{Z_{\sigma'}} [\dim Z_{\sigma'}])\simeq (\mathbb{D}D_{Z_\sigma} k_{Z_\sigma}) [-\dim Z_{\sigma'}]\\
& \simeq k_{Z_\sigma}[\dim Z_{\sigma}-\dim Z_{\sigma'}].
\end{align*}
Finally, this implies (if $Z_\sigma \subset \overline{Z_{\sigma'}}$)
\begin{align*}
\mathbb{R}Hom({\iota_\sigma}_! k_{Z_\sigma},{\iota_{\sigma'}}_! k_{Z_{\sigma'}})&\simeq \mathbb{R}R {\iota_\sigma}_* \mathbb{R}Hom(k_{Z_\sigma}, k_{Z_\sigma}[\dim Z_{\sigma}-\dim Z_{\sigma'}])\\ &\simeq \mathbb{R}R {\iota_\sigma}_* k_{Z_\sigma}[\dim Z_{\sigma}-\dim Z_{\sigma'}]\\ &\simeq k_{\overline{Z_\sigma}}[\dim Z_{\sigma}-\dim Z_{\sigma'}].
\end{align*}
\end{proof}
Nest, we first establish the following lemma, which is the ``sheaf hom'' version of \cite[Lemma 2.6]{BHHS22}. In loc.~cit., we considered the functor $\mathrm{Hom}$ instead of $\mathcal{H}om$ for compactly supported $\mathbb{R}$-constructible sheaves. However, we did not go into the details of the proof, but rather just indicated the induction to be performed. Here, we will give more details on the technique. We are grateful to Takuro Mochizuki for providing us the idea for this statement and its proof (including for Lemma~\ref{lemma:RHomSimplices}).
\begin{lemma}\label{lemma:HomRcCpt}
Let $L/K$ be a field extension. Let $X$ be a real analytic manifold and let $\mathcal{F},\mathcal{G}\in\ModRc{K_X}$. Assume that $\mathcal{F}$ and $\mathcal{G}$ have compact support. Then there is an isomorphism
$$\mathcal{H}om(L_X\otimes_{K_X}\mathcal{F},L_X\otimes_{K_X}\mathcal{G}) \simeq L_X\otimes_{K_X} \mathcal{H}om(\mathcal{F},\mathcal{G}).$$
\end{lemma}
\begin{proof}
We choose a locally finite subanalytic stratification $X=\bigsqcup_{\alpha\in A} X_\alpha$ such that $\mathcal{F}$ and $\mathcal{G}$ are locally constant on each stratum $X_\alpha$ (this is a common refinement of the stratifications that exist for $\mathcal{F}$ and $\mathcal{G}$ since they are $\mathbb{R}$-constructible, cf.\ \cite[Lemma 8.3.21]{KS90}).
By \cite[Proposition 8.2.5]{KS90}, there exists then a simplicial complex $\mathbf{S}$ with a homeomorphism $i\colon |\mathbf{S}|\overset{\sim}{\to} X$ such that in particular any $i(Z_\sigma)$ is contained in some $X_\alpha$, and hence the sheaves $i^{-1}\mathcal{F}$ and $i^{-1}\mathcal{G}$ are constant on $Z_\sigma$ for any simplex $\sigma$ of $\mathbf{S}$ (since simplices are contractible, local constancy already implies constancy). Let us write $\mathcal{S}\vcentcolon= |\mathbf{S}|$ and $F\vcentcolon=i^{-1}\mathcal{F}$, $G\vcentcolon=i^{-1}\mathcal{G}$. The statement we need to prove is equivalent to proving
\begin{equation*}
\mathcal{H}om(L_{\mathcal{S}}\otimes_{K_{\mathcal{S}}}F,L_\mathcal{S}\otimes_{K_\mathcal{S}}G) \simeq L_\mathcal{S}\otimes_{K_\mathcal{S}} \mathcal{H}om(F,G).
\end{equation*}
This will follow from the more general statement
\begin{equation}\label{eq:isoOnS}
\mathbb{R}Hom(L_{\mathcal{S}}\otimes_{K_{\mathcal{S}}}F,L_\mathcal{S}\otimes_{K_\mathcal{S}}G) \simeq L_\mathcal{S}\otimes_{K_\mathcal{S}} \mathbb{R}Hom(F,G).
\end{equation}
that we will prove now. (Remember that we still take $F,G$ to be sheaves, not elements of the derived category of sheaves.)
Assume first that $F={\iota_{\sigma}}_! K_{Z_\sigma}$ and $G$ is an arbitrary compactly supported sheaf on $\mathcal{S}$ which is constant on any $Z_\sigma$. We will argue by induction on the dimension of the support of $G$:
Lemma~\ref{lemma:RHomSimplices} shows that the isomorphism \eqref{eq:isoOnS} is true if $\dim\mathop{\mathrm{supp}} G=0$, since in this case $G$ is supported on finitely many points, i.e.\ $G\simeq \bigoplus_{i=1}^{k} {\iota_{\{p_i\}}}_! (K_{\{p_i\}})^{r_i}$ for some points ($0$-dimensional simplices) $p_1,\ldots, p_k\in \mathcal{S}$ and some $r_1,\ldots,r_k\in\mathbb{Z}_{>0}$, and $\mathbb{R}Hom$ as well as tensor products commute with direct sums.
Now suppose the isomorphism \eqref{eq:isoOnS} is proved for $\dim\mathop{\mathrm{supp}} G\leq n$ for some $n$, and consider a $G$ with $\dim\mathop{\mathrm{supp}} G=n+1$. Then let $Z_{n+1}$ be the (disjoint) union of all $n+1$-dimensional simplices in the support of $G$ (these are finitely many since the support of $G$ is compact), and denote by $Z_{\leq n}$ the union of all other simplices contained in the support of $G$, so that $\mathop{\mathrm{supp}} G = Z_{n+1}\sqcup Z_{\leq n}$. Note that $Z_{n+1}\subset \mathop{\mathrm{supp}} G$ is open. We therefore have a short exact sequence
$$0\longrightarrow G_{Z_{n+1}} \longrightarrow G \longrightarrow G_{Z_{\leq n}}\longrightarrow 0.$$
Considering the right derived functors of $\mathcal{H}_{F}^L\vcentcolon=\mathcal{H}om(L_X\otimes_{K_X}F, L_X\otimes_{K_X}(-))$ and ${}^L\mathcal{H}_{F}\vcentcolon= L_X\otimes_{K_X} \mathcal{H}om(F,-)$ as well as the natural morphism from the second to the first from Lemma~\ref{lemma:natMorph}, we obtain a commutative diagram in $\mathbb{D}bL{\mathcal{S}}$ whose rows are distinguished triangles:
$$\begin{tikzcd}
\mathbb{R}R\mathcal{H}_{F}^L(G_{Z_{n+1}})\arrow{r} & \mathbb{R}R\mathcal{H}_{F}^L(G)\arrow{r} & \mathbb{R}R\mathcal{H}_{F}^L(G_{Z_{\leq n}})\arrow{r}{+1} &\text{}\\
\mathbb{R}R{}^L\mathcal{H}_{F}(G_{Z_{n+1}})\arrow{r}\arrow{u} & \mathbb{R}R{}^L\mathcal{H}_{F}(G)\arrow{r}\arrow{u} & \mathbb{R}R{}^L\mathcal{H}_{F}(G_{Z_{\leq n}})\arrow{r}{+1}\arrow{u} & \text{}
\end{tikzcd}$$
The first vertical arrow is an isomorphism by Lemma~\ref{lemma:RHomSimplices} since $G_{Z_{n+1}}$ is direct sum of sheaves of the form ${\iota_{Z_\sigma}}_! (K_{Z_\sigma})^{r_\sigma}$ for some $(n+1)$-dimensional simplices $\sigma$ (the connected components of $Z_{n+1}$) and some $r_\sigma\in\mathbb{Z}_{>0}$. On the other hand, the third vertical arrow is an isomorphism by the induction hypothesis. This implies that the middle arrow is an isomorphism by the axioms of a triangulated category.
Finally, by an analogous induction on the dimension of the support of $F$, using the statement just proved instead of Lemma~\ref{lemma:RHomSimplices}, we conclude that the isomorphism \eqref{eq:isoOnS} holds for general $F$ and $G$ that are constant on each simplex.
\end{proof}
We can now deduce the following proposition. Its statement was also established in \cite{BHHS22}, where, however, this was deduced a posteriori from a similar result for ind-sheaves. We give a direct proof using the above Lemma~\ref{lemma:HomRcCpt}.
\begin{prop}[{see \cite[Lemma 2.7]{BHHS22}}]\label{prop:BHHSsheafhom}
Let $L/K$ be a field extension. Let $X$ be a real analytic manifold and let $\mathcal{F},\mathcal{G}\in\mathbb{D}bRcK{X}$. Then there is an isomorphism
$$\mathbb{R}Hom(L_X\otimes_{K_X}\mathcal{F},L_X\otimes_{K_X}\mathcal{G}) \simeq L_X\otimes_{K_X} \mathbb{R}Hom(\mathcal{F},\mathcal{G}).$$
\end{prop}
\begin{proof}
Let us first assume that $\mathcal{F},\mathcal{G}\in \ModRc{K_X}$. Then we will prove
\begin{equation}\label{eq:isoHomNC}
\mathcal{H}om(L_X\otimes_{K_X}\mathcal{F},L_X\otimes_{K_X}\mathcal{G}) \simeq L_X\otimes_{K_X} \mathcal{H}om(\mathcal{F},\mathcal{G}).
\end{equation}
There is a natural morphism between these two objects (from right to left, cf.\ Lemma~\ref{lemma:natMorph}), and it suffices to prove locally on an open covering that it is an isomorphism. Choose therefore an open covering $X=\bigcup_{i\in I} U_i$ by relatively compact open subsets $U_i\subseteq X$.
Now note that, writing $j_i\colon U_i\hookrightarrow X$ for the inclusion, we have
\begin{align*}
(L_X\otimes_{K_X} \mathcal{H}om(\mathcal{F},\mathcal{G}))|_{U_i} &\simeq L_{U_i}\otimes_{K_{U_i}} \mathcal{H}om(\mathcal{F}|_{U_i},\mathcal{G}|_{U_i})\\
&\simeq L_{U_i}\otimes_{K_{U_i}} \mathcal{H}om((\mathcal{F})_{U_i}|_{U_i},(\mathcal{G})_{U_i}|_{U_i})\\
&\simeq \big(L_X\otimes_{K_X} \mathcal{H}om((\mathcal{F})_{U_i},(\mathcal{G})_{U_i})\big)|_{U_i}
\end{align*}
because $(\mathcal{F})_{U_i}|_{U_i} \simeq j_i^{-1}{j_i}_!j_i^{-1}\mathcal{F}\simeq j_i^{-1}\mathcal{F}\simeq \mathcal{F}|_{U_i}$.
Similarly,
\begin{align*}
\mathcal{H}om(L_X\otimes_{K_X}\mathcal{F},L_X\otimes_{K_X}\mathcal{G})|_{U_i} &\simeq \mathcal{H}om(L_X\otimes_{K_X}\mathcal{F}_{U_i},L_X\otimes_{K_X}\mathcal{G}_{U_i})|_{U_i}.
\end{align*}
Since $\mathcal{F}_{U_i}$ and $\mathcal{G}_{U_i}$ have compact support (it is clear that $\mathop{\mathrm{supp}} \mathcal{F}_{U_i}=(\mathop{\mathrm{supp}}\mathcal{F})\cap \overline{U_i}$, i.e.\ the intersection of a closed and a compact subset of $X$), it follows from Lemma~\ref{lemma:HomRcCpt} that the restriction of \eqref{eq:isoHomNC} to each $U_i$ is an isomorphism, and hence that \eqref{eq:isoHomNC} itself is.
The statement of the proposition now follows by deriving functors.
\end{proof}
To deduce a statement about homomorphism spaces, (i.e.\ global sections of the isomorphism in Proposition~\ref{prop:BHHSsheafhom}), we need to study the behaviour of $\mathbb{R}$-constructible sheaves with respect to global sections. This is done in the following lemma.
\begin{lemma}\label{lemma:sectionsScalarextCpt}
Let $X$ be a compact real analytic manifold, let $L/K$ be a (not necessarily finite) field extension, and let $\mathcal{G}\in\ModRc{K_X}$ be an $\mathbb{R}$-constructible sheaf on $X$. Then there is an isomorphism
$$(L_X\otimes_{K_X}\mathcal{G})(X) \simeq L\otimes_K \mathcal{G}(X).$$
\end{lemma}
We prove this lemma after showing an auxiliary statement about the local behaviour of $\mathbb{R}$-constructible sheaves.
\begin{lemma}\label{lemma:ConstrStalkExt}
Let $k$ be a field and $\mathcal{G}\in\ModRc{k_X}$. Then for any $p\in X$ and for every $\delta>0$ there exists an open, contractible neighbourhood $U_p\subset B_\delta(p)$ of $p$ such that
$$\Gamma(U_p;\mathcal{G})\to \mathcal{G}_p$$
is an isomorphism. In particular, any element of the stalk can be uniquely extended to a section in a sufficiently small neighbourhood.
\end{lemma}
\begin{proof}
Let $p\in X$ and $\delta>0$, and set $V=B_\delta(p)$. Then $\mathcal{G}|_V$ is $\mathbb{R}$-constructible (see \cite[Proposition 8.4.10]{KS90}) and hence there exists a stratification $V=\bigsqcup_{\alpha\in A} V_\alpha$ such that $\mathcal{G}$ is locally constant on the strata (\cite[Lemma 8.3.21]{KS90}). We can, without loss of generality, assume that $\{p\}$ is a stratum, by passing to a refinement. There exists hence (see \cite[Proposition 8.2.5]{KS90}) a simplicial complex $\mathbf{S}$ with geometric realization $\mathcal{S}\vcentcolon=|\mathbf{S}|$ and a homeomorphism $i\colon |\mathbf{S}| \overset{\sim}{\to} V$ such that $i^{-1}(\mathcal{G}|_V)$ is constant on all the $Z_\sigma$. (In other words, $i^{-1}(\mathcal{G}|_V)$ is $\mathbf{S}$-constructible.)
We write $q\vcentcolon=i^{-1}(p)$, and this point is a vertex of the simplicial complex $\mathbf{S}$. Set $U\vcentcolon= U(\{q\})\subseteq \mathcal{S}$ (this is the union of all simplices with $q$ in their boundary, and it is an open subset of $\mathcal{S}$) and $F\vcentcolon= i^{-1}(\mathcal{G}|_V)$.
Then, by \cite[Proposition 8.1.4]{KS90}, the natural map $\Gamma(U;F)\to F_q$ is an isomorphism, and hence $\Gamma(U_p;\mathcal{G})\to\mathcal{G}_p$ is an isomorphism if we set $U_p\vcentcolon= i(U)$.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lemma:sectionsScalarextCpt}]
Let $p\in X$. Then, due to Lemma~\ref{lemma:ConstrStalkExt}, for any $\delta>0$, there exists $U_p\subset B_\delta(p)$ such that the morphisms
$$\Gamma(U_p;\mathcal{G})\to \mathcal{G}_p\quad \text{and}\quad \Gamma(U_p;L_X\otimes_{K_X}\mathcal{G})\to (L_X\otimes_{K_X}\mathcal{G})_p$$
is an isomorphism.
In other words, an element of the stalk extends uniquely to a section on a sufficiently small neighbourhood, regardless of a chosen upper bound for this neighbourhood.
Now, we know that $(L_X\otimes_{K_X} \mathcal{G})_p \simeq L\otimes_K \mathcal{G}_p$ and hence $\Gamma(U_p;L_X\otimes_{K_X}\mathcal{G})\simeq L\otimes_K \Gamma(U_p;\mathcal{G})$.
Now we choose such a neighbourhood $U^\delta_p$ for any $p\in X$ and $\delta>0$. Note that $\mathcal{T}\vcentcolon=\{U^\delta_p\mid p\in X,\delta>0\}$ is a basis of the topology of $X$, in particular, it is an open covering of $X$. Since $X$ is compact, there exists therefore a finite number of $U_i\in \mathcal{T}$, $i\in I$, that cover $X$, and there is an exact sequence
$$0\to\Gamma(X;\mathcal{G}) \to \prod_{i\in I} \Gamma(U_i;\mathcal{G}) \to \prod_{i,j\in I} \Gamma(U_i\cap U_j;\mathcal{G}).$$
We apply the functor $L\otimes_K(-)$ to this sequence and obtain (noting that tensor products commute with finite products and using the above observations)
$$0\to L\otimes_K\Gamma(X;\mathcal{G}) \to \prod_{i\in I} \Gamma(U_i;L_X\otimes_{K_X}\mathcal{G}) \to \prod_{i,j\in I} \Gamma(U_i\cap U_j;L_X\otimes_{K_X} \mathcal{G}).$$
On the other hand, due to the sheaf property of $L_X\otimes_{K_X} \mathcal{G}$, the first object of this sequence is isomorphic to $\Gamma(X;L_X\otimes_{K_X} \mathcal{G})$.
\end{proof}
With this in hand, we can now state the description of the space of homomorphisms between extensions of scalars.
\begin{prop}\label{prop:morphLificationCpt}
Let $L/K$ be a field extension and let $X$ be a compact real analytic manifold. Let $\mathcal{F},\mathcal{G}\in\mathbb{D}bRcK{X}$. Then there is an isomorphism
$$\mathrm{Hom}_{\mathbb{D}bRcL{X}}(L_X\otimes_{K_X}\mathcal{F}, L_X\otimes_{K_X}\mathcal{G}) \simeq L\otimes_{K}\mathrm{Hom}_{\mathbb{D}bRcK{X}}(\mathcal{F}, \mathcal{G}).$$
\end{prop}
\begin{proof}
By Lemma~\ref{lemma:sectionsScalarextCpt}, we get the isomorphism of functors $\Gamma\circ (L_X\otimes_{K_X} (-))\simeq L\otimes_K \Gamma(-)$ on $\ModRc{K_X}$, where $\Gamma=\Gamma(X;-)$ denotes the functor of global sections. By deriving functors, we get an isomorphism of functors $\mathbb{R}R \Gamma\circ (L_X\otimes_{K_X}(-))\simeq L\otimes_K\mathbb{R}R\Gamma(-)$ on $\mathbb{D}bRcK{X}$.
We then apply this functor to the object $\mathbb{R}Hom(\mathcal{F},\mathcal{G})$ (which is also in $\mathbb{D}bRcK{X}$, cf.\ \cite[Proposition 8.4.10]{KS90}) and get
\begin{align*}
L\otimes_K \mathbb{R}R \mathrm{Hom}(\mathcal{F},\mathcal{G}) &\simeq \mathbb{R}R\Gamma(L_X\otimes_{K_X}\mathbb{R}Hom(\mathcal{F},\mathcal{G}))\\
&\simeq \mathbb{R}R\Gamma(\mathbb{R}Hom(L_X\otimes_{K_X}\mathcal{F},L_X\otimes_{K_X}\mathcal{G}))\\
&\simeq \mathbb{R}R \mathrm{Hom}(L_X\otimes_{K_X}\mathcal{F},L_X\otimes_{K_X}\mathcal{G}).
\end{align*}
Here, the second isomorphism follows from Proposition~\ref{prop:BHHSsheafhom}. Then, the assertion follows by taking zeroth cohomology.
\end{proof}
Finally, we can use our results of this subsection to further ``complete'' the statement of Lemma~\ref{lemma:compatExt} with some more compatibilities in the case where $X$ is an oriented differentiable manifold by studying the duality functor. Note, however, that for the exceptional inverse image, we need an $\mathbb{R}$-constructibility assumption here.
\begin{cor}[of Proposition~\ref{prop:BHHSsheafhom}]\label{cor:dualityExt}
Let $L/K$ a field extension and let $X$ be a real analytic manifold.
\begin{itemize}
\item[(a)] Let $\mathcal{F}\in\mathbb{D}bK{X}$. If $L/K$ is finite, or if $\mathcal{F}\in\mathbb{D}bRcK{X}$, then there is an isomorphism $$\mathbb{D}D_X(L_X\otimes_{K_X}\mathcal{F})\simeq L_X\otimes_{K_X} \mathbb{D}D_X\mathcal{F}.$$
\item[(b)] Let $f\colon X\to Y$ be a morphism of real analytic manifolds, and let $\mathcal{F}\in \mathbb{D}bRcK{Y}$. Then there is an isomorphism $$f^!(L_Y\otimes_{K_Y}\mathcal{F})\simeq L_X\otimes_{K_X} f^!\mathcal{F}.$$
\item[(c)] If $X$ is a complex manifold and $\mathcal{G}\in\PervK{X}$, then $L_X\otimes_{K_X}\mathcal{G}$.
\end{itemize}
\end{cor}
\begin{proof}
\begin{itemize}
\item[(a)] It follows from \cite[Proposition 3.3.4]{KS90} that we have the relation $\omega_X^L\simeq L_X\otimes_{K_X}\omega_X^K$ between the dualizing complexes in $\mathbb{D}bK{X}$ and $\mathbb{D}bL{X}$. Then we get
\begin{align*}
\mathbb{D}D_X(L_X\otimes_{K_X}\mathcal{F}) &= \mathbb{R}Hom_{L_X}(L_X\otimes_{K_X}\mathcal{F},\omega_X^L)\\
&\simeq \mathbb{R}Hom_{L_X}(L_X\otimes_{K_X}\mathcal{F},L_X\otimes_{K_X} \omega^K_X)\\
&\simeq L_X\otimes_{K_X} \mathbb{R}Hom_{K_X}(\mathcal{F},\omega^K_X)=L_X\otimes_{K_X} \mathbb{D}D_X\mathcal{F}.
\end{align*}
Here, we have applied Proposition~\ref{prop:morphLification} (if $L/K$ is finite) or Proposition~\ref{prop:BHHSsheafhom} (if $\mathcal{F}\in\mathbb{D}bRcK{X}$) in the third line.
\item[(b)] Since $\mathcal{F}$ is a complex of $\mathbb{R}$-constructible sheaves (and so is $L_X\otimes_{K_X}\mathcal{F}$), we have $f^!\simeq \mathbb{D}D_X \circ f^{-1}\circ \mathbb{D}D_Y$ (cf.\ \cite[Proposition 8.4.9 and Exercise VIII.3]{KS90}). Hence, the statement follows from (a) and Lemma~\ref{lemma:compatExt}(b), since $\mathbb{D}_Y$ and $f^{-1}$ preserve $\mathbb{R}$-constructibility (see \cite[Propositions 8.4.9 and 8.4.10]{KS90}).
\item[(c)] It is clear by Lemma~\ref{lemma:compatExt}(c) that extension of scalars does not change the support of the cohomologies of $\mathcal{F}$. Therefore, $L_X\otimes_{K_X}\mathcal{F}$ and $\mathbb{D}D_X(L_X\otimes_{K_X} \mathcal{F})\overset{(a)}{\simeq}L_X\otimes_{K_X} \mathbb{D}D_X(\mathcal{F})$ satisfy the support condition if $\mathcal{F}$ and $\mathbb{D}D_X\mathcal{F}$ do.
\end{itemize}
\end{proof}
\subsection{$K$-structures and constructibility}
Let $L/K$ be a field extension, and let $X$ be a topological space.
We have seen in Lemma~\ref{lemma:compatExt} and Corollary~\ref{cor:dualityExt} that extension of scalars preserves constructibility and perversity. Here, we will show that constructiblity of a sheaf of $L$-vector spaces also descends to a $K$-structure.
\begin{lemma}\label{lemma:latticeLocSys}
Let $\mathcal{F}\in\Mod{L_X}$ be a local system of $L$-vector spaces (of finite rank), and let $\mathcal{G}\in\Mod{K_X}$ be a $K$-structure of $\mathcal{F}$. Then $\mathcal{G}$ is a local system of $K$-vector spaces (of finite rank).
\end{lemma}
\begin{proof}
We reproduce the proof in \cite{BHHS22} slightly differently here for completeness.
A local system is characterized by the fact that each $x\in X$ has an open neighbourhood $U\subseteq X$ such that for any $y\in U$ there is an isomorphism $\mathcal{F}_x\overset{\sim}{\to}\mathcal{F}_y$ such that the diagram
$$\begin{tikzcd}
&\mathcal{F}(U)\arrow{ld} \arrow{rd}\\
\mathcal{F}_x \arrow{rr}{\sim} & & \mathcal{F}_y
\end{tikzcd}$$
commutes. (This is a nice exercise in basic sheaf theory.) We are given this property for $\mathcal{F}$ and want to prove the analogous property for $\mathcal{G}$.
This follows from the commutative diagram
$$\begin{tikzcd}
&L\otimes_K \mathcal{G}(U)\arrow{ld} \arrow{rd}\arrow[bend left]{ddd}\\
L\otimes_K \mathcal{G}_x \arrow{d}{\sim} \arrow[dashed]{rr}{} & & L\otimes_K \mathcal{G}_y\arrow{d}{\sim}\\
\mathcal{F}_x \arrow{rr}{\sim} & & \mathcal{F}_y\\
&\mathcal{F}(U)\arrow{ru} \arrow{lu}
\end{tikzcd}$$
It remains to observe that the dashed morphism descends to a morphism $\mathcal{G}_x\to\mathcal{G}_y$ (we can shrink $U$ if necessary and then an element in $\mathcal{G}_x$ lifts to one in $\mathcal{G}(U)$ and hence its image is in $\mathcal{G}_y$), and that this is an isomorphism if and only if the dashed is so.
\end{proof}
\begin{cor}
Let $\mathcal{F}\in\mathbb{D}bL{X}$ and let $\mathcal{G}\in\mathbb{D}bK{X}$ be a $K$-lattice of $\mathcal{F}$.
\begin{itemize}
\item[(a)] If $\mathrm{H}^i\mathcal{F}$ is locally constant on some $Z\subseteq X$, so is $\mathrm{H}^i(\mathcal{G})$.
In particular, if $X$ is a real analytic manifold and $\mathcal{F}\in\mathbb{D}bRcL{X}$ (resp.\ $X$ is a complex manifold and $\mathcal{F}\in\mathbb{D}bCcL{X}$), then $\mathcal{G}\in \mathbb{D}bRcK{X}$ (resp.\ $\mathcal{G}\in\mathbb{D}bCcK{X}$).
\item[(b)] If $X$ is a complex manifold and $\mathcal{F}\in\PervL{X}$, then $\mathcal{G}\in\PervK{X}$.
\end{itemize}
\end{cor}
\begin{proof}
\begin{itemize}
\item[(a)] follows directly from Lemma~\ref{lemma:latticeLocSys}, since restriction and cohomology commute with extension of scalars (see Lemma~\ref{lemma:compatExt}), so $\mathrm{H}^i(\mathcal{F})|_Z\simeq L_{Z}\otimes_{K_Z} \mathrm{H}^i(\mathcal{G})|_Z$ being a local system of finite rank implies that $\mathrm{H}^i(\mathcal{G})|_Z$ is a local system of finite rank.
\item[(b)] By (a), the sheaf $\mathcal{G}$ is locally constant and nontrivial on the same covering as $\mathcal{F}$ (and the according statement holds for the duals due to Corollary~\ref{cor:dualityExt}). Hence the supports of the cohomologies of $\mathcal{F}$ (resp.\ $\mathbb{D}D_X\mathcal{F}$) are the same as those for $\mathcal{G}$ (resp.\ $\mathbb{D}D_X\mathcal{G}$) and hence the support condition for $\mathcal{G}$ and $\mathbb{D}D_X\mathcal{G}$ holds if it holds for $\mathcal{F}$ and $\mathbb{D}D_X\mathcal{F}$.
\end{itemize}
\end{proof}
\section{Galois descent for sheaves and their complexes}\label{sec:GaloisDescent}
Let $L/K$ be a finite Galois extension with Galois group $G$. In this section, we will formulate Galois descent for sheaves of vector spaces. Afterwards, we are going to investigate a similar procedure for derived categories of sheaves of vector spaces, and we will describe what kind of problems arise there and prevent us from obtaining an equally nice result. Finally, we restrict ourselves to a particularly nice subcategory of the derived category of sheaves of vector spaces, namely that of perverse sheaves, and we establish Galois descent for them, using a construction performed by A.\ Beilinson in \cite{Bei}.
\subsection{Galois descent for sheaves of vector spaces}
Galois descent for sheaves of vector spaces has already been studied in \cite{BHHS22}, but the statement was not formulated as an equivalence of categories in loc.~cit. We reformulate it here to fit into the framework set up in Section~\ref{sec:GaloisDescent}, using our results from Section~\ref{sec:Hom}.
\begin{prop}[{cf.\ \cite[Lemma 2.13]{BHHS22}}]\label{prop:GaloisDescentSheaves}
Let $L/K$ be a finite Galois extension. Let $X$ be a topological space.
For any $\mathcal{F}\in\Mod{L_X}$ equipped with a $G$-structure $(\varphi_g)_{g\in G}$, there exists a $K$-structure $\mathcal{G}\in\Mod{K_X}$ together with a natural isomorphism $\psi\colon L_X\otimes_{K_X} \mathcal{G}\overset{\sim}{\to} \mathcal{F}$ such that the natural $G$-structure on $L_X\otimes_{K_X}\mathcal{G}$ corresponds via this isomorphism to the given one on $\mathcal{F}$, i.e.\ such that, for any $g\in G$, the following diagram commutes:
$$\begin{tikzcd}
\mathcal{F}\arrow{rr}{\varphi_g} && \overline{\mathcal{F}}^g\\ \\
L_X\otimes_{K_X}\mathcal{G}\arrow{uu}{\psi}\arrow{rr}{g\otimes\mathrm{id}_\mathcal{G}} && \overline{L_X}^g\otimes_{K_X}\mathcal{G}\arrow{uu}{\overline{\psi}^g}
\end{tikzcd}$$
In particular, the functor $\Phi_{L/K}=L_X\otimes_{K_X}(-)$ induces an equivalence
$$\Phi_{L/K}^G\colon \Mod{K_X}\longrightarrow \Mod{L_X}^G.$$
\end{prop}
\begin{proof}
We proceed analogously to the construction for vector spaces. We consider $\mathcal{F}$ as a sheaf of $K$-vector spaces and each $\varphi_g$ as a $K$-linear automorphism of $\mathcal{F}$. Then $\mathcal{F}_K$ is defined as
$$\mathcal{F}_K\vcentcolon= \ker \big(\prod\limits_{g\in G} (\varphi_g-\mathrm{id})\colon \mathcal{F}\longrightarrow \prod_{g\in G} \mathcal{F}\big)\in\Mod{K_X}.$$
Since kernels of sheaves are computed sectionwise and sections of $L_X\otimes_{K_X} \mathcal{F}_K$ are also obtained by sectionwise applying $L\otimes_K(-)$ (see Lemma~\ref{lemma:sectionsScalarext}), the isomorphism $L_X\otimes_{K_X}\mathcal{F}_K\simeq \mathcal{F}$ is obtained from Galois descent for vector spaces. The commutation of the above square is also clear from the construction of $\mathcal{F}_K$ as a sheaf of invariants.
This proves in particular essential surjectivity of the functor $\Phi_{L/K}^G$.
Full faithfulness follows from Proposition~\ref{prop:morphLification}.
\end{proof}
\begin{rem}
We can also similarly establish a Galois descent statement for functor categories as in Example~\ref{ex:GConj}(b). They admit an obvious functor of extension of scalars $\mathrm{Funct}(\mathcal{C},\Vect{K})\to \mathrm{Funct}(\mathcal{C},\Vect{L})$. This yields then in particular Galois descent for presheaves of vector spaces (if we take $\mathcal{C}=\mathrm{Op}(X)^\mathrm{op}$), and to deduce Proposition~\ref{prop:GaloisDescentSheaves}, one just needs to check that the descent of a sheaf is still a sheaf. (The extension of scalars for sheaves is indeed the same as the one for presheaves in the finite Galois case due to Lemma~\ref{lemma:sectionsScalarext}.)
Choosing more complicated categories $\mathcal{C}$ (so-called categories of exit paths), one can also express certain categories of constructible sheaves as functor categories. This technique is known as \emph{exodromy} (see e.g.\ \cite{Treumann} and \cite{BGH}), and it has also been set up in the framework of quasi-categories (see \cite{Lurie}, and \cite{PT22} for a recent generalization). Such exodromy equivalences might serve as an alternative approach to our questions for certain constructible sheaves, and they might also lead to a clearer study of complexes of sheaves. We will, however, not take this viewpoint here. We are grateful to Jean-Baptiste Teyssier for drawing our attention to these constructions.
\end{rem}
\subsection{The case of derived categories of sheaves}
Having established the above equivalence (Theorem~\ref{prop:GaloisDescentSheaves}) for sheaves, one would, of course, like to generalize such a statement to derived categories of sheaves. Indeed, for a finite Galois extension $L/K$ with Galois group $G$, we still have the functor induced by extension of scalars
\begin{align*}
\Phi_{L/K}^G\colon \mathbb{D}bK{X} &\longrightarrow \mathbb{D}bL{X}^G.
\end{align*}
The following is deduced directly -- as above -- from Proposition~\ref{prop:morphLification}.
\begin{prop}\label{prop:ExtDerivedFF}
The functor $L_X\otimes_{K_X} (-)$ is fully faithful on $\mathbb{D}bK{X}$.
\end{prop}
\begin{rem}
Essential surjectivity does not seem to hold in this more general situation. An indication for why this makes sense is the following: To an object $\mathcal{F}^\bullet=\ldots\to\mathcal{F}_i\to \mathcal{F}_{i+1}\to\ldots$, the functor associates the object $L_X\otimes_{K_X}\mathcal{F}^\bullet = \ldots\to L_X\otimes_{K_X}\mathcal{F}_i\to L_X\otimes_{K_X}\mathcal{F}_{i+1}\to\ldots$ (due to the exactness of the tensor product), together with the natural $G$-structure given by that on $L_X$, i.e.\
$$\begin{tikzcd}
\ldots \arrow{r} & L_X\otimes_{K_X}\mathcal{F}_i \arrow{r}\arrow{d}{g\otimes\mathrm{id}} & L_X\otimes_{K_X}\mathcal{F}_{i+1} \arrow{r}\arrow{d}{g\otimes\mathrm{id}} & \ldots\\
\ldots \arrow{r} & \overline{L_X}^g\otimes_{K_X}\mathcal{F}_i \arrow{r} & \overline{L_X}^g\otimes_{K_X}\mathcal{F}_{i+1} \arrow{r} & \ldots
\end{tikzcd}$$
In particular, this $G$-structure is given by morphisms of complexes (rather than roofs, as is the general case for morphisms in the derived category). In general, morphisms $\varphi_g$ in the derived category can -- after choosing suitable resolutions -- be represented as morphisms of complexes, but the compatibilities that the $\varphi_g$ have to satisfy will only hold up to homotopy of morphisms of complexes, and hence in general such a $G$-structure will not be in the essential image.
\end{rem}
A standard technique for proofs of statements in derived categories is by induction on the amplitude of a complex. At least for complexes concentrated in two successive degrees, we can use this approach to deduce some existence statement of a $K$-lattice from the existence of a $G$-structure.
\begin{prop}\label{prop:descentDerived}
Let $X$ be a real analytic manifold and $\mathcal{F}^\bullet\in \mathbb{D}bL{X}$ with a $G$-structure $(\varphi_g)_{g\in G}$. Assume that $\mathcal{F}^\bullet$ is concentrated in degrees $a$ and $a+1$ (i.e.\ the only non-vanishing cohomologies are $H^a(\mathcal{F}^\bullet)$ and $H^{a+1}(\mathcal{F}^\bullet)$. Then there exists $\mathcal{F}^\bullet_K\in \mathbb{D}bK{X}$ and an isomorphism $L_X\otimes_{K_X} \mathcal{F}^\bullet_K \simeq \mathcal{F}^\bullet$.
\end{prop}
\begin{proof}
We know the statement for sheaves (i.e.\ complexes concentrated in one degree), so we know that $H^a(\mathcal{F}^\bullet)\simeq L_X\otimes_{K_X}\mathcal{G}_a$ and $H^{a+1}(\mathcal{F}^\bullet)\simeq L_X\otimes_{K_X}\mathcal{G}_{a+1}$ such that under these isomorphisms the $G$-structures on $H^i(\mathcal{F}^\bullet)$ induced by the one on $\mathcal{F}^\bullet$ coincide with those given by the natural $G$-structure on $L_X$.
Using the standard truncation functors for complexes (with respect to the standard t-structure on $\mathbb{D}bRcL{X}$), there is a distinguished triangle
$$H^{a+1}(\mathcal{F}^\bullet)[-a-1]\longrightarrow H^a(\mathcal{F}^\bullet)[-a] \longrightarrow \mathcal{F}^\bullet \longrightarrowPO$$
and for any $g\in G$ the $G$-structure on $\mathcal{F}^\bullet$ induces an isomorphism of distinguished triangles
\begin{equation}\label{eq:diagTruncation}\begin{tikzcd}
H^{a+1}(\mathcal{F}^\bullet)[-1]\arrow{r}{f}\arrow{d}[swap]{\simeq} & H^a(\mathcal{F}^\bullet) \arrow{r} \arrow{d}[swap]{\simeq} & \mathcal{F}^\bullet \arrow{r}{+1} \arrow{d}{\varphi_g}[swap]{\simeq} & \text{ }\\
\overline{H^{a+1}(\mathcal{F}^\bullet)[-1]}^g\arrow{r} & \overline{H^a(\mathcal{F}^\bullet)}^g \arrow{r} & \overline{\mathcal{F}^\bullet}^g \arrow{r}{+1} & \text{ }
\end{tikzcd}\end{equation}
(note that truncation and conjugation commute).
By Lemma~\ref{prop:morphLification}, there is a morphism $\tilde{f} \colon \mathcal{G}_{a+1}[-1]\to \mathcal{G}_{a}$ such that $f=g\otimes 1$. We can complete it to a distinguished triangle
$$\mathcal{G}_{a+1}[-1]\overset{\tilde{f}}{\longrightarrow} \mathcal{G}_{a} \longrightarrow \mathcal{G}^\bullet \longrightarrowPO$$
where is $\mathcal{G}^\bullet$ is unique up to (non-unique) isomorphism. Hence, there exists a (non-unique) isomorphism $\gamma\colon \mathcal{G}^\bullet\otimes_{K_X}L_X \overset{\sim}{\to} \mathcal{F}^\bullet$. In other words, we have completed \eqref{eq:diagTruncation} to a commutative diagram
$$\begin{tikzcd}[scale cd=0.7]
&[-15pt]\mathcal{G}_{a+1}[-a-1]\otimes_{K_X}L_X\arrow{ld}[swap]{\simeq}\arrow{dd}[crossing over]{} \arrow{rr}{g\otimes 1}&[-15pt]&[-15pt] \mathcal{G}_{a}[-a]\otimes_{K_X}L_X\arrow{ld}[swap]{\simeq}\arrow{dd}[crossing over]{} \arrow{rr} &[-15pt]&[-10pt] \mathcal{G}^\bullet\otimes_{K_X}L_X\arrow{ld}[swap]{\simeq}\arrow[dashed,crossing over]{dd} \arrow{r}{+1} &[-10pt] \text{ } \\
H^{a+1}(\mathcal{F}^\bullet)[-a-1]\arrow{rr}[near start]{f}\arrow{dd}[swap]{\simeq} && H^a(\mathcal{F}^\bullet)[-a] \arrow{rr} && \mathcal{F}^\bullet \arrow{r}{+1} & \text{ }\\
&\mathcal{G}_{a+1}[-a-1]\otimes_{K_X}\overline{L_X}^g\arrow{ld}[swap]{\simeq} \arrow{rr}{} && \mathcal{G}_{a}[-a]\otimes_{K_X}\overline{L_X}^g\arrow{ld}[swap]{\simeq} \arrow{rr}{} &&\mathcal{G}^\bullet\otimes_{K_X}\overline{L_X}^g\arrow{ld}[swap]{\simeq} \arrow{r}{+1} & \text{ } \\
\overline{H^{a+1}(\mathcal{F}^\bullet)[-a-1]}^g\arrow{rr} && \overline{H^a(\mathcal{F}^\bullet)[-a]}^g \arrow{rr} \arrow[crossing over,<-]{uu}[near end]{\simeq} && \overline{\mathcal{F}^\bullet}^g \arrow{r}{+1} \arrow[crossing over,<-]{uu}[near end]{\simeq}[swap,near end]{\varphi_g} & \text{ }
\end{tikzcd}$$
\end{proof}
\begin{rem}
The construction in the proof above is very non-canonical. This is due to the fact that the third objects and morphisms in distinguished triangles are not unique up to unique isomorphism. In particular, although we find an object $\mathcal{G}^\bullet$ with $L_X\otimes_{K_X}\mathcal{G}^\bullet\simeq \mathcal{F}^\bullet$ here, it seems not clear that the given $G$-structure corresponds to the natural one on $L_X\otimes_{K_X} \mathcal{G}^\bullet$. In other words, it is not clear that the dashed arrow in the big diagram is given by $\mathrm{id}_{\mathcal{G}^\bullet}\otimes g$. This has two implications:
\begin{itemize}
\item First, this theorem does not show that the object $(\mathcal{F},(\varphi_g)_{g\in G})$ is in the essential image of the functor $L_X\otimes_{K_X} (-)$. It just shows that $\mathcal{F}^\bullet$ can be realized as such a tensor product if there exists a $G$-structure on it (but not saying that exactly this $G$-structure comes through the tensor product).
\item Secondly, we cannot proceed inductively to get similar results for complexes concentrated in more than two degrees since the fact that the $G$-structure corresponds to the natural one on the tensor product is crucial for descending the morphism $f$ to $\tilde{f}$.
\end{itemize}
\end{rem}
\iffalse
\begin{prop}
Let $X$ be a complex manifold and $\mathcal{F}^\bullet\in \mathbb{D}bRcC{X}$ with an isomorphism $\varphi\colon \mathcal{F}^\bullet \overset{\sim}{\to} \overline{\mathcal{F}^\bullet}$. Then there exists $\mathcal{F}^\bullet_\mathbb{R}\in \mathbb{D}bRcR{X}$ and an isomorphism $\mathcal{F}^\bullet_\mathbb{R} \otimes_{\mathbb{R}_X} \mathbb{C}_X \simeq \mathcal{F}^\bullet$.
\end{prop}
\begin{proof}
Since we know the statement for sheaves (i.e.\ complexes concentrated in one degree), we want to proceed by induction on the amplitude of $\mathcal{F}^\bullet$, which we denote by $n=b-a$, where $\mathcal{F}^\bullet$ is concentrated in cohomological degrees between $a$ and $b$, $a<b$. Assume therefore that $\mathcal{F}^\bullet$ is such a complex and that the statement is known for amplitude $\leq n-1$.
Using the standard truncation functors for complexes (with respect to the standard t-structure on $\mathbb{D}bRcC{X}$), there is a distinguished triangle
$$\tau_{>a}\mathcal{F}^\bullet[-1]\longrightarrow \tau_{\leq a}\mathcal{F}^\bullet \longrightarrow \mathcal{F}^\bullet \longrightarrowPO$$
and $\varphi$ induces an isomorphism of distinguished triangles
\begin{equation}\label{eq:diagTruncation}\begin{tikzcd}
\tau_{>a}\mathcal{F}^\bullet[-1]\arrow{r}{f}\arrow{d}[swap]{\simeq} & \tau_{\leq a}\mathcal{F}^\bullet \arrow{r} \arrow{d}[swap]{\simeq} & \mathcal{F}^\bullet \arrow{r}{+1} \arrow{d}{\varphi}[swap]{\simeq} & \text{ }\\
\overline{\tau_{>a}\mathcal{F}^\bullet[-1]}\arrow{r} & \overline{\tau_{\leq a}\mathcal{F}^\bullet} \arrow{r} & \overline{\mathcal{F}^\bullet} \arrow{r}{+1} & \text{ }
\end{tikzcd}\end{equation}
(note that truncation and conjugation commute).
By the induction hypothesis, there exist $\mathcal{G}_{>a}, \mathcal{G}_{\leq a}\in \mathbb{D}bRcR{X}$ with $\tau_{>a}\mathcal{F}^\bullet\simeq \mathcal{G}_{>a}\otimes_{\mathbb{R}_X}\mathbb{C}_X$ and $\tau_{\leq a}\mathcal{F}^\bullet\simeq \mathcal{G}_{\leq a}\otimes_{\mathbb{R}_X}\mathbb{C}_X$.
By Lemma~\ref{lemma:morphComplexification}, there is a morphism $g\colon \mathcal{G}_{>a}[-1]\to \mathcal{G}_{\leq a}$ such that $f=g\otimes 1$. We can complete it to a distinguished triangle
$$\mathcal{G}_{>a}[-1]\overset{g}{\longrightarrow} \mathcal{G}_{\leq a} \longrightarrow \mathcal{G}^\bullet \longrightarrowPO$$
where is $\mathcal{G}^\bullet$ is unique up to (non-unique) isomorphism. Hence, there exists a (non-unique) isomorphism $\gamma\colon \mathcal{G}^\bullet\otimes_{\mathbb{R}_X}\mathbb{C}_X \overset{\sim}{\to} \mathcal{F}^\bullet$ such that the composition $\overline{\gamma}^{-1}\phi\gamma$ is the canonical morphism induced by complex conjugation. THIS IS NOT CLEAR. In other words, we have completed \eqref{eq:diagTruncation} to a commutative diagram
$$\begin{tikzcd}[scale cd=0.7]
&\mathcal{G}_{>a}\otimes_{\mathbb{R}_X}\mathbb{C}_X\arrow{ld}{\sim}\arrow{dd}[crossing over]{} \arrow{rr}{g\otimes 1}&& \mathcal{G}_{\leq a}\otimes_{\mathbb{R}_X}\mathbb{C}_X\arrow{ld}{\sim}\arrow{dd}[crossing over]{} \arrow{rr} && \mathcal{G}^\bullet\otimes_{\mathbb{R}_X}\mathbb{C}_X\arrow{ld}{\sim}\arrow{dd}[crossing over]{} \arrow{r}{+1} & \text{ } \\
\tau_{>a}\mathcal{F}^\bullet[-1]\arrow{rr}[near start]{f}\arrow{dd}[swap]{\simeq} && \tau_{\leq a}\mathcal{F}^\bullet \arrow{rr} \arrow{dd} && \mathcal{F}^\bullet \arrow{r}{+1} \arrow{dd}{\varphi}[swap]{\simeq} & \text{ }\\
&\mathcal{G}_{>a}\otimes_{\mathbb{R}_X}\overline{\mathbb{C}_X}\arrow{ld}{\sim} \arrow{rr}[crossing over]{} && \mathcal{G}_{\leq a}\otimes_{\mathbb{R}_X}\overline{\mathbb{C}_X}\arrow{ld}{\sim} \arrow{rr} [crossing over]{} &&\mathcal{G}^\bullet\otimes_{\mathbb{R}_X}\overline{\mathbb{C}_X}\arrow{ld}{\sim} \arrow{r}{+1} & \text{ } \\
\overline{\tau_{>a}\mathcal{F}^\bullet[-1]}\arrow{rr} && \overline{\tau_{\leq a}\mathcal{F}^\bullet} \arrow{rr} && \overline{\mathcal{F}^\bullet} \arrow{r}{+1} & \text{ }
\end{tikzcd}$$
\end{proof}
Denote by $\mathbb{D}bRcCG{X}$ the category whose objects are pairs $(\mathcal{F}^\bullet,\varphi)$ with $\mathcal{F}^\bullet\in\mathbb{D}bRcC{X}$ and $\varphi\colon\mathcal{F}^\bullet \overset{\sim}{\to} \overline{\mathcal{F}^\bullet}$ an isomorphism. Morphisms in $\mathbb{D}bRcCG{X}$ will be morphisms of the corresponding $\mathcal{F}^\bullet$ compatible with the $\varphi$.
\begin{thm}
The functor
\begin{align*}
\mathbb{D}bRcR{X} &\to \mathbb{D}bRcCG{X}\\
\mathcal{G}^\bullet &\mapsto \mathcal{G}^\bullet \otimes_{\mathbb{R}_X} \mathbb{C}_X
\end{align*}
is an equivalence of categories.
\end{thm}
\begin{proof}
First, it is not difficult to see that the functor is well-defined. Essential surjectivity is proved in Proposition~\ref{prop:descentDerived} and full faithfulness follows from Lemma~\ref{lemma:morphComplexification}.
\end{proof}
\begin{rem}
Due to the non-canonicity of the third object in a distinguished triangle, the quasi-inverse of complexification is not nicely described. To any complex $\mathcal{F}$, you would have to choose a real $\mathcal{G}$ and an isomorphism $\mathcal{G}\otimes_\mathbb{R} \mathbb{C}\simeq \mathcal{F}$. Then any complex morphism between $\mathcal{F}$s commuting with conjugation defines uniquely a morphism between $\mathcal{G}\otimes_\mathbb{R}\mathbb{C}$s commuting with conjugation and hence a unique morphism of $\mathcal{G}$s by the lemma.
\end{rem}
\fi
\subsection{Galois descent for perverse sheaves}
It is reasonable to expect that the results are closer to the statements for sheaves if we do not consider general objects of the derived category, but perverse sheaves.\footnote{Thinking in terms of Algebraic Analysis, perverse sheaves are the counterparts of regular holonomic D-modules -- objects concentrated in one degree -- via the Riemann--Hilbert correspondence (see \cite{KasRHreg}), so these are the objects to understand if one wants to understand topologically the category of regular holonomic D-modules.}
Let $L/K$ be a finite Galois extension, and let $X$ be a complex manifold.
The notion of $G$-conjugation on $\PervL{X}$ as well as the functors of extension and restriction of scalars on are inherited from those between $\mathbb{D}bK{X}$ and $\mathbb{D}bL{X}$.
Given a perverse sheaf $\mathcal{F}\in\PervL{X}$ together with a $G$-structure $(\varphi_g)_{g\in G}$, we can define the presumptive $K$-structure similarly to the case of vector spaces and sheaves, namely as an object of invariants. Contrarily to the case of derived categories, we dispose of the notion of kernels here, since perverse sheaves form an abelian category.
Consider the underlying $K$-perverse sheaf $\mathcal{F}^K\in\PervK{X}$ (which is the same for every $\overline{\mathcal{F}}^g$) with the automorphisms $\varphi^K_g\colon \mathcal{F}^K\to\mathcal{F}^K$ induced by the $\varphi_g$.
Then define $$\mathcal{G}\vcentcolon= \ker\big(\prod_{g\in G}(\varphi_g^K-\mathrm{id}_{\mathcal{F}^K})\colon \mathcal{F}^K\longrightarrow \prod_{g\in G} \mathcal{F}^K\big).$$
Clearly, we have a morphism $\mathcal{G}\to \mathcal{F}^K$, and hence by Lemma~\ref{lemma:ScalarExtAdj}, we have a morphism
\begin{equation}\label{eq:latticeComplPerv}
\mathcal{G}\otimes_{K_X} L_X \to \mathcal{F}.
\end{equation}
We will prove that it is an isomorphism. For this, we will use a description of perverse sheaves due to Beilinson \cite{Bei} (see also \cite{Rei} for some more details and complements on Beilinson's article).
\paragraph{Beilinson's equivalence} Let us recall the idea of Beilinson's ``gluing'' of perverse sheaves, and study the properties of his equivalence with respect to field extensions. We will not review all the details, for which we refer to \cite{Bei} and \cite{Rei}. From now on, we work in the context of complex analytic varieties, which is a generalization of (but often analogous to) the theory of complex manifolds.
Let $k$ be a field, $X$ a complex manifold and $f\colon X\to\mathbb{C}$ a holomorphic function. Write $Z\vcentcolon=f^{-1}(0)$ and $U\vcentcolon= X\setminus Z$ with inclusion $j\colon U\hookrightarrow X$. Moreover, we fix a generator $t$ of the fundamental group $\pi_1(\mathbb{C}\setminus\{0\})$.
There is a functor $\Psi^\mathrm{un}_f\colon \Pervk{U}\to \Pervk{Z}$ of \emph{unipotent nearby cycles}. By its construction, the fundamental group $\pi_1(\mathbb{C}\setminus\{0\})$ acts on $\Psi^\mathrm{un}_f(\mathcal{F}_U)$ for any $\mathcal{F}_U\in\Pervk{U}$. In particular, the fixed generator $t$ induces an endomorphism of any such $\Psi^\mathrm{un}_f(\mathcal{F}_U)$, which we will still denote by $t$.
There is also a functor $\Phi^\mathrm{un}_f\colon \Pervk{X}\to\Pervk{Z}$ of \emph{unipotent vanishing cycles}.
One defines the category of \emph{gluing data} $\mathcal{GD}_f(X,k)$ to be the category whose objects are tuples $(\mathcal{F}_U,\mathcal{F}_Z,u,v)$, where $\mathcal{F}_U\in\Pervk{U}$, $\mathcal{F}_Z\in\Pervk{Z}$ and $u$ and $v$ are morphisms
$$\Psi^\mathrm{un}_f(\mathcal{F}_U)\overset{u}{\to} \mathcal{F}_Z \overset{v}{\to} \Psi^\mathrm{un}_f(\mathcal{F}_U)$$
such that their composition coincides with the endomorphism $\mathrm{id}-t$ of $\Psi^\mathrm{un}_f(\mathcal{F}_U)$.
A morphism in $\mathcal{GD}_f(X,k)$ is defined in the obvious way, as two morphisms of perverse sheaves on $U$ and $Z$, respectively, making the natural diagram with the $u$ and $v$ commute.
It is shown (see \cite[Proposition 3.1]{Bei} or \cite[Theorem 3.6]{Rei}) that $\mathcal{GD}_f(X,k)$ is an abelian category and that there is an equivalence
$$\mathsf{F}_f\colon \Pervk{X}\overset{\sim}{\longrightarrow} \mathcal{GD}_f(X,k),$$
sending $\mathcal{F}\in\Pervk{X}$ to the tuple $(j^{-1}\mathcal{F},\Phi^\mathrm{un}_f(\mathcal{F}),u,v)$, where we will not go into detail with the construction of the maps $u$ and $v$. Also the quasi-inverse is explicitly described.
Let us now study this equivalence in the context of a finite field extension $L/K$. The construction of the nearby and vanishing cycles functor as well as of the maps $u$ and $v$ are completely topological and do not depend on the coefficient field (they could as well be performed on sheaves of sets). Therefore, the forgetful functor (restriction of scalars) $\mathsf{for}_{L/K}\colon \PervL{X}\to\PervK{X}$ corresponds to a forgetful functor $\mathcal{GD}_f(X,L)\to\mathcal{GD}_f(X,K)$ (given by the ones on perverse sheaves on $U$ and $Z$).
On the other hand, we have the following statement about extension of scalars.
\begin{lemma}\label{lemma:NCVCext}
For $\mathcal{A}\in\PervK{U}$, we have an isomorphism
$$\Psi^\mathrm{un}_f(L_U\otimes_{K_U}\mathcal{A})\simeq L_Z\otimes_{K_Z} \Psi^\mathrm{un}_f(\mathcal{A}).$$
For $\mathcal{B}\in\PervK{X}$, we have an isomorphism
$$\Phi^\mathrm{un}_f(L_X\otimes_{K_X}\mathcal{B})\simeq L_Z\otimes_{K_Z} \Phi^\mathrm{un}_f(\mathcal{B}).$$
The scalar extension functor $\Phi_{L/K}\colon \PervK{X}\to\PervL{X}$ corresponds to the functor
\begin{align*}
\mathcal{GD}_f(X,K)&\longrightarrow\mathcal{GD}_f(X,L)\\
(\mathcal{F}_U,\mathcal{F}_Z,u,v)&\longmapsto (L_U\otimes_{K_U}\mathcal{F}_U,L_Z\otimes_{K_Z}\mathcal{F}_Z,\mathrm{id}_{L_Z}\otimes u,\mathrm{id}_{L_Z}\otimes v)
\end{align*}
via Beilinson's equivalence $\mathsf{F}_f$.
\end{lemma}
\begin{proof}
The construction of $\Psi^\mathrm{un}_f(\mathcal{A})$ is performed as follows: One first defines the nearby cycles functor $\mathbb{R}R \varphi_f = i^{-1} \mathbb{R}R j_* \mathbb{R}R v_* v^{-1}$ (where $v\colon U\times_{\mathbb{C}\setminus\{0\}} \widetilde{\mathbb{C}\setminus\{0\}}\to U$ is the canonical map, with $\widetilde{\mathbb{C}\setminus\{0\}}$ the universal covering; this is, however, not important for what follows). Then one notices that $t$ acts naturally on $\mathbb{R}R \varphi_f(\mathcal{A})$ and that there is a decomposition $\mathbb{R}R \varphi_f(\mathcal{A})\simeq \mathbb{R}R \varphi_f^\mathrm{un}(\mathcal{A}) \mathrm{op}lus \mathbb{R}R \varphi_f^{\neq 1}(\mathcal{A})$, where $\mathrm{id}-t$ is nilpotent on the first and an automorphism on the second summand. Then one sets $\Psi^\mathrm{un}_f(\mathcal{A})\vcentcolon= \mathbb{R}R \varphi_f^\mathrm{un}(\mathcal{A})[-1]$.
It is clear that $\mathbb{R}R \varphi_f$ commutes with extension of scalars (see Lemma~\ref{lemma:compatExt} and Corollary~\ref{cor:directImage}). Moreover, the action of $t$ is induced purely topologically, i.e.\ the action of $t$ on $\mathbb{R}R \varphi_f(L_U\otimes_{K_U}\mathcal{A})\simeq L_Z\otimes_{K_Z} \mathbb{R}R \varphi_f(\mathcal{A})$ is induced by the one on $\mathbb{R}R \varphi_f(\mathcal{A})$. Hence, the part of $\mathbb{R}R \varphi_f(L_U\otimes_{K_U}\mathcal{A})$ on which $\mathrm{id}-t$ is nilpotent will be exactly $L_Z\otimes_{K_Z} \mathbb{R}R \varphi_f^\mathrm{un}(\mathcal{A})$. This proves the first statement.
The construction of $\Phi^\mathrm{un}_f(\mathcal{B})$ is roughly as follows: One first defines the \emph{maximal extension functor} $\Xi_f\colon \Pervk{U}\to \Pervk{X}$ and a complex
$$j_!j^{-1}\mathcal{B}\to \Xi_f(j^{-1}\mathcal{B})\mathrm{op}lus \mathcal{B}\to j_*j^{-1}\mathcal{B}.$$
Then one defines $\Phi^\mathrm{un}_f(\mathcal{B})$ as the cohomology of this complex and notes that it is supported on $Z$.
Without going too much into the details of the construction, let us just mention that the definition of $\Xi_f$ and the morphisms in the above complex is again just topological, i.e.\ using operations that do not depend upon the exact field of coefficients (such as natural morphisms $j_!\to j_*$, inclusions/projections, kernels etc.). Therefore and due to Lemma~\ref{lemma:compatExt} and Corollary~\ref{cor:directImage}, the complex associated to $L_X\otimes_{K_X} \mathcal{B}$ is $$L_X\otimes_{K_X} j_!j^{-1}\mathcal{B}\to L_X\otimes_{K_X} \big(\Xi_f(j^{-1}\mathcal{B})\mathrm{op}lus \mathcal{B}\big)\to L_X\otimes_{K_X} j_*j^{-1}\mathcal{B}$$ and finally, since extension of scalars is exact and hence commutes with taking cohomology, we get the second isomorphism of the lemma.
For the last statement, the arguments are similar, recognizing that the definition of the maps $u$ and $v$ is topological and can therefore be defined over the smaller field and just ``upgraded'' to $L$.
\end{proof}
Now let $L/K$ be a field extension and $G\vcentcolon=\mathrm{Aut}(L/K)$ (for our purposes, it will be a finite Galois extension with Galois group $G$). There is then an obvious $G$-conjugation on $\mathcal{GD}_f(X,L)$ defined by
$$\overline{(\mathcal{F}_U,\mathcal{F}_Z,u,v)}^g \vcentcolon= (\overline{\mathcal{F}_U}^g,\overline{\mathcal{F}_Z}^g,\overline{u}^g,\overline{v}^g)$$
for any $g\in G$, i.e.\ simply induced by the natural $G$-conjugations on $\PervL{U}$ and $\PervL{Z}$.
\begin{lemma}\label{lemma:NCVCconj}
For $\mathcal{A}\in\PervK{U}$, we have an isomorphism
$$\Psi^\mathrm{un}_f(\overline{\mathcal{A}}^g)\simeq \overline{\Psi^\mathrm{un}_f(\mathcal{A})}^g.$$
For $\mathcal{B}\in\PervK{X}$, we have an isomorphism
$$\Phi^\mathrm{un}_f(\overline{\mathcal{B}}^g)\simeq \overline{\Phi^\mathrm{un}_f(\mathcal{B})}^g.$$
Under Beilinson's equivalence $\mathsf{F}_f\colon \PervL{X}\overset{\sim}{\to} \mathcal{GD}_f(X,L)$, the natural $G$-conjugations on both categories correspond to each other, and hence a $G$-structure on $\mathcal{F}\in\PervL{X}$ induces a $G$-structure on $\mathsf{F}_f(\mathcal{F})$ and vice versa.
\end{lemma}
\begin{proof}
Similarly to the proof of Lemma~\ref{lemma:NCVCext}, the first statement follows mainly from the fact that conjugation is compatible with direct and inverse image functors. Moreover, since conjugation is an autoequivalence, the endomorphism $\mathrm{id}-t$ is nilpotent if and only if $\mathrm{id}-\overline{t}^g=\overline{\mathrm{id}-t}^g$ is.
The second and third statements are again due to the fact that the whole construction of $\Phi^\mathrm{un}_f$, $u$ and $v$ do not depend on the field structure and hence are the same if defined before or after applying $g$-conjugation.
\end{proof}
\paragraph{Application to Galois descent of perverse sheaves} We are now ready to prove that the object $\mathcal{G}$ constructed above is actually a $K$-structure of $\mathcal{F}$.
\begin{prop}
The morphism \eqref{eq:latticeComplPerv} is an isomorphism.
\end{prop}
\begin{proof}
Let $\mathcal{F}\in\PervL{X}$ be a perverse sheaf on a complex analytic variety $X$ and let $(\varphi_g)_{g\in G}$ be a $G$-structure on it. Let $\mathcal{G}\in\PervK{X}$ be the invariant $K$-perverse subsheaf (defined analogously as above) with its natural morphism $\mathcal{G}\to\mathcal{F}$.
Since $\mathcal{F}$ is perverse, it is in particular a complex of sheaves with $\mathbb{C}$-constructible cohomologies. For each of the (finitely many) cohomology sheaves $\mathrm{H}^i(\mathcal{F})$, there exists a locally finite covering $X=\bigcup_\alpha X^i_\alpha$ by $\mathbb{C}$-analytic subsets\footnote{We follow the terminology in \cite{KS90} here: A subset $Y\subset X$ is called complex analytic if for any point $x\in X$ there exists an open neighbourhood $U\subset X$ of $x$ and (finitely many) holomorphic functions $f_1,\ldots,f_n\in\mathcal{O}_X(U)$ such that $A\cap U=\{f_1=\ldots=f_n=0\}$. Moreover, $Y$ is called $\mathbb{C}$-analytic if $\overline{Y}$ and $\overline{Y}\setminus Y$ are complex analytic subsets of $X$.} on which $\mathrm{H}^i(\mathcal{F})$ is locally constant. Moreover, the problem is local (and restriction to an open subset is exact), so we can assume that the set of all $X^i_\alpha$ is finite.
If all the $X^i_\alpha$ are of maximal dimension, this means that every cohomology sheaf if locally constant on $X$. By the definition and basic properties of perverse sheaves, $\mathrm{H}^i(\mathcal{F})=0$ for $i<-\dim X$ and $\dim \mathop{\mathrm{supp}} \mathrm{H}^{-i}(\mathcal{F})\leq i$ for any $i\in \mathbb{Z}$. This implies that $\mathcal{F}$ is concentrated in cohomological degree $-\dim X$, i.e.\ $\mathcal{F}\simeq \mathcal{L}[\dim U]$ for some locally constant sheaf $\mathcal{L}\in\Mod{L_X}$. Hence, the statement follows from Proposition~\ref{prop:GaloisDescentSheaves} and we are done.
Now, assume that there exist $X_\alpha^i$ of non-maximal dimension.
Then each of the $X^i_\alpha$ not having maximal dimension is contained in the zero locus of an analytic function that is not identically zero (since $\overline{X^i_\alpha}$ is analytic and not equal to the whole space), yielding a finite family of functions $(f_k)_{k\in I}$, $I=\{1,\ldots,m\}$. We can multiply these functions to obtain $f\vcentcolon= f_1\cdot\ldots\cdot f_k$ whose zero locus contains all the $X^i_\alpha$ of non-maximal dimension.
Consider now $Z\vcentcolon= f^{-1}(0)$, $U\vcentcolon= X\setminus Z$ and the inclusion $j\colon U\hookrightarrow X$. By Beilinson's equivalence, the datum of $\mathcal{F}$ is equivalent to the tuple
$$(j^{-1}\mathcal{F}, \Phi^\mathrm{un}_f(\mathcal{F}), u, v)\in \mathcal{GD}_f(X,L).$$
By Lemma~\ref{lemma:NCVCconj}, it still comes equipped with a $G$-structure, which we denote by $(\widetilde{\varphi}_g)_{g\in G}$.
Accordingly, $\mathcal{G}$ corresponds to a tuple
$$(j^{-1}\mathcal{G},\Phi^\mathrm{un}_f(\mathcal{G}),u_K, v_K)\in \mathcal{GD}_f(X,K),$$
Due to the compatibility of Beilinson's equivalence with forgetful functors (restriction of scalars) $\mathsf{for}_{L/K}$ and kernels (equivalences of abelian categories are exact), we see that this object is actually the kernel of $$\prod_{g\in G} (\widetilde{\varphi}_g-\mathrm{id})\colon \mathsf{F}_f(\mathcal{F})^K\to \prod_{g\in G} \mathsf{F}_f(\mathcal{F})^K$$
in the category $\mathcal{GD}_f(X,K)$, and the morphism induced by $L_X\otimes_{K_X}\mathcal{G}\to\mathcal{F}$ via $\mathsf{F}_f$ is nothing but the natural morphism
$$(L_U\otimes_{K_U}j^{-1}\mathcal{G},L_Z\otimes_{K_Z}\Phi^\mathrm{un}_f(\mathcal{G}),\mathrm{id}_{L_Z}\otimes u_K, \mathrm{id}_{L_Z}\otimes v_K)\to (j^{-1}\mathcal{F}, \Phi^\mathrm{un}_f(\mathcal{F}), u, v)$$
by Lemma~\ref{lemma:NCVCext}.
To prove that it is an isomorphism, we need to prove that $L_U\otimes_{K_U}j^{-1}\mathcal{G}\to j^{-1}\mathcal{F}$ and $L_Z\otimes_{K_Z}\Phi^\mathrm{un}_f(\mathcal{G})\to \Phi^{\mathrm{un}}_f(\mathcal{F})$ are isomorphisms, where $j^{-1}\mathcal{G}$ (resp.\ $\Phi^\mathrm{un}_f(\mathcal{G})$) is the perverse sheaf of invariants of the induced $G$-structure on $j^{-1}\mathcal{F}$ (resp.\ $\Phi^\mathrm{un}_f(\mathcal{F})$), since $j^{-1}$ (resp.\ $\Phi^\mathrm{un}_f$) is exact and hence commutes with kernels.
For the first isomorphism, note that $j^{-1}\mathcal{F}$ is a complex of sheaves on $U$ whose cohomologies are all locally constant $L_U$-modules of finite rank. Then, with the same arguments as above, $j^{-1}\mathcal{F}\simeq \mathcal{L}[\dim U]$ for some locally constant sheaf $\mathcal{L}\in\Mod{L_U}$. Hence, the desired isomorphism follows from Proposition~\ref{prop:GaloisDescentSheaves}.
For the second isomorphism, note that $\Phi^\mathrm{un}_f(\mathcal{F})$ is a perverse sheaf on the complex analytic variety $Z$ and $\dim Z=\dim X - 1$. Hence, we can apply the same technique (determining, at least locally, a suitable covering of $Z$, choosing a suitable function that vanishes on all the sets of non-maximal dimension and applying Beilinson's equivalence) to this perverse sheaf. We continue this recursively, and the procedure will end if all elements of the covering are of maximal dimension, which will be the case at the latest when $\dim Z=0$, which shows that this inductive procedure terminates.
This concludes the proof.
\end{proof}
The statement just proved shows the essential surjectivity part of Galois descent for perverse sheaves. Full faithfulness is inherited from the derived category (Proposition~\ref{prop:ExtDerivedFF}). We have therefore proved the following statement. (We will formulate in a slightly more general setting than we did in the rest of this work, namely in the context of complex analytic varieties, since this is what we have actually proved.)
\begin{thm}
Let $X$ be a complex analytic variety. Then the functor of extension of scalars $\Phi_{L/K}\colon \PervK{X}\to\PervL{X}$ induces an equivalence
$$\Phi_{L/K}^G\colon \PervK{X}\to\PervL{X}^G.$$
\end{thm}
\noindent\textsc{Andreas Hohl\\ Université Paris Cité and Sorbonne Université, CNRS, IMJ-PRG, F-75013 Paris, France}\\ [email protected]
\end{document}
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\begin{document}
\begin{abstract}
Let $K$ be a finite extension of $\mathbf{Q}_p$. The field of norms of a $p$-adic Lie extension $K_\infty/K$ is a local field of characteristic $p$ which comes equipped with an action of $\mathrm{Gal}(K_\infty/K)$. When can we lift this action to characteristic $0$, along with a compatible Frobenius map? In this note, we formulate precisely this question, explain its relevance to the theory of $(\varphi,\Gamma)$-modules, and give a condition for the existence of certain types of lifts.
\end{abstract}
\title{Lifting the field of norms}
\tableofcontents
\setlength{\baselineskip}{18pt}
\section*{Introduction}\label{intro}
Let $K$ be a finite extension of $\mathbf{Q}_p$ and let $K_\infty/K$ be a totally ramified Galois extension whose Galois group $\Gamma_K$ is a $p$-adic Lie group (or, more generally, a ``strictly arithmetically profinite'' extension). Let $k_K$ denote the residue field of $K$. We can attach to $K_\infty/K$ its field of norms $X_K(K_\infty)$, a field of characteristic $p$ that is isomorphic to $k_K \dpar{\pi}$ and equipped with an action of $\Gamma_K$. Let $E$ be a finite extension of $\mathbf{Q}_p$ such that $k_E=k_K$. In this note, we consider the question: when can we lift the action of $\Gamma_K$ on $k_K \dpar{\pi}$ to the $p$-adic completion of $\mathcal{O}_E \dcroc{T}[1/T]$, which is a complete ring of characteristic $0$ that lifts $X_K(K_\infty)$, along with a compatible $\mathcal{O}_E$-linear Frobenius map $\varphi_q$? When it is possible to do so, we say that the action of $\Gamma_K$ is \emph{liftable}. In this case, Fontaine's construction of $(\varphi,\Gamma)$-modules applies, and we get the following well-known equivalence of categories (where $\mathbf{A}_K$ denotes the $p$-adic completion of $\mathcal{O}_E \dcroc{T}[1/T]$).
\begin{enonce*}{Theorem A}\label{pgmintro}
If the action of $\Gamma_K$ is liftable, then there is an equivalence of categories
\[ \text{$\{ (\varphi_q,\Gamma_K)$-modules on $\mathbf{A}_K \} \longleftrightarrow \{ \mathcal{O}_E$-linear representations of $G_K\}$}. \]
\end{enonce*}
Such a lift is possible when $K_\infty/K$ is the cyclotomic extension, or more generally when $K_\infty$ is generated by the torsion points of a Lubin-Tate formal $\mathcal{O}_F$-module for some $F \subset K$. In \S \ref{finht} of this note, we prove the following partial converse.
\begin{enonce*}{Theorem B}\label{liftintro}
If the action of $\Gamma_K$ is liftable with $\varphi_q(T) \in \mathcal{O}_E \dcroc{T}$, then $\Gamma_K$ is abelian, and there is an injective character $\Gamma_K \to \mathcal{O}_E^\times$, whose conjugates by $\mathrm{Emb}(E,\mathbf{Q}_pbar)$ are all de Rham with weights in $\mathbf{Z}_{\geqslant 0}$.
\end{enonce*}
At the end of \S \ref{finht}, we give some examples of constraints on the extension $K_\infty/K$ arising from the existence of such a character.
Some preliminary computations suggest that a similar result may hold in certain cases if we assume that $\varphi_q(T)$ is an overconvergent power series in $T$. However at this point, I do not know for which extensions we can expect the action of $\Gamma_K$ to be liftable in general.
The initial motivation for thinking about this problem was the question of whether there is a theory of ``anticyclotomic $(\varphi,\Gamma)$-modules'', that is a theory of $(\varphi,\Gamma)$-modules where $\Gamma$ is the Galois group of the anticyclotomic extension $K_\infty^{\mathrm{ac}}/K$ of $K = \mathbf{Q}_{p^2}$. Theorem B implies that there is no such theory if in addition we require that $\varphi_q(T) \in \mathcal{O}_K \dcroc{T}$.
\section{Lifting the field of norms}\label{fonsec}
Let $K$ be a finite extension of $\mathbf{Q}_p$ and let $K_\infty$ be an infinite and totally ramified Galois extension of $K$ that is ``strictly arithmetically profinite'' (see \S 1.2.1 of \cite{WCN} for the definition, which we don't use; \emph{arithmetically profinite} means that the ramification subgroups $\Gamma_K^u$ of $\Gamma_K$ are open and strictness is an additional condition). Note that if $\Gamma_K = \mathrm{Gal}(K_\infty/K)$ is a $p$-adic Lie group, then as recalled in \S 1.2.2 of \cite{WCN}, it follows from the main theorem of \cite{S72} that $K_\infty/K$ is strictly arithmetically profinite.
We can apply to $K_\infty/K$ the ``field of norms'' construction of \cite{FW,FW2} and \cite{WCN}, which we now recall. Let $\mathcal{F}$ denote the set of finite extensions $F$ of $K$ that are contained in $K_\infty$, and let $X_K(K_\infty)$ denote the set of sequences $(x_F)_{F \in \mathcal{F}}$ such that $\mathrm{N}_{F_2/F_1}(x_{F_2}) = x_{F_1}$ whenever $F_1 \subset F_2$. By the results of \S 2 of \cite{WCN}, one can endow $X_K(K_\infty)$ with the structure of a field, a field embedding of $k_K =k_{K_\infty}$ in $X_K(K_\infty)$, and a valuation $\mathrm{val}(\cdot)$ (where $\mathrm{val}(x)$ is the common value of the $\mathrm{val}_F(x_F)$ for $F \in \mathcal{F}$). We then have the following theorem (theorem 2.1.3 of \cite{WCN}).
\begin{theo}\label{cdnthm}
The field $X_K(K_\infty)$ is a complete valued field with residue field $k_K$.
\end{theo}
If $\pi_K$ denotes a uniformizer of $X_K(K_\infty)$, then $X_K(K_\infty) = k_K \dpar{\pi_K}$. The group $\Gamma_K$ acts on $X_K(K_\infty)$. If $q = \mathrm{Card}(k_K)$, then we have the $k_K$-linear Frobenius map $\varphi_q : X_K(K_\infty) \to X_K(K_\infty)$ given by $x \mapsto x^q$, and it commutes with the action of $\Gamma_K$.
Let $E$ be a finite extension of $\mathbf{Q}_p$ such that $k_E = k_K$, let $\varpi_E$ be a uniformizer of $E$ and let $\mathbf{A}_K$ denote the $\varpi_E$-adic completion of $\mathcal{O}_E \dcroc{T}[1/T]$. The ring $\mathbf{A}_K$ is a $\varpi_E$-Cohen ring for $X_K(K_\infty)$, that is a complete discrete valuation ring whose maximal ideal is generated by $\varpi_E$ and whose residue field is $X_K(K_\infty)$. The question that we want to ask is: when can we lift the action of $\Gamma_K$ on $X_K(K_\infty)$ to an $\mathcal{O}_E$-linear action on $\mathbf{A}_K$, along with a compatible $\mathcal{O}_E$-linear Frobenius lift?
\begin{enonce}{Question}\label{quest}
Are there power series $\{F_g(T)\}_{g \in \Gamma_K}$ and $P(T)$ in $\mathbf{A}_K$, such that
\begin{enumerate}
\item $\overline{F}_g(\pi_K) = g(\pi_K)$ and $\overline{P}(\pi_K) = \pi_K^q$ in $k_K \dpar{\pi_K}$,
\item $F_g \circ P = P \circ F_g$ and $F_h \circ F_g = F_{gh}$ whenever $g,h \in \Gamma_K$?
\end{enumerate}
If the answer to this question is ``yes'', then we say that the action of $\Gamma_K$ is \emph{liftable}.
\end{enonce}
If $K$ is unramified over $\mathbf{Q}_p$ and $K_\infty = K(\mu_{p^\infty})$ is the cyclotomic extension, then the action of $\Gamma_K$ is liftable, since for the uniformizer $\pi_K = ( (\zeta_{p^n}-1)_{K(\zeta_{p^n})} )_{n \geqslant 1}$ we can take $P(T)=(1+T)^p-1$ and $F_g(T)=(1+T)^{\chi_{\mathrm{cyc}}(g)}-1$. More generally, if $K_\infty/K$ is the extension generated by the torsion points of a Lubin-Tate formal $\mathcal{O}_F$-module, with $K/F$ unramified, then the action of $\Gamma_K$ is liftable with $P(T)=[\varpi_F](T)$ and $F_g(T)=[g](T)$ in appropriate coordinates.
Write $\mathbf{E}_K$ for the field $X_K(K_\infty)$ (this notation is somewhat standard, but unfortunate considering the fact that $\mathbf{E}_K$ depends on $K_\infty$ but not on $K$). Recall that every finite separable extension of $\mathbf{E}_K$ is of the form $\mathbf{E}_L$ where $L$ is a finite extension of $K$ (\S 3.2 of \cite{WCN}), and that to the extension $\mathbf{E}_L/\mathbf{E}_K$, there corresponds a unique \'etale extension of $\varpi_E$-rings $\mathbf{A}_L / \mathbf{A}_K$. Indeed, if $\mathbf{E}_L = \mathbf{E}_K[X] / Q(X)$, then we can take $\mathbf{A}_L = \mathbf{A}_K[X]/\widetilde{Q}(X)$, where $\widetilde{Q}$ is a unitary polynomial that lifts $Q$, and the resulting ring depends only on $Q(X)$ by Hensel's lemma.
\begin{theo}\label{life}
Let $L$ be a finite extension of $K$ and let $L_\infty=L K_\infty$. If the action of $\Gamma_K$ on $\mathbf{E}_K$ is liftable, then the action of $\Gamma_L$ on $\mathbf{E}_L$ is liftable.
\end{theo}
\begin{proof}
Note that $\Gamma_L$ injects into $\Gamma_K$. Since $\mathbf{A}_L / \mathbf{A}_K$ is \'etale, the Frobenius map $\varphi_q$ and the action of $g \in \Gamma_K$ extend to $\mathbf{A}_L$ (for exemple, if $x \in \mathbf{A}_L$ satisfies $Q(x)=0$ with $Q(X) \in \mathbf{A}_K[X]$ unitary, then $(gQ)(g(x)) = 0$ has a solution in $\mathbf{A}_L$ by Hensel's lemma). There exists an element $T_L \in \mathbf{A}_L$ lifting $\pi_L$ such that $\mathbf{A}_L$ is the $\varpi_E$-adic completion of $\mathcal{O}_E \dcroc{T_L}[1/T_L]$. We can take $F_g(T_L) = g(T_L)$ and $P(T_L)=\varphi_q(T_L)$. Note that if $L_\infty/K_\infty$ is not totally ramified, then we may need to replace $E$ by a larger unramified extension of degree $d$, and $\varphi_q$ by $\varphi_q^d$ accordingly.
\end{proof}
Even in the case of cyclotomic extensions, the series $F_g(T)$ and $P(T)$ can be quite complicated if $L/K$ is ramified. For example, suppose that $\pi_L=\pi_K^{1/n}$ with $p \nmid n$ (this corresponds to a tamely ramified extension $L/K$). We can then take $T_L = T_K^{1/n}$ and
\[ \varphi(T_L) = \left((1+T_K)^p-1\right)^{1/n} = T_L^p \cdot \left(1+\frac{p}{T_L^n} + \cdots + \frac{p}{T_L^{n(p-1)}} \right)^{1/n}, \]
so that $P(T_L)$ is overconvergent but does not belong to $\mathcal{O}_E \dcroc{T_L}$.
\begin{theo}\label{liftdown}
Let $F_\infty \subset K_\infty$ be a Galois subextension such that $K_\infty/F_\infty$ is finite, and let $\Gamma_F= \mathrm{Gal}(F_\infty/K)$. If the action of $\Gamma_K$ on $\mathbf{E}_K$ is liftable, then the action of $\Gamma_F$ on $\mathbf{E}_F$ is liftable.
\end{theo}
\begin{proof}
We check that the action of $\Gamma_F$ lifts to $\mathbf{A}_F = \mathbf{A}_K^{\mathrm{Gal}(K_\infty/F_\infty)}$. The ring $\mathbf{A}_F$ is stable under $\Gamma_F$ and $\varphi_q$ by construction, and its image in $\mathbf{E}_K$ is $\mathbf{E}_F$ since $\mathbf{A}_F$ contains both $\mathcal{O}_E$ and $\mathrm{N}_{K_\infty/F_\infty}(T)$.
\end{proof}
\section{Application to $(\varphi,\Gamma)$-modules}\label{pgmsec}
One reason for asking question \ref{quest} is that it is relevant to the theory of $(\varphi,\Gamma)$-modules for $\mathcal{O}_E$-representations of $G_K$. This theory has been developed in \cite{F90} when $K_\infty = K(\mu_{p^\infty})$, but it can easily be generalized to other extensions $K_\infty/K$ for which the action of $\Gamma_K$ is liftable, as was observed for example in \S 2.1 of \cite{AS06}. For instance, the generalization to Lubin-Tate extensions is explicitely carried out in \S 1 of \cite{KR09} and is further discussed in \cite{FX13} and \cite{CE13}. Let $\mathbf{A}$ be the $\varpi_E$-adic completion of $\varinjlim_L \mathbf{A}_L$, where $L$ runs through the set of finite extensions of $K$. Let $H_K = \mathrm{Gal}(\mathbf{Q}_pbar/K_\infty)$.
\begin{theo}\label{questpgm}
If the action of $\Gamma_K$ is liftable, then there is an equivalence of categories
\[ \text{$\{ (\varphi_q,\Gamma_K)$-modules on $\mathbf{A}_K \} \longleftrightarrow \{ \mathcal{O}_E$-linear representations of $G_K\}$}, \]
given by the mutually inverse functors $\mathrm{D} \mapsto (\mathbf{A} \otimes_{\mathbf{A}_K} \mathrm{D})^{\varphi_q=1}$ and $V \mapsto (\mathbf{A} \otimes_{\mathcal{O}_E} V)^{H_K}$.
\end{theo}
\begin{proof}
The proof follows \S A.1.2 and \S A.3.4 of \cite{F90} as well as \S 2.1 of \cite{AS06}, and we sketch it here. Note that $\mathbf{A}^{\varphi_q=1} = \mathcal{O}_E$ since $q=\mathrm{Card}(k_E)$. Let $\mathbf{E} = \mathbf{E}_K^{\mathrm{sep}}$, so that $\mathbf{A} / \varpi_E \mathbf{A} = \mathbf{E}$.
The theory of $\varphi$-modules tells us that if $M$ is a $\varphi_q$-module over $\mathbf{E}$, then $M = \mathbf{E} \otimes_{k_E} M^{\varphi_q=1}$ and that $1-\varphi_q : M \to M$ is surjective. These two facts imply that if $D$ is a $\varphi_q$-module over $\mathbf{A}_K$, then $\mathbf{A} \otimes_{\mathbf{A}_K} D = \mathbf{A} \otimes_{\mathcal{O}_E} V(D)$ with $V(D) = (\mathbf{A} \otimes_{\mathbf{A}_K} D)^{\varphi_q=1}$.
Conversely, Hilbert's theorem 90 says that $H^1(\mathrm{Gal}(\mathbf{E}/\mathbf{E}_K),\mathrm{GL}_d(\mathbf{E}))$ is trivial for all $d \geqslant 1$. The theory of the field of norms gives us an isomorphism between $\mathrm{Gal}(\mathbf{E}/\mathbf{E}_K)$ and $H_K$ (\S 3.2 of \cite{WCN}). By d\'evissage, this implies that if $V$ is an $\mathcal{O}_E$-representation of $H_K$, then $\mathbf{A} \otimes_{\mathcal{O}_E} V = \mathbf{A} \otimes_{\mathbf{A}_K} D(V)$ where $D(V) = (\mathbf{A} \otimes_{\mathcal{O}_E} V)^{H_K}$.
These two facts imply that the functors of the theorem are mutually inverse.
\end{proof}
\section{Embeddings into rings of periods}\label{atsec}
We now explain how to view the different rings whose construction we have recalled as subrings of some of Fontaine's rings of periods (constructed for example in \cite{FPP}). Let $I$ be the ideal of elements of $\mathcal{O}_{\mathbf{C}_p}$ with valuation at least $1/p$. Let $\widetilde{\mathbf{E}}$ denote the fraction field of $\widetilde{\mathbf{E}}plus=\varprojlim_{x \mapsto x^p} \mathcal{O}_{\mathbf{C}_p}/I$. Let $\widetilde{\mathbf{E}}_K=\widetilde{\mathbf{E}}^{H_K}$. By \S 4.2 of \cite{WCN}, there is a canonical $G_K$-equivariant embedding $X_K(K_\infty) \to \widetilde{\mathbf{E}}_K$ and we also denote its image by $\mathbf{E}_K$.
Let $W_E(\cdot) = \mathcal{O}_E \otimes_{\mathcal{O}_{E_0}} W(\cdot)$ denote the $\varpi_E$-Witt vectors. Let $\tilde{\mathbf{A}} = W_E(\widetilde{\mathbf{E}})$, and endow it with the $\mathcal{O}_E$-linear Frobenius map $\varphi_q$ and the $\mathcal{O}_E$-linear action of $G_K$ coming from those on $\widetilde{\mathbf{E}}$. These are well-defined since $E_0 \subset K$. Let $\tilde{\mathbf{A}}_K = \tilde{\mathbf{A}}^{H_K}$ so that $\tilde{\mathbf{A}}_K = W_E(\widetilde{\mathbf{E}}_K)$. If $\mathbf{A}_K$ is equipped with a lift of the action of $\Gamma_K$ and a commuting Frobenius map $\varphi_q$, then there is an embedding $\mathbf{A}_K \to \tilde{\mathbf{A}}_K$ that is compatible with $\varphi_q$, $\Gamma_K$-equivariant, and lifts the embedding $\mathbf{E}_K \to \widetilde{\mathbf{E}}_K$. See \S A.1.3 of \cite{F90} for a proof, or simply remark that since $\mathbf{A}_K$ is the $\varpi_E$-adic completion of $\mathcal{O}_E \dcroc{T}[1/T]$, it is enough to show that there exists one and only one element $v \in \tilde{\mathbf{A}}_K$ (the image of $T$) that lifts $\pi_K$ and satisfies $\varphi_q(v) = P(v)$. This now follows from the fact that if $S$ denotes the set of elements of $\tilde{\mathbf{A}}_K$ whose image in $\widetilde{\mathbf{E}}_K$ is $\pi_K$, then $x \mapsto \varphi_q^{-1} (P(x))$ is a contracting map on $S$. Let $v \in \tilde{\mathbf{A}}_K$ be the image of $T$ as above, so that $\varphi_q(v) = P(v)$ and $g(v) = F_g(v)$ for all $g \in \Gamma_K$. Note that $\overline{v} = \pi_K \in \widetilde{\mathbf{E}}plus$. Let $\tilde{\mathbf{A}}plus = W_E(\widetilde{\mathbf{E}}plus)$
\begin{lemm}\label{regvat}
If $P(T) \in \mathcal{O}_E \dcroc{T}$, then $v \in \tilde{\mathbf{A}}plus$.
\end{lemm}
\begin{proof}
We have $v - [\overline{v}] \in \varpi_E \tilde{\mathbf{A}}$ so that $v \in \tilde{\mathbf{A}}plus + \varpi_E \tilde{\mathbf{A}}$. Suppose that $v \in \tilde{\mathbf{A}}plus + \varpi_E^k \tilde{\mathbf{A}}$ for some $k \geqslant 1$. We have $P(T) \in T^q + \mathfrak{m}_E \dcroc{T}$. This implies that $P(v) \in \tilde{\mathbf{A}}plus + \varpi_E^{k+1} \tilde{\mathbf{A}}$ and hence also $v = \varphi_q^{-1} (P(v)) \in \tilde{\mathbf{A}}plus + \varpi_E^{k+1} \tilde{\mathbf{A}}$. By induction on $k$, we get $v \in \tilde{\mathbf{A}}plus$.
\end{proof}
In \S \ref{finht}, we use the fact that if $L$ contains $K$ and $E$, then $\tilde{\mathbf{A}}plus$ injects into $\mathbf{B}_{\mathrm{dR}}^+$ in a $G_L$-equivariant way. We also use the following lemma about $\mathbf{B}_{\mathrm{dR}}^+$.
\begin{lemm}\label{cvthbdr}
Let $E$ be a finite extension of $\mathbf{Q}_p$ and take $f(T) \in E \dcroc{T}$. If $x \in \mathbf{B}_{\mathrm{dR}}^+$, then the series $f(x)$ converges in $\mathbf{B}_{\mathrm{dR}}^+$ if and only if the series $f(\theta(x))$ converges in $\mathbf{C}_p$.
\end{lemm}
\begin{proof}
We prove that the series converges in $\mathbf{B}_{\mathrm{dR}}^+/t^k$ for all $k \geqslant 1$. Recall that $\mathbf{B}_{\mathrm{dR}}^+/t^k$ is a Banach space, the unit ball being the image of $\tilde{\mathbf{A}}plus \to \mathbf{B}_{\mathrm{dR}}^+ / t^k$. We can enlarge $E$ so that it contains an element of valuation $\mathrm{val}_p(\theta(x))$ and it is then enough to prove that if $\theta(x) \in \mathcal{O}_{\mathbf{C}_p}$, then $\{x^n\}_{n \geqslant 0}$ is bounded in $\mathbf{B}_{\mathrm{dR}}^+/t^k$. Let $\omega$ be a generator of $\ker(\theta : \tilde{\mathbf{A}}plus \to \mathcal{O}_{\mathbf{C}_p})$ and let $x_0$ be an element of $\tilde{\mathbf{A}}plus$ such that $\theta(x) = \theta(x_0)$. We can write $x=x_0+\omega y + t^k z$ where $y \in \tilde{\mathbf{A}}plus[1/p]$ and $z \in \mathbf{B}_{\mathrm{dR}}^+$. We then have
\[ x^n = x_0^n + \binom{n}{1} x_0^{n-1} \omega y + \cdots + \binom{n}{k-1} x_0^{n-(k-1)} (\omega y)^{k-1} + t^k z_k, \]
with $z_k \in \mathbf{B}_{\mathrm{dR}}^+$, so that $x^n \in (\tilde{\mathbf{A}}plus + y \tilde{\mathbf{A}}plus + \cdots + y^{k-1} \tilde{\mathbf{A}}plus) + t^k \mathbf{B}_{\mathrm{dR}}^+$ for all $n$.
\end{proof}
\section{Lifts of finite height}\label{finht}
In this section we prove theorem B, which we now recall.
\begin{theo}\label{carphiq}
If the action of $\Gamma_K$ is liftable with $\varphi_q(T) \in \mathcal{O}_E \dcroc{T}$, then $\Gamma_K$ is abelian, and there is an injective character $\Gamma_K \to \mathcal{O}_E^\times$, whose conjugates by $\mathrm{Emb}(E,\mathbf{Q}_pbar)$ are all de Rham with weights in $\mathbf{Z}_{\geqslant 0}$.
\end{theo}
Before proving theorem \ref{carphiq}, we give a number of intermediate results to the effect that if $P(T) = \varphi_q(T)$ belongs to $\mathcal{O}_E \dcroc{T}$, then one can improve the regularity of the power series $P(T)$ and $F_g(T)$ for $g \in \Gamma_K$.
\begin{prop}\label{colpow}
If $P(T) \in \mathcal{O}_E \dcroc{T}$, then $F_g(T) \in T \cdot \mathcal{O}_E \dcroc{T}$ for all $g \in \Gamma_K$.
\end{prop}
\begin{proof}
The ring $\mathbf{A}_K$ is a free $\varphi_q(\mathbf{A}_K)$-module of rank $q$. As in \S 2.3 of \cite{F90}, let $\mathcal{N} : \mathbf{A}_K \to \mathbf{A}_K$ denote the map
\[ \mathcal{N} : f(T) \mapsto \varphi_q^{-1} \circ \mathrm{N}_{\mathbf{A}_K/\varphi_q(\mathbf{A}_K)} (f(T)). \]
If $P(T) \in \mathcal{O}_E \dcroc{T}$, then $\mathcal{N}(\mathcal{O}_E \dcroc{T}) \subset \mathcal{O}_E \dcroc{T}$ since the ring $\mathcal{O}_E \dcroc{T}$ is a free $\mathcal{O}_E \dcroc{P(T)}$-module of rank $q$. Furthermore, we have $\mathcal{N}(1+\varpi_E^k \mathbf{A}_K) \subset 1+\varpi_E^{k+1} \mathbf{A}_K$ if $k \geqslant 1$ (see 2.3.2 of ibid). This implies that if $k \geqslant 1$, then
\[ \mathcal{N}(\mathcal{O}_E \dcroc{T}^\times+\varpi_E^k \mathbf{A}_K) \subset \mathcal{O}_E \dcroc{T}^\times+\varpi_E^{k+1} \mathbf{A}_K, \]
and likewise, since $\mathcal{N}(T)=T$ and $T$ is invertible in $\mathbf{A}_K$,
\[ \mathcal{N}(T \cdot \mathcal{O}_E \dcroc{T}^\times+\varpi_E^k \mathbf{A}_K) \subset T \cdot \mathcal{O}_E \dcroc{T}^\times+\varpi_E^{k+1} \mathbf{A}_K. \]
This implies, by induction on $k$, that $(T \cdot \mathcal{O}_E \dcroc{T}^\times+ \varpi_E \mathbf{A}_K)^{\mathcal{N}(x)=x} \subset T \cdot \mathcal{O}_E \dcroc{T}^\times$.
We have $F_g(T) \in T \cdot \mathcal{O}_E \dcroc{T}^\times+\varpi_E \mathbf{A}_K$ and since $\mathcal{N}$ commutes with the action of $\Gamma_K$, we have $\mathcal{N}(g(T))=g(T)$ and hence $F_g(T) \in (T \cdot \mathcal{O}_E \dcroc{T}^\times+\varpi_E \mathbf{A}_K)^{\mathcal{N}(x)=x} \subset T \cdot \mathcal{O}_E \dcroc{T}^\times$.
\end{proof}
\begin{rema}\label{colover}
The same proof implies that if $P(T)$ is overconvergent, then so is $F_g(T)$.
\end{rema}
\begin{lemm}\label{reduc1}
If $P(T) \in \mathcal{O}_E \dcroc{T}$, then there exists $a \in \mathfrak{m}_E$ such that if $T'=T-a$, then $\varphi_q(T')=Q(T')$ with $Q(T') \in T' \cdot \mathcal{O}_E \dcroc{T'}$.
\end{lemm}
\begin{proof}
Let $R(T)=P(T+a)$. We have $\varphi_q(T')=\varphi_q(T-a)=P(T)-a=R(T')-a$ so it is enough to find $a \in \mathfrak{m}_E$ such that $P(a)=a$. The Newton polygon of $P(T)-T$ starts with a segment of length $1$ and slope $-\mathrm{val}_p(P(0))$, which gives us such an $a$ with $\mathrm{val}_p(a)=\mathrm{val}_p(P(0))$.
\end{proof}
\begin{lemm}\label{reduc3}
If $P(T) \in T \cdot \mathcal{O}_E \dcroc{T}$, then $P'(0) \neq 0$.
\end{lemm}
\begin{proof}
By proposition \ref{colpow}, we have $F_g(T) \in T \cdot \mathcal{O}_E \dcroc{T}$ for all $g \in \Gamma_K$. Write $F_g(T)=f_1(g) T + \mathrm{O}( T^2 )$ and $P(T) = \pi_k T^k + \mathrm{O}(T^{k+1})$ with $\pi_k \neq 0$. Note that $g \mapsto f_1(g)$ is a character $f_1 : \Gamma_K \to \mathcal{O}_E^\times$. The fact that $F_g (P(T)) = P(F_g(T))$ implies that $f_1(g) \pi_k = \pi_k f_1(g)^k$ so that if $k \neq 1$, then $f_1(g)^{k-1}=1$.
In particular, taking $g$ in the open subgroup $f_1^{-1}(1+2p \mathcal{O}_E)$ of $\Gamma_K$, we must have $f_1(g)=1$. Take such a $g \in \Gamma_K \setminus \{ 1 \}$; since $\overline{F}_g(T) \neq T$, we can write $F_g(T) = T + T^i h(T)$ for some $i \geqslant 2$ with $h(0) \neq 0$. The equation $F_g (P(T)) = P(F_g(T))$ and the fact that $P(T + T^i h(T))=\sum_{j \geqslant 0} (T^i h(T))^j P^{(j)}(T)/j!$ imply that
\[ P(T) + P(T)^i h(P(T)) = P(T) + T^i h(T) P'(T) + \mathrm{O} (T^{2i+k-2}), \]
so that $P(T)^i h(P(T)) = T^i h(T) P'(T) + \mathrm{O} (T^{2i+k-2})$. The term of lowest degree of the LHS is of degree $ki$, while on the RHS it is of degree $i+k-1$. We therefore have $ki=i+k-1$, so that $(k-1)(i-1)=0$ and therefore $k=1$.
\end{proof}
\begin{proof}[Proof of theorem \ref{carphiq}]
By the preceding results, if $P(T) \in \mathcal{O}_E \dcroc{T}$, then we can make a change of variable so that $P(T) \in T \cdot \mathcal{O}_E \dcroc{T}$ and $F_g(T) \in T \cdot \mathcal{O}_E \dcroc{T}$ for all $g \in \Gamma_K$. Write $P(T) = \sum_{k \geqslant 1} \pi_k T^k$. By lemma \ref{reduc3}, we have $\pi_1 \neq 0$. If $A(T) = \sum_{k \geqslant 1} a_k T^k \in E \dcroc{T}$ with $a_1=1$, then the equation $A(P(T)) = \pi_1 \cdot A(T)$ is given by
\[ P(T) + a_2 P(T)^2 + \cdots = \pi_1 \cdot ( T + a_2 T^2 + \cdots). \]
Looking at the coefficient of $T^k$ in the above equation, we get the equation
\[ x_{k,1} a_1 + \cdots + x_{k,k-1} a_{k-1} = a_k(\pi_1-\pi_1^k), \]
where $x_{k,i}$ is the coefficient of $T^k$ in $P(T)^i$ and hence belongs to $\mathcal{O}_E$. This implies that the equation $A(P(T)) = \pi_1 \cdot A(T)$ has a unique solution in $E \dcroc{T}$, and that $a_k \in \pi_1^{1-k} \cdot \mathcal{O}_E$. In particular, the power series $A(T)$ belongs to $\mathcal{O}_E \dcroc{T/\pi_1}$ and so has a nonzero radius of convergence. If $g \in \Gamma_K$, then we have
\[ A(F_g(P(T))) = A(P(F_g(T))) = \pi_1 \cdot A(F_g(T)). \]
This implies that if $B(T) = f_1(g)^{-1} \cdot A(F_g(T))$, then $b_1=1$ and $B(P(T)) = \pi_1 \cdot B(T)$, so that $B(T)=A(T)$ and hence $A(F_g(T)) = f_1(g) \cdot A(T)$ for all $g \in \Gamma_K$. The map $g \mapsto f_1(g)$ is therefore injective, since $f_1(g)=1$ implies that $F_g(T)=T$ so that $g=1$.
Recall that in \S \ref{atsec}, we have seen that there is a map $\mathbf{A}_K \to \tilde{\mathbf{A}}$ that commutes with $\varphi_q$ and the action of $G_K$. Let $v \in \tilde{\mathbf{A}}$ be the image of $T$. By lemma \ref{regvat}, $v \in \tilde{\mathbf{A}}plus$. We have $\theta(v) \in \mathfrak{m}_{\mathbf{C}_p}$ and $\theta(\varphi_q^m(v)) = \theta (P \circ \cdots \circ P (v))$ so that there exists $m_0 \geqslant 0$ such that $\theta(\varphi_q^m(v))$ is in the domain of convergence of $A(T)$ if $m \geqslant m_0$. By lemma \ref{cvthbdr}, the series $A(\varphi_q^m(v))$ converges in $(\mathbf{B}_{\mathrm{dR}}^+)^{H_L}$ where $L=KE$ and if $g \in G_L$, then we have
\[ g( A(\varphi_q^m(v))) = A(F_g(\varphi_q^m(v))) = f_1(g) \cdot A(\varphi_q^m(v)). \]
We now show that $A(\varphi_q^m(v)) \neq 0$ for some $m \geqslant m_0$. If $\theta (\varphi_q^m(v)) = 0$ for some $m$, then $\varphi_q^m(v) \in \mathrm{Fil}^k \setminus \mathrm{Fil}^{k+1} \mathbf{B}_{\mathrm{dR}}^+$ for some $k \geqslant 1$, and then $A(\varphi_q^m(v)) \in \mathrm{Fil}^k \setminus \mathrm{Fil}^{k+1} \mathbf{B}_{\mathrm{dR}}^+$ as well, so that $A(\varphi_q^m(v)) \neq 0$. If $\theta (\varphi_q^m(v)) \neq 0$ for all $m \geqslant m_0$, then the sequence $\{ \theta(\varphi_q^m(v)) \}_{m \geqslant m_0}$ converges to zero in $\mathbf{C}_p$ and if $A(\varphi_q^m(v)) = 0$ for all $m \geqslant m_0$, then $A(\theta(\varphi_q^m(v))) = 0$ for all $m \geqslant m_0$ and this implies that $A(T)=0$ since $0$ would not be an isolated zero of $A(T)$. There is hence some $m \geqslant m_0$ such that $A(\varphi_q^m(v)) \neq 0$, and the fact that $g( A(\varphi_q^m(v))) = f_1(g) \cdot A(\varphi_q^m(v))$ if $g \in G_L$ implies that the character $g \mapsto f_1(g)$ is de Rham and that its weight is in $\mathbf{Z}_{\geqslant 0}$.
The conjugates of $g \mapsto f_1(g)$ are treated in the same way. If $h \in \mathrm{Emb}(E,\mathbf{Q}_pbar)$, then choose some $n(h) \in \mathbf{Z}$ such that $h = [x \mapsto x^p]^{n(h)}$ on $k_E$ so that $h=\varphi^{n(h)}$ on $\mathcal{O}_{E_0}$. Define an element $h(v) \in W_{h(E)}(\widetilde{\mathbf{E}}plus)$ by the formula $h(e \otimes a) = h(e) \otimes \varphi^{n(h)}(a)$. If $v \in W_E(\widetilde{\mathbf{E}}plus)$ satisfies $\varphi_q(v) = P(v)$ and $g(v) = F_g(v)$ for $g \in \Gamma_K$, then $\varphi_q(h(v)) = P^h (h(v))$ and $g(h(v)) = F^h_g(h(v))$. The same reasoning as above now implies that the character $g \mapsto h(f_1(g))$ is de Rham and that its weight is in $\mathbf{Z}_{\geqslant 0}$.
\end{proof}
\begin{exem}\label{ezpdrp}
If $E=\mathbf{Q}_p$, then theorem B implies that $K_\infty \subset \mathbf{Q}_p^{\mathrm{ab}} \cdot L$ where $L$ is a finite extension of $K$. Indeed, every de Rham character $\widetilde{\mathbf{E}}a : G_K \to \mathbf{Z}_p^\times$ is of the form $\chi_{\mathrm{cyc}}^r \cdot \mu$ for some $r \in \mathbf{Z}$ and some potentially unramified character $\mu$ (see \S 3.9 of \cite{FST}).
\end{exem}
More generally, the condition that there is an injective character $\widetilde{\mathbf{E}}a : \Gamma_K \to \mathcal{O}_E^\times$, whose conjugates by $\mathrm{Emb}(E,\mathbf{Q}_pbar)$ are all de Rham with weights in $\mathbf{Z}_{\geqslant 0}$, imposes some constraints on $K_\infty/K$. Here is a simple example (recall that $E$ is a finite extension of $\mathbf{Q}_p$ such that $k_E=k_K$).
\begin{prop}\label{kprim}
If $K$ is a Galois extension of $\mathbf{Q}_p$ of degree $d$, where $d$ is a prime number, and if $\widetilde{\mathbf{E}}a : \Gamma_K \to \mathcal{O}_E^\times$ is a de Rham character with weights in $\mathbf{Z}_{\geqslant 0}$, then the Lie algebra of the image of $\widetilde{\mathbf{E}}a$ is either $\{0\}$, $\mathbf{Q}_p$ or $K$.
\end{prop}
\begin{proof}
By local class field theory, $\Gamma_K$ can be realized as a quotient of $\mathcal{O}_K^\times$ and $\widetilde{\mathbf{E}}a$ can be seen as a character $\widetilde{\mathbf{E}}a : \mathcal{O}_K^\times \to \mathcal{O}_E^\times$. This character is then the product of a finite order character by $x \mapsto \prod_{h \in \mathrm{Gal}(K / \mathbf{Q}_p)} h(x)^{a_h}$ where $a_h$ is the weight of $\widetilde{\mathbf{E}}a$ at the embedding $h$, so that $a_h \in \mathbf{Z}_{\geqslant 0}$. It is therefore enough to prove that if $f : K \to K$ is defined by $f = \sum_{h \in \mathrm{Gal}(K / \mathbf{Q}_p)} a_h \cdot h$ with $a_h \in \mathbf{Z}_{\geqslant 0}$, then the image of $f$ is either $\{0\}$, $\mathbf{Q}_p$ or $K$.
Let $g$ be a generator of $\mathrm{Gal}(K / \mathbf{Q}_p)$ and write $a_i$ for $a_{g^i}$ if $i \in \mathbf{Z}/d\mathbf{Z}$. If $\sum_i a_i g^i(x) = 0$ for some $x \in K^\times$, then $\sum_i a_{i+j} g^i(x) = 0$ for all $j \in \mathbf{Z}/d\mathbf{Z}$. This implies that the circulant matrix $( a_{i+j} )_{i,j}$ is singular. Its determinant is $\prod_{j=0}^{d-1} \sum_{i=0}^{d-1} \zeta_d^{ij} a_i$ where $\zeta_d$ is a primitive $d$-th root of $1$. Since $d$ is a prime number and $a_i \in \mathbf{Z}_{\geqslant 0}$ for all $i$, we can have $\sum_{i=0}^{d-1} \zeta_d^{ij} a_i = 0$ for some $j$ if and only if all the $a_i$ are equal to each other. In this case, $f$ is equal to $a_0 \cdot \mathrm{Tr}_{K/\mathbf{Q}_p}(\cdot)$. Otherwise, $f : K \to K$ is bijective. This proves the proposition.
\end{proof}
\begin{coro}\label{anticyclex}
If $K = \mathbf{Q}_{p^2}$ and $K_\infty$ is the anticylotomic extension of $K$ and $E$ is a totally ramified extension of $\mathbf{Q}_{p^2}$, then it is not possible to find a lift for $\varphi_q$ and $\Gamma_K$ such that $\varphi_q(T) \in \mathcal{O}_E \dcroc{T}$.
\end{coro}
\begin{rema}\label{sr}
If $d$ is not a prime number, then the conclusion of proposition \ref{kprim} does not necessarily hold anymore. This is already the case if $\mathrm{Gal}(K/\mathbf{Q}_p) = \mathbf{Z}/4\mathbf{Z}$.
\end{rema}
\begin{rema}\label{lubin}
There is some similarity between our methods for proving theorem B and the constructions of \cite{JL94}. For example, the power series $A(T)$ constructed in the proof of theorem B is denoted by $\mathbf{L}_f$ in \S 1 of ibid.\ and called the \emph{logarithm}. Theorem B is then consistent with the suggestion on page 341 of ibid.\ that ``for an invertible series to commute with a noninvertible series, there must be a formal group somehow in the background''. Indeed, the existence of a de Rham character $\Gamma_K \to \mathcal{O}_E^\times$ with weights in $\mathbf{Z}_{\geqslant 0}$ indicates that the extension $K_\infty/K$ must in some sense ``come from geometry''.
\end{rema}
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\end{document}
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\begin{document}
\title{Preservation of loss in linear-optical processing}
\author{Dominic W. Berry}
\affiliation{Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada}
\author{A. I. Lvovsky}
\affiliation{Institute for Quantum Information Science, University of Calgary, Alberta T2N 1N4, Canada}
\begin{abstract}
We propose a measure of quantum efficiency of a multimode state of light that quantifies the amount of optical loss
this state has experienced, and prove that this efficiency cannot increase in any linear-optical processing with
destructive conditional measurements. Any loss that has affected a state can neither be removed nor redistributed so
as to further increase the efficiency in higher-efficiency modes at the expense of lower-efficiency modes. This
result eliminates the possibility of catalytically improving photon sources.
\end{abstract}
\pacs{42.50.Dv,42.50.Ex,03.67.-a}
\maketitle
\section{Introduction}
A leading approach to quantum information processing is via linear-optical quantum computing (LOQC),
first proposed in 2001 by Knill, Laflamme and Milburn \cite{KLM}. Major progress has been made both on the
theoretical and experimental fronts towards implementation of LOQC. Modifications have been proposed that greatly
reduce the overhead costs \cite{KokRMP}, a quantum error correction protocol has been introduced
\cite{Dawson06,Varnava}, and experimental implementation of primary gates has been demonstrated \cite{KLMexp}.
In spite of this progress, practical LOQC is still out of reach. Many of the difficulties arise because the
single-photon sources required for LOQC, as well as computational circuits themselves, suffer from losses. Although a
certain degree of tolerance to losses does exist in some LOQC schemes \cite{Varnava}, the efficiency of existing
single-photon sources \cite{NJPissue} as well as the quality of individual circuit elements and waveguides are far
below the required minima.
Under these circumstances it appears beneficial to develop a procedure that would reverse the effect of losses,
perhaps at a cost of introducing extra resources. It would be useful, for example, to employ the outputs of $N$
imperfect single-photon sources to obtain $K<N$ single-photon sources of improved quantum efficiency. Accomplishing this task
would be straightforward if nonlinear-optical interactions with single photons were readily available: for example, one could employ non-demolition photon number measurements to select only those modes that contain photons.
However, achieving such interactions is extremely technically challenging
\cite{KimblePhotonGate}, whereas linear-optical (LO) processing is easily achieved in the laboratory.
It is therefore important to investigate whether elimination of losses is possible under
LO processing. Under this processing we understand arbitrary interferometric transformations and
conditioning on results of arbitrary \emph{destructive} measurements on some of the optical modes involved. The
efforts to construct such a scheme began in 2004, mostly ending with various no-go results
\cite{Berry04a,Berry04b,Berry06,Berry07}. The most general result to date was obtained in Ref.~\cite{BL}.
In that work, we quantified the \emph{efficiency} of a quantum optical state by the amount of loss that state might have experienced.
We then proved that the efficiency in any single-mode optical state obtained through LO processing cannot exceed the quantum efficiency of the best available single-mode input \cite{BL}.
However, those previous results had limited application to multimode states.
First, as we show below, extending the definition of the efficiency of a quantum state to the multimode case is not straightforward, particularly when the loss has been ``mixed" among the modes by interferometric transformations.
Second, our earlier results do not provide any information on how the efficiencies can be distributed among the output modes, aside from the general upper bound mentioned above.
For example, they leave open the possibility of a ``catalytic'' scheme, in which some high-efficiency single photons are used to obtain additional high-efficiency single photons.
In the present work we generalize our study of the dynamics of optical losses to the multimode case. We introduce the notion of quantum
efficiency of a (possibly entangled) multimode state which quantifies the amount of loss this state may have
experienced. We show that this efficiency cannot increase under LO processing. That is, any loss that has occurred at
the input can neither be removed nor redistributed so as to improve the efficiency in some of the modes at the
expense of lower-efficiency modes. This means that there is a majorization relation between the efficiencies at the input and the output. The LO processing can act to average the efficiencies, but not to concentrate them. This rules out, in particular, any possibility of catalytic efficiency improvement.
\section{Single-mode measures of efficiency}
Before describing multimode measures of efficiency, we discuss the properties and relationships for single-mode measures of efficiency that have been previously proposed.
Usually efficiency is used to describe a process for producing a state.
However, it is also convenient to regard efficiency as a measure on the state itself, regardless of the process used to produce it \cite{Berry04a,Berry04b,Berry06,Berry07,BL}.
Specifically, Ref.~\cite{BL} uses the efficiency to quantify \emph{the maximum amount of loss an optical mode carrying the given state might have previously experienced}:
\begin{equation}\label{mostgen}
E(\hat\rho) := \inf \left\{ p |\ \exists \hat \rho_0\ge 0 ~:~ {\cal E}_{p} (\hat \rho_0)=\hat \rho \right\},
\end{equation}
where ${\cal E}_{p}$ is a loss channel with transmissivity $p$. That is, one considers all hypothetical methods of producing the given state $\hat\rho$ via loss from some valid initial quantum state $\hat\rho_0$.
We emphasize that the loss is just a way of mathematically quantifying the efficiency of the state.
It is not necessary that the state were created by such a process.
The efficiency is a measure on the state, and should not be regarded as an intrinsic feature of the mode.
Let us study a few examples. Ideally, single photon sources would produce a single photon state $\ket 1$ on demand.
In practice, such sources may with some probability fail to produce a photon, and there is no way to detect this failure without destructive measurement. Therefore the state produced by a generic single photon source may be approximated as
\begin{equation}
\label{eq:mix}
\hat\rho = p \ket{1}\bra{1} + (1-p) \ket{0}\bra{0}.
\end{equation}
Here the quantity $p$ is commonly referred to as the efficiency of the single photon source. In the context of our definition, state \eqref{eq:mix} can be obtained from the single-photon state by transmitting a (perfect) single photon through a loss channel with transmissivity $p$, and hence its efficiency equals $p$. In this way, the efficiency of state \eqref{eq:mix} according to our new definition is consistent with the traditional definition of the efficiency of a single-photon source.
Coherent states have efficiency exactly equal to zero, regardless of their amplitude.
This is because coherent states remain coherent states under loss.
A coherent state of amplitude $\alpha$ can be obtained from one of amplitude $\alpha/\sqrt{p}$ under a loss channel of transmissivity $p$. Although one cannot take $p=0$ (because complete loss always results in the vacuum state), possible values of $p$ form an open set with zero infimum.
On the other hand, any pure state \emph{other} than a coherent state (or the vacuum state) must have efficiency 1.
This is because a state under loss is a mixture of the original state, and the state with different numbers of photons lost.
That is, a pure state $\ket{\chi}$ becomes a mixture of $\ket{\chi}$, $\hat a\ket{\chi}$, $\hat a^2\ket{\chi}$, and so forth.
The only way in which the state after loss can remain pure is if $\ket{\chi}\propto\hat a\ket{\chi}$.
The only states for which this is true are eigenstates of the annihilation operator; i.e.\ coherent states.
Determining the efficiency of a known single-mode state is a straightforward computational task.
The loss channel ${\cal E}_{p}$ corresponds to a linear transformation known as the generalized Bernoulli transformation.
Provided the state $\hat\rho$ can be obtained via loss channel ${\cal E}_{p}$ from some initial operator, we can define the inverse map ${\cal E}_{p}^{-1}$, which can be calculated as in Ref.\ \cite{Herzog}.
Therefore, we need to find the infimum of the values of $p$ such that the inverse Bernoulli mapping ${\cal E}_{p}^{-1}(\hat\rho)$ exists and yields a valid quantum state, i.e.\ can be represented by a positive semidefinite density matrix.
A further interesting feature of a state's efficiency is that it equals zero if and only if the state is classical, i.e.\ it can be written as a statistical mixture of coherent states, or, equivalently, its Glauber-Sudarshan $P$ function has the properties of a probability density. As discussed above, any coherent state has efficiency zero, and hence so does any statistical mixture of coherent states. To prove the converse, let us suppose there exists a nonclassical state $\hat\rho$ such that $E(\hat\rho)=0$. Let $\Phi_{\hat\rho}(\eta)$ denote the Fourier transform of this state's $P$ function $P(\alpha)$ over the phase space. According to Bochner's theorem \cite{Bochner}, because $P(\alpha)$ is not a probability density, there exist two sets of $n$ complex numbers $\eta_k$ and $z_k$, such that
\begin{equation}\label{bochnereq}
\sum\limits_{i,j=1}^n \Phi_{\hat\rho}(\eta_i-\eta_j)z_iz_j^*<0.
\end{equation}
Because $E(\hat\rho)=0$, for any $p>0$ there exists state $\hat\rho_0$ such that $\hat \rho$ is obtained from $\hat\rho_0$ by means of attenuation by factor $p$. Because attenuation corresponds to ``shrinkage" of the $P$ function in the phase space \cite{Leonhardt}, we have $\Phi_{\hat\rho_0}(\eta)=\Phi_{\hat\rho}(\eta/\sqrt p)$ and hence
\begin{equation}\label{bochnereq0}
\sum\limits_{i,j=1}^n \Phi_{\hat\rho_0}(\eta'_i-\eta'_j)z_iz_j^*<0,
\end{equation}
where $\eta'_k=\eta_k\sqrt p$. By choosing $p$ close to zero, the set of arguments of function $\Phi_{\hat\rho_0}$ in the above equation can be upper bounded by an arbitrarily small value $A$.
Now recall that the Husimi $Q$ function of any quantum state must be non-negative. This means that the Fourier transform $\Psi_{\hat\rho_0}(\eta)$ of the $Q$ function of state $\hat\rho_0$ must obey
\begin{equation}\label{bochnerpsi}
\sum\limits_{i,j=1}^n \Psi_{\hat\rho_0}(\eta'_i-\eta'_j)z_iz_j^*\ge 0.
\end{equation}
But the $Q$ function is obtained from the $P$ function by convolving the latter with a Gaussian, $e^{-|\alpha|^2}/\pi$ \cite{Leonhardt}. This means that the Fourier transforms of these functions are connected by multiplication,
\begin{equation}\label{}
\Psi_{\hat\rho}(\eta)=\Phi_{\hat\rho}(\eta)e^{-|\eta|^2}.
\end{equation}
By choosing $p$ close to zero, one can make the factor $e^{-|\eta|^2}$ arbitrarily close to 1 within radius $A$. Accordingly, the left-hand sides of Eqs.~\eqref{bochnereq0} and \eqref{bochnerpsi} are equal in the limit $p\to 0$. We arrive at a contradiction, which means that any nonclassical state $\hat \rho$ must have a finite efficiency $E(\hat\rho)>0$.
\section{Multimode measures of efficiency}
\label{sec:mul}
Let us now generalize the notion of efficiency to an optical state carried by multiple modes. A direct generalization can be obtained by assuming that each mode has propagated through its own loss channel, and taking the sum of the transmissivities:
\begin{equation}\label{ED}
E_{\rm d}(\hat\rho,K) := \inf \left\{ \sum_{\ell=1}^K p^\downarrow_{\ell}\ |\ \exists \hat \rho_0\ge 0 ~:~ {\cal E}_{\vec p} (\hat \rho_0)=\hat \rho \right\}.
\end{equation}
The notation $p^\downarrow_\ell$ indicates the elements of the vector $\vec p$ sorted in non-increasing order. The value of $K$ can be less than the number of modes constituting state $\hat\rho$. In this way, the efficiency is defined not only for the entire state, but also for a subset of $K$ modes with the lowest losses. This extension facilitates comparison of efficiencies of states with different number of modes.
A drawback of this definition is that it does not adequately take account of loss that has been mixed between modes. For example, consider two polarization modes carrying a single-photon qubit in the state $\ket{\psi}=\ket{1_H}\ket {0_V}$. The efficiency of the state in the horizontally polarized mode is 1, and that in the vertically polarized mode 0, so $E_{\rm d}(\ketbra\psi\psi,2)=1$. On the other hand, writing the same state in terms of diagonal polarization modes, we find $\ket{\psi'}=(\ket{1_{+45^\circ}}\ket{0_{-45^\circ}}+\ket{0_{+45^\circ}}\ket{1_{-45^\circ}})/\sqrt 2$. This state cannot be obtained by independent loss in the two modes, and would have a different efficiency, $E_{\rm d}(\ketbra{\psi'}{\psi'},2)=2$, even though its utility for quantum information processing is exactly the same as that of $\ket\psi$.
An alternative approach to quantifying the efficiency is to treat each mode separately, and calculate the sum of single-mode efficiencies for $K$ highest-efficiency modes:
\begin{equation}\label{Ei}
E_{\rm s}(\hat\rho,K):= \sum_{\ell=1}^K E(\tr_{\forall k\ne \ell}\hat\rho)^\downarrow.
\end{equation}
This definition is also problematic. First, similarly to the d-efficiency \cite{FootNoteNom}, it depends on the choice of the mode basis. For the example above, $E_{\rm s}(\ketbra\psi\psi,1)=1$, but $E_{\rm s}(\ketbra{\psi'}{\psi'},1)=1/2$. Second, it may underestimate the efficiency in many cases. For example, the s-efficiency of the state $\ket\phi=\sqrt{1-p}\ket{00}+\sqrt{p}\ket{11}$ equals $E_{\rm s}(\ketbra\phi\phi,1)=p$, and can be very small. On the other hand, conditioning on detection of a photon in one of the modes of $\ket\phi$ results in a perfect single photon in the other mode, as is the case with producing heralded single photons via parametric down-conversion. State $\ket\phi$ is thus much more useful than, for example, single-mode state $\hat\sigma=(1-p)\ketbra{0}{0}+p\ketbra{1}{1}$, which has the same s-efficiency but cannot be processed to produce a high-quality single photon.
We aim to provide a definition of efficiency that would be invariant with respect to transformation of modes and adequately reflect the state's value for quantum information purposes.
To this end we modify the definition $E_{\rm d}$ by including an optimization over interferometers.
That is, we consider simultaneous loss channels on each of the modes ${\cal E}_{\vec p}$, followed by an arbitrary interferometer $W$, as shown in Fig.\ \ref{fig:int1}(a).
The efficiency is then the sum of the $K$ largest values of $p_\ell$:
\begin{equation}
\label{eq:kef} E_{\rm u}(\hat\rho,K):= \inf \left\{ \!\sum_{\ell=1}^K p^\downarrow_{\ell} \Big| \exists
\hat \rho_0\ge 0, W :W{\cal E}_{\vec p} (\hat \rho_0)=\hat \rho \right\}.
\end{equation}
An important property of the u-efficiency \eqref{eq:kef} is its invariance with respect to interferometric transformation of modes. Indeed, if state $\hat\rho'$ can be obtained from state $\hat\rho$ by applying interferometric transformation $U$, so that $\hat\rho'=U\hat\rho$, and we have $W{\cal E}_{\vec p} (\hat \rho_0)=\hat \rho$ in the context of \eeqref{eq:kef}, we also have $UW{\cal E}_{\vec p} (\hat \rho_0)=\hat \rho'$. But transformation $UW$ can be treated as a single interferometer, which means that $E_{\rm u}(\hat\rho',K)\leE_{\rm u}(\hat\rho,K)$. But because interferometric transformations are reversible, we also have $E_{\rm u}(\hat\rho,K)\leE_{\rm u}(\hat\rho',K)$ and hence $E_{\rm u}(\hat\rho',K)=E_{\rm u}(\hat\rho,K)$.
Similar to the case for the efficiency $E$, the Kf-efficiency can be calculated via inverting the channel.
In finite dimension, the channel given by the loss followed by the unitary operation $W$ may be represented by a matrix, which may be inverted to find $\hat\rho_0$.
The efficiency can then be found by a minimization over $\vec p$ and $W$ such that $\hat\rho_0$ is a valid quantum state.
For the other two efficiencies, the calculation is simpler.
For the d-efficiency, one only needs to minimize over $\vec p$, and for the s-efficiency one can just determine the single-mode efficiencies for the reduced density matrices in the individual modes.
Let us evaluate the multimode efficiency of the example states studied above. State $\ket\psi$ is a tensor product and has $E_{\rm s}(\ketbra\psi\psi,2)=E_{\rm s}(\ketbra\psi\psi,1)=1$. As we show below, the d-, s-, and u-efficiencies coincide for tensor product states, so we also have $E_{\rm u}(\ketbra\psi\psi,2)=E_{\rm u}(\ketbra\psi\psi,1)=1$. Since the u-efficiency is invariant under interferometric transformations, state $\ket{\psi'}$ has the same u-efficiency. Analyzing each of the modes of state $\ket{\psi'}$ on its own, we find them to carry the state $(\ket{1}\bra{1} + \ket{0}\bra{0})$/2, so $E_{\rm s}(\ketbra{\psi'}{\psi'},1)=1/2$ and $E_{\rm s}(\ketbra{\psi'}{\psi'},2)=1$. For state $\ket\phi$, both the u- and d-efficiencies equal 2. This is because, even if subjected to an interferometric transformation, it is a pure state that is not coherent, and hence cannot be obtained by attenuating another state.
\section{Proof that the Kf-efficiency cannot increase under LO processing}
In this section we show that it is impossible to increase the Kf-efficiency using LO processing.
A general LO scheme is shown in Fig.\ \ref{fig:int1}(b).
The input state $\hat\rho$, carried by $N$ optical modes with annihilation operators $\hat a_1,\ldots,\hat a_N$, is passed through a general interferometer which performs a unitary operation $Y$ on these mode operators.
We retain $M$ of the output modes $\hat a'_i$, and the remaining $N-M$ modes are subjected to a generalized destructive quantum measurement.
We consider postselection on a particular result of this measurement, and determine the Kf-efficiency of the state $\hat\rho_{\rm out}$ carried by the remaining output modes.
Our goal is to prove that
\begin{equation}\label{noincr}
E_{\rm u}(\hat\rho_{\rm out},K)\le E_{\rm u}(\hat\rho,K)
\end{equation}
for any $K\le M$.
\begin{figure}
\caption{\label{fig:int1}
\label{fig:int1}
\end{figure}
In accordance with definition \eqref{eq:kef}, we model the state $\hat\rho$ as being obtained from some initial state $\hat\rho_0$ by combining each of its modes, $\hat b_j$, with vacuum $\hat w_j$ on a beam splitter with transmissivity $p_j$ [Fig.\ \ref{fig:int1}(a)]:
\begin{equation}
\hat a^0_j = \sqrt{p_j} \hat b_j + \sqrt{1-p_j} \hat w_j,
\end{equation}
followed by interferometer $W$.
We assume that the settings are chosen such that, for some $\epsilon>0$,
\begin{equation}\label{Ep}
\sum_{\ell=1}^K p^\downarrow_\ell \le E_{\rm u}(\hat\rho,K)+\epsilon.
\end{equation}
The introduction of $\epsilon$ takes account of the possibility that there does not exist a setting which achieves the infimum.
Because interferometers $W$ and $Y$ are adjacent to each other, we can without loss of generality treat them as a single interferometer, corresponding to unitary transformation $U=YW$. The action of this interferometer can be written as
\begin{equation}\label{IntAct}
\hat a'_i = \sum_{j=1}^N U_{ij} \hat a^0_j =\sum_{j=1}^N U_{ij}\sqrt{p_j} \hat b_j + \sum_{j=1}^N U_{ij}\sqrt{1-p_j}
\hat w_j.
\end{equation}
We see that each vacuum mode contributes to each of the output modes, including those that are subjected to conditional measurements. These measurements may ``compromise" the vacuum contributions to the output state \cite{FootNoteComp}, so the output efficiency cannot be calculated directly from the matrix elements $U_{ij}$. We address this issue by performing an RQ decomposition on the matrix
$U_{ij}\sqrt{1-p_j}$ such that
\begin{equation}\label{Rmat}
U_{ij}\sqrt{1-p_j} = \sum_{\ell=1}^N R_{i\ell} Q_{\ell j},
\end{equation}
where $Q$ is unitary and $R$ is an upper triangular matrix, so $R_{i\ell}=0$ for $\ell<i$. Then we get
\begin{equation}
\label{eq:out} \hat a'_i = \sum\limits_{\ell=1}^N U_{i\ell}\sqrt{p_\ell} \hat b_\ell + \sum\limits_{\ell=1}^N
R_{i\ell} \hat v_\ell,
\end{equation}
where
\begin{equation}
\hat v_\ell := \sum_{j=1}^N Q_{\ell j}\hat w_j
\end{equation}
are obtained by transforming modes $\hat w_j$ in a fictitious interferometer $Q$. Because all the $\hat w_j$
correspond to vacuum states, so do the $\hat v_\ell$. The subset $\{\hat v_1,\ldots,\hat v_M\}$ of these modes does not contribute to the set of output modes $\{\hat a'_{M+1},\ldots, \hat a'_N\}$ that is subjected to measurement, and thus directly leads to the loss of efficiency in the output state.
Without loss of generality, we append another interferometer, $X$, acting on the $M$ output modes.
Because the Kf-efficiency is independent of linear interferometers, this interferometer does not affect the Kf-efficiency at the output.
To determine the interferometer to use, we perform a singular value decomposition on the upper left $M\times M$ block of $R$ such that
\begin{equation}\label{XRpQp}
R = X^\dagger R' Q',
\end{equation}
where the upper left $M\times M$ block of $R'$ is diagonal, and unitaries $X$ and $Q'$ are equal to the identity outside the upper left $M\times M$ block.
We choose the unitary matrix $X$ for the final interferometer to be that given by this decomposition.
Denoting the annihilation operators for the modes after the interferometer $X$ by $\hat a''_k$, we have, for $k\leM$,
\begin{align}
\label{eq:out2} & \hat a''_k = \sum_{i=1}^M X_{ki} \hat a'_i \nonumber \\
&=\sum_{i=1}^M X_{ki} \left( \sum_{\ell=1}^N U_{i\ell}\sqrt{p_\ell} \hat b_\ell
+ \sum_{\ell=1}^N R_{i\ell} \hat v_\ell \right) \nonumber \\
&=\sum_{i=1}^M \sum_{\ell=1}^N X_{ki}U_{i\ell}\sqrt{p_\ell} \hat b_\ell
+\sum_{i=1}^M \sum_{\ell=M+1}^N X_{ki} R_{i\ell} \hat v_\ell \nonumber \\
& \quad + \sum_{i=1}^M \sum_{\ell=1}^M X_{ki} \sum_{k',n=1}^M [X^\dagger]_{ik'} R'_{k'n} Q'_{n\ell} \hat v_\ell \nonumber \\
&= \sum_{i=1}^K\sum\limits_{\ell=1}^N X_{ki} U_{i\ell}\sqrt{p_\ell} \hat b_\ell
+ \sum_{i=1}^M \sum_{\ell=M+1}^N X_{ki} R_{il} \hat v_{\ell}
+ R'_{kk} \hat v''_k,
\end{align}
where
\begin{equation}
\hat v''_k := \sum_{\ell=1}^N Q'_{k\ell} \hat v_\ell.
\end{equation}
As the set $\{\hat v''_k\}$ may be regarded as being obtained from initial vacuum modes $\{\hat w_k\}$ via a unitary transformation, they represent an orthonormal set of bosonic modes in the vacuum state. Furthermore, those $\hat v''_k$ that contribute to $\hat a''_k$ do not contain any contribution from the ``compromised" vacuum modes. Indeed, they only contain contributions from $\hat v_\ell$ for $\ell\le M$, whereas the operators for the measured modes only contain contributions from $\hat v_\ell$ for $\ell>M$. As a result, these vacuum contributions are equivalent to loss.
To make this result explicit, we write the annihilation operator in the form $\hat a''_k = \hat B''_k + \hat
V''_k$, where
\begin{equation}
\hat B''_k = \sum_{i=1}^M\sum\limits_{\ell=1}^N X_{ki} U_{i\ell}\sqrt{p_\ell} \hat b_\ell + \sum_{i=1}^M
\sum_{\ell=M+1}^N X_{ki} R_{il} \hat v_{\ell} ,
\end{equation}
and
\begin{equation}
\label{eq:vdef} \hat V''_k = R'_{kk} \hat v''_k.
\end{equation}
We then find that
\begin{align}
[\hat V''_k , (\hat V''_{k'})^\dagger ] &= \delta_{kk'}|R'_{kk}|^2, \\ [\hat B''_k , (\hat B''_{k'})^\dagger ] &=
\delta_{kk'}(1-|R'_{kk}|^2).
\end{align}
The first line follows immediately from Eq.\ \eqref{eq:vdef}. The second line is obtained because $\hat B''_k =
\hat a''_k - \hat V''_k$ and $[\hat a''_k , (\hat a''_{k'})^\dagger ] = \delta_{kk'}$.
Defining
\begin{equation}
p''_k := 1-|R'_{kk}|^2, \qquad \hat b''_k := \hat B''_k/\sqrt{p''_k},
\end{equation}
we have
\begin{equation}
\hat a''_k = \sqrt{p''_k} \hat b''_k + \sqrt{1-p''_k}\hat v''_k.
\end{equation}
Therefore, the output state may be obtained by an interferometer that produces the modes with annihilation
operators $\hat b''_k$, then combining with vacua on beam splitters with transmissivities $p''_k$, as shown in Fig.\ \ref{fig:int2}.
\begin{figure}
\caption{\label{fig:int2}
\label{fig:int2}
\end{figure}
Without loss of generality, we can assume $X$ and $Q'$ to have been chosen such that the numbers $p''_k$ are in non-increasing
order.
The Kf-efficiency at the output is therefore upper bounded by
\begin{equation}\label{Epout}
E_{\rm u}(\hat\rho_{\rm out},K) \le \sum_{i=1}^K p''_k.
\end{equation}
To determine the sum \eqref{Epout}, we can define the unitaries
\begin{equation}
U' := XU, \qquad Q'' := Q' Q.
\end{equation}
It follows from \eeqref{XRpQp} that $R' = X R (Q')^\dagger$. Therefore, according to \eeqref{Rmat},
\begin{equation}
R'_{k\ell} = \sum_{m=1}^N U'_{km} \sqrt{1-p_m} (Q''_{\ell m})^*.
\end{equation}
Then we obtain
\begin{align}
&\sum_{k=1}^K p''_k \le K-\sum_{k=1}^K\sum_{\ell=1}^K |R'_{k\ell}|^2 \nonumber \\
&= K-\sum_{k=1}^N\sum_{\ell=1}^K\sum_{m,j=1}^N U'_{km}\sqrt{1-p_m} (Q''_{\ell m})^* \nonumber \\ &\quad \times (U'_{kj})^*\sqrt{1-p_j} Q''_{\ell j} \nonumber \\
&= K-\sum_{\ell=1}^K\sum_{m,j=1}^N \delta_{m j}\sqrt{1-p_m} (Q''_{\ell m})^*\sqrt{1-p_j} Q''_{\ell j} \nonumber \\
&= K-\sum_{\ell=1}^K\sum_{j=1}^N (1-p_j) |Q''_{\ell j}|^2 \nonumber \\ &= \sum_{\ell=1}^K\sum_{j=1}^N p_j |Q''_{\ell j}|^2 \le \sum_{\ell=1}^K p^\downarrow_\ell.
\label{longproof}
\end{align}
The last inequality in \eeqref{longproof} is obtained because $Q''_{ij}$ is unitary, so $|Q''_{ij}|^2$ is a doubly stochastic matrix, and thus vector $p_l$ majorizes vector \cite{NielsenChuang}
\begin{equation}
q_\ell := \sum_{j=1}^N p_j |Q''_{ij}|^2.
\end{equation}
Now, according to Eqs.~\eqref{Ep}, \eqref{Epout}, \eqref{longproof} and because we can choose $\epsilon$ to be arbitrarily close to zero, we obtain
\begin{equation}
\label{EKleEK}
E_{\rm u}(\hat\rho_{\rm out},K) \le E_{\rm u}(\hat\rho,K).
\end{equation}
This is the main result of this work: the universal measure of quantum efficiency of a multimode state, the
Kf-efficiency, cannot increase under LO processing.
\section{Comparison of efficiency measures}
We now use the above result to prove some additional properties of the different measures of multimode efficiency defined in Sec.~III. First, we show that these efficiencies are related according to
\begin{equation}\label{EffIneq}
E_{\rm s}(\hat\rho,K)\le E_{\rm u}(\hat\rho,K)\le E_{\rm d}(\hat\rho,K).
\end{equation}
To examine the s-efficiency, we can again assume that state $\hat\rho$ is obtained via a set of beam splitters with transmissivities
$p_j$ and an interferometer $W$ as in Fig.\ \ref{fig:int1}(a), such that the sum of the $K$ largest values of
$p_j$ is no more than $E_{\rm u}(\hat\rho,K)+\epsilon$.
Then the operators for the state $\hat\rho$ are given by
\begin{equation}
\hat a_j = \hat B_j + \hat V_j,
\end{equation}
with
\begin{equation}
\hat B_j :=\sum_{\ell=1}^N W_{j\ell}\sqrt{p_\ell}\hat b_\ell, \qquad \hat V_j :=\sum_{\ell=1}^N
W_{j\ell}\sqrt{1-p_\ell}\hat w_\ell,
\end{equation}
corresponding to operators carrying signal and vacuum fields, respectively.
To determine the s-efficiency, we determine the efficiency for each mode individually.
When determining the efficiency for mode $\hat a_j$, we can regard modes $\hat a_k$ for $k\ne j$ as being discarded.
The vacuum operators $\hat V_k$ for $k\ne j$ are not orthogonal to $\hat V_j$; however, since those modes are discarded, the addition of vacuum $\hat V_j$ is equivalent to loss.
Therefore, the efficiency of the state in mode $\hat a_j$ is no greater than
\begin{equation}
p'_j:=[\hat B_j,\hat B_j^\dagger]=\sum_{\ell=1}^N |W_{j\ell}|^2 p_\ell.
\end{equation}
The sum of the $K$ largest values of $p'_j$ upper bounds the s-efficiency; that is,
\begin{equation}
E_{\rm s}(\hat\rho,K)\le \sum_{j=1}^K {p'}^\downarrow_j.
\end{equation}
Because $W$ is unitary, $|W_{j\ell}|^2$ is a doubly stochastic matrix, and the vector of values $\vec p$
majorizes $\vec p'$. That means that
\begin{equation}
\sum_{j=1}^K {p'}^\downarrow_j \le \sum_{j=1}^K p_j^\downarrow \le E_{\rm u}(\hat\rho,K)+\epsilon.
\end{equation}
Because this holds for all $\epsilon>0$, we have $E_{\rm s}(\hat\rho,K)\le E_{\rm u}(\hat\rho,K)$.
The second inequality in \eeqref{EffIneq} is because the definition of $E_{\rm u}(\hat\rho,K)$ in \eeqref{eq:kef} looks for the minimum in a larger set of states than that of $E_{\rm d}(\hat\rho,K)$ in \eeqref{ED}.
For tensor product states, the s- and d-efficiencies are the same.
To see this, use the definition \eqref{Ei} on the tensor product state
\begin{equation}
\hat \rho = \bigotimes_{j=1}^N \hat\rho_j.
\end{equation}
One obtains
\begin{equation}
E_{\rm s}(\hat\rho,K)= \sum_{\ell=1}^K E(\hat\rho_{\ell})^\downarrow.
\end{equation}
Therefore, there exists a set of states $\hat\rho^0_j$ and transmissivities $p_j$, such that the sum of the $K$ largest values of $p_j$ is no more than $E_{\rm s}(\hat\rho,K)+\epsilon$, and the final states $\hat\rho_j$ may be obtained via loss channels with transmissivities $p_j$ from initial states $\hat\rho^0_j$.
This would also provide a scheme for producing $\hat\rho$ for the definition of $E_{\rm d}(\hat\rho,K)$, and therefore
\begin{equation}
E_{\rm d}(\hat\rho,K) \le \sum_{j=1}^K p_j^\downarrow \le E_{\rm s}(\hat\rho,K)+\epsilon.
\end{equation}
Because this is true for all $\epsilon>0$, we obtain $E_{\rm d}(\hat\rho,K) \le E_{\rm s}(\hat\rho,K)$.
Combining this with \eeqref{EffIneq}, we find that $E_{\rm d}(\hat\rho,K) = E_{\rm s}(\hat\rho,K)$ for tensor product states, and all inequalities in \eqref{EffIneq} saturate.
This result leads us to an important conclusion. Suppose we start with $N$ separable states (for example, imperfect single photons as in \eeqref{eq:mix}) with efficiencies $p_\ell$, which we subject to LO processing, resulting in a set of modes in which the efficiencies, when analyzed separately, are given by $p'_\ell$.
Using the result that LO processing cannot increase the u-efficiency, and \eeqref{EffIneq}, we have for any integer $K$,
\begin{equation}
\label{eq:cat}
\sum_{\ell=1}^K p'_\ell \le \sum_{\ell=1}^K p^\downarrow_\ell .
\end{equation}
In other words, the LO processing can act to average the efficiencies, but not to concentrate them.
One consequence is the exclusion of any possibility for ``catalytic'' efficiency improvement, in which some
highly efficient sources are used to increase the efficiency in other optical modes, without themselves suffering
from loss.
These results do not rule out increases in the individual efficiencies; for example, if the largest efficiency is decreased, it is possible for the second largest efficiency to be increased.
\section{Summary}
We have introduced a number of measures that enable us to quantify the efficiency in multimode systems.
The Kf-efficiency is a powerful general measure that takes account of how loss may have been mixed between the different modes.
It is unchanged under linear interferometers, and cannot increase under more general LO processing with destructive measurements.
We have used this result to show that catalytic improvement of photon sources is not possible with LO processing.
If one starts with independent optical sources (which produce a tensor product of states), then the efficiencies in the individual output modes are weakly majorized by the efficiencies in the input.
This means that it is not possible to concentrate the efficiencies, such that the sum of the highest $K$ output efficiencies is greater than the sum of the highest $K$ input efficiencies.
It is clearly possible to increase the Kf-efficiency if one uses \emph{non}linear optical elements.
For example, a standard method of producing single photons is via parametric downconversion (a nonlinear process), and postselection on detection of a photon in one of the output modes.
The initial beam is coherent, with efficiency zero, but the final output (ideally) has unit efficiency.
\acknowledgments
This work has been supported by NSERC, AIF, CIFAR and Quantum{\it Works}. We thank B.\ C.\ Sanders and H.\ M.\ Wiseman for stimulating
discussions.
\end{document}
|
\begin{document}
\numberwithin{theorem}{section}
\begin{abstract}
Recently Bowden, Hensel and Webb defined the {\em fine curve graph} for
surfaces, extending the notion of curve graphs for the study of
homeomorphism or diffeomorphism groups of surfaces.
Later Long, Margalit, Pham, Verberne and Yao proved that for a closed
surface of genus~$g\geqslant 2$,
the automorphism group of the fine graph is naturally isomorphic
to the homeomorphism group of the surface.
We extend this result to the torus case $g=1$; in fact our method works for
more general surfaces, compact or not, orientable or not.
We also discuss the case of a smooth version of the fine graph.
\end{abstract}
\title{Automorphisms of some variants of fine graphs}
\sloppy
\section{Introduction}
\subsection{Context and results}
For a connected, compact surface $\Sigma_g$ of genus $g\geqslant 1$,
Bowden, Hensel and Webb~\cite{BHW} recently introduced the
{\em fine curve graph} $\mathcal{C}^\dagger(\Sigma)$,
as the graph whose vertices are all the essential closed curves on
$\Sigma$, with an edge between two vertices $a$ and $b$ whenever
$a\cap b=\emptyset$, if $g\geqslant 2$, and whenever
$|a\cap b|\leqslant 1$ if $g=1$. They proved that for every $g\geqslant 1$,
the graph $\mathcal{C}^\dagger(\Sigma)$ is hyperbolic, and derived a
construction of an infinite dimensional family of quasi-morphisms on
$\mathrm{Hom}eo_0(\Sigma)$, thereby answering long standing questions of
Burago, Ivanov and Polterovich.
The ancestor of the fine graph is the
usual curve complex of a surface $\Sigma$, \textsl{i.e.}, the complex whose vertices
are the isotopy classes of essential curves, with an edge (or a simplex,
more generally) between some vertices if and only if they have disjoint
representatives. Since its introduction by Harvey~\cite{Harvey}, the curve
complex of a surface has been an extremely useful tool for the study of the
mapping class group $\mathrm{Mod}(\Sigma)$ of that surface, as it acts on it naturally.
In particular,
the fact that this complex is hyperbolic, discovered by Masur and Minsky,
has greatly improved the understanding of the mapping class groups (see~\cite{MM1,MM2}).
The result of Bowden, Hensel and Webb, promoting the
hyperbolicity of the curve complex to that of the fine curve graph, opens the
door both to the study of
what classical properties of usual curve complexes have counterparts in
the fine curve graph, and to the use of this graph to derive properties of
homeomorphism groups.
A first step in this direction was taken by
Bowden, Hensel, Mann, Militon and Webb~\cite{BHMMW}, who explored the metric
properties of the action of $\mathrm{Hom}eo(\Sigma)$ on this hyperbolic graph.
A classical theorem by Ivanov~\cite{Ivanov} states that, when $\Sigma$ is
a closed surface of genus $g\geqslant 2$, the natural map
$\mathrm{Mod}(\Sigma)\to\mathrm{Aut}(\mathcal{C}(\Sigma))$ is an isomorphism.
Recently Long, Margalit, Pham, Verberne and Yao~\cite{LMPVY} proved
the following natural counterpart of Ivanov's theorem for fine graphs:
provided $\Sigma$ is a compact orientable surface of genus $g\geqslant 2$, the natural map
\[ \mathrm{Hom}eo(\Sigma)\longrightarrow\mathrm{Aut}(\mathcal{C}^\dagger(\Sigma)) \]
is an isomorphism.
They also suggested that this map
(with the appropriate version of
$\mathcal{C}^\dagger$)
may also be an isomorphism when $g=1$, and
conjectured that the automorphism group of the fine curve graph
of smooth curves, should be nothing more than $\mathrm{Diff}(\Sigma)$.
In this article, we address both these questions.
Our motivation originates from the case of the torus: excited by~\cite{BHMMW},
we wanted to understand
more closely the relation between the rotation set of homeomorphisms isotopic
to the identity and the metric properties of their actions on the fine graph.
This subject
will be treated in another article, joint with Passeggi and
Sambarino~\cite{Fantome}. The methods developed in the present article are valid not only for the torus but for a large class of surfaces.
We work on nonspherical surfaces
(\textsl{i.e.}, surfaces not embeddable in the $2$-sphere, or equivalently,
containing at least one nonseparating simple closed curve), orientable or not,
compact or not.
We consider the graph $\mathbb{T}ransFin(\Sigma)$, whose vertices are
the nonseparating simple closed curves, and with an edge between two
vertices $a$ and $b$ whenever they are either disjoint, or have
exactly one, topologically transverse intersection point (see the beginning of Section~\ref{sec:Lemmes1-2-3} for more detail).
Our first result answers a problem raised in~\cite{LMPVY}.
\begin{theorem}\label{thm:AutC1}
Let $\Sigma$ be a connected, nonspherical surface, without boundary.
Then the natural map $\mathrm{Hom}eo(\Sigma)\to\mathrm{Aut}(\mathbb{T}ransFin(\Sigma))$
is an isomorphism.
\end{theorem}
Our second result concerns the smooth version of fine graphs.
We consider the graph $\mathbb{T}ransFinLisse(\Sigma)$ whose vertices are the
smooth nonseparating curves in $\Sigma$, with an edge between $a$
and $b$ if they
are disjoint or have one, transverse intersection point, in the differentiable
sense (in particular, $\mathbb{T}ransFinLisse(\Sigma)$ is not the subgraph of
$\mathbb{T}ransFin(\Sigma)$ induced by the vertices corresponding to smooth curves:
it has fewer edges). The following result partially confirms the
conjecture of~\cite{LMPVY}; here we restrict to the case of orientable surfaces for simplicity.
\begin{theorem}\label{thm:AutCFinLisse}
Let $\Sigma$ be a connected, orientable, nonspherical surface, without boundary.
Then all the automorphisms of $\mathbb{T}ransFinLisse(\Sigma)$ are realized
by homeomorphisms of~$\Sigma$.
\end{theorem}
In other words, if we denote by $\mathrm{Hom}eo_{\infty \pitchfork}(\Sigma)$ the subgroup of
$\mathrm{Hom}eo(\Sigma)$ preserving the collection of smooth curves and preserving
transversality, then the natural map
\[
\mathrm{Hom}eo_{\infty \pitchfork}(\Sigma) \longrightarrow \mathrm{Aut}(\mathbb{T}ransFinLisse(\Sigma))
\]
is an isomorphism. We were surprised to realize however that $\mathrm{Hom}eo_{\infty \pitchfork}(\Sigma)$ is strictly larger than $\mathrm{Diff}(\Sigma)$.
\begin{proposition}\label{prop:PasDiff}
Every surface $\Sigma$ admits a homeomorphism $f$
such that
$f$ and $f^{-1}$ preserve the set of smooth curves, and preserve
transversality, but such that neither $f$ nor $f^{-1}$ is differentiable.
In particular, the natural map
$$\mathrm{Diff}(\Sigma)\longrightarrow \mathrm{Aut}(\mathbb{T}ransFinLisse(\Sigma))$$
is not surjective.
\end{proposition}
\subsection{Idea of the proof of Theorem~\ref{thm:AutC1}}
The main step in this proof is the following.
\begin{proposition}\label{prop:Types}
If $\{a,b\}$, or $\{a,b,c\}$ is a $2$-clique or
a $3$-clique of $\mathbb{T}ransFin(\Sigma)$ then, from the graph
structure of $\mathbb{T}ransFin(\Sigma)$, we can tell the type of the clique.
\end{proposition}
If $\{a_1,\ldots, a_n\}$ is an $n$-clique in the graph
$\mathbb{T}ransFin(\Sigma)$, the homeomorphism type of the subset
$\cup_{j=1}^n a_j$ of $\Sigma$ will be called the {\em type} of the $n$-clique.
We will explore this only for $2$ and $3$-cliques. A $2$-clique
$\{a,b\}$, \textsl{i.e.}, an edge of the graph
$\mathbb{T}ransFin(\Sigma)$,
may have two distinct types: the intersection $a\cap b$ may be empty
or not. For a $3$-clique $\{a,b,c\}$, up to permuting the curves $a$, $b$
and $c$, the cardinals of the intersections
$a\cap b$, $a\cap c$ and $b\cap c$, respectively, may be $(1,1,1)$,
or $(1,1,0)$, or $(1,0,0)$, or $(0,0,0)$. This determines the type of
the $3$-clique, except in the case $(1,1,1)$, where the intersection
points $a\cap b$, $a\cap c$ and $b\cap c$ may be pairwise distinct, in
which case we will speak of a $3$-clique of type {\em necklace}, or these
intersection points may be equal, in which case we will speak of a
$3$-clique of type {\em bouquet}, see
Figure~\ref{fig:Bouquet-Collier}.
\begin{figure}
\caption{A bouquet (left) and a necklace (right) of three circles.}
\label{fig:Bouquet-Collier}
\end{figure}
The main bulk of the proof of Proposition~\ref{prop:Types} consists in
distinguishing the $3$-cliques of type
necklace from any other $3$-clique of $\mathbb{T}ransFin(\Sigma)$.
Here, the key is that among all the $3$-cliques, the cliques $\{a,b,c\}$ of
type necklace are exactly those such that the union $a\cup b\cup c$ contains
nonseparating simple closed curves other than $a$, $b$ and $c$.
In terms of the graph structure, this leads to the following property,
denoted by $N(a,b,c)$, which turns out to characterize these cliques:
{\em There exists
a finite set $F$ of at most $8$ vertices of $\mathbb{T}ransFin(\Sigma)$,
all distinct from $a$, $b$ and $c$, such that every vertex $d$ connected
to $a$, $b$ and $c$ in this graph, is connected to at least one element
of $F$.}
From this,
we will easily characterize all the configurations of $2$-cliques and $3$-cliques
in terms of similar statements in the first order logic of
the graph $\mathbb{T}ransFin(\Sigma)$.
Now, let $T(\mathbb{T}ransFin(\Sigma))$ denote the set of edges $\{a,b\}$ of
$\mathbb{T}ransFin(\Sigma)$ satisfying $|a\cap b|=1$. Then
we have a map
\[ \mathrm{Point}\colon \ T(\mathbb{T}ransFin(\Sigma))\to\Sigma, \]
which to each edge $\{a,b\}$ of $\mathbb{T}ransFin(\Sigma)$, associates the intersection point
$a\cap b$.
The next step in the the proof of Theorem~\ref{thm:AutC1} now consists in
characterizing the equality $\mathrm{Point}(a,b)=\mathrm{Point}(c,d)$ in terms of the
structure of the graph. This characterization shows that every automorphism of
$\mathbb{T}ransFin(\Sigma)$ is realized by some bijection of $\Sigma$; then we prove that such a
bijection is necessarily a homeomorphism (see Proposition~\ref{prop:BijHomeo}).
In order to characterize the equality $\mathrm{Point}(a,b)=\mathrm{Point}(c,d)$,
we introduce on $T(\mathbb{T}ransFin(\Sigma))$ the relation $\diamondvert$
generated, essentially (see section~\ref{sec:Adjacency} for details),
by
$(a,b)\diamondvert(b,c)$ if $(a,b,c)$ is a $3$-clique of type bouquet.
Obviously, if $(a,b)\diamondvert(c,d)$ then $\mathrm{Point}(a,b)=\mathrm{Point}(c,d)$.
Interestingly, the converse is false, but we can still use this idea in
order to characterize the points of $\Sigma$ in terms of the graph structure
of $\mathbb{T}ransFin(\Sigma)$.
This subtlety between the relation $\diamondvert$ and the equality of points
is related to the non smoothness of the curves involved, and more precisely,
to the fact that a curve may spiral infinitely with respect to another curve in a
neighborhood of a common point.
We think that this phenomenon is of independent interest and we investigate it in Section~\ref{sec:LocalSubgraphs}. In particular, we can easily state, in terms of the graph structure
of $\mathbb{T}ransFin(\Sigma)$, an
obstruction for a homeomorphism to be conjugate to a $C^1$-diffeomorphism,
see Section~\ref{ssec:Tourbillons}.
\subsection{Ideas of the proof of Theorem~\ref{thm:AutCFinLisse}}
In the smooth case, the adaptation of our proof of Theorem~\ref{thm:AutC1}
fails from the start: indeed, the closed curves contained in the union
$a\cup b\cup c$ of a necklace, and distinct from $a$, $b$ and $c$, are
not smooth.
This suggests the idea to use sequences of curves (at the expense of losing the
characterizations of configurations in terms of first order logic).
This time it is easiest to first characterize disjointness of curves (see Lemma~\ref{lem:DisjointLisse}), and then recover
the different types of $3$-cliques. Then the strategy follows the $C^0$ case.
Once we start to work with sequences, it is natural to say that a sequence
$(f_n)$
of curves not escaping to infinity
converges to $a$ in some weak sense, if for every vertex $d$ such
that $\{a,d\}$ is an edge of the graph, $\{f_n,d\}$ is also an edge for all
$n$ large enough.
As it turns out, this property implies convergence in $C^0$-sense to
$a$, and is implied by convergence in $C^1$-sense. But it is not equivalent
to the convergence in $C^1$-sense, and it is precisely this default of
$C^1$-convergence that enables us to distinguish between disjoint or
transverse pairs of curves.
Interestingly,
this simple criterion for disjointness has no counterpart in
the $C^0$-setting. Indeed, in that setting, no sequence of curves converges
in this weak sense:
given a curve $a$, and a sequence $(f_n)$ of curves with, say, some
accumulation point in $a$, we can build a curve $d$ intersecting $a$ once
transversally (topologically), but oscillating so much that it itersects every
$f_n$ several times.
From this perspective, none of our approaches in the
$C^0$-setting and in the $C^\infty$-setting are directly adaptable to the other.
\subsection{Further comments}
We can imagine many variants of fine graphs. For example, in the
arXiv version of~\cite{BHW}, for the case of the torus they worked with the
graph $\mathbb{T}ransFin(\Sigma)$ on which we are working here, whereas in the
published version, they changed to
a fine graph in which two curves $a$ and $b$ are still related by an edge
when they have one intersection, not necessarily transverse.
More generally, in the spirit of Ivanov's metaconjecture,
we expect that the group of
automorphisms should not change from
any reasonable variant to another. And indeed,
using the ideas of \cite[Section~2]{LMPVY} and those presented here,
we can navigate between various versions of fine graphs, and recover,
from elementary properties of one version, the configurations defining
the edges in another version, thus proving that their automorphism groups
are naturally isomorphic.
From this perspective, it seems satisfying to
recover the group of homeomorphisms of the surface as the automorphism
group of any reasonable variant of the fine graph.
In this vein, we should mention that the results of~\cite[Section~2]{LMPVY}
directly yield a natural map
$\mathrm{Aut}(\mathbb{C}fin{}(\Sigma))\to\mathrm{Aut}(\mathbb{T}ransFin(\Sigma))$, and from there, our
proof of Theorem~\ref{thm:AutC1} may be used as
an alternative proof of their main result.
All reasonable variants of the fine graphs should be quasi-isometric,
and a unifying theorem (yet out of reach today, as it seems to us)
would certainly be a counterpart of the theorem by
Rafi and Schleimer~\cite{RafiSchleimer}, which states
that every quasi-isometry of the usual curve graph is bounded distance
from an isometry.
\subsection{Organization of the article}
Section~\ref{sec:Lemmes1-2-3} is devoted to the recognition of the
$3$-cliques in the $C^0$-setting, and of some other configurations regarding
the nonorientable case. We encourage the reader to skip, at first reading,
everything that concerns the nonorientable case: these points shoud be easily
identified, and this halves the length of the proof.
In Section~\ref{sec:AutHomeo} we prove Theorem~\ref{thm:AutC1}.
In Section~\ref{sec:LocalSubgraphs} we characterize, from the topological
viewpoint, the relation $\diamondvert$ introduced above in terms of the graph
structure, and deduce our obstruction to differentiability.
Finally in Section~\ref{sec:AutCFinLisse} we prove Theorem~\ref{thm:AutCFinLisse}
and Proposition~\ref{prop:PasDiff}.
\subsection*{Acknowledgments}
We thank Kathryn Mann for encouraging discussions, and Dan Margalit for his extensive feedback on a preliminary version of this manuscript.
\section{Recognizing configurations of curves}\label{sec:Lemmes1-2-3}
\subsection{Standard facts and notation}
We will use, often without mention, the following easy or standard facts for curves
on surfaces.
The first is the classification of connected, topological surfaces with
boundary (not necessarily compact).
In particular, every topological surface admits a smooth structure.
Given a closed curve $a$ in a surface $\Sigma$, we can apply this classification
to $\Sigma\smallsetminus a$ and understand all possible configurations of simple
curves;
this is the so-called {\em change of coordinates
principle} in the vocabulary of the book of Farb and Margalit~\cite{FarbMargalit}.
In particular, every
closed curve $a$ has a neighborhood homeomorphic to an annulus or a M\"obius strip
in which $a$ is the ``central curve''.
Very often in this article, we will consider the curves $a'$ obtained by
deforming $a$ in such a neighborhood, so that $a'$ is disjoint from $a$ in the
first case, or intersects it once, transversely, in the second, as in
Figure~\ref{fig:FigureFacts}. We will say that $a'$ is obtained by {\em pushing
$a$ aside}.
The change of coordinates principle also
applies to
finite graphs embedded in $\Sigma$: there is a homeomorphism of $\Sigma$
that sends any given graph to
a smooth graph, such that all edges connected to a given vertex leave it in distinct directions.
In the simple case when the graph is the union of two or three simple closed curves that pairwise intersect at most once, this observation
justifies the description of the possible configurations of cliques in
the introduction. This also enables, provided two curves $a$ and $b$ intersect
at a single point (or more generally at a finite number of points), to speak
of a {\em transverse} (also called {\em essential}), or to the contrary
{\em inessential}, intersection point, as we did in the introduction.
Here are two other useful facts.
\begin{fact}\label{fact:NonSep}
A simple closed curve $a$ in a surface, is nonseparating if and only
if there exists a closed curve $b$, such that $a\cap b$ is a
single point and this intersection is transverse.
\end{fact}
\begin{fact}\label{fact:TroisArcsSep}
Let $p,q$ be two distinct points, and $x,x',x''$ three simple arcs,
each with end-points $p$ and $q$, such that
\[ x \cap x' = x' \cap x'' = x \cap x'' = \{p,q\}. \]
If two of the three curves $x \cup x', x' \cup x'', x'' \cup x$ are
separating, then the third one is also separating.
\end{fact}
\begin{proof}
Denote $y=x\smallsetminus\{p,q\}$, the arc $x$ without its ends, and
similarly, define $y'$ and $y''$. Suppose $x\cup x'$ and $x\cup x''$ are
separating.
Denote by $\Sigma_1, \Sigma_2$, resp. $\Sigma_3, \Sigma_4$,
the components of
$\Sigma\smallsetminus(x\cup x')$, resp. $\Sigma\smallsetminus(x\cup x'')$,
where $\Sigma_2$ contains $y''$ and $\Sigma_4$ contains $y'$.
By looking at neighborhoods of $p$ and $q$
(see Figure~\ref{fig:FigureFacts}, left), we see that
$\Sigma'=\Sigma_2\cap\Sigma_4$ is non empty, and that the arc $y$
bounds $\Sigma_1$ on one side, and $\Sigma_3$ on the other, so
$\Sigma''=\Sigma_1\cup y\cup \Sigma_3$ is a surface.
Now, $\Sigma\smallsetminus(x'\cup x'')=\Sigma'\cup\Sigma''$, and
$\Sigma'$ and $\Sigma''$ are disjoint by construction.
\end{proof}
\begin{figure}
\caption{Left: a neighborhood of $p$. Center: pushing a two-sided curve $a$.
Right: pushing a one-sided curve $a$.}
\label{fig:FigureFacts}
\end{figure}
\subsection{Properties characterizing geometric configurations}
Now we list the properties, in terms of the graph
$\mathbb{T}ransFin(\Sigma)$, that will be used as
characterizations of certain configurations of curves.
This allows us to specify the statement of Proposition~\ref{prop:Types}, which
will be proved in the next paragraph, and define the relation
$\diamondvert$ in terms of the graph $\mathbb{T}ransFin(\Sigma)$.
In the following, the letters $N, D, T, B$ respectively stand for
necklace, disjoint, transverse, and bouquet.
If $a,b,c$ are vertices of this graph, we will denote by:
\begin{itemize}
\item[$\bullet$] $N(a,b,c)$ the property that $\{a,b,c\}$ is a $3$-clique
of $\mathbb{T}ransFin(\Sigma)$ and there exists a finite set $F$ of at most $8$
vertices of $\mathbb{T}ransFin(\Sigma)$, all distinct from $a$, $b$ and $c$,
such that for every vertex $d$ such that $\{a,b,c,d\}$ is a $4$-clique,
there is an edge from $d$ to at least one element of $F$,
\item[$\bullet$] $D(a,b)$ the property that $\{a,b\}$ is an edge of
$\mathbb{T}ransFin(\Sigma)$ and there does not
exist a vertex $d$ such that $N(a,b,d)$ holds,
\item[$\bullet$] $T(a,b)$ the property that $\{a,b\}$ is an edge of
$\mathbb{T}ransFin(\Sigma)$ and $D(a,b)$ does not hold,
\item[$\bullet$] $B(a,b,c)$ the property that $T(a,b)$, $T(a,c)$, $T(b,c)$
all hold but $N(a,b,c)$ does not.
\end{itemize}
The following proposition is the main part of
Proposition~\ref{prop:Types}.
\begin{proposition}\label{prop:TypesPrecis}
Let $a$, $b$ and $c$ be vertices of the graph $\mathbb{T}ransFin(\Sigma)$.
Property $N(a,b,c)$ holds if and only if $\{a,b,c\}$
is a $3$-clique of type necklace of $\mathbb{T}ransFin(\Sigma)$.
\end{proposition}
The proof of Proposition~\ref{prop:TypesPrecis} will occupy the next
paragraph. The following corollary complements
Proposition~\ref{prop:TypesPrecis} and provides a precise version of
Proposition~\ref{prop:Types}.
\begin{corollary}\label{coro:TypesPrecis}~
\begin{itemize}
\item Property $D(a,b)$ holds if and only if the curves $a$ and
$b$ are disjoint.
\item Property $T(a,b)$ holds if and only if $a$ and $b$ have
a unique intersection point and the intersection is transverse.
\item Property $B(a,b,c)$ holds if and only if $\{a, b, c\}$
is a $3$-clique of type bouquet.
\end{itemize}
\end{corollary}
\begin{proof}
Let $a$ and $b$ be neighbors in the graph $\mathbb{T}ransFin(\Sigma)$.
Of course, if $a$ and $b$ are disjoint, then $D(a,b)$ holds: there does
not exist a curve $d$ such that $N(a,b,d)$, since this would mean
that $\{a,b,d\}$ is of type necklace and then, by definition, $a$ and $b$
would intersect.
Conversely, suppose that $a$ and $b$ are not disjoint, and let us prove
that $D(a,b)$ does not hold, \textsl{i.e.}, let us find a curve $c$
such that $\{a,b,c\}$ is a $3$-clique of type necklace.
In the case when one of $a$ or $b$ is one-sided, up to exchanging the two,
suppose $a$ is one-sided. Then we may push $a$ in order to find a curve $c$
which makes a $3$-clique of type necklace with $a$ and $b$, see
Figure~\ref{fig:OnComplete} (left). In the case when both $a$ and $b$ are
two-sided, then by the change of coordinates principle,
a regular neighborhood of $a\cup b$ is homeomorphic to a
one-holed torus, embedded in $\Sigma$, with a
choice of meridian and longitude
coming from $a$ and $b$.
In this torus, a curve $c$ with
slope $1$ will form a $3$-clique of type necklace with $a$ and $b$,
see Figure~\ref{fig:OnComplete}, right.
\begin{figure}
\caption{Completing $(a,b)$ to a $3$-clique of type necklace}
\label{fig:OnComplete}
\end{figure}
This proves the first point.
The second point is a straightforward
consequence of the first, and the third simply follows from the second
point together with Proposition~\ref{prop:TypesPrecis}.
\end{proof}
\subsection{Proof of Proposition~\ref{prop:TypesPrecis}}
The following lemma is a key step in the proof of the direct implication in Proposition~\ref{prop:TypesPrecis}.
\begin{lemma}\label{lem:drenccc}
Let $\{a,b,c\}$ be a $3$-clique of $\mathbb{T}ransFin(\Sigma)$ which is not
of type necklace. Then there exists a vertex $d$ of
$\mathbb{T}ransFin(\Sigma)$ such that $\{a,b,c,d\}$ is a $4$-clique of
$\mathbb{T}ransFin(\Sigma)$, and such that $d$ meets every connected
component of $\Sigma\smallsetminus(a\cup b\cup c)$.
\end{lemma}
Before entering the proof, we note that we cannot remove the hypothesis that
$\{a, b, c\}$ is not of type necklace. Indeed, in the flat torus
$\Sigma = \mathbb{R}^2/\mathbb{Z}^2$, consider three closed geodesics
$a,b,c$ respectively directed by $(1,0)$, $(0,1)$, and $(1, 1)$.
By pushing $c$ aside if necessary, we obtain a $3$-clique of type necklace.
The complement of $a \cup b \cup c$ in $\Sigma$ has three connected components,
and there is no curve $d$ satisfying the conclusion of the lemma.
\begin{proof}
In all the proof, we will denote
$\Sigma'=\Sigma\smallsetminus(a\cup b\cup c)$.
Up to permuting the curves $a$, $b$ and $c$, we may suppose that the triple
of cardinals of intersections, $(|a\cap b|, |a\cap c|, |b\cap c|)$, equals
$(1,1,1)$, or $(1,1,0)$, or $(1,0,0)$, or~$(0,0,0)$. We will deal with
these cases separately.
Let us begin with the case $(0, 0, 0)$. If $\Sigma'$ is connected,
then any curve $d$ making a $4$-clique with $(a,b,c)$ satisfies the Lemma.
Such a curve can be found, for example, by pushing $a$ aside.
If $\Sigma'$ has two connected components, denote them by $\Sigma_1$ and
$\Sigma_2$. Since $a$, $b$ and $c$ are each nonseparating, at least two of
the curves $a, b, c$ (say, $a$ and $b$) correspond to boundary components of
both $\Sigma_1'$ and $\Sigma_2'$.
Choose one point $x_a$ in $a$ and one point $x_b$ in $b$. For $i=1,2$, there
is an arc $\gamma_i$ connecting $x_a$ to $x_b$ in $\Sigma_i$, and disjoint
from the boundary of $\Sigma_i$ except at its ends. Then the
curve $d=\gamma_1\cup\gamma_2$ satisfies the lemma
(see Figure~\ref{fig:(0,0,0)}, left).
It may also happen that $\Sigma'$ has three connected components, in which
case we find a curve $d$ exactly in the same way, see
Figure~\ref{fig:(0,0,0)}, right.
\begin{figure}
\caption{Finding $d$ in the case $(0,0,0)$}
\label{fig:(0,0,0)}
\end{figure}
Next we deal with the case of intersections $(1, 0, 0)$. In this case,
$a$ and $b$ intersect transversely, once, and $c$ is disjoint from $a\cup b$.
By hypothesis, the curve $c$ is (globally) nonseparating.
Consider the union $a\cup b$. If $a$ or $b$ is two-sided, then
$a\cup b$ does not disconnect its regular neighborhoods.
This is seen by travelling
along a small band on one side of $a\cup b$,
(see Figure~\ref{fig:DeuxCourbes}, left).
In this case, $\Sigma'$ cannot have more connected components than
$\Sigma\smallsetminus c$, hence $\Sigma'$ is connected, and any
curve $d$ obtained by pushing $c$, as in the preceding case, satisfies
the lemma. If both $a$ and $b$ are one-sided, then $a\cup b$ is locally
disconnecting, so $\Sigma'$ may have up to two connected components.
In this case, a curve $d$ obtained by pushing
$a$ satisfies the lemma (See Figure~\ref{fig:DeuxCourbes}, right).
\begin{figure}
\caption{Finding $d$ in case $(1,0,0)$}
\label{fig:DeuxCourbes}
\end{figure}
Now assume we are in the case $(1, 1, 1)$ or $(1, 1, 0)$.
Since $\{a, b, c\}$ is not of type necklace, note that in any
case $b$ and $c$ do not meet outside $a$. We first treat the sub-case
when $a$ is two-sided. For this we consider any curve $d$ obtained by
pushing $a$ aside, and we claim that $d$ meets every connected component
of $\Sigma'$. Indeed, let $C$ be such a component.
Of course the closure of $C$ meets $a$ or $b$ or $c$.
Since both $b$ and $c$ meet $a$, it actually has to meet $a$, as we can see
by traveling along $b$ or $c$ in $C$. More precisely, by following $b$
or $c$ in both directions, we see that $C$ meets any neighborhood of $a$
from both sides. Thus it meets $d$.
It remains to treat the sub-case when $a$ is one-sided, first for the
$(1,1,1)$ case, and then for the $(1,1,0)$ case. In the $(1,1,1)$ case, the
curves $a$, $b$, $c$ play symmetric roles, and by the above argument it
just remains to consider the case when they are all one-sided.
Then the situation is depicted on Figure~\ref{fig:(1,1,1)Et(1,1,0)}, left:
$a \cup b \cup c$ disconnects its regular neighborhoods into three connected
components, and the figure shows a curve $d$, obtained by pushing $a$ aside,
which intersects all three components and such that
$\{a, b, c, d \}$ is a 4-clique.
In the remaining case the curves
$b$ and $c$ play symmetric roles,
and there are three different cases to
consider, regarding whether $b$ and $c$ are one or two-sided.
These three cases are pictured in Figure~\ref{fig:(1,1,1)Et(1,1,0)},
and in each case, we obtain $d$ by pushing $a$ aside.
\begin{figure}
\caption{Finding $d$ in cases $(1,1,1)$ and $(1,1,0)$}
\label{fig:(1,1,1)Et(1,1,0)}
\end{figure}
\end{proof}
We deduce the following.
\begin{lemma}\label{lem:CestLeBouquet}
Let $\{a,b,c\}$ be a $3$-clique of $\mathbb{T}ransFin(\Sigma)$, not of type
necklace. Let $(\alpha_1,\ldots,\alpha_n)$ be a finite family
of vertices of $\mathbb{T}ransFin(\Sigma)$, all distinct from $a$,
$b$ and~$c$. Then there exists a vertex $d$ of $\mathbb{T}ransFin(\Sigma)$, such that
\begin{itemize}
\item[$\bullet$] $\{a,b,c,d\}$ is a $4$-clique of $\mathbb{T}ransFin(\Sigma)$,
\item[$\bullet$] for all $j\in\{1,\ldots,n\}$, the intersection
$d\cap \alpha_j$ is infinite; in particular, $\{d, \alpha_{j}\}$ is not an
edge of $\mathbb{T}ransFin(\Sigma)$.
\end{itemize}
\end{lemma}
As a corollary, we get the direct implication in Proposition~\ref{prop:TypesPrecis}.
\begin{corollary}
If $\{a,b,c\}$ is a $3$-clique not of type necklace, then $N(a,b,c)$ does not
hold.
\end{corollary}
\begin{proof}[Proof of Lemma~\ref{lem:CestLeBouquet}]
The hypotheses that $\{a,b,c\}$ is not of type necklace, and
$\alpha_j\not\in\{a,b,c\}$, impose that for every $j$,
the curve $\alpha_j$ is not contained in the union $a\cup b\cup c$.
Hence, there exists a small subarc $\beta_j\subset\alpha_j$
lying in the complement of $a\cup b\cup c$, and we may
further suppose that these $n$ arcs are pairwise disjoint,
and choose a point $x_j$ in $\beta_j$ for each $j$.
Now, let $d_0$ be a vertex of $\mathbb{N}Cfin{\leqslant 1}(\Sigma)$ as from
Lemma~\ref{lem:drenccc}. Since $d_0$ meets every component of
$\Sigma'= \Sigma\smallsetminus(a\cup b\cup c)$,
we may perform a surgery on $d_0$, far from $a\cup b\cup c$,
to obtain a new curve $d_1$ such that $\{a,b,c,d_1\}$ is still a
$4$-clique, and $d_1$ still meets every component of
$\Sigma'$, and $d_1$ passes through $x_1$. We may iterate this
process, to get a curve $d_n$ which passes through $x_j$
for every $j$, and such that $\{a,b,c,d_n\}$ is a $4$-clique.
Finally, we may perform a last surgery on $d_n$, in the
neighborhood of all $x_j$, in order to obtain a curve $d$
such that for each $j$, $|\beta_j\cap d|$ is
infinite.
\end{proof}
It remains to prove the converse implication in
Proposition~\ref{prop:TypesPrecis}, which we restate as
Lemma~\ref{lem:CestLeCollier}.
\begin{lemma}\label{lem:CestLeCollier}
Let $\{a,b,c\}$ be a $3$-clique of $\mathbb{T}ransFin(\Sigma)$ of type
necklace. Then there exists a finite set $F$, of at most 8 vertices of
$\mathbb{T}ransFin(\Sigma)$ all distinct from $a$, $b$ and $c$, and
such that every $d$ such that $\{a,b,c,d\}$ is a $4$-clique of
$\mathbb{T}ransFin(\Sigma)$ is connected by an edge to some element
of~$F$.
\end{lemma}
In fact this set $F$ can be chosen explicitely, with cardinal at most~8,
as follows. If $\{a,b,c\}$ is a $3$-clique of type necklace, then there exists
a family of arcs $(x, X, y, Y, z, Z)$, all embedded in $\Sigma$, such that
$a=x\cup X$, $b=y\cup Y$, $c=z\cup Z$, and such that these
six arcs pairwise intersect at most at their ends. The union
$a\cup b\cup c$ may be viewed as a graph embedded in $\Sigma$,
and these six arcs are the edges of this embedded graph.
We let $F$ be the set of nonseparating curves, among the~8
curves $X\cup Y\cup Z$, $X\cup Y\cup z$,
$X\cup y\cup Z$ \textsl{etc.}, (there is one choice of
upper/lower case for each letter).
In the course of the proof of Lemma~\ref{lem:CestLeCollier}, we
will see that $F$ is nonempty, and satisfies the Lemma.
\begin{proof}[Proof of Lemma~\ref{lem:CestLeCollier}]
Let $\{a,b,c\}$ be a $3$-clique of type necklace.
Let $F$ be the set of nonseparating curves, as above, among all the~8
curves $x\cup y\cup z$, $X\cup y\cup z$, $X\cup Y\cup z$, etc.
Let $d$ be such that $\{a,b,c,d\}$ is a $4$-clique.
Up to permuting the curves $a$, $b$ and $c$, we may suppose that
$(|a\cap d|, |b\cap d|, |c\cap d|)$ equals
$(1,1,1)$, or $(1,1,0)$, or $(1,0,0)$ or $(0,0,0)$~; our proof proceeds
case by case.
The easiest case is $(1, 0, 0)$. In this case,
up to exchanging the arcs $X$ and $x$, we may suppose that
$d$ intersects $a$ at an interior point of $X$, and is disjoint from
all the other arcs. Consider $f=X\cup y\cup z$.
This curve intersects
$d$ at a unique point, transversely. It follows that $f$ is nonseparating.
Hence $f\in F$ and $f$ satisfies the conclusion of the lemma.
Now let us deal simultaneously with the cases $(1,1,1)$ and $(1,1,0)$.
Suppose first that the intersections of
$d$ with $a\cup b\cup c$ do not occur at the intersection points
$a\cap b$, $a\cap c$ or $b\cap c$. Up to exchanging $x$ with $X$,
$y$ with $Y$ and $z$ with $Z$, we may suppose that the intersections
occur in the interior of the arcs $X$, $Y$, and $Z$ in the case $(1,1,1)$,
and in the interior of the arcs $X$ and $Y$ in the case $(1,1,0)$.
Now the curve
$f=X\cup y\cup z$, for instance, satisfies the conclusion of the lemma.
Now suppose that $d$ contains one of the points $a\cap b$, $a\cap c$ or $b\cap c$.
In case $(1,1,1)$ we may suppose, up to permuting $a$, $b$ and $c$, that $d$
contains the point $a\cap b$, and in the case $(1,1,0)$, this is automatic,
as $d$ is disjoint from $c$.
Now in any case, $d$ cannot
contain $a\cap c$ nor $b\cap c$, because it intersects $a$ and $b$ only once.
Hence, up to exchanging $z$ with $Z$, we may suppose $d\cap z=\emptyset$.
In the neighborhood of the point $a\cap b$, up to homeomorphism, the configuration of
our curves is as depicted in
Figure~\ref{fig:dTransverse}, because all the intersections are supposed
to be transverse. Then, up to exchanging $X$ with $x$ or $Y$ with $y$,
we can suppose that the arc $X\cup Y$ has a transverse intersection with
$d$, and then the arc $f = X\cup Y \cup z$ satisfies the conclusion of the lemma.
\begin{figure}
\caption{A suitable choice of $X$ and $Y$}
\label{fig:dTransverse}
\end{figure}
We are left with the case $(0,0,0)$. In this case, any curve in $F$ will
satisfy the conclusion of the lemma, and hence all we have to do is to prove that $F$ is
nonempty. If $X\cup Y\cup Z$ and $X\cup Y\cup z$ were both separating,
then so would be $c=z\cup Z$, by Fact~\ref{fact:TroisArcsSep}.
Hence, among these two curves, at least one is nonseparating, and $F$ is
nonempty (in fact, it contains at least 4 elements).
\end{proof}
\subsection{One or two-sided curves, and extra bouquets}\label{ssec:xB}
In this last paragraph of this section, we will see how to recognize, from
the graph strucutre of $\mathbb{T}ransFin(\Sigma)$, some additional configurations.
We insist that the work in this paragraph is useful only in the case when
$\Sigma$ is non orientable; it is needed in order to make our proof of
Theorem~\ref{thm:AutC1} work in that case (see Remark~\ref{rmk:Klein} below).
We start with a simple characterization of one-sided and two-sided curves.
\begin{itemize}
\item[$\bullet$] $\mathrm{Two}(a)$ the property that for all $b$ such that $T(a,b)$
holds, there exists $c$ such that $T(b,c)$ and $D(a,c)$ both hold,
\item[$\bullet$] $\mathrm{One}(a)$ the negation of $\mathrm{Two}(a)$:
there exists $b$ such that $T(a,b)$ and such that there does not
exist $c$ satisfying $T(b,c)$ and $D(a,c)$.
\end{itemize}
\begin{observation}\label{obs:1-sided}
Let $a$ be a vertex of $\mathbb{T}ransFin(\Sigma)$. Then the curve $a$ is
one-sided, if and only if $\mathrm{One}(a)$ holds.
\end{observation}
\begin{proof}
If $a$ is two-sided, and $b$ satisfying $T(a,b)$, by pushing $a$
aside we find another curve $c$ as in the definition of $\mathrm{Two}(a)$.
This proves the revers implication.
If $a$ is one-sided, let $b$ be a curve obtained by pushing $a$ aside.
We have $T(a,b)$, and $a$ and $b$ bound a disk. Any curve $c$
disjoint from $a$, and with $T(b,c)$, has to enter this disk: but then,
it has to get out, which is impossible without touching $a$ and
without intersecting $b$ another time.
\end{proof}
Our next objective is to characterize when two one-sided curves $a$ and $b$
meet exactly once, non-transversely. We will do this in several steps.
\begin{lemma}\label{lem:acapbnonconnexe}
Let $a$, $b$ be one-sided simple curves of $\Sigma$. Suppose the intersection
$a\cap b$ is not connected.
Then there exists a vertex $c$ of $\mathbb{T}ransFin(\Sigma)$,
distinct from $a$ and $b$,
such that for every neighbor $d$ of both $a$ and $b$ in this graph,
and such that $D(a,d)$ or $D(b,d)$ (or both), the vertices $c$ and $d$ are
neighbors in this fine graph.
\end{lemma}
\begin{proof}
Let $x$ be a subarc of $b$, whose endpoints lie in $a$, and disjoint from $a$
otherwise. Since $a\cap b$ is disconnected, such an arc exists, and has two
distinct end points, $p$ and $q$. Let $x'$ and $x''$ be the two subarcs of $a$
whose ends are $p$ and $q$.
From Fact~\ref{fact:TroisArcsSep}, we know that $x\cup x'$ or $x\cup x''$
(or both) is a nonseparating curve; denote it by $c$. By construction,
we have $c\neq a$ and $c\neq b$.
Now let $d$ be a curve satisfying the hypothesis of the lemma. If $d$ is
disjoint from $a$ and $b$, then it is disjoint from $c$; otherwise $d$
intersects exactly one of $a$, $b$, far away from the other. So the intersection
between $d$ and $c$, if any, is still transverse, and $d$ is a neighbor of
$c$ in~$\mathbb{T}ransFin(\Sigma)$.
\end{proof}
This contrasts with the situation we want to characterize, as we see now.
\begin{lemma}\label{lem:I(a,b)-lemme1}
Let $a$ and $b$ be two one-sided curves, and suppose that $a\cap b$
consists in one, inessential intersection point. Then, for every
nonseparating curve $c\not\in\{a,b\}$, there exists $d$ such
that $T(a,d)$ and $D(b,d)$
hold but such that the intersection $c\cap d$ is infinite.
\end{lemma}
\begin{proof}
We first observe that $\Sigma'=\Sigma\smallsetminus(a\cup b)$ is connected.
This is seen by following the curves $a$ and $b$ in both directions: the
union $a\cup b$ does not disconnect its small neighborhoods.
Let $c$ be a curve as above. Then, we may consider a first curve $d_0$,
obtained by pushing $a$ aside, in such a way that $d_0$ is disjoint from
$b$ (this is possible since the intersection $a\cap b$ is inessential).
Since $c\not\in\{a,b\}$, the curve $c$ intersects $\Sigma'$.
Since $d_0$ meets every component of $\Sigma'$ (there is only one), we may
deform it into a curve $d$ which intersects $c$ infinitely many times,
exactly as in the proof of Lemma~\ref{lem:CestLeBouquet}.
\end{proof}
After these two lemmas, we have a simple sentence in terms of the graph
$\mathbb{T}ransFin(\Sigma)$, which holds when $a\cap b$ is a single inessential
intersection point, and which guarantees that $a\cap b$ is connected.
In order to upgrade this into a characterization of the first situation,
we need to be able to exclude as well the cases when $a\cap b$ is a
non degenerate arc. These cases fall into two subcases: the intersection
arc $a\cap b$ can be essential or inessential, exactly as an intersection
point. One way to formalize this, is to say that
the intersection $a\cap b$ is essential if $a$ cuts
a regular neighborhood of $a\cap b$ into two regions both containing a
subarc of $b$, and inessential otherwise.
\begin{lemma}\label{lem:ArcEssentiel}
Let $a$ and $b$ be one-sided curves.
Suppose that $a\cap b$ is a non degenerate arc, and suppose this intersection
is essential. Then there exist curves $\alpha,\alpha',\beta,\beta'$ obtained
by pushing $a$ aside, such that $B(a,\alpha,\beta)$, $B(b,\alpha,\beta)$,
$B(a,\alpha',\beta')$, $B(b,\alpha',\beta')$, and $N(a,\alpha,\beta')$.
\end{lemma}
\begin{proof}[Proof of Lemma~\ref{lem:ArcEssentiel}]
The curves $\alpha$, $\beta$, $\alpha'$ and $\beta'$ may be taken in a
neighborhood of $a\cup b$, as pictured in Figure~\ref{fig:PropI-1}.
\begin{figure}
\caption{The curves $\alpha,\beta,\alpha',\beta'$}
\label{fig:PropI-1}
\end{figure}
\end{proof}
Finally, we deal with inessential arcs.
\begin{lemma}\label{lem:IntNonEssentielle}
Let $a$ and $b$ be one-sided curves, such that $a\cap b$ is
an inessential arc or intersection point. Let $c$ be such that $T(a,c)$.
Then there exists $d$ such that $B(a,c,d)$ and $D(b,d)$, if and only
if the intersection point $a\cap c$ does not belong to~$b$.
\end{lemma}
\begin{proof}
If $a\cap c$ belongs to $b$, then obviously, any curve $d$ satisfying
$B(a,b,d)$ will meet $b$. Otherwise, we may push $a$ aside, in order to
obtain a curve $d$, as in Figure~\ref{fig:I-xB}, left.
\end{proof}
Putting all together, this yields the following characterization of inessential
intersection points between one-sided curves.
\begin{corollary}\label{cor:I(a,b)}
Let $a$ and $b$ be one-sided curves. Then, $a\cap b$ consists of one,
inessential intersection point, if and only if the following conditions are
satisfied:
\begin{enumerate}
\item $a$ and $b$ are not neighbors in $\mathbb{T}ransFin(\Sigma)$,
\item for every $c\not\in\{a,b\}$, there exists $d$ such that $d$ is a
neighbor of $a$ and $b$ in $\mathbb{T}ransFin(\Sigma)$, and $D(a,d)$ or $D(b,d)$
(or both), but $d$ is not a neighbor of $c$ in that graph,
\item there do not exist $\alpha,\beta,\alpha',\beta'$ such that
$B(a,\alpha,\beta)$, $B(b,\alpha,\beta)$, $B(a,\alpha',\beta')$,
$B(b,\alpha',\beta')$ and $N(a,\alpha,\beta')$ all hold,
\item there do not exist $c_1,c_2$ with $D(c_1,c_2)$ and with the following
property:
for $i=1,2$ we have: $T(a,c_i)$ and for all $d$, $B(a,c_i,d)$ and $D(b,d)$
do not both hold.
\end{enumerate}
\end{corollary}
This enumeration of conditions expressed only in terms of the graph structure
of $\mathbb{T}ransFin(\Sigma)$, with the addition of the conditions $\mathrm{One}(a)$ and
$\mathrm{One}(b)$, will be also denoted by $I(a,b)$, for {\em inessential
intersection} (of one-sided curves).
\begin{proof}
First, let us check that if $a$ and $b$ have one, inessential intersection
point then $I(a,b)$ holds. Condition~(1) holds by definition,
and~(2) follows from Lemma~\ref{lem:I(a,b)-lemme1}.
The negation of condition~(3) would imply that the cardinal of $a\cap b$ is at least~2.
Indeed, $B(a,\alpha,\beta)$ implies that $\alpha\cap\beta$ is a point lying
in $a$. Thus, the bouquet conditions imply that both $\alpha\cap\beta$ and
$\alpha'\cap\beta'$ lie in $a\cap b$. And the condition $N(a,\alpha,\beta')$
then implies that $a\cap\alpha$ and $a\cap\beta'$ are disjoint, hence
the two points $\alpha\cap\beta$ and $\alpha'\cap\beta'$ are distinct.
Finally, condition~(4) follows from Lemma~\ref{lem:IntNonEssentielle}.
Indeed, this lemma implies that the two curves $c_1$ and $c_2$ should
both contain a point of $a\cap b$, hence they cannot be disjoint.
Now, let $a$ and $b$ be any two nonseparating curves and suppose that $I(a,b)$.
By conditions~(1) and~(2), the intersection $a\cap b$ is non empty, and
connected. Along the lines of the proof of Lemma~\ref{lem:ArcEssentiel},
we can see that condition~(3) implies that $a\neq b$, so $a\cap b$
is an inessential intersection point, or an arc. Suppose for contradiction
that it is a nondegenerate arc. By Lemma~\ref{lem:ArcEssentiel} and
condition~(3), this intersection arc cannot be essential.
Now Figure~\ref{fig:I-xB}, right,
shows the desired contradiction with condition~(4).
\begin{figure}
\caption{Some configurations of curves for properties $I$ and $xB$}
\label{fig:I-xB}
\end{figure}
\end{proof}
Finally, we deal with extra bouquets.
We denote by $xB(a,b,c)$ the property that
$T(a,b)$, $T(a,c)$, $I(b,c)$ all hold and moreover:
for all $d$ such that $B(a,b,d)$ holds, $D(c,d)$ does not.
\begin{lemma}
Let $a$, $b$ and $c$ be such
that $T(a,b)$, $T(a,c)$ and $I(b,c)$, with $b$ and $c$ one-sided.
Then $xB(a,b,c)$ holds if and only if the intersection points
$a\cap b$, $a\cap c$ and $b\cap c$ coincide.
\end{lemma}
\begin{proof}
Of course if these points coincide, then property $xB(a,b,c)$
holds: every curve $d$ such that $B(a,b,d)$ holds, must contain
this point and hence cannot be disjoint from $c$.
Now suppose that these points do not coincide, hence, are three
pairwise distinct points. Then, we may push $b$ aside, in order
to find a curve $d$ which does not intersect $c$ any more, as
the intersection $b\cap c$ is not essential. This curve $d$,
obtained by pushing $b$, can be made to satisfy $T(b,d)$,
while crossing $b$ precisely at the point $a\cap b$, and
this intersection can be made transverse;
the illustration
of this situation is similar to Figure~\ref{fig:I-xB}, left,
and this time we leave it to the reader.
This yields a curve $d$
such that $B(a,b,d)$ holds and $d$ disjoint from $c$.
\end{proof}
\section{Proof of Theorem~\ref{thm:AutC1}}\label{sec:AutHomeo}
Here as above, $\Sigma$ is a connected surface admitting a nonseparating
closed curve.
\subsection{From bijections to homeomorphisms}
In order to prove Theorem~\ref{thm:AutC1}, it suffices to prove that every
automorphism of $\mathbb{T}ransFin(\Sigma)$ is supported by a bijection of the
surface, in virtue of the following observation.
\begin{proposition}\label{prop:BijHomeo}
Let $f\colon\Sigma\to\Sigma$ be a bijection. We suppose that for every
nonseparating simple closed curve $\alpha\subset\Sigma$, the sets $f(\alpha)$
and $f^{-1}(\alpha)$ are also nonseparating simple closed curves in $\Sigma$.
Then $f$ is a homeomorphism.
\end{proposition}
\begin{proof}
If our hypothesis was that $f$ and $f^{-1}$ are closed (\textsl{i.e.}, send
closed sets to closed sets), then $f$ would be a homeomorphism. So our
strategy is to use our hypothesis here in a similar fashion. We need only
prove that $f$ is continuous, the argument for $f^{-1}$ is symmetric.
Let $x\in\Sigma$ and suppose that $f$ is not continuous at $x$. Then there
exists a sequence $(x_n)_{n\geqslant 0}$ of distinct points converging
to $x$, and a neighborhood $V$ of $f(x)$, such that for all $n$, we have
$f(x_n)\not\in V$.
Notice that in the open unit disk of the plane, up to homeomorphism, there
is only one sequence of distinct points converging to the origin.
With this in head, we may construct an embedded arc, in $\Sigma$, with one
end at $x$, and which contains all the points $x_n$. Then we may construct a
nonseparating simple closed curve $\alpha$ containing this arc.
By hypothesis, $f(\alpha)$ is a nonseparating closed curve in $\Sigma$, which
contains $f(x)$. We may perform a surgery of $f(\alpha)$ inside $V$, to
obtain a nonseparating simple closed curve $\beta$, which coincides with
$f(\alpha)$ outside $V$ but which does not contain $f(x)$.
Now $f^{-1}(\beta)$ is, by hypothesis, a closed subset of $\Sigma$, which
contains all the points $x_n$ but not $x$. This is a contradiction.
\end{proof}
\subsection{The adjacency relation $\diamondvert$}\label{sec:Adjacency}
Let $E_T(\mathbb{T}ransFin(\Sigma))$ denote the set of edges $\{a,b\}$ of
$\mathbb{T}ransFin(\Sigma)$ satisfying $T(a,b)$. Then we have a map
\[ \mathrm{Point}\colon \ E_T(\mathbb{T}ransFin(\Sigma))\to\Sigma, \]
which to each edge $\{a,b\}$, associates the intersection point $a\cap b$.
The main part of proof of Theorem~\ref{thm:AutC1}
consists in showing that we can express the
equality
\[ \mathrm{Point}(a,b) = \mathrm{Point}(\alpha, \beta) \]
in terms of the graph.
For this we introduce the equivalence relation $\diamondvert$ on
$E_T(\mathbb{T}ransFin(\Sigma))$ as follows.
Let $\{a,b\}$ in $E_T(\mathbb{T}ransFin(\Sigma))$. If $\{a,b,c\}$
is a $3$-clique of $\mathbb{T}ransFin(\Sigma)$ of type bouquet we
set $\{a,b\}\diamondvert \{a,c\}$.
We also set
$\{a,b\}\diamondvert \{a,c\}$
if $a$, $b$ and $c$ satisfy the ``extra bouquet'' condition, denoted
above by $xB(a,b,c)$, see paragraph~\ref{ssec:xB} (this is void
when $\Sigma$ is orientable).
Then $\diamondvert$ is defined as the equivalence relation generated by these
relations.
When $\Sigma$ is orientable, the relation $\diamondvert$ corresponds to the
equivalence relation on triangles, generated by adjacency, in the subgraphs
of $\mathbb{T}ransFin(\Sigma)$ induced by curves passing through a common point.
This is what motivates our notation.
The relation $\{a,b\}\diamondvert \{a', b'\}$ obviously implies
$\mathrm{Point}(a,b) = \mathrm{Point}(a',b')$. We will see that the converse is not true,
and describe geometrically the equivalence classes in
Section~\ref{sec:LocalSubgraphs}, but for now we will only need the
following partial statement.
\begin{proposition}\label{prop:GermeAdjacent}
Let $a$, $b$, $a'$, $b'$ be such that $T(a,b)$ and $T(a',b')$.
Suppose that they have the same intersection point,
$x=\mathrm{Point}(a,b)=\mathrm{Point}(a',b')$, and suppose that the germs of $a$ and $a'$
coincide, \textsl{i.e.}, there exists a neighborhood $V$ of $x$ such
that $a\cap V=a'\cap V$.
Then $\{a,b\}\diamondvert\{a',b'\}$.
\end{proposition}
\begin{remark}\label{rmk:Klein}
If $\Sigma$ is a Klein bottle, there are no couples $\{a,b\}$ of two-sided
curves such that $T(a,b)$, and for every one-sided curve $b$, the curves
$c$ such that $T(b,c)$ holds fall into only two isotopy classes: that of $b$
and that of a two-sided curve, prescribed by $b$. It follows that, without
the extra bouquets in the definition of $\diamondvert$, there would
have been too many classes of $\diamondvert$, as such a class would remember
the isotopy class of a one-sided curve, and Proposition~\ref{prop:GermeAdjacent}
would not be true in this special case. These extra bouquets will be used
in the proof of Lemma~\ref{lem:SpheresConnexes} below.
\end{remark}
We postpone the proof of the proposition to the end of this section;
for now we will explain how it implies Theorem~\ref{thm:AutC1}.
\subsection{Proof of Theorem~\ref{thm:AutC1}}
If $a,b,c$ are vertices of $\mathbb{T}ransFin(\Sigma)$, we denote by $F(a,b,c)$ the
property that $T(a,b)$ holds, and there exists an edge $\{a',b'\}$ with
$\{a,b\}\diamondvert\{a',b'\}$ such that $\{a',b',c\}$ is a $3$-clique which is
not of type bouquet.
Note that this property $F(a,b,c)$ implies that $c$ does not contain the
point $\mathrm{Point}(a',b') = \mathrm{Point}(a,b)$. The next lemma asserts that $F(a,b,c)$
actually characterises this geometric property, and the letter $F$ stands for:
``$a\cap b$ is far from $c$''.
\begin{lemma}\label{lem:AppartenanceGraphe}
Let $a,b,c$ be vertices of $\mathbb{T}ransFin(\Sigma)$, and suppose $T(a,b)$ holds.
Then
\[ F(a, b, c) \Leftrightarrow \mathrm{Point}(a,b)\not\in c. \]
\end{lemma}
\begin{proof}
The direct implication follows directly from the
definitions; we have to prove
the converse implication. Suppose $\mathrm{Point}(a,b)\not\in c$.
Since $\Sigma$ is connected, there exists a regular neighborhood of $c$
containing the point $\mathrm{Point}(a,b)$. Depending on whether $c$ is one-sided or
two-sided, up to homeomorphism, this leads to only two distinct situations.
In Figure~\ref{fig:OnComplete2germes}, we represent in bold the germs of
the curves $a$ and $b$ near the point $a\cap b$, and show how to complete
these germs to new curves $a'$, $b'$ such that $\{a',b',c\}$ is a $3$-clique
not of type bouquet.
When $c$ is one-sided (see Figure~\ref{fig:OnComplete2germes}, left),
we may use two curves $a'$, $b'$ obtained by pushing $c$, while when $c$
is two-sided (see Figure~\ref{fig:OnComplete2germes}, right), we have to
use a curve $d$ which meets $c$ once transversely.
Now, by Proposition~\ref{prop:GermeAdjacent}, we have
$\{a,b\}\diamondvert\{a',b'\}$, and $\{a',b',c\}$ is a non-bouquet $3$-clique,
so we have $F(a,b,c)$ by definition.
\begin{figure}
\caption{Completing the germs of $a$ and $b$ to form a non-bouquet $3$-clique $(a',b',c)$}
\label{fig:OnComplete2germes}
\end{figure}
\end{proof}
\begin{corollary}\label{cor:AppartenanceGraphe}
Suppose $T(a,b)$ and $T(\alpha,\beta)$ hold.
Then $\mathrm{Point}(a,b)\neq\mathrm{Point}(\alpha,\beta)$ if and only if there
exists a nonseparating closed curve $c$ such that
$F(\alpha, \beta, c)$ holds but not $F(a,b,c)$.
\end{corollary}
The corollary is a direct consequence of Lemma~\ref{lem:AppartenanceGraphe}.
It follows that the equality $\mathrm{Point}(a,b)=\mathrm{Point}(\alpha,\beta)$ can be
expressed in terms of the graph structure of $\mathbb{T}ransFin(\Sigma)$, because,
as a consequence of Corollary~\ref{coro:TypesPrecis}, being a $3$-clique
not of type bouquet is also characterized in terms of this graph structure.
Now we can conclude the proof of Theorem~\ref{thm:AutC1}, provided
Proposition~\ref{prop:GermeAdjacent} holds.
\begin{proof}[Proof of Theorem~\ref{thm:AutC1}]
Let $\varphi$ be an automorphism of $\mathbb{T}ransFin(\Sigma)$.
Given a point $x$ in $\Sigma$, we choose two nonseparating simple closed
curves $a$, $b$ intersecting exactly once, transversely, at $x$,
and set $\varphi_\Sigma(x)=\mathrm{Point}(\varphi(a),\varphi(b))$. This
formula is valid because, by Proposition~\ref{prop:Types},
$\varphi(a)$ and $\varphi(b)$ are still nonseparating simple closed
curves intersecting exactly once. The point $\varphi_\Sigma(x)$
does not depend on the choice of $(a,b)$, because if $(\alpha,\beta)$
is another choice, the equalities $\mathrm{Point}(a,b)=\mathrm{Point}(\alpha,\beta)$
and $\mathrm{Point}(\varphi(a),\varphi(b))=\mathrm{Point}(\varphi(\alpha),\varphi(\beta))$
can be all expressed in terms of the graph structure of
$\mathbb{T}ransFin(\Sigma)$, thanks to Corollary~\ref{cor:AppartenanceGraphe}.
Thus, the map $\varphi_\Sigma$ is well-defined, and by following the
definitions we observe that the map
$(\varphi^{-1})_\Sigma$ is its inverse: hence $\varphi_\Sigma$
is a bijection of $\Sigma$. Finally, it follows from Lemma~\ref{lem:AppartenanceGraphe}
that for any nonseparating simple closed curve $\alpha$, the curve
$\varphi(\alpha)$ coincides with the set of points $\varphi_\Sigma(x)$
as $x$ describes $\alpha$. In other words, the automorphism $\varphi$
is realized by the bijection $\varphi_\Sigma$. Proposition~\ref{prop:BijHomeo}
concludes.
\end{proof}
\subsection{Connectedness of some arc graphs}
In order to finally prove
Proposition~\ref{prop:GermeAdjacent},
we will first need a couple of elementary results on
fine arc graphs.
\begin{lemma}\label{lem:ArcConnexe}
Let $S$ be a connected topological surface, with boundary, and let $x,y$ be
two distinct points of $\partial S$. Let $\mathcal{E}\Afin{}(S, x, y)$ be the graph whose
vertices are the simple arcs joining $x$ and $y$ and which meet $\partial S$ only at
their ends, with an edge between two such arcs if and only if they are
disjoint except at $x$ and $y$. Then the graph $\mathcal{E}\Afin{}(S, x, y)$ is connected.
\end{lemma}
Note that, when $x$ and $y$ are taken in the same connected component, we are
not requiring that the arcs be nonseparating; this is the reason why we use the
letter $\mathcal{E}$, for {\em extended}, in the same fashion as in~\cite{LMPVY}.
\begin{proof}
Let $a$, $b$ be two vertices of this graph. As a first case we suppose that
$a\cap b$ is made of a finite number of transverse intersection points:
we will prove by induction on the
cardinal of $a\cap b$ that, in this case, $a$ and $b$ are connected in
$\mathcal{E}\Afin{}(S,x,y)$. If $a\cap b$ is as small as possible, \textsl{i.e.}, is
equal to $\lbrace x,y\rbrace$, then $a$ and $b$ are neighbors in this graph.
Otherwise, if $a$ and $b$ intersect at other points than $x$ and $y$, we
may, in the spirit of~\cite{HPW},
pick a {\em unicorn path} $c$, made of one subarc of $a$ beginning at $x$,
and one subarc of $b$ ending at $y$. (For example, we may follow $a$
until it first meets $b$ after $x$, and then
continue along $b$).
Now we may push $c$ aside while fixing its ends,
at the appropriate side of $c$, to
obtain a new arc $c'$
with both $c'\cap a$ and $c'\cap b$ of cardinal strictly lower than that of
$a\cap b$.
This proves that arcs intersecting at finitely many points are
connected in $\mathcal{E}\Afin{}(S,x,y)$.
Now if the intersection $a\cap b$ is infinite, fix a differentiable structure
on the surface $S$. We may consider a smooth curve $a'$, neighbor of $a$, by
pushing $a$ aside while fixing its ends, and similarly, a smooth neighbor
$b'$ of $b$ similarly. Up to perturbing $b'$, we may suppose that $b'$ is
transverse to $a'$.
By the step above, $a'$ and $b'$ are
connected in $\mathcal{E}\Afin{}(S,x,y)$ and the Lemma is proved.
\end{proof}
We will also need a version for nonseparating arcs.
\begin{lemma}\label{lem:ArcConnexeBis}
Let $S$ be a connected surface containing a nonseparating curve. Suppose
$S$ has boundary, and let $x$ and $y$ be two distinct points in a same boundary
component of $S$. Let $\mathbb{N}Afin{}(S,x,y)$
be the set of nonseparating simple arcs
connecting $x$ to $y$, with an edge when they are disjoint away from $x$
and $y$. Then $\mathbb{N}Afin{}(S,x,y)$ is connected.
\end{lemma}
The points $x$ and $y$ add some technicality; let us first prove the following
simpler statement.
\begin{lemma}\label{lem:NonSepFinRelUnBordConnexe}
Let $S$ be a connected surface containing a nonseparating curve, and with
at least one boundary component, denoted $C$. Let $\mathbb{N}Afin{}(S,C)$ be the
graph whose vertices are the nonseparating arcs joining two distinct points
of $C$, and with an edge between two such vertices whenever they are disjoint.
Then this graph is connected.
\end{lemma}
This lemma is a variation on~\cite[Corollary~3.2]{LMPVY}; here we additionnaly
require that the arcs end at $C$.
In fact, in~\cite{LMPVY}, Corollary~3.3 is stated for surfaces with
$b>0$ boundary components, but proved only in the case $b=1$, which is the case needed in the proof of their main theorem.
Lemma~\ref{lem:NonSepFinRelUnBordConnexe} may be used to extend this corollary
to any $b>0$.
\begin{proof}
We begin with the observation that the graph $\mathbb{N}Afin{}(S,C)$ has no
isolated point. Indeed, if $\gamma$ is a vertex of $\mathbb{N}Afin{}(S,C)$,
by definition it is nonseparating. So we may consider a simple closed curve
$u$ with one, transverse intersection point with~$\gamma$. Obviously,
this curve $u$ is nonseparating; this follows from Fact~\ref{fact:NonSep},
applied to $u$, and a curve $v$ obtained by concatenation of $\gamma$
with some arc of $C$. Now we can perform a surgery on $u$,
and push its intersection point towards one end of $\gamma$ until we hit
$C$. This constructs an arc
$\alpha$,
which is now disjoint from $\gamma$, and which is also nonseparating.
Next, we claim that we can suppose, without loss of generality, that
the surface $S$ is compact. Indeed, if $\gamma_1,\gamma_2$ are vertices
of $\mathbb{N}Afin{}(S,C)$, and if $u$ is a nonseparating curve intersecting $\gamma_1$
as above, consider the set $K=C\cup\gamma_1\cup\gamma_2\cup u$. This set is
compact, hence there exists a compact topological subsurface $S'$ of $S$
containing~$K$.
This surface $S'$ contains nonseparating curves,
as it contains $u$ and $\gamma_1\cup C$, which may be used as above to
find two simple closed curves $u$, $v$ with one, essential intersection.
Now a path joining $\gamma_1$ to $\gamma_2$ in $S'$ is also a path joining
$\gamma_1$ to $\gamma_2$ in $S$.
So, until the end of the proof, $S$ is now supposed to be compact.
Next, observe that if two vertices $\gamma_1,\gamma_2$ of $\mathbb{N}Afin{}(S,C)$ are
isotopic (\textsl{i.e.}, there exists a continuous map $H\colon[0,1]^2\to S$
such that $\gamma_1(t)=H(0,t)$ and $\gamma_2(t)=H(1,t)$ for all $t$,
$H(s,0),H(s,1)\in C$ for all $s$ and the curve $H_s\colon t\to H(s,t)$ is
injective for all $s$),
then $\gamma_1$ and $\gamma_2$ are in the same component of $\mathbb{N}Cfin{}(S,C)$.
This argument is borrowed from~\cite{BHW}: for all $s$, the arc
$H_s$ has at least a neighbor $\alpha_s$ (by the first observation above),
and the set of $s'$ such that $H_{s'}$ is still a neighbor of $\alpha_s$ is
open in $[0,1]$. By compactness of $[0,1]$, there exist a finite number of
arcs $\alpha_1,\ldots,\alpha_n$, and a subdivision $0=t_0<t_1<\cdots<t_n=1$
such that $\alpha_j$ is disjoint from $H_t$ for all $t\in[t_{j-1},t_j]$ for
all $j$, and now $(\gamma_1,\alpha_1,\ldots,\alpha_n,\gamma_2)$ is a
path of $\mathbb{N}Cfin{}(S,C)$ joining $\gamma_1$ to $\gamma_2$.
As a result of this observation, we need only prove the connectedness of
the graph $\mathrm{NA}(S,C)$, whose vertices are the isotopy classes of
arcs between two distinct points of $C$, and with an edge between two
vertices whenever the corresponding classes admit disjoint representatives.
We proceed with the observation that the graph $\mathrm{A}(S,C)$, defined
exactly as $\mathrm{NA}(S,C)$ excpept we consider essential arcs, which may
be separating, is connected.
A simple way to do this is by using the idea of unicorn arcs exactly as
in the proof of the preceding lemma: if two arcs $a$ and $b$ are in
minimal position then their unicorn arcs are essential, and have fewer
intersections with both $a$ and $b$ than the number of points of $a\cap b$.
We will promote the connectedness of $\mathrm{A}(S,C)$ to that of
$\mathrm{NA}(S,C)$, by induction on the number of boundary components of $S$.
First, suppose that $S$ has only one boundary component,~$C$.
Let $\gamma_1,\gamma_2$ be two vertices of $\mathrm{NA}(S,C)$.
We may connect them by a path
$(\gamma_1,\alpha_1,\alpha_2,\ldots,\alpha_n,\gamma_2)$
in $\mathrm{A}(S,C)$, where each of the $\alpha_j$ may be separating;
consider such a path with minimal number of separating arcs.
For contradiction, and up to some relabeling, suppose $\alpha_1$ is
separating. Then it cuts $S$ in two components; denote by $S_1$ the
one containing $\gamma_1$
and $S_2$ the other. Then $\alpha_2$ is also contained in $S_1$,
otherwise we may delete $\alpha_1$ from our path. Since $S$ has
no boundary component other than $C$ and since the curve $\alpha_1$
is essential, the surface $S_2$ contains
a nonseparating arc, $\alpha_1'$. This arc
may be used instead of $\alpha_1$ in our initial path
from $\gamma_1$ to $\gamma_2$, contradicting the minimality of the
number of separating arcs. This proves that
$\mathrm{NA}(S,C)$ is connected if $S$ has no other boundary
component.
Now, we suppose, for inductive hypothesis, that
$\mathrm{NA}(S',C')$
is connected for every surface $S'$ with less boundary components
than $S$. Let $\gamma_1,\gamma_2$ be two vertices of $\mathrm{NA}(S,C)$.
As before, consider a path
$(\gamma_1,\alpha_1,\alpha_2,\ldots,\alpha_n,\gamma_2)$ in
$\mathrm{A}(S,C)$ between them, with minimal number of separating arcs.
For contradiction, and up to some relabeling, suppose $\alpha_1$ is
separating: it cuts $S$ into two subsurfaces, let $S_1$ be the one
containing $\gamma_1$, and, by hypothesis, must also contain $\alpha_2$,
and let $S_2$ be the other. If $S_2$ contains nonseparating arcs, we
conclude as before. If not, then $S_2$ contains some of the boundary
components of $S$, hence the surface with boundary $S'=S_1\cup\alpha_1$
has strictly less boundary components than $S$. One is $C'$, composed
by an arc of $C$ and the arc $\alpha_1$, and there may be others.
If $\alpha_2$ is nonseparating, then, by the induction hypothesis,
there is a path
$(\gamma_1,\beta_1,\ldots,\beta_k,\alpha_2)$ of $\mathrm{NA}(S',C')$
connected them. The arcs $\beta_1, \ldots, \beta_k$ may have end points
in $\alpha_1$, but we may perform a surgery in order to push all these
points to $C$, and obtain arcs $\beta_1',\ldots,\beta_k'$ which are
also vertices of $\mathrm{NA}(S,C)$, and we are done in this case.
Finally, if $\alpha_2$ is a separating arc (of $S$, or of $S_1$,
equivalently), then we may find an arc $\alpha_2'$ of $S_1$ which
is nonseparating and disjoint from $\alpha_2$. By following the
last case above, there exists a path
$(\gamma_1,\beta_1,\ldots,\beta_k,\alpha_2')$ in $\mathrm{NA}(S,C)$,
hence the path
$(\gamma_1,\beta_1,\ldots,\beta_k,\alpha_2',\alpha_2,\ldots,\alpha_n,\gamma_2)$
of $\mathrm{A}(S,C)$ has one less separating arc than the initial path.
This contradiction ends the proof.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lem:ArcConnexeBis}]
Let $\gamma_1,\gamma_2$ be two vertices of $\mathbb{N}Afin{}(S,x,y)$.
First, we may construct a neighbor $\gamma_2'$ in $\mathbb{N}Afin{}(S,x,y)$
of $\gamma_2$, which, in a neighborhood of $x$ (resp. $y$),
touches $\gamma_1$ only at $x$ (resp. $y$).
Indeed, there is a neighborhood $U_x$ of $x$ homeomorphic to the
closed half unit disk
\[ \{z, |z|\leqslant 1 \text{ and }\mathrm{Im}(z)\geqslant 0\}, \]
where the middle ray ($\mathrm{Re}(z)=0$) corresponds to the points
of $\gamma_1$. On either side of this ray, we may find an arc disjoint
from $\gamma_1$ and $\gamma_2$ except at $0$, arbitrarily close to
the boundary ($\mathrm{Im}(z)=0$), and joining $0$ to the unit circle,
and then this small arc may be continued to construct a curve $\gamma_2'$
which consists of pushing $\gamma_2$ aside.
So we may suppose that $\gamma_1$ and $\gamma_2$, close to $x$ and
$y$, intersect only at these points, and we may now find neighborhoods
$U_x$ and $U_y$ as above, such that their intersections with
$\gamma_1$ and $\gamma_2$ are along rays in this disk, in distinct
directions around $0$. Let $S'$ be the surface obtained by removing
the interiors of $U_x$ and $U_y$ from $S$. Then the path given by
applying Lemma~\ref{lem:NonSepFinRelUnBordConnexe} to $S$, yields
a path from $\gamma_1$ to $\gamma_2$ in $\mathbb{N}Afin{}(S,x,y)$, just by
adding some rays in $U_x$ and $U_y$ to the corresponding arcs.
\end{proof}
\subsection{Proof of Proposition~\ref{prop:GermeAdjacent}}
Let us go back to the proof of
Proposition~\ref{prop:GermeAdjacent}.
For the remaining of the section we fix a point $x\in\Sigma$.
Let $X$ denote the set of
nonseparating simple closed curves passing through~$x$.
\begin{lemma}\label{lem:SpheresConnexes}
Let $a,b,c\in X$.
Suppose that $T(a,b)$ and $T(a,c)$ hold.
Then $(a,b)\diamondvert(a,c)$.
\end{lemma}
\begin{proof}
Let $S$ be the surface obtained by cutting $\Sigma$ along $a$: it is
the surface with boundary obtained by gluing back two copies of the
curve $a$ to $\Sigma\smallsetminus a$. The point $x$ of $\Sigma$
yields two points, $p$ and $q$, of $\partial S$, and the curves
$b$ and $c$ define two arcs of $S$ joining $p$ and $q$.
By Lemma~\ref{lem:ArcConnexe}, there exists a finite sequence
$\gamma_0=b$, \ldots, $\gamma_n=c$, of arcs of $S$ joining $p$ and $q$,
with $\gamma_i$ and $\gamma_{i+1}$ disjoint except at $p$ and $q$.
For each $i$, the arc $\gamma_i$ defines a closed curve in $\Sigma$,
which has precisely one, transverse intersection with $a$; we will
still denote it by $\gamma_i$, abusively.
For every $i$, if $T(\gamma_i,\gamma_{i+1})$ holds, then we have
$(a,\gamma_i)\diamondvert(a,\gamma_{i+1})$, by definition.
If $T(\gamma_i,\gamma_{i+1})$ does not hold, then either $\gamma_i$
or $\gamma_{i+1}$ are both one-sided, or one of them is two-sided.
In the first case, the condition $xB(a,\gamma_i,\gamma_{i+1})$ holds,
by definition, and hence
$(a,\gamma_i)\diamondvert(a,\gamma_{i+1})$.
In the second, up to reversing the notation suppose $\gamma_i$ is
two-sided. Figure~\ref{fig:OnCompleteBouquets} shows how to insert
a curve $\delta$ such that
$B(a,\delta,\gamma_i)$ and $B(a,\delta,\gamma_{i+1})$ both hold,
and hence we still have $(a,\gamma_i)\diamondvert(a,\gamma_{i+1})$ in
this case.
\begin{figure}
\caption{Connecting the curves by common adjacency}
\label{fig:OnCompleteBouquets}
\end{figure}
By transitivity, we deduce that $(a,b)\diamondvert(a,c)$.
\end{proof}
The last ingredient for the proof of Proposition~\ref{prop:GermeAdjacent}
is the following observation.
\begin{observation}\label{obs:CCarc}
Let $a$, $a'$ be two nonseparating simple closed curves in $\Sigma$
such that $a\cap a'$ is an arc. Then, both sides of this arc lie
in the same connected component of $\Sigma\smallsetminus(a\cup a')$.
\end{observation}
\begin{proof}
\textsl{A priori}, the complement of $\Sigma\smallsetminus(a\cup a')$
may have up to four connected components, as suggested in
Figure~\ref{fig:CCarc}.
\begin{figure}
\caption{The arc $a\cap a'$ cannot disconnect}
\label{fig:CCarc}
\end{figure}
Suppose first that the intersection $a\cap a'$ is essential.
If $a$ (resp. $a'$) is one-sided, by following the curve $a$ (resp. $a'$)
we see that $A=B$. If both $a$ and $a'$ are two-sided, by following $a$ we
see that $A=D$ and $C=B$, while by following $a'$ we get $A=C$ and $B=D$,
so~$A=B$.
Now, suppose the intersection arc $a\cap a'$ is inessential.
By following $a$, we see that $C=D$, regardless of $a$ being one or
two-sided. Thus, if $A\neq B$, then one of $A$
or $B$, say $A$, is not connected from any of $B, C, D$. But this implies
that $a$ is separating, a contradiction.
\end{proof}
We are now in a position to prove~Proposition~\ref{prop:GermeAdjacent},
but instead we will
prove the following stronger statement, which will be more
convenient later in this article.
\begin{proposition}\label{prop:SemiGermeAdjacent}
Let $a$, $b$, $a'$, $b'$ be such that $T(a,b)$ and $T(a',b')$,
with intersection point
$x=\mathrm{Point}(a,b)=\mathrm{Point}(a',b')$, and suppose that $a$ and $a'$ locally ``half
coincide'' near $x$, \textsl{i.e.}, $a \cap a'$ contains a non degenerate arc
with endpoint $x$. Then $\{a,b\}\diamondvert\{a',b'\}$.
\end{proposition}
\begin{proof}[Proof of Proposition~\ref{prop:SemiGermeAdjacent}]
Suppose first that $a$ and $a'$ coincide along some arc with $x$ as
an end-point, and are
disjoint apart from this arc. By observation~\ref{obs:CCarc}, there exists
a curve $d$ passing through $x$ such that $T(a,d)$ and $T(a',d)$.
By Lemma~\ref{lem:SpheresConnexes}, this implies $(a,d)\diamondvert(a',d)$,
and by the same lemma we also have $(a,b)\diamondvert(a,d)$ and
$(a',b')\diamondvert(a',d)$. Hence $(a,b)\diamondvert(a',b')$.
Now we do not make the assumption any more that $a$ and $a'$ meet
only along an arc. Still, thanks to the hypothesis of the proposition,
we may choose a set $V$ homeomorphic to a closed disk, with $x$ on its
boundary, and such that $a\cap V=a'\cap V$ is an arc whose endpoints are $x$
and some other point $y$.
Lemma~\ref{lem:ArcConnexeBis}, applied to the surface
$\Sigma\smallsetminus \ring V$,
provides a
sequence $a_0=a$, \ldots, $a_n=a'$, of nonseparating curves such that
for all $i$, the curves $a_i$ and $a_{i+1}$ intersect only along
the arc $a\cap V$, hence we may conclude by applying iteratively
the reasoning above.
\end{proof}
\section{Local subgraphs}\label{sec:LocalSubgraphs}
In section~\ref{sec:Adjacency}, we considered
edges
$\{a,b\}, \{a',b'\}$ in the graph $\mathbb{T}ransFin(\Sigma)$
satisfying $|a\cap b|=|a'\cap b'|=1$, and $\mathrm{Point}(a,b) = \mathrm{Point}(a',b')$.
We defined and used the equivalence relation $\diamondvert$.
The aim of this section is to provide a geometric interpretation of the
equivalence classes. The results here are not used anywhere else in the paper.
In particular, this section is not used in the proof of our main results.
Nevertheless, we think it may help the reader to get a clear picture of
the situation.
\subsection{The graph of germs}
Let $x$ be a marked point in the surface $\Sigma$. In this section the we will study the local geometry of curves near $x$, so we may assume that $(\Sigma,x) = (\mathbb{R}^2, 0)$ whenever this is convenient. Given two simple arcs $a, a': [0,1] \to \Sigma$ with $a(0)=a'(0)=x$, we say that $a$ and $a'$ \emph{locally coincide} at $x$ if there exists a neighborhood $V$ of $x$ such that $a([0,1]) \cap V = a'([0,1]) \cap V$. This is an equivalence relation, whose equivalence classes are called \emph{germs of simple arcs at $x$}. The germ of $a$ is denoted $[a]_x$.
We say that $a$ and $a'$ \emph{locally intersect only at $x$} if there exists a neighborhood $V$ of $x$ such that
$a([0,1]) \cap a'([0,1]) \cap V = \{x\}$. This second relation obviously induces a relation on germs.
Let us consider the graph $\mathcal{A}(x)$ whose vertices are the germs of simple arcs at $x$, with an edge between the germs of $a$ and $a'$ whenever $a$ and $a'$ locally intersect only at $x$.
This graph is \emph{not} connected, in fact it has infinitely (uncountably) many connected components, as we will see below. We postpone the description of the connected components to explain the relation with the adjacency relation defined in section~\ref{sec:Adjacency}.
We say that two vertices $\alpha, \alpha'$ of the graph $\mathcal{A}(x)$ are \emph{comparable} if they belong to the same connected component of the graph.
\subsection{Germs and adjacency}
Given a point $x$ in $\Sigma$ and a simple closed curve $a$ in $\Sigma$ that contains $x$, we choose any one of the two germs of simple arc at $x$ included in $a$ and denote it by $\lfloor a \rfloor_{x}$. Which one of the two germs is chosen will not matter in what follows.
\begin{proposition}
\label{pro:AdjacencyGerms}
Let $a, b, a', b'$ be vertices in $\mathbb{T}ransFin(\Sigma)$ such that $T(a,b) $ and $T(a', b')$ hold, and assume
$\mathrm{Point}(a,b) = \mathrm{Point}(a',b')$. Then
$\{a,b\} \diamondvert \{a',b'\}$ if and only if the germs $\lfloor a\rfloor_x$ and $\lfloor a'\rfloor_x$ are comparable.
\end{proposition}
Proposition~\ref{pro:AdjacencyGerms} will be proved in section~\ref{sec:ProofAdjacencyGerms} below. The aim of the next two sections is to provide a simple characterization of distance, and connected components, in the graph of germs; see Proposition~\ref{Prop:DistanceAndWidth} below.
\subsection{Distance in local subgraphs}
In this section, we give a geometrical interpretation of the distance in three different graphs, which are very much like the graph of germs $\mathcal{A}(x)$.
Let $\Sigma$ be one of the following surfaces: (1) the compact annulus $\mathbb{S}^1 \times [0,1]$, (2) the open annulus $\mathbb{S}^1 \times \mathbb{R}$, or (3) the 2-torus $\mathbb{T}^2 =\mathbb{S}^1 \times \mathbb{S}^1$.
We consider non-oriented simple arcs in $\Sigma$, more precisely simple curves connecting both sides of the annulus in case (1), properly embedded images of the real line connecting both ends of the open annulus in case (2), or simple closed curves in a fixed homotopy class, say homotopic to $\{0\}\times \mathbb{S}^1$ in case (3).
Let $\mathcal{A}$ denote the graph whose vertices are one of the three above family of curves, with an edge between two curves whenever they are disjoint.
In order to express geometrically the distance in $\mathcal{A}$, let us consider the cyclic cover $p : \widetilde \Sigma \to \Sigma$, respectively in case (1), (2), (3)
$$
p:\mathbb{R} \times [0,1] \to \mathbb{R} /\mathbb{Z} \times [0,1],
\ \ \ \
p:\mathbb{R} \times \mathbb{R} \to \mathbb{R} /\mathbb{Z} \times \mathbb{R},
\ \ \ \
p:\mathbb{R} \times \mathbb{S}^1 \to \mathbb{R} /\mathbb{Z} \times \mathbb{S}^1
$$
given by the formula $p(x,y) = (x \text{ mod } 1, y)$.
Let $T$ be the deck transformation $(x, y) \to (x+1, y)$.
Now consider two curves $a,b$ which are vertices of the graph $\mathcal{A}$.
Let $\widetilde a, \widetilde b$ be respective lifts of $a,b$ under the covering map $p$.
Note that the set
$$
\{k \in \mathbb{Z}, \ \ T^k(\widetilde{a}) \cap \widetilde{b} \neq \emptyset \}
$$
is an interval of $\mathbb{Z}$, which is finite in the compact cases (1) and (3) but may be infinite in the open annulus case (2).
We define the \emph{relative width} $\mathrm{Width}(a,b)$ as the cardinal of this set. This is an element of $\{0, 1, \dots, +\infty \}$.
The reader may check easily that $\mathrm{Width}(a,b) = \mathrm{Width}(b,a)$.
\begin{proposition}
For every vertices $a \neq b$ of the graph $\mathcal{A}$, the distance in the graph
is given by
$$
d(a,b) = \mathrm{Width}(a,b) + 1.
$$
In cases (1) and (3), the graph $\mathcal{A}$ is connected.
In case (2), $a$ and $b$ are in the same connected component of $\mathcal{A}$ if
and only if $\mathrm{Width}(a,b) < +\infty$.
\end{proposition}
\begin{proof}
Let $a,b$ be as in the statement, and denote $w = \mathrm{Width}(a, b)$. We first assume that $w < +\infty$.
By Schoenflies' theorem (in case (2), applied in the two-point compactification of the annulus, which is a sphere),
we may assume that $a$ is a vertical curve whenever this makes our life easier.
We first note that if $w=0$ then $a$ and $b$ admit lifts that are disjoint from every $T$-translate of each other, which shows that $a$ and $b$ are disjoint, and thus $d(a,b)=1$.
Let us now assume $w>0$, and prove the two following key properties.
\begin{enumerate}
\item[(i)] For every vertex $a'$ of $\mathcal{A}$ such that $d(a,a')=1$,
$$\mathrm{Width}(a',b) \geq \omega-1.$$
\item[(ii)] There exists a vertex $a'$ of $\mathcal{A}$ such that $d(a,a')=1$ and
$$\mathrm{Width}(a',b) \leq w-1.$$
\end{enumerate}
To prove the first property, consider $a'$ such that $d(a,a')=1$.
By definition of the width $w$, we may find lifts $\widetilde{a}, \widetilde{b}$ of $a,b$ such that $\widetilde b$ is disjoint from $\widetilde{a}, T^{w+1}\widetilde{a}$ but meets
$T(\widetilde{a}), \dots, T^w(\widetilde{a})$. Since $a$ and $a'$ are disjoint, there is a lift $\widetilde{a'}$ of $a'$ which is between $\widetilde{a}$ and $T(\widetilde{a})$. Then the curves
$$
T(\widetilde{a'}), \dots, T^{w-1}(\widetilde{a'})
$$
are between the two curves $T(\widetilde{a})$ and $T^w(\widetilde{a})$, and those two curves are not in the same connected component of $\widetilde{\Sigma} \setminus T^i(\widetilde{a'})$, for $i=1, \dots, w-1$. Since the curve $\widetilde b$ is connected and meets the two curves
$T(\widetilde{a})$ and $T^w(\widetilde{a})$, it must meet all the $T^i(\widetilde{a'})$.
This proves that $\mathrm{Width}(a', b) \geq w-1$.
Let us prove the second property. We consider $\widetilde{a}, \widetilde{b}$ as above. Let $S$ denote the compact strip or annulus bounded by $\widetilde a \cup T(\widetilde{a})$.
Remember that $\widetilde{b}$ meets $T(\widetilde{a})$ but not $\widetilde{a}$. Thus $\widetilde{b} \cap S$ is included in a (maybe infinite) family of \emph{bigons}, \textsl{i.e.}, topological disks bounded by a simple closed curve made of a segment of the curve $T(\widetilde{a})$ and a segment of the curve $\widetilde{b}$. Let $S^+$ denote the union of these bigons.
Symmetrically, the curve $\widetilde{b'} := T^{-w}(\widetilde{b})$ meets $\widetilde{a}$ but not $T(\widetilde a)$. Thus $T^{-w}(\widetilde{b}) \cap S$ is included in a union $S^-$ of bigons formed by the curves $\widetilde{a}$ and $\widetilde{b'}$.
A key point is that the sets $S^-$ and $S^+$ are disjoint, because the curves $\widetilde b$ and $\widetilde b'$ are disjoint, since $b$ is simple. Thus we may construct a homeomorphism $H$ supported in $S$ such that $H(S^-)$ is included in an arbitrarily small neighborhood of $\widetilde a$, and $H(S^+)$ is included in an arbitrarily small neighborhood of $T(\widetilde{a})$. In particular, we may find a curve $\widetilde{a'}$, which is a lift of some element $a'$ of $\mathcal{A}$, included in the interior of $S$ and disjoint from both $S^-$ and $S^+$ (to be more explicit, take $\widetilde a' = H^{-1}(\{1/2\} \times [0,1])$ in the annulus case, in coordinates for which $a$ is the vertical curve $\{0\} \times [0,1]$). Note that $\widetilde{a'}$ is disjoint from $\widetilde{b}$ and $T^{-w}(\widetilde{b})$, and separate both curves, \textsl{i.e.}, the first one is on the right-hand side of $\widetilde{a'}$, and the second one is on the left-hand side. Thus the set
$$
\{k \in \mathbb{Z}, \ \ T^k(\widetilde{b}) \cap \widetilde{a'} \neq \emptyset \}
$$
has cardinality at most $w-1$. Which proves that $\mathrm{Width}(a', b) \leq w-1$, as wanted.
Using (i) and (ii), an induction on $n$ shows that $d(a,b)=n$ if and only if
$\mathrm{Width}(a,b)+1=n$, which completes the proof in the case when $\mathrm{Width}(a,b)$ is finite. When $\mathrm{Width}(a,b) = +\infty$,
an argument analogous to property (i) above shows that $\mathrm{Width}(a',b)=+\infty$ for every $a'$ such that $d(a,a')=1$. This shows that $a$ and $b$ are not in the same connected component of the graph. This completes the proof of the proposition.
\end{proof}
\subsection{Distance in the graph of germs}
Let us go back to the graph of germs $\mathcal{A}(x)$. Assume $(\Sigma,x) = (\mathbb{R}^2, 0)$. Given two vertices $a,b$ of $\mathcal{A}(x)$, we define their \emph{local relative width} $\mathrm{Width}(a,b)$ as follows. The plane minus the origin is identified with the open annulus $\mathbb{S}^1 \times \mathbb{R}$, and we consider the graph $\mathcal{A}$ from the previous section in the open annulus case. Then $\mathrm{Width}(a,b)$ is defined as the infimum of the quantity $\mathrm{Width}(A,B)$, where $A$ and $B$ are vertices of $\mathcal{A}$ whose germs respectively equal $a$ and $b$. Here is a more practical definition, which is easily seen to be equivalent. Consider the universal cover $p: \widetilde \Sigma \to \Sigma$ as above. Abuse the definition by still denoting $a,b:[0,1] \to \Sigma$ two curves with $a(0)=b(0)=0$ whose germs respectively equal $a,b$. Let $\widetilde a, \widetilde b$ denote lifts of (the restrictions to $(0,1]$ of) $a,b$ in $\widetilde \Sigma$.
Then the number $\mathrm{Width}(a,b)=w$ is characterized by the two following properties:
\begin{itemize}
\item[(i)] for every $t_0 \in (0,1]$, the restriction of $\widetilde a$ to $(0, t_0]$ meets at least $w$ integer translates of $\widetilde b$;
\item[(ii)] there exists $t_0 \in (0,1]$ such that the restriction of $\widetilde a$ to $(0, t_0]$ meets exactly $w$ integer translates of $\widetilde b$;
\end{itemize}
Analogously to the previous section, the distance in the graph of germs is characterized by the local relative width.
\begin{proposition}\label{Prop:DistanceAndWidth}
Let $a \neq b$ be two vertices of the graph $\mathcal{A}(x)$. Then $a$ and $b$ are in the same connected component of $\mathcal{A}(x)$ if and only if $\mathrm{Width}(a,b) < +\infty$.
In this case, the distance in the graph is given by
$$
d(a,b) = \mathrm{Width}(a,b) + 1.
$$
\end{proposition}
The proof is very similar to the proof in the previous section. Details are left to the reader.
\subsection{Proof of Proposition~\ref{pro:AdjacencyGerms}}
\label{sec:ProofAdjacencyGerms}
Let $a, b, a', b'$ be vertices of $\mathbb{T}ransFin(\Sigma)$ such that $T(a,b)$ and $T(a', b')$ hold, and assume
$\mathrm{Point}(a,b) = \mathrm{Point}(a',b')$. Denote $x$ the common intersection point.
If $c$ is another vertex such that $\{a, b, c\}$ is a 3-clique of type bouquet or extra bouquet, then the germs $\lfloor a\rfloor_x$ and $\lfloor c\rfloor_x$ are disjoint, thus obviously comparable. This entails the direct implication in Proposition~\ref{pro:AdjacencyGerms}.
Let us prove the converse implication. We assume that the germs $\lfloor a\rfloor_x$ and $\lfloor a'\rfloor_x$ are comparable. In other words, there exists arcs $\alpha_0, \dots, \alpha_n$ with $\alpha_i(0)=x$ and whose sequence of corresponding germs is a path from
$\lfloor a\rfloor_x$ to $\lfloor a'\rfloor_x$ in the graph of germs.
Note that each germ $\alpha_i$ may be extended to a non separating curve $a_i$, and we can find another non separating curve $b_i$ such that $T(a_i, b_i)$ holds. Thus the end of the proof is a direct consequence of the following lemma.
\begin{lemma}
Let $a, b, a', b'$ be vertices in $\mathbb{T}ransFin(\Sigma)$ such that $T(a,b)$ and $T(a', b')$ hold. Assume that for some choices $\lfloor a\rfloor_x, \lfloor a'\rfloor_x$ of arcs at $x$ included respectively in $a$ and $a'$, the germs $\lfloor a\rfloor_x, \lfloor a'\rfloor_x$ intersect only at $x$. Then $\{a, b\} \diamondvert \{a', b'\}$.
\end{lemma}
\begin{proof}[Proof of the lemma]
Let $c$ be an arc that contains $x$ in its interior and locally coincides with
$\lfloor a\rfloor_x \cup \lfloor a'\rfloor_x$. Extend $c$ into a non separating closed curve, still denoted $c$, and consider any other non separating curve $d$ such that $T(c,d)$ holds. Since $c$ locally ``half coincides'' near $x$ with both $a$ and $a'$, we may apply Proposition~\ref{prop:SemiGermeAdjacent} twice, and get that
$\{a,b\} \diamondvert \{c,d\} \diamondvert \{a', b'\}$.
\end{proof}
\subsection{Curves and diffeomorphisms}\label{ssec:Tourbillons}
In this short subsection we explain how one can use the fine curve graph to detect fundamental non differentiability.
Let $\Phi$ be an automorphism of $\mathbb{T}ransFin(\Sigma)$.
We introduce the following property $D(\Phi)$:
\emph{For every vertices $a$, $b$ of $\mathbb{T}ransFin(\Sigma)$ such that $T(a,b)$ holds, if $\mathrm{Point}(\Phi(a),\Phi(b)) = \mathrm{Point}(a,b)$ then there exists $a', b'$ such that $T(a', b')$ holds, $\mathrm{Point}(a',b') = \mathrm{Point}(a,b)$ and $\{\Phi(a'), \Phi(b') \} \diamondvert \{a', b'\}$.}
Note that this property is clearly invariant under conjugacy in the group of automorphisms. Let $h$ be a homemorphism of $\Sigma$, and denote $\Phi = \Phi_h$ the action of $h$ on the graph $\mathbb{T}ransFin(\Sigma)$.
\begin{observation}
If $h$ is differentiable everywhere, then property $D(\Phi_h)$ holds.
\end{observation}
Indeed, hypothesis $\mathrm{Point}(\Phi(a),\Phi(b)) = \mathrm{Point}(a,b)$ is equivalent to the fact that the point $x = \mathrm{Point}(a,b)$ is a fixed point of $h$. Since $h$ is differentiable at $x$, it is easy to check that every germ of smooth arc at $x$ is comparable to its image. Take any two smooth curves $a', b'$ such that $T(a', b')$ and $\mathrm{Point} (a', b') = \mathrm{Point}(a,b)$, then Proposition~\ref{pro:AdjacencyGerms} tells us that $\{\Phi(a'), \Phi(b') \} \diamondvert \{a', b'\}$.
Now consider a particular homeomorphism $h$ of $\Sigma$ and assume that $h$ admits a fixed point where, for some local polar coordinates, $h$ writes
$$
(r, \theta) \mapsto (r, \theta + \frac{1}{r}).
$$
\begin{observation}
Property $D(\Phi_h)$ does not hold.
\end{observation}
An easy proof of this is obtained by considering the
\emph{local rotation interval} of $h$ at $x$, as defined in~\cite{FredLocRot},
section 2.3.
Indeed, the local rotation interval of $h$ at $x$ equals $\{+\infty\}$,
which accounts for the fact that orbits turn faster and faster around $x$, in
the positive direction, as we get nearer and nearer to $x$ (the quickest way to
check this is to show that the \emph{local rotation set} of $h$ at $x$
is $\{+\infty\}$, and then to apply Théorème 3.9 of~\cite{FredLocRot} that
relates the local rotation set and the local rotation interval). We argue by
contradiction to show that property $D(\Phi_h)$ does not hold.
Assuming property $D(\Phi_h)$ holds, consider curves $a,b$ such that $T(a,b)$
holds and $\mathrm{Point}(a, b) = x$. Let $a', b'$ be given by property $D(\Phi_h)$,
such that $\{\Phi(a'), \Phi(b') \} \diamondvert \{a', b'\}$.
The reverse direction of Proposition~\ref{pro:AdjacencyGerms} tells us that the
germs of $h(a')$ and $a'$ are comparable at $x$. This entails easily, from the
definition, that the local rotation interval of $h$ at $s$ is a bounded
interval, a contradiction.
\section{Fine graph of smooth curves}\label{sec:AutCFinLisse}
In this section we address the case of smooth curves, and
prove Theorem~\ref{thm:AutCFinLisse} and Proposition~\ref{prop:PasDiff}.
In all the section, $\Sigma$ will be a connected, nonspherical
surface without boundary, endowed with a smooth structure.
In Section~\ref{sec:configSmoothCurves} we will restrict to the orientable case.
\subsection{From bijections to higher regularity}
One step in the proof of Theorem~\ref{thm:AutC1} was
Proposition~\ref{prop:BijHomeo}, in which we proved
that if an automorphism of $\mathbb{T}ransFin(\Sigma)$ is
supported by a bijection of $\Sigma$, then that
bijection is a homeomorphism of $\Sigma$.
We may ask the same question about automorphisms of
$\mathbb{T}ransFinLisse(\Sigma)$, and this paragraph is devoted
to the proof of the following two statements.
We denote by $\mathrm{Hom}eo_{\infty \pitchfork}(\Sigma)$ the group of
bijections of $\Sigma$ which preserve the family of
smooth, nonseparating closed curves, and preserve transversality
between such curves.
The first statment below justifies this notation. Here, for simplicity we
restrict to the case of orientable surfaces.
\begin{proposition}\label{prop:BijLisseHomeo}
Let $\Sigma$ be a connected, non spherical orientable surface.
The group $\mathrm{Hom}eo_{\infty \pitchfork}(\Sigma)$ is contained in
$\mathrm{Hom}eo(\Sigma)$.
\end{proposition}
\begin{proof}[Proof of Proposition~\ref{prop:BijLisseHomeo}]
Let $h \in \mathrm{Hom}eo_{\infty \pitchfork}(\Sigma)$. We will prove that the image under $h$
of any open set is an open set. This is the continuity of $h^{-1}$, and by
applying the argument to $h$ we also get the continuity of $h$.
To do this, we only need to consider the images of a family of sets that
generates the topology. Given three non separating curves $a,b,c$, we denote
$V(a; b, c)$ the union of all the non separating curves $d$ that meet $a$
and are disjoint from $b$ and $c$.
\begin{observation}
The set $V(a ; b,c)$ is the union of some of the connected components of the
complement of $b \cup c$ that meet $a$. In particular, it is an open set.
\end{observation}
Indeed, let $x$ be a point of $V(a; b,c)$. By definition there is a non
separating curve $d$ passing through $x$ and meeting $a$ but not $b$ nor $c$.
Consider another point $y$ that belongs to the connected component $V_x$ of
the complement of $b \cup c$ that contains $x$. By modifying $d$ using an arc
connecting $x$ to $y$ in $V_c$, we find another curve $d'$, isotopic to $d$,
still meeting $a$ but not $b$ nor $c$, and passing through $y$. This proves
that $V(a ; b,c)$ contains $V_x$, and the observation follows.
Now let $a$ be a nonseparating curve.
Let $a^+, a^-$ be obtained by pushing $a$ to both sides. Then $V(a; a^+, a^-)$
is a neighborhood of $a$, and by making $a^+$ and $a^-$ vary we get a basis
of neighborhoods ${\mathcal B}(a)$ of the curve $a$.
The union of all these families ${\mathcal B}(a)$ clearly
generates the topology of~$\Sigma$.
Thus it suffices to check that the image under $h$ of each set $V(a; b, c)$
is an open set. But since $h$ is a bijection, we have
\[ h(V(a; b, c)) = V( h(a); h(b), h(c)). \]
By hypothesis $h(a), h(b), h(c)$ are nonseparating closed curves, and by the
observation this set is open.
\end{proof}
We now prove Proposition~\ref{prop:PasDiff} stated in the introduction,
namely the existence of elements of $\mathrm{Hom}eo_{\infty \pitchfork}(\Sigma)$ that are not smooth.
\begin{proof}[Proof of Proposition~\ref{prop:PasDiff}]
We will construct a homeomorphism
$F\colon\mathbb{R}^2\to\mathbb{R}^2$, which is not differentiable at the origin,
but such that both $F$ and $F^{-1}$ send smooth curves to
smooth curves. The construction can easily be modified to make $F$ compactly
supported, and then be transported on our surface~$\Sigma$. It will be
clear from the constuction that this map preserves transversality.
Let $h\colon\mathbb{R}\to\mathbb{R}$ be a smooth diffeomorphism supported in the
segment $[1/2, 2]$. That is to say, $h(x)=x$ for all $x$ outside
$[1/2,2]$; we suppose however that $h(1)\neq 1$.
We consider the map $F$ defined by $F(x,y)=(x, xh(y/x))$ if
$x\neq 0$, and $F(x,y)=(x,y)$ otherwise. We claim that this map
has the desired property.
This map, as well as its inverse, is obviouly smooth in
restriction to $\mathbb{R}^2\smallsetminus\{(0,0)\}$. Direct computation
shows that $F$ has directional derivatives in all directions
around the origin,
but the ``differential'' fails to be linear:
both partial derivatives are
those of the identity, while the
directional derivative in the direction $(1,1)$ is not.
So $F$ is not differentiable at the origin.
Now, let $\gamma\colon\mathbb{R}\to\mathbb{R}^2$ be a smooth, proper embedding.
If $(0,0)$ is not in the image of $\gamma$, then of course,
$F\circ\gamma$ is still smooth. So suppose, say, that
$\gamma(0)=(0,0)$.
If $\gamma'(0)$ lies outside the two (opposite) sectors of
vectors of slopes between $1/2$ and $2$, then $F\circ\gamma$
and $\gamma$ have the same germ at $0$.
Otherwise,
and up to reparameterization,
we can write, near $0$,
$\gamma(t) = (t, \alpha(t))$ where $\alpha$ is a smooth map
(satisfying $\alpha(0)=0)$),
from a neighborhood of $0$, to~$\mathbb{R}$. This yields the formula
\[ F\circ\gamma(t) = \left(t, th \left(\frac{\alpha(t)}{t}\right)\right). \]
Now, the smoothness of $F\circ\gamma$
follows from the following elementary observation.
\noindent
{\bf Claim.}
{\em Let $\alpha\colon\mathbb{R}\to\mathbb{R}$ be a smooth map
satisfying $\alpha(0)=0$. Then the map $t\mapsto \frac{\alpha(t)}{t}$
when $t\neq 0$, and $\alpha'(0)$ when $t=0$, is smooth.}
Indeed, by the fundamental theorem of Calculus, for all $t\in\mathbb{R}^\ast$ we have
\[ \frac{\alpha(t)}{t} = \int_0^1 \alpha'(ts)ds, \]
and this integral with parameter can be differentiated
indefinitely.\footnote{We borrow this elegant argument from~\cite{stack}.}
\end{proof}
\subsection{A weak convergence for sequences of curves}
In order to prove~Theorem~\ref{thm:AutCFinLisse}, we now explain how to recognize configurations of smooth curves.
Given two vertices $a$ and $b$ of the graph $\mathbb{T}ransFinLisse(\Sigma)$,
we will denote by $a-b$ the property that they are neighbors in the graph.
If $(f_n)_{n\in\mathbb{N}}$ is a sequence of vertices, we denote by
$(f_n)_{n\in\mathbb{N}}-a$ the property that for all $n$ large enough, $f_n-a$.
The first property of sequences we may recover from the graph is the
distinction of what curves go to infinity.
\begin{lemma}\label{lem:fnSenVa}
Let $(f_n)_{n\in\mathbb{N}}$ be a sequence of vertices of $\mathbb{T}ransFinLisse(\Sigma)$.
The following are equivalent.
\begin{itemize}
\item[-] for all $d$, we have $(f_n)_{n\in\mathbb{N}}-d$;
\item[-] for every compact subset $K$ of $\Sigma$, for every $n$ large
enough, $K\cap f_n=\emptyset$.
\end{itemize}
\end{lemma}
\begin{proof}
The second statement obviously implies the first, as we may just
take $K=d$. Let us prove the converse implication by contraposition.
Suppose $K$ intersects infinitely many $f_n$. Since $K$ is compact,
there is a point $x\in K$, such that every neighborhood of $x$
intersects infinitely many $f_n$. We consider two open sets
$B_1$ and $B_2$ with $x\in B_1$ and $\overline{B_1}\subset B_2$,
and three bottle-shaped arcs, as in Figure~\ref{fig:3Bouteilles}.
These arcs may be continued to form three nonseparating closed curves,
$d_1$, $d_2$ and $d_3$. Now, let $n$ be such that $f_n$ enters $B_1$.
If $f_n$ does not enter nor leave $B_2$ through the neck of the bottle
corresponding to $d_1$, then we cannot have $f_n-d_1$, since $f_n$ and
$d_1$ have to intersect at least twice. Hence, $f_n$ passes
through the neck of $d_1$, and in order to impose that $f_n-d_2$, another
arc of $f_n$ has to get out of the bottle corresponding of $d_2$
through its neck. But then $f_n$ has to meet $d_3$ twice, and we cannot have
$f_n-d_3$. In other words, for
all $n$ such that $f_n$ enters $B_1$, we can't have $f_n-d_1$ and $f_n-d_2$
and $f_n-d_3$, hence the first statement is not true, and our
implication is proved.
\begin{figure}
\caption{Three bottles}
\label{fig:3Bouteilles}
\end{figure}
\end{proof}
Thus, we will say here that a sequence $(f_n)_{n\in\mathbb{N}}$ is {\em relevant}
if it has no subsequence $(f_{\varphi(n)})_{n\in\mathbb{N}}$ such that for
all $d$, $(f_{\varphi(n)})_{n\in\mathbb{N}}-d$.
We now explore, for such sequences, the following notion of convergence.
We say that a relevant sequence $(f_n)$ {\em converges in a weak sense}
to a curve $a$ if for every $d$ such that $a-d$, we have $(f_n)-d$.
We denote this property by~$W((f_n),a)$.
\begin{lemma}\label{lem:ConvFaible}
Let $(f_n)_{n\in\mathbb{N}}$ be a sequence of vertices, and $a$ be a vertex
of $\mathbb{T}ransFinLisse(\Sigma)$.
\begin{itemize}
\item[$\bullet$]
If $W((f_n),a)$, then the sequence $(f_n)_{n\in\mathbb{N}}$ converges
in the Hausdorff topology to $a$: for every neighborhood $V$ of the
curve $a$, for all $n$ large enough, we have $f_n\subset V$.
\item[$\bullet$]
If the sequence $(f_n)_{n\in\mathbb{N}}$ of curves, with some appropriate
parameterization, converges in $C^1$-topology to $a$, then
$W((f_n)_{n\in\mathbb{N}},a)$.
\end{itemize}
\end{lemma}
\begin{proof}
We first prove the first point. Let us first mention that the statement
we wrote is indeed equivalent to the Hausdorff convergence, because
any essential curve in a small enough neighbourhood of $a$ must pass close
to every point of $a$, and thus is Hausdorff-close to $a$.
Suppose for contradiction that
for some neighborhood $V$ of $a$, we have $f_n\not\subset V$
infinitely often. Then we may find a point $x$, not in $\overline{V}$,
such that every neighborhood of $x$ is visited by infinitely many
$f_n$. We may construct three bottle-shaped arcs around $x$ exactly
as in the proof of the preceding lemma, and complete these arcs
to non separating simple closed curves $d_1$, $d_2$, $d_3$, which
can be requested to satisfy $a-d_i$ for $i=1,2,3$. Then we cannot
have $f_n-d_i$ for all $i\in\{1,2,3\}$ for the same reason as in
this preceding proof, and this contradicts the hypothesis
that~$W((f_n),a)$ holds.
The second point is the well known stability of transversality in
the $C^1$-topology.
\end{proof}
\begin{remark}
In fact, the condition $W((f_n), a)$ implies $C^0$-convergence, in the
following sense. Given a parameterization $\alpha$ of $a$, we can choose the
parameterizations of the $f_n$'s yielding a sequence of parmaetrized curves
converging uniformly to~$\alpha$.
As we will not use this fact, we only sketch a quick argument. Let $V$ be
a small tubular neighborhood of $a$, and let $d_1,\ldots,d_N$ be simple
closed nonseparating curves, each meeting $a$ transversely at one point,
and cutting $V$ in small chunks $V_1,\ldots,V_N$ that are met by $\alpha$ in
that
cyclic order. Let $n$ be large enough so that $f_n\subset V$ and
$f_n-d_i$ for each $i$. Then $f_n\cap V_i$ is connected, and $f_n$
visits the pieces $V_1,\ldots,V_N$ in that order. Hence we may choose
a parameterization of $f_n$, say, $F_n$, in such a way that for all
$i\in\{1,\ldots,N\}$ and all $t$, $F_n(t)\in V_i$ if and only if
$\alpha(t)\in V_i$. This implies that $F_n$ is uniformly close to $\alpha$.
\end{remark}
This notion is actually somewhere strictly in between
$C^0$-convergence and $C^1$-convergence,
as we remark in the following
example. This construction will play a crucial role below in the proof of
Theorem~\ref{thm:AutCFinLisse}.
\begin{example}\label{eg:ConvFaible}
Let $a$ be a smooth nonseparating curve in $\Sigma$. We choose
a point $p$ in $a$, and a chart around one of its point, diffeomorphic
to $\mathbb{R}^2$, in such a way that $a$ corresponds to the axis of
equation $y=0$ in that plane. In this chart, we consider the
functions
$f_1\colon x\mapsto\frac{2}{1+x^2}$,
and for all $n\geqslant 2$,
$f_n\colon x\mapsto\frac{f_1(nx)}{n}$.
Abusively, we still denote their graphs by the same letters,
and then, we may extend these arcs, viewed in $\Sigma$, to
simple closed curves (consisting of pushing $a$ aside), that
converges $C^1$ to $a$ outside of the point $p$.
Abusively we still use the same letters $f_n$ to denote these
closed curves.
Obviously, the sequence $(f_n)_{n\geqslant 1}$ does not
converge $C^1$ to $a$, because $f_n$ has slope $-1$ at the
point $(1/n, 1/n)$.
Nonetheless, we claim that $W((f_n),a)$ holds.
Indeed, let $d$ be such that $a-d$. Since the sequence $(f_n)$
converges $C^1$ to $a$ everywhere except at the origin of this
$\mathbb{R}^2$ chart, the only case in which it is not already clear
that $(f_n)-d$ is when $d$ meets $a$ transversely at the origin.
If $d$ has a strictly positive slope there, then for $n$ large
enough, the intersection $f_n\cap d$ will be transverse because
the slopes of $f_n$ are all negative in the region $x>0$.
The case when $d$ has negative slope is symmetric, and if $d$
has vertical slope, it will be transverse with $f_n$ since these
have bounded slopes.
\end{example}
\subsection{Recognizing configurations of smooth curves}
\label{sec:configSmoothCurves}
In this last section we assume that our surface $\Sigma$ is orientable.
If $a$, $b$ are vertices of $\mathbb{T}ransFinLisse(\Sigma)$, we will
denote by $D_\infty(a,b)$ the condition that $a-b$ and
for all sequences $(f_n)$ and $(g_m)$ such that
$W((f_n),a)$ and $W((g_m),b)$, we have $f_n-g_m$ for all
$m, n$ large enough.
\begin{lemma}\label{lem:DisjointLisse}
Let $a$, $b$ be smooth nonseparating curves. Then
$D_\infty(a,b)$ holds if and only if $a$ and $b$ are disjoint.
\end{lemma}
\begin{proof}
Suppose first that $a$ and $b$ are disjoint. Then they admit
disjoint neighborhoods, $V_1$ and $V_2$. For any sequences
$(f_n)$ and $(g_m)$ with $W((f_n),a)$ and $W((g_m),b)$,
for all $m,n$ large enough we have $f_n\subset V_1$ and
$g_m\subset V_2$, by Lemma~\ref{lem:ConvFaible}. Hence,
$f_n-g_m$ for all $m,n$ large enough, and $D_\infty(a,b)$
holds indeed.
Now, suppose that $a$ and $b$ are not disjoint. Since
$a-b$, the curves $a$ and $b$ have a transverse intersection,
and in an appropriate chart diffeomorphic to $\mathbb{R}^2$, the
curves $a$ and $b$ correspond respectively to the axes
$y=0$ and $x=0$.
Then we may form a sequence $(f_n)$ such that $W((f_n),a)$
exactly as in example~\ref{eg:ConvFaible}, and for
$(g_n)$ we just exchange coordinates $x$ and $y$.
For all $n$, the curves $f_n$ and $g_n$ have
a non transverse intersection point (at $(1/n,1/n)$ in the
chart of Example~\ref{eg:ConvFaible}), hence the condition
$D_\infty(a,b)$ does not hold.
\end{proof}
In the end of the proof, the curves $f_n$ and $g_n$ were tangent
at their intersection point, hence not neighbors in the graph
$\mathbb{T}ransFinLisse(\Sigma)$. This may look accidental, but
upon changing the formula of $f_1$ in Example~\ref{eg:ConvFaible}
to $x\mapsto \frac{3}{2+x^2}$, for example, we get three intersection
points.
From now on, we restrict ourselves to the case of orientable
surfaces. One reason is that it would take more work to
recover the extra bouquets and not only the bouquets; one
other reason is that the next lemma works best when at least
one of $a$, $b$ or $c$ is two-sided.
\begin{lemma}\label{lem:BouquetLisse}
Suppose $\Sigma$ is orientable.
Let $\{a,b,c\}$ be a $3$-clique of $\mathbb{T}ransFinLisse(\Sigma)$,
and suppose that these three curves pairwise intersect.
Then the following are equivalent.
\begin{enumerate}
\item This $3$-clique is of type bouquet.
\item There exists a relevant sequence $(f_n)$ of vertices of
$\mathbb{T}ransFinLisse(\Sigma)$, which are all disjoint from $a$,
and such that for all $d$ disjoint from $b$ and
satisfying $c-d$, we have $(f_n)-d$.
\end{enumerate}
\end{lemma}
\begin{proof}
Suppose $\{a,b,c\}$ is of type bouquet. Then the sequence
$(f_n)_{n\in\mathbb{N}}$ can be constructed explicitly.
Let $p=b\cap c$. Fix a (smooth) metric on $\Sigma$,
we remove all points of the ball $B(p,\frac{1}{n})$ off the
curves $b$ and $c$, this gives two arcs. There is a natural
way of adding smooth subarcs of $B(p,\frac{1}{n})$ in order
to extend this union of two arcs, to a curve $f_n$ which
does not intersect $a$.
In a one-holed torus neighborhood of $b\cup c$, with
a choice of meridian and longitude coming from $b$ and $c$, these curves
$f_n$ have slope $1$, or $-1$; these are indeed nonseparating simple
closed curves.
Now if $d$ is disjoint from $b$ and satisfies $c-d$, then
either $d$ is disjoint from $c$, and then $f_n$ is disjoint from $d$
for all $n$ large enough, or $d$ has a transverse intersection
with $c$ at a point distinct from $p$, and we also have $f_n-d$ for all
$n$ large enough. Thus, (1) implies~(2).
Conversely, suppose~(2). We first claim that the sequence $(f_n)$ then
concentrates into neighborhoods of $b\cup c$.
For contradiction, suppose that we can find a neighborhood $V$ of
$b\cup c$, such that $f_n\not\subset V$ for infinitely many $n$. Then, there
exists a point $x$, with $x\not\in b\cup c$, and such that every
neighborhood of $x$ meets infinitely many $f_n$. Then we may choose
three bottle-shaped arcs around $x$, and complete them into curves
$d_1$, $d_2$ and $d_3$ disjoint from $b$ and satisfying $d_j-c$ for
$j=1,2,3$. Indeed, we may start with a curve $d_0$ obtained by
pushing $b$ aside, and then perform surgeries on $d_0$.
The same reasoning as in the proof of Lemma~\ref{lem:fnSenVa} shows
that $f_n\not -d_j$
for some $j\in\{1,2,3\}$ and for infinitely
many $n$,
contradicting the hypothesis~(2).
Now, suppose for contradiction that $\{a,b,c\}$ is a necklace. Then,
for a sufficiently small regular neighborhood $V$ of $b\cup c$, we
may observe that $V\smallsetminus a$ is contractible. Hence it cannot
contain any nonseparating simple closed curve $f_n$, and the
hypothesis~(2) cannot be fullfilled. This proves that (2) implies~(1).
\end{proof}
Now the proof of Theorem~\ref{thm:AutCFinLisse} is a straightforward
adaptation of the proof of Theorem~\ref{thm:AutC1}.
The statements about
connectedness of complexes of arcs, for example, are equivalent to
their counterparts with regularity, because of the argument of
homotopy recalled in the proof of
Lemma~\ref{lem:NonSepFinRelUnBordConnexe} and borrowed
from~\cite{BHW}.
\end{document}
|
\begin{document}
\title{Compactness by coarse-graining\ in long-range lattice systems}
\def{\cal L}{{\cal L}}
\noindent{\bf Abstract.} We consider energies on a periodic set ${\cal L}$ of the form
$\sum_{i,j\in{\cal L}} a^\varepsilon_{ij}|u_i-u_j|$, defined on spin functions $u_i\in\{0,1\}$, and we suppose that the typical range of the interactions is $R_\varepsilon$ with $R_\varepsilon\to +\infty$, i.e., if $|i-j|\le R_\varepsilon$ then $a^\varepsilon_{ij}\ge c>0$. In a discrete-to-continuum analysis, we prove that the overall behaviour as $\varepsilon\to 0$ of such functionals is that of an interfacial energy. The proof is performed using a coarse-graining procedure which associates to scaled functions defined on $\varepsilon{\cal L}$ with equibounded energy a family of sets with equibounded perimeter. This agrees with the case of equibounded $R_\varepsilon$ and can be seen as an extension of coerciveness result for short-range interactions, but is different from that of other long-range interaction energies, whose limit exits the class of surface energies. A computation of the limit energy is performed in the case ${\cal L}=\mathbb Z^d$.
\noindent{\bf Keywords.} Homogenization, lattice systems, long-range interactions, interfacial energies, coarse graining
\noindent{\bf MSC Classifications.} 49J45, 49Q20, 35B27, 82B20
\section{Introduction}
In this paper we give a contribution to the general problem of the asymptotic analysis of systems of lattice interactions of the form
\begin{equation}\label{1}
\sum_{i,j\in{\cal L}} a^\varepsilon_{ij}|u_i-u_j|
\varepsilonnd{equation}
where ${\cal L}$ is a periodic lattice in $\mathbb R^d$, $\varepsilon>0$ is a parameter tending to $0$, and $a^\varepsilon_{ij}$ are non-negative coefficients.
These functionals depend on (scalar) `spin functions' with $u_i\in\{0,1\}$, somehow related to ferromagnetic energies in the terminology of Statistical Mechanics (where usually $u_i\in\{-1,1\}$).
We investigate coerciveness properties related to such energies in a discrete-to-continuum process, where
the values $u^\varepsilon_i$ are identified as the values $u^\varepsilon(\varepsilon i)$ of a function defined on $\varepsilon{\cal L}$. In this way
a continuum limit of $u^\varepsilon$ can be defined as a limit of their piecewise-constant interpolations;
e.g., defined as $u^\varepsilon(x)= u^\varepsilon_i$ if the point of minimum distance of $\varepsilon{\cal L}$ from $x$ is $\varepsilon i$.
Coerciveness is established by exhibiting scales $s_\varepsilon$ such that if $u^\varepsilon_i$ are such that
\begin{equation}\label{2}
\sum_{i,j\in{\cal L}} a^\varepsilon_{ij}|u^\varepsilon_i-u^\varepsilon_j|\le s_\varepsilon,
\varepsilonnd{equation}
then the interpolations $u^\varepsilon$ are precompact in some topology and their limit points are in general non trivial. This can be expressed by proving that the domain of the $\Gamma$-limit of the scaled energies
\begin{equation}\label{2bis}
{1\over s_\varepsilon}\sum_{i,j\in{\cal L}} a^\varepsilon_{ij}|u^\varepsilon_i-u^\varepsilon_j|
\varepsilonnd{equation}
in that topology
is not trivial.
The simplest case that has been previously treated \cite{CDL,ABC} is nearest-neighbour interactions; i.e, when $a^\varepsilon_{ij}$ are strictly positive only when $i,j$ are nearest neighbours (n.n.~for short) in the Delaunay triangulation of ${\cal L}$ (e.g.,
$|i-j|=1$ if ${\cal L}=\mathbb Z^d$). In this case choosing $s_\varepsilon={\varepsilon^{1-d}}$ gives that the scaled energies
\begin{equation}\label{3}
\sum_{i,j\in{\cal L}\ i,j
\,\rm n.n.} \varepsilon^{d-1} a^\varepsilon_{ij}|u^\varepsilon_i-u^\varepsilon_j|
\varepsilonnd{equation}
can be directly seen as a (possibly anisotropic) perimeter of the sets $\{x: u^\varepsilon(x)=1\}$ defined through the piecewise-constant interpolation of $u^\varepsilon$ from the scaled lattice $\varepsilon{\cal L}$. Then, the compactness properties of sets of equibounded perimeter ensure the coerciveness in $L^1_{\rm loc}(\mathbb R^d)$ and the limits are characteristic functions. Moreover, the $\Gamma$-limit of the energies can be described
by an energy defined on sets of finite perimeter $A$, which, in the simplest homogenous case, takes the form
\begin{equation}\label{4}
\int_{\partial A} \varphi(\nu)d{\mathcal H}^{d-1}.
\varepsilonnd{equation}
The same scaling works for finite-range interactions; i.e., when $a^\varepsilon_{ij}$ is $0$ if $|i-j|>R$ for some $R$, even though the energies in that case must be interpreted as a non-local perimeter \cite{BP}.
The finiteness of the range of the interactions can be weakened to a decay condition that can be quantified as
\begin{equation}\label{4bis}
\sup\Bigl\{\sum_{j\in{\cal L}\setminus \{i\}}a^\varepsilon_{i j}|j-i|: i\in{\cal L},\ \varepsilon>0\Bigr\}<+\infty,
\varepsilonnd{equation}
even though the limit energies may have a non-local part if the `tails' of these series are not uniformly negligible \cite{AG}. We note that such analysis is valid beyond pair potentials and generalizes to classes of many-point interactions (see \cite{BK}).
If the decay assumptions \varepsilonqref{4bis} do not hold then the `natural' scaling for the energies may be different from the `surface scaling' $\varepsilon^{d-1}$, and we might exit the class of interfacial energies. An extreme case is that of `dense graphs', which is better stated in a bounded domain; i.e., when
considering energies
\begin{equation}\label{5}
\sum_{i,j\in{\cal L}\cap{1\over\varepsilon} Q} a^\varepsilon_{ij}|u_i-u_j|,
\varepsilonnd{equation}
with $Q$ a cube in $\mathbb R^d$, and suppose that $a^\varepsilon_{ij}\ge c>0$ for (a positive percentage of) all interactions.
In that case the scaling is $s_\varepsilon= {\varepsilon}^{-2d}$, and the limit behaviour is described by a more abstract limit functional called a `graphon' energy \cite{Benjamini,Borgs2012,Lovasz2006,Lovasz2012}, which can be viewed as a relaxation of a double integral on $(0,1)$ of the form
\begin{equation}\label{5}
\int_{(0,1)\times(0,1)}W(x,y)|v(x)-v(y)|\,dx\,dy
\varepsilonnd{equation}
defined on $BV((0,1);\{0,1\})$ after a complex and rather abstract relabeling procedure and identification of functions defined on $Q$ with functions defined on $(0,1)$ (see \cite{BCD}).
For sparse graphs (i.e., graphs which are not dense according to the definition above) and interactions not satisfying the decay conditions \varepsilonqref{4bis}, the correct scaling, the relative convergence and the form of the $\Gamma$-limit is a complex open problem. In \cite{BCS} an example is given of one-dimensional energies with range $R_\varepsilon={1/\sqrt\varepsilon}$ such that
a non-trivial $\Gamma$-limit exists for $s_\varepsilon={1/\sqrt\varepsilon}$ with respect to the $L^\infty$-weak$^*$ convergence, but it is defined on {\it all} functions of bounded variation with values in $[0,1]$ (and not only those with values in $\{0,1\}$). In that example a crucial issue is the topology of the graph of the connections where $a^\varepsilon_{ij}\neq0$.
In this paper we consider an intermediate case; i.e., when the decay condition described above does not hold, and $a^\varepsilon_{ij}\ge c>0$ when $|i-j|\le R_\varepsilon$ with $R_\varepsilon>\!>1$ but the topology of the interactions within that range is that of a `dense' graph.
We further make the assumption $\varepsilon R_\varepsilon<\!<\!1$ so that the discrete-to-continuum process makes sense. We note that this latter condition is not restrictive upon a redefinition of $\varepsilon$ in terms of $R_\varepsilon$; e.g.~taking $R_\varepsilon^{-1/2}$ in the place of $\varepsilon$. We keep the dependence of our system on $R_\varepsilon$ and $\varepsilon$ separate since these parameters may be defined independently in applications. Under these conditions we have
$$
s_\varepsilon={R_\varepsilon^{d+1}\over \varepsilon^{d-1}},
$$
and with this scaling functions of equi-bounded energy interpolated on the lattice $\varepsilon{\cal L}$ converge to a characteristic function of a set of finite perimeter. The main argument for obtaining this result is by coarse-graining. Namely, we average the values of $u^\varepsilon$ for interaction on cubes with side length of order $R_\varepsilon$, so that we can think of those averages as labelled on $\varepsilon R_\varepsilon\mathbb Z^d$. We prove first that those labels for which averages are not essentially close to $0$ and $1$ are negligible; hence, we may regard such functions as spin functions defined on a cubic lattice. Then, we show
that the arguments used for nearest-neighbour interactions of \cite{ABC} can be adapted for the interpolated functions
of the averages. Once a limit set of finite perimeter is obtained we can prove the convergence of the interpolations of the original functions to the same set.
As an application of this scaling argument we show that for
\begin{equation}\label{6}
a^\varepsilon_{ij}= a\Bigl({i-j\over R_\varepsilon}\Bigr),
\varepsilonnd{equation}
where $a$ is a positive function with $\int a(\xi)|\xi|d\xi$ finite,
the $\Gamma$-limit of the energies
\begin{equation}\label{7}
F_\varepsilon(u)={\varepsilon^{d-1}\over R_\varepsilon^{d+1}}\sum_{i,j\in\mathbb Z^d} a^\varepsilon_{ij}|u_i-u_j|,
\varepsilonnd{equation}
defined on the cubic lattice of $\mathbb R^d$,
is given by an energy as in \varepsilonqref{4} with
\begin{equation}\label{8}
\varphi(\nu)=\int_{\mathbb R^d} a(\xi)|\langle \xi,\nu\rangle|d\xi.
\varepsilonnd{equation}
In particular, if $a$ is radially symmetric then \varepsilonqref{4} is simply a multiple of the perimeter of $A$.
It is interesting to note that in a sense the case $R_\varepsilon\to+\infty$ can be seen as a limit of the case of $R_\varepsilon$ finite, for which the $\Gamma$-limit is of the form \varepsilonqref{4} with the integrand $\varphi(\nu)$ given by a discretization of the integral in \varepsilonqref{8}
(as seen in \cite{Ch,BG, BLB} in a slightly different context). This convergence can be re-obtained using the results in \cite{GS}, where transportation maps are used to transform discrete energies in convolution functionals.
\section{A compactness result}
We denote by $Q_R=[-R/2,R/2)^d$ the (semi-open) coordinate cube centered in $0$ and with side length $R$ in $\mathbb R^d$, by $B_R$ the open ball centered in $0$ and with side length $R$ in $\mathbb R^d$, and by $e_1,\dots, e_d$ the vectors of the canonical basis of $\mathbb R^d$. Moreover, $\mathcal H^{d-1}$ denotes the $d-1$-dimensional Hausdorff measure and $|\cdot|$ the Lebesgue $d$-dimensional measure.
Let ${\cal L}\subset\mathbb R^d$ be a discrete periodic set. We can suppose without loss of generality that it is periodic in the coordinate directions with period $1$; i.e., $${\cal L}+e_i={\cal L}\hbox{ for all }i\in\{1,\ldots,d\}.$$
The Voronoi cells of ${\cal L}$ are defined as
$$
V_i=\{x\in\mathbb R^d: |x-i|<|x-j|\hbox{ for all }j\in{\cal L}, \ j\neq i\,\}.
$$
By the periodicity of ${\cal L}$ there exists a constant $C_{\cal L}>0$ such that
\begin{equation}\label{Voro}
{1\over C_{\cal L}}\le |V_i|\le C_{\cal L},\qquad {1\over C_{\cal L}}\le{\mathcal H}^{d-1}(\partial V_i)\le C_{\cal L}.
\varepsilonnd{equation}
Each spin function $u\colon\varepsilon\mathbb {\cal L}\to\{0,1\}$ is identified with its piecewise-constant interpolation, which is the $L^\infty$ function defined by
$$
u(x)= u(\varepsilon i)\ \hbox{ if } x\in\varepsilon V_i\hbox{\,; i.e., } |x-\varepsilon i|<|x-\varepsilon j| \hbox{ for all }j\in{\cal L}, \ j\neq i\,.
$$
Note that by \varepsilonqref{Voro} the $L^1$ norm of such $u$ is equivalent to
$\varepsilon^d\#\{i: u_i\neq 0\}$.
In this section we prove coerciveness properties for energies $E_\varepsilon$ defined on
spin functions $u\colon\varepsilon{\cal L}\to\{0,1\}$ by
\begin{equation}\label{fe}
E_\varepsilon(u)=\frac{\varepsilon^{2d}}{\varepsilonta^{d+1}}\sum_{i,j\in{\cal L},\ i-j\in Q_{\varepsilonta/\varepsilon}} |u_i-u_j|,
\varepsilonnd{equation}
where we denote $u_i=u(\varepsilon i)$, and $\varepsilonta=\varepsilonta_\varepsilon$ are such that
\begin{equation}\label{eta}
\lim_{\varepsilon\to 0}\varepsilonta_\varepsilon=\lim_{\varepsilon\to0}{\varepsilon\over\varepsilonta_\varepsilon}=0.
\varepsilonnd{equation}
\begin{lemma}[Compactness]\label{lemma}
Let $u^\varepsilon$ be spin functions such that $E_\varepsilon(u^\varepsilon)$ is equibounded. Then, up to subsequences, the corresponding piecewise-constant interpolations, still denoted by $u^\varepsilon$, converge in $L^1_{\rm loc}(\mathbb R^d)$ to $u=\chi_A$, where $A$ is a set of finite perimeter.
\varepsilonnd{lemma}
\begin{proof} The idea of the proof is to subdivide the set of indices ${\cal L}$ into disjoint cubes of side-length $\varepsilonta/4\varepsilon$. The factor $4$ is chosen so that if we consider $i,j$ indices belonging to two neighbouring cubes with this side-length, respectively, then $i-j\in Q_{\varepsilonta/\varepsilon}$ so that they interact in energy $E_\varepsilon$. In such a way we can associate to each $u^\varepsilon$ and each such smaller cube the value $0$ or $1$ of the `majority phase', if such majority phase is sufficiently close to $0$ and $1$, respectively, while we prove that the remaining cubes can be neglected. In this way we will construct coarse-grained functions for which the energy $E_\varepsilon$ can be viewed as a standard nearest-neighbour ferromagnetic energy and the compactness then follows by interpreting spin functions as sets of finite perimeter.
For any $k\in\mathbb Z^d$ we set
$$Q_k^\varepsilon=\frac{\varepsilonta}{4\varepsilon}k+Q_{\frac{\varepsilonta}{4\varepsilon}}.$$
For $u:\varepsilon{\cal L}\to\{0,1\}$ we define
$$D(\varepsilon,k)(u)=\frac{\big|\#\{i\in Q_k^\varepsilon\cap {\cal L}: u_i=1\}-\#\{i\in Q_k^\varepsilon\cap {\cal L}: u_i=0\}\big|}{\# (Q_k^\varepsilon\cap {\cal L})}.$$
Note that $D(\varepsilon,k)(u)$ measures how much the function $u$ is close to its majority phase; more precisely, $D(\varepsilon,k)(u)=1$ if $u$ is constant on $Q_k^\varepsilon\cap {\cal L}$, while $D(\varepsilon,k)(u)=0$ if the values of $u$ are equally distributed between $0$ and $1$ in $Q_k^\varepsilon\cap {\cal L}$.
With fixed $\delta\in(0,1)$, we define
$$
\mathcal B^\varepsilon(u)=\{k\in\mathbb Z^d:D(\varepsilon,k)(u)<1-\delta\}.
$$
The $Q_k^\varepsilon$ corresponding to $k\in \mathcal B^\varepsilon(u)$ will be considered as the cubes where $u$ is not close to a phase $1$ or $0$. We will first show that such cubes are negligible.
Indeed, note that thanks to the first inequality in \varepsilonqref{Voro} the number of pairs of indices $i,j$ within $Q^\varepsilon_k$ are of order $(\varepsilonta/\varepsilon)^{2d}$ and hence there exists $C_\delta>0$ such that if $k\in \mathcal B^\varepsilon(u)$, then the number of `interactions within the cube' $Q_k^\varepsilon$ is at least
$C_\delta (\frac{\varepsilonta}{\varepsilon})^{2d}$; namely,
$$\#\{(i,j): i,j\in Q_k^\varepsilon\cap{\cal L}, u_i\neq u_j\}
\geq C_\delta \Big(\frac{\varepsilonta}{\varepsilon}\Big)^{2d}.$$
Hence, if $u^\varepsilon$ are as in the hypotheses of the lemma; that is, $F_\varepsilon(u^\varepsilon)\leq c$, we have
\begin{equation*}
\#\mathcal B^\varepsilon(u^\varepsilon)\leq\frac{c}{C_\delta}\varepsilonta^{1-d}.
\varepsilonnd{equation*}
We can estimate the measures
\begin{equation}\label{misura-diversi}
\Big|\bigcup_{k\in\mathcal B^\varepsilon(u^\varepsilon)} \varepsilon Q_k^\varepsilon\Big|=
\#\mathcal B^\varepsilon(u^\varepsilon){\varepsilonta^d\over 4^d}
\leq\frac{c}{4^dC_\delta}\varepsilonta,
\varepsilonnd{equation}
\begin{equation}\label{bordo-diversi}
{\mathcal H}^{d-1}\Bigl(\partial\bigcup_{k\in\mathcal B^\varepsilon(u^\varepsilon)} \varepsilon Q_k^\varepsilon\Big)=
\#\mathcal B^\varepsilon(u^\varepsilon){2d\,\varepsilonta^{d-1}\over 4^{d-1}}
\leq\frac{2cd}{4^{d-1}C_\delta}.
\varepsilonnd{equation}
As for the indices such that $D(\varepsilon,k)(u)\geq 1-\delta$, we subdivide them into the sets
\begin{eqnarray*}
\mathcal A^\varepsilon_1(u)=\{ k\in\mathbb Z^d: D(\varepsilon,k)(u)\geq 1-\delta, \#\{i\in Q_k^\varepsilon: u_i=1\}>\#\{i\in Q_k^\varepsilon: u_i=0\}\}\\
\mathcal A^\varepsilon_0(u)=\{ k\in\mathbb Z^d: D(\varepsilon,k)(u)\geq 1-\delta, \#\{i\in Q_k^\varepsilon: u_i=1\}<\#\{i\in Q_k^\varepsilon: u_i=0\}\}
\varepsilonnd{eqnarray*}
and define
\begin{eqnarray*}
K_j^\varepsilon(u)=\bigcup_{k\in \mathcal A^\varepsilon_j(u)} \varepsilon Q_k^\varepsilon\hbox{ for }j=0,1.
\varepsilonnd{eqnarray*}
In order to estimate the measure of the boundary of $K_1^\varepsilon(u)$ we estimate the number of cubes $Q_k^\varepsilon$ with
$k\in \mathcal A^\varepsilon_1(u)$ which have a side in common with a cube $Q_{k'}^\varepsilon$ with
$k'\in\mathcal A^\varepsilon_0(u)$, parameterized on the set
$$
\mathcal A^\varepsilon(u):=\{k\in\mathcal A_1^\varepsilon(u): k+e_j\in\mathcal A_0^\varepsilon(u) \hbox{ for some } j=1,\dots, d\}
$$
To that end, note that if $D(\varepsilon,k)(u)\ge 1-\delta$ and $k\in \mathcal A^\varepsilon_1(u)$ then
$$
\#\{i\in Q_k^\varepsilon\cap {\cal L}: u_i=1\}\ge \Bigl(1-{\delta\over 2}\Bigr)\#\{i\in Q_k^\varepsilon\cap {\cal L}\},
$$
so that, again recalling the first inequality in \varepsilonqref{Voro}, each site $i\in Q_k^\varepsilon$ such that $u_i=1$ interacts with $C'_\delta (\frac{\varepsilonta}{\varepsilon})^{d}$
and conversely for each site $i\in Q_{k'}^\varepsilon$ such that $u_i=0$. Hence, the
interacting pairs $(i,j)\in Q_k^\varepsilon\times Q_{k'}^\varepsilon$ are at least
$C'_\delta(\frac{\varepsilonta}{\varepsilon})^{2d}$.
Hence,
$$
E_\varepsilon(u)\ge \frac{\varepsilon^{2d}}{\varepsilonta^{d+1}}\#\mathcal A^\varepsilon(u)C''_\delta(\frac{\varepsilonta}{\varepsilon})^{2d}= C''_\delta\#\mathcal A^\varepsilon(u)\varepsilonta^{d-1}
$$
so that
$$
\#\mathcal A^\varepsilon(u)\le {1\over C''_\delta}E_\varepsilon(u)\varepsilonta^{1-d}.
$$
For the functions $u^\varepsilon$ we then obtain
$$
\#\mathcal A^\varepsilon(u^\varepsilon)\le \frac{c}{C''_\delta}\varepsilonta^{1-d},
$$
so that
\begin{eqnarray*}
{\mathcal H}^{d-1}(\partial K_1^\varepsilon(u^\varepsilon))&\le & 2d\biggl(
{\mathcal H}^{d-1}\Bigl(\bigcup_{k\in\mathcal A^\varepsilon(u^\varepsilon)} \varepsilon Q_k^\varepsilon\Bigr)
+{\mathcal H}^{d-1}\Bigl(\bigcup_{k\in\mathcal B^\varepsilon(u^\varepsilon)} \varepsilon Q_k^\varepsilon\Bigr)\biggr)\\
&\le& 2d\Bigl(\#\mathcal A^\varepsilon(u^\varepsilon) {2d\,\varepsilonta^{d-1}\over 4^{d-1}}+\frac{2cd}{4^{d-1}C_\delta}\Bigr)\le C^{'''}_\delta
\varepsilonnd{eqnarray*}
where $C^{'''}_\delta$ is a positive constant depending only on $d, c$ and $\delta$.
By the compactness of sets of equibounded perimeter this shows that the characteristic functions of the sets $K_1^\varepsilon(u^\varepsilon)$ are compact in $L^1_{\rm loc}(\mathbb R^d)$. The symmetric argument shows also that $K_0^\varepsilon(u^\varepsilon)$ are compact in $L^1_{\rm loc}(\mathbb R^d)$. Moreover, if we denote a limit of the sets $K_j^\varepsilon(u^\varepsilon)$ by $K_j$ then we have
\begin{equation}\label{tutto}|\mathbb R^d\setminus (K_0\cup K_1)|=0\varepsilonnd{equation} by
\varepsilonqref{misura-diversi}. We highlight the possible dependence of the sets obtained by this procedure on $\delta$ by renaming them
$K_1^\delta$ and $K_0^\delta$.
Note that if $\delta<\delta'$ then
$$
K_1^{\delta'}\subset K_1^\delta\hbox{ and }
K_0^{\delta'}\subset K_0^\delta.
$$
Since in both cases \varepsilonqref{tutto} holds, then we must have $K_1^{\delta'}=K_1^\delta$
and $K_0^{\delta'}=K_0^\delta$, so that these sets
are independent of $\delta$ and we may go back to denoting them by $K_1$
and $K_0$.
We can now prove the convergence of $u^\varepsilon$. Fixed $\delta<1$ as above, we write
$$
u^\varepsilon= u^\varepsilon\chi_{K_1^\varepsilon(u^\varepsilon)}+ u^\varepsilon\chi_{K_0^\varepsilon(u^\varepsilon)}+ u^\varepsilon\chi_{\mathbb R^d\setminus (K_1^\varepsilon(u^\varepsilon)\cup K_0^\varepsilon(u^\varepsilon))}.
$$
By \varepsilonqref{misura-diversi} the last term converges to $0$ in $L^1(\mathbb R^d)$
As for the other two terms we localize the convergence by restricting to a cube $Q_R$.
Note that for $k\in \mathcal A^\varepsilon_1(u^\varepsilon)$ we have
$$
\|u^\varepsilon- 1\|_{L^1(\varepsilon Q^\varepsilon_k)}\le C(1-\delta)\varepsilonta^d,
$$
so that
$$
\|u^\varepsilon\chi_{K_1^\varepsilon(u^\varepsilon)\cap Q_R}- \chi_{K_1^\varepsilon(u^\varepsilon)\cap Q_R}\|_{L^1(\mathbb R^d)}\le C(1-\delta)R^d
$$
where $C$ denotes a positive constant not depending on $\delta.$
Analogously for $k\in \mathcal A^\varepsilon_0(u^\varepsilon)$ we have
$$
\|u^\varepsilon\|_{L^1(\varepsilon Q^\varepsilon_k)}\le C(1-\delta)\varepsilonta^d,
$$
and hence
$$
\|u^\varepsilon\chi_{K_0^\varepsilon(u^\varepsilon)\cap Q_R}\|_{L^1(\mathbb R^d)}\le C(1-\delta)R^d.
$$
We then have, by the local convergence of $K^\varepsilon_j(u^\varepsilon)$,
\begin{eqnarray*}&&
\limsup_{\varepsilon\to0}\|u^\varepsilon \chi_{K_1^\varepsilon(u^\varepsilon)\cap Q_R}-\chi_{K_1\cap Q_R}\|_{L^1(\mathbb R^d)}
\\
&\le&
\limsup_{\varepsilon\to 0}\Bigl(\|u^\varepsilon \chi_{K_1^\varepsilon(u^\varepsilon)\cap Q_R}-\chi_{K^\varepsilon_1(u^\varepsilon)\cap Q_R}\|_{L^1(\mathbb R^d)}+
\| \chi_{K_1^\varepsilon(u^\varepsilon)\cap Q_R}-\chi_{K_1\cap Q_R}\|_{L^1(\mathbb R^d)}\Bigr)
\\
&\le&C(1-\delta)R^d,
\varepsilonnd{eqnarray*}
and
$$
\limsup_{\varepsilon\to0}\|u^\varepsilon \chi_{K_0^\varepsilon(u^\varepsilon)\cap Q_R}\|_{L^1(\mathbb R^d)}
\le C(1-\delta)R^d,
$$
so that, by the arbitrariness of $\delta$, $u^\varepsilon$ converge locally to $\chi_{K_1}$.
\varepsilonnd{proof}
\begin{remark}\rm
The proof of Lemma \ref{lemma} works exactly in the same way if we suppose that `almost all' pairs of indices of ${\cal L}$ within $Q_{\varepsilonta/\varepsilon}$ interact; namely, if in place of energy \varepsilonqref{fe} we consider
\begin{equation}\label{fe-a}
E_\varepsilon(u)=\frac{\varepsilon^{2d}}{\varepsilonta^{d+1}}\sum_{i,j\in{\cal L}:\ i-j\in Q_{\varepsilonta/\varepsilon}} a^\varepsilon_{ij} |u_i-u_j|,
\varepsilonnd{equation}
with the requirement that there exists $c>0$ such that
\begin{equation}\label{lim-a}
\lim_{\varepsilon\to 0}{\#\{(i,j):i,j\in x+Q_{\varepsilonta/\varepsilon}: a^\varepsilon_{ij}\ge c\}\over \#\{(i,j):i,j\in x+Q_{\varepsilonta/\varepsilon}\}}=1
\varepsilonnd{equation}
uniformly in $x\in\mathbb R^d$. Condition \varepsilonqref{lim-a} is trivially satisfied by energies \varepsilonqref{fe} for $c=1$.
Note that condition \varepsilonqref{lim-a} cannot be relaxed to `having a proportion' of pairs of indices of ${\cal L}$ within $Q_{\varepsilonta/\varepsilon}$ interacting, however large this proportion may be below $1$; i.e., it is not sufficient that
\begin{equation}\label{lim-a2}
\lim_{\varepsilon\to 0}{\#\{(i,j):i,j\in x+Q_{\varepsilonta/\varepsilon}: a^\varepsilon_{ij}\ge c\}\over \#\{(i,j):i,j\in x+Q_{\varepsilonta/\varepsilon}\}}\ge \lambda,
\varepsilonnd{equation}
for any $\lambda<1$. To check this, we may consider the following example: choose ${\cal L}=\mathbb R^d$, fix $N\in{\mathbb N}$, and define
$$
a^\varepsilon_{ij}=\begin{cases}
1 &\hbox{ if }i-j\in Q_{\varepsilonta/\varepsilon}\hbox{ and both }i,j\not\in N\mathbb Z\\
0 &\hbox{ otherwise.}
\varepsilonnd{cases}
$$
Then \varepsilonqref{lim-a2} holds for $\lambda= \bigl(1-{1\over N^d}\bigr)^2$ but, if we define
$$
u^\varepsilon_i=\begin{cases}
1 &\hbox{ if }i\not\in N\mathbb Z\\
0 &\hbox{ if }i\in N\mathbb Z,
\varepsilonnd{cases}
$$
then $u^\varepsilon$ converge weakly in $L^1_{\rm loc}(\mathbb R^d)$ to the constant $1-{1\over N^d}$.
Since $E_\varepsilon(u_\varepsilon)=0$ this shows that Lemma \ref{lemma} does not hold.
In this example the subset $N\mathbb Z^d$ of $\mathbb Z^d$ can be considered as a `perforation' of the domain and can be treated as such, considering convergence only of the restriction of the functions to $\mathbb Z^d\setminus N\mathbb Z^d$ (see Section 3 of \cite{BCPS}). However, the situation can be more complicated if we take
$$
a^\varepsilon_{ij}=\begin{cases}
1 &\hbox{ if }i-j\in Q_{\varepsilonta/\varepsilon}\hbox{ and both }i,j\not\in N\mathbb Z\\
c_\varepsilon &\hbox{ otherwise,}
\varepsilonnd{cases}
$$
that can be regarded as representing a `high-contrast medium', for which the effect of the `perforation' cannot be neglected and for some values of $c_\varepsilon$ may give a `double porosity' effect \cite{BCPS}.
\varepsilonnd{remark}
\section{Homogenization of long-range lattice systems}
Let $a:\mathbb R^d\to [0,+\infty)$ be such that $a(\xi)|\xi|$ is Riemann integrable on bounded sets and such that
\begin{equation}\label{a1}
\int_{\mathbb R^d} a(\xi)|\xi|\,d\xi<+\infty,
\varepsilonnd{equation}
and
\begin{equation}\label{a2}
a(\xi)\geq c_0 \hbox{ if } \ |\xi|\leq r_0
\varepsilonnd{equation}
for some $c_0,r_0>0$.
Given $\varepsilon,\varepsilonta=\varepsilonta_\varepsilon$ satisfying \varepsilonqref{eta} we define the coefficients
$$
a_{ij}^\varepsilon=a_{i-j}^\varepsilon=a\Big(\frac{\varepsilon (i-j)}{\varepsilonta}\Big)
$$
for $i,j\in\mathbb Z^d$,
and the energies
\begin{equation}\label{fe-a}
F_\varepsilon(u)=\frac{\varepsilon^{2d}}{\varepsilonta^{d+1}}\sum_{i,j\in\mathbb Z^d} a^\varepsilon_{i-j}|u_i-u_j|.
\varepsilonnd{equation}
\begin{definition}\label{conv}
A family $\{u^\varepsilon\}$ of functions $u^\varepsilon:\varepsilon\mathbb Z^d\to\{0,1\}$ {\rm converges} to a set $A\subset\mathbb R^d$ if the piecewise-constant interpolations of $u^\varepsilon$ converge to the characteristic function $\chi_A$ in $L^1_{\rm loc}(\mathbb R^d)$ as $\varepsilon\to0$.
\varepsilonnd{definition}
By hypothesis \varepsilonqref{a2} we may apply Compactness Lemma \ref{lemma}, obtaining that the family $\{F_\varepsilon\}$ is coercive with respect to this convergence.
\begin{proposition}
Let $\{u^\varepsilon\}$ be such that $\sup_\varepsilon F_\varepsilon(u^\varepsilon)<+\infty$. Then, up to subsequences, there exists a set of finite perimeter $A$ such that $u^\varepsilon$ converge to $A$ in the sense of Definition {\rm\ref{conv}}.
\varepsilonnd{proposition}
This coerciveness property justifies the computation of the $\Gamma$-limit of $F_\varepsilon$ with respect to the convergence in
Definition \ref{conv}. We use standard notation in the theory of sets of finite perimeter (see e.g.~\cite{LN98,Maggi}).
\begin{theorem}[Homogenization]\label{theorem}
The functionals defined in \varepsilonqref{fe-a} $\Gamma$-converge with respect to the convergence in Definition {\rm\ref{conv}} to the
functional $F$ defined on sets of finite perimeter by
\begin{equation}\label{f-a}
F(A)=\int_{\partial^* A} \varphi_a(\nu)d{\mathcal H}^{d-1},
\varepsilonnd{equation}
where $\partial^* A$ denotes the reduced boundary of $A$, $\nu$ the outer normal to $A$ and $\varphi_a$ is given by
\begin{equation}\label{fi-a}
\varphi_a(\nu)=\int_{\mathbb R^d} a(\xi)|\langle\xi,\nu\rangle|d\xi.
\varepsilonnd{equation}
\varepsilonnd{theorem}
\begin{proof}
In order to better illustrate the proof in the general $d$-dimensional case we first deal with the one-dimensional case, in which we may highlight the coarse-graining procedure without the technical complexities of the higher-order geometry.
In this case
we may rewrite the energies as
$$
F_\varepsilon(u)=\frac{\varepsilon^{2}}{\varepsilonta^{2}}\sum_{\xi\in\mathbb Z}\sum_{i\in\mathbb Z} a^\varepsilon_{\xi}|u_{i+\xi}-u_i|.
$$
The relevant
computation is that of the lower bound for the target $A=[0,+\infty)$. Let $u^\varepsilon$ converge to $A$. For each $\xi\in\mathbb Z\setminus\{0\}$ and $i\in\{1,\ldots,|\xi|\}$ we consider the function $u^\varepsilon$ restricted to $\varepsilon i+\varepsilon \xi\mathbb Z$. By the $L^1_{\rm loc}$ convergence, we may suppose that each such restriction changes value; i.e., there exists some $k_{i,\xi}\in\mathbb Z$ such that
$$
u^\varepsilon(\varepsilon i+\varepsilon k_{i,\xi}\xi)=0 \hbox{ and } u^\varepsilon(\varepsilon i+\varepsilon (k_{i,\xi}+1)\xi)=1.
$$
The set of $\xi$ and $i$ for which this does not hold is negligible for $\varepsilon\to0$; the precise proof is directly given for the $d$-dimensional functionals below. For each $\xi$ we then have
$$
\sum_{i=1}^{|\xi|} \sum_{k\in\mathbb Z}a^\varepsilon_{\xi}|u^\varepsilon_{i+(k+1)\xi}-u^\varepsilon_{i+k\xi}|\ge |\xi| a\Bigl({\varepsilon\over\varepsilonta}\xi\Bigr),
$$
so that
\begin{eqnarray*}
\liminf_{\varepsilon\to0} F_\varepsilon(u^\varepsilon)\ge\liminf_{\varepsilon\to0}{\varepsilon^2\over\varepsilonta^2}\sum_{\xi\in\mathbb Z} |\xi| a\Bigl({\varepsilon\over\varepsilonta}\xi\Bigr)=\liminf_{\varepsilon\to0}\sum_{\xi\in\mathbb Z} {\varepsilon\over\varepsilonta} a\Bigl({\varepsilon\over\varepsilonta}\xi\Bigr)\Bigl|{\varepsilon\over\varepsilonta}\xi\Bigr|,
\varepsilonnd{eqnarray*}
the latter being a Riemann sum giving the integral $\displaystylelaystyle\int_{\mathbb R}a(\xi)|\xi|\,d\xi$, which is $F(A)$.
We now deal with the $d$-dimensional case. The proof of the lower bound follows the argument above, but is more complex since we must take into account the direction of the interaction vectors $\xi$.
We prove the inequality by applying the blow-up technique (see \cite{fomu} and \cite{BMS}, and for instance \cite{BS,BP,NSS} for the discrete setting).
We assume that the sequence $\{F_\varepsilon(u^{\varepsilon})\}$ is equibounded and that $u^\varepsilon$ converge in $L^1_{\rm loc}(\mathbb R^d)$ to $u=\chi_A$, where $A$ is a set of finite perimeter. Up to subsequences, we can assume that $\liminf_{\varepsilon\to 0} F_\varepsilon(u^\varepsilon)=\lim_{\varepsilon\to 0}F_\varepsilon(u^\varepsilon)$.
We define the localized energy on an open set $U$ by
$$
F_\varepsilon(u^{\varepsilon};U)= \frac{\varepsilon^{2d}}{\varepsilonta^{d+1}}\sum_{i\in U} \sum_{j\in\mathbb Z^d}a^\varepsilon_{i-j}|u^\varepsilon_i-u^\varepsilon_j|,
$$
and define the measures $\mu_\varepsilon(U)=F_\varepsilon(u^{\varepsilon};U)$; since the family $\{\mu_\varepsilon\}$ is equibounded, we can assume that $\mu_\varepsilon\rightharpoonupstar \mu$ up to subsequences.
Now, let $\lambda=\mathcal H^{d-1}\restr \partial^\ast A$; the lower bound inequality
follows if we show that for $\mathcal H^{d-1}$-a.a.~$x\in\partial^\ast A$ we have
$$\frac{d\mu}{d \lambda}(x)\geq \varphi_a(\nu),$$
where $\frac{d\mu}{d \lambda}$ denotes the Radon-Nikodym derivative of $\mu$ with respect to the Hausdorff $d-1$-dimensional measure $\lambda$.
By the Besicovitch Derivation Theorem, for $\mathcal H^{d-1}$-a.a.~$x\in\partial^\ast A$ we have that
$$\frac{d\mu}{d \lambda}(x)=\lim_{\varrho\to 0}\frac{\mu(Q_\varrho^\nu(x))}{\lambda(Q_\varrho^\nu(x))},$$
where $\lambda$ is the measure $\mathcal H^{d-1}\restr \partial^\ast A$, $\nu$ is the normal vector to $\partial^\ast A$ at $x$ and
$Q_\varrho^\nu(x)$ is a cube centered in $x$ with side length $\varrho$ and a face orthogonal to $\nu$.
We can fix $x=0$ and denote $Q_\varrho^\nu(0)$ by $Q_\varrho^\nu$.
Hence, the lower bound follows if we show that
\begin{equation}\label{lower}
\lim_{\varrho\to 0}\liminf_{\varepsilon\to 0}\frac{1}{\varrho^{d-1}}F_\varepsilon(u^{\varepsilon};Q_\nu^\varrho) \ge \varphi_a(\nu).
\varepsilonnd{equation}
We may therefore assume that $\varrho=\varrho_\varepsilon$ be such that $\frac{\varepsilon}{\varrho}\to 0$ and the scaled functions $u^\varepsilon(\frac{\varepsilon}{\varrho}i)$ interpolated on the lattice $\frac{\varepsilon}{\varrho}\mathbb Z^d$ converge to the characteristic function of the half space $H^\nu=\{x: \langle x,\nu\rangle<0\}$ on $Q_\nu^1$. We define
$$
A_\varepsilon:=\{x\in Q_\nu^1: u^\varepsilon(x)\neq \chi_{H^\nu}(x)\},
$$
so that $|A_\varepsilon|\to 0$.
If we define
$$I_{\varepsilon/\varrho}^\xi=\Bigl\{i\in\mathbb Z^d: \frac{\varepsilon}{\varrho} i, \frac{\varepsilon}{\varrho} (i+\xi)\in Q_\nu^1\Bigr\}$$
then
$$
F_\varepsilon(u^{\varepsilon};Q_\nu^\varrho)\ge \frac{\varepsilon^{2d}}{\varepsilonta^{d+1}}\sum_{\xi\in\mathbb Z^d}a \Bigl(\frac{\varepsilon}{\varepsilonta}\xi\Bigr)
\sum_{i\in I_{\varepsilon/\varrho}^\xi}
\Bigl|u^\varepsilon(\frac{\varepsilon}{\varrho} (i+\xi))-u^\varepsilon(\frac{\varepsilon}{\varrho}i)\Bigr|.$$
\begin{figure}[h!]
\centerline{\includegraphics[width=0.6\textwidth]{Fig1bis}}
\caption{The set $R^{\alpha,\xi}$}
\label{Fig1}
\varepsilonnd{figure}
We begin by estimating
$$
\sum_{i\in I_{\varepsilon/\varrho}^\xi}
\Bigl|u^\varepsilon(\frac{\varepsilon}{\varrho} (i+\xi))-u^\varepsilon(\frac{\varepsilon}{\varrho}i)\Bigr|=\#\Bigl\{i\in I_{\varepsilon/\varrho}^\xi:
u^\varepsilon(\frac{\varepsilon}{\varrho} (i+\xi))\neq u^\varepsilon(\frac{\varepsilon}{\varrho}i)\Bigr\}.$$
With fixed $\alpha\in (0,1)$ for each $\xi\in\mathbb Z^d$ satisfying
\begin{equation}\label{alfa-xi}\Bigl|\langle \frac{\xi}{|\xi|},\nu\rangle\Bigr|\geq \frac{\alpha}{\sqrt{1+\alpha^2}},
\varepsilonnd{equation}
we define
$$P^{\alpha,\xi}=\Big\{y\in \Pi_\nu\cap Q_\nu^1: y\pm \frac{\alpha}{2|\langle \xi,\nu \rangle|} \xi\in Q_\nu^1\Big\},$$
which is not empty by \varepsilonqref{alfa-xi}, and
$$R^{\alpha,\xi}=\Big\{x\in Q_\nu^1: x=y+t \xi, \ y\in P^{\alpha,\xi}, \ -\frac{\alpha}{2|\langle \xi,\nu \rangle|}\leq t\leq \frac{\alpha}{2|\langle \xi,\nu \rangle|}\Big\}$$
(see Fig.~\ref{Fig1}).
Furthermore, we fix $\beta$ with
\begin{equation}\label{beta}\beta> \frac{\alpha}{\sqrt{1+\alpha^2}}.
\varepsilonnd{equation}
Since we will restrict our arguments to sets $P^{\alpha,\xi}$ and $R^{\alpha,\xi}$ above with $\xi$ satisfying
\begin{equation}\label{beta-xi}\Bigl|\langle \frac{\xi}{|\xi|},\nu\rangle\Bigr|\geq \beta;
\varepsilonnd{equation}
we omit the dependence of the sets $P^{\alpha,\xi}$ and $R^{\alpha,\xi}$ on $\nu$, since the estimates we will obtain will be independent on $\nu$.
As in the one-dimensional case we consider the functions restricted to the discrete lines $\frac{\varepsilon}{\varrho}i+\frac{\varepsilon}{\varrho}\xi\mathbb Z$.
The parameter $\alpha$ is introduced so as to estimate the number of sites of such discrete lines inside $Q_\nu^1$.
We then set
$$B^{\alpha,\xi}_{\varepsilon/\varrho}=\Bigl\{i\in \mathbb Z^d: \frac{\varepsilon}{\varrho}i\in R^{\alpha,\xi} \hbox{ and }
u^\varepsilon \hbox{ is not constant in } \Big(\frac{\varepsilon}{\varrho}i+\frac{\varepsilon}{\varrho}\xi\mathbb Z\Big)\cap R^{\alpha,\xi}\Bigr\}.$$
Note that if $i\in B^{\alpha,\xi}_{\varepsilon/\varrho}$ then $i+k\xi\in B^{\alpha,\xi}_{\varepsilon/\varrho}$ for all $k$ with $\frac{\varepsilon}{\varrho}(i+k\xi)\in R^{\alpha,\xi}$, so that, if we define the equivalence relation $i\sim i^\prime$ if $i-i^\prime\in \xi\mathbb Z$,
we may set
$$\widetilde B^{\alpha,\xi}_{\varepsilon/\varrho}=B^{\alpha,\xi}_{\varepsilon/\varrho}/\sim$$
getting
$$
\#\Bigl\{i\in I_{\varepsilon/\varrho}^\xi:
u^\varepsilon(\frac{\varepsilon}{\varrho} (i+\xi))\neq u^\varepsilon(\frac{\varepsilon}{\varrho}i)\Bigr\}\geq
\#\widetilde B^{\alpha,\xi}_{\varepsilon/\varrho}.$$
We can estimate the number of `discrete lines' intersecting $R^{\alpha,\xi}$ as
\begin{eqnarray*}
\#\Bigl(\Bigl\{i\in\mathbb Z^d: \frac{\varepsilon}{\varrho}i\in R^{\alpha,\xi}\Bigr\}/\sim\Bigr) &\geq&
\frac{|R^{\alpha,\xi}|}{\bigl(\frac{\varepsilon}{\varrho}\bigr)^d}\frac{1}{
\frac{\alpha}{\frac{\varepsilon}{\varrho}|\langle\xi,\nu\rangle|}
}-C_\alpha|\xi|\Bigl(\frac{\varepsilon}{\varrho}\Bigr)^{2-d} \\
&\geq&
\mathcal H^{d-1}(P^{\alpha,\xi})|\langle\xi,\nu\rangle|\Bigl(\frac{\varepsilon}{\varrho}\Bigr)^{1-d}-
C_\alpha|\xi|\Bigl(\frac{\varepsilon}{\varrho}\Bigr)^{2-d},
\varepsilonnd{eqnarray*}
where the last term is an error term accounting for the cubes intersecting the boundary of $ R^{\alpha,\xi}$.
Note that to every element of the complement of $\tilde B^{\alpha,\xi}_{\varepsilon/\varrho}$
there correspond at least
$\lfloor \frac{\alpha}{\frac{\varepsilon}{\varrho} |\langle \xi, \nu\rangle|}\rfloor$ points in ${\varepsilon\over\varrho}\mathbb Z^d\cap A_\varepsilon\}$, so that
for $\varepsilon$ sufficiently small we get
\begin{eqnarray*}
&&\#\Bigl(\Bigl\{i\in \mathbb Z^d: \frac{\varepsilon}{\varrho}i\in R^{\alpha,\xi} \hbox{ and }
u^\varepsilon \hbox{ constant in } \big(\frac{\varepsilon}{\varrho}i+\frac{\varepsilon}{\varrho}\xi\mathbb Z\big)\cap R^{\alpha,\xi}\Bigr\}
/\sim \Bigr)\\
&&\hspace{1cm}\leq \frac{|A_\varepsilon|}{\bigl(\frac{\varepsilon}{\varrho}\bigr)^d} \frac{\frac{\varepsilon}{\varrho}|\langle \xi, \nu\rangle|}{\alpha} + C'_\alpha|\xi|\Bigl(\frac{\varepsilon}{\varrho}\Bigr)^{2-d}\\
&&\hspace{1cm}= \frac{1}{\alpha}|A_\varepsilon| \Bigl(\frac{\varepsilon}{\varrho}\Bigr)^{1-d}
|\langle \xi, \nu\rangle| + C'_\alpha|\xi|\Bigl(\frac{\varepsilon}{\varrho}\Bigr)^{2-d},
\varepsilonnd{eqnarray*}
with $C'_\alpha$ again a positive constant accounting for boundary cubes,
and hence \begin{equation}\label{stima}
\#\widetilde B^{\alpha,\xi}_{\varepsilon/\varrho}\geq
\mathcal H^{d-1}(P^{\alpha,\xi})|\langle\xi,\nu\rangle|\Bigl(\frac{\varepsilon}{\varrho}\Bigr)^{1-d}-\frac{1}{\alpha}|A_\varepsilon| \Bigl(\frac{\varepsilon}{\varrho}\Bigr)^{1-d}
|\langle \xi, \nu\rangle|
-
(C_\alpha+C'_\alpha)|\xi|\Bigl(\frac{\varepsilon}{\varrho}\Bigr)^{2-d}.
\varepsilonnd{equation}
By \varepsilonqref{beta-xi} we can estimate
\begin{equation}\label{misuraP}
\mathcal H^{d-1}(P^{\alpha,\xi})\geq \Bigl(1-\frac{\alpha}{2|\langle\xi,\nu\rangle|}|\xi-\langle\xi,\nu\rangle\nu|\Bigr)^{d-1}
\geq \Big(1-\frac{\alpha}{2}\sqrt{\frac{1}{\beta^2}-1}\Big)^{d-1}
\varepsilonnd{equation}
(see also Fig.~\ref{Fig1}), and hence, upon fixing $R>0$ and introducing the set
$$\Xi_\varepsilon^\nu(R,\beta)=
\Bigl\{\xi\in \mathbb Z^d: |\xi|\leq \frac{\varepsilonta}{\varepsilon}R, \Bigl|\langle \frac{\xi}{|\xi|},\nu \rangle\Bigr|\geq \beta\Bigr\},$$
by \varepsilonqref{stima} and \varepsilonqref{misuraP} we have
\begin{eqnarray*}
\frac{1}{\varrho^{d-1}}F_\varepsilon(u^{\varepsilon};Q_\nu^\varrho) &\geq &
\frac{1}{\varrho^{d-1}}\frac{\varepsilon^{2d}}{\varepsilonta^{d+1}}\sum_{\xi\in\Xi_\varepsilon^\nu(R,\beta)}a \Bigl(\frac{\varepsilon}{\varepsilonta}\xi\Bigr)
\mathcal H^{d-1}(P^{\alpha,\xi})|\langle\xi,\nu\rangle|\Bigl(\frac{\varepsilon}{\varrho}\Bigr)^{1-d}\\
&&-\frac{1}{\varrho^{d-1}}\frac{\varepsilon^{2d}}{\varepsilonta^{d+1}}\sum_{\xi\in\Xi_\varepsilon^\nu(R,\beta)}a \Bigl(\frac{\varepsilon}{\varepsilonta}\xi\Bigr)
\frac{1}{\alpha}|A_\varepsilon| \Bigl(\frac{\varepsilon}{\varrho}\Bigr)^{1-d}
|\langle \xi, \nu\rangle|\\
&&-
(C_\alpha+C'_\alpha)\Bigl(\frac{\varepsilon}{\varrho}\Bigr)^{2-d}
\frac{1}{\varrho^{d-1}}\frac{\varepsilon^{2d}}{\varepsilonta^{d+1}}
\sum_{\xi\in\Xi_\varepsilon^\nu(R,\beta)}a \Bigl(\frac{\varepsilon}{\varepsilonta}\xi\Bigr)|\xi|\\
&\geq&
\Big(1-\frac{\alpha}{2}\sqrt{\frac{1}{\beta^2}-1}\Big)^{d-1}
\sum_{\xi\in\Xi_\varepsilon^\nu(R,\beta)}\Bigl(\frac{\varepsilon}{\varepsilonta}\Bigr)^{d}a \Bigl(\frac{\varepsilon}{\varepsilonta}\xi\Bigr)
|\langle \frac{\varepsilon}{\varepsilonta}\xi,\nu\rangle|\\
&&-\frac{1}{\alpha}|A_\varepsilon| \sum_{\xi\in\Xi_\varepsilon^\nu(R,\beta)}\Bigl(\frac{\varepsilon}{\varepsilonta}\Bigr)^{d}a \Bigl(\frac{\varepsilon}{\varepsilonta}\xi\Bigr)
|\langle \frac{\varepsilon}{\varepsilonta}\xi,\nu\rangle|\\
&&-
(C_\alpha+C'_\alpha)
\frac{\varepsilon}{\varrho}
\sum_{\xi\in\Xi_\varepsilon^\nu(R,\beta)}\Bigl(\frac{\varepsilon}{\varepsilonta}\Bigr)^{d}a \Bigl(\frac{\varepsilon}{\varepsilonta}\xi\Bigr)\Bigl|\frac{\varepsilon}{\varepsilonta}\xi\Bigr|.
\varepsilonnd{eqnarray*}
Since $|A_\varepsilon|\to 0$ and
\begin{eqnarray*}
\lim_{\varepsilon\to 0}\sum_{\xi\in\Xi_\varepsilon^\nu(R,\beta)}\Bigl(\frac{\varepsilon}{\varepsilonta}\Bigr)^{d}a \Bigl(\frac{\varepsilon}{\varepsilonta}\xi\Bigr)
\Bigl|\langle \frac{\varepsilon}{\varepsilonta}\xi,\nu\rangle\Bigr|
&=&\int_{\{|\xi|\leq R, |\langle \xi/|\xi|, \nu\rangle|\ge \beta\}}a(\xi)|\langle \xi,\nu\rangle|\, d\xi,\\
\lim_{\varepsilon\to 0}\sum_{\xi\in\Xi_\varepsilon^\nu(R,\beta)}\Bigl(\frac{\varepsilon}{\varepsilonta}\Bigr)^{d}a \Bigl(\frac{\varepsilon}{\varepsilonta}\xi\Bigr)
\Bigl|\frac{\varepsilon}{\varepsilonta}\xi\Bigr|
&=&\int_{\{|\xi|\leq R, |\langle \xi/|\xi|, \nu\rangle|\ge \beta\}}a(\xi)| \xi|\, d\xi,
\varepsilonnd{eqnarray*}
we get\begin{eqnarray*}\liminf_{\varepsilon\to0}
\frac{1}{\varrho^{d-1}}F_\varepsilon(u^{\varepsilon};Q_\nu^\varrho) &\geq &
\Big(1-\frac{\alpha}{2}\sqrt{\frac{1}{\beta^2}-1}\Big)^{d-1}
\int_{\{|\xi|\leq R, |\langle \xi/|\xi|, \nu\rangle|\ge\beta\}}a(\xi)|\langle\xi,\nu\rangle|\, d\xi .
\varepsilonnd{eqnarray*}
Note that by \varepsilonqref{beta} we may let first $\alpha\to 0$ and then $\beta\to 0$.
We eventually obtain
\begin{eqnarray*}
\liminf_{\varepsilon\to0}\frac{1}{\varrho^{d-1}}F_\varepsilon(u^{\varepsilon};Q_\nu^\varrho) \geq
\int_{\{|\xi|\le R\}}a(\xi)|\langle\xi,\nu\rangle|\, d\xi,
\varepsilonnd{eqnarray*}
which, by the arbitrariness of $R$, gives \varepsilonqref{lower}.
\begin{figure}[h!]
\centerline{\includegraphics[width=0.8\textwidth]{Fig2bis}}
\caption{Upper-bound construction}
\label{Fig2}
\varepsilonnd{figure}
The upper bound is obtained by a density argument (see \cite{GCB} Section 1.7). Hence, it suffices to treat the case of $A$ polyhedral. In this case it suffices to take (the interpolations) $u^\varepsilon_i=\chi_A(\varepsilon i)$ for $i\in\mathbb Z^d$.
Indeed,
we write $\partial A$ as a union of $N$ $d-1$-dimensional polytopes $\Sigma_k$ and we denote by $\nu_k$ the outer normal to $\Sigma_k$ and by $K$ the $d-2$-dimensional skeleton of $A$.
We note that there exists a constant $C$ depending only on $A$ such that, for any $\varepsilonta,R>0$,
after removing the closed neighborhood
$K+\overline B_{C \varepsilonta R}$ from $\partial A$,
we obtain a disjoint collection $\widetilde \Sigma_1,\dots, \widetilde \Sigma_N$
with $\widetilde \Sigma_k\subset \Sigma_k$ such that
$$\Big(\widetilde \Sigma_k+B_{\varepsilonta R} \Big)\cap \Big(\widetilde \Sigma_{k'}+B_{\varepsilonta R}\Big)=\varepsilonmptyset
\ \ \hbox{ for any } k\neq k'$$
(see Fig.~\ref{Fig2}).
Hence, for any $\xi\in\mathbb Z^d$ with $|\varepsilon\xi|\le \varepsilonta R$, $k\in\{1,\ldots,N\}$, and
$j\in\mathbb Z^d$ such that the line $\varepsilon j+\varepsilon\xi\mathbb R$ intersects $\widetilde \Sigma_k$,
the values $u^\varepsilon_i$ change only once on the points of the discrete lines $\varepsilon j+\varepsilon \xi\mathbb Z$ which lie in a $\varepsilonta R$ neighbourhood of $
\widetilde\Sigma_k$. We note that for lines intersecting $\Sigma_k$ at a point of distance not larger than $C\varepsilonta R$ from $K$, such changes of value are at most $N$; then, repeating the counting argument used in the lower bound, we obtain
\begin{eqnarray*}
&&\limsup_{\varepsilon\to 0}\frac{\varepsilon^{2d}}{\varepsilonta^{d+1}}\sum_{\substack{\xi\in\mathbb Z^d\\ |\varepsilon\xi|\leq\varepsilonta R}} \!\!a \Bigl(\frac{\varepsilon}{\varepsilonta}\xi\Bigr)\sum_{i\in\mathbb Z^d}|u^\varepsilon_{i+\xi}-u^\varepsilon_i|\\
&&\hspace{1cm}\leq\limsup_{\varepsilon\to 0}\Biggl(\sum_{k=1}^N \mathcal H^{d-1}(\Sigma_k)\frac{\varepsilon^{d}}{\varepsilonta^{d}}\sum_{\substack{\xi\in\mathbb Z^d\\ |\varepsilon\xi|\leq\varepsilonta R}} \!\!a \Bigl(\frac{\varepsilon}{\varepsilonta}\xi\Bigr)
\Bigr|\langle \frac{\varepsilon}{\varepsilonta}\xi, \nu_k\rangle\Bigl| +O(\varepsilonta R)\Biggr)\\
&&\hspace{1cm}\leq\limsup_{\varepsilon\to 0}\sum_{k=1}^N \mathcal H^{d-1}(\Sigma_k)\int_{ \{|\xi|\leq R\}} a(\xi)
|\langle \xi, \nu_k\rangle|\, d\xi \\
&&\hspace{1cm}\leq \int_{\partial A}\varphi_a(\nu)d{\mathcal H}^{d-1}.
\varepsilonnd{eqnarray*}
Since also for $|\varepsilon\xi|\ge \varepsilonta R$ the changes of value of $u^\varepsilon_i$ are at most $N$, we then get
$$\limsup_{\varepsilon\to 0}\frac{\varepsilon^{2d}}{\varepsilonta^{d+1}}\sum_{\substack{\xi\in\mathbb Z^d\\ |\varepsilon\xi|>\varepsilonta R}} \!\!a \Bigl(\frac{\varepsilon}{\varepsilonta}\xi\Bigr)\sum_{i\in\mathbb Z^d}|u^\varepsilon_{i+\xi}-u^\varepsilon_i|\leq
N{\mathcal H}^{d-1}(\partial A)\int_{\{|\xi|>R\}}a(\xi)|\xi|d\xi.
$$
Since this term vanishes as $R\to+\infty$ the upper bound follows.
\varepsilonnd{proof}
\begin{example}\label{example}\rm
If $a$ is radially symmetric, then we have
\begin{equation}\label{fe-symm}
F(A)= \sigma {\mathcal H}^{d-1}(\partial^*A),
\varepsilonnd{equation}
where $\sigma$ is given by
\begin{equation}\label{fe-symm-s}
\sigma=\int_{\mathbb R^d}a(\xi)|\xi_1|d\xi.
\varepsilonnd{equation}
In particular, we may take $a= \chi_{B_1}$ the characteristic function of the unit ball in $\mathbb R^d$. In this case the limit of
\begin{equation}\label{fe-ball}
F_\varepsilon(u)=\frac{\varepsilon^{2d}}{\varepsilonta^{d+1}}\sum_{i,j\in\mathbb Z^d\ |i-j|<\varepsilonta/\varepsilon} |u_i-u_j|.
\varepsilonnd{equation}
is given by
\begin{equation}\label{fe-ball-s}
\sigma=\int_{B_1}|\xi_1|d\xi.
\varepsilonnd{equation}
\varepsilonnd{example}
\begin{remark}[local version]\label{remark}\rm
If $\Omega\subset\mathbb R^d$ is an open set with Lipschitz boundary we may define
\begin{equation}\label{fe-a-omega}
F_\varepsilon(u)=\frac{\varepsilon^{2d}}{\varepsilonta^{d+1}}\sum_{i,j\in\mathbb Z^d\cap{1\over\varepsilon}\Omega} a^\varepsilon_{i-j}|u_i-u_j|.
\varepsilonnd{equation}
Then the $\Gamma$-limit is
\begin{equation}\label{f-a-omega}
F(A)=\int_{\Omega\cap\partial^* A} \varphi_a(\nu)d{\mathcal H}^{d-1},
\varepsilonnd{equation}
with minor modifications in the proof.
\varepsilonnd{remark}
\noindent{\bf Acknowledgments.}
Andrea Braides acknowledges the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006. Margherita Solci acknowledges
the project ``Fondo di Ateneo per la ricerca 2019", funded by the University of Sassari. We thank the anonymous referee of \cite{BCS}, who drew our attention to the problem in this paper.
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\begin{document}
\title{A Quasi-Polynomial Approximation for the Restricted Assignment Problem\thanks{
This article is an extended joint version of conference articles \cite{DBLP:conf/soda/JansenR17, DBLP:conf/ipco/JansenR17}
\begin{abstract}
The Restricted Assignment Problem is a prominent special case of Scheduling on Parallel
Unrelated Machines.
For the strongest known linear programming relaxation, the configuration LP,
we improve the non-constructive bound
on its integrality gap from $1.9142$ to $1.8334$ and significantly simplify the proof.
Then we give a constructive variant, yielding a
$1.8334$-approximation in quasi-polynomial time.
This is the first quasi-polynomial algorithm for this problem
improving on the long-standing approximation rate of $2$.
\end{abstract}
\section{Introduction}
We consider a special case of the problem \textsc{Scheduling on Unrelated Parallel Machines},
where the goal is to compute an allocation $\sigma: \mathcal J\rightarrow \mathcal M$
of the jobs $\mathcal J$ to the machines $\mathcal M$.
On machine $i$ the job $j$ has a processing time (size) $p_{ij}$.
We want to minimize the makespan, which is the maximum load
$\max_{i\in\mathcal M}\sum_{j\in\sigma^{-1}(i)} p_{ij}$.
The classical 2-approximation by Lenstra et al.~\cite{DBLP:journals/mp/LenstraST90}
is still the algorithm of choice for this problem. They also show that
no approximation ratio better than $3/2$ can be found in polynomial time unless $\mathrm{P} = \mathrm{NP}$.
Closing this gap appears in several lists of important open questions:
Schuurman and Woeginger~\cite{schuurman1999polynomial} include it in their influential
survey on open questions scheduling and
Shmoys and Williamson in their book on approximation algorithms~\cite{DBLP:books/daglib/0030297}.
While the general problem remains unclear, there has been
progress on a special case called \textsc{Restricted Assignment}.
Here each job $j$ has a processing time $p_j$, which is independent from the
machines, and a set of feasible machines $\Gamma(j)$. This means $j$
can only be assigned to one of the machines in $\Gamma(j)$.
Note that this is equivalent to the previous problem when $p_{ij}\in\{p_j,\infty\}$.
The lower bound of $3/2$ holds also in the restricted case and
even if given quasi-polynomial running time no better approximation ratio can be obtained,
unless $\mathrm{DTIME}(2^{\mathrm{polylog}(n)}) = \mathrm{NP}$,
which would contradict popular conjectures such as the Exponential Time Hypothesis.
In a seminal work, Svensson~\cite{DBLP:journals/siamcomp/Svensson12}
proved that the configuration LP, a natural
linear programming relaxation,
has an integrality gap of at most $33/17$.
By approximating the optimum of the configuration LP this yields an $(33/17 + \epsilon)$-estimation algorithm.
However, this proof is non-constructive and no polynomial
algorithm is known that can produce a solution of this quality.
For instances with only two processing times additional progress has been made.
Chakrabarty et al. gave a polynomial $(2-\delta)$-approximation
for a very small $\delta$~\cite{DBLP:conf/soda/ChakrabartyKL15}.
Annamalai improved this with a
$(17/9 + \epsilon)$-approximation for every $\epsilon > 0$~\cite{DBLP:journals/corr/Annamalai16}.
For this special case it was also shown that the integrality gap is at
most $5/3$~\cite{DBLP:conf/swat/JansenLM16}.
In \cite{DBLP:journals/siamcomp/Svensson12} and \cite{DBLP:conf/swat/JansenLM16}
the critical idea is to design a local search algorithm,
which is then shown to produce good solutions. However, the algorithm has a potentially high
running time; hence it could only be used to prove the existence of a good solution.
For the closely related \textsc{Santa Claus} problem, in which the minimum is maximized
instead of the maximum being minimized, similar algorithms were developed~\cite{DBLP:journals/talg/AsadpourFS12}.
There, a quasi-polynomial variant by Pol{\'{a}}cek et al.~\cite{DBLP:journals/talg/PolacekS16}
and a polynomial variant by Annamalai et al.~\cite{DBLP:journals/talg/AnnamalaiKS17} were later discovered.
In this article, we start by giving a simple quasi-polynomial time $(2+\epsilon)$-approximation
in order to introduce some ideas.
We then present a simpler variant of Svensson's non-constructive algorithm, which in
addition achieves a better approximation ratio; thereby we improve the bound on the
integrality gap of the configuration LP.
Finally, we combine both approaches in a
very sophisticated $(11/6+\epsilon)$-approximation algorithm that terminates in quasi-polynomial time.
This leads to the first better-than-2 approximation algorithm
for \textsc{Restricted Assignment}, which does not need exponential running time.
The algorithm is purely combinatorial and uses the configuration-LP only in the analysis.
The last algorithm implies the results of the previous two.
Nevertheless, they are significantly less complex and already
demonstrate many of the key techniques used in the last algorithm.
\paragraph*{Comparison to related algorithms}
In Svensson's algorithm~\cite{DBLP:journals/siamcomp/Svensson12} and ours,
jobs are moved until the desired allocation is found.
The presentations of the algorithms differ significantly, but on a high level Svensson's
algorithm is closely related to the second (exponential time) algorithm we give.
Considering simplified instances with only jobs of two sizes, both algorithms and
their analysis are essentially the same.
Our exponential time algorithm is a cleaner adaption to general instances,
which is much less technical and has an better approximation ratio.
The approach for obtaining a quasi-polynomial
running time resembles that of~\cite{DBLP:journals/talg/PolacekS16}, where it was done
for the restricted \textsc{Santa Claus} problem.
In the \textsc{Restricted Assignment} problem, however,
this turns out to be significantly more challenging.
The basic idea in both algorithms is to reduce the search depth to a logarithmic
value. A fundamental structure in both cases are chains of big jobs or resources.
Big jobs have a size greater than $1/2$ times the optimal makespan.
Informally, a chain of big jobs is a sequence $j_1,i_1,j_2,i_2,j_3,i_3\dotsc$, where $j_1$ is
a big job allowed to be placed on $i_1$; $j_2$ is a big job currently assigned to $i_1$, which is
also allowed on $i_2$; $j_3$ is a big job currently assigned to $i_2$, but also allowed on $i_3$; etc.
A similar situation can arise in the restricted \textsc{Santa Claus} problem, except
that big resources (the counter-part to jobs) have a size at least the desired solution value.
It turns out that these chains play a critical role and in the \textsc{Santa Claus} problem they
have a very simple structure. This is because on a player (the counter-part to a machine)
which has one big resource we do not place any other resources.
In the \textsc{Restricted Assignment} case it is necessary to place also other jobs on
machines that have big jobs. This means that the simple operation of moving every job of a chain
to the next machine works well in the \textsc{Santa Claus} case, but in the
\textsc{Restricted Assignment} case this can result in bad machines (machines that
have too much load), if we are not careful.
Perhaps this is also a reason why at this time there is no polynomial time
better-than-$2$ approximation algorithm known for \textsc{Restricted Assignment} and
the only progress in this direction is in the case where we have only one big job size.
Note that in this case, chains are again simple.
\paragraph*{Notation}
For a set of jobs $A\subseteq\jobs$, we write $p(A)$ in place
of $\sum_{j\in A} p_j$. For other variables indexed by jobs,
we may do the same.
An allocation is a function $\sigma: \jobs\rightarrow\machines$,
where $\sigma(j)\in\Gamma(j)$ for all $j\in\jobs$.
We write $\sigma^{-1}(i)$ for the set of
all jobs $j$ which have $\sigma(j) = i$.
\paragraph*{Configuration LP}
The configuration LP has an exponential size, but can be
solved approximately in polynomial time with a rate of $(1 + \epsilon)$
for every $\epsilon > 0$~\cite{DBLP:conf/stoc/BansalS06}.
For every machine $i$ and every $\tau \ge 0$ let
\begin{equation*}
\mathcal C(i, \tau) = \{ S \subseteq \jobs : p(S) \le \tau \text{ and for all $j\in S$}, i\in\Gamma(j)\} .
\end{equation*}
These are the configurations for machine $i$ and makespan $\tau$. They are a set of
jobs that have volume at most $\tau$ and can run on machine $i$.
The optimum $\OPT^*$ of the configuration LP is the lowest $\tau$ such that the following linear
program is feasible.
\begingroup
\floatname{algorithm}{Linear Program}
\begin{algorithm}
\caption{Primal of the configuration LP}
\begin{align*}
\sum_{C\in\mathcal C(i, \tau)} x_{i, C} &\le 1 & \forall i\in\machines \\
\sum_{i\in\machines}\sum_{C\in\mathcal C(i,\tau) : j\in C} x_{i, C} &\ge 1 & \forall j\in\jobs \\
x_{i, C} &\ge 0
\end{align*}
\end{algorithm}
This linear program assigns at least one configuration to every machine and makes sure that
every job is assigned at least once. We will also construct the dual after
adding the objective $\max \ (0,\dotsc,0) \cdot x$ to the configuration LP:
\begin{algorithm}
\caption{Dual of the configuration LP}
\begin{align*}
\min \sum_{i\in\machines} y_i &- \sum_{j\in\jobs} z_j \\
\sum_{j\in C} z_j &\le y_i &\forall i\in\machines, C\in\mathcal C(i, \tau) \\
y_i, z_j &\ge 0
\end{align*}
\end{algorithm}
\endgroup
Recall, the value $\tau$ is a constant in the LP and, if the configuration LP is
infeasible with $\tau$, this means $\OPT^* > \tau$.
Furthermore, we can derive the following condition from duality.
\begin{lemma}\label{lemma:condition-ra}
Let $y\in \mathbb R_{\ge 0}^\machines$ and $z\in \mathbb R_{\ge 0}^\jobs$ such
that $\sum_{i\in\machines} y_i < \sum_{j\in\jobs} z_j$ and
for every $i\in\machines$ and $C\in\mathcal C(i, \tau)$ it holds
that $\sum_{j\in C} z_j \le y_i$, then
$\OPT^* > \tau$.
\end{lemma}
It is easy to see that if such a solution $y, z$ exists, then every component can be scaled
by a constant to obtain a feasible solution lower than any given value. Hence, the
dual must be unbounded and therefore the primal must be infeasible.
\section{Simple algorithm}
In this section we present a quasi-polynomial time $(2+\epsilon)$-approximation algorithm.
It should be noted that there do exist clean and simple polynomial time $2$-approximation algorithms
for this problem and even for the general problem of scheduling on unrelated machines.
The purpose of this section is merely to introduce some concepts
and how they can be used to get a quasi-polynomial running time.
We use a dual approximation framework where we perform a binary search over variable
$\tau\in [p_{\max},\ n\cdot p_{\max}]$, where $p_{\max} = \max_{j\in\jobs} p_j$. In each iteration we either prove that $\OPT^* > \tau$ or find an allocation
with makespan at most $(1+\epsilon)\tau$. We stop the binary search once upper and lower bound
differ by less than a factor of $(1+\epsilon)$. This gives a $(1+\epsilon)^2$-approximation
in $\log_{1+\epsilon}(n) = O(1/\epsilon\cdot\log(n))$ iterations of the binary search.
By scaling down $\epsilon$ we can get a $(1+\epsilon)$-approximation in the same asymptotic time.
The algorithmic idea for the inner method is basically a breadth-first search:
Let $\sigma: \jobs\rightarrow\machines$ be an arbitrary allocation.
We use layers $L_0,\dotsc,L_\ell$, which are disjoint sets of machines. We write $L_{\le k}$
for the union over all machines in $L_0,\dotsc,L_k$.
Further, let $\tilde\jobs(L_{\le k}, \sigma)$ denote the set of jobs $j$ with $\sigma(j)\in L_{\le k}$.
We call a machine $i$ \emph{good}, if $p(\sigma^{-1}(i)) \le 2 + \epsilon$, and \emph{bad}, otherwise.
The algorithm (see Alg.~\ref{alg:simple}) initializes $L_0$ as the bad machines. Then
the subsequent layers $L_{k+1}$ are created as the union over all $\Gamma(j)\setminus L_{\le k}$ where
$j\in\tilde\jobs(L_{\le k}, \sigma)$. If there is a machine with a load at most $(1+\epsilon)\tau$
in some layer, we move a job from a lower layer to this layer and then start from the beginning.
\begin{algorithm}
\caption{Quasi-polynomial $(2+\epsilon)$-approximation algorithm}
\label{alg:simple}
\begin{algorithmic}
\STATE{let $\sigma$ be an arbitrary allocation}
\STATE{$\ell\gets 0$}
\STATE{let $L_0$ be the set of bad machines}
\WHILE{$L_0\neq\emptyset$}
\STATE{let $L_{\ell+1}$ be the union over all $\Gamma(j)\setminus L_{\le \ell}$ for $j\in\tilde\jobs(L_{\le i}, \sigma)$}
\IF{there is an $i\in L_{\ell+1}$ with $p(\sigma^{-1}(i)) \le (1+\epsilon)\tau$}
\STATE{find a job $j$ with $\sigma(j)\in L_{\le \ell}$ and $i\in\Gamma(j)$}
\STATE{$\sigma(j)\leftarrow i$}
\STATE{delete $L_0,\dotsc,L_{\ell+1}$}
\STATE{$\ell\gets 0$}
\STATE{let $L_0$ be the new set of bad machines}
\ELSE
\STATE{$\ell\gets\ell+1$}
\IF{$\ell \ge \log_{1+\epsilon}(|\machines|)$}
\RETURN "err"
\ENDIF
\ENDIF
\ENDWHILE
\end{algorithmic}
\end{algorithm}
\paragraph*{Running time} We consider two consecutive iterations
right before some job is moved and the layers are deleted. Let $\sigma$ and $\sigma'$ be
the allocations at the earlier and at the later iteration. Likewise, let $L_{\le\ell}$ and
$L'_{\le\ell'}$ be the layers.
We show that the vector
\begin{equation*}
(b', |\tilde\jobs(L'_{\le 0}, \sigma)|,\dotsc,|\tilde\jobs(L'_{\le \ell'}, \sigma)|,-1)
\end{equation*}
is lexicographically smaller than
\begin{equation*}
(b, |\tilde\jobs(L_{\le 0}, \sigma)|,\dotsc,|\tilde\jobs(L_{\le \ell}, \sigma)|,-1) ,
\end{equation*}
where $b'$ and $b$ are the number of bad machines for $\sigma'$ and $\sigma$.
Since the number of layers is at most
$\log_{1+\epsilon}(|\machines|) = O(1/\epsilon\cdot\log(|\machines|))$ and each component is bounded
by $|\jobs|$, the overall running time is $|\jobs|^{O(1/\epsilon\cdot\log(|\machines|))}$.
For the lexicographic decrease, notice that the algorithm moves a job $j$ to a machine $i$
only when $p(\sigma^{-1}(i))\le (1+\epsilon)\tau$. Hence, after adding $j$, the load on $i$
is $p(\sigma^{-1}(i)) + p_j \le (1+\epsilon)\tau + \tau \le (2+\epsilon)\tau$.
Thus, the algorithm never turns a good machine into a bad machine and $b'\le b$.
If $b' < b$ we are done. Otherwise, $b'=b$ and
since no jobs were moved to or from $L_{\le\ell-1}$,
$L'_{\le k} = L_{\le k}$ and $\tilde\jobs(L'_{\le k}, \sigma') = \tilde\jobs(L_{\le k}, \sigma)$ for all
$k\le \min\{\ell-1, \ell'\}$.
If $\ell'\le\ell-1$, then the $\ell'+1$-th component will be $-1$ and therefore smaller than
$|\tilde\jobs(L_{\le\ell'+1},\sigma)|$. Otherwise, the $\ell$-th component will be smaller because
we moved one job away from $L_{\le\ell}$.
\paragraph*{Correctness} We have to verify that if the algorithm returns "err", then $\OPT^* > \tau$.
We will do so using Lemma~\ref{lemma:condition-ra}.
Assume that $\ell\ge \log_{1+\epsilon}(|\machines|)$ and let $\sigma$ and $L_{\le\ell}$ be the
current allocation and layer structure.
For every $i\in L_k$ define
\begin{equation*}
y_i = (1+\epsilon)^{1-k} .
\end{equation*}
Furthermore, define $y_i = (1+\epsilon)^{-\log_{1+\epsilon}(|\machines|)}$, if $i\notin L_{\le\ell}$.
For jobs $j\in\jobs$ set
\begin{equation*}
z_j = (1+\epsilon)^{-k} \cdot p_j/\tau,
\end{equation*}
where $k$ is minimal with $j\in\tilde\jobs(L_{\le k}, \sigma)$ and $z_j = 0$ if there is no
such $k$.
We need to show that $\sum_{j\in\jobs} z_j > \sum_{i\in\machines} y_i$ and for all
$i\in\machines$ and $C\in\mathcal C(i, \tau)$ it holds that $z(C)\le y_i$.
For the former, we argue
\begin{multline*}
\sum_{j\in\jobs} z_j = \sum_{i\in\machines} z(\sigma^{-1}(i))
= \sum_{k=0}^{\ell} \sum_{i\in L_k} (1+\epsilon)^{-k} \frac{p(\sigma^{-1}(i))}{\tau} \\
> (2+\epsilon) |L_0| + \sum_{k=1}^{\ell} (1+\epsilon)^{-k} (1+\epsilon) |L_k|
\ge 1 + \sum_{k=0}^{\ell} (1+\epsilon)^{1-k} |L_k| \\
= |\machines| \cdot (1+\epsilon)^{-\log_{1+\epsilon}(|\machines|)} + \sum_{k=0}^{\ell} (1+\epsilon)^{1-k} |L_k|
\ge \sum_{i\in\machines} y_i .
\end{multline*}
For the latter condition, first consider a machine $i\in L_k$ and $C\in\mathcal C(i,\tau)$. There can be no job $j\in C$
with $j\in\tilde\jobs(L_{\le k-2}, \sigma)$, since otherwise $i$ would be in an earlier layer.
Therefore, $z_j \le (1+\epsilon)^{-(k-1)} p_j$ for all $j\in C$.
Thus,
\begin{equation*}
z(C) \le (1+\epsilon)^{1-k} \frac{p(C)}{\tau} \le (1+\epsilon)^{1-k} = y_i .
\end{equation*}
Now let $i\in\machines\setminus L_{\le\ell}$ and let $C\in\mathcal C(i,\tau)$.
No job in $C$ can be in $\tilde\jobs(L_{\le \ell-1}, \sigma)$.
This means, $z_j \le (1+\epsilon)^{-\ell} p_j$ for all $j\in C$ and
\begin{equation*}
z(C) \le (1+\epsilon)^{-\ell} \frac{p(C)}{\tau} \le (1+\epsilon)^{-\log_{1+\epsilon}(|\machines|)} = y_i .
\end{equation*}
Using the lemma this implies that $\OPT^* > \tau$.
\section{Non-constructive integrality gap bound}
In this section, we give an approximation algorithm with ratio $11/6$. The algorithm
is similar to the previous one, but it adds only single moves, instead of a whole layers of reachable
machines. This leads to an exponential running time bound. Hence, the algorithm in this section
only gives a non-constructive bound on the integrality gap of the configuration LP.
\subsection{Algorithm}
Given an allocation $\sigma : \jobs \rightarrow \machines$, we call a machine $i$
\emph{bad}, if $p(\sigma^{-1}(i)) > 11/6 \cdot \tau$.
A machine is \emph{good}, if it is not bad.
We define \emph{big} jobs to be those $j\in\jobs$ that have
$p_j > 1/2 \cdot \tau$ and small jobs all others.
As the previous one, this algorithm starts with an arbitrary allocation and moves jobs until
all machines are good, or it can prove that the configuration LP is
infeasible w.r.t. $\tau$. During this process, a machine that is already good
will never be made bad.
The central data structure of the algorithm is an ordered list of pending moves $P = (P_1,P_2,\dotsc,P_\ell)$.
Here, every component $P_k = (j, i)$, $j\in\jobs$ and $i\in\Gamma(j)$, stands for
a move the algorithm wants to perform.
It will not perform the move, if this would create a bad machine,
i.e., $p(\sigma^{-1}(i)) + p_j > 11/6 \cdot \tau$.
If it does not create a bad machine, we say that the move $(j, i)$ is valid.
For every $0\le k \le \ell$ define $L_{\le k} := (L_1,\dotsc,L_k)$, the first $k$ elements of $L$ (with $L_{\le 0}$ being the empty list).
Depending on the current allocation $\sigma$ and list of pending moves $P_{\le \ell}$,
we define a binary relation $R(P_{\le \ell}, \sigma)\subseteq \jobs\times\machines$.
For a pair $(j, i)\in R(P_{\le \ell}, \sigma)$ we say
machine $i$ \emph{repels} $j$ w.r.t. $P_{\le\ell}$. This does not
mention $\sigma$ and therefore slightly abuses notation, but during
the lifetime of $P_{\le\ell}$ the allocation $\sigma$ does
not change and is always clear from the context.
The definition of repelled jobs is given later.
The algorithm will only add a new move $(j, i)$ to the current list $P$, if
$j$ is repelled by its current machine and
not repelled by the target $i$ w.r.t. $P$ (see Alg.~\ref{alg:ra}).
In the algorithm we use a lexicographic order $(p_j, j, i)$ of the moves $(j, i)$.
Here, we assume that there is an arbitrary order on jobs and machines, which is consistent throughout
the iterations.
\begin{algorithm}
\caption{Algorithm for \textsc{Restricted Assignment}}
\label{alg:ra}
\begin{algorithmic}
\STATE{let $\sigma$ be an arbitrary allocation}
\STATE{$\ell \gets 0$}
\WHILE{there is a bad machine}
\STATE{choose a move $(j, i)\notin P_{\le\ell}$, $j\in\jobs$ and $i\in\Gamma(j)$, where
$j$ is repelled by $\sigma(j)$ and not repelled by $i$ w.r.t. $P_{\le\ell}$
and $(p_j, j, i)$ is lexicographically minimal among all candidates}
\STATE{$P_{\ell+1} \leftarrow (j, i)$}
\STATE{$\ell = \ell + 1$}
\IF{$p(\sigma^{-1}(i) + p_j \le 11/6\cdot\tau$}
\STATE{$\sigma(j) \leftarrow i$}
\STATE{delete $P_{1},\dotsc,P_{\ell}$}
\STATE{$\ell \leftarrow 0$}
\ENDIF
\ENDWHILE
\end{algorithmic}
\end{algorithm}
\paragraph*{Repelled jobs}
We define the repelled jobs of each machine inductively w.r.t. $P_{\le k}$, $k=0,1,\dotsc,\ell$.
\begin{description}
\item[(initialization)] If $k=0$, let every bad machine $i$ repel every job $j$ w.r.t. $P_{\le k}$.
\item[(monotonicity)] If $i$ repels $j$ w.r.t. $L_{\le k}$,
then let $i$ repel $j$ also w.r.t. $P_{\le k+1}$.
\end{description}
The remaining rules regard $k > 0$ and we let $(j_k, i_k) := P_k$, i.e.,
the last move added.
In order to make space for $j_k$, the machine $i_k$ should repel jobs.
\begin{description}
\item[(small-all)] If $j_k$ is small, let $i_k$ repel all jobs.
\end{description}
In the case that $j_k$ is big, we need to be more careful.
It helps to imagine that the algorithm is a lazy one: It
repels jobs only if it is really necessary.
For $i\in\machines$ let $S_i(P_{\le k-1}, \sigma)$ be those small jobs $j$ which have $\sigma(j) = i$
and which are repelled by all other potential machines, i.e., $\Gamma(j)\setminus\{i\}$,
w.r.t. $P_{\le k-1}$.
The intuition behind $S_i(P_{\le k-1}, \sigma)$ is that we do not expect that
$i$ can get rid of any of these jobs.
Next, define a threshold $W_0$ as the minimum
$W \ge 0$ such that the small jobs in $S_{i_k}(P_{\le k-1}, \sigma)$ and
all big jobs below this threshold are already too large to move $j_k$, i.e.,
\begin{equation*}
p(S_{i_k}(L_{\le k-1}, \sigma)) + p(\{j\in\sigma^{-1}(i_k) : 1/2 < p_j \le W\}) + p_{j_k} > 11/6 \cdot \tau .
\end{equation*}
Furthermore, define $W_0 = \infty$ if no such $W$ exists.
In order to make $(j_k, i_k)$ valid,
it is necessary (although not always sufficient) to remove one
of the big jobs with size at most $W_0$.
Hence, we define,
\begin{description}
\item[(big-all)] if $j_k$ is big and $W_0 = \infty$, then let $i_k$ repel all jobs w.r.t $P_{\le k}$ and
\item[(big-big)] if $j_k$ is big and $W_0 < \infty$, then let $i_k$ repel $S_{i_k}(L_{\le k-1}, \sigma)$ and all jobs $j$ with $1/2 < p_j \le W_0$.
\end{description}
Note that repelling $S_{i_k}(P_{\le k-1}, \sigma)$ seems unnecessary,
since those jobs do not have any machine to go to.
However, this definition simplifies the analysis.
It is also notable that the special case where $W_0 = 0$ is equivalent to
$p(S_{i_k}(L), \sigma) + p_{j_k} > 11/6 \cdot \tau$ and here the algorithm gives up making $(j_k, i_k)$ valid.
Finally, we want to highlight the following counter-intuitive (but intentional) aspect of the algorithm.
It might happen that some job of size greater
than $W_0$ is moved to $i_k$, but has to be moved again later in order to make $(j_k, i_k)$ valid.
\subsection{Analysis}
\begin{lemma}
If the configuration LP is feasible for $\tau$ and there is a bad machine,
the algorithm always finds a move to execute.
\end{lemma}
\begin{proof}
Suppose toward contradiction, there is a bad machine, no move in $P_{\le\ell}$ is valid and
no move can be added to $P_{\le\ell}$.
We will construct values $(z_j)_{j\in\jobs}$, $(y_i)_{i\in\machines}$ with the properties as in Lemma~\ref{lemma:condition-ra} and
thereby show that the configuration LP is infeasible.
During this proof, $\sigma$ and $P_{\le\ell}$ are constant and refer to the state at which the algorithm
is stuck. We omit $P_{\le\ell}$ when we say $i$ repels $j$.
Let $\tilde\jobs_i$ denote all jobs $j\in\sigma^{-1}(i)$
that are repelled by $i$.
We write $\tilde\jobs=\bigcup_{i\in\machines}\tilde\jobs_i$.
For every $j\in\jobs$ let
\begin{equation*}
z_j = \begin{cases}
\min\left\{\frac{p_j}{\tau}, \frac 5 6 \right\} &\text{if $j\in\tilde\jobs$ and} \\
z_j = 0 &\text{otherwise}.
\end{cases}
\end{equation*}
Let $y_i := 1$ if $i\in\machines$ repels all jobs and $y_i = z(\sigma^{-1}(i))$ otherwise.
Let $i\in\machines$ and
$C\in\mathcal C(i, \tau)$.
We need to show that $z(C) \le y_i$.
If $y_i = 1$ this follows immediately, because $z(C) \le p(C) / \tau \le 1$.
We assume w.l.o.g. that $i$ does not repel all jobs
and thus $y_i = z(\sigma^{-1}(i))$.
In particular, $i$ does not repel small jobs that are on other machines.
This means that $z_j = 0$ for every small job $j\in C\setminus\sigma^{-1}(j)$.
Otherwise, $(j, i)$ could be added to $P_{\le\ell}$.
If there are no big jobs in $C$, we therefore
get $z(C) \le z(\sigma^{-1}(i)) = y_i$.
Clearly, there can be at most one big job $j_B\in C$,
since such a job has $p_{j_B} > 1/2 \cdot \tau$ and $C$ cannot have a load greater than $\tau$.
If $z_{j_B} = 0$ or $\sigma(j_B) = i$ the argument above still holds.
We recap: The only interesting case is when $y_i = z(\sigma^{-1}(i))$, there is exactly one big job $j_B \in C\setminus\sigma^{-1}(i)$, and $z_{j_B} = \min\{p_{j_B}/\tau, 5/6\}$.
\begin{description}
\item[Case 1: $i$ repels $j_B$.]
Since $i$ is not the target of a small job move and it is good (otherwise it would repel all jobs),
there must be a big job move that causes $i$ to repel $j_B$.
In other words, there is a move $(j_k, i)=P_k$ such that $j_B\le W_0$, where $W_0$ is as in the definition of
repelled jobs for $P_{\le k}$.
Recall that $W_0 < \infty$ is the minimal $W$ with
\begin{equation*}
p(S_i(P_{\le k - 1}, \sigma)) + p(\{j\in\sigma^{-1}(i) : 1/2 < p_j \le W\}) + p_{j_k} > 11/6 \cdot \tau .
\end{equation*}
Since $W_0 \ge p_{j_B}$ and it is minimal,
there must be a big job $j_B'\in\sigma^{-1}(i)$ with
$p_{j_B'} = W_0 \ge p_{j_B}$ and $j_B'$ is also repelled by $i$ (because $p_{j'_B} \le W_0$).
We get
\begin{equation*}
z(C) \le z(C\setminus \{j_B\}) + z_{j_B} \le z(\sigma^{-1}(i)\setminus\{j_B'\}) + z_{j_B'} = z(\sigma^{-1}(i)) = y_i.
\end{equation*}
\item[Case 2: $i$ does not repel $j_B$.]
Since $(j_B, i)$ cannot be added to $P$, it must already be in $P$. Let $P_k = (j_B, i)$
and $W_0$ as in the definition of repelled edges w.r.t. $P_{\le k}$.
Then
\begin{equation*}
p(S_i(P_{\le k-1},\sigma)) + p(\{j\in\sigma^{-1}(i) : 1/2 < p_j \le W_0\}) + p_{j_B} > 11/6 \cdot \tau .
\end{equation*}
If $W_0\ge 5/6$, then there is some $j'_B\in\sigma^{-1}(i)$ with $p_{j'_B} = W_0 \ge 5/6$.
Similar to the previous case, it follows that
\begin{equation*}
z(C) \le z(C\setminus \{j_B\}) + z_{j_B} \le z(\sigma^{-1}(i)\setminus\{j'_B\}) + 5/6 = z(\sigma^{-1}(i)) = y_i.
\end{equation*}
If $W_0\le 5/6$, then all of the considered jobs have $z_j = p_j/\tau$, i.e.,
\begin{align*}
y_i &= z(\sigma^{-1}(i)) \\
&\ge z(S_i(P_{\le k-1},\sigma)) + z(\{j\in\sigma^{-1}(i) : 1/2 < p_j \le W_0\}) \\
&= p(S_i(P_{\le k-1},\sigma))/\tau + p(\{j\in\sigma^{-1}(i) : 1/2 < p_j \le W_0\})/\tau \\
&> 11/6 - p_{j_B}/\tau \ge 5/6 + (\tau-p_{j_B})/\tau \ge z(C) .
\end{align*}
The last inequality holds, because the $z_{j_B}$ is at most $5/6$ and the volume of the small jobs in $C$
(in particular, their value) is at most $(\tau-p_{j_B})/\tau$.
\end{description}
It remains to show that $\sum_{j\in\jobs} z_j > \sum_{i\in\machines} y_i$. We prove that, with amortization, good machines
satisfy $z(\sigma^{-1}(i)) \ge y_i$ and on bad machines
strict inequality holds.
Let $i$ be a bad machine.
Then $i$ repels all jobs (in particular, those in $\sigma^{-1}(i)$).
Hence,
\begin{equation*}
z(\sigma^{-1}(i)) \ge 5/6 \cdot p(\sigma^{-1}(i)) / \tau > 55/36 > 1 = y_i .
\end{equation*}
For good machines that do not repel all jobs, equality holds by definition.
We will partition those good machines that do repel all jobs
into those $i\in\machines$ which have $(j_S, i) \in P_{\le\ell}$ for a small
job $j_S$ and those that do not.
\begin{lemma}
\label{min-max-ratio}
At least half of the machines $i$ that repel all jobs are target of a small job, i.e.,
$(j_S, i)\in P_{\le\ell}$ for some small job $j_S$.
\end{lemma}
We argue that whenever a move $(j_B, i)=P_k$ of a big job $j_B$ is added, it is not valid, and
$W_0 = \infty$ in the definition of repelled jobs w.r.t. $P_{\le k}$, then there
is some small job move $(j_S, i')$ that can be added to $P$. Since the algorithm
prefers small job moves over big ones, the next move after $P_k$ will necessarily be a small job move.
Since no two small job moves can be added for the same target, the lemma follows.
If $W_0 = \infty$, this means that
\begin{equation*}
p(S_i(L_{\le k-1},\sigma)) + p(\{j\in\sigma^{-1}(i) : p_j > 1/2\}) + p_{j_B} \le 11/6\cdot\tau .
\end{equation*}
Since the move $(j_B, i)$ is not valid, however, we also have that
\begin{equation*}
p(\sigma^{-1}(i)) + p_{j_B} > 11/6\cdot\tau .
\end{equation*}
This implies that there must be a small job $j_S\in\sigma^{-1}(i)\setminus S_i(L_{\le k-1}\sigma)$.
In particular, there exists some $i'\in\Gamma(j_S)\setminus\{i\}$ by which $j_S$ is not repelled.
It is also not repelled by $i'$ w.r.t. $P_{\le k}$, since $P_k$ only adds
repelled jobs for $i$. Therefore, $(j_S, i')$ is a candidate for the next move to be added after $P_{\le k}$.
This concludes the proof for the lemma.
Let $i$ be a machine that repels all jobs, but is not target of a small job.
Then there is a big job $j_B$ with $(j_B, i) \in P_{\le\ell}$ and this move is not valid.
Either there is a job $j\in\sigma^{-1}(i)$ with $z_j = 5/6$ or $z_j = p_j / \tau$ for all $j\in\sigma^{-1}(i)$. Thus,
\begin{equation*}
z(\sigma^{-1}(i)) \ge \min\left\{\frac 5 6,\ \frac{p(\sigma^{-1}(i))}{\tau}\right\}
\ge \min\left\{\frac 5 6,\ \frac{11/6 - p_{j_B}}{\tau}\right\}
\ge \frac 5 6 = y_i - \frac 1 6 .
\end{equation*}
Next, let $i$ be a machine such that there exists a small job $j_S$ with $(j_S, i) \in P_{\le\ell}$.
This move is also not valid.
In the following, we distinguish between the cases where $\sigma^{-1}(i)$ has no job
$j$ with $z_j = 5/6$, one such job, or at least two. Note that all jobs have $p_j \le \tau$.
\begin{multline*}
z(\sigma^{-1}(i)) \ge \min\left\{\frac{p(\sigma^{-1}(i))}{\tau},\ \frac{p(\sigma^{-1}(i)) - \tau}{\tau} + \frac 5 6,\ \frac{10}{6}\right\} \\
\ge \min\left\{\frac{11}{6} - \frac{p_{j_S}}{\tau} - 1 + \frac 5 6,\ \frac{10}{6} \right\}
\ge \frac 7 6 = y_i + \frac 1 6 .
\end{multline*}
Because of Lemma~\ref{min-max-ratio}, we can amortize and get
\begin{equation*}
\sum_{j\in\jobs} z_j = \sum_{i\in\machines}z(\sigma^{-1}(i)) > \sum_{i\in\machines} y_i . \qedhere
\end{equation*}
\end{proof}
\begin{lemma}
The algorithm terminates.
\end{lemma}
\begin{proof}
Consider two consecutive iterations of the main loop right before a move is executed and
the list of moves $P$ is deleted.
Let $\sigma$, $P_{\le\ell}$ be the allocation and list of pending moves in the former iteration
and $\sigma'$, $P'_{\le\ell'}$ in the latter.
Let $b$ and $b'$ be the number of bad machines in $\sigma$ and $\sigma'$.
Recall, $R(P_{\le k}, \sigma)\subseteq \jobs\times\machines$ is the set of all $i, j$
where $j$ is repelled by $i$ w.r.t. $P_{\le k}$.
Further, let $\tilde\jobs(P_{\le k}, \sigma)$ denote all jobs $j$
that are repelled by $\sigma(j)$ w.r.t. $P_{\le k}$.
For each prefix of $P$ (and of $P'$)
we define a potential function
\begin{equation*}
\Phi(P_{\le k}, \sigma) = (p_{j_k}, j_k, i_k, |\jobs\times\machines| - |R(P_{\le k}, \sigma)|, |\tilde\jobs(P_{\le k}, \sigma)|)
\end{equation*}
We claim that the vector
\begin{equation*}
(b', |\tilde\jobs(P'_{\le 0}, \sigma')|, \Phi(P'_{\le 0}, \sigma'), \dotsc, \Phi(P'_{\le \ell'}, \sigma'), -1)
\end{equation*}
is lexicographically smaller than
\begin{equation*}
(b, |\tilde\jobs(P_{\le 0}, \sigma)|, \Phi(P_{\le 0}, \sigma), \dotsc, \Phi(P_{\le \ell}, \sigma), -1) .
\end{equation*}
Note that the length of the vector is bounded by $5\cdot |\machines|\cdot |\jobs| + 3$,
since no move appears twice in the list and every component can have at most
$|\jobs| \cdot |\machines| + 1$ different values. Thus, the number of possible vectors
is finite and hence the algorithm terminates.
Recall that the algorithm never turns a good machine bad, which means $b'\le b$.
If $b' < b$, we are done. Likewise, if $b=b'$ and some job is moved
from a bad machine to a good machine,
then $|\tilde\jobs(P'_{\le 0}, \sigma')| < |\tilde\jobs(P_{\le 0}, \sigma)|$
and again the first vector is lexicographically smaller. We can
therefore focus on the case $b' = b$,
$\tilde\jobs(P'_{\le 0}, \sigma') = \tilde\jobs(P_{\le 0}, \sigma)$,
and $\sigma(j) = \sigma'(j)$ for all $j\in\tilde\jobs(P_{\le 0}, \sigma)$.
The rest of the argument is by induction. Let $k\le\min\{\ell-1,\ell'\}$ and assume that
\begin{enumerate}
\item $P'_{\le k-1} = P_{\le k-1}$,
\item $R(P'_{\le k-1}, \sigma') = R(P_{\le k-1}, \sigma)$.
\item $\tilde\jobs(P'_{\le k-1}, \sigma') = \tilde\jobs(P_{\le k-1}, \sigma)$;
$\sigma'(j) = \sigma(j)$ for all $j\in\tilde\jobs(P_{\le k-1}, \sigma)$,
\end{enumerate}
We will show that $\Phi(L'_{\le k}) \le \Phi(L_{\le k})$ (lexicographically) and
if equality holds, then (1), (2), and (3) also hold for $k$.
This implies the lexicographical decrease:
If $\ell' < \ell$ it follows easily. This is because the prefix of the first
vector ending in $\Phi(P'_{\le \ell'})$ is lexicographically not bigger than
the prefix of the second vector ending in $\Phi(P_{\le \ell'})$. Furthermore,
the next component is $-1$ in
the first vector, but something non-negative in the other.
Now consider the case $\ell'\ge \ell$.
Let $(j_\ell, i_\ell)=P_\ell$ be the
move that was executed. Then $\sigma'(j_\ell) = i_\ell \neq \sigma(j_\ell)$.
Furthermore, $j_\ell$ is repelled by $\sigma(j_\ell)$ w.r.t. $L_{\le \ell-1}$.
Hence, (2) cannot hold with $k-1=\ell-1$ and thus
we cannot have $\Phi(L'_{\le k}) = \Phi(L_{\le k})$ for all $k$.
The approach for the induction is to show that if $\Phi(L'_{\le k}, \sigma')$ is not smaller
than $\Phi(L_{\le k}, \sigma)$, $(j_k, i_k) = P_k$ also has to be selected
as $P'_k$. In that case, no job can have been moved to $i_k$, if it is
repelled by $i_k$ w.r.t. $L_{\le k}$. From the way they are chosen,
this implies the jobs which $i_k$ repels as a consequence of $P_k$ are also repelled in
the rules for $P'_k$.
If they do not increase, the number of jobs $j$ with $\sigma(j)$ that are repelled by $i_k$
cannot increase. Let us now formalize this argument.
Notice that $(j_\ell, i_\ell) = P_\ell$ is the move that
was executed, i.e., $\sigma'(j_\ell) = i_\ell$
and by construction of $P_{\le\ell}$,
$j_\ell$ is not repelled by $i_\ell$ w.r.t. $P_{\le \ell-1}$
(in particular, not w.r.t. $P_{\le k}$ $(*)$).
Let $(j_k, i_k) = P_k$. By (1) we have that $(j_k, i_k)\notin P_{\le k-1} = P'_{\le k-1}$. Since $j_k$ is repelled by $\sigma(j_k)$ w.r.t. $L_{\le k}$, by (2) we have that
$\sigma'(j_k) = \sigma(j_k)$. By (3) it is not repelled by $i_k$ w.r.t. $L'_{\le k-1}$.
Therefore, $(j_k, i_k)$ is was a candidate for $P'_k$.
Either this or a move $(j, i)$, where $(p_j, j, i)$ is lexicographically smaller
than $(p_{j_k}, j_k, i_k)$ is chosen. In the latter case we have
$\Phi(L'_{\le k}) < \Phi(L_{\le k})$. Hence, assume that $P'_k = (j_k, i_k)$.
This means (1) holds for $k$.
Note that since (2) and (3) hold for $k-1$, we only have to check the consequences of $P'_k$ and $P_k$.
In other words, we have to check whether the rules for move $P'_k$ imply the same repelled jobs as $P_k$
and whether some job repelled due to this rule has been moved.
If $j_k$ is small, then $i_k$ repels all jobs w.r.t. $L'_{\le k}$ and $L_{\le k}$
and therefore (3) also holds for $k$. Job $j_\ell$ cannot have been moved to
$i_k$ (see $(*)$). If it was moved away from $i_k$, then
$\tilde\jobs(P'_{\le k}, \sigma') \subsetneq \tilde\jobs(P_{\le k-1}, \sigma)$.
Otherwise, equality holds and no job was moved in this set, i.e., (2) holds for $k$.
Now assume $j_k$ is big.
First, we argue that $S'_{i_k}(L'_{\le k-1}, \sigma') = S_{i_k}(L_{\le k-1}, \sigma)$.
Recall, $S_{i_k}(L_{\le k-1}, \sigma)$ are the small jobs $j$
with $\sigma(j) = i_k$ and $j$ is repelled by
all machines in $\Gamma(j)\setminus\{i_k\}$ w.r.t. $L_{\le k}$.
Let $j\in S'_{i_k}(L'_{\le k-1}, \sigma')$.
Assume toward contradiction that
$\sigma(j)\neq i_k = \sigma'(j)$. Then $j=j_\ell$ and $i_k=i_\ell$.
However, $\sigma(j_\ell)$ does not repel $j_\ell$ w.r.t. $P_{\le k-1}$. Otherwise
$(j_\ell, i_\ell)$ would have been chosen instead of $(j_k, i_k)$ as $P_k$.
By (3) $\sigma(j_\ell)$ also does not repel $j_\ell$ w.r.t. $P'_{\le k-1}$.
Hence, $j\notin S'_{i_k}(L'_{\le k-1},\sigma')$, a contradiction.
Consequently, $\sigma(j) = i_k$. By (3) it follows that $j\in S_{i_k}(L_{\le k-1},\sigma)$.
Let $j\in S_{i_k}(L_{\le k-1},\sigma)$. Since it is repelled by all potential machines w.r.t.
$P_{\le k-1}$, there cannot be a move for $j$ in a later layer. In particular $j_\ell$ cannot
be $j$. This means $\sigma'(j) = \sigma(j) = i_k$ and by (3)
$j\in S_{i_k}(L_{\le k-1})$.
We conclude that $S_{i_k}(L_{\le k-1})=S'_{i_k}(L'_{\le k-1})$.
Let $W_0$ be the minimal the minimal $W\ge 0$ with
\begin{equation*}
\underbrace{p(S_{i_k}(L_{\le k-1}, \sigma))}_{=p(S'_{i_k}(L'_{\le k-1},\sigma'))} + p(\{j\in\sigma^{-1}(i) : 1/2 < p_j \le W\}) + p_{j_k} > 11/6\cdot \tau ,
\end{equation*}
or $\infty$ if no such $W$ exists.
Since all jobs $j$ with $1/2 < p_j \le W_0$ are repelled by $i_k$ w.r.t. $P_{\le k}$,
it follows by $(*)$ that
\begin{equation*}
p(\{j\in\sigma^{\prime-1}(i) : 1/2 < p_j \le W\}) \le
p(\{j\in\sigma^{-1}(i) : 1/2 < p_j \le W\})
\end{equation*}
It follows that $W'_0$, the minimal $W\ge 0$ with
\begin{equation*}
p(S'_{i_k}(L'_{\le k-1},\sigma')) + p(\{j\in\sigma^{\prime-1}(i) : 1/2 < p_j \le W\}) + p_{j_k} > 11/6\cdot \tau ,
\end{equation*}
is at least as big as $W_0$. In particular, if $W_0 = \infty$, then $i_k$ repels all jobs w.r.t. $P_{\le k}$
and w.r.t. $P'_{\le k}$.
Otherwise, by definition $i_k$ repels all jobs in $S_i(L_{\le k-1},\sigma)$ and all
jobs $j$ with $1/2 < p_j \le W_0$ w.r.t. $L_{\le k}$.
By $W'_0 \ge W_0$ these jobs are also repelled by $i_k$ w.r.t. $L'_{\le k}$.
Hence $R(P'_{\le k}, \sigma') \supseteq R(P_{\le k}, \sigma)$.
If equality holds then because of $(*)$,
$\tilde\jobs(P'_{\le k}, \sigma') \supseteq \tilde\jobs(P_{\le k}, \sigma)$.
If equality also holds here, then none of these jobs are moved and (1), (2), and (3) hold.
If equality does not hold at some point, $\Phi(L'_{\le k-1},\sigma') < \Phi(L_{\le k-1},\sigma)$.
\end{proof}
\begin{theorem}\label{th-ra}
The integrality gap of the configuration LP for \textsc{Restricted Assignment}
is at most $11/6$.
\end{theorem}
\section{Quasi-polynomial time algorithm}
The previous running time bound is clearly exponential.
In this section, we will improve the running time to
$n^{O(1/\epsilon \cdot \log(n))}$ for an approximation rate of
$11/6 + 2\epsilon$,
where $n = |\machines| + |\jobs|$ and $\epsilon > 0$ can be chosen arbitrarily. Note that by scaling $\epsilon$, the coefficient of $2$ can be removed.
\subsection{Algorithm}
Our approach for turning the running time quasi-polynomial
is to combine the two algorithms presented in the previous sections.
Instead of adding only one candidate move at a time like in the exponential time algorithm,
we add them all. The set of these moves is called a layer. After a layer
is added, re-evaluate the repelled jobs and again construct a layer of all sensible moves.
This simple approach described has some major issues that we have to handle carefully
using more sophisticated techniques as described in the following.
First, however, we make some technical preparations.
We will call a machine $i$ \emph{bad}, if $p(\sigma^{-1}(i)) > (11/6 + 2\epsilon)\tau$ and \emph{good},
otherwise.
As before, we call jobs $j$ small, if $p_j \le 1/2 \cdot\tau$ and big otherwise.
Further, we distinguish big jobs into medium, which have $p_j\le 5/6\cdot\tau$, and huge,
which have $p_j > 5/6\cdot\tau$.
Unlike in the previous algorithms we establish the invariant
that at all times the current allocation
assigns at most one huge job to each machine.
An initial allocation that satisfies this can easily be found
via bipartite matching with the huge jobs
on one side and the machines on the other.
It is maintained by the following definition of a valid huge job move.
\begin{definition}[Valid huge move]
Let $j\in\jobs, i\in\Gamma(j)\setminus\{\sigma(j)\}$ with $p_j > 5/6\cdot\tau$. $(j, i)$ is a valid move,
if $p(\sigma^{-1}(i)) + p_j \le (11/6 + 2\epsilon)\tau$ and there is no huge job
in $\sigma^{-1}(i)$.
\end{definition}
For the remaining jobs we define the moves as follows.
\begin{definition}[Valid non-huge move]
Let $j\in\jobs, i\in\Gamma(j)\setminus\{\sigma(j)\}$ with $p_j \le 5/6\cdot \tau$. $(j, i)$ is a valid move, if
\begin{enumerate}
\item $p(\sigma^{-1}(i)) + p_j \le (11/6 + 2\epsilon)\tau$ and $\sigma^{-1}(i)$ contains no huge job, or
\item $p(\{j'\in \sigma^{-1}(i) : p_j \le 5/6\cdot \tau\}) + p_j \le (5/6 + 2\epsilon)\tau$ and $\sigma^{-1}(i)$ contains one huge job.\label{small-huge}
\end{enumerate}
\end{definition}
Each valid move $(j, i)$ satisfies $p(\sigma^{-1}(i)) + p_j \le (11/6 + 2\epsilon)\tau$,
i.e., each good machine stays good.
(\ref{small-huge}) needs further elaboration.
One could falsely assume that this establishes
an invariant which says the non-huge load is at most $(5/6+2\epsilon)\cdot\tau$ on a machine
with a huge job.
This would be a marvelous invariant, if it could be guaranteed.
However, a valid huge move can break this property.
Therefore, (\ref{small-huge}) only gives something weaker. It's purpose is that
when a machine has a huge job and a low non-huge load, then it will
stay this way for as long as the huge job remains on the machine.
This is to keep edges in the leap graph intact, a technique that will be elaborated later.
\paragraph*{Layers}
We will operate in layers $L_1, \dotsc, L_{\ell}$ like in the first algorithm.
These, however, will not only contain machines, but also fine grained moves
like in the second algorithm.
Again, we define a binary relation $R(L_{\le k}, \sigma)\subseteq \jobs\times\machines$,
which states that $i$ repels $j$ w.r.t. $L_{\le k}$, if $(j, i)\in R(L_{\le k}, \sigma)$.
In the previous algorithms,
the local search is mostly stateless, i.e., it searches for an
improvement of $\sigma$ without remembering anything from the past.
Here we make a small exception. We maintain an order $\pi$ on $\jobs\times\machines$.
When the algorithm adds certain critical moves, they are moved to the front of $\pi$.
This will hint at the algorithm to do the same in the next iteration.
It helps to argue about the running time, since the layers created this way
are more consistent throughout the iterations.
Finally, layers come in different forms.
They can be leap layers, critical layers, small layers, or non-critical layers.
This will become more clear in the actual description of the algorithm.
\paragraph*{Leaps}
A straight-forward example shows that if we are using only simple moves like in the previous
two algorithms, the number of layers needs to grow linearly.
This would be a problem for obtaining a quasi-polynomial running time.
\begin{figure}
\caption{Example for linear number of layers}
\label{fig:lin}
\end{figure}
In the example (see Fig.~\ref{fig:lin}) the leftmost machine has two jobs $j_1, j'_1$
of size $\tau$ assigned to it,
which make the machine bad. The jobs each have a chain of machines connected to it:
$j_1$ can go to a machine $i_1\in\Gamma(j_1)$. On $i_1$ there is another job $j_2$
of size $\tau$ which can go to a machine $i_2\in\Gamma(j_2)$, etc.
At some point this chain ends with an empty machine. The same construction is made
for $j'_1$.
In order to make the bad machine good, either the top chain of jobs or the bottom chain
has to be traversed. Hence, it seems like
the number of layers would be roughly half the jobs or machines.
It turns out that this problem only occurs with huge jobs
and we will carefully circumvent it.
The leap technique is intended for moving such a chain of huge jobs at once.
For an easy description we construct a directed bipartite graph $G(\sigma) = (V, E(\sigma))$
where $V = B\cup\machines$, i.e., the vertices are big jobs $B$ and the machines.
There is an edge $(j_B, i)\in E(\sigma)$ if $i\in\Gamma(j_B)\setminus\{\sigma(j_B)\}$ and
\begin{equation*}
p(\{j\in\sigma^{-1}(i) : p_{j} \le 5/6\}) + p_{j_B} \le (11/6 + 2\epsilon) \tau ,
\end{equation*}
i.e., if there is no huge job in $\sigma^{-1}(i)$ the move $(j_B, i)$ is
valid. Furthermore, we let $(i, j_H)\in E(\sigma)$, if $j_H$ is huge and $i = \sigma(j_H)$.
Suppose that there is some path $j_1, i_1, j_2, i_2,\dotsc, j_k, i_k$
in $G$, where no huge job is assigned to $i_k$.
Then we can move $j_1$ to $i_1$,
$j_2$ from $i_1$ to $i_2$, $j_3$ from $i_2$ to $i_3$, etc.
We will call this a \emph{leap}.
The general theme for using this graph is the following.
A big job $j_B$ is repelled by $\sigma(j_B)$. Then all machines that
are reachable by some path from $j_B$ should repel their huge jobs as well. When
one of them is removed, we can instantly free $i$ from $j_B$.
This way we are not going to put all moves of the path sequentially into the layers and avoid
making it unnecessarily long.
\begin{definition}[Valid leap]
Let $j_1,i_1,\dotsc,j_r,i_r$ be a path in the leap graph.
It is called valid leap, if $(j_r,i_r)$ is a valid move.
\end{definition}
By definition of the leap graph and the fact that every machine has at most one huge job,
for a valid leap the following moves are all valid if executed in reverse order, i.e.,
$(j_r, i_r), (j_{r-1}, i_{r-1}),\dots, (j_1,i_1)$.
Finally, we define a graph $G(L_{\le k}, \sigma)$ which has all edges from $G(\sigma)$
of the form $(i, j_H)$, but only the edges $(j_B, i)$ where $i$ does not repel $j_B$ w.r.t.
$L_{\le k}$. The definition of repelled edges will be given later.
\paragraph{Description of the algorithm}
We are now ready to state the algorithm (see Alg.~\ref{alg:ra2}).
It starts with an allocation $\sigma$ with the property that each machine
has at most one huge job assigned to it. The allocation of huge jobs
can be found using bipartite matching and the remaining jobs are assigned arbitrarily.
We initialize $\pi$ as an arbitrary permutation of $\jobs\times\machines$.
Until all machines are good the algorithm searches for valid moves or leaps that
improve $\sigma$. For this purpose we build layers. The layers are alternating
between leap layers, critical layers, small layers, and non-critical layers:
Layer $L_{4k+1}$ is always a leap layer; $L_{4k+2}$ is a critical layer;
$L_{4k+3}$ is a small layer; $L_{4k+3}$ is a non-critical layer.
A leap-layer $L_{\ell+1}$ consists of the machines that are reachable in the leap graph
$G(L_{\le\ell},\sigma)$ by a job that is repelled by its current machine w.r.t. $L_{\le\ell}$.
For the critical layer $L_{\ell+2}$ and non-critical layer $L_{\ell+4}$,
we select all $(j_B, i)$ where $j_B$ is a big job repelled by
$\sigma(j_B)$, but not by $i$ w.r.t. $L_{\le \ell+1}$. A subset of these
is taken in $L_{\ell+2}$.
We will define below precisely how they are chosen, but they depend on $\pi$
giving priority to the moves in the front.
After they are selected, the critical moves are pushed to the front of $\pi$.
All moves $(j_S, i)$, where $j_S$ is repelled by $\sigma(j_S)$, but not by $i$ w.r.t. $L_{\le \ell+2}$
are put in $L_{\ell+3}$.
Finally, the previously considered big job moves $(j_B, i)$ which were not taken in $L_{\ell+2}$
and where $i$ still does not repel $j_B$ (now w.r.t. $L_{\le\ell+3}$) are taken in
the non-critical layer $L_{\ell+4}$.
If at any point a valid leap or move is found, it is executed and the structure of
layers is reset. Note that the meaning of
the \emph{continue} statement in the pseudo-code is to
jump to the next iteration of the while loop.
\begin{algorithm}
\caption{Constructive algorithm for \textsc{Restricted Assignment}}
\label{alg:ra2}
\begin{algorithmic}
\STATE{let $\sigma$ be an allocation with at most one huge job on each machine}
\STATE{let $\pi$ be an arbitrary order on $\jobs\times\machines$}
\STATE{$\ell \gets 0$}
\WHILE{there is a bad machine}
\IF{$\ell \ge 4\lceil\log_{1+\epsilon}(4|\machines|)\rceil = O(1/\epsilon \cdot \log(|\machines|))$}
\RETURN "err"
\ENDIF
\STATE{let $L^L_{\mathrm{new}}$ be the set of machines reachable in the
leap graph $G(L_{\le\ell},\sigma)$
by a big job $j$ repelled by $\sigma(j)$ w.r.t. $L_{\le\ell}$}
\STATE{$L_{\ell+1}\gets (L^L_{\mathrm{new}}, \text{\textsc{leap}})$}
\IF{there is a machine $i$ in $L_{\ell+1}$ with no huge job in $\sigma^{-1}(i)$}
\STATE{let $j_1,i_1,\dotsc,j_r,i_r=i$ be a path in $G(L_{\le\ell},\sigma)$, where $j_1$ is repelled by $\sigma^{-1}(j_r)$ w.r.t. $L_{\le \ell}$}
\STATE{$\sigma(j_r)\gets i_r,\dotsc, \sigma(j_1)\gets i_1$}
\STATE{delete $L_{1},L_{2},\dotsc,L_{\ell+1}$}
\STATE{$\ell \gets 0$}
\STATE{continue}
\ENDIF
\STATE{let $L^B_{\mathrm{new}}$ be the set of all $(j_B, i)$, $j_B\in\jobs$ big and $i\in\Gamma(j_B)$, with
$j_B$ repelled by $\sigma(j_B)$ and not by $i$ w.r.t. $L_{\le \ell+1}$}
\STATE{$L^C_{\mathrm{new}}\gets \mathrm{CriticalMoves}(L^B_{\mathrm{new}}, \sigma, L_{\le\ell+1}, \pi)$}
\STATE{$L_{\ell+2}\gets (L^C_{\mathrm{new}}, \text{\textsc{critical}})$}
\STATE{move $L^C_{\mathrm{new}}$ to the front of $\pi$ (keeping their pairwise order)}
\IF{there exists a valid move $(j, i)$ in $L_{\ell+2}$}
\STATE{$\sigma(j) \leftarrow i$; delete $L_{1},L_{2},\dotsc,L_{\ell+2}$; $\ell \gets 0$}
\STATE{continue}
\ENDIF
\STATE{let $L^S_{\mathrm{new}}$ be the set of all $(j_S, i)$, $j_S\in\jobs$ small and $i\in\Gamma(j_S)$, with
$j_S$ repelled by $\sigma(j_S)$ and not by $i$ w.r.t. $L_{\le \ell+2}$}
\STATE{$L_{\ell+3}\gets (L^S_{\mathrm{new}}, \text{\textsc{small}})$}
\IF{there exists a valid move $(j, i)$ in $L_{\ell+3}$}
\STATE{$\sigma(j) \leftarrow i$; delete $L_{1},L_{2},\dotsc,L_{\ell+3}$; $\ell \gets 0$}
\STATE{continue}
\ENDIF
\STATE{let $L^{NC}_{\mathrm{new}}$ be the set of all $(j, i)\in L^B_{\mathrm{new}}\setminus L^C_{\mathrm{new}}$ where $i$ does not repel $j$ w.r.t. $L_{\le\ell+3}$}
\STATE{$L_{\ell+4}\gets (L^{NC}_{\mathrm{new}}, \text{\textsc{non-critical}})$}
\IF{there exists a valid move $(j, i)$ in $L_{\ell+4}$}
\STATE{$\sigma(j) \leftarrow i$; delete $L_{1},L_{2},\dotsc,L_{\ell+4}$; $\ell \gets 0$}
\STATE{continue}
\ENDIF
\STATE{$\ell\gets\ell+4$}
\ENDWHILE
\RETURN $\sigma$
\end{algorithmic}
\end{algorithm}
\paragraph*{Repelled jobs}
We define the repelled jobs of each machine inductively w.r.t. $L_{\le k}$, $k=0,1,\dotsc,\ell$.
\begin{description}
\item[(initialization)] Let the bad machine repel every job w.r.t. $L_{\le 0}$.
\item[(monotonicity)] If $i$ repels $j$ w.r.t. $L_{\le k}$,
then let $i$ repel $j$ also w.r.t. $L_{\le k + 1}$.
\end{description}
The remaining rules regard $k > 0$ and we define repelled jobs for a layer $L_k$.
A layer $L_k$ may be a leap layer, a critical layer, a small layer, or a non-critical layer.
In the first case, $L_k$ contains a set of machines reachable in the leap graph. We define:
\begin{description}
\item[(leap)] If $L_k$ is a leap layer, let every machine $i$ in $L_k$
repel all big jobs that are adjacent to $i$ in the leap graph $G(\sigma)$
---that is, all huge jobs in $\sigma^{-1}(i)$ and all big jobs $j_B$ with
$i\in\Gamma(j_B)$ and
\begin{equation*}
p(\{j'\in\sigma^{-1}(i) : p_{j'}\le 5/6\}) + p_{j_B} \le (11/6+\epsilon)\tau .
\end{equation*}
\end{description}
\begin{description}
\item[(critical)] If $L_k$ is a critical layer, for every move
$(j, i)$ in $L_k$ let $i$ repel all jobs.
\end{description}
\begin{description}
\item[(small)] If $L_k$ is a small layer, for every move
$(j, i)$ in $L_k$ let $i$ repel all jobs.
\end{description}
Now assume that $L_k$ is a non-critical-layer and consider a move $(j, i)$ in $L_k$.
In the non-critical case the algorithm is lazy:
It repels jobs only if it is really necessary.
We first identify a set of small jobs that is unlikely to be moved.
For $i\in\machines$ define $S_i(L_{\le k-1}, \sigma)$ to be the small jobs
$j\in\sigma^{-1}(i)$
which are repelled by all machines in $\Gamma(j)\setminus\{i\}$.
Next, define a threshold $W_0$ as the minimum
$W \ge 0$ such that the small jobs in $S_{i}(L_{\le k-1}, \sigma)$ and
all big jobs below this threshold
are already too large to add $j$, i.e.,
\begin{equation*}
p\left(\left\{j'\in\sigma^{-1}(i) : \frac 1 2 < p_{j'} \le W\right\}\right)
+ p(S_{i}(L_{\le k-1}, \sigma)) + p_{j}
> \left(\frac{11}{6} + 2\epsilon\right) \tau .
\end{equation*}
It will follow from the selection of critical moves that such a $W_0$ always exists
and, moreover, $W_0 \le 5/6$.
When none of the jobs in $S_{i}(L_{\le k-1}, \sigma)$ can be removed,
it is necessary (although not always sufficient) to remove one
of the big jobs with size at most $W_0$ in order to make $(j, i)$ valid.
Hence, we define,
\begin{description}
\item[(non-critical)] if $L_k$ is a non-critical layer then for every move $(j, i)$ let $i$
repel all jobs $j$ with $1/2 < p_j\le W_0$ (where $W_0$ is defined as above) and all jobs in $S_{i}(L_{\le k-1},\sigma)$.
\end{description}
It is notable that the corner case where $W_0 = 0$ is equivalent to
\begin{equation*}
p(S_{i}(L_{\le k-1},\sigma)) + p_{j} > (11/6 + 2\epsilon) \tau
\end{equation*}
and here the algorithm gives up making $(j, i)$ valid. In particular,
no additional big jobs will be repelled.
Finally, we want to highlight the following counter-intuitive (but intentional) aspect of the algorithm.
It might happen that some job of size greater
than $W_0$ is moved to $i$,
only to be removed again in a later iteration,
when $W_0$ has increased.
\paragraph*{Critical move selection}
Suppose we are given some layers $L_{\le\ell+1}$ and big job moves $(j, i) \in L^B_{\mathrm{new}}$
where $j$ is repelled
by $\sigma(j)$ w.r.t. $L_{\le\ell+1}$, but not by $i$. Which of these moves should be
critical? Recall that for critical moves $(j, i)$ the target machine $i$ always repels
all jobs.
As in the exponential time algorithm, we later need to amortize these moves with small job moves.
Hence, we should select critical moves in a way that they produce many small job moves.
\begin{figure}
\caption{Bottleneck for small jobs}
\label{sm-bottleneck}
\end{figure}
In the following let $\overline \machines$ denote the set of machines that repel all jobs
w.r.t. $L_{\le \ell+1}$.
As a prime example of a situation we want to avoid, consider the following:
There are a lot of critical moves $(j, i)$ where $p_j=1$, but
on $i$ there is a load of small jobs with volume
slightly above $(11/6 + 2\epsilon)\tau - p_j = (5/6 + 2\epsilon)\tau \ll \tau$ .
Moreover, these small jobs $j_S$ cannot go anywhere (meaning later layers will not have moves
for them), because
all their potential machines are in $\overline\machines$, i.e.,
$\Gamma(j_S)\setminus\{i\} \subseteq \overline \machines$.
Hence, there will not be any machines to amortize this
low load.
We should consider small jobs like this as blocked volume and when there is too much
blocked volume on $i$, a move $(j, i)$ should not be critical.
However, it is not enough to consider small jobs that have nowhere to go.
It might also be that a lot of small jobs have only very few machines to go to.
In the example above, imagine that all the small jobs share only one machine $i'\notin\overline\machines$ to which they could go (see Fig.~\ref{sm-bottleneck}).
Then the average load is still very low. This is also something we want to avoid.
So how do we avoid these situations?
We will make sure that for
non-valid critical moves $(j, i)$ where $j$ is big, $i$ has
one or more private machine $i'\in\overline\machines$,
which are reachable by some of its small jobs.
We select the critical edges sequentially (see Alg.~\ref{alg:critical}).
For the already added critical moves $(j, i)$, we consider all machines reachable by a small job on $i$
can go to as blocked as well, i.e., we add them to $\overline\machines$.
We add $(j, i)$ to the critical moves only when the blocked small jobs (as described above)
and the medium jobs on $i$ have a volume that allows $j$ to be added to $i$, if there
were no other jobs.
\begin{algorithm}
\caption{Selection of critical moves}
\label{alg:critical}
\begin{algorithmic}
\STATE{$C \gets \emptyset$}
\STATE{let $\overline\machines$ be the set of the machines that repel all jobs w.r.t. $L_{\le\ell+1}$}
\FOR{$(j, i)\in L^B_{\mathrm{new}}$ ordered by $\pi$}
\STATE{$\mathrm{Small} \gets p(\{j'\in\sigma^{-1}(i) : p_{j'} \le 1/2 \text{ and } \Gamma(j')\setminus\{i\} \subseteq \overline\machines\}$}
\STATE{$\mathrm{Med} \gets p(\{j'\in\sigma^{-1}(i) : 1/2 < p_{j'} \le 5/6 \})$}
\IF{$i\notin \overline\machines$ and
$\mathrm{Small} + \mathrm{Med} + p_{j} \le 11/6 + 3\epsilon$}
\STATE{$C\gets C\cup\{(j, i)\}$}
\STATE{$\overline\machines\gets \overline\machines\cup \{i\}$}
\FOR{$j'\in \sigma^{-1}(i)$ small}
\STATE{$\overline\machines\gets \overline\machines\cup \Gamma(j')$}
\ENDFOR
\ENDIF
\ENDFOR
\RETURN $C$
\end{algorithmic}
\end{algorithm}
\subsection{Analysis}
\begin{lemma}
If the configuration LP is feasible for $\tau$ and there remains a bad machine,
then within the first $\ell \le 4\lceil\log_{1+\epsilon}(4|\machines|)\rceil$ layers
there will be a valid leap or move.
\end{lemma}
\begin{proof}
Suppose toward contradiction, there are bad machines, no move in $L_{\le\ell}$ is valid, and $\ell = 4\lceil\log_{1+\epsilon}(4|\machines|)\rceil$.
We will construct values $(z_j)_{j\in\jobs}$, $(y_i)_{i\in\machines}$ with the properties as
in Lemma~\ref{lemma:condition-ra} and thereby show that the configuration LP is infeasible.
Throughout the proof the allocation $\sigma$ refers to the allocation
in the iteration where no move or leap is found.
First, we define values $z^{(k)}_j, y^{(k)}_i$ for all prefixes
of the layers ending in a leap layer,
i.e., for each $L_{\le 4k+1}$ with $0\le k < \ell/4$.
Furthermore, for technical reasons we define
the values $z^{(-1)}_j, y^{(-1)}_i$ as well as $y^{(\ell/4)}_i$.
Then $z_j$, $y_i$ will be set as a positive linear combination of these values.
Let $\tilde\jobs(L_{\le 4k+1})$ denote all jobs $j$
that are repelled by $\sigma(j)$ w.r.t. $L_{\le 4k+1}$.
For every $0 \le k < \ell/4$ and $j\in\jobs$ let
\begin{equation*}
z^{(k)}_j = \begin{cases}
\min\left\{\frac{p_j}{\tau}, \frac 5 6\right\} &\text{ if $j\in \tilde\jobs(L_{\le 4k+1})$}, \\
0 &\text{ otherwise}.
\end{cases}
\end{equation*}
Moreover, let $y^{(k)}_i := 1+\epsilon$,
if $i$ repels all jobs w.r.t. $L_{\le 4k+1}$ and
$y^{(k)}_i := z^{(k)}(\sigma^{-1}(i))$, otherwise.
Finally, define the corner cases $y^{(-1)}_i = 0$, $z^{(-1)}_j = 0$, and
$y^{(\ell/4)}_i := 1+\epsilon$ for all $i, j$.
Notice that $z^{(-1)}_j \le z^{(0)}_j \le \cdots \le z^{(\ell/4)}_j$ for all $j$ (and the same holds
for all $y^{(k)}_i$).
We set
\begin{align*}
z^{(\le k)}_j &= \sum_{k'=-1}^k \frac{1}{(1+\epsilon)^{k'}}\cdot z^{(k')}_j , \\
y^{(\le k)}_i &= \sum_{k'=-1}^{k} \frac{1}{(1+\epsilon)^{-k'}}\cdot y^{(k')}_i .
\end{align*}
The coefficients decrease exponentially with the layer number.
As we will see, this makes to the last values negligibly small (as in
the first algorithm).
Finally, set $z_j = z^{(\le\ell/4-1)}_j$ and $y_i = y^{(\le\ell/4)}_i$.
\begin{claim}\label{claim:constructive-feasibility}
Let $-1 \le k < \ell/4$, $i\in\machines$ and
$C\in\mathcal C(i, \tau)$.
Then
\begin{equation*}
z^{(\le k)}(C) \le y^{(\le k+1)}_i .
\end{equation*}
\end{claim}
In particular, this implies $z(C) = z^{(\le\ell/4-1)} \le y^{(\le\ell/4)}_i = y_i$ for all $i, C$.
\begin{claim}\label{claim:constructive-unbounded}
\begin{equation*}
\sum_{j\in\jobs} z_j > \sum_{i\in\machines} y_i .
\end{equation*}
\end{claim}
Together the claims imply $\tau < \OPT^*$, a contradiction.
\end{proof}
Before we prove the claims, we will state the following auxiliary lemmata.
\begin{lemma}\label{lemma:qp-aux1}
In an iteration where no valid move or leap is found
consider the set $L^B_{\mathrm{new}}$ selected in the algorithm after a leap layer $L_{\ell+1}$
is created
and let $(j_B, i)\in L^B_{\mathrm{new}}$.
Then
\begin{equation*}
p\left(\left\{j\in \sigma^{-1}(i) : p_j \le \frac 5 6 \right\}\right) + p_{j_B} > \left(\frac{11}{6} + 2\epsilon\right)\tau .
\end{equation*}
\end{lemma}
\begin{proof}
Suppose toward contradiction that this does not hold.
Then $(j_B, i)$ is in the leap graph $G(\sigma)$.
It is also in $G(L_{\le\ell}, \sigma)$, since $i$ does not repel $j_B$ w.r.t. $L_{\le\ell+1}$.
Otherwise, $(j_B, i)$ would not be in $L^B_{\mathrm{new}}$.
Obviously $i$ is reachable by $j_B$ in $G(L_{\le\ell}, \sigma)$.
We argue that $i$ is reachable by some big job repelled by its current machine w.r.t. $L_{\le\ell}$.
This implies that $i$ repels $j_B$ w.r.t. $L_{\le\ell+1}$ by definition of repelled edges
for a leap layer. This is a contradiction, since $(j_B, i)$ could not be in $L^B_{\mathrm{new}}$ then.
We know that $\sigma(j_B)$ repels $j_B$ w.r.t. $L_{\le \ell+1}$.
If it repels $j_B$ already w.r.t. $L_{\le \ell}$, this follows trivially.
Otherwise, $j_B$ is repelled by $i$ because of the (leap) rule in the definition of repelled edges
for $L_{\ell+1}$. This can only be when $i\in L_{\ell+1}$, which means it is reachable by some big job
repelled by its machine w.r.t. $L_{\le \ell}$.
\end{proof}
\begin{lemma}\label{lemma:W0-bound}
In an iteration where no valid move or leap is found
consider the set $L^B_{\mathrm{new}}$ selected in the algorithm after a leap layer $L_{\ell+1}$
is created
and let $(j_B, i)\in L^B_{\mathrm{new}}$.
Then
\begin{equation*}
p\left(\left\{j\in \sigma^{-1}(i) : \frac 1 2 < p_j \le \frac 5 6 \right\}\right) + p(S_i(L_{\le \ell+3},\sigma)) + p_{j_B} > \left(\frac{11}{6} + 2\epsilon\right)\tau ,
\end{equation*}
\end{lemma}
This lemma implies that the threshold $W_0$ chosen in the definition of repelled edges
always exists and $W_0\le 5/6$.
\begin{proof}
When $(j_B, i)$ is a critical moves in $L_{\ell+2}$ or when $(j_S, i)\in L_{\ell+3}$ for some small
job $j_S$, this is follows easily from the previous lemma,
since $S_i(L_{\le \ell+3},\sigma)$ contains all small jobs in $\sigma^{-1}(i)$.
Each other move $(j_B, i)$ would have been selected as a critical
move, if this inequality did not hold: Consider the set $\overline\machines$ in the selection
of critical moves at the time $(j_B, i)$ is considered. The algorithm adds $(j_B, i)$
to the critical moves, if $i\notin\overline\machines$ and
\begin{equation}
p\left(\left\{j\in \sigma^{-1}(i) : \frac 1 2 < p_j \le \frac 5 6 \right\}\right)
+ p(S'_i(\overline\machines, \sigma)) + p_{j_B} \le \left(\frac{11}{6} + 2\epsilon\right)\tau ,\label{criterion-critical}
\end{equation}
where $S'_i(\overline\machines, \sigma)$ is the set of small jobs $j_S\in\sigma^{-1}(i)$ with
$\Gamma(j_S)\setminus\{i\}\subseteq \overline\machines$.
Recall that all machines in $\overline\machines$ either repel all jobs w.r.t. $L_{\le\ell+1}$ or
they are reachable by a small job on a machine that is target of a critical move. The latter kind
must repel all jobs w.r.t. $L_{\le\ell+3}$, because it is target of a small job move in $L_{\ell+3}$.
Thus, $S'_i(\overline\machines, \sigma) \subseteq S_i(L_{\le \ell+3},\sigma)$ and (\ref{criterion-critical})
is satisfied. Furthermore, $i\notin\overline\machines$, since $i$ does not repel
all jobs w.r.t. $L_{\le\ell+1}$ and we assumed
that there is no small job $j_S$ with $(j_S, i)\in L_{\ell+3}$.
Thus, $(j_B, i)$ would have been selected as a critical move.
\end{proof}
\begin{proof}[Proof of Claim~\ref{claim:constructive-feasibility}]
We argue inductively.
The basis of the induction is trivial, since $z^{(\le -1)}(C) = 0 \le y^{(\le 0)}_i$.
Suppose that $k\ge 0$ and for all $k' < k$,
\begin{equation*}
z^{(\le k')}(C) \le y^{(\le k' + 1)}_i
\end{equation*}
If $y^{(k+1)}_i \ge 1+\epsilon$ then immediately
\begin{multline*}
z^{(\le k)}(C) = z^{(\le k-1)}(C) + (1+\epsilon)^{-k} z^{(k)}(C) \le y^{(\le k)}_i + (1+\epsilon)^{-k} \frac{p(C)}{\tau} \\
\le y^{(\le k)}_i + (1+\epsilon)^{-(k+1)} y^{(k+1)}_i = y^{(\le k+1)}_i .
\end{multline*}
We can therefore assume w.l.o.g. that $y^{(k+1)}_i = z^{(k+1)}(\sigma^{-1}(i))$.
Thus, $k<\ell/4$ and $i$ does not repel all jobs w.r.t. $L_{\le 4(k+1)+1}$.
Since by definition of repelled jobs, a machine that repels any small
job from another machine always repels all jobs, we
know that $i$ does not repel small jobs that are on other machines
w.r.t. $L_{\le 4(k+1)+1}$. Hence, for all small jobs $j_S\in C\setminus\sigma^{-1}(i)$ it holds that $z^{(k)}_{j_S} = 0$: If this was not true,
$\sigma(j_S)$ would repel $j_S$ w.r.t. $L_{4k+1}$,
in which case $(j_S, i)$ would have been added to
$L_{4k+3}$ and $i$ would repel all jobs, which is not true.
Consider the cases of big jobs in $C$.
If there is none, then obviously every jobs in $C\setminus\sigma^{-1}(i)$ is small.
Let $k' \le k$.
Then for all $j\in C\setminus\sigma^{-1}(i)$ it holds that $z^{(k')}_j \le z^{(k)}_j = 0$.
Consequently, $z^{(k')}(C) \le z^{(k')}(\sigma^{-1}(i)) = y^{(k')}_i$.
Hence,
\begin{equation*}
z^{(\le k)}(C) \le z^{(\le k)}(\sigma^{-1}(i)) = y^{(\le k)}_i \le y^{(\le k+1)})_i .
\end{equation*}
Clearly, there can be at most one big job $j_B\in C$,
since such a job has $p_{j_B} > 1/2 \cdot \tau$ and $C$ cannot have a volume greater than $\tau$.
If $z^{(k)}_{j_B} = 0$ or $j_B\in \sigma^{-1}(i)$, the argument above still works.
We recap: The crucial case is when $y^{(k+1)}_i = z^{(k+1)}(\sigma^{-1}(i))$,
there is exactly one big job $j_B \in C\setminus\sigma^{-1}(i)$,
and $z^{(k)}_{j_B} = \min\{p_{j_B}/\tau, 5/6\}$.
Let $k'\le k$ be minimal with $z^{(k')}_{j_B} = \min\{p_{j_B}/\tau, 5/6\}$.
In particular, $z^{(-1)}_{j_B} = z^{(0)}_{j_B} = \cdots = z^{(k'-1)}_{j_B} = 0$.
\begin{description}
\item[Case 1: $i$ repels $j_B$ w.r.t. $L_{\le 4k'+1}$.]
This can either be because of a leap layer or a move layer in $L_{\le 4k'+1}$.
In the former case, there has to be a huge job in $j_H\in\sigma^{-1}(i)$ which $i$
repels w.r.t. $L_{\le 4k'+1}$. Otherwise, there would be a valid path.
Thus, for all $k'' \ge k'$ it holds that $z^{(k'')}_{j_H} = 5/6 \ge z^{(k'')}_{j_B}$ and
\begin{multline*}
z^{(k'')}(C) = z^{(k'')}_{j_B} + z^{(k'')}(C\setminus\{j_B\})
\le z^{(k'')}_{j_H} + z^{(k'')}(\sigma^{-1}(i)\setminus\{j_H\}) \\
\le z^{(k'')}(\sigma^{-1}(i)) \le y^{(k'')}_i .
\end{multline*}
Furthermore, for all $k'' < k'$,
\begin{equation*}
z^{(k'')}(C) = z^{(k'')}(C\setminus\{j_B\})
\le z^{(k'')}(\sigma^{-1}(i)) \le y^{(k'')}_i .
\end{equation*}
Hence,
$z^{(\le k)}(C) \le y^{(\le k)}_i \le y^{(\le k+1)}_i$.
Now consider the case in which there is some move $(j_{k'}, i)$
which causes $i$ to repel $j_B$ is w.r.t. $L_{\le 4 k'+1}$.
The move $(j_{k'}, i)$ must be in a non-critical layer $L_{4k''+4}$, where $k'' < k'$,
since $i$ does not repel all jobs w.r.t. $L_{\le 4 k'+1}$.
Let $W_0$ as in the definition of repelled jobs in consequence of $(j_{k'}, i)$
and let
\begin{equation*}
R = \{j\in\sigma^{-1}(i) : 1/2 < p_j \le W_0\}\cup S_i(L_{\le 4(k''-1)+3},\sigma) ,
\end{equation*}
The edges repelled by $i$ because of $(j_{k'}, i)$ are exactly $S_i(L_{\le 4k''+3},\sigma)$ and
all those $j$ with $1/2 < p_j \le W_0$. Hence, $p_{j_B} \le W_0$.
Recall that $W_0$ is chosen minimal with $p(R) + p_{j_{k'}} > (11/6 + 2 \epsilon)\tau$.
There must be a job $j'_B\in\sigma^{-1}(i)$ with $p_{j'_B} = W_0$, since otherwise
$W_0$ would not be minimal.
Thus, for all $k'''$ it holds that $z^{(k''')}_{j'_B} \ge z^{(k''')}_{j_B}$ and
\begin{equation*}
z^{(k''')}(C) = z^{(k''')}_{j_B} + z^{(k''')}(C \setminus \{j_B\})
\le z^{(k''')}_{j'_B} + z^{(k''')}(\sigma^{-1}(i) \setminus \{j'_B\})
\le y^{(k''')}_i .
\end{equation*}
It follows conveniently that $z^{(\le k)}(C) \le y^{(\le k)}_i \le y^{(\le k+1)}_i$.
\item[Case 2: $i$ does not repel $j_B$ w.r.t. $L_{\le 4k'+1}$]
Since $z^{(k')}_{j_B} > 0$, $j_B$ is repelled by $\sigma(j_B)$
w.r.t. $L_{\le 4k' + 1}$. Machine $i$ does not repel all jobs w.r.t. $L_{\le 4k'+1}$, which
implies there is no move with target $i$ in $L_{4k' + 2}$ or $L_{4k' + 3}$.
Hence, $(j_B, i)$ must be a move in layer $L_{4k' + 4}$.
Let
\begin{equation*}
R = \{j\in\sigma^{-1}(i) : 1/2 < p_j \le W_0\}\cup S_i(L_{\le 4k'+1}, \sigma) ,
\end{equation*}
where $W_0$ is as in the definition of repelled jobs in consequence of $(j_B, i)$. Then
\begin{equation*}
p(R) + p_{j_B} > \left(\frac{11}{6} + 2\epsilon\right) \tau \ge p(C) + \left(\frac 5 6 + 2\epsilon\right) \tau .
\end{equation*}
Furthermore, all jobs in $R$ are repelled by $i$ w.r.t. $L_{\le 4k' + 4}$ and therefore in $\tilde\jobs(L_{\le 4(k' + 1) + 1})$.
Since for all $j'\in R$ it holds that $p_{j'} \le W_0 \le 5/6$ (see Lemma~\ref{lemma:W0-bound}),
it follows that $z^{(k'' + 1)}_{j'} = p_{j'} / \tau$ for all $k'' \ge k'$.
Thus,
\begin{align*}
z^{(k'')}(C) &= z^{(k'')}_{j_B} + z^{(k'')}(C\setminus\{j_B\}) \\
&\le z^{(k'')}_{j_B} + (p(C) - p_{j_B}) / \tau \\
&< z^{(k'')}_{j_B} + (p(R) - (5/6 + 2\epsilon) \tau) / \tau \\
&= z^{(k'')}_{j_B} + p(R)/\tau - 5/6 - 2\epsilon \\
&\le (1 - \epsilon) p(R) / \tau \\
&\le (1-\epsilon) z^{(k'' + 1)}(\sigma^{-1}(i))
\le \frac{z^{(k'' + 1)}(\sigma^{-1}(i))}{1+\epsilon} .
\end{align*}
Here we use that $i$ is a good machine and therefore
$p(R)\le p(\sigma^{-1}(i)) \le (11/6 + 2\epsilon)\tau < 2\tau$.
We conclude,
\begin{align*}
z^{(\le k)}(C) &= z^{(\le k' - 1)}(C)
+ \sum_{k'' = k'}^k (1 + \epsilon)^{-k''} z^{(k'')}(C) \\
&\le y^{(\le k')}_i
+ \sum_{k'' = k'}^k (1 + \epsilon)^{-k''} \frac{z^{(k''+1)}(\sigma^{-1}(i))}{1+\epsilon} \\
&\le y^{(\le k')}_i
+ \sum_{k'' = k'}^k (1 + \epsilon)^{-(k'' + 1)} y^{(k'' + 1)}_i
= y^{(\le k + 1)}_i . \qedhere
\end{align*}
\end{description}
\end{proof}
\begin{proof}[Proof of Claim~\ref{claim:constructive-unbounded}]
Let $i$ be a bad machine.
Then $i$ repels all jobs (in particular those in $\sigma^{-1}(i)$)
w.r.t. $L_{\le 0}$.
Hence, for every $0\le k < \ell/4$ and $j\in\sigma^{-1}(i)$,
$z^{(k)}_j = \min\{5/6, p_j/\tau\} \ge 5/6 \cdot p_j/\tau$
Thus,
\begin{equation*}
z^{(k)}(\sigma^{-1}(i)) \ge \frac 5 6 p(\sigma^{-1}(i))/\tau > \frac 5 6 \left(\frac {11} 6 + 2\epsilon\right) > \frac{55}{36} + \epsilon > 1+\epsilon + \frac 1 2 .
\end{equation*}
This implies
\begin{multline*}
y_i = \sum_{k=0}^{\ell/4} (1+\epsilon)^{-k} y^{(k)}_i = \sum_{k=0}^{\ell/4-1}[(1+\epsilon)^{-k} (1+\epsilon)] + (1+\epsilon)^{-(\ell/4-1)} \\
< \sum_{k=0}^{\ell/4-1} [(1+\epsilon)^{-k} \cdot z^{(k)}(\sigma^{-1}(i))] + (1+\epsilon)^{-(\ell/4-1)} - \frac 1 2 \sum_{k=0}^{\ell/4-1} (1+\epsilon)^{-k} \\
\le z(\sigma^{-1}(i)) - \frac 1 2 .
\end{multline*}
In the last inequation, we use the last two elements of the sum
$\sum_{k=0}^{\ell/4-1} (1+\epsilon)^{-k}$ to compensate for $(1+\epsilon)^{-(\ell/4-1)}$.
The inequation shows that $y_i$ is much smaller than $z(\sigma^{-1}(i))$.
If for all good machines $i$ and layers $k$ we had $y^{(k)}_i = z^{(k)}(\sigma^{-1}(i))$
(which is the case when $i$ does not repel all jobs w.r.t. $L_{\le k}$),
the proof would be easy: $\sum_{j\in\jobs} z^{(\le\ell/4-1)}_j = \sum_{i\in\machines}z^{(\le\ell/4-1)}(\sigma^{-1}(i))$
would be larger than $1/2 + \sum_{i\in\machines} y^{(\le\ell/4-1)}_i$.
The former is exactly $\sum_{j\in\jobs} z_j$ and the latter is
\begin{equation*}
1/2 + \sum_{i\in\machines} y_i - \sum_{i\in\machines} (1+\epsilon)^{-\ell/4} y^{(\ell/4)}_i < \sum_{i\in\machines} y_i .
\end{equation*}
Here we use that the decrease in the coefficient
makes $y^{(\ell/4)}_i$ neglectable, which we will explain in detail as we go
through the actual proof.
Of course, there can be machines that repel all jobs and
are set to $y^{(k)}_i = 1+\epsilon$.
We have to make sure that they do not have a negative effect.
Let $B_k$ be the machines $i$ with $(j_B, i)$ in the $k$-th critical layer for some $j_B$,
i.e., in $L_{4k+1}$.
Let $A_k$ be the machines $i$ with $(j_S, i)$ in the $k$-th small layer for some $j_S$,
i.e., in $L_{4k+3}$.
Let $k<\ell/4$ and $i\in B_k$. $i$ repels all jobs w.r.t. $L_{\le 4(k+1)+1}$. Thus,
$\sigma^{-1}(i)\subseteq \tilde\jobs(L_{\le 4(k+1)+1})$.
Let $(j_B, i)$ as above. This move is not valid.
Either there is a job $j\in\sigma^{-1}(i)$ with $z^{(k+1)}_j = 5/6$ or $z^{(k+1)}_j = p_j / \tau$ for all $j\in\sigma^{-1}(i)$. Thus,
\begin{multline*}
z^{(k+1)}(\sigma^{-1}(i)) \ge \min\{5/6,\ p(\sigma^{-1}(i)) / \tau\} \\
\ge \min\{5/6,\ 11/6 + 2\epsilon - p_{j_B} / \tau\}
\ge 5/6 \ge 1 + \epsilon - \epsilon - 1/6
= y^{(k+1)}_i - \epsilon - 1/6 .
\end{multline*}
Next, let $i\in A_k$. Then there is a move $(j_S, i) \in L_{\le 4k+1}$ with $j_S$ small. Of course, this move is not valid either. In the following, we distinguish between the cases where $\sigma^{-1}(i)$ has no huge job
or one huge job.
\begin{multline*}
z^{(k+1)}(\sigma^{-1}(i)) \ge \min\{p(\sigma^{-1}(i)) / \tau,\ (p(\sigma^{-1}(i)) - \tau) / \tau + 5/6\} \\
\ge 8/6 + 2\epsilon - 1 + 5/6
= \frac 7 6 + 2\epsilon \ge y^{(k+1)}_i + 1/6 + 2\epsilon .
\end{multline*}
The bounds above show that machines in $A_k$ have $z^{(k+1)}(\sigma^{-1}(i))$ above $y^{(k+1)}_i$
and machines in $B_k$ below.
In order to amortize the machines, we have to proof a bounded ratio between them:
We argue that for every $k < \ell/4$, $|A_{k}| \ge |B_{k}|$.
Notice that $B_{k} \dot\cup A_{k}$ are exactly the machines that are added to $\overline\machines$
in the selection of critical moves for $L_{4k+2}$.
Hence, it suffices to show that at most half of them are target of critical big job moves.
Consider a critical move $(j_B, i)$ for a big job $j_B$ that is added in the critical move selection.
By Lemma~\ref{lemma:qp-aux1}
\begin{equation*}
p\left(\left\{j\in\sigma^{-1}(i) : p_j \le \frac 5 6 \right\}\right) + p_{j_B} > \left(\frac{11}{6} + 2\epsilon\right)\tau .
\end{equation*}
Because the move $(j_B, i)$ is selected as a critical move, it holds that
\begin{equation*}
p\left(\left\{j\in\sigma^{-1}(i) : \frac 1 2 < p_j \le \frac 5 6 \right\}\right) + p(S'_i(\overline\machines, \sigma)) + p_{j_B} \le \left(\frac{11}{6} + 2\epsilon \right)\tau ,
\end{equation*}
where $S'_i(\overline\machines, \sigma)$ are the small jobs $j_S\in\sigma^{-1}(i)$
with $\Gamma(j_S)\setminus\{i\}\subseteq \overline\machines$ with $\overline\machines$ as
at the time before $(j_B, i)$ is selected.
Consequently, there is a small job $j_S\in\sigma^{-1}(i)\setminus S'_i(\overline\machines, \sigma)$.
It follows that there exists a machine $i'\in\Gamma(j_S)\setminus (\overline\machines\cup\{i\})$.
The algorithm adds $i$ and $i'$ to $\overline\machines$.
In other words, whenever the algorithm adds a machine $i$ to $B_k$, it adds at least one machine
$i'$ to $A_k$.
It follows that
\begin{align*}
\sum_{j\in\jobs} z_j
&= \sum_{k=0}^{\ell/4-1}\sum_{i\in\machines}(1+\epsilon)^{-k} z^{(k)}(\sigma^{-1}(i)) \\
&> \sum_{k=0}^{\ell/4-1} (1+\epsilon)^{-k} \bigg[ \left(\frac 1 6+\epsilon\right)\underbrace{(|A_{k}| - |B_k|)}_{\ge 0} + \sum_{i\in\machines} y^{(k)}_i \bigg] + \frac 1 2 \\
&\ge \sum_{i\in\machines} y_i
+ \underbrace{\frac 1 2 - \sum_{i\in\machines} (1+\epsilon)^{-\ell/4} y^{(\ell/4)}_i}_{\ge 0}
\ge \sum_{i\in\machines} y_i
\end{align*}
In the last inequality we use that
by choice of $\ell$, $(1+\epsilon)^{\ell/4} \le 4 |\machines|$, which implies
\begin{equation*}
\sum_{i\in\machines} (1+\epsilon)^{-\ell/4} y^{(\ell/4)}_i
\le 2 |\machines|(1+\epsilon)^{-\ell/4} \le \frac 1 2 . \qedhere
\end{equation*}
\end{proof}
\begin{lemma}
The algorithm terminates in time $n^{O(1/\epsilon \log(n))}$, where $n = |\jobs| + |\machines|$.
\end{lemma}
\begin{proof}
We are looking at the states of two consecutive iterations right before a move or leap
is performed.
Let $\sigma$ be the schedule in the former iteration and $\sigma'$ in the latter.
Likewise, define layers $L_{\le\ell}$ and $L'_{\le\ell'}$
right before they collapse.
Let $\tilde\jobs(L_{\le k},\sigma)$ be the jobs $j$ repelled by $\sigma(j)$ w.r.t. $L_{\le k}$.
Recall, $R(L_{\le k}, \sigma)$
is the set of all $(j, i)\in \jobs\times\machines$ where $i$ repels $j$ w.r.t. $L_{\le k}$.
We define a potential function $\Phi$ for each of the layers. Set
\begin{equation*}
\Phi(L_{\le k},\sigma) = \begin{cases}
|R(L_{\le k}, \sigma)| &\text{if $L_k$ is a leap layer}, \\
(|L_k|, |\jobs| - |\tilde\jobs(L_{\le k},\sigma)|) &\text{if $L_k$ is a critical layer,} \\
|\jobs| - |\tilde\jobs(L_{\le k},\sigma)| &\text{if $L_k$ is a small layer,} \\
(|R(L_{\le k}, \sigma)|, |\jobs| - |\tilde\jobs(L_{\le k},\sigma)|) &\text{if $L_k$ is a non-critical layer}.
\end{cases}
\end{equation*}
\begin{claim}\label{claim:lexicographic-decrease}
The vector
\begin{equation*}
(g', |\jobs| - |\tilde\jobs(L'_{\le 0},\sigma')|, \Phi(L'_{\le 1},\sigma'), \dotsc, \Phi(L'_{\le\ell'},\sigma'), \infty)
\end{equation*}
is lexicographically bigger than
\begin{equation*}
(g, |\jobs| - |\tilde\jobs(L_{\le 0},\sigma)|, \Phi(L_{\le 1},\sigma), \dotsc, \Phi(L_{\le\ell},\sigma), \infty)
\end{equation*}
\end{claim}
Since the number of layers is at most $O(1/\epsilon \log(n))$
and components can have only $O(n^3)$ different values,
the number of vectors is
is bounded by $n^{O(1/\epsilon \log(n))}$. Since for every move or leap it decreases lexicographically,
the lemma follows easily from the claim.
\end{proof}
\begin{proof}[Proof of Claim~\ref{claim:lexicographic-decrease}]
If the number of good machines increases, the claim follows immediately.
If it does not, but a job is moved from a bad machine to a good one, then
$\tilde\jobs'(L'_{\le 0},\sigma') \subsetneq \tilde\jobs(L_{\le 0},\sigma)$, i.e.,
the claim follows again.
Hence, assume neither case is true.
Let $1 \le k\le\min\{\ell-1,\ell'\}$ and:
\begin{enumerate}
\item $L_{\le k-1} = L'_{\le k-1}$;
\item $\tilde\jobs(L_{\le k-1}) = \tilde\jobs'(L'_{\le k-1})$ and $\sigma(j) = \sigma'(j)$ for all $j\in\tilde\jobs(L_{\le k-1})$;
\item $R(L'_{\le k-1}, \sigma') = R(L_{\le k}, \sigma)$.
\end{enumerate}
We will prove:
$\Phi(L'_{\le k}) \ge \Phi(L_{\le k})$ and
if equality holds, (1), (2) and (3) also hold for $k$.
This implies the claim by induction:
If $\ell' < \ell$, then the prefix of the first vector ending in $\Phi(L'_{\le \ell'},\sigma)$
is lexicographically not smaller than the prefix of the second one ending in
$\Phi(L_{\le \ell'},\sigma)$. Furthermore, the next component in the first vector is
$\infty$, whereas it is something finite in the second.
If $\ell' \ge \ell$, then we notice that (2) cannot hold for $k-1=\ell-1$. This is
because some leap or move in $L_{\ell}$ was executed and therefore a job $j$ that
is repelled by $\sigma(j)$ w.r.t. $L_{\le\ell-1}$ was moved.
\begin{description}
\item[Case 1: $L_k$ is a leap layer.]
We argue that every machine $i$ reachable in the leap graph $G(L_{\le k-1}, \sigma)$ by a
job big job $j_0$ repelled by $\sigma(j_0)$ w.r.t. $L_{\le k-1}$ is also reachable by $j_0$ in the leap
graph $G(L'_{\le k-1}, \sigma')$. Because of (2),
this means that the set of reachable machines in $\sigma$
by any such job, which is exactly $L_k$, is a subset of $L'_k$.
It suffices to show that every edge reachable by $j_0$ in $G(L_{\le k-1}, \sigma)$ is
also in the leap graph $G(L'_{\le k-1}, \sigma')$.
Because of (3) it suffices to show that it is in $G(\sigma')$.
An edge in the leap graph can be one of two kinds.
It can be from a machine to a huge job, i.e., $(i, j_H)$,
which exists because $\sigma(j_H) = i$.
We argue that $j_H$ was not not moved, which means $\sigma'(j_H) = \sigma(j_H) = i$
and therefore the edge is also in $G(\sigma')$.
Suppose toward contradiction that a move $(j_H, i')$ was executed.
Then there is no huge job in $\sigma^{-1}(i')$ and
$p(\sigma^{-1}(i')) + p_{j_H} \le (11/6 + 2\epsilon) \cdot \tau$. Therefore $i'$
would also be reachable by $j_0$ in $G(\sigma)$. Hence, the algorithm
would execute a leap in $L_k$, which it did not, since $\ell > k$.
Similarly, if $j_H$ was moved as part of a leap, then all machines in this leap
would be reachable already in $L_k$ and there would have been a valid leap already.
Now consider an edge of the form $(j_B, i)$. If $i$ had no huge job in $\sigma^{-1}(i)$, then
again, there would have been a valid leap in $L_k$, which cannot be.
The edge $(j_B, i)$ exists in the leap graph of $\sigma$
because $i\in\Gamma(j_B)\setminus\{\sigma(j_B)\}$ and
\begin{equation*}
p(\{j\in\sigma^{-1}(i) : p_{j} \le 5/6\}) + p_{j_B} \le (11/6 + 2\epsilon) \tau .
\end{equation*}
It could only be removed, if $j_B$ was moved to $i$
---this cannot be the case for the same reason as above---or
some job $j'$ with $p_{j'}\le 5/6$ is moved to $i$. By definition of a valid non-huge move, however, this means that
\begin{multline*}
(11/6 + 2\epsilon) \tau \ge p(\{j\in\sigma^{-1}(i) : p_{j} \le 5/6\}) + p_{j'} + 1 \\
\ge p(\{j\in\underbrace{\sigma^{-1}(i)\cup \{p_{j'}\}}_{=\sigma^{\prime-1}(i)} : p_{j} \le 5/6\}) + p_{j_B} .
\end{multline*}
Therefore, $(j_B, i)$ is also in $G(\sigma')$.
We conclude, all reachable machines in $L_k$ are also in $L'_k$, i.e., $L'_k \supseteq L_k$.
By the arguments above, every job adjacent to a machine
in $L_k$ in $G(L_{\le k-1}, \sigma)$ is also adjacent
to this machine in $L'_k$ in $G(L'_{\le k-1}, \sigma')$. Thus,
$R(L'_{\le k},\sigma')\supseteq R(L_{\le k},\sigma)$.
If equality does not hold, then $\Phi(L'_{\le k}, \sigma') > \Phi(L_{\le k}, \sigma)$.
Otherwise, (3) holds for $k$. (1) must also hold for $k$, because $L'_k \supsetneq L_k$ was true,
then the additional machine would repel at least one additional job (its huge job).
Finally, (2) holds, because no huge job on a reachable machine was moved as elaborated above.
\item[Case 2: $L_k$ is a critical layer.]
We show that every critical move in $L_k$ is also in $L'_k$.
By induction hypothesis, we know that the moves $(j, i)$, $i\in\Gamma(j)$, where $j$ is a big job
repelled by $\sigma(j) = \sigma'(j)$, but not by $i$,
w.r.t. $L_{\le k-1}$ and w.r.t. $L'_{\le k-1}$ are the same.
Therefore, the sets $L^B_{\mathrm{new}}$
from which the critical moves are selected are the same in both cases.
Recall that critical moves are added greedily in the order of $\pi'$ ($\pi$).
In $\pi'$ the moves $L^B_{\mathrm{new}}$ are ordered in a way that first the moves from $L_{k}$ appear
(in the order of $\pi$) and then all others. This is because in the main algorithm when $L_k$ was
created, all $(j, i)\in L_k$ were moved to the front of $\pi$. We just have to understand that
none of $L^B_{\mathrm{new}}\setminus L_k$ were moved to the front at a later time.
This is because there is no way that a move, which is not selected as critical, can be selected in a
later layer.
Let $(j_1, i_1), \dotsc, (j_{r-1}, i_{r-1})$ be the first $r-1$ critical moves selected in $L_k$.
Furthermore, let $\overline\machines_{r-1}$ and $\overline\machines_{r-1}'$
be as in the algorithm before the $r$-th critical move was added.
Note that before the first critical move was added, by (3) it holds that
$\overline\machines'_{0} = \overline\machines_{0}$, since these are the machines
that repel all jobs w.r.t. $L_{\le k-1}$.
We assume for induction that $\overline\machines'_{r-1} \subseteq \overline\machines_{r-1}$ and
that $(j_1, i_1), \dotsc, (j_{r-1}, i_{r-1})$ were also added to $L'_k$.
Because no move or leap in $L_{\le k-1}$ was executed and $i_r$ repels all jobs w.r.t. $L_{\le k}$,
we know that $\sigma^{\prime-1}(i_r)\subseteq \sigma^{-1}(i_r)$. In particular,
every medium job
in $\sigma^{\prime-1}(i_r)$ was already in $\sigma^{-1}(i_r)$. Moreover, every small job
$j_S\in \sigma^{\prime-1}(i_r)$ with $\Gamma(j_S)\setminus\{i_r\}\subseteq\overline\machines'_{r-1} \subseteq \overline\machines_{r-1}$ was also in $\sigma^{-1}(i_r)$ . Hence, the condition for adding
$(j_r, i_r)$ to $L'_k$ holds, since it did for $L_k$.
Finally, $\overline\machines'_r$ is the union of $\overline\machines'_{r-1}$, $\{i\}$,
and $\Gamma(j_S)$ for
every small $j_S\in\sigma^{\prime-1}(i_r)$. This is a subset of $\overline\machines_{r-1}$, $\{i\}$,
and $\Gamma(j_S)$ for every small $j_S\in\sigma^{-1}(i_r)\subseteq\sigma^{\prime-1}(i_r)$,
which is $\overline\machines_r$.
If $L'_k \supsetneq L_k$, nothing has to be shown, since $\Phi(L'_{\le k}) > \Phi(L_{\le k})$.
Otherwise, $L'_k = L_k$ and therefore (3) follows for $k$ directly.
If some job was moved away from a machine of a critical move, then again
$\Phi(L'_{\le k}) > \Phi(L_{\le k})$. Otherwise, (2) follows for $k$.
\item[Case 3: $L_k$ is a small layer.]
As in the previous case, we have that $L_k = L'_k$, i.e., (1) holds also for $k$.
(3) also holds for $k$, since in the rules of a small layer, every target of a move
repels every job. This is the same in $L'_{\le k}$ and $L_{\le k}$.
If some job was moved away from
a target machine of a move in $L_k$,
then $\tilde\jobs(L'_{\le k}) \subsetneq \tilde\jobs(L_{\le k})$
and therefore $\Phi(L'_{\le k}) > \Phi(L_{\le k})$. Otherwise, (2) follows for $k$ as well.
\item[Case 4: $L_k$ is a non-critical-layer.]
By the arguments in Case 2 we know that the previous critical moves and the moves
they are chosen from are the same
and therefore also for the non-critical moves $L'_k = L_k$.
Let $(j, i)\in L_k$.
We argue that
\begin{equation*}
S_i(L_{\le k-1}, \sigma) = S_i(L'_{\le k-1}, \sigma') .
\end{equation*}
Let $j_S\in S_i(L_{\le k-1}, \sigma)$. Then there cannot be a move $(j_S, i')$ in
some higher layer than $L_{k-1}$. This is because $j_S$ is repelled
by all $i'\in\Gamma(j_S)\setminus\{\sigma(j_S)\}$ w.r.t.
$L_{k-1}$. Hence, $\sigma'(j_S) = \sigma(j_S) = i$. With (3) it follows that
$j_S\in S_i(L'_{\le k-1}, \sigma')$.
Now let $j_S\in S_i(L'_{\le k-1}, \sigma')$. If $\sigma(j_S) = \sigma'(j_S) = i$, then
as above with (3) it follows that $j_S\in S_i(L_{\le k-1}, \sigma)$.
Now assume toward contradiction $\sigma(j_S) \neq i$.
By (2),
$j_S$ is not repelled by $\sigma(j_S)$ w.r.t. $L_{\le k-1}$;
By (3) this means that $j_S$ is also not repelled by $\sigma(j_S)\neq i$ w.r.t.
$L'_{\le k-1}$. Hence, $j_S\notin S_i(L'_{\le k-1}, \sigma')$, a contradiction.
Let $W_0$ be the minimal $W\ge 0$ such that
\begin{equation*}
p\left(\left\{j'\in\sigma^{-1}(i) : \frac 1 2 < p_{j'} \le W\right\}\right)
+ p(S_{i}(L_{\le k-1}, \sigma)) + p_{j}
> \left(\frac{11}{6} + 2\epsilon\right) \tau .
\end{equation*}
Since $i$ repels all jobs $j'$ with $1/2 < p_{j'} \le W$ w.r.t. $L_{\le k}$, we get
\begin{equation*}
\left\{j'\in\sigma^{'-1}(i) : \frac 1 2 < p_{j'} \le W\right\}
\subseteq \left\{j'\in\sigma^{-1}(i) : \frac 1 2 < p_{j'} \le W\right\} .
\end{equation*}
This implies that $W'_0$, the minimal $W\ge 0$ with
\begin{equation*}
p\left(\left\{j'\in\sigma^{\prime-1}(i) : \frac 1 2 < p_{j'} \le W\right\}\right)
+ p(S_{i}(L'_{\le k-1}, \sigma')) + p_{j}
> \left(\frac{11}{6} + 3\epsilon\right) \tau .
\end{equation*}
is at least as big as $W_0$, i.e., $W'_0\ge W_0$. This means all jobs repelled by $i$
w.r.t. $L_{\le k}$ are also repelled w.r.t. $L'_{\le k}$, which implies
$R(L'_{\le k}, \sigma') \supseteq R(L_{\le k}, \sigma)$. If equality does not hold,
then $\Phi(L'_{\le k}, \sigma') > \Phi(L_{\le k}, \sigma)$.
Otherwise (3) is fulfilled for $k$.
If one of the jobs repelled by $i$ is moved,
then $\tilde\jobs(L'_{\le k-1}) \subsetneq \tilde\jobs(L_{\le k-1})$. Otherwise,
equality holds and (2) follows for $k$.
\end{description}
\end{proof}
\begin{theorem}
We can find a $(11/6+\epsilon)$-approximate
solution for \textsc{Restricted Assignment} in time $n^{O(1/\epsilon \log(n))}$
for every $\epsilon > 0$, where $n = |\mathcal J| + |\mathcal M|$.
\end{theorem}
\end{document}
|
\begin{document}
\title{\bf Multivariate {\emph{q}}-P\'{o}lya and\\ inverse {\emph{q}}-P\'{o}lya distributions}
\author{Charalambos A. Charalambides\\
{\em Department of Mathematics, University of Athens}, \\
{\em Panepistemiopolis, GR-15784 Athens, Greece}}
\footnotetext{E-mail address: [email protected]}
\date{}
\maketitle
\begin{abstract}
An urn containing specified numbers of balls of distinct ordered colors is considered. A multiple $q$-P\'{o}lya urn model is introduced by assuming that the probability of $q$-drawing a ball of a specific color from the urn varies geometrically, with rate $q$, both with the number of drawings and the number of balls of the specific color, together with the total number of balls of the preceded colors, drawn in the previous $q$-drawings. Then, the joint distributions of the numbers of balls of distinct colors drawn (a) in a specific number of $q$-drawings and (b) until the occurrence of a specific number of balls of a certain color, are derived. These two distributions turned out to be $q$-analogues of the multivariate P\'{o}lya and inverse P\'{o}lya distributions, respectively. Also, the limiting distributions of the multivariate $q$-P\'{o}lya and inverse $q$-P\'{o}lya distributions, as the initial total number of balls in the urn tends to infinity, are shown to be $q$-multinomial and negative $q$-multinomial distributions, respectively.
\end{abstract}
{\em AMS(2000) subject classification}. Primary 60C05, Secondary 05A30.\\
{\em Keywords and phrases}: multivariate absorption distribution, multivariate inverse absorption distribution, multivariate inverse $q$-hypergeometric distribution, multivariate $q$-hyper\-geo\-metric distribution, negative $q$-multinomial distribution, $q$-multinomial distribution.
\section{Introduction}\label{sec1}
A $q$-P\'{o}lya urn model is introduced in Charalambides (2012, 2016) and $q$-P\'{o}lya and inverse $q$-P\'{o}lya distributions are studied. Moreover, their limiting distributions, as the number of balls in the urn, or the number of drawn balls of one of the two colors, tends to infinity are shown to be $q$-binomial and negative $q$-binomial distributions. Kupershmidt (2000) introduced a $q$-hypergeometric distribution and a $q$-P\'{o}lya distribution (under the name $q$-contagious distribution) and represented the corresponding random variable as a sum of two-valued dependent random variables. Kemp (2005) starting from the Chu-Vandermonde sum as a probability generating function obtained two $q$-confluent hypergeometric distributions. She also, deduced these distributions as steady-state birth and death Markov chains.
The aim of this article is to introduced and studied in detail multivariate $q$-P\'{o}lya and inverse $q$-P\'{o}lya distributions and also, examined some of their limiting distributions. Section 2 is devoted to $q$-multinomial convolutions. Precisely, multivariate $q$-Vandermonde and inverse $q$-Vandermonde formulae are presented. Also, a closely connected multivariate $q$-Cauchy formula is deduced. In section 3, the $q$-P\'{o}lya urn model is extended to a multiple $q$-P\'{o}lya urn model by considering successive $q$-drawings of balls from an urn containing specified numbers of balls of different colors and assuming that the probability of randomly $q$-drawing a ball of a specific color from the urn varies geometrically, with rate $q$. Then, on the stochastic model of a sequence of a specific number of random $q$-drawings of balls, a multivariate $q$-P\'{o}lya distribution is introduced and examined. Furthermore, in section 4, and on the stochastic model of a sequence of random $q$-drawings that is terminated with the occurrence of a specific number of balls of a given color, a multivariate inverse $q$-P\'{o}lya distribution is discussed.
\section{{\emph{q}}-Multinomial convolutions} \label{sec2}
\setcounter{equation}{0}
Let $x$ and $q$ be real numbers, with $q\neq 1$, and $r$ be an integer. The function of $x$, with parameter $q$, $[x]_q=(1-q^x)/(1-q)$ is called {\em $q$-number} and in particular $[r]_q$ is called {\em $q$-integer}. Also, the product $[x]_{r,q}=[x]_q[x-1]_q\cdots[x-r+1]_q$, $r=1,2,\ldots\,$, defines the {$q$-factorial of $x$ of order $r$. In particular, $[r]_q!=[1]_q[2]_q\cdots[r]_q$ is the {\em $q$-factorial} of $r$. The notion of $q$-factorial is extended to zero order by $[x]_{0,q}=1$ and to negative order by $[x]_{-r,q}=1/[x+r]_{r,q}$, $r=1,2,\ldots\,$.
The {\em $q$-multinomial coefficient} is defined by
\begin{eqnarray}\label{eq2.1}
\genfrac[]{0pt}{}{x}{r_1,r_2,\ldots,r_k}_q=\frac{[x]_{r_1+r_2+\cdots+r_k,q}}{[r_1]_q![r_2]_q!\cdots[r_k]_q!}, \ \ r_j=0,1,\ldots,
\ \ j=1,2,\ldots,k.
\end{eqnarray}
Notice that a $q^{-1}$-number is readily expressed into a $q$-number by $[x]_{q^{-1}}=q^{-x+1}[x]_q$. Consequently,
\[
[x]_{r,q^{-1}}=q^{-xr+\binom{r+1}{2}}[x]_{r,q}, \ \ [r]_{q^{-1}}!=q^{-\binom{r}{2}}[r]_q!.
\]
Furthermore, setting $s_j=\sum_{i=1}^jr_i$ and $m_j=\sum_{i=j}^kr_i$, for $j=1,2,\ldots,k$, and using the expression
\[
-xs_k+\binom{s_k+1}{2}+\sum_{j=1}^k\binom{r_j}{2}=-\sum_{j=1}^k r_j(x-m_j)=-\sum_{j=1}^k r_j(x-s_j),
\]
it follows that
\begin{align}\label{eq2.2}
\genfrac{[}{]}{0pt}{}{x}{r_1,r_2,\ldots,r_k}_{q^{-1}}
&=q^{-\sum_{j=1}^k r_j(x-m_j)}\genfrac{[}{]}{0pt}{}{x}{r_1,r_2,\ldots,r_k}_q\nonumber\\
&=q^{-\sum_{j=1}^k r_j(x-s_j)}\genfrac{[}{]}{0pt}{}{x}{r_1,r_2,\ldots,r_k}_q.
\end{align}
Therefore, a formula involving $q$-numbers, $q$-factorials, and $q$-multinomial coefficients in the base $q$, with $1<q<\infty$, can be converted, with respect to the base, into a similar formula in the base $p=q^{-1}$, with $0<p<1$. Two versions of a recurrence relation for the $q$-multinomial coefficients, useful in the sequel, are quoted here for easy reference.
The $q$-multinomial coefficient satisfies the recurrence relation
\begin{align}\label{eq2.3}
\genfrac{[}{]}{0pt}{}{x}{r_1,r_2,\ldots,r_k}_q&=\genfrac{[}{]}{0pt}{}{x-1}{r_1,r_2,\ldots,r_k}_q
+q^{x-m_1}\genfrac{[}{]}{0pt}{}{x-1}{r_1-1,r_2,\ldots,r_k}_q\nonumber \\
&+q^{x-m_2}\genfrac{[}{]}{0pt}{}{x-1}{r_1,r_2-1,\ldots,r_k}_q
+\cdots+q^{x-m_k}\genfrac{[}{]}{0pt}{}{x-1}{r_1,r_2,\ldots,r_k-1}_q,
\end{align}
and alternatively, the recurrence relation
\begin{align}\label{eq2.4}
\genfrac{[}{]}{0pt}{}{x}{r_1,r_2,\ldots,r_k}_q&=q^{s_k}\genfrac{[}{]}{0pt}{}{x-1}{r_1,r_2,\ldots,r_k}_q
+\genfrac{[}{]}{0pt}{}{x-1}{r_1-1,r_2,\ldots,r_k}_q\nonumber \\
&+q^{s_1}\genfrac{[}{]}{0pt}{}{x-1}{r_1,r_2-1,\ldots,r_k}_q
+\cdots+q^{s_{k-1}}\genfrac{[}{]}{0pt}{}{x-1}{r_1,r_2,\ldots,r_k-1}_q,
\end{align}
for $r_j=0,1,\ldots\,$ and $j=1,2,\ldots,k$, with $m_j=\sum_{i=j}^kr_i$ and $s_j=\sum_{i=1}^jr_i$.
Recurrence relations (\ref{eq2.3}) and (\ref{eq2.4}), by replacing the base $q$ by $q^{-1}$, and using the first and the second expression in (\ref{eq2.2}), respectively, are expressed as
\begin{align} \label{eq2.5}
\genfrac{[}{]}{0pt}{}{x}{r_1,r_2,\ldots,r_k}_q=\,&q^{m_1}\genfrac{[}{]}{0pt}{}{x-1}{r_1,r_2,\ldots,r_k}_q
+q^{m_2}\genfrac{[}{]}{0pt}{}{x-1}{r_1-1,r_2,\ldots,r_k}_q \nonumber\\
&+q^{m_3}\genfrac{[}{]}{0pt}{}{x-1}{r_1,r_2-1,\ldots,r_k}_q
+\cdots+\genfrac{[}{]}{0pt}{}{x-1}{r_1,r_2,\ldots,r_k-1}_q.
\end{align}
and
\begin{align} \label{eq2.6}
\genfrac{[}{]}{0pt}{}{x}{r_1,r_2,\ldots,r_k}_q&=\genfrac{[}{]}{0pt}{}{x-1}{r_1,r_2,\ldots,r_k}_q
+q^{x-s_1}\genfrac{[}{]}{0pt}{}{x-1}{r_1-1,r_2,\ldots,r_k}_q \nonumber\\
&+q^{x-s_2}\genfrac{[}{]}{0pt}{}{x-1}{r_1,r_2-1,\ldots,r_k}_q
+\cdots+q^{x-s_k}\genfrac{[}{]}{0pt}{}{x-1}{r_1,r_2,\ldots,r_k-1}_q,
\end{align}
respectively. It should be noticed that reversing the order of labeling the arguments of the $q$-multinomials, these expressions may be transformed to (\ref{eq2.4}) and (\ref{eq2.3}), respectively.
Two versions of a multivariate $q$-Vandermonde formula are derived in the next theorem.
\begin{thm}\label{thm2.1}
Let $n$ be a positive integer and let $x_j$, $j=1,2,\ldots,k+1$, and $q$ be real numbers, with $q\neq 1$. Then,
\begin{eqnarray}\label{eq2.7}
[x_1+x_2+\cdots+x_{k+1}]_{n,q}=\sum\genfrac{[}{]}{0pt}{}{n}{r_1,r_2,\ldots,r_k}_q
q^{\sum_{j=1}^{k}(n-s_j)(x_j-r_j)}\prod_{j=1}^{k+1}[x_j]_{r_j,q},
\end{eqnarray}
and, alternatively,
\begin{eqnarray}\label{eq2.7a}
[x_1\!+\!x_2\!+\!\cdots\!+\!x_{k+1}\!+\!n\!-\!1]_{n,q}\!=\!\sum\genfrac{[}{]}{0pt}{}{n}{r_1,r_2,\ldots,r_k}_q
q^{\sum_{j=1}^{k}r_jz_j}\prod_{j=1}^{k+1}[x_j\!+\!r_j\!-\!1]_{r_j,q}.
\end{eqnarray}
Also,
\begin{eqnarray}\label{eq2.8}
[x_1+x_2+\cdots+x_{k+1}]_{n,q}=\sum\genfrac{[}{]}{0pt}{}{n}{r_1,r_2,\ldots,r_k}_q
q^{\sum_{j=1}^{k}r_j(z_j-(n-s_j))}\prod_{j=1}^{k+1}[x_j]_{r_j,q},
\end{eqnarray}
and, alternatively,
\begin{eqnarray}\label{eq2.8a}
[x_1\!+\!x_2\!+\!\cdots\!+\!x_{k+1}\!+\!n\!-\!1]_{n,q}\!=\!\sum\genfrac{[}{]}{0pt}{}{n}{r_1,r_2,\ldots,r_k}_q
\!q^{\sum_{j=1}^{k}x_j(n\!-\!s_j)}\!\prod_{j=1}^{k+1}[x_j\!+\!r_j\!-\!1]_{r_j,q},
\end{eqnarray}
where $s_j=\sum_{i=1}^jr_i$, $z_j=\sum_{i=j+1}^{k+1}x_i$, $j=1,2,\ldots,k$, and $r_{k+1}=n-s_k$, and the summation, in all four sums, is extended over all $r_j=0,1,\ldots,n$, $j=1,2,\ldots,k$, with $\sum_{i=1}^kr_i\leq n$.
\end{thm}
{\bf Proof}. Consider the sequence of multiple sums
\[
s_n(x_1,x_2,\ldots,x_{k+1};q)=\sum\genfrac{[}{]}{0pt}{}{n}{r_1,r_2,\ldots,r_k}_q
q^{\sum_{j=1}^{k}(n-s_j)(x_j-r_j)}\prod_{j=1}^{k+1}[x_j]_{r_j,q},
\]
for $n=1,2\ldots\,$, with initial value
\[
s_1(x_1,x_2,\ldots,x_{k+1};q)=[x_1]_q+\sum_{j=2}^{k+1}q^{\sum_{i=1}^{j-1}x_i}[x_j]_q=[x_1+x_2+\cdots+x_{k+1}]_q.
\]
Using recurrence relation (\ref{eq2.6}), with $x=n$, the sequence may be expressed as
\begin{align*}
s_n(x_1,x_2,\ldots,x_{k+1};q)
&=\sum\genfrac{[}{]}{0pt}{}{n-1}{r_1,r_2,\ldots,r_k}_q q^{\sum_{j=1}^{k}(n-s_j)(x_j-r_j)}\prod_{j=1}^{k+1}[x_j]_{r_j,q}\\
&+\sum\genfrac{[}{]}{0pt}{}{n-1}{r_1-1,r_2,\ldots,r_k}_q q^{n-s_1+\sum_{j=1}^{k}(n-s_j)(x_j-r_j)}\prod_{j=1}^{k+1}[x_j]_{r_j,q}\\
+\cdots&+\sum\genfrac{[}{]}{0pt}{}{n-1}{r_1,r_2,\ldots,r_k-1}_q q^{n-s_k+\sum_{j=1}^{k}(n-s_j)(x_j-r_j)}\prod_{j=1}^{k+1}[x_j]_{r_j,q}.
\end{align*}
Replacing $r_j-1$ by $r_j$ in the $(j+1)$th multiple sum, for $j=1,2,\ldots,k$, and then executing the multiplications, summations, and cancelations in the exponents of $q$, we get the expression
\begin{align*}
s_n&(x_1,x_2,\ldots,x_{k+1};q)\\
&=\sum\genfrac{[}{]}{0pt}{}{n-1}{r_1,r_2,\ldots,r_k}_q q^{\sum_{j=1}^{k}(n-1-s_j)(x_j-r_j)+\sum_{j=1}^k(x_j-r_j)}[x_{k+1}-r_{k+1}+1]_q\prod_{j=1}^{k+1}[x_j]_{r_j,q}\\
&+\sum\genfrac{[}{]}{0pt}{}{n-1}{r_1,r_2,\ldots,r_k}_q q^{\sum_{j=1}^{k}(n-1-s_j)(x_j-r_j)}[x_1-r_1]_q\prod_{j=1}^{k+1}[x_j]_{r_j,q}+\cdots\\
&+\sum\genfrac{[}{]}{0pt}{}{n-1}{r_1,r_2,\ldots,r_k}_q q^{\sum_{j=1}^{k}(n-1-s_j)(x_j-r_j)+\sum_{j=1}^{k-1}(x_j-r_j)}[x_k-r_k]_q\prod_{j=1}^{k+1}[x_j]_{r_j,q}.
\end{align*}
which, since
\begin{align*}
[x_1-r_1]_q&+q^{x_1-r_1}[x_2-r_2]_q+\cdots+q^{\sum_{j=1}^{k-1}(x_j-r_j)}[x_k-r_k]_q\\
&+q^{\sum_{j=1}^{k}(x_j-r_j)}[x_{k+1}-r_{k+1}+1]_q=[x_1,x_2,\ldots,x_{k+1}-n+1]_q
\end{align*}
implies for the sequence $s_n(x_1,x_2,\ldots,x_{k+1};q)$, $n=1,2\ldots\,$, the first-order recurrence relation
\[
s_n(x_1,x_2,\ldots,x_{k+1};q)=[x_1,x_2,\ldots,x_{k+1}-n+1]_q s_{n-1}(x_1,x_2,\ldots,x_{k+1};q),
\]
for $n=1,2\ldots\,$, with initial condition $s_1(x_1,x_2,\ldots,x_{k+1};q)=[x_1+x_2+\cdots+x_{k+1}]_q$. Applying it successively, it follows that $s_n(x_1,x_2,\ldots,x_{k+1};q)=[x_1+x_2+\cdots+x_{k+1}]_{n,q}$, and so (\ref{eq2.7}) is shown.
Formula (\ref{eq2.8}) may be derived by following the steps of the derivation of (\ref{eq2.7}) and using recurrence relation (\ref{eq2.4}), with $x=n$, and the expression
\begin{align*}
[x_{k+1}-r_{k+1}+1]_q&+q^{(x_{k+1}-r_{k+1})+1}[x_k-r_k]_q+\cdots+q^{\sum_{j=3}^{k+1}(x_j-r_j)+1}[x_2-r_2]_q\\
&+q^{\sum_{j=2}^{k+1}(x_j-r_j)+1}[x_1-r_1]_q=[x_1,x_2,\ldots,x_{k+1}-n+1]_q.
\end{align*}
The alternative formulae (\ref{eq2.7a}) and (\ref{eq2.8a}) are deduced from (\ref{eq2.7}) and (\ref{eq2.8}), respectively, by replacing $x_j$ by $-x_j$, for $j=1,2,\ldots,k+1$ and using the relations
\[
[-x]_{r,q}=(-1)^xq^{-xr-\binom{r}{2}}[x+r-1]_{r,q}, \ \ \binom{n}{2}=\sum_{j=1}^{k+1}\binom{r_j}{2}+\sum_{j=1}^kr_j(n-s_j).
\]
Two versions of a multivariate $q$-Cauchy formula, which by virtue of
\[
\genfrac{[}{]}{0pt}{}{n}{r_1,r_2,\ldots,r_k}_q=\frac{[n]_q!}{[r_1]_q![r_2]_q!\cdots[r_k]_q![r_{k+1}]_q!},\ \
\genfrac{[}{]}{0pt}{}{x_j}{r_j}_q=\frac{[x_j]_{r_j,q}}{[r_j]_q!},
\]
constitute reformulations of the corresponding two versions of a multivariate $q$-Vander\-monde formula, are stated in the following corollary of Theorem 2.1.
\begin{cor}\label{cor2.1}
Let $n$ be a positive integer and let $x_j$, $j=1,2,\ldots,k+1$, and $q$ be real numbers, with $q\neq 1$. Then,
\begin{eqnarray}\label{eq2.9}
\genfrac{[}{]}{0pt}{}{x_1+x_2+\cdots+x_{k+1}}{n}_q=\sum q^{\sum_{j=1}^{k}(n-s_j)(x_j-r_j)}\prod_{j=1}^{k+1}\genfrac{[}{]}{0pt}{}{x_j}{r_j}_q
\end{eqnarray}
and, alternatively,
\begin{eqnarray}\label{eq2.9a}
\genfrac{[}{]}{0pt}{}{x_1+x_2+\cdots+x_{k+1}+n-1}{n}_q=\sum q^{\sum_{j=1}^{k}r_jz_j}\prod_{j=1}^{k+1}\genfrac{[}{]}{0pt}{}{x_j+r_j-1}{r_j}_q.
\end{eqnarray}
Also,
\begin{eqnarray}\label{eq2.10}
\genfrac{[}{]}{0pt}{}{x_1+x_2+\cdots+x_{k+1}}{n}_q=\sum q^{\sum_{j=1}^{k}r_j(z_j-(n-s_j))}\prod_{j=1}^{k+1}\genfrac{[}{]}{0pt}{}{x_j}{r_j}_q
\end{eqnarray}
and, alternatively,
\begin{eqnarray}\label{eq2.10a}
\genfrac{[}{]}{0pt}{}{x_1+x_2+\cdots+x_{k+1}+n-1}{n}_q
=\sum q^{\sum_{j=1}^{k}x_j(n\!-\!s_j)}\prod_{j=1}^{k+1}\genfrac{[}{]}{0pt}{}{x_j+r_j-1}{r_j}_q,
\end{eqnarray}
where $s_j=\sum_{i=1}^jr_i$, $z_j=\sum_{i=j+1}^{k+1}x_i$, $j=1,2,\ldots,k$, and $r_{k+1}=n-s_k$ and the summation, in all four sums, is extended over all $r_j=0,1,\ldots,n$, $j=1,2,\ldots,k$, with $\sum_{i=1}^kr_i\leq n$.
\end{cor}
\begin{rem}
Additional expressions of the multivariate $q$-Cauchy formulae. {\em The alter\-native expressions (\ref{eq2.9a}) and (\ref{eq2.10a}), which are useful in probability theory, may be rewritten as
\begin{eqnarray}\label{eq2.9b}
\genfrac{[}{]}{0pt}{}{r+k}{n+k}_q=\sum q^{\sum_{j=1}^{k}(r_j-x_j)(n-y_j+k-j+1)}\prod_{j=1}^{k+1}\genfrac{[}{]}{0pt}{}{r_j}{x_j}_q
\end{eqnarray}
and
\begin{eqnarray}\label{eq2.10b}
\genfrac{[}{]}{0pt}{}{r+k}{n+k}_q
=\sum q^{\sum_{j=1}^{k}(x_j+1)(n-s_j-r+y_j)}\prod_{j=1}^{k+1}\genfrac{[}{]}{0pt}{}{r_j}{x_j}_q,
\end{eqnarray}
respectively, where $s_j=\sum_{i=1}^jr_i$, $y_j=\sum_{i=1}^{j}x_i$, $j=1,2,\ldots,k$, $r_{k+1}=r-s_k$, $x_{k+1}=r-y_k$, and the summation, in both sums, is extended over all $r_j=x_j,x_j+1,\ldots,r$, $j=1,2,\ldots,k$, with $\sum_{i=1}^kr_i\leq r$. Indeed, replacing, the bound variable $r_j$ by $r_j-(x_j-1)$ and the constant $x_j$ by $x_j+1$, for all $j=1,2,\ldots,k+1$, formulae (\ref{eq2.9a}) and (\ref{eq2.10a}), after some algebra, are transformed to (\ref{eq2.9b}) and (\ref{eq2.10b}), respectively.
}
\end{rem}
Two versions of a multivariate inverse $q$-Vandermonde formula are derived in the following theorem.
\begin{thm}\label{thm2.2}
Let $n$ be a positive integer and let $x_j$, $j=1,2,\ldots,k+1$, and $q$ be real numbers, with $q\neq 1$. Then,
\begin{eqnarray}\label{eq2.11}
\frac{1}{[x_{k+1}]_{n,q}}\!=\!\sum\genfrac{[}{]}{0pt}{}{n+s_k-1}{r_1,r_2,\ldots,r_k}_q
q^{\sum_{j=1}^{k}(n\!+\!s_k\!-\!s_j)(x_j-r_j)}\frac{\prod_{j=1}^k[x_j]_{r_j,q}}{[x_1+x_2+\cdots+x_{k+1}]_{n+s_k,q}},
\end{eqnarray}
provided $|q^{-x_{k+1}}|<1$, and
\begin{eqnarray}\label{eq2.12}
\frac{1}{[x_{k+1}]_{n,q}}\!=\!\sum\genfrac{[}{]}{0pt}{}{n+s_k-1}{r_1,r_2,\ldots,r_k}_q
q^{\sum_{j=1}^{k}r_j(z_j\!-\!s_k\!+\!s_j\!-\!n\!+\!1)}\frac{\prod_{j=1}^k[x_j]_{r_j,q}}{[x_1+x_2+\cdots+x_{k+1}]_{n+s_k,q}},
\end{eqnarray}
provided $|q^{x_{k+1}}|<1$, where $s_j=\sum_{i=1}^jr_i$ and $z_j=\sum_{i=j+1}^{k+1}x_i$, for $j=1,2,\ldots,k$, and the summation, in both sums, is extended over all $r_j=0,1,\ldots\,$, $j=1,2,\ldots,k$.
\end{thm}
{\bf Proof}. According to an inverse $q$-Vandermonde formula (Charalambides (2016), p. 14), it holds true
\[
\frac{1}{[x_{k+1}]_{n,q}}=\sum_{r_k=0}^\infty\genfrac{[}{]}{0pt}{}{n+r_k-1}{r_k}_q
q^{n(x_k-r_k)}\frac{[x_k]_{r_k,q}}{[x_k+x_{k+1}]_{n+r_k,q}}.
\]
Similarly,
\[
\frac{1}{[x_k+x_{k+1}]_{n+r_k,q}}=\sum_{r_{k-1}=0}^\infty\genfrac{[}{]}{0pt}{}{n+r_k+r_{k-1}-1}{r_{k-1}}_q
\frac{q^{(n+r_k)(x_{k-1}-r_{k-1})}[x_{k-1}]_{r_{k-1},q}}{[x_{k-1}+x_k+x_{k+1}]_{n+r_k+r_{k-1},q}}
\]
and finally,
\[
\frac{1}{[x_2+x_3+\cdots+x_{k+1}]_{n+s_k-s_1,q}}=\sum_{r_1=0}^\infty\genfrac{[}{]}{0pt}{}{n+s_k-1}{r_1}_q
\frac{q^{(n+s_k-s_1)(x_1-r_1)}[x_1]_{r_1,q}}{[x_1+x_2+\cdots+x_{k+1}]_{n+s_k,q}}.
\]
Applying these $k$ expansions, one after the other in the inner sum of each step, and using the relation
\[
\genfrac{[}{]}{0pt}{}{n+r_k-1}{r_k}_q\genfrac{[}{]}{0pt}{}{n+r_k+r_{k-1}-1}{r_{k-1}}_q\genfrac{[}{]}{0pt}{}{n+s_k-1}{r_1}_q
=\genfrac{[}{]}{0pt}{}{n+s_k-1}{r_1,r_2,\ldots,r_k}_q,
\]
expansion (\ref{eq2.11}) is obtained. The alternative expansion (\ref{eq2.12}), is similarly deduced by using the following inverse $q$-Vandermonde expansions (Charalambides (2016), p. 14)
\[
\frac{1}{[x_{j+1}+\cdots+x_{k+1}]_{n+s_k-s_j,q}}=\sum_{r_j=0}^\infty\genfrac{[}{]}{0pt}{}{n+s_k-s_{j-1}-1}{r_j}_q
\frac{q^{r_j(z_j-s_k+s_j-n+1)}[x_j]_{r_j,q}}{[x_j+\cdots+x_{k+1}]_{n+s_k-s_{j-1},q}},
\]
for $j=1,2,\ldots,k$, with $s_0=0$.
\section{Multivariate {\emph{q}}-P\'{o}lya distribution}\label{sec3}
\setcounter{equation}{0}
A multiple $q$-P\'{o}lya urn model may be introduced, by first defining a $q$-analogue of the notion of a random drawing of a ball from an urn.
Consider an urn containing $r$ balls, $\{b_1,b_2,\ldots,b_r\}$, of $k+1$ different ordered colors, with $r_\nu$ distinct balls of color $c_\nu$, $\{b_{s_{\nu-1}+1},b_{s_{\nu-1}+2},\ldots,b_{s_\nu}\}$, for $\nu=1,2,\ldots,k+1$, where $s_0=0$, $s_\nu=\sum_{i=1}^\nu r_i$, for $\nu=1,2,\ldots,k+1$, with $s_{k+1}=r$. A {\em random $q$-drawing (or $q$-selection)} of a ball from the urn is carried out as follows. Assume that the balls in the urn are forced to pass through a random mechanism, one by one, in the order $(b_1,b_2,\ldots,b_r)$ or in the reverse order $(b_r,b_{r-1},\ldots,b_1)$. Also, suppose that each passing ball may or may not be caught by the mechanism, with probabilities $p=1-q$ and $q$, respectively. The first caught ball is drawn out of the urn. In the case all balls in the urn pass through the mechanism and no ball is caught, the ball passing procedure is repeated, with the same order. Clearly, the probability that ball $b_x$ is drawn from the urn is given by
\[
\sum^\infty_{k=0}(1-q)q^{(x-1)+rk}=(1-q)q^{x-1}\sum^\infty_{k=0}q^{rk}=\frac{q^{x-1}}{[r]_q},
\]
or by
\[
\sum^\infty_{k=0}(1-q)q^{(r-x)+rk}=\frac{q^{r-x}}{[r]_q}=\frac{q^{-(x-1)}}{[r]_{q^{-1}}},
\]
where $0<q<1$, according to whether the ball passing order is $(b_1,b_2,\ldots,b_r)$ or $(b_r,b_{r-1},\ldots,b_1)$. Consequently, the probability function of the number $N_r$ on the drawn ball is given by
\[
p_{r}(x;q)=P(N_r=x)=\frac{q^{x-1}}{[r]_q},\ \ x=1,2,\ldots,r,
\]
where $0<q<1$ or $1<q<\infty$. Note that this is the probability function of the discrete $q$-uniform distribution on the set $\{1,2,\ldots,r\}$. Also, the probability $P_r(r_\nu;q)$, that a ball of color $c_\nu$ is drawn from the urn is given by
\[
P_r(r_\nu;q)=P(s_{\nu-1}<N_r\leq s_\nu)=\frac{q^{s_{\nu-1}}[r_\nu]_q}{[r]_q}=\frac{q^{-(r-s_\nu)}[r_\nu]_{q^{-1}}}{[r]_{q^{-1}}},
\]
for $\nu=1,2,\ldots,k+1$, with $s_0=0$, where $0<q<1$ or $1<q<\infty$. As expected, the sum of these probabilities, on using successively the relation $[s]_q+q^s[r]_q=[s+r]_q$, is obtained as
\[
\sum_{\nu=1}^{k+1}P_r(r_\nu;q)=\frac{1}{[r]_q}\sum_{\nu=1}^{k+1}q^{s_{\nu-1}}[r_\nu]_q=\frac{[r_1+r_2+\cdots+r_{k+1}]_q}{[r]_q}=1,
\]
where $0<q<1$ or $1<q<\infty$. Finally, notice that a random $q$-drawing of a ball, for $q\rightarrow 1$ and since
\[
\lim_{q\rightarrow 1}P_{r}(r_\nu;q)=\frac{r_\nu}{r},\ \ \nu=1,2,\ldots,k+1,
\]
reduces to the usual random drawing of a ball from the urn.
Furthermore, assume that random $q$-drawings of balls are sequentially carried out, one after the other, from an urn, initially containing $r$ balls of $k+1$ different colors, with $r_\nu$ distinct balls of color $c_\nu$, for $\nu=1,2,\ldots,k+1$, according to the following scheme. After each $q$-drawing, the drawn ball is placed back in the urn together with $m$ balls of the same color. Then, the conditional probability of drawing a ball of color $c_\nu$ at the $i$th $q$-drawing, given that $j_\nu-1$ balls of color $c_\nu$ and a total of $i_{\nu-1}$ balls of colors $c_1,c_2,\ldots,c_{\nu-1}$ are drawn in the previous $i-1$ $q$-drawings, is given by
\begin{align}\label{eq3.1}
p_{i,j_\nu}(i_{\nu-1})&=\frac{q^{s_{\nu-1}+mi_{\nu-1}}\big(1-q^{r_\nu+m(j_\nu-1)}\big)}{1-q^{r+m(i-1)}}
=\frac{q^{-m(\beta_{\nu-1}-i_{\nu-1})}[\alpha_\nu-j_\nu+1]_{q^{-m}}}{[\alpha-i+1]_{q^{-m}}},
\end{align}
for $j_\nu=1,2,\ldots,i$, $i_\nu=0,1,\ldots,i-1$, $i=1,2,\ldots\,$, and $\nu=1,2,\ldots,k+1$, with $i_0=0$, where $0<q<1$ or $1<q<\infty$ and $\alpha=-r/m$, $\alpha_\nu=-r_\nu/m$, $\beta_\nu=-s_\nu/m$, $\nu=1,2,\ldots,k+1$, $\beta_0=0$. Note that $i_\nu=i_{\nu-1}+j_\nu=j_1+j_2+\cdots+j_\nu$, for $\nu=1,2,\ldots,k+1$. This model, which for $q\rightarrow 1$ and since
\[
p_{i,j_\nu}=\lim_{q\rightarrow 1}p_{i,j_\nu}(i_{\nu-1})=\frac{r_\nu+m(j_\nu-1)}{r+m(i-1)}
=\frac{\alpha_\nu-j_\nu+1}{\alpha-i+1},\ \ \alpha_\nu=-\frac{r_\nu}{m}, \ \ \alpha=-\frac{r}{m},
\]
for $j_\nu=1,2,\ldots,i$, $i=1,2,\ldots\,$, and $\nu=1,2,\ldots,k+1$, reduces to the (classical) multiple P\'{o}lya urn model, may be called {\em multiple $q$-P\'{o}lya urn model}.
\begin{Def}\label{def3.1}
Let $X_\nu$ be the number of balls of color $c_\nu$ drawn in $n$ $q$-drawings in a multiple $q$-P\'{o}lya urn model, with conditional probability of drawing a ball of color $c_\nu$ at the $i$th $q$-drawing, given that $j_\nu-1$ balls of color $c_\nu$ and a total of $i_{\nu-1}$ balls of colors $c_1,c_2,\ldots,c_{\nu-1}$ are drawn in the previous $i-1$ $q$-drawings, is given by (\ref{eq3.1}), for $\nu=1,2,\ldots,k$. The distribution of the random vector $(X_1,X_2,\ldots, X_k)$ is called $k$-variate $q$-P\'{o}lya distribution, with parameters $n$, $(\alpha_1,\alpha_2,\ldots,\alpha_k)$, $\alpha$, and $q$.
\end{Def}
The probability function of the $k$-variate $q$-P\'{o}lya distribution is obtained in the following theorem.
\begin{thm}\label{thm3.1}
The probability function of the $k$-variate $q$-P\'{o}lya distribution, with parameters $n$, $(\alpha_1,\alpha_2,\ldots,\alpha_k)$, $\alpha$, and $q$, is given by
\begin{align}\label{eq3.2}
P(X_1\!=\!x_1,X_2\!=\,&x_2,\ldots,X_k\!=\!x_k)=q^{-m\sum_{j=1}^{k}(n-y_j)(\alpha_j-x_j)}\prod_{j=1}^{k+1}\genfrac{[}{]}{0pt}{}{\alpha_j}{x_j}_{q^{-m}}
\bigg/\genfrac{[}{]}{0pt}{}{\alpha}{n}_{q^{-m}}\nonumber\\
&=\genfrac{[}{]}{0pt}{}{n}{x_1,x_2,\ldots, x_k}_{q^{-m}}
q^{-m\sum_{j=1}^{k}(n-y_j)(\alpha_j-x_j)}\frac{\prod_{j=1}^{k+1}[\alpha_j]_{x_j,q^{-m}}}{[\alpha]_{n,q^{-m}}},
\end{align}
for $x_j=0,1,\ldots,n$, $j=1,2,\ldots,k$, with $\sum_{j=1}^k x_j\leq n$, and $0<q<1$ or $1<q<\infty$, where $x_{k+1}=n-\sum_{j=1}^k x_j$, $\alpha_{k+1}=\alpha-\sum_{j=1}^k \alpha_j$, and $y_j=\sum_{i=1}^{j}x_i$, for $j=1,2,\ldots,k$.
\end{thm}
{\bf Proof}. The probability function $p_n(x_1,x_2,\ldots,x_k)=P(X_1\!=\!x_1,X_2\!=\!x_2,\ldots,X_k\!=\!x_k)$, on using the total probability theorem, satisfies the recurrence relation
\begin{align*}
p_n(x_1,x_2,\ldots,x_k)=&\,p_{n-1}(x_1,x_2,\ldots,x_k)\frac{q^{-m(\beta_k-y_k)}[\alpha_{k+1}-x_{k+1}+1]_{q^{-m}}}{[\alpha-n+1]_{q^{-m}}}\\
&+p_{n-1}(x_1-1,x_2,\ldots,x_k)\frac{[\alpha_1-x_1+1]_{q^{-m}}}{[\alpha-n+1]_{q^{-m}}}\\
&+p_{n-1}(x_1,x_2-1,\ldots,x_k)\frac{q^{-m(\beta_1-y_1)}[\alpha_2-x_2+1]_{q^{-m}}}{[\alpha-n+1]_{q^{-m}}}+\cdots\\
&+p_{n-1}(x_1,x_2,\ldots,x_k-1)\frac{q^{-m(\beta_{k-1}-y_{k-1})}[\alpha_k-x_k+1]_{q^{-m}}}{[\alpha-n+1]_{q^{-m}}},
\end{align*}
for $x_j=1,2,\ldots,n$, $j=1,2,\ldots,k$ and $n=1,2,\ldots\,$, with $\sum_{j=1}^kx_j\leq n$, $x_{k+1}=n-\sum_{j=1}^kx_j$, and
$p_0(0,0,\ldots,0)=1$. Also,
\[
p_n(0,0,\ldots,0)=\frac{\prod_{i=1}^nq^{-m\beta_k}[a_{k+1}-i+1]_{q^{-m}}}{\prod_{i=1}^n[a-i+1]_{q^{-m}}}
=\frac{q^{-mn\beta_k}[a_{k+1}]_{n,q^{-m}}}{[a]_{n,q^{-m}}}.
\]
Clearly, the sequence
\begin{eqnarray}\label{eq3.3}
c_n(x_1,x_2,\ldots,x_k)
=q^{m\sum_{j=1}^{k}(n-y_j)(\alpha_j-x_j)}\frac{[\alpha]_{n,q^{-m}}}{\prod_{j=1}^{k+1}[\alpha_j]_{x_j,q^{-m}}}p_n(x_1,x_2,\ldots,x_k)
\end{eqnarray}
satisfies the recurrence relation
\begin{align*}
c_n(x_1,x_2,\ldots,x_k)=c_{n-1}(x_1,x_2,\ldots,x_k)&+q^{-m(n-y_1)}c_{n-1}(x_1-1,x_2,\ldots,x_k)\\
&+q^{-m(n-y_2)}c_{n-1}(x_1,x_2-1,\ldots,x_k)\\
+\cdots&+q^{-m(n-y_k)}c_{n-1}(x_1,x_2,\ldots,x_k-1),
\end{align*}
for $x_j=1,2,\ldots,n$, $j=1,2,\ldots,k$ and $n=1,2,\ldots\,$, with $\sum_{j=1}^kx_j\leq n$, and $c_0(0,0,\ldots,0)\linebreak=1$. Since this recurrence relation, according to (\ref{eq2.6}), uniquely determines the $q$-multinomial coefficient,
\[
c_n(x_1,x_2,\ldots,x_k)=\genfrac{[}{]}{0pt}{}{n}{x_1,x_2,\ldots, x_k}_{q^{-m}},
\]
the second part of expression (\ref{eq3.2}) is readily deduced from (\ref{eq3.3}). The first part of (\ref{eq3.2}), which is a reformulation of the second, is deduced by using the expressions
\[
\genfrac{[}{]}{0pt}{}{n}{x_1,x_2,\ldots,x_k}_q=\frac{[n]_q!}{[x_1]_q![x_2]_q!\cdots[x_k]_q![x_{k+1}]_q!},\ \
\genfrac{[}{]}{0pt}{}{\alpha_j}{x_j}_q=\frac{[\alpha_j]_{x_j,q}}{[x_j]_q!}.
\]
Note that the multivariate $q$-Vandermonde formula (\ref{eq2.7}) and, equivalently, the multivariate $q$-Cauchy formula (\ref{eq2.9}), guarantees that the probabilities (\ref{eq3.2}) sum to unity.
Certain (and not any) marginal and conditional distributions of a $k$-variate $q$-P\'{o}lya distribution are derived in the next theorem.
\begin{thm}\label{thm3.2}
Suppose that the random vector $(X_1,X_2,\ldots,X_k)$ obeys a $k$-variate $q$-P\'{o}lya distribution, with parameters $n$, $(\alpha_1,\alpha_2,\ldots,\alpha_k)$, $\alpha$, and $q$. Then,
(a) the marginal distribution of the random vector $(X_1,X_2,\ldots,X_\nu)$ is a $\nu$-variate $q$-P\'{o}lya, with parameters $n$, $(\alpha_1,\alpha_2,\ldots,\alpha_\nu)$, $\alpha$, and $q$, for $\nu=1,2,\ldots,k$, and
(b) the conditional distribution of the random vector $(X_{\nu},X_{\nu+1},\ldots,X_{\nu+\kappa-1})$, given that $(X_1,X_2,\ldots,X_{\nu-1})=(x_1,x_2,\ldots,x_{\nu-1})$ is a $\kappa$-variate $q$-P\'{o}lya, with parameters $n$, $(\alpha_\nu,\alpha_{\nu+1},\ldots,\alpha_{\nu+\kappa-1})$, $\alpha-\alpha_1-\alpha_2-\cdots-\alpha_{\nu-1}$, and $q$, for $\kappa=1,2,\ldots,k-\nu+1$ and $\nu=1,2,\ldots,k$.
\end{thm}
{\bf Proof}. (a) Summing the probabilities (\ref{eq3.2}) for $x_j=0,1,\ldots,n-y_{\nu}$, $j=\nu+1,\linebreak\nu+2,\ldots,k$, with
$x_{\nu+1}+x_{\nu+2}+\cdots+x_k\leq n-y_\nu$, and using (\ref{eq2.9}), we get
\begin{align*}
P(X_1&=x_1,X_2=x_2,\ldots,X_{\nu}=x_{\nu})=q^{-m\sum_{j=1}^{\nu}(n-y_j)(\alpha_j-x_j)}
\prod_{j=1}^{\nu}\genfrac{[}{]}{0pt}{}{\alpha_j}{x_j}_{q^{-m}}\\
&\times\sum q^{-m\sum_{j=\nu+1}^{k}(n-y_j)(\alpha_j-x_j)}\prod_{j=\nu+1}^{k}\genfrac{[}{]}{0pt}{}{\alpha_j}{x_j}_{q^{-m}}
\genfrac{[}{]}{0pt}{}{\alpha-\alpha_1-\cdots-\alpha_k}{n-x_1-\cdots-x_k}_{q^{-m}}\bigg/\genfrac{[}{]}{0pt}{}{\alpha}{n}_{q^{-m}}\\
&\hspace{1.3cm}=q^{-m\sum_{j=1}^{\nu}(n-y_j)(\alpha_j-x_j)}\prod_{j=1}^{\nu}\genfrac{[}{]}{0pt}{}{\alpha_j}{x_j}_{q^{-m}}
\genfrac{[}{]}{0pt}{}{\alpha-\alpha_1-\cdots-\alpha_\nu}{n-x_1-\cdots-x_\nu}_{q^{-m}}
\bigg/\genfrac{[}{]}{0pt}{}{\alpha}{n}_{q^{-m}},
\end{align*}
which is the probability function of a $\nu$-variate $q$-P\'{o}lya distribution, with parameters $n$, $(\alpha_1,\alpha_2,\ldots,\alpha_\nu)$, $\alpha$, and $q$.
(b) The probability function of the conditional distribution of the random vector $(X_{\nu},X_{\nu+1},\ldots,X_{\nu+\kappa-1})$, given that $(X_1,X_2,\ldots,X_{\nu-1})=(x_1,x_2,\ldots,x_{\nu-1})$, using the expression
\begin{align*}
P(X_\nu=x_\nu,\ldots,X_{\nu+\kappa-1}&=x_{\nu+\kappa-1}|X_1=x_1,X_2=x_2,\ldots,X_{\nu-1}=x_{\nu-1})\\
&=\frac{P(X_1=x_1,X_2=x_2,\ldots,X_{\nu+\kappa-1}=x_{\nu+\kappa-1})}{P(X_1=x_1,X_2=x_2,\ldots,X_{\nu-1}=x_{\nu-1})},
\end{align*}
is obtained as
\begin{align*}
P(X_\nu&=x_\nu,\ldots,X_{\nu+\kappa-1}=x_{\nu+\kappa-1}|X_1=x_1,X_2=x_2,\ldots,X_{\nu-1}=x_{\nu-1})\\
&=\frac{q^{-m\sum_{j=1}^{\nu+\kappa-1}(n-y_j)(\alpha_j-x_j)}\prod_{j=1}^{\nu+\kappa-1}\genfrac{[}{]}{0pt}{}{\alpha_j}{x_j}_{q^{-m}}
\genfrac{[}{]}{0pt}{}{\alpha-\alpha_1-\cdots-\alpha_{\nu+\kappa-1}}{n-x_1-\cdots-x_{\nu+\kappa-1}}_{q^{-m}}
\big/\genfrac{[}{]}{0pt}{}{\alpha}{n}_{q^{-m}}}{q^{-m\sum_{j=1}^{\nu-1}(n-y_j)(\alpha_j-x_j)}\prod_{j=1}^{\nu-1}
\genfrac{[}{]}{0pt}{}{\alpha_j}{x_j}_{q^{-m}}\genfrac{[}{]}{0pt}{}{\alpha-\alpha_1-\cdots-\alpha_{\nu-1}}{n-x_1-\cdots-x_{\nu-1}}_{q^{-m}}
\big/\genfrac{[}{]}{0pt}{}{\alpha}{n}_{q^{-m}}}\\
&=q^{-m\sum_{j=\nu}^{\nu+\kappa-1}(n-y_j)(\alpha_j-x_j)}\prod_{j=\nu}^{\nu+\kappa-1}\genfrac{[}{]}{0pt}{}{\alpha_j}{x_j}_{q^{-m}}
\genfrac{[}{]}{0pt}{}{\alpha-\alpha_\nu-\cdots-\alpha_{\nu+\kappa-1}}{n-x_\nu-\cdots-x_{\nu+\kappa-1}}_{q^{-m}}
\bigg/\genfrac{[}{]}{0pt}{}{\alpha}{n}_{q^{-m}},
\end{align*}
which is the probability function of a $\kappa$-variate $q$-P\'{o}lya distribution, with parameters $n$, $(\alpha_\nu,\alpha_{\nu+1},\ldots,\alpha_{\nu+\kappa-1})$, $\alpha-\alpha_1-\alpha_2-\cdots-\alpha_{\nu-1}$, and $q$.
The multivariate $q$-P\'{o}lya distribution, for large $r$, can be approximated by a $q$-multinomial distribution of the second kind, which is introduced and studied in Charalambides (2020). Specifically, the following limiting theorem is derived.
\begin{thm}\label{thm3.3}
Consider the multivariate $q$-P\'{o}lya distribution, with probability function $p_{n}(r;m,q)=P(X_1=x_1,X_2=x_2,\ldots,X_k=x_k)$ given by (\ref{eq3.2}).
For $0<q<1$, assume that
\begin{eqnarray}\label{eq3.4}
\lim_{r\rightarrow\infty}\frac{[r-s_j]_{q{-1}}}{[r-s_{j-1}]_{q{-1}}}=\theta_j, \ \ s_j=\sum_{i=1}^jr_i, \ \ j=1,2,\ldots,k,\ \ s_0=0,
\end{eqnarray}
and in the case of a negative integer $m$ assume, in addition, that $\theta_j<q^{-m(\nu-1)}$, $j=1,2,\ldots,k$, for some positive integer $\nu$. Then,
\begin{eqnarray}\label{eq3.5}
\lim_{r\rightarrow\infty}p_{n}(r;m,q)=\genfrac[]{0pt}{}{n}{x_1,x_2,\ldots, x_k}_{q^{m}}
\prod_{j=1}^k\theta_j^{n-y_j}\prod_{i_j=1}^{x_j}(1-\theta_jq^{m(i_j-1)}),
\end{eqnarray}
for $x_j=0,1,\ldots,n$, $j=1,2,\ldots,k$, with $x_1+x_2+\cdots+x_k\leq n$, where $y_j=\sum_{i=1}^{j}x_i$, $0<q<1$ and $0<\theta_j<1$, $j=1,2,\ldots,k$, in the case $m$ is a positive integer, or $0<\theta_j<q^{-m(\nu-1)}$, $j=1,2,\ldots,k$, for some positive integer $\nu\ge n$, in the case $m$ is a negative integer.
Also, for $1<q<\infty$, assume that
\begin{eqnarray}\label{eq3.6}
\lim_{r\rightarrow\infty}\frac{[r_j]_{q}}{[r-s_{j-1}]_{q}}=\lambda_j, \ \ s_j=\sum_{i=1}^jr_i, \ \ j=1,2,\ldots,k,\ \ s_0=0,
\end{eqnarray}
and in the case of a negative integer $m$ assume, in addition, that $\lambda_j<q^{m(\nu-1)}$, $j=1,2,\ldots,k$, for some positive integer $\nu$. Then,
\begin{eqnarray}\label{eq3.7}
\lim_{r\rightarrow\infty}p_{n}(r;m,q)=\genfrac[]{0pt}{}{n}{x_1,x_2,\ldots, x_k}_{q^{-m}}
\prod_{j=1}^k\lambda_j^{x_j}\prod_{i_j=1}^{n-y_j}(1-\lambda_jq^{-m(i_j-1)}),
\end{eqnarray}
for $x_j=0,1,\ldots,n$, $j=1,2,\ldots,k$, with $x_1+x_2+\cdots+x_k\leq n$, where $y_j=\sum_{i=1}^{j}x_i$, $1<q<\infty$ and $0<\lambda_j<1$, in the case $m$ is a positive integer, or $0<\lambda_j<q^{m(\nu-1)}$, for some positive integer $\nu\ge n$, in the case $m$ is a negative integer.
\end{thm}
{\bf Proof}. For $0<q<1$, the probability function (\ref{eq3.2}),
\begin{align*}
p_{n}(r;m,q)=\genfrac{[}{]}{0pt}{}{n}{x_1,x_2,\ldots, x_k}_{q^{-m}}
\prod_{j=1}^{k}q^{-m(n-y_j)(\alpha_j-x_j)}
\frac{[\alpha_j]_{x_j,q^{-m}}[\alpha-\beta_j]_{n-y_j,q^{-m}}}{[\alpha-\beta_{j-1}]_{n-y_{j-1},q^{-m}}},
\end{align*}
using (\ref{eq2.2}), may be written as
\begin{align*}
p_{n}(r;m,q)&=\genfrac{[}{]}{0pt}{}{n}{x_1,x_2,\ldots, x_k}_{q^{m}}
\prod_{j=1}^{k}\prod_{i_j=1}^{x_j}(q^{-r_j-m(i_j-1)}-1)q^{-r+s_j+m(i_j-1)}\\
&\hspace{4.3cm}\frac{\prod_{i_j=1}^{n-y_j}(q^{-r+s_j-m(i_j-1)}-1)q^{m(i_j-1)}}{\prod_{i_{j-1}=1}^{n-y_{j-1}}(q^{-r+s_{j-1}-m(i_{j-1}-1)}-1)q^{m(i-1)}}.
\end{align*}
Moreover, by the assumption (\ref{eq3.4}), it follows that
\begin{align*}
\lim_{r\rightarrow\infty}\frac{(q^{-r_j-m(i_j-1)}-1)q^{-r+s_j+m(i_j-1)}}{q^{-r+s_{j-1}}-1}=1&-q^{m(i_j-1)}
\lim_{r\rightarrow\infty}\frac{q^{-r+s_j}-1}{q^{-r+s_{j-1}}-1}\\
&+\lim_{r\rightarrow\infty}\frac{q^{m(i_j-1)}-1}{q^{-r+s_{j-1}}-1}=1-\theta_jq^{m(i_j-1)},
\end{align*}
\begin{align*}
\lim_{r\rightarrow\infty}\frac{(q^{-r+s_j-m(i_j-1)}-1)q^{m(i_j-1)}}{q^{-r+s_{j-1}}-1}&=
\lim_{r\rightarrow\infty}\frac{q^{-r+s_j}-1}{q^{-r+s_{j-1}}-1}
-\lim_{r\rightarrow\infty}\frac{q^{m(i_j-1)}-1}{q^{-r+s_{j-1}}-1}=\theta_j,
\end{align*}
and
\begin{align*}
\lim_{r\rightarrow\infty}\frac{(q^{-r+s_{j-1}-m(i_j-1)}-1)q^{m(i_j-1)}}{q^{-r+s_{j-1}}-1}&=
1-\lim_{r\rightarrow\infty}\frac{q^{m(i_j-1)}-1}{q^{-r+s_{j-1}}-1}=1,
\end{align*}
for $j=1,2,\ldots,k$. Thus, dividing both the numerator and denominator of the $j$th factor in the last expression of the probability function (\ref{eq3.2}) by $(q^{-r+s_{j-1}}-1)^{n-y_{j-1}}$ and taking the limits as $r\rightarrow\infty$, the limiting expression (\ref{eq3.5}) is readily deduced.
For $1<q<\infty$, the probability function (\ref{eq3.2}),
\begin{align*}
p_{n}(r;m,q)=\genfrac{[}{]}{0pt}{}{n}{x_1,x_2,\ldots, x_k}_{q^{-m}}
\prod_{j=1}^{k}q^{-m(n-y_j)(\alpha_j-x_j)}
\frac{[\alpha_j]_{x_j,q^{-m}}[\alpha-\beta_j]_{n-y_j,q^{-m}}}{[\alpha-\beta_{j-1}]_{n-y_{j-1},q^{-m}}},
\end{align*}
may be written as
\begin{align*}
p_{n}(r;m,q)&=\genfrac{[}{]}{0pt}{}{n}{x_1,x_2,\ldots, x_k}_{q^{-m}}
\prod_{j=1}^{k}\prod_{i_j=1}^{x_j}(1-q^{r_j+m(i_j-1)})q^{-m(i_j-1)}\\
&\hspace{4.1cm}\times\frac{\prod_{i_j=1}^{n-y_j}(1-q^{r-s_j+m(i_j-1)})q^{r_j-m(i_j-1)}}
{\prod_{i_{j-1}=1}^{n-y_{j-1}}(1-q^{r-s_{j-1}+m(i_{j-1})})q^{-m(i_{j-1}-1)}}.
\end{align*}
Moreover, by the assumption (\ref{eq3.6}), it follows that
\begin{align*}
\lim_{r\rightarrow\infty}\frac{(q^{r_j+m(i_j-1)}-1)q^{-m(i_j-1)}}{q^{r-s_{j-1}}-1}=\lim_{r\rightarrow\infty}\frac{q^{r_j}-1}{q^{r-s_{j-1}}-1}
-\lim_{r\rightarrow\infty}\frac{q^{-m(i_j-1)}-1}{q^{r-s_{j-1}}-1}=\lambda_j,
\end{align*}
\begin{align*}
\lim_{r\rightarrow\infty}\frac{(q^{r-s_j+m(i_j-1)}-1)q^{r_j-m(i_j-1)}}{q^{r-s_{j-1}}-1}=1
&-q^{-m(i_j-1)}\lim_{r\rightarrow\infty}\frac{q^{r_j}-1}{q^{r-s_{j-1}}-1}\\
&-\lim_{r\rightarrow\infty}\frac{q^{-m(i_j-1)}-1}{q^{r-s_{j-1}}-1}=1-\lambda_jq^{-m(i_j-1)},
\end{align*}
and
\begin{align*}
\lim_{r\rightarrow\infty}\frac{(q^{r-s_{j-1}+m(i_j-1)}-1)q^{-m(i_j-1)}}{q^{r-s_{j-1}}-1}&=
1-\lim_{r\rightarrow\infty}\frac{q^{-m(i_j-1)}-1}{q^{r-s_{j-1}}-1}=1,
\end{align*}
for $j=1,2,\ldots,k$. Thus, dividing both the numerator and denominator of the $j$th factor in the last expression of the probability function (\ref{eq3.2}) by $(q^{r-s_{j-1}}-1)^{n-y_{j-1}}$ and taking the limits as $r\rightarrow\infty$, the limiting expression (\ref{eq3.7}) is readily deduced.
The multiple $q$-P\'{o}lya urn model in the particular case $m=0$ reduces to $q$-drawings with replacement and the distribution (\ref{eq3.2}) reduces to the classical multinomial distribution with probability of success of the $\nu$th kind $p_\nu=q^{s_{\nu-1}}[r_\nu]_q/[r]_q$, $\nu=1,2,\ldots,k+1$.
Also, for $m=-1$, the case corresponds to $q$-drawings without replacement and the probability function (\ref{eq3.2}) reduces to a
\begin{align}\label{eq3.8}
P(X_1=x_1,X_2=x_2,\ldots,&\,X_k=x_k)=q^{\sum_{j=1}^{k}(n-y_j)(r_j-x_j)}\prod_{j=1}^{k+1}\genfrac{[}{]}{0pt}{}{r_j}{x_j}_{q}
\bigg/\genfrac{[}{]}{0pt}{}{r}{n}_{q}\nonumber\\
&=\genfrac{[}{]}{0pt}{}{n}{x_1,x_2,\ldots, x_k}_{q}
q^{\sum_{j=1}^{k}(n-y_j)(r_j-x_j)}\frac{\prod_{j=1}^{k+1}[r_j]_{x_j,q}}{[r]_{n,q}},
\end{align}
for $x_j=0,1,\ldots,n$, $j=1,2,\ldots,k$, with $\sum_{j=1}^k x_j\leq n$, and $0<q<1$ or $1<q<\infty$, where $x_{k+1}=n-\sum_{j=1}^k x_j$, $r_{k+1}=r-\sum_{j=1}^k r_j$, and $y_j=\sum_{i=1}^{j}x_i$, $j=1,2,\ldots,k$. The distribution with probability function (\ref{eq3.8}) may be called {\em multivariate $q$-hypergeometric distribution}.
Furthermore, for $m=1$, the case to $q$-drawings with replacement and addition of another ball of the same color. The particular probability function may be deduced from (\ref{eq3.2}) by setting $\alpha_j=-r_j$, for $j=1,2\ldots,k$, $\alpha_{k+1}=-r+s_k$ and $x_{k+1}=n-y_k$, as
\[
P(X_1=x_1,X_2=x_2,\ldots,X_k=x_k)=\prod_{j=1}^{k}q^{(n-y_j)(r_j+x_j)}\genfrac{[}{]}{0pt}{}{-r_j}{x_j}_{q^{-1}}
\genfrac{[}{]}{0pt}{}{-r+s_k}{n-y_k}_{q^{-1}}\bigg/\genfrac{[}{]}{0pt}{}{-r}{n}_{q^{-1}}.
\]
which, using the expression
\[
\genfrac{[}{]}{0pt}{}{-r+s_k}{n-y_k}_{q^{-1}}\bigg/\genfrac{[}{]}{0pt}{}{-r}{n}_{q^{-1}}
=\prod_{j=1}^{k}\genfrac{[}{]}{0pt}{}{-r+s_j}{n-y_j}_{q^{-1}}\bigg/\genfrac{[}{]}{0pt}{}{-r+s_{j-1}}{n-y_{j-1}}_{q^{-1}},
\]
with $y_0=0$, becomes
\[
P(X_1=x_1,\ldots,X_k=x_k)=\prod_{j=1}^{k}q^{(n-y_j)(r_j+x_j)}\genfrac{[}{]}{0pt}{}{-r_j}{x_j}_{q^{-1}}
\genfrac{[}{]}{0pt}{}{-r+s_j}{n-y_j}_{q^{-1}}\bigg/\genfrac{[}{]}{0pt}{}{-r+s_{j-1}}{n-y_{j-1}}_{q^{-1}}.
\]
Then, since
\[
\genfrac{[}{]}{0pt}{}{-r_j}{x_j}_{q^{-1}}=(-1)^{x_j}q^{x_j+\binom{x_j}{2}}\genfrac{[}{]}{0pt}{}{r_j+x_j-1}{x_j}_q,
\]
\[
\genfrac{[}{]}{0pt}{}{-r+s_j}{n-y_j}_{q^{-1}}=(-1)^{n-y_j}q^{n-y_j+\binom{n-y_j}{2}}\genfrac{[}{]}{0pt}{}{r-s_j+n-y_j-1}{n-y_j}_{q^{-1}},
\]
and
\[
\binom{n-y_{j-1}}{2}=\binom{x_j+(n-y_j)}{2}=\binom{x_j}{2}+\binom{n-y_j}{2}+x_j(n-y_j),
\]
it takes the form
\[
P(X_1=x_1,X_2=x_2,\ldots,X_k=x_k)=\prod_{j=1}^{k}q^{r_j(n-y_j)}\frac{\genfrac{[}{]}{0pt}{}{r_j+x_j-1}{x_j}_q
\genfrac{[}{]}{0pt}{}{r-s_j+n-y_j-1}{n-y_j}_q}{\genfrac{[}{]}{0pt}{}{r-s_{j-1}+n-y_{j-1}-1}{n-y_{j-1}}_q},
\]
which after cancelations, reduces to
\begin{align}\label{eq3.9}
P(X_1\!=x_1,X_2\!=x_2,&\ldots,X_k\!=x_k)\!=\!q^{\sum_{j=1}^kr_j(n-y_j)}\prod_{j=1}^{k+1}\genfrac{[}{]}{0pt}{}{r_j+x_j-1}{x_j}_q
\bigg/\genfrac{[}{]}{0pt}{}{r+n-1}{n}_q\nonumber\\
&=\genfrac{[}{]}{0pt}{}{n}{x_1,x_2,\ldots, x_k}_{q}
q^{\sum_{j=1}^{k}r_j(n-y_j)}\frac{\prod_{j=1}^{k+1}[r_j+x_j-1]_{x_j,q}}{[r+n-1]_{n,q}},
\end{align}
for $x_j=0,1,\ldots,n$, $j=1,2,\ldots,k$, with $\sum_{j=1}^k x_j\leq n$, and $0<q<1$ or $1<q<\infty$, where $x_{k+1}=n-\sum_{j=1}^k x_j$, $r_{k+1}=r-\sum_{j=1}^k r_j$, and $y_j=\sum_{i=1}^{j}x_i$, $j=1,2,\ldots,k$. The distribution with probability function (\ref{eq3.9}) may be called {\em multivariate negative $q$-hypergeometric distribution}. Note that the probabilities (\ref{eq3.9}), according to (\ref{eq2.8a}), sum to unity.
\begin{exm}\label{exm3.1}
Distribution of the numbers of errors in the chapters of a manuscript. {\em Consider a manuscript of $k+1$ chapters (sections, parts), with chapter $c_\nu$ containing $r_\nu$ typographical errors, $\{e_{s_{\nu-1}+1},e_{s_{\nu-1}+2},\ldots,e_{s_\nu}\}$, for $\nu=1,2,\ldots,k+1$, where $s_0=0$, $s_\nu=\sum_{i=1}^\nu r_i$, for $\nu=1,2,\ldots,k+1$, with $s_{k+1}=r$. Assume that a proofreader reads the manuscript and when he/she finds an error corrects it and starts reading the manuscript from the beginning. Also, the proofreader starts reading the manuscript from the beginning when he/she reaches its end. Assume that the probability of finding any particular error is $p=1-q$. Clearly, the probability of finding an error in chapter $c_\nu$ at the first scan is given by $q^{s_{\nu-1}}[r_\nu]_q/[r]_q$, for $\nu=1,2,\ldots,k+1$, with $s_0=0$. Then, the conditional probability of finding (and correcting) an error in chapter $c_\nu$ at the $i$th scan, given that $j_\nu-1$ errors of chapter $c_\nu$ and a total of $i_{\nu-1}$ errors of chapters $c_1,c_2,\ldots,c_{\nu-1}$ are found in the previous $i-1$ scans, is given by
\begin{align*}
p_{i,j_\nu}(i_{\nu-1})=\frac{q^{s_{\nu-1}-i_{\nu-1}}[r_\nu-j_\nu+1]_q}{[r-i+1]_q},
\end{align*}
for $j_\nu=1,2,\ldots,i$, $i_\nu=0,1,\ldots,i-1$, $i=1,2,\ldots\,$, and $\nu=1,2,\ldots,k+1$, with $s_0=0$ and $i_0=0$, where $0<q<1$. Clearly, the joint distributions of the numbers $X_\nu$ of errors of chapter $c_\nu$ found (and corrected) in $n$ scans is the multivariate $q$-hypergeometric distribution, with probability function (\ref{eq3.8}).}
\end{exm}
\begin{exm}\label{exm3.2}
Random $q$-selection from a finite population. {\em Consider a finite population of $r$ people, classified into $k+1$ classes $c_j$, $j=1,2,\ldots,k$, with an unknown number of people in each class. Suppose that a sample of $n$ people is randomly $q$-selected from this population, without replacement. Let $x_j$ be the number of people of class $c_j$, for $j=1,2,\ldots,k$, in the sample. We are interested in the probability that the number of people of the population who belong in class $c_j$ equals $r_j$, for $j=1,2,\ldots,k$.
Let $X_j$ and $R_j$ be the numbers of people of class $c_j$, for $j=1,2,\ldots,k$, in the sample and the population, respectively. The conditional distribution of the random vector ${\textbf{X}}=(X_1,X_2,\ldots,X_k)$, given that the random vector ${\textbf{R}}=(R_1,R_2,\ldots,R_k)$ equals ${\textbf{r}}=(r_1,r_2,\ldots,r_k)$, is the multivariate $q$-hypergeometric distribution, with probability function (\ref{eq3.8}),
\[
f_{{\textbf{X}}|{\textbf{R}}}(x_1,x_2,\ldots,x_k|r_1,r_2,\ldots,r_k)=q^{\sum_{j=1}^{k}(n-y_j)(r_j-x_j)}
\prod_{j=1}^{k+1}\genfrac{[}{]}{0pt}{}{r_j}{x_j}_{q}\bigg/\genfrac{[}{]}{0pt}{}{r}{n}_{q},
\]
for $x_j=0,1,\ldots,n$, $j=1,2,\ldots,k$, with $\sum_{j=1}^k x_j\leq n$, and $0<q<1$ or $1<q<\infty$, where $x_{k+1}=n-\sum_{j=1}^k x_j$, $r_{k+1}=r-\sum_{j=1}^k r_j$, and $y_j=\sum_{i=1}^{j}x_i$, $j=1,2,\ldots,k$. The required probability is given by the value of the conditional probability function of the random vector ${\textbf{R}}=(R_1,R_2,\ldots,R_k)$, given ${\textbf{X}}=(X_1,X_2,\ldots,X_k)$, at the point $\textbf{r}=(r_1,r_2,\ldots,r_k)$. This conditional probability function is given
\begin{align*}
f_{{\textbf{R}}|{\textbf{X}}}(r_1,r_2,\ldots,r_k|x_1,x_2,\ldots,x_k)
&=\frac{f_{{\textbf{R}},{\textbf{X}}}(r_1,r_2,\ldots,r_k,x_1,x_2,\ldots,x_k)}{f_{{\textbf{X}}}(x_1,x_2,\ldots,x_k)}\\
&=\frac{f_{{\textbf{R}}}(r_1,r_2,\ldots,r_k)f_{{\textbf{X}}|{\textbf{R}}}(x_1,x_2,\ldots,x_k|r_1,r_2,\ldots,r_k)}
{f_{{\textbf{X}}}(x_1,x_2,\ldots,x_k)}
\end{align*}
and the probability function of the random vector ${\textbf{X}}=(X_1,X_2,\ldots,X_k)$ is
\[
f_{{\textbf{X}}}(x_1,x_2,\ldots,x_k)
=\sum f_{{\textbf{R}}}(r_1,r_2,\ldots,r_k)f_{{\textbf{X}}|{\textbf{R}}}(x_1,x_2,\ldots,x_k|r_1,r_2,\ldots,r_k),
\]
where the summation is extended over all $r_j=x_j,x_j+1,\ldots,r$, $j=1,2,\ldots,k$, with $\sum_{j=1}^kr_j\le r$. Thus, for the calculation of the probability in question, the additional knowledge of the distribution of the random vector ${\textbf{R}}=(R_1,R_2,\ldots,R_k)$ is required. Assume that this distribution is the $k$-variate discrete $q$-uniform with probability function ({\em Bose-Einstein $q$-stochastic model} ({\em $q$-statistic}))
\[
f_{{\textbf{R}}}(r_1,r_2,\ldots,r_k)=\prod_{j=1}^{k+1}q^{(k-j+1)r_j}\bigg/\genfrac{[}{]}{0pt}{}{r+k}{k}_{q},
\]
for $r_j=0,1,\ldots,r$, $j=1,2,\ldots,k$, with $\sum_{j=1}^kr_j\leq r$, where $r_{k+1}=r-\sum_{j=1}^kr_j$. Hence
\[
f_{{\textbf{X}}}(x_1,x_2,\ldots,x_k)
=\sum q^{\sum_{j=1}^{k}(n-y_j)(r_j-x_j)+(k-j+1)r_j}
\frac{\prod_{j=1}^{k+1}\genfrac{[}{]}{0pt}{}{r_j}{x_j}_{q}}{\genfrac{[}{]}{0pt}{}{r+k}{k}_{q}\genfrac{[}{]}{0pt}{}{r}{n}_{q}},
\]
where the summation is extended over all $r_j=x_j,x_j+1,\ldots,r$, $j=1,2,\ldots,k$, with $\sum_{j=1}^kr_j\le r$, and $0<q<1$ or $1<q<\infty$, where $r_{k+1}=r-\sum_{j=1}^kr_j$, $x_{k+1}=n-\sum_{j=1}^kx_j$, and $y_j=\sum_{i=1}^jx_i$. Then, using the $q$-Cauchy formula,
\[
\sum q^{\sum_{j=1}^{k}(r_j-x_j)(n-y_j+k-j+1)}
\prod_{j=1}^{k+1}\genfrac{[}{]}{0pt}{}{r_j}{x_j}_{q}=\genfrac{[}{]}{0pt}{}{r+k}{n+k}_{q}
\]
where the summation is extended over all $r_j=x_j,x_j+1,\ldots,r$, $j=1,2,\ldots,k$, with $\sum_{j=1}^kr_j\le r$, and $0<q<1$ or $1<q<\infty$, where $r_{k+1}=r-\sum_{j=1}^kr_j$, $x_{k+1}=n-\sum_{j=1}^kx_j$, and $y_j=\sum_{i=1}^jx_i$, the probability function of the random vector ${\textbf{X}}=(X_1,X_2,\ldots,X_k)$, is deduced as
\[
f_{{\textbf{X}}}(x_1,x_2,\ldots,x_k)=\prod_{j=1}^{k+1}q^{(k-j+1)x_j}\genfrac{[}{]}{0pt}{}{r+k}{n+k}_{q}
\bigg/\bigg(\genfrac{[}{]}{0pt}{}{r+k}{k}_{q}\genfrac{[}{]}{0pt}{}{r}{n}_{q}\bigg).
\]
Since
\[
\genfrac{[}{]}{0pt}{}{r+k}{k}_{q}\genfrac{[}{]}{0pt}{}{r}{n}_{q}=\genfrac{[}{]}{0pt}{}{r+k}{n+k}_{q}\genfrac{[}{]}{0pt}{}{n+k}{k}_{q},
\]
the last expression reduces to
\[
f_{{\textbf{X}}}(x_1,x_2,\ldots,x_k)=\prod_{j=1}^{k+1}q^{(k-j+1)x_j}\bigg/\genfrac{[}{]}{0pt}{}{n+k}{k}_{q},
\]
for $x_j=0,1,\ldots,n$, $j=1,2,\ldots,k$, with $\sum_{j=1}^kr_j\le r$, and $0<q<1$ or $1<q<\infty$, which is a $k$-variate discrete $q$-uniform probability function. Therefore, the required conditional probability function of the random vector ${\textbf{R}}=(R_1,R_2,\ldots,R_k)$, given that ${\textbf{X}}=(X_1,X_2,\ldots,X_k)$ is given by
\[
f_{{\textbf{R}}|{\textbf{X}}}(r_1,r_2,\ldots,r_k|x_1,x_2,\ldots,x_k)
=q^{\sum_{j=1}^{k}(n-y_j)(r_j-x_j)}
\prod_{j=1}^{k+1}\genfrac{[}{]}{0pt}{}{r_j}{x_j}_{q}\bigg/\genfrac{[}{]}{0pt}{}{r+k}{n+k}_{q},
\]
for $r_j=x_j,x_j+1,\ldots,r$, $j=1,2,\ldots,k$, with $\sum_{j=1}^kr_j\le r$, and $0<q<1$ or $1<q<\infty$, where $r_{k+1}=r-\sum_{j=1}^kr_j$, $x_{k+1}=n-\sum_{j=1}^kx_j$, and $y_j=\sum_{i=1}^jx_i$.}
\end{exm}
The multivariate $q$-hypergeometric distribution may be obtained as the conditional distribution of $k$ independent $q$-binomial distributions of the first kind, given their sum with another $q$-binomial distribution of the first kind independent of them. Precisely, the following theorem is shown.
\begin{thm}\label{thm3.4}
Consider a sequence of independent Bernoulli trials and assume that the probability of success at the $i$th trial is given by
\[
p_i=\frac{\theta q^{i-1}}{1+\theta q^{i-1}}, \ \ i=1,2,\ldots,\ \ 0<q<1 \ \ \text{or}\ \ 1<q<\infty.
\]
Let $X_j$ be the number of successes after the $(s_{j-1})$th trial and until the $(s_j)$th trial, for $j=1,2,\ldots,k+1$, with $s_0=0$, $s_j=\sum_{i=1}^jr_i$, $j=1,2,\ldots,k+1$, and $s_{k+1}=r$. Then, the conditional probability function of the random vector $(X_1,X_2,\ldots,X_k)$, given that $X_1+X_2+\cdots+X_{k+1}=n$, is the multivariate $q$-hypergeometric distribution with probability function (\ref{eq3.8}).
\end{thm}
{\bf Proof}. The random variables $X_j$, $j=1,2,\ldots,k+1$, are independent, with probability function, according to Theorem 2.1 in Charalambides (2016), is given by
\[
P(X_j=x_j)=\genfrac{[}{]}{0pt}{}{r_j}{x_j}_q\frac{(\theta q^{s_{j-1}})^{x_j}q^{\binom{x_j}{2}}}{\prod_{i=1}^{r_j}(1+\theta q^{s_{j-1}+i-1})},
\ \ x_j=0,1,\ldots,r_j, \ \ j=1,2,\ldots,k+1.
\]
Similarly, the probability function of the sum $Y=X_1+X_2+\cdots+X_{k+1}$, which is the number of successes in $r$ trials, is
\[
P(Y=n)=\genfrac{[}{]}{0pt}{}{r}{n}_q\frac{\theta^n q^{\binom{n}{2}}}{\prod_{i=1}^r(1+\theta q^{i-1})},\ \ n=0,1,\ldots,r.
\]
Then, the joint conditional probability function of the random vector $(X_1,X_2,\ldots,X_k)$, given that $Y=n$,
\[
P(X_1=x_1,\ldots,X_k=x_k|Y=n)=\frac{P(X_1=x_1)\cdots P(X_k=x_k)P(X_{k+1}=n-y_k)}{P(Y=n)},
\]
on using these expressions, is obtained as
\[
P(X_1=x_1,X_2=x_2,\ldots,X_k=x_k|Y=n)=q^{c_k}\prod_{j=1}^{k+1}\genfrac{[}{]}{0pt}{}{r_j}{x_j}_{q}
\bigg/\genfrac{[}{]}{0pt}{}{r}{n}_{q},
\]
where
\[
c_k=\sum_{i=1}^kx_is_{i-1}+(n-y_k)s_k+\sum_{j=1}^k\binom{x_j}{2}+\binom{n-y_k}{2}-\binom{n}{2}.
\]
Thus, after some algebraic manipulations, it reduces to
\begin{align*}
c_k&=n\sum_{j=1}^kr_j-\sum_{i=1}^kx_i(s_k-s_{i-1})+\sum_{j=1}^k\binom{x_j}{2}+\binom{y_k+1}{2}-ny_k\\
&=\sum_{j=1}^kr_j(n-y_j)-\sum_{j=1}^kx_j(n-y_j)=\sum_{j=1}^k(n-y_j)(y_j-x_j),
\end{align*}
and the derivation of (\ref{eq3.8}) is completed.
Furthermore, the multivariate negative $q$-hypergeometric distribution may be obtained as the conditional distribution of $k$ independent negative $q$-binomial distributions of the second kind, given their sum with another negative $q$-binomial distribution of the second kind independent of them, according to the following theorem.
\begin{thm}\label{thm3.5}
Consider a sequence of independent Bernoulli trials and assume that the conditional probability of success at a trial, given that $j-1$ successes occur in the previous trials, is given by
\[
p_j=1-\theta q^{j-1}, \ \ j=1,2,\ldots,\ \ 0<\theta<1, \ \ 0<q<1 \ \ \text{or}\ \ 1<q<\infty,
\]
where, for $1<q<\infty$, the number $j$ of successes is restricted by $j\leq m=-\log\theta/\log q$.
Let $W_j$ be the number of failures after the $(s_{j-1})$th success and until the occurrence of the $(s_j)$th success, for $j=1,2,\ldots,k+1$, with $s_0=0$, $s_j=\sum_{i=1}^jr_i$, $j=1,2,\ldots,k+1$, and $s_{k+1}=r$, where $r\le m$ in the case $1<q<\infty$. Then, the conditional probability function of the random vector $(W_1,W_2,\ldots,W_k)$, given that $W_1+W_2+\cdots+W_{k+1}=n$, is the multivariate negative $q$-hypergeometric distribution with probability function (\ref{eq3.9}).
\end{thm}
{\bf Proof}. The random variables $W_j$, $j=1,2,\ldots,k+1$, are independent, with probability function, according to Theorem 3.1 in Charalambides (2016), is given by
\[
P(W_j=w_j)=\genfrac{[}{]}{0pt}{}{r_j+w_j-1}{w_j}_q(\theta q^{s_{j-1}})^{w_j}\prod_{i=1}^{r_j}(1-\theta q^{s_{j-1}+i-1}),
\ \ w_j=0,1,\ldots,
\]
for all $j=1,2,\ldots,k+1$. Similarly, the probability function of the sum $U=W_1+W_2+\cdots+W_{k+1}$, which is the number of failures until the occurrence of the $r$th success, is
\[
P(U=n)=\genfrac{[}{]}{0pt}{}{r+n-1}{n}_q\theta^n \prod_{i=1}^r(1-\theta q^{i-1}),\ \ n=0,1,\ldots\,.
\]
Then, the joint conditional probability function of the random vector $(W_1,W_2,\ldots,W_k)$, given that $U=n$,
\[
P(W_1=w_1,\ldots,W_k=w_k|U=n)=\frac{P(W_1=w_1)\cdots P(W_k=w_k)P(W_{k+1}=n-u_k)}{P(U=n)},
\]
on using these expressions, is obtained as
\[
P(W_1=w_1,\ldots,W_k=w_k|U=n)=q^{c_k}\prod_{j=1}^{k+1}\genfrac{[}{]}{0pt}{}{r_j+w_j-1}{w_j}_q
\bigg/\genfrac{[}{]}{0pt}{}{r+n-1}{n}_q,
\]
where
\[
c_k=\sum_{i=1}^kw_is_{i-1}+(n-u_k)s_k, \ \ u_j=\sum_{i=1}^jw_i,\ \ j=1,2,\ldots,k.
\]
Thus, after some algebra, it reduces to
\[
c_k=n\sum_{j=1}^kr_j-\sum_{i=1}^kw_i(s_k-s_{i-1})=n\sum_{j=1}^kr_j-\sum_{j=1}^kr_ju_j=\sum_{j=1}^kr_j(n-u_j)
\]
and the derivation of (\ref{eq3.9}) is completed.
\section{Multivariate inverse {\emph{q}}-P\'{o}lya distribution}\label{sec4}
\setcounter{equation}{0}
Consider again the multiple $q$-P\'{o}lya urn model. Specifically, assume that random $q$-drawings of balls are sequentially carried out, one after the other, from an urn, initially containing $r_\nu$ balls of color $c_\nu$, for $\nu=1,2,\ldots,k+1$, according to the following scheme. After each $q$-drawing the drawn ball is placed back in the urn together with $k$ balls of the same color. Assume that the conditional probability of drawing a ball of color $c_\nu$ at the $i$th $q$-drawing, given that $j_\nu-1$ balls of color $c_\nu$ and a total of $i_{\nu-1}$ balls of colors $c_1,c_2,\ldots,c_{\nu-1}$ are drawn in the previous $i-1$ $q$-drawings, is given by (\ref{eq3.1}). In this section the interest is turned to the study of the particular numbers of balls of colors $c_1,c_2,\ldots,c_k$ drawn until the $n$th ball of color $c_{k+1}$ is drawn. For this reason, the following definition is introduced.
\begin{Def}\label{def4.1}
Let $W_\nu$ be the number of balls of color $c_\nu$ drawn until the $n$th ball of color $c_{k+1}$ is drawn in a multiple $q$-P\'{o}lya urn model, with the conditional probability of drawing a ball of color $c_\nu$ at the $i$th $q$-drawing, given that $j_\nu-1$ balls of color $c_\nu$ and a total of $i_{\nu-1}$ balls of colors $c_1,c_2,\ldots,c_{\nu-1}$ are drawn in the previous $i-1$ $q$-drawings, given by (\ref{eq3.1}), for $\nu=1,2,\ldots,k$. The distribution of the random vector
$(W_1, W_2,\ldots,W_k)$ is called multivariate inverse $q$-P\'{o}lya distribution, with parameters $n$, $(\alpha_1,\alpha_2,\ldots,\alpha_k)$, $\alpha$, and $q$.
\end{Def}
The probability function of the $k$-variate inverse $q$-P\'{o}lya distribution is obtained in the following theorem.
\begin{thm}\label{thm4.1}
The probability function of the $k$-variate inverse $q$-P\'{o}lya distribution, with parameters $n$, $(\alpha_1,\alpha_2,\ldots,\alpha_k)$, $\alpha$, and $q$, is given by
\begin{align}\label{eq4.1}
P&(W_1\!=\!w_1,W_2\!=w_2,\ldots,W_k\!=\!w_k)\nonumber\\
&=\genfrac{[}{]}{0pt}{}{n+u_k-1}{w_1,w_2,\ldots, w_k}_{q^{-m}}
q^{-m\sum_{j=1}^{k}(n+u_k-u_j)(\alpha_j-w_j)}\frac{\prod_{j=1}^k[\alpha_j]_{w_j,q^{-m}}[\alpha_{k+1}]_{n,q^{-m}}}
{[\alpha]_{n+u_k,q^{-m}}},
\end{align}
for $w_j=0,1,\ldots\,$, $j=1,2,\ldots,k$, and $0<q<1$ or $1<q<\infty$, where $\alpha_{k+1}=\alpha-\sum_{j=1}^k\alpha_j$, and $u_j=\sum_{i=1}^jw_i$, for $j=1,2,\ldots,k$.
\end{thm}
{\bf Proof}. The probability function of the $k$-variate inverse $q$-P\'{o}lya distribution is closely connected to the probability function $k$-variate $q$-P\'{o}lya distribution. Specifically,
\[
P(W_1=w_1,W_2=w_2,\ldots,W_k=w_k)=p_{n+u_k-1}(w_1,w_2,\ldots, w_k)p_{n+u_k,n},
\]
where $p_{n+u_k-1}(w_1,w_2,\ldots, w_k)$ is the probability of drawing $w_\nu$ balls of color $c_\nu$, for all $\nu=1,2,\ldots,k$, and $n-1$ balls of color $c_{k+1}$ in $n+u_k-1$ $q$-drawings and $p_{n+u_k,n}=q^{-m(\beta_k-u_k)}[a_{k+1}-n+1]_{q^{-m}}/[a-n-u_k+1]_{q^{-m}}$ is the conditional probability of drawing a ball of color $c_{k+1}$ at the $(n+u_k)$th $q$-drawing, given that $n-1$ balls of color $c_{k+1}$ and a total of $u_k$ balls of colors $c_1,c_2,\ldots,c_k$ are drawn in the previous $n+u_k-1$ $q$-drawings. Thus using (\ref{eq3.2}), expression (\ref{eq4.1}) is deduced. Note that the multivariate inverse $q$-Vandermonde formula (\ref{eq2.11}) guarantees that the probabilities (\ref{eq4.1}) sum to unity.
The multivariate inverse $q$-P\'{o}lya distribution, for large $r$, can be approximated by a negative $q$-multinomial distribution of the second kind, which is introduced and studied in Charalambides (2020). Specifically, the following limiting theorem is derived.
\begin{thm}\label{thm5.2}
Consider the multivariate inverse $q$-P\'{o}lya distribution, with probability function $q_{n}(r;m,q)=P(W_1=w_1,W_2=w_2,\ldots,W_k=w_k)$ given by (\ref{eq4.1}).
For $0<q<1$, assume that the limiting expression (\ref{eq3.4}) holds true. Then,
\begin{eqnarray}\label{eq4.2}
\lim_{r\rightarrow\infty}q_{n}(r;m,q)=\genfrac[]{0pt}{}{n+u_k-1}{w_1,w_2,\ldots, w_k}_{q^{m}}
\prod_{j=1}^k\theta_j^{n+u_k-u_j}q^{mw_j}\prod_{i_j=1}^{w_j}(1-\theta_jq^{m(i_j-1)}),
\end{eqnarray}
for $w_j=0,1,\ldots\,$, and $j=1,2,\ldots,k$, where $u_j=\sum_{i=1}^jw_i$, $0<q<1$ and $0<\theta_j<1$, $j=1,2,\ldots,k$, in the case $m$ is a positive integer, or $0<\theta_j<q^{-m(\nu-1)}$, $j=1,2,\ldots,k$, for some positive integer $\nu\ge n$, in the case $m$ is a negative integer.
Also, for $1<q<\infty$, assume that the limiting expression (\ref{eq3.6}) holds true. Then,
\begin{eqnarray}\label{eq4.3}
\lim_{r\rightarrow\infty}q_{n}(r;m,q)=\genfrac[]{0pt}{}{n+u_k-1}{w_1,w_2,\ldots, w_k}_{q^{-m}}
\prod_{j=1}^k\lambda_j^{w_j}\prod_{i_j=1}^{n+u_k-u_j}(1-\lambda_jq^{-m(i_j-1)}),
\end{eqnarray}
for $w_j=0,1,\ldots\,$, and $j=1,2,\ldots,k$, where $u_j=\sum_{i=1}^{j}w_i$, $1<q<\infty$ and $0<\lambda_j<1$, $j=1,2,\ldots,k$, in the case $m$ is a positive integer, or $0<\lambda_j<q^{m(\nu-1)}$, $j=1,2,\ldots,k$, for some positive integer $\nu\ge n$, in the case $m$ is a negative integer.
\end{thm}
{\bf Proof}. The probability function (\ref{eq4.1}), using (\ref{eq2.2}), may be written as
\begin{align*}
q_{n}(r;m,q)&=\genfrac{[}{]}{0pt}{}{n+u_k-1}{w_1,w_2,\ldots, w_k}_{q^{-m}}
\prod_{j=1}^{k}q^{-m(n+u_k-u_j)(\alpha_j-w_j)}[\alpha_j]_{w_j,q^{-m}}\\
&\hspace{4.6cm}\times\frac{[\alpha-\beta_j]_{n+u_k-u_j,q^{-m}}}{[\alpha-\beta_{j-1}]_{n+u_k-u_{j-1},q^{-m}}}\\
&=\genfrac{[}{]}{0pt}{}{n+u_k-1}{w_1,w_2,\ldots, w_k}_{q^{m}}
\prod_{j=1}^{k}q^{mw_j}\prod_{i_j=1}^{w_j}(q^{-r_j-m(i_j-1)}-1)q^{-r+s_j+m(i_j-1)}\\
&\hspace{4.3cm}\times\frac{\prod_{i_j=1}^{n+u_k-u_j}(q^{-r+s_j-m(i_j-1)}-1)q^{m(i_j-1)}}
{\prod_{i_{j-1}=1}^{n+u_k-u_{j-1}}(q^{-r+s_{j-1}-m(i_{j-1}-1)}-1)q^{m(i-1)}}.
\end{align*}
and, alternatively, as
\begin{align*}
q_{n}(r;m,q)&=\genfrac{[}{]}{0pt}{}{n+u_k-1}{w_1,w_2,\ldots, w_k}_{q^{-m}}
\prod_{j=1}^{k}q^{-m(n+u_k-u_j)(\alpha_j-w_j)}[\alpha_j]_{w_j,q^{-m}}\\
&\hspace{4.6cm}\times\frac{[\alpha-\beta_j]_{n+u_k-u_j,q^{-m}}}{[\alpha-\beta_{j-1}]_{n+u_k-u_{j-1},q^{-m}}}\\
&=\genfrac{[}{]}{0pt}{}{n+u_k-1}{w_1,w_2,\ldots, w_k}_{q^{-m}}
\prod_{j=1}^{k}\prod_{i_j=1}^{w_j}(1-q^{r_j+m(i_j-1)})q^{-m(i_j-1)}\\
&\hspace{4.3cm}\times\frac{\prod_{i_j=1}^{n+u_k-u_j}(1-q^{r-s_j+m(i_j-1)})q^{r_j-m(i_j-1)}}
{\prod_{i_{j-1}=1}^{n+u_k-u_{j-1}}(1-q^{r-s_{j-1}+m(i_{j-1}-1)})q^{-m(i-1)}}.
\end{align*}
Then, proceeding as in the derivation of Theorem 3.3, the required limiting expressions (\ref{eq4.2}) and (\ref{eq4.3}) are deduced.
\end{document}
|
\begin{document}
\input{amssym}
\begin{frontmatter}
\title{Symmetry Analysis of Telegraph Equation}
\author[]{Mehdi Nadjafikhah}\ead{m\[email protected]},
\author[MN]{Seyed Reza Hejazi}\ead{reza\[email protected]},
\address[MN]{School of Mathematics, Iran University of Science and Technology, Narmak-16, Tehran, I.R.Iran}
\begin{keyword}
Lie group analysis, Symmetry group, Optimal system, Invariant solution.
\end{keyword}
\begin{abstract}
Lie symmetry group method is applied to study the Telegraph
equation. The symmetry group and its optimal system are given, and
group invariant solutions associated to the symmetries are
obtained. Finally the structure of the Lie algebra symmetries is
determined.
\end{abstract}
\end{frontmatter}
\section{Introduction}
The telegrapher's equations (or just telegraph equations) are a
pair of linear differential equations which describe the voltage
and current on an electrical transmission line with distance and
time. The equations come from \textit{Oliver Heaviside} who
developed the transmission line model. Oliver Heaviside (May 18,
1850 – February 3, 1925) was a self-taught English electrical
engineer, mathematician, and physicist who adapted complex numbers
to the study of electrical circuits, invented mathematical
techniques to the solution of differential equations (later found
to be equivalent to Laplace transforms), reformulated Maxwell's
field equations in terms of electric and magnetic forces and
energy flux, and independently co-formulated vector analysis.
Although at odds with the scientific establishment for most of his
life, Heaviside changed the face of mathematics and science for
years to come the theory applies to high-frequency transmission
lines (such as telegraph wires and radio frequency conductors) but
is also important for designing high-voltage energy transmission
lines. The model demonstrates that the electromagnetic waves can
be reflected on the wire, and that wave patterns can appear along
the line. The telegrapher's equations can be understood as a
simplified case of Maxwell's equations. In a more practical
approach, one assumes that the conductors are composed of an
infinite series of two-port elementary components, each
representing an infinitesimally short segment of the transmission
line.
\section{Lie Symmetries of the Equation}
A PDE with $p-$independent and $q-$dependent variables has a Lie
point transformations
\begin{eqnarray*}
\widetilde{x}_i=x_i+\varepsilon\xi_i(x,u)+{\mathcal
O}(\varepsilon^2),\qquad
\widetilde{u}_{\alpha}=u_\alpha+\varepsilon\varphi_\alpha(x,u)+{\mathcal
O}(\varepsilon^2)
\end{eqnarray*}
where
$\displaystyle{\xi_i=\frac{\partial\widetilde{x}_i}{\partial\varepsilon}\Big|_{\varepsilon=0}}$
for $i=1,...,p$ and
$\displaystyle{\varphi_\alpha=\frac{\partial\widetilde{u}_\alpha}{\partial\varepsilon}\Big|_{\varepsilon=0}}$
for $\alpha=1,...,q$. The action of the Lie group can be
considered by its associated infinitesimal generator
\begin{eqnarray}\label{eq:18}
\textbf{v}=\sum_{i=1}^p\xi_i(x,u)\frac{\partial}{\partial{x_i}}+\sum_{\alpha=1}^q\varphi_\alpha(x,u)\frac{\partial}{\partial{u_\alpha}}
\end{eqnarray}
on the total space of PDE (the space containing independent and
dependent variables). Furthermore, the characteristic of the
vector field (\ref{eq:18}) is given by
\begin{eqnarray*}
Q^\alpha(x,u^{(1)})=\varphi_\alpha(x,u)-\sum_{i=1}^p\xi_i(x,u)\frac{\partial
u^\alpha}{\partial x_i},
\end{eqnarray*}
and its $n-$th prolongation is determined by
\begin{eqnarray*}
\textbf{v}^{(n)}=\sum_{i=1}^p\xi_i(x,u)\frac{\partial}{\partial
x_i}+\sum_{\alpha=1}^q\sum_{\sharp
J=j=0}^n\varphi^J_\alpha(x,u^{(j)})\frac{\partial}{\partial
u^\alpha_J},
\end{eqnarray*}
where
$\varphi^J_\alpha=D_JQ^\alpha+\sum_{i=1}^p\xi_iu^\alpha_{J,i}$.
($D_J$ is the total derivative operator describes in
(\ref{eq:19})).
The aim is to analysis the point symmetry structure of the
Telegraph equation, which is
\begin{equation}\label{eq:1}
u_{tt}+ku_t=a^2\Big[\frac{1}{r}\Big(r\frac{\partial u}{\partial
r}\Big)_r+\frac{1}{r^2}u_{xx}+u_{yy}\Big],
\end{equation}
where $u$ is a smooth function of $\displaystyle{(r,x,y,t)}$.
Let us consider a one-parameter Lie group of infinitesimal
transformations $(x,t,u)$ given by
\begin{eqnarray*}\begin{array}{lllll}
\widetilde{r}=r+\varepsilon\xi_1(r,x,y,t,u)+{\mathcal
O}(\varepsilon^2),&&
\widetilde{x}=x+\varepsilon\xi_2(r,x,y,t,u)+{\mathcal
O}(\varepsilon^2),&&
\widetilde{y}=y+\varepsilon\xi_3(r,x,y,t,u)+{\mathcal
O}(\varepsilon^2),\\
\widetilde{t}=t+\varepsilon\xi_4(r,x,y,t,u)+{\mathcal
O}(\varepsilon^2),&&
\widetilde{u}=u+\varepsilon\eta(r,x,y,t,u)+{\mathcal
O}(\varepsilon^2),\end{array}
\end{eqnarray*}
where $\varepsilon$ is the group parameter. Then one requires that
this transformations leaves invariant the set of solutions of the
Eq. (\ref{eq:1}). This yields to the linear system of equations
for the infinitesimals $\xi_1(r,x,y,t,u)$, $\xi_2(r,x,y,t,u)$,
$\xi_3(r,x,y,t,u)$, $\xi_4(r,x,y,t,u)$ and $\eta(r,x,y,t,u)$. The
Lie algebra of infinitesimal symmetries is the set of vector
fields in the form of
$\textbf{v}=\xi_1\partial_r+\xi_2\partial_x+\xi_3\partial_y+\xi_4\partial_t+\eta\partial_u$.
This vector field has the second prolongation
\begin{eqnarray*}
\textbf{v}^{(2)}=\textbf{v}+\varphi^r\partial_{r}+\varphi^x\partial_{x}+\varphi^y\partial_{y}+\varphi^t\partial_{t}+\varphi^{rr}\partial_{u_{rr}}
+\varphi^{rx}\partial_{u_{rx}}+\cdots+\varphi^{yy}\partial_{u_{yy}}+\varphi^{yt}\partial_{u_{yt}}+\varphi^{tt}\partial_{tt}
\end{eqnarray*}
with the coefficients
\begin{eqnarray*}\begin{array}{lll}
\varphi^r =D_rQ+\xi_1u_{rr}+\xi_2u_{rx}+\xi_3u_{ry}+\xi_4u_{rt},&&
\varphi^x =D_xQ+\xi_1u_{rx}+\xi_2u_{xx}+\xi_3u_{xy}+\xi_4u_{xt},\\
\varphi^y =D_yQ+\xi_1u_{ry}+\xi_2u_{xy}+\xi_3u_{yy}+\xi_4u_{yt},&&
\varphi^t =D_xQ+\xi_1u_{rt}+\xi_2u_{xt}+\xi_3u_{xt}+\xi_4u_{tt},\\
\varphi^{rr}=D^2_rQ+\xi_1u_{rrr}+\xi_2u_{rrx}+\xi_3u_{rry}+\xi_4u_{rrt},&&
\varphi^{rx}=D_rD_xQ+\xi_1u_{rxr}+\xi_2u_{rxx}+\xi_3u_{rxy}+\xi_4u_{rxt},\\
\varphi^{ry}=D_rD_yQ+\xi_1u_{ryr}+\xi_2u_{rxy}+\xi_3u_{ryy}+\xi_4u_{ryt}&&
\varphi^{rt}=D_rD_tQ+\xi_1u_{rtr}+\xi_2u_{rxt}+\xi_3u_{ryt}+\xi_4u_{rtt},\\
\varphi^{xx}=D^2_xQ+\xi_1u_{xxr}+\xi_2u_{xxx}+\xi_3u_{xxy}+\xi_4u_{xxt},&&
\varphi^{xy}=D_xD_yQ+\xi_1u_{xyr}+\xi_2u_{xxy}+\xi_3u_{xyy}+\xi_4u_{xyt},\\
\varphi^{xt}=D_xD_tQ+\xi_1u_{xrt}+\xi_2u_{xxt}+\xi_3u_{xyt}+\xi_4u_{xtt},&&
\varphi^{yy}=D^2_yQ+\xi_1u_{ryy}+\xi_2u_{xyy}+\xi_3u_{yyy}+\xi_4u_{yyt},\\
\varphi^{yt}=D_yD_tQ+\xi_1u_{ryt}+\xi_2u_{xyt}+\xi_3u_{yyt}+\xi_4u_{ytt},&&
\varphi^{tt}=D^2_tQ+\xi_1u_{rtt}+\xi_2u_{xtt}+\xi_3u_{ytt}+\xi_4u_{ttt},
\end{array}
\end{eqnarray*}
where the operators $D_r,D_x,D_y$ and $D_t$ denote the total
derivative with respect to $r,x,y$ and $t$:
\begin{eqnarray}\label{eq:19}\begin{array}{lll}
D_r=\partial_r+u_r\partial_u+u_{rr}\partial_{u_r}+u_{rx}\partial_{u_x}+\cdots,&&
D_x=\partial_x+u_x\partial_u+u_{xx}\partial_{u_x}+u_{rx}\partial_{u_r}+\cdots,\\
D_y=\partial_y+u_y\partial_u+u_{yy}\partial_{u_y}+u_{ry}\partial_{u_r}+\cdots,&&
D_t=\partial_t+u_t\partial_u+u_{tt}\partial_{u_t}+u_{rt}\partial_{u_r}+\cdots,\end{array}
\end{eqnarray}
Using the invariance condition, i.e., applying the second
prolongation $\textbf{v}^{(2)}$ to Eq. (\ref{eq:1}), the following
system of 27 determining equations yields:
\begin{eqnarray*}
\begin{array}{lclclclc}
{\xi_2}_u=0,&&{\xi_2}_{yy}=0,&&{\xi_3}_y=0,&&\hspace{-5cm}{\xi_3}_u=0,\\
{\xi_4}_t=0,&&{\xi_4}_u=0,&&{\xi_4}_{rr}=0,&&\hspace{-5cm}{\xi_4}_{xy}=0,\\
{\xi_4}_{yy}=0,&&{\xi_4}_{ry}=0,&&\eta_{tu}=0,&&\hspace{-5cm}\eta_{uu}=0\\
k{\xi_4}_y+2\eta_{ru}=0,&&\xi_1+r{\xi_2}_x=0,&&{\xi_2}_x+r{\xi_2}_{rx}=0,&&\hspace{-5cm}{\xi_2}_y+r{\xi_2}_{ry}=0,\\
{\xi_2}_{xx}-r{\xi_2}_r=0,&&k{\xi_4}_y+2\eta_{yu}=0,&&2{\xi_2}_r+r{\xi_2}_{rr}=0,&&\hspace{-5cm}{\xi_3}_r-r{\xi_2}_{xy}=0,\\
{\xi_3}_x+r^2{\xi_2}_y=0,&&{\xi_3}_t-a^2{\xi_4}_y=0,&&{\xi_4}_x-r{\xi_4}_{rx}=0,&&\hspace{-5cm}{\xi_4}_{xx}+r{\xi_4}_r=0,\\
r^2{\xi_2}_t-a^2{\xi_4}_x=0,&&k{\xi_4}_x+2\eta_{ux}=0,&&a^2r^2\eta_{rr}-kr^2\eta_t+a^2r\eta_r+a^2r^2\eta_{yy}-r^2\eta_{tt}+a^2\eta_{xx}=0.
\end{array}
\end{eqnarray*}
The solution of the above system gives the following coefficients
of the vector field $\textbf{v}$:
\begin{eqnarray*}
\xi_1&=&c_6\sin x-c_7\cos x-c_8y\cos x-c_9y\sin x+2c_{10}a^2t\sin x-2c_{11}a^2t\cos x,\\
\xi_2&=&c_1+c_6r^{-1}\cos x+c_7r^{-1}\sin x+c_8yr^{-1}\sin x-c_9yr^{-1}\sin x+2c_{10}a^2tr^{-1}\cos x-2c_{11}a^2tr^{-1}\sin x,\\
\xi_3&=&c_2+2c_5a^2t+c_8r\cos x+c_9r\sin x,\\
\xi_4&=&c_3+2c_5at+2c_{10}r\sin x-2c_{11}r\cos x,\qquad
\eta=c_4u-c_5kyu-c_{10}kru\sin x+c_{11}kru\cos x,
\end{eqnarray*}
where $c_1,...,c_{11}$ are arbitrary constants, thus the Lie
algebra ${\goth g}$ of the telegraph equation is spanned by the
seven vector fields
\begin{eqnarray*}
\begin{array}{lclc}
\textbf{v}_1=\partial_x,\qquad\textbf{v}_2=\partial_y,&&\hspace{-3cm}
\textbf{v}_3=\partial_t,\qquad\textbf{v}_4=u\partial_u,\\
\textbf{v}_5=2a^2t\partial_y+2y\partial_t-kyu\partial_u,&&\hspace{-3cm}\textbf{v}_6=\sin x\partial_r+r^{-1}\cos x\partial_x,\\
\textbf{v}_7=-\cos x\partial_r+r^{-1}\sin
x\partial_x,&&\hspace{-3cm} \textbf{v}_8=-y\cos
x\partial_r+r^{-1}y\sin x\partial_x+r\cos
x\partial_y,\\\textbf{v}_9=-y\sin x\partial_r-r^{-1}y\cos
x\partial_x+r\sin
x\partial_y,&&\hspace{-3cm}\textbf{v}_{10}=2a^2t\sin
x\partial_r+2a^2tr^{-1}\cos x\partial_x+2r\sin x\partial_t-kru\sin
x\partial_u,\\\textbf{v}_{11}=-2a^2t\cos
x\partial_r+2a^2tr^{-1}\sin x\partial_x-2r\cos x\partial_t+kru\cos
x\partial_u,
\end{array}
\end{eqnarray*}
which $\textbf{v}_1,\textbf{v}_2$ and $\textbf{v}_3$ are
translation on $x,t$ and $u$, $\textbf{v}_4$ is rotation on $u$
and $x$ and $\textbf{v}_7$ is scaling on $x,t$ and $u$. The
commutation relations between these vector fields is given by the
(\ref{table:1}), where entry in row $i$ and column $j$
representing $[\textbf{v}_i,\textbf{v}_j]$.
\begin{table}
\caption{Commutation relations of $\goth g$ }\label{table:1}
\begin{eqnarray*}\hspace{-0.75cm}\begin{array}{cccccccccccc}
\hline
[\,,\,] &\hspace{1cm}\textbf{v}_1 &\hspace{0.5cm}\textbf{v}_2 &\hspace{0.5cm}\textbf{v}_3 &\hspace{0.5cm}\textbf{v}_4 &\hspace{0.5cm}\textbf{v}_5 &\hspace{0.5cm}\textbf{v}_6 &\hspace{0.5cm}\textbf{v}_7 &\hspace{0.5cm}\textbf{v}_8 &\hspace{0.5cm}\textbf{v}_9 &\hspace{0.5cm}\textbf{v}_{10} &\hspace{0.5cm}\textbf{v}_{11} \\ \hline
\textbf{v}_1 &\hspace{1cm} 0 &\hspace{0.5cm} 0 &\hspace{0.5cm}0 &\hspace{0.5cm}0 &\hspace{0.5cm}0 &\hspace{0.3cm}-\textbf{v}_7 &\hspace{0.5cm}\textbf{v}_6 &\hspace{0.3cm}-\textbf{v}_9 &\hspace{0.5cm}\textbf{v}_8 &\hspace{0.3cm}-\textbf{v}_{11} &\hspace{0.5cm}\textbf{v}_{10} \\
\textbf{v}_2 &\hspace{1cm} 0 &\hspace{0.5cm} 0 &\hspace{0.5cm}0 &\hspace{0.5cm}0 &\hspace{0.5cm}2a^2\textbf{v}_3-k\textbf{v}_4 &\hspace{0.5cm}0 &\hspace{0.5cm}0 &\hspace{0.5cm}\textbf{v}_7 &\hspace{0.3cm}-\textbf{v}_6 &\hspace{0.5cm}0 &\hspace{0.5cm}0\\
\textbf{v}_3 &\hspace{1cm} 0 &\hspace{0.5cm} 0 &\hspace{0.5cm}0 &\hspace{0.5cm}0 &\hspace{0.5cm}\textbf{v}_2 &\hspace{0.5cm}0 &\hspace{0.5cm}0 &\hspace{0.5cm}0 &\hspace{0.5cm}0 &\hspace{0.5cm}\textbf{v}_6 &\hspace{0.5cm}\textbf{v}_7\\
\textbf{v}_4 &\hspace{1cm} 0 &\hspace{0.5cm} 0 &\hspace{0.5cm}0 &\hspace{0.5cm}0 &\hspace{0.5cm}0 &\hspace{0.5cm}0 &\hspace{0.5cm}0 &\hspace{0.5cm}0 &\hspace{0.5cm}0 &\hspace{0.5cm}0 &\hspace{0.5cm}0\\
\textbf{v}_5 &\hspace{1cm} 0 &\hspace{-0.1cm} -2a^2\textbf{v}_3+\textbf{v}_4&\hspace{0.3cm}-\textbf{v}_2 &\hspace{0.5cm}0 &\hspace{0.5cm}0 &\hspace{0.5cm}0 &\hspace{0.5cm}0 &\hspace{0.5cm}\textbf{v}_{11} &\hspace{0.3cm}-\textbf{v}_{10} &\hspace{0.3cm}-\textbf{v}_9 &\hspace{0.5cm}\textbf{v}_8\\
\textbf{v}_6 &\hspace{1cm} \textbf{v}_7 &\hspace{0.5cm} 0 &\hspace{0.5cm}0 &\hspace{0.5cm}0 &\hspace{0.5cm}0 &\hspace{0.5cm}0 &\hspace{0.5cm}0 &\hspace{0.5cm}\textbf{v}_7 &\hspace{0.3cm}-\textbf{v}_6 &\hspace{0.5cm}0 &\hspace{0.5cm}0\\
\textbf{v}_7 &\hspace{0.9cm} -\textbf{v}_6 &\hspace{0.5cm} 0 &\hspace{0.5cm}0 &\hspace{0.5cm}0 &\hspace{0.5cm}0 &\hspace{0.5cm}0 &\hspace{0.5cm}0 &\hspace{0.3cm}-\textbf{v}_2 &\hspace{0.5cm}0 &\hspace{0.5cm}0 &\hspace{-0.2cm}2a^2\textbf{v}_3-k\textbf{v}_4\\
\textbf{v}_8 &\hspace{1cm} \textbf{v}_9 &\hspace{0.3cm} -\textbf{v}_7 &\hspace{0.5cm}0 &\hspace{0.5cm}0 &\hspace{0.3cm}-\textbf{v}_{11} &\hspace{0.5cm}0 &\hspace{0.3cm}-\textbf{v}_2 &\hspace{0.5cm}0 &\hspace{0.3cm}-\textbf{v}_1 &\hspace{0.5cm}0 &\hspace{0.5cm}\textbf{v}_5\\
\textbf{v}_9 &\hspace{0.9cm} -\textbf{v}_8 &\hspace{0.5cm} \textbf{v}_6 &\hspace{0.5cm}0 &\hspace{0.5cm}0 &\hspace{0.5cm}\textbf{v}_{10} &\hspace{0.3cm}-\textbf{v}_2 &\hspace{0.5cm}0 &\hspace{0.5cm}\textbf{v}_1 &\hspace{0.5cm}0 &\hspace{0.3cm}-\textbf{v}_5 &\hspace{0.5cm}0\\
\textbf{v}_{10}&\hspace{1cm} \textbf{v}_{11} &\hspace{0.5cm} 0 &\hspace{0.3cm}-\textbf{v}_6 &\hspace{0.5cm}0 &\hspace{0.3cm}-\textbf{v}_9 &\hspace{-0.4cm}-2a^2\textbf{v}_3+k\textbf{v}_4 &\hspace{0.5cm}0 &\hspace{0.5cm}0 &\hspace{0.5cm}\textbf{v}_5 &\hspace{0.5cm}0 &\hspace{0.5cm}a^2\textbf{v}_1\\
\textbf{v}_{11}&\hspace{0.9cm} -\textbf{v}_{10}&\hspace{0.5cm} 0 &\hspace{0.3cm}-\textbf{v}_7 &\hspace{0.5cm}0 &\hspace{0.3cm}-\textbf{v}_8 &\hspace{0.5cm}0 &\hspace{-0.3cm}-2a^2\textbf{v}_3+k\textbf{v}_4&\hspace{0.3cm}-\textbf{v}_5 &\hspace{0.5cm}0 &\hspace{0.3cm}-a^2\textbf{v}_1 &\hspace{0.5cm}0\\
\hline\end{array}\end{eqnarray*}\end{table}
The one-parameter groups $G_i$ generated by the base of $\goth g$
are given in the following table.
\begin{eqnarray*}
g_1 &:&(r,x,y,t,u)\longmapsto(r,x+s,y,t,u),\qquad\qquad
g_2 :(r,x,y,t,u)\longmapsto(r,x,y+s,t,u),\\
g_3 &:&(r,x,y,t,u)\longmapsto(r,x,y,t+s,u),\qquad\qquad
g_4 :(r,x,y,t,u)\longmapsto(r,x,y,t,ue^s),\\
g_5 &:&(r,x,y,t,u)\longmapsto\Big(r,x,y+st,\frac{1}{a^2}sy+t,-\frac{1}{2a^2}skyu+u\Big),\\
g_6 &:&(r,x,y,t,u)\longmapsto\Big(s\sin x+r,\frac{s}{r}\cos x+x,y,t,u\Big),\\
g_7 &:&(r,x,y,t,u)\longmapsto\Big(-s\cos x+r,\frac{s}{r}\sin x+x,y,t,u\Big),\\
g_8 &:&(r,x,y,t,u)\longmapsto\Big(-sy\cos x+r,\frac{s}{r}y\sin x+x,sr\cos x+y,t,u\Big),\\
g_9 &:&(r,x,y,t,u)\longmapsto\Big(-sy\sin x+r,-\frac{s}{r}y\cos x+x,sr\sin x+y,t,u\Big),\\
g_{10}&:&(r,x,y,t,u)\longmapsto\Big(st\sin x+r,\frac{s}{r}t\cos x+x,y,\frac{s}{a^2}r\sin x+t-\frac{s}{2a^2}kru\sin x+u\Big),\\
g_{11}&:&(r,x,y,t,u)\longmapsto\Big(-st\cos x+r,\frac{s}{r}t\sin
x+x,y,-\frac{s}{a^2}r\cos x+t,\frac{s}{2a^2}kru\cos x+u\Big).
\end{eqnarray*}
Since each group $G_i$ is a symmetry group and if $u=f(r,x,y,t)$
is a solution of the Telegraph equation, so are the functions
\begin{eqnarray*}
u^1&=&U(r,x+\varepsilon,y,t),\qquad
u^2=U(r,x,y+\varepsilon,t),\qquad
u^3=U(r,x,y,t+\varepsilon),\qquad
u^4=e^{-\varepsilon}U(r,x,y,t),\\
u^5 &=&(2a^2+\varepsilon ky)U\Big(r,x,y+\varepsilon
t,\frac{1}{a^2}\varepsilon y+t\Big),\qquad
u^6 =U\Big(\varepsilon\sin x+r,\frac{\varepsilon}{r}\cos x+x,y,t\Big),\\
u^7 &=&U\Big(-\varepsilon\cos x+r,\frac{\varepsilon}{r}\sin
x+x,y,t\Big),\qquad\quad\;
u^8=U\Big(-\varepsilon y\cos x+r,\frac{\varepsilon}{r}y\sin x+x,\varepsilon r\cos x+y,t\Big),\\
u^9 &=&U\Big(-\varepsilon y\sin x+r,-\frac{\varepsilon}{r}y\cos x+x,\varepsilon r\sin x+y,t\Big),\\
u^{10}&=&(2a^2+\varepsilon kr\sin x)U\Big(\varepsilon t\sin x+r,\frac{\varepsilon}{r}t\cos x+x,y,\frac{\varepsilon}{a^2}r\sin x+t\Big),\\
u^{11}&=&(2a^2-\varepsilon kr\cos x)U\Big(-\varepsilon t\cos
x+r,\frac{\varepsilon}{r}t\sin x+x,y,-\frac{\varepsilon}{a^2}r\cos
x+t\Big).
\end{eqnarray*}
where $\varepsilon$ is a real number. Here we can find the general
group of the symmetries by considering a general linear
combination $c_1\textbf{v}_1+\cdots+c_1\textbf{v}_{11}$ of the
given vector fields. In particular if $g$ is the action of the
symmetry group near the identity, it can be represented in the
form
$g=\exp(\varepsilon_{11}\textbf{v}_{11})\cdots\exp(\varepsilon_1\textbf{v}_1)$.
\section{Optimal system of Telegraph equation}
$~~~~~$As is well known, the theoretical Lie group method plays an
important role in finding exact solutions and performing symmetry
reductions of differential equations. Since any linear combination
of infinitesimal generators is also an infinitesimal generator,
there are always infinitely many different symmetry subgroups for
the differential equation. So, a mean of determining which
subgroups would give essentially different types of solutions is
necessary and significant for a complete understanding of the
invariant solutions. As any transformation in the full symmetry
group maps a solution to another solution, it is sufficient to
find invariant solutions which are not related by transformations
in the full symmetry group, this has led to the concept of an
optimal system \cite{[6]}. The problem of finding an optimal
system of subgroups is equivalent to that of finding an optimal
system of subalgebras. For one-dimensional subalgebras, this
classification problem is essentially the same as the problem of
classifying the orbits of the adjoint representation. This problem
is attacked by the naive approach of taking a general element in
the Lie algebra and subjecting it to various adjoint
transformations so as to simplify it as much as possible. The idea
of using the adjoint representation for classifying
group-invariant solutions is due to \cite{[1-1],[2-1],[5],[6]}.
The adjoint action is given by the Lie series
\begin{eqnarray}\label{eq:9}
\mbox{Ad}(\exp(\varepsilon\textbf{v}_i)\textbf{v}_j)=\textbf{v}_j-\varepsilon[\textbf{v}_i,\textbf{v}_j]+\frac{\varepsilon^2}{2}[\textbf{v}_i,[\textbf{v}_i,\textbf{v}_j]]-\cdots,
\end{eqnarray}
where $[\textbf{v}_i,\textbf{v}_j]$ is the commutator for the Lie
algebra, $\varepsilon$ is a parameter, and $i,j=1,\cdots,11$. Let
$F^{\varepsilon}_i:{\goth g}\rightarrow{\goth g}$ defined by
$\textbf{v}\mapsto\mbox{Ad}(\exp(\varepsilon\textbf{v}_i)\textbf{v})$
is a linear map, for $i=1,\cdots,11$. The matrices
$M^\varepsilon_i$ of $F^\varepsilon_i$, $i=1,\cdots,11$, with
respect to basis $\{\textbf{v}_1,\cdots,\textbf{v}_{11}\}$ are
\begin{eqnarray*}
&\displaystyle
M^\varepsilon_1=\tiny\left(\begin{array}{ccccccccccc}
1&0&0&0&0&0&0&0&0&0&0\\0&1&0&0&0&0&0&0&0&0&0\\0&0&1&0&0&0&0&0&0&0&0\\0
&0&0&1&0&0&0&0&0&0&0\\0&0&0&0&1&0&0&0&0&0&0\\0&0&0&0&0&\cos s&\sin
s&0&0&0&0\\0&0&0&0&0&-\sin s&\cos s&0&0&0&0\\0&0&0&0&0&0&0&\cos
s&\sin s&0&0\\0&0&0&0&0&0&0&-\sin s&\cos
s&0&0\\0&0&0&0&0&0&0&0&0&\cos s&-\sin s\\0&0&0&0&0&0&0&0&0&-\sin
s&\cos s\end{array}\right),
M^\varepsilon_2=\tiny\left(\begin{array}{ccccccccccc}
1&0&0&0&0&0&0&0&0&0&0\\0&1&0&0&0&0&0&0&0&0&0\\0&0&1&0&0&0&0&0&0&0&0\\0
&0&0&1&0&0&0&0&0&0&0\\0&0&-2a^2s&ks&1&0&0&0&0&0&0\\0&0&0&0&0&1&0&0&0&0&0\\
0&0&0&0&0&0&1&0&0&0&0\\0&0&0&0&0&0&-s&1&0&0&0\\0&0&0&0&0&s&0&0&1&0&0\\0&0&0&0&0&0&0&0&0&1&0\\
0&0&0&0&0&0&0&0&0&0&1\end{array}\right), \qquad \cdots\\
&\displaystyle \mbox{7 matrices} \quad\cdots\quad
M^\varepsilon_{11}=\tiny\left(\begin{array}{ccccccccccc} \cosh
as&0&0&0&0&0&0&0&0&\frac{1}{a}\sinh
as&0\\0&1&0&0&0&0&0&0&0&0&0\\0&0&\cosh
\sqrt{2}as&\frac{k}{\sqrt{2}a}(1-\cosh
\sqrt{2}as)&0&0&\frac{1}{\sqrt{2}a}\sinh \sqrt{2}as&0&0&0&0\\0
&0&0&1&0&0&0&0&0&0&0\\0&0&0&0&\cosh as&0&0&\sinh as&0&0&0\\0&0&0&0&0&1&0&0&0&0&0\\
0&0&\sqrt{2}a\sinh \sqrt{2}as&-\frac{k}{\sqrt{2}a}\sinh
\sqrt{2}as&0&0&\cosh
\sqrt{2}as&0&0&0&0\\0&0&0&0&\sinh s&0&0&\cosh s&0&0&0\\0&0&0&0&0&0&0&0&1&0&0\\a\sinh as&0&0&0&0&0&0&0&0&\cosh as&0\\
0&0&0&0&0&0&0&0&0&0&1\end{array}\right).
\end{eqnarray*}
by acting these matrices on a vector field $\textbf{v}$
alternatively we can show that a one-dimensional optimal system
of ${\goth g}$ is given by
\begin{eqnarray*}\begin{array}{lll}
X_1=a_1\textbf{v}_1+a_2\textbf{v}_3+a_3\textbf{v}_4+a_4\textbf{v}_8&&
\hspace{-4cm}X_2=a_1\textbf{v}_1+a_2\textbf{v}_3+a_3\textbf{v}_4+a_4\textbf{v}_5-a_6\textbf{v}_9\\
X_3=a_1\textbf{v}_1+a_2\textbf{v}_2+a_3\textbf{v}_3+a_4\textbf{v}_4+a_6\textbf{v}_8&&
\hspace{-4cm}X_4=a_1\textbf{v}_1+a_2\textbf{v}_3+a_3\textbf{v}_4+a_4\textbf{v}_5-a_6\textbf{v}_8\\
X_5=a_1\textbf{v}_1+a_2\textbf{v}_3+a_3\textbf{v}_5+a_4\textbf{v}_6+a_6\textbf{v}_8&&
\hspace{-4cm}X_6=a_1\textbf{v}_1+\textbf{v}_2+a_2\textbf{v}_3+a_3\textbf{v}_4+a_4(\textbf{v}_6-\textbf{v}_{10})\\
X_7=a_1\textbf{v}_1+\textbf{v}_2+a_2\textbf{v}_3+a_3\textbf{v}_4+\textbf{v}_7-\textbf{v}_{11},&&
\hspace{-4cm}X_8=a_1\textbf{v}_1+a_2\textbf{v}_2+a_3\textbf{v}_3+a_4\textbf{v}_4+a_5\textbf{v}_5+a_6\textbf{v}_6,\\
X_9=a_1\textbf{v}_1+a_2\textbf{v}_2+\textbf{v}_3+\textbf{v}_4-(2a^2-k)\textbf{v}_5+\textbf{v}_7,&&
\hspace{-4cm}X_{10}=a_1\textbf{v}_1+a_2\textbf{v}_2+\textbf{v}_3+a_3\textbf{v}_4-(2a^2-k)\textbf{v}_5+\textbf{v}_6,\\
X_{11}=a_1\textbf{v}_1+a_2\textbf{v}_2+a_3\textbf{v}_3+a_4\textbf{v}_4+a_5\textbf{v}_5+a_6\textbf{v}_6+a_7\textbf{v}_{11},\\
X_{12}=a_1\textbf{v}_1+a_2\textbf{v}_2+a_3\textbf{v}_3+a_4\textbf{v}_4+a_5\textbf{v}_5+a_6\textbf{v}_6+a_7\textbf{v}_7+a_8\textbf{v}_{11},&&\\
X_{13}=a_1\textbf{v}_1+a_2\textbf{v}_2+a_3\textbf{v}_3+a_4\textbf{v}_4+a_5\textbf{v}_5+a_6\textbf{v}_6-a_7\textbf{v}_7-a_8\textbf{v}_9,&&\\
X_{14}=a_1\textbf{v}_1+a_2\textbf{v}_2+a_4\textbf{v}_3+a_4\textbf{v}_4+a_5\textbf{v}_5+a_6\textbf{v}_6+\textbf{v}_7-(2a+k)\textbf{v}_{11},&&\\
X_{15}=\frac{1}{2a^2-k}(\textbf{v}_1+\textbf{v}_8)+a_1\textbf{v}_2+a_2\textbf{v}_3+a_3\textbf{v}_4+a_5\textbf{v}_5+\textbf{v}_6+\textbf{v}_7,&&\\
X_{16}=a_1\textbf{v}_1+a_2\textbf{v}_2+\textbf{v}_3+\textbf{v}_4-(2a^2-k-1)\textbf{v}_5+a_3\textbf{v}_6+a_4\textbf{v}_7+a_5\textbf{v}_8+a_6\textbf{v}_9,&&
\end{array}\end{eqnarray*}
\section{Lie Algebra Structure}
$~~~~~$In this part, we determine the structure of symmetry Lie
algebra of the telegraph equation.\\
$\goth g$ has a \textit{Levi decomposition} in the form of ${\goth
g}={\goth r}\ltimes{\goth h}$, where ${\goth
r}=\langle\textbf{v}_2,\textbf{v}_3,\textbf{v}_4,\textbf{v}_5,\textbf{v}_6\rangle$
is the radical (the large solvable ideal) of $\goth g$ which is a
nilpotent nilradical of $\goth g$ and ${\goth
h}=\langle\textbf{v}_1,\textbf{v}_7,\textbf{v}_8,\textbf{v}_9,\textbf{v}_{10},\textbf{v}_{11}\rangle$
is a solvable and non-semisimple subalgebra of $\goth g$ with
centralizer $\langle\textbf{v}_4\rangle$ containing in the minimal
ideal
$\langle\textbf{v}_1,\textbf{v}_2,2a^2\textbf{v}_3-k\textbf{v}_4,\textbf{v}_5,...,\textbf{v}_{11}\rangle$.
Here we can find the quotient algebra generated from $\goth g$
such as
\begin{eqnarray}\label{eq:2}
{\goth g}_1={\goth g}/{\goth r},
\end{eqnarray}
with commutators table (\ref{table:2}), where
$\textbf{w}_i=\textbf{v}_i+{\goth r}$ for $i=1,...,11$ are members
of ${\goth g}_1$.
The (\ref{eq:2}) helps us to reduction differential equations. If
we want to integration an involutive distribution, the process
decomposes into two steps:
\begin{itemize}
\item integration of the involutive distribution with symmetry Lie
algebra ${\goth g}/{\goth r}$, and
\item integration on integral manifolds with symmetry algebra $\goth
r$.
\end{itemize}
$~~~~$First, applying this procedure to the radical $\goth r$ we
decompose the integration problem into two parts: the integration
of the distribution with semisimple algebra ${\goth g}/{\goth r}$,
then the integration of the restriction of distribution to the
integral manifold with the solvable symmetry algebra $\goth r$.\\
$~~~~$The last step can be performed by quadratures. Moreover,
every semisimple Lie algebra ${\goth g}/{\goth r}$ is a direct sum
of simple ones which are ideal in ${\goth g}/{\goth r}$. Thus, the
Lie-Bianchi theorem reduces the integration problem ti involutive
distributions equipped with simple algebras of symmetries.
$~~~~$Both $\goth g$ and ${\goth g}_1$ are non-solvable, because
if ${\goth
g}^{(1)}=\langle\textbf{v}_i,[\textbf{v}_i,\textbf{v}_j]\rangle=[\goth
g, \goth g]$, and ${\goth
g}_1^{(1)}=\langle\textbf{w}_i,[\textbf{w}_i,\textbf{w}_j]\rangle=[{\goth
g}_1,{\goth g}_1]$, be the derived subalgebra of $\goth g$ and
${\goth g}_1$ we have
\begin{eqnarray*}
{\goth g}^{(1)}=[{\goth g},{\goth g}] =\langle
\textbf{v}_1,\textbf{v}_2,2a^2\textbf{v}_3-k\textbf{v}_4,\textbf{v}_5,\textbf{v}_6,\textbf{v}_7,\textbf{v}_8,\textbf{v}_9,
\textbf{v}_{10},\textbf{v}_{11}\rangle=[{\goth g}^{(1)},{\goth
g}^{(1)}] ={\goth g}^{(2)},
\end{eqnarray*}
and
\begin{eqnarray*}
{\goth g}_1^{(1)}=[{\goth g}_1,{\goth g}_1]=\langle
\textbf{w}_1,\textbf{w}_2,\textbf{w}_3,\textbf{w}_4,\textbf{w}_5,\textbf{w}_6\rangle
={\goth g}_1.
\end{eqnarray*}
Thus, we have a chain of ideals ${\goth g}\supset{\goth
g}^{(1)}={\goth g}^{(2)}\neq 0$, ${\goth g}_1\supset{\goth
g}_1^{(1)}={\goth g}_1\neq 0$, which sows the non-solvability of
$\goth g$ and ${\goth g}_1$.
\begin{table}
\caption{Commutation relations of $\goth g$ }\label{table:2}
\begin{eqnarray*}\hspace{-0.75cm}\begin{array}{ccccccc}
\hline
[\,,\,] &\hspace{1cm}\textbf{w}_1 &\hspace{0.5cm}\textbf{w}_2 &\hspace{0.5cm}\textbf{w}_3 &\hspace{0.5cm}\textbf{w}_4 &\hspace{0.5cm}\textbf{w}_5 &\hspace{0.5cm}\textbf{w}_6 \\ \hline
\textbf{w}_1 &\hspace{1cm} 0 &\hspace{0.5cm} 0 &\hspace{0.5cm}-\textbf{w}_4 &\hspace{0.6cm}\textbf{w}_3 &\hspace{0.5cm}-\textbf{w}_6 &\hspace{0.5cm}\textbf{w}_5 \\
\textbf{w}_2 &\hspace{1cm} 0 &\hspace{0.5cm} 0 &\hspace{0.7cm}\textbf{w}_6 &\hspace{0.5cm}-\textbf{w}_5 &\hspace{0.5cm}-\textbf{w}_4 &\hspace{0.5cm}\textbf{w}_3 \\
\textbf{w}_3 &\hspace{0.9cm} \textbf{w}_4 &\hspace{0.3cm} -\textbf{w}_6 &\hspace{0.7cm}0 &\hspace{0.5cm}-\textbf{w}_1 &\hspace{0.7cm}0 &\hspace{0.5cm}\textbf{w}_2 \\
\textbf{w}_4 &\hspace{0.8cm} -\textbf{w}_3 &\hspace{0.4cm} \textbf{w}_5 &\hspace{0.7cm}\textbf{w}_1 &\hspace{0.7cm}0 &\hspace{0.5cm}-\textbf{w}_2 &\hspace{0.5cm}0 \\
\textbf{w}_5 &\hspace{0.9cm} \textbf{w}_6 &\hspace{0.4cm}\textbf{w}_4 &\hspace{0.7cm}0 &\hspace{0.7cm}\textbf{w}_2 &\hspace{0.7cm}0 &\hspace{0.4cm}a^2\textbf{w}_1 \\
\textbf{w}_6 &\hspace{0.8cm} -\textbf{w}_5 &\hspace{0.3cm} -\textbf{w}_3 &\hspace{0.5cm}-\textbf{w}_2 &\hspace{0.7cm}0 &\hspace{0.5cm}-a^2\textbf{w}_1 &\hspace{0.5cm}0 \\
\hline\end{array}\end{eqnarray*}\end{table}
\section{Conclusion}
In this article group classification of telegraph equation and the
algebraic structure of the symmetry group is considered.
Classification of one-dimensional subalgebra is determined by
constructing one-dimensional optimal system. The structure of Lie
algebra symmetries is analyzed.
\end{document}
|
\begin{document}
\begin{abstract}We introduce the notion of a poset scheme and study
the categories of quasi-coherent sheaves on such spaces.
We then show that smooth poset schemes may be used to obtain
categorical resolutions of singularities for usual singular schemes.
We prove that a singular variety $X$ possesses such a resolution if
and only if $X$ has Du Bois singularities. Finally we show that the de Rham-Du Bois
complex for an algebraic variety $Y$ may be defined using any smooth poset scheme which
satisfies the descent over $Y$ in the classical topology.
\end{abstract}
\maketitle
\tableofcontents
\section{Introduction}
\subsection{Categorical resolutions}
There is a good notion of smoothness for a DG algebra $A.$ Namely,
$A$ is called {\it smooth} if it is a perfect DG $A^{\operatorname{op}}\otimes
A$-module. This notion is Morita invariant: if DG algebras A and B
are derived equivalent (i.e. there exists a DG $A^{\operatorname{op}}\otimes
B$-module $M,$ such that the functor $(-)\stackrel{{\mathbf L}}{\otimes
}_AM:D(A)\to D(B)$ is an equivalence), then $A$ is smooth if and
only if $B$ is such. This allows one to define smoothness of derived
categories $D(A),$ and consequently of cocomplete triangulated
categories which possess a compact generator (and have an
enhancement). Examples of such categories are the derived categories
$D(X)$ of quasi-coherent sheaves on quasi-compact and separated
schemes X (see for example \cite{BoVdB}). The scheme $X$ is smooth
if and only if the category $D(X)$ is smooth in the above sense.
In the paper \cite{Lu2} we have introduced the concept of a
categorical resolution of singularities. Namely, given a DG algebra
$A,$ a categorical resolution of $D(A)$ is a pair $(B,M),$ where $B$
is a smooth DG algebra and $M$ is a DG $A^{\operatorname{op}}\otimes B$-module,
such that the functor $(-)\stackrel{{\mathbf L}}{\otimes }_AM:D(A)\to D(B)$
is full and faithful on the subcategory of perfect DG $A$-modules.
The main result of \cite{Lu2} is the following theorem.
\begin{theo}\label{prevmain} Let $X$ be a separated scheme of finite type
over a perfect field $k.$ Then
a) There exists a classical generator $E\in D^b(cohX)$, such that
the DG algebra $A={\mathbf R} \operatorname{op}eratorname{Ho}m (E,E)$ is smooth and hence the functor
$${\mathbf R} Hom (E,-):D(X)\to D(A)$$
is a categorical resolution.
b) Given any other classical generator $E^\prime \in D^b(cohX)$ with
$A^\prime ={\mathbf R} \operatorname{op}eratorname{Ho}m (E^\prime ,E^\prime)$, the DG algebras $A$ and
$A^\prime $ are derived equivalent.
\end{theo}
This theorem provides an intrinsic categorical resolution for
$D(X).$ This resolution has the flavor of Koszul duality. The
resolving DG algebra $A$ is Morita equivalent to its opposite
$A^{\operatorname{op}}$ and usually has unbounded cohomology.
\begin{example}\label{dualnumb} If in Theorem \ref{prevmain}
$X=Spec(k[\epsilon]/\epsilon ^2),$ and $E=k,$ then $A=k[t],$ where
$\deg (t)=1.$
\end{example}
We should note that the notion of categorical resolutions is
different from the usual resolution of singularities. Namely if $X$
is an algebraic variety and $\sigma :\tilde{X}\to X$ is its
resolution of singularities, then ${\mathbf L} \sigma ^*:D(X)\to
D(\tilde{X})$ is a categorical resolution if and only if $X$ has
rational singularities. If $D(X)\to D(A)$ is a categorical
resolution (and the singularities of $X$ are not rational), we find
that the category $D(A)$ has a closer relation to $D(X)$ than
$D(\tilde{X}).$ Also one may consider categorical resolutions of
nonreduced schemes.
\noindent{\bf Conjecture.} Let $X$ be a separated scheme of finite
type over a field. Then there exists a smooth DG algebra $A$ with
$H^i(A)=0$ for $\vert i\vert >>0$ and a functor $D(X)\to D(A)$ which
is a categorical resolution.
\subsection{Smooth poset schemes and Du Bois singularities}
In this article we introduce a new class of smooth categories, which
are constructed by "gluing" the categories $D(X)$ for smooth schemes
$X.$ Namely, we consider {\it poset schemes ${\mathcal X}$} which by
definition are diagrams of schemes $\{X_\alpha\} _{\alpha \in S}$
indexed by elements of a finite poset $S$ with a morphism $f_{\alpha
\beta}:X_\alpha \to X_\beta$ iff $\alpha \geq \beta.$ There is a
natural notion of a quasi-coherent sheaf on ${\mathcal X},$ which gives us
the abelian category $Qcoh {\mathcal X}$ and its derived category $D({\mathcal X}).$
This derived category is cocomplete and has a compact generator (if
all schemes $X_\alpha$ are separated and quasi-compact). So
$D({\mathcal X})\simeq D(A)$ for a DG algebra $A.$ The category $D({\mathcal X})$ is
smooth if the poset scheme ${\mathcal X}$ is smooth (i.e. all schemes
$X_{\alpha}$ are such). In any case the category $D({\mathcal X})$ has a
natural semi-orthogonal decomposition with semi-orthogonal summands
$D(X_\alpha), \alpha \in S.$ In this last sense we consider $D({\mathcal X})$
as a gluing of the categories $D(X_{\alpha})$ along the morphisms
$f_{\alpha \beta}.$
There is a natural notion of a morphism $\pi :{\mathcal X} \to X$ from a
poset scheme ${\mathcal X}$ to a scheme $X$ and the corresponding functor
${\mathbf L} \pi ^*:D(X)\to D({\mathcal X}).$ We say that $\pi $ is a categorical
resolution if ${\mathcal X}$ is smooth and ${\mathbf L} \pi ^*$ is a categorical
resolution. We prove the following theorem (=Theorem
\ref{posetres=DuBois}).
\begin{theo} Let $X$ be a reduced separated scheme
of finite type over a field of characteristic zero. Then $X$ has a
categorical resolution by a smooth poset scheme if and only if $X$
has Du Bois singularities.
\end{theo}
The "if" direction in the theorem is essentially the definition of
Du Bois singularities (plus the work \cite{LNM1335}), and the other
direction is a consequence of the general functorial formalism which
we develop. This theorem proves the above conjecture in the case of
Du Bois singularities.
\begin{cor} Let $X$ be a reduced separated scheme of finite
type over a field of characteristic zero. Assume that $X$ has Du
Bois singularities. Then there exists a smooth DG algebra $A$ and a
categorical resolution $D(X)\to D(A),$ such that
1) $H^i(A)=0$ for $\vert i\vert >>0;$
2) $D(A)$ has a finite semi-orthogonal decomposition with summands
$D(X_i)$ where each $X_i$ is smooth and $X_1$ is a usual resolution
of $X;$
3) If $X$ is proper, then each $X_i$ is also proper.
In particular in this case the DG algebra $A$ is proper (has finite
dimensional cohomology).
\end{cor}
Theorems
\ref{degeneration-standard},\ref{Hodge-to-deRham-degener-analytic},\ref{degeneration-algebraic},\ref{posetres=DuBois},\ref{descent}
may be viewed applications of our theory of smooth projective poset
schemes to the study of Du Bois singularities. In particular,
Theorem \ref{descent} asserts that the de Rham-Du Bois complex may
be defined by means of {\it any} smooth projective poset scheme
which satisfies the descent in the classical topology.
Our poset schemes are generalizations of {\it configuration schemes}
studied in \cite{Lu1}. (A configuration scheme is a poset scheme
where all the structure morphisms $f_{\alpha \beta}$ are closed
embeddings). Although the notion of a categorical resolution is not
present explicitly in \cite{Lu1} the ideas discussed in that paper
are similar to what we do here.
\subsection{Organization of the paper}
The paper consists of two parts. In the first one we develop in
detail the theory of poset schemes and discuss their relationship
with categorical resolutions. In the second part we prove three
results on degeneration of spectral sequences for smooth projective
poset schemes (Theorems
\ref{degeneration-standard},\ref{Hodge-to-deRham-degener-analytic},\ref{degeneration-algebraic})
These results are used to prove Theorem \ref{descent}. In Theorem
\ref{posetres=DuBois} we establish a connection between Du Bois
singularities and the existence of a categorical resolution by a
smooth poset scheme.
The appendix contains some general facts on functors between derived
categories of quasi-coherent sheaves.
In \cite{Lu2} we have collected some well known general categorical
facts about cocomplete triangulated categories, existence of compact
generators, smoothness of DG algebras, existence of enough
h-injectives in derived categories of Grothendieck abelian
categories, etc. These fact are not discussed in this article and we
refer the reader to \cite{Lu2} as needed.
I want to thank Tony Pantev who first suggested a connection between
categorical resolutions by poset schemes and Du Bois singularities.
A discussion of Du Bois singularities with Karl Schwede helped me
understand the subject. Finally I thank the participants of
algebraic geometry seminar in Steklov Institute in Moscow for their
interest in this work.
\part{Categorical resolutions by poset schemes}
\section{Quasi-coherent sheaves on poset schemes}
We fix a base field $k.$ A "scheme" means a separated quasi-compact
$k$-scheme, all morphisms of schemes are assumed to be separated and
quasi-compact. All the products and tensor products are taken over
$k$ unless specified otherwise. Throughout this article a "poset"
(=a partially ordered set) means a {\it finite} poset.
\begin{defi} Let $S=\{\alpha ,\beta ...\}$ be a poset which
we consider as a category: the set $\operatorname{op}eratorname{Ho}m (\alpha ,\beta )$ has a
unique element if $\alpha \geq \beta$ and is empty otherwise. Then
an {\rm $S$-scheme}, or an {\rm $S$-poset scheme}, or a {\rm poset
scheme} is simply a functor from $S$ to the category of schemes. In
other words, a poset scheme is a collection ${\mathcal X}=\{X_\alpha
,f_{\alpha \beta }\}_{\alpha \geq \beta \in S}$, where $X_\alpha$ is
a scheme and $f_{\alpha \beta }: X_\alpha \to X_\beta$ is a morphism
of schemes, such that $f_{\beta \gamma}f_{\alpha \beta}=f_{\alpha
\gamma}.$ We call ${\mathcal X}$ {\rm noetherian, regular, smooth, of finite
type, essentially of finite type, etc.} if all schemes $X_\alpha \in
{\mathcal X}$ are such.
\end{defi}
\begin{defi} Let ${\mathcal X}=\{X_{\alpha },f_{\alpha \beta}\}$ be a
poset scheme. A quasi-coherent sheaf on ${\mathcal X}$ is a collection $F
=\{F_\alpha \in Qcoh(X_{\alpha}), \varphi _{\alpha \beta}:f_{\alpha
\beta}^*F_\beta \to F_\alpha\}$ so that the morphisms $\varphi$
satisfy the usual cocycle condition: $\varphi _{\alpha
\gamma}=\varphi _{\alpha \beta}\cdot f^*_{\alpha \beta}(\varphi
_{\beta \gamma}).$ Quasi-coherent sheaves on ${\mathcal X}$ form a category
in the obvious way. We denote this category $Qcoh{\mathcal X}.$
\end{defi}
\begin{lemma} \label{abelian} The category $Qcoh {\mathcal X}$ is an abelian
category.
\end{lemma}
\begin{proof} Indeed, given a morphism $g:F\to G$ in $Qcoh {\mathcal X}$ we
define $\operatorname{op}eratorname{Ker} (g)$ and $\operatorname{Coker} (g)$ componentwise. Namely, put $\operatorname{op}eratorname{Ker}
(g)_\alpha := \operatorname{op}eratorname{Ker} (g_\alpha ),$ $\operatorname{Coker} (g)_\alpha :=\operatorname{Coker}
(g_\alpha).$ Note that $\operatorname{Coker} (g)$ is well defined since the
functors $f^*_{\alpha \beta}$ are right-exact.
\end{proof}
\begin{remark}\label{alternative-def} A quasi-coherent sheaf $F$
on a poset scheme ${\mathcal X} =\{X_\alpha ,f_{\alpha \beta}\}$ can be
equivalently defined as a collection $F =\{F_\alpha \in
Qcoh(X_{\alpha}), \psi _{\alpha \beta}:F_\beta \to f_{\alpha \beta
*}F_\alpha\},$ so that the morphisms $\psi$ satisfy the usual
cocycle condition: $\psi _{\alpha \gamma} =f_{\beta \gamma *}(\psi
_{\alpha \beta})\cdot \psi _{\beta \gamma}.$
\end{remark}
\begin{defi} The quasi-coherent sheaf ${\mathcal O} _{{\mathcal X}}=\{ {\mathcal O}
_{X_\alpha},\phi _{\alpha \beta}=\operatorname{op}eratorname{id}\}$ is called the {\rm structure
sheaf of ${\mathcal X}.$ } Also for each $i\geq 0$ we have the natural sheaf
${\mathcal O}mega ^i_{{\mathcal X}}$ - the i-th exterior power of the sheaf of Kahler
differentials ${\mathcal O}mega ^1_{{\mathcal X}}.$ Together these sheaves form the
{\rm deRham complex} ${\mathcal O}mega ^\bullet _{{\mathcal X}}$ (as usual the
differential in ${\mathcal O}mega ^\bullet _{{\mathcal X}}$ is not ${\mathcal O} _{{\mathcal X}}$-linear;
it is a differential operator of order 1).
\end{defi}
\subsection{Operations with quasi-coherent sheaves on poset
schemes}\label{operations}
Let $S$ be a finite poset and ${\mathcal X}$ be an $S$-scheme.
Denote for short ${\mathcal M} =Qcoh{\mathcal X}$ and ${\mathcal M} _\alpha =QcohX_\alpha$.
For $F\in {\mathcal M}$ define its support $\operatorname{op}eratorname{Supp}(F)=\{\alpha \in S\vert
F_{\alpha }\neq 0\}$.
Define a topology on $S$ by taking as a basis of open sets the
subsets $U_{\alpha }=\{\beta \in S\vert \beta \geq \alpha\}$.
Note that $Z_{\alpha }=\{ \gamma \in S\vert \gamma \leq \alpha \} $
is a closed subset in $S$.
Let $U\subset S$ be open and $Z=S-U$ -- the complementary closed.
Let ${\mathcal M}_U$ (resp. ${\mathcal M}_Z$) be the full subcategory of ${\mathcal M}$
consisting of objects $F$ with support in $U$ (resp. in $Z$). For
every object $F$ in ${\mathcal M}$ there is a natural short exact sequence
$$0\to F_U\to F\to F_Z\to 0,$$
where $F_U\in {\mathcal M}_U$, $F_Z\in {\mathcal M}_Z$. Indeed, take
$$(F_U)_{\alpha }=\begin{cases}
F_{\alpha}, &\ \text{if $\alpha\in U$}, \\
0, &\ \text{if $\alpha\in Z$}.
\end{cases}
$$
$$(F_Z)_{\alpha }=\begin{cases}
F_{\alpha }, &\ \text{if $\alpha\in Z$}, \\
0, &\ \text{if $\alpha\in U$}.
\end{cases}
$$
We may consider $U$ (resp.$Z$) as a subcategory of $S$ and restrict
the poset scheme ${\mathcal X}$ to $U$ (resp. to $Z$). Denote these
restrictions by ${\mathcal X} (U)$ and ${\mathcal X} (Z)$ and the corresponding
categories by ${\mathcal M} (U)$ and ${\mathcal M}(Z)$ respectively.
Denote by $j:U\operatorname{h}ookrightarrow S$ and $i:Z\operatorname{h}ookrightarrow S$ the
inclusions. We get the obvious restriction functors
$$j^*=j^!:{\mathcal M}\to{\mathcal M}(U),\operatorname{op}eratorname{qu}ad i^*:{\mathcal M} \to {\mathcal M}(Z).$$
Clearly these functors are exact. The functor $j^*$ has an exact
left adjoint $j_!:{\mathcal M}(U)\to {\mathcal M}$ (``extension by zero''). Its image
is the subcategory ${\mathcal M}_U$. The functor $i^*$ has an exact right
adjoint $i_*=i_!:{\mathcal M}(Z)\to {\mathcal M}$ (also ``extension by zero''). Its
image is the subcategory ${\mathcal M}_Z$. It follows that $j^*$ and $i_*$
preserve injectives (as right adjoints to exact functors). We have
$j^*j_!=Id$, $i^*i_*=Id$.
Note that the short exact sequence above is just
$$0\to j_!j^*F\to F\to i_*i^*F\to 0,$$
where the two middle arrows are the adjunction maps.
The functor $i_*$ also has a left-exact right adjoint functor $i^!$.
Namely $i^!F$ is the largest subobject of $F$ which is supported on
$Z$.
For $\alpha \in S$ denote by $j_{\alpha }:\{\alpha \}\operatorname{h}ookrightarrow
S$ the inclusion. The inverse image functor $j_{\alpha }^*:{\mathcal M} \to
{\mathcal M} _{\alpha } ,$ $F\mapsto F_{\alpha }$ has a right-exact left
adjoint $j_{\alpha +}$ defined as follows
$$(j_{\alpha +}P)_{\beta }=\begin{cases}
f^* _{\beta \alpha}P, &\ \text{if $\beta \geq \alpha$}, \\
0, &\ \text{otherwise}.
\end{cases}
$$
Thus for $P\in {\mathcal M} _\alpha $, $\operatorname{op}eratorname{Supp} j_{\alpha +}P\subset U_{\alpha
}$.
We also consider the ``extension by zero'' functor $j_{\alpha !}:{\mathcal M}
_\alpha \to {\mathcal M} $ defined by
$$j_{\alpha !}(P)_{\beta}=\begin{cases}
P, &\ \text{if $\alpha =\beta$}, \\
0, &\ \text{otherwise}.
\end{cases}
$$
\begin{lemma} The functor $j_{\alpha }^*:{\mathcal M}\to {\mathcal M}_\alpha $ has a
right adjoint $j_{\alpha *}$. This functor $j_{\alpha *}$ is
left-exact and preserves injectives. For $P\in {\mathcal M}_\alpha $ $\operatorname{op}eratorname{Supp}
(j_{\alpha
*}P) \subset Z_{\alpha }$.
\end{lemma}
\begin{proof}
Given $P\in {\mathcal M} _\alpha $ we set
$$j_{\alpha *}(P)_{\gamma}=\begin{cases}
f_{\alpha \gamma *}(P), &\ \text{if $\gamma \leq \alpha$}, \\
0, &\ \text{otherwise},
\end{cases}
$$
and the structure map
$$\varphi _{\gamma \delta } :f^* _{\gamma \delta }((j_{\alpha *}P)_{\delta })
\to (j_{\alpha *}P)_{\gamma }$$ is the adjunction map
$$f^* _{\gamma \delta }f _{\alpha \gamma * }P=
f^* _{\gamma \delta }f _{\gamma \delta * }f _{\alpha \gamma *}P \to
f _{\alpha \delta *}P$$ if $\delta \leq \gamma \leq \alpha$ and
$\varphi _{\gamma \delta }=0$ otherwise.
It is clear that $j_{\alpha *}$ is left-exact and that $\operatorname{op}eratorname{Supp}
(j_{\alpha
*}P)\subset Z_{\alpha }$.
Let us prove that $j_{\alpha *}$ is the right adjoint to
$j_{\alpha}^*$.
Let $P\in {\mathcal M}_\alpha $ and $M=\{ M_{\gamma},\varphi _{\gamma
\beta}\}\in {\mathcal M}$. Given $g_{\alpha }\in \operatorname{op}eratorname{Ho}m (M_{\alpha },P)$ for
each $\gamma \leq \alpha $ we obtain a map $g_{\alpha}\cdot\varphi
_{\alpha \gamma }: f^* _{\alpha \gamma }M_{\gamma }\to P$ and hence
by adjunction $g_{\gamma }:M_{\gamma }\to f _{\alpha \gamma *}P=
(j_{\alpha
*}P)_{\gamma }$.
The collection $g=\{g_{\gamma }\} $ is a morphism $g:M\to j_{\alpha
*}P$. It remains to show that the constructed map
$$\operatorname{op}eratorname{Ho}m (M_{\alpha },P)\to \operatorname{op}eratorname{Ho}m (M, j_{\alpha *}P)$$
is surjective or, equivalently, that the restriction map
$$\operatorname{op}eratorname{Ho}m (M,j_{\alpha *}P)\to \operatorname{op}eratorname{Ho}m (M_{\alpha },P),\operatorname{op}eratorname{qu}ad g\mapsto g_{\alpha }$$
is injective.
Assume that $0\neq g\in \operatorname{op}eratorname{Ho}m (M,j_{\alpha *}P)$, i.e. $g_{\gamma
}\neq 0$ for some $\gamma \leq \alpha $. By definition we have the
commutative diagram
$$
\begin{CD}
f^* _{ \alpha \gamma }M_{\gamma }@ >f^*_{\alpha \gamma}(g_{\gamma
})>>
f^* _{\alpha \gamma }f _{\alpha \gamma * }P\\
@V\varphi _{\alpha \gamma }VV @VV\epsilon_PV\\
M_{\alpha }@>g_{\alpha }>>P,
\end{CD}
$$
where $\epsilon_P$ is the adjunction morphism. Note that
$\epsilon_Pf^* _{\alpha \gamma }(g_{\gamma }):f^* _{ \alpha \gamma}
M_{\gamma }\to P$ is the morphism, which corresponds to $g_{\gamma
}: M_{\gamma }\to f _{\alpha \gamma * }P$ by the adjunction
property. Hence $\epsilon _Pf^* _{ \alpha \gamma }(g_{\gamma })\neq
0$. Therefore $g_{\alpha }\neq 0$. This shows the injectivity of the
restriction map $g\mapsto g_{\alpha }$ and proves that $j_{\alpha
*}$ is the right adjoint to $j_{\alpha }^*$. Finally, $j_{\alpha *}$
preserves injectives being the right adjoint to an exact functor.
\end{proof}
\begin{lemma}\label{Groth} The abelian category ${\mathcal M} $ is a Grothendieck
category. In particular it has enough injectives and the
corresponding category of complexes $C({\mathcal M})$ has enough h-injectives
\cite{KaSch},Thm.14.1.7.
\end{lemma}
\begin{proof} For a usual quasi-compact and quasi-separated scheme
$X $ the category $QcohX$ is known to be Grothendieck \cite{ThTr},
Appendix B. The category ${\mathcal M}$ is abelian \ref{abelian} and has
arbitrary direct sums (since the "gluing" functors $f_{\alpha
\beta}^*$ preserve direct sums), so it has arbitrary colimits.
Filtered colimits are exact, because the exactness is determined
locally on each $X_\alpha.$ It remains to prove the existence of a
generator for the abelian category ${\mathcal M}.$ For each $\alpha \in S$
choose a generator $M_\alpha \in Qcoh X_\alpha.$ We claim that $M:=
\operatorname{op}lus _\alpha (j_{\alpha +}M_\alpha)$ is a generator in ${\mathcal M}.$
Indeed, let $g:F\to G$ be a morphism in ${\mathcal M},$ such that $g_*:\operatorname{op}eratorname{Ho}m
(M,F)\to \operatorname{op}eratorname{Ho}m (M,G)$ is an isomorphism. We have
$$\operatorname{op}eratorname{Ho}m (M,-)=\operatorname{op}lus _\alpha \operatorname{op}eratorname{Ho}m
(j_{\alpha +}M_\alpha ,-)=\operatorname{op}lus _{\alpha} \operatorname{op}eratorname{Ho}m (M_\alpha
,(-)_{\alpha}).$$ So for each $\alpha $ the map $g_{\alpha *}:\operatorname{op}eratorname{Ho}m
(M_{\alpha},F_\alpha)\to \operatorname{op}eratorname{Ho}m (M_{\alpha},G_{\alpha})$ is an
isomorphism, hence $g_\alpha$ is an isomorphism. Thus $g$ is an
isomorphism.
\end{proof}
\subsection{Summary of functors and their properties}\label{summary-1}
For reader's
convenience we list all the
functors introduced so far together with their properties.
\noindent{\underline{Functors}:} $j^*=j^!, j_!, i^*, i_*=i_!, i^!,
j^*_{\alpha}, j_{\alpha +}, j_{\alpha *}.$
\noindent{\underline{Exactness}:} $j^*, j_!, i^*, i_*, j^*_{\alpha}$
- exact; $ i^!, j_{\alpha *}$ - left-exact; $ j_{\alpha +}$ -
right-exact.
\noindent{\underline{Adjunction}:} $(j_!,j^*), (i^*, i_*),
(i_*,i^!), (j_{\alpha +}, j^*_{\alpha}), (j^*_{\alpha}, j_{\alpha
*})$ are adjoint pairs.
\noindent{\underline{Preserve direct sums}:} All the above functors
preserve direct sums. (The functor
$j_{\alpha
*}$ preserves direct sums because the morphisms $f_{\alpha \beta}$ are
quasi-compact.)
\noindent{\underline{Preserve injectives}:} $j^*, i_*, i^!,
j_{\alpha
*}$ preserve injectives because they are right adjoint to exact
functors.
\noindent{\underline{Tensor product}:} The bifunctor $\otimes :{\mathcal M}
\times {\mathcal M} \to {\mathcal M}$ is defined componentwise: $(F\otimes G)_{\alpha}
=F_{\alpha }\otimes _{{\mathcal O} _{X_\alpha}}G_{\alpha}.$
\subsection{Cohomological dimension of poset schemes}
We keep the notation of Subsection \ref{operations}
\begin{prop}\label{finite-inj-res}
If the poset scheme ${\mathcal X}$ is regular noetherian, then ${\mathcal M}$ has
finite cohomological dimension.
\end{prop}
\begin{proof} The proposition asserts that any $F$ in ${\mathcal M}$ has a
finite injective resolution. Equivalently, a finite complex in ${\mathcal M}$
is quasi-isomorphic to a finite complex of injectives. We argue by
induction on the cardinality of $S,$ the case $\vert S\vert =1$ is
well known.
Let $\beta \in S$ be a biggest element. Put $U=U_{\beta }=\{\beta
\}$, $Z=S-U$. Let $j=j_{\beta}:U\operatorname{h}ookrightarrow S$ and $i:
Z\operatorname{h}ookrightarrow S$ be the corresponding open and closed embeddings.
Fix $F$ in ${\mathcal M};$ it suffices to find finite injective resolutions
for $j_!j^*F$ and $i_*i^*F$. Let $j^*F\to I_1$, $i^*F\to I_2$ be
such resolutions in categories ${\mathcal M}(U)$ and ${\mathcal M}(Z)$ respectively.
Then $i_*i^*F\to i_*I_2$ will be an injective resolution in ${\mathcal M}$.
Note that $j_*I_1$ is a (finite) complex of injectives in ${\mathcal M}$ and
that the cone $K$ of the natural morphism $j_*j^*F\to j_*I_1$ is
acyclic on $X_\beta$. Hence by the induction assumption $K$ is
quasi-isomorphic to $i_*J,$ where $J$ is a finite complex of
injectives in ${\mathcal M} (Z)$. Therefore the object $j_*j^*F$ has a finite
injective resolution in ${\mathcal M}.$
Consider the short exact sequence
$$0\to j_!j^*F\to j_*j^*F\to G\to 0.$$
Then $\operatorname{op}eratorname{Supp}(G)\subset Z$ and so by induction $G=i_*i^*G$ has a
finite injective resolution in ${\mathcal M}$. Therefore the same is true for
$j_!j^*F$.
\end{proof}
\section{Derived categories of poset schemes} Let $S$ be a
poset, ${\mathcal X}$ an $S$-scheme, ${\mathcal M}=Qcoh{\mathcal X},$ $C({\mathcal X})=C({\mathcal M})$ - the
abelian category of complexes in ${\mathcal M},$ $Ho({\mathcal X})=Ho({\mathcal M}),$
$D({\mathcal X})=D({\mathcal M})$ - its homotopy and derived category.
Let $U\stackrel{j}{\operatorname{h}ookrightarrow}S\stackrel{i}{\operatorname{h}ookleftarrow}Z$
be embeddings of an open $U$ and a complementary closed $Z$. The
exact functors $j^*,j_!,i^*,i_*, j^*_{\alpha }$ extend trivially to
corresponding functors between derived categories $D({\mathcal M}),$
$D({\mathcal M}(U)),$ $D({\mathcal M}(Z)),$ $D(X_{\alpha}).$ To define the derived
functors of the other functors we need h-injective and h-flat
objects in $C({\mathcal M}).$ (There are enough h-injectives by Lemma \ref{Groth})
\begin{defi} An object $F\in C({\mathcal M})$ is called {\rm h-flat} if for any
acyclic complex $S\in C({\mathcal M})$ the complex $F\otimes S$ is acyclic.
\end{defi}
Notice that for any $\alpha \in S$ the functor $j_{\alpha
*}:C(X_{\alpha})\to C({\mathcal X})$ preserves h-injectives. Indeed, its left
adjoint functor $j^*_{\alpha}$ preserves acyclic complexes. Denote
by $SI ({\mathcal X})\subset Ho({\mathcal X})$ the full triangulated subcategory
classically generated by objects $j_{\alpha
*}M,$ for h-injective $M\in C(X_{\alpha }).$ We
call objects of $SI({\mathcal X})$ {\it special h-injectives}. It is
sometimes convenient to use the following lemma.
\begin{lemma}\label{special-inj} There are enough special injectives in
$D({\mathcal X}).$
\end{lemma}
\begin{proof} Fix $F\in C({\mathcal X})$ and let $\beta \in S$ be a biggest
element such that the complex $F_{\beta}$ is not acyclic. Choose an
h-injective resolution $\rho : F_\beta \to I$ in $D(X_\beta).$ By
adjunction it induces a morphism $\sigma :F\to j_{\beta *}I.$ By
construction the cone $C_{\sigma}$ of the morphism $\sigma$ is
acyclic on $X_\gamma$ for all $\gamma \geq \beta.$ So by induction
we may assume that there exists a special h-injective $J$ and a
quasi-isomorphism $C_{\sigma}\to J.$ So $F$ is quasi-isomorphic to
the (shifted) cone of a morphism $j_{\beta *}I \to J.$
\end{proof}
It is known that for any quasi-compact separated scheme $X$ there
are enough h-flats in $D(X)$ \cite{AlJeLi}, Proposition 1.1. Clearly, an
object $F\in C({\mathcal X})$ is h-flat if and only if $F_{\alpha}\in
C(X_{\alpha})$ is h-flat for every $\alpha \in S.$ Let $M\in
C(X_{\alpha})$ be h-flat. Then $j_{\alpha +}M\in C({\mathcal X})$ is also
such. Indeed, the inverse image functors $f^*_{\beta \alpha}$
preserve h-flats \cite{Sp}, Proposition 5.4. Denote by $S F({\mathcal X})\subset
Ho({\mathcal X})$ the full triangulated subcategory classically generated by
objects $j_{\alpha +}M,$ where $M\in C(X_{\alpha})$ is h-flat. We
call objects of $SF({\mathcal X})$ {\it special h-flats}.
\begin{lemma} There are enough special h-flats in $D({\mathcal X}).$
\end{lemma}
\begin{proof} Similar to the proof of Lemma \ref{special-inj} but using the
adjoint pair $(j_{\alpha +},j^*_{\alpha})$ instead of $(j_{\alpha
}^*,j_{\alpha *}).$
\end{proof}
We now use h-injectives to define the right derived functors
$${\mathbf R} j_{\alpha
*}:D(X_\alpha)\to D({\mathcal X}), \ \ \ {\mathbf R} i^!:D({\mathcal X})\to D({\mathcal X}(Z)),$$
and h-flats to define the left derived functor
$${\mathbf L} j_{\alpha +}:D(X_{\alpha})\to D({\mathcal X})$$
and the derived functor $(-)\stackrel{{\mathbf L}}{\otimes }(-):D({\mathcal X})\times
D({\mathcal X})\to D({\mathcal X})$ (by resolving any of the two variables).
\subsection{Summary of functors and their properties} \
\noindent{\underline{Preserve h-flats and h-injectives}:} The
functors $j^*,j_!,i^*,i_*, j^*_{\alpha },j_{\alpha +}$ between the
categories $C({\mathcal X}),$ $C({\mathcal X}(U)),$ $C({\mathcal X}(Z)),$ $C(X_{\alpha})$
preserve h-flats. Also the functors $j^*,i_*,i^!,j_{\alpha *}$
preserve h-injective, since their left adjoint functors preserve
acyclic complexes.
\noindent{\underline{Derived functors}:} We have defined the
following triangulated functors between the derived categories
$D({\mathcal X}),$ $D({\mathcal X}(U)),$ $D({\mathcal X}(Z)),$ $D(X_{\alpha})$:
$j^*,j_!,i^*,i_*, {\mathbf R}
i^!,j^*_{\alpha },{\mathbf L} j_{\alpha +}, {\mathbf R} j_{\alpha *}.$
\noindent{\underline{Preserve direct sums}:} All the above functors
except possibly ${\mathbf R} i^!$ (${\mathbf R} j_{\alpha *}$ preserves direct sums
since the morphisms $f_{\alpha \beta}$ are quasi-compact and
separated [BoVdB],Cor.3.3.4).
\noindent{\underline{Adjunction}:} $(j_!,j^*), (i^*, i_*), (i_*,{\mathbf R}
i^!), (j^*_{\alpha}, {\mathbf R} j_{\alpha
*}), ({\mathbf L} j_{\alpha +}, j^*_{\alpha}), $ are adjoint pairs. This
follows (except for the last pair) from the adjunctions in
Subsection \ref{summary-1} above and the fact that the functors
$j^*,i_*,i^!,j_{\alpha *}$ preserve h-injectives. For the last pair
we need a lemma.
\begin{lemma} \label{adjunction} $({\mathbf L}
j_{\alpha +}, j^*_{\alpha})$ is an adjoint pair.
\end{lemma}
\begin{proof} Choose $M\in D(X_{\alpha})$ and $I\in D({\mathcal X}).$ We need
to show that ${\mathbf R} \operatorname{op}eratorname{Ho}m ({\mathbf L} j_{\alpha +}M,I)={\mathbf R} \operatorname{op}eratorname{Ho}m
(M,j^*_{\alpha}I).$ We may assume that $M$ is h-flat and $I$ is a
special h-injective (Lemma \ref{special-inj}). Moreover, we then may
assume that $I=j_{\beta *}K,\beta \leq \alpha$ where $K\in
C(X_{\beta})$ is h-injective. Then $j^*_{\alpha}I=f_{\beta \alpha
*}K$ and so
$$\operatorname{op}eratorname{Ho}m (M, j^*_{\alpha}I)={\mathbf R} \operatorname{op}eratorname{Ho}m (M,j^*_{\alpha}I)$$
by Corollary \ref{maps-to-h-inj} in Appendix. Therefore
$$\begin{array}{rcl} {\mathbf R} \operatorname{op}eratorname{Ho}m ({\mathbf L} j_{\alpha +}M,I) & = & \operatorname{op}eratorname{Ho}m ({\mathbf L} j_{\alpha +}M,I) \\
& = & \operatorname{op}eratorname{Ho}m (j_{\alpha +}M,I)\\
& = & \operatorname{op}eratorname{Ho}m (M,j^*_{\alpha}I)\\
& = & {\mathbf R} \operatorname{op}eratorname{Ho}m (M,j^*_{\alpha}I).
\end{array}
$$
\end{proof}
\begin{defi} For $F\in D({\mathcal X})$ we define the cohomology
$$H^i({\mathcal X}, F):={\mathbf R} ^i\operatorname{op}eratorname{Ho}m ({\mathcal O} _{{\mathcal X}},F).$$
\end{defi}
\subsection{Semi-orthogonal decompositions}
Recall that functors $j_!$ and $i_*$ identify categories ${\mathcal M}(U)$
and ${\mathcal M}(Z)$ with ${\mathcal M}_U$ and ${\mathcal M}_Z$ respectively. Denote by
$D_U({\mathcal M})$ and $D_Z({\mathcal M})$ the full subcategories of $D({\mathcal M})$
consisting of complexes with cohomologies in ${\mathcal M}_U$ and ${\mathcal M}_Z$
respectively.
\begin{lemma}\label{full-faithful} The functors $i_*:D({\mathcal M}(Z))\to D({\mathcal M})$
and $j_!:D({\mathcal M}(U))\to D({\mathcal M})$ are fully faithful. The essential
images of these functors are the full subcategories $D_Z({\mathcal M})$ and
$D_U({\mathcal M})$ respectively.
\end{lemma}
\begin{proof} Given $F \in D_Z({\mathcal M})$ (resp. $F \in
D_U({\mathcal M})$) the adjunction map $F \to i_*i^*F$ (resp. $j_!j^*F \to
F$) is a quasiisomorphism. This shows that the functors
$i_*:D({\mathcal M}(Z))\to D_Z({\mathcal M})$ and $j_!:D({\mathcal M}(U))\to D_U({\mathcal M})$ are
essentially surjective. Let us prove that they are fully faithful.
Let $F,G\in D({\mathcal M}(Z))$ and assume that $G$ is h-injective. Then
$i_*G$ is also h-injective and we have
$${\mathbf R} \operatorname{op}eratorname{Ho}m(i_*F,i_*G) = \operatorname{op}eratorname{Ho}m(i_*F,i_*G) = \operatorname{op}eratorname{Ho}m(i^*i_*F,G) = {\mathbf R} \operatorname{op}eratorname{Ho}m (F,G).$$
Similarly, let $F,G\in D({\mathcal M}(U))$ and choose a quasi-isomorphism
$j_!G\to I,$ where $I$ is h-injective. Then $j^*I$ is also
h-injective and quasi-isomorphic to $G.$ We have
$${\mathbf R} \operatorname{op}eratorname{Ho}m(j_!F,j_!G) = \operatorname{op}eratorname{Ho}m (j_!F,I) =\operatorname{op}eratorname{Ho}m(F,j^*I) = {\mathbf R} \operatorname{op}eratorname{Ho}m(F,G).$$
\end{proof}
We immediately obtain the following corollary
\begin{cor}\label{cor-full-faithful}
The categories $D({\mathcal M}(U))$ and $D({\mathcal M}(Z))$ are naturally equivalent
to $D_U({\mathcal M})$ and $D_Z({\mathcal M})$ respectively.
\end{cor}
\begin{cor}
Fix
$\alpha \in S$. Let $i:\{\alpha \}\operatorname{h}ookrightarrow U_{\alpha }$ and
$j:U_{\alpha }\operatorname{h}ookrightarrow S$ be the closed and the open
embeddings respectively. Then the functor
$$j_!\cdot i_*:D(X_{\alpha})\to D({\mathcal M})$$
is fully faithful. In particular, the derived category $D(X_\alpha)$
is naturally (equivalent to) a full subcategory of $D({\mathcal M})$.
\end{cor}
\begin{proof}
Indeed, by Lemma \ref{full-faithful} above the functors
$$i_*:D(X_\alpha )\to D({\mathcal M} (U_{\alpha }))$$
and
$$j_!:D({\mathcal M} (U_{\alpha }))\to D({\mathcal M} )$$
are fully faithful. So is their composition.
\end{proof}
Recall the following definitions from \cite{BoKa}.
\begin{defi}
Let ${\mathcal A}$ be a triangulated category, ${\mathcal B}\subset {\mathcal A}$ -- a full
triangulated subcategory. A right orthogonal to ${\mathcal B}$ in ${\mathcal A}$ is a full
subcategory ${\mathcal B}^{\perp}\subset {\mathcal A}$ consisting of all objects $C$
such that $\operatorname{op}eratorname{Ho}m(B,C[n])=0$ for all $n\in {\mathbb Z}$ and all $B\in {\mathcal B}$.
\end{defi}
\begin{defi}
Let ${\mathcal A}$ be a triangulated category, ${\mathcal B} \subset {\mathcal A}$ -- a full
triangulated subcategory. We say that ${\mathcal B}$ is {\it
right-admissible} if for each $X\in {\mathcal A}$ there exists an exact
triangle $B\to X\to C$ with $B\in {\mathcal B}$, $C\in {\mathcal B}^{\perp}$.
\end{defi}
Similarly one defines the left orthogonal to a full subcategory and
left admissible subcategories.
\begin{defi} Let ${\mathcal A}$ be a triangulated category, ${\mathcal B} , {\mathcal C} \subset {\mathcal A}$
-- two full triangulated subcategories. We say that ${\mathcal A}$ has the
semi-orthogonal decomposition ${\mathcal A} =\langle {\mathcal C} ,{\mathcal B} \rangle$ if
${\mathcal C} ={\mathcal B} ^\perp$ and ${\mathcal B}$ is right-admissible. More generally
given full triangulated subcategories ${\mathcal A} _1,...,{\mathcal A} _n\subset {\mathcal A}$
we say that ${\mathcal A}$ has the semi-orthogonal decomposition ${\mathcal A}
=\langle {\mathcal A} _1,{\mathcal A} _2,...,{\mathcal A} _n\rangle$ if
1) ${\mathcal A} _1$ is right-admissible;
2) the right orthogonal ${\mathcal A} _1^{\perp}$ is the category ${\mathcal D}$ which
is the triangulated envelop of the categories ${\mathcal A} _2,...,{\mathcal A} _n;$
3) there is a semi-orthogonal decomposition
${\mathcal D} =\langle {\mathcal A} _2,...,{\mathcal A} _n\rangle.$
\end{defi}
\begin{lemma}\label{semi-orth}
Consider the full subcategories $D_U({\mathcal M})$ and $D_Z({\mathcal M})$ of
$D({\mathcal M}).$ Then
i) $D_U({\mathcal M})^{\perp}=D_Z({\mathcal M})$,
ii) the subcategory $D_U({\mathcal M})\subset D({\mathcal M})$ is right-admissible.
\end{lemma}
\begin{proof}
i). Let $G\in D({\mathcal M})$. Then $G\in D_U({\mathcal M})^{\perp}\simeq (j_!D({\mathcal M}
(U)))^\perp $ iff
$Gj^*$ is acyclic, i.e. $G\in D_Z({\mathcal M})$.
ii). Given $X \in D({\mathcal M})$ the required exact triangle is $X _U\to
X\to X _Z$.
\end{proof}
\begin{cor}\label{final-semi-orth} a) In the notation of Lemma
\ref{semi-orth} we have the semi-orthogonal decomposition
$D({\mathcal M})=(D_Z({\mathcal M}),D_U({\mathcal M})).$
b) Choose a linear ordering $\alpha _1,...,\alpha _n$ of elements of
$S$ which is compatible with the given partial order. Using
Corollary \ref{cor-full-faithful} identify each category
$D(X_{\alpha _i})$ as a full subcategory of $D({\mathcal M})=D({\mathcal X}).$ Then
there is the semi-orthogonal decomposition
$$D({\mathcal X})=\langle D(X_{\alpha _1}),...,D(X_{\alpha _n})\rangle.$$
\end{cor}
\begin{proof} a). This follows directly from the definitions and
Lemma \ref{semi-orth}. b) Follows from a) by
induction on the cardinality of the poset $S.$
\end{proof}
\section{Compact objects and perfect complexes on poset schemes}
Let us first recall the situation with the usual schemes.
\begin{defi} Let $T$ be a triangulated category.
a) An object $K\in T$ is called compact if the functor $\operatorname{op}eratorname{Ho}m
_T(K,-)$ commutes with direct sums. We denote by $T^c\subset T$ the
full triangulated subcategory of compact objects.
b) An object $K\in T^c$ is called a compact generator of $T$ if
$$K^\perp =\{M\in T\vert \operatorname{op}eratorname{Ho}m (K,M[n])=0\ \text{for all $n$}\}=0.$$
\end{defi}
\begin{defi} Let $X$ be a scheme. An object $G\in D(X)$
is called {\rm perfect} if locally it is quasi-isomorphic to a
finite complex of free ${\mathcal O} _X$-modules of finite rank. We denote by
$\operatorname{op}eratorname{Perf} (X)\subset D(X)$ the full triangulated subcategory of perfect
objects.
\end{defi}
\begin{theo}\label{BoVdB} \cite{BoVdB} Let $X$ be a scheme. Then
a) $\operatorname{op}eratorname{Perf} (X)=D(X)^c,$
b) the category $D(X)$ has a compact generator.
\end{theo}
As a consequence of this theorem we obtain an equivalence of
categories $D(X)\simeq D(A)$ for a DG algebra $A.$ Namely, if $K\in
D(X)^c$ is a compact generator and $A={\mathbf R} \operatorname{op}eratorname{Ho}m (K,K),$ then the
functor
$${\mathbf R} \operatorname{op}eratorname{Ho}m (K, -):D(X)\to D(A)$$
is an equivalence (see for example \cite{Lu2}, Proposition 2.6).
We want to prove analogous results for poset schemes.
\begin{defi} Let ${\mathcal X}=\{X_\alpha ,f _{\alpha \beta}\}$ be a
poset scheme. We call a complex $F=\{F_\alpha\}\in D({\mathcal X})$ {\rm
perfect} if each $F_\alpha \in D(X_\alpha)$ is such. Denote by
$\operatorname{op}eratorname{Perf} ({\mathcal X})\subset D({\mathcal X})$ the full subcategory of perfect
complexes.
\end{defi}
\begin{remark} Notice that the functors $j^*, j_!, i^*, i_*,
j_{\alpha}^*, {\mathbf L} j_{\alpha +}$ preserve perfect complexes.
\end{remark}
\begin{prop}\label{comp=perf} $D({\mathcal X})^c=\operatorname{op}eratorname{Perf} ({\mathcal X}).$
\end{prop}
\begin{proof}
Fix a minimal element $\alpha \in S.$ Let $U=S-\{ \alpha \}$ and
denote by $j:U\operatorname{h}ookrightarrow S$ and $j_{\alpha }:\{ \alpha \}
\operatorname{h}ookrightarrow S$ the corresponding open and closed embeddings.
\begin{lemma} The functors $j_\alpha ^*,j_!,$ and ${\mathbf L} j_{\alpha
+}$ preserve compact objects.
\end{lemma}
\begin{proof} Indeed, their respective right adjoint
functors ${\mathbf R} j_{\alpha *},j^*,j_\alpha ^*$ preserve direct sums.
\end{proof}
By Theorem \ref{BoVdB} the proposition holds if $\vert S\vert =1.$
So by induction we may assume that it holds for $X_\alpha $ and ${\mathcal X}
(U).$
By Lemma \ref{full-faithful} the functor $j_!: D({\mathcal X} (U))\to D({\mathcal X})$
is full and faithful with the essential image $D_U({\mathcal X} ).$ Let $M\in
D_U({\mathcal X})$ be perfect. Then $j_!^{-1}M\in D({\mathcal X} (U))$ is also
perfect, hence compact by induction. Therefore $M=j_!(j_!^{-1}M)\in
D({\mathcal X})$ is also compact. Vice versa, let $M\in D({\mathcal X})^c\cap
D_U({\mathcal X}).$ Then $M\in D_U({\mathcal X})^c$ because the inclusion
$D_U({\mathcal X})\subset D({\mathcal X})$ preserves direct sums. So $j_!^{-1}(M)\in
D({\mathcal X} (U))^c.$ By induction $j^{-1}_!(M)$ is perfect, so $M$ is also
perfect. We proved that $D({\mathcal X})^c\cap D_U({\mathcal X})=\operatorname{op}eratorname{Perf} ({\mathcal X})\cap
D_U({\mathcal X}).$
Fix $F\in D({\mathcal X})^c.$ Then $F_\alpha =j^*_\alpha F\in D(X_\alpha)^c$,
hence $F_\alpha $ is perfect by induction. Then ${\mathbf L} j_{\alpha
+}j_\alpha ^*F$ is also compact and perfect. Hence the cone $C(g)$
of the canonical morphism $g:{\mathbf L} j_{\alpha +}j^*_\alpha F\to F$ is
compact. But $C(g)\in D_U( {\mathcal X}),$ so $C(g)\in \operatorname{op}eratorname{Perf} ({\mathcal X}).$ Thus
$F\in \operatorname{op}eratorname{Perf} ({\mathcal X}).$
Vice versa, let $F\in \operatorname{op}eratorname{Perf} ({\mathcal X})$. Then $j^*F\in \operatorname{op}eratorname{Perf} ({\mathcal X} (U)),$
$j_\alpha ^*F\in \operatorname{op}eratorname{Perf} (X_\alpha ).$ By induction $j^*F\in D( {\mathcal X}
(U))^c$ and so $j_!j^*F\in D({\mathcal X})^c.$ Also by induction $j^*_\alpha
F\in D(X_\alpha)^c.$ Consider the exact triangle
$$j_!j^*F\to F \to {\mathbf R} j_{\alpha *}j_\alpha ^*F.$$
It suffices to show that ${\mathbf R} j_{\alpha *}j^*_\alpha F$ is compact.
(Notice that ${\mathbf R} j_{\alpha *}j_\alpha ^*F$ is perfect because
$\alpha$ is a minimal element.) We know that ${\mathbf L} j_{\alpha
+}j_\alpha ^*F$ is perfect and compact. So the cone $C(p)$ of the
canonical morphism
$$p:{\mathbf L} j_{\alpha +}j^*_\alpha F \to {\mathbf R} j_{\alpha *}j^*_\alpha F$$
is perfect. Also $C(p)\in D_U({\mathcal X}).$ Hence $C(p)\in D({\mathcal X})^c$ and
so also ${\mathbf R} j_{\alpha *}j^*_\alpha F$ is compact.
\end{proof}
\subsection{Existence of a compact generator}
\begin{lemma} \label{existence-comp-gen}
The category $D({\mathcal X})$ has a compact generator.
\end{lemma}
\begin{proof} Choose a compact generator $E_\alpha \in D(
X_\alpha)$ for each $\alpha \in S.$ Put $E:=\operatorname{op}lus {\mathbf L} j_{\alpha
+}E_\alpha.$ Then $E\in D({\mathcal X})^c$, since the functor ${\mathbf L} j_{\alpha
+}$ preserves compact objects. For $M\in D( {\mathcal X})$ we have by
adjunction
$$\operatorname{op}eratorname{Ho}m (E,M)=\bigoplus _\alpha \operatorname{op}eratorname{Ho}m (E_\alpha ,M_\alpha).$$
So $\operatorname{op}eratorname{Ho}m (E[i],M)=0$ for all $i$ implies that $M=0.$
\end{proof}
\begin{defi} A compact generator $E\in D({\mathcal X})$ as constructed in the
proof of last lemma will be called {\rm special}.
\end{defi}
We get the following standard corollary.
\begin{cor} The category $D({\mathcal X})$ is equivalent to $D(A)$ for a
DG algebra $A.$
\end{cor}
\begin{proof} If $E$ is a compact generator of $D({\mathcal X})$ and $A={\mathbf R}
\operatorname{op}eratorname{Ho}m (E,E),$ then the functor
$${\mathbf R} \operatorname{op}eratorname{Ho}m (E,-):D({\mathcal X})\to D(A)$$
is an equivalence of categories.
\end{proof}
\section{Smoothness of poset schemes}
In this section we prove the following theorem.
\begin{theo} \label{smooth=smooth} Let $k$ be a perfect field, $S$ - a
(finite) poset and ${\mathcal X}$ a regular $S$-scheme essentially of finite
type. Then the derived category $D({\mathcal X})$ is smooth.
\end{theo}
\begin{proof} For each $\alpha \in S$ choose a compact generator
$E_\alpha$ for $D(X_\alpha)$. Then by (the proof of) Lemma
\ref{existence-comp-gen} the object
$$E:=\bigoplus_{\alpha \in S}{\mathbf L} j_{\alpha +}E_{\alpha}$$
is a compact generator for $D({\mathcal X}).$ Put $A:={\mathbf R} \operatorname{op}eratorname{Ho}m (E,E).$ It
suffices to prove that the DG algebra $A$ is smooth.
Choose a minimal element $\delta \in S,$ and consider the poset
$S^\prime :=S-\{\delta\}.$ Let ${\mathcal X} ^\prime :={\mathcal X} - X_{\delta}$ be
the corresponding $S^\prime$-scheme.
Since $({\mathbf L} j_{\alpha +}E_{\alpha
})\vert _{X_{\delta}}=0$ for each $\alpha \neq \delta,$ we may
consider
$$E^\prime:=\bigoplus _{\alpha \in S^\prime}{\mathbf L} j_{\alpha
+}E_{\alpha}$$ as a compact generator of $D({\mathcal X} ^\prime).$ Put
$A^\prime :={\mathbf R} \operatorname{op}eratorname{Ho}m (E^\prime ,E^\prime).$ (The quasi-isomorphism
type of $A^\prime$ is independent of where we compute this ${\mathbf R}
\operatorname{op}eratorname{Ho}m$: in $D({\mathcal X})$ or $D({\mathcal X} ^\prime).$)
By \cite{Lu2}, Proposition 3.13 and the induction on $\vert S\vert$
we may assume that $A^\prime$ is smooth. Denote
$$A_{\delta}:={\mathbf R} Hom ({\mathbf L} j_{\delta +}E_{\delta}, {\mathbf L} j_{\delta
+}E_{\delta})\simeq {\mathbf R} \operatorname{op}eratorname{Ho}m (E_{\delta},E_{\delta}).$$ Then
$A_{\delta}$ is also smooth for the same reason. Notice that ${\mathbf R}
\operatorname{op}eratorname{Ho}m ({\mathbf L} j_{\delta +}E_{\delta}, E^\prime)=0,$ hence $A$ is
quasi-isomorphic to the triangular DG algebra
$$\left( \begin{array}{cc}
A ^\prime & 0\\
{}_{A_{\delta}}N_{A^\prime} & A _{\delta}
\end{array} \right),
$$
where $N={\mathbf R} \operatorname{op}eratorname{Ho}m (E^\prime ,{\mathbf L} j_{\delta +}E_\delta).$ So by
\cite{Lu2}, Proposition 3.11 it suffices to show that the DG
$A^{\operatorname{op}}_{\delta}\otimes A^\prime$-module $N$ is perfect.
Consider the $S^\prime $-scheme ${\mathcal Y}={\mathcal X}^\prime \times X_\delta.$
That is ${\mathcal Y}$ consists of schemes $X_\alpha \times X_\delta$ for
$\alpha \in S^\prime$ and morphisms $f_{\alpha \beta}\times \operatorname{op}eratorname{id} :
X_\alpha \times X_\delta \to X_\beta \times X_\delta.$ We denote the
inclusion $X_\alpha \times X_\delta \to {\mathcal Y}$ by $j_{(\alpha
,\delta)}.$
Let $E_{\delta }^*:={\mathbf R} \operatorname{op}eratorname{Ho}m (E_{\delta},{\mathcal O} _{X_{\delta}})$ be the
dual compact generator of $D(X_{\delta}).$ Then ${\mathbf R} \operatorname{op}eratorname{Ho}m (E_{\delta
}^* ,E_{\delta }^*)\simeq A_{\delta}^{\operatorname{op}}$ \cite{Lu2}, Lemma 3.15.
For each $\alpha \in S^\prime$ $E_\alpha \boxtimes E^*_\delta$ is a
compact generator of $D(X_\alpha \times X_\delta)$ \cite{BoVdB},
Lemma 3.4.1. Thus
$$\tilde{E}:=\bigoplus_{\alpha \in S^\prime}{\mathbf L} j_{(\alpha ,\delta)
+}(E_\alpha \boxtimes E^*_{\delta})$$ is a special compact generator
for $D({\mathcal Y}).$
\begin{lemma} There is a natural quasi-isomorphism of DG algebras
$${\mathbf R} \operatorname{op}eratorname{Ho}m (\tilde{E},\tilde{E})\simeq A_\delta ^{\operatorname{op}}\otimes
A^\prime.$$
\end{lemma}
\begin{proof} We have
$$\begin{array}{rcl}
{\mathbf R} \operatorname{op}eratorname{Ho}m (\tilde{E},\tilde{E}) & \simeq & \bigoplus _{\alpha \geq
\beta}{\mathbf R} \operatorname{op}eratorname{Ho}m ({\mathbf L} j_{(\alpha ,\delta)+}(E_\alpha \boxtimes
E_{\delta }^*),
{\mathbf L} j_{(\beta ,\delta)+}(E_\beta \boxtimes E_{\delta }^*))\\
& \simeq & \bigoplus _{\alpha \geq
\beta}{\mathbf R} \operatorname{op}eratorname{Ho}m (E_\alpha \boxtimes E_{\delta }^*, {\mathbf L} (f_{\alpha
\beta}\times \operatorname{op}eratorname{id})^*(E_\beta \boxtimes E_{\delta }^*))\\
& \simeq & \bigoplus _{\alpha \geq
\beta}{\mathbf R} \operatorname{op}eratorname{Ho}m (E_\alpha \boxtimes E_{\delta }^*, {\mathbf L} f_{\alpha
\beta}^*E_\beta \boxtimes E_{\delta }^*).
\end{array}
$$
Now by \cite{Lu2}, Proposition 6.20
$$\begin{array}{cl}
& {\mathbf R} \operatorname{op}eratorname{Ho}m (E_\alpha \boxtimes E_{\delta }^*, {\mathbf L} f_{\alpha
\beta}^*E_\beta \boxtimes E_{\delta }^*)\\
\simeq & {\mathbf R} \operatorname{op}eratorname{Ho}m (E_\alpha ,{\mathbf L} f_{\alpha \beta }^*E_\beta)\otimes
{\mathbf R} \operatorname{op}eratorname{Ho}m (E_\delta ^*,E_\delta ^*)\\
\simeq & {\mathbf R} \operatorname{op}eratorname{Ho}m (E_\alpha ,{\mathbf L} f_{\alpha \beta }^*E_\beta)\otimes
A_\delta ^{\operatorname{op}}.
\end{array}
$$
Similarly,
$$\begin{array}{rcl}
{\mathbf R} \operatorname{op}eratorname{Ho}m (E^\prime ,E^\prime) & \simeq & \bigoplus _{\alpha \geq
\beta}
{\mathbf R} \operatorname{op}eratorname{Ho}m ({\mathbf L} j_{\alpha +}E_\alpha ,{\mathbf L} j_{\beta +}E_\beta)\\
& \simeq & \bigoplus_{\alpha \geq \beta} {\mathbf R} \operatorname{op}eratorname{Ho}m (E_\alpha ,{\mathbf L}
f_{\alpha \beta}^*E_\beta).
\end{array}
$$
This proves the lemma.
\end{proof}
It follows that the functor
$${\mathcal{P}}si _{\tilde{E}}(-):={\mathbf R} \operatorname{op}eratorname{Ho}m (\tilde{E},-):D({\mathcal Y})\to D(A_\delta
^{\operatorname{op}}\otimes A^\prime)$$ is an equivalence of categories.
For each $\alpha \in S^\prime,$ such that $\alpha >\delta$ denote by
${\mathcal G}amma (\alpha ,\delta)\subset X_\alpha \times X_\delta$ the graph
of the map $f_{\alpha ,\delta}:X_{\alpha}\to X_{\delta}.$ Define
the coherent sheaf $F$ on ${\mathcal Y}$ as follows. For $\alpha \in
S^\prime$ such that $\alpha >\delta$ put $F_{\alpha}:={\mathcal O} _{{\mathcal G}amma
(\alpha \delta)}\in coh (X_\alpha \times X_\delta).$ If $\delta
\nless \alpha,$ then put $F_{\alpha}=0.$ The structure morphism
$\phi _{\alpha \beta}:f^*_{\alpha \beta}F_\beta \to F_\alpha$ is the
canonical isomorphism.
\begin{lemma} We have ${\mathcal{P}}si _{\tilde{E}}(F)\simeq N.$
\end{lemma}
\begin{proof} By definition
$$\begin{array}{rcl}
N & = & {\mathbf R} \operatorname{op}eratorname{Ho}m _{{\mathcal X}}(E^\prime ,{\mathbf L} j_{\delta +}E_{\delta})\\
& = & \bigoplus _{\alpha \in S^\prime}
{\mathbf R} \operatorname{op}eratorname{Ho}m _{{\mathcal X}}({\mathbf L} j_{\alpha +}E_{\alpha} ,{\mathbf L} j_{\delta +}E_{\delta})\\
& = & \bigoplus _{\alpha \in S^\prime}
{\mathbf R} \operatorname{op}eratorname{Ho}m _{X_{\alpha}}(E_{\alpha} ,{\mathbf L} f^*_{\alpha \delta}E_{\delta})
\end{array}
$$
On the other hand
$$\begin{array}{rcl}
{\mathbf R} \operatorname{op}eratorname{Ho}m _{{\mathcal Y}}(\tilde{E} ,F) & = & \bigoplus _{\alpha \in S^\prime}
{\mathbf R} \operatorname{op}eratorname{Ho}m _{{\mathcal Y}}({\mathbf L} j_{(\alpha,\delta) +}(E_{\alpha}
\boxtimes E_{\delta}^*) ,F)\\
& = & \bigoplus _{\alpha \in S^\prime} {\mathbf R}
\operatorname{op}eratorname{Ho}m _{X_{\alpha} \times X_{\delta}}(E_{\alpha} \boxtimes
E_{\delta}^* ,{\mathcal O} _{{\mathcal G}amma (\alpha \delta)})\\
\end{array}$$
Let us analyze one summand in the last sum. Denote by $E_{\alpha}
\stackrel{p_{\alpha}}{\leftarrow} E_{\alpha}\times E_{\delta}
\stackrel{p_{\delta}}{\rightarrow} E_{\delta}$ and by $\gamma
:{\mathcal G}amma (\alpha \delta)\to X_{\delta}$ the obvious projections.
$$\begin{array}{rl}
& {\mathbf R}
\operatorname{op}eratorname{Ho}m (E_{\alpha} \boxtimes E_{\delta}^* ,{\mathcal O} _{{\mathcal G}amma (\alpha
\delta)}) \\ = & {\mathbf R} \operatorname{op}eratorname{Ho}m (p_{\alpha}^*E_{\alpha} \otimes
p^*_{\delta} {\mathbf R} {\mathcal H} om (E_{\delta}, {\mathcal O} _{X_{\delta}} ),{\mathcal O}
_{{\mathcal G}amma (\alpha
\delta)}) \\
= & {\mathbf R} \operatorname{op}eratorname{Ho}m (p_{\alpha}^*E_{\alpha}, {\mathbf R} {\mathcal H} om (p^*_{\delta} {\mathbf R}
{\mathcal H} om (E_{\delta}, {\mathcal O} _{X_{\delta}} ),{\mathcal O} _{{\mathcal G}amma (\alpha
\delta)}))\\
= & {\mathbf R} \operatorname{op}eratorname{Ho}m (p_{\alpha}^*E_{\alpha}, {\mathbf R} {\mathcal H} om _{{\mathcal G}amma (\alpha
\delta)} ({\mathbf L} \gamma ^*_{\delta} {\mathbf R} {\mathcal H} om (E_{\delta}, {\mathcal O}
_{X_{\delta}} ),{\mathcal O} _{{\mathcal G}amma (\alpha \delta)}))\\
= & {\mathbf R} \operatorname{op}eratorname{Ho}m (p_{\alpha}^*E_{\alpha}, {\mathbf R} {\mathcal H} om _{{\mathcal G}amma (\alpha
\delta)} ( {\mathbf R} {\mathcal H} om _{{\mathcal G}amma (\alpha \delta)}({\mathbf L} \gamma
^*E_{\delta},
{\mathcal O} _{{\mathcal G}amma (\alpha \delta)} ),{\mathcal O} _{{\mathcal G}amma (\alpha \delta)}))\\
= & {\mathbf R} \operatorname{op}eratorname{Ho}m (p_{\alpha
}^*E_{\alpha}, {\mathbf L} \gamma ^*E_{\delta})\\
= & {\mathbf R} \operatorname{op}eratorname{Ho}m (E_{\alpha},{\mathbf L} f^*_{\alpha
\delta}E_{\delta}).
\end{array}
$$
This proves the lemma.
\end{proof}
Since the poset scheme ${\mathcal Y}$ is regular the object $F\in D({\mathcal Y})$ is
compact by Proposition \ref{comp=perf}. Hence $N\simeq {\mathcal{P}}si
_{\tilde{E}} (F)\in D(A^{\operatorname{op}}_{\delta}\otimes A^\prime)$ is also
compact, i.e. is perfect. This proves Theorem \ref{smooth=smooth}.
\end{proof}
\section{Direct and inverse image functors for morphisms of poset schemes}
Let $S,$ $S^\prime$ be posets and $\tau :S\to S^\prime$ be an order
preserving map. Let ${\mathcal X}=\{X_\alpha ,f_{\alpha \beta}\}$ (resp. ${\mathcal X}
^\prime=\{X^\prime _{\alpha ^\prime},f_{\alpha ^\prime \beta
^\prime}\}$) be an $S$-scheme (resp. an $S^\prime$-scheme).
\begin{defi}\label{tau-morphism}
A $\tau$-morphism ${\mathcal F}:{\mathcal X} \to {\mathcal X} ^\prime $ is a collection of
morphisms $\{{\mathcal F}_{\alpha}:X_\alpha \to X^\prime _{\tau (\alpha)}\}
_{\alpha \in S}$ such that for each $\alpha \geq \beta $ the
following diagram commutes
$$
\begin{array}{ccc}
X_\alpha & \stackrel{f_{\alpha \beta}}{\to} & X_{\beta}\\
\downarrow {\mathcal F}_{\alpha} & & \downarrow {\mathcal F}_{\beta} \\
X^\prime _{\tau (\alpha)} & \stackrel{f^\prime _{\tau (\alpha) \tau
(\beta )}}{\to} & X^\prime _{\tau (\beta)}
\end{array}
$$
\end{defi}
Let ${\mathcal F}:{\mathcal X} \to {\mathcal X} ^\prime$ be a $\tau$-morphism and $G\in Qcoh
_{{\mathcal X} ^\prime}.$ We define ${\mathcal F} ^*G\in Qcoh {\mathcal X}$ as follows. For
$\alpha \in S$ put $({\mathcal F} ^*G)_{\alpha}={\mathcal F} _{\alpha}^*G_{\tau
(\alpha)}$ and define the structure morphism $\phi _{\alpha \beta}:
f^*_{\alpha \beta}{\mathcal F} _\beta ^*G_{\tau (\beta)}\to {\mathcal F}
^*_{\alpha}G_{\tau(\alpha )}$ as ${\mathcal F} ^*_{\alpha}\phi ^\prime
_{\tau(\alpha )\tau (\beta)},$ where $\phi ^\prime $ is the
structure morphism for $G.$ This defines a functor ${\mathcal F} ^*:Qcoh {\mathcal X}
^\prime \to Qcoh {\mathcal X} .$ We also consider its left derived functor
${\mathbf L} {\mathcal F} ^*:D({\mathcal X} ^\prime)\to D({\mathcal X})$ which is defined using the
h-flats.
Notice that the functor ${\mathcal F}^*$ preserves h-flats.
\begin{example} \label{pullback-differentials}
We have ${\mathcal F} ^*{\mathcal O} _{{\mathcal X} ^\prime}= {\mathbf L} {\mathcal F} ^*{\mathcal O} _{{\mathcal X} ^\prime}={\mathcal O}
_{{\mathcal X}}.$ Hence, for any $G\in D({\mathcal X}^\prime)$ we obtain the map ${\mathcal F}
^*:H^\bullet ({\mathcal X} ^\prime ,G) \to H^\bullet ({\mathcal X}, {\mathbf L}{\mathcal F} ^*G);$ in
particular we get the map ${\mathcal F} ^*:H^\bullet ({\mathcal X} ^\prime ,{\mathcal O} _{{\mathcal X}
^\prime}) \to H^\bullet ({\mathcal X}, {\mathcal O} _{{\mathcal X}}).$ Also the usual morphism
${\mathcal F} ^*{\mathcal O}mega ^i_{{\mathcal X} ^\prime}\to {\mathcal O}mega ^i_{{\mathcal X}}$ induces the map
$$H^\bullet ({\mathcal X} ^\prime ,{\mathcal O}mega ^i_{{\mathcal X} ^\prime})\to H^\bullet ({\mathcal X}
, {\mathcal O}mega ^i_{{\mathcal X}}).$$
\end{example}
Given another morphism of poset schemes ${\mathcal F} ^\prime :{\mathcal X} ^\prime
\to {\mathcal X} ^{\prime \prime}$ there are natural isomorphisms of functors
${\mathcal F} ^*{\mathcal F} ^{\prime *} \simeq ({\mathcal F} ^\prime {\mathcal F} )^*.$ Since the
functor ${\mathcal F} ^{\prime *}$ preserves h-flats we also have an
isomorphism ${\mathbf L} {\mathcal F} ^*\cdot {\mathbf L} {\mathcal F} ^{\prime
*} \simeq {\mathbf L} ({\mathcal F} ^\prime {\mathcal F} )^*.$
The functor ${\mathcal F} ^*$ has the right adjoint functor ${\mathcal F} _*$ which we
now describe.
We will use Remark \ref{alternative-def}
For $\alpha ^\prime \in S^\prime$ we put $\tau ^{-1}(\geq \alpha
^\prime):=\{ \gamma \in S\vert \tau (\gamma )\geq \alpha ^\prime\}.$
Fix $F\in Qcoh {\mathcal X}.$ If $\gamma \in \tau ^{-1}(\geq \alpha
^\prime),$ then $f^\prime _{\tau (\gamma)\alpha ^\prime *}({\mathcal F}
_{\gamma
*}F_{\gamma })\in Qcoh X^\prime _{\alpha ^\prime}.$ If $\delta \geq
\gamma, $ then the structure morphism $\psi _{\delta \gamma
}:F_{\gamma }\to f_{\delta \gamma *}F_{\delta}$ induces the morphism
$$f^\prime _{\tau (\gamma)\alpha ^\prime *}({\mathcal F}
_{\gamma *}F_{\gamma })\to f^\prime _{\tau (\delta)\alpha ^\prime
*}({\mathcal F} _{\delta *}F_{\delta }).$$
We define
$$({\mathcal F} _*F)_{\alpha ^\prime}=\lim _{\stackrel{\longleftarrow}{\gamma
\in \tau ^{-1}(\geq \alpha ^\prime)}}f^\prime _{\tau (\gamma)\alpha
^\prime *}({\mathcal F} _{\gamma *}F_{\gamma }).$$ If $\alpha ^\prime \geq
\beta ^\prime$ there is a natural morphism $\psi ^\prime _{\alpha
^\prime \beta ^\prime }:f^\prime _{\alpha ^\prime \beta ^\prime
*}({\mathcal F} _*F)_{\alpha ^\prime}\to ({\mathcal F} _*F)_{\beta ^\prime}.$ Thus
${\mathcal F} _*F\in Qcoh {\mathcal X} ^\prime$ and we get a functor ${\mathcal F} _*:Qcoh {\mathcal X}
\to Qcoh {\mathcal X} ^\prime.$ We define its right derived functor ${\mathbf R} {\mathcal F}
_*:D({\mathcal X} )\to D({\mathcal X} ^\prime)$ using the h-injectives. The pairs of
functors $({\mathcal F} ^*,{\mathcal F} _*)$ and $({\mathbf L} {\mathcal F} ^*,{\mathbf R} {\mathcal F} _*)$ and
adjoint.
Given another morphism of poset schemes ${\mathcal F} ^\prime :{\mathcal X} ^\prime
\to {\mathcal X} ^{\prime \prime}$ there are natural isomorphisms of functors
${\mathcal F} ^{\prime }_*{\mathcal F} _* \simeq ({\mathcal F} ^\prime {\mathcal F} )_*.$ Although the
functor ${\mathcal F} _*$ may not preserve h-injectives we still have a
natural isomorphism of functors ${\mathbf R} {\mathcal F} ^{\prime }_*\cdot {\mathbf R} {\mathcal F}
_* \simeq {\mathbf R} ({\mathcal F} ^\prime {\mathcal F} )_*$ (this follows by adjunction from
the isomorphism ${\mathbf L} {\mathcal F} ^*\cdot {\mathbf L} {\mathcal F} ^{\prime
*} \simeq {\mathbf L} ({\mathcal F} ^\prime {\mathcal F} )^*$).
The direct image functor may be computed fiberwise in case $\tau$ is
the projection of a product poset on one of the factors. Namely we
have the following lemma.
\begin{lemma}\label{direct-image-fiber} Assume that $T$ is a poset,
$S=S^\prime \times T$ is the product poset and $\tau :S\to
S^\prime$ is the projection. Then in the above notation for any
$\alpha ^\prime \in S^\prime$ we have
$$({\mathcal F} _*F)_{\alpha ^\prime}=\lim _{\stackrel{\longleftarrow}{\gamma
\in \tau ^{-1}(\alpha ^\prime)}}{\mathcal F} _{\gamma
*}F_{\gamma }.$$
\end{lemma}
\begin{proof} This is clear.
\end{proof}
\begin{example} Let $S^\prime$ consist of a single element $\alpha
^\prime$ and $X_{\alpha ^\prime}=pt.$ Then for $F\in D({\mathcal X})$
$${\mathbf R} ^i{\mathcal F} _*F=H^i({\mathcal X} ,F).$$
\end{example}
\section{Categorical resolutions by smooth poset schemes}
Let $S$ be a (finite) poset and let ${\mathcal X}$ be a smooth ($S-$)poset
scheme (so that the category $D({\mathcal X})$ is smooth by Theorem
\ref{smooth=smooth}). Let $Y$ be a scheme (which can be considered
as a poset scheme) and $\pi :{\mathcal X} \to Y$ be a morphism of poset
schemes (i.e. a $\tau$-morphism for $\tau :S\to pt,$ in the
terminology of the previous section).
\begin{defi} \label{def-cat-res-by-poset-schemes} We call the morphism $\pi$ a {\rm categorical
resolution} of $Y$ if the functor ${\mathbf L} \pi ^*:D(Y)\to D({\mathcal X})$ is a
categorical resolution, i.e. its restriction ${\mathbf L} \pi ^*:\operatorname{op}eratorname{Perf}
(Y)\to \operatorname{op}eratorname{Perf} ({\mathcal X})$ is full and faithful.
\end{defi}
We can localize the morphism $\pi$ over $Y$ in the obvious way.
Namely, given an open subset $W\subset Y$ we denote by ${\mathcal X} _W$ the
poset scheme which is the unverse image of $W$ under $\pi.$ Let $\pi
_W:{\mathcal X} _W\to W$ be the corresponding morphism. If $W$ is affine
$W=Spec B,$ then the $B$-module ${\mathbf R} ^j(\pi _W)_*{\mathcal O} _{{\mathcal X} _W}$ is
isomorphic to $H^j({\mathcal X} _W,{\mathcal O} _{{\mathcal X} _W}).$
\begin{prop} \label{criterion-cat-res} Let ${\mathcal X}$ be a smooth poset scheme, Y be a scheme and $\pi
:{\mathcal X} \to Y$ be a morphism. The following statements are equivalent.
1) $\pi$ is a categorical resolution;
2) the adjunction morphism ${\mathcal O} _Y\to {\mathbf R} \pi _*{\mathcal O} _{{\mathcal X}}$ is a
quasi-isomorphism;
3) for each affine open set $W\subset Y$ the map $H^0(W,{\mathcal O} _W)\to
H^0({\mathcal X} _W,{\mathcal O}_{{\mathcal X} _W})$ is an isomorphism and $H^j({\mathcal X} _W,{\mathcal O}_{{\mathcal X}
_W})=0$ for $j>0.$
\end{prop}
\begin{proof} The equivalence 2)$\Leftrightarrow$3) is clear. It
remains to prove the equivalence 1)$\Leftrightarrow$2).
\begin{lemma}\label{adjunction-categories} Let ${\mathcal C},{\mathcal D}$ be categories, $F:{\mathcal C} \to {\mathcal D}$ a
functor and $G:{\mathcal D} \to {\mathcal C}$ its right adjoint functor. Fix an object
$B\in {\mathcal C}$. Then the following assertions are equivalent
a) For any object $A\in {\mathcal C}$ the map $F:\operatorname{op}eratorname{Ho}m (A,B)\to \operatorname{op}eratorname{Ho}m
(F(A),F(B))$ is an isomorphism;
b) The adjunction morphism $I_B:B\to GF(B)$ is an isomorphism.
\end{lemma}
\begin{proof} The composition of the map $\operatorname{op}eratorname{Ho}m
(A,B)\stackrel{F}{\to}\operatorname{op}eratorname{Ho}m (F(A),F(B))$ with the canonical
isomorphism $\operatorname{op}eratorname{Ho}m (F(A),F(B))\simeq \operatorname{op}eratorname{Ho}m (A,GF(B))$ is equal to the
map $(I_B)_*:\operatorname{op}eratorname{Ho}m (A,B)\to \operatorname{op}eratorname{Ho}m (A, GF(B)).$
\end{proof}
Now we can prove the equivalence 1)$\Leftrightarrow$2).
Since the functor ${\mathbf L} \pi ^*:D(Y)\to D({\mathcal X})$ preserves direct sums
and perfect complexes (i.e. compact objects) it is easy to see that
it is full and faithful if and only if its restriction to the
subcategory $\operatorname{op}eratorname{Perf} (Y)$ is such. (Use the fact that $D(Y)$ is the
smallest triangulated subcategory of $D(Y)$ which contains $\operatorname{op}eratorname{Perf}
(Y)$ and is closed under direct sums.) Hence by Lemma
\ref{adjunction-categories} the functor ${\mathbf L} \pi ^*:\operatorname{op}eratorname{Perf} (Y)\to
\operatorname{op}eratorname{Perf} ({\mathcal X})$ is full and faithful if and only if for every $K\in
\operatorname{op}eratorname{Perf} (Y)$ the adjunction map $K\to {\mathbf R} \pi _*{\mathbf L} \pi ^*K$ is an
isomorphism. But the last assertion is local on $Y,$ and locally $K$
is isomorphic to a finite direct sum of shifted copies of the
structure sheaf.
\end{proof}
We give examples of categorical resolutions by smooth poset schemes
in Section \ref{examples} below.
\section{How to compute in $D({\mathcal X})$}\label{how-to-compute}
The restriction of an h-injective object $I\in D({\mathcal X})$ to $X_\alpha
\in {\mathcal X}$ may not be h-injective.
\begin{example} ${\mathcal X} =\{\operatorname{pt} \to {\mathbb A}^1\}$
and $I=j_*(k),$ where $j$ is the inclusion of the point $\operatorname{pt}$ in
${\mathcal X}.$ Then the object $I\in Qcoh{\mathcal X}$ is injective, hence
h-injective as an object in $D({\mathcal X}),$ but its restriction to ${\mathbb A}
^1$ is not.
\end{example}
Nevertheless if $I\in D({\mathcal X})$ is h-injective, then the object
$I_\alpha \in D(X_{\alpha})$ can be used to compute ${\mathbf R} \operatorname{op}eratorname{Ho}m
(M,-),$ if $M\in D(X_{\alpha})$ is h-flat.
\begin{lemma} \label{poset-maps-to-h-inj} Let $I\in D( {\mathcal X})$ be h-injective. Fix $\alpha \in
S$ and let $M\in D(X_{\alpha})$ be h-flat. Then the complex $\operatorname{op}eratorname{Ho}m
(M,I_{\alpha})$ is quasi-isomorphic to ${\mathbf R} \operatorname{op}eratorname{Ho}m (M,I_{\alpha}).$
\end{lemma}
\begin{proof} A proof of this lemma is contained in the proof of
Lemma \ref{adjunction} above.
\end{proof}
\begin{lemma} (a) Fix $\alpha \in S$ and let $F\in D({\mathcal X})$
be such that $F=j_{\alpha +}F_{\alpha}$ for an h-flat $F_{\alpha}\in
D(X_{\alpha}).$ Then for any $G\in D({\mathcal X})$ we have
$$\operatorname{op}eratorname{Ho}m _{D({\mathcal X})}(F,G)=\operatorname{op}eratorname{Ho}m
_{D(X_{\alpha})}(F_{\alpha},G_{\alpha}).$$
(b) Suppose that $\alpha \in S$ is the unique minimal element of
$S,$ i.e. $S=U_{\alpha}$ (Subsection \ref{operations}). Then for any
$G\in D({\mathcal X})$
$$H^\bullet({\mathcal X} ,G)=H^\bullet (X_{\alpha},G_{\alpha}).$$
\end{lemma}
\begin{proof} By Lemma \ref{adjunction} the functors $({\mathbf L} j_{\alpha
+},j_{\alpha}^*)$ are adjoint, which implies (a). Now (b) follows
because ${\mathcal O} _{{\mathcal X}}=j_{\alpha +}{\mathcal O} _{X_{\alpha}}.$
\end{proof}
The next proposition generalizes the last lemma.
\begin{prop}\label{prop-comput}
Suppose that a complex $F\in C({\mathcal X})$ has a resolution (in $C({\mathcal X})$)
\begin{equation} \label{resolution} 0\to K_{n}\to ...\to K_1 \to
K_0\to F\to 0\end{equation} where for each $i,$ $K_i=\operatorname{op}lus
_{\alpha}j_{\alpha +}M^i_{\alpha}$ with $M_{\alpha }^i\in
C(X_{\alpha})$ being h-flat. Let $I\in C({\mathcal X})$ be such that for
each $\alpha \in S$ and each $i,$ $\operatorname{op}eratorname{Ho}m (M^i_{\alpha},
I_{\alpha})={\mathbf R} \operatorname{op}eratorname{Ho}m (M^i_{\alpha}, I_{\alpha})$ (for example $I$ is
h-injective as in Lemma \ref{poset-maps-to-h-inj}).
Then the complex ${\mathbf R} \operatorname{op}eratorname{Ho}m (F,I)$ is
quasi-isomorphic to the total complex of the double complex
\begin{equation}\label{standard-double-complex} 0\to \operatorname{op}eratorname{Ho}m (K_0,I)\to
\operatorname{op}eratorname{Ho}m (K_1,I) \to ... \to \operatorname{op}eratorname{Ho}m (K_n,I)\to 0.\end{equation}
Moreover,
for each $i$
$$\operatorname{op}eratorname{Ho}m (K_i,I)=\bigoplus _{\alpha}\operatorname{op}eratorname{Ho}m ({\mathbf L} j_{\alpha *}M^i_{\alpha},I)=
\bigoplus _{\alpha }\operatorname{op}eratorname{Ho}m (M_{\alpha}^i,I_{\alpha})= \bigoplus
_{\alpha }{\mathbf R} \operatorname{op}eratorname{Ho}m (M_{\alpha}^i,I_{\alpha}).$$
Hence
in particular we obtain a spectral sequence which converges to $\operatorname{op}eratorname{Ext}
(F,I)$ with the $E_1$-term being the sum of groups $\operatorname{op}eratorname{Ext}
(M^i_{\alpha},I_{\alpha})$.
\end{prop}
\begin{proof} This follows from Lemma \ref{adjunction} and
Lemma \ref{poset-maps-to-h-inj}.
\end{proof}
The following example will be of primary interest to us.
\begin{example}\label{example} In case $F={\mathcal O} _{{\mathcal X}}$ one can take a
resolution \ref{resolution} with $K_i=\operatorname{op}lus _{\alpha }j_{\alpha
+}{\mathcal O} _{X_{\alpha}},$ i.e. $M^i_\alpha={\mathcal O} _{X_{\alpha}}.$ (The same
index $\alpha$ may appear in different $K_i$'s and it may also
appear more than once in a given $K_i.$) Given $G\in D({\mathcal X})$ choose
its h-injective replacement I. Then the double complex
\ref{standard-double-complex} consists of sums of spaces ${\mathcal G}amma
(X_{\alpha},I_{\alpha})$ and the $E_1$-term is the sum of groups
$H^\bullet (X_{\alpha },G_\alpha).$ The differential $d_1$ between
the cohomology groups is simply the sum of the maps induced by the
structure morphisms $\phi _{\alpha \beta}:f^*_{\alpha \beta}G_\beta
\to G_\alpha.$ In particular $d_1$ preserves the degree of the
cohomology groups $H^\bullet (X_{\alpha },G_\alpha).$
In case the complex $G\in D({\mathcal X})$ is bounded below we can use
instead of an h-injective $I$ the canonical Godement resolution
$G\to {\mathcal C} ^\bullet (G),$ such that for each $\alpha $ the complex
${\mathcal C} ^\bullet (G)$ consists of flabby sheaves. Notice that the
complex ${\mathcal C} ^\bullet(G)$ consists of ${\mathcal O} _{{\mathcal X}}$-modules which are
no longer quasi-coherent (see Section \ref{variants} below).
\end{example}
\begin{defi}\label{standard-sp-seq} We call any spectral sequence
converging to $H^\bullet ({\mathcal X} ,G)$ as in the above example a {\rm
standard} one. (It is not unique because one can choose different
resolutions \ref{resolution} of ${\mathcal O} _{{\mathcal X}}$.)
\end{defi}
\begin{example} Assume that a poset $S$ consists of 4 elements $\{
\alpha ,\beta _1,\beta _2,\beta _3\}$ where $\alpha \ge \beta _i$
for all $i$ and no other relations. Therefore we have 4 irreducible
open subsets $U_\alpha ,U_{\beta _i}\subset S.$ If ${\mathcal X}$ is an
$S$-scheme one can take for example the following resolution
\ref{resolution} of the structure sheaf ${\mathcal O} _{{\mathcal X}}$:
$$0\to K_1\to K_0\to {\mathcal O} _{{\mathcal X}} \to 0,$$
where $K_0=\operatorname{op}lus _ij_{\beta _i+}{\mathcal O} _{X_{\beta _i}}$ and $K_1=
(j_{\alpha +}{\mathcal O} _{X_\alpha})^{\operatorname{op}lus 2}.$ This gives a standard
spectral sequence converging to $H^\bullet ({\mathcal X} ,{\mathcal O} _{{\mathcal X}})$ with
the $E_1$-complex
$$\bigoplus _iH^\bullet (X_{\beta _i},{\mathcal O} _{X_{\beta _i}})\to
H^\bullet (X_\alpha ,{\mathcal O} _{{\mathcal X} _\alpha })^{\operatorname{op}lus 2}.$$
\end{example}
\part{Poset schemes and Du Bois singularities}
\section{Other variants of poset ringed spaces}\label{variants}
Besides poset schemes and quasi-coherent sheaves on them we can
consider "poset" versions of other usual structures. We give some
examples which will be used later. Let ${\mathcal X}$ be a poset scheme.
1) One may define an abelian category $\operatorname{op}eratorname{Mod} {\mathcal O} _{{\mathcal X}}$ just as we
defined $Qcoh {\mathcal X}$ by requiring the sheaves $F_\alpha $ to be
arbitrary ${\mathcal O} _{X_{\alpha}}$-modules and not necessarily
quasi-coherent ones. Moreover we may consider the abelian category
$\operatorname{op}eratorname{Sh}({\mathcal X})$ of sheaves of abelian groups on ${\mathcal X}.$ (That is we
consider each $X_\alpha$ as a ringed space with the structure sheaf
${\mathbb Z} _{X_\alpha},$ so that the gluing is by maps $\phi ^\prime
_{\alpha \beta}:f^{-1}_{\alpha \beta}F_\beta \to F_\alpha.$) Because
of the natural morphism $f_{\alpha \beta}^{-1}F_\beta \to f_{\alpha
\beta}^*F_\beta$ each object in $\operatorname{op}eratorname{Mod} {\mathcal X}$ defines an object of $\operatorname{op}eratorname{Sh}
({\mathcal X}).$
2) Denote by ${\mathcal X} ^{\operatorname{et}}$ the same diagram of schemes where we
consider each $X_\alpha$ with the etale topology. Let $\operatorname{op}eratorname{Sh} ({\mathcal X}
^{\operatorname{et}})$ denote the abelian category of sheaves of abelian groups on
${\mathcal X}.$ For a prime number $l$ and $n\geq 1$ let $\operatorname{op}eratorname{Sh} _{l^n} ({\mathcal X}
^{\operatorname{et}})\subset \operatorname{op}eratorname{Sh} ({\mathcal X} ^{\operatorname{et}})$ be the full subcategory of ${\mathbb Z}
/l^n$-modules.
3) If ${\mathcal X}$ is a complex poset scheme of finite type we may consider
the corresponding poset analytic space ${\mathcal X} ^{\operatorname{an}}.$ It comes with
the structure sheaf ${\mathcal O} _{{\mathcal X} ^{\operatorname{an}}}.$ (We will be interested in
${\mathcal X} ^{\operatorname{an}}$ only for projective ${\mathcal X}.$) Again we denote by $\operatorname{op}eratorname{Sh}
({\mathcal X} ^{\operatorname{an}})$ the abelian category of sheaves of abelian groups on
${\mathcal X} ^{\operatorname{an}}.$ As in the algebraic case, a sheaf of ${\mathcal O} _{{\mathcal X}
^{\operatorname{an}}}$-modules may be considered as an element of $\operatorname{op}eratorname{Sh} ({\mathcal X}
^{\operatorname{an}}).$ In particular the analytic deRham complex ${\mathcal O}mega ^\bullet
_{{\mathcal X} ^{\operatorname{an}}}$ is a complex in $\operatorname{op}eratorname{Sh} ({\mathcal X} ^{\operatorname{an}})$ which is a
resolution of the constant sheaf ${\mathbb C} _{{\mathcal X} ^{\operatorname{an}}}.$
All the functors defined in Section \ref{operations} for
quasi-coherent sheaves exist also in the categories described in
1),2),3) above. They have all the properties listed in Subsection
\ref{summary-1}.
\begin{lemma} \label{enough-injectives}
There are enough injectives in all the above categories $\operatorname{op}eratorname{Mod} {\mathcal O}
_{{\mathcal X}},$ $\operatorname{op}eratorname{Sh}({\mathcal X}),$ $\operatorname{op}eratorname{Sh} ({\mathcal X} ^{\operatorname{et}}),$ $\operatorname{op}eratorname{Sh} _{l^n} ({\mathcal X} ^{\operatorname{et}}),$
$\operatorname{op}eratorname{Sh} ({\mathcal X} ^{\operatorname{an}}),$ etc.
\end{lemma}
\begin{proof} The proof is essentially the same as the one of
Proposition \ref{finite-inj-res}.
\end{proof}
\begin{defi} Using the above lemma we may define for each bounded
below complex $L$ of sheaves in $\operatorname{op}eratorname{Sh} ({\mathcal X} ^?)$ its cohomology
$$H^\bullet ({\mathcal X} ^?, L)=\operatorname{op}eratorname{Ext} ^\bullet ({\mathbb Z} _{{\mathcal X} ^?},L)$$
\end{defi}
Let $L$ is a bounded above complex of sheaves in one of the
categories in Lemma \ref{enough-injectives}. There is a spectral
sequence converging to $H^\bullet ({\mathcal X} ^?,L)$ defined similarly to
Example \ref{example}. Namely, choose a resolution
$$0\to K_n\to ...\to K_0\to {\mathbb Z} _{{\mathcal X} ^?}\to 0$$
where each $K_i$ is a direct sum of objects $j_{\alpha +}{\mathbb Z}
_{X_{\alpha}},$ which are extensions by zero from irreducible open
subsets $U_{\alpha}$ of the constant sheaf ${\mathbb Z}.$ Choose also an
injective resolution $L\to I.$ Then exactly as in Section
\ref{how-to-compute} we get a spectral sequence which converges to
$H^\bullet({\mathcal X} ,L).$ The $E_0$-term consists of sums of spaces
$${\mathcal G}amma (X_{\alpha},I_{\alpha})=\operatorname{op}eratorname{Ho}m(j_{\alpha +}{\mathbb Z}
_{X_{\alpha}},I_{\alpha})$$ and the $E_1$-term is the sum of
cohomologies $H^\bullet (X_{\alpha},L_{\alpha}).$
Notice that instead of an injective resolution $L\to I$ we could use
the canonical flabby Godement resolution $L\to G(L).$ (Since the
Godement resolution of usual sheaves is functorial it extends to
poset sheaves in $\operatorname{op}eratorname{Sh} ({\mathcal X} ^?).$)
\begin{defi} \label{standard-sp-seq-general}
As in the case of quasi-coherent sheaves (Definition
\ref{standard-sp-seq}) we call the above spectral sequence
converging to $H^\bullet ({\mathcal X} ^?,L)$ a {\rm standard} one.
\end{defi}
\begin{remark} Assume that $L$ is a bounded below complex in $Qcoh
{\mathcal X} .$ By comparing the corresponding standard spectral sequences we
conclude that the cohomology of $L$ is the same whether we consider
$L$ as a complex over $Qcoh {\mathcal X}$ or over $\operatorname{op}eratorname{Sh} ({\mathcal X}).$
\end{remark}
\subsection{Poset GAGA}\label{GAGA}
Let $X$ be a complex projective variety, $X^{\operatorname{an}}$ - the
corresponding analytic space and $\iota :X^{\operatorname{an}}\to X$ the canonical
morphism of locally ringed spaces. For an ${\mathcal O} _X$-module $F$ we
denote by $F^{\operatorname{an}}=\iota ^*F$ its analytization. By adjunction we
obtain the canonical morphism of sheaves $a_F:F\to \iota _*F^{\operatorname{an}}.$
Let $Y$ be another complex projective variety and $f:X\to Y$ be a
morphism. The adjunction morphism $a_F$ induces a morphism of
sheaves $\theta _F:(f_*F)^{\operatorname{an}}\to f_*^{\operatorname{an}}F^{\operatorname{an}}.$ If $F$ is
coherent then it is known by \cite{SGAI}, Expose XII, Th. 4.2 (which
is an extension of GAGA) that this morphism $\theta _F$ induces a
quasi-isomorphism $({\mathbf R} f_*F)^{\operatorname{an}}\to {\mathbf R} f^{\operatorname{an}}_*F^{\operatorname{an}}.$ In
particular $H(X,F)=H(X^{\operatorname{an}},F^{\operatorname{an}})$ for a coherent sheaf $F.$
Let $S$ be a poset, let ${\mathcal X}$ be a complex projective $S$-scheme,
and $F\in \operatorname{op}eratorname{Mod} {\mathcal O} _{{\mathcal X}}.$ Again we denote by $F^{\operatorname{an}}$ - the
analytization of $F$ - the corresponding analytic sheaf on the poset
analytic space ${\mathcal X} ^{\operatorname{an}}.$ The poset analogue of the adjunction
map $a_F$ above induces a morphism of the standard spectral
sequences for $H^\bullet ({\mathcal X} ,F)$ and $H^\bullet ({\mathcal X}
^{\operatorname{an}},F^{\operatorname{an}}).$ If $F\in coh {\mathcal X}$ then it follows from the above
cited result in \cite{SGAI} that the induced morphism of $E_1$-terms
is an isomorphism. In particular for a coherent $F$ we have
$H^\bullet ({\mathcal X} ,F)=H^\bullet ({\mathcal X} ^{\operatorname{an}},F^{\operatorname{an}}).$ Moreover for
$F\in coh {\mathcal X}$ the standard spectral sequence for $H^\bullet ({\mathcal X}
,F)$ degenerates at $E_r$ for $r\geq 2$ if and only if the standard
spectral sequence for $H^\bullet ({\mathcal X} ^{\operatorname{an}} ,F^{\operatorname{an}})$ degenerates
at $E_r.$ All the above holds also for bounded below complexes of
coherent sheaves on ${\mathcal X}.$
Let $S^\prime $ be another poset and $\tau :S\to S^\prime$ - a map
of posets. Let ${\mathcal X} ^\prime$ be a complex projective
$S^\prime$-scheme and ${\mathcal F} :{\mathcal X} \to {\mathcal X} ^\prime$ - a $\tau$-morphism
(Definition \ref{tau-morphism}). Then for $F\in \operatorname{coh} {\mathcal X}$ there is a
natural quasi-isomorphism of complexes of sheaves on ${\mathcal X} ^{\prime
\operatorname{an}}$
$$({\mathbf R} {\mathcal F} _*F)^{\operatorname{an}}\stackrel{\sim}{\longrightarrow}{\mathbf R}
{\mathcal F}^{\operatorname{an}}_*F^{\operatorname{an}}.$$ In particular, for the deRham complex ${\mathcal O}mega
^\bullet _{{\mathcal X}}$ we have
$$({\mathbf R} {\mathcal F} _*{\mathcal O}mega ^\bullet _{{\mathcal X}})^{\operatorname{an}}\simeq {\mathbf R} {\mathcal F}^{\operatorname{an}}_*
{\mathcal O}mega^{\bullet}_{{\mathcal X} ^{\operatorname{an}}}.$$
\section{Degeneration of the standard spectral sequence for
$H^\bullet ({\mathcal X} ^{\operatorname{an}},{\mathbb C})$ when ${\mathcal X}$ is a smooth projective
poset scheme}
\begin{theo} \label{degeneration-standard} Let ${\mathcal X}$ be a smooth complex
projective poset scheme. Then the standard spectral sequence
converging to $H^\bullet({\mathcal X} ^{\operatorname{an}},{\mathbb C} )=H^\bullet({\mathcal X} ^{\operatorname{an}},{\mathbb C}
_{{\mathcal X} ^{\operatorname{an}}})$ degenerates at $E_2$ ($d_2=d_3=...0$). That is the
cohomology $H^\bullet({\mathcal X} ^{\operatorname{an}},{\mathbb C} )$ is isomorphic to the
cohomology of the complex
\begin{equation}\label{E-one}E_1=...\to \operatorname{op}lus H^\bullet
(X ^{\operatorname{an}}_{\beta},{\mathbb C} )\stackrel{\operatorname{op}lus f^*_{\alpha
\beta}}{\longrightarrow} \operatorname{op}lus H^\bullet (X ^{\operatorname{an}}_{\alpha},{\mathbb C}
)\to ...\end{equation}
\end{theo}
\begin{proof} We use Weil conjectures (Deligne's theorem)
\cite{De} to prove this. We follow the strategy of \cite{BBD},Ch.6
using canonical Godement flabby resolutions as in
\cite{FK},Ch.1,Sect.11,12. The argument has three steps: first we
pass from the analytic topology to the etale one, then pass to a
poset scheme over a finite field, and finally we use purity of the
Frobenius endomorphism on the etale $l$-adic cohomology of a smooth
projective scheme.
\noindent{\bf Step 1.} Choose a prime number $l.$ Since the fields
$\bar{{\mathbb Q}}_l$ and
${\mathbb C}$ are isomorphic, it suffices to prove the degeneration
of the analogous spectral sequence for the cohomology groups
$H^\bullet({\mathcal X} ^{\operatorname{an}},\bar{{\mathbb Q}}_{l}).$
Let $Y$ be a complex scheme. We have the natural morphism of topoi
$\iota:Y ^{\operatorname{an}}\to Y ^{\operatorname{et}}.$ This morphism induces the inverse
image functor between the corresponding categories of abelian
sheaves $\iota ^*:\operatorname{op}eratorname{Sh} (X^{\operatorname{et}})\to \operatorname{op}eratorname{Sh} (X^{\operatorname{an}}).$ It has the
following properties \cite{FK},Ch.1,Prop.11.4.
\begin{itemize}
\item Given a morphism of schemes $f:X\to Y$ there is a natural isomorphism
of functors $f^{\operatorname{an} *}\cdot \iota _Y^*=\iota _X^*\cdot f^{\operatorname{et} *}.$
In particular, $\iota ^*$ is an exact functor.
\item For any point $y\in Y^{\operatorname{an}}$ and any $F\in \operatorname{op}eratorname{Sh} (Y^{\operatorname{et}})$ the stalks
$F_y$ and $(\iota ^*F)_y$ are naturally isomorphic.
\item For a finite ring $R$ we have $\iota ^*(R_{Y^{\operatorname{et}}})=R_{Y^{\operatorname{an}}}$ and
it induces an isomorphism $H^\bullet (Y^{\operatorname{an}},R)=H^\bullet
(Y^{\operatorname{et}},R).$
\end{itemize}
Recall that the cohomology groups $H^\bullet(Y^{\operatorname{et}},\bar{{\mathbb Q}}_l)$
are defined as
$$H^\bullet(Y^{\operatorname{et}},\bar{{\mathbb Q}}_l):=
(\lim _{\leftarrow}H^\bullet(Y^{\operatorname{et}},{\mathbb Z} /l^n))\otimes _{{\mathbb Z}
_l}\bar{{\mathbb Q}}_l$$
It is known that the morphism $\iota$ induces an isomorphism $\iota
^*:H^\bullet (Y ^{\operatorname{et}},\bar{{\mathbb Q}}_l) \to H^\bullet (Y
^{\operatorname{an}},\bar{{\mathbb Q}}_l).$ We want to extend this result to poset
schemes.
Namely, let ${\mathcal X} ^{\operatorname{et}}$ denote the poset scheme ${\mathcal X}$ considered in
the etale topology. Similarly to the analytic case we define the
cohomology groups $H^\bullet ({\mathcal X} ^{\operatorname{et}},{\mathbb Z} /l^n)=\operatorname{op}eratorname{Ext}^\bullet
(({\mathbb Z}/l^n)_{{\mathcal X} ^{\operatorname{et}}},({\mathbb Z}/l^n)_{{\mathcal X} ^{\operatorname{et}}})$ and
$$H^\bullet({\mathcal X}^{\operatorname{et}},\bar{{\mathbb Q}}_l):=
(\lim _{\leftarrow}H^\bullet({\mathcal X}^{\operatorname{et}},{\mathbb Z} /l^n))\otimes _{{\mathbb Z}
_l}\bar{{\mathbb Q}}_l$$ Again there is an obvious standard spectral
sequence converging to $H^\bullet({\mathcal X}^{\operatorname{et}},\bar{{\mathbb Q}}_l).$
The morphism of topoi $\iota$ induces the corresponding morphism
$\iota :{\mathcal X} ^{\operatorname{an}}\to {\mathcal X} ^{\operatorname{et}}$ and the functor $\iota ^*:\operatorname{op}eratorname{Sh} ({\mathcal X}
^{\operatorname{et}})\to \operatorname{op}eratorname{Sh} ({\mathcal X} ^{\operatorname{an}}).$
\begin{lemma}\label{analytic=etale}
The morphism of topoi $\iota:{\mathcal X} ^{\operatorname{an}}\to {\mathcal X} ^{\operatorname{et}}$ induces an
isomorphism
$H^\bullet({\mathcal X}^{\operatorname{et}},\bar{{\mathbb Q}}_l)=H^\bullet({\mathcal X}^{\operatorname{an}},\bar{{\mathbb Q}}_l).$
More precisely, there is a natural morphism of standard spectral
sequences converging to $H^\bullet ({\mathcal X}^{\operatorname{et}},\bar{{\mathbb Q}}_l)$ and
$H^\bullet ({\mathcal X}^{\operatorname{an}},\bar{{\mathbb Q}}_l)$ respectively, which induces an
isomorphism of the corresponding $E_1$-complexes.
\end{lemma}
\begin{proof} For each $\alpha \in S$ and $n\in {\mathbb Z}$ denote by
$({\mathbb Z} /l^n)_{X_\alpha}\to G _{\alpha ,n}$ the canonical Godement
flabby resolution \cite{Go},\cite{FK},pp.129-130. Then naturally $G
_n= \{G _{\alpha n}\}$ is a complex in $\operatorname{op}eratorname{Sh} ({\mathcal X} ^{\operatorname{et}}).$ Moreover,
$G_{n+1}\otimes _{{\mathbb Z} /l^{n+1}}{\mathbb Z} /l^n=G_n.$ The cohomology
$H^\bullet ({\mathcal X} ^{\operatorname{et}},{\mathbb Z} /l^n)$ can be computed using the
resolution $G_n.$ In particular the standard spectral sequence
converging to $H^\bullet ({\mathcal X} ^{\operatorname{et}},{\mathbb Z} /l^n)$ is defined by the
double complex ${\mathcal G}amma (G_n)$ which consists of sums of groups
${\mathcal G}amma (X_\alpha ,G_n).$ These double complexes form an inverse
system
\begin{equation}\label{inverse-system} ...\to {\mathcal G}amma (G_2)\to {\mathcal G}amma (G_1)
\end{equation}
and the double complex
\begin{equation}\label{double-complex}
\lim _{\leftarrow}{\mathcal G}amma (G_n)\otimes _{{\mathbb Z} _l}\bar{{\mathbb Q} }_l
\end{equation}
computes the cohomology $H^\bullet ({\mathcal X} ^{\operatorname{et}},\bar{{\mathbb Q} }_l).$
Applying the functor $\iota ^*$ to the inverse system of complexes
$\{G_n\}$ provides the desider morphism of standard spectral
sequences for $H^\bullet({\mathcal X}^{\operatorname{et}},\bar{{\mathbb Q}}_l)$ and $
H^\bullet({\mathcal X}^{\operatorname{an}},\bar{{\mathbb Q}}_l)$ respectively. This morphism
induces an isomorphism of $E_1$-terms, because
$H^\bullet(X^{\operatorname{et}}_{\alpha},\bar{{\mathbb Q}}_l)=
H^\bullet(X^{\operatorname{an}}_{\alpha},\bar{{\mathbb Q}}_l)$ for each $\alpha.$
\end{proof}
So in order to prove the theorem it suffices to show the
degeneration of the standard spectral sequence for
$H^\bullet({\mathcal X}^{\operatorname{et}},\bar{{\mathbb Q}}_l).$
{\bf Step 2.} For any smooth complex scheme $Y$ we can find a
discrete valuation ring $V\subset {\mathbb C}$ whose residue field is the
algebraic closure of a finite field, and a smooth morphism $Y_V\to
SpecV,$ such that $Y$ is obtained by extension of scalars from
$Y_V.$ Let $Y_s$ be the closed fiber of $Y_V.$ We obtain the diagram
of schemes
$$Y\stackrel{u}{\longrightarrow}Y_V \stackrel{i}{\longleftarrow}Y_s.$$
These morphisms induce isomorphisms
$$H^\bullet (Y^{\operatorname{et}},\bar{{\mathbb Q}}_l)\stackrel{u^*}{\longleftarrow}
H^\bullet (Y^{\operatorname{et}}_V,\bar{{\mathbb Q}}_l) \stackrel{i^*}{\longrightarrow}
H^\bullet (Y^{\operatorname{et}}_s,\bar{{\mathbb Q}}_l).$$
This extends to smooth poset schemes. Namely, we can find $V$ as
above and a smooth poset scheme ${\mathcal X} _V$ over $SpecV,$ which gives
rise to ${\mathcal X}$ by extension of scalars. Let ${\mathcal X} _s$ again be the
closed fiber, which is a smooth poset scheme over $\bar{\bf F}_q.$
Consider the correspodning diagram of poset schemes
$${\mathcal X} \stackrel{u}{\longrightarrow}{\mathcal X} _V \stackrel{i}{\longleftarrow}
{\mathcal X} _s.$$
\begin{lemma} The morphisms $u,i$ induce isomorphisms
$$H^\bullet ({\mathcal X}^{\operatorname{et}},\bar{{\mathbb Q}}_l)\stackrel{u^*}{\longleftarrow}
H^\bullet ({\mathcal X}^{\operatorname{et}}_V,\bar{{\mathbb Q}}_l) \stackrel{i^*}{\longrightarrow}
H^\bullet ({\mathcal X}^{\operatorname{et}}_s,\bar{{\mathbb Q}}_l).$$ More precisely the morphisms
$u,i$ induce morphisms of the standard spectral sequences converging
to these groups. And these morphisms induces isomorphisms of the
corresponding $E_1$-terms.
\end{lemma}
\begin{proof} The proof is very similar to the proof of Lemma
\ref{analytic=etale}. Namely one considers the Godement resolution
$G_n$ of the constant sheaf ${\mathbb Z} /l^n$ on ${\mathcal X} _V$ and passes to
the inverse limit. We omit the details.
\end{proof}
So it suffices to prove the degeneration of the standard spectral
sequence for $$H^\bullet ({\mathcal X}^{\operatorname{et}}_s,\bar{{\mathbb Q}}_l).$$
\noindent{\bf Step 3.} The geometric Frobenius endomorphism $\operatorname{Fr}$
acts on the smooth poset scheme ${\mathcal X} _s$ and hence on the standard
spectral sequence which converges to $H^\bullet
({\mathcal X}^{\operatorname{et}}_s,\bar{{\mathbb Q}}_l).$ For each $\alpha \in S$ $\operatorname{Fr}$ acts on
$H^n (X^{\operatorname{et}}_{\alpha s},\bar{{\mathbb Q}}_l)$ with eigenvalues $\theta$
such that $\vert \theta \vert =q^{n/2}$ (Weil conjectures, see
\cite{De}).
In the standard spectral sequence each differential $d_r$ for $r\geq
2$ is a map between subquotients of $H^n (X^{\operatorname{et}}_{\alpha
s},\bar{{\mathbb Q}}_l)$ and $H^m (X^{\operatorname{et}}_{\beta s},\bar{{\mathbb Q}}_l)$ for
$n>m.$ Hence $d_r=0$ for $r\geq 2.$ This completes the proof of
Theorem \ref{degeneration-standard}.
\end{proof}
\section{Degeneration of Hodge to deRham spectral sequence for
smooth projective poset schemes.}
\begin{defi}
Let ${\mathcal X}$ be a smooth complex projective poset scheme. Recall that
the analytic deRham complex ${\mathcal O}mega ^\bullet _{{\mathcal X} ^{\operatorname{an}}}$ is a
resolution of the constant sheaf ${\mathbb C} _{{\mathcal X} ^{\operatorname{an}}}.$ As in the
case of a single smooth variety the "stupid" filtration $F^p{\mathcal O}mega
^\bullet _{{\mathcal X} ^{\operatorname{an}}}:=\operatorname{op}lus _{i\geq p}{\mathcal O}mega ^{i}_{{\mathcal X} ^{\operatorname{an}}}$
of this deRham complex gives rise to the {\it Hodge-to-deRham}
spectral sequence converging to $H^\bullet ({\mathcal X} ^{\operatorname{an}},{\mathbb C} ).$
\end{defi}
The following theorem is the poset scheme analogue of the well known
degeneration of the Hodge-to-deRham spectral sequence for smooth
projective varieties. The proof uses Theorem
\ref{degeneration-standard} above.
\begin{theo} \label{Hodge-to-deRham-degener-analytic}
Let ${\mathcal X}$ be a smooth complex projective poset scheme. Then the
Hodge-to-deRham spectral sequence degenerates at the $E_2$-term.
That is $d_2=d_3=...=0.$ Hence
\begin{equation}\label{Hodge-decomp}H^\bullet({\mathcal X} ^{\operatorname{an}},{\mathbb C} )=
\bigoplus _pH^{\bullet -p}({\mathcal X} ^{\operatorname{an}},{\mathcal O}mega ^p _{{\mathcal X} ^{\operatorname{an}}}).
\end{equation} In particular the map ${\mathbb C} _{{\mathcal X} ^{\operatorname{an}}}\to {\mathcal O}
_{{\mathcal X} ^{\operatorname{an}}}$ induces a surjection $H^\bullet ({\mathcal X} ^{\operatorname{an}},{\mathbb C} )\to
H^\bullet ({\mathcal X} ^{\operatorname{an}},{\mathcal O} _{{\mathcal X} ^{\operatorname{an}}})=H^\bullet ({\mathcal X} ,{\mathcal O}
_{{\mathcal X}}).$
The decomposition \ref{Hodge-decomp} is (contravariant) functorial
with respect to morphisms of smooth projective poset schemes.
\end{theo}
\begin{proof} The degeneration of the Hodge-to-deRham spectral
sequence follows by dimension counting from the isomorphism
\ref{Hodge-decomp}. The last assertion of the theorem is obvious. So
it suffices to prove \ref{Hodge-decomp}. To compute the cohomology
of ${\mathbb C} _{{\mathcal X} ^{\operatorname{an}}}$ we may use the Dolbeaut resolution ${\mathbb C}
_{{\mathcal X} ^{\operatorname{an}}}\stackrel{\sim}{\to} {\mathcal O}mega ^\bullet _{{\mathcal X} ^{\operatorname{an}}}\to
{\mathcal A} _{{\mathcal X} ^{\operatorname{an}}}^{\bullet \bullet},$ where ${\mathcal A} ^{p,q}$ is the
sheaf of $C^\infty$ $(p,q)$-forms. The canonical morphism of
complexes ${\mathcal O}mega _{{\mathcal X} ^{\operatorname{an}}}^\bullet \leftarrow {\mathcal O}mega ^{\geq
p}_{{\mathcal X} ^{\operatorname{an}}}\to {\mathcal O}mega ^p_{{\mathcal X} ^{\operatorname{an}}}$ lifts to a morphism of
the corresponding Dolbeaut resolutions ${\mathcal A} ^{\bullet \bullet}
\leftarrow {\mathcal A} ^{\geq p,\bullet}\to {\mathcal A} ^{p,\bullet}.$ Thus we
obtain the induced morphisms of standard spectral sequences
(Definition \ref{standard-sp-seq-general})
for ${\mathcal O}mega _{{\mathcal X} ^{\operatorname{an}}}^\bullet ,{\mathcal O}mega ^{\geq p}_{{\mathcal X} ^{\operatorname{an}}}$
and ${\mathcal O}mega ^p_{{\mathcal X} ^{\operatorname{an}}}$ respectively.
Using the usual Hodge decomposition for each $X_\alpha \in {\mathcal X}$ we
find that the $E_1$-term of the standard spectral sequence for
${\mathcal O}mega _{{\mathcal X} ^{\operatorname{an}}}^\bullet$ is the direct sum of complexes
$E_1^{(p,q)},$ where $E_1^{(p,q)}$ consists of summands
$H^{p,q}(X_\alpha ,{\mathbb C} ).$ Certainly the $E_1$ term of the standard
spectral sequence for the complex ${\mathcal O}mega ^{\geq p}_{{\mathcal X} ^{\operatorname{an}}}$
(resp. ${\mathcal O}mega ^{\leq p}_{{\mathcal X} ^{\operatorname{an}}},$ resp. ${\mathcal O}mega ^{p}_{{\mathcal X}
^{\operatorname{an}}}$) identifies as a direct summand of this complex which
consists of summands $H^{\geq p,\bullet}(X_\alpha {\mathbb C})$ (resp.
$H^{\leq p,\bullet}(X_{\alpha},{\mathbb C} ),$ resp.
$H^{p,\bullet}(X_{\alpha},{\mathbb C} )$). By Theorem
\ref{degeneration-standard} the standard spectral sequence for the
complex ${\mathcal O}mega ^\bullet _{{\mathcal X} ^{\operatorname{an}}}$ degenerates at $E_2.$
Applying the next lemma we conclude that the standard spectral
sequences for these other complexes also degenerate at $E_2.$ Now
using the dimension count we find the isomorphism
\ref{Hodge-decomp}, which proves the theorem.
\end{proof}
\begin{lemma} Let $A\to B$ be a morphism of bounded below double
complexes. Denote by $E_r(A)$ and $E_r(B)$ the $E_r$-terms of the
corresponding spectral sequences converging to $H^\bullet (Tot(A))$
and $H^\bullet(Tot(B))$ respectively.
i) Assume that the spectral sequence for $B$ degenerates at
$E_r(B),$ i.e. $0=d_r(B)=d_{r+1}(B)=...$ and the induced map of
complexes $E_r(A)\to E_r(B)$ is injective. Then the sequence for $A$
also degenerates at $E_r.$
ii) Assume that the sequence for $A$ degenerates at $E_r$ and the
map $E_r(A)\to E_r(B)$ is surjective. Then the sequence for $B$ also
degenerates at $E_r.$
\end{lemma}
\begin{proof} This is obvious.
\end{proof}
In the proof of the last theorem we also obtained the following
result.
\begin{prop}\label{analytic-degeneration} Let ${\mathcal X}$ be a smooth complex projective poset scheme.
Then the standard spectral sequences converging to the cohomology of
${\mathcal X} ^{\operatorname{an}}$ with coefficients respectively in ${\mathcal O}mega ^{\geq
p}_{{\mathcal X} ^{\operatorname{an}}}, {\mathcal O}mega ^{\leq p}_{{\mathcal X} ^{\operatorname{an}}},{\mathcal O}mega ^{p}_{{\mathcal X}
^{\operatorname{an}}}$ degenerate at $E_2$-terms.
\end{prop}
Now using GAGA we derive the corresponding statements in the
algebraic category. Namely let ${\mathcal X}$ be a smooth complex projective
poset scheme. We consider again the "stupid" filtration $F^p{\mathcal O}mega
^\bullet _{{\mathcal X}}:=\operatorname{op}lus _{i\geq p}{\mathcal O}mega ^{i}_{{\mathcal X} }$ of the
algebraic deRham complex. It gives rise to the spectral sequence
converging to $H^\bullet ({\mathcal X} ^{\operatorname{an}},{\mathcal O}mega ^\bullet _{{\mathcal X}} ).$ We
also call it "Hodge-to-de Rham".
\begin{theo} \label{degeneration-algebraic}
Let ${\mathcal X}$ be a smooth complex projective poset scheme.
a) The (algebraic) Hodge-to-de Rham spectral sequence degenerates at
the $E_2$-term. That is $d_2=d_3=...=0.$ Hence
\begin{equation}\label{algebraic-Hodge-decomp}
H^\bullet({\mathcal X} ,{\mathcal O}mega ^\bullet _{{\mathcal X}} )= \bigoplus _pH^{\bullet
-p}({\mathcal X} ,{\mathcal O}mega ^p _{{\mathcal X}}). \end{equation}
The decomposition \ref{algebraic-Hodge-decomp} is functorial with
respect to morphisms of poset schemes.
b) The standard spectral sequences converging to the cohomology of
${\mathcal X} $ with coefficients respectively in ${\mathcal O}mega ^{\geq p}_{{\mathcal X} },
{\mathcal O}mega ^{\leq p}_{{\mathcal X} },{\mathcal O}mega ^{p}_{{\mathcal X} }$ degenerate at
$E_2$-terms.
\end{theo}
\begin{proof} a) As in the analytic case everything follows from the
isomorphism \ref{algebraic-Hodge-decomp} by dimension counting. But
this isomorphism \ref{algebraic-Hodge-decomp} follows from the
isomorphism \ref{Hodge-decomp} and Subsection \ref{GAGA}.
b) This follows from Proposition \ref{analytic-degeneration} and
Subsection \ref{GAGA}.
\end{proof}
\begin{example} Let us give a simple example of a projective poset scheme
which is not smooth and for which the standard spectral sequence
converging to $H^\bullet ({\mathcal X} ,{\mathcal O} _{{\mathcal X}})$ does not degenerate at
$E_2.$ Namely, let $X$ be be a projective curve which is the union
of two projective lines $C_1$ and $C_2$ which intersect
transversally at 2 points $p_1$ and $p_2.$ Then $H^1(X,{\mathcal O} _X)$ has
dimension 1. Now take two copies of the curve $X=X_1=X_2,$ and let
the poset scheme ${\mathcal X}$ consist of $X_1,X_2,C_1,C_2,p_1,p_2$ with the
obvious maps from each of the $C$'s (resp. $p$'s) to each of the
$X$'s (resp. $C$'s). Then a standard spectral sequence converging to
$H^\bullet ({\mathcal X} ,{\mathcal O} _{{\mathcal X}})$ has for the $E_1$-term the natural
complex
$$0\to H^\bullet(X_1)\operatorname{op}lus H^\bullet (X_2)\to
H^\bullet(C_1)\operatorname{op}lus H^\bullet (C_2)\to H^\bullet(p_1)\operatorname{op}lus
H^\bullet (p_2)\to 0 $$ where $H^\bullet (Y)$ denotes $H^\bullet
(Y,{\mathcal O} _Y).$ Let $0\neq a\in H^1(X,{\mathcal O} _X).$ Then $(a,-a)$ is a
nonzero cycle in the above complex and it is not difficult to check
that $d_2(a,-a)\neq 0.$
\end{example}
\section{Cubical hyperresolutions and Du Bois singularities}
Cubical hyperresolutions are poset schemes of a certain type. Here
we briefly recall the definition and the main properties of cubical
hyperresolutions according to \cite{LNM1335},Ex.1.
For each integer $n\geq -1$ we denote by by $\square ^+_n$ the poset
which is the product of $n+1$ copies of the poset $\{0,1\}$. Thus
for $n=-1$ the poset $\square ^+_{-1}$ consists of one element and
$\square ^+_{0}=\{0,1\}.$ Let $\square _n$ denote the complement in
$\square ^+_{n}$ of the initial object $(0...0).$ For $\alpha
=(\alpha _0...\alpha _n)\in \square ^+_n$ we put $\vert \alpha \vert
=\alpha _0+...+\alpha _n.$
\begin{defi} Let $S$ be a (finite) poset,
${\mathcal X}$ be a reduced separated $S$-scheme
of finite type, and let ${\mathcal Z} $ be a reduced $\square ^+_1\times
S$-scheme. We call ${\mathcal Z} $ a 2-resolution of ${\mathcal X}$ if for each $\beta
\in S$ the commutative diagram
$$\begin{array}{ccc}
Z_{11\beta} & \to & Z_{01\beta}\\
\downarrow & & \downarrow f\\
Z_{10\beta } & \to & Z_{00\beta}
\end{array}
$$
has the following properties:
1) it is a cartesian square,
2) $Z_{00\beta }=X_{\beta},$
3) $Z_{01\beta}$ is smooth,
4) horizontal arrows are closed embeddings,
5) the morphism $f$ is proper,
6) $Z_{10\beta}$ contains the discriminant of $f.$ In other words
$f$ induces an isomorphism $f:Z_{01\beta }\backslash
Z_{11\beta}\stackrel{\sim}{\to} Z_{00\beta }\backslash Z_{10\beta}.$
\end{defi}
\begin{defi} Fix a poset $S$ and an integer $r\geq 1.$
Assume that for each $1\leq n \leq r$ we are given an
$\square ^+_{n}\times S$-scheme ${\mathcal X} ^n$ so that the $\square
^+_{n-1}\times S$ schemes ${\mathcal X} ^{n+1}_{00\bullet}$ and ${\mathcal X}
^n_{1\bullet}$ are equal. We define by induction on $r$ an $\square
^+_r\times S$-scheme ${\mathcal Z} =\rm{rd}({\mathcal X} ^1,{\mathcal X}^2,...,{\mathcal X} ^r),$ which
we call the reduction of $({\mathcal X} ^1 ,...,{\mathcal X} ^r ).$ Namely, if $r=1$
we put ${\mathcal Z} ={\mathcal X} ^1.$ If $r=2$ we define
$$Z _{\alpha \beta}= \left\{ \begin{array}{ll}
X^1_{0\beta}, & \text{if $\alpha =(00),$}\\
X^2_{\alpha \beta}, & \text{if $\alpha \in \square _1$}\\
\end{array} \right.
$$
for all $\beta \in \square ^+_0\times S.$ For $r>2$ we put
$${\mathcal Z} =\rm{rd}(\rm{rd}({\mathcal X} ^1 ,...,{\mathcal X} ^{r-1}),{\mathcal X} ^r).$$
\end{defi}
\begin{defi} Let $S$ be a poset and ${\mathcal X}$ be an $S$-scheme.
An {\rm augmented cubical hyperresolution} of ${\mathcal X}$ is an $\square
^+_r\times S$-scheme ${\mathcal Z} ^+$ such that
$${\mathcal Z} ^+=\rm{rd}({\mathcal X} ^1,...,{\mathcal X} ^r),$$
where
1) ${\mathcal X}^1$ is a 2-resolution of ${\mathcal X},$
2) for each $1\leq n \le r,$ ${\mathcal X} ^{n+1}$ is a 2-resolution of ${\mathcal X}
^n_{1\bullet},$ and
2) $Z_{\alpha }$ is smooth for each $\alpha \in \square _r.$
We will call the $\square _r$-scheme ${\mathcal Z} ={\mathcal Z} ^+ \backslash
Z_{(0,...,0)}$ a {\rm cubical hyperresolution} of ${\mathcal X}.$ It comes
with the augmentation morphism of poset schemes $\pi :{\mathcal Z} \to {\mathcal X},$
which is compatible with the projection of posets $\square _r\times
S\to S.$
\end{defi}
\begin{theo} Assume that the base field $k$ has characteristic zero.
Let $S$ be a poset and ${\mathcal X} $ be a separated reduced $S$-scheme of
finite type. Then there exists an augmented cubical hyperresolution
${\mathcal Z}$ of ${\mathcal X},$ such that $\dim Z_{\alpha}\leq \dim {\mathcal X} -\vert
\alpha \vert +1.$
\end{theo}
\begin{prop} \label{fiber-hyperres} Let $S$ be a poset,
${\mathcal X}$ an $S$-scheme and ${\mathcal Z}$ an
$\square ^+_r\times S$-scheme, which is an augmented cubical
hyperresolution of ${\mathcal X}.$ Then for each $\alpha \in S$ the $\square
^+_r$-scheme ${\mathcal Z} _{\bullet \alpha}$ is an augmented cubical
hyperresolution of $X_{\alpha}.$
\end{prop}
We refer the reader to \cite{LNM1335},Ex.1,Thm.2.15,Prop.2.14 for
the proof of the above theorem and proposition and also for the
study of the category of cubical hyperresolutions of $S$-schemes.
\begin{remark} Let $X$ be a reduced separated complex scheme of
finite type and let $\pi :{\mathcal Z} \to X$ be a cubical hyperresolution.
Then ${\mathbf R} \pi ^{\operatorname{an}}_*{\mathbb C} _{{\mathcal Z} ^{\operatorname{an}}}={\mathbb C} _{X^{\operatorname{an}}}.$ This
follows from \cite{LNM1335},Ex.1,Thm.6.1.
\end{remark}
\begin{defi} Let $X$ be a reduced separated scheme of finite type over a
field of characteristic zero. Choose its cubical hyperresolution
$\pi :{\mathcal Z} \to X.$ We say that $X$ has Du Bois singularities ($X$ is
Du Bois, for short) if the adjunction morphism ${\mathcal O} _X\to {\mathbf R} \pi
_*{\mathcal O} _{{\mathcal Z}}$ is a quasi-isomorphism.
\end{defi}
\begin{remark} The complex ${\mathbf R} \pi _*{\mathcal O} _{{\mathcal Z}}\in D(X)$ is independent
(up to a quasi-isomorphism) on the choice of a hyperresolution of
$X$ (\cite{LNM1335},Ex.3). So the notion of Du Bois singularities is
well defined.
\end{remark}
\begin{remark} \label{DuBois-sing} If $X$ has rational singularities (for example $X$ is
smooth), then $X$ is Du Bois. It was conjectured by Kollar \cite{Ko}
and recently proved by Kollar and Kovac \cite{KoKov} that if $X$ has
log canonical singularities, then $X$ is Du Bois.
\end{remark}
\begin{theo}\label{Kovac} Let $X$ be a reduced separated scheme of
finite type over a field of characteristic zero. Choose its
hyperresolution $\pi :{\mathcal Z} \to X.$ Assume that the adjunction map
${\mathcal O} _X\to {\mathbf R} \pi _*{\mathcal O} _{{\mathcal Z}}$ has a left inverse. Then $X$ is Du
Bois (i.e. this map is a quasi-isomorphism).
\end{theo}
\begin{proof} See \cite{Kov}.
\end{proof}
The notion of Du Bois singularities characterizes the existence of
categorical resolutions by smooth poset schemes as is shown in the
next theorem.
\begin{theo}\label{posetres=DuBois} Let $X$ be a reduced scheme of finite type over a field
of characteristic zero. Then there exists a categorical resolution
of $X$ by a smooth poset scheme (Definition
\ref{def-cat-res-by-poset-schemes}) if and only if $X$ has Du Bois
singularities.
\end{theo}
\begin{proof} One direction is clear: if $X$ has Du Bois singularities
and $\pi :{\mathcal Z} \to X$ is its hyperresolution then by Proposition
\ref{criterion-cat-res} $\pi$ is a
categorical resolution of $X$ by the smooth poset scheme ${\mathcal Z}.$
Vice versa, assume that $S$ is a poset, ${\mathcal X}$ is a smooth $S$-scheme
and $\sigma :{\mathcal X} \to X$ is a categorical resolution. Consider the
augmented $S^+:=S\cup \{0\}$-scheme ${\mathcal X} ^+$ defined by $\sigma$ (so
that $X_0=X$). A choice of a hyperresolution of $\pi :{\mathcal Y} \to {\mathcal X}
^+$ induces a commutative diagram of poset schemes
$$\begin{array}{ccc}
{\mathcal Y} & \stackrel{\pi}{\longrightarrow} & {\mathcal X} \\
\downarrow \tilde{\sigma}& & \downarrow \sigma\\
{\mathcal Y} _0 & \stackrel{\pi _0}{\longrightarrow} & X
\end{array}
$$
which is compatible with the diagram of projections of posets
$$\begin{array}{ccc}
\square _n\times S & \to & S\\
\downarrow & & \downarrow\\
\square _n & \to & \{0\}
\end{array}
$$
and such that $\pi _0$ (and $\pi $) are hyperresolutions
(Proposition \ref{fiber-hyperres}).
By our assumption the adjunction map ${\mathcal O} _{X}\to {\mathbf R} \sigma _*{\mathcal O}
_{{\mathcal X}}$ is an isomorphism, and we want to prove that the adjunction
morphism ${\mathcal O} _{X}\to {\mathbf R} (\pi _0)_*{\mathcal O} _{{\mathcal Y} _{0}}$ is an
isomorphism. By Theorem \ref{Kovac} it suffices to prove that this
last map has a left inverse.
Since the poset scheme ${\mathcal X}$ is smooth we conclude by Remark
\ref{DuBois-sing}, Proposition \ref{fiber-hyperres} and Lemma
\ref{direct-image-fiber} that the map ${\mathcal O} _{{\mathcal X}}\to {\mathbf R} \pi _{*}{\mathcal O}
_{{\mathcal Y} }$ is an isomorphism. Thus the adjunction map ${\mathcal O} _X\to {\mathbf R}
(\sigma \cdot \pi)_*{\mathcal O} _{{\mathcal Y}}={\mathbf R} (\pi _0\cdot \tilde{\sigma}){\mathcal O}
_{{\mathcal Y}}$ is an isomorphism. But this last map is the composition of
the adjunction maps ${\mathcal O} _X\to {\mathbf R} (\pi _0)_*{\mathcal O}_{{\mathcal Y} _0}\to {\mathbf R}
(\pi _0)_*\cdot{\mathbf R} (\tilde{\sigma})_*{\mathcal O}_{{\mathcal Y}}.$ Hence the map ${\mathcal O}
_{X}\to {\mathbf R} (\pi _0)_*{\mathcal O} _{{\mathcal Y} _{0}}$ has a left inverse. This
proves the theorem.
\end{proof}
Cubical hyperresolutions give more: one can define the de Rham
complex of a singular algebraic variety $X.$ Namely, choose a
hyperresolution $\pi :{\mathcal Z} \to X$ and define the de Rham-Du Bois
complex $\underline{{\mathcal O}mega}^\bullet _X:={\mathbf R} \pi _*{\mathcal O}mega ^\bullet
_{{\mathcal Z}}.$ This complex consists of ${\mathcal O} _X$-modules and has the
differential which is a differential operator of order 1. It has
coherent cohomology and is well defined (independent of the choice
of a hyperresolution) up to a quasi-isomorphism in the appropriate
derived category \cite{LNM1335},Ex.3. There exists a canonical
morphism of filtered complexes from the usual de Rham complex
${\mathcal O}mega ^\bullet _X$ to $\underline{{\mathcal O}mega}^\bullet _X$ which is a
quasi-isomorphism if $X$ is smooth.
If $X$ is a reduced separated complex scheme, then the analytization
$(\underline{{\mathcal O}mega}^\bullet _X)^{\operatorname{an}}=\underline{{\mathcal O}mega}^\bullet
_{X^{\operatorname{an}}}$ is a resolution of the constant sheaf ${\mathbb C} _{X^{\operatorname{an}}}.$
The stupid filtration of the complex ${\mathcal O}mega ^\bullet _{{\mathcal Z}}$
induces a filtration on the de Rham-Du Bois complex and
$\underline{{\mathcal O}mega}^\bullet _X$ is well defined even as a filtered
complex. The associated graded pieces are
$\underline{{\mathcal O}mega}^i_X:=\operatorname{gr} ^i\underline{{\mathcal O}mega}^\bullet _X={\mathbf R} \pi
_*{\mathcal O}mega ^i_{{\mathcal Z}}.$ If $X$ is proper then this filtration induces
the Hodge filtration on $H^\bullet (X^{\operatorname{an}},{\mathbb C}).$
We will prove in Theorem \ref{descent} below that for a reduced
complex projective scheme $X$ the filtered complex
$\underline{{\mathcal O}mega}^\bullet _X$ can be defined as ${\mathbf R} \sigma
_*{\mathcal O}mega ^\bullet _{{\mathcal X}},$ where ${\mathcal X}$ is a smooth complex
projective poset scheme and $\sigma :{\mathcal X} \to X$ is a morphism such
that ${\mathbf R} \sigma ^{\operatorname{an}}_*{\mathbb C} _{{\mathcal X} ^{\operatorname{an}}}={\mathbb C} _{X^{\operatorname{an}}}.$
\section{Examples of categorical resolutions by smooth poset
schemes}\label{examples}
Let $Y$ be a reducible scheme with irreducible components
$Y_1,...,Y_n.$ Assume that for each $1\leq k\leq n$ and each subset
$\alpha =\{i_1,...i_k\}\subset \{1,...n\}$ the scheme
$$X_{\alpha}:=\bigcap _{j=1}^kY_{i_j}$$
is smooth. (In particular the components $Y_i$ are smooth.) Let $S$
be the poset of nonempty subsets of $\{1,...,n\}$ with the natural
partial ordering by inclusion. Let ${\mathcal X} =\{ X_{\alpha }\}$ be the
corresponding smooth poset scheme with the maps $f_{\alpha
\beta}:X_{\alpha }\to X_{\beta}$ being the obvious inclusions. Let
$\pi :{\mathcal X} \to Y$ be the natural morphism.
\begin{prop} The functor ${\mathbf L} \pi ^*:D(Y)\to D({\mathcal X})$ is a
categorical resolution of singularities, i.e. the functor
$${\mathbf L} \pi^*:\operatorname{op}eratorname{Perf} (Y)\to \operatorname{op}eratorname{Perf} ({\mathcal X})$$
is full and faithful.
\end{prop}
\begin{proof} By Proposition \ref{criterion-cat-res}
we may assume that $Y$ is affine and we
only need to prove that the map $\operatorname{op}eratorname{Ext} ({\mathcal O} _Y,{\mathcal O} _Y) \to \operatorname{op}eratorname{Ext} ({\mathcal O}
_{{\mathcal X}},{\mathcal O} _{{\mathcal X}})$ is an isomorphism.
We have $\operatorname{op}eratorname{Ext} ^i({\mathcal O} _Y,{\mathcal O} _Y)=0$ for $i\neq 0.$ On the other hand
we have the obvious complex in $C({\mathcal X})$
$$C({\mathcal O} _{{\mathcal X}}):=... \to \bigoplus _{\vert \alpha \vert =2}
j_{\alpha +}({\mathcal O} _{{\mathcal X}})_{\alpha} \to \bigoplus _{\vert \beta \vert
=1}j_{\beta +}({\mathcal O} _{{\mathcal X}})_{\beta} \to 0,$$ which is a resolution of
${\mathcal O} _{{\mathcal X}}.$ Since all schemes $X_{\alpha}$ are affine we have
$\operatorname{op}eratorname{Ho}m (C({\mathcal O} _{{\mathcal X}}), {\mathcal O}_{{\mathcal X}})={\mathbf R} \operatorname{op}eratorname{Ho}m ({\mathcal O} _{{\mathcal X}},{\mathcal O} _{{\mathcal X}})$
(Example \ref{example}). But $\operatorname{op}eratorname{Ho}m (C({\mathcal O} _{{\mathcal X}}),{\mathcal O} _{{\mathcal X}})$ is
the complex
$$0\to \bigoplus _{\vert \beta \vert =1}H^0(X_\beta, {\mathcal O}
_{X_{\beta}}) \to \bigoplus _{\vert \alpha \vert =2}H^0(X_\alpha,
{\mathcal O} _{X_{\alpha}}) \to ...$$ which is quasi-isomorphic to $H^0(Y,
{\mathcal O}_Y).$
\end{proof}
\subsection{Categorical resolution of the cone over a plane cubic}
Here we show how smooth poset schemes can be used to construct a
categorical resolution of the simplest nonrational singularity - the
cone over a smooth plain cubic.
Let $C\subset {\mathbb P}^2$ be a smooth curve of degree 3 (and genus 1)
and $Y\subset {\mathbb P} ^3$ be the projective cone over $C.$ So $Y$ is a
cubic surface with a singular point $p$ - the vertex of the cone. We
have
$$H^i(Y,{\mathcal O} _Y)=\left\{ \begin{array}{ll}
k, & \text{if i=0}\\
0, & \text{otherwise.}\\
\end{array} \right.
$$
Let $f:X\to Y$ be the blowup of the vertex, so that $X$ is a smooth
ruled surface over the curve $C.$ Denote by
$i:E=f^{-1}(p)\operatorname{h}ookrightarrow X$ the inclusion of the exceptional
divisor. We have
$$H^i(X,{\mathcal O} _X)=\left\{ \begin{array}{ll}
k, & \text{if i=0,1}\\
0, & \text{otherwise,}\\
\end{array} \right.
$$
and the pullback map $i^*:H ^\bullet(X,{\mathcal O} _X)\to H ^\bullet (E,{\mathcal O}
_E)$ is an isomorphism.
Consider the following smooth poset scheme ${\mathcal X}$
$$\begin{array}{ccc}
E & \to & X\\
\downarrow & &\\
q & &
\end{array}
$$
where $q=Spec k,$ and the map $E\to X$ is the embedding $i.$ Denote
by $\pi :{\mathcal X} \to Y$ the obvious morphism which extends the blowup
$f:X\to Y.$
\begin{prop} ${\mathbf L} \pi^*:D(Y)\to D({\mathcal X})$ is a categorical resolution
of singularities, i.e. the functor
$${\mathbf L} \pi ^*:\operatorname{op}eratorname{Perf} (Y)\to \operatorname{op}eratorname{Perf} ({\mathcal X})$$
is full and faithful.
\end{prop}
\begin{proof} Note that the map $\pi $ is an isomorphism away from
the point $p\in Y.$ So we may replace $Y$ by the corresponding {\it
affine} cone $Y_0$ over $C,$ $f_0:X_0\to Y_0$ is still the blowup of
the vertex and the rest is the same. Denote the corresponding poset
scheme by ${\mathcal X} _0.$ Then it suffices to prove that the map
$H^\bullet (Y_0,{\mathcal O} _{Y_0})\to H^\bullet ({\mathcal X} _0,{\mathcal O} _{{\mathcal X} _0})$ is
an isomorphism. We have $H^i(Y_0,{\mathcal O}_{Y_0})=0$ for $i\neq 0.$ To
compute $H ({\mathcal X} _0,{\mathcal O} _{{\mathcal X} _0})$ we may use the spectral sequence
as in Example \ref{example}. Then the $E_1$-term is the sum of the
two complexes:
$$k\operatorname{op}lus {\mathcal G}amma (X _0,{\mathcal O}_{X_0}) \to {\mathcal G}amma (E,{\mathcal O} _E),\operatorname{op}eratorname{qu}ad \text{and}
\operatorname{op}eratorname{qu}ad H^1(X _0,{\mathcal O} _{X_0}) \to H^1(E,{\mathcal O} _E).$$ The second map is an
isomorphism, and the first one is surjective with the kernel ${\mathcal G}amma
(Y_0,{\mathcal O} _{Y_0}).$
\end{proof}
In view of Theorem \ref{posetres=DuBois} above the last example is a
special case of the following result of Du Bois
\cite{DuB},Prop.4.13.
\begin{prop} Let $W\subset {\mathbf P} ^m$ be a smooth variety such that for
all $i>0$ and $n>0$ the following holds
$$H^i(W,{\mathcal O} (n))=0.$$
Then the cone over $W$ has Du Bois singularities.
\end{prop}
\begin{remark} In fact, using the same construction as in the above
example of the
cone over a smooth cubic curve it is easy to see that the condition
in the last proposition is necessary for the cone over $W$ to be Du
Bois. For example if $W\subset {\mathbb P} ^2$ is a smooth curve of degree
$\geq 4,$ then the cone over $W$ is not Du Bois.
\end{remark}
Some other examples of Du Bois singularities are listed in
\cite{St}. For example if $X$ is a reduced curve, then $X$ is Du
Bois if and only if at every singular point of $X$ the branches are
smooth and their tangent directions are independent.
\section{Descent for Du Bois singularities}
\begin{theo}\label{descent} Let $X$ be a reduced complex projective scheme.
Let ${\mathcal X} $ be a smooth complex projective poset scheme and $\sigma
:{\mathcal X} \to X$ be a morphism such that the adjunction map ${\mathbb C}
_{X^{\operatorname{an}}}\to {\mathbf R} \sigma ^{\operatorname{an}}_*{\mathbb C} _{{\mathcal X} ^{\operatorname{an}}}$ is a
quasi-isomorphism. Consider the direct image ${\mathbf R} \sigma _*{\mathcal O}mega
^\bullet _{{\mathcal X}}.$ This complex has a filtration induced by the
stupid filtration of the de Rham complex ${\mathcal O}mega ^\bullet _{{\mathcal X}}.$
Then there exists a natural morphism of filtered complexes
$$\tau :\underline{{\mathcal O}mega}^\bullet _X\to {\mathbf R} \sigma _*{\mathcal O}mega ^\bullet _{{\mathcal X}}$$
which is a quasi-isomorphism. In particular, the map
$$\operatorname{gr} ^i\tau :\underline{{\mathcal O}mega}^i _X\stackrel{\sim}{\to} {\mathbf R} \sigma
_*{\mathcal O}mega ^i_{{\mathcal X}}$$ is a quasi-isomorphism for all $i\geq 0.$
So if $X$ has Du Bois
singularities, then $\underline{{\mathcal O}mega}^0_X\simeq {\mathcal O} _X \simeq {\mathbf R}
\sigma _*{\mathcal O} _{{\mathcal X}},$ i.e. the functor ${\mathbf L} \sigma ^*:D(X)\to
D({\mathcal X})$ is a categorical resolution of singularities.
\end{theo}
\begin{proof}
As in the proof of Theorem \ref{posetres=DuBois} choose a
commutative diagram
\begin{equation}\label{main-diag}\begin{array}{lcl}
{\mathcal Y} & \stackrel{\pi}{\longrightarrow} & {\mathcal X} \\
\downarrow \tilde{\sigma}& & \downarrow \sigma\\
{\mathcal Y} _0 & \stackrel{\pi _0}{\longrightarrow} & X
\end{array}
\end{equation}
where $\pi _0$ is a hyperresolution and for each scheme $X_\alpha
\in {\mathcal X}$ the induced morphism $\pi :\pi ^{-1}(X_\alpha)\to
X_{\alpha}$ is also a hyperresolution.
Since each $X_\alpha$ is smooth we have the quasi-isomorphism of
filtered complexes ${\mathcal O}mega ^\bullet_{{\mathcal X}}\stackrel{\sim}{\to}{\mathbf R} \pi
_*{\mathcal O}mega ^\bullet _{{\mathcal Y}}.$ It follows that ${\mathbf R} \sigma _*{\mathcal O}mega
^\bullet _{{\mathcal X}}\simeq {\mathbf R} (\sigma \cdot \pi)_*{\mathcal O}mega ^\bullet
_{{\mathcal Y}}={\mathbf R} (\pi _0 \cdot \tilde{\sigma})_*{\mathcal O}mega ^\bullet _{{\mathcal Y}}.$
On the other hand by definition ${\mathbf R} (\pi _0)_*{\mathcal O}mega ^\bullet _{{\mathcal Y}
_0}=\underline{{\mathcal O}mega}^\bullet _X.$ Hence the adjunction morphism
$\theta :{\mathcal O}mega ^\bullet _{{\mathcal Y} _0}\to {\mathbf R} \tilde{\sigma}_*{\mathcal O}mega
^\bullet _{{\mathcal Y}}$ induces the desired morphism of filtered complexes
\begin{equation}\label{morphism}
\tau :\underline{{\mathcal O}mega}^\bullet _X = {\mathbf R} (\pi _0)_*{\mathcal O}mega ^\bullet
_{{\mathcal Y} _0}\stackrel{{\mathbf R} (\pi _0)_*\theta}{\longrightarrow} {\mathbf R} (\pi_0
\cdot \tilde{\sigma})_*{\mathcal O}mega ^\bullet _{{\mathcal Y}}\simeq {\mathbf R} \sigma
_*{\mathcal O}mega ^\bullet _{{\mathcal X}}.
\end{equation}
We will prove that for each $i$ the map
$$\operatorname{gr} ^i\tau :\underline{{\mathcal O}mega}^i _X \to {\mathbf R} \sigma
_*{\mathcal O}mega ^i _{{\mathcal X}}$$ is a quasi-isomorphism (hence $\tau$ is a
quasi-isomorphism).
\begin{lemma} \label{isom-hypercohom} For each $i$
the morphism $\operatorname{gr} ^i\tau$ induces an isomorphism on the
hypercohomology
$$H^\bullet(\operatorname{gr} ^i\tau):H^\bullet (X,\underline{{\mathcal O}mega} ^i
_X)\to H^\bullet (X,{\mathbf R} \sigma _*{\mathcal O}mega ^i _{{\mathcal X}}).$$
\end{lemma}
\begin{proof} Note that the map $H^\bullet(\operatorname{gr} ^i\tau)$ coincides with
the inverse image map $H^\bullet ({\mathcal Y} _0,{\mathcal O}mega ^i _{{\mathcal Y} _0})\to
H^\bullet ({\mathcal Y} ,{\mathcal O}mega ^i _{{\mathcal Y} })=H^\bullet (X,{\mathbf R} \sigma _*{\mathcal O}mega
^i _{{\mathcal X}}).$
The diagram \ref{main-diag} induces the corresponding
diagram of analytic spaces
\begin{equation}\label{main-diag-analytic}\begin{array}{lcl}
{\mathcal Y}^{\operatorname{an}} & \stackrel{\pi ^{\operatorname{an}}}{\longrightarrow} & {\mathcal X} ^{\operatorname{an}} \\
\downarrow \tilde{\sigma}^{\operatorname{an}}& & \downarrow \sigma ^{\operatorname{an}}\\
{\mathcal Y} _0^{\operatorname{an}} & \stackrel{\pi _0^{\operatorname{an}}}{\longrightarrow} & X^{\operatorname{an}}
\end{array}
\end{equation}
By Subsection \ref{GAGA} it suffices to show that the corresponding
inverse image map $H^\bullet (\operatorname{gr} ^i\tau ^{ \operatorname{an}}):H^\bullet ({\mathcal Y}
_0^{\operatorname{an}},{\mathcal O}mega ^i _{{\mathcal Y} _0^{\operatorname{an}}})\to H^\bullet ({\mathcal Y}_{\operatorname{an}} ,{\mathcal O}mega
^i _{{\mathcal Y}^{\operatorname{an}}})$ is an isomorphism.
Since $\pi _0 $ and $\pi $ are cubical hyperresolutions we have
${\mathbf R} (\pi _0^{\operatorname{an}})_*{\mathbb C} _{{\mathcal Y}
_0^{\operatorname{an}}}={\mathbb C} _{X ^{\operatorname{an}}}$ and ${\mathbf R} (\pi ^{\operatorname{an}})_*{\mathbb C} _{{\mathcal Y}
^{\operatorname{an}}}={\mathbb C} _{{\mathcal X} ^{\operatorname{an}}}.$ Thus by our assumption ${\mathbf R} (\sigma
^{\operatorname{an}}\cdot \pi ^{\operatorname{an}})_*{\mathbb C} _{{\mathcal Y} ^{\operatorname{an}}}= {\mathbb C} _{X^{\operatorname{an}}}.$ As in
the case of the sheaves ${\mathcal O}mega ^i$ we obtain a natural morphism
$$\tau ^c:{\mathbf R} (\pi _0^{\operatorname{an}})_*{\mathbb C} _{{\mathcal Y} _0^{\operatorname{an}}}\to {\mathbf R} \sigma
^{\operatorname{an}}_*{\mathbb C} _{{\mathcal X} ^{\operatorname{an}}}$$ which is a quasi-isomorphism (both
sides are quasi-isomorphic to ${\mathbb C} _{X^{\operatorname{an}}}$). Hence the map
$$H^\bullet({\mathcal Y} _0^{\operatorname{an}},{\mathbb C} ) \stackrel{H^\bullet (\tau ^c)}
{\longrightarrow} H^\bullet ({\mathcal X} ^{\operatorname{an}},{\mathbb C} )=H^\bullet ({\mathcal Y}
^{\operatorname{an}},{\mathbb C} )$$ is an isomorphism.
By Theorem \ref{Hodge-to-deRham-degener-analytic}
\begin{equation}H^\bullet({\mathcal Y} _0 ^{\operatorname{an}},{\mathbb C} )=
\bigoplus _iH^{\bullet -i}({\mathcal Y} _0 ^{\operatorname{an}},{\mathcal O}mega ^i _{{\mathcal Y} _0^{\operatorname{an}}}).
\end{equation}
and similarly for
${\mathcal Y}.$ The map $H^\bullet (\tau ^c)$ respects this decomposition and
its restriction to the i-th summand is the map $H^\bullet (\operatorname{gr}
^i\tau ^{\operatorname{an}}).$ It follows that $H^\bullet (\operatorname{gr} ^i\tau ^{\operatorname{an}})$ is
also an isomorphism. This proves the lemma.
\end{proof}
\begin{lemma} \label{apply-serre} Let $Y$ be a complex projective scheme with an ample
line bundle $L.$ Let $u:K_1\to K_2$ be a morphism of complexes in
$D^b(cohY).$ Assume that for all $n>>0$ the map $u$ induces an
isomorphism of the hypercohomology
$$H^\bullet (Y,K_1\otimes L^n)\stackrel{\sim}{\longrightarrow}
H^\bullet (Y,K_2\otimes L^n).$$ Then $u$ is a quasi-isomorphism.
\end{lemma}
\begin{proof} See Lemma 3.4 in \cite{LNM1335} (p.139).
\end{proof}
We will prove that the morphism $\operatorname{gr} \tau ^i$ satisfies the
assumptions of Lemma \ref{apply-serre}, which will prove the
theorem.
\begin{prop} Let $L$ be an ample line bundle on $X.$ Then for any
$n\geq 1$ the map $\operatorname{gr} ^i\tau \otimes L^n:\operatorname{gr} ^i\tau
:\underline{{\mathcal O}mega}^i _X\otimes L^n \to ({\mathbf R} \sigma _*{\mathcal O}mega ^i
_{{\mathcal X}})\otimes L^n$ induces an isomorphism on hypercohomology
$$H^\bullet (X, \underline{{\mathcal O}mega}^i _X\otimes L^n) \to H^\bullet (X,
({\mathbf R} \sigma _*{\mathcal O}mega ^i _{{\mathcal X}})\otimes L^n).$$
\end{prop}
\begin{proof} We prove the proposition by induction on the dimension
of $X.$ If $\dim X=0,$ then the statement is equivalent to Lemma
\ref{isom-hypercohom}.
We denote by $L$ also the pullbacks of $L$ to the smooth poset
schemes ${\mathcal X}$ and ${\mathcal Y} _0.$ By the projection formula it suffices to
prove that the natural map
$$H^\bullet (X, {\mathbf R} \pi _{0 *}({\mathcal O}mega ^i _{{\mathcal Y} _0}\otimes L^n))
\to H^\bullet (X, {\mathbf R} \sigma _*({\mathcal O}mega ^i _{{\mathcal X}}\otimes L^n))$$ is
an isomorphism.
\begin{lemma} Let $Y$ be a smooth variety, $B\subset Y$ - a smooth
divisor, and $M$ - the corresponding line bundle. Then for each
$i\geq 1$ we have the exact sequences
$$0\to {\mathcal O}mega ^i_Y\to M\otimes {\mathcal O}mega ^i_Y\to M\otimes {\mathcal O}mega
^i_Y\otimes {\mathcal O} _B\to 0,$$
$$0\to {\mathcal O}mega _B^{i-1}\to M\otimes {\mathcal O}mega ^i_Y\otimes {\mathcal O} _B\to
M\otimes {\mathcal O}mega ^i_B \to 0.$$ These sequences are functorial with
respect to the pair $(Y,B).$
\end{lemma}
\begin{proof} \cite{LNM1335},p.136.
\end{proof}
Let $D\subset X$ be a general divisor corresponding to $L^n$ for
$n\geq 1.$ Let
\begin{equation}\label{main-diag-divisor}\begin{array}{lcl}
{\mathcal Z} & \stackrel{\pi}{\longrightarrow} & {\mathcal W} \\
\downarrow \tilde{\sigma}& & \downarrow \sigma\\
{\mathcal Z} _0 & \stackrel{\pi _0}{\longrightarrow} & D
\end{array}
\end{equation}
be the restriction of the diagram \ref{main-diag} to $D.$ Since $D$
is general this diagram has similar properties: ${\mathcal W}$ is a smooth
projective poset scheme, $\pi _0$ is a hyperresolution, and for each
scheme $W_\alpha \in {\mathcal W}$ the induced morphism $\pi :\pi
^{-1}(W_\alpha)\to W_{\alpha}$ is also a hyperresolution. Also the
adjunction morphism ${\mathbb C} _{D^{\operatorname{an}}}\to {\mathbb R} \sigma ^{\operatorname{an}}_*{\mathbb C}
_{{\mathcal W} ^{\operatorname{an}}}$ is a quasi-isomorphism.
The exact sequences in the last lemma give rise to similar exact
sequences on poset schemes ${\mathcal X}$ and ${\mathcal Y} _0$ respectively. Namely,
we have
\begin{equation}\label{d-1}\begin{array}{c}
0\to {\mathcal O}mega ^i_{{\mathcal Y} _0}\to L^n\otimes {\mathcal O}mega ^i_{{\mathcal Y} _0}\to
L^n\otimes
{\mathcal O}mega ^i_{{\mathcal Y} _0}\otimes {\mathcal O} _{{\mathcal Z} _0}\to 0,\\
0\to {\mathcal O}mega _{{\mathcal Z} _0}^{i-1}\to L^n\otimes {\mathcal O}mega ^i_{{\mathcal Y} _0} \otimes
{\mathcal O} _{{\mathcal Z} _0}\to L^n\otimes {\mathcal O}mega ^i_{{\mathcal Z} _0} \to 0, \end{array}
\end{equation} and
\begin{equation}\label{d-2}\begin{array}{c}
0\to {\mathcal O}mega ^i_{{\mathcal X}}\to L^n\otimes {\mathcal O}mega ^i_{{\mathcal X}}\to L^n\otimes
{\mathcal O}mega ^i_{{\mathcal X}}\otimes {\mathcal O} _{{\mathcal W}}\to 0,\\
0\to {\mathcal O}mega _{{\mathcal W}}^{i-1}\to L^n\otimes {\mathcal O}mega ^i_{{\mathcal X}} \otimes {\mathcal O}
_{{\mathcal W}}\to L^n\otimes {\mathcal O}mega ^i_{{\mathcal W}} \to 0.\end{array}
\end{equation}
We now push forward these diagrams \ref{d-1} and \ref{d-2} by the
functors ${\mathbf R} \pi _{0
*}$ and ${\mathbf R} \sigma _*$ respectively. By functoriality we have a
morphism between the resulting exact triangles on $X.$ On the
hypercohomology this morphism induces an isomorphism in the term
${\mathcal O}mega ^i_{{\mathcal Y} _0}$ by Lemma \ref{isom-hypercohom}. By induction it
also induces similar isomorphisms in the terms ${\mathcal O}mega _{{\mathcal Z}
_0}^{i-1}$ and $L^n\otimes {\mathcal O}mega ^i_{{\mathcal Z} _0}.$ Hence it induces an
isomorphism of hypercohomology in the term $L^n\otimes {\mathcal O}mega
^i_{{\mathcal Y} _0} \otimes {\mathcal O} _{{\mathcal Z} _0}$ and thus also in the term
$L^n\otimes {\mathcal O}mega ^i_{{\mathcal Y} _0}$ which proves the proposition and the
theorem.
\end{proof}
\end{proof}
\part{Appendix}
\section{Coherator and the functors ${\mathbf L} f^*,{\mathbf R} f_*$}
Probably this appendix contains nothing new but we decided to put
together some "well known" facts for convenience.
Let $X$ be a quasi-compact separated scheme. As usual $QcohX$
denotes the category of quasi-coherent sheaves on $X,$
$C(X)=C(QcohX)$ - the category of complexes over $QcohX,$
$D(X)=D(QcohX)$ - the derived category. We also consider the
category $\operatorname{op}eratorname{Mod} _X$ of {\it all} ${\mathcal O} _X$-modules, its category of
complexes $C(\operatorname{op}eratorname{Mod} _X)$ and the corresponding derived category
$D(\operatorname{op}eratorname{Mod} _X).$ Let $C_{\operatorname{qc}} (\operatorname{op}eratorname{Mod} _X)\subset C(\operatorname{op}eratorname{Mod} _X),$
$D_{\operatorname{qc}}(\operatorname{op}eratorname{Mod} _X)\subset D(\operatorname{op}eratorname{Mod} _X)$ be the full subcategories of
complexes with quasi-coherent cohomologies.
Both $Qcoh X$ and $\operatorname{op}eratorname{Mod} _X$ are Grothendieck categories.
The obvious exact functor $\phi: Qcoh X\to \operatorname{op}eratorname{Mod} _X$ preserves finite
limits and arbitrary colimits. It has a left-exact right adjoint
functor $Q_X=Q:\operatorname{op}eratorname{Mod} _X\to QcohX$ - the {\it coherator}. The functor
$Q$ preserves arbitrary limits and injective objects. The induced
functor $Q:C(\operatorname{op}eratorname{Mod} _X)\to C(X)$ preserves h-injectives. One defines
the right derived functor ${\mathbf R} Q :D(\operatorname{op}eratorname{Mod} _X)\to D(X)$ using the
h-injectives.
\begin{prop} \label{equivalences}
The functors $\phi,$ ${\mathbf R} Q$ induce mutually inverse equivalences of
categories
$$\phi : D(X)\to D_{\operatorname{qc}}(\operatorname{op}eratorname{Mod} _X),\operatorname{op}eratorname{qu}ad {\mathbf R} Q :D_{\operatorname{qc}}(\operatorname{op}eratorname{Mod} _X)\to
D(X).$$
\end{prop}
\begin{proof} See for example [AlJeLi],Prop.1.3.
\end{proof}
\begin{lemma} \label{preserve-h-flats}
The functor $\phi :C(X)\to C(\operatorname{op}eratorname{Mod} _X)$ preserves h-flats.
\end{lemma}
\begin{proof} Let $F\in C(X)$ be h-flat, $N\in C(\operatorname{op}eratorname{Mod} _X)$ be
acyclic, $x\in X.$ We need to show that the complex of ${\mathcal O}
_x$-modules $(F\otimes _{{\mathcal O} _X}N)_x=F_x\otimes _{{\mathcal O} _x}N_x$ is
acyclic. Let $i:Spec {\mathcal O} _x\to X$ be the inclusion and
$\tilde{N}_x\in C(Qcoh (Spec {\mathcal O} _x))$ be the sheafification of the
acyclic complex $N_x$ of ${\mathcal O} _x$-modules. Then $i_*\tilde {N}_x$ is
an acyclic complex of quasi-coherent sheaves on $X.$ Hence the
complex $F\otimes _{{\mathcal O} _X}i_*\tilde{N}_x$ is also acyclic. Thus
$F_x\otimes _{{\mathcal O} _x}N_x=(F\otimes _{{\mathcal O} _X}i_*\tilde {N}_x)_x$ is
also acyclic.
\end{proof}
Let $f:X\to Y$ be a quasi-compact separated morphism of
quasi-compact separated schemes. One defines the derived functors
$${\mathbf L} f^* :D(\operatorname{op}eratorname{Mod} _Y)\to D(\operatorname{op}eratorname{Mod} _X), \operatorname{op}eratorname{qu}ad {\mathbf R} f_*:D(\operatorname{op}eratorname{Mod} _X)\to
D(\operatorname{op}eratorname{Mod} _Y),$$ using h-flats and h-injectives in $C(\operatorname{op}eratorname{Mod} _Y)$ and
$C(\operatorname{op}eratorname{Mod} _X)$ respectively [Sp].
We can also define the derived functor ${\mathbf L} f^*:D(Y)\to D(X)$ using
the h-flats in $C(Y)$ (There are enough h-flats in $C(Y)$
[AlJeLi],Prop.1.1).
\begin{lemma} \label{commute-inverse-image}
There exists a natural isomorphism of functors
$${\mathbf L} f^*\cdot \phi _Y = \phi _X \cdot {\mathbf L} f^*:D(Y)\to D(\operatorname{op}eratorname{Mod} _X).$$
\end{lemma}
\begin{proof} Let $F\in D(Y)$ be h-flat. Then $\phi _X \cdot {\mathbf L}
f^*(F)=\phi _X\cdot f^*(F).$ On the other hand $\phi _Y(F)$ is
h-flat by Lemma \ref{preserve-h-flats}. Hence ${\mathbf L} f^*\cdot \phi _Y
(F)=f^* \cdot \phi _Y(F)=\phi _X\cdot f^*.$
\end{proof}
\begin{prop} a). The functors $({\mathbf L} f^*, {\mathbf R} f_*)$ between $D(\operatorname{op}eratorname{Mod}
_Y)$ and $D(\operatorname{op}eratorname{Mod} _X)$ are adjoint.
b). These functors preserve the subcategories $D_{\operatorname{qc}}(\operatorname{op}eratorname{Mod} _Y)$ and
$D_{\operatorname{qc}}(\operatorname{op}eratorname{Mod} _X).$
\end{prop}
\begin{proof} a). It is [Sp],Prop.6.7. b).
For the functor ${\mathbf L} f^* $ it follows from Proposition
\ref{equivalences} and Lemma \ref{commute-inverse-image} and for
the functor ${\mathbf R} f_*$ it is proved for example in [BoVdB],Thm.3.3.3
for the functor ${\mathbf R} f_*.$
\end{proof}
The functors $f^*:QcohY\to QcohX,$ $f_*:QcohX\to QcohY$ are well
defined and clearly $f^*\cdot \phi _Y =\phi _X \cdot f^*.$ Hence
also $f_*\cdot Q_X=Q_Y\cdot f_*$ by adjunction. One defines the
derived functor
$$ {\mathbf R} f_*:D(X)\to D(Y)$$
using h-injectives in $C(X).$
\begin{prop} \label{coherator-direct-image}
There exist a natural isomorphism of functor
$${\mathbf R} f_* \cdot {\mathbf R} Q_X\simeq {\mathbf R} Q_Y \cdot {\mathbf R} f_*:D_{\operatorname{qc}}(\operatorname{op}eratorname{Mod}
_X)\to D(Y).$$
\end{prop}
\begin{proof} Let $I\in D_{\operatorname{qc}}(\operatorname{op}eratorname{Mod} _X)$ be h-injective. Then ${\mathbf R}
Q_X(I)=Q_X(I)$ is h-injective in $D(X).$ Hence ${\mathbf R} f_*\cdot {\mathbf R}
Q_X(I)=f\cdot Q_X(I).$ also ${\mathbf R} f_*(I)=f_*(I).$ Since $f\cdot
Q_X(I)=Q_Y\cdot f(I)$ we get a morphism of functors
$$\theta :{\mathbf R} f_* \cdot {\mathbf R} Q_X \to {\mathbf R} Q_Y \cdot {\mathbf R} f_*.$$
We claim that $\theta $ is an isomorphism, i.e. $Q_Y \cdot
f_*(I)\simeq {\mathbf R} Q_Y \cdot f_*(I).$ We will use a lemma.
\begin{lemma} The functors ${\mathbf R} f_*:D_{\operatorname{qc}}(\operatorname{op}eratorname{Mod} _X)\to D_{\operatorname{qc}}(\operatorname{op}eratorname{Mod} _Y),$
${\mathbf R} f_*:D(X)\to D(Y),$ and ${\mathbf R} Q$ are way-out in both directions
(\cite{Ha}).
\end{lemma}
\begin{proof} Obviously all three functors are way-out left. The functor
${\mathbf R} f_*:D_{\operatorname{qc}}(\operatorname{op}eratorname{Mod} _X)\to D_{\operatorname{qc}}(\operatorname{op}eratorname{Mod} _Y)$ is way-out right by
[Li] (see also [BoVdB], Thm.3.3.3). For the functor ${\mathbf R} Q$ see for
example the proof of Proposition 1.3 in [AlJeLi].
Let us prove that the functor ${\mathbf R} f_*:D(X)\to D(Y)$ is way out
right. We may assume that $Y$ is affine and hence $f_*(-)={\mathcal G}amma
(X,-).$
Choose a finite affine open covering ${\mathcal U} =\{ U_i\}_{i=1}^n$ of $X.$
For $F\in C(X)$ denote by
$$C_{{\mathcal U}}(F):=0\to \operatorname{op}lus _{\vert I\vert =1}F_I\to
\operatorname{op}lus _{\vert I\vert =2}F_I \to ...$$ the corresponding (finite)
Cech resolution $F$ by alternating cochains. Here $I\subset
\{1,...,n\},$ $i:\cap _{i\in I}U_i\to X$ and $F_I=i_*i^*F\in C(X).$
The complex $F$ is quasi-isomorphic to $C_{{\mathcal U}}(F).$ Notice that
each complex $F_I$ is acyclic for ${\mathcal G}amma (X,-),$ i.e. ${\mathbf R} {\mathcal G}amma
(X,F_I)={\mathcal G}amma (X,F_I).$ This shows that if $F$ is in $D^{\leq
0}(X),$ then ${\mathbf R} f_*F\in D^{\leq n-1}(Y).$
\end{proof}
Using the lemma it suffices to prove that $\theta (M)$ is an
isomorphism for a single quasi-coherent sheaf $M$ on $X$
(\cite{Ha},Ch.1,Prop.7.1,(iii)). In other words we may assume that
$I$ is an (bounded below) injective resolution in $\operatorname{op}eratorname{Mod} _X$ of $\phi
(M)$ for $M\in Qcoh X.$ Then $Q_X(I)$ is an injective resolution of
$M$ in $QcohX.$ So $Q_Y\cdot f_*(I)=f_*\cdot Q_X(I)$ computes the
derived direct image of $M$ in the category of quasi-coherent
sheaves. On the other hand $f_*(I)$ computes the derived direct
image of $\phi (M).$ Since $f_*(I)\in D_{\operatorname{qc}}(\operatorname{op}eratorname{Mod} _Y)$ it is
quasi-isomorphic to ${\mathbf R} Q_Y\cdot f_*(I).$ So the needed assertion
becomes ${\mathbf R} f_*(M)\simeq {\mathbf R} f_*\cdot \phi (M).$ This is proved for
example in [ThTr],Appendix B,B.10.
\end{proof}
\begin{cor}\label{maps-to-h-inj} Let $I \in C(X)$ be h-injective and
$F\in C(Y)$ be h-flat. Then $$\operatorname{op}eratorname{Ho}m (F,f_*(I))=\operatorname{op}eratorname{Ho}m
_{D(X)}(F,f_*(I)).$$
\end{cor}
\begin{proof} An analogous statement for the category $D(\operatorname{op}eratorname{Mod} _X)$
is proved in \cite{Sp}.
We may assume that $I=Q_X(J)$ for an h-injective $J\in D(\operatorname{op}eratorname{Mod} _X).$
Then
$$\operatorname{op}eratorname{Ho}m (F,f_*\cdot Q_X(J))=\operatorname{op}eratorname{Ho}m (F,Q_Y\cdot f_*(J))=\operatorname{op}eratorname{Ho}m (\phi (F),
f_*(J)).$$ Since $\phi (F)$ is h-flat (Lemma \ref{preserve-h-flats})
by [Sp] we have
$$\operatorname{op}eratorname{Ho}m (\phi (F),f_*(J))=\operatorname{op}eratorname{Ho}m _{D(\operatorname{op}eratorname{Mod} _Y)}(\phi (F),f_*(J)),$$
and by adjunction $\operatorname{op}eratorname{Ho}m _{D(\operatorname{op}eratorname{Mod} _Y)}(\phi (F),f_*(J))= \operatorname{op}eratorname{Ho}m _{D(X)}
(F,{\mathbf R} Q_Y\cdot f_*(J)).$ But in the proof of Proposition
\ref{coherator-direct-image} we established a quasi-isomorphism ${\mathbf R}
Q_Y\cdot f_*(J)\simeq Q_Y\cdot f_*(J).$ This proves the lemma.
\end{proof}
\begin{cor} The functors ${\mathbf L} f^*:D(Y)\to D(X)$ and ${\mathbf R} f_*
:D(X)\to D(Y)$ are adjoint.
\end{cor}
\begin{proof} It follows immediately from Corollary
\ref{maps-to-h-inj}.
\end{proof}
\end{document}
|
\begin{document}
\maketitle
\renewcommand{Abstract}{Abstract}
\begin{abstract}
We study the $p$-rationality of real quadratic fields in terms of
generalized Fibonacci numbers and their periods modulo positive
integers.
\end{abstract}
\section{Introduction}
Let $K$ be a number field and $p$ an odd prime number. The field $K$
is said to be $p$-rational if the Galois group of the maximal
pro-$p$-extension of $K$ which is unramified outside $p$ is a free
pro-$p$-group of rank $r_2+1$, where $r_2$ is the number of pairs of
complex embedding of $K$. The notion of $p$-rational number fields
has been introduced by Movahhedi and Nguyen Quang Do
\cite{Movahhedi-Nguyen}, \cite{Movahhedi}, \cite{Movahhedi90}, and
is used for the construction of non-abelian extensions satisfying
Leopoldt's conjecture. Recently, R. Greenberg used complex abelian
$p$-rational number fields for the construction of $p$-adic Galois
representations with open images. In these paper we study the
$p$-rationality of real quadratic number fields. In fact, we give a
generalization of a result of Greenberg \cite[Corollary
4.1.5]{Greenberg} which relates the $p$-rationality of the field
$\mathbf{Q}(\sqrt{5})$ to properties of the classical Fibonacci
numbers. More precisely, let $d>0$ be a fundamental discriminant.
Denote by $\varepsilon_d$ and $h_d$ the fundamental unit and the
class number of the field $\mathbf{Q}(\sqrt{d})$ and let $N(.)$ be
the absolute norm. We associate to the field $\mathbf{Q}(\sqrt{d})$
a Fibonacci sequence
$F^{(\varepsilon_d+\overline{\varepsilon}_d,N(\varepsilon_d))}=(F_n)_{n\geq0}$
defined by $F_0=0$, $F_1=1$ and the recursion formula
$$F_{n+2}=(\varepsilon_d+\overline{\varepsilon}_d)F_{n+1}-N(\varepsilon_d)F_{n},\; for\;n\geq0.$$
The main result of this paper is the following theorem, which
describes the $p$-rationality in terms of Fibonacci-Wieferich prime
(see Definition \ref{fibonacci-wieferich} for Fibonacci-Wieferich
primes).
\begin{theo}\label{Main theorem}
Let $p\geq3$ be an odd prime number such that
$p\nmid(\varepsilon_d-\overline{\varepsilon}_d)^{2}h_d$. The
following assertions are equivalent:
\begin{enumerate}
\item the field $\mathbf{Q}(\sqrt{d})$ is $p$-rational,
\item $p$ is not a Fibonacci-Wieferich prime for
$\mathbf{Q}(\sqrt{d})$.
\end{enumerate}
\end{theo}
It is known that for every positive integer $m$, the reduction
modulo $m$ of the sequence $(F_{n})_n$ is periodic of period a
positive integer $k(m)$ \cite[Theorem 1.]{Wall}, \cite{Dan-Robert}.
Using this fact and properties of these periods, we give another
characterization of the $p$-rationality of $\mathbf{Q}(\sqrt{d})$ in
terms of the periods of the associated Fibonacci numbers.
\begin{pro}\label{periods and p-rationality}
Let $p\geq3$ be a prime number such that $p\nmid
(\varepsilon_{d}-\overline{\varepsilon_{d}})^2 h_d$. Then the
following assertions are equivalent:
\begin{enumerate}
\item the field $\mathbf{Q}(\sqrt{d})$ is $p$-rational,
\item $k(p)\neq k(p^2)$.
\end{enumerate}
\end{pro}
For the classical Fibonacci sequence, namely $a=b=1$, D.D. Wall is
the first to study these periods in \cite{Wall}, where he proved
many properties of these integers. One problem encountered by Wall
in his paper is the study of the hypothesis $k(p)\neq k(p^2)$. He
asked whether the equality $k(p)=k(p^2)$ is possible. This question
is still open with strong numerical evidence
\cite{Elsenhans-Jahnel}.
By Proposition \ref{periods and p-rationality}, it is equivalent to
whether the number field $\mathbf{Q}(\sqrt{5})$ is not $p$-rational
for some prime number $p$. It is generalized to Fibonacci sequences
$F^{(a,b)}$ where for some sequences we have an affirmative answer,
for example the Fibonacci sequence $F^{(2,-1)}$ gives that
$k(13)=k(13^2)$ and $k(31)=k(31^2)$, which means that the field
$\mathbf{Q}(\sqrt{2})$ is not $p$-rational for $p=13,31$. Under the
light of the above characterization of the $p$-rationality, the
conjecture of G. Gras \cite[Conjecture 7.9]{Gras} on the
$p$-rationality of real quadratic fields, it is suggested that for
almost all primes $p$ we have $k(p)\neq k(p^2)$.\\
\section{$p$-rational fields}
In this section we give a characterization of the $p$-rationality of
real quadratic fields in terms of values of the associated
$L$-functions at odd negative integers. In fact, the $p$-rationality
of totally real abelian number fields $K$ is intimately related to
special values of the associated zeta functions $\zeta_K$. The
relation is as follows. For any finite set $\Sigma$ of primes of
$K$, we denote by $G_{\Sigma}(K)$ the Galois group of the maximal
pro-$p$-extension of $K$ which is unramified outside $\Sigma$. Let
$S$ be the finite set of primes $S_{p}\cup S_{\infty}$, where
$S_{\infty}$ is the set of infinite primes of $K$ and $S_{p}$ is the
primes above $p$ in $K$. It is known that the group $G_{S_p}(K)$ is
a free pro-$p$-group on $r_2+1$ generators if and only if the second
Galois cohomology group $H^2(G_{S_p}(K),\mathbf{Z}/p\mathbf{Z})$
vanishes. This vanishing is related to special values of the zeta
function $\zeta_K$ via the conjecture of Lichtenbaum. More
precisely, let $\mathcal{G}_S$ be the Galois group of the maximal
extension of $K$ which is unramified outside $S$. The main
conjecture of Iwasawa theory (now a theorem of Wiles \cite{Wiles90})
relates the order of the group $H^2(\mathcal{G}_S,\mathbf{Z}_p(i))$,
for even integers $i$, to the $p$-adic valuation of $\zeta_K(1-i)$
by the $p$-adic equivalence:
\begin{equation}\label{Main conjecture}
w_{i}(K)\zeta_K(1-i)\sim_p |H^2(\mathcal{G}_S,\mathbf{Z}_{p}(i))|,
\end{equation}
where for any integer $i$, $w_i(F)$ is the order of the group
$H^0(G_F,\mathbf{Q}_p/\mathbf{Z}_p(i))$, and $\sim_p$ means having
the same $p$-adic valuation, see e.g \cite{Kolster}. Moreover, the
group $H^2(\mathcal{G}_S,\mathbf{Z}_{p}(i))$ vanishes if and only if
$H^2(\mathcal{G}_S,\mathbf{Z}/p\mathbf{Z}(i))$ vanishes. Let $\mu_p$
be the group of $p$-th unity. The periodicity of the groups
$H^2(\mathcal{G}_S,\mathbf{Z}/p\mathbf{Z}(i))$ modulo
$\delta=[K(\mu_p):K]$ gives that
$$H^2(\mathcal{G}_S,\mathbf{Z}/p\mathbf{Z}(i))\cong
H^2(\mathcal{G}_S,\mathbf{Z}/p\mathbf{Z}(i+j\delta)),$$ for any
integer $j$. In addition, since $p$ is odd, the vanishing of the
group $H^2(\mathcal{G}_S,\mathbf{Z}/p\mathbf{Z}(i))$ is equivalent
to the vanishing of the group
$H^2(G_{S_p}(K),\mathbf{Z}/p\mathbf{Z}(i))$. Number fields such that
$H^2(G_{S_p}(K),\mathbf{Z}/p\mathbf{Z}(i))=0$ are called
$(p,i)$-regular \cite{Assim}. In particular, the field $K$ is
$p$-rational if and only if $w_{p-1}(K)\zeta_K(2-p)\sim_{p} 1$. This
leads to the following characterization of the $p$-rationality of
totaly real number fields.
\begin{pro}\label{p-rational and L-fun}
Let $p$ be an odd prime number which is unramified in an abelian
totally real number field $K$. Then we have the equivalence
\begin{equation}
K\;\hbox{is p-rational}\;\Leftrightarrow\; L(2-p,\chi)\;\hbox{is a
p-adic unit},
\end{equation}
where $\chi$ is ranging over the set of irreducible characters of
$\mathrm{Gal}(K/\mathbf{Q})$.
\end{pro}
\noindent{\textbf{Proof}.} First, the zeta function $\zeta_{K}$
decomposes in the following way:
$$\zeta_{K}(2-p)=\zeta_{\mathbf{Q}}(2-p)\times\prod_{\chi\neq1}L(2-p,\chi).$$
Second, it is known that $\zeta_{\mathbf{Q}}(2-p)$ is of $p$-adic
valuation $-1$ and that $w_{p-1}(K)$ has $p$-adic valuation $1$,
giving that $w_{p-1}(K)\zeta_{\mathbf{Q}}(2-p)\sim_p 1$. Then from
(\ref{Main conjecture}) we obtain the formula
$$\prod_{\chi\neq1}L(2-p,\chi)\sim_p |H^2(\mathcal{G}_S,\mathbf{Z}_{p}(p-1))|.$$
Since, for every character $\chi\neq1$, the value $L(2-p,\chi)$ is a
$p$-integers \cite[Corollary 5.13]{Washington}, we have
$H^2(\mathcal{G}_S,\mathbf{Z}_p(p-1))=0$ if and only if for every
$\chi\neq1$, $L(2-p,\chi)$ is a $p$-adic unit. Furthermore, the
vanishing of the group $H^2(\mathcal{G}_S,\mathbf{Z}_{p}(p-1))$ is
equivalent to the vanishing of the group
$H^2(\mathcal{G}_S,\mathbf{Z}/p\mathbf{Z}(p-1))$, which turns out to
be equivalent to the vanishing of
$H^2(G_{S_p}(K),\mathbf{Z}/p\mathbf{Z})$ (by the above mentioned
periodicity statement). This last vanishing occurs exactly when the
field $K$ is $p$-rational.
$\blacksquare$\vskip 6pt
In the particular case of a real quadratic field
$K=\mathbf{Q}(\sqrt{d})$, we have the decomposition
$$\zeta_{K}(2-p)=\zeta_{\mathbf{Q}}(2-p)L(2-p,(\frac{d}{.})),$$
where $(\frac{d}{.})$ is the quadratic character associated to the
field $K=\mathbf{Q}(\sqrt{d})$.
\begin{coro}\label{L-function}
For every odd prime number $p\nmid d$, we have the equivalence
\begin{equation}
\mathbf{Q}(\sqrt{d})\;\hbox{is
p-rational}\;\Leftrightarrow\;L(2-p,(\frac{d}{.}))\not\equiv0\pmod{p}.
\end{equation}
\end{coro}
$\blacksquare$\vskip 6pt
\begin{rem}\label{remark 1}
The properties of special values of $p$-adic $L$-functions tells us
that the $p$-rationality is related to the class number and the
$p$-adic regulator. More precisely, let $K$ be a totally real number
field of degree $g$. Under the Leopoldt conjecture, class field
theory gives that
$G_{S_p}(K)^{ab}\cong\mathbf{Z}_{p}^{r_2+1}\times\mathcal{T}_K$,
where $\mathcal{T}_K$ is the $\mathbf{Z}_p$-torsion of
$G_{S_p}(K)^{ab}$. Then the field $K$ is $p$-rational precisely when
$\mathcal{T}_K=0$ \cite[Théor\`{e}me et Definition
1.2]{Movahhedi-Nguyen}. Moreover, the order of $\mathcal{T}_K$
satisfies
\begin{equation}
|\mathcal{T}_K|\sim_p
w(K(\mu_p))\prod_{v|p}(1-N(v)^{-1}).\frac{R_p(K).h_K}{\sqrt{|d_k|}},
\end{equation}
(\cite[app]{Coates}), where $h_K$ is the class number, $R_p(K)$ is
the $p$-adic regulator, $N(v)$ is the absolute norm of $v$,
$w(K(\mu_p))=|\mu(K(\mu_p))|$ the number of roots of unity of
$K(\mu_p)$ and $d_K$ is the discriminant of the number field $K$.
Hence for every odd prime number $p$ such that $(p,d_{K}h_K)=1$, the
field $K$ fails to be $p$-rational if and only if $v_p(R_p(K))>g-1$.
\end{rem}
Under the light of Remark \ref{remark 1}, for a real quadratic field
$\mathbf{Q}(\sqrt{d})$ we have the equivalence
\begin{equation}\label{regulator and p-rationality}
\mathbf{Q}(\sqrt{d})\;\hbox{is
p-rational}\;\Leftrightarrow\;R_p(\mathbf{Q}(\sqrt{d}))\not\equiv0\pmod{p^{2}}.
\end{equation}
Recall that $R_p(\mathbf{Q}(\sqrt{d}))=\log_{p}{(\varepsilon_d)}$,
where $\varepsilon_d$ is a fundamental unit of $K$ and $\log_p$ is
the $p$-adic logarithm.
\section{Fibonacci number}
The classical Fibonacci sequence is an interesting linear recurrence
sequence, in part because of its applications in several areas of
sciences. Here we consider a class of linear recurrence sequences
which arise from real quadratic fields and that we use for the study
of the $p$-rationality of these fields. As mentioned in the
introduction, Greenberg \cite[Corollary 4.1.5.]{Greenberg} used
classical Fibonacci numbers to give a characterization for the
$p$-rationality of the field $\mathbf{Q}(\sqrt{5})$. In this paper
we give a generalization of this result to any real quadratic field.
The Fibonacci numbers associated to real quadratic fields are given
as follows. Let $d>0$ be a fundamental discriminant and let $h_{d}$,
$\varepsilon_d$ be respectively the class number and the fundamental
unit of the field $\mathbf{Q}(\sqrt{d})$ with ring of integers
$\mathcal{O}_d$. We denote by $\overline{\varepsilon}_d$ the
conjugate of $\varepsilon_d$ and $N(.)$ the absolute norm. Define
the sequence
$F^{(\varepsilon_d+\overline{\varepsilon}_d,N(\varepsilon_d))}=(F_n)_n$
such that $F_0=0$, $F_1=1$ and
$$F_{n+2}=(\varepsilon_d+\overline{\varepsilon}_d)F_{n+1}-N(\varepsilon_d)F_n.$$
The Binet formula \cite[page 173]{Dan-Robert} gives that
$$F_n=\frac{\varepsilon_d^{n}-\overline{\varepsilon}_d^{n}}{\varepsilon_d-\overline{\varepsilon}_d},\quad\quad \forall n\geq0.$$
We establish a relation between Fibonacci numbers and the $p$-adic
regulator which allows us to prove the main result.
\begin{deft}
Let $a$ be a non trivial element of the ring of integers of the
field $\mathbf{Q}(\sqrt{d})$ such that $(a,p)=1$. Then the prime $p$
is said to be Wieferich of basis $a$ if the following congruence
holds:
$$a^{p^{r}-1}-1\equiv0\pmod{p^{2}},$$ where $r$ is the residue degree of $p$
in the quadratic field $\mathbf{Q}(\sqrt{d})$. Otherwise, the prime
number $p$ is said to be non-Wieferich of basis $a$.
\end{deft}
We have the following equality
$$\log_p{((\varepsilon_{d}^{p^{r}-1}-1)+1)}=(\varepsilon_{d}^{p^{r}-1}-1)-\frac{1}{2}(\varepsilon_{d}^{p^{r}-1}-1)^2+...$$
where $\log_p$ is the $p$-adic logarithm and as before $r$ is the
residue degree of $p$ in the quadratic field $\mathbf{Q}(\sqrt{d})$.
Since $R_p=\log_p{(\varepsilon_{d})}$ and the group
$(\mathcal{O}_d/p\mathcal{O}_d)^{\times}$ is cyclic of order
$p^r-1$, where $\mathcal{O}_d$ is the ring of integers of
$\mathbf{Q}(\sqrt{d})$, we obtain the equivalences
\begin{equation}\label{Wieferich-regulator}
\begin{array}{ccc}
\varepsilon_d^{p^{r}-1}-1\not\equiv0 \pmod{p^{2}} & \Leftrightarrow & R_p\equiv p\pmod{p^{2}}, \\
& \Leftrightarrow & R_p\not\equiv0\pmod{p^{2}}. \\
\end{array}
\end{equation}
Then combining this last equivalence with the equivalence
(\ref{regulator and p-rationality}) we obtain
\begin{pro}
Let $p$ be an odd prime number such that $p\nmid dh_d$. Then the
field $\mathbf{Q}(\sqrt{d})$ is $p$-rational if and only if $p$ is a
non-Wieferich prime of basis $\varepsilon_d$.
\end{pro}
$\blacksquare$\vskip 6pt
Very little is known about these primes and it is conjectured that
the set of Wieferich primes is of density zero \cite{Silverman}. In
the following we are interested with the set of Fibonacci-Wieferich
primes defined as follows.
\begin{deft}\label{fibonacci-wieferich}
A prime number $p$ is said to be a Fibonacci-Wieferich prime for the
field $\mathbf{Q}(\sqrt{d})$ if
$$F_{p-(\frac{d}{p})}\equiv0\pmod{p^2},$$
where $(\frac{d}{.})$ is the Legendre symbol associated to the
quadratic field $\mathbf{Q}(\sqrt{d})$.
\end{deft}
We give the main result of this section which describe the
$p$-rationality in terms of Fibonacci-Wieferich primes.
\begin{theo}\label{fibonacci}
Let $p\geq5$ be a prime number such that $p\nmid
(\varepsilon_{d}-\overline{\varepsilon_{d}})^2 h_d$. Then the
following assertions are equivalent:
\begin{enumerate}
\item the field $\mathbf{Q}(\sqrt{d})$ is $p$-rational,
\item $p$ is not a Fibonacci-Wieferich prime for
$\mathbf{Q}(\sqrt{d})$.
\end{enumerate}
\end{theo}
\noindent{\textbf{Proof.}} Using the equivalence
(\ref{Wieferich-regulator}), it suffices to prove that:
\begin{equation}
\varepsilon_d^{p^{r}-1}-1\not\equiv0\pmod{p^{2}}\;\Leftrightarrow\;F_{p-(\frac{d}{p})}\not\equiv0\pmod{p^{2}}.
\end{equation}
Let $Q_p(\varepsilon_d)$ be the residue class
$$\frac{\varepsilon_d^{p^{r}-1}-1}{p}\pmod{p}.$$
A prime number $p$ satisfying
$Q_p(\varepsilon_d)\not\equiv0\pmod{p}$ is non-Wieferich of basis
$\varepsilon_d$.\\
First suppose that $(\frac{d}{p})=1$. Then $r=1$ and
\begin{equation}
Q_p(\varepsilon_d)\equiv \frac{\varepsilon_d^{p-1}-1}{p}\pmod{p}.
\end{equation}
The Binet formula gives that
$$(\varepsilon_d-\overline{\varepsilon}_d)F_{p-1}=\varepsilon_d^{p-1}-\overline{\varepsilon}_d^{p-1}=
\varepsilon_d^{1-p}(\varepsilon_d^{(p-1)}-1)(\varepsilon_d^{(p-1)}+1).$$
Since $\varepsilon_d$ is a unit and $p\nmid
(\varepsilon_d-\overline{\varepsilon}_d)$, we have
$\varepsilon_d^{1-p}(\varepsilon_d^{(p-1)}+1)(\varepsilon_d-\overline{\varepsilon}_d)\not\equiv0\pmod{p}$.
Hence we obtain the equivalence
\begin{equation}
Q_p(\varepsilon_d)\not\equiv0\pmod{p}\Leftrightarrow
F_{p-1}\not\equiv0\pmod{p^2}.
\end{equation}
Second, suppose that the prime number $p$ is inert in the field
$\mathbf{Q}(\sqrt{d})$. Then we have
\begin{equation}
Q_p(\varepsilon_d)\equiv \frac{\varepsilon_d^{p^2-1}-1}{p}\pmod{p}.
\end{equation}
The Galois group of $\mathbf{Q}(\sqrt{d})/\mathbf{Q}$ is generated
by an element $\sigma$ of order two such that
$\sigma(\varepsilon_d)=\overline{\varepsilon}_d$. Since the group
$(\mathcal{O}_d/p\mathcal{O}_d)^{\times}$ is cyclic of order
$p^2-1$, we have $\varepsilon_d^{p+1}\equiv x\pmod{p}$ for some
$x\in\mathbf{Z}$. Hence $\overline{\varepsilon}_d^{p+1}\equiv
x\pmod{p}$ and $F_{p+1}\equiv0\pmod{p}$. Note that since
$$F_{p+1}=(\varepsilon_d-\overline{\varepsilon}_d)^{-1}\overline{\varepsilon}_d^{p+1}(\varepsilon_d^{2(p+1)}-1),$$
we have $$\varepsilon_d^{2(p+1)}-1\equiv0\pmod{p}.$$ Moreover,
$$Q_p(\varepsilon_d)=\frac{1}{p}(\varepsilon_d^{p^{2}-1}-1)=\frac{1}{p}((\varepsilon_d^{2(p-1)})^{\frac{p-3}{2}}-1)=
\frac{1}{p}(\varepsilon_d^{2(p+1)}-1)(\varepsilon_d^{2(p+1)\frac{p-1}{2}}+...+1).$$
Since $x^{2}\equiv1\pmod{p}$, we then obtain the congruence
$$Q_p(\varepsilon_d)\equiv\frac{1}{p}\frac{p-1}{2}(\varepsilon_d^{2(p+1)}-1)\pmod{p}.$$
Hence we have the equivalence
\begin{equation}
Q_p(\varepsilon_d)\not\equiv0\pmod{p}\Leftrightarrow
F_{p+1}\not\equiv0\pmod{p^2}.
\end{equation}
Then in all cases we obtain that the field $\mathbf{Q}(\sqrt{d})$ is
$p$-rational precisely when $p$ is not a Fibonacci-Wieferich prime.
$\blacksquare$\vskip 6pt
Using this characterization of the $p$-rationality on pariGP, we
obtain some numerical evidence for the primes $p$ for which a given
real quadratic number field is not $p$-rational.
$$
\begin{tabular}{|c|c|}
\hline
Discriminant & Primes$<10^9$\\
\hline
5& \\
\hline
8& 13, 31, 1546463 \\
\hline
12& 103 \\
\hline
13& 241 \\
\hline
17& \\
\hline
21& 46179311 \\
\hline
24& 7, 523 \\
\hline
28& \\
\hline
29& 3, 11 \\
\hline
33& 29, 37, 6713797 \\
\hline
37& 7, 89, 257, 631 \\
\hline
40& 191, 643, 134339, 25233137 \\
\hline
41& 29, 53, 7211 \\
\hline
44& \\
\hline
53& 5 \\
\hline
56& 6707879, 93140353 \\
\hline
57& 59, 28927, 1726079, 7480159 \\
\hline
60& 181, 1039, 2917, 2401457 \\
\hline
61& \\
\hline
65& 1327, 8831, 569831 \\
\hline
69& 5, 17, 52469057 \\
\hline
73& 5, 7, 41, 3947, 6079 \\
\hline
76& 79, 1271731, 13599893, 31352389\\
\hline
77& 3, 418270987 \\
\hline
85& 3, 204520559 \\
\hline
88& 73, 409, 43, 28477 \\
\hline
89& 5, 7, 13, 59 \\
\hline
92& 7, 733 \\
\hline
93& 13 \\
\hline
97& 17, 3331\\
\hline
\end{tabular}
$$\\
With the help of these results and further computations, we could
construct examples of multi-quadratic $p$-rational fields. The first
example is the field
$K_1=\mathbf{Q}(\sqrt{2},\sqrt{3},\sqrt{5},\sqrt{7},\sqrt{11},\sqrt{-1})$,
which is $p$-rational for all primes $100<p<1000$ except for
$p=103,173,181,191,199,227,251,269,\\409,523,571,577,643,859$.
Another example is the field
$K_2=\mathbf{Q}(\sqrt{13},\sqrt{17},\sqrt{19},\sqrt{23},\sqrt{29},\sqrt{-1})$.
The field $K_2$ is $p$-rational for all primes $100<p<1000$ except
for
$151,197,227,241,307,337,401,457,\\487,593,643,709,719,733,809,839$.
Hence for every prime $100<p<1000$ such that $p\neq227,643$, there
exist a $p$-rational field of degree $2^{t}$ for any $1\leq
t\leq6$.\\
The above examples are weak numerical evidence to a conjecture
proposed by Greenberg:
\begin{conj}(\cite[Conjecture
4.2.1.]{Greenberg}) For any odd prime $p$ and for any $t\geq1$,
there exists a $p$-rational field $K$ such that
$\mathrm{Gal}(K/\mathbf{Q})\cong(\mathbf{Z}/2\mathbf{Z})^{t}$.
\end{conj}
As an important consequence of this conjecture, Greenberg proved the
following proposition.
\begin{pro}\cite[Proposition 6.2.2]{Greenberg}
Suppose that $K$ is a complex $p$-rational number field and that
$\mathrm{Gal}(K/\mathbf{Q})$ is isomorphic to
$(\mathbf{Z}/2\mathbf{Z})^{t}$, where $t\geq4$. Let $n$ be an
integer such that $4\leq n\leq 2^{t-1}-3$. Then there exists a
continuous homomorphism $$\rho: G_{\mathbf{Q}}\rightarrow
GL_n(\mathbf{Z}_p),$$ with open image.
\end{pro}
Based on the above computations and Proposition $3.5$, we have the
following corollary.
\begin{coro}
For any integer $4\leq n\leq 2^{5}-3$ and any prime $100<p<1000$
such that $p\neq227,643$, there exists a $p$-adic Galois
representation
$$\rho: G_{\mathbf{Q}}\rightarrow GL_n(\mathbf{Z}_p),$$ with open
image.
\end{coro}
Another characterization of the $p$-rationality is given in terms of
periods of Fibonacci sequences modulo $p$ and $p^{2}$. Let
$F^{(a,b)}$ be a Fibonacci sequence and $m$ a positive integer such
that $(b,m)=1$. As mentioned above the sequence $F^{(a,b)}\pmod{m}$
is periodic of period $k(m)$. Wall studied these periods for
classical Fibonacci sequence and general results are obtained in
\cite[page 374-376]{Renault}. We describe the $p$-rationality of
real quadratic fields in terms of periods of Fibonacci sequence
associated to these fields.
\begin{theo}\label{periods}\cite[Proposition 3.2.4]{Elsenhans-Jahnel}
\begin{equation}
\hbox{The equality}\; k(p)=k(p^2)\;\hbox{holds}\; \hbox{if and only
if}\; F_{p-(\frac{d}{p})}\equiv0\pmod{p^2}.
\end{equation}
\end{theo}
Proposition \ref{periods and p-rationality} follows from Theorem
\ref{fibonacci} and Theorem \ref{periods}. For the classical
Fibonacci numbers $F_n$, the field $\mathbf{Q}(\sqrt{5})$ is
$p$-rational precisely when $p$ is not a Fibonacci-Wieferich prime
\cite[Corollary 4.1.5]{Greenberg}. It is known that up to
$6.7\times10^{15}$ there is no Fibonacci-Wieferich primes
\cite{François-Klyve}. Greenberg pointed out in \cite{Greenberg}
that such primes are quite rare, they have trivial density if we
assume G. Gras Conjecture, which asserts that a number field is
$p$-rational for almost all primes. Theorem \ref{periods}, gives
that the field $\mathbf{Q}(\sqrt{d})$ is $p$-rational if and only if
$k(p)\neq k(p^2)$. According to the table above there is fundamental
discriminants $d$ such that there exist primes $p$ for which
$k(p)=k(p^2)$. As an example we mentioned the case of
$\mathbf{Q}(\sqrt{2})$ where $k(13)=k(13^2)$. Note that up to
$10^{9}$, for some discriminants we still have no primes satisfying
the equality of Wall such as $17,28,44,61$.
\section{Williams Congruence}
Let $d$ be a positive fundamental discriminant and $p$ be an odd
prime number such that $p\nmid d$. We are interested with the
numbers $F_{p-(\frac{d}{p})}$. In the classical case, namely the
field $\mathbf{Q}(\sqrt{5})$, we have explicit formula for the
quotient $F_{p-(\frac{5}{p})}/p$ \cite[Theorem 4.1]{Williams}. For
the general case we have a result due to H.C. Williams in
\cite{Williams} which describes these quotients for any real
quadratic field. The results obtained in the above section, combined
with the formula proved by Williams gives another characterization
of the $p$-rationality of real quadratic fields. For an integer $n$,
let $\{n\}$ be the least non-negative residue of $n$ modulo $d$. The
integer $p'$ represents the inverse of $p$ modulo $d$ and
$(\frac{d}{.})$ is the Legendre symbol. Consider the following sum
of characters:
\begin{equation*}
\beta_p(i)=\sum_{j=1}^{\{p'i\}-1}(\frac{d}{j}).
\end{equation*}
Then the result of Williams is as follows:
\begin{theo}\label{Williams}\cite{Williams} Let $p$ be an odd prime number such
that $p\nmid d$. Then
\begin{equation}\label{Williams congruence}
h_dF_{p-(\frac{d}{p})}/p\equiv-2(\frac{d}{p})N^{\frac{(\frac{d}{p})-1}{2}}
\sum_{i=1}^{\frac{p-1}{2}}\beta_p(i)\frac{1}{i}\pmod{p},
\end{equation}
where $h_d$ is the class number of the field $\mathbf{Q}(\sqrt{d})$,
and $\frac{1}{i}$ is the inverse of $i$ modulo $p$.
\end{theo}
An interesting problem of combinatorics and additive number theory
is the study of sums of reciprocals in finite fields. Here we are
concerned with the linear combinations
$$\sum_{i=1}^{d}\beta_p(i)\alpha_p(i)\pmod{p},$$
where $$\alpha_p(i)=\sum_{\begin{array}{c}
1\leq k\leq \frac{p-1}{2} \\
k\equiv i\pmod{d}
\end{array}
}\frac{1}{k}.$$ We have the following description of the
$p$-rationality of the field $\mathbf{Q}(\sqrt{d})$.
\begin{theo}\label{sum of reciprocals}
If $p$ does not divides
$(\varepsilon_d-\overline{\varepsilon}_d)^2$, then
\begin{equation}
\mathbf{Q}(\sqrt{d})\;\hbox{is
p-rational}\;\Leftrightarrow\;\sum_{i=1}^{d}\beta_p(i)\alpha_p(i)\not\equiv0\pmod{p}.
\end{equation}
\end{theo}
\noindent{\textbf{Proof}.} It is known that
$F_{p-(\frac{d}{p})}\equiv0\pmod{p}$ \cite[page 431 formula
(1.2)]{Williams}. Then by Theorem \ref{Main theorem}, the field
$\mathbf{Q}(\sqrt{d})$ is $p$-rational if and only if
$h_{d}F_{p-(\frac{d}{p})}/p\not\equiv0\pmod{p}$. Using Theorem
\ref{Williams}, this occurs precisely when
$$-2(\frac{d}{p})N^{\frac{(\frac{d}{p})-1}{2}}
\sum_{i=1}^{\frac{p-1}{2}}\beta_p(i)\frac{1}{i}\not\equiv0\pmod{p}.$$
Since $p$ is an odd prime number and the term
$2(\frac{d}{p})N^{\frac{(\frac{d}{p})-1}{2}}$ equals $1$ or $2$, the
field $\mathbf{Q}(\sqrt{d})$ is $p$-rational if and only if
\begin{equation}\label{Main equivalence}
\sum_{i=1}^{\frac{p-1}{2}}\beta_p(i)\frac{1}{i}\not\equiv0\pmod{p}.
\end{equation}
Recall that for any integer $i$, $\{i\}$ is the least non-negative
residue class of $i$ modulo $d$. Hence by definition we have
$\{i+kd\}=\{i\}$ for any integer $k\geq0$ and the following equality
holds for any integer $i\in\{1,...,d\}$:
$$\beta_p(i+kd)=\beta_p(i).$$ Then the terms $\frac{1}{i}$ and
$\frac{1}{i+kd}$ of (\ref{Main equivalence}) have the same
coefficient $\beta_p(i)$. For $i\in\{1,...,d\}$ regrouping the
integers $\frac{1}{j}$ such that $j$ lies in the equivalence class
of $i$ modulo $d$ and $j\in\{1,...,\frac{p-1}{2}\}$, the sum in
(\ref{Main equivalence}) can be written
$$\sum_{i=1}^{\frac{p-1}{2}}\beta_p(i)\frac{1}{i}=\sum_{i=1}^{d}\beta_p(i)\alpha_p(i).$$
Then the field $\mathbf{Q}(\sqrt{d})$ is $p$-rational if and only if
$\sum_{i=1}^{d}\beta_p(i)\alpha_p(i)\not\equiv 0\pmod{p}$.
$\blacksquare$\vskip 6pt
As a consequence we have the following characterization of the
$p$-rationality of the field $\mathbf{Q}(\sqrt{5})$.
\begin{coro}
For every prime $p\equiv1\pmod{5}$, the field $\mathbf{Q}(\sqrt{5})$
is $p$-rational if and only if
\begin{equation}
\alpha_p(1)+\alpha_p(2)-\alpha_p(4)+2\alpha_p(5)\not\equiv0\pmod{p}.
\end{equation}
\end{coro}
\noindent{\textbf{Proof}.} Let $\ell$ be a prime number, then
$(\frac{5}{\ell})=1$ if and only if $\ell\equiv1,4\pmod{5}$, and
$(\frac{5}{\ell})=-1$ if and only if $\ell\equiv2,3\pmod{5}$. Since
$p\equiv1\pmod{5}$, we have for $i\in\{1,...,5\}$,
\begin{equation*}
\beta_p(i)=\sum_{j=1}^{i-1}(\frac{5}{j}),
\end{equation*}
such that $\beta_p(1)=1$, $\beta_p(2)=1$, $\beta_p(3)=0$,
$\beta_p(4)=-1$ and $\beta_p(5)=2$.
$\blacksquare$\vskip 6pt
If we fix the prime number $p$, we obtain a description of the set
of fundamental discriminants $d$ for which the field
$\mathbf{Q}(\sqrt{d})$ is $p$-rational. For the particular cases
$p=3$ and $p=5$ we obtain the following proposition.
\begin{pro}\label{3,5-rational}
Let $d$ be a fundamental discriminant such that $3,5\nmid
(\varepsilon_d-\overline{\varepsilon}_d)^{2}$ then we have the
equivalence:
\begin{enumerate}
\item $\mathbf{Q}(\sqrt{d})$\;\hbox{is
3-rational}\;$\Leftrightarrow$\;$\beta_3(1)\not\equiv0\pmod{3}$,
\item $\mathbf{Q}(\sqrt{d})$\;\hbox{is
5-rational}\;$\Leftrightarrow$\;$\beta_5(1)\not\equiv2\beta_5(2)\pmod{5}$.
\end{enumerate}
\end{pro}
\noindent{\textbf{Proof}.} By Theorem \ref{Williams}, we have for
$p=3$ the equality
$$\sum_{i=1}^{\frac{3-1}{2}}\beta_3(i)\frac{1}{i}=\beta_3(1),$$
and for $=5$,
$$\sum_{i=1}^{\frac{5-1}{2}}\beta_5(i)\frac{1}{i}=\beta_5(1)+3\beta_5(2).$$
Then the equivalences in $(1)$ and $(2)$ follow from Theorem
\ref{sum of reciprocals}.
$\blacksquare$\vskip 6pt
In general, given an odd prime number $p$, it is not known wether
there exist infinitely many real quadratic fields which are
$p$-rational. This is known for the cases of $p=3$ which is proved
by Dongho Byeon in \cite[Theorem 1]{Byeon}, and the other case is
$p=5$ (see \cite{Assim-Bouazzaoui}). Both cases are proved using
divisibility
properties of Fourier coefficients of half-integer weight modular forms.\\
\noindent{\textbf{Acknowledgement}.} I would like to thank my
advisor J.Assim for his guidance and patience during the preparation
of this paper. Many thanks goes to H.Cohen and B.Abombert for their
help on pariGP computations during the Atelier pariGP in
Besan\c{c}on.
\end{document}
|
\begin{document}
\title{A note on palindromic length of Sturmian sequences}
\author{Petr Ambro\v{z}, Edita Pelantov\'a\\[1mm]
Department of Mathematics FNSPE\\ Czech Technical University in Prague\\
Trojanova 13, 120 00 Praha 2, Czech Republic}
\date{}
\maketitle
\begin{abstract}
Frid, Puzynina and Zamboni (2013) defined the palindromic length of a finite word $w$ as the minimal number
of palindromes whose concatenation is equal to $w$. For an infinite word $\ensuremath{\boldsymbol{u}}$ we study $\PL{u}$, that is,
the function that assigns to each positive integer $n$, the maximal palindromic length of factors of length
$n$ in $\ensuremath{\boldsymbol{u}}$. Recently, Frid (2018) proved that $\limsup_{n\to\infty}\PL{u}(n)=+\infty$ for any Sturmian word $\ensuremath{\boldsymbol{u}}$.
We show that there is a constant $K>0$ such that $\PL{u}(n)\leq K\ln n$ for every Sturmian word $\ensuremath{\boldsymbol{u}}$, and
that for each non-decreasing function $f$ with property $\lim_{n\to\infty}f(n)=+\infty$ there is a Sturmian
word $\ensuremath{\boldsymbol{u}}$ such that $\PL{u}(n)=\mathcal{O}(f(n))$.
\end{abstract}
\section{Introduction}
Palindromic length of a word $v$, denoted by $|v|_{\text{pal}}$, is the minimal number $K$ of
palindromes $p_1,p_2,\ldots,p_K$ such that $v=p_1p_2\cdots p_K$. This notion has been introduced
by Frid, Puzynina and Zamboni~\cite{frid-puzynina-zamboni-aam-50} along with the following conjecture.
\begin{conj}
If there is a positive integer $P$ such that $|v|_{\text{pal}}\leq P$ for every factor $v$ of an infinite
word $\boldsymbol{w}$ then $\boldsymbol{w}$ is eventually periodic.
\end{conj}
Frid et al.\ proved validity of the conjecture for $r$-power-free infinite words, i.e., for words which
do not contain factors of the form $v^r=vv\cdots v$ ($r$ times for some integer $r\geq 2$). By result
of Mignosi~\cite{mignosi-tcs-82} the conjecture thus holds for any Sturmian word whose slope has bounded
coefficients in its continued fraction. Recently, Frid~\cite{frid-ejc-71} proved the conjecture for all Sturmian
words.
In this paper we study asymptotic growth of function $\PL{u}:\mathbb{N}\rightarrow\mathbb{N}$ defined for an infinite
word $\ensuremath{\boldsymbol{u}}$ by
\[
\PL{u}(n) = \max\{|v|_{\text{pal}} : \text{$v$ is factor of length $n$ in $\boldsymbol{u}$}\}.
\]
The aforementioned result by Frid can be stated, using function $\PL{u}$, in the form of the following theorem.
\begin{thm}[\cite{frid-ejc-71}]
Let $\boldsymbol{u}$ be a Sturmian word. Then $\limsup\limits_{n\to\infty}\PL{u}(n)=+\infty$.
\end{thm}
We prove the following two theorems about the rate of growth of function $\PL{u}$
for Sturmian words.
\begin{thm}\label{thm:libovolne_pomaly_rust}
Let $f:\mathbb{N}\rightarrow\mathbb{R}$ be a non-decreasing function with $\lim\limits_{n\to\infty}f(n)=+\infty$.
Then there is a Sturmian word $\ensuremath{\boldsymbol{u}}$ such that $\PL{u}(n)=o(f(n))$.
\end{thm}
\begin{thm}\label{thm:rust_pomalejsi_nez_ln}
There is a constant $K$ such that for every Sturmian word $\ensuremath{\boldsymbol{u}}$ we have $\PL{u}\leq K\ln n$.
\end{thm}
In other words, $\PL{u}$ may grow into infinity arbitrarily slow (Theorem~\ref{thm:libovolne_pomaly_rust})
and not faster than $\mathcal{O}(\ln n)$ (Theorem~\ref{thm:rust_pomalejsi_nez_ln}). Let us stress that
the constant $K$ in Theorem~\ref{thm:rust_pomalejsi_nez_ln} is universal for every Sturmian word.
Both theorems refer to upper
estimates on the growth of $\PL{u}$. Indeed, it is much more difficult to obtain a lower bound on the
growth, such bound is not known even for the Fibonacci word. Recently, Frid~\cite{frid-numeration-2018} considered
a certain sequence of prefixes of the Fibonacci word, denoted $(w^{(n)})$, and she formulated
a conjecture about the precise value of $|w^{(n)}|_{\text{pal}}$. This conjecture can be rephrased in the following
way (cf.~Remark~\ref{rem:frid_lower_bound_conjecture}).
\begin{conj}\label{conj:frid_lower_bound_conjecture}
Let $\boldsymbol{f}$ be the Fibonacci word, that is, the fixed point of the morphism $0\mapsto 01$, $1\mapsto 0$.
Then
\[
\limsup_{n\to\infty}\frac{\PL{f}(n)}{\ln n} \geq \frac{1}{3\ln\tau},
\]
where $\tau$ is the golden ratio.
\end{conj}
We propose (see Remark~\ref{rem:frid_lower_bound_conjecture} for more details) the following extension of this
so far unproved statement.
\begin{conj}
Let $\boldsymbol{u}$ be a Sturmian word whose slope has bounded coefficients in its continued fraction.
Then
\[
\limsup_{n\to\infty}\frac{\PL{u}(n)}{\ln n} > 0.
\]
\end{conj}
\section{Preliminaries}
An \emph{alphabet} $A$ is a finite set of \emph{letters}. A finite sequence of letters of $A$ is called
a (finite) \emph{word}. The \emph{length} of a word $w=w_1w_2\cdots w_n$, that is, the number of its
letters, is denoted $|w|=n$. The notation $|w|_a$ is used for the number of occurrences of the letter $a$ in $w$.
The \emph{empty word} is the unique word of length 0, denoted by $\varepsilon$. The set of all finite words over $A$
(including the empty word) is denoted by $A^*$, equipped with the operation of concatenation of words $A^*$ is
a free monoid with $\varepsilon$ as its neutral element.
We consider also \emph{infinite words} $\ensuremath{\boldsymbol{u}} = u_0u_1u_2\cdots$, the set of infinite words over $A$ is
denoted by $A^{\mathbb{N}}$.
A word $w$ is called a \emph{factor} of $v\in A^*$ if there exist words $w^{(1)},w^{(2)}\in A^*$ such that
$v = w^{(1)}ww^{(2)}$. The word $w$ is called a \emph{prefix} of $v$ if $w^{(1)}=\varepsilon$, it is called a
\emph{suffix} of $v$ if $w^{(2)}=\varepsilon$. The notions of factor and prefix can be easily extended to
infinite words. The set of all factors of an infinite word $\ensuremath{\boldsymbol{u}}$, called the \emph{language} of $\ensuremath{\boldsymbol{u}}$,
is denoted by $\mathcal{L}(\ensuremath{\boldsymbol{u}})$. Let $w$ be a prefix of $v$, that is, $v=wu$ for some word $u$.
Then we write $w^{-1}v=u$.
The \emph{slope} of a nonempty word $w\in\{0,1\}^*$ is the number $\pi(w)=\frac{|w|_1}{|w|}$.
Let $\ensuremath{\boldsymbol{u}}=(u_n)_{n\geq 0}$ be an infinite word. Then the limit
\begin{equation}\label{eq:def_slope}
\rho = \lim_{n\to\infty}\pi(u_0\cdots u_{n-1}) = \frac{|u_0\cdots u_{n-1}|_1}{n}
\end{equation}
is the \emph{slope} of the infinite word. Obviously, the slope of $\ensuremath{\boldsymbol{u}}$ is equal to the frequency of the
letter 1 in $\ensuremath{\boldsymbol{u}}$.
In this paper we are concerned with the so-called \emph{Sturmian words}~\cite{morse-hedlund-ajm-62}.
These are infinite words over
a binary alphabet that have exactly $n+1$ factors of length $n$ for each $n\geq 0$. Sturmian words admit
several equivalent definitions and have many interesting properties.
We will need the following two fact above all. The limit in~(\ref{eq:def_slope}) exists, and thus the
slope of a Sturmian word is well defined, and, moreover, it is an irrational number~\cite{lothaire2}.
Two Sturmian words have the same language if and only they have the same slope~\cite{mignosi-tcs-65}.
A \emph{morphism} of the free monoid $A^*$ is a map $\varphi:A^*\rightarrow A^*$ such that
$\varphi(vw)=\varphi(v)\varphi(w)$ for all $v,w\in A^*$.
A morphism $\varphi$ is called \emph{Sturmian} if $\varphi(\ensuremath{\boldsymbol{u}})$ is a Sturmian word for every
Sturmian word $\ensuremath{\boldsymbol{u}}$. The set of all Sturmian morphisms coincides with the so-called
\emph{monoid of Sturm}~\cite{mignosi-seebold-jtnb-5}, it is the monoid generated by the following
three morphisms
\[
E: \begin{aligned}0 &\mapsto 1 \\ 1&\mapsto 0\end{aligned}\,,\qquad
G: \begin{aligned}0 &\mapsto 0 \\ 1&\mapsto 01\end{aligned}\,,\qquad
\tilde{G}: \begin{aligned}0 &\mapsto 0 \\ 1&\mapsto 10\end{aligned}\,.
\]
\section{Images of Sturmian words}
In this section we study length and palindromic length of images of words under morphisms
$\psi_b:\{0,1\}^*\rightarrow\{0,1\}^*$, where $b\in\mathbb{N}$, $b\geq 1$ and
\begin{equation}\label{eq:morphism_psi}
\begin{split} \psi_b(0) & = 10^{b-1}, \\ \psi_b(1) &= 10^b. \end{split}
\end{equation}
Note that $\psi_b$ is a Sturmian morphism since $\psi_b = \tilde{G}^{b-1}\circ E\circ G$.
\begin{lem}\label{lem:len_of_image}
Let $b,c\in\mathbb{N}$, $b,c\geq 1$ and let $v\in\{0,1\}^*$. Then
\begin{enumerate}
\item
$|\psi_b(v)|\geq b|v|$,
\item
$|(\psi_c\circ\psi_b)(v)|\geq 2|v|$.
\end{enumerate}
\end{lem}
\begin{proof}
i) Let $x=|v|_0$ and $y=|v|_1$. Then $\psi_b(v)$ contains $x':=(b-1)x+by$ zeros and $y':=x+y$ ones.
Thus $|\psi_b(v)| = x'+y' = bx+(b+1)y \geq b(x+y) = b|v|$.
ii) The word $(\psi_c\circ\psi_b)(v)$ contains $x'':=(c-1)x'+cy'$ zeros and $y'':=x'+y'$ ones. Thus
$|(\psi_c\circ\psi_b)(v)|=x''+y''=cx'+(c+1)y'\geq x'+2y'\geq 2y'=2(x+y)=2|v|$.
\end{proof}
\begin{lem}\label{lem:palllen_of_image}
Let $b\in\mathbb{N}$, $b\geq 1$ and let $v\in\{0,1\}^*$, Then $|\psi_b(v)|_{\text{pal}}\leq|v|_{\text{pal}}+1$.
\end{lem}
\begin{proof}
One can easily check that if $p$ is a palindrome then both $\psi_b(p)1$ and $1^{-1}\psi_b(p)$ are
palindromes.
If $v=p_1p_2\cdots p_{2q}$, where all $p_i$ are palindromes, then
\[
\psi_b(v) =
\underbrace{\psi_b(p_1)1}_{p_1'}\cdot
\underbrace{1^{-1}\psi_b(p_2)}_{p_2'}\cdot
\underbrace{\psi_b(p_3)1}_{p_3'}\cdot
\underbrace{1^{-1}\psi_b(p_4)}_{p_4'}\cdots
\underbrace{\psi_b(p_{2q-1})1}_{p_{2q-1}'}\cdot
\underbrace{1^{-1}\psi_b(p_{2q})}_{p_{2q}'}
\]
is a factorization of $\psi_b(v)$ into $2q$ palindromes and therefore we have
$|\psi_b(v)|_{\text{pal}}\leq|v|_{\text{pal}}$.
On the other hand, if $|v|_{\text{pal}}$ is odd the factorization of $\psi_b(v)$ is almost the same with the only
exception that at the end there is (possibly non-palindromic) image of the last palindrome, i.e., $\psi_b(p_{2q+1})$.
The statement follows from the fact that $\psi_b(p_{2q+1}) = 1\cdot1^{-1}\psi_b(v_{2q+1})$.
\end{proof}
\begin{lem}\label{lem:cfe_of_image}
Let \ensuremath{\boldsymbol{u}}\ be a Sturmian word with slope $\alpha\in(0,1)$ and let $\alpha=[0,a_1,a_2,a_3,\ldots]$ be
its continued fraction. Then $\psi_b(\ensuremath{\boldsymbol{u}})$ is a Sturmian word with slope $\beta$,
where $\beta=[0,b,a_1,a_2,a_3,\ldots]$.
\end{lem}
\begin{proof}
Recall that $\alpha$ is the frequency of the letter 1 in \ensuremath{\boldsymbol{u}}, that is,
\[
\alpha = \lim_{|v|\to\infty}\frac{|v|_1}{|v|_0+|v|_1}, \quad \text{where $v\in\mathcal{L}(\ensuremath{\boldsymbol{u}})$}.
\]
Let us consider the image of $v\in\mathcal{L}(\ensuremath{\boldsymbol{u}})$ under $\psi_b$. We have
$|\psi_b(v)|_0 = (b-1)|v|_0 + b|v|_1$ and $|\psi_b(v)|_1 = |v|_0 + |v|_1$.
Therefore
\begin{align*}
\beta & = \lim_{|v|\to\infty}\frac{|\psi_b(v)|_1}{|\psi_b(v)|_0+|\psi_b(v)|_1} =
\lim_{|v|\to\infty}\frac{|v|_0 + |v|_1}{b|v|_0 + (b+1)|v|_1} = \\[2mm]
&= \lim_{|v|\to\infty}\frac{1}{b + \frac{|v|_1}{|v|_0+|v|_1}} = \frac{1}{b+\alpha}.
\qedhere
\end{align*}
\end{proof}
\begin{lem}\label{lem:preimage_and_its_pallen}
Let $v\in\{0,1\}^*$ be a factor of a Sturmian word $\ensuremath{\boldsymbol{u}}$ with slope $\beta=[0,b,a_1,a_2,a_3,\ldots]$
and let $|v|_1\geq 2$. Then there are words $v',v_L,v_R$ such that $v'\neq\varepsilon$ is a factor
of a Sturmian words with slope $\alpha=[0,a_1,a_2,a_3,\ldots]$, $v_L$ is a proper suffix of $\psi_k(x)$ and
$v_R$ is a proper prefix of $\psi_k(y)$ for some $x,y\in\{0,1\}$, and
\begin{enumerate}
\item
$v = v_L\psi_b(v')v_R$,
\item
$|v|_{\text{pal}} \leq 4 + |v'|_{\text{pal}}$.
\end{enumerate}
\end{lem}
\begin{proof}
i) Let $\ensuremath{\boldsymbol{u}}$ be a Sturmian word with slope $\alpha=[0,a_1,a_2,a_3,\ldots]$. By Lemma~\ref{lem:cfe_of_image},
$\psi_k(\ensuremath{\boldsymbol{u}})$ has slope $\beta=[0,b,a_1,a_2,a_3,\ldots]$. Recall that the language of a Sturmian word is
entirely determined by its slope, thus we have $v\in\mathcal{L}(\psi_b(\ensuremath{\boldsymbol{u}}))$. Since by assumption
$v$ contains at least two ones, we can unambiguously write it in the required form.
ii) This statement then follows from inequalities
$|v|_{\text{pal}} \leq |v_L|_{\text{pal}} + |\psi_b(v')|_{\text{pal}} + |v_R|_{\text{pal}}$, $|v_L|_{\text{pal}}\leq 1$,
$|v_R|_{\text{pal}}\leq 2$ and from Lemma~\ref{lem:palllen_of_image}.
\end{proof}
\section{Proofs of main Theorems}
Both proofs make use of the following idea. Let $\ensuremath{\boldsymbol{u}}$ be a Sturmian word with slope
$\alpha=[0,a_1,a_2,a_3,\ldots]$. Let $v=v^{(1)}\in\mathcal{L}(\ensuremath{\boldsymbol{u}})$. By successive application of
Lemma~\ref{lem:preimage_and_its_pallen} we find words $v^{(2)}, v^{(3)}, \ldots, v^{(j+1)}$ such that for
every $i=1,2,\ldots,j$ we have
\begin{enumerate}
\item
$v^{(i)}$ is a factor of a Sturmian word with slope $[0,a_i,a_{i+1},a_{i+2},\ldots]$,
\item
$|v^{(i)}|\geq |\psi_{a_i}(v^{(i+1)})|\geq a_i|v^{(i+1)}|$ \
(this follows from Lemmas~\ref{lem:preimage_and_its_pallen} and~\ref{lem:len_of_image}),
\item
$|v^{(i)}|_{\text{pal}} \leq 4 + |v^{(i+1)}|_{\text{pal}}$,
\item
$v^{(j+1)}$ does not contain two ones, in particular $|v^{(j+1)}|_{\text{pal}}\leq 2$ and
$|v^{(j+1)}|\geq 1$.
\end{enumerate}
Altogether we have
\begin{equation}\label{eq:pallen_from_iterations}
\begin{split}
|v| = |v^{(1)}| &\geq a_1a_2\cdots a_j, \\
|v|_{\text{pal}} &\leq 4j + 2.
\end{split}
\end{equation}
\begin{proof}[Proof of Theorem~\ref{thm:libovolne_pomaly_rust}]
Let $f:\mathbb{N}\rightarrow\mathbb{R}$ be a non-decreasing function with $\lim_{n\to\infty}f(n)=+\infty$. We find
$a_1\in\mathbb{N}$, $a_1\geq 2$ such that $f(a_1)\geq 1$, then $a_2\in\mathbb{N}$, $a_2\geq 2$ such that $f(a_1a_2)\geq 2^2$, and
so on, i.e., we proceed recurrently to find $a_k\in\mathbb{N}$, $a_k\geq 2$ such that
\begin{equation}\label{eq:f(a_1...a_k)}
f(a_1a_2\cdots a_k)\geq k^2\quad \text{ for all $k\in\mathbb{N}$, $k\geq 1$}.
\end{equation}
Using~(\ref{eq:pallen_from_iterations}),~(\ref{eq:f(a_1...a_k)}) and monotony of $f$ we can estimate
\[
\frac{|v|_{\text{pal}}}{f(|v|)} \leq \frac{4j+2}{f(a_1a_2\cdots a_j)} \leq \frac{4j+2}{j^2}.
\]
Obviously $j\to\infty$ as $|v|=n\to\infty$ and therefore
\[
\limsup_{n\to\infty} \frac{\PL{u}(n)}{f(n)} \leq \lim_{j\to\infty}\frac{4j+2}{j^2} = 0.
\qedhere
\]
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:rust_pomalejsi_nez_ln}]
The estimate $|v|\geq a_1a_2\cdots a_j$ is weak in the case where most of the coefficients of the
continued fraction are equal to 1. Therefore, we use the fact that $v^{(i)}$ contains factor
$(\psi_{a_i}\circ\psi_{a_{i+1}})(v^{(i+2)})$. By Lemma~\ref{lem:len_of_image} we have $|v^{(i)}|\geq 2|v^{(i+2)}|$
and thus $|v|\geq 2^{\lfloor\frac{j}{2}\rfloor}$. Using this estimate we get
\[
\frac{|v|_{\text{pal}}}{\ln |v|} \leq \frac{4j+2}{\frac{j-1}{2}\ln 2} \xrightarrow{\ j\to\infty\ }\frac{8}{\ln 2}.
\]
Statement of the theorem follows, using $K=\frac{8}{\ln2}$.
\end{proof}
\begin{rem}\label{rem:frid_lower_bound_conjecture}
In~\cite{frid-numeration-2018}, Frid defined the sequence $(w^{(n)})$ of prefixes of the Fibonacci word
$\boldsymbol{f}$, where $|w^{(n)}|$ has representation $(100)^{2n-1}101$ in the Ostrowski numeration system.
Using the Fibonacci sequence $(F_n)_{n\geq 0}$ (given by $F_0=1$, $F_2=2$ and $F_{n+2}=F_{n+1}+F_n$ for $n\in\mathbb{N}$)
one gets $|w^{(n)}|=F_0+F_2+\sum_{k=1}^{2n-1}F_{3k+2} < F_{6n}$. Frid proved that $|w^{(n)}|_{\text{pal}}\leq 2n+1$,
while she conjectured that the equality $|w^{(n)}|_{\text{pal}}= 2n+1$ holds. Since
$F_{n} = \tfrac{1}{\sqrt{5}}\tau^{n+2}(1+o(1))$, the validity of Frid's conjecture would imply
\begin{equation}\label{eq:frid_conjecture_rephrased}
\frac{|w^{(n)}|_{\text{pal}}}{\ln|w^{(n)}|} \geq \frac{2n+1}{\ln F_{6n}} = \frac{2n+1}{(6n+2)\ln\tau(1+o(1))}
\xrightarrow{\;n\to\infty\;}\frac{1}{3\ln\tau}
\end{equation}
as stated in Conjecture~\ref{conj:frid_lower_bound_conjecture}.
In her proof of the fact that for a Sturmian word $\ensuremath{\boldsymbol{u}}$ the function $\PL{u}(n)$ is not bounded,
Frid considered only prefixes of $\ensuremath{\boldsymbol{u}}$. This was made possible by the following result by
Saarela~\cite{saarela-words-2017}: for a factor $x$ of a word $y$ we have $|x|_{\text{pal}}\leq 2|y|_{\text{pal}}$.
Computer experiments do indicate that the prefixes $w^{(n)}$ have the highest possible ratio
$\frac{|w|_{\text{pal}}}{\ln|w|}$ (among all prefixes of $\boldsymbol{f}$). However, it is still possible that
there is a sequence of factors
(not prefixes) of $\boldsymbol{f}$ which can be used to enlarge the constant $\frac{1}{3\ln\tau}$
in~(\ref{eq:frid_conjecture_rephrased}).
\end{rem}
\end{document}
|
\begin{document}
\title{Using the existence of t-designs to prove Erd\H{o}
\begin{abstract}
In 1984, Wilson proved the Erd\H{o}s-Ko-Rado theorem for $t$-intersecting families of $k$-subsets
of an $n$-set: he showed that if $n\ge(t+1)(k-t+1)$ and $\cF$ is a family of $k$-subsets
of an $n$-set such that any two members of $\cF$ have at least $t$ elements in common, then
$|\cF|\le\binom{n-t}{k-t}$. His proof made essential use of a matrix whose origin is not obvious.
In this paper we show that this matrix can be derived, in a sense, as a projection of
$t$-$(n,k,1)$ design.
\end{abstract}
\section{Introduction}\label{sec:intro}
A family of sets is $t$-intersecting if every two sets in the family have at least $t$
elements in common. The Erd\H{o}s-Ko-Rado theorem states that if $\cF$ a $t$-intersecting family of
sets of size $k$ chosen from a set $N$ of size $n$ and $n\ge (t+1)(k-t+1)$, then
\[
|\cF| \ge\binom{n-t}{k-t}.
\]
If $n>(t+1)(k-t+1)$, equality holds if and only if $\cF$ consists of the $k$-subsets that contain a given
set of $t$ points from $V$. The lower bound on $n$ is necessary, because the result is false
when the bound fails. Subsequently Ahlswede and Khachatrian \cite{AhlKha96, AhlKha97} determined the maximal families
for all $n$. The result as just stated was proved by Wilson in 1984 \cite{Wilson1984}.
The goal of this paper is to motivate a key step in Wilson's proof. He introduces a ``magic matrix''
with rows and columns indexed by the $k$-subsets of a $v$-set; he then determines the eigenvalues
of this matrix and, given these, fairly standard machinery then leads to the proof of the EKR-bound.
From private discussions with Rick Wilson, it is clear that this matrix was the result of
a lot of calculation and a lot of inspiration. Our aim in this paper is to present a derivation
which requires less effort and less brilliance. To this end, we give a simpler formulation of this matrix and
show that it is equivalent to that of Wilson, using the recent proof of the existence of $t$-designs of Keevash \cite{ Kee14}.
\section{The Johnson Scheme}
Assume $N=\{1,\ldots,n\}$. The \textsl{Johnson scheme} $J(n,k)$ is a set of $01$-matrices
$\seq A0k$, with rows and columns indexed by the $k$-subsets of $N$, where $(A_r)_{\al,\be}=1$
if $|\al\cap\be|=k-r$ for $r = 0,\ldots,k$. We see that $A_0=I$. The matrices $\seq A1k$ are adjacency matrices
of graphs $\seq X1k$, where $X_1$ is the so-called \textsl{Johnson graph}. It can be shown that
two $k$-subsets are adjacent in $X_r$ if and only if they are at distance $k-r$ in the Johnson
graph. The Johnson scheme is discussed in detail in \cite[Chapter~6]{cg-km}, and anything we
state here without proof is treated there.
The matrices $A_r$ satisfy
\[
\sum_r A_r = J.
\]
Further, there are scalars $p_{i,j}(r)$ such that, for all $i$ and $j$,
\[
A_iA_j = \sum_r p_{i,j}(r) A_r.
\]
Since the product of two symmetric matrices is symmetric if and only if the matrices commute, it
follows that
the space of the matrices $A_r$ is a commutative matrix algebra. (To use the standard jargon,
the matrices $\seq A0k$ form a \textsl{symmetric association scheme}, and their span is known
as the \textsl{Bose-Mesner algebra} of the scheme.) All matrices that occur in Wilson's proof
lie in the Bose-Mesner algebra of the Johnson scheme.
To define his matrix, Wilson used another basis for the Bose-Mesner algebra of the Johnson scheme.
Let $W_{i,j}(n)$ denote the matrix with rows indexed by the $i$-subsets of $N$, columns
indexed by the $j$-subsets of $N$ and with $(\al,\be)$-entry equal to 1 if $\al\sbs\be$.
(So each row of $W_{i,j}(n)$ sums to $\binom{n-i}{j-i}$.)
Let $\bW_{i,j}(n)$ denote the matrix with rows indexed by the $i$-subsets of $N$, columns
indexed by the $j$-subsets of $N$ and with $(\al,\be)$-entry equal to 1 if $\al\cap\be=\emptyset$.
Now define matrices $\seq D0k$ by
\[
D_i = W_{i,k} \comp{W}_{i,k}^T
\]
(For details concerning these matrices see Wilson's paper \cite{}, or \cite[Section 6.4]{cg-km}.
Despite appeafances, these matrices are symmetric.)
The matrix $\Omega(n,k,t)$ is given by
\[
\Omega(n,k,t) = \sum_{i=0}^{k-i} (-1)^{t-1-i}
\frac{\binom{k-1-i}{k -t}}{\binom{n-k-t+1}{k-t}} D_{k-i}.
\]
The matrices $I+\Om(n,k,t)$ form the key to Wilson's proof of the EKR theorem. We define
\[
\nw(n,k,t) = I + \Om(n,k,t)
\]
and abbreviate $\nw(n,k,t)$ to $\nw$ where possible.
We use $M\circ N$ to denote the \textsl{Schur product} of two matrices of the same order, thus
\[
(M\circ N)_{i,j} = M_{i,j}N_{i,j}.
\]
Since the set $\{0,\seq A0k\}$ is closed inder the Schur product, it follows that the Bose-Mesner
algebra of the Johnson scheme is closed under Schur product.
The pertinent properties of $\nw$ are summarized in the following:
\begin{theorem}
The matrix $\nw(n,k,t)$ is positive semidefinite and lies in the span of the matrices
$\seq A{k-t+1}{k}$.\qed
\end{theorem}
Wilson's proof that $\nw$ is positive semidefinite is highly non-trivial; it is presented
at somewhat greater length, but with no essential improvement, in \cite[Chapter~8]{cg-km}.
\section{Projections on to matrix algebras}
We use $\elsm(M)$ to denote the sum of the entries of a matrix $M$. We note that
\[
\tr(M^TN) = \elsm(M\circ N)
\]
and so we have two expressions for the standard inner product on real matrices:
\[
\ip MN = \tr(M^TN) = \elsm(M\circ N).
\]
Relatve to this inner product, the Schur idempotents $\seq A0k$ form an orthogonal basis for the Bose-Mesner algebra. We also observe that
\[
\tr(M) = \ip{I}M,\quad \elsm(M) = \ip{J}M.
\]
We state a version of a result known as the clique-coclique bound. It is proved, for general
association schemes, as Lemma~3.8.1 in \cite{cg-km}.
\begin{lemma}
\label{lem:clqcoclq}
Assume $v=\binom{n}{k}$. If $M$ and $N$ are matrices in the Bose-Mesner algebra of the Johnson
scheme and
\begin{enumerate}[(a)]
\item
$M$ and $N$ are positive semidefinite, and
\item
for some constant $\ga$ we have $M\circ N=\ga I$,
\end{enumerate}
then
\[
\frac{\elsm(M)}{\tr(M)} \frac{\elsm(N)}{\tr(N)} \le v.\qed
\]
\end{lemma}
\begin{lemma}
The orthogonal projection of a positive semidefinite matrix onto a transpose-closed
real matrix algebra is positive semidefinite.
\end{lemma}
\noindent{{\sl Proof. }}
This is a special case of Tomiyama's theorem, see \cite{Tom59}.\qed
Given an orthogonal basis for the Bose-Mesner algebra, we can compute orthogonal projections of matrices onto it---if $M$ is an $\binom{n}{k}\times\binom{n}{k}$ matrix, its orthogonal projection $\Psi(M)$ is
given by Gram-Schmidt:
\[
\Psi(M) = \sum_i \frac{\ip{M}{A_i}}{\ip{A_i}{A_i}}A_i.
\]
Note that
\[
\ip{M-\Psi(M)}{A} = 0
\]
for any matrix $A$ in the Bose-Mesner algebra, and taking $A$ to be $J$ and $I$ in turn
yields that
\[
\elsm{\Psi(M)} = \tr(M),\quad \tr(\Psi(M)) = \tr(M).
\]
We consider an example. For any family $\cF$ of $k$-subsets of $N$, we denote by $N_{\cF}$ the
matrix $xx^T$ where $x$ is the characteristic vector of $\cF$. Let $\cF$ be a $t$-intersecting family of $k$-subsets.
Then
\[
\ip{A_r}{N_\cF} = \tr(AN_\cF) = x^TA_rx,
\]
which equals the number of pairs $(a\,\be)$ in $\cF\times\cF$ such that $|\al\cap\be|=k-r$.
Therefore $\ip{A_r}{N_\cF}=0$ if $r\ge k-t+1$.
\begin{lemma}
Let $\cF$ be a $t$-intersecting family. Then $\Psi(N_\cF)$ is a positive semidefinite matrix
lying in the span of $\seq A0{k-t}$ and
\[
\tr(\Psi(N_\cF)) = |\cF|, \quad \elsm(\Psi(N_\cF)) = |\cF|^2.
\]
\end{lemma}
\noindent{{\sl Proof. }}
Observe that if $A$ lies in the Bose-Mesner algebra of the Johnson scheme, then
\[
0 = \ip{M-\Psi(M)}{A} = \ip{M}{A} -\ip{\Psi(M)}{A},
\]
whence $\ip{\Psi(M)}{A}=\ip{M}{A}$. Therefore $\ip{\Psi(M)}{A_r}=\ip{M}{A_r}$, which proves
that $\Psi(N_\cF)$ lies in the span of $\seq A0{k-t}$. The remaining two claims follow
from the fact that $\Psi$ preserves trace.\qed
If we can show that the Bose-Mesner algebra of $J(n,k)$ contains a matrix $L$ such that:
\begin{enumerate}[(a)]
\item
$L$ is positive semidefinite,
\item
$\Psi(N_\cF)\circ L = \ga I$ for some $\ga$,
\item
$\elsm(L)/\tr(L)=\binom{n}{t}/\binom{n-t}{k-t}$,
\end{enumerate}
then Lemma~\ref{lem:clqcoclq} implies that
\[
|\cF| \le {\binom{n-t}{k-t}}.
\]
The key to Wilson's proof was to demonstrate that, provided
\[
n\le(t+1)(k-t+1),
\]
the matrix $I+\Om(n,k,t)$ satisfies these conditions.
Recall that a $t$-$(n,k,\la)$-design is a collection of subsets of size $k$ from an $n$-set such
that any any subset of $t$ points from $V$ lies in exactly $\la$ blocks (aka $k$-sets).
If $\la=1$, we call the design a \textsl{Steiner system}. The construction of
Steiner systems for large $t$ is something of a mystery (to which we shall return),
but projective and affine planes of finite order provide examples with $t=2$ and
M\"obius planes give examples with $t=3$.
\begin{lemma}
Let $\cD$ be a $t$-$(n,k,1)$ design. Then $\Psi(N_\cD)$ is a positive semidefinite
matrix lying in the span of $\seq A{k-t+1}{t}$ and
\[
\tr(\Psi(N_\cD)) = |\cD|,\quad \elsm(\Psi(N_\cD)) = |\cD|.
\]
\end{lemma}
\begin{lemma}
If a $t$-$(n,k,1)$-design exists, then a $t$-intersecting family
of $k$-subsets of a set of size $v$ has size at most $\binom{n-t}{k-t}$.\qed
\end{lemma}
We can compute $\Psi(N_\cD)$ explicitly. If $\la_i$ denotes the number of blocks of $\cD$
that contain a given set of $i$ points and $0\le i\le t$, then
\[
\la_i = \frac{\binom{n-i}{k-i}}{\binom{n-t}{k-t}}.
\]
If we define
\[
\ga_s = \sum_{i=s}^t (-1)^{i-s}\binom{i}{s}\binom{k}{i}(\la_i-1),
\]
then from Exercise~1 in Chapter 8 of \cite{cg-km}, we find that
\[
\Psi(N_\cD) = \sum_{s=0}^t \frac{\ga_s}{\binom{n-k}{k-s}\binom{k}{s}} A_{k-s}.
\]
For $n,k,t$, we will denote by $M(n,k,t)$ the following:
\[
M(n,k,t) = \sum_{s=0}^t \frac{\ga_s}{\binom{n-k}{k-s}\binom{k}{s}} A_{k-s}.
\]
If there exists a $t$-$(n,k,1)$-design $\cD$ exists, then $\Psi(N_\cD) = M(n,k,t) $.
Observe that the matrix $M(n,k,t)$ is always well-defined, whether or not the design exists.
We use Keevash's result \cite{Kee14} on the existence of $t$-designs to show that this
projection is equal to Wilson's matrix. For a second proof of Keevash's result, see \cite{GloKuhLoOs16}.
The following theorem is a restatement of Theorem 1.4 of \cite{Kee14} applied to $G=K_n^t$,
in the language of block designs instead of hypergraphs.
\begin{theorem}[Keevash]
\label{thm:keevash}
For fixed $k$ and $t$, there exists $N$ such that for $n > N$, if $\binom{k-i}{t-i}$
divides $\binom{n-i}{t-i}$ for $i = 0,\ldots,t-1$, then there exists a $t$-$(n,k,1)$
block design.
\end{theorem}
We are now able to prove the following.
\begin{theorem}
For any $n\geq k \geq t$, we have that $M(n,k,t) = \Omega(n,k,t) + I$.
\end{theorem}
\noindent{{\sl Proof. }}
Fix $k$ and $t$. Let \[f_r(n) = \theta_r(M(n,k,t))\] and \[g_r(n) = \theta_r(\Omega(n,k,t)) + 1.\]
If there exists a $t$-$(n,k,1)$-design $\cD$ exists, then $f_r(n) = g_r(n)$
for $r = 0,\ldots, t$. By Theorem~\ref{thm:keevash}, we have that $f_r(n)$ and $g_r(n)$ are equal
for infinitely many $n$. Consider $h_r(n) = f_r(n) - g_r(n)$. We see that $h_r(n)$ is a rational
function whose numerator $p(n)$ is a polynomial. Since $p(n) =0 $ infinitely often, we have that $p(n) = 0$
and so $h_r(n) =0$. We thus have that $f_r(n)= g_r(n)$ for all $n$. This shows that
$M(n,k,t) = \Omega(n,k,t) + I$ for all $n$.\qed
\end{document}
|
\begin{document}
\mathop{\mathrm{maj}}aketitle
\tableofcontents
\footnote{This work was supported by a scholarship from the NSERC.}
\begin{abstract}
This paper contains a partial answer to the open problem 3.11 of \cite{[H2008]}. That is to find an explicit bijection on Schr\"oder paths that inverts the statistics area and bounce. This paper started as an attempt to write the sum over $m$-Schr\"oder paths with a fix number of diagonal steps into Schur functions in the variables $q$ and $t$. Some results have been generalized to parking functions, and some bijections were made with standard Young tableaux giving way to partial combinatorial formulas in the basis $s_\mathop{\mathrm{maj}}u(q,t)s_\lambda(X)$ for $\nabla(e_n)$ (respectively, $\nabla^m(e_n)$), when $\mathop{\mathrm{maj}}u$ and $\lambda$ are hooks (respectively, $\mathop{\mathrm{maj}}u$ is of length one). We also give an explicit algorithm that gives all the Schr\"oder paths related to a Schur function $s_\mathop{\mathrm{maj}}u(q,t)$ when $\mathop{\mathrm{maj}}u$ is of length one. In a sense, it is a partial decomposition of Schr\"oder paths into crystals.
\end{abstract}
\section{Introduction}
In this paper, Proposition~\ref{Prop : prob 3.1} gives a partial answer to the open problem 3.11 of \cite{[H2008]}. The problem asks for an explicit bijection on Schr\"oder paths that inverts the statistic area and bounce. But the aim of this paper is to decompose parking functions and Schr\"oder paths in terms of the basis $s_\mathop{\mathrm{maj}}u(q,t)s_\lambda(X)$. It is then used in \cite{[Wal2019a]} to give explicit combinatorial formulas for the modules of multivariate diagonal harmonics. In other words, the combinatorics of parking functions are used to elevate the understanding of the structure of the modules of diagonal harmonics. This combinatorial representation was first known as the Shuffle Conjecture. It was introduced in \cite{[HHLRU2005]} and proven by Carlson and Mellit in \cite{[CM2015]}, \cite{[M2016]}. It was shown beforehand in \cite{[GH1996a]} and \cite{[H2002]} that the Frobenius transformation of its graded characters may be expressed as $\nabla^m(e_n)$, where $\nabla$ is the Macdonald eigenoperator introduced in \cite{[BG1999]}, and $e_n$ is the $n$-th elementary symmetric function, both recalled in Section~\ref{Sec : tools}, along with classical combinatorial tools.
More precisely, we will give a partial decomposition of parking functions and Schr\"oder paths in terms of the basis $s_\mathop{\mathrm{maj}}u(q,t)s_\lambda(X)$. By proving the following:
\begin{theo}\label{The : main} If $\mathop{\mathrm{maj}}u\in\{(d,1^{n-d})~|~1\leq d\leq n\}$ and $\nu\vdash n$, then:
\begin{equation}\label{Eq : 1 de the principale} \langle \nabla(e_n), s_{\mathop{\mathrm{maj}}u}\rangle|_{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{ht}}ooks}=\sum_{\tau\in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{SYT}}(\mathop{\mathrm{maj}}u)} s_{\mathop{\mathrm{maj}}(\tau)}(q,t)+\sum_{i=2}^{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(\tau)} s_{\mathop{\mathrm{maj}}(\tau)-i,1}(q,t),
\end{equation}
\begin{equation}\label{Eq : 2 de the principale}\langle \nabla^m(e_n), s_\nu \rangle|_{1 \mathop{\mathrm{Part}}}=\sum_{\tau\in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{SYT}}(\nu)} s_{m\binom{n}{2}-\mathop{\mathrm{maj}}(\tau')}(q,0)=\sum_{\tau\in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{SYT}}(\nu)} s_{m\binom{n}{2}-\mathop{\mathrm{maj}}(\tau')}(q,0),
\end{equation}
and:
\begin{equation}\label{Eq : 3 de the principale}\langle \nabla^m(e_n), e_n\rangle|_{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{ht}}ooks}= s_{m\binom{n}{2}}(q,t)+\sum_{i=2}^{n-1} s_{m\binom{n}{2}-i,1}(q,t).
\end{equation}
\end{theo}
This will be done by characterizing particular parking functions, in Section~\ref{Sec : parking}, which leads to Equation\eqref{Eq : 2 de the principale}. In Section~\ref{Sec : schroder}, we restrict the characterization on Schr\"oder paths. In Section~\ref{Sec : bijections}, we give bijections between subsets of Schr\"oder paths and Standard Young tableaux, and use them to prove Equation~\eqref{Eq : 1 de the principale} and Equation~\eqref{Eq : 3 de the principale}.
Moreover, in Section~\ref{Sec : algo}, we exhibit an explicit algorithm that gives all the Schr\"oder paths associated to a Schur function in the variables $q$ and $t$ when $\mathop{\mathrm{maj}}u$ is of length one. We will briefly explain, in Section~\ref{Sec : cristaux}, what it means in terms of Crystal decomposition. We end with a list of problems to solve in Section~\ref{Sec : conclu}. Finally, in Section~\ref{Sec : combi chemins}, we will recall notions on path combinatorics.
\section{Combinatorial Tools}\label{Sec : tools}
The notions discussed in this section are classical and are recalled to set notations.
An alphabet, $A$, is a set. The elements of that set are called \begin{bf}letters\end{bf}. A \begin{bf}word\end{bf} is a finite sequence of elements of $A$, we usually omit the parentheses and the commas. The empty word is denoted $\varepsilon$. The number of letters in a word $w$ is called the \begin{bf}length\end{bf}, denoted $|w|$, the number of occurrences of the letter $a$ in $w$ is denoted $|w|_a$. The set of words of length $n$ in the alphabet $A$ is denoted $A^n$, we denote $A^*$ the set $\cup_n A^n$. A \begin{bf}factor\end{bf} of $w$ is a consecutive subsequence of $w$. Additionally, if we are interested in word ending with a fixed factor $u$, we will denote the set $A^*u$, and $u$ is called a suffix. If we want those words to be of length, $n+|u|$ we will denote the set $A^nu$. Likewise, a factor at the beginning of a word is called a prefix, and the set of words with prefix $u$ is denoted $uA^*$. For a word $w=w_1w_2\cdots w_k$, $w^n$ is the concatenation on $m$ copies of $w$, and $w^{-1}=w_k\cdots w_2 w_1$. For two words $u$ and $w$ the set $u\shuffle w$ is the set containing all words such that $u$ and $w$ are subsequences. We call these words \begin{bf}shuffles\end{bf}.
A \begin{bf}permutation\end{bf} of $n$ can be represented as words of $\{1,\ldots,n\}^n$ with all distinct letters. The \begin{bf}descent set of a permutation $w=w_1\cdots w_n$, \end{bf} denoted $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(w)$, is the set of $i$'s such that $w_i>w_{i+1}$. The cardinality of the set will be denoted $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(w)$. The \begin{bf}major index of a permutation\end{bf}, denoted $\mathop{\mathrm{maj}}(w)$, is by definition $\mathop{\mathrm{maj}}(w)=\sum_{i\in\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(w)} i$. To avoid confusion we will write the \begin{bf}inverse of a permutation\end{bf} $w$, $\textrm{inv}(w)$.
A \begin{bf}partition\end{bf} of $n$ is a decreasing sequence of positive integers it can be represented by a Ferrers diagram (see Figure~\ref{Fig: diagram}). Each number in the sequence is called a \begin{bf}part\end{bf}, and, if it has $k$ parts, it is of \begin{bf}length\end{bf} $k$ denoted $\ell(\lambda)=k$. If $\lambda=\lambda_1,\cdots,\lambda_k$ and $n=\sum_i\lambda_i$, we say $\lambda$ is of \begin{bf}size\end{bf} $n$, denoted $|\lambda|=n$. Although the notation $|\cdot|$ is used for words and partitions, it will be clear by context which one is used. For $\lambda$ a \begin{bf}Ferrers diagram \end{bf} of shape $\lambda=\lambda_1,\lambda_2,\cdots,\lambda_k$ is a left justified pile of boxes having $\lambda_i$ boxes in the $i$-th row. We will use the French notation; hence, the second row lies on top of the first row (see Figure~\ref{Fig: diagram}). We can see them as a subset of $\mathop{\mathrm{maj}}athbb{N}\times\mathop{\mathrm{maj}}athbb{N}$ if we put the bottom left corner of the diagram to the origin. In this setting, we can associate the bottom left corner of a box to the coordinate it lies on. We say a partition is \begin{bf}hook-shaped\end{bf} if it has the shape $(a,1, \cdots,1)=(a,1^k)$, where $a, k \in \mathop{\mathrm{maj}}athbb{N}$. The \begin{bf}conjugate \end{bf} of a partition $\lambda$, (or a diagram) is denoted $\lambda'$, and is its reflection through the line $x=y$ (see Figures~\ref{Fig: conjugate}).
\begin{figure}
\caption{
$\lambda=42211$}
\label{Fig: diagram}
\caption{
$\lambda'=5311$}
\label{Fig: conjugate}
\caption{$\tau \in \mathop{\mathrm{maj}
\label{Fig : tableau}
\end{figure}
A \begin{bf}tableau\end{bf} is a filling of a diagram by positive integers, the number in each box is called an \begin{bf}entry\end{bf}. The \begin{bf}size\end{bf} of a tableau relates to the size of the diagram it fills. It is said to be a \begin{bf}semi-standard Young tableau\end{bf} if all entries are weakly increasing in rows and strictly increasing in columns. A \begin{bf}standard Young tableau\end{bf} is a tableaux of size $n$, such that all numbers from $1$ to $n$ appear exactly once and all entries are strictly increasing in rows and columns. If a tableau is a filling of the diagram associated to the partition $\lambda$, it is said to be of \begin{bf}shape\end{bf} $\lambda$. The set of standard Young tableaux of shape $\lambda$ is denoted $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{SYT}}(\lambda)$. The \begin{bf}descent set of a tableau\end{bf} $\tau$, denoted $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(\tau)$ is the set of entries $i$ such that $i+1$ lies in a higher row. The cardinality of the descent set of $\tau$ is denoted $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(\tau)$, and the sum of the elements in the descent set is the \begin{bf}major index\end{bf} denoted $\mathop{\mathrm{maj}}(\tau)$ (see Figure~\ref{Fig : tableau}). Again, it will be clear by context if the descent set and the major index are used on words or tableaux. Since each box of $\tau$ is associated to its own entry, we will write $c\in \tau$ when we refer to the entry $c$ in the tableau $\tau$. We will use the notation $x_\tau$ for the monomial $\prod_{c\in\tau}x_c$.
For a possibly infinite set of variables, $X=\{x_1,\ldots,x_n\}$, the \begin{bf}elementary symmetric functions\end{bf} $e_n(X)$ are the sum of all square-free monomials of degree $n$ in the set of variables $X$. The symmetric function $e_\lambda$ is simply $e_{\lambda_1}e_{\lambda_2}\cdots e_{\lambda_{\ell(\lambda)}}$. The elementary symmetric functions form a basis of the symmetric functions. Another basis is the set of Schur functions. For $\lambda$ a partition the \begin{bf}Schur function\end{bf} $s_\lambda(X)=\sum x_\tau$, where the sum is over all semi-standard Young tableau of shape $\lambda$. The Schur basis in the $X$ variables is self-dual for the Hall scalar product, denoted $\langle \--,\-- \rangle$. We will use this notation when we want to display the coefficient of a particular Schur function. Note that the Schur functions in the variables $q$ and $t$ are coefficients and can go in and out of the scalar product. We will sometimes call Schur functions index by partitions that are hook-shaped, hook-shaped Schur functions or, simply, hook Schur functions. It will also be useful to remember that $e_n=s_{1^n}$. Furthermore, the \begin{bf}complete homogeneous symmetric functions\end{bf} are a basis such that $h_n(X)=s_{(n)}(X)$ and $h_\lambda:=h_{\lambda_1}h_{\lambda_2}\cdots h_{\lambda_{\ell(\lambda)}}$. We simply write $e_\lambda$ for $e_n(X)$, $s_\lambda$ for $s_\lambda(X)$ and $h_\lambda$ for $h_\lambda(X)$ (not for $e_\lambda(q,t)$, $h_\lambda(q,t)$ or $s\lambda(q,t)$). A curious reader could look at \cite{[Mac1995]}.
The modified Macdonald polynomials $\tilde{H}_\mathop{\mathrm{maj}}u(X;q,t)$ form another base of the ring of symmetric functions. In \cite{[BG1999]} Bergeron and Garsia introduce the operator $\nabla$ defined with the modified Macdonald polynomials as eigenfunctions, with eigenvalues $\prod_{(i,j)\in\mathop{\mathrm{maj}}u}q^it^ j$. The Shuffle Theorem, proven by Carlson and Mellit (see \cite{[CM2015]} and \cite{[M2016]}), gives a combinatorial formula for $\nabla^m(e_n)$. This formula uses path combinatorics.
\section{Path Combinatorics}\label{Sec : combi chemins}
Before we can state the Shuffle Theorem, we need more classical definitions, relating to path combinatorics. More details on these classical notions can be found in \cite{[H2008]}.
The following $q$-analogues will be very useful:
\begin{equation*}
[n]_q:=1+q+q^2+\cdots+q^{n-1},
[n]!_q:=\prod{i=1}^n [i]_q, \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{ht}}space{20pt} \text{and}\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{ht}}space{20pt} \begin{bmatrix}n\\k\end{bmatrix}_q:=\frac{[n]!_q}{[n-k]!_q[k]!_q}.
\end{equation*}
Let $\mathop{\mathrm{maj}}athcal{C}^n_k$ be the set of paths composed of north and east steps, in an $(n-k)\times k$ grid starting at the bottom left corner. The \begin{bf}area\end{bf} of a path is the number of boxes under the path. A classical result relates the Gaussian Polynomials to path combinatorics. Indeed, $\begin{bmatrix}n\\k\end{bmatrix}_q=\sum_{\gamma\in \mathop{\mathrm{maj}}athcal{C}^n_k}q^{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}(\gamma)}$. A path $\gamma \in \mathop{\mathrm{maj}}athcal{C}_k^n$ can be identified with a word, $w_\gamma$ in $\{N,E\}^n$ such that $|w_\gamma|_E=k$. To facilitate reading we will frequently refer to $\gamma$ when we talk about $w_\gamma$ (see Figure~\ref{Fig : aire} for an example).
\begin{figure}
\caption{Path of area $6$ in a $5\times 4$ grid, with word representation $EENENNNNE$}
\label{Fig : aire}
\end{figure}
In an $n\times m$ grid, \begin{bf}the main diagonal\end{bf} is the diagonal starting at the bottom left corner and finishing at the top right corner. A \begin{bf}Dyck path\end{bf} of size $(n,m)$ is a path composed of north and east steps, starting at the bottom left corner in an $n\times m$ grid such that the path always stays over the main diagonal. The set of such paths is denoted $\mathop{\mathrm{maj}}athcal{D}_{n,m}$. A classical result makes it possible to represent Dyck paths by words in $\{N,E\}^*$ such that for all $\gamma_i$, prefix of $\gamma$, we have $|\gamma_i|_N\geq \frac{m}{n} |\gamma_i|_E$. The lines of the grid are numbered from bottom to top. A line $i$ is said to \begin{bf}contain an east step\end{bf} if the factor starting with the $i$-th north step and ending the letter before the $i+1$-th north step contains an east step. A \begin{bf}column of a path\end{bf}, $\gamma$, is a factor $N^jE^k$ such that $\gamma=uN^jE^kw$, where $u,w \in N\{N,E\}E\cup\{\varepsilon\}$. For example, in Figure~\ref{Fig : aire cat}, the path has 3 columns.
The \begin{bf}area of a Dyck path\end{bf} will be the number of boxes under the path and over the main diagonal (see Figure~\ref{Fig : aire cat} for an example). The area of the line $i$, denoted $a_i$, is the number of boxes in the line $i$ that are between the path and the main diagonal. Obviously, the area of a path is the sum of the $a_i$'s. The path $\gamma$ is said to have a \begin{bf}return to the main diagonal\end{bf} if there is $\gamma_i$, a non-trivial prefix of $\gamma$, such that $\gamma_i$ is a Dyck path and the end point of $\gamma_i$ lies on the main diagonal of $\gamma$. The \begin{bf}touch sequence of a path\end{bf} $\gamma$, denoted $\mathop{\mathrm{Touch}}(\gamma)$, is defined as a sequence $(\gamma_1,\ldots,\gamma_k)$ of factors of $\gamma$ such that $\gamma=\gamma_1\cdots\gamma_k$, all $\gamma_i$ are Dyck paths, all $\gamma_i$ contain no return to the main diagonal, and the end points of $\gamma_i$ returns to the main diagonal. Usually, the sequence $(\frac{1}{2}|\gamma_1|,\ldots,\frac{1}{2}|\gamma_k|)$ defines the touch vector $\mathop{\mathrm{touch}}(\gamma)$. The touch vector contains all the \begin{bf}touch points\end{bf}. For example, in Figure~\ref{Fig : aire cat}, $\mathop{\mathrm{Touch}}(NNENNEEENE)=(NNENNEEE,NE)$ and $\mathop{\mathrm{touch}}(NNENNEEENE)=(4,1)$. The \begin{bf}bounce path\end{bf} of a path $\gamma \in \mathop{\mathrm{maj}}athcal{D}_{n,n}$ will be the Dyck path that remains under the path $\gamma$ and changes direction if and only if it touches the path $\gamma$ or the main diagonal. The \begin{bf}bounce vector\end{bf} is the vector containing the positions of the return to the main diagonal, starting from the top, of the bounce path. For the bounce vector, the lines are numbered from the top starting at $0$. Finally, the \begin{bf}bounce statistic\end{bf} is the sum of the integer in the bounce vector minus $n$. It is, usually, simply called bounce (see Figure~\ref{Fig : bounce cat} for an example) . Note that the bounce statistic is not defined for Dyck paths in an $n\times m$ grid with $m\not=n$. In theses cases we use the diagonal inversion statistic which will be discussed at the end of this section.
\begin{figure}
\caption{Dyck path of area $4$, and word representation $NNENNEEENE$}
\label{Fig : aire cat}
\caption{Bounce vector is $(0,1,3,5)$ and bounce is $4$.}
\label{Fig : bounce cat}
\end{figure}
A \begin{bf}Schr\"oder path\end{bf} of size $(n,rn)$ is a path composed of north, east and diagonal steps in an $n\times rn$ grid such that the path always stay over the main diagonal starting at the bottom left corner. In respect to Cartesian coordinates, a diagonal step corresponds to adding $(1,1)$ . The set of paths containing $d$ diagonal steps is denoted $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}_{n,d}^{(r)}$. These paths can also be represented by words in the alphabet $\{N,E,D\}^*$ such that for all prefix $\gamma_i$ of $\gamma$ we have $|\gamma_i|_N\geq r|\gamma|_E$. Clearly $\mathop{\mathrm{maj}}athcal{D}_{n,rn}=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}_{n,0}^{(r)}$. Moreover, the path obtained by deleting all diagonal steps in a Schr\"oder path is a Dyck path. For a Schr\"oder path, $\pi$ this new path will be denoted $\Gamma(\pi)$. For example, the path $\pi$ in Figure ~\ref{Fig : aire sch}, is such that $\Gamma(\pi)$ is the path seen in Figure~\ref{Fig : aire cat}. We will also frequently use another subset of Schr\"oder paths:
\begin{equation*}
{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P}_{n,d}=\{ \gamma \in {\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}}_{n,d} ~|~ \gamma=wNE, w \in \{D,N,E\}^*\}={\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}}_{n,d}\cap\{D,N,E\}^*NE
\end{equation*}
The \begin{bf}area statistic of a Schr\"oder path\end{bf} is fairly the same as the other definitions of the area statistic. Instead of counting the squares, we count the number of \begin{bf}lower triangles\end{bf} under the path and over the main diagonal. Where a lower triangle is the lower half of a square cut in two starting by the botom left corner and ending at the top right corner (see Figure~\ref{Fig : aire sch} for an example).
In \cite{[H2008]}, Haglund defines a bounce statistic for Schr\"oder paths in an $n\times n$ grid. We first define the set of \begin{bf}peaks\end{bf} of the path, $\Gamma(\gamma)$. These are the set of lattice points at the beginning of an east step such that the bounce path of $\Gamma(\gamma)$ switches from a north step to an east step. By extension the peaks of $\gamma$ are the lattice points found by reinserting the diagonal steps in $\Gamma(\gamma)$. The number of peaks of the path $\gamma$, with multiplicity, that lie under each diagonal step is the statistic \begin{bf}numph\end{bf}, denoted $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{numph}}(\gamma)$. The \begin{bf}bounce statistic\end{bf} will be extended to a Schr\"oder path, $\gamma$, by the formula (see Figure~\ref{Fig : bounce sch} for an example):
\begin{equation*}
\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma)=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\Gamma(\gamma))+\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{numph}}(\gamma)
\end{equation*}
Finally, touch points can be defined for Schr\"oder path, simply change Dyck path for Schr\"oder paths in the definition.
\begin{figure}
\caption{Schr\"oder path of area $9$, and word representation $NDNENNEDDEEDNE$}
\label{Fig : aire sch}
\caption{For this path, $\gamma$, $\mathop{\mathrm{maj}
\label{Fig : bounce sch}
\end{figure}
The generating function of the Schr\"oder pathsis defined by:
\begin{equation*}
{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}}_{n,d}(q,t)=\sum_{\gamma\in {\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}}_{n,d}}q^{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma)}t^{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}(\gamma)},
\end{equation*}
and the generating function of the Schr\"oder paths ending with $NE$ is defined by:
\begin{equation*}
{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P}_{n,d}(q,t)=\sum_{\gamma\in {\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P}_{n,d}}q^{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma)}t^{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}(\gamma)}.
\end{equation*}
Since the subset $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P$ is chosen to work with the bounce statistic which is not defined for $n\times nm$ grids when $m\not=1$, we will not define $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}_{n,d}^{(m)}$. We will define ${\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P}_{n,d}^{(m)}(q,t)$ as follows:
\begin{equation*}
{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P}_{n,d}^{(m)}(q,t)=\sum_{k=d}^n (-1)^{k-d}{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P}_{n,k}^{(m)}(q,t)
\end{equation*}
The reason is du to the fact that $s_{d+1,1^{n-d-1}}=\sum_{k=d}^n (-1)^{k-d}e_{n-d}h_{d}$, which is used in Equation~\eqref{Eq : nabla schur} due to Haglund and Equation~\eqref{Eq : nabla m schur} due to Mellit.
An \begin{bf}$(n,mn)$-parking function\end{bf} is a pair consisting of and a $(n,mn)$-Dyck path and a permutation of $n$, $w$, for which we write $w_i$ on the line $i$ of the Dyck path. Moreover, all factors of $w$ in a given column of the path must contain no descents (see Figure~\ref{Fig : exemple park} and Figure~\ref{Fig : nonexemple park} for examples). The set of all $(n,mn)$-parking function is denoted $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athcal{P}}_{n,mn}$ .
\begin{figure}
\caption{Here a parking function with $w=183457692$. The factors in each column are $18$, $3$, $45$, $7$, $6$, $9$, $2$, and contain no descents.}
\label{Fig : exemple park}
\caption{This is NOT a parking function, since $w=831457692$. The factor in the first column is $83$ and has a descent.}
\label{Fig : nonexemple park}
\end{figure}
The \begin{bf}reading word\end{bf} is obtained by reading the letters of $w$ (which are written immediately to the right of each north step) in regard to the diagonals parallel to the main diagonal starting from top right corner to the bottom left corner and starting with the diagonal that is the farthest from the main diagonal. For example, the reading word in Figure~\ref{Fig : exemple park} is $675438291$. The reading word of the parking function $(\gamma,w)$ is denoted $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{read}}(\gamma,w)$.
The \begin{bf}area of a parking function\end{bf} is the area of its Dyck path. The \begin{bf}diagonal inversion statistic\end{bf}, of a parking function in $\mathop{\mathrm{maj}}athcal{P}_{n,mn}$, (sometimes called $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{dinv}}$ for short) is given by the formula $\sum_{i<j}d_i(j)$, where:
\begin{equation*}
d_i(j)=\begin{cases} \chi(w_i<w_j)\mathop{\mathrm{maj}}ax(0,r-|a_i-a_j|)+\chi(w_i>w_j)\mathop{\mathrm{maj}}ax(0,m-|a_j-a_i+1|) &\text{ if } i<j
\\ 0 &\text{ if } i\geq j.
\end{cases}
\end{equation*}
The diagonal inversion statistic of the parking function $(\gamma,w)$ is denoted $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{dinv}}(\gamma,w)$. Note that all the definition work if $w$ is not a permutation (some authors use words but these can be regrouped with permutations as representatives).
Equivalently, for a $(\gamma,w)\in\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athcal{P}}_{n,mn}$ we can consider the diagonal inversion of $(\tilde\gamma,\tilde w)$, where $\tilde\gamma$ is the $(mn,mn)$-Dyck path obtained by repeating all north steps $m$ times and for $w=w_1\cdots w_n$ we have $\tilde w=w_1^m\cdots w_n^m$ (here $\tilde w$ is not a permutation). In this case we can consider the sum $d_i(j)=\sum_{t=1}^m d_i^t(j)$, where $d_i^t(j)$ is calculated with $\tilde\gamma$, for the $t$-th copy of $w_i$.
A visual representation of the diagonal inversion statistic for $(\gamma,w)\in\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athcal{P}}_{n,n}$ is done considering one diagonal parallel to the main diagonal on each north step. For the north step on line $j$ if the diagonal crosses the north step on line $i$, with $i<j$ and $w_i<w_j$, then the pair $(i, j)$ contributes one to the diagonal inversion statistic. If the diagonal immediately over the diagonal crossing the north step on line $j$ crosses the line $i$, with $i<j$ and $w_i>w_j$, then the pair $(i, j)$ contributes one to the diagonal inversion statistic (see Figure~\ref{Fig : dinv diago} and Figure~\ref{Fig : dinv n-mn}).
\begin{figure}
\caption{The pair $(i,j)$ contributes $1$ if $i<j$ and $w_i>w_j$.}
\label{Fig : dinv diago}
\caption{The pair $(i,j)$ contributes $1$ if $i<j$ and $w_i<w_j$.}
\label{Fig : dinv n-mn}
\end{figure}
The Schr\"oder paths in an $n\times mn$ grid with $d$ diagonal steps can be represented by parking functions $(\gamma,w)$ such that $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{read}}(\gamma,w) \in \{n-d+1,\cdots,n\}\shuffle \{n-d,\cdots,1\}$. Indeed, by definition of parking functions, if $w_i$ is in $ \{n-d+1,\cdots,n\}$, then the north step at line $i$ is followed by an east step. Therefore, for all $w_i$ in $ \{n-d+1,\cdots,n\}$ one can change the factor $NE$ on line $i$ for a $D$ and unlabel the path. This procedure gives us a Schr\"oder path with $d$ diagonal steps. Conversely, all $D$ steps of a Schr\"oder path can be changed for $NE$ factors and tagged in the reading order by the letters in $ \{n-d+1,\cdots,n\}$ and all the north steps can be tagged in the reading order by letters in $\{n-d,\cdots,1\}$. This bijection will be mostly used for proofs. Hence, we will often refer to Schr\"oder paths by their parking function description.
In \cite{[TW2018]}, Thomas and Williams proved that the zeta map, denoted $\zeta$, is a bijection on rational parking functions, that preserves statistics. In the $n\times n$ case it is such that $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{dinv}}(\gamma,w)=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}(\zeta(\gamma,w))$ and $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}(\gamma,w)=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\zeta(\gamma,w))$. This will be used implicitly in the following way: if one as a decomposition of Schr\"oder paths with $d$ diagonal steps in Schur functions, in the variables $q$ and $t$, in terms of area and bounce, the decomposition in terms of diagonal inversions and area is the same.
For more on $m$-Schr\"oder paths see \cite{[H2008]} and \cite{[S2005]}.
In this paper we will give explicit decomposition in Schur function in the variables $q$ and $t$ for $\langle\nabla^m (e_n), e_{n} \rangle|_{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{ht}}ook}$, $\langle\nabla^m e_n,s_\mathop{\mathrm{maj}}u \rangle|_{1 \mathop{\mathrm{Part}}}$ and $\langle\nabla e_n,s_{d+1,1^{n-d-1}} \rangle|_{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{ht}}ook}$ by using Corollary 2.4 in \cite{[H2004]}:
\begin{theo*}[Haglund]\label{The : Hag} Let $n$, $d$ be positive integers such that $n\geq d$. Then:
\begin{equation}\label{Eq : nabla schur}{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P}_{n,d}(q,t)=\langle\nabla e_n,s_{d+1,1^{n-d-1}} \rangle,
\end{equation}
and:
\begin{equation*}{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}}_{n,d}(q,t)=\langle\nabla e_n,e_{n-d}h_d \rangle .
\end{equation*}
\end{theo*}
The following equalities will also be used and can be inferred from Mellit's proof found in \cite{[M2016]} of the compositional shuffle conjecture of \cite{[BGLX2016]}. Let $n$, $d$, $m$ be positive integer such that $n\geq d$. Then:
\begin{equation}\label{Eq : nabla m schur}{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P}_{n,d}^{(m)}(q,t)=\langle\nabla^m e_n,s_{d+1,1^{n-d-1}} \rangle
\end{equation}
and:
\begin{equation*}\nabla^m(e_n)=\sum_{(\gamma,w)\in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athcal{P}}_{n,nm}} t^{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{dinv}}(\gamma,w)}q^{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}(\gamma)}F_{\mathop{\mathrm{co}}(\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(\mathop{\mathrm{inv}}(\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{read}}(\gamma,w))))}(X),
\end{equation*}
where $F_c$ is the fundamental quasisymmetric function index by the composition $c$ and for $S$ a subset of $\{1,\ldots,n-1\}$, $\textrm{co}(S)$ is the composition associated to $S$.
We can infer the last result from \cite{[S1979]} and \cite{[H2002]}:
\begin{equation}\label{Eq : S-L,H}\nabla(e_n)|_{q=0}=\sum_{\tau\in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{SYT}}(n)} t^{\mathop{\mathrm{maj}}(\tau)} s_{\lambda(\tau)}
\end{equation}
\section{Algorithm on Schr\"oder Paths Related to Schur Functions Index by One Part Partitions}\label{Sec : algo}
It was proven in \cite{[H2002]} that $\nabla(e_n)$ is the character of the $GL_2\times \mathop{\mathrm{maj}}athbb{S}_n$-module of diagonal harmonics. Hence, the polynomials $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}_{n,d}(q,t)$ and $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P_{n,d}(q,t)$ are symmetric in $q,t$ and can be written as a sum of Schur functions evaluated in $q,t$.
The restriction of a symmetric function to the sum of Schur functions indexed by only one part (respectively,hook-shaped Schur functions) will be denoted by $|_{1 \mathop{\mathrm{Part}}}$ (respectively, $|_{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{ht}}ooks}$). For example, if $f=\sum_{\lambda\in C}c_\lambda s_\lambda$, then the restriction to one part is $f|_{1 \mathop{\mathrm{Part}}}= \sum_{\lambda\in C, \ell(\lambda)=1}c_\lambda s_\lambda$ (respectively, the restriction to hooks is $f|_{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{ht}}ooks}= \sum_{\lambda\in C, \lambda=(a,1^b)}c_\lambda s_\lambda$).
In this section, we will give a simple formula for the Schur functions indexed by one part partitions contained in the development of ${\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}}_{n,d}(q,t)$. This will be done by proving an algorithm that allows us to describe all the paths of $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P_{n,d}$ relating to the restrictions to Schur functions indexed by one part in the Schur function decomposition of $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P_{n,d}(q,t)$.
Let us first notice that Schur functions on a set of $k$ ordered variables are indexed by a partition with length smaller or equal to $k$. This follows from the combinatorial definition of Schur functions, since the filling of the first column of the semi-standard Young tableau must be strictly increasing and the first column as the same number of boxes to fill than the number of parts of the partition. Hence, Schur functions in two variables have at most two parts. Furthermore, a Schur function in two variables is such that $s_{a,b}(q,t)=q^{a-b}t^{a-b}(q^b+q^{b-1}t+\cdots+qt^{b-1}+t^b)$. Ergo, for $c$ an integer, the monomial $q^c$ as a non-zero coefficient in the decomposition of a symmetric function $f(q,t)$ if and only if the decomposition in Schur function contains the term $s_c(q,t)$.
This is equivalent to stating that the Schur functions in the variables $q$ and $t$ appearing in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}_{n,d}(q,t)|_{1 \mathop{\mathrm{Part}}}$ are the same than the Schur functions in the variable $q$ appearing in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}_{n,d}(q,0)$. The only paths that contribute to monomials with non-zero coefficients in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}_{n,d}(q,0)$ are the paths $\gamma$ such that $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}(\gamma)=0$. Hence, these are the set of paths $\{NE,D\}^n$.
\\
The algorithm $\varphi$ takes a path, $\gamma$ in $\{NE,D\}^*$ for input, and returns a sequence of paths $(\gamma_0, \gamma_1, \ldots,\gamma_{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma)})$, obtained as follows:
\begin{align*}
&\text{First set } \varphi(\gamma)=(\gamma_0), ~k=|\gamma|_E+|\gamma|_N+|\gamma|_D.
\\&\text{For $v=1$ to $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma)$;}
\\ &\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{ht}}space{20pt} \text{Let $\gamma_{v-1}=w_1w_2\cdots w_{k}$. }
\\ &\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{ht}}space{20pt} \text{Let $i$ be such that $w_i=E$, $w_{i+1}\not=E$ and $w_j=E$ implies $w_{j+1}=E$ or $j\leq i$.}
\\ &\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{ht}}space{20pt} \text{Set $\gamma_v=w_1\cdots w_{i-1}w_{i+1}w_iw_{i+2} \cdots w_{k}$.}
\\ &\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{ht}}space{20pt} \text{Append $\gamma_v$ to the sequence $\varphi(\gamma)$.}
\\&\text{repeat ;}
\\&\text{return } \varphi(\gamma).
\end{align*}
\begin{figure}
\caption{The sequence $\varphi(DDNEDNENE)$. }
\label{Fig : phi sur Schroder aire 0}
\end{figure}
We first need to prove this algorithm provides us with a sequence of Schr\"oder paths.
\begin{lem}For all $\gamma \in \{NE,D\}^*$, the elements of the sequence $\varphi(\gamma)$ are Schr\"oder paths. Moreover, if $\gamma \in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}_{n,d}$, then $\varphi(\gamma) \subseteq \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}_{n,d}$.
\end{lem}
\begin{proof}Recall that for a path $\gamma$ to be a Schr\"oder path we must have $|\omega|_N\geq|\omega|_E$ for all prefix $\omega$ of $\gamma$. Because $\gamma_0$ is a Schr\"oder path, it is sufficient to show that if $\gamma_i$ is a Schr\"oder path, then the path $\gamma_{i+1}$, obtained by parsing one time through the algorithm, is also a Schr\"oder path. Let $\gamma_i=w_1w_2\cdots w_k$, then $\gamma_{i+1}=w_1\cdots w_{i-1}w_{i+1}w_iw_{i+2} \cdots w_{k}$ and the only prefixes that are different are $w_1\cdots w_{i-1}w_{i+1}$ compared to $w_1\cdots w_{i-1}w_{i}$ and $w_1\cdots w_{i-1}w_{i+1}w_i$ compared to $w_1\cdots w_{i-1}w_iw_{i+1}$. The last pair has the same letters ordered differently, so $|w_1\cdots w_{i-1}w_{i+1}w_i|_N\geq |w_1\cdots w_{i-1}w_{i+1}w_i|_E$ if and only if $|w_1\cdots w_{i-1}w_iw_{i+1}|_N\geq|w_1\cdots w_{i-1}w_iw_{i+1}|_E$. Now for the first pair of prefixes we have:
\begin{align*}
|w_1\cdots w_{i-1}w_{i+1}|_N &\geq |w_1\cdots w_{i-1}w_{i}|_N, \text{ since $w_i=E$ and $w_{i+1}\in\{N,D\}$,}
\\ &\geq |w_1\cdots w_{i-1}w_{i}|_E, \text{ since $\gamma_i$ is a Schr\"oder path, }
\\ & > |w_1\cdots w_{i-1}w_{i+1}|_E, \text{ since $w_i=E$ and $w_{i+1}\not=E$.}
\end{align*}
Therefore, $\gamma_{i+1}$ is indeed a Schr\"oder path.
Finally, for $\gamma$ a Schr\"oder path we can move east steps to the left at least a number of times equal to bounce. Indeed, the peaks are associated to an east step and the numph statistic gives the number of diagonal steps over that east step (to the right of that $E$ in the word representation). Because the bounce path associated to $\Gamma(\gamma)$ changes direction at the peak when it hits an east step, except for the last entry, the vector associated to $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\Gamma(\gamma))$ gives the number of north steps over that east step (to the right of that $E$ in word representation).
\end{proof}
We give an example of $\varphi(\gamma)$ for $\gamma$ a Schr\"oder path not in $\{NE,D\}$.
\begin{figure}
\caption{The sequence $\varphi(DDNDNEENE)$. }
\label{Fig : phi sur Schroder quelconque}
\end{figure}
In Figure~\ref{Fig : phi sur Schroder quelconque} the $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}$ statistic does not decrease evenly throughout the iterations of the algorithm. In Figure~\ref{Fig : phi sur Schroder aire 0} each iteration increases the area statistic by exactly one and decreases the bounce statistic by exactly one. We will show in Lemma~\ref{Lem : +1-1} that this is not a coincidence if $\gamma_0$ is a Schr\"oder path of area $0$. But first, we need to show a result on the prefixes rellating to the paths in $\varphi(\gamma)$.
\begin{lem}\label{Lem : prefix algorithm}Let $\gamma$ be in $\{NE,D\}^*$ and $\varphi(\gamma)=(\gamma_0,\ldots,\gamma_{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma)})$. If $\gamma_i$ as a prefix $\alpha$ in $\{NE,D\}$, then $\alpha$ is a prefix of $\gamma$. Moreover, if $\alpha$ is the longest prefix of $\gamma_i$ such that $\alpha$ is in $\{NE,D\}$, then for $\beta$ such that $\alpha\beta=\gamma_i$ we have $\beta=\omega E^{|\beta|_E-1}$, where $\omega$ is a word in the alphabet $\{N,E,D\}^*$. Finally, if $\gamma_{i}=w_1\cdots w_{j-2}w_{j}w_{j-1}w_{j+1} \cdots w_{k}$ and $\gamma_{i-1}=w_1\cdots w_{j-2}w_{j-1}w_{j}w_{j+1} \cdots w_{k}$, then $w_{j-1}$ is a letter in $\omega$.
\end{lem}
\begin{proof}By induction on $i$. If $i=0$, then $\alpha=\gamma_0=\gamma$ and $\beta=\varepsilon$. For $i>0$ let $\alpha$ (respectively,$\alpha'$) be the longest prefix of $\gamma_i$ (respectively,$\gamma_{i-1}$) such that $\alpha$ is in $\{NE,D\}^*$ (respectively,$\alpha' \in \{NE,D\}^*$) and $\beta$ (respectively,$\beta'$) be such that $\alpha\beta=\gamma_i$ (respectively,$\alpha'\beta'=\gamma_{i-1}$). By induction we know that $\alpha'$ is a prefix of $\gamma$, and $\beta'=\omega'E^{|\beta'|_E-1}$.
By definition of each iteration of the algorithm, there is $j$ such that $\gamma_{i}=w_1\cdots w_{j-2}w_{j}w_{j-1}w_{j+1} \cdots w_{k}$ and $\gamma_{i-1}=w_1\cdots w_{j-2}w_{j-1}w_{j}w_{j+1} \cdots w_{k}$.
If $|\alpha'|=l\leq j-2$, then $\beta'=w_{l+1}\cdots w_{j-2}w_{j-1}w_{j}w_{j+1} \cdots w_{k}$ and $\alpha=\alpha'$. By definition of the algorithm $w_j\in \{N,D\}$. Due to $\beta'=\omega'E^{|\beta'|_E-1}$, we must have that $w_j$ and $w_{j-1}$ are both letters of $\omega'$. Ergo, the suffix $E^{|\beta'|_E-1}$ of $\beta'$ is unchanged in $\beta$. Hence, $\beta=\omega E^{|\beta'|_E-1}=\omega E^{|\beta|_E-1}$ and $|\omega|_E=|\omega'|_E$.
If $|\alpha'|=l > j-2$, then by definition of the algorithm $w_{j-1}=E$, because $\alpha' \in\{NE,D\}^*$, $w_{j-2}=N$. Consequently,$\gamma_{i}=w_1\cdots w_{j-3}Nw_{j}Ew_{j+1} \cdots w_{k}$ and $\alpha=w_1\cdots w_{j-3}$ is in $\{NE,D\}^*$ and is a prefix of $\alpha'$. Thus, it is a prefix of $\gamma$. This means $\beta=Nw_{j}Ew_{j+1} \cdots w_{k}$. Each iteration of the algorithm swaps the rightmost east step that is not followed by an east step to the right. In consequence, $w_{j}\in\{N,D\}$ and we must have $\omega$ such that $\beta=\omega E^{|\beta|_E-1}$, with $w_{j-1}$ is a letter of $\omega$; otherwise the letters $w_j$ and $w_{j-1}$ would not have been swapped.
\end{proof}
\begin{lem}\label{Lem : +1-1}Let $\gamma$ be in $\{NE,D\}^*$ and $\varphi(\gamma)=(\gamma_0,\ldots,\gamma_{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma)})$. Then, for all $i$, such that $0\leq i < \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma)$, the following equalities hold:
\begin{align*}\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}(\gamma_i)+1&=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}(\gamma_{i+1}),
\\ \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma_i)&=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma_{i+1})+1,
\\ \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}(\gamma_i)&=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma_{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma)-i}), \text{ and,}
\\ \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma_i)&=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}(\gamma_{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma)-i}).
\end{align*}
\end{lem}
\begin{proof}Let us first notice that the algorithm changes $EN$ for $NE$ or $ED$ for $DE$. In both cases, this adds exactly one lower triangle under the path. Therefore, $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}(\gamma_i)+1=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}(\gamma_{i+1})$.
For the second condition, let $\gamma_{i}=w_1\cdots w_{j-1}w_{j}w_{j+1}w_{j+2} \cdots w_{k}$. By definition of the algorithm $\varphi$, we know that $\gamma_{i+1}=w_1\cdots w_{j-1}w_{j+1}w_{j}w_{j+2} \cdots w_{k}$, $w_j=E$ and $w_{j+1}\in \{N,D\}$. By Lemma~\ref{Lem : prefix algorithm}, we know $\gamma_{i+1}=\alpha\beta$ with $\alpha\in\{NE,D\}^*$, so there is a return to the main diagonal of the bounce path between $\alpha$ and $\beta$. Hence, the following east step is associated to a peak. By Lemma~\ref{Lem : prefix algorithm}, $\beta=\omega E^{|\beta|_E-1}$, $w_j$ is a letter of $\omega$ and $\omega$ contains exactly one east step. Consequently, there is a peak at $w_j$ in $\gamma_{i+1}$. Let $\gamma_i=\alpha'\beta'$, if $w_j$ is a letter in $\omega'$, the same reasoning leads to a peak at $w_j$ in $\gamma_i$. If $w_j$ is a letter in $\alpha'$, there is a peak at $w_j$ in $\gamma_i$, since $\alpha'\in\{NE,D\}^*$. Thus, $NE=w_{j-1}w_j \in \mathop{\mathrm{Touch}}(\alpha)$.
If $w_{j+1}=D$, then $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\Gamma(\gamma_i))=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\Gamma(\gamma_{i+1}))$ because $\Gamma$ discards the diagonal steps. Recall that the peaks of a Schr\"oder path, $\gamma$, are also obtained from $\Gamma(\gamma)$, in consequence, the peaks in $\gamma_i$ and $\gamma_{i+1}$ are associated to the same east steps. Recall that numph is the number of diagonal steps, with multiplicity, positioned after a peak (higher if you consider the path itself rather than the word representation). The diagonal step $w_{j+1}$ is after the peak at $w_j$ in $\gamma_i$ and before the peak at $w_j$ in $\gamma_{i+1}$. All other peaks and diagonal steps remain unchanged. Hence, $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{numph}}(\gamma_i)=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{numph}}(\gamma_{i+1})+1$.
If $w_{j+1}=N$, then the peak at $w_j$ moves one position to the right in the word representation (one line higher if you consider the path itself). By definition, at this point, the bounce path returns to the main diagonal and goes north to the next east step,
which, by Lemma~\ref{Lem : prefix algorithm}, are all after $\omega$. Therefore, the peak at $w_j$ does not move to a line already containing a peak unless $\omega$ ends with $w_{j}D^l$. In this last case, the peak at $w_j$ contributed $1$ to $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma_i)$. In any case, the peak on the first line of $\Gamma(\gamma_{i+1})$, contributes $0$ to bounce. Consequently, in both cases, $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\Gamma(\gamma_i))=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\Gamma(\gamma_{i+1}))+1$. Additionally, all the east steps keep the same number of diagonal steps positioned after them. Hence, all the peaks keep the same number of diagonal steps positioned after them and $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{numph}}(\gamma_i)=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{numph}}(\gamma_{i+1})$.
Considering the path $\gamma$ in $\{NE,D\}^*$ has an area equal to zero, the third and fourth conditions follows from the first two conditions.
\end{proof}
We now present a map that will be useful for the discussion on crystals in Section~\ref{Sec : cristaux}.
For $\gamma$ in $\{NE,D\}^*$ we have $\varphi(\gamma)=(\gamma_0,\ldots,\gamma_{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma)})$. With this notation we define the map:
\begin{align*}\label{Eq : tilde varphi}\tilde\varphi : \{\gamma \in\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}{}_{n,d-1} ~|~\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}(\gamma)=0\}& \rightarrow\{\gamma \in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}{}_{n,d-1} ~|~\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}(\gamma)=1\}
\\ \gamma=\gamma_0&\mathop{\mathrm{maj}}apsto\gamma_1
\end{align*}
This next lemma will be used in the proof of Theorem~\ref{The : main}.
\begin{lem}\label{Lem : ens 1 part aire=1} Let $d$ be an integer such that $1\leq d\leq n-1$, then the image of the map $\tilde\varphi$ is given by the set $\{uD^jNNEE, vNDED^jNE ~|~ u\in \{NE,D\}^{n-d-2}, v\in\{NE,D\}^{n-d-1}\}$.
\end{lem}
\begin{proof}Follows from the definition of $\varphi$.
\end{proof}
In order to give the decomposition in Schur functions evaluated in the variables $q$ and $t$, for $\langle \nabla(e_n), s_{d+1,1^{n-d-1}}\rangle|_{1 \mathop{\mathrm{Part}}}$, we will show that for a path $\gamma$ in $\{NE,D\}^*$, the sum $\sum_{\pi\in\varphi(\gamma)}q^{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\pi)}t^{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}(\pi)}$ is a Schur function in the variables $q$ and $t$. For the general result to hold, we need the intersection of these sets to be empty.
\begin{lem}\label{Lem : intersection vide}Let $\gamma$, $\pi$ be in $\{NE,D\}^*$ such that $\gamma\not=\pi$ then $\varphi(\gamma) \cap \varphi(\pi)=\emptyset$.
\end{lem}
\begin{proof} Let us first notice that the algorithm changes $EN$ for $NE$ or $ED$ for $DE$. Therefore, the relative order of the north and diagonal steps does not change for all paths in $\varphi(\gamma)$ and $ \varphi(\pi)$. Hence, $\gamma_0$ and $\pi_0$ have the same relative order in regard to the north and diagonal steps which uniquely determine paths of $\{NE,D\}^*$. Consequently, $\varphi(\gamma) \cap \varphi(\pi)=\emptyset$.
\end{proof}
We can now display a bijection that inverts the statistics area and bounce. This partially solves open problem 3.11 of \cite{[H2008]}.
\begin{prop}\label{Prop : prob 3.1}Let $n$ be a positive integer and $\varphi(\{NE,D\}^n)$ be the set $\cup_{\gamma\in\{NE,D\}^n} \varphi(\gamma)$. There is a bijection, $\Omega_n$, of $\varphi(\{NE,D\}^n)$ onto itself. For all $n\geq 1$ we have, $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}(\gamma_i)=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\Omega_n(\gamma_i))$ and $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma_i)=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}(\Omega_n(\gamma_i))$.
\end{prop}
\begin{proof}By Lemma~\ref{Lem : intersection vide}, for $\gamma \in \varphi(\{NE,D\}^n)$ there is a unique $\gamma_0$ and a unique $i$ such that $\gamma \in \varphi(\gamma_0)$ and $\gamma=\gamma_i$. Thus, we can define $\Omega_n(\gamma)=\gamma_{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma_i)+\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}(\gamma_i)-i}$. The result is a consequence of Lemma~\ref{Lem : +1-1}.
\end{proof}
The following Lemma gives us a full set representatives of Schur functions indexed by one part.
\begin{lem}\label{Lem : bij schroder aire 0 et rectangle}
Let $A_d=\{ \gamma \in \{NE,D\}^n ~|~ |\gamma|_D=d \}$, there is a bijection $\theta: A_d \rightarrow \mathop{\mathrm{maj}}athcal{C}^n_d$ such that $\theta(NE)=N$ and $\theta(D)=E$. Moreover, for $\gamma \in A_d$ we have $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma)=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}(\theta(\gamma))+\binom{n-d}{2}$.
\end{lem}
\begin{proof}
In $A_d$, the factor $NE$, can be changed for a letter. A path $\gamma$ in $A_d$ is a word of length $n$ with $d$ occurrences of one letter and $n-d$ occurrences of the other letter. A path in $ \mathop{\mathrm{maj}}athcal{C}^n_d$ can be represented by a word with $d$ occurrences of the letter $E$ and $n-d$ occurrences of the letter $N$. Hence, $\theta$ merely relabels the letters and is a bijection.
Furthermore, in $A_d$ all east steps are associated to a peak; therefore, the $i$-th diagonal steps contribute the number of factors $NE$ before the $i$-th diagonal steps to $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{numph}}$. But that number is the number of boxes under the $i$-th north step in $\theta(\gamma)$. Thus, $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{numph}}(\gamma)=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}(\theta(\gamma))$. Finally, $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\Gamma(\gamma))=\binom{n-d}{2}$ for all paths $\gamma$ in $A_d$ because all the $n-d$ east steps return to the main diagonal.
\end{proof}
The next proposition will be generalized for parking functions by Proposition~\ref{Prop : 1 part parking} and generalized for the restriction to Schur functions indexed by a hook-shaped partition, evaluated in the variables $q$ and $t$ by Theorem~\ref{The : main}. Although the generalizations will not account for all the paths related to each Schur functions.
\begin{prop}\label{Prop : sum algo 1 part} For $\gamma$ in $\{NE,D\}^*$, we have:
\begin{equation}\label{Eq : bounce}
\sum_{\gamma_i \in \varphi(\gamma)}q^{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma_i)}t^{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}(\gamma_i)}=s_{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma)}(q,t),
\end{equation}
\begin{equation}\label{Eq : 1 part 1-schroder}
\langle\nabla e_n,e_{n-d}h_d\rangle|_{1 \mathop{\mathrm{Part}}}=\underset{|\gamma|_D=d}{\sum_{\gamma \in \{NE,D\}^{n}}} s_{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma)}(q,t)
=\sum_{\gamma\in \mathop{\mathrm{maj}}athcal{C}^{n}_d} s_{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}(\gamma)+\binom{n-d}{2}}(q,t), \text{ and,}
\end{equation}
\begin{equation}\label{Eq : 1 part 1-schroder schur}
\langle\nabla e_n,s_{d+1,1^{n-d-1}} \rangle|_{1 \mathop{\mathrm{Part}}}=\underset{|\gamma|_D=d}{\sum_{\gamma \in \{NE,D\}^{n-1}NE}} s_{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma)}(q,t)=\sum_{\gamma\in \mathop{\mathrm{maj}}athcal{C}^{n-1}_d} s_{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}(\gamma)+\binom{n-d}{2}}(q,t).
\end{equation}
\end{prop}
\begin{proof}Equation~\eqref{Eq : bounce} follows from Lemma~\ref{Lem : +1-1}. For the first equality of Equation~\eqref{Eq : 1 part 1-schroder}, we notice that $s_a(q,t)=q^a+q^{a-1}t+\cdots+qt^{a-1}+t^a$, and, thus, by \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{ht}}yperref[The : Hag]{Haglund's} Theorem, a Schur function indexed by a one part partition in $\langle\nabla e_n,e_{n-d}h_d \rangle$ can be associated to a path $\gamma$ in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}_{n,d}$ such that $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}(\gamma)=0$. But these are in $\{NE,D\}^n$ and have $d$ diagonal steps. For this reason, by Equation~\eqref{Eq : bounce}, the equality holds.
The second equality of Equation~\eqref{Eq : 1 part 1-schroder} Follows from Lemma~\ref{Lem : bij schroder aire 0 et rectangle}. Finally, for Equation~\eqref{Eq : 1 part 1-schroder schur} we only need to notice that paths of $\mathop{\mathrm{maj}}athcal{C}^n_d$ ending with a north step are in bijection with paths of $\mathop{\mathrm{maj}}athcal{C}^{n-1}_d$ and have the same area. The result is a consequence of \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{ht}}yperref[The : Hag]{Haglund's} Theorem, Lemma~\ref{Lem : bij schroder aire 0 et rectangle} and Equation~\eqref{Eq : bounce}.
\end{proof}
We end this section with a result needed for the generalization of Theorem~\ref{The : main}.
\begin{cor}\label{Cor : aire 1 vs 1 part}
Let $d$ be an integer and $\gamma$ be a path in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P_{n,d}$. Then, the path $\gamma$ is such that $\gamma=\gamma'NDED^jNE$ or $\gamma=\gamma'NED^jNNEE$, with $\gamma'\in \{NE,D\}^*$ if and only if $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}(\gamma)=1$ and $\gamma$ contributes to a Schur function indexed by a partition of length $1$ in $ \langle \nabla(e_n), s_{d+1,1^{n-d-1}}\rangle$.
\end{cor}
\begin{proof} If $\gamma=\gamma'NDED^jNE$ or $\gamma=\gamma'NED^jNNEE$), with $\gamma'\in \{NE,D\}^*$, then, by Lemma~\ref{Lem : ens 1 part aire=1}, $\gamma$ is in the image of $\tilde\varphi$ and the result follows from Proposition~\ref{Prop : sum algo 1 part}.
If $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}(\gamma)=1$ and $\gamma$ contributes to a Schur function indexed by a partition of length $1$, by Lemma~\ref{Lem : ens 1 part aire=1}, $\gamma$ is in the image of $\tilde\varphi$ and the result follows from Proposition~\ref{Prop : sum algo 1 part}.
\end{proof}
\section{From Parking Functions Formulas to Schur Functions}\label{Sec : parking}
The aim of this section is to give a combinatorial formula for $\nabla^m(e_n)$ restricted to Schur functions indexed by one part partitions in the variables $q$ and $t$. We will denote this restriction $\nabla^m(e_n)|_{1 \mathop{\mathrm{Part}}}$. In this section we will be using the diagonal inversion statistic, since bounce is not defined for parking functions. This is the main obstacle to knowing all path related to each Schur functions in the variables $q$ and $t$ in the formula.
To obtain a formula for $\nabla^m(e_n)|_{1 \mathop{\mathrm{Part}}}$, we will give the necessary and sufficient conditions for a parking function to have a diagonal inversion statistic of $0$. We will also determine the necessary and sufficient conditions for a parking function to have a diagonal inversion statistic of $1$ if $m$ is greater or equal to $2$.
The next three results are technical and will mostly be used to discard some reoccurring cases.
\begin{cl}\label{Cl : min dinv Park}Let $(\gamma,w)$ be in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athcal{P}}_{n,nm}$ such that $\gamma$ has a factor $\gamma'=NE^pN$, with $ 1\leq p\leq m$, and with its north steps associated to $w_i$ and $w_{i+1}$. Then, if $w_i>w_{i+1}$, $d_i(i+1)=p-1$ and if $w_i<w_{i+1}$, $d_i(i+1)=p$.
\end{cl}
\begin{proof} By definition:
\begin{equation*}
d_i(i+1)=\chi(w_i<w_{i+1})\mathop{\mathrm{maj}}ax(0,m-|a_i-a_{i+1}|)+\chi(w_i>w_{i+1})\mathop{\mathrm{maj}}ax(0,m-|a_{i+1}-a_i+1|)
\end{equation*}
Hence, by hypothesis $a_i=a_{i+1}+p-m$. Therefore:
\begin{equation*}
d_i(i+1)=\chi(w_i<w_{i+1})(p)+\chi(w_i>w_{i+1})(p-1)
\end{equation*}
\end{proof}
\begin{lem}\label{Lem : dinv Park p>m}Let $(\gamma,w)$ be in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athcal{P}}_{n,mn}$ such that $\gamma$ has a factor $\gamma'=NE^pN$, with $p > m$. Then, $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{dinv}}(\gamma,w)\geq m-1$.
\end{lem}
\begin{proof}Let $w_i$ and $w_{i+1}$ be the letters of $w$ associated to the factor $\gamma'$. Since the path is continuous and over the main diagonal, there exists $j_1,\ldots,j_m$ such that the $k$-th copy (from the top) of $w_{i+1}$ is to the north on the same diagonal than the north step associated to the letter $w_{j_k}$ in $w$. Note that the $j_k$'s are not necessarily distinct, ergo, $d_{j_k}(i+1)=d_s^t(i+1)$ for $s=j_k$ and $j_k$ is the $t$-th copy (from the top) of $w_{s}$. Which means that for $1\leq k\leq m-1$ when $w_{j_k}>w_{i+1}$ we get $d_{j_k}(i+1)\geq 1$ (see Figure~\ref{Fig : djk(i+1) meme diago}) and when $w_{j_k}<w_{i+1}$ the copy $k+1$ of $w_{i+1}$ is to the north and one diagonal lower than $w_{j_k}$. Therefore, $d_{j_k}(i+1)\geq 1$ (see Figure~\ref{Fig : djk(i+1) pas meme diago}). Hence:
\begin{equation*}\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{dinv}}(\gamma,w)= \sum_{s=1}^{n-1}\sum_{t=1}^m \sum_{r>l}^n d_s^t(r)\geq \sum_{k=1}^{m-1} d_{j_k}(i+1)\geq m-1.
\end{equation*}
\end{proof}
\begin{figure}\end{figure}
\begin{lem}\label{Lem : m=1 p>2 local}Let $(\gamma,w)$ be in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athcal{P}}_{n,mn}$. If there is a factor $\gamma$, $\gamma'=NE^pN$ associated to the lines $i-1$ and $i$ such that $p\geq 2$, then there is $k$ such that $d_k(i)=1$, if $m=1$ and $d_k(i)\geq 1$, if $m>1$.
\end{lem}
\begin{proof}We will work with $\tilde\gamma$ and $\tilde w$. Therefore we can use a Dyck path in an $mn\times mn$ grid. Let us suppose there is no such $k$.
Dyck paths have the property of always having more north steps than east steps for all prefixes. Hence, there is a line $j$ in $\gamma$ such that $j<i$ and the north step on line $j_s$ is on the same diagonal than the north step on line $i_1$. We can assume $j_s$ is the upper bound of such lines.
By hypothesis, $d_j(i)=0$ for $(\gamma,w)$, in consequence, $d_{j_s}(i_1)=0$ and we must have $\tilde w_{j,s}> \tilde w_{i,1}$, the contrary would lead to $d_{j_s}(i_1)=1$. Since $p\geq 2$, there is $l$ such that $j\leq l<i$ and the line $l_r$ is one diagonal over the diagonal passing through the north step at line $i_1$ in $(\tilde \gamma,\tilde w)$. We can assume $l$ is the smallest line satisfying these properties. Note that if $j=l$, then $r=s-1$ and if $j\not=l$, then $s=1$ (see Figure~\ref{Fig : p>2 j=l} and Figure~\ref{Fig : p>2 s=m}). Again, $\tilde w_{l,r} < \tilde w_{i,1}$ or else we would have $d_{l_r}(i_1)=1$. This means $l\not=j$ and $l\not=j+1$, since $w_j>w_{i}>w_l$. So there must be at least one east step between the lines $j_1$ and ${j+1}_m$. If there is just one, then the line ${j+1}_m$ is on the same diagonal than the lines $i_1$ and $j_1$ contradicting that $j$ is the upper bound. If there are two or more east steps between the lines $j_1$ and ${j+1}_m$, then the path goes under the diagonal passing through the north steps at the line $j_1$ and at the line $i_1$. But it must cross it again before the line $l_r$ because the path is continuous and the line $l_r$ is over the diagonal. Which contradicts again that $j$ is the upper bound.
\end{proof}
\begin{figure}\end{figure}
We can now state a first condition.
\begin{lem}\label{Lem : read=w-1 m=1}Let $(\gamma,w)$ be in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athcal{P}}_{n,n}$. If $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{dinv}}(\gamma,w)=0$, then $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{read}}(\gamma,w)=w^{-1}$. Additionally, all non-trivial factors of $\gamma$, $\gamma'=NE^pN$ are such that $p=1$.
\end{lem}
\begin{proof}If $p\leq 1$, then by Claim~\ref{Cl : min dinv Park}, for all factors $NE^pN$ of $\gamma$ we must have $p=1$ when $w_i>w_{i+1}$ and $p=0$ when $w_i<w_{i+1}$, since $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{dinv}}(\gamma,w)=0$.
If $\gamma'=NE^pN$ is a factor of $\gamma$ associated to lines $i$ and $i+1$ such that and $p>1$, then, by Lemma~\ref{Lem : m=1 p>2 local}, there is $k$ such that $d_k(i+1)=1$. Therefore, $p\not>1$ because $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{dinv}}(\gamma,w)=0$. Finally, $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{read}}(\gamma,w)=w^{-1}$, is a direct consequence of $p\leq 1$.
\end{proof}
The same result is also true for general parking functions when $m\geq2$. The following gives somewhat of a generalization.
\begin{lem}\label{Lem : read=w-1}Let $a$ and $m$ be integers such that $2\leq a \leq m$ and $(\gamma,w)$ be in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athcal{P}}_{n,nm}$. If $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{dinv}}(\gamma,w)=a-2$, then $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{read}}(\gamma,w)=w^{-1}$. Additionally, all factors of $\gamma$, $\gamma'=NE^pN$ are such that $p\leq m$.
\end{lem}
\begin{proof} By hypothesis $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{dinv}}(\gamma,w)=a-2$. If $p > m$, then, by Lemma~\ref{Lem : dinv Park p>m}, $a-2\geq m-1$, which contradicts $a\leq m$. Hence, all factors $\gamma'=NE^pN$ of $\gamma$ are such that $p\leq m$. Thereafter, all factors $\gamma_{i,j}=NE^{p_i}NE^{p_{i+1}}\cdots NE^{p_{j-1}}N$, with $i<j$, satisfy $|\gamma_{i,j}|_E =\sum_{k=i}^{j-1}p_k\leq (j-i)m=m|\gamma_{i,j}|_N$. Consequently, for all $i<j$ we read $w_j$ before $w_i$ in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{read}}(\gamma,w)$ and $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{read}}(\gamma,w)=w^{-1}$ as stated.
\end{proof}
Obviously, $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{read}}(\gamma,w)\not=w^{-1}$ in general. But sometimes $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{read}}(\gamma,w)=w^{-1}$ even without the condition on diagonal inversions. The last part of the proof gave us a weaker yet more general statement.
\begin{cl}\label{Cl : read=w-1} Let $m$ be an integer and $(\gamma,w)$ be in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athcal{P}}_{n,nm}$. If all factors of $\gamma$, $\gamma'=NE^pN$ are such that $p\leq m$, then $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{read}}(\gamma,w)=w^{-1}$.
\end{cl}
Sadly Lemma~\ref{Lem : read=w-1} does not apply for $m=2$, when the diagonal inversion statistic has value $1$. Therefore, we have to prove it separately.
\begin{lem}\label{Lem : read=w-1 m=2}Let $(\gamma,w)$ be in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athcal{P}}_{n,2n}$. If $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{dinv}}(\gamma,w)=1$, then $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{read}}(\gamma,w)=w^{-1}$. Additionally, all factors of $\gamma$, $\gamma'=NE^pN$ are such that $p\leq 2$.
\end{lem}
\begin{proof} Suppose there is a factor $NE^pN$ of $\gamma$ such that $p>2$, associated to lines ${j-1}$ and $j$. We can assume $j$ to be the smallest line satisfying that property. Let us consider $\tilde\gamma$ the path found by doubling each north step in $\gamma$ and $\tilde w$ the word $w_{1,1}w_{1,2}w_{2,1}w_{2,2}\cdots w_{n,1}w_{n,2}$. Considering the path is continuous and $p>2$, the path goes over the diagonal passing through $w_{j,1}$. The path ends under that diagonal, ergo there must be $i$ such that $w_{j,2}$ is on the same diagonal as $w_{i,1}$ or $w_{i,2}$ and $w_{i+1,1}$ is strictly over the diagonal passing through $w_{j,1}$. The three cases possible are illustrated by Figure~\ref{Fig : cas 1}, Figure~\ref{Fig : cas 2} and Figure~\ref{Fig : cas 3}.
For the first case, if $w_i <w_j$, then the pairs $(i_1,j_1)$ and $(i_2,j_2)$ both contribute to dinv. If $w_i >w_j$, then the pairs $(i+1_2,j_1)$ and $(i_1,j_2)$ both contribute to dinv. Thus, the diagonal inversion statistic cannot be equal to $1$.
For the second case, if $w_{i+1} <w_j$, then the pairs $(i+1_2,j_1)$ and $(i_1,j_2)$ both contribute to dinv, since $w_i<w_{i+1}$. If $w_{i+1} >w_j$, then the pairs $(i+1_1,j_1)$ and $(i+1_2,j_2)$ both contribute to dinv. Hence, the diagonal inversion statistic cannot be equal to $1$.
For the last case, if $w_i <w_j$, then the pairs $(i_1,j_1)$ and $(i_2,j_2)$ both contribute to dinv. If $w_i >w_j$, then the pairs $(i+1_2,j_2)$ and $(i_1,j_2)$ both contribute to dinv because $w_i<w_{i+1}$. Ergo, the diagonal inversion statistic cannot be equal to $1$.
Therefore, $p\leq 2$ for all factors $NE^pN$ of $\gamma$ and, by Claim~\ref{Cl : read=w-1}, we get $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{read}}(\gamma,w)=w^{-1}$.
\end{proof}
\begin{figure}
\caption{Case 1}
\label{Fig : cas 1}
\caption{Case 2}
\label{Fig : cas 2}
\caption{Case 3}
\label{Fig : cas 3}
\end{figure}
By definition, it is fairly easy to see that for $(\gamma,w)$ the descent set of $w$ is related to the number of columns of $\gamma$. We state the following claim is order to avoid repetition.
\begin{cl}\label{Cl : max descent par colonne}
Let $(\gamma,w)$ be in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athcal{P}}_{n,nm}$, $1\leq m$. Then, the number of descents of $w$ plus 1 is smaller or equal to the number of distinct columns. Additionally, the descents are at the top of a column.
\end{cl}
\begin{proof}If $w_i$ and $w_{i+1}$ are in the same columns, then by definition of parking functions we must have $w_i<w_{i+1}$. Therefore, descents must be at the top of a column. The last column cannot have descents, since the top of that column is $w_n$ and we know the last letter of a permutation cannot be a descent.
\end{proof}
The last result relates the number of distinct columns to the descent set of $w$. But the following relates the number of distinct columns to the descent set of $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{read}}(\gamma,w)^{-1}$.
\begin{lem}\label{Lem : dinv vs top colonnes}Let $m$ and $n$ be integers, $(\gamma,w)$ be in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athcal{P}}_{n,nm}$ and $T(\gamma)$ be the number of distinct columns. Let $\sigma$ be the permutation such that $\sigma.(\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{read}}(\gamma,w)^{-1})=w$. Then:
\begin{equation*}
\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{dinv}}(\gamma,w)\geq \begin{cases}T(\gamma)-\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{read}}(\gamma,w)^{-1})-1 &\text{ if } \sigma(n)=n,
\\ T(\gamma)-\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{read}}(\gamma,w)^{-1})-2 &\text{ if } \sigma(n)\not=n.
\end{cases}
\end{equation*}
\end{lem}
\begin{proof}We will show that the letter at the top of a column contribute at least 1 to $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{dinv}}$ unless they are in the descent set of $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{read}}(\gamma,w)^{-1}$, in the last column, or the last letter of $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{read}}(\gamma,w)^{-1}$. Notice that if $\sigma(n)=n$, then last letter of $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{read}}(\gamma,w)^{-1}$ is in the last column, so we only need to subtract it once.
Let $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{read}}(\gamma,w)^{-1}=v_1v_2\cdots v_n$ and let $v_i$ be at the top of a column. If $v_i$ is not in the last column and $i\not=n$, then by definition of the reading word and because the path is continuous, we have , in $(\tilde\gamma,\tilde w)$, these three cases: the last copy from the top of $v_{i+1}$ is to the north and on the same diagonal than a copy of $v_i$, let us say the $k$-th copy from the top (see Figure~\ref{Fig : cas 1 dans read}), the letter $v_{i+1}$ is to south and on one of the diagonals crossing one of the $m-1$ first copies of $v_i$ (see Figure~\ref{Fig : cas 2 dans read}), let us say the $p$-th copy, or $v_{i+1}$ is to the south one diagonal higher than the first copy of $v_i$ (see Figure~\ref{Fig : cas 3 dans read}).
Let $\sigma$ be the permutation that send $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{read}}(\gamma,w)^{-1}$ to $w$. When $i$ is not in the descent set of $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{read}}(\gamma,w)^{-1}$, our first case yields $d_{\sigma(i)}(\sigma(i+1)) = k$, by definition of the diagonal inversion statistics. The same reasoning shows $d_{\sigma(i+1)}(\sigma(i))= p+1$ for the second case and $d_{\sigma(i+1)}(\sigma(i)) = 1$ for the last case.
\end{proof}
\begin{figure}\end{figure}
We can now state necessary and sufficient conditions for the diagonal inversion statistic to be equal to zero.
\begin{prop}\label{Prop : critere dinv=0}Let $(\gamma,w)$ be in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athcal{P}}_{n,nm}$, $1\leq m$. Then, $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{dinv}}(\gamma,w)=0$ if and only if the following conditions applies:
\\
$\bullet$ The path $\gamma$ can be written as $\gamma'E^j$ where all factors of $\gamma'$ of length 2 have at most one east step.
\\
$\bullet$ If $\{i_1,\ldots,i_k,n\}$ is the set of all lines containing an east step, then $\{i_1,\ldots,i_k\}$ is the descent set.
\\
$\bullet$ $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{read}}(\gamma,w)=w^{-1}$.
\end{prop}
\begin{proof} If $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{dinv}}(\gamma,w)=0$, then by Claim~\ref{Cl : min dinv Park} and Lemma~\ref{Lem : dinv Park p>m}, we have the first condition. If $\gamma'=NE^pN$ is a factor of $\gamma$ associated to $w_i$ and $w_{i+1}$, then by now proven first condition and Claim~\ref{Cl : min dinv Park}, $w_i>w_{i+1}$. Hence, the position $i$ is a descent. But, by Claim~\ref{Cl : max descent par colonne}, we know that the number of descents plus $1$ is greater or equal to the number of columns and that $w_n$ contains an east step, ergo the second condition. The last condition follows from Lemma~\ref{Lem : read=w-1 m=1} and Lemma~\ref{Lem : read=w-1}. Therefore, $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{dinv}}(\gamma,w)=0$ does imply the stated conditions.
Conversely, by Claim ~\ref{Cl : max descent par colonne}, the descents are at the top of each column, since $n$ cannot be a descent. Therefore, by the first condition and by Claim~\ref{Cl : min dinv Park}, we know that, for all lines $i$, with an east step, we are at the top of a column and $d_i(i+1)=0$. For all lines $i$ with an east step, and, all lines $j$ such that $j>i+1$, we know that $a_i+(j-i)m\geq a_j\geq a_i+(j-i)(m-1)$, since there is at most one east step between each north step. This leads to:
\begin{equation*}
(j-i)m+1\geq |a_j-a_i+1|\geq (j-i)(m-1)+1\geq 2(m-1)+1\geq m,
\end{equation*}
and:
\begin{equation*}
(j-i)m\geq |a_j-a_i|\geq (j-i)(m-1)\geq 2(m-1)\geq m,
\end{equation*}
Therefore, $d_i(j)=0$ for all $i$ and $j$. Hence, $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{dinv}}(\gamma,w)=0$.
\end{proof}
Note that the second statement of the previous proposition implies that the number of descents in $w$ is equal to the number of distinct columns plus $1$.
Looking at the specialization $q=0$ is equivalent to looking only at the parking functions $(\gamma,w)$ such that $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{dinv}}(\gamma,w)=0$, and, thus, we need the area of theses parking functions.
\begin{prop}\label{Prop : aire parking}Let $(\gamma,w)$ be in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athcal{P}}_{n,nm}$, $1\leq m$ such that $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{dinv}}(\gamma,w)=0$, then:
\begin{equation*}
\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}(\gamma,w)=m\binom{n}{2}-\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(w)n+\mathop{\mathrm{maj}}(w)
\end{equation*}
\end{prop}
\begin{proof}Let $\{ i_1,\ldots,i_k,n\}$ be the lines with east steps. We know that these are tops of columns and by the previous proposition we know that $\{ i_1,\ldots,i_k\}$ is the descent set. By the previous proposition, we also know that:
\begin{align*}
\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}(\gamma,w) &=m\binom{n}{2}-\sum_{j=1}^k (n-i_j)
\\ &=m\binom{n}{2}-n\sum_{j=1}^k 1+\sum_{j=1}^ki_j
\\ &=m\binom{n}{2}-\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(w)n+\mathop{\mathrm{maj}}(w)
\end{align*}
\end{proof}
This last proposition allows us to give a proper formula for $\nabla^m(e_n)|_{q=0}$. Which is just an extension of the Stanley-Lusztig formula.
\begin{prop}\label{Prop : 1 part parking} For integers $n$, $m$ such that $1\leq n,m$
\begin{equation}\label{Eq : 1 part parking}\nabla^m(e_n)|_{1 \mathop{\mathrm{Part}}}=\sum_{\tau\in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{SYT}}(n)}s_{\mathop{\mathrm{maj}}(\tau)+(m-1)\binom{n}{2}}(q,t) s_{\lambda(\tau)}(X), \text{and},
\end{equation}
\begin{equation}\label{Eq : q=0 parking}\nabla^m e_n|_{q=0}=t^{(m-1)\binom{n}{2}}\sum_{w\in \mathop{\mathrm{maj}}athbb{S}_n} t^{\binom{n}{2}-\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(w)n+\mathop{\mathrm{maj}}(w)}F_{\mathop{\mathrm{co}}(\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(\mathop{\mathrm{inv}}(w^{-1})))}(X)=t^{(m-1)\binom{n}{2}}\sum_{\tau\in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{SYT}}(n)} t^{\mathop{\mathrm{maj}}(\tau)} s_{\lambda(\tau)}.
\end{equation}
\end{prop}
\begin{proof} The first equality of Equation~\eqref{Eq : q=0 parking} follows from Proposition~\ref{Prop : aire parking} and Proposition~\ref{Prop : critere dinv=0}. Consequently, by the Equation~inferred from \cite{[S1979]} and \cite{[H2002]} (see Equation~\eqref{Eq : S-L,H}), we have:
\begin{equation*}\sum_{w\in \mathop{\mathrm{maj}}athbb{S}_n} t^{\binom{n}{2}-\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(w)n+\mathop{\mathrm{maj}}(w)}F_{\mathop{\mathrm{co}}(\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(\mathop{\mathrm{inv}}(w^{-1})))}(X)=\sum_{\tau\in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{SYT}}(n)} t^{\mathop{\mathrm{maj}}(\tau)} s_{\lambda(\tau)}(X).
\end{equation*}
Therefore, the second equality of Equation~\eqref{Eq : q=0 parking} holds. For Equation~\eqref{Eq : 1 part parking}, we only need to notice that $\nabla^m(e_n)$ is symmetric in $q$,$t$ and $s_{\lambda}(q,t)=0$ if $\ell(\lambda)>2$ and $s_{a,b}(q,t)=(qt)^b(q^{a-b}+q^{a-b-1}t+\cdots +qt^{a-b-1}+t^{a-b}$. Hence, $s_{a,b}(0,t)=0$ if $b\not=0$ and $s_{a}(0,t)=t^a$. Ergo, we have the stated result by Equation~\eqref{Eq : q=0 parking}.
\end{proof}
From this last proposition and Proposition~\ref{Prop : sum algo 1 part} we can obtain the following $q$-analogues that are used in \cite{[Wal2019d]}.
\begin{cor}Let $n$, $m$, $d$ be integers, then:
\begin{align*}
\langle\nabla^m(e_n), s_{d+1,1^{n-d-1}}\rangle|_{t=0}&=q^{(m-1)\binom{n}{2}+\binom{n-d}{2}}\begin{bmatrix} n-1\\d\end{bmatrix}_q=q^{m\binom{n}{2}-\binom{d+1}{2}}\begin{bmatrix} n-1\\d\end{bmatrix}_{q^{-1}}
\end{align*}
\end{cor}
\begin{proof}We recall that $\begin{bmatrix}n\\k\end{bmatrix}_q=\sum_{\gamma\in \mathop{\mathrm{maj}}athcal{C}^n_k}q^{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}(\gamma)}$. In consequence, by Proposition~\ref{Prop : sum algo 1 part} and Proposition~\ref{Prop : 1 part parking}, we have the first equality. The second equality follows from $\binom{n}{2}-\binom{d+1}{2}=d(n-d-1)+\binom{n-d}{2}$ and $q^{d(n-d-1)}\begin{bmatrix} n-1\\d\end{bmatrix}_{q^{-1}}=\begin{bmatrix} n-1\\d\end{bmatrix}_q$.
\end{proof}
This next lemma emulates Proposition~\ref{Prop : critere dinv=0} for diagonal inversion statistics values of one.
\begin{lem}\label{Lem : m>2 dinv=1}
Let $(\gamma,w)$ be in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athcal{P}}_{n,nm}$, $2\leq m$ and $T(\gamma)$ be the number of distinct columns. Then, $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{dinv}}(\gamma,w)=1$ if and only if one of the following conditions applies:
\\
$\bullet$ All factors, $NE^pN$ of $\gamma$ are such that $p\leq1$ and $T(\gamma)=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(w)+2$.
\\
$\bullet$ Exactly one factor, $NE^pN$ of $\gamma$ is such that $p=2$ all other such factors satisfy $p\leq1$ and $T(\gamma)=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(w)+1$.
\end{lem}
\begin{proof}We will start by proving the statement for $m\geq3$.
If $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{dinv}}(\gamma,w)=1$, by Lemma~\ref{Lem : read=w-1}, we have $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{read}}(\gamma,w)=w^{-1}$. Thus,by Lemma~\ref{Lem : dinv vs top colonnes} and Claim~\ref{Cl : max descent par colonne}, $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(w)+1\leq T(\gamma)\leq \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(w)+2$.
By Claim~\ref{Cl : max descent par colonne}, if $T(\gamma)=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(w)+1$, then $w_i>w_{i+1}$ for all $i$ at the top of a column. Hence, by Lemma~\ref{Lem : read=w-1}, $p\leq m$. In consequence, by Claim~\ref{Cl : min dinv Park}, there is exactly one factor, $NE^pN$ of $\gamma$ is such that $p=2$, all other such factors satisfy $p\leq1$.
Again, by Claim~\ref{Cl : max descent par colonne}, if $T(\gamma)=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(w)+2$, there is exactly one line $i$ containing an east step such that $w_i<w_{i+1}$. Ergo, by Claim~\ref{Cl : min dinv Park}, all factors, $NE^pN$ of $\gamma$ are such that $p\leq1$.
If all factors, $NE^pN$ of $\gamma$ are such that $p\leq1$ and $T(\gamma)=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(w)+2$, then, by Claim~\ref{Cl : max descent par colonne}, we know there is exactly one $i$ at the top of a column such that $w_i<w_{i+1}$. Moreover, $p\leq1$ for all factors $NE^pN$ of $\gamma$, so, by Claim~\ref{Cl : min dinv Park}, $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{dinv}}(\gamma,w)=1$, if $d_i(j)=0$ for all $j\geq i+2$. But $p\leq 1$ means $m\geq a_{i+1}-a_i\geq m-1$. Hence, for $j>i$, $|a_j-a_i|\geq (m-1)(j-i)\geq m$ and $|a_j-a_i+1|\geq (m-1)(j-i)+1\geq m$, since $j-i\geq 2$ and $m\geq 3$. Consequently, $d_i(j)=0$ for all $j\geq i+2$ and $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{dinv}}(\gamma,w)=1$.
If exactly one factor, $NE^pN$ of $\gamma$ is such that $p=2$ all other such factors satisfy $p\leq1$ and $T(\gamma)=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(w)+1$. By Claim~\ref{Cl : max descent par colonne}, for all $i$ at the top of a column $w_i>w_{i+1}$. Additionally, exactly one factor, $NE^pN$ of $\gamma$ is such that $p=2$, all other such factors satisfy $p\leq1$. In consequence, by Claim~\ref{Cl : min dinv Park}, $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{dinv}}(\gamma,w)=1$ if $d_i(j)=0$ for all $j\geq i+2$. But $p\leq 1$ means $m\geq a_{i+1}-a_i\geq m-1$ and $p=2$ means $m\geq a_{i+1}-a_i\geq m-2$. Ergo, for $j>i$, $|a_j-a_i+1|\geq (m-1)(j-i)+1\geq m$ because $j-i\geq 2$ and $m\geq 3$. Thus, $d_i(j)=0$ for all $j\geq i+2$ and $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{dinv}}(\gamma,w)=1$.
For $m=2$ the proof is the same, we only need to change references of Lemma ~\ref{Lem : read=w-1} to Lemma ~\ref{Lem : read=w-1 m=2}.
\end{proof}
Note that
for $m=1$ nothing holds (see Figure~\ref{Fig : => contre-exemple m=1}, Figure~\ref{Fig : <= contre-exemple m=1 partie 1} and Figure~\ref{Fig : <= contre-exemple m=1 partie 2}) but, in Section~\ref{Sec : bijections}, we manage to obtain a Proposition~\ref{Prop : 1 part parking} type formula for the restriction to hook Schur functions in the variables $q$ and $t$, by using Schr\"oder paths and the $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}$ statistic.
\begin{figure}
\caption{The diagonal inversion statistic is $1$ yet $p>2$.}
\label{Fig : => contre-exemple m=1}
\caption{The diagonal inversion statistic is $2$ yet $p\leq1$ and $T(\gamma)=\mathop{\mathrm{maj}
\label{Fig : <= contre-exemple m=1 partie 1}
\end{figure}
\begin{figure}
\caption{The diagonal inversion statistic is $2$ yet exactly one factor $NE^pN$ is such that $p=2$ all others are such that $p\leq1$ and $T(\gamma)=\mathop{\mathrm{maj}
\label{Fig : <= contre-exemple m=1 partie 2}
\end{figure}
\section{Restriction to $m$-Schr\"oder Paths}\label{Sec : schroder}
This section is dedicated to the restriction to Schr\"oder paths. With this restriction we can give the necessary and sufficient conditions for a path to have a diagonal inversion statistic value of one.
Proposition~\ref{Prop : critere dinv=0} can be restated in terms of Schr\"oder paths.
\begin{cor}\label{Cor : Schroder->des=d}Let $(\gamma,w)$ be in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}_{n,d}^{(m)}$, $1\leq m$. Then, $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{dinv}}(\gamma,w)=0$ if and if, one of the following conditions applies:
\\
$\bullet$ The path $\gamma$ can be written as $\gamma'E^j$ where all factors of $\gamma'$ of length 2 have at most one east step.
\\
$\bullet$ If $\{i_1,\ldots,i_k,n\}$ is the set of all lines containing an east step, then $\{w_{i_1},\ldots,w_{i_k}\} \subseteq \{n-d+1,\ldots,n\}$.
\\
$\bullet$ $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{read}}(\gamma,w)=w^{-1}$.
\end{cor}
\begin{proof} Condition one and three are consequences of Proposition~\ref{Prop : critere dinv=0}. By definition of Schr\"oder paths $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{read}}(\gamma,w) \in \{n-d+1,\ldots,n\} \shuffle \{n-d,\ldots,1\}$. Therefore, $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{read}}(\gamma,w)^{-1}=w\in \{n,\ldots,n-d+1\} \shuffle \{1,\ldots,n-d\}$ and the descents of $w$ are the positions of $n-d+1,\ldots,n$ in $w$. Hence, the result follows from Proposition~\ref{Prop : critere dinv=0}.
\end{proof}
The restriction to $m$-Schr\"oder paths allow us to write the formula of Proposition ~\ref{Prop : 1 part parking} in terms of paths in a rectangular grid as we did for $m=1$ in Proposition~\ref{Prop : sum algo 1 part}.
\begin{cor} Let $n$, $d$ and $m$ be positive integer such that $n\geq d$ and $m>1$. Then:
\begin{align*}{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}}^m_{n,d}(q,t)|_{1 \mathop{\mathrm{Part}}}=&\langle\nabla^m e_n,e_{n-d}h_d \rangle|_{1 \mathop{\mathrm{Part}}}
\\ =&\sum_{\gamma\in \mathop{\mathrm{maj}}athcal{C}^{n-1}_{d-1}} s_{(m\binom{n}{2}-\binom{d}{2}-\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}(\gamma))}(q,t)+\sum_{\gamma\in \mathop{\mathrm{maj}}athcal{C}^{n-1}_{d}} s_{(m\binom{n}{2}-\binom{d+1}{2}-\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}(\gamma))}(q,t)
\end{align*}
Additionally:
\begin{equation}\label{Eq : 1 part m-schroder} {\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}}^m_{n,d}(q,t)|_{1 \mathop{\mathrm{Part}}}=\langle\nabla^m e_n,s_{d+1,1^{n-d-1}} \rangle|_{1 \mathop{\mathrm{Part}}}
=\sum_{\gamma\in \mathop{\mathrm{maj}}athcal{C}^{n-1}_{d}} s_{(m\binom{n}{2}-\binom{d+1}{2}-\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}(\gamma))}(q,t)
\end{equation}
\end{cor}
\begin{proof}Due to $\sum_{\gamma\in \mathop{\mathrm{maj}}athcal{C}^{n-1}_{d}}\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}(\gamma)=\sum_{\gamma\in \mathop{\mathrm{maj}}athcal{C}^{n-1}_{d}} (n-1-d)d-\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}(\gamma)$ and $\binom{n}{2}-\binom{d+1}{2}=\binom{n-d}{2}+(n-1-d)d$, the result follows from Proposition~\ref{Prop : sum algo 1 part} and Proposition~\ref{Prop : 1 part parking}.
\end{proof}
The main proof of this section is very technical case-by-case proof. It will be used to prove the main theorem via Corollary~\ref{Cor : m>=1 dinv=1}.
\begin{prop}\label{Prop : m=1 dinv=1}
Let $(\gamma,w)$ be in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}_{n,d}$ and let $T(\gamma)$ be the number of distinct columns. Then, $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{dinv}}(\gamma,w)=1$ if and only if one of the following conditions applies:
\\
$\bullet$ All factors, $NE^pN$ of $\gamma$ are such that $p\leq1$ and $T(\gamma)=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(w)+2$.
\\
$\bullet$ Exactly one factor, $\gamma'=NE^pN$ of $\gamma$ is such that $p=2$, and all other such factors satisfy $p\leq1$ and $T(\gamma)=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(w)+1$.
\\
$\bullet$ Exactly one factor, $\gamma'=NE^pN$ of $\gamma$ is such that $p > 2$ and $\gamma'$ is associated to lines $n-1$, $n$, $w_{n}\in \{n-d+1,\ldots,n\}$ and all other such factors satisfy $p\leq1$ and $T(\gamma)=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(w)+2$ if $w_{n-1}\in \{n-d,\ldots,1\}$ or $T(\gamma)=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(w)+1$ if $w_{n-1}\in \{n-d+1,\ldots,n\}$.
\end{prop}
\begin{proof}If $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{dinv}}(\gamma,w)=1$ and all factors $\gamma'=NE^pN$ are such that $p\leq 1$, then, by Claim~\ref{Cl : min dinv Park}, there is exactly one line $i$ at the top of a column such that $w_i<w_{i+1}$. Additionally, by Claim~\ref{Cl : max descent par colonne}, we know that $T(\gamma)\geq\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(w)+1$. By Claim~\ref{Cl : read=w-1} and Lemma~\ref{Lem : dinv vs top colonnes}, we have $T(\gamma)\leq\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(w)+2$. Since $w_n$ is at the top of its own column, $i$ and $n$ are not descents and the top of all the other columns are descents. Thus, $T(\gamma)=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(w)+2$.
For the remaining cases, if $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{dinv}}(\gamma,w)=1$, then, by Lemma~\ref{Lem : m=1 p>2 local}, we know there is at most one factor of $\gamma$ say $\gamma'=NE^pN$ associated to the lines $i$ and $i+1$ such that $p >1$ and there is $k$ such that $d_k(i+1)=1$.
If $w_k>w_{i+1}$, then the north step at line $k$ is on the diagonal above the north step at the line $i+1$. Moreover, the path is continuous and the north step at line $i$ is over the diagonal passing through the north steps at the line $k$ and $i+1$, and, thus, there exist $l<k$ such that the north step at the line $i+1$ and the north step at the line $l$ are on the same diagonal. Assuming $l$ is the biggest such $l$. We know, $w_l>w_{i+1}$ because $d_l(i+1)=0$. This means $w_k$ is read before $w_{i+1}$ and $w_{i+1}$ is read before $w_l$. By definition of Schr\"oder paths $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{read}}(\gamma,w)\in \{n-d+1,\ldots,n\}\shuffle \{n-d,\ldots,1\}$. Hence, $w_l\in\{n-d+1,\ldots,n\}$. There is at most one east step between the line $l$ and the line $l+1$, so $w_{l+1}$ is read before $w_{l}$. Ergo, $w_l>w_{l+1}$. Therefore, $w_{l+1}$ is not in the same column as $w_l$. Consequently, $w_{l+1}$ is on the same diagonal as $w_l$, since there is at most one factor $NE^rN$ with $r>1$. This contradicts that $l$ is the highest line such that $l<k$, $l$ and $i+1$ are on the same diagonal. (See Figure~\ref{Fig : k diago au-dessus i+1}.)
If $w_k<w_{i+1}$, $p\geq 2$ and there is an east step between the lines $k$ and $k+1$, then $d_k(i+1)=1$ implies the lines $k$ and $i+1$ are crossed by the same diagonal. By Lemma~\ref{Lem : m=1 p>2 local}, there is exactly one east step between the lines $k$ and $k+1$, and, thus, they are on the same diagonal and $k\not=i$. Considering $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{dinv}}(\gamma,w)=1$ we need $d_k(k+1)=0$ and $d_{k+1}(i+1)=0$. Therefore, $w_k>w_{k+1}$ and $w_{k+1}>w_{i+1}$ which is absurd. (See Figure~\ref{Fig : k,k+1,i meme diago}.)
For the case $w_k<w_{i+1}$, $p\geq 2$, $k=i-1$ and there is no east step between the lines $k$ and $k+1$. Notice that if $k=i-1$, there are $i+1-k=2$ north steps. Since $d_k(i+1)=1$, we know $k$ and $i+1$ are on the same diagonal. Hence, there is 2 east step between the north step at the line $k$ and the north step at the line $i+1$. In addition, letters on the same diagonal are separated by the same number of east steps than north steps. Thus, $p=2$.
Additionally, by Claim~\ref{Cl : max descent par colonne}, $T(\gamma)\geq\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(w)+1$. Moreover, $w_i$ is read before $w_{i+1}$, since they are separated by more than one east step and $d_i(i+1)=0$. Thus,$w_i>w_{i+1}$. Furthermore, all descent in $w$ contribute to a different column. Hence, if $T(\gamma)>\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(w)+1$ we must have a change of column at a line $l$, $l\not=i$, such that $l$ is not a descent. Ergo $w_l<w_{l+1}$ and because there is at most one east step between $w_l$ and $w_{l+1}$, by Lemma ~\ref{Lem : m=1 p>2 local}, we must have $d_l(l+1)=1$ which is absurd. Therefore, $T(\gamma)=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(w)+1$.
If $w_k<w_{i+1}$, $p\geq 2$, $k\not=i-1$ and there is no east step between the lines $k$ and $k+1$, then $k\not=i$ and $w_k<w_{k+1}$. Moreover, $d_k(i+1)=1$ implies the lines $k$ and $i+1$ are crossed by the same diagonal. Consequently, the north step at line $k+1$ is on the diagonal over the north step at the line $i+1$. Hence, $w_{k+1}<w_{i+1}$, since $d_{k+1}(i+1)=0$. Thus, $w_{k+1}$ is read before $w_{i+1}$ and $w_{i+1}$ is read before $w_k$ in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{read}}(\gamma,w)$. We know $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{read}}(\gamma,w)\in \{n-d+1,\ldots,n\}\shuffle\{n-d,\ldots,1\}$, ergo, $w_{i+1}\in \{n-d+1,\ldots,n\}$ and $w_{k+1},w_k \in \{n-d,\ldots,1\}$. If $i+1\not=n$ and $w_{i+2}$ is in the same column as $w_{i+1}$, then $w_{i+2}$ is read before $w_{i+1}$ and $w_{i+1}<w_{i+2}$. But this is impossible because $w_{i+1}$ is in the set $ \{n-d+1,\ldots,n\}$. Therefore, $w_{i+2}$ and $w_{i+1}$ are on the same diagonal and $w_{i+1}>w_{i+2}$, since $d_{i+1}(i+2)=0$. Due to $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{dinv}}(\gamma,w)=1$ and $d_k(i+1)=1$, we have $d_{k}(i+2)=0$. The north step at line $i+2$ is on the same diagonal as the north step at the line $k$, ergo $w_k>w_{i+2}$. For this reason, $w_{i+2}$ is read before $w_k$. But, $w_k \in \{n-d,\ldots,1\}$ means $w_{i+2}>w_k$ which is impossible. So, $i+1=n$. (See Figure~\ref{Fig : k sous k+1 et k, i+1 meme diago}.)
\begin{figure}
\caption{ }
\label{Fig : k diago au-dessus i+1}
\end{figure}
If $w_k<w_{i+1}$, $i+1=n$, $p>2$ and there is no east step between the lines $k$ and $k+1$, then, by Claim~\ref{Cl : max descent par colonne}, $T(\gamma)\geq\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(w)+1$. As in the previous case $w_{i+1}\in \{n-d+1,\ldots,n\}$. Additionally, $w_i$ is read before $w_{i+1}$, considering they are separated by more than one east step. Hence, $w_i<w_{i+1}$. Furthermore, all descent in $w$ contribute to a different column and the letters $w_{n-1}$, $w_{n}$ are in different columns (recall $i+1=n$ and there are $p$ east steps between $w_{n-1}$ and $w_n$). This means $T(\gamma)\geq\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(w)+2$ if $w_{n-1}\in \{n-d,\ldots,1\}$ and $T(\gamma)\geq\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(w)+1$ if $w_{n-1}\in \{n-d+1,\ldots,n\}$. In both cases if the inequality is strict, we must have a change of column at a line $l$ such that $l\not=n-1$ and $l$ is not a descent, ergo, $w_l<w_{l+1}$. Since there is at most one east step between $w_l$ and $w_{l+1}$, by Lemma ~\ref{Lem : m=1 p>2 local}, we must have $d_l(l+1)=1$ which is absurd. Therefore, $T(\gamma)=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(w)+2$ if $w_{n-1}\in \{n-d,\ldots,1\}$ and $T(\gamma)=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(w)+1$ if $w_{n-1}\in \{n-d+1,\ldots,n\}$.
Conversely, if all factors, $NE^pN$ of $\gamma$ are such that $p\leq1$ and $T(\gamma)=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(w)+2$, then there is exactly one line $i$ at the top of a column such that $i\not=n$ and $w_i<w_{i+1}$. By Claim~\ref{Cl : min dinv Park}, $d_i(i+1)=1$ and $d_l(l+1)=0$ for all $l\not=i$ because $p\leq 1$. For the same reason, all lines $j$ and $k$ such that $k-j\geq 2$, have a number of north steps greater or equal to the number on east steps between them and $d_j(k)=0$, unless the north step on lines $j$ and $k$ are on the same diagonal.
But when the north step at the line $k$ and the north step at the line $j$ are on the same diagonal, $k>j$ and $p\leq 1$ we know the line $j$ is associated to a factor $NEN$ of $\gamma$ and is at the top of a column. By Claim~\ref{Cl : read=w-1}, we have $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{read}}(\gamma,w)=w^{-1}$, and, thus, $w_j\in \{n-d+1,\ldots,n\}$ and $w_k$ is read before $w_j$. Consequently, $w_j>w_k$ and $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{dinv}}(\gamma,w)=1$.
If exactly one factor, $\gamma'=NE^pN$ of $\gamma$ is such that $p=2$, and all other such factors satisfy $p\leq1$ and $T(\gamma)=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(w)+1$, then by Claim~\ref{Cl : max descent par colonne}, all lines $j$ at the top of a column are such that $j$ is in the descent set of $w$. Hence, if $w_i$ and $w_{i+1}$ are associated to the factor $\gamma'=NE^2N$ of $\gamma$ $w_i$ is on the diagonal above $w_{i+1}$ and $w_i>w_{i+1}$, so $d_i(i+1)=1$. For the same reasons as in the previous case for all $j$ and $k$ such that $j\not=i$ and $k\not=i+1$, then $d_j(k)=0$. Therefore, $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{dinv}}(\gamma,w)=1$.
Let us now consider the case when exactly one factor, $\gamma'=NE^pN$ of $\gamma$ is such that $p > 2$ and $\gamma'$ is associated to lines $n-1$, $n$ and all other such factors satisfy $p\leq1$ and $T(\gamma)=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(w)+2$. If we take out the last north step and the last east step and call the new path $\tilde\gamma$, then, by Proposition~\ref{Prop : critere dinv=0}, $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{dinv}}(\tilde\gamma,w_1\cdots w_{n-1})=0$. Hence, for all $1\leq j<k\leq n-1$ we have $d_j(k)=0$. Because $d_j(k)$ is a local property, it is also true for $(\gamma,w)$. By continuity of the path, since $p>2$, there is a line $k$ such that $w_k$ and $w_{k+1}$ are in the same column and the north step a line $n$ is on the same diagonal than the north step at the line $k$. Thus, we read $w_{k+1}$ before $w_n$ and $w_n$ before $w_k$ in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{read}}(\gamma,w)$. Moreover, $w_{k+1}>w_k$, and, therefore, $w_k\in\{n-d,\ldots,1\}$ and $w_n>w_{k+1}>w_k$, is a consequence of $w_n\in\{n-d+1,\ldots,n\}$. So, $d_k(n)=1$ and $d_{k+1}(n)=0$.
All other factors $\gamma''=NE^{p'}N$ satisfy $p'\leq1$, if there is $j$ distinct from $k$ such that $w_j$ is on the same diagonal than $w_n$, then $w_j,w_{j+1},\ldots,w_k$ are all on the same diagonal and $w_j,w_{j+1},\ldots,w_{k-1}$ are at the top of their column. Furthermore, the condition $T(\gamma)=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(w)+2$ if $w_{i+1}\in \{n-d,\ldots,1\}$ or $T(\gamma)=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(w)+1$ if $w_{i+1}\in \{n-d+1,\ldots,n\}$ forces all letters of $w$ at the top of a column except for $w_{n-1}$ and $w_n$ to be in $\{n-d+1,\ldots,n\}$. Consequently, $w_j>w_{j+1}>\cdots>w_{k-1}>w_n>w_k$ and $d_l(n)=0$, for all $j\leq l\leq k-1$. Therefore, $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{dinv}}(\gamma,w)=1$.
\end{proof}
The next corollary can also be deduced from the more general Lemma~\ref{Lem : m>2 dinv=1}. We only state it here, so one can notice that Proposition~\ref{Prop : m=1 dinv=1} is hiding a general statement for $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}_{n,d}^{(m)}$.
\begin{cor}\label{Cor : m>1 dinv=1}
Let $m$ be an integer such that $m\geq 2$, $(\gamma,w)$ be in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}_{n,d}^{m}$ and let $T(\gamma)$ be the number of distinct columns. Then, $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{dinv}}(\gamma,w)=1$ if and only if one of the following conditions applies:
\\
$\bullet$ All factors, $NE^pN$ of $\gamma$ are such that $p\leq1$ and $T(\gamma)=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(w)+2$.
\\
$\bullet$ Exactly one factor, $\gamma'=NE^pN$ of $\gamma$ is such that $p=2$, and all other such factors satisfy $p\leq1$ and $T(\gamma)=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(w)+1$, then $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{dinv}}(\gamma,w)=1$.
\end{cor}
\begin{proof}The proof of the previous proposition can be extended to $\tilde\gamma$, since $w_i=w_j$ only if they are in the same column.
\end{proof}
The following is the restriction to unlabelled Dyck paths.
\begin{cor}\label{Cor : m>=1 dinv=1}
Let $m$ be an integer such that $m\geq 1$, $(\gamma,w)$ be in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}_{n,0}^{m}$ and let $T(\gamma)$ be the number of distinct columns. Then, $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{dinv}}(\gamma,w)=1$ if and only if one of the following conditions applies:
\\
$\bullet$ All factors, $NE^pN$ of $\gamma$ are such that $p\leq1$ and $T(\gamma)=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(w)+2$.
\\
$\bullet$ Exactly one factor, $\gamma'=NE^pN$ of $\gamma$ is such that $p=2$, and all other such factors satisfy $p\leq1$ and $T(\gamma)=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(w)+1$, then $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{dinv}}(\gamma,w)=1$.
\end{cor}
\begin{proof}For $m=1$, the proof is a direct consequence of $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{read}}(\gamma,w)=n\cdots1$ and Proposition~\ref{Prop : m=1 dinv=1}. For $m>1$ the proof follows from the last corollary.
Recall, from the proof of Corollary~\ref{Cor : Schroder->des=d}, that when $m\geq3$ a path $(\gamma,w)$ of $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}_{n,d}{(m)}$ is such that $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(w)=d$. Hence the last two corollaries can be stated with nicer formulas.
\end{proof}
\section{Bijections With Tableaux}\label{Sec : bijections}
From Equation~\eqref{Eq : q=0 parking} of Proposition~\ref{Prop : 1 part parking}, one could wonder what tableau is associated to what path. In this section, we will first show a bijection between standard Young tableaux of shape $(d,1^{n-d})$ and the subset of Schr\"oder paths $\{\gamma \in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P_{n,d-1} ~|~\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}(\gamma)=0\}$. Afterwards, we exhibit a bijection between the set of paths $\{\gamma \in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P_{n,d-1} ~|~\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}(\gamma)=1\}$ and pairs containing a standard Young tableaux of shape $(d,1^{n-d})$ and a number $i$, $0\leq i\leq n-d$. This last bijection will allow us to write the sum over these paths with the $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}$ and $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}$ statistics in terms of hook-shaped Schur functions in the variables $q$ and $t$. In other words, we will obtain an explicit combinatorial formula for the expansion in Schur functions of $ \langle \nabla(e_n), s_{\mathop{\mathrm{maj}}u}\rangle|_{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{ht}}ooks}$
Before we start, we shall also notice that these bijections could easily be extended to paths ending with a diagonal step, by using the bijection between paths with $d$ diagonal steps that end with the factor $NE$ and paths with $d+1$ diagonal steps that end with a $D$ step.
\\
Recall in Section~\ref{Sec : combi chemins} we defined the touch points of a path. Notice that for a path $\gamma$ if $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}(\gamma)=0$ and $\mathop{\mathrm{Touch}}(\gamma)=(\gamma_1,\gamma_2,\ldots,\gamma_k)$, then for all $i$, $\gamma_i$ is in $\{NE,D\}$. Let’s define the sets $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P_{n,d, (i)}$ by:
\begin{equation*}
\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P{}_{n,d, (i)}=\{\gamma \in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P{}_{n,d}~|~ \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}(\gamma)=i\}.
\end{equation*}
Let $\{\mathop{\mathrm{maj}}athcal{M}_{n,d}\}$ be a family of maps:
\begin{align*}\mathop{\mathrm{maj}}athcal{M}_{n,d}:\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{SYT}}(d,1^{n-d}) &\rightarrow \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P{}_{n,d-1, (0)}
\\ \tau &\mathop{\mathrm{maj}}apsto \gamma_1\gamma_2\cdots\gamma_n,
\end{align*}
with $\gamma_n=NE$, $\gamma_{n-i}=NE$ if $i\in\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(\tau)$ and $\gamma_{n-i}=D$ otherwise (see Figure~\ref{Fig : exemple de P} for an example).
Let $\{\mathop{\mathrm{maj}}athcal{R}_{n,d}\}$ be a family of maps:
\begin{align*}\mathop{\mathrm{maj}}athcal{R}_{n,d}: \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P{}_{n,d-1, (0)}&\rightarrow\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{SYT}}(d,1^{n-d})
\\ \gamma &\mathop{\mathrm{maj}}apsto \tau,
\end{align*}
with $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(\mathop{\mathrm{maj}}athcal{R}_{n,d}(\gamma))=\{n-i ~|~1\leq i \leq n-1, \gamma_i=NE \in \mathop{\mathrm{Touch}}(\gamma)\}$ (see Figure~\ref{Fig : exemple de R} for an example).
\begin{figure}
\caption{An example of the application of map $\mathop{\mathrm{maj}
\label{Fig : exemple de P}
\caption{An example of the application of map $\mathop{\mathrm{maj}
\label{Fig : exemple de R}
\end{figure}
\begin{lem}The families of maps $\{\mathop{\mathrm{maj}}athcal{M}_{n,d}\}$ and $\{\mathop{\mathrm{maj}}athcal{R}_{n,d}\}$ are well defined.
\end{lem}
\begin{proof}
We have already seen that $\gamma$ in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P_{n,d-1,(0)}$ is represented by a word in $\{NE,D\}^{n-1}NE$ such that $|\gamma|_D=d-1$. Moreover, $\mathop{\mathrm{maj}}athcal{M}_{n,d}(\tau)$ is in $\{NE,D\}^{n-1}NE$ by construction. For $\tau\in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{SYT}}(d,1^{n-d}) $ the descent set of the tableau $\tau$ is a subset of $n-d$ elements in $\{1,\ldots,n-1\}$, since for $j\not=1$ in the first column $j-1$ is lower or to the right of $j$ by definition of hook-shaped standard tableaux. Hence, there are $n-d+1$, $i$'s such that $\gamma_{n-i}=NE$ and $|\mathop{\mathrm{maj}}athcal{M}_{n,d}(\tau)|_D=d-1$.
For the map $\mathop{\mathrm{maj}}athcal{R}_{n,d}$, notice that hooked-shaped tableaux are uniquely defined by their descent set. Furthermore, there are $n-d+1$ north steps in $\gamma$ and $\gamma_n=NE\in \mathop{\mathrm{Touch}}(\gamma)$. Thus, for $1\leq i\leq n-1$ there are $n-d$ factors $\gamma_i$ such that $\gamma_i=NE$ and $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(\mathop{\mathrm{maj}}athcal{R}_{n,d}(\tau))=n-d$. Ergo, $\mathop{\mathrm{maj}}athcal{R}_{n,d}(\gamma)\in\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{SYT}}(d,1^{n-d})$.
\end{proof}
\begin{prop}\label{Prop : bij}Let $n$, $d$ be integers such that $0\leq d\leq n$. The map $\mathop{\mathrm{maj}}athcal{M}_{n,d}$ is a bijection from the set of standard tableaux of shape $(d,1^{n-d})$ to the subset of Schr\"oder paths $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P_{n,d-1,(0)} $, of inverse $\mathop{\mathrm{maj}}athcal{R}_{n,d}$. Moreover,the map $\mathop{\mathrm{maj}}athcal{M}_{n,d}$ is such that $\mathop{\mathrm{maj}}(\tau)=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\mathop{\mathrm{maj}}athcal{M}_{n,d}(\tau))$ and the map $\mathop{\mathrm{maj}}athcal{R}_{n,d}$ is such that $\mathop{\mathrm{maj}}(\mathop{\mathrm{maj}}athcal{R}_{n,d}(\gamma))=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma)$.
\end{prop}
\begin{proof}For the first statement, we only need to show that $\mathop{\mathrm{maj}}athcal{M}_{n,d}$ and $\mathop{\mathrm{maj}}athcal{R}_{n,d}$ are inverse maps. Let $\tau$ be in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{SYT}}(d,1^{n-d}) $ if $i$ is in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(\tau)$ (respectively, $i$ is not in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(\tau)$), then the map $\mathop{\mathrm{maj}}athcal{M}_{n,d}$ sends $i$ to $\gamma_{n-i}=NE$ in $\mathop{\mathrm{Touch}}(\mathop{\mathrm{maj}}athcal{M}_{n,d}(\tau))$ (respectively, to $\gamma_{n-i}=D$) and $\mathop{\mathrm{maj}}athcal{R}_{n,d}$ sends $\gamma_{n-i}=NE$ in $\mathop{\mathrm{Touch}}(\mathop{\mathrm{maj}}athcal{M}_{n,d}(\tau))$ to $i$ in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(\mathop{\mathrm{maj}}athcal{R}_{n,d}(\mathop{\mathrm{maj}}athcal{M}_{n,d}(\tau)))$ (respectively,to $i$ not in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(\mathop{\mathrm{maj}}athcal{R}_{n,d}(\mathop{\mathrm{maj}}athcal{M}_{n,d}(\tau)))$). Hence, $\mathop{\mathrm{maj}}athcal{R}_{n,d}(\mathop{\mathrm{maj}}athcal{M}_{n,d}(\tau))=\tau$, since hooked-shaped tableaux are uniquely determined by their descent set. For $\gamma$ in $ \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P_{n,d-1,(0)}$, the proof of $\mathop{\mathrm{maj}}athcal{M}_{n,d}(\mathop{\mathrm{maj}}athcal{R}_{n,d}(\gamma))=\gamma$ is similar.
Additionally, in $ \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P_{n,d-1,(0)}$ all east steps are associated to a peak. For $\gamma=\gamma_1\gamma_2\cdots\gamma_n $, if $\gamma_{n-i}=NE$ and there are $k$ diagonal steps after the factor $\gamma_{n-i}$, then the peak associated to that factor contributes $k$ to $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{numph}}$ and there is a return to the main diagonal after the factor $\gamma_{n-i}$ that contributes $i-k$ to $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\Gamma(\gamma))$. Moreover, the peak $\gamma_n=NE$ contributes nothing to $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma)$ because it is the end of the path. Hence, $\mathop{\mathrm{maj}}(\tau)=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\mathop{\mathrm{maj}}athcal{M}_{n,d}(\tau))$ and $\mathop{\mathrm{maj}}(\mathop{\mathrm{maj}}athcal{R}_{n,d}(\gamma))=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma)$, since both maps associate the factor $\gamma_{n-i}=NE$ in the touch sequence to $i$ in the descent set.
\end{proof}
This last proposition gives a combinatorial formula for $\langle\nabla e_n,s_{d,1^{n-d}} \rangle$ in terms of the major index as in Proposition~\ref{Prop : 1 part parking}. Although we now know to which paths the top weight are associated to (see Section~\ref{Sec : cristaux} for more on this).
\begin{cor} Let $n$, $d$ be positive integer such that $n\geq d$. Then:
\begin{equation*}\label{Eq : 1 part maj}\langle\nabla e_n,s_{d,1^{n-d}} \rangle|_{1 \mathop{\mathrm{Part}}}=\sum_{\tau\in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{SYT}}(d,1^{n-d})} s_{\mathop{\mathrm{maj}}(\tau)}(q,t)
\end{equation*}
\end{cor}
\begin{proof}By Proposition~\ref{Prop : sum algo 1 part} and Proposition~\ref{Prop : bij}.
\end{proof}
In order to get the same type of formula for the restriction on hook-shaped partitions, we will partition the set $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P_{n,d-1,(1)}$. We need maps to do so. Let $\tau$ be in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{SYT}}(d,1^{n-d})$, then we define the path $\gamma_\tau=\gamma_1\gamma_2\cdots\gamma_{n-1}$ where:
\begin{equation*}
\gamma_{n-i}= \begin{cases}NE & \text{ if } i=1 \text{ or } i\in\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(\tau)\backslash\{\mathop{\mathrm{maj}}ax(\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(\tau))\}
\\ D & \text{ if } i\not\in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(\tau)\cup\{1\}
\\ NNEE & \text{ if } i=\mathop{\mathrm{maj}}ax(\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(\tau)) \text{ and } 1\in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(\tau)
\\ NDE & \text{ if } i=\mathop{\mathrm{maj}}ax(\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(\tau)) \text{ and } 1\not\in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(\tau)
\end{cases}
\end{equation*}
Let $V_{n,d}$ be a collection of sets defined by:
\begin{equation*}
V_{n,d}=\{\gamma \in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P{}_{n,d-1} ~|~\gamma=D^jNNEEu \text{ or } \gamma=D^jNDEu, j\geq0, u\in \{NE,D\}^*NE \}.
\end{equation*}
Our first family of maps $\mathop{\mathrm{maj}}athcal{S}_{n,d}$ is defined as follows, for $n-d\geq 1$:
\begin{align*} \mathop{\mathrm{maj}}athcal{S}_{n,d} : \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{SYT}}(d,1^{n-d}) &\rightarrow V_{n,d}
\\ \tau & \mathop{\mathrm{maj}}apsto \gamma_\tau
\end{align*}
The second family of maps is defined as follows, for $n-d\geq 1$:
\begin{align*} \mathop{\mathrm{maj}}athcal{T}_{n,d} :V_{n,d} &\rightarrow \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{SYT}}(d,1^{n-d})
\\ \gamma & \mathop{\mathrm{maj}}apsto \tau_\gamma
\end{align*}
Where for $U=\{1\}$ if $\gamma=D^jNDEv$, $U=\emptyset$ if $\gamma=D^jNNEEu$, $u\in \{NE,D\}^*$, $v\in \{NE,D\}^*NE$ we have $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(\tau_\gamma)=\{n-i : \gamma_i=Nw\in \mathop{\mathrm{Touch}}(\gamma), w \in \{NEE,E,DE\}^*\}\backslash\{U\}$.
\begin{figure}
\caption{Example of the map $\mathop{\mathrm{maj}
\label{Fig : exemple de S}
\caption{Example of the map $\mathop{\mathrm{maj}
\label{Fig : exemple de T}
\end{figure}
\begin{lem}Let $n$, $d$ be positive integers such that $n-d\geq1$, the maps $\mathop{\mathrm{maj}}athcal{S}_{n,d}$ and $\mathop{\mathrm{maj}}athcal{T}_{n,d}$ are well defined.
\end{lem}
\begin{proof}The factors $\gamma_i$ are all Schr\"oder paths ans the concatenation of Schr\"oder paths is a Schr\"oder path. Moreover, the factor $\gamma_{n-1}$ of $\gamma_\tau$ contains a north step and all the other north steps are related to an element in the descent set. Hence, we have $n-d+1$ north steps and $\gamma_\tau$ is an element of $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P_{n,d-1}$. Additionally, the construction of $\gamma_\tau$ is based on four mutually exclusive conditions that define $\gamma_{n-i}$. If $i=\mathop{\mathrm{maj}}ax(\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(\tau))$, then $n-i\leq n-k$ for all $k\in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(\tau)$. Thus, by construction the path starts with $D^jNNEE$ or $D^jNDE$. Since there is only one maximum of the descent set the map $\mathop{\mathrm{maj}}athcal{S}_{n,d}$ is well defined.
Notice that for all path in $V_{n,d}$ there is exactly one factor in $\{NNEE,NDE\}$ and all other are in $\{NE,D\}$. Hence, $\mathop{\mathrm{Touch}}(\gamma)$ has $n-1$ factors because it is equivalent to counting the numbers of north and diagonal steps minus one. Therefore, the descent set is included in $\{1,\ldots,n-1\}$. Moreover, there are $n-d+1$ north step and one is in the factor $\gamma_{n-1}$. Ergo, $1\in\{n-i : \gamma_i=Nw\in \mathop{\mathrm{Touch}}(\gamma), w \in \{NEE,E,DE\}^*\}$.
If $\gamma=D^jNNEEu$, $u\in \{NE,D\}^*$, then there are $n-d+1$ north steps for $n-d$ factors containing a north step. Consequently, $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(\tau_\gamma)=n-d$ and it uniquely determines a hooked-shaped tableau in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{SYT}}(d,1^{n-d})$.
If $\gamma=D^jNDEv$, $v\in \{NE,D\}^*NE$, then there are $n-d+1$ north steps for $n-d+1$ factors containing a north step. But $U$ takes out one from the descent set. Hence, the descent set has $n-d$ elements and it uniquely determines a hooked-shaped tableau in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{SYT}}(d,1^{n-d})$. Consequently,$\mathop{\mathrm{maj}}athcal{T}_{n,d}$ is well defined.
\end{proof}
The interesting thing about these maps is that they preserve statistics as shown in the next lemma.
\begin{lem}\label{Lem : map bounce=maj-des}Let $n$, $d$ be positive integers such that $n-d\geq1$. The maps $\mathop{\mathrm{maj}}athcal{S}_{n,d}$ and $\mathop{\mathrm{maj}}athcal{T}_{n,d}$ are bijective maps and $\mathop{\mathrm{maj}}athcal{T}_{n,d}=\mathop{\mathrm{maj}}athcal{S}_{n,d}^{-1}$. Moreover, the maps $\mathop{\mathrm{maj}}athcal{T}_{n,d}$ and $\mathop{\mathrm{maj}}athcal{S}_{n,d}$ preserve statistics in the following way:
\begin{equation*}
\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma)=\mathop{\mathrm{maj}}(\mathop{\mathrm{maj}}athcal{T}_{n,d}(\gamma))-\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(\mathop{\mathrm{maj}}athcal{T}_{n,d}(\gamma)),
\end{equation*}
\begin{equation*}
\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\mathop{\mathrm{maj}}athcal{S}_{n,d}(\tau))=\mathop{\mathrm{maj}}(\tau)-\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(\tau).
\end{equation*}
\end{lem}
\begin{proof}Let $\tau$ be a Standard Young tableau in of shape $(d,1^{n-d})$, $\mathop{\mathrm{maj}}athcal{S}_{n,d}(\gamma)=\gamma_{\tau}$, $\gamma_\tau=\gamma_1\gamma_2\cdots\gamma_{n-1}$. For $i\geq 2$, if the factor $\gamma_{n-i}=NE$, then the map $\mathop{\mathrm{maj}}athcal{T}_{n,d}$ send that factor to $i\in\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(\mathop{\mathrm{maj}}athcal{T}_{n,d}(\gamma_\tau))$. But the map $\mathop{\mathrm{maj}}athcal{S}_{n,d}$ gives us $\gamma_{n-i}=NE$ when $i\in\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(\tau)$. If $\gamma_{n-i}=NNEE$, then the map $\mathop{\mathrm{maj}}athcal{T}_{n,d}$ sends that factor to $1, i\in\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(\mathop{\mathrm{maj}}athcal{T}_{n,d}(\gamma_\tau))$ and the map $\mathop{\mathrm{maj}}athcal{S}_{n,d}$ gives $\gamma_{n-i}=NNEE$ when $1, i\in\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(\mathop{\mathrm{maj}}athcal{T}_{n,d}(\gamma_\tau))$. Finally, if $\gamma_{n-i}=NDE$, then the map $\mathop{\mathrm{maj}}athcal{T}_{n,d}$ sends that factor to $i\in\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(\mathop{\mathrm{maj}}athcal{T}_{n,d}(\gamma_\tau))$ and the image of map $\mathop{\mathrm{maj}}athcal{S}_{n,d}$ is $\gamma_{n-i}=NDE$ when $i\in\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(\mathop{\mathrm{maj}}athcal{T}_{n,d}(\gamma_\tau))$. Thus,$\mathop{\mathrm{maj}}athcal{T}_{n,d}(\mathop{\mathrm{maj}}athcal{S}_{n,d}(\tau))=\tau$. The proof of $\mathop{\mathrm{maj}}athcal{S}_{n,d}(\mathop{\mathrm{maj}}athcal{T}_{n,d}(\gamma))=\gamma$ is similar.
The proof that $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\mathop{\mathrm{maj}}athcal{S}_{n,d}(\tau))=\mathop{\mathrm{maj}}(\tau)-\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(\tau)$ is almost the same as the proof in Proposition~\ref{Prop : bij}. Since there are only $\gamma_{n-1}$ factors, we need to subtract one to each contribution to $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\Gamma(\gamma))$. Moreover, if $\gamma_{n-i}=NNEE$ and there are $k$ diagonal steps after the factor $\gamma_{n-i}$, then the peak associated to that factor contributes $k$ to $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{numph}}$ and there is a return to the main diagonal after the factor $\gamma_{n-i}$ that contributes $i-k-1$ to $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\Gamma(\gamma))$. Finally, if $\gamma_{n-i}=NDE$ and there are $k$ diagonal steps after the factor $\gamma_{n-i}$, then the peak associated to that factor contributes $k$ to $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{numph}}$ and there is a return to the main diagonal after the factor $\gamma_{n-i}$ that contributes $i-k-1$ to $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\Gamma(\gamma))$. But $1\not \in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(\mathop{\mathrm{maj}}athcal{T}_{n,d}(\gamma)$ which mean we do not add $1-1=0$ to $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(\mathop{\mathrm{maj}}athcal{T}_{n,d}(\gamma)$. Hence, $\mathop{\mathrm{maj}}(\tau)-\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(\tau)=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\mathop{\mathrm{maj}}athcal{S}_{n,d}(\tau))$ and $\mathop{\mathrm{maj}}(\mathop{\mathrm{maj}}athcal{T}_{n,d}(\gamma))-\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(\mathop{\mathrm{maj}}athcal{T}_{n,d}(\gamma))=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma)$, because both maps associate the factor $\gamma_{n-i}=NE$ to $i$ in the descent set. Consequently, $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\mathop{\mathrm{maj}}athcal{S}_{n,d}(\tau))=\mathop{\mathrm{maj}}(\tau)-\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(\tau)$. The map $\mathop{\mathrm{maj}}athcal{S}_{n,d}$ is a bijection of inverse $\mathop{\mathrm{maj}}athcal{T}_{n,d}$, and, thus, we also have $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma)=\mathop{\mathrm{maj}}(\mathop{\mathrm{maj}}athcal{T}_{n,d}(\gamma))-\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(\mathop{\mathrm{maj}}athcal{T}_{n,d}(\gamma))$.
\end{proof}
To extend the maps $\mathop{\mathrm{maj}}athcal{S}_{n,d}$ and $\mathop{\mathrm{maj}}athcal{T}_{n,d}$ we need to partition the paths of $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P_{n,d,(1)}$. Notice that $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P_{n,n, (1)}=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P_{n,n-1, (1)}=\emptyset$ and $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P_{n,n-2, (1)}=\{D^{n-2}NNEE, D^iNDED^jNE ~|~ i+j=n-3\}$.
\\
For $d=n-2$, we define $\Pi_{n,d-1}$ to be the identity map. For $n-d+1\geq 3$, let
\begin{align*} \Pi{}_{n,d-1} : &\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P{}_{n,d-1, (1)} &&~\rightarrow~&& \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P{}_{n,d-1, (1)}
\\ & uNNEED^jNEv && \mathop{\mathrm{maj}}apsto && uNED^jNNEEv
\\ &uNDED^jNEv && \mathop{\mathrm{maj}}apsto & &uNED^jNDEv
\\ &uNNEED^jNE && \mathop{\mathrm{maj}}apsto & &uNED^jNNEE
\\ &D^iNEuD^jNNEE && \mathop{\mathrm{maj}}apsto & &D^iNNEEuD^jNE
\\ &D^iNEuNDED^jNE && \mathop{\mathrm{maj}}apsto & &D^iNDEuNED^jNE
\end{align*}
For $u$ in $\{NE,D\}^*$ and $v$ in $\{NE,D\}^*NE$.
\begin{figure}
\caption{Example of the map $\Pi_{6,3}
\label{Fig : exemple de pi-1}
\caption{Example of the map $\Pi_{6,3}
\label{Fig : exemple de pi-2}
\end{figure}
\begin{prop}\label{Prop : partition}For all integers $n$ and $d$ such that $n-d\geq 1$. The sets $\{ \Pi_{n,d-1}^k(\gamma_\tau)\}$ over all $\tau\in\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{SYT}}(d,1^{n-d})$ are a partition of the set $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P_{n,d-1, (1)}$. Additionally, $\Pi_{n,d-1}$ is cyclic of order $n-d$.
\end{prop}
\begin{proof}We shall first notice that the map is well defined, since there is exactly one factor $NNEE$ or $NDE$ in path of $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}$ 1. Secondly, $\gamma_\tau$ is of shape $D^iNNEED^jNEuNE$ or $D^iNDED^jNEuNE$ and the action of $\Pi_{n,d-1}$ is to exchange the factor $NNEE$ (respectively, $NDE$) with the factor next $NE$ factor. Because there are $n-d+1$ north steps in $\gamma_\tau$, if $NNEE$ is a factor of $\gamma_\tau$ (respectively, $NDE$) there are $n-d-1$ (respectively, $n-d$) factors $NE$ above the factor $NNEE$ (respectively, $NDE$) and $\Pi^{n-d-1}_{n,d-1}(\gamma_\tau)=D^iNED^jNEuNNEE$ (respectively, $\Pi^{n-d-1}_{n,d-1}(\gamma_\tau)=D^iNED^jNEu'NDED^kNE$, where $u=u'NED^k$). Therefore, $\Pi^{n-d}_{n,d-1}(\gamma_\tau)=\gamma_\tau$.
Let $\gamma$ be a path in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P_{n,d-1,(1)}$. Then, there is a unique factor $NNEE$ or $NDE$ which corresponds to the line of $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}$ $1$. Hence, $\gamma=D^jNEu'NNEEu''NE$ (respectively, $\gamma=D^jNEu'NDEu''NE$, $\gamma=D^jNEu'NNEE$), with $u'$ and $u''$ in $\{NE,D\}$. Let $k$ be the number of north steps before the factor $NNEE$ (respectively,$NDE$, $NNEE$, with $k=n-d-1$). Consequently, $\Pi_{n,d-1}^k(\gamma_0)=\gamma$, for $\gamma_0=D^jNNEEu'NEu''NE$ (respectively,$\gamma_0=D^jNDEu'NEu''NE$, $\gamma_0=D^jNNEEu'NE$). Thus, $\mathop{\mathrm{maj}}athcal{T}_{n,d}(\gamma_0)=\tau_{\gamma_0}$. So, $\gamma$ is in the set $\{ \Pi_{n,d-1}^k(\gamma_{\tau_{\gamma_0}})\}$.
Finally, let $\gamma$ be in $\{ \Pi_{n,d-1}^k(\gamma_\tau)\}\cap \{ \Pi_{n,d-1}^k(\gamma_\rho)\}$, then there is $k$ such that $\gamma=\Pi_{n,d-1}^k(\gamma_\tau)$ and $l$ such that $\gamma=\Pi_{n,d-1}^l(\gamma_\rho)$. Hence, $\Pi_{n,d-1}^{n-d-k+l}(\gamma_\rho)=\gamma_\tau$. By previous statements we know $k=l$ ; it is the number of north steps before the factor $NNEE$ of $NDE$ in $\gamma$. Ergo,$\gamma_\rho=\gamma_\tau$. Thus,by Lemma~\ref{Lem : map bounce=maj-des}, $\rho=\mathop{\mathrm{maj}}athcal{T}_{n,d}(\gamma_\rho)=\mathop{\mathrm{maj}}athcal{T}_{n,d}(\gamma_\tau)=\tau$. Therefore $\{ \Pi_{n,d-1}^k(\gamma_\tau)\}\cap \{ \Pi_{n,d-1}^k(\gamma_\rho)\}=\emptyset$ if $\rho\not=\tau$.
\end{proof}
One might see similarities between the next lemma and Lemma~\ref{Lem : +1-1}, since the bounce statistic increases by exactly one with each iteration of $\Pi_{n,d-1}$. But it is worth mentioning that in this case the area remains one. Thus, this is not an extension of the algorithm seen in Section~\ref{Sec : algo} for paths with a bounce statistic value of zero.
\begin{lem} Let $U=\{uNED^jNNEE, uNDED^jNE ~|~ u\in \{NE,D\}^*\}$. The map $\Pi_{n,d-1}$ is such that $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\Pi_{n,d-1}(\gamma))=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma)+1$ for all $\gamma \in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P_{n,d, (1)}\backslash U$.
\end{lem}
\begin{proof}If $\gamma$ as a factor $NNEE$, then the map $\Pi_{n,d-1}$ swaps the factor $NNEE$ with the next factor $NE$ and all factors return to the main diagonal in $\Pi_{n,d-1}(\gamma)$. Hence, the number of peak under each diagonal step is left unchanged and $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{numph}}(\gamma)=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{numph}}(\Pi_{n,d-1}(\gamma))$. Moreover, $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}(\gamma)=1$, and, therefore, there is some $k$ such that $\Gamma(\gamma)=(NE)^{n-d-k}NNEE(NE)^{k-1}$. Thus, we obtain $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\Gamma(\gamma))= \binom{n-d+2}{2}-k$ and $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\Gamma(\Pi_{n,d-1}(\gamma)))= \binom{n-d+2}{2}-k+1$.
If $\gamma$ as a factor $NDE$, then $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\Pi_{n,d-1}(\Gamma(\gamma)))=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\Gamma(\gamma))$. Since the map $\Pi_{n,d-1}$ swaps the factor $NDE$ with the next factor $NE$ and all factors return to the main diagonal, in $\Pi_{n,d-1}(\gamma)$ there is one more peak below the diagonal step coming from the factor $NDE$ than in $\gamma$. Moreover, the number of peaks below all the other diagonal steps remains unchanged. Ergo, $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{numph}}(\gamma)+1=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{numph}}(\Pi_{n,d-1}(\gamma))$.
\end{proof}
With this partition we can put forward a bijection between tableaux and Schr\"oder paths.
\begin{prop}\label{Prop : bij chemins tableaux}Let $\mathop{\mathrm{maj}}athcal{Q}_{n,d}$ be a map between the product set $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{SYT}}(d,1^{n-d})\times \{0,1,\cdots,n-d-1\}$ and the set $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P_{n,d-1, (1)}$, defined by $\mathop{\mathrm{maj}}athcal{Q}_{n,d}(\tau,i)=\Pi_{n,d-1}^i(\mathop{\mathrm{maj}}athcal{S}_{n,d}(\tau))$. Then, the map $\mathop{\mathrm{maj}}athcal{Q}_{n,d}$ is a bijection. Moreover, $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\mathop{\mathrm{maj}}athcal{Q}_{n,d}(\tau,i))=\mathop{\mathrm{maj}}(\tau)-\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(\tau)+i$ for all $i$ in $\{0,1,\cdots,n-d-2\}$.
\end{prop}
\begin{proof} By Lemma~\ref{Lem : map bounce=maj-des} and Proposition~\ref{Prop : partition}, this map is well defined. Furthermore, for $\gamma$ in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P_{n,d-1, (1)}$ there is a unique $\tau$ in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{SYT}}(d,1^{n-d})$ an a unique integer $i$ such that $0\leq i\leq n-d-1$ and $\Pi_{n,d-1}^i(\gamma_\tau)=\gamma$. Moreover, $\mathop{\mathrm{maj}}athcal{S}_{n,d}$ is a bijection of inverse $\mathop{\mathrm{maj}}athcal{T}_{n,d}$. Hence, $\mathop{\mathrm{maj}}athcal{T}_{n,d}(\gamma_\tau)=\tau$ and the pre-image of $\gamma$ is unique. Ergo $\mathop{\mathrm{maj}}athcal{Q}_{n,d}$ is a bijection.
By the previous lemma $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\Pi_{n,d-1}^i(\gamma_\tau))-i=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma_\tau)$ if $\Pi_{n,d-1}^i(\gamma_\tau)$ is not an element of $U=\{uNED^jNNEE, uNDED^jNE ~|~ u\in \{NE,D\}^*\}$. The proof of Proposition~\ref{Prop : partition} shows that $\Pi_{n,d-1}^i(\gamma_\tau)$ is an element of $U$ if and only if $i=n-d-1$. Hence, we only need to show $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma_\tau)=\mathop{\mathrm{maj}}(\tau)-\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(\tau)$. But this is true by Lemma~\ref{Lem : map bounce=maj-des}.
\end{proof}
We now restate and prove our main theorem.
\begin{theo*}[1] If $\mathop{\mathrm{maj}}u\in\{(d,1^{n-d})~|~1\leq d\leq n\}$ and $\nu\vdash n$, then:
\begin{equation} \langle \nabla(e_n), s_{\mathop{\mathrm{maj}}u}\rangle|_{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{ht}}ooks}=\sum_{\tau\in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{SYT}}(\mathop{\mathrm{maj}}u)} s_{\mathop{\mathrm{maj}}(\tau)}(q,t)+\sum_{i=2}^{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(\tau)} s_{\mathop{\mathrm{maj}}(\tau)-i,1}(q,t),
\end{equation}
\begin{equation}\langle \nabla^m(e_n), s_\nu \rangle|_{1 \mathop{\mathrm{Part}}}=\sum_{\tau\in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{SYT}}(\nu)} s_{m\binom{n}{2}-\mathop{\mathrm{maj}}(\tau')}(q,0)=\sum_{\tau\in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{SYT}}(\nu)} s_{m\binom{n}{2}-\mathop{\mathrm{maj}}(\tau')}(0,t),
\end{equation}
and:
\begin{equation}\langle \nabla^m(e_n), e_n\rangle|_{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{ht}}ooks}= s_{m\binom{n}{2}}(q,t)+\sum_{i=2}^{n-1} s_{m\binom{n}{2}-i,1}(q,t).
\end{equation}
\end{theo*}
Note that $\mathop{\mathrm{maj}}(1^n)=\binom{n}{2}$, $\binom{n}{2}-\mathop{\mathrm{maj}}(\tau')=\mathop{\mathrm{maj}}(\tau)$ and $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(\tau)=n-1-\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(\tau')$.
\begin{proof}Equation~\eqref{Eq : 2 de the principale} is true by Proposition~\ref{Prop : 1 part parking}. For $m=1$ Equation~\eqref{Eq : 3 de the principale} is a direct consequence of Equation~\eqref{Eq : 1 de the principale}. Notice that if $\gamma$ is in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}_{n,d}^{(1)}$, then $\gamma E^{(m-1)n}$ is in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}_{n,d}^{(m)}$. Furthermore, the difference between the area of $\gamma$ and the area of $\gamma E^{(m-1)n}$ is exactly $(m-1)\binom{n}{2}$. Ergo, for $m>1$ Equation~\eqref{Eq : 3 de the principale} follows from Corollary~\ref{Cor : m>=1 dinv=1}.
\\
The first sum on the right side of Equation~\eqref{Eq : 1 de the principale} follows from Proposition~\ref{Prop : 1 part parking}. For the second sum on the right side of Equation~\eqref{Eq : 1 de the principale}, notice that, by Proposition~\ref{Prop : bij chemins tableaux}, there is a bijection between paths of area value equal to one with $d$ diagonal steps ending with $NE$ or $NNEE$ and the product set of standard Young tableaux of shape $(d,1^{n-d})$ and the set $\{0,\ldots,n-d-1\}$.
Moreover,one can see that the cyclic action of the map $\Pi_{n,d-1}$, proven in Proposition~\ref{Prop : partition}, puts $\Pi_{n,d-1}^{n-d-1}(\mathop{\mathrm{maj}}athcal{S}_{n,d}(\tau))$ in $\{uNED^jNNEE, vNDED^jNE ~|~ u\in \{NE,D\}^{n-d-2}, v\in \{NE,D\}^{n-d-1}\}$, since it is of order $n-d$. So, by Corollary ~\ref{Cor : aire 1 vs 1 part} and Proposition~\ref{Prop : 1 part parking}, the set $\{\Pi_{n,d-1}^{n-d-1}(\mathop{\mathrm{maj}}athcal{S}_{n,d}(\tau))\}$ contains the only paths that contribute to Schur functions indexed by partition of length $1$. Consequently, we need only to consider the set $\{0,\ldots,n-d-2\}$ because the second sum relates to partition of length $2$. Additionally, $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(\tau)=n-d$, for $\tau\in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{SYT}}(d,1^{n-d})$, hence $2\leq i\leq \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(\tau)$ if and only if $n-d-2 \geq \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(\tau)-i\geq 0$. Therefore, we sum from $2$ to $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(\tau)$. Except for the paths already contributing to Schur function having only one part, the restriction to a value of one for the area corresponds to hook-shaped Schur functions. Indeed, $s_{a,b}(q,t)=(qt)^b(q^{a-b}+q^{a-b-1}t+\cdots+qt^{a-b-1}+t^{a-b})$, and, thus, the monomial $q^ct$ can only be found when $b\in\{0,1\}$. Ergo, by Proposition~\ref{Prop : bij chemins tableaux}, we have the stated result.
\end{proof}
We conjecture this is true for all $\mathop{\mathrm{maj}}u$ when $m=1$. We also know, by Lemma~\ref{Lem : m>2 dinv=1}, that this is not true for $m>1$.
\begin{conj}\label{Conj : nabla(e_n)} For all $\mathop{\mathrm{maj}}u\vdash n$:
\begin{equation*} \langle \nabla(e_n), s_{\mathop{\mathrm{maj}}u}\rangle|_{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{ht}}ooks}=\sum_{\tau\in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{SYT}}(\mathop{\mathrm{maj}}u)} s_{\mathop{\mathrm{maj}}(\tau)}(q,t)+\sum_{i=2}^{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(\tau)} s_{\mathop{\mathrm{maj}}(\tau)-i,1}(q,t),
\end{equation*}
\end{conj}
\section{Inclusion Exclusion}\label{Sec : inclu-exclu}
In this section we will see that half the paths in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P_{n,d,(1)}$ are related to $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P_{n,d-1,(1)}$ and the other half is related to $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P_{n,d+1,(1)}$. Thereafter, this will be used to find a positive formula for an alternating sum. The outcome is needed to prove results on multivariate diagonal harmonics in \cite{[Wal2019a]}.
Recall that $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P_{n,d, (1)}=\{\gamma \in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P{}_{n,d}~|~ \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}(\gamma)=1\}$ and $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P_{n,n, (1)}=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P_{n,n-1, (1)}=\emptyset$. Now, let:
\begin{equation*}
\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}T{}_{n,d}=\{\gamma \in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P{}_{n,d,(1)}~|~ \gamma=D^jNNEE\gamma'NENE \text{ or } \gamma=\gamma'NED^jNNEE\gamma'', j\geq 0 \},
\end{equation*}
and:
\begin{equation*}
\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}B{}_{n,d}=\{\gamma \in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P{}_{n,d,(1)}~|~ \gamma=D^jNNEE\gamma'DNE \text{ or } \gamma=\gamma'NDED^jNE\gamma'', j\geq 0\}.
\end{equation*}
\begin{lem}\label{Lem : top bottom}
For $1\leq d\leq n-4$ we have the following equality $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P_{n,d, (1)}=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}T{}_{n,d}\cup \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}B{}_{n,d}$. Additionally, $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P_{n,0, (1)}=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}T{}_{n,0}$, $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P_{n,n-3, (1)}=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}T{}_{n,n-3}\cup \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}B{}_{n,n-3}\cup \{D^{n-3}NNEENE\}$ and $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P_{n,n-2, (1)}= \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}B{}_{n,n-2}\cup\{D^{n-2}NNEE\}$. Furthermore, for all $d$, $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}T{}_{n,d}\cap \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}B{}_{n,d}=\emptyset$.
\end{lem}
\begin{proof}A simple check shows that the four cases of $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}T_{n,d}$ and $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}B_{n,d}$ are mutually exclusive. Hence, $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}T{}_{n,d}\cap \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}B{}_{n,d}=\emptyset$.
The cases $d=0$, $d=n-2$ and $d=n-3$ are related to the maximal number of north steps and diagonal steps. For $d$ general, $1\leq d\leq n-4$, let $\gamma$ be in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P_{n,d,(1)}$. When a Schr\"oder path has an area value of $1$, there is a factor $\pi=NNEE$ or $\pi=NDE$ such that $\gamma=u\pi v$, $u\in \{NE,D\}^*$ and $v\in \{NE,D\}^*NE\cup \{\varepsilon\}$. By definition of $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P_{n,d,(1)}$, if $v=\varepsilon$, then $\pi=NNEE$ and $\gamma$ is in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}T{}_{n,d}$. Moreover, if $\pi=NDE$, then there is a factor $NE$ in $v$ and $\gamma$ is in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}B{}_{n,d}$. If $\pi=NNEE$ and $u=D^j$, then $v$ has at least two factors $NE$, since $d\leq n-4$, and $v$ can end with $NENE$, in which case $\gamma$ is in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}T{}_{n,d}$ or $v$ end with $DNE$ and $\gamma$ is in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}B{}_{n,d}$. If $\pi=NNEE$ and $u\not=D^j$, then there is a factor $NE$ in $u$ and $\gamma$ is in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}T{}_{n,d}$. \end{proof}
Let $d$ be an integer such that $0\leq d \leq n-1$. For each $d$ let:
\begin{align*} \rho_{n,d} :&\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}T{}_{n,d}&&\rightarrow && \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}B{}_{n,d+1}
\\ & \gamma'NED^jNNEE\gamma''&&\mathop{\mathrm{maj}}apsto &&\gamma'NDED^jNE\gamma''
\\ &D^jNNEE\gamma'NENE&&\mathop{\mathrm{maj}}apsto&&D^jNNEE\gamma'DNE
\end{align*}
\begin{lem}\label{Lem : inclu exclu chemins}Let $d$ be an integer such that $0\leq d \leq n-1$. Then, for all $d$, $\rho_{n,d}$ is a bijection such that $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\rho(\gamma))=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma)-1$.
\end{lem}
\begin{proof}Notice that $\gamma'NED^jNNEE\gamma''\not=D^kNNEE\gamma'NENE$ for all $k$ and $j$, since one has a $NE$ factor before its $NNEE$ factor and not the other. The path as an area value of one, ergo there is only one factor $NNEE$ in $\gamma$. Moreover, the map $\rho_{n,d}$ increases the number of diagonal steps by one. Therefore, for all $d$ the map $\rho_{n,d}$ is well defined. For the same reasons the map $\rho_{n,d}^{-1}$ defined by $\rho_{n,d}^{-1}(\gamma'NDED^jNE\gamma'')=\gamma'NED^jNNEE\gamma''$ and $\rho_{n,d}^{-1}(D^jNNEE\gamma'DNE)=D^jNNEE\gamma'NENE$ is well defined. Thus, it is the inverse of $\rho_{n,d}$ and we have a bijection.
Recall that the numph statistic is related to the number of peaks that are in a lower row than the diagonals. When the area value is one, all factors $NE$ and $NNEE$ contain a peak; they always return to the diagonal. If we compare the path $\gamma=\gamma'NED^jNNEE\gamma''$ and the path $\rho_{n,d}(\gamma)$, we see that only one diagonal step was added in $\rho_{n,d}(\gamma)$ and the factor $NDE$ in which it was added has its peak above the diagonal step. Hence, $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{numph}}(\rho_{n,d}(\gamma))$ is equal to $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{numph}}(\gamma)$ plus the number of peaks in $\gamma'$. Considering that $\Gamma(\gamma'NED^jNNEE\gamma'')=(NE)^{|\gamma'|+1}NNEE(NE)^{|\gamma''|}$, the value of $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\Gamma(\gamma))$ is $\binom{n-d}{2}-|\gamma''|_N-1$. Moreover, $\Gamma(\gamma'NDED^jNE\gamma'')=(NE)^{n-d-1}$; therefore, $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\Gamma(\rho_{n,d}(\gamma)))=\binom{n-d-1}{2}$. Because the number of peaks in $\gamma'$ is equal to $|\gamma'|_E$ which is equal to $|\gamma'|_N$ and $|\gamma'|_N+|\gamma''|_N=n-d-3$, we obtain $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma)-\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\rho_{n,d}(\gamma))=1$.
Now comparing the paths $\gamma=D^jNNEE\gamma'NENE$ and $\rho_{n,d}(\gamma)$ we see that $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{numph}}(\rho_{n,d}(\gamma))$ is equal to $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{numph}}(\gamma)$ plus the number of peaks in $D^jNNEE\gamma'$. This is equivalent to counting the number of non-consecutive east steps in $D^jNNEE\gamma'$, that is $n-d-3$, since there are $n-d$ east steps in $\gamma$. We know $\Gamma(D^jNNEE\gamma'NENE)=NNEE(NE)^{n-d-2}$. The value of $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\Gamma(\gamma))$ is $\binom{n-d}{2}-(n-d-1)$. Moreover, in we consider the restriction to north steps and east steps is $\Gamma(D^jNNEE\gamma'DNE)=NNEE(NE)^{n-d-3}$, then $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\Gamma(\rho_{n,d}(\gamma)))=\binom{n-d-1}{2}-(n-d-2)$. Hence, $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma)-\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\rho_{n,d}(\gamma))=1$.
\end{proof}
We now know some Schr\"oder paths of area 1 are associated to Schur functions in the variables $q$ and $t$ indexed by partitions of length one. This corollary is merely to show that the bijection in the previous lemma sends paths associated to Schur functions indexed by a length one partition to another path associated to Schur functions indexed by a length one partition.
\begin{cor}\label{Cor : enlever chemin 1 part} Let $\gamma$ be a path of $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}T_{n,d}$. Then, $\gamma$ contributes to a Schur function indexed by a partition of length $1$ in $\langle \nabla(e_n), s_{d+1,1^{n-d-1}}\rangle$ if and only if $\rho_{n,d}(\gamma)$ contributes to a Schur function indexed by a partition of length $1$ in $\langle \nabla(e_n), s_{d+2,1^{n-d-2}}\rangle$. Moreover, if $\gamma$ satisfies this property, then $\gamma=\gamma'NED^jNNEE$ and $\rho_{n,d}(\gamma)=\gamma'NDED^jNE$, with $\gamma'\in \{NE,D\}^*$.
\end{cor}
\begin{proof}By Corollary~\ref{Cor : aire 1 vs 1 part}, the definition of $\rho_{n,d}$ and the definition of $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}T_{n,d}$.
\end{proof}
\begin{lem}\label{Lem : autre somme}
Let $d$ and $n$ be integers such that $0\leq d \leq n-2$. Then, $\mathop{\mathrm{maj}}athcal{T}_{n,d+2}\circ \rho_{n,d}\circ \mathop{\mathrm{maj}}athcal{S}_{n,d+1}$ is a bijection between the sets of tableaux:
\begin{equation*}
\{\tau \in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{SYT}}(d+1,1^{n-d-1})~|~ \{1,2\} \subseteq\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(\tau)\}\simeq \{\tau \in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{SYT}}(d+2,1^{n-d-2})~|~1\in\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(\tau), 2\not\in\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(\tau)\},
\end{equation*}
such that $\mathop{\mathrm{maj}}(\tau)-\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(\tau)=\mathop{\mathrm{maj}}(\mathop{\mathrm{maj}}athcal{T}_{n,d+2}\circ \rho_{n,d}\circ \mathop{\mathrm{maj}}athcal{S}_{n,d+1}(\tau))-\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(\mathop{\mathrm{maj}}athcal{T}_{n,d+2}\circ \rho_{n,d}\circ \mathop{\mathrm{maj}}athcal{S}_{n,d+1}(\tau))+1$.
Additionally, $\mathop{\mathrm{maj}}athcal{Q}_{n,d+2}^{-1}\circ \rho_{n,d}\circ \mathop{\mathrm{maj}}athcal{Q}_{n,d+1}$ is a bijection between the set :
\begin{equation*}
\{(\tau,i) \in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{SYT}}(d+1,1^{n-d-1})\times\{1,\ldots,n-d-3\}~|~ 1\in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(\tau)\},
\end{equation*}
and the set :
\begin{equation*}
\{(\tau,i) \in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{SYT}}(d+2,1^{n-d-2})\times\{0,\ldots,n-d-4\}~|~1\not\in\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(\tau)\},
\end{equation*}
such that $\mathop{\mathrm{maj}}(\tau)-i=\mathop{\mathrm{maj}}(\mathop{\mathrm{maj}}athcal{Q}_{n,d+2}^{-1}\circ \rho_{n,d}\circ \mathop{\mathrm{maj}}athcal{Q}_{n,d+1}(\tau,i))-(i-1)$.
\end{lem}
\begin{proof}By Corollary~\ref{Cor : aire 1 vs 1 part} we know paths of shape $\gamma'NDED^jNE$ or $\gamma'NED^jNNEE$ are associated to a Schur function of length one. Notice that by definition $\rho_{n,d}$ sends paths of theses shapes to other paths of theses shapes. Hence, $\mathop{\mathrm{maj}}athcal{Q}_{n,d}^{-1}\circ \rho_{n,d}\circ \mathop{\mathrm{maj}}athcal{Q}_{n,d}(\tau, n-d-2)=(\tau',n-d-3)$, where $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(\tau')=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(\tau)\backslash\{1\}$. At this point one only needs to notice that $\mathop{\mathrm{maj}}athcal{Q}_{n,d+2}^{-1}\circ \rho_{n,d}\circ \mathop{\mathrm{maj}}athcal{Q}_{n,d+1}(\tau,i)=(\tau',i-1)$, $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(\tau')=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(\tau)\backslash\{1\}$ and $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(\mathop{\mathrm{maj}}athcal{T}_{n,d+2}\circ \rho_{n,d}\circ \mathop{\mathrm{maj}}athcal{S}_{n,d+1}(\tau))=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(\tau)\backslash\{2\}$ . The rest of the proof follows Proposition~\ref{Prop : bij chemins tableaux} from the definition of the maps $\rho_{n,d}$, $\mathop{\mathrm{maj}}athcal{Q}_{n,d}$, $\mathop{\mathrm{maj}}athcal{S}_{n,d}$ and $\mathop{\mathrm{maj}}athcal{T}_{n,d}$ .
\end{proof}
\begin{prop}
Let $k$ be an integer such that $0\leq k\leq n-3$ and $\psi : \Lambda_\mathop{\mathrm{maj}}athbb{Q} \rightarrow \mathop{\mathrm{maj}}athbb{Q}[q,t]$ be a linear map defined by $\psi(s_\lambda)=q^{|\lambda|}t^{\ell(\lambda)-1}$. If:
\begin{equation*}
h_k(q):=\psi\left(\sum_{d=0}^{k} (-1)^{k-d} \langle \nabla(e_n), s_{d+1,1^{n-d-1}}\rangle |_\textrm{pure hook}\right)q^{-k+d}t^{-1},
\end{equation*}
then:
\begin{align*}h_k(q)&=\underset{\{1,2\}\subseteq \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(\tau)}{\sum_{\tau \in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{SYT}}(k+1,1^{n-k-1})}} q^{\mathop{\mathrm{maj}}(\tau)-\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(\tau)}
+\underset{1\in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(\tau)}{\sum_{\tau \in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{SYT}}(k+1,1^{n-k-1})}} \sum_{i=2}^{n-k-2} q^{\mathop{\mathrm{maj}}(\tau)-i},
\\ &=\underset{1\in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(\tau), 2\not\in\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(\tau)}{\sum_{\tau \in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{SYT}}(k+2,1^{n-k-2})}} q^{\mathop{\mathrm{maj}}(\tau)-\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(\tau)}
+\underset{1\not\in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Des}}(\tau)}{\sum_{\tau \in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{SYT}}(k+2,1^{n-k-2})}} \sum_{i=2}^{n-k-2} q^{\mathop{\mathrm{maj}}(\tau)-i},
\end{align*}
where the restriction $|_\textrm{pure hook}$ is the restriction to Schur function indexed by partitions $(a,1)$, with $a\geq 1$.
\end{prop}
\begin{proof}Let $\tilde h_k(q)=\psi\left(\sum_{d=0}^{k} (-1)^{k-d} \langle \nabla(e_n), s_{d+1,1^{n-d-1}}\rangle |_\textrm{hook}\right)q^{-k+d}t^{-1}$. Due to Haglund's theorem, Lemma~\ref{Lem : inclu exclu chemins} and Lemma~\ref{Lem : top bottom}, for an integer $k$ we have:
\begin{align*}
\tilde h_k(q)&=\sum_{d=0}^{k} (-1)^{k-d}\sum_{\gamma \in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}T_{n,d}} q^{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma)-k+d}+\sum_{\gamma \in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}B_{n,d}} q^{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma)-k+d},
\\ &=\sum_{d=0}^{k} (-1)^{k-d}\sum_{\gamma \in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}T_{n,d}} q^{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\rho_{n,d}(\gamma))+1-k+d}+\sum_{d=1}^{k} (-1)^{k-d}\sum_{\gamma \in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}B_{n,d}} q^{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma)-k+d},
\\ &=\sum_{d=0}^{k} (-1)^{k-d}\sum_{\gamma \in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}B_{n,d+1}} q^{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma)+1-k+d}-\sum_{d=1}^{k} (-1)^{k-d+1}\sum_{\gamma \in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}B_{n,d}} q^{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma)-k+d},
\\ &=\sum_{d=0}^{k} (-1)^{k-d}\sum_{\gamma \in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}B_{n,d+1}} q^{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma)+1-k+d}-\sum_{d=0}^{k-1} (-1)^{k-d}\sum_{\gamma \in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}B_{n,d+1}} q^{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma)-k+d+1},
\\ &=\sum_{\gamma \in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}B_{n,k+1}} q^{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma)+1},
\\ &=\sum_{\gamma \in \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}T_{n,k}} q^{\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma)}.
\end{align*}
The polynomial $h_k(q)$ only takes into account pure hooks, so we only need to consider the paths $\gamma'NED^jNNEE\gamma''$ and $D^jNNEE\gamma'NENE$, with $\gamma'$, $\gamma''\in \{NE,D\}$, as shown in Corollary~\ref{Cor : enlever chemin 1 part}. Thus, by Lemma~\ref{Lem : map bounce=maj-des}, the map $\mathop{\mathrm{maj}}athcal{T}_{n,k+1}$, for $\gamma=D^jNNEE\gamma'NENE$, gives us $\mathop{\mathrm{maj}}athcal{T}_{n,k+1}(\gamma)$ is a tableau containing $\{1,2\}$ in its descent set and such that $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma)=\mathop{\mathrm{maj}}(\mathop{\mathrm{maj}}athcal{T}_{n,k+1}(\gamma))-\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(\mathop{\mathrm{maj}}athcal{T}_{n,k+1}(\gamma))$. Additionally, for $\gamma=\gamma'NED^jNNEE\gamma''$, $\mathop{\mathrm{maj}}athcal{Q}_{n,k+1}^{-1}(\gamma)$ is a tableau containing $\{1\}$ in its descent set. By Proposition~\ref{Prop : bij chemins tableaux}, $\gamma$ is such that $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{bounce}}(\gamma)=\mathop{\mathrm{maj}}(\mathop{\mathrm{maj}}athcal{Q}_{n,k+1}^{-1}(\gamma))-i$, $2\leq i \leq \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(\mathop{\mathrm{maj}}athcal{Q}_{n,k+1}^{-1}(\gamma))$. Moreover, if $\{1\}$ is in the descent set of $\tau$, then the map $\mathop{\mathrm{maj}}athcal{Q}_{n,k+1}$ send $(\tau,0)$ to $D^jNNEE\gamma'NENE$ or $D^jNNEE\gamma'DNE$ the first one was already considered and the last one is in $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}B_{n,d}$. Finally, if $\{1\}$ is in the descent set of $\tau$, then the map $\mathop{\mathrm{maj}}athcal{Q}_{n,k+1}$ send $(\tau,i)$ to $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}T_{n,d}$ for all $1\leq i\leq n-d-3$. Hence, we sum over $2\leq i \leq \mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(\mathop{\mathrm{maj}}athcal{Q}_{n,k+1}^{-1}(\gamma))-1$. The second sum is a consequence of Lemma~\ref{Lem : autre somme}.
\end{proof}
Considering what is known about multivariate diagonal harmonics, it should be possible to extend the results of this section to the case for $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P_{n,d,(i)}$. This generalization would lead to more results on multivariate diagonal harmonics.
\section{Partial Crystal Decomposition}\label{Sec : cristaux}
This section is mainly to explain the underlying idea throughout this paper. We can see this as finding the crystal decomposition of the Schr\"oder paths and the parking function (since their weighted sum relate to modules). We basically found some of the top weight and for some of them gave a map that gives the remainder of the crystal. In that setting we can say that for $m=1$, we can describe all the crystals in the case where the Schur functions are indexed by partitions of length one. When $m>1$, we can characterize only the top weights. For hooked-shaped Schur functions, we can only depict the top weight, when $m=1$.
More precisely, the maps $\mathop{\mathrm{maj}}athcal{R}_{n,d}$ and $\mathop{\mathrm{maj}}athcal{T}_{n,d}$ determine in which crystal the paths lie. The maps $\tilde\varphi$, defined by the map $\varphi$ in Section~\ref{Sec : algo}, give the decomposition according to the top weight. This also is well defined, since for all $\gamma\in\{NE,D\}^{n-1}NE$ we have $\mathop{\mathrm{maj}}athcal{T}_{n,d}\circ\Pi\circ\tilde\varphi(\gamma)=\mathop{\mathrm{maj}}athcal{R}_{n,d}(\gamma)$ (see Figure~\ref{Fig : cristaux bien def} for an example). Notice that map $\mathop{\mathrm{maj}}athcal{M}_{n,d}$ (respectively,$\mathop{\mathrm{maj}}athcal{Q}_{n,d}$) tells us in which crystal component are the Schr\"oder paths of area value $0$ (respectively,area value $1$).
\begin{figure}\label{Fig : cristaux bien def}
\end{figure}
Using the zeta map, so far we know the top weights for all crystals containing a parking function having a diagonal inversion statistic value of $0$, and for all hook-shaped partitions we know all top weights for all crystals containing a parking function having a diagonal inversion statistic value of $1$. For crystals containing a parking function having a diagonal inversion statistic value of $0$ that are not associated to a hook-shaped partition, we do not know the exact paths. Although, we do know in which subset of parking functions the lowest weight lie. Figure~\ref{Fig : cristaux} gives an overview of what is known so far.
\begin{figure}
\caption{The nodes represent paths. Each chain is associated to a Schur function in the variables $q$ and $t$. The height of the first node determines which Schur function. The partitions determine the Schur function in the variables $X$. Each chain can be associated to a Standard Young tableau corresponding to the shape of the partition. More than one chain can be associated with the same tableau. When nodes are in black, we know which paths they relate to, in red we do not.}
\label{Fig : cristaux}
\end{figure}
\section{Conclusion and Further Questions}\label{Sec : conclu}
Proving Conjecture~\ref{Conj : nabla(e_n)} would be a great start. Moreover, can one describe nicely the algorithm described in Section~\ref{Sec : algo}, in terms of diagonal inversions and extend it to $m$-Schr\"oder paths? Here we are looking for more than just applying the zeta map. This would allow us to know exactly what paths contribute to each Schur function, even when $m\not=1$. It might be easier to start by the following problem:
\begin{prob} Using the bounce statistic, generalize the algorithm in Section~\ref{Sec : algo} to all Schr\"oder paths.
\end{prob}
This would give the Schr\"oder paths associated to all the Schur functions and not only the one with one part. It would also answer completely Haglund's open problem 3.11 of \cite{[H2008]}. One could also generalize the algorithm for labelled Dyck paths, relating to the Delta conjecture, and get a partial decomposition in Schur functions in the variables $q$ and $t$ indexed by partitions of length one.
Using Corollary~\ref{Cor : m>1 dinv=1} it should be possible to decompose $\nabla^m(e_n)$ into the basis $s_\mathop{\mathrm{maj}}u(q,t)s_\lambda(X)$.
The following problems could lead to finding a partial decomposition $\nabla^m(e_n)$ into Schur functions in the $X$, $s_\lambda(X)$, when $\lambda$ is not a hook. Which is a known hard problem.
\begin{prob}Using Lemma~\ref{Lem : m>2 dinv=1} decompose $\nabla^m(e_n)$ into the basis $s_\mathop{\mathrm{maj}}u(q,t)F_c(X)$, for $\mathop{\mathrm{maj}}u$ a hook and $c$ a composition.
\end{prob}
Even if the decomposition is in fundamental quasisymmetric functions, it would help get a partial decomposition $\nabla^m(e_n)$ into Schur functions, since it should be easier to regroup the fundamental quasisymmetric functions into Schur functions because there will be fewer coefficients.
Extending, the maps in this paper from $m$-Schr\"oder paths to tableaux with the multiplicity of the descent set, would help decompose completely the Schr\"oder paths into crystals. Of course the extended map must somewhat preserve the area and diagonal inversion statistic or area and bounce statistic through the major index and the number of descents. Another related problem is:
\begin{prob}\label{Prob : solution conj}Find a bijection between parking functions $(\gamma,w)$, in an $n\times n$, grid, having diagonal inversion statistic value of $1$ and Standard Young tableaux, $\tau$, with a multiplicity related to the major index of the tableau ($\mathop{\mathrm{maj}}(\tau, i)-i-1$) such that $\mathop{\mathrm{maj}}(\tau, i)-i=\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{area}}(\mathop{\mathrm{maj}}athcal{B}(\tau,i))$.
\end{prob}
Using the zeta map, we already have a bijection for such $n\times n$ parking functions $(\gamma,w)$ such that $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{read}}(\gamma,w)\in \{n-d+1,\ldots,n\}\shuffle\{n-d,\ldots,1\}$. The idea here is to “extend” that bijection, with multiplicity. We need the multiplicity because there are more than just the parking functions $(\gamma,w)$ such that $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{read}}(\gamma,w)\in \{n-d+1,\ldots,n\}\shuffle\{n-d,\ldots,1\}$, that contribute to $\nabla (e_n)$, seen as a sum of parking functions. The said paths are merely representatives. If the solution to Problem~\ref{Prob : solution conj} is indeed an extension of $\zeta\circ\mathop{\mathrm{maj}}athcal{Q}_{n,d}$ this would solve Conjecture~\ref{Conj : nabla(e_n)}.
As mentioned in Section~\ref{Sec : inclu-exclu} the insight coming from multivariate diagonal harmonics foresees a solution to the problem:
\begin{prob} Find a general map $\rho_{n,d}^{(i)}$ that partitions $\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{Sch}}P_{n,d,(i)}$.
\end{prob}
This generalization would lead to more results on combinatorial formulas for multivariate diagonal harmonics, like the one in \cite{[Wal2019d]}. Actually, any explicit decomposition in terms of Schur functions of parking functions could be lifted with the tools discussed in \cite{[Wal2019a]} (the long version of \cite{[Wal2019d]}) and give an explicit combinatorial formula for a partial Schur decomposition of the multivariate diagonal harmonics.
Finally, Proposition~\ref{Prop : 1 part parking} suggests a bijection between permutations and tableaux with a multiplicity such that $\binom{n}{2}-\mathop{\mathrm{maj}}athop{\mathop{\mathrm{maj}}athrm{des}}(w)n+\mathop{\mathrm{maj}}(w)=\mathop{\mathrm{maj}}(\tau)$. Research on this last problem could lead to a decomposition of $\nabla^m(e_n)$ altogether. Since it further our knowledge of how fundamental quasi-symmetric functions index by permutations relate to Schur functions.
\begin{prob}Find a combinatorial proof of Proposition~\ref{Prop : 1 part parking}.
\end{prob}
\section*{Acknowledgments}
Thank you to my advisor, Fran\c cois Bergeron, for introducing me to this beautiful subject and other helpful comments.
\end{document}
|
\betagin{document}
\betagin{frontmatter}
\title{Rosenthal-type inequalities for martingales in 2
-smooth Banach spaces}
\runtitle{Martingales in $2$-smooth spaces}
\betagin{aug}
kuthor{\mathbf{f}nms{Iosif} \snm{Pinelis}\thetaanksref{t2}\ead[label=e1]{[email protected]}
}
\thetaankstext{t2}{Supported by NSA grant H98230-12-1-0237}
\runauthor{Iosif Pinelis}
kddress{Department of Mathematical Sciences\\
Michigan Technological University\\
Houghton, Michigan 49931, USA\\
E-mail: \printead[[email protected]]{e1}}
\end{aug}
\betagin{abstract}
Certain previously known upper bounds on the moments of the norm of martingales in 2-smooth Banach spaces are improved. Some of these improvements hold even for sums of independent real-valued random variables. Applications to concentration of measure on product spaces for
separately Lipschitz functions are presented, including ones concerning the central moments of the norm of the sums of independent random vectors in any separable Banach space.
\end{abstract}
\betagin{keyword}[class=AMS]
\kwd{60E15}
\kwd{60B11}
\end{keyword}
\betagin{keyword}
\kwd{probability inequalities}
\kwd{bounds on moments}
\kwd{Rosenthal inequality}
\kwd{2-smooth Banach spaces}
\kwd{Hilbert spaces}
\kwd{martingales}
\kwd{sums of independent random variables}
\kwd{concentration of measure}
\kwd{separately Lipschitz functions}
\kwd{product spaces}
\end{keyword}
cations: Primary 60E15; secondary 46B09.
\end{frontmatter}
\searrowttocdepth{chapter}
\searrowttocdepth{subsubsection}
\thetaeoremstyle{plain}
\searrowction{Introduction, summary, and discussion}\lambdabel{intro}
Let $(S_i)_{i=0}^\infty$ be a martingale in a separable Banach space $({\mathfrak{X}},\|\cdot\|)$ relative to a filter $(\mathcal{F}_i)_{i=0}^\infty$ of $\sigma$-algebras.
Assume that $S_0=0$ and the differences $X_i:=S_i-S_{i-1}$
satisfy the condition
\betagin{equation}\lambdabel{eq:cond}
\operatorname{\mathsf{E}}_{i-1}\|X_i\|^2\leqslantslant b_i^2
\end{equation}
almost surely for all $i\in\intr1\infty$, where
$\operatorname{\mathsf{E}}_{i-1}$ denotes the conditional expectation given $\mathcal{F}_{i-1}$ and
the $b_i$'s are some positive real numbers.
Here and in what follows, for any $m$ and $n$ in $\mathbb{Z}\cup\{\infty\}$ we let $\intr mn:=\{i\in\mathbb{Z}\colon m\leqslantslant i\leqslantslant n\}$.
Also introduce
\betagin{equation*}
k it:=\operatorname{\mathsf{E}}\|X_i\|^t, \quad \mathcal{A} nt:=\sum\limits_{i\in\intr1n} a_i(t), \quad\text{and}\quad B_n:=\sqrt{\sum_{i\in\intr1n} b_i^2}
\end{equation*}
for all real $t\gammammae0$, $i\in\intr1\infty$, and $n\in\intr0\infty$; as usual, assume that the sum and product of an empty family are $0$ and $1$, respectively; let also $0^0:=1$.
Assume further that the Banach space $({\mathfrak{X}},\|\cdot\|)$ is 2-smooth \big(or, more exactly, $(2,D)$-smooth, for some $D = D({\mathfrak{X}} ) >0$\big), in the following sense:
\betagin{equation}\lambdabel{eq:2-smooth}
\| x+y\| ^2 + \| x-y\| ^2 \leqslantslant 2\| x\| ^2 +2D^2\| y\| ^2
\end{equation}
for all $x$ and $y$ in ${\mathfrak{X}}$.
The importance of the 2-smooth spaces was elucidated in
\cite{pisier75}: they play the same role with respect to the vector martingales as the spaces of type 2 do with respect to the sums of independent random vectors.
The definition of 2-smooth spaces assumed in this paper is the same as that in \cite{pin94}, which is slightly different from the one given
in \cite{pisier75} -- which latter required only that \eqref{eq:2-smooth} hold for an equivalent norm; the reason for the modified definition is that
we would like to follow the dependence of certain constants on $D$, the constant of 2-smoothness.
Substituting $\lambdambda x$ for $y$ in \eqref{eq:2-smooth}, where $\lambdambda \in \mathbb{R}$, one observes that necessarily $D\gammammae 1$, except for ${\mathfrak{X}}=\{0\}$.
As shown in \cite{pin94,wellner_nemir}, the space $L^p (T, {\cal A}, \nu)$ is $(2,\sqrt {p-1} \, )$-smooth for any $p\gammammae 2$ and any measure space
$(T,{\cal A}, \nu)$.
In particular, what is obvious and well-known, if ${\mathfrak{X}} $ is a Hilbert space then it is (2,1)-smooth.
\betagin{theorem}\lambdabel{th:}
Take any real $t\gammammae0$ and $n\in\intr0\infty$. Then
\betagin{equation}\lambdabel{eq:}
\betagin{aligned}
\operatorname{\mathsf{E}}\|S_n\|^t\leqslantslant
\sum_{j\in\intr0{m-1}}\c jt & \sum_{J\in\mathcal{J}_{n,j}} \mathcal{A}{\mu_n(J)-1}{t-2j} \prod_{i\in J}b_i^2 \\
+{\tilde{c}} mt & \sum_{J\in\mathcal{J}_{n,m}} \big(\mathcal{A}{\mu_n(J)-1}2\big)^{t/2-m} \prod_{i\in J}b_i^2,
\end{aligned}
\end{equation}
where $m:=\leqslantslantft\lfloorloor\mathbf{f}rac t2\right\rfloorloor$,
$\mathcal{J}_{n,j}$
denotes the set of all subsets of the set $\intr1n$ of cardinality $j$,
$\mu_n(J):=\min J$ if $J\ne\emptyset$, $\mu_n(\emptyset):=n+1$,
\betagin{align*}
\c jt:=
\mathbf{f}rac{t-2j-2+D^2}{t-2j-1}\,q(t-2j)
\prod_{k\in\intr0{j-1}}\mathbf{f}rac{(t-2k)(t-2k-2+D^2)\,p(t-2k)}2, \\
{\tilde{c}} mt:=
\prod_{j\in\intr0{m-1}}\mathbf{f}rac{(t-2j)(t-2j-2+D^2)\,p(t-2j)}2,
\end{align*}
and $p(\cdot)$ and $q(\cdot)$ are any functions such that
\betagin{equation}
\betagin{gathered}\lambdabel{eq:p(s),q(s)}
p(2)+q(2)\gammammae1,\quad p(2)>0,\quad q(2)>0, \\
p(s)\gammammae1 \quad\text{and}\quad q(s)\gammammae1\quad\text{for all}\quad s>2, \\
p(s)^{1/(3-s)}+q(s)^{1/(3-s)}\leqslantslant1\quad\text{for all}\quad s>3.
\end{gathered}
\end{equation}
\end{theorem}
Almost the same result was obtained in \cite{pin80} in the special case when ${\mathfrak{X}}$ is a Hilbert space (in which case one can take $D=1$) -- except that,
in place of the term $\mathcal{A}{\mu_n(J)-1}2$ in \eqref{eq:} above, the bound in \cite{pin80} contained the larger term $\mathcal{A} n2-\sum_{i\in J}k i2$.
The proofs, whenever necessary, are deferred to Section~\ref{proofs}.
\betagin{corollary}\lambdabel{cor:}
In the conditions of Theorem~\ref{th:}, let $t>2$ and take any positive real numbers $\lambda_j$ for all $j\in\intr0{m-1}$. Then
\betagin{equation}\lambdabel{eq:cor}
\operatorname{\mathsf{E}}\|S_n\|^t\leqslantslant C_A^{(t)}\mathcal{A} nt+C_B^{(t)}B_n^t,
\end{equation}
where
\betagin{equation}\lambdabel{eq:CA,CB}
\betagin{aligned}
C_A^{(t)}&:=\sum_{j\in\intr0{m-1}}\c jt \,\mathbf{f}rac{t-2j-2}{t-2}\mathbf{f}rac1{\lambda_j^{2j}j!}
\quad\text{and} \\
C_B^{(t)}&:={\tilde{c}} mt \prod_{j\in\intr1m}\mathbf{f}rac1{t/2-m+j} + \,\sum_{j\in\intr0{m-1}}\c jt\,\mathbf{f}rac{2j}{t-2}\mathbf{f}rac{\lambda_j^{t-2j-2}}{j!}.
\end{aligned}
\end{equation}
\end{corollary}
In the special case when ${\mathfrak{X}}$ is a Hilbert space, a similar but somewhat less precise bound was obtained in \cite[Corollary]{pin80}. Namely, in place of the terms $\mathbf{f}rac{t-2j-2}{t-2}$, $\mathbf{f}rac1{t/2-m+j}$, and $\mathbf{f}rac{2j}{t-2}$ in \eqref{eq:CA,CB}, the bound in \cite[Corollary]{pin80} contains the larger terms $1$, $\mathbf{f}rac1j$, and $1$, respectively. \\
Results somewhat similar to Corollary~\ref{cor:} were also obtained in \cite{pin94}, by rather different methods. Particularly, \cite[Theorem~4.1]{pin94} implies the ``spectrum'' of inequalities
\betagin{equation}\lambdabel{eq:pin94}
\operatorname{\mathsf{E}}\|S_n\|^t\leqslantslant K^t\big(c^t\mathcal{A} nt+c^{t/2}e^{t^2/c}D^tB_n^t\big)
\end{equation}
for some positive absolute constant $K$ and all real $t\gammammae2$, depending on the freely chosen value of the ``balancing'' parameter $c\in[1,t]$.
One can use this freedom to minimize or quasi-minimize the bound in \eqref{eq:pin94} in $c\in[1,t]$, depending on the value of the ratio $\mathcal{A} nt/B_n^t$ -- as is done in \cite{pin94}; see Theorem~6.1 and the definition of $B_p^*$ on page~1693 there.
In fact, \cite[Theorems~6.1 and 6.2]{pin94} show that
\betagin{enumerate}[(i)]
\item for each $t\gammammae2$ and each value of the ratio $\mathcal{A} nt/B_n^t$, the minimum in $c\in[1,t]$ of the upper bound in \eqref{eq:pin94} is an optimal (up to a factor of the form $K^t$) upper bound on $\operatorname{\mathsf{E}}\|S_n\|^t$;
\item the spectrum of the bounds in \eqref{eq:pin94} is also ``minimal'' in the sense that for each $c\in[1,t]$ there exists a value of the ratio $\mathcal{A} nt/B_n^t$ such that the corresponding ``individual'' bound in \eqref{eq:pin94} is the best possible (again up to a factor of the form $K^t$);
\item the above statements (i) and (ii) hold if the term(s) $\mathcal{A} nt$ and/or $B_n^t$ are/is replaced, respectively, by $\operatorname{\mathsf{E}}\max_{i\in\intr1n}\|X_i\|^t$ and/or $\operatorname{\mathsf{E}}\big(\sum_1^n\operatorname{\mathsf{E}}_{i-1}\|X_i\|^2
\big)^{t/2}$.
\end{enumerate}
One can similarly use the balancing parameters $\lambda_j$ in \eqref{eq:CA,CB} to minimize or quasi-minimize the bound in \eqref{eq:cor}.
Moreover,
by Remark~6.8 in \cite{pin94} (with details given by \cite[Proposition~9.2
]{pin94-arxiv}), the minimum of the bound on $\operatorname{\mathsf{E}}\|S_n\|^t$ in \cite[Theorem~1]{pin80} with respect to the corresponding balancing parameters is equivalent (once again up to a factor of the form $K^t$) to the minimum in $c\in[1,t]$ of the bound in \eqref{eq:pin94} -- at least when the condition \eqref{eq:cond} holds; hence, the same is true for the bounds in \eqref{eq:cor} and \eqref{eq:}, which therefore possess the same up-to-the-$K^t$-factor optimality property.
Thus, the main advantage of inequality \eqref{eq:pin94} over \eqref{eq:cor} and \eqref{eq:} is that the former does not require the condition \eqref{eq:cond}. On the other hand, the bounds in \eqref{eq:cor} and \eqref{eq:} are quite explicit and do not contain unspecified constants such as $K$ in \eqref{eq:pin94}.
If one follows the lines of the proof of \eqref{eq:pin94} in \cite{pin94} without serious efforts at modification, the resulting value of $K$ turns out to be large, namely equal $120$, quite in contrast with the constant factors in \eqref{eq:cor} and \eqref{eq:} -- cf.\ e.g.\ the bound in \eqref{eq:t=3} below, which is a particular case of \eqref{eq:cor}.
Moreover, \eqref{eq:cor} and \eqref{eq:} appear to provide a better dependence of the bounds on the 2-smoothness constant $D$.
As for the condition \eqref{eq:cond}, it turns out quite naturally satisfied in applications to concentration of measure on product spaces for separately Lipschitz functions and, in particular, for the norm of the sums of independent random vectors -- cf.\ e.g.\ \cite{re-center} and further bibliography there. Such an application is provided in Section~\ref{concentr} of the present note.
Special cases of Corollary~\ref{cor:} are the following inequalities:
\betagin{align}
\operatorname{\mathsf{E}}\|S_n\|^t &\leqslantslant\mathbf{f}rac{t-2+D^2}{t-1}\big(\mathcal{A} nt+(t-1)B_n^t\big)\quad\text{for all}\quad t\in(2,3]\quad\text{and} \lambdabel{eq:2<t<3} \\
\operatorname{\mathsf{E}}\|S_n\|^t &\leqslantslant\mathbf{f}rac{t-2+D^2}{t-1}\mathcal{B}ig(\mathbf{f}rac{\mathcal{A} nt}{klpha^{t-3}}
+(t-1)\mathbf{f}rac{B_n^t}{(1-klpha)^{t-3}}\mathcal{B}ig)\quad\text{for all}\quad t\in[3,4] \lambdabel{eq:3<t<4}
\end{align}
and $klpha\in(0,1)$.
Minimizing the upper bound in \eqref{eq:3<t<4} in $klpha\in(0,1)$, one can combine \eqref{eq:2<t<3} and \eqref{eq:3<t<4} into the inequality
\betagin{equation}\lambdabel{eq:2<t<4,min}
\operatorname{\mathsf{E}}\|S_n\|^t\leqslantslant\mathbf{f}rac{t-2+D^2}{t-1}\big[\big(\mathcal{A} nt\big)^{1/s_t}
+(t-1)^{1/s_t}B_n^{t/s_t}\big]^{s_t}\quad\text{for all}\quad t\in(2,4],
\end{equation}
where $s_t:=\max(1,t-2)$.
Note that the coefficient of $\mathcal{A} nt$ in \eqref{eq:2<t<3} cannot be less than $1$,
whatever the coefficient of $B_n^t$ in \eqref{eq:2<t<3} may be; for instance, take $n=1$ and let $X_1$ be such that $\operatorname{\mathsf{P}}(X_1=0)=1-2p$ and $\operatorname{\mathsf{P}}(X_1=1)=\operatorname{\mathsf{P}}(X_1=-1)=p$, with $p\mathrm{d}ownarrow0$.
So, the coefficient of $\mathcal{A} nt$ in \eqref{eq:2<t<3} takes the optimal value $1$ when $D=1$.
Consider the now the case when the latter condition holds, that is, when ${\mathfrak{X}}$ is a Hilbert space.
Then the behavior of the upper bound in \eqref{eq:2<t<4,min} depends mainly on the ratio $r:=r_n^{(t)}:=\mathcal{A} nt/B_n^t$.
If $r\to\infty$ \big(which happens e.g.\ in the situation described in the previous paragraph) then the upper bound in \eqref{eq:2<t<4,min} is asymptotic to $\mathcal{A} nt$ and thus is asymptotically optimal.
If $r\mathrm{d}ownarrow0$ \big(which happens e.g.\ in the important case when $n\to\infty$, the $X_i$'s are iid, $t$ is fixed, and $\operatorname{\mathsf{E}}|X_1|^t<\infty$\big) then the upper bound in \eqref{eq:2<t<4,min} is asymptotic to $(t-1)B_n^t$.
On the other hand, by the central limit theorem, this upper bound cannot be less than $\operatorname{\mathsf{E}}|Z|^t B_n^t$, where $Z$ is a standard normal r.v. Since $\operatorname{\mathsf{E}}|Z|^t=t-1$ for $t\in\{2,4\}$, it follows that the upper bound in \eqref{eq:2<t<4,min} is asymptotically optimal when $r\mathrm{d}ownarrow0$ and at that either $t\mathrm{d}ownarrow2$ or $t\to4$.
The graph of the ratio of $t-1$ to $\operatorname{\mathsf{E}}|Z|^t$ in Figure~\ref{fig:rat} shows that the asymptotic coefficient $t-1$ at $B_n^t$ in \eqref{eq:2<t<4,min} for $r\mathrm{d}ownarrow0$ is rather close to optimality for all $t\in(2,4]$.
\betagin{figure}[h]
\centering \includegraphics[width=.7\textwidth]{pic_rat.pdf}
\caption{Graph of the ratio of $t-1$ to $\operatorname{\mathsf{E}}|Z|^t$ for $t\in(2,4]$. }
\lambdabel{fig:rat}
\end{figure}
Again when ${\mathfrak{X}}$ is a Hilbert space, \eqref{eq:2<t<3} and \eqref{eq:3<t<4} with $klpha=\mathbf{f}rac12$ yield
\betagin{equation}\lambdabel{eq:2<t<4,H}
\operatorname{\mathsf{E}}\|S_n\|^t\leqslantslant2^{(t-3)_+}\big(\mathcal{A} nt+(t-1)B_n^t\big)\quad\text{for all}\quad t\in(2,4],
\end{equation}
where $u_+:=\max(0,u)$. This latter result was obtained, by a more Stein-like method, in \cite[Lemma~6.3]{chen-shao05} in the case when ${\mathfrak{X}}=\mathbb{R}$, $t\in(2,3]$, and the $X_i$'s are independent.
In the case when ${\mathfrak{X}}=\mathbb{R}$, $t=3$, and the $X_i$'s are iid, inequality \eqref{eq:2<t<4,H} was stated without proof in \cite[page~341]{novak05};
in the more general case when ${\mathfrak{X}}=\mathbb{R}$, $t\in(2,4]$, and the $X_i$'s are independent but not necessarily identically distributed, the bound
in \eqref{eq:2<t<4,H} was obtained in \cite[Lemma~13]{novak00}, but with the larger factor $t(t-1)2^{-t/2}$ in place of $t-1$.
In the case $t=3$, which is particularly important in applications to Berry--Esseen-type bounds (see e.g.\ \cite{chen-shao05,nonlinear}), \eqref{eq:2<t<3} or \eqref{eq:3<t<4}
yields
\betagin{equation}\lambdabel{eq:t=3}
\operatorname{\mathsf{E}}\|S_n\|^3\leqslantslant\tfrac{1+D^2}2\,(\mathcal{A} n3+2B_n^3).
\end{equation}
Rosenthal-type inequalities and related results can be found, among other papers, in \cite{rosenthal,burk,MR0443034,pin-utev84,utev-extr,sibam,ibr-shar97,latala-moments,gine-lat-zinn,
ibr-sankhya,bouch-etal
}.
\searrowction{Application to concentration of measure on product spaces for separately Lipschitz functions}\lambdabel{concentr}
In this section, let us re-define $X_1,\mathrm{d}ots,X_n$ to be independent r.v.'s with values in measurable spaces ${\mathfrak{X}}_1,\mathrm{d}ots,{\mathfrak{X}}_n$, respectively. Let $g\colon{\mathfrak{P}}\to\mathbb{R}$ be a measurable function on the product space ${\mathfrak{P}}:={\mathfrak{X}}_1\times\mathrm{d}ots\times{\mathfrak{X}}_n$.
Let us say (cf.\ \cite{bent-isr,normal}) that $g$ is {\em separately Lipschitz} if it satisfies a Lipschitz-type condition in each of its arguments:
\betagin{equation}\lambdabel{eq:Lip}
|g(x_1,\mathrm{d}ots,x_{i-1},\tilde x_i,x_{i+1},\mathrm{d}ots,x_n) -
g(x_1,\mathrm{d}ots,x_n)| \leqslantslant \rho_i(\tilde x_i,x_i)
\end{equation}
for some measurable functions $\rho_i\colon{\mathfrak{X}}_i\times{\mathfrak{X}}_i\to\mathbb{R}$ and
all $i\in\intr1n$, $(x_1,\mathrm{d}ots,x_n)\in{\mathfrak{P}}$, and $\tilde x_i\in{\mathfrak{X}}_i$.
Take now any separately Lipschitz function $g$ and let
$$Y:=g(X_1,\mathrm{d}ots,X_n).$$
Suppose that the r.v.\ $Y$ has a finite mean.
Take any real $t>2$.
\betagin{corollary}\lambdabel{cor:concentr}
For each $i\in\intr1n$, take any $x_i$ and $y_i$ in ${\mathfrak{X}}_i$.
Then
\betagin{equation*}
\operatorname{\mathsf{E}}|Y-\operatorname{\mathsf{E}} Y|^t
\leqslantslant C_t\, C_A^{(t)}\sum_1^n\operatorname{\mathsf{E}}\rho_i(X_i,x_i)^t+C_B^{(t)}\mathcal{B}ig(\sum_1^n\operatorname{\mathsf{E}}\rho_i(X_i,y_i)^2\mathcal{B}ig)^{t/2},
\end{equation*}
where $C_A^{(t)}$ and $C_B^{(t)}$ are as in \eqref{eq:CA,CB} with $D=1$,
\betagin{gather*}
C_t:=R(t,b_t),
\end{gather*}
and $b_t$ is the unique maximizer of
$$R(t,b):=(b^{t - 1} + (1 - b)^{t - 1}) \big(b^{\mathbf{f}rac1{t - 1}} + (1 - b)^{\mathbf{f}rac1{t - 1}}\big)^{t - 1}$$ over all $b\in[0,\mathbf{f}rac12]$.
\end{corollary}
This follows immediately from Corollary~\ref{cor:} and \cite[Corollary 3.1]{re-center}.
An example of separately Lipschitz functions $g:{\mathfrak{X}}^n\to\mathbb{R}$ is given by the formula
\betagin{equation*}
g(x_1,\mathrm{d}ots,x_n)=\|x_1+\mathrm{d}ots+x_n\|
\end{equation*}
for all $x_1,\mathrm{d}ots,x_n$ in any (not necessarily 2-smooth) separable Banach space $({\mathfrak{X}},\|\cdot\|)$.
In this case, one may take $\rho_i(\tilde x_i,x_i)\equiv\|\tilde x_i-x_i\|$.
Thus, one immediately obtains
\betagin{corollary}\lambdabel{cor:conc-sums}
Let $X_1,\mathrm{d}ots,X_n$ be independent random vectors in a separable Banach space $({\mathfrak{X}},\|\cdot\|)$.
Let here $Y:=\|X_1+\mathrm{d}ots+X_n\|$.
For each $i\in\intr1n$, take any $x_i$ and $y_i$ in ${\mathfrak{X}}$.
Then, with $C_A^{(t)}$, $C_B^{(t)}$, and $C_t$ as in Corollary~\ref{cor:concentr},
\betagin{equation*}
\operatorname{\mathsf{E}}|Y-\operatorname{\mathsf{E}} Y|^t
\leqslantslant C_t\, C_A^{(t)}\sum_1^n\operatorname{\mathsf{E}}\|X_i-x_i\|^t+C_B^{(t)}\mathcal{B}ig(\sum_1^n\operatorname{\mathsf{E}}\|X_i-y_i\|^2\mathcal{B}ig)^{t/2}.
\end{equation*}
In particular,
$C_3<1.316$ and hence, by \eqref{eq:t=3} with $D=1$,
\betagin{equation*}
\operatorname{\mathsf{E}}|Y-\operatorname{\mathsf{E}} Y|^3
\leqslantslant1.316\,\sum_1^n\operatorname{\mathsf{E}}\|X_i-x_i\|^3+2\mathcal{B}ig(\sum_1^n\operatorname{\mathsf{E}}\|X_i-y_i\|^2\mathcal{B}ig)^{3/2};
\end{equation*}
this improves the corresponding bound in \cite{re-center}, which had the constant factor $3$ in place of $2$.
\end{corollary}
The separate-Lipschitz
condition \eqref{eq:Lip} is easier to check than a joint-Lipschitz one. Also, the former is more generally applicable. On the other hand, when a joint-Lipschitz condition is satisfied, one can generally obtain better bounds. Literature on the concentration of measure phenomenon, almost all of it for joint-Lipschitz settings, is vast; let us mention here only \cite{ledoux-tala,ledoux_book,lat-olesz,
ledoux-olesz}.
\searrowction{Proofs}\lambdabel{proofs}
The proof of Theorem~\ref{th:} is almost the same as that of \cite[Theorem~1]{pin80};
instead of \cite[Lemma~1]{pin80} one should now use the following more general lemma, valid for general 2-smooth Banach spaces.
\betagin{lemma}\lambdabel{lem:1}
Suppose that $t\gammammae2$ and $p(t)$ and $q(t)$ satisfy conditions \eqref{eq:p(s),q(s)}.
Suppose also that the function $Q(\cdot):=\|\cdot\|^2$ is twice differentiable everywhere on ${\mathfrak{X}}$ --
which may be assumed without loss of generality in view of \cite[Remark~2.4]{pin94}.
Then for any $x$ and $y$ in ${\mathfrak{X}}$
\betagin{equation*}
\|x+y\|^t-\|x\|^t\leqslantslant\tfrac t2\|x\|^{t-2}Q'(x)(y)
+\tfrac{t(t-2+D^2)}2\, p(t)\|x\|^{t-2}\|y\|^2+\tfrac{t-2+D^2}{t-1}q(t)\|y\|^t.
\end{equation*}
\end{lemma}
\betagin{proof}[Proof of Lemma~\ref{lem:1}]
Take indeed any $x$ and $y$ in ${\mathfrak{X}}$ and introduce
$$f(\theta):=\|x_\theta\|^t-\|x\|^t=Q(x_\theta)-\|x\|^t,$$
where $\theta\in[0,1]$ and $x_\theta:=x+\theta y$, so that $f(0)=0$, $f'(0)=\tfrac t2\|x\|^{t-2}Q'(x)(y)$, and $f(1)=\|x+y\|^t-\|x\|^t$.
Moreover, since ${\mathfrak{X}}$ is $(2,D)$-smooth and $Q$ is twice differentiable on ${\mathfrak{X}}$, for all $\theta$ one has $Q''(x_\theta)(y,y)\leqslantslant2D^2\|y\|^2$ and $|Q'(x_\theta)(y)|\leqslantslant2\|x_\theta\| \|y\|$ (cf.\ \cite[Lemma~2.2 and Remark~2.4]{pin94}); so,
\betagin{align*}
f''(\theta)&=\tfrac t2\,(\tfrac t2-1)\, Q(x_\theta)^{t/2-2} \big(Q'(x_\theta)(y)\big)^2
+ \tfrac t2\, Q(x_\theta)^{t/2-1} Q''(x_\theta)(y,y) \\
&\leqslantslant t(t-2+D^2)\|x_\theta\|^{t-2}\|y\|^2 \\
&\leqslantslant t(t-2+D^2)\big(p(t)\|x\|^{t-2}\|y\|^2+q(t)\theta^{t-2}\|y\|^t\big),
\end{align*}
since $(klpha+\beta)^{t-2}\leqslantslant p(t)klpha^{t-2}+q(t)\beta^{t-2}$ for all $klpha$ and $\beta$ in $[0,\infty)$.
It remains to use the identities $$\|x+y\|^t-\|x\|^t=f(1)=f(0)+f'(0)+\textstyle{\int_0^1}(1-\theta)f''(\theta){\operatorname{d}}\theta.$$
\end{proof}
\betagin{proof}[Proof of Corollary~\ref{cor:}]
For any $j\in\intr0{m-1}$ and $\lambda_j>0$
\betagin{multline}\lambdabel{eq:sum1}
j!\,\sum_{J\in\mathcal{J}_{n,j}} \mathcal{A}{\mu_n(J)-1}{t-2j} \prod_{i\in J}b_i^2
\leqslantslant\mathcal{A} n{t-2j} B_n^{2j}
\leqslantslant(\mathcal{A} nt)^{\mathbf{f}rac{t-2j-2}{t-2}} \, (\mathcal{A} n2 B_n^{t-2})^{\mathbf{f}rac{2j}{t-2}} \\
\leqslantslant(\mathcal{A} nt)^{\mathbf{f}rac{t-2j-2}{t-2}} \, (B_n^t)^{\mathbf{f}rac{2j}{t-2}}
\leqslantslant\tfrac{t-2j-2}{t-2}\lambda_j^{-2j}\,\mathcal{A} nt + \tfrac{2j}{t-2}\,\lambda_j^{t-2j-2}\,B_n^t;
\end{multline}
the second inequality here follows by the log-convexity of $\mathcal{A} ns$ in $s$, and the fourth one follows by Young's inequality $x^{1/p}y^{1/q}\leqslantslant\mathbf{f}rac xp+\mathbf{f}rac yq$ for any $x$, $y$, $p$ and $q$ such that $x\gammammae0$, $y\gammammae0$, $p>0$, $q>0$, and $\mathbf{f}rac1p+\mathbf{f}rac1q=1$.
Next,
\betagin{multline}\lambdabel{eq:sum2}
\sum_{J\in\mathcal{J}_{n,m}} \big(\mathcal{A}{\mu_n(J)-1}2\big)^{t/2-m} \prod_{i\in J}b_i^2
\leqslantslant \sum_{J\in\mathcal{J}_{n,m}} B_{\mu_n(J)-1}^{t-2m} \prod_{i\in J}b_i^2 \\
=
\sum_{i_m\in\intr1n}b_{i_m}^2\sum_{i_{m-1}\in\intr1{i_m-1}}b_{i_{m-1}}^2\mathrm{d}ots\sum_{i_1\in\intr1{i_2-1}} b_{i_1}^2 B_{i_1-1}^{t-2m}
\leqslantslant\mathcal{B}ig(\prod_{j\in\intr1m}\mathbf{f}rac1{t/2-m+j}\mathcal{B}ig)B_n^t;
\end{multline}
the iterated sum here is bounded by induction, using the inequality
\betagin{equation}\lambdabel{eq:sum-int}
\sum_{i\in\intr1k} b_i^2 B_{i-1}^s\leqslantslant\mathbf{f}rac1{s/2+1}B_k^{s+2}
\end{equation}
for $s\gammammae0$ and $k\in\intr1\infty$; in turn, inequality \eqref{eq:sum-int} follows because its left-hand side is a left (and hence lower) Riemann sum for the integral $\int_0^{B_k^2}x^{s/2}{\operatorname{d}} x$ of the nondecreasing function $\cdot^{s/2}$
, whereas the right-hand side of \eqref{eq:sum-int} is the value of the integral.
Now Corollary~\ref{cor:} follows by \eqref{eq:}, \eqref{eq:sum1}, and \eqref{eq:sum2}.
\end{proof}
\end{document}
|
\begin{equation}gin{document}
\vglue-1cm \hskip1cm
\title[Modified Camassa-Holm Equation]{A comment about the paper On the instability of elliptic traveling wave solutions of the modified Camassa-Holm equation}
\begin{equation}gin{center}
\subjclass[2000]{76B25, 35Q51, 35Q53.}
\keywords{Orbital stability, modified Camassa-Holm equation, periodic traveling waves}
\maketitle
{\bf Renan H. Martins}
{Departamento de Matem\'atica - Universidade Estadual de Maring\'a\\
Avenida Colombo, 5790, CEP 87020-900, Maring\'a, PR, Brazil.}\\
{r3nan$\[email protected]}
{\bf F\'abio Natali}
{Departamento de Matem\'atica - Universidade Estadual de Maring\'a\\
Avenida Colombo, 5790, CEP 87020-900, Maring\'a, PR, Brazil.}\\
{[email protected]}
\end{center}
\begin{equation}gin{abstract}
We discuss the recent paper by A. Dar\'os and L.K. Arruda (On the instability of elliptic traveling wave solutions of the modified Camassa–Holm equation, J. Diff. Equat., 266 (2019), 1946-1968). Our intention is to correct some imperfections left by the authors and present the orbital stability of periodic snoidal waves in the energy space.
\end{abstract}
\section{Introduction}
The main goal of this note is to present a different result as determined in \cite{DLK} concerning the orbital stability of smooth periodic waves associated with the modified Camassa-Holm equation given by
\begin{equation}{\langle}bel{mCH}
u_t-u_{txx}=uu_{xxx}+2u_xu_{xx}-3u^2u_{x},
\end{equation}
where $u:\mathbb{R}\times{\mathbb R}\to{\mathbb R}$ is a real function which is $L-$periodic at space variable.\\
\indent Formally, equation $(\ref{mCH})$ admits the conserved quantities
\begin{equation}gin{equation}{\langle}bel{Eu}
E(u)=-\int_{0}^{L}\left[\frac{u^4}{4}+\frac{uu_x^2}{2}\right]dx,
\end{equation}
\begin{equation}gin{equation}{\langle}bel{Fu}
F(u)=\frac{1}{2}\int_{0}^{L} u^2+u_x^2dx,
\end{equation}
and
\begin{equation}gin{equation}{\langle}bel{Vu}
V(u)=\int_{0}^{L} udx.
\end{equation}
A smooth periodic traveling wave solution for \eqref{mCH} is a solution of the form $u(x,t)=\phi(x-c t)$, where $c$ is a positive real constant representing the wave speed and $\phi:\mathbb{R}\to{\mathbb R}$ is a smooth $L-$periodic function satisfying
$\phi^{(n)}(x+L)=\phi^{(n)}(x)$ for all $n\in\mathbb{N}$. Substituting this form into (\ref{mCH}), we obtain
\begin{equation}gin{equation}{\langle}bel{ode-wave}
(\phi-c)\phi''+\frac{\phi'^2}{2}-\phi^3+c\phi=A
\end{equation}
where $\phi_{c}:=\phi$ and $A_c:=A$ depend both on $c>0$. $A$ is a constant of integration.
In view of the conserved quantities $(\ref{Eu})$, $(\ref{Fu})$ and $(\ref{Vu})$, we may define the augmented Lyapunov functional,
\begin{equation}gin{equation}{\langle}bel{lyafun}
G(u)=E(u)+cF(u)-AV(u),
\end{equation}
and the symmetric linearized operator around the wave $\phi$ expressed by
\begin{equation}gin{equation}{\langle}bel{operator}
\mathcal{L}=G''(\phi)=(\phi-c)\partial_x^2+\phi'\partial_x+c-3\phi^2+\phi''.
\end{equation}
In addition, it is clear from $(\ref{ode-wave})$ that $G'(\phi)=0$.\\
\indent In \cite{DLK} the authors have been established the existence of periodic waves with the zero mean property associated with the equation $(\ref{ode-wave})$. They put forwarded an explicit solution given in terms of the Jacobi Elliptic Function with \textit{snoidal} type given by
\begin{equation}gin{equation}{\langle}bel{snoidal1}
\phi(x)=\alpha+\begin{equation}ta{\rm sn}^2\left(\frac{2K(k)x}{L};k\right),
\end{equation}
where $\alpha$ and $\begin{equation}ta$ are smooth functions depending on the period $L>0$ (which needs to be large enough) and the modulus $k\in(0,1)$. Here $K:=K(k)$ represents the complete elliptic integral of first kind.\\
\indent The results contained in \cite[Theorem 2]{DLK} do not bring any mention if the periodic wave has zero mean. In addition, the authors should use the implicit function theorem taking account this property in order to prove the existence of a smooth curve of periodic waves. This fact has been determined first in \cite{ABS} where the authors constructed snoidal periodic waves with zero mean and depending smoothly on the wave speed $c$ for the standard Korteweg-de Vries equations (in fact, they constructed cnoidal periodic waves, but to get the elliptic function depending on snoidal it makes necessary to use the basic equality $sn^2+cn^2=1$).\\
\indent On the other hand, the standard equality concerning Jacobi Elliptic Functions given by $k^2sn^2+dn^2=1$ can be used to deduce from $(\ref{snoidal1})$, a convenient solution given in terms of the Jacobi Elliptic Function with \textit{dnoidal} type as,
\begin{equation}gin{equation}{\langle}bel{dnoidal1}
\phi(x)=a+b\left({\rm dn}^2\left(\frac{2K(k)x}{L};k\right)-\frac{E(k)}{K(k)}\right),
\end{equation}
where $E$ is the complete elliptic integral of second kind. The main advantage of the formula $(\ref{dnoidal1})$ is that
$\frac{1}{L}\int_{0}^L\phi dx=a.$\\
\indent Substituting the solution $(\ref{dnoidal1})$ (or equivalently, $(\ref{snoidal1})$) in $(\ref{ode-wave})$, we obtain thanks to the terms $\phi'^2$ and $\phi^3$, a complicated equation given in short by
\begin{equation}gin{equation}{\langle}bel{powerSN}
\sum_{i=0}^{3}f_i(k,a,b,L,c,A){{\rm \,dn}}^{2i}\left(\frac{2K(k)x}{L};k\right)=0,
\end{equation}
where $f_i$, $i=0,1,2,3$, are smooth functions depending on the variables $k,b,a,L,c$, and $A$. Using again the equality $k^2sn^2+dn^2=1$, we get a similar equality as in $(\ref{powerSN})$ with $sn$ instead of $dn$. Since our intention is to get a smooth curve of periodic waves depending on the modulus $k\in(0,1)$ with fixed period $L>0$, we need to consider $f_i\equiv0$, $i=0,1,2,3$, to get $a,b,c$, and $A$ in terms of $k$ and $L$. This can be done and we have
\begin{equation}{\langle}bel{a}
\begin{equation}gin{array}{llll}a&=&\displaystyle-\frac{1}{3L^2}\left[-32(2-k^2)K(k)^2+96E(k)K(k)
+\frac{3}{2}L^2\right.\\\\
&-& \displaystyle\left.\frac{1}{2}\sqrt{9L^4-2048K(k)^4+2048K(k)^4k^2-2048K(k)^4k^4}\right],
\end{array}
\end{equation}
\begin{equation}{\langle}bel{b}
b=-\frac{32K(k)^2}{L^2},
\end{equation}
and
\begin{equation}{\langle}bel{c}
c=\frac{\frac{3}{2}L^2-\frac{1}{2}\sqrt{9L^4-2048K(k)^4+2048K(k)^4k^2-2048K(k)^4k^4}}{L^2}.
\end{equation}
The common term present in the square root appearing in equalities $(\ref{a})$ and $(\ref{c})$ gives us that the period $L$ needs to be considered large enough. In addition, the value of $c$ in $(\ref{c})$ is the same as in \cite{DLK}.\\
\indent It is clear from $(\ref{a})$ that if $\phi$ enjoys the zero mean property, one sees that $a=0$ and, in this case, $L$ can be seen as an implicit function in terms of the modulus $k\in(0,1)$. When the period is a function depending on the modulus $k$ (as a consequence, $L$ depends on the wave speed $c$), we get additional difficulties to apply classical arguments as in \cite{bona2}, \cite{bona}, \cite{DK}, and \cite{grillakis1} to conclude orbital stability/instability results. Indeed, we need to calculate the second derivative in terms of $c$ (consequently, in terms of $k$) associated with the one parameter function $d(c)=E(\phi)+cF(\phi)$ with the period $L$ depending on the modulus $k\in(0,1)$. The arguments in \cite{DLK} have showed the orbital instability of periodic waves with fixed periods only.\\
\indent As a consequence, since the periodic wave does not have necessarily zero mean, the orbital stability/instability can not be measured by analyzing only the sign of $d''(c)$, even though we have in hands, good spectral properties for the linearized operator $\mathcal{L}$ (namely, one negative eigevalue which is simple and zero being a simple eigenvalue with associated eigenfunction $\phi'$).\\
\indent Despite of the arguments established in \cite{DLK}, we prove that the periodic wave in $(\ref{dnoidal1})$ is orbitally stable in the energy space $H_{per}^1([0,L])$ without assuming that $\phi$ has zero mean. To do so, we employ the recent development in \cite{ANP} which gives a wide approach to deduce orbital stability results regarding periodic waves $\phi$ which are, at the same time, critical points and zero solutions of the modified Lyapunov function
\begin{equation}
\mathcal{B}(u)=G(u)-G(\phi)+N(Q(u)-Q(\phi))^2,
\end{equation}
where $N$ is a convenient positive constant and $Q$ is a convenient sum of the quantities $F(\phi)$ and $V(\phi)$. In some sense, our result recovers the orbital stability arguments as in \cite{hakka}.\\
\indent Our paper is organized as follows: next section is used to present the orbital stability of periodic waves associated with the model $(\ref{mCH})$, using the arguments in \cite{ANP}. Some important remarks concerning the orbital instability of periodic waves with zero mean are given in Section 3.
\section{Orbital Stability of Traveling Waves}{\langle}bel{OSPW1}
In this section, we present our stability results by using an simplification of the arguments in \cite{ANP}. First of all, in order to simplify the notation, the norm and inner product in $L_{per}^2([0,L])$ will be denoted by $||\cdot||$ and ${\langle}ngle\cdot,\cdot{\rangle}ngle$, respectively.
Before stating our main theorem, we need some preliminary results. Let $\rho$ be the semi-distance defined on the energy space $X=H_{per}^1([0,L])$.
\begin{equation}{\langle}bel{rho}
\rho(u,\phi)=\inf_{y\in\mathbb{R}}||u-\phi(\cdot+y)||_{X}.
\end{equation}
\begin{equation}gin{definition}{\langle}bel{defstab}
We say that a solitary wave solution $\phi$ is orbitally stable in $X$, by the flow of \eqref{mCH}, if for any ${\varepsilon}>0$ there exists $\delta>0$ such that for any $u_0\in X$ satisfying $\|u_0-\phi\|_X<\delta$, the solution $u(t)$ of \eqref{mCH} with initial data $u_0$ exists globally and satisfies
$
\rho(u(t),\phi)<{\varepsilon},
$
for all $t\geq0$.
\end{definition}
According with the arguments in \cite{hakka}, the Cauchy problem associated with the equation $(\ref{mCH})$ is locally well-posed in $H_{per}^s([0,L])$, for $s>\frac{3}{2}$. On the other hand, it is well known that some blow up results in finite time are expected for the same model (see again \cite{hakka} and references therein). Thus, since $F$ in $(\ref{Fu})$ is a conserved quantity, we can combine the local solution obtained in \cite{hakka} with the standard a priori estimate $F(u(t))=F(u_0)$ for all $t\geq0$ to get a \textit{conditional orbital stability result}.
For a given $\varepsilon>0$, we define the $\varepsilon$-neighborhood of the orbit $O_\phi=\{\phi(\cdot+y), y\in{\mathbb R}\}$ as
\begin{equation} {\langle}bel{tube}U_{\varepsilon} := \{u\in X;\ \rho(u,\phi) < \varepsilon\}.\end{equation}
To start our analysis, we first assume the existence of a smooth functional $Q:X\rightarrow \mathbb{R}$ which is conserved quantity in time, invariant by translations in the sense that $Q(u(\cdot+r))=Q(u)$, for all $r\in\mathbb{R}$, and satisfying ${\langle}ngle Q'(\phi),\phi'{\rangle}ngle=0$. Functional $Q$ plays an important role in our analysis since it inspires us the definition of the tangent space $\Upsilon_0=\{u\in X;\ {\langle}ngle Q'(\phi),u{\rangle}ngle=0\}$. In addition, $Q$ will be considered with a convenient form later on.
Before starting with the stability results, we need to prove some auxiliary results which are useful in our stability analysis. The first one concerns the existence of periodic waves for large periods $L>0$. In some sense, this fact has already presented in the introduction and for the sake of completeness we enunciate a full result of existence of periodic solutions.\\
\begin{equation}gin{lemma}{\langle}bel{lema12}
For $L>0$ sufficiently large, there exists $k_{1}\in(0,1)$ such that for all $k\in(0,k_1)$, the periodic traveling wave solution $\phi$ in $\ref{dnoidal1}$ is a solution of $(\ref{ode-wave})$ and depending smoothly on $k\in(0,k_1)$. Parameters $a$, $b$ and $c$ are given respectively by $(\ref{a})$, $(\ref{b})$ and $(\ref{c})$. The value of $A$ in $(\ref{ode-wave})$ can be expressed in terms of $k$ and $L$ by
\begin{equation}
\begin{equation}gin{array}{lllll}A&=&\displaystyle\frac{1}{27L^6}\left[(1280(-1+k^2-k^4)K(k)^4\right.\\\\
&+& \displaystyle\left.9L^4)
\sqrt{2048(-1+k^2-k^4)K(k)^4+9L^4}\right.\\\\
&+&\displaystyle\left.(-16384-16384k^6+24576k^2+24576k^4)K(k)^6\right.\\\\
&+&\displaystyle\left. 6912L^2(1-k^2+k^4)K(k)^4-27L^6\right].\end{array}
\end{equation}
Moreover, for $k\in(0,k_1)$ one has:\\
i) the strict inequality $c^2-3c+\frac{32\pi^4}{L^4}<0$ is always satisfied.\\
ii) $\phi-c<0$ in $[0,L]$.
\end{lemma}
\begin{equation}gin{flushright}
$\square$
\end{flushright}
The next result allows us to decide about the quantity and multiplicity of the first two negative eigenvalues of $\mathcal{L}$ in $(\ref{operator})$.\\
\begin{equation}gin{lemma}{\langle}bel{lema123}
Assume that conditions in Lemma $\ref{lema12}$ are satisfied. The linearized operator $\mathcal{L}$ defined in $(\ref{operator})$ has only one negative eigenvalue which is simple. Zero is a simple eigenvalue whose associated eigenfunction is $\phi'$.
\end{lemma}
\begin{equation}gin{proof}
See Propositions $2$ and $3$ in \cite{DLK}.
\end{proof}
We are in position to establish the following proposition which gives a sufficient condition for the positiveness of the quadratic form associated with $\mathcal{L}$.
\begin{equation}gin{proposition}{\langle}bel{prop2}
Suppose that conditions in Lemma $\ref{lema12}$ are satisfied. Assume the existence of $\Phi\in X$ such that ${\langle}ngle\mathcal{L}\Phi,\varphi{\rangle}ngle=0$, for all $\varphi\in \Upsilon_0$, and
$$
\mathcal{I}:={\langle}ngle\mathcal{L}\Phi,\Phi{\rangle}ngle<0\ \ \ \ \ \ \mbox{(Vakhitov-Kolokolov condition)}.
$$
Then, there exists a constant $c>0$ such that
${\langle}ngle\mathcal{L}v,v{\rangle}ngle\geq c||v||_{X}^2,$
for all $v\in \Upsilon_0\cap [\phi']^\perp$.
\end{proposition}
\begin{equation}gin{proof}
See \cite{ANP}.
\end{proof}
In our context, parameters $c$ and $A$ given by Lemma $\ref{lema12}$ depend on a third parameter $k\in(0,k_1)$. Next result gives us a sufficient condition to obtain a convenient formula for $\mathcal{I}$ in Lemma $\ref{prop2}$ in terms of $k$.
\begin{equation}gin{corollary}{\langle}bel{coro2}
Suppose that the assumptions in Lemma $\ref{lema12}$ are satisfied. The quantity $\mathcal{I}$ in Proposition $\ref{prop2}$ is given by:
\begin{equation}gin{equation}{\langle}bel{solcriterio}
\mathcal{I}=\frac{d A}{d k} \frac{d}{dk}V(\phi)-\frac{d c}{d k}\frac{d}{dk}F(\phi).
\end{equation}
Moreover, $\mathcal{I}<0$.
\end{corollary}
\begin{equation}gin{proof}
In Proposition $\ref{prop2}$ we define $\Phi=\frac{d}{dk}\phi$ and $Q(u)=\frac{d A}{d k}M(u)-\frac{dc}{dk}F(u)$ to get $(\ref{solcriterio})$. Now, we to need check that $\mathcal{I}<0$. Indeed, since $V(\phi)=aL$, we obtain from $(\ref{Fu})$ and $(\ref{solcriterio})$ that
\begin{equation}gin{equation}{\langle}bel{solcrit1}
\mathcal{I}=\frac{dA}{dk}\frac{da}{dk}L-\frac{1}{2}\frac{dc}{dk}\frac{d}{dk}\left(\int_0^{L}\phi'^2+\phi^2dx\right)
\end{equation}
The right-hand side of $(\ref{solcrit1})$ is a complicated function depending on $k\in(0,k_1)$ and $L>0$ large enough. We can show that $\mathcal{I}<0$ by plotting some pictures.
\begin{equation}gin{figure}[h!]
\centering
\begin{equation}gin{subfigure}[t]{0.5\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{fig3pi.jpg}
\end{subfigure}
\begin{equation}gin{subfigure}[t]{0.5\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{fig6pi.jpg}
\end{subfigure}
\caption{Graphics of $\mathcal{I}$ for $(k,L)\in(0,0.2]\times[3\pi,6\pi]$ (left) and for $(k,L)\in(0,0.8]\times[6\pi,10\pi]$ (right).}
\end{figure}
\end{proof}
Proposition \ref{prop2} and Corollary $\ref{coro2}$ are useful to establish the following result.
\begin{equation}gin{lemma}{\langle}bel{lemma1}
Under assumptions of Proposition $\ref{prop2}$, there exist $N>0$ and $\tau>0$ such that
$${\langle}ngle\mathcal{L}v,v{\rangle}ngle +2N{\langle}ngle Q'(\phi),v{\rangle}ngle^{2}\geq \tau||v||_{X}^2,$$
for all $v\in [\phi']^\perp$.
\end{lemma}
\begin{equation}gin{proof}
First, since ${\langle}ngle Q'(\phi),\phi'{\rangle}ngle=0$, we can write $v\in [\phi']^{\perp}$ as
$$v=\zeta w+z,$$
where $w=\frac{Q'(\phi)}{||Q'(\phi)||}$, $\zeta={\langle}ngle v,w{\rangle}ngle$ and $z\in \Upsilon_0$. Because $z\in\Upsilon_0\cap [\phi']^{\perp}$, Proposition \ref{prop2} implies
\begin{equation}gin{equation}{\langle}bel{eq01}
{\langle}ngle\mathcal{L}v,v{\rangle}ngle \geq \zeta^2{\langle}ngle\mathcal{L}w,w{\rangle}ngle + 2\zeta{\langle}ngle\mathcal{L}w,z{\rangle}ngle +C||z||_X^2.
\end{equation}
Using Cauchy-Schwartz and Young's inequalities, we have
\begin{equation}gin{equation}{\langle}bel{eq02}
2\zeta{\langle}ngle\mathcal{L}w,z{\rangle}ngle \leq \frac{C}{2}||z||_X^2 + \frac{2\zeta^2}{C}||\mathcal{L}w||_X^2.
\end{equation}
Furthermore, we may choose $N>0$ such that
\begin{equation}gin{equation}{\langle}bel{eq03}
{\langle}ngle\mathcal{L}w,w{\rangle}ngle -\frac{2}{c}||\mathcal{L}w||_X^2 +2N||Q'(\phi)||_X^2\geq \frac{C}{2}.
\end{equation}
We point out that $N$ depends only on $\phi$.
Therefore, using (\ref{eq01}), (\ref{eq02}) and (\ref{eq03}), we conclude
\begin{equation}gin{eqnarray}
{\langle}ngle\mathcal{L}v,v{\rangle}ngle +2N{\langle}ngle Q'(\phi),v{\rangle}ngle^{2}&=&{\langle}ngle\mathcal{L}v,v{\rangle}ngle + 2N\zeta^2||Q'(\phi)||_X^2 \nonumber \\
&\geq&\frac{C}{2}(\zeta^2+||z||_X^2) \nonumber \\
&=& \frac{C}{2}||v||_{X}^2. \nonumber
\end{eqnarray}
The proof is thus completed.
\end{proof}
Let $N>0$ be the constant obtained in the previous lemma. We define the functional $\mathcal{B}:X\rightarrow{\mathbb R}$ as
\begin{equation}gin{equation}{\langle}bel{functionalV1}
\mathcal{B}(u)=G(u)-G(\phi)+N(Q(u)-Q(\phi))^2,
\end{equation}
where $G$ is the augmented functional defined in (\ref{lyafun}) and $Q$ is the functional defined in Corollary $\ref{coro2}$. It is easy to see from $(\ref{functionalV1})$ and $(\ref{ode-wave})$ that $\mathcal{B}(\phi)=0$ and $\mathcal{B}'(\phi)=0$. In addition, since $G$ is a conserved quantity, $Q$ is also a conserved quantity and the Cauchy problem related to the equation $(\ref{mCH})$ is conditionally global well-posed in the energy space $X$, one has
\begin{equation}gin{equation}{\langle}bel{boundV}
\mathcal{B}(u(t))=\mathcal{B}(u_0),\ \ \ \ \ \ \mbox{for all}\ t\geq0.
\end{equation}
Thus, $\mathcal{B}(u(t))$ is finite for large values of $t$.\\
\begin{equation}gin{lemma}{\langle}bel{lemma2}
There exist $\alpha>0$ and $D>0$ such that
\begin{equation}gin{equation}{\langle}bel{eq003}
\mathcal{B}(u)\geq D\rho(u,\phi)^2\end{equation}
for all $u\in U_{\alpha}$
\end{lemma}
\begin{equation}gin{proof}
First, note that from the definition of $\mathcal{B}$ it follows that
$${\langle}ngle \mathcal{B}''(u)v,v{\rangle}ngle={\langle}ngle G''(u)v,v{\rangle}ngle+2N(Q(u)-Q(\phi)) {\langle}ngle Q''(u)v,v{\rangle}ngle+2N{\langle}ngle Q'(u),v{\rangle}ngle^2,$$
for all $u,v\in X$. In particular,
$${\langle}ngle \mathcal{B}''(\phi)v,v{\rangle}ngle={\langle}ngle \mathcal{L}v,v{\rangle}ngle+2N{\langle}ngle Q'(\phi),v{\rangle}ngle^2.$$
Consequently, from Lemma \ref{lemma1} we get
\begin{equation}gin{equation} {\langle}bel{eq001}
{\langle}ngle \mathcal{B}''(\phi)v,v{\rangle}ngle\geq\tau||v||_X^2,
\end{equation}
for all $v\in (\ker(\mathcal{L}))^\perp$.
On the other hand, a Taylor expansion of $\mathcal{B}$ around $\phi$ reveals that
\begin{equation}gin{equation}{\langle}bel{eq002}
\mathcal{B}(u)=\mathcal{B}(\phi)+ {\langle}ngle \mathcal{B}'(\phi),u-\phi{\rangle}ngle+\frac{1}{2} {\langle}ngle \mathcal{B}''(\phi)(u-\phi),u-\phi{\rangle}ngle +h(u),
\end{equation}
where $\lim\limits_{u\to\phi}\frac{|h(u)|}{||u-\phi||_X^2}=0$.
Thus, we can choose $\alpha_1>0$ such that
\begin{equation}gin{equation}{\langle}bel{limit1}|h(u)|\leq\frac{\tau}{4}||u-\phi||_X^2, \qquad \mbox{for all} \ u\in B_{\alpha_1}(\phi),\end{equation}
where $B_{\alpha_1}(\phi)=\left\{u\in X; ||u-\phi||_X <\alpha_1 \right\}$.
Since $\mathcal{B}(\phi)=0$ and $\mathcal{B}'(\phi)=0$, we have from (\ref{eq001}), (\ref{eq002}) and $(\ref{limit1})$ that
\begin{equation}gin{equation}{\langle}bel{eq0031}
\mathcal{B}(u)\geq \frac{\tau}{4}\rho(u,\phi)^2,
\end{equation}
for all $u\in B_{\alpha_1}(\phi)$ such that $(u-\phi)\in [\phi']^{\perp}.$\\
\indent There exists a continuously differentiable map $r:U_{\alpha}\rightarrow\mathbb{R}$, such that ${\langle}ngle u(\cdot -r(u)),\phi'{\rangle}ngle=0$ and $r(\phi)=0$, for all $u\in B_{\alpha}(\phi)$. In fact, let us define the smooth map $S:X\times \mathbb{R}\rightarrow\mathbb{R}$ given by $S(u,r)={\langle}ngle u(\cdot-r),\phi'{\rangle}ngle$. Since $S(\phi,0)={\langle}ngle\phi,\phi'{\rangle}ngle=0$ and
$S_r(\phi,0)=-{\langle}ngle\phi',\phi'{\rangle}ngle\neq0$, we guarantee, from the implicit function theorem, the existence of $\alpha_2>0$, an $\delta_0>0$ and a unique $C^1-$map $r:B_{\alpha_2}(\phi)\rightarrow(-\delta_0,\delta_0)$ such that $r(\phi)=0$ and $G(u,r(u))={\langle}ngle u(\cdot-r(u)),\phi'{\rangle}ngle=0$, for all $u\in B_{\alpha_2}(\phi)$. From continuity arguments and since $r(\phi)=0$, for a given $0<\varepsilon\leq\min\{\alpha_1,\alpha_2\}$, there exists $\alpha>0$ small enough (for instance, $0<\alpha\leq \varepsilon$) such that $||u(\cdot-r(u))-\phi||_X<\varepsilon$ with ${\langle}ngle u(\cdot-r(u)),\phi'{\rangle}ngle=0$, for all $u\in B_{\alpha}(\phi)$.\\
\indent From $(\ref{eq0031})$ and the arguments in the last paragraph, we obtain the existence of $D>0$ such that $\mathcal{B}(u)\geq D\rho(u,\phi)^2$, for all $u\in B_{\alpha}(\phi)$. The remainder of the proof follows from similar arguments as in \cite[Lemma 4.1]{bona2}.
\end{proof}
The above lemma is the key point to prove our main result. Roughly speaking, it says that $\mathcal{B}$ is a suitable Lyapunov function to handle with a our problem. Finally, we present our stability result.
\begin{equation}gin{theorem}{\langle}bel{teoest11} Suppose that assumptions in Lemma $\ref{lema12}$ hold. Then $\phi$ is orbitally stable in the sense of Definition $\ref{defstab}$.
\end{theorem}
\begin{equation}gin{proof}
Let $\alpha>0$ be the constant such that Lemma \ref{lemma2} holds. Since $\mathcal{B}$ is continuous at $\phi$, for a given $\varepsilon>0$, there exists $\delta\in (0,\alpha)$ such that if $||u_0-\phi||_X<\delta$ one has
\begin{equation}{\langle}bel{estepsilon2}
\mathcal{B}(u_0)\leq |\mathcal{B}(u_0)|=|\mathcal{B}(u_0)-\mathcal{B}(\phi)|<D\varepsilon^2,
\end{equation}
where $D>0$ is the constant in Lemma \ref{lemma2} and $C>0$ is a constant to be presented later.\\
\indent The continuity in time of the function $\rho(u(t),\phi)$ allows to choose $T_1>0$ such that \begin{equation}{\langle}bel{subalpha1}\rho(u(t),\phi)<\alpha,\ \ \ \mbox{for all}\ t\in [0,T_1).\end{equation}
Thus, one obtains $u(t)\in U_{\alpha}$, for all $t\in[0,T_1)$. From Lemma \ref{lemma2}, we have
\begin{equation}{\langle}bel{estepsilon12}
D\rho(u(t),\phi)^2\leq \mathcal{B}(u(t))=\mathcal{B}(u_0),\ \ \ \ \ \mbox{for all}\ t\in[0,T_1).
\end{equation}
\indent Next, we finally prove that $\rho(u(t),\phi)<\alpha$, for all $t\in [0,+\infty)$, from which one concludes the orbital stability. Indeed, let $T_2>0$ be the supremum of the values of $T_1>0$ for which $(\ref{subalpha1})$ holds. To obtain a contradiction, suppose that $T_2<+\infty$. By choosing $\varepsilon<\frac{\alpha}{2}$ we obtain, from $(\ref{estepsilon2})$ and $(\ref{estepsilon12})$ that
$$
\rho(u(t),\phi)<\frac{\alpha}{2}, \ \ \ \ \ \mbox{for all}\ t\in[0,T_2).
$$
Since $t\in(0,+\infty)\mapsto\rho(u(t),\phi)$ is continuous, there is $T_3>0$ such that
$\rho(u(t),\phi)<\frac{3}{4}\alpha<\alpha$, for $t\in [0,T_2+T_3)$, contradicting the maximality of $T_2$. Therefore, $T_2=+\infty$ and the theorem is established.
\end{proof}
\section{Remarks on the Orbital Instability of Periodic Waves with Zero Mean}
\indent In this section, we present some remarks concerning the orbital instability associated with a general Hamiltonian equation given by
\begin{equation}{\langle}bel{hamilt}
u_t=JE'(u),
\end{equation}
where $J$ is a skew-symmetric linear operator, $E$ is the energy functional related to the model and $E'$ represents the Fr\'echet derivative of $E$. We restrict ourselves to the case of the general equation
\begin{equation}{\langle}bel{gCH}
u_t+(p(u))_x-u_{xxt}=\left(q'(u)\frac{u_x^2}{2}+q(u)u_{xx}\right)_x,
\end{equation}
but our arguments can be extended for other equations. Here, $p$ and $q$ are smooth real functions with $p(0)=0$. When $E$ indicates the energy functional associated with the equation $(\ref{gCH})$ and $J=\partial_x(1-\partial_x^2)^{-1}$, particular cases of equation $(\ref{gCH})$ can be expressed as $(\ref{hamilt})$. In particlar, when $p(u)=u^3$ and $q(u)=u$, the general equation $(\ref{gCH})$ reduces to $(\ref{mCH})$ and it is possible to recover $(\ref{hamilt})$ in this specific case.\\
\indent It is well known that the classical instability theory as the one in \cite{grillakis1} can not be applied when $J$ is not one-to-one even though the eventual periodic wave of the form $u(x,t)=\phi(x-ct)$ enjoys the zero mean property (it is clear that $J=\partial_x(1-\partial_x^2)^{-1}$ is not one-to-one since $\ker(J)=[1]$). We believe that the instability analysis in \cite{grillakis1} over periodic Sobolev spaces can be done by restricting \textit{all the analysis} in the closed subspace
\begin{equation}gin{equation}{\langle}bel{zero}
Y_0=\Big\{f\in L^2_{\rm per}([0,L]);\ \int_{0}^{L} f(x) dx = 0 \Big\}.
\end{equation}
\indent To get a precise answer for this question, we could suggest the readers a study of a spectral instability result combined with a method where \textit{spectral instability} implies \textit{orbital instability}. As example: for the case of the generalized Korteweg-de Vries and Benjamin-Bona-Mahony equations, we could assume that the data-solution map $u_0\mapsto u(t)$ is smooth (see \cite{AN1}). We believe that such approach can be done for the general equation $(\ref{gCH})$ without further problems.\\
\indent A convenient spectral stability criterium can be determined. In fact, we first use the perturbation $u(x,t) = \phi(x-ct) + v(x-ct,t)$ in the equation $(\ref{gCH})$ and substituting the associated equation as in (\ref{ode-wave}) for the case of equation $(\ref{gCH})$ (when possible). If everything works fine, we obtain after some calculations the standard spectral problem
\begin{equation}gin{equation}{\langle}bel{vlinear}
v_t = \partial_x \mathcal{L} v,
\end{equation}
where $\mathcal{L}$ is the (self-adjoint) linearized operator associated with the equation $(\ref{gCH})$ around the wave $\phi$. Since $\phi$ depends only on $x$, a separation of variables in the form $v(x,t) = e^{{\langle}mbda t} \eta(x)$ with some ${\langle}mbda \in \mathbb{C}$
and $\eta : [0,L] \to \mathbb{C}$ reduces the linear equation (\ref{vlinear})
to the spectral stability problem
\begin{equation}gin{equation}
{\langle}bel{spectral-stab}
\partial_x \mathcal{L} \eta={\langle}mbda \eta.
\end{equation}
The spectral stability of the periodic wave $\phi$ is defined as follows.
\begin{equation}gin{definition}
{\langle}bel{defspe} The smooth periodic wave $\phi$ is said to be spectrally stable
with respect to perturbations of the same period if
$\sigma(\partial_x \mathcal{L}) \subset i\mathbb R$ in $L^2_{\rm per}([0,L])$.
Otherwise, that is, if $\sigma(\partial_x \mathcal{L})$ in $L^2_{\rm per}([0,L])$
contains a point ${\langle}mbda$ with $\mbox{\rm Re}({\langle}mbda)>0$, the periodic wave $\phi$ is said to be spectrally unstable.
\end{definition}
As we have already mentioned, we know that $\partial_x$ is not a one-to-one operator in periodic context. In order to overcome this difficult, a constrained spectral problem can be considered as
\begin{equation}gin{equation}{\langle}bel{modspecp1}
\partial_x \mathcal{L}\big|_{Y_0}\eta={\langle}mbda \eta,
\end{equation}
where $\mathcal{L}\big|_{Y_0}$ is
a restriction of $\mathcal{L}$ on the closed subspace $Y_0$ defined in $(\ref{zero})$. In addition, over $Y_0$, the linear operator $\partial_x$ has bounded inverse and this crucial fact could enable us to apply the arguments in \cite{grillakis1} to get orbital instability results (but it is necessary to perform suitable modifications in the mentioned theory). However, we need to analyze the quantity and multiplicity of the restricted linearized operator $\mathcal{L}\big|_{Y_0}$ instead of $\mathcal{L}$.\\
\indent To handle with $\mathcal{L}\big|_{Y_0}$, we need to count the quantity (and multiplicity) of non-positive eigenvalues associated with this restriction operator. If we assume that the kernel of $\mathcal{L}$ is simple and generated by $\phi'$, we can use the Morse Index Formula (see \cite{Pel-book}) as
\begin{equation}gin{equation}{\langle}bel{identnegLL}
\left\{ \begin{equation}gin{array}{l}
n(\mathcal{L} \big|_{Y_0}) = n(\mathcal{L}) - n({\langle}ngle \mathcal{L}^{-1}1,1{\rangle}ngle) - z({\langle}ngle \mathcal{L}^{-1}1,1{\rangle}ngle), \\
z(\mathcal{L} \big|_{Y_0}) = z(\mathcal{L}) + z({\langle}ngle \mathcal{L}^{-1}1,1{\rangle}ngle),
\end{array} \right.
\end{equation}
where $n(\mathcal{A})$ and $z(\mathcal{A})$ indicates, respectively, the number of negative eigenvalues (counting multiplicities) and the dimension of the kernel of a general linear operator $\mathcal{A}$. Since it has been assumed that $z(\mathcal{L})=1$, we have from $(\ref{identnegLL})$ that $z(\mathcal{L} \big|_{Y_0})=1+ z({\langle}ngle \mathcal{L}^{-1}1,1{\rangle}ngle)$. Moreover, if ${\langle}ngle \mathcal{L}^{-1}1,1{\rangle}ngle\neq0$, one sees that $z(\mathcal{L} \big|_{Y_0})=1$ and $n(\mathcal{L} \big|_{Y_0}) = n(\mathcal{L}) - n({\langle}ngle \mathcal{L}^{-1}1,1{\rangle}ngle)$.\\
\indent Let us assume that $F$ in $(\ref{Fu})$ is a conserved quantity associated to $(\ref{gCH})$. In this setting, the main result in \cite{DK} establishes a criterium for the (spectral) orbital stability of periodic waves by using the convenient formula for the Hamltonian Krein Index as
\begin{equation}{\langle}bel{krein}\mathcal{K}_{Ham}=n(\mathcal{L} \big|_{Y_0})-n(D)=n(\mathcal{L}) - n({\langle}ngle \mathcal{L}^{-1}1,1{\rangle}ngle)-n(D).\end{equation}
The periodic wave is orbitally (spectrally) unstable if $\mathcal{K}_{Ham}=1$ and orbitally (spectrally) stable if $\mathcal{K}_{Ham}=0$. Here, $D$ is the hessian determinant associated with $F(\phi)$ and $V(\phi)$ and it needs to be non-zero. If $\phi$ has fixed period, zero mean and depends smoothly on the wave speed $c$, one has $D=-\frac{1}{2}\frac{d}{dc}\int_0^L\left(\phi'^2+\phi^2\right)dx=-d''(c)$, provided that ${\langle}ngle \mathcal{L}^{-1}1,1{\rangle}ngle\neq0$. Therefore, if $n(\mathcal{L})=1$ and
${\langle}ngle \mathcal{L}^{-1}1,1{\rangle}ngle>0$, the periodic wave is (spectrally) stable if $d''(c)>0$ and (spectrally) unstable if $d''(c)<0$ by a direct application of $(\ref{krein})$. The last condition is exactly the same as requested in \cite{DLK} to conclude the orbital instability but the periodic wave $\phi$ determined by the authors only has zero mean whether $L$ depends on the modulus $k$. However, we could have $d''(c)<0$ with ${\langle}ngle \mathcal{L}^{-1}1,1{\rangle}ngle<0$ and using $(\ref{krein})$, we still have the (spectral) stability. In some particular cases, it is well known that if the Cauchy problem associated with the equation $(\ref{gCH})$ enjoys of a convenient global well-posedness result, the spectral stability implies the orbital stability. Therefore, \textit{in the case of a smooth curve of periodic waves $c\mapsto\phi$ with fixed period and zero mean}, we can not conclude a precise result of orbital instability only with $n(\mathcal{L})=1$, $\ker(\mathcal{L})=[\phi']$ and $d''(c)<0$ (using a combination of \cite{DK} and \cite{grillakis1}) as determined in \cite{DLK}.
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\end{document}
|
\begin{document}
\title[Levi's problem for complex homogeneous manifolds]
{Levi's problem for complex homogeneous manifolds}
\dedicatory{To the memory of R. Remmert}
\begin{abstract}
Suppose $G$ is a connected complex Lie group and $H$ is a closed complex subgroup.
Then there exists a closed complex subgroup $J$ of $G$ containing $H$ such that
the fibration $\pi:G/H \to G/J$ is the holomorphic reduction of $G/H$, i.e., $G/J$ is holomorphically
separable and ${\mathcal O}(G/H) \cong \pi^*{\mathcal O}(G/J)$.
In this paper we prove that if $G/H$ is pseudoconvex, i.e., if
$G/H$ admits a continuous plurisubharmonic exhaustion function,
then $G/J$ is Stein and $J/H$ has no non--constant holomorphic functions.
\end{abstract}
\thanks{This work was partially supported by an NSERC Discovery Grant.
We thank K. Oeljeklaus for pointing out Proposition \ref{Karl}.
We also thank A. Huckleberry for stimulating discussions concerning this paper,
C. Miebach for his constructive criticisms, and the referee for some further suggestions. }
\maketitle
\section{Introduction} \label{sect1}
The original Levi problem dealt with the characterization of domains of holomorphy in $\mathbb C^n$ having smooth boundary
in terms of conditions on that boundary.
Grauert \cite{Gr58} (resp. Narasimhan \cite{Nar62}) showed that a complex manifold (resp. complex space)
that admits a smooth (resp. continuous) strictly plurisubharmonic exhaustion function is Stein.
However, Grauert also pointed out that there exist pseudoconvex domains in tori all of whose holomorphic functions are constant, e.g., see \cite{Nar63}.
This shows that the Levi problem for pseudoconvex domains that are not strongly pseudoconvex fails, in general.
In this paper we restrict our attention to complex homogeneous manifolds $X := G/H$ with $G$ a connected complex Lie group and $H$ a closed complex subgroup,
and we seek conditions under which its holomorphic function algebra ${\mathcal O}(G/H)$ is Stein.
Setting
\[
J \; := \; \{ \ g \in G \ | \ f(gH) \; = \; f(eH) \mbox{ for all } f\in{\mathcal O}(G/H) \ \}
\]
yields a closed complex subgroup of $G$ containing $H$ and consequently one has
the holomorphic fibration $\pi : G/H \to G/J, \; gH \mapsto gJ$
which we call the \textbf{holomorphic reduction} of $G/H$ \cite{GH78}.
By construction one has ${\mathcal O}(G/H) \cong \pi^* {\mathcal O}(G/J)$.
The question then reduces to considerations of when the holomorphically separable complex homogeneous manifold $G/J$ is Stein.
Now it is known that holomorphic separability implies Steinness for complex Lie groups \cite{MM60},
for complex nilmanifolds \cite{GH78}, and for complex solvmanifolds \cite{HO86}.
But the situation is much different when the group acting is semisimple or reductive.
For example, $\mathbb C^n \setminus \{ 0 \}$ is not Stein whenever $n>1$.
What is known is that $G/H$ is holomorphically separable for $G$ reductive implies $H$ is an
algebraic subgroup of $G$ \cite{BO73} and such a $G/H$ is Stein if and only if $H$ is reductive \cite{Mat60} and \cite{On60}.
A reductive complex Lie group is the complexification of a totally real maximal compact subgroup and the
compactness of this subgroup is playing an essential role.
An excellent survey of the function theory on $G/H$ for $G$ reductive can be found in \cite[Chapter 5]{Akh95} and we refer
the interested reader to that book and the references listed therein.
The present paper analyzes the holomorphic function theory of \textbf{pseudoconvex} complex homogeneous manifolds,
where a complex manifold is pseudoconvex if it admits a continuous plurisubharmonic exhaustion function.
In \cite[Main Theorem]{GMO13} we proved that the base of the holomorphic reduction of any pseudoconvex complex
homogeneous manifold $G/H$ is Stein and its fiber has no non--constant holomorphic functions if $G$ is solvable (see also \cite[Theorem 8.5]{GMO13})
or if $G$ is reductive (see also \cite[Theorem 7.5]{GMO13}).
Here we prove that this result holds more generally for pseudoconvex homogeneous manifolds of mixed groups, where by a mixed group
we mean that $G$ is a connected, simply connected, complex Lie group that
has a Levi--Malcev decomposition $G = S \ltimes R$, where $R$ is the radical of $G$,
$S$ is a maximal semisimple subgroup, and both $\dim R > 0$ and $\dim S > 0$.
\begin{thm} \label{mnthm}
Let $G$ be a connected complex Lie group and $H$ a closed complex subgroup of $G$ such
that $G/H$ is pseudoconvex.
Suppose $G/H \to G/J$ is the holomorphic reduction of $G/H$.
Then $G/J$ is Stein and ${\mathcal O}(J/H) = \mathbb C$.
\end{thm}
This note is organized as follows.
Section two contains some technical results that are needed for the proof.
The third section presents the proof of Theorem \ref{mnthm}.
\section{Technical Preparations}
\subsection{The Hirschowitz annihilator} \label{Hirschowitz}
Every element $\xi\in\mathfrak g$, the Lie algebra of $G$, can be thought of as a right invariant vector field on $G$
and, as such, pushes down to a holomorphic vector field $\xi_X$ on any complex homogeneous manifold $X:=G/H$ of the group $G$.
An inner integral curve in such a homogeneous space $X$ is a non--constant holomorphic map $\mathbb C \to X$ with relatively compact image
in $X$ that is the integral curve of some vector field $\xi_X$ associated to some $\xi\in\mathfrak g$.
Hirschowitz \cite{Hir75} considered such concepts in the context of infinitesimally homogeneous manifolds, where a manifold
is infinitesimally homogeneous if every tangent space is generated by global holomorphic vector fields.
A complex manifold that is homogeneous under the action of a Lie group of holomorphic transformations is
infinitesimally homogeneous.
Hirschowitz showed that a pseudoconvex, infinitesimally homogeneous $X$ that does not contain
any inner integral curve is Stein \cite[Proposition 3.4]{Hir75}.
This is the starting point of our investigations of pseudoconvex homogeneous manifolds that are not Stein given in \cite{GMO13}.
By the maximum principle any plurisubharmonic function on a complex manifold $X$ is constant along every inner integral curve in $X$.
One has to determine the ``directions of degeneracy'' of plurisubharmonic functions in terms of a certain subset of
$\mathfrak g$ whose corresponding holomorphic vector fields ``kill them''.
So in \cite{GMO13} we define the Hirschowitz annihilator $\mathcal A$ to be the connected Lie subgroup of $G$ whose Lie algebra is given by
\[
\mathfrak a \; := \; \{ \ \xi\in\mathfrak g \ | \ \xi_X \varphi (x_0) \; = \; 0, \; \forall \varphi\in{\mathscr P(X)} \ \} ,
\]
where $\mathscr P(X)$ is the space of continuous plurisubharmonic functions on $X := G/H$.
For continuous functions the derivative is understood in the sense of distributions.
By definition $\mathfrak a$ is a complex vector subspace of $\mathfrak g$ and
it is also a Lie subalgebra of $\mathfrak g$ that properly contains $\mathfrak h$, if $X$ is not Stein \cite[Lemma 3.3]{GMO13}.
The corresponding (not necessarily closed) connected complex subgroup $\mathcal A$ of $G$ contains the identity component
$H^0$ of $H$ and defines a complex foliation $\mathscr{F} = \{ F_x\}_{x\in X}$, the {\bf Levi foliation} of the manifold $X$.
Every leaf $F_x$ of this foliation is a relatively compact immersed complex submanifold of $X$ containing every
inner integral curve in $X$ passing through the point $x$ and is an orbit of the group $\mathcal A$.
In general, complex foliations are rather difficult to understand.
But here the foliation arises from a subgroup strongly reflecting the underlying geometry of the homogeneous manifold
and is related to the existence of a plurisubharmonic exhaustion on it.
This allows a sufficiently good understanding in
order to analyze the structure from the point of view of its holomorphic function algebra.
If the leaves of the foliation are closed, then they are compact and $X$ is holomorphically convex \cite[Theorem 4.1]{GMO13}.
Indeed, the holomorphic reduction is then given by the Remmert reduction \cite{Rem56} and has compact fiber and Stein base.
The main difficulty occurs when this is not the case.
The following observation is essential in what follows.
\begin{remark}
A question of Serre \cite{Ser53} asks whether the total space of every holomorphic fibration with
Stein fiber and Stein base is itself Stein.
Counterexamples are known, some of which are even homogeneous, e.g., \cite{CL85}.
Among other things Lemma \ref{Hirsch} below asserts that Serre's question has an affirmative answer in the present setting.
\end{remark}
\begin{lemma} [\cite{Hir75} and Remark 2.5 in \cite{GMO13}] \label{Hirsch}
Suppose $Y$ is a pseudoconvex complex homogeneous manifold that admits a holomorphic fibration
\[
Y \; \stackrel{F}{\longrightarrow} \; B
\]
with both $F$ and $B$ being holomorphically separable manifolds.
Then $Y$ is Stein.
\end{lemma}
\begin{proof}
Note that every complex homogeneous manifold is infinitesimally homogeneous.
As neither $F$ nor $B$ can contain any inner integral curve,
the same is true of $Y$.
Therefore, $Y$ is Stein \cite[Proposition 3.4]{Hir75}.
\end{proof}
\subsection{Tits' Normalizer Fibration}
Suppose $H$ is a $k$--dimensional closed subgroup of an $n$--dimensional Lie group $G$.
The Lie algebra $\mathfrak h$ of $H$ is a Lie subalgebra of $\mathfrak g$ and can be considered
as a point in the Grassman manifold $Gr(k,n)$ of $k$--dimensional vector subspaces of $\mathfrak g$.
Since ${\rm ad}(G) \subset GL(\mathfrak g)$, there is a natural action of ${\rm ad}(G)$ on $Gr(k,n)$ and
the ${\rm ad}(G)$--orbit of the point $\mathfrak h$ can be identified with $G/N$, where $N := N_G(H^0)$,
i.e., the normalizer in $G$ of the identity component $H^0$ of $H$.
Via the Pl\"ucker embedding $Gr(k,n)$ can be realized as a submanifold of some projective space such
that the automorphisms of the Grassman are restrictions of those automorphisms of the projective space
that stabilize the embedded Grassman.
In this way we realize $G$ acting linearly on $G/N$ via the adjoint representation.
Since $H \subset N_G(H^0)$, we have the Tits normalizer fibration $G/H \to G/N$ \cite{Tits62}.
\begin{remark}
Let $I$ be a complex Lie subgroup of the complex Lie group $G$.
The normalizer $N_G(I^0)$, where $I^0$ denotes the connected component
of the identity of $I$ is a closed subgroup of $G$, since it is the
isotropy subgroup of a point in a complex projective space under the adjoint action of the group $G$.
\end{remark}
\subsection{Closure of orbits}
Let $G$ be a (real) Lie group and $H$ a closed subgroup of $G$.
For $I$ a normal Lie subgroup of $G$, set $F := cl_X(I.x_0)$,
where $x_0$ is the base point of $X := G/H$, and let $J := cl_G(I\cdot H)$.
\begin{lemma} \label{closure}
Then $J$ is a subgroup of $Stab_G(F)$.
\end{lemma}
\begin{proof}
Since $F$ is closed, $Stab_G(F)$ is closed.
Clearly $I\cdot H \subset Stab_G(F)$.
So $J \subset Stab_G(F)$.
\end{proof}
\begin{remark} \label{closurermk}
As a consequence, $F = J . x_0$.
Now for $x\in X$ define $[x] := cl_X(I.x)$ and note that if $x=g(x_0)$, then $[x] = g(F)$.
Thus the classes $[x]$ are the fibers of the homogeneous fibration $G/H \to G/J$.
\end{remark}
\subsection{Fibrations over Projective orbits}
We use the following notation for the derived series of the Lie group $G$
\[
G^{(0)} := G, \; G^{(1)} := G' = [G,G], \; \ldots, \; G^{(k)} := [G^{(k-1)},G^{(k-1)}] \mbox{ for all } k >1 .
\]
\begin{lemma} \label{flag1}
Let $X := G/H$ be an orbit of a connected complex Lie group
$G$ acting holomorphically and effectively on some projective space.
Assume $J$ is a connected, complex subgroup of $G$
that has positive dimensional orbits in $G/H$ that are relatively compact.
Then $J$ is a subgroup of every subgroup $G^{(m)}$ of the derived series of $G$.
\end{lemma}
\begin{proof}
By a result of Chevalley \cite{Chev51} the image of the commutator group $G'$
in the automorphism group of the ambient projective space is algebraically closed.
This means that $G'$ is acting as an algebraic group on the projective space and,
in particular, that its orbits in $G/H$ are closed.
Thus one has the commutator fibration $G/H \to G/H\cdot G'$.
Let $x_0\in G/H$ be the base point.
The Stein Abelian group $\overline{G}/\overline{G}_{x_0}\cdot \overline{G}'$ contains $G/H\cdot G'$ as a $G$--orbit \cite{HO81},
where the bar denotes the Zariski closure.
Therefore, $J.x_0$ is contained in the fiber of $G/H \to G/H\cdot G'$ by the maximum principle with
$J.x_0$ still being relatively compact in the closed fiber $G'/H\cap G'$.
Replace $G$ (resp. $H$) by $G^{(1)} :=G'$ (resp. $H^{(1)}:=H\cap G' = G'_{x_0}$) and
iterate the argument to see that $J$ is a subgroup of every group in the derived series of $G$.
\end{proof}
\begin{lemma} \label{flag3}
Let $G$ be a connected complex Lie group, $H$ a closed complex subgroup of $G$, and $I$ a closed complex subgroup of $G$ containing
$H$ with $G/I$ equivariantly embedded in the complex projective space $\mathbb P_N$.
Suppose that the fibration $G/H \to G/I$ is a covering.
If $J$ is a connected, normal, complex Lie subgroup of $G$ whose orbits in $G/H$ are relatively compact, then the $J$--orbits in $G/H$ are flag manifolds.
\end{lemma}
\begin{proof}
By Lemma \ref{flag1} the image of $J$ in the automorphism group of $G/I$
lies in the image of every subgroup $G^{(m)}$ of the derived series of $G$.
Since $G$ has finite dimension, one has $G^{(k)} = (G^{(k)})' = G^{(k+1)} = \ldots $ for some $k$.
As the $J$--orbits have positive dimension, $G^{(k)}$ is a positive dimensional perfect Lie group that is acting algebraically on $\mathbb P_N$.
Its radical $R_{G^{(k)}}$ is nilpotent \cite{Jac62}.
Since $J$ is normal, its radical $R_{J} = R_{G^{(k)}} \cap J$ \cite{Jac62}.
Thus $R_{J}$ is a connected complex subgroup of the unipotent group $R_{G^{(k)}}$ and so
is acting algebraically on $\mathbb P_N$ with each of its orbits a closed copy of $\mathbb C^q$ for some $q \ge 0$.
Since the $J$--orbits are relatively compact, we must have $q = 0$.
This implies that $R_J$ acts trivially and thus $J$ is acting algebraically as a semisimple group on $G/I$.
Now $A := {\rm cl}_{G}(J \cdot H) \subset K := {\rm cl}_{G}(J\cdot I)$, since $H \subset I$ by assumption.
By Remark \ref{closurermk} and the assumption that the $J$--orbits are relatively compact we get the following diagram
\[
\begin{array}{ccc} G/H & \longrightarrow & G/A \\
\downarrow & & \downarrow \\
G/I & \longrightarrow & G/K \end{array}
\]
The $J$--orbit through the base point in $G/I$ is contained in $K/I$ and the latter space is compact.
Thus the closure of this $J$--orbit lies in $K/I$.
Since $J$ is acting algebraically as a semisimple complex group, this orbit is Zariski open in its closure and
its boundary consists of $J$--orbits of strictly lower dimension.
Since $J \lhd G$, all orbits have the same dimension and so the boundary is empty, i.e., the $J$--orbits are closed and thus compact.
Compact orbits of a complex Lie group acting holomorphically on a projective space are flag manifolds.
Since flag manifolds are simply connected and the $J$--orbits in $G/H$ cover the $J$--orbits in $G/I$, it follows that
the $J$--orbits in $G/H$ are flag manifolds.
\end{proof}
\begin{cor} \label{flag2}
Let $G$ be a connected complex Lie group, $H$ a closed complex subgroup of $G$, and $I$ a closed complex subgroup of $G$ containing
$H$ with $G/I$ equivariantly embedded in the complex projective space $\mathbb P_N$
and the fibers of the fibration $G/H \to G/I$ holomorphically separable.
If $J$ is a connected, normal, complex Lie subgroup of $G$ whose orbits in $G/H$ are relatively compact, then the $J$--orbits in $G/H$ are flag manifolds.
\end{cor}
\begin{proof}
Since $I/H$ is holomorphically separable, the $J$--orbits intersect the fibers of the fibration $G/H \to G/I$ transversally and
so cover the corresponding $J$--orbits in $G/I$ which are necessarily relatively compact in $G/I$.
The result now follows from Lemma \ref{flag3}.
\end{proof}
\subsection{Normality under closure and complexification}
For $J$ a connected Lie subgroup of $G$ the complexification $J^{\mathbb C}$ of $J$ is the connected Lie subgroup of $G$
corresponding to the Lie algebra $\mathfrak j + i \mathfrak j$, where $\mathfrak j$ denotes the Lie algebra of $J$.
Subsequently we need to consider what happens to normality under closure and complexification and so the following is important.
\begin{lemma} \label{normal}
Suppose $I$ is a connected normal Lie subgroup of a connected Lie subgroup $J$ of a connected complex Lie group $G$.
Then
\begin{enumerate}
\item $I \lhd {\rm cl}_G(J)$
\item $I^{\mathbb C} \lhd J^{\mathbb C}$
\end{enumerate}
In particular, $I^{\mathbb C} \lhd \widetilde{J}$, where $\widetilde{J}$ is the smallest connected closed complex subgroup of $G$ that contains $J$.
\end{lemma}
\begin{proof}
(1) If $\lim g_n = g \in {\rm cl}_G(J)$,where $g_n\in J$, then $gIg^{-1} = \lim g_n I g_n^{-1} \subset I$.
\noindent
(2) This follows from the fact that
$ [\mathfrak i, \mathfrak j ] \subset \mathfrak i \Longrightarrow [ \mathfrak i + i \mathfrak i , \mathfrak j + i \mathfrak j ] \subset \mathfrak i + i \mathfrak i$.
\noindent
Note that $\widetilde{J}$ can be formed by alternately taking the complexification and closure of $J$.
Applying (1) and (2), as appropriate, completes the proof.
\end{proof}
\subsection{Existence of a fibration by a solvmanifold}
\begin{proposition}[personal communication from K. Oeljeklaus] \label{Karl}
Let $X := G/\Gamma$, where $\Gamma$ is a discrete subgroup of a connected, simply connected,
complex Lie group $G$ with Levi decomposition $G = S \ltimes R$ and $\dim R > 0$.
Then there is a connected, complex, solvable subgroup $H$ of $G$ normalized by
$\Gamma$, containing $R$, with $H\cdot \Gamma$ a closed subgroup of $G$.
In particular, one has the proper fibration (unless $S = \{ e \}$)
\begin{eqnarray} \label{radfibr}
G/\Gamma \; \longrightarrow \; G/H\cdot \Gamma \; = \; S/S\cap H\cdot \Gamma
\end{eqnarray}
that has the connected complex solvmanifold $H/H \cap \Gamma$ as its typical fiber.
\end{proposition}
\begin{proof}
If the $R$--orbits themselves are closed, set $H := R$.
If not, then the Zassenhaus Lemma \cite{Aus63} is used in \cite[Theorem 2]{Gi81} to show the existence of
a minimal connected complex solvable subgroup $H_1 \subset G$ normalized by $\Gamma$
and containing the identity component of ${\rm cl}_G(R\cdot\Gamma)$.
If $H_1\cdot\Gamma$ is closed in $G$, one has the desired result with $H := H_1$.
Otherwise, let $N_1 := N_G(H_1)$.
Since $\Gamma$ normalizes the identity component $N_1^0$ of $N_1$, it also normalizes its radical $R_1$.
Now $H_1 \subset R_1$, because $H_1$ is solvable and normal in $N_1$.
Either $R_1\cdot\Gamma$ is closed in $G$, or the identity component of ${\rm cl}_G(R_1\cdot\Gamma)$
is contained in $N_1^0$.
Applying the Zassenhaus Lemma again (in $N_1^0$) we see that this identity component is solvable
and normalized by $\Gamma$, since $N_1^0\cdot \Gamma = N_1$ is closed in $G$.
Let $H_2$ be the smallest connected closed complex subgroup of $G$ that contains
this identity component.
Then $H_2$ is solvable, normalized by $\Gamma$ and its dimension is strictly greater than the dimension of $H_1$.
A finite number of steps yields the desired connected complex solvable group $H$.
\end{proof}
\subsection{Existence of a tower}
\begin{lemma} \label{ind1}
Suppose $\Gamma$ is a cocompact, discrete subgroup of a (positive dimensional) connected solvable Lie group $L$ such that
$L^{\mathbb C}/\Gamma$ is Stein, where $L^{\mathbb C}$ is the complexification of $L$.
Then there exists a fibration by the center $Z$ of the nilradical of $L^{\mathbb C}$
\[
L^{\mathbb C}/\Gamma \; \stackrel{(\mathbb C^*)^k}{\longrightarrow} \; L^{\mathbb C}/Z\cdot\Gamma
\]
that has $(\mathbb C^*)^k$ as fiber with $k>0$.
\end{lemma}
\begin{proof}
Let $N_0$ be the nilradical of $L$ and $N$ the nilradical of $L^{\mathbb C}$.
Then $N_0$ has closed orbits in $L/\Gamma$ by a theorem of Mostow \cite{Mos54} or \cite{Mos71}
and thus $N$ has closed orbits in $L^{\mathbb C}/\Gamma$.
Let $Z$ be the center of $N$ (resp. $Z_0$ of $N_0$).
Incidentally, note that $\dim Z > 0$ \cite{Mat51}.
Since $N/N\cap\Gamma$ is a closed complex submanifold of the Stein manifold $L^{\mathbb C}/\Gamma$,
we see that $N/N\cap\Gamma$ is Stein.
Hence the subgroup $Z\cdot\Gamma$ is closed by a result of Barth--Otte \cite{BO69}; see also \cite[Theorem 4]{GH78}.
Therefore, we have the fibration
\[
L^{\mathbb C}/\Gamma \; \longrightarrow \; L^{\mathbb C}/Z\cdot\Gamma .
\]
It follows that the fibers of the fibration above are $(\mathbb C^*)^k$--orbits for some positive integer $k$; see \cite[Theorem 7]{GH78}.
\end{proof}
\begin{definition}
A $\mathbb C^*$ power tower of length {\it one} is simply the manifold $(\mathbb C^*)^p$ for some positive integer $p$.
For any integer $n>1$ a $\mathbb C^*$ power tower of length $n$ is a $(\mathbb C^*)^k$--bundle
over a $\mathbb C^*$ power tower of length $n-1$.
\end{definition}
\begin{remark}
Repeated application of Lemma \ref{ind1} shows that the space $L^{\mathbb C}/\Gamma$ is a $\mathbb C^*$ power tower of length $n$
for some positive integer $n$.
\end{remark}
\section{Proof of the main result}
\subsection{Formulation of the strategy of the proof}
\begin{remark} \label{connected}
There is a technical point that can arise in our construction,
in that an intermediary fibration whose fiber is not connected might be involved.
This is handled by a type of {\em Stein factorization for homogeneous fibrations}.
Suppose $G$ is a connected Lie group that contains a closed subgroup $I$ containing a closed subgroup $H$.
Let $\widetilde{I}$ be those connected components of $I$ that meet $H$.
Then $\widetilde{I}$ is a closed subgroup of $G$ containing $H$ and the fibration $G/H \to G/\widetilde{I}$ has connected fiber $\widetilde{I}/H$.
\end{remark}
\noindent
{\sc Strategy of the Proof:}
\noindent
Assume $G/H$ is a pseudoconvex homogeneous manifold that is not Stein.
In order to prove the theorem
we construct a closed complex subgroup $I$ of $G$ containing $H$ with $\dim I > \dim H$ and $I/H$ connected,
possibly with the aid of Remark \ref{connected}, such that
\begin{enumerate}
\item [] (i) ${\mathcal O}(I/H) = \mathbb C$ and
\item [] (ii) every continuous plurisubharmonic
exhaustion function on $G/H$ induces a continuous plurisubharmonic exhaustion function on $G/I$.
\end{enumerate}
Then if $G/H \to G/J$ is the holomorphic reduction, $I$ is a subgroup of $J$ because of (i), $G/I$ is pseudoconvex because of (ii)
and is either Stein (then $I=J$ and we are done) or not Stein and one applies the construction until
one does reach the holomorphic reduction; see also the last paragraph of the proof below for more details.
\begin{remark}
Here is a list of some complex homogeneous manifolds that do satisfy (ii),
whenever they occur as the fiber of a homogeneous fibration $G/H \to G/I$.
For (2) and (3), the basic tool to prove this is Kiselman's Minimum Principle \cite[Theorem 2.2]{Kis78}:
\begin{enumerate}
\item compact complex homogeneous manifolds,
\item Cousin groups, see \cite[Lemma 6.1 (1)]{GMO13},
\item the fibers of certain $\mathbb C^*$ power towers provided the exhaustion function is
constant on the underlying circle power tower, see below.
\end{enumerate}
\end{remark}
\subsection{The proof itself}
\begin{proof}
Let $G/H$ be a non--Stein pseudoconvex homogeneous manifold.
Then its Hirschowitz annihilator $\mathcal A$ satisfies $\dim \mathcal A > \dim H$.
We define subgroups by setting
\begin{enumerate}
\item $G_{1} := N_{G}(\mathcal A)$; note that $H$ is a subgroup of $N_G(\mathcal A)$ by \cite[p. 42]{GMO13} and
\item $G_{2} := N_{G_{1}}(H^0)$.
\end{enumerate}
Note that in (1), because the $\mathcal A$--orbits are positive dimensional and relatively compact, the fibration
$G/H \to G/N_{G}(\mathcal A)$ is not a covering and its fiber is not Stein.
Also because of the fact that $\mathcal A$ is normal in $G_1$, if the fibration $G_1/H \to G_1/N_{G_{1}}(H^0)$
is a covering (resp. has a Stein fiber), then the $\mathcal A$--orbits in $G_1/H$ are flag manifolds by Lemma \ref{flag3}
(resp. by Corollary \ref{flag2}).
We are done, since setting $I := \mathcal A \cdot H$ yields a fibration $G/H \to G/I$ with $I/H$ compact and thus satisfying (i) and (ii).
Thus we need only consider the setting where $G_2/H$ is a non--Stein pseudoconvex homogeneous manifold.
If the Hirschowitz annihilator $\mathcal A_2$ for $G_2/H$ is not normal in $G_2$,
we apply (1) again setting $G_3 := N_{G_{2}}(\mathcal A_2)$.
So $G_3 = N_{G_3}(\mathcal A_2) < N_{G_1}(H^0)$, the latter because $G_3$ is a subgroup of $G_2$.
Now set $\widehat{G} := G_3/H^0, \widehat{\mathcal A} := \mathcal A_2/H^0$ and $\Gamma := H/H^0$ and
note that $\widehat{\mathcal A} \lhd \widehat{G}$ and has positive dimensional orbits in $\widehat{G}/\Gamma$.
Furthermore, we may assume that $\widehat{G}$ is not solvable (resp. semisimple) because of
\cite[Theorem 8.5]{GMO13} (resp. \cite[Theorem 7.1]{GMO13}), since each of these settings directly
yields the desired subgroup $I$ satisfying (i) and (ii); in the first case it arises from a fibration by a Cousin group and in the second by a
compact complex manifold.
Hence we reduce to the case where $\widehat{G}$ is a mixed group.
The rest of the construction below produces a subgroup $\widehat{I}$ of $\widehat{G}$ containing $\Gamma$
with $\widehat{I}/\Gamma$ satisfying (i) and (ii).
Taking the preimage $I$ of $\widehat{I}$ via the quotient homomorphism $G_3 \to G_3/H^0$ gives us the desired fibration $G/H \to G/I$.
For notational convenience we suppress the hats from now on and write $G$ instead of $\widehat{G}$, etc.
Since $\mathcal A \lhd G$, we may apply Lemma \ref{closure} and Remark \ref{closurermk}.
Set $L := cl_{G}(\mathcal A\cdot \Gamma)$ and let $\widetilde{L}$ be the smallest connected, closed, complex subgroup of $G$ that contains $L$.
Since the $\mathcal A$--orbits are relatively compact, $L/\Gamma$ is compact.
As a consequence, ${\mathcal O}(\widetilde{L}/\Gamma) = \mathbb C$ by the maximum and identity principles.
We claim that we may further reduce to the setting where the group $L$ has a positive dimensional radical $R_L$, a fact that we will later use.
If $\widetilde{L}$ is semisimple, then one has the fibration $G/\Gamma \to G/\widetilde{L}$ and
$\widetilde{L}/\Gamma$ is pseudoconvex and thus holomorphically convex \cite[Theorem 7.1]{GMO13}.
This implies $\widetilde{L}/\Gamma$ is compact and we are again done with $I := \widetilde{L}$.
In particular, in the rest of the proof we assume that $\dim R_L > 0$.
We need to show that $\widetilde{L}/\Gamma$ satisfies (ii) in this setting.
By Proposition \ref{Karl} there is a fibration
\[
\widetilde{L}/\Gamma \; \longrightarrow \; \widetilde{L}/ \widetilde{H}\cdot\Gamma
\]
with $ \widetilde{H}\cdot\Gamma$ closed in $ \widetilde{L}$, where $ \widetilde{H}$ is a connected, solvable, complex Lie group
that contains the radical $R_{ \widetilde{L}}$ of $\widetilde{L}$ and is normalized by $\Gamma$.
Now we claim that we may further reduce to the setting where
the fiber $ \widetilde{H}\cdot\Gamma/\Gamma$ of the above fibration is Stein.
Otherwise, $ \widetilde{H}\cdot\Gamma/\Gamma$ would be a connected, pseudoconvex solvmanifold that is not Stein
and there would exist a closed complex subgroup $I$ of $ \widetilde{H}\cdot\Gamma$ containing $\Gamma$ with $I/\Gamma$
a positive dimensional Cousin group \cite[Theorem 8.5]{GMO13}.
Clearly, $I/\Gamma$ satisfies conditions (i) and (ii).
So from here on we may assume that $ \widetilde{H}\cdot\Gamma/\Gamma$ is Stein.
Now in order to finish the proof that $\widetilde{L}/\Gamma$ satisfies (ii) we have to analyze the structure of (at least part of)
the intersection of the compact orbit $L/\Gamma$ with the Stein orbit $\widetilde{H}/\widetilde{H}\cap\Gamma$.
Consider the connected, real, solvable group $B := (L \cap \widetilde{H})^0$.
Since $R_L \subset R_{\widetilde{L}}\subset\widetilde{H}$ by Lemma \ref{normal} and $R_L \subset L$, it follows that
$R_L \subset B$ and thus $\dim B >0$.
Let $B^{\mathbb C}$ be its complexification.
We have $\Gamma \subset N_{\widetilde{L}}(B^{\mathbb C})$,
since $\Gamma \subset L$ and $\Gamma$ normalizes $\widetilde{H}$ by Proposition \ref{Karl},
and thus we may consider the fibration $\widetilde{L}/\Gamma \to \widetilde{L}/N_{\widetilde{L}}(B^{\mathbb C})$.
Now we may assume that $N_{\widetilde{L}}(B^{\mathbb C})/\Gamma$ is not Stein,
for, otherwise, $\mathcal A$ would have compact orbits by Lemma \ref{flag2}, a case that could be easily handled, as above.
The Hirschowitz annihilator for the space $N_{\widetilde{L}}(B^{\mathbb C})/\Gamma$ need not be normal in $\widetilde{L}$.
So we begin the proof again with the space $N_{\widetilde{L}}(B^{\mathbb C})/\Gamma$ and run
at most a finite number of times (because $\dim G/H <\infty$) through all of its steps until
the only situation demanding further attention occurs when $N_{\widetilde{L}}(B^{\mathbb C})=\widetilde{L}$.
Hence we may assume that $B^{\mathbb C}$ is a connected, normal, solvable subgroup of ${\widetilde{L}}$.
As a consequence, $B^{\mathbb C} \cap L$ is normal in $L$ and this implies that $B \subset R_L$.
But $R_L \subset B$ and so $B=R_L$.
We claim that the $R_L$--orbits in $\widetilde{L}/\Gamma$ are compact.
As noted above, $R_L \subset R_{\widetilde{L}}$ by Lemma \ref{normal}.
So by the construction of $\widetilde{H}$, one has
\[
(R_L\cdot\Gamma)^0 \; \subset \; {\rm cl}_{\widetilde{L}}(R_L\cdot\Gamma)^0 \; \subset \;
((L\cap\widetilde{H})\cdot\Gamma)^0 \; = \; (R_L\cdot\Gamma)^0 ,
\]
where, as usual, the superscript denotes the connected component of the identity.
It follows that the $R_L$--orbits are closed and since the $L$--orbits are compact, the $R_L$--orbits are thus also compact.
Note that if ${\mathfrak r}_L \cap i {\mathfrak r}_L \not= (0)$, then the connected complex Lie group corresponding to
the complex ideal ${\mathfrak r}_L \cap i {\mathfrak r}_L $ has positive dimensional orbits in the compact $R_L$--orbits in the $\widetilde{H}$--orbits.
By the maximum principle this contradicts our reduction to the setting where the $\widetilde{H}$--orbits are Stein.
Thus one must have ${\mathfrak r}_L \cap i {\mathfrak r}_L = (0)$ and the $R_L$--orbits in $\widetilde{L}/\Gamma$ are totally real.
Now consider the complexification $R_L^{\mathbb C}$ of $R_L$ that has
Lie algebra ${\mathfrak r}_L^{\mathbb C} := {\mathfrak r}_L \oplus i {\mathfrak r}_L$.
According to a conjecture of Mostow \cite{Mos54} that was proved by Auslander--Tolimieri \cite{AT69} and Mostow \cite{Mos71} every solvmanifold has the
structure of a (real) vector bundle over a compact solvmanifold.
In general, the compact base is homogeneous with respect to a group that is not a subgroup of the original solvable Lie group acting on the manifold,
but rather lies in a certain algebraic hull of that group.
We claim that our setting is special in that $R_L \subset R_L^{\mathbb C}$ can be taken to be that subgroup.
Suppose
\[
R_L^{\mathbb C}/R_L^{\mathbb C}\cap\Gamma \; \stackrel{\mathbb R^k}{\longrightarrow} \; M
\]
is the vector bundle given by Mostow's conjecture.
Since $R_L^{\mathbb C}/R_L^{\mathbb C}\cap\Gamma$ is Stein, it follows from Serre's homology condition \cite{Ser53} that
$ \dim_{\mathbb C} R_L^{\mathbb C} \ge \dim_{\mathbb R} M \ge \dim_{\mathbb R} R_L$.
On the other hand we have $\dim_{\mathbb R} R_L = \dim_{\mathbb C} R_L^{\mathbb C}$, as noted above.
As a consequence, the compact base $M$ is diffeomorphic to $R_L/R_L\cap\Gamma$ and $R_L^{\mathbb C}\cap\Gamma = R_L\cap\Gamma$.
So the $R_L^{\mathbb C}$--orbits are closed in $\widetilde{L}/\Gamma$.
We may now apply Lemma \ref{ind1} to the triple $(R_L \cap\Gamma, R_L, R_L^{\mathbb C})$.
Now we have the fibration $\widetilde{L}/\Gamma \to \widetilde{L}/R_L^{\mathbb C}\cdot\Gamma$.
Any continuous plurisubharmonic exhaustion function $\varphi$ on $G/\Gamma$ is constant on each of the $\mathcal A$--orbits,
since these orbits lie in the level sets of the exhaustion function.
By continuity $\varphi$ is then constant on the orbits of the closure $L$ and thus on the orbits of its radical $R_L$.
The $(S^1)^k$--orbits that arise in fibration
\[
R_ L^{\mathbb C}/(R_L^{\mathbb C}\cap\Gamma) \;
\stackrel{(\mathbb C^*)^k}{\longrightarrow} \; R_L^{\mathbb C}/Z\cdot(R_L^{\mathbb C}\cap\Gamma)
\]
given by Lemma \ref{ind1} are part of the $R_L$--orbits.
Thus $\varphi$ is constant on the $(S^1)^k$--orbits.
As in \cite[Lemma 6.1 (2)]{GMO13} one can apply Kiselman's minimum principle \cite[Theorem 2.2]{Kis78}
and push $\varphi$ down to $R_L^{\mathbb C}/Z\cdot(R_L^{\mathbb C}\cap\Gamma)$.
As a consequence $ \widetilde{L}/R_L^{\mathbb C}\cdot\Gamma$ is pseudoconvex.
We continue in this fashion until a maximal semisimple group is acting transitively on the resulting quotient space $Z$.
But then $Z$ is pseudoconvex and thus holomorphically convex \cite[Theorem 7.1]{GMO13}.
Since $\mathcal{O} (\widetilde{L}/\Gamma) = \mathbb C$, it follows that $Z$ is compact and
we see that $\widetilde{L}/\Gamma$ satisfies (ii).
This completes the proof that $\widetilde{L}/\Gamma$ satisfies (i) and (ii).
In order to complete the proof of the Theorem, we assume that $G/H$ is a pseudoconvex homogeneous manifold
that is not Stein with ${\mathcal O}(G/H) \not= \mathbb C$ and we let $G/H \to G/J$ be its holomorphic reduction.
Note that $J/H$ cannot be Stein, since this would imply that $G/H$ itself would be Stein by Lemma \ref{Hirsch}.
We now choose the \emph{maximal} $I$ given by the construction above.
Then ${\mathcal O}(I/H) = \mathbb C$.
Moreover, $G/I$ is pseudoconvex due to the fact that $I$ satisfies (ii).
If $G/I$ were not Stein, then there would exist a subgroup $I_1$ with $\dim I_1 > \dim I$
with $I_1/I$ satisfying conditions (i) and (ii).
But this would imply that $I_1/H$ also satisfies these two conditions, contradicting the maximality of $I$.
Consequently, $I = J$ and shows that the holomorphic reduction has the desired properties.
\end{proof}
\end{document}
|
{\bar{e}}gin{document}
\title{K\"uchle fivefolds of type c5}
\author[A. Kuznetsov]{Alexander Kuznetsov}
\address{Algebra Section, Steklov Mathematical Institute,
8 Gubkin str., Moscow 119991 Russia}
\email{{\tt [email protected]}}
\thanks{This work is supported by the Russian Science Foundation under grant 14-50-00005.}
{\bar{e}}gin{abstract}
We show that K\"uchle fivefolds of type $(c5)$ --- subvarieties of the Grassmannian $\Gr(3,7)$ parameterizing 3-subspaces that are isotropic for a given 2-form and are annihilated by a given 4-form ---
are birational to hyperplane sections of the Lagrangian Grassmannian $\LGr(3,6)$ and describe in detail these birational transformations.
As an application, we show that the integral Chow motive of a K\"uchle fivefold of type $(c5)$ is of Lefschetz type.
We also discuss K\"uchle fourfolds of type $(c5)$ --- hyperplane sections of the corresponding K\"uchle fivefolds ---
an interesting class of Fano fourfolds, which is expected to be similar to the class of cubic fourfolds in many aspects.
\end{abstract}
\maketitle
\section{Introduction}
One of the most interesting classical questions of birational geometry is the question of rationality of cubic fourfolds.
In the Italian school of algebraic geometry it was believed \cite{morin1940} that a general cubic fourfold is rational, but the argument had a gap.
Now some families of rational cubic fourfolds are known \cite{hassett2000special} (such as Pfaffian cubics, and some cubics containing a plane),
but it is generally believed that a very general cubic fourfold is irrational.
However, in spite of many attempts, proving irrationality of a single cubic fourfold remains out of reach.
One of the points of view on (ir)rationality of cubic fourfolds is via the structure of their derived categories (see~\cite{kuznetsov2015rationality}).
It is known that the derived category of coherent sheaves on a cubic fourfold~$X$ has a semiorthogonal decomposition consisting of three exceptional objects and an additional
subcategory $\mathscr{A}_X$, whose properties resemble very much those of the derived category of a K3 surface (such categories are usually called noncommutative K3 surfaces).
It is conjectured (see~\cite{kuznetsov2010cubic,kuznetsov2015rationality}) that $X$ is rational if and only if $\mathscr{A}_X$ is equivalent to the derived category of a (commutative) K3 surface.
This conjecture is consistent with all known examples of rational cubic fourfolds --- for those $\mathscr{A}_X$ is equivalent
to the derived category of a K3 surface, while for a very general $X$ it is easy to show that $\mathscr{A}_X$ is not equivalent
to the derived category of any K3 surface.
While it is not clear how the above conjecture could be proved, it is quite interesting to investigate other families of fourfolds
which have similar properties (i.e.\ whose derived categories contain noncommutative K3 surfaces as semiorthogonal components).
One of such families, Gushel--Mukai fourfolds was investigated in \cite{debarre2015gushel} and \cite{kuznetsov2016gushel}.
This paper makes a first step to investigation of yet another family of fourfolds with similar properties.
In 1995 Oliver K\"uchle classified in~\cite{kuchle1995fano} all Fano fourfolds of index 1 that can be obtained as zero loci of regular global sections
of equivariant vector bundles on Grassmannians. The list of such fourfolds includes 20 families (in fact, originally there were
21 families in the list, but two of them were recently shown to be equivalent \cite{manivel2015}), and three of them --- types $(c7)$, $(d3)$, and $(c5)$ in K\"uchle's notation ---
judging by their Hodge numbers, might contain a noncommutative K3 surface as a component of their derived category.
The first two types were considered in~\cite{kuznetsov2015kuchle}, and were shown not to produce an interesting example.
A fourfold of type $(d3)$ was shown to be isomorphic to the blowup of $({\bf P}^1)^4$ with center in a K3 surface,
and a fourfold of type $(c7)$ to the blowup of a cubic fourfold with center in a Veronese surface.
However, we expect that the last of the three examples --- a fourfold of type $(c5)$ --- is new and interesting.
By definition such fourfolds can be constructed as follows. Consider the Grassmannian $\Gr(3,7)$ of 3-dimensional
vector subspaces in a 7-dimensional vector space. Let $\mathscr{U}_3$ and $\mathscr{U}_3^\perp$ be the tautological vector subbundles
on the Grassmannian, of ranks 3 and 4 respectively. Consider the following rank 8 vector bundle
{\bar{e}}gin{equation*}
\mathscr{U}_3^\perp(1) \oplus \mathscr{U}_3(1) \oplus \mathscr{O}(1).
\end{equation*}
Its global section is given by a triple $(\lambda,\mu,\nu)$, where $\lambda$ is a 4-form, $\mu$ is a 2-form, and $\nu$ is a 3-form
on the 7-dimensional vector space. The zero locus of such a section (provided it is sufficiently general) is a smooth Fano fourfold
{\bar{e}}gin{equation*}
X = X^4_{\lambda,\mu,\nu} \subset \Gr(3,7).
\end{equation*}
Its numerical invariants, computed by K\"uchle, are
{\bar{e}}gin{equation*}
K_X^4 = 66,
\qquad
h^0(X,\mathscr{O}(-K_X)) = 20,
\end{equation*}
and its Hodge diamond looks as
{\bar{e}}gin{equation*}
{\bar{e}}gin{smallmatrix}
&&&& 1 \\
&&& 0 && 0 \\
&& 0 && 1 && 0 \\
& 0 && 0 && 0 && 0 \\
0 && 1 && 24 && 1 && 0 \\
& 0 && 0 && 0 && 0 \\
&& 0 && 1 && 0 \\
&&& 0 && 0 \\
&&&& 1
\end{smallmatrix}
\end{equation*}
In particular, the Hodge diamond of a K3 surface is clearly seen in its center, so one can expect to find a noncommutative K3 category as a component of its derived category.
Of course, to prove something of this sort, we need to understand the geometry of this variety better. The goal of this paper is to do some steps in this direction.
Our approach is based on the following funny common feature of the three examples of Fano fourfolds, which have (or might have) a noncommutative K3 surface.
In fact, all of them are half-anticanonical sections of nice Fano fivefolds. The corresponding fivefolds are ${\bf P}^5$, a hyperplane section of $\Gr(2,5)$,
or the zero locus of the section $(\lambda,\mu)$ of the vector bundle $\mathscr{U}_3^\perp(1) \oplus \mathscr{U}_3(1)$ on $\Gr(3,7)$, which we denote by $X^5_{\lambda,\mu}$
(and call {\sf K\"uchle fivefolds of type $(c5)$}).
{\bar{e}}gin{equation*}
{\bar{e}}gin{array}{|l|c|c|c|}
\hline
\text{Fano fourfold} & \text{cubic fourfold} & \text{Gushel--Mukai fourfold} & \text{K\"uchle fourfold $X_{\lambda,\mu,\nu}$} \\
\hline
\text{Fano fivefold} & {\bf P}^5 & \Gr(2,5) \cap H & X^5_{\lambda,\mu} \\
\hline
\end{array}
\end{equation*}
The structure of the derived category of the fourfolds (at least of cubic and Gushel--Mukai fourfolds) is determined by the structure of the derived category of the corresponding fivefolds.
Both in case of ${\bf P}^5$ and $\Gr(2,5) \cap H$, the derived category has a rectangular Lefschetz decomposition with respect to (a fraction of) the half-anticanonical line bundle (see~\cite{kuznetsov2014icm}).
From this an existence of a noncommutative K3 category in a fourfold follows by \cite{kuznetsov2015calabi}.
By the way, two ``non-interesting'' examples $(d3)$ and $(c7)$ also share this feature --- the corresponding fivefolds are $({\bf P}^1)^5$ and the blowup of ${\bf P}^5$
with center in the Veronese surface both have a rectangular Lefschetz decomposition, see~\cite{kuznetsov2015kuchle}.
It is natural to expect that the same is true for K\"uchle fivefolds of type $(c5)$.
An attempt to construct a rectangular Lefschetz decomposition of $X^5_{\lambda,\mu}$ was the main motivation for this paper.
Although we have not succeeded in this yet, a geometrical construction we have found, allowed us to show that the Chow motive (with integral coefficients) of a general K\"uchle fivefold is of Lefschetz type.
This can be considered as an approximation to the derived category result we are up to.
Let us explain this geometrical construction.
Recall that by definition, $X^5_{\lambda,\mu}$ is a subvariety in the Grassmannian $\Gr(3,7)$ defined as the zero locus of a section $({\lambda,\mu})$
of the vector bundle $\mathscr{U}_3^\perp(1) \oplus \mathscr{U}_3(1)$ given by a 4-form $\lambda$ and a 2-form~$\mu$.
We associate with it a certain hyperplane section $\LGr(3,6) \cap H$ of the Lagrangian Grassmannian $\LGr(3,6)$
and a codimension 2 subvariety $Z \subset \LGr(3,6) \cap H$, that is isomorphic to a scroll (a ${\bf P}^1$-bundle) over a sextic del Pezzo surface.
Then we show that the blowup $\widetilde{X}^5_{\lambda,\mu}$ of $\LGr(3,6) \cap H$ with center in $Z$ can be also realized as a blowup of~$X^5_{\lambda,\mu}$.
Moreover, we show that the center $F$ of the blowup $\widetilde{X}^5_{\lambda,\mu} \to X^5_{\lambda,\mu}$ is isomorphic to the flag variety ${\bf F}l(1,2;3)$.
In other words, we have a simple birational transformation between $X^5_{\lambda,\mu}$ and a hyperplane section of the Lagrangian Grassmannian that can be expressed by a diagram
{\bar{e}}gin{equation*}
\xymatrix{
&& \widetilde{X}^5_{\lambda,\mu} \ar[dl] \ar[dr] \\
F \ar@{^{(}->}[r] & X^5_{\lambda,\mu} && \LGr(3,6) \cap H & Z \ar@{_{(}->}[l]
}
\end{equation*}
Everything in this construction can be described quite explicitly, see section~\ref{section:5folds} for more details.
We believe this description should be essential for understanding the geometry of K\"uchle fivefolds $X^5_{\lambda,\mu}$ and their hyperplane sections $X^4_{\lambda,\mu}n$.
We demonstrate its usefulness by applying it to the computation of the Chow motive of $X^5_{\lambda,\mu}$ in section~\ref{section:apps}.
The paper is organized as follows.
In section~\ref{section:preliminaries} we introduce some notation and prove a very basic, but rather useful result (a blowup lemma)
allowing in some cases to identify a subscheme in a projective bundle as a blowup of its base.
In section~\ref{section:forms-and-generality} we discuss the geometry of 3-forms on a 6-space and of 4-forms on a 7-space.
After that we introduce explicit generality assumptions on a pair $({\lambda,\mu})$ of a 4-form and a 2-form on a 7-space, under which we work later.
We explain the standard form, in which such a pair can be written, and introduce some useful geometric constructions related to this data.
Section~\ref{section:5folds} is the main part of the paper.
Here we construct the birational transformation between a K\"uchle 5-fold $X^5_{\lambda,\mu}$ and its associated hyperplane section of the Lagrangian Grassmannian $\LGr(3,6)$, and discuss the details of its geometry.
In section~\ref{section:apps} we give two applications of this description.
First, we show that a general K\"uchle fourfold $X^4_{\lambda,\mu}n \subset X^5_{\lambda,\mu}$ is birational to a singular (along a curve) quadratic section of the hyperplane section of the Lagrangian Grassmannian.
Second, we show that the integral Chow motive of a K\"uchle fivefold is of Lefschetz type.
Finally, in section~\ref{section:hyperplane-sgr} we prove that the integral Chow motive of any smooth hyperplane section of the Lagrangian Grassmannian $\LGr(3,6)$ is of Lefschetz type.
For this we introduce a new geometric construction --- we identify a certain ${\bf P}^2$-bundle over such a hyperplane section
with the blowup of the isotropic Grassmannian $\LGr(2,6)$ with center in the adjoint variety of the simple algebraic group of type ${\mathbb{G}_2}$.
\subsection*{Acknowledgements:}
I am very grateful to Atanas Iliev, Grzegorz and Micha\l{} Kapustka, Laurent Manivel, Dmitri Orlov, and Kristian Ranestad for useful discussions.
\section{Preliminaries}\label{section:preliminaries}
\subsection{Notations and conventions}\label{subsection:notations}
We work over an algebraically closed field $\mathbf k$ of characteristic~0.
For any vector space $V$ we denote by $\wedge$ the wedge product of skew forms and polyvectors and by $\,\lrcorner\,$ the convolution operation
{\bar{e}}gin{equation*}
\bw{p}V \otimes \bw{q}V^\vee \xrightarrow{\ \,\lrcorner\,\ } \bw{p-q}V
\qquad\qquad
\text{(if $p \ge q$),}
\end{equation*}
induced by the natural pairing $V \otimes V^\vee \to \mathbf k$.
If $p = n = \dim V$ and $0 \ne \varepsilonilon \in \det(V)$, the convolution with $\varepsilonilon$ gives an isomorphism
{\bar{e}}gin{equation}\label{eq:isomorphism-convolution}
\bw{q}V^\vee \simeqto \bw{n - q}V,
\qquad
\xi \mapsto \xi^\vee := \varepsilonilon \,\lrcorner\, \xi.
\end{equation}
This isomorphism is canonical up to rescaling (since $\varepsilonilon$ is unique up to rescaling). Note that
{\bar{e}}gin{equation}\label{eq:convolution-duality}
\omega \,\lrcorner\, \xi = (\xi^\vee) \,\lrcorner\, (\omega^\vee),
\end{equation}
where $\omega \in \bw{k}V$, $\xi \in \bw{p}V^\vee$ and $k \ge p$, and we use $\varepsilonilon^{-1} \in \det(V^\vee)$ to define $\omega^{-1}$.
We say that a $p$-form $\xi$ {\sf annihilates} a $k$-subspace $U \subset V$, if $k \le p$ and $\xi \,\lrcorner\, (\bw{k} U) = 0$.
Analogously, we say that $U \subset V$ is {\sf isotropic} for a $p$-form $\xi$ if $p \le k$ and $(\bw{k} U) \,\lrcorner\, \xi = 0$.
By~\eqref{eq:convolution-duality} a subspace $U \subset V$ is isotropic for $\xi$ if and only if $\xi^\vee$ annihilates $U^\perp := \Ker(V^\vee \to U^\vee)$.
For a vector space $V$ we denote by ${\bf P}(V)$ the projective space of one-dimensional subspaces in $V$ and by $\mathscr{O}(1)$ the very ample generator of its Picard group, so that $H^0({\bf P}(V),\mathscr{O}(1)) = V^\vee$.
Abusing the notation, we frequently consider nonzero vectors $v \in V$ as points of ${\bf P}(V)$ and vice versa.
In case we want to emphasize a difference, we denote the point, corresponding to a vector $v \in V$ by $[v] \in {\bf P}(V)$, and the corresponding one-dimensional subspace by $\mathbf k v \subset V$.
Analogously, for a vector bundle $\mathscr{V}$ on a scheme $S$ we denote by $\pr:{\bf P}_S(\mathscr{V}) \to S$ the projective bundle, parameterizing one-dimensional subspaces in the fibers of $\mathscr{V}$,
and by $\mathscr{O}(1)$ the ample generator of the relative Picard group such that $\pr_*\mathscr{O}(1) \isom \mathscr{V}^\vee$.
Sometimes, we refer to the divisor class of this line bundle as the {\sf relative hyperplane class}.
Note that although the projectivization of a vector bundle does not change if the vector bundle get twisted, the corresponding relative hyperplane class does.
We always denote by $W$ a vector space of dimension~7, and by ${\bf B}W$ a vector space of dimension 6.
In fact, further on the space ${\bf B}W$ will be a direct summand of~$W$, but for a moment this is irrelevant.
We use notation $e_0,e_1,\dots,e_6$ for a basis in $W$ and $e_1,\dots,e_6$ for a basis in~${\bf B}W$.
The dual bases in $W^\vee$ and ${\bf B}W^\vee$ are denoted by $x_0,x_1,\dots,x_6$ and $x_1,\dots,x_6$ respectively.
We usually abbreviate $x_{i_1} \wedge \dots \wedge x_{i_p}$ to $x_{i_1\dots i_p}$ and $e_{i_1} \wedge \dots \wedge e_{i_p}$ to $e_{i_1\dots i_p}$.
We denote by $\Gr(k,W)$ the Grassmannian of $k$-dimensional vector subspaces in $W$. The tautological vector subbundle of rank $k$ on it
is denoted by $\mathscr{U}_k \subset W \otimes \mathscr{O}_{\Gr(k,W)}$. The quotient bundle is denoted simply by $W/\mathscr{U}_k$, and for its dual we use the notation
{\bar{e}}gin{equation*}
\mathscr{U}_k^\perp := (W/\mathscr{U}_k)^\vee.
\end{equation*}
Analogously, we denote by ${\overline{\mathscr{U}}}_k$ the tautological subbundle on $\Gr(k,{\bf B}W)$, by ${\bf B}W/{\overline{\mathscr{U}}}_k$ the quotient bundle, and by ${\overline{\mathscr{U}}}_k^\perp$ its dual.
The point of the Grassmannian corresponding to a subspace $U_k \subset W$ is denoted by $[U_k]$, or even just $U_k$.
We recall that $\det \mathscr{U}_k \isom \det \mathscr{U}_k^\perp \isom \mathscr{O}(-1)$ is the antiample generator of ${\bf P}ic(\Gr(k,W))$, and analogously for $\Gr(k,{\bf B}W)$.
For a vector bundle $\mathscr{V}$ on a scheme $S$ we denote by $\Gr_S(k,\mathscr{V})$ the relative Grassmannian, parameterizing $k$-dimensional subspaces in the fibers of $\mathscr{V}$.
In particular, we consider the two-step flag variety
{\bar{e}}gin{equation*}
{\bf F}l(k_1,k_2;V) \isom \Gr_{\Gr(k_2,V)}(k_1,\mathscr{U}_{k_2}) \isom \Gr_{\Gr(k_1,V)}(k_2-k_1,V/\mathscr{U}_{k_1}).
\end{equation*}
We denote by $\mathscr{U}_{k_1} \hookrightarrow \mathscr{U}_{k_2} \hookrightarrow V \otimes \mathscr{O}$ the tautological flag of subbundles on ${\bf F}l(k_1,k_2;V)$.
In particular, we abuse the notation by using the same name for the tautological vector bundle on the Grassmannian and its pullback to the flag variety.
Given a morphism $\varphi:\mathscr{E} \to \mathscr{F}$ of vector bundles on a scheme $S$ we denote by $D_k(\varphi) \subset S$ its $k$-th degeneration scheme,
i.e.\ the subscheme of $S$ whose ideal is locally generated by all $(r+1-k)\times (r+1-k)$ minors of the matrix of $\varphi$ for $r = \min\{\rank(\mathscr{E}),\rank(\mathscr{F})\}$.
This is a closed subscheme in $S$, and $D_{k+1}(\varphi) \subset D_k(\varphi)$.
\subsection{A blowup Lemma}\label{subsection:blowup-lemma}
We will use several times the following observation, which is quite classical. Unfortunately, we were not able to find a reference for it, so we sketch a short proof.
{\bar{e}}gin{lemma}\label{lemma:blowup}
Let $\varphi:\mathscr{E} \to \mathscr{F}$ be a morphism of vector bundles of ranks $\rk(\mathscr{F}) = r$ and $\rk(\mathscr{E}) = r+1$ on a Cohen--Macaulay scheme $S$.
Denote by $D_k(\varphi)$ the $k$-th degeneracy locus of $\varphi$.
Consider the projectivization $p:{\bf P}_S(\mathscr{E}) \to S$, then $\varphi$ gives a global section of the vector bundle $p^*\mathscr{F} \otimes \mathscr{O}(1)$.
If $\codim D_k(\varphi) \ge k + 1$ for all $k \ge 1$ then the zero locus of $\varphi$ on ${\bf P}_S(\mathscr{E})$ is isomorphic to the blowup of $S$ with center in the degeneration locus $D_1(\varphi)$.
Moreover, in this case the line bundle corresponding to the exceptional divisor of the blowup is isomorphic to $\det(\mathscr{E}^\vee) \otimes \det(\mathscr{F}) \otimes \mathscr{O}(-1)$.
Finally, if the second degeneracy locus~$D_2(\varphi)$ is empty, then the exceptional divisor is isomorphic
to the projectivization of the vector bundle $\Ker(\varphi\vert_{D_1(\varphi)}:\mathscr{E}\vert_{D_1(\varphi)} \to \mathscr{F}\vert_{D_1(\varphi)})$.
\end{lemma}
{\bar{e}}gin{proof}
Consider the morphism $\varphi^\vee:\mathscr{F}^\vee \to \mathscr{E}^\vee$.
By assumption, it is generically injective, hence is a monomorphism of sheaves.
Let $\mathscr{C} := {\bf C}oker(\varphi^\vee)$ be its cokernel.
Let us show it is torsion free.
Indeed, dualizing the sequence $0 \to \mathscr{F}^\vee \to \mathscr{E}^\vee \to \mathscr{C} \to 0$ we obtain
{\bar{e}}gin{equation*}
0 \to \mathscr{C}^\vee \to \mathscr{E} \xrightarrow{\ \varphi\ } \mathscr{F} \to \operatorname{\mathscr{E}\!\mathit{xt}}^1(\mathscr{C},\mathscr{O}_S) \to 0.
\end{equation*}
The sheaf $\operatorname{\mathscr{E}\!\mathit{xt}}^1(\mathscr{C},\mathscr{O}_S)$ is supported on the degeneracy locus $D_1(\varphi)$, hence in codimension 2.
Therefore, by Cohen--Macaulay property,
we have $\operatorname{\mathscr{E}\!\mathit{xt}}^{\le 1}(\operatorname{\mathscr{E}\!\mathit{xt}}^1(\mathscr{C},\mathscr{O}_S),\mathscr{O}_S) = 0$,
so dualizing the above sequence we get an exact sequence
{\bar{e}}gin{equation*}
0 \to \mathscr{F}^\vee \xrightarrow{\ \varphi^\vee\ } \mathscr{E}^\vee \to \mathscr{C}^{\vee\vee} \to \operatorname{\mathscr{E}\!\mathit{xt}}^{2}(\operatorname{\mathscr{E}\!\mathit{xt}}^1(\mathscr{C},\mathscr{O}_S),\mathscr{O}_S).
\end{equation*}
Comparing it with the definition of the sheaf $\mathscr{C}$, we conclude that $\mathscr{C}$ embeds into the torsion free sheaf~$\mathscr{C}^{\vee\vee}$, hence is itself torsion free.
Next, consider the morphism
{\bar{e}}gin{equation*}
\mathscr{E}^\vee \isom \det(\mathscr{E}^\vee) \otimes \bw{r}\mathscr{E} \xrightarrow{\ \wedge^r\varphi\ } \det(\mathscr{E}^\vee) \otimes \bw{r}\mathscr{F} \isom \det(\mathscr{E}^\vee) \otimes \det(\mathscr{F}).
\end{equation*}
Its composition with $\varphi^\vee$ is zero, hence it induces a morphism $\mathscr{C} \to \det(\mathscr{E}^\vee) \otimes \det(\mathscr{F})$.
This morphism is an isomorphism away of $D_1(\varphi)$, so since $\mathscr{C}$ is torsion free, it is injective and we have a left exact sequence
{\bar{e}}gin{equation*}
0 \to \mathscr{F}^\vee \xrightarrow{\ \varphi^\vee\ } \mathscr{E}^\vee \xrightarrow{\ \wedge^r\varphi\ } \det(\mathscr{E}^\vee) \otimes \det(\mathscr{F}).
\end{equation*}
But since the scheme structure on $D_1(\varphi)$ is given by the minors of size $r$ of $\varphi$, i.e.\ by the entries of $\wedge^r\varphi$,
the image of $\wedge^r\varphi$ is the twist of the ideal of $D_1(\varphi)$, hence we have an exact sequence
{\bar{e}}gin{equation}\label{eq:exact-seq-varphi}
0 \to \mathscr{F}^\vee \xrightarrow{\ \varphi^\vee\ } \mathscr{E}^\vee \xrightarrow{\ \wedge^r\varphi\ } \det(\mathscr{E}^\vee) \otimes \det(\mathscr{F}) \to \det(\mathscr{E}^\vee) \otimes \det(\mathscr{F})\vert_{D_1(\varphi)} \to 0.
\end{equation}
Let $\tilde{S} \subset {\bf P}_S(\mathscr{E})$ be the zero locus of $\varphi$ on ${\bf P}_S(\mathscr{E})$.
Since $\tilde{S}$ is the zero locus of a rank~$r$ vector bundle on a Cohen--Macaulay variety ${\bf P}_S(\mathscr{E})$ of dimension $\dim(S) + r$, the dimension of any component of $\tilde{S}$ is greater or equal than~$\dim S$.
On the other hand, the fibers of $\tilde{S}$ over $D_k(\varphi) \smallsetminus D_{k+1}(\varphi)$ are projective spaces of dimension $k$,
hence $\tilde{S}$ has a stratification with strata of dimension $\dim(D_k(\varphi)) + k$ which is less than $\dim S$ for $k \ge 1$.
It follows that $\tilde{S}$ is irreducible of dimension $\dim\tilde{S} = \dim S$, and the map $p:\tilde{S} \to S$ is birational.
Now let us compute the pushforward to $S$ of the relatively very ample line bundle $\mathscr{O}(1)\vert_{\tilde{S}}$.
As we already have seen, the dimension of the zero locus of the section $\varphi$ of $p^*\mathscr{F} \otimes \mathscr{O}(1)$ on~${\bf P}_S(\mathscr{E})$ equals $\dim \tilde{S} = \dim {\bf P}_S(\mathscr{E}) - r$, hence the section is regular, so the Koszul complex
{\bar{e}}gin{equation*}
0 \to \bw{r}(p^*\mathscr{F}^\vee) \otimes \mathscr{O}(-r) \to \dots \to \bw2(p^*\mathscr{F}^\vee) \otimes \mathscr{O}(-2) \to p^*\mathscr{F}^\vee \otimes \mathscr{O}(-1) \to \mathscr{O} \to \mathscr{O}_{\tilde{S}} \to 0
\end{equation*}
is a resolution of the structure sheaf of $\tilde{S}$.
Twisting it by $\mathscr{O}(1)$ and pushing forward to $S$, we obtain an exact sequence
{\bar{e}}gin{equation*}
0 \to \mathscr{F}^\vee \xrightarrow{\ \varphi^\vee\ } \mathscr{E}^\vee \to p_*(\mathscr{O}_{\tilde{S}}(1)) \to 0.
\end{equation*}
Comparing it with~\eqref{eq:exact-seq-varphi} we deduce an isomorphism with a twisted ideal of $D_1(\varphi)$:
{\bar{e}}gin{equation*}
p_*(\mathscr{O}_{\tilde{S}}(1)) \isom \mathscr{I}_{D_1(\varphi)} \otimes \det(\mathscr{E}^\vee) \otimes \det(\mathscr{F}).
\end{equation*}
It follows that $p:\tilde{S} \to S$ is the blowup of the ideal $\mathscr{I}_{D_1(\varphi)}$.
Moreover, it follows also that the pushforward of $\det(\mathscr{E}) \otimes \det(\mathscr{F}^\vee) \otimes \mathscr{O}(1)$ is isomorphic to the ideal $\mathscr{I}_{D_1(\varphi)}$,
hence this line bundle corresponds to the minus exceptional divisor of the blowup.
This proves the second part of the Lemma.
The last part of the Lemma is evident.
\end{proof}
\section{Geometry of general skew forms and generality assumptions}\label{section:forms-and-generality}
We say that a skew-symmetric $p$-form on a vector space $V$ of dimension $n$ is {\sf general}, if its ${\bf P}GL(V)$-orbit in ${\bf P}(\bw{p}V^\vee)$ is open.
It is a classical fact, that a 2-form is general if and only if its rank is equal to $2\lfloor n/2 \rfloor$,
and that a general 3-form exists if and only if $n \le 8$.
In this section we remind a description of general 3-forms on vector spaces of dimensions 6 and 7 (we will not need the 8-dimensional case so we skip it here, however an interested reader can find a discussion of these in~\cite{kuznetsov2015kuchle}).
We also discuss some natural subschemes of Grassmannians associated with these forms.
After that we pass to the situation, which is the most important for the rest of the paper: a pair $(\lambda,\mu)$ consisting
of a 4-form and a 2-form on a vector space of dimension 7. We discuss how this pair looks under some generality assumptions
and give an explicit standard presentation for such a pair.
\subsection{A 3-form on a 6-space}\label{subsection:3form-6space}
Let ${\bf B}W$ be a vector space of dimension 6.
The following description of general 3-forms on ${\bf B}W$ is well known.
{\bar{e}}gin{lemma}\label{lemma:3form-general}
A $3$-form ${\bar{\lambda}} \in \bw3{\bf B}W^\vee$ is general if and only if there is a direct sum decomposition
{\bar{e}}gin{equation}\label{eq:bw-decomposition}
{\bf B}W = A_1 \oplus A_2,
\qquad\qquad
\dim A_1 = \dim A_2 = 3,
\end{equation}
and
{\bar{e}}gin{equation}\label{eq:blam-general}
{\bar{\lambda}} = {\bar{\lambda}}_1 + {\bar{\lambda}}_2
\end{equation}
for decomposable $3$-forms $0 \ne {\bar{\lambda}}_1 \in \bw3A_1^\perp$ and $0 \ne {\bar{\lambda}}_2 \in \bw3A_2^\perp$.
\end{lemma}
In appropriate coordinates a general 3-form can be written as
{\bar{e}}gin{equation}\label{eq:blam-explicit}
{\bar{\lambda}} = x_{123} + x_{456}.
\end{equation}
In the rest of the paper we always denote by $A_1$ and $A_2$ the summands of the canonical direct sum decomposition of ${\bf B}W$ associated with a general 3-form ${\bar{\lambda}}$.
{\bar{e}}gin{lemma}\label{lemma:gr26-3form}
Assume ${\bar{\lambda}}$ is a general $3$-form on ${\bf B}W$.
Consider the Grassmannian $\Gr(2,{\bf B}W)$ and let ${\overline{\mathscr{U}}}_2 \subset {\bf B}W \otimes \mathscr{O}$ be the tautological subbundle of rank $2$.
The zero locus of the global section~${\bar{\lambda}}$ of the vector bundle ${\overline{\mathscr{U}}}_2^\perp(1)$
is isomorphic to the product ${\bf P}(A_1) \times {\bf P}(A_2)$, and moreover
{\bar{e}}gin{equation}\label{eq:gr26-3form-bu2}
{\bar{e}}gin{aligned}
{\overline{\mathscr{U}}}_2\vert_{{\bf P}(A_1) \times {\bf P}(A_2)} &\isom \mathscr{O}(-h_1) \oplus \mathscr{O}(-h_2),
\\
{\overline{\mathscr{U}}}_2^\perp\vert_{{\bf P}(A_1) \times {\bf P}(A_2)} & \isom \Omega_{{\bf P}(A_1)}(h_1) \oplus \Omega_{{\bf P}(A_2)}(h_2),
\end{aligned}
\end{equation}
where $h_1$ and $h_2$ are the hyperplane classes on ${\bf P}(A_1)$ and ${\bf P}(A_2)$ respectively.
\end{lemma}
{\bar{e}}gin{proof}
The direct sum decomposition~\eqref{eq:bw-decomposition} induces a decomposition
{\bar{e}}gin{equation*}
\bw2{\bf B}W = \bw2A_1 \oplus \bw2A_2 \oplus (A_1 \otimes A_2).
\end{equation*}
The map $\bw2{\bf B}W \xrightarrow{\ {\bar{\lambda}}\ } {\bf B}W^\vee = A_1^\vee \oplus A_2^\vee$ is zero on the third summand
and gives isomorphisms of the first two summands with $A_1^\vee$ and $A_2^\vee$ respectively.
Hence the zero locus of ${\bar{\lambda}}$ on $\Gr(2,{\bf B}W)$ is the intersection $\Gr(2,{\bf B}W) \cap {\bf P}(A_1 \otimes A_2) \subset {\bf P}(\bw2{\bf B}W)$.
It remains to note that the restrictions of the Pl\"ucker quadrics from ${\bf P}(\bw2{\bf B}W)$ to ${\bf P}(A_1 \otimes A_2)$
are the Segre quadrics, cutting out ${\bf P}(A_1) \times {\bf P}(A_2) \subset {\bf P}(A_1 \otimes A_2)$.
Geometrically, this means that a 2-subspace ${\bf B}U_2 \subset {\bf B}W$ is in the zero locus of ${\bar{\lambda}}$ if and only if
it intersects both subspaces $A_1$ and $A_2$. This means that the restriction of the tautological bundle to ${\bf P}(A_1)\times{\bf P}(A_2)$
is the direct sum of the pullbacks of the tautological line bundles on ${\bf P}(A_1)$ and ${\bf P}(A_2)$, i.e.\ gives the first part of~\eqref{eq:gr26-3form-bu2}.
Further, it follows that for the quotient bundle we have
{\bar{e}}gin{equation*}
({\bf B}W/{\overline{\mathscr{U}}}_2)\vert_{{\bf P}(A_1) \times {\bf P}(A_2)} \isom
(A_1 \otimes \mathscr{O})/\mathscr{O}(-h_1) \oplus (A_2 \otimes \mathscr{O})/\mathscr{O}(-h_2),
\end{equation*}
so it is isomorphic to the direct sum of pullbacks of the twisted tangent bundles. Dualizing, we get the second part of~\eqref{eq:gr26-3form-bu2}.
\end{proof}
{\bar{e}}gin{corollary}\label{corollary:gr46-3form}
Assume ${\bar{\lambda}}$ is a general $3$-form on ${\bf B}W$.
Consider the Grassmannian $\Gr(4,{\bf B}W)$ and let ${\overline{\mathscr{U}}}_4 \subset {\bf B}W \otimes \mathscr{O}$ be the tautological subbundle of rank $4$.
The zero locus of the global section ${\bar{\lambda}}$ of the vector bundle $\bw3{\overline{\mathscr{U}}}_4^\vee$
is isomorphic to the product $\Gr(2,A_1) \times \Gr(2,A_2)$, and moreover
{\bar{e}}gin{equation}\label{eq:gr46-3form-bu4}
{\overline{\mathscr{U}}}_4\vert_{\Gr(2,A_1) \times \Gr(2,A_2)} \isom \mathscr{U}_{2,A_1} \oplus \mathscr{U}_{2,A_2},
\end{equation}
where $\mathscr{U}_{2,A_1}$ and $\mathscr{U}_{2,A_2}$ are the tautological bundles on $\Gr(2,A_1)$ and $\Gr(2,A_2)$ respectively.
\end{corollary}
{\bar{e}}gin{proof}
We have a canonical isomorphism $\Gr(4,{\bf B}W) \isom \Gr(2,{\bf B}W^\vee)$, that takes the bundle $\bw3{\overline{\mathscr{U}}}_4^\vee$ on the first Grassmannian to the bundle $({\overline{\mathscr{U}}}'_2)^\perp(1)$ on the second,
and the global section ${\bar{\lambda}}$ of the first to the global section ${\bar{\lambda}}^\vee$ of the second.
Hence the zero locus of ${\bar{\lambda}}$ in $\Gr(4,{\bf B}W)$ is isomorphic to the global section of ${\bar{\lambda}}^\vee$ in $\Gr(2,{\bf B}W^\vee)$.
Clearly, ${\bar{\lambda}}^\vee$ is a general 3-form on ${\bf B}W^\vee$ corresponding to the direct sum decomposition ${\bf B}W^\vee = A_1^\vee \oplus A_2^\vee$.
Hence by Lemma~\ref{lemma:gr26-3form} the zero locus is isomorphic to ${\bf P}(A_1^\vee) \times {\bf P}(A_2^\vee) \isom \Gr(2,A_1) \times \Gr(2,A_2)$.
The statement about the restriction of the tautological bundle follows from the second part of~\eqref{eq:gr26-3form-bu2}.
\end{proof}
Every 2-form $\mu \in \bw2{\bf B}W^\vee$ induces a pairing between the subspaces $A_1,A_2 \subset {\bf B}W$.
{\bar{e}}gin{lemma}\label{lemma:gr26-3form-2form}
Assume ${\bar{\lambda}}$ is a general $3$-form on ${\bf B}W$.
If the pairing between the subspaces $A_1$ and~$A_2$ induced by a $2$-form $\mu$ is nondegenerate then
the zero locus of the global section $({\bar{\lambda}},\mu)$ of the vector bundle ${\overline{\mathscr{U}}}_2^\perp(1) \oplus \mathscr{O}(1)$ on the Grassmannian $\Gr(2,{\bf B}W)$
is isomorphic to the flag variety ${\bf F}l(1,2;A_1) \isom {\bf F}l(1,2;A_2) \subset {\bf P}(A_1) \times {\bf P}(A_2)$.
\end{lemma}
{\bar{e}}gin{proof}
By Lemma~\ref{lemma:gr26-3form} the zero locus of ${\bar{\lambda}}$ is isomorphic to ${\bf P}(A_1) \times {\bf P}(A_2)$. By~\eqref{eq:gr26-3form-bu2} we have
{\bar{e}}gin{equation*}
\mathscr{O}(1)\vert_{{\bf P}(A_1)\times {\bf P}(A_2)} \isom
\det({\overline{\mathscr{U}}}_2^\vee)\vert_{{\bf P}(A_1)\times {\bf P}(A_2)} \isom
\mathscr{O}(h_1 + h_2),
\end{equation*}
hence the zero locus of $\mu$ in ${{\bf P}(A_1)\times {\bf P}(A_2)}$ is a divisor of bidegree $(1,1)$.
Clearly, this divisor corresponds to the pairing between $A_1$ and $A_2$ induced by the form $\mu$.
So, if this pairing is nondegenerate, it identifies $A_2$ with $A_1^\vee$, and the corresponding divisor
with the flag variety.
\end{proof}
\subsection{A 4-form on a 7-space}\label{subsection:4form-4space}
Let $W$ be a vector space of dimension 7.
Recall that with each $4$-form $\lambda$ on $W$ we can associate a 3-form $\lambda^\vee$ on the dual space.
Further on we will work more with 4-forms, but some geometric constructions are better adapted to 3-forms, so it is useful to keep this correspondence in mind.
Consider a 3-form $\lambda^\vee \in \bw3W$ on $W^\vee$ as a global section
of the vector bundle $\Omega^2_{{\bf P}(W^\vee)}(3)$. It gives a morphism of vector bundles
{\bar{e}}gin{equation*}
T_{{\bf P}(W^\vee)} \xrightarrow{\ \lambda^\vee\ } \Omega_{{\bf P}(W^\vee)}(3),
\end{equation*}
which is easily seen to be skew-symmetric (up to a twist). Since
{\bar{e}}gin{equation*}
\det(\Omega_{{\bf P}(W^\vee)}(3)) \otimes \det(T_{{\bf P}(W)})^{-1} \isom
\mathscr{O}(-7 + 3\cdot 6 - 7) =
\mathscr{O}(4),
\end{equation*}
the Pfaffian of the map $\lambda^\vee$ is a section of $\mathscr{O}(2)$, hence the degeneracy degeneracy locus of $\lambda^\vee$ is either a quadric, or the whole space.
We denote this degeneracy locus by $\mathbf{Q}^\vee_\lambda \subset {\bf P}(W^\vee)$ (geometrically, $\mathbf{Q}^\vee_\lambda$ is the set of vectors $w^\vee \in W^\vee$ such that the rank of the 2-form $\lambda^\vee \,\lrcorner\, w^\vee$ is less than 6)
and by $\mathbf q_\lambda \in \Sym^2 W$ its equation.
In case $\mathbf{Q}^\vee_\lambda$ is a smooth quadric, we denote by~$\mathbf{Q}_\lambda \subset {\bf P}(W)$ its projective dual, it is also smooth then and its equation is $\mathbf q^{-1}_\lambda \in \Sym^2W^\vee$.
{\bar{e}}gin{lemma}\label{lemma:4form-general}
The following conditions are equivalent:
{\bar{e}}gin{enumerate}\renewcommand{\roman{enumi}}{\roman{enumi}}
\item
a $4$-form $\lambda \in \bw4W^\vee$ is general;
\item
a $3$-form $\lambda^\vee \in \bw3W$ is general;
\item
the degeneracy locus $\mathbf{Q}^\vee_\lambda \subset {\bf P}(W^\vee)$ is a smooth quadric;
\item
$\lambda = x_{0123} + x_{0456} + x_{1256} + x_{1346} + x_{2345}$ in some basis;
\item
$\lambda^\vee = e_{456} + e_{123} + e_{034} + e_{025} + e_{016}$ in some basis.
\end{enumerate}
\end{lemma}
A straightforward computation shows that for $\lambda = x_{0123} + x_{0456} + x_{1256} + x_{1346} + x_{2345}$, the associated quadric $\mathbf{Q}_\lambda$ is
{\bar{e}}gin{equation}\label{eq:qlambda-explicit}
\mathbf{Q}_\lambda = \{ x_0^2 - x_1x_6 - x_2x_5 - x_3x_4 = 0 \} \subset {\bf P}(W).
\end{equation}
The stabilizer of a general 4-form $\lambda$ in $\GL(W)$ is the simple algebraic group ${\mathbb{G}_2}$, the space $W$ is one of its fundamental representations, and the action of ${\mathbb{G}_2}$ on ${\bf P}(W)$ has just two orbits.
The closed orbit is the quadric $\mathbf{Q}_\lambda$, and the open orbit is its complement.
In particular, $\mathbf{Q}_\lambda$ is one of the two minimal compact homogeneous spaces of ${\mathbb{G}_2}$.
{\bar{e}}gin{lemma}\label{lemma:la-w-gen}
Let $\lambda \in \bw4W^\vee$ be a general $4$-form.
For every vector $w \in W$ such that $\mathbf q^{-1}_\lambda(w,w) = 1$, let $w^\vee = \mathbf q^{-1}_\lambda(w) \in W^\vee$ be its polar covector.
Then there is a direct sum decomposition of the space $W$ and the corresponding decomposition of the form $\lambda$
{\bar{e}}gin{equation*}
W = \mathbf k w \oplus \Ker(w^\vee),
\qquad
\lambda = w^\vee \wedge {\bar{\lambda}} + \lambda',
\end{equation*}
with ${\bar{\lambda}} \in \bw3(\Ker(w^\vee))^\vee$ and $\lambda' \in \bw4(\Ker(w^\vee))^\vee$.
Moreover, ${\bar{\lambda}}$ is a general $3$-form on $\Ker(w^\vee)$, and if $\Ker(w^\vee) = A_1 \oplus A_2$
is the corresponding direct sum decomposition, then the form $\lambda'$ annihilates $\bw3A_1$ and $\bw3A_2$,
and defines a nondegenerate pairing between $\bw2A_1$ and $\bw2A_2$.
\end{lemma}
{\bar{e}}gin{proof}
Since the group ${\mathbb{G}_2}$ acts transitively on the complement of $\mathbf{Q}_\lambda$, it is enough to check all the properties for just one vector $w$.
So, choose a basis as in Lemma~\ref{lemma:4form-general}(iv), so that the quadric $\mathbf{Q}_\lambda$ is given by~\eqref{eq:qlambda-explicit},
and take $w = e_0$. Then $w^\vee = x_0$, and hence
{\bar{e}}gin{equation*}
{\bar{\lambda}} = x_{123} + x_{456}
\qquad \text{and}\qquad
\lambda' = x_{1256} + x_{1346} + x_{2345}.
\end{equation*}
Thus ${\bar{\lambda}}$ corresponds to the direct sum decomposition of ${\bf B}W = A_1 \oplus A_2$ with $A_1 = \langle e_1,e_2,e_3 \rangle$ and $A_2 = \langle e_4, e_5, e_6 \rangle$,
the form $\lambda'$ annihilates both $A_1$ and $A_2$ (i.e.\ $\lambda' \,\lrcorner\, e_{123} = \lambda' \,\lrcorner\, e_{456} = 0$),
and the pairing between the spaces $\bw2A_1 = \langle e_{12}, e_{13}, e_{23} \rangle$ and $\bw2A_2 = \langle e_{45}, e_{46}, e_{56} \rangle$ given by $\lambda'$ is nondegenerate.
\end{proof}
The other minimal compact homogeneous variety of ${\mathbb{G}_2}$ can be described as follows.
{\bar{e}}gin{lemma}[\cite{mukai1989fano}]\label{lemma:g2-grassmannian}
The zero locus of the global section $\lambda \in H^0(\Gr(5,W),\bw4\mathscr{U}_5^\vee)$ in $\Gr(5,W)$
is a minimal compact homogeneous variety of the group ${\mathbb{G}_2}$.
\end{lemma}
This homogeneous variety is usually called the {\sf adjoint variety} of group ${\mathbb{G}_2}$.
In what follows we will denote it by $\Gr_\lambda(5,W)$.
By duality, one can get another description of the adjoint variety.
Recall the canonical isomorphism $\Gr(5,W) \isom \Gr(2,W^\vee)$ (defined by $(U_5 \subset W) \mapsto (U_5^\perp \subset W^\vee)$).
By~\eqref{eq:convolution-duality} this isomorphism takes $\Gr_\lambda(5,W)$ to the subvariety of $\Gr(2,W^\vee)$ parameterizing all 2-subspaces $U'_2 \subset W^\vee$ annihilated by the 3-form $\lambda^\vee$ on $W^\vee$.
We denote this subvariety by $\Gr_{\lambda^\vee}(2,W^\vee)$.
{\bar{e}}gin{remark}\label{remark:gr27-g2}
Note that from this description it is clear that if $w^\vee \in {\bf P}(W^\vee) \smallsetminus \mathbf{Q}^\vee_\lambda$, then $w^\vee$ is not contained in any 2-subspace $U'_2 \subset W^\vee$, corresponding to a point of $\Gr_{\lambda^\vee}(2,W^\vee)$.
\end{remark}
\subsection{The structure of the data and genericity assumptions}\label{subsection:genericity-assumptions}
Let $W$ be a vector space of dimension 7, and
{\bar{e}}gin{equation*}
\lambda \in \bw4W^\vee,
\qquad
\mu \in \bw2W^\vee,
\end{equation*}
be a 4-form and a 2-form on $W$. In this section we discuss how a pair $(\lambda,\mu)$ looks under some genericity assumptions
and introduce some notions that will be actively used further on.
\underline{Assumption 1:} We assume that the form $\lambda$ is general (i.e.\ lies in the open ${\bf P}GL(W)$-orbit), the form $\mu$ is general (i.e.\ lies in the open ${\bf P}GL(W)$-orbit),
and the kernel of the form $\mu$ is in a general position with respect to the form $\lambda$.
As it was discussed above, these assumptions can be explicitly reformulated as
{\bar{e}}gin{equation}\label{assumption:lambda-mu-general}
{\bar{e}}gin{cases}
\text{the quadrics $\mathbf{Q}_\lambda^\vee \subset {\bf P}(W^\vee)$ and $\mathbf{Q}_\lambda \subset {\bf P}(W)$ are smooth;}\\
\rank(\mu) = 6;\\
w_0 \not\in \mathbf{Q}_\lambda \subset {\bf P}P(W).
\end{cases}
\end{equation}
Here $\mathbf{Q}_\lambda^\vee$ and $\mathbf{Q}_\lambda$ are the quadrics defined in section~\ref{subsection:4form-4space}, and $w_0$ is defined to be a generator of the one-dimensional kernel space of $\mu$ on $W$,
normalized by the condition $\mathbf q_\lambda^{-1}(w_0,w_0) = 1$.
As in Lemma~\ref{lemma:la-w-gen} we denote by $w_0^\vee = \mathbf q^{-1}_\lambda(w_0)$ the polar covector to the point $w_0$, so that
{\bar{e}}gin{equation*}\label{eq:w0v-w0}
w_0^\vee(w_0) = 1,
\end{equation*}
and consider the direct sum decomposition defined by the pair $(w_0,w_0^\vee)$:
{\bar{e}}gin{equation}\label{eq:bw}
W = \mathbf k w_0 \oplus {\bf B}W,
\qquad
{\bf B}W = \Ker w_0^\vee \subset W,
\end{equation}
and the induced decomposition of the form $\lambda$
{\bar{e}}gin{equation}\label{eq:la-la-la}
\lambda = w_0^\vee \wedge {\bar{\lambda}} + \lambda',
\qquad\qquad
\text{with ${\bar{\lambda}} \in \bw3{\bf B}W^\vee$ and $\lambda' \in \bw4{\bf B}W^\vee$.}
\end{equation}
Since $w_0$ generates the kernel of $\mu$, the form $\mu$ can be considered just as a form in $\bw2{\bf B}W^\vee$.
Thus, under assumption~\eqref{assumption:lambda-mu-general}, the data $({\lambda,\mu})$ reduces to the data $(\lambda',{\bar{\lambda}},\mu)$ of a 4-form, a 3-form, and a 2-form on a 6-dimensional vector space ${\bf B}W$,
such that the 2-form $\mu$ is non-degenerate, the 3-form ${\bar{\lambda}}$ is general and corresponds to a decomposition ${\bf B}W = A_1 \oplus A_2$,
and the 4-form $\lambda'$ annihilates the subspaces $A_1$ and $A_2$, and induces a nondegenerate pairing between the spaces $\bw2A_1$ and $\bw2A_2$ (see Lemma~\ref{lemma:la-w-gen}).
The 2-form $\mu \in \bw2{\bf B}W^\vee$ gives a 4-form $\mu^2 := \mu \wedge \mu \in \bw4{\bf B}W^\vee$.
We consider the pencil
{\bar{e}}gin{equation*}
t\lambda' + \mu^2 \in \bw4{\bf B}W^\vee,
\qquad
t \in \mathbf k
\end{equation*}
of 4-forms on ${\bf B}W$ generated by $\lambda'$ and $\mu^2$.
Both forms give a pairing between the spaces $\bw2A_1$ and $\bw2A_2$.
Thinking of these pairings as of linear maps $\bw2A_1 \to \bw2A_2^\vee$, that linearly depend on $t$,
we have a cubic polynomial
{\bar{e}}gin{equation*}
\chi_{A_1,A_2}^{\lambda',\mu^2}(t) := \det{\bf B}ig(t\lambda' + \mu^2 \colon \bw2A_1 \to \bw2A_2^\vee{\bf B}ig) \in \mathbf k[t].
\end{equation*}
\underline{Assumption 2:}
We assume that the polynomial $\chi_{A_1,A_2}^{\lambda',\mu^2}$ is general.
Explicitly, we assume
{\bar{e}}gin{equation}\label{assumption:pencil-regular}
\text{the polynomial $\chi_{A_1,A_2}^{\lambda',\mu^2}$ has three distinct roots.}
\end{equation}
This assumption allows to make the form of the data fairly explicit.
{\bar{e}}gin{lemma}\label{lemma:bases-a1-a2}
Assume a $3$-form ${\bar{\lambda}}$ on ${\bf B}W$ is general and corresponds to a direct sum decomposition ${\bf B}W = A_1 \oplus A_2$,
and the form $\lambda'$ annihilates $A_1$ and $A_2$ and defines a nondegenerate pairing between $\bw2A_1$ and $\bw2A_2$.
If~\eqref{assumption:pencil-regular} is satisfied then
one can choose a basis $e_1,e_2,e_3$ in~$A_1$ and a basis $e_4,e_5,e_6$ in $A_2$ such that
{\bar{e}}gin{equation}\label{eq:lambda-mu-explicit}
{\bar{e}}gin{aligned}
{\bar{\lambda}} & = x_{123} + x_{456},\\
\lambda' & = x_{1256} + x_{1346} + x_{2345},\\
\mu^2 & = (M_1x_{1456} + M_2x_{2456} + M_3x_{3456}) + (M_4x_{1234} + M_5x_{1235} + M_6x_{1236}) \\
& + (K_1x_{2345} + K_2x_{1346} + K_3x_{1256}),
\end{aligned}
\end{equation}
where $M_i, K_i \in \mathbf k$, and $K_i$ are pairwise distinct.
Such basis is unique up to rescaling and permuting.
\end{lemma}
{\bar{e}}gin{proof}
Since the pairing defined by the form $\lambda'$ is nondegenerate, it gives an isomorphism $\lambda':\bw2A_1 \simeqto \bw2A_2^\vee$, so we can use it to identify the two spaces.
Then $\mu^2$ becomes an endomorphism of $\bw2A_1$.
Condition~\eqref{assumption:pencil-regular} means that the characteristic polynomial of this endomorphism has three distinct roots, hence the endomorphism can be diagonalized in an appropriate basis.
Choosing such a basis in $\bw2A_1 \isom A_1^\vee$, considering its dual basis in $A_1$, and transferring the basis to $\bw2A_2^\vee \isom A_2$ via $\lambda'$,
we obtain bases in $A_1$ and $A_2$ such that $\lambda'$ has the required form and the pairing given by $\mu^2$ between $\bw2A_1$ and $\bw2A_2$ is diagonal.
Rescaling the bases we can also ensure that ${\bar{\lambda}}$ has the required form. Also it is clear that the bases defined in this way are unique up to rescaling and permuting.
\end{proof}
If the 2-form $\mu$ is nondegenerate, then the 4-form $\mu^2$, considered as a map ${\bf B}W^\vee \to {\bf B}W$, is inverse (up to a scalar factor) to~$\mu$, considered as a map ${\bf B}W \to {\bf B}W^\vee$.
Thus, $\mu$ can be reconstructed from $\mu^2$. A straightforward computation shows that
$\mu$ is proportional to the skew-symmetric matrix with the following upper-triangular part
{\bar{e}}gin{equation}\label{eq:mu-matrix}
\mu =
\left({\bar{e}}gin{array}{rrr|rrr}
0 & M_4K_3 & -M_5K_2 & M_1M_4 & M_1M_5 & K_2K_3 + M_1M_6 \\
& 0 & M_6K_1 & M_2M_4 & K_1K_3 + M_2M_5 & M_2M_6 \\
&& 0 & K_1K_2 + M_3M_4 & M_3M_5 & M_3M_6 \\
\hline
&&& 0 & M_1K_1 & -M_2K_2 \\
&&&& 0 & M_3K_3 \\
&&&&& 0
\end{array}\right)
\end{equation}
\underline{Assumption 3:}
We assume that all the coefficients of the matrix $\mu^2$ in~\eqref{eq:lambda-mu-explicit} are nonzero:
{\bar{e}}gin{equation}\label{assumption:m-nonzero}
\text{$M_i \ne 0$ for all $1 \le i \le 6$ and $K_i \ne 0$ for all $1 \le i \le 3$.}
\end{equation}
We also have some nonvanishings for free.
{\bar{e}}gin{lemma}
If the skew form $\mu$ is nondegenerate then
{\bar{e}}gin{equation}\label{eq:mmk-nonzero}
M_1M_6K_1 + M_2M_5K_2 + M_3M_4K_3 + K_1K_2K_3 \ne 0.
\end{equation}
If, moreover, \eqref{assumption:m-nonzero} holds, then the pairing between $A_1$ and $A_2$ defined by $\mu$ is nondegenerate.
\end{lemma}
{\bar{e}}gin{proof}
A straightforward computation shows that the Pfaffian of the matrix~\eqref{eq:mu-matrix} is equal to
{\bar{e}}gin{equation*}
{\bf P}f(\mu) = (M_1M_6K_1 + M_2M_5K_2 + M_3M_4K_3 + K_1K_2K_3 )^2.
\end{equation*}
In particular, the nondegeneracy of $\mu$ implies~\eqref{eq:mmk-nonzero}.
Furthermore, a computation of the determinant of the upper-right 3-by-3 block $\mu_{A_1,A_2}$ in~\eqref{eq:mu-matrix} gives
{\bar{e}}gin{equation*}
\det(\mu_{A_1,A_2}) = -K_1K_2K_3(M_1M_6K_1 + M_2M_5K_2 + M_3M_4K_3 + K_1K_2K_3),
\end{equation*}
so assuming that all $K_i$ are nonzero, we deduce that the pairing is nondegenerate.
\end{proof}
Later we will need the following consequence of assumption~\eqref{assumption:m-nonzero}.
{\bar{e}}gin{lemma}\label{lemma:no21cases}
Assume $\lambda'$ and $\mu$ are given by~\eqref{eq:lambda-mu-explicit} and~\eqref{eq:mu-matrix} with $K_i$ pairwise distinct, and assume that~\eqref{assumption:m-nonzero} holds.
\noindent$(a)$
If for a two-dimensional subspace $U_{2,A_1} \subset A_1$ and a one-dimensional subspace $U_{1,A_2} \subset A_2$ the subspace $U_{2,A_1} \oplus U_{1,A_2} \subset {\bf B}W$ is $\mu$-isotropic, then $\lambda'$ does not annihilate it.
\noindent$(b)$
If for a one-dimensional subspace $U_{1,A_1} \subset A_1$ and a two-dimensional subspace $U_{2,A_2} \subset A_2$ the subspace $U_{1,A_1} \oplus U_{2,A_2} \subset {\bf B}W$ is $\mu$-isotropic, then $\lambda'$ does not annihilate it.
\end{lemma}
{\bar{e}}gin{proof}
Let $\alpha_1 = s_1e_{23} - s_2e_{13} + s_3e_{12}$ be the bivector corresponding to the subspace $U_{2,A_1}$ and $a_2 = s_4e_4 + s_5e_5 +s_6e_6$ be the vector corresponding to $U_{1,A_2}$.
Assume that the subspace $U_{2,A_1} \oplus U_{1,A_2}$ is annihilated by $\lambda'$ and is $\mu$-isotropic.
It is easy to see that then it is also annihilated by $\mu^2$ (see the proof of Lemma~\ref{lemma:lambda-mu-shift} below).
Then
{\bar{e}}gin{align*}
\lambda' \,\lrcorner\, (\alpha_1 \wedge a_2) & = (s_1s_5 - s_2s_6)x_4 + (-s_1s_4 + s_3s_6)x_5 + (s_2s_4 - s_3s_5)x_6 && = 0,\\
\mu^2 \,\lrcorner\, (\alpha_1 \wedge a_2) & = (K_1s_1s_5 - K_2s_2s_6)x_4 + (-K_1s_1s_4 + K_3s_3s_6)x_5 + (K_2s_2s_4 - K_3s_3s_5)x_6 && = 0.
\end{align*}
Assume $s_4 \ne 0$.
Subtracting the second line from the first line multiplied by $K_3$ and considering the coefficients at $x_5$ and~$x_6$ gives $(K_1-K_3)s_1s_4 = (K_2 - K_3)s_2s_4 = 0$.
Since $K_i$ are pairwise distinct, it follows that $s_1 = s_2 = 0$, hence $\alpha_1 = s_3e_{12}$ and $\mu \,\lrcorner\, \alpha_1 = M_4K_3s_3 \ne 0$.
Analogously, if $s_5 \ne 0$ we conclude that $\alpha_1 = -s_2e_{13}$ and $\mu \,\lrcorner\, \alpha_1 = M_5K_2s_2 \ne 0$, and if $s_6 \ne 0$ we conclude that $\alpha_1 = s_1e_{23}$ and $\mu \,\lrcorner\, \alpha_1 = M_6K_1s_1 \ne 0$.
The second statement is proved similarly.
\end{proof}
\underline{Assumption 4:}
Consider the Lagrangian Grassmannian $\LGr_\mu(3,{\bf B}W)$
corresponding to the symplectic form $\mu$ on ${\bf B}W$. By definition
$\LGr_\mu(3,{\bf B}W) \subset \Gr(3,{\bf B}W) \subset {\bf P}(\bw3{\bf B}W)$,
hence the 3-form ${\bar{\lambda}} \in \bw3{\bf B}W^\vee$ gives its hyperplane section. We assume
{\bar{e}}gin{equation}\label{assumption:lgr-hyperplane}
\text{the hyperplane section $\LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}$ of $\LGr_\mu(3,{\bf B}W)$ given by ${\bar{\lambda}}$ is smooth.}
\end{equation}
In fact, it is possible that~\eqref{assumption:lgr-hyperplane} follows from the other assumptions we have made, but this seems to be not so easy to prove.
Now we want to summarize the results of this section.
{\bar{e}}gin{proposition}
For a triple $(\lambda',{\bar{\lambda}},\mu)$ consisting of a $4$-form, a $3$-form, and a $2$-form on ${\bf B}W$ such that assumptions~\eqref{assumption:lambda-mu-general} and~\eqref{assumption:pencil-regular} hold,
there is a basis in ${\bf B}W$ such that the forms are given by~\eqref{eq:lambda-mu-explicit} and~\eqref{eq:mu-matrix} with $K_i$ pairwise distinct and~\eqref{eq:mmk-nonzero} satisfied.
Conversely, if the forms are given by~\eqref{eq:lambda-mu-explicit} and~\eqref{eq:mu-matrix} with pairwise distinct $K_i$ and~\eqref{eq:mmk-nonzero} holds,
then assumptions~\eqref{assumption:lambda-mu-general} and~\eqref{assumption:pencil-regular} hold.
If, moreover, \eqref{assumption:m-nonzero} holds, then the pairing between the subspaces $A_1$ and $A_2$ defined by $\mu$ is nondegenerate and the assertion of Lemma~{\rm\ref{lemma:no21cases}} holds.
Finally, for general $M_i$ and $K_i$ assumption~\eqref{assumption:lgr-hyperplane} holds.
\end{proposition}
{\bar{e}}gin{proof}
This is clear since all the assumptions are open conditions.
\end{proof}
{\bar{e}}gin{remark}
The standard presentation~\eqref{eq:lambda-mu-explicit}, \eqref{eq:mu-matrix} of the data $(\lambda,\mu)$ allows to compute easily the number of parameters we have.
Indeed, we have a precise form of $\lambda$, and 9 parameters in $\mu$ (6 parameters $M_i$ and 3 parameters $K_i$).
On the other hand, we can rescale the form $\mu$, and also rescale and permute consistently the bases of $A_1$ and $A_2$ (i.e.\ act by the normalizer of a torus in~$\SL(A_1)$).
This allows to kill three parameters.
Moreover, as we will see soon (Lemma~\ref{lemma:lambda-mu-shift}), the variety $X^5_{\lambda,\mu}$ does not change if we replace the pair $(\lambda,\mu)$ by the pair $(\lambda - t\mu^2,\mu)$ for any $t \in \mathbf k$.
This allows to kill one more parameter (in fact, by using this we can assume $K_1 + K_2 + K_3 = 0$).
So, altogether, the moduli space of varieties $X^5_{\lambda,\mu}$ is 5-dimensional.
This feature makes the situation with K\"uchle fivefolds of type $(c5)$ rather different from other examples of Fano fivefolds (${\bf P}^5$, $\Gr(2,5) \cap H$, $({\bf P}^1)^5$, ${\bf B}l_{v_2({\bf P}^2)}{\bf P}^5$),
whose half-anticanonical section has a noncommutative K3 surface
\end{remark}
\section{A description of K\"uchle fivefolds}\label{section:5folds}
\subsection{The main Theorem}\label{subsection:main-theorem}
Recall the definition of the K\"uchle fivefold of type $(c5)$ from the introduction. We start with a 7-dimensional vector space $W$, a 2-form $\mu \in \bw2W^\vee$ and a 4-form $\lambda \in \bw4W^\vee$.
We consider the Grassmannian $\Gr(3,W)$ of 3-dimensional subspaces in $W$ with its tautological subbundles $\mathscr{U}_3 \subset W \otimes \mathscr{O}$ and $\mathscr{U}_3^\perp \subset W^\vee \otimes \mathscr{O}$.
By Bott Theorem $\mu$ can be considered as a global section of the vector bundle $\mathscr{U}_3(1) \isom \bw2\mathscr{U}_3^\vee$ and $\lambda$ can be considered as a global section of the vector bundle $\mathscr{U}_3^\perp(1) \isom \bw3(W/\mathscr{U}_3)$.
Then $X^5_{\lambda,\mu} \subset \Gr(2,W)$ is defined as the zero locus of $(\mu,\lambda)$.
In other words, it is the subvariety of $\Gr(3,W)$ parameterizing all 3-subspaces $U_3 \subset W$ annihilated by $\lambda$ and isotropic for $\mu$.
{\bar{e}}gin{lemma}\label{lemma:lambda-mu-shift}
For a general choice of $\lambda$ and $\mu$ the variety $X^5_{\lambda,\mu}$ is a smooth Fano fivefold of index~$2$.
Moreover, for any $t \in \mathbf k$ we have $X^5_{\lambda - t\mu^2,\mu} = X^5_{\lambda,\mu}$.
\end{lemma}
{\bar{e}}gin{proof}
The smoothness of general $X^5_{\lambda,\mu}$ follows from Bertini Theorem since the vector bundles $\mathscr{U}_3(1)$ and $\mathscr{U}_3^\perp(1)$ are globally generated.
The dimension of $X^5_{\lambda,\mu}$ is
{\bar{e}}gin{equation*}
\dim \Gr(3,W) - \rank(\mathscr{U}_3(1)) - \rank(\mathscr{U}_3^\perp(1)) = 3\cdot 4 - 3 - 4 = 5,
\end{equation*}
and by adjunction the canonical class of $X^5_{\lambda,\mu}$ is given by
{\bar{e}}gin{equation*}
\omega_{X^5_{\lambda,\mu}} \isom
(\omega_{\Gr(3,W)} \otimes \det(\mathscr{U}_3(1)) \otimes \det(\mathscr{U}_3^\perp(1)))\vert_{X^5_{\lambda,\mu}} \isom
(\mathscr{O}(-7) \otimes \mathscr{O}(2) \otimes \mathscr{O}(3))\vert_{X^5_{\lambda,\mu}} \isom
\mathscr{O}_{X^5_{\lambda,\mu}}(-2).
\end{equation*}
Thus $X^5_{\lambda,\mu}$ is a Fano fivefold of index 2.
If $u_1,u_2,u_3 \in U_3$ then
{\bar{e}}gin{equation*}
\mu^2 \,\lrcorner\, (u_1\wedge u_2 \wedge u_3) =
\mu(u_1,u_2) (\mu \,\lrcorner\, u_3) - \mu(u_1,u_3) (\mu \,\lrcorner\, u_2) + \mu(u_2,u_3) (\mu \,\lrcorner\, u_1).
\end{equation*}
Therefore, if $U_3$ is $\mu$-isotropic, it is annihilated by $\mu^2$.
This means that $X^5_{\lambda,\mu} = X^5_{\lambda - t\mu^2,\mu}$.
\end{proof}
Now we can state the main result of the paper.
{\bar{e}}gin{theorem}\label{theorem:main}
Assume the pair $(\lambda,\mu)$ satisfies assumptions~\eqref{assumption:lambda-mu-general}, \eqref{assumption:pencil-regular}, \eqref{assumption:m-nonzero}, and~\eqref{assumption:lgr-hyperplane}.
Then the K\"uchle fivefold $X^5_{\lambda,\mu}$ is smooth, and there is a smooth projective variety $\widetilde{X}^5_{\lambda,\mu}$ and a diagram
{\bar{e}}gin{equation}\label{diagram:fivefolds}
\vcenter{\xymatrix{
& E \ar[dl]_p \ar[r]^i & \widetilde{X}^5_{\lambda,\mu} \ar[dl]_\pi \ar[dr]^{\bar\pi} & {\overline{E}} \ar[l]_{\bar{\imath}} \ar[dr]^{\bar{p}} \\
F \ar[r] & X^5_{\lambda,\mu} && \overline{X}^5_{\lambda,\mu} & Z \ar[l]
}}
\end{equation}
where
{\bar{e}}gin{itemize}
\item
$\overline{X}^5_{\lambda,\mu} = \LGr_\mu(3,{\bf B}W)_{{\bar{\lambda}}}$ is a hyperplane section of the Lagrangian Grassmannian;
\item
$F \isom {\bf F}l(1,2;A_1) \isom {\bf F}l(1,2;A_2)$ is the flag variety;
\item
$Z$ is a smooth scroll over a del Pezzo surface of degree $6$;
\item
the maps $\pi$ and ${\bar\pi}$ are blowups with centers in $F$ and $Z$;
\item $E$ and ${\overline{E}}$ are the exceptional divisors of the blowups; and
\item the maps $i$ and ${\bar{\imath}}$ are the embeddings of the exceptional divisors.
\end{itemize}
\end{theorem}
The proof of the Theorem takes the rest of the section. More details about the structure of the diagram will come in the course of the proof.
Throughout the section we adopt generality assumptions~\eqref{assumption:lambda-mu-general}, \eqref{assumption:pencil-regular}, \eqref{assumption:m-nonzero}, and~\eqref{assumption:lgr-hyperplane}
and use the structural results from section~\ref{subsection:genericity-assumptions}.
\subsection{The odd and even symplectic Grassmannians}\label{subsection:odd-sgr}
We start by considering the zero locus of the section $\mu$ of the vector bundle $\mathscr{U}_3(1) \isom \bw2\mathscr{U}_3^\vee$ on $\Gr(3,W)$.
We denote this variety
{\bar{e}}gin{equation*}
\LGr_\mu(3,W) \subset \Gr(3,W).
\end{equation*}
It parameterizes $\mu$-isotropic subspaces of $W$ and, for $\mu$ of corank 1 it is smooth and is usually called an {\sf odd symplectic Grassmannian}.
There is a standard way to relate this variety to a usual (even) symplectic Grassmannian.
Recall that $w_0 \in W$ denotes a generator for the one-dimensional kernel space $\Ker\mu$ of the form $\mu$, and we have a direct sum decomposition $W = \mathbf k w_0 \oplus {\bf B}W$ of~\eqref{eq:bw}.
Note that the restriction of the form $\mu$ to ${\bf B}W$ is non-degenerate.
Denote by
{\bar{e}}gin{equation*}
\pr:W \to {\bf B}W
\end{equation*}
the projection along $w_0$. For each $\mu$-isotropic subspace $U_3 \subset W$ the subspace $\pr(U_3) \subset {\bf B}W$ is also $\mu$-isotropic, but its
dimension may be either 3 (typically), or 2 (if $U_3$ contains $w_0$). Thus, the map $\pr$ induces a rational map
{\bar{e}}gin{equation*}
\LGr_\mu(3,W) \dashrightarrow \LGr_\mu(3,{\bf B}W),
\end{equation*}
to the (even) symplectic Lagrangian Grassmannian. To resolve its indeterminacy it is natural to consider the following odd symplectic flag variety
{\bar{e}}gin{equation*}
\LFl_\mu(3,4;W) \subset {\bf F}l(3,4;W)
\end{equation*}
defined as the zero locus of the global section $\mu$ of the vector bundle $\bw2\mathscr{U}_4^\vee$ on ${\bf F}l(3,4;W)$
(i.e.\ the variety of flags $U_3 \subset U_4 \subset W$ such that $U_4$, and hence a fortiori $U_3$, is $\mu$-isotropic).
In what follows we denote by
{\bar{e}}gin{equation*}
h = c_1(\mathscr{U}_3^\vee) \in {\bf P}ic(\LGr_\mu(3,W))
\qquad\text{and}\qquad
{\bar{h}} = c_1({\overline{\mathscr{U}}}_3^\vee) \in {\bf P}ic(\LGr_\mu(3,{\bf B}W))
\end{equation*}
the ample generators of the Picard groups and their pullbacks to other varieties.
{\bar{e}}gin{lemma}\label{lemma:sfl}
The map $\pr$ induces a regular map
{\bar{e}}gin{equation*}
{\bar\pi} \colon \LFl_\mu(3,4;W) \to \LGr_\mu(3,{\bf B}W)
\end{equation*}
which is a ${\bf P}^3$-fibration. The forgetful map
{\bar{e}}gin{equation*}
\pi \colon \LFl_\mu(3,4;W) \to \LGr_\mu(3,W)
\end{equation*}
is the blowup with center in the subvariety isomorphic to $\LGr_\mu(2,{\bf B}W)$.
\end{lemma}
{\bar{e}}gin{proof}
Let $U_4 \subset W$ be a $\mu$-isotropic subspace.
Then the subspace $\pr(U_4) \subset {\bf B}W$ is also $\mu$-isotropic.
But since the form $\mu$ on ${\bf B}W$ is nondegenerate, we have $\dim(\pr(U_4)) \le 3$, hence any such $U_4$ contains $w_0$, and the space $\pr(U_4) \isom U_4/w_0 \subset {\bf B}W$ is Lagrangian.
This shows that the projection $\pr$ induces an isomorphism
{\bar{e}}gin{equation}\label{eq:lg4w-lg3bw}
\LGr_\mu(4,W) \isom \LGr_\mu(3,{\bf B}W),
\end{equation}
under which the tautological bundles are related by the (canonically split) exact sequence
{\bar{e}}gin{equation}\label{eq:u4-bu3}
0 \to \mathscr{O} \xrightarrow{\ w_0\ } \mathscr{U}_4 \xrightarrow{\quad} {\overline{\mathscr{U}}}_3 \to 0.
\end{equation}
Consequently, the projection $\LFl_\mu(3,4;W) \to \LGr_\mu(4,W)$ can be understood as a map
{\bar{e}}gin{equation*}
{\bar\pi}: \LFl_\mu(3,4;W) \to \LGr_\mu(3;{\bf B}W).
\end{equation*}
Since ${\bf F}l(3,4;W) \isom \Gr_{\Gr(4,W)}(3,\mathscr{U}_4)$ and the vector bundle $\bw2\mathscr{U}_4^\vee$ is a pullback from $\Gr(4,W)$, it follows that
{\bar{e}}gin{equation*}
\LFl_\mu(3,4;W) \isom
\Gr_{\LGr_\mu(3,{\bf B}W)}(3,\mathscr{O} \oplus {\overline{\mathscr{U}}}_3) \isom
{\bf P}_{\LGr_\mu(3,{\bf B}W)}(\bw3(\mathscr{O} \oplus {\overline{\mathscr{U}}}_3))
\end{equation*}
is a ${\bf P}^3$-bundle. So, it remains to describe the forgetful map $\pi$.
Consider the zero locus of the global section $w_0$ of the vector bundle $W/\mathscr{U}_3$ on $\LGr_\mu(3,W)$.
The above arguments show that the projection $\pr$ induces an isomorphism of this zero locus with the symplectic Grassmannian $\LGr_\mu(2,{\bf B}W)$ under which the tautological bundles are related by the (canonically split) exact sequence
{\bar{e}}gin{equation}\label{eq:u3-bu2}
0 \to \mathscr{O} \xrightarrow{\ w_0\ } w_0^*\mathscr{U}_3 \xrightarrow{\quad} {\overline{\mathscr{U}}}_2 \to 0,
\end{equation}
where we denote the corresponding embedding by
{\bar{e}}gin{equation*}\label{eq:map-w0}
w_0 \colon \LGr_\mu(2,{\bf B}W) \hookrightarrow \LGr_\mu(3,W).
\end{equation*}
Let us show that the map $\pi:\LFl_\mu(3,4;W) \to \LGr_\mu(3,W)$ is the blowup of the subvariety $w_0(\LGr_\mu(2,{\bf B}W)) \subset \LGr_\mu(3,W)$.
For this we use the blowup Lemma~\ref{lemma:blowup}.
Note that $\LFl_\mu(3,4;W)$ embeds into ${\bf P}_{\LGr_\mu(3,W)}(W/\mathscr{U}_3)$ as the zero locus of a section of the vector bundle $\mathscr{U}_3^\vee \otimes \mathscr{O}(1)$
corresponding to the morphism of vector bundles $\mu: W/\mathscr{U}_3 \to \mathscr{U}_3^\vee$ induced by the skew form $\mu$.
It is easy to see that the degeneracy locus $D(\mu)$ of this map is supported (at least set-theoretically) on $w_0(\LGr_\mu(2,{\bf B}W))$, hence has codimension 2, and that the second degeneracy locus $D_2(\mu)$ is empty.
Hence, Lemma~\ref{lemma:blowup} applies and shows that $\LFl_\mu(3,4;W)$ is the blowup of $\LGr_\mu(3,W)$ with center in $D(\mu)$,
and the exceptional divisor is isomorphic to the projectivization of the vector bundle $\Ker(W/\mathscr{U}_3 \xrightarrow{\ \mu\ } \mathscr{U}_3^\vee)$ on $D(\mu)$.
So, it remains to show that the degeneracy locus $D(\mu)$ of the morphism $W/\mathscr{U}_3 \to \mathscr{U}_3^\vee$ is equal to $w_0(\LGr_\mu(2,{\bf B}W))$ scheme-theoretically.
For this recall that, by the proof of Lemma~\ref{lemma:blowup}, the dual of this map extends to an exact sequence
{\bar{e}}gin{equation}\label{eq:deg-loc-sgr37}
0 \to \mathscr{U}_3 \to \mathscr{U}_3^\perp \to \mathscr{O} \to \mathscr{O}_D \to 0
\end{equation}
(since $\det(W/\mathscr{U}_3) \isom \det(\mathscr{U}_3^\vee)$), hence $D(\mu)$ is the zero locus of a global section of the vector bundle $W/\mathscr{U}_3$.
This section, evidently, corresponds to the vector $w_0$, and its zero locus was already shown to be equal to $w_0(\LGr_\mu(2,{\bf B}W))$.
\end{proof}
\subsection{The zero locus of $\lambda$}\label{subsection:zero-locus-lambda}
Recall that $X^5_{\lambda,\mu}$ is the zero locus of $\lambda$ considered as a global section of the bundle $\mathscr{U}_3^\perp(h)$ on $\LGr_\mu(3,W)$.
We define
{\bar{e}}gin{equation*}
\widetilde{X}^5_{\lambda,\mu} \subset \LFl_\mu(3,4;W)
\end{equation*}
as the zero locus of $\lambda$, considered as the global section of the pullback of this vector bundle to the odd symplectic flag variety.
Consider the composition of the maps
{\bar{e}}gin{equation*}
\widetilde{X}^5_{\lambda,\mu} \hookrightarrow \LFl_\mu(3,4;W) \isom {\bf P}_{\LGr_\mu(3,{\bf B}W)}(\bw3\mathscr{U}_4) \twoheadrightarrow \LGr_\mu(3,{\bf B}W),
\end{equation*}
where the isomorphism in the middle follows from~\eqref{eq:lg4w-lg3bw}.
{\bar{e}}gin{lemma}\label{lemma:tcx-zero-loc}
The map $\widetilde{X}^5_{\lambda,\mu} \to \LGr_\mu(3,{\bf B}W)$ factors as the composition
{\bar{e}}gin{equation*}
\widetilde{X}^5_{\lambda,\mu} \xrightarrow{\ {\bar\pi}\ } \LGr_\mu(3,{\bf B}W)_{\bar{\lambda}} \hookrightarrow \LGr_\mu(3,{\bf B}W),
\end{equation*}
where $\LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}$ is the hyperplane section of $\LGr_\mu(3,{\bf B}W)$ given by the $3$-form ${\bar{\lambda}} \in \bw3{\bf B}W^\vee$.
Moreover, $\widetilde{X}^5_{\lambda,\mu} \subset {\bf P}_{\LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}}(\bw3\mathscr{U}_4)$ is the zero locus of a global section of the bundle $\mathscr{U}_4^\perp(h)$.
\end{lemma}
{\bar{e}}gin{proof}
Dualizing the tautological sequence $0 \to \mathscr{U}_4/\mathscr{U}_3 \to W/\mathscr{U}_3 \to W/\mathscr{U}_4 \to 0$, twisting it by~$\mathscr{O}(h)$, and taking into account an isomorphism
{\bar{e}}gin{equation}\label{eq:u4-u3}
\mathscr{U}_4/\mathscr{U}_3 \isom \det(\mathscr{U}_4) \otimes \det(\mathscr{U}_3^\vee) \isom \det({\overline{\mathscr{U}}}_3) \otimes \det(\mathscr{U}_3^\vee) \isom \mathscr{O}(h - {\bar{h}})
\end{equation}
(with~\eqref{eq:u4-bu3} used in the second isomorphism), we get an exact sequence
{\bar{e}}gin{equation}\label{eq:u4p-u3p}
0 \to \mathscr{U}_4^\perp(h) \to \mathscr{U}_3^\perp(h) \to \mathscr{O}({\bar{h}}) \to 0.
\end{equation}
Since $\widetilde{X}^5_{\lambda,\mu} \subset \LFl_\mu(3,4;W) \isom {\bf P}_{\LGr_\mu(3,{\bf B}W)}(\bw3\mathscr{U}_4)$ is the zero locus of a section of $\mathscr{U}_3^\perp(h)$, it lies in the zero locus of the induced section of $\mathscr{O}({\bar{h}})$.
The line bundle $\mathscr{O}({\bar{h}})$ is a pullback from $\LGr_\mu(3,{\bf B}W)$, hence a zero locus of its section is the preimage of the zero locus of a hyperplane section of $\LGr_\mu(3,{\bf B}W)$.
By definition, evaluation of this section on the subspace ${\bf B}U_3 \subset {\bf B}W$ is equal to the evaluation of the 4-form $\lambda$ on the corresponding subspace $U_4 = \mathbf k w_0 \oplus {\bf B}U_3$ of $W$.
Since the convolution of $\lambda$ with $w_0$ is ${\bar{\lambda}}$, this is equal to the evaluation of ${\bar{\lambda}}$ on ${\bf B}U_3$.
Thus the hyperplane section of $\LGr_\mu(3,{\bf B}W)$ we are interested in is given by ${\bar{\lambda}}$.
It remains to note that when restricted to the zero locus ${\bf P}_{\LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}}(\bw3\mathscr{U}_4)$ of the section of $\mathscr{O}({\bar{h}})$, the section $\lambda$ of $\mathscr{U}_3^\perp(h)$ comes (via~\eqref{eq:u4p-u3p}) from a section of $\mathscr{U}_4^\perp(h)$,
and $\widetilde{X}^5_{\lambda,\mu}$ is the zero locus of this section.
\end{proof}
Consider the following diagram of vector bundles on $\LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}$.
{\bar{e}}gin{equation}\label{diagram:hlam}
\vcenter{\xymatrix{
& \bw3\mathscr{U}_4 \ar[r] \ar@{-->}[d]_{\hat{\lambda}} & \bw3W \otimes \mathscr{O} \ar[d]^\lambda \\
0 \ar[r] & \mathscr{U}_4^\perp \ar[r] & W^\vee \otimes \mathscr{O} \ar[r] & \mathscr{U}_4^\vee \ar[r] & 0
}}
\end{equation}
Here the bottom line is the tautological exact sequence, and since the composition of the top arrow with the map $\lambda$ and the projection $W^\vee \otimes \mathscr{O} \to \mathscr{U}_4^\vee$ vanishes
(by definition of $\LGr_\mu(3;{\bf B}W)_{\bar{\lambda}}$), there is a dashed arrow ${\hat{\lambda}}$ on the left, making the diagram commutative.
It is clear that the map
{\bar{e}}gin{equation}\label{eq:map-hlam}
{\hat{\lambda}} \colon \bw3\mathscr{U}_4 \to \mathscr{U}_4^\perp
\end{equation}
defined in this way, corresponds to the section of the vector bundle $\mathscr{U}_4^\perp(h)$ on ${\bf P}_{\LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}}(\bw3\mathscr{U}_4)$ cutting out $\widetilde{X}^5_{\lambda,\mu}$.
So, we can use the blowup Lemma~\ref{lemma:blowup} to describe $\widetilde{X}^5_{\lambda,\mu}$ as the blowup of $\LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}$.
For this, however, we need to control the degeneracy loci of ${\hat{\lambda}}$.
\subsection{Degeneracy loci of the morphism ${\hat{\lambda}}$}\label{subsection:deg-loc-hlam}
We start by discussing the degeneracy loci of the morphism ${\hat{\lambda}}$ on the hyperplane section $\Gr(4,W)_\lambda$ of the Grassmannian (where it is defined via the same diagram~\eqref{diagram:hlam}), and
later we will restrict to the Lagrangian Grassmannian $\LGr_\mu(3,{\bf B}W) \subset \Gr(4,W)$.
Recall the definition of the ${\mathbb{G}_2}$-adjoint variety $\Gr_\lambda(5,W) \subset \Gr(5,W)$ and the quadric $\mathbf{Q}^\vee_\lambda \subset {\bf P}(W^\vee)$ from section~\ref{subsection:4form-4space}.
{\bar{e}}gin{proposition}\label{proposition:gr-deg-loc}
Let $\Gr(4,W)_{\lambda,k} \subset \Gr(4,W)_\lambda$ be the $k$-th degeneracy locus of the map~\eqref{eq:map-hlam} on the Grassmannian hyperplane section $\Gr(4,W)_\lambda$.
Then $\Gr(4,W)_{\lambda,3} = \varnothing$ and
\noindent$(a)$ there is an isomorphism $\Gr(4,W)_{\lambda,2} \isom \mathbf{Q}^\vee_\lambda$;
\noindent$(b)$ there is a dominant regular birational morphism $\Gr_{\Gr_\lambda(5,W)}(4,\mathscr{U}_5) \to \Gr(4,W)_{\lambda,1}$, which is an isomorphism over the complement of $\Gr(4,W)_{\lambda,2}$.
\end{proposition}
{\bar{e}}gin{proof}
We start with part $(b)$.
Assume the map ${\hat{\lambda}}\colon \bw3\mathscr{U}_4 \to \mathscr{U}_4^\perp$ is degenerate at point $U_4$ of $\Gr(4,W)$.
Then its image is contained in a 2-dimensional subspace of $U_4^\perp$.
Every such subspace can be written as $U_5^\perp$ for some 5-dimensional subspace $U_5 \subset W$ containing $U_4$.
The condition ${\hat{\lambda}}(\bw3U_4) \subset U_5^\perp$ with the condition $\lambda(\bw4U_4) = 0$ (defining the hyperplane section $\Gr(4,W)_\lambda \subset \Gr(4,W)$), together imply that $U_5$ is $\lambda$-isotropic.
Thus $U_5 \in \Gr_\lambda(5,W)$.
Conversely, for every $U_5 \in \Gr_\lambda(5,W)$, any subspace $U_4 \subset U_5$ gives a point of $\Gr(4,W)_{\lambda,1}$.
This means that the natural map $\Gr_{\Gr_\lambda(5,W)}(4,\mathscr{U}_5) \to \Gr(4,W)$ is surjective onto the degeneracy locus $\Gr(4,W)_{\lambda,1}$.
Moreover, if $U_4 \in \Gr(4,W)_{\lambda,1} \smallsetminus \Gr(4,W)_{\lambda,2}$, then the subspace $U_5$ is determined uniquely by~$U_4$, hence the constructed map is an isomorphism over the complement of $\Gr(4,W)_{\lambda,2}$.
Now assume $U_4 \in \Gr(4,W)_{\lambda,2}$.
Then the same argument as above shows that there is a 6-dimensional subspace $U_6 \subset W$ such that for any $U_5$ such that $U_4 \subset U_5 \subset U_6$ the subspace $U_5$ is $\lambda$-isotropic.
Let $w^\vee \in {\bf P}(W^\vee)$ be the point corresponding to the subspace $U_6$.
Since $w^\vee \in U_5^\perp$, it follows from Remark~\ref{remark:gr27-g2} that $w^\vee \in \mathbf{Q}^\vee_\lambda$.
Conversely, if $U_6 \subset W$ is the subspace corresponding to a point $w^\vee \in \mathbf{Q}^\vee_\lambda$ then the bivector $\lambda^\vee \,\lrcorner\, w^\vee$ is degenerate, and moreover has rank 4.
Therefore, its kernel is 2-dimensional and can be written as $U_4^\perp$ for a unique subspace $U_4 \subset U_6$.
Then $U_5 \subset U_6$ is $\lambda$-isotropic if and only if $U_4 \subset U_5$.
Hence the image of ${\hat{\lambda}}$ at $U_4$ is $U_6^\perp$.
Altogether, this means that there is a surjective map $\mathbf{Q}^\vee_\lambda \to \Gr(4,W)_{\lambda,2}$, which is an isomorphism over the complement of $\Gr(4,W)_{\lambda,3}$.
So, it remains to show that $\Gr(4,W)_{\lambda,3} = \varnothing$.
Assume that $U_4 \in \Gr(4,W)_{\lambda,3}$, i.e.\ ${\hat{\lambda}} = 0$ at $U_4$.
Then every $U_5$ containing $U_4$ is $\lambda$-isotropic.
Therefore, the projective plane ${\bf P}(W/U_4) \subset \Gr(5,W)$ is contained in $\Gr_\lambda(5,W)$.
But the ${\mathbb{G}_2}$-adjoint variety $\Gr_\lambda(5,W)$ does not contain planes by \cite{kapustka2013genus10}.
This contradiction shows that $\Gr(4,W)_{\lambda,3} = \varnothing$ and thus completes the proof of the Proposition.
\end{proof}
Now we go back to the symplectic situation. Denote by $\LGr_\mu(3,{\bf B}W)_{{\bar{\lambda}},k}$ the $k$-th degeneracy locus of the morphism~\eqref{eq:map-hlam}.
Clearly,
{\bar{e}}gin{equation*}
\LGr_\mu(3,{\bf B}W)_{{\bar{\lambda}},k} = \LGr_\mu(3,{\bf B}W) \cap \Gr(4,W)_{\lambda,k}.
\end{equation*}
Our next goal is to describe these intersections. First, consider the case $k = 1$.
Instead of the degeneracy locus $\Gr(4,W)_{\lambda,k}$ consider its blowup $\Gr_{\Gr_\lambda(5,W)}(4,\mathscr{U}_5)$ and instead of the intersection consider the fiber product
{\bar{e}}gin{equation}\label{eq:z-def}
Z := \LGr_\mu(3,{\bf B}W) \times_{\Gr(4,W)} \Gr_{\Gr_\lambda(5,W)}(4,\mathscr{U}_5).
\end{equation}
Note that we have a canonical embedding
{\bar{e}}gin{equation*}
Z \hookrightarrow \Gr_{\Gr_\lambda(5,W)}(4,\mathscr{U}_5) \hookrightarrow {\bf F}l(4,5;W).
\end{equation*}
Thus $Z$ parameterizes some flags $(U_4,U_5)$ of subspaces of $W$ such that $U_5$ is $\lambda$-isotropic and $U_4$ is $\mu$-isotropic.
As usual, we denote by $\mathscr{U}_4 \hookrightarrow \mathscr{U}_5 \hookrightarrow W \otimes \mathscr{O}_Z$ the pullback to $Z$ of the tautological flag.
{\bar{e}}gin{proposition}\label{proposition:s-z}
The fiber product $Z$ is isomorphic to a ${\bf P}^1$-bundle over a sextic del Pezzo surface $S \subset {\bf P}(\bw2A_1) \times {\bf P}(\bw2A_2)$.
\end{proposition}
{\bar{e}}gin{proof}
By definition $Z$ parameterizes pairs $(U_4,U_5)$, where $U_5 \subset W$ is a $\lambda$-isotropic subspace, and $U_4 \subset U_5$ is a $\mu$-isotropic subspace.
In particular, $U_4$ and hence also $U_5$ contains $w_0$.
So, one can rephrase the definition of $Z$ by saying that it parameterizes all pairs $({\bf B}U_3,{\bf B}U_4)$, where ${\bf B}U_3 \subset {\bf B}U_4 \subset {\bf B}W$,
the subspace ${\bf B}U_4$ is both ${\bar{\lambda}}$ and $\lambda'$-isotropic, and ${\bf B}U_3$ is $\mu$-isotropic.
Since $\dim{\bf B}U_3 = 3$ and $\dim{\bf B}U_4 = 4$, the last condition means
that the restriction of $\mu$ to ${\bf B}U_4$ is degenerate, hence the 4-form $\mu^2 = \mu \wedge \mu$ vanishes on ${\bf B}U_4$, i.e.\ ${\bf B}U_4$ is also $\mu^2$-isotropic.
So, consider the subvariety
{\bar{e}}gin{equation*}
S \subset \Gr(4,{\bf B}W),
\end{equation*}
parameterizing subspaces ${\bf B}U_4 \subset {\bf B}W$ that are isotropic with respect to ${\bar{\lambda}}$, $\lambda'$ and $\mu^2$.
The above discussion shows that there is a map
{\bar{e}}gin{equation*}
\sigma: Z \to S,
\qquad
({\bf B}U_3,{\bf B}U_4) \mapsto {\bf B}U_4.
\end{equation*}
We will show first that $S$ is a sextic del Pezzo surface, and then that $\sigma$ is a ${\bf P}^1$-bundle.
First, the locus of ${\bar{\lambda}}$-isotropic subspaces ${\bf B}U_4 \subset {\bf B}W$ is the zero locus of the section ${\bar{\lambda}}$ of the vector bundle $\bw3{\overline{\mathscr{U}}}_4^\vee$ on $\Gr(4,{\bf B}W)$.
By Corollary~\ref{corollary:gr46-3form} it is isomorphic to
{\bar{e}}gin{equation*}
\Gr(2,A_1) \times \Gr(2,A_2) \isom {\bf P}(\bw2A_1) \times {\bf P}(\bw2A_2) \isom {\bf P}^2 \times {\bf P}^2.
\end{equation*}
By~\eqref{eq:gr46-3form-bu4} we have $\bw4{\overline{\mathscr{U}}}_4^\vee \isom \bw2\mathscr{U}_{2,A_1}^\vee \otimes \bw2\mathscr{U}_{2,A_2}^\vee \isom \mathscr{O}_{{\bf P}(\wedge^2A_1) \times {\bf P}(\wedge^2A_2)}(1,1)$.
Therefore, the zero loci of $\lambda'$ and $\mu^2$ on it are two divisors of bidegree $(1,1)$ in the above product corresponding to the pairings between $\bw2A_1$ and $\bw2A_2$ given by $\lambda'$ and $\mu^2$,
and $S$ is the intersection of these two divisors.
It is well known that an intersection of two such divisors is a smooth surface if and only if the line spanned by their equations in the space ${\bf P}(\bw2A_1^\vee \otimes \bw2A_2^\vee)$ is transversal to the divisor of degenerate pairings.
By assumption~\eqref{assumption:pencil-regular} transversality holds for the pencil generated by $\lambda'$ and $\mu^2$, hence $S$ is a smooth surface.
By adjunction its canonical class is the restriction of $\mathscr{O}(-1,-1)$, and its anticanonical degree equals the degree of ${\bf P}^2 \times {\bf P}^2$, which is equal to~6.
So, finally we see that $S$ is a sextic del Pezzo surface.
{\bar{e}}gin{remark}\label{remark:s}
Note that the projections $S \to {\bf P}(\bw2A_1)$ and $S \to {\bf P}(\bw2A_2)$ are birational and provide two representations of $S$ as a blowup of ${\bf P}^2$ in three points.
If we choose a basis of ${\bf B}W$ in which $\lambda'$ and $\mu^2$ have form~\eqref{eq:lambda-mu-explicit}, then the centers of the blowups are given
by the points $\{e_{12},e_{13},e_{23}\} \in {\bf P}(\bw2A_1)$ and $\{e_{45}, e_{46}, e_{56}\} \in {\bf P}(\bw2A_2)$ respectively.
We denote by ${\bar{h}}_1$ and ${\bar{h}}_2$ the pullbacks to $S$ of the hyperplane classes of ${\bf P}(\bw2A_1)$ and ${\bf P}(\bw2A_2)$ respectively.
Then
{\bar{e}}gin{equation}\label{eq:s-intersection}
{\bar{h}}_1^2 = {\bar{h}}_2^2 = 1,
\qquad
{\bar{h}}_1\cdot{\bar{h}}_2 = 2
\end{equation}
and ${\bar{h}}_1 + {\bar{h}}_2$ is the anticanonical class on $S$.
\end{remark}
Now consider the fiber of the map $\sigma:Z \to S$ over a point of $S$ corresponding to a subspace ${\bf B}U_4 \subset {\bf B}W$.
By definition of $Z$, it parameterizes all $\mu$-isotropic 3-subspaces in ${\bf B}U_4$.
The kernel of the restriction of $\mu$ to ${\bf B}U_4$ is equal to the intersection ${\bf B}U_4 \cap {\bf B}U_4^{\perp\mu}$, where ${\bf B}U_4^{\perp\mu} \subset {\bf B}W$ is the orthogonal of ${\bf B}U_4$ with respect to $\mu$.
Since $\mu$ is nondegenerate on ${\bf B}W$, the orthogonal is 2-dimensional.
On the other hand, the rank of $\mu$ on ${\bf B}U_4$ is even, hence the dimension of ${\bf B}U_4 \cap {\bf B}U_4^{\perp\mu}$ is even.
Thus $\mu$ is degenerate on ${\bf B}U_4$ if and only if ${\bf B}U_4^{\perp\mu} \subset {\bf B}U_4$ (i.e.\ ${\bf B}U_4$ is coisotropic), and then ${\bf B}U_4^{\perp\mu}$ is the kernel of the restriction of $\mu$ to ${\bf B}U_4$.
A 3-subspace in such ${\bf B}U_4$ is isotropic if and only if it contains the kernel ${\bf B}U_4^{\perp\mu}$ of the restriction of $\mu$, hence the fiber of $\sigma$ over ${\bf B}U_4$ is the projective line ${\bf P}({\bf B}U_4/{\bf B}U_4^{\perp\mu}) \isom {\bf P}^1$.
The argument above can be rephrased by saying that $Z$ is the projectivization of the following vector bundle over $S$.
Let $\mathscr{U}_{2,A_1}$ and $\mathscr{U}_{2,A_2}$ be the pullbacks to $S$ of the tautological vector bundles on $\Gr(2,A_1)$ and $\Gr(2,A_2)$.
Note that the corresponding quotient bundles are $\mathscr{O}({\bar{h}}_1)$ and $\mathscr{O}({\bar{h}}_2)$ respectively:
{\bar{e}}gin{equation*}
0 \to \mathscr{U}_{2,A_1} \to A_1 \otimes \mathscr{O} \to \mathscr{O}({\bar{h}}_1) \to 0,
\qquad
0 \to \mathscr{U}_{2,A_2} \to A_2 \otimes \mathscr{O} \to \mathscr{O}({\bar{h}}_2) \to 0.
\end{equation*}
By Corollary~\ref{corollary:gr46-3form} and~\eqref{eq:gr46-3form-bu4} we have ${\overline{\mathscr{U}}}_4 = \mathscr{U}_{2,A_1} \oplus \mathscr{U}_{2,A_2}$.
Summing up the above sequences we deduce that ${\bf B}W/{\overline{\mathscr{U}}}_4 \isom \mathscr{O}({\bar{h}}_1) \oplus \mathscr{O}({\bar{h}}_2)$,
hence ${\overline{\mathscr{U}}}_4^{\perp\mu} \isom \mathscr{O}(-{\bar{h}}_1) \oplus \mathscr{O}(-{\bar{h}}_2)$, and so
{\bar{e}}gin{equation}\label{eq:z-proj}
Z \isom {\bf P}_S(\mathscr{V}_S),
\end{equation}
where the vector bundle $\mathscr{V}_S$ is defined by an exact sequence
{\bar{e}}gin{equation}\label{eq:cvs}
0 \to \mathscr{O}(-{\bar{h}}_1) \oplus \mathscr{O}(-{\bar{h}}_2) \to \mathscr{U}_{2,A_1} \oplus \mathscr{U}_{2,A_2} \to \mathscr{V}_S \to 0
\end{equation}
on $S$.
Note also that the bundle $\mathscr{V}_S$ is self-dual.
\end{proof}
Let us fix some details of the description of $Z$ established in the proof of Proposition~\ref{proposition:s-z}.
First, the natural embedding $Z \to {\bf F}l(4,5;W)$ factors as the composition
{\bar{e}}gin{equation*}
Z \hookrightarrow {\bf F}l(3,4;{\bf B}W) \hookrightarrow {\bf F}l(4,5;W),
\end{equation*}
where the second map takes a flag $({\bf B}U_3,{\bf B}U_4)$ in ${\bf B}W$ to the flag $({\bf B}U_3 \oplus \mathbf k w_0, {\bf B}U_4 \oplus \mathbf k w_0)$ in $W$.
In particular, if ${\overline{\mathscr{U}}}_3 \hookrightarrow {\overline{\mathscr{U}}}_4 \hookrightarrow {\bf B}W \otimes \mathscr{O}$ is the restriction to $Z$ of the tautological flag on ${\bf F}l(3,4;{\bf B}W)$, then
the restriction to $Z$ of the tautological flag from ${\bf F}l(4,5;W)$ is given by
{\bar{e}}gin{equation}\label{eq:z-u4-u5}
\mathscr{U}_4 \isom {\overline{\mathscr{U}}}_3 \oplus \mathscr{O}
\qquad\text{and}\qquad
\mathscr{U}_5 \isom {\overline{\mathscr{U}}}_4 \oplus \mathscr{O}.
\end{equation}
Moreover,
there is a commutative diagram
{\bar{e}}gin{equation*}
\vcenter{\xymatrix{
Z \ar@{^{(}->}[rr] \ar[d]_\sigma && {\bf F}l(3,4;{\bf B}W) \ar[d] \\
S \ar@{^{(}->}[r] & \Gr(2,A_1) \times \Gr(2,A_2) \ar@{^{(}->}[r] & \Gr(4,{\bf B}W)
}}
\end{equation*}
and if $\mathscr{U}_{2,A_1}$ and $\mathscr{U}_{2,A_2}$ denote the tautological bundles on $\Gr(2,A_1)$ and $\Gr(2,A_2)$ then
{\bar{e}}gin{equation}\label{eq:z-bu4}
{\overline{\mathscr{U}}}_4 \isom \mathscr{U}_{2,A_1} \oplus \mathscr{U}_{2,A_2}.
\end{equation}
Denote by ${\mathsf{v}}_Z$ the hyperplane class for the projectivization $Z \isom {\bf P}_S(\mathscr{V}_S)$, see~\eqref{eq:z-proj}. Then we have an exact sequence
{\bar{e}}gin{equation}\label{eq:bcu3-1}
0 \to \mathscr{O}(-{\bar{h}}_1) \oplus \mathscr{O}(-{\bar{h}}_2) \to {\overline{\mathscr{U}}}_3 \to \mathscr{O}(-{\mathsf{v}}_Z) \to 0
\end{equation}
and since $\mathscr{V}_S$ is self-dual, also
{\bar{e}}gin{equation}\label{eq:bcu3-2}
0 \to {\overline{\mathscr{U}}}_3 \to {\overline{\mathscr{U}}}_4 \to \mathscr{O}({\mathsf{v}}_Z) \to 0.
\end{equation}
\subsection{A description of the map ${\hat{\lambda}}$ on $Z$}\label{subsection:hlam-z}
In this section we discuss the pullback to $Z$ of the map ${\hat{\lambda}}$ defined by~\eqref{eq:map-hlam}.
In particular, we show that on $Z$ it has a constant rank equal to~2.
As we will see, this gives a description of the degeneracy loci of ${\hat{\lambda}}$ on $\LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}$.
{\bar{e}}gin{lemma}\label{lemma:deg-loc-u2-vs}
On the surface $S \subset \Gr(2,A_1) \times \Gr(2,A_2)$ consider the compositions of the maps
{\bar{e}}gin{equation}\label{eq:maps-l1-l2}
\mathscr{U}_{2,A_1} \hookrightarrow \mathscr{U}_{2,A_1} \oplus \mathscr{U}_{2,A_2} \twoheadrightarrow \mathscr{V}_S
\qquad\text{and}\qquad
\mathscr{U}_{2,A_2} \hookrightarrow \mathscr{U}_{2,A_1} \oplus \mathscr{U}_{2,A_2} \twoheadrightarrow \mathscr{V}_S
\end{equation}
and let ${\overline{C}}_1 \subset S$ and ${\overline{C}}_2 \subset S$ be their degeneracy loci.
Then ${\overline{C}}_1$ is a smooth rational curve in the linear system $|{\bar{h}}_1|$, ${\overline{C}}_2$ is a smooth rational curve in the linear system $|{\bar{h}}_2|$,
and the ranks of the maps~\eqref{eq:maps-l1-l2} on ${\overline{C}}_1$ and ${\overline{C}}_2$ are equal to $1$.
\end{lemma}
{\bar{e}}gin{proof}
Consider the first map (the second map is dealt with analogously).
Since the kernel of the map $\mathscr{U}_{2,A_1} \oplus \mathscr{U}_{2,A_2} \to \mathscr{V}_S$ by definition of $\mathscr{V}_S$ is the kernel of the restriction of the skew-form $\mu$ to ${\overline{\mathscr{U}}}_4$,
the subscheme ${\overline{C}}_1 \subset S$, parameterizes pairs $(U_{2,A_1},U_{2,A_2})$ such that $U_{2,A_1}$ intersects the kernel of $\mu$.
If this holds, then of course $U_{2,A_1}$ is $\mu$-isotropic.
Conversely, if $U_{2,A_1}$ is $\mu$-isotropic, it has to intersect the kernel space of $\mu$ on ${\overline{\mathscr{U}}}_4$ (since otherwise $\mu$ would be identically zero on~${\overline{\mathscr{U}}}_4$, and this contradicts to the no-degeneracy of $\mu$ on ${\bf B}W$).
Thus ${\overline{C}}_1$ is the locus of points of $S$, such that $U_{2,A_1}$ is $\mu$-isotropic.
This condition, in its turn, is equivalent to the condition $\mu_1 \in U_{2,A_1}$, where $\mu_1 \in A_1$ is the kernel vector of the restriction of $\mu$ to $A_1$.
Therefore, ${\overline{C}}_1$ is the preimage of the line in ${\bf P}(\bw2A_1)$ corresponding to the vector $\mu_1 \in A_1 \isom \bw2A_1^\vee$ under the projection $S \to {\bf P}(\bw2A_1)$.
Hence it is a curve in the linear system~$|{\bar{h}}_1|$.
Note that the line corresponding to $\mu_1$ does not pass through the centers of the blowup $S \to {\bf P}(\bw2A_1)$, hence ${\overline{C}}_1$ is a smooth curve.
Indeed, by Remark~\ref{remark:s} in a standard basis the center of this blowup is the triple of points $\{e_{12},e_{13},e_{23}\} \in {\bf P}(\bw2A_1)$, while the line is given by the equation
$\mu_1 = M_6K_1e_1 + M_5K_2e_2 + M_4K_3e_3 = M_6K_1x_{23} - M_5K_2x_{13} + M_4K_3x_{12}$.
By~\eqref{assumption:m-nonzero} all coefficients are nonzero.
Finally, note that the subspace $U_{2,A_1}$ cannot be equal to the kernel of $\mu$ on~${\overline{\mathscr{U}}}_4$, since otherwise the subspaces $U_{2,A_1} \subset A_1$ and $U_{2,A_2} \subset A_2$
would be $\mu$-orthogonal, which would contradict to the nondegeneracy of the pairing between $A_1$ and $A_2$ induced by $\mu$.
Therefore, the rank of the map $\mathscr{U}_{2,A_1} \to \mathscr{V}_S$ is always nonzero, so on ${\overline{C}}_1$ it is identically equal to~1.
\end{proof}
Recall that ${\mathsf{v}}_Z$ denotes the relative hyperplane class of $Z = {\bf P}_S(\mathscr{V}_S)$.
{\bar{e}}gin{corollary}\label{corollary:zero-u2-vz}
On $Z$ consider the compositions
{\bar{e}}gin{equation*}
\mathscr{U}_{2,A_1} \to \sigma^*\mathscr{V}_S \twoheadrightarrow \mathscr{O}_Z({\mathsf{v}}_Z)
\qquad\text{and}\qquad
\mathscr{U}_{2,A_2} \to \sigma^*\mathscr{V}_S \twoheadrightarrow \mathscr{O}_Z({\mathsf{v}}_Z)
\end{equation*}
of the maps~\eqref{eq:maps-l1-l2} with the canonical epimorphisms, and let $C_1 \subset Z$ and $C_2 \subset Z_2$ be their zero loci.
Then the map $\sigma:Z \to S$ induces isomorphisms
{\bar{e}}gin{equation*}
C_1 \isom {\overline{C}}_1
\qquad\text{and}\qquad
C_2 \isom {\overline{C}}_2,
\end{equation*}
so $C_1$ and $C_2$ are smooth rational curves. Moreover, $C_1 \cap C_2 = \varnothing$.
\end{corollary}
{\bar{e}}gin{proof}
The first statement follows from the fact that the maps~\eqref{eq:maps-l1-l2} both have rank 1 on ${\overline{C}}_1$ and~${\overline{C}}_2$ respectively.
For the second, note that a point of intersection $C_1 \cap C_2$ should correspond to a subspace ${\bf B}U_3 \subset {\bf B}U_4$ such that $U_{2,A_1} \subset {\bf B}U_3$ and $U_{2,A_2} \subset {\bf B}U_3$.
But this is of course impossible since $\dim{\bf B}U_3 = 3$.
\end{proof}
Consider the pullback of the map ${\hat{\lambda}}$ to $Z$.
Using~\eqref{eq:z-u4-u5} we see that
{\bar{e}}gin{equation*}
\bw3\mathscr{U}_4 \isom \bw3({\overline{\mathscr{U}}}_3 \oplus \mathscr{O}) \isom \bw3{\overline{\mathscr{U}}}_3 \oplus \bw2{\overline{\mathscr{U}}}_3
\end{equation*}
(where the second summand embeds into $\bw3\mathscr{U}_4$ by the wedge product with $w_0$), and
{\bar{e}}gin{equation*}
\mathscr{U}_5^\perp \isom {\overline{\mathscr{U}}}_4^\perp \isom \mathscr{U}_{2,A_1}^\perp \oplus \mathscr{U}_{2,A_2}^\perp \isom \mathscr{O}(-{\bar{h}}_1) \oplus \mathscr{O}(-{\bar{h}}_2).
\end{equation*}
Therefore, the map ${\hat{\lambda}}$ can be rewritten as
{\bar{e}}gin{equation}\label{eq:hlam-z-rewritten}
{\hat{\lambda}}\vert_Z = \lambda' \oplus {\bar{\lambda}} : \bw3{\overline{\mathscr{U}}}_3 \oplus \bw2{\overline{\mathscr{U}}}_3 \xrightarrow{\ \qquad\ } \mathscr{O}(-{\bar{h}}_1) \oplus \mathscr{O}(-{\bar{h}}_2)
\end{equation}
where $\lambda'$ is considered as a map from the first summand in the left hand side, and ${\bar{\lambda}}$ as a map from the second summand.
We will prove that the map ${\hat{\lambda}}$ is surjective (hence its rank is equal to~2) everywhere on $Z$, and this will give a complete description of the degeneracy loci of ${\hat{\lambda}}$ on $\LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}$.
For this we first analyze the rank and the cokernel of the component~${\bar{\lambda}}$, and then we show that the component $\lambda'$ maps surjectively onto the cokernel of ${\bar{\lambda}}$.
{\bar{e}}gin{lemma}\label{lemma:deg-loc-blam}
The map ${\bar{\lambda}}: \bw2{\overline{\mathscr{U}}}_3 \to \mathscr{O}(-{\bar{h}}_1) \oplus \mathscr{O}(-{\bar{h}}_2)$ on $Z$ is surjective away of the complement of the curve $C_1 \sqcup C_2$.
Moreover, it extends to an exact sequence
{\bar{e}}gin{equation}\label{eq:exact-seq-blam}
0 \to \mathscr{O}(-2{\mathsf{v}}_Z - {\bar{h}}_1 - {\bar{h}}_2) \to \bw2{\overline{\mathscr{U}}}_3 \xrightarrow{\ {\bar{\lambda}}\ } \mathscr{O}(-{\bar{h}}_1) \oplus \mathscr{O}(-{\bar{h}}_2) \to \mathscr{O}_{C_2}(-2) \oplus \mathscr{O}_{C_1}(-2) \to 0,
\end{equation}
and the right arrow is the direct sum of two restriction maps $\mathscr{O}(-{\bar{h}}_1) \to \mathscr{O}(-{\bar{h}}_1)\vert_{C_2} \isom \mathscr{O}_{C_2}(-2)$ and $\mathscr{O}(-{\bar{h}}_2) \to \mathscr{O}(-{\bar{h}}_2)\vert_{C_1} \isom \mathscr{O}_{C_1}(-2)$.
\end{lemma}
{\bar{e}}gin{proof}
Recall that the 3-form ${\bar{\lambda}}$ is the sum~\eqref{eq:blam-general} of two summands ${\bar{\lambda}}_1 \in \bw3A_1^\perp$ and ${\bar{\lambda}}_2 \in \bw3A_2^\perp$.
It is easy to see that the component ${\bar{\lambda}}_1$ of the map ${\bar{\lambda}} : \bw2{\overline{\mathscr{U}}}_3 \to \mathscr{O}(-{\bar{h}}_1) \oplus \mathscr{O}(-{\bar{h}}_2)$ can be written as the composition of the map
{\bar{e}}gin{equation*}
\bw2{\overline{\mathscr{U}}}_3 \hookrightarrow \bw2{\overline{\mathscr{U}}}_4 \isom \bw2(\mathscr{U}_{2,A_1} \oplus \mathscr{U}_{2,A_2}) \twoheadrightarrow \bw2\mathscr{U}_{2,A_1} \isom \mathscr{O}(-{\bar{h}}_1)
\end{equation*}
with the embedding $\mathscr{O}(-{\bar{h}}_1) \hookrightarrow \mathscr{O}(-{\bar{h}}_1) \oplus \mathscr{O}(-{\bar{h}}_2)$.
Therefore, the zero locus of ${\bar{\lambda}}_1$ is the locus of points, where the fiber of $\mathscr{U}_{2,A_2}$ is contained in the fiber of ${\overline{\mathscr{U}}}_3$,
hence is equal to the zero locus of the composition
{\bar{e}}gin{equation*}
\mathscr{U}_{2,A_2} \hookrightarrow {\overline{\mathscr{U}}}_4 \twoheadrightarrow {\overline{\mathscr{U}}}_4/{\overline{\mathscr{U}}}_3.
\end{equation*}
By~\eqref{eq:bcu3-2} the quotient ${\overline{\mathscr{U}}}_4/{\overline{\mathscr{U}}}_3$ is isomorphic to $\mathscr{O}({\mathsf{v}}_Z)$ and it is easy to see that the composition of the maps above coincides with the map of Corollary~\ref{corollary:zero-u2-vz}.
Hence its zero locus (and thus also the zero locus of the map ${\bar{\lambda}}_1:\bw2{\overline{\mathscr{U}}}_3 \to \mathscr{O}(-{\bar{h}}_1)$ is equal to the curve $C_2$.
Analogously, the zero locus of the map ${\bar{\lambda}}_2:\bw2{\overline{\mathscr{U}}}_3 \to \mathscr{O}(-{\bar{h}}_2)$ is the curve $C_1$.
Since the curves do not intersect, the rank of the map ${\bar{\lambda}} : \bw2{\overline{\mathscr{U}}}_3 \to \mathscr{O}(-{\bar{h}}_1) \oplus \mathscr{O}(-{\bar{h}}_2)$ on $C_1 \sqcup C_2$ is 1.
Further, let us show that away of $C_1 \sqcup C_2$ the map has rank 2.
It follows from the above discussion that the kernel of its first component at point $(U_{2,A_1},U_{2,A_2},{\bf B}U_3)$ is equal to $({\bf B}U_3 \cap U_{2,A_2}) \wedge {\bf B}U_3 \subset \bw3{\bf B}U_3$,
and the kernel of the second component is $({\bf B}U_3 \cap U_{2,A_1}) \wedge {\bf B}U_3 \subset \bw3{\bf B}U_3$.
Since both intersections ${\bf B}U_3 \cap U_{2,A_i}$ are 1-dimensional away of $C_1 \sqcup C_2$, it follows that the kernels are 2-dimensional and distinct,
hence their intersection is 1-dimensional (in fact, it is equal to $({\bf B}U_3 \cap U_{2,A_1}) \wedge ({\bf B}U_3 \cap U_{2,A_2}) \subset \bw3{\bf B}U_3$), hence the rank of the map ${\bar{\lambda}}$ is 2.
So far, we have proved the first statement. For the second, we note that the kernel of the map ${\bar{\lambda}}$ is a reflexive sheaf of rank 1, hence a line bundle.
Moreover, since the map ${\bar{\lambda}}$ is surjective away of a codimension 2 locus, the first Chern class of the kernel is
{\bar{e}}gin{equation*}
c_1(\bw2{\overline{\mathscr{U}}}_3) - c_1(\mathscr{O}(-{\bar{h}}_1) \oplus \mathscr{O}(-{\bar{h}}_2)) = (-{\bar{h}}_1-{\bar{h}}_2) + (-{\bar{h}}_1-{\mathsf{v}}_Z) + (-{\bar{h}}_2-{\mathsf{v}}_Z) + ({\bar{h}}_1 + {\bar{h}}_2)
\end{equation*}
(we use~\eqref{eq:bcu3-1} to compute the first Chern class of $\bw2{\overline{\mathscr{U}}}_3$),
hence the kernel has the form as in~\eqref{eq:exact-seq-blam}.
Finally, as we have seen earlier, the cokernel of the map ${\bar{\lambda}}$ equals the direct sum of the cokernels of ${\bar{\lambda}}_1$ and ${\bar{\lambda}}_2$, hence is equal to $\mathscr{O}(-{\bar{h}}_1)\vert_{C_2} \oplus \mathscr{O}(-{\bar{h}}_2)\vert_{C_1}$.
But since the curve $C_2$ projects by $\sigma$ to the curve ${\overline{C}}_2$ in the linear system $|{\bar{h}}_2|$, and the intersection product ${\bar{h}}_1\cdot{\bar{h}}_2$ on $S$ is equal to 2, the first summand is $\mathscr{O}_{C_2}(-2)$.
Analogously, the second summand is $\mathscr{O}_{C_1}(-2)$.
\end{proof}
{\bar{e}}gin{lemma}
On $Z$ the composition of the maps
{\bar{e}}gin{equation*}
\bw3{\overline{\mathscr{U}}}_3 \xrightarrow{\ \lambda'\ } \mathscr{O}(-{\bar{h}}_1) \oplus \mathscr{O}(-{\bar{h}}_2) \twoheadrightarrow {\bf C}oker({\bar{\lambda}}) \isom \mathscr{O}_{C_2}(-2) \oplus \mathscr{O}_{C_1}(-2)
\end{equation*}
is surjective. In particular, the map ${\hat{\lambda}}$ is surjective on all $Z$.
\end{lemma}
{\bar{e}}gin{proof}
The summand $\mathscr{O}_{C_2}(-2)$ in the right hand side is a quotient of $\mathscr{O}(-{\bar{h}}_1)$.
So, for the surjectivity over it, it is enough to show that the map
{\bar{e}}gin{equation*}
\lambda':\bw3{\bf B}U_3 \to U_{2,A_1}^\perp \oplus U_{2,A_2}^\perp
\end{equation*}
has a nonzero first component.
Recall that on $C_2$ the space ${\bf B}U_3$ contains $U_{2,A_2}$, and since the form $\lambda'$ annihilates $A_2$, it follows that the image of $\lambda'$ is contained in $U_{2,A_1}^\perp$, so it is enough to show that $\lambda'$ is nonzero on $C_2$.
But this follows immediately from Lemma~\ref{lemma:no21cases}, since vanishing of $\lambda'$ on ${\bf B}U_3$ contradicts to the $\mu$-isotropicity.
The surjectivity of $\lambda'$ on $C_1$ is proved analogously, and the surjectivity of ${\hat{\lambda}}$ follows.
\end{proof}
Recall that $Z$ by~\eqref{eq:z-def} comes with a projection to $\LGr_\mu(3,{\bf B}W) \cap \Gr(4,W)_{\lambda} = \LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}$.
{\bar{e}}gin{corollary}\label{corollary:z-deg-loc}
The map $Z \to \LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}$ is a closed embedding and gives an isomorphism
{\bar{e}}gin{equation}\label{eq:z-deg-locus}
Z \isom \LGr_\mu(3,{\bf B}W)_{\lambda,1}.
\end{equation}
Moreover, $\LGr_\mu(3,{\bf B}W)_{\lambda,2} = \varnothing$.
\end{corollary}
{\bar{e}}gin{proof}
First, by definition of $Z$ and Proposition~\ref{proposition:gr-deg-loc} the map $Z \to \LGr_\mu(3,{\bf B}W)_{\lambda,1}$ is surjective.
Further, since the rank of ${\hat{\lambda}}$ on $Z$ is identically equal to 2, so the corank is 1, it follows that $\LGr_\mu(3,{\bf B}W)_{\lambda,2}$ is not in the image, hence is empty.
Finally, by Proposition~\ref{proposition:gr-deg-loc} the map is an isomorphism over the complement of $\LGr_\mu(3,{\bf B}W)_{\lambda,2}$, hence is an isomorphism.
\end{proof}
\subsection{Proof of the main Theorem}\label{subsection:proof-main-theorem}
Recall that the subvariety $\widetilde{X}^5_{\lambda,\mu} \subset \LFl_\mu(3,4;W)$ was defined as the zero locus of the section $\lambda$ of the vector bundle $\mathscr{U}_3^\perp(h)$.
In particular, the projections $\LFl_\mu(3,4;W) \to \LGr_\mu(3,W)$ and $\LFl_\mu(3,4;W) \to \LGr_\mu(4,W) \isom \LGr_\mu(3,{\bf B}W)$ (see~\eqref{eq:lg4w-lg3bw})
induce two maps
$\pi:\widetilde{X}^5_{\lambda,\mu} \to X^5_{\lambda,\mu}$ and ${\bar\pi}:\widetilde{X}^5_{\lambda,\mu} \to \LGr_\mu(4,{\bf B}W)_{\bar{\lambda}}$.
{\bar{e}}gin{proposition}\label{proposition:blowups}
The map ${\bar\pi}:\widetilde{X}^5_{\lambda,\mu} \to \LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}$ is the blowup with center in $Z$.
Furthermore, $X^5_{\lambda,\mu}$ is smooth and the map $\pi:\widetilde{X}^5_{\lambda,\mu} \to X^5_{\lambda,\mu}$ is the blowup with center in a subvariety $F \subset X^5_{\lambda,\mu}$ isomorphic to the flag variety ${\bf F}l(1,2;3)$.
\end{proposition}
{\bar{e}}gin{proof}
For the first statement we apply the blowup Lemma~\ref{lemma:blowup}.
By Lemma~\ref{lemma:tcx-zero-loc} we know that $\widetilde{X}^5_{\lambda,\mu}$ is the zero locus of a global section of the vector bundle $\mathscr{U}_4^\perp(h)$ on ${\bf P}_{\LGr_\mu(3;{\bf B}W)_{\bar{\lambda}}}(\bw3\mathscr{U}_4)$
that corresponds to the morphism~\eqref{eq:map-hlam} from a rank 4 vector bundle to a rank 3 vector bundle.
Moreover, by Corollary~\ref{corollary:z-deg-loc} its degeneracy locus is equal to the subscheme $Z$ and thus has codimension 2 (and the higher degeneracy loci are empty).
Hence Lemma~\ref{lemma:blowup} applies and proves that $\widetilde{X}^5_{\lambda,\mu}$ is the blowup of $\LGr_\mu(3;{\bf B}W)_{\bar{\lambda}}$ with center in $Z$.
Since $\LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}$ is smooth by~\eqref{assumption:lgr-hyperplane} and $Z$ is smooth by Proposition~\ref{proposition:s-z}, it follows that~$\widetilde{X}^5_{\lambda,\mu}$ is smooth.
Moreover, $\dim\widetilde{X}^5_{\lambda,\mu} = 5$, and hence
$\dim X^5_{\lambda,\mu} \le 5$.
On the other hand, since~$X^5_{\lambda,\mu}$ is the zero locus of a section of a rank 7 vector bundle on a smooth 12-dimensional variety (see the proof of Lemma~\ref{lemma:lambda-mu-shift}),
its dimension is greater or equal than 5. Combining these two observations, we conclude that the dimension is 5 and $X^5_{\lambda,\mu}$ is Cohen--Macaulay.
Now let us show that $\pi$ is also a blowup.
By definition
{\bar{e}}gin{equation}\label{eq:tcx-fiber-product}
\widetilde{X}^5_{\lambda,\mu} = \LFl_\mu(3,4;W) \times_{\LGr_\mu(3,W)} X^5_{\lambda,\mu},
\end{equation}
and the map $\LFl_\mu(3,4;W) \to \LGr_\mu(3,W)$ is the blowup with center in $w_0(\LGr_\mu(2,{\bf B}W))$ by Lemma~\ref{lemma:sfl}, so it is enough to show that $X^5_{\lambda,\mu}$ intersects transversally with $w_0(\LGr_\mu(2,{\bf B}W))$.
Since $X^5_{\lambda,\mu}$ is the zero locus of the section $\lambda$ of $\mathscr{U}_3^\perp(1)$ on $\LGr_\mu(3,W)$, the intersection
{\bar{e}}gin{equation*}
F := X^5_{\lambda,\mu} \cap w_0(\LGr_\mu(2,{\bf B}W))
\end{equation*}
is the zero locus of the induced section of the vector bundle $w_0^*(\mathscr{U}_3^\perp(1))$ on $\LGr_\mu(2,{\bf B}W)$.
By~\eqref{eq:u3-bu2} we have an isomorphism $w_0^*(W/\mathscr{U}_3) \isom {\bf B}W/{\overline{\mathscr{U}}}_2$, and hence
{\bar{e}}gin{equation*}
w_0^*\mathscr{U}_3^\perp \isom {\overline{\mathscr{U}}}_2^\perp.
\end{equation*}
Therefore $F$ is the zero locus of the induced by~$\lambda$ section of ${\overline{\mathscr{U}}}_2^\perp(1)$ on $\LGr_\mu(2,{\bf B}W)$.
It is easy to see that this section is given by the 3-form $\lambda \,\lrcorner\, w_0 = {\bar{\lambda}}$.
By Lemma~\ref{lemma:gr26-3form-2form} we deduce that $F \isom {\bf F}l(1,2;A_1) \isom {\bf F}l(1,2;A_2)$.
In particular, $\dim(F) = 3$ and so
{\bar{e}}gin{equation*}
\codim_{\LGr_\mu(2,{\bf B}W)}(F) = 4 = \codim_{\LGr_\mu(3,W)}(X^5_{\lambda,\mu}),
\end{equation*}
hence the intersection is transversal.
It follows that the fiber product~\eqref{eq:tcx-fiber-product} is the blowup of $X^5_{\lambda,\mu}$ with center in $F$.
Finally, by base change and the proof of Lemma~\ref{lemma:sfl} it follows that
$F$ in $X^5_{\lambda,\mu}$ is a locally complete intersection.
Thus, smoothness of $F$ implies smoothness of $X^5_{\lambda,\mu}$ along $F$.
On the other hand, away of $F$ the morphism $\pi$ is an isomorphism, hence $X^5_{\lambda,\mu}$ is smooth everywhere.
\end{proof}
Theorem~\ref{theorem:main} is a combination of Proposition~\ref{proposition:blowups} and Proposition~\ref{proposition:s-z}.
\subsection{More details}\label{subsection:details}
We finish this section with some details of the geometry of the diagram~\eqref{diagram:fivefolds}.
First, note that there are three ${\bf P}^1$-bundles in the picture: $p:E \to F$, $\sigma:Z \to S$ and ${\bar{p}}:{\overline{E}} \to Z$.
Each of them is a projectivization of a rank 2 vector bundle
{\bar{e}}gin{equation*}
E \isom {\bf P}_F(\mathscr{V}_F),
\qquad
Z \isom {\bf P}_S(\mathscr{V}_S),
\qquad
{\overline{E}} \isom {\bf P}_Z(\mathscr{V}_Z)
\end{equation*}
respectively. These vector bundles can be described explicitly.
{\bar{e}}gin{lemma}\label{lemma:vector-bundles}
The bundles $\mathscr{V}_F$ and $\mathscr{V}_S$ can be represented as cohomology bundles of the monads
{\bar{e}}gin{equation}\label{eq:cvf}
0 \to \mathscr{O}(-h_1) \oplus \mathscr{O}(-h_2) \xrightarrow{\ \ \ } (A_1 \oplus A_2) \otimes \mathscr{O} \xrightarrow{\ \mu\ } \mathscr{O}(h_1) \oplus \mathscr{O}(h_2) \to 0
\end{equation}
on $F$, and
{\bar{e}}gin{equation}\label{eq:cvs-monad}
0 \to \mathscr{O}(-{\bar{h}}_1) \oplus \mathscr{O}(-{\bar{h}}_2) \xrightarrow{\ \ \ } (\bw2A_1 \oplus \bw2A_2) \otimes \mathscr{O} \xrightarrow{\ \mu^2\ } \mathscr{O}({\bar{h}}_1) \oplus \mathscr{O}({\bar{h}}_2) \to 0
\end{equation}
on $S$ respectively, and the bundle $\mathscr{V}_Z$ fits into exact sequences
{\bar{e}}gin{equation}\label{eq:cvz}
0 \to \mathscr{V}_Z \xrightarrow{\ \ \ \ } \bw3{\overline{\mathscr{U}}}_3 \oplus \bw2{\overline{\mathscr{U}}}_3 \xrightarrow{\ {\hat{\lambda}}\vert_Z\ } \mathscr{O}(-{\bar{h}}_1) \oplus \mathscr{O}(-{\bar{h}}_2) \to 0
\end{equation}
and
{\bar{e}}gin{equation}\label{eq:exact-seq-cvz}
0 \to \mathscr{O}(-2{\mathsf{v}}_Z - {\bar{h}}_1 - {\bar{h}}_2) \to \mathscr{V}_Z \to \mathscr{O}(-{\bar{h}}) \to \mathscr{O}_{C_2}(-2) \oplus \mathscr{O}_{C_1}(-2) \to 0.
\end{equation}
\end{lemma}
{\bar{e}}gin{proof}
By the proof of Lemma~\ref{lemma:sfl} the bundle $\mathscr{V}_F$ is the kernel of the map $\mu:W/\mathscr{U}_3 \to \mathscr{U}_3^\vee$ on~$F$.
Moreover, by Lemma~\ref{lemma:gr26-3form} we have
{\bar{e}}gin{equation*}
{\bar{e}}gin{aligned}
\mathscr{U}_3^\vee &\isom \mathscr{O} \oplus {\overline{\mathscr{U}}}_2^\vee &&\isom \mathscr{O} \oplus \mathscr{O}(h_1) \oplus \mathscr{O}(h_2),\\
W/\mathscr{U}_3 &\isom {\bf B}W/{\overline{\mathscr{U}}}_2 &&\isom ((A_1 \oplus A_2) \otimes \mathscr{O}) / (\mathscr{O}(-h_1) \oplus \mathscr{O}(-h_2)),
\end{aligned}
\end{equation*}
and the map $\mu$ factors through $\mathscr{O}(h_1) \oplus \mathscr{O}(h_1) \subset \mathscr{U}_3^\vee$.
Altogether, this identifies $\mathscr{V}_F$ with the cohomology bundle of the monad~\eqref{eq:cvf}.
Furthermore, monadic representation~\eqref{eq:cvs-monad} is just a reformulation of~\eqref{eq:cvs}.
By Lemma~\ref{lemma:blowup} and the proof of Proposition~\ref{proposition:blowups} the bundle $\mathscr{V}_Z$ is the kernel of the morphism ${\hat{\lambda}}:\bw2\mathscr{U}_4 \to \mathscr{U}_4^\perp$ on $Z$.
Furthermore, as it was explained in section~\ref{subsection:hlam-z}, the map ${\hat{\lambda}}$ factors through a surjection~\eqref{eq:hlam-z-rewritten}. This proves~\eqref{eq:cvz}.
Finally, consider the composition
$\mathscr{V}_Z \hookrightarrow \bw3{\overline{\mathscr{U}}}_3 \oplus \bw2{\overline{\mathscr{U}}}_3 \twoheadrightarrow \bw3{\overline{\mathscr{U}}}_3 \isom \mathscr{O}(-{\bar{h}})$
By~\eqref{eq:cvz} its kernel and cokernel are isomorphic to the kernel and the cokernel of the map ${\bar{\lambda}}: \bw2{\overline{\mathscr{U}}}_3 \to \mathscr{O}(-{\bar{h}}_1) \oplus \mathscr{O}(-{\bar{h}}_2)$.
So, applying~\eqref{eq:exact-seq-blam} we deduce~\eqref{eq:exact-seq-cvz}.
\end{proof}
Recall the notation for various divisorial classes introduced earlier:
{\bar{e}}gin{itemize}
\item
$h_1$ and $h_2$ are the hyperplane classes on ${\bf P}(A_1)$ and ${\bf P}(A_2)$;
\item
${\bar{h}}_1$ and ${\bar{h}}_2$ are the hyperplane classes on ${\bf P}(\bw2A_1)$ and ${\bf P}(\bw2A_2)$;
\item
${\mathsf{v}}_E \in {\bf P}ic(E)$ is the hyperplane class on $E = {\bf P}_F(\mathscr{V}_F)$;
\item
${\mathsf{v}}_Z \in {\bf P}ic(Z)$ is the hyperplane class on $Z = {\bf P}_S(\mathscr{V}_S)$;
\item
${\mathsf{v}}_{\overline{E}} \in {\bf P}ic({\overline{E}})$ is the hyperplane class on ${\overline{E}} = {\bf P}_Z(\mathscr{V}_Z)$;
\item
$h$ is the hyperplane class on $\Gr(3,W)$ and its restriction to $X^5_{\lambda,\mu}$;
\item
${\bar{h}}$ is the hyperplane class on $\LGr_\mu(2,{\bf B}W)_{\bar{\lambda}}$;
\item
$e$ and ${\bar{e}}$ are the classes of the exceptional divisors $E$ and ${\overline{E}}$ in $\widetilde{X}^5_{\lambda,\mu}$.
\end{itemize}
As usually, when we have a natural map between two varieties, we suppress the pullback notation for the pullbacks of divisorial classes.
{\bar{e}}gin{lemma}\label{lemma:picard}
We have the following relations in the Picard groups. First, we have
{\bar{e}}gin{equation}\label{eq:pic-relations}
{\bar{e}}gin{cases}
{\bar{h}} = h - e\\
{\bar{e}} = h - 2e
\end{cases}
\qquad\qquad
{\bar{e}}gin{cases}
h = 2{\bar{h}} -{\bar{e}}\\
e = {\bar{h}} - {\bar{e}}
\end{cases}
\qquad\qquad\text{in ${\bf P}ic(\widetilde{X}^5_{\lambda,\mu})$.}
\end{equation}
Furthermore, we have
{\bar{e}}gin{align*}
h &= h_1 + h_2 & \text{in ${\bf P}ic(F)$,} &&
{\mathsf{v}}_E &= -e & \text{in ${\bf P}ic(E)$,} \\
{\bar{h}} &= {\mathsf{v}}_Z + {\bar{h}}_1 + {\bar{h}}_2 & \text{in ${\bf P}ic(Z)$,} &&
{\mathsf{v}}_{\overline{E}} &= h & \text{in ${\bf P}ic({\overline{E}})$.}
\end{align*}
\end{lemma}
{\bar{e}}gin{proof}
To prove~\eqref{eq:pic-relations} it is enough to express the classes of the exceptional divisors $e$ and ${\bar{e}}$ in terms of $h$ and ${\bar{h}}$ using the blowup Lemma~\ref{lemma:blowup}.
To express $e$ we recall that by the argument of Lemma~\ref{lemma:sfl} the blowup $\pi$ is realized as a subvariety in ${\bf P}_{X^5_{\lambda,\mu}}(W/\mathscr{U}_3)$ corresponding to the morphism $\mu:W/\mathscr{U}_3 \to \mathscr{U}_3^\vee$.
Since $\det(W/\mathscr{U}_3) \isom \det(\mathscr{U}_3^\vee)$, it follows that $-e$ is the relative hyperplane class for this projectivization.
On the other hand, the relative hyperplane class corresponds to the line bundle $(\mathscr{U}_4/\mathscr{U}_3)^\vee$, that by~\eqref{eq:u4-u3} is isomorphic to $\mathscr{O}({\bar{h}} - h)$.
Thus $e = h - {\bar{h}}$.
Similarly, the blowup ${\bar\pi}$ is realized as a subvariety in ${\bf P}_{\LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}}(\bw3\mathscr{U}_4)$ corresponding to the morphism ${\hat{\lambda}}:\bw3\mathscr{U}_4 \to \mathscr{U}_4^\perp$.
Since $\det(\bw3\mathscr{U}_4) \isom \mathscr{O}(-3{\bar{h}})$ and $\det(\mathscr{U}_4^\perp) \isom \mathscr{O}(-{\bar{h}})$, it follows that $2{\bar{h}} - {\bar{e}}$ is the relative hyperplane class for this projectivization.
On the other hand, the relative hyperplane class corresponds to the line bundle $(\bw3\mathscr{U}_3)^\vee \isom \mathscr{O}(h)$, hence ${\bar{e}} = 2{\bar{h}} - h$.
Now all the relations in~\eqref{eq:pic-relations} easily follow.
The relation for $h\vert_F$ follows from~\eqref{eq:u3-bu2} and~\eqref{eq:gr26-3form-bu2} and the relation for ${\bar{h}}\vert_Z$ from~\eqref{eq:bcu3-1}.
Finally, since the bundle $\mathscr{V}_F$ is a subbundle in $W/\mathscr{U}_3$, the relative hyperplane class of $E = {\bf P}_F(\mathscr{V}_F)$ equals the restriction of the relative hyperplane class of ${\bf P}_{X^5_{\lambda,\mu}}(W/\mathscr{U}_3)$, that was just shown to be equal to $-e$.
This implies ${\mathsf{v}}_E = -e$.
Analogously, $\mathscr{V}_Z$ is a subbundle in $\bw3\mathscr{U}_4$, hence the relative hyperplane class of ${\overline{E}} = {\bf P}_Z(\mathscr{V}_Z)$ equals the restriction of the relative hyperplane class of ${\bf P}_{\LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}}(\bw3\mathscr{U}_4)$, that was shown to be equal to $h$.
This implies ${\mathsf{v}}_E = h$.
\end{proof}
Note that, the above relations show that the fibers of $Z$ over $S$ are lines on $\LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}$, so $Z$ is a scroll.
{\bar{e}}gin{lemma}\label{lemma:intersection}
We have the following relations in the Chow rings:
{\bar{e}}gin{align*}
{\mathsf{v}}_E^2 &= -h_1h_2 && \text{in ${\bf C}H^2(E)$},\\
{\mathsf{v}}_Z^2 &= -{\bar{h}}_1{\bar{h}}_2 && \text{in ${\bf C}H^2(Z)$},\\
{\mathsf{v}}_{\overline{E}}^2 &= (3{\mathsf{v}}_Z + 2{\bar{h}}_1 + 2{\bar{h}}_2){\mathsf{v}}_{\overline{E}} - 4{\mathsf{v}}_Z({\bar{h}}_1 + {\bar{h}}_2) && \text{in ${\bf C}H^2({\overline{E}})$}.
\end{align*}
\end{lemma}
{\bar{e}}gin{proof}
These are just Grothendieck relations.
So, we only have to compute the Chern classes of the bundles $\mathscr{V}_F$, $\mathscr{V}_S$, and $\mathscr{V}_Z$.
For this we use~\eqref{eq:cvf}, \eqref{eq:cvs-monad}, and~\eqref{eq:cvz} respectively.
\end{proof}
\section{Applications}\label{section:apps}
\subsection{Geometry of K\"uchle fourfolds}\label{subsection:4folds}
In this section we assume that the corresponding K\"uchle fivefold $X^5_{\lambda,\mu}$ satisfies the generality assumptions of section~\ref{subsection:genericity-assumptions}.
Recall that a K\"uchle fourfold is a hyperplane section of a K\"uchle fivefold.
Let $H_\nu \subset {\bf P}(\bw3W)$ be a hyperplane corresponding to a 3-form $\nu$ (defined up to a 3-form of type $(\lambda \,\lrcorner\, w) + (\mu \wedge f)$, where $w \in W$ and $f \in W^\vee$).
We denote the corresponding K\"uchle fourfold by
{\bar{e}}gin{equation*}
X^4_{\lambda,\mu}n := X^5_{\lambda,\mu} \cap H_\nu \subset X^5_{\lambda,\mu} \subset \Gr(3,W).
\end{equation*}
{\bar{e}}gin{theorem}
Assume the pair $({\lambda,\mu})$ satisfies the generality assumptions~\eqref{assumption:lambda-mu-general}, \eqref{assumption:pencil-regular}, \eqref{assumption:m-nonzero}, and~\eqref{assumption:lgr-hyperplane} and $\nu$ is a general $3$-form on $W$.
Then there is a diagram
{\bar{e}}gin{equation}\label{diagram:4folds}
\vcenter{\xymatrix{
&
D \ar[dl]_p \ar[r]^i &
\widetilde{X}^4_{\lambda,\mu}n \ar[dl]_\pi \ar[dr]^{\bar\pi} &
{\overline{D}} \ar[l]_{\bar{\imath}} \ar[dr]^{\bar{p}}
\\
\Sigma \ar[r] &
X^4_{\lambda,\mu}n &&
\overline{X}^4_{\lambda,\mu}n &
\Gamma \ar[l]
}}
\end{equation}
where
{\bar{e}}gin{itemize}
\item
$\Sigma = F \cap H_\nu$ is a sextic del Pezzo surface;
\item
$\overline{X}^4_{\lambda,\mu}n \subset \LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}$ is a quadratic section containing the scroll $Z$ and with singularities along a curve $\Gamma$;
\item
$\Gamma \subset Z \subset \overline{X}^4_{\lambda,\mu}n$ is a curve of genus $37$, a section of the map $\sigma:Z \to S$ over a smooth curve in the linear system $-4K_S$ on the sextic del Pezzo surface $S$;
\item
the map $\pi$ is the blowup with center in $\Sigma$;
\item
the map ${\bar\pi}$ is the blowup of the Weil divisor $Z$ on $\overline{X}^4_{\lambda,\mu}n$;
\item
$D \isom {\bf P}_\Sigma(\mathscr{V}_F\vert_\Sigma)$ is the exceptional divisor of $\pi$;
\item
${\overline{D}} \isom {\bf P}_\Gamma(\mathscr{V}_Z\vert_\Gamma)$ is the exceptional locus of ${\bar\pi}$;
\item
the maps $i$ and ${\bar{\imath}}$ are the embeddings of $D$ and ${\overline{D}}$.
\end{itemize}
\end{theorem}
{\bar{e}}gin{proof}
We define $\widetilde{X}^4_{\lambda,\mu}n = \widetilde{X}^5_{\lambda,\mu} \times_{X^5_{\lambda,\mu}} X^4_{\lambda,\mu}n$ to be the full preimage of $X^4_{\lambda,\mu}n$ under the blowup $\pi:\widetilde{X}^5_{\lambda,\mu} \to X^5_{\lambda,\mu}$.
Recall that the center of this blowup is the flag variety $F \subset {\bf P}(A_1) \times {\bf P}(A_2)$ given by the pairing between $A_1$ and $A_2$ induced by the 2-form $\mu$.
Clearly, the hyperplane section $F \cap H_\nu$ is given by one more $(1,1)$-divisor in ${\bf P}(A_1) \times {\bf P}(A_2)$, corresponding to the pairing given by the 2-form $\nu \,\lrcorner\, w_0$ on ${\bf B}W$.
So, assuming that the pencil of pairings between $A_1$ and $A_2$ given by the pencil of 2-forms $\langle \mu, \nu \,\lrcorner\, w_0 \rangle$ is regular (this is the first genericity assumption),
we conclude that $\Sigma := F \cap H_\nu$ is a smooth del Pezzo surface of degree 6.
Moreover, in this case the intersection $F \cap H_\nu$ is transversal, hence the map $\widetilde{X}^4_{\lambda,\mu}n \to X^4_{\lambda,\mu}n$ is the blowup with center in $\Sigma$.
We denote the exceptional divisor of this blowup by $D$.
Clearly, we have
{\bar{e}}gin{equation*}
D = \Sigma \times_F E \isom {\bf P}_\Sigma(\mathscr{V}_F\vert_\Sigma).
\end{equation*}
Abusing notation we denote by $\pi$, $p$, and $i$ the maps $\widetilde{X}^4_{\lambda,\mu}n \to X^4_{\lambda,\mu}n$, $D \to \Sigma$ and $D \to \widetilde{X}^4_{\lambda,\mu}n$ induced by the same named maps in~\eqref{diagram:fivefolds}.
Now recall that by Lemma~\ref{lemma:picard} in the Picard group of $\widetilde{X}^5_{\lambda,\mu}$ we have a relation $h = 2{\bar{h}}-{\bar{e}}$.
It means that the pullback to $\widetilde{X}^5_{\lambda,\mu}$ of the hyperplane $H_\nu$ corresponds to a quadratic section $\overline{X}^4_{\lambda,\mu}n$ of $\LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}$ containing $Z$.
Again by Lemma~\ref{lemma:picard} the restriction of $h$ to the divisor ${\overline{E}} = {\bf P}_Z(\mathscr{V}_Z)$ corresponds to a section of the vector bundle $\mathscr{V}_Z^\vee$.
Since $c_2(\mathscr{V}_Z^\vee) = 4{\mathsf{v}}_Z({\bar{h}}_1 + {\bar{h}}_2)$ by Lemma~\ref{lemma:intersection}, it follows that for general $\nu$ this section vanishes on a curve $\Gamma \subset Z$ with the class $4{\mathsf{v}}_Z({\bar{h}}_1 + {\bar{h}}_2)$ in ${\bf C}H^2(Z)$.
The projection of the curve to $S$ is a general curve ${\overline{\Gamma}}$ in the linear system $4({\bar{h}}_1 + {\bar{h}}_2) = -4K_S$, hence of genus~37, and $\Gamma$ gives a section of $\sigma$ over ${\overline{\Gamma}}$.
\end{proof}
The description of the K\"uchle fourfold $X^4_{\lambda,\mu}n$ provided by this Theorem is not as useful as the description of the fivefold in Theorem~\ref{theorem:main}, because of the singularities.
On the other hand, it looks as if the contraction ${\bar\pi}$ is a flopping contraction, so it is interesting to consider the flop $(\widetilde{X}^4_{\lambda,\mu}n)^+$ of $\widetilde{X}^4_{\lambda,\mu}n$ and continue the two-rays game for it.
\subsection{The Chow motive of K\"uchle fivefolds}\label{subsection:motive-5folds}
For a smooth projective variety $X$ we denote by $\Mot(X)$ its Chow motive, and by $\Lef$ the Lefschetz motive.
We say that a motive is {\sf of Lefschetz type} if it is a direct sum of powers of the Lefschetz motive.
Note that we consider motives with integer coefficients.
To compute the motive of $X^5_{\lambda,\mu}$ we use the birational transformation of Theorem~\ref{theorem:main}
and the following formula for the motive of the hyperplane section of the Lagrangian Grassmannian:
{\bar{e}}gin{equation}\label{eq:motive-lgrh}
\Mot(\LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}) = {\mathbf{1}} \oplus \Lef \oplus \Lef^2 \oplus \Lef^3 \oplus \Lef^4 \oplus \Lef^5.
\end{equation}
We prove this formula in section~\ref{section:hyperplane-sgr} by a geometric argument (see Corollary~\ref{corollary:motive-lgr-blam}),
and now we use it to prove the following result.
{\bar{e}}gin{theorem}\label{theorem:motive}
The Chow motive of $X^5_{\lambda,\mu}$ is of Lefschetz type:
{\bar{e}}gin{equation*}
\Mot(X^5_{\lambda,\mu}) = {\mathbf{1}} \oplus \Lef \oplus (\Lef^2)^{\oplus 4} \oplus (\Lef^3)^{\oplus 4} \oplus \Lef^4 \oplus \Lef^5.
\end{equation*}
\end{theorem}
{\bar{e}}gin{proof}
Note that the integral motive of $S$ is of Lefschetz type, since $S$ is isomorphic to a blowup of ${\bf P}^2$ in three points.
Explicitly, we have
{\bar{e}}gin{equation*}
\Mot(S) = {\mathbf{1}} \oplus (\Lef)^{\oplus 4} \oplus \Lef^2.
\end{equation*}
Applying the projective bundle formula to $Z \isom {\bf P}_S(\mathscr{V}_S)$, we deduce
{\bar{e}}gin{equation*}
\Mot(Z) = \Mot(S) \oplus \Mot(S) \otimes \Lef = {\mathbf{1}} \oplus (\Lef)^{\oplus 5} \oplus (\Lef^2)^{\oplus 5} \oplus \Lef^3.
\end{equation*}
Applying~\eqref{eq:motive-lgrh} and the blowup formula to the morphism ${\bar\pi}:\widetilde{X}^5_{\lambda,\mu} \to \LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}$, we deduce
{\bar{e}}gin{multline*}
\Mot(\widetilde{X}^5_{\lambda,\mu}) = \Mot(\LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}) \oplus \Mot(Z) \otimes \Lef =
{\mathbf{1}} \oplus (\Lef)^{\oplus 2} \oplus (\Lef^2)^{\oplus 6} \oplus (\Lef^3)^{\oplus 6} \oplus (\Lef^4)^{\oplus 2} \oplus \Lef^5.
\end{multline*}
On the other hand, the blowup formula for the morphism $\pi:\widetilde{X}^5_{\lambda,\mu} \to X^5_{\lambda,\mu}$ gives
{\bar{e}}gin{equation*}
\Mot(\widetilde{X}^5_{\lambda,\mu}) = \Mot(X^5_{\lambda,\mu}) \oplus \Mot(F) \otimes \Lef.
\end{equation*}
Since a direct summand of a motive of Lefschetz type is itself a motive of Lefschetz type, and $\Mot(F) = {\mathbf{1}} \oplus (\Lef)^{\oplus 2} \oplus (\Lef^2)^{\oplus 2} \oplus \Lef^3$,
we deduce the required formula for $\Mot(X^5_{\lambda,\mu})$.
\end{proof}
Note, that a similar result for the motive with rational coefficients follows immediately from the existence of an exceptional collection (see~\cite{marcolli2015exceptional}).
However, no analogue of this result for integral coefficients is known, so some geometric argument to prove this seems necessary
The following is an immediate consequence of Theorem~\ref{theorem:motive}.
{\bar{e}}gin{corollary}
The Hodge diamond of $X^5_{\lambda,\mu}$ is diagonal with
{\bar{e}}gin{equation*}
h^{0,0}(X^5_{\lambda,\mu}) = h^{1,1}(X^5_{\lambda,\mu}) = h^{4,4}(X^5_{\lambda,\mu}) = h^{5,5}(X^5_{\lambda,\mu}) = 1,
\qquad
h^{2,2}(X^5_{\lambda,\mu}) = h^{3,3}(X^5_{\lambda,\mu}) = 4,
\end{equation*}
and the Chow groups of the K\"uchle fivefold are free abelian groups
{\bar{e}}gin{equation*}
{\bf C}H^0(X^5_{\lambda,\mu}) \isom {\bf C}H^1(X^5_{\lambda,\mu}) \isom {\bf Z}, \
{\bf C}H^2(X^5_{\lambda,\mu}) \isom {\bf C}H^3(X^5_{\lambda,\mu}) \isom {\bf Z}^4,\
{\bf C}H^4(X^5_{\lambda,\mu}) \isom {\bf C}H^5(X^5_{\lambda,\mu}) \isom {\bf Z}.
\end{equation*}
\end{corollary}
One can use the description of Theorem~\ref{theorem:main} to find explicit generators of the Chow groups.
\section{A hyperplane section of the Lagrangian Grassmannian}\label{section:hyperplane-sgr}
The goal of this section is to describe the geometry of a smooth hyperplane section of the Lagrangian Grassmannian $\LGr(3,6)$.
In particular, we show that its motive is a sum of Lefschetz motives, and describe its Hilbert scheme of lines.
See~\cite{iliev2011fano} for another approach.
As usual, we denote by ${\bf B}W$ a vector space of dimension 6 and by $\mu \in \bw2{\bf B}W^\vee$ a symplectic form.
The linear span of the Lagrangian Grassmannian $\LGr_\mu(3,{\bf B}W)$ in ${\bf P}(\bw3{\bf B}W)$ is the subspace
{\bar{e}}gin{equation*}
\bww3\mu{\bf B}W := \Ker(\bw3{\bf B}W \xrightarrow{\ -\,\lrcorner\,\mu\ } {\bf B}W).
\end{equation*}
The space $\bw3{\bf B}W$ has a canonical direct sum decomposition
$\bw3{\bf B}W = \bww3\mu{\bf B}W \oplus {\bf B}W$,
where the second summand is embedded by the wedge product with the bivector $\mu^{-1} \in \bw2{\bf B}W$.
Consequently, there is a direct sum decomposition
{\bar{e}}gin{equation*}
\bw3{\bf B}W^\vee = \bww3\mu{\bf B}W^\vee \oplus {\bf B}W^\vee.
\end{equation*}
For a 3-form ${\bar{\lambda}} \in \bw3{\bf B}W^\vee$, the hyperplane section $\LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}$ depends only on the projection of ${\bar{\lambda}}$ to the summand $\bww3\mu{\bf B}W^\vee$, so, we may safely assume that
{\bar{e}}gin{equation}
{\bar{\lambda}} \in \bww3\mu{\bf B}W^\vee.
\end{equation}
We will keep this assumption from now on.
Consider a 7-dimensional vector space
{\bar{e}}gin{equation*}
W = \mathbf k \oplus {\bf B}W,
\end{equation*}
and pack the data of the 2-form $\mu$ and the 3-form ${\bar{\lambda}}$ on ${\bf B}W$ into a single 3-form $\xi$ on $W$ as follows.
Denote by $w_0 \in W$ the base vector of the summand $\mathbf k$ and by $w_0^\vee$ a linear function on~$W$ which is zero on ${\bf B}W$ and satisfies $w_0^\vee(w_0) = 1$, and set
{\bar{e}}gin{equation}\label{eq:xi-def}
\xi := {\bar{\lambda}} + w_0^\vee \wedge \mu \in \bw3W^\vee.
\end{equation}
The following observation is crucial.
{\bar{e}}gin{lemma}\label{lemma:xi-general}
If $\mu \in \bw2{\bf B}W^\vee$ is non-degenerate and ${\bar{\lambda}} \in \bww3\mu{\bf B}W^\vee$ is a $3$-form such that the hyperplane section $\LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}$ of the Lagrangian Grassmannian is smooth,
then the $3$-form $\xi \in \bw3W^\vee$ defined by~\eqref{eq:xi-def} is general.
\end{lemma}
{\bar{e}}gin{proof}
There is a well-known necessary and sufficient condition for a 3-form ${\bar{\lambda}} \in \bww3\mu{\bf B}W^\vee$ to give a smooth hyperplane section (see~\cite{landsberg-manivel} or~\cite{iliev-ranestad}).
It is in fact equivalent to existence of a {\em Lagrangian} (with respect to the 2-form $\mu$) direct sum decomposition
{\bar{e}}gin{equation*}
{\bf B}W = L_1 \oplus L_2
\end{equation*}
such that ${\bar{\lambda}} = {\bar{\lambda}}_1 + {\bar{\lambda}}_2$ with ${\bar{\lambda}}_i$ being generators of the subspace $\bw3L_1^\perp \subset \bww3\mu{\bf B}W^\vee$.
Choosing the bases $\{e_1,e_2,e_3\}$ and $\{e_4,e_5,e_6\}$ of the vector spaces $L_i$ appropriately, we can thus assume that
{\bar{e}}gin{equation*}
{\bar{\lambda}} = x_{123} + x_{456}
\qquad\text{and}\qquad
\mu = x_{16} + x_{25} + x_{34}.
\end{equation*}
Setting $e_0 = w_0$, so that $w_0^\vee = x_0$, we then have
{\bar{e}}gin{equation*}
\xi = x_{123} + x_{456} + x_{016} + x_{025} + x_{034},
\end{equation*}
hence $\xi$ is general by Lemma~\ref{lemma:4form-general}(v).
\end{proof}
Recall that if a 3-form $\xi \in \bw3W^\vee$ is general, then its stabilizer in ${\bf P}GL(W)$ is the simple algebraic group ${\mathbb{G}_2}$,
and one can define its homogeneous spaces
{\bar{e}}gin{equation*}
\mathbf{Q}_\xi \subset {\bf P}(W)
\qquad\text{and}\qquad
\Gr_\xi(2,W) \subset \Gr(2,W).
\end{equation*}
The quadric $\mathbf{Q}_\xi$ parameterizes all vectors $w \in W$ such that the rank of the 2-form $\xi \,\lrcorner\, w$ is less than 6,
and $\Gr_\xi(2,W)$ parameterizes all 2-subspaces $U_2 \subset W$ that are annihilated by $\xi$.
{\bar{e}}gin{remark}\label{remark:w0-notin-gr}
If the 3-form $\xi$ is defined by~\eqref{eq:xi-def} then $w_0 \not\in \mathbf{Q}_\xi$. Indeed, this is immediate since
{\bar{e}}gin{equation}\label{eq:xi-w0}
\xi \,\lrcorner\, w_0 = \mu
\end{equation}
has rank 6.
In particular, by Remark~\ref{remark:gr27-g2} the vector $w_0$ is not contained in any $U_2 \in \Gr_\xi(2,W)$.
\end{remark}
The composition ${\overline{\mathscr{U}}}_2 \hookrightarrow {\bf B}W \otimes \mathscr{O} \xrightarrow{\ \mu\ } {\bf B}W^\vee \otimes \mathscr{O} \twoheadrightarrow {\overline{\mathscr{U}}}_2^\vee$ is zero on $\LGr_\mu(2,{\bf B}W)$,
hence the composition of the first two arrows factors through the subbundle ${\overline{\mathscr{U}}}_2^\perp \subset {\bf B}W^\vee \otimes \mathscr{O}$.
This allows to consider ${\overline{\mathscr{U}}}_2$ as a subbundle in ${\overline{\mathscr{U}}}_2^\perp$, and to define the quotient bundle ${\overline{\mathscr{U}}}_2^\perp/{\overline{\mathscr{U}}}_2$.
Analogously, the composition $\bw2{\overline{\mathscr{U}}}_2 \hookrightarrow \bw2{\bf B}W \otimes \mathscr{O} \xrightarrow{\ {\bar{\lambda}}\ } {\bf B}W^\vee \otimes \mathscr{O} \twoheadrightarrow {\overline{\mathscr{U}}}_2^\vee$ is zero on the hyperplane section $\LGr_\mu(2,{\bf B}W)_{\bar{\lambda}}$,
hence the composition of the first two arrows again factors through the subbundle ${\overline{\mathscr{U}}}_2^\perp \subset {\bf B}W^\vee \otimes \mathscr{O}$.
Therefore, the composition
{\bar{e}}gin{equation*}
\mathscr{O}(-1) \isom \bw2{\overline{\mathscr{U}}}_2 \xrightarrow{\ {\bar{\lambda}}\ } {\overline{\mathscr{U}}}_2^\perp \twoheadrightarrow {\overline{\mathscr{U}}}_2^\perp/{\overline{\mathscr{U}}}_2
\end{equation*}
defines a global section of the vector bundle $({\overline{\mathscr{U}}}_2^\perp/{\overline{\mathscr{U}}}_2)(1)$ on $\LGr_\mu(2,{\bf B}W)$.
We denote by
{\bar{e}}gin{equation*}
D_{{\bar{\lambda}},\mu} \subset \LGr_\mu(2,{\bf B}W)
\end{equation*}
its zero locus.
{\bar{e}}gin{proposition}\label{proposition:dlm-grxi}
Let $W = \mathbf k \oplus {\bf B}W$ and let $\xi$ be the $3$-form on $W$ defined by~\eqref{eq:xi-def}.
If $\xi$ is general then
{\bar{e}}gin{equation*}
D_{{\bar{\lambda}},\mu} \isom \Gr_\xi(2,W)
\end{equation*}
is the ${\mathbb{G}_2}$-adjoint variety corresponding to the $3$-form $\xi$. In particular, $D_{{\bar{\lambda}},\mu}$ is a smooth fivefold.
\end{proposition}
{\bar{e}}gin{proof}
Consider the embedding $w_0:\LGr_\mu(2,{\bf B}W) \to \Gr(3,W)$ defined by taking a subspace ${\bf B}U_2 \subset {\bf B}W$ to the subspace $U_3 = \mathbf k w_0 \oplus {\bf B}U_2 \subset W$.
It follows from~\eqref{eq:xi-w0} that the image lies in the hyperplane section of $\Gr(3,W)$ given by the 3-form $\xi$
and in fact is identified with the subvariety of this hyperplane section $\Gr(3,W)_\xi$ parameterizing all 3-subspaces containing $w_0$.
Define on $\Gr(3,W)_\xi$ a morphism
{\bar{e}}gin{equation*}
{\hat{\xi}}:\bw2\mathscr{U}_3 \to \mathscr{U}_3^\perp,
\end{equation*}
analogously to the definition of the morphism ${\hat{\lambda}}$ in~\eqref{eq:map-hlam}.
We restrict this morphism to $\LGr_\mu(2,{\bf B}W)$ and consider its degeneracy loci $D_k({\hat{\xi}}) \subset \LGr_\mu(2,{\bf B}W)$.
We will show that $D_2({\hat{\xi}}) = \varnothing$ and $D_1({\hat{\xi}}) \isom \Gr_\xi(2,W)$.
Indeed, assume first that $U_3 \subset W$ is a $\xi$-isotropic subspace containing $w_0$ such that the corank of the map ${\hat{\xi}}$ at~$U_3$ is~1.
This means that the map ${\hat{\xi}}:\bw2U_3 \to U_3^\perp$ is not injective.
Then its kernel is generated by a bivector in $U_3$, which is necessarily decomposable, and thus corresponds to a 2-subspace $U_2 \subset U_3$.
Then the condition that $\bw2U_2 \subset \bw2U_3$ is in the kernel of ${\hat{\xi}}$ means that $\xi$ annihilates~$U_2$.
Therefore, $U_2$ is a point of the ${\mathbb{G}_2}$-adjoint variety $\Gr_\xi(2,W)$.
Since $U_3$ also contains $w_0$ and $w_0 \not\in U_2$ by Remark~\ref{remark:w0-notin-gr}, it follows that
{\bar{e}}gin{equation*}
U_3 = \mathbf k w_0 \oplus U_2.
\end{equation*}
Thus $D_1({\hat{\xi}})$ parameterizes all subspaces representable as the sum of the line generated by $w_0$ and a 2-subspace $U_2$ annihilated by $\xi$.
In particular, $D_1({\hat{\xi}})$ is equal to the the image of the regular map $\Gr_\xi(2,W) \to \Gr(2,{\bf B}W)$ induced by the linear projection $\pr:W \to {\bf B}W$ along $w_0$.
Now assume that the corank of ${\hat{\xi}}$ at $U_2$ is 2 or more.
Then the subspace $U_3$ contains a pencil of 2-dimensional subspaces $U_2 \subset U_3$ annihilated by $\xi$.
Clearly, at least one of subspaces in the pencil contains $w_0$, which contradicts Remark~\ref{remark:w0-notin-gr}.
This shows that $D_2({\hat{\xi}}) = \varnothing$. Moreover, this also shows that for $U_3 \in D_1({\hat{\xi}})$ the subspace $U_2 \subset U_3$ annihilated by $\xi$ is unique,
hence the map $\pr:\Gr_\xi(2,W) \to D_1({\hat{\xi}}) \subset \LGr_\mu(2,{\bf B}W)$ is an isomorphism.
Now, finally, let us show that $D_1({\hat{\xi}}) = D_{{\bar{\lambda}},\mu}$. Indeed, assume ${\bf B}U_2 \subset {\bf B}W$ is a 2-subspace, such that ${\hat{\xi}}$ is degenerate at $U_3 = \mathbf k w_0 \oplus {\bf B}U_2$.
If $\{u',u''\}$ is a basis of ${\bf B}U_2$ then $\{w_0\wedge u', w_0\wedge u'', u' \wedge u''\}$ is a basis of $\bw2U_3$ and by~\eqref{eq:xi-def} we have
{\bar{e}}gin{equation*}
{\bar{e}}gin{aligned}
\xi \,\lrcorner\, (w_0\wedge u') &= \mu \,\lrcorner\, u',\\
\xi \,\lrcorner\, (w_0\wedge u'') & = \mu \,\lrcorner\, u'',\\
\xi \,\lrcorner\, (u' \wedge u'') &= {\bar{\lambda}} \,\lrcorner\, (u' \wedge u'') + \mu(u',u'')w_0^\vee.
\end{aligned}
\end{equation*}
Since the 2-form $\mu$ is nondegenerate, the two linear functions $\mu\,\lrcorner\, u'$ and $\mu \,\lrcorner\, u''$ are linearly independent.
Moreover, they vanish on $w_0$.
Hence the degeneracy condition means that $\mu(u',u'') = 0$ and the linear function ${\bar{\lambda}} \,\lrcorner\, (u' \wedge u'')$ is a linear combination of $\mu \,\lrcorner\, u'$ and $\mu \,\lrcorner\, u''$.
The first means that ${\bf B}U_2$ is $\mu$-isotropic, and the second means that ${\bar{\lambda}}$, considered as a section of $({\overline{\mathscr{U}}}_2^\perp/{\overline{\mathscr{U}}}_2)(1)$, vanishes at ${\bf B}U_2$.
Thus $D_1(\xi) \subset D_{{\bar{\lambda}},\mu}$.
The converse statement can be proved by the same computation.
\end{proof}
Now we are ready to give an alternative geometric description of $\LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}$.
Consider the relative Grassmannian
{\bar{e}}gin{equation*}
{\bf P}_{\LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}}(\bw2{\overline{\mathscr{U}}}_3) \isom \Gr_{\LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}}(2,{\overline{\mathscr{U}}}_3) \isom \LFl_\mu(2,3;{\bf B}W) \times_{\LGr_\mu(3,{\bf B}W)} \LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}.
\end{equation*}
Being embedded into the symplectic flag variety, it comes with the tautological flag of vector bundles ${\overline{\mathscr{U}}}_2 \hookrightarrow {\overline{\mathscr{U}}}_3 \hookrightarrow {\bf B}W \otimes \mathscr{O}$.
We denote by
{\bar{e}}gin{equation}\label{diagram:gr-lgr}
\vcenter{\xymatrix{
& \Gr_{\LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}}(2,{\overline{\mathscr{U}}}_3) \ar[dl]_{\rho_2} \ar[dr]^{\rho_3} \\
\LGr_\mu(2,{\bf B}W) && \LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}
}}
\end{equation}
the natural projections, induced by the projections of the symplectic flag variety.
{\bar{e}}gin{theorem}\label{theorem:lgrh}
The map $\rho_3$ is a ${\bf P}^2$-bundle.
If the hyperplane section $\LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}$ of $\LGr_\mu(3,{\bf B}W)$ is smooth, then
the map $\rho_2:\Gr_{\LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}}(2,{\overline{\mathscr{U}}}_3) \to \LGr_\mu(2,{\bf B}W)$ is the blowup with center in the subvariety $D_{{\bar{\lambda}}.\mu} \isom \Gr_\xi(2,W)$.
\end{theorem}
{\bar{e}}gin{proof}
The first part is evident. For the second part,
we use again the blowup Lemma~\ref{lemma:blowup}.
We note that the symplectic flag variety $\LFl_\mu(2,3;{\bf B}W)$ can be represented as the projectivization of the (self-dual) vector bundle ${\overline{\mathscr{U}}}_2^\perp/{\overline{\mathscr{U}}}_2$ on $\LGr_\mu(2,{\bf B}W)$,
and that the subvariety $\Gr_{\LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}}(2,{\overline{\mathscr{U}}}_3) \subset \LFl_\mu(2,3;{\bf B}W)$ is the zero locus of the global section of the line bundle $\mathscr{O}({\bar{h}}) \isom \rho_3^*\mathscr{O}(1)$,
which is a hyperplane class for the projection $\LFl_\mu(2,3;{\bf B}W) \to \LGr_\mu(2,{\bf B}W)$, corresponding to the twisted bundle $({\overline{\mathscr{U}}}_2^\perp/{\overline{\mathscr{U}}}_2)(1)$.
Therefore, the 3-form ${\bar{\lambda}}$ gives a global section of this bundle, and it is easy to see that this section is the same as the section, defining the subvariety $D_{{\bar{\lambda}},\mu} \subset \LGr_\mu(2,{\bf B}W)$.
If the hyperplane section $\LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}$ is smooth, then by Lemma~\ref{lemma:xi-general} the 3-form $\xi$ defined by~\eqref{eq:xi-def} is general,
and hence by Proposition~\ref{proposition:dlm-grxi} the zero locus $D_{{\bar{\lambda}},\mu}$ of ${\bar{\lambda}}$ on $\LGr_\mu(2,{\bf B}W)$ is isomorphic to the ${\mathbb{G}_2}$-adjoint variety $\Gr_\xi(2,W)$.
Its codimension in $\LGr_\mu(2,{\bf B}W)$ is $7-5 = 2$, hence Lemma~\ref{lemma:blowup} applies and shows that the map $\rho_2$ is the blowup with center in $\Gr_\xi(2,W)$.
\end{proof}
This Theorem has the following nice consequences.
{\bar{e}}gin{corollary}\label{corollary:motive-lgr-blam}
The Chow motive of $\LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}$ is of Lefschetz type:
{\bar{e}}gin{equation*}
\Mot(\LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}) = {\mathbf{1}} \oplus \Lef \oplus \Lef^2 \oplus \Lef^3 \oplus \Lef^4 \oplus \Lef^5.
\end{equation*}
\end{corollary}
{\bar{e}}gin{proof}
Note that both $\LGr_\mu(2,{\bf B}W)$ and $\Gr_\xi(2,W)$ are homogeneous spaces (for the groups $\Sp(W)$ and ${\mathbb{G}_2}$ respectively),
hence their Chow motives are of Lefschetz type:
{\bar{e}}gin{align*}
\Mot(\LGr_\mu(2,{\bf B}W)) &= {\mathbf{1}} \oplus \Lef \oplus (\Lef^2)^{\oplus 2} \oplus (\Lef^3)^{\oplus 2} \oplus (\Lef^4)^{\oplus 2} \oplus (\Lef^5)^{\oplus 2} \oplus \Lef^6 \oplus \Lef^7,\\
\Mot(\Gr_\xi(2,W)) &= {\mathbf{1}} \oplus \Lef \oplus \Lef^2 \oplus \Lef^3 \oplus \Lef^4 \oplus \Lef^5.
\end{align*}
By the blowup formula, we have
{\bar{e}}gin{equation*}
\Mot(\Gr_{\LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}}(2,{\overline{\mathscr{U}}}_3)) = {\mathbf{1}} \oplus (\Lef)^{\oplus 2} \oplus (\Lef^2)^{\oplus 3} \oplus (\Lef^3)^{\oplus 3} \oplus (\Lef^4)^{\oplus 3} \oplus (\Lef^5)^{\oplus 3} \oplus (\Lef^6)^{\oplus 2} \oplus \Lef^7.
\end{equation*}
On the other hand, by the projective bundle formula we have
{\bar{e}}gin{equation*}
\Mot(\Gr_{\LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}}(2,{\overline{\mathscr{U}}}_3)) = \Mot(\LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}) \otimes ({\mathbf{1}} \oplus \Lef \oplus \Lef^2).
\end{equation*}
In particular, $\Mot(\Gr_{\LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}}(2,{\overline{\mathscr{U}}}_3))$ is a direct summand of a motive of Lefschetz type, hence is itself a motive of Lefschetz type.
The explicit form of its decomposition easily follows.
\end{proof}
{\bar{e}}gin{remark}
Of course, it follows from Corollary~\ref{corollary:motive-lgr-blam} that the Chow groups of $\LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}$ are free abelian of rank 1:
{\bar{e}}gin{equation*}
{\bf C}H^p(\LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}) \isom {\bf Z},
\qquad
0 \le p \le 5.
\end{equation*}
Moreover, one can find explicit generators of those groups --- these are the fundamental class, the class of a hyperplane section, $c_2({\overline{\mathscr{U}}}_3)$, $c_3({\overline{\mathscr{U}}}_3)$, the class of a line, and the class of a point.
An interesting feature of this example is that Chern classes of the full exceptional collection do not generate the Chow ring (the class of a line is not generated).
\end{remark}
Another immediate application of the geometric construction of Theorem~\ref{theorem:lgrh} is the following description of the Hilbert scheme of lines.
{\bar{e}}gin{corollary}
The Hilbert scheme of lines on $\LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}$ is isomorphic to $\Gr_\xi(2,W)$.
Moreover, the exceptional divisor of the blowup $\rho_2$ in~\eqref{diagram:gr-lgr} is the universal family of lines.
\end{corollary}
{\bar{e}}gin{proof}
The Hilbert scheme of lines on the Lagrangian Grassmannian $\LGr_\mu(3,{\bf B}W)$ is well-known to be isomorphic to the isotropic Grassmannian $\LGr_\mu(2,{\bf B}W)$
with the universal family of lines provided by the isotropic flag variety
{\bar{e}}gin{equation*}
\LFl_\mu(2,3;{\bf B}W) \isom {\bf P}_{\LGr_\mu(2,{\bf B}W)}({\overline{\mathscr{U}}}_2^\perp/{\overline{\mathscr{U}}}_2).
\end{equation*}
It follows that the Hilbert scheme of lines on the hyperplane section $\LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}$ given by a 3-form ${\bar{\lambda}}$ is the zero locus of the section of $({\overline{\mathscr{U}}}_2^\perp/{\overline{\mathscr{U}}}_2)(1)$ given by ${\bar{\lambda}}$,
i.e.\ coincides with the subscheme $D_{{\bar{\lambda}},\mu} \subset \LGr_\mu(2,{\bf B}W)$.
So, Proposition~\ref{proposition:dlm-grxi} applies.
\end{proof}
{\bar{e}}gin{remark}
One can modify the construction of this section to get a birational description of a hyperplane section $\LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}$ of the Lagrangian Grassmannian.
Consider a linear function $f \in {\bf B}W^\vee$ and the zero locus $M$ of $f$, considered
as a global section of ${\overline{\mathscr{U}}}_2^\vee$ on $\Gr_{\LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}}(2,{\overline{\mathscr{U}}}_3)$.
Then it is easy to see that the projection $\rho_3$ maps $M$ to $\LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}$ birationally (and in fact identifies $M$ with the blowup of $\LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}$ with center in a quadric surface),
and the projection $\rho_2$ maps $M$ birationally onto a hyperplane section $\Gr_\mu(2,5)$ of $\Gr(2,5)$ (and in fact identifies $M$ with the blowup of $\Gr_\mu(2,5)$ with center in the flag variety ${\bf F}l(1,2;3)$).
This construction gives an alternative way to describe the motive of $\LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}$ from the relation
$\Mot(\LGr_\mu(3,{\bf B}W)_{\bar{\lambda}}) \oplus \Mot(Q^2) \otimes (\Lef \oplus \Lef^2) = \Mot(\Gr_\mu(2,5)) \oplus \Mot({\bf F}l(1,2;3)) \otimes \Lef$.
\end{remark}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
\providecommand{\MRhref}[2]{
\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
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\providecommand{\href}[2]{#2}
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\end{thebibliography}
\end{document}
|
\begin{document}
\title{A matched alternating direction implicit (ADI) method for solving
the heat equation with interfaces}
\author{Shan Zhao$^{1,2,}$\footnote{
Corresponding author. Tel: 1-205-3485155, Fax: 1-205-3487067,
Email: [email protected]}\\
$^1$ Department of Mathematics,
University of Alabama, Tuscaloosa, AL 35487, USA. \\
$^2$ Beijing Computational Science Research Center,
Beijing 100084, PR China.
}
\date{\today}
\maketitle
\begin{abstract}
A novel Douglas alternating direction implicit (ADI) method is proposed in this
work to solve a two-dimensional (2D) heat equation with interfaces.
The ADI scheme is a powerful finite difference method for solving parabolic equations,
due to its unconditional stability and high efficiency.
However, it suffers from a serious accuracy reduction in space for interface problems
with different materials and nonsmooth solutions.
If the jumps in a function and its derivatives are known across the interface,
rigorous ADI schemes have been successfully constructed in the literature
based on the immersed interface
method (IIM) so that the spatial accuracy can be restored.
Nevertheless, the development of accurate and stable ADI methods
for general parabolic interface problems with
physical interface conditions that describe jumps of a function and its flux,
remains unsolved.
To overcome this difficulty, a novel tensor product decomposition is proposed
in this paper to decouple 2D jump conditions into essentially one-dimensional
(1D) ones. These 1D conditions can then be incorporated into the ADI
central difference discretization, using the matched interface and boundary
(MIB) technique.
Fast algebraic solvers for perturbed tridiagonal
systems are developed to maintain the computational efficiency.
Stability analysis is conducted through eigenvalue spectrum analysis, which
numerically demonstrates the unconditional stability of the proposed ADI method.
The matched ADI scheme achieves the first order of accuracy in time
and second order of accuracy in space
in all tested parabolic interface problems with
complex geometries and spatial-temporal dependent jump conditions.
\noindent {\bf Keyword:}
Heat equation; Parabolic interface problem;
Jump conditions;
Alternating direction implicit (ADI);
Matched interface and boundary (MIB).
\noindent {\bf MSC:}
65M06,
65M12,
35K05.
\end{abstract}
\section{Introduction}
In this paper, we propose a new alternating direction implicit (ADI) method for solving
two-dimensional (2D) parabolic interface problems
\begin{equation}\label{heat}
\frac{\partial u}{\partial t} = \nabla \cdot ( \alpha \nabla u) + f, \quad \mbox{in}~~
\Omega=\Omega^- \cup \Omega^+
\end{equation}
where $u(x,y,t)$ is a function of interests, e.g. the temperature,
$\alpha$ is the diffusion coefficient and $f(x,y,t)$ is a source.
For simplicity, the domain
$\Omega$ is assumed to be a rectangle one, with
proper boundary conditions prescribed for $u$ on $\partial \Omega$.
Across a material interface $\Gamma$ separating two media $\Omega^-$ and
$\Omega^+$, the diffusion coefficient $\alpha$ is discontinuous, while the source term
$f(x,y,t)$ may be even singular.
Physically, the solution $u$ on both sides of $\Gamma$
is related analytically via jump conditions
\begin{equation}\label{jump}
[u] = u^+ - u^- = \phi(s,t), \quad
[\alpha u_n] = \alpha^+ \frac{\partial u^+}{\partial n}
- \alpha^- \frac{\partial u^-}{\partial n} = \psi(s,t),
\end{equation}
where $s$ is the arc-length parametrization
of the interface $\Gamma$, and $n$ the unit normal direction.
The superscript, $-$ or $+$, denotes the limiting value of a function from one side or
the other of the interface.
We note that (\ref{jump}) takes a quite general form, while for many applications,
we have simply $\phi=\psi=0$.
The parabolic interface problem governed by (\ref{heat}) and (\ref{jump})
appears in many physical and engineering applications,
such as the continuous casting in the metallurgical industry,
the freezing process of perishable foodstuffs in the food engineering,
and the magnetic fluid hyperthermia treatment of cancer.
The analysis of conductive heat transfer process over composite media is
indispensable in these applications.
Since the physical solution is non-smooth
or even discontinuous across the interface, the standard numerical methods
often perform poorly for the parabolic interface problem.
To restore the accuracy near the interface, the jump conditions (\ref{jump})
have to be incorporated into the numerical discretization in certain manner.
For finite element methods
\cite{ChenZou,Sinha05,Sinha09,Attanayake}
and finite volume methods \cite{Wang10},
many rigorous interface treatments have been proposed to deliver high accuracy
in solving parabolic interface problems.
As one of the most successful finite difference methods for solving material
interface problems, the immersed interface method (IIM) was originally introduced
by LeVeque and Li \cite{LevLi} for solving elliptic equations with discontinuous
coefficients and singular sources.
By rigorously imposing jump conditions via local Taylor expansions,
the IIM achieves the second order of accuracy for complex elliptic interfaces.
The development of the IIM for solving parabolic equations has been considered in
\cite{Kandilarov04,Kandilarov07,Bouchon10a,Bouchon10b,AdamsLi}.
In \cite{Kandilarov04,Kandilarov07}, the authors constructed the IIM together with the
implicit Euler time integration for parabolic equations with singular own sources.
The IIM has been applied in \cite{Bouchon10a,Bouchon10b} for solving the Poisson
equation over moving irregular domains or
over fixed domains with the Neumann boundary condition.
The maximum principle preserving IIM has been proposed in \cite{AdamsLi} by
Adams and Li for solving the convection-diffusion equation with general jump conditions.
In this scheme, the advection term is discretized explicitly while the
diffusion term is treated implicitly. A fast multigrid method is implemented to
efficiently solve the linear system of equations for the implicit time stepping \cite{AdamsLi}.
As one of the most successful finite difference methods for solving parabolic
equations, the classical ADI method \cite{DougPeace,Douglas,PeaceRach} can
be written as some perturbations of multidimensional implicit methods, such as the
Crank-Nicolson and backward Euler. In general, the ADI method is unconditionally stable
for parabolic problems without interfaces so that a large time increment is admissible,
which in turn can lead to a faster simulation.
The major attraction of the ADI method, as compared with other implicit methods,
is that it reduces a multidimensional problem
to sets of independent one-dimensional (1D) problems of tridiagonal structures,
and such matrices can be efficiently solved using the Thomas algorithm \cite{ADIbook}.
Moreover, the essentially 1D feature of the ADI computations allows
a tremendously efficient parallelization,
including on modern Graphics Processing Units (GPUs) \cite{Tay,Wei13}.
Neglecting jump conditions, the classical ADI method has been applied to
solve interface problems with sharp or smeared interfaces, see for example
\cite{Bates09,Chen11,Geng13,Zhao14,Tian14}.
However, for sharp interface problems, the central finite difference approximation
degrades to the first order of accuracy in space
and the precious unconditional stability may be lost \cite{Geng13,Zhao14}.
Therefore, it is highly desired to construct novel ADI methods
for solving parabolic interface problems
without compromising stability and spatial accuracy.
The first rigorous interface treatment in the ADI framework was due to
Li and Mayo \cite{LiMayo}, in which a homogeneous 2D heat equation
with a constant diffusion coefficient and a singular source is solved.
For such an interface setting,
the jump conditions (\ref{jump}) take a simpler form with $\alpha=1$. Thus,
higher order jump conditions can be simply derived \cite{LiMayo}.
This allows the construction of a second order accurate IIM-ADI method by
adding some correction terms into the classical ADI scheme
for irregular points near the interface.
Considering the same jump conditions, Liu and Zheng have extended
the IIM-ADI method to solve a 2D homogeneous
convection-diffusion equation \cite{Liu13}
and a three-dimensional (3D) homogeneous heat equation \cite{Liu14}.
However, grand difficulties are encountered when the IIM-ADI method
\cite{LiMayo,Liu13,Liu14} is generalized in \cite{LiShen}
to solve a 2D heat equation with nonhomogeneous media,
i.e., $\alpha$ being a piecewise constant.
By still assuming simple jump conditions with prescribed function and
derivative jumps, i.e., given $[u]$ and $[u_n]$ values,
the second order jump condition for $[u_{nn}]$ cannot be simply derived
from the governing equation now.
A rather complicated second order jump condition is considered in \cite{LiShen}
so that the SOR iterative method has to be used to solve a 2D linear
system in each step of the Crank-Nicolson time integration.
We note that, on the other hand, if the general jump conditions like (\ref{jump})
are used, the second order jump conditions could be naturally derived.
Nevertheless, the construction of the IIM-ADI scheme through introducing
correction terms remains a challenge for general jump conditions.
Therefore, the development of accurate and stable ADI methods for
the parabolic interface problem (\ref{heat}) and (\ref{jump})
is essentially an open problem.
The objective of this paper is to propose a novel matched ADI method to
overcome the aforementioned difficulties for solving
general parabolic interface problems.
The proposed matched ADI method is formulated based on our previous
interface scheme, the matched interface and boundary (MIB) method,
originally developed for solving elliptic and hyperbolic interface problems
\cite{Zhao04,Zhou06}.
One distinction between the MIB and IIM is that the MIB interface modeling
just needs zeroth and first order jump conditions, i.e., (\ref{jump}) so that the difficulty
associated with the second order jump conditions of the IIM is simply
bypassed in the MIB approach.
However, the MIB scheme has never been applied to a parabolic interface
problem before.
Moreover, the existing MIB scheme cannot be directly utilized in the ADI
formulation, because the 2D MIB interface treatment will couple $x$ and $y$
directions simultaneously.
The most significant contribution of this work is the introduction
of a novel tensor-product
decomposition of jump conditions (\ref{jump}), which decouples 2D
jump conditions into 1D ones, in the same spirit of the ADI method.
Then, 1D MIB interface treatments will be developed in space to secure
a second order of accuracy.
Fast algebraic solvers based on the Thomas algorithm will be developed
to solve 1D linear systems efficiently.
The stability proof of the matched ADI algorithm is highly non-trivial,
because the finite difference weights of the MIB discretization
depend on the interface geometry in an unpredictable manner.
In the present study, through calculating the spectral radius,
the proposed matched ADI method is numerically verified to be unconditionally stable.
The rest of this paper is organized as follows.
Section \ref{Sec:theory} is devoted to the theory and algorithm
of the proposed matched ADI method.
Numerical tests are carried out to validate the proposed method by
considering various particular forms for the jump conditions (\ref{jump}).
Finally, a conclusion ends this paper.
\section{Theory and Algorithm}\label{Sec:theory}
Consider an interface problem, in which $\Omega^-$ is interior to $\Omega^+$.
Define a uniform mesh partition of the computational domain $\Omega$.
Without the loss of generality, we assume that the grid spacing $h$ in
both $x$ and $y$ directions is the same and
one grid line cuts the interface $\Gamma$ at most twice.
Denote the time increment to be $\Delta t$ and take
$N_x$ and $N_y$ as the number of grid points in each direction.
To facilitate the following discussions, we adopt a
notation at node $(x_i,y_j,t_k)$:
$u^k_{i,j}=u(x_i,y_j,t_k)$.
\subsection{Temporal discretization}
We first rewrite the heat equation (\ref{heat2}) by dividing $\alpha$ throughout
\begin{equation}\label{heat2}
\frac{1}{\alpha} \frac{\partial u}{ \partial t}
=\frac{\partial^2 u }{\partial x^2} + \frac{\partial^2 u }{\partial y^2}
+\frac{f}{\alpha}, \quad \mbox{in}~~ \Omega^- ~~\mbox{or}~~ \Omega^+,
\end{equation}
If the jump conditions (\ref{jump}) are rigorously enforced in the numerical
discretization, the numerical solution to (\ref{heat2}) will be identical to
that of (\ref{heat}), whereas
(\ref{heat2}) allows an easier formulation for the ADI method.
The semi-discretization of (\ref{heat2}) using the implicit Euler time
integration at a general spacial node $(x_i,y_j)$ reads
\begin{equation}\label{ADIsemi}
\frac{u^{k+1}_{i,j} - u^k_{i,j} }{\alpha \Delta t}
=\delta_{xx} u^{k+1}_{i,j} + \delta_{yy} u^{k+1}_{i,j}
+\frac{f^{k+1}_{i,j}}{\alpha},
\end{equation}
which is first order accurate in time.
Here $\delta_{xx}$ and $\delta_{yy}$ are discrete operators for
finite difference approximations in $x$ and $y$ directions.
We propose a first order Douglas ADI method for solving (\ref{heat2}),
\begin{align}
(\frac{1}{\alpha} - \Delta t \delta_{xx}) u^*_{i,j} & =
(\frac{1}{\alpha} + \Delta t \delta_{yy}) u^k_{i,j}
+ \frac{\Delta t}{\alpha} f^{k+1}_{i,j}, \nonumber \\
(\frac{1}{\alpha} - \Delta t \delta_{yy}) u^{k+1}_{i,j} & =
\frac{1}{\alpha} u^*_{i,j} - \Delta t \delta_{yy} u^k_{i,j}. \label{ADI}
\end{align}
To see the connection between (\ref{ADIsemi}) and (\ref{ADI}), we can
eliminate $u^*_{i,j}$ in (\ref{ADI}),
\begin{equation}\label{Doug2}
(\frac{1}{\alpha}-\Delta t \delta_{xx})
(\frac{1}{\alpha}-\Delta t \delta_{yy}) u^{k+1}_{i,j}
= (\frac{1}{\alpha} +\Delta t \delta_{yy}) \frac{1}{\alpha} u^k_{i,j}
- (\frac{1}{\alpha}-\Delta t \delta_{xx}) \Delta t \delta_{yy} u^k_{i,j}
+ \frac{\Delta t}{\alpha^2} f^{k+1}_{i,j}.
\end{equation}
After fully expanding terms, Eq. (\ref{Doug2}) can be written into the form
\begin{equation}\label{Doug3}
(\frac{1}{\alpha}-\Delta t \delta_{xx} -\Delta t \delta_{yy}
+ \alpha \Delta t^2 \delta_{xx} \delta_{yy} ) u^{k+1}_{i,j}
= (\frac{1}{\alpha} + \alpha \Delta t^2 \delta_{xx} \delta_{yy})u^{k}_{i,j}
+ \frac{\Delta t}{\alpha} f^{k+1}_{i,j}.
\end{equation}
If we drop the higher order perturbation term
$\alpha \Delta t^2 \delta_{xx} \delta_{yy} u$ on both hand sides of
(\ref{Doug3}), we actually obtain an equivalent form of the implicit
Euler scheme (\ref{ADIsemi})
\begin{equation}\label{imEuler}
(\frac{1}{\alpha}-\Delta t \delta_{xx} -\Delta t \delta_{yy}) u^{k+1}_{i,j}
= \frac{1}{\alpha} u^{k}_{i,j} + \frac{\Delta t}{\alpha} f^{k+1}_{i,j}.
\end{equation}
In this work, the Douglas scheme (\ref{ADI}) will be employed in all
ADI computations, while the implicit Euler scheme (\ref{imEuler}) can be
used in the theoretical analysis.
\subsection{Spatial discretization}
We next consider the spatial discretization.
For nodes away from the interface, a central difference approximation
is used, e.g.,
\begin{equation}\label{uyy}
\delta_{yy} u^{k+1}_{i,j}
:=\frac{1}{h^2} (u^{k+1}_{i,j-1} - 2 u^{k+1}_{i,j} + u^{k+1}_{i,j+1}).
\end{equation}
In the present study, a proper Dirichlet or Neumann
boundary condition for $u$ is assumed to be given
on the boundary $\partial \Omega$. Such a boundary condition is
implemented as in the classical ADI schemes
\cite{DougPeace,Douglas,PeaceRach}.
If a nontrivial boundary condition is encountered, the advanced
MIB boundary closure method \cite{Zhao07,Zhao09}
can be utilized to enclose such a
condition into the finite difference discretization.
For nodes near the interface $\Gamma$, novel interface treatments will
be developed to correct discrete finite difference operators
$\delta_{xx}$ and $\delta_{yy}$ via rigorously imposing the jump
conditions (\ref{jump}).
To this end, we consider some tensor product decompositions of jump
conditions (\ref{jump}) in the ADI framework.
At an interface point, we denote the outer normal and tangential directions
as $n$ and $\tau$, respectively.
Denote the angle between $n$ and the $x$-axis as $\theta$.
Coordinate transformations can be employed to convert between the
derivatives
\begin{equation}
\frac{\partial}{\partial n} = \cos \theta \frac{\partial}{\partial x}
+ \sin \theta \frac{\partial}{\partial y}, \quad
\frac{\partial}{\partial \tau} = -\sin \theta \frac{\partial}{\partial x}
+ \cos \theta \frac{\partial}{\partial y}.
\end{equation}
Based on (\ref{jump}), one more jump condition can be derived by
differentiating along the interface $\Gamma$:
$[u_{\tau}] = \frac{\partial \phi}{\partial \tau} = \phi_{\tau}$.
We thus have three zeroth and first order jump conditions
\begin{equation}\label{jump2}
[u] = \phi, \quad [u_{\tau}] = \phi_{\tau}, \quad
[\alpha u_n] = \psi.
\end{equation}
However, jump conditions given in (\ref{jump2}) cannot be applied in a 1D
manner in the ADI algorithm. To illustrate this, we transform the flux
jump condition into Cartesian directions
\begin{equation}\label{jump-xy}
\cos \theta [\alpha u_x] + \sin \theta [\alpha u_y] = \psi.
\end{equation}
A strong coupling in $x$ and $y$ directions is clearly seen.
In this paper, we propose to decompose the 2D jump conditions (\ref{jump2})
into two sets of essentially 1D jump conditions.
We illustrate the idea by considering the $x$ direction formulation.
The $y$ direction can be similarly treated.
Consider an interface point $(x_{\Gamma},y_j)$ which is the
intersection point between one $x$ grid line and $\Gamma$. If the normal
direction $n$ at this point happens to be along the $x$ direction, we have simply
1D jump conditions
$[u]=\phi$ and $[\alpha u_x]= \pm \psi$. For more general scenario, in which
$n$ is not along the $x$ direction, we will analytically derive a hybrid jump condition
using $x$ and $\tau$.
In particular, we have $u_y=\sec \theta u_{\tau} + \tan \theta u_x$. By
substituting this into (\ref{jump-xy}), we arrive at
\begin{equation}\label{jump-xtau}
[\alpha u_x] + \sin \theta [\alpha u_{\tau}] = \cos \theta \psi.
\end{equation}
Even though Eq. (\ref{jump-xtau}) looks similar to Eq. (\ref{jump-xy}),
Eq. (\ref{jump-xtau}) allows a 1D numerical approximation, whereas
Eq. (\ref{jump-xy}) dose not. This is because the jump value of $[u_{\tau}]$ is known,
while that of $[u_y]$ is not.
In fact, Eq. (\ref{jump-xtau}) can be further rewritten as
\begin{equation}\label{jump-x}
[\alpha u_x] = \cos \theta \psi - \sin \theta (\alpha^+ - \alpha^-) u^+_{\tau}
- \sin \theta \alpha^- \phi_{\tau} = \bar{\psi},
\end{equation}
where $u^+_{\tau}$ shall be evaluated based on finite difference approximations
using grid nodes exclusively from the positive side of $\Gamma$, i.e., $\Omega^+$.
Once $u^+_{\tau}$ is accurately estimated, $\bar{\psi}$ is known.
We thus have essentially 1D jump conditions
$[u]=\phi$ and $[\alpha u_x]=\bar{\psi}$. Similarly, we derive the following
essentially 1D jump conditions in the $y$ direction,
\begin{equation}\label{jump-y}
[u]=\phi, \quad
[\alpha u_y] = \sin \theta \psi + \cos \theta (\alpha^+ - \alpha^-) u^+_{\tau}
+ \cos \theta \alpha^- \phi_{\tau} = \hat{\psi}.
\end{equation}
We note that $\bar{\psi}$ and $\hat{\psi}$ can also be evaluated through calculating
$u^-_{\tau}$ from the negative side of the interface.
We propose a new MIB scheme to impose the decomposed 1D jump
conditions in the vicinity of the interface $\Gamma$.
Comparing with other established interface methods, the MIB method
\cite{Zhao04,Zhou06,Zhao09} is ideally-suited to the present problem because
in the MIB, jump condition enforcement is disassociated with the derivative
discretization and can be conducted in a 1D manner along each Cartesian
direction.
Here, we consider the MIB modification to $\delta_{yy} u^{k+1}_{i,j}$
in the ADI scheme (\ref{ADI}) as an example,
where $(x_i,y_j)$ is an irregular node near the
interface $\Gamma$.
A typical situation is shown in Fig. \ref{fig.grid} (a).
In the MIB scheme, to approximate function or its derivatives on one side of
interface, one never directly refers to function values from the other side.
Instead, fictitious values from the other side of the interface will be supplied.
Referring to Fig. \ref{fig.grid} (a), $\delta_{yy} u^{k+1}_{i,j}$ will be corrected as
\begin{equation}\label{uyy2}
\delta_{yy} u^{k+1}_{i,j}
=\frac{1}{h^2} (u^{k+1}_{i,j-1} - 2 u^{k+1}_{i,j} + \tilde{u}^{k+1}_{i,j+1}),
\end{equation}
where $\tilde{u}^{k+1}_{i,j+1}$ is a fictitious value at the node
$(x_i,y_{j+1})$.
Similarly, $\delta_{yy} u^{k+1}_{i,j+1}$ will be modified.
This calls for two fictitious values $\tilde{u}^{k+1}_{i,j}$ and
$\tilde{u}^{k+1}_{i,j+1}$, which will be resolved based on the jump conditions
(\ref{jump-y}).
\begin{figure*}
\caption{Illustration of the MIB grid partitions.
(a). For a regular interface; (b). For a corner case.
In both charts, the jump conditions will be discretized by using
fictitious values (open circles) and function values (filled
circles).
In (a), the approximation of $u^+_{\tau}
\label{fig.grid}
\end{figure*}
In order to impose the jump conditions (\ref{jump-y}), we first need to
approximate $u^+_{\tau}$.
Consider the situation shown in Fig. \ref{fig.grid} (a).
We calculate $u^+_{\tau}$ in two steps.
First, we calculate the intersection points between the tangential
line $\tau$ and two grid lines $x=x_{i-1}$ and $x=x_{i+1}$.
There two auxiliary nodes are shown as open squares in Fig. \ref{fig.grid} (a).
A central difference is conducted to approximate the
$\tau$ derivative at the interface point $(x_i,y_{\Gamma})$
using two auxiliary values of $u$.
Second, each of these two auxiliary values will be
further interpolated by using three on-grid function values, all selecting
from $\Omega^+$.
These six nodes are shown as filled squares in Fig. \ref{fig.grid} (a).
In this manner, $u^+_{\tau}$ is actually approximated by six grid values of $u$,
with the spatial accuracy being second order.
In the present study, we will make use of the known values of $u$ at
the current time instant $t_k$ to estimate $u^+_{\tau}$, which avoids the
introduction of a coupling among different $y$ grid lines at the future
time instant $t_{k+1}$. Otherwise, the 1D linear systems of the ADI algorithm
are not independent, which cannot be solved efficiently.
We note that
the present approximation is of first order accurate in time when
it is applied to correct $\delta_{yy} u^{k+1}_{i,j}$ and
$\delta_{xx} u^*_{i,j}$ in (\ref{ADI}).
This is acceptable, since the temporal order of the ADI scheme (\ref{ADI})
is also one.
In summary, the calculated $u^+_{\tau}$ will depend on six $u^{k}_{i,j}$ values
for some nearby nodes $(x_i,y_j)$.
With the calculated $u^+_{\tau}$,
$\hat{\psi}$ is then known at the interface point $(x_i,y_{\Gamma})$.
We will next determine $\tilde{u}^{k+1}_{i,j}$ and
$\tilde{u}^{k+1}_{i,j+1}$ by using four function values at the future time
$t_{k+1}$, i.e.,
$u^{k+1}_{i,j-1}$, $u^{k+1}_{i,j}$, $u^{k+1}_{i,j+1}$, and
$u^{k+1}_{i,j+2}$, see Fig. \ref{fig.grid} (a).
For this purpose, (\ref{jump-y}) will be discretized in the same manner of
(\ref{uyy2}), i.e., never referring to function values across the
interface $\Gamma$
\begin{align}
& w_{0,1}^+ \tilde{u}^{k+1}_{i,j}
+ w_{0,2}^+ u^{k+1}_{i,j+1} + w_{0,3}^+ u^{k+1}_{i,j+2} \label{MIB} \\
= & w_{0,1}^- u^{k+1}_{i,j-1} + w_{0,2}^- u^{k+1}_{i,j}
+ w_{0,3}^- \tilde{u}^{k+1}_{i,j+1} + \phi, \nonumber \\
& \alpha^+ \Big( w_{1,1}^+ \tilde{u}^{k+1}_{i,j}
+ w_{1,2}^+ u^{k+1}_{i,j+1} + w_{1,3}^+ u^{k+1}_{i,j+2} \Big) \nonumber \\
= & \alpha^- \Big( w_{1,1}^- u^{k+1}_{i,j-1} + w_{1,2}^- u^{k+1}_{i,j}
+ w_{1,3}^- \tilde{u}^{k+1}_{i,j+1} \Big)+ \hat{\psi}, \nonumber
\end{align}
where $w^-_{I,J}$ and $w^+_{I,J}$ for $I=0,1$ and $J=1,2,3$ are one-sided
finite difference weights, respectively, for left and right subdomains.
Here the subscript $I$ represents interpolation ($I=0$) and the first
derivative approximation ($I=1$), and $J$ is for grid index.
After the discretization, (\ref{MIB}) actually represents two algebraic equations.
By solving (\ref{MIB}), one can determine $\tilde{u}^{k+1}_{i,j}$ and
$\tilde{u}^{k+1}_{i,j+1}$ as linear combinations of
$u^{k+1}_{i,j-1}$, $u^{k+1}_{i,j}$, $u^{k+1}_{i,j+1}$, $u^{k+1}_{i,j+2}$,
$\phi$, and $\hat{\psi}$.
By substituting such combinations into (\ref{uyy2}) to eliminate the
fictitious value and applying the definition of $\hat{\psi}$ given in
(\ref{jump-y}),
$\delta_{yy} u^{k+1}_{i,j}$ is now a spatially second order
accurate finite difference approximation to the double $y$ derivative,
involving four $u^{k+1}_{i,j}$ values along the $y$ direction,
six $u^{k}_{i,j}$ values nearby, and two nonhomogeneous values $\phi$ and $\psi$.
In the present study, $\phi$ and $\psi$ will be evaluated at the time
instant $t_k$.
When one grid line intersects the interface $\Gamma$ near a rounded or
sharp corner, the interface could be cut twice within a short distance.
If in between these two intersection
points there is no grid node, a grid refinement is necessary. If there are at least
two grid nodes, the aforementioned matched ADI algorithm can be conducted.
However, additional corner treatments are called for the case where
only one node locates in between two intersection points, see
Fig. \ref{fig.grid} (b).
A MIB corner scheme is proposed to solve this problem.
Denote two intersection points as $(x_i,y_{\Gamma 1})$ and
$(x_i,y_{\Gamma 2})$ with $y_{\Gamma 1}< y_j < y_{\Gamma 2}$.
At $(x_i,y_{\Gamma 1})$ and $(x_i,y_{\Gamma 2})$,
$u^+_{\tau}$ can be calculated individually as outlined above, so that
the nonhomogeneous values $(\phi_1,\hat{\psi}_{1})$ and
$(\phi_2,\hat{\psi}_{2})$ are known, respectively.
The jump conditions (\ref{jump-y}) can then be imposed at
$(x_i,y_{\Gamma 1})$ and $(x_i,y_{\Gamma 2})$ to form four algebraic equations.
Nevertheless, only three fictitious values are needed to correct
$\delta_{yy} u$, i.e., $\tilde{u}^{k+1}_{i,j-1}$, $\tilde{u}^{k+1}_{i,j}$, and
$\tilde{u}^{k+1}_{i,j+1}$.
This difficulty can be trivially avoided by introducing one more fictitious value.
Referring to Fig. \ref{fig.grid} (b), we denote $D_1 = |y_{\Gamma 1} - y_{j-1}|$
and $D_2= | y_{j+1} - y_{\Gamma 2} |$. If $D_1 < D_2$, the fourth fictitious
value is chosen as $\tilde{u}^{k+1}_{i,j-2}$. Otherwise, it is selected as
$\tilde{u}^{k+1}_{i,j+2}$. With such a grid partition, each term of the
jump conditions (\ref{jump-y}) is approximated through a third order finite
difference approximation involving four points. This ensures the overall
accuracy of the matched ADI scheme, since the solution usually undergoes
a rapid change near the corner.
In particular, for the case shown in Fig. \ref{fig.grid} (b), the positive
and negative terms at $(x_i,y_{\Gamma 1})$ are approximated based on
$(u^{k+1}_{i,j-2}, u^{k+1}_{i,j-1}, \tilde{u}^{k+1}_{i,j}, u^{k+1}_{i,j+1})$ and
$(\tilde{u}^{k+1}_{i,j-1}, u^{k+1}_{i,j},\tilde{u}^{k+1}_{i,j+1},\tilde{u}^{k+1}_{i,j+2})$,
respectively,
while those at $(x_i,y_{\Gamma 2})$ are based on
$(u^{k+1}_{i,j-1}, \tilde{u}^{k+1}_{i,j},u^{k+1}_{i,j+1},u^{k+1}_{i,j+2})$ and
$(\tilde{u}^{k+1}_{i,j-1}, u^{k+1}_{i,j},\tilde{u}^{k+1}_{i,j+1},\tilde{u}^{k+1}_{i,j+2})$,
respectively.
The details of the discretization are omitted here. By substituting the solved linear
combinations into (\ref{uyy2}), $\delta_{yy} u^{k+1}_{i,j}$ is approximated by
five function values
$(u^{k+1}_{i,j-2}, u^{k+1}_{i,j-1},u^{k+1}_{i,j}, u^{k+1}_{i,j+1}, u^{k+1}_{i,j+2})$
and four nonhomogeneous values
$(\phi_1,\hat{\psi}_{1}, \phi_2,\hat{\psi}_{2})$.
Furthermore, since $u^+_{\tau}$ is evaluated at at two interface points,
$\delta_{yy} u^{k+1}_{i,j}$ involves up to 12 nearby $u^k_{i,j}$ values,
and four jump values $(\phi_1,\psi_{1}, \phi_2,\psi_{2})$ evaluating at time $t_k$.
We note that the proposed MIB discretization for the regular interface case and
the corner case need to be conducted only once
at the beginning of the simulation, because the geometric domain, grid, and
finite difference approximations of the jump conditions are all time invariant.
In fact, the entries of the discrete operators $\delta_{xx} u$ and
$\delta_{yy} u$ are all time independent and can be pre-determined.
At each time step, one just needs to update nonhomogeneous values
$\phi$, $\bar{\psi}$ and $\hat{\psi}$ for time dependent jump conditions.
\subsection{Fast algebraic solution}
In the proposed matched ADI algorithm, the 1D linear systems underlying
(\ref{ADI}) are actually independent from each other,
so that we can solve them separately.
Without the loss of generality, we denote the 1D linear system to be solved
in one ADI step as
\begin{equation}\label{Axb}
{\bf A} {\bf x} = {\bf b},
\end{equation}
where ${\bf x}$ represents unknown $u^{k+1}$ or $u^*$ values on one
$x$ or $y$ grid line. The matrix ${\bf A}$ is of dimension $N$ by $N$, where
$N$ could be $N=N_x$ or $N=N_y$.
The vector ${\bf b}$ contains all the right hand side terms.
Nevertheless, ${\bf A}$ will be non-tridiagonal after the MIB treatment
so that new algebraic solvers have to be developed to maintain the
overall computational efficiency.
In particular, for a regular interface case,
one grid line cuts the interface $\Gamma$ twice at a well separated distance.
There are totally four irregular nodes sandwiched the interface.
For each of them, the actual band width becomes four,
while two nonhomogeneous values $\phi$ and $\psi$ shall be added into ${\bf b}$.
A typical matrix structure of ${\bf A}$ is shown in Fig. \ref{fig.matrix} (a).
For the corner interface case, a similar analysis indicates that
the band width of three consecutive irregular nodes
are changed from three to five, see Fig. \ref{fig.matrix} (b).
\begin{figure*}
\caption{Matrix structures of the matched ADI algorithm.
(a). For a regular interface; (b). For a corner case.
}
\label{fig.matrix}
\end{figure*}
Since the change of the band-structure is not too dramatic,
the linear system (\ref{Axb}) could be
solved by the Woodbury formula \cite{NR}.
We consider the regular interface case as an example.
Denote the indices of four irregular nodes to be $I$, $I+1$, $J$, and $J+1$.
See Fig. \ref{fig.matrix} (a).
The extra coefficients of ${\bf A}$ can be accounted for by defining two
$N \times 2$ matrices ${\bf P}$ and ${\bf Q}$ with four nonzero elements each:
\begin{align*}
& {\bf P}_{I,1}=1, \quad {\bf P}_{I+1,1}=1, \quad {\bf P}_{J,2}=1,
\quad {\bf P}_{J+1,2}=1, \\
& {\bf Q}_{I-1,1}={\bf A}_{I+1,I-1}, \quad {\bf Q}_{I+2,1}={\bf A}_{I,I+2}, \quad
{\bf Q}_{J-1,2}={\bf A}_{J+1,J-1}, \quad {\bf Q}_{J+2,2}={\bf A}_{J,J+2}.
\end{align*}
We then have ${\bf A}={\bf T} + {\bf P}{\bf Q}^T$, where ${\bf T}$
is a tridiagonal matrix. Thus, we have analytically
\begin{equation}
{\bf A}^{-1}=({\bf T} + {\bf P}{\bf Q}^T)^{-1}
={\bf T}^{-1} - [ {\bf T}^{-1} {\bf P}
({\bf 1} + {\bf Q}^T {\bf T}^{-1} {\bf P})^{-1} {\bf Q}^T {\bf T}^{-1}],
\end{equation}
by the Woodbury formula \cite{NR}.
In other words, the inversion of ${\bf A}$ can be carried out through applying the
Thomas algorithm three times by solving some auxiliary systems about ${\bf T}$.
However, the Woodbury formula is not used in our computation,
because a more efficient algebraic procedure is available.
For example, for the regular interface case,
four elementary row operations are simply conducted.
In particular, by denoting the $I^{\rm th}$ row of ${\bf A}$ as ${\bf R}_I$,
the elementary row operation
${\bf R}_{I+1} - {\bf A}_{I+1,I-1}/{\bf A}_{I,I-1} {\bf R}_{I}$
will zero the entry at the position $(I+1,I-1)$.
Similar row operations are conducted to vanish other three positions, so that
a tridiagonal system is formed.
In a similar manner, the matrix ${\bf A}$ of the corner interface can be
treated by six row operations to reduce to a tridiagonal one.
After the Gauss elimination,
the Thomas algorithm \cite{ADIbook}
is applied only once to solve the adjusted system.
The proposed matched ADI algorithm is very efficient.
To solve the 1D linear system within each inner ADI step,
the flop counts are essentially due to the Thomas algorithm,
i.e., on the order of $O(N)$,
because the overhead for the Gauss elimination is very small and does
not grow with $N$.
Thus, like the standard 2D ADI algorithm, the complexity
of advancing one time step in the proposed matched ADI schemes is about
$O(N^2)$.
Moreover, due to the excellent stability of the matched ADI scheme,
one can simply fix $\Delta t$ to be on the order of $h$.
Consequently, the complexity of
entire time integration will be on the order of $O(N^3)$ for solving
parabolic interface problems with $N^2$ unknowns.
\subsection{Stability analysis}\label{sec.stab}
As discussed above that the Douglas ADI scheme (\ref{ADI}) is a higher
order perturbation of the implicit Euler scheme (\ref{imEuler}).
In the present subsection, we will analyze the stability of the
implicit Euler scheme with the proposed MIB spatial discretization,
because in this case, it is relatively easier
to construct 2D matrices for the spectrum analysis.
The stability of the matched ADI scheme is essentially determined by
that of the matched Euler scheme.
We first establish a vector notation
for the proposed MIB spatial discretization.
Denote ${\bf U}^{k} = [ u^k_{1,1}, u^k_{2,1}, \ldots, u^k_{N_x,1},
u^k_{1,2}, u^k_{2,2}, \ldots, u^k_{N_x,2}, \ldots]^T$,
which is a vector of the length $N_x \times N_y$,
containing all $u$ values at the time $t_k$.
The second order $x$ derivative of ${\bf U}^{k+1}$ can be expressed as
\begin{equation}\label{Mat_x}
\frac{\partial ^2}{\partial x^2} {\bf U}^{k+1} \approx
{\bf D}_{xx} {\bf U}^{k+1} + {\bf \bar{B}} {\bf U}^{k} + {\bf \bar{\Phi}}^k,
\end{equation}
where ${\bf D}_{xx} $ and ${\bf \bar{B}}$ are matrices of the dimension
$N_x \times N_y$ by $N_x \times N_y$, while ${\bf \bar{\Phi}}^k$ is a vector
of the length $N_x \times N_y$.
The matrix ${\bf D}_{xx} $ is a perturbation of the standard matrix for the central
difference approximation, while ${\bf \bar{B}}$ is due to the approximation of
$u^+_{\tau}$ by some $u^k_{i,j}$ values.
Here ${\bf \bar{\Phi}}^k$ is a correction term,
based on linear combinations of the nonhomogeneous values $\phi$ and $\psi$.
For a regular node $(x_i,y_j)$, the corresponding rows of ${\bf \bar{B}}$
and ${\bf \bar{\Phi}}^k$ have only zero entries, while that of ${\bf D}_{xx} $ has three non-zero
entries centered at the diagonal, i.e., $1/h^2$, $-2/h^2$ and $1/h^2$.
For an irregular node in the non-corner case, the corresponding row of
${\bf D}_{xx} $ and ${\bf \bar{B}}$ has four and six non-zero elements, respectively,
whereas in a corner case, there are five and twelve non-zero elements, respectively,
for ${\bf D}_{xx} $ and ${\bf \bar{B}}$ .
We note that sparse elements of ${\bf \bar{B}}$ are distributed in a rather random
fashion -- their locations depend on the interface geometry and grid size. Our MIB
code will automatically calculate them.
However, the sparse structure of ${\bf D}_{xx} $ can be well predicted. Essentially,
${\bf D}_{xx} $ has $N_y$ nonzero blocks along the diagonal.
When the grid line $x=x_i$ does not cut the interface $\Gamma$,
the corresponding diagonal block of ${\bf D}_{xx} $ is simply a tridiagonal sub-matrix.
Otherwise,
the corresponding diagonal block will take the form showing in either
Fig. \ref{fig.matrix} (a) or (b), depending on whether this is a corner case or not.
Similarly, the second order $y$ derivative of ${\bf U}^{k+1}$ is approximated as
\begin{equation}\label{Mat_y}
\frac{\partial ^2}{\partial y^2} {\bf U}^{k+1} \approx
{\bf D}_{yy} {\bf U}^{k+1} + {\bf \hat{B}} {\bf U}^{k} + {\bf \hat{\Phi}}^k,
\end{equation}
with ${\bf D}_{yy}$, ${\bf \hat{B}}$, and $ {\bf \hat{\Phi}}^k$ being
appropriately defined.
The vector form of the implicit Euler scheme (\ref{imEuler}) can then be given as
\begin{equation}\label{stab1}
\left( \frac{1}{\alpha} {\bf I} - \Delta t {\bf D}_{xx}- \Delta t {\bf D}_{yy} \right)
{\bf U}^{k+1} = \left( \frac{1}{\alpha} {\bf I} + \Delta t {\bf \bar{B}}
+ \Delta t {\bf \hat{B}} \right) {\bf U}^{k} + \Delta t {\bf \bar{\Phi}}^k
+ \Delta t {\bf \hat{\Phi}}^k + \frac{\Delta t }{\alpha} {\bf F}^{k+1},
\end{equation}
where ${\bf I}$ is the identity matrix and ${\bf F}^{k+1}$ represents the
source term. We can rewrite (\ref{stab1}) into a more compact form
\begin{equation}\label{stab2}
{\bf D} {\bf U}^{k+1} = {\bf B} {\bf U}^{k} + {\bf C},
\end{equation}
where ${\bf D} =\frac{1}{\alpha} {\bf I} - \Delta t {\bf D}_{xx}- \Delta t {\bf D}_{yy}$,
${\bf B}=\frac{1}{\alpha} {\bf I} + \Delta t {\bf \bar{B}} + \Delta t {\bf \hat{B}}$, and
${\bf C}=\Delta t {\bf \bar{\Phi}}^k +
\Delta t {\bf \hat{\Phi}}^k + \frac{\Delta t }{\alpha} {\bf F}^{k+1}$.
Recall again that ${\bf D}$ is slightly modified from the standard matrix for the central
finite difference, while ${\bf B}$ is resulting from the approximation of $u^+_{\tau}$
by $u^k_{i,j}$ values at various interface points.
By taking an inverse, (\ref{stab2}) becomes
\begin{equation}\label{stab3}
{\bf U}^{k+1} = {\bf D}^{-1} {\bf B} {\bf U}^{k} + {\bf D}^{-1} {\bf C}
={\bf M} {\bf U}^{k} + {\bf D}^{-1} {\bf C}.
\end{equation}
Thus, the stability of the entire spatial-temporal discretization depends on the
magnifying matrix ${\bf M} = {\bf D}^{-1} {\bf B}$.
In particular, it depends on the spectral radius $\rho$ of ${\bf M}$, which
is defined to be
\begin{equation}
\rho ({\bf M}) = \max_j | \lambda_j |,
\end{equation}
where $ \lambda_j$ are eigenvalues of ${\bf M}$.
In general, an analytical spectrum analysis of ${\bf M}$ is extremely difficult,
because the finite difference formulas underlying ${\bf D}$ and ${\bf B}$
depend on the positions of grid nodes and interface intersection points. Thus,
in the present study, the leading eigenvalues of ${\bf M}$ will be calculated
numerically. This enables us to directly examine the impact of
parameters, such as $h$, $\Delta t$ and $\alpha$, on the stability.
Moreover, we will also investigate how a complex geometry could affect the
stability.
The stability of the MIB spatial discretization combined with explicit time
integrations has been analyzed in \cite{Zhao04,Zhao07,Zhao09}
for solving both interface and boundary closure problems.
It is known that an instability could occur, if a too asymmetric MIB
finite difference approximation is involved \cite{Zhao04,Zhao09},
while a severely asymmetric approximation will produce
spurious modes in the spectrum analysis \cite{Zhao07}.
When a complicated interface $\Gamma$ is studied,
a severely asymmetric approximation could be encountered, if
$\Gamma$ and/or its tangential lines cut a grid line at
a point that is very close to a node.
However, one cannot predict when this will happen,
because $\Gamma$ will intersect the grid in a random manner.
To evaluate the impact of such a geometrical effect on the stability,
we will consider a fixed interface and test various
different mesh sizes in the next section.
This allows us to see if the complex geometry will compromise
the stability of the present implicit time stepping method.
\section{Numerical experiments}\label{Sec:Numerical}
In this section, we investigate the stability and accuracy of
the proposed matched ADI algorithm for
solving 2D parabolic interface problems with different jump conditions
and interface geometries.
Piecewisely defined analytical solutions will be constructed in each example.
The initial solution is chosen according the analytical solution at $t=0$.
The ADI time stepping will be carried out until a stopping time $t=T$.
Without the loss of generality, a square domain $[-D,D]\times[-D,D]$
with the Dirichlet boundary condition is considered in all examples.
Here, the boundary data is simply given by the analytical solutions.
Similarly, the jump conditions defining function and flux jumps
across the interfaces are also calculated according to the given
analytical solutions.
For simplicity, the mesh sizes in both $x$ and $y$ directions are chosen
to be the same, i.e., $N=N_x=N_y$ with $h=\Delta x =\Delta y$.
The domain size $D$ is usually chosen as a non-integer so that
the corner interface case could be encountered in a coarse grid.
This enables us to fully validate the proposed matched ADI algorithm.
Numerical errors in $L_\infty$ and $L_2$ norms are
reported in all examples.
In all examples, the proposed matched ADI method is found to be
unconditionally stable, through both direct numerical verifications and
eigenvalue stability analysis. To save the space, the detailed stability
analysis will be presented only for the last numerical example,
in which the most complicated interface geometry and the most complicated
jump conditions will be studied.
{\bf Example 1.} We first study a circular interface problem with a
continuous solution.
Consider a square domain $[-D,D]\times[-D,D]$
with a circular interface $r^2=x^2+y^2=1$.
The piecewise coefficient is defined to be $\alpha^-=1$ and $\alpha^+=10$,
respectively, for $r<1$ and $r \ge 1$.
The analytical solution to the heat equation is designed to be
\begin{equation*}
u(x,y,t)= \begin{cases}
\Big( \frac{r^6 -1}{\alpha^-} - \frac{3}{\alpha^+} \Big) \cos(t),
\quad \mbox{ if } r <1 \\
-\frac{3}{\alpha^+ r^2} \cos(t),
\quad \quad \quad \quad \mbox{ if } r \ge 1,
\end{cases}
\end{equation*}
so the the jump conditions are simply $[u]=0$ and $[\alpha u_n]=0$.
The source term is then given as
\begin{equation*}
f(x,y,t)= \begin{cases}
-\Big( \frac{r^6 -1}{\alpha^-} - \frac{3}{\alpha^+} \Big) \sin(t)
-36 r^4 \cos(t),
\quad \mbox{ if } r <1 \\
\frac{3}{\alpha^+ r^2} \sin(t) +12 r^{-4} \cos(t),
\quad \quad \quad \quad \quad \mbox{ if } r \ge 1.
\end{cases}
\end{equation*}
In this example, the domain size is set to be $D=1.99$ and
the stopping time is fixed as $T=2$.
\begin{figure*}
\caption{Temporal convergence tests. (a). Example 1;
(b). Example 2.
}
\label{fig.ex12}
\end{figure*}
We first examine the temporal convergence.
The proposed matched ADI algorithm is found to be unconditionally stable for
all tested $h$ and $\Delta t$.
By taking $N=321$,
the numerical errors generated by using different $\Delta t$ are shown in
Fig. \ref{fig.ex12} (a).
A similar pattern can be observed for both $L_\infty$ and $L_2$ errors.
i.e., the temporal errors become smaller and smaller
until they are limited by the accuracies of the spatial discretization.
The temporal convergence order of the matched ADI algorithm
can be analyzed via considering
errors before reaching the limiting precision.
For these errors, a linear least-squares fitting \cite{Zhao09} is conducted
in the log-log scale. The fitted convergence lines are shown as solid lines in
Fig. \ref{fig.ex12} (a). Moreover, the fitted slope essentially represents
the numerical convergence rate $r$ of the scheme.
The temporal order in $L_\infty$ and $L_2$ norms is found to be
$r=1.58$ and $r=1.48$, respectively.
In other word, the numerically detected temporal oder of the matched ADI algorithm
is about half order higher than its theoretical design.
This is perhaps because the present solution is continuous.
The superconvergence of the Douglas ADI scheme for smooth solutions has also
been observed in other literature studies \cite{Geng13,Zhao14}.
\begin{table}[!t]
\caption{Spatial convergence tests for first two examples. }
\label{table.ex12}
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|}
\hline
& \multicolumn{4}{c|}{Example 1} & \multicolumn{4}{c|}{Example 2} \\
\cline{2-5} \cline{6-9}
& \multicolumn{2}{c|}{$L_\infty$} & \multicolumn{2}{c|}{$L^2$}
& \multicolumn{2}{c|}{$L_\infty$} & \multicolumn{2}{c|}{$L^2$} \\
\cline{2-3} \cline{4-5} \cline{6-7} \cline{8-9}
$N$ & error & order & error & order & error & order & error & order \\
\hline
21 & 1.92E-2 & & 6.75E-3 & & 1.78E-4 & & 9.15E-5 & \\
41 & 4.49E-3 & 2.09 & 1.51E-3 & 2.16 & 4.38E-5 & 2.02 & 2.76E-5 & 1.73 \\
81 & 1.34E-3 & 1.74 & 4.97E-4 & 1.61 & 1.77E-5 & 1.31 & 1.13E-5 & 1.28 \\
161 & 3.78E-4 & 1.83 & 1.46E-4 & 1.77 & 3.07E-6 & 2.53 & 2.08E-6 & 2.45 \\
321 & 9.47E-5 & 2.00 & 3.62E-5 & 2.01 & 5.38E-7 & 2.51 & 3.01E-7 & 2.79\\
\hline
\end{tabular}
\end{center}
\end{table}
We next quantitatively examine the spatial accuracy.
A small enough $\Delta t= 10^{-4}$ is employed so that the temporal error
can be neglected in the present study.
The numerical errors of the matched ADI method
for different mesh size $N$ are listed in Table \ref{table.ex12}.
Based on successive mesh refinements,
the numerically calculated convergence rates are also reported for both
error measurements.
It can be seen that the matched ADI algorithm achieves the second order of
accuracy in both $L_\infty$ and $L_2$ norms
for the present parabolic interface problem.
{\bf Example 2.}
We next consider a circular interface problem with constant jump values.
The interface $\Gamma$ is defined as $r=0.5$ and the domain size is
set as $D=0.99$.
The diffusion coefficient is chosen as $\alpha^- =2$ and $\alpha^+=10$,
respectively, for $r < 0.5$ and $r \ge 0.5$.
The source term is defined to be
\begin{equation*}
f(x,y,t)= \begin{cases}
-\sin(t) - 4 \alpha^-, \quad \quad \quad \mbox{ if } r <0.5 \\
-\sin(t) - 8 r^2 -4,
\quad \quad \mbox{ if } r \ge 0.5.
\end{cases}
\end{equation*}
The analytical solution can then be given as
\begin{equation*}
u(x,y,t)= \begin{cases}
\cos(t) + r^2 -1, \quad \quad \quad \quad \quad
\quad \quad \quad \quad \quad \mbox{ if } r <0.5 \\
\cos(t) + \frac{1}{4}(1- \frac{9}{8 \alpha^+}) +
\frac{1}{\alpha^+} (\frac{r^4}{2} + r^2),
\quad \mbox{ if } r \ge 0.5.
\end{cases}
\end{equation*}
It can be verified that the jump values are constants along the interface $\Gamma$
and are time invariant.
In particular, we have $[u]=1$ and $[\alpha u_n]=-0.75$.
For all tested $h$ and $\Delta t$, the matched ADI method is again found to
be unconditionally stable for this example.
By choosing $N=321$ and $T=1$,
the temporal accuracies are analyzed in
Fig. \ref{fig.ex12} (b). It can be seen that both $L_\infty$ and $L_2$ errors
decrease uniformly until the limiting precisions of the spatial discretization are
reached. After that, by using a smaller $\Delta t$, the error become slightly
larger. The least-squares error analysis is also conducted. With a rate of $1.11$
for both error norms, the matched ADI method clearly attains the first order of
accuracy in time for this example with a discontinuous solution.
By using a sufficiently small $\Delta t=10^{-4}$, the spatial accuracies of the
matched ADI method are investigated in Table \ref{table.ex12}.
The spatial convergence is not very uniform for this example.
However, the overall order of the matched ADI method is still around two
in both error measurements.
{\bf Example 3.}
To further explore the potential of the proposed ADI method, we consider
a circular interface problem with general jump values.
The interface is also defined as $r=0.5$ with domain size $D=0.99$.
By taking $\alpha^- = 1$ and $\alpha^+=10$, the analytical solution is chosen as
\begin{equation*}
u(x,y,t)= \begin{cases}
\cos(t) + \exp(x^2+y^2), \quad \quad \mbox{ if } r <0.5 \\
\cos(t) + \sin(kx) \cos(ky),
\quad \mbox{ if } r \ge 0.5,
\end{cases}
\end{equation*}
where the wavenumber is chosen as $k=2$.
The source term is given as
\begin{equation*}
f(x,y,t)= \begin{cases}
-\sin(t) - 4 \alpha^- \exp(x^2+y^2)(x^2+y^2+1), \quad \mbox{ if } r <0.5 \\
-\sin(t) + 2 \alpha^+ k^2 \sin(kx)\cos(ky),
\quad \quad \quad \quad \quad \mbox{ if } r \ge 0.5.
\end{cases}
\end{equation*}
The jump conditions at a interface point
$(x,y)=(\frac{1}{2} \cos \theta, \frac{1}{2} \sin \theta)$
can be derived from the analytical solution
\begin{align*}
[u] =& \sin (\frac{k}{2} \cos \theta) \cos (\frac{k}{2} \sin \theta) - \exp(\frac{1}{4}), \\
[\alpha u_n] =&
\alpha^+ k \cos \theta \cos (\frac{k}{2} \cos \theta) \cos (\frac{k}{2} \sin \theta) \\
- & \alpha^+ k \sin \theta \sin (\frac{k}{2} \cos \theta) \sin (\frac{k}{2} \sin \theta)
-\alpha^- \exp(\frac{1}{4}), \\
[u_{\tau}] =&
- k \sin \theta \cos (\frac{k}{2} \cos \theta) \cos (\frac{k}{2} \sin \theta)
- k \cos \theta \sin (\frac{k}{2} \cos \theta) \sin (\frac{k}{2} \sin \theta) ,
\end{align*}
where the third jump condition $[u_{\tau}]=\phi_{\tau}$ is derived from the function
jump $[u]=\phi$.
We note that the present jump conditions are quite general in the sense that
all jump values are functions of space, even though they are time independent.
\begin{figure*}
\caption{Temporal convergence tests. (a). Example 3;
(b). Example 4.
}
\label{fig.ex34}
\end{figure*}
Again, the matched ADI algorithm is unconditionally stable for all
tested $h$ and $\Delta t$ in this example. By using $N=321$ and $T=1$,
both $L_\infty$ and $L_2$ errors immediately begin to decay
as $\Delta t$ becomes smaller,
and the limiting precisions are approached when $\Delta t$ is small enough.
It can be seen from Fig. \ref{fig.ex34} (a) that the slope for the $L_\infty$ and $L_2$
error curve is, respectively, 1.07 and 1.06. Thus, the matched ADI delivers the first
order of accuracy in time for parabolic interface problems with general jump values.
By taking $\Delta t=10^{-4}$, the spatial errors are reported in
Table \ref{table.ex34}. The matched ADI method clearly achieves second order
in space for this example.
The matched ADI solution based on a mesh $N=81$ at the time $T=1$ is
shown in Fig. \ref{fig.solu34} (a). The jump values clearly change with respect to
the angle $\theta$.
\begin{table}[!t]
\caption{Spatial convergence tests for Example 3 and Example 4. }
\label{table.ex34}
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|}
\hline
& \multicolumn{4}{c|}{Example 3} & \multicolumn{4}{c|}{Example 4} \\
\cline{2-5} \cline{6-9}
& \multicolumn{2}{c|}{$L_\infty$} & \multicolumn{2}{c|}{$L^2$}
& \multicolumn{2}{c|}{$L_\infty$} & \multicolumn{2}{c|}{$L^2$} \\
\cline{2-3} \cline{4-5} \cline{6-7} \cline{8-9}
$N$ & error & order & error & order & error & order & error & order \\
\hline
21 & 9.12E-3 & & 1.61E-3 & & 4.77E-3 & & 8.54E-4 & \\
41 & 2.51E-3 & 1.86 & 3.76E-4 & 2.10 & 1.32E-3 & 1.85 & 1.96E-4 & 2.12 \\
81 & 4.93E-4 & 2.35 & 7.24E-5 & 2.38 & 2.56E-4 & 2.37 & 3.57E-5 & 2.46 \\
161 & 7.47E-5 & 2.72 & 1.39E-5 & 2.38 & 3.91E-5 & 2.71 & 6.66E-6 & 2.42 \\
321 & 1.36E-5 & 2.45 & 3.04E-6 & 2.19 & 7.22E-6 & 2.44 & 1.57E-6 & 2.09\\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{figure*}
\caption{Numerical solution with $N=81$ at $T=1$.
(a). Example 3; (b). Example 4.
}
\label{fig.solu34}
\end{figure*}
{\bf Example 4.}
We next extend the Example 3 to a more general situation, by considering
time dependent jumps.
The parameters for the interface and domain are fixed to be $r=0.5$, $D=0.99$,
$\alpha^- = 1$, and $\alpha^+=10$.
The analytical solution is constructed as
\begin{equation*}
u(x,y,t)= \begin{cases}
\sin(kx) \cos(ky) \cos(t), \quad \mbox{ if } r <0.5 \\
\cos(kx) \sin(ky) \cos(t), \quad \mbox{ if } r \ge 0.5,
\end{cases}
\end{equation*}
where the wavenumber is chosen as $k=2$.
The source term is given as
\begin{equation*}
f(x,y,t)= \begin{cases}
(2k^2 \alpha^- \cos(t) -\sin(t)) \sin(kx)\cos(ky), \quad \mbox{ if } r <0.5 \\
(2k^2 \alpha^+ \cos(t) -\sin(t)) \cos(kx)\sin(ky), \quad \mbox{ if } r \ge 0.5.
\end{cases}
\end{equation*}
Now, the jump conditions at a interface point
$(x,y)=(\frac{1}{2} \cos \theta, \frac{1}{2} \sin \theta)$
depend on both space and time
\begin{align*}
[u] =& \cos (\frac{k}{2} \cos \theta) \sin (\frac{k}{2} \sin \theta) \cos(t)-
\sin (\frac{k}{2} \cos \theta) \cos (\frac{k}{2} \sin \theta) \cos(t), \\
[\alpha u_n] =&
k \cos(t) (\alpha^- \sin \theta - \alpha^+ \cos \theta)
\sin (\frac{k}{2} \cos \theta) \sin (\frac{k}{2} \sin \theta) \\
+& k \cos(t) (\alpha^+ \sin \theta - \alpha^- \cos \theta)
\cos (\frac{k}{2} \cos \theta) \cos (\frac{k}{2} \sin \theta), \\
[u_{\tau}] =&
k \cos(t) (\cos \theta + \sin \theta)
(\cos (\frac{k}{2} \cos \theta) \cos (\frac{k}{2} \sin \theta)
+ \sin (\frac{k}{2} \cos \theta) \sin (\frac{k}{2} \sin \theta) ).
\end{align*}
Such jump conditions are the most general ones for parabolic
interface problems.
\begin{figure*}
\caption{Bounded numerical errors with $N=321$
and $T=10^4 \Delta t$.
(a). Example 4; (b). Example 5, four leaves case.
}
\label{fig.large}
\end{figure*}
The temporal convergence pattern now turns out to be significantly different
from those of the previous examples.
By using $N=321$ and $T=1$, the $L_\infty$ and $L_2$ errors
are depicted in Fig. \ref{fig.ex34} (b).
It can be observed that the error curve does not immediately decay
for large $\Delta t$ values. In other word, the temporal convergence
of the matched ADI method is somehow polluted by the time dependent
jump conditions. To relieve concerns about a potential instability for a
large $\Delta t$, the contaminated errors are depicted in Fig. \ref{fig.large} (a)
for $\Delta t$ values up to $\Delta t=5$.
For each $\Delta t$, we choose $N=321$ and $T=10^4 \Delta t$.
This picture shows that
after $10^4$ time steps, all errors remain to be bounded.
This demonstrates the unconditional stability of the matched ADI
algorithm for solving time dependent jump conditions.
As shown in Fig. \ref{fig.ex34} (b), only
when a rather small $\Delta t= 10^{-3}$ is employed,
the matched ADI method begins to converge. Nevertheless, once the
convergence starts, the rate is pretty high. The least-squares fitting shows
that the rate for the descending parts is $r=1.88$ and $r=1.91$, respectively,
for the $L_\infty$ and $L_2$ errors.
Thus, excluding the polluted region,
the matched ADI method yields a second order of accuracy in time.
On the other hand, if the contaminated region was included in the least-squares
analysis, the overall temporal order would become about one.
The spatial orders are not affected by the time dependent jumps.
By using a sufficiently small $\Delta t = 10^{-6}$, the numerical errors
based on different meshes are listed in Table \ref{table.ex34}.
The numerical orders are all around two and are
comparable to those of the Example 3.
This demonstrates the robustness of the proposed MIB interface
treatment in solving time dependent jumps.
The matched ADI solution with $N=81$ and $T=1$ is plotted
in Fig \ref{fig.solu34} (b). The jump values shown in the figure will
oscillate with respect to the time $t$.
\begin{figure*}
\caption{Contour plots of numerical solutions in Example 5.
Here $N=81$ and $T=1$.
(a). Two leaves case;
(b). Four leaves case.
}
\label{fig.solu5}
\end{figure*}
{\bf Example 5.}
At last, we explore the performance of the proposed matched ADI algorithm
for interfaces of general shape.
To this end, the following interface which is parameterized with the polar angle $s$
will be studied
\begin{equation}
\Gamma: \quad r= \frac{1}{2} + b \sin (m s), \quad s \in [0,2 \pi].
\end{equation}
Here the parameter $m$ determines the number of ``leaves'' of the core region
$\Omega^-$ and $b$ controls the magnitude of the curvature.
Two independent cases with parameters $(m,b)=(2,1/4)$ and $(4,1/10)$
are considered. A square domain with $D=0.99$ is also employed.
The resulting configurations of the two leaves and four leaves cases
can be seen from the contour plots of numerical
solutions given in Fig. \ref{fig.solu5}.
It is clear that concave segments or negative curvatures are involved in the
present interfaces.
The analytical solution is constructed as in the Example 4
\begin{equation*}
u(x,y,t)= \begin{cases}
\sin(kx) \cos(ky) \cos(t), \quad \mbox{in } \Omega^- \\
\cos(kx) \sin(ky) \cos(t), \quad \mbox{in } \Omega^+,
\end{cases}
\end{equation*}
with $k=2$.
The source term is also given as
\begin{equation*}
f(x,y,t)= \begin{cases}
(2k^2 \alpha^- \cos(t) -\sin(t)) \sin(kx)\cos(ky), \quad \mbox{in } \Omega^- \\
(2k^2 \alpha^+ \cos(t) -\sin(t)) \cos(kx)\sin(ky), \quad \mbox{in } \Omega^+.
\end{cases}
\end{equation*}
The jump conditions can be similarly calculated according to the analytical solution.
The details are omitted here.
We will rigorously examine the stability of the proposed matched ADI method by
using the four leaves case.
For this purpose, we numerically calculate the leading eigenvalues of the
magnifying matrix ${\bf M}={\bf D}^{-1} {\bf B}$ with the largest magnitudes.
For a given $h$ and $\Delta t$, both ${\bf D}$ and ${\bf B}$ are saved in
a sparse matrix format. The inverse of ${\bf D}$ is carried out by using a
biconjugate gradient iterative solver,
while the eigenvalues are computed by the eigenvalue package ARPACK.
The tolerance is set to be $10^{-14}$ in these algebraic solvers.
Because there are usually multiple leading eigenvalues whose magnitudes
are the same, we will report the largest ten eigenvalues in magnitude,
instead of just one spectral radius. Without the loss of generality, we denote
these ten eigenvalues to be $\lambda_i$ with $i=1,\ldots,10$ and
$|\lambda_i| \ge |\lambda_{i+1}|$.
The spatial-temporal discretization can be
claimed to be stable, if the magnitudes of these leading eigenvalues are all
less than or equal to one, i.e., $|\lambda_i| \le 1$.
\begin{figure*}
\caption{Leading eigenvalues of ${\bf M}
\label{fig.eig}
\end{figure*}
We first study the impact of $\Delta t$ on the stability.
By taking $\alpha^- = 1$, $\alpha^+=10$, and $N=41$, the leading
eigenvalues are shown in Fig. \ref{fig.eig} (a) for seven $\Delta t$ values.
For $\Delta t=1$, $\Delta t=0.1$, and $\Delta t=0.01$, respectively,
there are four, five, and six eigenvalues whose magnitudes equal to one.
The rest leading eigenvalues take much smaller magnitudes. However,
for $\Delta t \le 10^{-3}$, all ten $\lambda_i$ values have almost the
same height in Fig. \ref{fig.eig} (a). Actually, there are just three
$\lambda_i$ with $|\lambda_i|=1$. The magnitudes of other eigenvalues
are strictly less than one, but are very close to one.
Since $|\lambda_i| \le 1$ in all cases,
the corresponding ADI computations are always stable.
We then carry out the similar tests by considering a much larger
$\alpha^+=1000$. The other parameters are chosen to be the same.
It can be seen from Fig. \ref{fig.eig} (b) that the leading eigenvalues
are very similar to those in Fig. \ref{fig.eig} (a).
The only minor change is that
for $\Delta t=1$, $\Delta t=0.1$, and $\Delta t=0.01$, respectively,
there are four, six, and eight eigenvalues whose magnitudes equal to one.
The present analysis validates the stability of the
matched ADI scheme for a large $\alpha^+$.
This study also demonstrates the robustness of the MIB interface method in
handling large jump ratios.
\begin{figure*}
\caption{Leading eigenvalues of ${\bf M}
\label{fig.eig2}
\end{figure*}
We next calculate leading eigenvalues for a different $N$ with a fixed
$\Delta t=1$. The results with $\alpha^- = 1$ and $\alpha^+=10$ will be
reported, while those of $\alpha^+=1000$ are found to be similar.
By considering 20 mesh sizes starting from $N=31$, the eigenvalues are
shown in Fig. \ref{fig.eig2}. With a different $N$, the number of the
largest eigenvalues with the same magnitude
is at least three, and could sometimes be more than ten.
In general, the dependence of this number with respect to $N$
is quite random.
This agrees with our previous discussion that
the geometry of a complicated interface $\Gamma$ will
affect the spectrum of ${\bf M}$ in a random way,
because the intersection of $\Gamma$ with a mesh of size $N$ by $N$ is
quite arbitrary. With $|\lambda_i| \le 1$ for all $N$ values,
the present study demonstrates that such a geometry effect will not
compromise the unconditional stability of the matched ADI method.
\begin{figure*}
\caption{Temporal convergence tests of Example 5.
(a). Two leaves case;
(b). Four leaves case.
}
\label{fig.ex5}
\end{figure*}
After establishing the stability of the matched ADI scheme,
we next examine the temporal convergence by considering
$\alpha^-=1$ and $\alpha^+=10$.
By using $N=321$, $T=1$ and different $\Delta t$ values,
the $L_\infty$ and $L_2$ errors for both
two leaves and four leaves cases are shown in Fig. \ref{fig.ex5}.
The temporal convergences of these two cases are very similar to that of Example 4,
because the present jump conditions are also time dependent.
In particular, the convergence is polluted for large $\Delta t$ values.
Similarly, the contaminated errors remain to be bounded for a
long time-stepping, see Fig \ref{fig.large}. (b).
As shown in Fig. \ref{fig.ex5},
the temporal convergence begins only when $\Delta t$ is small enough
and is of second order once occurs.
The overall temporal order would be also around one, if the polluted region was also
included in the least-squares analysis.
\begin{table}[!t]
\caption{Spatial convergence tests for Example 5. }
\label{table.ex5}
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|}
\hline
& \multicolumn{4}{c|}{Two leaves case} & \multicolumn{4}{c|}{Four leaves case} \\
\cline{2-5} \cline{6-9}
& \multicolumn{2}{c|}{$L_\infty$} & \multicolumn{2}{c|}{$L^2$}
& \multicolumn{2}{c|}{$L_\infty$} & \multicolumn{2}{c|}{$L^2$} \\
\cline{2-3} \cline{4-5} \cline{6-7} \cline{8-9}
$N$ & error & order & error & order & error & order & error & order \\
\hline
21 & 3.06E-3 & & 7.49E-4 & & 5.97E-3 & & 1.48E-3 & \\
41 & 5.37E-4 & 2.51 & 1.47E-4 & 2.35 & 2.48E-3 & 1.27 & 5.83E-4 & 1.34 \\
81 & 1.80E-4 & 1.58 & 4.59E-5 & 1.68 & 9.95E-4 & 1.32 & 1.63E-4 & 1.84 \\
161 & 3.92E-5 & 2.20 & 1.03E-5 & 2.16 & 1.32E-4 & 2.91 & 1.86E-5 & 3.13 \\
321 & 1.08E-5 & 1.86 & 2.55E-6 & 2.01 & 4.01E-5 & 1.72 & 6.57E-6 & 1.50\\
\hline
\end{tabular}
\end{center}
\end{table}
We finally study the spatial convergence with $\alpha^-=1$ and $\alpha^+=10$.
By taking $\Delta t=2.5 \times 10^{-6}$ and $\Delta t= 10^{-5}$, respectively, for the
two leaves case and four leaves case, the spatial errors are reported in
Table \ref{table.ex5}.
It can be observed that when the geometrical structure
becomes more complicated or the number of leaves $m$ is larger,
the convergence pattern of the matched ADI method becomes more oscillatory
and the overall numerical order becomes slightly smaller.
Nevertheless, the proposed matched ADI method still can secure a second order
of accuracy for these challenging parabolic interface problems of complicated
geometry.
The contour plots of the matched ADI solutions based on a mesh $N=81$
at time $T=1$ are illustrated in Fig. \ref{fig.solu5}. The solutions clearly undergo
sharp changes across the interface $\Gamma$, and such changes are time
variant.
\section{Conclusion}\label{Sec:Conclusion}
This paper presents a novel matched ADI method for solving parabolic interface
problems with general jump conditions and complex geometries.
The development of accurate and stable ADI schemes for such interface problems
is essentially an open problem, because the existing IIM-ADI schemes
require the second order jump conditions and have difficulties to correct finite difference
approximations based on general flux jumps.
The second order jump conditions are not needed in our interface treatment.
Moreover, a novel tensor product decomposition is proposed
to decouple 2D jump conditions into essentially 1D ones. By enforcing these 1D conditions,
rigorous 1D MIB schemes are developed to treat regular interfaces and corner interfaces.
The resulting matched ADI scheme achieves second order of accuracy in space
and first order of accuracy in time for interfaces of different shapes.
The efficiency of the ADI scheme is well maintained, because the MIB interface treatment
needs to be conducted only once at the beginning of the computation and fast algebraic
solvers are developed for perturbed tridiagonal systems.
Stability analysis by means of the numerical spectrum analysis of the magnifying
matrix is conducted to examine the impact of geometry and various parameters.
The matched ADI scheme is found to be unconditional stable with all
numerical eigenvalues having magnitude less than or equal to one.
However, the stability proof of the matched ADI method
remains to be an open question,
because the finite difference weights of the MIB discretization
depend on the interface geometry in an unpredictable manner.
In our preliminary studies,
the direct application of the present MIB spatial discretization with the
Peaceman-Rachford ADI method is found to be conditionally stable.
The development of robust matched ADI schemes with second order in time and
for more general parabolic equations is currently under our investigation.
\centerline{\bf Acknowledgment}
\noindent
This work was supported in part by NSF grants DMS-1016579 and DMS-1318898,
and the University of Alabama Research Stimulation Program (RSP) award.
\end{document}
|
\begin{document}
\title{Generalized guidance equation for peaked quantum solitons: the single
particle case\footnote{To appear in the annales de la Fondation de Broglie in 2017, under the title ``de Broglie double solution and self-gravitation.''}}
\date{}
\author{}
\maketitle
\centerline{Thomas Durt\footnote{ Aix Marseille Universit\'e, CNRS,
Centrale Marseille, Institut Fresnel UMR 7249,13013 Marseille, France.email: [email protected]}}
\abstract{ We study the Schr\"odinger-Newton equation (a generalisation of the linear Schr\"odinger equation, which contains a self-focusing non-linear interaction of gravitational nature) in the light of de Broglie's double solution program. In particular we consider solutions of the Schr\"odinger-Newton equation which obey the so-called factorisation ansatz \cite{new} according to which the full wave function is the product of a smoothly varying function with a peaked self-collapsed soliton. We show that these solitons obey a generalized de Broglie-Bohm guidance equation where the smooth function plays the role of the pilot-wave. We derive an Ehrenfest-like theorem for the Schr\"odinger-Newton equation and conjecture the existence of a stochastic subquantum medium in order to explain departures from classical trajectories.}
\section{Introduction}
{\bf Self-gravitational interaction.}
The so-called Schr\"odinger-Newton (S-N) equation\footnote{This equation is also often referred to as the (attractive) Schr\"odinger-Poisson equation \cite{Brezzi, Illner, Arriola} or the gravitational Schr\"odinger equation \cite{Jones}. } \cite{Jones} reads
\begin{equation}
{i}\hbar\frac{\partial\Psi(t,{{\bf x}})}{\partial t}=-\hbar^2\frac{\Delta\Psi(t,{{\bf x}})}{2m}
-Gm^2\int {d}^3 x'(\frac{|\Psi(t,{{\bf x}'})|^2}{|{{\bf x} -x'}|})\Psi(t,{{\bf x}}),\langlebel{NS}
\end{equation}where $G$ represents the Newton gravitational constant and $m$ the mass of a quantum object.
It has been intensively studied in the past, due to the self-focusing character of the non-linear potential which could possibly explain the wave function collapse, seen in this context as a self-localisation process \cite{diosi84,penrose}.
Our prior motivation is to study the manifestations of self-gravitation in presence of an external potential $V^L$ where $V^L$ represents the external potentials that are commonly considered when solving the linear Schr\"odinger equation (for instance electro-magnetic potentials). Contrary to the self-gravitational interaction which non-linearily depends on $\Psi$, $V^L$ does not depend on $\Psi$. It is thus represented by a self-adjoint operator, linearily acting on the Hilbert space, as usually.
We shall thus assume that at the same time an external potential $V^{L}$ acts on the particle together with a non-linear self-focusing potential $V^{NL}$ of gravitational nature:
\begin{equation}
{i}\hbar\frac{\partial\Psi(t,{{\bf x}})}{\partial t}=-\hbar^2\frac{\Delta\Psi(t,{{\bf x}})}{2m}
+V^{L}(t,{{\bf x}})\Psi(t,{{\bf x}})+V^{NL}(\Psi)\Psi(t,{{\bf x}}),\langlebel{nonfreeNL}
\end{equation}
with $V^{NL}(\Psi)=-Gm^2\int {d}^3 x'(\frac{|\Psi(t,{{\bf x}'})|^2}{|{{\bf x} -{{\bf x'}}}|})$, in the case of elementary particles\footnote{\langlebel{coco}It is worth noting that other choices of non-linear self-interaction are possible \cite{Fargue,old,CDW}, which lead to essentially the same results as those derived in the present paper. For instance, in the case of an homogeneous sphere, whenever the mean width of the center-of-mass wave function is small enough in comparison to the size of the sphere, the gravitational self-interaction reduces, in a first approximation, to a non-linear harmonic potential (see \cite{diosi84,Chen,CDW}). Indeed, $-Gm^2/|{{\bf x} -{{\bf x'}}}|$ must be replaced in this case by
$-G({M\over {4\pi R^3\over 3}})^2\int_{|\tilde x| \leq R, |\tilde x'| \leq R} {d}^3 \tilde x {d}^3 \tilde x'\frac{1}{|{{\bf x}}_{CM} +\tilde{{\bf x}}-({{\bf x}}_{CM}'+\tilde {{\bf x}}')|}$
$\approx {GM^2 \over R}(-\frac{6}{5}+\frac{1}{2}(\frac{|{{\bf x}}_{CM}-{{\bf x}}'_{CM}|}{R})^2+{\cal O}((\frac{|{{\bf x}}_{CM}-{{\bf x}}'_{CM}|}{R})^3))$.}.
If, for instance, we consider an electron, the self-gravitational potential is usually considered to be very weak and treated as a perturbation. The self-collapsed ground state (\ref{Choquard}) of (\ref{NS}) is usually predicted (see details in appendix) to be normalized to unity, in which case its size of the order of $\hbar^2/Gm^3$ (more or less 10$^{32}$ meter in the case of an electron). The corresponding ground state energy (\ref{Choquard-energy}) is of the order of $G^2m^5/\hbar^2$ which is very small (for instance, in the case of an electron it is very small compared to the usual energies of electronic orbitals in an atom). Now, this result is obtained by assuming that the norm of the ground state is equal to unity, which, following de Broglie, we consider to be a superfetatory condition (as discussed with more detail in the last section). Making use of the well-known scaling properties of equation (\ref{NS}) (see the review paper by S. Colin, T.D. and R. Willox (same issue) and also \cite{CDW,vanMeter}) , it is actually possible to reduce arbitrarily the size of the self-collapsed ground state; at the same time, its norm will increase to plus infinity and its energy will tend to minus infinity.
{\bf Non-standard normalisation.}
The first non-standard ingredient of our work is that we choose not to normalize to unity the L$_2$ norm of the wave function. Instead, we impose to begin with that the size of the self-collapsed ground state is very small (of the order of the Schwarzschild radius $Gm/c^2$, thus of the order of 10$^{-57}$ meter in the case of an electron\footnote{We were led to this choice by studying certain implications of our work that go beyond the scope of the present paper \cite{new}. Making use of the scaling properties of (\ref{NS}) \cite{CDW}, the norm of the ground state is of the order of $(mc^2\hbar^2/G^2m^5)^{1/3}\approx 10^{30}$ in the case of an electron.}). The ground state energy is then of the order of $-mc^2$. Contrary to the usual approach, in which external potentials are strong compared to the self-gravitational potential, in our case, the external potentials are supposedly weak, while the self-gravitational potential is strong and we treat the former as a perturbation. In particular the ground state will supposedly be very stable, because the stability analysis of the S-N equation in absence of external potential (\ref{NS}) shows that the non-linearity will inhibit the spreading unless the kinetic energy is at least of the order of the ground state energy \cite{Arriola,vanMeter,CDW}. We shall take for granted, without demonstration, that this is still approximately true in presence of an external potential $V^L$. Therefore we expect that in first approximation the state of the particle is, up to galilean boosts and translations, the static ground state, solution of (\ref{Choquard}). Moreover, the spectrum of negative energy solutions of (\ref{Choquard}) is discrete (see \cite{Bernstein} and appendix), so that at usual temperatures, the transitions to excited static self-collapsed states are frozen.
What we are looking for is thus a solution of (\ref{nonfreeNL}) in the form of a soliton (solitary wave) which, in first approximation, looks like the self-collapsed ground state of (\ref{NS}). In the rest of the paper we shall identify this highly concentrated peak of energy with the quantum particle itself, which is consistent with the aforementioned stability criterion: stability is menaced whenever an energy of the order of $mc^2$ is communicated to the system. It is worth noting that in our approach, contrary to the mainstream approach to self-localisation \cite{diosi84,penrose,CDW,arxiv}, the system is supposedly collapsed to begin with, since arbitrary long times.
This picture is reminiscent of the so-called de Broglie-Bohm (dB-B) causal interpretation of quantum mechanics. In particular, the fact that the solution is expected to have a very large norm and amplitude is reminiscent of de Broglie's double solution program\footnote{Louis de Broglie proposed in 1927 a realistic interpretation of the quantum theory in which particles are guided by the solution of the linear Schr\"odinger equation ($\Psi_L$), in accordance with the so-called guidance equation \cite{debrogliebook,debroglieend}. The theory was generalised by David Bohm in 1952 \cite{bohm521,bohm522}. Certain ingredients of de Broglie's original idea disappeared in Bohm's formulation, in particular the double solution program, according to which the particle is associated to a wave $u$ distinct from the pilot-wave $\Psi_L$. This program was never fully achieved, $u$ being sometimes treated as a moving singularity \cite{vigier}, and sometimes as a solution $\phi_{NL}$ of a non-linear equation (see \cite{debroglieend,Fargue} and the papers Fargue, and of Colin, Durt and Willox, same issue).}, who wrote \cite{debrogliebook}
{\it`` ... a set of two coupled solutions of the wave equation: one, the $\Psi$ wave, definite in phase, but, because of the continuous character of its amplitude, having only a statistical and subjective meaning; the other, the $u$ wave of the same phase as the $\Psi$ wave but with an amplitude having very large values around a point in space and which ($\cdots$) can be used to describe the particle objectively."}
The fact that in our approach the particle, represented by $\phi_{NL}$, has a very small size is reminiscent of Bohm's description of particles as material points.
Our approach is also reminiscent of Poincar\'e's attempts \cite{Poincar} to explain the stability of the electron in terms of an internal self-attraction (the so-called Poincar\'e pressure), aimed at counterbalancing Coulomb self-repulsion. In our model, self-gravitation plays the role of the Poincar\'e pressure, and it counterbalances the spread of the soliton that we identify with the quantum particle\footnote{Several physicists of the de Broglie school, Fer, Lochak, Andrade e Silva, Lochak and others developed in the past models mixing Poincar\'e and de Broglie views on the stability of particles (see for instance \cite{debroglieend,Fargue,IHP} and references therein as well as the papers of Fargue, Drezet and Colin, Durt and Willox, same issue)). These ideas are in a sense unavoidable whenever we try and describing particles as localised waves.}.
{\bf Factorization ansatz.}
In a first step, we tried to find a double solution {\it \`a la} de Broglie in the form of the sum of a wave function $\Psi_L$ (where $\Psi_L$ is a solution of the linear Schr\"odinger equation (\ref{S1V})) and of a soliton $\phi_{NL}$. However, due to the intrinsic non-linearity of (\ref{nonfreeNL}), we did not manage to derive interesting results.
This brings us to the second non-standard ingredient of our paper which is that we tried to solve (\ref{nonfreeNL}) with an ansatz solution $\Psi$ which factorizes (\ref{ansatz}) into the product of two functions $\Psi_L$ and $\phi_{NL}$:
\begin{equation}\Psi(t,{{\bf x}})=\Psi_L(t,{{\bf x}})\cdot \phi_{NL}(t,{{\bf x}}), \langlebel{ansatz}\end{equation}
for which we imposed that $\Psi_L$, the linear wave, is a solution of the linear Schr\"odinger equation (\ref{S1V}):
\begin{equation}a &&{i}\hbar\cdot \frac{\partial \Psi_L(t,{\bf x})}{\partial t}=
-\frac{\hbar^2}{2m}\Delta\Psi_L(t,{{\bf x}})+V^{L}(t,{{\bf x}})\Psi_L(t,{{\bf x}}),\langlebel{S1V}\end{equation}a
The factorization ansatz results from the recognition that, due to the fundamental non-linearity of the wave dynamics, a linear partition of the type $\Psi(t,{{\bf x}})=\Psi_L(t,{{\bf x}})+ \phi_{NL}(t,{{\bf x}})$ is irrelevant. From this point of view the factorization ansatz incorporates non-linearity from the beginning.
Originally, this ansatz has been introduced by us \cite{new} in order to describe the phenomenology of ``walkers'' (also called bouncing oil droplets\footnote{These are macroscopic objects that exhibit certain quantum-like features. In particular their average trajectories seemingly obey a pseudo dB-B dynamics. In ref.\cite{new}, we simulated the properties of bouncing oil droplets by representing through $\Psi_L(t,{{\bf x}})$ the medium (oil bath) on which droplets propagate and through $\phi_{NL}(t,{{\bf x}})$ the droplets themselves. We derived in that paper an expression for the pseudo-gravitational interaction between two droplets, assuming from the beginning that dB-B guidance equation (\ref{dBB}) was satisfied. In the present paper, we focus on the single particle (droplet) case. We aim here at deriving the dB-B guidance equation from the ansatz (\ref{ansatz}).}). In the case of droplets, our basic motivation for imposing the factorization ansatz is that walkers prepared at different positions and represented by $\phi^i_{NL}(t,{{\bf x}}) (i=1,2...)$ always see the same bath (environment) represented by $\Psi_L(t,{{\bf x}})$. In the same paper \cite{new}, we extended this idea to arbitrary quantum systems, for instance to elementary particles and/or atoms molecules and so on.
In our (wave monist) approach, $\Psi(t,{{\bf x}})$ is assumed to represent the full reality of the quantum system. According to our factorization ansatz $\Psi(t,{{\bf x}})$ can be split into the particle represented by $\phi_{NL}(t,{{\bf x}})$ and in the ``linear'' wave represented by $\Psi_L(t,{{\bf x}})$. The spatial size of the particle being assumed to be extremely small (every particle is a tiny black hole in our approach \cite{new}), the experimenter has supposedly no direct access to/control on their location. In our view, identical experimental preparations however result in the same value for $\Psi_L(t,{{\bf x}})$, which, considered so, does not represent the full information about the system but the information accessible to and controllable by the experimentalist. As we shall show, $\Psi_L$ can be interpreted as a pilot-wave, while $\phi_{NL}$ behaves as a solitary wave moving, in good approximation, in accordance with a generalized dB-B guidance equation (\ref{dBB}). Roughly summarized, our main results are the following:
{\bf Property 1}
whenever $\phi_{NL}$ remains peaked throughout time in a sufficiently small region, its barycentre (from now on denoted ${\bf x_0}$) obeys, in good approximation, the generalized guidance equation
\begin{equation}a{\bf v}_{drift}&=&{\hbar \over m}{{\bf \bigtriangledown}}\varphi_L({{\bf x_0}}(t),t)+{<\phi_{NL}| {\hbar\over i m}{\bf \bigtriangledown}|\phi_{NL}>\over <\phi_{NL}|\phi_{NL}>} \nonumber \\&=&{\bf v}_{dB-B}+{\bf v}_{int.},\langlebel{drift}\end{equation}a
which contains the well-known Madelung-de Broglie-Bohm contribution (${\bf v}_{dB-B}={\hbar \over m}{{\bf \bigtriangledown}}\varphi_L({{\bf x_0}}(t),t)$) plus a new contribution due to the internal structure of the soliton (${\bf v}_{int.}={<\phi_{NL}| {\hbar\over i m}{\bf \bigtriangledown}|\phi_{NL}>\over <\phi_{NL}|\phi_{NL}>}$).
{\bf Property 2}
Denoting $\Psi_L= A_Le^{i\varphi_L}$, where $A_L$ is a real amplitude and $\varphi_L$ a real phase, and defining $\phi'_{NL}\equiv\phi_{NL}/A_L$ we find that the L$_2$ norm of $\phi'_{NL}$ remains constant throughout time in good approximation, while
the solution $\Psi$ obeys
\begin{equation}a \Psi(t,{\bf x})\approx e^{i\varphi_L(t,{\bf x})}\phi'_{NL}(t,{\bf x}),\end{equation}a
which confirms indirectly de Broglie's double solution program according to which the linear wave $\Psi_L$ does not represent the particle. Here $\Psi$ represents the particle and it is essentially equal to the product of the soliton $\phi'_{NL}$ (which plays the role of de Broglie's second solution here) with $e^{i\varphi_L}$, the phase of the ``pilot-wave'' $\Psi_L$. $A_L$, the amplitude of the linear wave function, plays here the role of an auxiliary computation tool.
What we shall not prove rigorously in the present paper is the stability of the soliton, in the sense that we assume from the beginning that the soliton remains peaked throughout time, due to the self-focusing nature of the non-linear self-interaction to which it is submitted. However, if this stability condition is satisfied, then the properties 1 and 2 can be established by lengthy but straightforward computations that we shall detail in the core of the paper. Actually, we independently established by numerical simulations, in a special case (homogeneous self-gravitating sphere in the case where the extent of the ground state is quite smaller than the radius of the sphere), that stability is de facto guaranteed while properties 1 and 2 are satisfied in very good approximation.
The paper is structured as follows.
In section \ref{notfree} we derive the aforementioned properties 1 and 2, concerning the velocity of the barycentre of $\phi_{NL}$ and its scaling. In section \ref{numeric} we present the confirmations of properties 1 and 2 obtained from numerical simulations. Those simulations also confirm the stability of the soliton, provided the self-focusing is strong enough. Moreover they show that in the classical, non-relativistic regime, the trajectories of the solitons are classical and do not obey the de Broglie guidance law. This property can be explained in terms of a generalized Ehrenfest's theorem. This leads us to formulate in section \ref{conjecture} a conjecture (main conjecture) according to which de Broglie guidance equation is valid ``in average'' due to the presence of an external stochastic field acting at the level of individual velocities. The last section is devoted to discussions and conclusions. In appendix (section \ref{Diracsect}), we attempt to generalize the previous results to Dirac's equation.
\section{\langlebel{notfree}Factorisability ansatz, solitary waves and generalized dB-B guidance.}
We now assume that at the same time an external potential acts on the particle together with a non-linear self-focusing potential of a gravitational nature. We simultaneously impose the factorisability ansatz. Therefore equations (\ref{nonfreeNL},\ref{ansatz}) are valid.
Substituting (\ref{ansatz}) in (\ref{nonfreeNL}) we get
\begin{equation}a &&{i}\hbar\cdot (( \frac{\partial \Psi_L(t,{\bf x})}{\partial t})\phi_{NL}(t,{\bf x})+\Psi_L(t,{{\bf x})}\cdot (\frac{\partial \phi_{NL}(t,{{\bf x}})}{\partial t}))=\nonumber\\ \langlebel{ansatz2}
&&-\frac{\hbar^2}{2m}\Delta\Psi_L(t,{{\bf x}})\cdot \phi_{NL}(t,{{\bf x}})\nonumber \\
&&-\frac{\hbar^2}{2m}(2{\bf \bigtriangledown} \Psi_L(t,{{\bf x}}) \cdot {\bf \bigtriangledown} \phi_{NL}(t,{{\bf x}}) +\Psi_L(t,{{\bf x}})\cdot \Delta\phi_{NL}(t,{{\bf x}}))\nonumber \\&+&V^L\Psi(t,{{\bf x}})+V^{NL}(\Psi)\Psi(t,{{\bf x}}),\end{equation}a
that, making use of the identity
${\bf \bigtriangledown} \Psi_L(t,{{\bf x}})=({\bf \bigtriangledown} A_L(t,{{\bf x}}))e^{i\varphi_L(t,{{\bf x}})}+\Psi_L(t,{{\bf x}})i{\bf \bigtriangledown}\varphi_L(t,{{\bf x}})$, we replace by a system of two equations\footnote{This replacement is not one to one in the sense that there could exist solutions of equation (\ref{ansatz2}) that do not fulfill the system (\ref{S1V},\ref{S2}). In any case, we focus on a particular class of solutions here.}:
-the linear Schr\"odinger equation
\begin{equation}a &&{i}\hbar\cdot \frac{\partial \Psi_L(t,{\bf x})}{\partial t}=
-\frac{\hbar^2}{2m}\Delta\Psi_L(t,{{\bf x}})+V^{L}(t,{{\bf x}})\Psi_L(t,{{\bf x}}),\nonumber\end{equation}a
and the non-linear equation
\begin{equation}a &&{i}\hbar\cdot \frac{\partial \phi_{NL}(t,{{\bf x}})}{\partial t}=\nonumber
-\frac{\hbar^2}{2m}\cdot \Delta\phi_{NL}(t,{{\bf x}}) \\& -&\frac{\hbar^2}{m}\cdot (i{\bf \bigtriangledown} \varphi_L(t,{{\bf x}}) \cdot {\bf \bigtriangledown} \phi_{NL}(t,{{\bf x}})+\frac{{\bf \bigtriangledown} A_L(t,{{\bf x}})}{A_L(t,{{\bf x}})} \cdot {\bf \bigtriangledown} \phi_{NL}(t,{{\bf x}}))\nonumber\\ &+&V^{NL}(\Psi)\phi_{NL}(t,{{\bf x}})\langlebel{S2} \end{equation}a
In order to solve the system of equations (\ref{S1V},\ref{S2}), it is worth noting that while the L$_2$ norm of the linear wave $ \Psi_L$ is preserved throughout time, because (\ref{S1V}) is unitary, this is no longer true in the case of the non-linear wave $\phi_{NL}$, because the terms mixing $\Psi_{L}$ and $\phi_{NL}$ are not hermitian.
By a straightforward but lengthy computation that we reproduce integrally in appendix, we established the following result:
The change of norm of $\phi_{NL}$ obeys
\begin{equation}a{d <\phi_{NL}|\phi_{NL}>\over dt}\approx \frac{\hbar}{m}{ \Delta} \varphi_L(t,{{\bf x_0}}) \cdot <\phi_{NL} |\phi_{NL}>\nonumber\\ -2\frac{{\bf \bigtriangledown} A_L(t,{{\bf x_0}})}{A_L(t,{{\bf x_0}})}\cdot
\int d^3{\bf x} (\phi_{NL}(t,{{\bf x}}))^*\frac{\hbar\bf \bigtriangledown} {mi } \cdot \phi_{NL}(t,{{\bf x}}).\langlebel{normchangebis}\end{equation}a
\subsection{Property 1}
Let us now consider the barycentre ${\bf x_0}$ of the soliton: ${\bf x_0}\equiv { <\phi_{NL}| {\bf x}|\phi_{NL}>\over <\phi_{NL}|\phi_{NL}>} $ in order to estimate its velocity ${\bf v}_{drift}$:
\begin{equation}a {\bf v}_{drift}\equiv{d({ <\phi_{NL}| {\bf x}|\phi_{NL}>\over <\phi_{NL}|\phi_{NL}>} ) \over dt }\langlebel{defdrift}\end{equation}a
For instance, if we consider its $z$ component:
$z_0={<\phi_{NL}|z|\phi_{NL}>\over <\phi_{NL}|\phi_{NL}>}$ and
${d z_0\over dt}={1\over <\phi_{NL}|\phi_{NL}>}{d<\phi_{NL}|z|\phi_{NL}>\over dt}-{z_0\over <\phi_{NL}|\phi_{NL}>}{d<\phi_{NL}|\phi_{NL}>\over dt},$
so that we find (making use of (\ref{normchangebis}) as well as of results in appendix, section \ref{appendicit})
\begin{equation}a &&{d z_0\over dt}= \nonumber {1\over <\phi_{NL}|\phi_{NL}>} \int d^3{\bf x} (\phi_{NL}(t,{{\bf x}}))^*\frac{\hbar\bf \bigtriangledown_z} {mi } \cdot \phi_{NL}(t,{{\bf x}})
\\ &+&{1\over <\phi_{NL}|\phi_{NL}>} \int d^3{\bf x} (\phi_{NL}(t,{{\bf x}}))^*(\frac{\hbar\bf \bigtriangledown_z} {m } \cdot \varphi_{L}(t,{{\bf x}})) \phi_{NL}(t,{{\bf x}})\nonumber
\\&+& \nonumber {1\over <\phi_{NL}|\phi_{NL}>} <\phi_{NL}|(\frac{\hbar}{m}{ \Delta} \varphi_L(t,{{\bf x}}) ) \cdot z|\phi_{NL}>
\\ &+& {1\over <\phi_{NL}|\phi_{NL}>} \frac{\hbar}{im}\int d^3{\bf x} \frac{{\bf \bigtriangledown} A_L(t,{{\bf x}})}{A_L(t,{{\bf x}})} \cdot {\bf \bigtriangledown} (\phi_{NL}(t,{{\bf x}}))^*\cdot z \cdot \phi_{NL}(t,{{\bf x}})
\nonumber
\\ &-& {1\over <\phi_{NL}|\phi_{NL}>} \frac{\hbar}{im}\int d^3{\bf x} (\phi_{NL}(t,{{\bf x}}))^*\cdot z \cdot \frac{{\bf \bigtriangledown} A_L(t,{{\bf x}})}{A_L(t,{{\bf x}})} \cdot {\bf \bigtriangledown} \phi_{NL}(t,{{\bf x}})) \nonumber
\\&-&{z_0\over <\phi_{NL}|\phi_{NL}>}\cdot (\frac{\hbar}{m}){ \Delta} \varphi_L(t,{{\bf x_0}}) \cdot<\phi_{NL} |\phi_{NL}>\langlebel{trip}
\\ \nonumber&+&2{z_0\over <\phi_{NL}|\phi_{NL}>}\frac{{\bf \bigtriangledown} A_L(t,{{\bf x_0}})}{A_L(t,{{\bf x_0}})}\cdot \int d^3{\bf x} (\phi_{NL}(t,{{\bf x}}))^*\frac{\hbar\bf \bigtriangledown} {mi } \cdot \phi_{NL}(t,{{\bf x}}))\end{equation}a
Now, $\frac{\hbar}{im}\int d^3{\bf x}(\phi_{NL}(t,{{\bf x}}))^*\cdot z \cdot \frac{{\bf \bigtriangledown} A_L(t,{{\bf x}})}{A_L(t,{{\bf x}})} \cdot {\bf \bigtriangledown} \phi_{NL}(t,{{\bf x}})$
$\approx z_0 \frac{{\bf \bigtriangledown} A_L(t,{{\bf x_0}})}{A_L(t,{{\bf x_0}})}\int d^3{\bf x} (\phi_{NL}(t,{{\bf x}}))^*\frac{\hbar\bf \bigtriangledown} {mi } \cdot \phi_{NL}(t,{{\bf x}})) $,
while
$<\phi_{NL}|(\frac{\hbar}{m}){ \Delta} \varphi_L(t,{{\bf x}}) \cdot z|\phi_{NL}>$ $\approx z_0\cdot (\frac{\hbar}{m}){ \Delta} \varphi_L(t,{{\bf x_0}}) \cdot <\phi_{NL}|\phi_{NL}>$ and so on so that finally only the two first lines of (\ref{trip}) survive. We get thus the generalized dB-B guidance equation (\ref{drift}), which constitutes the
{\bf Property 1}:
\begin{equation}a{\bf v}_{drift}&=&{\hbar \over m}{{\bf \bigtriangledown}}\varphi_L({{\bf x_0}}(t),t)+{<\phi_{NL}| {\hbar\over i m}{\bf \bigtriangledown}|\phi_{NL}>\over <\phi_{NL}|\phi_{NL}>} \nonumber \\&=&{\bf v}_{dB-B}+{\bf v}_{int.}.\nonumber\end{equation}a
${\bf v}_{drift}$ contains the de Broglie-Bohm velocity
\begin{equation}a {\bf v}_{dB-B}\equiv{\hbar \over m}{{\bf \bigtriangledown}}\varphi_L({{\bf x_0}}(t),t),\langlebel{dBB},\end{equation}a
and the internal velocity
\begin{equation}a{\bf v}_{int.}\equiv{<\phi_{NL}| {\hbar\over i m}{\bf \bigtriangledown}|\phi_{NL}>\over <\phi_{NL}|\phi_{NL}>}.\langlebel{int.}\end{equation}a
(\ref{dBB}) is nothing else than de Broglie-Bohm's guidance equation \cite{Holland}, while ${\bf v}_{int.}$ can be considered as a contribution to the average velocity originating from the internal structure of the soliton. Both contributions to the drift are evaluated at the barycentre of the soliton, ${\bf x_0}$.
\subsection{Property 2}
Let us now consider the change of norm of $\phi_{NL}$.
To do so, we introduce the total time derivative of $A_L$ (${d A_L\over dt}={\partial A_L\over \partial t}+{\bf v}_{drift}\cdot {{\bf \bigtriangledown}} A_L$)
where ${\bf v}_{drift}={{d <\phi_{NL}| {\bf x}|\phi_{NL}> \over <\phi_{NL}|\phi_{NL}> }\over dt}$ obeys the generalized dB-B guidance equation (\ref{drift}).
By a direct computation, we find
\begin{equation}a {{d A_L\over dt}\over A_L}={1\over A_L}({\partial A_L\over \partial t}+{{\bf \bigtriangledown}} A_L\cdot\frac{\hbar\bf \bigtriangledown} {m } \cdot \varphi_{L}(t,{{\bf x_0}}))+\nonumber\\ {1\over A_L}{{\bf \bigtriangledown}} A_L\cdot{1\over <\phi_{NL}|\phi_{NL}>} \int d^3{\bf x} (\phi_{NL}(t,{{\bf x}}))^*\frac{\hbar\bf \bigtriangledown} {mi } \cdot \phi_{NL}(t,{{\bf x}})\langlebel{step}\end{equation}a
Making use of the conservation equation of the linear Schr\"odinger equation ${\partial A^2_L\over \partial t}=-div(A^2\frac{\hbar\bf \bigtriangledown} {m } \cdot \varphi_{L}(t,{{\bf x_0}}))$ we find
${1\over A_L}({\partial A_L\over \partial t}+{{\bf \bigtriangledown}} A_L\cdot\frac{\hbar\bf \bigtriangledown} {m } \cdot \varphi_{L}(t,{{\bf x_0}}))={-1\over 2}div(\frac{\hbar\bf \bigtriangledown} {m } \cdot \varphi_{L}(t,{{\bf x_0}}))$ and we can rewrite (\ref{step}) as follows:
\begin{equation}a {{d A_L\over dt}\over A_L}={-1\over 2}\frac{\hbar}{m}{ \Delta} \varphi_L(t,{{\bf x_0}})\nonumber\\+{{{\bf \bigtriangledown}} A_L\over A_L}\cdot{1\over <\phi_{NL}|\phi_{NL}>} \int d^3{\bf x} (\phi_{NL}(t,{{\bf x}}))^*\frac{\hbar\bf \bigtriangledown} {mi } \cdot \phi_{NL}(t,{{\bf x}})\langlebel{step2}\end{equation}a
Making use of (\ref{normchangebis}) (derived in appendix, section \ref{appendicit}), we obtain at the end
${{d A_L\over dt}\over A_L}={-1\over 2}{1\over <\phi_{NL}|\phi_{NL}>}{d <\phi_{NL}|\phi_{NL}>\over dt}$ so that, finally,
\begin{equation}{{d<\phi_{NL}|\phi_{NL}>\over dt}\over <\phi_{NL}|\phi_{NL}>}= -2{{d A_L\over dt}\over A_L}.\langlebel{scalcons}\end{equation}
From the constraint (\ref{scalcons}) we infer the
{\bf Property 2}
\begin{equation}a{<\phi_{NL}|\phi_{NL}>(t)\over <\phi_{NL}|\phi_{NL}>(t=0)}={A^2_L(t=0)\over A^2_L(t)},\end{equation}a where we evaluate $A^2_L(t)$ at the barycentre of $\phi_{NL}$, which moves according to the generalized dB-B guidance equation (\ref{drift}).
Let us rescale $\phi_{NL}(t,{{\bf x}})$ by defining $\phi'_{NL}$ through $\phi_{NL}(t,{{\bf x}})\equiv \phi'_{NL}(t,{{\bf x}})/A_L$;
we can thus predict in general that, if it exists and remains peaked during its evolution, the solution of (\ref{S2}) has the form
\begin{equation}\Psi(x,y,z,t) \approx \phi'_{NL}({\bf x},t)e^{i\varphi_L({\bf x},t)},\end{equation} where $\phi'_{NL}({\bf x},t)$ is centred in $ {\bf x}_0(t=0)+\int_0^t dt {\bf v}_{drift}$ and is of constant L$_2$ norm.
Now, $V^{NL}(\Psi)=V^{NL}(\phi'_{NL}(t,{{\bf x}}))$, in virtue of (\ref{Coulomb}) (appendix) and (\ref{S2}) can then be cast in the form
\begin{equation}a &&{i}\hbar\cdot \frac{\partial ( \phi'_{NL}(t,{{\bf x}})/A_L(t,{{\bf x_0}}))}{\partial t}\nonumber
=\\ \nonumber
&&-\frac{\hbar^2}{2m}\cdot \Delta ( \phi'_{NL}(t,{{\bf x}})/A_L(t,{{\bf x_0}}))+V^{NL}(\phi'_{NL})( \phi'_{NL}(t,{{\bf x}})/A_L(t,{{\bf x_0}}))\nonumber\\ &&-\frac{\hbar^2}{m}\cdot i{\bf \bigtriangledown} \varphi_L(t,{{\bf x}}) \cdot {\bf \bigtriangledown} (\phi'_{NL}(t,{{\bf x}})/A_L(t,{{\bf x_0}})) \nonumber\\ &&-\frac{\hbar^2}{m}\cdot\frac{{\bf \bigtriangledown} A_L(t,{{\bf x}})}{A_L(t,{{\bf x}})} \cdot {\bf \bigtriangledown} ( \phi'_{NL}(t,{{\bf x}})/A_L(t,{{\bf x_0}})).
\langlebel{S3}\end{equation}a
In order to say more about $\phi'_{NL}({\bf x},t)$ we must solve equation (\ref{S3}) which is a complicated problem, beyond the scope of our paper.
{\bf Remark.}
As has been noted in \cite{old}, when there is no external potential ($V^L$), which means that the particle is only submitted to its self-interaction,
we find the exact solution $\phi^0_{NL}({\bf x}-{\bf v}\cdot t)e^{-i((E_0+\hbar \omega)\cdot t-\hbar {\bf k}\cdot {\bf x})/\hbar}$. This is a plane wave moving at velocity $v_{dB-B}$. As already noted by Fargue \cite{Fargue} many years ago, such solutions exist for a large class of different non-linearities (e.g. a non-linearity proportional to $|\Psi|^2$), and they all agree with the principle of phase concordance proposed by de Broglie in 1927. Actually, this class of solution is well-known and it can be generated from the static solution $\phi^0_{NL}({\bf x})e^{-iE_0 t/\hbar}$ by a Galilean boost. In the present case, this property is seen to be a direct consequence of the Galilean invariance of equation (\ref{NS}).
\section{Main conjecture.}
\subsection{Numerical simulations.\langlebel{numeric}}
{\bf Normalisation and Born rule.}
One could object that in order to fit to the constraints required by our model, in particular in order to ensure that the size of the soliton is quite smaller than the size of the linear wave, the coupling constant ought to be huge. However this is not true. We are free to rescale the solitonic solutions of (\ref{Choquard}) without being constrained by the normalisation to unity of the wave function, which is a condition imposed by the Born rule. In our case, the Born rule is not postulated to begin with, it should rather be derived from the so-called equilibrium condition according to which the statistical distribution of the positions of the particles asymptotically converges in time to the Born distribution in $|\Psi_L|^2$. In our eyes, the equilibrium condition ought to be derived from the generalized dB-B dynamics as is done in e.g. classical chaos theory\footnote{There exist serious attempts to derive the onset of quantum equilibrium from the de Broglie-Bohm mechanics \cite{tony,grec,colin,ColinStruyve,grec2,ab}, but this is a deep and complex problem, reminiscent of the H-theorem of Boltzmann, which opens the door to a Pandora box that we do not wish to open here (see papers of Colin, Durt and Willox, of Drezet and of Efthymiopoulos, same issue).}. It is well-known for instance from the study of deterministic chaotic systems that the sensitivity to initial conditions is an essential ingredient for generating stochasticity and impredictability. This ingredient is present in the dB-B \cite{butterfly} dynamics too.
{\bf Self-gravitating nanosphere in an external one-dimensional harmonic trap.}
We shall now briefly mention some results obtained by us regarding the properties of a self-gravitating nanosphere placed in an harmonic trap, relaxing the constraint on the normalisation of the wave function. For convenience, we treat the system as an homogeneous sphere, in the limit where the extent of the wave function of its center of mass is quite smaller than its radius. In this case, $V^L=k^{ext.} x^2/2$ while (see \cite{diosi84,Chen,CDW} and discussion of footnote \ref{coco}).
\begin{equation}a V^{NL}(x)= {GM^2 \over R}\int {d} x' |\Psi(t,x')|^2(-\frac{6}{5}+\frac{1}{2}(\frac{|x-x'|}{R})^2+{\cal O}((\frac{|x-x'|}{R})^3))\nonumber \\ \approx N^2({-6GM^2 \over 5R}+{GM^2 \over 2R^3}(<x^2>-<x>^2))+{GM^2 \over 2R^3}N^2(x-<x>)^2,\end{equation}a
with $N^2=\int {d} x' |\Psi(t,x')|^2$,
$<x>=\int {d} x' |\Psi(t,x')|^2 x'/N^2$ and $<x^2>=\int {d} x' |\Psi(t,x')|^2 (x')^2/N^2$.
Denoting $k^{self}={GM^2 \over 2R^3}N^2$ and reexpressing the Hamiltonian up to irrelevant position-independent factors, we get the following evolution equation:
\begin{equation}
{i}\hbar\frac{\partial\Psi(t,x)}{\partial t}=-({\hbar^2\over 2m})\frac{\partial^2\Psi(t,x)}{\partial x^2}
+({k^{ext.}\over 2} x^2+{k^{self}\over 2}(x-<x>)^2)\Psi(t,x).\langlebel{Carlos}
\end{equation}
{\bf Numerical simulations.}
The main advantage of this simple model is that it is gaussian: initially gaussian states remain gaussian throughout their temporal evolution under (\ref{Carlos}). Therefore we were able to numerically solve it with high accuracy, which would be impossible with for instance the single particle N-S equation (\ref{NS}) \cite{ducan}.
$N^2$, the square of the $L_2$-norm of $\Psi$ ($N^2=\int {d} x' |\Psi(t,x')|^2$) is a conserved quantity under the evolution (\ref{Carlos}); it is however a free parameter in our approach (in agreement with the discussion of the previous paragraph), which means that we are essentially free to choose the value of the effective self-gravitating constant $k^{self}$. Varying this free parameter, we studied the behavior of gaussian solutions (denoted $\Psi^{NL}$) of the non-linear equation (\ref{Carlos}). We also decomposed the initial gaussian states $\Psi^{NL}(t,x=0)$ according to the ansatz (\ref{ansatz}):
$\Psi^{NL}(x,t=0)=\Psi^{L}(x,t=0)\cdot \phi_{NL}(x-<x>_0,t=0)$,
where $\phi_{NL}(x-<x>_0,t=0)$ is a gaussian real state, of which the shape is close to the shape of the ground state $\phi^0_{NL}(x)$ of the ``free'' equation ${i}\hbar\frac{\partial\Psi(t,x)}{\partial t}=-({\hbar^2\over 2m})\frac{\partial^2\Psi(t,x)}{\partial x^2}
+({k^{self}\over 2}(x-<x>)^2)\Psi(t,x),$ which is \cite{CDW} a real gaussian state centered in $<x>_0$ of extent $\sqrt{\hbar \over \sqrt{k^{self}\cdot m}}$. $\Psi^{L}(x,t)$ and $\Psi^{NL}(x,t)$ were obtained by integrating respectively the linear equation ${i}\hbar\frac{\partial}{\partial t}\Psi^L(t,x)=-({\hbar^2\over 2m})\frac{\partial^2}{\partial x^2}\Psi^L(t,x)
+({k^{ext.}\over 2} x^2)\Psi^L(t,x)\langlebel{linear}$ and equation (\ref{Carlos}). $<x>_0$ was chosen freely, excepted that we imposed that $\Psi^L(x,t=0)=\Psi^{NL}(x,t=0)/\phi^{NL}(x,t=0)$ was a normalisable (gaussian) state. This allowed us to estimate the solitonic wave function $\phi^{NL}(x,t)=\Psi^{NL}(x,t)/\Psi^L(x,t)$ at all times. Our first result concerns stability: our criterion for stability is that $\phi^{NL}(x,t)=\Psi^{NL}(x,t)/\Psi^L(x,t)$ is a normalisable (gaussian) state. This means that the soliton remains peaked during time. We noticed that when $k^{self}/k^{ext.}$ was high enough (larger than 10 was obviously sufficient), stability was ensured. Not surprisingly, in this case, the self-gravitation forces the soliton to oscillate around its center, without ever escaping to the self-focusing gravitational potential ${k^{self}\over 2}(x-<x>)^2$.
The results of our numerical study are encapsulated in the figure \ref{Momo} \cite{Momo} which represents the velocity $v_{drift}\equiv{d\over dt}(<x_{NL}>)$ (with $<x_{NL}>=\int {d}^3 x' |\phi^{NL}(t,x')|^2 x'/N_{NL}^2$) of the centre of the soliton $\phi^{NL}(x,t)$, in function of time.
\begin{figure}
\caption{Plot of $v_{dBB}
\end{figure}
To derive the plot \ref{Momo}, we imposed the following constraints:
$k^{self}/k^{ext.}= 10^3$,
$(<x_{NL}^2>-<x_{NL}>^2)_{t=0}= \sqrt{\hbar \over \sqrt{(k^{ext}+k^{self})\cdot m}}$,
and $(<x_{NL}^2>-<x_{NL}>^2)_{t=0}=10^{-3}(<x_{L}^2>-<x_{L}>^2)_{t=0}$.
We observed that
(i) In the limit where $k^{self}/k^{ext.}$ is large enough, and provided at time $t=0$ the extent of the soliton $(<x_{NL}^2>-<x_{NL}>^2)_{t=0}$ is comparable to the extent of the self-focused ground state ($\sqrt{\hbar \over \sqrt{k^{self}\cdot m}}$), and also quite smaller than the extent of the linear wave ($(<x_{NL}^2>-<x_{NL}>^2)_{t=0}$ $<<$ ($<x_{L}^2>-<x_{L}>^2)_{t=0}$), $\phi^{NL}(x,t)$ remains strongly peaked and oscillates around its central value throughout time.
(ii) In good approximation (the quality of the approximation being an increasing function of the ratio $k^{self}/k^{ext.}$), property 1 is confirmed by our numerical estimates. Indeed, as can be seen in figure \ref{Momo}, $v_{drift}$ is indistinguishable from $v_{dBB}+ v_{int.}$.
(iii) In these conditions, property 2 is automatically satisfied.
(iv) We also noted that in good approximation (the quality of the approximation being an increasing function of the ratio $k^{self}/k^{ext.}$) the soliton $\phi^{NL}(x,t)$, as well as $\Psi^{NL}(x,t)$, follow classical trajectories.
\subsection{Main conjecture.\langlebel{conjecture}}
The numerical results presented in the previous chapter (in particular observation (iv)) brought us to demonstrate the
{\bf Generalized Ehrenfest theorem.}
To do so, let us consider the evolution equation
\begin{equation}
{i}\hbar\frac{\partial\Psi(t,{{\bf x}})}{\partial t}=(-\hbar^2\frac{\Delta}{2m}+V^{L}(t,{{\bf x}})
+V^{NL}(t,{{\bf x}}))\Psi(t,{{\bf x}})\langlebel{NSgen},\end{equation} where $V^{NL}t,{{\bf x}})=-Gm^2\int {d}^3 x' |\Psi(t,{{\bf x}'})|^2F(|{\bf x} -{\bf x'}|)$, with $F$ a real function of $|{\bf x} -{\bf x'}|$.
Then the centre of its solution, denoted $<x>$ ($<x>=\int {d}^3 x |\Psi(t,x)|^2 x/N^2$ with $N^2=\int {d}^3 x' |\Psi(t,x')|^2$) obeys the laws of classical dynamics in the limit where $<x^2>-<x>^2$ goes to zero.
To prove it, it suffices to note that $N$ remains constant throughout time, while
${d <x>\over d t}=<{i\hbar\over m}{\bf \nabla}>$, and
${d^2 <x>\over d t^2}={1\over m}<[{\bf \nabla},V^{L}+V^{NL}]>={1\over m}<{\bf \nabla}V^{L}+{\bf \nabla}V^{NL}>.$
Now, $<{\bf \nabla}V^{NL}>$
$=\int {d}^3 x' |\Psi(t,x')|^2\int {d}^3 x |\Psi(t,x)|^2({\bf x} -{\bf x'}) {dF\over du}_{|u=|{\bf x} -{\bf x'}|}=0,$
by symmetry so that
${d^2 <x>\over d t^2}={1\over m}<{\bf \nabla}V^{L}>$.
In the limit where $<x^2>-<x>^2$ goes to zero, $<{\bf \nabla}V^{L}>\approx {\bf \nabla}V^{L}(<x>)$. Note that when $V^{L}$ is a quadratic function of the position (harmonic oscillator), $<{\bf \nabla}V^{L}>$ is always exactly equal to ${\bf \nabla}V^{L}(<x>).$
At this level, we face a serious problem: our original goal \cite{old} was to derive dB-B dynamics from the non-linear dynamics (\ref{nonfreeNL}) and the factorization ansatz (\ref{ansatz}). However, the trajectory of the soliton departs from dB-B dynamics, due to the presence of the residual contribution ${\bf v}_{int.}={\bf v}_{drift}-{\bf v}_{dB-B}$. Our simulations reveal that, in order to ensure the constraints imposed by Eherenfest's theorem, ${\bf v}_{int.}$ ``conspires'' in order to be equal to ${\bf v}_{classical}-{\bf v}_{dB-B}$. This compensation is fragile however in the sense that ${\bf v}_{classical}-{\bf v}_{dB-B}$ contains the memory of the initial preparation process that eventually took place a long time ago. We shall conjecture here that in the practice some stochasticity is present that will wash out this memory and decorrelate ${\bf v}_{int.}$ from ${\bf v}_{classical}$. Averaging over this stochastic contribution we expect thus (and this is our main conjecture) that $<<{\bf v}_{int.}>>=0$ so that $<<{\bf v}_{drift}>>=<<{\bf v}_{dB-B}>>$, where the bracket $<<,>>$ represents an averaging over this extra-stochastic perturbation of the velocities that we conjecture to be present in nature.
Note that in the past Bohm, Vigier \cite{BV} and de Broglie \cite{debroglieend} suspected the existence of a stochastic noise superposed to the quantum potential, necessary in their eyes in order to explain the irreversible in time convergence to quantum equilibrium. de Broglie wrote for instance in \cite{debroglieend} (chapter XI: On the necessary introduction of a random element in the double solution theory. The hidden thermostat and the Brownian motion of the particle in its wave) the following sentences
{\it ...Finally, the particle's motion is the combination of a regular motion defined by the guidance formula, with a random motion of Brownian character... any particle, even isolated, has to be imagined as in continuous ``energetic contact'' with a hidden medium, which constitutes a concealed thermostat. This hypothesis was brought forward some fifteen years ago by Bohm and Vigier \cite{BV}, who named this invisible thermostat the ``subquantum medium''....If a hidden sub-quantum medium is assumed, knowledge of its nature would seem desirable...}
Our conjecture is a re-expression of the subquantum medium hypothesis invoked by Bohm and Vigier. At this level its deep nature is not clear yet: it could be the manifestation of a relativistic effect similar to the zitterbewegung (footnote \ref{footnotezitter} in appendix) or it could be of a fundamental nature \cite{Sonego}. It is out of the scope of the present paper to study these possibilities. It is worth noting however that in the case of bouncing oil droplets\footnote{Of course we do not expect that Planck's constant plays a role in the case of walkers, because instead of Schr\"odinger equation, one should consider the classical wave equation that describes the dynamics of waves at the surface of the oil bath to begin with, which is outside of the scope of the present paper. We expect however that the techniques developed in the present paper are still valid in the classical regime in the case of droplets.
} \cite{new} such a stochastic disturbance of velocities is always present, due to the periodical forcing imposed to the bath. This explains why dB-B trajectories are never observed directly at the level of droplets but result from the averaging of a large number of trajectories \cite{fort,fort2,Bush}.
\section{Discussion and Conclusions.}
{\bf Discussion: experimental validation.}
The overwhelming majority of experiments \cite{smolin,Chen,arxiv,bili,optom2} proposed so far in order to reveal the existence of intrinsic non-linearities at the quantum level (like e.g. the self-gravity interaction) is a priori doomed to fail, for what concerns our model, because their conceivers systematically took for granted that the wave function was normalised to unity\footnote{Even in this case, the generalized Ehrenfest theorem demonstrated in section \ref{conjecture} is valid, which predicts that in the limit of massive enough objects trajectories become classical. This could open the way to new experimental tests for falsifying self-gravity. It also sheds a new light onto the problem of the classical limit (see paper of Matzkin same issue).}. In the same line of thought, if our conjecture is correct, then trajectories of the self-collapsed solitons are very close to de Broglie-Bohm trajectories. Therefore we expect that our model cannot easily be falsified, to the same extent that the dB-B interpretation leads to the same predictions as the standard quantum theory (through the Born rule).
Even if our model is not relevant, and in the last resort its relevance ought to be confirmed or falsified by experiments, it could appear to be useful as a phenomenological tool.
For instance it was already applied by us \cite{new} in the past\footnote{In the present paper we essentially studied the feedback of the pilot wave $\Psi_L$ on the soliton or particle $\phi_{NL}$, postulating to begin with that the factorisation ansatz is satisfied and that a self-focusing non-linearity is present, that prohibits the spread of the soliton. In ref. \cite{new}, which is complementary to the present paper we studied the feedback of the dB-B solitons on the pilot wave. We predicted the appearance, when several droplets are present simultaneously, of an effective gravitational potential and formulated certain predictions that are likely to be tested in the laboratory but this is another story. } to the phenomenology of bouncing oil droplets \cite{fort,fort2,Bush}. It is at this level that we expect that the validity of the approximations and hypotheses of our model could be confirmed or falsified.
The dynamics outlined here might also be relevant in the field of cold atoms physics \cite{Kaiser} where effective non-linear self-focusing equations properly describe collective excitations of the atomic density \cite{anas}. As far as we know, no de Broglie like trajectory has yet been observed during such experiments.
In the same order of ideas, it would be highly interesting to investigate whether de Broglie like trajectories are good tools for describing optical and/or rogue waves \cite{akhme,koon}. After all our factorisability ansatz could be applied to classical non-linear wave equations too.
{\bf Conclusions.}
We have studied particular solutions of the S-N equation, in presence of an external potential, that behave as peaked solitons, due to the self focusing nature of self-gravitation. We showed that, if they exist and are stable, their shape is close in first approximation to the shape of the self-collapsed ground state, solution of (\ref{Choquard}) and, provided they remain peaked throughout time, they move in accordance with a generalized dB-B dynamics. These predictions were confirmed by numerical simulations. These simulations also showed that in the non-relativistic limit, the generalized dB-B dynamics is equivalent to classical dynamics, which is confirmed by a generalized Ehrenfest theorem. We conjecture however that if we add a stochastic field to the velocity field, dB-B dynamics will be restored. In last resort, this field could be a manifestation of the relativistic zitter bewegung (see appendix). In our approach, the linear wave function plays the role of an auxiliary computation tool (a pilot wave), and does not represent the particle which is represented by the soliton, which fits well into the double solution program of de Broglie.
We hope that the rather simple models treated in this paper will convince the reader that de Broglie's ideas were maybe not that much surrealistic \cite{surreal} and deprived of consistence. Our results indeed reinforces the dB-B picture according to which the particle non-locally and contextually explores its environment thanks to the nearly immaterial tentacles provided by the solution of the linear Schr\"odinger equation. This picture is not comfortable but it is maybe the price to pay to restore wave monism\footnote{An advantage of wave monism is that, contrary to the Bohmian dynamics formulated in terms of material points, wave monism does not violate the No Singularity Principle formulated by Gouesbet \cite{Gouesbet}. Everything is continuous in our approach, even if very different spatial scales coexist.}. Our conjecture (section \ref{conjecture}) also reemphasises the necessity to invoke the existence of a subquantum medium in order to add to the regular and deterministic dB-B trajectories a random motion of brownian character. The brief incursion in the relativistic domain sketched in appendix suggests that the origin of this brownian motion could be linked to zitterbewegung \cite{Hestenes,Messiah}. It would be highly interesting to try and reveal the existence of this subquantum medium experimentally, which is seemingly not an impossible task as suggested in ref.\cite{Thrust}.
Last but not least, our analysis also confirms several prophetic intuitions originally presented by Louis de Broglie during the Solvay conference of 1927 \cite{1927}.
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\section{Appendices.}
\subsection{The Schr\"odinger-Newton equation.\langlebel{didon2}}
One can look for a ``ground-state" solution to \eqref{NS} in the form
\begin{equation}
\psi({\bf x}, t) = {e}^{-\frac{{i}E_0 t}{\hbar}} \phi^0_{NL}({\bf x})~\!,\langlebel{redtoChoq}
\end{equation}
This leads to a stationary equation for $\phi^0_{NL}$
\begin{equation}\frac{-\hbar^2}{2M} \Delta \phi^0_{NL}({\bf x}) + G M^2\! \int\!d^3y ~\!\frac{\big| \phi^0_{NL}({\bf y})\big|^2}{|{\bf x}-{\bf y}|}~\! \phi^0_{NL}({\bf x}) = E_0 \phi^0_{NL}({\bf x})~\!,\langlebel{Choquard}
\end{equation}
which was studied in astrophysics and is known under the name of the {\em Choquard} equation \cite{Lieb}. In \cite{Lieb}, Lieb showed that the energy functional
\begin{equation}
E(\phi) = \frac{\hbar^2}{2 M} \int\!{d}^3x~\big|\nabla\phi({\bf x})\big|^2 - \frac{G M^2}{2} \iint\!{d}^3x {d}^3y~\!\frac{~\big|\phi({\bf y})\big|^2}{|{\bf x}-{\bf y}|}~\!\big|\phi({\bf x})\big|^2~\!,\langlebel{Choquard-energy}
\end{equation}
is minimized by a unique solution $ \phi^0_{NL}({\bf x})$ of the Choquard equation \eqref{Choquard} for a given $L_2$ norm $N( \phi^0_{NL}({\bf x}))$. However no analytical expression is known for this ground state. Numerical treatments established that this ground state has a quasi-gaussian shape, and that its extent is, in the case $N( \phi^0_{NL}({\bf x}))=1$, of the order of ${\hbar^2\over G M^3}$. Then its energy is of the order of $G^2M^5/\hbar^2$. Otherwise, energy scales like $N( \phi^0_{NL}({\bf x}))^3$ \cite{CDW}. Numerical evidence suggests the existence of a discrete spectrum of values for the energy (\ref{Choquard-energy}) of the bound states solutions of (\ref{Choquard}). They can be written in the form $-\mathrm{e}_n \frac{G^2 M^5}{\hbar^2}$, described in terms of dimensionless constants $\mathrm{e}_n$ as :
\begin{equation}
\mathrm{e}_n = \frac{a}{(n+b)^c}~\!,\quad n= 0, 1, 2, \hdots~\!,\langlebel{Choquard-spectrum}
\end{equation}
for approximated constants \cite{Bernstein}
\begin{equation}
a = 0.096\pm0.01~,\quad b = 0.76\pm 0.01~,\quad c= 2.00\pm 0.01~\!.
\end{equation}
The ``ground state" for the Choquard equation corresponds, at $N=1$, to $n=0$ (see \cite{CDW} and references therein for more details).
In summary, the S-N potential exhibits several interesting features that play an important role in our paper: it is self-focusing and possess a localized ground state of the type $\phi^0_{NL}({\bf x})e^{-iE_0 t/\hbar}$ which behaves as a static bright soliton. The spectrum of negative energy eigenstates solutions of (\ref{Choquard}) is discrete. Moreover this potential scale like a Coulomb and/or Newtonian self-interaction in the sense that
\begin{equation}a V^{NL}(\langlembda \Psi)=|\langlembda|^2V^{NL}(\Psi)\langlebel{Coulomb}\end{equation}a
\subsection{Change of norm\langlebel{appendicit}}
Let us denote $H_L$ the linear part of the the full Hamiltonian in (\ref{S2}). It is not hermitian, so that $\sqrt{<\phi_{NL}|\phi_{NL}>}$, the L$_2$ norm of its solution $\phi_{NL}(t,{{\bf x}}))$ is not constant throughout time. The non-linear potentials considered by us preserve the L$_2$ norm however. We can thus evaluate the time derivative of $<\phi_{NL}|\phi_{NL}>$ by direct computation, either integrating by parts, or making use of the formula
\begin{equation}a &&{d <\phi|O|\phi>\over dt}=\nonumber \\&&<\phi|{\partial O\over \partial t}|\phi>+{1\over i\hbar}<\phi|O H_L-H_L^\dagger O|\phi>\nonumber \\&&=<\phi|{\partial O\over \partial t}|\phi>+{1\over i\hbar}( <\phi|[O, Re.H_L]_-|\phi>\\ \nonumber &&+{1\over \hbar}( <\phi|[O, Im.H_L]_+|\phi>),\end{equation}a
where $O$ is an arbitrary observable, described by a self-adjoint operator, while $Re.H_L$ and $Im.H_L$, the real and imaginary parts of $H_L$ are self-adjoint operators defined through $2\cdot Re.H_L=H_L+H_L^\dagger$ and $2i\cdot Im.H_L=H_L-H_L^\dagger$. Here, the symbol $[,]_-$ ($[,]_+$) represents the (anti)commutator.
We find by direct computation that
\begin{equation}a Re. (-\frac{\hbar^2}{m}i{\bf \bigtriangledown} \varphi_L(t,{{\bf x}}) \cdot {\bf \bigtriangledown})\nonumber \\=
(-\frac{\hbar^2}{m}i{\bf \bigtriangledown} \varphi_L(t,{{\bf x}}) \cdot {\bf \bigtriangledown})-(\frac{\hbar^2}{2m}i{ \Delta} \varphi_L(t,{{\bf x}}) )\end{equation}a and $Im. (-\frac{\hbar^2}{m}i{\bf \bigtriangledown} \varphi_L(t,{{\bf x}}) \cdot {\bf \bigtriangledown})=(\frac{\hbar^2}{2m}{ \Delta} \varphi_L(t,{{\bf x}}) )$.
Therefore the guidance potential contributes to
${d <\phi_{NL}|\phi_{NL}>\over dt}={d <\phi_{NL}|1|\phi_{NL}>\over dt}$ by a quantity
\begin{equation}a <\phi_{NL}| (\frac{\hbar}{m}{ \Delta} \varphi_L(t,{{\bf x}}) ) |\phi_{NL}>\approx (\frac{\hbar}{m}{ \Delta} \varphi_L(t,{{\bf x}}) )<\phi_{NL} |\phi_{NL}>),\nonumber\end{equation}a due to the fact that, over the size of the soliton, $\varphi_L(t,{{\bf x}}) $ and its derivatives are supposed to vary so slowly that we can consistently neglect their variation and put them in front of the L$_2$ integral.
The contribution of the $A_L-\phi_{NL}$ coupling to ${d <\phi_{NL}|\phi_{NL}>\over dt}$ is
$\frac{\hbar^2}{m}{1\over i\hbar}\int d^3{\bf x} (\frac{{\bf \bigtriangledown} A_L(t,{{\bf x}})}{A_L(t,{{\bf x}})} \cdot {\bf \bigtriangledown} (\phi_{NL}(t,{{\bf x}}))^*\phi_{NL}(t,{{\bf x}})- (\phi_{NL}(t,{{\bf x}}))^*\frac{{\bf \bigtriangledown} A_L(t,{{\bf x}})}{A_L(t,{{\bf x}})} \cdot {\bf \bigtriangledown} \phi_{NL}(t,{{\bf x}})).$
We now suppose that we are in right to neglect the variation of $\frac{{\bf \bigtriangledown} A_L(t,{{\bf x}})}{A_L(t,{{\bf x}})}$ in the integral above and we find, integrating by parts, a contribution $-2\frac{{\bf \bigtriangledown} A_L(t,{{\bf x_0}})}{A_L(t,{{\bf x_0}})}\cdot
\int d^3{\bf x} (\phi_{NL}(t,{{\bf x}}))^*\frac{\hbar\bf \bigtriangledown} {mi } \cdot \phi_{NL}(t,{{\bf x}})$
Putting all these results together, we find that
\begin{equation}a{d <\phi_{NL}|\phi_{NL}>\over dt}\approx \frac{\hbar}{m}{ \Delta} \varphi_L(t,{{\bf x_0}}) \cdot <\phi_{NL} |\phi_{NL}>\nonumber \\ -2\frac{{\bf \bigtriangledown} A_L(t,{{\bf x_0}})}{A_L(t,{{\bf x_0}})}\cdot
\int d^3{\bf x} (\phi_{NL}(t,{{\bf x}}))^*\frac{\hbar\bf \bigtriangledown} {mi } \cdot \phi_{NL}(t,{{\bf x}}).\langlebel{normchange}\end{equation}a
\subsection{The Dirac spinor.\langlebel{Diracsect}}
In our view the S-N equation is a good candidate for fulfilling the de Broglie double solution program of 1927, in the same sense that Schr\"odinger's equation was a good candidate for realizing de Broglie's wave mechanics program of 1925. In the same line of thought, it is natural to look for a non-linear relativistic equation that would be to the S-N equations what are the Klein-Gordon or Dirac equations to the non-relativistic Schr\"odinger equation.
To do so let us consider the non-linear Dirac equation
\begin{equation}a \beta i\hbar \partial_t {\bf \Psi}(t,{\bf x})-\beta{\bf \alpha}c{\hbar\over i}{\bf \bigtriangledown}{\bf \Psi}(t,{\bf x})-(mc^2+V_L){\bf \Psi}(t,{\bf x})-V_{NL}{\bf \Psi}(t,{\bf x})=0.\langlebel{NLD} \end{equation}a
where $\alpha$ and $\beta$ are the well-known Dirac matrices and \begin{equation}a V_{NL}{\bf \Psi}(t,{\bf x})=-Gm^2\int {d}^3 x'(\frac{|\Psi(t,{{\bf x}'})|_4^2}{|{{\bf x} -{\bf x}'}|})\Psi(t,{{\bf x}}).\langlebel{Newton}\end{equation}a
Here, $|\Psi(t,{{\bf x}'})|_4^2$ represents the local Dirac density:
$|\Psi(t,{{\bf x}'})|_4^2$=$({\bf \Psi}(t,{\bf x}))^\dagger \cdot_4 {\bf \Psi}(t,{\bf x}))$, where the lower index $_4$ refers to the spinorial in-product.
Let us solve (\ref{NLD}) imposing the ansatz
\begin{equation}a {\bf \Psi}=\left(\begin{array}{c}
\Psi_0(t,{{\bf x}}) \\
\Psi_1(t,{{\bf x}}) \\
\Psi_2(t,{{\bf x}}) \\
\Psi_3(t,{{\bf x}}) \\
\end{array}\right)=\left(\begin{array}{c}
\Psi^L_0(t,{{\bf x}}) \\
\Psi^L_1(t,{{\bf x}}) \\
\Psi^L_2(t,{{\bf x}}) \\
\Psi^L_3(t,{{\bf x}}) \\
\end{array}\right)\phi_{NL}(t,{{\bf x}})={\bf \Psi_L}\phi_{NL}\langlebel{ansatzD} \end{equation}a where $\phi_{NL}(t,{{\bf x}})$ is a Lorentz scalar. We find
\begin{equation}a (\phi_{NL}(t,{{\bf x}}))(\beta i\hbar \partial_t {\bf \Psi^L}(t,{\bf x})-\beta{\bf \alpha}c{\hbar\over i}{\bf \bigtriangledown}{\bf \Psi^L}(t,{\bf x})-(mc^2+V_L){\bf \Psi^L}(t,{\bf x}))\nonumber\\
+(\beta {\bf \Psi^L}(t,{\bf x}) {i\hbar \partial \phi_{NL}(t,{{\bf x}})\over \partial t} -\beta{\bf \alpha}c {\bf \Psi^L}(t,{\bf x}){\hbar \over i}{\bf \bigtriangledown}\phi_{NL}(t,{{\bf x}}) \nonumber\\
+{\bf \Psi^L}(t,{\bf x})V_{NL}(|{\bf \Psi^L}\phi_{NL}|) \phi_{NL}(t,{{\bf x}}))=0,\end{equation}a
that we solve in the same way as in the section 2 by requiring two constraints to be fulfilled:
\begin{equation}a\beta i\hbar \partial_t {\bf \Psi^L}(t,{\bf x})-\beta{\bf \alpha}c{\hbar\over i}{\bf \bigtriangledown}{\bf \Psi^L}(t,{\bf x})-(mc^2+V_L){\bf \Psi^L}(t,{\bf x})=0,\end{equation}a which is is the usual, linear, Dirac equation, and the non-linear constraint
\begin{equation}a\beta {\bf \Psi^L}(t,{\bf x}) {i\hbar \partial \phi_{NL}(t,{{\bf x}})\over \partial t} -\beta{\bf \alpha}c {\bf \Psi^L}(t,{\bf x}){\hbar \over i}{\bf \bigtriangledown}\phi_{NL}(t,{{\bf x}}) \nonumber\\+{\bf \Psi^L}(t,{\bf x})V_{NL}(|\Psi(t,{{\bf x}})|_4) \phi_{NL}(t,{{\bf x}}))=0.\langlebel{unknwonw}\end{equation}a
In agreement with (\ref{Coulomb}), $V_{NL}$ supposedly scales like a Newtonian self interaction:
\begin{equation}a V_{NL}({\bf \Psi^L}\phi_{NL}) =A_L^2V_{NL}(({\bf \Psi^L}(t,{\bf x})/A_L)\cdot \phi_{NL}(t,{{\bf x}})) \phi_{NL}(t,{{\bf x}})\langlebel{scalingV}\end{equation}a
where
\begin{equation}a A_L=\sqrt{| \Psi^L_0(t,{{\bf x}})|^2+| \Psi^L_1(t,{{\bf x}})|^2+| \Psi^L_2(t,{{\bf x}})|^2+| \Psi^L_3(t,{{\bf x}})|^2}.\end{equation}a
At this level, a serious problem appears: equation (\ref{unknwonw}) consists of four equations. In the non-relativistic limit \cite{Messiah} and in absence of external magnetic field, spin decouples and can be factorized, but in general the four equations implicitly contained in equation (\ref{unknwonw}) are NOT equivalent to each other.
A posteriori, we feel free to formally tackle the problem by proceeding as follows. We firstly decompose ${\bf \Psi}$ into its projection along ${\bf \Psi_L}$ and its projection along the 4-spinors orthogonal to ${\bf \Psi_L}$:
\begin{equation}a {\bf \Psi}={\bf \Psi_L}({\bf \Psi_L}^\dagger {\bf \Psi})+(1-{\bf \Psi_L}\cdot {\bf \Psi_L}^\dagger ) {\bf \Psi}\langlebel{ansatzDgen}\end{equation}a
Let us introduce $\phi_{NL}(t,{{\bf x}})$ through ${\bf \Psi_L}({\bf \Psi_L}^\dagger {\bf \Psi})$ = ${\bf \Psi_L}\phi_{NL}(t,{{\bf x}})$, as well as the auxillary spinor $ {\bf \delta \Psi}$=$(1-{\bf \Psi_L}\cdot {\bf \Psi_L}^\dagger ) {\bf \Psi}$, which allows us to rewrite (\ref{ansatzDgen}) as follows:
\begin{equation}a {\bf \Psi}={\bf \Psi_L}\phi_{NL}+ {\bf \delta \Psi} \langlebel{ansatzDdoublegen} \end{equation}a
In a second time, let us simply neglect ${\bf \delta \Psi}$, in a grossly coarse-grained approach, in order to gain more insight about the physics of the problem, but keeping in mind however that there is a serious difficulty hidden under the rug at this level.
Retrospectively, our approximation justifies the ansatz (\ref{ansatzD}).
4-multiplying (\ref{unknwonw}) by $({\bf \Psi^L}(t,{\bf x}))^\dagger \beta$ and dividing by $({\bf \Psi^L}(t,{\bf x}))^\dagger \cdot_4 {\bf \Psi^L}(t,{\bf x}))=A_L^2$ we get
\begin{equation}a{i\hbar \partial \phi_{NL}(t,{{\bf x}})\over \partial t}=\langlebel{guidance''} {\bf v^L_{Dirac}}(t,{{\bf x}}){\hbar \over i}{\bf \bigtriangledown} \phi_{NL}(t,{{\bf x}})\\ \nonumber +<\beta>_4 \cdot A_L^2\cdot V_{NL}(\phi_{NL}) \phi_{NL}(t,{{\bf x}}). \end{equation}a
where $<\beta>_4$=$({\bf \Psi^L}(t,{\bf x}))^\dagger \beta {\bf \Psi^L}(t,{\bf x}))$/$({\bf \Psi^L}(t,{\bf x}))^\dagger \cdot_4 {\bf \Psi^L}(t,{\bf x}))$
while ${\bf v^L_{Dirac}}$ obeys
\begin{equation}a {\bf v^L_{Dirac}}\equiv{({\bf \Psi^L})^\dagger \alpha c {\bf \Psi^L}\over ({\bf \Psi^L})^\dagger {\bf \Psi^L}}\langlebel{exact}\end{equation}a
As is well-known, the conservation equation associated to the linear Dirac equation is ${\partial A^2_L\over \partial t}=-div(A_L^2 \cdot {\bf v_{Dirac}}(t,{{\bf x_0}}))$. The dB-B guidance equation is thus nothing else than (\ref{exact}) \cite{Takaba,Holland}. Now, if we suppress in (\ref{S2}) the kinetic energy and the factor proportional to $A_L$, we find an equation which is formally equivalent to (\ref{guidance''}). Therefore, by repeating computations similar to those made in section 2, one can easily show that the dB-B guidance equation linked to Dirac's equation (\ref{exact}) is exactly satisfied.
Furthermore, resorting to the conservation equation ${\partial A^2_L\over \partial t}=-div(A_L^2 \cdot {\bf v_{Dirac}}(t,{{\bf x_0}}))$, it is straightforward to establish the validity of property 2 as in the non-relativistic case studied before. For instance ${{d A_L\over dt}\over A_L}={-1\over 2}{1\over <\phi_{NL}|\phi_{NL}>}{d <\phi_{NL}|\phi_{NL}>\over dt}$
Consequently, we are entitled to look for peaked wave functions of the form
\begin{equation}a {\bf \Psi}=\left(\begin{array}{c}
\Psi_0(t,{{\bf x}}) \\
\Psi_1(t,{{\bf x}}) \\
\Psi_2(t,{{\bf x}}) \\
\Psi_3(t,{{\bf x}}) \\
\end{array}\right) ={1\over A_L}\cdot \left(\begin{array}{c}
\Psi^L_0(t,{{\bf x}}) \\
\Psi^L_1(t,{{\bf x}}) \\
\Psi^L_2(t,{{\bf x}}) \\
\Psi^L_3(t,{{\bf x}}) \\
\end{array}\right)\cdot \phi'_{NL}(t,{{\bf x}}),\end{equation}a
for which we know that
(Property 1) the barycentre of $ \phi'_{NL}$ moves along the hydrodynamical flow lines of Dirac's linear equation, at velocity ${\bf v_{Dirac}}$.
(Property 2) the L$_2$ norm of $ \phi'_{NL}(t,{{\bf x}})$ is constant throughout time.
As in the non-relativistic case, the amplitude $A_L$ is an auxiliary function that disappears at the end of the computation of ${\bf \Psi}$.
It is worth noting that we did not demonstrate the existence of a ground state of the type $\phi^{0}_{NL}$ in the relativistic case\footnote{\langlebel{footnotezitter}Setting $V_L=0$ in (\ref{NLD}) an mutiplying it by $(\beta i\hbar \partial_t -\beta{\bf \alpha}c{\hbar\over i}{\bf \bigtriangledown}-mc^2(1+\phi_G^{self}/c^2))$ where we denoted $\phi_G^{self}=-Gm\int {d}^3 x'(\frac{|\Psi(t,{{\bf x'}})|^2}{|{{\bf x} -{\bf x'}}|})$, we get the non-linear Klein-Gordon equation
\begin{equation}a \hbar^2c^2({1\over c^2}({\partial^2\over \partial t^2}-\Delta)) {\bf \Psi}(t,{\bf x}) + m^2c^4(1+\phi_G^{self}/c^2)^2{\bf \Psi}(t,{\bf x}) =0 \end{equation}a
Then the static version of the non-linear Klein-Gordon equation reads \begin{equation}a -\hbar^2c^2\Delta {\bf \Psi}({\bf x}) + m^2c^4(1+\phi_G^{self}/c^2)^2{\bf \Psi}({\bf x}) =\hbar^2 E^2 {\bf \Psi}({\bf x}) \langlebel{KGNL} \end{equation}a In the weak potential limit ($\phi^{self}_G<<c^2$) the static solution of (\ref{KGNL}) also satisfies Choquard equation (\ref{Choquard}), up to rescaling. In the strong field limit, we have no guarantee about the existence of such a solution, but if it exists then the non-linear Klein-Gordon equation and its static counterpart, the non-linear Choquard equation constitute an interesting toy-model for self-gravity, even in the relativistic regime. We suspect however the appearance of difficulties in the same limit, related to zitter bewegung \cite{Messiah,Hestenes}, because in general the spin does not decouple from $\phi_{NL}$ in (\ref{unknwonw}).}. We also suspect that the self-gravitational potential $V_{NL}$ proposed in (\ref{Newton}) is not the right candidate in a relativistic context because it is obviously not Lorentz covariant. In any case, it is beyond the scope of our paper to characterize in detail what ``really'' happens at the level of the soliton. We expect for instance \cite{new} that in the case of an electron, the size of the soliton is of the order of 10$^{-57}$ meter, well beyond the Planck scale. Our main aim in the present section was merely to suggest that hopefully properties 1 and 2 exhibit some structural invariance when we pass from the non-relativistic to the relativistic regime.
\end{document}
|
\begin{equation}gin{document}
\begin{equation}gin{abstract}
The ``drum problem''---finding the eigenvalues and eigenfunctions of
the Laplacian with Dirichlet boundary condition---has many applications,
yet remains challenging for general domains
when high accuracy or high frequency is needed.
Boundary integral equations
are appealing for large-scale problems,
yet certain difficulties have limited their use.
We introduce two ideas to remedy this:
1) We solve the resulting nonlinear eigenvalue problem
using Boyd's method for analytic root-finding applied to the
Fredholm determinant.
We show that this is
many times faster than the usual iterative
minimization of a singular value.
2) We fix the problem of spurious {\em exterior resonances}
via a combined-field representation.
This also provides the first robust boundary integral eigenvalue method
for non-simply-connected domains.
We implement the new method in two dimensions using
spectrally accurate Nystr\"om product quadrature.
We prove exponential convergence of the determinant at roots
for domains with analytic boundary.
We demonstrate 13-digit accuracy, and improved
efficiency, in a variety of domain shapes including
ones with strong exterior resonances.
\end{abstract}
\maketitle
\section{Introduction}
Eigenvalue problems (EVPs) for linear PDEs have a wealth of applications
\cite{babosrev}
to modeling vibration problems, acoustic, electromagnetic and quantum cavity resonances,
as well as in modern areas such as nano-scale devices \cite{qdots},
micro-optical resonators for high-power lasers \cite{hakan05},
accelerator design \cite{akcelik}, and
data analysis \cite{saito}.
The paradigm is the Dirichlet eigenvalue problem:
given a bounded connected
domain $\Omega \subset \mathbb{R}^2$ with boundary $\Gamma$,
to find eigenvalues $\kappa^2$ and corresponding nontrivial eigenfunctions $u$
that satisfy
\begin{equation}a
(\Delta + \kappa^2)u &=& 0 \qquad \mbox{ in } \Omega,
\label{helm}
\\
u &=& 0 \qquad \mbox{ on } \Gamma,
\label{bc}
\end{equation}a
where $\Delta := \partial^2/\partial x_1^2 + \partial^2/\partial x_2^2$
is the Laplacian.
We refer to $\kappa$ as the eigenfrequency,
and label the allowable set $\kappa_1<\kappa_2\le \kappa_3 \le \cdots
\nearrow +\infty$,
counting multiplicities.
$u_j$ will refer to an eigenfunction for the eigenfrequency $\kappa_j$.
A numerical solution is necessary in all but a few special shapes
(in 2D, ellipses and rectangles) where the Laplacian is separable
\cite{CoHi53}.
This and related EVPs are also of interest in mathematical areas such as
quantum chaos \cite{nonnenrev}.
This is covered in
excellent reviews by Kuttler--Sigillito \cite{KS} and Grebenkov--Nguyen
\cite{grebenkov}.
Numerical solution of \eqref{helm}--\eqref{bc}
falls broadly into two categories:
A) direct discretization, using finite differencing or finite elements
to give a sparse linear EVP where $\kappa^2$ is the eigenvalue;
vs
B) reformulation as a boundary integral equation (BIE)
\cite{backerbim},
discretized using the Galerkin or Nystr\"om methods,
resulting in a highly {\em nonlinear} EVP, again for the eigenvalue $\kappa^2$
or eigenfrequency $\kappa$.
The nonlinearity with respect to $\kappa$
comes from that of the fundamental solution to the Helmholtz
equation \eqref{helm}.
The advantages of type B include:
a huge reduction in the number of unknowns
(due to the decrease in dimensionality by one) especially at high frequency,
and
increased accuracy (since finite element high-frequency
``pollution'' \cite{pollution}
is absent). High-order or spectral accuracy is not hard to achieve,
at least in two dimensions (2D).
Yet, as pointed out by B\"acker \cite[Sec.~3.3.6]{backerbim}, the standard
BIE method
is not even robust for a simply-connected domain,
due to the possibility of spurious exterior resonances.
Recently, type-B methods which approximately {\em linearize}
the nonlinearity, hence boost efficiency at high frequency,
have been created,
but these are limited to moderate $\kappa$ \cite{kirkup},
to heuristic methods with low accuracy \cite{hakansca,veblemonza},
or to domains that are star-shaped \cite{v+s,que,mush,sca}.
This motivates the need for a robust type-B method that
applies to all domain shapes, including multiply-connected ones,
and remains efficient up to at least medium-high frequencies.
In this work we solve two of these issues:
(1)
The standard approach to solve the nonlinear EVP is
by searching for ``V-shaped'' minima of a smallest singular value
\cite{tref06};
we boost efficiency by
turning this into a search for the roots of an {\em analytic} function,
which can be done with less function evaluations and without the
expensive computation of the SVD.
(2) We solve the exterior resonance problem, and at the same
time the case of multiply-connected domains,
using a combined field integral equation \eqref{CFIE}.
We also provide several analysis results that place our method
on a rigorous footing.
The outline of this paper is as follows. In Section~\ref{s:bie}
we review the use of potential theory to reformulate
the eigenvalue problem as a BIE,
and give a discretization of the BIE due to Kress
\cite{kress91} that achieves spectral accuracy for smooth domains.
To tackle issue (1) above,
in Section \ref{s:fred} we introduce the {\em Fredholm determinant},
\begin{equation}
f(\kappa) = \det ( I - 2 D(\kappa))
\label{fk}
\end{equation}
where $D$ is the double-layer operator (defined by \eqref{dlp} below),
whose roots are precisely the eigenfrequencies $\kappa_j$ for simply connected domains.
Following Bornemann \cite{bornemann},
we approximate this with the determinant of a Nystr\"om matrix.
Our main Theorem~\ref{t:main},
in Section~\ref{s:thm}, states that
this approximation convergences exponentially to zero at the true
eigenfrequencies, if the domain has analytic boundary.
Since $f(\kappa)$ is analytic for $\kappa$ nonzero, we
propose in Section~\ref{s:boyd} applying Boyd's method to find its roots,
an application we have not seen in the literature before.
In Section ~\ref{s:res} we prove, and demonstrate numerically,
that the CFIE \eqref{CFIE}
is robust for domains with exterior resonances or interior holes.
It is well known that finding roots becomes ill-conditioned when they are close,
hence we explain in Section~\ref{s:svd}
how we retain robustness in the case of nearby eigenvalues
by reverting to the (more expensive) SVD method in these (rare) case.
Section \ref{s:c} gives numerical performance tests of the entire scheme,
achieving 13 digits the first 100 eigenfrequencies of
a general domain, and a domain with exterior resonances, showing that
our method is competitive in terms of both accuracy and timing.
\section{Boundary integral formulation and quadrature scheme}
\label{s:bie}
Now we lay the foundation of our method for computing eigenfrequencies
by describing the boundary integral formulation and its analyticity
properties for analytic domains,
and then its numerical treatment in 2D, which is standard \cite{kress91}.
\subsection{Integral equation formulation}
For a bounded domain $\Omega$ with twice continuously differentiable boundary $\Gamma$, we explicitly construct solutions to the Helmholtz equation by layer potentials using the fundamental solution as the kernel. The fundamental solution is given by
\begin{equation}gin{equation}
\Phi(x,y):= \frac{i}{4}H^{(1)}_{0}(\kappa \vert x-y \vert), \hspace{0.5cm} x \neq y,
\qquad x,y\in\mathbb{R}^2,
\end{equation}
where $H^{(1)}_{0}$ is the first-kind Hankel function of order zero.
For a continuous function $\varphi$ on $\Gamma$, the single layer operator $\mathcal{S} : C(\Gamma) \rightarrow C(\mathbb{R}^2 \begin{align}ckslash \Gamma)$ is defined as follows, with $v$ denoting the resulting single layer potential
\begin{equation}gin{equation}
v(x) = \mathcal{S} \varphi (x):= \int_{\Gamma} \Phi(x,y) \varphi(y)ds(y), \hspace{0.5cm} x\in \mathbb{R}^{2} \begin{align}ckslash \Gamma,
\end{equation}
where $ds(y)$ is the arc-length element on $\Gamma$.
Note that the domain of $v$ excludes $\Gamma$,
The corresponding boundary operator $S : C(\Gamma) \rightarrow C(\Gamma) $ is
\begin{equation}gin{equation}
S \varphi (x):= \int_{\Gamma} \Phi(x,y) \varphi(y)ds(y), \hspace{0.5cm} x\in \Gamma.
\end{equation}
The double layer operator $\mathcal{D}: C(\Gamma) \rightarrow C(\mathbb{R}^2 \begin{align}ckslash \Gamma)$ with associated double layer potential $u$ is given by
\begin{equation}gin{equation}
u(x) = \mathcal{D} \varphi (x):= \int_{\Gamma} \frac{\partial\Phi(x,y)}{\partial n(y)} \varphi(y)ds(y), \hspace{0.5cm} x \in \mathbb{R}^{2} \begin{align}ckslash \Gamma.
\label{DLP}
\end{equation}
where $n(y)$ is the unit normal vector at $y\in\Gamma$ directed to the exterior of the domain. Again, because the integral exists for $x \in \Gamma$, one may define
a boundary operator $D: C(\Gamma) \rightarrow C(\Gamma)$ by
\begin{equation}gin{equation}
D \varphi (x):= \int_{\Gamma} \frac{\partial\Phi(x,y)}{\partial n(y)} \varphi(y)ds(y), \hspace{0.5cm} x \in \Gamma.
\label{dlp}
\end{equation}
The above operators depends on the frequency $\kappa$, and we will
indicate this only when needed.
Both $u$ and $v$ as defined are solutions to the Helmholtz equation and can be continuously extended, either from the interior or the exterior of $\Omega$, to the boundary by taking limits in the following sense:
\begin{equation}
u^{\pm}(x):=\lim_{h\rightarrow 0+} u(x \pm hn(x)),
\qquad
u_n^\pm(x):=\lim_{h\rightarrow 0+} n(x) \cdot \nabla u(x \pm hn(x)),
\qquad x \in \Gamma,
\end{equation}
and analogously for $v$.
These limits relate to the boundary operators via the jump relations
\cite{coltonkress}
\begin{equation}a
v^{\pm}(x) = S \varphi (x), \hspace{0.5cm} x \in \Gamma,
\label{JR1}
\\
v_n^{\pm}(x) = (D^T \mp \half) \varphi (x), \hspace{0.5cm} x \in \Gamma,
\label{JR2}
\\
u^{\pm}(x)= (D \pm \half) \varphi(x), \hspace{0.5cm} x \in \Gamma.
\label{JR3}
\\
u_n^{\pm}(x) = T \varphi (x), \hspace{0.5cm} x \in \Gamma,
\label{JR4}
\end{equation}a
where $D^{T}$ is given by
\begin{equation}gin{equation}
D^T \varphi (x):= \int_{\Gamma} \frac{\partial\Phi(x,y)}{\partial n(x)} \varphi(y)ds(y), \hspace{0.5cm} x \in \Gamma.
\end{equation}
and the hypersingular operator $T$ is defined by
\begin{equation}gin{equation}
T \varphi (x):= \frac{\partial}{\partial n(x)}\int_{\Gamma} \frac{\partial\Phi(x,y)}{\partial n(y)} \varphi(y)ds(y), \hspace{0.5cm} x \in \Gamma,
\end{equation}
When $u$ is given by a double-layer potential with density $\varphi$,
enforcing the Dirichlet boundary condition \eqref{bc} gives
\begin{equation}gin{equation}
\label{eq: BIEwD}
(I -2D(\kappa)) \varphi = 0 .
\end{equation}
Thus we might hope that the (nonlinear) eigenvalue problem that
$I-2D(\kappa)$ has a nontrivial nullspace is equivalent to
the (linear) eigenvalue problem \eqref{helm}-\eqref{bc}.
For a domain of general connectivity there is {\em not} such an equivalence;
we merely have the following.
\begin{equation}gin{lemma}
Let $\Omega$ be a (possibly non-simply connected)
bounded domain with twice continuously differentiable boundary $\Gamma$.
Then if $\kappa^2$ is a Dirichlet eigenvalue of $\Omega$,
$I-2D(\kappa)$ has a nontrivial nullspace.
\label{l:nullspace}
\end{lemma}
\begin{equation}gin{proof}
Green's representation theorem \cite[Theorem 2.1]{coltonkress} states
that if $(\Delta +\kappa^2)u=0$ in $\Omega$, then
\begin{equation}
{\mathcal S}u_n^- - {\mathcal D} u^- \; = \;
\left\{\begin{equation}gin{array}{ll} u, & \qquad \mbox{ in } \Omega ~, \\
0, & \qquad
\mbox{ in } \mathbb{R}^2\begin{align}ckslash\overline{\Omega}~.\end{array}\right.
\label{GRF}
\end{equation}
Applying this to $u$ an eigenfunction at frequency $\kappa_j$, and
taking its derivative on $\Gamma$ using \eqref{JR2} gives
$u_n^- = (D^T + \half)u_n^-$. Since $u_n^-$ is nontrivial,
the compactness of $D$ and the Fredholm alternative proves
$I -2D(\kappa_j)$ has a nontrivial nullspace.
\end{proof}
The consequence for a multiply-connected domain is that it is
possible that there are {\em spurious} frequencies where $I-2D(\kappa)$ has
a nontrivial nullspace but $\kappa^2$ is not a Dirichlet eigenvalue
(we will characterize these frequencies in Lemma~\ref{l:hole}).
Only for the case of $\Omega$ simply-connected does
equivalence hold, as the following well-known theorem states.
\begin{equation}gin{theorem}\cite{CK83}
\label{kress}
Let $\Omega$ be a bounded, simply-connected
domain with twice continuously differentiable boundary $\Gamma$.
Then for each $\kappa \in \mathbb{C}\begin{align}ckslash\{0\}$ with $\operatorname{Im}\kappa \geq 0$, $\kappa^2$ is a Dirichlet eigenvalue of $\Omega$
if and only if $I-2D(\kappa)$ has a nontrivial nullspace.
Moreover, the dimension of the eigenspace is the same as that of the nullspace.
\end{theorem}
For the case of Lipschitz boundary, see Mitrea \cite{mitrea}.
This motivates integral equations as a robust approach to the Dirichlet
eigenvalue problem for simply-connected domains;
later in Section~\ref{s:res} we will show how to handle
multiply-connected domains.
\subsection{Splitting of the kernel}
We will discuss a quadrature scheme for Helmholtz kernels
that is highly accurate for smooth boundaries \cite{kress91};
for this an analytic splitting is needed.
Assume $\Gamma$ is analytic and has a regular parametrization $x(t) = (x_{1}(t), x_{2}(t)), 0 \leq t \leq 2\pi$. We transform \eqref{eq: BIEwD} into the parametric form
\begin{equation}gin{equation}
\label{paraBIE}
\psi (t) - \int_{0}^{2\pi}L(t,s) \psi (s) ds = 0, \hspace{0.5cm} 0 \leq t \leq 2\pi
\end{equation}
where $\psi (t):= \varphi(x(t))$ and the kernel of the reparametrized
operator $2D(\kappa)$ is given by
\begin{equation}gin{eqnarray}
L(t,s) &:=&
\frac{\partial\Phi(x(t),x(s))}{\partial n(x(s))} |x'(s)|
\label{L}
\\
&=&\frac{i\kappa}{2} \{ x'_{2}(s)[x_{1}(t)-x_{1}(s)]-x'_{1}(s)[x_{2}(t)-x_{2}(s)]\}\frac{H_{1}^{(1)}(\kappa r(t,s))}{r(t,s)}
\end{eqnarray}
with the distance function $r(t,s) := \{ [x_{1}(t) - x_{1}(s)]^{2} + [x_{2}(t) - x_{2}(s)]^{2}\}^{\frac{1}{2}}$.
With a slight abuse of notation, at each $\kappa$ we use $L(\kappa): C[0, 2\pi] \rightarrow C[0, 2\pi]$ to denote the integral operator with $L(t,s)$ as its kernel, that is, the reparametrized operator $2D$.
We will sometimes drop the explicit dependence on $\kappa$ and write $L$.
The kernel $L$ is continuous but not analytic,
so one splits the kernel into
\begin{equation}gin{equation}
L(t,s) = L^{(1)}(t,s) \ln{\begin{equation}gin{itemize}gl(4\sin^2{\frac{t-s}{2}}\begin{equation}gin{itemize}gr)} + L^{(2)}(t,s)
\end{equation}
where
\begin{equation}gin{equation}
L^{(1)}(t,s):=-\frac{\kappa}{2\pi} \{ x'_{2}(s)[x_{1}(t)-x_{1}(s)]-x'_{1}(s)[x_{2}(t)-x_{2}(s)]\}\frac{J_{1}(\kappa r(t,s))}{r(t,s)}
\label{L1}
\end{equation}
\begin{equation}gin{equation}
L^{(2)}(t,s):=L(t,s)-L^{(1)}(t,s)\ln \begin{equation}gin{itemize}gl(4\sin^2{\frac{t-s}{2}}\begin{equation}gin{itemize}gr)
\label{L2}
\end{equation}
Both $L^{(1)}$ and $L^{(2)}$ are analytic, provided that $\Gamma$ is analytic
\cite{kress91}.
In that case we get the following.
\begin{equation}gin{lemma}
Let $\Omega$ have analytic boundary.
Then any density function $\psi(s)$ solving \eqref{paraBIE}
is an analytic function of the parameter $s$.
\label{l:anal}
\end{lemma}
This follows from the argument
of \cite[Prob.\ 12.4, p.~217]{LIE}, namely that the operator $L$ is compact in the space of $2\pi$-periodic analytic
functions in a complex strip $\mathbb{R}\times (-a,a)$ for some $a>0$,
and the Fredholm alternative.
\subsection{Quadrature and Nystr\"om method}
\label{s:quad}
We choose a set of quadrature points equidistant in parameter,
$s_{k} := \frac{2\pi k}{N}$, $k = 0,1,..., N-1$,
where $N$ is an even number, with equal weights $2\pi/N$,
and insert this quadrature into \eqref{paraBIE}
to get the approximation
\begin{equation}gin{equation}
\label{eq: FDAppr}
\psi^{(N)}(t) - \sum_{k=0}^{N-1} \{ R^{(N)}_{k}(t)L^{(1)}(t, s_{k}) + \frac{2\pi}{N} L^{(2)}(t, s_{k})\} \psi^{(N)}(s_{k}) = 0, \hspace{0.5cm} 0\leq t \leq 2\pi.
\end{equation}
Here the second term inside the curly brackets arises from the usual quadrature
rule, whereas the first term arises from a spectrally-accurate
product quadrature
scheme for the periodized log singularity (reviewed in \cite[Sec.~6]{hao}),
with weights
\begin{equation}gin{equation}
R^{(N)}_k(t) = -\frac{4\pi}{N} \sum_{m=1}^{\frac{N}{2}-1}\frac{1}{m}\cos{m(t-s_{k})} - \frac{4\pi}{N^{2}} \cos{\frac{N}{2}(t-s_{k})}, \hspace{0.5cm}
k = 0,\ldots,N-1
\end{equation}
Define $L_N$ to be the Nystr\"om interpolant from \eqref{eq: FDAppr},
which maps $\psi \in C[0, 2\pi]$ to
\begin{equation}gin{equation}
\label{defl}
L_{N} \psi (t) = \sum_{k=0}^{N-1} \{ R^{(N)}_{k}(t)L^{(1)}(t, s_{k}) + \frac{2\pi}{N} L^{(2)}(t, s_{k})\} \psi(s_{k})
\end{equation}
Kress \cite[Sec.~12.3]{LIE}
showed that, when $L^{(1)}$ and $L^{(2)}$ have analytic kernels,
interpolation of analytic functions
with this product quadrature convergences exponentially with $N$ in the $L_{\infty}$-norm.
Thus for each analytic $\psi$, $\Vert L_{N} \psi - L\psi \Vert _{\infty} \leq Ce^{-aN}$ for some constants $C$ and $a$ depending on $\psi$ \cite[p.~185]{LIE}.
By setting $t$ to $s_i$ in \eqref{defl}, one obtains the Nystr\"om matrix $M_N$ with elements
\begin{equation}gin{equation}
\label{nm}
(M_N)_{ij} := R^{(N)}_{|i-j|}(0) L^{(1)}(s_{i},s_{j})+\frac{2\pi}{N}L^{(2)}(s_{i},s_{j})
\qquad i,j = 0,\dots, N-1.
\end{equation}
The condition \eqref{paraBIE} that $I-2D(\kappa)$, and hence $I-L(\kappa)$,
is singular
can now be approximated with exponentially small error
by the condition that the matrix $I-M_N(\kappa)$ is singular.
Each null vector of $I-L_N$ is exactly reconstructed
by applying the interpolant on the right-hand side of \eqref{defl} to the
corresponding null vector of the matrix.
By the analysis in \cite[Sec.~12.2-12.3]{LIE} in the homogeneous case,
this reconstructs the desired null vectors of $I-L$ to exponential
accuracy.
\section{The Fredholm determinant}
\label{s:fred}
As we have seen,
for simply-connected domains,
$\kappa$ is an eigenfrequency if and only if the boundary integral operator $I-L{(\kappa)}$ has nontrivial kernel.
We now convert this to a condition on a Fredholm determinant.
The following theorem says that we can study the invertibility of $I-L$ on $L_{2}[0, 2\pi]$ instead of $C[0, 2\pi]$.
\begin{equation}gin{theorem}\cite{hahner} \cite[p.91]{LIE}
Let $A$ be an integral operator with weakly singular kernel, then the nullspaces of $I-A$ in $C[0, 2\pi]$ and $L_{2}[0, 2\pi]$ coincide.
\end{theorem}
This implies that $L$ has the same set of nonzero eigenvalues, counting multiplicities, in $C[0, 2\pi]$ as in $L_{2}[0, 2\pi]$.
Thus from now on we need not specify in which space we consider these eigenvalues.
Let $\mathcal{J}_{1}(L_{2}[0, 2\pi])$ be the space of {\em trace-class}
operators in $L_{2}[0, 2\pi]$.
This space is defined by finiteness of the
operator norm
$\|A\|_{\mathcal{J}_1}$, which is the sum of the operator singular values
\cite{bornemann}; this insures that the sum of the eigenvalues is also bounded.
\begin{equation}gin{lemma}
$L$ with kernel given by \eqref{L} is a trace-class operator.
\label{l:tr}
\end{lemma}
\begin{equation}gin{proof}
Using the Bessel function asymptotic \cite[10.8.1]{DLMF},
the leading non-analytic term in $L(t,s)$ is
$O((t-s)^2\log |t-s|)$ for small $t-s$,
thus $L(t,s)$ and the partial derivative
$\partial_{s}L(t,s)$ are continuous on $[0, 2\pi]^2$,
thus $L$ is trace class on $L_{2}[0,2\pi]$ \cite{bornemann}.
\end{proof}
For trace-class operators, the Fredholm determinant as a linear functional can be constructed in several equivalent ways; we take the approach of
Gohberg and Krein \cite[p.~157]{GK69}.
Let $\mathcal H$ be a Hilbert space. For $A \in \mathcal{J}_{1}(\mathcal{H)}$ with nonzero eigenvalues $\lambda_{1}(A), \lambda_{2}(A),\ldots$ (counting multiplicities), the Fredholm determinant of $I-A$ is defined by
\begin{equation}gin{equation}
\label{fd1}
\det(I-A) : = \displaystyle \prod_{j=1}^\infty(I-\lambda_{j}(A))
\end{equation}
One important property of the Fredholm determinant is that it completely describes when $I-A$ is invertible:
\begin{equation}gin{theorem}\cite[p.~34]{simon}
\label{invTr}
For $A \in \mathcal{J}_{1}(\mathcal{H)}$, $\det(I-A)\neq 0$ if and only if $I-A$ is invertible.
\end{theorem}
\begin{equation}gin{corollary}
\label{iffL}
L with kernel given by \eqref{L} satisfies $\det{(I-L)}=0$ if and only if $I-L$ has nontrivial kernel space.
\end{corollary}
\begin{equation}gin{proof}
The third Riesz theorem \cite[page 11]{CK83} says if $I-L$ is not surjective, it is not injective.
The claim follows from Theorem \ref{invTr}.
\end{proof}
As we will see in Section \ref{s:thm}, the nonzero eigenvalues of $L$ will be approximated numerically by the nonzero eigenvalues of $L_{N}$ in $C[0, 2\pi]$. The following lemma connects those nonzero eigenvalues of $L_{N}$ to the ones of the Nystr\"om matrix, making accurate numerical approximation of $\det(I-L)$ possible.
\begin{equation}gin{lemma}
\label{fnm}
The collection of nonzero eigenvalues, counting multiplicities, of $ L_{N}$, defined in \eqref{defl}, is the same as the nonzero eigenvalues of the associated Nystr\"om matrix $M_{N}$ as defined in \eqref{nm}.
\end{lemma}
\begin{equation}gin{proof}
If $\lambda$ is a nonzero eigenvalue of $L_{N}$, then there exists a finite dimensional eigenspace with
basis $\{\varphi_{i}\}$ such that $L_{N}\varphi_{i} = \lambda \varphi_{i}$ holds on $[0, 2\pi]$. Certainly it holds on all the quadrature nodes,
meaning $M[\varphi_{i}(s_{k})]_{k =0}^{N-1}=\lambda [\varphi_{i}(s_{k})]_{k=0}^{N-1}$,
where $[\varphi_{i}(s_{k})]_{k =0}^{N-1}$ indicates a column vector.
It cannot be true that $\varphi_{i}$ is simultaneously zero at all quadrature nodes, since then by ~\eqref{defl}, $\varphi_{i}$ is identically zero on $[0, 2\pi]$. By the same reasoning the set of $[\varphi_{i}(s_{k})]_{k=0}^{N-1}$ for all $i$
is a linearly independent set of eigenvectors of $M_{N}$ with eigenvalue $\lambda$.
If on the other hand $\lambda$ is a nonzero eigenvalue of $M_{N}$, then there exists a finite dimensional eigenspace with a basis spanned by the vectors $\{[\phi_{i,k}]_{k=0}^{N-1}\}$. For each $i$ we can construct $\varphi_{i} (t) = \frac{1}{\lambda}\sum_{k=0}^{N-1} \{ R^{(N)}_{k}(t)L^{(1)}(t, s_{k}) + \frac{2\pi}{N} L^{(2)}(t, s_{k})\} \phi_{i,k}$, then $[\varphi_{i}(s_{k})]_{k=0}^{N-1} = \frac{1}{\lambda} M_{N} [\phi_{i,k}]_{k=0}^{N-1} = [\phi_{i,k}]_{k=0}^{N-1}$. One sees that $\varphi_{i}$ is an eigenfunction of $L_{N}$ with eigenvalue $\lambda$, and $\{\varphi_{i}\}$ is a linearly independent set because the set $\{[\phi_{i,k}]_{k=0}^{N-1}\}$ is.
\end{proof}
The Fredholm determinant is a function of $\kappa$, and we use the
notation \eqref{fk} for the determinant of the exact operator.
Similarly we use, for the matrix determinant of the associated Nystr\"om matrix,
\begin{equation}gin{equation}
\label{fnk}
f_{N}(\kappa) := \det(I -M_{N}{(\kappa)}) ~.
\end{equation}
\begin{equation}gin{remark}
In fact, from Lemma \ref{fnm}, it follows that $f_{N}$ is the Fredholm determinant of $I-L_{N}$ as a finite dimensional operator on $C[0, 2\pi]$. The definition of Fredholm determinant for certain operators on a Banach space and more can be found in \cite{gohberg}.
\end{remark}
\begin{equation}gin{figure}[!ht]
\centering
\includegraphics[width=0.4\textwidth]{proof}
\caption{Illustration of proof idea for the main Theorem~\ref{t:main}, showing spectrum of $L{(\kappa_{j})}$, and circles of radius $r_{0}$ and $\varepsilonilon$.}
\label{fig: proof}
\end{figure}
\section{Error analysis of the Fredholm determinant}
\label{s:thm}
We prove our main error analysis result in this section. The approximation sequence $\{L_{N}\}$ converges pointwise to the integral operator $L$ on $C[0, 2\pi]$, and is collectively compact \cite[p.~202]{LIE}. The following two theorems of Atkinson describe the convergence of eigenvalues of $L_{N}$ to the ones of $L$.
\begin{equation}gin{theorem}\cite{atkinspec}
\label{Atkinson1}
Let $K$ be an integral operator on a Banach space and $\{K_N\}$ be a collectively compact sequence of numerical integral operators approximating $K$ pointwise, and let $R$ and $\varepsilonilon$ be arbitrary small positive numbers. Then there is an $N_{0}$ such that for $N \geq N_{0}$, any eigenvalue $\lambda$ of $K_{N}$ satisfying $\vert \lambda \vert \geq R$ is within $\varepsilonilon$ of an eigenvalue $\lambda_{0}$ of $K$ with $\vert \lambda_{0} \vert \geq R$. Furthermore let $\sigma_{N}$ be the set of eigenvalues of $K_{N}$ within distance $\varepsilonilon$ from a fixed $\lambda_{0}$, then the sum of multiplicities of $\lambda$ in $\sigma_{N}$ equals the multiplicity of $\lambda_{0}$.
\end{theorem}
To summarize, outside of an arbitrarily small disk eigenvalues of $L_{n}$ approximate the eigenvalues of $L$ with correct multiplicities. We also have a guarantee that the convergence rate of $L_{n}$ is carried over to the eigenvalues.
\begin{equation}gin{theorem}\cite{atkinrate}
\label{Atkinson2}
With the same assumption as in the above theorem, let $\lambda_{0}$ be of index $\nu$, i.e., $\nu$ is the smallest integer for which
\begin{equation}gin{equation}
\ker( (\lambda_{0}-K)^{\nu} )= \ker((\lambda_{0}-K)^{\nu+1}),
\end{equation}
where ker means the kernel space.
Then for some $c >0$ and all sufficiently large $n$,
\begin{equation}gin{equation}
\vert \lambda - \lambda_{0} \vert \leq c \max \{ \Vert K \varphi_{i} -K_{n} \varphi_{i} \Vert ^{\frac{1}{\nu}} \vert 1 \leq i \leq m\}
\end{equation}
for all $\lambda \in \sigma_{n}$, and the set $\{ \varphi_{1},...\varphi_{m}\}$ is a basis for $\ker ((\lambda_{0}-K)^{\nu})$.
\end{theorem}
When $\lambda_{0} =1$, then index is also called the Riesz number \cite[p.11]{CK83} of $K$. The Riesz number of our $L$ is 1 \cite[p.84]{CK83}. We can now
prove the main theorem that the determinant of $I-M_{N}{(\kappa)}$ at an eigenfrequency $\kappa=\kappa_{j}$ vanishes exponentially with $N$.
\begin{equation}gin{theorem}
Let $\kappa_{j}^{2}$ be a
Dirichlet eigenvalue of a bounded domain $\Omega$ with analytic boundary. Then
there exists an $N_{0}$ such that
\begin{equation}
\vert f_N(\kappa_{j}) \vert \leq Ce^{-\alpha N}
\qquad \mbox{for all } N>N_0,
\end{equation}
where $C$ and $\alpha>0$ are constants depending on $\Omega$ and $\kappa_{j}$.
\label{t:main}
\end{theorem}
\remark{This theorem includes the case of $\Omega$ non-simply connected,
although later we will show that a modification to the
definition of $f(\kappa)$ and $f_N(\kappa)$ is needed to
make a robust method for this case.}
\begin{equation}gin{proof}
Let $\{ \lambda_{i}^{(N)}\}_{i=1}^{N'}$
be the set of nonzero eigenvalues of $L_N(\kappa_j)$, counting multiplicities,
where $N'$ is at most $N$.
Let $\{\lambda_{i}\}$ be nonzero eigenvalues of $L{(\kappa_{j})}$.
If $\kappa_{j}$ is an eigenfrequency for \eqref{helm}--\eqref{bc},
then according to Lemma~\ref{l:nullspace}, $I - L{(\kappa_{j})}$ has
nontrivial kernel. Based on Corollary~\ref{iffL}, $1$ is an eigenvalue of $L$, which we can label
$\lambda_{1}=1$.
Theorem \ref{Atkinson1} implies that we can pick an ordering of $
\lambda_{i}^{(N)}$ so that $\{\lambda_{1}^{(N)}\}$ converges to
$\lambda_{1}$ as $N \rightarrow \infty$, and there might be multiple
such sequences, depending on the multiplicity of $\lambda_{1}$, i.e.,
essentially, the number of sequences with $\lambda_{1}$ as the limit
is the same as the multiplicity of $\lambda_{1}$. We only need the
existence of one such sequence for the following proof to hold.
Theorem \ref{Atkinson1} also implies that if we let $r_{0}$ be
a constant with $r_{0} \leq \frac{1}{2}$, then there exists $N_{1}$
such that for $N \geq N_{1}$, the number of $\lambda_{i}^{(N)}$ with
$\vert \lambda_{i}^{(N)} \vert \geq r_{0}$ equals the number of
$\lambda_{i}$ with $\vert \lambda_{i} \vert \geq r_{0}$, and we can
relabel $\{\lambda_{i}^{(N)}\}_{i=1}^{N'}$ in such a way that $\displaystyle
\lim_{N \rightarrow \infty} \lambda_{i}^{(N)} = \lambda_{i}$ for all
$i$ with $\lambda_{i}^{(N_{1})} \geq r_{0}$.
Since by Lemma \ref{fnm}, $\{ \lambda_{i}^{(N)} \vert 1\leq i \leq N' \}$
is also the set of
nonzero eigenvalues of $M_{N}{(\kappa_{j})}$, we write the matrix determinant of $I - M_{N}{(\kappa_{j})}$ as a product of three factors as follows,
\begin{equation}
\det(I - M_{N}{(\kappa_{j})}) \;=\; \prod _{i=1}^{N'} (1 - \lambda_{i}^{(N)})
\;=\; (1 - \lambda_{1}^{(N)}) \prod_{\vert \lambda^{(N)}_{i} \vert \geq r_{0} \mbox{, } i \neq 1} (1 - \lambda_{i}^{(N)}) \prod_{\vert \lambda^{(N)}_{i} \vert < r_{0}} (1 - \lambda_{i}^{(N)})
\end{equation}
Then, for the second factor, since there are a finite number of terms,
\begin{equation}
\lim_{N\rightarrow \infty} \prod_{\vert \lambda^{(N)}_{i} \vert \geq r_{0} \mbox{, } i \neq 1} (1 - \lambda_{i}^{(N)})
\; = \;
\prod_{\vert \lambda^{(N)}_{i} \vert \geq r_{0} \mbox{, } i \neq 1} \lim_{N \rightarrow \infty} (1 - \lambda_{i}^{(N)})
\; = \;
\prod_{\vert \lambda_{i} \vert \geq r_{0} \mbox{, } i \neq 1} (1 - \lambda_{i})
\end{equation}
Thus there exists $N_{2}$ and constant $C_{1}$ such that for $N \geq N_{2}$,
\begin{equation}gin{equation}
\begin{equation}gin{itemize}ggl\vert \prod_{\vert \lambda^{(N)}_{i} \vert \geq r_{0} \mbox{, } i \neq 1} (1 - \lambda_{i}^{(N)} ) \begin{equation}gin{itemize}ggr\vert \leq C_{1}
\nonumber
\end{equation}
For the third factor
\begin{equation}
\prod_{\vert \lambda^{(N)}_{i} \vert < r_{0}} \vert1 - \lambda_{i}^{(N)} \vert
\;=\;
\exp (\sum_{\vert \lambda^{(N)}_{i} \vert < r_{0}} \log \vert1 - \lambda_{i}^{(N)} \vert )
\; \le \;
\exp (\sum_{\vert \lambda^{(N)}_{i} \vert < r_{0}} 2 \vert \lambda_{i}^{(N)} \vert )
\end{equation}
Choose $\varepsilonilon \in (0,r_0)$, and
let $m_{N}(\varepsilonilon)$, $m_{N}(r_{0})$ be the number of $\lambda^{(N)}_i$ with $\vert \lambda^{(N)}_i \vert \geq \varepsilonilon$,
$\vert \lambda^{(N)}_i \vert \geq r_{0}$, respectively; see Fig.~\ref{fig: proof}.
In Theorem~\ref{Atkinson1}, pick $R = \varepsilonilon$, then there exists $N_{3}$ such that for $N \geq N_{3}$, all $ \lambda^{(N)}_{i} $ with $\vert \lambda_{i}^{(N)} \vert \geq \varepsilonilon$ are within distance $\varepsilonilon$ of some $\lambda_{i}$,
and each $\lambda_{i}$ with $\vert \lambda_{i} \vert \geq \varepsilonilon$ has exactly one sequence $\{ \lambda_{i}^{(N)} \}$ approaching it, i.e., we have $|\lambda_i^{(N)}-\lambda_i|<\varepsilonilon$ and $\vert \lambda_{i}^{(N)} \vert \geq \varepsilonilon$ for $N \geq N_{3}$.
Then for $N\ge N_3$, we bound
\begin{equation}gin{align*}
\sum_{\vert \lambda^{(N)}_{i} \vert < r_{0}} \vert \lambda_{i}^{(N)} \vert & = \sum_{\vert \lambda^{(N)}_{i} \vert < \varepsilonilon} \vert \lambda_{i}^{(N)} \vert + \sum_{\varepsilonilon \leq \vert \lambda^{(N)}_{i} \vert < r_{0}} \vert \lambda_{i}^{(N)} \vert \\
& \leq (N' - m_{N}(\varepsilonilon) )\varepsilonilon + \sum_{\varepsilonilon \leq \vert \lambda_{i} \vert < r_{0}} \vert \lambda_{i} \vert + (m_{N}(\varepsilonilon) -m_{N}(r_{0}))\varepsilonilon \\
& = ( N'- m_{N}(r_{0})) \varepsilonilon + \sum_{\varepsilonilon \leq \vert \lambda_{i} \vert < r_{0}} \vert \lambda_{i} \vert \\
& \leq N' \varepsilonilon + \sum_{\varepsilonilon \leq \vert \lambda_{i} \vert < r_{0}} \vert \lambda_{i} \vert
\end{align*}
where
\begin{equation}gin{equation}
\sum_{\varepsilonilon \leq \vert \lambda_{i} \vert < r_{0}} \vert \lambda_{i} \vert \leq \sum_{i} \vert \lambda_{i} \vert \leq \Vert L{(\kappa_{j})} \Vert _{\mathcal{J}_{1}}
\nonumber
\end{equation}
which is bounded since by Lemma~\ref{l:tr} $L$ is in trace class.
For $\varphi \in \ker(I-L)$, from Lemma~\ref{l:anal}, $\varphi$ is analytic thus,
as discussed in Section~\ref{s:quad},
our quadrature scheme has $\Vert L_{N} \varphi - L \varphi \Vert _{\infty} \leq Ce^{-a_{0}N}$ for $N$ sufficiently large, where $a_{0}>0$ and $C$ are constants which only depend on $\varphi$. $\ker(I-L)$ is finite dimensional so by theorem \ref{Atkinson2}, so there exists $N_{4}$, $a >0$ and $C_{2}$ such that for $N \geq N_{4}$, $\vert 1-\lambda_{1}^{(N)}\vert \leq C_{2} e^{-a N}$,
Let $N_{0}= \max \{N_{1}, N_{2}, N_{3}, N_{4}\}$ then for $N \geq N_{0}$,
since $N'\le N$,
\begin{equation}
\vert \det(I - M_{N}{(\kappa_{j})}) \vert
\;\leq\;
C_{2} e^{-aN} C_{1} \exp (2N\varepsilonilon +2 \Vert L{(\kappa_{j})} \Vert _{\mathcal{J}_{1}})
\end{equation}
Now let $C := C_{1}C_{2}\exp(2 \Vert L{(\kappa_{j})} \Vert _{\mathcal{J}_{1}})$,
then
$\vert \det(I - M_{N}^{(\kappa_{j})}) \vert \leq C e^{-(\alpha-2 \varepsilonilon)N}$,
so we may choose any positive $\alpha<a - 2\varepsilonilon$ to finish the proof.
\end{proof}
\remark{From the above proof, it is clear that when $\ker(I-L)$ is one-dimensional the rate $\alpha$
may be chosen arbitrarily close to $a$, the width of the
strip in which the null-vector $\varphi$ (density generating the
eigenfunction) is analytic.
Similar result holds for $\ker(I-L)$ higher-dimensional. }
\remark{When the boundary $\Gamma$ is merely $C^\infty$ smooth
(not necessarily analytic), we expect that
$\ker(I-L)$ is in $C^{\infty}[0, 2\pi]$, and that the
determinant converges to zero super-algebraically at eigenfrequencies.
We leave a proof of this to future work.}
\section{Boyd's method for finding roots of the determinant}
\label{s:boyd}
Here we describe a new approach to finding eigenvalues efficiently,
using Theorem \ref{invTr} to equate these with the roots of the
Fredhold determinant $f(\kappa)$.
Our method is inspired by the following fact.
\begin{equation}gin{lemma}
$f(\kappa) = \det(I-L{(\kappa)})$ is analytic with respect to $\kappa$
for $\kappa \in \mathbb{C} \begin{align}ckslash \{0\}$.
\end{lemma}
\begin{equation}gin{proof}
For $L \in C[0, 2\pi]^{2}$, $\det(I-L) = \sum_{m=0}^{\infty}\frac{(-1)^{m}}{m!} \int_{0}^{2\pi}...\int_{0}^{2\pi} \det(L(t_{p},t_{q})_{p,q=1}^{m})dt_{1}...dt_{m}$ \cite[page 112]{gohberg}. $L(t_{p},t_{q})$ is analytic in $\kappa$ on $\mathbb{C} \begin{align}ckslash \{0\}$ by construction. Define $L_{m}:=\det(L(t_{p},t_{q})_{p,q=1}^{m})$ then $L_{m}$ is analytic in $\kappa$ on $\mathbb{C} \begin{align}ckslash \{0\}$. The idea is to show that $\det(I-L)$ is the uniform limit of the sequence of analytic functions $\{L_{m}\}$ on any compact set in
$\mathbb{C} \begin{align}ckslash \{0\}$.
\begin{equation}gin{align*}
R_{M} &:= \begin{equation}gin{itemize}ggl|\sum_{m=M}^{\infty} \frac{(-1)^{m}}{m!} \int_{[0, 2\pi]^{m} }L_{m}(t_{1},....,t_{m})dt_{1}...dt_{m} \begin{equation}gin{itemize}ggr| \\
& \leq \sum_{m=M}^{\infty} \frac{1}{m!} (2\pi)^{m} \Vert L_{m} \Vert_{L^{\infty}} \\
& \leq \sum_{m=M}^{\infty} \frac{1}{m!} m^{\frac{m}{2}} (2\pi \Vert L \Vert_{L^{\infty}})^{m}
\end{align*}
The second inequality comes from Hadamard's Inequality.
As proved in \cite{bornemann}, the power series $\Phi(z) = \sum_{m=1}^{\infty} \frac{m^{(m+2)/2}}{m!}z^{m}$ defines an entire function on $\mathbb{C}$, together with the fact that $\Vert L \Vert_{L_{\infty}}$ is uniformly continuous in $\kappa$ on any compact set in $\mathbb{C} \begin{align}ckslash \{0\}$, we have $R_{M} \rightarrow 0$ as $M \rightarrow \infty$ locally uniformly in $\kappa$ on $\mathbb{C} \begin{align}ckslash \{0\}$. Thus $\det(I-L)$ is the locally uniformly convergent limit of a sequence of analytic functions in $\kappa$ on $\mathbb{C} \begin{align}ckslash \{0\}$. The claim follows.
\end{proof}
An analogous statement holds for our numerical approximation, namely
that $f_N(\kappa)$ is analytic in $\kappa$ close enough
to the positive real axis.
This follows from Lemma \eqref{fnm}, which says $f_{N}(\kappa) = \det(I-M_N(\kappa))$, an $N$-dimensional matrix determinant, and the fact that matrix entries are linear combinations of Hankel functions.
From Theorem~\ref{t:main}, $f_N$ vanishes exponentially fast at
each eigefrequency $\kappa_j$,
and thus, if we assume that the derivative $f'_N(\kappa_j)$ is bounded away from
zero for sufficiently large $N$,
the roots of $f_N$ approach the true eigenfrequencies with accuracy
exponential in $N$.
\remark{We do not prove that $f_N(\kappa)$ converges to $f(\kappa)$
exponentially for all $\kappa$;
indeed the numerical evidence (Section~\ref{s:num}) will be
that this convergence is merely algebraic for $\kappa$
away from eigenfrequencies.}
All that is now needed is an efficient method to find good approximations to the
real roots of the numerical Fredholm determinant $f_N(\kappa)$.
We propose Boyd's ``degree-doubling'' method \cite{boyd},
which, given that our function is analytic on the real axis,
is spectrally accurate in the number of function evaluations \cite{boyd}.
Thus just a few evaluations per root found will be enough
to approach machine accuracy.
Say we wish to find roots of $f_N$ in an interval $\kappa\in[a,b]$.
We change variable to
$\kappa(\theta) = \frac{b+a}{2} + \frac{b-a}{2} \cos \theta$,
choose a small number $M$, and evaluate the function on
a regular grid in $\theta$, i.e.\ $f_j = f_N(\kappa(\pi j/M))$, $j=1,\ldots,2M$.
Note that only $M+1$ evaluations are needed since $\kappa(2\pi-\theta)
=\kappa(\theta)$.
Since $f_N(\kappa(\theta))$ is a $2\pi$-periodic function of $\theta$
analytic in a neighborhood of the real axis,
the Fourier representation
\begin{equation}
f_N(\kappa(\theta)) \approx \sum_{m=-M}^M c_m e^{im\theta}
\label{fou}
\end{equation}
is exponentially convergent in $M$.
(This is equivalent to a Chebyshev expansion in the variable $\kappa$.)
The coefficients $\{c_m\}$ are
computed via the fast Fourier transform of the vector $\{f_j\}$.
In practice we start with $M=4$, and double $M$, reusing previous
$f_j$ values, until $|c_M/c_0|\le 10^{-12}$.
Writing $z=e^{i\theta}$, \eqref{fou} is a Laurent expansion in $z$,
hence
$$
q(z) \;:=\;
z^M \!\! \sum_{m=-M}^M c_m z^m
$$
is a degree-$2M$ Taylor series with the same nonzero roots.
These roots are found by insertion of the vector $\{c_m\}$
into a companion matrix \cite{companion}
and finding its eigenvalues $\mu_i$ at a cost of $O(M^3)$
(although we note that evaluation of $f_j$ dominates over this
cost by far).
Finally, only the eigenvalues $\mu_i$ within $\varepsilonilon$ of the unit
circle are kept; these are converted back to give the roots
$\kappa_i = \frac{b+a}{2} + \frac{b-a}{2} \re \mu_i$.
The imaginary parts
\begin{equation}
\label{eq: betai}
\begin{equation}ta_i := \frac{b-a}{2} \im \mu_{i}
\end{equation}
we observe are good indicators of of the size of errors in the roots.
This algorithm is available in {\tt MPSpack}\ \cite{mpspack}
as \verb?@utils/intervalrootsboyd.m?
Finally, if the above criterion for Fourier series decay is not
met with $M=512$, or if it turns out that $|\begin{equation}ta_i|>\begin{equation}ta$, where $\begin{equation}ta$
is a fixed algorithm parameter,
then the interval $[a,b]$ is instead subdivided
and the process repeated on the smaller intervals.
\section{Numerical results for a simply-connected domain}
\label{s:num}
\subsection{Convergence of the Fredholm determinant}
To demonstrate the convergence of $f_{N}(\kappa)$ given by \eqref{fnk}
as a function of $N$, the number of quadrature nodes on $\Gamma$,
we use the non-symmetric planar domain described in Fig.~\ref{fig: ns}.
We test $\kappa$ values near the 100th eigenfrequency $\kappa_{100}$.
As the graph in Fig.~\ref{fig: ns} shows, for $\kappa=\kappa_{100}$,
convergence to zero is at least exponential.
However, as $\kappa$ moves away from the eigenfrequency,
the colorscale plot shows that the initial exponential
convergence deteriorates to much slower algebraic convergence.
We believe the latter is of third order,
although we do not have a proof of this.
(A possible explanation for third-order convergence is that it is
what a naive Nystr\"om method without Kress' analytic split would
give for the operator $I-2D$.)
\begin{equation}gin{figure}[!ht]
(a) \raisebox{-1.6in}{\includegraphics[width=0.25\textwidth]{NS}}
(b) \raisebox{-1.8in}{\includegraphics[width=0.45\textwidth]{convRdetsNS}}\\
\centering
(c) \raisebox{-1.3in}{\includegraphics[width=0.95\textwidth]{convRdetExNS}}
\caption{
(a) Domain defined by $r(\theta) = 1+0.2\cos{3\theta}+0.3\sin{2\theta}$.
(b) $\log_{10}{f_{N}(\kappa)}$ near $\kappa_{100} = 20.43009417604$ (converged value).
the vertical axis shows $\log_{10}{(\kappa-\kappa_{100})}$;
(c) the convergence of $f_{N}(\kappa_{100})$ to zero. $N$ is the number of quadrature nodes on the boundary.}
\label{fig: ns}
\end{figure}
\subsection{Convergence of the determinant roots to the eigenfrequencies}
With the same domain as above, we now verify the claim of the previous
section that a root converges as fast as the rate of vanishing
of the determinant at a true eigenfrequency.
We solve for roots of $f_{N}(\kappa)$ on the interval $[20.4, 20.5]$
containing $\kappa_{100}$ using the method of Section \ref{s:boyd}.
Fig.~\ref{fig: zeroConvNS}
shows at least exponential convergence of the numerical root
to its converged value $\kappa_{100}$.
Note that 14-digit accuracy (15-digit relative accuracy)
is achieved using only $N=180$.
\begin{equation}gin{figure}[!ht]
\includegraphics[width=0.9\textwidth]{convspec}
\caption{Convergence of the eigenfrequency error with $N$.
The vertical axis shows the error
(relative to its converged value)
of the root found by the method of Sec.~\ref{s:boyd} at each $N$.
$N$ is the number of quadrature nodes on the boundary.
}
\label{fig: zeroConvNS}
\end{figure}
\begin{equation}gin{figure}[!ht]
(a) \raisebox{-2.4in}{\includegraphics[width=0.45\textwidth]{annulus}}
(b) \raisebox{-2.4in}{\includegraphics[width=0.45\textwidth]{crescent}}
\caption{
(a) Annular domain with boundary curves $\gamma_{2}: r(\theta) = 1+0.2\cos{3\theta}+0.3\sin{2\theta}$, and $\gamma_{1}: r(\theta) = 0.5+0.1\cos{3\theta}+0.15\sin{2\theta}$, $0 \leq \theta \leq 2\pi$.
(b) Crescent-shaped domain with strong exterior resonances,
with polar parametric description
$r(s) = \frac{0.2}{1+\exp{(4(s-3\pi/2)(s-\pi/2)})}+0.4$,
$\theta(s) = -\frac{49}{50}\pi\sin{s}$,
$0 \leq s \leq 2\pi$.
}
\label{fig:annularSector}
\end{figure}
\section{The resonance phenomenon and multiply-connected domains}
\label{s:res}
If the domain $\Omega$ has a hole,
Theorem~\ref{kress} does not apply,
and we cannot therefore know that every root of the Fredholm determinant
$f(\kappa)$ indicates a
Dirichlet eigenfrequency of $\Omega$.
The following lemma characterizes this new scenario.
We denote the inner boundary $\gamma_{1}$ and outer boundary $\gamma_{2}$.
Also let $\Omega_{1}$ be the domain that $\gamma_{1}$ encloses;
see Fig.~\ref{fig:annularSector}(a).
\begin{equation}gin{lemma}
Let $\Omega$ be a domain with a hole $\Omega_{1}$, and boundary
$\Gamma = \gamma_{1} \cup \gamma_{2}$.
Then the operator $I-2D$ on $\Gamma$ has a nontrivial nullspace if $\kappa$ is a Neumann eigenfrequency of $\Omega_{1}$.
\label{l:hole}
\end{lemma}
Recall that Neumann eigenfrequencies are the discrete $\kappa$ values where
nontrivial solutions to \eqref{helm} with $u_n=0$ on $\Gamma$ exist.
Such eigenfrequencies generally do not coincide with the desired Dirichlet
eigenfrequencies, thus our method of double layer potential produces incorrect roots for domains not simply connected.
The obvious generalization of the lemma
to domains with multiple holes also holds.
\begin{equation}gin{proof}
$\Omega_{1}$ has countably many interior Neumann eigenmodes. For any such eigenmode with boundary data $u$ and $u_{n}\equiv 0$ on $\gamma_{1}$,
we can first extend $u$ to $\tilde{u}$ defined on $\gamma_{1} \cup \gamma_{2}$ by setting $\tilde{u}=0$ on $\gamma_{2}$. We construct the double-layer potential
$\mu:=\mathcal{D} \tilde{u}$ at the corresponding eigenfrequency. Thus $\mu$ is a solution to the Helmholtz equation on $\mathbb{R}^2 \begin{align}ckslash \gamma_{1}$. Furthermore, for $x \in \mathbb{R}^{2} \begin{align}ckslash \overline {\Omega_{1}}$,
$\mu (x) = \mathcal{D} \tilde{u}
= \mathcal{D} u = \mathcal{D}u - \mathcal{S} u_{n} = 0$
by Green's representation theorem \eqref{GRF}
applied to the exterior of $\Omega_1$.
Consider the continuous extension of $\mu$ from inside $\Omega$ to $\gamma_{1}$, from the jump relation \eqref{JR3},
we see $(D-\frac{1}{2})\tilde{u} = 0$, i.e. the integral equation has a nontrivial solution.
\end{proof}
For such a domain, if one solves for the roots of $f_{N}(\kappa)$, one gets not only the Dirichlet eigenfrequencies of $\Omega$, but also the Neumann eigenfreqencies of the enclosed domain $\Omega_{1}$, which we call the spurious roots.
This has an important consequence:
even for a simply connected domain, as the geometry becomes more concave,
spurious roots may show up numerically
(first observed in this context by B\"acker \cite[Sec.~3.3.6]{backerbim}).
The operator $I-2D(\kappa)$ becomes singular for a $\kappa$ very close
to the real axis, resulting in a determinant very close to zero for a real
$\kappa$.
Any root-finding method working in finite precision thus cannot distinguish those $\kappa$ from true eigenfrequencies. Physically, this corresponds to a resonance of the exterior Neumann boundary-value problem for the domain's boundary,
since the operator $I-2D$ also arises in the potential-theoretic solution of this problem.
In \cite{ellipseres} it is proved, via an elliptical cavity domain,
that such boundary value problem resonances may exist with $\im \kappa$
becoming exponentially small as $\re \kappa$ grows.
We now demonstrate this problem, using the concave domain of Fig.~\ref{fig:annularSector}(b). It closely resembles, and can be viewed as a smooth approximation of, an annular sector with inner radius $0.4$, outer radius $0.6 $ and angular ``openness'' parameter $\frac{49}{50}\pi$. The disk with radius $0.4$ has a Neumann eigenfrequency $\kappa_{N} = 26.2996521844$. And indeed, for the
cresent domain, our
root-finding method returns a spurious root $\kappa_{0} = 26.30048303974$,
clearly visible in Fig.~\ref{fig: sweep}(b). This $\kappa_{0}$ is not exactly $\kappa_{N}$ because the crescent domain is not an exact annulus.
\subsection{A new representation for the Dirichlet eigenvalue problem}
We can remedy the above non-robustness by constructing the boundary integral equation using the {\em combined field} potential,
$$u : = \mathcal{D}\varphi+i \eta \mathcal{S}\varphi~,$$
where $\eta$ is a real parameter which, following \cite{coltonkress}, we
set to be $\kappa$.
This is standard in the acoustic scattering literature, but to our
knowledge has not been used for the eigenvalue problem before.
(The idea was suggested in one sentence of \cite[Sec.~3.3.6]{backerbim}.)
Enforcing the Dirchlet boundary condition \eqref{bc} on the combined field potential gives the CFIE
\begin{equation}gin{equation}
\label{CFIE}
(I-2D-2i\eta S)\varphi= 0
\end{equation}
For the CFIE we have the following equivalence relation;
in contrast to Theorem~\ref{kress}, it does not require simply connectedness
of the domain.
\begin{equation}gin{theorem}
\label{thm: CFIE}
Let $\Omega$ be a bounded domain with twice continuously differentiable boundary $\Gamma$.
For each $\kappa \in \mathbb{C}\begin{align}ckslash\{0\}$ with $\operatorname{Im}\kappa \geq 0$,
$\kappa^2$ is a Dirichlet eigenvalue of $\Omega$ if and only if
$I-2D(\kappa)-2i\eta S(\kappa)$ has a nontrivial nullspace,
where $\eta \neq 0$ is an arbitrary real number with
$\eta \operatorname{Re} \kappa \geq 0$.
\end{theorem}
\begin{equation}gin{proof}
"$\Rightarrow$" Suppose $u$ is an eigenfunction,
using the same argument as in Lemma \ref{l:nullspace} we have
$(1-2D^T)u_{n}^{-}=0$. Green's representation theorem \ref{GRF} says
$Su_{n}^{-}=0$. Thus $(1-2D^T-2i\eta S)u_{n}^{-}=0$
In the dual system
$\langle C(\Gamma), C(\Gamma) \rangle$ with the bilinear form
$\langle \varphi, \psi \rangle:= \int_{\Gamma} \varphi(x) \psi(x) dx$,
$S$ is self-adjoint and the adjoint of $D$ is $D^T$ \cite[p. 41]{LIE}.
By the Fredholm alternative, $I-2D-2i\eta S$ has a nontrivial nullspace.
"$\Leftarrow$" Suppose $\varphi\in \Null(I-2D-2i\eta S)$ and $\varphi$ is not identically zero.
Consider $\mu:=(\mathcal{D}+i\eta \mathcal{S})\varphi\in C^2(\mathbb{R}^2\begin{align}ckslash\Gamma)$,
then $\mu$ satisfies \eqref{helm} by construction.
We look at $\mu^{\pm}$ and $\mu_{n}^{\pm}$ using the jump relations \eqref{JR1} through \eqref{JR4}.
First, $\mu$ satisfies the zero Dirichlet boundary condition for the interior problem since
$\mu^{-}=(D-\frac{1}{2}+i\eta S)\varphi =0$.
So now we need only show that $\mu$ is nontrivial.
Suppose $\mu$ is identically zero in $\Omega$, then
$\mu_{n}^{-}=[T+i\eta (D^T+\frac{1}{2})]\varphi=0$.
Thus we have
$\mu^{+} = (D+\frac{1}{2} +i\eta S)\varphi = \varphi$, and
$\mu_{n}^{+} = [T+i\eta(D^T-\frac{1}{2})]\varphi = -i\eta \varphi$.
Therefore $\mu$ is a solution to \eqref{helm} on $\Omega_{+}$ with the impedance boundary condition
\begin{equation}
i\eta \mu+\mu_{n}\;=\;0 \qquad \mbox{ on } \hspace{0.5cm} \Gamma
~,
\label{imp}
\end{equation}
and $\mu$ is radiative in
the exterior component containing infinity.
In this infinite component $\mu$ has a unique solution when $\eta \operatorname{Re} \kappa \geq 0$ ~\cite[p.~97]{CK83},
thus $\mu\equiv 0$ in this component.
So $\varphi$ must be identically zero on the
boundary of this component. If $\Omega$ has no holes, we have reached a contradiction.
Otherwise, let $\Omega_{1}$ be any of the holes in $\Omega$, with boundary $\gamma_1$.
Let $n'$ be the unit normal vector pointing to the exterior of $\Omega_{1}$, then
$n' = -n|_{\gamma_{1}}$. $\mu$ is a solution to \eqref{helm} on $\Omega_{1}$
with boundary condition
\begin{equation}
i\eta \mu-\mu_{n'}\;=\;0 \qquad \mbox{ on } \hspace{0.5cm} \gamma_{1}
~,
\label{imp1}
\end{equation}
Multiplying each side of \eqref{helm} by $\overline{\mu}$,
integrating over $\Omega_{1}$ and applying Green's first identity
and \eqref{imp1}, we get
\begin{equation}gin{equation}
\kappa^{2} \| \mu \|^{2}_{L^{2}(\Omega_{1})} = \| \nabla \mu \|^{2}_{L^{2}(\Omega_{1})}-i \eta \| \mu \|^{2}_{L^{2}(\gamma_{1})}
~.
\end{equation}
Taking the imaginary we have
$2\operatorname{Re} \kappa \operatorname{Im}\kappa \| \mu \|^{2}_{L^{2}(\Omega_{1})} = -\eta \| \mu \|^{2}_{L^{2}(\gamma_{1})}$, which is impossible given all the conditions on $\kappa$ and $\eta$
unless $\mu$ vanishes on $\gamma_{1}$.
Hence $\mu_{n'}$ vanishes on $\gamma_1$ by \eqref{imp1}. By Green's representation theorem,
$\mu$ is identically zero in $\Omega_{1}$.
We have shown $\mu\equiv0$ in all of $\mathbb{R}^2\begin{align}ckslash\overline{\Omega}$
and this means $\varphi$ is identically zero on $\Gamma$,
which is a contradiction.
So $\mu$ is a nontrivial solution to \eqref{helm} -\eqref{bc}, hence an eigenfunction.
\end{proof}
Thus by adopting the combined field integral equation, we
have a robust method with no spurious frequencies where the
boundary operator is singular.
We show this in Fig.~\ref{fig: sweep},
where we show the minimum singular value of (the Nystr\"om approximation to)
the original operator and of the CFIE,
for (a) a doubly-connected domain and (b) a simply-connected domain with
strong exterior resonances.
In both cases this shows that the CFIE removes the spurious roots.
We now mention numerical implementation issues for the
CFIE formulation.
For the spectrally-accurate discretization of
the single-layer operator, we use the same method
as for the double-layer, replacing
$L$ by $Q(t,s) = \Phi(x(t),x(s)) |x'(s)|$,
and replacing the logarithmically singular term
\eqref{L1} by \cite[Eq.~(2.6)]{kress91},
\begin{equation}
Q^{(1)}(t,s):=-\frac{1}{2\pi}
J_0(\kappa r(t,s)) \sqrt{x_1'(s)^2+x_2'(s)^2}
\label{Q1}
\end{equation}
and defining $Q^{(2)}$ as before by the difference \eqref{L2}.
The resulting matrix we call $Q_{N}{(\kappa)}$.
For each $N$ the determinant of the $N$-node Nystr\"om discretization matrix
$I-M_{N}(\kappa)-i\eta Q_{N}(\kappa)$ is analytic in $\kappa$. Thus we are able to apply the same
root-finding method to it as before, and achieve rapid convergence with $N$
for the roots, hence eigenvalues found.
\begin{equation}gin{remark}
Note that for $\eta \neq 0$, $D+i\eta S$ is no longer in $\mathcal{J}_{1}(L^{2}[0, 2\pi])$,
so the main convergence theorem \ref{t:main} does not readily apply.
Instead let $\mathcal{J}_{2}(L^{2}[0, 2\pi])$ be the space of {\em Hilbert-Schmidt operators} on $L^{2}[0, 2\pi]$,
which is the collection of all linear operators with square summable singular values,
then $D+i\eta S$ is in $\mathcal{J}_{2}(L^{2}[0, 2\pi])$.
Given $A \in \mathcal{J}_{2}(L^{2}[0, 2\pi])$, $ \displaystyle \prod_{j=1}^\infty(I-\lambda_{j}(A))$
is not necessarily convergent. However, we expect that numerically,
the convergence theorem \ref{t:main} should be {\em close} to holding.
Since the singular values of $S$ decay like $\frac{1}{j}$,
their sum only diverges logarithmically.
In addition, our experiments show that $\det(I-M_{N}(\kappa)-i\eta Q_{N}(\kappa))$
converges to zero as $N \rightarrow \infty$ if and only if $\kappa=\kappa_{j}$.
\end{remark}
\begin{equation}gin{figure}[!ht]
(a)\raisebox{-1.8in}{\includegraphics[width=0.95\textwidth]{sweepannulusSVD}}
\\
(b)\raisebox{-1.8in}{\includegraphics[width=0.95\textwidth]{sweepcrescentSVD}}
\caption{
Lowest singular values vs frequency $\kappa$, for:
(a) the annular domain Fig.~\ref{fig:annularSector}(a);
(b) the crescent domain Fig.~\ref{fig:annularSector}(b).
$\sigma_{min} $ denotes the smallest singular value
of the discretized
$I-2D(\kappa)$ (red), and CFIE $I-2D(\kappa)-2i\eta S(\kappa)$ (blue).
The true eigenfrequencies are shown by the blue diamonds.
$N$ is the number of quadrature nodes on the boundary}
\label{fig: sweep}
\end{figure}
\section{Close eigenfrequencies and the singular value method}
\label{s:svd}
Our root-finding method worsens in accuracy when
$f(\kappa)$ has close roots, or roots with multiplicity higher than one.
\footnote{Note that we do not expect this to occur too often, since
for a generic domain eigenvalues are all simple \cite{KS}.}
In this section we discuss how we overcome this problem if it does occur,
by reverting to the standard SVD method.
Indeed, no method that relies on evaluating the Fredholm determinant
$f(\kappa)$ alone could succeed in this case, because the root-finding
problem is well known to be ill-conditioned with respect to perturbations in
the function
(eg, for a polynomial, perturbations in its coefficients).
We discuss the case of two close eigenfrequencies
$\kappa_j \approx \kappa_{j+1}$.
Then $f(\kappa)=t(\kappa)(\kappa-\kappa_j)(\kappa-\kappa_{j+1})$
for some locally smooth function $t(\kappa)$.
For simplicity, let $f$ be perturbed by a constant value $\varepsilon$;
then, for small $\varepsilon$, the change induced in the root $\kappa_j$
is of size
\begin{equation}
\delta\kappa \approx \begin{equation}gin{itemize}ggl|\frac{\varepsilon}{f'(\kappa_j)}\begin{equation}gin{itemize}ggr| =
\begin{equation}gin{itemize}ggl|\frac{\varepsilon}{(\kappa_{j+1}-\kappa_j)t(\kappa_j)}\begin{equation}gin{itemize}ggr|
~,
\label{cond}
\end{equation}
which blows up inversely with the gap between the eigenfrequencies.
This particular perturbation demonstrates the ill-conditioning; other perturbations lead generically to a similar effect.
Even for $\varepsilon \approx 10^{-16}$ we may only retain accuracy $O(\varepsilon^{1/2})$
as two roots approach each other, and more if there are more close
roots or a higher-order degeneracy.
To remedy this, when two roots are found closer than $s \approx 10^{m} \varepsilon$,
where $m$ is the desired number of digits of accuracy in rootfinding,
we propose switching to a more expensive method based on the SVD.
This requires finding the lowest singular values of the CFIE Nystr\"om
matrix $I-M_N(\kappa) - i\eta Q_N(\kappa)$,
and is very similar to existing eigenvalue solvers
\cite{backerbim,gsvd}.
We only use the SVD when forced to do so
since, due to the high cost of the SVD, and the increased number of
function evaluations required to find each root,
we will show that it is
an order of magnitude less efficient than our proposed method.
Thus the choice of the parameter $s$ affects the robustness and the speed of the algorithm. The smaller it is, the less often roots less than $s$ apart will
occur, and thus the faster the computation.
However, smaller $s$ causes a worsening of the accuracy of close roots.
This is more severe for multiple roots:
for $n>1$, an order-$n$ root has error on the order of $\varepsilon^{\frac{1}{n}}$,
Thus to obtain desired accuracy, $s$ has to be set to be large enough.
In practice we fix $s = 10^{-3}$.
Once we switch to using the SVD on an interval of frequency $\kappa$,
the smallest singular value $\sigma_{\min}(I-M_{N}(\kappa)-i\eta Q_{N}(\kappa))$ is far from analytic in $\kappa$ (see Fig.~\ref{fig: sweep} which shows the typical W-shaped function), so the Boyd's method is not useful.
Instead we use recursive subdivision starting on a regular grid of values, followed by iterative parabolic fitting of $\sigma_{\min}^2(I-M_{N}(\kappa)-i\eta Q_{N}(\kappa))$
as detailed in \cite[Appendix~B]{sca}. This algorithm is available in {\tt MPSpack}\ \cite{mpspack}
as \verb?@evp/gridminfit.m?
To demonstrate the higher accuracy of the SVD method over the Boyd's method in the presence of close eigenfrequencies, we choose an ellipse domain,
and vary its eccentricity to cause a near-degeneracy of controllable separation $\kappa_{j+1}-\kappa_j$.
Fig.~\ref{fig:ellipse} shows the eigenfrequencies passing through
each other as a function of the eccentricity,
solved by the determinant (red) and by the SVD methods (blue).
Errors of absolute size around $10^{-7}$ appear in the determinant
method but not the SVD method.
As expected from \eqref{cond}, we see the errors $\delta\kappa$ blow up like $\frac{1}{|\kappa_{j+1}-\kappa_{j}|}$.
\begin{equation}gin{figure}[h!]
\includegraphics[width=1\textwidth]{ellipboydsvd}
\caption{Two close eigenfrequencies of an ellipse crossing as a function of
the eccentricity.
Red shows values computed by Boyd's method applied to the determinant on the frequency interval $\kappa\in[7, 9]$. Blue shows values computed by the SVD method
of Sec.~\ref{s:svd}. $\eta$ is set to be zero since we expect and observe
no exterior resonances.}
\label{fig:ellipse}
\end{figure}
\section{Numerical performance of the solver}
\label{s:ps}
In this section we demonstrate the
improved efficiency of our solver, the Boyd's method with determinant, compared
to an existing boundary-integral solver, namely the
SVD method described in the previous section.
We used a Linux workstation with two quad-core E5-2643 3.3GHz Xeon CPUs,
running MATLAB R2013b,
except for Hankel function evaluations which use Rokhlin's
fortran code {\tt hank103.f} (eg see \cite{mpspack}).
\subsection{Non-resonant domain solved via pure double-layer representation}
We computed the first 100 eigenfrequencies for the domain in Fig.~\ref{fig: ns}(a)
using both the Boyd's method and the standard SVD method
as shown on the first two rows of table \ref{tab:t}, respectively. For both methods,
the initial number of quadrature nodes is scaled by setting $N = \max{(150, 100+5\kappa)}$.
For the Boyd's method, the initial interval used was $[2, 5]$, converged $\kappa_{100} = 20.4300941760382$
and the largest $N$ is 202.
For moderate eigenfrequencies, as shown on the last two rows of table \ref{tab:t},
we solved the 6 eigenfrequencies in the interval $[100, 100.1]$ using 750 quadrature nodes
using both methods. We used a pure double-layer potential ($\eta=0$)
since this domain is simply-connected and has no problem with
exterior resonances. The error parameter from Section~\ref{s:boyd} is set to $\begin{equation}ta=10^{-14}$.
For the Boyd's method,
the error $\varepsilonilon$ of each eigenfrequency is estimated using the
magnitude of the imaginary part of
the root found, as explained in Sec.~\ref{s:boyd}.
For the SVD method, error $\varepsilonilon$ is estimated as follows.
From Theorem~1 in \cite{bnds}, the distance of any fixed $\kappa_{0}^{2}$ to the true spectrum
can be bounded by $C\kappa_{0} t[u]$, where $C$ is a constant depending only on $\Omega$, $u$ is a solution to \eqref{helm} with $\kappa = \kappa_{0}$,
and $t[u] := \| u \|_{L^{2}(\partial \Omega)}/ \| u \|_{L^{2}(\Omega)}$
is a measure of the relative boundary error.
Since our domain is star-shaped, we can use \cite[(6.1)--(6.2)]{bnds}
to give an explicit estimate for $C$ of approximately 3.5.
By representing $u$ as double layer potential with density $\varphi$, we have $u |_{\partial \Omega} = (D - \frac{1}{2}) \varphi$ and $ u |_{\Omega} = \mathcal{D}\varphi$. Numerically $t[u]$ can be bounded by $\frac{\sigma_{min}(I-M_{N})}{2\| \mathcal{D}\hat{\varphi}\|_{L^{2}(\Omega)}}$, where $\hat{\varphi}$ is the associated right singular vector of $\sigma_{min}(I-M_{N})$. Thus we estimate the relative error in $\kappa$ to be $\frac{C\sigma_{min}(I-M_{N})}{2\kappa \| \mathcal{D}\hat{\varphi}\|_{L^{2}(\Omega)}}$, where $ \| \mathcal{D}\hat{\varphi}\|_{L^{2}(\Omega)}$ is estimated using
crude quadrature scheme in the interior of $\Omega$.
\begin{equation}gin{table}[!ht]
\begin{equation}gin{tabular}{|c| c | c | c | c | c | c | c | c |}
\hline
task & method & $\max{\operatorname{Im}{\tilde{\kappa}}}$ & mean ${\operatorname{Im}{\tilde{\kappa}}}$ & $\max{\sigma_{\min}}$ & mean $\sigma_{\min}$ & $\max{\varepsilonilon}$ &mean $\varepsilonilon$ & Time (s)\\ \hline
\multirow{2}{*}{$\kappa \le 20.5$}
&Boyd's &7.3e-15 & 1.4e-15 & 1.7e-14 & 2.1e-15 & 3.8e-14 & 6.2e-15 & 20\\ \cline{2-9}
&SVD & - & - & 6.8e-11 & 1.6e-12 &1.1e-10 & 2.6e-12 & 42\\ \hline
\multirow{2}{*}{$\kappa \sim 100$}
&Boyd's &1.6e-15 & 7.4e-16 & 6.1e-15 & 3.2e-15 & 5.5e-14 & 3.3e-14 & 16\\ \cline{2-9}
&SVD & - & - & 3.1e-11 & 5.5e-12 &1.1e-11 & 2.0e-12 & 151\\ \hline
\end{tabular}
\caption{Performance data for the nonsymmetric domain in Fig.~\ref{fig: ns}(a)}
\label{tab:t}
\end{table}
\subsection{Crescent-shaped domain solved via the CFIE}
For an example requiring the combined field potential
for a robust solution,
we test the highly-resonant crescent domain in Fig.~\ref{fig:annularSector}(b). Computation is done again for the first 100 eigenfrequencies.
In both methods, the number of quadrature nodes is given by $N = \max{(350, 100+7\kappa)}$.
For the Boyd's method, the initial interval used was $[15, 17]$,
converged $\kappa_{100} = 50.17535680154$
and the largest $N$ is 456.
The error parameter is set to $\begin{equation}ta=10^{-12}$.
For error estimate, the $C$ value for this highly concave domain is not known but we expect it to be $O(1)$ based on discussion in \cite{bnds}.
Thus we computed $\frac{\sigma_{min}(I-M_{N}-i\eta Q_{N})}{2\| \mathcal{D}\hat{\varphi}\|_{L^{2}(\Omega)}}$,
where $\hat{\varphi}$ is the associated right singular vector of $\sigma_{min}(I-M_{N}-i\eta Q_{N})$, as an estimate for
the relative error $\varepsilonilon$ in $\kappa$, up to the constant factor $C$.
\begin{equation}gin{table}[!ht]
\begin{equation}gin{tabular}{| c | c | c | c | c | c | c | c |}
\hline
method & $\max{\operatorname{Im}{\tilde{\kappa}}}$ & mean ${\operatorname{Im}{\tilde{\kappa}}}$ & $\max{\sigma_{\min}}$ & mean $\sigma_{\min}$ & $\max{\varepsilonilon / C}$ &mean $\varepsilonilon/ C$ & Time (s)\\ \hline
Boyd's & 6.7e-13 & 1.7e-14 & 4.9e-13 & 1.6e-14 & 2.1e-13 & 9.0e-15 & 98 \\ \hline
SVD & - & - & 3.5e-6 & 5.0e-8 & 1.4e-11& 1.7e-13 & 368 \\ \hline
\end{tabular}
\caption{Performance data for the crescent domain in Fig.~\ref{fig:annularSector}(b)}
\end{table}
\begin{equation}gin{remark}
Boyd's rooting search method is sufficient to find the first 100 eigenfrequencies to at least 12 digits accuracy for those two examples,
i.e., adjacent roots were never closer than $10^{-3}$ so the SVD was never needed
to replace Boyd's method.
\end{remark}
Finally, we show some eigenmodes of the crescent domain in Fig.~\ref{fig:annularSector}(b),
computed as follows.
Once we obtain an eigenfrequency $\kappa_{j}$, we can extract the normal
derivative data
from the left kernel of the Nystr\"om matrix $I-M_{N}(\kappa_{j})$
then use Green's representation
formula \eqref{GRF} to reconstruct the eigenmode.
Fig.~\ref{f:modes} shows the first 100 such modes;
they are close to the separation-of-variable forms
which would result for an annular sector.
\begin{equation}gin{figure}[!ht]
\includegraphics[width=0.9\textwidth]{emCrescent}
\caption{
Modes $u_1$ to $u_{100}$ of the crescent domain, computed via the CFIE
method of this paper, as discussed in Sec.~\ref{s:ps}.
\label{f:modes}
}
\end{figure}
\section{Conclusions}
\label{s:c}
We have developed a robust method to compute Dirichlet eigenvalues for 2D domains with high accuracy and high efficiency compared to the traditional SVD root-finding method.
We applied Boyd's root-finding method, exploiting the analyticity with respect to frequency of the Fredholm determinant of the boundary integral operator.
This is approximated by the determinant of a Nystr\"om matrix
derived using as spectrally-accurate product quadrature.
Since the determinant is cheap to evaluate, and Boyd's
method requires only around 5 evaluations per eigenvalue found,
we show that the method is 2-10 times faster than existing
SVD-based methods.
In the case of an analytic boundary, we proved that our determinant has
exponential convergence to zero at the true eigenvalues,
and show that this rapid convergence carries over to the computed eigenvalues.
Hence we are able to achieve 13 digits of relative accuracy for
all eigenvalues computed for a star-shaped domain and 12 digits for a highly concave domain, with small numbers of boundary nodes.
For multiply-connected domains or those with exterior resonances,
we introduce a combined-field representation, prove that it
is robust, and show that it eliminates spurious solutions that are
present in the standard approach.
In the case of close eigenfrequencies, we revert to the SVD-based method;
this is not a common occurrence.
We expected that corners, and thus very general domains, can be handled
with a corner-refined quadrature scheme.
One challenge remaining is to analyze a regularization of the CFIE
(case $\eta>0$) in which the Fredholm determinant is not infinite;
the $S$ operator we currently use in the CFIE is not in trace class.
For this we suggest considering $\mathcal{D}+i\eta \mathcal{S}^{2}$.
\begin{equation}gin{itemize}bliographystyle{abbrv}
\begin{equation}gin{itemize}bliography{alex}
\end{document}
|
\begin{document}
\title{Exploiting Non-Markovianity of the Environment for Quantum Control}
\author{Daniel M. Reich}
\affiliation{Theoretische Physik, Universit\"{a}t Kassel,
Heinrich-Plett-Str. 40, D-34132 Kassel, Germany}
\author{Nadav Katz}
\affiliation{Racah Institute of Physics, The Hebrew University of
Jerusalem, Jerusalem 91904, Israel}
\author{Christiane P. Koch}
\affiliation{Theoretische Physik, Universit\"{a}t Kassel,
Heinrich-Plett-Str. 40, D-34132 Kassel, Germany}
\email{[email protected]}
\date{\today}
\begin{abstract}
When the environment of an open quantum system is non-Markovian,
amplitude and phase
flow not only from the system into the environment but
also back. Here we show that this feature can be exploited to
carry out quantum control tasks that could not be realized if the system
was isolated. Inspired by recent experiments on superconducting
phase circuits, we consider an anharmonic ladder with resonant amplitude
control only. This restricts realizable operations to SO(N). The
ladder is immersed in an environment of two-level systems.
Strongly coupled two-level systems lead to non-Markovian
effects, whereas the weakly coupled ones result in
single-exponential decay. Presence of the environment allows for
implementing diagonal unitaries that, together with SO(N), yield the
full group SU(N). Using optimal control theory, we obtain
errors that are solely $T_1$-limited.
\end{abstract}
\pacs{03.65.Yz,02.30.Yy,85.25.Dq}
\maketitle
Quantum control, employing external fields to steer the outcome of a
dynamical process~\cite{RiceBook,ShapiroBook}, holds the promise of
utilizing entanglement and matter interference as cornerstones of
future technologies. Accurate and reliable control solutions may be
identified by optimal control, provided the control target is
reachable. This question is addressed by
controllability analysis. For closed quantum systems, the answer is
determined solely by symmetries in the Hamiltonian and the available
resources such as power or bandwidth of the controls~\cite{DAlessandroBook}.
Controllability and control strategies for non-Markovian open quantum
systems remain largely uncharted territory. Non-Markovianity refers to
memory effects in the environment and the built-up of non-negligible
correlations between system and environment~\cite{BreuerJPB12}. It
is generic for condensed phase settings encountered e.g. in
light harvesting or solid-state devices. Non-Markovianity can be
measured in terms of information flowing from the environment back into
the system~\cite{BreuerPRL09}, increase of correlations if the system
is bi- or multipartite~\cite{RivasPRL10}, or re-expansion of the volume of
accessible states in Liouville space~\cite{LorenzoPRA13}. Each of
these measures
holds a promise for improved control for non-Markovian compared to
Markovian open systems: Partial recovery of coherence or growth of
correlations or a
larger accessible state space volume should all clearly facilitate
control. Indeed, correlations between system and environment
may improve fidelities of single qubit gates~\cite{RebentrostPRL09},
cooperative effects of control and dissipation may allow for entropy
export and thus cooling~\cite{SchmidtPRL11}, and
harnessing non-Markovianity may enhance the efficiency of quantum
information processing and communication~\cite{LaineSciRep14,BylickaSciRep14}.
Here, we go beyond merely improving a given figure of merit
and show that a non-Markovian environment may enable the implementation
of quantum operations that could not be realized without presence of
the environment. Our approach is based on separating the environment
into potentially beneficial and potentially detrimental parts, with the
latter setting the timescale for (almost) unitary operations. We
employ optimal control theory (OCT) to best exploit the beneficial
non-Markovian part of the environment while beating decoherence due to
the detrimental Markovian part. For a four-level anharmonic ladder
system with resonant amplitude control only, which by itself is
SO(4)-controllable, we demonstrate that full SU(4)-controllability
can be achieved due to the
presence of the environment. The fidelities are only limited by the
Markovian decay.
We investigate quantum control for a four-level system since
analytical solutions to the problem of population inversions can be
obtained by Pythagorean coupling~\cite{SuchowskiPRA11} which allow
for realizing arbitrary operations in SO(4).
A recent experimental demonstration employed resonant amplitude control in a
flux-biased Josephson phase circuit~\cite{Svetitsky14}.
The simplest way to construct an arbitrary element
of SU(N), provided that one is able to implement any
element of SO(N), is obtained by the Cartan decomposition.
It results in a decomposition
of all unitaries $U\in\,$SU(N) into local operations, $k_1,k_2\in\,$SO(N),
and a diagonal, unitary matrix $A$
such that $U = k_1 A k_2$~\cite{DAlessandroBook}.
The task to achieve full unitary controllability on the $N$-level
system therefore reduces to implementing an
arbitrary diagonal unitary. This is the problem we address in the
following, employing OCT.
We consider an anharmonic $N$-level system that interacts, possibly
strongly, with an environment. This interaction leads to
(i) pure dephasing due to long-time memory,
low-frequency noise;
(ii) energy relaxation due to weakly coupled near-resonant environmental
nodes; and (iii) visible splittings in the systems's energy levels due
to strongly coupled near-resonant
environmental nodes.
The strongly coupled modes are best accounted for
explicitly ("primary bath"), and we assume here that they can be
modeled by two-level systems
(TLS)~\cite{BaerJCP97,KochJCP02,GelmanJCP04,GualdiPRA13}. Both $N$-level
system and primary bath are weakly coupled to a thermal reservoir
("secondary bath") to account for effects (i) and (ii)~\footnote{
Strictly speaking, the low-frequency noise can also lead to
non-Markovian effects~\cite{Galperin2006}. However, purely
transversal coupling to a single environmental TLS can often fully
reproduce the experimentally observed behavior,
see e.g. Ref.~\cite{Lisenfeld2010}. We therefore absorb the effect
of the low-frequency modes into the phenomenological $T_{1}$ and $T_{2}$
times.}.
This is modelled by a Markovian master equation ($\hbar=1$)
for the joint state of system ("Q") and primary bath ("P"),
\begin{equation}
\label{eq:EoM}
\frac{d\rho_{QP}}{dt}=-i[H_{QP},\rho_{QP}] + \mathcal{L}_S(\rho_{QP})\,,
\end{equation}
with the Hamiltonian $H_{QP}$ generating the coherent evolution and
the Liouvillian $\mathcal{L}_S$ capturing the effect of the
secondary bath ("S"). The state of the system alone, $\rho_Q$,
is obtained by integrating over the primary bath
modes~\cite{BaerJCP97,KochJCP02} that can give rise to non-Markovian
effects. For $n_P$ TLS in the primary bath, $H_{QP}$ reads
\begin{equation}
\label{eq:Ham}
H_{QP} = H_Q + \sum_{i=1}^{n_P} H^{(i)}_{P} + \sum_{i=1}^{n_P}
H_{int}^{(i)}\,,
\end{equation}
with $H_Q$ describing an
anharmonic ladder, $E_n = n\omega_Q + \beta n(n+1)/2$,
with base frequency $\omega_Q$ and anharmonicity $\beta$ plus
control by an external field.
The $i$th TLS is characterized by the splitting
$\omega_i$, $H^{(i)}_P=\omega_i\sigma^{z}_{i}$, and couples
transversally to the $N$-level system,
\begin{equation}
\label{eq:coupling}
H_{int}^{(i)} = \frac{S^{(i)}}{2}\left(a\sigma_{i}^+ +
a^\dagger\sigma_{i}^-\right)\,,
\end{equation}
with $a^+$ ($a$) the creation (annihilation) operator of the $N$-level
system, and
the coupling constant $S^{(i)}$ corresponding to the
system's energy level splitting when on resonance with the $i$th TLS.
The Liouvillian models decay of system and primary bath,
\begin{equation}
\label{eq:lindblad}
\mathcal{L}_S (\rho) = \sum_k \left(A_k \rho A_k^{\dagger}
- \frac{1}{2} \left[A_k^{\dagger} A_k ,\rho \right]_+\right)\,,
\end{equation}
with $A_{n}=\sqrt{n/T_{1}}\Ket{n-1}\Bra{n}$ and
$A_i=\sqrt{1/T_{1}^{(i)}}\sigma_{i}^-$.
In order to limit the number of parameters,
we restrict our model to a $T_1$-limited environment. We have verified
that it effectively captures both loss and dephasing, i.e.,
adding pure dephasing characterized by $T_2^*$ behaves, in terms of
the final fidelities, similarly to Eq.~\eqref{eq:lindblad} with
increased $T_1$. A good realization of this model is
given by superconducting circuits where the TLS
correspond to dielectric defects~\cite{MartinisPRL05} and the thermal
bath can be taken at $T=0\,$K~\footnote{
For superconducting circuits, non-Markovian effects of purely
dephasing $1/f$ noise are overshadowed by the strongly transversally
coupled TLS, while the Markovian part of the $1/f$ noise leads to a
further reduction of $T_{1}$ and $T_{2}$.}.
In particular, the TLS can be characterized experimentally in terms of
their splitting, coupling to the $N$-level system, and
$T_1$~\cite{ShaliboPRL10,ShaliboPhD}; and the upper bound of modelling
both Markovian loss and pure dephasing by an effective $T_1$
becomes tight since
$T_2$ is typically close to $T_2^*$~\cite{BarendsPRL13}.
The $N$-level system is subjected to an external control $u(t)$ that
shifts its energy levels. This can be achieved, for example, by low-frequency
steering of the bias flux in phase qudits~\cite{ShaliboPhD}.
For low anharmonicity, the shift is harmonic,
\begin{equation}
\label{eq:control}
H_{c}\left[u(t)\right] = \sum_{n=0}^{N-1} u(t)\; n \omega_Q
\Ket{n}\Bra{n}\,.
\end{equation}
In case of the bias flux on the phase qudit, this corresponds to
neglecting terms that oscillate strongly
on the timescale of $\omega_Q$.
It is those terms that, for $N=4$,
yield SO(4) operations via the Pythagorean
coupling~\cite{Svetitsky14}.
Consequently, the two control
mechanisms, high-frequency steering on the one hand and low-frequency
steering on the other, do not interfere.
Moreover, our low-frequency control does not induce transitions to
levels with $n>4$
since all operators in the Hamiltonian~\eqref{eq:Ham} conserve
the occupation number of the joint state of system and primary bath.
In the absence of the primary bath, the control
Hamiltonian~\eqref{eq:control} does not allow for realizing arbitrary
diagonal unitaries in the four-level subspace. This is best analyzed in
terms of the dynamic Lie algebra. It represents the Hilbert space
directions along which the system can evolve and is formed by nested
commutators of control and drift Hamiltonian. Since $H_c$ and $H_Q$
commute, evolution along only a single direction is possible. The
scenario changes once the strongly coupled TLS of the primary bath
come into play. In fact, a single strongly coupled TLS is sufficient
to provide the remaining $N-1$ Hilbert space directions, required for
realizing an arbitrary diagonal unitary.
This is due to $H_{c}$ not commuting with
$H_{int}^{(i)}$. In more physical terms, $H_{int}^{(i)}$ allows for
the system wave function to be transferred to the TLS and back,
after acquiring the desired non-local phases.
These considerations of controllability hold, however, only for
unitary evolution. The secondary bath leads to irreversible loss of
energy and phase of both system and primary bath TLS. The only control
strategy that is available for such Markovian dynamics is to beat
decoherence (unless a protected region in Hilbert space exists
in which the desired dynamics can be generated). It is thus crucial to
carry out all operations as fast as possible. Since
OCT allows for identifying controls that operate at the speed
limit~\cite{GoerzJPB11}, we use it here, employing a recent
variant for unitary gates in open quantum systems~\cite{GoerzNJP14}.
Our optimization target is
$U_1 = \mathrm{diag}(1,-1,1,1)$, and we quantify success in terms of
the error, $1-F_{average}$~\cite{NielsenChuang}.
$U_1$ is a particularly difficult unitary to
implement, as exemplified by an error of over 40\%
in the absence of any strongly coupled TLS. While we discuss in the
following only $U_1$, we have verified that optimization towards
diagonal unitaries with random phases yield very similar errors~\footnote{
For example, optimization for 20 random diagonal unitaries, using
the parameters of Fig.~\ref{fig:controls}, yields
errors between $1.390\cdot 10^{-2}$ and $1.804\cdot
10^{-2}$, differing from that for $U_1$ by less than a
factor of 1.2.}. This suggests full SU(4)-controllability, once
implementation of $U_1$ is successful.
\begin{figure}
\caption{(color online)
Error after optimization for $\mathrm{diag}
\label{fig:markov}
\end{figure}
Figure~\ref{fig:markov} demonstrates the interplay of Markovian and
non-Markovian effects by plotting the error for $U_1$
as a function of the $T_1$ times of qudit
and one TLS: Errors below 1\% can be reached even for $T_1$ times
of the order of a few microseconds.
Due to increasing decoherence rate with increasing
excitation, short $T_1$ times of the qudit have a slightly more
severe effect than short $T_1$ times of the TLS.
A multitude of controls lead to the results shown in
Fig.~\ref{fig:markov}.
Two examples of optimized controls, obtained using different
constraints, are displayed in
Fig.~\ref{fig:controls}(a):
\begin{figure}
\caption{(color online)
(a): Optimized amplitudes with
the control shown in blue following a fixed ramp of $\pm 500\,$MHz over
$2.5\,$ns at the beginning and end and the
red dashed line obtained without imposing a ramp.
(b): Liouville space determinant of the system evolution --
increase of the determinant indicates non-Markovianity.
}
\label{fig:controls}
\end{figure}
The control can be restricted to low bandwidth by ramping it
into and out of resonance at the beginning and end of the
optimization time interval
(blue solid line in Fig.~\ref{fig:controls}(a)),
whereas fast oscillating controls are obtained without imposing a
ramp (red dashed line in Fig.~\ref{fig:controls}(a)).
The different controls all share the mechanism of moving the
qudit close to resonance with the TLS, picking up a non-local phase
due to the enhanced interaction, and moving the qudit back off
resonance. This sequence is repeated several times in order to
properly align all the phases in the four-level subspace. A
visualization of the dynamics is provided as supplementary material.
While both controls lead to similar
errors, the ramped control is easier to implement experimentally
and also fulfills the low-frequency approximation
used to derive the control Hamiltonian~\eqref{eq:control}.
All further calculations therefore employ ramped controls.
Both solutions shown in Fig.~\ref{fig:controls}(a) use
non-Markovianity of the time evolution as core resource for control.
This is seen in Fig.~\ref{fig:controls}(b) which plots the determinant
of the volume of reachable system states~\cite{LorenzoPRA13}, a
non-Markovianity measure that is easily evaluated numerically:
Any increase in the determinant indicates non-Markovianity.
Use of the environment as a resource
is further illustrated in Fig.~\ref{fig:control_nodiss} which explores
the dependence of the best possible error on qudit anharmonicity
and coupling strength between qudit and TLS: For very small coupling
no solution can be found and the error remains of the order one.
\begin{figure*}
\caption{(color online)
Error after optimization for $\mathrm{diag}
\label{fig:control_nodiss}
\end{figure*}
On the other hand,
a single, only moderately coupled TLS in the primary bath
is sufficient to yield good fidelities even for weak or zero
anharmonicity. In the latter case (Fig.~\ref{fig:control_nodiss}(a,d)),
the desired diagonal unitaries can be
realized if the operation time is sufficiently long. This
can only be exploited
for good $T_1$ times, utilizing the level-dependent coupling strengths.
The control problem becomes much easier for non-zero anharmonicity,
with a subtle interplay between the requirements of resolving the
qudit levels and sufficient interaction with all qudit levels.
The latter corresponds to small anharmonicity
(Fig.~\ref{fig:control_nodiss}(b,e)) and subsequently allows good
results even for weak coupling, whereas energy resolution is best for
larger anharmonicity (Fig.~\ref{fig:control_nodiss}(c,f)), which in
turn allows for very short operation times.
For fixed anharmonicity, one expects larger
coupling strengths and longer gate times to allow for better
fidelities. A few exceptions to this rule, which are observed
in Fig.~\ref{fig:control_nodiss}, can be attributed to the
numerical nature of our controllability analysis.
The observation that for a very weakly coupled TLS there exists
no anharmonicity and no gate time that lead to even moderate
fidelities is clear evidence that the primary bath TLS is
essential for the generation of arbitrary diagonal unitaries.
\begin{table}
\centering
\begin{tabular}{|c|c|c|c|}
\hline
$\Delta^{(2)}$ &$S^{(2)}$ &$T^{(2)}_1$ &
error \\
\hline
50 MHz &40 MHz &2000 ns & $3.076
\cdot 10^{-2}$ \\
50 MHz &40 MHz &200 ns & $4.052
\cdot 10^{-2}$ \\
50 MHz &40 MHz &40 ns & $7.867
\cdot 10^{-2}$ \\ \hline
50 MHz &10 MHz &2000 ns & $3.196
\cdot 10^{-2}$ \\
50 MHz &10 MHz &200 ns & $3.564
\cdot 10^{-2}$ \\
50 MHz &10 MHz &40 ns & $4.241
\cdot 10^{-2}$ \\ \hline
450 MHz &40 MHz &2000 ns & $1.659
\cdot 10^{-2}$ \\
450 MHz &40 MHz &200 ns & $1.652
\cdot 10^{-2}$ \\
450 MHz &40 MHz &40 ns & $1.758
\cdot 10^{-2}$ \\ \hline
450 MHz &10 MHz &2000 ns & $1.663
\cdot 10^{-2}$ \\
450 MHz &10 MHz &200 ns & $1.674
\cdot 10^{-2}$ \\
450 MHz &10 MHz &40 ns & $1.675
\cdot 10^{-2}$ \\
\hline
\end{tabular}
\caption{Error after optimization for $\mathrm{diag}(1,-1,1,1)$
with two primary bath TLS (parameters for qudit and first TLS as in
Fig.~\ref{fig:controls}, second TLS positioned $\Delta^{(2)}$ below
$\omega^{(1)}$). For comparison, the error obtained for
a single TLS is $1.652\cdot 10^{-2}$.
}
\label{tab:MutliTLS}
\end{table}
While the primary bath may provide interactions with the
system that can be used as a resource for control, it can also have
detrimental effects on the system, in particular when more than one
TLS comes into
play. This is likely to happen since number, position and coupling
strength of the TLS cannot be controlled in the preparation
of the actual devices. We therefore analyze
the presence of an additional primary bath TLS in our optimizations,
cf. Table~\ref{tab:MutliTLS}. If the TLS are not too close to each
other, a suitable control can suppress the effect of
the additional TLS even if it is strongly coupled and very noisy. On
the other hand, and not surprisingly so, the stronger a closely lying
second TLS is coupled to the qudit, the more difficult it is to
maintain good fidelities.
This is due to the fact that the gate time needs to be sufficiently
long to resolve the energy difference between the two TLS.
Adding more TLS to the primary bath does not change the picture shown
in Table~\ref{tab:MutliTLS}: In optimizations with as many as four
strongly coupled primary bath TLS, the error is increased by less than
a factor of 2 compared to the error for a single TLS
if none of the additional TLS is close to the favourable one
and less than a factor of 4 if a moderately lossy TLS is in its
vicinity.
In summary, we have shown that a non-Markovian environment can be
exploited for quantum control, enabling realization of all quantum
operations in SU(4) where the system alone allows only for SO(4). The
enhanced controllability results from an
effective control over the system-bath coupling by moving the system into
and out of resonance with a selected bath mode.
Fast implementations of this control scheme were obtained with OCT
such that the
errors are solely $T_1$-limited. Our model and results are directly
applicable to superconducting phase and transmon circuits for which we
predict, with reasonably simple controls, errors below one per cent
for state of the art decoherence times.
More generally, our results provide a new perspective on open quantum
systems -- the environment can act as a resource for (almost) unitary
quantum control which can be exploited using OCT to get the details of
the dynamics right. It requires one or a few environmental modes to be
sufficiently isolated and sufficiently strongly coupled to the
system. These conditions are met for a variety of solid-state
devices other than superconducting circuits, for example NV centers in
nanodiamonds or nanomechanical oscillators. In addition,
on an abstract level, our work calls for a comprehensive investigation
of controllability of open quantum systems, in order to gain a
rigorous understanding of when and how non-Markovianity is beneficial
for quantum control.
\begin{acknowledgments}
We would like to thank Ronnie Kosloff for fruitful
discussions. Financial support by the DAAD
and the ISF (Bikura Grant No. 1567/12)
is gratefully acknowledged.
\end{acknowledgments}
\begin{thebibliography}{26}
\expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi
\expandafter\ifx\csname bibnamefont\endcsname\relax
\def\bibnamefont#1{#1}\fi
\expandafter\ifx\csname bibfnamefont\endcsname\relax
\def\bibfnamefont#1{#1}\fi
\expandafter\ifx\csname citenamefont\endcsname\relax
\def\citenamefont#1{#1}\fi
\expandafter\ifx\csname url\endcsname\relax
\def\url#1{\texttt{#1}}\fi
\expandafter\ifx\csname urlprefix\endcsname\relax\defURL {URL }\fi
\providecommand{\bibinfo}[2]{#2}
\providecommand{\eprint}[2][]{\url{#2}}
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\end{document}
|
\begin{document}
\newcommand{{p_1,\dots,p_n}rotect\ref}{{p_1,\dots,p_n}rotect\refotect\ref}
\newcommand{\subseteq}{\subseteqbseteq}
\newcommand{{{p_1,\dots,p_n}artial}}{{{{p_1,\dots,p_n}artial}rtial}}
\newcommand{{\mathcal C}}{{\mathcal C}}
\newcommand{{\mathcal A}}{{\mathcal A}}
\newcommand{{{\Bbb B}bb R}}{{{\Bbb B}bb R}}
\newcommand{{\Bbb B}}{{{\Bbb B}bb B}}
\newcommand{{{\Bbb R}^3}}{{{{\Bbb B}bb R}^3}}
\newcommand{{\Bbb G}}{{{\Bbb B}bb G}}
\newcommand{{\Bbb Z}}{{{\Bbb B}bb Z}}
\newcommand{{Imm(F,\E)}}{{Imm(F,{{\Bbb R}^3})}}
\newcommand{{{\epsilon}quiv}}{{{{\epsilon}quiv}uiv}}
\newcommand{{\epsilon}}{{{\epsilon}psilon}}
\newcommand{{p_1,\dots,p_n}}{{p_1,\dots,p_n}}
\newcommand{{\lambda}}{{\lambda}}
\newcommand{{\dots}}{{\dots}}
\newcounter{numb}
\title{Order One Invariants of Immersions}
\author{Tahl Nowik}
\address{Department of Mathematics and Computer Science, Bar-Ilan University, Ramat-Gan 52900, Israel.}
{\epsilon}mail{tahl@@macs.biu.ac.il}
\date{March 25, 2001}
\begin{abstract}
We classify all order one invariants of immersions
of a closed orientable surface $F$ into ${{\Bbb R}^3}$, with values in an arbitrary
Abelian group ${\Bbb G}$.
We show that for any $F$ and ${\Bbb G}$ and any regular homotopy class ${\mathcal A}$ of immersions of $F$ into ${{\Bbb R}^3}$,
the group of all order one invariants on ${\mathcal A}$ is isomorphic to
${\Bbb G}^{\aleph_0} \oplus {\Bbb B} \oplus {\Bbb B}$ where ${\Bbb G}^{\aleph_0}$ is the group of all functions from a set of cardinality $\aleph_0$ into ${\Bbb G}$ and
${\Bbb B}=\{ x\in{\Bbb G} : 2x=0 \}$.
Our work includes foundations for the study of finite order
invariants of immersions of a closed orientable surface into ${{\Bbb R}^3}$,
analogous to chord diagrams and the
1-term and 4-term relations of knot theory.
{\epsilon}nd{abstract}
\maketitle
\section{Introduction}\label{0}
The notion of finite order invariants has developed in knot theory.
We extend it here to the setting of immersions of a closed orientable surface
into ${{\Bbb R}^3}$. We give a general analysis of finite order invariants, finally classifying all order one invariants (Theorem {p_1,\dots,p_n}rotect\ref{t1}).
A small subclass of order one invariants has previously been studied, namely, the ``local invariants'', studied in [G], [N1], [N3].
The structure of the paper is as follows:
In Section {p_1,\dots,p_n}rotect\ref{A} we describe the self-intersection appearing in immersions which correspond to the codimension 1 (abbreviated codim 1) strata in the space of immersions. They will be the basis of our definition of finite order invariants.
In Section {p_1,\dots,p_n}rotect\ref{B} we show that there is no continuous choice of co-orientation
for the codim 1 strata in the space of immersions. We show that this is not an obstacle for defining finite order invariants; we construct the combinatorial object on which order $n$ invariants are defined up to order $n-1$ invariants.
In Section {p_1,\dots,p_n}rotect\ref{C} we analyze the relations on invariants arising from local
2-parameter families of immersions.
In Section {p_1,\dots,p_n}rotect\ref{D} we classify all order one invariants.
In Section {p_1,\dots,p_n}rotect\ref{E} we show that for ${\Bbb G}={{\Bbb B}bb Z}/2$,
the analogous classification does not hold for invariants of order $n>1$;
in that we also complete the analysis of [N2].
\section{Self intersection}\label{A}
The space of all immersions of a closed orientable surface $F$ into ${{\Bbb R}^3}$ is naturally stratified by the types of self intersection appearing in the immersions; see [HK] for
local analysis. Parts of our work will also be local;
given an immersion $i:F\to{{\Bbb R}^3}$ and a self intersection point of $i$ we will look at all deformations of $i$ which move $F$ only in a small neighborhood of that intersection point. The {\epsilon}mph{local stratification} will then mean the stratification of this smaller space of immersion and {\epsilon}mph{local codimension} of the local strata will again refer to this smaller space.
We will take a close look at the local stratification up to codim 2.
We discuss the codim 0 and 1 strata here.
The codim 2 strata will be discussed in Section {p_1,\dots,p_n}rotect\ref{C}.
The codim 0 strata corresponds to stable self intersection, which are double lines and triple points.
The codim 1 strata divide into four types
which (following [G]) we call: $E$, $H$, $T$, $Q$.
In the notation of [HK] they are respectively
$A_0^2|A_1^+$, $A_0^2|A_1^-$, $A_0^3|A_1$, $A_0^4$.
The four types may be demonstrated by the following local representatives, where formulae
in 3-space defining the different sheets involved in the self intersection, are given. A representative of the codim 1 strata is obtained from the formulae below by setting ${\lambda}=0$. Letting ${\lambda}$ vary, we obtain a 1 parameter family of immersions which is transverse to this codim 1 stratum.
$E$: \ \ $z=0$, \ \ $z=x^2+y^2+{\lambda}$. See Figure {p_1,\dots,p_n}rotect\ref{fet}, ignoring the vertical plane.
$H$: \ \ $z=0$, \ \ $z=x^2-y^2+{\lambda}$. See Figure {p_1,\dots,p_n}rotect\ref{fht}, ignoring the vertical plane.
$T$: \ \ $z=0$, \ \ $y=0$, \ \ $z=y+x^2+{\lambda}$. See Figure {p_1,\dots,p_n}rotect\ref{ftq}, ignoring the vertical plane.
$Q$: \ \ $z=0$, \ \ $y=0$, \ \ $x=0$, \ \ $z=x+y+{\lambda}$. This is simply four planes passing through one point, no three of which intersect in a line
A self intersection of local codim 1 (namely $E$, $H$, $T$ or $Q$) will be called a CE point (after the first and last letters of ``co-dimension one''). By CE point we will also refer to the point in ${{\Bbb R}^3}$ where this self-intersection takes place.
Once an orientation is chosen for the surface, the above four types of self intersection split into twelve types, as we explain below.
A choice of one of the two sides of the local codim 1 stratum at a given point of the stratum, is represented by the choice of ${\lambda}<0$ or ${\lambda}>0$ in the formulae above. We will refer to such a choice as a
{\epsilon}mph{co-orientation for the configuration of the self intersection.}
A completely different notion of co-orientation that we will encounter is the following:
If $i:F\to{{\Bbb R}^3}$ is an immersion and $F$ is oriented then its orientation and that of ${{\Bbb R}^3}$ induce a co-orientation for $i(F)$ in ${{\Bbb R}^3}$.
To avoid confusion between the two notions,
we will use the term ``co-orientation'' only for the former.
For the latter we will speak of the ``preferred side of $i(F)$ in ${{\Bbb R}^3}$''.
We now present the twelve types of self intersection in the oriented setting and specify a co-orientation wherever possible:
Type $E$: the configuration of the self intersection at the two sides of the stratum is distinct even with no orientation on the surface, namely, for ${\lambda}<0$ there is an additional 2-sphere in the image of the immersion, and we choose this side (${\lambda}<0$) as our positive side for the co-orientation. Now, this additional 2-sphere is made of two 2-cells; if the surface is oriented then we may distinguish three different types of $E$ self intersections, which we denote by $E^a$ where $0\leq a \leq 2$ denotes the number of such 2-cells
for which the little 3-cell bounded by the 2-sphere lies on their non-preferred side in ${{\Bbb R}^3}$.
Type $H$: the configuration of the self intersection on the two sides of this stratum are indistinguishable with no orientations. If the surface is oriented
then we may compare the orientations of the two sheets at the time of tangency; if they coincide then the configurations of intersection on the two sides of the stratum are still indistinguishable. We will denote by $H^1$ this type of self intersection. As explained in Remark {p_1,\dots,p_n}rotect\ref{r1}
below, this local symmetry of the $H^1$
configuration implies a global one-sidedness of the $H^1$ strata in the space of immersions.
If on the other hand the orientation of the two sheets disagree at time of tangency, we may distinguish the two sides of the stratum; we will consider as positive, the side for which the region in ${{\Bbb R}^3}$ between
the two sheets is on the preferred side of both of them.
We denote by $H^2$ this type of self intersection.
Type $T$: This is similar to type $E$ in the sense that the two sides of the stratum are distinguishable even in the un-oriented setting, by the additional sphere which appears on one side and which we will consider as the positive side (${\lambda}<0$ in the formula above). This time the 2-sphere is composed of three 2-cells.
With orientation on the surface we distinguish four types of $T$ intersection, which we name $T^a$ where $0\leq a \leq 3$ is (as for $E$) the number of 2-cells
for which the little 3-cell bounded by the 2-sphere lies on their non-preferred side in ${{\Bbb R}^3}$.
Type $Q$: Here as for $H$, the two sides may not be distinguished without orientation; a sphere with four faces (bounding a simplex)
is formed on both sides of the stratum.
If the surface is oriented, then the numbers of faces for which the simplex is not on the preferred side in ${{\Bbb R}^3}$, are complementary; i.e. if it is $a$ on one side of the stratum, then it is $4-a$ on the other; so they appear in pairs $(4,0)$, $(3,1)$, $(2,2)$;
we name the configurations, respectively $Q^4$, $Q^3$, $Q^2$. For $Q^4$ and $Q^3$ the two sides of the stratum may be distinguished and we choose the positive side to be the side where a larger number of 2-cells have the simplex on their non-preferred side.
On the other hand, for $Q^2$ the two sides are indistinguishable, and as for $H^1$, the problem is global; the $Q^2$ strata are one-sided in the space of immersions (Remark {p_1,\dots,p_n}rotect\ref{r1}).
\section{Finite Order Invariants}\label{B}
We fix once and for all a closed oriented surface $F$ and a regular homotopy class ${\mathcal A}$ of immersions of $F$ into ${{\Bbb R}^3}$.
We denote by $I_n\subseteq {\mathcal A}$ ($n\geq 0$) the space of all immersions in ${\mathcal A}$ which have precisely $n$ CE points (the self intersection being elsewhere stable).
In particular, $I_0$ is the space of all stable immersions in ${\mathcal A}$.
Let ${\Bbb G}$ be any Abelian group and let $f:I_0\to{\Bbb G}$ be an invariant, i.e. a function which is constant on each connected component of $I_0$.
Given an immersion $i\in I_n$ we make an arbitrary choice of co-orientation for the configuration of intersection, at each of the $n$ CE points of $i$, and define $f^{TCO}(i)$ (Where ``TCO'' stands for ``Temporary Co-Orientation) as follows:
Let $i_1,\dots,i_{2^n}$ be the $2^n$ immersions in $I_0$ obtained from $i$ by slightly deforming it in the $2^n$ possible ways. We define:
$$f^{TCO}(i)=\subseteqm_{m=1}^{2^n} {\epsilon}psilon^m_1\cdots{\epsilon}psilon^m_n f(i_m) $$
where ${\epsilon}psilon^m_k$ is $1$ or $-1$ according to whether in order to obtain
$i_m$ we deformed the configuration of $i$
at $p_k$ positively or negatively, according to the temporary
co-orientation chosen for $i$ at $p_k$.
The following is clear:
\begin{lemma}\label{l0}
If $i\in I_n$ and $TCO_1$, $TCO_2$ are two temporary co-orientations for $i$ which differ at precisely one CE of $i$ then $f^{TCO_1}(i) = -f^{TCO_2}(i)$.
{\epsilon}nd{lemma}
By Lemma {p_1,\dots,p_n}rotect\ref{l0},
the statement $f^{TCO}(i)=0$ is independent of the temporary co-orientation and we may simply write $f(i)=0$.
\begin{dfn}\label{d3}
An invariant $f:I_0\to{\Bbb G}$ will be called {\epsilon}mph{of finite order} if
there is an $n$ such that $f(i)=0$ for all $i\in I_{n+1}$.
The minimal such $n$ will be called the {\epsilon}mph{order} of $f$.
{\epsilon}nd{dfn}
Let $i:F\to{{\Bbb R}^3}$ be an immersion having a CE located at $p\in{{\Bbb R}^3}$. As in [N2], we define
the degree $d_p(i)$ of the CE at $p$ as follows: Let $B$ be a tiny ball in ${{\Bbb R}^3}$
centered at $p$.
$i^{-1}(B)$ is a union of some (two, three or four) disks in $F$ which pass $p$.
Let $\hat{i}:F\to {{p_1,\dots,p_n}artial} B$ be the map obtained from $i$ as follows:
On $F-i^{-1}(B)$ we define $\hat{i}$ by radial projection (centered at $p$).
Now, if $D$ is one of the disks in
$i^{-1}(B)$ then $i(D)$ cuts ${{p_1,\dots,p_n}artial} B$ into two hemispheres; $\hat{i}$ is defined to
map $D$ onto the hemisphere which lies on the preferred side of $i(D)$ in ${{\Bbb R}^3}$.
(Recall that $F$ is oriented.)
Finally we define $d_p(i)$ as the degree of the map $\hat{i}:F\to {{p_1,\dots,p_n}artial} B$ (the orientation on ${{p_1,\dots,p_n}artial} B$ being that induced to it from $B$.)
Let $C_p(i)$ be the expression $R^a_m$ where $R^a$
is the configuration of the CE of $i$ at $p$ (one of our twelve symbols
e.g. $E^0$) and $m=d_p(i)$.
Let ${\mathcal C}_n$ denote the set of all
{\epsilon}mph{un-ordered} $n$-tuples of such expressions $R^a_m$.
Finally, we define a map $C:I_n \to {\mathcal C}_n$ as follows:
If $i\in I_n$ with CEs
located at $p_1,\dots,p_n\in{{\Bbb R}^3}$, then we define $C(i)\in {\mathcal C}_n$ to be
the un-ordered n-tuple $[C_{p_1}(i),\dots, C_{p_n}(i)]$.
\begin{dfn}\label{d1}
\begin{enumerate}
\item A regular homotopy $H_t:F\to{{\Bbb R}^3}$ will be called {\epsilon}mph{of type A} if it is of the form $H_t = U_t \circ i \circ V_t$ where $U_t:{{\Bbb R}^3}\to{{\Bbb R}^3}$ and
$V_t:F\to F$ are isotopies.
\item A regular homotopy $H_t:F\to{{\Bbb R}^3}$ between immersions $i,j\in I_n$
will be called {\epsilon}mph{of type B} if it is of the following form:
If $B_1,\dots,B_n\subseteq{{\Bbb R}^3}$ are little balls centered at the $n$ CE points
of $i$ and $U=i^{-1}(\bigcup_k B_k)$ then
$H_t$ fixes $U$ and moves $F-U$ within ${{\Bbb R}^3} - \bigcup B_k$.
\item Two immersions $i,j\in I_n$ will be called {\epsilon}mph{AB equivalent} if
there is a regular homotopy $H_t$ between $i$ and $j$ which is alternatingly
of type A and B.
Such a regular homotopy will be called an {\epsilon}mph{AB equivalence}.
{\epsilon}nd{enumerate}
{\epsilon}nd{dfn}
A proof of the following proposition appears in [N2] for immersions including
only quadruple points; the proof for the general case is identical; we include it
here for completeness.
\begin{prop}\label{p1}
Let $i,j\in I_n$, then $i$ and $j$ are AB equivalent iff $C(i)=C(j)$.
{\epsilon}nd{prop}
\begin{pf}
If $i$ and $j$ are AB equivalent then clearly $C(i)=C(j)$. For the converse, assume $C(i)=C(j)$. One can order the CEs of $i$ and $j$, respectively $p'_1,\dots,p'_n$ and $p_1,\dots,p_n$, such that $C_{p'_k}(i)=C_{p_k}(j)$, $k=1,\dots,n$. This means in particular, that if $B'_1,\dots,B'_n$ and $B_1,\dots,B_n$ are neighborhoods of the $p'_k$s and $p_k$s respectively, then for each $k$ there is an orientation preserving diffeomorphism from $B'_k$ to $B_k$ which takes each sheet of $i(F)\cap B'_k$ orientation
preservingly onto the corresponding sheet of $j(F)\cap B_k$.
These diffeomorphisms may all be realized by one ambient isotopy $U_t:{{\Bbb R}^3}\to{{\Bbb R}^3}$. There is then an isotopy $V_t:F\to F$ such that the
final immersion $i'$ of the regular homotopy $U_t\circ i \circ V_t$
satisfies that $i'$ and $j$ have the same $n$ CE points $p_1,\dots,p_n\in{{\Bbb R}^3}$, ${i'}^{-1}(\bigcup_k B_k) = j^{-1}(\bigcup_k B_k)$
which we name $U$
and $i'|_U = j|_U$.
Also $d_{p_k}(i') = d_{p_k}(j)$ for $k=1,\dots,n$.
Now $U$ is a union of some disks $D_1,\dots,D_r$. We construct the following
handle decomposition of $F$. $D_1,\dots D_r$ will be the 0-handles. If $g$ is the
genus of $F$ we will have 1-handles $h_1,\dots,h_{2g+r-1}$
as follows: $h_1,\dots,h_{2g}$ will
each have both ends glued to $D_1$ such that $D_1$ with $h_1,\dots,h_{2g}$ will
decompose $F$ in the standard way.
Then for $k=1,\dots r-1$, $h_{2g+k}$ will have one end glued to $D_k$ and the other to $D_{k+1}$.
The complement of the
0- and 1-handles is one disk which will be the unique 2-handle.
We will now construct a regular homotopy of the form $i'\circ V'_t$
($V'_t:F\to F$ an isotopy)
from $i'$ to an immersion $i''$ which will have the property that the restrictions of $i''$ and $j$ to all 1-handles, are regularly homotopic
keeping all 0-handles fixed.
Since $i'$ and $j$ are
regularly homotopic (recall $i,j\in I_n \subseteq {\mathcal A}$),
this is already true for $h_1,\dots,h_{2g}$.
Now take $h_{2g+1}$. If $i'|_{h_{2g+1}}$ and $j|_{h_{2g+1}}$ are not regularly homotopic keeping $D_1$ and $D_2$ fixed, then $V'_t$ performs one full rotation
of $D_2$, creating a Dehn twist in a thin annulus around $D_2$ in $F$. $h_{2g+1}$ will now satisfy the needed property. Note also that this rotation of $D_2$ moves only $h_{2g+1}$ and $h_{2g+2}$, keeping all other 0- and 1-handles
fixed. We continue this way along the chain of 1-handles, rotating $D_{k+1}$ if necessary for the sake of $h_{2g+k}$.
For $k<r-1$ this will also move $h_{2g+k+1}$,
but we never need to move 1-handles that have previously been taken care of.
Also $d_{p_k}(i'')=d_{p_k}(i)=d_{p_k}(j)$ for all $k=1,\dots, n$.
We now perform a regular homotopy $H_t$ on
the union of 0- and 1-handles which fixes the 0-handles,
and regularly homotopes each 1-handle $h$, from $i''|_{h}$ to
$j|_{h}$, avoiding $\bigcup_k B_k$. This is possible by the construction
of $i''$.
Denote our 2-handle by $D$. So far we have constructed $H_t$ only on $F-D$. By means of [S], $H_t$ may be extended to $D$, still avoiding $\bigcup_k B_k$, arriving at an immersion $i'''$. And so, we are left with regularly homotoping $i'''|_D$ to $j|_{D}$ (relative ${{p_1,\dots,p_n}artial} D$). Since $d_{p_k}(i''')=d_{p_k}(j)$ for all $k=1,\dots,n$, these maps are homotopic in ${{\Bbb R}^3}-\bigcup_k B_k$. It then follows from the Smale-Hirsch Theorem ([H]), that they are also
{\epsilon}mph{regularly} homotopic in ${{\Bbb R}^3}-\bigcup_k B_k$ (since the obstruction
to that would lie in ${p_1,\dots,p_n}i_2(SO_3)=0$).
The regular homotopy from $i$ to $i''$ was of type A, and that from $i''$ to $j$ was of type B.
{\epsilon}nd{pf}
\begin{prop}\label{p2}
Let $f:I_0\to{\Bbb G}$ be an invariant of order $n$. Let $i\in I_n$ be an immersion
with CEs $p_1,\dots,p_n$ and assume the configuration of $i$ at $p_1$ is of type $H^1$ or $Q^2$. Then $f^{TCO}(i) = -f^{TCO}(i)$ in ${\Bbb G}$.
It follows by Lemma {p_1,\dots,p_n}rotect\ref{l0} that in this case $f^{TCO}(i)$ is
independent of the temporary co-orientation.
{\epsilon}nd{prop}
\begin{pf}
For $k=1,\dots,n$ let $B_k$ be a small neighborhood of $p_k$ in ${{\Bbb R}^3}$.
Since the CE at $p_1$ is of type $H^1$ or $Q^2$,
there is an orientation preserving diffeomorphism from $B_1$ to itself which
maps $i(F)\cap B_1$ onto itself, permuting the sheets, being orientation preserving on each sheet,
but reversing the co-orientation of the configuration at $p_1$.
We use this self diffeomorphism of $B_1$ and the identity map on $B_2,\dots,B_n$, in the proof of Proposition {p_1,\dots,p_n}rotect\ref{p1}, getting an AB equivalence $H_t$ from $i$ {\epsilon}mph{to itself} which reverses the co-orientation of the configuration in $B_1$ and fixes $F$ in $B_2,\dots,B_n$. We choose a temporary co-orientation for $i$ at $p_1,\dots,p_n$ and we carry it along $H_t$. If at some time $t_0$, $H_t$ passes through an $n+1$th CE at $p\in{{\Bbb R}^3}$, we choose an arbitrary co-orientation for $H_{t_0}$ at $p$. Together with the co-orientations we are carrying from $i$ this gives a temporary co-orientation for the $n+1$
CEs of $H_{t_0}$.
Since $f$ is of order $n$, $f^{TCO}(H_{t_0})=0$ which means that
$f^{TCO}(H_{t_0-{\epsilon}})=f^{TCO}(H_{t_0+{\epsilon}})$ where the TCOs at $t_0-{\epsilon}$ and $t_0+{\epsilon}$ are those carried from $i$ along $H_t$. Finally we arrive back at $i$ but with opposite co-orientation than the original one at $p_1$ and the same co-orientation at $p_2,\dots,p_n$. By Lemma {p_1,\dots,p_n}rotect\ref{l0} we get
$f^{TCO}(i)=-f^{TCO}(i)$.
{\epsilon}nd{pf}
\begin{remark}\label{r1}
We have seen in the proof of Proposition {p_1,\dots,p_n}rotect\ref{p2} that if $i\in I_n$
has a CE of type $H^1$ or $Q^2$ then there is an AB equivalence from $i$ to itself such that if we follow that CE along the AB equivalence, then
it returns to itself but with opposite co-orientation.
If one seeks a globally defined co-orientation, it is a minimal requirement that it be continuously defined along AB equivalences; so we see that such globally defined co-orientation does not exist for CEs of type $H^1$ and $Q^2$.
{\epsilon}nd{remark}
If $f$ is an invariant of order $n$ then Proposition {p_1,\dots,p_n}rotect\ref{p2} enables us to extend $f$ to $I_n$. We do it as follows: We choose once and for all a {\epsilon}mph{permanent}
co-orientation for the ten configurations which allow it;
in fact we choose those co-orientations given in Section {p_1,\dots,p_n}rotect\ref{A} above.
Now if $i\in I_n$ and at least one of the CEs of $i$ is of configuration $H^1$ or $Q^2$ then by Proposition {p_1,\dots,p_n}rotect\ref{p2}
$f(i)$ is well defined, independent of a TCO.
If all $n$ CEs of $i$ are not of configuration $H^1$ and $Q^2$ then we define $f(i)$ using our permanent co-orientation for each of the CEs.
We will assume from now on without mention that any $f$ of order $n$ is extended to $I_n$ in this way.
(Note that if $f$ is of order $n$ then we are not extending $f$ to $I_k$ for
$0<k<n$).
\begin{prop}\label{p3}
Let $f$ be an invariant of order $n$ and $i,j\in I_n$.
If $C(i)=C(j)$ then $f(i)=f(j)$.
{\epsilon}nd{prop}
\begin{pf}
By Proposition {p_1,\dots,p_n}rotect\ref{p1} $i$ and $j$ are AB equivalent.
As in the proof of Proposition {p_1,\dots,p_n}rotect\ref{p2}, $f$ is unchanged whenever we pass an $n+1$th CE and so $f(i)=f(j)$.
{\epsilon}nd{pf}
By Proposition {p_1,\dots,p_n}rotect\ref{p3}
(and since $C:I_n\to{\mathcal C}_n$ is clearly surjective),
any order $n$ invariant $f$ induces a well defined
function $u(f):{\mathcal C}_n\to{\Bbb G}$; if $u(f) = u(g)$ then $f$ and $g$ differ by an invariant of order at most $n-1$.
And so if $V_n$ denotes the space of all invariants on ${\mathcal A}$
of order at most $n$ then
$f\mapsto u(f)$ induces an injection $u:V_n / V_{n-1} \to {\mathcal C}_n^*$
where ${\mathcal C}_n^*$ is the space of all function from ${\mathcal C}_n$ to ${\Bbb G}$.
The purpose of the next section is to demonstrate
a space $\Delta_n \subseteq {\mathcal C}_n^*$ which contains
the image of $u$.
$\Delta_n$ will be a subspace of ${\mathcal C}_n^*$ which is determined by
the restriction of Proposition {p_1,\dots,p_n}rotect\ref{p2} above
and by relations obtained by looking at local
2-parameter families of immersions.
Comparing to knot theory,
${\mathcal C}_n$ is analogous to the set of all chord diagrams of order $n$; the analogy is made clear by Propositions {p_1,\dots,p_n}rotect\ref{p1} and {p_1,\dots,p_n}rotect\ref{p3} above.
$\Delta_n$ which we define below, will be analogous to the
space of functions on chord diagrams which satisfy the 1-term and 4-term
relations.
In Section {p_1,\dots,p_n}rotect\ref{D} we will show that
$u:V_1/V_0\to \Delta_1$ is surjective; by this we classify all order one
invariants (Theorem {p_1,\dots,p_n}rotect\ref{t1}).
We will show in Section {p_1,\dots,p_n}rotect\ref{E} that for ${\Bbb G}={{\Bbb B}bb Z}/2$ and $n>1$,
$u:V_n/V_{n-1}\to\Delta_n$ is {\epsilon}mph{not} surjective; this we do by demonstrating one particular function in $\Delta_n$ which is not attained by $u$.
\begin{ques}\label{qu}
What is the image of $u$ for $n>1$?
(given $F$, ${\mathcal A}$ and ${\Bbb G}$.)
{\epsilon}nd{ques}
\section{Local analysis}\label{C}
Let $i\in {\mathcal A}$ be an immersion with a self intersection of local
codim 2 at $p_1$ and
$n-1$ additional self-intersections of local codim 1 (i.e. CEs) at $p_2,\dots,p_n$.
We look at a 2-parameter family of immersions which moves $F$ only in a neighborhood of $p_1$, such that the immersion $i$ corresponds to
parameters $(0,0)$ and such that this 2-parameter family is transverse to the local codim 2 stratum at $i$.
In this 2-parameter family of immersions we look at a loop which circles the point of intersection with the codim 2 strata.
(This corresponds to a circle around the origin in the parameter plane.)
This circle crosses the local codim 1 strata some $r$ times.
Between each two intersections we have an immersion in $I_{n-1}$
with the same $n-1$ CEs, at $p_2,\dots,p_n$.
At each intersection with the local codim 1 strata, an $n$th CE is added,
obtaining an immersion in $I_n$. Let $i_1,\dots,i_r$ be the $r$ immersions in $I_n$ so obtained and let ${\epsilon}_k$, $k=1,\dots,r$ be $1$ or $-1$ according to whether we are passing the $n$th CE of $i_k$ in the direction of its
permanent co-orientation, if it has one; if the CE is of type $H^1$ or $Q^2$ then ${\epsilon}_k$ is arbitrarily chosen.
Now let $f:I_0\to{\Bbb G}$ be an invariant of order $n$,
then it is easy to see that
$\subseteqm_{k=1}^r {\epsilon}_k f(i_k) = 0$.
Looking now at $u:V_n / V_{n-1} \to {\mathcal C}_n^*$ we obtain relations that must be satisfied by a function in ${\mathcal C}_n^*$ in order for it to lie in the image of $u$.
In this section we will find all relations on ${\mathcal C}_n^*$ obtained in this way.
The relations on a $g\in{\mathcal C}_n^*$ will be written as relations on the symbols $R^a_m$, e.g.
$0 = T^a_m - T^{3-a}_m$ will stand for the set of all relations of the form
$0 = g([T^a_m, R^{a_2}_{d_2},\dots,R^{a_n}_{d_n}]) -
g([T^{3-a}_m, R^{a_2}_{d_2},\dots,R^{a_n}_{d_n}])$ with arbitrary $R^{a_2}_{d_2},\dots,R^{a_n}_{d_n}$.
It will be convenient for the analysis of this section
to extend the set ${\mathcal C}_n$ to a set $\widetilde{{\mathcal C}_n}$
by also allowing the symbols $H^0$, $Q^0$, $Q^1$.
The symbol $H^0$ will represent the same configuration as $H^2$ only with opposite co-orientation. Similarly $Q^a$ ($a=0,1$) will represent the same configuration as $Q^{4-a}$ only with opposite
co-orientation.
Note that our
co-orientation for the $H$ and $Q$ configurations
under the different labelings
may be given the following characterization: The positive side of $H^a$ is the side where $a$ of the two sheets involved have the region between the sheets on their preferred side.
The positive side of $Q^a$ is the side where $a$ of the four faces of the simplex have the simplex on their non-preferred side.
${\mathcal C}_n^*$ will now be identified with the space of functions on $\widetilde{{\mathcal C}_n}$
which satisfy the relations $H^0_m=-H^2_m$, $Q^0_m=-Q^4_m$, $Q^1_m = -Q^3_m$
(these represent relations in the above sense).
We now need to look at local 2-parameter families of immersions which are transverse to the local codim 2 strata.
These may be divided into six types (see [HK]) which we will name by the types of CEs appearing in them:
$EH$, $TT$, $ET$, $HT$, $TQ$, $QQ$. In the notation of [HK] they are respectively: $A_0^2|A_2$, $A_0^3|A_2$, $(A_0^2|A_1^+)(A_0)$,
$(A_0^2|A_1^-)(A_0)$, $(A_0^3|A_1)(A_0)$, $A_0^5$.
For each of the first five types we give the following: 1. Formula for a local representative. 2. Sketch of the configuration for some value $({\lambda}_1,{\lambda}_2)$ of the parameters (not $(0,0)$). 3. Diagram of the 2 dimensional parameter space, where intersection with the codim 1 strata is depicted, including their
co-orientations (this is called a bifurcation diagram).
4. The relation arising.
Note that Proposition {p_1,\dots,p_n}rotect\ref{p2} takes care of the cases when there is actually no co-orientation.
For these five types,
the bifurcation diagram is obtained from the sketch and formula in a straight forward manner, by following the required loop of immersions. Whenever the plane $x=0$ appears in a
configuration below, we assume
(by rotating the configuration if necessary)
that its preferred side is $x>0$.
The integer $m$ in terms of which the degrees of the CEs are given,
is the degree of the central cod 2 immersion at its cod 2 selfintersection. (We have originally defined degree only for CEs, but the same definition applies to any self-intersection.)
We then go on to type $QQ$; it requires special analysis which will be done in detail.
\begin{figure}[h]
\scalebox{0.6}{\includegraphics{EH.eps}}
\caption{$EH$ configuration}\label{feh}
{\epsilon}nd{figure}
$EH$: \ \ $z=0$, \ \ $z=y^2 + x^3+{\lambda}_1 x + {\lambda}_2$.
\begin{equation}\label{eeh}
0 = E^a_m - H^a_m
{\epsilon}nd{equation}
\begin{figure}[h]
\scalebox{0.6}{\includegraphics{TT.eps}}
\caption{$TT$ configuration}\label{ftt}
{\epsilon}nd{figure}
$TT$: \ \ $z=0$, \ \ $y=0$, \ \ $z=y+x^3+{\lambda}_1 x + {\lambda}_2$.
\begin{equation}\label{ett}
0 = T^a_m - T^{3-a}_m
{\epsilon}nd{equation}
\begin{figure}[h]
\scalebox{0.6}{\includegraphics{ET.eps}}
\caption{$ET$ configuration}\label{fet}
{\epsilon}nd{figure}
$ET$: \ \ $z=0$, \ \ $x=0$, \ \ $z=(x-{\lambda}_1)^2 + y^2 + {\lambda}_2$.
\begin{equation}\label{eet}
0 = T^a_m - T^{a+1}_m - E^a_{m-1} + E^a_m
{\epsilon}nd{equation}
\begin{figure}[h]
\scalebox{0.6}{\includegraphics{HT.eps}}
\caption{$HT$ configuration}\label{fht}
{\epsilon}nd{figure}
$HT$: \ \ $z=0$, \ \ $x=0$, \ \ $z=(x-{\lambda}_1)^2 - y^2 + {\lambda}_2$.
\begin{equation}\label{eht}
0 = -T^{a+1}_m + T^a_m - H^a_{m-1} + H^a_m
{\epsilon}nd{equation}
\begin{figure}[h]
\scalebox{0.6}{\includegraphics{TQ.eps}}
\caption{$TQ$ configuration}\label{ftq}
{\epsilon}nd{figure}
$TQ$: \ \ $z=0$, \ \ $y=0$, \ \ $x=0$, \ \ $z=y+(x-{\lambda}_1)^2 + {\lambda}_2$
\begin{equation}\label{etq}
0 = Q^a_m - Q^{a+1}_m - T^a_{m-1} + T^a_m
{\epsilon}nd{equation}
$QQ$: This configuration is a quintuple point, i.e. five sheets passing through a point, each three of which intersect generically.
We construct a 2-parameter family of quintuplets of oriented planes
which represents a local 2-parameter family of immersions which is transverse to the local quintuple point stratum.
Let $P_1,\dots,P_5$ be five oriented planes through the origin in ${{\Bbb R}^3}$ which are in general position, i.e. no three of them intersect in a line.
For $k=1,{\dots},5$ let $u_k\in{{\Bbb R}^3}$ be the unit vector which is perpendicular to $P_k$ and pointing into the preferred side of $P_k$ in ${{\Bbb R}^3}$; any three of the vectors
$u_1,{\dots},u_5$ are independent.
A vector $({\lambda}_1,{\dots},{\lambda}_5)\in{{\Bbb B}bb R}^5$ will represent the quintuplet of planes
$P_1^{{\lambda}_1},\dots,P_5^{{\lambda}_5}$ in ${{\Bbb R}^3}$ where
$P_k^{{\lambda}_k}=\{ x\in{{\Bbb R}^3} : x\cdot u_k = {\lambda}_k \}$.
(In particular, for each $k$: $P^0_k = P_k$).
If ${\lambda}\in{{\Bbb B}bb R}$ and $v\in{{\Bbb R}^3}$ then $P_k^{{\lambda}+v\cdot u_k}$ is the translate by the vector $v$ of the plane $P_k^{\lambda}$. Let $V\subseteq{{\Bbb B}bb R}^5$ be the
3-dimensional subspace defined by
$V=\{ (v\cdot u_1, {\dots} , v\cdot u_5) : v\in {{\Bbb R}^3} \}$ then
$({\lambda}_1,\dots,{\lambda}_5)\in V$ iff $P_1^{{\lambda}_1},\dots,P_5^{{\lambda}_5}$ all meet at a point.
If $U$ is a direct summand of $V$ in ${{\Bbb B}bb R}^5$ then $U$ represents all configurations of quintuplets
of planes which are respectively parallel to $P_1,{\dots},P_5$, up to a common translation of the five planes. Such a $U$ is then a representative local 2-parameter family of immersions which is transverse to the local codim 2 stratum of quintuple points.
We make the choice $U=V^{p_1,\dots,p_n}erp$. If $A$ is the $5\times 3$ matrix whose rows are $u_1 , {\dots} , u_5$
then the columns of $A$ span $V$ and so
$U=V^{p_1,\dots,p_n}erp$ is the left kernel of $A$
i.e. $U=\{ ({\lambda}_1,{\dots},{\lambda}_5) : {\lambda}_1 u_1 +\cdots + {\lambda}_5 u_5 = 0 \}$.
We now find the points in $U$ which represent configurations involving a
CE, which in this case must be a quadruple point. Let $l_k = U\cap ( V + {{\Bbb B}bb R} {\epsilon}_k)$ where ${\epsilon}_1,{\dots},{\epsilon}_5$ is the standard basis of ${{\Bbb B}bb R}^5$ (i.e. ${\epsilon}_1=(1,0,0,0,0)$ etc.);
then $l_k$ represents all configurations in $U$ where the planes
$\{P_j : j\neq k \}$ meet at a quadruple point; indeed, an element of $l_k$ is
an element of $U$ which is
obtained from an element of $V$ i.e. a quintuple point,
by adding some $r{\epsilon}_k$ i.e. pushing away the plane $P_k$.
For a given $l_k$ we would like to determine the configuration of the quadruple point represented by it. Take say $l_5$. We must look at the configuration of the four planes $P_1^{{\lambda}_1},{\dots},P_4^{{\lambda}_4}$ given by points
$({\lambda}_1,\dots,{\lambda}_5)$ on the two sides of $l_5$ in $U$,
and see how many of the planes $P_1^{{\lambda}_1},{\dots},P_4^{{\lambda}_4}$
have the simplex created by them, on their non-preferred side.
It is enough to check one point on each side of $l_5$ in $U$ and as we shall see, it will be most convenient to look at the points of
$l_5^{p_1,\dots,p_n}erp$ (here ${p_1,\dots,p_n}erp$ means the orthogonal complement in $U$).
Now $l_5^{p_1,\dots,p_n}erp = {\epsilon}_5^{p_1,\dots,p_n}erp \cap U$ (the ${p_1,\dots,p_n}erp$ on the right is
the orthogonal complement in ${{\Bbb B}bb R}^5$);
i.e. $l_5^{p_1,\dots,p_n}erp
= \{ ({\lambda}_1,{\dots} ,{\lambda}_4, 0 ) \in {{\Bbb B}bb R}^5 : {\lambda}_1 u_1 + \cdots + {\lambda}_4 u_4 = 0 \}$.
For such points we can determine the configuration of the simplex
using the following lemma:
\begin{lemma}\label{l1}
Let $v_1,{\dots},v_4\in{{\Bbb R}^3}$ be four vectors such that any three of them are independent and $v_1 + \cdots + v_4 =0$.
If $\mu_1,{\dots},\mu_4$ are {\epsilon}mph{positive} real numbers
then the origin $0\in {{\Bbb R}^3}$ lies in the interior of the simplex determined by the four planes
$\{x\in{{\Bbb R}^3} : x\cdot v_k = \mu_k \}$.
{\epsilon}nd{lemma}
\begin{pf}
Since $0\cdot v_k = 0 < \mu_k$, the domain determined by the four planes
in which the origin lies is:
$D=\{ x\in{{\Bbb R}^3} : x\cdot v_k \leq \mu_k \ \ \hbox{for all} \ \ 1\leq k \leq 4 \}$.
This is a convex domain in ${{\Bbb R}^3}$. If it is not the simplex determined by the four planes then it is unbounded and so there
is a ray based at 0 which is contained in $D$, i.e. there is a vector $v\neq 0$ such that $rv \cdot v_k \leq \mu_k$ for every $1\leq k \leq 4$ and any $r>0$.
It follows that for each $k$: $v \cdot v_k \leq 0$.
If for some $k$, $v\cdot v_k < 0$ then $v \cdot (v_1 + \cdots + v_4) < 0$
contradicting $v_1 + \cdots + v_4 =0$. So $v\cdot v_k =0$ for all $k$, contradicting the fact that any three of $v_1,{\dots},v_4$ are independent.
{\epsilon}nd{pf}
Back to our $l_5^{p_1,\dots,p_n}erp$,
we use Lemma {p_1,\dots,p_n}rotect\ref{l1} with $v_k = {\lambda}_k u_k$ and $\mu_k = ({\lambda}_k)^2$, $k=1,{\dots},4$;
obtaining that 0 lies in the interior of the simplex determined by the equations
$x\cdot {\lambda}_k u_k = ({\lambda}_k)^2$ ($k=1,{\dots},4$) which is the same as
$x\cdot u_k = {\lambda}_k$ i.e. the planes $P_1^{{\lambda}_1}, {\dots} , P_4^{{\lambda}_4}$.
Now, the vector $u_k$, when based at $P_k^{{\lambda}_k}$, points into the preferred
side of $P_k^{{\lambda}_k}$ in ${{\Bbb R}^3}$. On the other hand,
$u_k$ points away from the side where the origin lies, which is the side
where the simplex lies, iff ${\lambda}_k > 0$.
We conclude that if
$({\lambda}_1,{\dots},{\lambda}_4,0)\in l_5^{p_1,\dots,p_n}erp$ then the number of faces of the simplex determined by $P_1^{{\lambda}_1},{\dots},P_4^{{\lambda}_4}$
which have the simplex on their non-preferred side, is precisely the number of positive numbers among ${\lambda}_1,{\dots},{\lambda}_4$. (Note that ${\lambda}_1,{\dots},{\lambda}_4$ are all non-zero since each three $u_k$s are independent).
The origin of $U$ splits $l_5^{p_1,\dots,p_n}erp$ into two half lines; clearly the number $p$ of plus signs is constant on such a half-line and is $4-p$ on the other half line.
So for given quintuple point, our task is to find the number $p$ of plus signs
in each of the ten half lines of the $l_k^{p_1,\dots,p_n}erp$s.
Now $l_k^{p_1,\dots,p_n}erp$ has ${\lambda}_k=0$, so it partitions $U$ into the domains where
${\lambda}_k>0$ and ${\lambda}_k<0$; we thus use it to determine the sign of ${\lambda}_k$ in the half-lines of $l_j^{p_1,\dots,p_n}erp$ for $j\neq k$.
Examples of such analysis appear in Figure {p_1,\dots,p_n}rotect\ref{fqq}, which we now explain:
Each of the short thick lines in a diagram represents an $l_k^{p_1,\dots,p_n}erp$. The arrows on each $l_k^{p_1,\dots,p_n}erp$ point to the
side of it in $U$ where ${\lambda}_k$ is positive.
To determine the number $p$ of positive ${\lambda}_j$s corresponding to a given half line of $l_k^{p_1,\dots,p_n}erp$ we need to count for how many $l_j^{p_1,\dots,p_n}erp$s ($j\neq k$)
this half line lies on their positive side, i.e. the side designated by the arrows.
This is the number appearing in the circle located on the given half $l_k^{p_1,\dots,p_n}erp$ in the diagram (at the tip of the short thick line).
Finally the longer thinner lines are the $l_k$s themselves (each drawn perpendicular to the corresponding $l_k^{p_1,\dots,p_n}erp$).
The pair of numbers at each tip of $l_k$ is simply copied
from the corresponding sides of $l_k^{p_1,\dots,p_n}erp$. When passing an $l_k$, this pair of numbers (which is of the form $p, 4-p$) tells us the type of quadruple point we are passing, and the co-orientation with which we are passing it, namely,
the side chosen by our permanent co-orientation is
the side where the larger number of the pair appears.
(As before, Proposition {p_1,\dots,p_n}rotect\ref{p2} takes care of the case when there is no
co-orientation.)
\begin{figure}[ht]
\scalebox{0.7}{\includegraphics{QQ.eps}}
\caption{$QQ$ configuration}\label{fqq}
{\epsilon}nd{figure}
Finally we need to determine the degrees $d_p(i)$ at each of the ten quadruple points. Let $m$ be the degree of the quintuple point and look say at $l_1$.
We claim that the half of $l_1$ which has ${\lambda}_1>0$ represents quadruple points with degree $m$ whereas the half with ${\lambda}_1<0$ represents quadruple points with degree $m-1$.
To establish this we need to show that a quadruple point represented by a point in the half of $l_1$ with ${\lambda}_1>0$ is obtained from a translate of our quintuple point by pushing $P_1$ into its preferred side i.e. the side pointed at by $u_1$. Recall that a point $({\lambda}_1,\dots,{\lambda}_5)\in l_1$ is of the
form $r{\epsilon}_1 + v$ where $v$ is perpendicular to $U$ and
represents a common translation of all five planes;
so $P_1^{{\lambda}_1}$ is obtained by pushing $P_1$ away from the quintuple
point represented by $v$, and this push is into the preferred side of $P_1$
iff $r>0$.
It remains to notice that the sign of ${\lambda}_1$ is the same as that of $r$,
since $({\lambda}_1,\dots,{\lambda}_5)$ is the orthogonal projection of $(r,0,0,0,0)$ to $U$. The integers $m$ or $m-1$ appearing at the tips of each $l_k$ in the diagrams are the degrees.
Once we have all this information registered at the tips of the $l_k$s, we can read off the diagram the relation determined by the given quintuple point.
We claim that for any configuration of lines and arrows as above,
eight of the terms cancel, always leaving the same relation:
\begin{equation}\label{eqq}
Q^2_m = Q^2_{m-1}
{\epsilon}nd{equation}
We offer two ways of seeing this: The first way is to verify that the four diagrams appearing in Figures {p_1,\dots,p_n}rotect\ref{fqq} are in fact all essentially distinct ways for choosing such distribution of arrows; then explicitly write out the relation obtained by each of them. We only remark that there is no loss of generality by the fact that the diagrams are sketched with equal angles between the $l_k$s, since all that is relevant to us is the cyclic ordering of the $l_k^{p_1,\dots,p_n}erp$s and not the common cyclic ordering of all $l_1,{\dots},l_5,l_1^{p_1,\dots,p_n}erp,{\dots},l_5^{p_1,\dots,p_n}erp$.
A second way to see that the same relation $Q^2_m=Q^2_{m-1}$ is always obtained is as follows: Since we have five arrows pointing clockwise and five counter-clockwise, there must be two consecutive arrows which are pointing at each other.
These half $l_k^{p_1,\dots,p_n}erp$s must have the same number at their tip, since they are on the same side of any other $l_j^{p_1,\dots,p_n}erp$ and are both on the positive side of each other (i.e. the side designated by the arrow). We now look at the corresponding positive half $l_k$s. They both have degree $m$ and will both have the same number copied from $l_k^{p_1,\dots,p_n}erp$ next to them, but written on the opposite side. So their contribution to the relation is precisely the negative of each other;
the same will be true for the negative halves of these $l_k$s (only now with degrees $m-1$.)
We now erase these two lines from the diagram and argue by (a two step) induction that the formal property analogous to the one we are proving, holds for the remaining three line diagram (checking that it holds for a one line diagram.) It remains to observe that once we re-insert the two erased lines, they jointly add precisely 1 to the number at the end of all other short thick lines and so the property holds for the five line diagram.
We denote by $\Delta_n = \Delta_n({\Bbb G})$
the subspace of ${\mathcal C}_n^*$ satisfying relations {p_1,\dots,p_n}rotect\ref{eeh}-{p_1,\dots,p_n}rotect\ref{eqq}
above and the restriction of Proposition {p_1,\dots,p_n}rotect\ref{p2}, which we may write as
$0= 2H^1_m = 2Q^2_m$. Recall also that by definition of ${\mathcal C}_n^*$ when using
the set $\widetilde{{\mathcal C}_n}$, we also have the relations $H^0_m=-H^2_m$
and $Q^a_m=-Q^{4-a}_m$ ($a=0,1$).
Now that we have obtained our relations,
we return to our original ${\mathcal C}_n$ with only twelve symbols
i.e. we dispose of the redundant symbols $H^0$, $Q^0$ and $Q^1$. Let ${\Bbb B}\subseteq{\Bbb G}$ be defined by
${\Bbb B}=\{ x\in {\Bbb G} : 2x=0\}$.
After some simplification, the relations defining $\Delta_n$
may finally be presented as follows:
\begin{itemize}
\item $E^2_m = - E^0_m = H^2_m$, \ \ $E^1_m = H^1_m$.
\item $T^0_m = T^3_m$, \ \ $T^1_m = T^2_m$.
\item $H^1_m = H^1_{m-1} \in {\Bbb B}$.
\item $Q^2_m = Q^2_{m-1} \in {\Bbb B}$.
\item $H^2_m - H^2_{m-1} = T^3_m - T^2_m$
\item $Q^4_m - Q^3_m = T^3_m - T^3_{m-1}$, \ \ $Q^3_m - Q^2_m = T^2_m - T^2_{m-1}$
{\epsilon}nd{itemize}
\section{Order One Invariants}\label{D}
In this section we will show that the injection $u:V_1 / V_0 \to \Delta_1$ is surjective.
Let us first give an explicit presentation of $\Delta_1$. We see from the presentation of the relations
appearing in the end of the previous section, that a function $g\in\Delta_1$
may be assigned arbitrary values in ${\Bbb G}$ for the symbols
$\{T^2_m\}_{m\in{{\Bbb B}bb Z}}$, $\{T^3_m\}_{m\in{{\Bbb B}bb Z}}$, $H^2_0$
and arbitrary values in ${\Bbb B}$ for the two symbols $H^1_0$, $Q^2_0$. Once this is done then the value of $g$ on all other symbols is uniquely determined; namely:
\begin{enumerate}
\item $H^1_m = H^1_0$ for all $m$.
\item $H^2_m = H^2_0 + \subseteqm_{k=1}^m (T^3_k - T^2_k)$ for $m\geq 0$.
\item $H^2_m = H^2_0 - \subseteqm_{k=m+1}^0 (T^3_k - T^2_k)$ for $m<0$.
\item $E^0_m = -H^2_m$, \ \ $E^1_m = H^1_m$, \ \ $E^2_m = H^2_m$ for all $m$.
\item $T^0_m = T^3_m$, \ \ $T^1_m = T^2_m$ for all $m$.
\item $Q^2_m = Q^2_0$ for all $m$.
\item $Q^3_m = Q^2_m + T^2_m - T^2_{m-1}$ for all $m$.
\item $Q^4_m = Q^3_m + T^3_m - T^3_{m-1}$ for all $m$.
{\epsilon}nd{enumerate}
Let $X$ denote the set of symbols
$\{T^2_m\}_{m\in{{\Bbb B}bb Z}}\cup\{T^3_m\}_{m\in{{\Bbb B}bb Z}}\cup\{H^2_0\}$
then we have obtained that
$\Delta_1({\Bbb G})\cong{\Bbb G}^X \oplus {\Bbb B} \oplus {\Bbb B}$ where
${\Bbb G}^X$ denotes the group of all functions from $X$ to ${\Bbb G}$.
We can also present $\Delta_1({\Bbb G})$ through a universal object as follows:
We define a universal Abelian group ${\Bbb G}_U$ by the Abelian group presentation
${\Bbb G}_U = \left< \{t^a_m\}_{a=2,3, m\in{{\Bbb B}bb Z}}, h^2_0, h^1_0, q^2_0 \ | \
2h^1_0 = 2q^2_0 = 0 \right>$.
Then we define the universal element $g_U\in\Delta_1({\Bbb G}_U)$ by $g_U(T^a_m) = t^a_m,
g_U(H^2_0)=h^2_0, g_U(H^1_0)=h^1_0, g_U(Q^2_0)=q^2_0$
and the value of $g_U$ on the other symbols of ${\mathcal C}_1$ is determined by
formulae 1-8 above, so indeed $g_U\in\Delta_1({\Bbb G}_U)$.
Then for arbitrary Abelian group ${\Bbb G}$ we have
$\Delta_1({\Bbb G}) \cong Hom({\Bbb G}_U , {\Bbb G})$ where the isomorphism
maps a homomorphism ${p_1,\dots,p_n}hi:{\Bbb G}_U \to {\Bbb G}$ to the function ${p_1,\dots,p_n}hi \circ g_U \in \Delta_1({\Bbb G})$.
We will show that there is an order 1
invariant $f_U:I_0\to{\Bbb G}_U$ such that its extension
to $I_1$ induces $g_U$ on ${\mathcal C}_1$, i.e. $u(f_U)=g_U$. It will follow that for any group ${\Bbb G}$, $u:V_1/V_0\to \Delta_1({\Bbb G})$ is surjective, since if $g\in\Delta_1({\Bbb G})$ and $g={p_1,\dots,p_n}hi \circ g_U$ where ${p_1,\dots,p_n}hi\in Hom({\Bbb G}_U , {\Bbb G})$
then $u({p_1,\dots,p_n}hi\circ f_U) = g$.
We choose a base immersion $i_0\in I_0$ once and for all. Given any $i\in I_0$ we take a generic regular homotopy from $i_0$ to $i$ i.e. a path in ${\mathcal A}$ from $i_0$ to $i$ transverse with respect to the global stratification. The value of $f_U$ on $i$ will be the sum with signs, of the values of $g_U$
on the CEs that we pass along our path from $i_0$ to $i$, where the signs are determined by whether we are passing the given CE
in the direction of its co-orientation.
(Note that whenever there is no co-orientation then the element we are adding
is of order 2 in ${\Bbb G}_U$.)
We must show that this sum is independent of our choice of path, or equivalently that it is 0 along any closed path.
First we observe that it is 0 on null-homotopic paths. Indeed, by slightly deforming the null-homotopy, we may assume it too is transverse with respect to the stratification. If we then break the null-homotopy into small pieces and look at the value going around each little piece,
then each is 0 since $g_U\in\Delta_1({\Bbb G}_U)$. Note that the
symbols encountered when going around a global codim 2 stratum which corresponds to two local codim 1 intersections occurring at distinct places, always adds up to 0.
Once we have established that this sum in ${\Bbb G}_U$ is 0 for any null-homotopic loop, we have a well defined homomorphism ${p_1,\dots,p_n}i_1({\mathcal A})\to{\Bbb G}_U$ and we must show that this homomorphism is 0.
We first show this for the case $F=S^2$ in which case there is only one regular homotopy class, which we still name ${\mathcal A}$. Now ${p_1,\dots,p_n}i_1({\mathcal A})={\Bbb Z} \oplus {\Bbb Z}/2$ where the unique order 2 element is represented by one full rotation of $S^2$
in ${{\Bbb R}^3}$.
Such rotation passes through no CEs at all, and so gives 0 as needed.
Let $K\subseteq {p_1,\dots,p_n}i_1({\mathcal A})$ be the subgroup generated by this order 2 element in ${p_1,\dots,p_n}i_1({\mathcal A})$ and let $P=P(S^2)={p_1,\dots,p_n}i_1({\mathcal A}) / K$; then we are left with showing that the map induced on $P$ ($\cong {\Bbb Z}$) is 0.
If $H_t:S^2\to{{\Bbb R}^3}$ is a regular homotopy, then attaching to $H_t$ one or more
of the superscripts ${{\Bbb R}^3}$, $S^2$ and $T$ will denote respectively, composition from the left with the antipodal map of ${{\Bbb R}^3}$, composition from the right with the antipodal map of $S^2$ and reversal of time. Note that
these three operations on $H_t$ commute with each other.
Let $p:S^2\to {{\Bbb B}bb R} P^2$ be the double covering and let $i:{{\Bbb B}bb R} P^2\to{{\Bbb R}^3}$ be some immersion. Let $s=i\circ p$ then $s:S^2\to{{\Bbb R}^3}$ is an immersion satisfying
$s(-x)=s(x)$. Let $H_t$ be a regular homotopy from the inclusion $e:S^2\to {{\Bbb R}^3}$ to $s$. Let $G_t = H_t *(H_t^{S^2,T})$ where $*$ denotes concatenation from left to right, then $G_t$ is a regular homotopy from $e$ to $-e$ i.e. an eversion of the sphere.
Also $G_t^{S^2,T} = G_t$.
We now define $F_t = G_t * G_t^{{\Bbb R}^3} = G_t * (G_t^{S^2,T})^{{\Bbb R}^3} = G_t * (G_t^{S^2,{{\Bbb R}^3}})^T$.
From the presentation of $F_t$ as $G_t * G_t^{{\Bbb R}^3}$ we can see that
there are isotopies $U_t:{{\Bbb R}^3}\to{{\Bbb R}^3}$ and $V_t:S^2\to S^2$ such that
$U_t \circ F_t \circ V_t$ satisfies the conditions of [N1] Proposition 2.1.
It follows that $F_t$ represents an odd power of the generator of $P$.
We also have $F_t=G_t * (G_t^{S^2,{{\Bbb R}^3}})^T$ and let $G'_t$ be a generic
regular homotopy which is a slight perturbation of $G_t$
($G_t$ was highly non-generic), then
$F'_t=G'_t * ({G'}_t^{S^2,{{\Bbb R}^3}})^T$ will still represent an odd power of the generator of $P$ (though it will not be precisely equal to $G'_t * {G'}_t^{{\Bbb R}^3}$).
We now observe that reversing the orientations of both $S^2$ and ${{\Bbb R}^3}$ simultaneously, preserves all our definitions of types, degrees and co-orientations of CEs and so ${G'}_t^{S^2,{{\Bbb R}^3}}$ produces precisely the same element in ${\Bbb G}_U$ as $G'_t$ and so $({G'}_t^{S^2,{{\Bbb R}^3}})^T$ produces the negative of that element. So we get that the value on the loop $F'_t$ is 0.
Since $F'_t$ represents an odd power of the generator of $P$ and since in ${\Bbb G}_U$
an odd multiple of any non-zero element is still non-zero, we get that $g_U$ is 0 on the generator of $P$ and so on all $P$.
This completed the proof for the case $F=S^2$.
Now let $F$ be of higher genus and let ${\mathcal A}$ be a given regular homotopy class.
Again ${p_1,\dots,p_n}i_1({\mathcal A}) \cong {\Bbb Z} \oplus {\Bbb Z}/2$. Again our claim is clear for
$K\subseteq {p_1,\dots,p_n}i_1({\mathcal A})$ which is defined as above and so again it is enough to look at the map induced on $P=P(F,{\mathcal A})={p_1,\dots,p_n}i_1({\mathcal A}) / K$.
Looking back for a moment at $S^2$ let $H_t$ be a loop representing a generator of $P(S^2)$ and we may assume $H_t$ fixes a given disc $D\subseteq S^2$. Choose a point $x$ on a ray perpendicular to the fixed image of $D$ in ${{\Bbb R}^3}$, $x$ being far enough so that the image of $H_t$ never passes $x$. Now change the constant embedding of $D$ to be one with a thin ``thorn'' pulled out of $D$ and embedded along this ray, reaching $x$.
The new loop $H'_t$ obtained in this way
also represents the generator of $P(S^2)$
and has the property that a given point in the fixed disc $D$ (namely, the tip of the thorn) does not participate in any self intersections throughout $H'_t$.
Since we have already proved our result for $S^2$, we know that the symbols encountered along the loop $H'_t$ add up to 0 in the group $G_U$.
Now take a tiny immersion of $F$ in our regular homotopy class ${\mathcal A}$,
located near the tip of our fixed thorn, and connect sum it with $H'_t$ for all $t$, obtaining a loop $F_t:F\to{{\Bbb R}^3}$ in ${\mathcal A}$.
It follows from the proof of [N1] Theorem 3.4
that this loop $F_t:F\to{{\Bbb R}^3}$ represents a generator of $P(F,{\mathcal A})$.
Since the tip of the thorn was far away from any self intersections occurring
in $H'_t$, the tiny immersion will not participate in any CEs occurring in $F_t$ and so will not add to the sum of symbols attained by $H'_t$, which we know is 0 in $G_U$.
We conclude that $f_U$ is well defined; it is evident that $f_U$ is of order 1 and
$u(f_U)=g_U$.
This completes the proof of our main result:
\begin{thm}\label{t1}
For any closed orientable surface $F$, regular homotopy class ${\mathcal A}$ of immersions of $F$ into ${{\Bbb R}^3}$ and Abelian group ${\Bbb G}$, the injection
$u:V_1 / V_0 \to \Delta_1$ is surjective.
{\epsilon}nd{thm}
\section{Higher order invariants}\label{E}
In this section we show that for ${\Bbb G}={\Bbb Z}/2$ and $n>1$ the map
$u:V_n / V_{n-1} \to \Delta_n$ is not surjective.
We define a function $g:{\mathcal C}_n\to{\Bbb Z}/2$ as follows:
If $x\in {\mathcal C}_n$ includes at least one symbol which is not of type $Q$, then $g(x)=0$ and if all symbols in $x$ are of type $Q$ then $g(x)=1$. One verifies directly that $g$ satisfies the relations appearing in the end of Section {p_1,\dots,p_n}rotect\ref{C}
and so $g\in \Delta_n({\Bbb Z}_2)$. We will show that there is no order $n$ invariant $f$ such that $u(f)=g$. Let $f:I_0\to {\Bbb Z}/2$ be an invariant of order $n$ and assume $u(f)=g$.
Since ${\Bbb G}={\Bbb Z}/2$, there is no need for co-orientations for the extension of $f$ and so $f$ extends to $I_k$ for any $k>0$, in particular to $I_{n-1}$.
Let $i,j\in I_{n-1}$ be two AB equivalent immersions such that all $n-1$ CEs of $i$ and $j$ are of type $Q$.
If $H_t$ is a generic AB equivalence between $i$ and $j$ then there are some finitely many times along $H_t$ for which an $n$th CE appears. By definition of $g$, the number mod 2 of such $n$th CEs which are of type $Q$ is
$f(i)-f(j)\in{\Bbb Z}/2$.
In particular if $i=j$ then the number of such occurrences of an $n$th quadruple point is 0 mod 2.
Now take some $i\in I_{n-1}$ with $n-1$ CEs located at $p_1,\dots,p_{n-1}$ all of type $Q$ and furthermore the CE at $p_1$ is of type $Q^2$.
Let $H_t$ be an AB equivalence from $i$ to itself which fixes $F$ in neighborhoods of
$p_2,\dots,p_{n-1}$ and reverses
the co-orientation of the CE at $p_1$
(as in the proof of Proposition {p_1,\dots,p_n}rotect\ref{p2}).
By the above discussion, the number mod 2 of quadruple point occurrences
along $H_t$ is $0$.
Let $i'\in I_0$ be an immersion which is obtained from $i$ by slightly deforming $i$ in a neighborhood of each $p_k$, $k=1,\dots,n-1$.
$H_t$ induces a regular homotopy $H'_t$
from $i'$ to an immersion which differs from $i'$ only in a neighborhood of $p_1$ and there are 0 mod 2 quadruple point occurrences along $H'_t$, precisely those occurring during $H_t$.
We complete $H'_t$ to a closed loop ending at $i'$ causing one more quadruple point to occur, namely that at $p_1$.
And so we have constructed a generic closed loop in ${\mathcal A}$ with 1 mod 2 quadruple point
occurrences. This would imply that the analogous order 1 invariant does not exist; but we have shown (Theorem {p_1,\dots,p_n}rotect\ref{t1})
that for order 1, all invariants defined in this way do exist; a contradiction.
We conclude that
there is no order $n$ invariant such that $u(f)=g$ and so for ${\Bbb G}={\Bbb Z}/2$
and $n>1$ the map $u:V_n / V_{n-1} \to \Delta_n({\Bbb Z} / 2)$ is not surjective.
We remark that the order 1 invariant whose existence has been used in the above proof, namely that order 1 invariant which counts the number mod 2 of quadruple points occurring along generic regular homotopies, has been studies in [N1] and [N3]. In [N1] its existence has been established for all closed surfaces (not only orientable) and also
the existence of the analogous invariant in general 3-manifolds,
under certain conditions. In [N3] an explicit formula has been given
for the value of this invariant on embeddings.
We conclude with another remark. In [N2] the notion of a $q$-invariant has been defined. This is an invariant $f:I_0\to{\Bbb Z}/2$ such that its extension to $I_1$ satisfies that $f(i)=0$ if the CE of $i$ is not a quadruple point.
It follows that $f(i)=0$ for any
$i\in I_k$ ($k\geq 1$)
which includes at least one CE which is not a quadruple point.
If $f$ is of order $n$ then this means that $u(f)(x)=0$ for any $x\in{\mathcal C}_n$ which includes at least one symbol which is not of type $Q$.
It has been shown in [N2] and also follows from the relations defining $\Delta_n$
(end of Section {p_1,\dots,p_n}rotect\ref{C}), that for such $f$ of order $n$, $f(x)=1$ for any $x\in{\mathcal C}_n$
for which all symbols are of type $Q$.
So what we have shown in this section is that $q$-invariants of order $n>1$ do not exist;
by this completing the work of [N2].
(The unique $q$-invariant of order 1 is the invariant discussed in the
previous paragraph.)
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{\epsilon}mph{Topology} 39 (2000) 1069-1088.
\bibitem[N2]{N2}
T. Nowik:
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{\epsilon}mph{Mathematische Zeitschrift} 236 (2001) 215-221.
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{\epsilon}nd{thebibliography}
{\epsilon}nd{document}
|
\begin{document}
\begin{abstract}
We prove several consistency results concerning the notion of $\omega$-strongly measurable cardinal in $\HOD$. In particular, we show that is it consistent, relative to a large cardinal hypothesis weaker than $o(\kappa) = \kappa$, that every successor of a regular cardinal is $\omega$-strongly measurable in $\HOD$.
\end{abstract}
\title{On $\omega$-Strongly Measurable Cardinals}
\noindent
\section{Introduction}
A prominent line of research in set theory is the study of the set theoretic universe $V$ (i.e., model of the axioms of set theory, $\mathrm{ZFC}$) by considering canonical inner model $M \subseteq V$ with additional strong features, which approximates $V$.
The concept builds on the suggestion that if $M$ is sufficiently ``close'' to $V$, then some of the properties of $M$ may lift to $V$, and allow us to derive new consequences about models of set theory. \\
The prospects of this approach are demonstrated in the theory of G\"{o}del's constructible universe $L \subseteq V$, and Jensen's Covering Theorem (\cite{JensenCovering}), which asserts that under the anti-large cardinal assumption of the nonexistence of $0^{\#}$, the covering property holds for $L \subseteq V$.\footnote{I.e., every set of ordinals $x \in V$ is contained in a set $y \in L$ such that $|y| {\langle}eq |x| + \aleph_1^V$.}
The combination of covering for $L \subseteq V$ together with the rigid structure of $L$, has been shown to have many implications both on cardinal arithmetic in $V$,\footnote{E.g., it implies that the Singular Cardinal Hypothesis ($\mathrm{SCH}$) holds in $V$} as well as on the existence of incompactness phenomena in $V$ such as an Abelian group $G \in V$ of size $\aleph_{\omega+1}$ which is not free, although every subgroup $H \triangleleft G$ of smaller cardinality is free.
Jensen's Covering Lemma describes one side of a sharp dichotomy: If $0^{\#}$ does not exists $L$ covers subsets of $V$ successfully. By Silver, if $0^{\#}$ does exist, $L$ fails to approximate even the most basic features of $V$. For example, in the presence of $0^{\#}$, $L$ never computes successor cardinals correctly.
Jensen's Covering Lemma can be utilized in order to obtain a lower bound for the consistency strength of set theoretical statements, which not necessarily mention large cardinals. But it is quite restrictive: in order to be able to obtain various lower bounds, one has to construct canonical models which can accommodate stronger large cardinal axioms. The construction of such inner models is the subject of a prominent program in set theory, known as the Inner Model program. For a large cardinal property $\Phi$,\footnote{E.g., the existence of a cardinal $\kappa$ with a large cardinal property such as a measurable cardinal, a strong cardinal, a Woodin cardinal, or a supercompact cardinal.}
one would like to to construct a canonical $L$-like inner model $K$ which is maximal with respect to inner models which do not satisfy the large cardinal property $\Phi$ (see Schimmerling-Steel \cite{SchimmerlingSteelMaximality} for the precise statement). This maximality property couples with a covering lemma: assuming there is no inner model of $V$ with the property $\Phi$, $K$ approximates the universe $V$ by satisfying a certain covering property. For example, for $\Phi$ being the existence of $0^{\#}$, $K = L$.
For stronger $\Phi$, the existence of such an inner model $K$ would allow us to extend Jensen's sharp dichotomy. Namely, either the large cardinal property $\Phi$ holds, or $V$ is close to $K$ and therefore inherits various combinatorial properties such as the existence of certain incompactness phenomena.
Starting at around the 1970s, inner models for increasing large cardinal properties $\Phi$ have been constructed. Starting from the seminal studies of Kunen, Silver, and Solovay on a model $L[U]$ with a measurable cardinal (\cite{Kunen1970},\cite{Silver1971}), extended by Dodd-Jensen (\cite{DoddJensen1981}) and Mitchell (\cite{Mitchell1984}), to large cardinal properties $\Phi$ involving coherent sequences of normal measures and many measurable cardinals. Then, following major developments and the introduction of iteration trees in Martin and Steel (\cite{MartinSteel1994}), Mitchell and Steel (\cite{MitchellSteel1994}), and Steel (\cite{Steel1996}), the theory was extended to the level of Woodin cardinals.
The program took a significant turn after Woodin showed that there cannot be a single maximal inner model, in an absolute sense, past a Woodin cardinal. This has sparked new lines of study, involving forms of the $K^c$ construction (see Jensen, Schimmerling, Schindler, and Steel \cite{Stacking2009} and Andereta, Neeman and Steel \cite{Domestic2001}).
Another seminal development was the introduction of The Core Model Induction method, first introduced by Woodin and extensively developed by Steel, Schindler, Sargsyan, Trang and many others (\cite{CMI}). The method establishes new consistency results for stronger large cardinal properties by incorporation ideas form descriptive set theory with various local construction methods. The relevant large cardinal properties are often described in terms of expansions of the Axiom of Determinacy ($\mathrm{AD}$) in inner models $M$ of $\mathrm{ZF}$, and can be further translated to inner models of $\mathrm{ZFC}$ with large cardinal properties.
First results on fine structural inner models for finite levels of supercompact cardinals were obtained by Neeman and Steel, \cite{neemansteel}, and by Woodin. It is still unknown whether similar constructions could lead to an inner model with a (full) supercompact cardinal, and some recent results of Woodin suggest that major obstructions appear past the level of finite supercompactness, \cite{Woodin-Midrasha}. There are many excellent resources for the introduction of the inner model theory, the inner model program and its development.
We refer the reader to \cite{JensenHistory, MitchellHistory, NeemanLongGames, SargysyanBSL, SchimerlingABC, SteelHB, Zeman}.
The inner model of Hereditarily Ordinal Definable sets (HOD) plays a significant role in many of the recent advancements in the Inner Model program.
\begin{definition}
Let $M$ be a model of set theory. A set $x \in M$ is hereditarily ordinal definable in $M$ if both $x$ and every set in the transitive closure of $x$ is definable in $V$ using some formula with ordinal parameters. The class of all hereditarily ordinal definable sets in a model $M$ is denoted by $\HOD^M$.
We write $\HOD$ for $\HOD^V \subseteq V$.
\end{definition}
The class $\HOD$ was first introduced by G\"{o}del, and has been extensively studied (for example, see \cite{MyhillScott}). The study of $\HOD$ in inner models of strong forms of $\mathrm{AD}$ and the associated strategic-extender models plays a critical role in Descriptive Inner Model Theory.
See \cite{SargsyanTrang2016, SargysyanTrangLSA, WoodinSteel2016}.
In \cite{Woodin-SuitableExtendersI}, Woodin presents a new approach of addressing inner model problem for all large cardinals.
Woodin analyses the possible properties and limitations of some of the current methods, and introduces the seminal notions of a suitable extender model $N$ for a supercompact cardinal $\delta$ (in $V$), which in addition to several properties similar to well-known inner models, requires that $N$ captures witnessing $\delta$-supercompact measures in $V$.\footnote{Namely, for all ${\langle}anglembda > \delta$ there is a supercompact measure $U$ for $P_\delta({\langle}anglembda)$ such that $N \cap P_\delta({\langle}anglembda) \in U$ and $U \cap N \in N$.}
In a following work (see \cite{Woodin-Midrasha}), Woodin presents the ``V = Ultimate-L'' axiom, to assert (roughly) that $\Sigma_2$-definable properties of the universe are satisfied in canonical strategic-extender models of the form $\HOD^{L(A,\mathbb{R})} \cap V_{\Theta}$, for some Universally Baire set $A \subseteq \mathbb{R}$.
Combining the above notions, Woodin has formulated the ``Ultimate-L'' conjecture, asserting that there exists a suitable extender model $N \subseteq \HOD$
which satisfies the axiom ``V = Ultimate-L''.
Following the search for some $N$,
the theory established in \cite{Woodin-SuitableExtendersI} studies the possibility of $\HOD \subseteq V$ being a suitable extender model, and possible implication.
For this, Woodin introduces a new assumption known as the \emph{HOD-conjecture} (Conjecture {\rangle}ef{conj-HOD} below) and shows that remarkably, if the $\HOD$-conjecture is true then a sufficiently strong large cardinal assumption (e.g., an extendible cardinal) guarantees a version of the covering lemma for $\HOD \subseteq V$. On the other hand, if the $\HOD$-conjecture fails in the presence of sufficiently large cardinals then $\HOD$ is very far from $V$, just like the smaller inner models. See Theorem {\rangle}ef{thm:hod-dichotomy} for an exact formulation.
We remark that in general, the inner model $\HOD$ of an arbitrary model of $\mathrm{ZFC}$ can be easily modified by forcing. Nevertheless, it contains every canonical inner model and thus, the $\HOD$-conjecture might be a consequence of the covering theorem for some extremely large canonical inner model, \cite{WoodinUltimateL}.
An appealing aspect of the $\HOD$-conjecture is that it is a combinatorial statement about the $\HOD$ and $V$, and does not rely on inner model theory. Even without any further development in the inner model program, Woodin established that the $\HOD$-conjecture poses many significant limitations on the consistency of large cardinals in the choice-less context. Moreover, large cardinals beyond choice, if consistent, form a hierarchy of failures of the $\HOD$-conjecture, \cite{BKW}.
The $\HOD$-conjecture centers around the notion of $\omega$-strongly measurable cardinals in $\HOD$. Not much was know about this notion, and previously, Woodin has raised the question (\cite{Woodin-Midrasha}) of whether more than three $\omega$-strongly measurables in $\HOD$ can exist. In this work, we study the notion of $\omega$-strongly measurable cardinals in $\HOD$, we prove several consistency results concerning this notion, and establish the consistency of a model where all successors of regular cardinal are $\omega$-strongly measurable in $\HOD$.
\begin{definition}{\langle}anglebel{def:omega.strongly.measurable}
Let $\kappa$ be an uncountable regular cardinal, and $S$ a stationary subset of $\kappa$.
We say that $\kappa$ is strongly measurable in $\HOD$ with respect to $S$ if there exists some $\eta < \kappa$ such that
$(2^\eta)^{\HOD} < \kappa$ and there is no partition ${\langle} S_\alpha \mid \alpha < \eta{\rangle} \in \HOD$ of $S$ into sets, all stationary sets in $V$.
We say that $\kappa$ is $\omega$-strongly measurable in $\HOD$ if it is strongly measurable in $\HOD$ with respect to the set $S = \kappa \cap \mathbb{C}f(\omega)$, and that it is strongly measurable in $\HOD$ if it is strongly measurable in $\HOD$ with respect to $S = \kappa$.
\end{definition}
In general, one might replace $\HOD$ with any other inner model of $V$, $M$, obtaining a meaningful notion of strong measurability in $M$. Since in this paper we will be interested solely in strong measurability in $\HOD$, we will occasionally omit the emphasis ``in $\HOD$'' and say simply that $\kappa$ is strongly measurable.
It is shown in \cite{Woodin-SuitableExtendersI} that if $\kappa$ is an $\omega$-strongly measurable in $\HOD$ then there are stationary sets $S \subseteq \kappa \cap \mathbb{C}f(\omega)$ for which the restriction of the filter $\mathcal{C}UB_\kappa restriction S$ to $\HOD$, forms a measure on $\kappa$ in $\HOD$.
On the other hand, Woodin shows (\cite{Woodin-SuitableExtendersI}) that the existence of a class of regular cardinals which are not $\omega$-strongly measurable in $\HOD$, together with the existence of a $\HOD$-supercompact, implies that $\HOD$ satisfies many appealing approximation properties with respect to $V$. The results promote Woodin's $\HOD$-conjecture.
\begin{conjecture}[$\HOD$ conjecture, {\cite[Definition 191]{Woodin-SuitableExtendersI}}]{\langle}anglebel{conj-HOD}
There is a proper class of regular uncountable cardinals $\kappa$ which are not
$\omega$-strongly measurable in $\HOD$.
\end{conjecture}
In light of the $\HOD$-conjecture, it is natural to attempt forming models with as many as possible $\omega$-strongly measurable cardinals in $\HOD$.
Woodin has established the consistency (relative to large cardinals) of models with up to three $\omega$-strong measurable cardinals (see \cite[Remark 3.43]{Woodin-Midrasha}).
The main purpose of this work is to prove that many strongly measurable cardinals can be obtained from relatively mild large cardinal assumption of hyper-measurability.
\begin{theorem}{\langle}anglebel{THM1}
It is consistent relative to the existence an inaccessible cardinal $\theta$ for which $\{ o(\kappa) \mid \kappa < \theta\}$ is unbounded in $\theta$,
that every successor of a regular cardinal is strongly measurable in $\HOD$.\end{theorem}
Cummings, Friedman, and Golshani (\cite{CFG}) have established the consistency of a model where $(\alpha^+)^{\HOD} < \alpha^+$ for every infinite cardinal $\alpha$.
In \cite[Theorem 2.2]{GitikMerimovich}, Gitik and Merimovich prove that it is consistent relative to large cardinals that every regular uncountable cardinal is measurable in $\HOD$. A similar result is obtained using a different technique in \cite[Theorem 1.4]{bunger}. Perhaps, more related to our work is \cite[Theorem 1.3]{bunger}, in which a club of cardinals which are measurable in $\HOD$ is obtained from a large cardinal axiom weaker than $o(\kappa) = \kappa$.
In those models there are no $\omega$-strongly measurable successor cardinals.
We note that these results do not apply to models where there is an extendible cardinal. The existence of an extendible cardinal in $V$ derives a sharp dichotomy between $\HOD$ being either very close or very far from $V$, as shown by Woodin's $\HOD$-Dichotomy Theorem (\cite{Woodin-SuitableExtendersI}).
\begin{theorem}[The $\HOD$-Dichotomy, Woodin, {\cite{WJD-HOD-Dichotomy-Survey}}]{\langle}anglebel{thm:hod-dichotomy}
Let $\delta$ be an extendible cardinal. Then one of the following holds:
\begin{enumerate}
\item Every cardinal $\eta$ above $\delta$ which is singular in $V$, is singular in $\HOD$ and
$(\eta^+)^{\HOD} = \eta^+$.
\item Every regular cardinal above $\delta$ is $\omega$-strongly measurable in $\HOD$.
\end{enumerate}
\end{theorem}
\vskip
amount
In the last part of this work, we prove a consistency result regarding strong measurability at successors of singular cardinals.
Woodin (\cite{Woodin-SuitableExtendersI}) establishes the consistency of a successor of a singular cardinal ${\langle}anglembda$, which is $\omega$-strong measurable cardinal in $\HOD$, from the large cardinal assumption $I_0$.
Here, we prove a weaker consistency result from a weaker large cardinal assumption.
\begin{theorem}{\langle}anglebel{thm:singular.strong.meas}
Suppose that $\kappa < {\langle}anglembda$ are cardinals, $\kappa$ is ${\langle}anglembda$-supercompact and ${\langle}anglembda$ is measurable. Then, there is a generic extension in which $\kappa$ is a singular cardinal of cofinality $\omega$, and ${\langle}anglembda = \kappa^+$ is
strongly measurable in $\HOD$ with respect to $S$, for some stationary subset
$S \subseteq {\langle}anglembda \cap \mathbb{C}f(\omega)$.
\end{theorem}
\vskip
amount
\emph{A brief summary of this paper}. In \textbf{section {\rangle}ef{section:strong measurability}} we review some basic facts about strong measurability which will be central in the proof of the main theorem.
In the following sections, we gradually develop the forcing methods used to prove our main results (Theorems {\rangle}ef{THM1} and {\rangle}ef{thm:singular.strong.meas}):
In \textbf{section {\rangle}ef{Sec-omega1}} we show how to obtain a model where $\omega_1$ is strongly measurable in $\HOD$ starting with a single measurable cardinal. The case of $\kappa = \omega_1$ is different from the general case as it does not require incorporating posets for changing cofinalities. It can also be seen as a warm-up for the general case.
In \textbf{section {\rangle}ef{Sec-Cforcing}} we further develop the ideas from the previous section and combine them with a suitable iteration for changing cofinalities. As a result, we establish the consistency of a strongly measurable cardinal which is a successor of an arbitrary regular cardinal ${\langle}anglembda$, from the large cardinal assumption of $o(\kappa) = {\langle}anglembda+1$.
In \textbf{section {\rangle}ef{Sec-PrikryEmbedding}} we introduce a method to construct a Prikry-type poset which is equivalent to the forcing from the previous section, and has a direct extension order that is ${\langle}anglembda$-closed. This is utilized in
\textbf{section {\rangle}ef{Sec-Many}} to form iterations of the single cardinal forcing, thus obtaining models with many strongly measurable cardinals.
In \textbf{section {\rangle}ef{Sec:suc.of.sing}} we prove our theorem concerning successors of singular cardinals. The results of this section do not depend on the other sections past our preliminaries.
In the appendix we cite and prove
some useful results related to
homogeneous forcings and their iterations (including Prikry type forcings),
and homogeneous iterations for changing cofinalities.
Our notations are mostly standard. We follow the Jerusalem forcing convention in which for two conditions $p,p'$ in a poset $\mathbb{P}$, the fact $p'$ is stronger (more informative) than $p$ is denoted by $p' \geq p$.
\section{Variations of Strong Measurability}{\langle}anglebel{section:strong measurability}
We start with several observations concerning a natural generalization of the notion of $\omega$-strongly measurability.
\begin{definition}
Let $S \subseteq \kappa$, $S \in \HOD$ stationary and $\eta$ a cardinal in $\HOD$. $\kappa$ is $(S,<\eta)$-strongly measurable if there is no partition in $\HOD$ of $S$ into $\eta$ many disjoint stationary sets. $\kappa$ is $(S,\eta)$-strongly measurable if it is $(S, <(\eta^{+})^{\HOD})$-strongly measurable.
\end{definition}
\begin{definition}
A cardinal $\kappa$ is $S$-strongly measurable if $S \in \HOD$ and $\kappa$ is $(S, <\eta)$-strongly measurable for some $\eta$ such that $(2^\eta)^{\HOD} < \kappa$. We say that $\kappa$ is strongly measurable if $\kappa$ is $\kappa$-strongly measurable.
\end{definition}
Therefore a cardinal $\kappa$ is $\omega$-strongly measurable if it is $(S^{\kappa}_{\omega},\eta)$-strongly measurable for $\eta$ such that ${\langle}eft(2^\eta{\rangle}ight)^{\HOD} < \kappa$. Note that if $S \subseteq T$ are stationary subsets of $\kappa$ in $\HOD$ and $\kappa$ is $T$-strongly measurable then it is $S$-strongly measurable. In particular, every strongly measurable cardinal is $\omega$-strongly measurable.
\begin{theorem}[Woodin]
Let $\delta$ be an extendible cardinal.
Then the following are equivalent:
\begin{enumerate}
\item There is a regular cardinal $\kappa \geq \delta$ which is not $\omega$-strongly measurable.
\item There is a regular cardinal $\kappa \geq \delta$ which is not $(S, \delta)$-strongly measurable for some $S \in \HOD$ which consists of singular ordinals of cofinality $<\delta$.
\item The $\HOD$-conjecture.
\item There is no regular $\omega$-strongly measurable cardinal above $\delta$.
\end{enumerate}
\end{theorem}
For the proof see \cite[Theorems 197, 212, 213]{Woodin-SuitableExtendersI}.
Without the assumption that the ordinals of $S$ have fixed $V$-cofinality the equivalence might fail.
The next result provides a necessary and sufficient condition for a cardinal $\kappa$ to be $\omega$-strongly measurable in $\HOD$.
This observation will guide us in devising the main forcing construction, which will be used to prove theorem {\rangle}ef{THM1}.
\begin{lemma}{\langle}anglebel{lem:sufficient.omega.strongly}
A cardinal $\kappa$ is strongly measurable with respect to $S \in \HOD$ if and only if
$\kappa$ is an inaccessible cardinal in $\HOD$ and the restriction of the club filter on $S$ to $\HOD$ is the intersection of $\eta$ normal measures from $\HOD$, ${\langle} U_{\kappa,i} \mid i < \eta{\rangle} \in \HOD$, for some $\eta < \kappa$.
\end{lemma}
\begin{proof}
For the backwards implication, since $\kappa$ is inaccessible in $\HOD$ and $\eta < \kappa$, ${\langle}eft(2^\eta{\rangle}ight)^{\HOD} < \kappa$.
Let ${\langle}anglengle T_\alpha \mid \alpha < (\eta^+)^{\HOD}{\rangle}anglengle\in \HOD$ be a decomposition of $S$ into stationary sets. By the assumption, for each $\alpha$ there is a measure $U_{\kappa,i}$ in $\HOD$ such that $T_\alpha \in U_{\kappa,i}$. Since the sets $T_\alpha$ are pairwise disjoint, it is impossible for $\alpha \neq \beta$ to belong to the same $U_{\kappa,i}$. Thus, we obtain an injective function from $(\eta^+)^{\HOD}$ to $\eta$ in $\HOD$---a contradiction.
Let us assume now that $\kappa$ is strongly measurable with respect to $S \in \HOD$. In particular, $\kappa$ is inaccessible in $\HOD$. Let $\mathcal{S} \in \HOD$ be a maximal collection of pairwise disjoint stationary subsets of $S$, in $\HOD$, such that for all $T \in \mathcal S$, the club filter restricted to $T$ is an ultrafilter in $\HOD$. Let us denote this ultrafilter by $U_T$. Since this collection is a partition of $S$ into stationary sets, $|\mathcal S| < \kappa$.
If $\bigcap \{U_T \mid T \in \mathcal S\}$ is not the club filter restricted to $S$ in $\HOD$, then it contains a set $S \setminus S'$, where $S' \subseteq S$ stationary, $S' \in \HOD$. In particular, $S' \notin U_T$ for all $T \in \mathcal S$, so $S' \cap T$ is non-stationary for all $T \in \mathcal S$, and $S'' = S' \setminus \bigcup_{T \in \mathcal S} (S' \cap T)$ is a stationary subset of $S$, disjoint from all members of $\mathcal S$. Since $\kappa$ is strongly measurable with respect to $S$, it is also strongly measurable with respect to $S''$, and thus there is some $T' \subseteq S''$ stationary such that the club filter restricted to $T'$ is an ultrafilter. But this contradicts the maximality of $\mathcal{S}$.
\end{proof}
\begin{corollary}{\langle}anglebel{lem:necessary.omega.strongly}
Let $\kappa$ be $(S,{<}\kappa)$-strongly measurable. Then $S$ is contained in the regular cardinals of $\HOD$, up to a non-stationary error.
\end{corollary}
\section{$\omega_1$ is strongly measurable from one measurable cardinal}{\langle}anglebel{Sec-omega1}
In this section, we would like to present a forcing that forces $\omega_1$ to be strongly measurable. By Lemma {\rangle}ef{lem:sufficient.omega.strongly}, this means that in $\HOD$, the club filter of $\omega_1$ is an intersection of countably many normal measures. In the case of $\omega_1$, we can take a single measure. So, we would like to collapse a measurable cardinal $\kappa$ with a normal measure to be $\omega_1$ and then using a Mathias-type forcing, to add a club that diagonalizes the normal measure. In order to show that this works, we need to show two things. First, we must show that the iteration is cone homogeneous. This is done in Lemma {\rangle}ef{Lem0:CHomog}. Second, we need to show that it does not collapse $\omega_1$. This amounts to show that the second step of the iteration is $\sigma$-distributive, which in turn requires us to be able to add a $U$-generic point to the generic club, see Lemma {\rangle}ef{Lem01} for the precise formulation.
Let us present the forcing. Suppose that $\kappa$ is a measurable cardinal in a model $V$ and $U$ is a normal measure on $\kappa$.
Force with Levy collapse poset $\mathbb{C}l(\omega,<\kappa)$ over $V$. Let $H$ be a $V$-generic filter.
Working in the generic extension $V[H]$, let $\mathbb{C}_U$ be the poset consisting of pairs $x = {\langle} c, A{\rangle}$, where $c \subseteq \kappa$ is a bounded closed subset of $\kappa$ and $A \in U$. The condition $x' = {\langle} c',A'{\rangle}$ extends $x$ if $c'$ is an end extension of $c$, $A' \subseteq A$, and $c'\setminus c \subseteq A$.
It is clear that if $x = {\langle} c,A{\rangle}$ and $x' = {\langle} c',A'{\rangle}$ are two conditions with the same bounded closed set $c = c'$ then $x,x'$ are compatible. Since
$\kappa^{<\kappa} = \kappa$ in $V[H]$ then $\mathbb{C}_U$ satisfies $\kappa^+$-c.c.\ (which is $\aleph_2^{V[H]}$-c.c.).
The forcing $\mathbb{C}_U$ adds a diagonalizing club to $U$. It has also been studied in \cite{Rinot2009} in the context of well-behaved posets which can introduce square sequences, and was found useful in other contexts.
The following lemma is the key ingredient in the proof of the distributivity of $\mathbb{C}_U$.
\begin{lemma}{\langle}anglebel{Lem01}
Work in $V[H]$ and fix some regular cardinal $\theta > \kappa^+$. There exists a stationary set of structures $M \prec H_\theta$ of size $|M| < \kappa$, with the property that $\sup(M \cap \kappa) \in A$ for every $A \in U \cap M$.
\end{lemma}
\begin{proof}
Fix any $f\mathbb{C}lon H_\theta^{<\omega} \to H_\theta$ in $V[H]$. We would like to show that there exists some $M \subseteq H_\theta$ which is closed under $f$ and satisfies the conditions in the statement of the lemma.
Fix in $V$ a name $\mathunderaccent\tilde-3 {f}$ for $f$ and let $f' \mathbb{C}lon\mathbb{C}l(\omega, <\kappa) \times (H_\theta^V)^{<\omega} \to H_{\theta}^V$ be a function that sends $(p, x) \in \mathbb{C}l(\omega,<\kappa)\times (H_\theta)^{<\omega}$ to $y$ if $p \Vdash \mathunderaccent\tilde-3 {f}(x) = \check{y}$. Note that $x$ is a finite sequence of names.
By the definition of $f'$, if $M' \prec H_\theta^V$
is closed under $f'$, $M' \cap \kappa \in \kappa$, and $\kappa, U \in M'$,
then $M' = M \cap H_{\theta}^V$ for some
$M \subseteq H_\theta$ which is closed under $f$.
Indeed, we may take $M = M'[H \cap M]$. By the chain condition of $\mathbb{C}l(\omega,<\kappa)$,
every name for a ground model object that belongs to $M'$, can be refined to
a nice name which is contained in $M'$.
Since $U \subseteq H_{\theta}^V$, it is therefore sufficient to prove that there exists some $M' \subseteq H_\theta^V$ which is closed under $f'$ and satisfies
$|M'| < \kappa$ and $\sup(M' \cap \kappa) \in A$ for all $A \in M' \cap U$.
Working in $V$, take an elementary substructure $N \prec H_\theta^V$ satisfying $f'[N] \subseteq N$, $N^{<\kappa} \subseteq N$, $|N| = \kappa$, $\kappa \in N$.
Let $j \mathbb{C}lon V \to W \mathbb{C}ng \mathcal{U}lt(V,U)$ be the ultrapower embedding induced by $U$.
Consider the structure $\tilde{M}' = j" N \prec j(H_\theta^V)$. $\tilde{M}' \in W$, is closed under $j(f')$, and $\tilde{M}' \cap j(\kappa) \in j(\kappa)$.
It follows that $0_{j(\mathbb{C}l(\omega,<\kappa))}$ forces $\tilde{M}'$ to be closed under $j(\mathunderaccent\tilde-3 {f})$.
Finally,
for every $A \in \tilde{M}' \cap j(U)$, $A = j(b^*r{A})$ for some $b^*r{A}\in U$ and therefore $\kappa \in A$.
So $\tilde{M}$ satisfies the conclusion of the lemma in $W$, and it is closed under $j(f')$. By the elementarity of $j$, there is $M' \in V$ satisfying the conclusion of the lemma, and closed under $f'$,
and thus $M'[H \cap M'] = M$ satisfies the requirements of the lemma.
\end{proof}
\begin{proposition}{\langle}anglebel{prop01}
$\mathbb{C}_{U}$ is $\kappa$-distributive.
\end{proposition}
\begin{proof}
Since $\kappa = \aleph_1$ in $V[H]$, we need to check that the intersection of a countable family $\{ D_n \mid n<\omega\}$ of dense open subsets of $\mathbb{C}_{U}$ is dense. Pick some regular cardinal $\theta > \kappa^+$ such that $\mathbb{C}_U, \{D_n \mid n<\omega\} \in H_\theta$.
By lemma {\rangle}ef{Lem01}, for every condition $x \in \mathbb{C}_{U}$ there exists an elementary substructure $M \prec H_\theta$ of size $|M| = \aleph_0$, with $x, \mathbb{P},\mathbb{C}_{U},\{D_n \mid n < \omega\} \in M$ and further satisfies that $\sup(M \cap \kappa) \in A$ for every $A \in M \cap \kappa$. We may also assume that $M = M'[H \cap M']$ for $M' \in V$, $M' \prec H_\theta^V$.
Denote $\sup(M \cap \kappa)$ by $\alpha$ and pick a cofinal sequence ${\langle} \alpha_n \mid n < \omega{\rangle}$ in $\alpha$.
We can construct an increasing sequence of extensions ${\langle} x_n \mid n<\omega{\rangle} \subseteq M$ of $x$, $x_n = {\langle} c_n,A_n{\rangle}$ such that $x_{n+1} \in D_n$ and $\max(c_n) \geq \alpha_n$ for every $n<\omega$. Without loss of generality, we may assume that for every $A \in U \cap M$ there is $n < \omega$ such that $A_n \subseteq A$.
Since $x_n = {\langle} c_n,A_n{\rangle} \in M$ then $\alpha \in A_n$ for all $n < \omega$.
It follows that $x^* = {\langle} \{ \alpha \} \cup (\bigcup_n c_n), \bigcap_n A_n{\rangle}$ is a condition in $\mathbb{C}_U$, which is clearly an upper bound of ${\langle} x_n \mid n < \omega{\rangle}$.\footnote{Note that $\bigcap A_n = \bigcap_{A \in M' \cap U} A$.}
We conclude that there exists $x^*$ extending our given condition $x$ such that $x^* \in \bigcap_n D_n$.
\end{proof}
\begin{lemma}{\langle}anglebel{Lem0:CHomog}
$\mathbb{C}_{U}$ is cone homogeneous.
\end{lemma}
\begin{proof}
Let $x_1 = {\langle} c_1,A_1{\rangle}$, $x_2 = {\langle} c_2,A_2{\rangle}$ be two conditions of $\mathbb{C}_{U}$. Take $\nu \in A_1 \cap A_2$ above $\max(c_1),\max(c_2)$
and consider the extensions $y_1 = {\langle} c_1 \cup \{\nu\}, (A_1 \cap A_2) \setminus (\nu+1){\rangle}$, $y_2 = {\langle}angle c_2 \cup \{\nu\}, (A_1 \cap A_2) \setminus (\nu+1){\rangle}$ of $x_1$ Define a cone isomorphism $\sigma\mathbb{C}lon \mathbb{C}_U/y_1 \to \mathbb{C}_U/y_2$ by
\[\sigma( {\langle} c, A{\rangle}) = {\langle} c_2 \cup (c\setminus \nu), A{\rangle} \]
$\sigma$ is clearly order preserving map onto $\mathbb{C}_U/y_1$, and has an order preserving inverse which is given by
\[
\sigma^{-1}( {\langle} c, A{\rangle}) = {\langle} c_1 \cup (c\setminus \nu), A{\rangle}
\]
\end{proof}
\begin{theorem}{\langle}anglebel{Thm-0stmeas}
Suppose $C \subseteq \mathbb{C}_U$ is a $V[H]$-generic filter. Then, in $V[H \ast C]$,
$\kappa = \aleph_1^{V[H \ast C]}$ is strongly measurable.
\end{theorem}
\begin{proof}
By Lemma {\rangle}ef{corr: OD implies homogeneous}, $\mathbb{C}l(\omega,<\kappa) * \mathbb{C}_U$ is cone homogeneous, and therefore $\HOD^{V[H \ast C]} \subseteq V$.
It is clear from the definition of $\mathbb{C}_U$ that for every subset $S \subseteq \kappa$ in $V$, $S$ is stationary in $V[H \ast C]$ if and only if
$S \in U$. It follows that the closed unbounded filter on $\kappa = \aleph_1^{V[H \ast C]}$ in $V[H \ast C]$ is a $\HOD$-ultrafilter. Therefore $V[H \ast C]\triangleleftdels \kappa$ is strongly measurable.
\end{proof}
\section{Strongly measurable successor of a regular cardinal}{\langle}anglebel{Sec-Cforcing}
In this section we would like to force a successor of an uncountable regular cardinal, $\kappa = {\langle}anglembda^+$ to be strongly measurable. There are a few difficulties that arise. First, there is a definable splitting of the ordinals below $\kappa$, according to the cofinalities, so the club filter cannot be an ultrafilter but rather an intersection of a few normal measures. This means that we should fix a collection of normal measures, that their intersection is indented to become the club filter. Moreover, when killing a stationary set which is small with respect to the designated filter, we are forcing a club through the previous regulars, which are now going to change cofinalities to various possibilities. This means that a Levy collapse by itself would not provide all the cofinality changes that we need, and we must use a more complicated method of changing cofinalities in a homogeneous way.
Suppose that ${\langle}anglembda < \kappa$ are two cardinals such that ${\langle}anglembda$ is regular and $\kappa$ is measurable with $o(\kappa) = {\langle}anglembda+1$. Let $\mathcal{U} = {\langle} U_{\alpha,\tau} \mid {\langle}anglembda < \alpha {\langle}eq \kappa, \tau < o^{\mathcal{U}}(\alpha){\rangle}$ be a coherent sequence of normal measures with $o^{\mathcal{U}}(\kappa) = {\langle}anglembda + 1$.
Let $\mathbb{P}^\mathcal{U}_\kappa = {\langle} \mathbb{P}_\alpha, \mathbb{Q}_\alpha \mid \alpha < \kappa{\rangle}$ be the homogeneous iteration of subsection {\rangle}ef{ssec-non-stationary-iteration}. In the next two sections, $\mathbb{P}$ stands for $\mathbb{P}^\mathcal{U}_\kappa$. For the main properties of $\mathbb{P}$, we refer the reader to Fact {\rangle}ef{fact:summary-properties-of-PU}. We will explicitly need the following additional property of the iteration.
\begin{remark}{\langle}anglebel{rmk1}
We note that it follows at once from the definition of $\mathbb{Q}_\alpha$ that every $V$-set $A \in \bigcap_{i<o^{\mathcal{U}}(\alpha)} U_{\alpha,i}$ contains a tail of the cofinal sequence $b_\alpha$. This is because
every condition $q = {\langle} t, T{\rangle} \in \mathbb{Q}_\alpha$ has a direct extension $q^A = {\langle} t, T^A{\rangle}$ of $q$, which satisfies that $\suc_T(s) \subseteq A$ for all $s \in T$.
\end{remark}
\begin{definition}
Let $G \subseteq \mathbb{P}$ be a $V$-generic filter and $H \subseteq \mathbb{C}l({\langle}anglembda,<\kappa)$ be the Levy collapse generic over $V[G]$.
Working in $V[G \ast H]$ we consider the filter $\mathcal{F}_\kappa$ generated by $\bigcap_{i {\langle}eq {\langle}anglembda} U_{\kappa,i}$,
\[ \mathcal{F}_\kappa = \{ A \subseteq \kappa \mid \exists B \in \bigcap_{i{\langle}eq {\langle}anglembda} U_{\kappa,i}, B \subseteq A\}. \]
\end{definition}
The filter $\mathcal{F}_\kappa$ is going to generate the club filter in $\HOD$ in the generic extension.
\begin{lemma}{\langle}anglebel{Lem1}
$\mathcal{F}_\kappa$ is a $\kappa$-complete filter in $V[G\ast H]$.
\end{lemma}
\begin{proof}
Suppose that ${\langle} A_\nu \mid \nu < \beta{\rangle} \in V[G\ast H]$ is a sequence of $\beta < \kappa$ many sets of $\mathcal{F}_\kappa$. We would like to show that $\bigcap_{\nu < \beta} A_\nu$ belongs to $\mathcal{F}_\kappa$. We may assume that $A_\nu \in \bigcap_{i {\langle}eq {\langle}anglembda} U_{\kappa_i}$ for all $\nu < \beta$.
In order to prove the claim, we move from $V[G\ast H]$ to $V[G]$, and then to $V$.
Working in $V[G]$, let $\mathunderaccent\tilde-3 {A_\nu}$ be a $\mathbb{C}l({\langle}anglembda,<\kappa)$-name for the $V$-set in $\bigcap_{i{\langle}eq{\langle}anglembda} U_{\kappa,i}$. Since $\mathbb{C}l({\langle}anglembda,<\kappa)$ satisfies $\kappa$-c.c., there exists a family of $V$-sets $X_\nu \subseteq \bigcap_{i{\langle}eq{\langle}anglembda} U_{\kappa,i}$, $X_\nu \in V[G]$, of size $<\kappa$ such that $\Vdash\mathunderaccent\tilde-3 {A_\nu} \in \check{X}_\nu$. Fix in $V$ a $\mathbb{P}$-name $\mathunderaccent\tilde-3 {X}_\nu$ for each $X_\nu$.
Let $p \in G$ be a condition forcing the above. Moving back to $V$, the fusion lemma for non-stationary support iteration of Prikry type forcings, \cite[Lemma 3.6]{bunger}, guarantees
that there exists some $q \in G$ and a sequence of sets ${\langle} Y_\nu \mid \nu < \beta{\rangle}$ in $V$,
so that for each $\nu < \beta$, $Y_\nu \subseteq \bigcap_{i {\langle}eq {\langle}anglembda}U_{\kappa,i}$ has size $|Y_\nu| < \kappa$,
and $q\Vdash \mathunderaccent\tilde-3 {X}_\nu \subseteq \check{Y}_\nu$.
For each $\nu$, let $A'_\nu = \bigcap Y_\nu$, and $A' = \bigcap_{\nu < \beta} A'_\nu$.
Since $U_{\kappa,i}$ is $\kappa$-complete for all $i {\langle}eq {\langle}anglembda$, we have in $V[G\ast H]$ that
$A' \in \mathcal{F}_\kappa$ and $A' \subseteq \bigcap_{\nu < \beta} A_\nu$.
\end{proof}
To produce a model where $\kappa$ is $\omega$-strongly measurable, we will force over $V[G\ast H]$ to add
a closed unbounded set $C \subseteq \kappa$ which is almost contained in every set $A \in \mathcal{F}_\kappa$.
\begin{definition}
Working in a $V$-generic extension $V[G\ast H]$ by $G\ast H \subseteq \mathbb{P}*\mathbb{C}l({\langle}anglembda,<\kappa)$,
we define the forcing $\mathbb{C}_{\mathcal{F}_\kappa}$. Conditions $x \in \mathbb{C}_{\mathcal{F}_\kappa}$ are pairs $x = {\langle} c, A{\rangle}$ where
$c$ is a closed and bounded subset of $\kappa$ and $A \in \mathcal{F}_\kappa$.
A condition $x' = {\langle} c', A'{\rangle} \in \mathbb{C}_{\mathcal{F}_\kappa}$ extends $x$ (denoted $x' \geq x$) if
\begin{itemize}
\item[(i)] $c' \cap \max(c) = c$,
\item[(ii)] $A' \subseteq A$, and
\item[(iii)] $c' \setminus c \subseteq A$.
\end{itemize}
\end{definition}
For conditions $x = {\langle} c,A{\rangle} \in \mathbb{C}_{\mathcal{F}_{\kappa}}$ we will frequently denote $c$ and $A$ by $c^x$ and $A^x$ respectively.
It is clear that if $R \subseteq \mathbb{C}_{\mathcal{F}_\kappa}$ is generic then the union $C = \bigcup\{ c^x \mid x \in R\}$ is a closed and unbounded subset of $\kappa$ which is almost contained in every $A \in \mathcal{F}_{\kappa}$.
Since $\mathcal{F}_\kappa$ is a filter and $\kappa^{<\kappa} = \kappa$, the forcing $\mathbb{C}_{\mathcal{F}_\kappa}$ is $\kappa$-centered and therefore satisfies $\kappa^+$-chain condition.
The following lemma is a parallel of Lemma {\rangle}ef{Lem01}. From this lemma we will infer the distributivity of the forcing $\mathbb{C}_{\mathcal{F}_\kappa}$.
\begin{lemma}{\langle}anglebel{Lem2}
Working in $V[G\ast H]$, for any regular cardinal $\theta > \kappa^+$ and $\tau {\langle}eq {\langle}anglembda$, there exists a stationary set of structures $M \prec H_\theta$ with $\sup(M \cap \kappa) = \alpha$ which satisfy
\begin{itemize}
\item[(i)] $M^{<\tau} \subseteq M$;
\item[(ii)] $o^{\mathcal{U}}(\alpha) = \tau$;
\item[(iii)] For every $A \in \mathcal{F}_\kappa \cap M$, $\alpha \in A$ and moreover $b_\alpha \subseteq^* A$ (namely $b_\alpha \setminus A$ is bounded in $\alpha$).
\end{itemize}
\end{lemma}
\begin{proof}
Fix a function $f\mathbb{C}lon [H_{\theta}]^{<\omega} \to H_{\theta}$ in $V[G\ast H]$. Back in the ground model $V$, let $\mathunderaccent\tilde-3 {f}$ be a $\mathbb{P} * \mathbb{C}l({\langle}anglembda,<\kappa)$-name for $f$. Since $\mathbb{C}l({\langle}anglembda,<\kappa)$ is $\kappa$-c.c., there exists a $\mathbb{P}$-name function $\mathunderaccent\tilde-3 {F} \mathbb{C}lon [H_\theta^V]^{<\omega} \to [H_\theta^V]^{<\kappa}$ such that $\mathunderaccent\tilde-3 {f}(x)$ is forced to be a member of $\mathunderaccent\tilde-3 {F}(x)$ for every $x \in H_\theta^V$.
Let us consider our ability to approximate $\mathunderaccent\tilde-3 {F}$ in $V$. Let $N \prec H_\theta^V$ be an elementary substructure of size $\kappa$ with $N^{<\kappa} \subseteq N$ and $\kappa, \mathbb{P},\mathunderaccent\tilde-3 {F}\in N$.
\begin{claim}{\langle}anglebel{claim:properness}
Let $N$ be as above and $p \in \mathbb{P} \cap N$. Then, there is $p^* {\langle}eq p$ which is \emph{$N$-generic}, namely for every name for an ordinal $\mathunderaccent\tilde-3 \sigma \in N$, there is set of ordinals $S \in N$ such that $S \subseteq N$ and $p^* \Vdash \mathunderaccent\tilde-3 \sigma \in S$.
\end{claim}
\begin{proof}
By a standard argument concerning capturing dense open sets in Prikry-type forcings and fat-trees (e.g., see \cite{Gitik-HB})
for every dense open set $D$ of $\mathbb{P}$, $p \in \mathbb{P}$ there exists a direct extension $p' \geq^* p$ which reduces capturing $D$ to a dense subset of $\mathbb{P}_\mu$ for some $\mu < \kappa$, namely the set of all $r \in \mathbb{P}_\mu$ such that $r ^\smallfrown p' restriction [\mu,\kappa) \in D$ is dense below $p' restriction \mu$.
Moreover, given $\nu < \kappa$ we can also make the direct extension $p'$ to agree with $p$ up to $\nu+1$ (i.e., $p'restriction\nu+1 = prestriction\nu+1$) in which case $\mu > \nu$.
Given an initial condition $p \in \mathbb{P}$, we can list the dense open sets in $N$,
${\langle} D_i \mid i < \kappa{\rangle}$, and form an increasing sequence of direct extensions of $p$, ${\langle} p^i \mid i < \kappa{\rangle}$, together with a closed unbounded set $C^* = {\langle} \nu_i \mid i <\kappa{\rangle}$ such that for every successor ordinal $i = i'+1$, $p^i \in N$ reduces the dense set $D_{i'}$ of $\mathbb{P}$ to a bounded dense set $D_{i'}'$ of $\mathbb{P}_{\mu_i}$ for some $\nu_i <\mu_i < \kappa$, and $p^irestriction \nu_i+1 = p^{i'}restriction \nu_i+1$.
By a standard argument concerning non-stationary support iterations (e.g., see the fusion argument in the proof of \cite[Lemma 2.2]{bunger}), the sequence of direct extensions ${\langle} p^i \mid i < \kappa{\rangle}$ has an upper bound $p^* \geq^* p$.
It follows that for every $\mathbb{P}$-name $\mathunderaccent\tilde-3 {\sigma} \in N$ of an element of $H_\theta^V$, there exists some $\mu < \kappa$ and
a $\mathbb{P}_\mu$-name $\mathunderaccent\tilde-3 {\sigma}' \in N$ such that $p^* \Vdash \mathunderaccent\tilde-3 {\sigma} = \mathunderaccent\tilde-3 {\sigma}'$.
In particular, for
each such name $\mathunderaccent\tilde-3 {\sigma}$, $p^*$ forces that it can take $<\kappa$ many values in $H_\theta^V$, all of which are in $N$. This follows from the elementarity of $N$ in $H_\theta$ and the fact $\kappa+1 \subseteq N$.
\end{proof}
Let $j_\tau\mathbb{C}lon V \to M_\tau$ be the ultrapower embedding by $U_{\kappa,\tau}$ and $M' = j_{\tau}" N \prec j_\tau(H_\theta^V)$, $M' \in M_\tau$.
\begin{claim}{\langle}anglebel{claim:closure of j_tau''N}
$j_\tau(p^*)$ forces that $M'$ is closed under $j(\mathunderaccent\tilde-3 {F})$.
\end{claim}
\begin{proof}
Indeed
if $G^* \subseteq j_\tau(\mathbb{P})$ is $M_\tau$-generic with $j_\tau(p^*) \in G^*$
then for each
$\mu < \kappa$, $G^*_\mu = \{ prestriction \mu \mid p \in G^*\}$ is a $V$-generic filter for $\mathbb{P}_\mu$.
For every $y = j_\tau(\mathunderaccent\tilde-3 {\sigma})_{G^*} \in M' \cap j_\tau(H_\theta^V)$ and $F^* = j_\tau(\mathunderaccent\tilde-3 {F})_{G^*}$, $F^*(y) = j_\tau(\mathunderaccent\tilde-3 {F}){\langle}eft( j_\tau(\mathunderaccent\tilde-3 {\sigma}) {\rangle}ight)_{G^*}$ is the $G^*$-generic interpretation of the $j_\tau(\mathbb{P})$-name $j_\tau{\langle}eft( \mathunderaccent\tilde-3 {F}(\mathunderaccent\tilde-3 {\sigma}) {\rangle}ight)$. As $p^*$ forces $\mathunderaccent\tilde-3 {F}(\mathunderaccent\tilde-3 {\sigma}) = \mathunderaccent\tilde-3 {\sigma}'$ for some $\mathunderaccent\tilde-3 {\sigma}' \in N$
which is a $\mathbb{P}_\mu$-name for some $\mu < \kappa$,
we see that $j_\tau(p^*)$ forces $j_\tau{\langle}eft( \mathunderaccent\tilde-3 {F}(\mathunderaccent\tilde-3 {\sigma}) {\rangle}ight)= j_\tau(\mathunderaccent\tilde-3 {\sigma}')$, where $j_\tau(\mathunderaccent\tilde-3 \sigma')$ is a $j_\tau(\mathbb{P}_\mu) = \mathbb{P}_\mu$-name.
If $q \in G_\mu$ and $z \in N \cap H_\theta^V$ are such that $q \Vdash_{\mathbb{P}_\mu} \mathunderaccent\tilde-3 {\sigma}' = \check{z}$ then
$j_\tau(q) = q \Vdash j_\tau(\mathunderaccent\tilde-3 {\sigma}') = j_\tau(\check{z})$. We conclude that
$F^*(y) = z \in M'$.
\end{proof}
We now return to prove the statement of the lemma. It
is sufficient to prove that in $V[G]$ there exists some
$M' \subseteq H_\theta^V$ which is closed under $F$ and satisfies
requirements (i)-(iii).
Let $p \in \mathbb{P}$ be a condition.
By a standard density argument there are $N \prec H_\theta^V$ and $p^* \in G$ which is $N$-generic, with $p^* \geq^* p$.\footnote{This is true, since for every $q \geq p$ there is $r \geq^* p$ such that $q \geq r$ is a finite Prikry extension. Let $q^* \geq^* q$ be $N$-generic. Then, there is $\alpha$ such that $q restriction [\alpha, \kappa) = r restriction [\alpha, \kappa)$. So, the condition $p^* = r restriction \alpha \cup q^* restriction [\alpha, \kappa)$ is an $N$-generic direct extension of $p$, from $G$.}
By Claim {\rangle}ef{claim:closure of j_tau''N}, $j_\tau(p^*)$ forces that
$M' = j_\tau" N$ is closed under $j_\tau(F)$.
It is now clear that $M'$ satisfies condition (ii) in the ultrapower,
as $o^{j_\tau(\mathcal{U})}(\kappa) = \tau$ and $M' \cap j_\tau(\kappa) = \kappa$.
Condition (i) holds as well, since $j(\mathbb{P}_\kappa) / \mathbb{P}_\kappa$ does not
introduce new ${<}\tau$-sequences to $j_\tau " N$.
Therefore, it remains to verify that $j_\tau(p^*)$
forces $M'$ to satisfy condition (iii).
For every $A \in M' \cap j_\tau(\mathcal{F}_\kappa)$ there is some $B \in \mathcal{F}_\kappa$ such that $A = j_\tau(B)$. In particular, $A \cap \kappa = B \in \mathcal{F}_\kappa$ and $\kappa \in A$.
Since $\mathcal{F}_\kappa \subseteq \bigcap_{i {\langle}eq \tau} U_{\kappa,i}$ (which is $\mathcal{F}_\kappa^{M_\tau}$), it follows form remark {\rangle}ef{rmk1} that for every generic filter $G^* \subseteq j_\tau(\mathbb{P})$
over $M_\tau$, if $b_\kappa$ is the $G^*$-induced $\mathbb{Q}_\kappa^\tau$ cofinal generic sequence, then it is almost contained in $B = A \cap \kappa$.
\end{proof}
\begin{proposition}{\langle}anglebel{prop1}
$\mathbb{C}_{\mathcal{F}_\kappa}$ is $\kappa$-distributive.
\end{proposition}
\begin{proof}
Since $\kappa = {\langle}anglembda^+$ in $V[G\ast H]$, we need to check that the intersection of every set $\{ D_i \mid i < {\langle}anglembda\}$ of ${\langle}anglembda$-many dense open subsets of $\mathbb{C}_{\mathcal{F}_\kappa}$ is dense. Pick some regular cardinal $\theta > \kappa^+$ such that $\mathbb{P}, \mathbb{C}_{\mathcal{F}_\kappa}, \{D_i \mid i < {\langle}anglembda\} \in H_\theta$.
By Lemma {\rangle}ef{Lem2}, for every condition $x \in \mathbb{C}_{\mathcal{F}}$ there exists an elementary substructure $M \prec H_\theta$ of cardinality $< \kappa$, with $x, \mathbb{P},\mathbb{C}_{\mathcal{F}_\kappa},\{D_i \mid i < {\langle}anglembda\} \in M$ and which further satisfies (i) $M^{<{\langle}anglembda} \subseteq M$; (ii) $\sup(M \cap \kappa) = \alpha$ has $o^{\mathcal{U}}(\alpha) = {\langle}anglembda$; and
(iii) $\alpha \in A$ and $b_\alpha$ is almost contained in $A$ for every $A \in \mathcal{F}_\kappa \cap M$.
Let ${\langle} \alpha_i \mid i < {\langle}anglembda{\rangle}$ be an increasing enumeration of $b_\alpha$.
We construct by induction an increasing sequence of extensions ${\langle} x_j \mid j < {\langle}anglembda{\rangle}$ of $x$, together with an increasing sub-sequence ${\langle} \alpha_{i_j} \mid j < {\langle}anglembda{\rangle}$ of $b_\alpha$ such that $x_{j+1} \in D_j$ for every $j < {\langle}anglembda$, and $\{\alpha\} \cup \{ \alpha_{i_j} \mid j > j^*\} \subseteq A^{x_{j^*}}$ for all $j^* < {\langle}anglembda$.
For notational simplicity, denote $x$ by $x_{-1}$. Given a condition $x_j \in M$ with a suitable $\alpha_j$ as above,
we take $x_{j+1}\in D_{j+1}$ to be an extension of $x_j$ with $\max(c^{x_{j+1}}) > \alpha_{i_j}$. Since $A^{x_{j+1}} \in M \cap \mathcal{F}_{\kappa}$ we can use (iii) and get that $\alpha \in A^{x_{j+1}}$ and
there exist some $i' > i_j$ such that $\{ \alpha_i \mid i >i'\} \subseteq A^{x_{j+1}}$. Take $i_{j+1} < {\langle}anglembda$ to be the minimal such $i' > i_j$.
It remains to show that the construction goes through at limit stages $\delta {\langle}eq {\langle}anglembda$.
Given ${\langle} x_j \mid j < \delta{\rangle}$ we define $i_\delta = \sup_{j < \delta} \alpha_{i_j}$. It is clear from our construction at successor steps that
$\alpha_{i_\delta} = \sup_{j < \delta} \max(c^{x_j})$ and $\alpha_{i_\delta} \in A^{x_j}$ for every $j < \delta$.
It follows that the condition $x_\delta = {\langle} \{ \alpha_\delta \} \cup \bigcup_{j < \delta} c^{x_j}, \bigcap_{j<\delta} A^{x_j}{\rangle}$ satisfies the desirable conditions. Moreover if $\delta < {\langle}anglembda$ then $x_\delta \in M$ since $M$ is closed under $<{\langle}anglembda$-sequences.
Since the limit construction goes through at stage ${\langle}anglembda$ as well (although not producing a condition in $M$), the limit condition $x_{\langle}anglembda$ is an extension of $x$, and belongs to $\bigcap_{j <{\langle}anglembda} D_j$.
\end{proof}
The argument of the proof of lemma {\rangle}ef{Lem0:CHomog} for $\mathbb{C}_U$ applies to $\mathbb{C}_{\mathcal{F}_\kappa}$ as well.
\begin{lemma}{\langle}anglebel{Lem:CHomog}
$\mathbb{C}_{\mathcal{F}_\kappa}$ is cone homogeneous.
\end{lemma}
\begin{theorem}{\langle}anglebel{Thm-1stmeas}
In the generic extension by $\mathbb{P} * \mathbb{C}l({\langle}anglembda,<\kappa) * \mathbb{C}_{\mathcal{F}_\kappa}$, $\kappa$ is strongly measurable.
\end{theorem}
\begin{proof}
Suppose $G(\mathbb{C}_{\mathcal{F}_\kappa}) \subseteq\mathbb{C}_{\mathcal{F}_\kappa}$ be a generic filter over $V[G\ast H]$.
We may identify $G(\mathbb{C}_{\mathcal{F}_\kappa})$ with its derived generic closed and unbounded set \[C = \bigcup \{ c \mid \exists A {\langle} c,A{\rangle} \in G(\mathbb{C}_{\mathcal{F}_\kappa})\}.\]
By a standard density argument we have that for every set $X \subseteq \kappa$ in $V$,
if $X \notin U_{\kappa,\tau}$ for all $\tau {\langle}eq {\langle}anglembda$ then $|C \cap X| < \kappa$.
We conclude that for $X \subseteq\kappa$ in $V$ to be stationary in $V[G \ast H \ast C]$ it must belong to $U_{\kappa,\tau}$ for some $\tau {\langle}eq {\langle}anglembda$.
It follows that if ${\langle} S_i \mid i <\eta{\rangle} \subseteq V$ is a partition of $\kappa$ into disjoint sets which are stationary in $V[G \ast H \ast C]$ then $|\eta| {\langle}eq {\langle}anglembda$.
Moreover, since $\kappa$ is inaccessible in $V$ we have $(2^\eta)^V < \kappa$.
Finally, we know that each poset $\mathbb{P}$, $\mathbb{C}l({\langle}anglembda,<\kappa)$, and $\mathbb{C}_{\mathcal{F}_\kappa}$ is forced in turn to be cone homogeneous and clearly definable using parameters from the ground model. Therefore
$\mathbb{P}* \mathbb{C}l({\langle}anglembda,<\kappa) \ast \mathbb{C}_{\mathcal{F}_\kappa}$ is cone homogeneous, and therefore $\HOD^{V[G \ast H \ast C]} \subseteq V$. The claim follows.
\end{proof}
The result in this section is weaker than the result of section {\rangle}ef{Sec-omega1}, since the club filter is not an ultrafilter in $\HOD$. Since the club filter restricted to $S^{\omega_2}_{\omega}$ is an ultrafilter in a model of $\mathrm{AD} + V = L(\mathbb{R})$, one can force with the $\mathbb{P}_{max}$ forcing and obtain a generic extension in which the club filter restricted to $S^{\omega_2}_{\omega}$ is an ultrafilter in $\HOD$.\footnote{We would like to thank the referee for pointing us to this fact.}
\begin{question}
Is it consistent that the club filter restricted to $S^{{\langle}anglembda}_\omega$ is an ultrafilter in $\HOD$ for a regular cardinal ${\langle}anglembda > \aleph_2$?
\end{question}
By the general behavior of covering arguments, it is possible that the consistency strength of $\omega_2$ being $\omega$-strongly measurable in $\HOD$ might be much lower than the same property for other successor of a regular cardinal and even be as low as a single measurable cardinal.
\begin{question}
What is the consistency strength of $\omega_2$ being $\omega$-strongly measurable in $\HOD$?
\end{question}
\section{Many $\omega$-strongly measurable cardinals}{\langle}anglebel{Sec-Many}
Suppose that $\mathcal{U}$ is a coherent sequence of normal measures so that ${\langle}anglembda < \kappa$ are regular cardinals and $o^{\mathcal{U}}(\kappa) = {\langle}anglembda + 1$ and that the first measure in $\mathcal{U}$ is on a cardinal strictly greater than ${\langle}anglembda$.
Let $\mathbb{P}^{\mathcal{U}}$ be the non-stationary support iteration of Prikry/Magidor forcing from \cite{bunger}, and $\mathbb{C}_{\mathcal{F}^{\mathcal{U}}_\kappa}$ be the $\mathbb{P}^{\mathcal{U}}* \mathbb{C}l({\langle}anglembda,<\kappa)$-name of the associated
diagonalizing club forcing for the filter $\mathcal{F}^{\mathcal{U}}_\kappa = \bigcap_{\tau {\langle}eq {\langle}anglembda} U_{\kappa,{\langle}anglembda}$ on $\kappa$.
In the next section we construct a $\mathbb{P}^{\mathcal{U}} * \mathbb{C}l({\langle}anglembda,<\kappa)$-name of a Prikry-type forcing notion $b^*r{\mathbb{C}}_{\mathcal{F}^{\mathcal{U}}_\kappa}$, which is equivalent to $\mathbb{C}_{\mathcal{F}_\kappa}$, and its direct extension order is ${\langle}anglembda$-closed. We will use that as a black box in this section.
\begin{definition}
Denote the post $\mathbb{P}^{\mathcal{U}} * \mathbb{C}l({\langle}anglembda,<\kappa) * b^*r{\mathbb{C}}_{\mathcal{F}_\kappa}$ by $\mathbb{Q}[\mathcal{U}]$.
\end{definition}
We have shown in the previous section that $\mathbb{Q}[\mathcal{U}]$ is cone homogeneous and equivalent as a forcing notion to
the iteration $\mathbb{P}^{\mathcal{U}} \ast \mathbb{C}l({\langle}anglembda,<\kappa) \ast \mathbb{C}_{\mathcal{F}_\kappa}$.
By theorem {\rangle}ef{Thm-1stmeas} we conclude that $\kappa$ is strongly measurable in the generic extension by $\mathbb{Q}[\mathcal{U}]$. \vskip
amount
In what follows, we would like to view $\mathbb{Q}[\mathcal{U}]$ as a Prikry-type forcing whose direct extension order is ${\langle}anglembda$-closed.
This is easily possible since $\mathbb{Q}[\mathcal{U}]$ is an iteration of three posets, each of which can be seen as a Prikry-forcing whose direct extension order is ${\langle}anglembda$-closed (for $\mathbb{C}l({\langle}anglembda,<\kappa)$ we identify the direct extension order with the standard order of the poset). \vskip
amount
We finally turn to prove our main result.
\begin{proof}(Theorem {\rangle}ef{THM1})\vskip
amount
To simplify our arguments, we work over a minimal Mitchell model $V = L[\mathcal{U}]$ with a coherent sequence of measures $\mathcal{U}$ witnessing the assumed large cardinal assumption. Therefore
$\theta$ is the least inaccessible cardinal in $V$ for which $\{ o(\kappa) \mid \kappa < \theta\}$ is unbounded in $\theta$.
We note that all normal measures in this model appear on the main sequence $\mathcal{U}$, in particular, $o(\kappa) = o^{\mathcal{U}}(\kappa)$ for all $\kappa$.
We also record here that by Mitchell Covering Theorem and the fact $\theta$ is not measurable, there is no generic extension of $V = L[\mathcal{U}]$ which preserves the cardinals below $\theta$ and changes the cofinality of $\theta$.
Similarly, the Mitchell Covering Theorem guarantees that generic extensions of $V = L[\mathcal{U}]$ satisfy the Weak Covering Lemma with respect to $V$, which implies that successors of singular cardinals cannot be collapsed.
Let ${\langle}anglengle \kappa_\alpha \mid \alpha < \theta{\rangle}anglengle$ be an increasing sequence of cardinals below $\theta$, which satisfies the following conditions:
\begin{enumerate}
\item $\kappa_0 = \omega$, $\kappa_1$ is the least measurable,
\item for a limit ordinal $\alpha$, $\kappa_\alpha = {\langle}eft(\sup_{\beta < \alpha} \kappa_{\beta}{\rangle}ight)^+$,
\item for a successor ordinal $\alpha + 1$ let $\kappa_{\alpha + 1}$ is the least cardinal such that $o^{\mathcal{U^\alpha}}(\kappa_{\alpha + 1}) = \kappa_{\alpha} + 1$, for the coherent sequence of measures $\mathcal{U}^\alpha = \mathcal{U}restriction_{(\kappa_\alpha,\kappa_{\alpha+1}]}$. In particular, the first measure of the sequence $\mathcal{U}^\alpha$ has critical point $> \kappa_\alpha$.
\end{enumerate}
We define by induction on $\alpha < \theta$ a Magidor iteration $\mathbb{P} = {\langle} \mathbb{P}_\alpha, \mathbb{Q}_\alpha \mid \alpha < \theta{\rangle}$ of Prikry type forcings. Our description of the Magidor style iteration follows Gitik's handbook chapter, \cite{Gitik-HB}. We recall that conditions are sequences of the form ${\langle}anglengle q_\alpha \mid \alpha < \theta{\rangle}anglengle$ where only finitely many coordinates are not a direct extension of the weakest condition $0_{\mathbb{Q}_\alpha}$.
Let $\mathbb{Q}_{0}$ be $\mathbb{C}l(\omega, <\kappa_1) * \mathbb{C}^*_{\mathcal{F}_{\kappa_1}}$, where $\mathcal{F}_{\kappa_1}$ is the filter generated from the normal measure on $\kappa_1$. For $\alpha > 0$, we define $\mathbb{Q}_{\alpha} = \mathbb{Q}[\mathcal{U}^\alpha]$.
The coherent sequence $\mathcal{U}^\alpha$ from $L[\mathcal{U}]$ uniquely extends in a generic extension by $\mathbb{P}_\alpha$, and can therefore be used to force with $\mathbb{Q}[\mathcal{U}^\alpha]$. This is because as $L[\mathcal{U}]$ satisfies the $\GCH$, we have that $|\mathbb{P}_\alpha| {\langle}eq \kappa_\alpha$ and all measures of $\mathcal{U}^\alpha$ are assumed to have critical points strictly above $\kappa_\alpha$.
It is clear from our definitions that $\mathbb{Q}_\alpha$ satisfies the Prikry Property, that its direct extension order is $\kappa_\alpha$-closed, and that $\mathbb{Q}_\alpha$ is forced to be cone homogeneous.
By the general theory of Magidor iteration of Prikry type posets, the iteration $\mathbb{P}_{\theta}/ \mathbb{P}_1$ also satisfy the Prikry Property. Moreover, for every $\alpha < \theta$, $\mathbb{P}_{\theta} / \mathbb{P}_\alpha$ has the Prikry Property in the generic extension by $\mathbb{P}_{\alpha}$, and its direct extension order is $\kappa_\alpha$-closed (see \cite{Gitik-HB} for details).
\begin{claim}
Every bounded subset of $\kappa_{\alpha}$ is introduced by $\mathbb{P}_{\alpha}$. Moreover, in the generic extension by $\mathbb{P}_{\theta}$, $\kappa_{\alpha}$ is a regular cardinal for all $\alpha < \theta$.
\end{claim}
\begin{proof}
The first assertion is an immediate consequence of the fact $\mathbb{P}_\theta/ \mathbb{P}_\alpha$ satisfies the Prikry Property and its direct extension order is $\kappa_\alpha$ closed.
It follows that in order to show that all cardinal $\kappa_\alpha$ remain regular in a generic extension by $\mathbb{P}_\theta$, it suffices to show that
$\kappa_\alpha$ remains regular in the intermediate generic extension by $\mathbb{P}_\alpha$. We prove the last assertion by induction on $\alpha < \theta$.
For a limit ordinal $\alpha$, the assertion follows from the fact that the generic extension by $\mathbb{P}_\alpha$ satisfies the Weak Covering property with respect to the ground model $V = L[\mathcal{U}]$. Indeed, $\kappa_{\alpha} = {\langle}eft(\sup_{\beta < \alpha} \kappa_{\beta}{\rangle}ight)^+$ cannot be collapsed without collapsing a tail of the cardinals $\kappa_\beta$, $\beta < \alpha$, which would contradict our inductive assumption.
Suppose now that $\alpha$ is a successor ordinal. Then the forcing $\mathbb{P}_{\alpha}$ naturally breaks into two parts $\mathbb{P}_\alpha \mathbb{C}ng \mathbb{P}_{\alpha - 1} * \mathbb{Q}_{\alpha-1}$. The size of $\mathbb{P}_{\alpha - 1}$ is ${\langle}eft(2^{\kappa_{\alpha-1}}{\rangle}ight)^V < \kappa_\alpha$, and cannot singularize $\kappa_{\alpha}$. The second poset $\mathbb{Q}_{\alpha-1}$ does not collapse $\kappa_{\alpha}$ by Proposition {\rangle}ef{prop1}. Note that in order to apply the result of Proposition {\rangle}ef{prop1} we use our inductive hypothesis that $\kappa_{\alpha-1}$ remain regular in a generic extension by $\mathbb{P}_{\alpha-1}$.
\end{proof}
\begin{claim}
In the generic extension $\theta$ is regular.
\end{claim}
\begin{proof}
This follows from the Mitchell Covering Theorem and the smallness assumption of $\theta$, as was mentioned at the beginning of the proof.
\end{proof}
\begin{claim}
$\mathbb{P}$ is cone homogeneous.
\end{claim}
\begin{proof}
It suffices to verify conditions (i),(ii) of Lemma {\rangle}ef{FACT:homogiter} hold for every $\alpha < \kappa$.
(i) holds for $\mathbb{Q}_\alpha = \mathbb{Q}[\mathcal{U}restriction_{(\kappa_\alpha,\kappa_{\alpha+1}]}]$ since $\mathbb{Q}_\alpha,{\langle}eq_{\mathbb{Q}_\alpha},{\langle}eq^*_{\mathbb{Q}_\alpha}$ are clearly definable in $V = L[\mathcal{U}]$ from $\mathcal{U}$, $\kappa_\alpha$, and $\kappa_{\alpha+1}$.
The fact $(\mathbb{Q}_\alpha,{\langle}eq_{\mathbb{Q}_\alpha},{\langle}eq^*_{\mathbb{Q}_\alpha})$ is an immediate consequence of Lemma {\rangle}ef{Lem:Q[U].weakly.homogeneous}.
\end{proof}
Let $G_\theta \subseteq \mathbb{P}_\theta$ be a generic filter over $V$. We conclude that $\HOD^{V[G_\theta]} \subseteq V$.\footnote{As a matter of fact $\HOD^{V[G_\theta]} = V$ since $V = L[\mathcal{U}]$ is ordinal definable in $V[G_\theta]$. We will not used this fact here.} Moreover, for each $\alpha < \kappa$, the $\mathbb{Q}[\mathcal{U}^\alpha]$ generic filter induced by $G_\theta$ guarantees that $\kappa_{\alpha+1} = (\kappa_\alpha^+)^{V[G_\theta]}$ and that $\kappa_\alpha$ is strongly measurable in $\HOD^{V[G_\theta]}$.
It follows that all successors of regular cardinals below $\theta$ in $V[G_\theta]$ are strongly measurable in $\HOD$. Since $\theta$ remains strongly inaccessible in $V[G_\theta]$ and all the relevant witnessing objects clearly belong to $V_\theta^{V[G_\theta]}$, we conclude that
in $V_\theta^{V[G_\theta]}$, all successors of regular cardinals are strongly measurable.
\end{proof}
\section{Embedding $\mathbb{C}_{\mathcal{F}_\kappa}$ in suitable Prikry-type forcings}{\langle}anglebel{Sec-PrikryEmbedding}
The method of the previous section can be iterated finitely many times in order to get finitely many successive $\omega$-strongly measurable cardinals. In order to get a global result (or even just infinitely many $\omega$-strongly measurables) we need to have a preservation of distributivity under iterations.
This is, in general, a difficult task. One way to obtain this is by shifting our goal from preserving distributivity into preserving the Prikry Property. There are several ways to iterate Prikry type forcings and preserve the Prikry Property as well as the closure properties of the direct extension. Thus, embedding the distributive forcings into a Prikry type forcing can be used in order to get a suitable distributivity of the iteration. Usually, in order to achieve this, some strong compactness assumption is made that enables one to embed any sufficiently distributive forcing into a Prikry type forcing. See \cite{Gitik-Compact, BenHamouHayutGitik}, for some examples for the consistency strength of such constructions.
Our goal is to embed $\mathbb{C}_{\mathcal{F}_\kappa}$ into a Prikry type forcing without increasing our large cardinal hypothesis from $o(\kappa)= {\langle}anglembda+1$. For this, our approach follows the finer technique, introduced by Gitik in \cite{Gitik-ClubOnRegs}.
This section is devoted to prove the following technical lemma:
\begin{proposition}{\langle}anglebel{proposition:embedding-C-into-prikry-type}
Let us assume that $\mathcal{U}$ is a coherent measure sequence witnessing $o(\kappa) = {\langle}anglembda+1$. Let $\mathbb{P}^\mathcal{U}$ be the non-stationary support iteration of Subsection {\rangle}ef{ssec-non-stationary-iteration}. Then in $\mathbb{P}^{\mathcal{U}} \ast \mathbb{C}l({\langle}anglembda,<\kappa)$ there is a Prikry-type forcing notion $b^*r{\mathbb{C}}_{\mathcal{F}_\kappa}$, whose direct extension order is ${\langle}anglembda$-closed and it has a dense subset isomorphic to $\mathbb{C}_{\mathcal{F}_\kappa}$.
Moreover, both orders ${\langle}eq$ and ${\langle}eq^*$ witness the forcing $\mathbb{P}^{\mathcal{U}} \ast \mathbb{C}l({\langle}anglembda,<\kappa) \ast b^*r{\mathbb{C}}_{\mathcal{F}_\kappa}$ to be cone homogeneous.
\end{proposition}
The proof of the first part of the proposition is given in Corollary {\rangle}ef{cor:bar-C-prikry-type} and the proof of the moreover part appears in Lemma {\rangle}ef{Lem:Q[U].weakly.homogeneous}.
Let us sketch the main ideas behind to proof of the proposition. In order to construct a Prikry type forcing that projects onto $\mathbb{C}_{\mathcal{F}_\kappa}$ we first work in the generic extension in which $\kappa$ is singularized to be of cofinality ${\langle}anglembda$. In this model, the Magidor sequence is already a closed unbounded set that diagonalizes the filter $\mathcal{F}_\kappa$, so we can use it as a guide to the generic of $\mathbb{C}_{\mathcal{F}_\kappa}$. This means that there is a projection from the generic extension by the singularizing forcing iterated by a ${\langle}anglembda$-closed forcing onto $\mathbb{C}_{\mathcal{F}_\kappa}$. $b^*r{\mathbb{C}}_{\mathcal{F}_\kappa}$ is obtained by "forgetting" the Magidor sequence and keeping the diagonalizing club. A technical issue that arise when trying to pull up this strategy is that the singularizing forcing must be defined \emph{after} the cardinals between ${\langle}anglembda$ and $\kappa$ were collapsed, and a major part of this section is devoted to developing this forcing.
The rest of this section is organized as follows: in subsection {\rangle}ef{subsection:q-star} we review the basic construction and properties of the tree Prikry-type forcings $\mathbb{Q}^\tau_\kappa$, $\tau {\langle}eq o^{\mathcal{U}}(\kappa)$ which is defined in the generic extension by $\mathbb{P}$. Then, we introduce a filter-based variant $\mathbb{Q}_{\kappa,\tau}^*$ to be forced over a generic extension $V[G \ast H]$ by $\mathbb{P} \ast \mathbb{C}l({\langle}anglembda,<\kappa)$ extension $V[G \ast H]$ of $V$, where $\kappa = {\langle}anglembda^+$ is no longer measurable.
In subsection {\rangle}ef{subsection:c-star}, we use the posets $\mathbb{Q}_{\kappa,\tau}^*$, $\tau {\langle}eq {\langle}anglembda$ in order to introduce a forcing equivalent
$b^*r{\mathbb{C}}_{\mathcal{F}_\kappa}$ of $\mathbb{C}_{\mathcal{F}_\kappa}$ with a dense Prikry-type sub-forcing $\mathbb{C}^*_{\mathcal{F}_\kappa}$ whose direct extension order is ${\langle}anglembda$-closed.
This completes the proof of Section {\rangle}ef{Sec-Many}, as the posets $b^*r{\mathbb{C}}_{\mathcal{F}_\kappa}$ can be iterated on different cardinals to construct models with many $\omega$-strongly measurable cardinals.
\subsection{The forcing $\mathbb{Q}^*_{\kappa,\tau}$}{\langle}anglebel{subsection:q-star}
We turn back to consider our forcing scenario with $\mathcal{F}_{\kappa}$ over $V[G\ast H]$, where $G \subseteq \mathbb{P}$ is generic over $V$, and $H \subseteq \mathbb{C}l({\langle}anglembda,<\kappa)$ is generic over
$V[G]$. Recall that $o^{\mathcal{U}}(\kappa) = {\langle}anglembda+1$ and that each measure $U_{\kappa,\tau}$, $\tau {\langle}eq {\langle}anglembda$ in $V$, extends in $V[G]$ to $U_{\kappa,\tau}(t)$, where $t$ is $\tau$-coherent.
For each such $\tau {\langle}eq {\langle}anglembda$ and $t$, let $j_{\kappa,\tau,t}\mathbb{C}lon V[G] \to M_{\kappa,\tau,t}$ be the ultrapower embedding of $V[G]$ by $U_{\kappa,\tau}(t)$.
Moving to the further generic extension $V[G \ast H]$ of $V[G]$, $\kappa$ is no longer measurable. Let $F_{\kappa,\tau}(t)$ denote the filter generated by
$U_{\kappa,\tau}(t)$ on $\mathcal{P}(\kappa)^{V[G]}$ and $F_{\kappa,\tau}(t)^+$ denote the poset on $\mathcal{P}(\kappa)^{V[G]}$ of $F_{\kappa,\tau}(t)$ positive sets where a set $A$ is stronger than
$B$ if $A \setminus B$ belongs to the dual ideal of $F_{\kappa,\tau}(t)$.
By further forcing with the collapse quotient \[\mathbb{R}_{\tau} = \mathbb{C}l({\langle}anglembda,<j_{\kappa,\tau,t}(\kappa))/H \mathbb{C}ng \mathbb{C}l({\langle}anglembda, [\kappa, j_{\kappa,\tau,t}(\kappa)),\] over $V[G \ast H]$, producing a generic filter $H_{\tau}^* \subseteq \mathbb{C}l({\langle}anglembda,<j_{\kappa,\tau,t}(\kappa))$, with $H_{\tau}^* restriction \mathbb{C}l({\langle}anglembda,<\kappa) = H$, the elementary embedding $j_{\kappa,\tau,t}$ extends into
\[j^*_{\kappa,\tau,t} \mathbb{C}lon V[G \ast H] \to M_{\kappa,\tau,t}[H_\tau^*].\]
In turn, the embedding $j^*_{\kappa,\tau,t}$ generates a $V[G \ast H]$ ultrafilter $U_{\kappa,\tau}(t)^* \subseteq F_{\kappa,\tau}(t)^+$, which is an $F_{\kappa,\tau}(t)^+$-generic ultrafilter over $V[G \ast H]$, by standard arguments connecting forcing with positive sets and generic ultrapowers.\footnote{Indeed, one can verify that the trivial condition in $\mathbb{C}l({\langle}anglembda, [\kappa, j_{\kappa,\tau,t}(\kappa))$ forces $\kappa \in j(\dot{X})$ if and only if there is a subset of $\dot{X}$ in $U_{\kappa,\tau}(t)$.}
Since the poset $\mathbb{R}_{\tau} = \mathbb{C}l({\langle}anglembda,<j_{\kappa,\tau,t}(\kappa))/H$ is ${\langle}anglembda$-closed in $V[G \ast H]$, we have that $F_{\kappa,\tau}(t)^+$ has a ${\langle}anglembda$-closed dense sub-forcing $D_{\kappa,\tau,t}$. Other examples of applications of Prikry-type forcings generated by ideals can be found in \cite{MSI} and \cite{MSLB}.
It would be useful for our purposes to work with a concrete description of the sets in $D_{\kappa,\tau,t}$.
We proceed to introduce the relevant notions.
\begin{definition}
Let $G \subseteq \mathbb{P}$ be a generic filter over $V$. For each cardinal $\nu < \kappa$ with $o^{\mathcal{U}}(\nu) > 0$, let $b_\nu$ be the $G$-induced generic cofinal sequence in $\nu$.
\begin{enumerate}
\item Recall that every finite coherent sequence $t = {\langle} \nu_0,\dots, \nu_{k-1}{\rangle} \in [\kappa]^{<\omega}$ in $V[G]$, has an assigned closed unbounded set $b_t = \cup_{i<k} (b_{\nu_i} \cup \{\nu_i\})$.
For a coherent sequence $t$ and a finite set of ordinals $s \in [\min(t)]^{<\omega}$, we define
$\pi^s(t) = \min(b_t \setminus (\max(s)+1))$.
When $t = {\langle} \nu{\rangle}$ has a single element, we will often abuse this definition and write $\pi^s(\nu)$ for $\pi^s({\langle} \nu {\rangle})$. \vskip
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For every $\eta$, the function
\[\pi^s_{\eta}(\nu) = \min( \{ \mu \in b_\nu \setminus (\max(s)+1) \mid o^{\mathcal{U}}(\mu) = \eta\})\]
defines a Rudin-Keisler projection from
$U_{\kappa,\tau}(s)$ to $U_{\kappa,\eta}(s)$, for all $ \tau > \eta$. In particular $\pi^s = \pi^s_0 \mathbb{C}lon \kappa \to \kappa$ is a Rudin-Keisler projection of $U_{\kappa,\tau}(s)$ to its normal projected measure
$U_{\kappa,0}(s)$, for every $\tau \geq 0$. \vskip
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\item Let $T \subseteq [\kappa]^{<\omega}$ a tree, $t \in [\kappa]^{<\omega}$, and $Q\mathbb{C}lon T \to \mathbb{C}l({\langle}anglembda,<\kappa)$ be a function.
We say that $Q$ is \textbf{$(T,t)$-suitable} if for every
$s \in T$ we have
\begin{itemize}
\item $Q(s) \in \mathbb{C}l({\langle}anglembda,<\kappa)$, and
\item for every $s' \in T$ that extends $s$, $Q(s')restriction {\langle}anglembda \times \pi_0^{t {}^\frown s}(s') = Q(s)$.
\end{itemize}
For every $s \in T$ we define $Q_s$ to be the induced function on $T_s = \{ r \in [\kappa]^{<\omega} \mid s {}^\frown r \in T\}$, given by
\[Q_s(r) = Q(s {}^\frown r).\]
\item Suppose that $H \subseteq \mathbb{C}l({\langle}anglembda,<\kappa)$ is generic over $V[G]$,
$T,Q \in V[G]$ as above, and let $A = \suc_T(\emptyset)$. We define in $V[G][H]$ the set $Q$-generic restriction of $A$ with respect to $H$, to be the set
\[A^H_Q = \{ \nu \in A \mid Q(\nu) \in H\}.\]
\end{enumerate}
\end{definition}
Let $G \subseteq \mathbb{P}$ be generic over $V$ and $H \subseteq \mathbb{C}l({\langle}anglembda,<\kappa)$ be a generic over $V[G]$.
In $V[G]$, for all $\tau {\langle}eq {\langle}anglembda$ and $t \in [\kappa]^{<\omega}$,
the function $\pi^t = \pi^t_0$ represents $\kappa$ in the ultrapower by $U_{\kappa,\tau}(t)$.
It is therefore immediate from our definition of $D_{\kappa,\tau,t} \subseteq F_{\kappa,\tau}(t)^+$ that sets in $D_{\kappa,\tau,t}$ are of the form
as $A^H_{r} = \{ \nu \in A \mid r(\nu) \in H\}$ where
$A \in U_{\kappa,\tau}(t)$, and $r\mathbb{C}lon A \to \mathbb{C}l({\langle}anglembda,<\kappa)$
satisfying $r(\nu) \in \mathbb{C}l({\langle}anglembda,[\pi^t(\nu),\kappa))$ for all $\nu \in A$.\footnote{i.e., $\dom(r(\nu)) \subseteq {\langle}anglembda \times (\kappa \setminus \pi^t(\nu))$.}
We use these facts to introduce a variant of Gitik's forcing $\mathbb{Q}_{\kappa,\tau}$ in $V[G \ast H]$. The following poset $\mathbb{Q}^*_{\kappa,\tau}$ collapses cardinals up to $\kappa^+$ to ${\langle}anglembda$ and adds a cofinal Magidor sequence $b_{\kappa}$ of length $\omega^{\tau}$ to $\kappa$, which diagonalizes the filter $\bigcap_{\tau' < \tau} U_{\kappa,\tau'}$ (i.e., $b_\kappa$ is almost contained in each filter set).
\begin{definition}{\langle}anglebel{def:Q*poset}
In $V[G \ast H]$ the forcing $\mathbb{Q}^*_{\kappa,\tau}$ consists of all $(t, T, Q) \in V[G]$ such that:
\begin{enumerate}
\item $t$ is a $\tau$-coherent finite sequence of ordinals below $\kappa$,
\item $T$ is a tree of $\tau$-coherent finite sequences with stem $t$,
\item $Q$ is a $(T,t)$-suitable function,
\item $Q(\emptyset) \in H$, and
\item{\langle}anglebel{condition:agreement} For every $s, s' \in T$, if $b_{t {}^\frown s} = b_{t {}^\frown s'}$ then $Q(s) = Q(s')$.
\end{enumerate}
\end{definition}
As in $\mathbb{Q}_{\kappa,\tau}$, we identify two condition $(t,T,Q),(t',T,Q) \in \mathbb{Q}_{\kappa,\tau}^*$, whenever $b_t= b_{t'}$.
The \textbf{direct extension} ordering of $\mathbb{Q}^*_{\kappa,\tau}$ naturally extends the direct extension ordering of $\mathbb{Q}_{\kappa,\tau}$. Namely, for two conditions $(t,T,Q)$, $(t^*,T^*,Q^*)$ of $\mathbb{Q}^*_{\kappa,\tau}$, we have $(t,T,Q) {\langle}eq^* (t^*,T^*,Q^*)$ if $(t,T){\langle}eq^*_{\mathbb{Q}_{\kappa,\tau}} (t^*,T^*)$ and $Q^*(s) \geq Q(s)$ for every $s \in T^*$. \vskip
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We observe that the direct extension order ${\langle}eq^*$ is ${\langle}anglembda$-closed in $V[G\ast H]$. For this, note that it is immediate from the definition above that
the partial order $\tilde{\langle}eq\in V[G]$, obtained from ${\langle}eq^*$ by removing the requirement $Q(\emptyset)\in H$, belongs to $V[G]$ and is clearly ${\langle}anglembda$-closed in both $V[G]$ and $V[G\ast H]$ (note that the two generic extensions agree on sequences of length $<{\langle}anglembda$).
Then, as ${\langle}eq^*$ is equivalent to the restriction of $\tilde{{\langle}eq}$ to a ${\langle}anglembda$-closed set (which is essentially $H$), it remains ${\langle}anglembda$-closed in $V[G\ast H]$.\vskip
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The \textbf{end-extension} ordering of $\mathbb{Q}^*_{\kappa,\tau}$ is based the restriction of the end-extension of $\mathbb{Q}_{\kappa,\tau}$ to the $Q$-generic restriction of $T$ with respect to $H$. Namely, for a condition $p = (t, T, Q)$, \textbf{the only values $\nu \in \suc_T(\emptyset)$ which are allowed to used when taking a one-point extension, are $\nu \in \suc_T(\emptyset)^H_Q$.}\footnote{I.e., $\nu \in A^H_Q$ for $A=\suc_T(\emptyset)$.} In this case, the resulting one-point extension is defined to be $p {}^\frown {\langle} \nu{\rangle} = (t \cup \{\nu\}, T_{{\langle} \nu {\rangle}}, Q_{{\langle} \nu {\rangle}})$. In general, for a sequence $r = {\langle} \nu_0,\dots, \nu_{k-1} {\rangle} \in T$, the end extension of $p$ by $r$, denoted $p {}^\frown r$, is the one obtained by taking a sequence of one-point extensions by $\nu_0,\dots,\nu_{k-1}$, in turn.
\vskip
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We note that although $\mathbb{Q}_{\kappa,\tau}^*$ depends on the collapse generic $H$, and is fully defined only in $V[G\ast H]$, we still have that $\mathbb{Q}_{\kappa,\tau}^* \subseteq V[G]$. Moreover, dropping the requirement $Q(\emptyset) \in H$ in the definition of conditions $p =(t,T,Q) \in \mathbb{Q}_{\kappa,\tau}^*$ allows us to examine conditions $(t,T,Q)$ and evaluate possible direct extensions and one-point extensions in $V[G]$.
For example, working in $V[G]$, we can consider possible one-point extensions $p {}^\frown {\langle} \nu {\rangle}$ of a condition $p= (t,T,Q)$ by an arbitrary $\nu \in \suc_T(\emptyset)$. Although eventually, in $V[G\ast H]$, $p$ will be a valid condition only if $Q(\emptyset) \in H$, and $p {}^\frown {\langle} \nu {\rangle}$ will form valid extensions of $p$ only for a $D_{\kappa,\tau,t}$-positive set of ordinals $\nu \in \suc_T(\emptyset)$, it is still possible to decide certain properties of such extensions on a measure one set of $U_{\kappa,\tau}(t)$ in $V[G]$.
This approach of arguing from $V[G]$ about the poset $\mathbb{Q}_{\kappa,\tau}^*$ in $V[G\ast H]$ plays a significant role in our proof below, showing that $\mathbb{Q}_{\kappa,\tau}^*$ satisfies the Prikry Property.
\begin{remark}
The forcing $\mathbb{C}l(\kappa < {\langle}anglembda) * \mathbb{Q}^*_{\kappa, \tau}$ is isomorphic to the collection of all $(t, T, Q)$ that satisfy all requirements of Definition {\rangle}ef{def:Q*poset} expect the fourth one, $Q(\emptyset) \in H$. Nevertheless, the decomposition into the collapse part and the singularization part would be more appropriate for our construction, as eventually $\mathbb{Q}^*_{\kappa,\tau}$ is used as merely an auxiliary forcing.
\end{remark}
The coherency requirements in definition {\rangle}ef{def:Q*poset} allow us to obtain a natural amalgamation property, similar to the one satisfied by the $V[G]$ poset $\mathbb{Q}_{\kappa,\tau}$.
\begin{lemma}{\langle}anglebel{Lem.amalgamation.in.Q*}
Work in $V[G]$. Let $p = (t,T,Q)$, forced to be a condition in $\mathbb{Q}^*_{\kappa,\tau}$ by $q = Q(\emptyset)$, and $A = \suc_T(\emptyset)$.
For each $\eta < \tau$,
denote \[A(\eta) = A \cap \{ \nu \in A \mid o^{\mathcal{U}}(\nu) = \eta\} \in U_{\kappa,\eta}(t) .\]
Suppose that there are $b^*r{\eta} < \tau$, a set $A'(b^*r{\eta}) \subseteq A(b^*r{\eta})$ with $A'(b^*r{\eta}) \in U_{\kappa,b^*r{\eta}}(t)$, and a sequence of conditions
${\langle} (t\cup\{\nu\} , T^\nu,Q^\nu) \mid \nu \in A'(b^*r{\eta}) {\rangle}$, such that
$(t\cup\{\nu\} , T^\nu,Q^\nu)$ is a direct extension of $p {}^\frown {\langle} \nu {\rangle}$ for all $\nu \in A'(b^*r{\eta})$.
Then there exists a direct extension $p^* \geq^* p$, $p^* = (t,T^*,Q^*)$,
such that the set $\{ (t{}^\frown {\langle} \nu {\rangle},T^\nu,Q^\nu) \mid \nu \in A'(b^*r{\eta}) \}$ is forced by $Q^*(\emptyset)$ to be predense above $p^*$.
\end{lemma}
\begin{remark}{\langle}anglebel{RMK:Qconstruction}
In the proof of the lemma we make use of several construction arguments involving trees $T$ associated to conditions $(t, T)$ in the poset $\mathbb{Q}_{\kappa,\tau}$
from \cite{gitik-nonstionary-ideal}. We list these arguments and refer the reader to \cite{gitik-nonstionary-ideal} for proofs. \vskip
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For a finite sequence $s$, we write $o(s) = \max(\{o(\nu) \mid \nu \in s\} )$.\vskip
amount
\begin{enumerate}
\item Suppose that $(t,T)$ is a condition of $\mathbb{Q}_{\kappa,\tau}$ and $A'(\eta) \subseteq \suc_T(\emptyset)$ belongs to
$U_{\kappa,\eta}(t)$ for some $\eta < \tau$. Then there exists a sub-tree $T'$ of $T$, so that $(t,T') \in \mathbb{Q}_{\kappa,\tau}$ is a direct extension of
$(t,T)$, and
\[ \{ \nu \in \suc_{T'}(\emptyset) \mid o(\nu) = \eta\} \subseteq A'(\eta) .\]
Similarly, for every $s \in T$ and $A'_s(\eta) \subseteq \suc_T(s)$ which
belongs to
$U_{\kappa,\eta}(t{}^\frown s)$ there is a direct extension $(t,T')$ of $(t,T)$,
which only requires shrinking the tree $T$ above $s$ (i.e., shrinking $T_s$) and in particular $s\in T'$,
so that $\{ \nu \in \suc_{T'}(s) \mid o(\nu) = \eta\} \subseteq A'_s(\eta)$. \vskip
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Furthermore, this construction can be naturally combined over different values $s \in T$. Namely, given a family $\{ A'_s(\eta) \mid s \in T \}$ of sets as above we can apply the same procedure, level by level, to the tree $T$, and obtain a sub-tree $T' \subseteq T$ with the property that
$(t,T') \in \mathbb{Q}_{\kappa,\tau}$ and for every $s \in T'$,
\[ \{ \nu \in \suc_{T'}(s) \mid o(\nu) = \eta\} \subseteq A'_s(\eta). \]
\item For a condition $(t,T)$, $s \in T$, and $\eta < \tau$, there exists a direct extension $(t,T') \geq^* (t,T)$, which only requires shrinking the tree $T$ above $s$ such that
for all $s' \in T$ which end extends $s$, if there exists $\nu \in b_{s'}\setminus b_s$ such that $o(\nu) = \eta$,\footnote{this is equivalent to the existence of $\mu \in s'\setminus s$ such that $o(\mu) \geq \eta$.} then $\nu' = \pi^{t {}^\frown s}_{\eta}(s')$ (the minimal such $\nu$) belongs to $\suc_{T'}(s)$.
Repeating this construction, level by level, produces a direct extension $(t,T')$ of $(t,T)$ satisfying that for every $s \in T'$ and $s' \in T'$ which extends $s$, $\nu' = \pi_\eta^{t {}^\frown s}(s') \in \suc_{T'}(s) \cap \{ \nu < \kappa \mid o(\nu) = \eta\}$. \vskip
amount
We note that if $s \in T'$ satisfies that $o(\mu) < \eta$ for all $\mu \in s$, then for every $\mu \in \suc_{T'}(s)$ with $o(\mu) \geq \eta$, we have $\pi^{t{}^\frown s}_\eta(\mu) = \pi^t_\eta(s {}^\frown {\langle}\mu{\rangle})$, which by our assumption of $T'$ (applied to $s' = s {}^\frown {\langle} \mu{\rangle}$), implies that $\pi^t_\eta(s {}^\frown {\langle}\mu{\rangle}) \in \suc_{T'}(\emptyset)$. It follows that \[\suc_{T'}(s) \cap \{\nu \mid o(\nu) = \eta\} \subseteq \suc_{T'}(\emptyset) \cap \{\nu \mid o(\nu) = \eta\}.\]
Since the former set belongs to $U_{\kappa,\eta}(t {}^\frown s)$ we conclude that
\[\suc_{T'}(\emptyset) \cap \{\nu \mid o(\nu) = \eta\} \in U_{\kappa,\eta}(t {}^\frown s)\]
as well.
The same consideration applies to any $s \in T'$ and $s'\in T'$ which extends $s$, and for which $o(\nu) < \eta$ for every $\nu \in s'\setminus s$,
and implies that \[\suc_{T'}(s) \cap \{\nu \mid o(\nu) = \eta\} \in U_{\kappa,\eta}(t {}^\frown s').\] \vskip
amount
\item Let $(t,T')$ be a condition as in the previous clause.
There exists a direct extension $(t,T^*)$ of $(t,T')$ such that for every $s' \in T^*$ for which $\nu' = \pi^{t}_{\eta}(s') \in b_{s'}\setminus b_t$ is defined, not only that \[\nu' \in \suc_{T^*}(\emptyset) \cap \{\nu \mid o(\nu) = \eta\},\] but further, there is some $s'' \in T^*$ which extends ${\langle} \nu'{\rangle}$ such that $b_{t {}^\frown s''} = b_{t {}^\frown s'}$ and $T^*_{s'} \subseteq T^*_{s''}$. We note that it implies that the set \[\{ (t \cup \{ \nu \}, T^*_{{\langle}angle \nu {\rangle}angle}) \mid \nu \in \suc_{T^*}(\emptyset) , o(\nu) = \eta\}\] is predense above $(t,T^*)$.\vskip
amount
\noindent
\end{enumerate}
We turn to the proof of Lemma {\rangle}ef{Lem.amalgamation.in.Q*}.
\end{remark}
\begin{proof}(Lemma {\rangle}ef{Lem.amalgamation.in.Q*})\\
For $s \in T$ define $o(s) = \max(\{o(\nu) \mid \nu \in s\})$,
and for a tree $T \subseteq [\kappa]^{<\omega}$, and $\eta$, $T(<\eta) = \{ s \in T \mid o(s) < \eta\}$.
Let $p = (t,T,Q)$, $A'(b^*r{\eta})$, and ${\langle} (t\cup\{\nu\} , T^\nu,Q^\nu) \mid \nu \in A'(b^*r{\eta}) {\rangle}$, as in the statement of the Lemma.
By part (1) of Remark {\rangle}ef{RMK:Qconstruction} above, we may assume (by reducing to a suitable sub-tree) that $\suc_T(\emptyset) \cap \{ \nu \mid o(\nu) = b^*r{\eta}\}
\subseteq A'(b^*r{\eta})$.
Furthermore, by part (2) of the remark, we may further assume that $A'(b^*r{\eta}) \in U_{\kappa,b^*r{\eta}}(t {}^\frown s)$ for every $s \in T$ with $o(s) < b^*r{\eta}$.\vskip
amount
\noindent
Recall that for each sequence $s \in T$, $o(s) < b^*r{\eta}$, the function $\pi^{t {}^\frown s}(\nu)$ is a normal projection
of $U_{\kappa,b^*r{\eta}}(t {}^\frown s)$ to $U_{\kappa,0}(t {}^\frown s)$. Since $Q^\nu(\emptyset)restriction {\langle}anglembda \times \pi^{t {}^\frown s}(\nu)$ is bounded in $\pi^{t {}^\frown s}(\nu)$,\footnote{i.e., $Q^\nu(\emptyset)restriction {\langle}anglembda \times \pi^{t {}^\frown s}(\nu)\in V_{\pi^{t {}^\frown s}(\nu)}$ and $\pi^{t {}^\frown s}(\nu)$ is an inaccessible cardinal} we can press down on its value, and find a subset $A'_{s}(b^*r{\eta}) \in U_{\kappa,b^*r{\eta}}(t {}^\frown s)$ of $A'(b^*r{\eta})$, and a collapse condition $Q'(s)$ such that $Q^\nu(\emptyset)restriction \pi^{t {}^\frown s}(\nu) = Q'(s)$ for all $\nu \in A'_{s}(b^*r{\eta})$.\vskip
amount
\noindent
By applying the construction arguments of Remark {\rangle}ef{RMK:Qconstruction}, we may find direct extension $(t,T')$ of $(t,T)$ having both properties from parts (1), (2) of the remark,
where (1) is applied with respect to the sets $A'_s(b^*r{\eta})$, $s \in T'$, $o(s) < b^*r{\eta}$, given by the pressing down
process above, by which $Q'(s)$ is defined.
We note that, as mentioned at the end of part (2) of the remark, for every $s' \in T'(<b^*r{\eta}) = \{ s \in T' \mid o(s) < b^*r{\eta}\}$ which end extends $s$,
\[\suc_{T'}(s) \cap \{ \nu \mid o(\nu) = b^*r{\eta} \} \in U_{\kappa,b^*r{\eta}}(t {}^\frown s').\]
Moreover, since
\[\suc_{T'}(s) \cap \{ \nu \mid o(\nu) = b^*r{\eta}\} \subseteq A'_{s}(b^*r{\eta}),\]
we conclude that $A'_{s}(b^*r{\eta}) \in U_{\kappa,b^*r{\eta}}(s')$. It follows that
$A'_s(b^*r{\eta}) \cap A'_{s'}(b^*r{\eta}) \neq\emptyset$, which in turn, implies that
\[Q'(s) = Q'(s')restriction {\langle}anglembda \times \pi^{t {}^\frown s}(s')\]
(as witnessed by $Q^\nu(\emptyset)$ for any $\nu \in A'_s(b^*r{\eta}) \cap A'_{s'}(b^*r{\eta})$).\vskip
amount
Finally, we form a sub-tree $T''$ of $T'$ by intersecting $T'_{{\langle} \nu {\rangle}}$ with $T^\nu$, for each $\nu \in \suc_{T'}(\emptyset) \cap \{\nu \mid o(\nu) = b^*r{\eta}\} \subseteq A'(b^*r{\eta})$. By appealing to part (3) of the previous remark, we can find a direct extension $(t,T^*)$ of $(t,T'')$ which further satisfies that
for every $s \in T^*$ for which $\nu_{s} := \pi^{t}_{\eta}(s) \in b_{s}\setminus b_t$ is defined, $\nu_{s} \in \suc_{T^*}(\emptyset) \cap \{\nu \mid o(\nu) = \eta\}$ and there exists some $s' \in T^*$ which end extends ${\langle} \nu_s{\rangle}$, such that $b_{t {}^\frown s'} = b_{t {}^\frown s}$ and $T^*_{s} \subseteq T^*_{s'}$.
Let $b^*r{s} = s'\setminus \nu_s+1$.
$Q^{\nu_s}(b^*r{s})$ is defined, since $T^*_{{\langle} \nu_{s}{\rangle}} \subseteq T^{\nu_{s}} = \dom(Q^{\nu_{s}})$. Moreover,
since $(t\cup\{\nu_s\},T^{\nu_s},Q^{\nu_s})$ is assumed to be a condition in $\mathbb{Q}^*_{\kappa,\tau}$, the value
$Q^{\nu_s}(b^*r{s})$ does not depend on the choice of a sequence $s'$, and its associated sub-sequence $b^*r{s} \in T^{\nu_s}$ satisfying $b_{t {}^\frown{\langle} \nu_s{\rangle} {}^\frown b^*r{s}} = b_{t {}^\frown s'} = b_{t {}^\frown s}$. \vskip
amount
We turn to define the function $Q^*$ on $T^*$. We follow the convention from the last paragraph, where for $s \in T^*$ with $o(s) \geq b^*r{\eta}$, we denote $\nu_s = \pi^t_{b^*r{\eta}}(s)$. We set
\[
Q^*(s) =
\begin{cases}
Q^{\nu_s}(b^*r{s}) &\mbox{ if } o(s) \geq b^*r{\eta}, b^*r{s} \in T^{\nu_s} \text{ satisfies } b_{t {}^\frown {\langle} \nu_s {\rangle} {}^\frown b^*r{s}} = b_{t {}^\frown s}\\
Q'(s) &\mbox{ if } o(s) < b^*r{\eta}
\end{cases}
\]
We claim that $Q^*(\emptyset)$ forces that $(t,T^*,Q^*)$ is a condition that extends $(t,T,Q)$.
We show first that $Q^*(\emptyset)$ forces $(t,T^*,Q^*)$ is a condition of $\mathbb{Q}^*_{\kappa,\tau}$, which requires verifying the first three conditions
in the definition of the poset. Conditions (i), (ii) are clearly satisfied as $(t,T^*) \in \mathbb{Q}_{\kappa,\tau}$.
To verify condition (iii), we need to check that for every $s,s' \in T^*$, if $s'$ extends $s$ then $Q^*(s) = Q^*(s')restriction {\langle}anglembda \times \pi^{t {}^\frown s}(s')$.
The verification breaks down to {three cases}. \vskip
amount
\noindent
\textbf{Case I:} If $o(s),o(s') < b^*r{\eta}$, then $s,s' \in T^*(<b^*r{\eta})$ and, as described above, the result is an immediate consequence of the fact $s' \in T'(<b^*r{\eta})$ end extends $s$. \vskip
amount
\noindent
\textbf{Case II:}
If $o(s) < b^*r{\eta}$ and $o(s') \geq b^*r{\eta}$ then $\nu' = \pi^{t{}^\frown s}_{b^*r\eta}(s') \in A'_s(b^*r{\eta})$. As
$\pi^{t {}^\frown s}(s') = \pi^{t {}^\frown s}(\nu')$ and $(t \cup \{\nu'\}, T^\nu,Q^\nu) \in \mathbb{Q}^*_{\kappa,\tau}$,
it follows that
\[
\begin{matrix}
Q^*(s')restriction {\langle}anglembda \times \pi^{t {}^\frown s}(s') = & \\
Q^{\nu'}(b^*r{s'}) restriction {\langle}anglembda \times \pi^{t {}^\frown s}(s') = & \\
Q^{\nu'}(\emptyset) restriction {\langle}anglembda \times \pi^{t {}^\frown s}(\nu') = & Q'(s) = Q^*(s)
\end{matrix}\]
\noindent
\textbf{Case III:} If $o(s) \geq b^*r{\eta}$ then there exists some $b^*r{s} \in T^*_{\nu_{s}} \subseteq T^{\nu_s}$
such that $b_{t {}^\frown {\langle} \nu_s {\rangle} {}^\frown b^*r{s}} = b_{t {}^\frown s}$ and $T^*_s \subseteq T^*_{{\langle} \nu_s {\rangle} {}^\frown b^*r{s}}$. In particular, $Q^*(s) = Q^*(b^*r{s})$ and $s' \in T^*_{{\langle} \nu_s {\rangle} {}^\frown b^*r{s}} \subseteq T^{\nu_s}_{b^*r{s}}$ . Since $Q^{\nu_s}$ is $(t \cup \{\nu_s\}, T^{\nu_s})$-coherent, we conclude that
\[
\begin{matrix}
Q^*(s')restriction {\langle}anglembda \times \pi^{t {}^\frown s}(s') = & \\
Q^{\nu_s}(s') restriction {\langle}anglembda \times \pi^{t {}^\frown s}(s') = &
Q^{\nu_{s}}(b^*r{s}) = Q^*(s)
\end{matrix}
\]
This concludes the proof that $p^* = (t,T^*,Q^*)$ satisfies the property (iii) of the definition of $\mathbb{Q}^*_{\kappa,\tau}$, and thus, that $Q^*(\emptyset) \in \mathbb{C}l({\langle}anglembda,<\kappa)$ forces it is a condition of $\mathbb{Q}^*_{\kappa,\tau}$.
It is immediate from its definition that $p^*$ is a direct extension of $p$.
Finally, our choice of the tree $T^*$ above, obtain from $T'$ using fact (3) from Remark {\rangle}ef{RMK:Qconstruction} above, implies at once that
$Q^*(\emptyset)$ forces
$\{ (t{}^\frown {\langle} \nu {\rangle},T^\nu,Q^\nu) \mid \nu \in A'(b^*r{\eta}) \}$ to be predense above $p^*$.
\end{proof}
\begin{lemma}
$\mathbb{Q}^*_{\kappa,\tau}$ satisfies the Prikry Property.
\end{lemma}
\begin{proof}
Suppose otherwise. Working in $V[G \ast H]$, let $(t,T,Q)$, $\sigma$ be condition and statement of $\mathbb{Q}^*_{\kappa,\tau}$, respectively, such that no direct extension of $(t,T,Q)$ decides $\sigma$.
Back in $V[G]$, let $q \in H$ be a condition which forces this statement about $(t,T,Q)$ and $\sigma$.
Since $q$ forces $(t,T,Q)$ to be a condition of $\mathbb{Q}^*_{\kappa,\tau}$ we have that $q \geq Q(\emptyset)$. Therefore, by moving to a
direct extension of $(t,T,Q)$, we may assume that $q = Q(\emptyset)$.
For notational simplicity, we make the assumption that $t = \emptyset$. The proof for an arbitrary sequence $t$ is similar.
Let $A = \suc_T(\emptyset)$. We may assume that $q\in V_{\mu_0}$, where $\mu_0 = \min(\{ \pi_0^\emptyset(\nu) \mid \nu \in A\})$.
For each $\nu \in A$, we choose a condition $q(\nu) \in \mathbb{C}l({\langle}anglembda,<\kappa)$, extending $q \cup Q(\nu)$, which decides the $\mathbb{C}l({\langle}anglembda,<\kappa)$ statement of whether there exists a direct extension
$p^{\nu} = ({\langle} \nu {\rangle},T^\nu,Q^\nu)$
of $p {}^\frown {\langle} \nu {\rangle}$ which decides $\sigma$, and if so, whether $p^\nu$ forces $\sigma$ or $\neg\sigma$.
Let $A^0$ be the sets of $\nu \in A$ for which $q(\nu)$ forces $p^\nu$ exists, and ``$p^\nu \Vdash \sigma$''.
Similarly, let $A^1 \subseteq A$ consists of $\nu$ such that $q(\nu)$ forces $p^\nu$ exists, and ``$p^\nu \Vdash \neg\sigma$'',
and $A^2 = A \setminus (A^0 \uplus A^1)$. The proof splits now into three main cases: \vskip
amount
\noindent
\textbf{Case 0:} There exists some $b^*r{\eta} < \tau$ such that $A^0 \in U_{\kappa,b^*r{\eta}}(\emptyset)$.\vskip
amount
Let $A'(b^*r{\eta}) = A^0 \cap \{ \nu \mid o(\nu) = b^*r{\eta}\}$. By applying Lemma {\rangle}ef{Lem.amalgamation.in.Q*} with respect to the family of conditions
$\{ p^{\nu} =( {\langle} \nu {\rangle},T^\nu,Q^\nu) \mid \nu \in A'(b^*r{\eta}) \}$, we can find a
$p^* = (\emptyset,T^*,Q^*)$ which is forced by $Q^*(\emptyset)$ to be a direct extension of $(\emptyset,T,Q)$, and to have
$\{ p^{\nu} \mid \nu \in A'(b^*r{\eta}) \}$ be a predense in $\mathbb{Q}^*_{\kappa,\tau}/(\emptyset,T^*,Q^*)$.
It follows that $Q^*(\emptyset) \geq Q(\emptyset) = q$ forces $p^* = (\emptyset,T^*,Q^*)$ is a direct extension of $p$ which decides $\sigma$. Contradicting our assumption. \vskip
amount
\noindent
\textbf{Case 1:} There exists some $b^*r{\eta} < \tau$ such that $A^1 \in U_{\kappa,b^*r{\eta}}(\emptyset)$.\vskip
amount
The argument for this case is similar to the previous one, and leads to an extension $q^* \geq q$ in $\mathbb{C}l({\langle}anglembda,<\kappa)$, and a direct extension $b^*r{p} \geq^* p$, such that $q^*$ forces
$b^*r{p} \Vdash_{\mathbb{Q}^*_{\kappa,\tau}}\neg\sigma$. Contradiction.\vskip
amount
\noindent
\textbf{Case 2:} $A^2 \in \bigcap_{\eta < \tau} U_{\kappa,\eta}(\emptyset)$. \vskip
amount
\noindent
Let $A^2(0) = A^2 \cap \{ \nu \mid o(\nu) = 0\}$, and apply Lemma {\rangle}ef{Lem.amalgamation.in.Q*} with respect to
$A^2(0)$ and $\{ p {}^\frown {\langle} \nu {\rangle} \mid \nu \in A^2(0)\}$,
to obtain a direct extension
$p_0^* = (\emptyset, T_0^*,Q_0^*)$ of $p$, with $\{ p{}^\frown {\langle}{\nu}{\rangle} \mid \nu \in A^2(0) \}$ being predense in $\mathbb{Q}^*_{\kappa,\tau}/p_0^*$.
Denoting $q_0^* = Q_0^*(\emptyset)$, we define $b^*r{A}^2$ to be the set of all $\nu \in A^2$ for which $q^*_0$ forces there is no direct extension $p^\nu \geq^* p' {}^\frown \nu$ which decides $\sigma$.
Note that $b^*r{A}^2$ must belong to $\bigcap_{\eta < \tau}U_{\kappa,\eta}(\emptyset)$, since otherwise, there would be some $\eta < \tau$, and a set $A'(\eta) \subseteq b^*r{A}^2$ consisting of $\nu$ for which some $q^*(\nu) \geq q^*_0$ forces there exists a direct extension of $p^*_0 {}^\frown \nu$ which decides $\sigma$. This, in turn would allow us to repeat the construction of one of the previous cases 0 and 1, to show that there is $q^* \geq^* q^*_0$ which forces some direct extension $p^*$ of $p^*_0$ to force either $\sigma$ or $\neg\sigma$, contradicting the choice of $q^*_0$.
Let $q^1:=q^*_0$ and $p^1 = ( \emptyset, T^1, Q^1)$ be the direct extension of $p^*_0$ obtained by shrinking $\suc_{T^{p_0^*}}(\emptyset)$ to points in $b^*r{A}^2$. It follows from the construction $q^1$ forces that for all $\nu \in \suc_{T^1}(\emptyset)$, $p^1 {}^\frown {\langle} \nu {\rangle}$ does not have a direct extension in $\mathbb{Q}^*_{\kappa,\tau}$ which decides $\sigma$.
Next, we move up to the second level of the tree. To each $\mu \in \suc_{T^1}(\emptyset)$, we can repeat the above analysis with respect to $q(\mu) = q^1 \cup Q^1(\mu)$ and $p^1 {}^\frown {\langle}\mu{\rangle} = ( {\langle} \mu {\rangle}, T^1_{{\langle} \mu {\rangle}}, Q^1_{{\langle} \mu {\rangle}})$. Accordingly, we split $B = \suc_{T^1}(\emptyset)$ into three sets, $B^0, B^1, B^2$, based on whether the analysis for
$q(\mu)$ and $p^1 {}^\frown {\langle}\mu{\rangle}$ has produced an extension $q^*(\mu) \geq q(\mu)$ which forces some direct extension $p^{1,\mu} \geq^* p^1 {}^\frown {\langle}\mu{\rangle}$ to decide $\sigma$ ($B^0$ and $B^1$ for forcing $\sigma$ and $\neg\sigma$, respectively), or not.
The argument above shows that if $B^0$ or $B^1$ belong to $U_{\kappa,\eta}(\emptyset)$ for some $\eta < \tau$ then there exists some $q^* \geq q^1$ which forces that $p^1$ has a direct extension which decides $\sigma$, contradicting our assumptions.
It follows that $B^2 \in \bigcap_{\eta < \tau}U_{\kappa,\eta}(\emptyset)$, and by repeating the argument from the beginning of Case 2 for each $p^1{}^\frown {\langle}\mu{\rangle}$, $\mu \in B^2$, we can find for each $\mu \in B^2$, conditions $q^*(\mu) \geq q^1(\mu)$, $q^*(\mu) \in \mathbb{C}l({\langle}anglembda,\pi^\emptyset_0(\mu))$, and $p^*_\mu = ( {\langle} \mu {\rangle}, T^*_\mu, Q^1_\mu) \geq^* p^1 {}^\frown \mu$, such that $q^*(\mu)$ forces there is no direct extension of $p^*_\mu {}^\frown \nu$ which decides $\sigma$, for any $\nu \in \suc_{T^*_\mu}(\emptyset)$.
We may assume $q^*(\mu) = Q^1_\mu(\emptyset)$ and apply Lemma {\rangle}ef{Lem.amalgamation.in.Q*} with respect to
$B^2(0)$ and $\{ p^*_{\mu} \mid \mu \in B^2(0)\}$, to conclude, similarly to the above, that there are extensions $q^2 \geq q^1$ and $p^2 = (\emptyset,T^2,Q^2) \geq^* p^1$, such that $q^2$ forces that for any pair ${\langle} \nu_0,\nu_1{\rangle} \in T^2$, $p^2_{{\langle} \nu_0,\nu_1 {\rangle}}$ does not have a direct extension which decides $\sigma$.
The construction is now repeated level by level, for all $n < \omega$. This produces sequences of extensions $q = q^0 {\langle}eq q^1 {\langle}eq \cdots {\langle}eq q^n \cdots$ in $\mathbb{C}l({\langle}anglembda,<\kappa)$ and $p = p^0 {\langle}eq^* p^1 {\langle}eq^* \cdots {\langle}eq^* p^n \cdots$ in $\mathbb{Q}^*_{\kappa,\tau}$, such that for each $n < \omega$, writing $p^n = (\emptyset, T^n,Q^n)$, we have that
$q^n$ forces that for all sequences $s \in T^n$ of length $|s| {\langle}eq n$, $p^n {}^\frown s$ does not have a direct extension which decides $\sigma$.
Finally, let $q^\omega \in \mathbb{C}l({\langle}anglembda,<\kappa)$ be an union of all $q^n$, $n < \omega$, and $p^* \in \mathbb{Q}^*_{\kappa,\tau}$ be a direct extension of $p^n$ for all $n < \omega$. Writing $p^* = (\emptyset, T^*,Q^*)$, it follows from the construction that $p^\omega$ forces that for no $s \in T^*$ such that $p^* {}^\frown s$ has a direct extension which decides $\sigma$. This is of course absurd.
\end{proof}
We conclude that $( \mathbb{Q}^*_{\kappa,\tau},{\langle}eq,{\langle}eq^*)$ is a Prikry type forcing whose direct extension order ${\langle}eq^*$ is ${\langle}anglembda$-closed. In particular, it does not add bounded subsets to ${\langle}anglembda$.
Moreover, like $\mathbb{Q}^\tau_\kappa$, $(\mathbb{Q}^*_{\kappa,\tau},{\langle}eq)$ introduces a generic club $b^\tau_\kappa \subseteq \kappa$ of order-type $\mathop{\mathrm{ot}}(b^\tau_\kappa) = \omega^\tau$.
Finally, since $U_{\kappa,\tau'} \subseteq F_{\kappa,\tau'}(t)$ is clearly contained in $D_{\kappa,\tau',t}$ for every coherent sequence $t$,
it follows from a standard density argument that if $b_{\kappa}^\tau$ is a $\mathbb{Q}^*_{\kappa,\tau}$-generic sequence over $V[G\ast H]$, then for every $V$-set $A \in \mathcal{F}_\kappa$ there exists some $\beta < \kappa$ such that $b^\tau_\kappa \setminus \beta \subseteq A$.
\subsection{The forcing $b^*r{\mathbb{C}}_{\mathcal{F}_{\kappa}}$}{\langle}anglebel{subsection:c-star}
Our goal now is to introduce a poset $b^*r{\mathbb{C}}_{\mathcal{F}_\kappa}$ which is equivalent to $\mathbb{C}_{\mathcal{F}_\kappa}$, and further has a dense sub-forcing $\mathbb{C}^*_{\mathcal{F}_\kappa}$, which is of Prikry-type, and its direct extension order is ${\langle}anglembda$-closed.
We first introduce the poset $\mathbb{C}^*_{\mathcal{F}_\kappa}$, obtained from $\mathbb{Q}^*_{\kappa,{\langle}anglembda}$ and $\mathbb{C}_{\mathcal{F}_\kappa}$.
\noindent
Recall that $G \ast H$ is $V$ generic for $\mathbb{P} \ast \mathbb{C}l({\langle}anglembda,<\kappa)$.
Working in $V[G \ast H]$ we consider the two-step iterations
$\mathbb{Q}^*_{\kappa,{\langle}anglembda} \ast \mathbb{C}_{\mathcal{F}_\kappa}$ consisting of conditions $(q, \mathunderaccent\tilde-3 {x})$ so that $q \Vdash_{\mathbb{Q}^*_{\kappa,{\langle}anglembda}} \mathunderaccent\tilde-3 {x} \in \check{\mathbb{C}_{\mathcal{F}_\kappa}}$.
Note that when forcing with $\mathbb{C}_{\mathcal{F}_\kappa}$ over a $V[G \ast H]$-generic extension by $\mathbb{Q}^*_{\kappa,{\langle}anglembda}$, we require that the bounded closed sets $c \subseteq\kappa$ in the conditions
$x = {\langle} c,A{\rangle} \in \mathbb{C}_{\mathcal{F}_\kappa}$ are actually ground model sets, from $V[G \ast H]$.
In particular, for every condition $(q,\mathunderaccent\tilde-3 {x})$ there exists an extension $q' \geq q$ and a pair $x \in \mathbb{C}_{\mathcal{F}_\kappa}$ so that $q' \Vdash \mathunderaccent\tilde-3 {x} = \check{x}$. \vskip
amount
Let $b^{\langle}anglembda_\kappa \subseteq \kappa$ be a $\mathbb{Q}^*_{\kappa,{\langle}anglembda}$ generic club in $\kappa$. We know that $\mathop{\mathrm{ot}}(b^{\langle}anglembda_\kappa) = {\langle}anglembda$ and that $b^{\langle}anglembda_\kappa$ is almost contained in every set $A \in \mathcal{F}_{\kappa}$.
Working in a $\mathbb{Q}_{\kappa,{\langle}anglembda}^*$ generic extension $V[G \ast H \ast b^{\langle}anglembda_\kappa]$ of $V[G \ast H]$ we see that for every condition $x = {\langle} c, A{\rangle}$ in $\mathbb{C}_{\mathcal{F}_\kappa}$ there exists some $\beta \in A \setminus (\max c + 1)$ such that $x' = {\langle} c', A{\rangle}$, with $c' = c \cup \{\beta\}$, extends $x$ and satisfies that $b^{\langle}anglembda_\kappa \setminus (\max c'+1) \subseteq A$.
\begin{definition}[$\mathbb{C}^{\langle}anglembda_{\mathcal{F}_\kappa}$] ${}$\vskip
amount
Working in a $\mathbb{Q}_{\kappa,{\langle}anglembda}^*$ generic extension $V[G \ast H \ast b^{\langle}anglembda_\kappa]$ of $V[G \ast H]$,
let $\mathbb{C}^{\langle}anglembda_{\mathcal{F}_\kappa}$ denote the subset of $\mathbb{C}_{\mathcal{F}_\kappa}$, consisting of conditions $x' = {\langle} c', A'{\rangle}$ so that
$\max(c') \in b^{\langle}anglembda_\kappa$, and $b^{\langle}anglembda_\kappa \setminus (\max(c')+1) \subseteq A'$.
\end{definition}
It follows from the above that $\mathbb{C}^{\langle}anglembda_{\mathcal{F}_\kappa}$ is a dense subset of $\mathbb{C}_{\mathcal{F}_\kappa}$.
Since $b^{\langle}anglembda_\kappa \subseteq\kappa$ is closed of order-type ${\langle}anglembda = \cf(\kappa)^{V[G \ast H \ast b^{\langle}anglembda_\kappa]}$, and no sequences of ordinals of length $<{\langle}anglembda$ are introduced by $b^{\langle}anglembda_\kappa$, it follows that the restriction of the $\mathbb{C}_{\mathcal{F}_\kappa}$ order to $\mathbb{C}^{\langle}anglembda_{\mathcal{F}_\kappa}$ is ${\langle}anglembda$-closed. \vskip
amount
With this observation, we move back to $V[G \ast H]$ to define the poset $\mathbb{C}^*_{\mathcal{F}_\kappa}$.
\begin{definition}[$\mathbb{C}^*_{\mathcal{F}_\kappa}$]${}$\vskip
amount
Let $\mathbb{C}^*_{\mathcal{F}_\kappa}$ be the two step iteration $\mathbb{C}^*_{\mathcal{F}_\kappa} = \mathbb{Q}^*_{\kappa,{\langle}anglembda} \ast \mathbb{C}^{\langle}anglembda_{\mathcal{F}_\kappa}$.
We define the direct extension ordering ${\langle}eq^*$ of $\mathbb{C}^*_{\mathcal{F}_\kappa}$ to be the extension of the usual direct extension order of $\mathbb{Q}^{\langle}anglembda_\kappa$ with the standard order on the
second $\mathbb{C}^{\langle}anglembda_{\mathcal{F}_\kappa}$ component.
\end{definition}
\begin{corollary}{\langle}anglebel{cor:bar-C-prikry-type}
$\mathbb{C}^*_{\mathcal{F}_\kappa}$ is a dense sub-forcing of $\mathbb{Q}^*_{\kappa,{\langle}anglembda} \ast \mathbb{C}_{\mathcal{F}_\kappa}$ which satisfies the Prikry Property, and its direct extension order is ${\langle}anglembda$-closed.
\end{corollary}
Note that for every dense subset $D$ of $\mathbb{C}_{\mathcal{F}_\kappa}$ and a condition $( q, \mathunderaccent\tilde-3 {x}) \in \mathbb{C}^*_{\mathcal{F}_\kappa}$ there exists a direct extension $(q^*,\mathunderaccent\tilde-3 {x}^*) \geq^* (q,\mathunderaccent\tilde-3 {x})$ such that $q^* \Vdash \mathunderaccent\tilde-3 {x}^* \in \check{D}$. \vskip
amount
Similarly, it is clear that the set of conditions $(q',\check{x'}) \in \mathbb{C}^*_{\mathcal{F}_\kappa} $, for which the second component is a canonical name $\check{x'}$ of a condition $x' \in \mathbb{C}_{\mathcal{F}_\kappa}$, is dense in $\mathbb{C}^*_{\mathcal{F}_\kappa}$.
The map $(q',\check{x'}) \mapsto x'$ defined on this dense set naturally induces a forcing projection $\pi$ from $\mathbb{C}^*_{\mathcal{F}_\kappa}$ to the boolean completion of $\mathbb{C}_{\mathcal{F}_\kappa}$. This projection sends a condition of the form ${\langle}anglengle q, \mathunderaccent\tilde-3 {x}{\rangle}anglengle$ to the join of the collection of all $y \in \mathbb{C}_{\mathcal{F}_\kappa}$ such that there is some extension of $q$, $q'$ that forces $\mathunderaccent\tilde-3 {x} =\check{y}$.
\vskip
amount
Next, we follow Gitik's machinery from \cite{Gitik-HB}, to form a Prikry-type forcing notion $b^*r{\mathbb{C}}_{\mathcal{F}_\kappa}$ which is equivalent to $\mathbb{C}_{\mathcal{F}_\kappa}$, from $\mathbb{C}^*_{\mathcal{F}_\kappa}$.
\begin{definition}[$b^*r{\mathbb{C}}_{\mathcal{F}_\kappa}$]${}$\vskip
amount
We define a Prikry-type forcing notion $(b^*r{\mathbb{C}}_{\mathcal{F}_\kappa},{\langle}eq',{\langle}eq^*)$ as follows.
\begin{itemize}
\item
$b^*r{\mathbb{C}}_{\mathcal{F}_\kappa}= \mathbb{C}^*_{\mathcal{F}_\kappa}$,
\item the partial ordering ${\langle}eq'$ is defined by
$p' \geq' p$ if $\pi(p') \geq \pi(p)$, and
\item ${\langle}eq^*$ is taken to be the same direct extension order of $\mathbb{C}^*_{\mathcal{F}_\kappa}$
\end{itemize}
\end{definition}
It is immediate from the definition that $(b^*r{\mathbb{C}}_{\mathcal{F}_\kappa}, {\langle}eq')$ is equivalent as a forcing notion to $(\mathbb{C}_{\mathcal{F}_\kappa},{\langle}eq)$ and that the direct extension order ${\langle}eq^*$ of $b^*r{\mathbb{C}}_{\mathcal{F}_\kappa}$ is ${\langle}anglembda$-closed. \vskip
amount
To show that $(b^*r{\mathbb{C}}_{\mathcal{F}_\kappa},{\langle}eq',{\langle}eq^*)$ satisfies the Prikry Property, it suffices to verify that
for every statement $\sigma$ in the forcing language of $\mathbb{C}_{\mathcal{F}_\kappa}$ and every condition $p \in b^*r{\mathbb{C}}_{\mathcal{F}_\kappa}$, there is a direct extension $p^* \geq^* p$ such that $\pi(p^*)$ decides $\sigma$.
Indeed, defining $D_0 = \{ p' \in \mathbb{C}^*_{\mathcal{F}_\kappa} \mid \pi(p') \Vdash \sigma\}$ and $D_1 = \{ p' \in \mathbb{C}^*_{\mathcal{F}_\kappa} \mid \pi(p') \Vdash \neg\sigma\}$, it is clear that $D_0 \cup D_1$ is dense in $\mathbb{C}^*_{\mathcal{F}_\kappa}$ and that a generic filter $G^*$ of $\mathbb{C}^*_{\mathcal{F}_\kappa}$ will have a nontrivial intersection with exactly one of the two sets. Let $\sigma^*\mathbb{C}lon \mathunderaccent\tilde-3 {G}^* \cap D_0 \neq\emptyset$. Then $\sigma^*$ is a statement for the forcing language of $\mathbb{C}^*_{\mathcal{F}_\kappa}$. Moreover, it is clear from our construction that for a condition $p^* \in \mathbb{C}^*_{\mathcal{F}_\kappa}$ which decides $\sigma^*$ we have that $p^* \Vdash \sigma^*$ implies that $\pi(p^*) \Vdash \sigma$, and
$p^* \Vdash \neg\sigma^*$ implies that $\pi(p^*)\Vdash \sigma$. Since $\mathbb{C}^*_{\mathcal{F}_\kappa}$ satisfies the Prikry Property every condition $p$ has a direct extension $p^* \geq^* p$ which decides $\sigma^*$. \vskip
amount
We note that similarly to $\mathbb{P} \ast \mathbb{C}l({\langle}anglembda,<\kappa) \ast \mathbb{C}_{\mathcal{F}_\kappa}$, the forcing $\mathbb{P} \ast \mathbb{C}l({\langle}anglembda,<\kappa) \ast \mathbb{C}^*_{\mathcal{F}_\kappa}$ is cone homogeneous. In most applications of homogeneity, moving to an equivalent forcing does not change the main properties of its iterations.
In order
to apply the results from section {\rangle}ef{Sec:homog} and argue that the iteration
$\mathbb{P} \ast \mathbb{C}l({\langle}anglembda,<\kappa) \ast b^*r{\mathbb{C}}_{\mathcal{F}_\kappa}$
is cone homogeneous, we need to verify that this posets meets the assumptions of Lemma {\rangle}ef{FACT:homogiter}.
\begin{lemma}{\langle}anglebel{Lem:Q[U].weakly.homogeneous}
Denote $\mathbb{P} \ast \mathbb{C}l({\langle}anglembda,<\kappa) \ast \mathbb{C}^*_{\mathcal{F}_\kappa}$ by $\mathbb{W}$, and its regular and direct extension orders by ${\langle}eq_{\mathbb{W}}$ and ${\langle}eq^*_{\mathbb{W}}$ respectively.\\
For every $w_0, w_1 \in \mathbb{W}$ there are direct extensions $w_0^*,w_1^*$ of $w_0,w_1$ respectively,
and a cone isomorphism $\varphi \mathbb{C}lon \mathbb{W}/w_0^* \to \mathbb{W}/w_1^*$ which respects the direct extension order ${\langle}eq^*_{\mathbb{W}}$.
In particular, the forcing $\mathbb{P} \ast \mathbb{C}l({\langle}anglembda,<\kappa)\ast b^*r{\mathbb{C}}_{\mathcal{F}_\kappa}$ is cone homogeneous.
\end{lemma}
We observe that assuming the coherent sequence $\mathcal{U}$ (by which $\mathbb{W} = \mathbb{P} \ast \mathbb{C}l({\langle}anglembda,<\kappa) \ast \mathbb{C}^*_{\mathcal{F}_\kappa}$ is defined) is ordinal definable in $V$,
then the statement of the lemma guarantees that $\mathbb{W}$ satisfies the requirements of the iterated poset $\mathbb{Q}_\alpha$ from Lemma {\rangle}ef{FACT:homogiter}.
\begin{proof}
Let us start with the last assertion. Since the identity is a projection from $\mathbb{C}^*_{\mathcal{F}_\kappa}$ to $b^*r{\mathbb{C}}_{\mathcal{F}_\kappa}$, an isomorphism of a cone of elements in $\mathbb{C}^*_{\mathcal{F}_\kappa}$ naturally induces an isomorphism of the corresponding cone in $b^*r\mathbb{C}_{\mathcal{F}_\kappa}$.
Let $w_0 = {\langle}anglengle p_0, c_0, {\langle}anglengle q_0, \mathunderaccent\tilde-3 {x}_0{\rangle}anglengle{\rangle}anglengle$ and
$w_1 = {\langle}anglengle p_1, c_1, {\langle}anglengle q_1, \mathunderaccent\tilde-3 {x}_1{\rangle}anglengle{\rangle}anglengle$, where the conditions $q_0 = {\langle}anglengle t_0, T_0, Q_0{\rangle}anglengle$ and $q_1 = {\langle}anglengle t_1, T_1, Q_1{\rangle}anglengle$ belong to $\mathbb{Q}^*_{\kappa,{\langle}anglembda}$.
By \cite[Theorem 4.6]{bunger} applied to the iteration $\mathbb{P} \ast \mathbb{Q}_{\kappa,{\langle}anglembda}$, there are direct extensions ${\langle} p_0^*, (t_0,T) {\rangle}$ of
${\langle} p_0, (t_0,T_0) {\rangle}$, and ${\langle} p_1^*, (t_1,T) {\rangle}$ of ${\langle} p_1, (t_1,T_1) {\rangle}$, with a common top tree $T$, and an
isomorphism $\psi$ between the cone below
${\langle} p_0^*, (t_0,T) {\rangle}$ and the cone below
${\langle} p_1^*, (t_1,T) {\rangle}$.
We record here that the map $\psi$ constructed in the proof of \cite[Theorem 4.6]{bunger} satisfies two additional
properties: First, it does not make any changes to the $\mathcal{F}_{\kappa}$-trees $S$ appearing in conditions $p {}^\frown {\langle} s,S {\rangle} \in \mathbb{P} \ast \mathbb{Q}_{\kappa,{\langle}anglembda}$.
Second, it respects the direct extension order of $\mathbb{P} \ast \mathbb{Q}_{\kappa,{\langle}anglembda}$. \vskip
amount
Next, we move to examine the Levy collapse condition and the suitable functions in the conditions from $\mathbb{W}$.
Since the collapsing forcing $\mathbb{C}l({\langle}anglembda, <\kappa)$ is evaluated in the generic extension by $\mathbb{P}$, $\psi$ naturally acts also on the $\mathbb{P}$-names $c_0$ and $Q_0$ that appear in $w_0$. As usual, we denote the resulting names by $c_0^\psi$ and $Q_0^\psi$.
Let $\tau_\emptyset$ be a $\mathbb{P}$-name of an automorphism of the Levy collapse poset which maps an extension $c_0'$ of $c_0^{\psi}$ to an extension $c_1^*$ of $c_1$,
and define $c_0^* = (c_0')^{\psi^{-1}}$.
Note that since $c_i$ forces $Q_i(\emptyset) \in \mathunderaccent\tilde-3 {H}$, we may extend $Q_0(\emptyset),Q_1(\emptyset)$ to $Q_0^*(\emptyset),Q_1^*(\emptyset)$ so that $c_i = Q_i^*(\emptyset)$ for $i = 0,1$.
Next, for each $s \in T$ and $i = 0,1$, define $\dom_1(Q_i(s)) = \{ \alpha < \kappa \mid Q_i(s)restriction {\langle}anglembda \times \{\alpha\} \neq \emptyset\}$
and ${\rangle}ho^i_s = \sup(\dom_1(Q_i(s)))$. Set ${\rangle}ho_s = \max({\rangle}ho_s^0,{\rangle}ho_s^1)$.
By moving to a direct extension tree $T^*$ of $T$, we may assume that
for every $s \in T^*$ and $\nu \in \suc_{T^*}(s)$, $\pi^{t {}^\frown s}_0(\nu) > {\rangle}ho_s$ where the projection is computed in both generic extensions.
This leaves enough space between the conditions $Q_i(s), Q_i(s {}^\frown {\langle} \nu {\rangle})$, $i = 0,1$, to define autormorphisms taking an extension $Q_0^*(s)$ of $Q_0(s)$ to an extension $Q_1^*(s)$ of $Q_1(s)$, without conflicting with $Q_i(s {}^\frown {\langle} \nu {\rangle})$, $i = 0,1$. We can therefore define by induction on the lexicographic order $<_{lex}$ on $T^*$ (where two sequences are compared from their top elements down)
automorphisms $\tau_s$, $s \in T^*$, of $\mathbb{C}l({\langle}anglembda,<\kappa)$, and collapse extensions $Q_i^*(s) \geq Q_i(s)$, $i = 0,1$,
with the following properties: For all $s \in T^*$,
\begin{itemize}
\item $\tau_s$ is supported in $\mathbb{C}l({\langle}anglembda,<{\rangle}ho_s)$,\footnote{I.e., for every $p \in \mathbb{C}l({\langle}anglembda,<\kappa)$, if $p = p_0 \cup p_1$ where $p_0 = p restriction {\langle}anglembda \times {\rangle}ho_s$, then $\tau_s(p) = \tau_s(p_0) \cup p_1$.}
\item $\tau_s((Q_0^*)^\psi(s)) =Q^*_1(s)$,
\item If $s' \in T^*_s$ then $\tau_{s'}restriction \mathbb{C}l({\langle}anglembda,<\pi_0^{t {}^\frown s}(s)) = \tau_s$,
\item If $b^*r{s} \in T^*$ and $b_{t_0 {}^\frown s} = b_{ t_0 {}^\frown b^*r{s}}$ then $\tau_{s} = \tau_{b^*r{s}}$.
\end{itemize}
Let $b^*r{\mathbb{W}} = \mathbb{P} \ast \mathbb{C}l({\langle}anglembda,<\kappa) \ast \mathbb{Q}_{\kappa,{\langle}anglembda}^*$ be the initial forcing iteration of
\[\mathbb{W} = \mathbb{P} \ast \mathbb{C}l({\langle}anglembda,<\kappa) \ast \mathbb{C}^*_{\mathcal{F}_\kappa} = \mathbb{P} \ast \mathbb{C}l({\langle}anglembda,<\kappa) \ast (\mathbb{Q}_{\kappa,{\langle}anglembda}^* \ast \mathbb{C}^{\langle}anglembda_\kappa).\]
Let $w_0restriction b^*r{\mathbb{W}} = {\langle} p_0, c_0, {\langle} t_0,T_0,Q_0{\rangle} {\rangle}$ and $w_1restriction b^*r{\mathbb{W}} = {\langle} p_1,c_1,{\langle} t_1,T_1,Q_1{\rangle} {\rangle}$ be the restrictions of $w_0,w_1$ to $b^*r{\mathbb{W}}$
and consider their direct extensions $b^*r{w_o},b^*r{w_1}$ in $b^*r{\mathbb{W}}$, defined by
$b^*r{w_i} = {\langle} p^*_0, c^*_0, {\langle} t_0,T^*,Q^*_0{\rangle} {\rangle}$, $i = 0,1$. \vskip
amount
Our choice of cone isomorphism $\psi$ for $\mathbb{P} \ast \mathbb{Q}_{\kappa,{\langle}anglembda}$ together with
the collection of Levy-Collapse automorphisms $\vec{\tau} = \{\tau_s\}_{s \in T^*}$, naturally induces a function $b^*r{\varphi}$ on the cone $b^*r{\mathbb{W}}/ b^*r{w_0}$, defined as follows.
For a condition
$b^*r{w} = {\langle} p, c,{\langle} s , S, Q{\rangle}{\rangle} $, we set $b^*r{\varphi}(b^*r{w}) = {\langle} p',c',{\langle} s', S,Q'{\rangle}{\rangle}$ to be:
\[
{\langle} p',{\langle} s',S{\rangle}{\rangle} = \psi({\langle} p',{\langle} s,S{\rangle}{\rangle} ) , \quad c' = \tau_{s}(c^\psi), \quad Q'(s') = \tau_{s {}^\frown s'}(Q(s'))
\]
We claim that ${\langle} p',c',{\langle} s',S, Q'{\rangle}{\rangle}$ is a condition in $b^*r{\mathbb{W}}$.
First, it is immediate from our choice of $p,c$ that ${\langle} p',c'{\rangle} \in \mathbb{P} * \mathbb{C}l({\langle}anglembda,<\kappa)$. It therefore remains to verify that
${\langle} s',S,Q'{\rangle}$ is forced by ${\langle} p',c'{\rangle}$ to be a condition in $\mathbb{Q}^*_{{\langle}anglembda,<\kappa}$.
The fact that ${\langle} s',S,Q'{\rangle}$ satisfies requirements $(1)$ and $(2)$ of definition {\rangle}ef{def:Q*poset} is immediate.
To verify the coherency requirement $(3)$ of definition {\rangle}ef{def:Q*poset}, we note that $Q'$ is forced by $p'$ to be $(s, S)$-suitable.
Indeed, it follows from our choice of $\tau_{s'}$ that its support is bounded below the projection $\pi^{s {}^\frown s'}_0(\nu)$, for any $\nu \in \suc_{T^*}(s')$.
Property $(4)$ follows from the fact that the statement ``$c \Vdash_{\mathbb{C}l({\langle}anglembda,<\kappa)} Q(\emptyset) \in \mathunderaccent\tilde-3 {H}$'' is forced by $p \in \mathbb{P}$,
which implies that $\tau_\emptyset(c^\psi) \Vdash \tau_\emptyset(Q(\emptyset)^\psi) \in \tau_\emptyset(\mathunderaccent\tilde-3 {H}^\psi)$ is forced by $p'$. This, combined with definition of
$c'$ and $Q'$, and the fact that the name $\mathunderaccent\tilde-3 {H}$ is a fixed point of both $\tau_\emptyset$ and $\psi$, guarantees that requirement $(4)$ is satisfied.
Next, $(s',S,Q')$ satisfies requirement $(5)$ of definition {\rangle}ef{def:Q*poset} by a similar argument to the previous one, using the fact $(s, S, Q)$ satisfies property $(5)$ together with the last property listed above for $\{\tau_s\}_{s\in T}$.
Having verified that ${\langle} p',c',{\langle} s',S, Q'{\rangle}{\rangle}$ is a condition in $b^*r{\mathbb{W}}$ it is straightforward to check that it extends $b^*r{w_1}$ and
thus $b^*r{\varphi}\mathbb{C}lon b^*r{\mathbb{W}}/b^*r{w_0} \to b^*r{\mathbb{W}}/b^*r{w_1}$ is a well defined function. In order to show that it is cone isomorphism we need to show that it is order-preserving.
Let us remark that the automorphism $\psi$ modifies the values of $b_s$ for $s \in T^*$ by changing the value of their initial segments. Since those initial segments do not affect the definition of $\tau_{b_s}$, we will ignore it and write always $b_s$ instead $b_s^\psi$.
Let $b^*r{w}_1 = {\langle} p_1, c_1, {\langle} s_1, S_1, Q_1{\rangle} {\rangle},\, b^*r{w}_2 ={\langle} p_2, c_2, {\langle} s_2, S_2, Q_2{\rangle} {\rangle}$ be pair of conditions in the cone above $b^*r{w}_0$. We need to show that $b^*r{\varphi}(b^*r{w}_1) {\langle}eq b^*r{\varphi}(b^*r{w}_2)$ if and only if $b^*r{w}_1 {\langle}eq b^*r{w}_2$.
For direct extensions, this is clear, as the tree $S_1$ does not move under $b^*r\varphi$. Let us assume that $b^*r{w}_2$ is a one-point extension of $b^*r{w}_1$, by the point ${\langle}\nu{\rangle}$. By moving to a dense subset, $c_2 \geq c_1, Q_1({\langle} \nu {\rangle})$ and $b_{s_2} = b_{s_1 {}^\frown {\langle}\nu{\rangle}}$. Let us apply $b^*r{\varphi}$ on $b^*r{w}_1, b^*r{w}_2$. The trees $S_1$ and $S_2$ do not move, so we must verify that ${\langle}\nu{\rangle}$ is still a legitimate choice for an one-point extension of $b^*r\varphi(b^*r{w}_1)$. Indeed, $\tau_{b_{s_2}}(c_2)$ is (by the definition of $\tau_{b_{s_2}}$) stronger than $\tau_{{\langle} \nu{\rangle}}(Q_1({\langle}\nu{\rangle})$. Thus, we conclude that $b^*r\varphi(b^*r w_2)$ is an one-point extension of $b^*r\varphi(b^*r w_2)$ by ${\langle}\nu{\rangle}$. The other direction is the same.\vskip
amount
Finally, to obtain a desirable cone isomorphism $\varphi$ for $\mathbb{W} = b^*r{\mathbb{W}} \ast \mathbb{C}^{\langle}anglembda_\kappa$, it remains to extend $b^*r{\varphi}$ to the final additional components $\mathunderaccent\tilde-3 {x_0},\mathunderaccent\tilde-3 {x_1}$ of $\mathbb{C}^{\langle}anglembda_{\mathcal{F}_\kappa}$.
The proof Lemma {\rangle}ef{Lem:CHomog} shows that there are $b^*r{\mathbb{W}}$-names $\mathunderaccent\tilde-3 {y}_0',\mathunderaccent\tilde-3 {y}_1$ of extensions of $\mathunderaccent\tilde-3 {x}_0^{b^*r{\varphi}},\mathunderaccent\tilde-3 {x}_1$ respectively,
and a name of a cone isomorphism $\sigma\mathbb{C}lon \mathbb{C}^{\langle}anglembda_\kappa/ \mathunderaccent\tilde-3 {y}_0' \to \mathbb{C}^{\langle}anglembda_\kappa/\mathunderaccent\tilde-3 {y}_1$.
Accordingly, we set $\mathunderaccent\tilde-3 {y}_0 = (\mathunderaccent\tilde-3 {y}_0')^{b^*r{\varphi}^{-1}}$ and
define direct extensions $w_0^* \geq^* w_0,w_1^*\geq^* w_1$ and a map $\varphi\mathbb{C}lon \mathbb{W}/w_0^* \to \mathbb{W}/w_1^*$ by
$w_i^* = b^*r{w}_i {}^\frown \mathunderaccent\tilde-3 {y}_i$ and
\[ \varphi( {\langle} b^*r{w},\mathunderaccent\tilde-3 {y} {\rangle}) = {\langle} b^*r{\varphi}(b^*r{w}),\sigma(\mathunderaccent\tilde-3 {y}^{b^*r{\varphi}}){\rangle}.\]
The fact $w_0^*,w_1^*,\varphi$ satisfy the result stated in the lemma is an immediate consequence of the fact $b^*r{w}_0,b^*r{w}_1,b^*r{\varphi}$ satisfy similar properties for $b^*r{\mathbb{W}}$ and our choice of $\varphi$,$\mathunderaccent\tilde-3 {y}_0,\mathunderaccent\tilde-3 {y}_1$.
\end{proof}
\section{Strong Measurability at Successors of Singulars}{\langle}anglebel{Sec:suc.of.sing}
Suppose that $V = \HOD$, $\kappa$ is a supercompact cardinal and ${\langle}anglembda > \kappa$ is a measurable cardinal with a normal measure $\mathcal{U}$.
We would like to construct a cone homogeneous poset in $V$ which will collapse ${\langle}anglembda$ to be the successor of $\kappa$, change the cofinality of $\kappa$ to $\omega$, and add a closed unbounded subset
of ${\langle}anglembda$ whose restriction to the set of $\{ \alpha < {\langle}anglembda \mid \alpha \text{ is regular in } V\}$ is almost contained in every set $A \in \mathcal{U}$.\vskip
amount
It is natural to attempt obtaining this result by starting with an indestructible supercompact cardinal $\kappa$, and forcing with a Levy collapse of ${\langle}anglembda$ to $\kappa^+$ followed by a Prikry forcing at $\kappa$ and a club forcing at ${\langle}anglembda$. The difficulty with this approach is in its second step, where the choice of the measure on $\kappa$ depends on the generic filter for the Levy collapse and might lead to a Prikry generic sequence which will introduce to $\HOD$ information about the collapse of ${\langle}anglembda$ to $\kappa^+$, and in particular prevent from $\HOD$ to witness that ${\langle}anglembda$ is a measurable cardinal.
Instead, our approach will be based on recent use of the supercompact extender based forcing, introduced by Merimovich (\cite{Merimovich3}). Given a supercompact cardinal $\kappa$, we derive a $(\kappa,{\langle}anglembda)$-supercompact extender $E$ from a supercompact embedding $j\mathbb{C}lon V \to M$ for which ${}^{{\langle}anglembda}M \subseteq M$. Let $\mathbb{P}_{E}$ be the supercompact extender based forcing associated to the extender $E$ of \cite{Merimovich3}. The conditions of $\mathbb{P}_E$ are pairs of the form ${\langle}anglengle f, T{\rangle}anglengle$ where $f$ is roughly a condition in the Cohen forcing and $T$ is a tree, with large splittings. We denote by $\mathbb{P}_E^*$ the Cohen part.
The forcing $\mathbb{P}_E$ preserves ${\langle}anglembda$ and singularizes all the regular cardinals in the interval $[\kappa, {\langle}anglembda)$. We will follow the definitions and notations of \cite{Merimovich3}.
In \cite{GitikMerimovich}, Gitik and Merimovich show that this forcing is weakly homogeneous (and therefore cone homogeneous).
Let $\mathcal U$ be a normal measure on ${\langle}anglembda$ in the ground model. We would like to force a club to diagonalize $\mathcal{U}$ relative to the set of $V$-regular cardinals below ${\langle}anglembda$.
Note that the ordinals of uncountable cofinality below ${\langle}anglembda$ in the extender based forcing extension are of measure zero in $\mathcal{U}$. Therefore, our club shooting poset has to allow $V$-singular ordinals as well.
Moreover, since the set of previous inaccessible cardinals below ${\langle}anglembda$ does not reflect at its complement, it is impossible for the generic club to avoid ground model singular cardinals of countable cofinality.
Thus, we restrict our club forcing poset to diagonalize $\mathcal{U}$ only relative to the set of
the regular cardinals in $V$.
To make this precise, we denote by $\Sing$ the set of all ground model singular cardinals below ${\langle}anglembda$, and define
$b^*r{\mathcal{U}}= \{ \Sing \cup A \mid A \in \mathcal{U}\}$. We force with the poset $\mathbb{C}_{b^*r{\mathcal{U}}}$, consisting of pairs $(c,B)$ where $c \subseteq {\langle}anglembda$ is a closed bounded set and $B \in b^*r{\mathcal{U}}$. The extension order is as in the previous section.\vskip
amount
We start by recalling a fundamental and useful fact, which lies in the heart of the proof of the Prikry Property of $\mathbb{P}_E$.
\begin{lemma}{\langle}anglebel{lem:extender.properness}
Let $M$ be an elementary sub-model of $H_\chi$ for some large $\chi$ such that $M \cap {\langle}anglembda = \delta \in {\langle}anglembda$ is inaccessible cardinal and $M^{<\delta} \subseteq M$. Let $p\in M \cap \mathbb{P}_E$.
Then, there is a condition $f^* \in \mathbb{P}_E^{*}$ which is $M$-generic (namely, it belongs to every dense open subset of $\mathbb{P}_E^*$ in $M$) and $\dom f^* = M \cap {\langle}anglembda$. Moreover, if $p^* = {\langle}anglengle f^*, T{\rangle}anglengle$ is a condition in $\mathbb{P}_E$, then there is $T^* \subseteq T$, $E(f^*)$-large such that $T^* \subseteq M$ and $D \in M$ is a dense open subset of $\mathbb{P}_E$ then there is a natural number $n$ such that for every ${\langle}anglengle \nu_0, \dots, \nu_{n-1}{\rangle}anglengle$ in the $n$-th level of $T*$, $p^*_{{\langle}anglengle \nu_0, \dots, \nu_{n-1}{\rangle}anglengle} \in D$.
\end{lemma}
\begin{proof}
The first claim follows from the closure of $\mathbb{P}_E^*$. Let us focus in the second part.
Let $f^*$ be as in the lemma. Let $D\in M$ be dense open. For each ${\langle}anglengle \nu_0, \dots, \nu_{n-1}{\rangle}anglengle \in M$ and for each $g\in \mathbb{P}_E^* \cap M$, we can ask whether there is a condition $q \in D$ of the form ${\langle}anglengle h, S{\rangle}anglengle \in D$ such that $h \geq^* g_{{\langle}anglengle \nu_0, \dots, \nu_{n-1}{\rangle}anglengle}$. The set of conditions that decide this statement is dense open and definable in $M$ and thus $f^*$ decides whether there is such extension or not (for each possible ${\langle}anglengle \nu_0, \dots, \nu_{n-1}{\rangle}anglengle$). Let $D_{\vec{\nu}}$ be this set and let us split it into two parts $D_{\vec{\nu}}^0 \cup D_{\vec\nu}^1$ according to the decision, where conditions in $D_{\vec\nu}^0$ are direct extensions that enter $D$ after the non-direct extension.
Let $p^* = {\langle}anglengle f^*, T{\rangle}anglengle$.
Since a typical point $\nu$ in a measure one tree $T$, associate with the measures $E(f^*)$ is a finite sequence of elements contained in $M$ each has size $|\nu| < \kappa$, we may assume that $T \subseteq M$. There is an extension $q \geq p^*$ in $D$. By the definition of the order of $\mathbb{P}_E$, $q$ is obtained by taking first some Prikry extension and then a direct extension, and therefore the Prikry extension is done using some $\vec{\nu} \in M$. Thus, for this specific Prikry extension, $f^* \in D_{\vec{\nu}}^0$. We conclude that already $p^*_{\vec{\nu}} \in D$.
We can now shrink $T$ in order to stabilize the length of the extensions that enter $D$.
\end{proof}
\begin{lemma}
$b^*r{\mathcal{U}}$ extends to a ${\langle}anglembda$-complete filter in the generic extension by $\mathbb{P}_E$.
\end{lemma}
\begin{proof}
Assume that this is not the case. Since $\kappa$ is singular, the closure of $b^*r{\mathcal{U}}$ must drop to some cardinal ${\rangle}ho < \kappa$. Let ${\langle}anglengle \mathunderaccent\tilde-3 {A}_i \mid i < {\rangle}ho{\rangle}anglengle$ be a sequence of names of elements in $b^*r{\mathcal{U}}$ which are forced to have non-measure one intersection.
Using the strong Prikry Property, we can find a sequence of direct extensions $p_i$, and natural numbers $n_i$ such that any $n_i$-length Prikry extension of $p_i$ decides the value of $\mathunderaccent\tilde-3 {A}_i$. Since there are fewer than ${\langle}anglembda$ many such extensions, we can find a set $B_i \in b^*r{\mathcal{U}}$ such that $p_i \Vdash B_i \subseteq \mathunderaccent\tilde-3 {A}_i$. In particular, $p_{{\rangle}ho} \Vdash \bigcap B_i \subseteq \bigcap \mathunderaccent\tilde-3 {A}_i$, but $\bigcap B_i \in b^*r{\mathcal{U}}$.
\end{proof}
\begin{lemma}
$\mathbb{C}_{b^*r{\mathcal{U}}}$ is ${\langle}anglembda$-distributive in the generic extension by $\mathbb{P}_E$.
\end{lemma}
\begin{proof}
Since $\kappa$ is singular in the extension by $\mathbb{P}_E$, it is enough to show that the forcing $\mathbb{C}_{b^*r{\mathcal{U}}}$ is ${\rangle}ho$-distributive for every ${\rangle}ho < \kappa$.
We first work in $V$.
Let $\mathunderaccent\tilde-3 {\vec{D}} = {\langle}anglengle \mathunderaccent\tilde-3 {D}_i \mid i < {\rangle}ho{\rangle}anglengle$ be a sequence of $\mathbb{P}_E$-names for dense open subsets of $\mathbb{C}_{b^*r{\mathcal{U}}}$, ${\rangle}ho < \kappa$. Let ${\langle}anglengle p,q{\rangle}anglengle$ be a condition in $\mathbb{P}_E * \mathbb{C}_{b^*r{\mathcal{U}}}$. Let us define an increasing sequence of models ${\langle}anglengle M_i \mid i < {\rangle}ho{\rangle}anglengle$ such that:
\begin{itemize}
\item $\mathunderaccent\tilde-3 {\vec{D}}, \mathbb{P}_E, \mathbb{C}_{b^*r{\mathcal{U}}} \in M_0$.
\item $M_i \prec H_\chi$ for some large $\chi$, $M_i \cap {\langle}anglembda = \delta_i \in {\langle}anglembda$ inaccessible.
\item $M_{i}^{<\delta_{i}} \subseteq M_{i}$ and $\delta_{i} \in \bigcap \{A \in \mathcal{U} \cap M_{i}\}$.
\item ${\langle}anglengle M_j \mid j < i{\rangle}anglengle \in M_i$.
\end{itemize}
This chain of models can be easily obtained using the same argument as in Lemma {\rangle}ef{Lem01}.
Next, let us pick by induction, for each $i < {\rangle}ho$, an $M_i$-generic condition $f^*_i\in \mathbb{P}_E^*$ such that $f^*_i \in M_{i+1}$, and $f^*_i \subseteq f^*_j$ for $i < j$. We will define a sequence of names $\mathunderaccent\tilde-3 {q}_i$ and a sequence of conditions $p_i$ such that:
\begin{itemize}
\item $p_i = {\langle}anglengle f^*_i, T_i{\rangle}anglengle \in M_{i+1}$, $q_i \in M_i$.
\item $p_{i+1} \Vdash q_{i+1} \in \mathunderaccent\tilde-3 {D}_i$.
\item The sequence of conditions $p_i$ is ${\langle}eq^*$-increasing. Let $p_{{\rangle}ho}$ be their limit.
\item $p_{{\rangle}ho}$ forces that the conditions $q_i$ are increasing and they have a limit $q_{\rangle}ho$.
\end{itemize}
In $M_{i}$, let $D'_i$ be the dense open set in $\mathbb{P}_E$ of all extensions of $p_i$ that force for some condition $q = (c^q, B^q) \geq q_i$ to be in $\mathunderaccent\tilde-3 {D}_i$, and decide its maximum and its large set $B^q$ from $b^*r{\mathcal{U}}$. By applying Lemma {\rangle}ef{lem:extender.properness} inside $M_{i}$, we conclude that there is an $E(f^*_i)$-large tree $T_i\subseteq M_i$ and a natural number $n_i$ such that for the condition $p_{i} = {\langle}anglengle f^*_i, T_i{\rangle}anglengle$, for every $\vec{\nu} \in Lev_{n_i}(T_i)$, $(p_i)_{\vec{\nu}}\in D'_i$. In particular, it picks a condition $q_{i+1, \vec{\nu}} \geq q_i$ from $\mathbb{C}_{b^*r{\mathcal{U}}}$, which is going to be in $M_{i}$. Since this condition is in $M_i$, it is going to be bounded below $\delta_{i}$ and its large set belongs to $b^*r{\mathcal{U}} \cap M_{i}$.
Note that the collection of all $n$-step extensions of a fixed condition in $\mathbb{P}_E$ is always a maximal anti-chain above this condition and thus, we can define $\mathunderaccent\tilde-3 {q}_{i+1}'$ to be equal to $q_{i+1, \vec{\nu}}$ above $(p_i)_{\vec{\nu}}$, and trivial below any condition which is incompatible with $p_i$. Finally, we define $\mathunderaccent\tilde-3 {q}_{i+1}$ to be the extension of $\mathunderaccent\tilde-3 {q}_{i+1}'$ by the single ordinal $\delta_{i}$. By the construction, this is indeed an extension, as $\delta_{i} \in B$ for all $B \in b^*r{\mathcal{U}} \cap M_{i}$.
At limit steps, we define $\mathunderaccent\tilde-3 {q}_i$ to be the limit of previous conditions. This is possible, since the filter $b^*r{\mathcal{U}}$ is still ${\langle}anglembda$-complete and since the maximal element of the closed set in $\mathunderaccent\tilde-3 {q}_j$ is forced to be $\delta_j$ and therefore, the maximal element of $\mathunderaccent\tilde-3 {q}_i$ is $\delta_i$ which is singular strong limit cardinal in the limit case.
\end{proof}
\begin{lemma}{\langle}anglebel{Lem: preservation of stationarity}
Let $B' \in \mathcal{U}$. Then $B'$ is stationary in $\mathbb{P}_E * \mathbb{C}_{b^*r{\mathcal{U}}}$.
\end{lemma}
\begin{proof}
Let $\mathunderaccent\tilde-3 {C}$ be a name for a club. We show that every condition $q \in \mathbb{C}_{b^*r{\mathcal{U}}}$ has an extension which forces that $\mathunderaccent\tilde-3 {C} \cap B' \neq \emptyset$.
Working in $V$,
let $M \prec H_\chi$, such that $M \cap {\langle}anglembda = \delta$, $\mathbb{P}_E, \mathbb{C}_{b^*r{\mathcal{U}}}, \mathunderaccent\tilde-3 {C}, q,B' \in M$, and $\delta \in B'$ is inaccessible. Moreover, let us assume that $M$ is obtained as a union of a chain of models of length $\delta$, $M_i$, such that $M_i \cap {\langle}anglembda = \delta_i$ and $M_{i+1}^{<\delta_{i+1}} \subseteq M_{i+1}$ and $\delta_{i+1} \in \bigcap (\mathcal{U} \cap M_{i+1})$.
For each $i$, let $f^*_i$ be $M_i$-generic for $\mathbb{P}_E^*$, such that $f_i^* \subseteq f_j^*$ for $i < j$. Let $f^* = \bigcup f_i^*$.
Let $G \subseteq \mathbb{P}_E$ be a generic filter that contains a condition $p^* = {\langle}anglengle f^*, A{\rangle}anglengle$, $A \subseteq M$. In $V[G]$, $\cf \delta = \omega$. Let ${\langle}anglengle \delta_n \mid n < \omega{\rangle}anglengle$ be a cofinal sequence in $\delta$. For each $n$, for sufficiently large $\xi < \delta$, $M_\xi$ contains the dense set of conditions in $\mathbb{P}_E$ that decide on some condition $q \in \mathbb{C}_{b^*r{\mathcal{U}}}$ that forces some ordinal $\gamma_n \geq \delta_n$ to be in $\mathunderaccent\tilde-3 {C}$.
Following the same arguments as in the previous lemma, we can define a condition $\mathunderaccent\tilde-3 {q}_n$ by going over some maximal anti-chain. The maximum of the closed set of $\mathunderaccent\tilde-3 {q}_n$ is always $\delta_{n + 1}$.
Finally, the sequence of conditions $q_n^G$ has an upper bound, by attaching $\delta$ on top of the union. Let $q_\omega$ be the upper bound. Clearly, $q_\omega$ forces $\delta \in \mathunderaccent\tilde-3 {C}$, as wanted.
\end{proof}
Finally, the following proposition finishes the proof of Theorem {\rangle}ef{thm:singular.strong.meas}.
\begin{proposition}
Let $\kappa$ be ${\langle}anglembda$-supercompact, where ${\langle}anglembda$ is measurable. Then, there is a generic extension in which $\cf \kappa = \omega$, $\kappa$ is a cardinal, ${\langle}anglembda = \kappa^+$ and it is $({\langle}eft(S_{reg}^{\langle}anglembda{\rangle}ight)^V, 1)$-strongly measurable cardinal.
\end{proposition}
We can now finish the proof of theorem {\rangle}ef{thm:singular.strong.meas}.
\begin{proof}[Proof of Theorem {\rangle}ef{thm:singular.strong.meas}]
The iteration $\mathbb{P}_E * \mathbb{C}_{b^*r{\mathcal{U}}}$ is cone homogeneous as an iteration of two cone homogeneous, ordinal definable forcing notions. Since $\mathbb{P}_E$ preserves cardinals below $\kappa$ and $\geq{\langle}anglembda$ and $\mathbb{C}_{b^*r{\mathcal{U}}}$ preserves cardinals, the result follows. The set $S_{reg}^{\langle}anglembda$ is stationary by Lemma {\rangle}ef{Lem: preservation of stationarity}.
\end{proof}
The result that we obtain for the successor of a singular cardinal is weaker than the result for a successor of a regular cardinal. The reason is that in order to get the closed unbounded filter to be sets from the intersection of some ground model normal measures we will have to obtain a situation in which the regular cardinals between the supercompact cardinal $\kappa$ and the measurable cardinal ${\langle}anglembda$ are going to change cofinalities into values which differ from the cofinality of $\kappa$ in the generic extension. This is also the reason that such a method cannot work for getting $\omega$-strongly measurable successor of a singular cardinal of uncountable cofinality.
We remark that Woodin in \cite{Woodin-SuitableExtendersI}, proved that it is consistent relative to the large cardinal axiom $I_0$ that a successor of a singular cardinal is $\omega$-strongly measurable.
\begin{question}
Is it consistent that there is an $\omega$-strongly measurable cardinal ${\langle}anglembda^+$, where $\cf {\langle}anglembda > \omega$ is a limit cardinal?
\end{question}
\begin{question}
Is it consistent that there is a cardinal ${\langle}anglembda^+$, where $\cf {\langle}anglembda > \omega$ is a limit cardinal and ${\langle}eft(S^{{\langle}anglembda^{+}}_{reg}{\rangle}ight)^{\HOD}$ contains a club in $V$?
\end{question}
\section{Appendix - Homogeneity}{\langle}anglebel{appendix:homogeneity}
In this section we review some basic facts related to homogeneity and develop some basic tools in order to preserve homogeneity of iterations of Prikry type forcings.
\subsection{Homogeneity and $\HOD$}{\langle}anglebel{Sec:homog}
When dealing with $\HOD$, we would like to modify the universe (via forcing) while not adding objects to $\HOD$. The main method to obtain this is to force with posets which satisfy certain weak homogeneity property. The main results of this work will focus on the notion of cone homogeneous posets.
\begin{definition}{\langle}anglebel{Def:ConeHom}
We say that a poset $\mathbb{P}$ is \textbf{cone homogeneous} if for every $p,q \in \mathbb{P}$ there are extensions $p^*,q^*$ of $p,q$ respectively, and a forcing isomorphism $\varphi$ from the cone $\mathbb{P}/p^*$ (i.e., of conditions extending $p^*$) to the cone $\mathbb{P}/q^*$.
\end{definition}
This notion can also be found under different names in the literature concerning weak forms of homogeneity. Our terminology follow Dorbinen and Friedman, \cite{DrobinenFriedman}, for the most part. It is easy to see that cone homogeneous posets satisfy most standard properties of homogeneous posets concerning ordinal definability sets. In particular, the following well-known result holds.
\begin{fact}[Levy, \cite{Levy}]{\langle}anglebel{fact:Levy}
If $\mathbb{P}$ is cone homogeneous and belongs to $\HOD$, and $G \subseteq \mathbb{P}$ is generic over $V$ then $\HOD^{V[G]} \subseteq \HOD^V$.
\end{fact}
If $\varphi$ is an isomorphism of two cones $\mathbb{P} /p_0$ and $\mathbb{P}/p_1$ and $\sigma$ is a $\mathbb{P} /p_0$ name, then by recursively applying $\varphi$ we obtain a $\mathbb{P} /p_1$-name, which we denote by $\sigma^\varphi$.
Let $\mathbb{P} = \mathbb{P}_\kappa$ where ${\langle} \mathbb{P}_\alpha,\mathbb{Q}_\alpha \mid \alpha < \kappa{\rangle}$ is an iteration of cone homogeneous posets $\mathbb{Q}_\alpha$ and moreover let us assume that all cone automorphisms of $\mathbb{P}_\alpha$ do not modify $\mathbb{Q}_\alpha$ as a poset. For simplicity, we may assume that $\mathbb{Q}_\alpha$ and its order are ordinal definable.
Given two conditions $\vec{p} = {\langle} p_\alpha \mid \alpha < \kappa{\rangle},
\vec{q} = {\langle} q_\alpha \mid \alpha < \kappa{\rangle}$ in $\mathbb{P}$, it is natural to try forming extensions $\vec{p^*} = {\langle} p^*_\alpha \mid \alpha < \kappa {\rangle} \geq \vec{p}$, $\vec{q^*} = {\langle} q^*_\alpha \mid \alpha < \kappa {\rangle} \geq \vec{q}$, and an isomorphism $\varphi\mathbb{C}lon \mathbb{P}/\vec{p^*} \to \mathbb{P}/\vec{q^*}$ as follows:
By induction on $\beta {\langle}eq \kappa$, we attempt defining extensions $\vec{p}^\beta = {\langle} p^*_\alpha \mid \alpha < \beta{\rangle} $ of $\vec{p}restriction \beta$, and $\vec{q}^\beta = {\langle} q^*_\alpha \mid \alpha < \beta{\rangle}$ of $\vec{q}restriction \beta$, and an isomorphism $\varphi_\beta \mathbb{C}lon \mathbb{P}_\beta/\vec{p}^\beta \to \mathbb{P}_\beta/\vec{q}^\beta$.
Our inductive assumptions further include $\vec{p}^{\beta_1}restriction \beta_0 = \vec{p}^{\beta_0}$,
$\vec{q}^{\beta_1}restriction \beta_0 = \vec{q}^{\beta_0}$, and $\varphi_{\beta_1} restriction \mathbb{P}_{\beta_0}/\vec{p}^{\beta_0} = \varphi_{\beta_0}$,\footnote{i.e., the restriction
$\varphi_{\beta_1} restriction \mathbb{P}_{\beta_0}/\vec{p}^{\beta_0}$ is obtained by identifying conditions $\vec{r}^{\beta_0} \in \mathbb{P}_{\beta_0}/\vec{p}^{\beta_0}$ with their extension
$\vec{r}^{\beta_1} = \vec{r}^{\beta_0} {}^\frown (\vec{p}^{\beta_1}restriction_{[\beta_0,\beta_1)})$.}
for all $\beta_0 < \beta_1$.
For $\beta=0$, where $\mathbb{P}_0 = \{ 0_{\mathbb{P}_0}\}$ is a trivial forcing we take $\varphi_0$ to be the identity.
At a successor step, assuming $\vec{p}^\beta,\vec{q}^\beta$ and $\varphi_\beta$ have been defined, we have that $\varphi_\beta(\vec{p}^\beta) = \vec{q}^\beta$ forces that
$p_\beta^{\varphi_\beta}$ and $q_\beta$ are conditions of the cone homogeneous poset $\mathbb{Q}_\beta$. There are therefore $\mathbb{P}_\beta$-names
$p'_\beta$ and $q'_\beta$ of extensions of $p_\beta^{\varphi_\beta}$ and $q_\beta$, respectively, and a name of a cone isomorphism
$\psi_\beta \mathbb{C}lon \mathbb{Q}_\beta/p'_\beta \to \mathbb{Q}_\beta/q'_\beta$. We stress that we use the maximality principle, and do not extend the conditions $\vec{p}^\beta$ and $\vec{q}^\beta$ in order to determine the values of $p'_\beta, q'_\beta$ and $\psi_\beta$.
Let $p^*_\beta = (p'_\beta)^{\varphi_\beta}$ and $q^*_\beta = q'_\beta$. Clearly, $\vec{p}^\beta$,$\vec{q}^\beta$ force that $p^*_\beta,q^*_\beta$ extend $p_\beta, q_\beta$, respectively. We set $\vec{p}^{\beta+1} = \vec{p}^\beta {}^\frown {\langle} p^*_\beta{\rangle}$, $\vec{q}^{\beta+1} = \vec{q}^\beta {}^\frown {\langle} q^*_\beta{\rangle}$, and define
$\varphi_{\beta+1}\mathbb{C}lon \mathbb{P}_{\beta+1}/\vec{p}^{\beta+1} \to \mathbb{P}_{\beta+1}/\vec{q}^{\beta+1}$ by mapping a condition
$\vec{r} = \vec{r}restriction\beta {}^\frown {\langle} r_\beta{\rangle} \in \mathbb{P}_{\beta+1}/\vec{p}^{\beta+1}$ to
\[
\varphi_{\beta+1}( \vec{r}) = \varphi_\beta(\vec{r}restriction\beta) {}^\frown {\langle} \psi_\beta(r_\beta^{\varphi_\beta}) {\rangle}.
\]
It is immediate from our assumption of $\varphi_{\beta}$ and choice of $\psi_\beta$ that $\varphi_{\beta+1}$ is an isomorphism.
Finally, for a limit ordinal $\delta {\langle}eq \kappa$, $\vec{p}^{\delta}$ (similarly $\vec{q}^\delta$) is determined by the requirement
$\vec{p}^{\delta}restriction\beta = \vec{p}^\beta$ for all $\beta < \delta$ (similarly for $\vec{q}^{\delta}$),
and $\varphi_\delta$ by the requirement $\varphi_\deltarestriction \mathbb{P}_\beta/\vec{p}^\beta = \varphi_\beta$ for all $\beta < \delta$. See \cite{DrobinenFriedman} for more detailed proof for the validity of this construction. \vskip
amount
We conclude that for this construction to succeed the following conditions need to hold for all $\beta{\langle}eq \kappa$:
(i) $\vec{p}^\beta,\vec{q}^\beta$ are well-defined conditions in $\mathbb{P}_\beta$ which extend $\vec{p}restriction\beta,\vec{q}restriction\beta$ respectively,
and (ii) $\varphi_\beta$ is a well-defined cone isomorphism. \vskip
amount
If the construction succeeds throughout all stages $\beta {\langle}eq \kappa$, then the final conditions $\vec{p^*} = \vec{p}^\kappa$,
$\vec{q^*} = \vec{q}^\kappa$ and cone isomorphism $\varphi = \varphi_\kappa$, satisfy the required properties.
It is easy to see that condition (i) and (ii) may only fail at limit stages $\delta {\langle}eq \kappa$, where the precise formation of the iteration (e.g., its support)
may prevent $\vec{p}^\delta$ to be a condition in $\mathbb{P}_\delta$.
Similarly, the definition of the limit order ${\langle}eq_{\mathbb{P}_\delta}$ might prevent the defined map $\varphi_\delta$ to be an isomorphism.
This problem does not occur for finite iteration:
\begin{lemma}[{\cite{DrobinenFriedman}}]{\langle}anglebel{corr: OD implies homogeneous}
A finite iteration of ordinal definable cone homogeneous forcings is cone homogeneous.
\end{lemma}
Since our proof of theorem {\rangle}ef{THM1} is based on a construction of a \textbf{Magidor Iteration} $\mathbb{P} = {\langle} \mathbb{P}_\alpha,\mathbb{Q}_\alpha \mid \alpha < \theta{\rangle}$ of Prikry-type forcings $(\mathbb{Q}_\alpha,{\langle}eq_{\mathbb{Q}_\alpha},{\langle}eq^*_{\mathbb{Q}_\alpha})$, we conclude this section with a description of a specific variant of cone homogeneity for the posets $\mathbb{Q}_\alpha$, which guarantees that the Magidor iteration $\mathbb{P}$ is cone-homogeneous as well.
\begin{definition}[{Prikry type forcing, \cite{Gitik-HB}}]
${\langle}anglengle \mathbb{P}, {\langle}eq, {\langle}eq^*{\rangle}anglengle$ is a Prikry type forcing if
\begin{itemize}
\item ${\langle}eq \supseteq {\langle}eq^*$ are partial orders on $\mathbb{P}$ and
\item (the Prikry Property) for every statement $\sigma$ in the forcing language for ${\langle}anglengle \mathbb{P}, {\langle}eq{\rangle}anglengle$, and a condition $p$ there is a condition $p^*$, $p {\langle}eq^* p^*$ such that $p^* \Vdash \sigma$ or $p^* \Vdash \neg \sigma$.
\end{itemize}
\end{definition}
Conditions in the Magidor iteration $\mathbb{P} = {\langle} \mathbb{P}_\alpha,\mathbb{Q}_\alpha \mid \alpha < \kappa{\rangle}$ of Prikry type posets ${\langle} \mathbb{Q}_\alpha, {\langle}eq_{\mathbb{Q}_\alpha},{\langle}eq^*_{\mathbb{Q}_\alpha}{\rangle}$ are sequences $\vec{p} = {\langle} p_\alpha \mid \alpha < \kappa{\rangle}$ which beyond the standard requirement of $prestriction\alpha \Vdash p_\alpha \in \mathbb{Q}_\alpha$, also satisfy that for all but finitely many ordinals $\alpha < \kappa$, $\vec{p} restriction \alpha \Vdash p_\alpha \geq^*_{\mathbb{Q}_\alpha} 0_{\mathbb{Q}_\alpha}$. We note that in particular, the definition allows using full-support condition, as long as almost all components $p_\alpha$ are direct extensions of the trivial conditions. Similarly for the definition of the ordering ${\langle}eq_{\mathbb{P}}$, we have that $\vec{p'} \geq \vec{p}$ requires
both that $\vec{p'}restriction\alpha \Vdash p'_\alpha \geq_{\mathbb{Q}_\alpha} p_\alpha$ for all $\alpha$ and that
for all but finitely many ordinals $\alpha < \kappa$, $\vec{p'}restriction \alpha \Vdash p'_\alpha \geq^* p_\alpha$.
See \cite{Gitik-HB} for a comprehensive description of the Magidor iteration style and its main properties.
\begin{lemma}{\langle}anglebel{FACT:homogiter}
Suppose that $\mathbb{P} = {\langle} \mathbb{P}_\alpha,\mathbb{Q}_\alpha \mid \alpha < \kappa{\rangle}$ is a Magidor iteration of Prikry-type posets ${\langle} \mathbb{Q}_\alpha, {\langle}eq_{\mathbb{Q}_\alpha},
{\langle}eq^*_{\mathbb{Q}_\alpha}{\rangle}$ so that the following conditions hold for each $\alpha < \kappa$:
\begin{itemize}
\item[(i)]$\mathbb{Q}_\alpha$, ${\langle}eq_{\mathbb{Q}_\alpha}$ and ${\langle}eq^*_{\mathbb{Q}_\alpha}$ are ordinal definable in $V$, and
\item[(ii)] it is forced by $0_{\mathbb{P}_{\alpha}}$ that for every two conditions $p,q \in \mathbb{Q}_\alpha$ there are $p^* \geq_{\mathbb{Q}_\alpha}^* p$ and
$q^* \geq^*_{\mathbb{Q}_\alpha} q$ and a cone isomorphism $\psi_\alpha\mathbb{C}lon \mathbb{Q}_\alpha/p^* \to \mathbb{Q}_\alpha/q^*$ which respects the direct extension order ${\langle}eq^*_{\mathbb{Q}_\alpha}$.
\end{itemize}
Then $\mathbb{P}$ is cone homogeneous.
\end{lemma}
\begin{proof}
Let $\vec{p},\vec{q} \in \mathbb{P}$, and $(\vec{p}^\beta,\vec{q}^\beta,\varphi_\beta \mid \beta {\langle}eq \kappa)$ be the sequence obtained form the procedure described above.
It suffices to verify inductively, that conditions (i) and (ii) are satisfied by the sequence.
We note that in the successor step construction of $p'_\beta$, $p^*_\beta = (p'_\beta)^{\varphi_\beta^{-1}}$, $q^*_\beta = q'_\beta$, and $\psi_\beta$,
we may assume that $\varphi_\beta(\vec{p}^\beta) \Vdash p'_\beta \geq^*_{\mathbb{Q}_\beta} p_\beta^{\varphi_\beta}$, $\vec{q}^\beta\Vdash q'_\beta \geq^* q_\beta$, and that $\psi_\beta$ is ${\langle}eq^*_{\mathbb{Q}_\beta}$-preserving. Since ${\langle}eq^*_{\mathbb{Q}_\beta}$ is ordinal definable in $V$, $0_{\mathbb{P}_{\beta}} \Vdash {\langle}eq^*_{\mathbb{Q}_\beta} = ({\langle}eq^*_{\mathbb{Q}_\beta})^{\varphi_\beta^{-1}}$, and therefore by applying the automorphism $\varphi_\beta^{-1}$ we get
$\vec{p}^\beta \Vdash p^*_\beta \geq^*_{\mathbb{Q}_\beta} p_\beta$, and $p \mapsto \psi_{\beta}(p^{\varphi_\beta})$ is forced by $\vec{p}^\beta$ to be ${\langle}eq^*_{\mathbb{Q}_\beta}$-preserving in the cone below $p^*_\beta$.
In particular, assuming $\varphi_\beta$ is order preserving and $\vec{r}^{\beta+1} \geq \vec{s}^{\beta+1} \in \mathbb{P}_{\beta+1}/\vec{p}^{\beta+1}$,
$\vec{r}^{\beta+1} = \vec{r}^\beta {}^\frown {\langle} r_\beta{\rangle}$, $\vec{s}^{\beta+1} = \vec{s}^\beta {}^\frown {\langle} s_\beta{\rangle}$,
we have that if $\vec{r}^\beta \Vdash r_\beta \geq^*_{\mathbb{Q}_\beta} s_\beta$ then
$\varphi_{\beta}(\vec{r}^\beta) \Vdash \psi_{\beta}(r_\beta^{\varphi_\beta}) \geq^*_{\mathbb{Q}_\beta} \psi_{\beta}(r_\beta^{\varphi_\beta}) = \varphi_{\beta+1}(\vec{s}^{\beta+1})_\beta$.
The same conclusion holds for ${\langle}eq_{\mathbb{Q}_\beta}$ which is also ordinal definable in $V$.
\vskip
amount
We conclude that, first, $p^*_\beta, q_\beta^*$ are forced to be direct extensions of $0_{\mathbb{Q}_\beta}$ whenever $p_\beta,q_\beta$ are, which in turn, implies that
$\vec{p}^\alpha,\vec{q}^\alpha$ are conditions of $\mathbb{P}_\alpha$ for all $\alpha {\langle}eq \kappa$. Hence, (i) is satisfied.
Second, for every $\vec{r}^\alpha,\vec{s}^\alpha \in \mathbb{P}_\alpha/\vec{p}^\alpha$ and $\beta < \alpha$, if
$\vec{r}^\alpha restriction \beta \Vdash r_\beta \geq^*_{\mathbb{Q}_\beta} s_\beta$
then $\varphi_\alpha(\vec{r}^\alpha)restriction\beta \Vdash \varphi_\alpha(\vec{r}^\alpha)_\beta \geq^*_{\mathbb{Q}_\beta} \varphi_\alpha(\vec{s}^\alpha)_\beta$, and similarly, when replacing replacing ${\langle}eq^*_{\mathbb{Q}_\beta}$ with ${\langle}eq_{\mathbb{Q}_\beta}$.
It follows at once from this and the definition of the ordering ${\langle}eq_{\mathbb{P}_\alpha}$ of the Magidor iteration that $\varphi_\alpha$ is a cone isomorphism. Hence (ii) holds.
\end{proof}
\subsection{Homogeneous change of cofinalities}{\langle}anglebel{ssec-non-stationary-iteration}
Our approach to construct a model with an $\omega$-strongly measurable cardinal $\kappa$, is to force over a ground model satisfying $V = \HOD$ with a weakly homogeneous poset (i.e., therefore also cone-homogeneous) to form a generic extension $V[G]$ with
a cardinal $\kappa$, which satisfies the conditions of lemma {\rangle}ef{lem:sufficient.omega.strongly}.
In light of lemma {\rangle}ef{lem:necessary.omega.strongly} above, we see that many regular cardinals in $V$ need to change their cofinality in $V[G]$.
The main challenge in that regard, is to change the cofinality of many cardinals with a weakly homogeneous forcing.
Fortunately, such forcing has been constructed in \cite{bunger}, where the theory of non-stationary support iteration of Prikry-type forcings is developed, and employed to form a weakly-homogeneous variant of the Gitik iteration (\cite{gitik-nonstionary-ideal}). We note that as opposed to an Easton-style version of the Gitik iteration, which has a good chain condition (i.e., $\kappa$-c.c. when iterating up to a Mahlo cardinal $\kappa$), the non-stationary support variant of \cite{bunger} has a weaker, fusion-type property. \vskip
amount
We briefly describe the construction of the non-stationary support iteration $\mathbb{P}$ of iteration of Prikry-type forcings $\mathbb{Q}_{\alpha}$ from \cite{bunger}.
The iteration, which is based on the given coherent sequence of measures ${\langle} U_{\alpha,\tau}\mid \alpha <\kappa, \tau <o^{\mathcal{U}}(\alpha){\rangle}$ is nontrivial at each $\alpha< \kappa$, $o^{\mathcal{U}}(\alpha) > 0$.
As this $\alpha$, the forcing $\mathbb{Q}_\alpha$ adds a cofinal closed unbounded set $b_{\alpha}$ to $\alpha$ of order-type $\omega^{o(\alpha)}$ (ordinal exponentiation).
More specifically, given a $V$-generic filter $G_\alpha \subseteq\mathbb{P}_\alpha$, which adds clubs $b_\beta$, for $\beta< \alpha$, $o^{\mathcal{U}}(\beta) > 0$,
one considers finite sequences $t = {\langle}\nu_0,\dots,\nu_{k-1}{\rangle}$ with the property that for every $i < k-1$, if $o^{\mathcal{U}}(\nu_i) < o^{\mathcal{U}}(\nu_{i+1})$ then
$\nu_i \in b_{\nu_{i+1}}$ and $b_{\nu_{i+1}} \cap \nu_i = b_{\nu_i}$. Such sequences are called coherent (with respect to $G_\alpha$). If ${\rangle}ho$ is an ordinal so that $o(\nu_i) < {\rangle}ho$
for all $i < k$ then we say $t$ is ${\rangle}ho$-coherent. Otherwise, we denote by $trestriction {\rangle}ho$ to be the sub-sequence of $\nu_i \in t$ so that $o(\nu_i) < {\rangle}ho$.
Working at $V[G_\alpha]$, one constructs posets $\mathbb{Q}_{\alpha,\tau}$, $\tau {\langle}eq o^{\mathcal{U}}(\alpha)$, and simultaneously shows by induction on $\tau {\langle}eq o^{\mathcal{U}}(\alpha)$ that
for each $\tau$-coherent sequence $t$, $U_{\alpha,\tau} \in \mathcal{U}$ extends to $U_{\alpha,\tau}(t)$.
We define $\mathbb{Q}^0_{\alpha}$ to be the trivial poset, and
given that the measures $U_{\alpha,\tau'}(t')$, have been defined for every $\tau' < \tau$, and every $\tau'$-coherent sequence $t'$,
the forcing $\mathbb{Q}^\tau_{\alpha}$ consists of pairs $q = {\langle} t, T{\rangle}$ where $t$ is $\tau$-coherent, $T \subseteq[\alpha]^{<\omega}$ is a tree whose stem is $\emptyset$
and for every $s \in T$, $\suc_T(s) := \{ \mu < \kappa \mid s {}^\frown {\langle} \mu {\rangle} \in T\} \in \bigcap_{\tau' < \tau} U_{\alpha,\tau'}( t{}^\frown srestriction\tau')$.
Direct extensions and end extensions of $\mathbb{Q}_{\alpha}^\tau$ are defined as usual. $\mathbb{Q}_{\alpha}^\tau$ is a Prikry type forcing whose direct extension is $\alpha$-closed.
With $\mathbb{Q}_{\alpha,\tau}$ determined we consider the $V$-ultrapower by $U_{\alpha,\tau}$,
by taking $j_{\alpha,\tau}\mathbb{C}lon V \to M_{\alpha,\tau} \mathbb{C}ng Ult(V,U_{\alpha,\tau})$, and define for each $\tau$-coherent sequence $t$ a $V[G_\alpha]$ measure
$U_{\alpha,\tau}(t)$ by $X = \mathunderaccent\tilde-3 {X}_{G_\alpha} \in U_{\alpha,\tau}(t)$ if there exist $p \in G_\alpha$ and a valid tree $T$ such that
\[ p {}^\frown {\langle} t,T{\rangle} {}^\frown j_{\alpha,\tau}(p)\setminus(\alpha+1) \Vdash_{j_{\alpha,\tau}(\mathbb{P})} \check{\alpha} \in j_{\alpha,\tau}(\mathunderaccent\tilde-3 {X}).\]
\begin{fact}{\langle}anglebel{fact:summary-properties-of-PU} ${}$
\vskip
amount
\begin{enumerate}
\item For each $\alpha$ such that $o^{\mathcal{U}}(\alpha) > 0$, $b_\alpha$ is a cofinal sequence at $\alpha$ of order type $\omega^{o^{\mathcal{U}}(\alpha)}$ (ordinal exponentiation).
\item For each $\alpha {\langle}eq \kappa$, $(\mathbb{P}_\alpha,{\langle}eq,{\langle}eq^*)$ is a Prikry-type forcing.
\item For every $\gamma < \alpha {\langle}eq \kappa$, the quotient $(\mathbb{P}_\alpha/\mathbb{P}_\gamma, {\langle}eq,{\langle}eq^*)$ is a Prikry-type forcing whose direct extension order ${\langle}eq^*$ is $\gamma$-closed. In particular, the quotient $\mathbb{P}_\alpha/\mathbb{P}_\gamma$ does not add new bounded subsets to $\gamma$.
\item For every $\gamma < \alpha$, the iteration $\mathbb{P}_\alpha / \mathbb{P}_{\gamma+1}$ is weakly homogeneous.
\end{enumerate}
\end{fact}
\section{acknowledgments}
We are grateful to Sandra M\"{u}ller and Grigor Sargsyan for valuable conversations concerning the Inner Model Program and the study of $\HOD$.
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\begin{document}
\begin{abstract}
We consider the interaction between passing to finite covers
and ergodic properties of the straight-line flow on finite area translation surfaces with infinite topological type. Infinite type provides for a rich family of degree $d$ covers for any integer $d>1$. We give examples which demonstrate that passing to a finite cover can destroy ergodicity, but we also provide evidence that this phenomenon is rare. We define a natural notion of a random degree $d$ cover and show that, in many cases, ergodicity and unique ergodicity are preserved under passing to random covers. This work provides a new context for exploring the relationship between recurrence of the Teichm\"uller flow and ergodic properties of the straight-line flow.
\end{abstract}
\dedicatory{
\begin{flushleft}The private lives of surfaces \\
are innocent, not devious. \\
\hspace{1in} - Kay Ryan
\end{flushleft}}
\maketitle
\thispagestyle{empty}
\section{Introduction}
A {\em translation surface} is a pair $(S, \alpha)$, where $S$ is a Riemann surface and
$\alpha$ is a holomorphic 1-form on $S$. Let $Z \subset S$ denote the zeros of $\alpha$.
The $1$-form $\alpha$ endows $S \smallsetminus Z$ with local coordinates to the plane: for any
$p$ we have the locally defined {\em coordinate chart} to $\C$ given by the local homeomorphism
$q \mapsto \int_p^q \alpha$. These coordinate charts differ locally only by translation.
A translation surface inherits a metric by pulling back the Euclidean metric on the plane along the coordinate charts.
Points in $S \smallsetminus Z$ are locally isometric to the plane, while points in $Z$ are cone singularities with cone angle $2(k+1) \pi$
where $k \geq 1$ is the degree of the zero of $\alpha$.
We'll say a translation surface is {\em classical} if $S$ is a closed surface.
Here, there is a well known interplay between two types of dynamical systems:
{\em (1)} the dynamics of the (horizontal) {\em translation flow} on a translation surface $(S,\alpha)$ given in local coordinates
by
$$F^t:S \to S; \quad (x,y) \mapsto (x+t,y),$$
and {\em (2)} the Teichm\"uller deformation $g^t$ on the moduli space of translation surfaces, where
$g^t(S,\alpha)$ is obtained from $(S,\alpha)$ by postcomposing the coordinate charts with the affine coordinate change
$g^t(x,y)=(e^{-t} x, e^t y)$. Namely, the Teichm\"uller deformation renormalizes the translation flow. A famous consequence of this relationship is given by Masur's Criterion: if the forward orbit of the Teichm\"uller deformation, $\{g^t(S,\alpha)\}_{t \geq 0}$, has a convergent subsequence $g^{t_n}(S,\alpha)$ with $t_n \to \infty$ (i.e., the orbit is {\em non-divergent}), then the translation flow is uniquely ergodic \cite{M92}.
A surface has {\em infinite topological type} if its fundamental group is not finitely generated. This includes infinite genus surfaces.
Recently, many results similar in spirit to Masur's criterion have been proven in special cases for translation surfaces of infinite
topological type \cite{Hinf,HW13, rodrigo:erg, trevino:bratMasur}.
Reasoning about infinite type translation surfaces introduces several difficulties:
\begin{enumerate}
\item Translation flow trajectories on a infinite type surface may not be defined for all time. In fact, there are interesting examples where trajectories are defined nowhere for some fixed time, for example the Icicled surface of Randecker \cite[Example 5.5]{Randecker}. In order to consider ergodicity, we require the translation flow to be {\em defined for all time almost everywhere} (i.e., for all $t$, the time $t$ map $F^t$ is defined a.e.).
\item While the translation flow is continuous, the domain is not compact.
\item There is no well established moduli space of infinite-type translation surfaces. (In this direction, the preprints \cite{HooperImmersions1} and \cite{HooperImmersions2} propose a topology for the space of translation surfaces with basepoint.)
\end{enumerate}
In the classical case (1) and (2) can be easily resolved (via standard coding arguments) and moduli space is a finite dimension orbifold.
The translation flow can often be easily shown to be defined for all time almost everywhere, and we will in fact be assuming stronger hypotheses. Difficulty (2) simply must be worked around. A standard approach to work around (3) involves defining some topological space of surfaces where $\SL(2,\R)$ acts and prove that non-divergence of an orbit $t \mapsto g^t(S, \alpha)$ within this space entails consequences for the translation flow on $(S,\alpha)$. To date, the primary mechanism for
building such a topological space of surfaces uses affine symmetries of the translation surface $(S,\alpha)$, as we will now explain.
Two translation surfaces
$(S,\alpha)$ and $(S', \alpha')$ are {\em translation equivalent} if there is a homeomorphism $h:S \to S'$ which is a translation in local coordinates.
Let $\SL_\pm(2,\R)$ denote the group of $2 \times 2$ real matrices with determinant $\pm 1$.
The group $\SL_\pm(2,\R)$ acts on the collection of all translation surfaces by simultaneous postcomposition with all local charts;
see \S \ref{sect:background}.
We define $\sO(S,\alpha)$ to be the $\SL_\pm(2,\R)$ orbit:
\begin{equation}
\label{eq:sO}
\sO(S,\alpha)=\{A(S,\alpha)~:~A \in \SL_\pm(2,\R)\}/\text{translation equivalence}.
\end{equation}
The orbit $\sO(S,\alpha)$ is parameterized by a choice of $A \in \SL_\pm(2,\R)$
and thus inherits a topology as a topological quotient space. This space can be described concretely
as the quotient of $\SL_\pm(2,\R)$ by a subgroup, namely the surface's {\em Veech group},
$$V(S,\alpha) = \{A \in \SL_\pm(2,\R)~:~\text{$A(S,\alpha)$ is translation equivalent to $(S,\alpha)$}\}.$$
Observe that two surfaces $A_1(S,\alpha)$ and $A_2(S,\alpha)$ with $A_1,A_2 \in \SL_\pm(2,\R)$ are
translation equivalent if and only if $A_1$ and $A_2$ determine the same left coset (or equivalently, if $A_2^{-1} A_1 \in V(S,\alpha)$). Thus, there is a natural identification between $\sO(S,\alpha)$ and the left coset space
$\SL_\pm(2,\R)/V(S,\alpha)$.
In particular, this structure allows us to say
that the Teichm\"uller trajectory $g^t(S,\alpha)$ is
{\em non-divergent in $\sO(S,\alpha)$} if there is a sequence $t_n \to +\infty$ so that
$g^{t_n}(S,\alpha)$ converges in $\sO(S,\alpha)$.
Work of the second author \cite[Theorem 2]{rodrigo:erg} (in this paper as Corollary \ref{cor:rodrigo ergodicity from Veech group})
shows that if a finite area translation surface of possibly infinite genus $(S,\alpha)$ has a non-divergent Teichm\"uller trajectory in $\sO(S,\alpha)$, then the translation flow is ergodic on $(S,\alpha)$.
It seems reasonable to conjecture that in fact unique ergodicity holds under these hypotheses: Separate works of the two co-authors each prove results to this effect in special cases; see Example F.2 and Appendix H of \cite{Hinf} and the main result of \cite{trevino:bratMasur}.
Suppose $(S,\alpha)$ is a finite area translation surface with infinite topological type which has a translation flow which is defined for all time almost everywhere and is ergodic.
Any cover of $S$ inherits a translation structure by pulling back the $1$-form $\alpha$ under the covering map.
When endowed with this structure, the cover's translation flow will also be defined for all time almost everywhere.
In this paper we study the ergodicity of the translation flow on finite unbranched covers $(\tilde S, \tilde \alpha)$
of $(S,\alpha)$. We introduce some spaces of covers and following the above paradigm study how the behavior of Teichm\"uller trajectories through this space influences the ergodicity of the translation flow on a cover.
In order to say something constructive, we pick some integer $d \geq 2$, and restrict attention to covers of degree $d$. Choose an arbitrary non-singular basepoint on the surface $S$. Let $(\tilde S, \tilde \alpha)$ be a cover of $(S,\alpha)$, where the flat structure on $(\tilde S, \tilde \alpha)$ is lifted from the one on $(S,\alpha)$. The fiber of the basepoint can be identified in an arbitrary way with the set $\{1,2, \ldots, d\}$. The {\em monodromy action} is the natural right action of the fundamental group of $S$ on the fiber of the basepoint. Our identification of the fiber determines
a {\em monodromy representation} $\pi_1(S) \to \Pi_d$, where $\Pi_d$ is the permutation group of $\{1, \ldots, d\}$. We note that a cover can be reconstructed from its monodromy representation, and two covers are
isomorphic (in the sense of covering theory) if and only if these monodromy representations differ by conjugation by an element of $\Pi_d$. These ideas are reviewed in \S \ref{sec:spaces}.
Now fix a subgroup $G \subset \Pi_d$. We say a cover $(\tilde S, \tilde \alpha)$
has {\em monodromy in $G$} if it can be realized by a monodromy representation to $G$. From the above remarks, we note that the space of covers of $(S,\alpha)$ with monodromy in $G$ up to cover isomorphism is identified with
$$\Pi_d \bs \Hom\big(\pi_1(S),G\big),$$
where $\Pi_d$ acts on $\Hom\big(\pi_1(S),G\big)$ by conjugation.
We endow $\Hom\big(\pi_1(S),G\big)$ with the product topology by viewing it as a subset of $G^{\pi_1(S)}$,
where the finite set $G$ is given the discrete topology. In particular, $\Hom\big(\pi_1(S),G\big)$ is homeomorphic to a Cantor set, since $\pi_1(S)$ is a free group with countably many generators; see \S \ref{sec:spaces}.
We define
$\text{Cov}_G(S,\alpha)$ to be the collection of all covers of $(S,\alpha)$ with monodromy in $G$ up to translation equivalence.
Translation equivalence is a coarser notion of equivalence than covering isomorphism, and we give $\text{Cov}_G(S,\alpha)$ the quotient topology by viewing this space of covers as a
topological quotient of $\Pi_d \bs \Hom\big(\pi_1(S),G).$
The choice of $h \in \Hom\big(\pi_1(S),G)$ determines a cover $(\tilde S_h, \tilde \alpha_h)$ with $h$ describing the monodromy of the cover. We note that the cover need not be connected. In fact $(\tilde S_h, \tilde \alpha_h)$ is connected
if and only if the image of the monodromy representation, $h\big(\pi_1(S)\big)$ acts transitively on $\{1, \ldots, d\}$.
The main object of interest to this paper is the union of $\SL_\pm(2, \R)$ orbits of
covers of $(S,\alpha)$ with monodromy in $G$:
\begin{equation}
\label{eq:covers cocycle}
\tilde \sO_G(S,\alpha)=\{A (\tilde S, \tilde \alpha):~
\text{$A \in \SL_\pm(2,\R)$ and $(\tilde S, \tilde \alpha) \in \Cov_G(S,\alpha)$}\}
/\sim,
\end{equation}
where $\sim$ denotes translation equivalence.
We call the action of the diagonal subgroup $g^t$ of $\SL_\pm(2, \R)$ on $\tilde \sO_G(S,\alpha)$ the {\em cover cocycle}. Again, $\tilde \sO_G(S,\alpha)$ inherits a topology because it can be considered a topological quotient
of $\SL_\pm(2,\R) \times \Cov_G(S,\alpha)$. This is a natural space in which to study Teichm\"uller deformations
of covers of $(S, \alpha)$. Like $\Cov_G(S,\alpha)$, this space contains both connected and disconnected surfaces
provided $G$ acts transitively on $\{1,\ldots,d\}$. We prove the following:
\begin{theorem}[Connected accumulation point implies ergodicity]
\label{thm:1}
Let $d \geq 2$ be an integer and let $G \subset \Pi_d$ be a subgroup which acts transitively on $\{1,\ldots, d\}$.
Let $(S,\alpha)$ be a finite area translation surface
with infinite topological type, and let $(\tilde S, \tilde \alpha)$ be a cover with monodromy in $G$.
Then, the translation flow on $(\tilde S, \tilde \alpha)$ is defined for all time almost everywhere and is ergodic if
the Teichm\"uller trajectory $g^t(\tilde S, \tilde \alpha)$ has an
$\omega$-limit point in $\tilde \sO_G(S,\alpha)$ representing a connected surface.
\end{theorem}
Theorem \ref{thm:1} is proved in \S \ref{sec:erg}.
We note that ergodicity of the translation flow on a finite cover has consequences for classifying the invariant measures.
\begin{proposition}[Ergodicity and unique lifts of measures]
\label{prop:lifting measures}
Suppose that $(\tilde S, \tilde \alpha)$ is a finite cover of the finite area translation surface $(S, \alpha)$ with infinite topological type. Also suppose that both surfaces have translation flows which are defined for all time almost everywhere and are ergodic.
Then, Lebesgue measure on $(\tilde S,\tilde \alpha)$ is the unique translation flow-invariant measure which projects to Lebesgue measure on $(S, \alpha)$ under the covering map.
\end{proposition}
See \S \ref{sect:proofs} for the brief proof.
\begin{corollary}[Lifting unique ergodicity]
\label{cor:lifting}
Suppose the conditions of Proposition \ref{prop:lifting measures} are satisfied
and additionally the translation flow on $(S, \alpha)$ is uniquely ergodic.
Then, the translation flow on the cover $(\tilde S,\tilde \alpha)$ is uniquely ergodic.
\end{corollary}
\begin{proof}
Suppose $\mu$ is a translation flow-invariant probability measure on $(\tilde S,\tilde \alpha)$.
Then its push forward under the covering map, $p_\ast \mu$ is a translation-flow invariant measure on $(S, \alpha)$, and by uniqueness is necessarily Lebesgue measure. The proposition guarantees that $\mu$
is also Lebesgue measure.
\end{proof}
The existence of a Veech group for the surface $(S,\alpha)$ gives a mechanism for generating the type of
accumulation we need for applying Theorem \ref{thm:1}:
\begin{proposition}
\label{prop:accumulation points}
Suppose $(S,\alpha)$ is a finite area translation surface with infinite topological type, and let $(\tilde S, \tilde \alpha)$ be a cover with monodromy in $G$.
If $(S,\alpha)$ has a non-divergent Teichm\"uller trajectory in $\sO(S,\alpha)$, then the Teichm\"uller trajectory of the cover $g^t(\tilde S, \tilde \alpha)$ has an $\omega$-limit point in $\tilde \sO_G(S,\alpha)$.
\end{proposition}
Again, see \S \ref{sect:proofs} for the proof. In order to use the conclusions of Theorem \ref{thm:1},
we would need to know that there is a connected $\omega$-limit point. We expect that it is
difficult in general to determine precisely when this is true. However, we will introduce
a notion under which most covers have connected accumulation points.
\begin{comment}
We will give a brief overview
here, but the topic is formally treated in \S \ref{sect:cocycle}.
We'd like to parameterize covers of surfaces affinely equivalent to $(S,\alpha)$
with monodromy in $G$.
Such a cover can be specified by a pair $\big(A, (\tilde S, \tilde \alpha)\big)$, where
$A \in \SL_\pm(2,\R)$ and $(\tilde S, \tilde \alpha) \in \text{Cov}_G(S,\alpha)$. This pair
would then represent the surface $A(\tilde S, \tilde \alpha)$ which covers $A(S,\alpha)$.
The Veech group $V(S, \alpha)$ acts naturally on $\text{Cov}_G(S,\alpha)$.
Observe that if $R \in V(S, \alpha)$, then the pair $\big(AR^{-1}, R(\tilde S, \tilde \alpha)\big)$ represents the same surface. Therefore, the quotient space
\begin{equation}
\label{eq:tilde O}
\tilde \sO_G(S,\alpha)=\big(\SL_\pm(2,\R) \times \text{Cov}_G(S,\alpha)\big)/V(S,\alpha)
\end{equation}
is canonically identified with the collection of all surfaces which are affinely equivalent to a surface in $\text{Cov}_G(S,\alpha)$.
(Here, $W$ acts simultaneously on both factors as just described.)
Further, $\SL_\pm(2,\R)$ has a well defined left action, which descends from left multiplication on the $\SL_\pm(2,\R)$
factor of the product $\SL_\pm(2,\R) \times \text{Cov}_G(S,\alpha)$.
We call $\tilde \sO_G(S,\alpha)$ the {\em cover cocycle} associated to $G$,
and endow it with the quotient topology.
We use $G^t$ to denote the left-action of the diagonal subgroup on $\tilde \sO_G(S,\alpha)$.
We say the Teichm\"uller trajectory of a cover $(\tilde S, \tilde \alpha) \in \Cov_G(S,\alpha)$ has a {\em connected accumulation point}
if there is a sequence of times $t_n$ tending to $+\infty$ so that $G^{t_n}[Id, (\tilde S, \tilde \alpha)]$ converges to a
limit in $\tilde \sO_G(S,\alpha)$ that parameterizes a connected surface.
$$\tilde \sO_G(S,\alpha)=\bigcup_{(S',\alpha') \in \sO(S,\alpha)} \text{Cov}_G(S',\alpha'),$$
the space of covers of surfaces in $\sO(S,\alpha)$ with monodromy in $G$. This space is a lamination\footnote{I would call this a foliated space in the sense of Moore and Schochet. Why lamination?}: There is local product structure coming from the
natural projection
$p:\tilde \sO_G(S,\alpha) \to \sO(S,\alpha)$ and from the locally natural identifications of between fibers.
Each fiber $p^{-1}(S', \alpha')$ is precisely $\text{Cov}_G(S',\alpha')$, which is homeomorphic to
a Cantor set. In order to understand the space $\tilde \sO_G(S,\alpha)$, and the $g^t$ action on it,
we introduce the notion of the
\end{comment}
Let $G \subset \Pi_d$ be as above, and let $(S,\alpha)$ be a finite area translation surface with infinite topological type. We explain in \S \ref{sect:measures} that there is a natural Borel probability measure $m_G$ on the space $\text{Cov}_G(S,\alpha)$ of covers with monodromy in $G$ which is invariant under automorphisms of the base surface $S$. This gives us a notion of a ``random cover.''
Informally, the measure $m_G$ corresponds to the notion of random cover
obtained by flipping a fair coin to determine the images
of the generators of the fundamental group under the monodromy representation.
\begin{theorem}[Random covers accumulate on connected covers]
\label{thm:2}
Let $(S,\alpha)$ be a finite area translation surface
with infinite topological type, and suppose that it has a non-divergent Teichm\"uller trajectory in $\sO(S,\alpha)$.
Suppose $G$ is a transitive subgroup of the permutation group $\Pi_d$.
Then $m_G$-almost every cover $(\tilde S, \tilde \alpha) \in \Cov_G(S, \alpha)$ has
a Teichm\"uller trajectory with a connected accumulation point.
\end{theorem}
The proof lies in \S \ref{sect:proofs}. As a consequence of Theorem \ref{thm:1}, we see:
\begin{corollary}[Ergodicity of random covers]
\label{cor:2}
With the hypotheses of Theorem \ref{thm:2}, $m_G$-almost every cover of $(S, \alpha)$ with monodromy in $G$
has ergodic translation flow.
\end{corollary}
\begin{comment}
\compat{I'm not sure we want to keep this paragraph. It is not edited yet. Note that if we strike it, we need to remove the corresponding sentence from the abstract.}
The existence and naturality of the measure $m_G$ on the space of covers $\text{Cov}_G(S,\alpha)$ has a nice consequence in terms of producing Teichm\"uller flow-invariant probability measures on cover cocycles. Suppose $W$ is a discrete subgroup of the Veech group of $(S,\alpha)$, and that $\mu$ is a Borel probability measure on $\SL_\pm(2,\R)/W$ which is invariant under left multiplication
by elements of the diagonal subgroup, $\{g^t\}$. Choose a Borel fundamental domain $D \subset \SL_\pm(2,\R)$ for the $W$-action. Then, a Borel subset $E \subset \tilde \sO_G(S,\alpha)$ can be written in a unique way as a disjoint union
$$E=\bigcup_{A \in D} (\{A\} \times E_A)/W,$$
where $E_A \subset \text{Cov}_G(S,\alpha)$ are Borel. The measure $\tilde \mu_G$ on $\tilde \sO_G(S,\alpha)$ defined by
$$\tilde \mu_G(E)=\int_{D} m_G(E_A)~d \mu(A),$$
is a $g^t$-invariant probability measure on $\tilde \sO_G(S,\alpha)$.
\end{comment}
Corollary \ref{cor:2} has applications to certain types of finite skew product extensions of $n$-adic odometers as follows.
For any integer $n \geq 2$, let $X_n = \{0,\dots, n-1\}^\mathbb{N}$ be given the product topology.
The \emph{$n$-adic odometer} is the map $\Omega_n: X_n \rightarrow X_n$ defined as addition by $1$ with infinite carry to the right:
\begin{equation}
\label{eq:odometer}
\Omega_n : x = (x_1,x_2, \dots) \mapsto (0, 0, \dots, x_{\kappa(x)} + 1, x_{\kappa(x) + 1}, x_{\kappa(x) + 2}, \dots ),
\end{equation}
where $\kappa(x) \in \N \cup \{+\infty\}$ is the smallest index $k \in \N$ so that $x_k \neq n-1$ or $+\infty$ if no such index exists.
Denote by $\Gamma^+$ the free group on the countably infinite set of generators $\{\gamma_i:~i \in \N\}$.
Let $n \geq 2$ be an integer and $G$ a subgroup of $\Pi_d$. For any $\psi_+ \in \mathrm{Hom}(\Gamma^+,G)$, we define the skew product over the $n$-adic odometer $E_{\psi_+}: X_n \times \{1,\dots, d\} \rightarrow X_n \times \{1,\dots, d\}$ as
\begin{equation}
\label{eqn:skew}
E_{\psi_+}(x,m) = \big(\Omega_n(x), \psi_+(\gamma_{\kappa(x)})\,(m)\big),
\end{equation}
where $\kappa(x)$ is as above. The image $E_{\psi_+}(x,m)$ is well defined unless $\kappa(x)=+\infty$, but we ignore this issue because points have well defined orbits off a countable set.
The space $\mathrm{Hom}(\Gamma^+,G)$ parameterizes these skew products and comes with a natural product measure as in Definition \ref{def:random cover} in \S \ref{sect:measures}. We will denote this measure by $\mu_+$.
\begin{theorem}
\label{thm:skew}
Let $n \geq 2$ be an integer and $G$ be a subgroup of $\Pi_d$ that acts transitively on $\{1,\ldots, d\}$. Then for $\mu_+$-almost every $\psi_+ \in \mathrm{Hom}(\Gamma^+,G)$, the skew product $E_{\psi_+}$ is uniquely ergodic.
\end{theorem}
The connection between translation flows and $n$-adic odometers comes from the fact that the suspension flow over an $n$-adic odometer is measurably isomorphic to the translation flow on the infinite genus surface first studied by Chamanara. The skew-products described here have a suspension given by translation flow on a cover of Chamanara's surface. As such, the theorem follows as a consequence of Corollaries \ref{cor:lifting} and \ref{cor:2}. We prove this result at the beginning of \S \ref{sect:Chamanara}.
\subsection*{Devious covers}
The above discussion has left open a natural question: What happens if $g^t(S,\alpha)$ is non-divergent in $\sO(S,\alpha)$, but the Teichm\"uller orbit of the connected cover $(\tilde S, \tilde \alpha)$ only accumulates
on disconnected covers in $\tilde \sO_G(S,\alpha)$. We call such covers {\em devious}.
Apparently there is no {\em a priori} reason why the translation flow on a devious cover should be ergodic or non-ergodic. We illustrate this by example in \S \ref{sect:evil covers}.
In \S \ref{sect:Chamanara} after proving Theorem \ref{thm:skew}, we
concentrate on devious covers of the surface $(S_2,\alpha_2)$
of Chamanara which is related to the $2$-adic odometer. Devious $G$-covers with non-ergodic translation flow
occur for all $G$ for fairly trivial reasons in this case. (The surface $(S_2,\alpha_2)$ has horizontal saddle connections which can connect the surface but are not seen by the translation flow.)
However we also exhibit devious covers of $(S_2,\alpha_2)$ whose translation flow is uniquely ergodic. Our construction of such covers works as long as the subgroup $G \subset \Pi_d$ is large enough to contain two subgroups $H_1$ and $H_2$ which fail to act transitively on $\{1,\ldots, d\}$ but so that $H_1 \cup H_2$ generates a subgroup of $G$ which does act transitively.
The translation flow on Chamanara's surface is special for many reasons, including that the Teichm\"uller trajectory of this surface is periodic in $\sO(S_2, \alpha_2)$, and so we consider a different surface in \S \ref{sect:ladder surface}. We consider double covers of affine images of a translation surface we call the ladder surface, $(S_\ladder,\alpha_\ladder)$. (See Figure \ref{fig:ladder_surface} on page \pageref{fig:ladder_surface}.)
The ladder surface has a non-elementary Veech group containing a parabolic and a reflection.
We produce several possible behaviors of the translation flow on double covers of $A(S_\ladder,\alpha_\ladder)$:
\begin{enumerate}
\item[(L1)] For some $A \in \SL_\pm(2,\R)$, there are no devious double covers of $A(S_\ladder,\alpha_\ladder)$, i.e.
for every connected double cover $(\tilde S,\tilde \alpha)$, the forward orbit $g^t A(\tilde S,\tilde \alpha)$ accumulates on a connected cover in $\tilde \sO_{\Z_2}(S_L,\alpha_L)$ so that $A(\tilde S,\tilde \alpha)$ always has ergodic translation flow.
\item[(L2)] For some $A \in \SL_\pm(2,\R)$, there are devious double covers of $A(S_\ladder,\alpha_\ladder)$ but nonetheless every connected double cover has ergodic translation flow.
\item[(L3)] For some $A \in \SL_\pm(2,\R)$, there are devious double covers of $A(S_\ladder,\alpha_\ladder)$ and every devious double cover has non-ergodic translation flow.
\end{enumerate}
Let us briefly summarize how to distinguish the cases above. We use $V' \subset \SL_\pm(2,\R)$ to denote the
known Veech group of $(S_L,\alpha_L)$. In all the cases we assume that $g^t A V'$ is non-divergent in $\SL_\pm(2,\R) / V'$, which implies that $g^t A(S_L, \alpha_L)$ is non-divergent in $\sO(S_L,\alpha_L)$.
We find an infinite index subgroup $\tilde V' \subset V'$ with the property that $\tilde V' \subset V(\tilde S, \tilde \alpha)$ for every double cover $(\tilde S, \tilde \alpha)$ of $(S_L, \alpha_L)$.
We are in case (L1) if $g^t A \tilde V'$ is non-divergent in $\SL_\pm(2,\R) / \tilde V'$. (Statement (L1) follows from Proposition \ref{prop:non-divergence covers} of \S \ref{sect:ladder surface}.) Now assume $g^t A \tilde V'$ is divergent.
We describe in Theorem \ref{thm:main ladder} a combinatorial measurement $v(A)$ of the divergence rate
of the trajectory $g^t A \tilde V'$ in $\SL_\pm(2,\R)/\tilde V'$. As $v(A)$ decreases the trajectory diverges quicker. We show that if $v(A)>\varphi^2$ where $\varphi$ is the golden mean then we are in case (L2). On the other hand if $v(A)<\varphi^2$, then we are in case (L3). (Statements (L2) and (L3) appear again in Theorem \ref{thm:main ladder}.) The sharpness of this transition demonstrates the power of the methods used to distinguish ergodic and non-ergodic behavior. We wonder:
\begin{enumerate}
\item[(L2$\frac{\text{1}}{\text{2}}$)] Could there be an $A \in \SL_\pm(2,\R)$ (with $v(A)=\varphi^2$) so that some devious double covers
of $(S_\ladder,\alpha_\ladder)$ exhibit ergodic translation flow while others exhibit non-ergodic translation flow?
\end{enumerate}
\begin{comment}
On the other hand, it is not hard to produce examples of skew products as above which are not ergodic (and we do so
in \S \ref{sect:Chamanara}). Such skew products are associated to covers which only accumulate on disconnected covers.
But accumulating only on disconnected covers does not guarantee non-ergodicity.
In the proof of Theorem \ref{thm:Reza slow} of \S \ref{sect:Chamanara}, we explicitly construct covers of Chamanara's
surface which only accumulate on disconnected covers but nonetheless have uniquely ergodic translation flow.
In the proof we make use of the main result of \cite{rodrigo:erg}, restated here as Theorem
\ref{thm:integrability}. The mechanism for ergodicity here has an analog in the classical setting of closed translation surfaces: if a surface diverges sufficiently slowly under Teichm\"uller flow, then the translation flow is uniquely ergodic
\cite{CE07} \cite{rodrigo:erg}.
\end{comment}
\begin{comment}
I'm not sure how relevant this is!
We also have a mechanism to produce $g^t$-invariant probability measures on the space $\tilde \sO_G(S,\alpha)$.
Let $\mu$ be a $g^t$-invariant Borel probability measure on $\sO(S,\alpha)$. (We note that if the Veech group is a non-elementary Fuchsian group then there are a myriad of such measures, as each corresponds to a geodesic flow invariant probability measure on $\H^2/V(S,\alpha)$.) In \red{WHERE}, we produce
a canonical $g^t$-invariant Borel probability measure $\tilde \mu$ on $\tilde \sO_G(S,\alpha)$
so that $\mu=\tilde \mu \circ p^{-1}$, and so that the support of $\tilde \mu$ is the collection of all covers
over surfaces in the support of $\mu$. Then as a consequence of Poincar\'e recurrence, we see that $\tilde \mu$-almost every surface is
\end{comment}
\subsection*{Acknowledgments}
The authors would like thank the anonymous referee for numerous useful comments which greatly improved the exposition. This project began at ICERM which provided a stimulating environment.
W. P. H. was supported by N.S.F. Grants DMS-1101233 and DMS-1101233 as well as by a PSC-CUNY Award (funded by The Professional Staff Congress and The City University of New York). R. T. was supported by BSF grant 2010428, ERC starting grant DLGAPS 279893, NSF Postdoctoral Fellowship DMS-1204008, and an AMS-Simons Travel Grant. The quote from on the first page is taken from \emph{Surfaces} in \cite{RyanPoems}.
\section{Context on finite area flat surfaces of infinite topological type}
There has been an increased interest in the study of the dynamics and geometry of flat surfaces of infinite genus.
Unlike classical flat surfaces (which are compact flat surfaces of finite type), there is no natural space for
parametrizing flat metrics for all surfaces of a given topological type. This gives the first obstacle to utilizing tools from the theory of compact surfaces. Different techniques have been developed to overcome this fundamental shortcoming, which prevents us from developing a theory to answer one of the most basic questions: whether the translation flow on a given flat surface of infinite type and finite area is ergodic or not.
Most studies concentrate on the case of the surface having finite or infinite area (a notably exception being \cite{Hinf}, where the methods work for surfaces of finite and infinite area). Such choice has great implications to the tools used, the results obtained, and the method of construction used to produce examples or to define some ``spaces of surfaces''. A common tool in both contexts is the use of Veech groups, which are a sort of symmetry groups of the surface. For compact flat surfaces, these are always discrete subgroups of $SL(2,\mathbb{R})$. Flat surfaces of finite type with non-trivial Veech groups are part of a very deep theory
which has grown out of the foundational contributions of Thurston \cite{T88} and Veech \cite{V}, so the fact that they can be used in the infinite type setting is encouraging.
We will concentrate here on the development of the theory of flat surfaces of infinite type and finite area, though there is also a rapidly developing theory of flat surfaces of infinite type and infinite area (where often interest centers on infinite abelian covers of finite type flat surfaces).
To our knowledge, the first papers on dynamics on flat surfaces of infinite type are those which come out of the infinite step polygonal billiards introduced in \cite{troubetzkoy:infinite, infinite-step} through the unfolding procedure. All such surfaces considered were of finite area and came from ``rational'' polygons, i.e., the angles of the billiard from which the surfaces were constructed satisfied some rationality conditions. Ergodic properties as well as topological results were obtained for a large class of these types of surfaces. The approach of these articles is very different from the approach here, because arguments do not make use of Veech groups.
The seminal paper of Chamanara \cite{Chamanara04} introduced a 1-parameter family of flat surfaces of infinite type and of finite area with a non-trivial Veech group. The main results of that paper discussed the Veech groups that appear. Most importantly, even though the surfaces constructed possess many symmetries, the Veech group is never a lattice for any surface arising in this construction. We review Chamanara's construction in section \ref{sect:evil covers} and apply some of our results to spaces of covers thereof.
Another study of flat surfaces of infinite type and finite area has been the construction of Bowman which extends the Arnoux-Yoccoz family of flat surfaces to include a surface of infinite genus and finite area \cite{Bowman13}. This surface of infinite genus and finite area admits a affine automorphism with hyperbolic derivative. Moreover, it was found that the Veech group of this surface is isomorphic to $\mathbb{Z}\times \mathbb{Z}_2$ and that the directions preserved by the hyperbolic automorphism correspond to uniquely ergodic translation flows.
In \cite{Hinf}, a construction of Thurston is modified to produce infinite genus flat surfaces with non-elementary Veech groups. The construction sometimes produces finite area surfaces,
and in this case the translation flow can be shown to be uniquely ergodic on various affine images of the surfaces.
The work \cite{rodrigo:erg} addresses the question of ergodicity of translation flows for surfaces of finite area. The hypotheses of the main results are
independent of topological type and therefore can be used to determine when the translation flow of a flat surface of infinite genus and finite area is ergodic or uniquely ergodic. We use this criterion in section \ref{sec:erg} to study ergodic properties of translation flows on covers of infinite translation surfaces.
Finally, in \cite{LT:models} a way of constructing flat surfaces of finite area and infinite genus is developed through a connection to adic and cutting and stacking transformations,
generalizing constructions of Bufetov in the classical case \cite{Bufetov13}.
Among other things, it is shown there that translation flows on surfaces of infinite genus and finite area can exhibit behavior which does not occur for translation flows on compact flat surfaces.
\compat{I'm commenting out this. I think it is not true unless we place more hypotheses on the compact surface. I think it is true if we assumed the base compact surface recurs modulo its Veech group. ---- In the present article, we also obtain results which are also not present for translation flows on compact surfaces, namely the existence of devious (unbranched) covers.}
In general it is not well understood what Veech groups can arise for a flat surface of infinite type and finite area. In particular, it is unknown whether there exists a flat surface of infinite type and finite area whose Veech group is a lattice in $\SL_\pm(2,\mathbb{R})$.
(In contrast, it is known that for practically any subgroup $G$ of $\SL_\pm(2,\mathbb{R})$, there is a flat surface of infinite type and infinite area whose Veech group is $G$ \cite{PSV11}.)
As it can be seen, Veech groups have played a significant role in the study of translation flows. The properties of translation flows so far obtained for surfaces of infinite genus and finite area are mostly similar to those of compact flat surfaces.
In this paper, we broaden the scope of known phenomena of translation flows on flat surfaces by utilizing the structure provided by coverings to produce spaces of flat surfaces of infinite type and finite area which are invariant under Teichm\"uller deformations, and provide sufficiently interesting Teichm\"uller dynamics so that varied
phenomena appear for corresponding translation flows. This construction allows us to determine when the lift of a translation flow to a cover retains ergodic properties which the base surface possesses, such as unique ergodicity. This approach thus
overcomes some of the problems created by not having a natural parameter space
for infinite type flat surfaces.
\section{Background on translation surfaces}
\label{sect:background}
\begin{comment}A {\em translation surface} is a pair $(S, \alpha)$, where $S$ is a Riemann surface and
$\alpha$ is a holomorphic 1-form on $S$. Let $Z \subset S$ denote the zeros of $\alpha$.
The $1$-form $\alpha$ endows $S \smallsetminus Z$ with local coordinates to the plane: for any
$p$ we have the locally defined {\em coordinate chart} to $\C$ given by
$q \mapsto \int_p^q \alpha$. The choice of path is irrelevant: since $\alpha$ is closed, it is locally exact in any small disc containing $p$.
Furthermore, the transition function between such charts is given locally by translations.
The {\em flat metric} on $(S,\alpha)$ is obtained by pulling back the flat metric on $\mathbb{C}$ via the coordinate charts.
\end{comment}
\begin{comment}
No longer needed for my work. -Pat
As in the introduction, a translation surfaces can be presented as a pair $(S, \alpha)$, where $S$ is a Riemann surface and $\alpha$ is a holomorphic 1-form on $S$. Alternately,
a translation surface can also be presented as a (possibly infinite) collection of polygons in $\C$ with edges glued by translations. In this case, an equivalence classes of glued vertices can be considered as part of the surface
only when there are only finitely many vertices of polygons in the equivalence class.
The surface inherits a Riemann surface structure by pulling back the complex structure from the polygons, and we
get a holomorphic $1$-form from the coordinates provided by viewing the polygons and their edges in $\C$.
\end{comment}
Let $(S, \alpha)$ be a translation surface and let $Z \subset S$ denote the zeros of $\alpha$. The {\em straight line flow} on $(S,\alpha)$ in direction $\theta$ is the flow
$F^t_\theta:S \to S$ defined for $t \in \R$
given in local coordinates by $F^t_\theta(x,y)=(x+t \cos \theta, y +t \sin \theta)$ away from the zeros of $\alpha$.
We reserve the name {\em translation flow} for the straight line flow $F^t_\theta$ where
$\theta=0$. This is the flow on $S$ determined by the rightward unit vector field $X$. We will use $Y$ denote the upward unit vector field.
A more global definition of the straight line flows can be done as follows. Since $\alpha$ is holomorphic, the 1-forms $\Re(\alpha)$ and $\Im(\alpha)$ are harmonic, and thus closed. Therefore, the distributions $\mathrm{ker}\, \Im(\alpha)$ and $\mathrm{ker}\, \Re(\alpha)$ define a pair of foliations away from $Z$, the \emph{horizontal and vertical} foliations. The generators of the distributions in the unit tangent bundle of $S-Z$ are the vector fields $X_\alpha$ and $Y_\alpha$ which generate, respectively, the \emph{horizontal and vertical} flows. The translation flow, as defined above, then corresponds to the horizontal flow. From this point of view, we will denote by $\varphi_t^{X_\alpha}$ and $\varphi_t^{Y_\alpha}$, respectively, the horizontal and vertical flows.
There is a group of deformations of the flat metric on $(S,\alpha)$ which is parametrized by the group $GL(2,\mathbb{R})$. We will mostly be interested in the action of the area preserving subgroup $\SL_\pm(2,\R) \subset \GL(2,\R)$, and this is the $\SL_\pm(2,\R)$ action mentioned in the introduction.
Fix a matrix $A \in \GL(2,\R)$. We get new (non-conformal) local coordinate charts to the plane by postcomposing the charts on $(S,\alpha)$ to $\C$ by the real-linear map
$$A:\C \to \C; \quad x+iy \mapsto a_{1,1} x + a_{1,2}y + i (a_{2,1} x+a_{2,2} y).$$
Then, we get a new Riemann surface structure $S'$ on $S$ by pulling back the complex structure using these deformed
charts, and the charts determine a new holomorphic $1$-form $\alpha'$ on $S'$. We define $A(S,\alpha)=(S', \alpha')$. The action of the rotation subgroup $\SO(2,\mathbb{R})$ parametrizes the directional flows on a given surface $(S,\alpha)$: for $\theta \in \SO(2,\mathbb{R}) \simeq S^1$, the horizontal flow on $\theta(S,\alpha)$ corresponds to the straight line flow on $(S,\alpha)$ in direction $\theta$.
\compat{Please read this paragraph for correctness and clarity.}
Let $(S,\alpha)$ and $(S',\alpha')$ be translation surfaces. We say they are {\em translation equivalent} if there is a homeomorphism $h:S \to S'$ which is a locally a translation in local coordinate charts provided by the $1$-forms.
The {\em Veech group} of $(S,\alpha)$ is the subgroup $V(S,\alpha) \subset \SL_\pm(2,\R)$ of elements $A \in \SL_\pm(2,\R)$ so that
$A(S,\alpha)$ and $(S,\alpha)$ are translation equivalent. An {\em affine homeomorphism} from a translation surface $(S,\alpha)$ to another
translation surface $(S',\alpha')$ is a diffeomorphism $\phi:S \to S'$ whose derivative $D(\phi)$ is constant when measured using local coordinates provided by the $1$-forms. By considering the derivative $D(\phi)$ to be a real linear map we have $D(\phi) \in \GL(2, \R)$.
We can describe this condition in terms of the horizontal and vertical vectors on $(S,\alpha)$ and $(S,\alpha')$ using the equation
$$\left( \begin{array}{r}
\phi_\ast X_\alpha \\
\phi_\ast Y_\alpha
\end{array}\right)=A \left( \begin{array}{r}
X_{\alpha'} \\
Y_{\alpha'}
\end{array}\right),
$$
where $\phi_\ast$ denotes the push forward action on vector fields and
$A=D(\phi)$ is a $2 \times 2$ matrix (the {\em derivative}) .
Observe that the statement that $A(S,\alpha)$ and $(S',\alpha')$ are translation equivalent is equivalent to the statement that there is a affine homeomorphism $\phi:(S,\alpha) \to (S', \alpha')$ with derivative $A$.
The {\em affine automorphism group of $(S,\alpha)$}, $\Aff(S,\alpha)$ is the group of all affine homeomorphisms
from $(S,\alpha)$ to itself. By the prior observation, we have
$$D\big(\Aff(S,\alpha)\big)=V(S,\alpha).$$
\begin{comment}
The quotient
$$ Q(S,\alpha) = \SL(2,\R) / V(S,\alpha)$$
parameterizes the space of translation surfaces which are affinely equivalent to $(S,\alpha)$. Here we will only consider cases where $V(S,\alpha)$ is discrete so that $Q(S,\alpha)$ is a reasonable space. If we identify $(S,\alpha)$ with the identity in $Q(S,\alpha)$, then the 1 parameter family of deformations given by the Teichm\"{u}ller deformation $g^t$ can be identified with the geodesic orbit of the identity in $Q(S,\alpha)$. Therefore we will call the geodesic orbit of the identity in $Q(S,\alpha)$ the \emph{Teichm\"{u}ller orbit} $g^t (S,\alpha)$ of $(S,\alpha)$ in $Q(S,\alpha)$.
\end{comment}
\begin{comment}
For the purposes of this article, a {\em translation surface} $S$ is a collection of polygons with edges identified in an by translations.
We call our translation surface {\em infinite} if it is constructed from infinitely many polygons.
Before we identify edges, we remove the vertices from all polygons. This guarantees that every point in $S$ has a chart to the plane so that transition functions are translation.
These charts provide {\em local coordinates} on the surface.
Observe that the tangent bundle of the plane $T\R^2$ with its usual metric has a translation-invariant trivialization (i.e., we can identify each tangent plane to $\R^2$ with the plane),
$T\R^2=\R^2 \times \R^2$. Since our transition functions are translations, there is a canonical lift of this trivialization to a translation surface,
$TS=S \times \R^2$.
Let $S_1$ and $S_2$ be translation surfaces.
An {\em affine homeomorphism} from $S_1$ to $S_2$ is a homeomorphism $\phi:S_1 \to S_2$ which is affine in the local coordinates. In these coordinates,
$$\phi(x,y)=(ax+by+t_1,cx+dy+t_2).$$
Because of our translation-invariant trivialization of the unit tangent bundle, we can observe that the
matrix
$$D \phi=\left[\begin{array}{rr} a & b \\ c & d \end{array}\right] \in \GL(2,\R)$$
is independent of our choice of local coordinates. We call this the {\em derivative} of $\phi$.
\end{comment}
\section{Finite covers of infinite surfaces}
\label{sec:covers}
In this section, we work out the theory of spaces of finite degree covers of a translation surface $(S,\alpha)$ of infinite topological type. We describe the topology of the space of covers $\text{Cov}_G(S,\alpha)$, mentioned in the introduction, in subsection \ref{sec:spaces}. In subsection \ref{sect:measures}, we place a natural Borel measure on this space.
Subsection \ref{sect:disconnected} discusses why disconnected covers should be considered rare.
\subsection{Spaces of covers}
\label{sec:spaces}
Let $S$ be a topological surface without boundary having infinite topological type (i.e., with infinitely generated fundamental group) which is realizable as a Riemann surface. By Rad\'o's Theorem $S$ is second-countable
\cite{Rado1925} \cite[\S 1.3]{HubbardTeich1}.
Choose a basepoint $s_0 \in S$.
It follows from Richards' classification of surfaces \cite[Theorem 3]{Richards}
that the fundamental group $\pi_1(S,s_0)$ is isomorphic to the free group with countably many generators.
\compat{The paper ON FENCHEL-NIELSEN COORDINATES ON TEICHM ULLER
SPACES OF SURFACES OF INFINITE TYPE
by
D. ALESSANDRINI, L. LIU, A. PAPADOPOULOS, W. SU, AND Z. SUN
takes a similar point of view of this fact. See the first few paragraphs of \S 4.}
A {\em covering} of $S$ is a pair $(p, \tilde S)$, where $\tilde S$ is a topological surface and $p: \tilde S \to S$ is a topological covering. Two covers $(p_1, \tilde S_1)$ and $(p_2, \tilde S_2)$ are {\em isomorphic} if there is a homeomorphism $h:\tilde S_1 \to \tilde S_2$ so that $p_1=p_2 \circ h$.
We recall idea of the monodromy action from covering space theory. Let $p:\tilde S \to S$ be a covering map.
The {\em monodromy action} is the right action on the fiber over the basepoint, $p^{-1}(s_0)$,
defined by
$$p^{-1}(s_0) \times \pi_1(S,s_0) \to p^{-1}(s_0); \quad \tilde s \cdot \gamma := \tilde \beta(1),$$
where $\beta:[0,1] \to S$ is a loop in the class of $\gamma \in \pi_1(S,s_0)$, and $\tilde \beta:[0,1] \to \tilde S$
is a lift (i.e., $p \circ \tilde \beta=\beta$) so that $\tilde \beta(0)=\tilde s$. It should be noted that the definition of $\tilde s \cdot \gamma$
is independent of the choice of $\beta$. (Once $\beta$ is chosen,
its lift $\tilde \beta$ is determined based on the condition that
$\tilde \beta(0)=\tilde s$, and $\tilde \beta(1)$ depends only on $\gamma$.)
It is a basic observation from covering space theory that the monodromy action determines the cover up to isomorphism. This includes
disconnected covers of the connected surface $S$. We will briefly show how to build a cover from an action. Concretely, given any right action of $\pi_1(S,s_0)$
on a discrete set $J$, we can build a cover of $S$ with this action as the monodromy action. To see this, fix such an
action. For each $j \in J$, let $\text{Stab}(j) \subset \pi_1(S,s_0)$ be the stabilizer of $j$.
We can then build a cover $\tilde S_j$ as the quotient of the universal cover of $S$ by $\text{Stab}(j)$. In the cover $\tilde S_j$, the lifts of the basepoint are then naturally identified with elements of the orbit $[j]$ of $j$ under the action of $\pi_1(S,s_0)$.
It can be observed that if $k \in [j]$, then there is a cover isomorphism from $\tilde S_j$ to $\tilde S_k$ which respects the identification between $[j]$ and the lifts of the basepoint on each surface. It may be observed that by taking the disjoint union of such surfaces over all orbits under the action, we obtain a cover,
\begin{equation}
\label{eq:built cover}
\tilde S=\bigsqcup_{[j] \subset J} \tilde S_j
\end{equation}
of $S$ with the desired monodromy action. (Here we are picking a representative $j$ from each orbit [j].) Furthermore,
actions on two discrete sets $J$ and $J'$ determine isomorphic covers if and only if the actions are conjugate up to bijection,
i.e., if there is a bijection $f:J \to J'$
so that $f(j \cdot \gamma)=f(j) \cdot \gamma$ for all $\gamma \in \pi_1(S, s_0)$ and $j \in J$.
We will now specialize this discussion to finite covers of $S$.
Suppose $p:\tilde S \to S$ is a covering map of degree $d$.
Let $\Pi_d$ be the symmetric group acting by permutations of
$\{1, \ldots, d\}$, and let
$$\ell:\{1, \ldots, d\} \to p^{-1}(s_0)$$
be a labeling (a bijection) to the fiber. The associated
{\em monodromy representation} (which depends on the labeling) is the group homomorphism
$M_\ell: \pi_1(S,s_0) \to \Pi_d$ defined so that
$$M_\ell(\gamma)(i)=\ell^{-1} \big(\ell(i) \cdot \gamma^{-1}\big) \quad
\text{for all $\gamma \in \pi_1(S,s_0)$ and all $i \in \{1, \ldots, d\}$,}
$$
where we have multiplied on the right by $\gamma^{-1}$ to make this map into a homomorphism (as opposed to an anti-homomorphism).
Conversely, such a representation determines an action on $\{1,\ldots, d\}$ and so, from the above discussion,
a choice of a monodromy representation $\pi_1(S,s_0) \to \Pi_d$ determines
a $d-$fold cover of $S$. Given two such representations, the covers are isomorphic if and only if they differ by conjugation by an element of $\Pi_d$, which has the effect of changing the labeling function. Thus, the space of $d$-fold covers of $S$ up to isomorphism is canonically identified with
$$\Pi_d \bs \Hom\big(\pi_1(S,s_0),\Pi_d\big),
\quad \text{where $\Pi_d$ is acting by conjugation.}$$
We endow $\Hom\big(\pi_1(S,s_0),\Pi_d\big)$ with the topology of pointwise convergence (or equivalently, the subspace topology coming from the inclusion
of $\Hom\big(\pi_1(S,s_0),\Pi_d\big)$ into the product space $\Pi_d^{\pi_1(S,s_0)}$), and this space of covers gets the quotient topology.
As in the introduction if $G$ is a subgroup of $\Pi_d$
for an integer $d \geq 2$, we say that a cover $\tilde S$ has {\em monodromy in $G$} if there is a representation $\pi_1(S,s_0) \to G$ which determines a cover isomorphic to $\tilde S$.
Note that this concept is independent of the basepoint. We define $\Covt_G(S)$ to be the collection of all covers of $S$ with monodromy in $G$
up to isomorphism of covers. Such covers are thus determined by elements of
$\Hom\big(\pi_1(S,s_0),G\big) \subset \Hom\big(\pi_1(S,s_0),\Pi_d\big)$.
So, the space $\Covt_G(S)$ is in bijective correspondence with
\begin{equation}
\label{eq:space of covers}
\Pi_d \bs \Hom\big(\pi_1(S,s_0),G\big).
\end{equation}
We use the identification with this quotient space to topologize $\Covt_G(S)$.
\begin{proposition}
\label{prop:topological covers}
The space $\Covt_G(S)$ is homeomorphic to a Cantor set.
\end{proposition}
\begin{proof}
We use the characterization of Cantor sets as non-empty, perfect, compact, totally disconnected, and metrizable.
We already know that $\Hom\big(\pi_1(S,s_0),G\big)$ is a Cantor set and thus satisfies these properties.
Because $\Covt_G(S)$ is a quotient of the Cantor set $\Hom\big(\pi_1(S,s_0),G\big)$, we see it is non-empty
and compact. Because the equivalence classes have finite size, we see that because $\Hom\big(\pi_1(S,s_0),G\big)$ is perfect, so is $\Covt_G(S)$.
Also because of this, we can use a metric on $\Hom\big(\pi_1(S,s_0),G\big)$ and the use the Hausdorff metric restricted to equivalence classes to put a metric
on $\Covt_G(S)$. To see the space is totally disconnected, suppose $h_1, h_2 \in \Hom\big(\pi_1(S,s_0),G\big)$ have distinct images
in $\Covt_G(S)$. Then for any permutation $p\in \Pi_d$, there is an element $\gamma_{p} \in \pi_1(S,s_0)$
so that $h_1(\gamma_{p}) \neq p \cdot h_2(\gamma_{p}) \cdot p^{-1}$.
For each permutation $q \in \Pi_d$, consider the following sets $U_q \subset \Hom\big(\pi_1(S,s_0),G\big)$:
$$U_q=\{h~:~\text{$h_1(\gamma_{p})=q \cdot h(\gamma_p) \cdot q^{-1}$ for all $p \in \Pi_d$}\}.$$
The set $U_q$ is clopen because it is a finite union of cylinder sets. (To see this, we write each $\gamma_p$ as a product of generators. Then there are finitely many values $h$ can take on the generators used so that each $h(\gamma_p)$ equals $q^{-1} h_1(\gamma_p) q$, i.e., so that $h \in U_q$.) Also observe $h_1 \in U_e$, where
$e \in \Pi_d$ is the identity element, and $h_2 \not \in U_q$ for any $q$ from the remarks above.
Thus the two complimentary sets
$$U=\bigcup_{q \in \Pi_d} U_q \quad \text{and} \quad V=\bigcap_{q \in \Pi_d} \Hom\big(\pi_1(S,s_0),G\big) \smallsetminus U_q$$
are both clopen while $h_1 \in U$ and $h_2 \in V$. Finally, we observe they are invariant under conjugation since
$$U=\bigcup_{q \in U_q} q^{-1} U_e q.$$
Thus $U$ and $V$ descend to clopen sets in $\Pi_d \bs \Hom\big(\pi_1(S,s_0),G\big)$ which separate
$[h_1]$ and $[h_2]$, proving that this quotient is totally disconnected.
\end{proof}
\compat{Please read this paragraph and the remark below. The referee asked for clarification here.}
Now suppose that we give the topological surface $S$ a translation structure,
$(S,\alpha)$.
As in the introduction, we use $\Cov_G(S,\alpha)$ to denote
the space of covers of $(S,\alpha)$ with monodromy in $G$ up to translation equivalence. There is a natural map
\begin{equation}
\label{eq:top cover to cover}
\Covt_G(S) \to \Cov_G(S,\alpha)
\end{equation}
which sends a topological covering $p:\tilde S \to S$ to $(\tilde S, \tilde \alpha)$
where $\tilde \alpha$ is obtained by pulling back $\alpha$ under $p$.
This map is clearly surjective, and we endow $\Cov_G(S,\alpha)$ with the finest topology so that this natural map is continuous. In other words, translation equivalence induces an equivalence relation on $\Covt_G(S)$, and $\Cov_G(S,\alpha)$
as a topological space is naturally identified with the resulting quotient.
\begin{remark}
\label{rem:primitivity}
Note that our definition of translation automorphism does no consider basepoints.
So, it is possible for two covers $(\tilde S_1, \tilde \alpha_1)$ and $(\tilde S_2, \tilde \alpha_2)$ of $(S,\alpha)$ to be translation equivalent without arising from an isomorphism of covering maps. However such pairs of can be ruled out if the surface $(S,\alpha)$ is geometrically primitive in the sense the deck group of the universal cover of $S$ is the same as the translation automorphism group of the universal cover (when endowed with the pullback translation structure).
\end{remark}
Observe that a homeomorphism $\phi:S \to S'$ between topological surfaces induces a homeomorphism
between their spaces of covers with monodromy in $G$ up to cover isomorphism. This is easily seen through the fundamental group.
In order to consider their fundamental groups, we choose basepoints $s_0$ and $s_0'$. To identify the fundamental groups,
we make a choice of a curve
$\beta:[0,1] \to S'$ so that $\beta(0)=\phi(s_0)$ and $\beta(1)=s'_0$.
The choice of $\beta$ gives rise to a group homomorphism
\begin{equation}
\label{eq:curve needed}
\phi_\beta:\pi_1(S,s_0) \to \pi_1(S',s'_0); [\gamma] \mapsto [\beta^{-1} \bullet (\phi \circ \gamma) \bullet \beta],
\end{equation}
where $\bullet$ denotes path concatenation (with the path on the left being traversed first). This action allows $\phi$ to act on
the spaces of topological covers with monodromy in $G$. By identifying these spaces of covers as
in equation \ref{eq:space of covers}, we see that this action is given by
\begin{equation}
\label{eq:homeomorphism acting on covers}
\Covt_G(S) \to \Covt_G(S'); \quad
[h] \mapsto [h \circ \phi_\beta^{-1}],
\end{equation}
where $[h]$ is the $\Pi_d$-conjugacy class of a homomorphism
$h:\pi_1(S,s_0) \to G$.
Note that while the action of $\phi$ on the fundamental group depended on the choice of the curve $\beta$,
the action on this space of covers does not, since the map $h \mapsto h \circ \phi_\beta^{-1}$
only changes by post-conjugation by a permutation when $\beta$ is changed.
A homeomorphism $(S,\alpha) \to (S',\alpha')$ between translation surfaces does not necessarily induce a homeomorphism
from $\Cov_G(S,\alpha)$ to $\Cov_G(S',\alpha')$. (This fails for instance, if $(S,\alpha)$ admits translation automorphisms and $(S',\alpha')$ does not.) However, an affine homeomorphism does induce such a homeomorphism.
As above, this homeomorphism is induced by the map $h \mapsto h \circ \phi_\beta^{-1}$. We summarize this observation below.
\begin{proposition}
\label{prop:action on covers}
Let $\phi:(S,\alpha) \to (S',\alpha')$ be an affine homeomorphism with derivative $A \in \GL(2,\R)$,
and let $\phi_\beta:\pi_1(S,s_0) \to \pi_1(S,s'_0)$ be the group homomorphism as defined
above for some choice of curve $\beta$. If $(\tilde S, \tilde \alpha)$ is a cover
of $(S,\alpha)$ with monodromy homomorphism $h:\pi_1(S,s_0) \to G$, then
$A(\tilde S, \tilde \alpha)$ is translation equivalent
to the cover of $(S',\alpha')$ with monodromy homomorphism $h \circ \phi_\beta^{-1}$.
Thus, map $h \mapsto h \circ \phi_\beta^{-1}$ induces a homeomorphism
$A_\ast:\Cov_G(S,\alpha) \to \Cov_G(S',\alpha')$, which depends only on the derivative
$A$ of the affine homeomorphism $\phi$.
\end{proposition}
As a consequence of this proposition, we observe that the Veech group of $(S,\alpha)$ acts on $\Cov_G(S,\alpha)$:
\begin{corollary}[The Veech group acts on covers]
Let $A \in \SL_\pm(2,\R)$ be an element of the Veech group of $(S,\alpha)$, and let $(\tilde S, \tilde \alpha) \in \Cov_G(S,\alpha)$. Then $A (\tilde S, \tilde \alpha)$ is also in $\Cov_G(S,\alpha)$.
\end{corollary}
\begin{comment}
Let
$\phi:S \to S$ be a homeomorphism (such as an affine automorphism). We will explain how $\phi$ acts on the space
of isomorphism classes of covers with monodromy in $G$, $\Pi_d \bs \Hom\big(\pi_1(S,s_0),G\big)$, and observe how it factors through the outer automorphism group of $\pi_1(S,s_0)$.
Most naturally, a homeomorphism $\phi$ induces an action on paths, which in turn induces an isomorphism between the fundamental group of $S$ with different basepoints.
To get an automorphism of $\pi_1(S,s_0)$ to itself, we need to make a choice of a curve $\delta$ joining $s_0$ to $\phi(s_0)$. With this choice, we define
\begin{equation}
\label{eq:phi beta}
\phi_\delta: \pi_1(S,s_0) \to \pi_1(S,s_0); \quad
[\beta] \mapsto \big[\delta \bullet (\phi \circ \beta) \bullet \delta^{-1} \big],
\end{equation}
where $\beta:[0,1] \to S$ is a loop with $\beta(0)=\beta(1)=s_0$ and $\bullet$ denotes path concatenation. This action depends on $\delta$, but the outer automorphism class of $\phi_\delta$ is independent of the choice of $\delta$. Since outer automorphisms act on
$\Pi_d \bs \Hom\big(\pi_1(S,s_0),G\big)$, we get a well defined
action of $\phi$ on this space. Furthermore by Corollary \ref{cor:outer invariance}, we see that $\nu_G'$ is invariant under
the action of $\phi$.
We now consider the naturality of the space $\Cov_G(S,\alpha)$ under the affine group.
Let $A \in \SL(2,\R)$
and suppose $A(S,\alpha)$ is translation equivalent to $(S',\alpha')$.
\begin{proposition}[Affine naturality of spaces of covers]
\label{prop:affine action}
Let $A \in \SL(2,\R)$ and suppose $A(S,\alpha)$ is translation equivalent to $(S',\alpha')$.
Let $G \subset \Pi_d$ for an integer $d \geq 2$. Then, whenever $(\tilde S, \tilde \alpha) \in \Cov_G(S,\alpha)$, we have
$A (\tilde S, \tilde \alpha) \in \Cov_G(S',\alpha')$. Moreover, there is a group homomorphism
$\eta:\pi_1(S,s_0) \to \pi_1(S',s'_0)$ so that whenever $(\tilde S_h, \tilde \alpha_h)$ is the cover of $(S,\alpha)$ with monodromy homomorphism $h:\pi_1(S,s_0) \to G$, the surface $A(\tilde S_h, \tilde \alpha_h)$ is translation equivalent
to the cover of $(S',\alpha')$ with monodromy homomorphism $h \circ \eta$.
\end{proposition}
\begin{proof}
Choose basepoints $s_0$ and $s'_0$ for the translation surfaces $(S,\alpha)$ and $(S',\alpha')$.
By hypothesis, we have a affine homeomorphism $\phi:(S,\alpha) \to (S',\alpha')$ with derivative $A$.
Let
Let $(\tilde S, \tilde \alpha) \in \Cov_G(S,\alpha)$
and let $p:\tilde S \to S$ be the covering map. Let $(\tilde S', \tilde \alpha')= A (\tilde S, \tilde \alpha)$,
and let $\tilde \phi:(\tilde S, \tilde \alpha) \to (\tilde S', \tilde \alpha')$ be the canonical identification
determined by viewing $A$ as deformation of geometric structures. Then, $\tilde \phi$ is also an affine homeomorphism with derivative $A$. The surface $(\tilde S', \tilde \alpha')$ covers $(S', \alpha')$ because $\phi \circ p \circ \tilde \phi^{-1}$
gives a covering map respecting the translation structures. There is an identification between lifts
of the basepoint $s_0 \in S$ to $\tilde S$ and lifts of the basepoint $s'_0 \in S'$. This is given by
$\tilde s_0 \mapsto \tilde s'_0$, where $\tilde s'_0$ is the endpoint $\tilde \beta(1)$
where $\tilde \beta$ is the unique lift of $\beta$ to $\tilde S$ where $\tilde \beta(0)=\phi(\tilde s_0)$.
It follows that up to conjugation by an element of $\Pi_d$, the monodromy representation $h'$ of
the cover $(\tilde S', \tilde \alpha')$ is given by $h \circ \eta^{-1}$ where $h$ is the monodromy representation
coming from $(\tilde S, \tilde \alpha)$. Since a monodromy representation uniquely determines the cover,
we see $(\tilde S', \tilde \alpha')$ is translation equivalent to the cover of $(S',\alpha')$ with monodromy representation
$h \circ \eta^{-1}$.
\end{proof}
\end{comment}
\subsection{Measures on spaces of covers}
\label{sect:measures}
In this subsection, we will construct some natural
measures on our spaces of covers.
We begin by describing an abstract construction. Later in the subsection, we will specialize the discussion to
our setting of translation surfaces.
Let $\Gamma^+$ be the non-abelian free group with a countable generating set $\{\gamma_i~:~i \in \N\}$, and let $G \subset \Pi_d$ as above. We endow the space $\Hom(\Gamma^+,G)$ with its natural product topology,
which makes the $\Hom(\Gamma^+,G)$ homeomorphic to a Cantor set.
This is the coarsest topology so that
for each $\eta \in \Gamma^+$ and each $\sigma \in G$, the set of the form
$$\{h~:~\Hom(\Gamma^+,G)~:~h(\eta)=\sigma\}.$$
is open. Two ordered $k$-tuples of distinct elements $(e_1,\ldots,e_k) \in \N^k$ and $(\sigma_1 ,\ldots, \sigma_k) \in G^k$, determine a
{\em cylinder set} in $\Hom(\Gamma^+,G)$,
\begin{equation}
\label{eqn:cylinders}
\Cyl(e_1,\ldots,e_k; \sigma_1,\ldots,\sigma_k)=\{h~:~\Hom(\Gamma^+,G)~:~
\text{$h(\gamma_{e_i})=\sigma_i$ for $i=1,\ldots,k$}\}.
\end{equation}
Each cylinder set is both closed and open in the product topology, and the collection of cylinder sets generate the topology.
To characterize a Borel measure on $\Hom(\Gamma^+,G)$, it suffices to describe the measures of
the cylinder sets.
\begin{definition}
\label{def:random cover}
The {\em product measure} $\mu$ on $\Hom(\Gamma^+,G)$
is defined so that for every cylinder set we have
$$\mu\big(\Cyl(e_1,\ldots,e_k; \sigma_1,\ldots,\sigma_k)\big)=\frac{1}{|G|^k}.$$
This is the product measure induced on $\Hom(\Gamma^+,G)$
by the counting measure on $G$.
\end{definition}
We remark that this measure $\mu$ is interesting even in the case when $\Gamma^+$ is a finitely generated free group, and related questions remain open \cite{Puder13}.
Automorphisms of $\Gamma^+$ act on $\Hom(\Gamma^+,G)$. Concretely, if $\phi: \Gamma^+ \to \Gamma^+$ is an automorphism, then we can define
\begin{equation}
\label{eqn:homAction}
\phi_\ast:\Hom(\Gamma^+, G) \to \Hom(\Gamma^+, G); \quad h \mapsto h \circ \phi^{-1}.
\end{equation}
\begin{lemma}
\label{lem:measPres}
The action of any automorphism of $\Gamma^+$ preserves
the product measure $\mu$ on $\Hom(\Gamma^+,G)$.
In particular, the measure $\mu$ is independent
of our choice of generating set.
\end{lemma}
\begin{remark}[Proof in abelian case]
In the case when $G$ is abelian, $\Hom(\Gamma^+,G)$
can be identified with the topological group $G^\N$, and
$\mu$ is Haar measure. In this case, the proposition follows from the naturality of Haar measure.
\end{remark}
\begin{proof}
Let $\phi: \Gamma^+ \to \Gamma^+$ be an automorphism, and let $\phi_\ast$ be its action
on $\Hom(\Gamma^+, G)$:
$$\phi_\ast:\Hom(\Gamma^+, G) \to \Hom(\Gamma^+, G); \quad
h \mapsto h \circ \phi^{-1}.$$
We will prove that $\phi_\ast$ preserves the product measure $\mu$ on $\Hom(\Gamma^+, G)$.
It suffices to prove that the measures of cylinder sets are preserved.
Let $\Cyl=\Cyl(e_1,\ldots,e_k; \sigma_1,\ldots,\sigma_k)$
be a cylinder set. We will prove that $\mu \circ \phi_\ast(\Cyl)=1/|G|^k$.
Let
$X=\langle \gamma_{e_1},\ldots, \gamma_{e_k} \rangle \subset \Gamma^+$. Since $\Gamma^+=\langle \gamma_e~:~e \in \N\rangle$, there is a finite set $\{e_1',\ldots, e_m'\} \subset \N$ so that
$$\phi^{-1}(X) \subset \langle \gamma_{e_1'}, \ldots, \gamma_{e'_m} \rangle.$$
We'll call the subgroup on the right hand side of the equation $Y$. By viewing $\mu$ as the product of counting measures, we see
\begin{equation}
\label{eq:count}
\mu \circ \phi^{\ast}(\Cyl)=\frac{\#\{h \in \Hom(Y,G)~:~h \circ \phi^{-1}(\gamma_{e_i})=\sigma_i \quad \text{for $1 \leq i \leq k$}\}}{|G|^m}.
\end{equation}
So it suffices to show that the number of homomorphisms in the numerator is $|G|^{m-k}$.
As above, we can find a finite set $\{e_1'', \ldots, e_n''\} \subset \N$ so that
$$\phi(Y) \subset \langle \gamma_{e_1''},\ldots, \gamma_{e_n''}\rangle.$$
Call the set on the right hand side $Z$. Note that $X \subset Z$.
We recall some basic definitions from the theory of free groups.
A {\em basis} of a free group $F$ is a set $x_1,\ldots,x_k$ so that $F=\langle x_1 \rangle \ast \ldots \ast \langle x_k \rangle$.
A subgroup $H$ of a free group $F$ is a {\em free factor} if every (equivalently, some) basis of $H$ can be extended to a basis of $F$.
Consider $X$, $Y$, and $Z$ as above.
Observe that $X$ is a free factor in $Z$.
So, $X$ is a free factor in $\phi(Y)$ \cite[Claim 2.5]{Puder13}.
That is,
we can extend $\{\gamma_{e_1},\ldots, \gamma_{e_k}\}$ to a free generating set of $\phi(Y)$.
Using $\phi^{-1}$, we can pull this back to a generating set of $Y$. So, we have $k \leq n$,
and there is a free generating set of $Y$ given by $\beta_1,\ldots, \beta_m$ so that
$$\beta_i=\phi^{-1}(\gamma_{e_i}) \quad \text{for $1 \leq i \leq k$.}$$
Since this set generates $Y$, we see that $\mathrm{Hom}(Y,G)$ is in bijective correspondence
with the possible images of $\{\beta_1, \ldots, \beta_m\}$. The last $m-k$ elements in this
basis are irrelevant to the values of $\phi^{-1}(\gamma_{e_i})$, so we see
that there are exactly $|G|^{m-k}$ possible values which give homomorphisms
in the numerator of equation \ref{eq:count}.
\end{proof}
Recall that whenever $S$ is a topological surface of infinite topological type,
then its fundamental group $\pi_1(S,s_0)$ is isomorphic to a countably generated free group. It is a simple observation that conjugation
by a permutation preserves the measure $\mu_G$ constructed above. By identifying
$\Hom(\Gamma^+,G)$ with $\Hom\big(\pi_1(S,s_0),G\big)$ and
$\Covt_G(S)$ with $\Pi_d \bs \Hom\big(\pi_1(S,s_0),G\big)$, we obtain a measure
$\nu_G$ on $\Covt_G(S)$, which we call the {\em product measure} on $\Covt_G(S)$.
\begin{corollary}
\label{cor:action on topological covers}
Let $\phi:S \to S'$ be a homeomorphism between two topological surfaces of infinite topological type. Let $G \subset \Pi_d$ for $d \geq 2$, and let $\nu_G$
and $\nu_G'$ be the product measures on $\Covt_G(S)$
and $\Covt_G(S')$, respectively. Then, $\nu_G'$ is the pushforward
of the measure $\nu_G$ under the map $\Covt_G(S) \to \Covt_G(S')$
induced by $\phi$ as in equation \ref{eq:homeomorphism acting on covers}.
\end{corollary}
\begin{proof}
We can identify each space
of covers with $\Hom(\Gamma^+,G)$
using isomorphisms to the fundamental groups.
As noted in equation \ref{eq:homeomorphism acting on covers},
the action of a homeomorphism on the space of covers is induced by a group isomorphism between the fundamental groups and via our identifications, an automorphism of $\Gamma^+$.
Lemma \ref{lem:measPres}
tells us that the measure $\mu_G$ is invariant under such automorphisms.
Also the automorphism commutes with the (partially defined) $\Pi_d$-action.
Thus, our measures $\nu_G$
and $\nu_G'$ are the same in view of the identification of each space
of covers with $\Pi_d \bs \Hom(\Gamma^+,G)$.
\end{proof}
Now let $(S,\alpha)$ be a translation surface of infinite topological type.
Since $\Cov_G(S, \alpha)$ is a quotient of $\Covt_G(S)$, we obtain an
measure $m_G$ on $\Cov_G(S, \alpha)$ as the pushforward of $\nu_G$.
We call $m_G$ the {\em product measure} on $\Cov_G(S, \alpha)$.
Recall that Proposition \ref{prop:action on covers} says that when $A \in \SL(2,\R)$ and $A(S,\alpha)$ is translation equivalent to $(S',\alpha')$, there is an induced homeomorphism $A_\ast$
from $\Cov_G(S,\alpha)$ to $\Cov_G(S',\alpha')$. This homeomorphism respects the product measures
on these spaces:
\begin{corollary}[Affine naturality of measures]
Let $(S,\alpha)$ be a translation surface of infinite topological type.
Let $A \in \SL_\pm(2,\R)$ and let $(S',\alpha')=A(S,\alpha)$. Then
$m_G'=m_G \circ A_\ast^{-1}$ where $m_G$ and $m_G'$ are the product measures
on $\Cov_G(S,\alpha)$ and $\Cov_G(S',\alpha')$.
\end{corollary}
\begin{proof}
Because $A(S,\alpha)=(S', \alpha')$, there must be an affine homeomorphism
$\phi:(S,\alpha) \to (S',\alpha')$ with derivative $A$.
The homeomorphism $A_\ast:\Cov_G(S,\alpha) \to \Cov_G(S',\alpha')$ lifts to a homeomorphism
$\Phi:\Covt_G(S,\alpha) \to \Covt_G(S',\alpha')$ by Proposition \ref{prop:action on covers}.
Since $\Phi_\ast(\nu_G)=\nu_G'$ by Corollary \ref{cor:action on topological covers}
and $m_G$ and $m_G'$ are obtained as images of $\nu_G$ and $\nu_G'$, we see
$m_G'=m_G \circ A_\ast^{-1}$.
\end{proof}
\subsection{Disconnected covers}
\label{sect:disconnected}
Let $G \subset \Pi_d$ and let $h \in \Hom\big(\pi_1(S,s_0),G\big)$. By interpreting $h$ as the monodromy action of the fundamental group of $S$ on the fibers of the basepoint, we obtain a cover $\tilde S$ of $S$ as in \S \ref{sec:spaces}.
This cover is explicitly described by equation (\ref{eq:built cover}), and we can see the following:
\begin{proposition}
\label{prop:connectivity}
The cover associated to $h \in \Hom\big(\pi_1(S,s_0),G\big)$ is connected if and only if the image
$h\big(\pi_1(S,s_0)\big)$ acts transitively on $\{1,2, \ldots, d\}$.
\end{proposition}
In particular, in order to have connected covers of $(S,\alpha)$ with monodromy in $G$, the subgroup $G \subset \Pi_d$ must act transitively on $\{1,2, \ldots, d\}$. The goal of this subsection is to
formulate the following precise version of the statement that the collection of all disconnected covers is small.
\begin{proposition}
\label{prop:measure zero}
Let $G \subset \Pi_d$ be a subgroup which acts transitively on $\{1,2, \ldots, d\}$. Let $S$ be a topological surface of infinite topological type. Then, $\nu_G$-almost every cover
in $\Covt_G(S)$ is connected.
\end{proposition}
\begin{proof}
Let $\sH$ denote the collection of all subgroups of $H \subset G$ so that $H$ does not act transitively on $\{1,\ldots, d\}$. Note that
$\sH$ is a finite set. Consider the set $\sD \subset \text{Cov}_G(S,\alpha)$ of disconnected covers with monodromy in $G$.
Recall that $\text{Cov}_G(S,\alpha)$ is a quotient of
$\Hom\big(\pi_1(S,s_0),G\big)$.
Let $\tilde \sD \subset \Hom\big(\pi_1(S,s_0),G\big)$ be the lift of $\sD$.
By definition of $\nu_G$, we have $\nu_G(\sD)=\mu_G(\tilde \sD)$.
Proposition \ref{prop:connectivity} tells us that
$$\tilde \sD=\bigcup_{H \in \sH} \Hom\big(\pi_1(S,s_0),H\big).$$
Let $\tilde \sD_H=\Hom\big(\pi_1(S,s_0),H\big)$. By subadditivity of measures, it suffices to prove that
$\mu_G(\tilde \sD_H)=0$ for all $H \in \sH$.
Fix $H \in \sH$. Observe that $H$ is a proper subgroup of $G$, since $H$ does not act transitively while $G$ does.
Fix some $\epsilon>0$. We will show that $\mu_G(\tilde \sD_H)<\epsilon$. Since $H$ is a proper subset of $G$, we can find a $k$ so that $(\frac{|H|}{|G|})^k<\epsilon$. Observe that $\tilde \sD_H$ is contained in the union of cylinder sets
$$\bigcup_{(h_1, \ldots, h_k) \in H^k}
\Cyl(1,\ldots,k; h_1,\ldots,h_k),$$
where we are using notation from equation \ref{eqn:cylinders}.
Observe that by monotonicity and by Definition \ref{def:random cover} of $\mu_G$, we have
$$\mu_G(\tilde \sD_H) \leq \sum_{(h_1, \ldots, h_k) \in H^k} \mu_G\big(\Cyl(1,\ldots,k; h_1,\ldots,h_k)\big)=
\frac{|H|^k}{|G|^k}<\epsilon.$$
This proves that $\mu_G(\tilde \sD_H)=0$, and thus $\nu_G(\sD)=0$ by the remarks in the previous paragraph.
\end{proof}
\section{Ergodicity}
\label{sec:erg}
Let $(S,\alpha)$ be a flat surface and let $G$ be a subgroup of the permutation group $\Pi_d$ for some integer $d \geq 2$. The group $\SL_\pm(2,\R)$ acts on the
the space of affine deformations of covers with monodromy in $G$, $\tilde \sO_G(S,\alpha)$,
and the action of the diagonal subgroup, $g^t$, is the cover cocycle.
(See (\ref{eq:covers cocycle}).)
In this section, we prove Theorem \ref{thm:1}, which pertains to a
connected cover $(\tilde S, \tilde \alpha) \in \Cov_G(S,\alpha)$: If the Teichm\"uller trajectory (covers cocycle orbit) $g^t(\tilde S, \tilde \alpha)$ has an accumulation point in $\tilde \sO_G(S,\alpha)$ representing a connected surface, then the translation flow on $(\tilde S, \tilde \alpha)$ is defined for all time almost everywhere and is ergodic.
\compat{Please check this paragraph.}
We will see that Theorem \ref{thm:1} is a consequence of the following result,
which gives a criterion for ergodicity in terms of the geometries realized under the Teichm\"uller deformation. Before stating the Theorem we establish some notation. Let $\Sigma\subset \bar{S}$ be the subset of the metric completion of $S$ defined as the union of the zeros of $\alpha$ and the points in $\bar S \smallsetminus S$.
For $t \in \R$ we define $\mbox{dist}_t$ to be the metric on $S$ obtained by pulling back the flat metric on $g^t(S,\alpha)$ under the affine homeomorphism $(S,\alpha) \to g^t(S,\alpha)$ with derivative $g_t$.
\begin{theorem}[{\cite[Theorem 3]{rodrigo:erg}}]
\label{thm:integrability}
Let $(S,\alpha)$ be a flat surface of finite area. Suppose that for any $\eta>0$ there exist a function $t\mapsto \varepsilon(t)>0$, a one-parameter family of subsets
$$S_{t} = \bigsqcup_{i=1}^{C_t}S_t^i$$
of $S$ made up of $C_t < \infty$ path-connected components, each homeomorphic to a closed orientable surface with boundary, and functions $t\mapsto \mathcal{D}_t^i>0$, for $1\leq i \leq C_t$, such that for
$$\Gamma_t^{i,j} = \{\mbox{paths connecting }\partial S_t^i \mbox{ to }\partial S_t^j\}$$
and
\begin{equation}
\label{eqn:systole}
\delta_t = \min_{i\neq j} \sup_{\gamma\in\Gamma_t^{i,j} }\mbox{dist}_t(\gamma,\Sigma)
\end{equation}
the following hold:
\begin{enumerate}
\item $\mathrm{Area}(S\backslash S_{t}) < \eta\,\mathrm{Area}(S)$ for all $t>0$,
\item $\mbox{dist}_t(\partial S_{t},\Sigma) > \varepsilon(t)$ for all $t>0$,
\item the diameter of each $S_t^i$, measured using $\dist_t$, is bounded above by $\mathcal{D}_t^i$ and
\begin{equation}
\label{eqn:integrability2}
\int_0^\infty \left( \varepsilon(t)^{-2}\sum_{i=1}^{C_t}\mathcal{D}_t^i + \frac{C_t-1}{\delta_t}\right)^{-2}\, dt = +\infty.
\end{equation}
\end{enumerate}
Then the translation flow is defined for all time almost everywhere and is ergodic.
\end{theorem}
The theorem above is a geometric criterion for ergodicity. The spirit of the theorem is that if, as ones deforms a flat surface $(S,\alpha)$ using the Teichm\"{u}ller deformation $g^t$, the geometry of the surface does not deteriorate too quickly (as measured by the diameter of big components, among other things), the translation flow is ergodic.
In \cite{rodrigo:erg}, this theorem was proved with the additional hypothesis that the set of points whose trajectories leave every compact subset of $S$ has zero measure. This is equivalent to the statement that the translation flow is defined for all time almost everywhere (which we state as a conclusion above). To get the version above we need to prove that the hypothesis is unnecessary:
\begin{proof}[Proof that the translation flow is defined for all time almost everywhere]
We will show that the geometric conditions listed above force the translation flow to be defined for all time almost everywhere.
Suppose $(S,\alpha)$ is a finite area translation surface and satisfies the list of geometric conditions given in the Theorem. Assume to the contrary that the translation flow $F^s$ is not defined for all time almost everywhere. Then there is a time $s_0$ and a measurable subset $X \subset S$ of Lebesgue measure $m>0$ so that $F^{s_0}(x)$ is undefined for all $x \in X$. There is a geometric consequence to lying in $X$: For any $x \in X$ there is a $s_x\in \R$ with $0<s_x\leq s_0$ so that $F^s(x)$ is defined for $s \in [0,s_x)$ and
$$\lim_{s \to {s_x}^-} F^s(x) \in \Sigma.$$
In particular, the distance from $x$ to $\Sigma$ measured with $\dist_t$ is no larger than $s_x e^{-t} \leq s_0 e^{-t}$.
Select an $\eta$ so that $m>\eta\,\mathrm{Area}(S)$. Then any subsurface $S_t$ satisfying condition (1) of the theorem must satisfy $S_t \cap X \neq \nullset$. As a consequence of condition (2) and remarks above, we must have $\varepsilon(t)<s_0 e^{-t}$.
We will draw a contradiction to the integral \eqref{eqn:integrability2} being infinite. To do this it suffices to get some control of the sum of the diameters. We partition the set $\{t \in \R:~t \geq 0\}$ into two pieces ${\mathcal S}$ and ${\mathcal L}$ (for ``small'' and ``large''). We declare $t$ to lie in ${\mathcal S}$ if all components $S_t^i$ have diameter less than or equal to $\varepsilon(t)$,
and declare $t$ to lie in ${\mathcal L}$ otherwise. We will control the sum $\sum_i {\mathcal D}_t^i$ by separate arguments on the two sets.
Suppose $t \in {\mathcal S}$. Then all components of $S_t$ have diameter less than $\varepsilon(t)$. Let $x_i \in S_t^i$ be a point in one of the components. Since $\dist_t(x_i,\Sigma) > \varepsilon(t)$ by hypothesis, we can embed a Euclidean $B_i$ ball of radius
${\mathcal D}_t^i<\varepsilon(t)$ about $x_i$ within $S$ using the distance $\dist_t$.
Then we have $S_t^i \subset B_i$, so
$$\mathrm{Area}(S_t^i) \leq \mathrm{Area}(B_i) = \pi ({\mathcal D}_t^i)^2.$$
Observe this holds for all $i$ and so it follows that the sum $\sum_i {\mathcal D}_t^i$ is bounded from below by a constant:
\begin{equation}
\label{eq:diameter bound 1}
\Big(\sum_i {\mathcal D}_t^i\Big)^2 \geq \sum_i ({\mathcal D}_t^i)^2 \geq \frac{4}{\pi} \sum_i \mathrm{Area}(S_{t}^i)=
\frac{1}{\pi} \mathrm{Area}(S_{t}) >\frac{1}{\pi} (1-\eta) \mathrm{Area}(S).
\end{equation}
Now suppose $t \in {\mathcal L}$. Then $S_t$ has a component of diameter at least $\varepsilon(t)$.
In this case we can use the very na\"ive bound
\begin{equation}
\label{eq:diameter bound 2}
\sum_i {\mathcal D}_t^i \geq \varepsilon(t).
\end{equation}
Combining \eqref{eq:diameter bound 1} and \eqref{eq:diameter bound 2}, we see that $\sum_i {\mathcal D}_t^i \geq D(t)$
where $D(t)$ is defined by
$$D(t)=\min~\left\{\varepsilon(t), \sqrt{\frac{1}{\pi} (1-\eta) \mathrm{Area}(S)}\right\}.$$
The quantity on the right is a constant while $\varepsilon(t)$ tends to zero since $\varepsilon(t)<s_0 e^{-t}$.
Thus $D(t)=\varepsilon(t)$ for $t>t_\ast$ for some $t_\ast \geq 0$. For $t>t_\ast$, $\sum_i {\mathcal D}_t^i \geq \varepsilon(t)$
and so the quantity being integrated in \eqref{eqn:integrability2} satisfies
$$\left( \varepsilon(t)^{-2}\sum_{i=1}^{C_t}\mathcal{D}_t^i + \frac{C_t-1}{\delta_t}\right)^{-2} \leq \left(\varepsilon(t)^{-2} \varepsilon(t)\right)^{-2}=\varepsilon(t)^2.$$
Recalling $\varepsilon(t)<s_0 e^{-t}$, we see the total integral \eqref{eqn:integrability2} is bounded by
$$\int_0^{t_\ast} \left( \varepsilon(t)^{-2}\sum_{i=1}^{C_t}\mathcal{D}_t^i + \frac{C_t-1}{\delta_t}\right)^{-2}\,dt+
s_0^2 \int_{t_\ast}^\infty e^{-2t}\,dt,$$
which is finite. This is our contradiction.
\end{proof}
A key consequence for us is the following which was mentioned in the introduction:
\begin{corollary}[{\cite[Theorem 2]{rodrigo:erg}}]
\label{cor:rodrigo ergodicity from Veech group}
Suppose $(S,\alpha)$ is a finite area translation surface of infinite topological type. Let $a \in \SL_\pm(2,\R)$. Then, if the trajectory $g^t a (S,\alpha)$ is non-divergent in $\sO(S,\alpha)$, then the translation flow on $a (S,\alpha)$ is defined for all time almost everywhere and is ergodic.
\end{corollary}
It is worth noting that in \cite{rodrigo:erg} this was proved independently from Theorem \ref{thm:integrability}.
We will prove this here using Theorem \ref{thm:integrability} because we will use some of the same ideas in the proof of Theorem \ref{thm:1}.
\begin{proof}
By replacing $(S,\alpha)$ with $a(S,\alpha)$, we can assume that $a$ is the identity.
We will use square brackets to denote the translation equivalence class of a translation surface in $\sO(S,\alpha)$.
Non-divergence guarantees that there is sequence of times $t_k$ tending to $+\infty$ so that the translation equivalence class
$[g^{t_k} (S,\alpha)]$ converges in $\sO(S,\alpha)$ to some limit $[b(S, \alpha)]$.
Since $\N \cup \{+\infty\}$ with its usual topology is compact and the map $\SL(2,\R) \to \sO(S,\alpha)$ given by $a \mapsto [a(S,\alpha)]$
is continuous, $K=\{[g^{t_k} (S,\alpha)]~:~k \in \N\} \cup \{[b(S, \alpha)]\}$ is compact in $\sO(S,\alpha)$.
Fix any $\eta>0$. Select a compact connected subsurface with boundary $L \subset S$ whose area is greater than $1- \eta$ times the area of $S$, and so that $L$ does not include any zeros of $\alpha$. Fix any $\epsilon>0$. For $t \in [t_k-\epsilon, t_k +\epsilon]$ select the subsurface $S_t=g^{-t_k} b_k(L)$ (or any such subsurface if $t$ belongs to multiple such intervals).
Then since $g^{t} (S,\alpha)$ is translation equivalent to $g^{t-t_k} b_k(S, \alpha)$, there is an isometry from $S$ with the metric $\dist_t$
to the translation surface $g^{t-t_k} b_k(S,\alpha)$ which carries $S_t$ to the image of $L$ under $g^{t-t_k} b_k$.
So for all $t$ in any $[t_k-\epsilon, t_k +\epsilon]$, the subsurface we have selected is isometric to the image
of $m(L)$ under an $m \in g^{[-\epsilon,\epsilon]} K$ viewed as a subsurface of $m(S,\alpha)$.
Observe that the quantities in Theorem \ref{thm:integrability} vary continuously in $m$ as we deform the metric in this way.
Thus, the quantity being integrated over times $t\in \bigcup_k [t_k-\epsilon, t_k +\epsilon]$ is bounded uniformly from below by a uniform positive constant. Since $t_k$ is an infinite sequence tending to $+\infty$, the integral is infinite.
\end{proof}
\begin{comment}
\begin{remark}
\label{rem:area}
Without loss of generality, we can assume that any surface to which we apply Theorem \ref{thm:integrability} has area 1. The point is that if we can control the deforming geometry on a subset with measure arbitrarily close to the total measure of the surface (which is finite) then we obtain ergodicity.
\end{remark}
\end{comment}
\begin{proof}[Proof of Theorem \ref{thm:1}]
\compat{This proof was completely rewritten. Please read.}
Let $(S,\alpha)$ be a finite area translation surface with infinite topological type,
and let $(\tilde S, \tilde \alpha)$ be a cover with monodromy in $G \subset \Pi_d$.
We assume that $[g^t(\tilde S, \tilde \alpha)]$ has an $\omega$-limit point in $\tilde \sO_G(S,\alpha)$ which is the translation equivalence class of a connected surface. Let $t_k$ be a sequence of times tending to $+\infty$ for which
$[g^{t_k}(\tilde S, \tilde \alpha)]$ approaches this $\omega$-limit.
Then by definition of the topology on $\tilde \sO_G(S,\alpha)$, there is a sequence
$$\big(b_k, [(\tilde S_k, \tilde \alpha_k)]\big) \in \SL(2,\R) \times \Cov_G(S,\alpha)$$
so that for all $k$ the surface
$b_k (\tilde S_k, \tilde \alpha_k)$ is translation equivalent to $g^{t_k}(\tilde S, \tilde \alpha)$,
the sequence $b_k$ converges to some $b \in \SL(2,\R)$ and the sequence $[(\tilde S_k, \tilde \alpha_k)]$ converges to some connected $[(\tilde S_\infty, \tilde \alpha_\infty)] \in \Cov_G(S,\alpha)$.
We will briefly discuss the convergence of $[(\tilde S_k, \tilde \alpha_k)]$
to the connected cover $[(\tilde S_\infty, \tilde \alpha_\infty)]$ within $\Cov_G(S,\alpha)$. Fix a basepoint $s_0 \in S$ which is not a zero. Choose a representative cover $(\tilde S_\infty, \tilde \alpha_\infty)$ from $[(\tilde S_\infty, \tilde \alpha_\infty)]$.
Since $[(\tilde S_\infty, \tilde \alpha_\infty)] \in \Cov_G(S,\alpha)$, we can
denote the lifts of $s_0$ by $s_\infty^1,\ldots, s_\infty^d \in \tilde S_\infty$
obtaining a monodromy homomorphism
$h_\infty:\pi_1(S,s_0) \to G \subset \Pi_d$. From the definition of the topology $\Cov_G(S,\alpha)$,
for all $k$ we can select covers $(\tilde S_k, \tilde \alpha_k)$ from the equivalence classes $[(\tilde S_k, \tilde \alpha_k)]$ and
denote the lifts of the basepoint $s_0$ by $s_k^1,\ldots, s_k^d \in \tilde S_k$ in such a way so that
the corresponding monodromy homomorphisms $h_k:\pi_1(S,s_0) \to G$ converge
to $h_\infty$ within $\Hom\big(\pi_1(S,s_0), G\big)$.
We will be using Theorem \ref{thm:integrability}.
Fix an $\epsilon>0$. As in the prior proof we will only bother to choose a subsurface when $t \in [t_k-\epsilon,t_k+\epsilon]$ for some $k$.
We will now describe how we choose these subsurfaces. As in the prior proof, we can let $L \subset S$ be a compact connected subsurface with boundary whose area is more than $1-\eta$ times the area of $S$ so that $L$ contains no zeros of $\alpha$. A compact surface with boundary
has finite genus, so by removing small open neighborhoods of a maximal collection of smooth disjoint arcs joining $\partial L$ to itself whose collective complement in $L$ is connected, we may assume that $L$ has all these properties and is a homeomorphic to a closed topological disk. We can also assume by possibly adding a bit to the subsurface that $s_0 \in L$. From above
$(\tilde S, \tilde \alpha)$ is translation equivalent to $g^{-t_k} b_k(\tilde S_k, \tilde \alpha_k)$ and so these surfaces cover $g^{-t_k} b_k(S, \alpha)$ by Proposition \ref{prop:action on covers}.
Let $L_k \subset (\tilde S_k, \tilde \alpha_k)$ be the collection of all lifts of $L$ under the covering map to $S$.
The components of $L_k$ are naturally labeled $L_k^1, \ldots, L_k^d$ so that the lift of the basepoint $s^i_k$ lies in $L_k^i$ for all $i \in \{1,\ldots, d\}$.
For $t \in [t_k-\epsilon,t_k+\epsilon]$ and $i \in \{1,\ldots, d\}$
we define $S_t^i \subset \tilde S$ to be image of $g^{-t_k} b_k(L_k^i)$ under a translation isomorphism
$g^{-t_k} b_k (\tilde S_k, \tilde \alpha_k) \to (\tilde S, \tilde \alpha)$. We define $S_t=\bigcup_{i=1}^d S_t^i$.
Observe that with this definition:
\begin{itemize}
\item The surface $S_t$ has a number of components $C_t$ equal to the degree $d$ of the covering maps.
\end{itemize}
Consider geometric quantities of $S_t \subset \tilde S$ measured with $\dist_t$. The surface $\tilde S$ with this metric is isometric
to $g^t(\tilde S, \tilde \alpha)$ which is translation equivalent to $g^{t-t_k} b_k (\tilde S_k, \tilde \alpha_k)$.
So by definition of $S_t$ these geometric quantities are the same as for the subsurface $g^{t-t_k} b_k(L_k)$
of $g^{t-t_k} b_k (\tilde S_k, \tilde \alpha_k)$. Observe that $g^{t-t_k} b_k$ lies in the compact set $g^{[-\epsilon,\epsilon]} K$ where $K=\{b_k:~k \in \N\} \cup \{b\}$ is compact as in the prior proof. Observe:
\begin{itemize}
\item The quantities $\epsilon(t)$ and $\mathcal{D}_t^i$ used to measure the components of $g^{t-t_k} b_k(L_k)$ as a subsurface
of $g^{t-t_k} b_k (\tilde S_k, \tilde \alpha_k)$ are exactly the same as the quantities for $g^{t-t_k} b_k(L)$ viewed as a subsurface of
$g^{t-t_k} b_k (S, \tilde \alpha)$, because these quantities are covering map invariant.
\end{itemize}
In particular, this means that $\varepsilon(t)$ can be bounded uniformly away from zero when $t \in \bigcup_k [t_k-\epsilon,t_k+\epsilon]$
and $\mathcal{D}_t^i$ can be bounded uniformly away from $+\infty$ as in the proof of Corollary \ref{cor:rodrigo ergodicity from Veech group}.
It remains to control the quantity $\delta_t$. We must first construct the curves $\Gamma^{i,j}_t$. We utilize the convergence
of $h_k$ to $h_\infty$.
Since $S_\infty$ is connected, we can select for all distinct $i,j \in \{1, \ldots, d\}$ a path $\Gamma_\infty^{i,j}$ in $\tilde S_\infty$ disjoint from the zeros joining $s_\infty^i$ to $s_\infty^j$. Let $\tilde \delta_0>0$ be the minimum over all pair $(i,j)$ of
the distance from $\Gamma_\infty^{i,j}$ to the set of points of the completion of $\tilde S$ which are either zeros or added in the completion. Let $\gamma^{i,j}$ be the loop in $S$ based at $s_0$ which is obtained as the image of $\Gamma_\infty^{i,j}$ under the covering map $\tilde S_\infty \to S$. Since $h_k$ tends to $h_\infty$, for $k$ sufficiently large $h_k(\gamma^{i,j})=h_\infty(\gamma^{i,j})$ for all $i$ and $j$. We will assume by dropping finitely many $k$ that this holds for all $k$. Then for each $k$, the lift $\tilde \gamma^{i,j}_k$ of $\gamma^{i,j}$ to $\tilde S_k$ which begins at $s_k^i$ ends at $s_k^j$. Observe that the minimal distance over all pairs $(i,j)$ of
the distance of $\gamma^{i,j}$ to zeros or points added in the completion is still $\tilde \delta_0$.
A subpath $\tilde p^{i,j}_k \subset \tilde \gamma^{i,j}_k$ joins $\partial L^i_k$ to $\partial L^j_k$ since the path joins $s_k^i \in L^i_k$ to $s_k^j \in L^j_k$.
For $t \in [t_k-\epsilon,t_k+\epsilon]$, we define $\Gamma_t^{i,j} \subset S$ to be the image of $g^{-t_k} b_k(\tilde p^{i,j}_k)$
under the translation isomorphism $g^{-t_k} b_k (\tilde S_k, \tilde \alpha_k) \to (\tilde S, \tilde \alpha)$ (as used above to define $S_t$). The quantity $\delta_t$ is the minimal distance of $\Gamma_t^{i,j}$ to $\Sigma \subset \bar S$ taken over pairs $(i,j)$
and measured with $\dist_t$.
As above $S$ with metric $\dist_t$ is isometric to $g^t(\tilde S, \tilde \alpha)$ which in turn is translation isomorphic to $g^{t-t_k} b_k (\tilde S_k, \tilde \alpha_k)$. Thus we get the same value of $\delta_t$ by looking at the path $g^{t-t_k} b_k(\tilde p^{i,j}_k)$ in the translation surface $g^{t-t_k} b_k (\tilde S_k, \tilde \alpha_k)$. Thus $\delta_t$ is bounded from below by
$\frac{1}{z} \tilde \delta_0$ when $t \in \bigcup_k [t_k-\epsilon,t_k+\epsilon]$ where $z$ is the maximal operator norm of $m^{-1}$ taken over $m$ taken from the compact set $g^{[-\epsilon,\epsilon]} K$ as above.
We have shown that the quantity being integrated in \eqref{eqn:integrability2} has a positive lower bound when $t \in \bigcup_k [t_k-\epsilon,t_k+\epsilon]$. Since $t_k \to +\infty$, this means that the integral is infinite, so Theorem \ref{thm:integrability}
guarantees that the translation flow is defined for all time almost everywhere and is ergodic.
\end{proof}
\begin{comment}
Now let us review the strategy of the proof of Theorem \ref{thm:1}. The Teichm\"{u}ller orbit of $(S,\alpha)$ in $\sO(S,\alpha)$ corresponds to the Teichm\"{u}ller deformation of the surface $g^t(S,\alpha)$. By assumption, it has a converging subsequence, i.e.,
there is an $a \in \SL(2,\R)$ and a sequence of times $t_k \rightarrow \infty$ such that $g^{t_k}(S,\alpha) \rightarrow a(S,\alpha)$ in the sense of conformal structures.
This means that for any neighborhood of the identity $U \subset \SL(2,\R)$,
there is an $r \in V(S,\alpha)$
and a $k$ so that $g^{t_k} r^{-1} a^{-1} \in U$. Informally, this says we do not need to deform much to move from $a (S,\alpha)$ to $g^{t_k} r^{-1}(S,\alpha)$, the later of which is translation equivalent
to $g^{t_k} (S,\alpha)$.
Through any subsequence $t_k \rightarrow \infty$ as above we can control the geometry of the surface $g^t (S,\alpha)$ at times nearly $t_k$. We also gain some control at these times over covers $(\tilde S, \tilde \alpha)$ with monodromy in $G$
through the use of renormalizing elements $r_k$ of the Veech group $V(S,\alpha)$. The Veech group acts on the spaces of covers,
and we can control the geometry of the deformed cover $g^t(\tilde S,\tilde \alpha)$
by considering its convergence along a subsequence to a connected cover
$a(\tilde S^\ast, \tilde \alpha^\ast)$ in $\tilde \sO_G(S,\alpha)$.
The convergence of $g^{t_k}(S,\alpha)$ to $a (S,\alpha)$ controls the deforming geometry of $g^t(S,\alpha)$, and we also have subsequential convergence in the fiber of covers
of $a (S,\alpha)$ to the connected cover $a(\tilde S^\ast, \tilde \alpha^\ast)$. Together
these ideas allow us to control the geometry of $g^t(\tilde S, \tilde \alpha)$
for times $t$ near each $t_k$. This control is sufficient to verify the hypothesis of Theorem \ref{thm:integrability}
and conclude that the horizontal flow on $(\tilde S,\tilde \alpha)$ is ergodic.
\begin{proof}[Proof of Theorem \ref{thm:1}]
Let $(S,\alpha)$ be a unit area translation surface with infinite topological type,
and let $(\tilde S, \tilde \alpha)$ be a cover with monodromy in $G \subset \Pi_d$.
We assume that $g^t(\tilde S, \tilde \alpha)$ has an $\omega$-limit point
$(\tilde L, \tilde \beta)=a(\tilde S^\ast, \tilde \alpha^\ast)$ in
$\tilde \sO(S, \alpha)$, where $(\tilde S^\ast, \tilde \alpha^\ast)$ is another
cover with monodromy in $G$ and $a \in \SL(2,\R)$.
The limit point $(\tilde L, \tilde \beta)$ is a cover of the surface
$(L,\beta)=a(S,\alpha)$.
Recall that the Veech group $V(S,\alpha)$ acts on the space of covers with monodromy in $G$, $\Cov_G(S,\alpha)$.
By definition of the topology of $\tilde \sO_G(S,\alpha)$, the existence of this $\omega$-limit point
means we can find a sequence
of times $t_k\rightarrow \infty$, a sequence of Veech group elements $r_k\in V(S,\alpha)$,
and a sequence $w_k \in \SL(2,\R)$ tending to the identity such that
$g^{t_k} r_k^{-1}=w_k a$ and $r_k(\tilde S, \tilde \alpha) \rightarrow (\tilde S^\ast, \tilde \alpha^\ast)$ in $\Cov_G(S,\alpha)$. Here, $w_k$ deforms
the limit $(L,\beta)$ into the approximate
$g^{t_k} r_k^{-1}(S,\alpha)$, which is translation equivalent to $g^{t_k}(S,\alpha)$.
Recall that $\Sigma$ denotes the set points added in the metric completion of a surface as well as the zeros of the holomorphic form defining the translation structure. Let
$$L_{\varepsilon} := \{z\in L: \mathrm{dist}_{\beta}(z,\Sigma) \geq \varepsilon\},$$
where the distance function $\mathrm{dist}_{\beta}(\cdot,\cdot)$ is defined with respect to the flat metric on $L$ given by $\beta$.
Fix $\eta > 0$. The sets $L_{\varepsilon}$ form an exhaustion of $L$, i.e.,
the sets are increasing as $\varepsilon$ decreases and their union is all of $L$.
Since $L$ is connected, it follows that there is an $\varepsilon_\eta>0$ so that
$L_{\varepsilon}$ has a connected component
of area at least $1-\eta$ when $\varepsilon<\varepsilon_\eta$. For $\varepsilon<\varepsilon_\eta$, let $\varepsilon \mapsto X_\varepsilon$ be a nested family of choices of such a component. Then we can choose $\varepsilon<\varepsilon'<\varepsilon_\eta$, and find a connected compact subsurface
$L^\circ \subset L$ with smooth boundary satisfying
$X_{\varepsilon} \supset L^\circ \supset X_{\varepsilon'}.$
By possibly removing some bits of this subsurface (e.g., by removing small open neighborhood of a union $U$ of a maximal collection of disjoint smooth simple curves so that $L^\circ\smallsetminus U$ is connected), we can arrange that $L^\circ$
is a topological disk, contained in $X_{\varepsilon}$ (but not necessarily containing
$X_{\varepsilon'}$), and that $\mathrm{Area}\, L^\circ>1-\eta$ is still satisfied.
Let ${\mathcal D}$ denote the diameter of the topological disk $L^\circ$ measured with respect to $\dist_\beta$.
Consider the cover $(\tilde L, \tilde \beta)$ of $(L,\beta)$, and let $p_L:\tilde L \to L$
denote the covering map. Let $\tilde L^\circ = p_L^{-1}(L^\circ)$. Since $L^\circ$
is a topological disk, the preimage $\tilde L^\circ$ has $d$ connected components,
each of which is a copy of $L^\circ$. Select a basepoint $\ell \in L^\circ$,
and choose a labeling of the lifts of the basepoints $\{\ell_1, \ldots, \ell_d\}= p_L^{-1}(\ell)$. This allows us to describe our cover $\tilde L$ as the cover determined
by monodromy representation $h_{\tilde L}:\pi_1(L,\ell) \to G$. For each pair of distinct
$i,j \in \{1,\ldots,d\}$, choose a curve $\gamma_{i,j} \subset \tilde L$ joining $\ell_i$ to $\ell_j$ which does not intersect $\Sigma$ (which includes zeros). Let $\Gamma = \bigcup_{i,j} p_{L}(\gamma_{i,j}) \subset L$ and let
$$\delta = \min_{i,j} \mathrm{dist}_{\tilde \beta}(\gamma_{i,j},\Sigma) = \mathrm{dist}_{\beta}(\Gamma,\Sigma).$$
Observe that $\delta>0$ since $\Gamma$ is compact.
Observe that $(L,\beta)$ is translation equivalent to $w_k^{-1} g^{t_k}(S,\alpha)$ since we chose these group elements so that $w_k^{-1} g^{t_k} r_k^{-1}=a$ and $r_k \in V(S,\alpha)$. We define
$$(\tilde L_k, \tilde \beta_k)=w_k^{-1} g^{t_k}(\tilde S, \tilde \alpha),$$
which is also a cover of $(L,\beta)$. Let $p_k:\tilde L_k \to L$ be the covering maps,
and define $\tilde L^\circ_k=p_k^{-1}(L^\circ)$. Again these surfaces have $d$ components, and the normalized area satisfies
\begin{equation}
\label{eqn:areaGood}
\mathrm{Area}(\tilde L^\circ_k) \geq 1-\eta.
\end{equation}
Now recall that in $\Cov_G(S,\alpha)$, we have $r_k(\tilde S, \tilde \alpha) \rightarrow (\tilde S^\ast, \tilde \alpha^\ast)$. It follows that
\begin{equation}
\label{eq:L k cover}
(\tilde L_k, \tilde \beta_k) = w_k^{-1} g^{t_k} (\tilde S, \tilde \alpha) \rightarrow (\tilde L, \tilde{\beta})
\end{equation}
within $\Cov_G(L,\beta)$. Since $\Cov_G(L,\beta)$ is a subspace of a topological quotient
of the space of monodromy representations, $\Hom\big(\pi_1(L,\ell),\Pi_d)$, we know that we can choose labels for the fibers for the basepoint $p_k^{-1}(\ell) \subset \tilde L_k$ so that the induced representations to a conjugate of $G$, $h_k:\pi_1(L,\ell) \to \Pi_d$ converge in the product topology to
$h_{\tilde L}$. This means that there is a $K$ so that for $k>K$, we have
\begin{equation}
\label{eq:same permutations}
h_k(\gamma_{i,j})=h_{\tilde L}(\gamma_{i,j}) \quad \text{for all distinct $i,j \in \{1, \ldots, d\}.$}
\end{equation}
We can label the $d$ components of $\tilde L^\circ_k$ according to the label of the basepoint
each subsurface contains. We observe that (\ref{eq:same permutations})
tells us that for each $i \neq j$, there is a lift of $p_L(\gamma_{i,j})$
under the projection $p_k:\tilde L_k \to L$ which joins the point in $p_k^{-1}(\ell)$
labeled $i$ to the the point labeled $j$. Taken together, these lifts
give a collection of curves joining all pairs of components of $\tilde L^\circ$.
We let $\Gamma_k \subset \tilde L_k$ denote the union of lifted curves.
These curves in $\tilde L_k$ also get no closer than $\delta$ to the completion locus $\Sigma$.
Now choose a flow box $B$ about the identity for the flow $m \mapsto g^t m$ in $\SL(2,\R)$.
That is, we choose $B \subset \SL(2,\R)$ to be a compact neighborhood of the identity so that
whenever $a \in B$, the maximal interval, $I(a)$, containing zero and contained in the set
$$\{t \in \R~:~g^t a \in B\}$$
has length $|I(a)|=\tau>0$.
\compat{Justification requested by referee:}
(The box $B$ can be constructed by choosing a closed $2$-dimensional topological disk in $\SL(2,\R)$ which contains the identity and is everywhere transverse to the $g^t$ orbits, and then flowing that disk under $g^t$ for $t \in [-\frac{\tau}{2},\frac{\tau}{2}]$ for sufficiently small $\tau>0$.) Recall that $w_k=g^{t_k} r_k^{-1} a^{-1}$ converges
to the identity, and so by possibly dropping the first few terms in our sequences, we can
assume that each $w_k$ lies in our flow box $B$. For each $k$, define
$$I_k=I(w_k)+t_k.$$
Then $I_k$ is the maximal interval of times $t$ contiguous with $t_k$ so that
$g^t r_k^{-1} a^{-1} \in B$. Since $t_k \rightarrow \infty$ and the length of these intervals is always $\tau$, by passing to a subsequence, we can assume that the intervals $I_k$
are all disjoint.
Our flow box $B$ is compact, so we can find constants $0<m<1<M$ so that for any non-zero vector $\v \in \R^2$,
\begin{equation}
\label{eq:geometry flow box}
0<m|\v|<|b(\v)|<M|\v| \quad \text{for all $b \in B$}.
\end{equation}
These constants control how metrics deform when move within the flow box.
We have quite explicit control of the geometry of the covers $(\tilde L_k,\tilde \beta_k)$.
We have chosen a subsurface $\tilde L^\circ_k$, which is a union of $d$ topological disks
joined by curves which lie in the set $\Gamma_k$. The diameter of each of these components of $\tilde L^\circ_k$
is always precisely $\sD$, the distance from $\tilde L^\circ_k$ to $\Sigma$ is more than $\varepsilon$, and the curves in $\Gamma_k$ are always distance at least $\delta$ from $\Sigma$.
Recalling the definition of $(\tilde L_k,\tilde \beta_k)$ in (\ref{eq:L k cover}), we see
that
$g^{t_k} (\tilde S, \tilde \alpha)=w_k(\tilde L_k, \tilde \beta_k).$
Then, for $t \in I_k$, we have $g^{t-t_k} w_k \in B$ and
$$g^{t} (\tilde S, \tilde \alpha)=g^{t-t_k} w_k(\tilde L_k, \tilde \beta_k).$$
Since each $g^{t-t_k} w_k \in B$, using (\ref{eq:geometry flow box}), we obtain
control on the geometry of $g^{t} (\tilde S, \tilde \alpha)$, through the subsurfaces
$\tilde S^\circ_t=g^{t-t_k} w_k(\tilde L^\circ_k)$ and curves in $\Gamma_t=g^{t-t_k} w_k(\Gamma_k)$. Namely, for $t \in I_k$,
\begin{itemize}
\item The subsurface $\tilde S^\circ_t$ has $d$ components each with diameter no more than
$M{\mathcal D}$ measured in the metric of $g^{t} (\tilde S, \tilde \alpha)$.
\item The distance from $\tilde S^\circ_t$ to $\Sigma$ to $\Sigma$ is more than $\varepsilon m$.
\item The minimal distance from the curves in $\Gamma_t$ to $\Sigma$ is more than $\delta m$.
\end{itemize}
Using this, we observe that the quantity being integrated in Theorem \ref{thm:integrability} is always at least as large as the constant
$$\left((\varepsilon m)^{-2} d (M\sD)+\frac{d-1}{\delta m}\right)^{-2}.$$
Since this is a positive constant independent of $k$, and the total length of $\bigcup_k I_k$
is infinite, we see that the integral in Theorem \ref{thm:integrability} is infinite.
We conclude that the translation flow on $(\tilde S, \tilde \alpha)$ is ergodic with respect to Lebesgue measure on this cover.
\end{proof}
\end{comment}
\section{Further proofs}
\label{sect:proofs}
In this section, we prove Proposition \ref{prop:lifting measures}, Proposition \ref{prop:accumulation points} and Theorem \ref{thm:2} from the introduction.
\begin{proof}[Proof of Proposition \ref{prop:lifting measures} (Ergodicity and unique lifts of measures)]
Let $(S,\alpha)$ be a translation surface, let $(\tilde S, \tilde \alpha)$ be a degree $d$ cover, and
let $p:\tilde S \to S$ denote the covering map. We assume that the translation flow is ergodic on both of these surfaces.
We will prove that Lebesgue measure is the unique translation flow invariant measure which projects to Lebesgue measure on $(S,\alpha)$ under the covering map.
Fix a non-singular basepoint $s \in S$,
and let $h:\pi_1(S,s) \to \Pi_d$ be the monodromy representation.
We will consider the regular (or normal) cover of $S$ associated to the subgroup $\ker~h \subset \pi_1(S,s)$.
Let $(\hat S, \hat \alpha)$ denote this cover. Because the subgroup $\ker~h$ is normal, there is a covering group action
of $\Delta=\pi_1(S,s)/\ker~h$ on $\hat S$, and the quotient $\hat S/\Delta$ is naturally identified with $S$.
The covering $\hat S \to S$ factors through $\tilde S$. That is, there is a subgroup $\Gamma \subset \Delta$ so that
the $\tilde S$ is isomorphic as a cover to $\hat S/\Gamma$. We let $\hat p:\hat S \to \tilde S$ denote the covering
obtained by identifying $\tilde S$ with $\hat S/\Gamma$.
Now suppose that $\tilde \mu$ is a measure on $\tilde S$ which is invariant for the translation flow,
and satisfies $p_\ast(\tilde \mu)=\lambda$, where $\lambda$ denotes Lebesgue measure on $(S,\alpha)$.
Note that the measure $\tilde \mu$ lifts to a unique measure $\hat \mu$ on $(\hat S, \hat \alpha)$ so that $\hat p_\ast(\hat \mu)=\tilde \mu$
and so that
$\gamma_\ast(\hat \mu)=\hat \mu$ for all $\gamma \in \Gamma$.
The key point in the proof is that because $\hat \mu$ projects through to Lebesgue measure on $S$, we know that if we average
the push forwards of $\hat \mu$ under the covering group $\Delta$, we get Lebesgue measure on $(\hat S, \hat \alpha)$, which we denote by $\hat \lambda$. That is,
$$\frac{1}{|\Delta|} \sum_{\delta \in \Delta} \delta_\ast(\hat \mu)=\hat \lambda.$$
Now consider the push-forward of these measures under the covering map $\hat p$. Since $p_\ast(\hat \mu)=\tilde \mu$, we see that
$$\frac{1}{|\Delta|} \left(\tilde \mu+\sum_{\delta \in \Delta \smallsetminus \{e\}} \hat p_\ast \circ \delta_\ast(\hat \mu)\right)=\tilde \lambda,$$
where $\tilde \lambda$ is the Lebesgue measure on $(\tilde S, \tilde \alpha)$. Finally, $\tilde \lambda$ is ergodic,
so each of the probability measures in the above convex combination must equal $\tilde \lambda$. In particular,
$\tilde \mu=\tilde \lambda$, which concludes the proof.
\end{proof}
\begin{proof}[Proof of Proposition \ref{prop:accumulation points}]
We assume that $(S,\alpha)$ is a finite area translation surface with infinite topological type
and that it has Teichm\"uller trajectory which is non-divergent in $\sO(S,\alpha)$.
By non-divergence, there is a subsequence of times $t_n \to \infty$
so that $g^{t_n}(S,\alpha)$ tends to some $A_\infty (S,\alpha) \in \sO(S,\alpha)$,
where $A_\infty \in \SL_\pm(2,\R)$.
Because the topology on $\sO(S,\alpha)$ arises as a quotient of the topology on $\SL_\pm(2,\R)$,
we see that there is a sequence $A_n \in \SL(2,\R)$ tending to $A_\infty$ so that $g^{t_n}(S,\alpha)$ is translation equivalent to $A_n (S, \alpha)$. It then follows that there is a sequence
of elements $R_n$ of the Veech group of $(S,\alpha)$ so that
$g^{t_n}=A_n R_n.$
Now consider a cover $(\tilde S, \tilde \alpha)$
with monodromy in $G$.
Proposition \ref{prop:action on covers} explains that the Veech group acts on the space of covers with monodromy in $G$. In particular, for each $n$, there is a cover $(\tilde S_n, \tilde \alpha_n) \in \Cov_G(S, \alpha)$ which is translation equivalent to $R_n (\tilde S, \tilde \alpha)$.
Then, in the space $\tilde \sO_G(S,\alpha)$, we have that
$$g^{t_n}(\tilde S, \tilde \alpha)=A_n R_n (\tilde S, \tilde \alpha)=A_n (\tilde S_n, \tilde \alpha_n).$$
The key observation is that $\Cov_G(S, \alpha)$ is a quotient of a Cantor set and thus sequentially compact,
so there must be a limit point $(\tilde S_\infty, \tilde \alpha_\infty)$ for the sequence $(\tilde S_n, \tilde \alpha_n) \in \Cov_G(S, \alpha)$. Since $A_n$ tends to $A_\infty$ in $\SL_\pm(2,\R)$,
we see that $g^{t_n}(\tilde S, \tilde \alpha)$ tends to $A_\infty (\tilde S_\infty, \tilde \alpha_\infty)$, which
is our desired accumulation point.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:2} (Random covers accumulate on connected covers)]
Let $(S,\alpha)$ be a finite area translation surface with infinite topological type
and a Teichm\"uller trajectory which is non-divergent in $\sO(S,\alpha)$.
As in the previous proof, this guarantees that there is a sequence $t_n \to \infty$ so that
$g^{t_n}=A_n R_n$ where $\{A_n \in \SL_\pm(2,\R)\}$ is a sequence tending to $A_\infty \in \SL_\pm(2,\R)$, and $R_n \in V(S,\alpha)$.
We will work with the space $\Covt_G(S)$ of topological covers of $(S,\alpha)$ which comes equipped with a measure $\nu_G$. In order to do this, observe that for each Veech group element, $R_n \in V(S,\alpha)$ we can find an affine homeomorphism $\phi_n:(S,\alpha) \to (S,\alpha)$ so that $D(\phi_n)=R_n$.
Recall that we denote elements of $\Covt_G(S)$ by pairs $(p, \tilde S)$ where
$p:\tilde S \to S$ is a covering map.
With an additional choice of a curve for each $n$, we obtain from $\phi_n$ an action
$$\Phi_n:\Covt_G(S) \to \Covt_G(S)$$
as in equation \ref{eq:homeomorphism acting on covers}.
Let ${\mathcal T}:\Covt_G(S) \to \Cov_G(S,\alpha)$ be the map defined in \eqref{eq:top cover to cover}. Then for each $(p, \tilde S) \in \Covt_G(S)$ we have that
$$R_n(\tilde S,\tilde \alpha)=(\tilde S_n, \tilde \alpha_n) \quad \text{where
$(\tilde S,\tilde \alpha)={\mathcal T}(p,\tilde S)$ and
$(\tilde S_n,\tilde \alpha_n)={\mathcal T}\circ \Phi_n(p,\tilde S)$.}$$
This follows from Proposition \ref{prop:action on covers}
and equation \ref{eq:homeomorphism acting on covers}.
In particular, with these hypotheses, we have that $g^{t_n}(\tilde S, \tilde \alpha)$ is translation equivalent to $A_n (\tilde S_n, \tilde \alpha_n)$. Since the sequence $\{A_n\}$ converges in $\SL_\pm(2,\R)$, it suffices to find a connected
accumulation point of the sequence $\{\Phi_n(p, \tilde S)\}$ for $\nu_G$-almost every $(p, \tilde S) \in \Covt_G(S)$.
(This is because our measure $m_G$ on the space of covers up to translation equivalence, $\Cov_G(S,\alpha)$, is the push forward of $\nu_G$ under the projection $\Covt_G(S) \to \Cov_G(S,\alpha)$.)
Let $\sD \subset \Covt_G(S)$ denote the collection of disconnected surfaces.
Note that these surfaces have $\nu_G$-measure zero by Proposition \ref{prop:measure zero}.
Let $\sE \subset \Covt_G(S)$ denote the collection of covers $(p, \tilde S) \in \Covt_G(S)$
so that every accumulation point of $\{\Phi_n(p, \tilde S)\}$ is disconnected.
Then if $\sU \subset \Covt_G(S)$ is open and contains $\sD$, we have that
for every $(p, \tilde S) \in \sE$,
there must be an $N$ so that $\Phi_n(p, \tilde S) \in \sU$ for all $n > N$.
Indeed, if this were not true, then infinitely many $\Phi_n(p, \tilde S)$ lie in
the complement of $\sU$, which is sequentially compact since $\Cov_G(S,\alpha)$ is a Cantor set by Proposition \ref{prop:topological covers}. In other words, we have
$$\sE \subset \bigcup_N \bigcap_{n>N} \Phi_n^{-1}(\sU)= \liminf_{n \to \infty} \Phi_n^{-1}(\sU).$$
Now fix some $\epsilon>0$. We will show that $\nu_G(\sE)<\epsilon$.
Because $\nu_G$ is a Borel probability measure on a Cantor set,
$\nu_G$ is regular. Thus because $\nu_G(\sD)=0$, we can find an open set $\sU$ containing $\sD$
so that $\nu_G(\sU)<\epsilon$. Then from the above, we have
$$\nu_G(\sE) \leq \nu_G \left( \liminf_{n \to \infty} \Phi_n^{-1}(\sU) \right) \leq \liminf_{n \to \infty} \nu_G \circ \Phi_n^{-1}(\sU).$$
But, Corollary \ref{cor:action on topological covers} tells us that
$\nu_G$ is $\Phi_n$-invariant. Thus, the above sequence of inequalities
tells us that $\nu_G(\sE) \leq \nu_G(\sU) < \epsilon$. Since $\epsilon$
was arbitrary, we conclude that $\nu_G(\sE)=0$ as desired.
\end{proof}
\section{Examples of devious covers}
\label{sect:evil covers}
\subsection{Chamanara's surfaces}
\label{sect:Chamanara}
\compat{The surfaces studied here should all be named $(S_n,\alpha_n)$ with the primary case being $n=2$. Only a generic surface should be $(S,\alpha)$.}
We introduce a surface $(S_2,\alpha_2)$ first studied by Chamanara in \cite{Chamanara04}. (See also the related work \cite{CGL}.)
The surface is built from a closed $1 \times 1$ square with each of the edges subdivided into intervals of length $\frac{1}{2^k}$ for $k \in \N$ as indicated in left side of Figure \ref{fig:Chamanara}. The vertical intervals of equal length are then glued together by translation, and we do the same to the horizontal intervals. The intervals being identified have been labeled by the same integers in the figure. The endpoints of these intervals being glued
and the corners of the square are discarded to give the space a translation structure.
\begin{figure}
\caption{Chamanara's surfaces $(S_2,\alpha_2)$ and $(S_3, \alpha_3)$.}
\label{fig:Chamanara}
\end{figure}
The surface $(S_2,\alpha_2)$ has an affine automorphism $\phi_2$ whose derivative is
$$D(\phi_2)=\left[\begin{array}{rr} \frac{1}{2} & 0 \\ 0 & 2 \end{array}\right] \in V(S_2,\alpha_2).$$
To see this observe that the image of the square under this linear map is a $\frac{1}{2} \times 2$ rectangle. If we push this rectangle into the surface allowing the rectangle only to pass through the edge labeled zero in the figure,
we see the identifications are respected and thus this map determines
an affine automorphism $\phi_2$ of the surface.
The vertical and horizontal flows on Chamanara's surface $(S_2,\alpha_2)$ can be seen as the suspension flow over the dyadic odometer
as defined in \eqref{eq:odometer}, except that our construction introduces countably many singularities through which the flows are not defined. To see this, it suffices to consider the dyadic odometer as an infinite interval exchange map on $[0,1]$ defined for $x\in[0,1]$, as the dyadic odometer on the dyadic expansion $(x_1,x_2,\dots) \in X_2=\{0,1\}^\N$ of $x = \sum x_i 2^{-i}$. As such, the map induced on a transversal $T_2$ running from the bottom to the top of the square making up $(S_2,\alpha_2)$ such as the one depicted in Figure \ref{fig:skew}
is isomorphic to the dyadic odometer. See \cite[\S 2]{LT:models} for a thorough description of the dyadic odometer and its relationship to Chamanara's surface.
More generally, for integers $n \geq 2$, one can construct a homeomorphic translation surface $(S_n,\alpha_n)$ by letting the identified sides in Figure \ref{fig:Chamanara} be of length $\frac{n-1}{n^k}$ for $k\in\mathbb{N}$. The right side of Figure \ref{fig:Chamanara}
illustrates the case of $n=3$. The surface $(S_n, \alpha_n)$ admits
an affine automorphisms $\phi_n$ with diagonal derivative and eigenvalues of $n$ and $\frac{1}{n}$. As in the case of the dyadic odometer, this surface admits a section which is the $n$-adic odometer.
\begin{remark}[Veech groups]
The Veech group $V(S_n,\alpha_n)$ is known to be generated by two parabolics. See \cite{HR:chamanara}.
\end{remark}
\compat{Moved this from later in the section.}
We will now introduce some more notation which will be useful for the Proof of Theorem \ref{thm:skew} and for work later in this section.
Let $G$ be a subgroup of the symmetric group $\Pi_d$ with $d \geq 2$ which acts simply transitively on $\{1,\ldots, d\}$.
For a general surface $(S,\alpha)$ the space $\Cov_G(S,\alpha)$ of covers with monodromy in $G$ is a quotient of the space of topological covers $\Covt_G(S)$ by the translation equivalence relation; see \eqref{eq:top cover to cover}.
However it is not hard to see that by Remark \ref{rem:primitivity} that this equivalence relation is trivial in the case of $(S_n, \alpha_n)$. (The surface $(S_n,\alpha_n)$ is formed by identifying the boundary edges of the square along an IET. The deck group of the universal cover of $(S_n,\alpha_n)$ acts transitively on lifts of these squares, and the collection of these lifts must be preserved by translation automorphisms because of the singularities in their boundary.) \compat{Detail added above to address referee's comment.}
Thus by recalling \eqref{eq:space of covers} we see
that the space of covers of $(S_n,\alpha_n)$ with monodromy in $G$ can be thought of as
\begin{equation}
\label{eq:cov chamanara}
\Cov_G(S_n,\alpha_n)=\Pi_d \bs \Hom\big(\pi_1(S_n,s_0),G\big),
\end{equation}
where $s_0$ is a basepoint of $S_n$.
If $h$ is a homomorphism from $\pi_1(S_n,s_0)$ to $G$, we use $[h]$ to denote its equivalence class in $\Cov_G(S_n,\alpha_n)$.
Recall that formally, the affine automorphism $\phi^{-1}_n$ does not act on the fundamental group of $S_n$.
We choose a basepoint $s_0$ in the interior of the square near the southwest corner of the square in Figure \ref{fig:Chamanara}.
To get an action on the fundamental group, we need to select
a curve joining $s_0$ to $\phi_n^{-1}(s_0)$ as described by equation \ref{eq:curve needed}. Because we chose $s_0$ in the interior of the square near the southwest corner, its image $\phi_n^{-1}(s_0)$ will also lie near the southwest corner and in the interior of the square. We specify $\beta$ to be a curve joining $\phi^{-1}(s_0)$ to $s_0$ while not leaving the interior of the square. Then as in \eqref{eq:curve needed} we define the group automorphism
\begin{equation}
\label{eq:phi beta}
\phi^{-1}_\beta:\pi_1(S_n,s_0) \to \pi_1(S_n,s_0); \quad [\gamma] \mapsto [\beta^{-1} \bullet (\phi_n^{-1} \circ \gamma) \bullet \beta].
\end{equation}
We now see that Theorem \ref{thm:skew} is a consequence of Corollary \ref{cor:2}.
\begin{proof}[Proof of Theorem \ref{thm:skew}]
Recall that the $n$-adic odometer is uniquely ergodic, because it can be understood as a minimal rotation of a compact abelian group.
Therefore, the translation flow on the surface $(S_n, \alpha_n)$ is also uniquely ergodic.
Observe that the Teichm\"uller flow is periodic, because the affine automorphism $\phi_n$ has diagonal derivative.
We begin with some definitions which are illustrated in Figure \ref{fig:skew}. Since this figure illustrates $n=2$, we will just discuss this case for this paragraph. The dotted line $T_2$ on $S_2$, isometric to $[0,1]$, is a transversal to the translation flow. We selected this transversal to be vertical and to pass through the basepoint $s_0$ of $S_2$. The return map to the transversal is isomorphic to the dyadic odometer.
We create an infinite set of closed loops $\gamma_i$ on $S_2$ indexed by $\mathbb{N}$. For each $i\in\mathbb{N}$, the loop $\gamma_i$
moves from the basepoint to a point $x\in T_2$ within $T_2$, follows the translation flow until it first returns to $T_2$,
and then travels back to the basepoint within $T_2$ (shown in Figure \ref{fig:skew} as three dashed line segments).
In defining $\gamma_i$, we insist that the dyadic expansion $(x_1,x_2,x_3,\dots)$ of $x$ has the property that $x_j = 1$ for all $j<i$ and $x_i = 0$. Doing this for every $i\in \mathbb{N}$ we create the countable set $\{\gamma_i\}$. Let $\Gamma^+ = \langle\gamma_1,\gamma_2,\dots\rangle$ be the free group generated by the $\gamma_i$ which is a subgroup of $\Gamma = \pi_1(S_2,s_0)$.
\begin{figure}
\caption{{\em Bottom:}
\label{fig:skew}
\end{figure}
The general case is not qualitatively different. Following the same ideas in the previous paragraph, we define $\gamma_i$, $\Gamma^+$
and $\Gamma$ for all integers $n \geq 2$.
Fix a $d \geq 2$ and a subgroup $G \subset \Pi_d$ that acts transitively on $\{1,\ldots, d\}$.
Corollaries \ref{cor:lifting} and \ref{cor:2} imply that $m_G$-almost every cover has a uniquely ergodic translation flow,
where $m_G$ is the probability measure on $\Cov_G(S_n,\alpha_n)$ from \S \ref{sect:measures}.
Recall that $m_G$ is the measure induced by the quotient \eqref{eq:cov chamanara} from the product measure $\mu$ on
$\Hom(\Gamma,G)$; see Definition \ref{def:random cover} of in \S \ref{sect:measures}.
Thus, for $\mu$-a.e. $h \in \Hom(\Gamma,G)$, the corresponding cover
$(\tilde S_h, \tilde \alpha_h)$ has uniquely ergodic translation flow.
Consider the automorphism
$\phi^{-1}_\beta$ of $\pi_1(S_n, \alpha_n)$ given in \eqref{eq:phi beta}. We observe by inspecting the action
of $\phi^{-1}$ that
$$\phi^{-1}_\beta(\gamma_i)=\gamma_1^{n-1} \gamma_{i+1} \quad \text{for all $i \geq 1$},$$
where calculations are done in the fundamental group.
This implies that $\phi^{-1}_\beta(\Gamma^+) \subset \Gamma^+$.
The inclusion $i:\Gamma^+\rightarrow\Gamma$ then induces a surjective map $i^*: \Hom(\Gamma,G)\rightarrow \Hom(\Gamma^+,G)$ which commutes with the actions of $\phi^{-1}_\beta$.
The set $\Hom(\Gamma^+,G)$ is a Cantor set with a product measure $\mu_{+}$ as defined in Definition \ref{def:random cover}. It is straight forward to see from the definitions of these measures that
$\mu_{+} = (i^*)_*\mu$.
Let $i^*(\psi) = \psi_+\in\Hom(\Gamma^+,G)$ for some $\psi \in\Hom(\Gamma,G)$ and let
$(\tilde S_{\psi},\tilde \alpha_{\psi})$ be the $d$-cover of $(S_n,\alpha_n)$ determined by $\psi$.
The transversal $T_2$ lifts to $\tilde{T}_2$, a transversal on $\tilde S_{\psi}$
for the translation flow.
It is the union of the $d$ copies $T_2^1,\dots, T_2^d$ of $T_2$, which are also illustrated using dotted lines in the figure. As such, the translation flow on a cover $(\tilde S_{\psi}, \tilde \alpha_{\psi})$ is canonically a suspension flow of the skew product over the $n$-adic odometer.
One can observe that this skew product is precisely $E_{\psi_+}$ given in \eqref{eqn:skew} of the introduction.
Whenever the translation flow on $(\tilde S_{\psi}, \tilde \alpha_{\psi})$ is uniquely ergodic,
the skew product $E_{\psi_+}$ must be as well. Since this holds for $\mu$ a.e. $\psi$ and $\mu_{+} = (i^*)_*\mu$,
we see that for $\mu_+$ a.e. $\psi_+$, $E_{\psi_+}$ is uniquely ergodic.
\end{proof}
\subsection*{Pathological covers}
For the remainder of the section, we will concentrate on the simplest of Chamanara's surfaces,
$(S_2,\alpha_2)$. \compat{Note that previously this section called $(S_2,\alpha_2)$ simply $(S,\alpha)$. The referee objected so hopefully I have eliminated all $(S,\alpha)$ from this section.}
We will simplify the notation for the hyperbolic automorphism
$\phi_2$ by denoting it by $\phi$.
Our goal with the remainder of this section is to investigate what happens when we have a connected
cover $(\tilde S_2, \tilde \alpha_2)$ which when iterated by application of $\phi$
only accumulates on disconnected covers. The disconnected covers in $\Cov_G(S_n,\alpha_n)$ are given by
$$\bigcup_{H \in \sH} \Pi_d \bs \Hom(\Gamma,H),$$
where $\Gamma=\pi_1(S_2,s_0)$ and $\sH$ denotes the collection of all subgroups of $G$ which fail to act transitively on $\{1,\ldots, d\}$.
It can be observed that the connectivity of $(\tilde S, \tilde \alpha)$
does not guarantee the ergodicity of the (horizontal) translation flow, since the connectivity can be arranged with only the gluings of horizontal edges when building the cover as $d$ copies of the unit square with edge identifications. It is not surprising then that such connected devious covers are dense
inside of the space of covers:
\begin{theorem}[Non-ergodic covers]
\label{thm:Reza non-ergodic}
Let $h \in \Hom(\Gamma,G)$. The translation flow on the cover $(\tilde S_h,\tilde \alpha_h)$
of $(S_2,\alpha_2)$ associated to $[h]$ is non-ergodic whenever there is an $H \in \sH$ so that every accumulation point of $h \circ \phi_\beta^{-n}$ (as $n \to +\infty$) lies in $\Hom(\Gamma,H)$. For any $H \in \sH$, there exists a
collection of connected covers dense in $\Cov_G(S_2,\alpha_2)$
so that every accumulation point lies in $\Pi_d \bs \Hom(\Gamma,H)$
but no accumulation point lies in $\Pi_d \bs \Hom(\Gamma,H')$ for any proper subgroup $H' \subset H$.
\end{theorem}
It is interesting to consider whether there are connected covers which accumulate only on disconnected covers under the Teichm\"uller deformation but whose translation flow is nonetheless uniquely ergodic.
We show such covers exist.
This is analogous to sufficiently slow divergence of the Teichm\"uller deformation giving rise to unique ergodicity in the classical setting of closed translation surfaces as in \cite{CE07} \cite{rodrigo:erg}.
\begin{theorem}
\label{thm:Reza slow}
Suppose $G$ is a subgroup of $\Pi_d$, and $H_1, H_2 \subset G$ are subgroups which do not act transitively
on $\{1,\ldots,d\}$, but the group generated by the elements of $H_1 \cup H_2$ does act transitively.
Then, there are finite covers of $(S_2,\alpha_2)$ with monodromy in $G$
whose translation flow is uniquely ergodic, but whose orbit under $\phi$ accumulates only on surfaces in the collection of disconnected covers,
$$\Pi_d \bs \big(\Hom(\Gamma,H_1) \cup \Hom(\Gamma,H_2)\big).$$
\end{theorem}
\begin{remark}
It seems likely that there are also non-ergodic covers whose orbits under $\phi$ accumulate as in Theorem \ref{thm:Reza slow}. We do not investigate this question.
\end{remark}
The key to proving these results is an understanding of the action on $\Hom(\Gamma,G)$ given by
$$h \mapsto h \circ \phi^{-1}_\beta,$$
where $\phi^{-1}_\beta: \Gamma \to \Gamma$ is the automorphism from \eqref{eq:phi beta}.
In order to describe this action, we select a generating set for $\Gamma$. For each integer
$n$, we will let $\gamma_n \in \Gamma$ be a homotopy class of curves which start and end at the basepoint.
If $n \leq 0$, we define $\gamma_n$ to contain the curves which move downward from the basepoint passing through the horizontal edge labeled $n$ and returning to the basepoint without passing through any other labeled edges. We similarly define $\gamma_n$ for $n>0$ to contain the curves which move rightward over the vertical edge labeled $n$.
(This is compatible with the definition of $\gamma_n$ for $n \geq 1$ in the Proof of Theorem \ref{thm:skew},
and depicted in Figure \ref{fig:skew}.)
Observe:
\begin{equation}
\label{eq:generated}
\Gamma=\pi_1(S_2,s_0) \quad \text{is freely generated by} \quad \{\gamma_n~:~n \in \Z\}.
\end{equation}
(The curves $\gamma_n$ can be taken to be pairwise disjoint except at the basepoint and so the union of these curves is a bouquet of countably many circles. There is a deformation retraction of the surface to this bouquet.)
The dynamics of $\phi_\beta$ action on $\Hom(\Gamma,G)$ turn out to be conjugate to the shift on $G^\Z$:
\begin{lemma}
\label{lem:conjugacy}
The map $\mathfrak{g}: \Hom\big(\Gamma,G\big) \to G^\Z$ defined by
$$\mathfrak{g}(h)_m=h \circ \phi_\beta^{-m}(\gamma_1)$$
is a homeomorphism. Let $\sigma:G^\Z \to G^\Z$ be the shift map $\sigma(\g)_i=\g_{i+1}$. Then
$$\mathfrak{g}(h \circ \phi_\beta^{-1})=\sigma \circ \mathfrak{g}(h) \quad \text{for all $h \in \Hom\big(\Gamma,G\big)$.}$$
\end{lemma}
We call $\mathfrak{g}(h)$ the {\em $G$-sequence} of $h$. As a first step to proving this theorem, we work out the action
of $\phi_\beta^{-1}$ and its inverse $\phi_\beta$ on $\Gamma$:
\begin{proposition}
For each $n \in \N$, we have
$$\phi^{-1}_\beta(\gamma_n)=\begin{cases}
\gamma_{n+1} \gamma_1^{-1} & \text{if $n < 0$} \\
\gamma_1 & \text{if $n=0$} \\
\gamma_1 \gamma_{n+1} & \text{if $n>0$,}
\end{cases}
\and
\phi_\beta(\gamma_n)=\begin{cases}
\gamma_{n-1} \gamma_{0} & \text{if $n \leq 0$}\\
\gamma_0 & \text{if $n=1$}\\
\gamma_0^{-1} \gamma_{n-1} & \text{if $n>1$.}
\end{cases}
$$
\end{proposition}
This proposition may be proved by inspecting the action of $\phi^{-1}_\beta$ as defined in
\eqref{eq:phi beta}. We leave the details to the reader.
The following lemma describes how to recover information about the $\phi_\beta$-orbit of $h$ from its $G$-sequence.
\begin{lemma}
\label{lem:G-sequence}
Let $\g=\mathfrak{g}(h)$ be the $G$-sequence of a homomorphisms $h:\Gamma \to G$. Then, for each $k, n \in \Z$,
$$h \circ \phi_\beta^{-k}(\gamma_n)=\begin{cases}
\g_{k+n-1}\g_{k+n}^{-1} \g_{k+n+1}^{-1} \ldots \g_{k-1}^{-1} & \text{if $n<0$}\\
\g_{k-1} & \text{if $n=0$}\\
\g_k & \text{if $n=1$} \\
\g_k^{-1} \g_{k+1}^{-1} \ldots \g_{k+n-2}^{-1} \g_{k+n-1} & \text{if $n>1$.}
\end{cases}$$
\end{lemma}
\begin{proof}
Observe that by definition $h \circ \phi_\beta^{-k}(\gamma_1)=\g_k$ for all $k \in \Z$. This case of $n=1$ will serve as a base case for proving the statement holds when $n \geq 1$. Note that the formula given in the case of $n>1$ can be extended to hold for $n=1$ if one allows the (empty) product of inverses $\g_k^{-1} \g_{k+1}^{-1} \ldots \g_{k+n-2}^{-1}$
to be the identity when $n=1$. So, suppose our formula holds for some $n \geq 1$ and all $k$,
we will show it holds for $n+1$ and all $k$. Using the proposition, we observe that for $n \geq 1$,
$$h \circ \phi_\beta^{-k-1}(\gamma_n)=h\circ \phi_\beta^{-k}(\gamma_1 \gamma_{n+1})=\g_k \cdot h\circ \phi_\beta^{-k}(\gamma_{n+1}).$$
By our inductive hypothesis applied to the left side, we see that
$$\g_{k+1}^{-1} \g_{k+2}^{-1} \ldots \g_{k+n-1}^{-1} \g_{k+n}=\g_k \cdot h\circ \phi_\beta^{-k}(\gamma_{n+1})$$
which gives us that $\phi_\beta^{-k}(\gamma_{n+1})=\g_k^{-1} \ldots \g_{k+n-1}^{-1} \g_{k+n}$. This proves
our formula for $n+1$ and and all $k$. So, by induction, the statement holds for $n \geq 1$.
Now we consider the base case of $n=0$. Observe that
$$h \circ \phi_\beta^{-k}(\gamma_0)=h \circ \phi_\beta^{-k+1} \circ \phi^{-1}_\beta(\gamma_0)=h \circ \phi_\beta^{-k+1}(\gamma_1)=\g_{k-1}.$$
We will proceed by induction to cover cases with $n<0$. The case of $n=0$ can serve as our base case since
when $n=0$ we have $\g_{k+n-1}\g_{k+n}^{-1} \g_{k+n+1}^{-1} \ldots \g_{k-1}^{-1}=\g_{k-1}$ by convention as above since the product of negations
$\g_{k+n}^{-1} \g_{k+n+1}^{-1}\ldots \g_{k-1}^{-1}$ is an empty product (as $k+n>k-1$ here).
Now suppose the formula holds for some $n\leq 0$ and all $k$. Then, by the proposition and the base case
$$h \circ \phi_\beta^{-k+1}(\gamma_n)=h \circ \phi_\beta^{-k}(\gamma_{n-1}\gamma_0)=h \circ \phi_\beta^{-k}(\gamma_{n-1}) \cdot \g_{k-1}.$$
By inductive hypothesis, we see
$$h \circ \phi_\beta^{-k}(\gamma_{n-1})=h \circ \phi_\beta^{-k+1}(\gamma_n) \cdot \g_{k-1}^{-1}=
(\g_{k+n-2}\g_{k+n-1}^{-1} \ldots \g_{k}^{-1}) \g_{k-1}^{-1}.$$
This completes the inductive step, proving the statement for all $n \leq 0$.
\end{proof}
This Lemma allow us to prove that $\mathfrak{g}$ is a topological conjugacy:
\begin{proof}[Proof of Lemma \ref{lem:conjugacy}]
The map $\mathfrak{g}$ is clearly continuous. The identity provided in Lemma \ref{lem:G-sequence} considered in the special case of $k=0$ allows us to evaluate $h$ on the generators $\{\gamma_n\}$ of $\Gamma$ in terms of $\mathfrak{g}(h)$. This therefore gives an inverse $\mathfrak{g}^{-1}$, and we observe that it is continuous. The conjugacy equation is easily verified:
$$\mathfrak{g}(h \circ \phi_\beta^{-1})_m=h \circ \phi_\beta^{-1} \circ \phi_\beta^{-m}(\gamma_1)=h \circ \phi_\beta^{-m-1}(\gamma_1)=\mathfrak{g}(h)_{m+1}=\big(\sigma \circ \mathfrak{g}(h)\big)_m$$
for all $m \in \Z$.
\end{proof}
Now we prove Theorem \ref{thm:Reza non-ergodic} on the non-ergodicity of covers.
\begin{proof}[Proof of Theorem \ref{thm:Reza non-ergodic}]
Let $h:\Gamma \to G$ be a homomorphism, and suppose that all accumulation points of
$h \circ \phi_\beta^{-k}$ as $k \to \infty$ lie in $\Hom(\Gamma,H)$ for some subgroup $H \subset G$,
where $H$ does not act transitively on $\{1,\ldots, d\}$.
We will show that the translation flow on $(\tilde S_h, \tilde \alpha_h)$ is not ergodic. By compactness of $\Hom(\Gamma,H)$, there is a $K$ so that
for each $k \geq K$, $h \circ \phi_\beta^{-k}(\gamma_1) \in H$. In terms of the $G$-sequence $\g=\mathfrak{g}(h)$, we see that $\g_k \in H$ for $k \geq K$. It can be observed from the formula in Lemma \ref{lem:G-sequence}
that it follows that $h \circ \phi_\beta^{-K}(\gamma_n) \in H$ for all $n \geq 1$. Observe that the horizontal straight-line flow on the surface $(S_2,\alpha_2)$ only crosses
the intervals with positive label in Figure \ref{fig:Chamanara}. Now consider the cover associated to
$h \circ \phi_\beta^{-K}$. The cover can be built from copies of the square indexed by $\{1,\ldots, d\}$ with edges identified according to
$h \circ \phi_\beta^{-K}$. In particular, the intervals with positive label are glued only according to elements of $H$. Thus, points in copy $i \in \{1,\dots, d\}$ only can reach the copies of the square indexed by elements
of the orbit $H(i)$, and by assumption $H(i) \neq \{1,\ldots, d\}$. In particular the union of the squares
indexed by $H(i)$ gives an invariant set with measure strictly between zero and full measure. Note that $\phi_\beta^K$ induces an affine homeomorphism with diagonal derivative from the cover associated to $h$ to the cover associated to $h \circ \phi_\beta^{-K}$, so pulling back this invariant set gives an invariant subset of the straight line flow for the cover associated to $h$ with intermediate measure as desired.
We will now construct a dense set of devious covers as described in second sentence of the Theorem.
Consider the set $X_H \subset G^\Z$ consisting of those $\g \in G^\Z$ which satisfy the following statements:
\begin{enumerate}
\item The subgroup of $G$ generated by $\{\g_k:~k \in \Z\}$ acts transitively on $\{1,\ldots, d\}$.
\item There is a $K \in \Z$ so that $\g_k \in H$ for $k>K$.
\item There is a $B>0$ so that for any $k>K$, the subgroup $H \subset G$ is generated by
$\{\g_k, \g_{k+1}, \ldots, \g_{k+B-1}\}.$
\end{enumerate}
It should be clear that $X_H$ is dense in $G^\Z$. Since $\mathfrak{g}$ is a homeomorphism by Lemma \ref{lem:conjugacy}, we know $\mathfrak{g}^{-1}(X_H)$ is dense in $\Hom(\Gamma,G)$. Now let $h \in \mathfrak{g}^{-1}(X_H)$ and consider the cover $(\tilde S_h, \tilde \alpha_h)$ of $(S_2, \alpha_2)$. By Lemma \ref{lem:G-sequence},
the group generated by $\{\g_k\}$ coincides with the image $h(\Gamma)$ and so statement (1) implies that $\tilde S_h$ is connected. Now let $h'$ be any accumulation point of $h \circ \phi_\beta^{-n}$ taken as $n \to +\infty$.
Let $\g'=\mathfrak{g}(h')$. Then by Lemma \ref{lem:conjugacy}, $\g'$ is an accumulation point of $\sigma^n(\g)$.
Statement (2) then implies that $\g' \in H^\Z$ and so $h' \in \Hom(\Gamma,H)$ by Lemma \ref{lem:G-sequence}.
Statement (3) implies that $\{\g'_1, \g'_2, \ldots, \g'_B\}$ generates $H$ and Lemma \ref{lem:G-sequence}
then guarantees that $h'(\Gamma)=H$ so in particular $h' \not \in \Hom(\Gamma,H')$ for any proper subgroup $H' \subset H$.
\end{proof}
Now we will move toward proving Theorem \ref{thm:Reza non-ergodic} about the existence of covers with ergodic translation flow whose $\phi_\beta$-orbits accumulate only on disconnected covers. Recall that this theorem dealt with the situation when we have two subgroups $H_1$ and $H_2$ of $G$ which do not act transitively on $\{1,\ldots, d\}$ but which together act transitively.
We now introduce a combinatorial tool that will allow us
to find examples of $\g=\mathfrak{g}(h)$ where the sequence of covers associated to $h \circ \phi_\beta^{-k}$ only has disconnected limits.
Let $a \in \Z$ and $A=\{n \in \Z~:~n \geq a\}.$ Let $p:A \to \{0, 1, 2\}$ be an arbitrary function
with the property that $p(n)+p(n+1) \neq 3$ for each $n \in A$ (or equivalently $p(n)=1$ implies $p(n+1) \neq 2$ and
$p(n)=2$ implies $p(n+1)\neq 1$.) For such a map $p$ and an $n \in A$, we define the {\em preimage
interval} containing $n$, $I(p,n) \subset \Z$, to be the maximal collection of consecutive elements of $A$ so that $n \in I(p,n)$ and $p$ is constant on $I(p,n)$. We'll say that $\g \in G^\Z$ is
{\em $p$-ready} \label{def:p-ready} if the following two statements hold for each $n \in A$:
\begin{itemize}
\item If $p(n)=0$, then $\g_n \in H_1 \cap H_2$.
\item If $p(n) \in \{1,2\}$, then $\g_n \in H_{p(n)}$ and $\{\g_k~:~k \in I(p,n)\}$ generates $H_{p(n)}$.
\end{itemize}
Let $|I(p,n)|$ denote the number of integers in $I(p,n)$.
\begin{proposition}[Criterion for disconnected accumulation points]
\label{prop:disconnected}
Let $h \in \Hom(\Gamma,G)$ and suppose $\g=\mathfrak{g}(h)$ is $p$-ready. Suppose that
\begin{equation}
\label{eq:growth of zeros}
\lim_{n \to \infty; ~p(n)=0} |I(p,n)|=\infty,
\end{equation}
(with the limit taken only over those $n$ so that $p(n)=0$).
Then every accumulation point of $h \circ \phi_\beta^{-k}$ as $k \to \infty$ corresponds to a disconnected cover. Moreover, these accumulation points all lie in
$\Hom(\Gamma,H_1) \cup \Hom(\Gamma,H_2).$
\end{proposition}
\begin{proof}
Let $h_\infty$ be an accumulation point of $h \circ \phi_\beta^{-k}$ as $k \to \infty$.
Then there is an increasing sequence of integers $k_j$ so that $h_\infty=\lim_{j \to \infty} h \circ \phi_\beta^{-k_j}$. Let $\g^\infty=\mathfrak{g}(h_\infty)$. By Lemma \ref{lem:conjugacy}, we have
$$\g^\infty = \lim_{j \to \infty} \sigma^{k_j}(\g).$$
We claim there is an $i \in \{1,2\}$ so that
$\g^\infty_m \in H_i$ for all $m \in \Z$. If not, there are two indices $n_1$ and $n_2$ so that $\g^\infty_{n_1} \in H_1 \smallsetminus H_2$ and $\g^\infty_{n_2} \in H_2 \smallsetminus H_1$. Then for $j$ large enough,
$\sigma^{k_j}(\g)_{n_1}=\g_{k_j+n_1}$ and $\sigma^{k_j}(\g)_{n_2}=\g_{k_j+n_2}$ have the same property.
From the definition of $p$-ready we have $p(k_j+n_1)=1$ and $p(k_j+n_2)=2$ so between $k_j+n_1$ and $k_j+n_2$ there is a $m_j \in \Z$ so that $p(m_j)=0$ and we must have $|I(p,m_j)| < |n_1-n_2|$. This violates \eqref{eq:growth of zeros}.
We have shown $\g^\infty \in H_i^\Z$. Lemma \ref{lem:G-sequence} implies that
$h_\infty \in \Hom(\Gamma, H_i)$.
\end{proof}
Our proof of Theorem \ref{thm:Reza slow} will combine the above with the following lemma, which will allow us to apply the ergodicity
criterion given in Theorem \ref{thm:integrability}.
\begin{lemma}
\label{lem:contribution}
Let $G \subset \Pi_d$ be a subgroup which acts transitively on $\{1,\ldots, d\}$
for some $d \geq 2$.
For each finite subset $J \subset \Z$ and for each $\eta>0$, there is a constant $c=c(\eta, J)>0$ so that
for each $h \in \Hom(\Gamma,G)$ that satisfies the condition that the subgroup generated by
$\{h(\gamma_j)~:~j \in J\}$ acts transitively on $\{1, \ldots, d\}$, there is a connected subsurface $\tilde U_h$ of the cover $(\tilde S_h, \tilde \alpha_h)$ associated to $h$, so that $\text{Area}(\tilde S_h \smallsetminus \tilde U_h)<\eta \text{Area}(\tilde S_h)$ and
$$\epsilon(t)^4 \sD_t^{-2} > c \quad \text{for each
$t \in \R$ with $|t| \leq \frac{\ln(2)}{2}$,}$$
where $\epsilon(t)$ represents a lower bound for the distance from points in $\tilde U_t$ to points in $\Sigma$
and $\sD_t$ is an upper bound for the diameter of $\tilde U_h$ both measured using the pullback metric $\dist_t$
as in Theorem \ref{thm:integrability}.
\end{lemma}
We note that the expression $\epsilon(t)^4 \sD_t^{-2}$ is the function inside the integral in Theorem \ref{thm:integrability} in the special case of considering a single connected subsurface (i.e., $C_t \equiv 1$).
\begin{proof}
We will explicitly describe how to build $\tilde U_h \subset (\tilde S_h, \tilde \alpha_h)$. We will define
$U$ to be a subsurface of $(S_2,\alpha_2)$, which only depends on $J$ and $\eta$. Recall that $(S_2,\alpha_2)$ was built from a single unit area square with edge identifications. To build $U$, start with a smaller concentric square with area equal to $1-\eta$. Then for each $j \in J$, consider the edge of the surface labeled $j$. Attach to the concentric square a ``handle'' passing through the edge, which stays a bounded distance away from $\Sigma$ (the points added in the metric completion). See Figure \ref{fig:reza u} for an example. Then, we define $\tilde U_h$ to be the preimage of $U$
under the covering map $\tilde S_h \to S$. We note that $\tilde U_h$ is connected because of our condition that $\{h(\gamma_j)~:~j \in J\}$ acts transitively.
\begin{figure}
\caption{A subsurface $U$ when $J=\{-1,1,3\}
\label{fig:reza u}
\end{figure}
The $\dist_t$ distances $\epsilon(t)$ and ${\mathcal D}_t$ are the same as the corresponding distances measured
using $g^t(\tilde U)$ in $g^t(\tilde S_h, \tilde \alpha_h)$. Note that the minimal distance from the boundary of $g^t(\tilde U_h)$ to the metric completion in $g^t(\tilde S_h, \tilde \alpha_h)$ is the same as the minimal distance from $g^t(U)$ to the metric completion in $g^t(S_2,\alpha_2)$. This
distance is always positive and varies continuously in $t$, so we can get a uniform lower bound
which holds when $|t| \leq \frac{1}{2} \ln 2$. Similarly,
the diameter of $g^t(\tilde U_h)$ varies continuously in $t$, so we can get an upper bound on the diameter which is uniform in $t$ with $|t| \leq \frac{1}{2} \ln 2$. Finally observe that up to translation equivalence there are only finitely many $\tilde U_h$ as we vary $h$. (The geometry of $\tilde U_h$ only depends on the restriction of $h$ to $J$.)
Thus, we can get an upper bound on the diameter of which is uniform in both $h$ satisfying our condition
and $t$ satisfying $|t| \leq \frac{1}{2} \ln 2$.
\end{proof}
We need a mechanism to ensure that $h$
satisfies the condition of the lemma (i.e., for some fixed $J \subset \Z$ the subgroup generated by $\{ h(\gamma_j)~:~j \in J\}$ acts transitively on $\{1,\ldots, d\}$),
given conditions on the $G$-sequence $\mathfrak{g}(h)$. This mechanism is a corollary of Lemma \ref{lem:G-sequence}.
\begin{corollary}
\label{cor:conversion}
Suppose $h$ has $G$-sequence $\g=\mathfrak{g}(h)$. Let $m, k \in \Z$ with $m>0$. Then, if
the subgroup generated by $\{\g_j~:~k-m\leq j \leq k+m\}$ acts transitively on $\{1,\ldots,d\}$, then so does
the subgroup generated by $\{h \circ \phi_\beta^{-k} (\gamma_j)~:~-m+1\leq j \leq m+1\}$.
\end{corollary}
\begin{proof}
Using Lemma \ref{lem:G-sequence}, we can find an expression for each $\g_j$ with $k-m\leq j \leq k+m$
in terms of a $\{h \circ \phi_\beta^{-k} (\gamma_j)~:~-m+1\leq j \leq m+1\}$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:Reza slow}]
We will exhibit an ergodic cover $(\tilde S_h, \tilde \alpha_h)$ by verifying the conditions of Theorem \ref{thm:integrability}. As stated, this theorem involves verifying that for each $\eta>0$, a certain integral is infinite. However, it really only involves proving it for arbitrarily small $\eta$, since if the statement is true for $\eta>0$ then it is also true for any $\eta'>\eta$ with the same subsurfaces and geometric functions chosen.
Hence, it suffices to verify that this integral is infinite for each of $\eta_i>0$ in a sequence tending to zero.
The main tool to get infinite integrals is Lemma \ref{lem:contribution}. We will only apply this Lemma to sets $J$ of the form
$$J_m=\{n \in \Z~:~|n| \leq m \}.$$
Let
$\Hom_m \subset \Hom(\Gamma,G)$ be the collection of all $h$ so that
the subgroup generated by $\{h(\gamma_n):~n \in J_m\}$ acts transitively on $\{1,\ldots, d\}$.
Then the conditions needed for applying Lemma \ref{lem:contribution} with $J=J_m$ can be succinctly stated as $h \in \Hom_m$. Let $c_{i,m}=c(\eta_i, J_m)$ be the constants given by Lemma \ref{lem:contribution}.
Observe that $D(\phi)=g^{\ln 2}$. This means that
$$g^{k \ln 2}(\tilde S_h, \tilde \alpha_h)=(\tilde S_{h \circ \phi^{-k}}, \tilde \alpha_{h \circ \phi^{-k}}).$$
So, assuming that $h \circ \phi^{-k} \in \Hom_m$, the contribution to the integral in Theorem \ref{thm:integrability}
taken with $\eta=\eta_i$ and $t \in \big[(k-\half) \ln 2, (k+\half) \ln 2\big]$ is at least $c_{i,m} \ln 2$.
We need the total contribution to the integrals for each $\eta_i$ to be infinite.
For each $i$, choose a sequence $N_{i,m}$ so that
$$\sum_m^\infty N_{i,m} c_{i,m} \ln 2=+\infty.$$
(The number $N_{i,m}$ representing a number of times we contribute $c_{i,m} \ln 2$ to the integral using the method of the previous paragraph.) Now define the sequence
\begin{equation}
\label{eq:integral as sum}
N_m = \max \{ N_{1,m}, N_{2,m}, \ldots, N_{m,m}\}
\quad \text{and observe}\quad
\sum_m N_{m} c_{i,m} \ln 2=+\infty \quad \text{for all $i$.}
\end{equation}
Observe that we have the following criterion for ergodicity:
{\bf Claim.} Fix $h$. If there is a sequence of pairwise disjoint subsets $K_m \subset \N$ defined for $m \geq m_0$
for some $m_0>0$ so that $|K_m| \geq N_m$ and
$h \circ \phi^{-k} \in \Hom_m$ for each $k \in K_m$, then the translation flow on $(\tilde S_h, \tilde \alpha_h)$ is ergodic.
The proof of the claim is that by using Lemma \ref{lem:contribution} with $J=J_m$ on $h \circ \phi^{-k}$ at times
$t \in \big[(k-\frac{1}{2}) \ln 2, (k+\frac{1}{2}) \ln 2\big]$ where $k \in K_m$ we get a contribution of $c_{i,m} \ln 2$ to the integral associated to $\eta_i$. These contributions occur at least $N_m$ times when $m \geq m_0$,
so each of the integrals is infinite by \eqref{eq:integral as sum}.
Now we will explain how to meet the hypotheses of this claim. First we prefer to work with the $G$-sequence
$\g=\mathfrak{g}(h)$ than with $h$ directly. By Corollary \ref{cor:conversion}, we can guarantee that $h \circ \phi^{-k} \in \Hom_{m}$ if $\sigma^k(\g) \in \mathit{Gen(m)}$ where
$\mathit{Gen(m)}$ denotes the set of $\g' \in G^\Z$ so that the subgroup
generated by $\{\g'_j:~|j|<m\}$ acts transitively on $\{1,\ldots, d\}$. Hence we can replace
the condition $h \circ \phi^{-k} \in \Hom_m$ in our Claim with the condition that $\sigma^k(\g) \in \mathit{Gen(m)}$.
We will now produce a $\g$ together with subsets $K_m$ as above so that $\sigma^k(\g) \in \mathit{Gen(m)}$ when $k \in K_m$. To do this we only need to specify a tail for $\g$; i.e., the values for $\g_k$ for $k$ large.
Say that a {\em word} is an element $w \in G^l$ for some $l \in \N$. We use $|w|$ to denote the length of $w$.
We will express this tail for $\g$ as an infinite concatenation of words. For $i \in \{1,2\}$ choose words
$w_i$ so that the group elements appearing in the words generate the subgroup $H_i$ as in the statement of the Theorem. Let $e$ denote the identity element of $G$, and let $e^j$ denote word formed by repeating the identity $j$ times. Observe that if $\ell+|w_1|+|w_2|=2m-1$ and the word
\begin{equation}
\label{eq:word form}
\g'_{-m+1} \g'_{-m+2} \ldots \g'_{m-1} \quad \text{equals} \quad w_1 e^\ell w_2 \quad \text{or} \quad w_2 e^\ell w_1
\end{equation}
then $\g' \in \mathit{Gen(m)}$ since by hypothesis the group generated by $H_1 \cup H_2$ acts transitively on $\{1,\ldots, d\}$. Let $m_0$ be such that $2 m_0-1 \geq 1+|w_1|+|w_2|$. Let $(m_j)$ be any sequence in the set $\{m \in \N:~m \geq m_0\}$ so that each $m \geq m_0$ appears $N_m$ times in the sequence.
For each $j$ define $\ell_j$ so that $\ell_j+|w_1|+|w_2|=2m_j-1$.
Now assume $\g$ has a tail of the form
\begin{equation}
\label{eq:tail}
w_1 e^{\ell_1} w_2 e^{\ell_2} w_1 e^{\ell_3} w_2 e^{\ell_4} w_1 \ldots.
\end{equation}
Then for each $j$ there is a $k_j$ so that when $\g'=\sigma^{k_j}(\g)$ the word of \eqref{eq:word form}
taken with $m=m_j$ has the form of either $w_1 e^{\ell_j} w_2$ or $w_2 e^{\ell_j} w_1$ depending on the parity of $j$. In particular, $\sigma^{k_j}(\g) \in \mathit{Gen(m_j)}$ for each $j$. By construction, $m$ appears $N_m$ times in the sequence $(m_j)$, so by application of the Claim we see that if $\mathfrak{g}(h)=\g$ then
the translation flow on $(\tilde S_h, \tilde \alpha_h)$ is ergodic.
Recall that we can improve ergodicity to unique ergodicity using Corollary \ref{cor:lifting} since the $2$-adic odometer is uniquely ergodic.
It remains to explain that the orbit of $(\tilde S_h, \tilde \alpha_h)$ under $\phi$ only accumulates on disconnected covers as stated in the Theorem. Observe that the $\g$ we constructed is $p$-ready, where $p:\{n \in \Z:~n \geq a\} \to \{0,1,2\}$ is defined so that $a$ represents the position of the first letter in the tail \eqref{eq:tail},
and $p(a+n)$ is determined by the $n+1$-st position in this tail: If the $n+1$-st position belongs to an $e^{\ell_\ast}$ word we define $p(a+n)=0$ and if it belongs to a $w_i$ word we define $p(a+n)=i$. Since each $m$
appears only finitely many times in the sequence $m_j$, we see that the length of the intervals $I(p,n)$ where $p(n)=0$ tends to infinity. Thus as an application of Proposition \ref{prop:disconnected}, $(\tilde S_h, \tilde \alpha_h)$ only accumulates on disconnected covers as desired.
\end{proof}
\subsection{The ladder surface}
\label{sect:ladder surface}
For this subsection, we let $(S_L,\alpha_L)$ denote the infinite genus translation surface of Figure \ref{fig:ladder_surface}, which we call the {\em ladder surface}. It can be constructed from a region in the plane bounded by countably many horizontal and vertical edges. We have labeled the edges using the set $\Z^\ast=\Z \smallsetminus \{0\}$. Edges with the same label are glued by translation to form the surface, which does not include endpoints of these edges or the limiting point in the top right of the region. Let $\varphi$ denote the golden ratio, $\frac{\sqrt{5}+1}{2}$. An edge with label $j \in \Z^\ast$ has length given by
$\varphi^{-|j|}.$
This information specifies the geometry of the surface. We have also selected a basepoint $s_0$ for our surface in the figure.
The surface is built using Thurston's construction \cite[\S 6]{T88} from the graph $\sG$ depicted in Figure \ref{fig:ladder_surface}. We will briefly review this construction. The surface $(S_L, \alpha_L)$ has horizontal and vertical cylinder decompositions with cylinders bounded by dotted lines in Figure \ref{fig:ladder_surface}.
From these cylinders one can build a bipartite graph: the white vertices consist of the horizontal cylinders, the gray vertices consist of the vertical cylinders and an edge is drawn between two vertices for each intersection between the corresponding two cylinders.
The vertices of $\sG$ have been labeled in the figure to match the label of the edge passed through by the corresponding cylinder.
(Two adjacent cylinders pass through edges labeled $-1$ and $1$.)
For such a connected graph, for any $A>0$ there is at most one way to assign widths to the cylinders so that the area of the resulting surface is $A$ and the moduli (ratio of widths to circumferences) of all these cylinders are equal: this assignment viewed as a real valued function defined on the vertices of the graph must be an eigenfunction for the adjacency operator. (This eigenfunction always exists for finite graphs by the Perron-Frobenius theorem, while uniqueness is guaranteed in the infinite case by an analogous result \cite[Theorem 2.2]{V68}.)
This forces the lengths of edges of the region defining our surface to be as mentioned above (up to uniform scaling).
\begin{figure}
\caption{{\em Left:}
\label{fig:ladder_surface}
\end{figure}
Once the moduli of the cylinders are all equal, an observation of Thurston
guarantees existence of two affine automorphisms of $(S_L, \alpha_L)$ with parabolic derivative and horizontal and vertical eigenvectors respectively preserving the horizontal and vertical cylinders in the decompositions. In the case of our surface, there is an additional reflective symmetry in a line of slope $1$. To generate these three affine automorphisms it is sufficient to use $\psi$ which performs a single left Dehn twist in each of the countably many vertical cylinders, and the reflective symmetry $\rho$. The derivatives are given by
$$D(\psi)=\left[\begin{array}{rr}
1 & 0 \\
2 \varphi & 1
\end{array}\right] \and
D(\rho)=\left[\begin{array}{rr}
0 & 1 \\
1 & 0
\end{array}\right].
$$
The composition $\rho \circ \psi \circ \rho$ performs a right Dehn twist in each of the horizontal cylinders. All these affine automorphisms fix the basepoint $s_0$
depicted in Figure \ref{fig:ladder_surface}.
We define $\Aff'$ to be the subgroup of $\Aff(S_L,\alpha_L)$ generated by $\psi$ and $\rho$. The group $\Aff'$ is isomorphic to the free product $\Z \ast (\Z/2\Z)$ and contains no translation automorphisms.
\begin{remark}
\label{rem:unknown affine group}
It is unknown to the authors if $\Aff'=\Aff(S_L,\alpha_L)$.
\end{remark}
Recall that $\sO(S,\alpha)$ is naturally identified with $\SL_\pm(2,\R)/V(S,\alpha)$. See the discussion below \eqref{eq:sO}.
Let $V'=D(\Aff')$. Since $\Aff'$ contains no translation automorphisms, the derivative map $D:\Aff' \to V'$ is a group isomorphism.
Unfortunately we do not know if $V'=V(S,\alpha)$, so all we get is a covering map:
\begin{equation}
\label{eq:covering O 1}
\SL_\pm(2,\R)/V' \to \sO(S,\alpha).
\end{equation}
Fortunately, if a geodesic recurs in $\SL_\pm(2,\R)/V'$, then its image in $\sO(S,\alpha)$ also recurs.
So non-divergence of $g^t A V'$ implies ergodicity of the translation flow on $A(S,\alpha)$ by Corollary \ref{cor:rodrigo ergodicity from Veech group}.
We will be working throughout this section without knowing precise information about the Veech group and this will create
difficulties which will be dealt with. It turns out we will be able to get by only knowing the following:
\begin{proposition}
\label{prop:discrete}
The Veech group $V(S_L,\alpha_L)$ is discrete.
\end{proposition}
\begin{proof}
Consider the maximal vertical cylinder $C_{-2}$ in $S_L$ which passes through the edge labeled $-2$ in Figure \ref{fig:ladder_surface}. We chose this because it has the longest circumference of all vertical cylinders.
Suppose $A \in \SL(2,\R)$ is close enough to the identity so that a closed vertical segment of length equal to the circumference of $C_{-2}$ immerses inside of $A(C_{-2})$. Now suppose that $A \in V(S_L,\alpha_L)$. Then there is a translation isomorphism $e:A(S_L,\alpha_L) \to (S_L,\alpha_L)$.
The restriction $e|_{\ell}$ must not given an embedding of $\ell$ because all vertical trajectories on $(S_L,\alpha_L)$ close up at or before this length. Since $e|_{C_{-2}}$ is an embedding we see that $\ell$ must close up on $A(C_{-2})$. Thus $A(C_{-2})$ must be a vertical cylinder.
Because $C_{-2}$ has singularities in its boundary so must $A(C_{-2})$, so that $A(C_{-2})$ is a maximal cylinder.
Area considerations show that $A(C_{-2})=C_{-2}$. We conclude that $A$ has a vertical eigenvector with eigenvalue $1$. A symmetric argument implies that
if $A$ is taken from a small enough open neighborhood of the identity then
$A$ has a horizontal eigenvector with eigenvalue $1$. By choosing our neighborhood small enough, both arguments hold and we see that the only element of $V(S_L,\alpha_L)$ in this neighborhood is the identity.
\end{proof}
We will need to understand $\SL_\pm(2,\R)/V'$ for later arguments.
The group $\SL_\pm(2,\R)$ can be identified with the flag bundle of the hyperbolic plane (where a flag is a choice of a unit tangent vector
and an orthogonal unit tangent vector taken from the same tangent plane). Here the hyperbolic plane is $O(2) \bs \SL_\pm(2,\R)$.
The group $\SL_\pm(2,\R)$ then acts on the right by isometry. The unit speed geodesics have the form $t \mapsto O(2) g^t A$
for some $A \in \SL_\pm(2,\R)$.
The boundary of the hyperbolic plane can be identified with $\R \P^1=\R^2/(\R \smallsetminus \{0\})$.
The geodesic $t \mapsto O(2) g^t A$ converges as $t \to +\infty$ to the boundary point representing the vectors which are contracted
to zero as $t \to \infty$, the projective equivalence class of
$$\left(\begin{array}{r} x \\ y \end{array}\right)=A^{-1}\left(\begin{array}{r} 1 \\ 0 \end{array}\right).$$
We identify the hyperbolic plane $O(2) \bs \SL_\pm(2,\R)$ with the upper half plane in such a way so that
the projective class of $(x,y)$ viewed as a boundary point is identified with the slope $y/x \in \R \cup \{\infty\}$.
\begin{figure}
\caption{A fundamental domain for the action of $V'$ on the upper half plane is shown in gray (it extends vertically to $\infty$).
The geodesic between $-1$ and $1$ is preserved by the reflection $\rho$, and
$\psi$ acts by translation by $2 \varphi$. The convex core is in dark gray and extends upward.}
\label{fig:fundamental domain}
\end{figure}
We let $V'$ denote the group of derivatives of $\Aff'$.
The group $V'$ acts discretely on the hyperbolic plane, and Figure \ref{fig:fundamental domain} shows a fundamental domain for the action.
A geodesic ray $t \mapsto O(2) g^t A V'$ in $\SL_\pm(2,\R)/V'$ for $t>0$
is non-divergent if and only if the endpoint of the geodesic in the hyperbolic plane lies
in the horospherical limit set, which in this case is the limit set of $V'$ with fixed points of parabolics removed
\cite[Theorem 2]{BM74}. The limit set in this case is a Cantor set of Hausdorff dimension larger than $\frac{1}{2}$
\cite[Remark 4.1]{Hinf}. The {\em convex core} is the convex hull of the limit set in the hyperbolic plane projected into the surface $O(2) \bs \SL_\pm(2,\R)/V'$.
A geodesic ray in this surface is non-divergent if and only if it has an $\omega$-limit point in the convex core.
The convex core of this surface is depicted in Figure \ref{fig:fundamental domain}.
\begin{remark}[Boundary of convex core]
The geodesic in the surface $O(2) \bs \SL_\pm(2,\R) / V'$ with monodromy given by the derivative of the commutator $D([\rho, \psi])$ is the boundary of the convex core of this surface.
\end{remark}
\begin{theorem}
\label{thm:prior work}
Let $A \in \SL_\pm(2,\R)$, and $(S_L,\alpha_L)$ be the ladder surface. Consider the geodesic ray
$t \mapsto O(2) g^t A V'$ in the surface $O(2) \bs \SL_\pm(2,\R) / V'$.
\begin{itemize}
\item If the ray has an $\omega$-limit point (which must be in the convex core) then
the translation flow on $A (S_L,\alpha_L)$ is defined for all time almost everywhere and is ergodic.
\item If the ray has an $\omega$-limit point in the interior of the convex core then
the translation flow on $A(S_L,\alpha_L)$ is uniquely ergodic.
\end{itemize}
\end{theorem}
\begin{proof}
To see the first statement observe that if the geodesic has an $\omega$-limit point then so does the trajectory
$t \mapsto g^t A (S_L,\alpha_L)$ in $\sO(S_L,\alpha_L)$ because of the covering map \eqref{eq:covering O 1}. The conclusion is then given by
Corollary \ref{cor:rodrigo ergodicity from Veech group}.
The second statement is an application of Theorem H.5 of \cite{Hinf}. To use the theorem we need to verify several hypotheses.
First $(S_L,\alpha_L)$ arises from Thurston's construction using a graph with no vertices of valence one
as with the graph here; see Figure \ref{fig:ladder_surface}. Second in the language of that article we need the trajectory
$g^t A$ not be asymptotic to an endpoint of a maximal interval of the compliment of the limit set of the subgroup of $V(S_L,\alpha_L)$ generated by the horizontal and vertical parabolics. The subgroup generated by these parabolics is the same as $V'$ up to finite index,
so the two limit sets are identical. This condition that the trajectory is not asymptotic to an endpoint of a complimentary interval
is equivalent to $O(2) g^t A V'$ not being asymptotic to the boundary of the convex core in $O(2) \bs \SL_\pm(2,\R) / V'$.
It is impossible for a geodesic ray with an $\omega$-limit point in the interior of the convex core to be asymptotic to the convex core boundary.
\end{proof}
We will be considering double covers of the ladder surface.
For $i \in \Z^\ast$, we let $\gamma_i$ be one of the two homotopy class of loops which start at the basepoint, cross only the edge labeled $i$, and return to the basepoint. These two homotopy classes are inverses in $\Gamma=\pi_1(S,s_0)$. We make the choice of $\gamma_i$ so $\gamma_i$ moves rightward across the vertical edge labeled $i$ if $i>0$,
and moves upward over the horizontal edge labeled $i$ if $i<0$. These curves freely generate the fundamental group,
which we denote by $\Gamma=\pi_1(S,s_0)$. (As with the the Chamanara surface in the previous section, the curves $\gamma_i$ can be chosen so that $S_L$ retracts onto the union of the curves which is homeomorphic to a countable bouquet of circles.)
Let $\Z_2=\Z/2\Z$. Observe that $\Z_2$ acts trivially on $\Hom(\Gamma, \Z_2)$ by conjugation so that
$\Covt_{\Z_2}(S_L)=\Hom(\Gamma, \Z_2)$; see \eqref{eq:space of covers}. It can be seen from Remark \ref{rem:primitivity}
that $\Cov_{\Z_2}(S_L,\alpha_L)=\Covt_{\Z_2}(S_L)$, i.e., \eqref{eq:top cover to cover} is a bijection in this case.
So in summary,
\begin{equation}
\label{eq:covers equals hom}
\Cov_{\Z_2}(S_L,\alpha_L)=\Hom(\Gamma, \Z_2).
\end{equation}
Since our basepoint is fixed by the affine automorphisms in $\Aff'$, we have a canonical action of each
element in $\Aff'$ on the space of covers. For example, if $h \in \Hom(\Gamma, \Z_2)$, then
$\psi_\ast(h)=h \circ \psi^{-1}$ and $\rho_\ast(h)=h \circ \rho^{-1}$. We will work out this action below.
\begin{notation}
Let $h \in \Hom(\Gamma, \Z_2)$.
We will write $h(i)$ to abbreviate $h(\gamma_i)$ for $i \in \Z^\ast$.
(We have a bijection between elements of $\Hom(\Gamma, \Z_2)$
and the set of all functions $\Z^\ast \to \Z_2$.)
\end{notation}
It can be observed that the generators of
$\Aff'$ act on $\Hom(\Gamma, \Z_2)$ as follows:
\begin{equation}
\label{eq:psi ladder}
\big(\psi_\ast(h)\big)(i)=\begin{cases}
h(i) & \text{if $i <0$,} \\
h(i)+h({-2}) & \text{if $i=1$,} \\
h(i)+h({-2})+h({-3}) & \text{if $i=2$,} \\
h(i)+h({-i-1})+h({-i})+h({-i+1}) & \text{if $i>2$.}
\end{cases}
\end{equation}
\begin{equation}
\label{eq:rho ladder}
\big(\rho_\ast(h)\big)(i)=h(-i).
\end{equation}
\begin{comment}
This is for the action on $\Z/n\Z$-marked covers:
$$\big(\phi_\ast(h)\big)(\gamma_i)=\begin{cases}
h(i) & \text{if $i >0$,} \\
h(i)+2h(1)+h(2) & \text{if $i=-1$,} \\
h(i)+2h(1)+h(2)+h(3) & \text{if $i=-2$,} \\
h(i)+h({-i-1})+h({-i})+h({-i+1}) & \text{if $i<-2$.}
\end{cases}$$
$$\big(\psi_\ast(h)\big)(i)=\begin{cases}
h(i) & \text{if $i <0$,} \\
h(i)+2h({-1})+h({-2}) & \text{if $i=1$,} \\
h(i)+2h({-1})+h({-2})+h({-3}) & \text{if $i=2$,} \\
h(i)+h({-i-1})+h({-i})+h({-i+1}) & \text{if $i>2$.}
\end{cases}$$
\end{comment}
A key observation is the following:
\begin{proposition}
\label{prop:psi squared trivial}
The square $\psi^2_\ast$ acts trivially on
$\Hom(\Gamma, \Z_2)$.
\end{proposition}
\begin{proof}
The action of $\psi^2_\ast$ preserves the value of $h(i)$ for $i<0$, and adds a sum of values of $h$ evaluated
at negative integers to the values of $h(i)$ for $i>0$.
Since adding the same number twice is the same as adding zero
in $\Z_2$, $\psi_\ast^2$ acts trivially.
\end{proof}
We have the following trivial consequence:
\begin{corollary}
\label{cor:veech group of cover}
The subgroup of $V'$ given by
\begin{equation}
\label{eq:tilde V'}
\tilde V'=\langle R\, D(\psi^2) R^{-1} ~|~R \in V' \rangle
\end{equation}
is a subgroup of the Veech group of any $(\tilde S_h, \tilde \alpha_h) \in \Cov_{\Z_2}(S_L,\alpha_L)$.
\end{corollary}
From this we get a covering map (similar to \eqref{eq:covering O 1}):
\begin{equation}
\label{eq:covering O 2}
\SL_\pm(2,\R)/ \tilde V' \to \sO(\tilde S_h, \tilde \alpha_h).
\end{equation}
Observe that a non-divergent geodesic in $\SL_\pm(2,\R)/ \tilde V'$ will descend to a non-divergent geodesic in $\sO(\tilde S_h, \tilde \alpha_h).$ We have recovered statement (L1) of the introduction:
\begin{proposition}
\label{prop:non-divergence covers}
Suppose that $g^t A \tilde V'$ is non-divergent in $\SL_\pm(2,\R)/ \tilde V'$. Then
for any connected $(\tilde S_h, \tilde \alpha_h) \in \Cov_{\Z_2}(S_L,\alpha_L)$,
the double cover $A(\tilde S_h, \tilde \alpha_h)$ of $A(S_L,\alpha_L)$ is not devious
and every $A(\tilde S_h, \tilde \alpha_h)$ has uniquely ergodic translation flow.
\end{proposition}
\begin{proof}
Fix $A$ satisfying the first statement and any cover $(\tilde S_h, \tilde \alpha_h)$.
From the covering map \eqref{eq:covering O 2}, we know $g^t A(\tilde S_h, \tilde \alpha_h)$ is non-divergent
in $\sO(\tilde S_h, \tilde \alpha_h)$. Thus this cover is not devious and ergodicity follows from Corollary \ref{cor:rodrigo ergodicity from Veech group}.
To prove unique ergodicity, it suffices in light of Corollary \ref{cor:lifting} to show that the translation flow on the surface $A(S_L,\alpha_L)$ is uniquely ergodic. This holds unless $O(2) g^t A V'$ is forward asymptotic to the convex core boundary of $O(2)\bs \SL_\pm(2,\R) /V'$ by Theorem \ref{thm:prior work}. But this convex core boundary lifts to a pair of divergent geodesics in $O(2)\bs \SL_\pm(2,\R) /\tilde V'$. (This will be justified more below; the wavy lines between light and dark gray regions in Figure \ref{fig:periodic disk} denote the lifted boundary of the convex core.)
\end{proof}
Our main goal of the section is to find some devious covers, so from the above proposition we will need to consider
elements $A \in \SL_\pm(2,\R)$ so that $g^t A \tilde V'$ is divergent in $\SL_\pm(2,\R)/ \tilde V'$.
We will show that the existence of such covers is related to a combinatorial rate of divergence of this trajectory. To measure this we need to further develop our understanding of $\tilde V'$ and the surface $O(2) \bs \SL_\pm(2,\R)/ \tilde V'$.
From \eqref{eq:tilde V'}, the group $\tilde V'$ is evidently normal in $V'$. Let $\Delta=V' / \tilde V'$ be the quotient. This group has a right action as the Deck group of the covering maps
\begin{equation}
\label{eq:cover}
\SL_\pm(2,\R)/ \tilde V' \to \SL_\pm(2,\R)/ V'
\quad \text{and} \quad
O(2) \bs \SL_\pm(2,\R)/ \tilde V' \to O(2) \bs \SL_\pm(2,\R)/ V'.
\end{equation}
Since as a group $V'=\langle D(\psi), D(\rho)\rangle$ is isomorphic to the free product $\Z \ast \Z_2$, and
$\psi^2$ is in $\tilde V'$, the quotient $\Delta$ is isomorphic to the infinite dihedral group which we think of as $\Isom(\Z)$, where $\Z$ is given the standard metric. A homomorphism $\delta:V' \to \Isom(\Z)$ which factors through $\Delta$ is given by
\begin{equation}
\label{eq:delta}\delta \circ D(\psi):n \mapsto -n \and \delta \circ D(\rho):n \mapsto 1-n.
\end{equation}
The action of the Veech group on $\Cov_{\Z_2}(S_L, \alpha_L)=\Hom(\Gamma, \Z_2)$ induces an action of $\Isom(\Z)$, where
\begin{equation}
\label{eq:ast action}
\big(\delta \circ D(\psi)\big)_\ast=\psi_\ast \and \big(\delta \circ D(\rho)\big)_\ast=\rho_\ast.
\end{equation}
We will now give a cartoon description of the surface $O(2) \bs \SL_\pm(2,\R)/ \tilde V'$.
We can deform our fundamental domain for the $V'$ action so that it looks like the left side of Figure \ref{fig:periodic disk}. The advantage here is that two copies of this domain can be joined together by a Euclidean rotation by $180^\circ$ about the point $\infty$. This rotation is then representing the order two action on $\SL_\pm(2,\R)/ \tilde V'$ given by the right action of $D(\psi)$. The surface $O(2) \bs \SL_\pm(2,\R)/ \tilde V'$ can be tiled by copies of this fundamental domain as shown on the right side of the figure. We have drawn this in such a way so that the action of the Deck group $\Isom(\Z)$ appears natural. In particular, for each $n \in \Z$, we can join together two copies of the fundamental domain together about the point labeled $\infty_n$. We call this region $F_n$; these regions are depicted in the figure.
The isometry
$\delta \circ D(\psi):n \mapsto -n$ acts by a rotation by $180^\circ$ about the point $\infty_0$.
The isometry $\delta \circ D(\rho):n \mapsto 1-n$ acts by a Euclidean reflection in the line forming the boundary between $F_0$ and $F_1$. We observe that $\Isom(\Z)$ acts in a natural way on these regions as labeled by $\Z$. Namely, we can think of these regions as subsets of $O(2) \bs \SL_\pm(2,\R) / \tilde V'$ and
\begin{equation}
\label{eq:action on regions}
F_n R^{-1} = F_{\delta(R)(n)} \quad \text{for all $R \in V'$ and $n \in \Z$}.
\end{equation}
The actual action is somewhat subtle: the translation
\begin{equation}
\label{eq:translation action}
\tau^m=\delta \circ D\big((\rho \circ \psi)^m\big):n \mapsto n+m
\end{equation}
acts by translation by $m$ in the figure
when $m$ is even, but when $m$ is odd $\tau^m$ acts as a glide reflection carrying each $F_n$ to $F_{n+m}$.
\begin{figure}
\caption{{\em Left:}
\label{fig:periodic disk}
\end{figure}
We use the regions $F_n$ in $\SL_\pm(2,\R)/ \tilde V'$ to code $g^t$-trajectories in $\SL_\pm(2,\R)/ \tilde V'$ arising from non-divergent trajectories in $\SL_\pm(2,\R)/ V'$:
\begin{proposition}[Coding walk]
\label{prop:walk}
Let $A \in \SL(2,\R)$.
If the trajectory $g^t A V'$ is non-divergent in forward time on $\SL_\pm(2,\R)/ V'$, then there is a sequence of countably many times
$$0=t_0 < t_1<t_2< \ldots$$
with $\lim_{k \to \infty} t_k=+\infty$ and a sequence of integers $\{n_k~:~k=0,1,2, \ldots\}$
so that
$$O(2) g^t A \tilde V' \in F_{n_k} \quad \text{whenever $t_k<t<t_{k+1}$}.$$
Furthermore, $n_{k+1} \in \{n_k +1, n_k-1\}$ for each $k \geq 0$.
\end{proposition}
In other words, the lift of a non-divergent geodesic on $O(2) \bs \SL_\pm(2,\R)/ V'$ to a geodesic on $O(2) \bs \SL_\pm(2,\R)/ \tilde V'$ gives rise to a walk on the integers. We'll call this walk the {\em coding walk} of the geodesic $g^t A \tilde V'$.
\begin{proof}
If this is not true, then the geodesic change regions only finitely many times.
Then the geodesic eventually stays in one region, say $F_n$. But, we can see that the only geodesics that stay within $F_n$ forever are geodesics which exit the cusp $\infty_n$ or which exit the preimage of the convex core of
$O(2) \bs \SL_\pm(2,\R)/ V'$ in $F_n$. (The region $F_n$ is an annulus with parabolic monodromy.) Both possibilities contradict the non-divergence of $g^t A V'$ in $\SL_\pm(2,\R)/V'$.
\end{proof}
The following is a tool which will be useful to obtain convergence in $\tilde O_{\Z_2}(S_L,\alpha_L)$
using the coding walk.
\begin{lemma}
\label{lem:convergence}
Let $U \subset O(2) \bs \SL_\pm(2,\R) / V'$ be a cusp neighborhood given by the points in the fundamental domain
of Figure \ref{fig:fundamental domain} with imaginary part larger than $2$. Let $C_d$ be a metric $d$-neighborhood of the convex core of $\subset O(2) \bs \SL_\pm(2,\R) / V'$. Let $A \in \SL(2,\R)$ and assume $g^t A V'$ is non-divergent in forward time on $\SL_\pm(2,\R)/V'$. Let $\{n_k\}$ be the associated coding walk. Let $(\tilde S_h,\tilde \alpha_h) \in \Cov_{\Z_2}(S_L,\alpha_L)$. There is a compact
set $K \subset \SL_\pm(2,\R)$ so that if $t$ lies in the interval $[t_k, t_{k+1}]$ associated to $n_k$ (as in Proposition \ref{prop:walk})
and $O(2) g^t A V' \in C_d \smallsetminus U$ then there is a $M \in K$ so that
$$g^t A (\tilde S_h,\tilde \alpha_h) = M \left(\tilde S_{\tau^{-n_k}_\ast(h)},\tilde \alpha_{\tau^{-n_k}_\ast(h)}\right)
\quad \text{in $\tilde O_{\Z_2}(S_L,\alpha_L)$.}
$$
Moreover, for any $k$ sufficiently large, there exists such a $t \in [t_k, t_{k+1}]$.
\end{lemma}
\begin{proof}
Recall there is a covering map $O(2) \bs \SL_\pm(2,\R) / \tilde V' \to O(2) \bs \SL_\pm(2,\R) / V'$.
Consider the region $F_0$ in the domain. Let $F_0' \subset O(2) \bs \SL_\pm(2,\R) / \tilde V'$ be the $d$-neighborhood of preimage of the convex core of
$O(2) \bs \SL_\pm(2,\R) / V'$ in $F_0$ with the preimage of $U$ removed. Then $F_0'$ is compact and this covering map sends $F_0'$ onto $C_d \smallsetminus U$. Let
$\tilde F_0' \subset \SL_\pm(2,\R) / \tilde V'$ be the preimage of $F_0'$ under projection to $O(2) \bs \SL_\pm(2,\R) / \tilde V'$. Then $\tilde F_0'$ is also compact and so there is a compact set $K \subset \SL_\pm(2,\R)$ which projects to $\tilde F_0'$ under the quotient map $\SL_\pm(2,\R) \to \SL_\pm(2,\R) / \tilde V'$.
Now consider a $t \in [t_k, t_{k+1}]$ so that $O(2) g^t A V' \in C_d \smallsetminus U$. Then by definition of the coding sequence, $O(2) g^t A \tilde V' \in F_{n_k}$. Then
\begin{equation}
\label{eq:tau image}
\tau^{-n_k} \big(O(2) g^t A \tilde V'\big) \in F_0;
\end{equation}
see \eqref{eq:action on regions} and \eqref{eq:translation action}. Since $O(2) g^t A V'$ is asymptotic to the convex core in $O(2) \bs \SL_\pm(2,\R) / V$,
for $k$ sufficiently large the point in \eqref{eq:tau image} actually lies in $F_0'$.
Observe also that
$$\tau^{-n_k} \big(O(2) g^t A \tilde V'\big)=O(2) g^t A D\big((\rho \circ \psi)^{n_k}\big)\tilde V',$$
so that from the above there is an $R \in \tilde V'$ and an $M \in K$ so that
$$g^t A D\big((\rho \circ \psi)^{n_k}\big) R = M.$$
Now observe that
$$g^t A (\tilde S_h,\tilde \alpha_h)=M\cdot \Big(R^{-1} \circ D\big((\rho \circ \psi)^{-n_k}\big)(\tilde S_h,\tilde \alpha_h) \quad \text{in $\tilde O_{\Z_2}(S_L,\alpha_L)$,}
$$
where we think of the expression to the right of $M\cdot$ as the action of the Veech group of $(S_L,\alpha_L)$
on $\Cov_{\Z_2}(S_L,\alpha_L)$. Here $\Cov_{\Z_2}(S_L,\alpha_L)=\Hom(\Gamma,\Z_2)$; see \eqref{eq:covers equals hom}.
Thus, the action of $V'$ on $\Hom(\Gamma,\Z_2)$ factors through $\Delta=V'/\tilde V'$ and in particular,
$$\Big(R^{-1} \circ D\big((\rho \circ \psi)^{-n_k}\big)(\tilde S_h,\tilde \alpha_h)=(\tilde S_{\tau^{-n_k}_\ast(h)},\tilde \alpha_{\tau^{-n_k}_\ast(h)}) \quad \text{in $\Cov_{\Z_2}(S_L,\alpha_L)$.}$$
Finally, to obtain such a $t \in [t_k, t_{k+1}]$ for $k$ sufficiently large, observe that because $g^t A V'$ is asymptotic to the convex core in $\SL_\pm(2,\R)/V'$, it eventually lies in $C_d$. Values of $t=t_k$ and $t=t_{k+1}$ satisfy the statement because at these times
$O(2) g^{t_k} A \tilde V'$ lies in the boundary of a region, so the corresponding point $O(2) g^{t_k} A V'$ in $O(2) \bs \SL_\pm(2,\R) / V'$ lies on the geodesic joining $-1$ to $1$ in
Figure \ref{fig:fundamental domain} and in particular this point does not lie in $U$.
\end{proof}
We already understand the ergodic properties of the translation flow of covers when $g^t A \tilde V'$ is non-divergent; see Proposition \ref{prop:non-divergence covers}. This non-divergence can be captured by the coding walk:
\begin{proposition}
\label{prop:non-divergence in tilde V'}
Assume $g^t A V'$ is non-divergent in forward time on $\SL_\pm(2,\R)/ V'$.
The trajectory $g^t A \tilde V'$ is non-divergent in $\SL_\pm(2,\R) / \tilde V'$ if and only if the coding walk $\langle n_k\rangle$ is recurrent, i.e., there is an $n \in \Z$ so that $n_k=n$ for infinitely many $k$.
\end{proposition}
\begin{proof}
First suppose $g^t A \tilde V'$ is non-divergent. Then $g^t A \tilde V'$ has an $\omega$-limit point in $\SL_\pm(2,\R) / \tilde V'$. This limit point must lie in some region $F_n$. Proposition \ref{prop:walk} tells us we must visit $F_n$ infinitely often, so infinitely many $n_k=n$. Now suppose the coding walk recurs to $n$. Then infinitely many times the trajectory $g^t A \tilde V'$ moves from $F_{n-1}$ to $F_n$ or from $F_{n+1}$ to $F_n$. This means that the trajectory $g^t A \tilde V'$ infinitely often visits a compact neighborhood of the common boundary between two adjacent regions. But then the trajectory must have an $\omega$-limit point in this compact neighborhood,
and so the trajectory is non-divergent.
\end{proof}
The proposition above tells us that the only way devious covers could exist
is if the coding walk does not recur. In this case $\lim n_k=+\infty$ or $\lim n_k=-\infty$. Theorem \ref{thm:1} tells us that covers which are not devious
have ergodic translation flow. So, it remains to understand the devious covers
when $\lim n_k= \pm \infty$. We now state our main result of the section.
\begin{theorem}[Main result on double covers of the ladder surface]
\label{thm:main ladder}
Let $A \in \SL(2,\R)$ and suppose that the geodesic $g^t A V'$ is non-divergent in $\SL_\pm(2,\R) / V'$. Let $\langle n_k \rangle$ be the coding walk of the geodesic $g^t A \tilde V'$, and assume this walk does not recur.
Then $\lim_{k \to \infty} n_k=s \infty$ where $s\in\{\pm 1\}$ is a sign.
In this case, there exist covers $(\tilde S_h, \tilde \alpha_h) \in \Cov_{\Z_2}(S_L,\alpha_L)$ so that $A(\tilde S_h, \tilde \alpha_h)$ is a devious double cover
of $A(S_L,\alpha_L)$. We define the {\em growth exponent of the visit count} to be
$$v=\limsup_{N \to s \infty} \big(\# \{k:~n_k=N\}\big)^\frac{1}{|N|}.$$
Then the following hold:
\begin{enumerate}
\item[(L2)] If $v>\varphi^2$, then all devious double covers $A(\tilde S_h, \tilde \alpha_h)$ have ergodic
translation flow.
\item[(L3)] If $v< \varphi^2$, then all devious double covers $A(\tilde S_h, \tilde \alpha_h)$ have non-ergodic
translation flow.
\end{enumerate}
\end{theorem}
Our first order of business in proving this theorem is to characterize the devious double covers $A(\tilde S_h, \tilde \alpha_h)$ of $A(S_L,\alpha_L)$.
We will prove the following:
\begin{lemma}[Criterion for deviousness]
\label{lem:deviousness}
In the context of the theorem above, suppose that $\langle n_k \rangle$ limits
to $s\infty$ with $s \in \{\pm 1\}$.
Let $A \in \SL_\pm(2,\R)$ and $h \in \Hom(\Gamma,\Z_2)$ be chosen so that
$h \neq \0$, where $\0 \in \Hom(\Gamma,\Z_2)$ denotes the trivial homomorphism.
Then, the cover $A(\tilde S_h, \tilde \alpha_h)$ of $A(S_L,\alpha_L)$ is devious if and only if
$$\lim_{m \to +\infty} \tau^{-sm}_\ast(h)= \0.$$
\end{lemma}
\begin{remark}[Reduction to case $s=-1$]
\label{rem:reflection simplification}
It suffices to prove Lemma \ref{lem:deviousness} in the case of $s=-1$.
To see this fix $A$ and suppose $\lim n_k=+\infty$.
Observe that $A(\tilde S_h, \tilde \alpha_h)$ is a devious double cover of $A(S_L,\alpha_L)$ if and only if $A'(\tilde S_{\psi_\ast(h)}, \tilde \alpha_{\psi_\ast(h)})$ is a devious double cover of $A'(S_L,\alpha_L)$, since the surfaces are translation equivalent. Let $A'=D(\psi)A$ and $h'=\psi_\ast(h)$.
Use $A'$ to define the coding walk $\langle n_k' \rangle$. Because of the action of $D(\psi)$ on regions (see \eqref{eq:delta} and \eqref{eq:action on regions}),
we have $n_k'=-n_k$ so that $\lim n_k=-\infty$. Finally observe that
$\lim_{m \to \infty} \tau^{-m}_\ast(h)=\0$ if and only if
$\lim_{n \to \infty} \tau^{-n}_\ast(h')=\0$ since
$$\tau^{-n}_\ast(h')=\tau^{-n} \circ \psi_\ast(h)=
\psi_\ast \circ \tau_\ast^{n}(h) \quad \text{for all $n$}$$
as the action factors through the $\Isom(\Z)$; see \eqref{eq:ast action}.
Similar observations demonstrate that we can assume $s=-1$ in the proof of Theorem \ref{thm:main ladder} as well.
\end{remark}
One direction of Lemma \ref{lem:deviousness} is fairly easy:
\begin{proof}[Proof of the ``only if'' part of Lemma \ref{lem:deviousness}]
From the remark above, we can assume $s=-1$. Observe that a cover $(\tilde S_{h'},\tilde \alpha_{h'})$ is connected if and only if $h' \neq \0$.
So, assume $\lim n_k=-\infty$
and there is an $h' \neq \0$ which is an accumulation point of $\tau^m_\ast(h)$ as $m \to +\infty$. Let $m(j)$ be an increasing sequence of integers so that $\tau^{m(j)}_\ast(h) \to h'$.
We will show that $g^t A(\tilde S_h, \tilde \alpha_h)$ has a connected accumulation point in $\tilde \sO_{\Z_2}(S_L,\alpha_L)$.
Since the walk $\langle n_k \rangle$ tends to $-\infty$, for sufficiently large integers $j$, there is a $k(j)$ so that $n_{k(j)}=-m(j)$. From Lemma \ref{lem:convergence}, there is a sequence of times $t^j$ taken from the intervals associated to $n_{k(j)}$, a compact subset $K \subset \SL_\pm(2,\R)$, and $M_j \in K$ so that
$$g^{t^j} A (\tilde S_h, \tilde \alpha_h) = M_j \left(\tilde S_{\tau^{-n_{k(j)}}_\ast(h)},\tilde \alpha_{\tau^{-n_{k(j)}}_\ast(h)}\right).$$
Since $n_{k(j)}=-m(j)$ we know $\lim_{k \to \infty} \tau^{-n_{k(j)}}_\ast(h)=h'$. Since $K$ is compact, we find a subsequence $M_{j(i)}$ converging to some $M \in K$. Then
$$\lim_{i \to \infty} M_{j(i)} \left(\tilde S_{\tau^{-n_{k \circ j(i)}}_\ast(h)},\tilde \alpha_{\tau^{-n_{k \circ j(i)}}_\ast(h)}\right)=M (\tilde S_{h'}, \tilde \alpha_{h'}) \quad \text{in $\tilde \sO(S_L,\alpha_L)$.}$$
This is our desired connected accumulation point.
\end{proof}
The other direction is more difficult in part because we do not know if $V'=V(S_L,\alpha_L)$. We need to show that if $\lim_{m \to +\infty} \tau^{-sm}_\ast(h)= \0$ then every accumulation point of $g^t A(\tilde S_h, \tilde \alpha_h)$ in $\tilde \sO_{\Z_2}(S_L,\alpha_L)$ is disconnected. But $\tilde \sO_{\Z_2}(S_L,\alpha_L)$ is a quotient of $\SL(2,\R) \times \Cov_{\Z_2}(S_L,\alpha_L)$ by $V(S_L,\alpha)$ which we do not know.
Fortunately, we are restricting attention to $A \in \SL(2,\R)$ so that $g^t A V'$ is non-divergent in $\SL_\pm(2,\R)/V'$. Then the geodesic $g^t A V'$ is asymptotic to the convex core.
We will use this and a compactness argument to argue that if $g^{t_i} A(\tilde S_h, \tilde \alpha_h)$ converges then there is a subsequence converging to a disconnected cover. Assuming that the space $\tilde \sO_{\Z_2}(S_L,\alpha_L)$ is Hausdorff, the two limit points must be the same showing that every accumulation point is disconnected. So we need:
\begin{proposition}
\label{prop:Hausdorff}
The space $\tilde \sO_{\Z_2}(S_L,\alpha_L)$ is Hausdorff.
\end{proposition}
\begin{proof}
Let $A(\tilde S_h, \tilde \alpha_h)$ and $A'(\tilde S_{h'}, \tilde \alpha_{h'})$
be distinct points in $\tilde \sO_{\Z_2}(S_L,\alpha_L)$. We will find an open sets that isolate them from each other.
Observe that there is a natural continuous map $\tilde \sO_{\Z_2}(S_L,\alpha_L) \to \sO(S_L,\alpha_L)$ which sends $A(\tilde S_h, \tilde \alpha_h)$
to the quotient of $A(\tilde S_h, \tilde \alpha_h)$ by its translation automorphisms. Since all double covers are regular and $(S_L,\alpha_L)$ has no translation automorphisms, there is always a $\Z_2$ action on $A(\tilde S_h, \tilde \alpha_h)$ by its translation automorphisms. If under this map
$A(\tilde S_h, \tilde \alpha_h)$ and $A'(\tilde S_{h'}, \tilde \alpha_{h'})$ project to different points, then we can build disjoint open sets by lifting disjoint open sets from $\sO(S_L,\alpha_L)$. Recall that $\sO(S_L,\alpha_L)$ is naturally identified with $\SL_\pm(2,\R)/V(S_L,\alpha_L)$ which is Hausdorff because $V(S_L,\alpha_L)$ is discrete by Proposition \ref{prop:discrete}.
Now suppose that $A(\tilde S_h, \tilde \alpha_h)$ and $A'(\tilde S_{h'}, \tilde \alpha_{h'})$ both cover $A(S_L,\alpha_L)$. By left multiplying both surfaces by $A^{-1}$ we can assume without loss of generality that both surfaces cover
$(S_L,\alpha_L)$. Then they both lie in the $\Cov_{\Z_2}(S_L,\alpha_L)$.
Recall $\Cov_{\Z_2}(S_L,\alpha_L)=\Hom(\Gamma,\Z_2)$; see \eqref{eq:covers equals hom}. This space is a Cantor set and so is Hausdorff. So, we can find open sets $U_1$ and $U_2$ that isolate the two surfaces within $\Cov_{\Z_2}(S_L,\alpha_L)$. Since the Veech groups are discrete, we can find a neighborhood $N \subset \SL_\pm(2,\R)$ so that no non-trivial element of $V(S_L,\alpha_L)$ has the form $B_1^{-1} B_2$ where $B_1,B_2 \in N$. We claim that the images of
$N \times U_1$ and $N \times U_2$ are disjoint in $\tilde \sO_{\Z_2}(S_L,\alpha_L)$. Let $B_1 (S_{h_1},\alpha_{h_1})$ and $B_2 (S_{h_2},\alpha_{h_2})$ be two surfaces taken from these neighborhoods. Then they are translation equivalent if and only if $(S_{h_1},\alpha_{h_1})$ is equivalent to $B_1^{-1} B_2 (S_{h_2},\alpha_{h_2})$. If they are translation equivalent, we must have
$B_1^{-1} B_2= I $ from arguments in the previous paragraph together with the definition of $N$. But $(S_{h_1},\alpha_{h_1}) \in U_1$ and $(S_{h_2},\alpha_{h_2}) \in U_2$, so they can not be translation equivalent.
\end{proof}
The following will finish our proof of Lemma \ref{lem:deviousness}.
\begin{proof}[Proof of the ``if'' part of Lemma \ref{lem:deviousness}]
Suppose that $\lim n_k=-\infty$ and $\lim_{m \to +\infty} \tau^{m}_\ast(h)= \0$. We must show that the cover $A(\tilde S_{h}, \tilde \alpha_h)$ of $A(S_L, \alpha_L)$ is devious.
We will show every $\omega$-limit point of $g^t A (\tilde S_{h}, \tilde \alpha_{h})$ in $\tilde \sO_{\Z_2}(S_L,\alpha_L)$ is disconnected.
Suppose that $t_i$ is a sequence of times tending to $+\infty$ and
$$\lim_{i \to \infty} g^{t_i} A (\tilde S_{h}, \tilde \alpha_{h})=B(\tilde S_{h'}, \tilde \alpha_{h'}) \quad \text{in $\tilde O_{\Z_2}(S_L,\alpha_L)$}.$$
Consider the sequence of points $p_i=O(2) g^{t_i} A V'$ in $O(2) \bs \SL_\pm(2,\R) / V'$. We claim that there is a cusp neighborhood $U$ of $O(2) \bs \SL_\pm(2,\R) / V'$ (given by $\mathrm{Im}(z)>c$ for some $c \geq 2$ in the fundamental domain of Figure \ref{fig:fundamental domain}) so that $p_i \not \in U$ for all $i$. Otherwise there is a subsequence where $p_{i_j}$ exists the cusp. However, in this case the injectivity radius of the surface $g^{t_{i_j}} A (S_L,\alpha_L)$ would tend to zero, because $g^{t_{i_j}} A$ would contract the circumferences of some cylinder decomposition (which up the action of $V'$ is vertical) more an more as we move up the cusp, and thus $g^{t_{i_j}} A (S_L,\alpha_L)$ would have injectivity radius tending to zero as $j \to \infty$. For the same reason,
the injectivity radius of the surfaces $g^{t_{i_j}} A (\tilde S_h \tilde \alpha_h)$ would have to tend to zero. But this contradicts that this sequence limits to $B(\tilde S_{h'}, \tilde \alpha_{h'})$.
Let $C_d$ be a $d$-neighborhood of the compact core of $O(2) \bs \SL_\pm(2,\R) / V'$. Since by hypothesis $g^t A V'$ is non-divergent in $\SL_\pm(2,\R)/V'$, we know that this trajectory is asymptotic to the convex core. Combining this with this the previous paragraph, we see that
$$O(2) g^{t_i} A V' \in C_d \smallsetminus U \quad \text{for $i$ sufficiently large.}$$
By compactness of the fiber over $C_d \smallsetminus U$, we can assume that after passing to a subsequence that
$g^{t_i} A V'$ converges in $\SL_\pm(2,\R)/V'$.
Now consider the sequence $g^{t_i} A \tilde V'$ in $\SL_\pm(2,\R) / \tilde V'$. Let $k(i)$ be such that $t_i \in [t_{k(i)}, t_{k(i)+1}]$. Defining
$$\tilde p_i=O(2) g^{t_i} A \tilde V' \quad \text{we see} \quad
\tilde p_i \in F_{n_{k(i)}} \quad \text{for each $i$,}$$
and so $\tau^{-n_{k(i)}}(\tilde p_i) \in F_{0}$ for each $i$ where
$\tau^{-n_{k(i)}}$ is acting as an element of the deck group of the covering \eqref{eq:cover} as in \eqref{eq:translation action}. Since each point of
$O(2) \bs \SL_\pm(2,\R) / V'$ has only two lifts to $F_0$, we see that
by passing to a subsequence we can assume
$\tau^{-n_{k(i)}}(g^{t_i} A \tilde V')$ converges in $\SL_\pm(2,\R) / \tilde V'$
above $F_0$. Recalling how $\tau$ acts (see \eqref{eq:ast action} and \eqref{eq:translation action}), this means there is a sequence $R_i \in \tilde V'$ so that
\begin{equation}
\label{eq:convergence of matrices}
g^{t_i} A D(\rho \circ \psi)^{n_{k(i)}} R_i \quad \text{converges to some limit $L \in \SL_\pm(2,\R)$}.
\end{equation}
Observe that in $\tilde \sO_{\Z_2}(S_L,\alpha_L)$ we have
$$\begin{array}{rcl}
\displaystyle g^{t_i} A (\tilde S_{h}, \tilde \alpha_{h}) & = &
\displaystyle g^{t_i} A D(\rho \circ \psi)^{n_{k(i)}} R_i \cdot \big(R_i^{-1} D(\rho \circ \psi)^{-n_{k(i)}} (\tilde S_{h}, \tilde \alpha_{h})\big) \\
& = & \displaystyle g^{t_i} A D(\rho \circ \psi)^{n_{k(i)}} R_i
\big(\tilde S_{\tau^{-n_{k(i)}}_\ast(h)}, \tilde \alpha_{\tau^{-n_{k(i)}}_\ast(h)}\big).
\end{array}$$
The $\SL_\pm(2,\R)$ part of this last expression converges to $L$. The surface part converges to $(\tilde S_{\0}, \tilde \alpha_\0)$ because $\lim n_k=-\infty$ and $\lim_{m \to +\infty} \tau^{m}_\ast(h)= \0$. So, because $\tilde O_{\Z_2}(S_L,\alpha_L)$ is Hausdorff by Proposition \ref{prop:Hausdorff}, we must have
$$B(\tilde S_{h'}, \tilde \alpha_{h'})=\lim_{i \to \infty} \displaystyle g^{t_i} A (\tilde S_{h}, \tilde \alpha_{h})=L (\tilde S_{\0}, \tilde \alpha_\0).$$
In particular,
$B(\tilde S_{h'}, \tilde \alpha_{h'})$ is disconnected.
\end{proof}
Now we will study devious covers by studying those $h$ so that $\tau^m_\ast(h)$ decays to $\0$. A formula for $\tau_\ast$ is given below.
\begin{proposition}
\label{prop:tau 1}
The action of translation by one on $\Hom(\Gamma, \Z_2)$ is given by
\begin{equation*}
\big(\tau_\ast(h)\big)(i)=\begin{cases}
h(-i)+h({i-1})+h({i})+h({i+1}) & \text{if $i<-2$,} \\
h(-i)+h({-2})+h({-3}) & \text{if $i=-2$,} \\
h(-i)+h({-2}) & \text{if $i=-1$,} \\
h(-i) & \text{if $i>0$.}
\end{cases}
\end{equation*}
The action of translation by negative one is given by:
\begin{equation*}
\big(\tau^{-1}_\ast(h)\big)(i)=\begin{cases}
h(-i) & \text{if $i <0$,} \\
h(-i)+h({2}) & \text{if $i=1$,} \\
h(-i)+h({2})+h({3}) & \text{if $i=2$,} \\
h(-i)+h({i-1})+h({i})+h({i+1}) & \text{if $i>2$.}
\end{cases}
\end{equation*}
\end{proposition}
\begin{proof}
This follows from the fact that $\tau=\delta \circ D(\rho \circ \psi)$ and
$\tau^{-1}=\delta \circ D(\psi \circ \rho)$; see equation \ref{eq:delta}. The actions of $\psi$ and $\rho$ on $\Hom(\Gamma, \Z_2)$
are given in equations \ref{eq:psi ladder} and \ref{eq:rho ladder}, respectively.
\end{proof}
To analyze the case when $\lim_{m \to +\infty} \tau^m_\ast(h)=\0$,
we introduce the {\em proximity function} which measures how close a homomorphism is to the trivial homomorphism $\0$:
\begin{equation}
\label{eq:proximity}
P:\Hom(\Gamma, \Z_2) \to \{1,2,3, \ldots, \infty\};
\quad
P(h)= \begin{cases}
\infty & \text{if $h=\0$,}\\
\min \{|i|~:~h(i) \neq 0\} & \text{otherwise}.
\end{cases}
\end{equation}
Observe a sequence $h_n$ tends to $\0$ if and only if $P(h_n) \to \infty$.
For each integer $k \geq 0$, we define a cylinder set in $\Hom(\Gamma, \Z_2)$ by
$$C_k=\{h~:~\text{$h(k)=h(-k-1)=1$ and $h(i)=0$ when $-k-1<i<k$}\}.$$
Note that $h \in C_k$ implies $P(h)=k$.
These cylinder sets turn out to be very important, for understanding which elements of $\Hom(\Gamma, \Z_2)$ are limit to the trivial homomorphism $\0$
under translation.
\begin{lemma}
\label{lem:limits to zero}
The collection of $h \in \Hom(\Gamma, \Z_2)$ so that $\lim_{n \to +\infty} \tau^n_\ast(h)=\0$ is given by
$$
\{\0\} \cup \bigcup_{j \geq 0} \tau^{-j}_\ast(Z)
\quad \text{where}
\quad Z=\bigcup_{k \geq 2} \bigcap_{m \geq 0} \tau^{-m}_\ast(C_{k+m}).$$
Furthermore, for any function $f:\{i~:~i \geq 2\} \to \Z_2$ which is not identically zero,
there is an $h \in Z$ so that $h(i)=f(i)$ for all $i \geq 2$.
\end{lemma}
The second statement says that there are a number of elements of $\Hom(\Gamma, \Z_2)$ which limit on $\0$. (Informally, $Z$ has half the information entropy of $\Hom(\Gamma, \Z_2)$, or with an appropriate natural metric, $Z$ has half the Hausdorff dimension of $\Hom(\Gamma, \Z_2)$.)
This together with Lemma \ref{lem:deviousness} prove the existence of devious covers as stated in Theorem \ref{thm:main ladder}.b
The following proposition is the main ingredient in the proof of the lemma above.
\begin{proposition}
\label{prop:cylinder}
\begin{enumerate}
\item If $k \geq 2$ and $h \in C_k$, then $P\big(\tau_\ast(h)\big)=k+1$.
\item If $k \geq 2$ is an integer, then $\tau^{-1}_\ast(C_{k+1}) \subset C_k$.
\item If $k \geq 2$, $P(h)=k$ and $h \not \in C_k$, then either
$P\big(\tau_\ast(h)\big)=k-1$ and $\tau_\ast(h) \not \in C_{k-1}$
or $P\big(\tau^2_\ast(h)\big)=k-1$ and $\tau^2_\ast(h) \not \in C_{k-1}$.
\end{enumerate}
\end{proposition}
\begin{proof}
We prove each of the statements below.
{\bf (1) } Suppose that $h \in C_k$ and $k \geq 2$. Then $h(k)=h(-k-1)=1$ while $h(i)=0$ for $-k \leq i \leq k-1$. Then $P\big(\tau_\ast(h)\big)\leq k+1$, because
$\tau_\ast(h)(k+1)=h(-k-1)=1$
by Proposition \ref{prop:tau 1}. For $0<i \leq k$, we have
$\tau_\ast(h)(i)=h(-i)=0$. Also observe that
$$\tau_\ast(h)(-k)=h(k)+h(-k-1)=1+1=0,$$
while the other terms from Proposition \ref{prop:tau 1} vanish. Finally, all terms in the expression for $\tau_\ast(h)(i)$ vanish when $-k<i<0$. Therefore,
$P\big(\tau_\ast(h)\big)= k+1$ as claimed.
{\bf (2) } Suppose $h \in C_{k+1}$ with $k \geq 2$. Then $h(k+1)=h(-k-2)=1$ while $h(i)=0$ for $-k-1 \leq i \leq k$. By Proposition \ref{prop:tau 1}, we see that
$\tau^{-1}_\ast(h)(-k-1)=h(k+1)=1.$ Also,
$$\tau^{-1}_\ast(h)(k)=\begin{cases}
h(-k)+h(2)+h(3)=0+0+1=1 & \text{if $k=2$,}\\
h(-k)+h(k-1)+h(k)+h(k+1)=0+0+0+1=1 & \text{if $k>2$}.
\end{cases}
$$
Now suppose $-k \leq i <0$. Here we have
$\tau^{-1}_\ast(h)(i)=h(-i)=0.$
Finally consider the case when $0 < i < k$. All terms vanish from
the expression for the expression $\tau^{-1}_\ast(h)(i)$ given in Proposition \ref{prop:tau 1}, so $\tau^{-1}_\ast(h)(i)=0$.
Taken together, we see $\tau^{-1}_\ast(h) \in C_k$.
{\bf (3) } Suppose $k \geq 2$, $P(h)=k$ and $h \not \in C_k$.
Since $P(h)=k$, we know that $h(i)=0$ when $|i|<k$. Observe that the set of $h$ with $P(h)=k$ and
$h \not \in C_k$ is the union of the two pieces:
$$A=\{h~:~\text{$P(h)=k$ and $h(-k)=1$}\},$$
$$B=\{h~:~\text{$P(h)=k$, $h(-k)=0$, $h(k)=1$, and $h(-k-1)=0$}\}.$$
First assume that $h \in A$ so that $h(-k)=1$ but $h(i)=0$ when $|i|<k$. Then,
$\tau^{1}_\ast(h)(-k+1)$ is given by one of the expressions
$$
\begin{cases}
h(k-1)+h(-k)+h(-k+1)+h(-k+2)=0+1+0+0=1 & \text{if $k>3$},\\
h(k-1)+h(-2)+h(-3)=0+0+1=1 & \text{if $k=3$},\\
h(k-1)+h(-2)=0+1=1 & \text{if $k=2$}.
\end{cases}$$
Since proximity can decrease by at most one when applying $\tau^{1}_\ast$,
we see $P\big(\tau^{1}_\ast(h)\big)=k-1$. Also, we have by definition of $C_{k-1}$
that $\tau^{1}_\ast(h) \not \in C_{k-1}$.
Now consider the case when $h \in B$. Then $h(i)=0$ when $-k-1 \leq i<k$
and $h(k)=1$. In this case $\tau^{1}_\ast(h)(i)=0$ when $|i|<k-1$ since proximity can decrease by at most one. We observe $\tau^{1}_\ast(h)(k-1)=h(-k+1)=0.$
All terms vanish in the expression for $\tau^{1}_\ast(h)(-k+1)$ given by
Proposition \ref{prop:tau 1}, so $\tau^{1}_\ast(h)(-k+1)=0.$
On the other hand,
$$\tau^{1}_\ast(h)(-k)=\begin{cases}
h(k)+h({-k-1})+h({-k})+h({-k+1})=1+0+0+0=1 & \text{if $k>2$,} \\
h(k)+h({-2})+h({-3})=1+0+0=1 & \text{if $k=2$.}
\end{cases}
$$
Since $\tau^{1}_\ast(h)(-k)=1$, we see that $\tau^{1}_\ast(h) \in A$.
Because of our discussion of what happens for elements of $A$, we
see that $P\big(\tau^2_\ast(h)\big)=k-1$ and $\tau^2_\ast(h) \not \in C_{k-1}$.
\end{proof}
In the proof of Lemma \ref{lem:limits to zero}, it is useful to note the following Corollary to the Proposition above.
\begin{corollary}
Let $k \geq 1$ be an integer.
If $P(h)=k$ but $h \not \in C_k$, then there is an $n \geq 0$ so that $P\big(\tau^n_\ast(h)\big)=1$.
\end{corollary}
\begin{proof}
We prove this by induction in $k$.
This holds with $n=0$ when $k=1$. When $k=2$, this follows from statement (3) of Proposition \ref{prop:cylinder}. Now let $k \geq 3$ and suppose the conclusion holds when $P(h)=k-1$ and $h \not \in C_{k-1}$. Suppose $P(h)=k$ and $h \not \in C_k$. Again using statement (3) of Proposition \ref{prop:cylinder}, we see that
either $P\big(\tau_\ast(h)\big)=k-1$ and $\tau_\ast(h) \not \in C_{k-1}$
or $P\big(\tau^2_\ast(h)\big)=k-1$ and $\tau^2_\ast(h) \not \in C_{k-1}$. Using our induction hypothesis applied to these cases, we see the conclusion holds, and the whole statement holds by induction.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem:limits to zero}]
Let $Z=\bigcup_{k \geq 2} \bigcap_{m \geq 0} \tau^{-m}_\ast(C_{k+m})$ as in the theorem. Let $h \in Z$. Then there is a $k \geq 2$ so that $\tau^m_\ast(h) \in C_{k+m}$ for all $m \geq 0$. Then $P\big(\tau^m_\ast(h)\big)=k+m$ tends to $\infty$ as $m \to \infty$. Thus, $\lim_{m \to \infty} \tau^m_\ast(h)=\0$. From this it follows that everything in the set listed in the theorem,
$\{\0\} \cup \bigcup_{j \geq 0} \tau^{-j}_\ast(Z)$, limits to $\0$.
Conversely, suppose $h \in \Hom(\Gamma, \Z_2)$ satisfies $\lim_{n \to \infty} \tau^n_\ast(h)=\0$ and $h \neq \0$. We know that $P \big(\tau^n_\ast(h)\big) \to \infty$ as $n \to \infty$. In particular, we see that for any $k \geq 1$, there are only finitely many $n$ so that
\begin{equation}
\label{eq:proximity2}
P \big(\tau^n_\ast(h)\big)=k.
\end{equation}
Let $K \geq 2$ be the smallest value of $k$ larger than one so that the above equation has a solution for $n$, and let $N$ be the maximal $n$ satisfying the equation when $k=K$. We claim that
\begin{equation}
\label{eq:h}
\tau^N_\ast(h) \in \bigcap_{n \geq 0} \tau^{-n}_\ast(C_{K+n}).
\end{equation}
Otherwise, there is a smallest $n \geq 0$ so that $\tau^{N+n}_\ast(h) \not \in C_{K+n}$. Then by the corollary above, we see that there is an $m \geq N+n$
so that $P\big(\tau^m_\ast(h)\big)=1$. But since $P \big(\tau^n_\ast(h)\big)$ tends to $\infty$ and as $n$ increases the proximity can increase by at most one,
we see that there is an $m'>m$ so that \eqref{eq:proximity2} is satisfied for $n=m'$. But this contradicts the definition of $N$. Therefore, \eqref{eq:h} is true after all. Observe that the equation implies that
$\tau^N_\ast(h) \in Z$, and thus $h \in \bigcup_{j \geq 0} \tau^{-j}_\ast(Z)$
as desired.
It remains to prove the last sentence of the theorem which guarantees there are a lot of $h$ so that $\lim_{n \to \infty} \tau^n_\ast(h)=\0$.
A main point here is that for each $k \geq 2$, the set
$A_k=\bigcap_{m \geq 0} \tau^{-m}_\ast(C_{k+m})$ is non-empty.
Indeed, as continuous images of cylinder sets, each $\tau^{-m}_\ast(C_{k+m})$ is compact. The sets being intersected are nested in the sense that
$$\tau^{-m-1}_\ast(C_{k+m+1}) \subset \tau^{-m}_\ast(C_{k+m})$$
by statement (2) of the lemma. Therefore, we can make a choice of an $\tilde h_k \in A_k$ for every $k \geq 2$.
Now suppose that $f: \{i \in \Z~:~i \geq 2\} \to \Z_2$ is defined and not identically zero. Let $k=\min \{i~:~f(i)\neq 0\}$.
We will inductively define a sequence of functions $h_j \in \Hom(\Gamma, \Z_2)$
for $j \geq k$ so that the sequence converges to an extension of $f$.
Our functions will all lie in $A_k$. Since $A_k$ is closed, this will suffice
to prove that the limiting function lies in $A_k$. We will also ensure that
\begin{equation}
\label{eq:ast}
h_j(i)=f(i) \quad \text{when $2 \leq i \leq j$}.
\end{equation}
To ensure convergence of the sequence $h_j$, we will also have that
\begin{equation}
\label{eq:prior}
h_j(i)=h_{j-1}(i) \quad \text{when $|i|<j$.}
\end{equation}
Observe that $f(k)=1$, thus we can define $h_k=\tilde h_k \in A_k$. This serves as our base case. Now assume that $j > k$ and $h_{j-1}$ is defined and satisfies the hypotheses above. We will define $h_{j}$. If $h_{j-1}(j)=f(j)$,
then we can take $h_j=h_{j-1}$. Otherwise, we define
$h_{j}=h_{j-1}+\tilde h_j$. Since $\tilde h_j \in C_{j}$, we know that $\tilde h_j(j)=1$. Therefore, we must have $h_{j}(j)=f(j)$. When $|i|<j$, we have
$\tilde h_j(i)=0$ because $h_j \in C_j$, therefore
$$h_j(i)=h_{j-1}(i) \quad \text{when $|i|<j$}$$
by inductive hypothesis. This simultaneously ensures both equation (\ref{eq:prior})
holds and verifies equation (\ref{eq:ast}) for $i<j$ since by hypothesis
$h_{j-1}(i)=f(i)$ when $2 \leq i \leq j-1$.
From the inductive hypothesis and definition of $\tilde h_j$, we have
$$\tau^m_\ast(h_{j-1}) \in C_{k+m} \and
\tau^m_\ast(\tilde h_{j}) \in C_{j+m}$$
for each integer $m \geq 0$. Observe that when $j>k$, the sum of an element
in $C_{k+m}$ and $C_{j+m}$ lies in $C_{k+m}$. Therefore, we see by linearity of $\tau^m_\ast$ that
$$\tau^m_\ast(h_{j})=\tau^m_\ast(h_{j-1})+\tau^m_\ast(\tilde h_{j}) \in C_{k+m}$$
for every integer $m \geq 0$. Thus, $h_j \in A_k$ as desired. This completes the inductive step.
\end{proof}
\compat{Change (a) and (b) to (L2) and (L3)}
\begin{proof}[Proof of statement {\em (L2)} of Theorem \ref{thm:main ladder}]
We will consider the translation flow on the devious double cover $A(\tilde S_h, \tilde \alpha_h)$ of $A(S_L,\alpha_L)$ under several assumptions.
We assume the trajectory $g^t A V'$ is non-divergent in $\SL_\pm(2,\R)/V'$.
From the geodesic $g^t A \tilde V'$, we define the coding walk $\{n_k\}$.
We assume that the number of visits to any integer is finite, but that the
growth exponent of the visit count satisfies $v>\varphi^2$.
As allowed by Remark \ref{rem:reflection simplification},
we assume that $\lim_{k \to \infty} n_k = -\infty$.
By Lemma \ref{lem:deviousness}, we know that $\lim_{m \to +\infty} \tau^m_\ast(h)=\0.$
In particular, $\lim_{k \to \infty} \tau^{-n_k}_\ast(h)=\0.$
We will prove ergodicity holds for the translation flow on $A(\tilde S_h, \tilde \alpha_h)$
by appealing to Theorem \ref{thm:integrability}.
(We upgrade to unique ergodicity at the end of the proof.)
Let $\eta>0$.
We will find subsurfaces of $X_t=g^t A(\tilde S_h, \tilde \alpha_h)$ and related geometric quantities so that the integral \eqref{eqn:integrability2} is infinite.
Now recall the definitions of the coding walk; see Proposition \ref{prop:walk}.
There is a sequence of times $t_k$ so that
$$g^t A \tilde V' \in F_{n_k} \quad \text{when $t_k<t<t_{k+1}$}.$$
Let $U$ and $C_d$ be subsets of $O(2) \bs \SL_\pm(2,\R) / V'$ as in Lemma \ref{lem:convergence}. Let
$$J_k=\{t \in (t_k,t_{k+1}):~O(2) g^t A V' \in C_d \smallsetminus U\}.$$
Since the geodesic $O(2) g^t A V'$ is non-divergent it is asymptotic to the convex core and so
there is a ${\text \j}>0$ so that
the constant
\begin{equation}
\label{eq:j}
{\text \j} \geq \textit{length}(J_k) \quad \text{for $k$ sufficiently large}.
\end{equation}
(In fact, $\textit{length}(J_k)$ has a uniform lower bound once the geodesic is within distance $d$
from the convex core.) By Lemma \ref{lem:convergence}, there is a compact set $K \subset \SL_\pm(2,\R)$ so that for any $t \in J_k$ there is a matrix $M_t \in K$ so that
\begin{equation}
\label{eq:X t}
X_t=M_t (\tilde S_{\tau^{-n_k}_\ast(h)}, \tilde \alpha_{\tau^{-n_k}_\ast(h)}\big) \quad \text{as elements of $\tilde \sO_{\Z_2}(S_L,\alpha_L)$}.
\end{equation}
We will now explain how to find subsurfaces. The surfaces will always be
obtained by lifting two copies of a disk in $(S_L,\alpha_L)$; so we will always
have $C_t=2$ in the language of the Theorem \ref{thm:integrability}.
We think of $(S_L,\alpha_L)$ as depicted by Figure \ref{fig:ladder_surface}: the surface is a topological disk in the plane with edge identifications. For small $\kappa>0$, consider the subset $U_0(\kappa)$ of this disk consisting of points whose distance from the boundary of the disk is greater than $\kappa$. Then let $U(\kappa) \subset U_0(\kappa)$
be the subsurface of the largest area. Since these regions exhaust the disk as $\kappa \to 0$, we can
choose a $\kappa$ so that $U=U(\kappa)$ contains more than a factor of $1-\eta$ of the disk's area.
See Figure \ref{fig:ladder_surface_subsurface} for an example of $U(\kappa)$.
We can think of $U$ as lying in the surface $(S_L,\alpha_L)$.
\begin{figure}
\caption{An example $U(\kappa)$ in the disk making up $(S_L,\alpha_L)$.}
\label{fig:ladder_surface_subsurface}
\end{figure}
In order to define our surfaces $S_t \subset A(\tilde S_h, \tilde \alpha_t)$,
observe that $A(\tilde S_h, \tilde \alpha_t)=g^{-t}(X_t)$
when $t \in \bigcup_k J_k$
and consider $X_t$ as in \eqref{eq:X t}. (We will not bother to estimate the value being integrated when $t \not \in \bigcup_k J_k$.)
Let $\tilde U_t^1$ and $\tilde U_t^2$ be the two lifts of $U$ to
$(\tilde S_{\tau^{-n_k}_\ast(h)}, \tilde \alpha_{\tau^{-n_k}_\ast(h)}\big)$. Then we define
$S_t=g^{-t} M_t(\tilde U_t^1 \cup \tilde U_t^2).$ Evaluating geometric quantities of $S_t \subset A(\tilde S_h, \tilde \alpha_t)$ with $\dist_t$ is the same
as evaluating the same quantities for the subsurface $M_t(\tilde U_t^1 \cup \tilde U_t^2)$ of $X_t$. Observe that for $t \in \bigcup_k J_k$:
\begin{itemize}
\item $C_t=2$.
\item The distance from $M_t(\tilde U_t^1 \cup \tilde U_t^2)$ to the singular set has a uniform lower bound, i.e., $\epsilon(t)$ can be taken to be a positive constant, since $\epsilon(t)$ is bounded by $\kappa$ divided by the operator norm of $M_t^{-1}$ which has a uniform upper bound since $M_t \in K$ and $K$ is compact.
\item The diameters of the components have a uniform upper bound given by the diameter of $U$ times the maximal operator norm of $M \in K$. So ${\mathcal D}_t^i$ can be taken to be a constant independent of $t$ and $i$.
\end{itemize}
Finally, we need to consider the maximum over all curves joining
our two subsurfaces of the minimum $\dist_t$ distance from a point on the curve to a singularity. We will choose a canonical curve which depends mostly on the proximity of $\tau^{-n_k}_\ast(h)$ to zero, $P \circ \tau^{-n_k}_\ast(h)$, which was defined in equation \eqref{eq:proximity}. We will first define a curve joining
$\tilde U_t^1$ and $\tilde U_t^2$ in $(\tilde S_{\tau^{-n_k}_\ast(h)}, \tilde \alpha_{\tau^{-n_k}_\ast(h)}\big)$, then we will push this curve under $M_t$ into $X_t$.
(Again it suffices to measure things in $X_t$.)
Consider the cover $(\tilde S_{\tau^{-n_k}_\ast(h)}, \tilde \alpha_{\tau^{-n_k}_\ast(h)}\big)$ as a pair of infinite polygons with edges labeled as in Figure \ref{fig:ladder_surface} and identified in some way. The edge labels with the smallest absolute values which are glued so as to join up the two polygons
are of the form $p=\pm P \circ \tau^{-n_k}_\ast(h)$ in Figure \ref{fig:ladder_surface}.
We choose a curve to leave the first subsurface and move upward along the slope $1$ line of symmetry until it reaches the height of the edge $p$, then it changes trajectory by $\pm 45^\circ$ and travels through the midpoint of the edge connecting to the other disk. The curve continues until it hits the line of symmetry in the second disk, and it returns to the second subsurface along the line of symmetry of the disk. See Figure \ref{fig:ladder_surface_path}. It should be observed that there are positive constants
$a$ and $b$ so that
the distance from this curve to metric completion are of the form
$$\min~\{a, b \varphi^{-|P \circ \tau^{-n_k}_\ast(h)|}\}.$$
(This is a consequence of the apparent self-similarity of the surface.)
Then using the fact that $M_t$ is taken from a compact set, we see we can take
\begin{itemize}
\item $\delta_t=c \min~\{a, b \varphi^{-|P \circ \tau^{-n_k}_\ast(h)|}\}$ for some $c>0$.
\end{itemize}
\begin{figure}
\caption{The path joining the two subsurfaces when $P(h_\ast)=5$ and $h_\ast(5)=1$.
The letters $a$ and $b$ denote edges joining the two disks defining the cover in this case.}
\label{fig:ladder_surface_path}
\end{figure}
We now have control of all the geometric quantities and can bound the integral.
From the above definitions $d=\epsilon(t)^{-2} \sum_{i=1}^2 {\mathcal D}_t^i$ is a positive constant. For $t \in J_k$, the quantity being integrated is of the form
$$\Big(d+\frac{1}{c \min~\{a, b \varphi^{-|P \circ \tau^{-n_k}_\ast(h)|}\}}\Big)^{-2}.$$
Now we invoke the hypothesis that $\lim_{N \to \infty} \tau^N_\ast(h)=\0$. By Lemma \ref{lem:limits to zero},
we know that there is an $M>0$ and a $L \in \Z$ so that
$\tau^N_\ast(h)$ lies in the cylinder set $C_{L+N}$ for all integers $N>M$. In particular,
$P\big(\tau^N_\ast(h)\big)=L+N$. Then for sufficiently large values of $N$, say $N \geq N_0$, we
can arrange that when $n_k=-N$,
$$\min~\{a, b \varphi^{-|P \circ \tau^{-n_k}_\ast(h)|}\}=b \varphi^{-N-L}.$$
Let $V_N=\#\{k:~-n_k=N\}$. Recall \eqref{eq:j}, the length of each $J_j$ is bounded from below by ${\text \j}>0$. This allows us to write a lower bounds for the integral as
$${\text \j} \sum_{N=N_0}^\infty V_N \big(d+\frac{1}{b c} \varphi^{N+L}\big)^{-2}.$$
An application of the root test tells us that this series diverges if
$$v=\limsup_{N \to \infty} V_N> \varphi^2.$$
This was precisely our hypothesis, and Theorem \ref{thm:integrability} gives us ergodicity.
We will now show that ergodicity implies unique ergodicity in statement (L2).
By Corollary \ref{cor:lifting}, we just need to know that the translation flow on $A(S_L,\alpha_L)$ is uniquely ergodic. By Theorem \ref{thm:prior work}, we have unique ergodicity unless $O(2)g^t A V'$ is asymptotic to the convex core boundary. But then the lifted geodesic $O(2)g^t A \tilde V'$ on $O(2) \bs \SL_\pm(2,\R) / \tilde V'$ is asymptotic to the lifted convex core boundary, which is depicted as the boundary between the light and dark gray regions on the right side of Figure \ref{fig:periodic disk}. But then the coding walk $\langle n_k \rangle$ must grow asymptotically linearly, i.e., there is a $K$ so that for all $k>K$ we have $n_{k+1}=n_k+1$ or for all $k>K$ we have $n_{k+1}=n_k+1$. But then the growth exponent is $v=1$ which is not allowed in case (L2).
\end{proof}
Now we will consider how to obtain non-ergodic covers. We will make use of ideas
of Masur and Smillie which first appeared in \cite[Theorem 2.1]{MS91}.
The criterion developed there for non-ergodicity carries over from the closed surface case to the infinite type case.
We will state the (only slightly different) version from \cite[Theorem 3.3]{MT} in our setting. For the following theorem recall that the {\em vertical holonomy} of a curve $\gamma$ in a translation surface is the imaginary part of $\int_\gamma \alpha$.
\begin{theorem}[Masur-Smillie \cite{MS91}]
\label{thm:Masur-Smillie}
Let $(S,\alpha)$ be a unit area translation surface of possibly infinite topological type,
and assume the translation flow is defined for all time almost everywhere.
Suppose there is a sequence of directions $\theta_n$ tending to the horizontal
and a sequence of partitions of the surface into two pieces, $S=A_n \sqcup B_n$,
so that the common boundary consists of a countable union
of line segments in direction $\theta_n$.
Assume further that the absolute values of the vertical holonomies of the segments sum to $h_n<\infty$. Suppose also that:
\begin{enumerate}
\item[(i)] $\lim_{n \to \infty} h_n=0$.
\item[(ii)] There are constants $c$ and $c'$, so that $0<c<\mu(A_n)<c'<1$
for each $n$, where $\mu$ is Lebesgue measure on $(S_L,\alpha_L)$.
\item[(iii)] $\sum_{n=1}^\infty \mu(A_n \Delta A_{n+1})<\infty$, where $\Delta$ denotes symmetric difference.
\end{enumerate}
Then, the translation flow (horizontal straight-line flow) on $(S_L,\alpha_L)$ is not ergodic.
\end{theorem}
The proof as provided in \cite[Theorem 3.3]{MT} works in our setting, so we only explain the main ideas of the proof.
Through a measure theoretic argument, one can see that the set $A_\infty=\liminf A_n$ must be almost flow invariant in the sense that for any $t$ the symmetric difference of $A_\infty$ with its image under translation flow for time $t$ has Lebesgue measure zero.
The set $A_\infty$ has measure bounded away from zero and one. An argument involving Fubini's theorem then produces an invariant set which is the same up to sets of Lebesgue measure zero which proves non-ergodicity. The arguments make no use of compactness or any other properties which do not hold in our setting.
\begin{proof}[Proof of statement {\em (L3)} of Theorem \ref{thm:main ladder}]
Let $A \in \SL(2,\R)$ and $h \in \Hom(\Gamma,\Z_2)$ determine a devious cover $A(\tilde S_h,\tilde \alpha_h)$ of $A(S_L,\alpha_L)$. We also make several other hypotheses. We consider the coding walk of the geodesic $g^t A \tilde V'$ in $O(2) \bs \SL_\pm(2,\R)/ \tilde V'$,
and consider its coding walk $\{n_k\}$. Statement (L3) concerns the case when the coding walk diverges quickly
in the sense that the growth exponent $v$ is smaller than $\varphi^2$. In light of Remark \ref{rem:reflection simplification}, we can assume that $\lim_{k \to \infty} n_k=-\infty$. By Lemma \ref{lem:deviousness}, we know that $\lim_{m \to +\infty} \tau_\ast^{m}(h)=\0$.
We will prove that the translation flow on $A(\tilde S_h, \tilde \alpha_h)$ is not ergodic. To do this, we will find a sequence of partitions satisfying the criteria set out by Masur and Smillie (Theorem \ref{thm:Masur-Smillie}). Our partitions will be obtained by pulling back partitions
of the deformed surface $g^t A(\tilde S_h, \tilde \alpha_h)$ along a sequence of times tending to $+\infty$.
It will be important for us to label the lifts of the basepoint to our double covers. We label the lifts
by the elements of $\Z_2$. Then when we deform our surfaces or apply linear maps, we will respect the labels of the basepoint. (Note that all double covers are normal and so admit a translation automorphism swapping the lifts of the basepoint.)
We introduce the following setup. Consider the region $F_0$ in $O(2) \bs \SL_\pm(2,\R) / \tilde V'$.
Since the geodesic $g_t A V'$ is asymptotic to the convex core of $O(2) \bs \SL_\pm(2,\R) / V'$, there is a $d>0$ so that
this geodesic is contained entirely in the $d$-neighborhood of this convex core.
Let $F_0' \subset F_0$ be the closed $d$-neighborhood of the lift of the convex core.
Choose an $M_0 \in \SL_\pm(2,\R)$ so that $M_0 \tilde V'$ lies in $F_0'$. Now consider another point $B \tilde V'$ of $F_0'$. Select a path
$\gamma_B$ in $F_0'$ joining $M_0 \tilde V'$ to $B \tilde V'$. We can lift this path in $\SL_\pm(2,\R)/ \tilde V'$
to a path in $\SL(2,\R)$ beginning at $M_0$. We let $E(B) \in \SL(2,\R)$ denote the endpoint of this path.
Observe that $E(B)$ lies in the coset $B \tilde V'$. The selection of $E(B)$ is not quite canonical; it
does not depend on the choice of $B$ from the class $B \tilde V'$, but it depends on the homotopy class of the curve $\gamma_B$. Since $F_0'$ is topologically an annulus, a different choice of path might give us a different $E(B)$, and difference is explained by monodromy around the cusp. Let $\Psi=\langle D(\psi)^2 \rangle \subset \SL(2,\R)$ be the monodromy group of $F_0$. We see that while $E(B)$ is not canonical, the coset $E(B) \Psi$ is. We will make choices only depending on $E(B)\Psi$.
Now consider an affine image of our cover,
$g^t A(\tilde S_{h}, \tilde \alpha_{h})$, and suppose that $g^t A \tilde V'$ lies in the region $F_n$. Select
an element $R \in V'$ which projects to $\tau^{-n} \in V'/\tilde V'$
so that $g^t A \tilde V' R^{-1}=g^t A R^{-1} \tilde V'$ lies
in the region $F_0'$. The action of $R$ on $\Hom(\Gamma, \Z_2)$ is given by $\delta(R)=\tau^{-n}_\ast$.
Therefore,
$$g^t A(\tilde S_{h}, \tilde \alpha_{h})=g^t A R^{-1} (\tilde S_{\tau^{-n}_\ast(h)}, \tilde \alpha_{\tau^{-n}_\ast(h)}),$$
where equality denotes translation equivalence respecting marked points.
Now consider $E(g^t A R^{-1})$ as in the prior paragraph. We have $E(g^t A R^{-1})=g^t A R^{-1} \tilde R^{-1}$
for some $\tilde R \in \tilde V'$. Since $\tilde V'$ acts trivially on $\Hom(\Gamma,\Z_2)$, we see that
\begin{equation}
\label{eq:identification}
g^t A(\tilde S_{h}, \tilde \alpha_{h})=E(g^t A R^{-1}) (\tilde S_{\tau^{-n}_\ast(h)}, \tilde \alpha_{\tau^{-n}_\ast(h)})
\quad \text{in $\tilde \sO_{\Z_2}(S_L,\alpha_L)$.}
\end{equation}
A partition of the cover $(\tilde S_{\tau^{-n}_\ast(h)}, \tilde \alpha_{\tau^{-n}_\ast(h)})$
will then pull back under this translation isomorphism followed by $g^{-t}$ to a partition of our original surface $A(\tilde S_{h}, \tilde \alpha_{h})$.
The cover $(\tilde S_{\tau^{-n}_\ast(h)}, \tilde \alpha_{\tau^{-n}_\ast(h)})$ can be thought of two copies of the infinite polygon used to define $(S_L,\alpha_L)$ labeled $\sR_0$ and $\sR_1$ and glued together according to $\tau^{-n}_\ast(h) \in \Hom(\Gamma,\Z_2)$.
The index of the regions is determined by the index of the lift of the basepoint the region contains.
We will partition the cover into two subsurfaces $\sA$ and $\sB$. From our point of view, there are three types of vertical cylinders
on the cover $(\tilde S_{\tau^{-n}_\ast(h)}, \tilde \alpha_{\tau^{-n}_\ast(h)})$:
\begin{enumerate}
\item vertical cylinders that stay entirely in region $\sR_0$.
\item vertical cylinders that stay entirely in region $\sR_1$.
\item vertical cylinders that pass through both $\sR_0$ and $\sR_1$.
\end{enumerate}
Figure \ref{fig:ladder_surface_cylinders} illustrates the two regions and the cylinder types.
For any integer $j<0$, we have either $\tau^{-n}_\ast(h)(j)=0$ or $\tau^{-n}_\ast(h)(j)=1$. In the first
case, we get vertical cylinders of types (1) and (2) passing through lifts of the edge labeled $j$
of the base surface (as depicted in Figure \ref{fig:ladder_surface}),
and in the second case, there is a single vertical cylinder of type (3) which passes through both the lifts of edges labeled $j$. We define the surface $\sA_n \subset (\tilde S_{\tau^{-n}_\ast(h)}, \tilde \alpha_{\tau^{-n}_\ast(h)})$ to
be the union of all vertical cylinders of type (1), and we define $\sB_n$ to be the union of cylinders of the remaining
two types. We can push this partition onto the surface $g^t A (\tilde S_{h}, \tilde \alpha_{h})$
by applying the affine map $E(g^t A R^{-1})$ and using the identification given in equation \ref{eq:identification}.
This gives us a partition $(\sA^t, \sB^t)$ of $g^t A (\tilde S_{h}, \tilde \alpha_{h})$.
Note that while $E(g^t A R^{-1})$ is not quite canonical as noted in the prior paragraph, it is well defined
up to a power of $D(\psi)^2$. Note that the partition of $g^t A (\tilde S_{h}, \tilde \alpha_{h})$
obtained is the same no mater which element of $E(g^t A R^{-1})\Psi$ we choose, because the action of $D(\psi^2) \in \tilde V'$ preserves each vertical cylinder.
\begin{figure}
\caption{A double cover of $(S_L,\alpha_L)$ with some edge gluings of horizontal edges labeled. There are two vertical cylinders of type (1) shown in gray with white waves, two of type (2) with a checkerboard pattern, and two of type (3) shown in light gray.}
\label{fig:ladder_surface_cylinders}
\end{figure}
For each $t$, we can pullback the partition $(\sA^t, \sB^t)$ of $g^t A (\tilde S_{h}, \tilde \alpha_{h})$
to a partition $g^{-t}(\sA^t, \sB^t)$ of $A (\tilde S_{h}, \tilde \alpha_{h})$.
This is actually a sequence of partitions, as we now explain.
When $g^t A \tilde V'$ lies in the region $F_n$, we have that $t$ lies in some interval
$(t_k, t_{k+1})$ where $n_k=n$. We claim the partition $g^{-t}(\sA^t, \sB^t)$ is independent of the choice
of $t$ from this interval. This is because the coset $g^{-t} E(g^t A R^{-1})\Psi$
is constant on the interval, since given a path to $g^t A R^{-1}$ we can get a path to
$g^{t_\ast} A R^{-1}$ by traveling along the geodesic, and this change is canceled by the application of $g^{-t}$.
Thus the partition only depends on $k$, and we define
$\sA'_k=g^{-t}(\sA^t)$ and $\sB'_k=g^{-t}(\sA^t)$ for any $t$ satisfying $t_k<t<t_{k+1}$.
These subsurfaces partition $A(\tilde S_{h}, \tilde \alpha_{h})$.
It remains to check that our sequence of partitions $(\sA'_k, \sB'_k)$ satisfy the criterion of Masur and Smillie.
Consider statement (i), we need to show the total vertical holonomy of $\partial \sA'_k$ tends to zero as $k \to \infty$. This can be based on the observation that the total length of the boundary of the partitions $(\sA_n, \sB_n)$ of
the surface $(\tilde S_{\tau^{-n}_\ast(h)}, \tilde \alpha_{\tau^{-n}_\ast(h)})$ can be bounded from above independent of
$n$ and $\tau^{-n}(h)$. Indeed, for any double cover, the total length of all boundaries of all vertical cylinders is a finite constant independent of the cover. It is twice the corresponding constant for the base surface $(S_L,\alpha_L)$,
and the constant there is finite because the cylinders decay in size exponentially. We can choose $R \in V'$
as above so that $g^t A R^{-1} \tilde V'$ lies in the region $F_0'$ when $t_k<t<t_{k+1}$.
Select $t$ so that $g^t A R^{-1} \tilde V'$ lies in a fixed compact set of $F_0'$, which can be taken to be a neighborhood
of the convex core with a cusp neighborhood removed. Note then that by definition of $E$, the quantity
$E(g^t A R^{-1})\Psi$ is taken from a compact subset of $\SL(2,\R)/ \Psi$. So, in particular we get a uniform upper bound $L<\infty$ on the length of the boundary of the partition $(\sA^t, \sB^t)$ of the surface $g^t A(\tilde S_{h}, \tilde \alpha_{h})$. (This uses the identification of equation \ref{eq:identification}.) In particular,
the vertical component of this length is bounded from above by $L$. Pulling back via $g^{-t}$, we see
that the vertical component of the length of the common boundary of $\sA'_k$ and $\sB'_k$ is bounded from above by $e^{-t} L$, which tends to zero as $k \to \infty$ because there is a uniform lower bound on $t_{k+1}-t_k$. This verifies statement (i).
To prove statement (ii), we need to make use of the assumptions that $\lim_{k \to \infty} n_k=+\infty$
and $\lim_{m \to \infty} \tau^m_\ast(h)=\0$. By Lemma \ref{lem:limits to zero},
there is an $M \geq 0$ and an integer $j$ so that $m>M$ implies that $P\big(\tau^m_\ast(h)\big)=j+m,$
where $P$ denotes proximity; see equation \ref{eq:proximity}.
Since $\{n_k\}$ tends to $+\infty$, there is a $K$ so that $k>K$ implies $P\big(\tau^{-n_k}_\ast(h)\big)>1$.
Since the proximity is larger than one, $\tau^{-n_k}_\ast(h)(-1)=0$. This means that there are two vertical cylinders in the surface $(\tilde S_{\tau^{-n_k}_\ast(h)}, \tilde \alpha_{\tau^{-n_k}_\ast(h)})$ passing through lifts of the edge labeled $-1$. One of these cylinders lies in $\sA_{n_k}$ and the other lies in $\sB_{n_k}$. Thus when $k>K$,
we obtain upper and lower bounds on the area of $\sA_{n_k}$. Since the partition $(\sA'_k, \sB'_k)$
was obtained by pulling back along an area preserving map, the same bounds hold here. (We only need to verify statement (ii) holds only for all but finitely many $k$.)
In order to understand (iii), we need to investigate how our partition changes
when the geodesic $g^t A \tilde V'$ passes from a region $F_n$ to $F_{n+1}$. Consider a time
$t$ so that $g^t A \tilde V'$ lies in the common boundary between $F_n$ and $F_{n+1}$. This is the point at which the partition changes. First consider this point $g^t A \tilde V'$ as part of $F_n$. Then,
we build a partition $(\sA_{n}, \sB_{n})$ of the cover $(\tilde S_{\tau^{-n}_\ast(h)}, \tilde \alpha_{\tau^{-n}_\ast(h)})$ as described above,
and apply $g^{-t} E(g^t A R_n^{-1})$, where $R_n \in V'$ is any Veech group element carrying $F_0$ to $F_n$,
to partition our original surface.
Here $E(g^t AR_n^{-1}) \in \SL(2,\R)$ is determined by lifting a path joining the $M_0 \tilde V'$ to $g^t AR_n^{-1} \tilde V'$ within $F_0'$. Now we consider what happens when we consider $g^t A \tilde V'$ as part of $F_{n+1}$.
We construct a partition $(\sA_{n+1}, \sB_{n+1})$ of the cover $(\tilde S_{\tau^{-n-1}_\ast(h)}, \tilde \alpha_{\tau^{-n-1}_\ast(h)})$. We apply $g^{-t} E(g^t A R_{n+1}^{-1})$, where $R_{n+1}$ is some Veech group element carrying $F_0$ to $F_{n+1}$, to partition our original surface.
We need to understand the area of the symmetric difference of the pulled back partitions. It is equivalent to find the area of the symmetric difference between the partition $(\sA_{n}, \sB_{n})$ and the other partition of $(\tilde S_{\tau^{-n}_\ast(h)}, \tilde \alpha_{\tau^{-n}_\ast(h)})$ obtained as the image of $(\sA_{n+1}, \sB_{n+1})$ under
$$R_\ast=\big(g^{-t} E(g^t A R_n^{-1})\big)^{-1}g^{-t} E(g^t A R_{n+1}^{-1})=E(g^t A R_n^{-1})^{-1} E(g^t A R_{n+1}^{-1}).$$
The elements $E(g^t A R_n^{-1})$ and $E(g^t A R_{n+1}^{-1})$ are determined based on lifting paths $\gamma_{n}$ and $\gamma_{n+1}$
in $\SL_\pm(2,\R) /\tilde V'$ to paths $\tilde \gamma_n$ and $\tilde \gamma_{n+1}$ respectively.
The value of $R_\ast \in \SL(2,\R)$ can then be determined by lifting the path obtained by first following $\gamma_1$
and then following the translated path $E(g^t A R_n^{-1})(\gamma_2)$ backward. The result is a path which passes once through the common boundary between $F_0$ and $F_1$, and joins the equivalence class of the $M_0$ to $D(\rho \circ \psi) \tilde V'$. The value of $R_\ast$ is the endpoint of this path lifted to $\SL(2,\R)$, which we see lies in the double coset
$$\Psi D(\rho \circ \psi)\Psi \in \Psi \bs \SL(2,\R) / \Psi.$$
As the action of an element $\Psi$ does not change our partitions, we can choose to work with the simplest element
from our point of view, $R_\ast=D(\rho \circ \psi)$.
We need to estimate the area of the symmetric difference
$\sA_n \Delta R_\ast(\sA_{n+1})$.
Here, we interpret $R_\ast$ as an affine homeomorphism
$$R_\ast:(\tilde S_{\tau^{-n-1}_\ast(h)}, \tilde \alpha_{\tau^{-n-1}_\ast(h)}) \to (\tilde S_{\tau^{-n}_\ast(h)}, \tilde \alpha_{\tau^{-n}_\ast(h)})$$
which is characterized by its derivative and respects the labels of lifts of the basepoint. Observe that
the action of the matrix $R_\ast$ carries vertical cylinders to horizontal cylinders and preserves their widths.
It also respects the labeling of the lifts of the basepoints.
Suppose that the proximity $P\big(\tau^{-n-1}_\ast(h)\big)=p$. For any integer $e<0$ with $-e<p$,
we have $\tau^{-n-1}_\ast(h)(e)=0$, and thus the
vertical cylinder in $(\tilde S_{\tau^{-n-1}_\ast(h)}, \tilde \alpha_{\tau^{-n-1}_\ast(h)})$
starting in regions $\sR_0$ and passing through a lift of the edge labeled $e$ in Figure \ref{fig:ladder_surface}
lies in the subsurface $\sA_{n+1}$. Consider the union $U \subset \sA_{n+1}$ of all such vertical cylinders in $\sA_{n+1}$
with $-p<e<0$. We observe that because basepoint labels are preserved, $R_\ast(U) \subset \sA_n$, with each
vertical cylinder through $e$ being sent to a horizontal cylinder in $\sA_n$ passing through a vertical edge labeled $-e$. Let $\nu$ be the Lebesgue probability measure on $(\tilde S_{\tau^{-n}_\ast(h)}, \tilde \alpha_{\tau^{-n}_\ast(h)})$. We get the upper bound on the area of the symmetric difference
$$\nu \big(\sA_n \Delta R_\ast(\sA_{n+1})\big) \leq \nu(\sA_n)+\nu\big(R_\ast(\sA_{n+1})\big)-2 \nu(U) \leq 1 -2 \nu(U).$$
Observe that successively smaller cylinders in $(S_L,\alpha_L)$ decrease in area by a factor of $\varphi^2$.
It follows that there is a constant $\alpha>0$ so that $1-2 \nu(U)< \alpha \varphi^{-2 p}$. So, we see
$$\nu \big(\sA_n \Delta R_\ast(\sA_{n+1})\big)< \alpha \varphi^{-2 p},$$
where $p=P\big(\tau^{-n-1}_\ast(h)\big)$ as above.
Let $\mu$ be the normalized Lebesgue measure on the surface $A (\tilde S_h, \tilde \alpha_h)$.
The prior paragraph gives an upper bound on $\mu(\sA'_k \Delta \sA'_{k+1})$ when $n_{k+1}=n_k+1$
in terms of the proximity $p(n_{k+1})=P\big(\tau^{-n_k-1}_\ast(h)\big)$.
A similar bound holds for the case when $n_{k+1}=n_k-1$; there is a constant $\alpha'>\alpha$ so that
$$\mu(\sA'_{k} \Delta \sA'_{k+1})<\alpha' \varphi^{-2 p(n_{k+1})}$$
regardless if $n_{k+1}$ equals $n_k+1$ or $n_k-1$.
Now we will consider the total sum of the symmetric differences to verify statement (iii). Let
$V_N=\# \{k:~n_k=N\}$ for integers $N$. By assumption, each $V_N$ is finite, and $V_N=0$ for $N<N_0$
for some $N_0 \in \Z$. Then,
$$\sum_k \mu(\sA'_{k} \Delta \sA'_{k+1}) \leq \sum_{N=N_0}^\infty V_N \varphi^{-2 p(N)}.$$
Now we incorporate the assumption that $\lim_{N \to +\infty} \tau^N_\ast(h)=\0$.
From Lemma \ref{lem:limits to zero},
we obtain an $M \geq 0$ and an integer $j$ so that $N>M$ implies that $P\big(\tau^N_\ast(h)\big)=j+N.$
Let $X < \infty$ be the sum over the terms with $N \leq M$, then we see that
$$\sum_k \mu(\sA'_{k} \Delta \sA'_{k+1}) \leq X + \sum_{N=M+1}^\infty V_N \varphi^{-2 (j+N)}.$$
An application of the root test tells us this sum converges if $\limsup_{N \to \infty} V_N^{1/N}<\varphi^2$.
Since this was a hypothesis, we have verified statement (iii). As an application of Theorem \ref{thm:Masur-Smillie},
we see that the translation flow on $A (\tilde S_h, \tilde \alpha_h)$ is not ergodic.
\end{proof}
\end{document}
|
\begin{document}
\author{Mohammadreza~Chamanbaz,~\IEEEmembership{Member,~IEEE,}
Fabrizio Dabbene,~\IEEEmembership{Senior~Member,~IEEE,}, and
Constantino Lagoa,~\IEEEmembership{Member,~IEEE}
\thanks{M. Chamanbaz is with iTrust Center for Research in Cyber Security, Singapore University of Technology and Design, 8 Somapah Road Singapore 487372, (E-mail: [email protected], [email protected]).}
\thanks{F. Dabbene is with CNR-IEIIT, Corso Duca degli Abruzzi 24, 12129 Torino, Italy (E-mail:[email protected]). }
\thanks{C. Lagoa is with the Department of Electrical Engineering and Computer Science, Penn State University, University Park, PA 16802 USA, (E-mail: [email protected]).}
\thanks{This work was supported in part by the CNR International Joint Lab COOPS, the National Science Foundation under grant \mbox{CNS-1329422}, and the Singapore National Research Foundation (NRF) grant
under the ASPIRE project, grant No NCR-NCR001-040.}}
\markboth{IEEE Transactions on Control of Network Systems}{}
\title{\LARGE \bf{Probabilistically Robust AC Optimal Power Flow}
\begin{abstract}
The increasing penetration of renewable energy resources, paired with the fact that load can vary significantly, introduce a high degree of uncertainty in the behavior of modern power grids. Given that classical dispatch solutions are ``rigid," their performance in such an uncertain environment is in general far from optimal. For this reason, in this paper, we consider AC optimal power flow (AC-OPF) problems in the presence of uncertain loads and (uncertain) renewable energy generators. The goal of AC-OPF design is to guarantee that controllable generation is dispatched at minimum cost, while satisfying constraints on generation and transmission for almost all realizations of the uncertainty.
We propose an approach based on a randomized technique recently developed, named \textit{scenario with certificates},
which allows us to tackle the problem without the conservative parameterizations on the uncertainty used in currently available approaches.
The proposed solution can exploit the usually available probabilistic description of the uncertainty and variability, and provides solutions with \textit{a-priori} probabilistic guarantees on the risk of violating the constraints on generation and transmission.
\end{abstract}
\section{Introduction}
Modern power grids are characterized by increasing penetration of renewable energy sources, such as solar photovoltaic and wind power.
This trend is expected to increase in the near future, as also testified by strict commitments to large renewable power penetration being made by major countries worldwide; e.g., see \cite{SET2016,GWEC2016}.
While the advantages of renewable energy in terms of environmental safeguard are indisputable, its introduction does not come without a cost. Indeed, renewable energy generation technologies are highly variable and not fully dispatchable, thus imposing novel challenges to the existing power system operational paradigm. As discussed in e.g. \cite{Bienstock2014}, when uncontrollable resources fluctuate, classical optimal power flow (OPF) solutions can provide very inefficient power generation policies, that result in line overloads and, potentially, cascading outrages.
Despite the increasingly larger investments, which are costly and subject to several regulatory and policy limitations,
power outages due to the uncertainty introduced by renewable power generation still occur frequently.
This situation clearly shows that a strategy based only on investments in technological improvements of the transmission lines and controllable generation capacity---as those discussed e.g.\ in \cite{
Conejo2010}---is not
sufficient anymore. Instead, radically new dispatch philosophies need to be devised, able to cope with the increasing \textit{uncertainty}, due to unpredictable fluctuations in renewable output and time-varying loads.
Indeed, classical OPF dispatch is typically computed based on simple predictions of expected loads and generation levels for the upcoming time window. Although these predictions can be fairly precise for the case of traditional generators and loads, they may be highly unreliable in the case of renewable generators, thus explaining its failure in these latter situations.
It follows that one of the major problems in today's power grids is the following: \textit{Given the high level of uncertainty introduced by renewable energy sources, design a dispatch policy that {{\text{i}}t i)} minimizes generation costs and {{\text{i}}t ii)} does not violate generation and transmission constraints for all admissible values of renewable power and variable demand. In other words, one would like to design an optimal dispatch policy that is robust against uncertainty.}
However, such robust policies might be very conservative. Power networks can tolerate temporary violations of their generation and transmission constraints. Moreover, being robust with respect to any possible value of the uncertainty may lead to very inefficient policies, since some scenarios are very unlikely. Hence, in this paper we take a different approach: instead of requiring that the network constraints are satisfied for all possible values of uncertainty, we allow for a small well-defined risk of constraint violation.
More precisely, we start by assuming that one can adapt the power generated by conventional generators, based on real-time information about current values of renewable power generation and demand. Under this assumption, we aim at minimizing cost of power generation while meeting demand and allowing a small well defined risk of violating network constraints.
We re-formulate the problem by highlighting the fundamental difference between control and state variables. Then, to approximate the solution of the resulting (complex) optimization problem we leverage recent results on convex relaxations of the optimal power flow problem~\cite{Lavaei:2012},
and show that such a relaxation is exact for a restrictive class of networks even in the presence of uncertainty.
Finally, together with the above mentioned relaxation, we use a novel way of addressing probabilistic robust optimization problems known as \emph{scenario with certificates} (SwC) \cite{SWC-TAC} to optimize generation cost subject to a small risk of network failure. In this way, we derive a convex problem that i) is efficiently solvable, ii) provides a good approximation of the optimal power generation under the above mentioned risk constraints and iii) for some special classes of networks, it provides an exact solution.
\subsection{Previous Results}
\subsubsection{Nominal OPF}
The optimal power flow problem is known to be \mbox{NP-hard} even in the absence of uncertainty; see e.g.,~\cite{NP-Hard-1,FERC-1,NP-Hard-2} and references therein. Hence, several numerical approaches propose approximations of OPF, based e.g.\ on Newton methods~\cite{DommelTinney1968}, interior point based methods~\cite{YanQuintana1999} or global optimizations heuristics \cite{Abido2002,LaiMaYokoyamaEtAl1997}.
In particular, several approximations have been introduced to recover convexity and make the problem numerically tractable. The most common is the so-called DC approximation, see \cite{Coffrin:2014} and references therein, in which the AC OPF problem is linearized. However, the solution is in general sub-optimal and, more importantly, it may not be feasible, in the sense that it may not satisfy the original nonlinear power flow equations.
Also, as noted in \cite{Coffrin:2014}, the fact that DC approximation fixes voltage magnitudes and ignores reactive power, makes the solution not applicable in several important practical situations.
Motivated by the above considerations, semidefinite programming (SDP) relaxations of the AC OPF problem have been recently introduced to alleviate the computational burden, in \cite{Jabr:06} for radial networks and in \cite{Bai:08} for general networks---see \cite{Low:2014a} for a detailed review. These convex relaxations
have recently received renewed interest thanks to \cite{Lavaei:2012}, that analyzes the optimality properties of the relaxation, showing how in many practical situation these relaxations turn out to be non-conservative---in the sense that the solution of the relaxed problem coincides with that of the original non-convex one.
This result has sparked an {interesting literature} analyzing specific cases in which the SDP relaxations can be proven to be exact, see e.g. \cite{Farivar:13}, and showing how graph sparsity can be exploited to simplify the SDP relaxation of OPF, see e.g.\ \cite{Bai:11}.
However, while these works have reached a good level of maturity, (see for instance the recent two-part tutorial \cite{Low:2014a,Low:2014b}),
there is still very little research analyzing if and how these relaxations could be extended to the problem considered in this paper, namely AC OPF in the presence of possibly large and unpredictable uncertainties, as detailed next.
\subsubsection{DC-based approaches to uncertain OPF}
Most literature on uncertain OPF gets around the nonlinearities by recurring to the DC-based approximation.
These assumptions reduce the optimization problem to a quadratic program subject to uncertain linear equalities and inequalities, which still represents a challenging problem, at least for general probability distributions.
First approaches in this direction have been based on scenario-tree generation methods, see for instance \cite{Yong2000}. These techniques suffer from severe computational complexity limitations, and do not offer theoretical guarantees on the probability of satisfaction of the constraints of the found solution.
In the case uncertainties are assumed to be Gaussian, the problem can be written in closed form \cite{Andersson2013}, or is amenable to a second-order cone-program \cite{Bienstock2014}, for which efficient solutions exist. This approach has been extended to ambiguous densities introducing the so-called robustified chance constraints in \cite{Lubin2016}. Similar ideas are at the basis of the approaches in \cite{Morari2015,Xie2018}, which consider distributionally robust approaches. That is, the
solution has to be valid for all uncertainties whose probability density functions (pdfs) belong to a
family of distribution functions sharing the same mean and variance, leading to the so-called conditional value at risk (CVar) optimization, which is again a convex problem.
The problem becomes much more difficult for general uncertainty distributions (indeed, it is known that the distribution of wind power is not Gaussian \cite{hodge_wind_2012}). In this case, a very promising approach is the application of recent results based on the so-called scenario approach, which are based on random generation of uncertainty samples. This is the approach followed for instance in \cite{ETH-PMAPS,ETH-TPS,ETH-Springer,ETH-PSCC}.
\subsubsection{Convex relaxation approaches for uncertain OPF}
The limitations inherent to the DC approximation have motivated a few recent approaches to uncertain OPF, which employ more sophisticated relaxations able to capture also the reactive components of the power equations.
The work \cite{Perninge2013} makes use of second-order approximations of the stability boundary to approximate the probability of line violations.
In \cite{ETH-AC} the SDP based relaxation introduced in \cite{Lavaei:2012} is exploited to derive a solution based on the scenario approach.
However, in order to guarantee solvability of the robust problem, the authors need to parameterize the dependence of the state variables on the uncertainty, see Remark \ref{crac-remark} for a detailed discussion.
Inspired by these recent works, in this paper we provide a less conservative approach, that does not require dependent variable parameterization. Instead, we exploit a recently developed approach to probabilistic robust optimization \emph{scenario with certificates} \cite{SWC-TAC} to develop less conservative, efficient relaxations of the OPF in the presence of uncertainty. As compared to previous results, in this paper we provide an approach to the robust optimal power flow problem that: i) is exact for a (small) class of networks;
ii) is computationally more efficient than other approaches in the literature and
iii) provides solutions leading to low risk of network failure even for networks that do not satisfy the theoretical requirements, as exemplified in the simulations provided.
\subsection{The Sequel}
The paper is organized as follows: In Section~\ref{sec:prob formulation} we precisely formulate the AC-OPF problem. We also describe the adopted dispatch policy to be used in real time to cope with uncertainty. Then, we introduce a precise formulation of the Robust AC-OPF problem, which divides the set of optimization variables into two distinct classes: i) \emph{independent/control variables} (those that can be controlled by the operator), and ii) \textit{dependent/state variables} (representing the state of the system).
This reformulation represents one of the main contributions of our work, allowing to represent in a non-conservative way the problem of AC-OPF design in the presence of uncertainty.
Also, in Section~\ref{sec:relaxation}, we show how this reformulation can be combined with the convex relaxation of~\cite{Lavaei:2012}, and we formally prove that this allows to recover its properties in terms of exactness of the relaxation in some special cases.
Moreover, in Section \ref{sec:scenario} we show how this new formulation directly translates in the scenario with certificates paradigm. Numerical examples illustrating the performance of the proposed approach are provided in Section~\ref{sec:examples}. Finally, in Section~\ref{sec: conclusions}, some concluding remarks are presented.
\section{Optimal Power Flow Allocation Problem under Uncertainty}
\label{sec:prob formulation}
In this section, we briefly summarize the adopted power flow model, which takes into explicit account load uncertainties and variable generators, and we formalize the ensuing optimization problem, arising from the necessity of robustly guaranteeing that safety limits are not exceeded.
{
\subsection{Robust AC-OPF}
We consider a power network with graph representation
$
\mathbb{G}=\{\mathcal{N},\mathcal{L}\},
$
where $\mathcal{N}\doteq\{1,2,\ldots,n\}$ denotes the set of buses (which can be represented as nodes of the graph), and $\mathcal{L}\subseteq\mathcal{N}\times\mathcal{N}$ denotes the set of electrical lines connecting the different buses in the network (represented as edges of the graph). The set of conventional generator buses is denoted by $\mathcal{G}\subseteq\mathcal{N}$, and its cardinality is $n_g$. As a convention, it is assumed that the bus indices are ordered so that the first $n_g$ buses are generators, i.e. $\mathcal{G}\doteq\{1,2,\ldots,n_g\}$.
Each generator (we assume for ease of notation that no more than one generator is present on each generator bus) connected to the bus $k{\text{i}}n\mathcal{G}$ provides complex power
$
\bar{P}^G_{k}+ \bar{Q}^G_{k} {\text{i}}$,
where $\bar{P}^G_{k}$ is the active power generated by $k$-th generator, and $\bar{Q}^G_{k}$ is the corresponding reactive power. Goal of the network manager is to guarantee that the output of generators is such that the network operates safely and, if possible, at minimal cost.
We now elaborate on the different constraints that should be satisfied in order
for the network to operate safely.
In the framework considered in this paper, we assume both the existence of
renewable energy sources and uncertain load.
A renewable energy generator connected to bus $k{\text{i}}n\mathcal{N}$ provides an
uncertain complex power
\begin{equation}\label{eq: renewable uncertainty}
P^R_{k}(\delta^R_{k})+Q^R_{k}(\delta^R_{k}){\text{i}}=P^{R,0}_{k}+Q^{R,0}_{k}{\text{i}}+\delta^R_{k},
\end{equation}
with $P^{R,0}_{k}+Q^{R,0}_{k}{\text{i}}$ being the nominal (predicted) power generated
by the \FD{RES}\footnote{We use the convention that $P^R_{k}=0$,
$Q^R_{k}=0$ if no renewable generator is connected to node $k$.}, and
$\delta^R_{k}{\text{i}}n\boldsymbol{\Delta}^R_{k}\subset\mathbb{C}$ representing an
uncertain complex fluctuation, which mainly depends on the environmental
conditions, such as wind speed in the case of wind generators.
The uncertain demand in bus $k{\text{i}}n\mathcal{N}$ is also represented as
\begin{equation}\label{eq: load uncertainty }
P^L_{k}(\delta^L_{k})+Q^L_{k}(\delta^L_{k}){\text{i}}=P^{L,0}_{k}+Q^{L,0}_{k}{\text{i}}+\delta^L_{k}
\end{equation}
where $P^{L,0}_{k}$ and $Q^{L,0}_{k}$ denote the expected active and reactive
load and $\delta^L_{k}{\text{i}}n\boldsymbol{\Delta}^L_{k}\subset\mathbb{C}$ is the
complex fluctuation in the demand at bus $k{\text{i}}n\mathcal{N}$.
The support set is the point $\{0\}$ if no uncertainty (i.e. no renewable
generator or variable load) is present in bus $k$.
To simplify the notation, we collect the different sources of uncertainty by
introducing \textit{uncertainty vector}
$
\boldsymbol{\delta} \doteq
[\delta^L_{1}\,\cdots\,\delta^L_{n}\,\delta^R_{1}\,\cdots\,\delta^R_{n}]^T,
$
which varies in the set
$
\boldsymbol{\Delta} \doteq \boldsymbol{\Delta}^L_{1} \times \cdots
\times\boldsymbol{\Delta}^L_{n} \times
\boldsymbol{\Delta}^R_{1} \times \cdots \times \boldsymbol{\Delta}^R_{n}.
$
To deal in a rigorous way with the uncertainties introduced above, we adopt a
modification of the so-called frequency control (primary and secondary
control), similar to that discussed in \cite{Bienstock2014} in the
context of DC-OPF. In classical OPF,
this approach is used to distribute to the generators the difference between
real-time (actual) and predicted demand, through some coefficients which are
generator specific. However, these coefficients are in general decided \textit{a-priori}
in an ad-hoc fashion.
This approach worked well in cases where the amount of power mismatch was not
significant; however, once renewable generators are in the power network, this
difference may become large, thus leading to line overloads in the network.
The approach presented in \cite{Bienstock2014} specifically incorporates
these distribution parameters in the OPF optimization problem. In the setup
proposed in this paper, we follow a similar strategy, and formally introduce a
\textit{deployment vector}
$
\boldsymbol{\alpha}\doteq[\alpha_1,\ldots,\alpha_{n_g}]^T,
$
with
$\sum_{k{\text{i}}n\mathcal{G}}\alpha_k=1$, $\alpha_k \geq 0$ for all $k {\text{i}}n
\mathcal{G}$,
whose purpose is to distribute among the available generators
the power mismatch created by the uncertain generators and loads.
To the best of our knowledge, the use of a deployment vector was originally
introduced in the context of DC-approximations in
\cite{ETH-PMAPS} and \cite{ETH-Springer}, where it is referred to as ``distribution vector".
The same concept is present in many other works under different terminologies, such as
``participation factor" in \cite{Jabr2013}, ``corrective control" in \cite{Jabr2015}, or ``affine control" in
\cite{Morari2013}.
During operation, the active generation output of each generator is adapted
according to the realizations of the uncertain loads and renewable power (which are
assumed to be measured on-line) as follows
\begin{align}\label{eq: affine control}
\bar{P}^G_{k} &=
P^G_{k}+\alpha_k\left(\sum_{j{\text{i}}n\mathcal{N}}\text{Re}\{\delta^L_{j}\}-\sum_{k{\text{i}}n\mathcal{R}}\text{Re}\{\delta^R_{k}\}\right)\\
&= P^G_{k}+\alpha_k \mathbf{s}^T \text{Re}\{\boldsymbol{\delta} \},\quad \forall k {\text{i}}n \mathcal{G}
\nonumber
\end{align}
with $\mathbf{s}^T\doteq[\mathbf{1}_n^T,\,-\mathbf{1}_n^T]$.
Now, consider the line $(l,m){\text{i}}n \mathcal{L}$, which is the line connecting buses $l$ and $m$. Let $y_{lm}$ be the (complex) admittance of the line and $V_k=|V_k|\angle{\theta_k}$ be the (complex) voltage at bus $k$, with magnitude $|V_k|$
and phase angle $\theta_k$. Then, the following \textit{Balance Equations} should be satisfied at all times
{\small \begin{subequations}\label{eq:unc_bal}
\begin{align}
\label{eq:unc_act_bal}
&P^G_{k}+\alpha_k \mathbf{s}^T \text{Re}\{\boldsymbol{\delta} \} + P^R_{k}(\boldsymbol{\delta} ) -
P^L_{k}(\boldsymbol{\delta} ) = \\ \nonumber
&\hspace{1 in}\sum_{l{\text{i}}n\mathcal{N}_k}
\text{Re}\left\{V_k(V_k-V_l)^*y_{kl}^*\right\}, \quad \forall k {\text{i}}n
\mathcal{N}\\
\label{eq:unc_reac_bal}
&Q^G_{k} + Q^R_{k}(\boldsymbol{\delta} ) - Q^L_{k}(\boldsymbol{\delta} ) = \\ \nonumber & \hspace{1
in}\sum_{l{\text{i}}n\mathcal{N}_k}
\text{Im}\left\{V_k(V_k-V_l)^*y_{kl}^*\right\},\quad \forall k {\text{i}}n
\mathcal{N},\\
&\mathbf{1}^T\boldsymbol{\alpha}=1 \label{eq:unc_bal alpha}
\end{align}
\end{subequations}}
where $\mathcal{N}_k$ is the set of all neighboring buses directly connected to bus $k$.
Also, at generator bus $k{\text{i}}n \mathcal{G}$, one has the so-called \textit{Power Constraints}, which restrict the active and reactive power, and have the form
\begin{subequations}\label{eq:unc power constraints}
\begin{align}
\label{eq:unc power constraints active generation bound}
&P_{k\,\min}\leq P^G_{k}+\alpha_k \mathbf{s}^T \text{Re}\{\boldsymbol{\delta} \} \leq
P_{k\,\max},\quad \forall k {\text{i}}n \mathcal{G} \\
\label{eq:unc power constraints reactive generation bound}
&Q_{k\,\min}\leq Q^G_{k} \leq Q_{k\,\max},\quad \forall k {\text{i}}n \mathcal{G},\\
&\alpha_k \geq 0, \quad \forall k {\text{i}}n \mathcal{G}.
\end{align}
\end{subequations}
Finally, the voltages should satisfy the following \textit{Voltage Constraints}
\begin{subequations}\label{eq: voltage constraints}
\begin{align}
\label{eq: voltage bound 1}
&V_{k\,\min}\leq |V_k|\leq V_{k\,\max}, \quad \forall k {\text{i}}n \mathcal{N}, \\
\label{eq: voltage bound 2}
& |V_l-V_m|\le \Delta V_{lm}^{\max},
\quad \forall (l,m) {\text{i}}n \mathcal{L}.
\end{align}
\end{subequations}
In \cite{Lavaei:2012}, the constraints in \eqref{eq: voltage bound 2} have been proven to be practically equivalent to the more classical bounds
\[
\left | V_l(V_l-V_m)^*y_{lm}^*\right | \le S_{lm\, \max},\quad\forall(l,m){\text{i}}n\mathcal{L},
\]
where $S_{lm\, \max}$ is the maximum apparent power flow which can path through the line $(l,m){\text{i}}n\mathcal{L}$.
\subsection{Control and State Variables}
\label{sec:control-state}
In the power-systems literature, the variables appearing in formulation above are usually divided into two classes.
Indeed, already in \cite{Carpentier:1979aa}, the distinction between control and state variables is explicitly made. \emph{Control variables} are those used by the network operator to set the operating condition of the network. \emph{State variables} are \emph{dependent} variables that represent the state of a power network, see also the recent survey \cite{Capitanescu:2016}
In particular, control and state variables are defined differently depending on the type of bus. In a generator bus $k{\text{i}}n\mathcal{G}$, usually referred to as \textit{PV bus} (see e.g.\ \cite[Remark 1]{Low:2014a}) mean active power $P^G_{k}$ of the generator, deployment factor $\alpha_k$ and magnitude $|V_k|$ of the complex bus voltage represent the control variables,
while phase angle $\theta_k$ of bus voltage and generator reactive power $Q^G_{k}$ are the state variables. In a load bus, or \textit{PQ bus}, the active and reactive power of the load $P^L_{k},Q^L_{k}$ and the active and reactive power of the renewable generators $P^R_{k},Q^R_{k}$ are given (their values are known to the network operator)
while magnitude and phase angle of bus voltage $|V_k|,\theta_k$ are state variables. A node to which both a generator and loads are connected is to be considered as a generator bus.
{\begin{remark}[Slack bus]
In power flow studies, usually bus $0$ is considered as \emph{reference} or \emph{slack} bus. The role of the slack bus is to balance the active and reactive power in the power grid. The slack bus must include a generator. In most cases, the voltage magnitude and phase are fixed at the slack bus, whereas active and reactive generator power are variables. Therefore, in a slack bus there is no control variable, while active and reactive generator power $P^G_k$ and $Q^G_k$ are state variables. For simplicity of notation, we do not include slack bus in the formulation,. However, its introduction is seamless, and slack bus is considered in the numerical simulation.
\end{remark}
}
To emphasize the inherent difference between control and state variables, we introduce the notation
\begin{align}\label{eq: control variables}
\mathbf{u} &\doteq \{P^G_1\, \cdots \,P^G_{n_g},
|V_1| \cdots |V_{n_g}|, \alpha_1,\ldots,\alpha_n\}\\
\label{eq: state variables}
\mathbf{x}_{\boldsymbol{\delta} } & \doteq \{Q^G_1\, \cdots \,Q^G_{n_g},
|V_{n_g+1}|,\ldots,|V_n|,\theta_1,\ldots,\theta_n\}
\end{align}
Note that in the notation above, we emphasize the fact that the state variables
depend on the realization of the uncertainty.
We are now ready to provide a formal statement of the \emph{robust} version of the optimal power flow problem. To this end, denote by $g(\mathbf{u},\mathbf{x}_{\boldsymbol{\delta} },\boldsymbol{\delta} ) = 0$
the uncertain equality constraints collected in \eqref{eq:unc_bal}, and by $h(\mathbf{u},\mathbf{x}_{\boldsymbol{\delta} },\boldsymbol{\delta} ) \le 0$ the uncertain inequalities collected in \eqref{eq:unc power constraints} and \eqref{eq: voltage constraints}.
\noindent{\bf Robust AC-OPF}
\begin{eqnarray} \label{eq: Robust AC-OPF}
\underset{\mathbf{u}}{\min} \
\ f(\mathbf{u})
\ \text{s.t.:}&
\text{for all } \boldsymbol{\delta} {\text{i}}n\boldsymbol{\Delta} ,
\text{ there exist } \mathbf{x}_{\boldsymbol{\delta} } \text{ such that }\ \ \ \\ \nonumber
&\qquad g(\mathbf{u},\mathbf{x}_{\boldsymbol{\delta} },\boldsymbol{\delta} ) = 0 \text{ and } h(\mathbf{u},\mathbf{x}_{\boldsymbol{\delta} },\boldsymbol{\delta} ) \le 0.
\end{eqnarray}
}
In the above formulation of the robust OPF problem, seemingly first introduced in \cite{Zhang2011,Wada:2014}, the objective is to optimize the values of the nominal generated power, the bus voltage magnitude at generator node and the deployment vector so that i) the network operates safely for all values of the uncertainty and ii) the generation cost of the network is minimized.
In fact, if a solution to the problem above exists, we guarantee that for any admissible uncertainty, there exists a network state $\mathbf{x}_{\boldsymbol{\delta} }$ satisfying the operational constraints.
Note that problem \eqref{eq: Robust AC-OPF} is computationally very hard.
Indeed, even when no uncertainty is present, the AC-OPF problem is known to be non-convex, due to the presence of the nonlinear (quadratic) equality constraints \eqref{eq:unc_act_bal} and \eqref{eq:unc_reac_bal}, and the nonconvex line constraints \eqref{eq: voltage constraints}.
On top of that, these nonconvex constraints should be guaranteed over all the uncertainty set $\boldsymbol{\Delta} $, which is an infinite set (actually, uncountable). Hence, the optimization problem \eqref{eq: Robust AC-OPF} is a nonlinear/nonconvex semi-infinite optimization problem.
We remark that taking a robust approach, i.e.\ enforcing the constraints in~\eqref{eq: Robust AC-OPF} for ``all'' possible values of the uncertain parameters, is in many cases excessive, and leads to conservative results, with consequent degradation of the cost function (in our case, higher generation cost).
Hence, in the remainder of this paper, we follow a chance-constrained approach, in which a probabilistic description of the uncertainty is assumed, and a solution is sought which is valid for the entire set of uncertainty except for a (small) subset having probability measure smaller than a desired (small) risk level~$\varepsilon$. This approach is suitable for problems where ``occasional'' violation of constraints can be tolerated. One can argue that this is the case in power networks, since violation of line flow constraints does not necessarily lead to immediate line tripping. Rather, the line gradually heats up until a critical condition is reached and only then the line is disconnected.
Therefore, if line overload happens with low probability, this will not lead to line tripping nor it will damage the network.
Also, in the robust case, one clearly needs to assume that the uncertainty set $\boldsymbol{\Delta} $ is bounded, and to know this bound. In a probabilistic setup, however, one can also consider cases where $\boldsymbol{\Delta} $ is unbounded, that is where the distribution of $\boldsymbol{\delta} $ has infinite support, as e.g.\ the Gaussian case.
Formally, in the sequel, we assume that the uncertainty vector $\boldsymbol{\delta} $ is random with possibly unbounded support $\boldsymbol{\Delta} $. Then, given a (small) risk level~$\varepsilon{\text{i}}n(0,1)$, the chance constrained version of the optimal power flow problem is stated as follows
\noindent
\textbf{Chance-constrained AC-OPF}
\begin{eqnarray} \label{eq: OPF CCP }
\underset{\mathbf{u}}{\min}
\ f(\mathbf{u})
\ \text{s.t.:}&
\Pr\Big\{\boldsymbol{\delta} {\text{i}}n\boldsymbol{\Delta} \text{ for which }
\nexists\, \mathbf{x}_{\boldsymbol{\delta} } \text{ such that } \ \ \ \ \ \ \\ \nonumber
& g(\mathbf{u},\mathbf{x}_{\boldsymbol{\delta} },\boldsymbol{\delta} ) = 0 \text{ and } h(\mathbf{u},\mathbf{x}_{\boldsymbol{\delta} },\boldsymbol{\delta} ) \le 0 \Big\}\leq \varepsilon. \nonumber
\end{eqnarray}
We remark that, from a numerical point of view, the above chance-constrained optimization problem is not tractable. On one hand, the original problem \eqref{eq: Robust AC-OPF}, which does not have probabilistic constraints, is, as mentioned before, non-convex. On the other hand, the presence of probabilistic constraints in \eqref{eq: OPF CCP } requires the solution of hard multi-dimensional integration problems.
\FD{To circumvent these numerical difficulties, in the next section we introduce an integrated solution approach that allows to i) relax the non-convex problem \eqref{eq: Robust AC-OPF} to a convex one and ii) derive randomized algorithms for solving the chance constraint optimization problem.}
\section{Efficient Numerical Relaxation}
\label{sec:relaxation}
\subsection{Approximation of Non-Convex Terms}
In this subsection, we extend the relaxation technique discussed in \cite{Bai:08,Lavaei:2012} to the robust case circumventing the non-convexity associated with the optimal power flow problem \eqref{eq: Robust AC-OPF}. We stress that the only source of non-convexity of \eqref{eq: Robust AC-OPF}, is due to non-linear terms $V_kV_l$'s appearing in \emph{Balance Equations} \eqref{eq:unc_bal}, and \emph{Voltage Constraints} \eqref{eq: voltage constraints}. However, the quadratic constraints can be reformulated as linear ones by introducing a new variable $\mathbf{W}=\mathbf{V}\mathbf{V}^*$ where $\mathbf{V}$ is the vector of complex bus voltages $\mathbf{V} \doteq[V_1,\ldots,V_n]^T$. In order to replace $\mathbf{V}\mathbf{V}^*$ with the new variable $\mathbf{W}$, two additional constraints need to be included: i) the matrix $\mathbf{W}$ needs to be positive semi definite, i.e. the following \textit{Positivity Constraint} should hold
$\mathbf{W}\succeq 0,$
and ii) its rank should be one, i.e.\ the following \textit{Rank Constraint} should hold
$\text{rank}\{\mathbf{W}\}=1.$
An important observation is that, by introducing matrix $\mathbf{W}$, the only source of nonconvexity is captured by the rank constraint. Indeed, as shown first in \cite{Lavaei:2012} and subsequently in \cite{madani2015convex}, in most cases this constraint can be dropped without affecting the OPF solution.
To formally define the \textit{convexified version} of the Robust AC-OPF, we note that bus voltage $\mathbf{V}$ appears in \emph{Voltage Constraints} \eqref{eq: voltage constraints} and \emph{Balance Equations} \eqref{eq:unc_bal}. Therefore, these constraints are redefined in terms of the new variable $\mathbf{W}$ as
\begin{subequations}\label{eq: voltage constraints W}
\begin{align}
\label{eq: voltage bound W}
& (V_{k\,\min})^2\leq W_{kk}\leq (V_{k\,\max})^2, \,\, \forall k {\text{i}}n \mathcal{N}\\
\label{eq: voltage bound 2W}
& W_{ll}+W_{mm}-W_{lm}-W_{ml}\le (\Delta V_{lm}^{\max})^2,\,\,\forall(l,m){\text{i}}n\mathcal{L}
\end{align}
\end{subequations}
\small \begin{subequations}\label{eq:unc_bal W}
\begin{align}
\nonumber
&P^G_{k}+\alpha_k \mathbf{s}^T \text{Re}\{\boldsymbol{\delta} \} + P^R_{k}(\boldsymbol{\delta} ) - P^L_{k}(\boldsymbol{\delta} ) =\\ \label{eq:unc_act_bal W}
& \qquad\sum_{l{\text{i}}n\mathcal{N}_k} \text{Re}\left\{(W_{kk}-W_{kl})^*y_{kl}^*\right\} \\% \quad \forall k {\text{i}}n \mathcal{N}\\
\label{eq:unc_reac_bal W}
&Q^G_{k} + Q^R_{k}(\boldsymbol{\delta} ) - Q^L_{k}(\boldsymbol{\delta} ) =
\sum_{l{\text{i}}n\mathcal{N}_k} \text{Im}\left\{(W_{kk}-W_{kl})^*y_{kl}^*\right\},~\forall k {\text{i}}n \mathcal{N}
\end{align}
\end{subequations}
\normalsize
respectively.
Recalling that the first $n_g$ buses are generator buses and the remaining ones are load buses, we observe that the matrix $\mathbf{W}$ can be written as
\small
\begin{equation}
\mathbf{W}=\mathbf{V}\mathbf{V}^* = \left[\begin{array}{cccccc}
|V_{1}|^2 & V_{1}V_{2}^* & \ldots & V_{1}V_{n_g}^* & \ldots & V_{1}V_{n}^* \\
& |V_{2}|^2 & V_{2}V_{3}^* & \ldots & \ldots & V_2V_n^*\\
& & \boldsymbol{\delta} ots & & & \\
& & & |V_{n_g}|^2 & & \\
& & & & \boldsymbol{\delta} ots & \\
& & & & & |V_{n}|^2 \\
\end{array}\right].
\end{equation}
\normalsize
It is immediately noticed that some elements of $\mathbf{W}$ involve the control variables $|V_{1}|,\ldots,|V_{n_g}|$,
while others are dependent variables corresponding to the voltage magnitude $|V_{n_g+1}|,\ldots,|V_n|$ at non-generator nodes, and the voltage phases $\theta_1,\ldots,\theta_n$.
In order to distinguish between control and state variables appearing in it, we ``decompose'' $\mathbf{W}$ into the sum of two submatrices $\mathbf{W}^\mathbf{u}$ and $\mathbf{W}^\mathbf{x}$ as follows
\begin{equation} \label{eq: W_u}
\mathbf{W}^\mathbf{u} \doteq \text{diag}(|V_1|^2, \ldots,|V_{n_g}|^2, 0,\ldots),
\mathbf{W}^\mathbf{x} \doteq \mathbf{W}-\mathbf{W}^\mathbf{u}.
\end{equation}
In this decomposition, we have a matrix $\mathbf{W}^\mathbf{u}$ that includes the diagonal elements of $\mathbf{W}$ corresponding to the generator nodes only, while the remaining elements of $\mathbf{W}$ are collected in $\mathbf{W}^\mathbf{x}$. With this in mind, the control and state variables are redefined as
$\mathbf{u} \doteq \{\mathbf{P}^{G}, \boldsymbol{\alpha},\mathbf{W}^\mathbf{u}\}$
and
$\mathbf{x}_{\boldsymbol{\delta} } \doteq \{\mathbf{Q}^G,\mathbf{W}^\mathbf{x} \}$
respectively.
With this notation settled, we are in the position to formally introduce the convexified version of the robust AC optimal power flow problem as follows\\
\vskip -0.1in
\noindent{\bf Convexified Robust AC-OPF (CR-AC-OPF)}
{\small
\begin{align} \label{eq: CR-AC-OPF}
&\underset{\mathbf{P}^{G}, \boldsymbol{\alpha},\mathbf{W}_\mathbf{u}}{\text{minimize }}
\sum_{k{\text{i}}n\mathcal{G}} f_k(P^G_{k})\\ \nonumber
&\text{subject to: }
\text{for all } \boldsymbol{\delta} {\text{i}}n\boldsymbol{\Delta} ,
\text{ there exist } \mathbf{Q}^G=\mathbf{Q}^G(\boldsymbol{\delta} ),
\mathbf{W}^\mathbf{x}=\mathbf{W}^\mathbf{x}(\boldsymbol{\delta} ) \\\nonumber
&\text{such that } \\ \nonumber
&\quad \nonumber
\mathbf{W}=\mathbf{W}^\mathbf{u}+\mathbf{W}^\mathbf{x},
\quad \mathbf{1}^T\boldsymbol{\alpha}=1, \quad \mathbf{W}\succeq 0,\quad \alpha_k\geq0, \quad \forall k{\text{i}}n\mathcal{G} \\
&\quad \nonumber
P^G_{k}+\alpha_k \mathbf{s}^T \text{Re}\{\boldsymbol{\delta} \} + P^R_{k}(\boldsymbol{\delta} ) - P^L_{k}(\boldsymbol{\delta} ) =\\ \nonumber &\qquad\qquad\qquad\sum_{l{\text{i}}n\mathcal{N}_k} \text{Re}\left\{(W_{kk}-W_{kl})^*y_{kl}^*\right\}, \; \forall k {\text{i}}n \mathcal{N}\\
&\quad \nonumber
Q^G_{k} + Q^R_{k}(\boldsymbol{\delta} ) - Q^L_{k}(\boldsymbol{\delta} ) =
\sum_{l{\text{i}}n\mathcal{N}_k} \text{Im}\left\{(W_{kk}-W_{kl})^*y_{kl}^*\right\}, \forall k {\text{i}}n \mathcal{N}\\
&\quad \nonumber
P_{k\,\min}\leq P^G_{k}+\alpha_k \mathbf{s}^T \text{Re}\{\boldsymbol{\delta} \} \leq P_{k\,\max}, \quad \forall k {\text{i}}n \mathcal{G} \\
&\quad \nonumber
Q_{k\,\min}\leq Q^G_{k}\leq Q_{k\,\max}, \quad \forall k {\text{i}}n \mathcal{G}\\
&\quad \nonumber
\left(V_{k\,\min}\right)^2\leq W_{kk}\leq \left(V_{k\,\max}\right)^2, \quad \forall k {\text{i}}n \mathcal{N}\\
&\quad \nonumber
W_{ll}+W_{mm}-W_{lm}-W_{ml}\le (\Delta V_{lm}^{\max})^2,\ \quad\forall(l,m){\text{i}}n\mathcal{L}.
\end{align}} \vskip -.2in
\begin{remark}[About \textbf{CR-AC-OPF} formulation]
\label{crac-remark}
To the best of the authors knowledge, the formulation of the \textbf{CR-AC-OPF} is original, and it represents the first main contribution of the present paper. First, the approach clearly differs from the formulations based on DC power flow. Second, it improves upon the formulation based on convex AC-formulation in \cite{ETH-AC}.
Indeed, in that work, the authors cope with the need of guaranteeing the existence of a different value of $\mathbf{W}$ for different values of the uncertainty $\boldsymbol{\delta} $ by imposing a specific dependence on $\mathbf{W}$ from the uncertainty. In our notation, \cite{ETH-AC} introduces the following finite (linear) parameterization
\begin{equation}
\label{eq:W-param}
\mathbf{W}(\boldsymbol{\delta} )=A+\sum_k B_k\boldsymbol{\delta} _k,
\end{equation}
where $A, B_1,\ldots,B_n$ become design variables in the optimization problem.
{
We note that the choice of introducing some type of parameterization of the voltages with respect to uncertainty is unavoidable if a certificates approach is not adopted. this is due to the equality constraints in the Balance Equations.
In DC-based results, these become linear equalities, and hence they can be explicitly solved, as in as \cite{Bienstock2014,Lubin2016}.
In nonlinear approaches, this parameterization is always found, and goes under different names;
for instance it is referred to as \textit{affine control}
\cite{Morari2015},
\textit{affinely adjustable robust counterpart} in \cite{Jabr2013}, \textit{affine feedback policies} in \cite{Guo2018}, and first-order Taylor approximation in \cite{Andersson2018}.
With respect to these formulation, we remark the following:
i) the \textbf{CR-AC-OPF} formulation is surely less conservative, since it does not impose a \textit{specific dependence} on $\mathbf{W}$. The gain in terms of performance with respect to \cite{ETH-AC} is reported in the numerical experiments in Section \ref{sec:examples},
ii)
the parameterization in \eqref{eq:W-param} requires modification of the voltage magnitude at the generators during operation. Indeed, as previously noted, $\mathbf{W}$ involves not only the state variables, but also the control variables corresponding to generator voltages $|V_{1}|,\ldots,|V_{n_g}|$ which are not supposed to change during the operation. In the \textbf{CR-AC-OPF}, contrary, all control variables---including generators voltage magnitude---are designed during the optimization phase and do not need to be changed when demand/renewable power fluctuations occur.}
\end{remark}
Moreover, although the \textbf{CR-AC-OPF} formulation represents a relaxation of the robust optimal power flow problem, we can show that there are special cases in which this relaxation is indeed exact. In particular, we introduce next a class of networks where \textbf{CR-AC-OPF} does provide the optimal solution to the Robust AC-OPF problem. To this end, we first briefly recall some concepts from graph theory that are central for the results to follow. For a more detailed discussion on the following definitions and their interpretation in the context of power networks the reader is referred to \cite{madani2015convex}.
\begin{definition}
A network/graph is called \emph{weakly cyclic} if every edge belongs to at most one cycle.
\end{definition}
\begin{definition}
A network is said to be \emph{lossless} if
\mbox{$
\text{Re}\{y_{lm}\} = 0$,} $ \forall (l,m) {\text{i}}n \mathcal{L}.
$
\end{definition}
With these definitions at hand, we can now describe a class of networks for which the relaxed formulation above provides an exact solution. This theorem represents a natural extension of the results presented in \cite{madani2015convex} to the robust case and the proof is given in appendix.
\begin{theorem}
\label{theo-OPF-robust}
Consider a lossless weakly-cyclic network with cycles of size 3, and assume $Q_k^{\min}=-{\text{i}}nfty$ for every $k{\text{i}}n\mathcal{G}$. Then, the convex relaxation {\bf CR-AC-OPF} is exact.
\end{theorem}
\noindent
\textbf{Proof:} See Appendix \ref{sec: appendix}.
\begin{remark}
In the theorem above, by exact we mean that i) the robust convex relaxation has an optimal value equal to the one of the Robust AC-OPF and ii) for any admissible value of the uncertainty $\boldsymbol{\delta} $, there exists a rank one matrix $\mathbf{W}_\delta$ that satisfies the constraints of the convex relaxation and, hence, there exists a $\mathbf{V}_\delta$ that satisfies the constraints of the Robust AC-OPF.
\FD{Hence, if the convex relaxation \textbf{CR-AC-OPF} is feasible, then Theorem~1 guarantees that the ensuing optimal values of $\mathbf{P}^{G}$, $\boldsymbol{\alpha}$,$\mathbf{W}_\mathbf{u}$ are such that for every value of the uncertainty there exists a rank one $\mathbf{W}$ solving the problem.
Hence, for those cases when a solution to \textbf{CR-AC-OPF} is found, the Theorem provides a very strong guarantee.
On the other hand, it should be remarked that the theoretical guarantees of Theorem 1 clearly do not automatically translate for those cases when the robust solution of \textbf{CR-AC-OPF} is {prohibitive from a computational viewpoint}.
In these cases, the existence of a rank one solution for {some uncertainty instances} may not imply the existence of a rank one solution for an unseen {instance of the uncertainty}.
}\end{remark}
We should remark that the class considered in Theorem~\ref{theo-OPF-robust} is not fully realistic. However,
we feel that this result is important for several reasons. First, it shows that the exactness of the relaxation for a particular class of networks---proven for the nominal case in \cite{madani2015convex}---carries over to the robust formulation we introduced in \eqref{eq: CR-AC-OPF}. This shows that {\bf CR-AC-OPF} represents indeed the right way to formulate the robust counterpart of the AC-OPF problem. Second,
several results of the same spirit have been obtained in the literature for deterministic (no uncertainty) networks, see for instance \cite{Low:2014b}: we believe that these results can be extended to the convexified robust formulation introduced here. This will the subject of further research.
\subsection{Penalized and robust costs}\label{sec: worst case cost}
Note that, besides the configurations where the relaxation has been proven to be exact, in the general case it is important to obtain solutions $\mathbf{W}$ with low rank. To this end, we adopt a penalization technique similar to that introduced in \cite{Madani_promises,madani2015convex}. More precisely, it is observed there that maximizing the weighted sum of off-diagonal entries of $\mathbf{W}$ often results in a low-rank solution.
To this end, one can augment the objective function with a weighted sum of generators' reactive power---for lossless network---and apparent power loss over the series impedance of some of the lines of the network (the so-called problematic lines $\mathcal{L}_{\text{prob}}$)---for lossy network---to increase the weighted sum of the real parts of off-diagonal elements of $\mathbf{W}$ and hence obtain a low rank solution.
Formally, let $L_{lm} \doteq|S_{lm}+S_{ml}|$ denote the apparent power loss over the line $(l,m)$ with $S_{lm}\doteq|(W_{ll}-W_{lm})^*y_{lm}^*|$. Given nonnegative factors $\gamma_b$ and $\gamma_\ell$, in \cite{Madani_promises,madani2015convex} the following penalized cost was considered
\[
f_\mathrm{pen}(\mathbf{P}^G,\mathbf{Q}^G,\mathbf{W})\doteq
\sum_{k{\text{i}}n\mathcal{G}} f_k(P^G_{k}\!)+ \gamma_b \!\!\sum_{k{\text{i}}n\mathcal{G}}\!\! Q^G_{k}+ \!\gamma_\ell \!\!\!\!\sum_{(l,m){\text{i}}n\mathcal{L}_{\text{prob}}}\!\! \!\!\!\!\!\!L_{lm}.
\]
Since, in the proposed approach, $\mathbf{Q}^G$ and $\mathbf{W}$ depend on the value of the uncertainty $\boldsymbol{\delta} $, the formulation of the robust optimal power flow problem needs to reflect this fact. More precisely, the right cost function to consider is
\[
\max_{\boldsymbol{\delta} {\text{i}}n \boldsymbol{\Delta} } f_\mathrm{pen}(\mathbf{P}^G,\mathbf{Q}^G,\mathbf{W})\doteq
\]
\[
\max_{\boldsymbol{\delta} {\text{i}}n \boldsymbol{\Delta} } \left(
\sum_{k{\text{i}}n\mathcal{G}} f_k(P^G_{k}\!)+ \gamma_b \!\!\sum_{k{\text{i}}n\mathcal{G}}\!\! Q^G_{k}+ \!\gamma_\ell \!\!\!\!\sum_{(l,m){\text{i}}n\mathcal{L}_{\text{prob}}}\!\! \!\!\!\!\!\!L_{lm}\right).
\]
The constraints remain the same as in~\eqref{eq: CR-AC-OPF}. This modification of the cost provides a way of ``encouraging'' low rank solutions and it has been shown to work well in practice.
\section{A randomized approach to \textbf{CR-AC-OPF}}
\label{sec:scenario}
In this section, we first briefly summarize the main features of the scenario with certificates problem, and then show how this approach can be used to tackle in an efficient way the \textbf{CR-AC-OPF} problem introduced in the previous section.
\subsection{Scenario with Certificates}
In \cite{SWC-TAC} a \FD{class of robust optimization problems ``with certificates" was formally introduced. This definition refers to robust optimization problems in which a clear distinction can be made between so-called \textit{design} variables $\theta$ and \textit{certificates} $\xi$.
While the design variables $\theta$ play the role of classical optimization variable, the certificates are variables for which we are not interested in, as long as we are guaranteed that for any possible value of the uncertainty there exists a value of $\xi$ guaranteeing feasibility of the problem.}
\FD{Formally, we assume that the design variables $\theta$ belong to a given convex set $\Theta\subseteq\mbox{\rm Rel}eal{n_\theta}$,
and the certificates $\xi$ belong to a compact (possibly nonconvex) set $X\subseteq\mbox{\rm Rel}eal{n_\xi}$. Then, we
consider a function $f(\theta,\xi,\boldsymbol{\delta} )$ which is jointly convex in $\theta$ and $\xi$, for any given $\boldsymbol{\delta} {\text{i}}n\boldsymbol{\Delta} $. Then, we can define the so-called \textit{robust optimization problem with certificates} introduced as follows}
\begin{eqnarray}
\min_{\theta} && c^T \theta \label{eq:certificates_opt}\\
\text{subject to} && \forall \boldsymbol{\delta} {\text{i}}n\boldsymbol{\Delta} \ \exists \xi=\xi(\boldsymbol{\delta} )
\text{ satisfying } f(\theta,\xi,\boldsymbol{\delta} )\le 0. \nonumber
\end{eqnarray}
Then, it is shown that problem \eqref{eq:certificates_opt} can be approximated by introducing an appropriate randomized counterpart, based on the extraction of $N$ random samples of the uncertainty.
Formally, we extract an \textit{uncertainty multisample}
\[
\{\boldsymbol{\delta} ^{(1)},\ldots,\boldsymbol{\delta} ^{(N)}\}
\]
and construct the following \textit{scenario problem with certificates}
\begin{eqnarray}
\theta\ped{SO}C=\arg\min_{x,\xi_{1},\ldots,\xi_{N}}&& c^T \theta \label{eq:scenario_cert}\\
\text{subject to}&& f(\theta,\xi_{i},\boldsymbol{\delta} ^{(i)}) \leq 0, \ i=1,\ldots,N.\nonumber
\end{eqnarray}
Note that
contrary to the classical scenario problem,
in SwC \textit{a new certificate variable $\xi_{i}$ is created for every sample $\boldsymbol{\delta} ^{(i)}$}. In this way, one implicitly constructs an "uncertainty dependent" certificate, without assuming any \textit{a-priori} explicit functional dependence on $\boldsymbol{\delta} $.
To present the properties of the solution $\theta\ped{SO}C$, we first introduce the violation probability of a given design $\theta$ as follows
\[
\mbox{\rm Viol}(\theta)=
\Pr\Bigl\{\exists \boldsymbol{\delta} {\text{i}}n\boldsymbol{\Delta} | \nexists \xi \text{ satisfying } f(\theta,\xi,\boldsymbol{\delta} )\le 0 \Bigr\}.
\]
Then, the main result regarding the scenario optimization with certificates is recalled next. The result was derived in \cite{SWC-TAC} as an extension to the classical scenario approach developed in~\cite{calafiore2006scenario}.
\vskip .2in
\begin{theorem}
\label{them:SwC}
Assume that, for any multisample extraction, problem \eqref{eq:scenario_cert} is feasible and attains a unique optimal solution.
Then, given an accuracy level $\varepsilon{\text{i}}n(0,1)$ and a confidence level $\beta{\text{i}}n(0,1)$, if the number of samples is chosen as
\begin{equation}
\label{eq: N_SwC}
N\ge N\ped{SwC} = \frac{\mathrm{e}}{\varepsilon(\mathrm{e}-1)}\left(\ln\frac{1}{\beta}+n_{\theta}-1\right)
\end{equation}
where $n_\theta$ is the dimension of $\theta$, and e is the Euler number. Then, with probability at least $1-\beta$ , the solution $\theta\ped{SO}C$ of problem \eqref{eq:scenario_cert} satisfies
$
\mbox{\rm Viol}(\theta\ped{SO}C)\le\varepsilon.
$
\end{theorem}
Note that, in the above theorem, we guaranteed with high confidence $(1-\beta)$ that the solution returned by the SwC has a risk of violation of constraints less than the predefined (small) risk level~$\varepsilon$.
\subsection{SwC Solution to \textbf{CR-AC-OPF}}
Clearly, problem \textbf{CR-AC-OPF} represents a
robust optimization problem with certificates, in which the design variables are those "controllable" by the network manager, i.e.
$\theta\equiv \mathbf{u}$
while the certificates represent the quantities that can be "adjusted" to guarantee constraint satisfaction, i.e.
$\xi \equiv \mathbf{x}_{\boldsymbol{\delta} }$.
Based on this consideration, in this paper we propose the following solution strategy to the \textbf{CR-AC-OPF} problem: \\
\noindent{\bf SwC-AC-OPF Optimization Procedure}
\begin{itemize}
{\text{i}}tem[i)] Given probabilistic levels $\varepsilon$, and $\beta$ compute $N\ped{SwC}$ according to \eqref{eq: N_SwC}.
{\text{i}}tem[ii)] Generate $N\ge N\ped{SwC}$ sampled scenarios $\boldsymbol{\delta} ^{(1)},\ldots,\boldsymbol{\delta} ^{(N)}$, where the uncertainty is drawn according to its known probability density.
{\text{i}}tem[iii)]
Solve the following convex optimization problem, which returns the control variables $\mathbf{P}^{G}, \mathbf{W^u}, \boldsymbol{\alpha}$.\\
\noindent{\bf SwC-AC-OPF} {\small
\begin{align} \label{eq: SwC Original}
&\underset{\tiny
\mathbf{P}^{G}, \mathbf{W^u}, \boldsymbol{\alpha},
\mathbf{Q}^{G,[1]},\ldots,\mathbf{Q}^{G,[N]},
\mathbf{W}^{\mathbf{x},[1]},\ldots, \mathbf{W}^{\mathbf{x},[N]}
}{\text{minimize}}
\ {\boldsymbol{\gamma}}\\
&\text{subject to:} \text{ for } i=1,\ldots,N\nonumber \\
&\nonumber \mathbf{W}^{[i]}=\mathbf{W}^\mathbf{u}+\mathbf{W}^{\mathbf{x},[i]}, \quad \mathbf{W}^{[i]}\succeq 0, \quad \alpha_k\geq0. \quad \forall k{\text{i}}n\mathcal{G}\\
&\nonumber
L_{lm}^{[i]} =
|(W^{[i]}_{ll}-W^{[i]}_{lm})^*y_{lm}^*|
+
|(W^{[i]}_{mm}-W^{[i]}_{ml})^*y_{lm}^*|\\
&\sum_{k{\text{i}}n\mathcal{G}} f_k(P^G_{k}\!)+ \gamma_b \!\!\sum_{k{\text{i}}n\mathcal{G}}\!\! Q^{G,[i]}_{k}+ \!\gamma_\ell \!\!\!\!\sum_{(l,m){\text{i}}n\mathcal{L}^{\text{prob}}}\!\! \!\!\!\!\!\!L_{lm}^{[i]} \le \gamma
\nonumber\\
&\nonumber
P^G_{k}+\alpha_k \mathbf{s}^T \text{Re}\{\boldsymbol{\delta} ^{(i)}\} + P^R_{k}(\boldsymbol{\delta} ^{(i)}) - P^L_{k}(\boldsymbol{\delta} ^{(i)}) = \\ \nonumber
& \qquad\qquad\qquad\sum_{l{\text{i}}n\mathcal{N}_k} \text{Re}\left\{(W_{kk}^{[i]}-W_{kl}^{[i]})^*y_{kl}^*\right\}, \quad \forall k {\text{i}}n \mathcal{N}\\
&\nonumber
Q^{G,[i]}_{k} + Q^R_{k}(\boldsymbol{\delta} ^{(i)}) - Q^L_{k}(\boldsymbol{\delta} ^{(i)}) = \\ \nonumber
& \qquad\qquad\qquad\sum_{l{\text{i}}n\mathcal{N}_k} \text{Im}\left\{(W_{kk}^{[i]}-W_{kl}^{[i]})^*y_{kl}^*\right\}, \quad \forall k {\text{i}}n \mathcal{N}\\
&\nonumber
P_{k\,\min}\leq P^G_{k}+\alpha_k \mathbf{s}^T \text{Re}\{\boldsymbol{\delta} ^{(i)}\} \leq P_{k\,\max}, \quad \forall k {\text{i}}n \mathcal{G} \\
&\nonumber
Q_{k\,\min}\leq Q^{G,[i]}_{k}\leq Q_{k\,\max}, \quad \forall k {\text{i}}n \mathcal{G}\\
&\nonumber
\left(V_{k\,\min}\right)^2\leq W_{kk}^{[i]}\leq \left(V_{k\,\max}\right)^2, \quad \forall k {\text{i}}n \mathcal{N}\\
&\nonumber
W_{ll}^{[i]}+W_{mm}^{[i]}-W_{lm}^{[i]}-W_{ml}^{[i]}\le (\Delta V_{lm}^{\max})^2,\ \quad\forall(l,m){\text{i}}n\mathcal{L}
\end{align}}
{\text{i}}tem[iv)] During operation, measure uncertainty in generations and loads $\boldsymbol{\delta} $, and accommodate the $k$-th controllable generator as
$\bar{P}^G_{k} =P^G_{k}+\alpha_k \mathbf{s}^T \text{Re}\{\boldsymbol{\delta} \},~
|V_{k}| =\sqrt{W_{kk}},~k {\text{i}}n \mathcal{G}$.
\end{itemize}
We remark that Theorem~\ref{them:SwC} guarantees that the \textbf{SwC-AC-OPF} optimization procedure is such that the probabilistic constraints of the Chance-Constrained AC-OPF \eqref{eq: OPF CCP } are satisfied with high confidence~$(1-\beta)$.
In other words, one has an \textit{a-priori} guarantee that the risk of violating the constrains is bounded, and one can accurately bound this violation level by choosing the probabilistic parameters $\varepsilon,\beta$. The above described procedure represents the main result of this paper. In Section \ref{sec:examples}, we demonstrate how this procedure outperforms existing ones in terms of guarantees of lower line-violations.
\begin{remark}[Using available data on uncertainty]
\FD{It is frequently the case that several measurements of the uncertainty are available to the system designer.
Several approaches aiming at designing uncertainty sets based on this available data have been hence recently proposed in the literature, and employed in approaches such as Robust Sample Average Approximation and Distributionally Robust methods, see \cite{Guo2018} and references therein.
We remark that it is not directly necessary in a sample-based approach---as the one presented here---to design the uncertainty set. Indeed, the data itself can be used as ``scenarios" to be fed to the SwC problem, thus avoiding this critical point. }
\end{remark}
\subsection{Handling Security Constraints}
An important observation is that the approach introduced in this paper can be readily extended to handle security constraint. In particular, the popular $N-1$ security requirements discussed, for instance, in \cite{ETH-AC} can be directly included in the robust optimization problem \eqref{eq: Robust AC-OPF}, and in its subsequent derivations.
We recall that, in the $N-1$ security constrained OPF framework, only the outages of \textit{a single component} are taken into account. A list of $N_\mathrm{out}$ possible outages, $\mathcal{I}^\mathrm{out}=\left\{0,1,\ldots,N_\mathrm{out}\right\}$, is formed (with $0$ corresponding to the case of no outages). Then, a
large optimization problem is constructed, with $N_\mathrm{out}$ instances of the constraints, where the $i$-th instance corresponds to removing the $i$-th component form the equations, with $i{\text{i}}n\mathcal{I}^\mathrm{out}$.
The exact same approach can be replicated here.
However, we remark that the SwC formulation allows for a more precise and realistic handling of the possible outages. Indeed, we note that in a real network some buses may have a larger probability of incurring into an outage, for instance because they are located in a specific geographical position, or because they employ older technologies.
To reflect this scenario, we associate to each element $i{\text{i}}n\mathcal{I}^\mathrm{out}$ in the network a given \textit{probability of outage}
\[
p_i^\mathrm{out}{\text{i}}n\left[0,1\right],
\]
which can be different for each component,
and is supposed to be known to the network manager.
Large values of $p_i$ correspond to large probability of an outage occurring in the $i$-th component.
Then, in our framework, outages can be treated in the same way as uncertainties.
Formally, we can associate to each component a random variable
$\delta^\mathrm{out}_i,$ $ i{\text{i}}n\mathcal{I}^\mathrm{out}$,
with Bernoulli density with mean $1-p_i^\mathrm{out}$. Then, the scenario with certificates approach is applied considering the extended uncertainty
$\left\{\boldsymbol{\delta} ,\boldsymbol{\delta} ^\mathrm{out}\right\}$.
In practice, when constructing the \textbf{SwC-AC-OPF} problem, for each sampling instance the $i$-th component is removed with probability $p_i^\mathrm{out}$. It is important to note that this approach goes beyond the standard $N-1$ security constrained setup, since: i) \textit{multiple} simultaneous outages are automatically taken into account, ii) it allows weighting differently the different component.
\section{Numerical Examples} \label{sec:examples}
In order to examine the effectiveness of the proposed method, we perform extensive simulations using New England 39-bus system case adopted from \cite{matpower2011}. The network has $39$ buses, $46$ lines and $10$ conventional generation units, and it is modified to include $4$ wind generators connected to buses $5$, $6$, $14$ and $17$. The renewable energy generators and all loads connected to different buses are considered to be uncertain. In total, there are $46$ uncertain parameters in the network. The goal is to design active power and voltage amplitude of all controllable generators as well as the distribution vector $\boldsymbol{\alpha}$ such that the generation cost is minimized while all the constraint of the network i.e. line flow, bus voltage and generators output constraint are respected with high probability.
\subsection{Numerical Results}
We use the methodology presented in Section \ref{sec:scenario} to solve the robust optimal power flow problem for New England 39 bus system.
We consider a $24$ hour demand pattern shown in Fig. \ref{fig: 24hour results}-(a) and solve the optimal power flow problem for each hour by minimizing the nominal cost. The error probability distribution for wind generator and load is chosen based on Pearson system \cite{pearson1895contributions}---as suggested in \cite{hodge2013short,hodge_wind_2012}. Pearson system is represented by the mean $\mu$ (first moment), variance~$\sigma^2$ (second moment), skewness $\gamma$ (third moment) and
kurtosis $\kappa$ (forth moment). In the simulation, we set $\sigma=0.2\times\text{(predicted value)},~\gamma=0,$ and, $\kappa=3.5$ leading to a leptokurtic distribution with heavier tail than Gaussian. The selected probabilistic accuracy and confidence levels $\varepsilon$ and $\beta$ are $0.02$ and $1\times10^{-15}$ respectively resulting in $5,105$ number of scenario samples. We note that the number of design (control) variables is $31$ in our formulation.
The penetration level is chosen to be $30\%$.
The penetration level indicates how much of the total demand is provided by the wind generators.
A $30\%$ penetration level means that, in total, the wind generators provide $30\%$ of the nominal load. We also assume that each renewable generator contributes equally to provide this power.
The optimization problem is formulated and then solved using YALMIP \cite{loefberg_yalmip_2004} and Mosek \cite{andersen2000mosek} respectively, and returned the control variables $\mathbf{P}^{G}$, $\mathbf{W^u}$ and $\boldsymbol{\alpha}$.
In order to examine robustness of the obtained solutions, we run an \textit{a-posteriori} analysis based on Monte Carlo simulation. To this end, we generated $10,000$ random samples from the uncertainty set---corresponding to uncertain active and reactive generated wind power and load---and for each sample, modified the network to include wind power generators, replaced the load vector by its uncertain counterpart, computed the power mismatch, distributed the mismatch to all conventional generators by using (\ref{eq: affine control}) and solved the power flow problem whose constraint sets include the non-linear power balance equation, bus voltage constraints and generator power constraints. For each sample of the Monte-Carlo simulation, we solved the non-linear non-convex feasibility problem to see if there exist feasible state variables that respect the constraints. We then measured the number of infeasible samples as a measure of robustness of the designed control variables: the smaller the number of infeasible samples, the more robust are the control variables. This simulation is formally formulated in Algorithm \ref{alg: MonteCarlo} where the nonlinear feasibility problems were solved using the function \texttt{fmincon} of Matlab\textregistered.
It should be noted that the matrices $\mathbf{W}$ obtained were mostly not rank 1, but as it can be seen in the simulation, the control variable values obtained work well in practice.
Running Algorithm \ref{alg: MonteCarlo} with the control variables designed using the proposed SwC method results in only $14$ out of $10,000$ infeasible samples while the nominal control variables---the control variables designed without considering uncertainty---results in $5,374$ infeasible samples. Hence, the probability of
joint feasibility of all constraints is simply approximated by [number of feasible instances]$/N$. In our experiment this probability turned out to be $9,986/10,0000=0.9986$, with a consequent probability of failure less than $.002$ (and hence much less than the considered $\epsilon=0.02$).
On the contrary, the same analysis on the nominal design led to a joint violation probability of $5374/10000=.5374$, hence more than $50\%$. The empirical violation is shown in Table \ref{tab: comarison of empirical violation}. We remark that both control variables are designed for the peak demand level which is at hour $14$. This proves that the control variables designed using SwC method are indeed robust compared to design variables for which no uncertainty has been considered.
\renewcommand{\textbf{Input:}}{\textbf{Input:}}
\renewcommand{\textbf{Output:}}{\textbf{Output:}}
\begin{algorithm}[H]
\begin{algorithmic}[1] \begin{footnotesize}
\caption{Monte-Carlo Simulation}
\label{alg: MonteCarlo}
\STATE\textbf{Input:}{ Control variables $\{P^G_1\, \cdots \,P^G_{n_g},
|V_1| \cdots |V_{n_g}|, \alpha_1,\ldots,\alpha_n\}$}
\STATE\textbf{Output:}{\texttt{Infeas\_Counter}}\\
{\bf Initialization:}
\STATE Set \texttt{Infeas\_Counter = 0}\\
{\bf Evolution:}\\
Extract $10,000$ i.i.d samples $\boldsymbol{\delta} ^{(1)},\ldots,\boldsymbol{\delta} ^{(10,000)}$\\
\FOR{$i = 1$ \TO $10,000$}
\STATE{ Solve
\begin{align*}
\text{find } \mathbf{x}_{\boldsymbol{\delta} }& \text{ s.t.}\\
&\nonumber
P^G_{k}+\alpha_k \mathbf{s}^T \text{Re}\{\boldsymbol{\delta} ^{(i)}\} + P^R_{k}(\boldsymbol{\delta} ^{(i)}) - P^L_{k}(\boldsymbol{\delta} ^{(i)}) = \\ \nonumber
& \qquad\qquad\qquad\sum_{l{\text{i}}n\mathcal{N}_k} \text{Re}\left\{V_k(V_k-V_l)^*y_{kl}^*\right\}, \quad \forall k {\text{i}}n \mathcal{N}\\
&\nonumber
Q^{G}_{k} + Q^R_{k}(\boldsymbol{\delta} ^{(i)}) - Q^L_{k}(\boldsymbol{\delta} ^{(i)}) = \\ \nonumber
& \qquad\qquad\qquad\sum_{l{\text{i}}n\mathcal{N}_k} \text{Im}\left\{V_k(V_k-V_l)^*y_{kl}^*\right\}, \quad \forall k {\text{i}}n \mathcal{N}\\
&\nonumber
P_{k\,\min}\leq P^G_{k}+\alpha_k \mathbf{s}^T \text{Re}\{\boldsymbol{\delta} ^{(i)}\} \leq P_{k\,\max}, \quad \forall k {\text{i}}n \mathcal{G} \\
&\nonumber
Q_{k\,\min}\leq Q^{G}_{k}\leq Q_{k\,\max}, \quad \forall k {\text{i}}n \mathcal{G}\\
&\nonumber
V_{k\,\min}\leq V_{k}\leq V_{k\,\max}, \quad \forall k {\text{i}}n \mathcal{N}\\
&\nonumber
|V_l(V_l-V_m)^*y_{lm}^*|\leq S_{lm \max}, \quad \forall(l,m){\text{i}}n\mathcal{L}
\end{align*}
\IF{$\mathbf{x}_{\boldsymbol{\delta} }==\emptyset$}
\STATE{\texttt{Infeas\_Counter = Infeas\_Counter+1}}\\
\ENDIF
}
\ENDFOR \end{footnotesize}
\end{algorithmic}
\end{algorithm}
\begin{table}[t]
\caption{Empirical violation of the Monte Carlo Algorithm \ref{alg: MonteCarlo} for the SwC solution and the nominal one---where no uncertainty is considered in the design phase.}
\begin{center}
\begin{tabular}{c||c}
\toprule
Approach & Empirical Violation\tabularnewline
\midrule
SwC & $0.0014$ \tabularnewline
\midrule
Nominal & $0.5374$ \tabularnewline
\bottomrule
\end{tabular}
\end{center}
\label{tab: comarison of empirical violation}
\end{table}
The rate of violation of network constraints can not be identified by the Monte Carlo simulation presented in Algorithm \ref{alg: MonteCarlo}. An infeasible optimization problem does not define if the infeasibility is due to violation of bus voltages, line ratings or generator active and reactive generated power. Furthermore, even if the constraint(s) causing infeasibility becomes clear, it remains unclear how much the constraint(s) exceed their limits. For this reason, we used \texttt{runpf} command of MATPOWER \cite{matpower2011} to derive line flows. We remark that Matpower power flow solver does not respect voltage and generator power constraints and only solves the non-linear balance equation. On the other hand---due to non-linearity of balance equation---the solution to power flow equation may not be unique \cite{korsak1972question,johnson1977extraneous,wang2003existence}. Hence, one may not obtain a solution that respects voltage and power constraints. For this reason we focus on line flows derived from MATPOWER.
The empirical violation of line flows is computed by counting the number of times each line exceeds its limit in the Monte Carlo simulation and dividing this value by $10,000$. Figure \ref{fig: 24hour results}(b) shows the $24$ hour empirical violation of line flow for all lines of the network. The mean value of generation cost associated with the $24$ hour demand is also shown in Fig. \ref{fig: 24hour results}(a). We note that this graph is obtained by averaging the generation cost over all $10,000$ samples.
\begin{figure}
\caption{\footnotesize (a): $24$ hours demand pattern and average generation cost computed in the posteriori analysis based on Monte Carlo simulation. (b): $24$ hours empirical violation of line flow computed in the posteriori analysis based on Monte Carlo simulation. }
\label{fig: 24hour results}
\end{figure}
\begin{figure*}
\caption{\footnotesize (a): Line loading percentage for $10,000$ random samples extracted from the uncertainty set, results of scenario with certificates approach, (b) results of standard OPF design where no uncertainty is taken into account in designing the control parameters;
In the boxplots, the red line represents the median value, edges of each box correspond to the $25^\text{th}
\label{fig: OPF SImulation}
\end{figure*}
In Fig. \ref{fig: OPF SImulation}, we compare line loading percentage for scenario with certificates approach against standard optimal power flow design where no uncertainty is taken into account while designing the control parameters. Demand level
is $6,406$ MVA. This proves the necessity of adopting a robust strategy for the new power grids containing renewable generators. In such a network if we rely on the classical OPF design where no uncertainty is taken into account in designing the control variables, it results in very frequent overloads, as shown in Fig. \ref{fig: OPF SImulation}(b), and hence frequent line tripping or even cascading outage. On the other hand, the robust strategy presented in the current paper successfully designs the control variables such that only very occasional violation happens during the operation of the network in the presence of large number of uncertain parameters.
We also compare the proposed strategy against the one presented in \cite{ETH-AC}. In this comparison, we only consider uncertainty in renewable energy generators. Loads are considered to be known exactly. This is because \cite{ETH-AC} can only handle uncertainty in renewable energy generator.
In \cite{ETH-AC}, the number of design variables appearing in the optimization problem is much larger than SwC approach. This is due to the linear parametrization \eqref{eq:W-param} where some additional design variables are introduced. Since the number of scenario samples $N_\ped{SwC}$ depends on the number of design variables $n_\theta$, additional design variables lead to significant growth in the number of scenario samples as compared to SwC approach.
The worst-case cost is minimized in the OPF formulation---see subsection \ref{sec: worst case cost}. The penetration level is assumed to be $33\%$ and demand level is $7,112$ MVA corresponding to the peak demand in Fig. \ref{fig: 24hour results}.
Choosing $\varepsilon=0.1,~\beta=1\times10^{-10}$, the number of scenario samples for \cite{ETH-AC} is $11,838$---with $896$ design variables---and for scenario with certificates is $839$---with only $31$ design variables. Large number of scenario samples associated with the approach presented in \cite{ETH-AC} results in much higher computational cost compared to SwC approach.
Indeed, the optimization problem formulated based on \cite{ETH-AC} takes more than $51$ hours to be solved on a workstation with $12$ cores and $48$ GB of RAM, while the solution of the SwC approach takes less than $49$ minutes. We remark that the computational time refers to a non-optimized implementation of the problem.
We also remark that the alternative approach proposed in \cite{ETH-AC}, and followed by recent works as \cite{Rostampour2017},
based on computing a probabilistically guaranteed hyper-rectangle, may not be practically viable. Indeed,
these require to solve a problem involving all the vertices of the uncertainty sets, whose number grows exponentially in the size of the uncertainty. For instance, in our numerical example, it would amount at imposing the constraints on $2^{46}\approx 7\times10^{13}$ vertices.
Finally, we note that sequential approaches as those proposed in \cite{Chamanbaz_TAC_2016}, and the
sophisticated techniques of Sparsity Pattern Decomposition presented in \cite{Rostampour2017},
may be adopted to significantly reduce computation times.
\begin{figure}
\caption{\footnotesize The probability of line flow violation compared with \cite{ETH-AC}
\label{fig: ETHCompare}
\end{figure}
\begin{table}[!t]
\caption{\footnotesize Comparison between the average generation cost and average computation time -- over $10,000$ Monte Carlo simulation -- achieved using SwC with the one obtained by~\cite{ETH-AC}.}
\begin{center}
\begin{tabular}{c||c||c}
\toprule
Approach &Generation Cost [\$]& Computation Time [min]\tabularnewline
\midrule
SwC & $25,280$ & 49\tabularnewline
\midrule
\cite{ETH-AC} & $25,359$ & 3,074\tabularnewline
\bottomrule
\end{tabular}
\end{center}
\label{tab: comarison of average cost}
\end{table}
Finally, we run an \textit{a-posteriori} analysis based on Monte Carlo simulation to estimate the probability of line flow violation. In the posteriori analysis, we use exactly the same set of samples---different from the design samples of course---to evaluate performance of the two approaches. Figure \ref{fig: ETHCompare}
compares the probability of line flow violation for the two approaches.
This comparison shows that the linear parameterization adopted in \cite{ETH-AC} is conservative leading to larger violation level compared to SwC approach. The average---over $10,000$ samples---generation cost is also compared in Table \ref{tab: comarison of average cost}. Therefore, the paradigm presented in this paper improves upon \cite{ETH-AC} in computational complexity, violation probability, and generation cost.
\section{Concluding Remarks}\label{sec: conclusions}
In this paper, we proposed a novel approach to the AC optimal power flow problem in the presence of uncertain renewable energy sources and uncertain load. Assuming that the probability distribution of the uncertainty is available, we aim at optimizing the nominal power generation subject to a small well defined risk of violating generation and transmission constraints. To tackle this complex NP-hard problem, we propose a randomized algorithm based on the novel concept of scenario with certificates and on convex relaxations of power flow problems. The effectiveness of the proposed solution is illustrated via numerical examples, where it is shown that one can significantly decrease the probability of constraint violation without a significant impact on the nominal power generation cost. Moreover, the approach is shown to be very efficient from a computational viewpoint.
Future research aims at exploiting the flexibility of the methodology to extend its application.
For instance, we can consider the case of optimizing with respect to the slack bus voltage, to improve the objective function and even the safety. Since the slack bus voltage is a discrete variable, the resulting OPF problem will be of mixed-integer type, as in \cite{MoGaLi:17}.
\appendices
\section{Proof of Theorem~\ref{theo-OPF-robust}}\label{sec: appendix}
\noindent
Let $P_{k,opt}^G$ $\alpha_{opt}$ and $\mathbf{W}_{\mathbf{u},opt}$ be achievers of the solution of {\bf CR-AC-OPF}. Now, take any $\boldsymbol{\delta} {\text{i}}n\boldsymbol{\Delta} $ and consider the following optimization problem
{\small \begin{align*}
&\max_{\mathbf{W}\succeq 0} \ \sum_{k{\text{i}}n \mathcal{G}} Q_k \\
&\text{such that: }
P^G_{k,opt}+\alpha_{k,opt} \mathbf{s}^T \text{Re}\{\boldsymbol{\delta} \}\leq - P^R_{k}(\boldsymbol{\delta} ) + P^L_{k}(\boldsymbol{\delta} ) +\\ &\sum_{l{\text{i}}n\mathcal{N}_k} \text{Re}\left\{(W_{kk}-W_{kl})^*y_{kl}^*\right\} \leq
P^G_{k,opt}+\alpha_{k,opt} \mathbf{s}^T \text{Re}\{\boldsymbol{\delta} \},\,\, \forall k {\text{i}}n \mathcal{N}\\
& Q_{k\,\min} \leq - Q^R_{k}(\boldsymbol{\delta} ) + Q^L_{k}(\boldsymbol{\delta} ) =\\ &\qquad\sum_{l{\text{i}}n\mathcal{N}_k} \text{Im}\left\{(W_{kk}-W_{kl})^*y_{kl}^*\right\} \leq Q_{k\,\max}, \quad \forall k {\text{i}}n \mathcal{N}\\
& (\mathbf{W}^{\mathbf{u}}_{opt})_{kk} \leq W_{kk} \leq (\mathbf{W}^{\mathbf{u}}_{opt})_{kk} , \quad \forall k {\text{i}}n \mathcal{G};\\
& \left(V_{k\,\min}\right)^2\leq W_{kk}\leq \left(V_{k\,\max}\right)^2, \quad \forall k {\text{i}}n \mathcal{N}/\mathcal{G}\\
&
W_{ll}+W_{mm}-W_{lm}-W_{ml}\le (\Delta V_{lm}^{\max})^2,\ \quad\forall(l,m){\text{i}}n\mathcal{L}.
\end{align*} } \vskip -.2in
Note that, for the given value of the uncertainty, the solution of the optimization problem above has exactly the same generation cost as the solution of the {\bf CR-AC-OPF}.
The optimization problem above is of the same form as that used in the proof of part (a) of Theorem~2 in~\cite{madani2015convex}. Hence, the same reasoning can be applied to show that there exists a rank one solution for the problem above.
Since, as mentioned before, the solution of the convex relaxation and the one of the original optimal power flow problem coincide when $\text{rank}(\mathbf{W})=1$, this implies that the values of control variables obtained using {\bf CR-AC-OPF} are a feasible power allocation for any value of the uncertainty $\boldsymbol{\delta} {\text{i}}n\boldsymbol{\Delta} $. Moreover, for all $\boldsymbol{\delta} {\text{i}}n\boldsymbol{\Delta} $, the generation cost is equal to the optimal value of {\bf CR-AC-OPF.}
This, together with the fact that the solution of {\bf CR-AC-OPF} is optimal for at least one $\boldsymbol{\delta} ^*{\text{i}}n\boldsymbol{\Delta} $, leads to the conclusion that {\bf CR-AC-OPF} does provide the best worst-case solution or, in other words, the robustly optimal power generation allocation.
$\square$\\
\vskip -.15in
\begin{IEEEbiography}
[{{\text{i}}ncludegraphics[width=1in,height=1.25in,clip,keepaspectratio]{Mohammadreza2.jpg}}]{Mohammadreza~Chamanbaz} received his BSc in Electrical Engineering from Shiraz University of Technology. In 2014 he received his PhD from the Department of Electrical \& Computer Engineering, National University of Singapore in control science. Dr. Chamanbaz was with Data Storage Institute, Singapore as research scholar from 2010 to 2014. From 2014 to 2017, he was postdoctoral research fellow in Singapore University of Technology and Design. He was Assistant Professor in Arak University of Technology, from Jan 2017 to Jan 2019. Dr. Chamanbaz is now senior research fellow in iTrust Center for Research in Cyber Security, Singapore.
His research activities are mainly focused on probabilistic and randomized algorithms for analysis and control of uncertain systems, robust and distributed optimization, and secure control of cyber-physical systems.
\end{IEEEbiography}
\begin{IEEEbiography}
[{{\text{i}}ncludegraphics[width=1in,height=1.25in,clip,keepaspectratio]{Dabbene-picture.jpg}}]{Fabrizio~Dabbene} received the Laurea degree in 1995 and the Ph.D. degree in 1999, both from Politecnico di Torino, Italy. He is currently Senior Researcher at the CNR-IEIIT institute. His research interests include randomized and robust methods for systems and control, and modeling of environmental systems. He published more than 100 research papers and two books, and is recipient of the 2010 EurAgeng Outstanding Paper Award. He served as Associate Editor for Automatica (2008-2014) and IEEE Transactions on Automatic Control (2008-2012). Dr. Dabbene is a Senior Member of the IEEE, and has taken various responsibilities within the IEEE-CSS: he served as elected member of the Board of Governors (2014-2016) and as Vice President for Publications (2015-2016).
\end{IEEEbiography}
\begin{IEEEbiography}
[{{\text{i}}ncludegraphics[width=1in,height=1.25in,clip,keepaspectratio]{Lagoa-picture.pdf}}]{Constantino~M.~Lagoa}
received the B.S. and M.S. degrees from the
Instituto Superior Tecnico, Tech- ical University of Lisbon, Portugal
in 1991 and 1994, respectively, and the Ph.D. degree from the University
of Wisconsin at Madison in 1998. He joined the Electrical Engineering
Department of Pennsylvania State University, University Park, PA, in
August 1998, where he currently holds the position of Professor. He
has a wide range of research interests including robust optimization and
control, chance constrained optimization, controller design under risk
specifications, system identification and control of computer networks. Dr. Lagoa has served as Associate
Editor of IEEE Transactions on Automatic Control (2012-2017) and IEEE Transactions on Control systems Technology (2009-2013) and he is currently Associate Editor of Automatica.
\end{IEEEbiography}
\end{document}
|
\begin{document}
\title{Semidefinite representations of gauge functions
for structured low-rank matrix decomposition}
\begin{abstract}
This paper presents generalizations of semidefinite programming
formulations of 1-norm optimization problems over infinite
dictionaries of vectors of complex exponentials, which were
recently proposed for superresolution, gridless compressed sensing,
and other applications in signal processing.
Results related to the generalized Kalman-Yakubovich-Popov
lemma in linear system theory provide simple, constructive proofs of the
semidefinite representations of the penalty functions used in these
applications. The connection leads to several extensions to
gauge functions and atomic norms for sets of vectors
parameterized via the nullspace of matrix pencils.
The techniques are illustrated with examples of
low-rank matrix approximation problems arising in spectral estimation
and array processing.
\end{abstract}
\section{Introduction} \label{s-intro}
The notion of atomic norm introduced in \cite{CRPW:12} gives a
unified description of convex penalty functions that extend
the $\ell_1$-norm penalty, used to promote sparsity in
the solution of an optimization problem,
to various other types of structure.
The atomic norm associated with a non-empty set $C$ is defined as
the gauge of its convex hull, {\it i.e.}, the convex function
\begin{eqnarray}
g(x)
& = & \inf {\{t \geq 0 \mid x \in t \conv{C}\}} \nonumber \\
&= & \inf {\{\sum_{k=1}^r \theta_k \mid
x =\sum_{k=1}^r \theta_k a_k, \; \theta_k \geq 0, \; a_k\in C\}}.
\label{e-atomic}
\end{eqnarray}
This function is convex, nonnegative, positively homogeneous,
and zero if $x=0$. It is not necessarily a norm,
but it is common to use the term `atomic norm' even when $g$
is not a norm.
When used as a regularization term in an optimization problem,
the function $g(x)$ defined in~(\ref{e-atomic}) promotes the property
that $x$ can be expressed as
a nonnegative linear combination of a small number of elements
(or `atoms') of $C$.
The best known examples of atomic norms are the vector $\ell_1$-norm
and the matrix trace norm.
The $\ell_1$-norm of a real or complex $n$-vector is the atomic norm
associated with $C = \{ se_k \mid |s|=1, \; k=1,\ldots,n\}$,
where $e_k$ is the $k$th unit vector of length $n$.
The matrix trace norm (or nuclear norm)
is the atomic norm for the set of rank-1 matrices with unit
norm. Specifically, the trace norm on ${\mbox{\bf C}}^{n\times m}$
is the atomic norm for $C=\{vw^H \mid \|v\| =\|w\|=1\}$,
where $w^H$ is the conjugate transpose and $\|\cdot\|$ denotes
the Euclidean norm.
Many other examples are discussed in \cite{CRPW:12,BTR12,TBSR:13}.
The atomic norm associated with the set
\begin{equation} \label{e-Ce}
C_\mathrm{e}
= \{ \gamma\, (1, e^{\mathrm{j}\omega}, \ldots, e^{\mathrm{j} (n-1) \omega})
\in {\mbox{\bf C}}^n \mid \omega \in [0,2\pi), \; |\gamma|=1/\sqrt n\},
\end{equation}
where $\mathrm{j} = \sqrt{-1}$, has been studied extensively in recent
research in signal processing
\cite{dCG12,dCGH15,CaF:14,BTR12,TBSR:13,YX14,LC14,YX15,CC15}.
It is known that the atomic norm for this set is the optimal value of
the semidefinite program (SDP)
\begin{equation} \label{e-Ce-sdp}
\begin{array}{ll}
\mbox{minimize} & (\mathop{\bf tr} V + w)/2 \\*[1ex]
\mbox{subject to} & \left[\begin{array}{cc}
V & x \\ x^H & w \end{array}\right] \succeq 0 \\*[1ex]
& \mbox{$V$ is Toeplitz},
\end{array}
\end{equation}
with variables $w$ and $V\in\mathbf{H}^n$ (the $n\times n$ Hermitian matrices).
This result can be proved via convex duality and
semidefinite characterizations of bounded trigonometric
polynomials \cite{dCG12}, or directly by referring to
Carath\'eodory's decomposition of positive semidefinite
Toeplitz matrices \cite{TBSR:13}.
More generally, one can consider the atomic norm of the set of matrices
\[
C = \{ vw^H \in{\mbox{\bf C}}^{n\times m} \mid
v \in C_\mathrm{e}, \; \|w\| =1 \}.
\]
The atomic norm for this set, evaluated at a matrix
$X\in{\mbox{\bf C}}^{n\times m}$, is the optimal value of
the SDP
\begin{equation} \label{e-Ce-sdp-matrix}
\begin{array}{ll}
\mbox{minimize} & (\mathop{\bf tr} V + \mathop{\bf tr} W)/2 \\*[1ex]
\mbox{subject to} & \left[\begin{array}{cc}
V & X \\ X^H & W \end{array}\right] \succeq 0 \\*[1ex]
& \mbox{$V$ is Toeplitz},
\end{array}
\end{equation}
with variables $V\in\mathbf{H}^n$ and $W\in\mathbf{H}^m$; see \cite{YX14,LC14,Gra15}.
Further extensions, that place restrictions on the parameter
$\omega$ in the definition~(\ref{e-Ce}), can be found in
\cite{MCKX14,MCKX15}.
In this paper we discuss extensions of the SDP
representations~(\ref{e-Ce-sdp}) and~(\ref{e-Ce-sdp-matrix}) to
a larger class of atomic norms and gauge functions.
The starting point is the observation that $C_\mathrm e$ can be
parameterized as
\begin{equation} \label{e-Ce-def}
C_\mathrm e = \{ a \mid (\lambda G - F) a = 0, \; \lambda \in
\mathcal C, \; \|a\|=1\}
\end{equation}
where $\mathcal C$ is the unit circle in the complex plane, and
$F$ and $G$ are the $(n-1)\times n$ matrices
\[
F = \left[\begin{array}{cc} 0 & I_{n-1} \end{array}\right], \qquad
G = \left[\begin{array}{cc} I_{n-1} & 0 \end{array}\right].
\]
We generalize~(\ref{e-Ce-def}) in three ways and derive semidefinite
representations of the corresponding atomic norms.
The first generalization is to replace $\lambda G - F$ with an arbitrary
matrix pencil. Second, we allow $\mathcal C$ to be an arbitrary circle or
line in the complex plane, or a segment of a line or a circle.
Third, we replace the normalization $\|a\|=1$ with
a condition of the type $\|Ea\|\leq 1$ where $E$ is not necessarily
full column rank.
Specific examples of these extensions, with different choices
of $F$, $G$, and $\mathcal C$, are discussed in
sections~\ref{s-conic-examples}--\ref{s-rational}.
We present direct, constructive proofs, based on elementary matrix
algebra, of the semidefinite representations of the atomic norms.
These results are the subject of
sections~\ref{s-factorization} and~\ref{s-sdp},
and appendix~\ref{s-matrix-fact}.
In section~\ref{s-duality} we derive the convex conjugates of the
atomic norms and gauge functions, and discuss the relation between
the dual SDP representations and the Kalman-Yakubovich-Popov
lemma from linear system theory.
Appendix~\ref{s-slater} contains a discussion of the properties
of the matrix pencil $\lambda F-G$ that are needed to ensure
strong duality in the dual problems.
In section~\ref{s-sp-ex} the SDP formulations are illustrated with several
applications in signal processing.
\section{Positive semidefinite matrix factorization}
\label{s-factorization}
Throughout the paper we assume that $F$ and $G$ are complex matrices
of size $p\times n$, and $\Phi$ and $\Psi$ are Hermitian $2\times 2$
matrices with $\mathrm{d}et\Phi < 0$.
We define
\begin{equation} \label{e-dict}
\mathcal A = \{ a \in {\mbox{\bf C}}^n \mid
(\mu G-\nu F) a = 0, \; (\mu,\nu) \in \mathcal C\},
\end{equation}
where
\begin{equation} \label{e-mC}
\mathcal C = \left\{(\mu,\nu) \in{\mbox{\bf C}}^2 \mid
(\mu,\nu) \neq 0, \; q_\Phi(\mu,\nu) =0, \;
q_\Psi(\mu,\nu) \leq 0 \right\}.
\end{equation}
Here $q_\Phi$, $q_\Psi$ are the quadratic forms defined by $\Phi$ and
$\Psi$:
\begin{equation} \label{e-g-def}
q_\Phi(\mu,\nu)
= \left[\begin{array}{c} \mu \\ \nu \end{array}\right]^H \Phi
\left[\begin{array}{c} \mu \\ \nu \end{array}\right], \qquad
q_\Psi(\mu,\nu)
= \left[\begin{array}{c} \mu \\ \nu \end{array}\right]^H \Psi
\left[\begin{array}{c} \mu \\ \nu \end{array}\right].
\end{equation}
The set $\mathcal C$ is a subset of a line or circle in the complex
plane, expressed in homogeneous coordinates,
as explained in appendix~\ref{s-regions}.
If $\Phi_{11} \neq 0$ or $\Psi_{11} > 0$, then
$\nu\neq 0$ for all elements $(\mu,\nu) \in \mathcal C$, and we
can simplify the definition of $\mathcal A$ as
\begin{equation} \label{e-dict-finite}
\mathcal A = \{ a \in {\mbox{\bf C}}^n \mid
(\lambda G- F) a = 0, \; (\lambda,1) \in \mathcal C\}.
\end{equation}
If $\Phi_{11} = 0$ and $\Psi_{11} \leq 0$, then the pair $(1,0)$
is also in $\mathcal C$ and the set $\mathcal A$ in~(\ref{e-dict})
is the union of the right-hand side of~(\ref{e-dict-finite})
and the nullspace of $G$.
Examples of sets $\mathcal A$ are given in
sections~\ref{s-conic-examples}--\ref{s-rational}.
The purpose of this section is to discuss a semidefinite representation of
the convex hull of the set of matrices $aa^H$ with $a\in\mathcal A$,
{\it i.e.}, the set
\begin{equation} \label{e-KA}
\conv{ \{ aa^H \mid a\in\mathcal A\}}
= \{ \sum_{k=1}^m a_ka_k^H \mid a\in\mathcal A\}.
\end{equation}
\subsection{Conic decomposition}
The key decomposition result (Theorem~\ref{t-decomp}) is known under
various forms
in system theory, signal processing, and moment theory
\cite{KaS:66,KrN:77,GrS:84}.
Our purpose is to give a simple semidefinite formulation that encompasses
a wide variety of interesting special cases,
and to present a constructive proof
that can be implemented using the basic decompositions of
numerical linear algebra (specifically, symmetric eigenvalue,
singular value, and Schur decompositions).
\begin{theorem} \label{t-decomp}
Let $\mathcal A$ be defined by~(\ref{e-dict}) and~(\ref{e-mC}),
where $F$, $G\in{\mbox{\bf C}}^{p\times n}$ and $\Phi$, $\Psi\in\mathbf{H}^2$
with $\mathrm{d}et\Phi < 0$.
If $X\in\mathbf{H}^n$ is a positive semidefinite matrix of rank $r\geq 1$
that satisfies
\begin{eqnarray}
&
\Phi_{11} FXF^H + \Phi_{21} FXG^H + \Phi_{12} GXF^H + \Phi_{22} GXG^H = 0
& \label{e-phi} \\
&
\Psi_{11} FXF^H + \Psi_{21} FXG^H + \Psi_{12} GXF^H + \Psi_{22} GXG^H
\preceq 0,
& \label{e-psi}
\end{eqnarray}
then $X$ can be decomposed as
\begin{equation} \label{e-X-decomp}
X = \sum\limits_{k=1}^r a_ka_k^H,
\end{equation}
with linearly independent vectors
$a_1$, \ldots, $a_r \in \mathcal A$.
\end{theorem}
\noindent\emph{Proof.}\ \
We start from any factorization $X=YY^H$ where
$Y\in{\mbox{\bf C}}^{n\times r}$ has rank $r$.
It follows from Lemma~\ref{l-quad-eq-ineq-general}
in appendix~\ref{s-matrix-fact}, applied to the
matrices $U=FY$ and $V=GY$, that
there exist a matrix $W\in{\mbox{\bf C}}^{p\times r}$,
a unitary matrix $Q\in{\mbox{\bf C}}^{r\times r}$,
and two vectors $\mu, \nu \in {\mbox{\bf C}}^r$ such that
\begin{equation} \label{e-dcmp-pf}
FYQ = W \mathop{\bf diag}(\mu), \qquad GYQ = W\mathop{\bf diag}(\nu), \qquad
(\mu_i, \nu_i) \in \mathcal C, \quad i=1,\ldots,r.
\end{equation}
Choosing $a_k$ equal to the $k$th column of $YQ$
gives the decomposition~(\ref{e-X-decomp}).
$\Box$
Viewed geometrically, the theorem says that
(\ref{e-KA})~is the set of positive semidefinite
matrices $X$ that satisfy~(\ref{e-phi}) and~(\ref{e-psi}).
It is useful to note that the proof of Lemma~\ref{l-quad-eq-ineq-general}
in the appendix is constructive and gives a simple algorithm, based
on singular value and Schur decompositions, for computing
the matrices $W$, $Q$ and the vectors $\mu$, $\nu$.
In the following three sections we illustrate the decomposition
in Theorem~\ref{t-decomp} with different choices of
$F$, $G$, $\Phi$, $\Psi$.
\subsection{Trigonometric polynomials} \label{s-conic-examples}
\paragraph{Complex exponentials}
As a first example, we take $p=n-1$,
\begin{equation} \label{e-FG-toep}
F = \left[\begin{array}{cc} 0 & I_{n-1} \end{array}\right],
\qquad
G = \left[\begin{array}{cc}
I_{n-1} & 0 \end{array}\right],
\qquad
\Phi = \Phi_\mathrm u =
\left[\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right], \qquad
\Psi = 0.
\end{equation}
A nonzero pair $(\mu,\nu)$ satisfies
$q_\Phi(\mu,\nu) = |\mu|^2 - |\nu|^2 =0$
only if $\mu$ and $\nu$ are nonzero and $\lambda = \mu/\nu$ is on the
unit circle.
The condition $(\lambda G - F)a = 0$ in the definition of $\mathcal A$
gives a recursion
\[
\lambda a_1 = a_2, \qquad
\lambda a_2 = a_3, \qquad \ldots, \qquad \lambda a_{n-1} = a_n.
\]
Defining $\exp(\mathrm{j}\omega)= \lambda$, we find that $\mathcal A$ contains
the vectors
\begin{equation} \label{e-ak-toep}
a = c \, (1, e^{\mathrm{j}\omega}, e^{\mathrm{j} 2\omega}, \ldots, e^{\mathrm{j} (n-1)\omega}),
\end{equation}
for all $\omega \in [0,2\pi)$ and $c\in{\mbox{\bf C}}$.
The matrix constraints~(\ref{e-phi})--(\ref{e-psi}) reduce to
$FXF^H = GXG^H$,
{\it i.e.}, $X$ is a Toeplitz matrix.
Theorem~\ref{t-decomp} therefore states that every $n\times n$ positive
semidefinite Toeplitz matrix can be decomposed as
\begin{equation} \label{e-carath}
X = \sum_{k=1}^r |c_k|^2
\left[\begin{array}{c}
1 \\ e^{\mathrm{j}\omega_k} \\ e^{\mathrm{j} 2\omega_k} \\ \vdots \\
e^{\mathrm{j} (n-1)\omega_k} \end{array}\right]
\left[\begin{array}{c}
1 \\ e^{\mathrm{j}\omega_k} \\ e^{\mathrm{j} 2\omega_k} \\ \vdots \\
e^{\mathrm{j} (n-1)\omega_k} \end{array}\right]^H,
\end{equation}
with $c_k\neq 0$ and
distinct $\omega_1$, \ldots, $\omega_r$.
This is often called the Carath\'eodory parameterization of
positive semidefinite Toeplitz matrices \cite[page 170]{StM:97}.
For this example, the algorithm outlined in the proof of
Theorem~\ref{t-decomp} and Lemma~\ref{l-quad-eq-ineq-general}
reduces to the following.
Compute a factorization $X = YY^H$ where $Y\in{\mbox{\bf C}}^{n\times r}$
with rows $y_k^H$, $k=1,\ldots,n$.
Then find a unitary $r\times r$ matrix $\Lambda$
that satisfies
\[
\left[\begin{array}{c} y_2^H \\ \vdots \\ y_n^H \end{array}\right]
= \left[\begin{array}{c} y_1^H \\ \vdots \\ y_{n-1}^H \end{array}\right]
\Lambda,
\]
and compute a Schur decomposition $\Lambda = Q \mathop{\bf diag}(\lambda)Q^H$.
The eigenvalues give $\lambda_k = \exp(\mathrm{j}\omega_k)$, $k=1,\ldots,r$, and
the columns of $YQ$ are the vectors $a_k$.
\paragraph{Restricted complex exponentials}
Define $F$, $G$, $\Phi$ as in~(\ref{e-FG-toep}), and
\[
\Psi = \left[\begin{array}{cc} 0 & -e^{\mathrm{j}\alpha} \\
-e^{-\mathrm{j}\alpha} & 2\cos\beta\end{array}\right]
\]
with $\alpha \in [0, 2\pi)$ and $\beta\in [0,\pi)$.
The elements $a\in\mathcal A$ have the same general
form~(\ref{e-ak-toep}), with the added constraint that
$\cos\beta \leq \cos(\omega-\alpha)$.
Since we can restrict $\omega$ to the interval
$[\alpha - \pi, \alpha + \pi]$, this
is equivalent to $|\omega - \alpha| \leq \beta$.
The constraints~(\ref{e-phi})--(\ref{e-psi}) specify that
$X$ is Toeplitz and satisfies the matrix inequality
\begin{equation} \label{e-toep-restricted}
-e^{-\mathrm{j}\alpha} FXG^H - e^{\mathrm{j}\alpha}GXF^H + 2 (\cos\beta) GXG^H \preceq 0.
\end{equation}
The theorem states that a positive semidefinite Toeplitz matrix
of rank $r$
satisfies~(\ref{e-toep-restricted}) if and only if it can be
decomposed as~(\ref{e-carath}) with nonzero $c_k$ and
$|\omega_k - \alpha| \leq \beta$ for $k=1,\ldots,r$.
\paragraph{Real trigonometric functions}
Next consider $p=n-1$,
\[
G = \left[\begin{array}{cccccc}
1 & 0 & 0 & \cdots & 0 & 0 \\
0 & 2 & 0 & \cdots & 0 & 0 \\
0 & 0 & 2 & \cdots & 0 & 0 \\
\vdots & \vdots & \vdots & \mathrm{d}dots & \vdots & \vdots \\
0 & 0 & 0 & \cdots & 2 & 0
\end{array}\right], \qquad
F = \left[\begin{array}{ccccccc}
0 & 1 & 0 & \cdots & 0 & 0 & 0 \\
1 & 0 & 1 & \cdots & 0 & 0 & 0 \\
0 & 1 & 0 & \cdots & 0 & 0 & 0 \\
\vdots & \vdots & \vdots & & \vdots & \vdots &\vdots \\
0 & 0 & 0 & \cdots & 1 & 0 & 1
\end{array}\right],
\]
and
\[
\Phi = \Phi_\mathrm r = \left[\begin{array}{cc}
0 & \mathrm{j} \\ -\mathrm{j} & 0 \end{array}\right], \qquad
\Psi = \Phi_\mathrm u = \left[\begin{array}{cc}
1 & 0 \\ 0 & -1 \end{array}\right].
\]
A nonzero pair $(\mu,\nu)$ satisfies
$q_\Phi(\mu,\nu) = \mathrm{j}(\bar\mu\nu - \mu\bar \nu) = 0$
and $q_\Psi(\mu,\nu) = |\mu|^2 - |\nu|^2 \leq 0$
only if $\nu \neq 0$ and $\lambda = \mu/\nu$ is real with
$|\lambda| \leq 1$.
The condition $(\lambda G - F)a = 0$ gives a recursion
\[
\lambda a_1 = a_2, \qquad
2\lambda a_2 = a_1+a_3, \qquad
\ldots, \qquad
2\lambda a_{n-1} = a_{n-2} +a_n.
\]
If we write $\lambda = \cos\omega$, we recognize the recursion
$2\cos\omega \cos{k\omega} = \cos{(k-1)\omega} + \cos{(k+1)\omega}$
and find that $\mathcal A$ contains the vectors
\[
a = c \, (1,\, \cos\omega,\, \cos{2\omega},\, \ldots, \,
\cos{(n-1)\omega}),
\]
for all $\omega\in[0,2\pi)$ and all $c$.
With the same $F$ and
$G = [\begin{array}{cc} 2I_{n-1} & 0 \end{array}]$,
the condition $(\lambda G-F)a = 0$ reduces to
\[
2\lambda a_1 = a_2, \qquad
2\lambda a_2 = a_1 + a_3, \qquad
\ldots, \qquad
2\lambda a_{n-1} = a_{n-2} + a_n.
\]
If we write $\lambda = \cos\omega$, the solutions are the vectors
\[
a = c\; (1, \frac{\sin{2\omega}}{\sin{\omega}}, \,
\frac{\sin{3\omega}}{\sin{\omega}}, \, \ldots, \,
\frac{\sin{n\omega}}{\sin{\omega}}),
\]
for all $\omega \in [0,2\pi)$ and all $c$.
\paragraph{Trigonometric vector polynomials}
We take $p=(k-1)l$, $n=kl$,
and replace $F$ and $G$ in~(\ref{e-FG-toep})
with
\[
F = \left[\begin{array}{ccccc}
0 & I & 0 & \cdots & 0 \\
0 & 0 & I & \cdots & 0 \\
\vdots & \vdots & \vdots & \mathrm{d}dots & \vdots \\
0 & 0 & 0 & \cdots & I \end{array}\right],
\qquad
G = \left[\begin{array}{ccccc}
I & 0 & \cdots & 0 & 0\\
0 & I & \cdots & 0 & 0 \\
\vdots & \vdots & \mathrm{d}dots & \vdots & \vdots \\
0 & 0 & \cdots & I & 0 \end{array}\right],
\]
and blocks of size $l\times l$.
Then $\mathcal A$ contains the vectors of the form
\[
a = (1,\, e^{\mathrm{j}\omega}, \, e^{\mathrm{j} 2\omega}, \, \ldots, \,
e^{\mathrm{j} (k-1)\omega} )\otimes c,
\]
for all $c\in{\mbox{\bf C}}^l$ and $\omega \in [0,2\pi)$,
where $\otimes$ denotes Kronecker product.
\subsection{Polynomials}
\paragraph{Real powers}
Next, define $F$, $G$ as in~(\ref{e-FG-toep}), and
\begin{equation} \label{e-power-moments}
\Phi = \Phi_\mathrm r =
\left[\begin{array}{cc} 0 & \mathrm{j} \\ -\mathrm{j} & 0 \end{array} \right], \qquad
\Psi = 0.
\end{equation}
A pair $(\mu,\nu)$ satisfies $q_\Phi(\mu,\nu) = 0$ if and only if
$\bar \mu\nu$ is real.
If $(\mu,\nu) \neq 0$, we either have $\nu=0$ and $\mu$ arbitrary,
or $\nu\neq 0$ and $\lambda = \mu/\nu$ real.
The set $\mathcal A$ therefore contains the vectors
\[
a = c \, (1, \lambda, \lambda^2 , \ldots, \lambda^{n-1}), \qquad
a = c \, (0, 0, \ldots, 0, 1)
\]
for all $\lambda\in{\mbox{\bf R}}$ and $c$.
The matrix constraints~(\ref{e-phi})--(\ref{e-psi}) reduce to
$FXG^H = GXF^H$,
{\it i.e.}, $X$ is a symmetric (real) Hankel matrix.
Hence, a real symmetric positive semidefinite Hankel matrix of rank
$r$ can be decomposed in one of two forms
\[
X = \sum_{k=1}^r
c_k^2
\left[\begin{array}{c} 1 \\ \lambda_k \\ \vdots \\ \lambda_k^{n-2}
\\ \lambda_k^{n-1}
\end{array}\right]
\left[\begin{array}{c} 1 \\ \lambda_k \\ \vdots \\ \lambda_k^{n-2}
\\ \lambda_k^{n-1}
\end{array}\right]^T, \qquad
X = \sum_{k=1}^{r-1}
c_k^2
\left[\begin{array}{c} 1 \\ \lambda_k \\ \vdots \\ \lambda_k^{n-2}
\\ \lambda_k^{n-1}
\end{array}\right]
\left[\begin{array}{c} 1 \\ \lambda_k \\ \vdots \\ \lambda_k^{n-2}
\\ \lambda_k^{n-1}
\end{array}\right]^T + |c_r|^2
\left[\begin{array}{c} 0 \\ 0 \\ \vdots \\ 0 \\ 1 \end{array}\right]
\left[\begin{array}{c} 0 \\ 0 \\ \vdots \\ 0 \\ 1 \end{array}\right]^T,
\]
with distinct real $\lambda_k$ and nonzero $c_k$.
\paragraph{Restricted polynomials}
If $\Psi=0$ in~(\ref{e-power-moments}) is replaced by
\[
\Psi = \left[\begin{array}{cc}
2 & -(\alpha + \beta) \\ -(\alpha +\beta) & 2\alpha \beta
\end{array}\right]
\]
where $-\infty < \alpha < \beta < \infty$, then $\mathcal A$ contains
all vectors $a = c (1, \lambda, \ldots, \lambda^{n-1})$
with $\lambda \in [\alpha, \beta]$.
The matrix constraints require $X$ to be a real symmetric Hankel matrix
that satisfies
\[
2FXF^H - (\alpha+\beta) (FXG^H + GXF^H) + 2\alpha\beta GXG^H \preceq 0.
\]
\paragraph{Orthogonal polynomials}
Let $p_0(\lambda)$, $p_1(\lambda)$, $p_2(\lambda)$,~\ldots\ be
a sequence of real polynomials on ${\mbox{\bf R}}$, with $p_i$ of degree $i$.
It is well known that the polynomials
are orthonormal with respect to an inner product that satisfies the
property
\begin{equation} \label{e-shift}
\langle f(\lambda), \lambda g(\lambda) \rangle =
\langle \lambda f(\lambda), g(\lambda)\rangle
\end{equation}
(for example, an inner product of the form
$\langle f, g\rangle =
\int f(\lambda) g(\lambda) w(\lambda) d\lambda$ with $w(\lambda) \geq 0$)
if and only if the polynomials satisfy
a three-term recursion
\begin{equation} \label{e-3-term}
\beta_{i+1}p_{i+1}(\lambda) =
(\lambda-\alpha_i) p_i(\lambda) - \beta_i p_{i-1}(\lambda),
\end{equation}
where $p_{-1}(\lambda) = 0$ and
$p_0(\lambda) = 1/d_0$ where $d_0^2 = \langle 1,1\rangle$.
This can be seen as follows \cite{GoK:83}.
Suppose $p_0$, \ldots, $p_{n-1}$ is any set of polynomials, with $p_i$ of
degree $i$. Then $\lambda p_i(\lambda)$ can be expressed as a linear
combination of the polynomials $p_0(\lambda)$, \ldots,
$p_{i+1}(\lambda)$, and therefore
\begin{equation} \label{e-3term-pf}
\lambda \left[\begin{array}{c}
p_0(\lambda) \\ p_1(\lambda) \\ \vdots \\ p_{n-2}(\lambda)
\end{array}\right]
= \left[\begin{array}{cc} J & \beta_{n-1} e_{n-1} \end{array}\right]
\left[\begin{array}{c}
p_0(\lambda) \\ p_1(\lambda) \\ \vdots \\ p_{n-1}(\lambda)
\end{array}\right]
\end{equation}
for some lower-Hessenberg matrix $J$
({\it i.e.}, satisfying $J_{ij} = 0$ for $j>i+1$).
Let $\langle \cdot, \cdot \rangle$ be an inner product
on the space of polynomials of degree $n-1$ or less.
Taking inner products on both sides of~(\ref{e-3term-pf}), we find that
\[
H = JG + \beta_{n-1} e_{n-1}g^T
\]
where
\[
H_{ij} = \langle \lambda p_{i-1}(\lambda), p_{j-1}(\lambda)\rangle,
\qquad
G_{ij} = \langle p_{i-1}(\lambda), p_{j-1}(\lambda)\rangle,
\qquad
g_j = \langle p_{n-1}(\lambda), p_{j-1}(\lambda)\rangle,
\]
for $i,j = 1, \ldots,n-1$. The polynomials are orthonormal for the
inner product if and only if $G=I$ and $g=0$.
The inner product satisfies the property~(\ref{e-shift})
if and only if $H$ is symmetric.
Hence if the polynomials are orthonormal for an inner product
that satisfies~(\ref{e-shift}), then $J$ is a symmetric tridiagonal
matrix.
If we use the notation
\begin{equation} \label{e-jacobi}
J = \left[\begin{array}{ccccccc}
\alpha_0 & \beta_1 & 0 & \cdots & 0 & 0 \\
\beta_1 & \alpha_1 & \beta_2 & \cdots & 0 & 0 \\
0 & \beta_2 & \alpha_2 & \cdots & 0 & 0 \\
\vdots & \vdots & \vdots & \mathrm{d}dots & \vdots & \vdots \\
0 & 0 & 0 & \cdots & \alpha_{n-3} & \beta_{n-2} \\
0 & 0 & 0 & \cdots & \beta_{n-2} & \alpha_{n-2}
\end{array}\right],
\end{equation}
the recursion~(\ref{e-3-term}) follows.
Conversely, if the three-term recursion holds, and we define
the inner product by setting $G=I$, $g=0$, then $H$ is symmetric and
the inner product satisfies~(\ref{e-shift}).
Now consider~(\ref{e-dict}) and~(\ref{e-mC}), with $p=n-1$ and
\[
\Phi = \Phi_\mathrm r, \qquad \Psi = 0, \qquad
G = \left[\begin{array}{cc}
I_{n-1} & 0
\end{array}\right], \qquad
F = \left[\begin{array}{cc}
J & \beta_{n-1}e_{n-1} \end{array}\right],
\]
where $J$ is the Jacobi matrix~(\ref{e-jacobi}) of a system of
orthogonal polynomials.
Then $(\mu,\nu) \in \mathcal C$ if and only if either
$\nu\neq 0$ and $\lambda = \mu/\nu \in {\mbox{\bf R}}$, or $\nu=0$.
The set contains the vectors $a$ of the form
\[
a = c \, (p_0(\lambda),\, p_1(\lambda), \, p_2(\lambda),\ldots,
\, p_{n-1}(\lambda)), \qquad
a = c \, (0,0, \ldots, 0,1)
\]
for all $\lambda \in{\mbox{\bf R}}$.
\subsection{Rational functions} \label{s-rational}
As a final example, we consider the controllability pencil of a linear
system:
\begin{equation} \label{e-state-space}
G = \left[\begin{array}{cc} I & 0 \end{array}\right],
\qquad
F = \left[\begin{array}{cc} A & B \end{array}\right],
\end{equation}
where $A \in {\mbox{\bf C}}^{n_\mathrm s \times n_\mathrm s}$
and $B\in {\mbox{\bf C}}^{n_\mathrm s \times m}$.
With this choice, $\mathcal A$ contains the vectors $a =(x,u)$
that satisfy the equality $(\mu I - \nu A) x = \nu Bu$ for some
$(\mu, \nu) \in \mathcal C$.
Since $(\mu,\nu) \neq 0$, we either have $\nu= 0$ and $x=0$,
or $\nu\neq 0$ and $((\mu/\nu) I - A) x= Bu$.
If $A$ has no eigenvalues $\lambda$ that satisfy
$(\lambda, 1) \in \mathcal C$, then $\mathcal A$ contains the vectors
\[
a = \left[\begin{array}{c} (\lambda I -A)^{-1}B u \\ u \end{array}\right]
\]
for all $(\lambda, 1)\in\mathcal C$ and all $u\in{\mbox{\bf C}}^m$.
If $\mathcal C$ includes the point $(1,0)$ at infinity, then
$\mathcal A$ also contains the vectors $(0,u)$ for all $u\in{\mbox{\bf C}}^m$.
This can be extended to the controllability pencil of a descriptor system
\[
G = \left[\begin{array}{cc} E & 0 \end{array}\right],
\qquad
F = \left[\begin{array}{cc} A & B \end{array}\right],
\]
where $E\in{\mbox{\bf C}}^{n_\mathrm s \times n_\mathrm s}$ is possibly
singular.
With this choice, $\mathcal A$ contains the vectors $a =(x,u)$
that satisfy the equality $(\mu E - \nu A) x = \nu Bu$ for some
$(\mu, \nu) \in \mathcal C$.
If $\mathrm{d}et(\mu E - \nu A) \neq 0$ for all $(\mu,\nu) \in \mathcal C$,
then $\mathcal A$ contains all vectors
\[
a = \left[\begin{array}{c} (\lambda E -A)^{-1}B u \\ u \end{array}\right]
\]
for all $(\lambda, 1)\in \mathcal C$ and all $u\in{\mbox{\bf C}}^m$.
If $(0,1) \in \mathcal C$, then $\mathcal A$ also contains the points
$(0,u)$ for all $u\in{\mbox{\bf C}}^m$.
\section{Semidefinite representation of gauges and atomic norms}
\label{s-sdp}
A function $g$ is called a \emph{gauge} if it is convex,
positively homogeneous ($g(tx) = tg(x)$ for $t>0$), nonnegative,
and vanishes at the origin
\cite[section 15]{Roc:70}, \cite[chapter 1]{KrN:77}.
Examples are the \emph{(Minkowski) gauges} of nonempty convex sets $C$,
which are defined as
\[
g(x) = \inf{\{ t \geq 0 \mid x \in t C\}}.
\]
Conversely, if $g$ is a gauge, then it is the Minkowski gauge of the set
$C = \{ x \mid g(x) \leq 1\}$.
A gauge is a norm if it is defined everywhere, positive except at
the origin, and symmetric ($g(x) = g(-x)$).
The gauge of the convex hull $\conv C$ of a set $C$ can be expressed as
\[
g(x) = \inf{\{\sum_{k=1}^r \theta_k \mid
x = \sum_{k=1}^r \theta_k x_k, \; \theta_k \geq 0, \; x_k \in C, \;
k=1,\ldots,r \}}.
\]
The minimum is over all possible decompositions of $x$ as a nonnegative
combination of a finite number of elements of $C$.
The gauge of the convex hull of a compact set is
also called the \emph{atomic norm} associated with the set
\cite{CRPW:12}.
\subsection{Symmetric matrices} \label{s-gauge}
Let $F$, $G$, $\Phi$, $\Psi$ be defined as in Theorem~\ref{t-decomp}.
We assume that the set $\mathcal C$ defined in~(\ref{e-mC}) is not empty.
In this section we discuss the gauge of the convex hull of the set
\[
C = \{ aa^H \in \mathbf{H}^n \mid a \in \mathcal A, \; \|a\|=1\},
\]
where $\mathcal A$ is defined in~(\ref{e-dict}).
The gauge of the convex hull of $C$ is the function
\begin{eqnarray}
g(X) & = & \inf{\{\sum_{k=1}^r \theta_k \mid
X = \sum_{k=1}^r \theta_k a_ka_k^H, \; \theta_k \geq 0, \;
a_k \in \mathcal A, \; \|a_k\| =1, \; k=1,\ldots, r\}}
\label{e-gammaC-def-a} \\
& = & \inf{\{\sum_{k=1}^r \|a_k\|^2 \mid
X = \sum_{k=1}^r a_ka_k^H, \; a_k \in \mathcal A, \; k=1,\ldots, r\}}.
\label{e-gammaC-def}
\end{eqnarray}
The second expression follows from the fact that if $a \in \mathcal A$
then $\beta a\in \mathcal A$ for all $\beta$.
The expressions $\sum_k \theta_k$ and $\sum_k \|a_k\|^2$ in these
minimizations take only two possible values: $\mathop{\bf tr} X$ if $X$
can be decomposed as in~(\ref{e-gammaC-def-a})
and~(\ref{e-gammaC-def}), and $+\infty$ otherwise.
Theorem~\ref{t-decomp} tells us that a decomposition exists
if only if $X$ is positive semidefinite and satisfies the two
constraints~(\ref{e-phi}),~(\ref{e-psi}).
Therefore
\begin{equation} \label{e-gammaC-lmi}
g(X) = \left\{ \begin{array}{ll}
\mathop{\bf tr} X & \mbox{$X\succeq 0$, (\ref{e-phi}),~(\ref{e-psi})} \\
+ \infty & \mbox{otherwise.}
\end{array}\right.
\end{equation}
Now consider an optimization problem in which we minimize the sum of a
function $f:\mathbf{H}^n \rightarrow{\mbox{\bf R}}$ and the gauge
defined in~(\ref{e-gammaC-def}) and~(\ref{e-gammaC-lmi}),
\begin{equation} \label{e-f-plus-gamma}
\begin{array}{ll}
\mbox{minimize} & f(X) + g(X).
\end{array}
\end{equation}
If we substitute the definition~(\ref{e-gammaC-def}),
this can be written as
\begin{equation} \label{e-f-plus-gauge}
\begin{array}{ll}
\mbox{minimize} & f(X) + \sum\limits_{k=1}^r \|a_k\|^2 \\*[1ex]
\mbox{subject to} & X = \sum\limits_{k=1}^r a_ka_k^H \\*[2ex]
& a_k \in \mathcal A, \; k=1,\ldots, r.
\end{array}
\end{equation}
The variables are $X$ and the parameters $a_1$, \ldots, $a_r$, and
$r$ of the decomposition of $X$.
This formulation shows that the function $g(X)$
in~(\ref{e-f-plus-gamma}) acts as a regularization term that
promotes a structured low rank property in $X$.
If we substitute the expression~(\ref{e-gammaC-lmi}) we obtain the
equivalent formulation
\begin{equation} \label{e-gauge-sdp-rep}
\begin{array}{ll}
\mbox{minimize} & f(X) + \mathop{\bf tr} X \\
\mbox{subject to}
& \Phi_{11} FXF^H + \Phi_{21} FXG^H +
\Phi_{12} GXF^H + \Phi_{22} GXG^H = 0\\
& \Psi_{11} FXF^H + \Psi_{21} FXG^H +
\Psi_{12} GXF^H + \Psi_{22} GXG^H \preceq 0 \\
& X \succeq 0.
\end{array}
\end{equation}
This problem is convex if $f$ is convex.
A useful generalization of~(\ref{e-gammaC-def}) is the
gauge of the convex hull of
\[
C = \{ aa^H \mid a\in \mathcal A, \; \|Ea\| \leq 1\}
\]
where $E$ may have rank less than $n$.
The gauge of $\conv{C}$ is
\begin{equation} \label{e-gammaCP-def}
g(X) = \inf{\{ \sum_{k=1}^r \theta_k \mid
X =\sum_{k=1}^r \theta_k a_ka_k^H, \; \theta_k\geq 0, \;
a_k\in\mathcal A, \; \|Ea_k\| \leq 1, \; k=1,\ldots, r \}}.
\end{equation}
The variables $\theta_k$ in this definition can be eliminated
by making the following observation.
Suppose that the directions of the vectors $a_k$ in the decomposition
of $X$ in~(\ref{e-gammaCP-def}) are given, but not their norms or
the coefficients $\theta_k$.
If $0 < \|Ea_k\| < 1$, we can decrease $\theta_k$ by scaling
$a_k$ until $\|Ea_k\| = 1$.
If $Ea_k = 0$, $\theta_k$ can be made arbitrarily small by
scaling $a_k$.
Hence, we obtain the same result if we use $\sqrt \theta_k a_k$ as
variables and write the infimum as:
\begin{equation} \label{e-gammaC-def-weighted}
g(X)
= \inf{\{ \sum_{k=1}^r \|Ea_k\|^2 \mid X = \sum_{k=1}^r a_ka_k^H, \;
a_k \in \mathcal A, \; k=1,\ldots, r\}}.
\end{equation}
Therefore $g(X) = \sum_k \|Ea_k\|^2 = \mathop{\bf tr}(EXE^H)$ if $X$
can be decomposed as in~(\ref{e-gammaC-def-weighted}) and
$+\infty$ otherwise.
Using Theorem~\ref{t-decomp} we can express this result as
\begin{equation} \label{e-gammaCP-lmi}
g(X)
= \left\{ \begin{array}{ll}
\mathop{\bf tr}(EXE^H) & \mbox{$X\succeq 0$, (\ref{e-phi}), (\ref{e-psi})} \\
+ \infty & \mbox{otherwise.} \end{array}\right.
\end{equation}
Minimizing $f(X) + g(X)$ is equivalent to
the optimization problem
\begin{equation} \label{e-gauge-sdp-rep-E}
\begin{array}{ll}
\mbox{minimize} & f(X) + \sum\limits_{k=1}^r \|Ea_k\|^2 \\*[1ex]
\mbox{subject to} & X = \sum\limits_{k=1}^r a_ka_k^H \\*[1.5ex]
& a_k \in \mathcal A, \;\; k=1,\ldots, r,
\end{array}
\end{equation}
with variables $X$ and the parameters $a_1$, \ldots, $a_r$, $r$
of the decomposition of $X$.
When $E^HE=I$ this is the same as~(\ref{e-f-plus-gauge}).
By choosing different $E$ we assign different weights
to the vectors $a_k$.
Using the expression~(\ref{e-gammaCP-lmi}), the
problem~(\ref{e-gauge-sdp-rep-E}) can be written as
\begin{equation} \label{e-gauge-sdp-2}
\begin{array}{ll}
\mbox{minimize} & f(X) + \mathop{\bf tr}{(EXE^H)} \\
\mbox{subject to}
& \Phi_{11} FXF^H + \Phi_{21} FXG^H +
\Phi_{12} GXF^H + \Phi_{22} GXG^H = 0\\
& \Psi_{11} FXF^H + \Psi_{21} FXG^H +
\Psi_{12} GXF^H + \Psi_{22} GXG^H \preceq 0 \\
& X \succeq 0.
\end{array}
\end{equation}
\paragraph{Example}
Parametric line spectrum estimation is concerned with fitting
signal models of the form
\begin{equation} \label{e-sinusoids-in-noise}
y(t) = \sum_{k=1}^r c_k e^{\mathrm{j} \omega_k t} + v(t),
\end{equation}
where $v(t)$ is noise.
If the phase angles of $c_k$ are independent random variables,
uniformly distributed on $[-\pi,\pi]$, and $v(t)$ is
circular white noise with $\mathop{\bf E{}} |v(t)|^2 = \sigma^2$, then
the covariance matrix of $y(t)$ of order $n$ is given by
\begin{equation} \label{e-line-spec-covariance}
\left[\begin{array}{cccc}
r_0 & r_{-1} & \cdots & r_{-n+1} \\
r_1 & r_0 & \cdots & r_{-n+2} \\
\vdots & \vdots & \mathrm{d}dots & \vdots \\
r_{n-1} & r_{n-2} & \cdots & r_0
\end{array}\right]
=
\sigma^2 I + \sum\limits_{k=1}^r |c_k|^2
\left[\begin{array}{c}
1 \\ e^{\mathrm{j}\omega_k} \\ \vdots \\ e^{\mathrm{j} (n-1)\omega_k}
\end{array}\right]
\left[\begin{array}{c}
1 \\ e^{\mathrm{j}\omega_k} \\ \vdots \\ e^{\mathrm{j} (n-1)\omega_k}
\end{array}\right]^H,
\end{equation}
where $r_k = \mathop{\bf E{}}{(y(t) \overline{y(t-k)})}$
\cite[section 4.1]{StM:97}\cite[section 12.5]{PrM:96}.
Classical methods, such as MUSIC and ESPRIT,
are based on the eigenvalue decomposition of an
estimated covariance matrix.
With the formulation outlined in this section one can solve
related but more general covariance fitting problems, expressed as
\[
\begin{array}{ll}
\mbox{minimize} & f(R) +
n \sum\limits_{k=1}^r |c_k|^2 \\
\mbox{subject to} &
R = \sigma^2 I + \sum\limits_{k=1}^r |c_k|^2
\left[\begin{array}{c}
1 \\ e^{\mathrm{j}\omega_k} \\ \vdots \\
e^{\mathrm{j} (n-1)\omega_k} \end{array}\right]
\left[\begin{array}{c}
1 \\ e^{\mathrm{j}\omega_k} \\ \vdots \\
e^{\mathrm{j} (n-1)\omega_k} \end{array}\right]^H,
\end{array}
\]
with variables $R\in\mathbf{H}^n$, $\sigma^2$, $|c_k|$, $\omega_k$, and $r$,
where $f$ is a convex penalty or indicator function that measures
the quality of the fit between $R$ and the estimated covariance matrix.
This is equivalent to the convex optimization problem
\[
\begin{array}{ll}
\mbox{minimize} & f(X + tI) + \mathop{\bf tr} X \\
\mbox{subject to} & X \succeq 0, \; t\geq 0 \\
& \mbox{$X$ is Toeplitz}.
\end{array}
\]
A numerical example is given in section~\ref{s-sp-ex}.
\subsection{Non-symmetric matrices} \label{ss-main}
We define $F$, $G$, $E$, $\Phi$, $\Psi$, and $\mathcal A$ as in the
previous section, but add the assumption that the matrices $F$, $G$, and
$E$ are block-diagonal:
\begin{equation} \label{e-GF-diag}
G = \left[\begin{array}{cc}
G_1 & 0 \\ 0 & G_2 \end{array}\right], \qquad
F = \left[\begin{array}{cc}
F_1 & 0 \\ 0 & F_2 \end{array}\right], \qquad
E = \left[\begin{array}{cc}
E_1 & 0 \\ 0 & E_2 \end{array}\right].
\end{equation}
Here $F_1$, $G_1 \in {\mbox{\bf C}}^{p_1 \times n_1}$ and
$F_2$, $G_2 \in {\mbox{\bf C}}^{p_2 \times n_2}$
(possibly with $p_1$ or $p_2$ equal to zero).
The matrices $E_1$ and $E_2$ have $n_1$ and $n_2$ columns,
respectively.
In this section we discuss the function
\[
h(Y) = \frac{1}{2} \inf_{V,W} g(\left[\begin{array}{cc}
V & Y \\ Y^H & W \end{array}\right])
\]
of $Y\in {\mbox{\bf C}}^{n_1\times n_2}$, where $g$ is the function defined
in~(\ref{e-gammaC-def-weighted}) and~(\ref{e-gammaCP-lmi}).
Using~(\ref{e-gammaC-def-weighted}) we can write $h(Y)$ as
\begin{equation}
h(Y)
= \inf{ \{ \frac{1}{2} \sum_{k=1}^r (\|E_1v_k\|^2 + \|E_2w_k\|^2) \mid
Y = \sum_{k=1}^r v_k w_k^H, \; (v_k,w_k) \in \mathcal A\}},
\label{e-gauge-nonsymm-d}
\end{equation}
while the equivalent characterization~(\ref{e-gammaCP-lmi})
shows that $h(Y)$ is the optimal value of the SDP
\begin{equation} \label{e-gauge-nonsymm-sdp}
\begin{array}{ll}
\mbox{minimize} & \left(\mathop{\bf tr}(E_1VE_1^H) + \mathop{\bf tr}(E_2WE_2^H) \right)/2\\*[1ex]
\mbox{subject to}
& \Phi_{11} FXF^H + \Phi_{21} FXG^H +
\Phi_{12} GXF^H + \Phi_{22} GXG^H = 0\\*[.5ex]
& \Psi_{11} FXF^H + \Psi_{21} FXG^H +
\Psi_{12} GXF^H + \Psi_{22} GXG^H \preceq 0 \\*[.5ex]
& X = \left[\begin{array}{cc}
V & Y \\ Y^H & W \end{array}\right] \succeq 0,
\end{array}
\end{equation}
with $V$ and $W$ as variables.
This can be seen as an extension of the well-known SDP formulation of
the trace norm of a rectangular matrix.
If we take $F$ and $G$ to have zero row dimensions (equivalently,
define $\mathcal A = {\mbox{\bf C}}^{n_1} \times {\mbox{\bf C}}^{n_2}$ and
omit the first two constraints in~(\ref{e-gauge-nonsymm-sdp}))
and choose identity matrices for $E_1$ and $E_2$,
then $h(Y) = \|Y\|_*$, the trace norm of $Y$.
The block-diagonal structure of $F$ and $G$ implies that if
$(v,w)\in\mathcal A$, then $(\alpha v, \beta w)\in \mathcal A$
for all $\alpha$, $\beta$.
This observation leads to a number of useful equivalent expressions
for~(\ref{e-gauge-nonsymm-d}).
First, we note that $h(Y)$ can be written as
\begin{equation}
h(Y)
= \inf{ \{ \sum_{k=1}^r \|E_1v_k\| \|E_2w_k\| \mid
Y = \sum_{k=1}^r v_k w_k^H, \; (v_k,w_k) \in \mathcal A\}}.
\label{e-gauge-nonsymm-c}
\end{equation}
This follows from the fact
$\|E_1v_k\|^2 + \|E_2w_k\|^2 \geq 2\|E_1v_k\| \|E_2w_k\|$,
with equality if $\|E_1v_k\| = \|E_2w_k\|$.
If the decomposition of $Y$ in~(\ref{e-gauge-nonsymm-d})
involves a term $v_k w_k^H$ with $E_1v_k$ and $E_2w_k$ nonzero,
then replacing $v_k$ and $w_k$ with
\[
\tilde v_k = \frac{\|E_2w_k\|^{1/2}}{\|E_1v_k\|^{1/2}} v_k, \qquad
\tilde w_k = \frac{\|E_1v_k\|^{1/2}}{\|E_2w_k\|^{1/2}} w_k
\]
gives another valid decomposition with
\[
\frac{1}{2} (\|E_1 \tilde v_k\|^2 + \|E_2 \tilde w_k\|^2)
= \|E_1v_k\| \|E_2 w_k\|
\leq \frac{1}{2} (\|E_1 v_k\|^2 + \|E_2 w_k\|^2).
\]
If $E_1v_k = 0$ and $E_2w_k \neq 0$, then replacing
$v_k$ and $w_k$ with $\tilde v_k = \alpha v_k$,
$\tilde w_k = (1/\alpha) w_k$
gives an equivalent decomposition with
\[
\frac{1}{2} (\|E_1 \tilde v_k\|^2 + \|E_2 \tilde w_k\|^2)
= \frac{1}{2\alpha^2} \|E_2 w_k\|^2 \rightarrow 0
\]
as $\alpha$ goes to infinity.
The same argument applies when $E_1v_k \neq 0$ and $E_2w_k = 0$.
In all cases, therefore, the two
expressions~(\ref{e-gauge-nonsymm-d}) and~(\ref{e-gauge-nonsymm-c}) give
the same result.
From~(\ref{e-gauge-nonsymm-c}) we obtain two other useful expressions:
\begin{eqnarray}
h(Y) & = & \inf{ \{ \sum_{k=1}^r \|E_1v_k\| \mid
Y = \sum_{k=1}^r v_k w_k^H, \;
(v_k, w_k) \in \mathcal A, \; \|E_2w_k\| \leq 1\}}
\label{e-gauge-nonsymm-a} \\
& = & \inf{ \{ \sum_{k=1}^r \|E_2w_k\| \mid
Y = \sum_{k=1}^r v_k w_k^H, \;
(v_k,w_k) \in \mathcal A, \; \|E_1v_k\| \leq 1\}}.
\label{e-gauge-nonsymm-b}
\end{eqnarray}
This again follows from the property that the two components of
elements $(v_k, w_k)$ in $\mathcal A$ can be scaled independently.
At the optimal decomposition in~(\ref{e-gauge-nonsymm-a}), all
terms in the decomposition satisfy $E_2w_k = 0$ or $\|E_2w_k\|=1$.
In~(\ref{e-gauge-nonsymm-b}), all terms satisfy
$E_1v_k = 0$ or $\|E_1v_k\|=1$.
A final interpretation of $h$ is
\begin{equation}
h(Y) =
\inf{ \{ \sum_{k=1}^r \theta_k \mid
Y = \sum_{k=1}^r \theta_k v_k w_k^H, \; \theta_k \geq 0, \;
(v_k,w_k) \in \mathcal A, \; \|E_1v_k\| \leq 1, \; \|E_2w_k\| \leq 1
\}}.
\label{e-gauge-nonsymm-weighted}
\end{equation}
The equivalence with~(\ref{e-gauge-nonsymm-c}) follows from the fact
that if the optimal decomposition of $Y$
in~(\ref{e-gauge-nonsymm-weighted}) involves
the term $v_kw_k^H$, then the norms $\|E_1v_k\|$ and $\|E_2w_k\|$ will be
either zero or one.
(If $0 < \|E_1v_k\| < 1$ we can decrease $\theta_k$ by scaling $v_k$
until $\|E_1v_k\| = 1$, and similarly for $w_k$.)
The expression~(\ref{e-gauge-nonsymm-weighted}) shows that
$h(Y)$ is the gauge of the convex hull of the set
\begin{equation} \label{e-uv-dict2}
\{ vw^H \in{\mbox{\bf C}}^{n_1\times n_2} \mid
(v,w) \in \mathcal A, \, \|E_1v\| \leq 1, \; \|E_2w\| \leq 1\}.
\end{equation}
The SDP representation of $h$ in~(\ref{e-gauge-nonsymm-sdp})
allows us to reformulate problems
\begin{equation} \label{e-f-plus-gamma-nonsymm}
\mbox{minimize} \quad f(Y) + h(Y),
\end{equation}
where $f$ is convex and $h$ is the
gauge~(\ref{e-gauge-nonsymm-d})--(\ref{e-gauge-nonsymm-weighted}),
as a convex problem.
Minimizing $f(Y) + h(Y)$ is equivalent to
\begin{equation} \label{e-f+g-nonsymm}
\begin{array}{ll}
\mbox{minimize} &
f(Y) + \sum\limits_{k=1}^r \|E_1v_k\| \|E_2w_k\| \\
\mbox{subject to} & Y = \sum\limits_{k=1}^r v_k w_k^H \\
& (v_k, w_k) \in \mathcal A, \; k=1,\ldots, r.
\end{array}
\end{equation}
Alternatively, one can replace the second term in the objective
with $\sum_k \|E_2w_k\|$ and add constraints $\|E_1v_k \| \leq 1$,
as in
\begin{equation} \label{e-f+g-nonsymm-2}
\begin{array}{ll}
\mbox{minimize} &
f(Y) + \sum\limits_{k=1}^r \|E_2w_k\| \\
\mbox{subject to} & Y = \sum\limits_{k=1}^r v_k w_k^H \\[.5ex]
& (v_k, w_k) \in \mathcal A, \; k=1,\ldots, r \\[.5ex]
& \|E_1v_k\| \leq 1, \; k = 1,\ldots, r,
\end{array}
\end{equation}
or vice versa.
When $E_1$ and $E_2$ are identity matrices, we can interpret
$h(Y)$ as a convex penalty that promotes a structured low-rank
property of $Y$. The outer products $v_kw_k^H$ are constrained
by the set $\mathcal A$; the penalty term in the objective
is the sum of the norms $\|v_kw_k^H\|_2 = \|v_k\| \|w_k\|$.
The matrices $E_1$ and $E_2$ can be chosen to assign a different
weight to different terms $v_k w_k^H$.
Problems~(\ref{e-f+g-nonsymm}) and~(\ref{e-f+g-nonsymm-2}) can be
reformulated as
\begin{equation} \label{e-min-f+g-general}
\begin{array}{ll}
\mbox{minimize}
& f(Y) + (\mathop{\bf tr}(E_1VE_1^H) + \mathop{\bf tr}(E_2WE_2^H))/2 \\[.5ex]
\mbox{subject to}
& \Phi_{11} FXF^H + \Phi_{21} FXG^H + \Phi_{12} GXF^H + \Phi_{22} GXG^H
= 0\\[.5ex]
& \Psi_{11} FXF^H + \Psi_{21} FXG^H +
\Psi_{12} GXF^H + \Psi_{22} GXG^H \preceq 0 \\[.5ex]
& X = \left[\begin{array}{cc}
V & Y \\ Y^H & W \end{array}\right] \succeq 0.
\end{array}
\end{equation}
\paragraph{Example: column structure}
When $p_2=0$, the matrices $F$ and $G$ in~(\ref{e-GF-diag}) have the
form
$F = [\begin{array}{cc} F_1 & 0 \end{array}]$ and
$G = [\begin{array}{cc} G_1 & 0 \end{array}]$.
This means that $\mathcal A = \mathcal A_1 \times {\mbox{\bf C}}^{n_2}$ where
\[
\mathcal A_1
= \{v\in{\mbox{\bf C}}^{n_1} \mid
(\mu G_1 - \nu F_1) v = 0, \; (\mu, \nu) \in \mathcal C\}.
\]
There are no restrictions on the $w$-component of elements
$(v,w) \in \mathcal A$.
Problem~(\ref{e-f+g-nonsymm}) simplifies to
\begin{equation} \label{e-f+g-nonsymm-column}
\begin{array}{ll}
\mbox{minimize} &
f(Y) + \sum\limits_{k=1}^r \|E_1v_k\| \|E_2w_k\| \\
\mbox{subject to} & Y = \sum\limits_{k=1}^r v_k w_k^H \\
& v_k \in \mathcal A_1, \; k=1,\ldots, r,
\end{array}
\end{equation}
and the equivalent semidefinite formulation~(\ref{e-min-f+g-general}) to
\[
\begin{array}{ll}
\mbox{minimize}
& f(Y) + (\mathop{\bf tr}(E_1VE_1^H) + \mathop{\bf tr}(E_2WE_2^H))/2 \\[.5ex]
\mbox{subject to}
& \Phi_{11} F_1VF_1^H + \Phi_{21} F_1VG_1^H +
\Phi_{12} G_1VF_1^H + \Phi_{22} G_1VG_1^H = 0\\[.5ex]
& \Psi_{11} F_1VF_1^H + \Psi_{21} F_1VG_1^H +
\Psi_{12} G_1VF_1^H + \Psi_{22} G_1VG_1^H \preceq 0 \\[.5ex]
& \left[\begin{array}{cc}
V & Y \\ Y^H & W \end{array}\right] \succeq 0.
\end{array}
\]
As an example, we again consider the signal
model~(\ref{e-sinusoids-in-noise}).
A natural idea for estimating the parameters $\omega_k$ and $c_k$
is to solve a nonlinear least squares problem
\[
\mbox{minimize} \quad
\sum_{t=0}^{n-1} | y_\mathrm m(t) - \sum_{k=1}^r c_k e^{\mathrm{j} \omega_k t}|^2,
\]
where $y_\mathrm m(t)$ is the observed signal.
This problem is not convex and difficult to solve iteratively without
a good starting point \cite[page 148]{StM:97}.
However, suppose that, instead of fixing $r$, we impose a penalty on
$\sum_k |c_k|$, and consider the optimization problem
\begin{equation} \label{e-nlls}
\begin{array}{ll}
\mbox{minimize} & \gamma \|y - y_\mathrm m\|^2 + \sum\limits_{k=1}^r |c_k|
\\[1ex]
\mbox{subject to}
& y = \sum\limits_{k=1}^r c_k
\left[\begin{array}{c}
1 \\ e^{\mathrm{j}\omega_k} \\ \vdots \\ e^{\mathrm{j} (n-1)\omega_k} \end{array}
\right].
\end{array}
\end{equation}
The optimization variables are $y$ and the parameters
$c_k$, $\omega_k$, $r$ in the decomposition of $y$.
The vector $y_\mathrm m$ has elements $y_\mathrm m(0)$, \ldots,
$y_\mathrm m(n-1)$.
This is a special case of~(\ref{e-f+g-nonsymm-2}) with
$f(y) = \gamma \|y-y_\mathrm m\|^2$,
$n_1 = n$, $n_2=1$,
\[
E_1 = \frac{1}{\sqrt n} I, \qquad E_2 = 1, \qquad
F_1 = \left[\begin{array}{cc} 0 & I_{n_1-1}\end{array}\right], \qquad
G_1 = \left[\begin{array}{cc} I_{n_1-1} & 0 \end{array}\right],
\]
and $\Phi = \Phi_\mathrm u$, $\Psi = 0$, so that
$\mathcal A_1$ is the set of all multiples of the vectors
$(1, e^{\mathrm{j}\omega}, \ldots, e^{\mathrm{j}(n-1)\omega})$.
The problem is therefore equivalent to the convex problem
\[
\begin{array}{ll}
\mbox{minimize} & \gamma \|y-y_\mathrm m\|^2 + (\mathop{\bf tr} V)/(2n) + w/2
\\*[1ex]
\mbox{subject to} &
\left[ \begin{array}{cc} V & y \\ y^H & w \end{array}\right]
\succeq 0 \\*[2ex]
& \mbox{$V$ is Toeplitz}.
\end{array}
\]
A related numerical example will be given in section~\ref{s-huber}.
\paragraph{Example: joint column and row structure}
To illustrate the general problem~(\ref{e-f+g-nonsymm}), we
consider a variation on the previous example.
Suppose we arrange the observations in an $n\times m$ Hankel matrix
\[
Y_\mathrm m =
\left[\begin{array}{ccccc}
y_\mathrm{m}(0) & y_\mathrm{m} (1) & \cdots & y_\mathrm{m}(m-1) \\
y_\mathrm{m}(1) & y_\mathrm{m} (2) & \cdots & y_\mathrm{m}(m) \\
\vdots & \vdots & & \vdots \\
y_\mathrm{m}(n-1) & y_\mathrm{m} (n) & \cdots & y_\mathrm{m}(m+n-1)
\end{array}\right],
\]
and we fit to this matrix a matrix $Y$ with the same Hankel
structure and with elements $y(t) = \sum_{k=1}^r c_k \exp(\mathrm{j} \omega_k t)$.
We formulate the problem as
\begin{equation} \label{e-hankel-structure}
\begin{array}{ll}
\mbox{minimize} & \gamma \|Y - Y_\mathrm m\|_F^2
+ \sum\limits_{k=1}^r |c_k| \\
\mbox{subject to} &
Y = \sum\limits_{k=1}^r
c_k \left[\begin{array}{c}
1 \\ e^{\mathrm{j}\omega_k} \\ \vdots \\ e^{\mathrm{j} (n-1)\omega_k} \end{array}\right]
\left[\begin{array}{c}
1 \\ e^{-\mathrm{j}\omega_k} \\ \vdots \\ e^{-\mathrm{j} (m-1)\omega_k}
\end{array}\right]^H.
\end{array}
\end{equation}
This is an instance of~(\ref{e-f+g-nonsymm}) with
$n_1=n$, $n_2 =m$,
$E_1= (1/\sqrt n) I$,
$E_2= (1/\sqrt m) I$,
and
\[
G_1 = \left[\begin{array}{cc}
I_{m-1} & 0
\end{array}\right], \qquad
F_1 = \left[\begin{array}{cc}
0 & I_{m-1}
\end{array}\right], \qquad
G_2 = \left[\begin{array}{cc}
0 & I_{n-1}
\end{array}\right], \qquad
F_2 = \left[\begin{array}{cc}
I_{n-1} & 0
\end{array}\right].
\]
With these parameters, the set $\mathcal A$ contains the pairs $(v,w)$
of the form
\[
v = \alpha (1, e^{\mathrm{j}\omega}, \ldots, e^{\mathrm{j} (m-1)\omega}), \qquad
w = \beta (1, e^{-\mathrm{j}\omega}, \ldots, e^{-\mathrm{j} (n-1)\omega}).
\]
The convex formulation is
\[
\begin{array}{ll}
\mbox{minimize} & \gamma \|Y-Y_\mathrm m\|_F^2
+ (\mathop{\bf tr} V) / (2n) + (\mathop{\bf tr} W)/(2m) \\*[1ex]
\mbox{subject to} & \left[\begin{array}{cc}
V & Y \\ Y^H & W \end{array}\right] \succeq 0 \\*[1ex]
&\left[\begin{array}{cc} F_1 & 0 \\ 0 & F_2 \end{array}\right]
\left[\begin{array}{cc} V & Y \\ Y^H & W \end{array}\right]
\left[\begin{array}{cc} F_1 & 0 \\ 0 & F_2 \end{array}\right]^T
= \left[\begin{array}{cc} G_1 & 0 \\ 0 & G_2 \end{array}\right]
\left[\begin{array}{cc} V & Y \\ Y^H & W \end{array}\right]
\left[\begin{array}{cc} G_1 & 0 \\ 0 & G_2 \end{array}\right]^T.
\end{array}
\]
An example is discussed in section~\ref{s-huber}.
\section{Duality} \label{s-duality}
In this section we derive the conjugates of the gauge functions defined in
section~\ref{s-sdp} and show that they can be interpreted as
indicator functions of sets of nonnegative or bounded generalized
polynomials. This gives a useful interpretation of the dual problems
for~(\ref{e-f-plus-gamma}) and~(\ref{e-f-plus-gamma-nonsymm}).
We assume that the subset of the complex
plane represented by $\mathcal C$ in~(\ref{e-mC}) is one-dimensional,
{\it i.e.}, $\mathcal C$ is not a singleton and not the empty set.
Equivalently, the inequality $q_\Psi(\mu,\nu) \leq 0$
in the definition is either redundant (and $\mathcal C$ represents a line
or circle), or it is not redundant and then there exist elements of
$\mathcal C$ with $q_\Psi(\mu, \nu) < 0$.
When stating and analyzing the dual problems, we will need to
distinguish these two cases ($q_\Psi(\mu,\nu) \leq 0$
is redundant or not).
For the sake of brevity we only give the formulas for the case where
the inequality is not redundant.
The dual problems for the other case follow by setting $\Psi=0$
and making obvious simplifications.
We also assume that $\mu G - \nu F$ has full row rank
($\mathop{\bf rank}(\mu G - \nu F) = p$) for all nonzero $(\mu,\nu)$).
This condition will serve as a
`constraint qualification' that guarantees strong duality.
\subsection{Symmetric matrix gauge} \label{s-duality-symm}
We first consider the conjugate of the function $g$ defined
in~(\ref{e-gammaCP-lmi}). The conjugate is defined as
\[
g^*(Z) = \sup_{X} {(\mathop{\bf tr}(XZ) - g(X))},
\]
{\it i.e.}, the optimal value of the SDP
\begin{equation} \label{e-g-sdp}
\begin{array}{ll}
\mbox{maximize} & \mathop{\bf tr}{((Z-E^HE)X)} \\
\mbox{subject to}
& X\succeq 0 \\
& \Phi_{11} FXF^H + \Phi_{21} FXG^H + \Phi_{12} GXF^H + \Phi_{22} GXG^H
= 0 \\
& \Psi_{11} FXF^H + \Psi_{21} FXG^H + \Psi_{12} GXF^H + \Psi_{22} GXG^H
\preceq 0.
\end{array}
\end{equation}
The dual of this problem is
\begin{equation} \label{e-g-conj}
\begin{array}{ll}
\mbox{minimize} & 0 \\
\mbox{subject to} &
Z - \left[\begin{array}{c} F \\ G \end{array}\right]^H
(\Phi \otimes P + \Psi \otimes Q)
\left[\begin{array}{c} F \\ G \end{array}\right] \preceq E^HE \\
& Q \succeq 0,
\end{array}
\end{equation}
with variables $P, Q\in \mathbf{H}^p$.
It is shown in appendix~\ref{s-slater} that strong duality holds
(under the assumptions listed at the top of section~\ref{s-duality}).
If strong duality holds, then $g^*(Z)$ is the optimal value
of~(\ref{e-g-conj}), {\it i.e.}, equal to zero if there exist $P$, $Q$
that satisfy the constraints in~(\ref{e-g-conj}), and $+\infty$
otherwise. We now show that this can be expressed as
\begin{equation} \label{e-g-conj-pos-pol}
g^*(Z) = \left\{\begin{array}{ll}
0 & a^H Z a \leq \|Ea\|^2 \mbox{\ for all $a\in \mathcal A$} \\
+\infty & \mbox{otherwise.}
\end{array}\right.
\end{equation}
Suppose $P$ and $Q$ are feasible in~(\ref{e-g-conj}).
Consider any $a\in\mathcal A$ and $(\mu,\nu) \in \mathcal C$
with $\mu Ga = \nu Fa$. Define $y = (1/\nu)Ga$ if
$\nu \neq 0$ and $y = (1/\mu)Fa$ otherwise.
Then
\begin{eqnarray*}
a^H Za - \|Ea\|^2
& \leq & \left[\begin{array}{c} Fa \\ Ga \end{array}\right]^H
(\Phi \otimes P + \Psi \otimes Q)
\left[\begin{array}{c} Fa \\ Ga \end{array}\right] \\
& = & \left[\begin{array}{c} \mu y \\ \nu y \end{array}\right]^H
(\Phi \otimes P + \Psi \otimes Q)
\left[\begin{array}{c} \mu y \\ \nu y \end{array}\right] \\
& = & (y^H P y) q_\Phi(\mu,\nu) + (y^HQy) q_\Psi(\mu,\nu) \\
& \leq & 0.
\end{eqnarray*}
The last line follows from $Q\succeq 0$ and $q_\Phi(\mu,\nu) = 0$,
$q_\Psi(\mu,\nu) \leq 0$.
Conversely, if problem~(\ref{e-g-conj}) is infeasible,
then the optimal value is $+\infty$ and, since strong duality holds,
there exist matrices $X$ that are feasible
for~(\ref{e-g-sdp}) with $\mathop{\bf tr}((Z-E^HE)X) > 0$.
Applying Theorem~\ref{t-decomp} we see that there exist
$a_1, \ldots, a_r \in\mathcal A$ with
\[
\sum_{k=1}^r (a_k^HZa_k - \|Ea_k\|^2) > 0.
\]
Therefore $a_k^H Z a_k > \|Ea_k\|^2$ for at least one $a_k$.
The interpretation of the conjugate gives useful insight in
problem~(\ref{e-f-plus-gamma}), where $g$ is
defined in~(\ref{e-gammaCP-lmi}).
The dual problem is
\[
\mbox{maximize} \quad -f^*(Z) - g^*(-Z).
\]
Expanding $g^*(-Z)$ using~(\ref{e-g-conj}) gives the equivalent problem
\begin{equation}\label{e-f-plus-gauge-daul-kyp}
\begin{array}{ll}
\mbox{maximize} & -f^*(Z) \\
\mbox{subject to} &
-Z - \left[\begin{array}{c} F \\ G \end{array}\right]^H
(\Phi \otimes P + \Psi \otimes Q)
\left[\begin{array}{c} F \\ G \end{array}\right] \preceq E^HE \\
& Q \succeq 0,
\end{array}
\end{equation}
with variables $Z$, $P$, $Q$,
and using the expression~(\ref{e-g-conj-pos-pol}) we can put the
constraints in this problem more succinctly as
\begin{equation} \label{e-f-plus-gauge-dual}
\begin{array}{ll}
\mbox{maximize} & -f^*(Z) \\
\mbox{subject to} & \|Ea\|^2 + a^HZ a \geq 0
\quad\mbox{for all $a\in\mathcal A$}.
\end{array}
\end{equation}
This last form leads to an interesting set of optimality conditions.
Suppose $X$ and $Z$ are feasible for~(\ref{e-gauge-sdp-rep-E})
and~(\ref{e-f-plus-gauge-dual}), respectively.
Then
\begin{eqnarray*}
f(X) + \sum_{k=1}^r \|Ea_k\|^2
& \geq & -f^*(Z) + \mathop{\bf tr}(XZ) + \sum_{k=1}^r \|Ea_k\|^2\\
& = & -f^*(Z) + \sum_{k=1}^r (\|Ea_k\|^2 + a_k^H Za_k) \\
& \geq & -f^*(Z).
\end{eqnarray*}
The first inequality follows by definition of $f^*(Z) =
\sup_{X}{(\mathop{\bf tr}(ZX) - f(X))}$, and the second and third line from
primal and dual feasibility.
If $X$ and $Z$ are optimal and strong duality holds, then $f(X) + \sum_k \|Ea_k\|^2 = -f^*(Z)$.
This is only possible if $f(X) + f^*(Z) = \mathop{\bf tr}(XZ)$
and
\[
\|Ea_k\|^2 +a_k^H Z a_k = 0, \quad k=1,\ldots, r.
\]
Hence only the vectors $a\in\mathcal A$ at which the inequality
in~(\ref{e-f-plus-gauge-dual}) is active, can be used to form an
optimal $X = \sum_k a_ka_k^H$.
\paragraph{Example: Generalized Kalman-Yakubovich-Popov lemma}
When specialized to the controllability pencil~(\ref{e-state-space}),
the equivalence between the constraints
in~(\ref{e-f-plus-gauge-dual})
and~(\ref{e-f-plus-gauge-daul-kyp}) is known as the
(generalized) Kalman-Yakubovich-Popov lemma
\cite{Kal:63,Yak:62,Pop:62,Sch:06,IwH:05}.
We assume that $A$ has no eigenvalues $\lambda$ with $(\lambda, 1)
\in \mathcal C$, and that the pair $(A,B)$ is controllable, so
the pencil satisfies the rank condition
that $\mathop{\bf rank}(\lambda F - G) = n_\mathrm s$ for all $\lambda$.
The dual problem~(\ref{e-f-plus-gauge-dual})
becomes
\[
\begin{array}{ll}
\mbox{maximize} & -f^*(Z) \\
\mbox{subject to} &
\mathcal F(\lambda, Z) \succeq 0 \quad
\mbox{for all $(\lambda, 1)\in \mathcal C$} \\
& M_{22} + Z_{22} \succeq 0 \quad \mbox{if $(1,0) \in \mathcal C$}
\end{array}
\]
where
\[
\mathcal F(\lambda, Z) =
\left[\begin{array}{c} (\lambda I - A)^{-1} B \\ I \end{array}\right]^H
\left[\begin{array}{cc} M_{11} + Z_{11} & M_{12} + Z_{12}
\\ M_{21} +Z_{21} & M_{22} + Z_{22}
\end{array}\right]
\left[\begin{array}{c} (\lambda I - A)^{-1} B \\ I \end{array}\right]
\]
and $M= E^HE$.
The function $\mathcal F$ is called the
\emph{Popov function} with central matrix $M+Z$ \cite{IOW:99,HSK:99}.
\subsection{Non-symmetric matrix gauge}\label{s-duality-nonsymm}
Next we consider the conjugate of the
gauge defined in~(\ref{e-gauge-nonsymm-d})--(\ref{e-gauge-nonsymm-a}).
We have
\[
h^*(Z) = \sup_Y{(\mathop{\bf tr}(Z^TY) - h(Y))}
\]
where $h(Y)$ is the optimal value of~(\ref{e-gauge-nonsymm-sdp}).
Therefore $h^*(Z)$ is the optimal value of the SDP
\begin{equation} \label{e-g-conj-nonsymm-prim}
\begin{array}{ll}
\mbox{maximize} & \mathrm{d}isplaystyle \frac{1}{2} \mathop{\bf tr}(\left[\begin{array}{cc}
-E_1^H E_1 & Z \\ Z^H & -E_2^HE_2 \end{array}\right] X) \\*[2ex]
\mbox{subject to} &
\Phi_{11} FXF^H + \Phi_{21} FXG^H + \Phi_{12}GXF^H + \Phi_{22} GXG^H = 0
\\
& \Psi_{11} FXF^H + \Psi_{21} FXG^H + \Psi_{12}GXF^H + \Psi_{22} GXG^H
\preceq 0 \\
& X \succeq 0.
\end{array}
\end{equation}
The dual of this problem is
\begin{equation} \label{e-g-conj-nonsymm}
\begin{array}{ll}
\mbox{minimize} & 0 \\
\mbox{subject to} &
\left[\begin{array}{cc}
0 & Z \\ Z^H & 0 \end{array}\right]
- \left[\begin{array}{c} F \\ G \end{array}\right]^H
(\Phi \otimes P + \Psi \otimes Q)
\left[\begin{array}{c} F \\ G \end{array}\right]
\preceq \left[\begin{array}{cc}
E_1^HE_1 & 0 \\ 0 & E_2^H E_2 \end{array}\right] \\
& Q \succeq 0.
\end{array}
\end{equation}
As in the previous section, it follows from appendix~\ref{s-slater}
that strong duality holds. Therefore $h^*(Z)$
is equal to the optimal value of~(\ref{e-g-conj-nonsymm}), {\it i.e.}, zero
if there exists $P$ and $Q$ that satisfy the constraints of this problem,
and $+\infty$ otherwise.
This will now be shown to be equivalent to
\begin{eqnarray}
h^*(Z) & = & \left\{\begin{array}{ll}
0 & \newRe{(v^H Z w)} \leq (\|E_1v\|^2 + \|E_2w\|^2)/2
\quad \mbox{for all $(v,w)\in\mathcal A$} \\
+\infty & \mbox{otherwise} \end{array} \right. \nonumber \\
& = & \left\{\begin{array}{ll}
0 & \newRe{(v^H Z w)} \leq \|E_1v\| \|E_2w\|
\quad \mbox{for all $(v,w)\in\mathcal A$} \\
+\infty & \mbox{otherwise.} \end{array} \right.
\label{e-g-conj-nonsymm-pos}
\end{eqnarray}
To see this, first assume $P$ and $Q$ are feasible
in~(\ref{e-g-conj-nonsymm}), and $a = (v,w) \in \mathcal A$
satisfies $(\mu G - \nu F)a=0$ with $(\mu,\nu)\in\mathcal C$. Then
\begin{eqnarray*}
v^H Z w + w^H Z^H v - \|E_1u\|^2 - \|E_2v\|^2
& \leq & \left[\begin{array}{c} Fa \\ Ga \end{array}\right]^H
(\Phi \otimes P + \Psi \otimes Q)
\left[\begin{array}{c} Fa \\ Ga \end{array}\right] \\
& = & (y^H Py) q_\Phi(\mu,\nu) + (y^HQy) q_\Psi(\mu,\nu) \\
& \leq & 0,
\end{eqnarray*}
where we defined $y = (1/\nu) Ga$ if $\nu \neq 0$ and $y=(1/\mu)Fa$
otherwise.
Conversely, if problem~(\ref{e-g-conj-nonsymm}) is infeasible,
then~(\ref{e-g-conj-nonsymm-prim}) is unbounded above,
so there exists a feasible $X$ with positive objective value.
If we decompose $X$ as in Theorem~\ref{t-decomp},
with $a_k = (v_k,w_k)$, we find that
\begin{eqnarray*}
0 & < &
\mathop{\bf tr}( \left[\begin{array}{cc}
-E_1^H E_1 & Z \\ Z^H & -E_2^HE_2 \end{array}\right]
\sum_{k=1}^r \left[\begin{array}{c} v_k \\ w_k \end{array}\right]
\left[\begin{array}{c} v_k \\ w_k \end{array}\right]^H) \\
& = &
\sum_{k=1}^r (v_k^HZw_k + w_k^H Z^H v_k - \|E_1 v_k\|^2 - \|E_2w_k\|^2)
\end{eqnarray*}
so at least one term in the sum is positive.
The second expression for $h^*(Z)$ in~(\ref{e-g-conj-nonsymm-pos})
follows from the block diagonal structure of $F$ and $G$.
The interpretation of the conjugate $h^*$ can be applied to interpret
the dual of~(\ref{e-f-plus-gamma-nonsymm}), {\it i.e.},
\[
\begin{array}{ll}
\mbox{maximize} & -f^*(Z) - h^*(-Z).
\end{array}
\]
Substituting the expression~(\ref{e-g-conj-nonsymm}) for $h^*(-Z)$,
one can write this as
\[
\begin{array}{ll}
\mbox{maximize} & -f^*(Z) \\
\mbox{subject to} &
\left[\begin{array}{cc}
0 & -Z \\ -Z^H & 0 \end{array}\right]
- \left[\begin{array}{c} F \\ G \end{array}\right]^H
(\Phi \otimes P + \Psi \otimes Q)
\left[\begin{array}{c} F \\ G \end{array}\right]
\preceq \left[\begin{array}{cc} E_1^H E_1 & 0 \\ 0 & E_2^H E_2
\end{array}\right] \\
& Q \succeq 0,
\end{array}
\]
with variables $Z$, $P$, $Q$.
Substituting~(\ref{e-g-conj-nonsymm-pos}) we obtain
\[
\begin{array}{ll}
\mbox{maximize} & -f^*(Z) \\
\mbox{subject to} &
\newRe{(v^H Z w)} \leq \|E_1v\| \|E_2w\|
\quad \mbox{for all $(v,w)\in\mathcal A$}.
\end{array}
\]
As in the previous section, the primal-dual optimality conditions
provide a useful set of complementary slackness relations
between primal optimal $Y$ and dual optimal $Z$.
The optimal $Y$ can be decomposed as $Y = \sum_k v_k w_k^H$ with
elements $(v_k,w_k) \in \mathcal A$ at which
$\newRe{(v_k^H Z w_k)} = \|E_1 v_k\| \|E_2 w_k\|$.
\paragraph{Example}
Suppose
$A \in {\mbox{\bf C}}^{n_\mathrm s\times n_\mathrm s}$,
$B \in {\mbox{\bf C}}^{n_\mathrm s\times m}$,
$C\in{\mbox{\bf C}}^{l\times n_\mathrm s}$,
$D\in{\mbox{\bf C}}^{l\times m}$
are matrices in a state-space model,
and $A$ has no eigenvalues that satisfy $(\lambda, 1) \in \mathcal C$.
We take $p_1=0$, $n_1 = l$, $p_2 = n_\mathrm s$,
$n_2 = n_\mathrm s + m$,
\[
G_2 = \left[\begin{array}{cc} I & 0 \end{array}\right], \qquad
F_2 = \left[\begin{array}{cc} A & B \end{array}\right], \qquad
E_1= I, \qquad
E_2 = \left[\begin{array}{cc} 0 & I \end{array}\right].
\]
With this choice of parameters, $\mathcal A = {\mbox{\bf C}}^l \times
\mathcal A_2$, where $\mathcal A_2$ contains the vectors of the form
\[
w = \left[\begin{array}{c}
(\lambda I - A)^{-1}B u \\ u \end{array}\right]
\]
for all $u\in{\mbox{\bf C}}^{m}$ and all $(\lambda,1)\in\mathcal C$,
plus the vectors $(0,u)$ if $(0,1) \in\mathcal C$.
Since $v$ is arbitrary and $E_1 = I$, the inequality
in~(\ref{e-g-conj-nonsymm-pos}) reduces to
$\|Zw\|_2 \leq \|E_2w\|$ for all $w\in\mathcal A_2$.
If $Z$ is partitioned as $Z = [\begin{array}{cc} C & D\end{array}]$,
this is equivalent to a bound on the transfer function
\begin{equation} \label{e-tf-bound}
\|D + C(\lambda I - A)^{-1}B \|_2 \leq 1 \quad \mbox{
for all $(\lambda, 1) \in \mathcal C$}, \qquad
\|D\|_2 \leq 1 \quad \mbox{if $(1,0) \in \mathcal C$}.
\end{equation}
\section{Examples}
\label{s-sp-ex}
The formulations in section~\ref{s-sdp} will now be illustrated
with a few examples from signal processing.
The convex optimization problems in the examples were solved
with CVX \cite{GrB:14}.
\subsection{Line spectrum estimation by Toeplitz covariance fitting}
\label{s-ex-covariance}
In this example we fit a covariance matrix of the
form~(\ref{e-line-spec-covariance}) to an estimated covariance
matrix~$R_\mathrm m$.
The estimate $R_\mathrm m$ is constructed from $N=150$ samples
of the time series $y(t)$ defined in~(\ref{e-sinusoids-in-noise}),
with $r=3$, and frequencies $\omega_k$ and magnitudes $|c_k|$ shown in
figure~\ref{f-cov-fitting}. The noise is Gaussian white noise with variance
$\sigma^2 = 64$.
The sample covariance matrix is constructed as
\[
R_\mathrm m = \frac{1}{N-n+1} Y Y^H
\]
where $Y$ is the $n\times (N-n+1)$ Hankel matrix
with $y(1)$, \ldots, $y(N-n+1)$ in its first row.
\begin{figure}
\caption{Line spectrum estimation by Toeplitz covariance fitting
(section~\ref{s-ex-covariance}
\label{f-cov-fitting}
\end{figure}
To estimate the model parameters we solve the optimization problem
\begin{equation} \label{e-cov-fitting}
\begin{array}{ll}
\mbox{minimize} & \gamma \|R-R_\mathrm{m}\|_2 +
\sum\limits_{k=1}^r |c_k|^2 \\
\mbox{subject to} & R = \sigma^2 I + \sum\limits_{k=1}^r |c_k|^2
\left[\begin{array}{c}
1 \\ e^{\mathrm{j}\omega_k} \\ \vdots \\
e^{\mathrm{j} (n-1)\omega_k} \end{array}\right]
\left[\begin{array}{c}
1 \\ e^{\mathrm{j}\omega_k} \\ \vdots \\
e^{\mathrm{j} (n-1)\omega_k} \end{array}\right]^H,
\end{array}
\end{equation}
with variables $|c_k|^2$, $\omega_k$, $r$, and $R$.
The norm $\|\cdot\|_2$ in the objective is the spectral norm.
The regularization parameter $\gamma$ is set to $0.25$.
This problem is equivalent to the convex problem
\[
\begin{array}{ll}
\mbox{minimize} & \gamma \|tI + X -R_\mathrm{m}\|_2
+ (1/n) \mathop{\bf tr} X\\
\mbox{subject to} & X\succeq 0, \quad t\geq 0 \\
& FXF^H - GXG^H = 0
\end{array}
\]
with variables $X$ and $t$, and $F$ and $G$ defined in~(\ref{e-FG-toep}).
As can be seen from Figure~\ref{f-cov-fitting},
the recovered parameters $\omega_k$ and $|c_k|$ are quite accurate,
despite the very low signal-to-noise ratio.
The estimated noise variance $t$ is $79.6$.
The semidefinite optimization approach allows us to fit
a covariance matrix with the structure prescribed
in~(\ref{e-line-spec-covariance}) to a sample covariance
matrix that may not be Toeplitz or positive semidefinite.
The formulation can also be extended to applications where the noise
$v(t)$ is modeled as a moving-average process, by combining it with
the formulation in \cite{Geo:06}.
\subsection{Line spectrum estimation by penalty approximation}
\label{s-huber}
This example is a variation on problem~(\ref{e-nlls}).
We take $n=50$ consecutive measurements of the signal
defined in~\eqref{e-sinusoids-in-noise}. There are
three sinusoids with frequencies and magnitudes shown in
figure~\ref{f-huber}.
The noise $v(t)$ is a superposition of white noise and
a sparse corruption of $20$ elements (see Figure~\ref{f-huber-data}).
\begin{figure}
\caption{The input data for the example in section~\ref{s-huber}
\label{f-huber-data}
\end{figure}
The model parameters are estimated by solving the problem
\begin{equation} \label{e-huber}
\begin{array}{ll}
\mbox{minimize} &
\gamma \sum\limits_{i=1}^n \phi(y_i - y_{\mathrm m,i}) +
\sum\limits_{k=1}^r |c_k| \\
\mbox{subject to}
& y = \sum\limits_{k=1}^r c_k
\left[\begin{array}{c}
1 \\ e^{\mathrm{j}\omega_k} \\ \vdots \\ e^{\mathrm{j} (n-1)\omega_k} \end{array}
\right] \\
& |\omega_k| \leq \omega_\mathrm c, \quad k = 1,\ldots,r,
\end{array}
\end{equation}
where $\phi$ is the Huber penalty,
$\gamma = 0.071$,
and $\omega_\mathrm c = \pi/6$.
The variables in this problem are the $n$-vector $y$, and
the parameters $r$, $c_k$, $\omega_k$ in the decomposition of $y$.
The problem is equivalent to the convex problem
\begin{equation} \label{e-huber-sdp}
\begin{array}{ll}
\mbox{minimize} &
\gamma \sum\limits_{i=1}^n \phi(y_i-y_{\mathrm m,i}) +
(\mathop{\bf tr} V)/(2n) + w/2 \\*[1ex]
\mbox{subject to} &
\left[ \begin{array}{cc} V & y \\ y^H & w \end{array}\right]
\succeq 0 \\*[2ex]
& FVF^H - GVG^H = 0 \\
& -FVG^H - GXF^H + 2 (\cos \omega_\mathrm c) GVG^H \preceq 0
\end{array}
\end{equation}
with $F$ and $G$ defined in~(\ref{e-FG-toep}).
The variables are the $n$-vector $y$,
the Hermitian $n\times n$ matrix $V$, and the scalar $w$.
The results are shown in Figure~\ref{f-huber}.
\begin{figure}
\caption{Line spectrum models estimated from the signal in
Figure~\ref{f-huber-data}
\label{f-huber}
\end{figure}
The second figure shows the result of a simple
implementation of the matrix pencil method
with a $30\times 21$ Hankel matrix constructed from the
measurements~\cite{HS88}.
The comparison shows the importance of the prior frequency constraints
in the formulation~(\ref{e-huber}).
It is interesting to note that
problem~(\ref{e-huber}) can be equivalently formulated as
\begin{equation} \label{e-huber-matrix}
\begin{array}{ll}
\mbox{minimize} &
\gamma \sum\limits_{i=1}^{n} \phi(y_i - y_{\mathrm m,i})
+ \sum\limits_{k=1}^r |c_k| \\
\mbox{subject to} &
\left[\begin{array}{ccccc}
y_1 & y_2 & \cdots & y_{n_2} \\
y_2 & y_3 & \cdots & y_{n_2-1} \\
\vdots & \vdots & & \vdots \\
y_{n_1} & y_{n_1-1} & \cdots & y_{n_1+n_2-1}
\end{array}\right]
= \sum\limits_{k=1}^r
c_k \left[\begin{array}{c}
1 \\ e^{\mathrm{j}\omega_k} \\ \vdots \\ e^{\mathrm{j} (n_1-1)\omega_k} \end{array}\right]
\left[\begin{array}{c}
1 \\ e^{-\mathrm{j}\omega_k} \\ \vdots \\ e^{-\mathrm{j} (n_2-1)\omega_k}
\end{array}\right]^H \\*[5ex]
& |\omega_k| \leq \omega_\mathrm c,\quad k = 1,\ldots,r,
\end{array}
\end{equation}
where $n_1+n_2-1=n$.
This problem is equivalent to
\begin{equation} \label{e-huber-matrix-sdp}
\begin{array}{ll}
\mbox{minimize} & \gamma\sum\limits_{i=1}^m \phi(y_i - y_{\mathrm m,i})
+ (\mathop{\bf tr} V)/(2n_1) + (\mathop{\bf tr} W)/(2n_2) \\
\mbox{subject to} & X = \left[\begin{array}{cc}
V & Y \\ Y^H & W \end{array}\right] \succeq 0 \\*[2ex]
& F X F^T = GXG^T \\
& - FXG^T - GXF^T + 2\cos\omega_\mathrm c GXG^T \preceq 0
\end{array}
\end{equation}
where $G$ and $F$ are block diagonal with blocks
\[
G_1 = \left[\begin{array}{cc}
I_{n_1-1} & 0
\end{array}\right], \qquad
F_1 = \left[\begin{array}{cc}
0 & I_{n_1-1}
\end{array}\right], \qquad
G_2 = \left[\begin{array}{cc}
0 & I_{n_2-1}
\end{array}\right], \qquad
F_2 = \left[\begin{array}{cc} I_{n_2-1} & 0 \end{array}\right].
\]
The variables in~(\ref{e-huber-matrix-sdp}) are the matrices
$V$, $Y$, $W$. The elements $y_i$ in the objective are the elements in
the first row and last column of the matrix variable $Y$.
The two SDP~(\ref{e-huber-sdp})
and~(\ref{e-huber-matrix-sdp}) give the same result $y$,
but may have different numerical properties (in terms of accuracy or
complexity).
\subsection{Direction of arrival estimation} \label{s-doa}
This example illustrates the use of frequency interval
constraints in direction of arrival estimation.
We consider the example described in~\cite[section 3.1]{ChV:16}:
\begin{equation} \label{e-doa-bp}
\begin{array}{ll}
\mbox{minimize} &
\sum\limits_{j=1}^3 \sum\limits_{k=1}^{r_j} |x_{jk}| \\*[-.1ex]
\mbox{subject to}
& y_j = \sum\limits_{k=1}^{r_j} x_{jk}\!\left[\begin{array}{c}
1 \\ e^{\mathrm{j}\pi\sin \theta_{jk}} \\ \vdots \\ e^{\mathrm{j} (n-1)
\pi \sin \theta_{jk}} \end{array}\right]
\\*[4.7ex]
& \theta_{jk} \in \Theta_j, \quad k=1,\ldots, r_j, \quad j=1,2,3
\\*[.5ex]
& (y_1 + y_2)_{I_1} = b_1, \quad (y_2 + y_3)_{I_2} = b_2.
\end{array}
\end{equation}
The vectors $b_1$ and $b_2$ contain the outputs of two subsets
of the elements in a linear array of $n$ non-isotropic antennas.
Elements in the first group, indexed by the index set $I_1$,
measure input signals arriving from angles in
$\Theta_1\cup\Theta_2 = [-\pi/2, -\pi/6] \cup [-\pi/6, \pi/6]$.
Elements in the second group, indexed by the index set $I_1$,
measure input signals arriving from
$\Theta_2\cup\Theta_3=[-\pi/6, \pi/6] \cup [\pi/6, \pi/2]$.
The convex formulation of this problem can be found in
\cite{ChV:16}.
Figure~\ref{f-doa} shows the results of an instance with $n=500$
elements in the array,
but using only a total of $40$ randomly selected measurements
($|I_1| = |I_2| = 20$).
The red dots show the angles and magnitudes of 7 signals used
to compute the measurement vectors $b_1$, $b_2$.
The estimated angles and coefficients $|c_{jk}|$ are shown with blue lines.
The right-hand plot shows the solution if we omit the
interval constraints in~(\ref{e-doa-bp}).
\begin{figure}
\caption{Directional of arrival estimation with and without interval
constraints (section~\ref{s-doa}
\label{f-doa}
\end{figure}
Figure~\ref{f-recovery} shows the success rate as a function of
the number $|I_1| + |I_2|$ of available measurements, for an example with
$n=50$ elements, and the same angles as in~\cite{ChV:16} and
figure~\ref{f-doa}.
Each data point is the average of $100$ trials,
with different, randomly generated coefficients,
and different random selections of the two sensor groups.
We observe that solving the optimization problem with the interval
constraints has a higher rate of exact recovery.
For example, with $30$ available measurements, including the interval
constraints gave the exact answer in all instances, whereas the method
without the interval constraints was successful in only about 25\% of
the instances.
\begin{figure}
\caption{Comparison of recovery rate for different number of
available measurements with interval constraints (red) and without (blue),
in the example of section~\ref{s-doa}
\label{f-recovery}
\end{figure}
\subsection{Direction of arrival from multiple measurement vectors}
\label{s-doa-multiple}
This example demonstrates the advantage of using multiple measurement
vectors (or snapshots), as pointed out in~\cite{LC14,YX14}.
Suppose we have $K$ omnidirectional sensors placed at randomly chosen
positions of a linear grid of length $n$.
The measurements of the $K$ sensors at one time instance form one
measurement vector.
We collect $m$ of these measurement vectors, at $m$ different times,
and assume that the directions of arrival and the source magnitudes
remain constant while the measurements are taken.
The problem is formulated as
\begin{equation} \label{e-doa-snapshot}
\begin{array}{ll}
\mbox{minimize} & \sum\limits_{k=1}^r \|c_k\| \\
\mbox{subject to}
& Y = \sum\limits_{k=1}^r
\left[\begin{array}{c}
1 \\ e^{\mathrm{j}\alpha\sin\theta_k} \\ \vdots \\
e^{\mathrm{j} (n-1)\alpha\sin\theta_k} \end{array}
\right] c_k^H \\
& Y_I = B \\
& |\theta_k| \leq \theta_\mathrm c, \quad k = 1,\ldots,r,
\end{array}
\end{equation}
with variables $Y\in{\mbox{\bf C}}^{n\times m}$, $c_k\in{\mbox{\bf C}}^m$,
$\omega_k$, and $r$.
Here $\alpha = 2\pi d / \lambda_\mathrm c$, where $d$ is the distance between the grid points
and $\lambda_\mathrm c$ is the signal wavelength,
and $\theta_\mathrm c$ is a given cutoff angle.
The columns of the $K\times m$ vector $B$ are the measurement
vectors. The matrix $Y_I$ is the submatrix of $Y$
containing the rows indexed by $I$.
The problem can be interpreted as identifying a continuous form
of group sparsity~\cite{Gra15}.
The convex formulation is
\[
\begin{array}{ll}
\mbox{minimize} & (\mathop{\bf tr} V)/(2n) + (\mathop{\bf tr} W)/2 \\*[1ex]
\mbox{subject to}
& \left[\begin{array}{cc}
V & Y \\ Y^H & W \end{array}\right] \succeq 0 \\*[2ex]
& FVF^H - GVG^H = 0 \\
& -FVG^H - GVF^H + 2 \cos \omega_\mathrm c GVG^H \preceq 0 \\
& Y_I = B
\end{array}
\]
with $F$ and $G$ defined in~(\ref{e-FG-toep}) and $\omega_\mathrm c = \alpha\sin\theta_\mathrm c$.
Figure~\ref{f-doa-snapshots} shows
an example with $n=30$, $K=7$, $\alpha = 2$,
and $\theta_\mathrm c = \pi/4$.
We show the solution for $m=1$, $m=15$, $m=30$.
The blue lines show the values of $\omega_k$ and $\|c_k\|/\sqrt m$
computed by solving problem~(\ref{e-doa-snapshot}).
\begin{figure}
\caption{From top to bottom are shown the results of recovery
with $1$, $15$, $30$ measurement vectors in the DOA estimation
problem of section~\ref{s-doa-multiple}
\label{f-doa-snapshots}
\end{figure}
\section{Conclusion}
In this paper we developed semidefinite representations of a class
of gauge functions and atomic norms for sets
parameterized by linear matrix pencils.
The formulations extend the semidefinite representation
of the atomic norm associated with the trigonometric moment curve,
which underlies recent results in continuous or `off-the-grid'
compressed sensing.
The main contribution is a self-contained constructive proof of
the semidefinite representations, using techniques developed
in the literature on the Kalman-Yakubovich-Popov lemma.
In addition to opening new possible areas of applications in
system theory and control,
the connection with the KYP lemma is important for numerical
algorithms. Specialized techniques for solving SDPs derived from
the KYP lemma, for example, by exploiting
real symmetries and rank-one structure
\cite{GHNV:03,LoP:04,RoV:06,LiV:07,HaV:14}, should be useful in the
development of fast solvers for the SDPs discussed in this paper.
\appendix
\section{Subsets of the complex plane} \label{s-regions}
In this appendix we explain the notation used in
equation~(\ref{e-mC}) to describe subsets of the closed complex plane.
Recall that we use the notation
\[
q_\Theta(\mu,\nu) = \left[\begin{array}{c}
\mu \\ \nu \end{array}\right]^H
\left[\begin{array}{cc}
\Theta_{11} & \Theta_{12} \\ \Theta_{21} & \Theta_{22}
\end{array}\right]
\left[\begin{array}{c} \mu \\ \nu \end{array}\right]
\]
for the quadratic form defined by a Hermitian $2\times 2$ matrix $\Theta$.
\paragraph{Lines and circles}
If $\Phi$ is a $2\times 2$ Hermitian matrix with $\mathrm{d}et \Phi < 0$,
then the quadratic equation
\begin{equation} \label{e-quad-eq}
q_\Phi (\lambda,1) = 0
\end{equation}
defines a straight line (if $\Phi_{11} = 0$) or a circle (if $\Phi_{11}
\neq 0$) in the complex plane.
Three important special cases are
\[
\Phi_\mathrm u =
\left[\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right], \qquad
\Phi_\mathrm i =
\left[\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right], \qquad
\Phi_\mathrm r =
\left[\begin{array}{cc} 0 & \mathrm{j} \\ -\mathrm{j} & 0 \end{array} \right],
\]
for the unit circle, imaginary axis, and real axis, respectively.
Curves defined by two different matrices $\Phi$, $\tilde \Phi$ can
be mapped to one another by applying a
nonsingular congruence transformation $\tilde \Phi = R\Phi R^H$.
When $\Phi_{11} = 0$, we include the point $\lambda = \infty$ in the
solution set of~(\ref{e-quad-eq}). Alternatively, one can define
points in the closed complex plane as directions $(\mu,\nu)\neq 0$.
If $\nu \neq 0$, the pair $(\mu,\nu)$ represents the complex
number $\lambda = \mu/\nu$. If $\nu=0$, it represents the point
at infinity. Using this notation, a circle or line in the closed
complex plane is defined as the nonzero solution set of a quadratic
equation
\[
q_\Phi (\mu,\nu) =
\left[\begin{array}{c} \mu \\ \nu \end{array}\right]^H
\Phi
\left[\begin{array}{c} \mu \\ \nu \end{array}\right] =0,
\]
with $\mathrm{d}et\Phi < 0$.
A congruence transformation $\tilde \Phi = R \Phi R^H$ corresponds to
a linear transformation between the sets associated with the
matrices $\Phi$ and $\tilde\Phi$.
\paragraph{Segments of lines and circles}
The second type of set we encounter is defined by a quadratic equality
and inequality
\begin{equation} \label{e-quad-eq-ineq}
q_\Phi(\lambda,1) = 0, \qquad
q_\Psi(\lambda,1) \leq 0.
\end{equation}
We assume that $\mathrm{d}et \Phi < 0$.
If the inequality is redundant ({\it e.g.}, $\Psi = 0$)
the solution set of~(\ref{e-quad-eq-ineq}) is the line or circle
defined by the equality.
Otherwise it is an arc of a circle, a closed interval of a line, or the
complement of an open interval of a line.
It includes the point at infinity if $\Phi_{11} = 0$ and
$\Psi_{11} \leq 0$.
Alternatively, one can use homogeneous coordinates and consider
sets of points $(\mu,\nu)$ that satisfy
\begin{equation} \label{e-quad-eq-ineq-hom}
q_\Phi (\mu,\nu) = 0, \qquad
q_\Psi (\mu,\nu) \leq 0, \qquad
(\mu,\nu) \neq 0.
\end{equation}
For easy reference, we list the most common combinations of
$\Phi$ and $\Psi$ in tables~\ref{t-phid}--\ref{t-phir}
\cite{IwH:03,IwH:05}.
\begin{table}
\begin{center}
\begin{tabular}{ccc}\toprule
$\angle{\lambda}$ & $\Psi$ & Assumptions \\
\midrule
$[a-b, a+b]$ &
$\left[\begin{array}{cc} 0 & -e^{\mathrm{j} a} \\
-e^{-\mathrm{j} a} & 2\cos b\end{array}\right]$ &
$0 \leq b \leq \pi$ \\
$[a, 2\pi-a]$ &
$\left[\begin{array}{cc}0 & 1 \\ 1 & -2\cos a\end{array}\right]$
& $0\leq a\leq\pi$ \\ \bottomrule
\end{tabular}
\end{center}
\caption{Common choices of $\Psi$ with $\Phi = \Phi_\mathrm u$
($\lambda$ on the unit circle).}
\label{t-phid}
\end{table}
\begin{table}
\begin{center}
\begin{tabular}{ccc}\toprule
$\newIm \lambda$ & $\Psi$ & Assumptions \\
\midrule
$[a, b]$ &
$\left[\begin{array}{cc} 2 & -\mathrm{j}(a+b)\\ \mathrm{j}(a+b)& 2ab
\end{array}\right]$ & $a\leq b$ \\
$[-\infty, -a]\cup [a,\infty]$&
$\left[\begin{array}{cc} -1 & 0 \\ 0 & a^2\end{array}\right]$
& $a\geq 0$ \\
\bottomrule
\end{tabular}
\end{center}
\caption{Common choices of $\Psi$ with $\Phi = \Phi_\mathrm i$
($\lambda$ imaginary). } \label{t-phic}
\end{table}
\begin{table}
\begin{center}
\begin{tabular}{ccc}\toprule
$\lambda$ & $\Psi$ & Assumptions \\
\midrule
$[a, b]$ &
$\left[\begin{array}{cc} 2 & -(a+b)\\ -(a+b) & 2ab
\end{array}\right]$ & $a \leq b$ \\
$[-\infty, a]\cup [b,\infty]$&
$\left[\begin{array}{cc} -2 & a+b \\ a+b & -2ab\end{array}\right]$
& $a\leq b$
\\
$[a, \infty]$ &
$\left[\begin{array}{cc} 0 & -1 \\ -1 & 2a\end{array}\right]$ \\
$[-\infty, a]$ &
$\left[\begin{array}{cc} 0 & 1 \\ 1 & -2a\end{array}\right]$
\\ \bottomrule
\end{tabular}
\end{center}
\caption{Common choices of $\Psi$ with $\Phi = \Phi_\mathrm r$
($\lambda$ real).} \label{t-phir}
\end{table}
As for circles and lines, we can apply a congruence transformation
to reduce~(\ref{e-quad-eq-ineq}) to a simple canonical case.
We mention two examples.
Iwasaki and Hara \cite[lemma 2]{IwH:05} show
that for every $\Phi$, $\Psi$ with $\mathrm{d}et \Phi < 0$,
there exists a nonsingular $R$ such that
\begin{equation} \label{e-IwH-canonical}
\Phi = R^H \Phi_\mathrm i R, \qquad
\Psi = R^H \left[\begin{array}{cc} \alpha & \beta \\ \beta
& \gamma \end{array}\right] R
\end{equation}
with $\alpha, \beta, \gamma$ real, and $\alpha \geq \gamma$.
To see this, we first apply a congruence transformation
$\Phi = R_1^H \Phi_\mathrm i R_1$ to
transform $\Phi$ to $\Phi_\mathrm i$.
Define
\[
R_1^{-H}\Psi R_1^{-1}
= \left[\begin{array}{cc}
x & \beta + \mathrm{j} z \\ \beta - \mathrm{j} z & y \end{array}\right]
\]
with real $x$, $y$, $z$, $\beta$, and consider the eigenvalue
decomposition
\begin{equation} \label{e-IwH-evd}
\left[\begin{array}{cc} x & \mathrm{j} z \\ -\mathrm{j} z & y \end{array}\right]
= Q
\left[ \begin{array}{cc} \alpha & 0 \\ 0 & \gamma \end{array}
\right]Q^H,
\end{equation}
with eigenvalues sorted as $\alpha \geq \gamma$.
Since the $2,1$ element of the matrix on the left-hand side
of~(\ref{e-IwH-evd}) is purely imaginary,
the columns of $Q$ can be normalized to be of the form
\[
Q = \left[\begin{array}{cc}
u & \mathrm{j} v \\
\mathrm{j} v & u \end{array}\right]
\]
with $u$ and $v$ real, and $u^2 + v^2 = 1$.
This implies that $Q \Phi_\mathrm i Q^H = Q^H \Phi_\mathrm i Q =
\Phi_\mathrm i$ and
\[
Q^H \left[\begin{array}{cc} x & \beta +\mathrm{j} z \\
\beta - \mathrm{j} z & y \end{array}\right]Q
= Q^H \left[\begin{array}{cc} x & \mathrm{j} z \\
- \mathrm{j} z & y \end{array}\right]Q
+ \left[\begin{array}{cc}
0 & \beta \\ \beta & 0 \end{array}\right]
= \left[\begin{array}{cc} \alpha & \beta \\
\beta & \gamma \end{array}\right].
\]
The transformation~(\ref{e-IwH-canonical}) now follows by
taking $R = Q^H R_1$.
Applying the congruence defined by $R$, we can reduce the
conditions~(\ref{e-quad-eq-ineq-hom}) to an equivalent system
\begin{equation} \label{e-IwH-transformed-hom}
\left[\begin{array}{c} \mu' \\ \nu' \end{array}\right]^H
\left[\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right]
\left[\begin{array}{c} \mu' \\ \nu' \end{array}\right]
= 0, \qquad
\left[\begin{array}{c} \mu' \\ \nu' \end{array}\right]^H
\left[\begin{array}{cc} \alpha & 0 \\ 0 & \gamma \end{array}\right]
\left[\begin{array}{c} \mu' \\ \nu' \end{array}\right]
\leq 0, \qquad
(\mu', \nu') \neq 0,
\end{equation}
where $(\mu', \nu') = R (\mu,\nu)$. In non-homogeneous coordinates,
\begin{equation} \label{e-IwH-transformed}
\newRe{\lambda'} = 0, \qquad \alpha |\lambda'|^2 +\gamma \leq 0.
\end{equation}
Keeping in mind that $\alpha \geq \gamma$, we can distinguish four
cases.
If $0 < \gamma \leq \alpha$ the solution set
of~(\ref{e-IwH-transformed}) is empty.
If $\gamma = 0 < \alpha$ the solution set is a singleton $\{0\}$.
If $\gamma < 0 < \alpha$, the solution set
of~(\ref{e-IwH-transformed}) is the interval of the imaginary
axis defined by $|\lambda'| \leq (-\gamma/\alpha)^{1/2}$.
If $\gamma \leq \alpha \leq 0$, the inequality is redundant and
the solution set is the entire imaginary axis.
Another useful canonical form of~(\ref{e-quad-eq-ineq})
is obtained by transforming the solution set to a subset of the unit
circle. If we define
\[
T = \frac{1}{\sqrt 2} \left[\begin{array}{cc}
1 & 1 \\ -1 & 1 \end{array}\right] R,
\qquad
\epsilon = \frac{1}{2} (\alpha + \gamma),
\qquad
\mathrm{d}elta = \frac{1}{2} (\alpha - \gamma),
\qquad
\eta = \beta.
\]
then it follows from from~(\ref{e-IwH-canonical}) that
\[
\Phi = T^H \Phi_\mathrm u T, \qquad
\Psi = T^H \left[\begin{array}{cc} \epsilon +\eta &
-\mathrm{d}elta \\ -\mathrm{d}elta & \epsilon - \eta \end{array}\right] T.
\]
The coefficients $\epsilon$, $\mathrm{d}elta$, $\eta$ are real,
with $\mathrm{d}elta \geq 0$.
The congruence defined by $T$ therefore transforms
the conditions~(\ref{e-quad-eq-ineq-hom}) to an equivalent system
\[
\left[\begin{array}{c} \mu' \\ \nu' \end{array}\right]^H
\left[\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right]
\left[\begin{array}{c} \mu' \\ \nu' \end{array}\right] = 0, \qquad
\left[\begin{array}{c} \mu' \\ \nu' \end{array}\right]^H
\left[\begin{array}{cc} 0 & -\mathrm{d}elta \\ -\mathrm{d}elta & 2\epsilon
\end{array}\right]
\left[\begin{array}{c} \mu' \\ \nu' \end{array}\right] \leq 0, \qquad
\]
where $(\mu', \nu') = T (\mu,\nu)$. In non-homogeneous coordinates,
this is
\[
|\lambda'|^2 = 1, \qquad \mathrm{d}elta \newRe{\lambda'} \geq
\epsilon.
\]
The solution set is empty if $\epsilon > \mathrm{d}elta$.
It is the unit circle if $\epsilon \leq -\mathrm{d}elta$.
It is the singleton $\{1\}$ if $\epsilon = \mathrm{d}elta > 0$.
It is a segment of the unit circle if $-\mathrm{d}elta < \epsilon < \mathrm{d}elta$.
\section{Matrix factorization results}
\label{s-matrix-fact}
This appendix contains a self-contained proof of
Lemma~\ref{l-quad-eq-ineq-general}, needed in the proof
of Theorem~\ref{t-decomp}, and some other matrix factorization
results that have appeared in papers on the Kalman-Yakubovich-Popov (KYP)
lemma \cite{Ran:96,IMF:00,BaV:02,BaV:03,PiV:11}.
We include the proofs because their constructive character is
important for the result in Theorem~\ref{t-decomp}.
Lemma~\ref{l-rantzer} is based on
\cite[lemma 3]{Ran:96} and~\cite[lemma 5]{IwH:05}.
Lemma~\ref{l-quad-eq-ineq-general} can be found in
\cite[corollary 1]{PiV:11}.
\begin{lemma} \label{l-rantzer}
Let $U$ and $V$ be two matrices in ${\mbox{\bf C}}^{p\times r}$.
\begin{itemize}
\item If $UU^H = VV^H$, then $U=V\Lambda$ for some unitary matrix
$\Lambda\in{\mbox{\bf C}}^{r\times r}$.
\item If $UU^H = VV^H$ and $UV^H + VU^H = 0$,
then $U=V\Lambda$ for some unitary and skew-Hermitian matrix
$\Lambda\in{\mbox{\bf C}}^{r\times r}$.
\item If $UU^H \preceq VV^H$ and $UV^H + VU^H = 0$,
then $U=V\Lambda$ for some skew-Hermitian matrix
$\Lambda\in{\mbox{\bf C}}^{r\times r}$ with $\|\Lambda\|_2 \leq 1$.
\end{itemize}
\end{lemma}
\noindent\emph{Proof.}\ \
If $UU^H = VV^H$, then $U$ and $V$ have singular value decompositions
of the form
\[
U = P \Sigma Q_u^H, \qquad V = P\Sigma Q_v^H,
\]
with unitary matrices $P\in{\mbox{\bf C}}^{p\times p}$,
diagonal $\Sigma \in {\mbox{\bf C}}^{p\times r}$,
and unitary $Q_u, Q_v \in {\mbox{\bf C}}^{r\times r}$.
The unitary matrix $\Lambda = Q_vQ_u^H$ satisfies $U = V\Lambda$.
To show the second part of the lemma, we substitute the singular value
decompositions of $U$ and $V$ in the equation $UV^H+V^HU=0$:
\[
\Sigma (Q_u^H Q_v + Q_v^H Q_u) \Sigma^T = 0.
\]
We define $\tilde\Lambda = Q_u^HQ_v$ (a unitary $r\times r$ matrix)
and write this as
\[
\left[\begin{array}{cc} \Sigma_1 & 0 \\ 0 & 0 \end{array}\right]
\left[\begin{array}{cc}
\tilde \Lambda_{11} + \tilde \Lambda_{11}^H &
\tilde \Lambda_{12} + \tilde \Lambda_{21}^H \\
\tilde \Lambda_{21} + \tilde \Lambda_{12}^H &
\tilde \Lambda_{22} + \tilde \Lambda_{22}^H
\end{array}\right]
\left[\begin{array}{cc} \Sigma_1 & 0 \\ 0 & 0 \end{array}\right]
= 0
\]
with $\Sigma_1$ positive diagonal of size $q\times q$, where
$q =\mathop{\bf rank}(U) = \mathop{\bf rank}(V)$,
and $\tilde \Lambda_{11}$ the $q\times q$ leading diagonal block of
$\tilde \Lambda$.
This shows that $\tilde\Lambda_{11} + \tilde\Lambda_{11}^H =0$,
so $\tilde \Lambda$ is unitary with a skew-Hermitian $1,1$ block.
Since $\tilde \Lambda_{11}$ is skew-Hermitian it has a Schur decomposition
$\tilde \Lambda_{11} = Q \Delta Q^H$
with unitary $Q\in{\mbox{\bf C}}^{q\times q}$, and
$\Delta$ a diagonal and purely imaginary matrix.
Moreover $\Delta\Delta^H \preceq I$ because $\tilde \Lambda_{11}$ is a
submatrix of the unitary matrix $\tilde \Lambda$.
Partition $Q$ and $\Delta$ as
\[
\tilde \Lambda_{11} =
\left[\begin{array}{cc} Q_1 & Q_2 \end{array}\right]
\left[\begin{array}{cc} \Delta_1 & 0\\ 0 & \Delta_2 \end{array}\right]
\left[\begin{array}{cc} Q_1 & Q_2 \end{array}\right]^H
\]
with $\Delta_1\Delta_1^H \prec I$ and $\Delta_2 \Delta_2^H = I$.
Since $\tilde \Lambda$ is unitary, we have
\begin{eqnarray*}
\tilde \Lambda_{12}\tilde \Lambda_{12}^H
& = & I - \tilde \Lambda_{11}\tilde \Lambda_{11}^H \\
& = & Q_1Q_1^H + Q_2 Q_2^H - Q_1 \Delta_1 \Delta_1^H Q_1^H
- Q_2 \Delta_2 \Delta_2^H Q_2^H \\
& = & Q_1(I-\Delta_1\Delta_1^H) Q_1^H,
\end{eqnarray*}
and, by the first part of the lemma,
$\tilde \Lambda_{12} = Q_1 (I-\Delta_1\Delta_1^H)^{1/2} W$ for
some unitary matrix $W$.
Therefore the matrix
\begin{eqnarray*}
\lefteqn{
\left[\begin{array}{cc}
\tilde \Lambda_{11} & \tilde\Lambda_{12} \\
-\tilde\Lambda_{12}^H & W^H \Delta_1^H W
\end{array}\right] }\\
& = & \left[\begin{array}{ccc}
Q_1 & Q_2 & 0 \\ 0 & 0 & W^H \end{array}\right]
\left[\begin{array}{ccc}
\Delta_1 & 0 & (I-\Delta_1\Delta_1^H)^{1/2} \\
0 & \Delta_2 & 0 \\ -(I-\Delta_1\Delta_1^H)^{1/2} & 0 & \Delta_1^H
\end{array}\right]
\left[\begin{array}{cc} Q_1^H & 0 \\ Q_2^H & 0 \\ 0 & W
\end{array}\right]
\end{eqnarray*}
is skew-Hermitian (from the expression on the left-hand side and
the fact that $\tilde \Lambda_{11}$ is skew-Hermitian
and $\Delta_1$ is purely imaginary)
and unitary (the right-hand side is a product of three unitary matrices).
If we now define
\[
\Lambda = Q_v \left[\begin{array}{cc}
\tilde \Lambda_{11} & \tilde \Lambda_{12} \\
-\tilde \Lambda_{12}^H & W^H \Delta_1^H W \end{array}\right] Q_v^H
\]
then $\Lambda$ is unitary and skew-Hermitian, and
\begin{eqnarray*}
U
& = & P \left[\begin{array}{cc} \Sigma_1 & 0 \\ 0 & 0 \end{array}\right]
\left[\begin{array}{cc}
\tilde \Lambda_{11} & \tilde \Lambda_{12} \\
\tilde \Lambda_{21} & \tilde \Lambda_{22} \end{array}\right] Q_v^H \\
& = & P \left[\begin{array}{cc} \Sigma_1 & 0 \\ 0 & 0 \end{array}\right]
\left[\begin{array}{cc}
\tilde \Lambda_{11} & \tilde \Lambda_{12} \\
-\tilde \Lambda_{12}^H &
W^H \Delta_1^H W \end{array}\right] Q_v^H \\
& = & P \left[\begin{array}{cc} \Sigma_1 & 0 \\ 0 & 0 \end{array}\right]
Q_v^H \Lambda \\
& = & V \Lambda.
\end{eqnarray*}
This proves part two of the lemma.
Assume $UU^H \preceq VV^H$ and $VV^H-UU^H$ has rank $s$.
We use any factorization $VV^H - UU^H = \tilde U\tilde U^H$ with
$\tilde U\in{\mbox{\bf C}}^{p\times s}$ and
write $UU^H \preceq VV^H$ and $UV^H+VU^H= 0$ as
\[
\left[\begin{array}{cc} U & \tilde U \end{array}\right]
\left[\begin{array}{cc} U & \tilde U \end{array}\right]^H
= \left[\begin{array}{cc} V & 0 \end{array}\right]
\left[\begin{array}{cc} V & 0 \end{array}\right]^H
\]
and
\[
\left[\begin{array}{cc} U & \tilde U \end{array}\right]
\left[\begin{array}{cc} V & 0 \end{array}\right]^H
+
\left[\begin{array}{cc} V & 0 \end{array}\right]
\left[\begin{array}{cc} U & \tilde U \end{array}\right]^H = 0.
\]
It follows from part 2 that
\[
\left[\begin{array}{cc} U & \tilde U \end{array}\right]
= \left[\begin{array}{cc} V & 0 \end{array}\right]
\left[\begin{array}{cc}
\tilde \Lambda_{11} & \tilde \Lambda_{12} \\
\tilde \Lambda_{21} & \tilde \Lambda_{22}
\end{array}\right]
\]
with $\tilde \Lambda$ unitary and skew-Hermitian. The subblock
$\Lambda= \tilde \Lambda_{11}$ satisfies $U=V\Lambda$, $\Lambda +
\Lambda^H = 0$ and
$\Lambda^H \Lambda \preceq I$.
$\Box$
\begin{lemma} \label{l-quad-eq-ineq-general}
Let $\Phi$, $\Psi\in\mathbf{H}^2$ with $\mathrm{d}et\Phi < 0$.
If $U, V\in{\mbox{\bf C}}^{p\times r}$ satisfy
\begin{eqnarray}
\Phi_{11} UU^H + \Phi_{21} UV^H + \Phi_{12} VU^H + \Phi_{22} VV^H
& = & 0, \label{e-quadeq-1}\\
\Psi_{11} UU^H + \Psi_{21} UV^H + \Psi_{12} VU^H + \Psi_{22} VV^H
& \preceq & 0, \label{e-quadeq-2}
\end{eqnarray}
then there exist a matrix $W\in{\mbox{\bf C}}^{p\times r}$,
a unitary matrix $Q\in{\mbox{\bf C}}^{r\times r}$, and vectors
$\mu,\nu\in{\mbox{\bf C}}^r$ such that
\begin{equation} \label{e-quadeq-facts}
U = W\mathop{\bf diag}(\mu)Q^H, \qquad V = W\mathop{\bf diag}(\nu)Q^H, \qquad
\end{equation}
and
\begin{equation} \label{e-quadeq-C}
q_\Phi(\mu_i,\nu_i) =0, \qquad
q_\Psi(\mu_i,\nu_i) \leq 0, \qquad
(\mu_i,\nu_i) \neq 0, \qquad i=1,\ldots, r.
\end{equation}
\end{lemma}
\noindent\emph{Proof.}\ \
Suppose $U$ and $V$ are $p\times r$ matrices that
satisfy~(\ref{e-quadeq-1}) and~(\ref{e-quadeq-2}).
As explained in appendix~\ref{s-regions}, there exists a nonsingular $R$
such that
\[
\Phi = R^H \left[\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right]R,
\qquad
\Psi = R^H \left[\begin{array}{cc} \alpha & \beta \\ \beta & \gamma
\end{array}\right]R
\]
with $\beta$ real and $\gamma \leq \alpha$.
Define $S = R_{11} U + R_{12} V$ and $T=R_{21} U + R_{22} V$.
From~(\ref{e-quadeq-1}) and~(\ref{e-quadeq-2}),
\[
\left[\begin{array}{cc} S & T \end{array}\right]
\left[\begin{array}{cc} 0 & I \\ I & 0 \end{array}\right]
\left[\begin{array}{c} S^H \\ T^H \end{array}\right]
= \left[\begin{array}{cc} U & V \end{array} \right]
\left[\begin{array}{cc}
\Phi_{11} I & \Phi_{21} I \\ \Phi_{12} I & \Phi_{22} I \end{array}
\right]
\left[\begin{array}{c} U^H \\ V^H \end{array}\right] = 0
\]
and
\[
\left[\begin{array}{cc} S & T \end{array}\right]
\left[\begin{array}{cc} \alpha I & \beta I \\ \beta I & \gamma I
\end{array}\right]
\left[\begin{array}{c} S^H \\ T^H \end{array}\right]
= \left[\begin{array}{cc}
U & V \end{array}\right] \left[\begin{array}{cc}
\Psi_{11} I & \Psi_{21} I \\ \Psi_{12} I & \Psi_{22} I \end{array}
\right]
\left[\begin{array}{c} U^H \\ V^H \end{array}\right] \preceq 0.
\]
Therefore
\begin{equation} \label{e-2eqs-transf}
ST^H + TS^H = 0, \qquad \alpha SS^H + \gamma TT^H \preceq 0.
\end{equation}
We show that this implies that
\begin{equation}\label{e-2eqs-transf-fact-a}
S = W\mathop{\bf diag}(s)Q^H, \qquad T = W\mathop{\bf diag}(t)Q^H,
\end{equation}
for some $W\in{\mbox{\bf C}}^{p\times r}$, unitary $Q\in{\mbox{\bf C}}^{r\times r}$,
and vectors $s,t\in{\mbox{\bf C}}^{r}$ that satisfy
\begin{equation}\label{e-2eqs-transf-fact-b}
s_i \bar t_i + \bar s_i t_i = 0, \qquad
\alpha |s_i|^2 + \gamma |t_i|^2 \leq 0, \qquad
(s_i,t_i) \neq 0, \qquad i=1,\ldots,r.
\end{equation}
The result is trivial if $S$ and $T$ are zero, since in that
case we can choose
$W$ zero, and arbitrary $Q$, $s$, $t$.
If at least one of the two matrices is nonzero, then the inequality
in~(\ref{e-2eqs-transf}), combined with $\alpha \geq \gamma$,
implies that $\gamma\leq 0$.
Therefore there are three cases to consider.
\begin{itemize}
\item
If $\alpha \leq 0$, we write the equality in~(\ref{e-2eqs-transf})
as
\[
(S+T)(S+T)^H = (S-T)(S-T)^H.
\]
From Lemma~\ref{l-rantzer}, this implies that
$S+ T = (S-T) \Lambda$ with $\Lambda$ unitary. Let
$\Lambda = Q\mathop{\bf diag}(\rho)Q^H$ be the Schur decomposition of $\Lambda$,
with $|\rho_i| = 1$ for $i=1,\ldots, r$.
Define
\[
W = (S-T)Q, \qquad s = \frac{1}{2} (\rho + \mathbf 1), \qquad
t = \frac{1}{2} (\rho - \mathbf 1).
\]
\item
If $\gamma = 0 < \alpha$, then $S=0$, and we can take $Q=I$
\[
W=T, \qquad s=0, \qquad t=\mathbf 1.
\]
\item
If $\gamma < 0 < \alpha$, then from Lemma~\ref{l-rantzer},
we have $S = (-\gamma/\alpha)^{1/2} T\Lambda$
for some skew-Hermitian $\Lambda$ with $\Lambda^H\Lambda \preceq I$.
This matrix has a Schur decomposition $\Lambda = Q\mathop{\bf diag}(\rho) Q^H$
with $|\rho_i| \leq 1$ for $j=1,\ldots,r$.
Define
\[
W = TQ, \qquad s = (-\gamma/\alpha)^{1/2} \rho, \qquad t =\mathbf 1.
\]
\end{itemize}
The factorizations of $U$ and $V$ now follow from
\[
\left[\begin{array}{c}
U \\ V\end{array}\right]
= (R^{-1} \otimes I) \left[\begin{array}{c} S \\ T\end{array}\right]
= (R^{-1} \otimes I) \left[\begin{array}{c} W\mathop{\bf diag}(s) \\
W\mathop{\bf diag}(t) \end{array}\right]Q^H
= \left[\begin{array}{c} W\mathop{\bf diag}(\mu) \\
W\mathop{\bf diag}(\nu) \end{array}\right]Q^H
\]
where $\mu$ and $\nu$ are defined as
\[
\left[\begin{array}{c} \mu_i \\ \nu_i \end{array}\right]
= R^{-1} \left[\begin{array}{c} s_i \\ t_i \end{array}\right],
\quad i=1,\ldots,r.
\]
These pairs $(\mu_i,\nu_i)$ are nonzero and satisfy
\[
\left[\begin{array}{c} \mu_i \\ \nu_i \end{array}\right]^H
\Phi
\left[\begin{array}{c} \mu_i \\ \nu_i \end{array}\right]
= \left[\begin{array}{c} s_i \\ t_i \end{array}\right]^H
\left[\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right]
\left[\begin{array}{c} s_i \\ t_i \end{array}\right]
= \bar s_i t_i + s_i \bar t_i
= 0
\]
and
\[
\left[\begin{array}{c} \mu_i \\ \nu_i \end{array}\right]^H
\Psi
\left[\begin{array}{c} \mu_i \\ \nu_i \end{array}\right]
= \left[\begin{array}{c} s_i \\ t_i \end{array}\right]^H
\left[\begin{array}{cc} \alpha & \beta \\ \beta & \gamma
\end{array}\right]
\left[\begin{array}{c} s_i \\ t_i \end{array}\right]
= \alpha |s_i|^2 + \beta(\bar s_i t_i + s_i\bar t_i)
+ \gamma |t_i|^2
\leq 0.
\]
$\Box$
\section{Strict feasibility}
\label{s-slater}
In this appendix we discuss strict feasibility of the
constraints $X\succeq 0$, (\ref{e-phi}), (\ref{e-psi})
in Theorem~\ref{t-decomp}.
We assume that the set $\mathcal C$ defined in~(\ref{e-mC})
is not empty and not a singleton.
This means that if the inequality $q_\Psi(\mu,\nu) \leq 0$
in the definition is not redundant, then there exist points in $\mathcal C$
with $q_\Psi(\mu,\nu) < 0$.
We will distinguish these two cases.
\begin{itemize}
\item \emph{Line or circle.}
If the inequality $q_\Psi(\mu,\nu) \leq 0$ in the definition is redundant,
we have
\[
\mathcal C = \{ (\mu,\nu) \in {\mbox{\bf C}}^2 \mid (\mu,\nu) \neq 0,
\; q_\Phi(\mu,\nu) = 0 \},
\]
and $\mathcal C$ is a line or circle in homogeneous coordinates.
In this case we understand by strict feasibility of $X$ that
\begin{equation} \label{e-strict-feas-case-1}
X \succ 0, \qquad
\Phi_{11} FXF^H + \Phi_{21} FXG^H + \Phi_{12} GXF^H + \Phi_{22} GXG^H = 0.
\end{equation}
We also define $\mathcal C^\circ = \mathcal C$.
\item \emph{Segment of line or circle.}
In the second case, $\mathcal C$ is a proper one-dimensional subset of
the line or circle defined by $q_\Psi(\mu,\nu) = 0$.
In this case we define strict feasibility of $X$ as
\begin{equation}\label{e-strict-feas-case-2}
\mbox{(\ref{e-strict-feas-case-1})}, \qquad
\Psi_{11} FXF^H + \Psi_{21} FXG^H + \Psi_{12} GXF^H + \Psi_{22} GXG^H
\prec 0.
\end{equation}
We also define
$\mathcal C^\circ = \{ (\mu,\nu) \neq 0 \mid
q_\Phi(\mu,\nu) = 0, \; q_\Psi(\mu,\nu) < 0\}$.
\end{itemize}
The conditions on $F$ and $G$ that guarantee strict feasibility
will be expressed in terms of the Kronecker structure of the
matrix pencil $\lambda G - F$ \cite{Gan:05,Van:79}.
For every matrix pencil there exist nonsingular matrices $P$ and $Q$
such that
\begin{eqnarray}
\lefteqn{P( \lambda G - F) Q } \nonumber \\
& = & \left[\begin{array}{ccccccccccc}
L_{\eta_1}(\lambda)^T & 0 & \cdots & 0 & 0 & 0 & 0 & \cdots & 0 \\
0 & L_{\eta_2}(\lambda)^T & \cdots & 0 & 0 & 0 & 0 & \cdots & 0 \\
\vdots & \vdots & \mathrm{d}dots & \vdots & \vdots & \vdots &
\vdots & & \vdots \\
0 & 0 & \cdots & L_{\eta_l}(\lambda)^T & 0 & 0 & 0 & \cdots & 0 \\
0 & 0 & \cdots & 0 & \lambda B - A & 0 & 0 & \cdots & 0 \\
0 & 0 & \cdots & 0 & 0 & L_{\epsilon_1}(\lambda) & 0 & \cdots & 0 \\
0 & 0 & \cdots & 0 & 0 & 0 & L_{\epsilon_2} (\lambda) &\cdots & 0 \\
\vdots & \vdots & & \vdots & \vdots & \vdots & \vdots & \mathrm{d}dots &
\vdots \\
0 & 0 & \cdots & 0 & 0 & 0 & 0 & \cdots & L_{\epsilon_r}(\lambda)
\end{array}\right]
\label{e-kronecker}
\end{eqnarray}
where
$L_\epsilon(\lambda)$ is the $\epsilon \times (\epsilon +1)$ pencil
\[
L_\epsilon(\lambda) = \left[\begin{array}{cccccc}
\lambda & -1 & 0 & \cdots & 0 & 0 \\
0 & \lambda & -1 & \cdots & 0 & 0 \\
\vdots & \vdots & \vdots & & 0 & 0 \\
0 & 0 & 0 & \cdots & -1 & 0 \\
0 & 0 & 0 & \cdots & \lambda & -1
\end{array}\right],
\]
and $\lambda B - A$ is a regular pencil, {\it i.e.}, it is square and
$\mathrm{d}et(\lambda B - A)$ is not identically zero.
The generalized eigenvalues of $\lambda B - A$
are sometimes referred to as the generalized eigenvalues of the pencil
$\lambda G - F$ \cite[page 16]{IOW:99}.
The parameters $\epsilon_1$, \ldots, $\epsilon_r$ are the
\emph{right Kronecker indices} of the pencil and
the parameters $\eta_1$, \ldots, $\eta_l$ are the
\emph{left Kronecker indices}.
The \emph{normal rank} of the pencil is equal to $p-l$,
where $p$ is the row dimension of $F$ and $G$.
We show that there exists a strictly feasible $X$ if and
only if the following two conditions hold.
\begin{enumerate}
\item The normal rank of $\lambda G - F$ is $p$.
This means that $l =0$ in~(\ref{e-kronecker}).
\item The generalized eigenvalues of the pencil $\lambda G - F$
(defined as the generalized eigenvalues of $\lambda B -A$)
are \emph{nondefective}, {\it i.e.}, their algebraic multiplicity is equal to
the geometric multiplicity, and lie in $\mathcal C^\circ$.
(More accurately, if $\lambda$ is a finite generalized eigenvalue,
then $(\lambda, 1) \in \mathcal C^\circ$. If it is an infinite
generalized eigenvalue, then $(1,0) \in \mathcal C^\circ$.).
\end{enumerate}
A sufficient but more easily verified condition is that
$\mathop{\bf rank}{(\mu G - \nu F)} = p$ for all $(\mu, \nu) \neq 0$,
{\it i.e.}, $l=0$ and the block $\lambda B-A$ in~(\ref{e-kronecker})
is not present.
\noindent\emph{Proof.}\ \
Without loss of generality we can assume that the
pencil is in the Kronecker canonical form ($P=I$, $Q=I$
in~(\ref{e-kronecker})) and that $\Phi = \Phi_\mathrm u$, so the
equality constraint in~(\ref{e-strict-feas-case-1}) is
\begin{equation} \label{e-FXF=GXG}
FXF^H = GXG^H.
\end{equation}
We first show that the two conditions are necessary.
Assume $X$ is strictly feasible.
Partition $X$ as an $(l+1+r)\times (l+1+r)$ block matrix, with block
dimensions equal to the column dimensions of the $l+1+r$ block columns
in~(\ref{e-kronecker}).
Suppose $l\geq 1$ and consider the $k$th diagonal block
$X_{kk}$ with $1 \leq k\leq l$.
The $k$th diagonal block of the pencil is
\[
\lambda G_k - F_k = L_{\eta_k}(\lambda)^T =
\lambda \left[\begin{array}{c} I_{\eta_k} \\ 0_{1\times \eta_k}
\end{array}\right] -
\left[\begin{array}{c} 0_{1\times \eta_k} \\ I_{\eta_k}
\end{array}\right].
\]
The $k$th diagonal block of~(\ref{e-FXF=GXG}) is
$F_k X_{kk} F_k^H = G_kX_{kk} G_k^H$ or
\[
\left[\begin{array}{c} 0_{1\times\eta_k} \\ I_{\eta_k}
\end{array}\right]
X_{kk}
\left[\begin{array}{cc} 0_{\eta_k\times 1} & I_{\eta_k}
\end{array}\right]
=
\left[\begin{array}{c} I_{\eta_k} \\ 0_{1\times\eta_k}
\end{array}\right]
X_{kk}
\left[\begin{array}{cc} I_{\eta_k} &
0_{\eta_k \times 1} \end{array}\right].
\]
This is impossible since $X_{kk} \succ 0$.
Hence, if~(\ref{e-FXF=GXG}) holds with $X\succ 0$, then $l=0$.
Next suppose $\mathrm{d}et(\mu B - \nu A)=0$ for some $(\mu,\nu) \neq 0$.
If $\nu\neq 0$, then $\mu/\nu$ is a finite generalized eigenvalue
of the pencil $\lambda B-A$; if $\nu=0$ then the pencil has a
generalized eigenvalue at infinity.
Let $y$ be a corresponding left generalized eigenvector, {\it i.e.},
$y^H(\mu B - \nu A) = 0$, while $y^H B$ and $y^HA$ are not both zero
(since $y^HB = y^HA = 0$ would imply that the pencil $\lambda B - A$
is singular).
Define $u^H = y^HB$ if $\nu\neq 0$ and $u^H = y^HA$ otherwise.
This is a nonzero vector.
The first diagonal block of~(\ref{e-FXF=GXG}) is
\begin{equation} \label{e-AXA}
AX_{11}A^H = BX_{11}B^H.
\end{equation}
From this it follows that $|\mu|^2 u^H X_{11} u = |\nu|^2 u^HX_{11} u$,
and, since $X_{11} \succ 0$, we have
$q_\Phi(\mu,\nu) = |\mu|^2 - |\nu|^2 = 0$,
{\it i.e.}, the generalized eigenvalues are on the unit circle.
In addition, if the inequality in~(\ref{e-strict-feas-case-2})
holds, then
\[
\Psi_{11} AX_{11}A^H + \Psi_{21} AX_{11}B^H + \Psi_{12} BX_{11}A^H
+ \Psi_{22} BX_{11}B^H \prec 0
\]
and from this, $q_\Psi(\mu,\nu) (u^HX_{11}u) < 0$.
This is only possible if $q_\Psi(\mu,\nu) < 0$.
We conclude that if $\mathrm{d}et (\mu B - \nu A) = 0$ for nonzero $(\mu,\nu)$,
then $(\mu, \nu) \in \mathcal C^\circ$.
Next we show that the generalized eigenvalues of the pencil
$\lambda B - A$ are nondefective. Since $C^\circ$ is the unit circle
or a subset of the unit circle, there are no infinite generalized
eigenvalues.
Assume the pencil is in Weierstrass canonical form, {\it i.e.},
\[
\lambda B-A =
\left[\begin{array}{cccc}
(\lambda - \rho_1) I - J_{s_1} & 0 & \cdots & 0 \\
0 & (\lambda - \rho_2) I - J_{s_2} & \cdots & 0 \\
\vdots & \vdots & \mathrm{d}dots & \vdots \\
0 & 0 & \cdots & (\lambda - \rho_t) I - J_{s_t}
\end{array}\right],
\]
where $\rho_1$, \ldots, $\rho_t$ are the generalized eigenvalues
(which satisfy $|\rho_i|=1$), and $J_s$ is the $s\times s$ matrix
\[
J_s =
\left[\begin{array}{cccccc}
0 & 1 & 0 & \cdots & 0 & 0 \\
0 & 0 & 1 & \cdots & 0 & 0 \\
0 & 0 & 0 & \cdots & 0 & 0 \\
\vdots & \vdots & \vdots & & \vdots & \vdots \\
0 & 0 & 0 & \cdots & 0 & 1\\
0 & 0 & 0 & \cdots & 0 & 0
\end{array} \right].
\]
Then~(\ref{e-AXA}) implies that
\[
(\rho_i - J_{s_i}) X_{11,i}
(\rho_i - J_{s_i})^H = X_{11,i}
\]
where $X_{11, i}$ is the $i$th diagonal block of $X_{11}$,
if we partition $X_{11}$ as a $t\times t$ block matrix
with $i,j$ block of size of $s_i \times s_j$.
Expanding this gives
\[
|\rho_i|^2 X_{11,i} - \rho_i X_{11,i} J_{s_i}^T
- \bar\rho_i J_{s_i} X_{11,i} + J_{s_i}X_{11,i} J_{s_i}^T = X_{11,i}.
\]
Since $|\rho_i|=1$ this simplifies to
\[
\rho_i X_{11,i} J_{s_i}^T +
\bar\rho_i J_{s_i} X_{11,i} = J_{s_i}X_{11,i} J_{s_i}^T.
\]
The last row of the second matrix on the left-hand side and the
last row of the matrix on the right-hand side are zero.
Therefore the last row of the first matrix on the left is zero.
However the element in column $s_i-1$ is the last diagonal element
of the positive definite matrix $X_{11,i}$.
Hence, we have a contradiction unless $s_i=1$, {\it i.e.}, the generalized
eigenvalue $\rho_i$ is nondefective.
We conclude that the two conditions are necessary.
It remains to show that the conditions are sufficient.
If the two conditions hold, then
$\lambda G-F$ has the Kronecker canonical form
\[
\lambda G - F
= \left[\begin{array}{cccccc}
\lambda - \rho_1 & \cdots & 0 & 0 & \cdots & 0 \\
\vdots & \mathrm{d}dots & \vdots & \vdots & & \vdots \\
0 & \cdots & \lambda - \rho_t & 0 & \cdots & 0 \\
0 & \cdots & 0 & L_{\epsilon_1}(\lambda) & \cdots & 0 \\
\vdots & & \vdots & \vdots & \mathrm{d}dots & \vdots \\
0 & \cdots & 0 & 0 & \cdots & L_{\epsilon_r}(\lambda)
\end{array}\right]
\]
with $\rho_i\in\mathcal C^\circ$ for $i=1,\ldots, t$.
Define a block diagonal matrix
\[
X =
\left[\begin{array}{cccccc}
1 & \cdots & 0 & 0 & \cdots & 0 \\
\vdots & \mathrm{d}dots & \vdots & \vdots & & \vdots \\
0 & \cdots & 1 & 0 & \cdots & 0 \\
0 & \cdots & 0 & X_{11} & \cdots & 0 \\
\vdots & & \vdots & \vdots & \mathrm{d}dots & \vdots \\
0 & \cdots & 0 & 0 & \cdots & X_{rr}
\end{array}\right]
\]
with
diagonal blocks
\[
X_{kk} = \sum_{i=1}^{\epsilon_k+1}
\left[\begin{array}{c}
1 \\ \lambda_{ki} \\ \lambda_{ki}^2 \\ \vdots \\
\lambda_{ki}^{\epsilon_k} \end{array}\right]
\left[\begin{array}{c}
1 \\ \lambda_{ki} \\ \lambda_{ki}^2 \\ \vdots \\
\lambda_{ki}^{\epsilon_k} \end{array}\right]^H
\]
for $k=1,\ldots, r$, where $\lambda_{k1}$, \ldots,
$\lambda_{k,\epsilon_k+1}$ are distinct elements of $\mathcal C^\circ$.
This matrix $X$ is strictly feasible.
$\Box$
\end{document}
|
\begin{document}
\title[]{Generalized Bernstein operators defined by increasing nodes}
\author{J. M. Aldaz and H. Render}
\address{Instituto de Ciencias Matem\'aticas (CSIC-UAM-UC3M-UCM) and Departamento de
Matem\'aticas,
Universidad Aut\'onoma de Madrid, Cantoblanco 28049, Madrid, Spain.}
\email{[email protected]}
\email{[email protected]}
\address{H. Render: School of Mathematical Sciences, University College
Dublin, Dublin 4, Ireland.}
\email{[email protected]}
\thanks{2010 Mathematics Subject Classification: \emph{Primary: 41A10}}
\thanks{Key words and phrases: \emph{Bernstein polynomial, Bernstein operator.}}
\thanks{The first named author was partially supported by Grant MTM2015-65792-P of the
MINECO of Spain, and also by by ICMAT Severo Ochoa project SEV-2015-0554 (MINECO)}
\begin{abstract}
We study certain generalizations of the classical Bernstein operators,
defined via increasing sequences of nodes. Such operators are required to
fix
two functions, $f_0$ and $f_1$, such that $f_0 > 0$ and $f_1/ f_0$ is increasing
on an interval $[a,b]$.
A characterization regarding when this can be done is presented. From it we
obtain, under rather general circumstances, the following necessary condition
for existence: if nodes are non-«decreasing, then $(f_1/f_0)^\prime >0 $ on $(a,b)$,
while if nodes are strictly increasing, then $(f_1/f_0)^\prime >0 $ on $[a,b]$.
\end{abstract}
\maketitle
\markboth{J. M. Aldaz, H. Render}{Generalized Bernstein Operators}
\section{Introduction}
Let $\mathbb{P}_{n} = \mathbb{P}_{n}[a,b]$ denote the space of polynomials of degree
bounded by $n$, over the interval $[a,b]$. In recent years there has been a continued interest in finding generalizations or modifications of the
classical Bernstein operators
$B_{n}:C
\left[ a,b\right] \rightarrow \mathbb{P}_{n}[a,b]$, defined by
\begin{equation}
B_{n}f\left( x\right)
=
\sum_{k=0}^{n}f\left( a+\frac{k}{n}\left( b-a\right)
\right) \binom{n}{k}\frac{\left( x-a\right) ^{k}\left(
b-x\right) ^{n-k}}{\left( b-a\right) ^{n}}, \label{defBPr}
\end{equation}
to more general spaces of functions, but still reproducing a two dimensional
subspace, say $\operatorname{Span}\{f_0, f_1\}$, with
$f_0 > 0$ and $f_1/f_0$ injective. Within the realm of
polynomial spaces, one asks when the exact reproduction of functions, other than the affine ones, is possible.
Also, similar questions have been asked about related positive
operators, (cf. for instance \cite{AcArGo}).
Sometimes fixing a subspace different from the affine functions
is achieved by modifying the Bernstein
bases
(consider, for instance, the nowadays called King's operators,
after \cite{Ki}).
Within the line of research followed here (cf. \cite{MoNe00}, \cite{AKR07},
\cite{AKR08b}, \cite{AKR08}, \cite{KR07b}, \cite{AR}, \cite{Ma}, \cite{AiMa},
\cite{AlRe18})
fixing $f_0$ and $f_1$ is achieved,
when possible, by modifying the location of the nodes $t_{n, k }$
(instead of having $t_{n, k } = a+\frac{k}{n}\left( b-a\right)$ as in (\ref{defBPr})). A motivation for this approach
is that it allows us to keep the Bernstein bases unchanged, a desirable feature given
their several optimality properties, cf. for instance \cite{Fa}. Multiplying by $-1$ if needed,
we may assume that $f_1/f_0$ is strictly increasing.
The situation regarding the existence of generalized Bernstein operators,
defined by strictly increasing sequences of nodes, is well understood in the context of
extended Chebyshev spaces, cf. \cite{AKR08b}: one considers a two
dimensional extended Chebyshev space $U_1$, for which a generalized
Bernstein operator fixing it can always be defined with increasing
nodes (since they are the endpoints of the interval),
and inductively, via the interlacing property of nodes (cf. \cite[Theorem 6]{AKR08b})
this definition is extended to
$U_1 \subset U_2 \subset \cdots \subset U_n$, where each $U_k$ is a
$k + 1$-dimensional extended Chebyshev space.
But this framework is insufficient even for spaces of polynomials,
since for instance, it cannot handle the case where we have
$U_1: = \operatorname{Span}\{\mathbf{1}, x^3\}$ over $[a,b]$, with
$a < 0 < b$ (cf. Example \ref{E2}). It is thus natural to try to go beyond
chains of extended Chebyshev spaces. Now, a salient difference between
various definitions of generalized Bernstein operators appearing in the literature,
is whether one should
require the sequence of nodes to be strictly increasing
(as in \cite{Ma}),
or not (as in \cite{AKR08b}, \cite{AKR08}). Of course, having
strictly increasing nodes leads to better properties from the point of view of
shape preservation, but existence will be obtained in fewer cases.
We shed light on this issue by characterizing, in terms of the
spaces $U_n$ and $D_{f_{0}}U_{n}:=\left\{ \frac{d}{dx}\left( \frac{f}{f_{0}}\right) :f\in
U_{n}\right\}$, when the sequence of nodes is non-decreasing, and
when it is strictly increasing, cf.
Theorem \ref{ThmBern2} below for full details.
This Theorem improves on
\cite[Theorem1]{AKR07} (the main result of \cite{AKR07}) and generalises
\cite[Theorem 3.2]{AlRe18}, which deals exclusively with the polynomial case
$\mathbb{P}_{n}[a,b]$. From Theorem \ref{ThmBern2}
the following necessary condition is obtained:
if both spaces $U_n$ and $D_{f_{0}}U_{n}:=\left\{ \frac{d}{dx}\left( \frac{f}{f_{0}}\right) :f\in
U_{n}\right\}$ have positive Bernstein bases (a hypothesis weaker than
being extended Chebyshev spaces) then the existence of a generalized
Bernstein operator having non-decreasing nodes entails
that $(f_1/f_0)^\prime >0 $ on $(a,b)$,
while if nodes are strictly increasing, then $(f_1/f_0)^\prime >0 $ on $[a,b]$, cf. Corollary \ref{antimaz}.
Since this result contradicts some statements
made in \cite[Section 7.2]{Ma}, in an effort to clarify these issues
we have emphasized concrete examples
and explicit computations thoroughout the paper.
To sum up, the difference between the cases where $f_1^\prime$ vanishes
at some point inside $(a,b)$, and where $f_1^\prime >0$
on $(a,b)$, turns out to be very important
from the point of view of the ordering of the nodes, and hence, of shape preservation
and of the existence of generalized Bernstein operators.
\section{Definitions and motivating examples.}
\begin{definition} Let $U_{n}$ be an $n+1$ dimensional subspace of
$C^n\left( \left[
a,b\right], \mathbb{K}\right)$, where $\mathbb{K} = \mathbb{R}$ or
$\mathbb{K} = \mathbb{C}$. A
Bernstein basis $\{p_{n,k}: k=0,\dots,n\}$ of $U_n$ is a basis
with the property that each $p_{n,k}$ has a zero of order $k$ at $a$, and
a zero of order
$n-k$ at $b$. The function $p_{n,k}$ might have additional zeros inside
$\left(a,b\right) $; this is not excluded by the preceding definition. A Bernstein basis
is
{\em non-negative} if for all
$k= 0, \dots, n$, $p_{n,k} \ge 0$ on $\left[ a,b\right]$, and
{\em positive} if $p_{n,k} > 0$ on $\left(a,b\right)$. Finally,
a non-negative Bernstein basis is {\em normalized} if $\sum_{k=0}^n p_{n,k} \equiv 1$.
\end{definition}
It is easy to check that non-negative Bernstein bases are unique
up to multiplication by a positive scalar, and that normalized
Bernstein bases are unique.
\begin{definition} If $U_{n}$ has a non-negative Bernstein basis $\{p_{n,k}: k=0,\dots,n\}$, we define a
{\em generalized Bernstein operator} $B_{n}:C\left[ a,b\right] \rightarrow U_{n}$
by setting
\begin{equation}
B_{n}\left( f\right) =\sum_{k=0}^{n}f\left( t_{n,k}\right) \alpha
_{n,k}p_{n,k}, \label{eqBern}
\end{equation}
where the nodes $t_{n,0},...,t_{n,n}$ belong to the interval
$\left[ a,b\right]$, and the weights $\alpha_{n,0},...,\alpha_{n,n}$ are positive.
\end{definition}
Non-negativity of the functions $p_{n,k}$ and positivity of the weights $\alpha_{n,0},...,\alpha_{n,n}$ are required
so that the resulting operator is positive, a natural property from the viewpoint of
shape preservation. Strict positivity
of the weights entails that all the basis functions are used in
the definition of the operator, something useful if we want families of
operators to converge to the identity. Finally,
the nodes must belong to $\left[ a,b\right]$; otherwise, the operator will not be well defined on
$C\left[ a,b\right]$. But no requirement is made about the
ordering of the nodes, and in particular, we do not ask that they be {\em strictly increasing},
i.e.,
that $t_{n,0} < t_{n,1} < \cdots < t_{n,n}$. When we only have
$t_{n,0} \le t_{n,1} \le \cdots \le t_{n,n}$ we say that the sequence of nodes is
{\em increasing}, or equivalently, {\em non-decreasing}.
\vskip .2 cm
The problem of
existence, as studied in \cite{AKR08b} and \cite{AKR08}, arises when we choose two functions $f_{0},f_{1}\in U_{n}$, such that $f_{0} >0 $,
$f_{1}/f_{0}$ is strictly increasing, and we require that
\begin{equation}
B_{n}\left( f_{0}\right) =f_{0}\text{ and }B_{n}\left( f_{1}\right) =f_{1}.
\label{eqBern2}
\end{equation}
If these equalities can be satisfied, they uniquely determine the location of the
nodes and the values of the coefficients, cf. \cite[Lemma 5]{AKR08b}; in other words, there is at most
one Bernstein operator $B_{n}$ of the form (\ref{eqBern}) satisfying (\ref
{eqBern2}).
We will consistently use the following notation. Assume that $p_{n,k},$
$k=0,...,n$, is a Bernstein basis of the space $U_{n}$. Given
$f_{0},f_{1}\in U_{n}$, there exist coefficients $\beta_{n,0},...,\beta_{n,n}$
and $\gamma_{n,0},...,\gamma_{n,n}$ such that
\begin{equation}
f_{0}\left( x\right) =\sum_{k=0}^{n}\beta_{n,k}p_{n,k}\left( x\right) \text{
and }f_{1}\left( x\right) =\sum_{k=0}^{n}\gamma_{n,k}p_{n,k}\left( x\right) .
\label{eqeq}
\end{equation}
The following elementary fact regarding bases will be used throughout
(cf. \cite[Lemma5]{AKR08b}):
If there exists a generalized Bernstein operator $B_n$ of the form
(\ref{eqBern}), fixing $f_0$ and $f_1$,
then it must be the case that for each $k= 0, \dots, n$,
\begin{equation}
\beta_{n,k} = f_0 (t_{n,k}) \ \alpha_{n,k} \text{ \ \ \
and \ \ \ } \gamma_{n,k} = f_1 (t_{n,k}) \ \alpha_{n,k}.
\label{bases}
\end{equation}
Note that since by hypothesis $f_0 > 0$ and $\alpha_{n,k} > 0$,
if $B_n$ exists then $\beta_{n,k} > 0$.
Now, using the injectivity of $f_1/f_0$,
the
nodes are uniquely determined by
\begin{equation}
t_{n,k}:=\left( \frac{f_{1}}{f_{0}}\right) ^{-1}\left( \frac{\gamma_{n,k}}{
\beta_{n,k}}\right), \label{nodes}
\end{equation}
and the weights, by
\begin{equation}
\alpha_{n,k}:=\frac{\beta_{n,k}}{f_{0}(t_{n,k})}. \label{coeff}
\end{equation}
Finally, when the Bernstein basis is non-negative and normalized,
and $f_0 = \mathbf{1}$, we have that
\begin{equation} \label{one}
1 = \alpha_{n,k} = \beta_{n,k}
\text{ \ \ \
and \ \ \ } t_{n,k}= f_{1}^{-1}\left(\gamma_{n,k}\right).
\end{equation}
Suppose that instead of fixing $ \mathbf{1} $ and $x$ over $[a,b]$, we want a
generalized Bernstein operator that reproduces $f_0 = \mathbf{1} $ and some strictly
increasing function other than $x$. Possibly the simplest choice is
to fix $f_1 (x) = x^3$, since $x^2$ is not increasing over arbitrary intervals.
Already in this case we observe a wide range of behavior, depending
on the values of $a$ and $b$.
Recall that $\mathbb{P}_n [a,b]$ denotes the space of polynomials
on $[a,b]$, of degree bounded
by $n$. In this case the Bernstein bases are given by
$p_{n,k} (x) = \binom{n}{k}
\frac{(x - a)^{k}(b-x)^{n-k}}{(b - a)^n}$.
\begin{example}\label{E1} Consider $\mathbb{E}_1 =
\operatorname{Span}\{\mathbf{1},x^3\}$ over
$[-1,1]$. Then $\{p_{1,0}(x) := (1-x^3)/2, p_{1,1}(x) := (1 + x^3)/2\}$ is the unique normalized
Bernstein basis for $\mathbb{E}_1$.
Define, as in \cite[Formula (21)]{AKR08b},
\begin{equation} \label{dim1}
B_{1}f:= f\left( a\right) p_{1,0}+f\left( b\right) p_{1,1}.
\end{equation}
Then it is clear that $B_1 \mathbf{1} = p_{1,0}+p_{1,1} = \mathbf{1}$ and
$B_1 f_1 = - p_{1,0}+p_{1,1} =x^3$. Note that
$\operatorname{Span}\{\mathbf{1},x^3\}$ is not
an extended Chebyshev space, a notion defined next.
\end{example}
\begin{definition} \label{ECS} An \emph{extended Chebyshev
space} $U_{n}$ of dimension $n+1$ over the interval $\left[ a,b\right] $
is an $n+1$ dimensional subspace of $C^{n}\left( \left[
a,b\right] \right) $ such that each $f\in U_{n}$ has at most $n$ zeros in
$\left[ a,b\right] $, counting multiplicities, unless $f$ vanishes
identically.
\end{definition}
Extended Chebyshev
spaces of dimension $n+1$ generalize
the space of polynomials of degree at most $n$ by retaining the bound on the
number of zeros.
It is well known that extended Chebyshev spaces always have
positive Bernstein bases.
\begin{example}\label{E2} Let $\mathbb{E}_2 =\operatorname{Span}\{\mathbf{1},x, x^3\}$ over
$[a,b] = [-1,1]$. In this case it is impossible to obtain a non-negative Bernstein basis
for $\mathbb{E}_2 $, whence the corresponding generalized Bernstein operator
cannot be defined. To see why this is true, note that if $p_{2,1} (x) = a + b x + c x^3$
has one zero at $-1$ and another at $1$, then
$a - b -c = a + b + c = 0$, so $a = 0$ and $b = -c$, with $b \ne 0$. But for any such
$b$, $p_{2,1} (x) = b (x - x^3)$ crosses the $y$-axis at $0$.
Note that Bernstein bases do exist for $\mathbb{E}_2$:
One such
basis is given by
$\{p_{2,0} (x) = 2 - 3 x + x^3, p_{2,1} (x) = x - x^3, p_{2,2} (x) = 2 + 3 x - x^3\}$.
Let us now consider $\mathbb{E}_2 = \operatorname{Span}\{\mathbf{1},x, x^3\}$ over
$[-1,2]$. In this case it is impossible to obtain a Bernstein basis
for $\mathbb{E}_2 $, even allowing for changes of signs. Suppose
there is such a basis, and let us try to compute $p_{2,2}$. Note that the
coefficient of $x^3$ cannot be zero, since $ p_{2,2} (x) $ has degree at least
two (hence three); dividing by the said coefficient, we may assume that
$p_{2,2} (x) = a + b x + x^3 = ( x + 1)^2 ( x + c) = c + (1 + 2c) x + (2 + c) x^2 + x^3$.
Equating coefficients we see that $c = - 2$. Hence $p_{2,2} ( 2) = 0$, which is a contradiction.
\end{example}
\begin{example}\label{P3} Next we consider
$\mathbb{P}_3[a,b] =\operatorname{Span}\{\mathbf{1},x, x^2, x^3\}$, with the standard
Bernstein bases over
$[a, b] = [-1,1]$ and over $[a, b] = [-1,2]$. In
the first case a generalized Bernstein operator fixing
$f_0 = \mathbf{1}$ and
$ f_1 (x) = x^3$ exists, but the
sequence of nodes fails to be increasing. In fact, this must be
the case, by Corollary \ref{antimaz} below, since
$ f_1^\prime (0) = 0$.
More explicitly, solving for
the coefficients $\gamma_{3,k}$ of $x^3$ we have
$\gamma_{3,0} = -1$, $\gamma_{3,1}= 1$, $\gamma_{3,2} = -1$, and
$\gamma_{3,3} = 1$, so in this particular instance the coordinates and the nodes take the
same values, oscillating between $-1$ and $1$. Note that
$B_3$ is just the projection from $C\left[ a,b\right]$ onto
$\operatorname{Span}\{\mathbf{1}, x^3\}$. This can be seen by observing
that for $k \ge 0$, $B_3 x^{2 k} = 1$ and $B_3 x^{2 k + 1} = x^3$.
Alternatively, given $f \in C\left[ a,b\right]$, if we simplify the
expression for $B_3 f(x)$, we find that
$B_3 f(x) \in \operatorname{Span}\{\mathbf{1}, x^3\}$.
When $[a,b] =[-1,2]$, a generalized Bernstein operator fixing
$\mathbf{1}$ and $x^3$ does not exist on $\mathbb{P}_3 [a,b]$, since solving for
the coefficients $\gamma_{3,k}$ of $x^3$ we find that
$\gamma_{3,0} = -1$, $\gamma_{3,1}= 2$, $\gamma_{3,2} = -4$, $\gamma_{3,3} = 8$,
so the node
$t_{3,2} = (-4)^{1/3}$ falls outside $[-1,2]$.
However,
a generalized Bernstein operator fixing
$\mathbf{1}$ and $x^3$ does exist on $\mathbb{P}_4 [-1,2]$, for now the coefficients $\gamma_{4,k}$ of $x^3$ are
$\gamma_{4,0} = -1$, $\gamma_{4,1}= 5/4 $,
$\gamma_{4,2} = -1$, $\gamma_{4,3} = - 1$
and $\gamma_{4,4}= 8$, so all the nodes fall inside $[-1 , 2]$.
Thus, not only the cases where $f_1$ is strictly increasing and where
$f_1^\prime > 0$ on $(a,b)$ are different, but also the (relative) location of the possible
zeros of $f_1^\prime$ is relevant; a more extreme instance of this phenomenon
can be found in \cite[Theorem 5.2]{AlRe18}
\end{example}
Next we present some counterexamples. The results are analogous to the
instances seen so far, but spaces and bases are chosen to specifically address
some claims made in \cite[pages 121-122 ]{Ma}, where it is stated ``Our purpose here is not to develop a comprehensive theory on Bernstein-type operators, but to convince the
reader via a few relevant examples, that there do exist similar operators in more general situations." While Bernstein-type operators can be defined in more general situations,
they do not exist in some of the relevant examples presented there.
The following spaces are considered in \cite[page 123]{Ma}:
Let $a < 0 < b$, and
let $n\ge 4$. Consider the
sequence
$
\mathbb{E}_1\subset \mathbb{E}_2\subset \mathbb{P}_{3} \subset \cdots\subset \mathbb{P}_{n-1}\subset \mathbb{E}_n,
$
where $\mathbb{E}_1 := \operatorname{Span}\{ \mathbf{1}, x^3\}$, $\mathbb{E}_2 :=
\operatorname{Span}\{ \mathbf{1}, x, x^3\}$, and for $n \ge 4$, $\mathbb{E}_n :=
\operatorname{Span}\{ \mathbf{1}, x, \dots, x^{n-1}, x^{n+ 2}\}$.
The domain of definition of these functions is taken to be the interval $[a,b]$.
In \cite[Definitions 3.1 and 3.2]{Ma} ``Bernstein-like operators" are required
to have strictly increasing sequences of nodes.
Now in \cite[Example 7.1]{Ma}, the
existence of ``Bernstein-like operators"
$B_{n}:C\left[ a,b\right] \rightarrow \mathbb{E}_n$
for $a < 0 < b$, fixing
$\mathbf{1}$ and $x^3$, is asserted. We show here, by explicit computation, that
when $n= 4$ and $[a,b]= [-1,2]$, a generalized Bernstein operator
fixing $\mathbf{1}$ and $x^3$
does not exist.
When $[a,b]= [-1,1]$, such an operator exists, but one must give up
the condition of increasing nodes.
\begin{example} \label{ex1} First we take $[a,b] =[-1,1]$. In this case a generalized Bernstein operator $B_{4}:C\left[ a,b\right] \rightarrow \mathbb{E}_4$ fixing
$ \mathbf{1}$ and $x^3$ does exist, but it is not defined
via an increasing sequence of nodes.
The normalized Bernstein basis on $\mathbb{E}_4$ can simply be found
by writing arbitrary linear combinations of the functions
$\{ \mathbf{1}, x, x^2, x^3, x^6\}$, and imposing the conditions of having precisely 4 zeros,
$k$ of them at $-1$ and the other
$4-k$ at $1$. Multiplying by $-1$ if needed, these basis functions can
be assumed to be non-negative at 0 (in fact, we shall see that they are positive
inside $(-1, 1)$). Imposing the additional condition that they
add up to 1, we find the
unique normalized Bernstein basis $\{p_{4,0}, \dots,p_{4,4}\}$
on $\left[ -1,1\right] $, where
\begin{eqnarray*}
p_{4,0}\left( x\right) &=&\frac{5}{56} - \frac{9}{28} x + \frac{45}{112} x^2 - \frac{5}{28}
x^3 + \frac{1}{112} x^6, \\
p_{4,1}\left( x\right) &=&\frac{2}{7} - \frac{3}{7} x - \frac{3}{14} x^2 + \frac{3}{7}
x^3 - \frac{1}{14} x^6, \\
p_{4,2}\left( x\right) &=&\frac{1}{4} - \frac{3}{8} x^2 + \frac{1}{8} x^6, \\
p_{4,3}\left( x\right) &=&\frac{2}{7} + \frac{3}{7} x - \frac{3}{14} x^2 - \frac{3}{7}
x^3 - \frac{1}{14} x^6,\\
p_{4,4}\left( x\right) &=&\frac{5}{56} + \frac{9}{28} x + \frac{45}{112} x^2 + \frac{5}{28}
x^3 + \frac{1}{112} x^6.
\end{eqnarray*}
These functions are positive at zero, add up to 1, and have the
correct number of
zeros at the endpoints. To see
that they form a positive basis, since $p_{4,k}(0) > 0$ for
$k=0,\dots, 4$,
it suffices to show that they have no additional zeros inside
$(-1,1)$. But this is easily checked, for we already know
the location of four zeros of each $p_{4,k}$. Using the division
algorithm, we factor all the corresponding linear terms $(x +1)$ and
$(x - 1)$, and are left
in each case with a second degree polynomial having no real roots.
Once we have found the normalized Bernstein bases, we use the condition
(\ref{eqBern2}) to determine nodes:
Equating coefficients in
$
x^3 =\sum_{k=0}^{4}\gamma_{4,k}p_{4,k}\left( x\right)
$
and solving for $\gamma_{4,k}$ we find that
$\gamma_{4,0} = -1$, $\gamma_{4,1}= 3/4$, $\gamma_{4,2} = 0$, $\gamma_{4,3} = -3/4$
and $\gamma_{4,4}= 1$.
Since the nodes are the cube roots of these coordinates, it follows
that $t_{4,0} <
t_{4,3} < t_{4,2} < t_{4,1} < t_{4,4}$, and we see that the nodes do not form an increasing sequence in $k$.
\end{example}
\begin{example} \label{ex2} Let us now take $[a,b] =[-1,2]$. In this case a generalized Bernstein operator $B_{4}:C\left[ a,b\right] \rightarrow \mathbb{E}_4$ fixing
$\mathbf{1}$ and $x^3$, cannot be defined.
Using the same steps as in the preceding example, we find the following
Bernstein basis functions:
\begin{eqnarray*}
p_{4,0}\left( x\right) &=& \frac{640}{2673} - \frac{128}{297} x + \frac{80}{297} x^2 - \frac{160}{2673}
x^3 + \frac{1}{2673} x^6,
\\
p_{4,1}\left( x\right) &=& \frac{5776}{13365} - \frac{152}{1485} x - \frac{532}{1485} x^2 + \frac{2318}{13365}
x^3 - \frac{38}{13365} x^6,
\\
p_{4,2}\left( x\right) &=& \frac{98}{405} + \frac{14}{45} x - \frac{7}{90} x^2 - \frac{56}{405}
x^3 + \frac{7}{810} x^6,
\\
p_{4,3}\left( x\right) &=& \frac{16}{243} + \frac{4}{27} x + \frac{2}{27} x^2 - \frac{4}{243}
x^3 - \frac{2}{243} x^6,
\\
p_{4,4}\left( x \right) &=& \frac{5}{243} + \frac{2}{27} x + \frac{5}{54} x^2 +
\frac{10}{243} x^3 + \frac{1}{486} x^6.
\end{eqnarray*}
Positivity of these functions on $(-1,2)$ is obtained by noticing, first, that
$p_{4,k} (0) = 1$, and second, that after factoring the linear terms $(x +1)$ and
$(x - 2)$,
in each case we are left with a second degree polynomial having no real roots.
Adding up we see that the basis is normalized,
so it is enough to compute the coordinates $\gamma_{4, k}$ of $x^3$. Doing so,
we find that $\gamma_{4, 2} = - 16/7 < -1$,
so the node $t_{4, 2} \notin [-1, 2]$.
\end{example}
\section{Characterizing when nodes increase for general spaces.}
The following technical results,
used to prove
Theorem \ref{ThmBern2}., come from \cite{AKR08}. Proposition \ref{PropABL} appears in
\cite[Proposition 3]{AKR08}, while Lemma \ref{LemA} is a less
general version of \cite[Lemma 6]{AKR08}.
\begin{proposition}
\label{PropABL} Assume that $U_{n}\subset C^n([a,b], \mathbb{K})$ has a Bernstein basis $p_{n,k},k=0,...,n$.
Let $f_{0}\in U_{n}$ be strictly positive and suppose that $D_{f_{0}}U_{n}:=\left\{ \frac{d}{dx}\left( \frac{f}{f_{0}}\right) :f\in
U_{n}\right\}$
has a Bernstein basis $q_{n-1,k}$, $k=0,...,n-1$. Set $c_0 := 0$, $q_{n-1, -
1} := 0$, $d_n:= 0$, and $q_{n-1, n} := 0$. For $k=1,...,n$, define the
non-zero numbers
\begin{equation}
c_{k}:= \frac{1}{f_{0}\left( a\right) }\lim_{x\downarrow a}\frac{\frac{d}{dx}
p_{n,k}\left( x\right) }{q_{n-1,k-1}\left(x\right)} = \frac{1}{f_{0}\left(
a\right) }\frac{p_{n,k}^{\left( k\right) }\left( a\right) }{
q_{n-1,k-1}^{\left( k-1\right) }\left( a\right) } \label{eqPR1}
\end{equation}
and for $k=0,...,n-1$, the non-zero numbers
\begin{equation}
d_{k}:=\frac{1}{f_{0}\left( b\right) }\lim_{x\uparrow b}\frac{\frac{d}{dx}
p_{n,k}\left( x\right) }{q_{n-1,k}\left(x\right)} = \frac{1}{
f_{0}\left(b\right) }\frac{p_{n,k}^{\left( n-k\right) }\left( b\right) }{
q_{n-1,k}^{\left( n-1-k\right) }\left( b\right) }. \label{eqPR}
\end{equation}
Then for every $k=0,...,n$,
\begin{equation}
\frac{d}{dx}\frac{p_{n,k}\left( x\right) }{f_{0}\left( x\right) }
=c_{k}q_{n-1,k-1}\left( x\right) +d_{k}q_{n-1,k}\left( x\right).
\label{eqPREC}
\end{equation}
\end{proposition}
\begin{lemma}
\label{LemA} Let $p_{n,k}$, $k=0,...,n$, be a non-negative Bernstein
basis of $U_{n}\subset C^n([a,b], \mathbb{K})$. Then there exists a $\delta >0$ such that
$
p_{n,k}^{\prime }\left( x\right) <0$ for all $x\in \left[ b-\delta ,b\right]
$ and all $k=0,...,n-1$, while $p_{n,k}^{\prime }\left( x\right) > 0$ for
all $x\in \left[a, a + \delta \right]$ and all $k=1,...,n$. Thus, the
numbers $c_{k}$ defined in (\ref{eqPR1}) for $k = 1, \dots, n$ are positive,
and the numbers $d_{k}$ defined in (\ref{eqPR}) for $k = 0, \dots, n-1$ are
negative.
\end{lemma}
\begin{theorem}
\label{ThmBern2} Assume that both $U_{n}\subset C^n([a,b], \mathbb{K})$ and
$D_{f_{0}}U_{n}:=\left\{ \frac{d}{dx}\left( \frac{f}{f_{0}}\right) :f\in
U_{n}\right\}$ possess
non-negative Bernstein basis $p_{n,k}$, for $k=0, \dots , n$, and $q_{n-1,k}$,
for $k=0, \dots , n-1$, respectively. Suppose $f_{0},f_{1}\in U_{n}$ are such that $f_{0}>0$,
its coordinates $\beta_{n, k}$ satisfy
$\beta_{n, k} > 0$, and
$f_{1}/f_{0}$ is strictly increasing on $\left[ a,b\right] $. Then the
following statements are equivalent:
a) There exists a generalized Bernstein operator $B_{n}:C\left[ a,b\right]
\rightarrow U_n$ defined by a sequence of non-decreasing
(resp. strictly increasing) nodes, and
fixing both $f_{0}$ and $f_{1}.$
b) For $k=0, \dots , n$, the numbers $\frac{\gamma_{n,k}}{\beta_{n, k}}$ are non-decreasing
(resp. strictly increasing).
c) The coefficients $w_{k}$, defined by
\begin{equation}
\frac{d}{dx}\frac{f_{1}}{f_{0}}=\sum_{k=0}^{n-1}w_{k}q_{n-1,k}
\label{eqA}
\end{equation}
for $k=0, \dots , n-1$,
are non-negative (resp. strictly positive).
\end{theorem}
\begin{proof}
We begin with some technical preliminaries, under
the assumption that for $k=0, \dots , n$, the coordinates $\beta_{n, k}$ of $f_0$
are strictly positive. Let $k_{0}\in\left\{
0, \dots ,n-1\right\} .$ Since $p_{n,k},k=0,...,n$, is a basis, there exists
numbers $\delta_{1},...,\delta_{n}$ such that
\begin{equation}
\psi_{k_{0}}:=f_{1}-\frac{\gamma_{n,k_0}}{\beta_{n,k_0}}f_{0}=\sum_{k=0}
^{n}\delta_{k}p_{n,k}.
\label{eqfneu}
\end{equation}
From (\ref{eqeq}) we get
\begin{equation*}
\delta_{k}=\gamma_{n,k}-\frac{\gamma_{n,k_0}}{\beta_{n,k_0}}\beta_{n, k}
\end{equation*}
for $k=0,...,n.$ Setting $k=k_{0}$ we obtain
\begin{equation}
\label{berob}
\delta_{k_{0}}=\gamma_{n,k_0}-\frac{\gamma_{n,k_0}}{\beta_{n,k_0}}
\beta_{n, k_{0}}=0.
\end{equation}
Let us write
$$
\frac{f_{1}}{f_{0}}-\frac{\gamma_{n,k_0}}{\beta_{n,k_0}}
=
\frac{\psi_{k_{0}}}{f_{0}}
=
\sum_{k=1}^{n}\delta_{k}\frac{p_{n,k}}{f_{0}}.
$$
Differentiating we get
\[
\frac{d}{dx}\frac{f_{1}}{f_{0}}=\sum_{k=0}^{n}\delta_{k}\frac{d}{dx}\left(
\frac{p_{n,k}}{f_{0}}\right) .
\]
Proposition \ref{PropABL} together with Lemma \ref{LemA} show that
\[
\frac{d}{dx}\frac{f_{1}}{f_{0}}=\sum_{k=0}^{n}\delta_{k}\left[
c_{k}q_{n-1,k-1}+d_{k}q_{n-1,k}\right],
\]
where $c_0 = 0$ and $c_k > 0$ for $k=0, \dots , n$, while
$d_k < 0$ for $k=0, \dots , n-1$ and $d_n = 0$.
Thus,
\[
\frac{d}{dx}\frac{f_{1}}{f_{0}}=
\delta_{0} d_{0}q_{n-1,0}
+
\sum_{k=1}^{n-1}\delta_{k}\left[
c_{k}q_{n-1,k-1}+d_{k}q_{n-1,k}\right] +c_{n}\delta_{n}q_{n-1,n-1}
\]
\[
=
\sum_{k=1}^{n}\delta_{k}
c_{k} q_{n-1,k-1}
+
\sum_{k=0}^{n-1}\delta_{k} d_{k}q_{n-1,k}
=
\sum_{k=0}^{n-1}\left( \delta_{k+1}c_{k+1}+\delta_{k}d_{k}\right) q_{n-1,k}.
\]
Using (\ref{eqA}) we conclude that
\begin{equation}
c_{k+1}\delta_{k+1}=w_{k}-\delta_{k}d_{k}
\label{eqck}
\end{equation}
for $k= 0 ,\dots, n-1$. Inserting $k=k_{0}$ in (\ref{eqck}), from (\ref{berob}) we get
\begin{equation}
c_{k_{0}+1}\delta_{k_{0}+1}=w_{k_{0}}-\delta_{k_{0}}d_{k_{0}}=w_{k_{0}}
\label{eqnew11}
\end{equation}
whenever $k_{0}\in\left\{
0, \dots ,n-1\right\} .$
Now the result is easily obtained from this equality. We mention only the non-decreasing case,
since the strictly increasing one is handled in an identical manner.
First we prove that a) and b) are
equivalent. It follows from (\ref{eqeq}) that
$\frac{f_1 (a) }{f_0 (a) } = \frac{\gamma_{n,0}}{\beta_{n, 0}}$
and $\frac{f_1 (b) }{f_0 (b) } = \frac{\gamma_{n,n}}{\beta_{n, n}}$.
By (\ref{nodes}), the nodes are non-decreasing (and hence they belong to $[a,b]$)
if and only if so
are the numbers $\frac{\gamma_{n,k}}{\beta_{n, k}}$.
Regarding the equivalence between b) and c), by (\ref{eqnew11}) we have
$w_{k_{0}}\geq 0$ if and only if $\delta_{k_{0}+1}=\gamma_{k_{0}+1}-\frac{\gamma
_{k_{0}}}{\beta_{n,k_0}}\beta_{k_{0}+1}\geq0$, which is clearly equivalent to
$\frac{\gamma_{n,k_0}}{\beta_{n,k_0}}\leq\frac{\gamma_{k_{0}+1}}{\beta
_{k_{0}+1}}$.
\end{proof}
\begin{corollary}\label{antimaz}
Let $f_{0}>0$, let $f_{1}/f_{0}$ be strictly increasing on $\left[ a,b\right]
$, and let $f_{0},f_{1}\in U_{n}\subset C^n([a,b], \mathbb{K})$. Suppose that $U_{n}$
has a non-negative Bernstein basis, and that $D_{f_{0}}U_{n}$ possesses a positive Bernstein basis. If there exists a generalized Bernstein operator
$B_{n}:C\left[ a,b\right]
\rightarrow U_n$
with non-decreasing nodes (resp. strictly increasing nodes), fixing $f_{0}$ and $f_{1}$, then
\[
\left( \frac{f_{1}}{f_{0}}\right) ^{\prime}\left( x\right) >0 \text{ for
all }x\in\left( a,b\right) (\text{resp. for
all }x\in\left[ a,b\right]) .
\]
\end{corollary}
\begin{proof} We consider the non-decreasing case.
Recall that if $B_n$ exists, by hypothesis $f_0 > 0$ and $\alpha_{n,k} > 0$,
so $\beta_{n,k} = f_0 (t_{n,k}) \ \alpha_{n,k} \ > 0$.
Thus, we can use the implication a) $\implies$ c)
from the preceding Theorem. Writing
\[
\frac{d}{dx}\frac{f_{1}}{f_{0}}=\sum_{k=0}^{n-1}w_{k}q_{n-1,k},
\]
we have $w_{k} \ge 0$ for $k = 0, \dots , n-1$.
Since $f_{1}/f_{0}$ is strictly increasing, it is non-constant, so
$\frac{d}{dx}\frac{f_{1}}{f_{0}}\neq0$ and thus some coefficient $w_{j}$ is strictly
positive. But now
$$
\frac{d}{dx}\frac{f_{1}}{f_{0}} \ge w_j q_{n-1,j} > 0
$$ on $\left( a,b\right)$.
\end{proof}
\begin{example} The condition $
\left( \frac{f_{1}}{f_{0}}\right) ^{\prime} >0$ on $\left[ a,b\right]$
is not sufficient to ensure non-decreasing nodes, as \cite[Example 4.1]{AKR08b}
shows: consider $\mathbb{P}_3[0,1]$ with the standard Bernstein basis,
let $f_0 = \mathbf{1}$, and let $f_1 (x) = 3 x /8 - x^2/2 + x^3/3$.
Then $f_1^\prime (x) := (x - 1/2)^2 + 1/8
= 3 p_{2,0} (x) /8 - p_{2,1} (x) /8 + 3 p_{2,2} (x) /8$, so by Theorem \ref{ThmBern2}
no generalized Bernstein operator $B_3 : C([a,b], \mathbb{K}) \to \mathbb{P}_3[a,b]$, fixing $\mathbf{1}$ and $f_1$,
can be defined via a non-decreasing sequence of nodes.
\end{example}
\begin{example} To finish, we revisit Example \ref{ex1} under the
light of the preceding results. A practical advantage of applying,
for instance, Corollary \ref{antimaz},
is that one does not need to determine whether or not $U_{n}$
has a non-negative Bernstein basis. If it does not,
a generalized Bernstein operator does not exist, and nothing
else needs to be done. So it is enough to check that $D_{f_{0}}U_{n}$ possesses a positive Bernstein basis, a task a priori simpler, since
$D_{f_{0}}U_{n}$ has one fewer dimension.
As before, we can find
a positive Bernstein basis of
$$
D_{\mathbf{1}} \mathbb{E}_4 =
\operatorname{Span}\{ 1, x, x^2, x^{5}\}
$$
over $[a,b] =[-1,1]$, by writing arbitrary linear combinations of the functions
$\{ 1, x, x^2, x^5\}$, and imposing the conditions of having precisely 3 zeros,
$k$ of them at $-1$ and the other
$3-k$ at $1$. Multiplying by $-1$ if needed, these basis functions can
be assumed to be non-negative at 0 (we shall see that they are
actually positive
inside $(-1, 1)$). In this way we find
the following Bernstein basis $\{p_{3,0}, \dots,p_{3,3}\}$
on $\left[ -1,1\right] $:
\begin{eqnarray*}
p_{3,0}\left( x\right) &=& 1 - \frac{5}{2} x + \frac{5}{3} x^2 - \frac{1}{6}
x^5, \\
p_{3,1}\left( x\right) &=& 1 - \frac{1}{2} x - x^2 + \frac{1}{2}
x^5, \\
p_{3,2}\left( x\right) &=& 1 + \frac{1}{2} x - x^2 - \frac{1}{2} x^5, \\
p_{3,3}\left( x\right) &=& 1 + \frac{5}{2} x + \frac{5}{3} x^2 + \frac{1}{6}
x^5.
\end{eqnarray*}
These functions have the
correct number of
zeros at the endpoints. To see
that they form a positive basis, since $p_{3,k}(0) = 1 > 0$ for
$k=0,\dots, 3$,
it suffices to show that they have no additional zeros inside
$(-1,1)$, which follows by factoring all the corresponding linear terms $(x +1)$ and
$(x - 1)$. In each case we are left
with a second degree polynomial having no real roots. So we are within the
realm of
Theorem \ref{ThmBern2} or Corollary \ref{antimaz}. Since the derivative
of $f_1 (x) = x^3$ vanishes at 0, we conclude that
no generalized Bernstein operator $B_3 : C([-1,1], \mathbb{K}) \to \mathbb{E}_4 [-1,1]$, fixing $\mathbf{1}$ and $f_1$,
can be defined via a non-decreasing sequence of nodes.
Observe however
that this result is less informative than Example \ref{ex1},
since it does not tell us whether a generalized Bernstein operator
can be defined, by dropping the requirement that nodes be non-decreasing.
\end{example}
\end{document}
|
\begin{document}
\title{Flavor entanglement in neutrino oscillations in the wave packet description}
\section{Introduction}
The study of entanglement in elementary particle systems belongs to the current frontiers of research in Physics~\cite{Bert1,Bert2,DiDomenico:2011fq,NoiPRD,NoiEPL,Blasone:2013yoa,Kayser10,Smirnov11,Boyanovsky:2011xq,Lello:2013bva,NoiEPL2,NoiHindawi,EPLBrazil,Bernardini:2012uf}. In fact, the study of quantum correlations in these systems can offer a complementary
and enlightening viewpoint for understanding fundamental phenomena.
Furthermore, it represents a prerequisite for a subsequent analysis in applicative contexts
concerning, for instance, the implementation of quantum information protocols in the framework
of elementary particle physics and quantum field theory.
The phenomenon of flavor oscillations associated with the neutrino mixing must be considered
a significant instance of dynamical entangled system
characterized by single-particle, multi-mode entanglement~\cite{NoiPRD,NoiEPL,NoiEPL2,NoiHindawi,EPLBrazil,Alok:2014gya}.
By using proper entanglement measures, it has been shown that flavor entanglement can be written
in terms of the flavor transition probabilities;
furthermore, it has been also
proposed an experimental scheme that allows the transfer
to spatially delocalized two-flavor charged lepton states of the quantum information encoded in neutrino states~\cite{NoiEPL}.
The analysis inherent the quantification and characterization of multi-mode flavor entanglement
in the oscillating neutrino system has been carried out both in a quantum-mechanical setting~\cite{NoiPRD,NoiEPL,EPLBrazil}
and in the framework of quantum field theory~\cite{NoiEPL2,NoiHindawi}, while decoherence effects have been investigated in Refs.~\cite{NoiPRD,EPLBrazil}.
In these articles, the multipartite entanglement in the dynamics of flavor oscillations was analyzed by using, as reference space,
the (three-qubit) Hilbert space associated with neutrino mass eigenstates.
By adopting the wave-packet description for the mass eigenstates, it was shown a strict connection
between the decoherence effects and the spatial behavior of quantum entanglement; in fact, due to the differences
among the neutrino masses, the corresponding wave packets propagate at different group velocities, thus resulting
in a mutual spatial separation that increases during the propagation. Consequently, the free evolution leads to a
progressive loss of the coherent interference effects, that are connected with the destruction of the oscillation
phenomenon and with the vanishing of the multipartite quantum entanglement among mass modes.
On the other hand, the (anyway interesting) approach based on the mass eigenstates involves some drawbacks,
since it deals with a not directly observable context.
In the present paper we accomplish a further step forward in the analysis of the dynamic of quantum
correlations by considering the propagation of flavor neutrino states, which directly enter into the production and
detection processes.
By using the wave packet approach, we analyze the behavior of multipartite entanglement relative to the
(three-qubit) Hilbert space of flavor neutrino eigenstates.
{Although the neutrino system is described in principle by a pure state, we obtain a mixed state after a time integration has been performed, following a standard procedure. Consequently, in order to quantify the entanglement content of such a mixed state, we exploit two measures, the concurrence and the logarithmic negativity, that were specifically devised for this case.
In detail, we consider the content of entanglement shared by two given flavors after a partial trace has been performed with
respect to the third flavor (concurrence), and the content of entanglement in bipartitions of the three-flavor system
(logarithmic negativity).}
We show that the transition probabilities provide only partial information about the dynamics of quantum correlations,
and that the use of quantum information methods leads to a deeper insight.
Indeed, such correlations are not fully encoded into flavor oscillation transition probabilities, but can be unveiled by
means of appropriate experimental protocols. In particular, it is worth to be remarked that the combined exploitation of the two, operationally different, measures provides indications about the distribution of the entanglement among the different flavors.
The paper is organized as follows. First we review the wave-packet approach for free propagating neutrinos, and
the main aspects related to flavor oscillations. After introducing the suitable entanglement measures, we move
to the results by investigating the behavior of the entanglement associated with flavor oscillations.
Finally, we draw our conclusions.
\section{Neutrino flavor oscillations: wave packet approach}
The standard theory of neutrino oscillations, which describes the free evolution dynamics,
is developed using the plane-wave approximation \cite{neutroscillplanewave}.
Of course, such an approximation cuts off all the effects due to localization.
In order to recover a more realistic description of the phenomenon,
one has to resort to the wave packet approach~\cite{Nussinov,GiuntiKim,Giunti2,Giunti:2008cf}
(for reviews see Refs.~\cite{Beuthe,Giunti:2007ry}).
Following the procedure reported in Refs.~\cite{GiuntiKim,Giunti2,Giunti:2008cf},
a neutrino with definite flavor, which propagates along the $x$ direction, can be described by the state:
{
\begin{equation}
|\nu_{\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha}(x,t)\rangle \,=\, \sum_{j} U_{\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha j}^* \, \psi_{j}(x,t) \,|\nu_{j}\rangle \,,
\lambda}\def\La{\Lambda}\def\si{\sigma}\def\Si{\Sigmabel{neutwvpack}
\end{equation}}
where $|\nu_{j}\rangle$ is the mass eigenstate of mass $m_{j}$, and $\psi_{j}(x,t)$ is its wave function;
the $U_{\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha j}$ denotes the corresponding element of the PMNS mixing matrix in the standard form $U(\tilde{\theta},\delta}\def\De{\Delta}\def\ep{\epsilonlta)$,
with $(\tilde{\theta},\delta}\def\De{\Delta}\def\ep{\epsilonlta)\equiv (\theta_{12},\theta_{13},\theta_{23};\delta}\def\De{\Delta}\def\ep{\epsilonlta)$, being $\theta_{ij}$ the mixing angles
and $\delta}\def\De{\Delta}\def\ep{\epsilonlta$ the CP-violating phase (see {\it Eq.~(1)} of Ref.~\cite{NoiEPL}).
Assuming a Gaussian distribution $\psi_{j}(p)$ for the momentum of the massive neutrino $|\nu_{j}\rangle$
\begin{equation}
\psi_{j}(x,t) \,=\, \frac{1}{\sqrt{2\pi}} \,
\int \,dp \, \psi_{j}(p) \, e^{i p x -i E_{j}(p) t} \,,
\lambda}\def\La{\Lambda}\def\si{\sigma}\def\Si{\Sigmabel{wavfunc1}
\end{equation}
the wave function writes:
\begin{equation}
\psi_{j}(p) \,=\, \frac{1}{(2\pi \sigma_{p}^{2})^{1/4}} \,
e^{-\frac{1}{4\sigma_{p}^{2}}(p-p_{j})^{2}} \,,
\lambda}\def\La{\Lambda}\def\si{\sigma}\def\Si{\Sigmabel{wavfunc2}
\end{equation}
where $p_{j}$ is the average momentum, $\sigma_{p}$ is
the momentum uncertainty, and $E_{j}(p) \,=\, \sqrt{p^{2}+m_{j}^{2}}$.
The density matrix associated with the pure state, Eq.~(\ref{neutwvpack}), is given by:
\begin{equation}
\rho_{\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha}(x,t) \,=\, |\nu_{\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha}(x,t)\rangle \lambda}\def\La{\Lambda}\def\si{\sigma}\def\Si{\Sigmangle \nu_{\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha}(x,t)| \,.
\lambda}\def\La{\Lambda}\def\si{\sigma}\def\Si{\Sigmabel{densmatwvpack}
\end{equation}
By assuming the condition $\sigma_{p}\ll E_{j}^{2}(p_{j})/m_{j}$,
the energy $E_{j}(p)$ can be approximated by $E_{j}(p)\simeq E_{j}+v_{j}(p-p_{j})$,
with $E_{j}\equiv \sqrt{p_{j}^{2}+m_{j}^{2}}$, and
$v_{j}\equiv \frac{\partial E_{j}(p)}{\partial p}\big|_{p=p_{j}} \,=\,\frac{p_{j}}{E_{j}} $
is the group velocity of the wave packet of the massive neutrino $|\nu_{j}\rangle$.
With such an approximation, the Gaussian integration over $p$ in Eq.~(\ref{wavfunc1}) can be easily performed
(see Ref. \cite{Giunti2} for details).
In the instance of extremely relativistic neutrinos, one can exploit the following further approximations:
\begin{equation}
E_{j} \,\simeq \, E , \quad
p_{j} \,\simeq \, E-\frac{m_{j}^{2}}{2E} , \quad
v_{j} \,\simeq \, 1-\frac{m_{j}^{2}}{2E_{j}^{2}}
\lambda}\def\La{\Lambda}\def\si{\sigma}\def\Si{\Sigmabel{relapprox}
\end{equation}
where $E$ is the neutrino energy in the limit of zero mass \cite{GiuntiKim,Giunti2}.
The resulting density matrix provides a space-time description of neutrino's dynamics.
Due to the long time exposure of the detectors,
it is convenient to consider an average in time of $\rho_{\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha}(x,t)$,
i.e. a further Gaussian integration over the time, yielding the final density matrix \cite{Giunti2}:
\begin{eqnarray}
&&\rho_{\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha}(x) \,=\, \sum_{k,j} U_{\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha k} U_{\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha j}^{*}
\; f_{jk}(x) \;
|\nu_{j}\rangle \lambda}\def\La{\Lambda}\def\si{\sigma}\def\Si{\Sigmangle\nu_{k}| \,,
\lambda}\def\La{\Lambda}\def\si{\sigma}\def\Si{\Sigmabel{statwavepack}
\\
&& f_{jk}(x) \equiv \exp\left[ -i \frac{\Delta m_{jk}^{2} x}{2E}
-\left( \frac{\Delta m_{jk}^{2} x}{4\sqrt{2}E^{2}\sigma_{x}}\right}\delta}\def\De{\Delta}\def\ep{\epsilonf\ti{\tilde}\delta}\def\De{\Delta}\def\ep{\epsilonf\we{\wedge}\delta}\def\De{\Delta}\def\ep{\epsilonf\wti{\widetildeght)^{2}\right}\delta}\def\De{\Delta}\def\ep{\epsilonf\ti{\tilde}\delta}\def\De{\Delta}\def\ep{\epsilonf\we{\wedge}\delta}\def\De{\Delta}\def\ep{\epsilonf\wti{\widetildeght] ,
\lambda}\def\La{\Lambda}\def\si{\sigma}\def\Si{\Sigmabel{statwavepack2}
\end{eqnarray}
where $\sigma_{x} \,=\, (2\sigma_{p})^{-1}$, and $\Delta m_{jk}^{2} \,=\, m_{j}^{2}-m_{k}^{2}$.
The parameters in Eq.~(\ref{statwavepack}), i.e. the mixing angles $\theta_{ij}$
and the squared mass differences $\Delta m_{jk}^{2}$,
and the parameters $E$ and $\sigma_{p}$, are fixed to the experimental values (see Ref.~\cite{Fogli:2006yq}):
\begin{eqnarray}
&&\sin^{2}\theta_{12} = 0.314 \;,\; \sin^{2}\theta_{13} =0.8 \times 10^{-2}\;, \nonumber}\delta}\def\De{\Delta}\def\ep{\epsilonf\ot{\otimes}\delta}\def\De{\Delta}\def\ep{\epsilonf\pa{\partial}\delta}\def\De{\Delta}\def\ep{\epsilonf\ran{\rangleumber
\\
&&\sin^{2}\theta_{23} = 0.45 \;, \nonumber}\delta}\def\De{\Delta}\def\ep{\epsilonf\ot{\otimes}\delta}\def\De{\Delta}\def\ep{\epsilonf\pa{\partial}\delta}\def\De{\Delta}\def\ep{\epsilonf\ran{\rangleumber \\
&&\Delta m_{21}^{2} = \delta}\def\De{\Delta}\def\ep{\epsilonlta m^{2} = 7.92 \times 10^{-5} \, eV^{2} \;, \\
&&\Delta m_{31}^{2} = \Delta m^{2} + \frac{\delta}\def\De{\Delta}\def\ep{\epsilonlta m^{2}}{2} \,, \qquad
\Delta m_{32}^{2} \,=\, \Delta m^{2} - \frac{\delta}\def\De{\Delta}\def\ep{\epsilonlta m^{2}}{2} \,, \nonumber}\delta}\def\De{\Delta}\def\ep{\epsilonf\ot{\otimes}\delta}\def\De{\Delta}\def\ep{\epsilonf\pa{\partial}\delta}\def\De{\Delta}\def\ep{\epsilonf\ran{\rangleumber \\
&&
\Delta m^{2} \,=\, 2.6 \times 10^{-3} \, eV^{2} \;, E = 10 \,GeV \;, \sigma_{p} = 1 \,GeV \nonumber}\delta}\def\De{\Delta}\def\ep{\epsilonf\ot{\otimes}\delta}\def\De{\Delta}\def\ep{\epsilonf\pa{\partial}\delta}\def\De{\Delta}\def\ep{\epsilonf\ran{\rangleumber \,.
\lambda}\def\La{\Lambda}\def\si{\sigma}\def\Si{\Sigmabel{sqmassdiffpar}
\end{eqnarray}
Quantum entanglement is a physical quantity that depends on the chosen observables,
and that is endowed with an operational meaning determined by the selected observables and subsystems.
In Refs.~\cite{NoiPRD,EPLBrazil}, by establishing the identification
$|\nu_{i}\rangle \,=\, |\delta}\def\De{\Delta}\def\ep{\epsilonlta_{i1}\rangle_{1}|\delta}\def\De{\Delta}\def\ep{\epsilonlta_{i2}\rangle_{2}|\delta}\def\De{\Delta}\def\ep{\epsilonlta_{i3}\rangle_{3}
\equiv |\delta}\def\De{\Delta}\def\ep{\epsilonlta_{i1}\delta}\def\De{\Delta}\def\ep{\epsilonlta_{i2}\delta}\def\De{\Delta}\def\ep{\epsilonlta_{i3}\rangle$ $(i=1,2,3)$,
the coherence of the quantum superposition of the neutrino mass eigenstates
has been investigated in terms of the spatial behavior of the multipartite entanglement
of the state (\ref{statwavepack}).
However, from an experimental point of view, it would be preferable to consider the three-qubit Hilbert space
associated with the three flavors, i.e. through the alternative identification
$|\nu_{\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha}\rangle \,=\, |\delta}\def\De{\Delta}\def\ep{\epsilonlta_{\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha e}\rangle_{e}|\delta}\def\De{\Delta}\def\ep{\epsilonlta_{\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha \mu}\rangle_{\mu}|\delta}\def\De{\Delta}\def\ep{\epsilonlta_{\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha \tau}\rangle_{\tau}
\equiv |\delta}\def\De{\Delta}\def\ep{\epsilonlta_{\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha e}\delta}\def\De{\Delta}\def\ep{\epsilonlta_{\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha \mu}\delta}\def\De{\Delta}\def\ep{\epsilonlta_{\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha \tau}\rangle$ $(\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha \,=\, e,\mu,\tau)$.
To this aim, by using the relation {$|\nu_i\rangle \,=\, \sum_{\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha=e,\mu,\tau} \, U_{\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha i} \, |\nu_{\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha}\rangle$},
we rewrite the $\rho_{\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha}(x)$ in the form:{
\begin{eqnarray}
&&\rho_{\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha}(x) \,=\, \sum_{\beta,\gamma} \,F_{\beta \gamma}^{(\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha)}(x) \,
|\delta}\def\De{\Delta}\def\ep{\epsilonlta_{\beta e}\delta}\def\De{\Delta}\def\ep{\epsilonlta_{\beta \mu}\delta}\def\De{\Delta}\def\ep{\epsilonlta_{\beta \tau}\rangle \lambda}\def\La{\Lambda}\def\si{\sigma}\def\Si{\Sigmangle \delta}\def\De{\Delta}\def\ep{\epsilonlta_{\gamma e}\delta}\def\De{\Delta}\def\ep{\epsilonlta_{\gamma \mu}\delta}\def\De{\Delta}\def\ep{\epsilonlta_{\gamma \tau}| \,,
\lambda}\def\La{\Lambda}\def\si{\sigma}\def\Si{\Sigmabel{rhoxfinal}
\\
&&{F_{\beta \gamma}^{(\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha)}(x)} \equiv \sum_{k,j} \; U_{\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha j}^* U_{\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha k}\, f_{jk}(x) \, U_{\beta j}
U_{\gamma k}^{*} \,,
\end{eqnarray}}
with $k,j=1,2,3$ and $\beta,\gamma=e,\mu,\tau$.
The transition probability for the neutrino state $\rho_{\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha}(x)$ to be in the flavor $\eta$ at position $x$
is given by:
\begin{equation}
P_{\nu_\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha\longrightarrow\nu_\eta}(x) \,=\, Tr[\lambda}\def\La{\Lambda}\def\si{\sigma}\def\Si{\Sigmangle \nu_\eta| \rho_{\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha}(x) |\nu_\eta\rangle] \,=\, {F_{\eta\eta}^{(\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha)}(x)} \,.
\lambda}\def\La{\Lambda}\def\si{\sigma}\def\Si{\Sigmabel{TransProb}
\end{equation}
In Fig.~\ref{figTransProb}, we plot $P_{\nu_\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha\longrightarrow\nu_\eta}(x)$ (for $\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha=e,\,\mu$) as a function of $x$.
\begin{figure}
\caption{(Color online) The transition probabilities
$P_{\nu_\alpha}
\end{figure}
We observe that the transition probabilities are all characterized by a common feature:
if their behavior is analyzed with respect to the spatial dimension,
after an initial more stable trend they undergo an intermediate phase of wide and rapidly decreasing oscillations,
until achieving a final stationary value.
From definition (\ref{TransProb}), we see that the transition probabilities provide only
partial information; in fact, these probabilities are given by the diagonal elements $F_{\eta\eta}^{(\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha)}$.
They do not provide any information about quantum correlations between two generic flavors
or between two generic subsystems, which are encoded in the off-diagonal elements $F_{\eta\eta'}^{(\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha)}$,
$\eta\neq\eta'$, $\eta,\eta'=e,\mu,\tau$.
Therefore we resort to quantum information tools to gain additional insights.
\section{Entanglement measures}
The density matrix written in the form (\ref{rhoxfinal}) represents, in general, an entangled
mixed state, whose entanglement content can be quantified by means of properly devised measures.
Thus, we provide a brief recall of the definitions of the entanglement measures used in the present analysis
(for recent and detailed reviews on the qualification, quantification, and applications of entanglement,
see Refs. \cite{EntRevFazio,EntRevHorodecki}).
In the instance of mixed states, the entropic measures, such as the von Neumann entropy and the linear entropy,
cannot be used.
In order to characterize the bipartite entanglement of mixed states, several entanglement measures have been proposed, see e.g. Refs. \cite{EntFormDistill,EntRelEntr,Negativity,CoffKundWoot}. In this work, we are concerned with three-partite states, that is three flavor (three-qubit) states. In order to characterize the bipartite entanglement of such multipartite mixed states we will use two computable entanglement monotones: the concurrence \cite{CoffKundWoot} and the logarithmic negativity \cite{Negativity}. Conceptually, the logarithmic negativity is of particular interest because it has been showed that it is a full entanglement monotone notwithstanding the fact that it is not convex \cite{Plenio}.
We anticipate that, in order to investigate the entire structure of multipartite entanglement in three-flavor mixed states, in future work \cite{noifuture}, we will study the behavior of the genuine multipartite conccurence and of the three-tangle, which were introduced and discussed in Refs. \cite{Ma,Eltschka1,Eltschka2}.
The first measure is closely related to the entanglement of formation which is the minimal amount of entanglement
needed for the production of a mixed state described by a given density matrix.
We denote by $\rho$ the density operator, corresponding to an arbitrary $N$-qubit state,
that describes a system partitioned into $N$ parties.
The reduced density operator $\rho^{(\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha,\,\beta)}$ associated with the $\rho$ is defined as:
\begin{equation}
\rho^{(\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha,\,\beta)} \,=\, Tr_{\gamma\neq\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha,\beta}[\rho] \,,
\lambda}\def\La{\Lambda}\def\si{\sigma}\def\Si{\Sigmabel{mixedrho}
\end{equation}
where the trace operation is made over all the parties different from $\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha$ and $\beta$.
The spin-flipped state $\tilde{\rho}^{(\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha,\,\beta)}$ reads:
\begin{equation}
\tilde{\rho}^{(\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha,\,\beta)} = (\sigma_y\otimes\sigma_y)\rho^{(\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha,\,\beta)*}(\sigma_y\otimes\sigma_y),
\end{equation}
where the complex conjugate is taken in the standard basis
$\left\{|00\rangle, |01\rangle, |10\rangle, |11\rangle \right}\delta}\def\De{\Delta}\def\ep{\epsilonf\ti{\tilde}\delta}\def\De{\Delta}\def\ep{\epsilonf\we{\wedge}\delta}\def\De{\Delta}\def\ep{\epsilonf\wti{\widetildeght\}$.
Then the concurrence is given by:
\begin{equation}
C(\rho^{(\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha,\,\beta)})=\max\{0,\lambda}\def\La{\Lambda}\def\si{\sigma}\def\Si{\Sigmambda_1-\lambda}\def\La{\Lambda}\def\si{\sigma}\def\Si{\Sigmambda_2-\lambda}\def\La{\Lambda}\def\si{\sigma}\def\Si{\Sigmambda_3-\lambda}\def\La{\Lambda}\def\si{\sigma}\def\Si{\Sigmambda_4\},
\lambda}\def\La{\Lambda}\def\si{\sigma}\def\Si{\Sigmabel{Concurrence}
\end{equation}
where $\{\lambda}\def\La{\Lambda}\def\si{\sigma}\def\Si{\Sigmambda_i\}_{i=1}^4$ are the square roots
of the four eigenvalues of the non-Hermitian matrix $\rho^{(\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha,\,\beta)}\tilde{\rho}^{(\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha,\,\beta)}$,
and are non-negative real numbers taken in decreasing order with respect to the index $i$.
Therefore the concurrence can be used to measure the entanglement between two flavors in the neutrino three flavor state,
after tracing the third flavor.
We thus compute the quantities $C\left(\rho_\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha^{(\beta,\gamma)} (x)\right}\delta}\def\De{\Delta}\def\ep{\epsilonf\ti{\tilde}\delta}\def\De{\Delta}\def\ep{\epsilonf\we{\wedge}\delta}\def\De{\Delta}\def\ep{\epsilonf\wti{\widetildeght) \equiv C_\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha^{(\beta,\gamma)}$
where $\rho_\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha^{(\beta,\gamma)} (x)= Tr_{\eta\neq\beta,\gamma}\left[\rho_\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha (x) \right}\delta}\def\De{\Delta}\def\ep{\epsilonf\ti{\tilde}\delta}\def\De{\Delta}\def\ep{\epsilonf\we{\wedge}\delta}\def\De{\Delta}\def\ep{\epsilonf\wti{\widetildeght]$,
and with $\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha,\beta,\gamma,\eta=e,\mu,\tau$.
Since the state $\rho_\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha (x)$ possesses multipartite components,
in order to obtain a complete characterization of the entanglement content we exploit
a measure belonging to the typology of global measure of entanglement
\cite{Wallach,Brennen,Scott,Oliveira,Pascazio}.
The global measure approach relies on the construction of the set of all possible bipartitions of the total system,
that are able to encompass both bipartite and multipartite contributions.
We define a global entanglement measure for mixed states based on the logarithmic negativity
as proper measure for each bipartition.
Let $\rho$ be a multipartite mixed state associated with a system $S$,
partitioned into $N$ parties. Again, we consider the bipartition of the
$N$-partite system $S$ into two subsystems $S_{A_{n}}$ and $S_{B_{N-n}}$.
We denote by
\begin{equation}
\tilde{\rho}_{A_{n}} \equiv \rho^{PT\, B_{N-n} }
\,=\, \rho^{PT\, j_{1},j_{2},\ldots,j_{N-n} }
\lambda}\def\La{\Lambda}\def\si{\sigma}\def\Si{\Sigmabel{rhoPT}
\end{equation}
the {\it bona fide} density matrix, obtained by the partial transposition of $\rho$
with respect to the parties belonging to the subsystem $S_{B_{N-n}}$.
The logarithmic negativity associated with the fixed bipartition will be given by
{
\begin{equation}
E_{\mathcal{N}}^{(A_{n};B_{N-n})} \,=\,
\log_{2} \parallel \tilde{\rho}_{A_{n}} \parallel_{1} \,,
\lambda}\def\La{\Lambda}\def\si{\sigma}\def\Si{\Sigmabel{lognegAn}
\end{equation}}
where $\parallel \cdot \parallel_1$ denotes the trace norm.
Finally, we define the average logarithmic negativity
\begin{equation}
\lambda}\def\La{\Lambda}\def\si{\sigma}\def\Si{\Sigmangle E_{\mathcal{N}}^{(n:N-n)} \rangle \,=\, \left(
\begin{array}{c}
N \\
n \\
\end{array}
\right}\delta}\def\De{\Delta}\def\ep{\epsilonf\ti{\tilde}\delta}\def\De{\Delta}\def\ep{\epsilonf\we{\wedge}\delta}\def\De{\Delta}\def\ep{\epsilonf\wti{\widetildeght)^{-1} \; \sum_{A_{n}} E_{\mathcal{N}}^{(A_{n};B_{N-n})} \,,
\lambda}\def\La{\Lambda}\def\si{\sigma}\def\Si{\Sigmabel{avnlogneg}
\end{equation}
where the sum is intended over all the possible bipartitions of the system.
Of course we can easily construct from Eq.~(\ref{statwavepack}) the matrix with elements
$\lambda}\def\La{\Lambda}\def\si{\sigma}\def\Si{\Sigmangle lmn|\rho_{\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha}(x)|ijk \rangle$, where $i,j,k,l,m,n \,=\, 0,1$,
and analytically compute the quantities $E_{\mathcal{N}\,\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha}^{(\beta,\gamma;\eta)}$,
for $\beta,\gamma,\eta=e,\mu,\tau$ and $\beta\neq \gamma\neq \eta$, and the average logarithmic negativity
$\lambda}\def\La{\Lambda}\def\si{\sigma}\def\Si{\Sigmangle E_{\mathcal{N}\,\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha}^{(2:1)} \rangle$, for the neutrino state $\rho_\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha (x)$
with flavor $\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha$.
\section{Results}
We analytically compute both the concurrence $C_{\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha}^{(\beta,\gamma)}$
and the logarithmic negativity $E_{\mathcal{N}\,\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha}^{(\beta,\gamma;\eta)}$
associated with the state $\rho_\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha(x)$:
{
\begin{eqnarray}
&&\hspace{-.8cm} C_{\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha}^{(\beta,\gamma)}(x) \,=\, 2\,|F_{\beta\gamma}^{(\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha)}(x)| \,,
\lambda}\def\La{\Lambda}\def\si{\sigma}\def\Si{\Sigmabel{Concurr} \\
&&\hspace{-.8cm} E_{\mathcal{N}\,\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha}^{(\beta,\gamma;\eta)}(x) \,=\,
\log_2 \left[ 1+2\sqrt{|F_{\beta\eta}^{(\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha)}(x)|^2+|F_{\gamma\eta}^{(\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha)}(x)|^2} \right}\delta}\def\De{\Delta}\def\ep{\epsilonf\ti{\tilde}\delta}\def\De{\Delta}\def\ep{\epsilonf\we{\wedge}\delta}\def\De{\Delta}\def\ep{\epsilonf\wti{\widetildeght]
\lambda}\def\La{\Lambda}\def\si{\sigma}\def\Si{\Sigmabel{LogNegat}
\end{eqnarray}}
We observe that the above quantities are expressed in terms of off-diagonal terms.
Fig.~\ref{figConca} contains the plots of the concurrence $C_{\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha}^{(\beta,\gamma)}$
for $\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha=e,\,\mu$.
\begin{figure}
\caption{(Color online) The concurrences $C_{\alpha}
\end{figure}
Looking at the curves in Fig.~\ref{figConca}, we can investigate the distribution of the entanglement
between two specific flavors. For instance in the panel I of the same figure,
the entanglement is initially distributed between the couples $(\nu_e,\nu_\mu)$ and $(\nu_e,\nu_\tau)$, while,
for greater distance, also the component $(\nu_\mu,\nu_\tau)$ acquires a growing, and then oscillating, weight.
On the contrary, in the panel II, $(\nu_\mu,\nu_\tau)$ is initially associated with the greatest component;
subsequently the entanglement is distributed quite evenly among all the components,
exhibiting an oscillatory behavior till a stabilization at a final constant value.
Nevertheless, being based on the trace performed on one flavor,
the concurrence cannot provide a complete description of the entanglement distribution.
Therefore, we exploit the global entanglement measure, built of the logarithmic negativities.
In Fig.~\ref{figLogNega} it is plotted the logarithmic negativity
$E_{\mathcal{N}\,\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha}^{(\beta,\gamma;\eta)}$ for $\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha=e,\,\mu$.
\begin{figure}
\caption{(Color online) The logarithmic negativities $E_{\mathcal{N}
\end{figure}
The curves in Fig.~\ref{figLogNega} allow to guess some indications about the distribution
of entanglement between the three bipartitions $(\beta,\gamma;\eta)$.
For example, in panel I of the figure, we can observe that all the bipartitions
possess a similar amount of entanglement as $x$ varies.
Instead in panel II, the bipartitions $(e,\mu;\tau)$ and $(e,\tau;\mu)$
exhibit a quite identical oscillatory behavior, with most of the entanglement content of the state.
The set of measures plotted in Figs.~\ref{figConca} and \ref{figLogNega}
can be exploited in a complementary way to get a clear picture of the entanglement concentration
between the flavors $(\beta,\gamma)$ and the bipartitions $(\beta,\gamma;\eta)$.
In fact, for certain regions of $x$, the entanglement is equally distributed both in the flavors and the bipartitions.
As an example, looking at the space interval $[10^8,\,10^9]$ in panel I of Fig. \ref{figConca} and
in panel I of Fig. \ref{figLogNega},
we observe that both the concurrences and the logarithmic negativities exhibit,
for some limited regions, high and comparable levels.
This evidence can be considered a signature of tripartite entanglement.
A very important aspect is that, by comparing Fig. \ref{figTransProb}
with Figs. \ref{figConca} and \ref{figLogNega}, the behavior of entanglement measures
is evidently not monotone with respect to that of transition probabilities.
This implies that the off-diagonal correlations add further insight to the understanding of the evolution of flavor neutrino states.
We see that, analogously to the transition probabilities, both $C_{\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha}^{(\beta;\gamma)}$ and $E_{\mathcal{N}\,\alpha}\delta}\def\De{\Delta}\def\ep{\epsilonf\bt{\beta}\delta}\def\De{\Delta}\def\ep{\epsilonf\ga{\gamma}\delta}\def\De{\Delta}\def\ep{\epsilonf\Ga{\Gammapha}^{(\beta,\gamma;\eta)}$
tend to constant values for sufficiently high distances $x$.
This fact is due to the spatial separation of the wave packets;
indeed, as a consequence, the interference effects are destroyed by the decoherence due to localization
(i.e. damping of the oscillations).
In this regime, the only surviving entanglement is the static intrinsic one \cite{NoiPRD},
due to the peculiarity of systems exhibiting particle mixing.
\section{Conclusions}
In order to gain further insight in the physics of the evolution of neutrino oscillations, we have
investigated the flavor entanglement content, and the corresponding quantum correlations when the neutrino state
is described by a time-averaged wave-packet.
The description of the neutrino state is carried out in its natural and directly observable playground,
i.e. the flavor Hilbert space. The flavor entanglement provides further information about quantum correlations
between different flavors and between different bipartitions of the three-flavor system.
The main aspects concerning the entanglement shared by flavors and/or flavor subsystems have
been analyzed in terms of space distribution. In particular, the entanglement distribution,
i.e. the entanglement shared by flavors and/or flavor bipartitions, shows a peculiar behavior as a function of distance,
which cannot be inferred by looking only at transition probabilities.
It is worth to remark that an uniform distribution of entanglement among the three parties is observed in some regions of the space variable $x$. The aim of such investigations is twofold: to provide a deeper understanding of fundamental phenomena, and to lay the theoretical basis for possible practical implementations of quantum information protocols in the framework of elementary particle physics and quantum field theory.
\end{document}
|
\begin{document}
\title{Kernels and Small Quasi-Kernels in Digraphs}
\begin{abstract}
A directed graph $D=(V(D),A(D))$ has a kernel if there exists an
independent set $K\subseteq V(D)$ such that every vertex $v\in V(D)-K$
has an ingoing arc $u\mathbin{\longrightarrow}v$ for some $u\in K$. There are
directed graphs that do not have a kernel (e.g. a 3-cycle). A quasi-kernel
is an independent set $Q$ such that every vertex can be reached in at most
two steps from $Q$. Every directed graph has a quasi-kernel. A conjecture
by P.L. Erd\H{o}s and L.A. Sz\'ekely (cf. A. Kostochka, R. Luo, and, S. Shan,
\href{https://arxiv.org/abs/2001.04003v1}{\color{blue}arxiv:2001.04003v1}, 2020) postulates
that every source-free directed graph has a
quasi-kernel of size at most $|V(D)|/2$, where source-free refers to every vertex
having in-degree at least one. In this note it is shown that every source-free directed graph
that has a kernel also has a quasi-kernel of size at most $|V(D)|/2$, by
means of an induction proof. In addition, all definitions and proofs in
this note are formally verified by means of the Coq proof assistant.
\end{abstract}
In this note, all directed graphs $D=(V(D),A(D))$ are assumed to be finite
and without self-loops. The notation $u\mathbin{\longrightarrow}v$ is used
to denote $(u,v)\in A(D)$. A set $I\subseteq V(D)$ is \emph{independent} if
there are no two vertices $u,v\in I$ connected by an arc in any direction. A
\emph{kernel} $K\subseteq V(D)$ is an independent set such that every vertex
in $V(D)$ can be reached from a vertex in $K$ in at most one step. A
\emph{quasi-kernel} $Q\subseteq V(D)$ is a weakening of the concept of kernel by
requiring that every vertex in $V(D)$ can be reached in at most two steps from
a vertex in $Q$. A source in a directed graph is a vertex $u\in V(D)$ having
only outgoing arcs. A directed graph is therefore said to be \emph{source-free}
if every vertex has at least one ingoing arc.
Based on these definitions, it is clear that every kernel is also a quasi-kernel.
However, there do exist source-free directed graphs that do not have a kernel,
for instance an odd directed cycle. Moreover, there are source-free directed graphs
that do have a kernel, but none less than or equal to $|V(D)|/2$ in size, as shown
by the examples in Figure \ref{fig:examples}. It is straightforward to prove
that every directed graph has a quasi-kernel \cite{chvatal}.
The following conjecture is attributed to P.L. Erd\H{o}s and L.A. Sz\'ekely
(cf. \cite{fete,kostochka,web}).
\begin{conjecture}
\label{con:es}
Every source-free digraph has a quasi-kernel of size at most $|V(D)|/2$.
\end{conjecture}
\begin{figure}
\caption{Two directed source-free graphs that do not possess a kernel of size
less than or equal to $|V(D)|/2$. The black vertices indicate smallest possible
kernels in both examples.}
\label{fig:examples}
\end{figure}
The purpose of this note is to prove Conjecture \ref{con:es} in the presence
of a kernel, by means of an induction proof of moderate technicality. A formal
verification of this proof by means of the Coq proof assistant is also supplied
with this paper\footnote{This proof has been checked using version 8.4pl4 of the
Coq proof assistant, and is not guaranteed to work in earlier or later versions
of Coq.}.
A kernel in a directed graph is somewhat related to the concept of an
independent dominating set in the context of undirected graphs \cite{goddard},
and some general terminology in this note is inherited from the theory of domination
in undirected graphs \cite{cockayne}.
A digraph has a kernel if and only if it does not have a directed odd
cycle \cite{richardson}. Existence of a quasi-kernel of size at most
$|V(D)|/2$ is guaranteed in case $D$ is a tournament, semicomplete multipartite,
or locally semicomplete \cite{heard}, or an orientation of a graph with
chromatic number at most four \cite{kostochka}.
The two definitions below are followed by a brief outline of
the solution proposed in this note. It is assumed that $D$ is a directed
graph in these two definitions.
\begin{definition}
\label{def:epon}
For $S\subseteq V(D)$ and $u\in S$, an out-neighbor $v\in V(D)-S$ of $u$ is
said to be an \textit{external private out-neighbor} (\textsc{epon}) with
regard to $S$, if for all $w\in S$ such that $w\mathbin{\longrightarrow}v$ it holds
that $u=w$.
\end{definition}
As an example, the right-most vertex in Figure \ref{fig:examples} is an
\textsc{epon} with regard to the black vertices. The directed graph on the
left does not have an \textsc{epon}.
\begin{definition}
\label{def:in_dom}
A quasi-kernel $Q\subseteq V(D)$ is \textit{inward dominated} if for all
$w\in Q$ and $v\in V(D)-Q$ such that $v\mathbin{\longrightarrow}w$, there exists a
$u\in Q$ such that $u\mathbin{\longrightarrow}v$.
\end{definition}
The induction proof in Theorem \ref{thm:main} is outlined briefly here.
First, Lemma \ref{lem:in_dom} is used to derive existence of an inward
dominated quasi-kernel $K$, based on the assumption of a kernel being
present. If every vertex $u\in K$ has an \textsc{epon}, then the conclusion
$|K|\leq|V(D)|/2$ follows directly from Lemma \ref{lem:epon_leq}. If there
is some vertex $u\in K$ that does not have an \textsc{epon}, then the vertex
can be removed, and this process clearly terminates in finitely many steps.
The technicality of this solution lies entirely in proving that the premisses
for the induction hypothesis are satisfied. In particular, it is not entirely
trivial to show that $K-\{u\}$ is again an inward dominated quasi-kernel.
Theorem \ref{thm:main} is supported by the three lemmas below. These are
proved here in detailed form, in order to achieve closer resemblance to
the way they are formalized in the Coq proof. It is assumed that $D$ is a
directed graph in these three lemmas.
\begin{lemma}
\label{lem:in_dom}
If $K\subseteq V(D)$ is a kernel then $K$ is an inward dominated quasi-kernel.
\end{lemma}
\begin{proof}
From the definition it is immediate that $K$ is a quasi-kernel. Assume
that $w\in K$ and $v\in V(D)-K$ such that $v\mathbin{\longrightarrow}w$. Then,
since $K$ is a kernel, there exists a $u\in K$ such that
$u\mathbin{\longrightarrow}v$. Lemma \ref{lem:in_dom} is formalized as
\texttt{Lemma kernel\_qkernel} and \texttt{Lemma kernel\_in\_dom} in
the Coq proof.
\end{proof}
\begin{lemma}
\label{lem:epon_incl}
If $u$ has an \textsc{epon} with regard to $T\subseteq V(D)$ and
$S\subseteq T$ such that $u\in S$ then $u$ has an \textsc{epon} with
regard to $S$.
\end{lemma}
\begin{proof}
Suppose that $u\in S$ has an \textsc{epon} $v\in V(D)-T$ with regard to
$T$ and assume towards a contradiction that there exists a $w\in S$ such
that $w\not=u$ and $w\mathbin{\longrightarrow}v$. Then clearly $w\in T$ and therefore
$v$ cannot be an \textsc{epon} with regard to $T$. Lemma \ref{lem:epon_incl}
is encoded as \texttt{Lemma has\_epon\_incl} in the Coq proof.
\end{proof}
\begin{lemma}
\label{lem:epon_leq}
If all vertices in $S\subseteq V(D)$ have an \textsc{epon} with regard
to $S$, then $|S|\leq |V(D)|/2$.
\end{lemma}
\begin{proof}
In general, say that $R\subseteq X\times Y$ is a binary total injective relation if
\begin{enumerate}[(1)]
\item for all $x\in X$ there exists a $y\in Y$ such that $R\,(x,y)$, and
\item for all $x,x'\in X$ and $y\in Y$ such that $R\,(x,y)$ and $R\,(x',y)$ it
holds that $x=x'$.
\end{enumerate}
Clearly, $|X|\leq |Y|$ holds here due to injectivity.
Say that $T\subseteq V(D)$ contains the vertices that are an \textsc{epon}
with regard to the set $S$, as given in the statement of this lemma. Define
$R\subseteq
S\times T$ as
\begin{center}
\begin{math}
R=\{(u,v)\in S\times T\mid
v\,\,\textrm{is an \textsc{epon} of}\,\,u\,\,\textrm{with regard to}\,\,S\}
\end{math}
\end{center}
\noindent and observe that $R$ is a binary total injective relation. It then
follows that $|S|\leq |T|$ and hence, as $S$ and $T$ are disjoint,
$|S|+|T|\leq |V(D)|$ and thus $2|S|\leq |V(D)|$. Lemma \ref{lem:epon_leq}
is encoded as \texttt{Lemma all\_epon\_half\_size} (using \texttt{Lemma inj\_leq})
in the Coq proof.
\end{proof}
Conjecture \ref{con:es}, in case a kernel is present, is proved in
Theorem \ref{thm:main}. This part of the proof is formalized as
\texttt{Theorem main} at the end of the Coq code.
\begin{theorem}
\label{thm:main}
If a source-free digraph $D$ has a kernel, then $D$ has a quasi-kernel
of size at most $|V(D)|/2$.
\end{theorem}
\begin{proof}
Assume $D$ is a source-free directed graph that has a kernel $K$. From
Lemma \ref{lem:in_dom} it is clear that $K$ is also an inward dominated
quasi-kernel. By induction, the following will be shown: if $K$ is an
inward dominated quasi-kernel, then there exists a quasi-kernel of size
at most $|V(D)|/2$. For this purpose, define $S\subseteq V(D)$ as follows:
\begin{center}
\begin{math}
S = \{u\in K\mid u\,\,\textrm{does not have an \textsc{epon} with regard to}\,\,K\},
\end{math}
\end{center}
\noindent and apply induction towards $|S|$, thereby generalizing over all
other variables.
If $|S|=0$ then every vertex in $K$ has an \textsc{epon} and by Lemma
\ref{lem:epon_leq} it then follows that $|K|\leq |V(D)|/2$.
Assume that $|S|>0$ and assume that there exists a vertex $u\in S$
such that $u$ does not have an \textsc{epon} with regard to $S$ in $D$.
Now define $R\subseteq K-\{u\}$ as follows:
\begin{center}
\begin{math}
R = \{v\in K-\{u\}\mid v\,\,\textrm{does not have an \textsc{epon} with regard to}\,\,K-\{u\}\}.
\end{math}
\end{center}
Now, three premisses are required to be able to apply the induction
hypothesis and thereby complete the proof: (1) $|R|<|S|$, (2) $K-\{u\}$
is a quasi-kernel, and, (3) $K-\{u\}$ is inward dominated. These will
be proved here one by one.
\begin{enumerate}[(1)]
\item Clearly, it holds that $u\in S$ and $u\not\in R$, and therefore
it suffices to show that $R\subseteq S$. Assume that $v\in R$ such
that $v$ does not have an \textsc{epon} with regard to $K-\{u\}$.
Then, by contraposition of Lemma \ref{lem:epon_incl}, $v$ cannot
have an \textsc{epon} with regard to $K$, hence $v\in S$.
\item It is immediate that $K-\{u\}$ is also independent. Assume that
$v\in V(D)$ and distinguish between the following cases.
\begin{itemize}
\item If $u=v$ then, as $D$ is source-free, there must exist a vertex
$u'\in V(D)$ such that $u'\mathbin{\longrightarrow}u$ and $u'\not\in K$. As $K$ is an
inward dominated quasi-kernel, there must exist a vertex $w\in K$
such that $w\mathbin{\longrightarrow}u'$. Clearly, only the case $w=u$
is relevant. Assume there does not exist some alternative vertex
$w'\in K$ such that $w'\mathbin{\longrightarrow}u'$. Then, $u'$ is an
\textsc{epon} of $u$ with regard to $K$, thereby contradicting
the assumption $u\in S$.
\item Now consider the case $u\not=v$, and distinguish between the
cases following from the assumption that $K$ is a quasi-kernel.
\begin{itemize}
\item If $v\in K$ then $v\in K-\{u\}$.
\item Now suppose that $v\not\in K$ and there exists a vertex $w\in K$
such that $w\mathbin{\longrightarrow}v$. If $w=u$ then either there
exists an alternative vertex $w'\in K$ such that $w'\mathbin{\longrightarrow}v$,
or $v$ is an \textsc{epon} of $u$ with regard to $K$.
\item For the final case corresponding to $v\not\in K$, assume there exist
vertices $x\in K$ and $w\not\in K$ such that $x\mathbin{\longrightarrow}w$ and
$w\mathbin{\longrightarrow}v$. If $x=u$ then either there exists an alternative
vertex $x'\in K$ such that $x'\mathbin{\longrightarrow}w$, or $w$ is an
\textsc{epon} of $u$ with regard to $K$.
\end{itemize}
\end{itemize}
\item Assume that $w\in V(D)-(K-\{u\})$ and $v\in K-\{u\}$ such that
$w\mathbin{\longrightarrow}v$. Since $K$ is inward-dominated, there must
exist some vertex $x\in K$ such that $x\mathbin{\longrightarrow}w$. Now
assume $x=u$. If there does not exist some alternative vertex $y\in K$
such that $y\mathbin{\longrightarrow}w$, then $w$ is an \textsc{epon} of $u$
with regard to $K$, contradicting $u\in S$.
\end{enumerate}
\end{proof}
Conjecture \ref{con:es} remains open if there is no kernel present. Results
concerning the presence of multiple distinct quasi-kernels in this situation
are known \cite{jacob, gutin}. However, these distinct quasi-kernels are
not necessarily disjoint. The reasonable complexity of the proof in this
note leads to the suggestion that Conjecture \ref{con:es}, in its
unconditional form, is perhaps also accessible via elementary methods.
\end{document}
|
\begin{document}
\title{Prehomogeneous geometries vs. \\
bundles with connections}
\author{Erc\"{u}ment H. Orta\c{c}gil}
\maketitle
\begin{abstract}
The below discussion is in three sections A, B, C, each section in two parts
I, II, I representing the standpoint of bundles with connections and II
representing the standpoint of prehomogeneous geometries (phg's). In A, our
object of study is a fibered manifold $\mathcal{E\rightarrow }M$, in B its $
k $-jet bundle $J^{k}(\mathcal{E})\rightarrow M$ and in C a general
nonlinear PDE $\mathcal{H}^{k}\subset J^{k}(\mathcal{E}).$
\end{abstract}
\section{A.I.}
Let $\pi :\mathcal{E\rightarrow }M$ be a smooth fibered manifold fibering as
$(x,y)\rightarrow x$ in coordinates ([P2], Chapter II). Suppose we want to
define a structure on $\pi :\mathcal{E\rightarrow }M$ which will enable us
to make the following decision: For any $q\in \mathcal{E}$ with $\pi (q)=p,$
there esists a \textit{unique} local section $s$ defined near $p$ satisfying
$s(p)=q.$ This is the well known framework of connections: First, we
consider the $1$-jet bundle $J^{1}(\mathcal{E)\rightarrow E}$ whose fiber $
J^{1}(\mathcal{E)}_{q}$ over $q\in \mathcal{E}$ is the set of all $1$-jets
of local sections $s$ of $\pi :\mathcal{E\rightarrow }M$ at $p=\pi (q).$ Now
$J^{1}(\mathcal{E)\rightarrow E}$ is an affine bundle modeled on the vector
bundle $T^{\ast }(M)\otimes V(\mathcal{E})\rightarrow \mathcal{E}$ (where $
T^{\ast }(M)\rightarrow M$ is pulled back to $\mathcal{E)}.$ In particular,
the fibers of $J^{1}(\mathcal{E)\rightarrow E}$ are contractible and $J^{1}(
\mathcal{E)\rightarrow E}$ admits a global crossection $c:\mathcal{
E\rightarrow }J^{1}(\mathcal{E)}.$
\begin{definition}
A crossection $c:\mathcal{E\rightarrow }J^{1}(\mathcal{E)}$ of the affine
bundle $J^{1}(\mathcal{E)\rightarrow E}$ is a connection on $\pi :\mathcal{
E\rightarrow }M.$
\end{definition}
\textbf{Remark 1. }Some bundles $\pi :\mathcal{E\rightarrow }M$ may carry
some extra structure and it is natural to expect a connection to respect
this structure. With such a restriction a connection may not exist. We will
turn back to this issue below.
It turns out that the structure we search for is a flat connection, i.e., a
connection with vanishing curvature. Though well known, it is instructive to
see this construction explicitly in coordinates for our future purpose. So
let $q=(\overline{x},\overline{y})=(\overline{x}^{i},\overline{y}^{\alpha
})\in $ $U\times V,$ $1\leq i\leq n=\dim M,$ $1\leq \alpha \leq k=\dim
\mathcal{E}_{p}$ and $s(x)$ a local section defined on $U$ satisfying $s(
\overline{x})=\overline{y}.$ Now $(j^{1}s)(\overline{x})=(\overline{x}^{i},
\overline{y}^{\alpha },\frac{\partial s^{\alpha }(\overline{x})}{\partial
x^{i}})\overset{def}{=}(\overline{x}^{i},\overline{y}^{\alpha },\overline{y}
_{j}^{\alpha })$ and the values $\overline{y}_{i}^{\alpha }$ as $s(x)$
ranges over all such local section define the fiber $J^{1}(\mathcal{E)}_{q}.$
So a connection $c$ is of the form $(x^{i},y^{\alpha },c_{j}^{\alpha
}(x,y)). $ Now by the definition of $c,$ given $(\overline{x}^{i},\overline{y
}^{\alpha },c_{j}^{\alpha }(\overline{x},\overline{y})),$ there exists a
local section $s(x)$ with $s(\overline{x})=\overline{y}$ satisfying
\begin{equation}
(\overline{x}^{i},\overline{y}^{\alpha },\frac{\partial s^{\alpha }(
\overline{x})}{\partial x^{j}})=(\overline{x}^{i},\overline{y}^{\alpha
},c_{j}^{\alpha }(\overline{x},\overline{y}))
\end{equation}
Now $s(x)$ defines also the section $(x^{i},y^{\alpha },\frac{\partial
s^{\alpha }(x)}{\partial x^{j}}),$ $x\in Dom(s)$ but we need not have
\begin{equation}
(x^{i},y^{\alpha },\frac{\partial s^{\alpha }(x)}{\partial x^{j}}
)=(x^{i},y^{\alpha },c_{j}^{\alpha }(x,y))\text{ \ \ \ }x\in Dom(s)
\end{equation}
even though (2) holds at $x=\overline{x}$ by (1). The key fact here is that
the section in (1) depends on the point $(\overline{x},\overline{y})$ and
the same section may not work for all near points. We will write (2) shortly
as
\begin{equation}
\frac{\partial y^{\alpha }(x)}{\partial x^{j}}=c_{j}^{\alpha }(x,y)
\end{equation}
Now (3) is a first order system of PDE's with initial conditions, i.e.,
given the initial condition $(\overline{x},\overline{y}),$ we search for a
local section $y(x)$ solving (3) for $x\in U$ and satisfying $y(\overline{x}
)=\overline{y}.$ This can be done if and only if the integrability
conditions of (3) are identically satisfied. To find these conditions, we
differentiate (3) with respect to $x^{k},$ substitute back from (3) and
alternate $j,k.$ The result is
\begin{equation}
\mathcal{R}_{jk}^{\alpha }(x,y)\overset{def}{=}\left[ \frac{\partial
c_{j}^{\alpha }(x,y)}{\partial x^{k}}+\frac{\partial c_{j}^{\alpha }(x,y)}{
\partial y^{\beta }}c_{k}^{\beta }(x,y)\right] _{[jk]}=0
\end{equation}
It is easy to check that $\mathcal{R}$ is a $2$-form with values in the
vertical bundle $V(\mathcal{E})\rightarrow M,$ i.e., given $(\overline{x},
\overline{y})\in \mathcal{E}$ and the two tangent vectors $\xi ,\eta $ at $
\overline{x},$ $\mathcal{R}_{ab}^{\alpha }(\overline{x},\overline{y})\xi
^{a}\eta ^{b}$ is a vertical tangent vector at $(\overline{x},\overline{y}).$
We recall here the well known coordinatefree interpretation of $c$ and $
\mathcal{R}:$ $c$ gives a decomposition of the tangent space $T_{q}(\mathcal{
E})$ as a direct sum of horizontal and vertical components as
\begin{eqnarray}
T_{q}(\mathcal{E}) &=&H_{q}\oplus V(\mathcal{E})_{q} \\
(v^{i},v^{\alpha }) &=&(v^{i},v^{\alpha }-v^{s}c_{s}^{\alpha
})+(0,v^{s}c_{s}^{\alpha }) \notag
\end{eqnarray}
and the horizontal distribution of dimension $n=\dim M$ defined by (5) is
integrable if and only if $\mathcal{R}=0.$
As the solution of our problem, we make the following
\begin{definition}
The unique local section $s(x)$ satisfying the initial condition $s(
\overline{x})=\overline{y}$ is called a solution of the flat connection $c.$
\end{definition}
\section{A.II.}
Clearly $c(\mathcal{E})\subset J^{1}(\mathcal{E})$ by the definition of the
connection $c.$
\begin{definition}
The submanifold $c(\mathcal{E})\subset J^{1}(\mathcal{E})$ is a
prehomogeneous geometry (phg) of order one on the fibered manifold $\mathcal{
E\rightarrow }M.$
\end{definition}
We observe that $c$ is viewed as part of the definition of a geometric
structure called a phg according to Definition 3. At first sight, a phg
seems to be nothing but a bundle with a connection on it but we will shortly
see that the situation is quite more subtle. In the language of [KS], [Gs],
[KLV], [P1], [P2], $c(\mathcal{E})$ is a nonlinear system of PDE's of order
one on $\mathcal{E\rightarrow }M.$ Note that the restriction of the jet
projection $\pi :J^{1}(\mathcal{E})\rightarrow \mathcal{E}$ gives a
bijection $\pi :c(\mathcal{E})\simeq \mathcal{E}$ so that $c(\mathcal{E}
)\subset J^{1}(\mathcal{E})$ is a very special PDE. We recall ([P2], Chapter
3) that the prolongation $\varrho (c(\mathcal{E}))$ of $c(\mathcal{E})$ is
defined by
\begin{equation}
\varrho (c(\mathcal{E}))\overset{def}{=}J^{1}(c(\mathcal{E}))\cap J^{2}(
\mathcal{E})\subset J^{1}J^{1}(\mathcal{E})
\end{equation}
Since $J^{1}(c(\mathcal{E}))$ surjects onto $c(\mathcal{E})$ by the jet
projection, (6) gives the map
\begin{equation}
\varrho (c(\mathcal{E}))\rightarrow c(\mathcal{E})
\end{equation}
which need not be onto.
\begin{proposition}
\textit{(7) is onto if and only if }$\mathcal{R}=0.$
\end{proposition}
To see this, we first recall the canonical injection ([P2], Chapter 2, Lemma
4))
\begin{equation}
J^{2}(\mathcal{E})\subset J^{1}J^{1}(\mathcal{E})
\end{equation}
Now $(x^{i},y^{\alpha },y_{j}^{\alpha },y_{jk}^{\alpha })$ are coordinates
on $J^{2}(\mathcal{E}),$ $(x^{i},y^{\alpha },y_{j}^{\alpha }\mid
y_{,k}^{\alpha },y_{j,k}^{\alpha })$ are coordinates on $J^{1}J^{1}(\mathcal{
E})$ and (8) is given by
\begin{equation}
(x^{i},y^{\alpha },y_{j}^{\alpha },y_{jk}^{\alpha })\rightarrow
(x^{i},y^{\alpha },y_{i}^{\alpha }\mid y_{j}^{\alpha },y_{jk}^{\alpha })
\end{equation}
i.e., the image of $J^{2}(\mathcal{E})$ in $J^{1}J^{1}(\mathcal{E})$ is
defined by the equations $y_{,k}^{\alpha }=y_{k}^{\alpha },$ $
y_{j,k}^{\alpha }=y_{jk}^{\alpha }.$ Clearly $J^{1}(c(\mathcal{E}))\subset
J^{1}J^{1}(\mathcal{E})$ since $c(\mathcal{E})\subset J^{1}(\mathcal{E})$
and therefore $J^{1}(c(\mathcal{E}))$ is given by
\begin{equation}
J^{1}(c(\mathcal{E})):(x^{i},y^{\alpha },c_{j}^{\alpha }(x,y)\mid
y_{,k}^{\alpha },\frac{\partial c_{j}^{\alpha }(x,y)}{\partial x^{k}}+\frac{
\partial c_{j}^{\alpha }(x,y)}{\partial y^{\beta }}y_{,k}^{\beta })
\end{equation}
where $y_{,k}^{\alpha }=\frac{\partial y^{\alpha }}{\partial x^{k}}.$ It
follows from (10) that $\varrho (c(\mathcal{E}))=$ $J^{1}(c(\mathcal{E}
))\cap J^{2}(\mathcal{E})$ surjects onto $c(\mathcal{E})$ if and only if the
substitution $y_{,k}^{\alpha }=c_{j}^{\alpha }(x,y)$ makes $y_{jk}^{\alpha }
\overset{def}{=}$\ $\frac{\partial c_{j}^{\alpha }(x,y)}{\partial x^{k}}+
\frac{\partial c_{j}^{\alpha }(x,y)}{\partial y^{\beta }}y_{,k}^{\beta }$\
symmetric in $j,k,$\ i.e, if and only if $\mathcal{R}=0.$
\textbf{Remark 2. }A very important fiber bundle in geometry is the
following: Consider the projection on the first factor $\pi :$ $M\times
M\rightarrow M$ whose local sections are local maps $M\rightarrow M.$ We
restrict our attention to sections which are local diffeomorphisms. With
this agreed, $J^{1}(M\times M)$ is $1$-jets of local diffeomorphisms called $
1$-arrows in [Or1]. Now a connection $c$ (denoted by $\varepsilon $ in
[Or1]) assigns to the pair $(x,y)\in M\times M$ a $1$-arrow from $x$ to $y.$
These $1$-arrows have the structure of a groupoid and it is natural to
assume that connections respect this groupoid structure. Now such
connections exist if and only if $M$ is parallelizable and the first two
parts of [Or1] is devoted to the study of these phg's of order one (called
order zero in [Or1]. It turns out that the theory of Lie groups and Lie
algebras can be deduced from the study of these particular phg's. The name
prehomogeneous is motivated by this case because these structures make $M$
locally homogeneous (or endow $M$ with a geometric structure in the sense of
[T], [Gm]) when their curvatures vanish.
\section{B.I.}
Now we specialize our fiber bundles. Let $\pi :\mathcal{E}\rightarrow M$ be
a fibered manifold and our object of study in this section is the $k$-jet
bundle $J^{k}(\mathcal{E})\rightarrow M$ whose fiber over $q\in \mathcal{E}$
is $k$-jets of local sections $s(x)$ of $\mathcal{E}\rightarrow M$ with $
s(p)=q.$ Now $J^{1}(J^{k}(\mathcal{E}))\rightarrow $ $J^{k}(\mathcal{E})$ is
an affine bundle modelled on $T^{\ast }(M)\otimes J^{k}(V(\mathcal{E}
))\simeq T^{\ast }(M)\otimes V(J^{k}(\mathcal{E}))$ ([P2], Propositions 3,
5) and therefore admits a global crossection $c.$ If the curvature of this
connection vanishes, then for any $r\in J^{k}(\mathcal{E}),$ there exists a
unique local section $s_{k}(x)$ of $J^{k}(\mathcal{E})\rightarrow M$ passing
through $r.$ It is very crucial to observe that this section is not
necessarily holonomic, i.e., it may not satisfy $j^{k}(s)(p)=r$ where $p$ is
the projection of $r.$ However, we are interested in holonomic sections
because it is such sections which are true solutions of PDE's. To see this
point clearly, it is instructive to look at $c$ in coordinates. For
simplicity of notation, we assume $k=1.$ Now we have
\begin{eqnarray}
J^{1}(\mathcal{E}) &:&(x^{i},y^{\alpha },y_{j}^{\alpha }) \notag \\
J^{1}J^{1}(\mathcal{E)} &:&(x^{i},y^{\alpha },y_{j}^{\alpha }\mid
y_{,j}^{\alpha },y_{k,j}^{\alpha }) \\
c &:&(x^{i},y^{\alpha },y_{j}^{\alpha }\mid c_{,j}^{\alpha
}(x,y_{1}),c_{k,j}^{\alpha }(x,y_{1}) \notag
\end{eqnarray}
where $(x,y_{1})\overset{def}{=}(x^{s},y^{\beta },y_{m}^{\beta }).$ If $r=(
\overline{x}^{i},\overline{y}^{\alpha },\overline{y}_{j}^{\alpha })\in J^{1}(
\mathcal{E}),$ a section $s$ of $J^{1}(\mathcal{E})\rightarrow M$ passing
through $r$ is of the form $s(x)=(x^{i},y^{\alpha }(x),y_{j}^{\alpha }(x))$
where $y^{\alpha }(\overline{x})=\overline{y}^{\alpha },y_{j}^{\alpha }(
\overline{x})=\overline{y}_{j}^{\alpha }$ and the PDE for such sections is
\begin{eqnarray}
\frac{\partial y^{\alpha }}{\partial x^{j}} &=&c_{,j}^{\alpha }(x,y_{1}) \\
\frac{\partial y_{j}^{\alpha }(x)}{\partial x^{k}} &=&c_{k,j}^{\alpha
}(x,y_{1}) \notag
\end{eqnarray}
The integrability conditions of (12) are abtained by differentiating (12),
substituting back from (12) and alternating. They are given by
\begin{equation*}
\mathcal{R}_{rj}^{\alpha }(x,y_{1})\overset{def}{=}
\end{equation*}
\begin{equation}
\left[ \frac{\partial c_{,j}^{\alpha }(x,y_{1})}{\partial x^{r}}+\frac{
\partial c_{,j}^{\alpha }(x,y_{1})}{\partial y^{\beta }}c_{,r}^{\beta
}(x,y_{1})+\frac{\partial c_{,j}^{\alpha }(x,y_{1})}{\partial y_{a}^{\beta }}
c_{r,a}^{\beta }(x,y_{1})\right] _{[rj]}=0 \notag
\end{equation}
\begin{equation}
\mathcal{R}_{rj,k}^{\alpha }(x,y_{1})\overset{def}{=}
\end{equation}
\qquad
\begin{equation*}
\left[ \frac{\partial c_{k,j}^{\alpha }(x,y_{1})}{\partial x^{r}}+\frac{
\partial c_{k,j}^{\alpha }(x,y_{1})}{\partial y^{\beta }}c_{,r}^{\beta
}(x,y_{1})+\frac{\partial c_{k,j}^{\alpha }(x,y_{1})}{\partial y_{a}^{\beta }
}c_{r,a}^{\beta }(x,y_{1})\right] _{[rj]}=0
\end{equation*}
and as in (4), the total curvature $\mathcal{R}(x,y_{1})\overset{def}{=}$ $(
\mathcal{R}_{rj}^{\alpha }(x,y_{1}),\mathcal{R}_{rj,k}^{\alpha }(x,y_{1}))$
defined on $J^{1}(\mathcal{E})$ is easily seen to be a $2$-form on $M$ with
values in the vertical bundle $V(J^{1}(\mathcal{E}))\simeq J^{1}(V(\mathcal{E
})).$ The computation for general $J^{k}(\mathcal{E})$ is similar and the
formulas can be written in compact form using the multi index notation $
(x^{i},y^{\alpha },y_{j_{1}}^{\alpha },y_{j_{1}j_{2}}^{\alpha
},...,y_{j_{1}...j_{k}}^{\alpha })=(x^{i},y_{\mu }^{\alpha }),$ $0\leq
\left\vert \mu \right\vert \leq k,$ $=(x,y_{k}).$ Now if $\mathcal{R=}0,$
then $s(x)=(x^{i},y^{\alpha }(x),y_{j}^{\alpha }(x))$ satisfying (12) and
the above initial condition is unique but this solution does not necessarily
satisfy $y_{j}^{\alpha }(x)=\frac{\partial y^{\alpha }(x)}{\partial x^{j}},$
i.e., it is not necessarily holonomic. Thus the following question is of
fundamental importance.
\textbf{Q. }What property must the flat connection posess so that its
solutions are holonomic?
\section{B.II.}
We recall that $J^{k+1}(\mathcal{E})\rightarrow J^{k}(\mathcal{E})$ is an
affine bundle modelled on $S^{k+1}(T^{\ast }(M))\otimes V(\mathcal{E})$ and
therefore admits a global section $c.$ Now (8) generalizes to the canonical
inclusion $J^{k+1}(\mathcal{E})\subset J^{1}(J^{k}(\mathcal{E}))$ ([P2],
Chapter 2, Lemma 4). Therefore $c$ is also a section of $J^{1}(J^{k}(
\mathcal{E}))\rightarrow J^{k}(\mathcal{E}),$ i.e., it is a connection on $
J^{k}(\mathcal{E}).$
\begin{definition}
Connections on $J^{k}(\mathcal{E})\rightarrow M,$ i.e., crossections of $
J^{1}(J^{k}(\mathcal{E}))\rightarrow J^{k}(\mathcal{E}),$ which arise from
the canonical inclusion $J^{k+1}(\mathcal{E})\subset J^{1}(J^{k}(\mathcal{E}
))$ are geometric.
\end{definition}
\textbf{Remark 3. }All connections in A.I. are clearly geometric. For the
groupoids $J^{k}(M\times M),$ geometric connections exist only for $k=0$
(absolute parallelizm), $k=2,$ $\dim M$ arbitrary (affine) and $k=3,$ $\dim
M=1$ (projective) because jet groups split only for these values.
The following proposition answers \textbf{Q.}
\begin{proposition}
Solutions of flat geometric connections are holonomic.
\end{proposition}
To see what is involved in Proposition 6, let us look at the case $k=1$
above. It follows from (9) that a geometric connection $c$ satisfies
\begin{eqnarray}
c_{,j}^{\alpha }(x,y_{1}) &=&y_{j}^{\alpha } \\
c_{k,j}^{\alpha }(x,y_{1}) &=&y_{kj}^{\alpha } \notag
\end{eqnarray}
In particular note that $c_{k,j}^{\alpha }(x,y_{1})$ is symmetric in $k,j.$
\textbf{Remark 4. }The above symmetry condition is interpreted as
torsionfreeness of $c$ which is meaningful only for $k=1$ ([Or3]). It is not
easy to detect this fact for large $k$ using the sophisticated formulation
(5).
Therefore $s(x)=(x^{i},y^{\alpha }(x),y_{j}^{\alpha }(x))$ solves
\begin{eqnarray}
\frac{\partial y^{\alpha }}{\partial x^{j}} &=&y_{j}^{\alpha }(x) \\
\frac{\partial y_{j}^{\alpha }(x)}{\partial x^{k}} &=&c_{kj}^{\alpha
}(x,y_{1}) \notag
\end{eqnarray}
The integrability conditions of (15) are given \textit{only }by the second
formula of (13) and the first equation of (15) shows that the solution is
holonomic, in fact, $\frac{\partial ^{2}y^{\alpha }}{\partial x^{k}\partial
x^{j}}=c_{kj}^{\alpha }.$ For general $k,$ all lower order integrability
conditions in (15) are satisfied except the top order which gives the
curvature. This is due to the fact that we work with $J^{k}(\mathcal{E})$
and not with a general nonlinear PDE $\mathcal{H}^{k}\subset J^{k}(\mathcal{E
})$ which we will come to below.
\begin{definition}
Let $c$ be a geometric connection on $J^{k}(\mathcal{E}).$ Then the
nonlinear PDE $c(J^{k}(\mathcal{E}))\subset J^{k+1}(\mathcal{E})$ is a phg
of order $k+1.$
\end{definition}
As before, note that $c$ is now part of the definition of a geometric
structure. The following proposition, whose proof is identical to the
computation at the end of A.II, should not come as a surprise.
\begin{proposition}
The following are equivalent for a geometric connection $c.$
i) $\mathcal{R}=0$
ii) The prolongation map $\varrho (c(J^{k}(\mathcal{E})))\rightarrow $ $
c(J^{k}(\mathcal{E}))$ is surjective.
\end{proposition}
As long as we work with connections on $J^{k}(\mathcal{E}),$ geometric ones
are clearly preferable in view of Proposition 6. From this standpoint, a phg
emerges as an alternative to a bundle with connection and it seems at this
stage that we are at a point of departure from connections. Furthermore,
Proposition 8, ii) hints at the possibility of defining the curvature of a
phg independently of the curvature of any connection. Below we will see that
this departure is not only a technical possibility, but also a very natural
alternative choice.
Below we will specialize to a fibered submanifold $\mathcal{H}^{k}\subset
J^{k}(\mathcal{E}),$ i.e., a general nonlinear system of PDE's. This is a
vast class and a highly sophisticated machinery is developed in [S], [KS],
[Gm], [KLV], [Ol1], [P1], [P2] (see also the references in these works) to
study the formal properties of these PDE's. Unfortunately this machinery is
not widely known and used among the differential geometers. The reason for
this state of affairs, we believe, is that this theory has not been able to
produce significant global examples and their invariants for large $k$ and
the well known examples for small $k$ (like affine, riemannian, projective,
conformal structures) can be thoroughly studied by the classical methods
without this machinery. In [Or4], we gave a completely elementary method to
construct phg's (see Definition 9 below) for arbitrarily large $k$ using the
irreducable representations of semi simple Lie algebras. For instance, using
$SL(2,\mathbb{R}),$ it is possible to construct phg's for arbitrarily large $
k$ on all $4$-manifolds. The question whether there exists a flat one among
all these phg's is quite relevant as Poincare Conjecture is a problem of
this type (see Remark 1 and [Or1], Chapter 13) Motivated by these facts, we
believe that jet theory has the potential to open new horizons in geometry
for large $k$ and has something new to offer also in the study of the well
known classical structures.
\section{C.I.}
The tale of connections is the same: We choose a crossection of $J^{1}(
\mathcal{H}^{k})\rightarrow \mathcal{H}^{k},$ define its curvature which
vanishes if and only if there exist unique not necessarily holonomic
sections of $\mathcal{H}^{k}\rightarrow M$ satisfying the given initial
conditions.
\section{C.II}
The restriction of $J^{k+1}(\mathcal{E})\rightarrow J^{k}(\mathcal{E})$ to $
\mathcal{H}^{k}\subset J^{k}(\mathcal{E})$ gives the fibered manifold $
J^{k+1}(\mathcal{E})_{\mid \mathcal{H}^{k}}\rightarrow \mathcal{H}^{k}$ with
contractible fibers and therefore admits a crossection $\varepsilon .$
\begin{definition}
The nonlinear PDE $\mathcal{H}^{k}\simeq \varepsilon (\mathcal{H}
^{k})\subset J^{k+1}(\mathcal{E})$ of order $k+1$ is a phg of order $k+1.$
\end{definition}
Therefore any nonlinear PDE $\mathcal{H}^{k}\subset J^{k}(\mathcal{E})$ of
order $k$ can be extended to a phg $\varepsilon (\mathcal{H}^{k})\subset
J^{k+1}(\mathcal{E})$ of order $k+1.$ \textit{The arbitrariness in the
choice of }$\varepsilon $\textit{\ is quite subtle and reflects the
"modelfreeness of the geometry" ([B], [Or1]).} Now even though $J^{k+1}(
\mathcal{E})\subset J^{1}(J^{k}(\mathcal{E}))$ we do not necessarily have $
\varepsilon (\mathcal{H}^{k})\subset J^{1}(\mathcal{H}^{k}),$ i.e., $
\varepsilon $\textit{\ need not be a connection !! }Now $\varrho (\mathcal{H}
^{k})=J^{1}(\mathcal{H}^{k})\cap J^{k+1}(\mathcal{E)}$ (we should not
confuse $\varrho (\mathcal{H}^{k})$ and $\varrho (\varepsilon (\mathcal{H}
^{k}))=J^{1}(\varepsilon (\mathcal{H}^{k}))\cap J^{k+2}(\mathcal{E}))).$
Therefore $\varepsilon (\mathcal{H}^{k})\subset J^{1}(\mathcal{H}^{k}),$
i.e., $\varepsilon $ is a connection $\Leftrightarrow $ $\varepsilon (
\mathcal{H}^{k})\subset \varrho (\mathcal{H}^{k}).$ If we stay on the side
of I, the question is whether we can choose $\varepsilon $ to be a
connection, and if so, whether it is unique,...etc (the answers depend, of
course, on $\mathcal{E}\rightarrow M$ we work with). If we choose the
alternative path II, the problem is to define the curvature of $\varepsilon (
\mathcal{H}^{k})$ as we may not have any connection with its curvature at
our disposal. Of course, the solution is hinted by Propositions 4, 8 and
their proofs.
\begin{proposition}
The nonlinear PDE $\varepsilon (\mathcal{H}^{k})\subset J^{k+1}(\mathcal{E})$
has unique holonomic solutions with arbitrary inital conditions if and only
if the map $\varrho (\varepsilon (\mathcal{H}^{k}))\rightarrow \varepsilon (
\mathcal{H}^{k})$ is surjective. In fact, there exists a vector bundle $
F_{1} $ over $\varepsilon (\mathcal{H}^{k})$ and a section $\mathcal{R}$ of
this bundle satisfying the following diagram which is exact at the middle
term
\begin{equation}
\varrho (\varepsilon (\mathcal{H}^{k}))\overset{\pi _{k+1}^{k+2}}{
\longrightarrow }\varepsilon (\mathcal{H}^{k})\overset{\mathcal{R}}{
\longrightarrow }F_{1}
\end{equation}
\end{proposition}
So $\mathcal{R=}0$ if and only if $\varrho (\varepsilon (\mathcal{H}
^{k}))\rightarrow \varepsilon (\mathcal{H}^{k})$ is onto.
\begin{definition}
$\mathcal{R}$ is the curvature of the phg $\varepsilon (\mathcal{H}
^{k})\subset J^{k+1}(\mathcal{E}).$
\end{definition}
Proposition 10 is a special case of Theorem 1, pg. 95, [P2], where $
\varepsilon (\mathcal{H}^{k})$ is replaced by a general nonlinear PDE $
\mathcal{K}^{k+1}\subset J^{k+1}(\mathcal{E})$ and $\varrho (\varepsilon (
\mathcal{H}^{k}))$ by its prolongation $\varrho (\mathcal{K}^{k+1}).$ The
proof of this general case is rather technical and not easy to verify even
on concrete examples, as the author states. Furthermore, $\mathcal{R}$ is
only the first of a sequence of curvatures $\mathcal{R}_{1},\mathcal{R}
_{2},...,$ whose vanishing implies surjectivity of higher order
prolongations. These methods imply only formal integrability in smooth
category. Now a phg, as we remarked above, is a very special PDE and its
definition is motivated by the stabilization order of Klein geometries $
(G,H) $ ([Or1], [Ol2]). For instance, the symbol of a phg vanishes, i.e., it
is of type zero and therefore trivially $k$-acyclic for any $k$ and
involutive. Therefore all the assumptions of Theorem 1 [P2] are satisfied by
a phg. Another simplification is that formal integrability can be replaced
by strong integrability as stated by Proposition 10 whose proof, as
expected, reduces to Frobenius Theorem.
It remains to check Proposition 10 on projective and conformal structures as
a preperation for higher order structures ([Or5]) .
\textbf{References}
[B] A.D.Blaom: Geometric structures as deformed infinitesimal symmetries,
Trams. Amer. Math. Soc., 358, 2006, 2651-71
[Gm] W.M.Goldman: Geometric Structures on Manifolds, AMS, Graduate Studies
in Math., 227, 2022
[Gs] H. Goldschmidt: Integrability criterion for systems of nonlinear
partial differential equations, J. Differential Geom. 1, (1969), 269-307
[KS] A.Kumpera, D.C.Spencer: Lie Equations, Ann. Math. Studies, 73,
Princeton Univ. Press, Princeton, New Jersey, 1972
[KLV] I.S.Krasil'shchik, V.V.Lychagin, A.M.Vinogradov: Geometry of Jet
Spaces and Nonlinear Partial Differential Equations, Gordon and Breach, 1986
[Ol1] P.J.Olver: Applications of Lie Groups to Differential Equations,
Springer, Berlin, 1986
[Ol2] \ \ \ \ \ \ " \ \ \ \ \ \ : Equivalence, Invariants and Symmetry,
Cambridge University Press, 1995
[Or1] E.H.Orta\c{c}gil: An Alternative Approach to Lie Groups and Geometric
Structures, OUP, 2018
[Or2] \ \ " \ \ \ : Curvature without connection, arXiv 2003.06593
[Or3] \ \ " \ \ : The mystery of torsion in differential geometry,
Researchgate
[Or4] \ \ " \ \ : Klein geometries of high order, arXiv 2211.02355
[Or5] \ \ " \ \ : Projective and conformal prehomogeneous geometries, in
progress
[P1] J.F.Pommaret : Systems of Partial Differential Equations and Lie
Pseudogroups, Gordon and Breach, London, New York, 1978
[P2] \ \ \ " \ \ \ : Partial Differential Equations and Group Theory, New
Perspectives for Applications, Kluwer Academic Publishers, Mathematics and
its Applications, Vol. 293, 1994
[S] D.C.Spencer: Overdetermined systems of partial\c{s} differential
equations, Bull. Amer. Math. Soc. 75 (1965), 1-114
[T] W.P.Thurston: Three-Dimensional Geometry and Topology, Vol.1, Princeton
University Press, 1997
[email protected]
\end{document}
|
\begin{document}
\frontmatter
\title[Layer Potentials and BVPs in Besov Spaces]{Layer Potentials and Boundary-Value Problems for Second Order Elliptic Operators with Data in Besov Spaces}
\author{Ariel Barton}
\address{Ariel Barton, 202 Math Sciences Bldg., University of Missouri, Columbia, MO 65211}
\email{[email protected]}
\author{Svitlana Mayboroda}
\address{Svitlana Mayboroda, School of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church St.\ SE, Minneapolis, Minnesota 55455}
\email{[email protected]}
\date{}
\subjclass[2010]{Primary
35J25,
Secondary
31B20,
35C15,
46E35
}
\keywords{Elliptic equation, boundary-value problem, Besov space, weighted Sobolev space}
\begin{abstract} This monograph presents a comprehensive treatment of second order divergence form elliptic operators with bounded measurable $t$-independent coefficients in spaces of fractional smoothness, in Besov and weighted $L^p$ classes. We establish:
\begin{enumerate}
\item Mapping properties for the double and single layer potentials, as well as the Newton potential;
\item Extrapolation-type solvability results: the fact that solvability of the Dirichlet or Neumann boundary value problem at any given $L^p$ space automatically assures their solvability in an extended range of Besov spaces;
\item Well-posedness for the non-homogeneous boundary value problems.
\end{enumerate}
In particular, we prove well-posedness of the non-homogeneous Dirichlet problem with data in Besov spaces for operators with real, not necessarily symmetric coefficients.
\end{abstract}
\maketitle
\tableofcontents
\addtocounter{tocdepth}{1}
\mainmatter
\input sec-1-introduction
\input sec-2-dfn
\input sec-3-main
\input sec-4-background
\input sec-5-bounded
\input sec-6-trace
\input sec-7-sobolev
\input sec-8-green
\input sec-9-invertible
\input sec-10-besov
\backmatter
\end{document}
|
\mathfrak{b}egin{document}
\mathfrak{b}egin{abstract}We study the space of period polynomials associated with modular forms of
integral weight for finite index subgroups of the modular group. For the modular group, this space is endowed
with a pairing, corresponding to the Petersson inner product on modular forms via a formula of
Haberland, and with an action of Hecke operators, defined algebraically by Zagier. We generalize
Haberland's formula to (not necessarily cuspidal) modular forms for finite index subgroups, and we show that it
conceals two stronger formulas. We extend the action of Hecke operators to period polynomials of modular forms, we
show that the pairing on period polynomials appearing in Haberland's formula is nondegenerate, and we determine
the adjoints of Hecke operators with respect to it. We give a few applications for $\Gamma_1(N)$: an extension of
the Eichler-Shimura isomorphism to the entire space of modular forms; the determination of the
relations satisfied by the even and odd parts of period polynomials associated with cusp forms,
which are independent of the period relations; and an explicit formula for Fourier coefficients of Hecke eigenforms
in terms of their period polynomials, generalizing the Coefficients Theorem of Manin.
\end{abstract}
\title{Modular forms and period polynomials}
\section{Introduction}
Let $\Gamma$ be a finite index subgroup of $\Gamma_1=\mathrm{SL}_2(\mathbb{Z})$, and
let $S_k(\Gamma)$ be the space of cusp forms of integer weight $k\mathfrak{g}e 2$ for $\Gamma$. Let $V_w$ be
the $\Gamma$-module of complex polynomials of degree at most $w=k-2$.
Each form $f\in S_k(\Gamma)$ determines a collection of polynomials $\rho_f: \Gamma\mathfrak{b}ackslash
\Gamma_1\rightarrow V_w$ given by
\[
\rho_f(A)=\int_0^{i\infty} f|_k A(t) (t-X)^w dt.
\]
The object $\rho_f$ belongs to the induced $\Gamma_1$-module $\mathrm{Ind}_\Gamma^{\Gamma_1}
V_w$, and
we call it the (multiple) period polynomial associated to $f$. The goal of this paper is to
investigate the structure of the space of period polynomials, reflecting the Petersson inner
product and the Hecke operators on modular forms. Working inside the subspace of
period polynomials $W_w^\Gamma\subset \mathrm{Ind}_\Gamma^{\Gamma_1} V_w$, we
show that the Petersson product, and the action of Hecke operators, can be
stated in a
simple way in terms of period polynomials. On an abstract level, this is explained by the fact
that the parabolic cohomology class associated to $f$
is completely determined by $\rho_f$, as reviewed in Section \ref{sec2} where we restate the
Eichler-Shimura isomorphism in terms of period polynomials. Our results can be
interpreted as translating the cup product and the action of Hecke operators on cohomology,
into a pairing and a Hecke action on the space of period polynomials.
An essential ingredient in our approach is a generalization of a formula of Haberland
expressing the Petersson product of two cusp forms for the modular group in terms of a pairing on their period
polynomials \cite{H,KZ}. In Section \ref{sechab} we show that Haberland's formula can be extended to finite
index subgroups of $\Gamma_1$. More importantly, using an involution on period polynomials that corresponds to
complex conjugation, we show that Haberland's formula splits in two simpler formulas, pairing
the opposite sign parts (respectively the same sign parts) of the period polynomials of
the two forms when $k$ is even (respectively when $k$ is odd). For the full
modular group, the stronger formulas were proved by different means in \cite{Po}.
When $k=2$, a generalization of Haberland's formula was given by Merel \cite{M09}, and a proof for finite
index subgroups and arbitrary weight was very recently given by Cohen \cite{Co}.
Our proof is simplified by the use of Stokes' theorem on a fundamental domain for $\Gamma(2)$,
which clarifies the appearence of the period polynomial pairing in the formula.
The action of Hecke operators on period polynomials was defined algebraically by Zagier for
the full modular group \cite{Z90,Za93,CZ}. It was extended by Diamantis to operators of index coprime with the
level for the congruence subgroups $\Gamma_0(N)$ \cite{Di01}. We show in Section
\ref{sec4} that the same elements as in the full level case, which go back to work of Manin
\cite{M73}, have actions on period polynomials that correspond to actions of a large class of double coset
operators on modular forms, including Hecke and Atkin-Lehner operators for $\Gamma_1(N)$. We also determine
the adjoint of the Hecke action with respect to the pairing on period polynomials appearing in Haberland's formula.
The proof of Hecke equivariance given here relies on the generalization of Haberland's formula, and a completely
algebraic proof is given in \cite{PP12a}.
As an application of the action of Hecke operators on period polynomials, we obtain a generalization of the
Coefficients Theorem of Manin, giving the Fourier coefficients of a Hecke eigenform for $\Gamma_1(N)$ in terms of
its even period polynomial. We also give a simple proof of the rationality of period polynomials of Hecke
eigenforms for $\Gamma_1(N)$ in \S\ref{sec5.1}. We discuss period polynomials of cusp forms with
nontrivial Nebentypus in \S\ref{sec5.2}. Our results can be used to efficiently compute period polynomials of Hecke
eigenforms numerically, as well as Hecke eigenvalues and Petersson norms, and we give an example in
\S\ref{sec5.3}.
We give two applications of the stronger form of Haberland's formula for cusp forms: in Section
\ref{sec6} we prove a decomposition of cusp forms in terms of Poincar\'e series generators;
while in Section \ref{sec7} we obtain the extra relations satisfied
by the even and odd period polynomials of cusp forms, obtained by Kohnen and Zagier in the
full level case \cite{KZ}. For $\Gamma=\Gamma_0(N)$, we characterize those $N$ for which the
extra relations involve only the even parts of period polynomials just like in the full
level case, that is those $N$ for which the map $\rho^-:S_k(\Gamma)\rightarrow (W_w^\Gamma)^-$ is an isomorphism
(Prop. \ref{prop7.2}). The extra relations are explicit once the period polynomials
of the generators with rational periods are computed, as partially done in \cite{An}, \cite{FY}.
For small $N$ that is enough to give completely explicit relations, and we illustrate this
for $\Gamma_0(2)$.
In the last section, we define the space $\widehat{W}_w^\Gamma$ of period polynomials of all modular forms, following the
construction for the modular group in \cite{Z91}. We generalize Haberland's formula and its
refinement to this larger space, and we show that the pairing appearing in Haberland's formula is nondegenerate on
$\widehat{W}_w^\Gamma$. In contrast, on $W_w^\Gamma$ the radical of this pairing consists of the ``coboundary
polynomials'', of dimension equal to the dimension of the Eisenstein subspace of $M_k(\Gamma)$, as shown in Section
\ref{sec_cw}.
For $\Gamma=\Gamma_1(N)$ and $k>2$, we show
that the plus and minus period polynomial maps extend to isomorphisms $\rho^{\mathfrak{p}m}:M_k(\Gamma)
\rightarrow(\widehat{W}_w^\Gamma)^\mathfrak{p}m$. This can be seen as an extension of the
Eichler-Shimura isomorphism to the entire space of modular forms. Surprisingly, when
$k=2$ the two maps are not always isomorphisms: for $\Gamma=\Gamma_0(N)$ with $N$ square free
with at least two prime factors, precisely one of the two maps is an isomorphism (Proposition
\ref{p7.4} and Remark \ref{r7.5}). In this context we point out that Haberland's formula has been generalized
to weakly holomorphic modular forms of full level in \cite{BGKO}, and it would be interesting to investigate if
the results proved here for modular forms hold in that setting as well.
In \S\ref{sec7.1} we extend the action of Hecke operators to the space of period polynomials of all modular
forms, and we show that the adjoints of Hecke operators on the larger space $\widehat{W}_w^\Gamma$ are the same as on
$W_w^\Gamma$. As an application, for $\Gamma=\Gamma_1(N)$ we show that for $(n,N)=1$
\[\mathrm{Tr} (W_w^\Gamma |_\mathcal{D}elta \widetilde{T}_n ) = \mathrm{Tr} ( M_k(\Gamma)| T_n)+ \mathrm{Tr} ( S_k(\Gamma)| T_n)
\]
where the $|_\mathcal{D}elta \widetilde{T}_n$ is the action on period polynomials corresponding to the action of the Hecke operator
$T_n$ on modular forms. A similar statement holds for modular forms with Nebentypus, and for traces
of Atkin-Lehner operators on $\Gamma_0(N)$. This fact is used in upcoming joint work of
the second author with Don
Zagier to give a simple proof of the Eichler-Selberg trace formula for modular forms on $\Gamma_0(N)$ with
Nebentypus.
A second method of obtaining explicit extra relations among periods of cusp forms
is sketched in Section \ref{sec7.2}. It generalizes the $\Gamma_1$ approach in \cite{KZ}, by
using period polynomials of Eisenstein series and Haberland's formula for arbitrary modular
forms.
We note that period polynomials are dual to modular symbols, in the sense that the coefficients
of period polynomials are values of the integration pairing between modular forms and Manin
symbols (the duality between cohomology and homology). The results of this paper are
therefore parallel to the modular symbol formalism developed by Merel \cite{Me}, and they also lead
to efficient algorithms for modular form computations, as shown in \S\ref{sec5.3}. An additional structure that
we introduce here is the extended pairing on $\widehat{W}_w^\Gamma$, whose nondegeneracy and Hecke equivariance
properties are important even if one is interested only in period polynomials of cusp forms. For example, the
properties of the extended pairing were used in the computation of the Hecke and Atkin-Lehner traces on
$W_w^\Gamma$ mentioned above, and in the proof of rationality of $\rho_f^\mathfrak{p}m$ for newforms $f\in S_k(\Gamma_1(N))$
(Prop. \ref{p5.7}).
The paper is self-contained, except for using the fact that the dimension of the parabolic cohomology group
$H^1_P(\Gamma, V_w)$ equals twice the dimension of $S_k(\Gamma)$, a consequence of the Eichler-Shimura
isomorphism.
\comment{
The paper is organized as follows. In Section \ref{sec2} we restate the
Eichler-Shimura isomorphism in terms of period polynomials, and introduce most of the
notations used throughout. In Sections \ref{sechab}-\ref{sec7} we consider
only period polynomials of cusp forms, while in Section \ref{sec8} we generalize most of the
results of the previous sections to period polynomials of arbitrary modular forms. }
\section{Period polynomials and the Eichler-Shimura isomorphism}\,\langle\,bel{sec2}
The theory of period polynomials for $\Gamma_0(N)$ has been treated in
\cite{sk90,An,Di01}. We review it here in a general setting, and interpret the Eichler-Shimura
isomorphism in terms of period polynomials. We use the properties of the pairing on period
polynomials introduced in Section \ref{sechab} to prove injectivity of the Eichler-Shimura
map. In this section we fix notations in use throughout the paper.
Let $\Gamma$ be a finite index subgroup of $\Gamma_1=\mathrm{SL}_2(\mathbb{Z})$, and denote by
${\overline{\Gamma}}=\Gamma/(\Gamma\cap \{\mathfrak{p}m 1\})$ the projectivisation of $\Gamma$. Throughout
the paper, the weight $k\mathfrak{g}e 2$ is an integer, and we set $w=k-2$. Let $V_w$ be the module
of complex polynomials of degree at most $w$, with (right) $\Gamma_1$-action by the $|_{-w}$
operator: $P|_{-w} g (z)= P(gz) j(g,z)^w$ where $j(g,z)=cz+d$ for $g=\leqslantft(\mathfrak{b}egin{smallmatrix}
* & * \\ c & d \end{smallmatrix}\right)\in \mathfrak{g}l_2(\mathbb{R})$. Since this is the only
action on polynomials, we will omit the subscript.
Viewing $V_w$ as a $\Gamma$-module, let $\widetilde{V}_w^\Gamma$ be the induced
$\Gamma_1$-module $\mathrm{Ind}_{\Gamma}^{\Gamma_1}(V_w)$.
Since $V_w$ is also a $\Gamma_1$-module,
we can identify $\widetilde{V}_w^\Gamma$ with the space of maps $ P :\Gamma\mathfrak{b}ackslash \Gamma_1
\rightarrow V_w$ with $\Gamma_1$ action:
\[
P|g (A) = P(Ag^{-1})|_{-w}g.
\]
By the Shapiro isomorphism, we have $H_P^1(\Gamma, V_w)\simeq H_P^1(\Gamma_1,
\widetilde{V}_w^\Gamma)$ (parabolic cohomology groups). For background on Shapiro's lemma and induced
modules, see \cite[p.59]{NSW}.
Letting $J=\leqslantft(\mathfrak{b}egin{smallmatrix} -1 & 0 \\ 0 & -1
\end{smallmatrix}\right)$, for any cocycle $\sigma: \Gamma_1\rightarrow
V_w^\Gamma$ we have $\sigma(g)|(1-J)=\sigma(J)|(1-g)$ for all $g\in \Gamma_1$(which follows from
$\sigma(Jg)=\sigma(gJ)$). It follows that the cocycle
$\tilde{\sigma}={\scriptscriptstyle F}rac{\sigma+\sigma|J}{2}$ is in the same cohomology class as $\sigma$,
where $\sigma|J(g):=\sigma(g)|J$. Since the cocycle $\tilde{\sigma}$ takes values in the
subspace
$$
V_w^\Gamma:=\{P\in \widetilde{V}_w^\Gamma: P|J=P, \text{ that is } P(A)=(-1)^w P(-A) \}
$$
we have $H_P^1(\Gamma_1, \widetilde{V}_w^\Gamma)\simeq H_P^1(\Gamma_1, V_w^\Gamma) $ and from now on we
will only work inside the space $V_w^\Gamma$. Note that when $k$ is even, $V_w$ is both a
${\overline{\Gamma}}$ and a ${\overline{\Gamma}}_1$ module, and the space $V_w^\Gamma$ can be identified with
$\mathrm{Ind}_{{\overline{\Gamma}}}^{{\overline{\Gamma}}_1}(V_w)$.
Let now $f\in S_k(\Gamma)$, and define a cocycle $\sigma_f:\Gamma_1\rightarrow V_w^\Gamma$
by:
\[
\sigma_f(g) (A) =\int_{g^{-1}i\infty}^{i\infty} f|A(t) (t-X)^w dt,
\]
where the stroke operator acting on modular forms of weight $k$ is $f|g=f|_k g$ for
$g\in GL_2(\mathbb{R})^+$. The action of
the coset $A$ is defined
by acting with any coset representative; this is independent of the representative chosen since
$f|_k\mathfrak{g}ammamma=f$ for $\mathfrak{g}ammamma\in \Gamma$. We will show at the end of this section that
$\sigma_f$ satisfies the cocycle relation
\[
\sigma_f(g_1 g_2)=\sigma_f(g_2)+\sigma_f(g_1)|g_2.
\]
Let $S=\leqslantft(\mathfrak{b}egin{smallmatrix} 0 & -1 \\ 1 & 0 \end{smallmatrix}\right), \
T=\leqslantft(\mathfrak{b}egin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}\right)
$, and let $U=TS$, so that $U^3=J$. Clearly $\sigma_f(\mathfrak{p}m T^n)$ vanishes for $n\in \mathbb{Z}$,
and it is easy to see (by a change of variables) that it is a coboundary for other
parabolic elements of $\Gamma_1$, hence $\sigma_f$ defines an element $[\sigma_f]
\in H_P^1(\Gamma_1,
V_w^\Gamma)$. Since $T$, $S$ and $J$ generate $\Gamma_1$, it follows that the cohomology class $[\sigma_f]$ is
completely determined by the value $\rho_f=\sigma_f(S)\in V_w^\Gamma$, which is the
multiple period polynomial attached to $f$ in the introduction. Using the fact that
$\sigma_f(S^2)=\sigma_f(U^3)=\sigma_f(US)=0$ and the cocycle
relation, it follows that $\rho_f$ satisfies the period polynomial relations:
\[
\rho_f|(1+S)=0, \ \ \ \rho_f|(1+U+U^2)=0.
\]
We also have $\rho_f(-A)=(-1)^w\rho_f(A)$, so $\rho_f|J= \rho_f$. Therefore the image of
the map $f\rightarrow \rho_f$ is contained in the subspace{\scriptscriptstyle F}ootnote{The condition $P|J=P$ is
part of the definition of $V_w^\Gamma$, but we include it for clarity.}
$$
W_w^\Gamma=\{P\in V_w^\Gamma\ :\ P|(1+S)=0, \ \ P|(1+U+U^2)=0, \ \ P|J= P \}
$$
whose elements we call \emph{period polynomials} (each element is in fact a collection of
$[\Gamma_1:\Gamma]$ polynomials belonging to $V_w$).
In fact, setting $C_w^\Gamma=\{P|(1-S) : P \in
V_w^\Gamma,\ \ P|T=P\}\subset W_w^\Gamma,$ we have an isomorphism
\mathfrak{b}egin{equation}\,\langle\,bel{2.2}
H_P^1 (\Gamma_1, V_w^\Gamma)\simeq W_w^\Gamma/C_w^\Gamma,
\end{equation}
obtained by choosing representative cocycles $\sigma$ such that $\sigma( T)=\sigma(J)=0$, and
sending $[\sigma]$ to $\sigma(S)\in W_w^\Gamma$. The space $C_w^\Gamma$ is the image
of coboundaries, and we show in Lemma \ref{L7.1} that its dimension equals the dimension of
the Eisenstein subspace $\mathcal{E}E_k(\Gamma)$ of $M_k(\Gamma)$.
Assume now that $\Gamma$ is normalized by
$\epsilon=\mathfrak{b}ig(\mathfrak{b}egin{smallmatrix} -1 & 0 \\
0 & 1 \end{smallmatrix}\mathfrak{b}ig)$. The matrix $\epsilon$ acts on $P\in V_w^\Gamma$ by
\mathfrak{b}egin{equation}\,\langle\,bel{eps}
P|\epsilon (A) = P(A')|_{-w} \epsilon,
\end{equation}
where $A'=\epsilon A \epsilon$. This action is compatible with the action of $\Gamma_1$: $P| g
|
\epsilon = P|\epsilon|\epsilon g \epsilon$ for all $g\in
\Gamma_1$. If $f^*\in S_k(\Gamma)$ denotes the form $f^*(z)=\overline{f(-\overline{z})}$, then
\mathfrak{b}egin{equation}\,\langle\,bel{2.1}
\overline{\rho_{f^*}}=(-1)^{w+1}\rho_f|\epsilon
\end{equation}
where $\overline{P}(A)$ is obtained by taking the complex conjugates of the coefficients
of $P(A)$.
Under the action of $\epsilon$, the space $V_w^\Gamma$ breaks into $\mathfrak{p}m
1$-eigenspaces, denoted by $(V_w^\Gamma)^\mathfrak{p}m$. For $P\in V_w^\Gamma$ we denote its
+1 and -1-components by $P^+$ and $P^-$ respectively:
\mathfrak{b}egin{equation}\,\langle\,bel{e_ev}
P^{\mathfrak{p}m}={\scriptscriptstyle F}rac{1}{2}( P \mathfrak{p}m P|\epsilon ) \in (V_w^\Gamma)^\mathfrak{p}m.
\end{equation}
We call $P^+$ the \emph{even part} and $P^-$ the \emph{odd part} of $P$,
which is justified by the fact that $P(I)^+$ is an even polynomial, and $P(I)^-$ is an odd
polynomial ($I$ is
the identity coset). For $P\in W_w^\Gamma$, it is easily checked that $P|\epsilon \in
W_w^\Gamma$ as well. Therefore $P^+, P^-\in W_w^\Gamma$, and the space $W_w^\Gamma$ also
decomposes into eigenspaces $(W_w^\Gamma)^\mathfrak{p}m$.
Making use of the pairing on period polynomials introduced in Section \ref{sechab}, we restate
the Eichler-Shimura isomorphism in terms of period polynomials as follows.
\mathfrak{b}egin{theorem}[Eichler-Shimura] \,\langle\,bel{thm2.1}
The two maps $\rho^{\mathfrak{p}m}:S_k(\Gamma) \rightarrow
(W_w^\Gamma)^{\mathfrak{p}m}$, $f\mapsto \rho_f^\mathfrak{p}m$, give rise to
isomorphisms, denoted by the same symbols:
\mathfrak{b}egin{equation}\,\langle\,bel{7.1}
\rho^\mathfrak{p}m:S_k(\Gamma) \longrightarrow (W_w^\Gamma)^{\mathfrak{p}m}/(C_w^\Gamma)^{\mathfrak{p}m}.
\end{equation}
\end{theorem}
\mathfrak{b}egin{proof}
By the stronger version of Haberland's formula (Theorem \ref{thm_main}), the two
maps $\rho^{\mathfrak{p}m}:S_k(\Gamma) \rightarrow (W_w^\Gamma)^{\mathfrak{p}m}$ are injective. Moreover, their
images intersect trivially with $C_w^\Gamma$ by Lemma \ref{l4.4}, so the two maps in \eqref{7.1} are
also injective. Using \eqref{2.2} and the
Eichler-Shimura isomorphism \cite[Ch. 8]{Sh} we have $\dim W_w^\Gamma= 2\dim S_k(\Gamma)+\dim
C_w^\Gamma$, and we conclude that $\rho^\mathfrak{p}m$ in \eqref{7.1} are isomorphisms.
\end{proof}
We now show that $\sigma_f$ satisfies the cocycle relation, while also giving another
construction of the associated period polynomial. In analogy with the $\Gamma_1$ case, the
``Eichler
integral'' associated with $f\in S_k(\Gamma)$ is a function
$\widetilde{f}:\Gamma\mathfrak{b}ackslash\Gamma_1\rightarrow
\mathcal{A}$, where $\mathcal{A}$ is the space of holomorphic functions on the upper half plane, given by:
\mathfrak{b}egin{equation}\,\langle\,bel{eq_eich}
\widetilde{f}(A) (z)=\int_z^{i\infty} f|A(t) (t-z)^w dt,
\end{equation}
with $\Gamma_1$-action as on period polynomials:
$\widetilde{f}|g (A)= \widetilde{f}(Ag^{-1})|_{-w}g$ for $g\in \Gamma_1$. By a change of variables we
see that $ \widetilde{f}|(1-g)=\sigma_f(g)$, which implies that $\sigma_f$ satisfies the cocycle
relation. Note that this provides another
construction for the period polynomial $\rho_f$ attached to $f$, which we record for further
use:
\mathfrak{b}egin{equation}\,\langle\,bel{e_int}
\rho_f=\widetilde{f}|(1-S).
\end{equation}
\mathfrak{b}egin{remark}
A similar construction will be used in Section \ref{sec8} to define period polynomials of
arbitrary modular forms, by means of an Eichler integral $\widetilde{f}$ of $f\in M_k(\Gamma)$,
which has the property that $\widetilde{f}| (1-T)=0$ and $\widetilde{f}|(1-S)$ is the (extended) period
polynomial
attached to $f$. As pointed out in \cite{DIT}, the construction of period polynomials of
cusp forms using their higher order integrals goes back to Poincar\'e.
\end{remark}
\section{Generalization of Haberland's formula} \,\langle\,bel{sechab}
In \cite{H}, Haberland proved a formula expressing the Petersson product of two
cusp forms for the full modular group in terms of a pairing on their period polynomials.
In this section we extend Haberland's formula to a finite index subgroup
$\Gamma$ of $\Gamma_1$, and we prove a stronger version for subgroups normalized by
$\epsilon$.
For $f,g\in S_k(\Gamma)$, define the Petersson scalar product:
\[(f,g)={\scriptscriptstyle F}rac{1}{[{\overline{\Gamma}}_1:{\overline{\Gamma}}]}\int_{\Gamma\mathfrak{b}ackslash\mathcal{H}} f(z)
\overline{g(z)} y^k
{\scriptscriptstyle F}rac{dx dy}{y^2}.
\]
On $V_w\times V_w$ we have a natural pairing
$\,\langle\, \textstyle\sum a_n x^n, \sum b_n x^{n}\,\rangle\, =\sum (-1)^{w-n} \mathfrak{b}inom{w}{n}^{-1} a_n b_{w-n},$
satisfying $\,\langle\, P,Q \,\rangle\,=(-1)^w\,\langle\, Q,P\,\rangle\,$. We will mostly use the equivalent formulation
\mathfrak{b}egin{equation}\,\langle\,bel{3.3}
\,\langle\, (aX+b)^w,(cX+d)^w\,\rangle\, =(ad-bc)^w.
\end{equation}
An easy consequence of \eqref{3.3} is that $\,\langle\, P|g,Q\,\rangle\,=\,\langle\, P, Q|g^\vee\,\rangle\,$ for $g\in
\mathfrak{g}l_2(\mathbb{R})$, where $g^\vee=g^{-1} \deltat g$; in particular the pairing is $\mathrm{SL}_2(\mathbb{R})$-invariant.
We define a similar pairing on $V_w^\Gamma \times V_w^\Gamma$:
\mathfrak{b}egin{equation}\,\langle\,bel{3.0}
\,\,\langle\,ngle\!\,\langle\,ngle\, P, Q \,\,\rangle\,ngle\!\,\rangle\,ngle\,={\scriptscriptstyle F}rac{1}{[{\overline{\Gamma}}_1:{\overline{\Gamma}}]}\sum_{A\in {\overline{\Gamma}}\mathfrak{b}ackslash {\overline{\Gamma}}_1} \,\langle\, P(A),Q(A)\,\rangle\, \ \
\text{ for } P,Q \in
V_w^\Gamma.
\end{equation}
\mathfrak{b}egin{remark} \,\langle\,bel{r_sign}
For odd $k$ there is a sign ambiguity in defining $P(A)$, $Q(A)$ for $P,Q \in V_w^\Gamma$ and
$A\in {\overline{\Gamma}}\mathfrak{b}ackslash{\overline{\Gamma}}_1$, but the pairing is well-defined since $P(-A)=(-1)^w P(A)$,
$Q(-A)=(-1)^w Q(A)$. For the same reason, one can replace the range by $A\in
\Gamma\mathfrak{b}ackslash \Gamma_1$ and the normalizing factor by ${\scriptscriptstyle F}rac{1}{[\Gamma_1:\Gamma]}$ without
changing the pairing. A similar observation applies below, when $f|A$ always appears paired
with $\ov{g}|A$, for $f,g\in S_k(\Gamma)$.
\end{remark}
This pairing is $\Gamma_1$-invariant: $ \,\,\langle\,ngle\!\,\langle\,ngle\, P|g, Q|g \,\,\rangle\,ngle\!\,\rangle\,ngle\,=\,\,\langle\,ngle\!\,\langle\,ngle\, P,Q\,\,\rangle\,ngle\!\,\rangle\,ngle\,$, for all $P,Q \in
V_w^\Gamma $
and $g\in \Gamma_1$. It is normalized such that if $f,g \in S_k(\Gamma)$ and $\Gamma'\subset
\Gamma$ then $\,\,\langle\,ngle\!\,\langle\,ngle\,\rho_f, \rho_g \,\,\rangle\,ngle\!\,\rangle\,ngle\,_{\Gamma} = \,\,\langle\,ngle\!\,\langle\,ngle\,\rho_f, \rho_g \,\,\rangle\,ngle\!\,\rangle\,ngle\,_{\Gamma'}$.
Define also the modified pairing on $V_w^\Gamma \times V_w^\Gamma$:
\mathfrak{b}egin{equation}\,\langle\,bel{pairing}
\{P,Q\}=\,\,\langle\,ngle\!\,\langle\,ngle\, P|T-T^{-1}, Q\,\,\rangle\,ngle\!\,\rangle\,ngle\,,
\end{equation}
which satisfies $\{P,Q\}=(-1)^{w+1}\{Q,P\}$.
Part a) of the following
theorem generalizes Haberland's formula \cite{H,KZ}. Part b) follows easily from the
proof of part a), although to our knowledge it has not appeared previously in the literature
(except for $\Gamma_1$ in \cite{Po}, but the proof there is more complicated).
\mathfrak{b}egin{theorem}\,\langle\,bel{thm_hab} (a) For $f,g\in S_k(\Gamma)$, we have
\[
6 C_{ k} \cdot(f,g)=\{ \rho_f , \overline{\rho_g}\},
\]
where complex conjugation acts coefficientwise on polynomials and $C_{k}= -(2i)^{k-1}$.
(b) For $f,g\in S_k(\Gamma)$, we have $\{ \rho_f , \rho_g\}=0$.
\end{theorem}
\mathfrak{b}egin{proof}
(a) The proof is based on Stokes' theorem, as in
\cite{KZ}, except that we apply it to a fundamental domain for $\Gamma(2)$, namely the
quadrilateral region $\mathcal{D}$ with vertices $i\infty, -1, 0, 1$ and with sides the geodesics
connecting the points in that order. The region $\mathcal{D}$ consists of six copies of the fundamental
domain for $\Gamma_1$, which explains the constant 6 appearing in the formula. Therefore we
have:
\mathfrak{b}egin{equation}\,\langle\,bel{3.1}
\mathfrak{b}egin{aligned}
6 C_{k}[{\overline{\Gamma}}_1:{\overline{\Gamma}}]\cdot (f,g) &= 6 \int_{\Gamma\mathfrak{b}ackslash \mathcal{H}} f(z)
\overline{g(z)}(z-\overline{z})^w
dz d\overline{z}\\
&= \sum_{A\in{\overline{\Gamma}} \mathfrak{b}ackslash {\overline{\Gamma}}_1} \int_\mathcal{D} f|A(z) \overline{g|A} (z)
(z-\overline{z})^w dz d\overline{z}\\
&=\sum_{A\in{\overline{\Gamma}} \mathfrak{b}ackslash {\overline{\Gamma}}_1} \int_{\mathfrak{p}artial\mathcal{D}} F_A(z) \overline{g|A} (z)
d\overline{z}
\end{aligned}
\end{equation}
where $F_A(z)=\int_{i\infty}^{z} f|_k A(t)(t-\overline{z})^w dt$, so that
${\scriptscriptstyle F}rac{\mathfrak{p}artial F_A}{\mathfrak{p}artial z}=f|A(z) (z-\overline{z})^w$, and the last line follows from
Stokes' theorem. For $B\in \mathrm{SL}_2(\mathbb{Z})$ a change of variables shows that
\mathfrak{b}egin{equation}\,\langle\,bel{3.2}
j(B, \overline{z})^w F_A (Bz)=F_{AB}(z) - \int_{i\infty}^{B^{-1} i\infty} f|AB
(t)(t-\overline{z})^w dt.
\end{equation}
We denote by $\int_a^b$ the integral over the geodesic arc from the cusp
$a$ to the cusp $b$. A change of variables $z=T^2 \tau$ and \eqref{3.2}
yields:
\[
\int_{1}^{i\infty} F_A(z) \overline{g|A} (z)
d\overline{z} = \int_{-1}^{ i\infty} F_{A T^2}(\tau) \overline{g|AT^2} (\tau)
d\overline{\tau},
\]
and it follows that the sum of integrals over the vertical sides of $\mathcal{D}$ vanishes.
A change of variables $z=S\tau$ and \eqref{3.2} yields:
\[
\int_{-1}^{0} F_A(z) \overline{g|A} (z)
d\overline{z} = \int_{1}^{i\infty} F_{A S}(\tau) \overline{g|A S} (\tau)
d\overline{\tau} + \int_{1}^{i \infty} \int_0^{i\infty} f|AS (t) (t-\overline{\tau})^w
\overline{g|AS(\tau)} dt d\overline{\tau}.
\]
\[
\int_{0}^{1} F_A(z) \overline{g|A} (z)
d\overline{z} = \int_{i\infty}^{-1} F_{A S}(\tau) \overline{g|A S} (\tau)
d\overline{\tau} - \int_{-1}^{i \infty} \int_0^{i\infty} f|AS(t) (t-\overline{\tau})^w
\overline{g|AS(\tau)} dt d\overline{\tau}.
\]
When adding the last two equations and summing over $A \in{\overline{\Gamma}} \mathfrak{b}ackslash {\overline{\Gamma}}_1$,
the single integrals cancel as before and \eqref{3.1} becomes
\[
6 C_{k} [{\overline{\Gamma}}_1:{\overline{\Gamma}}]\cdot (f,g)=\sum_{A\in{\overline{\Gamma}} \mathfrak{b}ackslash {\overline{\Gamma}}_1}
\int_{1}^{i \infty}
\int_0^{i\infty}- \int_{-1}^{i \infty} \int_0^{i\infty}
f|A (t) (t-\overline{\tau})^w \overline{g|A(\tau)} dt d\overline{\tau}.
\]
To write the double integrals in terms of the period polynomial pairing, we use \eqref{3.3}.
After a change of variables $\tau=Tz$ the first double
integral becomes
\[
\int_{0}^{i \infty} \int_0^{i\infty}f|A (t) \leqslantft\,\langle\,ngle (t-X)^w, (\overline{Tz}-X)^w
\right\,\rangle\,ngle
\overline{g|AT(z)}\ d t d\overline{z}=\,\langle\, \rho_f(A), \overline{\rho_g}
(AT)|T^{-1}\,\rangle\, .
\]
The second integral yields the same result, with $T$ replaced by $T^{-1}$, and
the conclusion follows from the fact that the pairing $\,\,\langle\,ngle\!\,\langle\,ngle\, , \,\,\rangle\,ngle\!\,\rangle\,ngle\,$ is $\Gamma_1$ invariant.
\noindent (b) Going backwards in the proof of part (a) we have:
\[
\{\rho_f, \rho_g\} = {\scriptscriptstyle F}rac{1}{[{\overline{\Gamma}}_1:{\overline{\Gamma}}]} \sum_{A\in{\overline{\Gamma}} \mathfrak{b}ackslash {\overline{\Gamma}}_1} \int_{\mathfrak{p}artial\mathcal{D}}
H_A(z) g|A (z)
d z
\]
where $H_A(z)=\int_{i\infty}^{z} f|_k A(t)(t-z)^w dt=-\widetilde{f}(A)(z)$. Since the integrand is now
holomorphic and vanishes exponentially at the cusps, it follows that each integral above
vanishes.
\end{proof}
Now let $\Gamma$ be a congruence subgroup normalized by $\epsilon$. The pairing $\{\cdot,
\cdot \}$ satisfies
\mathfrak{b}egin{equation}\,\langle\,bel{conj1}
\{P|\epsilon, Q|\epsilon\} = (-1)^{w+1}\{P,Q\},
\end{equation}
hence $\{P,Q\}=0$ if $k$ is even and $P, Q\in V_w^\Gamma$ have the same parity, or if $k$ is
odd and $P,Q$ have opposite parity. We have the following stronger
version of Haberland's theorem, generalizing the result for the full modular group from
\cite{Po}.
\mathfrak{b}egin{theorem} \,\langle\,bel{thm_main}
Let $\Gamma$ be a subgroup of finite index in $\Gamma_1$, normalized by $\epsilon$. For $f,g\in
S_k(\Gamma)$:
\[
3 C_{k}\cdot (f,g)=\{\rho_f^{\kappa_1} , \overline{\rho_g^{\kappa_2}}\}
\]
for any $\kappa_1, \kappa_2\in\{+, -\}$ with $\kappa_1\ne \kappa_2$ if $k$ even
and $\kappa_1=\kappa_2$ if $k$ odd.
\end{theorem}
\mathfrak{b}egin{proof} We assume $k$ even, the case $k$ odd being entirely
similar. In view of Theorem \ref{thm_hab} (a) and \eqref{conj1}, it is enough to show that
$\{\rho_f^+ , \overline{\rho_g^-}\}=\{\rho_f^- , \overline{\rho_g^+}\}$. By \eqref{2.1}, we
have $\overline{\rho_g^+}=(-1)^{w+1}\rho_{g^*}^+$, $\overline{\rho_g^-}=(-1)^w\rho_{g^*}^-$,
and the previous
equality reduces to $\{\rho_f , \rho_{g^*}\}=0$, which is Theorem \ref{thm_hab} b).
\end{proof}
\section{Coboundary polynomials}\,\langle\,bel{sec_cw}
In this section we show that the space of coboundary polynomials
$$C_w^\Gamma=\{P|(1-S): P \in V_w^\Gamma \cap \ker(1-T) \}
$$ is the radical
of the bilinear form $\{\cdot, \cdot\}$ on $W_w^\Gamma$. The dimension of $C_w^\Gamma$ equals
the dimension of the Eisenstein subspace $\mathcal{E}E_k(\Gamma)\subset M_k(\Gamma)$. For
$\Gamma=\Gamma_0(N)$, we characterize those $N$ for which
$(C_w^\Gamma)^-$ is trivial, namely those $N$ for which the map $\rho^-:S_k(\Gamma)\rightarrow (W_w^\Gamma)^-$ is
an isomorphism, as in the full level case.
\mathfrak{b}egin{lemma}\,\langle\,bel{l4.4} Let $\Gamma$ be a finite index subgroup of $\Gamma_1$. The period
polynomials $W_w^\Gamma$ are
orthogonal to the coboundary polynomials $C_w^\Gamma\subset W_w^\Gamma$ with respect to the
pairing $\{\cdot , \cdot \}$.
\end{lemma}
\mathfrak{b}egin{proof}
Let $P|(1-S)\in C_w^\Gamma$ with $P|1-T=0$, and let $Q\in W_w^\Gamma$. Then $\,\,\langle\,ngle\!\,\langle\,ngle\,
P|(1-S)(T-T^{-1}),Q \,\,\rangle\,ngle\!\,\rangle\,ngle\,=0$ follows from the following
relation in $\mathbb{Z}[{\overline{\Gamma}}_1]$
\mathfrak{b}egin{equation}\,\langle\,bel{4.8}
(1-S)(T-T^{-1})=(T-1)(2+T^{-1}+ST-S)+(1+TS+ST^{-1})(1-S)
\end{equation}
using the $\Gamma_1$ invariance of the pairing $\,\,\langle\,ngle\!\,\langle\,ngle\,\cdot , \cdot \,\,\rangle\,ngle\!\,\rangle\,ngle\,$ (recall $U=TS$).
\end{proof}
Let $e_\infty(\Gamma)$, $e_\infty^\mathrm{reg}(\Gamma)$ denote the number of inequivalent cusps,
respectively regular cusps \cite[Ch. 3]{DS}. The next lemma shows that $\dim
C_w^\Gamma=\dim \mathcal{E}E_k(\Gamma)$.
\mathfrak{b}egin{lemma} \,\langle\,bel{L7.1}
Let $\Gamma$ be any finite index subgroup of $\Gamma_1$. The dimension of $C_w^\Gamma$ equals:
$e_\infty(\Gamma)$ if $k>2$ is even; $e_\infty(\Gamma)$-1 if $k=2$; $e_\infty^\mathrm{reg}(\Gamma)$ if
$k>2$ is odd.
\end{lemma}
\mathfrak{b}egin{proof}
Let $P\in V_w^\Gamma \cap \ker(1-T)$.
Then $P|T^n= P$ for every $n\in \mathbb{Z}$, that is $P(A T^{-n})|T^n =P(A)$ for $A\in
\Gamma\mathfrak{b}ackslash\Gamma_1$. Since $\Gamma\mathfrak{b}ackslash\Gamma_1$ is finite, there is $n$ such
that $AT^{-n}=A$, and $P(A)(X+n)=P(A)(X)$. Since the only periodic polynomials are the
constants, it follows that $P(A)(X)=c_A\in \mathbb{C}$, with $c_{AT}=c_A$. Since $P|J=P$ we also
have $c_{AJ}=(-1)^w c_A$.
Hence we have:
\mathfrak{b}egin{equation}\,\langle\,bel{cw}
C_w^\Gamma=\{(c_A-c_{AS^{-1}}X^w)_A\in V_w^\Gamma: c_A\in \mathbb{C}, \ c_{AT}=c_A, \ c_{AJ}=(-1)^w c_A
\}.
\end{equation}
If $k>2$ is even it follows that $ \dim C_w^\Gamma=|\Gamma\mathfrak{b}ackslash \Gamma_1/
\Gamma_{1\infty}|$,
with $\Gamma_{1\infty}=\{\mathfrak{p}m T^n: n\in \mathbb{Z}\}$ the stabilizer of $\infty$. Since the map
\[
\Gamma\mathfrak{b}ackslash \Gamma_1/ \Gamma_{1\infty}\rightarrow \Gamma \mathfrak{b}ackslash \mathbb{P}^1(\mathbb{Q}),
\quad [\mathfrak{g}ammamma]\rightarrow [\mathfrak{g}ammamma\infty]
\]
is a bijection and $|\Gamma \mathfrak{b}ackslash \mathbb{P}^1(\mathbb{Q})|=e_\infty(\Gamma)$, the claim
follows.
If $k=2$, we identify $\mathbb{C}^{e_\infty(\Gamma)}$ with the vector space $\{(c_A)_{A\in
\Gamma\mathfrak{b}ackslash \Gamma_1}: c_{AT}=c_A=c_{AJ}\in\mathbb{C} \}$ and define the (surjective) map
$\mathbb{C}^{e_\infty(\Gamma)}\rightarrow C_w^\Gamma$ by $(c_A)_A \rightarrow (c_A-c_{AS})_A $. Its
kernel consists of those vectors $(c_A)_A$ with $c_{AT}=c_{AS}=c_{AJ}=c_{A}$. Since $S,T,J$
generate $\Gamma_1$, it follows that $c_A=c$ for all $A\in \Gamma\mathfrak{b}ackslash\Gamma_1$, so the
kernel is isomorphic to $\mathbb{C}$. Therefore $ \dim C_w^\Gamma=e_\infty(\Gamma)-1$.
If $k>2$ is odd (so $-1\notin \Gamma$), we have $c_{AJ}=-c_A$. Therefore $\dim C_w^\Gamma$
equals the number of classes $[A] \in \Gamma\mathfrak{b}ackslash \Gamma_1/\Gamma_{1\infty}$, such that
the two associated classes $[A]^+, [AJ]^+\in \Gamma\mathfrak{b}ackslash\Gamma_1/\Gamma_{1\infty}^+$ are
distinct, where $\Gamma_{1\infty}^+=\{T^n:n\in \mathbb{Z}\}$ (when $[A]^+=[AJ]^+$ then clearly
$c_A=c_{AJ}=0$). But $[A]^+\ne[AJ]^+$ precisely when $[A]$ corresponds to a regular cusp of
$\Gamma$ since $[A]^+=[AJ]^+$ means that $A^{-1}\mathfrak{g}ammamma A=-T^n$ for some $\mathfrak{g}ammamma\in \Gamma$,
$n>1$, so the cusp $A\infty$ is irregular. We conclude that $\dim
C_w^\Gamma=e_\infty^\mathrm{reg}(\Gamma)$.
\end{proof}
\mathfrak{b}egin{definition}\,\langle\,bel{d_cusp}
Following the proof of Lemma \ref{L7.1}, we call \emph{cusp} a double coset $\mathcal{C}\in
\Gamma\mathfrak{b}ackslash
\Gamma_1/\Gamma_{1\infty}$, which corresponds to the $\Gamma$-equivalence class of the usual
cusp $A\infty$ for any representative $A\in\mathcal{C}$. We call \emph{regular} those
cusps $\mathcal{C}=[A]$ such that the double cosets $[A]^+, [AJ]^+ \in
\Gamma\mathfrak{b}ackslash\Gamma_1/\Gamma_{1\infty}^+$ are distinct. The terminology agrees with the
usual one for the cusp $A\infty$ by the last paragraph in the proof of the lemma.
\end{definition}
For $\Gamma_0(N)$ it turns out that $(C_w^\Gamma)^-$ is often
trivial, in which case $\rho^-:S_k(\Gamma)\rightarrow (W_w^\Gamma)^-$ is an isomorphism just like for $\Gamma_1$.
The following proposition was discovered using SAGE \cite{Sg}.
\mathfrak{b}egin{prop}\,\langle\,bel{prop7.2} Let $\Gamma=\Gamma_0(N)$. Then $(C_w^\Gamma)^-=\{0\}$
if and
only if $N=2^e N'$ with $N'$ odd square free and $0\leqslant e\leqslant 3$.
\end{prop}
\mathfrak{b}egin{proof} From the proof of Lemma \ref{L7.1} we identify $(C_w^\Gamma)^-$ with the
space $(\mathbb{C}^{e_\infty(\Gamma)})^-$ of vectors $(c_A)_{A\in{\overline{\Gamma}}\mathfrak{b}ackslash{\overline{\Gamma}}_1}$ with $c_A=c_{AT}$
and $c_A=-c_{A'}$ (including for $k=2$).
Assume $N$ does not satisfy the conditions, so there exists $t\mathfrak{g}e 3$ with
$t^2|N$. We claim that $[A]\ne [A']$ for $A= \leqslantft(\mathfrak{b}egin{smallmatrix} x & y \\
t & z \end{smallmatrix}\right)\in \Gamma_1$, so $(C_w^\Gamma)^-\ne\{0\}$. Assuming by
contradiction that $\mathfrak{g}ammamma A T^s=A'$ for $\mathfrak{g}ammamma=\leqslantft(\mathfrak{b}egin{smallmatrix} * & * \\
c & d \end{smallmatrix}\right)\in \Gamma$, it follows that $cx+dt=-t, c(y+sx)+d(z+st)=z$. The
first equation implies that $t|d+1$ while the second that $t|d-1$, a contradiction with $t\mathfrak{g}e
3$.
Assuming $N$ satisfies the conditions, let $(c_A)_{A\in{\overline{\Gamma}}\mathfrak{b}ackslash{\overline{\Gamma}}_1}$ with $c_A=c_{AT}$
and $c_A=-c_{A'}$. Identifying the coset space $\Gamma_0(N)\mathfrak{b}ackslash\Gamma_1$ with
$\mathbb{P}^1(\mathbb{Z}/N\mathbb{Z})$, it follows that
$c_{(a:b)}=c_{(a:a+b)}, \ c_{(a:b)}=-c_{(-a:b)}.
$
The second relation implies $c_{(0:1)}=0$. Let $N=d d'$ and $k\in \mathbb{Z}$, $(k,d)=1$. We will
show that $c_{(d:k)}=0$. We have:
\mathfrak{b}egin{equation*}
c_{(d:k)} =c_{(d:k+ad)} = -c_{(-d:k)}=-c_{((bd'-1)d: k)}
\end{equation*}
and it is enough to find $ a,b\in \mathbb{Z}$ with $(bd'-1,N)=1$ and $k\equiv (bd'-1)(k+ad)
\mathfrak{p}mod{N}$. The latter equation can be written
$ k(bd'-2)\equiv ad \mathfrak{p}mod{d d'}$
and the hypothesis on $N$ ensures that $(d,d')|2$, so that we can find $b,u$ such that
$bd'-2=du$. Taking $a\equiv ku \mathfrak{p}mod{d'}$, it follows that $(d:k+ad)=((bd'-1)d:k)$, which
implies that $c_{(d:k)}=0$. It follows that $c_A=0$ for all $A\in {\overline{\Gamma}}\mathfrak{b}ackslash{\overline{\Gamma}}_1$,
finishing the proof.
\end{proof}
\section{Hecke operators}\,\langle\,bel{sec4}
Following the Eichler integral method sketched in \cite{Za93} for the modular group, we show that the same
elements $\widetilde{T}_n$ as in the full level case define actions on period polynomials corresponding to a large class of
double coset operators on modular forms, including the Hecke operators and Atkin-Lehner operators. On modular
symbols, the action of the same type of double coset operators was determined by Merel, and the results of
$\S$\ref{sec5.11} parallel those of \cite{Me}. We find the adjoints of these operators with respect to the pairing
$\{\cdot ,\cdot\}$ in $\S$\ref{sec5.12}.{\scriptscriptstyle F}ootnote{For $\Gamma_1$, the Hecke equivariance of the pairing is
mentioned without proof in \cite[p.96]{GKZ}.} In $\S$\ref{sec5.1} we give a new proof of the rationality of the
plus and minus parts of period polynomials of newforms, while in $\S$\ref{sec5.2} we discuss modular forms with
Nebentypus. We end with a discussion of the numerical computation of period polynomials,
Hecke eigenvalues, and Petersson norms of newforms.
\subsection{The universal Hecke operators}\,\langle\,bel{sec5.11}
Let $M_n$ be the set of integer matrices of determinant $n$, set $\ov{M}_n=M_n/\{\mathfrak{p}m I\}$, and let
$R_n=\mathbb{Q}[\ov{M}_n]$. Thus ${\overline{\Gamma}}_1$ acts on $R_n$ by left and right multiplication. Let
$$M_n^\infty=\mathfrak{b}ig\{\leqslantft(\mathfrak{b}egin{smallmatrix} a & b \\
0 & d \end{smallmatrix}\right):n=ad, 0\leqslant b<d\mathfrak{b}ig\}$$ be the usual system of representatives for
$\Gamma_1\mathfrak{b}ackslash M_n$, and $T_n^\infty=\sum_{M\in M_n^\infty} M \in R_n$. Following
\cite{CZ}, let $\widetilde{T}_n, Y_n\in R_n$ be such that
\mathfrak{b}egin{equation}\,\langle\,bel{hecke}
T_n^{\infty}(1-S)=(1-S)\widetilde{T}_n+(1-T)Y_n .
\end{equation}
We will show that for all $n$, the elements $\widetilde{T}_n$ define Hecke operators on period
polynomials for a large class of congruence subgroups. Note that the elements $\widetilde{T}_n$ are universal, not depending
on the weight or level.
Let $\Gamma$ be a finite index subgroup of $\Gamma_1$ normalized by $\epsilon$, which we will often specialize to
be $\Gamma_1(N)$ or $\Gamma_0(N)$. Fix an integer $n\mathfrak{g}e 1$, and let $\Sigma_n\subset M_n$ such that
$\Sigma_n=\Gamma \Sigma_n =\Sigma_n \Gamma$ and $\Sigma_n$
is a disjoint finite union of cosets $\Gamma \sigma$. Such double cosets $\Sigma_n$ define
operators $[\Sigma_n]$ on $f\in M_k(\Gamma)$
\[
f|[\Sigma_n]=n^{w+1} \sum_{\sigma \in \Gamma \mathfrak{b}ackslash \Sigma_n} f|_k \sigma.
\]
Setting $g^\vee=g^{-1}\deltat g$, the adjoint of $[\Sigma_n]$ with respect to the Petersson product is given by
$\Sigma_n^\vee=\{g^\vee: g\in \Sigma_n\}$, namely $(f|[\Sigma_n], g)=(f,g|[\Sigma_n^\vee] )$.
We now define an action of $M_n$ on $V_w^\Gamma$ which depends on $\Sigma_n$, and which is based on
the following property of the pair $(\Gamma,\Sigma_n)$
\mathfrak{b}egin{equation} \,\langle\,bel{eq_star} \tag{H}
\text{The map $\Gamma\mathfrak{b}ackslash
\Sigma_n\rightarrow \Gamma_1\mathfrak{b}ackslash\Gamma_1\Sigma_n, \ \ \ $ $\Gamma \sigma \mapsto \Gamma_1 \sigma $
is bijective.}
\end{equation}
See \cite[Prop. 3.36]{Sh} for a large class of congruence subgroups when this property holds.
For $M\in M_n$, $A\in \Gamma_1$ and $M A^{-1}\in \Gamma_1 \Sigma_n
$, there exists a decomposition $M A^{-1}=A_M^{-1} M_A$, with
$M_A\in \Sigma_n$, $A_M\in \Gamma_1$, and by \eqref{eq_star} the coset $\Gamma A_M$ is independent of the
decomposition. Moreover $\Gamma A_M$ depends only on $\Gamma A$, since $\Gamma \Sigma_n=\Sigma_n \Gamma$.
For $P\in V_w^\Gamma$ we define an element $P|_{\Sigma}M =P|_{\Sigma_n}M\in V_w^\Gamma$
by{\scriptscriptstyle F}ootnote{With the notation $P|_{\Sigma}M$, the dependence on $n$ is recorded in the fact that $M\in M_n$.}
\mathfrak{b}egin{equation}\,\langle\,bel{eq_act}
P|_{\Sigma}M(A)=\mathfrak{b}egin{cases} P(A_M)|_{-w}M & \text{ if } M A^{-1} \in \Gamma_1 \Sigma_n \\
0 & \text{ otherwise.}
\end{cases}
\end{equation}
Note that both $M$ and $JM$ act in the same way (recall $J=-I$),
so the action of $\ov{M}_n=M_n/ \{\mathfrak{p}m I\}$ is also well defined. This action extends linearly to an action of
$R_n$ on $V_w^\Gamma$. In the same way we define an action of $\ov{M}_n$ and $R_n$ on the Eichler integrals $\widetilde{f}$
in \eqref{eq_eich}.
We are interested in the following double cosets $\Sigma_n$ satisfying \eqref{eq_star}, for $\Gamma=
\Gamma_1(N)$ or $\Gamma_0(N)$.
\mathfrak{b}egin{enumerate}
\item[(1)] The double coset $\mathcal{D}elta_n$ consisting of $\leqslantft(\mathfrak{b}egin{smallmatrix} a & b \\
c & d \end{smallmatrix}\right)\in M_n$ with $N|c$, and $a\equiv 1 \mathfrak{p}mod{N}$ if $\Gamma=\Gamma_1(N)$, or $(a,N)=1$
if $\Gamma=\Gamma_0(N)$. The operator $[\mathcal{D}elta_n]$ is the usual
Hecke operator, denoted by $T_n$, and Property \eqref{eq_star} follows from \cite[Prop. 3.36]{Sh}. Note that for
$(n,N)=1$ we have $\Gamma_1 \mathcal{D}elta_n=M_n$, so the second case in \eqref{eq_act} does not occur.
\item[(2)] The double coset $\mathcal{D}elta_n^\vee$ for $(n,N)=1$. The operator $[\mathcal{D}elta_n^\vee]$ is denoted by $T_n^*$,
the adjoint of
$T_n$ with respect to the Petersson inner product. Property \eqref{eq_star} can be checked directly.
\item[(3)] The double coset $\Theta_n=\Gamma w_n \Gamma $ for $N=n n'$, $(n,n')=1$, where
$w_n=\leqslantft(\mathfrak{b}egin{smallmatrix} nx & y \\
Nz & nt \end{smallmatrix}\right)\in M_n$ with $x,y,z,t\in \mathbb{Z}$ for $\Gamma=\Gamma_0(N)$, while for
$\Gamma=\Gamma_1(N)$ we impose the extra conditions $nx\equiv 1 \mathfrak{p}mod{n'}$, $y\equiv 1 \mathfrak{p}mod{n}.$ Then
$\Gamma\mathfrak{b}ackslash \Theta_n$ has one element so Property \eqref{eq_star} is trivially satisfied. The operator
$[\Theta_n]$ is denoted by $W_n$, and if $W_n^*=[\Theta_n^\vee]$ denotes its adjoint we have $f|W_n|W_n^*=n^w f$.
For $\Gamma=\Gamma_0(N)$ (so that $W_n=W_n^*$), the usual Atkin-Lehner
involution is $W_n/n^{w/2}$ due to our choice of normalization.
\item[(4)] The double coset $\Theta_n^\vee$ with $\Theta_n$ as in (3). If $\Gamma=\Gamma_0(N)$ then
$[\Theta_n^\vee]=[\Theta_n]$,
and if
$\Gamma=\Gamma_1(N)$ then $[\Theta_n^\vee]=\,\langle\, d \,\rangle\,[\Theta_n]$ for $d\equiv -1
\mathfrak{p}mod{n}$, $nd\equiv 1 \mathfrak{p}mod{n'}$, where $\,\langle\, d\,\rangle\,$ denotes the diamond operator.
\end{enumerate}
\mathfrak{b}egin{remark}\,\langle\,bel{r5.1} Assume $\Gamma=\Gamma_1(N)$ and $\Sigma_n$ is as in (1) or (3). The action of $M_n$
on $V_w^\Gamma$ can be determined as follows. Applying adjoint we have that
$MA^{-1}\in\Gamma_1 \Sigma_n$ if and only if
$$
\leqslantft(\mathfrak{b}egin{smallmatrix} a & b \\
c & d \end{smallmatrix}\right)=AM^\vee=M_A^\vee A_M \in \Sigma_n^\vee\Gamma_1.
$$
For $\Sigma_n=\mathcal{D}elta_n$, the inclusion is satisfied if and only if $\mathfrak{g}cd(c,d,N)=1$, and in that case
$A_M= \leqslantft(\mathfrak{b}egin{smallmatrix} * & * \\
c' & d' \end{smallmatrix}\right)\in \Gamma_1$ has $c'\equiv c, d'\equiv d\mathfrak{p}mod{N}$, which
uniquely defines its class in $\Gamma\mathfrak{b}ackslash \Gamma_1$.
For $\Sigma_n=\Theta_n$, let $N=n n'$ with $n\|N$. The inclusion is satisfied if and only if $n|c, n|d$, and
the class of $A_M=\leqslantft(\mathfrak{b}egin{smallmatrix} * & * \\
c' & d' \end{smallmatrix}\right)$ in $\Gamma\mathfrak{b}ackslash \Gamma_1$ is determined by
\[c'\equiv -a ,\ d'\equiv -b \mathfrak{p}mod{n}, \quad c'\equiv c/n ,\ d'\equiv d/n \mathfrak{p}mod{n'}.
\]
When $\Gamma=\Gamma_0(N)$ and $\Sigma_n=\Theta_n$, the last congruences are replaced by $yc'\equiv a, yd'\equiv b
\mathfrak{p}mod{n}$ and $tc'\equiv c/n ,\ td'\equiv d/n \mathfrak{p}mod{n'}$ where $t,y$ are any integers such that $nxt-n'yz=1$
(taking $M_A^\vee=\leqslantft(\mathfrak{b}egin{smallmatrix} nx & y \\
Nz & nt \end{smallmatrix}\right)\in M_n$). The class of $(c',d')$ in $P^1(\mathbb{Z}/N\mathbb{Z})\sim \Gamma\mathfrak{b}ackslash\Gamma_1$ is
then independent of $y,t$.
\end{remark}
While the operation $P|_{\Sigma}M$ is not a proper action, it is compatible with the action of $\Gamma_1$ and of
$\epsilon$. For
$h\in \Gamma_1$, $M\in M_n$ a formal computation shows that
\mathfrak{b}egin{equation}\,\langle\,bel{Ecom}
P|_{\Sigma}M|h=P|_{\Sigma}Mh, \ \ P|h|_{\Sigma} M=P|_{\Sigma}hM, \ \
P|_{\Sigma}M|\epsilon=P|\epsilon|_{\Sigma'}\epsilon M\epsilon,
\end{equation}
where $\Sigma_n'=\{g'=\epsilon g\epsilon : g\in \Sigma_n \}$. Using the compatibility with the $\Gamma_1$ action,
we
show that the universal operators $\widetilde{T}_n$ play on period polynomials the role of \emph{all} double coset operators
$[\Sigma_n]$ on modular forms, as long as the pair $(\Gamma, \Sigma_n)$ satisfies \eqref{eq_star}.
\mathfrak{b}egin{prop} \,\langle\,bel{p4.2}
Assume the pair $(\Gamma, \Sigma_n)$ satisfies \eqref{eq_star}. If
$\widetilde{T}_n\in R_n$ satisfies \eqref{hecke} then for every $f\in S_k(\Gamma)$
$$\rho_{f|[\Sigma_n]}=\rho_f |_{\Sigma} \widetilde{T}_n.$$
\noindent In particular for $\Gamma=\Gamma_1(N)$, we have $\rho_{f|T_n}=\rho_f |_{\mathcal{D}elta} \widetilde{T}_n$ and, if $n\|N$,
$\rho_{f|W_n}=\rho_f |_{\Theta} \widetilde{T}_n$.
\end{prop}
\mathfrak{b}egin{proof} Using the fact that $\rho_f=\widetilde{f}|(1-S)$
and $\tilde{f}|(1-T)=\sigma_f(T)=0$, we have as in \cite{Za93}
\[
\rho_f|_{\Sigma}\widetilde{T}_n = \widetilde{f}|_{\Sigma}(1-S)\widetilde{T}_n=
\tilde{f}|_{\Sigma}T_n^{\infty}(1-S)=\widetilde{f|[\Sigma_n]}|(1-S)=\rho_{f|[\Sigma_n]},
\]
once we show that $\tilde{f}|_{\Sigma}T_n^{\infty}= \widetilde{f| [\Sigma]}$.
For $M\in M_n^\infty$ and $A\in\Gamma_1$, let $M A^{-1}=A_M^{-1} M_{A}$ with
$A_M\in\Gamma_1$, $M_{A}\in \Sigma_n$. By \eqref{eq_act}
\mathfrak{b}egin{equation}\,\langle\,bel{4.3}
\mathfrak{b}egin{split}
\widetilde{f}|_{\Sigma}T_n^{\infty}(A)&=\sum_{M\in M_n^\infty\cap \Gamma_1 \Sigma_n A }\int_{M z}^{i \infty} f|A_M
(t)(t-Mz)^w j(M,z)^w dt\\
&=n^{w+1} \sum_{M\in M_n^\infty \cap \Gamma_1 \Sigma_n A}\int_{z}^{i \infty}f|M_{A} A
(u)(u-z)^w du
\end{split}
\end{equation}
where we made a change of variables $t=M u$. For fixed $A$, the map
$$ M_n^\infty\cap \Gamma_1 \Sigma_n A\rightarrow \Gamma\mathfrak{b}ackslash\Sigma_n, \quad M\mapsto M_{A}$$
is well defined by \eqref{eq_star}. It is easy to check that the map is bijective,
hence the last expression equals $\widetilde{f|[\Sigma_n]} (A)$ finishing the proof.
\end{proof}
\mathfrak{b}egin{corollary}\,\langle\,bel{c4.3} If in addition to the hypotheses of Proposition \ref{p4.2} we assume that $\Gamma$ is
normalized by $\epsilon$ and that $\Sigma_n'=\Sigma_n$ then $$\rho_{f|[\Sigma_n]}^\mathfrak{p}m=\rho_f^\mathfrak{p}m |_{\Sigma}
\widetilde{T}_n.$$
In particular if $\Gamma=\Gamma_1(N)$, then
$\rho_{f|T_n}^\mathfrak{p}m=\rho_f^\mathfrak{p}m |_{\mathcal{D}elta} \widetilde{T}_n$, and if $\Gamma=\Gamma_0(N)$, then
$\rho_{f|W_n}^\mathfrak{p}m=\rho_f^\mathfrak{p}m |_{\Theta} \widetilde{T}_n$.
\end{corollary}
\mathfrak{b}egin{proof} By the last compatibility in \eqref{Ecom}, it is enough to show that $\epsilon \widetilde{T}_n \epsilon$ also
satisfied \eqref{hecke} (for a different $Y_n$).
This follows from conjugating \eqref{hecke} by $\epsilon$, and using that
$T_n^{\infty}-\epsilon T_n^{\infty}\epsilon \in (1-T) R_n$.
We have $\mathcal{D}elta_n'=\mathcal{D}elta_n$ and, for $\Gamma=\Gamma_0(N)$, $\Theta_n^{\mathfrak{p}rime}=\Theta_n$. Note that if
$\Gamma=\Gamma_1(N)$, then $[\Theta_n^{\mathfrak{p}rime}]= \,\langle\, d \,\rangle\, [\Theta_n]$ with $d\equiv -1\mathfrak{p}mod{n}$, $d\equiv
1\mathfrak{p}mod{n'}$, so the action $|_{\Theta} \widetilde{T}_n$ on $(W_w^\Gamma)^\mathfrak{p}m$ does not correspond to $W_n$ in this
situation.
\end{proof}
Taking $\Sigma_n=\mathcal{D}elta_n$ in Corollary \ref{c4.3}, we obtain explicit formulas for the Fourier coefficients of a
Hecke eigenform $f\in S_k(\Gamma)$ in terms of the polynomials $\rho_f^\mathfrak{p}m$, generalizing the Coefficients Theorem
of
Manin \cite{M73}. The formulas can be used for fast computation of Hecke eigenvalues, as explained in
$\S$\ref{sec5.3}, and they also yield an explicit inverse of the Eichler-Shimura maps $\rho^\mathfrak{p}m$ of Theorem
\ref{thm2.1}.
We state the formula corresponding to $\rho_f^+$. For $\Gamma=\Gamma_1(N)$ we identify a coset representative
$A\in \Gamma\mathfrak{b}ackslash \Gamma_1$ with the element $ (c_A,d_A)\in E_N$, where $c_A$, $d_A$ are the lower left,
respectively lower right
entries of $A$ and
\[
E_N=\mathfrak{b}ig\{(c,d) \in (\mathbb{Z}/N\mathbb{Z})^2 : (c,d,N)=1\mathfrak{b}ig\}.
\]
For $P\in W_w^\Gamma$, we write $P(A)=P(c_A,d_A)\in V_w$.
\mathfrak{b}egin{prop}\,\langle\,bel{p5.4}
Let $f\in S_k(\Gamma_1(N))$ be an eigenform for $T_n$ with eigenvalue $\,\langle\,mbda_n$, and let $\widetilde{T}_n=\sum_{M\in M_n}
\mathfrak{a}lphapha(M) M$ be any operator satisfying \eqref{hecke}.
$\mathrm{(a)}$ Assume $r_{I,w}(f)\ne 0$ (which is satisfied if $k>2$ and $f$ is a newform), and let $P_f^+
=\rho_f^+/r_{I,w}(f)$. Then
\[
\,\langle\,mbda_n=\sum_{\substack{M\in M_n\\(c_M,a_M,N)=1}} \mathfrak{a}lphapha(M) P_f^+(-c_M, a_M)|M(0)
\]
where $M=\leqslantft(\mathfrak{b}egin{smallmatrix} a_M & b_M \\
c_M & d_M \end{smallmatrix}\right) $. When $(n,N)=1$ the congruence condition in the sum can be omitted.
$\mathrm{(b)}$ If $k=2$, let $(x,y)\in E_N$ represent a coset in $ \Gamma\mathfrak{b}ackslash \Gamma_1$ with $\rho_f^+(x,y)
\ne 0$ and let $P_f^+=\rho_f^+/\rho_f^+(x,y)$. For each $M\in M_n$, let $x_M=xd_M-yc_M$, $y_M=-xb_M+ya_M$. Then
\[
\,\langle\,mbda_n=\sum_{\substack{M\in M_n\\(x_M,y_M,N)=1}} \mathfrak{a}lphapha(M) P_f^+(x_M, y_M).
\]
\end{prop}
\mathfrak{b}egin{remark}
By Proposition \ref{p5.7}, when $f$ is a newform the polynomials $P_f^+$ have coefficients in the field $K_f$ of
coefficients of $f$. When $f$ has known Nebentypus, we obtain simpler
formulas using the results of $\S$\ref{sec5.2}.
\end{remark}
\mathfrak{b}egin{proof}
By Corollary \ref{c4.3}, $P_f^+$ is an eigenform of $\widetilde{T}_n$ with eigenvalue $\,\langle\,mbda_n$, and the conclusion
follows by writing the action of $\widetilde{T}_n$ on $P_f^+$ explicitly, using Remark \ref{r5.1}.
\end{proof}
Elements $\widetilde{T}_n$ satisfying condition \eqref{hecke} go back to work of Manin \cite{M73},
and particular examples are given in \cite{CZ,Z90}. The element
$\widetilde{T}_n$ is unique, up to addition of any element in the right $\Gamma_1$-module
\mathfrak{b}egin{equation}\,\langle\,bel{ideal}
\mathcal{I}=(1+S)R_n+(1+U+U^2)R_n.
\end{equation}
\mathfrak{b}egin{remark}\,\langle\,bel{r5.4} In \cite{Me} Merel gives several examples of elements $\widetilde{\mathfrak{T}}_n\in
R_n$ acting on modular symbols, satisfying a condition that plays the same role as \eqref{hecke}. It can
be shown that the elements
$\widetilde{\mathfrak{T}}_n^\vee$ (with the notation explained in the next paragraph) satisfy Property
\eqref{hecke}, reflecting the fact that the action of $\Gamma_1$ on modular symbols is on the left, and on period
polynomials is on the right.
\end{remark}
\subsection{Adjoints of Hecke operators} \,\langle\,bel{sec5.12}
Next we determine the adjoint of the action of Hecke and Atkin-Lehner operators for the pairing on
$W_w^\Gamma$ defined in \eqref{pairing}. For $g\in M_n$ we denote by $g^\vee=g^{-1}\deltat g $
the adjoint of $g$, and we apply this notation to all elements of $R_n$ by linearity.
Recall that $\,\langle\, P|g,Q \,\rangle\,=\,\langle\, P,Q|g^\vee\,\rangle\,$ for $P,Q\in V_w$, $g\in GL_2(\mathbb{C})$.
\mathfrak{b}egin{lemma} Assume that both $(\Gamma,\Sigma_n)$ and $(\Gamma,\Sigma_n^\vee)$
satisfy \eqref{eq_star}. For $P,Q\in V_w^\Gamma, M\in M_n$ we have
\mathfrak{b}egin{equation}\,\langle\,bel{3.4}
\,\,\langle\,ngle\!\,\langle\,ngle\, P|_{\Sigma} M,Q \,\,\rangle\,ngle\!\,\rangle\,ngle\,=\,\,\langle\,ngle\!\,\langle\,ngle\, P,Q|_{\Sigma^\vee} M^\vee\,\,\rangle\,ngle\!\,\rangle\,ngle\, .
\end{equation}
In particular, \eqref{3.4} holds for $\Gamma=\Gamma_1(N)$ and $\Sigma_n=\mathcal{D}elta_n$ with $(n,N)=1$, or
$\Sigma_n=\Theta_n$ for $n\|N$.
\end{lemma}
\mathfrak{b}egin{proof} For $A\in \Gamma_1$ and $M\in \Gamma_1 \Sigma_n A$, let $M A^{-1}=A_M^{-1}
M_A$ with $A_M\in\Gamma_1, M_A\in \Sigma_n$. We have (see Remark \ref{r_sign})
\[
[\Gamma_1:\Gamma] \,\,\langle\,ngle\!\,\langle\,ngle\, P|M,Q \,\,\rangle\,ngle\!\,\rangle\,ngle\,=\sum_{\substack{A\in \Gamma\mathfrak{b}ackslash \Gamma_1\\
MA^{-1}\in \Gamma_1 \Sigma_n}} \,\langle\, P(A_M), Q(A)|M^\vee \,\rangle\,.
\]
Taking adjoint we have
$M^\vee A_M^{-1}=A^{-1}M_A^\vee$ with $M_A^\vee\in \Sigma_n^\vee$, hence $MA^{-1}\in \Gamma_1 \Sigma_n$ if and
only if $M^\vee A_M^{-1}\in \Gamma_1 \Sigma_n^\vee$. Moreover the map $A\mapsto A_M$ is injective, by Property
\eqref{eq_star} applied to $\Sigma_n^\vee$. Summing over $A_M$ instead of $A$ finishes the
proof.
\end{proof}
Next we give two proofs of the Hecke equivariance of the period polynomial pairing, one of
them requiring the following lemma.
\mathfrak{b}egin{lemma}\,\langle\,bel{l4.40}
The space $C_w^\Gamma$ is preserved by the Hecke operators $\widetilde{T}_n$, whenever an
action of $M_n$ on $V_w^\Gamma$ satisfying \eqref{Ecom} can be defined.
\end{lemma}
\mathfrak{b}egin{proof}
Let $P|(1-S)\in C_w^\Gamma$ with $P|1-T=0$. By \eqref{hecke},
$P(1-S)|_{\Sigma}\widetilde{T}_n=P|_{\Sigma}T_n^\infty (1-S)$, and the latter is an element of
$C_w^\Gamma$, since $T_n^\infty(1-T)\in (1-T)R_n$, so $P|_{\Sigma}T_n^\infty(1-T)=0$.
\end{proof}
\mathfrak{b}egin{theorem} \,\langle\,bel{thm_equiv} Assume that both $(\Gamma,\Sigma_n)$ and $(\Gamma,\Sigma_n^\vee)$
satisfy property \eqref{eq_star}. For $P,Q\in W_w^\Gamma$ and any $\widetilde{T}_n$ as
in \eqref{hecke} we have:
\[
\{ P|_{\Sigma}\widetilde{T}_n, Q \} =\{P, Q|_{\Sigma^\vee} \widetilde{T}_n\}.
\]
\end{theorem}
\mathfrak{b}egin{proof} We give two proofs. For the first we assume that both $\Sigma_n, \Sigma_n^\vee$ satisfy
\eqref{eq_star}, and in addition $\Sigma_n=\Sigma_n'$. From Theorem
\ref{thm_main} and Corollary \ref{c4.3}, it follows that the claim is true for $P=\rho_f^{\mathfrak{p}m}$ and
$Q=\rho_g^\mp$, for any $f,g\in S_k(\Gamma)$. By \eqref{7.1}, any $P\in W_w^\Gamma$ can be written $P=\rho_f^+
+\rho_g^- + Q$ with $Q\in C_w^\Gamma$. Since $\widetilde{T}_n$ preserves $(W_w^\Gamma)^\mathfrak{p}m$ and $C_w^\Gamma$, the claim
follows taking into account Lemmas \ref{l4.4}.
The second proof is purely algebraic and we only assume that $\Sigma_n, \Sigma_n^\vee$ satisfy
\eqref{eq_star}. Via \eqref{3.4} the equality to prove is equivalent to
\[\,\,\langle\,ngle\!\,\langle\,ngle\, P|_{\Sigma}\mathfrak{b}ig[\widetilde{T}_n(T-T^{-1})+(T^{-1}-T)\widetilde{T}_n^\vee\mathfrak{b}ig], Q\,\,\rangle\,ngle\!\,\rangle\,ngle\,=0.
\]
Since $P,Q\in W_w^\Gamma$, we are reduced to proving the next theorem.
\end{proof}
\mathfrak{b}egin{theorem}\,\langle\,bel{thm_hecke}
For any element $\widetilde{T}_n\in R_n$ satisfying property \eqref{hecke} we have
\[
\widetilde{T}_n(T-T^{-1})+(T^{-1}-T)\widetilde{T}_n^\vee \in \mathcal{I}+\mathcal{I}^\vee,
\]
where $\mathcal{I}$ is defined in \eqref{ideal}.
\end{theorem}
\noindent The proof is quite involved, and we give it in the short article \cite{PP12a}.
\subsection{Rationality of period polynomials of Hecke eigenforms}\,\langle\,bel{sec5.1}
Let $\Gamma=\Gamma_1(N)$ and for a character $\chi$ modulo $N$ let $S_k(N,\chi)\subset S_k(\Gamma)$ be the
subspace of forms of Nebentypus $\chi$. The
following proposition is well-known, although the precise statement is hard to find in the literature. For the
trivial character, an equivalent statement to part (a) was given in terms of modular symbols in \cite{GS}. The
proof requires the extension of the pairing $\{ \cdot ,\cdot\}$ to the whole space of modular forms and its
properties proved in Section \ref{sec8}.
\mathfrak{b}egin{prop} \,\langle\,bel{p5.7} Assume that $f\in S_k(N, \chi)$ is a newform (a normalized eigenform for all Hecke
operators which does not come from lower levels) and let $K_f$ be the
field of coefficients of $f$.
$\mathrm{(a)}$ There exist nonzero complex numbers $\omega_f^+$ and $\omega_f^{-}$ such that all the polynomial
components of $\rho_f^\mathfrak{p}m/\omegaega_f^{\mathfrak{p}m}$ have coefficients in $K_f$.
$\mathrm{(b)}$ We have
\[ {\scriptscriptstyle F}rac{\omegaega_f^+ \overline{\omegaega_f^-}}{i(2\mathfrak{p}i)^{k-1} (f,f)} \in K_f \text{ if $k$ is even}, \quad \quad
{\scriptscriptstyle F}rac{|\omegaega_f^\mathfrak{p}m|^2 }{(2\mathfrak{p}i)^{k-1} (f,f)} \in K_f \text{ if $k$ is odd}.
\]
In particular, if $k$ is even one can choose $\omegaega_f^\mathfrak{p}m$ such that $\omega_f^+\overline{\omega_f^-}= i (2\mathfrak{p}i )^{k-1}
(f,f)$.
$\mathrm{(c)}$ If $f$ has real Fourier coefficients at infinity, then $\omega_f^+\in i^{k+1} \mathbb{R} $ and $\omega_f^-\in
i^k\mathbb{R}$.
\end{prop}
\mathfrak{b}egin{proof} (a) We view $f\in S_k(\Gamma)$ for $\Gamma=\Gamma_1(N)$. By multiplicity
one and Corollary \ref{c4.3}, the polynomials $\rho_f^\mathfrak{p}m$ are the unique (up to scalars) elements in
$\rho^\mathfrak{p}m(S_k(\Gamma))\subset W_w^\Gamma$ with the same eigenvalues under $|_\mathcal{D}elta\widetilde{T}_n$ as those of $f$.
Moreover on $C_w^\Gamma$ there is no eigenvector for $|_{\mathcal{D}elta}\widetilde{T}_n$ with the same eigenvalues $\,\langle\,mbda_n$ as
those of $f$ for every $n$ coprime to $N$. Indeed, $|\,\langle\,mbda_n|\ll n^{(k-1)/2+\epsilon} $ by the Ramanujan bounds
(even a nontrivial upper bound would suffice), and by Corollary \ref{c8.9} the eigenvalues of $|_\mathcal{D}elta\widetilde{T}_n$ on
$C_w^\Gamma$ are the same as the eigenvalues of an Eisenstein series, which are of order $n^{k-1}$ if $(n,N)=1$.
Therefore the common eigenspace of $|_{\mathcal{D}elta}\widetilde{T}_n$ for $(n,N)=1$ acting on $(W_w^\Gamma)^\mathfrak{p}m$ having the same
eigenvalues as those of $f$ is one dimensional, generated by $\rho_f^\mathfrak{p}m$. Since the matrix of $\widetilde{T}_n$ has
integer coefficients with respect to a basis of $V_w^\Gamma$, and $(W_w^\Gamma)^\mathfrak{p}m$ are subspaces cut by relations
with integer coefficients, it follows that there is a common eigenvector defined over $K_f$.
(b) The claim follows from (a) and Theorem \ref{thm_main}.
(c) The conclusion follows from \eqref{2.1}.
\end{proof}
The coefficients of $\rho_f^\mathfrak{p}m(I)$ are related to the critical values of the $L$-function of $L(s,f)$, so the
proposition implies well-known rationality statements about critical values. With the notation of
\eqref{5.1}, we have
\mathfrak{b}egin{equation}\,\langle\,bel{5.7}
r_{I,n}(f)=i^{n+1}\Lambda(n+1,f), \text{ with } \Lambda(s,f):=(2\mathfrak{p}i)^{-s}\Gamma(s)L(s,f)
\end{equation}
the completed $L$-function. This proves the following special case of Theorem 1 in \cite{Sh77}.
\mathfrak{b}egin{corollary} Assume that $f\in S_k(N, \chi)$ is a newform, and let $K_f$ be the field of
coefficients of $f$. Then with the periods $\omega_f^\mathfrak{p}m$ defined in the
Proposition \ref{p5.7} we have
\[ {\scriptscriptstyle F}rac{i^{n}\Lambda(n,f)}{\omega_f^\mathfrak{p}m}\in K_f \text{ for } 0<n<k, \quad (-1)^{n}=\mathfrak{p}m (-1)^{k-1}.
\]
\end{corollary}
\noindent When $k$ is even, our convention regarding the signs of the transcendental factors differs from
the usual one eg. in \cite{Sh77}, since $\omegaega_f^+$ is the normalizing factor for \emph{odd} critical values (and
for \emph{even} period polynomials).
\mathfrak{b}egin{remark}
When $k$ is odd it follows from Prop. \ref{p5.7} (b) that ${\scriptscriptstyle F}rac{|\omegaega_f^+|^2}{|\omegaega_f^-|^2}\in K_f$. In fact,
when $K_f$ is a real field, it follows from the
functional equation $N^{s/2}\Lambda(s,f)=\mathfrak{p}m N^{(k-s)/2} \Lambda(k-s,f)$ that ${\scriptscriptstyle F}rac{\omegaega_f^+}{i\omegaega_f^-}\in
K_f(\sqrt{N})$. From \eqref{5.7} and the functional equation, when $K_f$ is real we can choose
$$\omegaega_f^-=-i \Lambda(1,f),\quad \omegaega_f^+=\mathfrak{p}m i^{w+1} N^{(k-1)/2}\Lambda(k-1,f)$$
so that ${\scriptscriptstyle F}rac{\omegaega_f^+}{i\omegaega_f^-}=\sqrt{N}$.
\end{remark}
\subsection{Modular forms with Nebentypus} \,\langle\,bel{sec5.2}
Let $\Gamma=\Gamma_1(N)$, $\Gamma'=\Gamma_0(N)$ and $\chi$ a Dirichlet character modulo $N$. For $f\in S_k(N,
\chi)\subset S_k(\Gamma) $ a cusp form with Nebentypus $\chi$, we view the period polynomial $\rho_f$ as an
element of $W_w^{\Gamma}$. It clearly
satisfies $\rho_f(BC)=\chi(B)\rho_f(C)$ where $B$ runs through a fixed system of coset representatives for
$\Gamma\mathfrak{b}ackslash\Gamma'$
and $C$ runs through a fixed system of coset representatives for
$\Gamma'\mathfrak{b}ackslash\Gamma_1$. Here $\chi(B)=\chi(d_B)$ where $d_B$ is
the lower right entry of $B$ ($d_B$ is well defined modulo $N$). Let therefore $W_w^{\Gamma,\chi}$,
$C_w^{\Gamma,\chi}$ be the subspaces of $W_w^{\Gamma}$, $C_w^{\Gamma}$ consisting of elements $P$ with
$P(BC)=\chi(B)P(C)$ with $B,C$ as above. We have orthogonal decompositions with respect to the pairing
$\{\cdot , \cdot\}$
\[W_w^{\Gamma}=\mathfrak{b}igoplus_{\chi} W_w^{\Gamma,\chi}, \quad C_w^{\Gamma}=\mathfrak{b}igoplus_\chi C_w^{\Gamma,\chi}
\]
where the direct sums are over all characters modulo $N$. It is easily checked that the spaces
$W_w^{\Gamma,\chi}$, $C_w^{\Gamma,\chi}$ are preserved by the action of Hecke operators $\widetilde{T}_n$, and the dimension
of $C_w^{\Gamma,\chi}$ equals the dimension of the Eisenstein subspace $\mathcal{E}E_k(N, \chi)\subset M_k(\Gamma) $.
An indirect proof that the dimensions are equal is provided by Prop. \ref{l7.3}, which shows that
$C_w^{\Gamma,\chi}$ is dual to the space of period polynomials of Eisenstein series
$\widehat{E}_w^{\Gamma, \ov{\chi}} $ with respect to an extension of $\{ \cdot, \cdot \}$ to the space $\widehat{W}_w^{\Gamma}$ of
extended period polynomials (see also Prop. \ref{p8.9}).
The action of $\epsilon$ also preserves these subspaces and we have the following generalization of the
Eichler-Shimura isomorphism theorem \ref{thm2.1}.
\mathfrak{b}egin{theorem}[(Eichler-Shimura)] The two maps $\rho^{\mathfrak{p}m}:S_k(N, \chi) \rightarrow
(W_w^{\Gamma,\chi})^{\mathfrak{p}m}$, $f\mapsto \rho_f^\mathfrak{p}m$, give rise to
isomorphisms, denoted by the same symbols:
\mathfrak{b}egin{equation*}
\rho^\mathfrak{p}m:S_k(N,\chi) \longrightarrow (W_w^{\Gamma,\chi})^{\mathfrak{p}m}/(C_w^{\Gamma,\chi})^{\mathfrak{p}m}.
\end{equation*}
\end{theorem}
\mathfrak{b}egin{proof}
Injectivity follows from the refinement of Haberland's formula (Theorem \ref{thm_main}) as in the proof of
Theorem \ref{thm2.1}. Surjectivity follows by considering the maps above for all characters $\chi$, and using the
fact that the sum of the dimensions of their domains equals the sum of the dimensions of their ranges.
\end{proof}
When computing spaces of modular forms, it is much more efficient to fix coset representatives $C_1, \ldots, C_d$
for $\Gamma'\mathfrak{b}ackslash \Gamma_1$, with $d=[\Gamma_1:\Gamma']$, and to view $P\in W_w^{\Gamma,\chi}$ as a vector
$(P(C_1),\ldots, P(C_d))\in V_w^d$, since the values at other cosets in $\Gamma\mathfrak{b}ackslash \Gamma_1$ are determined
by those above. Note however that this representation depends on the coset representatives chosen, which are fixed
once for all.
The action of $\mathfrak{g}ammamma\in\Gamma_1$ on $P\in V_w^d$ can be described as follows. Let $C_i \mathfrak{g}ammamma^{-1}=B_i
C_{\sigma(i)}$ where $B_i$ are coset representatives for $\Gamma\mathfrak{b}ackslash\Gamma'$ and $\sigma=\sigma_{\mathfrak{g}ammamma}$ is
a permutation of $\{1, \ldots,d\}$. Then
\[
P|\mathfrak{g}ammamma(C_i)= \chi(B_i)P(C_{\sigma(i)})|\mathfrak{g}ammamma.
\]
Similarly one can compute the actions $P|_{\Sigma} \widetilde{T}_n$ with $\Sigma_n$ the double cosets giving the action of
Hecke or Atkin-Lehner operators.
\comment{
To define the action of the Hecke operator $\widetilde{T}_n$ on $P\in V_w^d$, we use Remark \ref{r5.1}. Let $M\in M_n$.
If $C_i M^\vee$ has lower row elements $c,d$ with $\mathfrak{g}cd(c,d,N)=1$, then we can write
\[
C_i M^\vee=M_A^\vee \mathfrak{g}ammamma_i C_{\sigma(i)}
\]
where $M_A\in M_n^{\Gamma}$, $\sigma(i)\in\{1,\ldots , d\}$ and $\mathfrak{g}ammamma_i$ are coset representative for
$\Gamma\mathfrak{b}ackslash\Gamma'$. In this case we have $P|M(C_i)=\chi(\mathfrak{g}ammamma_i)P(C_{\sigma(i)})|M$, while if
$\mathfrak{g}cd(c,d,N)>1$ then $P|M(C_i)=0$. }
\subsection{Computing period polynomials numerically} \,\langle\,bel{sec5.3}
The period polynomials of a newform $f\in S_k(\Gamma_0(N),\chi)$ can be easily computed numerically using
the results of this chapter. One finds the space of period polynomials as the kernel of the matrix of period
relations, and then its plus and minus subspaces. Assuming the Hecke eigenvalues of $f$ at primes $p\nmid N$ are
known, then one can find a rational common eigenvector of enough operators $\widetilde{T}_p$ acting on $(W_w^\Gamma)^\mathfrak{p}m$
such that the resulting eigenspace is one dimensional (often one operator $\widetilde{T}_p$ for the smallest prime $p\nmid N$
suffices). The period polynomials $\rho_f^\mathfrak{p}m$ are thus determined up to scalars. The normalizing factors
can be determined from the critical values $L(w+1,f)$ (nonzero if $k>2$) and $L(w,f)$ (nonzero if $k>2$, $k\ne 4$),
unless they vanish and one has to determine $L(n,f|A)$ for some $A\in \Gamma\mathfrak{b}ackslash\Gamma_1$. Once the
polynomials $\rho_f^\mathfrak{p}m$ are determined, Theorem \ref{thm_main} gives the Petersson product $(f,f)$, and
Proposition \ref{p5.4} gives the Hecke eigenvalue $\,\langle\,mbda_n$ of $f$ for every $n$.
We have implemented the procedure sketched above in MAGMA \cite{Mgm} (available upon request). To give an idea
of the running time, when $f=q + 12q^3 + 88q^7 + \ldots\in S_6(\Gamma_0(100)) $ is
a newform with rational coefficients, the computation of $\rho_f^\mathfrak{p}m$ takes 1 sec (in this case
$\dim(W_w^\Gamma)^+=78$, $\dim(W_w^\Gamma)^-=72$, $\dim S_k(\Gamma)=66$, and $[\Gamma_1:\Gamma]=180$).
Once the polynomial $\rho_f^+$ is computed, the Hecke eigenvalue for $p=10037$ is computed in less than 1 sec using
Prop. \ref{p5.4}. The `Coefficient' command in MAGMA computes the same eigenvalue in 140 sec, while the
`coefficients' command in SAGE takes 3 sec for the same computation (on the same machine). We used the efficient
implementation by W. Stein of the coset space $ P^1(\mathbb{Z}/N\mathbb{Z})\simeq \Gamma_0(N)\mathfrak{b}ackslash\Gamma_1$ via `P1Classes',
and the Hecke elements given by `HeilbronnCremona' in MAGMA (which give the action of Hecke operators on modular
symbols, but see Remark \ref{r5.4}).
\noindent{\mathfrak{b}f An example.} Let $\Gamma=\Gamma_0(5)$, $k=4$, and $f=q-4q^2+2q^3+8q^4-5q^5\ldots \in S_k(\Gamma)$ the
unique normalized cusp form. In the following table, the first row contains the elements of $P^1(\mathbb{Z}/5\mathbb{Z})$, namely
the bottom rows of a system of representatives for $\Gamma \mathfrak{b}ackslash \Gamma_1$, while the second and third rows
list the components of rational generators $P_f^\mathfrak{p}m$
of the Hecke eigenspace of $(W_w^\Gamma)^\mathfrak{p}m$ that corresponds to the Hecke eigenform $f$ (here $P_f^-$ generates
$(W_w^\Gamma)^-$, and only the eigenvalue of $T_2$ is needed for the computation of $P_f^+$).
\mathfrak{b}egin{table}[ht]{\scriptscriptstyle F}ootnotesize\tabcolsep=0.21cm
\mathfrak{b}egin{tabular}{c c c c c c c}
\ & (0 1) & (1 1) & (1 3) & (1 2) &(1 4) & (1 0) \\ \hline \noalign{
}
$P_f^+$ & $-5X^2 + 1$ & $-5X^2 + 5$ & $8X^2 + 13X - 8$ & $8X^2 - 13X - 8$ & $-5X^2 + 5$ & $-X^2 + 5$\\
$P_f^-$ & $X$ & $X^2 + 2X + 1$ & $2X^2 - 3X - 2$ & $-2X^2 - 3X + 2$ & $-X^2 + 2X - 1$ & $ X $
\end{tabular}
\end{table}
\noindent We have $\rho_f^\mathfrak{p}m=\omega_f^\mathfrak{p}m P_f^\mathfrak{p}m$ with
\[\omega_f^+=r_{I,w}(f)= -0.0051365773 i, \quad
\omega_f^-=-wr_{I,w-1}(f)= 0.0208651386, \]
computed via \eqref{5.7} using the numerical computation of $L$-series implemented by Tim Dokchitser in MAGMA.
Theorem \ref{thm_main} then gives $(f,f)=0.00014513335$, in agreement with the value obtained analytically in
\cite{Co}.
Notice that $r_{A,n}^+(f)/r_{I,w}(f) \equiv 0 \mathfrak{p}mod{13}$ for all $n$ with $0<n<w$, and all $A\in
\Gamma\mathfrak{b}ackslash\Gamma_1$ (see \eqref{5.1} for notation).
As in Manin's proof of Ramanujan's congruence mod 691 \cite{M73}, this suggests that there should be a congruence
between $f$ and an Eisenstein series. Indeed by looking at the first few coefficients{\scriptscriptstyle F}ootnote{By Sturm's result
checking the first three coefficients of both sides is enough to prove the congruence.} we obtain that
$$f \equiv E_4(z)-E_4(5z)\mathfrak{p}mod{13}$$
where $E_4$ is the weight 4 Eisenstein series of full level, normalized to have the coefficient of $q$ equal to 1.
A generalization of Ramanujan's congruence to Hecke eigenforms and Eisenstein series for $\Gamma_0(N)$ is in
progress.
\section{Rational decomposition of modular forms}\,\langle\,bel{sec6}
For $\Gamma$ a finite index subgroup of $\Gamma_1$ normalized by $\epsilon$, we
give an explicit decomposition of $f\in S_k(\Gamma)$ in terms of explicit generators,
generalizing the result in the full level case \cite{Po}. For $\Gamma_1$, these are
the generators with rational periods studied in \cite{KZ}, where their periods were
first computed. For $\Gamma_0(N)$, the periods of these generators were computed for $N$
square free in \cite{An}, and for arbitrary $N$ in \cite{FY}. These generators
have explicit formulas as Poincar\'e series when $k>2$.
To define these generators, for $A\in \Gamma\mathfrak{b}ackslash\Gamma_1$ , $0\leqslant
n\leqslant w$, define the periods $r_{A,n}(f)$ by
\mathfrak{b}egin{equation}\,\langle\,bel{5.1}
\rho_f(A)(X)=\sum_{n=0}^w (-1)^{w-n}\mathfrak{b}inom{w}{n}r_{A,n}(f) X^{w-n}
\end{equation}
and similarly define $r_{A,n}^{\mathfrak{p}m}(f)$ with $\rho_f$ replaced by $\rho_f^{\mathfrak{p}m}$. Let
$R_{A,n}\in S_k(\Gamma)$ be the dual of the linear functional
$f\rightarrow {\scriptscriptstyle F}rac{1}{[{\overline{\Gamma}}_1:{\overline{\Gamma}}]}r_{A,n}(f)$, with respect to the Petersson product:
\[ (f, R_{A,n})={\scriptscriptstyle F}rac{r_{A,n}(f)}{[{\overline{\Gamma}}_1:{\overline{\Gamma}}]}, \text{ for all } f\in S_k(\Gamma),
\]
and similarly define $R_{A,n}^{+}$, $R_{A,n}^-$. For $\Gamma=\Gamma_0(N)$, the
polynomials $\rho^-(R_{A,n}^{+})$, $\rho^+({R_{A,n}^-})$ have rational coefficients.
For $\kappa\in \{+, -\}$ and $0\leqslant j\leqslant w$, define the
linear combinations of periods:
\[
s_{A,j}^\kappa(f)=\sum_{n=0}^j \mathfrak{b}inom{j}{n} (-1)^{j-n} r_{A,j}^{\kappa}(f).
\]
We then have the following generalization of Theorem 1.1 in \cite{Po}, which gives explicit
inverses of the Eichler-Shimura maps \eqref{7.1}.
\mathfrak{b}egin{theorem}\,\langle\,bel{thm6.1} Let $\Gamma$ be a finite index subgroup of $\Gamma_1$ normalized
by $\epsilon$, and let $\kappa_1, \kappa_2\in\{+, -\}$ with $\kappa_1\ne \kappa_2$ if $k$ even
and $\kappa_1=\kappa_2$ if $k$ odd. For $f\in S_k(\Gamma)$
\mathfrak{b}egin{equation*}
{\scriptscriptstyle F}rac{-3 C_{k}}{2}\cdot f=\sum_{A\in {\overline{\Gamma}}\mathfrak{b}ackslash{\overline{\Gamma}}_1}
\sum_{n=0}^w
\mathfrak{b}inom{w}{n}s_{AU^{-1},n}^{\kappa_1}(f) R_{A,n}^{\kappa_2} .
\end{equation*}
\end{theorem}
\mathfrak{b}egin{proof}
Let $P\in (W_w^\Gamma)^{\kappa_1}$, $Q\in (W_w^\Gamma)^{\kappa_2}$. Then
\[\{P,\overline{Q}\}=\,\,\langle\,ngle\!\,\langle\,ngle\, P|US-SU^2,\overline{Q}\,\,\rangle\,ngle\!\,\rangle\,ngle\,
=\,\,\langle\,ngle\!\,\langle\,ngle\, P|U^2-U,\overline{Q} \,\,\rangle\,ngle\!\,\rangle\,ngle\,
=-2 \,\,\langle\,ngle\!\,\langle\,ngle\, P|U, \overline{Q} \,\,\rangle\,ngle\!\,\rangle\,ngle\,
\]
where the second equality follows since $P|S=-P, Q|S=-Q$, while the third from
$P|U^2-U=P|-1-2U$, together with \eqref{conj1}.
If $R(X)=\sum_{n=0}^w (-1)^{w-n}\mathfrak{b}inom{w}{n}r_n X^{w-n}\in V_w$ then
$$
R|U(X)=\sum_{j=0}^w\mathfrak{b}inom{w}{j} s_j X^j, \text{ with } s_j=\sum_{n=0}^j (-1)^{j-n}\mathfrak{b}inom{j}{n}
r_n.
$$
By Theorem \ref{thm_main} and the preceding computations it follows that for
$f,g\in S_{k}(\Gamma)$
\[
3 C_{k}(f,g)=\{\rho_f^{\kappa_1} ,
\overline{\rho_g^{\kappa_2}}\}=-{\scriptscriptstyle F}rac{2}{[{\overline{\Gamma}}_1:{\overline{\Gamma}}]} \sum_{A\in {\overline{\Gamma}}\mathfrak{b}ackslash
{\overline{\Gamma}}_1}\sum_{j=0}^w
\mathfrak{b}inom{w}{j}
s_{AU^{-1},j}^{\kappa_1}(f) \overline{r_{A,j}^{\kappa_2}(g)}.
\]
Since $\overline{r_{A,j}^+(g)}=[{\overline{\Gamma}}_1:{\overline{\Gamma}}]\cdot (R_{A,j}^+, g)$, the claim follows.
\end{proof}
\section{Extra relations satisfied by period polynomials of cusp forms}\,\langle\,bel{sec7}
In this section, we determine the image of the maps
$\rho^\mathfrak{p}m:S_k(\Gamma)\rightarrow (W_w^\Gamma)^\mathfrak{p}m$, namely the extra relations satisfied by
$\rho_f^\mathfrak{p}m$ for $f\in S_k(\Gamma)$ which are independent of the period relations. To be
explicit, the extra relations we
obtain require the determination of the periods $r_{B,m}^\mp(R_{A,w}^\mathfrak{p}m)$ of the generators
defined in the previous section. For $\Gamma_1$, these periods were computed in \cite{KZ},
and for $\Gamma_0(N)$, they were computed in \cite{An} (for $N$ square free, and not quite in
closed form), and in \cite{FY} (only the principal periods $r_{I,m}^\mp ({R_{I,n}^\mathfrak{p}m})$).
For $\Gamma_0(N)$ with small $N$, the computations in \cite{FY} are sufficient to make
completely explicit the extra relations, and we illustrate this for the case
$\Gamma=\Gamma_0(2)$. The relations are similar to the relation found by
Kohnen and Zagier in the full-level case \cite{KZ} (see also \cite[Sec. 2]{Po}).
We first define bases of $(C_w^\Gamma)^\mathfrak{p}m$, using the terminology in Definition
\ref{d_cusp}. For each cusp $\mathcal{C}\in \Gamma\mathfrak{b}ackslash \Gamma_1/\Gamma_{1\infty}$,
which is regular
if $k$ is odd, define $P_{\mathcal{C}}\in C_w^\Gamma$ as
in \eqref{cw} by
fixing $A_\mathcal{C}\in\mathcal{C}$ a representative, and setting $c_A=(-1)^w c_{AJ}=1$ if
$[A]^+=[A_\mathcal{C}]^+$, and $c_A=0$ if $[A]\ne [A_\mathcal{C}]$. Then $\{P_{\mathcal{C}}\}$ form
a basis of $C_w^\Gamma$ if $k>2$, and if $k=2$ there is only one relation $\sum_\mathcal{C} P_\mathcal{C}=0$.
Assume that $\Gamma$ is normalized by $\epsilon$, and denote $A'=\epsilon
A\epsilon$. Note that if $\mathcal{C}=[A]$ is a cusp, then $\mathcal{C}'=[A']$ is well-defined.
Since $P_{\mathcal{C}}|\epsilon=P_{\mathcal{C}'}$, we have $ P_{\mathcal{C}}^+=P_{\mathcal{C}'}^+$ and
$P_{\mathcal{C}}^-=-P_{\mathcal{C}'}^-$. Therefore a basis of
$(C_w^\Gamma)^-$ consists of $P_{\mathcal{C}}^-$ for each unordered pair $( \mathcal{C},\mathcal{C}')$ of cusps
with $\mathcal{C}\ne \mathcal{C}'$, and a basis of $(C_w^\Gamma)^+$ consists of $P_{\mathcal{C}}^+$ for each
unordered pair $( \mathcal{C},\mathcal{C}')$ of cusps (when $k$ is odd only pairs of regular cusps are
considered; note that $\mathcal{C}$ is regular iff $\mathcal{C}'$ is regular).
We now fix $A\in \mathcal{C}$ for each (regular if $k$ odd) cusp
$\mathcal{C}\in\Gamma\mathfrak{b}ackslash
\Gamma_1/\Gamma_{1\infty}$ and we write, by a slight abuse of notation,
$R_{\mathcal{C},n}=R_{A,n}$, $r_{\mathcal{C},n}=r_{A,n}$ with the notation of the previous section
($R_{\mathcal{C},n}$ does depend on the choice of representative $A$, but we fix such a choice).
For each unordered pair of cusps $(\mathcal{C},\mathcal{C}')$ we take $g=R_{\mathcal{C},w}^+$ in Theorem
\ref{thm_main}, and if
$\mathcal{C}\ne \mathcal{C}'$ we take also $g=R_{\mathcal{C},w}^-$ (again only for regular cusps if $k$ is odd),
obtaining the following linear relations satisfied by all $f\in S_k(\Gamma)$ {\scriptscriptstyle F}ootnote{In this
section we occasionally write $\rho(f)$ instead of $\rho_f$ to simplify notation.}
\mathfrak{b}egin{equation}\,\langle\,bel{e_rel}
{\scriptscriptstyle F}rac{3 C_k}{ [{\overline{\Gamma}}_1:{\overline{\Gamma}}] } r_{\mathcal{C},w}^+(f)= \{ \rho^+(f), \overline{\rho^-({R_{\mathcal{C},w}^+})}\},
\quad
{\scriptscriptstyle F}rac{3 C_k}{ [{\overline{\Gamma}}_1:{\overline{\Gamma}}] } r_{\mathcal{C},w}^-(f)= \{ \rho^-(f),
\overline{\rho^+({R_{\mathcal{C},w}^-})}\}.
\end{equation}
The linear forms appearing in these relations can be applied to all $P\in
(W_w^\Gamma)^\mathfrak{p}m$,
and putting together the relations involving $\rho^+(f)$ into a map $\,\langle\,mbda_+$, and the
relations involving $\rho^-(f)$ (for pairs with $\mathcal{C}\ne\mathcal{C}'$) into a map $\,\langle\,mbda_{-}$, we
obtain two linear maps $\,\langle\,mbda_\mathfrak{p}m:(W_w^\Gamma)^\mathfrak{p}m\rightarrow
\mathbb{C}^{d_\mathfrak{p}m}$ with $d_\mathfrak{p}m=\dim(C_w^\Gamma)^\mathfrak{p}m$. If $d_{-}=0$ the map $\,\langle\,mbda_{-}$ is trivial.
\mathfrak{b}egin{prop}\,\langle\,bel{p6.2} Assume $k \mathfrak{g}e 3$ and let $\Gamma$ be a finite index subgroup of
$\Gamma_1$ normalized by $\epsilon$. With $\,\langle\,mbda_\mathfrak{p}m$ defined above, we have exact sequences:
\[
0\rightarrow S_k(\Gamma)\overset{\rho^{\mathfrak{p}m}}{\longrightarrow}
(W_w^\Gamma)^\mathfrak{p}m\overset{\,\langle\,mbda_\mathfrak{p}m}{\longrightarrow} \mathbb{C}^{d_\mathfrak{p}m}\rightarrow 0.
\]
\end{prop}
\mathfrak{b}egin{proof} We have $\im \rho^{\mathfrak{p}m}\subset \ker \,\langle\,mbda_\mathfrak{p}m$ by construction.
Note that the first relation in \eqref{e_rel} is not satisfied by $P_{\mathcal{C}}^+$, while
the second is not satisfied by $P_{\mathcal{C}}^-$ if $\mathcal{C}\ne \mathcal{C}'$, since the LHS is nonzero,
while the RHS vanishes by Lemma \ref{l4.4} a). Since $\{P_{\mathcal{C}}^\mathfrak{p}m \}$ form a basis of
$(C_w^\Gamma)^\mathfrak{p}m$, the conclusion follows.
\end{proof}
\mathfrak{b}egin{remark} For $\Gamma=\Gamma_0(N)$, Proposition \ref{prop7.2} characterizes those $N$ for
which the extra relations involve only the even parts of the period polynomials.
\end{remark}
For example, if $\Gamma=\Gamma_0(p)$ with $p$ prime, there are only two cusps $[I]$ and $[S]$,
and for all $P\in W_w^\Gamma$ we have $P(S)=-P(I)|S$ by the period relations. In
particular $r_{S,w}(f)=-r_{I,0}(f)$. Noting also that $P(I)^+$, $P(S)^+$ are the even parts of
$P(I), P(S)$, so that $R_{I,n}^+= R_{I,n} $ for $n$ even, we have the
following simpler version
of Proposition \ref{p6.2}.
\mathfrak{b}egin{corollary}\,\langle\,bel{c6.4} Let $\Gamma=\Gamma_0(p)$ with $p$ prime, and let $k>2$ even. Then
the two extra relations satisfied by all even period polynomials $\rho^+(f)$ for $f\in
S_k(\Gamma)$ are
\[{\scriptscriptstyle F}rac{3 C_k}{ [{\overline{\Gamma}}_1:{\overline{\Gamma}}] } r_{I,a}(f)= \{ \rho^+(f),
\overline{\rho^-({R_{I,a}})}\},\quad
\text{ for } a=0,w.
\]
\end{corollary}
For small values of $N$ (eg $N=2,3,4,5$), the polynomials $\rho_f$ are completely determined
by
the principal parts $\rho_f(I)$, so that the relations above are completely explicit via the
computation of $\rho^-(R_{I,a})(I)$ in \cite{FY}. In the remainder of this section, we discuss
in detail the case of period polynomials for $\Gamma=\Gamma_0(2)$, which have been studied in
\cite{IK},\cite{FY}, \cite{KT}.
We take as coset representatives for $\Gamma\mathfrak{b}ackslash \Gamma_1$ the set
$\{I,U,U^2\}$. Denoting by $\overline{A}$ the coset
$\Gamma A$, we have $\ov{S}=\ov{U}$, $\ov{US}=\ov{I}$, $\ov{U^2 S}=\ov{U^2}$. For
$P\in W_w^\Gamma$, the
period relations $P|1+S=0$, $P|1+U+U^2=0$ reduce to
\mathfrak{b}egin{equation}\,\langle\,bel{7.2}
P(U)+P(I)|S=0,\quad P(U^2)|1+S=0, \quad P(U^2)+ P(U)|U+P(I)|U^2=0.
\end{equation}
The polynomials $P(U), P(U^2)$ are therefore determined by $P(I)$ which satisfies the relation
\mathfrak{b}egin{equation}\,\langle\,bel{per2}
P(I)|(ST-ST^{-1})(1+S)=0.
\end{equation}
Let $U_w\subset V_w$ denote the set of polynomials satisfying \eqref{per2}, so that we can
identify $W_w^\Gamma$ with $U_w$ via $P\rightarrow P(I)$. Conjugation by
$\epsilon$ leaves unchanged each coset $\ov{I}, \ov{U}, \ov{U^2}$, hence $P^+, P^-$
correspond to the even and odd parts of the polynomial $P(I)$ in this identification.
To express the formula in Theorem \ref{thm_main} in terms of $\rho_f(I), \rho_g(I)$ alone,
let $P\in (W_w^\Gamma)^+, Q\in
(W_w^\Gamma)^-$. We have $\,\,\langle\,ngle\!\,\langle\,ngle\, P|T-T^{-1}, Q \,\,\rangle\,ngle\!\,\rangle\,ngle\,=-2\,\,\langle\,ngle\!\,\langle\,ngle\, P|U, Q \,\,\rangle\,ngle\!\,\rangle\,ngle\,$ as in the proof of
Theorem
\ref{thm6.1}, and using \eqref{7.2} we obtain
\[
\,\,\langle\,ngle\!\,\langle\,ngle\, P|U, Q \,\,\rangle\,ngle\!\,\rangle\,ngle\,={\scriptscriptstyle F}rac{1}{3}<P(I)|2T^{-1}-2I-T, Q(I)>=-{\scriptscriptstyle F}rac{1}{2} <P(I)|T-T^{-1}, Q(I)>
\]
where $\,\langle\, P(I),Q(I)\,\rangle\,=\,\langle\, P(I)|T+T^{-1}, Q(I)\,\rangle\, =0$ since $P(I),Q(I)$ have opposite
parity. We can therefore restate Theorem \ref{thm_main} as follows, setting $P_f=\rho_f(I)\in
U_w$
\mathfrak{b}egin{equation}\,\langle\,bel{7.5}
3 C_{k}\cdot (f,g)=\,\langle\, P_f^+|T-T^{-1}, \ov{P_g^-}\,\rangle\,.
\end{equation}
By Proposition \ref{prop7.2} and the Eichler-Shimura isomorphism \eqref{7.1}, the map $f
\rightarrow P_f^{-}$ gives an isomorphism
$S_{k}(\Gamma_0(2))\simeq U_w^-$, while the image of the map $f\rightarrow P_f^+$ is a
codimension 2 subspace of $U_w^+$. {\scriptscriptstyle F}ootnote{A direct proof in this case is contained in
\cite[Theorem
4]{KT}.} To simplify notation, let $r_n(f)=r_{I,n}(f)$, and $R_n=R_{I,n}$ for $0\leqslant n
\leqslant w$.
\mathfrak{b}egin{corollary}\,\langle\,bel{c6.5}
For $f$ in $S_k(\Gamma_0(2))$, let
$
s_n(f)=\displaystyle\sum_{\substack{j=0\\ n-j \text{ odd}}}^n\mathfrak{b}inom{n}{j} r_j(f)
$. The extra relations satisfied by the even periods of $f$ are
\[
r_a(f)=\sum_{\substack{n=0\\ n \text{ odd}}}^w\mathfrak{b}inom{w}{n} s_{w-n}(f)
{\scriptscriptstyle F}rac{2}{C_{k}} r_{a}(R_{n})\quad \text{ for } a=0,w.
\]
\end{corollary}
\noindent From \cite{FY}, for $0<w<n$, $n$ odd, we have
$r_{w}(R_{n})=-\displaystyle{\scriptscriptstyle F}rac{r_0(R_{\,\widetilde{n}})}{N^n}$ and
\[
{\scriptscriptstyle F}rac{2}{C_{k}}r_0(R_{n})=-N^{\,\widetilde{n}}{\scriptscriptstyle F}rac{B_{\,\widetilde{n}+1}}{\,\widetilde{n}+1}+{\scriptscriptstyle F}rac{k}{B_k}{\scriptscriptstyle F}rac{B_{ n+1 } }
{ n
+1} {\scriptscriptstyle F}rac{B_{\,\widetilde{n}+1}}{\,\widetilde{n}+1}{\scriptscriptstyle F}rac{\mathfrak{a}lphapha_{N,k}(n)}{N}+{\scriptscriptstyle F}rac{\deltalta_{w,n+1}}{w},
\]
where $\,\widetilde{n}=w-n$, $\mathfrak{a}lphapha_{N,k}(n)={\scriptscriptstyle F}rac{1-N^{-n-1}}{1-N^{-k}}$ (recall $N=2$), and $B_m$ are
the Bernoulli numbers. Note that there is a
minus sign missing in the normalization of the generators denoted by $R_{\Gamma,w,n}$ in
\cite[Def. 1.1]{FY}, and with this correction we have $R_n=-{\scriptscriptstyle F}rac{C_{k}}{2}
R_{\Gamma,w,n}$.
\mathfrak{b}egin{proof}
Since $P_f|T-T^{-1}(X) =-2 \sum_{n=0}^w (-1)^n \mathfrak{b}inom{w}{n}s_{n}(f) X^{w-n},$
and $\ov{r_{n}(R_{0})}=r_0(R_{n})$, the claim follows from Corollary \ref{c6.4}.
\end{proof}
The periods $r_n(f)$ are related to the critical values of the $L$-series
associated to $f$, and when $f$ is a newform they can be readily computed using MAGMA \cite{Mgm}. The relations
in Corollary \ref{c6.5} have been checked numerically for $k=8,10,14$.
\section{Period polynomials of arbitrary modular forms}\,\langle\,bel{sec8}
In this section we define period polynomials for noncuspidal modular forms, and extend
Haberland's formula and the action of Hecke operators to the larger space of
period polynomials of all modular forms. An important feature of the
larger space $\widehat{W}_w^\Gamma$ is that the
the pairing $\{ \cdot,\cdot \}$ has a natural nondegenerate extension to it, while
on $W_w^\Gamma$ it is degenerate (its radical is $C_w^\Gamma$). If $\Gamma=\Gamma_1(N)$ and
$k>2$, the period polynomial maps $\rho^{\mathfrak{p}m}$ extend naturally to the larger space, and
they give isomorphisms between $M_k(\Gamma)$ and $(\widehat{W}_w^\Gamma)^\mathfrak{p}m$. Surprisingly, when
$k=2$, $\Gamma=\Gamma_0(N)$ and $N$ is squarefree with at least two prime factors, only one of
the two maps is an isomorphism.
For the full modular
group, period polynomials of Eisenstein series were defined in \cite{KZ}, using the description
of periods as special values of the associated $L$-function, and the enlarged space of period
polynomials was introduced in \cite{Z91}. A different construction using an Eichler integral
was given more recently in \cite{BGKO}, in the more general context of weakly holomorphic
modular forms. We extend both the Eichler integral and the $L$-function approach to a finite
index subgroup $\Gamma$ of $\Gamma_1$.
For $f\in M_k(\Gamma)$, we define $\widehat{\rho}_f=\widetilde{f}|1-S$ as in \eqref{e_int} (with the same action as
on period polynomials), where the Eichler
integral $\widetilde{f}:\Gamma \mathfrak{b}ackslash \Gamma_1 \rightarrow \mathcal{A}$ is given by
\mathfrak{b}egin{equation}\,\langle\,bel{a1}
\widetilde{f}(A)(z)=\int_{z}^{i\infty}[f|A(t)-a_0(f|A)] (t-z)^w dt
\end{equation}
with $a_0(f|A)$ the constant term of the Fourier expansion of $f|A$. Note that
$a_0(f|A)=a_0(f|AT)$, so $\widetilde{f}|1-T=0$. Let
$$\widehat{V}_w:=\mathbb{B}ig\{\sum_{-1\leqslant i \leqslant w+1} a_i X^{i}: a_i\in \mathbb{C} \mathbb{B}ig\},$$
and let $\widehat{V}_w^\Gamma$ be the space of functions $P:\Gamma \mathfrak{b}ackslash \Gamma_1 \rightarrow
\widehat{V}_w$ with $P(-A)=(-1)^wP(A)$ for $A \in \Gamma \mathfrak{b}ackslash \Gamma_1$.
We define an action of $g\in \Gamma_1$ on $P\in \widehat{V}_w^\Gamma$ by $P|g(A)=P(Ag^{-1})|_{-w} g$
as before. Note that this is no longer well defined in general, as elements of
$\Gamma_1$ do not preserve the space $\widehat{V}_w$. However one can still define the subspace
$$
\widehat{W}_w^\Gamma=\{P\in \widehat{V}_w^\Gamma: P|1+S=P|1+U+U^2=0 \}.
$$
We will show below that $\widehat{\rho}_f \in \widehat{W}_w^\Gamma$. That it satisfies the period relations is
immediate: we have $\widehat{\rho}_f|1+S=0$, and $\widehat{\rho}_f|1-T=0$ from the definition, while
\[
\widehat{\rho}_f|1+U+U^2 = \widetilde{f} | (1-S)(1+U+U^2)= \widetilde{f}|(1-T^{-1})(1+U+U^2)=0.
\]
It remains to show that $\widehat{\rho}_f\in \widehat{V}_w^\Gamma$, and we will do this by relating it with the
polynomial $\rho_f \in V_w^\Gamma$ defined by \eqref{5.1}, where
\mathfrak{b}egin{equation}\,\langle\,bel{e7.2}
r_{A,n}(f)=(-1)^{n+1} {\scriptscriptstyle F}rac{\Gamma(n+1)}{(2\mathfrak{p}i i)^{n+1}} L(n+1,f|A).
\end{equation}
The $L$-function $L(s,f)=\sum_{n=1}^\infty a_n(f) n^{-s}$
is given, if $\re(s)>k$, by the Mellin transform
$$
(-1)^s (2\mathfrak{p}i i)^{-s}\Gamma(s) L(s,f)=\int_0^{i\infty} [f(t)-a_0(f)]t^{s-1} dt,$$
and it can be extended meromorphically to $\mathbb{C}$ by fixing $z_0\in\mathcal{H}$, decomposing
$\int_0^{i\infty}=\int_0^{z_0}+\int_{z_0}^{i\infty}$, and making a change of
variables $t=S u $ in the first integral. We obtain a meromorphic function with at most
simple
poles at $s=0$ and $s=k$:
\mathfrak{b}egin{equation*}
\mathfrak{b}egin{split}
{\scriptscriptstyle F}rac{(-1)^{s}\Gamma(s)}{(2\mathfrak{p}i i)^s}L(s,f)=\int_{z_0}^{i\infty} [f(t)-a_0(f)]t^{s-1}
dt+(-1)^s\int_{S z_0}^{i\infty} [f|S(t)-a_0(f|S)]t^{k-s-1} dt-\\
-a_0(f){\scriptscriptstyle F}rac{z_0^s}{s}-(-1)^s a_0(f|S) {\scriptscriptstyle F}rac{(Sz_0)^{k-s}}{k-s}.
\end{split}
\end{equation*}
Introducing as in \cite{KZ} the function $H_{z_0}\in V_w^\Gamma$ defined for $A\in
\Gamma\mathfrak{b}ackslash \Gamma_1$ by
\mathfrak{b}egin{equation}\,\langle\,bel{a10}
H_{z_0}(A)=\int_{z_0}^{i\infty} [f|A(t)-a_0(f|A)](t-X)^w
dt-a_0(f|A)\int_0^{z_0} (t-X)^w dt\in V_w
\end{equation}
we obtain from \eqref{5.1} and the analytic continuation above
\mathfrak{b}egin{equation}\,\langle\,bel{a11}
\rho_f(A)=H_{z_0}(A)-H_{S z_0}(AS^{-1})|S,
\end{equation}
namely $\rho_f=H_{z_0}-H_{S z_0}|S$.
We now determine the relation between $\widehat{\rho}_f$ and $\rho_f$, which also shows that $\widehat{\rho}_f\in
\widehat{W}_w^\Gamma$.
\mathfrak{b}egin{prop} \,\langle\,bel{pa1} For $f\in M_k(\Gamma)$, let $\rho_f^0 \in \widehat{V}_w^\Gamma$ be given by
$\rho_f^0(A)=(-1)^w{\scriptscriptstyle F}rac{a_0(f|A)}{w+1} X^{w+1}$. We have
\[\widehat{\rho}_f=\rho_f+ \rho_f^0|(1-S) ,
\]
namely $\widehat{\rho}_f(A)=\rho_f(A)+(-1)^w{\scriptscriptstyle F}rac{a_0(f|A)}{w+1} X^{w+1}+{\scriptscriptstyle F}rac{a_0(f|AS^{-1})}{w+1}
X^{-1}$.
\end{prop}
\mathfrak{b}egin{proof} Fixing $z_0\in \mathcal{H}$ and decomposing the integral in \eqref{a1} as
$\int_{z_0}^{i\infty}+\int_{z}^{z_0}$ we have
\[\widetilde{f}(A)(z)=H_{z_0}(A)(z)+\int_{z}^{z_0} f|A(t) (t-z)^w dt +a_0(f|A)\int_0^{z}(t-z)^w dt.
\]
Using the same relation for $\widetilde{f}(AS^{-1})(Sz)j(S, z)^w$ with $Sz_0$ in
place of $z_0$ we obtain, after a change of variables $u=St$ in the first integral above
\mathfrak{b}egin{equation*}
\mathfrak{b}egin{aligned}
(\widetilde{f}|1-S) (A)(z)=& H_{z_0}(A)(z)-H_{Sz_0}(AS^{-1})|S (z)+\\
&\quad +a_0(f|A)\int_0^{z} (t-z)^w dt -a_0(f|AS^{-1})\int_0^{Sz} (t-Sz)^w j(S,z)^w dt.
\end{aligned}
\end{equation*}
Computing the last integrals and comparing with \eqref{a11} yields the conclusion.
\end{proof}
We now determine the exact relationship between $\widehat{W}_w^\Gamma$ and $W_w^\Gamma$. For $\widehat{P}\in
\widehat{W}_w^\Gamma$ write $\widehat{P}=P+P_0$ where $P\in V_w^\Gamma$ and $P_0(A)=c_A X^{w+1}+d_A X^{-1}$
for $A\in \Gamma\mathfrak{b}ackslash\Gamma_1$.
From $\widehat{P}|1+S=0$ we obtain $d_A=c_{AS}$. From $\widehat{P}|1+U+U^2=0$, it follows that $P_0|1+U+U^2\in
V_w^\Gamma$, which implies that $c_A=c_{AT}$ for all $A\in \Gamma_1$. Therefore we have
$P_0=P^0|1-S$, where $P^0(A)=c_A X^{w+1}$, with $c_A=c_{AT}$. In conclusion, letting
\mathfrak{b}egin{equation}\,\langle\,bel{a2}
D_w^\Gamma=\{(c_A X ^{w+1})_A | (1-S) : c_A=c_{AT}=(-1)^w c_{AJ}\in \mathbb{C} \}\subset
\widehat{V}_w^\Gamma
\end{equation}
we have a unique decomposition of $\widehat{P}\in\widehat{W}_w^\Gamma$ as above
\mathfrak{b}egin{equation}\,\langle\,bel{a3}
\widehat{P}=P+P^0|(1-S) , \quad P\in V_w^\Gamma, \ P^0|(1-S)\in D_w^\Gamma .
\end{equation}
For $\widehat{\rho}_f$ this is the decomposition in Proposition \ref{pa1},
since $a_0(f|A)=a_0(f|AT)=(-1)^w a_0(f|AJ)$. As in the proof of Lemma \ref{L7.1}, note that
$\dim D_w^\Gamma$ equals $e_\infty(\Gamma)$ or $e_\infty^\mathrm{reg}(\Gamma)$, depending on whether
$k$ is even or odd respectively.
When $k=2$ there is an extra relation satisfied by the coefficients of $P^0$ in \eqref{a3}.
Letting $P(A)=d_A\in\mathbb{C}$, $P^0(A)=c_A X^{w+1}$, the period relations now imply that
\[
d_A+d_{AU}+d_{AU^2}+2(c_A+c_{AU}+c_{AU^2})=0, \text{ with } d_A+d_{AS}=0, c_A=c_{AT}
\]
for all $A\in \Gamma\mathfrak{b}ackslash \Gamma_1$. The relation $d_A+d_{AS}=0$ implies $\sum_A d_A=0$, and then the first
relation above implies that $\sum_{A} c_A=0$ as well, where the sum is over a complete system of representatives
for $\Gamma\mathfrak{b}ackslash\Gamma_1$. From Proposition \ref{pa1} it follows that $\sum_A a_0(f|A)=0$ for all $f\in
M_2(\Gamma)$.
\mathfrak{b}egin{prop}\,\langle\,bel{pa2} a) If $k \mathfrak{g}e 3$ there is an exact sequence
\[
0 \rightarrow W_w^\Gamma \rightarrow \widehat{W}_w^\Gamma \rightarrow D_w^\Gamma\rightarrow 0
\]
where the first map is inclusion, and the second is the map $\widehat{P}\rightarrow P^0|1-S$ defined
above.
b) If $k=2$ there is an exact sequence
\[
0 \rightarrow W_w^\Gamma \rightarrow \widehat{W}_w^\Gamma \rightarrow D_w^\Gamma\rightarrow
\mathbb{C}\rightarrow 0
\]
where the last map takes $P^0|1-S\in D_w^\Gamma$ with $P^0(A)=c_A X^{w+1}$ to $\sum_{A\in
\Gamma\mathfrak{b}ackslash \Gamma_1} c_A$.
\end{prop}
\mathfrak{b}egin{proof} Exactness at $\widehat{W}_w^\Gamma$ follows from the definition. If $k\mathfrak{g}e 3$,
surjectivity of the second map
follows from Proposition \ref{a1}, and the fact that there is a basis of Eisenstein series
$E_k^{\mathcal{C}}\in M_k(\Gamma)$ for $\mathcal{C}=[A_\mathcal{C}]$ a complete system of representatives for the
(regular if $k$ is odd) cusps in $\Gamma\mathfrak{b}ackslash \Gamma_1/\Gamma_{1\infty}$, such that
$a_0(E_k^{\mathcal{C}}|A)=(-1)^w a_0(E_k^{\mathcal{C}}|AJ)$ equals 1 if $[A]^+=[A_\mathcal{C}]^+$, and 0 if $[A]\ne\mathcal{C}$
(see Definition \ref{d_cusp} for notation). If $k=2$ the Eisenstein subspace of
$M_2(\Gamma)$ is spanned by
modular forms
$f$ which are nonzero only at a fixed pair of nonequivalent cusps and are zero at other cusps,
and such that $\sum_{A\in \Gamma\mathfrak{b}ackslash\Gamma_1} a_0(f|A)=0$. Their images in $D_w^\Gamma$
span the kernel of the last map, proving exactness at $D_w^\Gamma$.
\end{proof}
The previous proposition shows that $\dim \widehat{W}_w^\Gamma=2 \dim M_k(\Gamma)$. From the
proof we conclude that there is a direct sum decomposition
\mathfrak{b}egin{equation}\,\langle\,bel{7.6}
\widehat{W}_w^\Gamma=W_w^\Gamma\oplus\widehat{E}_w^\Gamma
\end{equation}
where $\widehat{E}_w^\Gamma$ is the image of the Eisenstein subspace
$\mathcal{E}E_k(\Gamma)\subset M_k(\Gamma)$ under the map $f\rightarrow \widehat{\rho}_f$.
The pairing $\{\cdot, \cdot \}$ extends to a pairing on $\widehat{W}_w^\Gamma \times\widehat{W}_w^\Gamma$, by
decomposing $\widehat{P},\widehat{Q}\in \widehat{W}_w^\Gamma$ as in \eqref{a3} and setting:
\mathfrak{b}egin{equation}\,\langle\,bel{a7}
\{\widehat{P},\widehat{Q} \}=\,\,\langle\,ngle\!\,\langle\,ngle\, P|T-T^{-1}, Q \,\,\rangle\,ngle\!\,\rangle\,ngle\, +\,\,\langle\,ngle\!\,\langle\,ngle\, 2P^0|T-T^{-1},Q \,\,\rangle\,ngle\!\,\rangle\,ngle\,+ \,\,\langle\,ngle\!\,\langle\,ngle\, P, 2 Q^0|T^{-1}-T
\,\,\rangle\,ngle\!\,\rangle\,ngle\,+I_k(P^0, Q^0),
\end{equation}
where $I_k(P^0, Q^0)=0$ if $k$ even and $I_k(P^0, Q^0)={\scriptscriptstyle F}rac{6 (k-1)
}{k[\Gamma_1:\Gamma]}\sum_{A\in \Gamma\mathfrak{b}ackslash \Gamma_1}{c_A c_A'}$ if $k$ is odd, where $P^0(A)=c_A
X^{w+1}$, $Q^0(A)=c_A' X^{w+1}$.
Since $P^0|T-T^{-1}\in V_w^\Gamma$ this pairing is well-defined, and it is easily checked
that it behaves as in \eqref{conj1} under the action of $\epsilon$ defined as in \eqref{eps}.
We will show below that this definition is natural for two reasons:
Haberland's formula generalizes to arbitrary modular forms, if the Petersson product is
extended in a natural way to all modular forms, and this pairing is Hecke equivariant
for $\Gamma=\Gamma_1(N)$, with the same action of Hecke operators on $\widehat{W}_w^\Gamma$ as on
$W_w^\Gamma$.
Recall that on $W_w^\Gamma$ the pairing $\{\cdot, \cdot \}$ is degenerate, its radical being
$C_w^\Gamma$ (Lemma \ref{l4.4}). We now show that the extended pairing is nondegenerate on
$\widehat{W}_w^\Gamma$, more precisely that the dual of $C_w^\Gamma$ inside $\widehat{W}_w^\Gamma$ is the space
$\widehat{E}_w^\Gamma$.
\mathfrak{b}egin{prop}\,\langle\,bel{l7.3}
$\mathrm{(a)}$ Let $P=P'|1-S \in C_w^\Gamma$ and $\widehat{Q}=Q+Q^0|1-S \in \widehat{W}_w^\Gamma$, and let
$P'(A)=c_A'$, $Q^0 (A)=(-1)^w c_A {\scriptscriptstyle F}rac{X^{w+1}}{w+1}$ for $A\in
\Gamma\mathfrak{b}ackslash\Gamma_1$ (so that $c_A'=c_{AT}'=(-1)^w c_{AJ}',
c_A=c_{AT}=(-1)^w c_{AJ}$). Then
\[
\{ P, \widehat{Q}\}=-{\scriptscriptstyle F}rac{6}{[{\overline{\Gamma}}_1:{\overline{\Gamma}}]} \sum_{A\in {\overline{\Gamma}}\mathfrak{b}ackslash {\overline{\Gamma}}_1} c_A' c_A .
\]
$\mathrm{(b)}$ The pairing $\{ \cdot, \cdot\}$ is nondegenerate on $\widehat{W}_w^\Gamma$, and the dual of $C_w^\Gamma$
is $\widehat{E}_w^\Gamma$.
\end{prop}
\mathfrak{b}egin{proof}
(a) As in the proof of Lemma \ref{l4.4}, we use the formal relation \eqref{4.8}, together
with the relation $(1-S)(1+U+U^2)=(1-T^{-1})(1+U+U^2)$ and the $\Gamma_1$ invariance of the pairing $\,\,\langle\,ngle\!\,\langle\,ngle\, \cdot
,\cdot \,\,\rangle\,ngle\!\,\rangle\,ngle\,$:
\mathfrak{b}egin{equation*}
\mathfrak{b}egin{aligned}
\{P,\widehat{Q}\}&=\,\,\langle\,ngle\!\,\langle\,ngle\, P'|(1-S)(T-T^{-1}), Q \,\,\rangle\,ngle\!\,\rangle\,ngle\,+2\,\,\langle\,ngle\!\,\langle\,ngle\, P', Q^0|(T^{-1}-T)(1-S) \,\,\rangle\,ngle\!\,\rangle\,ngle\, \\
&= 2\,\,\langle\,ngle\!\,\langle\,ngle\, P', Q|1+U+U^2 \,\,\rangle\,ngle\!\,\rangle\,ngle\, + 2\,\,\langle\,ngle\!\,\langle\,ngle\, P', Q^0|(T^{-1}-T)(1-S) \,\,\rangle\,ngle\!\,\rangle\,ngle\, \\
&=-2\,\,\langle\,ngle\!\,\langle\,ngle\, P', Q^0|[(1-T^{-1})(1+U+U^2)+(T-T^{-1})(1-S)] \,\,\rangle\,ngle\!\,\rangle\,ngle\, \\
&=-2\,\,\langle\,ngle\!\,\langle\,ngle\, P', Q^0|[2(1-T^{-1})+T-1 ]\,\,\rangle\,ngle\!\,\rangle\,ngle\, \\
&=-{\scriptscriptstyle F}rac{6}{[{\overline{\Gamma}}_1:{\overline{\Gamma}}]} \sum_A c_A' c_A.
\end{aligned}
\end{equation*}
(b) We choose a basis of $\widehat{W}_w^\Gamma$ by concatenating bases for $C_w^\Gamma,
\rho^-(S_k(\Gamma))+ \rho^+(S_k(\Gamma)), \widehat{E}_w^{\Gamma}$ in this order.
The block form matrix of the pairing $\{\cdot , \cdot\}$ with respect to this basis is
\[\leqslantft(\mathfrak{b}egin{smallmatrix} 0 & 0& \mathcal{A} \\0 & B & 0 \\(-1)^{w+1}\mathcal{A}^t & 0& C
\end{smallmatrix}\right),
\]
so it is enough to show that $\mathcal{A}$ is nonsingular ($B$ is nonsingular by Theorem
\ref{thm_main}). When $k>2$ this is obvious from part (a). For $k=2$, we fix a cusp $\mathcal{C}_0$ in
$\Gamma\mathfrak{b}ackslash\Gamma_1/\Gamma_{1\infty}=\{\mathcal{C}_0, \mathcal{C}_1, \ldots, \mathcal{C}_n\}$ and we let a basis
of $C_w^\Gamma$ consist of $P_i$, $1\leqslant i \leqslant n$, as in the statement of the lemma, with the
constants $c_{iA}'=1$ if $[A]\in \mathcal{C}_i$, and $c_{iA}'=0$ otherwise (note that
$P_0=-\sum_{i=1}^nP_i$). Letting $l_i=\#\{A\in {\overline{\Gamma}}\mathfrak{b}ackslash{\overline{\Gamma}}_1: [A]\in \mathcal{C}_i \}$ (the width
of the cusp $\mathcal{C}_i$), we take a basis of $\widehat{E}_w^\Gamma$ to consist of $\widehat{Q}_i$ as in the
statement, $1\leqslant i \leqslant n$, with $c_{iA}=1$ if $[A]\in \mathcal{C}_i$, $c_{iA}=-{\scriptscriptstyle F}rac{l_i}{l_0}$ if
$[A]\in \mathcal{C}_0$, and $c_{iA}=0$ otherwise. With respect to this basis the matrix $\mathcal{A}$ is
diagonal, so the pairing is nondegenerate.
\end{proof}
Assume now that $\Gamma$ is normalized by $\epsilon$. Since
the action of $\epsilon$ given by \eqref{eps} preserves $\widehat{W}_w^\Gamma$ and $ D_w^\Gamma$,
passing to the $\mathfrak{p}m 1$ eigenspaces in Proposition \ref{pa2} gives exact sequences
\mathfrak{b}egin{equation}\,\langle\,bel{7.7}
0 \rightarrow (W_w^\Gamma)^\mathfrak{p}m \rightarrow (\widehat{W}_w^\Gamma)^\mathfrak{p}m \rightarrow
(D_w^\Gamma)^\mathfrak{p}m\rightarrow \mathbb{C} \rightarrow 0,
\end{equation}
where the last map is nontrivial only if $k=2$ and the sign is minus, when it is defined in
Proposition \ref{pa2} b) (when $k=2$ and $P^0|1-S \in (D_w^\Gamma)^+$ with $ P^0(A)=c_A
x^{w+1}$, then $c_A=-c_{A'}$ and $\sum_A c_A=0$ automatically).
From \eqref{cw} and \eqref{a2} we see that $\dim (D_w^\Gamma)^+=\dim (C_w^\Gamma)^-$ for all
$k$; $\dim (D_w^\Gamma)^-=\dim (C_w^\Gamma)^+$ for $k\mathfrak{g}e 3$; and $\dim
(D_w^\Gamma)^- -1=\dim(C_w^\Gamma)^+$ for $k=2$. Combined with the Eichler-Shimura
isomorphism \eqref{7.1} and Lemma \ref{L7.1}, this implies that $\dim (\widehat{W}_w^\Gamma)^\mathfrak{p}m =\dim
M_k(\Gamma)$.
The next Proposition can be seen as a extension of the Eichler-Shimura isomorphism \eqref{7.1}
to the entire space of modular forms.
\mathfrak{b}egin{prop}\,\langle\,bel{p7.4} $\mathrm{(a)}$ Assume that $k, \Gamma$ are such that the extended Petersson scalar
product on $M_k(\Gamma)$ defined in $\S$\ref{s7.1} is nondegenerate (see Remark \ref{r7.5}).
Then the maps $\widehat{\rho}^\mathfrak{p}m:
M_k(\Gamma) \rightarrow (\widehat{W}_w^\Gamma)^\mathfrak{p}m$, $f\rightarrow \widehat{\rho}_f^\mathfrak{p}m$, are isomorphisms.
$\mathrm{(b)}$ Assume that $(C_w^\Gamma)^-=0$ (for example $\Gamma=\Gamma_0(N)$ with $N$ as in
Proposition \ref{prop7.2}). Then $\widehat{\rho}^-$ is an isomorphism.
\end{prop}
\mathfrak{b}egin{remark}\,\langle\,bel{r7.5}
It is shown in \cite{PP12} that the extended Petersson product is nondegenerate for
$\Gamma_1(N)$ (and therefore also for $\Gamma_0(N)$) when $k>2$. When $k=2$ the extended
Petersson product is nondegenerate for $\Gamma_1(p)$ or $\Gamma_0(p)$
with $p$ prime, while it is degenerate for $\Gamma_0(N)$ with $N$ squarefree with at least two
prime factors. This implies that in the latter case the map $\widehat{\rho}^+$ is not an isomorphism;
indeed, part b) shows that $\widehat{\rho}^-$ is an isomorphism, and if both $\widehat{\rho}^{\mathfrak{p}m}$ were
isomorphisms, then the
Petersson product would be nondegenerate by Theorem \ref{pa4} c) below, since the pairing
$\{\cdot, \cdot\}$ is nondegenerate.
\end{remark}
\mathfrak{b}egin{proof} a) Since the dimensions of the spaces are equal, we only have to prove
injectivity.
If $\widehat{\rho}_f^\mathfrak{p}m=0$, it follows from Theorem \ref{pa4} c) that $f=0$, when
the extended Petersson product on $M_k(\Gamma)$ is nondegenerate.
b) If $(C_w^\Gamma)^-=0$, then $(D_w^\Gamma)^+=0$ as well. Assuming $\widehat{\rho}_f^-=0$ for $f\in
M_k(\Gamma)$, it follows that $\rho_f^0=0$ so $f$ is a cusp form, hence $f=0$ by Theorem
\ref{thm_main}.
\end{proof}
\subsection{An example} As an example, we check directly that the map
$\widehat{\rho}^+$ is not an isomorphism for $k=2$ and $\Gamma=\Gamma_0(6)$. As explained in Remark
\ref{r7.5}, this gives an alternative proof that the extended Petersson product is degenerate
in this case, in agreement with \cite{PP12}.
As representatives $A_j$ for $\Gamma\mathfrak{b}ackslash \Gamma_1$ we take the matrices
\[
ST^{-i}S\{I, U^2, U\}, \quad i=0,1,2,3
\]
in this order (namely $(A_1,\ldots, A_{12} )= (I, U^2, U, ST^{-1}S, ST^{-1}S U^2, \ldots,
ST^{-3}S U )$), obtained from the set of representatives provided by the command
`CosetRepresentatives' in MAGMA. There are four cusps $\mathcal{C}_i\in
\Gamma \mathfrak{b}ackslash \Gamma_1/\Gamma_{1\infty} $, and we have $\mathcal{C}_1=[A_1]$,
$\mathcal{C}_2=[A_9]=[A_{12}]$,
$\mathcal{C}_3=[A_6]=[A_7]=[A_{11}]$, while the remaining six matrices are in the class $\mathcal{C}_4$.
Since there are no cusp forms of weight two on $\Gamma_0(6)$, we have $W_w^\Gamma=C_w^\Gamma$.
The latter space is spanned by polynomials $P_i$ supported at the class $\mathcal{C}_i$, $i=1,2,
3$, namely $P_i=(c_A)_A|1-S$ with $c_A=c_{AT}$ and $c_A=1$ if $[A]=\mathcal{C}_i$, $c_A=0$ otherwise.
We identify a polynomial $P\in C_w^\Gamma$ with a vector ${\mathfrak{b}f d}=(d_i)\in \mathbb{C}^{12}$ with
$P(A_i)=d_i$. Let $\sigma \in \mathcal{S}_{12}$ be the permutation such that $A_j S=A_{\sigma j}$. We
have $\sigma= (3, 4, 1, 2, 7, 10, 5, 12, 11, 6, 9, 8), $
and it follows that the vectors ${\mathfrak{b}f d}$ corresponding to the polynomials $P_1$, $P_2$, $P_3$
are given respectively by (the entries not specified are equal to 0):
\[
d_1=1, d_3=-1;\quad d_9=d_{12}=1, d_{11}=d_{8}=-1; \quad d_6=d_7=d_{11}=1,
d_{10}=d_5=d_{9}=-1.
\]
Therefore in order to decompose a polynomial $P\in C_w^\Gamma$ with respect to the basis
$\{P_1, P_2, P_3\}$ it is enough to know $d_1$, $d_{12}$ and $d_9$.
The space $M_2(\Gamma_0(6))$ is spanned by the Eisenstein series $E_2^t(z)=E_2(z)-tE_2(tz)$,
for $t=2,3,6$, where $E_2(z)=-{\scriptscriptstyle F}rac{1}{24}+\sum_{n\mathfrak{g}e 1} \sigma_1(n) e^{2\mathfrak{p}i i nz}$. Since
$(D_w^\Gamma)^+=(C_w^\Gamma)^-=0$, we have $\widehat{\rho}^+(E_2^t)=\rho^+(E_2^t)\in C_w^\Gamma$. Letting
$\rho(E_2^t)(A_i)=e_i$, $\rho(E_2^t)(A_i')=e_i'$, we have
$\rho^+(E_2^t)(A_i)={\scriptscriptstyle F}rac{e_i+e_i'}{2}=d_i$, where $e_j'=e_{\tau j}$ with $\tau=(1,
4, 3, 2, 10, 7, 6, 8, 9, 5, 11,12)\in \mathcal{S}_{12}$.
We now determine the constants $d_i$ for
each of the three Eisenstein series. Taking into account that $L(s, E_2)=\zeta(s)\zeta(s-1)$
and $L(s, E_2^t)=\zeta(s)\zeta(s-1)\mathfrak{b}ig(1-{\scriptscriptstyle F}rac{1}{t^{s-1}}\mathfrak{b}ig)$,
the constant $d_1$ is given by \eqref{e7.2}:
$$ d_1= C \ln(t), \quad \text{ where }C= -{\scriptscriptstyle F}rac{\zeta(0)}{2\mathfrak{p}i i}.$$
Since $E_2^2\in M_2(\Gamma_0(2))$, and $A_6,A_7, A_{11}\in \Gamma_0(2) $ it follows that
$e_1=e_6=e_7=e_{11}$. We also have $e_6'=e_7$, $e_{11}'=e_{11}$, hence
$d_1=d_6=d_7=d_{11}$. We obtain $\rho^+(E_2^2)=C \ln(2)(P_1+P_3).$
Since $E_2^3\in M_2(\Gamma_0(3))$, and $A_9,A_{12}\in \Gamma_0(3) $, we have $d_1=d_9=d_{12}$
and $\rho^+(E_2^3)=C \ln(3)(P_1+P_2). $
For $E_2^6$, in order to determine $d_9, d_{12}$ we easily find
\comment{
so writing $E_2^6=E_2^3+ 3 E_2^2 |
\mathfrak{b}ig(\mathfrak{b}egin{smallmatrix} 3 & 0 \\ 0 & 1 \end{smallmatrix}\mathfrak{b}ig),$ it follows that
\[E_2^6|A_9=E_2^3+ 3 (E_2^2 |S)|
\mathfrak{b}ig(\mathfrak{b}egin{smallmatrix} 3 & 0 \\ 0 & 1\end{smallmatrix}\mathfrak{b}ig), \quad E_2^6|A_{12}=E_2^3+ 3
(E_2^2 |ST)|
\mathfrak{b}ig(\mathfrak{b}egin{smallmatrix} 3 & 0 \\ 0 & 1\end{smallmatrix}\mathfrak{b}ig).
\]
From the transformation properties of $E_2$ under $\Gamma_1$ we have
$E_2|S(z)=E_2(z)-{\scriptscriptstyle F}rac{1}{2}E_2\mathfrak{b}ig({\scriptscriptstyle F}rac{z}{2}\mathfrak{b}ig)$, so
$E_2^6|A_9(z)=E_2(z)-{\scriptscriptstyle F}rac{3}{2}E_2\mathfrak{b}ig({\scriptscriptstyle F}rac{3z}{2}\mathfrak{b}ig)$ and
$E_2^6|A_{12}=E_2(z)-{\scriptscriptstyle F}rac{3}{2}E_2\mathfrak{b}ig({\scriptscriptstyle F}rac{3z+1}{2}\mathfrak{b}ig)$, obtaining
}
\[L(s, E_2^6|A_9)=\zeta(s)\zeta(s-1) \mathfrak{b}ig(1-\textstyle{\scriptscriptstyle F}rac{2^{s-1}}{3^{s-1}}\mathfrak{b}ig), \ \ L(s,
E_2^6|A_{12})=\zeta(s)\zeta(s-1)
\mathfrak{b}ig(1-3^{2-s}+ {\scriptscriptstyle F}rac{2^{s-1}}{3^{s-1}}+6^{1-s}\mathfrak{b}ig) .
\]
By \eqref{e7.2} it follows $d_9=C (\ln(3)-\ln(2))$ , $d_{12}= C \ln 3$, so that
\[
\rho^+(E_2^6)=C( P_1\ln 6 +P_2\ln 3 + P_3\ln 2 )= \rho^+(E_2^2)+\rho^+(E_2^3)
\]
concluding that $\widehat{\rho}^+$ is not surjective.
\subsection{Haberland's formula for arbitrary modular forms}\,\langle\,bel{s7.1}
The Petersson scalar product of two Eisenstein series of full level is defined by Zagier
in \cite{Z81}. Let
$\mathcal{F}$ be the fundamental domain $\{z\in \mathcal{H}: |z|\mathfrak{g}e 1, |\re\, z |\leqslant 1/2 \}$ for $\Gamma_1$,
and for $T>1$ let $\mathcal{F}_T$ be the truncated domain for which $\im\, z <T$. Since $\sum_A
f|A(z)\ov{g}|A(z) y^k$ is a $\Gamma_1$-invariant, renormalizable function in the sense of
\cite{Z81}, we can define for $f,g\in M_k(\Gamma)$
\mathfrak{b}egin{equation}\,\langle\,bel{petersson}
(f,g)={\scriptscriptstyle F}rac{1}{[{\overline{\Gamma}}_1:{\overline{\Gamma}}]}\lim_{T\rightarrow \infty}\sum_A\mathbb{B}ig[ \int_{\mathcal{F}_T}
f|A(z)\overline{g|A(z)} y^w dx dy -{\scriptscriptstyle F}rac{T^{k-1}}{k-1} a_0(f|A)\overline{a_0(g|A) }\mathbb{B}ig]
\end{equation}
where the sum is over a complete system of representatives $A\in{\overline{\Gamma}}\mathfrak{b}ackslash {\overline{\Gamma}}_1$. As in
\cite{Z81}, it can be shown that the extended Petersson product equals $\mathrm{Res }_{s=k} L(s, f,
\overline{g})$ up to a nonzero constant, where $ L(s, f,
\overline{g})=\sum_{n\mathfrak{g}e 1} {\scriptscriptstyle F}rac{a_n(f)\overline{a_n(g)}}{n^s} $. Using this fact, we show in
\cite{PP12} that for $\Gamma=\Gamma_1(N)$ the extended Petersson product is nondegenerate
when $k>2$, while for $k=2$ it may or may not be degenerate.
We have the following generalization of Theorems \ref{thm_hab} and \ref{thm_main}.
\mathfrak{b}egin{theorem} \,\langle\,bel{pa4} Assume $k\mathfrak{g}e 2$, and $\Gamma$ is a finite index subgroup of
$\Gamma_1$. Let $f,g\in M_k(\Gamma)$.
a) We have:
$\ \
6 C_k\cdot (f,g) = \{\widehat{\rho}_f, \overline{\widehat{\rho}_g}\},\ \
$ where $C_k=-(2i)^{k-1}$.
b) We have: $\ \{ \widehat{\rho}_f, \widehat{\rho}_g \}=0.$
c) Assuming further that $\Gamma$ is normalized by $\epsilon$, and letting $\kappa_1,
\kappa_2\in\{+,-\}$ as in Theorem \ref{thm_main}:
$$
3 C_k\cdot (f,g) = \{\widehat{\rho}_f^{\kappa_1}, \overline{\widehat{\rho}_g^{\kappa_2}}\}.
$$
\end{theorem}
\mathfrak{b}egin{proof} a) If one of $f,g$ is a cusp form, then we can apply Stokes' theorem over the
fundamental domain $\mathcal{D}$ for $\Gamma(2)$ as in the proof of Theorem \ref{thm_hab}, and
easily obtain the desired identity. When both $f,g$ have nonzero constant terms, this approach
is complicated by the fact that both $f$ and $g$ blow up at the cusps $-1,0,1$, and we prefer
to apply Stokes' theorem to the domain $\mathcal{F}$ as in \cite{KZ}. We use the following
abbreviations: $f_A=f|A$, $g_A=g|A$, $a_A=a_0(f|A)$, $b_A=a_0(g|A)$, $C_\Gamma=[{\overline{\Gamma}}_1:{\overline{\Gamma}}]$,
$C_k=-(2i)^{k-1}$. Sums over $A$ are over systems of representatives
$A\in{\overline{\Gamma}}\mathfrak{b}ackslash
{\overline{\Gamma}}_1$. For all $T>1$ we have
\[
C_k C_\Gamma (f,g)= \sum_A \int_{\mathcal{F}} [f_A(z) \ov{g_A}(z)-a_A\ov{b_A}](z-\ov{z})^w dz
d\ov{z}+a_A \ov{b_A}\mathbb{B}ig[\int_{\mathcal{F}_T} (z-\ov{z})^w dz d\ov{z}-C_k{\scriptscriptstyle F}rac{T^{k-1}}{k-1} \mathbb{B}ig]
\]
By Stokes' theorem we find $\int_{\mathcal{F}_T} (z-\ov{z})^w dz d\ov{z}=C_k{\scriptscriptstyle F}rac{T^{w+1}}{w+1}+
{\scriptscriptstyle F}rac{1}{w+1}\int_{\rho^2}^\rho (z-\ov z)^{w+1} d\ov{z}$. In the first integral we apply
Stokes' theorem after writing $f_A \ov{g_A}-a_A\ov{b_A}= (f_A- a_A)\ov{g_A}+a_A
(\ov{g_A}-\ov{b_A})$ to get
\[
C_k C_\Gamma (f,g)=\sum_{A} \int_{\mathfrak{p}artial \mathcal{F}} -F_A(z) \ov{g_A}(z) +
a_A[\ov{g_A}(z)-\ov{b_A}] {\scriptscriptstyle F}rac{(z-\ov{z})^{w+1}}{w+1} d\ov{z}+
{\scriptscriptstyle F}rac{a_A\ov{b_A}}{w+1}\int_{\rho^2}^\rho (z-\ov{z})^{w+1} d\ov{z}
\]
where $F_A(z)=\int_{z}^{i\infty} [f_A(t)-a_A] (t-\ov{z})^w dt$. Since $F_{AT}(z)=F_A(Tz)$, the
integrals over the vertical sides of $\mathcal{F}$ cancel (after summing over $A$) and setting
$\widetilde{F}_A(z)=F_A(z)-a_A\int_0^z(t-\ov{z})^w dt$ we obtain:
\[
C_k C_\Gamma (f,g)=\sum_{A} \int_\rho^{\rho^2} \widetilde{F}_A(z) \ov{g_A}(z)d\ov{z}
+(-1)^w{\scriptscriptstyle F}rac{a_A}{w+1} \int_{\rho}^{\rho^2} \ov{g_A}(z) \ov{z}^{w+1} d\ov{z}.
\]
In the first integral we change variables $z\rightarrow Sz$, which reverses the order of
integration. As in the proof of Proposition \ref{pa1} we have
$\widetilde{F}_A(z)-\widetilde{F}_{AS^{-1}}|_{-w} S (z)=\rho_f(A)(\ov{z})$ obtaining
\mathfrak{b}egin{equation}\,\langle\,bel{7.11}
C_k C_\Gamma (f,g)= \sum_{A}{\scriptscriptstyle F}rac{1}{2} \int_\rho^{\rho^2} \rho_{f}(A)(\ov{z})
\ov{g_A}(z)d\ov{z}
+(-1)^w{\scriptscriptstyle F}rac{a_A}{w+1}\int_{\rho}^{\rho^2} \ov{g_A}(z) \ov{z}^{w+1} d\ov{z}.
\end{equation}
We now proceed to write the result in terms of the pairing $\,\,\langle\,ngle\!\,\langle\,ngle\, \cdot ,\cdot \,\,\rangle\,ngle\!\,\rangle\,ngle\,$ on
$V_w^\Gamma$. Define $H_{z_0}\in V_w^\Gamma$ as in \eqref{a10}, with $g$ in
place of $f$. Using $\int_{\rho}^{\rho^2} \ov{g_A}(z) (\ov{z}-X)^w
d\ov{z}= \ov{H}_{\rho}(A)-\ov{H}_{\rho^2}(A)$ and \eqref{3.3} we get
$$
\int_{\rho}^{\rho^2} \rho_{f}(A)(\ov{z})
\ov{g_A}(z)d\ov{z}=\mathbb{B}ig\,\langle\,ngle\rho_{f}(A), \int_{\rho}^{\rho^2} \ov{g_A}(z) (\ov{z}-X)^w
d\ov{z}\mathbb{B}ig\,\rangle\,ngle=\,\langle\, \rho_{f}(A), \ov{H}_{\rho}(A)-\ov{H}_{\rho^2}(A) \,\rangle\,.
$$
The second integral in \eqref{7.11} can be written
\[
\int_{\rho}^{\rho^2} \ov{g_A}(z) \ov{z}^{w+1} d\ov{z}=
\ov{b_A}\int_{\rho}^{\rho^2}\ov{z}^{w+1} d\ov{z}+\int_{\rho}^{i \infty} -\int_{\rho^2}^{i
\infty} (\ov{g_A}(z)-\ov{b_A})\ov{z}^{w+1}d\ov{z}
\]
and changing variables $z=t-1$ in the last integral, recalling that
$\rho_f^0(A)=(-1)^w a_A{\scriptscriptstyle F}rac{X^{w+1}}{w+1}$ (=$\rho_f^0(AT)$), and using
\eqref{3.3} we obtain
\mathfrak{b}egin{gather*}
\sum_A (-1)^w{\scriptscriptstyle F}rac{a_A}{w+1}\int_{\rho}^{\rho^2} \ov{g_A}(z) \ov{z}^{w+1} d\ov{z}= \sum_A
\,\langle\, \rho_f^0(A)|1-T^{-1} , \ov{H}_\rho(A) \,\rangle\, +\\
\sum_A a_A\ov{b_A} \mathbb{B}ig[\int_{0}^1 \int_{0}^\rho
(t-\ov{z})^w d\ov{z} dt +(-1)^w\int_{\rho}^{\rho^2}{\scriptscriptstyle F}rac{\ov{z}^{w+1}}{w+1} d\ov{z} \mathbb{B}ig].
\end{gather*}
The expression inside square brackets equals ${\scriptscriptstyle F}rac{1}{(k-1)k}$, and setting $I(f,g)=
{\scriptscriptstyle F}rac{2}{k(k-1)C_\Gamma}\sum_A a_A\ov{b_A}$ we get
\[
2 C_k(f,g)=\,\,\langle\,ngle\!\,\langle\,ngle\, \rho_f, \ov{H}_\rho-\ov{H}_{\rho^2} \,\,\rangle\,ngle\!\,\rangle\,ngle\, + 2\,\,\langle\,ngle\!\,\langle\,ngle\, \rho_f^0|1-T^{-1},
\ov{H}_\rho\,\,\rangle\,ngle\!\,\rangle\,ngle\,+I(f,g).
\]
Now we use
\mathfrak{b}egin{equation}\,\langle\,bel{8.12}
H_{\rho^2}=H_{\rho}|T-\rho_g^0|(1-T), \quad
\rho_g=H_\rho-H_{\rho^2}|S=H_\rho|(1-TS) +\rho_g^0|(1-T)S .\end{equation}
Taking into account the relation
$(1+U^2)={\scriptscriptstyle F}rac{1}{3}(U^2-U)(1-U^{-1})+{\scriptscriptstyle F}rac{2}{3}(1+U+U^2)$ in $\mathbb{Q}[{\overline{\Gamma}}_1]$ we have
\mathfrak{b}egin{equation*}
\mathfrak{b}egin{aligned}
\,\,\langle\,ngle\!\,\langle\,ngle\, \rho_f, \ov{H}_\rho|1-T \,\,\rangle\,ngle\!\,\rangle\,ngle\,&=\,\,\langle\,ngle\!\,\langle\,ngle\, \rho_f|1-T^{-1}, \ov{H}_\rho \,\,\rangle\,ngle\!\,\rangle\,ngle\, = \,\,\langle\,ngle\!\,\langle\,ngle\, \rho_f|1+S
T^{-1}, \ov{H}_\rho \,\,\rangle\,ngle\!\,\rangle\,ngle\, = \\
&={\scriptscriptstyle F}rac{1}{3}\,\,\langle\,ngle\!\,\langle\,ngle\, \rho_f|U^2-U, \ov{H}_\rho|1-U \,\,\rangle\,ngle\!\,\rangle\,ngle\,+{\scriptscriptstyle F}rac{2}{3} \,\,\langle\,ngle\!\,\langle\,ngle\, \rho_f|1+U+U^2,
\ov{H}_\rho\,\,\rangle\,ngle\!\,\rangle\,ngle\,\\
&={\scriptscriptstyle F}rac{1}{3}\,\,\langle\,ngle\!\,\langle\,ngle\, \rho_f|\,-T^{-1}\,-TS, \ov{\rho_g}-\ov{\rho_g^0}|(1-T)S \,\,\rangle\,ngle\!\,\rangle\,ngle\,+{\scriptscriptstyle F}rac{2}{3} \,\,\langle\,ngle\!\,\langle\,ngle\,
\rho_f|1+U+U^2, \ov{H}_\rho\,\,\rangle\,ngle\!\,\rangle\,ngle\, \\
&={\scriptscriptstyle F}rac{1}{3}\,\,\langle\,ngle\!\,\langle\,ngle\, \rho_f |T-T^{-1}, \ov{\rho_g} \,\,\rangle\,ngle\!\,\rangle\,ngle\, +{\scriptscriptstyle F}rac{1}{3}\,\,\langle\,ngle\!\,\langle\,ngle\, \rho_f|T+T^{-1}S,
\ov{\rho_g^0}|1-T \,\,\rangle\,ngle\!\,\rangle\,ngle\,-\\
&\ \quad\ \quad\ \quad-{\scriptscriptstyle F}rac{2}{3} \,\,\langle\,ngle\!\,\langle\,ngle\,
\rho_f^0|(1-T^{-1})(1+U+U^2), \ov{H}_\rho\,\,\rangle\,ngle\!\,\rangle\,ngle\,
\end{aligned}
\end{equation*}
(on the last line we used Proposition \ref{pa1}) and collecting terms we get
\mathfrak{b}egin{equation*}
\mathfrak{b}egin{split}
6 C_k(f,g)=\,\,\langle\,ngle\!\,\langle\,ngle\, \rho_f |T-T^{-1}, \ov{\rho_g} \,\,\rangle\,ngle\!\,\rangle\,ngle\,+\,\,\langle\,ngle\!\,\langle\,ngle\,\rho_f|3+T^{-1}S +T, \ov
\rho_g^0|1-T\,\,\rangle\,ngle\!\,\rangle\,ngle\,+\\
+2\,\,\langle\,ngle\!\,\langle\,ngle\, \rho_f^0|(1-T^{-1})(2-U-U^2), \ov{H}_\rho \,\,\rangle\,ngle\!\,\rangle\,ngle\,+3I(f,g)
\end{split}
\end{equation*}
In the second term we use the relation
$$(1-T)(3+ST+T^{-1})=2(T^{-1}-T)+(1-T)S[1+U+U^2-U^2(1+S)]S,$$
while in the third we use $2-U-U^2=(1-U)(1-U^2)$ and \eqref{8.12}:
\mathfrak{b}egin{equation}\,\langle\,bel{8.13}
\mathfrak{b}egin{split}
6 C_k(f,g)=\,\,\langle\,ngle\!\,\langle\,ngle\, \rho_f|T-T^{-1}, \ov{\rho_g} \,\,\rangle\,ngle\!\,\rangle\,ngle\, +\,\,\langle\,ngle\!\,\langle\,ngle\, \rho_f,
2\ov{\rho_g^0}|(T^{-1}-T) \,\,\rangle\,ngle\!\,\rangle\,ngle\,+\,\,\langle\,ngle\!\,\langle\,ngle\, 2\rho_f^0|(T-T^{-1}), \ov{\rho_g}\,\,\rangle\,ngle\!\,\rangle\,ngle\,+\\
+\,\,\langle\,ngle\!\,\langle\,ngle\, \rho_f^0|(1-T)(T^{-1}S-ST-3), \ov{\rho_g^0}|1-T \,\,\rangle\,ngle\!\,\rangle\,ngle\,+3I(f,g)
\end{split}
\end{equation}
Let $p(X)={\scriptscriptstyle F}rac{X^{w+1}}{w+1}|1-T$, $q(X)={\scriptscriptstyle F}rac{X^{w+1}}{w+1}|1-T^{-1}$. Then
\[
\,\,\langle\,ngle\!\,\langle\,ngle\, \rho_f^0|(1-T)(T^{-1}S-ST), \ov{\rho_g^0}|1-T \,\,\rangle\,ngle\!\,\rangle\,ngle\,={\scriptscriptstyle F}rac{1}{C_\Gamma}\sum_A (a_A
\ov{b_{AS^{-1}}}-(-1)^w a_{AS^{-1}}\ov{b_A}) \,\langle\, p, q|S\,\rangle\,=0,
\]
where we changed $A$ to $AS^{-1}$ in one of the sums and used that $a_{AJ}=(-1)^w a_A$. We
also have
\[\,\,\langle\,ngle\!\,\langle\,ngle\, \rho_f^0|(1-T),\ov{\rho_g^0}|1-T \,\,\rangle\,ngle\!\,\rangle\,ngle\,={\scriptscriptstyle F}rac{1}{C_\Gamma}\sum_{A} a_A\ov{b_A}\,\langle\,
p,p \,\rangle\,={\scriptscriptstyle F}rac{1+(-1)^w}{k(k-1)C_\Gamma}\sum_{A} a_A\ov{b_A},
\]
which vanishes if $w$ is odd, and equals $I(f,g)$ if $w$ is even. Therefore the second line in
\eqref{8.13} vanishes if $k$ is even, and it equals $3 I(f,g)=I_k(\rho_f^0, \ov{\rho}_g^0)$ if
$k$ is
odd, finishing the proof.
b) Going backwards in the proof of part a) up to the first equation after applying Stokes'
theorem, we obtain
\[
C_\Gamma\{\widehat{\rho}_f,\widehat{\rho}_g\}=-6 \sum_{A} \int_{\mathfrak{p}artial \mathcal{F}} \widetilde{f}(A)(z) g_A(z)d z
\]
with $\widetilde{f}$ defined in \eqref{a1}. Since the integrand is holomorphic and vanishes at
$i\infty$, each term vanishes.
c) Since the extended pairing $\{\cdot, \cdot\}$ behaves as the original one under the action
of $\epsilon$, the claim follows
from a) and b) as in the proof of Theorem \ref{thm_main}.
\end{proof}
\subsection{Hecke operators.} \,\langle\,bel{sec7.1}
For a finite index subgroup $\Gamma$ and a double coset $\Sigma_n$ satisfying \eqref{eq_star},
we define the operation $|_{\Sigma}$ of the Hecke operators $\widetilde{T}_n$ on
$\widehat{W}_w^\Gamma$ as in Section \ref{sec4}. Although matrices in the
definition of $\widetilde{T}_n$ do not preserve $\widehat{V}_w^\Gamma$, we have the
following generalization of Proposition \ref{p4.2} and of Corollary \ref{c4.3}.
\mathfrak{b}egin{prop} \,\langle\,bel{pa5} Assume the pair $(\Gamma, \Sigma_n)$ satisfy \eqref{eq_star},
and for part (b) assume that $\Sigma_n=\Sigma_n'$ and $\Gamma$ is normalized by $\epsilon$. Let $\widetilde{T}_n\in R_n$ be
any element satisying \eqref{hecke}.
$\mathrm{(a)}$ We have $\widehat{\rho}_{f|[\Sigma_n]}=\widehat{\rho}_f |_{\Sigma} \widetilde{T}_n$ for $f\in M_k(\Gamma)$.
$\mathrm{(b)}$ We have ${\widehat{\rho}_{f|[\Sigma_n]}}^\mathfrak{p}m={\widehat{\rho}_f}^\mathfrak{p}m |_{\Sigma} \widetilde{T}_n$ for $f\in M_k(\Gamma)$.
$\mathrm{(c)}$ The operators $\widetilde{T}_n$ preserve the space $\widehat{W}_w^\Gamma$.
\end{prop}
\mathfrak{b}egin{proof} (a) The proof is the same as of Prop. \ref{p4.2}, once we show that
$\widetilde{f}|_{\Sigma}T_n^{\infty}=\widetilde{f| [\Sigma_n]}$. Equation \eqref{4.3} becomes
\mathfrak{b}egin{equation*}\,\langle\,bel{a5}
\mathfrak{b}egin{split}
\widetilde{f}|_{\Sigma}T_n^{\infty}(A)&=\sum_{M\in M_n^\infty\cap \Gamma_1 \Sigma_n A }\int_{M z}^{i \infty}
[f|A_M (t)-a_0(f|A_M)](t-Mz)^w j(M,z)^w dt\\
&=n^{w+1} \sum_{M\in M_n^\infty \cap \Gamma_1 \Sigma_n A}\int_{z}^{i \infty}\mathfrak{b}ig[f|M_{A} A
(u)-a_0(f|A_M)j(M,u)^{-k}\mathfrak{b}ig](u-z)^w du
\end{split}
\end{equation*}
As in Proposition \ref{p4.2}, we obtain $\widetilde{f}|_{\Sigma}T_n^{\infty}(A)=\int_{z}^{i
\infty}\mathfrak{b}ig[(f|[\Sigma_n])|A-c(n,f,A) \mathfrak{b}ig](u-z)^w du$ where $c(n,f,A)$ is the sum of the terms
involving $a_0(f|A_M)$ (which is independent of $u$ since $j(M,u)=d_M$ for $M\in M_n^\infty$). Since
the integral converges, we must have $c(n,f,A)=a_0(f|[\Sigma_n]|A)$ (which can alsp be proved directly, using
Prop. \ref{pa1}), hence the last expression equals $\widetilde{f|[\Sigma_n]} (A)$.
(b) The proof is the same as of Corollary \ref{c4.3}.
(c) This follows from part (a) and the decomposition \eqref{7.6}
\end{proof}
\mathfrak{b}egin{prop}\,\langle\,bel{pa6} Assume the hypotheses of Prop. \ref{pa5} and furthermore that $\Sigma_n^\vee$
satisfies \eqref{eq_star}. We have for all $\widehat{P},\widehat{Q}\in\widehat{W}_w^\Gamma$
\[
\{\widehat{P}|_{\Sigma}\widetilde{T}_n, \widehat{Q}\} = \{\widehat{P}, \widehat{Q}|_{\Sigma^\vee}\widetilde{T}_n \}.
\]
\end{prop}
\mathfrak{b}egin{proof}
As in the first proof of Theorem \ref{thm_equiv}, we decompose
$\widehat{P}=R+\rho_f^+ +\rho_g^-+\widehat{\rho}_e$ with $R\in C_w^\Gamma$, $f,g\in S_k(\Gamma)$, $e\in M_k(\Gamma)$. Taking into
account Theorem \ref{pa4}, Proposition \ref{pa5} and the fact that the adjoint of the operator $[\Sigma_n]$ is
$[\Sigma_n^\vee]$ with respect to the extended Petersson inner product on $M_k(\Gamma)$ \cite{PP12}, it remains to
show that
\mathfrak{b}egin{equation}\,\langle\,bel{8.14}
\{R|_{\Sigma}\widetilde{T}_n, \widehat{\rho}_e\} = \{R, \widehat{\rho}_{e|[\Sigma_n^\vee]} \}.
\end{equation}
We use Prop. \ref{l7.3}. Let $R=R'|(1-S)$, with $R'(A)=c(A)\in \mathbb{C}$ and $R'|(1-T)=0$. By \eqref{hecke}
$R|_{\Sigma}\widetilde{T}_n= R'|_{\Sigma} T_n^\infty (1-S)$, and for $A\in \Gamma_1$ we have
$R'|_{\Sigma} T_n^\infty(A)= \sum_{M\in M_n^\infty} c(A_M)d_M^w $
where $d_M$ is the lower right entry of $M$, and $MA^{-1}=A_M^{-1} M_A$ with $A_M\in \Gamma_1$, $M_A\in \Sigma_n$.
The left side of \eqref{8.14} becomes (up to a constant which we ignore in the right side as well)
\[
\text{LHS}= \sum_{M\in M_n^\infty} \sum_{\substack{A\in \Gamma\mathfrak{b}ackslash \Gamma_1\\
MA^{-1}\in A_M^{-1}\Sigma_n }}c(A_M) a_0(e|A)d_M^w.
\]
Since $c(AT)=c(A)$, $a_0(e|AT)=a_0(e|A)$, we can replace $M$ by $T^i M T^j$ in the interior sum without changing
it. Therefore, if we write $M=M_{a,d,b}=\leqslantft(\mathfrak{b}egin{smallmatrix}
a & b \\ 0 & d \end{smallmatrix}\right)$ and fix $d$, the interior sum depends only on $b$ modulo
$g_{a,d}=\mathfrak{g}cd(a,d)$ and we have
\[\text{LHS}= \sum_{\substack{d|n \\ b \text{ mod } {g_{a,d}}}} \sum_{\substack{A\in \Gamma\mathfrak{b}ackslash \Gamma_1\\
M_{a,d,b}A^{-1}\in A_{M}^{-1}\Sigma_n }}c(A_M) a_0(e|A){\scriptscriptstyle F}rac{d^{w+1}}{g_{a,d}}.
\]
For the right side of \eqref{8.14}, from the proof of Prop. \ref{pa5} we have for $B\in \Gamma_1$
$$a_0(e|[\Sigma_n^\vee]|B)=\sum_{\substack{M\in M_n^\infty\\ MB^{-1}\in B_M^{-1} \Sigma_n^\vee }}
a_0(e|B_M){\scriptscriptstyle F}rac{n^{w+1}}{d_M^k} .$$
Since $MB^{-1}\in B_M^{-1} \Sigma_n^\vee\iff M^\vee B_M^{-1} \in B^{-1} \Sigma_n$, the right side of \eqref{8.14}
becomes, after interchanging $B_M$ and $B$
\mathfrak{b}egin{equation*}
\mathfrak{b}egin{split}
\text{RHS} &=\sum_{M\in M_n^\infty}\sum_{\substack{B\in \Gamma\mathfrak{b}ackslash \Gamma_1\\
M^\vee B^{-1}\in B_M^{-1}\Sigma_n }}c(B_M) a_0(e|B) {\scriptscriptstyle F}rac{n^{w+1}}{d_M^k} \\
&=\sum_{\substack{d|n \\ b \text{ mod } {g_{a,d}}}}\sum_{\substack{B\in \Gamma\mathfrak{b}ackslash \Gamma_1\\
M_{a,d,b}^\vee B^{-1}\in B_{M}^{-1}\Sigma_n }} c(B_M) a_0(e|B) {\scriptscriptstyle F}rac{a^{w+1}}{d} {\scriptscriptstyle F}rac{d}{g_{a,d}}
\end{split}
\end{equation*}
where the second line follows by writing $M=M_{a,d,b}$ as before. Comparing the expressions obtained for RHS and
LHS finishes the proof of \eqref{8.14}.
\end{proof}
From the duality in Proposition \ref{l7.3} and from Propositions \ref{pa5}, \ref{pa6} we immediately obtain:
\mathfrak{b}egin{corollary}\,\langle\,bel{c8.9}
Assume that both $\Sigma_n$ and
$\Sigma_n^\vee$ satisfy property \eqref{eq_star}. There exist bases of $\mathcal{E}_k(\Gamma)$ and $C_w^\Gamma$
such that the matrix of the operator $[\Sigma_n^\vee]$ acting on $\mathcal{E}_k(\Gamma)$ is the same as the matrix
of the operator $|_\Sigma \tilde{T}_n$ acting on $C_w^\Gamma$.
\end{corollary}
As an application, we let $\Gamma=\Gamma_1(N)$, and $\chi$ a character modulo $N$, and we show
that the trace of Hecke operators $T_n$ on the Eisenstein subspace
$\mathcal{E}E_k(N, \chi)\subset M_k(N,\chi)$ is the same as the trace of $\widetilde{T}_n$ on $C_w^{\Gamma,\chi}$ (see
$\S$\ref{sec5.2} for the notation). For $\Gamma_1$, when $C_w^{\Gamma_1}=<X^w-1>$, a direct proof is immediate, but
for $\Gamma=\Gamma_1(N)$ it seems difficult to prove the statement without using the dual space $\widehat{E}_w^\Gamma$ and
the pairing $\{\cdot, \cdot\}$.
\mathfrak{b}egin{prop}\,\langle\,bel{p8.9} $\mathrm{(a)}$ Let $\Gamma=\Gamma_1(N)$ and
let $\widehat{E}_w^{\Gamma,\chi}\subset \widehat{W}_w^\Gamma$ be the image of the Eisenstein subspace
$\mathcal{E}E_k(N,\chi)\subset M_k(\Gamma)$ under the map $f\rightarrow \widehat{\rho}_f$. For $(n,N)=1$ we have
\[
\mathrm{Tr}(\mathcal{E}E_k(N, \chi)|T_n)=\mathrm{Tr}(\widehat{E}_w^{\Gamma,\chi}|_{\mathcal{D}elta} \widetilde{T}_n)=\mathrm{Tr}(C_w^{\Gamma,\chi}|_{\mathcal{D}elta} \widetilde{T}_n).
\]
$\mathrm{(b)}$ For $\Gamma=\Gamma_0(N)$ and $n\|N$, let $\Theta_n$ be the double coset and let $W_n$ be
the Atkin-Lehner operator defined in Section \ref{sec5.11}. We have
\[
\mathrm{Tr}(\mathcal{E}E_k(\Gamma)|W_n)=\mathrm{Tr}(\widehat{E}_w^{\Gamma}|_{\Theta} \widetilde{T}_n)=\mathrm{Tr}(C_w^{\Gamma}|_{\Theta} \widetilde{T}_n).
\]
\end{prop}
\mathfrak{b}egin{proof}
(a) The duality in Prop. \ref{l7.3} between $C_w^\Gamma$ and $\widehat{E}_w^{\Gamma}$
with respect to the pairing $\{\cdot, \cdot\}$ implies dualities between $C_w^{\Gamma,\chi}$ and $\widehat{E}_w^{\Gamma,
\ov{\chi}}$. Therefore, taking $\Sigma_n=\mathcal{D}elta_n$ for $(n,N)=1$ in Corollary \ref{c8.9}, it follows that the
eigenvalues of $|_\mathcal{D}elta \widetilde{T}_n$ on $C_w^{\Gamma,\chi}$ are the same as the eigenvalues of $T_n^*=[\mathcal{D}elta_n^\vee]$
on $\mathcal{E}E_k(N, \ov{\chi})$, which are the same as the eigenvalues of $T_n$ on $\mathcal{E}E_k(N, \chi)$ (the latter
space has a basis of eigenforms for $T_n$ with $(n,N)=1$).
(b) The claim follows from Corollary \ref{c8.9}, using the fact that $\Theta_n=\Theta_n^\vee$.
\end{proof}
\mathfrak{b}egin{remark} \,\langle\,bel{r8.10}
Prop \ref{p8.9} shows that for $\Gamma=\Gamma_1(N)$ and $(n,N)=1 $ we have
$$\mathrm{Tr}(W_w^{\Gamma,\chi}|_{\mathcal{D}elta}\widetilde{T}_n)=\mathrm{Tr}(M_k(N,\chi)|T_n)+\mathrm{Tr}(S_k(N, \chi)|T_n),$$
and the same for Atkin-Lehner operators on $\Gamma_0(N)$.
For $\Gamma=\Gamma_1$, this fact was an ingredient used by Zagier to sketch an elementary proof of
the Eichler-Selberg trace formula, by computing directly the left side for an
appropriately chosen $\widetilde{T}_n$ \cite{Za93}. A generalization of this approach giving a simple trace
formula for $M_k(N,\chi)$ is work in progress of the second author and Don Zagier.
\end{remark}
\subsection{Extra relations revisited}\,\langle\,bel{sec7.2}
Theorem \ref{pa4} gives another way of determining the extra relations satisfied by all
period polynomials of cusp forms which are independent of the period relations.
Assuming that $\widehat{\rho}^-$ is an isomorphism (see Proposition \ref{p7.4}), it follows that there
exist $g\in \mathcal{E}E_k(\Gamma)$ such that $\widehat{\rho}^-_g$ form a basis for $(\widehat{E}_w^\Gamma)^-$. Since the
pairing $\{\cdot ,\cdot \}$ is nondegenerate, it follows that the linear relations
$\{P, \overline{\widehat{\rho}^-_g} \}=0$ are satisfied by $P=\rho_f^{+}$, for all $f\in S_k(\Gamma)$,
but they are not satisfied by some $P\in (C_w^\Gamma)^+$. A similar argument applies to
determine the relations satisfied by $\rho_f^{-}$, when $(C_w^\Gamma)^-\ne 0$ and $\widehat{\rho}^+$ is an
isomorphism. These linear relations can be used to define other versions of the linear forms
$\,\langle\,mbda_+, \,\langle\,mbda_-$ in Proposition
\ref{p6.2}, which are entirely explicit once the period polynomials of Eisenstein series are
determined.
As an example we take $\Gamma=\Gamma_1(N)$, and we assume $k\mathfrak{g}e 3$. Then the Eisenstein
subspace $\mathcal{E}E_k(\Gamma)$ has a basis of Eisenstein series which are Hecke eigenforms for the
Hecke operators of index coprime with the level \cite[Ch. 5]{DS}. Their period polynomials for
the identity coset can be determined in terms of special values of Dirichlet $L$-functions by
Proposition \ref{pa1}. For other cosets $A\in \Gamma\mathfrak{b}ackslash \Gamma_1$ the period
polynomials of the Hecke eigenforms are harder to compute. Instead, consider a second
basis, consisting of Eisenstein series which vanish at all but one cusp, so that the action
$|A$ permutes the elements of this basis. The elements of the second basis can be decomposed in
terms of Hecke eigenforms, so their period polynomials corresponding to all cosets $A$ can be
determined explicitly. For $\Gamma_0(N)$ we are planning
to return to this question in a future work.
\noindent\textbf{\small Acknowledgments.} Part of this work was completed at the
Max Planck Institute in Bonn, which provided financial support and a great working
environment. We would like to thank Don Zagier for inspiring conversations. The
first author was partially supported by the CNCSIS grant PN-II-RU-TE-2012-3-0455. The second
author was partially supported by the European Community grant PIRG05-GA-2009-248569.
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\end{document}
|
\begin{document}
\title[AR-comp. of $\kbp$ for a cl.t. alg of type $\widetilde{A}$]{The
Auslander-Reiten components of $\kbp$ for a
cluster-tilted algebra of type $\widetilde{A}$}
\author{Kristin Krogh Arnesen}
\author{Yvonne Grimeland}
\address{Institutt for matematiske fag\\ NTNU\\ 7491 Trondheim\\ Norway}
\email{[email protected]}
\email{yvonne\[email protected]}
\begin{abstract}
We classify the Auslander-Reiten components of $\kbp$, where $\Lambda$ is
a cluster tilted algebra of type $\widetilde{A}$. The main tool is the
combinatoric description of the indecomposable complexes in $\kbp$ via
homotopy strings and homotopy bands.
\end{abstract}
\maketitle
\section{Introduction}
The derived category of an abelian category was introduced by
Grothen\-dieck and Verdier in the early 1960s, and in the 1980s,
Happel started studying the derived category of a finite dimensional
algebra \cite{happel}. The module category is embedded into the
derived category of the algebra, and this expansion to larger
categories provided a new tool for comparing and distinguishing the
module categories of two algebras.
Let $\Lambda$ be a finite dimensional $k$-algebra, where $k$ is an
algebraically closed field, and let $\mathrm{mod}~ \Lambda$ denote the
category of finitely generated left $\Lambda$-modules. The derived
category of $\Lambda$ is denoted by $D(\operatorname{mod}\nolimits \Lambda)$, with suspension
functor called shift and denoted by $[1]$. Two important subcategories
is the bounded derived category, denoted by $D^b(\operatorname{mod}\nolimits \Lambda)$, and
its subcategory $\kbp$, the bounded homotopy category of finitely
generated projective $\Lambda$-modules. One way of describing $D(\operatorname{mod}\nolimits
\Lambda)$ is to describe its Auslander-Reiten (hereafter abbreviated
AR) structure. In general, the subcategory $\kbp$ has AR-triangles
whenever $\Lambda$ has finite Gorenstein dimension. In particular,
there is an explicit description of the AR-structure when $\Lambda$ is
gentle \cite{bob}.
The gentle algebras form a subclass of the special biserial
algebras, introduced by Skowro\'nski and Waschb\"{u}sch in 1983
\cite{specialbiserial}. The AR-structure of the module category of a
gentle algebra $\Lambda\cong kQ/I$, where $I$ is an admissible ideal,
can be combinatorially described in terms of $Q$ and $I$
\cite{tamebiserial}. More recently, the AR-structure of $\kbp$ has
also been given combinatorially for gentle algebras. This was done in
2011, when Bobi\'nski gave a combinatorial algorithm for computing the
AR-triangle starting in any given indecomposable object of $\kbp$,
where $\Lambda$ is gentle \cite{bob}.
The foundation for Bobi\'nski's algorithm is diverse. In 2003,
Bekkert and Merklen showed that the indecomposable objects of $\kbp$
are the complexes arising from so called homotopy strings and
homotopy bands (in Bobi\'nski's renaming) \cite{bekkert}. Moreover,
Bobi\'nski takes advantage of the Happel functor, introduced by
Happel in \cite{happel}. The Happel functor is an exact functor of
triangulated categories $\Psi \! : D^b(\operatorname{mod}\nolimits \Lambda) \rightarrow
\underline{\mathrm{mod}}~\hat{\Lambda}$, where the latter category is
the stable module category of the repetitive algebra of
$\Lambda$. When $\Lambda$ is gentle, the repetitive algebra
$\hat{\Lambda}$ is combinatorially described in terms of strings, see
Ringel \cite{ringel}. Since the repetitive algebra of a gentle
algebra is special biserial, it is even possible to describe the
AR-sequences of $\ensuremath{\underline{\mathrm{mod}}~\hat{\Lambda}}$, using the methods of Wald and Waschb\"usch
\cite{tamebiserial}.
In its original appearance, the Happel functor is rather abstract and
no explicit construction is given. Nevertheless, some of its
properties are quite remarkable: It is always full and faithful, and
if $\Lambda$ is of finite global dimension, it is a
triangle-equivalence. It also extends the inclusion functor embedding
$\mathrm{mod}~\Lambda$ into $\ensuremath{\underline{\mathrm{mod}}~\hat{\Lambda}}$. By using all the known structure
of $\Lambda$, $\mathrm{mod}~\Lambda$, $\ensuremath{\underline{\mathrm{mod}}~\hat{\Lambda}}$ and $\kbp$ when
$\Lambda$ is gentle, Bobi\'nski constructed a formula for the Happel
functor in the gentle case.
Special classes of gentle algebras include some \emph{cluster-tilted
algebras}. The cluster-tilted algebras were introduced in 2007 by
Buan, Marsh and Reiten \cite{BMR}. In 2010, Assem, Br\"ustle,
Charbonneau-Jodoin and Plamondon showed that cluster-tilted algebras
of type $A$ and $\widetilde{A}$ are gentle \cite{abcp}. The mutation
classes of type $\widetilde{A}$ and the derived equivalences between
cluster-tilted algebras of type $\widetilde{A}$ are described by
Bastian \cite{bastian}.
Also worth noting is the derived invariant described by
Avella-Alaminos and Geiss \cite{diana}. Using this invariant one can
find an upper bound for the number of AR-components containing
sequences with only one middle term.
In this paper, we classify the AR-components of $\kbp$, where
$\Lambda$ is a cluster-tilted algebra of type $\widetilde{A}$. Our
main tool is Bobi\'nski's algorithm for computing AR-triangles in
$\kbp$.
Parallel to this paper, a paper in progress by Fedra Babaei describes
the AR-structure of $\kbp$ where $\Lambda$ is a
cluster-tilted algebra of type $A$.
The paper is organized as follows: In Section 2, we state the main
result of the paper. Section 3 is an overview of the background theory
needed. In Section 4, we give the details needed from Bastian's
description of the mutation classes of type $\widetilde{A}$. In
Section 5, we introduce the combinatorial concepts of walks and
reductions, which we use to restate Bobi\'nski's algorithm for our
class of algebras in Section 6. The main result is proved in Section
7. Finally, in Section 8, we give an example. Some technical
results are proved in the appendices.
We would like to thank the referee for very helpful comments.
\section{Main results} \label{sec:main}
In this section, we state the main result. By a quiver, we mean a pair
$Q = (Q_0, Q_1)$ where $Q_0$ is a set of vertices and $Q_1$ is a set
of arrows, together with functions $s,t: Q_1 \rightarrow Q_0$
returning the starting and ending vertex of an arrow, respectively.
Let $Q$ be the fixed quiver given by the parameters $x$, $y$, $x'$ and
$y'$, as shown in Figure \ref{firstquiver}. We require that at least
one of $x, y$ and at least one of $x',y'$ are non-zero.
We then define the double quiver $Q'$ to be the quiver with vertices
$Q'_0 = Q_0$ and arrows $Q'_1 = Q_1 \cup Q_1^{-1}$. Let a
\emph{homotopy string} denote a path in $Q'$ with no subpath of the
form $\alpha\alpha^{-1}$ or $\alpha^{-1}\alpha$ for any $\alpha \in
Q_1$. By a \emph{central homotopy string}, we mean a homotopy string
both starting and ending in a vertex marked with $\vartriangle$ or
$\blacktriangle$, excluding the homotopy strings starting with the
arrows $\alpha_1$ or $\beta_1$ and the homotopy strings ending with
the arrows $\alpha_1^{-1}$ or $\beta_1^{-1}$, and the trivial homotopy
strings for the vertices marked with $\blacktriangle$. (Note that in
case one or more of the parameters are zero, then any vertex which is
adjacent to both an oriented cycle of length $3$ and an arrow
$\alpha_i$ or $\beta_j$ should be marked with $\blacktriangle$, and
that any vertex which is adjacent to two oriented cycles of length $3$
should be marked with $\vartriangle$.) A \emph{homotopy band} is a
central homotopy string starting and ending in the same vertex, with
some additional constraints (see Section \ref{subhomstring}).
\begin{figure}
\caption{The quiver $Q$ given by $x$, $y$, $x'$ and $y'$.}
\label{firstquiver}
\end{figure}
Let $\Lambda$ be the algebra $kQ/I$, where $I$ is the ideal generated
by all compositions of two arrows in each directed cycle of length $3$
in $Q$, and $kQ$ is the path algebra of $Q$. This is in fact a
cluster-tilted algebra of type $\ensuremath{\widetilde{A}_n}$, and all cluster-tilted algebras
of type $\ensuremath{\widetilde{A}_n}$ are of this form, up to derived equivalence
\cite{bastian}. We will now describe the AR-components of $\kbp$.
First we state their types and numbers in the following theorem. Let
$\tau$ denote the AR-translate in $\kbp$.
\begin{theorem}\label{mainresult1}
Let $Q$ be a quiver as in Figure \ref{firstquiver}, and let $\Lambda
= kQ/I$ where $I$ is as described above. Then the AR-quiver of
$\kbp$ consists of:
\begin{enumerate}[(i)]
\item A class of tubes of rank one (homogeneous tubes), where up to
shift, the tubes are parametrized by the set of pairs consisting of one
homotopy band and one element of $k$.
\item A class of components given by the parameters $x$ and $y$. If
$x = 0$, we get up to shift a tube of rank $y$. If $x >
0$, we get $x$ components of type $\ensuremath{\mathbb{Z}A_{\infty}}$ with $\tau^{x+y} = [x]$.
\item A class of components given by the parameters $x'$ and $y'$.
If $x' = 0$, we get up to shift a tube of rank $y'$. If $x' > 0$,
we get $x'$ components of type $\ensuremath{\mathbb{Z}A_{\infty}}$ with $\tau^{x'+y'} = [x']$.
\item Up to shift, one $\ensuremath{\mathbb{Z}A_{\infty}^{\infty}}$-component containing all the stalk
complexes corresponding to vertices marked by $\Box$ and
$\blacktriangle$.
\item Up to shift, a class of $\ensuremath{\mathbb{Z}A_{\infty}^{\infty}}$-components, parametrized by the
central homotopy strings.
\end{enumerate}
\end{theorem}
For any quiver as in Figure \ref{firstquiver} the edge of the
components in (ii) can be described easily in terms of the
quiver. Figure \ref{edge} shows the edge of a $\ensuremath{\mathbb{Z}A_{\infty}}$-component
where $\tau^{x+y} = [x]$. The edge of a $\ensuremath{\mathbb{Z}A_{\infty}}$-component
where $\tau^{x'+y'} = [x']$ can be found symmetrically.
\begin{figure}
\caption{The edge of an AR-component of type $\ensuremath{\mathbb{Z}
\label{edge}
\end{figure}
\begin{exmp}\label{ex:hovedeks}
We now consider the quiver $Q$ given in Figure \ref{firstex}, and
the path algebra $\Lambda = kQ/I$ where $I =\left\langle ih, gi, hg,
ed, fe, df, ba, cb, ac, ts, ut, su, qp, rq, pr \right\rangle$.
Figures \ref{edge1} and \ref{edge2} show the edges of two
AR-components of type $\ensuremath{\mathbb{Z}A_{\infty}}$, one given by $x$ and $y$, and one
given by $x'$ and $y'$. The first component has the property
$\tau^5 = [3]$ and the second component has the property $\tau^6 =
[2]$.
\begin{figure}
\caption{The quiver from Example \ref{ex:hovedeks}
\label{firstex}
\end{figure}
\begin{figure}
\caption{The edge of a $\ensuremath{\mathbb{Z}
\label{edge1}
\end{figure}
\begin{figure}
\caption{The edge of a $\ensuremath{\mathbb{Z}
\label{edge2}
\end{figure}
\begin{figure}
\caption{The special $\ensuremath{\mathbb{Z}
\label{uglycomp}
\end{figure}
\end{exmp}
Figures \ref{edge1} and \ref{edge2} show the edges of two
AR-components of type $\ensuremath{\mathbb{Z}A_{\infty}}$, one given by $x$ and $y$, and one
given by $x'$ and $y'$. The first component has the property $\tau^5
= [3]$ and the second component has the property $\tau^6 = [2]$.
Figure \ref{uglycomp} shows part of the structure in the special
$\ensuremath{\mathbb{Z}A_{\infty}^{\infty}}$-component. In particular, it shows the irreducible maps
between the stalk complexes.
The remaining $\ensuremath{\mathbb{Z}A_{\infty}^{\infty}}$-components are parametrized by the central
homotopy strings, that is, each component contains exactly one central
homotopy string. Note that also the trivial homotopy strings
corresponding to the vertices $4$, $2$, $1$ and $14$ are central
homotopy strings.
\section{Background}
In this section we give an overview of the theory needed to prove
Theorem \ref{mainresult1}. First we give the definition of a gentle
algebra. Then we describe $\kbp$ via homotopy strings and homotopy
bands, for the gentle algebra $\Lambda$. Finally, we state some
results about the almost split triangles and components in the
AR-quiver of $\kbp$.
\subsection{Gentle algebras}\label{gentledef}
Let $\Lambda$ be isomorphic to $kQ/I$, for some quiver $Q$ and some
admissible ideal $I$. Then $\Lambda$ is called \textit{special
biserial} ~\cite{specialbiserial} if the following are satisfied:
\begin{enumerate}[(a)]
\item for each vertex $x$ of $Q$ there are at most two arrows
starting in $x$ and at most two arrows ending in $x$, and
\item for any arrow $\alpha$ in $Q$ there is at most one arrow
$\beta$ such that $\alpha\beta \notin I$ and at most
one arrow $\gamma$ such that $\gamma \alpha \notin I$.
\end{enumerate}
Furthermore, if $I$ consists of only zero-relations, then $\Lambda$ is
a \textit{string algebra}, and if in addition all the relations in $I$
have length $2$ and for any arrow $\alpha$ in $Q$ there is at most
one arrow $\beta$ such that $\alpha\beta \in I$ and at most one arrow
$\gamma$ such that $\gamma \alpha \in I$, then $\Lambda$ is a
\textit{gentle} algebra.
We now state an equivalent definition of a gentle algebra, see
~\cite{bob}. This definition will be used later in the paper.
\begin{definition}\label{gentlealternativ}
A finite dimensional algebra $\Lambda = kQ/I$ is \emph{gentle} if
there exist two functions $S,T: Q_1 \rightarrow \{-1,1\}$ satisfying
the following:
\begin{enumerate}[(a)]
\item if $\alpha \neq \beta$ start in the same vertex, then
$S\alpha = -S\beta$,
\item if $\alpha \neq \beta$ end in the same vertex, then $T\alpha
= -T\beta$,
\item if $\alpha$ starts in the vertex where $\beta$ ends, and
$\alpha \beta$ is not in $I$, then $S\alpha = -T\beta$,
\item if $\alpha$ starts in the vertex where $\beta$ ends, and
$\alpha \beta$ is in $I$, then $S\alpha = T\beta$.
\end{enumerate}
\end{definition}
\subsection{Homotopy strings and the category
$\kbp$}\label{subhomstring}
In this subsection we first introduce the concept of homotopy strings
for a gentle algebra $\Lambda$, and given a homotopy string explain
how one can construct an associated string complex in $\kbp$. We also
discuss some special homotopy strings called homotopy bands, which in
addition to string complexes give rise to band complexes in $\kbp$.
Finally, we state a result giving the connection between
indecomposable objects of $\kbp$ and homotopy strings and bands.
Let $\Lambda\cong kQ/I$ be a gentle algebra. We now want to explain
what the homotopy strings associated with $\Lambda$ are. First we
define the double quiver $Q'$ of $Q$: Let $Q_0' = Q_0$, and $Q_1' =
Q_1 \cup Q_1^{-1}$, where $Q_1^{-1}$ is the set of formal inverses of
the arrows of $Q$; that is, for each $\alpha: x \rightarrow y$ in $Q$,
we have $\alpha^{-1}: y \rightarrow x$ in $Q'$. We also add formal
inverses of the trivial paths of $Q$: For a trivial path $1_x$ in $Q$
we add the inverse $1_x^{-1}$ to $Q'$.
We define $(\alpha^{-1})^{-1} = \alpha$ and $(1_x^{-1})^{-1} = 1_x$
and extend the functions $s,t:Q'_1 \rightarrow Q_0$ to include
$s(1_x^{\varepsilon}) = t(1_x^{\varepsilon}) = x$ for all vertices $x$
and $\varepsilon \in \{-1,1\}$. We define the \emph{homotopy strings
associated with $\Lambda$} to be all paths in $Q'$ which contain no
subpath of the form $\alpha\alpha^{-1}$ or $\alpha^{-1}\alpha$ for
$\alpha\in Q_1$. Note that each vertex $x$ in $Q_0$ gives rise to two
\emph{trivial homotopy strings}, namely the paths $1_x$ and $1_x^{-1}$
in $Q'$. We also consider the empty homotopy string, denoted by
$\emptyset$.
A non-trivial, non-empty homotopy string $\omega$ can be written as
$\omega = \alpha_l \alpha_{l-1} \cdots \alpha_1$ where for each $1\leq
i \leq l$ the \emph{i}'th \emph{letter} $\alpha_i(\omega)=\alpha_i$ is
one arrow or the inverse of one arrow, and $l(\omega)=l$ is the number
of letters, called the \emph{length} of $\omega$. If $\omega$ is a
trivial or empty homotopy string, then $l(\omega)=0$. A homotopy
string $\omega$ is called \emph{direct} if all of the letters in
$\omega$ are arrows, and \emph{inverse} if all of the letters are
inverse arrows. We define $\omega^{-1} = \alpha_1^{-1} \cdots
\alpha_l^{-1}$.
We now state when \emph{composition} of non-trivial and non-empty
homotopy strings is defined; two homotopy strings
$\omega=\alpha_l\cdots\alpha_1$ and
$\omega'=\alpha'_{l'}\cdots\alpha'_1$ where $l,l' \geq 1$, can be
composed if $s(\omega) = t(\omega')$ and one of the following
statements holds:
\begin{itemize}
\item $\alpha_1$ is direct and $\alpha'_{l'}$ is inverse and $\alpha_1^{-1}
\neq \alpha'_{l'}$,
\item $\alpha_1$ is inverse and $\alpha'_{l'}$ is direct and $\alpha_1^{-1}
\neq \alpha'_{l'}$,
\item $\alpha_1$ and $\alpha'_{l'}$ are both direct and
$\alpha_1\alpha'_{l'}$ is in $I$, or
\item $\alpha_1$ and $\alpha'_{l'}$ are both inverse and
${\alpha'}_{l'}^{-1}\alpha_1^{-1}$ is in $I$.
\end{itemize}
The composition $\omega\cdot\omega'$ is the path $\omega\omega'$ in
$kQ'$, which is also a homotopy string.
We now define composition of homotopy strings involving trivial
homotopy strings. To do this we first need to extend the functions
$S,T$ from Definition \ref{gentlealternativ} to homotopy strings; for
any arrow $\alpha$ in $Q_1$, we define $S\alpha^{-1}=T\alpha$ and
$T\alpha^{-1}=S\alpha$. Furthermore, define $S1_x^{\varepsilon} =
\varepsilon$ and $T1_x^{\varepsilon}=-\varepsilon$ for $\varepsilon\in
\{-1,1\}$ and $x \in Q_0$. Let $\omega$ be a non-trivial and non-empty
homotopy string, then the composition $\omega \cdot 1_x^{\varepsilon}$
is defined (and equals $\omega$) if $x = s\omega$, and one of the
following statements holds:
\begin{itemize}
\item $\varepsilon = S(\alpha_1(\omega))$ and $\alpha_1(\omega)$ is an
arrow, or
\item $\varepsilon = -S(\alpha_1(\omega))$ and $\alpha_1(\omega)$ is
an inverse arrow.
\end{itemize}
Similarly, the composition $1_x^{\varepsilon}\cdot \omega$ is defined
(and equals $\omega$) if $x=t\omega$, and one of the following
statements holds:
\begin{itemize}
\item $\varepsilon = T(\alpha_l(\omega))$ and $\alpha_l(\omega)$ is an
arrow, or
\item $\varepsilon = -T(\alpha_l(\omega))$ and $\alpha_l(\omega)$ is
an inverse arrow.
\end{itemize}
The composition $1_x^{\varepsilon}\cdot 1_{x'}^{\varepsilon'}$ is
defined (and equals $1_x^{\varepsilon}$) if $x=x'$ and $\varepsilon =
\varepsilon'$. Note that for any non-empty homotopy string $\omega$,
we have $\emptyset \cdot \omega = \omega$ and $\omega \cdot \emptyset
= \omega$.
For a homotopy string $\omega$ of positive length, there is a unique
partition $\omega=\sigma_L \cdots \sigma_1$ where each $\sigma_i$ is a
homotopy string of positive length; for each $1\leq i\leq L$ the
homotopy strings $\sigma_i$ and $\sigma_{i-1}$ can be composed as
homotopy strings; and none of the homotopy strings $\sigma_i$ can be
partitioned into non-empty non-trivial homotopy strings. We call this
the \emph{homotopy partition} of $\omega$, and the $\sigma_j$'s are
called \emph{homotopy letters}. Define $\omega^{[i]}=\sigma_L\cdots
\sigma_{L-i+1}$ for $i>0$ and $\omega^{[0]}$ to be the trivial
homotopy string $1_{t\omega}^{\varepsilon}$ such that
$1_{t\omega}^{\varepsilon}\cdot \omega$ is defined as composition of
homotopy strings.
Furthermore, we define the \emph{degree} of $\omega$, denoted by
$\deg(\omega)$, to be the number of direct homotopy letters of
$\omega$ minus the number of inverse homotopy letters of $\omega$.
The degree of a trivial homotopy string is defined to be $0$.
A non-trivial and non-empty homotopy string $\omega=\sigma_L\cdots
\sigma_1$ with $s\omega=t\omega$ is called a \emph{homotopy band} if
$\deg(\omega)=0$; either $\sigma_{L}$ and $\sigma^{-1}_1$ are both
direct homotopy letters or $\sigma_{L}^{-1}$ and $\sigma_1$ are both
direct homotopy letters; $\sigma_1\cdot\sigma_{L}$ is defined as
composition of homotopy strings and there is no homotopy string
$\widetilde{\omega}$ with $l(\widetilde{\omega})<l(\omega)$ such that
$s\widetilde{\omega}=t\widetilde{\omega}$ and
$\omega=\widetilde{\omega}^n$ for some positive integer $n$. If
$\omega= \sigma_L\cdots \sigma_1$ is a homotopy band, then the
homotopy string $\omega' = \sigma_{j-1}\cdots \sigma_1\sigma_L \cdots
\sigma_j$ is a \emph{rotation} of $\omega$ if $\omega'$ is a homotopy
band.
We now give an explicit description of how to construct a complex
$X_{m,\omega}$ in $\kbp$ from a homotopy string $\omega$ associated to
$\Lambda$ and an integer $m$. Let $P_x$ denote the indecomposable
projective in vertex $x$. If $\omega = \emptyset$, then the complex
$X_{m,\omega}$ is the zero complex for all integers $m$. If $\omega$
is trivial, that is, $\omega=1_x^{\varepsilon}$ for $\varepsilon \in
\{-1,1\}$, then the complex $X_{m,\omega}$ is the stalk complex
\[
\xymatrix{
\cdots \ar[r] &0 \ar[r] & P_x \ar[r] &0 \ar[r] &\cdots
}
\]
with $P_x$ in degree $m$. If $l(\omega) > 0$ we have the homotopy
partition $\omega = \sigma_L \cdots \sigma_1$ with $L \geq 1$. Let
$\sigma_i^*$ be the direct homotopy string in $\{\sigma_i,
\sigma_i^{-1}\}$. Then $\sigma_i^*$ gives rise to a map
\[P_{t\sigma_i^*} \stackrel{p_{\sigma_i^*}}{\longrightarrow}
P_{s\sigma_i^*}~,\] sending $e_{t\sigma_i^*}$ to $\sigma_i^*
e_{s\sigma_i^*} = \sigma_i^*$, where $e_x$ is the primitive idempotent
corresponding to the vertex $x$ in $Q$.
For $m'\in\mathbb Z$ define an index set $\mathcal
I_{m'}(m,\omega)$ by
\[
\mathcal I _{m'}(m,\omega) = \{ i \in [0,L] ~|~ \deg(\omega^{[i]}) + m
= m' \} ~~.
\]
The object in degree $m'$ of $X_{m,\omega}$ is the direct sum
\[\bigoplus_{i\in \mathcal I_{m'}(m,\omega)} P_{s (\omega^{[i]})}\ .\]
The differentials are defined componentwise, if
$\delta_X^{m'}$ is the differential from degree $m'$ to degree $m'+1$,
we define
\[
(\delta_X^{m'})_{i,j} = \left\{\begin{array}{ll}
p_{\sigma_{L-i}^{-1}} & i = j-1 \text{ and } \sigma_{L-i} \text{ is
inverse}\\
p_{\sigma_{L-j}} & i = j+1 \text{ and } \sigma_{L-j} \text{ is
direct}\\
0 & \text{otherwise}
\end{array}\right.
\]
for $j \in \mathcal I _{m'}(m,\omega)$ and $i \in \mathcal
I_{m'+1}(m,\omega)$. The complexes $X_{m,\omega}$ constructed in this
way are called string complexes. Observe that $X_{m,\omega} \cong
X_{m+\deg \omega,\omega^{-1}}$.
\begin{exmp}
Consider the algebra $\Lambda = kQ/I$ given in Example
\ref{ex:hovedeks}. The homotopy string $\omega = u^{-1}cbf$ associated with
$\Lambda$ has homotopy partition $\omega = u^{-1}
\cdot c \cdot bf$. We compute the string complex $X_{0,\omega}$ as
follows:
We have $\mathcal I_{-1}(0,\omega) = \{1\}$; $\mathcal
I_{0}(0,\omega) = \{0,2\}$ and $\mathcal I_1(0,\omega) = \{3\}$.
For the differentials in the complex, we get $(\delta_X^{-1})_{0,1}
= p_{u}$; $(\delta_X^{-1})_{2,1} = p_{c}$; $(\delta_X^{0})_{3,0}
= 0$ and $(\delta_X^{0})_{3,2} = p_{bf}$. Hence, the complex
$X_{0,\omega}$ is
\[
\xymatrix{
\cdots \ar[r] &0 \ar[r] &P_1 \ar[r]^(0.35){\left(\begin{smallmatrix}p_u \\
p_c \end{smallmatrix}\right)} &P_{16} \oplus P_3
\ar[r]^(0.6){\left(\begin{smallmatrix}
0 & p_{bf} \end{smallmatrix}\right)} &P_5
\ar[r] &0 \ar[r] &\cdots
}
\]
with $P_1$ in degree $-1$.
\end{exmp}
Since homotopy bands are homotopy strings they give rise to
complexes as described above, and in addition each homotopy band
$\omega$ also gives rise to a family of band complexes
$Y_{m,\omega,\mu}$ in $\kbp$, where $m\in\mathbb Z$ and $\mu$ is an
indecomposable automorphism of a finite dimensional vector space.
Consider the equivalence relation on the set of all homotopy strings
generated by inverting a homotopy string, and let $\mathfrak W$ be a
complete class of representatives under this equivalence relation.
Similarly, we consider the equivalence relation on the set of all
homotopy bands generated by inverting a homotopy band and identifying
each homotopy band with its rotations, and let $\mathfrak B$ be a
complete set of representatives under this equivalence relation.
\begin{proposition}[{\cite[Theorem 3]{bekkert}}, see also
{\cite[Proposition 3.1]{bob}}]
Let $\Lambda\cong kQ/I$ be a gentle algebra. Then the
indecomposable objects of $\kbp$ are exactly the string complexes
$X_{m,\omega}$ for $m \in \mathbb{Z}$ and $\omega \in \mathfrak W$,
and the band complexes $Y_{m,\omega,\mu}$ for $m \in \mathbb{Z}$,
$\omega \in \mathfrak B$ and $\mu$ an indecomposable automorphism of a
finite dimensional vector space.
\end{proposition}
\subsection{Almost split triangles in $\kbp$} \label{sec:ast} Before
we state Bobi\'{n}ski's main result, giving the connection between
homotopy strings and almost split sequences, we need a result about
the almost split sequences in the category $\mathcal C$ of
indecomposable automorphisms of finite-dimensional vector spaces over
$k$. Let $\mu: V\rightarrow V$ be an indecomposable object of
$\mathcal C$ where $\dim_kV = n>0$. Since $\mu$ is indecomposable, it
is similar to a Jordan matrix $J_n(\lambda)$ consisting of one Jordan
block, where $\lambda \in k$ is the eigenvalue of $\mu$. Hence the
object $\mu$ of $\mathcal C$ can be represented by the pair
$(\lambda,n)$ and we denote it by $V_n(\lambda)$, as in \cite{karin}.
The following lemma from \cite{karin} gives the AR-sequence starting
in $V_n(\lambda)$ in $\mathcal C$.
\begin{lemma}
Let $\mu=V_n(\lambda)$ be an indecomposable object in the category
$\mathcal C$. Then there is an AR-sequence
\[ \psi: \ \ 0\rightarrow V_n(\lambda) \rightarrow
V_{n-1}(\lambda)\oplus V_{n+1}(\lambda) \rightarrow V_n(\lambda)
\rightarrow 0 \] in $\mathcal C$, where $V_0(\lambda) = 0$.
\end{lemma}
In particular, the AR-structure of $\mathcal C$ consists of
homogeneous tubes, parametrized by the eigenvalues $\lambda$ of the
indecomposable automorphisms.
For each homotopy string $\omega$ we will give combinatorial
definitions of homotopy strings $\omega_+$, $\omega^+$ and
$\omega^+_+$ and integers $m'(\omega)$ and $m''(\omega)$, see Section
\ref{almostsplittriangels} and Appendix \ref{bobinskiappendix}. With
these definitions we can state the following:
\begin{theorem}[\cite{bob}, Main Theorem]\label{bobinskimain}$ _{}$
\begin{itemize}
\item[i)] Let $\omega$ be a homotopy band, $m\in \mathbb{Z}$, $\lambda
\in k$ and $\mu=V_n(\lambda)$ an indecomposable automorphism of
a finite dimensional vector space. Then we have an almost split
triangle in $\kbp$ of the form
\[
Y_{m,\omega,\mu} \rightarrow Y_{m,\omega,\mu_1} \oplus
Y_{m,\omega,\mu_2} \rightarrow Y_{m,\omega,\mu}\rightarrow Y_{m,\omega,\mu}[1]
\]
where $\mu_1 = V_{n-1}(\lambda)$ and $\mu_2=V_{n+1}(\lambda)$.
\item[ii)] Let $\omega$ be a homotopy string, and $m \in
\mathbb{Z}$. Then we have an almost split triangle in $\kbp$ of the
form
\[
X_{m,\omega} \rightarrow X_{m+m'(\omega),\omega_+} \oplus
X_{m,\omega^+} \rightarrow X_{m+m''(\omega),\omega^+_+}\rightarrow
X_{m-1,\omega} ~~~.
\]
\end{itemize}
\end{theorem}
From now on, we will denote the string complex $X_{m,\omega}$ by
$\omega[m]$. We call the integer $m$ the \emph{degree} of the string
complex.
\subsection{Components of the AR-quiver of
$\kbp$}\label{sec:components}
We define the number of middle terms in an AR-triangle $\chi$ to be
$\alpha(\chi)$. Note that from Theorem \ref{bobinskimain}, we have
$\alpha(\chi) \leq 2$ for all AR-triangles $\chi$ in $\kbp$. By
\cite{riedtmann} we know that any component in a stable translation
quiver is of the form $\mathbb Z{\operatorname{D}\nolimits}elta/G$ where ${\operatorname{D}\nolimits}elta$ is a directed
tree and $G$ is an admissible group of automorphisms on
$\mathbb{Z}{\operatorname{D}\nolimits}elta$. It is clear that since any component in the
AR-quiver $\kbp$ is a stable translation quiver, and $\alpha(\chi)
\leq 2$ for all AR-triangles, ${\operatorname{D}\nolimits}elta$ is either $A_n$, $A_{\infty}$ or
$A_{\infty}^{\infty}$ (see also \cite{butlerringel}).
\section{Derived equivalence classes of
$\ensuremath{\widetilde{A}_n}$-quivers} \label{sec:bastian}
In this section we describe representatives of the derived equivalence
classes of the cluster-tilted algebras of type $\ensuremath{\widetilde{A}_n}$. These algebras
are known to be gentle by \cite{abcp}. We now
introduce some notation.
We start by fixing an embedding into the plane, to be able to make the
distinction between the clockwise and the counterclockwise
direction.
Let $\mathcal Q_n^{\star}$ be the class of quivers of the form as in
Figure \ref{normalform}, for some non-negative integers $r_1$, $r_2$,
$s_1$, $s_2$ where $r = r_1 + r_2 > 0$ and $s = s_1 + s_2 > 0$, and $r
+ s = n$, \cite{bastian}. For $Q$ in $\mathcal Q_n^{\star}$ we
denote by $\Lambda_Q$ the path algebra of $Q$ modulo the ideal
generated by all zero-relations which are compositions of two arrows
in the same 3-cycle of $Q$. We have the following:
\begin{theorem}[\cite{bastian}] \label{derivertekvivalent} $\ $
\begin{itemize}
\item[(1)] If $\Lambda$ is a cluster-tilted algebra of type $\ensuremath{\widetilde{A}_n}$, then
there exists $Q$ in $\mathcal Q_n^{\star}$ such that $\Lambda$ and
$\Lambda_Q$ are derived equivalent.
\item[(2)] If $Q$ and $Q'$ belong to $\mathcal Q_n^{\star}$, then
$\Lambda_Q$ and $\Lambda_{Q'}$ are derived equivalent if and only if
they have the same parameters (up to changing the roles of $r_i$
and $s_i$ for $i \in \{1,2\}$).
\end{itemize}
\end{theorem}
\begin{figure}
\caption{Normal form of a quiver with all parameter values non-zero.}
\label{normalform}
\end{figure}
For a quiver $Q$ in $\mathcal Q_n^{\star}$, a clockwise oriented 3-cycle in
$Q$ will be called an $\emph{r-cycle}$ and a counter-clockwise
oriented 3-cycle an $\emph{s-cycle}$. The arrows
$\alpha_{r_2+1},\ldots,\alpha_r$ are called r-arrows, and similarly
the arrows $\beta_{s_2+1},\ldots, \beta_s$ are called s-arrows.
The vertices of a quiver $Q$ in $\mathcal{Q}_n^{\star}$ can be divided into
ten disjoint sets as follows, see Figure \ref{normalform}:
\begin{itemize}
\item $A=\left\{x\in Q_0~|~ x\text{ has valency 2 and is part of an
r-cycle} \right\}$
\item $B=\left\{x\in Q_0~|~ x\text{ is part of two r-cycles}\right\}$
\item $C=\left\{x \in Q_0 ~|~ x\text{ has valency 2 or 3, with
$\alpha_r$ and $\beta_s$ ending here}\right\}$
\item $\widetilde C=\left\{x \in Q_0 ~|~ x\text{ has valency 2 and is
a source}\right\}$
\item $D = \left\{x\in Q_0 ~|~ x\text{ is the starting vertex of an
r-arrow} \right\}$
\item $F=\left\{x\in Q_0 ~|~ x \text{ has valency 4 } \alpha_1 \text{
starting here}\right\}$
\end{itemize}
The sets $A', B'$ and $D'$ are defined similarly as $A, B$ and $D$,
respectively, by replacing r-cycle with s-cycle and r-arrow with
s-arrow. The set $F'$ is defined similarly as $F$ by replacing
``$\alpha_1$ starting here'' with ``$\alpha_r$ ending here''. We will
use the notation $D_{\geq 1}$ for the subset $D\setminus \{D_0\}$.
The types of vertices occurring in a quiver $Q$ in
$\mathcal{Q}_n^{\star}$ depends on the values of the parameters
$r_1,r_2,s_1$ and $s_2$. In particular one should note that if there
is a vertex of type $C,\widetilde{C}, F$ or $F'$ in $Q$,
then there is only one vertex of the respective type. Also, a vertex
of type $F'$ will only occur in the case when both $r_1=0$ and
$s_1=0$. Vertices of type $B$ will only occur when $r_2>1$ and
vertices of type $D$ will only occur when $r_1>0$.
Moreover, as Theorem \ref{derivertekvivalent} states, the roles of
$r_i$ and $s_i$ can be interchanged while preserving the derived
equivalence. Therefore, it is sufficient to consider the cases listed
in Table \ref{parameterverdier}. Figure \ref{types} shows examples of
quivers in the cases 2--5. Case 6 is illustrated in Figure
\ref{normalform}. The first case is hereditary, and will not be
considered in this paper since its derived category is well-known
~\cite{happel}.
\begin{table}[h]
\caption{Variations of the normal form}\label{parameterverdier}
{\renewcommand{1.4}{1.4}
\begin{tabular}{|c|c|c|c|c|l|}
\hline
\textnumero & $r_1$ & $r_2$ & $s_1$ & $s_2$ & Possible vertices \\
\hline
1 & $\neq 0$ & $0$ & $\neq 0$ & $0$ & $\widetilde C, C, D, D'$ \\
\hline
2 & $0$ & $\neq 0$ & $0$ & $\neq 0$ & $A, A', B, B', F, F'$ \\
\hline
3 & $0$ & $\neq 0$ & $\neq 0$ & $0$ & $A,B,C,D'$ \\
\hline
4 & $\neq 0$ & $\neq 0$ & $0$ & $\neq 0$ & $A, A', B, B', C,
D, F$ \\
\hline
5 & $\neq 0$ & $\neq 0$ & $\neq 0$ & $0$ & $A,B,C,D,D'$ \\
\hline
6 & $\neq 0$ & $\neq 0$ & $\neq 0$ & $\neq 0$ &
$A, A',B, B',C,D,D',F$ \\
\hline
\end{tabular}}
\end{table}
\begin{figure}
\caption{\textnumero~~2}
\label{Qtype2}
\caption{\textnumero~~3}
\label{Qtype3}
\caption{\textnumero~~4}
\label{Qtype4}
\caption{\textnumero~~5}
\label{Qtype5}
\caption{Some quivers in $\mathcal{Q}
\label{types}
\end{figure}
\section{{\rm r}-walks and {\rm s}-walks}
In this section we define r- and s-walks, which are special ways of
traversing a quiver in $\mathcal Q_n^{\star}$. We will also define
reduction of homotopy strings. The walks and reductions will later be
used to describe AR-triangles in $\kbp$.
Let $Q$ be a quiver in $\mathcal Q_n^{\star}$. For a vertex $x\in
Q_0$ let a \textit{walk starting in $x$} be a possibly infinite series
of homotopy strings $[w_1, w_2, \ldots]$ with $sw_1 = x$, $sw_i
=tw_{i-1}$ for $i>1$ and such that $w_n\cdots w_2w_1$ for any $n>1$ is
a homotopy string. However, it is not necessary that $w_i\cdot
w_{i-1}$ is defined as composition of homotopy strings.
We define a clockwise r-walk $W = [w_1, w_2, \ldots]$ starting in a
vertex $x$ in $A\cup D$: It is recursively defined by the
function $cw\_r(x)$, where if $x$ is the vertex
\begin{itemize}
\item $A_i$ for $i> 1$: then $cw\_r(x) =
\gamma_{2(i-1)}\gamma_{2i-1}$, going from $A_i$ to $A_{i-1}$
\item $A_1$: $\left\{ \begin{array}{ll} r_1 = 0 & \text{then
$cw\_r(x) = \gamma_{2r_2}\beta_s\cdots \beta_1\gamma_1$, going from $A_1$ to $A_{r_2}$ } \\
r_1 > 0 & \text{then $cw\_r(x) = \alpha_r^{-1} \beta_s
\cdots \beta_1 \gamma_1$, going from $A_1$ to $D_{r_1-1}$} \\
\end{array} \right.$
\item $D_i$ for $i>0$: then $cw\_r(x) = \alpha_{r_2 +i}^{-1}$, going
from $D_i$ to $D_{i-1}$
\item $D_0$: $\left\{ \begin{array}{ll} r_2 = 0 & \text{then $cw\_r(x)
=
\alpha_r^{-1}\beta_s\cdots \beta_1$, going from $D_0$ to $D_{r_1-1}$} \\
r_2 > 0 & \text{then $cw\_r(x) = \gamma_{2r_2}$, going from $D_0$ to $A_{r_2}$} \\
\end{array} \right.$
\end{itemize}
where the vertices $A_i$ and $D_i$ are as shown in Figure
\ref{normalform}. We define $w_1 = cw\_r(x)$. Observe that the vertex
$t(w_1)$ is always in $A\cup D$. Further, we define the $i$th step of
the clockwise r-walk to be $w_i = cw\_r(t(w_{i-1}))$ for
$i>1$. Observe that $w_i = w_{i+r}$ for $i \geq 1$. Note that
$cw\_r(x)$ is always the shortest homotopy string from $x$ to $t(w_1)$
in the clockwise direction.
A clockwise r-walk is illustrated in Figure \ref{cwr}. The vertices of
type $A \cup D$ are marked with $\star$, and the paths between the
$\star$'s in the clockwise direction are the steps of the walk.
Now we extend the function $cw\_r(x)$ to vertices $x$ of type
$B,B',C,D',F$ or $F'$. The \emph{clockwise r-prefix} is the shortest
clockwise homotopy string $w$ with $s(w) = x$, such that there exists
some homotopy string $w'$ such that $ww' = cw\_r(y)$ for some vertex
$y \in A\cup D$. From the above, it follows that $w$ is unique, and we
denote it by $cw\_r\_p(x)$. The extended definition of the clockwise
r-walk is then
\[
cw\_r(x) = \begin{cases}
cw\_r(x) & x \in A \cup D \\
cw\_r\_p(x) & x \in B \cup B' \cup C \cup D' \cup F \cup F'
\end{cases}
\]
Note that according to Table \ref{parameterverdier}, we always have
$r_2 > 0$. However, we have included the case $r_2 = 0$ in the
definition of clockwise r-walk, as this is needed to give a complete
definition of counter-clockwise s-walk which will be defined as a
mirror image of the clockwise r-walk.
Next we define a \emph{counter-clockwise r-walk} $V =
[v_1,v_2,\ldots]$ starting in a vertex $x$ in $A \cup C \cup D_{\geq
1}$. It is recursively defined by the function $ccw\_r(x)$, given by
\begin{itemize}
\item $A_i$ for $i < r_2$: then $ccw\_r(x) =
\gamma_{(2(i+1)-1)}^{-1}\gamma_{2i}^{-1}$, going from $A_i$ to $A_{i+1}$
\item $A_{r_2}$: $\left\{ \begin{array}{ll}
r_1 = 0 & \text{then $ccw\_r(x)
=\gamma_1^{-1}\beta_1^{-1}\cdots\beta_{s}^{-1}\gamma_{2r_2}^{-1}$,
going from $A_{r_2}$ to $A_1$} \\
r_1 > 0 & \text{then $ccw\_r(x) =
\alpha_{r_2+1}\gamma_{2r_2}^{-1}$,}\\
& \text{going from $A_{r_2}$ to $D_1$, or from $A_{r_2}$ to $C$ } \\
\end{array} \right.$
\item $C$: $\left\{ \begin{array}{ll} r_2 = 0 & \text{then $ccw\_r(x)
=
\alpha_{1}\beta_{1}^{-1}\cdots \beta_s^{-1}$,}\\ &\text{going from $C$ to $C$, or from $C$ to $D_1$} \\
r_2 > 0 & \text{then, $ccw\_r(x) =
\gamma_1^{-1}\beta_1^{-1}\cdots \beta_s^{-1}$, going from $C$ to $A_{1}$} \\
\end{array} \right.$
\item $D_i$ for $1 \leq i \leq r_1 -1$: then $ccw\_r(x) = \alpha_{r_2
+ i + 1}$, going from $D_i$ to $D_{i+1}$, or from $D_i$ to $C$
\end{itemize}
As for the clockwise case, we define $v_1 = ccw\_r(x)$, and the $i$th
step of the counter-clockwise r-walk is defined by $v_i =
ccw\_r(t(v_{i-1}))$ for $i > 1$. See Figure \ref{ccwr} for an
illustration of the steps in a counter-clockwise r-walk.
The \emph{counter-clockwise r-prefix} for a vertex $x$ in
$B,B',D_0,D',F$ or $F'$ is defined as follows: It is the shortest
counter-clockwise homotopy string $v$ with $s(v) = x$, such that there
exists some homotopy string $v'$ such that $vv' = ccw\_r(y)$ for some
vertex $y \in A\cup C \cup D_{\geq 1}$. Again, $v$ is
unique, and we denote it by $ccw\_r\_p(x)$. We extend the
counter-clockwise r-walk as follows:
\[
ccw\_r(x) = \begin{cases}
ccw\_r(x) & x \in A \cup C \cup D_{\geq 1} \\
ccw\_r\_p(x) & x \in B \cup B' \cup D_0 \cup D' \cup F \cup F'
\end{cases}
\]
For the same reason as for the clockwise r-walk, we have included
the case when $r_2 = 0$.
A \emph{counter-clockwise s-walk} is defined to be the mirror image of
a clockwise r-walk, see Figure \ref{ccws}. A \emph{clockwise s-walk}
is defined as the mirror image of a counter-clockwise r-walk, see
Figure \ref{cws}.
\begin{figure}
\caption{A clockwise r-walk.}
\caption{A counter-clockwise r-walk.}
\caption{A clockwise s-walk.}
\caption{A counter-clockwise s-walk.}
\label{cwr}
\label{ccwr}
\label{cws}
\label{ccws}
\label{fig:}
\label{walks}
\end{figure}
\subsection{Reduction of a homotopy string}\label{redsubkap}
Let $\omega$ be a non-trivial and non-empty homotopy string with one
of the following properties:
\begin{enumerate}[(i)]
\item $t(\omega) \in A \cup D$, and such that
$\alpha_l(\omega)$ is the last letter in the $r$th step of the
clockwise r-walk starting in $t(\omega)$,\label{condition1}
\item $t(\omega) \in A' \cup D'$, and such that
$\alpha_l(\omega)$ is the last letter in the $s$th step of the
counter-clockwise s-walk starting in $t(\omega)$,\label{condition2}
\item $t(\omega) \in A \cup C \cup D_{\geq 1}$, and such that
$\alpha_l(\omega)$ is the last letter in the $r$th step of the
counter-clockwise r-walk starting in $t(\omega)$,\label{condition3}
\item $t(\omega) \in A' \cup C \cup D_{\geq 1}'$, and such that
$\alpha_l(\omega)$ is the last letter in the $s$th step of the
clockwise s-walk starting in $t(\omega)$.\label{condition4}
\end{enumerate}
Observe that a homotopy string $\omega$ satisfies at most one of these
properties.
\begin{definition}\label{reduction1}
Let $\omega$ be a homotopy string satisfying property (i). Let
$w_r$ be the $r$th step of the clockwise r-walk starting
in $t(\omega)$. We define the \emph{clockwise r-reduction} of
$\omega$ to be $\omega'$, where $\omega = \sigma \omega'$ for a
non-trivial homotopy string $\sigma$ satisfying
\begin{itemize}
\item $w_r = \sigma \hat{w}$ for some homotopy string $\hat{w}$, and
\item there is no $\sigma'$ such that $\omega = \sigma' \omega''$
and $w_r = \sigma' \widetilde{w}$ with $l(\sigma') > l(\sigma)$.
\end{itemize}
\end{definition}
Similarly, we define \emph{counter-clockwise r-reduction} by replacing
property (i) by property (iii), and by letting $w_r$ be the $r$th step
of the counter-clockwise r-walk starting in $t(\omega)$. The
\emph{clockwise s-reduction} and \emph{counter-clockwise s-reduction}
are defined analogously.
Note that for some homotopy strings $\omega$, the reduction
removes $\omega$ itself -- that is, $\omega = \sigma \omega'$ where
$\sigma = \omega$. In this case, $\omega'$ is the trivial homotopy
string $1_{s\omega}^{\varepsilon}$ such that $\omega \cdot
1_{s\omega}^{\varepsilon}$ is defined as composition of homotopy
strings. To do this, we fix the string functions $S$ and $T$
described in Definition \ref{gentlealternativ}. See Appendix
\ref{SandT} for details.
\begin{exmp}
Recall Example \ref{ex:hovedeks}. Consider the homotopy string
$\omega = bfed$ associated with $\Lambda$. It is clear that
$\omega$ satisfies property (i). Then the homotopy string $ed$ is
the clockwise r-reduction of $\omega$. Note that this homotopy
string also satisfies property (i), and has the clockwise
r-reduction $d$. In the last case, the homotopy string we remove
is the clockwise r-prefix for vertex $4$.
Moreover, the homotopy string $\nu = jgd$ satisfies property (ii),
and has counter-clockwise r-reduction $gd$. Here, the homotopy
string we remove is the counter-clockwise r-prefix for vertex $6$.
Note that the homotopy string $gd$ does not satisfy properties
(i)--(iv) and hence does not have a reduction.
\end{exmp}
\section{Almost split triangles for string
complexes}\label{almostsplittriangels}
Let $\Lambda$ be a cluster-tilted algebra of type $\ensuremath{\widetilde{A}_n}$, and let $\omega[m]$
be a string complex in $\kbp$. In this section, we give explicit
calculations of the AR-triangle starting in $\omega[m]$ and
the AR-triangle ending in $\omega[m]$.
Theorem \ref{bobinskimain} states that the almost split sequence
starting in $\omega[m]$ is of the form
\[
\omega[m] \rightarrow \omega^+[m+m'(\omega)]\oplus
\omega_+[m] \rightarrow \omega^+_+[m+m''(\omega)] \rightarrow \omega[m-1]
\]
where all the involved homotopy strings and integers can be found
combinatorially by using Bobi\'nski's algorithm (see Appendix
\ref{bobinskiappendix}). Figure \ref{omeganeighbours} shows the
AR-triangle ending in $\omega[m]$, and the AR-triangle starting in
$\omega[m]$.
\begin{figure}
\caption{The AR-triangles starting and ending in $\omega[m]$.}
\label{omeganeighbours}
\end{figure}
\begin{lemma}[\cite{bob}]\label{omegaminuslemma}
We have that $\omega_+ = ((\omega^{-1})^+)^{-1}$ and $\omega^- =
((\omega^{-1})_-)^{-1}$.
\end{lemma}
Hence, if we have a combinatorial description of $\omega^+$ and
$\omega_-$, then we also have a combinatorial description of all
homotopy strings shown in Figure \ref{omeganeighbours}. The
combinatorial descriptions of $\omega^+$ and $\omega_-$ are given in
Tables \ref{omegaplus}--\ref{omegaminustrivial}. In Tables
\ref{omegaplus} and \ref{omegaplustrivial} we include the integer
$m'(\omega)$. The integer $m''(\omega)$ is equal to $m'(\omega)$ if
$\omega^+$ is non-empty, and otherwise equal to $m'(\omega_+)$.
\begin{proposition}\label{tabellproposisjon}
Let $\omega[m]$ be a string complex. The middle term
$\omega^+[m+m'(\omega)]$ in the AR-triangle starting in $\omega$ is
given by the entries in Tables \ref{omegaplus} and
\ref{omegaplustrivial}. The middle term $\omega_-[m-m'(\omega_-)]$
in the AR-triangle ending in $\omega$ is given by the entries in
Tables \ref{omegaminus} and \ref{omegaminustrivial}.
\end{proposition}
\begin{proof}
See Appendix \ref{bobinskiappendix}.
\end{proof}
\renewcommand{1.4}{1.2}
\begin{longtable}[c]{|p{0.05\textwidth}l|p{0.1\textwidth}|p{0.24\textwidth}|m{0.2\textwidth}|}
\caption{$\omega^+$ for a non-trivial and non-empty homotopy string
$\omega$\label{omegaplus}}\\
\hline
$\alpha_l(\omega)$ & & condition & $\omega^+$ & $m'(\omega)$ \\
\hline
\endfirsthead
\multicolumn{5}{l}
{\tablename\ \thetable\ -- \textit{Continued from previous page}} \\
\hline
$\alpha_l(\omega)$ & & condition & $\omega^+$ & $m'(\omega)$ \\
\hline
\endhead
\hline \multicolumn{5}{l}{\textit{Continued on next page}} \\
\endfoot
\endlastfoot
$\alpha_i$, & $1 \leq i \leq r_2$ & & $cw\_r(t\omega)\cdot \omega$ &
$-1$ \\ \hline \pagebreak[3]
$\alpha_i$, & $r_2 + 1 \leq i \leq r$ & $l(\omega) = 1$ & $\emptyset$ &
-- \\ \hline \pagebreak[3]
$\alpha_i$, & $r_2 + 1 \leq i \leq r$ & $l(\omega) > 1$ & ccw
r-reduction of $\omega$ & 0 \\ \hline \pagebreak[3]
$\beta_i$, & $1 \leq i \leq s_2$ & & $ccw\_s(t\omega)\cdot \omega$ &
$-1$ \\ \hline \pagebreak[3]
$\beta_i$, & $s_2 + 1 \leq i \leq s$ & $l(\omega) = 1$ & $\emptyset$ &
-- \\ \hline \pagebreak[3]
$\beta_i$, & $s_2 + 1 \leq i \leq s$ & $l(\omega) > 1$ & cw
s-reduction of $\omega$ & 0 \\ \hline \pagebreak[3]
$\alpha_i^{-1}$, & $1 \leq i \leq r$ & & $cw\_r(t\omega)\cdot \omega$ &
\parbox[t]{\columnwidth}{$-1$ if $2 \leq i \leq r_2 +1$ \\
$0$ if $r_2 + 2 \leq i \leq r$\\
$\phi(r_1)$ if $i=1$
}
\\ \hline \pagebreak[3]
$\beta_i^{-1}$, & $1 \leq i \leq s$ & & $ccw\_s(t\omega)\cdot \omega$ & $0$ or $-1^*$
\\ \hline \pagebreak[3]
$\gamma_{2i}$, & $1 \leq i \leq r_2$ & & $cw\_r(t\omega)\cdot \omega$
& \parbox[t]{\columnwidth}{
$0$ if $i=1$ and $r_1\! >\! 0$\! \\
$-1$ otherwise
}\\ \hline \pagebreak[3]
$\delta_{2i}$, & $1 \leq i \leq s_2$ & & $ccw\_s(t\omega)\cdot \omega$
& $0$ or $-1^*$ \\ \hline \pagebreak[3]
$\gamma_{2i-1}$, & $1 \leq i \leq r_2$ & & $ccw\_s(t\omega)\cdot
\omega$ & \parbox[t]{\columnwidth}{$\phi(s_1)$ if $i>1$\\
$\phi(r_1)$ if $i\! =\! 1$ and $s_2\! >\! 0$ \\
$0$ otherwise
} \\ \hline \pagebreak[3]
$\delta_{2i-1}$, & $1 \leq i \leq s_2$ & & $cw\_r(t\omega)\cdot
\omega$ & $0$ or $-1^*$ \\ \hline \pagebreak[3]
$\gamma_{2i}^{-1}$, & $1 \leq i \leq r_2$ & & $ccw\_s(t\omega)\cdot
\omega$ & $\phi(s_1)$ \\ \hline \pagebreak[3]
$\delta_{2i}^{-1}$, & $1 \leq i \leq s_2$ & & $cw\_r(t\omega)\cdot
\omega$ & $0$ or $-1^*$ \\ \hline \pagebreak[3]
$\gamma_{2i-1}^{-1}$, & $1 \leq i \leq r_2$ & & ccw r-reduction of
$\omega$ & $-1$ \\ \hline \pagebreak[3]
$\delta_{2i-1}^{-1}$, & $1 \leq i \leq s_2$ & & cw s-reduction of
$\omega$ & $-1$ \\ \hline \pagebreak[3]
\multicolumn{5}{p{\linewidth}}{\parbox[t]{\linewidth}{\small ${}^{*}$ as in above row, but
interchanging $r$ and $s$.\\ $\phi: \mathbb{Z}_{\geq 0} \rightarrow
\{-1,0\}$ is defined by $\phi(a) = -1$ if $a=0$, otherwise $\phi(a) = 0$. }} \\
\end{longtable}
\begin{longtable}[c]{|ll|l|l|}
\caption{$\omega_-$ for a non-trivial and non-empty homotopy string
$\omega$\label{omegaminus}}\\
\hline
$\alpha_l(\omega)$ & & conditions & $\omega_-$ \\
\hline
\endfirsthead
\multicolumn{4}{l}
{\tablename\ \thetable\ -- \textit{Continued from previous page}} \\
\hline
$\alpha_l(\omega)$ & & conditions & $\omega_-$ \\
\endhead
\hline \multicolumn{4}{l}{\textit{Continued on next page}} \\
\endfoot
\endlastfoot
$\alpha_i$, & $1 \leq i \leq r$ & & $ccw\_r(t\omega)\cdot \omega$
\\
\hline
$\beta_i$, & $1 \leq i \leq s$ & & $cw\_s(t\omega)\cdot \omega$ \\
\hline
$\alpha_i^{-1}$, & $1 \leq i \leq r_2$ & & $ccw\_r(t\omega)\cdot
\omega$ \\
\hline
$\alpha_i^{-1}$, & $r_2 + 1 \leq i \leq r$ & $l(\omega) = 1$ &
$\emptyset$ \\
\hline
$\alpha_i^{-1}$, & $r_2 + 1 \leq i \leq r$ & $l(\omega) > 1$ & cw
r-reduction of $\omega$ \\
\hline
$\beta_i^{-1}$, & $1 \leq i \leq s_2$ & & $cw\_s(t\omega)\cdot
\omega$ \\
\hline
$\beta_i^{-1}$, & $s_2 + 1 \leq i \leq s$ & $l(\omega) = 1$ &
$\emptyset$ \\
\hline
$\beta_i^{-1}$, & $s_2 + 1 \leq i \leq s$ & $l(\omega) > 1$ & ccw
s-reduction of $\omega$ \\
\hline
$\gamma_{2i}$, & $1 \leq i \leq r_2$ & & cw r-reduction of $\omega$
\\
\hline
$\gamma_{2i-1}$, & $1 \leq i \leq r_2$ & & $cw\_s(t\omega)\cdot
\omega$ \\
\hline
$\delta_{2i}$, & $1 \leq i \leq s_2$ & & ccw s-reduction of $\omega$
\\
\hline
$\delta_{2i-1}$, & $1 \leq i \leq s_2$ & & $ccw\_r(t\omega)\cdot
\omega$ \\
\hline
$\gamma_{2i}^{-1}$, & $1 \leq i \leq r_2$ & & $cw\_s(t\omega)\cdot
\omega$ \\
\hline
$\gamma_{2i-1}^{-1}$, & $1 \leq i \leq r_2$ & & $ccw\_r(t\omega)
\cdot \omega$ \\
\hline
$\delta_{2i}^{-1}$, & $1 \leq i \leq s_2$ & & $ccw\_r(t\omega)\cdot
\omega$ \\
\hline
$\delta_{2i-1}^{-1}$, & $1 \leq i \leq s_2$ & & $cw\_s(t\omega)
\cdot \omega$ \\
\hline
\end{longtable}
\renewcommand{1.4}{1.05}
\begin{longtable}[c]{|>{\centering\arraybackslash}m{0.14\linewidth}|m{0.20\linewidth}|m{0.18\linewidth}|m{0.22\linewidth}|m{0.07\linewidth}|}
\caption{\small $\omega^+$ for a trivial homotopy string\label{omegaplustrivial}}\\
\hline
$x$ in & $\omega^+$ for $\omega=1_{x}$ & $m'(\omega)$ &$\omega^+$ for
$\omega = 1_{x}^{-1}$ & $m'(\omega)$ \\
\hline
\endfirsthead
\multicolumn{5}{l}
{\tablename\ \thetable\ -- \textit{Continued from previous page}} \\
\hline
$x$ in & $\omega^+$ for $\omega=1_{x}$ & $m'(\omega)$ & $\omega^+$ for
$\omega = 1_{x}^{-1}$ & $m'(\omega)$ \\
\hline
\endhead
\hline \multicolumn{5}{l}{\textit{Continued on next page}} \\
\endfoot
\endlastfoot
$A$ & \multirow{2}{*}{$cw\_r(x)$} &$\phi(r_1)$ if $i\!=\!1$& \multirow{2}{*}{$\emptyset$} & \multirow{2}{*}{--}\\
(so $x=A_i$) & &$0$ otherwise & & \\
\hline\pagebreak[3]
$A'$ & \multirow{2}{*}{$ccw\_s(x)$} & $\phi(s_1)$ if $i\!=\!1$ & \multirow{2}{*}{$\emptyset$} & \multirow{2}{*}{--}\\
(so $x=A'_i$) & &$0$ otherwise & & \\
\hline\pagebreak[3]
$B$ & $cw\_r(x)$ & $-1$ & $ccw\_s(x)$ & $\phi(s_1)$ \\
\hline\pagebreak[3]
$B'$ & $ccw\_s(x)$ &$-1$ & $cw\_r(x)$ & $\phi(r_1)$\\
\hline\pagebreak[3] \multirow{2}{*}{$C$} & $1_{D_{r_1-1}} $ if $r_1 >
0$ & $0$ & $1_{D'_{s_1-1}}$ if $s_1 > 0$
&$0$ \\
& $cw\_r(x) $ if $r_1 = 0$ & $-1$ & $ccw\_s(x)$ if $s_1 = 0$ & $-1$ \\
\hline\pagebreak[3]
$D$ & $1_{D_{i-1}} $ if $i > 0$ & $0$ & \multirow{2}{*}{$ccw\_s(x)$} & \multirow{2}{*}{$\phi(s_1)$} \\
(so $x = D_i$) & $cw\_r(x)$ if $i = 0$ &$-1$ & & \\
\hline\pagebreak[3]
\multirow{2}{*}{\parbox[c][1.3cm][c]{\linewidth}{
\centering $D'$\\(so $x = D'_i$)}}
& $1_{D'_{i-1}} $ if $i > 0$ & $0$& \multirow{3}{*}{$cw\_r(x)$} & \multirow{3}{*}{$\phi(r_1)$}\\
& $ccw\_s(x)$ if $i = 0$ &
$\!\left\{\! \begin{array}{l} 0 \text{ if } s_2\!=\!0 \\
-1 \text{ otherwise} \end{array} \right. $ & & \\
\hline\pagebreak[3]
$F$ & $ccw\_s(x)$ & $\phi(s_1)$ & $cw\_r(x)$& $\phi(r_1)$\\
\hline
$F'$ & $cw\_s(x)$ & $-1$ & $ccw\_r(x)$& $-1$ \\
\hline \multicolumn{5}{p{0.85\linewidth}}{\small $\phi: \mathbb{Z}_{\geq 0} \rightarrow
\{-1,0\}$ is defined by $\phi(a) = -1$ if $a=0$, otherwise $\phi(a) = 0$.} \\
\end{longtable}
\begin{longtable}[c]{|c|p{0.35\linewidth}|p{0.20\linewidth}|}
\caption{\small $\omega_-$ for a trivial homotopy string\label{omegaminustrivial}}\\
\hline
$x$ in & $\omega_-$ for $\omega = 1_{x}$ & $\omega_-$ for $\omega = 1_{x}^{-1}$ \\
\hline
\endfirsthead
\multicolumn{3}{l}
{\tablename\ \thetable\ -- \textit{Continued from previous page}} \\
\hline
$x$ in & $\omega_-$ for $\omega = 1_{x}$ & $\omega_-$ for $\omega = 1_{x}^{-1}$ \\
\hline
\endhead
\hline \multicolumn{3}{l}{\textit{Continued on next page}} \\
\endfoot
\endlastfoot
$A$ & $\emptyset$ & $ccw\_r(x)$\\
\hline\pagebreak[3]
$A'$ & $\emptyset$ & $cw\_s(x)$\\
\hline\pagebreak[3]
$B$ & $ccw\_r\_p(x)$ & $cw\_s\_p(x)$\\
\hline\pagebreak[3]
$B'$ & $cw\_s\_p(x)$ & $ccw\_r\_p(x)$\\
\hline\pagebreak[3]
$C$ & $ccw\_r(x)$ & $cw\_s(x)$\\
\hline\pagebreak[3]
$D_i$ & $1_{D_{i+1}}$ if $i < r_1-1$ & \multirow{2}{*}{$cw\_s(x)$}\\
(so $x = D_i$) & $1_C$ if $i = r_1 - 1$ and $s_1 > 0$ & \\
\hline\pagebreak[3]
$D'_i$ & $1_{D'_{i+1}}$ if $i < s_1-1$ & \multirow{2}{*}{$ccw\_r(x)$}\\
(so $x = D'_i$) & $1_C^{-1}$ if $i = s_1 - 1$ and $r_1 > 0$ & \\
\hline\pagebreak[3]
$F$ & $cw\_r\_p(x)$ & $ccw\_s\_p(x)$\\
\hline
$F'$ & $ccw\_r\_p(x)$ & $cw\_s\_p(x)$\\
\hline
\end{longtable}
For the remaining part of this section, we will only consider
properties of homotopy strings, and not complexes in $\kbp$. Note that
it will never happen that $\omega$ is equal to any of $\omega^+$,
$\omega_+$, $\omega^-$ and $\omega_-$.
We start by defining four diagonals for a given homotopy string
$\omega$. The \emph{upper right diagonal} of $\omega$ is the
sequence $(\omega, \omega^+, (\omega^+)^+, \ldots)$. Furthermore, we
define the \emph{lower right diagonal} of $\omega$ to be the
sequence $(\omega, \omega_+, (\omega_+)_+, \ldots)$. The upper and
lower left diagonals of $\omega$ are defined similarly.
Next, let a \emph{$Q$-walk} denote one of the following walks:
clockwise r-walk, clockwise s-walk, counter-clockwise r-walk,
counter-clockwise s-walk. Note that it follows from Proposition
\ref{tabellproposisjon}, that the condition $l(\omega^+)>l(\omega)$
implies that $\omega^+$ is of the form $\sigma\omega$ for a step
$\sigma$ in a $Q$-walk.
\begin{corollary}\label{diagonaler}
Let $\omega$ be a homotopy string, let $\omega^*$ be either
$\omega^+$ or $\omega_-$, and let $\mathcal{D}$ be the diagonal
$(\omega, \omega^*, \ldots)$. If $\omega^*$ is of
the form $\sigma \omega$, where $\sigma$ is a step in a
$Q$-walk $W$, then the homotopy string in $\mathcal D$ succeeding
$\omega^*$ is $\sigma' \omega^*$, where $\sigma'$ is the step
succeeding $\sigma$ in $W$.
\end{corollary}
\begin{proof}
This follows from the definition of walks and Tables
\ref{omegaplus}--\ref{omegaminustrivial}. For example, assume that
$\omega^{*} = \omega^+$ and that $\omega^+ = \sigma \omega$ where
$\sigma$ is a step in a clockwise r-walk. Then $t(\omega^{+}) \in A
\cup D$, such that $\alpha_l(t(\omega^+))$ is an arrow $\gamma_{2i}$
if $t(\omega^+) \in A$, and an inverse r-arrow otherwise. In all
these cases, $(\omega^+)^+ = cw\_r(t(\omega^+))\cdot \omega^+$.
\end{proof}
\begin{proposition}\label{todiagonaler}
(1) Let $\omega$ be a homotopy string. Assume that
$\omega^{+}=\sigma\omega$ and $\omega_+=\omega\sigma'$, where
$\sigma$ is a step in a $Q$-walk $W$ and $\sigma'^{-1}$ is a step in a
$Q$-walk $W'$. If $\widetilde{\omega}$ is in the lower right
diagonal of $\omega$, and $\widehat{\omega}$ is in the upper
right diagonal of $\omega$, then
\[\widetilde{\omega}^+ = \sigma\widetilde{\omega} ~~\text{ and }~~ \widehat{\omega}_+ =
\widehat{\omega}\sigma' ~.\]
(2) Let $\omega$ be a homotopy string. Assume that
$\omega^{+}=\sigma\omega$ and $\omega^-=\omega\sigma'$, where
$\sigma$ is a step in a $Q$-walk $W$ and $\sigma'^{-1}$ is a step in a
$Q$-walk $W'$. If $\widetilde{\omega}$ is in the upper left
diagonal of $\omega$, and $\widehat{\omega}$ is in the upper
right diagonal of $\omega$, then
\[\widetilde{\omega}^+ = \sigma\widetilde{\omega} ~~\text{ and }~~ \widehat{\omega}^- =
\widehat{\omega}\sigma' ~.\]
(3) Let $\omega$ be a homotopy string. Assume that
$\omega_{-}=\sigma\omega$ and $\omega^-=\omega\sigma'$, where
$\sigma$ is a step in a $Q$-walk $W$ and $\sigma'^{-1}$ is a step in a
$Q$-walk $W'$. If $\widetilde{\omega}$ is in the upper left
diagonal of $\omega$, and $\widehat{\omega}$ is in the lower
left diagonal of $\omega$, then
\[\widetilde{\omega}_- = \sigma\widetilde{\omega} ~~\text{ and }~~ \widehat{\omega}^- =
\widehat{\omega}\sigma' ~.\] (4) Let $\omega$ be a homotopy
string. Assume that $\omega_-=\sigma\omega$ and
$\omega_+=\omega\sigma'$, where $\sigma$ is a step in a $Q$-walk $W$
and $\sigma'^{-1}$ is a step in a $Q$-walk $W'$. If
$\widetilde{\omega}$ is in the lower left diagonal of $\omega$, and
$\widehat{\omega}$ is in the lower right diagonal of $\omega$, then
\[\widetilde{\omega}_+ = \widetilde{\omega}\sigma' ~~\text{ and }~~ \widehat{\omega}_- =
\sigma\widehat{\omega} ~.\]
\end{proposition}
\begin{proof}
We prove case (1). The proofs for cases (2)--(4) are similar. If
$\omega \neq \emptyset$ is non-trivial, then by Corollary
\ref{diagonaler}, we have that $\alpha_1(\widehat{\omega}) =
\alpha_1(\omega)$ and that
$\alpha_{l(\widetilde{\omega})}(\widetilde{\omega}) =
\alpha_{l(\omega)}(\omega)$; and $\widehat{\omega}_+$ is determined
by $\alpha_1(\widehat{\omega})$, and $\widetilde{\omega}^+$ is
determined by $\alpha_{l(\widetilde{\omega})}$.
Assume now that $\omega$ is trivial, that is, $\omega =
1_x^{\varepsilon}$ for some vertex $x$ and some $\varepsilon \in
\{-1,1\}$. Then, by Table \ref{omegaplustrivial}, it follows that
$x$ is in $B \cup B' \cup D_0 \cup D_0' \cup F \cup F'$, since these are the
only vertex types where both $1_x^+$ and $(1_x^{-1})^+$ are of
length greater than $\omega$. It is easy to verify by Tables
\ref{omegaplus} and \ref{omegaplustrivial} that we are in this
situation:
\[
\begin{tikzpicture}
\node (1) at (-1,0) [tvertex] {$1_x^{\varepsilon}$};
\node (2) at (0,1) [tvertex] {$\sigma$};
\node (3) at (0,-1) [tvertex] {$\sigma'$};
\node (4) at (1,0) [tvertex] {$\sigma \sigma'$};
\draw [->] (1)--(2);
\draw [->] (1)--(3);
\draw [->] (2)--(4);
\draw [->] (3)--(4);
\end{tikzpicture}
\]
We can now consider the upper right diagonal of $\sigma$ and the
lower right diagonal of $\sigma'$, but then we are in the
non-trivial case.
\end{proof}
\section{Classification of the AR-components}
In this section, we give a complete classification of all the
AR-components in $\kbp$ where $\Lambda \cong kQ/I$ is a fixed
cluster-tilted algebra of type $\ensuremath{\widetilde{A}_n}$ with $Q$ in $\mathcal
Q_n^{\star}$.
We start by defining admissible reduction for homotopy strings and we
show that there are three types of homotopy strings that can not be
admissibly reduced. Moreover, any other homotopy string can be
admissibly reduced to a homotopy string of one of these three types.
In Section \ref{sec:charcomp} we show that one of the classes of
homotopy strings that can not be admissibly reduced, gives rise to the
$\ensuremath{\mathbb{Z}A_{\infty}}$-components and tubes containing string complexes. In Section
\ref{sec:zainfinf} we parametrize the $\ensuremath{\mathbb{Z}A_{\infty}^{\infty}}$-components arising
from the second class of homotopy strings that can not be admissibly
reduced, and we describe the $\ensuremath{\mathbb{Z}A_{\infty}^{\infty}}$-components arising from the
third class of homotopy strings that can not be admissibly reduced.
In Section \ref{sec:band} we consider the AR-components containing
band complexes. Finally, Section \ref{summary} provides a summary of
the main results.
In the remainder of this section, all homotopy strings and homotopy
bands are associated with the fixed algebra $\Lambda$.
\begin{definition}\label{admissible}
Let $\omega$ be a homotopy string which is neither an r-arrow, nor an
s-arrow, nor the inverse of such an arrow. If $\omega'$ is either the
clockwise r-reduction of $\omega$, or the counter-clockwise
r-reduction of $\omega$, or the clockwise s-reduction of $\omega$, or
the counter-clockwise s-reduction of $\omega$, then we call $\omega'$
a \emph{left admissible reduction} of $\omega$.
We define a \emph{right admissible reduction} of $\omega$ to be $\omega''$,
where $(\omega'')^{-1}$ is a left admissible reduction of
$\omega^{-1}$.
\end{definition}
If there exists a left or right admissible reduction of a homotopy
string $\omega$, then we say that $\omega$ can be admissibly reduced,
and the operation performed is an admissible reduction.
\begin{lemma}\label{admredlemma}
Let $\omega$ be a homotopy string. If $\omega'$ is a left admissible
reduction of $\omega$, then either $\omega'=\omega^+$ or
$\omega'=\omega_-$. Moreover,
\begin{itemize}
\item if $\omega'$ is a clockwise r-reduction or a counter-clockwise
s-reduction of $\omega$, then $\omega'=\omega_-$, and
\item if $\omega'$ is a clockwise s-reduction or a counter-clockwise
r-reduction of $\omega$, then $\omega'=\omega^+$.
\end{itemize}
Similarly, if $\omega'$ is a right admissible reduction of $\omega$,
then $\omega'$ is either ${\omega_+}$ or $\omega^-$.
\end{lemma}
\begin{proof}
Let $\omega$ be a homotopy string and assume it can be left
admissibly reduced and that $\omega'$ is the left admissible
reduction of $\omega$. The left admissible reduction of $\omega$ is
either a clockwise r-reduction of $\omega$, a counter-clockwise
r-reduction of $\omega$, a clockwise s-reduction of $\omega$ or a
counter-clockwise s-reduction of $\omega$.
If the reduction is a clockwise r-reduction, then $\omega$ satisfies
condition (\ref{condition1}) in chapter \ref{redsubkap}. In
particular $\alpha_l(\omega)$ is either an arrow $\gamma_{2i}$ for
some $1 \leq i \leq r_2$, or an inverse r-arrow. Thus we need to
check that in all these cases $\omega'=\omega_-$. The entries in
Table \ref{omegaminus} clearly show that this is so, except for the
case when $l(\omega)=1$ and $\omega$ is the inverse of an
r-arrow. The clockwise r-reduction of the exceptions are defined,
however the reductions are not admissible reductions, and therefore
are not cases we need to consider, as we have assumed that $\omega$ can
be admissibly reduced.
The three other cases follow by similar arguments.
\end{proof}
\begin{corollary}\label{admarcorollar}
Let $\omega$ be a homotopy string. Then $\omega'$ is a left
admissible reduction of $\omega$ if and only if $\omega'\neq
\emptyset$ and $\omega'$ is one of $\{\omega^+, \omega_-\}$ such
that $l(\omega') < l(\omega)$.
\end{corollary}
\begin{proof}
Let $\omega$ be a homotopy string.
Assume that $\omega$ has an admissible reduction $\omega'$. By the
definition of admissible reduction it is clear that
$l(\omega')<l(\omega)$ and $\omega' \neq \emptyset$, and by Lemma
\ref{admredlemma} either $\omega'=\omega^+$ or $\omega'=\omega_-$.
Assume that $\omega' \neq\emptyset$ is either $\omega^+$ or $\omega_-$
and that $l(\omega')<l(\omega)$. By Tables \ref{omegaplus} and
\ref{omegaminus} it is clear that in any of these cases $\omega^+$ and
$\omega_-$ is an admissible reduction.
\end{proof}
It is clear from the definition of a right admissible reduction, that
there is a similar result as Corollary \ref{admarcorollar} for right
admissible reductions. We now give the definition of central homotopy
strings, which we will show form one of the classes of homotopy strings
that can not be admissibly reduced.
\begin{definition}
A \emph{central homotopy string} is either a trivial homotopy string
corresponding to a vertex of type $B, B', F$ or $F'$, or a homotopy
string $\omega$ where
\begin{itemize}
\item $\alpha_1(\omega)\in\left\{\alpha_i\,, \alpha_i^{-1}\,, \beta_j\,,
\beta_j^{-1}\,, \gamma_{2i}\,, \gamma_{2i-1}^{-1}\,, \delta_{2j}\,,
\delta_{2j-1}^{-1}~|\,1\leq i\leq r_2, 1\leq j\leq s_2 \right\}$
\item $\alpha_l(\omega)\in\left\{\alpha_i\,, \alpha_i^{-1}\,, \beta_j\,,
\beta_j^{-1}\,, \gamma_{2i-1}\,, \gamma_{2i}^{-1}\,, \delta_{2j-1}\,,
\delta_{2j}^{-1}~|\,1\leq i\leq r_2, 1\leq j\leq s_2 \right\}.$
\end{itemize}
\end{definition}
\begin{lemma}\label{centralnoreduction}
If $\omega$ is a central homotopy string, then $\omega$ can not be
admissibly reduced.
\end{lemma}
\begin{proof}
This follows from Tables \ref{omegaplus} and \ref{omegaminus}.
\end{proof}
We have now seen that there are three classes of homotopy strings
which can not be admissibly reduced: The central homotopy strings (by
the above lemma), the non-central trivial homotopy strings (because no
trivial homotopy string can be reduced, by Definition
\ref{reduction1}), and the r- and s-arrows and their inverses (by
Definition \ref{admissible}). The following lemma shows that these
classes of homotopy strings are the only ones which can not be
admissibly reduced.
\begin{lemma}\label{reduksjonstyper}
Let $\omega \neq \emptyset$ be a homotopy string. By a series of
left or right admissible reductions, $\omega$ can be reduced to a
homotopy string which is of one of the following types:
\begin{itemize}
\item[(i)] an r- or s-arrow or an inverse of such an arrow or a
trivial homotopy string corresponding to a vertex of type $A$ or
$A'$, or
\item[(ii)] a central homotopy string, or
\item[(iii)] a trivial homotopy string corresponding to a vertex
of type $C$, $D$ or $D'$.
\end{itemize}
\end{lemma}
\begin{proof}
Let $\omega \neq \emptyset$ be a homotopy string, and assume that
$\omega$ is neither of type $(i)$, nor $(ii)$, nor $(iii)$. Then
$l(\omega) > 0$. Denote by $X$ the complement of
\[\left\{\alpha_i\,, \alpha_i^{-1}\,, \beta_j\,, \beta_j^{-1}\,,
\gamma_{2i-1}\,, \gamma_{2i}^{-1}\,, \delta_{2j-1}\,,
\delta_{2j}^{-1}~|\,1\leq i\leq r_2, 1\leq j\leq s_2 \right\}\] in
$Q'_1$. Let $\widehat{\omega} = \omega$ if $\alpha_l(\omega) \in
X$, otherwise let $\widehat{\omega} = \omega^{-1}$. It is clear
that $\alpha_l(\widehat{\omega}) \in X$, since $\omega$ is not a
central homotopy string. In any case, either $\widehat{\omega}^+$
or $\widehat{\omega}_-$ will be an admissible reduction of
$\widehat{\omega}$.
For instance, assume that $\alpha_l(\widehat{\omega}) = \gamma_{2i}$
for some $1\leq i \leq r_2$. Then by Table \ref{omegaminus},
$\widehat{\omega}_-$ is an admissible reduction of
$\widehat{\omega}$.
Next, let $\widehat{\widehat{\omega}}$ be the admissible reduction
of $\widehat{\omega}$. Repeat the above step for the homotopy
string $\widehat{\widehat{\omega}}$.
\end{proof}
\subsection{Characteristic components containing string complexes}
\label{sec:charcomp}
By a characteristic component, we mean an AR-component containing
AR-triangles with only one middle term. In this section, we will
consider characteristic components containing string complexes.
These components are dependent on the parameters of the quiver of the
cluster-tilted algebra.
A similar result as Proposition \ref{randproposisjon} holds by
exchanging the parameters $s_1$ and $s_2$ with the parameters $r_1$
and $r_2$.
\begin{proposition}\label{randproposisjon}
If $s_2 = 0$ then, for each $i \in \mathbb Z$, there is a characteristic
component with the following edge:
\[
\begin{tikzpicture}
\node (1) at (0,0) [kvertex] {$\beta_s^{-1}[i]$};
\node (2) at (2,0) [kvertex] {$\beta_{s-1}^{-1}[i]$};
\node (3) at (3.25,0) [kvertex] {$\cdots$};
\node (4) at (4.5,0) [kvertex] {$\beta_1^{-1}[i]$};
\node (5) at (6.5,0) [kvertex] {$\beta_s^{-1}[i]$};
\node (9) at (3.25,1) [kvertex] {$\cdots$};
\draw [->] (1)--(0.9,1);
\draw [->] (1.1,1)--(2);
\draw [->] (2) -- (2.9,1);
\draw [->] (3.6,1)--(4);
\draw [->] (4)--(5.4,1);
\draw [->] (5.6,1)--(5);
\draw [dotted] (0,0.3)--(0,1);
\draw [dotted] (6.5,0.3)--(6.5,1);
\end{tikzpicture}
\]
If $s_2 \neq 0$, there is a class of $s_2$ AR-components with the
following edges:
\[
\begin{tikzpicture}
\node (0) at (-0.5,0.5) [kvertex] {$\cdots$};
\node (1) at (0,0) [kvertex] {$1_{A'_{s_2}}$};
\node (s1) at (0,-0.4) [kvertex] {$[i]$};
\node (2) at (2,0) [kvertex] {$1_{A'_{s_2-1}}$};
\node (s2) at (2,-0.4) [kvertex] {$[i-1]$};
\node (3) at (3.25,0) [kvertex] {$\cdots$};
\node (4) at (4.5,0) [kvertex] {$1_{A'_1}$};
\node (s4) at (4.5,-0.4) [kvertex] {$[i-s_2+1]$};
\node (5) at (6.5,0) [kvertex] {$\beta_s^{-1}$};
\node (s5) at (6.5,-0.4) [kvertex] {$[i-s_2+1]$};
\node (7) at (7.75,0) [kvertex] {$\cdots$};
\node (6) at (9,0) [kvertex] {$\beta_1^{-1}$};
\node (s6) at (9,-0.4) [kvertex] {$[i-s_2+1]$};
\node (8) at (11,0) [kvertex] {$1_{A'_{s_2}}$};
\node (s8) at (11,-0.4) [kvertex] {$[i-s_2]$};
\node (9) at (3.25,1) [kvertex] {$\cdots$};
\node (10) at (7.75,1) [kvertex] {$\cdots$};
\node (11) at (11.25, 0.5) [kvertex] {$\cdots$};
\draw [->] (1)--(0.9,1);
\draw [->] (1.1,1)--(2);
\draw [->] (2) -- (2.9,1);
\draw [->] (3.6,1)--(4);
\draw [->] (4)--(5.4,1);
\draw [->] (5.6,1)--(5);
\draw [->] (5)--(7.4,1);
\draw [->] (8.1,1)--(6);
\draw [->] (6)--(9.9,1);
\draw [->] (10.1,1)--(8);
\end{tikzpicture}
\]
\end{proposition}
\begin{proof}
From Proposition \ref{tabellproposisjon}, it is clear that for any
of the homotopy strings $1_{A'_i}$ for $1 \leq i \leq s_2$ and
$\beta_j^{-1}$ for $s_2+1 \leq j \leq s$, we have
$\omega_+=\emptyset$. Hence the string complexes shown in the above
figures are all in some characteristic component. The rest of the
result follows from direct calculations.
\end{proof}
The characteristic components described in Proposition
\ref{randproposisjon} are called s-components. Similarly, the
characteristic components depending on the parameters $r_1$ and $r_2$
are called r-components.
In the next corollaries we show that the r- and s-components are
tubes or $\ensuremath{\mathbb{Z}A_{\infty}}$-components, and that they are exactly the
characteristic AR-components containing string complexes. We also
describe the string complexes occurring in such components.
\begin{corollary}\label{allcharacteristicstrings}
Let $\omega[i]$ be a string complex occurring in an s-component.
Then $\omega$ is of the following form: $\omega = w_k\cdots w_1
\omega'$ where $\omega'[j]$ is on the edge of the component (for
some $j\in \mathbb{Z}$), and where $w_1, \cdots, w_k$ are the $k$
first consecutive steps of the counter-clockwise s-walk starting
in $t\omega'$.
Similarly, let $\omega[i]$ be a string complex occurring in an
r-component. Then $\omega$ is of the form $\omega = w_k \cdots
w_1 \omega'$ where $\omega'[j]$ is on the edge of the component (for
some $j \in \mathbb{Z}$), and where $w_1, \cdots, w_k$ are the $k$
first consecutive steps of the clockwise r-walk starting in
$t\omega'$.
\end{corollary}
\begin{proof}
From Tables \ref{omegaplus}--\ref{omegaminustrivial}, we know that
for a string complex $\omega[i]$ on the edge of an s-component, we
have that $\omega^+ = ccw\_s(t\omega)\cdot \omega$. Then, from
Corollary \ref{diagonaler}, we know that the upper right diagonal of
$\omega$ will continue to grow with succeeding steps in a
counter-clockwise s-walk.
The other case is proved similarly.
\end{proof}
\begin{corollary}
If $s_2=0$, then the s-components are tubes of rank $s_1$. If
$s_2 >0$, then the s-components are of type $\ensuremath{\mathbb{Z}A_{\infty}}$ with $\tau^s
= [s_2]$.
\end{corollary}
\begin{corollary}\label{allcomp}
The r- and s-components are exactly the characteristic
components containing string complexes.
\end{corollary}
\begin{proof}
This is clear, as in Tables
\ref{omegaplus}--\ref{omegaminustrivial} the empty homotopy string
$\emptyset$ occurs only for the homotopy strings listed in
Proposition \ref{randproposisjon}.
\end{proof}
\begin{corollary}\label{nocentrals}
Let $\omega[i]$ be a string complex in a characteristic
component. Then $\omega$ is not a central homotopy string.
\end{corollary}
\begin{proof} If $\omega[i]$ is a string complex on the edge of a
characteristic component, then $\omega$ is not a central homotopy
string. Let $\omega'[j]$ be a string complex in a characteristic
component, which is not on the edge. Then by Corollary
\ref{allcharacteristicstrings} it is clear that $\omega'$ can be
admissibly reduced. By Lemma \ref{centralnoreduction} no central
homotopy string can be in this characteristic component.
\end{proof}
From Corollary \ref{allcharacteristicstrings}, it is clear that no
string complex $\omega[i]$ where $\omega$ is of type (iii) in Lemma
\ref{reduksjonstyper} can be in a characteristic component.
\subsection{The $\ensuremath{\mathbb{Z}A_{\infty}^{\infty}}$-components}
\label{sec:zainfinf}
From Corollary \ref{allcomp}, Corollary
\ref{nocentrals} and Section \ref{sec:components}, it is clear that if
$\omega[i]$ is a string complex where $\omega$ is of type (ii) or
(iii) in Lemma \ref{reduksjonstyper}, then $\omega[i]$ is in a
component of type $\ensuremath{\mathbb{Z}A_{\infty}^{\infty}}/G$ where $G$ is an admissible group of
automorphisms on $\ensuremath{\mathbb{Z}A_{\infty}^{\infty}}$.
Let $\omega[i]$ be a string complex where $\omega$ is a central
homotopy string. In the following lemma, we show that if $\upsilon$
is another central homotopy string, then the string complex
$\upsilon[j]$ can not be in the same AR-component as $\omega[i]$.
\begin{lemma}\label{lem:centralhomstr}
Let $\omega[i]$ and $\upsilon[j]$ be string complexes where $\omega$
and $\upsilon$ are central homotopy strings. If $\omega[i] \ncong
\upsilon[j]$, then $\omega[i]$ and $\upsilon[j]$ are in different
AR-components.
\end{lemma}
\begin{proof}
Let $\omega$ be a central homotopy string. From Tables
\ref{omegaplus}--\ref{omegaminustrivial}, it is easy to verify that
each of $\omega^+$, $\omega_+$, $\omega^-$ and $\omega_-$ is
obtained by adding (possibly the inverse of) a step of some $Q$-walk
to $\omega$, and that the four $Q$-walks involved are of four
different types. By Corollary \ref{diagonaler} and Proposition
\ref{todiagonaler}, there can be no central homotopy string
different from $\omega$ in the AR-component.
\end{proof}
It follows from the proof of Lemma \ref{lem:centralhomstr} that the
string complex $\omega[i]$, where $\omega$ is a central homotopy
string, never occurs twice in the same component. Hence, the only
possibility for $G$ is the trivial group, and thus there is a class of
$\ensuremath{\mathbb{Z}A_{\infty}^{\infty}}$-components which up to shift are parametrized by the
central homotopy strings. Note that for each $\ensuremath{\mathbb{Z}A_{\infty}^{\infty}}$-component in
this class, the corresponding central homotopy string $\omega$ is of
strictly smaller length than any homotopy string $\nu$ where $\nu[j]$
is in the same component for some $j \in \mathbb Z$.
We now consider string complexes of the form $\omega[i]$ where
$\omega$ is of type (iii) in Lemma \ref{reduksjonstyper}, that is a
trivial homotopy string corresponding to a vertex of type $C, D$ or
$D'$.
For a string complex $\omega[i]$ in an AR-component of type
$\ensuremath{\mathbb{Z}A_{\infty}^{\infty}}$, we define the upper right diagonal of $\omega[i]$ to be
the sequence $(\omega[i],\omega^+[i'],
(\omega^+)^+[i''],\ldots)$ where $i' = i+m(\omega)$ and $i'' =
i'+m(\omega^+)$. Similarly, we define the lower right diagonal of
$\omega[i]$, the upper left diagonal of $\omega[i]$ and the lower left
diagonal of $\omega[i]$.
The following lemma is illustrated in Example 2.1 revisited in
Section \ref{eksempler}.
\begin{lemma}\label{lem:uglycomp}
All stalk complexes $\omega[i]$ where $\omega$ is a trivial homotopy
string corresponding to a vertex of type $C, D$ or $D'$ and $i$ is a
fixed integer, are in the same $\ensuremath{\mathbb{Z}A_{\infty}^{\infty}}$-component. There are
irreducible maps, which depend on the values of the parameters $r_1$
and $s_1$, as described below:
\begin{itemize}
\item If $r_1> 0$ the irreducible maps in the lower
right diagonal of $1_C^{-1}[i]$ are
\[
1_C^{-1}[i] \rightarrow 1_{D_{r_1-1}}^{-1}[i] \rightarrow\cdots
1_{D_1}^{-1}[i]\rightarrow 1_{D_0}^{-1}[i].
\]
\item If $s_1 > 0$ the irreducible maps in the upper right diagonal
of $1_C^{-1}[i]$ are
\[1_C^{-1}[i]\rightarrow 1_{D'_{r_1-1}}[i]\rightarrow\cdots
1_{D'_1}[i]\rightarrow 1_{D'_0}[i].
\]
\end{itemize}
Furthermore, for each $j$ in $\mathbb{Z}$ with $j \neq i$ and
$\omega$ as above, $\omega[j]$ and $\omega[i]$ are not in the same
component.
\end{lemma}
\begin{proof}
We have seen that the string complexes arising from trivial homotopy
strings corresponding to a vertex of type $C, D$ or $D'$ are in a
component of the form $\ensuremath{\mathbb{Z}A_{\infty}^{\infty}}/G$. By a similar argument as in the
proof of Lemma \ref{lem:centralhomstr} and its subsequent comment,
$G$ is trivial. The irreducible maps follow from Tables
\ref{omegaplustrivial} and \ref{omegaminustrivial}.
\end{proof}
Hence, if at least one of $r_1$ and $s_1$ is non-zero, we get a class
of $\ensuremath{\mathbb{Z}A_{\infty}^{\infty}}$-components parametrized by $\mathbb Z$. If $r_1 = s_1
= 0$, then we have no vertices of type $C, D$ or $D'$, and hence no
AR-component of type $\ensuremath{\mathbb{Z}A_{\infty}^{\infty}}$ as in Lemma \ref{lem:uglycomp}.
\subsection{Characteristic components containing band complexes}
\label{sec:band}
For any $Q$ in $\mathcal Q_n^{\star}$ there is always a homotopy band
$\omega=\beta_1^{-1}\cdots\beta_s^{-1}\alpha_r\cdots\alpha_1$, called
\emph{the central homotopy band of $Q$}. Note that it follows from the
definition of a homotopy band that there can be no homotopy bands
starting in a vertex of type $A, A', D_{\geq 1}$ and $D_{\geq 1}'$.
\begin{proposition}
Let $\Lambda\cong kQ/I$ be a cluster-tilted algebra of type
$\widetilde{A}_n$. There are finitely many homotopy bands associated
with $\Lambda$ if and only $\Lambda$ is hereditary.
\end{proposition}
\begin{proof}
It is clear that for the hereditary case, the central homotopy band
is the only homotopy band. Assume now that $\Lambda$ is not
hereditary, that is, we have $r_2 > 0$ (see Table
\ref{parameterverdier}). We can then construct one class consisting
of infinitely many homotopy bands for the case when $r=r_2=1$, and
one class consisting of infinitely many homotopy bands for the case
when $r>1$.
First, let $r=r_2=1$. Then the homotopy string \[\omega =
\beta_1^{-1}\cdots
\beta_s^{-1}\gamma_2^{-1}\gamma_1^{-1}\alpha_1^{-1}\beta_s \cdots
\beta_1\gamma_1\gamma_2\alpha_1\] is a homotopy band. Let
$\omega_c$ denote the
central homotopy band. Then the homotopy string
$
\omega_n = \omega \cdot (\omega_c)^n
$
is a homotopy band for all positive integers $n$.
Now let $r>1$. Then the homotopy string \[ \omega = \beta_1^{-1}
\cdots \beta_s^{-1} \alpha_r \cdots \alpha_2
\gamma_2^{-1}\gamma_1^{-1}\alpha_1^{-1}\cdots \alpha_r^{-1}\beta_s
\cdots \beta_1 \gamma_1 \gamma_2 \alpha_1
\]
is a homotopy band. Again, let $\omega_c$ denote the central homotopy
band. Then the homotopy string
$
\omega_n = \omega \cdot (\omega_c)^n
$
is a homotopy band for all positive integers $n$. Thus, it is clear
that any non-hereditary cluster-tilted algebra of type $\ensuremath{\widetilde{A}_n}$ will
have infinitely many homotopy bands associated with itself.
\end{proof}
Note that for most quivers in $\mathcal Q_n^{\star}$, there are also
ways of constructing classes of infinitely many homotopy bands other
than in the proof.
\subsection{Summary}
\label{summary}
In the following theorem we give a full overview of all the
AR-components of $\kbp$. Note that we always assume that $r_2 > 0$
(see Table \ref{parameterverdier}).
\begin{theorem}\label{finalthm}
Let $\mathfrak C$ be the set of central homotopy strings associated
with $\Lambda$, and $\mathfrak B$ the set of homotopy bands
associated with $\Lambda$. The AR-quiver of $\kbp$
consists of:
\begin{itemize}
\item A class of homogeneous tubes, parametrized by $\mathfrak B
\times k \times \mathbb Z$.
\item A class of s-components. If $s_2 = 0$, we get a class of
tubes of rank $s_1$ parametrized by $\mathbb Z$. If $s_2 >
0$, we get $s_2$ components of type $\ensuremath{\mathbb{Z}A_{\infty}}$ with $\tau^{s} =
[s_2]$.
\item $r_2$ components of type $\ensuremath{\mathbb{Z}A_{\infty}}$ with $\tau^{r} =
[r_2]$.
\item A class of $\ensuremath{\mathbb{Z}A_{\infty}^{\infty}}$-components containing all the stalk
complexes corresponding to a vertex of type $C$, $D$ or $D'$,
parametrized by $\mathbb Z$.
\item A class of $\ensuremath{\mathbb{Z}A_{\infty}^{\infty}}$ components parametrized by $\mathfrak
C \times \mathbb Z$.
\end{itemize}
\end{theorem}
\begin{proof}
By Lemma \ref{reduksjonstyper}, any homotopy string admissibly
reduces to one of three types. Hence, by Corollary \ref{allcomp},
Lemma \ref{lem:centralhomstr} and Lemma \ref{lem:uglycomp}, the list
in the theorem gives all AR-components.
\end{proof}
\section{Example} \label{eksempler}
In this section, we revisit Example \ref{ex:hovedeks}. The following
notation will be used: The arrows of the quivers are denoted by small
letters (e.g. $a$, $b$), and for an arrow $a$ the inverse is denoted
by $\overline a$.
Recall that $\Lambda = kQ/I$ is a cluster-tilted algebra of type
$\widetilde{A}_{15}$, where $Q$ is the quiver in Figure \ref{16kogger}
and \[ I =\left\langle ih, gi, hg, ed, fe, df, ba, cb, ac, ts, ut, su,
qp, rq, pr \right\rangle.\] The parameters of $Q$ are $r_1 = 2$, $r_2
= 3$, $s_1 = 4$ and $s_2 = 2$. We will now give part of the
AR-structure of $\kbp$.
\begin{figure}
\caption{The quiver $Q$.}
\label{16kogger}
\end{figure}
Recall from Section \ref{sec:bastian} that the 16 vertices can be
divided into disjoint sets as follows: $A = \{3,5,7\}$, $A' =
\{15,16\}$, $B = \{2,4\}$, $B' = \{14\}$, $C = \{9\}$, $D = \{6,8\}$,
$D' = \{10,11,12,13\}$ and $F = \{1\}$.
The steps of the clockwise r-walk starting in vertex $7$ are \[ [ei, bf,
\overline{k}lmnopsc, \overline{j}, h, ei, bf, \ldots]. \] The steps of
the clockwise r-walk starting in vertex $5$ consists of the same
steps as for $7$, but deleting the first step.
By Theorem \ref{finalthm}, we know that there are two classes of
characteristic components containing string complexes, or more
precisely, one class of s-components and one of
r-components. Moreover, since neither
of $r_2$ and $s_2$ are zero, we know that both classes consists of
$\ensuremath{\mathbb{Z}A_{\infty}}$-components. For an r-component, the edge is given by
Proposition \ref{randproposisjon}. We look at the component including
the stalk complex $1_7[0]$.
For any complex on the edge of this component, the upper right
diagonal of the complex is given by Corollary
\ref{allcharacteristicstrings}. In particular, the upper right
diagonal of $1_7[0]$ is $(1_7[0], ei[-1], bfei[-2], \ldots)$. Figure
\ref{komponent1} shows the three lower rows of the AR-component. Note
that this is the same component as in Figure \ref{edge1}.
\begin{figure}
\caption{The lower rows of the AR-component containing $1_7[0]$.}
\label{komponent1}
\end{figure}
Next, we give the $\ensuremath{\mathbb{Z}A_{\infty}^{\infty}}$-components, up to shift, including the
trivial homotopy strings corresponding to vertices of type $C$, $D$,
and $D'$. This is the special $\ensuremath{\mathbb{Z}A_{\infty}^{\infty}}$-component described in
Lemma \ref{lem:uglycomp}. The component is given in Figure
\ref{uglycomp}.
Further, by Tables \ref{omegaplus}--\ref{omegaminustrivial}, we get
that the upper right diagonals starting in the stalk complexes
$1_8^{-1}[0]$, $1_6^{-1}[0]$ and $1_{13}[0]$ consists of subsequent
steps of a counter-clockwise s-walk; the lower left diagonals
starting in $1_9^{-1}[0]$, $1_8^{-1}[0]$ and $1_6^{-1}[0]$ consists of
subsequent steps of a clockwise s-walk; the lower right diagonals
starting in $1_6^{-1}[0]$, $1_{10}[0]$, $1_{11}[0]$, $1_{12}[0]$ and
$1_{13}[0]$ are inverse steps of a clockwise r-walk; and finally
that the upper left diagonals starting in $1_9^{-1}[0]$, $1_{10}[0]$,
$1_{11}[0]$, $1_{12}[0]$ and $1_{13}[0]$ are inverse steps of a
counter-clockwise r-walk. Using this, we have a complete
description of the AR-component. In Figure \ref{spesialkomp}, a
part containing the stalk complexes is shown.
Some examples of homotopy bands associated with $\Lambda$ are
$\overline{s}\overline{t}\overline{u}\overline{s}\overline{p}\overline{o}\overline{n}\overline{m}\overline{l}kjgdacba$
and $\overline{s}\overline{t}\overline{u}cba$.
\begin{figure}
\caption{The special \ensuremath{\mathbb{Z}
\label{spesialkomp}
\end{figure}
\appendix
\section{Assignments of $S$ and $T$}\label{SandT}
Recall Definition \ref{gentlealternativ} of a gentle algebra. We fix
the functions $S,T:Q_1 \rightarrow \{-1,1\}$ for a quiver $Q$ in
$\mathcal Q_n^{\star}$. For the arrows $\alpha_i$, where $1 < i \leq
r$, we set $S\alpha_i = -1$ and $T\alpha_i = 1$. Similarly, for the
arrows $\beta_j$, we set $S\beta_j = -1$ and $T\beta_j = 1$ for $1
\leq j < s$. For any 3-cycle containing neither $\alpha_1$ nor
$\beta_s$, we assign values of $S$ and $T$ as shown in Figure
\ref{standard}, where the arrow marked $a$ is either $\alpha_i$ or
$\beta_j$ for some $1 < i \leq r$ or $1\leq j <s$ (note that
$Sa$ and $Ta$ are already taken care of by the above assignments). For
the arrow $\alpha_1$, we set $S\alpha_1 = 1 = T\alpha_1$, and for the
arrow $\beta_s$, we set $S\beta_s = -1 = T\beta_s$. In the case where
$\alpha_1$, respectively $\beta_s$, is part of a 3-cycle, the values
of $S$ and $T$ for the remaining arrows of the 3-cycle are shown in
Figures \ref{alpha1} and \ref{betas}, respectively.
\begin{figure}
\caption{Assignment of $S$ and $T$ to the arrows of a
3-cycle.}
\label{ST}
\end{figure}
\begin{lemma}\label{stringfunctions}
If $\Lambda \cong kQ/I$ is a cluster-tilted algebra of type
$\ensuremath{\widetilde{A}_n}$, then the above assignments of $S$ and $T$ to the quiver $Q$ in $\mathcal
Q_n^{\star}$ satisfies the conditions given in Definition \ref{gentlealternativ}.
\end{lemma}
The proof is straight-forward. Note that this assignment of $S$ and
$T$ is not unique. A different assignment of the functions will still
give the same final result, but to give examples and technical
results in an unequivocally manner, we need to fix explicitly given
functions.
\section{The longest antipath $\theta_{x,\varepsilon}$}
An \emph{antipath} in a gentle algebra is either a trivial homotopy
string, or a direct homotopy string $\rho$ such that for any two
consecutive arrows $\alpha$ and $\beta$ in $\rho$, we have that their
composition is a relation. Bobi\'nski defines, for each vertex $x \in
Q_0$ and each $\varepsilon \in \{-1,1\}$, the set
$\Theta_{x,\varepsilon}$ consisting of all antipaths $\theta$ such
that $t\theta = x$ and $T\theta = \varepsilon$. Moreover, he defines
$\theta_{x,\varepsilon}$ to be the maximal element of
$\Theta_{x,\varepsilon}$ if such element exists; otherwise, he defines
$\theta_{x,\varepsilon} = \emptyset$.
We will now consider the possible values of $\theta_{x,\varepsilon}$
when $Q$ is a quiver in $\mathcal Q_n^{\star}$. The cases where
$\theta_{x,\varepsilon}$ is not the empty string are the following:
\begin{itemize}
\item $\theta_{A_i,-1} = 1_{A_i}$ for $1 \leq i \leq r_2$,
\item $\theta_{C,1}=\alpha_r$ when $r_1 > 0$,
\item $\theta_{C,-1}=\beta_s$ when $s_1 > 0$,
\item $\theta_{D_i,1}=\alpha_{i+r_2}$ for $1 \leq i \leq r_1-1$,
\item $\theta_{D_i,-1}=1_{D_i}$ for $0 \leq i \leq r_1-1$ and $r_2 >
0$,
\item $\theta_{D_i,-1}=1_{D_i}$ for $1 \leq i \leq r_1-1$ and $r_2 = 0$,
\item $\theta_{D'_j,1}=\beta_{j+s_2}$ for $1 \leq j \leq s_1-1$,
\item $\theta_{D'_j,-1}=1_{D'_i}$ for $0 \leq j \leq s_1-1$.
\end{itemize}
For the rest of the $\theta_{x,\varepsilon}$'s, the set
$\Theta_{x,\varepsilon}$ is infinite with no maximal element. This is
the case when there exists an arrow $\alpha$ which is part of a
3-cycle and such that $t\alpha = x$ and $T\alpha = \varepsilon$.
\section{Proof of Proposition
\ref{tabellproposisjon}} \label{bobinskiappendix}
We now give a proof of Proposition \ref{tabellproposisjon} in section
\ref{almostsplittriangels}. In this proof we will use the algorithm
presented by Bobi\'{n}ski in ~\cite[Section 6]{bob}, which we will now
recall. Let $\omega=\alpha_l \cdots \alpha_1$ be a homotopy string
with homotopy partition $\omega = \sigma_L \cdots \sigma_1$. Note
that our convention of labeling the letters and homotopy letters is
opposite of the convention used in \cite{bob}.
We now state Bobi\'nski's algorithm for finding $\omega^+$. If
$\alpha_l$ is a direct letter, let $\rho(\omega)$ denote the maximal
integer $i \in [1,l]$ such that the $i$ last letters of $\omega$, that
is $\alpha_l \cdots \alpha_{l-i+1}$, is an antipath. If $\alpha_l$ is
an inverse letter or $\omega$ is a trivial homotopy string, we let
$\rho(\omega) = 0$. Next, we define the substring $\omega'$ of
$\omega$ to be
\[
\omega' =
\begin{cases}
\omega & \text{if $\rho(\omega)= 0$} \\
1_{s\omega}^{S\omega} & \text{if $\rho(\omega) = l$}\\
\alpha_{l-\rho(\omega)}\cdots \alpha_1 & \text{if $0<\rho(\omega)<l$}\\
\end{cases}
\]
Note that $\omega'$ is obtained by removing the longest possible
antipath from the end of $\omega$, and that $\rho(\omega)$ denotes the
number of letters removed.
Let $\sigma$ denote the maximal path of $Q$ with no subpath in $I$
such that $\sigma\cdot \omega$ is defined as composition of homotopy
strings. Now, we have 6 cases for computing $\omega^+$, as listed
below:
\[
\omega^+ = \left\{
{\renewcommand{1.4}{1.4}
\begin{array}{lll}
\theta^{-1}_{t\sigma,-T\sigma}\sigma \omega & l(\sigma) > 0 & (1) \\
\theta^{-1}_{t\omega',-T\omega'}\omega' & l(\sigma) = 0\text{, }~ l(\theta_{t\omega',
-T\omega'}) > 0\text{, and }~ l(\omega') > 0 & (2) \\
({}^{[1]}\theta_{t\omega',-T\omega'})^{-1} & l(\sigma) = 0\text{, }~ l(\theta_{t\omega',
-T\omega'}) > 0\text{, and }~ l(\omega') = 0 & (3) \\
{}^{[1]}\omega' & l(\sigma) = 0\text{, }~ l(\theta_{t\omega',
-T\omega'}) = 0{, }~ l(\omega') > 0\text{,} & \\
& \text{and } \alpha_{l(\omega')}(\omega') \text{ is an inverse arrow} & (4) \\
\omega' & l(\sigma) = 0\text{, }~ l(\theta_{t\omega',
-T\omega'}) = 0{, }~ l(\omega') > 0\text{,} & \\
& \text{and } \alpha_{l(\omega')}(\omega') \text{ is an arrow} & (5)\\
\emptyset & l(\sigma) = 0\text{, }~ l(\theta_{t\omega',
-T\omega'}) = 0\text{, and }~ l(\omega') = 0 & (6)
\end{array}}\right.
\]
Finally, for an integer $m$, we compute
\[
m'(\omega) = \left\{ \begin{array}{ll}
l(\theta_{t\sigma,-T\sigma}) - 1 & l(\sigma) > 0 \\
l(\theta_{t\omega',-T\omega'}) + \rho(\omega) -1 & l(\sigma) = 0
\end{array}\right.
\]
and this completely describes how we find the pair
$(m'(\omega),\omega^+)$ from the pair $(m, \omega)$. This concludes
Bobi\'nski's algorithm. We will now state again Proposition
\ref{tabellproposisjon} and then use the above algorithm to give the
proof.
\begin{proposition}Let $\Lambda$ be a cluster-tilted algebra of type
$\ensuremath{\widetilde{A}_n}$, and let $\omega[m]$ be a string complex in $\kbp$. The
middle term $\omega^+[m+m'(\omega)]$ in the AR-triangle starting in
$\omega$ is given by the entries in Tables \ref{omegaplus} and
\ref{omegaplustrivial}. The middle term $\omega_-[m-m'(\omega_-)]$
in the AR-triangle ending in $\omega$ is given by the entries in
Tables \ref{omegaminus} and \ref{omegaminustrivial}.
\end{proposition}
\begin{proof}
The proof of this proposition can be done by direct calculations, and
thus we only include part of the calculations. We include below the
calculations for $\omega^+$ when $\alpha_l(\omega)=\alpha_i^{-1}$
for $1\leq i \leq r$. There are then the following cases to
consider:
\begin{itemize}
\item If $2\leq i\leq r_2+1$, then $\sigma=\gamma_{2i-2}$ so we are in
the first case of the algorithm. Now
$\theta_{t\sigma,-T\sigma}=\theta_{A_{i-1,-1}}=1_{A_i}$, thus
$\omega^+=\gamma_{2i-2}\omega=cw\_r(t\omega)\omega$ and
$m'(\omega)=-1$.
\item If $r_2+2\leq i\leq r$, then it is clear that $\sigma=\emptyset$
and $\rho(\omega)=0$ so $\omega'=\omega$. Now
$\theta_{t\omega',-T\omega'}=\theta_{D_{i-r_2-1},1}=\alpha_{i-1}$ so
we are in case $(2)$ of the algorithm, and
$\omega^+=\alpha_{r_2+j-1}^{-1}\omega=cw\_r(t\omega)\omega$ and
$m'(\omega)=0$.
\item If $i=1$ and $r_2 = 0$, then $\alpha_l(\omega) = \alpha_1^{-1}$
and $\sigma = \beta_s \cdots \beta_1$. Hence we are in case $(1)$
of the algorithm, and $\theta_{t\sigma, -T\sigma} = \alpha_r$. Then
$\omega^+ = \alpha_r^{-1}\beta_s \cdots \beta_1 \omega =
cw\_r(t\omega)\omega$ and $m'(\omega)=0$.
\item If $i=1$ and $r_2 > 0$, then $\alpha_l(\omega) = \alpha_1^{-1}$
and
\[
\sigma = \begin{cases}
\beta_s \cdots \beta_1 & r_1 > 0 \\
\gamma_{2r_2} \beta_s \cdots \beta_1 & r_1 = 0 ~.
\end{cases}
\]
In both cases, we are in case $(1)$ of the algorithm, and
\[
\theta_{t\sigma, -T\sigma} = \begin{cases}
\alpha_r & r_1 > 0 \\
1_{A_{r_2}} & r_1 = 0 ~.
\end{cases}
\]
In both cases, we get $\omega^+ = cw\_r(t\omega)\omega$, with
$m'(\omega) = 0$ if $r_1 > 0$ and $m'(\omega) = -1$ if $r_1 = 0$.
\end{itemize}
The calculations for Tables \ref{omegaplus} and \ref{omegaplustrivial}
can be done similarly.
For the proof of Table \ref{omegaminus} and \ref{omegaminustrivial},
the following procedure is applied: For every entry of the table, let
$\omega$ be a homotopy string satisfying the given description of the
entry and let $\widehat{\omega}$ be the homotopy string arising by
performing the operation from the corresponding entry in the third
column (i.e. $\widehat{\omega} = \omega_-$). Next, consider all
possible $\widetilde{\omega}$ such that $\widetilde{\omega}^+ =
\omega$ (we use Tables \ref{omegaplus} and \ref{omegaplustrivial} for
this). Then, for each such $\widetilde{\omega}$, verify that
$\widetilde{\omega}=\widehat{\omega}$. In this proof, we shall only
do this for the entry $\alpha_l(\omega) = \gamma_{2i}$ in Table
\ref{omegaminus}. The rest is left to the reader.
Assume that $\omega$ is a homotopy string with $\alpha_l(\omega) =
\gamma_{2i}$ for some $1 \leq i \leq r_2$. We examine Tables
\ref{omegaplus} and \ref{omegaplustrivial} to find
$\widetilde{\omega}$ such that $\widetilde{\omega}^+$ can be a
homotopy string ending with such an arrow. The possibilities for
$\widetilde{\omega}$ are:
\begin{itemize}
\item $\alpha_l(\widetilde{\omega}) = \alpha_i$ for $1 \leq i \leq
r_2$
\item $\alpha_l(\widetilde{\omega}) = \alpha_i^{-1}$ for $2 \leq i
\leq r_2 + 1$
\item $\alpha_l(\widetilde{\omega}) = \gamma_{2i} $ for $2 \leq i \leq
r_2$ or $i = 1$ and $r_1 = 0$
\item $\alpha_l(\widetilde{\omega}) = \delta_{2i-1}$ for $1 \leq i
\leq s_2$ and $r_1 = 0$
\item $\alpha_l(\widetilde{\omega}) = \delta_{2i}^{-1}$ for $1 \leq i
\leq s_2$ and $r_1 = 0$
\end{itemize}
It is easy to see that those are all homotopy strings such that
$\widetilde{\omega}^{+}$ is
$cw\_r(t\widetilde{\omega})\widetilde{\omega}$. We have that
$\widehat{\omega}$ is the clockwise r-reduction of $\omega$, an since
this operation undoes adding a step of a clockwise r-walk, we will
always get $\widetilde{\omega} = \widehat{\omega}$.
\end{proof}
\end{document}
|
\begin{document}
\begin{abstract}{It is established that non-isotropic vector field with Jacobi operator
of maximal rank is an obstacle for the existence of non-trivial second-order symmetric
parallel tensor field. It follows that such manifold
as pseudo-Riemannian manifold is locally non-reducible. In particular result is applied
to widely studied classes of almost para-contact metric manifolds -
para-contact metric, para-cosymplectic or para-Kenmotsu manifolds
satisfying nullity and generalized nullity conditions. As corollary we have the
following theorem: almost para-contact metric manifold with
maximal rank Jacobi operator of characteristic vector field is locally non-isometric
to Riemann product. }
\end{abstract}
\title{Rank of Jacobi operator and existence of quadratic parallel differential form, with
applications to geometry of almost para-contact metric manifolds}
\section{Introduction}
Many authors has recently studied the problem of the existence of non-trivial parallel
quadratic form on almost para-contact metric manifold which satisfies generalized nullity conditions.
Let us recall it is said that almost para-contact metric manifold
satisfies generalized nullity condition if
\begin{multline}
\label{nullcond}
R_{XY}\xi = \kappa (\eta(Y)X-\eta(X)Y)+
\mu (\eta(Y)hX-\eta(X)Y)+ \\ \nu(\eta(Y)h'X-\eta(X)h'Y),
\end{multline}
$R$ is the operator of the Riemann curvature, and $\kappa$, $\mu$, $\nu$ are functions,
$\xi$ is the characteristic vector field. The more elaborate terminology is that characteristic vector
field belongs to generalized nullity distribution.
For the studied classes of manifolds the results are that in generic case
there is no parallel quadratic form, different from metric tensor up to non-zero multiplier.
The proofs of these results are based
on properties of Jacobi operator $X\mapsto J_\xi=R_{X\xi}\xi$,
particularly its algebraic form.
However more careful analysis shows that, for above mentioned generic cases, $J_\xi$
has maximal rank. Going into this direction we have
\begin{proposition}
There is no non-trivial parallel quadratic form if Jacobi operator
of characteristic vector field has maximal rank. Manifold as Riemannian
manifold is locally irreducible.
\end{proposition}
The latter sentence comes from the fact that Riemann product always admit
non-trivial parallel differential quadratic form.
Let's denote by $\chi(\xi,x)$ the characteristic polynomial of the Jacobi operator
\begin{equation}
\chi(\xi,x)=x^n-\omega_1(\xi)x^{n-1}+\ldots (\pm 1)^{n-1}\omega_{n-1}(\xi)x,
\end{equation}
we will show the following
\begin{proposition}
Assuming $\xi$ is non-isotropic, $J_\xi$ has maximal rank if and only if the coefficient
$\omega_{n-1}$ at the lowest power term is non-zero $\omega_{n-1}\neq 0$.
\end{proposition}
Of course this assumption is always satisfied for characteristic vector field on almost
para-contact metric manifold. Yet our results are remain valid in wider framework
of pseudo-Riemannian manifolds. Then fact that $\xi$ is non-isotropic is essential.
Author would like to express his gratitude to Professor Quanxiang Pan for our fruitfull discussion.
\section{Preliminaries}
Let $(\mathcal M, g)$ be pseudo-Riemannian $n$-dimensional manifold, $\nabla$ denote the Levi-Civita connection, and $R_{XY}Z$ its curvature
\begin{equation}
R_{XY}Z = \nabla_X\nabla_Y Z -\nabla_Y\nabla_X Z - \nabla_{[X,Y]} Z,
\end{equation}
it is assumed that letters $U$, $V$, $X$,... are used to denote
vector fields, if it is not stated otherwise. Our convention for the Riemann covariant curvature tensor is
\begin{equation}
R(X,Y,Z,W) = g(R_{XY}Z,W).
\end{equation}
For vector field $\xi$ on $\mathcal M$, $(1,1)$-tensor field
$X \mapsto J_\xi X=R_{X \xi}\xi$, is called Jacobi operator.
From the properties of the curvature $J_\xi$ is $g$-self-adjoint and uppper limit of its rank is
$$
g(J_\xi X,Y)=g(J_\xi Y,X),\quad r < {\rm dim}\,\mathcal M.
$$
If $g$ is definite, $J_\xi$ is semi-simple: eigenvalues are real and tangent space splits
into orthogonal direct sum of corresponding eigen-spaces.
Given Riemannian manifold $\mathcal M$, A. Gray \cite{Gray}, considered so-called k-nullity distribution $\mathcal N_k$, $k=const.$,
distribution where the curvature has algebraic form as curvature of constant sectional curvature manifold
\begin{equation}
R_{XY}Z = k ( g(Y,Z)X - g(X,Z)Y),
\end{equation}
where $Z$ is section of $\mathcal N_k$.
\subsection{Almost para-contact metric manifolds}(\cite{Blair}, \cite{Zam})
Let $\mathcal M$ be $(2n+1)$-dimensional manifold.
Almost para-contact metric structure $(\varphi, \xi,\eta,g)$ is quadruple of tensor fields: $(1,1)$-tensor field (affinor) $\varphi$,
characteristic (or Reeb) vector field $\xi$, characteristic 1-form $\eta$,
and pseudo-Riemannian metric $g$, such that, $\varepsilon=\pm 1$,
\begin{eqnarray}
\label{def1} & \varphi^2 = \varepsilon(Id -\eta\otimes \xi), \quad \eta(\xi) =1, & \\[+4pt]
\label {def2} & g(\varphi X, \varphi Y) = \varepsilon(g(X,Y) -\eta(X)\eta(Y)). &
\end{eqnarray}
For $\varepsilon=-1$, structure is customary called almost contact metric.
Eigenvalues of $\varphi$ are imaginary, spectrum contains $\{0,-i,i\}$. It is assumed that the metric
is strictly positive. The latter condition implies that eigenvalues $-i$, $i$ have the same multiplicity
$n$.
The triple $(\varphi, \xi, \eta)$ is called almost contact structure.
For $\varepsilon=+1$, structure is called almost para-contact metric.
Eigenvalues of $\varphi$ are real, the spectrum is $\{-1,0,+1\}$. Tangent space decomposes into direct sum of one-dimensional kernel,
and $n$-dimensional eigen-spaces $\mathcal V(\pm 1)$.
From definition it follows that restriction of the metric to any of eigen-space is null tensor,
and $\mathcal V(\pm 1)$ are maximal in dimension isotropic subspaces. Signature of $g$ is
\begin{equation}
\underbrace{-1,\ldots,-1}_{n}, \underbrace{+1,\ldots,+1}_{n+1}
\end{equation}
Operators
$P_\pm=\varphi \pm Id$, are orthogonal projectors $P_\pm^2=P_\pm $ , $P_{+}P_{-}=P_{-}P_{+}=0$,
onto eigen-spaces $\mathcal V(\pm 1)$.
We have ${\rm Im}\,P_\pm = \mathcal V(\mp 1)$.
In particular
$g(P_\pm X, P_\pm Y)=0$, $g(P_{\pm}X,P_\mp Y)=g(X,Y)$, for $\eta(X)=\eta(Y)=0$.
Per analogy the triple $(\varphi, \xi, \eta)$, is called almost para-contact structure.
From now on we adopt in this paper the convention where both almost contact
metric and almost para-contact metric structures are all together called
almost para-contact metric manifolds. We just follow the line where some authors
use term pseudo-Riemannian manifold in wider sense: manifold equipped with non-degenerate
quadratic differential form. So the reader should be aware of this.
For almost para-contact metric structure tensor field $\varPhi(X,Y) = g(X,\varphi Y)$, is skew-symmetric form called fundamental form.
It satisfies
\begin{equation}
\eta\wedge \varPhi ^n \neq 0,
\end{equation}
at every point, so $\varPhi$ has maximal rank everywhere, its kernel is spanned by characteristic vector field $\xi$.
Manifold equipped with almost para-contact metric structure is called almost para-contact metric manifold.
Such manifold is always orientable.
An important notion is normality. Almost para-contact metric manifold is called normal if
\begin{equation}
[\varphi,\varphi](X,Y)-2\varepsilon\, d\eta\otimes\xi =0,
\end{equation}
where $[\varphi,\varphi]$ denotes Nijenhuis torsion of $\varphi$
\begin{equation*}
[\varphi X,\varphi Y]=\varphi^2 [X,Y]+[\varphi X, \varphi Y]-\varphi ([\varphi X,Y]+[X,\varphi Y]).
\end{equation*}
Non-degenerate hypersurface of almost para-Hermitian manifold can be equipped with
almost para-contact metric structure. Thus such hypersurfaces are one of the fundamental examples of almost para-contact
metric manifolds.
We just mention some classes of almost para-contact metric manifolds.
\begin{definition}
\label{almcont}
{\rm (\cite{Blair}, \cite{MonNicYud}, \cite{DilPas})}.
{\rm Almost para-contact metric
manifold $(\mathcal M, \varphi,\xi,\eta, g)$ is called }
\begin{enumerate}
\item para-contact metric
\begin{equation*}
d\eta = \varPhi,
\end{equation*}
\item almost para-cosymplectic (or almost para-coKaehler)
\begin{equation*}
d\eta =0, \quad d\Phi =0,
\end{equation*}
\item almost para-Kenmotsu
\begin{equation*}
d\eta =0, \quad d\Phi = 2\eta\wedge\Phi.
\end{equation*}
\end{enumerate}
\end{definition}
Assuming additionally normality
we obtain following classes of manifolds: para-Sasakian\footnote{Contact metric and normal, etc},
para-cosymplectic (or para-coKaehler) and para-Kenmotsu.
Let define
$$
h=\frac{1}{2}\mathcal L_\xi \varphi, \quad h'=h\circ\varphi ,
$$
$\mathcal L_\xi$ denotes the Lie derivative along $\xi$. Applying $\mathcal L_\xi$ to identity $\varphi \xi=0$,
we obtain $h\xi=0$ ($h'\xi=0$ is evident).
For given appropriate functions $\kappa$, $\mu$, $\nu$
- the choice depends on the structure in question - almost para-contact metric manifold, such that
\begin{multline}
R_{XY}\xi = \kappa(\eta(Y)X-\eta(X)Y)+\mu(\eta(Y)hX -\eta(X)hY) + \\
\nu(\eta(Y)h'X-\eta(X)h'Y),
\end{multline}
is called $(\kappa,\mu,\nu)$-space or $(\kappa,\mu,\nu)$-almost para-contact metric manifold,
etc. here authors
adopt different naming conventions.
The Jacobi
operator of characteristic vector field on almost contact metric $(\kappa,\mu,\nu)$-space is
\begin{equation}
J_\xi: X \mapsto R_{X\xi}\xi = -\kappa\eta(X)\xi + (\kappa Id+\mu h +\nu h')X ,
\end{equation}
cf. \cite{BlKoufPap}, \cite{Boeckx}, \cite{DacOl}, \cite{DilPas}. Note that $J_\xi$ has
maximal rank if and only if
$$
J_\xi |_{ \eta = 0} = \kappa Id +\mu h +\nu h',
$$
is invertible on $ \eta = 0$.
\section{Symmetric parallel tensors of pseudo-Riemannian manifolds}
The goal of this section is to prove the following result.
\begin{proposition}
\label{symm}
Let $(\mathcal M,g)$ be $(n+1)$-dimensional pseudo-Riemannian manifold. Let assume there is non-isotropic vector field
$\xi$, $g(\xi,\xi)\neq 0$, with Jacobi operator of maximal rank. If $\alpha \neq 0$ is parallel differential quadratic form,
then it is proportional to metric tensor $\alpha=c g$.
\end{proposition}
\begin{proof}
We may assume $g(\xi,\xi)=\epsilon$, $\epsilon =\pm 1$. So there is $(0,2)$-tensor $\alpha(X,Y)=\alpha(Y,X)$ symmetric and parallel $\nabla \alpha =0$, such that our quadratic differential form is just $X \mapsto \alpha(X,X)$.
Moreover by assumption $r={\rm dim (Im}\,J_\xi)=n$.
We set
\begin{equation}
i_\xi\alpha(X)=\alpha(\xi,X), \quad i_\xi g(X)=g(\xi,X).
\end{equation}
Let consider the case $i_\xi\alpha\neq 0$. For
$\alpha$ and $g$ are both parallel
\begin{equation}
i_\xi\alpha(J_\xi \,\cdot) =0, \quad i_\xi g(J_\xi\, \cdot)=0,
\end{equation}
hence
$$
{\rm Im }\,J_\xi \subset ( \iota_\xi\alpha =0 ) \cap ( \iota_\xi g=0 ),
$$
and assumption that $J_\xi$ has maximal rank, implies
$$
{\rm Im }\,J_\xi = ( \iota_\xi\alpha =0 ) = ( \iota_\xi g=0 ),
$$
in particular forms $i_\xi\alpha$ and $i_\xi g$ are
collinear
$i_\xi\alpha = \epsilon\alpha(\xi,\xi)i_\xi g$.
We apply now the second covariant derivatives $\nabla_{Y,\xi}^2$, and $\nabla_{\xi,Y}^2$ resp. to the both sides of the identity
\begin{equation}
\label{coll}
\alpha(\xi,X)=\epsilon \alpha(\xi,\xi)g(\xi,X),
\end{equation}
we have to take into account that by assumption $\alpha$ is covariant constant.
Subtracting resulting equations, having in
mind that $R_{Y\xi}\xi = \nabla_{Y,\xi}^2-\nabla_{\xi,Y}^2$, we obtain
\begin{equation}
\alpha(J_\xi Y, X) =\epsilon\alpha(\xi,\xi)g( J_\xi Y, X),
\end{equation}
which by maximality follows
\begin{equation}
\alpha = c g, \quad c=\epsilon\alpha(\xi, \xi) \neq 0,
\end{equation}
and by $0=\nabla \alpha = dc\otimes g $, we have $c=const$.
In the case $i_\xi\alpha =0$, in the similar way as above we find
$\alpha(X,Y) = 0$, $g(\xi,Y)=0$. For $\xi$ is non-isotropic and
$i_\xi\alpha =0$, $\alpha$ must vanish which contradicts our assumption $\alpha \neq 0$.
Hence the case $i_\xi\alpha =0$ is not possible.
\end{proof}
\subsection{Trace form}
Let $\mathcal M$ be $(n+1)$-dimensional pseudo-Riemannian manifold. Let recall formula
for characteristic polynomial
\begin{equation}
\chi(\xi,x)=det(x Id -J_{\xi})=x^{n+1} -\omega_1(\xi) x^{n}+\ldots (-1)^{n}\omega_{n}(\xi)x.
\end{equation}
Note that $\omega_i$'s as a functions
$$
: \xi \mapsto \omega_{i}(\xi)
$$
are all smooth differential forms. Indeed, let
$J^{i}_{\xi}$ denotes the $i$-th exterior power of $J_{\xi}$.
It is $(i,i)$-tensor field - interpreted as endomorphism acting on $i$-th degree polivectors
on $\mathcal M$. By definition on simple polivector $\mathcal W = V_1\wedge\ldots V_i$
$$
J^{i}_{\xi}(V_1\wedge\ldots V_i) = J_{\xi}(V_1)\wedge\ldots J_{\xi}(V_i),
$$
then $\omega_{i}=(-1)^{n} tr(J^{i}_{\xi})$. From the
definition of Ricci tensor we have $Ric(\xi,\xi)=\omega_{1}(\xi)$. So
we may think of $\omega_{i}$, $i > 1$, as a kind of Ricci tensors of higher
degrees.
Let us recall the metric tensor gives rise to canonical symmetric bilinear form on
polivectors: we denote it by $g^{\wedge k}$
\begin{equation}
g^{\wedge k}(X_1\wedge\ldots X_k) = det [ g(X_i,X_j)].
\end{equation}
The metric $g^{\wedge k}$ and $J^{k}_{\xi}$ are compatible in the sense
that the latter is $g^{\wedge k}$-self-adjoint.
\begin{proposition}
Let assume $J^{n}_{\xi}\neq 0$. Then $J^n_{\xi}$ has rank one. In particular there is $n$-form $\tau_{\xi}$ and
$n$-multivector $\mathcal W_{\xi}$, such that
\begin{equation}
J^{n}_{\xi} = \tau_{\xi}\otimes \mathcal W_{\xi},\quad \omega_{n}(\xi)=\tau_{\xi}(\mathcal W_{\xi}),
\end{equation}
and $\omega_{n}(\xi) = 0$ if and only if $\xi$ is isotropic.
\end{proposition}
\begin{proof}
Let extend $\xi$ to local frame $(\xi, X_1,\ldots,X_{n})$, then
\begin{equation}
\label{base}
\mathcal W_i = \xi\wedge X_{1}\ldots \wedge \hat{ X}_{i}\wedge\ldots X_n, \quad 1 \leq i \leq n,
\end{equation}
span the kernel of $J^{n}_\xi$,
we see that dimension of kernel is $n$, so
if $J^{n}_\xi \neq 0$ its image is one-dimensional subspace, so
there is polivector $\mathcal W$ and $n$-form $\tau$, such that
\begin{equation}
J^{n}_\xi = \tau\otimes \mathcal W,\quad
\omega_{n}=tr(J^{n}_\xi) = \tau(\mathcal W),
\end{equation}
$\tau$ and $\mathcal W$ are determined up to re-scaling
\begin{equation}
\tau \mapsto f\tau, \quad \mathcal W \mapsto f^{-1}\mathcal W,
\end{equation}
where $f$ is arbitrary non-vanishing function.
Condition $tr(J^{n}_{\xi})=\tau(\mathcal W)=0$ means that $\mathcal W$ itself belongs to
the kernel of $J^{n}_{\xi}$. Every element of the kernel is linear
combination of polivectors as in (\ref{base}), therefore $\mathcal W$
has decomposition
$$
\mathcal W = \xi\wedge \mathcal W_{0},
$$
For some simple polivector $X_{1}\wedge\ldots X_{n}$ we have
$$
J^{n}_\xi (X_1\wedge\ldots X_{n}) = \mathcal W,
$$
Then
\begin{equation}
\xi\wedge \mathcal W_0 = \mathcal W = J^{n}_\xi (X_1\wedge\ldots X_{n})
= (R_{X_1\xi}\xi)\wedge\ldots (R_{X_{n}\xi}\xi),
\end{equation}
which follows that up to reoder
\begin{equation}
R_{X_1\xi}\xi = c_{0} \xi + c_{1}X_{1}+\ldots c_{n}X_{n}, \quad c_0 \neq 0,
\end{equation}
from other hand
$$
0=g(R_{X_{1}\xi}\xi,\xi)=c_{0}g(\xi,\xi)+\sum\limits_{i=1}^{n}c_{i}g(X_{i},\xi),
$$
If $g(\xi,\xi)\neq 0$, we
would take all $X_{1},\ldots X_{n}$, such that $g(X_{i},\xi)=0$
then from the above equation we will have $c_{0}=0$ - contradiction.
Hence $g(\xi,\xi)=0$.
\end{proof}
Note Jacobi operator has maximal rank if and only if its exterior power $J^n_\xi \neq 0$.
\begin{corollary}
Jacobi operator of non-isotropic vector field has maximal rank if and only if coefficient at lowest term
of its characteristic polynomial is non-zero.
\end{corollary}
Note there are non-trivial
parallel quadratic differential forms on Riemannian products.
\begin{corollary}
Let $(\mathcal M, g)$ be pseudo-Riemannian manifold. If every point admits
locally defined vector field with maximal rank Jacobi operator, then
$\mathcal M$ is locally irreducible.
\end{corollary}
\section{Applications to almost para-contact metric manifolds}
Applications are direct. Assume $(\mathcal M, \varphi,\eta,g)$ is almost
para-contact metric manifold.
\begin{proposition}
If Jacobi operator of characteristic vector field has maximal rank, then
non-zero parallel second order differential form is proportional to
pseudo-length form. In particular manifold is locally irreducible.
\end{proposition}
In case of $\kappa$-nullity spaces, $\kappa \neq 0$, we just rephrase above result.
\begin{corollary} Non-zero parallel second order differential form on
$\kappa$-nullity almost para-contact metric manifold is proportional to pseudo-length form
provided $\kappa \neq 0$.
\end{corollary}
The case of $(\kappa,\mu)$-spaces requires study of singular values of
operator $\kappa Id +\mu h$. Let denote by
$\omega(x)=\sum\limits_{i=0}c_ix^{n-i}$, $c_0=1$, the characteristic polynomial of $h$.
Those values are solutions $(\kappa, \mu)$ of polynomial equation
\begin{equation}
det(\kappa Id+\mu h) = \sum\limits_{i=0}c_i \kappa^{n-i}\mu^i=0,
\end{equation}
are singular values.
The case of almost contact
metric manifold is simpler due to fact that $h$ is
diagonalizable. Denoting by $\lbrace \lambda_1,\ldots\lambda_k \rbrace$ spectrum of $h$,
we see that
$(\kappa,\mu)$ are singular if they satisfy one of the equations
\begin{equation}
\kappa+\lambda_i \mu =0,\quad i=1,\ldots k.
\end{equation}
Let have a look at some particular classes of manifolds where we posses more
detailed information concerning operator $h$.
\noindent{\bf Example 1.}
Almost Kenmotsu $(\kappa,\mu)$-nullity manifolds. There is strong result which
asserts that $\kappa = -1$, and $\mu=0$.
So for such class of manifolds Jacobi operator of $\xi$ is of maximal rank.
If we take instead
$h'=h\circ \varphi$ (generalized $(\kappa,\mu)'$-nullity spaces),
then $\kappa \leq -1$ , for $\kappa =-1$, $h'=0$ and for
$\kappa < -1$, $\mu =- 2$, eigenvalues of $h'$ are $0$, $\pm\sqrt{-k-1}$,
from these conditions there is one-point singularity $(-2,-2)$.
Beyond that pair of values vector field $\xi$ has maximal Jacobi operator.
\noindent{\bf Example 2.} Contact metric $(\kappa,\mu)$-nullity spaces.
For such manifold $\kappa \leq 1$, if $\kappa =1$, then $h=0$. For $\kappa <1 $,
eigenvalues of $h$ are $0$, $\pm\sqrt{1-k}$ and singular values are
pairs $(\kappa,\mu)$ which satisfy $\kappa^2 -(1-k)\mu^2=0$.
So, beyond these points Jacobi operator has maximal rank.
\footnote{Minimal polynomial of $h$ is $x(x^2-(1-\kappa))$}
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\end{document}
|
\begin{document}
\title[a]{Mapping multiple photonic qubits into and out of one solid-state atomic ensemble}
\pacs{}
\author{Imam Usmani}
\author{Mikael Afzelius}
\email{[email protected]}
\author{Hugues de Riedmatten}
\author{Nicolas Gisin}
\affiliation{Group of Applied Physics, University of Geneva, CH-1211 Geneva 4, Switzerland}
\date{\today}
\begin{abstract}
The future challenge of quantum communication are scalable quantum
networks, which require coherent and reversible mapping of photonic qubits onto stationary
atomic systems (quantum memories). A crucial
requirement for realistic networks is the ability to efficiently
store multiple qubits in one quantum memory.
Here we demonstrate coherent and reversible mapping of 64 optical modes
at the single photon level in the time domain onto one solid-state
ensemble of rare-earth ions. Our light-matter interface is based
on a high-bandwidth (100 MHz) atomic frequency comb, with a
pre-determined storage time $\gtrsim$ 1 $\mu$s. We can then encode
many qubits in short $<$10 ns temporal modes (time-bin qubits). We
show the good coherence of the mapping by simultaneously storing
and analyzing multiple time-bin qubits.
\end{abstract}
\maketitle
Quantum communication \cite{Gisin2007a} offers the possibility of secure transmission of
messages using quantum key distribution (QKD) \cite{Gisin2002} and teleportation of unknown quantum states \cite{Bennett1993}. Quantum communication relies on creation, manipulation and transmission of qubits in photonic channels. Photons have proven to be robust carriers of quantum information. Yet, the transmission of photons through a fiber link, for instance, is inherently a lossy process. This leads to a probabilistic nature of the outcome of experiments. In large-scale quantum networks \cite{Kimble2008} the possibility of synchronizing independent and probabilistic quantum channels will be required for scalability \cite{Duan2001,Sangouard2009a}. A quantum memory enables this by momentarily holding a photon and then releasing it when another part of the network is ready. In order to reach reasonable rates in a realistic network it will be necessary to use multiplexing \cite{Simon2007}, which requires quantum memories capable of storing many single photons in different modes.
A quantum memory requires a coherent medium with strong coupling
to a light mode. Strong and coherent interactions can been found
in ensembles of atoms \cite{Hammerer2008}, for instance alkali
atoms or rare-earth (RE) ions doped into crystals. The latter are
attractive for quantum storage applications, as they provide
solid-state systems with a large number of stationary atoms with
excellent coherence properties. Optical coherence times of up to
milliseconds \cite{Sun2002} and spin coherence times $>$ seconds
\cite{Longdell2005} have been demonstrated at low temperature
($\lesssim$ 4 K).
A quantum memory also requires a scheme for achieving efficient
and reversible mapping of the photonic qubit onto the atomic
ensemble. Techniques investigated include stopped light based on
electromagnetically induced transparency (EIT)
\cite{Chaneliere2005,Eisaman2005,Choi2008}, Raman interactions
\cite{Julsgaard2004,Nunn2007,Hosseini2009} or photon-echo based
schemes
\cite{Moiseev2001,Kraus2006,Hetet2008,Afzelius2009a,Riedmatten2008}.
Much progress has been made in terms of quantum memory efficiency
\cite{Choi2008,Simon2007a}, and storage time
\cite{Zhao2009,Zhao2009a}. Storage of multiple qubits is
challenging, however, because it requires a quantum memory that
can store many optical modes into which qubits can be encoded.
Note that each pair of modes can encode a different qubit, or more
generally, d modes encode a qudit. A mode can be defined in time \cite{Simon2007}, space \cite{Lan2009}, or frequency. Time multiplexing as used in classical communication has the great advantage of requiring only a single spatial mode
\cite{Afzelius2009a,Nunn2008}, hence a single quantum memory.
Moreover, temporal modes can be used to define time-bin qubits
\cite{Gisin2007a}, which are widely used in fiber-based quantum
communication due to their resilience against polarization
decoherence in fibers.
This type of temporal multimode storage is difficult, however, due
to the scaling of the number of stored modes $N_m$ as a function
of optical depth $d$ of the storage medium
\cite{Afzelius2009a,Nunn2008}. For EIT and Raman interactions
$N_m$ scales as $\sqrt{d}$ \cite{Nunn2008}, making it very
difficult to store many modes. Recently we proposed
\cite{Afzelius2009a} a multimode storage scheme based on atomic
frequency combs (AFC) with high intrinsic temporal multimode
capacity \cite{Afzelius2009a,Nunn2008}. Using this method we
recently demonstrated \cite{Riedmatten2008} that a weak coherent
state $|\alpha\rangle_L$ with mean photon number
$\overline{n}=|\alpha|^2$ $<$ 1, can be coherently and reversibly
mapped onto a YVO$_4$ crystal doped with neodymium ions. Later
experiments \cite{chaneliere-2009,Afzelius2009b,Amari2009a} in
other RE-doped materials have improved the overall storage
efficiency (35\%) and storage time (20$\mu$s). Yet, in these
experiments at most 4 modes have actually been stored at the single photon level, thus the
predicted \cite{Afzelius2009a,Nunn2008} high multimode capacity
has yet to be shown experimentally.
\begin{figure*}
\caption{(a) Simplified level scheme of the Nd ions doped into
Y$_2$SiO$_5$. We use the optical transition at 883nm between the
$^4$I$_\frac{9}
\label{fig:AFC}
\end{figure*}
\section{Results}
Here we demonstrate reversible mapping of 64 temporal modes
containing weak coherent states at the single photon level onto
one atomic ensemble in a single spatial mode using an AFC-based
light-matter interface \cite{Afzelius2009a}. An AFC is based on a
periodic modulation (with periodicity $\Delta$) of the absorption
profile of an inhomogeneously broadened optical transition
$|g\rangle \rightarrow |e\rangle$ (see Fig. \ref{fig:AFC}). The
modulation should ideally consist of sharp teeth (with full-width
at half-maximum $\gamma$) having high peak absorption depth $d$,
cf. Figure \ref{fig:AFC}b. Such a modulation can be created by
optical pumping techniques (see Figure \ref{fig:AFC} and Methods).
This requires, however, an atomic ensemble with a static
inhomogeneous broadening and many independently addressable
spectral channels. This can be found in RE-doped solids where
inhomogeneous broadening is of order 1-10 GHz and the homogeneous
linewidth is of order 1-100 kHz when cooled $<$ 4K. When a weak
photonic coherent state $|\alpha\rangle_L$ with $\overline{n}<1$
is absorbed by the atoms in the comb, the state of the atoms can
be written as
$|\alpha\rangle_A=|G\rangle+\alpha|W\rangle+O(\alpha^2)$. Here
$|G\rangle=|g_1\cdot\cdot\cdot g_N \rangle$ represents the ground
atomic state and
\begin{equation}
|W\rangle=\sum_n c_n e^{i2\pi\delta_nt}e^{-ikz_n} |g\cdot\cdot\cdot e_n\cdot\cdot\cdot g \rangle
\label{eq_etatW}
\end{equation}
\noindent represents one induced optical excitation delocalized
over all the $N$ atoms in the comb. In Eq. (\ref{eq_etatW}) $z_n$
is the position of atom $n$, $k$ is the wave-number of the
single-mode light field, $\delta_n$ the detuning of the atom with
respect to the laser frequency and the amplitudes $c_n$ depend on
the frequency and on the spatial position of the particular atom
$n$. The initial (at $t$=0) collective strong coupling between the
light mode and atoms is rapidly lost due to inhomogeneous
dephasing caused by the $\exp(i2\pi\delta_nt)$ phase factors. If
we assume that the peaks are narrow as compared to the periodicity
(i.e. a high comb finesse $F=\Delta/\gamma$), then
$\delta_n\approx m_n\Delta$ and the W state will rephase after a
pre-programmed time $1/\Delta$. The rephased collective state W
will cause a strong emission in the forward direction (as defined
by the absorbed light).
\begin{figure}
\caption{The output from a frequency-stabilized ($<$100kHz) diode laser was split into two beams using a polarization beam splitter (PBS). Each beam could be amplitude, frequency and phase modulated using a double-pass acousto-optic modulators (AOM). One beam was used for creating the preparation pulses (see text), and
the other one for creating the weak pulses to be stored (strongly attenuated using neutral density (ND) filters). In the weak path an additional electro-optic amplitude modulator (EOM) was used to create short input pulses for the multimode storage
experiments. The paths were mode overlapped using a fiber-coupled beam combiner. The light was sent through the crystal, again in free space, in a double-pass setup using a Faraday rotator (FR) and a PBS. The output light was
collected with a multimode fiber and detected by an Si single-photon counter (APD). Two synchronized mechanical choppers (MC) blocked the detector during the preparation sequence and blocked the preparation beam during the storage sequence, respectively. See Supplementary Information for more details.}
\label{fig:setup}
\end{figure}
This type of photon-echo emission is also
observed in accumulated or spectrally programmed photon echoes
\cite{Hesselink1979,Carlson1984,Mitsunaga1991,Schwoerer1994}, which inspired our proposal. Spectral atomic gratings have also been proposed \cite{Merkel1996} and demonstrated \cite{Tian2001} for coherent optical delay of streams of strong classical pulses. The interest in spectral gratings was recently renewed in the context of quantum memories, when it was realized how to achieve a much more efficient spectral grating than previously possible. This is possible due to the highly absorbing and sharp peaks in the AFC structure \cite{Afzelius2009a}. In practice the finite finesse of the
comb still needs to be accounted for, which causes a partial loss
of the collective state. But in Ref. \cite{Afzelius2009a} we
show theoretically that $F$=10 induces a negligible loss, which in
combination with a high optical depth $d$ makes the AFC scheme
very efficient. High-efficiency mapping using high-finesse combs
have been shown experimentally \cite{chaneliere-2009,Amari2009a}.
These experiments and the present work stores light for a
pre-determined time given by $1/\Delta$. We thus emphasize that we
also proposed \cite{Afzelius2009a} and experimentally
demonstrated \cite{Afzelius2009b} a way to achieve on-demand
readout by combining AFC with spin-wave storage. On-demand readout is a crucial resource for applications in quantum networks in order to render different quantum channels independent.
The multimode property of an AFC memory can easily be understood
qualitatively. For a periodicity $\Delta$ and $N_p$ peaks, its
total bandwidth is of order $\sim N_p \Delta$ meaning that a pulse
of duration $\tau \sim 1/(N_p \Delta)$ can be stored. The
multimode capacity stems from the fact that the grating can absorb
a train of weak pulses before the first pulse is re-emitted after
$T=1/\Delta$ (cf. Fig. \ref{fig:AFC}c). This simple calculation
gives a multimode capacity $N_m \propto T/\tau \propto N_p$. Thus
a comb with many peaks $N_p$ allows us to create a highly
multimode memory in the temporal domain. In this context RE-doped
solids are particularly interesting due to their high spectral
channel density.
\begin{figure}
\caption{(a) Experimental combs created using preparation
sequences with either single (solid line) or five (dashed line)
simultaneous pump frequencies. The frequency shifted sequences
allows us to enlarge the frequency range over which the optical
pumping is efficient and thereby creating a wide 100 MHz comb. (b)
Efficiency as a function of the duration (FWHM) of the input pulse
for a single (circles) and five (squares) frequency preparation.
As the duration decreases the bandwidth of the input pulse
increases. The decrease in efficiency for short pulses is due to
bandwidth mismatch for large bandwidths when using a single
preparation frequency. This experiment clearly illustrates the
gain in bandwidth in the extended preparation sequence for which
only a small decrease in efficiency is observed. (c) Pulse sequence for atomic frequency comb
preparation (see text). In order to increase the bandwidth, the pulses are
repeated with shifted frequencies. This pulse sequence was used
for most of our experiments. Here it creates a comb of 100MHz
bandwidth and a periodicity of 1MHz. The total sequence takes
16$\mu$s and it is repeated around 2000 times in order to prepare
the AFC.}
\label{fig:memory bandwidth}
\end{figure}
Here we work with a neodymium-doped Y$_2$SiO$_5$ crystal having a transition
wavelength at 883 nm with good coherence properties (see Methods
for the spectroscopic information). This wavelength is convenient
since we can work with a diode laser and Si based single photon
counters having low noise (300 Hz) and high efficiency (32\%).
\begin{figure*}
\caption{Storage of an input state composed of 64 temporal modes.
The input (left part) is a random sequence of full and empty time
bins, where the mean photon number in the full ones is
$\overline{n}
\label{fig:64modes}
\end{figure*}
The comb is prepared on the $|g\rangle$ - $|e\rangle$ transition by frequency selectively pumping atoms into an auxiliary state $|aux\rangle$ (see Fig. \ref{fig:AFC}). There are different techniques for achieving
this. For instance, by creating a large spectral hole, and then
transferring back atoms from an auxiliary state to create a comb,
as used in \cite{Afzelius2009b}. Here we use a similar technique
to \cite{Riedmatten2008}, where a series of pulses separated by a
time $\tau$ pump atoms from $|g\rangle$ to $|aux\rangle$ (through
$|e\rangle$) with a power spectrum having a periodicity
$1/\tau=\Delta$. This technique is also frequently used in accumulated photon echo techniques \cite{Hesselink1979,Tian2001}. Here each pulse sequence consisted of three pulses where the central pulse is $\pi$-dephased, which has a
power spectrum with "holes" (see Fig. \ref{fig:memory bandwidth}c). A straightforward calculation shows
that the width of the holes in the power spectrum decreases with
the number of pulses in the sequence. In this experiment three
pulses were enough to reach the optimal comb finesse (F$\approx$3) to achieve the maximal efficiency for a given optical depth (see Methods). Note that
the sum of the amplitudes of the side pulses (here two) should
correspond to the amplitude of the central $\pi$-dephased pulse,
in order to obtain the appropriate power spectrum. This rule also
holds for sequences with more pulses. To increase the depth of the
comb the sequence was repeated 2000 times. More details can be found in the
Supplementary Information.
The experimental sequence is divided into two parts: the
preparation of the AFC (cf. above) and the storage of the weak pulses. The
preparation lasts 100 ms, which is followed by a delay of 5 ms( $\approx 17
T_1$) to avoid fluorescence noise from atoms left in the excited state. During the storage sequence, 1000 independent
trials are performed at a repetition rate of 200 kHz. The entire
sequence preparation plus storage is then repeated with a
repetition rate of 5 Hz. An overview of the experimental set up is shown in Fig. \ref{fig:setup}.
In Fig. \ref{fig:AFC}c we show storage experiments with pre-determined storage times of
$T$=100ns and 1$\mu$s, for a single temporal mode. The overall
in-out mapping efficiencies, defined as the ratio of the output
pulse counts to the input pulse counts, are $\sim 6\%$ and $\sim
1\%$ respectively (see inset of Fig. \ref{fig:AFC}c). In the
Methods section we present a theoretical analysis of the
efficiency performance. The efficiency for single-mode storage is
currently lower than has been achieved in the best-performance
single-mode memories, e.g.
\cite{Choi2008,Hetet2008,Hosseini2009,chaneliere-2009,Amari2009a}.
But as explained later our interface compares very favorably to
these experiments in terms of potential multimode storage
efficiency.
The main goal of the present work is to demonstrate high multimode
storage, as theoretically predicted in
\cite{Afzelius2009a,Nunn2008}. Following the discussion above, we
should maximize the number of peaks in the comb. This can be done
by increasing the density of peaks in a given spectral region
(i.e. increasing the storage time $T$) or by changing the width of
the AFC (i.e. increasing the bandwidth). Here we fix the storage
time to $T$=1.3 $\mu$s where we reach an efficiency of $\gtrsim$
1\% and concentrate our efforts on increasing the bandwidth. The
spectral width of the grating is essentially given by the width of
the power spectrum of the preparation sequence, which here results
in a width of about 20-30 MHz. We can however substantially
increase the total width by inserting more pulses in the
preparation sequence, which are shifted in frequency (see
Methods). We thus optically pump atoms over a much larger
frequency range. Note that the frequency shift should be a
multiple of $\Delta$ in order to form a grating without
discontinuities. In this way we managed to extend the bandwidth of
the interface to 100 MHz as shown in Fig. \ref{fig:memory
bandwidth}a, without significantly affecting the AFC echo
efficiency. This is illustrated in Figure \ref{fig:memory
bandwidth}b, where we show storage efficiency as a function of the
duration of the input pulse when the preparation sequence contains
a single or five frequencies. The maximum bandwidth allows us to
map short $<$10 ns pulses into the memory.
\begin{figure}
\caption{The coherence of the multimode storage was measured via
an interference measurement. (a) The output signal (solid line)
generated by the double-AFC scheme (see text), which causes an
interference between consecutive modes. The input sequence (not
shown) is a series of weak coherent states
($\overline{n}
\label{fig:interference}
\end{figure}
In addition to the present motivation for multimode storage, a
large bandwidth is equally interesting for interfacing a memory
with non-classical single-photon or photon pair sources. These
usually have large intrinsic bandwidth which requires extensive
filtering for matching bandwidths. In the present case our
extended bandwidth ($\times$5) would require a corresponding
factor of less filtering.
We show the high multimode capacity of our interface by storing 64
temporal modes during a pre-determined time of 1.3 microseconds
(see Fig. \ref{fig:64modes}), with an overall efficiency of 1.3\%.
This capacity is more than an order of magnitude higher than
previously achieved for multiplexing a quantum memory in a single
spatial mode \cite{Riedmatten2008,Hosseini2009}. As shown we can
store a random sequence of weak coherent states. Storage of random
trains of single photon states has been proposed for multiplexing
long-distance quantum communication systems based on so called
quantum repeaters \cite{Duan2001,Sangouard2009a}. The maximum rate
of communication would then be proportional to the number of modes
that can be stored \cite{Simon2007}. Our experiment clearly shows
the gain that can be made using an AFC-based quantum memory. It
thus opens up a route towards achieving efficient quantum
communication using quantum repeaters.
It is now possible to use consecutive temporal modes, e.g. modes
$|k\rangle$ and $|k+1\rangle$, to encode time-bin qubits
$c_k|k\rangle + c_m e^{i\phi_{km}}|m\rangle$, in which case a good
coherence between modes is crucial. The coherence can be measured
by preparing superposition states and performing projective
measurements using an interferometric set up. Projective
measurements on time-bin qubits is usually performed using an
unbalanced Mach-Zehnder interferometer (MZI) where consecutive
time-bin are interfering \cite{Gisin2007a}. We can perform the
same task with our light-matter interface by using a double-AFC
scheme (with $\Delta_1$ and $\Delta_2$) as shown in
\cite{Riedmatten2008}. In short, the difference in delay
$1/\Delta_1-1/\Delta_2$ plays the role of the delay in a
unbalanced MZI. We observe excellent coherence over all modes with
an average visibility of $V=86\%\pm3\%$, see Fig.
\ref{fig:interference}, corresponding to a conditional qubit
fidelity of $F=(1+V)/2\approx93\%$.
To further illustrate our ability to store multimode light states
we create a light pulse with a random amplitude modulation. As
shown in Fig. \ref{fig:arbitrary} we can faithfully store this
kind of light pulses. The possibility of storing weak arbitrary
light states using photon-echo based schemes was pointed out
already by Kraus et al. \cite{Kraus2006}. We believe that this
work, where complex phase and amplitude information are reversible
and coherently mapped onto one atomic ensemble, is the first
experimental realization showing these properties at the single
photon level.
\begin{figure}
\caption{Mapping of a 1 $\mu$s long input pulse with randomly
varying amplitude. As seen the overlap between the normalized
input (dashed line) and output (solid line) pulses is excellent.
The total average number of photons in the input pulse is
$\overline{n}
\label{fig:arbitrary}
\end{figure}
\section{Discussion}
For multimode storage the efficiency of our
interface would outperform the current EIT and Raman based quantum
memories in homogeneously broadened media, although impressive
efficiencies have been achieved for single-mode storage
\cite{Choi2008,Simon2007a,Hammerer2008}. This is due to the poor
scaling of the efficiency as a function of the number of modes for
a given optical depth \cite{Nunn2008}. It also compares favorably
to the recent few modes storage experiment \cite{Hosseini2009}
using the gradient echo memory (GEM), another echo based storage
scheme, also due to the scaling of mode capacity for a given
optical depth (N$_m\sim d$)\cite{Simon2007,Nunn2008}. Still, an
increase in storage efficiency and on-demand read out is necessary
for applications in quantum communication.
The next grand challenge is to combine multimode storage, high efficiency
\cite{Amari2009a} and on-demand read out \cite{Afzelius2009b} in
one experiment. The immediate efforts will most probably by devoted to
praseodymium and europium-doped Y$_2$SiO$_5$ crystals, where the ground state manifold has the necessary number of spin levels (three levels) for implementing the on-demand readout. The recent achievements in Pr-doped Y$_2$SiO$_5$ crystals are very encouraging \cite{Afzelius2009b,Amari2009a}, although the bandwidth was limited to a few MHz due to the hyperfin level splitting. Europium-doped Y$_2$SiO$_5$ has the potential of offering higher bandwidths (up to 70 MHz) and narrower comb peaks, which results in higher multimode capacity \cite{Afzelius2009a}. In order to exploit the high-bandwidth results reported in this work, using neodymium-doped crystals, one needs to find a third spin level with a long spin coherence lifetime. An interesting path forward is to investigate neodymium isotopes with a hyperfine structure ($^{143}$Nd and $^{145}$Nd) \cite{Macfarlane1998}. Recent results on a similar system \cite{Bertaina2007}, $^{167}$Er$^{3+}$:CaWO$_4$, show coherence times approaching 100 $\mu$s for hyperfine transitions. Clearly this path requires extensive spectroscopic studies in order to optimize the spin population and coherence lifetimes. But it is very interesting since it opens up several material candidates (e.g. doped with Erbium \cite{Lauritzen2009} and Neodymium) for quantum memory applications.
To summarize we have demonstrated the reversible mapping of up to
64 optical temporal modes at the single photon level onto one
solid state atomic ensemble. We have demonstrated that the quantum
coherence of the stored modes is preserved to a high extent. The
different modes can then be used to encode multiple time-bin
photonic qubits. Alternatively, they could also be considered as
high-dimensional qudits states. This opens up possibilities to
store higher dimensional quantum states such as entangled qudits
encoded in time bin bases. Our experiment opens the way to
multi-qubit quantum memories, which are a crucial requirement for
realistic quantum networks.
\section{Methods}
\footnotesize{
\textbf{Sample}
The sample is a 10mm long neodymium-doped yttrium orthosilicate
crystal(Nd$^{3+}$:Y$_2$SiO$_5$) with a low Nd$^{3+}$ concentration
of 30 ppm. The inhomogeneous broadening of the
$^4$I$_\frac{9}{2}$-$^4$F$_\frac{3}{2}$ absorption line is around
6GHz and the optical depth 1.5 for this sample. By using a
double-pass set up through the crystal we could increase the
optical depth to 3. We measured an excited state lifetime of $T_1$=300$\mu$s using fluorescence
spectroscopy and stimulated photon echoes. With conventional
photon echoes (two-pulse) we measure a homogeneous linewidth of
3.4kHz (T$_2=92.7\mu$s). Each level is a Kramer's doublet which
split into two spin states in a magnetic field. For the field
orientation used in this experiment we measured g factors of
$g_g=2.6$ and $g_e=0.5$. In a 300mT magnetic field, the excited
states were separated with 2GHz. We measured a ground state Zeeman
population relaxation lifetime, by spectral hole burning (SHB), of
around $T_{1Z}$=100ms. In the SHB measurements we also observed a
superhyperfine interaction of Nd ions with yttrium. This causes
additional spectral side holes at around 640kHz (for the present
magnetic field), thus the effective homogeneous linewidth is
around 1 MHz. This was our main limitation for the efficiency of
our light-matter interface since it affected our ability to create
a good comb for the longer storage times ($1/\Delta\approx1\mu$s).
\textbf{Storage efficiency analysis}
The efficiency can be calculated theoretically using the formula
\cite{Afzelius2009a,Riedmatten2008} $\eta \approx (d/F)^2 e^{-d/F}
e^{-7/F^2} e^{-d_0}$. The different terms can be given a
qualitative understanding. The first term represents the
collective coupling, the second the re-absorption of the
re-emitted light, the third is an intrinsic dephasing factor due
to the finesse and the last term a loss due to an absorption
background $d_0$. For the comb with $\Delta$=10MHz we measure
$d\approx1.7$,$F\approx2.7$ and $d_0\approx0.5$ (see Fig.
\ref{fig:AFC}b), resulting in a theoretical efficiency of $\eta
\approx5\%$ in close agreement with the experiment (see Fig.
\ref{fig:AFC}c). The major limiting factor here is $d_0$ (caused
by imperfect preparation of the comb) and then the optical depth
of the comb $d$ (the finesse being close to optimum for this $d$
\cite{Afzelius2009a}). The decrease in efficiency for longer
storage times (see inset of Fig. \ref{fig:AFC}c) is principally
due to an increase in the background absorption $d_0$ and an
accompanying decrease in the peak absorption $d$. This in turn is
caused by the effective spectral resolution of 1 MHz in the
optical pumping, which is a limitation of the present material
(see Methods above). However, the storage efficiency is between
one and two orders of magnitude higher than what we achieved in
the material Nd:YVO$_4$ \cite{Riedmatten2008}, which we attribute
to an improvement in optical pumping in this Nd-doped material.
\textbf{Acknowledgements} We acknowledge financial support from
the Swiss NCCR Quantum Photonics, the EC projects Qubit
Applications (QAP) and ERC Advanced Grant (QORE). We also
acknowledge useful discussions with Christoph Simon and Nicolas
Sangouard.
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\appendix
\section{Supplementary Information}
We provide additional information for our manuscript. This
material includes a detailed description of the experimental setup
and of the atomic frequency comb preparation, as well as
additional results.
\section{Experimental details}
\subsection{Setup}
The experimental setup is shown in Fig.\ref{fig:setup}. The light
is emitted by an external cavity diode laser (Toptica) at 883nm
and is then split with a polarizing beam splitter (PBS) into two
paths. In both paths, the amplitude, frequency and phase of the
light can be modulated using acousto-optic modulators (AOM,
AA-Opto-Electronics) in double pass configuration. The path on the
top is used to prepare an AFC into the sample. To achieve this,
the AOM creates a series of pulses with a characteristic spectrum
explained in the next section. The other path is used to create
the weak pulses of light at the single photon level that are going
to be mapped onto the crystal. With an AOM and neutral density(ND)
filters we create pulses containing less than one photon in
average. The AOMs have a good extinction ratio ($>$ 40 dB) and
allows us to shift the frequency or the phase of the light.
However, to create very short pulses(few ns), we use an
additional a 20 GHz electro-optical modulator(EOM, EO Space).
\begin{figure}
\caption{\footnotesize{
Experimental setup : see text for details. The following abbreviations are used : polarizing beam splitter (PBS), acousto-optic modulator (AOM), neutral density filter(ND filter), mechanical chopper (MC), polarization controller (PC), electro-optical modulator (EOM), Faraday rotator (FR) and avalanche photo-diode (APD).}
\label{fig:setup}
\end{figure}
Both path are then combined using a 90/10 fiber coupler and sent
into free space into the sample cooled at 3K using a pulse tube
refrigerator (Oxford Instruments). The PBS combined to a $\lambda
/2$ wave-plate allow us to adjust the polarization of the light
parallel to the D1 crystallographic axis (optical axis). In order
to increase the available optical depth, we implemented a
double-pass setup using a Faraday Rotator(FR). In this way, we
have a configuration where the polarization of the light remains
constant when propagating in double pass through the crystal, but
is rotated by 90 degrees after the second pass in the FR, such
that the the light emitted by the crystal can be separated from
the input light, thanks to a PBS.
The light is then coupled to a multimode optical fiber and
detected by a single photon Si Avalanche Photo-Diode (APD) with
32\% efficiency. The mechanical chopper(MC) in the preparation
path is used to block the leakage of the preparation AOM when the
APD is detecting echoes. The second MC is used to protect the APD
when strong preparation pulses are used. The transmission between
the input of the cryostat and the APD was typically between 25$\%$
and 30 $\%$. Finally, a third path (partially shown), is used to
actively frequency stabilize the laser to less than 100kHz using
the Pound-Drever-Hall technique. The frequency reference is given
by a spectral hole in the crystal. The stabilization light is sent
into the same sample, but with a slightly different angle.
The experiment is repeated every 200ms, allowing sufficient time
for the ground state population to relax between experiments. The
first 100ms were used for the preparation sequence to create an
AFC. Then, we waited 5ms to avoid fluorescence during the storage
and retrieval sequence, due to atoms left in the excited state
after the preparation of the AFC. After that, we perform $\sim$
1000 independent storage trials separated by 5$\mu$s. Each trial
contains the particular pulse sequence to be stored and the
re-emitted AFC echoes. Depending on the mean photon number per
pulses, the total integration time to accumulate sufficient
statistics varied from 30s to 10min.
\subsection{Comb Preparation}
We now explain in more detail the preparation sequence allowing us
to create a desired AFC. Similarly to \cite{deRiedmatten2008}, we
send series of pulses separated by a time $\tau$, which create a
periodicity of $\Delta=1/\tau$ in the absorption profile. By
sending repeatedly such sequence, this modulation will take the
form of a comb with sharp peaks. However, instead of sending only
pairs of pulses as in \cite{deRiedmatten2008}, we extended the
method by allowing the possibility of sending N pulses. The
spectrum of this series of N pulses is a a comb of periodicity
$\Delta=1/\tau$. However, we want to remove from the initial state
only atoms that are not in the desired comb. The spectrum of the
sequence must thus be the inverse of a comb : a series of holes
separated by $\Delta$. To achieve this, the central pulse must be
$\pi$-dephased, and with a pulse area equal to the sum of the
other pulses (see Fig.2).
The property of the comb can be determined by the characteristic
of the pulses sequence. As already mentioned, the periodicity
$\Delta$ is given by the time $\tau$ between pulses. The duration
of one pulse(or its spectrum) will determine the bandwidth of the
whole comb. Finally, the width of each peak will be inversely
proportional to the duration of the whole pulse sequence.
Similarly, we can say that the finesse of the comb will increase
with the number of pulses (F$\sim$ N$_{\mathrm{pulses}}$). Note
that these rules are true for the spectrum of the light, but the
process of spectral hole burning being more complex, the property
of the AFC can be different. For example, the width of each peak
is limited by the crystal properties, such as homogeneous
linewidth and superhyperfine interaction.
The comb preparation sequence for most of our experiments is shown
in Fig.\ref{fig:preparation}. It takes 16$\mu$s and it is repeated around 2000 times. The available optical depth being
relatively small (d=3), the optimal finesse to get the maximum
storage end retrieval efficiency was about 3 \cite{Afzelius2009}.
It was thus sufficient to send only series of three pulses (N=3)to
create the desired comb. The spectrum of the comb created in this
way was around 20MHz. It was possible to extend this bandwidth by
sending other series of pulses at a shifted
frequency($\pm$20MHz,$\pm$40MHz,etc\dots). Since there is no
interference between different frequencies, the pulses can be sent
within the coherence time $T_2$. This means that the whole
preparation sequence take the same time as with a single
frequency, and does not require more Rabi frequency.
\begin{figure}
\caption{\footnotesize{Pulse sequence for atomic frequency comb
preparation. In order to increase the bandwidth, the pulses are
repeated with shifted frequencies. This pulse sequence was used
for most of our experiments. Here it creates a comb of 100MHz
bandwidth and a periodicity of 1MHz. The total sequence takes
16$\mu$s and it is repeated around 2000 times in order to prepare
the AFC.}
\label{fig:preparation}
\end{figure}
\subsection{Double read-out using double AFC}
For the interference experiment, the stored excitations must be
read-out at different times. For that purpose, we need to create
multiple AFCs, with different periodicity $\Delta_i$ and possibly
different phases. This can be realized by using a preparation
pulse sequences with different pulse separations. Suppose that we
want to create two AFCs with periodicity $\Delta_1$ and
$\Delta_2$. The necessary sequence is shown in
Fig.\ref{fig:preparation2AFC}. We send a pulse sequence with N
pulses separated by $\tau_1$ (with the central pulse at t=0),
superposed with N pulses separated by $\tau_2$ (with the central
pulse also at t=0). Thus, the total number of pulses will be 2N-1,
and the pulse area of the central one is equal to the sum of the
others. When we send a storage pulse in the sample, two echo will
be emitted, after a time $\tau_1$ and $\tau_2$. In our case, we
use N=3, such that the total number of pulses for creating two
AFCs with different periodicity is 5 (for one pumping frequency).
Note that when we superpose two combs, some peaks of the two AFCs
can be at the same position and be summed. This means that some
peaks will be higher than others. Thus, using this method the two
AFCs cannot be created independently (i.e. with two independent
pulses series) or the absorption profile will not correspond to
the sum of two AFCs and we will face additional echoes. So we must
create directly two AFCs in one sequence.
Finally, we would like to be able to induce a phase $\phi$ in one
of the AFCs, in other world to shift the frequency of the comb
\cite {Afzelius2009}. To do so, we must add a phase in each pulse
preparing the corresponding comb (See Fig. 3). If we label the
pulses on the right of the central one with k=1,2,3\dots and the
pulses on the left with k=-1,-2,-3\dots, then the phase in the
pulse k must be k$\phi$.
\begin{figure}
\caption{\footnotesize{Pulse sequence for the preparation of two
atomic frequency combs, as required for the double read out for
the interference experiments. In this example, the two AFCs have a
periodicity $1/\tau_1$ and $1/\tau_2$. The first AFC will also get
a phase $\phi$.}
\label{fig:preparation2AFC}
\end{figure}
\section{Additional results}
We present here some additional results that illustrate the
multimode capacity of AFC. In Fig. \ref{fig:combs} we show
experimental combs with 1MHz peak separation (corresponding to 1
$\mu s$ storage time) and 20 MHz and 100 MHz bandwidth,
respectively. As noted, before the enlargement of the bandwidth
does not affect the center of the comb. Thus it allows us to
increase the number of mode for a fixed storage time without
affecting the efficiency (see Fig.2 of the manuscript). However,
compared to combs with bigger peak separation(shorter storage
time), the height of the peaks has decreased, and the absorption
background has increased, which explained the decay of efficiency
with storage time. As explained in the paper, the number of mode
is proportional to the number of peaks in the comb. Here we create
more than 100 peaks which illustrate the multimode performance of
our experiments.
\begin{figure}
\caption{\footnotesize{Experimental atomic frequency combs with 1
MHz peak separation for single(green line) or five(black line)
different simultaneous pump frequencies.}
\label{fig:combs}
\end{figure}
Finally, we show two more example of storage of 64 temporal modes
with different inputs. In Fig.\ref{fig:64modesbis}, all the modes
are full, which allows us in principle to store 32 time-bin
qubits. In Fig.\ref{fig:64modesbis2}, we modulated the amplitude
of the input pulses. We note that even if the input mode changes,
we do not need to adapt the AFC preparation. Indeed, the
efficiency is constant for all modes, and the echo always closely
follow the input pulses.
\begin{figure*}
\caption{\footnotesize{Storage of 64 consecutive pulses.
An efficiency of 1.4\% was measured. The mean photon number per pulse is $\bar{n}
\label{fig:64modesbis}
\end{figure*}
\begin{figure*}
\caption{\footnotesize{Storage of 64 pulses with a modulating amplitude.
An efficiency of 1.6\% was measured. The mean photon number in the biggest pulses is $\bar{n}
\label{fig:64modesbis2}
\end{figure*}
\end{document}
|
\begin{document}
\title{A Non-Oracle Quantum Search Algorithm and Its Experimental Implementation}
\author{Nanyang Xu,$^{1}$ Jin Zhu,$^{1}$ Xinhua Peng,$^{1}$ Xianyi Zhou,$^{1}$ Jiangfeng Du$^{1\ast}$\\
\normalsize{$^{1}$Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern
Physics,University of Science and Technology of China, Hefei, Anhui 230026, People's Republic of China}\\
\normalsize{$^\ast$To whom correspondence should be addressed; E-mail: [email protected]}}
\begin{abstract}
Grover's algorithm has achieved great success. But quantum search algorithms still are not complete algorithms
because of Grover's Oracle. We concerned on this problem and present a new quantum search algorithm
in adiabatic model without Oracle. We analyze the general difficulties in quantum search algorithms and show how to
solve them in the present algorithm. As well this algorithm could deal with both single-solution and
multi-solution searches without modification. We also implement this algorithm on NMR quantum computer. It is the
first experiment which perform a real quantum database search rather than a marked-state search.
\end{abstract}
\pacs{03.67.Lx, 89.70.-a, 03.65.-w}
\maketitle
Quantum computation is a promising way to solve classical hard problems. Although large-scale quantum hardware
has yet been built, quantum computation model in analog to classical circuit is well developed during the last
few years. Based on this model, several quantum algorithms have been designed to perform classical algorithms
with remarkable speedups. The most splendid one among these is Shor's Algorithm\cite{Shor_algorithm}, which can
factorize a big number using a running time only polynomial in the size of the number, while all known classical
algorithms need a exponential time\cite{QCQI}. Another important algorithm\cite{Grover_search}, named after its
inventor Grover, concerns the problem of searching for a required item in a unsorted database. One common example
for this unsorted database search is to find a person's name in a phone book (the items are sorted by names) with
only knowing his phone number. Classically, the only way to achieve this is brute-force search\cite{Ju_2007}
which for $N$ entries in the phone book requires an average of $ \frac{N}{2}$ quires. However, if the information
is stored in a quantum database, to find the right name with Grover's algorithm costs only a time of order
$\sqrt{N}$, providing a quadratic speedup.
While quantum algorithms are presented in standard circuit
model(\textit{i.e.}, using a sequence of discrete quantum gates), a
new model of quantum computation show up where the states of quantum
computer evolves continuously and adiabatically under a certain
time-dependent Hamiltonian. This new idea was firstly brought out by
Farhi and co-workers\cite{Farhi}. In this new computation model, a
problem Hamiltonian is well designed whose ground state encodes the
unknown solution to the problem. Then this adiabatic evolution can
be used to switch gradually from an initial Hamiltonian whose ground
state is known, to the problem Hamiltonian. If this evolution
evolves slowly enough, the system will stay near its instantaneous
ground state\cite{Messiah}. So in the end of evolution, the system
will on the solution state of the problem. This method has been
applied to the database search problem. However, this adiabatic
search algorithm results in a complexity of order $N$, which is the
same order with classical algorithms. More recently, Roland and
Cerf\cite{Roland_2002} improved the performance of adiabatic search
to order $\sqrt{N}$, the same with Grover's algorithm, by applying
adiabatic evolution locally.
Although these quantum search algorithms seems brilliant as they
have already done, they are still incomplete algorithms. Grover's
algorithm utilized a Oracle (\emph{i.e.}, a blackbox) , which gets
an input state $\arrowvert i \rangle$, checks the quantum database,
and changes the state to $-\arrowvert i\rangle$ if the $i$-th value
in the database satisfies the search condition and does nothing
otherwise. It is easy to implement such operations in classical
database cases, but up to now there's no efficient universal method
to design this Oracle in quantum circuit. And in adiabatic
algorithms, the solution of the problem is encoded to the problem
Hamiltonian. Since the mechanics of the Oracle remains unknown, the
encoding process of the Hamiltonian in the adiabatic algorithm is
unclear. Instead, just like what previous
experiments\cite{Chuang_1998search, Dodd_2003,
Vandersypen_3bitsearch, Brickman_2005} of Grover's algorithm did,
the adiabatic search algorithm forms the Hamiltonian directly from
the solution state, which means we have to know the state \emph{in
prior} and then perform a algorithm to show it. Obviously this
marked-state search algorithm is not a real database search. Thus
the main problem with current quantum search algorithms is the
existence of Grover's Oracle.
In this article, we present a new adiabatic algorithm for quantum
search. By encoding the database to quantum format and forming the
problem Hamiltonian from target value, this adiabatic search
algorithm solves Grover's problem without Oracles. Furthermore, we
experimentally implement this non-Oracle quantum search algorithm in
NMR quantum computer. Because of the reasons mentioned before, this
is the first time implementing a real quantum unsorted database
search in experiment. We also analyze the general difficulties in
quantum search and show how to solve them in our algorithm.
As a new quantum computation model, adiabatic algorithm was brought out by Fahi \textit{et al.}\cite{Farhi} and
soon became a rapidly expanding field. This new computing model relies on the \textit{Adiabatic Theorem} which
states as follows: \newtheorem*{adthm}{Adiabatic Theorem}
\begin{adthm}
A physical system remains in its instantaneous eigenstate if a given perturbation is acting on it slowly enough
and if there is a gap between the eigenvalue and the rest of the Hamiltonian's spectrum.
\end{adthm}
The idea of adiabatic quantum computation is straightforward. First
find a complex Hamiltonian whose ground state describes the solution
to the problem of interest, Next, prepare a system with a simple
Hamiltonian and initialized to the ground state. Finally, the simple
Hamiltonian adiabatically switches to the complex Hamiltonian.
According to adiabatic theorem, the system stays in the ground
state, so in the end the state of the system describes the solution
to the problem. The time dependent Hamiltonian is usually
constructed as follows,
\begin{equation} \label{ht}
H(t)=[1-s(t)]H_{i}+s(t)H_{p},
\end{equation}
where $H_{i}$ is the initial Hamiltonian whose ground state is easy
to know and $H_{p}$ is the complex problem Hamiltonian whose ground
state describe the solution to our problem, and the monotonic
function $s(t)$ fulfills $s(0) = 0$ and $s(T)=1$.
Several adiabatic algorithms have been designed for solving
computational hard problems\cite{Farhi_2000,Peng_2008}. And a simple
proof has been given to show that adiabatic model is equivalent to
circuit mode in quantum computation\cite{Mizel_2007}. Moreover,
since adiabatic computation only involves the ground state, it keeps
the system at a low temperature. Thus the system appears lower
sensitive to some perturbations and have a improved robustness
against dephasing, environmental noise and some unitary control
errors\cite{Childs_2002,Roland_robust}.
As mentioned before, the key part of an adiabatic algorithm is how
to describe the solution to a specific problem in the problem
Hamiltonian. Here let's focus on the database search problem. To be
simplified, we now consider a phone book which contains $N$ (assume
$N = 2^{n}$) entries with each entry a pair of telephone number and
person's name. Usually, the entries are sorted by name. The database
search problem here is to find a specific name in the book whose
telephone number is given. To solve this problem in our model, the
names are encoded to $n$-qubit states and the phone numbers
represented as integers (in fact, any real numbers are permit). An
example for $N=4$ is shown in Table \ref{phone-book}. We encode the
names and the phone numbers as in Table \ref{encoder}. Thus the
database could be stored by the state-integer pairs like
$\{(|0\rangle,4),(|1\rangle,3),(|2\rangle,1),(|3\rangle,2)\}$. If we
want to find the name which connect to the number 3601003 which is
encoded as 3, state $|1\rangle$ should be returned from our quantum
search machine.
\begin{table}[h]
\caption{phone book example} \label{phone-book}
\begin{center}
\begin{tabular}{ccc}
\multicolumn{1}{c}{} &Name& Number \\
\hline
&Alex & 3601004 \\
&Bob & 3601003 \\
&Cherry & 3601001\\
&David & 3601002 \\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[h]
\caption{encodor table} \label{encoder}
\begin{center}
\begin{tabular}{ccc||cc}
\multicolumn{1}{c}{} & Name & State & Number & Integer \\
\hline
&Alex & $|0\rangle$ & 3601001 & 1 \\
&Bob & $|1\rangle$ &3601002 & 2 \\
&Cherry & $|2\rangle$ &3601003 & 3 \\
&David & $|3\rangle$ &3601004 & 4 \\
\hline
\end{tabular}
\end{center}
\end{table}
After encoding classical database to quantum database, next step is to design the problem Hamiltonian $H_{p}$.
For this problem, $H_{p}$ is.
\begin{equation}
H_{p} = (\sum_{i=0}^{N-1} number_{i}|name_{i}\rangle \langle name_{i}|-target \cdot I ^{\otimes n})^{2}
\end{equation}
where \textit{target} is the code of the phone number which we want to search for. Obviously, the ground state of
$H_{p}$ is the state which connects to the target number. In general, for an encoded database where entries are
pairs as $(i,value_{i})$ and sorted by $i$, we write $H_{p}$ as
\begin{eqnarray}
\label{hpd}
H_{p} &=& (\sum_{i=0}^{N-1} value_{i}|i\rangle \langle i| - target\cdot I ^{\otimes n})^{2}\nonumber\\
&=& \mathcal{D}^{2}-2\cdot target\cdot\mathcal{D} + target^2,
\end{eqnarray}
where $\mathcal{D}=\sum_{i=0}^{N-1} value_{i}|i\rangle \langle i|$ is the database operator and could be
formulated separately.
Next, we will choose an initial Hamiltonian $H_{i}$. Conventionlly\cite{Farhi_2000}, $H_{i}$ is chosen to be
noncommutative
with $H_{p}$ to avoid crossing of energy levels. Thus we write $H_{i}$ as:
\begin{eqnarray}
H_{i} & = & g(\sigma_{x}^{0}+\sigma_{x}^{1}+\cdots + \sigma_{x}^{n-1}).
\end{eqnarray}
which means the qubits coulpling with a magnetic field at the $x$-direction and the coupling strength is $g$. the
gound state of this Hamiltonian is simple and they are,
\begin{eqnarray}
|\psi_{0}\rangle & = &\frac{|0^{(n-1)}\rangle-|1^{(n-1)}\rangle}{\sqrt{2}}
\otimes \frac{|0^{(n-2)}\rangle-|1^{(n-2)}\rangle}{\sqrt{2}} \otimes \cdots \nonumber\\
&\otimes& \frac{|0^{(0)}\rangle-|1^{(0)}\rangle}{\sqrt{2}} \nonumber \\ & =&
\frac{1}{\sqrt{N}}\sum_{j=0}^{N}(-1)^{b(j)}|j\rangle,
\label{Int_state}
\end{eqnarray}
where $b(j)$ is the Hamming distance between $j$ and $0$.
In the adiabatic evolution, the system Hamiltonian interpolates from $H_{i}$ to $H_{p}$ (\textit{i.e.,} see Eq
\ref{ht}) and the state of the system evolves according to the Schr\"{o}dinger equation:
\begin{eqnarray}
i\frac{d}{dt}|\psi(t)\rangle=H(t)|\psi(t)\rangle \\
|\psi(0)\rangle=|\psi_{0}\rangle \label{e.Schord}.
\end{eqnarray}
If this evolution acts slow enough (\textit{i.e.,} the total evolution time $T$ is large enough), the
\textit{\textbf{Adiabatic Theorem}} ensures the system will always stay on the ground state of $H(t)$ and in the
end the solution of our problem will show up.
Again, we take the phone book in Table \ref{phone-book} as example. If we want to find number 3601002 in the
database and using the encoder in Table \ref{encoder}, we will get the Hamiltonians as follows,
\begin{eqnarray}
H_{i} & = & g(\sigma_{x}^{0}+\sigma_{x}^{1})\nonumber\\
H_{p}&=&(4|0\rangle \langle0| + 3|1\rangle \langle1| + 1|2\rangle \langle2| + 2|3\rangle \langle3| - 2I)^{2} \nonumber\\
&=&\dfrac{3}{2}I +\sigma_{z}^{(0)}+\sigma_{z}^{(1)}+\dfrac{\sigma_{z}^{(1)}\otimes \sigma_{z}^{(0)}}{2},
\label{david}
\end{eqnarray}
And the eigenvalues of time dependent Hamiltonian $H(t)$ (see EQ.(\ref{ht}) ) are plotted in FIG.\ref{energy}. By
the adiabatic theorem, the state will stay on the lowest energy level during the adiabatic evolution. And finally
we will get the state on the basis $|3\rangle$. After measurement and decoding we will get the name connecting to
the number 3601002 which is \textit{David}.
\begin{figure}
\caption{Energy diagram for searching an item (\textit{i.e.}
\label{energy}
\end{figure}
To demonstrate this Non-Oracle search algorithm, we selected $^{13}$C-labeled CHCl$_3$ as a physical system for
our experiments. The two qubits are represented by $^{13}$C and $^1$H. Its natural Hamiltonian in the multiply
rotating frame is
\begin{equation}
\mathcal{H}_{sys}=\omega_1 I_z^1+\omega_2 I_z^2+2\pi J I_z^1 I_z^2,
\end{equation}
where $\omega_1$ and $\omega_2$ are Larmour frequencies, J is the spin-spin coupling constant $J=214.5Hz$.
Experiments were performed at room temperature using a standard 400MHz NMR spectrometer (AV-400 Bruker
instrument).
\begin{figure}
\caption{The quantum network for adiabatic search using NMR interferometry. The input state is $|00\rangle$.
$\theta_s=0.95*(1-s/10)$, $\tau_s=0.95*(s/10)/\pi J$. }
\label{net}
\end{figure}
The experiments was devided into three parts, shown as Fig.\ref{net}: the first part consists of preparation of
the state of the initial Hamiltonian. The second part is the adiabatic evolution, and the third part is the
tomography of the resultant state. To prepare the initial ground state [Eq. \ref{Int_state}], we first created a
pseudopure state (PPS) \cite{Gershenfeld:1997aa,Cory:1997aa} $\rho _{00}=\frac{1-\epsilon }{4}\mathbf{I}+\epsilon
|00\rangle \langle 00|$, where $\epsilon \approx 10^{-5}$ describes the thermal polarization of the system and I
is a unit matrix, using the method of spatial averaging. Then $\pi/2$ pulses along the $-y$ axis was applied to
prepare the ground state.
Discretizing a continuous Hamiltonian is a straightforward process and changes the run time T of the adiabatic
algorithm only polynomially. Simply, let the discrete time Hamiltonian $H(s)$ be a linear interpolation from some
beginning Hamiltonian $H(0)=H_0$ to some final problem Hamiltonian $H(S)=H_1$ such that
$H(s)=(s/S)H_1+(1-s/S)H_0$. The unitary evolution of the discrete algorithm can be written as
\begin{equation}
U=\prod_{s} U_s=\prod_{s} e^{-iH(s)\tau},
\end{equation}
where $\tau=T/(S+1)$, $T$ is the total duration of the adiabatic passage and $S+1$ is the total number of
discretization steps. When both $T, S \to \infty$ and $\tau \to 0$, the adiabatic limit is achieved.
For our example shown in Eq.\ref{david}, an optimized set of parameters was set as $g=1$, $T=10.45$ and $S=10$,
so $\tau=T/(S+1)=0.95$. This set of parameters yields an adiabatic evolution that finds the solution in a
relatively efficient way. Using the Trotter formula, we can approximate $U_s$ to second order
\begin{equation}
U_{s} \approx U'_s= e^{-i (1-\frac{s}{S})H_0\frac{\tau}{2}} e^{-i\frac{s}{S}H_{p}\tau} e^{-i
(1-\frac{s}{S})H_0\frac{\tau}{2}} + \mathcal{O}(\tau ^2), \label{trotter}
\end{equation}
the fidelity of $U_s \to U'_s$ is all above $0.996$ and overall fidelity is $0.991$. For the implementation of
$U'_s$, $e^{-i (1-\frac{s}{S})H_0\frac{\tau}{2}}$ can be simply realized using a $\theta_s$ pulses around $x$
axis for both H and C nuclei, $\theta_s=\tau*(1-s/S)$, shown in Fig. \ref{net}. And the evolution under $H_1$ can
be simulated by a free evolution $\tau_s$ under the Hamiltonian $\mathcal{H}_{sys}$, the identity term of $H_1$
does not cause any evolution of the state and so it can be omitted, $\tau_s=\tau*(s/S)/\pi J$.
\begin{figure}
\caption{The tomography of theoretically expected and experimentally obtained density matrices for the search
states in adiabatic search algorithm. The density matrices consist of just a real term on the diagonal
corresponding to the population of the state that has been searched.}
\label{result}
\end{figure}
The third stage of the experiment is the tomography of the final density matrix after the adiabatic evolution.
The result was shown as Fig. \ref{result}. Theoretically, the four state $|00\rangle$, $|01\rangle$, $|10\rangle$
and $|11\rangle$ should be find at the probability of 0, 0.014, 0.014 and 0.972. Our experiments show that the
probability is 0.037, 0.032, 0.006 and 0.925. The fidelity of the experiment is 0.985.
The errors in adiabatic algorithms may be caused by three parts. Firstly, the total time of evolution in
adiabatic algorithms should be infinite. Actually the evolution is terminated when the state is supposed to reach
our expected high probability. Secondly, the error is due to neglect of $\mathcal{O}(\tau^2)$ terms in the
Trotter Formula (Eq. \ref{trotter}). The third part of error is due to decoherence effects of the NMR system and
imperfect pulses.
Unlike previous standard quantum algorithms only using qubits as
registers to store information, our algorithm represents the $value$
field by the strength of interactions in the operator and the
$index$ field by qubits. This is because that if both the fields are
represented in qubits, $2n$ qubits are needed for a database with
$N$ items, which result in the failure that the optimal running time
would be scaled from order $\sqrt{N}$ to $N$\cite{Roland_2002}, the
same performance as classical algorithms. Since the construction
function is simply quadratic, the interaction strength in problem
Hamiltonian grows with the database's size. For further
consideration, the algorithm may be improve to suppress the
interaction strength in the problem Hamiltonian by choosing a better
construction function in Eq. \ref{hpd}.
All practical quantum search algorithms must face the problem that
the database is unsorted, thus quantum operators would traverse all
the items in the database to learn the complete information and
after measurement all information in the states are destroyed. It is
a hard problem to efficiently implement the quantum operators
concerning the database. The first effort was reported by Ju and
coworkers\cite{Ju_2007} when they tried to implement Grover's Oracle
in quantum circuit. They have to spend $N$ steps to build the
relation in the database to the states for each query, such that the
circuit design is not efficient. In this algorithm, we describe the
database information in a single operator($\mathcal{D}$ in
Eq.\ref{hpd}), thus this operator may be analyzed and formulate
separately for each database. On the other side, if it could not be
formulated efficiently in some cases, approximate implementation is
another possible solution according to recent works on geometric
quantum computation\cite{Nielsen_geometry}, of which detailed
consideration is beyond the scope of this article.
As an end of this section, we will give a simple analysis of the multi-solution search case in our algorithm. If
there're $m > 1$ entries in the database satisfying the search condition, the problem Hamiltonian will have
ground states with $m$ times degenerated. And because of symmetry, the state would finally evolute to an average
superposition of all the ground states. Thus without any modification, our algorithm could also deal with
\emph{multi-solution search}.
To be concluded, we present a new kind of adiabatic search algorithm
to solve Grover's problem without Oracles and give a demonstrative
experiment on NMR quantum computer. The result of experiment agrees
well with theoretical expectation. This is a new style of quantum
search algorithm which utilize both quantum registers and
interaction strength to store information. This algorithm aims at
general difficulties of quantum search algorithms and give a
promising way to solve them utimately.
We thank Zeyang Liao for initial discussion and help. This work was
supported by National Nature Science Foundation of China, the CAS,
Ministry of Education of PRC, the National Fundamental Research
Program, and the DFG through Su 192/19-1. For this article, any
comment is welcome.
\end{document}
|
\begin{document}
\title[Pontryagin principles]{Pontryagin principles in infinite horizon in presence of asymptotical constraints}
\author[BLOT and NGO]
{Jo\"{e}l BLOT and Thoi Nhan NGO}
\address{Jo\"{e}l Blot: Laboratoire SAMM EA 4543,\newline
Universit\'{e} Paris 1 Panth\'{e}on-Sorbonne, centre P.M.F.,\newline
90 rue de Tolbiac, 75634 Paris cedex 13,
France.}
{\epsilon}mail{[email protected]}
\address{Thoi Nhan Ngo: Laboratoire SAMM EA 4543,\newline
Universit\'{e} Paris 1 Panth\'{e}on-Sorbonne, centre P.M.F.,\newline
90 rue de Tolbiac, 75634 Paris cedex 13,
France.}
{\epsilon}mail{[email protected]}
\date{November 5, 2015}
\begin{abstract} We establish necessary conditions of optimality for discrete-time infinite-horizon optimal control in presence of constraints at infinity. These necessary conditions are in form of weak and strong Pontryagin principles. We use a functional analytic framework and multipliers rules in Banach (sequence) spaces. We establish new properties on Nemytskii operators in sequence spaces. We also provide sufficient conditions of optimality.
{\epsilon}nd{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\noindent
{MSC 2010:} 49J21, 65K05, 39A99.\\
{Key words:} infinite-horizon optimal control, discrete time
\section{Introduction}
The aim of this paper is to establish necessary conditions of optimality in the form of Pontryagin principles for the following Optimal Control problem
\[
(P)
\left\{
\begin{array}{rl}
{\rm Maximize} & K(\underline{y}, \underline{u}) := \sum_{t = 0}^{+ \infty} \beta^t \psi(y_t,u_t)\\
{\rm when} & \underline{y} :=(y_t)_{t \in {\mathbb{N}}} \in ({\mathbb{R}}^n)^{{\mathbb{N}}}, \underline{u} :=(u_t)_{t \in {\mathbb{N}}} \in U^{{\mathbb{N}}}\\
\null & y_0 = {\epsilon}ta, \; \lim_{t \rightarrow + \infty} y_t = y_{\infty}, \; \underline{u} \; {\rm is} \; {\rm bounded}\\
\null & \forall t \in {\mathbb{N}}, y_{t+1} = g(y_t, u_t)
{\epsilon}nd{array}
\right.
\]
where $\beta \in (0,1)$, $U \subset {\mathbb{R}}^d$ is nonempty, $\psi : {\mathbb{R}}^n \times U \rightarrow {\mathbb{R}}$ is a function, ${\epsilon}ta$ and $y_{\infty}$ are fixed vectors of ${\mathbb{R}}^n$, $g : {\mathbb{R}}^n \times U \rightarrow {\mathbb{R}}^n$ is a function, and $({\mathbb{R}}^n)^{{\mathbb{N}}}$ (respectively $U^{{\mathbb{N}}}$) denotes the set of the sequences in ${\mathbb{R}}^n$ (respectively $U$). In comparison with existing results on bounded processes, the specificity of the present work is the presence of the asymptotical constraint on the state variable: $\lim_{t \rightarrow + \infty} y_t = y_{\infty}$; its meaning is that the optimal state of the problem stays near a "good" state value on the long run.
\vskip2mm
Such problem in discrete time and infinite horizon arises in several fields of applications, for instance in optimal growth macroeconomic theory and in optimal management of forests and fisheries; see the references in \cite{BH2}.
\vskip1mm
Our approach is functional analytic; we translate our problems as static of optimization in suitable Banach sequence spaces.
\vskip2mm
Now we describe the contents of the paper. In Section 2 we introduce a problem of optimal control which is equivalent to the initial problem in order to use classical sequence spaces: $c_0({\mathbb{N}},{\mathbb{R}}^n)$ the space of the sequences into ${\mathbb{R}}^n$ which converge to zero at infinity, and ${{\epsilon}ll}^{\infty}({\mathbb{N}},U)$ the space of the sequences into $U$ which are bounded.\\
In Section 3 we study properties of operators and functionals on sequence spaces. A first novelty is a characterization of the operators which send $c_0({\mathbb{N}},{\mathbb{R}}^n) \times {{\epsilon}ll}^{\infty}({\mathbb{N}},U)$ into $c_0({\mathbb{N}},{\mathbb{R}}^n)$ (Theorem \ref{th31}). The other results use this characterization and existing results on Nemytskii operators from $ {{\epsilon}ll}^{\infty}({\mathbb{N}},{\mathbb{R}}^n) \times {{\epsilon}ll}^{\infty}({\mathbb{N}},U)$ into $ {{\epsilon}ll}^{\infty}({\mathbb{N}},{\mathbb{R}}^m)$. \\
Section 4 is devoted to the solutions which converge toward zero of a linear difference equation. These results are useful to establish regularity properties of the differential of operators which formalize the nonlinear difference equation which governs the system.\\
In Section 5 we establish a variation of a Karush-Kuhn-Tucker theorem which is useful for weak Pontryagin principles and we recall a result which is useful for strong Pontryagin principles.
\vskip1mm
\noindent
In Section 6 and Section 7 (respectively Section 8 and Section 9) we establish weak (respectively strong) Pontryagin principles. \\
In Section 10 and Section 11, we establish results of sufficient condition of optimality.
\section{An equivalent problem}
In this section we formulate a problem which is equivalent to Problem (P) for which we can work in classical Banach sequence spaces.\\
We consider the following Optimal Control problem
\[
( P1)
\left\{
\begin{array}{rl}
{\rm Maximize} & J(\underline{x}, \underline{u}) := \sum_{t = 0}^{+ \infty} \beta^t \phi(x_t,u_t)\\
{\rm when} & \underline{x} :=(x_t)_{t \in {\mathbb{N}}} \in c_0({\mathbb{N}},{\mathbb{R}}^n), \underline{u} :=(u_t)_{t \in {\mathbb{N}}} \in {{\epsilon}ll}^{\infty}({\mathbb{N}},U)\\
\null & x_0 = \sigma\\
\null & \forall t \in {\mathbb{N}}, x_{t+1} = f(x_t, u_t).
{\epsilon}nd{array}
\right.
\]
When we choose $\phi : {\mathbb{R}}^n \times U \rightarrow {\mathbb{R}}$ as $\phi(x,u) = \psi(x + y_{\infty},u)$, $f(x,u) = g(x + y_{\infty}, u) - y_{\infty}$, $x_t = y_t - y_{\infty}$ for all $t \in {\mathbb{N}}$, $\sigma = {\epsilon}ta - y_{\infty}$, Problem $({\mathcal P1})
$ is equivalent to Problem $({\mathcal P})$. And so our strategy for the sequel of the paper is to work on (P1) and to translate the results on $({\mathcal P1})$ into results on (P).\\
For the properties of $c_0({\mathbb{N}}, {\mathbb{R}}^n)$ we refer to Section 15.3 in \cite{AB}, and for those of the space ${{\epsilon}ll}^{\infty}({\mathbb{N}}, U)$ we refer to Section 15.7 in \cite{AB}.
\section{Nonlinear operators and functionals}
This section is devoted to the study of several operators between sequence spaces; notably the Nemytskii operators (also called superposition operators), and to the study of the functionals which define the criterium of our maximization problems. We establish results of continuity and of Fr\'echet differentiability.
\vskip1mm
\begin{theorem}\label{th31}
Let $X$, $V$, $W$ be three real normed spaces, $U$ be a nonempty subset of $V$, and $F : X \times U \rightarrow W$ be a mapping such that, for all $x \in X$, the partial mapping $F(x, \cdot)$ transforms the bounded subsets of $U$ into bounded subsets of $W$. Then the following assertions are equivalent.
\begin{enumerate}
\item [(i)] $\forall \underline{x} \in c_0({\mathbb{N}}, X)$, $\forall \underline{u} \in {{\epsilon}ll}^{\infty}({\mathbb{N}}, U)$, $(F(x_t,u_t))_{t \in {\mathbb{N}}} \in c_0({\mathbb{N}},W)$.
\item[(ii)] For all nonempty bounded subset $B$ in $U$, $\lim_{x \rightarrow 0}( \sup_{u \in B} \Vert F(x, u) \Vert) = 0$.
{\epsilon}nd{enumerate}
{\epsilon}nd{theorem}
\begin{proof} ${\bf (i \Longrightarrow ii)}$ Let $B$ be a nonempty bounded subset of $U$. Let $\underline{x} \in c_0({\mathbb{N}}, X)$. From the assumption on $F$, we know that, for all $t \in {\mathbb{N}}$, we have $\sup_{u \in B} \Vert F(x_t, u) \Vert < + \infty$. Therefore, for all $t \in {\mathbb{N}}$, there exists $u_t \in B$ such that
$$0 \leq \sup_{u \in B} \Vert F(x_t, u) \Vert \leq \Vert F(x_t, u_t) \Vert + \frac{1}{t+1}.$$
Since, for all $t \in {\mathbb{N}}$, $u_t \in B$, we have $\underline{u} \in {{\epsilon}ll}^{\infty}({\mathbb{N}}, U)$. Then using (i), we obtain \\
$\lim_{t \rightarrow + \infty} \Vert F(x_t,u_t) \Vert = 0$, and from the previous inequality we obtain \\
$\lim_{t \rightarrow + \infty} (\sup_{u \in B} \Vert F(x_t, u) \Vert) = 0$, and since we work in normed spaces we can use the sequential characterization of the limit and assert that we obtain (ii).\\
${\bf (ii \Longrightarrow i)}$ Let $\underline{x} \in c_0({\mathbb{N}}, X)$ and $\underline{u} \in {{\epsilon}ll}^{\infty}({\mathbb{N}}, U)$. Then the subset $B := \{ u_t : t \in {\mathbb{N}} \}$ is bounded, and, for all $t \in {\mathbb{N}}$, the following inequality holds:
$$0 \leq \Vert F(x_t, u_t) \Vert \leq \sup_{u \in B} \Vert F(x_t, u) \Vert,$$
and from (ii), since $\lim_{t \rightarrow + \infty} x_t = 0$, we obtain $\lim_{t \rightarrow + \infty} (\sup_{u \in B} \Vert F(x_t, u) \Vert) = 0$, and from the previous inequality we deduce $\lim_{t \rightarrow + \infty} \Vert F(x_t,u_t) \Vert = 0$, i.e. the sequence $(F(x_t,u_t))_{t \in {\mathbb{N}}}$ belongs to $ c_0({\mathbb{N}},Y)$.
{\epsilon}nd{proof}
\begin{remark}\label{rem32} Assertion (i) of Theorem \ref{th31} permits to define the Nemytskii operator
$$N_F : c_0({\mathbb{N}}, X) \times {{\epsilon}ll}^{\infty}({\mathbb{N}}, U) \rightarrow c_0({\mathbb{N}},W), N_F((x_t)_{t \in {\mathbb{N}}}, (u_t)_{t \in {\mathbb{N}}}) := (F(x_t,u_t))_{t \in {\mathbb{N}}}.$$
{\epsilon}nd{remark}
\begin{remark}\label{rem33}
We set $B_R := \{ v \in V : \Vert v \Vert \leq R \}$ when $R \in (0, + \infty)$.
In the setting of Theorem \ref{th31}, the assumption on $F$ is equivalent to the following condition: $\forall x \in X$, $\forall R \in (0, + \infty)$, $\sup_{ u \in B_R \cap U} \Vert F(x, u) \Vert < + \infty$, and the assertion (ii) is equivalent to: $\forall R \in (0, + \infty)$, $\lim_{x \rightarrow 0} (\sup_{u \in B_R \cap U } \Vert F(x, u) \Vert) = 0$. \\
Also note that assumption (ii) and the continuity of $F( \cdot, u)$ for all $u \in U$ imply $F(0,u) = 0$ for all $u \in U$, since, for all $u \in {\mathbb{R}}^d$, $\{ u\}$ is a nonempty bounded subset and $0 = \lim_{x \rightarrow 0} \Vert F(x,u) \Vert = \Vert F(0,u) \Vert$.
{\epsilon}nd{remark}
\begin{remark}\label{rem34} In the setting of Theorem \ref{th31}, if in addition we assume that dim$V < + \infty$ and $U$ is closed, using the relative compactness of bounded subsets of $U$, if $F(x, \cdot) \in C^0( U,W)$ (the space of continuous mappings from $U$ into $W$), $F(x, \cdot)$ transforms the bounded sets into bounded sets.
{\epsilon}nd{remark}
\begin{theorem}\label{th35} Let $U$ be a nonempty closed subset of ${\mathbb{R}}^d$. Let $F \in C^0({\mathbb{R}}^n \times U, {\mathbb{R}}^m)$ such that $\lim_{x \rightarrow 0}( \sup_{u \in B_R \cap U } \Vert F(x, u) \Vert ) = 0$ for all $R \in (0, + \infty)$.\\
Then we have the continuity of the Nemytskii operator on $F$, i.e.\\
$N_F \in C^0(c_0({\mathbb{N}}, {\mathbb{R}}^n) \times {{\epsilon}ll}^{\infty}({\mathbb{N}}, U), c_0({\mathbb{N}},{\mathbb{R}}^m))$.
{\epsilon}nd{theorem}
\begin{proof}
First using Remark \ref{rem34}, the assumption of Theorem \ref{th31} is fulfilled. Using Remark \ref{rem33}, assertion (ii) of Theorem \ref{th31} is fulfilled, and using Theorem \ref{th31}, and Remark \ref{rem32}, the operator $N_F$ is well defined from $c_0({\mathbb{N}}, {\mathbb{R}}^n) \times {{\epsilon}ll}^{\infty}({\mathbb{N}}, U)$ into $c_0({\mathbb{N}},{\mathbb{R}}^m)$.
Since the bounded subsets of ${\mathbb{R}}^n \times U$ are relatively compacts, we can defined the other Nemytskii operator
$$N^1_F : {{\epsilon}ll}^{\infty}({\mathbb{N}}, {\mathbb{R}}^n) \times {{\epsilon}ll}^{\infty}({\mathbb{N}}, U) \rightarrow {{\epsilon}ll}^{\infty}({\mathbb{N}},{\mathbb{R}}^m), N^1_F((x_t)_{t \in {\mathbb{N}}}, (u_t)_{t \in {\mathbb{N}}}) := (F(x_t,u_t))_{t \in {\mathbb{N}}}.$$
Since ${{\epsilon}ll}^{\infty}({\mathbb{N}}, {\mathbb{R}}^n) \times {{\epsilon}ll}^{\infty}({\mathbb{N}}, U)$ is isometrically isomorphic to
${{\epsilon}ll}^{\infty}({\mathbb{N}}, {\mathbb{R}}^n \times U)$, as a consequence of Theorem A1.1 in \cite{BCr} (p. 22), we can assert that $N^1_F$ is continuous, and then $N_F$ is continuous as a restriction of a continuous operator.
{\epsilon}nd{proof}
Let $X$, $V$, $W$ be real Banach spaces, and $U$ be a nonempty subset of $V$. Let $F : X \times U \rightarrow W$ be a mapping. we say that $F$ is of class $C^1$ on $ X \times U$ when there exist an open subset $U_1$ in $ V$ such that $U \subset U_1$ and a mapping $F_1 \in C^1(X \times U_1 , W)$ such that ${F_1}_{\mid_{ X \times U}} = F$. Such a definition is common in the differential theories; see e.g. \cite{Mi} (p. 1).
\begin{remark}\label{rem36}
When $F_1, F_2 \in C^1(X \times U_1 , W)$ such that ${F_1}_{\mid_{ X \times U}} = {F_2}_{\mid_{ X \times U}} = F$, when $U$ is star-shaped with respect to $u^0$, when $u, u^0 \in U$ and $x^0 \in X$, note that, for all $\theta \in (0,1)$, we have $F_1(x^0, (1 - \theta) u^0 + \theta u) = F_2(x^0, (1 - \theta) u^0 + \theta u) = F (x^0, (1 - \theta) u^0 + \theta u)$. Therefore we have $D_2 F_1(x^0,u^0)(u - u^0) ={ \frac{d}{d \theta}}_{\mid_{\theta = 0}}F_1(x^0, (1 - \theta) u^0 + \theta u) = { \frac{d}{d \theta}}_{\mid_{\theta = 0}}F_2(x^0, (1 - \theta) u^0 + \theta u)= D_2 F_2(x^0, u^0)(u-u^0)$, and so $D_2F(x^0,u^0)(u-u^0)$ does not depend of the extension of $F$.\\
Recall that $U$ is star-shaper with respect to $u^0$ means that, for all $u \in U$, the segment $[u^0,u]$ is included in $U$, \cite{Sp} (p. 93).
{\epsilon}nd{remark}
When $V$ and $W$ are normed spaces, ${\mathfrak L}(V,W)$ denotes the space of the linear continuous functions from $V$ into $W$, and when $L \in {\mathfrak L}(V,W)$, we write $\Vert L \Vert_{\mathfrak L} := \sup \{ \Vert Lv \Vert : v \in V, \Vert v \Vert \leq 1 \}$.
\vskip1mm
\begin{theorem}\label{th37}
Let $U$ be a nonempty closed subset of ${\mathbb{R}}^d$.
Let $F : {\mathbb{R}}^n \times U \rightarrow {\mathbb{R}}^m$ be a mapping which satisfies the following assumptions.
\begin{enumerate}
\item[(i)] $F \in C^1({\mathbb{R}}^n \times U , {\mathbb{R}}^m)$.
\item[(ii)] There exists $u^0 \in U$ such that $F(0,u^0) = 0$ and $U$ is star-shaped with respect to $u^0$.
\item[(iii)] $\lim_{x \rightarrow 0} (\sup_{u \in B} \Vert DF(x,u) \Vert_{\mathfrak L} ) = 0$ for all nonempty bounded subset $B \subset U$.
{\epsilon}nd{enumerate}
Then $N_F \in C^1(c_0({\mathbb{N}}, {\mathbb{R}}^n) \times {{\epsilon}ll}^{\infty}({\mathbb{N}}, U), c_0({\mathbb{N}},{\mathbb{R}}^m))$, and for all $\underline{x} \in c_0({\mathbb{N}}, {\mathbb{R}}^n)$, $\underline{u} \in {{\epsilon}ll}^{\infty}({\mathbb{N}}, U)$, $\underline{\delta x} \in c_0({\mathbb{N}}, {\mathbb{R}}^n) $, $\underline{\delta u} \in {{\epsilon}ll}^{\infty}({\mathbb{N}}, {\mathbb{R}}^d)$, we have
\[
\begin{array}{ccl}
DN_F(\underline{x}, \underline{u})(\underline{\delta x},\underline{ \delta u}) & = & (DF(x_t,u_t)(\delta x_t, \delta u_t))_{t \in {\mathbb{N}}}\\
\null & = & (D_1F(x_t,u_t)\delta x_t + D_2F(x_t,u_t)\delta u_t)_{t \in {\mathbb{N}}}
{\epsilon}nd{array}
\]
where $D_1$ and $D_2$ denote the partial Fr\'echet differentiations.
{\epsilon}nd{theorem}
\begin{proof}
Let $B \subset U$ be nonempty and bounded. Let $R \in (0, + \infty)$ such that $\Vert u \Vert \leq R$ when $u \in B \cup \{ u^0 \}$. Using the mean value theorem, we have, for all $x \in {\mathbb{R}}^n$ and for all $u \in B$,
\[
\begin{array}{rcl}
\Vert F(x,u) \Vert & \leq & \Vert F(x,u^0) \Vert + \sup_{v \in [u^0,u]} \Vert D_2 F(x,v) \Vert \cdot \Vert v \Vert\\
\null & \leq & \Vert F(x,u^0) \Vert + \sup_{ v \in B_R \cap U} \Vert D_2F (x,v) \Vert \cdot \Vert v \Vert\\
\null & \leq & \Vert F(x,u^0) \Vert + \sup_{ v \in B_R \cap U} \Vert D F(x,v) \Vert \cdot R
{\epsilon}nd{array}
\]
which implies
$$\sup_{u \in B}\Vert F(x,u) \Vert \leq \Vert F(x,u^0) \Vert + \sup_{ v \in B_R \cap U} \Vert D F(x,v) \Vert \cdot R,$$
and therefore, using assumptions (iii) and (ii) and the continuity of $F$, we obtain
\begin{equation}\label{eq31}
\lim_{ x \rightarrow 0} (\sup_{u \in B}\Vert F(x,u) \Vert ) = 0 \; {\rm when} \; B \neq {\epsilon}mptyset, B \subset {\mathbb{R}}^d \; {\rm is} \; {\rm bounded}.
{\epsilon}nd{equation}
Since $F$ is continuously Fr\'echet differentiable, $F$ is continuous, and then, with (\ref{eq31}), we can apply Theorem \ref{th35} to $F$ and assert that $N_F$ is well defined from $c_0({\mathbb{N}}, {\mathbb{R}}^n) \times {{\epsilon}ll}^{\infty}({\mathbb{N}}, U)$ into $c_0({\mathbb{N}},{\mathbb{R}}^m)$ and it is continuous, i.e.
\begin{equation}\label{eq32}
N_F \in C^0(c_0({\mathbb{N}}, {\mathbb{R}}^n) \times {{\epsilon}ll}^{\infty}({\mathbb{N}}, U), c_0({\mathbb{N}},{\mathbb{R}}^m)).
{\epsilon}nd{equation}
Using Theorem A1.2 of \cite{BCr} (p. 24) to the operator $N^1_F$ defined in the proof of the previous theorem, we can assert that $N^1_F$ is continuously Fr\'echet differentiable from ${{\epsilon}ll}^{\infty}({\mathbb{N}}, {\mathbb{R}}^n) \times {{\epsilon}ll}^{\infty}({\mathbb{N}}, U)$ into ${{\epsilon}ll}^{\infty}({\mathbb{N}}, {\mathbb{R}}^m)$ and, for all $\underline{x} \in {{\epsilon}ll}^{\infty}({\mathbb{N}}, {\mathbb{R}}^n)$, $\underline{u} \in {{\epsilon}ll}^{\infty}({\mathbb{N}}, U)$, $ \underline{ \delta x} \in {{\epsilon}ll}^{\infty}({\mathbb{N}}, {\mathbb{R}}^n) $, $\underline{ \delta u} \in {{\epsilon}ll}^{\infty}({\mathbb{N}}, {\mathbb{R}}^d)$, we have $
DF(\underline{x}, \underline{u})( \underline{ \delta x}, \underline{ \delta u}) = (DF(x_t,u_t)(\delta x_t, \delta u_t))_{t \in {\mathbb{N}}}$. Since $N_F$ is a restriction to a vector subspace of $N^1_F$, we obtain that $N_F$ is continuously Fr\'echet differentiable from $c_0({\mathbb{N}}, {\mathbb{R}}^n) \times {{\epsilon}ll}^{\infty}({\mathbb{N}}, U)$ into $c_0({\mathbb{N}},{\mathbb{R}}^m)$
and the formula of its differential is identical to this one of $N^1_F$.
{\epsilon}nd{proof}
The following result is useful to translate the properties of the dynamical system which governs (P1) into the language of operators between sequence spaces.
\begin{corollary}\label{cor38}
Let $U$ be a nonempty closed subset of ${\mathbb{R}}^d$.
Let $f : {\mathbb{R}}^n \times U \rightarrow {\mathbb{R}}^n$ be a mapping which satisfies the assumptions (i, ii, iii) of Theorem \ref{th37}.
We consider the operator ${\mathcal T}(\underline{x}, \underline{u}) := (x_{t+1} -f(x_t,u_t))_{t \in {\mathbb{N}}}$. \\
Then ${\mathcal T} \in C^1(c_0({\mathbb{N}}, {\mathbb{R}}^n) \times {{\epsilon}ll}^{\infty}({\mathbb{N}}, U), c_0({\mathbb{N}}, {\mathbb{R}}^n))$ and for all $\underline{x} \in c_0({\mathbb{N}}, {\mathbb{R}}^n)$, $\underline{u} \in {{\epsilon}ll}^{\infty}({\mathbb{N}}, U)$, $\underline{\delta x} \in c_0({\mathbb{N}}, {\mathbb{R}}^n) $, $\underline{\delta u} \in {{\epsilon}ll}^{\infty}({\mathbb{N}}, {\mathbb{R}}^d)$, we have
\[
\begin{array}{rcl}
D{\mathcal T}(\underline{x}, \underline{u})(\underline{\delta x}, \underline{\delta u}) & = & (\delta x_{t+1} -Df(x_t,u_t)(\delta x_t, \delta u_t) )_{t \in {\mathbb{N}}}\\
\null & = & (\delta x_{t+1} - D_1f(x_t,u_t)\delta x_t - D_2f(x_t,u_t) \delta u_t )_{t \in {\mathbb{N}}}.
{\epsilon}nd{array}
\]
{\epsilon}nd{corollary}
\begin{proof}
Since $f$ satisfies the assumptions of Theorem \ref{th37}, we have $N_f \in C^1(c_0({\mathbb{N}}, {\mathbb{R}}^n) \times {{\epsilon}ll}^{\infty}({\mathbb{N}}, U), c_0({\mathbb{N}}, {\mathbb{R}}^n))$. We set $\Lambda (\underline{x}, \underline{u}) := (x_{t+1})_{t \in {\mathbb{N}}}$ when $\underline{x} \in c_0({\mathbb{N}}, {\mathbb{R}}^n)$ and $\underline{u} \in {{\epsilon}ll}^{\infty}({\mathbb{N}}, U)$. Then $\Lambda$ is well defined and is linear. Since
$\Vert \Lambda (\underline{x}, \underline{u}) \Vert_{\infty} \leq \Vert \underline{x} \Vert_{\infty} \leq \Vert \underline{x} \Vert_{\infty} + \Vert \underline{u} \Vert_{\infty}$, $\Lambda$ is continuous, and consequently it is of class $C^1$. Moreover we have for all $\underline{x} \in c_0({\mathbb{N}}, {\mathbb{R}}^n)$, $\underline{u} \in {{\epsilon}ll}^{\infty}({\mathbb{N}}, U)$, $\underline{\delta x} \in c_0({\mathbb{N}}, {\mathbb{R}}^n) $, $\delta\underline{u} \in {{\epsilon}ll}^{\infty}({\mathbb{N}}, {\mathbb{R}}^d)$, $D \Lambda (\underline{x}, \underline{u}) (\underline{\delta x}, \underline{\delta u}) = \Lambda (\underline{\delta x}, \underline{\delta u}) = ( \delta x_{t+1})_{t \in {\mathbb{N}}}$. Note that ${\mathcal T} = \Lambda - N_f $ which implies that ${\mathcal T}$ is continuously Fr\'echet differentiable, and using Theorem \ref{th37} we obtain
\[
\begin{array}{rcl}
D {\mathcal T} (\underline{x}, \underline{u}) (\underline{\delta x}, \underline{\delta u}) &=& D \Lambda (\underline{x}, \underline{u}) (\underline{\delta x}, \underline{\delta u}) -D N_f (\underline{x}, \underline{u}) (\underline{\delta x}, \underline{\delta u}) \\
\null &=& ( \delta x_{t+1} - Df(x_t,u_t)(\delta x_t, \delta u_t) )_{t \in {\mathbb{N}}}.
{\epsilon}nd{array}
\]
{\epsilon}nd{proof}
\begin{remark}\label{rem39} We consider the operator ${\mathcal F} : c_0({\mathbb{N}}_*, {\mathbb{R}}^n) \rightarrow c_0({\mathbb{N}}, {\mathbb{R}}^n)$ defined by ${\mathcal F} ( \underline{x'}) := \underline{x}$ where $x_0 := 0$ and $x_t := x'_t$ when $t \in {\mathbb{N}}_*$. We introduce the sequence $\underline{\sigma} \in c_0({\mathbb{N}}, {\mathbb{R}}^n)$ by setting $\sigma_0 := \sigma$ and $\sigma_t := 0$ when $t \in {\mathbb{N}}_*$. We consider the operator ${\mathcal E} : c_0({\mathbb{N}}_*, {\mathbb{R}}^n) \rightarrow c_0({\mathbb{N}}, {\mathbb{R}}^n)$ defined by ${\mathcal E} (\underline{x'}) := {\mathcal F} (\underline{x'}) + \underline{\sigma}$. Then ${\mathcal F}$ is linear continuous, ${\mathcal E}$ is affine continuous, and consequently these operators are continuously Fr\'echet differentiable, and for all $\underline{x'} \in c_0({\mathbb{N}}_*, {\mathbb{R}}^n)$ and $ \underline{\delta x'} \in c_0({\mathbb{N}}_*, {\mathbb{R}}^n)$, we have $D {\mathcal E} (\underline{x'}) \underline{\delta x'} = D {\mathcal F} (\underline{x'}) \underline{\delta x'} = {\mathcal F} ( \underline{\delta x'} )= (0, \delta x'_1, \delta x'_2, \cdot \cdot \cdot)$.
{\epsilon}nd{remark}
\begin{proposition}\label{prop310}
Let $U$ be a nonempty closed subset of ${\mathbb{R}}^d$.
let $f : {\mathbb{R}}^n \times U \rightarrow {\mathbb{R}}^n$ be a mapping which satisfies the following properties.
\begin{itemize}
\item[({\bf a})] $f \in C^0( {\mathbb{R}}^n \times U, {\mathbb{R}}^n)$.
\item[({\bf b})] $f(0,u) = 0$ for all $u \in U$.
\item[({\bf c})] For all $u \in U$, for all $x \in {\mathbb{R}}^n$, $D_1f(x,u)$ exists and \\
$D_1f( \cdot , u) \in C^0( {\mathbb{R}}^n, {\mathfrak L}({\mathbb{R}}^n, {\mathbb{R}}^n))$.
\item[({\bf d})] $D_1f$ transforms the nonempty bounded subsets of ${\mathbb{R}}^n \times U$ into bounded subsets of ${\mathfrak L}({\mathbb{R}}^n, {\mathbb{R}}^n)$.
\item[({\bf e})] $\lim_{x \rightarrow 0}( \sup_{u \in B} \Vert D_1 f(x,u) \Vert_{\mathfrak L} ) = 0$ for all nonempty bounded subset $B \subset U$.
{\epsilon}nd{itemize}
Then the operator ${\mathcal T}(\underline{x}, \underline{u}) := (x_{t+1} -f(x_t,u_t))_{t \in {\mathbb{N}}}$ satisfies the following properties.
\begin{itemize}
\item[($\alpha$)] ${\mathcal T} \in C^0(c_0({\mathbb{N}}, {\mathbb{R}}^n) \times {{\epsilon}ll}^{\infty}({\mathbb{N}}, U), c_0({\mathbb{N}}, {\mathbb{R}}^n))$
\item[($\beta$)] For all $( \underline{x}, \underline{u}) \in c_0({\mathbb{N}}, {\mathbb{R}}^n) \times {{\epsilon}ll}^{\infty}({\mathbb{N}}, U)$, $D_1 {\mathcal T}( \underline{x}, \underline{u})$ exists and for all
$\underline{u} \in {{\epsilon}ll}^{\infty}({\mathbb{N}}, U)$, $D_1 {\mathcal T}( \cdot , \underline{u}) \in C^0(c_0({\mathbb{N}}, {\mathbb{R}}^n) , {\mathfrak L}(c_0({\mathbb{N}}, {\mathbb{R}}^n), c_0({\mathbb{N}}, {\mathbb{R}}^n)))$.
{\epsilon}nd{itemize}
{\epsilon}nd{proposition}
\begin{proof}
Let $B$ be a nonempty bounded subset of $U$. We fix $R \in (0, + \infty)$. For all $x \in {\mathbb{R}}^n$ such that $\Vert x \Vert \leq R$, using (b), (c) and the mean value theorem we obtain
$$\Vert f(x,u) \Vert \leq \Vert f(0,u) \Vert + \sup_{z \in [0,x]} \Vert D_1f(z,u) \Vert_{\mathfrak L} \cdot \Vert x \Vert \leq \sup_{z \in [0,x]} \Vert D_1f(z,u) \Vert_{\mathfrak L} \cdot \Vert x \Vert \Longrightarrow $$
$$\sup_{u \in B} \Vert f(x,u) \Vert \leq \sup_{ \Vert z \Vert \leq R} \sup_{u \in B} \Vert D_1f(z,u) \Vert_{\mathfrak L} \cdot \Vert x \Vert$$
which implies, using (d), the following property.
\begin{equation}\label{eq33}
\lim_{ x \rightarrow 0} (\sup_{u \in B} \Vert f(x,u) \Vert ) = 0.
{\epsilon}nd{equation}
Therefore, from (a) and (\ref{eq33}) we obtain the conclusion ($\alpha$).
When we fix $\underline{u} \in {{\epsilon}ll}^{\infty}({\mathbb{N}}, U)$, using Theorem A1.2 in \cite{BCr}, we obtain the conclusion ($\beta$).
{\epsilon}nd{proof}
\vskip1mm
After the operators, we consider the criterion of Problem (P1).
\vskip2mm
\begin{proposition}\label{prop311}
Let $U$ be a nonempty closed subset of ${\mathbb{R}}^d$.
Let $\phi \in C^1({\mathbb{R}}^n \times U, {\mathbb{R}})$ and $\beta \in (0,1)$. We consider $J(\underline{x}, \underline{u}) := \sum_{ t=0}^{+ \infty} \beta^t \phi(x_t,u_t)$ when $\underline{x} \in c_0({\mathbb{N}}, {\mathbb{R}}^n)$ and $\underline{u} \in {{\epsilon}ll}^{\infty}({\mathbb{N}}, U)$. Then $J \in C^1(c_0({\mathbb{N}}, {\mathbb{R}}^n) \times {{\epsilon}ll}^{\infty}({\mathbb{N}}, U), {\mathbb{R}})$ and for all $\underline{x} \in c_0({\mathbb{N}}, {\mathbb{R}}^n)$, $\underline{u} \in {{\epsilon}ll}^{\infty}({\mathbb{N}}, U)$, $ \underline{\delta x} \in c_0({\mathbb{N}}, {\mathbb{R}}^n) $, $ \underline{\delta u} \in {{\epsilon}ll}^{\infty}({\mathbb{N}}, {\mathbb{R}}^d)$, we have
$$DJ(\underline{x}, \underline{u})( \underline{\delta x}, \underline{\delta u}) = \sum_{t = 0}^{\infty} \beta^t ( D_1 \phi(x_t,u_t) \delta x_t + D_2 \phi(x_t,u_t) \delta u_t).$$
{\epsilon}nd{proposition}
\begin{proof}
We consider the Nemytskii operator
$$N^1_{\phi} : {{\epsilon}ll}^{\infty}({\mathbb{N}}, {\mathbb{R}}^n) \times {{\epsilon}ll}^{\infty} ({\mathbb{N}}, U) \rightarrow {{\epsilon}ll}^{\infty}({\mathbb{N}}, {\mathbb{R}}), N^1_{\phi}(\underline{x}, \underline{u}) = (\phi(x_t,u_t))_{t \in {\mathbb{N}}}.$$
Using Theorem A1.2 in \cite{BCr}, we know that $N^1_{\phi} \in C^1({{\epsilon}ll}^{\infty}({\mathbb{N}}, {\mathbb{R}}^n) \times {{\epsilon}ll}^{\infty} ({\mathbb{N}}, U), {{\epsilon}ll}^{\infty}({\mathbb{N}}, {\mathbb{R}}))$ and for all $\underline{x} \in {{\epsilon}ll}^{\infty}({\mathbb{N}}, {\mathbb{R}}^n)$, $\underline{u} \in {{\epsilon}ll}^{\infty}({\mathbb{N}}, U)$, $ \underline{\delta x} \in {{\epsilon}ll}^{\infty}({\mathbb{N}}, {\mathbb{R}}^n) $, $ \underline{\delta u} \in {{\epsilon}ll}^{\infty}({\mathbb{N}}, {\mathbb{R}}^d)$, we have
$$D N^1_{\phi}(\underline{x}, \underline{u})( \underline{\delta x}, \underline{\delta u}) = (D_1 \phi (x_t,u_t) \delta x_t + D_2 \phi (x_t,u_t) \delta u_t)_{ t \in {\mathbb{N}}}.$$
We also consider the other Nemytskii operator
$$N_{\phi} : c_0({\mathbb{N}}, {\mathbb{R}}^n) \times {{\epsilon}ll}^{\infty} ({\mathbb{N}}, U) \rightarrow {{\epsilon}ll}^{\infty}({\mathbb{N}}, {\mathbb{R}}), N_{\phi}(\underline{x}, \underline{u}) = (\phi(x_t,u_t))_{t \in {\mathbb{N}}}.$$
Since $N_{\phi}$ is a restriction of $N^1_{\phi}$ we have $N_{\phi} \in C^1( c_0({\mathbb{N}}, {\mathbb{R}}^n) \times {{\epsilon}ll}^{\infty} ({\mathbb{N}}, U), {{\epsilon}ll}^{\infty}({\mathbb{N}}, {\mathbb{R}}))$
and for all $\underline{x} \in c_0({\mathbb{N}}, {\mathbb{R}}^n)$, $\underline{u} \in {{\epsilon}ll}^{\infty}({\mathbb{N}}, U)$, $ \underline{\delta x} \in c_0({\mathbb{N}}, {\mathbb{R}}^n) $, $ \underline{\delta u} \in {{\epsilon}ll}^{\infty}({\mathbb{N}}, {\mathbb{R}}^d)$, we have
$$D N_{\phi}(\underline{x}, \underline{u})( \underline{\delta x}, \underline{\delta u}) = (D_1 \phi (x_t,u_t) \delta x_t + D_2 \phi (x_t,u_t) \delta u_t)_{ t \in {\mathbb{N}}}.$$
Since $\beta \in (0,1)$, $(\beta^t)_{t \in {\mathbb{N}}} \in {{\epsilon}ll}^1({\mathbb{N}}, {\mathbb{R}})$ (the space of the absolutely convergent real series). We define the linear functional
$$L(\underline{z}) := \sum_{t = 0}^{+ \infty} \beta^t z_t = \langle (\beta^t)_{t \in {\mathbb{N}}}, \underline{z} \rangle_{{{\epsilon}ll}^1, {{\epsilon}ll}^{\infty}}$$
where $\underline{z} \in {{\epsilon}ll}^{\infty}({\mathbb{N}}, {\mathbb{R}})$ and $\langle \cdot , \cdot \rangle_{{{\epsilon}ll}^1, {{\epsilon}ll}^{\infty}}$ denotes the duality bracket between ${{\epsilon}ll}^1({\mathbb{N}}, {\mathbb{R}})$ and ${{\epsilon}ll}^{\infty}({\mathbb{N}}, {\mathbb{R}})$. Using \cite{AB} (Theorem 15.22, p. 503), we know that $L$ is linear continuous on ${{\epsilon}ll}^{\infty}({\mathbb{N}}, {\mathbb{R}})$, and consequently we have $L \in C^1({{\epsilon}ll}^{\infty}({\mathbb{N}}, {\mathbb{R}}), {\mathbb{R}})$, and for all $\underline{z}$ and $ \underline{\delta z}$ in ${{\epsilon}ll}^{\infty}({\mathbb{N}}, {\mathbb{R}})$, we have $D L(\underline{z}) \underline{ \delta z} = L (\underline{\delta z}) = \sum_{t = 0}^{+ \infty} (\beta^t \cdot \delta z_t)$. Since $J = L \circ N_{\phi}$, $J$ is continuously differentiable as a composition of continuously differentiable mappings, and using the chain rule of the differential calculus, for all $\underline{x} \in c_0({\mathbb{N}}, {\mathbb{R}}^n)$, $\underline{u} \in {{\epsilon}ll}^{\infty}({\mathbb{N}}, U)$, $ \underline{\delta x} \in c_0({\mathbb{N}}, {\mathbb{R}}^n) $, $\underline{\delta u} \in {{\epsilon}ll}^{\infty}({\mathbb{N}}, {\mathbb{R}}^d)$, we obtain
\[
\begin{array}{rcl}
D J(\underline{x}, \underline{u})( \underline{\delta x}, \underline{\delta u}) & = & D L(N_{\phi}(\underline{x}, \underline{u})) D (N_{\phi}(\underline{x}, \underline{u})( \underline{\delta x}, \underline{\delta u})\\
\null & = & L(D (N_{\phi}(\underline{x}, \underline{u})( \underline{\delta x}, \underline{\delta u})\\
\null & = & \sum_{t=0}^{+ \infty} \beta^t (D_1 \phi (x_t,u_t) \delta x_t + D_2 \phi (x_t,u_t) \delta u_t).
{\epsilon}nd{array}
\]
{\epsilon}nd{proof}
Using similar arguments we establish the following result.
\vskip1mm
\begin{proposition}\label{prop312}
Let $U$ be a nonempty subset of ${\mathbb{R}}^d$, $\beta \in (0,1)$ and $\phi \in C^0({\mathbb{R}}^n \times U, {\mathbb{R}})$ such that $D_1 \phi(x,u)$ exists for all $(x,u) \in {\mathbb{R}}^n \times U$ and, for all $u \in U$, $D_1 \phi( \cdot, u) \in C^0({\mathbb{R}}^n, {\mathfrak L}({\mathbb{R}}^n, {\mathbb{R}}^n))$.\\
Then $J \in C^0(c_0({\mathbb{N}}, {\mathbb{R}}^n) \times {{\epsilon}ll}^{\infty}({\mathbb{N}}, U), {\mathbb{R}})$, and for all $\underline{x} \in c_0({\mathbb{N}}, {\mathbb{R}}^n)$, for all $\underline{u} \in {{\epsilon}ll}^{\infty}({\mathbb{N}}, U)$, $D_1J(\underline{x}, \underline{u})$ exists and $D_1J(\cdot , \underline{u}) \in C^0(c_0({\mathbb{N}}, {\mathbb{R}}^n), {\mathfrak L}(c_0({\mathbb{N}}, {\mathbb{R}}^n), {\mathbb{R}}))$. Moreover, for all $\underline{x} \in c_0({\mathbb{N}}, {\mathbb{R}}^n)$, for all $\underline{u} \in {{\epsilon}ll}^{\infty}({\mathbb{N}}, U)$, for all $\underline{\delta x} \in c_0({\mathbb{N}}, {\mathbb{R}}^n)$, we have
$$D_1J(\underline{x}, \underline{u}) \underline{\delta x} = \sum_{t = 0}^{+ \infty} \beta^t D_1 \phi(x_t, u_t) \delta x_t.$$
{\epsilon}nd{proposition}
\section{ Linear difference equations}
We establish a result on the existence of a solution of a nonhomogeneous linear equation which belongs to $c_0({\mathbb{N}}_*, {\mathbb{R}}^n)$ when the second member belongs to $c_0({\mathbb{N}}_*, {\mathbb{R}}^n)$. These results permit to obtain useful properties on the operator which represents the dynamical system of Problem (P1).
\vskip1mm
\begin{proposition}\label{prop41}
Let $(A_t)_{t \in {\mathbb{N}}_*}$ be a sequence in ${\mathfrak L}({\mathbb{R}}^n, {\mathbb{R}}^n)$ and $e \in c_0({\mathbb{N}}_*, {\mathbb{R}}^n)$. We consider the following Cauchy problem
\[
(DE)
\left\{
\begin{array}{rcl}
z_{t+1} & = & A_t z_t + e_t\\
z_1 & = & \zeta.
{\epsilon}nd{array}
\right.
\]
We assume that $\sup_{t \in {\mathbb{N}}_*} \Vert A_t \Vert_{\mathfrak L} < 1$.
Then the solution of (DE) belongs to $c_0({\mathbb{N}}_*, {\mathbb{R}}^n)$.
{\epsilon}nd{proposition}
\begin{proof}
We denote by $\underline{z}$ the solution of (DE). Doing a straightforward calculation we obtain, for all $t \in {\mathbb{N}}$, that
$$z_{t+1} = (A_t \cdot \cdot \cdot A_1)\zeta + \sum_{i= 1}^{t-1} (A_t \cdot \cdot \cdot A_{i+1}) e_i + e_t.$$
Let $M > 0$ such that $\sup_{t \in {\mathbb{N}}_*} \Vert A_t \Vert_{\mathfrak L} \leq M < 1$. Therefore we have
\[
\begin{array}{rcl}
\Vert z_{t+1} \Vert & \leq & (\prod_{i=1}^t \Vert A_i \Vert_{\mathfrak L}) \Vert \zeta \Vert + \sum_{i=1}^{t-1} (\prod_{j = i + 1}^{t-1} \Vert A_j \Vert_{\mathfrak L}) \Vert e_i \Vert + \Vert e_t \Vert \\
\null & \leq & M^{t} \Vert \zeta \Vert + (\sum_{i=1}^{t-1} M^{t-i}) \Vert e \Vert_{\infty} + \Vert e \Vert_{\infty} \\
\null & = & M^{t} \Vert \zeta \Vert + (\sum_{k= 0}^{t-1} M^k ) \Vert e \Vert_{\infty} \\
\null & \leq & \max \{ \Vert \zeta \Vert, \Vert e \Vert_{\infty} \} \sum_{k= 0}^{t} M^k \leq \max \{ \Vert \zeta \Vert, \Vert e \Vert_{\infty} \} \frac{1}{1-M} < + \infty
{\epsilon}nd{array}
\]
which proves that $\underline{z} \in {{\epsilon}ll}^{\infty}({\mathbb{N}}_*, {\mathbb{R}}^n)$.
\vskip1mm
From the definition of $\underline{z}$, using $ \limsup_{t \rightarrow + \infty} \Vert z_t \Vert < + \infty$, we deduce
\[
\begin{array}{l}
\Vert z_{t+1} \Vert \leq \Vert A_t \Vert_{\mathfrak L} \cdot \Vert z_t \Vert + \Vert e_t \Vert \leq M \cdot \Vert z_t \Vert + \Vert e_t \Vert \Longrightarrow \\
\limsup_{t \rightarrow + \infty} \Vert z_t \Vert = \limsup_{t \rightarrow + \infty} \Vert z_{t+1} \Vert \leq M \cdot \limsup_{t \rightarrow + \infty} \Vert z_t \Vert + 0 \Longrightarrow \\
(1-M) \limsup_{t \rightarrow + \infty} \Vert z_t \Vert \leq 0 \Longrightarrow \limsup_{t \rightarrow + \infty} \Vert z_t \Vert = 0
{\epsilon}nd{array}
\]
since $1-M > 0$, and therefore we obtain $ \lim_{t \rightarrow + \infty} z_t = 0$.
{\epsilon}nd{proof}
\begin{corollary}\label{cor42}
Let $(B_t)_{t \in {\mathbb{N}}_*}$ be a sequence in ${\mathfrak L}({\mathbb{R}}^n, {\mathbb{R}}^n)$ and $d \in c_0({\mathbb{N}}_*, {\mathbb{R}}^n)$. We consider the following Cauchy problem
\[
(DE1)
\left\{
\begin{array}{rcl}
w_{t+1} & = & B_t w_t + d_t\\
w_1 & = & \xi.
{\epsilon}nd{array}
\right.
\]
We assume that there exists $t_* \in {\mathbb{N}}_*$ such that $\sup_{t \geq t_*} \Vert B_t \Vert_{\mathfrak L} < 1$.\\
Then the solution of (DE1) belongs to $c_0({\mathbb{N}}_*, {\mathbb{R}}^n)$.
{\epsilon}nd{corollary}
\begin{proof}
For all $t \in {\mathbb{N}}_*$, we set $A_t := B_{t + t_*}$ and $e_t := d_{t + t_*}$.\\
Then we have $\sup_{ t \in {\mathbb{N}}} \Vert A_t \Vert_{\mathfrak L} < 1$.
We denote by $\underline{w}$ the solution of (DE1). We set $z_t := w_{t + t_*}$ for all $t \in {\mathbb{N}}$. Then we have $z_{t+1} = w_{t + 1 + t_*} = B_{t+t_*} w_{t+ t_*} + d_{t+t_*} = A_t z_t + e_t$ for all $t \in {\mathbb{N}}$ and $z_1 = w_{t_* +1}$. Using Proposition \ref{prop41} we obtain $\lim_{t \rightarrow + \infty} z_t = 0$, i.e. $\lim_{t \rightarrow + \infty} w_{t + t_*} = 0$ which implies $\lim_{t \rightarrow + \infty}w_t = 0$.
{\epsilon}nd{proof}
\section{Static optimization}
In this section we establish a result in the form of a Karush-Kuhn-Tucker theorem in abstract Banach spaces, and we recall a result issued from the book of Ioffe and Tihomirov \cite{IT}. The first result is useful to prove our weak Pontryagin principles, and the second one is useful to prove our strong Pontryagin principles.
\begin{lemma}\label{lem51} Let ${\mathcal X}$, ${\mathcal V}$, ${\mathcal W}$ be real Banach spaces, and ${\mathcal U}$ be a nonempty subset of ${\mathcal V}$. Let ${\mathcal J} \in C^1({\mathcal X} \times {\mathcal U}, {\mathbb{R}})$ and $\Gamma \in C^1({\mathcal X} \times {\mathcal U}, {\mathcal W})$. Let $(\hat{x}, \hat{u})$ be a solution of the following optimization problem
\[
\left\{
\begin{array}{rl}
{\rm Maximize} & {\mathcal J}(x,u)\\
{\rm when} & x \in {\mathcal X}, u \in {\mathcal U}, \Gamma(x,u) = 0.
{\epsilon}nd{array}
\right.
\]
We assume that $D_1 \Gamma(\hat{x}, \hat{u})$ is invertible and that ${\mathcal U}$ is star-shaped with rerspect to $\hat{u}$. Then there exists $M \in {\mathcal W}^*$ which satisfies the following conditions.
\begin{itemize}
\item[(i)] $D_1 {\mathcal J}(\hat{x}, \hat{u}) + M \circ D_1 \Gamma(\hat{x}, \hat{u}) = 0$.
\item[(ii)] $\forall u \in {\mathcal U}$, $\langle D_2 {\mathcal J}(\hat{x}, \hat{u}) + M \circ D_2 \Gamma(\hat{x}, \hat{u}), u- \hat{u} \rangle \leq 0$.
{\epsilon}nd{itemize}
{\epsilon}nd{lemma}
\begin{proof}
Let ${\mathcal U}_1$ be an open subset of ${\mathcal V}$ such that ${\mathcal U} \subset {\mathcal U}_1$ and such that there exists $\Gamma_1 \in C^1({\mathcal X} \times {\mathcal U}_1, {\mathcal W})$ such that ${ \Gamma_1}_{\mid_{{\mathcal X} \times {\mathcal U}}} = \Gamma$. Since $D_1 \Gamma_1(\hat{x}, \hat{u}) = D_1 \Gamma (\hat{x}, \hat{u})$ is invertible, we can use the implicit function theorem and assert that there exist ${\mathcal N}_{\hat{x}}$ an open neighborhood of $\hat{x}$ in ${\mathcal X}$, ${\mathcal N}_{\hat{u}}$ an open convex neighborhood of $\hat{u}$ in ${\mathcal U}_1$, and a mapping $\pi \in C^1({\mathcal N}_{\hat{u}}, {\mathcal N}_{\hat{x}})$ such that
$$\{ (x,u) \in {\mathcal N}_{\hat{x}} \times {\mathcal N}_{\hat{u}} : \Gamma_1(x,u) = 0 \} = \{ (\pi(u), u) : u \in {\mathcal N}_{\hat{u}} \}.$$
Differentiating $\Gamma_1(\pi(u),u) = 0$ at $\hat{u}$ we obtain $D_1 \Gamma_1 (\hat{x}, \hat{u}) \circ D \pi(\hat{u}) + D_2 \Gamma_1 (\hat{x}, \hat{u}) = 0$ which implies
\begin{equation}\label{eq51}
D \pi (\hat{u}) = - (D_1 \Gamma (\hat{x}, \hat{u}))^{-1} \circ D_2 \Gamma (\hat{x}, \hat{u}).
{\epsilon}nd{equation}
Since $(\hat{x}, \hat{u})$ is a solution of the initial problem, $\hat{u}$ is a solution of the following problem
\[
\left\{
\begin{array}{rl}
{\rm Maximize} & {\mathcal B}(u) \\
{\rm when} & u \in {\mathcal N}_{\hat{u}} \cap {\mathcal U}
{\epsilon}nd{array}
\right.
\]
where ${\mathcal B}(u) = {\mathcal J}(\pi(u),u)$. Since ${\mathcal B}$ is differentiable (as a composition of differentiable mappings) and ${\mathcal N}_{\hat{u}} \cap {\mathcal U}$ is also star-shaped with respect to $\hat{u}$, a necessary condition of optimality for the last problem is
\begin{equation}\label{eq52}
\forall u \in {\mathcal N}_{\hat{u}} \cap {\mathcal U}, \langle D{\mathcal B}(\hat{u}), u - \hat{u} \rangle \leq 0
{\epsilon}nd{equation}
since $0 \geq \lim_{\theta \rightarrow 0+} \frac{1}{\theta}({\mathcal B}(\hat{u} + \theta (u - \hat{u})) - {\mathcal B}(\hat{u})) = \langle D{\mathcal B}(\hat{u}), u - \hat{u} \rangle$. When $u \in {\mathcal U}$, there exists $\theta_u \in (0,1)$ such that $(1- \theta_u) \hat{u}+ \theta_u u \in {\mathcal N}_{\hat{u}} \cap {\mathcal U}$. Using (\ref{eq52}) we obtain
$$\theta_u \cdot \langle D{\mathcal B}(\hat{u}), u - \hat{u} \rangle ) = \langle D{\mathcal B}(\hat{u}), \theta_u(u - \hat{u}) \rangle = \langle D{\mathcal B}(\hat{u}), [(1 - \theta_u) \hat{u} + \theta_u u] - \hat{u} \rangle \leq 0,$$
and so we obtain
\begin{equation}\label{eq53}
\forall u \in {\mathcal U}, \langle D{\mathcal B}(\hat{u}), u - \hat{u} \rangle \leq 0.
{\epsilon}nd{equation}
Using the chain rule we obtain
\begin{equation}\label{eq54}
D {\mathcal B}(\hat{u}) = D_1 {\mathcal J}(\hat{x}, \hat{u}) \circ D \pi( \hat{u}) + D_2 {\mathcal J}(\hat{x}, \hat{u}).
{\epsilon}nd{equation}
We define
\begin{equation}\label{eq55}
M := - D_1 {\mathcal J}(\hat{x}, \hat{u}) \circ (D_1 \Gamma (\hat{x}, \hat{u}))^{-1} \in {\mathcal W}^*.
{\epsilon}nd{equation}
From (\ref{eq55}) we obtain
\begin{equation}\label{56}
D_1 {\mathcal J}(\hat{x}, \hat{u}) + M \circ D_1 \Gamma (\hat{x}, \hat{u}) = 0.
{\epsilon}nd{equation}
Using (\ref{eq54}) and (\ref{eq51}) we obtain $D {\mathcal B}(\hat{u}) = - D_1 {\mathcal J}(\hat{x}, \hat{u}) \circ (D_1 \Gamma (\hat{x}, \hat{u}))^{-1} \circ D_2 \Gamma (\hat{x}, \hat{u}) + D_2 {\mathcal J}(\hat{x}, \hat{u}) = M \circ D_2 \Gamma (\hat{x}, \hat{u}) + D_2 {\mathcal J}(\hat{x}, \hat{u})$, and therefore, from (\ref{eq53}) we obtain
\begin{equation}\label{57}
\forall u \in {\mathcal U}, \langle D_2 {\mathcal J}(\hat{x}, \hat{u}) + M \circ D_2 \Gamma (\hat{x}, \hat{u}) , u - \hat{u} \rangle \leq 0.
{\epsilon}nd{equation}
{\epsilon}nd{proof}
\begin{remark}\label{rem52}
There exist several results like this one in the books \cite{Co} and \cite{We} which use the convexity of $U$. In the necessary conditions of optimality we prefer to avoid the convexity of the sets; it is why we have established this lemma.
{\epsilon}nd{remark}
As a corollary of the extremal principle in mixed problems (Theorem 3, p. 71 in \cite{IT}), we obtain the following result.
\begin{lemma}\label{lem53}
Let ${\mathcal X}$, ${\mathcal V}$, ${\mathcal W}$ be real Banach spaces, and ${\mathcal U}$ be a nonempty subset of ${\mathcal V}$. Let ${\mathcal J} : {\mathcal X} \times {\mathcal U} \rightarrow {\mathbb{R}}$ and $\Gamma : {\mathcal X} \times {\mathcal U} \rightarrow {\mathcal W}$ be mappings. Let $(\hat{x}, \hat{u})$ be a solution of the following optimization problem
\[
\left\{
\begin{array}{rl}
{\rm Maximize} & {\mathcal J}(x,u)\\
{\rm when} & x \in {\mathcal X}, u \in {\mathcal U}, \Gamma(x,u) = 0.
{\epsilon}nd{array}
\right.
\]
We assume that the following conditions are fulfilled.
\begin{itemize}
\item[(a)] For all $u \in {\mathcal U}$, $[x \mapsto \Gamma(x,u)]$ and $[x \mapsto {\mathcal J}(x,u)]$ are of class $C^1$ at $\hat{x}$.
\item[(b)] There exists a neighborhood ${\mathcal N}$ of $\hat{x}$ in ${\mathcal X}$ such that, for all $x \in {\mathcal N}$, for all $u', u'' \in {\mathcal U}$, for all $\theta \in [0,1]$, there exists $u \in {\mathcal U}$ which satisfies the following conditions
\[
\left\{
\begin{array}{ccl}
\Gamma(x,u) & =& (1- \theta) \Gamma(x,u') + \theta \Gamma(x, u'') \\
{\mathcal J}(x,u) & \geq & (1- \theta) {\mathcal J}(x,u') + \theta {\mathcal J}(x,u'').
{\epsilon}nd{array}
\right.
\]
\item[(c)] The codimension of Im$D_1 \Gamma(\hat{x}, \hat{u})$ in ${\mathcal W}$ is finite.
\item[(d)] The set $\{ D_1 \Gamma(\hat{x}, \hat{u}) x + \Gamma(\hat{x}, u) : x \in {\mathcal X}, u \in {\mathcal U} \}$ contains a neighborhood of the origine of ${\mathcal W}$.
{\epsilon}nd{itemize}
Then there exists $M \in {\mathcal W}^*$ which satisfies the two following conditions.
\begin{itemize}
\item[(i)] $D_1 {\mathcal J}(\hat{x}, \hat{u}) + M \circ D_1 \Gamma(\hat{x}, \hat{u}) = 0$.
\item[(ii)] For all $u \in {\mathcal U}$, $ {\mathcal J}(\hat{x}, \hat{u}) + M \Gamma(\hat{x}, \hat{u}) \geq {\mathcal J}(\hat{x}, u) + M \Gamma(\hat{x}, u)$.
{\epsilon}nd{itemize}
{\epsilon}nd{lemma}
\section{Weak Pontryagin principle for (P1)}
We start by a translation of Problem (P1) into a more simple abstract optimization problem in Banach spaces.
We define the functional $J_1(\underline{x'}, \underline{u}) := J({\mathcal E}(\underline{x'}), \underline{u})$ and the nonlinear operator ${\mathcal T}_1(\underline{x'}, \underline{u}) := {\mathcal T}({\mathcal E}(\underline{x'}), \underline{u})$. Then we can translate (P1) into the following problem.
\[
(P2)
\left\{
\begin{array}{rl}
{\rm Maximize} & J_1(\underline{x'}, \underline{u})\\
{\rm when} & \underline{x'} \in c_0({\mathbb{N}}_*, {\mathbb{R}}^n), \underline{u} \in {{\epsilon}ll}^{\infty}({\mathbb{N}}, U)\\
\null & {\mathcal T}_1(\underline{x'}, \underline{u}) = \underline{0}.
{\epsilon}nd{array}
\right.
\]
We consider the following list of assumptions.
\begin{itemize}
\item[(A1)] $U$ is a nonempty closed subset of ${\mathbb{R}}^d$.
\item[(A2)] $\phi \in C^1({\mathbb{R}}^n \times U, {\mathbb{R}})$ and $f \in C^1({\mathbb{R}}^n \times U, {\mathbb{R}}^n)$.
\item[(A3)] There exists $u^0 \in U$ such that $f(0, u^0) = 0$ and $U$ is star-shaped with respect to $u^0$.
\item[(A4)] $\lim_{x \rightarrow 0}(\sup_{u \in B} \Vert Df(x,u) \Vert_{\mathfrak L}) = 0$ for all nonempty bounded subset $B \subset U$.
{\epsilon}nd{itemize}
\vskip1mm
\noindent
Recall that ${{\epsilon}ll}^1({\mathbb{N}}, {\mathbb{R}}^{n*})$ can be assimilated to the dual topological space of $c_0({\mathbb{N}}, {\mathbb{R}}^n)$, i.e. an element of ${{\epsilon}ll}^1({\mathbb{N}}, {\mathbb{R}}^{n*})$ can be considered as a continuous linear functional on $c_0({\mathbb{N}}, {\mathbb{R}}^n)$, \cite{AB} (Theorem 15.9, p. 498).
\vskip1mm
\begin{lemma}\label{lem61}
We assume (A1-A4) fulfilled. Let $(\hat{\underline{x'}}, \hat{\underline{u}})$ be a solution of (P2). \\
Then there exists $\underline{q} \in {{\epsilon}ll}^1({\mathbb{N}}, {\mathbb{R}}^{n*})$ which satisfies the two following conditions.
\begin{itemize}
\item[(i)] $D_1J_1(\hat{\underline{x'}}, \hat{\underline{u}}) + \underline{q} \circ D_1 {\mathcal T}_1 (\hat{\underline{x'}}, \hat{\underline{u}}) = \underline{0}$.
\item[(ii)] For all $\underline{u} \in {{\epsilon}ll}^{\infty}({\mathbb{N}}, U)$, $\langle D_2J_1(\hat{\underline{x'}}, \hat{\underline{u}}) + \underline{q} \circ D_2 {\mathcal T}_1 (\hat{\underline{x'}}, \hat{\underline{u}}), \underline{u} - \hat{\underline{u}} \rangle \leq 0$.
{\epsilon}nd{itemize}
{\epsilon}nd{lemma}
\begin{proof}
Using Remark \ref{rem39} and Proposition \ref{prop311}, $J_1$ is of class $C^1$ as a composition of mappings of class $C^1$. Using Remark \ref{rem39} and Corollary \ref{cor38}, ${\mathcal T}_1$ is of class $C^1$ as a composition of operators of class $C^1$.
\vskip1mm
We set $\hat{B} := \{ \hat{u}_t : t \in {\mathbb{N}} \}$. Then $\hat{B}$ is nonempty bounded in $U$ since $\hat{\underline u} \in {{\epsilon}ll}^{\infty}({\mathbb{N}}, U)$. For all $t \in {\mathbb{N}}_*$, we have
$$\Vert D_1 f(\hat{x'}_t, \hat{u}_t) \Vert_{\mathfrak L} \leq \Vert D f(\hat{x'}_t, \hat{u}_t) \Vert_{\mathfrak L} \leq \sup_{u \in \hat{B}} \Vert D f(\hat{x}'_t,u) \Vert_{\mathfrak L}$$
and therefore, using (A4), we obtain $\lim_{ t \rightarrow + \infty} \Vert D_1 f(\hat{x'}_t, \hat{u}_t) \Vert_{\mathfrak L} = 0$. Therefore there exists $t_* \in {\mathbb{N}}$ such that $\sup_{ t \geq t_*} \Vert D_1 f(\hat{x'}_t, \hat{u}_t) \Vert_{\mathfrak L} < 1$.
Note that to solve equation (DE1) of Section 4, with $B_t := D_1f(\hat{x'}_t, \hat{u}_t)$ when $t \in {\mathbb{N}}_*$, is equivalent to solve the equation $D_1 {\mathcal T}_1(\hat{\underline{x'}}, \hat{\underline{u}}) \underline{\delta x'} = \underline{e}$ where $\underline{e} \in c_0({\mathbb{N}}, {\mathbb{R}}^n)$ and the unknown variable is $\underline{\delta x'} \in c_0({\mathbb{N}}_*, {\mathbb{R}}^n)$. We can use Corollary \ref{cor42} and assert that $D_1 {\mathcal T}_1(\hat{\underline{x'}}, \hat{\underline{u}})$ is surjective and it is clearly injective, and consequently $D {\mathcal T}_1(\hat{\underline{x'}}, \hat{\underline{u}})$ is also invertible. Therefore we can use Lemma \ref{lem51} and assert that there exists a Lagrange multiplier $\underline{q} \in c_0({\mathbb{N}}_ù, {\mathbb{R}}^n)^* = {{\epsilon}ll}^1({\mathbb{N}}_ù, {\mathbb{R}}^{n*})$ which satisfies the announced conclusions.
{\epsilon}nd{proof}
\begin{theorem}\label{th62}
We assume (A1-A4) fulfilled. Let $(\hat{\underline{x}}, \hat{\underline{u}})$ be a solution of (P1).
Then there exists $\underline{p} \in {{\epsilon}ll}^1({\mathbb{N}}_*, {\mathbb{R}}^{n*})$ such that the following relations hold.
\vskip1mm
\noindent
{\bf (AE1)} \hskip3mm $p_t = p_{t+1} \circ D_1f(\hat{x}_t, \hat{u}_t) + D_1 \phi(\hat{x}_t, \hat{u}_t) $ for all $t \in {\mathbb{N}}_*$\\
{\bf (WM1)} \hskip3mm $\langle p_{t+1} \circ D_2f(\hat{x}_t, \hat{u}_t) + D_2 \phi(\hat{x}_t, \hat{u}_t) , u - \hat{u}_t \rangle \leq 0$ for all $u \in U$, for all $t \in {\mathbb{N}}$.
{\epsilon}nd{theorem}
\begin{proof}
We define $\hat{\underline{x'}}$ by setting $\hat{x'}_t := \hat{x}_t$ when $t \in {\mathbb{N}}_*$. Since $(\hat{\underline{x}}, \hat{\underline{u}})$ is a solution of (P1), $(\hat{\underline{x'}}, \hat{\underline{u}})$ is a solution of (P2). Then Lemma \ref{lem51} provides $\underline{q} \in {{\epsilon}ll}^1({\mathbb{N}}, {\mathbb{R}}^{n*})$ such that
\begin{equation}\label{eq61}
\left.
\begin{array}{rcl}
D_1J_1(\hat{\underline{x'}}, \hat{\underline{u}}) + \underline{q} \circ D_1{\mathcal T}_1 (\hat{\underline{x'}}, \hat{\underline{u}}) &=& 0\\
\langle D_2J_1(\hat{\underline{x'}}, \hat{\underline{u}}) + \underline{q} \circ D_2{\mathcal T}_1 (\hat{\underline{x'}}, \hat{\underline{u}}), \underline{u} - \hat{\underline{u}} \rangle &\leq& 0
{\epsilon}nd{array}
\right\}
{\epsilon}nd{equation}
for all $ \underline{u} \in {{\epsilon}ll}^{\infty}({\mathbb{N}}, U)$.
Now we translate these conditions to obtain the conclusions of our theorem.
Using Remark \ref{rem39}, Proposition \ref{prop311} and the chain rule we obtain
\[
\begin{array}{rcl}
D_1 J_1(\hat{\underline{x'}}, \hat{\underline{u}}) \underline{\delta x'} &= & D_1 J( {\mathcal E}(\hat{\underline{x'}}),\hat{\underline{u}}) D {\mathcal E}(\hat{\underline{x'}})\underline{\delta x'} = D_1 J( \hat{\underline{x}},\hat{\underline{u}}) {\mathcal F}(\underline{\delta x'})\\
\null &=& \beta^0 D_1 \phi(\sigma, \hat{u}_0) 0 + \sum_{t=1}^{+\infty} \beta^t D_1 \phi(\hat{x}_t, \hat{u}_t) \delta x_t,
{\epsilon}nd{array}
\]
and therefore we have
\begin{equation}\label{eq62}
D_1 J_1(\hat{\underline{x'}}, \hat{\underline{u}}) \underline{\delta x'} = \sum_{t=1}^{+\infty} \beta^t D_1 \phi(\hat{x}_t, \hat{u}_t) \delta x'_t.
{\epsilon}nd{equation}
Using the same arguments we have
$D_2 J_1(\hat{\underline{x'}}, \hat{\underline{u}}) \underline{\delta u} = D_2 J ( {\mathcal E}(\hat{\underline{x'}}),\hat{\underline{u}}) \underline{\delta u}$
which implies
\begin{equation}\label{eq63}
D_2 J_1(\hat{\underline{x'}}, \hat{\underline{u}}) \underline{\delta u} = \sum_{t = 0}^{+\infty} \beta^t D_2 \phi (\hat{x}_t, \hat{u}_t) \delta u_t.
{\epsilon}nd{equation}
Using Corollary \ref{cor38} and Remark \ref{rem39} and the chain rule we obtain
\[
\begin{array}{rcl}
D_1 {\mathcal T}_1(\hat{\underline{x'}}, \hat{\underline{u}}) \underline{\delta x'} &=& D_1{\mathcal T}({\mathcal E}(\hat{\underline{x'}}), \hat{\underline{u}})D{\mathcal E}(\hat{\underline{x'}})\underline{\delta x'}\\
\null & = & D_1{\mathcal T}(\hat{\underline{x}}, \hat{\underline{u}}) {\mathcal F}(\underline{\delta x'})\\
\null & = & (\delta x_1 - D_1 f(\sigma, \hat{u}_0)0, (\delta x_{t+1} - D_1f( \hat{x}_t, \hat{u}_t) \delta x_t)_{ t \in {\mathbb{N}}_*}),
{\epsilon}nd{array}
\]
and therefore we have
\begin{equation}\label{eq64}
D_1 {\mathcal T}_1(\hat{\underline{x'}}, \hat{\underline{u}}) \underline{\delta x'} = (\delta x'_1, (\delta x'_{t+1} - D_1f( \hat{x}_t, \hat{u}_t) \delta x'_t)_{ t \in {\mathbb{N}}_*}).
{\epsilon}nd{equation}
Using the same arguments, we obtain $D_2 {\mathcal T}_1(\hat{\underline{x'}}, \hat{\underline{u}}) \underline{\delta u} = D_2 {\mathcal T}({\mathcal E}(\hat{\underline{x'}}), \hat{\underline{u}}) \underline{\delta u} = $\\
$ D_2 {\mathcal T}(\hat{\underline{x}}, \hat{\underline{u}}) \underline{\delta u}$ which implies
\begin{equation}\label{eq65}
D_2 {\mathcal T}_1(\hat{\underline{x'}}, \hat{\underline{u}}) \underline{\delta u} = (-D_2 f(\hat{x}_t, \hat{u}_t) \delta u_t)_{ t \in {\mathbb{N}}}.
{\epsilon}nd{equation}
Using (\ref{eq61}) we calculate $\underline{q} \circ D_1 {\mathcal T}_1 (\hat{\underline{x'}}, \hat{\underline{u}}) \underline{\delta x'} = \langle q_0, \delta x'_1 \rangle +
\sum_{t=1}^{+ \infty} \langle q_t, \delta x'_{t+1} \rangle - \sum_{t=1}^{+ \infty} q_t \circ D_1 f(\hat{x}_t, \hat{u}_t) \delta x'_t = \sum_{t=0}^{+ \infty} \langle q_t, \delta x'_{t+1} \rangle - \sum_{t=1}^{+ \infty} q_t \circ D_1 f(\hat{x}_t, \hat{u}_t) \delta x'_t =
\sum_{t=1}^{+\infty} \langle q_{t-1}, \delta x'_t \rangle - \sum_{t=1}^{+ \infty} \langle q_t \circ D_1 f(\hat{x}_t, \hat{u}_t), \delta x'_t \rangle$ which implies
\begin{equation}\label{eq66}
\underline{q} \circ D_1 {\mathcal T}_1 (\hat{\underline{x'}}, \hat{\underline{u}}) \underline{\delta x'} =
\sum_{t=1}^{+\infty} \langle q_{t-1} -q_t \circ D_1 f(\hat{x}_t, \hat{u}_t), \delta x'_t \rangle.
{\epsilon}nd{equation}
Using (\ref{eq61}), (\ref{eq62}) and (\ref{eq66}) we obtain
\begin{equation}\label{eq67}
\sum_{t=1}^{+\infty} \beta^t D_1 \phi(\hat{x}_t, \hat{u}_t) \delta x'_t = \sum_{t=1}^{+\infty} \langle q_{t-1} -q_t \circ D_1 f(\hat{x}_t, \hat{u}_t), \delta x'_t \rangle.
{\epsilon}nd{equation}
We fix $t \in {\mathbb{N}}_*$, we set $\delta x'_s = 0$ when $s \neq t$ and $\delta x'_t$ varies in ${\mathbb{R}}^n$, then from the last equation we obtain $\beta^t D_1 \phi(\hat{x}_t, \hat{u}_t) = q_{t-1} -q_t \circ D_1 f(\hat{x}_t, \hat{u}_t)$, which implies, for all $t \in {\mathbb{N}}_*$,
\begin{equation}\label{eq68}
q_{t-1} = q_t \circ D_1 f(\hat{x}_t, \hat{u}_t) + \beta^t D_1 \phi(\hat{x}_t, \hat{u}_t).
{\epsilon}nd{equation}
We define $\underline{p} \in {{\epsilon}ll}^1({\mathbb{N}}_*, {\mathbb{R}}^{n*})$ by setting $p_t := q_{t-1}$. Then (\ref{eq68}) implies (AE1).
\vskip1mm
From (\ref{eq65}) we obtain
$$\sum_{t=0}^{+ \infty} \langle q_t \circ D_2 f(\hat{x}_t, \hat{u}_t), u_t - \hat{u}_t \rangle + \langle \underline{q} \circ D_2 {\mathcal T}_1(\hat{\underline{x'}}, \hat{\underline{u}}) \underline{u} - \hat{\underline{u}} \rangle \leq 0 $$
for all $\underline{u} \in {{\epsilon}ll}^{\infty}({\mathbb{N}}, U)$, therefore from (\ref{eq61}) and (\ref{eq63}) we obtain
$$\sum_{t=0}^{+ \infty} \langle q_t \circ D_2 f(\hat{x}_t, \hat{u}_t), u_t - \hat{u}_t \rangle + \sum_{t = 0}^{+\infty} \beta^t \langle D_2 \phi (\hat{x}_t, \hat{u}_t), u_t - \hat{u}_t \rangle \leq 0$$
for all $\underline{u} \in {{\epsilon}ll}^{\infty}({\mathbb{N}}, U)$. We fix $t \in {\mathbb{N}}$, we take $u_s = \hat{u}_s$ when $s \neq t$, and $u_t$ varies in $U$. Then we obtain $\langle q_t \circ D_2 f(\hat{x}_t, \hat{u}_t) + \beta^t D_2 \phi (\hat{x}_t, \hat{u}_t), u_t - \hat{u}_t \rangle \leq 0$ which implies, for all $t \in {\mathbb{N}}$ and for all $u_t \in U$
\begin{equation}\label{eq69}
\langle q_t \circ D_2 f(\hat{x}_t, \hat{u}_t) + \beta^t D_2 \phi (\hat{x}_t, \hat{u}_t), u_t - \hat{u}_t \rangle \leq 0.
{\epsilon}nd{equation}
Replacing $q_t$ by $p_{t+1}$ in this last equation we obtain (WM1).
{\epsilon}nd{proof}
\begin{remark}\label{rem63}
In Theorem \ref{th62}, (AE1) means Adjoint Equation for (P1), (WM1) means Weak Maximum principle for (P1). Since $\underline{p} \in {{\epsilon}ll}^1({\mathbb{N}}_*, {\mathbb{R}}^{n*})$, note that and the transversality condition at infinity for problem (P1), $\lim_{t \rightarrow + \infty} p_t = 0$, is satisfied.
{\epsilon}nd{remark}
\section{Weak Pontryagin principle for (P)}
In this section we translate the main result of Section 6 on (P1) into a result on (P).\\
We introduce the following conditions
\begin{itemize}
\item[(B1)] $U$ is a nonempty closed subset of ${\mathbb{R}}^d$.
\item[(B2)] $\psi \in C^1({\mathbb{R}}^n \times U, {\mathbb{R}})$ and $g \in C^1({\mathbb{R}}^n \times U, {\mathbb{R}}^n)$.
\item[(B3)] There exists $u^0 \in U$ such that $g(y_{\infty},u^0) = y_{\infty}$ and $U$ is star-shaped with respect to $u^0$.
\item[(B4)] $\lim_{ y \rightarrow y_{\infty}} ( \sup_{u \in B} \Vert Dg(y,u) \Vert) = 0$ for all nonempty bounded subset $B \subset U$.
{\epsilon}nd{itemize}
\begin{theorem}\label{th71}
We assume (B1-B4) fulfilled. let $(\hat{\underline y}, \hat{\underline u})$ be a solution of Problem (P).
Then there exists $\underline{p} \in {{\epsilon}ll}^1({\mathbb{N}}_*, {\mathbb{R}}^{n*})$ such that the following relations hold.
\vskip1mm
\noindent
{\bf (AE)} \hskip3mm $p_t = p_{t+1} \circ D_1 g(\hat{y}_t, \hat{u}_t) + \beta^t D_1 \psi(\hat{y}_t, \hat{u}_t)$ for all $t \in {\mathbb{N}}_*$.\\
{\bf (WM)} \hskip3mm $\langle p_{t+1} \circ D_2 g(\hat{y}_t, \hat{u}_t) + \beta^t D_2 \psi(\hat{y}_t, \hat{u}_t) , u - \hat{u} \rangle \leq 0$ for all $u \in U$, for all $t \in {\mathbb{N}}$
{\epsilon}nd{theorem}
\begin{proof}
Using Section 2, since $(\hat{\underline y}, \hat{\underline u})$ is a solution of (P), $(\hat{\underline x}, \hat{\underline u})$ is a solution of (P1) with $\hat{x}_t = \hat{y}_t - y_{\infty}$. For all $j \in \{ 1,2,3,4 \}$, (Bj) implies (Aj) and so the conclusions of Theorem \ref{th62} hold. We conserve the same $\underline{p}$, and we translate to see that (AE1) implies (AE) and (WM1) implies (WM).
{\epsilon}nd{proof}
\section{Strong Pontryagin principle for (P1)}
First we introduce the Hamiltonian of Pontryagin which is defined, for all $t \in {\mathbb{N}}$, as follows
$$H_t : {\mathbb{R}}^n \times U \times {\mathbb{R}}^{n*} \rightarrow {\mathbb{R}}, H_t(x,u,p) := \beta^t \phi(x,u) + \langle p, f(x,u) \rangle.$$
Note that the condition (WM1) of Theorem \ref{th82} is equivalent to the condition
$$\langle D_2 H_t(\hat{x}_t, \hat{u}_t, p_{t+1}), u_t - \hat{u}_t \rangle \leq 0$$
for all $u \in U$ and for all $t \in {\mathbb{N}}$. In this section we want replace (WM1) by the strengthened condition $ H_t(\hat{x}_t, \hat{u}_t, p_{t+1}) = \max_{u \in U} H_t(\hat{x}_t, u, p_{t+1})$ for all $t \in {\mathbb{N}}$. Note that (WM1) can be viewed as a first-order necessary condition of the optimality of $ H_t(\hat{x}_t, \cdot , p_{t+1})$ at $\hat{u}_t$ on $U$.
\vskip1mm
We consider the following conditions
\begin{itemize}
\item[(C1)] $U$ is a nonempty compact subset of ${\mathbb{R}}^d$.
\item[(C2)] $\phi \in C^0({\mathbb{R}}^n \times U, {\mathbb{R}})$ and $f \in C^0({\mathbb{R}}^n \times U, {\mathbb{R}}^n)$.
\item[(C3)] For all $u \in U$, $f(0,u) = 0$.
\item[(C4)] For all $u \in U$, $D_1f(x,u)$ and $D_1\phi (x,u)$ exist for all $x \in {\mathbb{R}}^n$, and $D_1f( \cdot, u) \in C^0({\mathbb{R}}^n, {\mathfrak L}({\mathbb{R}}^n, {\mathbb{R}}^n))$, and $D_1 \phi( \cdot, u) \in C^0({\mathbb{R}}^n,{\mathbb{R}}^{n*})$.
\item[(C5)] $D_1f$ transforms the nonempty bounded subsets of ${\mathbb{R}}^n \times U$ into bounded subsets of ${\mathfrak L}({\mathbb{R}}^n, {\mathbb{R}}^n)$.
\item[(C6)] For all nonempty bounded subset $B \subset U$, $\lim_{x \rightarrow 0}( \sup_{u \in B} \Vert D_1 f(x,u) \Vert_{\mathfrak L}) = 0$.
\item[(C7)] For all $t \in {\mathbb{N}}$, for all $x_t \in {\mathbb{R}}^n$, for all $u'_t$, $u''_t \in U$ and for all $\theta \in (0,1)$, there exists $u_t \in U$ such that
\[
\left\{
\begin{array}{rcl}
\phi(x_t, u_t) & \geq & (1- \theta) \phi(x_t, u'_t) + \theta \phi (x_t, u''_t)\\
f(x_t, u_t) & = & (1- \theta) f(x_t, u'_t) + \theta f (x_t, u''_t).
{\epsilon}nd{array}
\right.
\]
{\epsilon}nd{itemize}
\begin{lemma}\label{lem81}
Under the assumptions (C1-C7) let $(\hat{\underline{x'}}, \hat{\underline{u}})$ be a solution of Problem (P2) defined in Section 5. Then there exists $\underline{q} \in {{\epsilon}ll}^1({\mathbb{N}}, {\mathbb{R}}^{n*})$ which satisfies the following properties.
\begin{itemize}
\item[(1)] $D_1 J_1(\hat{\underline{x'}}, \hat{\underline{u}}) + \underline{q} D_1 {\mathcal T}_1(\hat{\underline{x'}}, \hat{\underline{u}}) = \underline{0}$.
\item[(2)] $ J_1(\hat{\underline{x'}}, \hat{\underline{u}}) + \langle \underline{q}, N'_f(\hat{\underline{x'}}, \hat{\underline{u}}) \rangle_{c_0, {{\epsilon}ll}^1} = \max_{\underline{u} \in {{\epsilon}ll}^{\infty}({\mathbb{N}}, U)} ( J_1(\hat{\underline{x'}}, \underline{u}) + \langle \underline{q}, N'_f(\hat{\underline{x'}}, \underline{u}) \rangle_{c_0, {{\epsilon}ll}^1})$.
{\epsilon}nd{itemize}
{\epsilon}nd{lemma}
\begin{proof}
We want to use Lemma \ref{lem53} with ${\mathcal J} = J_1$, $\Gamma = {\mathcal T}_1$.\\
Since (C1-C6) imply that $U$ is closed and that the conditions (a, b, c, d, e) of Proposition \ref{prop310} hold, we obtain that ${\mathcal T}$ and $D_1 {\mathcal T}( \cdot, \underline{u})$ are continuous, and using Remark \ref{rem39} we obtain that ${\mathcal T}_1$ and $D_1 {\mathcal T}_1( \cdot, \underline{u})$ are continuous. Using Proposition \ref{prop312}, from (C2) and (C4) we obtain that $J$ and $D_1J( \cdot, \underline{u})$ are continuous, and using Remark \ref{rem39} we obtain that $J_1$ and $D_1J_1( \cdot, \underline{u})$ are continuous. And so the assumption (a) of Lemma \ref{lem53} is fulfilled.\\
Since $U$ is bounded, from (C7) we obtain assumption (b) of Lemma \ref{lem53}.\\
Proceeding as in the proof of Lemma \ref{lem61}, from (C6), with $B_t := D_1f(\hat{x}_t, \hat{u}_t)$, we obtain the assumptions of Corollary \ref{cor42} which implies that\\
$D_1 {\mathcal T}_1(\hat{\underline{x'}}, \hat{\underline{u}})$ is surjective from $c_0({\mathbb{N}}_*, {\mathbb{R}}^n)$ onto $c_0({\mathbb{N}}, {\mathbb{R}}^n)$, and since it is clearly injective, it is invertible. Using the Isomorphism Theorem of Banach, this invertibility implies the assumptions (c) and (d) of Lemma \ref{lem53}. \\
Consequently we can use Lemma \ref{lem53} and we obtain the conclusions with $\underline{q} = M$.
{\epsilon}nd{proof}
\begin{theorem}\label{th82}
Under the assumptions (C1-C7), let $(\hat{\underline{x}}, \hat{\underline{u}})$ be a solution of Problem (P) defined in Section 5. Then there exists $\underline{p} \in {{\epsilon}ll}^1({\mathbb{N}}_*, {\mathbb{R}}^{n*})$ which satisfies the following properties.
\vskip1mm
\noindent
{\bf (AE1)} \hskip3mm $D_1 \phi(\hat{x}_t, \hat{u}_t) + p_{t+1} \circ D_1 f(\hat{x}_t, \hat{u}_t) = 0$ for all $t \in {\mathbb{N}}_*$.\\
{\bf (MP1)} \hskip3mm $ \phi(\hat{x}_t, \hat{u}_t) + \langle p_{t+1},f(\hat{x}_t, \hat{u}_t) \rangle = \max_{u \in U} ( \phi(\hat{x}_t, u) + \langle p_{t+1}, f(\hat{x}_t,u) \rangle )$ for all $t \in {\mathbb{N}}$.
{\epsilon}nd{theorem}
\begin{proof}
Proceeding as in the proof of Theorem \ref{th62}, conclusion (1) of Lemma \ref{lem81} implies (AE1). A straightforward translation of conclusion (2) of Lemma \ref{lem81} provides (MP1).
{\epsilon}nd{proof}
\section{Strong Pontryagin principle for (P)}
In this section we translate the strong Pontryagin principle on (P1) into a result on (P).\\
We consider the following conditions.
\begin{itemize}
\item[(D1)] $U$ is a nonempty compact subset of ${\mathbb{R}}^d$.
\item[(D2)] $\psi \in C^0({\mathbb{R}}^n \times U, {\mathbb{R}})$ and $g \in C^0({\mathbb{R}}^n \times U, {\mathbb{R}}^n)$.
\item[(D3)] For all $u \in U$, $g(y_{\infty}, u) = y_{\infty}$.
\item[(D4)] For all $(y,u) \in {\mathbb{R}}^n \times U$, $D_1 \psi (y,u)$ and $D_1g(y,u)$ exist and, for all $u \in U$, $D_1 \psi ( \cdot, u) \in C^0({\mathbb{R}}^n, {\mathbb{R}}^{n*})$, $D_1 g( \cdot , u) \in C^0( {\mathbb{R}}^n, {\mathfrak L}({\mathbb{R}}^n, {\mathbb{R}}^n))$.
\item[(D5)] $D_1 g$ transforms the nonempty bounded subsets of ${\mathbb{R}}^n \times U$ in bounded subsets of ${\mathfrak L}({\mathbb{R}}^n, {\mathbb{R}}^n)$.
\item[(D6)] $\lim_{y \rightarrow y_{\infty}} (\sup_{u \in B} \Vert D_1 g(y,u) \Vert_{\mathfrak L} ) = 0$ for all nonempty bounded subset $B$ of $U$.
\item[(D7)] For all $t \in {\mathbb{N}}$, for all $y_t \in {\mathbb{R}}^n$, for all $u'_t$, $u''_t \in U$ and for all $\theta \in (0,1)$, there exists $u_t \in U$ such that
\[
\left\{
\begin{array}{rcl}
\psi(y_t, u_t) & \geq & (1- \theta) \psi(y_t, u'_t) + \theta \psi (y_t, u''_t)\\
g(y_t, u_t) & = & (1- \theta) g(y_t, u'_t) + \theta g (y_t, u''_t).
{\epsilon}nd{array}
\right.
\]
{\epsilon}nd{itemize}
\begin{theorem}\label{th91}
Under the assumptions (D1-D7) let $(\hat{\underline{y}}, \hat{\underline{u}})$ be a solution of Problem (P). Then there exists $\underline{p} \in {{\epsilon}ll}^1({\mathbb{N}}_*, {\mathbb{R}}^{n*})$ which satisfies the following properties.
\vskip1mm
\noindent
{\bf (AE)} $D_1 \psi(\hat{y}_t, \hat{u}_t) + p_{t+1} \circ D_1 g(\hat{y}_t, \hat{u}_t) = 0$ for all $t \in {\mathbb{N}}_*$.\\
{\bf (MP)} $ \psi(\hat{y}_t, \hat{u}_t) + \langle p_{t+1}, g(\hat{y}_t, \hat{u}_t) \rangle = \max_{u \in U} ( \psi(\hat{y}_t, u) + \langle p_{t+1}, g(\hat{y}_t,u) \rangle )$ for all $t \in {\mathbb{N}}$.
{\epsilon}nd{theorem}
\begin{proof}
Using Section 2, since $(\hat{\underline{y}}, \hat{\underline{u}})$ is a solution of Problem (P), $(\hat{\underline{x}}, \hat{\underline{u}})$ is a solution of Problem (P1). For all $j \in \{ 1,...,7 \}$ note that (Cj) implies (Dj). Therefore the assumptions of Theorem \ref{th82} are fulfilled, and so its conclusions hold. Using Section 2, we conserve $\hat{\underline{u}}$ and $\underline{p}$, we set $\hat{x}_t = \hat{y}_t - y_{\infty}$ for all $t \in {\mathbb{N}}$, and the translation of (AE1) gives (AE) and the translation of (MP1) gives (MP).
{\epsilon}nd{proof}
\section{Sufficient conditions for (P1)}
In this section we establish a resullt of sufficient condition of optimality which uses the adjoint equation and the weak maximum principle and the concavity of the Hamiltonian with respect the state variable and the control variable.
\begin{theorem}\label{th101}
Let $U$ be a nonempty convex subset of ${\mathbb{R}}^d$, $\beta \in (0,1)$, $\sigma \in {\mathbb{R}}^n$ and two mappings $\phi : {\mathbb{R}}^n \times U \rightarrow {\mathbb{R}}$ and $f : {\mathbb{R}}^n \times U \rightarrow {\mathbb{R}}^n$. \\
Let $(\hat{\underline{x}}, \hat{\underline{u}}) \in c_0({\mathbb{N}}, {\mathbb{R}}^n) \times {{\epsilon}ll}^{\infty}({\mathbb{N}}, U)$ and $\underline{p} \in {{\epsilon}ll}^1({\mathbb{N}}_*, {\mathbb{R}}^{n*})$. Assume that the following conditions hold.
\begin{itemize}
\item[(i)] $\hat{x}_{t+1} = f(\hat{x}_t, \hat{u}_t)$ for all $t \in {\mathbb{N}}$, and $\hat{x}_0 = \sigma$.
\item[(ii)] $\phi \in C^1( {\mathbb{R}}^n \times U , {\mathbb{R}})$ and $f \in C^1({\mathbb{R}}^n \times U , {\mathbb{R}}^n)$.
\item[(iii)] $\phi$ transforms bounded subsets of ${\mathbb{R}}^n \times U$ into bounded subsets of ${\mathbb{R}}$.
\item[(iv)] $p_t = p_{t+1} \circ D_1f(\hat{x}_t, \hat{u}_t) + \beta^t D_1 \phi(\hat{x}_t, \hat{u}_t)$ for all $t \in {\mathbb{N}}_*$.
\item[(v)] $\langle p_{t+1} \circ D_2f(\hat{x}_t, \hat{u}_t) + D_2 \phi(\hat{x}_t, \hat{u}_t) , u - \hat{u}_t \rangle \leq 0$ for all $u \in U$, for all $t \in {\mathbb{N}}$.
\item[(vi)] The function $[(x,u) \mapsto \langle p_{t+1}, f(x,u) \rangle + \beta^t \phi(x,u))]$ is concave on ${\mathbb{R}}^n \times U$ for all $t \in {\mathbb{N}}$.
{\epsilon}nd{itemize}
Then $(\hat{\underline{x}}, \hat{\underline{u}})$ is a solution of (P1).
{\epsilon}nd{theorem}
\begin{proof}
Let $(\underline{x}, \underline{u})$ be an admissible process for (P1), i.e. $\underline{x} \in c_0({\mathbb{N}}, {\mathbb{R}}^n)$, $\underline{u} \in {{\epsilon}ll}^{\infty}({\mathbb{N}}, U)$, $x_{t+1} = f(x_t,u_t)$ for all $t \in {\mathbb{N}}$, and $x_0 = \sigma$. From (iii), since $\{ \phi(x_t,u_t) : t \in {\mathbb{N}} \}$ is bounded, $J(\underline{x}, \underline{u}) = \sum_{t = 0}^{+\infty} \beta^t \phi(x_t,u_t)$ exists in ${\mathbb{R}}$.
From (ii) and (iv) we obtain
\begin{equation}\label{eq10.1}
D_1 H_t(\hat{x}_t, \hat{u}_t, p_{t+1}) = p_t.
{\epsilon}nd{equation}
From (vi) we obtain, for all $t \in {\mathbb{N}}$,
\begin{equation}\label{eq10.3}
\left.
\begin{array}{r}
H_t((\hat{x}_t, \hat{u}_t, p_{t+1}) - H_t(x_t,u_t,p_{t+1}) \\
- \langle D_1 H_t(\hat{x}_t, \hat{u}_t, p_{t+1}), \hat{x}_t - x_t \rangle - \langle D_2 H_t(\hat{x}_t, \hat{u}_t, p_{t+1}), \hat{u}_t - u_t \rangle \geq 0.
{\epsilon}nd{array}
\right\}
{\epsilon}nd{equation}
From (v) the following relation holds for all $t \in {\mathbb{N}}$
\begin{equation}\label{eq10.4}
\langle D_2 H_t(\hat{x}_t, \hat{u}_t, p_{t+1}), \hat{u}_t - u_t \rangle \geq 0.
{\epsilon}nd{equation}
For all $t \in {\mathbb{N}}$ we have
\[
\begin{array}{rcl}
\beta^t \phi (\hat{x}_t, \hat{u}_t) - \beta^t \phi(x_t,u_t) &=& H_t( \hat{x}_t, \hat{u}_t, p_{t+1}) - \langle p_{t+1}, f( \hat{x}_t, \hat{u}_t) \rangle \\
\null & \null & -H_t(x_t,u_t, p_{t+1}) + \langle p_{t+1}, f(x_t,u_t)\\
\null& =& H_t( \hat{x}_t, \hat{u}_t, p_{t+1}) - H_t(x_t,u_t, p_{t+1}) \\
\null & \null & -\langle p_{t+1}, \hat{x}_{t+1} - x_{t+1} \rangle.
{\epsilon}nd{array}
\]
Then, using (\ref{eq10.1}) and (\ref{eq10.4}) we obtain
\[
\begin{array}{rcl}
\beta^t \phi (\hat{x}_t, \hat{u}_t) - \beta^t \phi(x_t,u_t) &\geq & H_t( \hat{x}_t, \hat{u}_t, p_{t+1}) -H_t(x_t,u_t, p_{t+1}) \\
\null & \null & - \langle D_2H_t( \hat{x}_t, \hat{u}_t, p_{t+1}), \hat{u}_t - u_t \rangle\\
\null & \null & - \langle D_1H_{t+1}( \hat{x}_{t+1}, \hat{u}_{t+1}, p_{t+2}), \hat{x}_{t+1} - x_{t+1} \rangle
{\epsilon}nd{array}
\]
which implies
\[
\begin{array}{rcl}
\beta^t \phi (\hat{x}_t, \hat{u}_t) - \beta^t \phi(x_t,u_t) &\geq & [H_t( \hat{x}_t, \hat{u}_t, p_{t+1}) -H_t(x_t,u_t, p_{t+1}) \\
\null & \null & - \langle D_1H_t( \hat{x}_t, \hat{u}_t, p_{t+1}), \hat{x}_t - x_t \rangle\\
\null & \null & - \langle D_2H_t( \hat{x}_t, \hat{u}_t, p_{t+1}), \hat{u}_t - u_t \rangle ] \\
\null & \null & + [ \langle D_1H_t( \hat{x}_t, \hat{u}_t, p_{t+1}), \hat{x}_t - x_t \rangle\\
\null & \null & - \langle D_1H_{t+1}( \hat{x}_{t+1}, \hat{u}_{t+1}, p_{t+2}), \hat{x}_{t+1} - x_{t+1} \rangle ]
{\epsilon}nd{array}
\]
and using (\ref{eq10.3}) we obtain
\[
\begin{array}{rcl}
\beta^t \phi (\hat{x}_t, \hat{u}_t) - \beta^t \phi(x_t,u_t) &\geq &[ \langle D_1H_t( \hat{x}_t, \hat{u}_t, p_{t+1}), \hat{x}_t - x_t \rangle\\
\null & \null & - \langle D_1H_{t+1}( \hat{x}_{t+1}, \hat{u}_{t+1}, p_{t+2}), \hat{x}_{t+1} - x_{t+1} \rangle ]
{\epsilon}nd{array}
\]
Therefore, using (\ref{eq10.1}), we obtain, for all $T \in {\mathbb{N}}_*$,
\[
\begin{array}{rcl}
\sum_{t=0}^T \beta^t \phi (\hat{x}_t, \hat{u}_t) -\sum_{t=0}^T \beta^t \phi(x_t,u_t) &\geq &\langle D_1 H_0(\sigma, \hat{u}_0, p_1), \sigma - \sigma \rangle \\
\null & \null &- \langle p_{T+1}, \hat{x}_{T+1} - x_{T+1} \rangle \Longrightarrow
{\epsilon}nd{array}
\]
\begin{equation}\label{eq10.5}
\sum_{t=0}^T \beta^t \phi (\hat{x}_t, \hat{u}_t) -\sum_{t=0}^T \beta^t \phi(x_t,u_t) \geq - \langle p_{T+1}, \hat{x}_{T+1} - x_{T+1} \rangle.
{\epsilon}nd{equation}
Since $\underline{p} \in {{\epsilon}ll}^1({\mathbb{N}}_*, {\mathbb{R}}^{n*})$, we have $\lim_{T \rightarrow + \infty} p_{T+1} = 0$, and since $\hat{\underline{x}}, \underline{x} \in c_0({\mathbb{N}}, {\mathbb{R}}^n)$ we have
$\lim_{T \rightarrow + \infty} (\hat{x}_{T+1} - x_{T+1}) = 0$ which implies $\lim_{T \rightarrow + \infty}( - \langle p_{T+1}, \hat{x}_{T+1} - x_{T+1} \rangle) = 0$, and then, from
(\ref{eq10.5}), doing $T \rightarrow + \infty$ we obtain $J(\hat{\underline{x}}, \hat{\underline{u}}) - J (\underline{x}, \underline{u}) \geq 0$. And so we have proven that $(\hat{\underline{x}}, \hat{\underline{u}})$ is a solution of (P1).
{\epsilon}nd{proof}
\begin{remark}\label{rem102}
The structure of the previous proof is inspired by the proof of Theorem 5.1 in \cite{BH2}. Note that our assumption (iii) permits to avoid to assume that $U$ is compact. Moreover note that we can replace the assumption (iii) by the condition: $U$ is closed.
{\epsilon}nd{remark}
\begin{remark}\label{rem103}
Note that under our assumptions, the process $(\hat{\underline{x}}, \hat{\underline{u}})$ is also solution of the following problem
\[
\left\{
\begin{array}{cl}
{\rm Mawimize} & \sum_{t=0}^{+ \infty} \beta^t \phi(x_t,u_t)\\
{\rm when}& \underline{x} \in {{\epsilon}ll}^{\infty}({\mathbb{N}}, {\mathbb{R}}^n), \underline{u} \in {{\epsilon}ll}^{\infty}({\mathbb{N}}, U)\\
{\rm and}& \forall t \in {\mathbb{N}}, x_{t+1} = f(x_t,u_t), x_0 = \sigma
{\epsilon}nd{array}
\right.
\]
since, in the previous proof, when we obtain (\ref{eq10.5}), having $\hat{\underline{x}}$ and $\underline{x}$ bounded is sufficient to obtain $\lim_{T \rightarrow + \infty}( - \langle p_{T+1}, \hat{x}_{T+1} - x_{T+1} \rangle) = 0$ and consequently to have the optimality of $(\hat{\underline{x}}, \hat{\underline{u}})$ for the last problem.
{\epsilon}nd{remark}
\section{Sufficient conditions for (P)}
This section is devoted to the translation of the result of sufficient condition of optimality on (P1) into an analogous result on (P).\\
When $y_{\infty} \in {\mathbb{R}}^n$, we denotes by $c_{y_{\infty}}({\mathbb{N}}, {\mathbb{R}}^n)$ the set of the sequences $\underline{y}$ in ${\mathbb{R}}^n$ such that $\lim_{t \rightarrow + \infty} y_t = y_{\infty}$. It is a complete affine subset of ${{\epsilon}ll}^{\infty}({\mathbb{N}}, {\mathbb{R}}^n)$.
\vskip1mm
\begin{theorem}\label{th111}
Let $U$ be a nonempty convex subset of ${\mathbb{R}}^d$, $\beta \in (0,1)$, ${\epsilon}ta, y_{\infty} \in {\mathbb{R}}^n$, and two mappings $\psi : {\mathbb{R}}^n \times U \rightarrow {\mathbb{R}}$ and $g : {\mathbb{R}}^n \times U \rightarrow {\mathbb{R}}^n$. Let $(\hat{\underline{y}}, \hat{\underline{u}}) \in c_{y_{\infty}}({\mathbb{N}}, {\mathbb{R}}^n) \times {{\epsilon}ll}^{\infty}({\mathbb{N}}, U)$ and $\underline{p} \in {{\epsilon}ll}^1({\mathbb{N}}_*, {\mathbb{R}}^{n*})$ which satisfy the following conditions.
\begin{itemize}
\item[(i)] For all $t \in {\mathbb{N}}$, $\hat{y}_{t+1} = g(\hat{y}_t, \hat{u}_t)$, and $\hat{y}_0 = {\epsilon}ta$.
\item[(ii)] $\psi \in C^1({\mathbb{R}}^n \times U, {\mathbb{R}})$ and $g \in C^1({\mathbb{R}}^n \times U, {\mathbb{R}}^n)$.
\item[(iii)] $\psi$ transforms bounded subsets of ${\mathbb{R}}^n \times U$ into bounded subsets of ${\mathbb{R}}$.
\item[(v)] $p_t = p_{t+1} \circ D_1g(\hat{y}_t, \hat{u}_t) + \beta^t D_1 \psi (\hat{y}_t, \hat{u}_t)$ for all $t \in {\mathbb{N}}_*$.
\item[(vi)] $\langle p_{t+1} \circ D_2 g(\hat{y}_t, \hat{u}_t) + \beta^t D_2 \psi(\hat{y}_t, \hat{u}_t) , u - \hat{u} \rangle \leq 0$ for all $u \in U$, for all $t \in {\mathbb{N}}$
\item[(vii)] The function $[(y,u) \mapsto \langle p_{t+1}, g(y, u) \rangle + \beta^t \psi (y,u)]$ is concave on ${\mathbb{R}}^n \times U$ for all $t \in {\mathbb{N}}$.
{\epsilon}nd{itemize}
Then $(\hat{\underline{y}}, \hat{\underline{u}})$ is a solution of (P).
{\epsilon}nd{theorem}
\begin{proof}
Using Section 2, $\hat{x}_t = \hat{y}_t - y_{\infty}$ for all $t \in {\mathbb{N}}$, we see that $(\hat{\underline{x}}, \hat{\underline{u}}) \in c_0({\mathbb{N}}, {\mathbb{R}}^n) \times {{\epsilon}ll}^{\infty}({\mathbb{N}}, U)$ satisfies all the assumptions of Theorem \ref{th101}. And so $(\hat{\underline{x}}, \hat{\underline{u}})$ is a solution of (P1) which implies that $(\hat{\underline{y}}, \hat{\underline{u}})$ is a solution of (P).
{\epsilon}nd{proof}
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{\epsilon}nd{document}
|
\begin{document}
\title{Analytic families of eigenfunctions\\
on a reductive symmetric space}
\author{E.P. van den Ban and H. Schlichtkrull}
\date{}
\maketitle
\begin{abstract}
The asymptotic behavior of holomorphic families of generalized
eigenfunctions on a reductive symmetric space is studied.
The family parameter is a complex character on the split component of a
parabolic subgroup. The main result asserts
that the family vanishes if a particular asymptotic coefficient
does. This allows an induction of relations between families
that will be applied in forthcoming
work on the Plancherel and the Paley-Wiener theorem.
\end{abstract}
\tableofcontents
\section*{Introduction}
In harmonic analysis on a reductive symmetric space
${\rm X}$ an important role is played
by families of generalized eigenfunctions for the algebra
${\msy D}X$ of invariant differential
operators. Such families arise for instance as matrix coefficients of
representations
that come in series, such as the (generalized) principal series.
In particular, relations between such families are of great interest.
We recall that a real reductive group $G,$ equipped with the left times
right multiplication action, is a reductive symmetric space.
In the case of the group, examples of the mentioned relations are
functional equations for Eisenstein integrals,
see \bib{HCEis} and \bib{HCIII}, or Arthur-Campoli relations
for Eisenstein integrals, see \bib{Arthur}, \bib{Campoli}.
In this paper we develop a general tool to
establish relations of this kind.
We show that they can be derived from similar relations satisfied
by the family of functions obtained by taking
one particular coefficient in a
certain asymptotic expansion. Since the functions in the
family so obtained are eigenfunctions on symmetric spaces of lower split rank,
this yields a powerful inductive method; we call it
{\em induction of relations.} In the case of the group, a closely related
lifting theorem by Casselman was used by Arthur in the proof of
the Paley-Wiener theorem, see \bib{Arthur}, Thm.\ II.4.1. However,
no proof seems yet to have appeared of Casselman's theorem.
The tools developed in this paper are used in \bib{BSfi},
and they will also be applied
in the forthcoming papers \bib{BSpl} and \bib{BSpw}.
For example, it is the induction
of relations that allows us to establish symmetry properties
of certain integral kernels appearing in a Fourier inversion formula in
\bib{BSfi}. Also in \bib{BSfi}, the induction of relations is used
to define generalized Eisenstein integrals corresponding to
non-minimal principal series. In \bib{BSpl}, the results of this paper will
be applied to identify these `formal' Eisenstein integrals
with those defined in Delorme \bib{DelEis}. This is a key
step towards the Plancherel decomposition.
The results will also be applied to
establish functional equations for the Eisenstein integrals.
Applied in this manner our technique serves as a replacement
for the use of the Maass-Selberg relations as in Harish-Chandra \bib{HCIII}
and \bib{DelEis}. On the other hand, in \bib{BSpw}
we apply our tool to show that Arthur-Campoli relations
satisfied by normalized Eisenstein integrals of spaces of lower split
rank induce similar relations for normalized
Eisenstein integrals of ${\rm X}.$ This result is then used to prove
a Paley-Wiener theorem for ${\rm X}$ that generalizes Arthur's theorem
for the group. In particular, the missing proof of Casselman's
theorem will then be circumvented by means of a technique of
the present paper.
It should be mentioned that in the case of
the group, induction of Arthur-Campoli relations for unnormalized
Eisenstein integrals is easily derived from their
integral representations (see \bib{Arthur}, p.\ 77, proof of Lemma 2.3).
For normalized Eisenstein integrals, which are
not representable by integrals, the result seems to be much
deeper, also in the group case.
One of the interesting
features of the theory is that it also deals with families of functions
that are not necessarily globally defined on the space
${\rm X}$ but on a suitable open dense subset.
Asymptotic behavior of eigenfunctions on a symmetric space has been
studied at many other places in the literature. The following
papers hold results that are related to some of the ideas
of the present paper
\bib{HCsf}, \bib{Gang}, \bib{TroVar}, \bib{HCI}, \bib{HCIII}, \bib{Helg},
\bib{KKMOOT}, \bib{OS}, \bib{CM},
\bib{Wal}, \bib{Arthur}, \bib{Osh}, \bib{BS}, \bib{Carmona}.
The core results of this paper were found and announced
in the fall of 1995, when both authors were guests at the
Mittag-Leffler Institute. In the same period Delorme announced
his proof of the Plancherel
theorem, which has now appeared in \bib{Delpl}.
We shall now explain the contents of this paper in more detail.
The space ${\rm X}$ is of the form $G/H,$ with $G$ a real reductive
Lie group of Harish-Chandra's class, and $H$ an open subgroup
of the set of fixed points for an involution $\sigma$ of~$G.$
The group $G$ has a $\sigma$-stable maximal compact subgroup $K,$
let ${\msy C}artan$ be the associated Cartan
involution of $G$.
Let $P_0 = M_0 A_0 N_0$ be a fixed minimal
$\sigma \,{\scriptstyle\circ}\, {\msy C}artan$-invariant parabolic subgroup of $G,$
with the indicated Langlands decomposition.
The Lie algebra
${\mathfrak a}_0$ of $A_0$ is invariant
under the infinitesimal involution
$\sigma;$ we denote the associated $-1$ eigenspace
in ${\mathfrak a}_0$ by ${\mathfrak a}q.$ Its dimension is called
the split rank of ${\rm X}.$
Let $A_\iq$ be
the vectorial subgroup of $G$ with Lie algebra ${\mathfrak a}q$
and let $A_\iq^{\rm reg}$ be the set of regular points relative to the adjoint
action of $A_\iq$ in ${\mathfrak g}$. Then ${\rm X}p\mathchar"303A= K A_\iq^{\rm reg} H$ is a
$K$-invariant dense open subset of ${\rm X}$. Let
$A_\iq^+$
be the open chamber in $A_\iq$ determined by $P_0.$
Then
${\rm X}p$ is a finite union of disjoint sets of the form
$KA_\iq^+ vH,$ with $v$ in the normalizer of ${\mathfrak a}q$ in $K.$
In this introduction we assume, for simplicity of exposition,
that ${\rm X}p = KA_\iq^+ H.$ This assumption is actually fulfilled in the
case that ${\rm X}$ is a group.
Let $(\tau, V_\tau)$ be a finite
dimensional continuous representation of $K.$
Then by ${\msy C}i({\rm X}p \,:\, \tau)$
we denote the space of smooth functions
$f: {\rm X}p \rightarrow V_\tau$ that are $\tau$-spherical,
i.e., $f(kx) = \tau(k)f(x),$ for all $x \in {\rm X}p$ and $k \in K.$
Let $\cP_\gs$ denote the (finite) set of $\sigma\,{\scriptstyle\circ}\, {\msy C}artan$-invariant
parabolic subgroups of $G$ containing $A_\iq.$
Let $Q = M_Q A_Q N_Q$ be an element of $\cP_\gs.$
Then $\sigma$ restricts to an involution
of ${\mathfrak a}Q,$ the Lie algebra of $A_Q;$ we denote its $-1$ eigenspace
by ${\mathfrak a}Qq.$
In the first part of the paper we study
a family $f$ of the following type (cf.\ Definition
\ref{d: anfamQY newer}).
The family is a smooth map of the form
$$
f: \Omega \times {\rm X}p \rightarrow V_\tau,
$$
with $\Omega$ an open subset of ${\mathfrak a}Qqdc,$
the complexified linear dual of ${\mathfrak a}Qq.$ It is assumed
that $f$ is holomorphic in its first variable.
Moreover, for every $\lambda \in \Omega$ the function
$f_\lambda:= f(\lambda, \,\cdot\,)$ belongs
to ${\msy C}i({\rm X}p \,:\, \tau).$ It is furthermore assumed that the functions
$f_\lambda$ allow suitable exponential polynomial
expansions along $A_\iq^+.$ More precisely, we assume,
for $m \in M_0$ and $a \in A_\iq^+,$ that
\begin{equation}
\label{e: prototype expansion}
f_\lambda (ma) = \sum_{s \in W/W_Q}
a^{s\lambda - \rho_{P_0}} \sum_{\xi \in - sW_QY + {\msy N}\Sigma(P_0)} a^{-\xi} q_{s,\xi}
(\lambda, \log a, m).
\end{equation}
Here $W$ is the Weyl group of $\Sigma = \Sigma({\mathfrak g}, {\mathfrak a}q)$ and
$W_Q$ is the centralizer of ${\mathfrak a}Qq$ in $W.$
Moreover, $\Sigma(P_0)$ denotes the collection of
roots from $\Sigma$ occurring
in $N_0$ and $Y$ is a finite subset of ${}^*\fa_{Q\iq}dc,$ the annihilator
of ${\mathfrak a}Qq$ in ${\mathfrak a}qdc.$
Finally, the
$q_{s, \xi}$ are smooth functions, holomorphic in the
first and polynomial in the second variable. Thus, we impose a
limitation on the set of exponents and assume that the
coefficients depend holomorphically on the parameter~$\lambda.$
The type of convergence that we impose on the expansion
(\ref{e: prototype expansion}) is described in general terms in the
preliminary Section \ref{s: exp pol series}.
We show that the functions
$f_\lambda$ actually allow exponential polynomial expansions
similar to (\ref{e: prototype expansion}) along
any (possibly non-minimal) $P \in \cP_\gs.$
These expansions are investigated in detail in Sections \ref{s: asymp walls}
and \ref{s: analytic families}. Their coefficients are families of
$\tau|_{M_P\cap K}$-spherical functions on ${\rm X}Pp,$ the analogue of
${\rm X}p$ for
the lower split rank symmetric space ${\rm X}P:= M_P/M_P \cap H.$
The operators from ${\msy D}X$ do also allow expansions along every
$P\in \cP_\gs.$
In Section \ref{s: diff op along walls}
this is shown by investigating a radial decomposition that
reflects the decomposition
$G = K M_P A_{Pq} H.$
It is of importance that the coefficients in these expansions
are globally defined smooth functions on $M_P,$
see Prop.\ \ref{p: radial deco with MQ}
and Cor.\ \ref{c: cor on ringQ}.
{}From the expansions we derive that the algebra ${\msy D}X$ acts on the space
of families of the above type, see Prop.\ \ref{p: D on families}.
In Section \ref{s: asymptotic globality} we
introduce the notion of asymptotic
$s$-globality of a family along $P.$
Losely speaking, it means that the coefficients $q_{s, \xi}(\lambda, \log a , \,\cdot\,)$ of the expansion along $P$
extend smoothly from ${\rm X}Pp$ to the full space ${\rm X}P,$ for every
$\xi \in (sW_Q Y - {\msy N}\Sigma(P))|_{{\mathfrak a}Pq}.$ This notion is proved to be stable under the action of ${\msy D}X.$
In Section \ref{s: vanishing thm} we impose three other conditions on the family.
The first is that each
member satisfies a system of differential equations of the form
$$
Df_\lambda = 0\quad\quad (D \in I_{\gd, \gl}).
$$
Here $I_{\gd, \gl}$ is a certain cofinite ideal in the algebra ${\msy D}X$
depending polynomially on $\lambda\in {\mathfrak a}Qqdc$ in a suitable way.
Accordingly, $\lambda$ is called
the spectral parameter of the family.
The second condition imposed is a suitable condition of asymptotic globality along certain
parabolic subgroups $P$ with $\dim({\mathfrak a}q/{\mathfrak a}Pq) =1.$
Thirdly, it is required that the domain $\Omega$ for the
parameter $\lambda$ is unbounded in certain directions
(see Defn.~\ref{d: Q-distinguished}).
The first main result of the paper is then the following vanishing theorem, see
Theorem \ref{t: vanishing theorem new}.
\medbreak\noindent
{\bf The vanishing theorem.\ }\sl
Let $f$ be a family as above, and assume that the coefficient of $\lambda - \rho_Q$
in the expansion along $Q$ vanishes for $\lambda$ in a non-empty open subset of $\Omega.$
Then the family $f$ is identically zero.\rm
\medbreak
In the proof the globality assumption is needed to link suitably many asymptotic coefficients
together; the vanishing of one of them then inductively causes the vanishing of others.
In the induction step a key role is played by the observation that a symmetric space
cannot have a continuum of discrete series.
The importance of the vanishing theorem is that it applies to
many families that naturally
arise in representation theory.
In the present paper we show that this is so for Eisenstein integrals
associated with the minimal principal series for ${\rm X};$ in \bib{BSpl}
we will show that Eisenstein integrals obtained by parabolic induction from discrete
series form a family of the above type. The idea is that the latter Eisenstein
integrals can be obtained from those associated with the minimal principal
series by the application of residual operators with respect
to the spectral parameter. Such residual operators occur
in our papers \bib{BSres} and \bib{BSfi}.
A suitable class of operators
containing the residual operators is formed by the Laurent operators.
In the second half of the paper we study the application of
them
to suitable families
of eigenfunctions, with respect to the spectral parameter. The Laurent operators
are best described by means of Laurent functionals, see Sections \ref{s: Laurent functionals}
and \ref{s: Laurent operators}.
In Section
\ref{s: spec fam new} we introduce a special type of families
$g$ of eigenfunctions.
It is of the above type, with $\Omega$ dense in ${\mathfrak a}Pqdc,$
$P$ a minimal parabolic subgroup in $\cP_\gs,$ and satisfies some additional
requirements, see Definition \ref{d: defi cEhyp Q Y gd}. One of these is that
the family and its asymptotic expansions should depend meromorphically on the spectral
parameter $\lambda \in {\mathfrak a}Pqdc$ with singularities along translated root hyperplanes.
This allows the application of Laurent functionals with respect to the spectral parameter.
More precisely, let $Q \in \cP_\gs$ contain $P,$
and let $\cL$ be a Laurent functional on ${}^*\fa_{Q\iq}dc.$
{}From the family $g$
a new family $f = {\cal L}_* g,$ with a spectral
parameter from ${\mathfrak a}Qqdc,$
is obtained by the application of
${\cal L}$ to the ${}^*\fa_{Q\iq}dc$-component
of the spectral parameter.
In Theorem \ref{t: source of functions by Lau new} it is shown that
the resulting family ${\cal L}_* g$ satisfies the requirements
of the vanishing theorem, provided the special family $g$ satisfies certain holomorphic asymptotic
globality conditions.
In Section \ref{s: partial Eisenstein integrals} we introduce partial Eisenstein
integrals associated with a minimal parabolic subgroup $P$ from $\cP_\gs.$
The partial Eisenstein integrals
are spherical generalized eigenfunctions on ${\rm X}p$ obtained from the normalized
Eisenstein integral ${E^\circ}(P\,:\, \lambda),$ ($\lambda \in {\mathfrak a}qdc$ generic), by splitting it
according to its exponential polynomial expansion along $P.$
More precisely, the exponents
of ${E^\circ}(P\,:\, \lambda)$ are contained in $W\lambda - \rho_P -{\msy N}\Sigma(P);$ the partial Eisenstein
integrals $E_{+,s}(P\,:\, \lambda),$ for $s \in W,$ are the smooth spherical functions on ${\rm X}p$
determined by the requirements that
$$
{E^\circ}(P\,:\, \lambda) = \sum_{s \in W} E_{+,s}(P\,:\, \lambda)
$$
and the set of exponents of $E_{+,s}(P \,:\, \lambda)$ along $P$ should be contained in
$s\lambda - \rho_P -{\msy N}\Sigma(P).$
It is then shown that the partial Eisenstein integrals yield examples
of the special families mentioned above. Moreover, if $Q\in \cP_\gs,$ $Q\supset P,$
let $W^Q$ be the collection of minimal length
(with respect to $\Sigma(P)$) coset representatives for $W/W_Q$ in $W.$
Then it is shown that for each $t\in W_Q$ the family
\begin{equation}
\label{e: g as sum of partial Eis}
f_t = \sum_{s \in W^Q} E_{+,st}(P\,:\, \,\cdot\,)
\end{equation}
satisfies the additional holomorphic asympotic globality property
guaranteeing that ${\cal L}_* f_t$ satisfies the hypothesis of
the vanishing theorem, for $\cL$ a Laurent
functional on ${}^*\fa_{Q\iq}dc.$
In Section \ref{s: asymptotics of partial Eisenstein integrals}
the asymptotic behavior of ${\cal L}_* f_t$ is investigated, and the
coefficient of $a^{\lambda - \rho_Q}$ in the expansion along $Q$ is expressed
in terms of partial Eisenstein integrals of ${\rm X}Q.$
The above preparations pave the way for the induction of relations
in Section \ref{s: induction of relations}. The idea is as follows.
Let $f_t$ be the family defined by
(\ref{e: g as sum of partial Eis}), and
let a Laurent functional ${\cal L}_t$ on ${}^*\fa_{Q\iq}dc$ be given for each $t\in W_Q$.
Then by the vanishing theorem
a relation of the form $\sum_t {\cal L}_t f_t=0$ is valid if a similar
relation is valid for the $(\lambda - \rho_Q)$-coefficients along $Q;$
this in turn may be expressed as a similar relation between partial Eisenstein
integrals for the lower split rank space ${\rm X}Q.$
In this setting, taking the $(\lambda -\rho_Q)$-coefficient along $Q$
essentially inverts the procedure
of parabolic induction from $Q$ to $G.$ This motivaties our choice
of terminology. The precise result is formulated in Theorem
\ref{t: induction of relations, new}. An equivalent result, closer
to the formulation of Casselman's theorem in \bib{Arthur} is stated
at the end of the section.
\section{Exponential polynomial series}
\label{s: exp pol series}
Let $A$ be a vectorial group and ${\mathfrak a}$
its Lie algebra. The exponential map
$\exp: {\mathfrak a} \rightarrow A$ is a diffeomorphism;
we denote its inverse by $\log.$
If $\xi$ belongs to ${\mathfrak a}dc,$ the complexified linear dual of ${\mathfrak a},$
then we define the function
$e^\xi: a :to a^\xi$ on $A$ by $a^\xi = e^{\xi(\log a)}.$
Let $P({\mathfrak a})$ denote the algebra of
polynomial functions ${\mathfrak a} \rightarrow {\msy C}.$ If $d \in {\msy N},$ let
$P_d({\mathfrak a})$ denote the (finite dimensional) subspace
of polynomials of degree at most $d.$ Let ${\msy D}elta$ be
a set of linearly independent vectors in ${\mathfrak a}$
(we do not require
this set to span ${\mathfrak a}$).
We put
$$
{\mathfrak a}^+ = {\mathfrak a}^+({\msy D}elta) := \{ X \in {\mathfrak a} \mid \alpha(X) > 0, \;\;\;\forall\;\alpha \in {\msy D}elta\},
$$
and $A^+ = A^+({\msy D}elta) = \exp ({\mathfrak a}^+).$
We define ${\msy N} \Delta$ to be the ${\msy N}$-span of $\Delta;$ if $\Delta = \emptyset$ then
${\msy N}\Delta = \{0\}.$
Moreover, if $X$ is a subset of ${\mathfrak a}dc,$ we denote by $X- {\msy N} \Delta$ the
vectorial sum of $X$ and ${\msy N} \Delta.$
Let $V$ be a complete locally convex space; here and in the following we
will always assume such a space to be Hausdorff.
If $\xi \in {\mathfrak a}dc,$ then by a $V$-valued $\xi$-exponential polynomial function on $A$
we mean a function $A \rightarrow V$ of the form $a :to a^\xi q(\log a),$ with
$q \in P({\mathfrak a})\otimes V.$
\begin{defi}
\label{d: exp pol series}
By a ${\msy D}elta$-exponential polynomial series on $A$
with coefficients in $V$ we mean a formal series $F$
of exponential polynomial functions of the form
\begin{equation}
\label{e: intro exp pol series}
\sum_{\xi \in {\mathfrak a}dc} a^\xi \,q_\xi (\log a),
\end{equation}
with $\xi :to q_\xi$ a map ${\mathfrak a}dc \rightarrow P({\mathfrak a}) \otimes V,$ such that
\begin{enumerate}
\item[{\rm (a)}]
there exists a finite subset $X \subset {\mathfrak a}dc$ such that $q_\xi = 0$ for $\xi \notin X - {\msy N} \Delta;$
\minspace\item[{\rm (b)}]
there exists a constant $d \in {\msy N}$ such that $q_\xi \in P_d({\mathfrak a}) \otimes V$
for all $\xi\in {\mathfrak a}dc.$
\end{enumerate}
The smallest $d \in {\msy N}$ with property (b) will be called the polynomial degree of the series;
this number is denoted by $\deg(F).$
The collection of all $\Delta$-exponential polynomial series
with coefficients in $V$ is denoted by
${\cal F}ep(A,V) = {\cal F}ep_\Delta(A,V).$
\end{defi}
If $F \in {\cal F}ep(A,V)$ is an expansion of the form (\ref{e: intro exp pol series})
then, for every $\xi \in {\mathfrak a}dc,$
we write $q_\xi(F)$ for $q_\xi.$ Moreover, we write $q(F)$ for the map
$\xi :to q_\xi(F)$
from ${\mathfrak a}dc$ to $P_d({\mathfrak a})\otimes V.$ Then
$F :to q(F)$
defines a bijection from ${\cal F}ep(A,V)$ onto a linear subspace of
$(P_d({\mathfrak a})\otimes V)^{{\mathfrak a}dc},$ the space of maps ${\mathfrak a}qdc \rightarrow P_d({\mathfrak a}) \otimes V.$
Via this bijection we equip ${\cal F}ep(A,V)$ with the structure of a linear space.
If $F \in {\cal F}ep(A,V),$ then
$$
{\rm Exp}(F): = \{ \xi \in {\mathfrak a}dc \mid q_\xi(F) \not= 0 \}
$$
is called the set
of exponents of $F.$ If $F_1, F_2 \in {\cal F}ep(A,V),$
we call $F_1$ a subseries of $F_2$ if
$q_\xi(F_2) = q_\xi(F_1)$
for all $\xi \in {\rm Exp}(F_1).$
The series (\ref{e: intro exp pol series})
is said to converge absolutely
in a fixed point $a_0 \in A$
if the series
$$
\sum_{\xi \in {\rm Exp}(F)} a_0^{\xi} q_\xi(\log a_0)
$$
with coefficients in $V$
converges absolutely. It is said to converge absolutely on a subset $\Omega \subset A$
if it converges absolutely in every point $a_0 \in \Omega.$ In this case
pointwise summation of the series defines a function $\Omega \rightarrow V.$
We will also need a more special type of convergence for the series
(\ref{e: intro exp pol series}).
\begin{defi}
\label{d: neat convergence}
The series (\ref{e: intro exp pol series})
is said to converge neatly at a fixed point $a_0 \in A$
if for every continuous seminorm $s$ on $P_d({\mathfrak a}) \otimes V,$ where $d = \deg(F),$
the series
$$
\sum_{\xi \in {\rm Exp}(F)} s(q_\xi) a_0^{{\msy R}e \xi}
$$
converges.
The series (\ref{e: intro exp pol series}) is said to converge neatly on a
subset $\Omega$ of $A$ if it converges neatly at every
point of $\Omega.$
\end{defi}
\begin{rem}
If the series (\ref{e: intro exp pol series}) converges neatly at a point
$a_0 \in A,$ then so does every subseries. Moreover, neat convergence at $a_0$
implies absolute convergence in $a_0.$ However, we should warn the reader
that neat convergence at $a_0$ cannot be seen from the
series with coefficients in $V$ arising from (\ref{e: intro exp pol series}) by evaluation
of its terms at $a = a_0,$ since this type of convergence involves
the global behavior of the polynomials $q_\xi.$ In particular, it is
possible that the series (\ref{e: intro exp pol series})
does not converge neatly at $a_0,$ whereas
its evaluation in $a_0$ is identically zero.
The motivation for the definition of neat convergence is provided
later by Lemmas \ref{l: neat conv of exp pol series} and
\ref{l: formal application of Ufa}, which
express that neat convergence of the series (\ref{e: intro exp pol series})
on an open subset $\Omega \subset A$ guarantees that
(a) the function $f: \Omega \rightarrow V$
defined by (\ref{e: intro exp pol series})
is real analytic on $\Omega;$ (b) its derivatives
are given by series obtained by termwise differentiation
from (\ref{e: intro exp pol series}).
\end{rem}
By a ${\msy D}elta$-power series on $A,$ with coefficients
in $V,$
we mean a $\Delta$-exponential polynomial series $F$ with $\deg F = 0$
and ${\rm Exp}(F) \subset - {\msy N} \Delta,$ i.e.,
\begin{equation}
\label{e: Delta power series}
F = \sum_{\xi \in -{\msy N}{\msy D}elta} a^\xi c_\xi,
\end{equation}
with $c_\xi \in V,$ for $\xi \in -{\msy N}{\msy D}elta.$
Note that for a ${\msy D}elta$-power series the notion of neat convergence
at a point $a_0 \in A$
coincides with the notion of absolute convergence in the point $a_0.$
The terminology `power series' is motivated
by the following consideration. If $\mu \in {\msy N}\Delta,$ we put $\mu = \sum_{\alpha \in \Delta} \mu_\alpha \alpha,$
with $\mu_\alpha \in {\msy N}.$
For $z \in {\msy C}^{{\msy D}elta},$ we write
$$
z^\mu = {\rm pr}od_{\alpha \in {\msy D}elta} z_\alpha^{\mu_\alpha}.
$$
Finally, to the series (\ref{e: Delta power series}) we associate the power series
\begin{equation}
\label{e: power series in z}
\sum_{\mu \in {\msy N} {\msy D}elta} z^\mu c_{-\mu}
\end{equation}
with coefficients in $V.$
Let ${\underline z}: A \rightarrow {\msy C}^{\msy D}elta$ be the map defined by
${\underline z}(a)_\alpha = a^{-\alpha}.$ Then it is obvious that the series (\ref{e: Delta power series})
converges with sum $S$ for $a = a_0$ if and only if the power series
(\ref{e: power series in z}) converges with sum $S$ for $z = {\underline z}(a_0).$
If $r \in \,]\,0,\infty\,[^{\msy D}elta$ we write $D(0,r)$ for the polydisc in ${\msy C}^{\msy D}elta$
consisting of the points $z$ with $|z_\alpha| < r_\alpha$ for all $\alpha \in {\msy D}elta.$
Note that the preimage of this set in $A$ under the map ${\underline z}$ is given
by
$$
A^+({\msy D}elta, r) := \{ a \in A \mid a^{-\alpha} < r_\alpha, \;\; \forall \; \alpha \in {\msy D}elta\}.
$$
If $R >0,$ we also agree to write $A^+(\Delta, R)$ for $A^+(\Delta,r)$ with $r$ defined
by $r_\alpha = R$ for all $\alpha \in \Delta.$
Finally, if $a_0 \in A,$ we write $A^+(\Delta,a_0) := A^+(\Delta, {\underline z}(a_0)).$ Thus,
\begin{equation}
\label{e: Ap Delta a zero}
A^+({\msy D}elta, a_0) := \{ a \in A \mid a^{\alpha} > a_0^\alpha, \;\; \forall \; \alpha \in {\msy D}elta\}
= A^+ a_0.
\end{equation}
We now note that if (\ref{e: Delta power series}) converges absolutely for $a=a_0,$
then the power series (\ref{e: power series in z})
converges absolutely for $z = z(a_0),$ hence uniformly absolutely
on the closure of the polydisc $D(0,{\underline z}(a_0)).$ It follows that the series
(\ref{e: Delta power series}) then converges uniformly absolutely on
the closure of $A^+(\Delta, a_0).$
Let $a_0 \in A.$ By ${\cal O}(A^+({\msy D}elta, a_0), V)$ we denote the space
of functions $f: A^+({\msy D}elta, a_0) \rightarrow V$ that are given by an absolutely converging
series of the form (\ref{e: Delta power series}).
For such a function the associated power series (\ref{e: power series in z})
converges absolutely on the polydisc $D(0,r),$ with $r = {\underline z}(a_0);$
let $\tilde f: D(0,r) \rightarrow V$ be the holomorphic function defined by it.
Then obviously
$$
f(a) = \tilde f({\underline z}(a)),\quad\quad (a \in A^+(\Delta, a_0)).
$$
We see that the $\Delta$-power series representing $f \in {\cal O}(A^+(\Delta,a_0)$ is
unique. Moreover, let ${\cal O}(D(0,r), V)$ denote the space of holomorphic
functions $D(0,r) \rightarrow V,$ then it follows that the map
$$
f :to \tilde f,\quad {\cal O}(A^+(\Delta, r),V) \rightarrow {\cal O}(D(0,r),V)
$$
is a linear isomorphism.
In particular it follows that every $f\in {\cal O}(A^+(\Delta,r),V)$
is real analytic on $A^+(\Delta, r).$ Moreover, its $\Delta$-power series
converges uniformly absolutely on every set of the form
$A^+(\Delta, \rho),$ where $\rho \in \,]\,0, \infty\,[^\Delta,$ $\rho_\alpha < r_\alpha$
for all $\alpha \in \Delta.$
If $\mathfrak v$ is a real linear space, then by $S(\mathfrak v)$ we denote the symmetric
algebra of its complexification $\mathfrak v_{\scriptscriptstyle \C}.$
Via the right regular
action we identify $S({\mathfrak a})$ with the algebra of invariant differential
operators on $A.$ If $f\in {\cal O}(A^+(\Delta,r),V)$ and $u \in S({\mathfrak a}),$
then $uf$ belongs to ${\cal O}(A^+(\Delta, r), V)$
again; its series may be obtained from the series of $f$ by termwise application of
$u.$
We now return to the more general exponential polynomial series
(\ref{e: intro exp pol series})
with
coefficients in $V.$
Let $d {\rm ep}silonq \deg(F).$
Fix a basis $\Lambda$ of ${\mathfrak a}.$ For $m \in {\msy N}\Lambda$ we write
$m = \sum_{\lambda \in \Lambda} m_\lambda \lambda$ and $|m| = \sum_{\lambda} m_\lambda.$
For such $m$ we define the polynomial function
$X :to X^m$ on ${\mathfrak a}$ by
$$
X^m = {\rm pr}od_{\lambda \in \Lambda} \lambda(X)^{m_\lambda}.
$$
These polynomial functions with $|m|\leq d$ constitute a basis
for $P_d({\mathfrak a}).$ Accordingly, we may write:
\begin{equation}
\label{e: q and coefficients}
q_\xi(X) = \sum_{|m| \leq d} X^m c_{\xi,m},
\end{equation}
with $c_{\xi,m} \in V.$
\begin{lemma}
\label{l: neat convergence each m}
The series (\ref{e: intro exp pol series})
converges neatly
on a set $\Omega\subset A$ if and only if for every $m \in {\msy N}\Lambda$
with $|m| \leq d$ the series
$$
\sum_{\xi \in {\rm Exp}(F)} a^\xi c_{\xi,m}
$$
with coefficients in $V$
converges absolutely
for all $a \in \Omega.$
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
This is a straightforward consequence of the definition of neat
convergence and the finite dimensionality of the space $P_d({\mathfrak a}).$
~
$\square$\medbreak\noindent\medbreak
We define a partial ordering ${\rm pr}eceq_{\msy D}elta$ on ${\mathfrak a}dc$
by
\begin{equation}
\label{e: partial ordering preceq gD}
\xi_1 {\rm pr}ecD \xi_2 \iff \xi_2 - \xi_1 \in {\msy N} {\msy D}elta.
\end{equation}
Moreover, we define the relation of ${\msy D}elta$-integral equivalence on ${\mathfrak a}dc$ by
$$
\xi_1 \sim_\Delta \xi_2 \iff \xi_2 - \xi_1 \in {\msy Z}{\msy D}elta.
$$
Let $F \in {\cal F}ep(A,V)$ be as in (\ref{e: intro exp pol series}) and
have polynomial degree at most $d.$
In view of condition (a) of Definition \ref{d: exp pol series},
the restriction of the relation $\sim_\Delta$ to the set ${\rm Exp}(F)$ induces a
finite partition of it. Every class $\omega$ in this
partition has a least ${\rm pr}ecD$-upper bound $s(\omega)$ in ${\mathfrak a}dc.$
Let $S = S_F$ be the set of these upper bounds.
For every $s \in S$ and every $m \in {\msy N}\Lambda$ with $|m| \leq d$
we define the $\Delta$-power series
\begin{equation}
\label{e: series for f s m}
f_{s,m}(a) = \sum_{\mu \in {\msy N} {\msy D}elta} a^{- \mu} c_{s -\mu ,m},
\end{equation}
with coefficients determined by (\ref{e: q and coefficients}).
\begin{lemma}
\label{l: neat conv of exp pol series}
Let the series (\ref{e: intro exp pol series}) be
neatly convergent at the point $a_0 \in A.$
Then the series (\ref{e: intro exp pol series}) and,
for every
$s \in S= S_F$ and $m \in {\msy N}\Lambda$ with $|m|\leq d,$ the series
(\ref{e: series for f s m}) is neatly convergent
on the closure of the set $A^+(\Delta, a_0).$
The functions $f_{s,m},$ defined by (\ref{e: series for f s m}),
belong to ${\cal O}(A^+(\Delta, a_0), V).$ Moreover, let $f: A^+(\Delta, a_0) \rightarrow V$ be the function
defined by the summation of (\ref{e: intro exp pol series}). Then
\begin{equation}
\label{e: f as sum fsm}
f(a) = \sum_{ s \in S \atop |m| \leq d }
a^s (\log a)^m f_{s,m}(a),\quad\quad
(a \in A^+({\msy D}elta, a_0)).
\end{equation}
In particular, the function $f: A^+({\msy D}elta,a_0) \rightarrow V$ is real analytic.
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
{}From the neat convergence of (\ref{e: intro exp pol series}) at $a_0$
it follows by Lemma \ref{l: neat convergence each m} that for every
$s$ and $m$ the series $\sum_{\mu \in {\msy N}{\msy D}elta} a^{s -\mu}c_{s - \mu, m}$
converges absolutely for $a = a_0.$ This implies that
the ${\msy D}elta$-power series (\ref{e: series for f s m}) converges
absolutely for $a = a_0.$ Hence it converges (uniformly) absolutely
on the closure of $A^+(\Delta, a_0);$ in particular it converges neatly on that set.
It follows from this that $f_{s,m} \in {\cal O}(A^+(\Delta, a_0),V),$ for $s \in S$ and $m \in {\msy N} \Lambda$ with
$|m| \leq d.$ Moreover,
\begin{equation}
\label{e: series for a s log a m times f s m}
a^s (\log a)^m \,f_{s,m} = \sum_{\xi \in s - {\msy N}\Delta} a^\xi (\log a)^m c_{\xi, m}
\end{equation}
where the $\Delta$-exponential polynomial series on the right-hand side converges
neatly on the closure of $A^+(\Delta, a_0).$ The
series (\ref{e: series for a s log a m times f s m}), for $s \in S$ and
$m\in {\msy N} \Lambda$ with $|m|\leq d$ add up to the series
(\ref{e: intro exp pol series}), which is therefore neatly convergent as well.
Moreover, (\ref{e: f as sum fsm}) follows. This in turn implies the real
analyticity of the function $f.$
~
$\square$\medbreak\noindent\medbreak
\begin{rem}
\label{r: 1.5 bis}
Let ${\mathfrak a}_\Delta: = \cap_{\alpha \in \Delta} \ker \alpha$ and $A_\Delta: = \exp ({\mathfrak a}_\Delta).$
Then the functions $f_{s,m},$ defined by
(\ref{e: series for f s m}) satisfy $f_{s,m}(a a_\Delta) = f_{s,m}(a)$
for all $a \in A,\, a_\Delta \in A_\Delta.$ In particular, the function $f$ of
(\ref{e: f as sum fsm}) generates
a finite dimensional $A_\Delta$-module with respect to the right regular action.
Thus, if $\Delta = \emptyset,$ then $f$ is an exponential polynomial function.
\end{rem}
\begin{lemma}
{\rm (Uniqueness of asymptotics)\ }
\label{l: uniqueness of asymp}
Let $a_0 \in A,$ and assume that the ${\msy D}elta$-exponential polynomial
series
(\ref{e: intro exp pol series})
converges neatly on $A^+(\Delta,a_0).$
If the sum of the series is zero for all $a \in A^+(\Delta,a_0),$
then $q_\xi = 0$ for all $\xi \in {\mathfrak a}dc.$
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
Let $f: A^+(\Delta,a_0) \rightarrow V$ be defined by summation
of the series (\ref{e: intro exp pol series}).
Then it follows from Lemma \ref{l: neat conv of exp pol series}
that the series (\ref{e: intro exp pol series}) is
an asymptotic expansion for $f$ in the sense
of \bib{BS}, Sect.\ 3.
Hence, if $f=0,$ then by uniqueness of asymptotics, see \bib{HCsf}, p.\ 305, Cor.\ and
\bib{BS}, Prop.\ 3.1, it follows that the series vanishes identically.
~
$\square$\medbreak\noindent\medbreak
\begin{defi}
\label{d: defi Cep A a zero}
Let $a_0 \in A.$ By ${\msy C}ep(A^+(\Delta,a_0), V)$
we denote the space of functions
$f: A^+({\msy D}elta,a_0) \rightarrow V$ that are given by the summation of a
(necessarily unique)
neatly converging
${\msy D}elta$-exponential
polynomial series of the form (\ref{e: intro exp pol series}).
If $f \in {\msy C}ep(A^+(\Delta,a_0), V),$ then by ${\rm ep}(f)$ we denote
the unique series from ${\cal F}ep(A,V)$ whose summation gives $f.$
Moreover, the asymptotic degree of $f$
is defined to be the number
$$
{\rm deg}_{\rm a}(f): = \deg ({\rm ep}(f)).
$$
\end{defi}
Note that the map
$$
{\rm ep}: {\msy C}ep(A^+(\Delta, a_0),V) \rightarrow {\cal F}ep(A,V),
$$
defined above, is a linear embedding.
Let $f \in {\msy C}ep(A^+(\Delta, a_0),V).$
We briefly write ${\rm Exp}(f)$ for the set
${\rm Exp}({\rm ep}(f));$ its elements are called the exponents
of $f.$
We put $q_\xi(f,\,\cdot\,):= q_\xi({\rm ep}(f),\,\cdot\,),$ for $\xi \in {\mathfrak a}dc.$
Then $\xi \in {\rm Exp}(f) \iff q_\xi(f)\neq 0.$
The ${\rm pr}ecD$-maximal elements in ${\rm Exp}(f)$ are called
the (${\msy D}elta$-)leading exponents of $f$ (or of the expansion). The set
of these is denoted by ${\rm Exp}L(f).$
By the formal application of $S({\mathfrak a})$ to ${\cal F}^{\rm ep}(A, V)$ we shall mean the linear
map
$$
S({\mathfrak a}) \otimes {\cal F}^{\rm ep}(A, V) \rightarrow {\cal F}^{\rm ep}(A, V)
$$
induced by termwise differentiation
(recall that $S({\mathfrak a})$ acts on $C^\infty(A)$ via the right regular action). The image of
an element $u\otimes F$ under this map will be denoted by $uF.$
\begin{lemma}
\label{l: formal application of Ufa}
Let $a_0 \in A$ and let $f\in C^{\rm ep}(A^+(\Delta, a_0),V).$ If $u \in S({\mathfrak a})$ then the function
$uf: a :to R_uf(a)$ belongs to $C^{\rm ep}(A^+(\Delta, a_0),V).$ Moreover,
$$
{\rm ep} (uf) = u \, {\rm ep}(f).
$$
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
We may assume that $u \in {\mathfrak a}.$
Express $f$ as in (\ref{e: f as sum fsm}). For
each $s, m$ the function
$u f_{s,m}$ belongs to ${\cal O}(A^+(\Delta, a_0), V);$ its expansion is obtained
from ${\rm ep}(f_{s,m})$ by termwise application of $u,$ hence by the formal
application of $u.$
~
$\square$\medbreak\noindent\medbreak
We shall also need a second type of formal application.
Suppose that complete locally convex spaces $U$ and $W$ are given,
and a continous bilinear map $U \times V \rightarrow W,$ denoted by $(u,v) :to uv.$
By the formal application of ${\cal F}^{\rm ep}(A, U)$
to ${\cal F}^{\rm ep}(A, V)$ we mean the linear map
$$
{\cal F}^{\rm ep}(A, U) \otimes {\cal F}^{\rm ep}(A, V) \rightarrow {\cal F}^{\rm ep}(A, W),
$$
given by
\begin{equation}
\label{e: defi second formal appl}
\sum_{\xi\in {\mathfrak a}dc} a^\xi p_\xi(\log a) \otimes
\sum_{\eta\in {\mathfrak a}qd} a^\eta q_\eta(\log a)
:to
\sum_{\nu \in {\mathfrak a}dc}
a^\nu \sum_{\xi + \eta = \nu} p_\xi(\log a) q_\eta(\log a).
\end{equation}
This map is indeed well defined. To see this, let $F$ denote the first
series and $G$ the second. Then for every $\nu \in {\mathfrak a}qdc,$ the collection
$S_\nu$ of $(\xi, \eta) \in {\rm Exp}(F)\times {\rm Exp}(G)$ with $\xi + \eta = \nu$ is finite.
Hence the $W$-valued polynomial function
$$
r_\nu: X:to \sum_{(\xi, \eta)\in S_\nu} p_\xi(X)q_\eta(X)
$$
has degree at most $\deg(F) + \deg(G).$
Moreover, let $X_1, X_2 \subset {\mathfrak a}dc$ be finite subsets such that
${\rm Exp}(F) \subset X_1 - {\msy N}\Delta$
and ${\rm Exp}(G) \subset X_2 - {\msy N}\Delta$ and put $X = X_1 + X_2.$ Then for
$\nu \in {\mathfrak a}dc \setminus [X - {\msy N}\Delta]$ the collection $S_\nu$ is empty,
hence $r_\nu = 0.$
Therefore, the formal series on the right-hand side
of (\ref{e: defi second formal appl}) satisfies
the conditions of Definition \ref{d: exp pol series}.
The image of an element $F \otimes G$ under the map
(\ref{e: defi second formal appl})
is denoted
by $FG.$
Again we have a lemma relating the formal
application with neat convergence.
\begin{lemma}
\label{l: formal application of hom valued series}
Let $U \times V \rightarrow W, (u,v) :to uv$ be a continuous bilinear map
of complete locally convex spaces.
Let $a_0 \in A$ and let $f\in C^{\rm ep}(A(\Delta, a_0),U)$ and
$g \in C^{\rm ep}(A(\Delta, a_0), V).$
Then the function $fg: a :to f(a)g(a)$ belongs
to $C^{\rm ep}(A(\Delta, a_0),W).$
Moreover, its $\Delta$-exponential polynomial expansion is given by
$$
{\rm ep}(fg) = {\rm ep}(f)\, {\rm ep}(g).
$$
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
This follows by a straightforward application of
Lemma \ref{l: neat conv of exp pol series}.
~
$\square$\medbreak\noindent\medbreak
\section{Basic notation, spherical functions}
In this section we study spherical functions that are defined
on a certain open dense
subset ${\rm X}p$ of the symmetric space ${\rm X},$
and are (radially) given by
exponential polynomial series. This class of functions
will play an important role in the paper. Later
we will see that ${\msy D}GH$-finite spherical funtions
belong to this class.
Throughout this paper, we assume that ${\rm X}$ is a reductive symmetric space of
Harish-Chandra's class, i.e., ${\rm X}= G/H$ with $G$ a real reductive group of Harish-Chandra's
class and $H$ an open subgroup of $G^\sigma,$
the group of fixed points for an involution $\sigma$ of $G.$
There exists a Cartan involution ${\msy C}artan$ of $G,$
commuting with $\sigma.$ The associated fixed point group $K$ is a $\sigma$-stable
maximal compact subgroup.
We adopt the usual convention to denote Lie groups by Roman capitals
and their Lie algebras by the corresponding Gothic lower cases.
The infinitesimal involutions ${\msy C}artan$ and $\sigma$ of ${\mathfrak g}$ commute;
let
\begin{equation}
\label{e: Cartan decos}
{\mathfrak g} = {\mathfrak k} \oplus {\mathfrak p} = {\mathfrak h} \oplus {\mathfrak q}
\end{equation}
be the associated decompositions into $+1$ and $-1$ eigenspaces
for ${\msy C}artan$ and $\sigma$, respectively.
We equip ${\mathfrak g}$ with a positive definite inner product $\inp{\,\cdot\,}{\,\cdot\,}$
that is invariant under the compact group of automorphisms generated by
${\rm Ad}(K),$ $e^{i {\rm ad}\,({\mathfrak p})},$ ${\msy C}artan$ and $\sigma.$ Then the decompositions
(\ref{e: Cartan decos}) are orthogonal.
Let ${\mathfrak a}q$ be a maximal abelian subspace of ${\mathfrak p} \cap {\mathfrak q}.$
We equip ${\mathfrak a}q$ with the restricted inner product $\inp{\,\cdot\,}{\,\cdot\,}$
and its dual ${\mathfrak a}qd$ with the dual inner product. The latter is
extended to a complex bilinear form, also denoted $\inp{\,\cdot\,}{\,\cdot\,},$
on the complexified dual ${\mathfrak a}qdc.$
The exponential
map is a diffeomorphism from ${\mathfrak a}q$ onto a vectorial subgroup $A_\iq$ of $G.$
We recall that $G = KA_\iq H.$
Let $\Sigma$ be the restricted root system of ${\mathfrak a}q$ in ${\mathfrak g};$
we recall that the associated Weyl group $W$ is naturally isomorphic
to ${\msy N}Kaq/ {\msy Z}Kaq,$ the normalizer modulo the centralizer of ${\mathfrak a}q$ in $K.$
Let ${\mathfrak a}q^{\rm reg}$ denote the associated set of regular elements in ${\mathfrak a}q,$
i.e., the complement of the union of the root hyperplanes $\ker \alpha,$
as $\alpha \in \Sigma.$ We put $A_\iq^{\rm reg}\mathchar"303A=\exp({\mathfrak a}q^{\rm reg})$ and define
the dense subset ${\rm X}p$ of ${\rm X}$ by
$${\rm X}p=K A_\iq^{\rm reg} H.$$
If $Q$ is a parabolic subgroup of $G,$ we denote its Langlands decomposition
by $Q = M_Q A_Q N_Q.$
By a $\sigma$-parabolic subgroup of $G$ we mean a parabolic subgroup
that is invariant under the composition $\sigma \,{\scriptstyle\circ}\, {\msy C}artan.$
It follows from \bib{Bprser1}, Lemmas 2.5 and 2.6, that
the collection $\cP_\gs$ of $\sigma$-parabolic
subgroups of $G$ containing $A_\iq$ is finite.
If $Q$ is a $\sigma$-parabolic subgroup then
the Lie algebra ${\mathfrak a}Q$ of its split component
is $\sigma$-stable, hence decomposes as
${\mathfrak a}Q = {\mathfrak a}_{Q{\rm h}} \oplus {\mathfrak a}Qq,$ the vector sum of the associated
$+1$ and $-1$ eigenspaces
of $\sigma|_{{\mathfrak a}Q},$ respectively.
We write
$A_{Q\iq} := \exp {\mathfrak a}Qq$ and $M_Qgs:= M_Q (A_Q\cap H);$
the decomposition
$$
Q = M_Qgs A_{Q\iq} N_Q
$$
is called the $\sigma$-Langlands decomposition of $Q.$
If $Q \in \cP_\gs,$ then $M_{1Q} = Q \cap {\msy C}artan(Q)$ contains $A_\iq.$ Hence
${\mathfrak a}Qq$ is contained in ${\mathfrak p} \cap {\mathfrak q}$ and centralizes ${\mathfrak a}q;$ it follows
that ${\mathfrak a}Qq \subset {\mathfrak a}q.$ By $\Sigma(Q)$ we denote the set of roots of $\Sigma$ occurring
in ${\mathfrak n}_Q.$ Obviously,
$$
{\mathfrak n}_Q = \oplus_{\alpha \in \Sigma(Q)} \;{\mathfrak g}_\alpha.
$$
Let $\cP_\gs^{\rm min}$ denote the collection of elements of $\cP_\gs$ that
are minimal with respect to inclusion. An element $P \in \cP_\gs$
belongs to $\cP_\gs^{\rm min}$ if and only if ${\mathfrak a}Pq = {\mathfrak a}q,$ see
\bib{Bprser1}, Cor.\ 2.7.
This implies that the
associated groups $M_P$ and $A_P$ are independent of $P \in \cP_\gs^{\rm min}.$
We denote them by $M$ and $A,$ respectively.
From the maximality of
${\mathfrak a}q$ in ${\mathfrak p}\cap {\mathfrak q}$ it follows that ${\mathfrak m} \cap {\mathfrak p} \subset {\mathfrak h}.$
Thus, if $K_\iM:= K \cap M$ and $H_\iM:= H \cap M,$
then the inclusion map $K_\iM \rightarrow M$
induces a diffeomorphism
\begin{equation}
\label{e: isomorphism for M mod HM}
K_\iM/K_\iM\cap H\;\;{\buildrel \simeq \over \longrightarrow} \;\; M/H_\iM.
\end{equation}
In particular, the symmetric space $M/H_\iM$ is compact.
According to \bib{Bprser1}, Lemma 2.8, the map $P :to \Sigma(P)$
induces a bijective map from $\cP_\gs^{\rm min}$ onto the collection of positive
systems for $\Sigma.$ If $\Phi$ is a positive system for $\Sigma,$
then the associated element $P \in \cP_\gs^{\rm min}$ is given by the following
characterization of its Lie algebra:
${\rm Lie}\,(P) = {\mathfrak m} + {\mathfrak a} + \sum_{\alpha \in \Phi}{\mathfrak g}_\alpha.$ From this we see that
${\msy N}Kaq$ acts on $\cP_\gs^{\rm min}$ by conjugation; moreover, the action
commutes with the map $P :to \Sigma(P).$ Accordingly, the action
factors to a free transitive action of $W$ on $\cP_\gs^{\rm min},$
see also \bib{Bprser1}, Lemma 2.8.
If $P \in \cP_\gs^{\rm min},$
then the collection of simple roots for the
positive system $\Sigma(P)$ is denoted by ${\msy D}P;$
the associated positive chamber
in ${\mathfrak a}q$ is denoted by ${\mathfrak a}qp(P)$ and
the corresponding chamber in $A_\iq$ by $A_\iq^+(P).$
Thus, we see that $A_\iq^{\rm reg}$
is the disjoint union of the chambers
$A_\iq^+(P),$ as $P \in \cP_\gs^{\rm min}.$
More generally, if $Q \in \cP_\gs,$ we write
\begin{equation}
\label{e: defi faQqp}
{\mathfrak a}Qqp: = \{X \in {\mathfrak a}Qq \mid \alpha( X) > 0 \text{for} \alpha \in \Sigma(Q) \}.
\end{equation}
It follows from \bib{Bprser1}, Lemmas 2.5 and 2.6, that ${\mathfrak a}Qqp \not= \emptyset.$
Moreover, if $X \in {\mathfrak a}Qqp,$ then the parabolic subgroup $Q$ is determined
by the following characterization of its Lie algebra
\begin{equation}
\label{e: char Lie Q}
{\rm Lie}\, (Q) = {\mathfrak m} \oplus {\mathfrak a} \oplus \bigoplus_{\alpha \in \Sigma\atop \alpha(X) {\rm ep}silonq 0} {\mathfrak g}_\alpha.
\end{equation}
Conversely, if $X$ is any element of ${\mathfrak a}q,$ then (\ref{e: char Lie Q})
defines the Lie
algebra of a group $Q$ from $\cP_\gs;$ moreover, $X \in {\mathfrak a}Qqp.$
{}From this we readily see that conjugation
induces an action of ${\msy N}Kaq$ on $\cP_\gs,$ which factors to an action of
$W.$
By a straightforward calculation involving root spaces,
it follows that the multiplication
map $K \times A_\iq^{\rm reg} \rightarrow {\rm X}$ induces a diffeomorphism
$$
K \times_{{\msy N}Kaq \cap H} A_\iq^{\rm reg} \;\;{\buildrel \simeq \over \longrightarrow} \;\; {\rm X}p.
$$
In particular, it follows that ${\rm X}p$ is an open dense subset of ${\rm X}$.
Let $W_{K \cap H}$ denote the canonical image of ${\msy N}Kaq \cap H$ in $W$
and let ${\cal W}$ be a complete set of representatives for $W/W_{K \cap H}$ in ${\msy N}Kaq.$
If $P \in \cP_\gs^{\rm min},$ then it follows that
\begin{equation}
\label{e: space X plus as union}
{\rm X}p = \cup_{w \in {\cal W}} \;\;KA_\iq^+(P)wH \quad \text{(disjoint union).}
\end{equation}
Moreover, for each $w \in {\cal W}$ the multiplication map $(k,a) :to kawH$
induces a diffeomorphism
\begin{equation}
\label{e: diffeo onto KAqpPwH}
K \times_{K_\iM \cap wHw^{-1}} A_\iq^+(P)
\;\;{\buildrel \simeq \over \longrightarrow} \;\; KA_\iq^+(P)wH.
\end{equation}
Here we have written $K_\iM = K \cap M;$ in (\ref{e: diffeo onto KAqpPwH})
the set on the right
is an open
subset of ${\rm X}.$
Let $(\tau, V_\tau)$ be a smooth representation
of $K$ in a complete locally convex space.
For later applications it will be crucial that we allow
$\tau$ to be infinite dimensional (see
the proof of Theorem \ref{t: behavior along the walls for families new}).
By $\Ci({\rm X}p \col \tau)$ we denote the space of smooth functions
$f: {\rm X}p \rightarrow V_\tau$ that are $\tau$-spherical, i.e.,
\begin{equation}
\label{e: spherical transformation rule}
f(kx) = \tau(k)f(x),
\end{equation}
for $x \in {\rm X}p, \, k\in K.$
The space ${\msy C}i({\rm X}\,:\, \tau)$ of smooth $\tau$-spherical
functions
on ${\rm X}$ will be identified with the subspace
of functions in $\Ci({\rm X}p \col \tau)$ that extend smoothly to all of ${\rm X}.$
In the following we assume that $P \in \cP_\gs^{\rm min}$
is fixed.
If $w \in {\msy N}Kaq,$
then by $C^\infty_{P,w}({\rm X}p\,:\, \tau)$ or
$C^\infty_w({\rm X}p\,:\, \tau)$
we denote the space of functions $f \in {\msy C}i({\rm X}p\,:\, \tau)$
with support contained in $KA_\iq^+(P)wH.$
{}From (\ref{e: space X plus as union}) we see that
$$
{\msy C}i({\rm X}p\,:\, \tau) = \oplus_{w\in {\cal W}} \;\;C^\infty_w({\rm X}p\,:\, \tau).
$$
Let $w \in {\msy N}Kaq$ be fixed for the moment.
For $f \in {\msy C}i({\rm X}p \,:\, \tau)$ we define the
function $T_P^\downarroww f \in {\msy C}i(A_\iq^+(P), V_\tau^{K_\iMwH})$ by
$$
T_P^\downarroww f (a) = f(awH).
$$
Since (\ref{e: diffeo onto KAqpPwH}) is a diffeomorphism,
the restriction of $T_P^\downarroww$ to
$C^\infty_w({\rm X}p\,:\, \tau)$
is an isomorphism of complete locally convex spaces
onto the space ${\msy C}i(A_\iq^+(P), V_\tau^{K_\iMwH}).$
Taking the direct sum of the maps
$T_P^\downarroww,$ as $w \in {\cal W},$
we therefore obtain an isomorphism of complete locally convex spaces
\begin{equation}
\label{e: the iso T down P cW}
T_P^\downarrowcW:
\;\;{\msy C}i({\rm X}p \,:\, \tau)
\;\;{\buildrel \simeq \over \longrightarrow} \;\;
\oplus_{w \in {\cal W}}\;\; {\msy C}i(A_\iq^+(P), V_\tau^{K_\iMwH}).
\end{equation}
\begin{defi}
\label{d: Cep spXp tau}
We denote by $\Cep({\rm X}p\col \tau)$ the space of functions
$f \in {\msy C}iXptau$ such that for every $w \in {\cal W}$ the
function $T_P^\downarroww f$ belongs to ${\msy C}ep(A_\iq^+(P),V_\tauKwH),$
where the latter space is defined as in Definition
\ref{d: defi Cep A a zero},
with ${\mathfrak a},$ $a_0$ and $\Delta$ replaced
by ${\mathfrak a}q,$ $e$ and $\Delta(P),$ respectively.
If $f \in \Cep({\rm X}p\col \tau),$ we define its asymptotic degree to be the number
$$
{\rm deg}_{\rm a} (f): = \max_{w \in {\cal W}}\; \deg (T_P^\downarroww f).
$$
\end{defi}
\noindent
It follows from the above definition that
restriction of $T_P^\downarrowcW$ induces a linear isomorphism
\begin{equation}
\label{e: isomorphism of exppol}
\Cep({\rm X}p\col \tau) \simeq
\oplus_{w \in {\cal W}}\;\; {\msy C}ep(A_\iq^+(P), V_\tau^{K_\iMwH}).
\end{equation}
Using conjugations by elements of ${\msy N}Kaq$
it is readily seen
that the space $\Cep({\rm X}p\col \tau)$
and the map ${\rm deg}_{\rm a}: \Cep({\rm X}p\col \tau)\rightarrow {\msy N}$
are independent of the
particular choices of $P$ and ${\cal W}.$ In particular, if $P \in \cP_\gs^{\rm min}$
and $w \in {\msy N}Kaq,$ then $T_P^\downarroww f \in {\msy C}ep(A_\iq^+(P), V_\tauKwH)$
and
$\deg (T_P^\downarroww f) \leq {\rm deg}_{\rm a} (f).$
We put
$$
{\rm Exp}(P,w\,|\, f) := {\rm Exp}(T_P^\downarroww f),\quad \text{and}
\quad{\rm Exp}L(P,w\,|\, f) := {\rm Exp}L(T_P^\downarroww f).
$$
Moreover, for all $\xi \in {\mathfrak a}qdc$ we define
${\underline q}_\xi(P,w \,|\, f) = q_\xi(T_P^\downarroww f).$
Then, for every $a \in A_\iq^+(P),$
\begin{equation}
\label{e: exp pol expression f a w}
f(aw) = \sum_{\xi \in {\rm Exp}(P,w\,|\, f)} a^\xi\,{\underline q}_\xi(P,w\,|\, f, \log a),
\end{equation}
where the $\Delta(P)$-exponential polynomial series on the right-hand side
neatly converges on $A_\iq^+(P).$
For $w \in {\msy N}Kaq,$ we will use the notation
\begin{equation}
\label{e: defi spXzerow}
{\rm X}zerow := M /M \cap wHw^{-1};
\end{equation}
moreover,
we
put $\tau_{\iM}:= \tau_{K_\iM}$ and
write
$
C^\infty({\rm X}zerow\,:\, \tau_{\iM})
$
for the space of
$\tau_{\iM}$-spherical
$C^\infty$ functions from
${\rm X}zerow$ to $V_\tau,$ i.e., the space of functions $f \in {\msy C}i({\rm X}zerow, V_\tau)$ satisfying the
rule (\ref{e: spherical transformation rule}) for $k \in K_\iM$ and $x \in {\rm X}zerow.$
From (\ref{e: isomorphism for M mod HM}) with $wHw^{-1}$ in place of $H$ we see that
the inclusion $K_\iM \rightarrow M$ induces a diffemorphism from $K_\iM/K_\iM \cap wHw^{-1}$ onto ${\rm X}zerow.$
Hence evaluation at the point $e(M\cap wHw^{-1})$ induces a linear isomorphism
from $C^\infty({\rm X}zerow \,:\, \tau_{\iM})$ onto $V_\tau_{\iM}KwH.$ Thus, if
$f \in \Cep({\rm X}p\col \tau),$
then for every $\xi \in {\mathfrak a}qc$ there exists a unique
$C^\infty({\rm X}zerow \,:\,\tau_{\iM})$-valued
polynomial function $q_\xi(P,w\,|\, f)$ on ${\mathfrak a}q$ such that
$$
q_\xi(P,w \,|\, f, X, e)= {\underline q}_\xi(P,w\,|\, f)(X)
\quad\quad (X \in {\mathfrak a}q).
$$
Using sphericality of the function $f$
we obtain from (\ref{e: exp pol expression f a w})
that
\begin{equation}
\label{e: expansion f on PqPw with m}
f(maw) = \sum_{\xi \in {\rm Exp}(P,w\,|\, f)} a^\xi q_\xi(P,w\,|\, f , \log a, m),
\end{equation}
for $m \in M,\; a \in A_\iq^+(P).$ The series on the right-hand side
is a $\Delta(P)$-exponential polynomial series in the variable
$a,$ with coefficients
in ${\msy C}i({\rm X}zerow \,:\, \tau_{\iM}),$ relative to the variable $m.$ As
such it converges neatly on $A_\iq^+(P).$
We shall now discuss a lemma whose main purpose is to enable us to
reduce on the set of exponents in certain proofs, in order to simplify
the exposition.
\begin{lemma}
\label{l: splitting lemma}
Let $P \in \cP_\gs^{\rm min}$ and let ${\cal W} \subset {\msy N}Kaq$ be a
complete set of representatives
of $W/W_{K \cap H}.$ Assume that
$f \in \Cep({\rm X}p\col \tau).$
There exists a finite set $S \subset {\mathfrak a}qdc$
of mutually $\Delta(P)$-integrally inequivalent elements such that
${\rm Exp}(P,v\,|\, f) \subset S - {\msy N}\Delta(P)$
for every $v \in {\cal W}.$
If $S$ is a set as above, then there exist unique
functions $f_s \in \Cep({\rm X}p\col \tau),$ for $s \in S,$
such that
$$
f = \sum_{s \in S} f_s,
$$
and such that ${\rm Exp}(P,v\,|\, f_s) \subset s - {\msy N} \Delta(P),$
for every $v \in {\cal W}.$
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
There exists a finite set $X \subset {\mathfrak a}qdc$ such that
${\rm Exp}(P,v\,|\, f) \subset
X - {\msy N} \Delta(P)$ for all $v \in {\cal W}.$ Obviously there exists
a finite set $S$ as
required, such that $X - {\msy N} \Delta(P) \subset S - {\msy N} \Delta(P).$
If $S$ is such as mentioned, then for
$s \in S$ and $v \in {\cal W}$ we define the function
$f_{s,v}: A_\iq^+(P) \rightarrow V_\tauKvH$
by
$$
f_{s,v}(a) = \sum_{\nu \in {\msy N}\Delta(P)} a^{s -\nu} q_{s -\nu} (P,v\mid f,\log a,e);
$$
here the exponential polynomial series is neatly convergent,
hence $f_{s,v} $ belongs to the space ${\msy C}ep(A_\iq^+(P), V_\tauKvH),$ for every $v \in {\cal W}.$
By the isomorphism (\ref{e: isomorphism of exppol}) there exists a unique
function $f_s \in {\msy C}ep({\rm X}p\,:\, \tau)$
such that $f_s(av) = f_{s,v}(a)$ for $v \in {\cal W},\, a \in A_\iq^+(P).$
By the hypothesis on $S$ the sets $s - {\msy N}\Delta(P),$ for $s \in S$, are disjoint.
Hence $f = \sum_{s \in S} f_s$ on $A_\iq^+(P)v,$ for every $v \in {\cal W}.$
By (\ref{e: space X plus as union}) and
sphericality this equality holds on all of ${\rm X}p.$
~
$\square$\medbreak\noindent\medbreak
\section{Asymptotic behavior along walls}
\label{s: asymp walls}
In this section we study the asymptotic behavior along walls
of functions from $\Cep({\rm X}p\col \tau);$ here $\tau$
is a smooth representation
in a complete locally convex space $V_\tau.$
Let $P \in \cP_\gs^{\rm min}$ and let
$Q $ be a $\sigma$-parabolic subgroup with Langlands decomposition
$Q = M_Q A_Q N_Q,$ containing $P.$
In addition to the notation introduced in the beginning of the previous
section, the following notation will also be convenient.
We agree to write
$K_Q:= K \cap M_Q$ and $H_Q: = H \cap M_Q.$ Moreover,
$W_Q$ denotes the centralizer of ${\mathfrak a}Qq$ in $W.$ Then $W_Q \simeq N_{K_Q}({\mathfrak a}q)/Z_{K_Q}({\mathfrak a}q).$
On the other hand $W_Q$ is also the subgroup of $W$ generated by the reflections in the roots
from the set
$$
{\msy D}subQP := \{ \alpha \in {\msy D}P \mid \alpha|_{{\mathfrak a}Qq}
= 0 \}.
$$
We note that $\Sigma(Q) = \Sigma(P)\setminus {\msy N}{\msy D}elta_Q(P).$ Moreover, let
$\Sigma_r(Q)$ denote the collection of ${\mathfrak a}Qq$-weights in ${\mathfrak n}_Q.$
Then
$$
\Sigma_r(Q) = \{ \alpha|_{{\mathfrak a}Qq} \mid \alpha \in \Sigma(Q)\}.
$$
Let ${\msy D}rQ$ be the collection of weights from the set $\Sigma_r(Q)$ that cannot be written as the
sum of two weights from that set; then one readily verifies that
${\msy D}rQ$ equals the set of restrictions of elements from ${\msy D}elta(P)\setminus {\msy D}elta_Q(P)$
to ${\mathfrak a}Qq.$ In
particular, the elements of ${\msy D}rQ$ are linearly independent.
Given $a_0 \in A_{Q\iq}$ we shall briefly write $A_{Q\iq}^+(a_0)$ for the set
$A_{Q\iq}^+({\msy D}rQ, a_0)$ defined as
in (\ref{e: Ap Delta a zero}) with ${\mathfrak a}Qq$ and ${\msy D}rQ$ in place of ${\mathfrak a}$ and ${\msy D}elta,$
respectively.
Similarly, if $\rho \in \,]\,0,\infty\,[^{{\msy D}rQ},$ we briefly write
$$
A_{Q\iq}^+(\rho):= A_{Q\iq}^+({\msy D}rQ, \rho) =
\{a \in A_{Q\iq} \mid a^{-\alpha} < \rho_\alpha,\;\;\forall \alpha \in {\msy D}rQ \}.
$$
If $R>0,$ we write $A_{Q\iq}^+(R)$ for $A_{Q\iq}^+(\rho),$
where $\rho$ is defined by $\rho_\alpha = R$
for every $\alpha \in {\msy D}rQ.$ Note that
$A_{Q\iq}^+(1)$ equals the positive chamber $A_{Q\iq}^+: = \exp({\mathfrak a}Qqp),$
see (\ref{e: defi faQqp}).
If $v \in {\msy N}Kaq,$ we define
\begin{equation}
\label{e: defi XoneQv}
{\rm X}oneQv := M_{1Q} / M_{1Q} \cap vHv^{-1}.
\end{equation}
This is a symmetric space for the involution $\sigma^v$ of $M_{1Q}$ defined
by $\sigma^v(m) = v\sigma(v^{-1} m v)v^{-1}.$ Note that this involution commutes with
the Cartan involution $\theta|_{M_{1Q}}.$
Note also that ${\mathfrak a}q$ is a maximal
abelian subspace of $ {\rm Ad}(v)({\mathfrak p} \cap {\mathfrak q}) = {\mathfrak p} \cap {\rm Ad}(v){\mathfrak q}.$ Hence it
is the analogue of ${\mathfrak a}q$ for the triple $(M_{1Q}, K_Q, M_{1Q} \cap v H v^{-1}).$
The corresponding group $A_\iq$ may naturally be identified with
a subspace of ${\rm X}oneQv.$
The image of $M_Q$ in ${\rm X}oneQv$ may be identified
with
$$
\spX_{Q,v} := M_Q/M_Q \cap vHv^{-1},
$$
the symmetric space for the involution $\sigma^v|_{M_Q}.$
It follows from the characterization of $\cP_\gs$ expressed by
(\ref{e: char Lie Q}) that
\begin{equation}
\label{e: allparabsv}
\cP_\gs = \cP_\gsv
\end{equation}
Hence $Q$
is a $\sigma^v$-parabolic subgroup as well. Hence ${\mathfrak a}Q \cap {\rm Ad}(v) {\mathfrak q} =
{\mathfrak a}Q \cap {\mathfrak a}q = {\mathfrak a}Qq,$ and we deduce that the inclusion $ A_{Q\iq} \rightarrow A_Q$
induces a diffeomorphism $A_{Q\iq} \simeq A_Q/A_Q\cap vHv^{-1}.$ From this
we conclude that the multiplication map $M_Q \times A_{Q\iq} \rightarrow M_{1Q}$
induces the decomposition
\begin{equation}
\label{e: deco spXoneQv}
{\rm X}oneQv \simeq {\rm X}Qv \times A_{Q\iq}.
\end{equation}
\begin{rem}
\label{r: extreme cases subspaces}
In particular,
the
above definitions cover the two extreme cases that $Q$ is minimal
and that it equals $G.$
In the case that $Q \in \cP_\gs^{\rm min},$ we have $Q = M A N_Q,$ and ${\rm X}Qv$ equals
the space ${\rm X}zerov$ defined in (\ref{e: defi spXzerow}).
Moreover, ${\rm X}oneQv \simeq {\rm X}zerov \times A_\iq.$
In the other extreme case we have ${\rm X}_{1G,v} = G/vHv^{-1}.$ This symmetric space will
also be denoted by ${\rm X}v.$
Note that right multiplication by $v$ induces an isomorphism of ${\rm X}_v$ onto ${\rm X}.$
Note also that $M_G$ equals ${}^\circ G,$
the intersection of $\ker \chi,$ as $\chi$ ranges over the positive characters
of $G.$ Hence ${\rm X}_{G,v} = {}^\circ G / {}^\circ G \cap vH v^{-1}.$
Finally, ${\rm X}_v \simeq {\rm X}_{G,v} \times A_{G{\rm q}},$
where $A_{G{\rm q}}$ is the image under $\exp$ of the space
${\mathfrak a}_{G{\rm q}},$ which in turn is the intersection
of the root hyperplanes $\ker \alpha$ as $\alpha \in \Sigma.$
\end{rem}
Let
$\bar\fn_Q: = {\msy C}artan {\mathfrak n}_Q$ be equipped with the restriction
of the inner product $\inp{\,\cdot\,}{\,\cdot\,}$ from ${\mathfrak g}.$
If $Q \neq G$ we define the function
$R_{Q,v}: M_{1Q} \rightarrow ]0,\infty[$ by
$$
R_{Q,v}(m) = \|{\rm Ad}(m \sigma^v(m)^{-1})|_{\bar\fn_Q}\|_{\rm op}^{1/2},
$$
where $\|\,\cdot\,\|_{\rm op}$ denotes the operator norm. We
define $R_{G,v}$ to be the constant function $1.$
The
function $R_{Q,v}$
is right $M_{1Q} \cap v H v^{-1}$-invariant. It
may therefore also be viewed as a function on ${\rm X}oneQv.$
We shall describe the function $R_{Q,v}$ in more detail below.
The orthocomplement of ${\mathfrak a}Qq$ in ${\mathfrak a}q$ is denoted by ${}^*\fa_{Q\iq}.$
Note that
\begin{equation}
\label{e: char staQq}
{}^*\fa_{Q\iq} = {\mathfrak m}_Q \cap {\mathfrak a}q;
\end{equation}
hence ${}^*\fa_{Q\iq}$ is the analogue of
${\mathfrak a}q$ for the triple $(M_Q, K_Q, H_Q).$
We recall from the text following (\ref{e: defi XoneQv})
that ${\mathfrak a}q$ is maximal abelian in ${\mathfrak p} \cap {\rm Ad}(v){\mathfrak q}$ hence is the analogue
of ${\mathfrak a}q$ for the triple $(G, K, vHv^{-1}).$ Accordingly, ${}^*\fa_{Q\iq}$ is also
the analogue of ${\mathfrak a}q$ for the triple $(M_Q, K_Q , M_Q \cap vHv^{-1}).$
In view of
(\ref{e: allparabsv}), the group
${}^*P = P \cap M_Q$ is readily seen to be
a minimal $\sigma^v$-parabolic subgroup for $M_Q;$
the associated positive chamber
in ${}^*\!A_{Q\iq} = \exp ({}^*\fa_{Q\iq})$ is denoted by ${}^*\!A_{Q\iq}p({}^*P).$
Let ${\cal W}_{Q,v}$ be an analogue for $\spX_{Q,v}$ of ${\cal W},$ that is, ${\cal W}_{Q,v}$
is a complete set of representatives in ${\msy N}KQaq$ for the quotient
$W_Q/W_{K_Q \cap vHv^{-1}}.$ Let $\spX_{Q,v}p$ be the analogue for $\spX_{Q,v}$
of the open dense subset
${\rm X}p$ of ${\rm X}.$ According to (\ref{e: space X plus as union}) this set may
be expressed as the following disjoint union of open subsets of $\spX_{Q,v}$
\begin{equation}
\label{e: deco XQvp}
\spX_{Q,v}p: = \bigcup_{u \in {\cal W}Qv} K_Q {}^*\!A_{Q\iq}p({}^*P)\, u\, (M_Q \cap vHv^{-1})
\quad\quad \text{(disjoint union).}
\end{equation}
Let
${\rm X}oneQvp$ be the analogue of ${\rm X}p$ for ${\rm X}oneQv;$ then from
(\ref{e: deco spXoneQv}) we see that
$
{\rm X}oneQvp \simeq {\rm X}Qvp \times A_{Q\iq}.
$
In terms of this decomposition and (\ref{e: deco XQvp})
the function $R_{Q,v}$ may be expressed as
follows.
\begin{lemma}
\label{l: properties RQv}
The function $R_{Q,v}: M_{1Q} \rightarrow ]0,\infty [$ is continuous, and
right $M_{1Q} \cap vHv^{-1}$- and
left $K_Q$-invariant.
Moreover, if $Q \not=G$ and if $a \in A_\iq$ and $u \in {\msy N}KQaq,$ then
\begin{equation}
\label{e: value RQv}
R_{Q,v}(au) = \max_{\alpha\in \Sigma(Q)} a^{-\alpha}.
\end{equation}
Finally, $R_{Q,v} {\rm ep}silonq 1$ on $\spX_{Q,v}.$
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
Since $R_{G,v}$ is the constant function $1,$ we may as well assume that $Q \not= G.$
Continuity of the function $R_{Q,v}$ is obvious from its definition.
The group $K_Q$ is $\sigma^v$ invariant and acts unitarily on $\bar\fn_Q;$
hence the left $K_Q$-invariance is obvious from the definition.
If $a \in A_\iq,$ then $a\sigma^v(a)^{-1} =a^2.$
Hence the operator norm
of ${\rm Ad}(a\sigma^v(a)^{-1})$ on $\bar\fn_Q$ equals the maximal value of $a^{-2\alpha}$ as
$\alpha \in \Sigma(Q).$ This implies (\ref{e: value RQv}) for $u = 1.$
The element $u \in {\msy N}KQaq$ belongs to $M_Q,$ hence ${\rm Ad}(u)$
normalizes
${\mathfrak n}_Q.$ Therefore, ${\rm Ad}(u)$ leaves the collection
$\Sigma(Q)$ of ${\mathfrak a}q$-roots in ${\mathfrak n}_Q$ invariant.
Put $a' = u^{-1} a u.$ Then
$
R_{Q,v}(a u) = R_{Q,v}(a') = \max_{\alpha \in \Sigma(Q)} (a')^{-\alpha}.$
Since ${\rm Ad}(u)$ leaves $\Sigma(Q)$ invariant, (\ref{e: value RQv}) follows.
If $\alpha \in \Sigma,$ let $h_\alpha$ be the element of ${\mathfrak a}q$ determined by
$\alpha(X) = \inp{h_\alpha}{X},$ for $X \in {\mathfrak a}q.$ Then the closure of ${{}^*\fa_{Q\iq}p({}^*P)}$ is
contained in the closed convex cone
generated by the elements $h_\beta,$ for $\beta \in {\msy D}elta_Q(P).$
If $\alpha \in {\msy D}P\setminus {\msy D}elta_Q(P),$ then
$\alpha(h_\beta) = \inp{\alpha}{\beta} \leq 0,$
for $\beta \in {\msy D}elta_Q(P);$ hence $\alpha \leq 0$ on ${}^*\fa_{Q\iq}p({}^*P).$
But ${\msy D}P\setminus {\msy D}elta_Q(P)\subset \Sigma(Q),$ hence
it follows that $R_{Q,v} {\rm ep}silonq 1$ on ${}^*\!A_{Q\iq}p({}^*P)u,$
for every $u \in {\cal W}Qv.$
The final assertion follows from combining this observation
with (\ref{e: deco XQvp}), the left $K_Q$-invariance of $R_{Q,v}$
and density of ${\rm X}Qvp$ in ${\rm X}Qv.$
~
$\square$\medbreak\noindent\medbreak
If $1\leq R \leq \infty$ we define
\begin{equation}
\label{e: defi spXQvb R}
\spX_{Q,v}b{R}:= \{m \in \spX_{Q,v} \mid R_{Q,v}(m) < R\}.
\end{equation}
Note that $\spX_{Q,v}b{1} = \emptyset$ and
$\spX_{Q,v}b{\infty} = {\rm X}Qv;$ moreover,
$R_1 < R_2 \Rightarrow \spX_{Q,v}b{R_1} \subset \spX_{Q,v}b{R_2}.$
Finally, the union of the sets $\spX_{Q,v}b{R}$ as $1 \leq R < \infty$ equals $\spX_{Q,v}.$
In accordance
with (\ref{e: defi spXQvb R}) we define
$\spX_{Q,v}pb{R}: = {\rm X}Qvp \cap \spX_{Q,v}b{R},$ for
$1 \leq R \leq \infty.$
Moreover, we also put
$$
{}^*\!A_{Q\iq}p({}^*P)\lbr{R}: =
\{ a \in {}^*\!A_{Q\iq}p({}^*P)\mid a^{-\alpha} < R,\;\;\forall \alpha \in \Sigma(Q)\}.
$$
Note that, if $\alpha \in \Sigma(P)\setminus\Sigma(Q),$ then $a^{-\alpha} < 1 \leq R$ for
all $a \in {}^*\!A_{Q\iq}p({}^*P).$ Hence
$$
{}^*\!A_{Q\iq}p({}^*P)\lbr{R} = {}^*\!A_{Q\iq}p({}^*P) \cap A_\iq^+(P, R).
$$
It follows from (\ref{e: deco XQvp})
and Lemma \ref{l: properties RQv} that
\begin{equation}
\label{e: deco XQvp added R}
\quad\spX_{Q,v}pb{R} = \bigcup_{u \in {\cal W}Qv} K_Q {}^*\!A_{Q\iq}p({}^*P)\lbr{R}\, u\, (M_Q \cap vHv^{-1})
\quad \text{(disjoint union).}
\end{equation}
The function $R_{Q,v}$ plays a role in the description
of the asymptotic behavior of a function $f \in {\msy C}ep({\rm X}p\,:\, \tau)$
along `the wall' $A_{Q\iq}^+ v.$ This behaviour is described
in terms of an expansion of $f(mav)$ in the variable $a \in A_{Q\iq}^+,$
for $m \in {\rm X}Qvp.$ Thus, it is of interest to know when $mavH$ belongs
to $ {\rm X}p,$ the domain of $f.$
\begin{lemma}\label{l: inclusions for asymp}
\hbox{\hspace{1mm}}
\break
\begin{enumerate}
\item[{\rm (a)}]
If $b \in {}^*\!A_{Q\iq}p({}^*P)$ and $a \in A_{Q\iq}^+(R_{Q,v}(b)^{-1})$
then $ba\in A_\iq^+(P).$
\minspace\item[{\rm (b)}]
Let $m \in {\rm X}Qvp.$ Then $mavH \in {\rm X}p$ for all $a \in A_{Q\iq}^+(R_{Q,v}(m)^{-1}).$
\minspace\item[{\rm (c)}]
Let $R {\rm ep}silonq 1.$ Then $\spX_{Q,v}pb{R} A_{Q\iq}^+(R^{-1}) v H \subset {\rm X}p.$
\end{enumerate}
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
Let $b$ and $a$ fulfill the hypotheses of (a).
If $\alpha \in \Delta_Q(P),$ then $(ba)^{-\alpha} = b^{-\alpha} <1.$
On the other hand we have,
for $\alpha \in \Delta(P)\setminus \Delta_Q(P),$ that $\alpha \in \Sigma(Q),$ hence
$(ba)^{-\alpha} \leq R_{Q,v}(b) a^{-\alpha} < 1,$ by Lemma \ref{l: properties RQv}.
Hence $ba \in A_\iq^+(P),$ and (a) is proved.
Let $m$ be as in (b), and let $a \in A_{Q\iq}^+(R_{Q,v}(m)^{-1}).$
In view of (\ref{e: deco XQvp})
we may write $m = kbuh$ with $k \in K_Q,$ $b \in {}^*\!A_{Q\iq}p({}^*P),$
$u \in {\cal W}_{Q,v}$ and $h \in M_Q \cap vHv^{-1}.$ Now
$mavH = kbuhavH= kbauvH.$ Thus, it suffices to show that
$ba \in A_\iq^+(P).$ This follows from (a) and the observation that
$R_{Q,v}(b) = R_{Q,v}(m),$ by Lemma \ref{l: properties RQv}.
Finally, (c) is a straightforward consequence of (b).
~
$\square$\medbreak\noindent\medbreak
If
$Q \in \cP_\gs$
we put $\tau_Q:= \tau|_{K_Q}.$ Then, for
$v \in {\msy N}Kaq,$
the space ${\msy C}ep({\rm X}Qvp \,:\, \tau_Q)$ is defined as
above (\ref{e: isomorphism of exppol}) with
${\rm X}Qv$ and
$\tau_Q$
in place of ${\rm X}$ and $\tau,$ respectively.
\begin{thm}
\label{t: expansion along the walls}
Let $f \in \Cep({\rm X}p\col \tau).$
Let $Q \in \cP_\gs$
and $v \in {\msy N}Kaq.$
\begin{enumerate}
\item[{\rm (a)}]
There exist a constant $k\in{\msy N}$, a
finite set $Y \subset {\mathfrak a}Qqdc,$ and
for each $\eta \in Y - {\msy N} {\msy D}rQ$ a
$C(\spX_{Q,v}p, V_\tau)$-valued
polynomial function $q_\eta = q_{\eta}(Q,v\,|\, f)$ on ${\mathfrak a}Qq$ of degree at most
$k,$ such that for every $m \in \spX_{Q,v}p$
\begin{equation}
\label{e: expansion f along Q v}
f(mav) = \sum_{\eta \in Y - {\msy N} {\msy D}rQ} a^\eta q_\eta(\log a, m),
\quad\quad(a \in A_{Q\iq}^+(R_{Q,v}(m)^{-1})),
\end{equation}
where the ${\msy D}rQ$-exponential polynomial series with coefficients in $V_\tau$
converges neatly on the indicated subset of $A_{Q\iq}.$
\minspace\item[{\rm (b)}]
The set
${\rm Exp}(Q,v\,|\, f):=\{\eta \in Y - {\msy N} {\msy D}rQ \mid q_\eta \neq 0 \}$ is uniquely determined.
Moreover, the functions $q_\eta,$ where $\eta \in Y - {\msy N} {\msy D}rQ,$ are unique and
belong to $P_d({\mathfrak a}Qq) \otimes \Cep(\spX_{Q,v}p \,:\, \tau_Q),$ where $d:={\rm deg}_{\rm a}(f)$.
Finally,
if $R> 1,$ then the
the series on the right-hand side of (\ref{e: expansion f along Q v})
converges neatly on
$A_{Q\iq}^+(R^{-1})$ as a
${\msy D}rQ$-exponential polynomial series with coefficients in
${\msy C}i(\spX_{Q,v}pb{R}\,:\, \tau_Q).$
\end{enumerate}
\end{thm}
\par\noindent{\bf Proof:}{\ }{\ }
We will establish existence. Uniqueness then follows from
uniqueness of asymptotics, see Lemma \ref{l: uniqueness of asymp}.
Fix $P \in \cP_\gs^{\rm min}$ with $P \subset Q.$
Select a complete set ${\cal W}Qv \subset {\msy N}KQaq$ of representatives
for $W_Q/W_Q\cap W_{K\cap \vH}.$
The set ${\cal W}Qv v$ maps injectively into the coset space $W/W_{K \cap H}.$
Hence it may be extended to a complete set ${\cal W}$ of representatives
in ${\msy N}Kaq$ for
$W/W_{K \cap H}.$ In view of Lemma \ref{l: splitting lemma} we may therefore decompose
$f,$ if necessary, so that we arrive in the situation
that there exists a $s \in {\mathfrak a}qdc$ such that ${\rm Exp}(P,uv\,|\, f) \subset
s - {\msy N} \Delta(P),$ for all $u \in {\cal W}Qv.$
We put $s_Q = s|{\mathfrak a}Qq.$
Let $u \in {\cal W}Qv.$ Then the function
$f_{uv}: a :to f(auv)$ has a (unique) ${\msy D}elta(P)$-exponential
polynomial expansion on $A_\iq^+(P)$
of the following type:
\begin{equation}
\label{e: expansion with p u xi}
f_{uv}(a) = f(auv) = \sum_{\xi \in s - {\msy N}\Delta(P)} q_{u,\xi}(\log a) a^\xi.
\end{equation}
Here $q_{u,\xi}(\,\cdot\,) = q_\xi(P,uv\,|\, f, \,\cdot\,,e)$
belongs to $P_d({\mathfrak a}q)\otimes V_\tauuvH$.
Let $\partial \in S({\mathfrak a}q).$ Then according to Lemma
\ref{l: formal application of Ufa}, the function $\partial f_{uv}$ is given on $A_\iq^+(P)$
by a neatly convergent $\Delta(P)$-exponential
polynomial series that is obtained from (\ref{e: expansion with p u xi})
by term by term application
of $\partial.$ That is,
\begin{equation}
\label{e: series with diffop}
\partial f_{uv} (a) = \sum_{\xi \in s - {\msy N}\Delta(P)} q_{\partial, u,\xi}(\log a) a^\xi,
\end{equation}
where $q_{\partial, u, \xi}$ is the $V_\tauuvH$-valued polynomial function
on ${\mathfrak a}q$ of degree at most $d$ given by
$$
q_{\partial, u, \xi}(X) =
e^{-\xi(X)} \partial [e^{\xi(\,\cdot\,)} q_{u,\xi}](X)\quad\quad (X \in {\mathfrak a}q).
$$
Let now $R >1$ and let $\cK$ and $\cK'$ be compact subsets of ${}^*\! A_{Q\iq}^+({}^*P)_{[R]}$
and $A_{Q\iq}^+(R^{-1}),$ respectively. Then $\cK' \cK$ is a compact
subset of $A_\iq^+(P),$ by Lemma \ref{l: inclusions for asymp} (a).
Thus, if $a \in \cK'$ and $b \in \cK,$ then
the series in
(\ref{e: series with diffop}) with $ba$ in place of $a$ converges
absolutely, and may be rearranged as follows:
\begin{equation}
\label{e: rearranged expansion for f}
\partial f_{uv}(ab) =
\sum_{\eta \in s_Q - {\msy N}{\msy D}rQ}
a^\eta \sum_{\xi \in s - {\msy N} {\msy D}P\atop {\xi |{{\mathfrak a}Qq} = \eta} }
b^\xi\, q_{\partial, u,\xi}(\log b + \log a).
\end{equation}
In view of Lemma \ref{l: neat conv of exp pol series}, the convergence is
absolutely uniformly for $(a,b) \in \cK' \times \cK.$
By a similar reasoning it follows from the neat convergence of the series
(\ref{e: series with diffop})
that, for any continuous seminorm $\sigma_0$ on $P_d({\mathfrak a}q) \otimes V_\tau,$
the series
\begin{equation}
\label{e: series TAG}
\sum_{\eta\in s_Q - {\msy N} {\msy D}rQ}
a^{{\msy R}e \eta}
\sum_{\xi \in s- {\msy N} {\msy D}P \atop \xi|{{\mathfrak a}Qq} = \eta}
b^{{\msy R}e \xi} \,
\sigma_0(q_{\partial, u, \xi})
\end{equation}
converges uniformly for $a \in \cK'$ and $b \in \cK.$
Let now $\eta \in s_Q - {\msy N} {\msy D}elta_r(Q)$ and let $b \in {}^*\!A_{Q\iq}p({}^*P)$
and $a \in A_{Q\iq}^+(R_{Q,v}(b)^{-1}).$
Then there exists a $R>1$ such that $b \in {}^*\! A_{Q\iq}^+({}^*P)_{[R]}$
and $a \in A_{Q\iq}^+(R^{-1}).$ Hence the series (\ref{e: series TAG}) converges, and by positivity
of all of its terms we infer that the series
\begin{equation}
\label{e: series with gs zero and diffop}
\sum_{\xi \in s - {\msy N} {\msy D}P\atop {\xi | {\mathfrak a}Qq = \eta} }
b^{{\msy R}e \xi} \,\sigma_0(q_{\partial, u,\xi})
\end{equation}
converges for every continuous seminorm $\sigma_0$ on
$P_{d}({\mathfrak a}q) \otimes V_\tau,$ for every $b \in {}^*\!A_{Q\iq}p({}^*P).$
We now specialize to $\partial = 1$ and note that $q_{1,u,\xi} = q_{u, \xi}.$
Let $X \in {\mathfrak a}Qq.$
We define the linear endomorphism $T_X$ of $P_d({\mathfrak a}q) \otimes V_\tau$
by $T_Xp(H) = p(X + H).$ This endomorphism is continuous linear by finite dimensionality.
Combining this with the convergence of
(\ref{e: series with gs zero and diffop}) we infer,
for every $X \in {\mathfrak a}Qq,$ that
\begin{equation}
\label{e: expansion q eta}
q_{Q,u,\eta}(X, b) := \sum_{\xi \in s - {\msy N} {\msy D}P\atop {\xi |{\mathfrak a}Qq = \eta} }
b^\xi \,T_X (q_{u,\xi})(\log b)
\end{equation}
is a function of $b$ defined by a neatly convergent ${\msy D}elta_Q(P)$-exponential polynomial series
on ${}^*\!A_{Q\iq}p({}^*P).$ It is polynomial in $X$ of degree at most $d,$
and real analytic in $b \in {}^*\! A_{Q\iq}^+({}^*P).$
Moreover, its values are in the space $V_\tauuvH.$ Thus
$q_{Q,u, \eta} \in P_d({\mathfrak a}Qq) \otimes {\msy C}ep({}^*\!A_{Q\iq}p({}^*P), V_\tauuvH).$
In view of the isomorphism (\ref{e: isomorphism of exppol}) for
$\spX_{Q,v}p$, $\tau_Q$, ${\cal W}Qv$ in
place of ${\rm X}p$, $\tau$, ${\cal W},$ see also the decomposition
(\ref{e: deco XQvp}),
there exists a unique polynomial function
$q_\eta = q_{\eta}(Q,v\,|\, f)$ on ${\mathfrak a}Qq$ with values in ${\msy C}ep(\spX_{Q,v}p \,:\, \tau_Q)$
such that
\begin{equation}
\label{e: defi q eta in proof}
q_\eta(X, bu) = q_{Q, u,\eta}(X, b),
\quad\quad (X \in {\mathfrak a}Qq, u \in {\cal W}Qv, b \in {}^*\!A_{Q\iq}p({}^*P)).
\end{equation}
The degree of $q_\eta$ as a polynomial function
on ${\mathfrak a}Qq$ is at most $d.$
Combining this
with (\ref{e: expansion q eta}) and (\ref{e: rearranged expansion for f})
and using that $R_{Q,v}(bu) = R_{Q,v}(b),$ we arrive at the
expansion (\ref{e: expansion f along Q v}) for $m = bu$ and
$a \in A_{Q\iq}^+(R_{Q,v}(m)^{-1}).$
Using the left $K_Q$-invariance of $R_{Q,v}$ and the sphericality
of $f$ and the functions $m :to q_\eta(\log a, m),$ we now obtain
(\ref{e: expansion f along Q v}) with absolute convergence;
the first two assertions of (b) follow as well. The assertion
of neat convergence in (a) is a consequence of the final assertion in (b),
which we will now proceed to establish.
Let $u \in {\cal W}Qv$ and
$R > 1$ be fixed.
Then in view of the union (\ref{e: deco XQvp added R}) it
suffices to prove the neat
convergence of
the series (\ref{e: expansion f along Q v}) as a
${\msy D}rQ$-exponential polynomial series with coefficients
in $C^\infty(K_Q {}^*\!A_{Q\iq}p({}^*P)\lbr{R} \,u(M_Q \cap vHv^{-1})\,:\, \tau_Q).$ The
map $(k, a) :to kau(M_Q\cap vHv^{-1})$ induces a diffeomorphism from
$K_Q/(K_Q \cap vHv^{-1}) \times {}^*\!A_{Q\iq}p({}^*P)\lbr{R}$ onto the open subset
$K_Q {}^*\!A_{Q\iq}p({}^*P)\lbr{R}\,u(M_Q \cap vHv^{-1})$ of $\spX_{Q,v}p.$
By sphericality of the coefficients of the series
(\ref{e: expansion f along Q v}) we see
that it suffices to prove that
$$
\sum_{\eta \in s_Q - {\msy N} {\msy D}rQ} a^\eta \sigma_1(q_{Q,u,\eta})
$$
converges absolutely, for $a \in A_{Q\iq}^+(R^{-1})$ and for
$\sigma_1$ any continuous seminorm on $P_d({\mathfrak a}Qq) \otimes {\msy C}i({}^*\!A_{Q\iq}p({}^*P)\lbr{R}, V_\tauuvH).$
Fix $X \in {\mathfrak a}Qq,$ $\partial \in S({}^*\fa_{Q\iq}),$
$a \in A_{Q\iq}^+(R^{-1})$ and ${\cal K}\subset {}^*\!A_{Q\iq}p({}^*P)\lbr{R}$
a compact subset.
Then it suffices to prove that
\begin{equation}
\label{e: series with stdiffop to be estimated}
\sum_{\eta \in s_Q - {\msy N} {\msy D}rQ} a^\eta \sup_{\cal K} \| \partial ( q_{Q,u,\eta}(X, \,\cdot\,))\|
\end{equation}
converges absolutely.
{}From the neat convergence of the series (\ref{e: expansion q eta}),
for $b \in {}^*\!A_{Q\iq}({}^*P),$
it follows
that term by term differentiation is allowed. Since $\partial \in S({}^*\fa_{Q\iq}),$
whereas $X \in {\mathfrak a}Qq,$
we have
$$
b^{-\xi}\partial ( b^\xi T_X (q_{u,\xi})(\log b))
=
q_{\partial, u, \xi}(X + \log b).
$$
Hence, for every $\eta \in s_Q - {\msy N}{\msy D}rQ,$
\begin{equation}
\label{e: series for stdiffop q}
\partial ( q_{Q,u,\eta}(X, \,\cdot\,))(b) =
\sum_{\xi \in s - {\msy N} {\msy D}P\atop \xi|{\mathfrak a}Qq = \eta}
b^\xi q_{\partial, u, \xi}(X + \log b).
\end{equation}
There exists a continuous seminorm $\sigma_2$ on
$P_{d}({\mathfrak a}q) \otimes V_\tau,$
such that, for every $b \in {\cal K}$ and all
$q \in P_{d}({\mathfrak a}q) \otimes V_\tau,$
$$
\|q(X + \log b) \| \leq \sigma_2(q).
$$
In particular, this implies that
\begin{equation}
\label{e: estimate q on gs two}
\| q_{\partial, u, \xi}(X + \log b)\| \leq
\sigma_2(q_{\partial, u, \xi}),
\end{equation}
for every $b \in {\cal K}.$
Combining
(\ref{e: series for stdiffop q})
with (\ref{e: estimate q on gs two})
we now obtain
$$
| a^\eta | \sup_{{\cal K}} \| \partial(q_{Q,u,\eta})(X, \,\cdot\,)\|
\leq
\sum_{\xi \in s - {\msy N}{\msy D}P\atop \xi|{\mathfrak a}Qq = \eta}
a^{{\msy R}e \eta} b^{{\msy R}e \xi}\, \sigma_2(q_{\partial, u, \xi}).
$$
Thus, the absolute convergence of
(\ref{e: series with stdiffop to be estimated})
follows from the uniform convergence of
(\ref{e: series TAG}), $b \in {\cal K}.$
~
$\square$\medbreak\noindent\medbreak
Let $f \in \Cep({\rm X}p\col \tau)$
and let
$Q \in \cP_\gs$ and $v \in {\msy N}Kaq.$ Moreover,
let the set $Y \subset {\mathfrak a}Qqdc$ and the polynomials $q_\eta = q_\eta(Q,v\,|\, f),$
for $\eta \in Y- {\msy N}{\msy D}rQ$
be as in Theorem \ref{t: expansion along the walls}. As in that
theorem, we define
$$
{\rm Exp}(Q,v\,|\, f) = \{ \eta \in Y - {\msy N}{\msy D}rQ \mid q_\eta \neq 0\}
$$
and call the elements of this set the exponents
of $f$ along $(Q,v).$
If $\eta \in {\mathfrak a}Qqdc$ does not belong to ${\rm Exp}(Q,v\,|\, f),$ we agree to write
$q_\eta(Q,v\,|\, f) = 0.$
Let now $P \in \cP_\gs^{\rm min}$ be contained in
$Q$ and put ${}^*P:= P \cap M_Q.$ Then, for $u \in {\msy N}KQaq,$ we define
$$
{\rm Exp}(Q,v\,|\, f)_{P, u} = \{ \eta \in {\mathfrak a}Qqdc \mid q_{\eta} \neq 0 \text{on}
{\mathfrak a}Qq \times K_Q {}^*\! A_{Q\iq}^+({}^*P) u (M_Q \cap vHv^{-1}) \}.
$$
The elements of this set are called the $(Q,v)$-exponents
of $f$ on ${}^*\!A_{Q\iq}p({}^*P)u.$ Let ${\cal W}Qv \subset N_{K_Q}({\mathfrak a}q)$ be a complete
set of
representatives
of $W_Q / W_Q \cap W_{K \cap vHv^{-1}}.$ Then it follows from
(\ref{e: deco XQvp}) that
\begin{equation}
\label{e: Exp Q v f as union over index set cWQv}
{\rm Exp}(Q,v\,|\, f) = \bigcup_{u \in {\cal W}Qv} {\rm Exp}(Q,v\,|\, f)_{P, u}.
\end{equation}
We now have the following result.
\begin{thm}
\label{t: transitivity of asymptotics}
{\rm (Transitivity of asymptotics)\ }
Let $f \in \Cep({\rm X}p\col \tau).$
Let $P,Q \in \cP_\gs,$ assume that $P$ is minimal
and $P \subset Q$ and put ${}^*P = P \cap M_Q.$
Then for all $v \in {\msy N}Kaq$ and $u \in {\msy N}KQaq$ we
have:
\begin{equation}
\label{e: equality of exponents}
{\rm Exp}(Q,v \,|\, f)_{P,u} = {\rm Exp}(P, uv\,|\, f )\,|_{{\mathfrak a}Qq}.
\end{equation}
Moreover, if $\eta\in {\rm Exp}(P, uv\,|\, f ) |_{ {\mathfrak a}Qq},$
then for every $b \in {}^*\! A_{Q\iq}^+({}^*P),$ $X \in {\mathfrak a}Qq,$ and $m \in M,$
\begin{equation}
\label{e: q eta by transitivity}
q_{\eta}(Q,v \,|\, f , X, mbu) =
\sum_{\xi \in {\rm Exp}(P,uv\,|\, f ) \atop {\xi |_{{\mathfrak a}Qq} = \eta }}
b^\xi q_{\xi}(P, uv \,|\, f, X + \log b, m),
\end{equation}
where the ${\msy D}QP$-exponential polynomial series (in the variable $b$)
on the right is
neatly convergent on ${}^*\! A_{Q\iq}^+({}^*P).$
Furthermore,
the series
\begin{equation}
\label{e: series for q eta without m}
\sum_{\xi \in {\rm Exp}(P,uv\,|\, f ) \atop {\xi |_{ {\mathfrak a}Qq} = \eta }}
b^\xi q_{\xi}(P, uv \,|\, f, X + \log b)
\end{equation}
converges neatly as a ${\msy D}elta_Q(P)$-exponential polynomial series
in the variable $b \in {}^*\! A_{Q\iq}^+({}^*P)$
with coefficients in ${\msy C}i({\rm X}_{0,uv}\,:\, \tau_{\iM}).$
\end{thm}
\par\noindent{\bf Proof:}{\ }{\ } Let $v \in {\msy N}Kaq$ and $u \in {\msy N}KQaq$ be fixed.
Fix a set ${\cal W}_{Q,v}$ such as in the beginning of the proof
of Theorem \ref{t: expansion along the walls}, and such that it contains $u.$
Moreover, we select a set ${\cal W}$ of representatives for $W/W_{K \cap H}$ in ${\msy N}Kaq$
containing ${\cal W}_{Q,v}v.$ As in the proof of the mentioned
theorem we may restrict ourselves to the situation
that ${\rm Exp}(P, u'v\,|\, f) \subset s - {\msy N} {\msy D}P,$ for some $s \in {\mathfrak a}qdc$
and all $u' \in {\cal W}_{Q,v}.$ In the following we may now use the notation
and results of the proof
of Theorem \ref{t: expansion along the walls}.
Let $\eta \in s_Q - {\msy N} {\msy D}rQ.$ Then from
(\ref{e: defi q eta in proof}) and (\ref{e: expansion q eta}) we infer that, for
every $X \in {\mathfrak a}Qq,$
$$
q_\eta(Q,v\,|\, f, X, bu) = \sum_{\xi \in s - {\msy N} {\msy D}elta(P)\atop \xi|{\mathfrak a}Qq = \eta} b^\xi
q_\xi(P, uv\,|\, f, X + \log b, e), \quad\quad (b \in {}^*\!A_{Q\iq}p({}^*P));
$$
the series on the left-hand side converges neatly as a ${\msy D}QP$-exponential polynomial
series in the variable $b \in {}^*\!A_{Q\iq}p({}^*P).$
The function $m :to q_\eta(Q,v\,|\, f, X, mbu)$ belongs to ${\msy C}i({\rm X}_{0, uv}\,:\, \tau_{\iM}),$
and so does the function $m :to q_\xi(P, uv\,|\, f, X + \log b, m),$ for
every $\xi \in s - {\msy N}{\msy D}P.$ Evaluation at $e$ induces
a topological linear isomorphism
${\msy C}i(X_{0,w}\,:\, \tau_{\iM}) \simeq V_\tau^{M \cap w H w^{-1}},$ for every
$w \in {\msy N}Kaq,$ hence in particular
for $w = uv.$ Thus, it follows from the above that
(\ref{e: q eta by transitivity}) holds, with the asserted convergence. In addition,
it follows that the series (\ref{e: series for q eta without m}) converges as asserted.
In the proof of Theorem \ref{t: expansion along the walls} we saw that
${\rm Exp}(Q,v\,|\, f) \subset s_Q - {\msy N}{\msy D}rQ.$
It follows from the derived expansion (\ref{e: q eta by transitivity}) that
(\ref{e: equality of exponents}) holds with the inclusion `$\subset$'
in place of the equality sign.
For the converse inclusion, let $\xi_0 \in {\rm Exp}(P, uv\,|\, f )$ and
put $\eta = \xi_0 |_{ {\mathfrak a}Qq}.$ We select $X \in {\mathfrak a}Qq$ such
that the function $b :to q_{\xi_0}(P, uv \,|\, f, X + \log b, e)$ does not
vanish identically on ${}^*\!A_{Q\iq}.$ The equality (\ref{e: q eta by transitivity})
holds for all $b \in {}^*\!A_{Q\iq}p({}^*P)$ with a ${\msy D}elta_Q(P)$-exponential polynomial series
that converges neatly on ${}^*\!A_{Q\iq}p({}^*P).$ Any
exponent $\xi$ of this series
coincides with $\eta = \xi_0|_{{\mathfrak a}Qq}$ on ${\mathfrak a}Qq;$ if it also coincides with
$\xi_0$ on ${}^*\fa_{Q\iq},$ then $\xi = \xi_0.$
Therefore, the function of $b$ defined by the series
on the right-hand side of (\ref{e: q eta by transitivity})
is non-zero. Hence $q_\eta(Q,v\,|\, f)$ does not vanish identically
on ${\mathfrak a}Qq \times {}^*\!A_{Q\iq}p({}^*P)u$ and we conclude that
$\eta \in {\rm Exp}(Q,v \,|\, f)_{P,u}.$
~
$\square$\medbreak\noindent\medbreak
We proceed by discussing
some useful transformation properties
for the coefficients in the expansion (\ref{e: expansion f along Q v}).
If
$u \in {\msy N}Kaq$ it will sometimes be convenient to write $u X := {\rm Ad}(u) X$
for $X \in {\mathfrak a}q.$ Similarly, we will write $u\xi : = \xi \,{\scriptstyle\circ}\, {\rm Ad}(u)^{-1},$
for $\xi \in {\mathfrak a}qdc.$
If $u,v \in {\msy N}Kaq$ and $Q \in \cP_\gs,$ then conjugation by $u$ induces
a diffeomorphism $\gamma_{u}$
from the space ${\rm X}_{Q,v}$ onto ${\rm X}_{uQu^{-1}, uv};$
we note that $\gamma_u$ maps ${\rm X}Qvp$ onto ${\rm X}_{uQu^{-1}, uv, +}.$
It is easily seen that $R_{uQu^{-1},uv}(\gamma_u(m)) = R_{Q,v}(m),$
for $m \in {\rm X}Qv.$
For
$\varphi \in {\msy C}i({\rm X}_{Q,v,+}\,:\, \tau_Q),$ we define the function
$\rho_{\tau,u}\varphi: {\rm X}_{uQu^{-1}, uv, +} \rightarrow V_\tau$ by
\begin{equation}
\label{e: defi rho tau u}
\rho_{\tau,u}\varphi(x) = \tau(u) \varphi(\gamma_u^{-1}(x)).
\end{equation}
Then $\rho_{\tau,u}$ is a topological linear isomorphism from
the space
${\msy C}i({\rm X}_{Q,v,+}\,:\, \tau_Q)$ onto
the space
${\msy C}i({\rm X}_{uQu^{-1},uv,+} \,:\, \tau_{uQu^{-1}}).$
Likewise, by similar definitions we obtain a topological linear isomorphism
from
${\msy C}i({\rm X}_{1Q,v,+}\,:\, \tau_Q)$ onto
${\msy C}i({\rm X}_{1\,uQu^{-1},uv,+} \,:\, \tau_{uQu^{-1}}),$
also denoted by $\rho_{\tau,u}.$
\begin{lemma}
\label{l: transformation of coeffs}
Let $f \in \Cep({\rm X}p\,:\,\tau),$
let $Q\in\cP_\gs$ and $u,v \in {\msy N}Kaq.$
Then
$$
{\rm Exp}(uQu^{-1},uv\,|\, f) = u\, {\rm Exp}(Q,v\,|\, f).
$$
Moreover, for every $\eta \in {\rm Exp}(Q,v\,|\, f),$
$$
q_{u\eta} (uQu^{-1} ,uv\,|\, f) = [{\rm Ad}(u^{-1})^* \otimes \rho_{\tau,u}]\, q_\eta(Q,v\,|\, f).
$$
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
Put $Q' = uQu^{-1}.$
Let $m \in {\rm X}_{Q', uv,+}.$
Then, by Theorem \ref{t: expansion along the walls},
\begin{equation}
\label{e: first series transf rule}
f(ma uv) = \sum_{\eta \in {\rm Exp}(Q',uv\,|\, f)}
a^\eta q_\eta(Q',uv\,|\, f)(\log a, m),
\end{equation}
for $a \in A_{Q' {\rm q}}^+(R_{Q',uv}(m)^{-1}),$
where the series on the right-hand side is neatly convergent.
On the other hand, from $f(ma uv) = \tau(u) f(\gamma_u^{-1}(m ) u^{-1}au \, v)$
we see, using Theorem \ref{t: expansion along the walls} again, that
\begin{equation}
\label{e: second series transf rule}
f(mauv) = \tau(u) \sum_{\zeta \in {\rm Exp}(Q,v\,|\, f)}
a^{u\zeta} q_\zeta (Q,v\,|\, f)({\rm Ad}(u)^{-1}\log a, \gamma_u^{-1}(m)),
\end{equation}
for $u^{-1} a u \in A_{Q\iq}^+(R_{Q,v}(\gamma_u^{-1}(m))^{-1}).$
We now note that the latter condition is equivalent to
$$
a \in A_{Q'{\rm q}}^+(R_{Q,v}(\gamma_{u}^{-1}(m))^{-1})
= A_{Q'{\rm q}}^+(R_{Q', uv}(m)^{-1}).
$$
Hence the series (\ref{e: first series transf rule}) and
(\ref{e: second series transf rule}) both converge neatly for
$a \in A_{Q'{\rm q}}(R_{Q', uv}(m)^{-1}).$
All assertions now follow by uniqueness of asymptotics.
~
$\square$\medbreak\noindent\medbreak
For later purposes,
we also need another type of transformation property.
Recall from Remark \ref{r: extreme cases subspaces}
that for $u \in {\msy N}Kaq$ we write ${\rm X}u= {\rm X}_{1G,u} = G/uHu^{-1};$
let ${\rm X}up$ denote the analogue of ${\rm X}p$ for this symmetric space.
We note that right multiplication by $u$ induces a diffeomorphism $ r_u$
from ${\rm X}u$ onto ${\rm X},$ mapping ${\rm X}up$ onto ${\rm X}p.$
Hence pull-back by $r_u$ the topological
linear isomorphism $R_u: = r_u^*$ from
${\msy C}i({\rm X}p\,:\, \tau)$ onto ${\msy C}i({\rm X}up \,:\, \tau);$ it is given by $R_u f(x) = f(xu).$
We note that the map $R_u$ coincides with the map $\rho_{\tau, u},$
introduced in the text above Lemma \ref{l: transformation of coeffs},
by sphericality of the functions involved.
The following result
is now an immediate consequence of the definitions.
\begin{lemma}
\label{l: q of Rv f}
Let $f \in C^{\rm ep}({\rm X}p\,:\, \tau)$ and $u \in {\msy N}Kaq.$
Then $R_u f \in C^{\rm ep}({\rm X}up \,:\, \tau).$ Moreover, for each $Q \in \cP_\gs$
and every $v \in {\msy N}Kaq,$ the set ${\rm Exp}(Q,vu \,|\, f)$ equals ${\rm Exp}(Q, v \,|\, R_uf).$
Finally, if $\xi \in {\rm Exp}(Q,vu \,|\, f),$ then
$$
q_\xi(Q,vu\,|\, f) = q_\xi(Q,v\,|\, R_u f).
$$
\end{lemma}
\section{Behavior of differential operators along walls}
\label{s: diff op along walls}
We assume that $Q\in \cP_\gs$ is fixed.
The purpose of this section is to study a $Q$-radial decomposition
of invariant differential operators on ${\rm X}.$ This leads to
a series expansion of such operators along $(Q,e),$
with coefficients
that turn out to be globally defined on the group $M_Qgs.$
This will be of crucial importance for the applications
later on (see Proposition \ref{p: stability of globality new}).
The involution ${\msy C}artan\sigma$ fixes ${\mathfrak a}q$ pointwise, hence leaves every root space
${\mathfrak g}_\alpha,$ for $\alpha \in \Sigma,$ invariant. We denote the
associated
eigenspaces
of ${\msy C}artan\sigma|_{{\mathfrak g}_\alpha}$
for the eigenvalues $+1$ and $-1$ by ${\mathfrak g}_\alpha^+$ and ${\mathfrak g}_\alpha^-,$
respectively. Moreover, we put $m_\alpha^\pm := \dim {\mathfrak g}_\alpha^\pm.$
We recall that $K_Q = K \cap M_Q$ and $H_Q = H \cap M_Q.$
Define
$H_{1Q} := H \cap M_{1Q};$
then $H_{1Q} = H_Q (A_Q \cap H).$ Note that $K_Q = K \cap M_{1Q}.$
The group $M_{1Q}$ admits the Cartan decomposition
$M_{1Q} = K_Q A_\iq H_{1Q}$ and
normalizes the subalgebra
$\bar \fn_Q.$
For $m \in M_{1Q}$ we define the endomorphism $A(m) = A_Q(m)\in {\rm End}(\bar\fn_Q)$
by
\begin{equation}
\label{e: defi A m}
A(m):= \sigma \,{\scriptstyle\circ}\, {\rm Ad}(m^{-1}) \,{\scriptstyle\circ}\, {\msy C}artan \,{\scriptstyle\circ}\,{\rm Ad}(m).
\end{equation}
Moreover, we define the real analytic function $\delta = \delta_Q: M_{1Q} \rightarrow {\msy R}$
by
\begin{equation}
\label{e: defi gdQ}
\delta(m) = \det(I - A(m)).
\end{equation}
Finally, we define the following subset of $M_{1Q}$
\begin{equation}
\label{e: defi MoneQpr}
M_{1Q}pr := M_{1Q}\setminus \delta^{-1}(0).
\end{equation}
\begin{lemma}
\label{l: first lemma on A m}{\ }
\begin{enumerate}
\item[{\rm (a)}]
Let $m \in M_{1Q},$ $k \in K_Q$ and $h \in H_{1Q}.$ Then
$A(kmh) = {\rm Ad}(h^{-1}) \,{\scriptstyle\circ}\, A(m) \,{\scriptstyle\circ}\, {\rm Ad}(h).$
\minspace\item[{\rm (b)}]
The endomorphism $A(m) \in {\rm End}(\bar\fn_Q)$
is diagonalizable, for every $m \in M_{1Q}.$
The eigenvalues are given as follows.
Let $m = k a h,$ with $k \in K_Q,$ $a \in A_\iq$ and $h \in H_{1Q}.$
Then the eigenvalues of $A(m)$ are $\pm a^{-2\alpha},$ $\alpha \in \Sigma(Q),$
with multiplicities $m_{\alpha}^\pm.$
\minspace\item[{\rm (c)}]
The operator norm of $A(m)$
is given by $\|A(m)\|_{\rm op} = R_{Q,1}(m)^2.$
\end{enumerate}
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
(a) is an immediate consequence of (\ref{e: defi A m}).
Hence, for (b) we may assume that $m = a \in A_\iq.$ It is easily seen that
$A(a)|_{{\mathfrak g}^\pm_{-\alpha}} = \pm a^{-2\alpha} I$ for $\alpha\in\Sigma(Q)$.
Finally, (c) is an immediate consequence of (b) and
(\ref{e: value RQv})
with $v = 1.$
~
$\square$\medbreak\noindent\medbreak
\begin{cor}
If $k\in K_Q,$ $a \in A_\iq,$ $h \in H_{1Q}$ then
$$
\delta(kah) = {\rm pr}od_{\alpha \in \Sigma(Q)} (1 - a^{-2\alpha})^{m_\alpha^+}(1 + a^{-2\alpha})^{m_\alpha^-}.
$$
The set $M_{1Q}pr$ is left $K_Q$- and right $H_{1Q}$-invariant, and open dense
in $M_{1Q}.$
\end{cor}
\par\noindent{\bf Proof:}{\ }{\ }
This follows immediately from Lemma \ref{l: first lemma on A m} combined with
(\ref{e: defi gdQ}) and (\ref{e: defi MoneQpr}).
~
$\square$\medbreak\noindent\medbreak
We define the linear subspace $\fk{\scriptstyle(Q)}$ of ${\mathfrak k}$ by
$\fk{\scriptstyle(Q)}: = {\mathfrak k} \cap ({\mathfrak n}_Q + \bar \fn_Q).$ Then the
map $(I +{\msy C}artan): X :to X + {\msy C}artan X$
is a linear isomorphism from $\bar \fn_Q$ onto $\fk{\scriptstyle(Q)}.$
\begin{lemma}
\label{l: Ad m k and h deco}
{\ }
\begin{enumerate}
\item[{\rm (a)}]
If $m \in M_{1Q},$ then ${\rm Ad}(m^{-1}) \fk{\scriptstyle(Q)} + {\mathfrak h} \subset \bar\fn_Q + {\mathfrak h}.$
\minspace\item[{\rm (b)}]
If $m \in M_{1Q}pr,$ then ${\rm Ad}(m^{-1}) \fk{\scriptstyle(Q)} \oplus {\mathfrak h} = \bar\fn_Q + {\mathfrak h}.$
\end{enumerate}
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
(a) Since $\fk{\scriptstyle(Q)}\subset \bar\fn_Q + {\mathfrak n}_Q \subset \bar\fn_Q + {\mathfrak h},$
we have, for all $m \in M_{1Q},$
$$
{\rm Ad}(m^{-1}) \fk{\scriptstyle(Q)} \subset {\rm Ad}(m^{-1}) (\bar\fn_Q + {\mathfrak n}_Q) =
\bar\fn_Q + {\mathfrak n}_Q \subset \bar\fn_Q + {\mathfrak h}.
$$
(b)
The dimension of ${\rm Ad}(m^{-1}) \fk{\scriptstyle(Q)}$ equals that of $\fk{\scriptstyle(Q)},$
which in turn equals that of $\bar\fn_Q.$
Hence it suffices to prove, for $m \in M_{1Q}pr,$ that
${\rm Ad}(m^{-1}) \fk{\scriptstyle(Q)} \cap {\mathfrak h} = 0.$
Let $X \in {\rm Ad}(m^{-1})\fk{\scriptstyle(Q)} \cap {\mathfrak h}.$ Then ${\msy C}artan {\rm Ad}(m)X = {\rm Ad}(m) X$
and $\sigma X = X,$ and we see that $(I - A(m))X = 0.$ If $m \in M_{1Q}pr$
then $\det(I - {\rm Ad}(m)) = \delta(m) \not= 0 $ and it follows that $X = 0.$
~
$\square$\medbreak\noindent\medbreak
{}From Lemma \ref{l: Ad m k and h deco}(b) we see that for $m \in M_{1Q}pr$ we
may define linear maps ${P,s}i(m) = {P,s}i_Q(m) \in {\rm Hom}(\bar\fn_Q, \fk{\scriptstyle(Q)})$ and
$R(m) = R_Q(m) \in {\rm Hom}(\bar\fn_Q, {\mathfrak h})$ by
\begin{equation}
\label{e: defi Psi and R}
X = {\rm Ad}(m^{-1}) {P,s}i(m) X + R(m) X.
\end{equation}
\begin{lemma}
Let $m \in M_{1Q}pr,$ $k \in K_Q$ and $h \in H_{1Q}.$ Then
\begin{eqnarray*}
{P,s}i(kmh) &= & {\rm Ad}(k)\,{\scriptstyle\circ}\, {P,s}i(m) \,{\scriptstyle\circ}\, {\rm Ad}(h),\\
R(kmh) &=& {\rm Ad}(h^{-1}) \,{\scriptstyle\circ}\, R(m) \,{\scriptstyle\circ}\, {\rm Ad}(h).
\end{eqnarray*}
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
This is an immediate consequence of (\ref{e: defi Psi and R})
combined with Lemma \ref{l: Ad m k and h deco}(b).
~
$\square$\medbreak\noindent\medbreak
\begin{lemma}
\label{l: formulas for Psi and R}
Let $m \in M_{1Q}pr.$
Then
\begin{eqnarray*}
{P,s}i(m) \,{\scriptstyle\circ}\, (I - A(m)) &=& (I + {\msy C}artan) \,{\scriptstyle\circ}\, {\rm Ad}(m),\\
R(m) \,{\scriptstyle\circ}\, (I - A(m))\, &=& - (I + \sigma) \,{\scriptstyle\circ}\, A(m).
\end{eqnarray*}
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
{}From (\ref{e: defi A m}) it follows that
$$
I + \sigma \,{\scriptstyle\circ}\, A(m) = {\rm Ad}(m^{-1})\,{\scriptstyle\circ}\, (I + {\msy C}artan) \,{\scriptstyle\circ}\, {\rm Ad}(m).
$$
This implies in turn that
\begin{equation}
\label{e: expression I min A m}
I - A(m) = {\rm Ad}(m^{-1})\,{\scriptstyle\circ}\, (I + {\msy C}artan) \,{\scriptstyle\circ}\, {\rm Ad}(m) - (I + \sigma) \,{\scriptstyle\circ}\, A(m).
\end{equation}
Since $I + {\msy C}artan$ and $I + \sigma$ map $\bar\fn_Q$ into
$\fk{\scriptstyle(Q)}$ and ${\mathfrak h},$ respectively, the lemma follows from
combining (\ref{e: expression I min A m}) with (\ref{e: defi Psi and R}).
~
$\square$\medbreak\noindent\medbreak
\begin{cor}
The functions ${P,s}i: M_{1Q}pr \rightarrow {\rm Hom}(\bar\fn_Q, \fk{\scriptstyle(Q)})$
and $R: M_{1Q}pr \rightarrow {\rm Hom}(\bar\fn_Q, {\mathfrak h})$ are real analytic.
Moreover, the functions $\delta\,{P,s}i$ and $\delta \,R$ extend to
real analytic functions on $M_{1Q}.$
\end{cor}
\par\noindent{\bf Proof:}{\ }{\ }
{}From (\ref{e: defi gdQ}) and (\ref{e: defi MoneQpr}) we see
that $I - A(m)$ is an invertible endomorphism of $\bar\fn_Q,$ for $m \in M_{1Q}pr.$
Since ${\rm Ad}(m)$ and $A(m)$ depend real analytically on $m \in M_{1Q},$ all statements
now follow from Lemma \ref{l: formulas for Psi and R}.
~
$\square$\medbreak\noindent\medbreak
If $R >0,$ then in accordance with (\ref{e: defi spXQvb R}) we define
$$
M_{1Q}\br{R}: = \{ m\in M_{1Q} \mid R_{Q,1}(m) < R\}.
$$
Moreover, we set $M_Qgs\br{R}: = M_Qgs \cap M_{1Q}\br{R}.$
\begin{lemma}{\ }
\label{l: about MoneQpr}
\begin{enumerate}
\item[{\rm (a)}]
$M_{1Q}b{1} \subset M_{1Q}pr.$
\minspace\item[{\rm (b)}]
Let $R_1,R_2 > 0.$ Then $M_Qgsb{R_1}\,A_{Q\iq}^+(R_2) \subset M_{1Q}b{R_1 R_2}.$
\end{enumerate}
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
Let $m \in M_{1Q}\br{1}.$ Then $\|A(m)\|_{\rm op}<1$ by Lemma
\ref{l: first lemma on A m}(c), and hence $\delta(m)\neq0$.
This establishes (a).
Assume that $m \in M_Qgsb{R_1}$ and $a \in A_{Q\iq}^+(R_2).$
Write $m = k b h$ with $k \in K_Q,$ $b \in {}^*\!A_{Q\iq}$
and $h \in H_{1Q}.$ Then $ma = k (ab) h,$ hence $R_{Q,1}(ma) = \max_{\alpha \in \Sigma(Q)}
a^{-\alpha} b^{-\alpha} < R_2 R_{Q,1}(m)< R_1R_2.$ It follows that $ma \in M_{1Q}b{R_1 R_2}.$
~
$\square$\medbreak\noindent\medbreak
\begin{prop}
\label{p: series expansions for Psi and R}
There exist unique real analytic functions ${P,s}i_\mu, R_\mu: M_{Q\sigma} \rightarrow {\rm End}(\bar\fn_Q),$
for $\mu \in {\msy N} {\msy D}rQ,$ such that for every $m \in M_{Q\sigma}$ and every
$a \in A_{Q\iq}^+(R_{Q,1}(m)^{-1}),$
\begin{eqnarray*}
{P,s}i(ma) &=&
(1 + {\msy C}artan) \,{\scriptstyle\circ}\, \sum_{\mu \in {\msy N} {\msy D}rQ} a^{-\mu} {P,s}i_\mu(m), \\
R(ma) &=& (1 + \sigma) \,{\scriptstyle\circ}\, \sum_{\mu \in {\msy N} {\msy D}rQ} a^{-\mu} R_\mu(m),
\end{eqnarray*}
with absolutely convergent series.
For every $R>1$ the above series converge neatly on $A_{Q\iq}^+(R^{-1})$
as ${\msy D}rQ$-power series with coefficients in ${\msy C}i(M_Qgs[R], {\rm End}(\bar\fn_Q)).$
\end{prop}
\par\noindent{\bf Proof:}{\ }{\ }
Let $m \in M_Qgs$ and $a \in A_{Q\iq}^+(R_{Q,1}(m)^{-1}).$ It
follows from Lemma \ref{l: about MoneQpr}
that $ma \in M_{1Q}[1] \subset M_{1Q}pr.$
Hence ${P,s}i(ma)$ and $R(ma)$ are defined.
It follows from Lemma \ref{l: first lemma on A m} that $\|A(ma)\|_{\rm op} < 1.$
Hence the series
$$
(I - A(ma))^{-1} = \sum_{n=0}^\infty A(ma)^n
$$
converges absolutely. Let $\alpha \in \Sigma_r(Q).$ Then $A(m)$ leaves the space
${\mathfrak g}_{-\alpha}$ invariant, and $A(ma)|_{{\mathfrak g}_{-\alpha}} = a^{-2\alpha} A(m)|_{{\mathfrak g}_{-\alpha}}.$
Hence, in view of Lemma \ref{l: formulas for Psi and R},
$$
{P,s}i(ma)|_{{\mathfrak g}_{-\alpha}} =
(I + {\msy C}artan) \,{\scriptstyle\circ}\, {\rm Ad}(m) \,{\scriptstyle\circ}\, \sum_{n=0}^\infty a^{-(2n +1)\alpha}
A(m)^n |_{{\mathfrak g}_{-\alpha}}
$$
and
$$
R(ma)|_{{\mathfrak g}_{-\alpha}} = - (I + \sigma) \,{\scriptstyle\circ}\, \sum_{n=1}^\infty a^{-2n\alpha}
A(m)^n |_{{\mathfrak g}_{-\alpha}}.
$$
It is now easy to complete the proof.
~
$\square$\medbreak\noindent\medbreak
We denote by $\cR_Qplus$ the algebra of functions on $M_{1Q}pr$ generated by
the functions $\xi \,{\scriptstyle\circ}\, {P,s}i_Q,$ where $\xi \in {\rm Hom}(\bar \fnQ, \fk{\scriptstyle(Q)})^*,$
and by the functions $\eta \,{\scriptstyle\circ}\, {\rm R}Q,$ where $\eta \in {\rm Hom}(\bar \fnQ, {\mathfrak h})^*.$
By $\cR_Q$ we denote the algebra of functions generated by $1$ and $\cR_Qplus.$
Note that $\cR_Qplus$ is an ideal in $\cR_Q.$
\begin{cor}
\label{c: cor on ringQ}
The elements of $\cR_Q$ are left $K_Q$- and right $H_{1Q}$-finite functions
on $M_{1Q}pr.$
Let $\varphi \in \cR_Q.$ There exists a $k \in {\msy N}$ such that
$\gd_Q^k \varphi$ extends to a real analytic function on $M_{1Q}.$
Moreover, there exist unique real analytic functions $\varphi_\xi$
on $M_Qgs,$ for $\xi \in {\msy N}{\msy D}rQ,$ such that for every
$m \in M_Qgs$ and every $a \in A_{Q\iq}(R_{Q,1}(m)^{-1}),$
\begin{equation}
\label{e: series for phi}
\varphi(ma) =\sum_{\xi \in {\msy N}{\msy D}rQ} a^{-\xi} \varphi_\xi(m).
\end{equation}
Let $R{\rm ep}silonq 1.$ Then the series (\ref{e: series for phi})
converges neatly on $A_{Q\iq}(R^{-1}),$ as an exponential polynomial
series with coefficients in ${\msy C}i(M_Qgsb{R}).$
Finally, if $\varphi \in \cR_Qplus,$ then (\ref{e: series for phi})
holds with $\varphi_0 = 0.$
\end{cor}
\par\noindent{\bf Proof:}{\ }{\ }
Uniqueness of the functions $\varphi_\xi$ is obvious. Therefore it
suffices to prove existence and the remaining assertions.
One readily checks that it suffices to prove the
assertions for a collection of generators of the algebra $\cR_Qplus.$
Such a collection of generators is formed by the functions of
the form $\varphi = \xi \,{\scriptstyle\circ}\, {P,s}i,$ with $\xi \in {\rm Hom}(\bar \fnQ, \fk{\scriptstyle(Q)})^*,$
and by the functions of the form $\varphi = \eta \,{\scriptstyle\circ}\, {\rm R}Q,$ where $\eta \in {\rm Hom}(\bar \fnQ, {\mathfrak h})^*.$
For both types of generators all assertions follow immediately from
Proposition \ref{p: series expansions for Psi and R}.
~
$\square$\medbreak\noindent\medbreak
As is the previous section we assume that
$\tau$ is a smooth representation of $K$ in a locally convex space
$V_\tau.$
The space of continuous linear endomorphisms of $V_\tau$ is denoted
by ${\rm End}(V_\tau).$
If an element $u$ of the space
\begin{equation}
\label{e: defi cDoneQ}
{\cal D}oneQ : = \ringQ \otimes \End(\Vtau) \otimes U(\fmoneQ)
\end{equation}
is of the form
$ \varphi \otimes L \otimes v,$ with $\varphi \in \cR_Q,$ $L \in {\rm End}(V_\tau),$ and
$v \in U({\mathfrak m}oneQ),$ then we define the differential operator
$u_*$ on ${\msy C}i(M_{1Q}pr, V_\tau)$ by $u_*f = \varphi L\,{\scriptstyle\circ}\, [R_v f];$
here $R$ denotes the right regular representation.
The map $u :to u_*$ extends to an injective linear map from
${\cal D}oneQ$ to the
space of smooth ${\rm End}(V_\tau)$-valued differential operators of ${\msy C}i(M_{1Q}pr, V_\tau).$
We also define the subspace
$$
{\cal D}oneQplus := \ringQ \otimes \End(\Vtau) \otimes U(\fmoneQ)plus.
$$
Via the map $u :to 1 \otimes I \otimes u$ we identify $U({\mathfrak m}oneQ)$ with
a subspace of ${\cal D}oneQ.$ Then $u_* = R_u$ for $u \in U({\mathfrak m}oneQ).$
Let $M_{1Q}p$ be the preimage in $M_{1Q}$ of the set ${\rm X}_{1Q,1,+}$
(see below (\ref{e: deco XQvp})).
The set
$$
M_{1Q}ppr:= M_{1Q}p \cap M_{1Q}pr
$$
is an open dense subset
of $M_{1Q}$ that is left $K_Q$- and right $H_{1Q}$-invariant.
In view of the decomposition ${\mathfrak g} = \bar \fn_Q \oplus ({\mathfrak m}oneQ + {\mathfrak h}),$
there exists,
for every $D \in U({\mathfrak g}),$ an element $D_0 \in U({\mathfrak m}oneQ)$
with $\deg(D_0) \leq \deg(D),$ such that
\begin{equation}
\label{e: characterization D zero}
D - D_0 \in \bar \fnQ U({\mathfrak g}) + U({\mathfrak g}){\mathfrak h}.
\end{equation}
The element $D_0$ is uniquely determined
modulo $U({\mathfrak m}oneQ){\mathfrak h}oneQ.$
We recall from \bib{B91}, Sect.\ 2, see also \bib{BSft}, p.\ 548-549,
that the assignment $D :to D_0$ induces an algebra homomorphism
$\mu_Q'={}^\backprime\mu_{\bar Q}: {\msy D}X\rightarrow {\msy D}(M_{1Q}/H_{1Q}),$
and that the homomorphism
$\mu_Q: {\msy D}X\rightarrow {\msy D}(M_{1Q}/H_{1Q})$, defined by
$\mu_Q(D)= d_Q\,{\scriptstyle\circ}\, \mu_Q'(D) \,{\scriptstyle\circ}\, d_Q^{-1}$ with
$d_Q(m):=|\det({\rm Ad}(m)|_{{\mathfrak n}_Q})|^{1/2}$ for $m\inM_{1Q}$,
only depends on $Q$ through the Levi component $M_{1Q}$.
\begin{prop}
\label{p: radial deco with MQ}
Let $D \in {\msy D}X.$
There exists a $u_+ \in {\cal D}oneQplus$ of degree $\deg(u_+) < \deg (D)$
such that,
for every $f \in {\msy C}i({\rm X}p\,:\, \tau),$
$$
Df|_M_{1Q}ppr = [\mu_Q'(D) + u_{+*}](f|_M_{1Q}ppr).
$$
\end{prop}
\par\noindent{\bf Proof:}{\ }{\ }
By induction on the degree we will first establish the following assertion
for an element $D$ of $U({\mathfrak g}).$ Let $D_0 \in U({\mathfrak m}oneQ)$ satisfy
(\ref{e: characterization D zero}).
Then there exist finitely many $\varphi_i \in \cR_Qplus,$ $u_i \in U({\mathfrak k}),$
and $v_i \in U({\mathfrak m}oneQ),$ for $1\leq i \leq n,$ such that
$\deg(u_i) + \deg(v_i) < \deg(D),$ and such that
\begin{equation}
\label{e: general radial expression for D}
D - D_0 \equiv \sum_{i=1}^n \;\varphi_i(m)\, [{\rm Ad}(m)^{-1}u_i]\, v_i
\quad \text{mod} U({\mathfrak g}){\mathfrak h},
\end{equation}
for every $m\in M_{1Q}pr.$
The assertion is trivially true for $ D$ constant.
Thus, assume that
$D$ is not constant and that the assertion has been established for
$D$ of strictly smaller degree. Let $ D_0 \in U({\mathfrak m}oneQ)$ be as above.
Then, modulo $U({\mathfrak g}){\mathfrak h},$ $D - D_0$ equals a finite sum of terms
of the form $X D_1,$ with $X \in \bar \fnQ$ and $D_1 \in U(\bar \fn_Q \oplus {\mathfrak m}oneQ)$
such that $\deg D_1 < \deg D.$
For $m \in M_{1Q}ppr$ we have
$X = {\rm Ad}(m)^{-1} {P,s}i(m)X + {\rm R}Q(m)X;$ hence
$$
X D_1 \equiv ( {\rm Ad}(m)^{-1} {P,s}i(m)X) D_1 + [{\rm R}Q(m)X, D_1] \quad \text{mod} U({\mathfrak g}){\mathfrak h}.
$$
Now ${\rm Ad}(m)^{-1} {P,s}i(m)X$ is a finite sum of terms of
the form $\varphi(m) [{\rm Ad}(m)^{-1} u]$ with $u \in \fk{\scriptstyle(Q)}$ and $\varphi \in \cR_Qplus.$
Applying the induction hypothesis to $D_1$ we see that $[{\rm Ad}(m)^{-1}{P,s}i(m)X]D_1$
may be expressed as a sum similar to the one on the right-hand side of
(\ref{e: general radial expression for D}).
On the other hand, $[{\rm R}Q(m)X, D_1]$
is a finite sum of elements of the form $\psi(m) D_2,$
with $\psi \in \cR_Qplus$ and $D_2 \in U({\mathfrak g}),$ $\deg D_2 < \deg D.$
Applying the induction
hypothesis to $D_2,$ we see that $[{\rm R}Q(m)X, D_1]$ may also be expressed
as a sum of the form (\ref{e: general radial expression for D}).
This establishes the assertion involving
(\ref{e: general radial expression for D}) of the beginning of the proof.
Let now $D \in {\msy D}X.$ By abuse of notation we use the same symbol
$ D$ for a representative of $D$ in $U({\mathfrak g})^H,$ and
let $ D_0$ be as above. Then $\mu_Q'(D)$ equals the canonical image of
$ D_0$ in $U({\mathfrak m}oneQ)^{H_{1Q}}.$ Let $\varphi_i, u_i, v_i$ be as above and such that
(\ref{e: general radial expression for D}) holds.
Then for every $f \in {\msy C}i({\rm X}p \,:\, \tau)$ and all $m\in M_{1Q}ppr$ we have
\begin{eqnarray*}
Df(m) & =& \mu_Q'(D) (f|_M_{1Q}ppr)(m) + \sum_{i=1}^n \varphi_i(m) R_{{\rm Ad}(m)^{-1}u_i}R_{v_i} f(m)
\\
&=& \mu_Q'(D) (f|_M_{1Q}ppr)(m) + \sum_{i=1}^n \varphi_i(m) \tau(\check u_i)R_{v_i} f(m)
\end{eqnarray*}
where we have used that $R_{{\rm Ad}(m)^{-1}u_i}R_{v_i} f(m) =
L_{\check u_i} R_{v_i} f(m) =
\tau(u_i) R_{v_i} f(m).$
Thus, we obtain the desired expression with
$
u_+ = \sum_{i=1}^n \varphi_i \otimes \tau(u_i) \otimes v_i.
$
~
$\square$\medbreak\noindent\medbreak
Let $U \subset M_{Q\sigma}$ be an open subset.
It will be convenient to be able to refer to a `formal application' of
elements of the space
${\cal D}oneQ,$ defined in (\ref{e: defi cDoneQ}),
to
${\cal F}^{\rm ep}(A_{Q\iq}, C^\infty(U, V_\tau)),$ the space of (formal) ${\msy D}rQ$-exponential polynomial series
with coefficients in $C^\infty(U, V_\tau),$ see the definition preceding
Lemma \ref{l: formal application of Ufa}. There is
a natural way to define a formal application that is
compatible with the expansions of Corollary \ref{c: cor on ringQ} and with
the map $u :to u_*,$ defined in the text following
(\ref{e: defi cDoneQ}). The motivation for the following somewhat
tedious chain of definitions will become clear in Lemma
\ref{l: formal application of cDoneQ}.
The product decomposition $M_{1Q} \simeq M_{Q\sigma} \times A_{Q\iq}$ induces
a natural isomorphism
from $U({\mathfrak m}oneQ)$ onto $U({\mathfrak m}Qgs) \otimes U({\mathfrak a}Qq),$
by which we shall identify.
Accordingly we have a natural isomorphism
\begin{equation}
\label{e: tensor prod deco of cDoneQ}
{\cal D}oneQ \simeq {\cal D}oneQgs \otimes U({\mathfrak a}Qq),
\end{equation}
where ${\cal D}oneQgs:= {\cal R}_Q \otimes {\rm End}(V_\tau) \otimes U({\mathfrak m}_{Q\sigma}).$
To each element $\varphi \in {\cal R}_Q$ we may associate its ${\msy D}rQ$-exponential polynomial
series of the form (\ref{e: series for phi}); this induces a linear
embedding ${\cal R}_Q \rightarrow {\cal F}^{\rm ep}(A_{Q\iq}, C^\infty(M_Qgs))$ which by identity on the
other tensor components may be extended to a linear embedding
$$
{\cal D}oneQgs \rightarrow {\cal F}^{\rm ep}(A_{Q\iq}, {\cal D}Qgs),
$$
where
${\cal D}Qgs := C^\infty(M_Qgs) \otimes {\rm End}(V_\tau) \otimes U({\mathfrak m}Qgs).$
By identity on the second tensor component in (\ref{e: tensor prod deco of cDoneQ})
this embedding extends to a
linear embedding
\begin{equation}
\label{e: the embedding ep}
{\rm ep}:\;\;{\cal D}oneQ \rightarrow {\cal F}^{\rm ep}(A_{Q\iq}, {\cal D}Qgs) \otimes U({\mathfrak a}Qq).
\end{equation}
The image ${\rm ep}(u)$ of an element $u \in {\cal D}oneQ$ under this embedding will be called
the ${\msy D}rQ$-exponential polynomial expansion of $u.$
Via the right regular action of $U({\mathfrak m}Qgs)$ we may naturally
identify ${\cal D}Qgs$
with the space of $C^\infty$-differential operators
acting on $C^\infty(M_Qgs, V_\tau).$
Accordingly, we have a continuous bilinear pairing
${\cal D}_{Q\sigma} \times {\msy C}i(U, V_\tau) \rightarrow {\msy C}i(U, V_\tau).$
This induces a formal application map
from ${\cal F}^{\rm ep}(A_{Q\iq}, {\cal D}Qgs)\otimes {\cal F}^{\rm ep}(A_{Q\iq}, C^\infty(U, V_\tau))$ to
${\cal F}^{\rm ep}(A_{Q\iq}, C^\infty(U, V_\tau))$ in the fashion described above Lemma
\ref{l: formal application of hom valued series}. The image of an element of the form
$u \otimes f$ under this map will be denoted by $uf.$
On the other hand,
in Lemma \ref{l: formal application of Ufa}
we described the formal
application map
$U({\mathfrak a}Qq) \otimes {\cal F}^{\rm ep}(A_{Q\iq}, C^\infty(U, V_\tau)) \rightarrow {\cal F}^{\rm ep}(A_{Q\iq}, C^\infty(U, V_\tau)).$
The image of an element of the
form $v \otimes f$ under this map is denoted by $vf.$
Combination of the above formal application maps
leads to the formal application map
$$
[{\cal F}^{\rm ep}(A_{Q\iq}, {\cal D}Qgs) \otimes U({\mathfrak a}Qq)] \otimes {\cal F}^{\rm ep}(A_{Q\iq}, C^\infty(U, V_\tau)) \rightarrow
{\cal F}^{\rm ep}(A_{Q\iq}, C^\infty(U, V_\tau)),
$$
given by $(u \otimes v) \otimes f :to (u\otimes v)f:= u(vf),$
for $u \in {\cal F}^{\rm ep}(A_{Q\iq}, {\cal D}Qgs),$ $v\in U({\mathfrak a}Qq)$
and $f \in {\cal F}^{\rm ep}(A_{Q\iq}, C^\infty(U, V_\tau)).$
Composing with the embedding (\ref{e: the embedding ep}) we finally obtain the linear map
$$
{\cal D}oneQ
\otimes {\cal F}^{\rm ep}(A_{Q\iq}, C^\infty(U, V_\tau)) \rightarrow
{\cal F}^{\rm ep}(A_{Q\iq}, C^\infty(U, V_\tau))
$$
given by $u \otimes f :to uf:= {\rm ep}(u)f,$ for $u \in {\cal D}oneQ$
and $f \in {\cal F}^{\rm ep}(A_{Q\iq}, C^\infty(U, V_\tau)).$ We shall call this map the
formal application of ${\cal D}oneQ$
to ${\cal F}^{\rm ep}(A_{Q\iq}, C^\infty(U, V_\tau)).$
Let now $R{\rm ep}silonq 1$ and let $U \subset M_Qgsb{R}$ be an open subset.
We use the obvious natural isomorphism to identify the space
$C^{\rm ep}(A_{Q\iq}^+(R^{-1}), C^\infty(U, V_\tau))$ with
a subspace of $C^\infty(UA_{Q\iq}^+(R^{-1}), V_\tau).$ If
$u \in {\cal D}oneQ,$ then the associated differential operator
$u_*$ induces a map from the first space into the latter.
\begin{lemma}
\label{l: formal application of cDoneQ}
Let $u \in {\cal D}oneQ,$ let $R{\rm ep}silonq 1$ and let $U\subset M_Qgsb{R} $ be an open
subset.
Then $u_*$ maps the space $C^{\rm ep}(A_{Q\iq}^+(R^{-1}), C^\infty(U, V_\tau))$
into itself. Moreover, if $f$ belongs to that space,
then the ${\msy D}rQ$-exponential polynomial expansion of $u_*f$ is obtained
from the formal application of $u$ to the exponential polynomial expansion of $f.$
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
This follows from retracing the definitions of $u_*$ and of the formal application
of $u$ given above
and applying Corollary \ref{c: cor on ringQ} and Lemmas
\ref{l: formal application of Ufa} and
\ref{l: formal application of hom valued series}.
~
$\square$\medbreak\noindent\medbreak
Given $v\in{\msy N}Kaq$ we define
$\mu_{Q,v}: {\msy D}X\rightarrow {\msy D}({\rm X}oneQv)={\msy D}(M_{1Q} / M_{1Q} \cap vHv^{-1})$
by
$$\mu_{Q,v}={\rm Ad}(v)\,{\scriptstyle\circ}\,\mu_{v^{-1}Qv},$$
where ${\rm Ad}(v): {\msy D}({\rm X}_{1v^{-1}Qv,e})\rightarrow
{\msy D}({\rm X}oneQv)$ is induced by the restriction to $U({\mathfrak m}_{1v^{-1}Qv})$
of ${\rm Ad}(v)$ on $U({\mathfrak g})$.
Then $\mu_{Q,v}$ depends on $Q$ only through $M_{1Q}$. It is easily seen
that
\begin{equation}
\label{e: muQv}
\mu_{Q,v}=\mu_Q^v\,{\scriptstyle\circ}\,{\rm Ad}(v)
\end{equation}
where $\mu_Q^v: {\msy D}Xv={\msy D}(G/vHv^{-1})\rightarrow {\msy D}({\rm X}oneQv)=
{\msy D}(M_{1Q} / M_{1Q} \cap vHv^{-1})$ is defined similarly as $\mu_Q$, but
with $H$ replaced by $vHv^{-1}$, and where ${\rm Ad}(v): {\msy D}X\rightarrow
{\msy D}Xv$ is induced by ${\rm Ad}(v)$ on $U({\mathfrak g})$.
Let $M_Qgsp=M_Qgs\capM_{1Q}p$ and, for $R{\rm ep}silonq 1$,
$M_Qgsp[R]=M_Qgs[R]\capM_{1Q}p$.
\begin{lemma}
\label{l: radial component applied to expansion}
Let $f\in {\msy C}ep({\rm X}p\,:\, \tau)$ and let $D \in {\msy D}X$.
Then $Df \in {\msy C}ep({\rm X}p\,:\, \tau).$
Let $Q \in \cP_\gs$ and let
$u_+ \in {\cal R}_Q^+ \otimes {\rm End}(V_\tau) \otimes U({\mathfrak m}oneQ)$
be associated with $D$ as in Proposition \ref{p: radial deco with MQ}.
Then the following holds.
\begin{enumerate}
\item[{\rm (a)}]
The ${\msy D}rQ$-exponential expansion of $Df$ along $(Q,e)$ is
obtained by the formal application of $\mu_Q'(D) + u_+$ to the ${\msy D}rQ$-exponential polynomial expansion
of $f$ along $(Q,e).$
\minspace\item[{\rm (b)}]
Let $v \in {\msy N}Kaq$, then
$
{\rm Exp}(Q,v\,|\, Df) \subset {\rm Exp}(Q,v\,|\, f) - {\msy N}{\msy D}rQ.
$
\minspace\item[{\rm (c)}]
If $\xi$ is a leading exponent of $f$ along $(Q,v),$ then
\begin{equation}
\label{e: poly of Df}
a^{\xi+\rho_Q} q_\xi(Q,v \,|\, Df, \log a, m) = [\mu_{Q,v}(D) \varphi](m a),
\quad\quad (m \in M_Qgsp,\;a\in A_{Q\iq}),
\end{equation}
where the function $\varphi: M_{1Q}p \rightarrow V_\tau$ is defined by
$\varphi(ma) = a^{\xi+\rho_Q} q_\xi(Q,v\,|\, f, \log a, m),$
for $m \in M_Qgsp$ and $a\in A_{Q\iq}.$
\end{enumerate}
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
Let $R{\rm ep}silonq 1$ and let $\underline f$ be the function $A_{Q\iq}^+(R^{-1}) \rightarrow
{\msy C}i(M_Qgsp[R], V_\tau)$
defined by $\underline f (a, m) = f(ma).$ It follows from the hypothesis on $f$
and Theorem \ref{t: expansion along the walls} that
$\underline f(a,m)$ belongs to
$C^{\rm ep}(A_{Q\iq}^+(R^{-1}), {\msy C}i(M_Qgsp[R], V_\tau)).$
Moreover, its ${\msy D}rQ$-exponential polynomial expansion coincides with the expansion
of $f$ along $(Q,e).$
Put $u = \mu_Q'(D) + u_+.$
Then it follows from the previous lemma that $u_*\underline f$ belongs to
$C^{\rm ep}(A_{Q\iq}^+(R^{-1}), {\msy C}i(M_Qgsp[R], V_\tau));$ its expansion is obtained from the
formal application of $u$ to the $(Q,e)$-expansion of $f.$ It follows from
Theorem \ref{t: expansion along the walls}
that the expansion is independent of $R$ and that its coefficients
are functions in $C^\infty(M_Qgsp, V_\tau).$
On the other hand, it follows from
Proposition \ref{p: radial deco with MQ} that $u_*\underline f(a, m) = Df(ma).$ This implies that
$Df$ has a ${\msy D}rQ$-exponential polynomial expansion along $(Q,e)$ with coefficients in
$C^\infty(M_Qgsp, V_\tau).$ Since $Df$ is right $H$-invariant, the coefficients
are actually functions in $C^\infty({\rm X}Qep, V_\tau).$ Moreover,
the expansion is independent of $R$ and converges neatly on $A_{Q\iq}^+(R^{-1})$ as an expansion with
coefficients in ${\msy C}i(\XQepb{R},V_\tau).$ In particular this holds for every
minimal parabolic
subgroup $Q;$
hence $Df \in C^{\rm ep}({\rm X}p\,:\, \tau).$
In the above we have established assertion (a). It follows from this assertion that
(b) holds with $v =1$ for every $Q \in \cP_\gs.$ By Lemma \ref{l: transformation of coeffs}
it also holds for arbitrary
$Q\in \cP_\gs$ and $v \in {\msy N}Kaq.$
It remains to establish (c). Assume first that $v=e$.
Fix $\xi \in {\rm Exp}L(Q,e\,|\, f).$
Then by (a), $a^{\xi} q_\xi(Q,e \,|\, Df, \log a, m)$ is the term with
exponent $\xi$ in the series that arises from the formal application
of $\mu_Q'(D) + u_+$ to the $(Q,e)$-expansion of $f.$
The exponents of the expansion ${\rm ep}(u_+)$ of $u_+$ all belong to
$-[{\msy N} {\msy D}rQ]\setminus \{0\}.$ The application of
$u_+$ therefore
gives rise to an expansion with exponents in ${\rm Exp}(Q,e\,|\, f) -
[{\msy N} {\msy D}rQ]\setminus \{0\}.$
The latter set does not contain $\xi,$ since $\xi$ is leading.
Hence $a^{\xi} q_\xi(Q,e \,|\, Df, \log a, m)$
is the term with exponent $\xi$ in the
expansion that arises from the formal application of $\mu_Q'(D)$ to the
$(Q,e)$-expansion of $f.$
Now $\mu_Q'(D) \in U({\mathfrak m}oneQ) \simeq U({\mathfrak m}Qgs) \otimes U({\mathfrak a}Qq)$
and we see that the formal
application of $\mu_Q'(D)$ to the $(Q,e)$ expansion of $f$
is induced by term by term differentiation
in the $A_{Q\iq}$ and the $M_Qgs$ variables. This implies that
$a^{\xi} q_\xi(Q,e \,|\, Df, \log a, m) = [\mu_{Q}'(D) \varphi'](m a),$
where $\varphi'(ma) = a^{\xi} q_\xi(Q,e\,|\, f, \log a, m).$ This implies
(\ref{e: poly of Df}) for $v=e$.
Let now $v \in {\msy N}Kaq$ be arbitrary, and put $f^v=R_vf$. We shall apply the
version of (\ref{e: poly of Df}) just established to the expansion
along $(Q,e)$ of the function $f^v$ on ${\rm X}v$. Let $\xi$
be a leading exponent of $f$ along $(Q,v),$ then it follows from Lemma
\ref{l: q of Rv f} that $\xi$ is also a
leading exponent of $f^v$ along $(Q,e)$. Moreover, let $D\in{\msy D}X$, then
$(Df)^v=D^vf^v$ where $D^v:={\rm Ad}(v)D\in{\msy D}Xv$. Hence
\begin{equation}
\label{e: poly of Dvfv}
a^{\xi+\rho_Q} q_\xi(Q,e \,|\, (Df)^v, \log a, m) =
[\mu_Q^v(D^v) \varphi](m a),
\end{equation}
for $m \in M_Qgsp,\;a\in A_{Q\iq}$,
where $\varphi(ma) = a^{\xi+\rho_Q} q_\xi(Q,e\,|\, f^v, \log a, m)$.
It follows from Lemma \ref{l: q of Rv f} that
$\varphi(ma) = a^{\xi+\rho_Q} q_\xi(Q,v\,|\, f, \log a, m),$
and
$q_\xi(Q,e \,|\, (Df)^v)=q_\xi(Q,v \,|\, Df)$.
Now (\ref{e: poly of Df}) follows from (\ref{e: poly of Dvfv}) and
(\ref{e: muQv}).
~
$\square$\medbreak\noindent\medbreak
\begin{lemma}
\label{l: action of DX on splitting}
Let $P \in \cP_\gs^{\rm min}$ and assume that
$f \in \Cep({\rm X}p\col \tau).$ Let $S \subset {\mathfrak a}qdc$ be a finite
set as in Lemma \ref{l: splitting lemma}, and let $D \in {\msy D}X.$
Then ${\rm Exp}(P,v\,|\, Df) \subset S -{\msy N}\Delta$ for every $v \in{\msy N}Kaq$
and,
with notation as in Lemma \ref{l: splitting lemma},
\begin{equation}
\label{e: D of fs}
(Df)_s = D(f_s).
\end{equation}
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
It follows immediately from
Lemma \ref{l: radial component applied to expansion}(b)
that ${\rm Exp}(P,v\,|\, Df) \subset S -{\msy N}\Delta$ and that
${\rm Exp}(P,v\,|\, D(f_s)) \subset s -{\msy N}\Delta$ for $s\in S$.
Now (\ref{e: D of fs}) follows from Lemma \ref{l: splitting lemma}.
~
$\square$\medbreak\noindent\medbreak
\section{Spherical eigenfunctions}
In this section we assume that $(\tau, V_\tau)$ is a
finite dimensional continuous representation of $K.$
Let $I$ be a cofinite ideal of the algebra
${\msy D}GH.$ Then by $\Ci({\rm X}p \col \tau)I$
we denote the
space of $f \in \Ci({\rm X}p \col \tau)$ satisfying the system
of differential equations
$$
Df = 0, \quad\quad (D \in I).
$$
\begin{rem}
\label{r: extended results}
Many results of \bib{B87} that are formulated
for ${\msy D}GH$-finite $\tau$-spherical functions
on ${\rm X}$ are actually valid for the bigger class
of ${\msy D}GH$-finite
functions in $\Ci({\rm X}p \col \tau)$ as well, since their proofs
only involve behavior of functions and operators on
${\rm X}p.$ If such extended results are used in the
text, we may give a reference to the present remark.
\end{rem}
\begin{rem}
\label{r: right translate by v}
Let $v \in N_K({\mathfrak a}q).$
We recall from the text preceding Lemma \ref{l: q of Rv f}
that right translation by $v$ induces a topological linear
isomorphism $R_v$ from $\Ci({\rm X}p \col \tau)$ onto the space $C^\infty(\spX_{v,+} \col \tau).$
It maps the subspace of ${\msy D}GH$-finite functions onto
the subspace of ${\msy D}Xv$-finite functions. Thus,
if $f \in \Ci({\rm X}p \col \tau)$ is a ${\msy D}X$-finite function, then the theory
of \bib{B87} may be applied to the ${\msy D}Xv$-finite function $R_vf;$ the results
are then easily reformulated in terms of the function $f.$
\end{rem}
\begin{lemma}
\label{l: DX finite in exppol}
Let $I \subset {\msy D}X$ be a cofinite ideal. Then
$\Ci({\rm X}p \col \tau)I \subset \Cep({\rm X}p\col \tau).$
In particular, the elements of $\Ci({\rm X}p \col \tau)I$ are real analytic
functions on ${\rm X}p$.
Moreover,
there exists a finite set $X_I\subset {\mathfrak a}qdc$
such that ${\rm Exp}L(P,v\,|\, f)\subset X_I,$ for all
$f \in \Ci({\rm X}p \col \tau)I,$ $P\in \cP_\gs^{\rm min}$
and $v \in {\msy N}Kaq.$
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
Let $Q \in \cP_\gs^{\rm min}.$ Applying Theorem
2.5 of \bib{B87}, see Remark \ref{r: extended results},
we obtain that $f|A_\iq^+(Q)$ is given by a neatly converging
$\Delta(Q)$-exponential polynomial expansion
for each $f \in \Ci({\rm X}p \col \tau)I$. Moreover, by Theorem
2.4 of \bib{B87}, there exists a finite set $X_{I,Q,e}\subset {\mathfrak a}qdc$,
such that ${\rm Exp}L(f|A_\iq^+(Q))\subset X_{I,Q,e}.$
Let $w \in {\cal W}.$ Applying the above argument to $R_wf,$
cf.\ Remark \ref{r: right translate by v}, we see, more generally,
that $T^\downarrow_{Q,w} f$ is given by the same type of expansion with
leading exponents in a finite set $X_{I,Q,w}\subset{\mathfrak a}qdc$
independent of $f$.
This implies that $f \in \Cep({\rm X}p\col \tau),$ with
${\rm Exp}L(P,v\,|\, f)\subset X_I:=\cup_{Q,w} X_{I,Q,w}$,
for all $P\in \cP_\gs^{\rm min}$ and $v \in {\cal W}.$
Finally, if $v \in {\msy N}Kaq$ is arbitrary,
there exists $w\in{\cal W}$, $m\inK_\iM$ and
$h\in N_{K\cap H}({\mathfrak a}q)$ such that $v=mwh$, and then
${\rm Exp}L(P,v\,|\, f)={\rm Exp}L(P,w\,|\, f)\subset X_I$.
~
$\square$\medbreak\noindent\medbreak
\begin{cor}
Let $P \in \cP_\gs^{\rm min}$ and let ${\cal W} \subset {\msy N}Kaq$ be a
complete set of representatives
of $W/W_{K \cap H}.$
Let $I$ be a cofinite ideal in ${\msy D}X.$
Then there exists a finite set $S= S_I$ satisfying the properties of
Lemma \ref{l: splitting lemma} for every $f \in {\msy C}epXpI.$ Moreover,
if $S_I$ is any such set, then $f_s \in {\msy C}epXpI$
for every $f\in {\msy C}epXpI$ and all $s \in S_I.$
\end{cor}
\par\noindent{\bf Proof:}{\ }{\ }
This is an immediate consequence of Lemmas
\ref{l: DX finite in exppol}
and
\ref{l: action of DX on splitting}.
~
$\square$\medbreak\noindent\medbreak
The set $X_I$ in Lemma \ref{l: DX finite in exppol}
can be described more explicitly if the ideal $I$ has codimension $1.$
Let ${\mathfrak b}$ be a maximal abelian subspace of ${\mathfrak q}$ containing ${\mathfrak a}q,$
let $\Sigma({\mathfrak b})$ be the restricted root system of ${\mathfrak b}$ in ${\mathfrak g}_{\scriptscriptstyle \C},$
and let $W({\mathfrak b})$ be the associated reflection group.
Let $\gamma$ be the Harish-Chandra isomorphism from ${\msy D}X$
onto the algebra $I({\mathfrak b})$ of $W({\mathfrak b})$-invariants in $S({\mathfrak b}),$
see \bib{B91}, Sect.\ 2.
To an element $\nu \in {\mathfrak b}dc$ we associate the character
$D :to \gamma(D\,:\, \nu)$ of ${\msy D}X$ and denote its kernel by $I_\nu.$
Then $I_\nu$ is
an ideal of codimension one in ${\msy D}X;$ in fact,
any codimension one ideal is of this form.
Let $W_0({\mathfrak b})$
be the normalizer of ${\mathfrak a}q$ in $W({\mathfrak b}).$ Then restriction
to ${\mathfrak a}q$ induces an epimorphism from $W_0({\mathfrak b})$ onto $W,$ cf.\ \bib{B91},
Lemma 4.6.
We put ${\mathfrak b}_{\rm k}:= {\mathfrak b} \cap {\mathfrak k}.$ Then ${\mathfrak b} = {\mathfrak b}k \oplus {\mathfrak a}q.$
Moreover, this decomposition is invariant under $W_0({\mathfrak b}).$
\begin{lemma}
\label{l: restriction on leading exponents}
There exists a finite subset ${\cal L} = {\cal L}_\tau$ of ${\mathfrak b}kdc$ with the following
property.
Let $\nu \in {\mathfrak b}dc$ and $f \in {\msy C}i({\rm X}p\,:\, \tau\,:\, I_\nu).$
Let $P \in \cP_\gs^{\rm min}, v \in {\msy N}Kaq$ and assume that
$\xi \in {\rm Exp}L(P,v\,|\, f).$
Then
$$
\nu \in W({\mathfrak b})({\cal L} + \xi + \rho_P).
$$
\end{lemma}
The proof is based on the following result, which will be proved first.
\begin{lemma}
\label{l: eigenfunctions on M1}
There exists a finite subset ${\cal L} = {\cal L}_\tau$ of ${\mathfrak b}kdc$ with the following
property. Let $\nu \in {\mathfrak b}dc$ and $\varphi\in{\msy C}i(M_{1}/H_\iMone\,:\,\tau)$,
and assume that
$$
\mu_P(D) \varphi = \gamma(D\,:\, \nu) \varphi
$$
for all $D\in{\msy D}GH$, where $\mu_P: {\msy D}GH\rightarrow {\msy D}(M_{1}/H_\iMone)$ is as defined
above Proposition \ref{p: radial deco with MQ}, with $P\in\cP_\gs^{\rm min}$.
Then $\varphi|_{A_\iq}$ is a linear combination of exponential polynomials of
the form $a:to p(\log a) a^{w\nu}$, where $p\in P({\mathfrak a}q)$ and where
$w\in W({\mathfrak b})$ satisfies $w\nu|_{{\mathfrak b}k}\in{\cal L}$.
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
The algebra ${\msy D}(M/H_\iM)$ acts semisimply on
${\msy C}i(M/H_\iM \,:\, \tau),$ see \bib{B91}, Lemma 4.8;
let ${\cal L}$ be the (finite) set of $\Lambda \in {\mathfrak b}kdc$ such that
the associated character of ${\msy D}(M/H_\iM)$ occurs.
We may assume that $\varphi$ is a joint eigenfunction for ${\msy D}(M/H_\iM)$,
with eigenvalue character given by $\Lambda\in{\cal L}$.
It follows that
$$ (D\varphi)|_{A_\iq}= \gamma_{{\rm M}_1}(D\,:\,\Lambda+\,\cdot\,)(\varphi|_{A_\iq})$$
for $D\in{\msy D}MoneH\simeq{\msy D}(M/H_\iM)\otimes S({\mathfrak a}q)$.
Here $\gamma_{{\rm M}_1}$ denotes the Harish-Chandra isomorphism
from ${\msy D}MoneH$ into $S({\mathfrak b}),$ defined as in \bib{BSmc}, above eq.~(2.11),
and $\gamma_{{\rm M}_1}(D\,:\,\Lambda+\,\cdot\,)\in S({\mathfrak a}q)$ is considered as a
differential operator on $A_\iq$. Combining this identity with the
assumption on $\varphi$, the identity
$\alphamma_{{\rm M}_1}\,{\scriptstyle\circ}\,\mu_P=\gamma$, and the surjectivity of
$\alphamma:{\msy D}GH \rightarrow S({\mathfrak b})^{W({\mathfrak b})},$
it follows that
$$u(\Lambda+\,\cdot\,)(\varphi|_{A_\iq})=u(\nu)\varphi|_{A_\iq}$$
for all $u\in S({\mathfrak b})^{W({\mathfrak b})}.$ Let $\tilde\varphi\in C^\infty({\mathfrak b})$ be
defined by $\tilde\varphi(X+Y)=e^{\Lambda(X)}\varphi(\exp Y)$ for
$X\in{\mathfrak b}k$, $Y\in{\mathfrak a}q$, then $u\tilde\varphi=u(\nu)\tilde\varphi$.
This implies that $\tilde\varphi$ is a linear combination of exponential
polynomials of the form $p\, e^{w\nu}$, where $p\in P({\mathfrak b})$ and
$w\in W({\mathfrak b})$, see \bib{Helgason}, Thm.\ III.3.13. However, from the
definition of $\tilde\varphi$ it is readily seen that $w$ only contributes
if $w\nu|_{{\mathfrak b}k}=\Lambda$.
~
$\square$\medbreak\noindent\medbreak
{\bf Proof of Lemma \ref{l: restriction on leading exponents}:}
We define the
$\tau_{\iM}$-spherical function
$\varphi: M_{1}/M_1\cap vHv^{-1} \simeq M/M\cap vHv^{-1}
\times A_\iq \rightarrow V_\tau$ by
$$
\varphi(ma) = a^{\rho_P + \xi} q_\xi(P,v \,|\, f)(\log a, m).
$$
Then it follows from the equation $Df = \gamma(D\,:\, \nu) f$
and Lemma \ref{l: radial component applied to expansion} (c) applied
to $D - \gamma(D\,:\, \nu)$ in place of $D,$
that
$$
\mu_{P,v}(D) \varphi = \gamma(D\,:\, \nu) \varphi.
$$
Since $\varphi$ is $\tau$-spherical and non-zero, its restriction to $A_\iq$
does not vanish.
Let first $v=e$, and let ${\cal L}$ be as in
Lemma \ref{l: eigenfunctions on M1}.
It then follows immediately from that lemma
that there exists $w\in W({\mathfrak b})$ such that $w\nu|_{{\mathfrak b}k}\in {\cal L}$
and $w\nu|_{{\mathfrak a}q}=\xi+\rho_P$.
For general $v \in {\msy N}Kaq$ we also obtain the result from
Lemma \ref{l: eigenfunctions on M1}, by applying it to the function
$\varphi^v:=\rho_{\tau,v^{-1}}\varphi$. Indeed, it follows from the definition of
$\mu_{P,v}$ that $\varphi^v$
satisfies the assumption of the lemma. Hence there exists $w\in W({\mathfrak b})$
such that $w\nu|_{{\mathfrak b}k}\in{\cal L}$ and $w\nu|_{{\mathfrak a}q}=v^{-1}(\xi+\rho_P)$.
Let $v'\in W_0({\mathfrak b})$ be such that $v'Y=vY$ for all $Y\in{\mathfrak a}q$,
then $\nu\in (v'w)^{-1}(v'{\cal L}+\xi+\rho_P).$
~
$\square$\medbreak\noindent\medbreak
We will also need a result on leading coefficients along non-minimal parabolic subgroups.
\begin{lemma}
\label{l: D finiteness of leading coefficient}
Let $f \in C^{\rm ep}({\rm X}p\,:\, \tau)$ be a ${\msy D}X$-finite function. Let $Q\in \cP_\gs,$ $v\in {\msy N}Kaq$
and assume that $\xi \in {\rm Exp}L(Q,v\,|\, f).$
Then the function $\varphi: {\rm X}oneQvp \rightarrow V_\tau$ defined
by
$$
\varphi(ma) = a^{\xi+\rho_Q}
q_\xi(Q,v \,|\, f, \log a, m)\quad\quad (m \in {\rm X}Qvp,\;a \in A_{Q\iq}),
$$
is ${\msy D}({\rm X}oneQv)$-finite.
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
Let $I$ be the annihilator of $f$ in the algebra ${\msy D}X.$
Then it follows from
Lemma \ref{l: radial component applied to expansion} (c)
that $\mu_{Q,v}(D) \varphi = 0$ for all $D \in I.$
The algebra ${\msy D}({\rm X}oneQv)$ is a finite
module over the image of the homorphism $\mu_{Q,v};$ see
\bib{B91}, p.\ 342, and apply conjugation by $v$.
Hence $\mu_{Q,v}(I)$ generates a cofinite ideal in
${\msy D}({\rm X}oneQv).$ This establishes the result.
~
$\square$\medbreak\noindent\medbreak
We end this section with a result that limits the asymptotic exponents occurring
in discrete series representations to a countable set.
Later
we will apply this result to exclude the possibility
of a `continuum of discrete series'
(see the proof of Lemma \ref{l: second step vanishing thm}).
To
formulate the result we need
to define asymptotic exponents for a $K$-finite rather than a $\tau$-spherical function.
We denote by $\widehat K$ the collection of equivalence classes of
irreducible continuous representations of $K.$
If $\vartheta\subset \widehat K$ is a finite subset, then by ${\msy C}i({\rm X}p)_\vartheta$ we denote
the space of smooth $K$-finite functions in ${\msy C}i({\rm X}p)$ all of whose $K$-types
belong to $\vartheta.$ By ${V}_\types:=C(K)_\vartheta$ we denote the space of left $K$-finite continuous functions
on $K$ all of whose left $K$-types belong to $\vartheta.$ Moreover, by $\tau_\vartheta$ we
denote the restriction of the right regular representation to ${V}_\types.$
If $f \in {\msy C}i({\rm X}p)_\vartheta,$ then the function $\varsigma_\vartheta(f): {\rm X} \rightarrow {V}_\types,$
defined by $\varsigma_\vartheta(f)(x)(k) = f(kx)$ for $x \in {\rm X}p, k\in K$
belongs to ${\msy C}i({\rm X}p \,:\, \tau_\vartheta).$
The map $\varsigma: =\varsigma_\vartheta$ is a topological linear isomorphism from
${\msy C}i({\rm X}p)_\vartheta $ onto ${\msy C}i({\rm X}p \,:\, \tau_\vartheta),$ intertwining the ${\msy D}X$-actions
on these spaces. Moreover, $\varsigma$ maps the closed subspace ${\msy C}i({\rm X})_\vartheta$ of globally
defined smooth functions onto the similar subspace ${\msy C}i({\rm X} \,:\, \tau_\vartheta).$
We denote by ${\msy C}ep({\rm X}p)_\vartheta$ the preimage of ${\msy C}ep({\rm X}p\,:\, \tau_\vartheta)$ under $\varsigma.$
It follows from Lemma \ref{l: DX finite in exppol}
that ${\msy D}X$-finite funcitons in ${\msy C}i({\rm X}p)_\vartheta$
belong to ${\msy C}ep({\rm X}p)_\vartheta.$
Let $f \in {\msy C}ep({\rm X}p)_\vartheta;$ then for $P \in \cP_\gs$ and $v \in {\msy N}Kaq$ we
define the set of exponents of $f$ along $(P,v)$ by
$$
{\rm Exp}(P,v\,|\, f):= {\rm Exp}(P, v\,|\, \varsigma(f)).
$$
Note that this collection is the union for $k \in K$ and $m\in {\rm X}Pvp$ of the collections
of exponents
occurring in the ${\msy D}P$-exponential polynomial expansions of $a :to f(kamv).$
Let ${\cal C}({\rm X})$ denote the space of Schwartz functions on ${\rm X}$, see
\bib{BSmc}, Section 6, and let ${\cal A}_2({\rm X})_K$ denote the space of $K$-finite
and ${\msy D}GH$-finite functions $f\in{\cal C}({\rm X}).$ These functions are real
analytic and belong to $L^2({\rm X})$, cf.\ \bib{B87}, Thm.\ 7.3.
\begin{lemma}
\label{l: limitation on exponents of a Schwartz function}
Assume that the center of $G$ is compact. Then
$$\{\xi\in{\rm Exp}(P,v\,|\, f) \mid
{P \in \cP_\gs^{\rm min},v \in {\msy N}Kaq,f \in {\cal A}_2({\rm X})_K}\}$$
is a countable subset of ${\mathfrak a}qdc$.
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
Let $\widehat\spX_d$ denote the set of equivalence classes of discrete series
representations of the symmetric space ${\rm X}.$ This set is
countable, since $L^2({\rm X})$ is a separable Hilbert space.
Given $\omega \in \widehat\spX_d$ we denote by $L^2({\rm X})_\omega$ the collection of
functions $f \in L^2({\rm X})$ whose closed $G$-span in $L^2({\rm X})$ is equivalent
to a finite direct sum of copies of $\omega.$
Let $\widehat K$ denote the countable set of equivalence classes of irreducible
representations of $K.$ Given $\omega \in \widehat\spX_d$ and $\delta \in \widehat K,$ we
denote by $L^2({\rm X})_{\omega, \delta}$ the collection of $K$-finite elements
of type $\delta$ in $L^2({\rm X})_\omega.$ It follows from \bib{B87}, Thm.\ 7.3,
that $L^2({\rm X})_{\omega, \delta}$ is a subspace of ${\cal A}_2({\rm X})_K,$ and from
\bib{B87b}, Lemma 3.9, that this subspace is finite dimensional.
On the other hand, let $f \in {\cal A}_2({\rm X})_K$, and let $V\subset L^2({\rm X})$
denote the closed $G$-span of $f$. It follows from
\bib{B87b}, Lemma 3.9, that $V$ is admissible. Since $V$ is finitely generated,
it must then be a finite direct sum of irreducible
representations. This implies that
$f$ belongs to a finite direct sum of spaces $L^2({\rm X})_{\omega, \delta}.$
{}From the above
we conclude that ${\cal A}_2({\rm X})_K$ equals the following countable
algebraic direct sum:
\begin{equation}
\label{e: cF as a direct sum}
{\cal A}_2({\rm X})_K = \oplus_{\omega \in \widehat\spX_d,\,\delta \in \widehat K }\;\;\;
L^2({\rm X})_{\omega, \delta}.
\end{equation}
Let $\omega \in \widehat\spX_d$ and $\delta \in \widehat K.$
Then it follows from Lemma \ref{l: DX finite in exppol}
and the finite
dimensionality of $L^2({\rm X})_{\omega, \delta}$ that there exists a countable
subset ${\cal E}_{\omega, \delta} \subset {\mathfrak a}qdc$ such that
$$
{\rm Exp}(P, v\,|\, f) \subset {\cal E}_{\omega, \delta}
$$
for all $f \in L^2({\rm X})_{\omega, \delta}, P \in \cP_\gs^{\rm min}, v \in {\msy N}Kaq.$
Combining this observation with (\ref{e: cF as a direct sum}), we obtain
the desired result.
~
$\square$\medbreak\noindent\medbreak
\section{Separation of exponents}
\label{s: separation}
Let $Q\in\cP_\gs$.
In the next section we shall consider functions
$f_\lambda\in{\msy C}ep({\rm X}p\,:\, \tau)$, with parameter $\lambda\in{\mathfrak a}Qqdc$, whose
exponents along $P\in\cP_\gs^{\rm min}$ lie in sets of the form
$W\lambda+S-{\msy N}{\msy D}elta(P)$, where $S\subset{\mathfrak a}qdc$ is a finite set.
In general, given $\xi\in W\lambda+S-{\msy N}{\msy D}elta(P)$, the elements
$s\in W/W_Q$ and $\eta\in S-{\msy N}{\msy D}elta(P)$, such that $\xi=s\lambda+\eta$,
are not unique. In the present section we define a condition on $\lambda$
that allows this unique determination for all $\xi$. In particular,
the condition is valid for generic $\lambda\in{\mathfrak a}Qqdc$.
We consider also the case where $P$ is non-minimal.
Let $P,Q \in \cP_\gs.$ We define the equivalence
relation
$\sim_{P|Q}$ on $W$ by
\begin{equation}
\label{e: equivalence relation PQ}
s \sim_{P|Q} t \;\; \iff\;\; \forall \lambda \in {\mathfrak a}Qqd:\;\; s\lambda|_{{\mathfrak a}Pq} =
t\lambda|_{{\mathfrak a}Pq}.
\end{equation}
The associated quotient is denoted by $W/\!\sim_{P|Q}.$
We note that the classes in $W/\!\sim_{P|Q}$ are left
$W_P$- and right $W_Q$-invariant.
Thus, $W/\!\sim_{P|Q}$ may also be viewed as a quotient of $W_P\backslash W/W_Q.$
If $s,t \in W$ then one readily sees that $s \sim_{P|Q} t \iff s^{-1} \sim_{Q|P} t^{-1}.$
Hence the anti-automorphism $s :to s^{-1}$ of $W$ factors to
a bijection from $W/\!\sim_{P|Q}$ onto $W/\!\sim_{Q|P},$ which we denote by
$\sigma :to \sigma^{-1}.$
If $s \in W$ and $\lambda \in {\mathfrak a}Qqdc,$ then
the restriction $s\lambda|_{{\mathfrak a}Pq}$ depends on $s$ through its class
$[s]$ in $W/\!\sim_{P|Q}.$ We therefore agree to write
$$
[s] \lambda |_{{\mathfrak a}Pq} := s\lambda |_{{\mathfrak a}Pq}.
$$
\begin{defi}
\label{d: a zero sets newer}
For $S \subset {\mathfrak a}qdc$ a finite subset, we define
${\mathfrak a}Qqdzero(P,S)$ to be the subset of
${\mathfrak a}Qqdc$ consisting of elements $\lambda$ such that,
for all $s_1,s_2 \in W,$
$$
(s_1\lambda - s_2\lambda)|_{{\mathfrak a}Pq} \in [S + (-S)]|_{{\mathfrak a}Pq} + {\msy Z} {\msy D}rP
\;\;\;\Rightarrow \;\;\;s_1 \sim_{P|Q} s_2.
$$
\end{defi}
\begin{lemma}
\label{l: exponents disjoint}
Let $S\subset {\mathfrak a}qdc$ be finite. Then, for $\lambda \in {\mathfrak a}Qqdc,$
$$
W\lambda|_{{\mathfrak a}Pq} + (S - {\msy N} {\msy D}P )|_{{\mathfrak a}Pq} = \bigcup_{\sigma \in W/\!\sim_{P|Q} } \;
\left(\sigma\lambda|_{{\mathfrak a}Pq} + (S - {\msy N} {\msy D}P )|_{{\mathfrak a}Pq}\right).
$$
Moreover, the union is disjoint if and only if $\lambda \in {\mathfrak a}Qqdzero(P,S).$
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
Straightforward.
~
$\square$\medbreak\noindent\medbreak
\begin{lemma}
\label{l: azero is full}
Let $Q,P \in \cP_\gs,$ and let $S$ be a finite subset of
${\mathfrak a}qdc.$ Then ${\mathfrak a}Qqdzero(P,S)$ equals the complement of
the union of a locally finite collection of proper
affine subspaces in ${\mathfrak a}Qqdc.$
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
Let
$p: {\mathfrak a}qdc \rightarrow {\mathfrak a}Pqdc$ denote the map induced by restriction
to ${\mathfrak a}Pq.$ Let $\Pi$ be the complement of the diagonal in the set $W/\!\sim_{P|Q} \times W/\!\sim_{P|Q}.$
Then for every $\sigma =(\sigma_1, \sigma_2) \in \Pi$ and every $\eta \in {\mathfrak a}Pqdc$ we write
${\cal A}_{\sigma, \eta} =\{\lambda \in {\mathfrak a}Qqdc\mid p(\sigma_1\lambda - \sigma_2\lambda) = \eta\}.$
Note that ${\cal A}_{\sigma, 0}$ is a proper affine subspace of ${\mathfrak a}Qqdc.$
If $\lambda \in {\cal A}_{\sigma, \eta}$ then ${\cal A}_{\sigma, \eta}$ equals
$ \lambda + {\cal A}_{\sigma, 0};$ hence the set ${\cal A}_{\sigma, \eta}$ is either empty or a proper
affine subspace.
Let ${\cal A}$ be the collection of subsets of the form
${\cal A}_{\sigma, \xi},$ for $\sigma\in \Pi$
and $\xi \in p(S + (-S)) + {\msy Z}{\msy D}rP.$
Then ${\mathfrak a}Qqdzero(P,S)$ equals the
complement of $\cup{\cal A}$ in ${\mathfrak a}Qqdc.$ Thus, it remains to show that
the collection ${\cal A}$ is locally finite.
Let ${\cal C}$ be a compact subset of ${\mathfrak a}Qqdc$ and let $X$ be the collection of
$\xi \in p(S + (-S)) + {\msy Z}{\msy D}rP$ such that ${\cal C} \cap {\cal A}_{\sigma, \xi} \not= \emptyset$
for some $\sigma\in \Pi.$ Then it suffices to show that $X$ is finite.
Let ${\cal C}'\subset {\mathfrak a}Pqdc$ be the image of $\Pi \times {\cal C}$ under the map
$(\sigma, \lambda):to p(\sigma_1 \lambda - \sigma_2 \lambda).$ Then $X$ equals the intersection of
${\cal C}'$ with $p(S + (-S)) + {\msy Z} {\msy D}rP.$ The latter set is discrete since $S$ is finite,
whereas the elements of ${\msy D}rP$
are linearly independent. It follows that $X$ is finite.
~
$\square$\medbreak\noindent\medbreak
\begin{rem}
In particular, it follows from the above lemma that ${\mathfrak a}Qqdczero(P,S)$ is
a full open subset of ${\mathfrak a}Qqdc;$ see Appendix B for the notion of full.
\end{rem}
\begin{lemma}
\label{l: WPQ as cosets}
Let $Q,P \in \cP_\gs.$ If either ${\mathfrak a}Qq$ or ${\mathfrak a}Pq$ has codimension
at most $1$ in ${\mathfrak a}q,$ then the natural projection
$W_P\backslash W / W_Q \rightarrow W/\!\sim_{P|Q}$ is a bijection.
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
It suffices to prove injectivity of the map.
Since $s :to s^{-1}$ induces a bijection from $W/\!\sim_{P|Q}$ onto $W/\!\sim_{Q|P},$
it suffices to prove this when ${\mathfrak a}Pq$ has codimension
at most $1.$ We assume the latter to hold.
For $s \in W,$ let $[s]$ denote its canonical image in $W/\!\sim_{P|Q}.$
Assume that $s,t \in W$ and that $[s] = [t].$ Then for every $\lambda \in {\mathfrak a}Qqd$
we have $s\lambda = t\lambda$ on ${\mathfrak a}Pq.$ If ${\mathfrak a}Pq = {\mathfrak a}q,$ this implies
that $s =t$ on ${\mathfrak a}Qqd,$ hence $s W_Q = t W_Q,$ and since $W_P$ is trivial
in this case, the proof is finished.
Thus, we may as well assume that ${\mathfrak a}Pq$ has codimension $1.$ Then there
exists a root $\alpha \in \Sigma$ such that ${\mathfrak a}Pq = \ker \alpha .$
Note that $W_P = \{1, s_\alpha\}.$
For every $\lambda \in {\mathfrak a}Qqd$ the Weyl group images $s\lambda$ and $t\lambda$
have equal length in ${\mathfrak a}qd$ and equal image under the orthogonal
projection to $\alpha^\perp.$ Hence there exists a constant $\eta \in \{0,1\}$
such that
$\inp{s\lambda}{\alpha} = (-1)^\eta\inp{t\lambda}{\alpha}$ for all $\lambda \in {\mathfrak a}Qqd.$
It follows that $s\lambda = s_\alpha^{\eta} t \lambda$ for all $\lambda \in {\mathfrak a}Qqd;$
hence $sW_Q= s_\alpha^\eta t W_Q,$ from which it follows in turn
that $s$ and $t$ have the same image in $W_P\backslash W / W_Q.$
~
$\square$\medbreak\noindent\medbreak
In particular, if $P$ is minimal, then
the natural map
$W/W_Q \rightarrow W/\!\sim_{P|Q}$ is a bijection; we shall use it to identify
the sets involved.
\section{Analytic families of spherical functions}
\label{s: analytic families}
In this section we assume that $(\tau,V_\tau)$ is
a finite dimensional continuous representation of $K.$
Let $Q \in \cP_\gs$ and let $Y$ be a finite subset of ${}^*\fa_{Q\iq}dc,$ see
(\ref{e: char staQq}).
In the following definition we introduce a space of analytic families of
$\tau$-spherical functions that will play a crucial role in the rest of this
paper.
\begin{defi}
\label{d: anfamQY newer}
Let $Q,Y$ be as above and let $\Omega \subset {\mathfrak a}Qqdc$ be an open subset.
We define
\begin{equation}
\label{e: anfamQY new}
\CepQY({\rm X}p\col \tau \col \Omega)
\end{equation}
to be the space of $C^\infty$-functions
$f: \Omega \times {\rm X}p \rightarrow V_\tau$
satisfying the following conditions.
\begin{enumerate}
\item[{\rm (a)}]
For every $\lambda \in \Omega$ the function $f_\lambda: x :to f(\lambda, x)$
belongs to $C^\infty({\rm X}p\,:\, \tau).$
\minspace\item[{\rm (b)}]
There exists a constant $k \in {\msy N},$
and, for every $P \in \cP_\gs^{\rm min}$ and $v\in {\msy N}Kaq,$
a collection of functions
$q_{s, \xi}(P,v\,|\, f)\in P_k({\mathfrak a}q)\otimes {\cal O}(\Omega, {\msy C}i(\spX_{0,v}\,:\, \tau_{\iM})),$
for $s \in W/W_Q$ and
$\xi \in - sW_Q Y + {\msy N}{\msy D}P,$ with the following property.
For all $\lambda \in \Omega,$ $m \in {\rm X}zerov$ and $a\in A_\iq^+(P),$
\begin{equation}
\label{e: expansion f from anfamQY new}
f_\lambda(mav) = \sum_{s \in W/W_Q} a^{s\lambda - \rho_P}
\sum_{\xi \in -sW_Q Y + {\msy N} {\msy D}P} a^{-\xi} q_{s, \xi}(P,v\,|\, f, \log a)(\lambda, m),
\quad\quad
\end{equation}
where the ${\msy D}P$-exponential polynomial series with coefficients in $V_\tau$
is neatly convergent on $A_\iq^+(P).$
\minspace\item[{\rm (c)}]
For every $P\in \cP_\gs^{\rm min}, v\in {\msy N}Kaq$ and $s \in W/W_Q,$ the series
$$
\sum_{\xi \in - s W_Q Y + {\msy N}{\msy D}P}
a^{-\xi} q_{s, \xi}(P,v\,|\, f, \log a)
$$
converges neatly on $A_\iq^+(P)$ as a ${\msy D}P$-exponential polynomial
series with coefficients in
${\cal O}(\Omega, {\msy C}i(\spX_{0,v}\,:\,\tau_{\iM})).$
\end{enumerate}
If $f\in C^{\rm ep}_{Q,Y}({\rm X}p\,:\, \tau\,:\, \Omega),$ we
define the asymptotic degree of $f,$ denoted ${\rm deg}_{\rm a} f,$ to be the smallest number
$k \in {\msy N}$ for which the above condition (b) is fulfilled.
\end{defi}
\begin{rem}
\label{r: on defi anfamQY new}
We note that the space (\ref{e: anfamQY new}) depends on $Q$ through its
$\sigma$-split component $A_{Q\iq}.$
Moreover, from Lemma
\ref{l: transformation of coeffs}
we see that in the above definition it suffices
to require (b) and (c)
for a fixed given $P \in \cP_\gs^{\rm min}$ and for each $v$
in a given set ${\cal W}\subset {\msy N}Kaq$ of representatives
for $W/W_{K \cap H}.$ Alternatively, by the same lemma it suffices to require (b) and (c)
for a fixed given $v\in {\msy N}Kaq$ and arbitrary $P\in \cP_\gs^{\rm min}.$
\end{rem}
\begin{lemma}
\label{l: pointwise expansion of family}
Let $f \in \CepQY({\rm X}p\col \tau \col \Omega).$ Then $f_\lambda \in {\msy C}ep({\rm X}p\,:\, \tau)$
and
\begin{equation}
\label{e: exponents family}
{\rm Exp}(P,v\,|\, f_\lambda) \subset W(\lambda + Y) - \rho_P - {\msy N} {\msy D}P
\end{equation}
for all $\lambda \in \Omega,$ $P \in \cP_\gs^{\rm min},$ and
$v \in {\msy N}Kaq.$ Moreover, let $\Omega':= \Omega \cap {\mathfrak a}Qqdczero(P, WY)$
(see Definition \ref{d: a zero sets newer}).
Then $\Omega'$ is open dense in $\Omega$ and
\begin{equation}
\label{e: q-functions unique}
q_{s, \xi}(P,v\,|\, f, X, \lambda) = q_{s\lambda -\rho_P - \xi}(P,v\,|\, f_\lambda, X)
\end{equation}
for every $s \in W/W_Q$, $\xi \in - sW_Q Y + {\msy N}{\msy D}P,$
$X \in {\mathfrak a}q$ and $\lambda \in \Omega'.$
In particular, the functions
$q_{s, \xi}(P,v\,|\, f)$ are uniquely determined.
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ } The first statement and (\ref{e: exponents family}) follow
immediately from condition (b) in the above definition. The set
$\Omega'$ is open dense in $\Omega$ by Lemma \ref{l: azero is full},
and it follows from Lemmas \ref{l: exponents disjoint} and
\ref{l: WPQ as cosets} that if $\lambda \in \Omega'$ then the sets
$s(\lambda + W_Q Y) - \rho_P - {\msy N}{\msy D}P$, $s \in W/W_Q,$ are mutually disjoint.
Then (\ref{e: q-functions unique}) holds by uniqueness of asymptotics.
~
$\square$\medbreak\noindent\medbreak
The following result shows that an element of $\CepQY({\rm X}p\col \tau \col \Omega)$ may
be viewed as an analytic family of spherical functions.
\begin{lemma}
\label{l: the family is analytic}
Let $f \in \CepQY({\rm X}p\col \tau \col \Omega).$ Then $\lambda :to f_\lambda$ is a holomorphic
function on $\Omega$ with values in $C^\infty({\rm X}p\,:\,\tau).$
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
Let ${\cal W}\subset {\msy N}Kaq$ be a complete set of representatives for $W/W_{K \cap H}.$
Note that for $v \in {\cal W}$ the $V_\tau^{K_\iM \cap vHv^{-1}}$-valued
function $T_P^\downarrowv f_\lambda$
on $A_\iq^+(P)$ is given by the series on the right-hand side of
(\ref{e: expansion f from anfamQY new}) with $m=e.$
It follows from condition (c) of Definition \ref{d: anfamQY newer}
that $a :to T_P^\downarrowv f_\lambda (a)$ defines a smooth function on
$A_\iq^+(P)$ with values in ${\cal O}(\Omega) \otimes V_\tau^{K_\iM \cap vHv^{-1}}.$
According to
Appendix A, the function $\lambda :to T_P^\downarrowv f_\lambda(\,\cdot\,)$ is
a holomorphic function on $\Omega$
with values in ${\msy C}i(A_\iq^+(P), V_\tau^{K_\iM\cap vHv^{-1}}).$
Hence $\lambda :to T_P^\downarrowcW (f_\lambda)$ is a holomorphic function
on $\Omega$ with values in ${\msy C}i(A_\iq^+(P), \oplus_{v \in {\cal W}} V_\tau^{K_\iM \cap vHv^{-1}}).$
The conclusion of the lemma now follows by application of the isomorphism
(\ref{e: the iso T down P cW}).
~
$\square$\medbreak\noindent\medbreak
If $\Omega', \Omega$ are open subsets of ${\mathfrak a}qdc$ with $\Omega' \subset \Omega,$
then restriction from $\Omega \times {\rm X}p$ to $\Omega' \times {\rm X}p$
obviously induces a linear map
\begin{equation}
\label{e: restriction map presheaf}
\rho^{\Omega}_{\Omega'}:
C^{\rm ep}_{Q,Y}({\rm X}p\,:\,\tau \,:\, \Omega) \rightarrow C^{\rm ep}_{Q,Y}({\rm X}p\,:\,\tau \,:\, \Omega').
\end{equation}
Accordingly, the assignment
\begin{equation}
\label{e: presheaf CepQY}
\Omega :to C^{\rm ep}_{Q,Y}({\rm X}p\,:\,\tau \,:\, \Omega),
\end{equation}
defines a presheaf of complex linear spaces on ${\mathfrak a}Qqdc.$
Here we agree that (\ref{e: presheaf CepQY}) assigns the
trivial space to the empty set.
The following lemma will be useful at a later stage.
\begin{lemma}
\label{l: CepQY is sheaf}
Let $Q \in \cP_\gs$ and $Y \subset{}^*\fa_{Q\iq}dc$ a finite subset.
\begin{enumerate}
\item[{\rm (a)}]
If $\Omega', \Omega$ are open subsets of ${\mathfrak a}Qqdc$ with $\Omega' \neq \emptyset,$
$\Omega$ connected and $\Omega' \subset \Omega,$ then the restriction map
(\ref{e: restriction map presheaf}) is injective.
Moreover, ${\rm deg}_{\rm a}( \rho^{\Omega}_{\Omega'} f) = {\rm deg}_{\rm a} ( f)$ for all $f \in
C^{\rm ep}_{Q,Y}({\rm X}p\,:\,\tau \,:\, \Omega).$
\minspace\item[{\rm (b)}]
The presheaf (\ref{e: presheaf CepQY}) is a sheaf.
\end{enumerate}
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
The injectivity of the restriction map follows by analytic continuation,
in view of Lemma \ref{l: the family is analytic}.
Let $f'=\rho^{\Omega}_{\Omega'} f$.
Let $P \in \cP_\gs^{\rm min},$ $v \in {\msy N}Kaq,$ $s \in W/W_Q$ and $\xi \in -sW_Q \lambda + {\msy N}{\msy D}P$
then it follows from (\ref{e: q-functions unique}) that
\begin{equation}
\label{e: restriction of q}
q_{s, \xi}(P, v\,|\, f', \,\cdot\,, \lambda) = q_{s, \xi}(P, v\,|\,
f, \,\cdot\,, \lambda)
\end{equation}
for $\lambda$ in a dense open subset of $\Omega'$, hence for all $\lambda\in\Omega'$.
In particular this implies that the polynomial degree of the function
on the left-hand side of the equation is bounded by ${\rm deg}_{\rm a} (f);$ hence
${\rm deg}_{\rm a}(f') \leq {\rm deg}_{\rm a} (f).$
To prove the converse inequality, we note that the polynomial on the left-hand
side of (\ref{e: restriction of q})
is of degree at most $k':= {\rm deg}_{\rm a} (f') $ by the definition
of the latter number. Since $\Omega$ is connected, it follows
by analytic continuation that $\deg q_{s, \xi}(P, v\,|\, f, \,\cdot\,, \lambda)
\leq k'$
for all $\lambda \in \Omega.$ Since this holds for all $P,v,\sigma, \xi,$
it follows that ${\rm deg}_{\rm a} (f ) \leq k'$ and we obtain (a).
Assertion (b) is equivalent with the assertion that the presheaf satisfies
the localization property (see \bib{mybook}, p.\ 9). This is
established in a straightforward manner, by using (a).
~
$\square$\medbreak\noindent\medbreak
We shall now discuss the action of invariant differential
operators on families.
If $f$ is a family in ${\msy C}epQY({\rm X}p\,:\, \tau\,:\, \Omega),$ and $D \in {\msy D}X,$ then
we define the family
$Df: \Omega \times {\rm X}p \rightarrow V_\tau$
by
\begin{equation}
\label{e: action D on family}
(Df)_\lambda = D(f_\lambda),\quad\quad (\lambda \in \Omega).
\end{equation}
\begin{prop}
\label{p: D on families}
Let $f \in \CepQY({\rm X}p\col \tau \col \Omega).$ Then, for every $D \in {\msy D}X,$
the family $Df$ belongs to $\CepQY({\rm X}p\col \tau \col \Omega);$ moreover, ${\rm deg}_{\rm a} (Df) \leq {\rm deg}_{\rm a} (f).$
\end{prop}
\par\noindent{\bf Proof:}{\ }{\ }
Let $D \in {\msy D}X.$ Then $g = Df$
is a smooth function $\Omega \times {\rm X}p \rightarrow V_\tau;$
moreover, for $\lambda \in \Omega$ the function $(Dg)_\lambda = Df_\lambda$
is $\tau$-spherical. Thus, $g$ satisfies
condition (a) of Definition \ref{d: anfamQY newer} and it remains
to establish properties (b) and (c). In view of Remark \ref{r: on defi anfamQY new}
it suffices to do this for $v =e$ and arbitrary $P \in \cP_\gs^{\rm min}.$
Let $k := {\rm deg}_{\rm a} f.$
It follows from condition (b) of Definition \ref{d: anfamQY newer}
that, for $\lambda \in \Omega,$ the function $f_\lambda$ belongs to
${\msy C}ep({\rm X}p\,:\, \tau);$
moreover, its $(P,e)$-expansion is given by
\begin{equation}
\label{e: Pe expansion of f gl}
f_\lambda(ma) = \sum_{s \in W/W_Q} a^{s\lambda - \rho_P}
\sum_{\xi \in -sW_Q Y + {\msy N} \Delta(P)} a^{-\xi} q_{s,\xi}(P,e\,|\, f, \log a)(\lambda, m),
\end{equation}
for $a \in A_\iq^+(P)$ and $m\in M.$
Let $u:= \mu_P'(D) + u_+$ be the element of ${\cal D}_{1P}$ associated with $D$ as in
Proposition \ref{p: radial deco with MQ} with $P$ in place of $Q.$
In view of
Corollary \ref{c: cor on ringQ} its expansion ${\rm ep}(u),$ defined as in
(\ref{e: the embedding ep}),
is the sum, as $i$ ranges over a finite index set $I,$ of series of the form
$$
{\rm ep}(u)_i = \sum_{\nu \in {\msy N}{\msy D}P} a^{-\nu} \varphi_{i,\nu}\otimes S_{i,\nu} \otimes u_{i,\nu} \otimes
v_{i,\nu}.
$$
Here $\varphi_{i,\nu} \in C^\infty(M_\sigma),$ $S_{i,\nu} \in {\rm End}(V_\tau),$
$u_{i,\nu} \in U({\mathfrak m}_{\sigma})$
and $v_{i,\nu} \in U({\mathfrak a}q),$
and $\deg(u_{i,\nu}) + \deg(v_{i,\nu}) \leq d:=\deg(D)$
for
all $i,\nu.$
By Lemma \ref{l: radial component applied to expansion},
the function
$g_\lambda$ belongs to ${\msy C}ep({\rm X}p\,:\, \tau),$ for $\lambda \in \Omega,$ and its
$(P,e)$ expansion results from (\ref{e: Pe expansion of f gl})
by the formal application of the element ${\rm ep}(u).$
This
gives, for $\lambda \in \Omega,$ $m \in M$ and $a \in A_\iq^+(P),$
the neatly converging exponential polynomial expansion
$$
g_\lambda(m a) = \sum_{s \in W/W_Q} a^{s\lambda - \rho_P} \sum_{\eta \in -sW_Q Y + {\msy N} {\msy D}P}
a^{-\eta} \tilde q_{s, \eta}(\log a)(\lambda, m),
$$
where $\tilde q_{s, \eta}$ is given by the following finite sum
$$
\tilde q_{s, \eta}(X)(\lambda, m)
:= \sum_{i\in I}
\sum_{{\nu \in {\msy N} {\msy D}P \atop {\xi \in -sW_Q Y + {\msy N} {\msy D}P}}\atop \nu + \xi = \eta}
\varphi_{i, \nu}(m) S_{i, \nu}[\;
q_{s, \xi}(P,e\,|\, f, X; T_{s\lambda - \rho_P -\xi}(v_{i, \nu}), \lambda, m ; u_{i,\nu})\;],
$$
for $\lambda \in \Omega, $ $X \in {\mathfrak a}q$ and $m \in M.$
Here
we have used Harish-Chandra's convention to indicate by a semicolon on the
left or right-hand side of a Lie group variable the differentiation
on the corresponding side, with respect to that variable,
by elements of the appropriate universal enveloping algebra.
Moreover,
given $\gamma \in {\mathfrak a}qdc$ we have denoted by $T_\gamma$ the automorphism of $U({\mathfrak a}q)$
determined by $T_\gamma(X) = X + \gamma(X)$ for $X \in {\mathfrak a}q.$
{}From the above formula it readily follows
that $\tilde q_{s, \eta}(X,\lambda)$ is a smooth function of $(X,\lambda)$ with
values in ${\msy C}i(M, V_\tau);$ moreover, it is polynomial in $X$ of degree at most
$k$ and holomorphic in $\lambda\in \Omega.$ This establishes condition (b)
of Definition \ref{d: anfamQY newer} with $v=e,$ arbitrary $P\in \cP_\gs^{\rm min},$
and with
$$
q_{s, \eta}(P,e\,|\, g) = \tilde q_{s, \eta},
\quad\quad (s \in W/W_Q, \;\eta \in -sW_QY + {\msy N} \Delta(P)).
$$
For condition (c) we note that the series
\begin{equation}
\label{e: series for g}
\sum_{\eta \in -sW_Q Y + {\msy N} {\msy D}P} a^{-\eta} q_{s, \eta}(P,e\,|\, g, \log a)
\end{equation}
arises from the series
\begin{equation}
\label{e: similar series for f}
\sum_{\xi \in -sW_Q Y + {\msy N} {\msy D}P} a^{-\xi} q_{s, \xi}(P,e\,|\, f, \log a)
\end{equation}
by the formal application of ${\rm ep}(u)$
conjugated with multiplication by $a^{-s\lambda + \rho_P}.$ {}From this we see that
(\ref{e: series for g}) arises from (\ref{e: similar series for f})
by the formal application of the series
$$
\sum_{\nu \in {\msy N} {\msy D}P} a^{-\nu} \sum_{i \in I} \varphi_{i, \nu} \otimes S_{i, \nu} \otimes u_{i, \nu}
\otimes v_{i, \nu}(\lambda),
$$
with $v_{i, \nu}(\lambda) = T_{s\lambda - \rho_P} (v_{i, \nu}).$
We now observe that $\lambda :to T_{s\lambda - \rho_P}|_{U_d({\mathfrak a}Qq)}$ is a polynomial
${\rm End}(U_d({\mathfrak a}Qq))$-valued function, of degree at most $d.$
Hence there exists a finite set $J$ and elements
$p_j \in P_d({\mathfrak a}Qqd)$ and $T_j \in {\rm End}(U_d({\mathfrak a}Qq)),$ for $j \in J,$
such that
$$
T_{s\lambda - \rho_P}|_{U_d({\mathfrak a}Qq)} = \sum_{j \in J} p_j(\lambda) T_j.
$$
Let $B_{i,\nu, j}$ be the continuous endomorphism of ${\cal O}(\Omega, {\msy C}i(M_{\sigma}, V_\tau))$ defined
by
$$
B_{i, \nu, j} \psi(\lambda)(m) = p_j(\lambda) \varphi_{i, \nu}(m) S_{i,\nu}
[\psi(\lambda)(m; u_{i,\nu})].
$$
Then the series (\ref{e: series for g}) arises from
the formal application of
the series
$$
\sum_{\nu \in {\msy N} {\msy D}P} a^{-\nu} \sum_{i\in I\atop j \in J} B_{i,\nu,j} \otimes T_j(v_{i,\nu})
$$
with coefficients in
${\rm End}({\cal O}(\Omega, C^\infty(M_{\sigma}, V_\tau)))\otimes U({\mathfrak a}Qq)$
to (\ref{e: similar series for f}), viewed as a
series with coefficients in ${\cal O}(\Omega, C^\infty(M_{\sigma}, V_\tau)).$
It follows from Lemmas \ref{l: formal application of Ufa} and
\ref{l: formal application of hom valued series} that the resulting series
is neatly convergent as a series
on $A_\iq^+(P)$ with coefficients in ${\cal O}(\Omega, C^\infty(M_{\sigma}, V_\tau)).$
This establishes (c) with $v =e$ and arbitrary $P \in \cP_\gs^{\rm min}.$
~
$\square$\medbreak\noindent\medbreak
We will now describe the asymptotic behavior along walls for a family.
If $P,Q\in\cP_\gs$
and $\sigma \in W/\!\sim_{P|Q}$ (see (\ref{e: equivalence relation PQ})),
then for every subset $Y \subset {\mathfrak a}qdc$ we put
\begin{equation}
\label{e: defi gs cdot Y}
\sigma \cdot Y :=
\{ s \eta|_{{\mathfrak a}Pq} \mid s \in W, [s] = \sigma, \; \eta \in Y \}.
\end{equation}
\begin{thm}{\rm (Behavior along the walls).\ }
\label{t: behavior along the walls for families new}
Let $Q\in \cP_\gs,$ $\Omega \subset {\mathfrak a}Qqdc$ a non-empty open subset and
$Y \subset {}^*\fa_{Q\iq}dc$
a finite subset.
Let $f \in \CepQY({\rm X}p\col \tau \col \Omega)$
and let $k= {\rm deg}_{\rm a}(f).$
Let $P \in \cP_\gs$ and $v \in {\msy N}Kaq.$
Then ${\rm Exp}(P,v\,|\, f_\lambda) \subset W(\lambda + Y)|_{{\mathfrak a}Pq} -\rho_P- {\msy N} {\msy D}rP$
for every $\lambda \in \Omega.$ Moreover,
there exist unique
functions
$$
q_{\sigma, \xi}(P,v\,|\, f) \in P_k({\mathfrak a}Pq) \otimes {\cal O}(\Omega, {\msy C}i({\rm X}Pvp\,:\, \tau_P)),
$$
for $\sigma \in W/\!\sim_{P|Q}$ and $\xi \in - \sigma\cdot Y + {\msy N} {\msy D}rP,$
with the following property. For all
$\lambda \in \Omega,$ $m \in {\rm X}Pvp$ and $a \in A_{P\iq}^+(R_{P,v}(m)^{-1}),$
\begin{equation}
\label{e: series for fam along P}
f_\lambda(mav) = \sum_{\sigma \in W/\!\sim_{P|Q}} a^{\sigma\lambda - \rho_P}
\sum_{\xi \in - \sigma\cdot Y + {\msy N} {\msy D}rP}
a^{-\xi}\, q_{\sigma, \xi} (P, v\,|\, f ,\log a)( \lambda, m),
\end{equation}
where the ${\msy D}rP$-exponential polynomial series with coefficients in
$V_\tau$ is neatly convergent on $A_{P\iq}^+(R_{P,v}(m)^{-1}).$
In particular, if $\lambda\in\Omega':=\Omega \cap {\mathfrak a}Qqdczero(P, WY)$
then
\begin{equation}
\label{e: q of f in gl versus q of fgl new}
q_{\sigma, \xi}(P, v\,|\, f)(X,\lambda) =
q_{\sigma \lambda|_{{\mathfrak a}Pq }- \rho_P -\xi} (P, v\,|\, f_\lambda, X),
\end{equation}
for
$X\in {\mathfrak a}Pq.$
Finally, for each $\sigma \in W/\!\sim_{P|Q}$ and every
$R >1,$
the series
\begin{equation}
\label{e: series with q gs xi}
\sum_{\xi \in - \sigma \cdot Y + {\msy N} {\msy D}rP}
a^{-\xi} q_{\sigma, \xi} (P, v \,|\, f,\log a)
\end{equation}
converges neatly on $A_{P\iq}^+(R^{-1})$ as a ${\msy D}rP$-exponential
polynomial series with coefficients
in ${\cal O}(\Omega, {\msy C}i({\rm X}Pvp[R]\,:\, \tau_P)).$
\end{thm}
\par\noindent{\bf Proof:}{\ }{\ }
Let $P \in \cP_\gs$ and let $v \in {\msy N}Kaq.$
Fix a minimal parabolic subgroup
${P_1} \in \cP_\gs^{\rm min},$ contained
in $P.$
Fix a set ${\cal W}_{P,v}\subset {\msy N}KPaq$ of representatives
for $W_P/ W_P \cap v W_{K\cap H} v^{-1}.$ Then the natural map ${\msy N}Kaq \rightarrow W$
induces an embedding ${\cal W}_{P,v}v \hookrightarrows W/W_{K \cap H}.$ Therefore, we may fix
a set ${\cal W} \subset {\msy N}Kaq$ of representatives for $W/W_{K \cap H}$ containing
${\cal W}_{P, v} v.$
Fix $\lambda \in \Omega$ for the moment. Then by
Lemma \ref{l: pointwise expansion of family},
the function $f_\lambda$ belongs to ${\msy C}ep({\rm X}p\,:\, \tau),$ and
${\rm Exp}(P_1, w\,|\, f_\lambda) \subset W(\lambda + Y) - \rho_{P_1} - {\msy N} \Delta(P_1),$
for every $w \in {\msy N}Kaq.$
According to
Theorem \ref{t: transitivity of asymptotics},
for every $u \in {\cal W}_{P,v},$
the set ${\rm Exp}(P,v\,|\, f_\lambda)_{{P_1}, u}$ is
contained in ${\rm Exp}({P_1}, uv\,|\, f_\lambda)|_{{\mathfrak a}Pq}.$ Hence, by
of (\ref{e: Exp Q v f as union over index set cWQv}) with $P$ and $P_1$
in place of $Q$ and $P,$ respectively, we infer that
\begin{eqnarray}
{\rm Exp}(P,v\,|\, f_\lambda)
&\subset & [W(\lambda + Y) - \rho_{{P_1}} - {\msy N} \Delta({P_1})]|_{{\mathfrak a}Pq}
\nonumber\\
&= & W(\lambda + Y)|_{{\mathfrak a}Pq} - \rho_P - {\msy N} {\msy D}rP.
\label{e: exponents f gl along P v in this set}
\end{eqnarray}
Notice that (\ref{e: q of f in gl versus q of fgl new}) is a consequence
of (\ref{e: series for fam along P}), by Lemma \ref{l: exponents disjoint}.
Therefore the functions $q_{\sigma, \xi}(P,v\,|\, f)$ are unique. We will
now establish their existence.
It follows form (\ref{e: exponents f gl along P v in this set})
that the elements of ${\rm Exp}(P,v\,|\, f_\lambda)$ are all of the
form $\sigma \lambda|_{{\mathfrak a}Pq} - \rho_P - \xi ,$ with $\sigma \in W/\!\sim_{P|Q}$
and $\xi \in -\sigma\cdot Y + {\msy N}{\msy D}rP.$
Fix such elements $\sigma$ and $\xi.$
Then by transitivity of asymptotics,
cf.\ Theorem \ref{t: transitivity of asymptotics},
we have,
for $u \in {\cal W}_{P,v},$ $X \in {\mathfrak a}Pq,$ $m \in M$ and $b \in {}^*\!A_{P\iq}^+({}^*Pmin),$
that
\begin{equation}
\label{e: first series for q gs f gl}
q_{\sigma \lambda|_{{\mathfrak a}Pq }- \rho_P -\xi} (P, v\,|\, f_\lambda, X, m bu)
=
\sum_{\zeta \in {\rm Exp}(P_1, uv \,|\, f_\lambda) \atop
\zeta|_{{\mathfrak a}Pq} = \sigma \lambda|_{{\mathfrak a}Pq} - \rho_P - \xi}
b^{\zeta} q_\zeta({P_1}, uv \,|\, f_\lambda, X + \log b, m),
\end{equation}
where the $\Delta_P(P_1)$-exponential polynomial series
in the variable $b$ converges neatly on ${}^*\!A_{P\iq}^+({}^*Pmin).$
It follows from condition (b) in Definition \ref{d: anfamQY newer} that,
for $\zeta \in {\rm Exp}(P_1, uv \,|\, f_\lambda),$
\def\mu{\mu}
\begin{equation}
\label{e: formula for q xi in series}
q_\zeta({P_1}, uv \,|\, f_\lambda, X + \log b, m) =
\sum_{{s \in W/W_Q \atop \mu \in -sW_QY + {\msy N} \Delta(P_1)}\atop
s \lambda - \rho_{P_1} - \mu = \zeta }
q_{s, \mu} (P_1, uv \,|\, f, X + \log b)(\lambda, m).
\end{equation}
Now assume that $\lambda$ is contained in the full (cf.\ Lemma
\ref{l: azero is full}) subset $ \Omega'$ of $\Omega.$ Then, if
$s \in W$ and $\mu \in - sW_QY + {\msy N}\Delta(P_1)$ satisfy
$[s\lambda -\rho_{P_1} - \mu]|_{{\mathfrak a}Pq} = \sigma \lambda|_{{\mathfrak a}Pq } - \rho_P -\xi$,
it follows that $[s] = \sigma$ and $\mu|_{{\mathfrak a}Pq} = \xi,$
see Lemma \ref{l: exponents disjoint}. Hence,
combining (\ref{e: first series for q gs f gl})
and (\ref{e: formula for q xi in series}) we infer that
for $\lambda\in\Omega'$, $u \in {\cal W}_{P,v},$ $X \in {\mathfrak a}Pq,$ $m \in M$ and
$b \in {}^*\!A_{P\iq}^+({}^*Pmin),$
\begin{eqnarray}
\nonumber
\lefteqn{
q_{\sigma \lambda|_{{\mathfrak a}Pq }- \rho_P -\xi} (P, v\,|\, f_\lambda, X, m bu)=\quad}\\
\label{e: series for q gs f gl}
&=& \label{e: guiding formula}
\sum_{s \in W/W_Q\atop [s]= \sigma} b^{s\lambda - \rho_{{P_1}}}
\sum_{\mu \in - sW_Q Y + {\msy N} \Delta({P_1})\atop
\mu|_{{\mathfrak a}Pq} = \xi}
b^{- \mu} q_{s, \mu}({P_1}, uv \,|\, f, X + \log b, \lambda)(m).
\end{eqnarray}
It will be seen below that each inner sum in
(\ref{e: guiding formula}) converges neatly, so that the separation
of terms by the outer sum is justified.
This formula will guide us towards
the definition of the functions $q_{\sigma, \xi}(P,v\,|\, f).$
In the following we assume that $s \in W/W_Q$ and $[s]= \sigma.$
For $w \in {\cal W}$
we define the function
$F_{s,w}: A_\iq^+({P_1}) \times \Omega \rightarrow V_\tau^{K_\iM\cap wHw^{-1}}$ by
$$
F_{s,w}(a,\lambda) =
\sum_{\mu \in - sW_Q Y + {\msy N} \Delta({P_1})}
a^{- \mu} q_{s, \mu}({P_1}, w \,|\, f, \log a, \lambda)(e),
$$
for $a \in A_\iq^+({P_1}),$ $\lambda \in \Omega.$
The representation $\tilde \tau:= 1\otimes \tau$ of $K$
on the complete locally convex space ${\cal O}(\Omega) \otimes V_\tau$ is smooth.
We shall apply the results of Section \ref{s: asymp walls}, with $\tilde \tau$ in place
of $\tau.$
The series defining $F_{s,w}$ is a $\Delta({P_1})$-exponential polynomial series
with coefficients in ${\cal O}(\Omega) \otimes V_\tau.$ By condition (c)
of Definition \ref{d: anfamQY newer} it converges
neatly on $A_\iq^+({P_1});$ hence
$F_{s,w}$ may be viewed as an element of
$C^{\rm ep}(A_\iq^+({P_1}), [{\cal O}(\Omega) \otimes V_\tau]^{K_\iM \cap wHw^{-1}}).$
In view of the isomorphism (\ref{e: isomorphism of exppol}),
there exists a unique
function $F_s \in C^{\rm ep}({\rm X}p \,:\, \tilde \tau)$ such that
$T_P^\downarrowminw(F_s)(a) = F_s(aw) = F_{s,w}(a),$
for $w \in {\cal W}$ and $a \in A_\iq^+({P_1}).$
{}From the definition of $F_s$ it follows that
${\rm Exp}({P_1},w\,|\, F_s) \subset sW_Q Y - {\msy N} \Delta({P_1}),$ for every $w \in {\cal W}.$
Moreover, for every $w \in {\cal W}$ and every $\mu \in - sW_QY + {\msy N} \Delta({P_1}),$
\begin{equation}
\label{e: q of Fs versus q of f}
q_{-\mu}({P_1}, w \,|\, F_s,X,m)(\lambda)
= q_{s,\mu}({P_1}, w\,|\, f,X)(\lambda, m),
\end{equation}
for $X \in {\mathfrak a}Pq,$ $m \in {\rm X}_{0,w}$ and $\lambda \in \Omega.$
By transitivity of asymptotics, cf.\ Theorem \ref{t: transitivity of asymptotics},
applied to $F_s,$ we have that
${\rm Exp}(P, v \,|\, F_s)_{{P_1}, u} \subset \sigma\cdot Y - {\msy N} {\msy D}rP,$
for $u \in {\cal W}_{P,v}.$
Moreover, by the same result it follows that,
for $\xi \in - \sigma \cdot Y + {\msy N} {\msy D}rP,$
\begin{equation}
\label{e: series for q minus xi FW}
q_{-\xi}(P, v\,|\, F_s)(X, m bu) =
\sum_{\mu \in - sW_Q Y + {\msy N} \Delta({P_1})\atop
\mu|_{{\mathfrak a}Pq} = \xi}
b^{- \mu} q_{-\mu}({P_1}, uv \,|\, F_s, X + \log b, m ),
\end{equation}
where the series on the right-hand side converges neatly as
a $\Delta_P({P_1})$-exponential polynomial series in the variable
$b \in {}^*\!A_{P\iq}^+({}^*Pmin),$ with coefficients in
${\msy C}i({\rm X}_{0, uv}\,:\, \tilde\tau_{\rm M}).$
In particular, the asserted convergence of (\ref{e: guiding formula})
follows.
Substituting (\ref{e: q of Fs versus q of f}) in the right-hand side of
(\ref{e: series for q gs f gl}) and using (\ref{e: series for q minus xi FW})
we find, for $\lambda \in \Omega',$ that
\begin{equation}
\label{e: q of f gl as s sum}
q_{\sigma \lambda|_{{\mathfrak a}Pq }- \rho_P -\xi} (P, v\,|\, f_\lambda, X, m bu)=
\sum_{s \in W/W_Q\atop [s] =\sigma}
b^{s\lambda- \rho_{{P_1}}} q_{-\xi}(P, v\,|\, F_s)(X, m bu)(\lambda).
\end{equation}
We
are now ready to define the functions $q_{\sigma, \xi}(P,v\,|\, f).$
Let $1$ denote the trivial representation of $K$ in ${\msy C},$
and $1_P$ its restriction to $K_P.$
If $s \in W/W_Q,$ we define the function
$\varphi_s \in {\cal O}({\mathfrak a}Qqdc,{\msy C}i({\rm X}Pvp\,:\, 1_P))$
by
\begin{equation}
\label{e: defi gf s}
\varphi_s(\lambda, kbu) = b^{s\lambda-\rho_{{P_1}}},
\end{equation}
for
$\lambda \in {\mathfrak a}Qqdc,$ $u \in {\cal W}_{P,v},$ $k \in K_P$ and $b \in {}^*\!A_{P\iq}^+({}^*Pmin).$
Moreover,
for $\sigma \in W/\!\sim_{P|Q}$ and $\xi \in - \sigma\cdot Y + {\msy N} {\msy D}rP$ we
define the function
$q_{\sigma, \xi}(P, v\,|\, f): {\mathfrak a}Pq \times \Omega \rightarrow {\msy C}i({\rm X}Pvp\,:\,\tau_P)$
by
\begin{equation}
\label{e: q of f as sum of q of Fs new}
q_{\sigma,\xi}(P,v\,|\, f,X,\lambda)(m) =
\sum_{s \in W/W_Q\atop [s] = \sigma} \varphi_s(\lambda, m) \,q_{-\xi}(P,v\,|\, F_s, X,m)(\lambda),
\end{equation}
for $X \in {\mathfrak a}Pq,$ $\lambda \in \Omega$ and $m \in {\rm X}Pvp.$
If
$1 < R \leq \infty,$ then the locally convex space
${\msy C}i({\rm X}Pvp[R], {\cal O}(\Omega) \otimes V_\tau)$
is naturally isomorphic with ${\cal O}(\Omega, {\msy C}i({\rm X}Pvp[R], V_\tau)),$ see
Appendix A{}.
The isomorphism induces in turn a natural isomorphism of locally convex spaces
\begin{equation}
\label{e: spaces cO Omega Ci}
{\msy C}i({\rm X}Pvp[R] \,:\, \tilde\tau_P)) \simeq {\cal O}(\Omega, {\msy C}i({\rm X}Pvp[R] \,:\, \tau_P)).
\end{equation}
In particular, for $R=\infty,$ we obtain that
${\msy C}i({\rm X}Pvp \,:\, \tilde\tau_P)$ is naturally isomorphic with
${\cal O}(\Omega, {\msy C}i({\rm X}Pvp \,:\, \tau_P)).$
Thus, from (\ref{e: q of f as sum of q of Fs new})
we deduce that $q_{\sigma, \xi}(P, v\,|\, f)$
is an element
of $P_k({\mathfrak a}Pq) \otimes {\cal O}(\Omega, {\msy C}i({\rm X}Pvp\,:\, \tau_P)).$
Combining (\ref{e: q of f gl as s sum}), (\ref{e: defi gf s}) and
(\ref{e: q of f as sum of q of Fs new}) we infer that
(\ref{e: q of f in gl versus q of fgl new})
holds for $X\in {\mathfrak a}Pq,$ $\lambda \in \Omega'.$
On the other hand,
if $\lambda \in \Omega',$ then it follows from
(\ref{e: expansion f along Q v})
with $P$ and $f_\lambda$ in place of $Q$ and $f,$
that, for $R > 1,$ $m \in {\rm X}Pvp[R]$ and $a \in A_{P\iq}^+(R^{-1}),$
\begin{equation}
\label{e: series for f gl in proof}
f_\lambda(mav) = \sum_{\sigma \in W/\!\sim_{P|Q}}a^{\sigma \lambda -\rho_P}
\sum_{\xi \in -\sigma \cdot Y + {\msy N} {\msy D}rP}
a^{-\xi} q_{\sigma\lambda|_{{\mathfrak a}Pq} - \rho_P - \xi}(P, v\,|\, f_\lambda, \log a)(m),
\end{equation}
where the series converges neatly on $A_{P\iq}^+(R^{-1}),$ as
a ${\msy D}rP$-exponential polynomial series with coefficients in $V_\tau$
(use (\ref{e: exponents f gl along P v in this set})
and Lemma \ref{l: exponents disjoint}).
Substituting (\ref{e: q of f in gl versus q of fgl new}) in (\ref{e: series for f gl in proof}) we obtain
the identity (\ref{e: series for fam along P}) for $\lambda \in \Omega',$
$m \in {\rm X}Pvp[R]$ and $a \in A_{P\iq}^+(R^{-1}),$ with the convergence as asserted.
Thus, it remains to show that the identity (\ref{e: series for fam along P})
extends to all $\lambda \in \Omega$ and that the final assertion of the theorem holds.
We will first establish the final assertion.
It follows from Theorem
\ref{t: transitivity of asymptotics} that the series
\begin{equation}
\label{e: series with gf s q}
\sum_{\xi \in -sW_Q Y|_{{\mathfrak a}Pq} + {\msy N} {\msy D}rP} a^{-\xi} q_{-\xi}(P,v\,|\, F_s,\log a)
\end{equation}
converges neatly on $A_{P\iq}^+(R^{-1})$ as a ${\msy D}rP$-exponential polynomial series
with coefficients in the space (\ref{e: spaces cO Omega Ci}).
The series (\ref{e: series with q gs xi})
arises as the sum over $s \in W/W_Q$ with $[s] = \sigma$ of the series
in (\ref{e: series with gf s q}) multiplied by $\varphi_s.$ Since multiplication
by $\varphi_s$ induces a continuous linear endomorphism of the space
(\ref{e: spaces cO Omega Ci}), this establishes the final assertion of the theorem.
{}From the final assertion it follows that, for every $R > 1,$
the series on the right-hand side of (\ref{e: series for fam along P})
defines a holomorphic function of $\lambda \in \Omega,$ for
every $m \in {\rm X}Pvp[R]$ and $a \in A_{P\iq}^+(R^{-1}).$ For such $m,a$
the function $\lambda :to f_\lambda(mav)$ is holomorphic in $\lambda \in \Omega$
by Lemma \ref{l: the family is analytic}; hence the identity
(\ref{e: series for fam along P}) extends to all $\lambda \in \Omega,$
by density of $\Omega'$ in $\Omega.$
~
$\square$\medbreak\noindent\medbreak
\begin{thm}{\rm (Transitivity of asymptotics).\ }
\label{t: transitivity of asymptotics for families new}
Let $Q$, $\Omega$, $Y$, $f$, $P$ and $v$ be as in Theorem
\ref{t: behavior along the walls for families new}.
Let ${P_1} \in \cP_\gs^{\rm min}$ be contained
in $P.$
Let $\sigma \in W/\!\sim_{P|Q}$ and $\xi \in - \sigma \cdot Y +{\msy N} {\msy D}rP.$
Then for every $X \in {\mathfrak a}Pq,$ all $u \in {\msy N}KPaq,$
$b\in {}^*\!A_{P\iq}^+({}^*Pmin),$
$m \in M$
and $\lambda \in \Omega,$
\begin{equation}
\label{e: series in transitivity for families}
q_{\sigma,\xi}(P,v \,|\, f, X)(\lambda, mbu)
=
\sum_{s \in W/W_Q\atop [s] = \sigma} b^{s\lambda - \rho_{{P_1}}}
\sum_{\mu \in - sW_Q Y + {\msy N} {\msy D}Pmin \atop \mu|_{{\mathfrak a}Pq} = \xi}
b^{-\mu}\; q_{s, \mu} ({P_1}, uv\,|\, f, X + \log b)(\lambda, m).
\end{equation}
Moreover, for every $s \in W/W_Q$ with $[s] = \sigma$ and every
$X \in {\mathfrak a}Pq,$ the series
\begin{equation}
\label{e: second series in transitivity for families}
\sum_{\mu \in - sW_Q Y + {\msy N} {\msy D}Pmin \atop \mu|_{{\mathfrak a}Pq} = \xi}
b^{-\mu}\; q_{s, \mu} ({P_1}, uv\,|\, f, X + \log b)
\end{equation}
converges neatly on ${}^*\!A_{P\iq}^+({}^*Pmin)$
as a ${\msy D}Pmin$-exponential polynomial series in the variable $b$
with coefficients in ${\cal O}(\Omega, {\msy C}i({\rm X}_{0,uv}\,:\, \tau_{\iM})).$
\end{thm}
\par\noindent{\bf Proof:}{\ }{\ }
Fix $u \in {\msy N}KPaq.$ Moreover,
we fix a set ${\cal W}_{P,v}$
as in the beginning of the proof of
Theorem \ref{t: behavior along the walls for families new}
such that it contains the element $u.$
We will also use the remaining notation of the proof of the mentioned
theorem.
Using (\ref{e: q of Fs versus q of f}) we see that, via the natural
isomorphism of ${\cal O}(\Omega, {\msy C}i({\rm X}_{0, uv}\,:\, \tau_{\iM}))$ with
${\msy C}i({\rm X}_{0,uv} \,:\, \tilde \tau_{\iM}),$ the series
(\ref{e: second series in transitivity for families})
may be identified with the series
with coefficients in ${\msy C}i({\rm X}_{0,uv} \,:\, \tilde \tau_{\iM})$
that arises from the series on the right-hand side of
(\ref{e: series for q minus xi FW})
by omitting the evaluation at $m.$
The neat convergence of the latter series was noted already.
Moreover, the identity (\ref{e: series in transitivity for families})
follows by insertion of (\ref{e: series for q minus xi FW})
in the definition (\ref{e: q of f as sum of q of Fs new})
of $q_{\sigma,\xi}$.
~
$\square$\medbreak\noindent\medbreak
The following result is an important consequence of
`holomorphy of asymptotics.'
\begin{lemma}
\label{l: holo of asymp}
Let $Q\in \cP_\gs,$ $Y\subset {}^*\fa_{Q\iq}dc$ a finite subset and
$\Omega \subset {\mathfrak a}Qqdc$ a non-empty open subset.
Let $f\in {\msy C}epQY({\rm X}p\,:\,\tau \,:\, \Omega)$
and let $P\in \cP_\gs,$ $v\in {\msy N}Kaq,$ and
$\sigma\in W/\!\sim_{P|Q}.$
Let $\xi \in -\sigma \cdot Y + {\msy N} {\msy D}rP$ and assume that there exists
a $\lambda_0\in {\mathfrak a}Qqdczero(P, WY) \cap \Omega$ such that
\begin{equation}
\label{e: expression for gl zero is an exponent}
\sigma \lambda_0|_{{\mathfrak a}Pq} - \rho_P - \xi \in {\rm Exp}(P,v\,|\, f_{\lambda_0}).
\end{equation}
Then there exists a full open subset $\Omega_0$ of $\Omega$ such that
$$
\sigma \lambda|_{{\mathfrak a}Pq} - \rho_P - \xi \in {\rm Exp}(P,v\,|\, f_{\lambda}),\quad\quad (\forall \lambda \in \Omega_0).
$$
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
{}From (\ref{e: expression for gl zero is an exponent}) combined with
(\ref{e: q of f in gl versus q of fgl new}) it follows that the
$P_k({\mathfrak a}Pq) \otimes {\msy C}i({\rm X}Pvp\,:\, \tau_P)$-valued holomorphic
function $q: \lambda :to q_{\sigma,\xi}(P,v\,|\, f, \,\cdot\,, \lambda)$
does not vanish at $\lambda= \lambda_0.$
Hence there exists
a full open subset $\Omega_1 \subset \Omega$ such that $q(\lambda)\neq 0$ for all $\lambda \in \Omega.$
Let $\Omega_0:= \Omega_1 \cap {\mathfrak a}Qqdczero(P, WY),$ then the conclusion follows
by application of (\ref{e: q of f in gl versus q of fgl new}).
~
$\square$\medbreak\noindent\medbreak
We end this section with a result describing the behavior of the functions
$q_{\sigma,\xi}$ under the action of ${\msy N}Kaq.$
Let $Q,P \in \cP_\gs$
and $u \in {\msy N}Kaq,$ and put $P' =uPu^{-1}.$
The left multiplication by $u$ naturally induces
a map $W/\!\sim_{P|Q} \rightarrow W/\sim_{P' \mid Q},$ which we denote
by $\sigma :to u \sigma.$ Moreover, the endomorphism ${\rm Ad}(u^{-1})^*$
of ${\mathfrak a}qdc$ restricts to a linear map ${\mathfrak a}Pqdc \rightarrow {\mathfrak a}_{P'{\rm q}{\scriptscriptstyle \C}}^*,$
which we denote by $\eta :to u \eta.$ With these notations,
if $Y \subset {}^*\fa_{Q\iq}dc$ is a finite subset and $\sigma \in W/\!\sim_{P|Q},$ then
$$
u(\sigma\cdot Y) = (u\sigma)\cdot Y;
$$
see also (\ref{e: defi gs cdot Y}). For $v \in {\msy N}Kaq,$ let the map
$\rho_{\tau,u}: {\msy C}i({\rm X}Pvp\,:\, \tau_P) \rightarrow
{\msy C}i({\rm X}_{P', uv, +}\,:\, \tau_{P'})$
be defined by (\ref{e: defi rho tau u}) with $P$ in place of $Q.$
If $\Omega \subset {\mathfrak a}Qqdc$ is an open subset, let
${\rm Ad}(u^{-1})^* \otimes 1 \otimes \rho_{\tau,u}$ denote the naturally induced map from
$P({\mathfrak a}Pq) \otimes {\cal O}(\Omega, {\msy C}i({\rm X}Pvp\,:\, \tau_P))$
to
$P({\mathfrak a}_{P'{\rm q}})
\otimes {\cal O}(\Omega, {\msy C}i({\rm X}_{P', uv, +}\,:\, \tau_{P'})).
$
\begin{lemma}
\label{l: transformation holo coeffs}
Let $Q \in \cP_\gs,$ $Y \subset {}^*\fa_{Q\iq}dc$ a finite subset
and $\Omega \subset {\mathfrak a}Qqdc$ a non-empty open subset.
Let $f\in C^{\rm ep}_{Q,Y}({\rm X}p\,:\, \tau \,:\, \Omega).$
If $P \in \cP_\gs$ and $u,v\in {\msy N}Kaq,$ then for all $\sigma \in W/\!\sim_{P|Q}$
and $\xi \in \sigma\cdot Y,$
$$
q_{u\sigma, u\xi}(uPu^{-1}, uv \,|\, f) = [{\rm Ad}(u^{-1})^* \otimes 1 \otimes \rho_{\tau,u}]
q_{\sigma, \xi}(P, v\,|\, f).
$$
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
{}From combining (\ref{e: q of f in gl versus q of fgl new}) and
Lemma \ref{l: transformation of coeffs}
it follows that there exists a full open subset $\Omega_0$ of $\Omega$ such that,
for
$\lambda \in \Omega_0,$
$$
q_{u\sigma, u\xi}(uPu^{-1}, uv \,|\, f,\,\cdot\,,\lambda)
= [{\rm Ad}(u^{-1})^* \otimes \rho_{\tau,u}]
q_{\sigma, \xi}(P, v\,|\, f,\,\cdot\,,\lambda).
$$
The result now follows by holomorphy of the above expressions in $\lambda$
and density of $\Omega_0.$
~
$\square$\medbreak\noindent\medbreak
\section{Asymptotic globality}
\label{s: asymptotic globality}
In this section we introduce the notion of asymptotic globality
of a spherical function on ${\rm X}p$ and of an analytic family of such functions.
We discuss properties needed in the statement and proof of the vanishing theorem
in the next section.
\begin{defi}
\label{d: globality property of exponents new}
Let $P \in \cP_\gs$ and $v\in {\msy N}Kaq.$
A function $f\in \Cep({\rm X}p\,:\, \tau)$ is said to be
asymptotically global along $(P,v)$ at an element
$\xi \in {\mathfrak a}Pqdc$ if, for every $X \in {\mathfrak a}Pq,$ the $V_\tau$-valued smooth function
$q_\xi(P,v\,|\, f,X)$
has a $C^\infty$-extension from ${\rm X}Pvp$ to ${\rm X}Pv.$
\end{defi}
\begin{rem}
\label{r: after defi 7.1}
Since $q_\xi(P,v\,|\, f,X)$ is polynomial in $X,$ with values in
${\msy C}i({\rm X}Pvp\,:\, \tau_P),$
the requirement on $q_\xi$ implies that $q_\xi(P,v\,|\, f)$ is a polynomial
${\msy C}i({\rm X}Pv\,:\, \tau_P)$-valued function on ${\mathfrak a}Pq.$
Note that for $P$ minimal the condition of asymptotic globality along
$(P,v)$ is automatically fulfilled, since ${\rm X}Pvp = {\rm X}Pv.$
Finally, if $f \in {\msy C}ep({\rm X}p\,:\, \tau),$ then $f$ is asymptotically
global along $(G,e)$ at every exponent if and only if $f$ extends
smoothly to ${\rm X}$ (use Remark \ref{r: 1.5 bis}).
\end{rem}
The property of asymptotic globality is preserved under the action
of ${\msy D}X$ in the following fashion.
If $P \in \cP_\gs,$ then by ${\rm pr}eceq_{{\msy D}rP}$ we denote the partial ordering
on ${\mathfrak a}Pqdc,$ defined as in (\ref{e: partial ordering preceq gD}),
with ${\mathfrak a}Pq$ and ${\msy D}rP$ in place of ${\mathfrak a}$ and $\Delta,$
respectively.
\begin{prop}
\label{p: stability of globality new}
Let $f \in \Cep({\rm X}p\,:\, \tau)$ and $D \in {\msy D}GH.$
Let $P \in \cP_\gs,$ $v\in {\msy N}Kaq$ and $\xi_0 \in {\mathfrak a}Pqdc.$
If
$f$ is asymptotically global along $(P,v)$ at
every exponent $\xi \in {\rm Exp}(P,v\,|\, f)$ with $\xi_0 {\rm pr}eceq_{{\msy D}rP} \xi,$
then $Df$ is asymptotically global along $(P,v)$ at $\xi_0.$
\end{prop}
\par\noindent{\bf Proof:}{\ }{\ }
Let
$u := \mu_P'(D) + u_+$ be the element of ${\cal D}_{1P}$ associated with $D$ as
in Proposition \ref{p: radial deco with MQ},
with $P$ in place of $Q.$ The key idea in the present proof
is that $u$ has a ${\msy D}rP$-exponential polynomial expansion with coefficients
that are globally defined smooth functions on $M_{P\gs},$ by
Cor. \ref{c: cor on ringQ}.
More precisely, the expansion ${\rm ep}(u)$ is a finite sum, as $i$
ranges over a finite index set $I,$
of terms of the form
$$
{\rm ep}(u)_i = \sum_{\nu \in {\msy N}{\msy D}rP} a^{-\nu} \varphi_{i,\nu}\otimes S_{i,\nu} \otimes u_{i,\nu} \otimes
v_{i,\nu}.
$$
Here $\varphi_{i,\nu} \in C^\infty(M_{P\gs}),$ $S_{i,\nu} \in {\rm End}(V_\tau),$ $u_{i,\nu} \in U({\mathfrak m}_{P\sigma})$
and $v_{i,\nu} \in U({\mathfrak a}Pq),$ and $\deg(u_{i,\nu}) + \deg(v_{i,\nu}) \leq \deg(D)$ for
all $i,\nu.$
By Lemma \ref{l: radial component applied to expansion}, $Df$ belongs to
${\msy C}ep({\rm X}p\,:\, \tau)$
and its $(P,e)$-expansion results from the $(P,e)$-expansion of $f$
by the formal application of the element ${\rm ep}(u).$
Hence the asymptotic coefficient of $\xi_0$
is given by the finite sum
$$
q_{\xi_0}(P,e \,|\, Df)(X , m) = \sum_{{\xi \in {\rm Exp}(P,e\,|\, f)
\atop \nu \in{\msy N} {\msy D}rP}\atop
\xi - \nu = \xi_0}
\sum_{i\in I}
\varphi_{i,\nu}(m)
S_{i,\nu}[\, q_\xi(P,e\,|\, f)(X; T_\xi(v_{i,\nu}), m ; u_{i,\nu})\,].
$$
Let now $f$ satisfy the hypothesis of the proposition.
The $\xi's$ occurring in the above sum
belong to $\xi_0 + {\msy N}{\msy D}rP,$ hence satisfy $\xi_0 {\rm pr}eceq_{{\msy D}rP} \xi.$
By hypothesis, the associated coefficients $q_\xi(P,e\,|\, f)$ all extend smoothly
to ${\mathfrak a}Pq \times M_{P\gs},$ see Remark \ref{r: after defi 7.1}.
Therefore, so does $q_{\xi_0}(P,e \,|\, Df).$
This establishes the result for arbitrary $P \in \cP_\gs$ and the special choice $v=e.$
The result with general $v \in {\msy N}Kaq$ now follows by application of Lemma
\ref{l: transformation of coeffs} (cf.\ Lemma
\ref{l: transformation of globality} (a)).
~
$\square$\medbreak\noindent\medbreak
We shall also introduce a notion of asymptotic globality for families
from the space
$\CepQY({\rm X}p\col \tau \col \Omega)$ introduced in the previous section, with $\Omega \subset {\mathfrak a}Qqdc$
an open subset.
\begin{defi}
\label{d: s globality new}
Let $Q \in \cP_\gs,$ $Y $ a finite subset of
${}^*\fa_{Q\iq}dc$ and $\Omega \subset {\mathfrak a}Qqdc$ a non-empty open subset.
Let $P \in \cP_\gs,$ $v \in {\msy N}Kaq$ and $\sigma \in W/\!\sim_{P|Q}.$
We will say that a family $f \in {\msy C}epQY({\rm X}p\,:\, \tau\,:\, \Omega)$
is $\sigma$-global along $(P,v),$ if there exists a dense open
subset $\Omega_0$ of $\Omega,$
such that, for every $\lambda \in \Omega_0,$ the function $f_\lambda$
is asymptotically global along $(P,v)$ at each exponent
$\xi \in \sigma\lambda|_{{\mathfrak a}Pq} + \sigma\cdot Y -\rho_P - {\msy N}{\msy D}rP.$
\end{defi}
\begin{rem}
\label{r: globality independent of Y}
If $Y_1$ and $Y_2$ are finite subsets of ${}^*\fa_{Q\iq}dc$ with $Y_1\subset Y_2,$
then obviously
$$
C^{\rm ep}_{Q, Y_1} ({\rm X}p \,:\, \tau \,:\, \Omega)
\subset
C^{\rm ep}_{Q, Y_2} ({\rm X}p \,:\, \tau \,:\, \Omega).
$$
If $f$ belongs to the first of these spaces, then the condition of $\sigma$-globality
along $(P,v)$ relative to $Y_1$ is equivalent to the similar condition relative
to $Y_2.$ This is readily seen by using Lemmas
\ref{l: exponents disjoint} and \ref{l: azero is full}.
{}From this we see that the notion of $\sigma$-globality along $(P,v)$ extends to the space
$$
C^{\rm ep}_{Q} ({\rm X}p \,:\, \tau \,:\, \Omega) :=
\bigcup_{Y\subset {}^*\fa_{Q\iq}dc \; {\rm finite}}
C^{\rm ep}_{Q, Y} ({\rm X}p \,:\, \tau \,:\, \Omega)
$$
\end{rem}
The
property of asymptotic globality for families is also stable under the action
of ${\msy D}X.$
\begin{cor}
\label{c: stability of globality}
Let $Q \in \cP_\gs,$ $Y $ a finite subset of
${}^*\fa_{Q\iq}dc$ and $\Omega \subset {\mathfrak a}Qqdc$ a non-empty open subset.
Let $P \in \cP_\gs,$ $v \in {\msy N}Kaq$ and $\sigma \in W/\!\sim_{P|Q}.$
Let $f \in {\msy C}epQY({\rm X}p\,:\, \tau\,:\, \Omega)$
be $\sigma$-global along $(P,v).$ Then for every $D \in {\msy D}X$ the family
$Df \in {\msy C}epQY({\rm X}p\,:\, \tau\,:\, \Omega)$ is $\sigma$-global
along $(P,v)$ as well.
\end{cor}
\par\noindent{\bf Proof:}{\ }{\ }
It
follows from Proposition \ref{p: D on families} that
$Df$ belongs to ${\msy C}epQY({\rm X}p\,:\, \tau\,:\, \Omega).$
According to Theorem \ref{t: behavior along the walls for families new}, both sets
${\rm Exp}(P, v\,|\, f_\lambda)$ and ${\rm Exp}(P, v \,|\, Df_\lambda)$ are contained
in the set $E_\lambda:= W(\lambda + Y)|_{{\mathfrak a}Pq} - \rho_P - {\msy N} {\msy D}rP,$
for every $\lambda \in \Omega.$
Let $\Omega_0$ be as in Definition \ref{d: s globality new}. Then the set
$\Omega_0': = \Omega_0 \cap {\mathfrak a}Qqdczero(P, WY)$ is open dense in $\Omega$
by Lemma \ref{l: azero is full}.
Let $\lambda \in \Omega_0'$ and let
$\xi_0 \in \sigma \lambda|_{{\mathfrak a}Pq} + \sigma \cdot Y - \rho_P - {\msy N} {\msy D}rP.$
If $\xi \in {\rm Exp}(P,v\,|\, f_\lambda)$ satisfies $\xi_0 {\rm pr}eceq \xi,$
then $\xi \in \sigma \lambda |_{{\mathfrak a}Pq} + \sigma \cdot Y - \rho_P - {\msy N} {\msy D}rP$
by Lemma \ref{l: exponents disjoint}.
By hypothesis, $f_\lambda$ is asymptotically global along $(P,v)$ at
the exponent $\xi.$ It now follows by application of
Proposition \ref{p: stability of globality new} that $Df_\lambda$
is asymptotically global along $(P,v)$ at $\xi_0.$
~
$\square$\medbreak\noindent\medbreak
The following lemma describes the behavior of asymptotic
globality under the action of ${\msy N}Kaq.$
\begin{lemma}
\label{l: transformation of globality}
Let $P\in \cP_\gs$ and $u,v \in {\msy N}Kaq$. Put $P'=u P u^{-1}$
and $v'=uv$.
\begin{enumerate}
\item[{\rm (a)}]
Let $f \in C^{{\rm ep}}({\rm X}p \,:\, \tau )$ and $\xi\in{\mathfrak a}Pqdc$.
If $f$ is asymptotically global along $(P,v)$ at $\xi$, then
$f$ is asymptotically global along $(P',v')$ at $u\xi$.
\minspace\item[{\rm (b)}]
Let $Q \in \cP_\gs,$ $\Omega \subset {\mathfrak a}Qqdc$ a non-empty open subset,
$Y\subset {}^*\fa_{Q\iq}dc$ a finite subset,
$f \in C^{{\rm ep}}_{Q,Y} ({\rm X}p \,:\, \tau \,:\, \Omega)$ and $\sigma \in W/\!\sim_{P|Q}.$
If $f$ is $\sigma$-global along $(P,v),$ then
$f$ is $u\sigma$-global along $(P',v')$.
\end{enumerate}
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
{}From (\ref{e: defi rho tau u}) with $P$ in place of $Q$
it is readily seen that $\rho_{\tau, u}$ maps
${\msy C}i({\rm X}Pv\,:\, \tau_P)$ to ${\msy C}i({\rm X}_{P',v'} \,:\, \tau_{P'}).$
Then (a) and (b) follow immediately from Lemmas
\ref{l: transformation of coeffs}
and \ref{l: transformation holo coeffs}, respectively.
~
$\square$\medbreak\noindent\medbreak
We end this section with the following result, which shows that the
globality condition is fulfilled for a certain natural class
of $\tau$-spherical functions. {}From
the text preceding Lemma \ref{l: restriction on leading exponents} we recall that
${\mathfrak b}$ is a maximal abelian subspace of ${\mathfrak q}$ containing
${\mathfrak a}q$ and that if $\mu \in {\mathfrak b}dc,$ then by $I_\mu$ we denote the kernel
of the character $\gamma(\,\cdot\, \,:\, \mu)$ of ${\msy D}GH.$ Thus $I_\mu$ is an
ideal in ${\msy D}X$ of codimension one (over ${\msy C}$).
\begin{prop}
\label{p: global eigen implies as global}
Let $\mu\in{\mathfrak b}dc$ and let $f\in {\cal E}({\rm X}\,:\,\tau\,:\, I_\mu)$. Then
$f|_{{\rm X}p}\in C^{\rm ep}({\rm X}p\,:\,\tau)$. Moreover, this function
is asymptotically global along all pairs $(P,v)\in\cP_\gs\times{\msy N}Kaq$
and at all exponents $\xi\in{\mathfrak a}Pqdc$.
\end{prop}
\par\noindent{\bf Proof:}{\ }{\ }
The first statement follows immediately from Lemma
\ref{l: DX finite in exppol}. By Lemma
\ref{l: transformation of globality} (a) it suffices to consider
$v=e$ and arbitrary $P\in\cP_\gs$. Let $\psi\in V_\tau$ be fixed.
Then it suffices to prove that the scalar valued function
$m:to q_\xi(X,m):= \hinp{q_\xi(P,e|f,X,m)}{\psi}$
on ${\rm X}_{P,+}$ has a $C^\infty$ extension to ${\rm X}_P$, for each
$\xi\in{\mathfrak a}Pqdc$, $X\in{\mathfrak a}Pq$. It follows from
Theorem \ref{t: expansion along the walls} that
\begin{equation}
\label{e: expansion of f(ma)psi}
\hinp{f(ma)}{\psi}=\sum_{\xi\in Y-{\msy N}{\msy D}elta_r(P)}a^\xi q_\xi(\log a,m).
\end{equation}
On the other hand, it follows from \bib{B91}, Lemma 12.3, that
\bib{B91}, Thm.\ 12.8 can be applied to the $K$-finite function
$F\,:\,on x:to \hinp{f(x)}{\psi}$. By uniqueness of asymptotics
(see Lemma \ref{l: uniqueness of asymp} and its proof) the expansion
(\ref{e: expansion of f(ma)psi}) coincides with that of
\bib{B91}, Thm.\ 12.8. We conclude that, in the notation of loc.\ cit.,
$q_\xi(X,m)=p_{\mu|_{{\mathfrak a}q},\xi}(P|F,m,X)$ for all $X\in{\mathfrak a}Pq$,
$m\in{\rm X}_{P,+}$. The function $x:to p_{\mu|_{{\mathfrak a}q},\xi}(P|F,x,X)$
is smooth on $G$.
{}From this the smooth extension of $q_\xi(X,m)$
follows immediately.~
$\square$\medbreak\noindent\medbreak
\section{A vanishing theorem}
\label{s: vanishing thm}
In this section we formulate and prove the vanishing theorem.
We assume that $Q$ is a $\sigma$-parabolic subgroup containing $A_\iq.$
As before, let ${\mathfrak b}$ be a maximal abelian subspace of ${\mathfrak q}$ containing
${\mathfrak a}q.$ By ${}^*\fa_{Q\iq}$ and ${}^*\fb_Q$
we denote the orthocomplements of ${\mathfrak a}Qq$ in ${\mathfrak a}q$ and ${\mathfrak b},$ respectively.
Let ${\mathfrak b}k: = {\mathfrak b} \cap {\mathfrak k};$ then
$$
{}^*\fb_Q = {\mathfrak b}k \oplus {}^*\fa_{Q\iq}.
$$
We write ${\msy D}Qmaps$ for the collection of functions
$\deltamap: {}^*\fb_Qdc \rightarrow {\msy N}$ with finite support ${\rm supp}\, \deltamap.$
For $\delta \in {\msy D}Qmaps$ we put
$$
|\delta|:= \sum_{\nu \in {\rm supp}\, \delta} \delta(\nu).
$$
For $\deltamap \in {\msy D}Qmaps$ and $\lambda \in {\mathfrak a}Qqdc$ we define
the ideal $I_{\gd, \gl}$ in ${\msy D}GH$ as the following product of ideals
\begin{equation}
\label{e: ideal I gl gdmap}
I_{\gd, \gl} := {\rm pr}od_{\nu \in {\rm supp}\,{\deltamap}} (I_{\nu+ \lambda})^{\deltamap(\nu)}.
\end{equation}
If $\deltamap = 0,$ this ideal is understood to be the full ring ${\msy D}GH.$
Being a product of cofinite ideals in the Noetherian ring ${\msy D}GH,$ the
ideal $I_{\gd, \gl}$ is cofinite.
\begin{defi}
\label{d: defi cE Q Y revised new}
Let $\Omega \subset {\mathfrak a}Qqdc$ be a non-empty open subset and
$\delta\in{\msy D}Qmaps.$ For every finite subset $Y \subset {}^*\fa_{Q\iq}dc$
we define
\begin{equation}
\label{e: anfamQY gd}
{\cal E}QY({\rm X}p\,:\, \tau \,:\, \Omega \,:\, \deltamap)
\end{equation}
to be the space of families
$f \in \CepQY({\rm X}p\col \tau \col \Omega)$ (cf.\ Def.\ \ref{d: anfamQY newer}) such that
for every $\lambda \in \Omega$ the function $f_\lambda: x :to f(\lambda, x)$
is annihilated by the cofinite ideal (\ref{e: ideal I gl gdmap}).
Moreover, we define
$$
\cE_Q({\rm X}p\col\tau\col\Omega\col \gd):=
\bigcup_{Y\subset {}^*\fa_{Q\iq}dc \;\;{\rm finite}}\;\;
{\cal E}_{Q,Y}({\rm X}p\,:\, \tau \,:\, \Omega\,:\,\deltamap).
$$
\end{defi}
Note that the space (\ref{e: anfamQY gd}) depends on $Q$ through its
$\sigma$-split component $A_{Q\iq}.$
If $\nu \in {}^*\fb_Qdc,$ we denote by $\delta_\nu$ the characteristic
function of the set $\{\nu\}.$ Then $\delta_\nu \in {\msy D}Qmaps.$ Moreover,
if $\delta \in {\msy D}Qmaps$ and $\nu \in {\rm supp}\, \delta,$ then $\delta - \delta_\nu \in {\msy D}Qmaps$
and $|\delta - \delta_\nu| = |\delta| -1.$
\begin{lemma}
\label{l: stability family under D new}
Let $f \in {\cal E}QY({\rm X}p\,:\, \tau \,:\, \Omega \,:\, \deltamap).$
\begin{enumerate}
\item[{\rm (a)}]
If $D \in {\msy D}X$ then $Df \in {\cal E}QY({\rm X}p\,:\, \tau \,:\, \Omega \,:\, \deltamap).$
\minspace\item[{\rm (b)}]
If $D \in {\msy D}X$ and $\nu \in {\rm supp}\, \delta$ then the function
$g: \Omega \times {\rm X}p \rightarrow V_\tau$
defined by
\begin{equation}
\label{e: defi g as D minus gamma f}
g(\lambda,x): = [D - \gamma(D\,:\, \nu + \lambda)]f_\lambda(x),\quad\quad (\lambda \in \Omega, \; x \in {\rm X}p),
\end{equation}
belongs to ${\cal E}QY({\rm X}p\,:\, \tau \,:\, \Omega \,:\, \deltamap- \delta_\nu).$
\end{enumerate}
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
Let $D \in {\msy D}X.$
By Proposition \ref{p: D on families}, the family $Df$ belongs to
$C^{{\rm ep}}_{Q,Y}({\rm X}p\,:\, \tau\,:\, \Omega).$ Moreover, if $\lambda\in\Omega$ and
$D ' \in I_{\gd, \gl},$ then $D' (Df)_\lambda = D'D f_\lambda = D D' f_\lambda = 0$
and we see that assertion (a) holds.
The function $\lambda :to \gamma(D\,:\, \nu + \lambda)$ is polynomial on ${\mathfrak a}Qqdc,$
hence holomorphic
on $\Omega$ and it follows that $G: (\lambda, x) :to \gamma(D\,:\, \nu +\lambda) f(\lambda, x)$
belongs to $C^{\rm ep}_{Q,Y}({\rm X}p\,:\, \tau\,:\, \Omega).$ Hence
$g = Df - G$ belongs to the latter space as well.
Furthermore, if $D' \in I_{\delta - \delta_\nu, \lambda},$ then
$D'':= D' (D - \gamma(D\,:\, \nu + \lambda)) \in I_{\gd, \gl},$ and we see that
$D'g_\lambda = D'' f_\lambda = 0.$ Hence (b) holds.
~
$\square$\medbreak\noindent\medbreak
\begin{rem}
It follows from Lemma \ref{l: stability family under D new} (a)
that (\ref{e: action D on family}) defines a representation of ${\msy D}X$
in
${\cal E}Q({\rm X}p\,:\, \tau \,:\,\Omega\,:\, \deltamap),$ leaving the subspaces
${\cal E}QY({\rm X}p\,:\, \tau \,:\, \Omega\,:\,\deltamap)$
invariant.
\end{rem}
\begin{lemma}
\label{l: locally killed by I gl gd}
Let $Q \in \cP_\gs,$ $\delta\in {\msy D}Qmaps$
and $\Omega$ a connected non-empty open subset of ${\mathfrak a}Qqdc.$
Assume that $f \in C^{\rm ep}_Q({\rm X}p\,:\, \tau\,:\, \Omega).$
If $f_\lambda$ is annihilated by $I_{\gd, \gl}$ for $\lambda$ in a non-empty open
subset $\Omega'$ of $\Omega,$ then
$f \in {\cal E}Q({\rm X}p\,:\, \tau \,:\, \Omega\,:\, \delta).$
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
Fix a finite subset $Y\subset {}^*\fa_{Q\iq}dc$ such that
$f \in C^{{\rm ep}}_{Q,Y}({\rm X}p\,:\, \tau\,:\, \Omega).$
We proceed by induction on $|\delta|.$
First, assume that $|\delta| = 0.$ Then $I_{\delta, \lambda} = {\msy D}X$ for all $\lambda$
and hence $f|_{\Omega'\times{\rm X}p}=0.$
Since $\Omega$ is connected, this implies that $f = 0,$
see Lemma \ref{l: the family is analytic}.
Next assume that $|\delta| = k {\rm ep}silonq 1$ and assume the result
has already been established for all $\delta \in {\msy D}Qmaps$ with
$|\delta| < k.$
Fix $\nu \in {\rm supp}\, \delta$ and put $\delta' = \delta - \delta_\nu,$ then $|\delta'| < k.$
Let $D \in {\msy D}X$ and define $g$ as in (\ref{e: defi g as D minus gamma f}).
Then $g \in C^{\rm ep}_{Q,Y}({\rm X}p \,:\, \tau \,:\, \Omega)$, as seen
in the proof of Lemma \ref{l: stability family under D new}.
On the other hand, it follows from (b) of that lemma
that $g|_{\Omega'\times{\rm X}p}\in {\cal E}QY({\rm X}p\,:\, \tau \,:\, \Omega' \,:\, \delta')$.
Hence $g \in {\cal E}Q({\rm X}p\,:\, \tau \,:\, \Omega\,:\, \delta')$
by the induction hypothesis. Fix $\lambda \in \Omega.$ Then it follows,
for $D' \in I_{\delta', \lambda},$ that $D'(D - \gamma(D\,:\, \nu + \lambda)) f_\lambda = D'g_\lambda = 0.$
Since $D$ was arbitrary, we conclude that $f_\lambda$ is annihilated
by the ideal $I_{\delta', \lambda} I_{\nu+\lambda} = I_{\gd, \gl}.$
~
$\square$\medbreak\noindent\medbreak
We define the following subset of $\cP_\gs,$ consisting of the
parabolic subgroups whose $\sigma$-split rank is of codimension one,
$$
\cP_\gs^1:= \{P \in \cP_\gs \mid \dim {\mathfrak a}q / \dim {\mathfrak a}Pq = 1\}.
$$
\begin{defi}
\label{d: family for the vanishing thm newer}
Let $Q \in \cP_\gs,$ $\Omega\subset {\mathfrak a}Qqdc$
a non-empty open
subset and $\delta \in {\msy D}Qmaps.$
By $\cE_Q({\rm X}p\col\tau\col\Omega\col \gd)P$ we denote the space of functions
$f \in \cE_Q({\rm X}p\col\tau\col\Omega\col \gd)$
satisfying the following condition.
\hbox{\hspace{-12pt}
\vbox{
\begin{enumerate}
\item[]
For every $s \in W$ and every $P \in \cP_\gs^1$ with $ s ({\mathfrak a}Qq) \not\subset {\mathfrak a}Pq,$
the family $f$ is $[s]$-global along $(P,v),$ for all $v \in {\msy N}Kaq;$
here $[s]$ denotes the image of $s$ in $W/{\sim_{P|Q}}= W_P \backslash W / W_Q.$
\end{enumerate}
}}
\noindent
If $Y \subset {}^*\fa_{Q\iq}dc$ is a finite subset, we define
$$
\cE_{Q,Y}({\rm X}p\col \tau\col \Omega\col \gd)P: = \cE_{Q,Y}({\rm X}p\col \tau\col \Omega\col \gd) \cap \cE_Q({\rm X}p\col\tau\col\Omega\col \gd)P.
$$
\end{defi}
\begin{rem}
\label{r: W Pga Q new}
Note that $\cE_Q({\rm X}p\col\tau\col\Omega\col \gd)P$ depends on $Q$ through its $\sigma$-split component ${\mathfrak a}Qq.$
The equality $W/{\sim_{P|Q}} = W_P \backslash W/ W_Q$ follows from
Lemma \ref{l: WPQ as cosets}.
Note that the condition $s({\mathfrak a}Qq) \not \subset {\mathfrak a}Pq$ on $s$
factors to a condition on its class in $W_P\backslash W/ W_Q$.
\end{rem}
The following result reduces the globality condition of Definition
\ref{d: family for the vanishing thm newer} to a condition involving
a smaller set of $(s, P) \in W \times \cP_\gs^1.$ Its formulation requires
some more notation.
Let
$\Delta$ be a fixed basis for the root system $\Sigma,$ let $\Sigma^+$ be
the associated system of positive roots and ${\mathfrak a}qp$ the associated open positive
chamber. Let ${}^*Po$ be the unique element of $\cP_\gs^{\rm min}$ with $\Delta({}^*Po) = \Delta.$
A $\sigma$-parabolic subgroup $Q$ is said to be standard if it contains
${}^*Po;$ of course then $Q \in \cP_\gs.$
Given such a $Q,$ we write ${\msy D}elta_Q$ for the subset of $\Delta$ consisting
of the roots vanishing
on ${\mathfrak a}Qq$ and ${\msy D}elta(Q)$ for its complement.
If $\alpha$ is any root in $\Delta,$ we write
$
{\mathfrak n}_\alpha
$
for the sum of the root spaces ${\mathfrak g}_\beta$ where $\beta$ ranges over the set
$\Sigma^+\setminus {\msy N} \alpha.$
Moreover, we put $N_\alpha: = \exp({\mathfrak n}_\alpha)$ and write $M_{1\alpha}$ for
the centralizer in $G$ of the root hyperplane $\ker \alpha.$ Then $P_\alpha = M_{1\alpha} N_\alpha$
is the standard parabolic subgroup with ${\msy D}elta_{P_\alpha}=\{\alpha\}$.
We write $P_\alpha = M_\alpha A_\alpha N_\alpha$ and $P_\alpha = M_{\sigma \alpha} A_{\alpha {\rm q}} N_\alpha$
for the Langlands and $\sigma$-Langlands decompositions of $P_\alpha,$ respectively.
Accordingly, ${\mathfrak a}gaq = \ker \alpha$ and ${}^*\fagaq = (\ker \alpha)^\perp.$
Finally, we write $W_\alpha = W_{P_\alpha}$ for the centralizer of $\ker \alpha$ in $W.$
\begin{lemma}
\label{l: minimal condition for glob new}
Let $Q\in \cP_\gs$ be a standard parabolic subgroup,
$\Omega \subset {\mathfrak a}Qqdc$ a non-empty open subset,
$\delta \in {\msy D}Qmaps$ and $f \in {\cal E}_Q({\rm X}p\,:\, \tau \,:\, \Omega\,:\, \delta).$
Then $f$ belongs to ${\cal E}_Q({\rm X}p\,:\, \tau \,:\, \Omega\,:\, \delta)_\lambdaob$
if and only if the following condition is fulfilled.
\hbox{\hspace{-12pt}
\vbox{
\begin{enumerate}
\item[]
For every $s \in W$ and every $\alpha \in \Delta$ with $s^{-1}\alpha|_{{\mathfrak a}Qq} \neq 0,$
the family $f$ is $[s]$-global along $({}^*Pga,v),$ for all $v \in {\msy N}Kaq;$
here $[s]$ denotes the image of $s$ in $W/{\sim_{{}^*Pga|Q}}= W_\alpha \backslash W / W_Q.$
\end{enumerate}
}}
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
We must show that the condition of Definition \ref{d: family for the vanishing thm newer}
is fulfilled if and only if
the above condition holds.
For this we first observe that for $\alpha \in \Delta$ and $s \in W,$
$$
s^{-1}\alpha|_{{\mathfrak a}Qq} \neq 0 \iff s({\mathfrak a}Qq) \not\subset {\mathfrak a}_{\alpha{\rm q}}.
$$
The `only if part' is now immediate.
For the `if part', assume that the above condition is fulfilled.
Let $(s, P) \in W \times \cP_\gs^1$ be such that $s({\mathfrak a}Qq) \not\subset {\mathfrak a}Pq.$
There exist $\alpha \in \Delta$ and $t \in W$ such that $tP t^{-1} = P_\alpha.$
It follows that $ts({\mathfrak a}Qq) \not\subset t{\mathfrak a}Pq = \ker \alpha,$ hence $(ts)^{-1} \alpha = \alpha \,{\scriptstyle\circ}\, (ts) $ is not
identically zero on ${\mathfrak a}Qq.$
From the
hypothesis it now follows
that $f$ is $[ts]$-global along $(t P t^{-1}, v),$ for all $v \in {\msy N}Kaq.$
By Lemma \ref{l: transformation of globality}
it follows that $f$ is $[s]$-global along $(P, w),$ for all
$w \in {\msy N}Kaq.$
~
$\square$\medbreak\noindent\medbreak
\begin{lemma}
\label{l: stability family with globality under D}
Let $Q \in \cP_\gs,$ $\Omega \subset {\mathfrak a}Qqdc$ a non-empty open subset
and $\delta\in {\msy D}Qmaps.$
Then the space $\cE_Q({\rm X}p\col\tau\col\Omega\col \gd)P$ is ${\msy D}X$-invariant. Moreover, $\cE_{Q,Y}({\rm X}p\col \tau\col \Omega\col \gd)P$ is a ${\msy D}X$-submodule,
for every finite subset $Y \subset {}^*\fa_{Q\iq}dc.$
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
This follows from combining the ${\msy D}X$-invariance of the space
$\cE_{Q,Y}({\rm X}p\col \tau\col \Omega\col \gd)$ with Proposition \ref{p: stability of globality new}.
~
$\square$\medbreak\noindent\medbreak
\begin{defi}\label{d: Q-distinguished}
Let $Q\in \cP_\gs.$
An open subset $\Omega$ of ${\mathfrak a}Qqdc$ will be called
$Q$-distinguished if it is connected and if
for every $\alpha \in \Sigma(Q)$ the function
$\lambda :to \inp{{\msy R}e \lambda}{\alpha}$ is not bounded from above
on $\Omega.$
\end{defi}
In particular, a connected open dense subset of ${\mathfrak a}Qqdc$ is
$Q$-distinguished.
In the following theorem we assume that ${}^Q\cW \subset {\msy N}Kaq$ is a complete
set of representatives for $W_Q\backslash W / W_{K \cap H}.$
\begin{thm}
{\rm (Vanishing theorem).\ }
\label{t: vanishing theorem new}
Let $Q \in \cP_\gs$
and $\deltamap \in {\msy D}Qmaps.$
Let $\Omega \subset {\mathfrak a}Qqdc$
be a $Q$-distinguished open subset and let
$f \in {\cal E}_Q({\rm X}p\,:\,\tau\,:\,\Omega\,:\, \delta)_\lambdaob.$
Assume that there exists a non-empty open subset
$\Omega' \subset \Omega$ such that, for each $v \in {}^Q\cW, $
\begin{equation}
\label{e: hypothesis vanishing thm new}
\lambda - \rho_Q \notin {\rm Exp}(Q,v \,|\, f_\lambda), \quad\quad (\lambda \in \Omega').
\end{equation}
Then $f = 0.$
\end{thm}
The proof of this theorem will be given after the following lemmas on which it is
based. We may and shall assume that $Q$ is standard. Thus, $Q$ contains
the minimal standard $\sigma$-parabolic subgroup $P_0$ which will
be denoted by $P$ in the rest of this section.
\begin{lemma}
\label{l: first step vanishing thm}
Let $\Omega \subset {\mathfrak a}Qqdc $ be a non-empty connected open subset,
$\delta \in {\msy D}Qmaps$ and assume that $|\delta| =1.$ Let $Y \subset {}^*\fa_{Q\iq}dc$ be a finite subset and
let $f \in {\cal E}_{Q,Y}({\rm X}p\,:\,\tau\,:\,\Omega\,:\, \delta).$
Moreover, let $v \in {\msy N}Kaq$ and
assume that
there
exist $t \in W_Q,\, \eta \in Y,$ $\mu \in {\msy N}{\msy D}elta$ and $u \in {\msy N}KQaq$
such that
\begin{equation}
\label{e: gl s eta mu in Exp new}
\lambda + t\eta - \rho - \mu \in {\rm Exp} (P, uv \,|\, f_\lambda)
\end{equation}
for $\lambda$ in some non-empty open subset of $ \Omega.$
Then there exists a full open subset $\Omega_0 \subset \Omega$ such that
$$
\lambda - \rho_Q \in {\rm Exp}(Q,v \,|\, f_\lambda), \quad\quad (\lambda \in \Omega_0).
$$
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
Let $\nu\in{}^*\fb_Qdc$ be the unique element such that ${\rm supp}\,\delta=\{\nu\}$.
Fix $t, \eta, \mu$ and $u$ with the mentioned property.
Replacing $\mu$ by a ${\rm pr}eceq_{\Delta}$-smaller
element if necessary
we may in addition assume that $\mu$ is ${\rm pr}eceq_{\Delta}$-minimal subject to the condition
that (\ref{e: gl s eta mu in Exp new}) holds for
$\lambda$ in some non-empty open subset of $\Omega.$
By holomorphy of asymptotics, see
Lemma \ref{l: holo of asymp},
it follows that (\ref{e: gl s eta mu in Exp new}) holds
for $\lambda$ in a full open subset $\Omega'$ of $\Omega.$
Moreover, using the minimality of $\mu$ and applying
Lemma \ref{l: exponents disjoint} we see that
for every $\lambda$ in the full open subset $\Omega_0:= \Omega' \cap {\mathfrak a}Qqdzero(P, WY)$ of $\Omega,$
$$
\lambda + t\eta - \rho - \mu \in {\rm Exp}L(P,uv\,|\, f_\lambda).
$$
Since $f_\lambda$ is annihilated by $I_{\gd, \gl} = I_{\nu + \lambda},$ this implies,
in view of Lemma \ref{l: restriction on leading exponents},
that there exists a finite subset ${\cal L} \subset {\mathfrak b}kdc$ such
that
$$
\nu + \lambda \in
W({\mathfrak b}) ( {\cal L} + \lambda + t \eta - \mu),\quad\quad (\lambda \in \Omega_0).
$$
For $\Lambda_0 \in {\cal L}, \, w \in W({\mathfrak b})$
we define
$\Omega_0(\Lambda_0, w)$ to be the set
of $\lambda \in \Omega_0$ satisfying
\begin{equation}
\label{e: defining relation for Omega 1 gL zero w}
\nu + \lambda = w(\Lambda_0 + \lambda + t \eta -\mu).
\end{equation}
The union of these sets, as $\Lambda_0 \in {\cal L},$ $w \in W({\mathfrak b}),$ equals $\Omega_0.$ By finiteness
of the union, we may select $\Lambda_0$ and $w$ such that $\Omega_0(\Lambda_0,w)$
has a non-empty interior in $\Omega_0.$ Since $\Omega_0(\Lambda_0,w)$
is also the intersection of $\Omega_0$ with an affine linear subspace of ${\mathfrak b}dc,$
it must be all of $\Omega_0.$
Hence for all $\lambda_1,\lambda_2 \in \Omega_0$
we have $w(\lambda_1 - \lambda_2) = \lambda_1 - \lambda_2.$ Since $\Omega_0$ is
a non-empty open subset of ${\mathfrak a}Qqdc$ this implies that
$w$ belongs to $W_Q({\mathfrak b}),$ the centralizer of ${\mathfrak a}Qq$ in $W({\mathfrak b}).$
{}From (\ref{e: defining relation for Omega 1 gL zero w}) we now deduce that
$-w\mu = \nu -w\Lambda_0 -w t \eta.$ The expression on the right-hand side of this
equality
has zero restriction to ${\mathfrak a}Qq.$ Therefore, so has $w\mu,$
and we conclude that also $\mu|_{{\mathfrak a}Qq} = 0.$
Combining
this fact with (\ref{e: gl s eta mu in Exp new})
and transitivity of asymptotics, see Theorem \ref{t: transitivity of asymptotics},
we
conclude that
$$
\lambda - \rho_Q =
[\lambda + t\eta - \rho - \mu]|_{{\mathfrak a}Qq} \in {\rm Exp}(Q,v\,|\, f_\lambda),
$$
for
all $\lambda \in \Omega_0.$
~
$\square$\medbreak\noindent\medbreak
For the formulation of the next lemma, we need the following definition.
\begin{defi}
\label{d: special W set}
Let $\Omega\subset{\mathfrak a}Qqdc$ and $s_0\in W$ be given.
The subset $W(\Omega,s_0)$ of $W$ is defined as follows. Let $s'\in W$.
Then $s'\in W(\Omega,s_0)$ if and only if
there exists a chain $s_1, \ldots, s_k=s'$ of elements in $W$,
with $s_j s_{j-1}^{-1}=s_{\alpha_j}$
a simple reflection, such that the following condition
(\ref{e: condition of unboundedness}) holds
for each of the pairs $(s,\alpha)=(s_{j-1},\alpha_j)\in W\times\Delta$,
$j=1,\ldots,k$.
\begin{equation}
\label{e: condition of unboundedness}
\text{If $s^{-1}\alpha|_{{\mathfrak a}Qq}\neq0$ then $\lambda:to{\msy R}e\inp{s\lambda}{\alpha}$ is
not bounded from below on $\Omega$.}
\end{equation}
\end{defi}
Notice that if $\Omega$ is dense in ${\mathfrak a}Qqdc$, then $W(\Omega,s_0)=W$
for all $s_0\in W$. Indeed, (\ref{e: condition of unboundedness}) is
then fulfilled by all
elements $\alpha\in\Delta$. Hence, in order to verify the conditions
of Definition \ref{d: special W set} for $s'\in W$ arbitrary,
we may choose as $s_{\alpha_1},\ldots,s_{\alpha_k}$
the elements in a reduced expression
$s's_0^{-1}=s_{\alpha_k}\cdots s_{\alpha_1}$.
\begin{lemma}
\label{l: second step vanishing thm}
Let $\Omega\subset {\mathfrak a}Qqdc$ be a non-empty connected open subset,
$Y \subset {}^*\fa_{Q\iq}dc$ a finite subset, and $\delta \in {\msy D}Qmaps.$
Let $f \in {\cal E}_{Q,Y}({\rm X}p\,:\,\tau\,:\,\Omega\,:\, \delta)_\lambdaob$
and $s \in W.$ Assume that there exist
$t \in W_Q,\, \eta \in Y$, $\mu \in {\msy N}{\msy D}elta$ and
$w \in {\msy N}Kaq$
such that
\begin{equation}
\label{e: exponents in Exp P w f new}
s\lambda + st \eta - \rho - \mu \in {\rm Exp}(P, w\,|\, f_\lambda),
\end{equation}
for all $\lambda$ in some non-empty open subset of $\Omega.$
Then for every $s_1\in W(\Omega,s)$
there exist
$t_1 \in W_Q,\, \eta_1 \in Y,$ $\mu_1 \in {\msy N}{\msy D}elta$ and $w_1 \in {\msy N}Kaq,$
such that
\begin{equation}
\label{e: one exponents in Exp P w f new}
s_1\lambda + s_1t_1 \eta_1 - \rho - \mu_1 \in {\rm Exp}(P, w_1\,|\, f_\lambda),
\end{equation}
for all $\lambda$ in a full open subset of $\Omega.$
In particular, if $\Omega$ is dense in ${\mathfrak a}Qqdc$, then the above conclusion
holds for every $s_1\in W$.
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
In the proof we will frequently use the following consequence of
Lemma \ref{l: holo of asymp}, based on holomorphy of asymptotics.
If $s_1\in W,$
$t_1 \in W_Q,\, \eta_1 \in Y,$ $\mu_1 \in {\msy N}{\msy D}elta$ and $w_1 \in {\msy N}Kaq,$
then (\ref{e: one exponents in Exp P w f new}) holds for $\lambda$ in a full open
subset of $\Omega$ as soon as it holds for a fixed $\lambda$ in the full open subset
$\Omega \cap {\mathfrak a}Qqdczero(P, WY)$ of $\Omega.$ We now turn to the proof.
If $s_1 = s,$ or more generally, if $s_1\in sW_Q$,
then the conclusion readily follows by the previous remark.
By Definition \ref{d: special W set} we now see that it suffices to
prove the lemma for $s_1 = s_\alpha s,$ with $\alpha\in {\msy D}elta$ such that
(\ref{e: condition of unboundedness}) holds.
There are two cases to consider, namely that $s^{-1} \alpha |_{ {\mathfrak a}Qqd}$
equals zero or not.
In the first case, $s_1 = s s_{s^{-1} \alpha}\in sW_Q$ and the conclusion
is valid.
We may thus assume that we are in the second case, i.e., $s_1=s_\alpha s$ with
\begin{equation}
\label{e: s inv ga in min gsQ new}
s^{-1} \alpha |_{ {\mathfrak a}Qqd}\neq 0.
\end{equation}
We will complete the proof by showing that the following assumption leads to a contradiction.
{\bf Assumption:} for all $t_1 \in W_Q,$ $\eta_1 \in Y,$ $\mu_1\in {\msy N} {\msy D}elta$ and
$w_1 \in {\msy N}Kaq$ there exists
no non-empty open subset $\Omega'$ of $\Omega$ such that
(\ref{e: one exponents in Exp P w f new}) holds for $\lambda \in \Omega'.$
Let $\Xi$ be the set of elements $(st\eta - \mu)|_{{\mathfrak a}gaq}$ with
$t \in W_Q, \eta \in Y, \mu \in {\msy N} {\msy D}elta$ such that
(\ref{e: exponents in Exp P w f new}) holds
for $\lambda$ in a non-empty open subset of $\Omega$,
for some $w\in {\msy N}Kaq$. Then $\Xi$ is a non-empty subset
of ${\mathfrak a}gaqdc$ contained in a set of the form $X - {\msy N} {\msy D}r({{}^*Pga}),$ with
$X \subset {\mathfrak a}gaqdc$ finite. Hence we may select $t \in W_Q, \eta \in Y$ and
$\mu \in {\msy N} \Delta$ such that $(st\eta -\mu)|_{{\mathfrak a}gaq}$ is
${\rm pr}eceq_{{\msy D}r({{}^*Pga})}$-maximal in $\Xi.$
According to the first paragraph of the proof,
there exists $w\in{\msy N}Kaq$ such that
(\ref{e: exponents in Exp P w f new}) is
valid for $\lambda$ in a full open subset $\Omega_0$ of $\Omega.$
For $\lambda \in \Omega_0$ we put
$$
\xi(\lambda) = [s\lambda + st \eta - \rho - \mu]|_{{\mathfrak a}gaq}.
$$
Then by transitivity of asymptotics, see
Theorem \ref{t: transitivity of asymptotics},
it follows that
$$
\xi(\lambda) \in {\rm Exp}({{}^*Pga}, w\,|\, f_\lambda)
$$
for $\lambda \in \Omega_0.$ In the following we shall investigate the
coefficient of the expansion of $f_\lambda$ along
$({{}^*Pga}, w),$ for $\lambda \in \Omega_0,$ given by
$$
\varphi_\lambda (m) := q_{\xi(\lambda)}({{}^*Pga},w\,|\, f_\lambda, \,\cdot\,, m).
$$
Here $\varphi_\lambda$ is a non-trivial $\tau_Qga$-spherical function
on ${\rm X}gawp$ with values in $P_k({\mathfrak a}gaq),$ for $k={\rm deg}_{\rm a} f$, see
Thm.\ \ref{t: expansion along the walls} (b).
It follows from (\ref{e: s inv ga in min gsQ new}) and
the asymptotic globality assumption on $f,$ see
Lemma \ref{l: minimal condition for glob new},
that actually $\varphi_\lambda$ extends
to a smooth function on ${\rm X}gaw,$ for every $\lambda$ in an dense open subset
$\Omega_0'$ of $\Omega_0.$
This observation will play a crucial role at a later stage of this proof.
Let
$$
\Omega_1 := \Omega_0' \cap {\mathfrak a}Qqdczero(P, WY) \cap {\mathfrak a}Qqdczero({}^*Pga, WY).
$$
The second and third set in this intersection
are full open subset of ${\mathfrak a}Qqdc,$
see Lemma \ref{l: azero is full}.
Hence $\Omega_1$ is a dense open subset of $\Omega.$
We claim that for $\lambda \in \Omega_1$ the following holds.
If $s' \in W,\, t' \in W_Q, \,\eta' \in Y,$ $\mu' \in {\msy N}\Delta$ and $w'\in {\msy N}Kaq$
are such that
\begin{equation}
\label{e: hypothesis claim on exponents new}
\left\{
\begin{array}{l}
s'\lambda + s't'\eta' -\rho -\mu' \in {\rm Exp}(P, w'\,|\, f_\lambda)
\text{and}\\
\xi(\lambda) {\rm pr}eceq_{{\msy D}r({{}^*Pga})} (s'\lambda + s't'\eta' -\rho -\mu')|_{\mathfrak a}gaq,
\end{array}\right.
\end{equation}
then
\begin{equation}
\label{e: conclusion claim on exponents}
s' \in sW_Q \text{and} (s't'\eta' - \mu')|_{{\mathfrak a}gaq} = (st \eta - \mu)|_{{\mathfrak a}gaq}.
\end{equation}
To prove the claim, let $s',t',\eta',\mu',w'$ satisfy (\ref{e: hypothesis claim on exponents new}).
Then there exists a $\nu \in {\msy N}\Delta({{}^*Pga})$
such that $s'\lambda + s't'\eta' -\rho -\mu'- \nu$ and $s\lambda + st \eta -\rho - \mu$
have the same restriction $\xi(\lambda)$ to ${\mathfrak a}gaq.$ By the definition of $\Omega_1$
this implies that $s'$ and $s$ define the same class in $W/\sim_{{}^*Pga|Q},$
see Lemma \ref{l: exponents disjoint}.
The latter set equals $W_\alpha\backslash W/W_Q,$ by Lemma \ref{l: WPQ as cosets},
hence $s'$ belongs to $s_\alpha s W_Q=s_1W_Q$ or to $ s W_Q.$ In the first case
it follows that $s' \lambda= s_1 \lambda,$ hence
$s_1 \lambda + s_1 t''\eta' -\rho -\mu' \in {\rm Exp}(P, w'\,|\, f_\lambda)$
for some $t''\in W_Q$. This assertion
then
holds for $\lambda$ in a full open subset of $\Omega_1,$ contradicting the above
assumption.
It follows that we are in the second case $s' \in s W_Q,$
hence $s' = s t''$ for some $t'' \in W_Q.$
The element
$(s't'\eta' - \mu')|_{{\mathfrak a}gaq} = (st''t'\eta' - \mu')|_{{\mathfrak a}gaq}$
therefore belongs to $\Xi;$ from (\ref{e: hypothesis claim on exponents new})
it follows that it dominates
the maximal element $(st \eta -\mu)|_{{\mathfrak a}gaq},$ hence is equal to that
element. This implies (\ref{e: conclusion claim on exponents}), hence establishes
the claim.
It follows from the above claim that, for $\lambda \in \Omega_1,$
the exponent $\xi(\lambda)$ is
actually a leading exponent of $f_\lambda$ along $({{}^*Pga}, w).$
To see this, let $\lambda \in \Omega_1$ and let $\xi \in {\rm Exp}({{}^*Pga}, w \,|\, f_\lambda)$
be an exponent with $\xi(\lambda) {\rm pr}eceq_{{\msy D}r({{}^*Pga})} \xi.$
Then, in view of Theorem \ref{t: transitivity of asymptotics},
there exist $s' \in W,\, t' \in W_Q, \,\eta' \in Y$ and $\mu' \in {\msy N}\Delta$
such that the element $s'\lambda + s't'\eta' -\rho -\mu'$ restricts to $\xi$ on ${\mathfrak a}gaq$
and belongs to ${\rm Exp}(P, w'\,|\, f_\lambda)$
for some $w' \in {\cal W}.$ It now follows from
the claim established above that $\xi = \xi(\lambda).$
Thus, we see that $\xi(\lambda)$ is a leading exponent indeed.
Consequently, by
Lemma \ref{l: D finiteness of leading coefficient}
the function $\varphi_\lambda$ is ${\msy D}({\rm X}gaw)$-finite, for every $\lambda \in \Omega_1.$
We proceed by investigating the exponents of its expansion.
Select a complete set ${\cal W}_{\alpha, w}$ of representatives for
$W_\alpha / (W_\alpha \cap W_{K \cap wHw^{-1}})$ in ${\msy N}Kaq.$
We put ${}^*P = P \cap M_\alpha.$ Then by transitivity
of asymptotics, cf.\ Theorem \ref{t: transitivity of asymptotics},
we see that for
the set of $({}^*P, u)$-exponents of $\varphi_\lambda,$ as $u\in {\cal W}_{\alpha,w},$
the following inclusion
holds:
$$
{\rm Exp}({}^*P, u\,|\, \varphi_\lambda) \subset \{ \xi|_{{}^*\fagaq}\mid \xi \in {\rm Exp}(P, uw\,|\, f_\lambda)
\;\;\; \xi | {\mathfrak a}gaq= \xi(\lambda) | {\mathfrak a}gaq \}.
$$
Hence, for $\lambda\in \Omega_1,$
every exponent in ${\rm Exp}({}^*P,u \,|\, \varphi_\lambda)$
is of the form
$(s' \lambda + s't'\eta' - \rho - \mu')|_{{}^*\fagaq} $ with
$s' \in W, \;t' \in W_Q, \;\eta' \in Y$ and $\mu' \in {\msy N}{\msy D}elta$
satisfying
$$
\left\{
\begin{array}{l}
s'\lambda + s't'\eta' -\rho -\mu' \in {\rm Exp}(P, u w\,|\, f_\lambda),\\{}
[s'\lambda + s't'\eta' - \rho - \mu']|_{{\mathfrak a}gaq}
= \xi(\lambda)|_{{\mathfrak a}gaq}.
\end{array}
\right.
$$
It follows from the claim established above that
(\ref{e: conclusion claim on exponents}) holds.
We have thus shown that for every $\lambda \in \Omega_1$ the exponents in
${\rm Exp}({}^*P,u \,|\, \varphi_\lambda)$
are of the form
$ (s \lambda + st'\eta' - \rho - \mu' )|_{{}^*\fagaq}$ with
$t' \in W_Q,\; \eta' \in Y,\;\mu' \in {\msy N}{\msy D}elta$ satisfying
$$
[st'\eta' - \mu'] |_{{\mathfrak a}gaq} = [st \eta - \mu ]|_{{\mathfrak a}gaq}.
$$
{}From this it follows that
the restriction $\mu'|_{{\mathfrak a}gaq}$ of the $\mu'$ occurring
runs through a finite subset of
${\msy N}{\msy D}r({}^*Pga)= {\msy N}[\Delta\setminus\{\alpha\}]|_{\mathfrak a}gaq,$ independent of $\lambda$.
Hence there exists a finite subset $S' \subset {\msy N}\Delta$
such that $\mu'$ runs through $S' - {\msy N} \alpha.$ We thus see
that there exists
a finite subset $S \subset {}^*\fagaqdc$
such that, for every $\lambda \in \Omega_1,$
\begin{equation}
\label{e: inclusions for Exp starP u gf}
\cup_{u \in {\cal W}_{\alpha,w}}\;
{\rm Exp}({}^*P,u\,|\, \varphi_\lambda) \subset s\lambda |_{{}^*\fagaq} + S - {\msy N}\alpha.
\end{equation}
{}From (\ref{e: condition of unboundedness}) and
(\ref{e: s inv ga in min gsQ new}) it now follows that we may select
a non-empty open
subset $\Omega_2$ of the dense open subset $\Omega_1$ of $\Omega$
such that, for every
$\lambda \in \Omega_2,$ each $u \in {\cal W}_{\alpha,w}$
and all $\xi \in {\rm Exp}({}^*P, u\,|\, \varphi_\lambda),$
$$
\inp{{\msy R}e \xi + {}^* \rho }{\alpha} < 0.
$$
Since $\varphi_\lambda$ is ${\msy D}({\rm X}gaw)$-finite this implies that
$\varphi_\lambda$ is square integrable on ${\rm X}gaw,$
see \bib{B87}, Thm. 6.4 with $p =2;$ hence $\varphi_\lambda$ a Schwartz function
for $\lambda \in \Omega_2,$
see \bib{B87}, Thm.\ 7.3.
On
the other hand, from (\ref{e: s inv ga in min gsQ new}) it follows that
the linear map $\lambda :to s\lambda |_{{}^*\fagaq}$ is surjective
from ${\mathfrak a}Qqdc$ onto ${}^*\fagaqdc.$
Therefore, the set
$\{s\lambda|_{{}^*\fagaq}\mid \lambda \in \Omega_2\}$ has a non-empty interior
in ${}^*\fagaqdc.$ Combining this observation
with (\ref{e: inclusions for Exp starP u gf}) we infer that there exists a non-empty
open subset $\Omega_3 \subset \Omega_2,$ such that the sets
$\cup_{u \in {\cal W}_{\alpha,w}}\;
{\rm Exp}({}^*P,u\,|\, \varphi_\lambda),$ for $\lambda \in \Omega_3,$ are mutually disjoint.
Now these sets are non-empty, since $\varphi_\lambda \neq 0,$ for $\lambda\in \Omega_3.$
Therefore, the union of these sets, as $\lambda \in \Omega_3,$ is
uncountable. This contradicts Lemma
\ref{l: limitation on exponents of a Schwartz function},
applied to the space ${\rm X}gaw.$
~
$\square$\medbreak\noindent\medbreak
\begin{lemma}
\label{l: e in W set}
Assume that $\Omega\subset{\mathfrak a}Qqdc$ is $Q$-distinguished.
Then $e\in W(\Omega,s_0)$ for all $s_0\in W$.
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ } Let $k=l(s_0)$ denote the length of $s_0$, and let
$s_0=s_{\alpha_1}\cdots s_{\alpha_k}$ be a reduced expression for $s_0$.
Put $s_j=s_{\alpha_j}\cdots s_{\alpha_1}s_0=s_{\alpha_{j+1}}\cdots s_{\alpha_k}$
for $j=1,\ldots,k$, then $s_k=e$. We claim that
(\ref{e: condition of unboundedness}) holds for each pair
$(s,\alpha)=(s_{j-1},\alpha_j)$. Since $l(s_j)=l(s_{j-1})-1$, the root
$s_{j-1}^{-1}\alpha_j$ must be negative. Hence the restriction of
this root to ${\mathfrak a}Qq$ is zero or belongs to $-\Sigma(Q)$. Now
(\ref{e: condition of unboundedness})
follows immediately from Definition \ref{d: Q-distinguished}.~
$\square$\medbreak\noindent\medbreak
{\bf Proof of Theorem \ref{t: vanishing theorem new}:\ }
We prove the result by induction on $|\deltamap|.$
If $\delta = 0,$ then for $\lambda \in {\mathfrak a}Qqdc$ the ideal
$I_{\gd, \gl}$ equals ${\msy D}GH;$
hence ${\cal E}_Q({\rm X}p\,:\, \tau \,:\, \Omega \,:\, \delta)_\lambdaob = 0$ and the result follows.
Let now $|\delta| =1,$ let
$f \in {\cal E}_Q({\rm X}p\,:\, \tau\,:\, \Omega\,:\, \delta)_\lambdaob$ and let
(\ref{e: hypothesis vanishing thm new}) be fulfilled for all $v \in {}^Q\cW.$
Assume that $f \neq 0.$ We will show that this assumption leads to a contradiction.
There exists a finite subset $Y\subset {}^*\fa_{Q\iq}dc$
such that $f \in {\cal E}QY({\rm X}p\,:\, \tau\,:\, \Omega\,:\, \delta)_\lambdaob$ and
a $\lambda_0 \in \Omega \cap {\mathfrak a}Qqdczero(P, WY)$
such that $f_{\lambda_0} \neq 0.$ Let ${\cal W}$ be a complete set of representatives
of $W/W_{K \cap H}$ in ${\msy N}Kaq$ containing ${}^Q\cW.$ Then
${\rm Exp}(P, w\,|\, f_{\lambda_0}) \neq \emptyset$ for some $w \in {\cal W}.$
In view of (\ref{e: exponents family})
it follows that there exist
$s \in W,$ $t\in W_Q,$ $\eta \in Y$ and $\mu \in {\msy N} \Delta,$
such that
\begin{equation}
\label{e: exponent for gl is gl zero}
s \lambda+ st \eta - \rho -\mu \in {\rm Exp}(P, w\,|\, f_\lambda),
\end{equation}
for $\lambda = \lambda_0.$ From Lemma \ref{l: holo of asymp}
it follows that (\ref{e: exponent for gl is gl zero}) is valid for $\lambda$
in a full
open subset of $\Omega.$
By Lemmas \ref{l: second step vanishing thm} and \ref{l: e in W set}
this implies that there exist
$t_1 \in W_Q, $ $\eta_1 \in Y,$ $\mu_1 \in {\msy N}\Delta$ and $w_1 \in {\msy N}Kaq,$ such that
$\lambda + t_1 \eta_1 - \rho -\mu_1 \in {\rm Exp}(P, w_1 \,|\, f_\lambda)$
for $\lambda$ in a full open subset of $\Omega.$
Let $v\in{}^Q\cW$ be the representative of $W_Qw_1W_{K\cap H}$.
By Lemma \ref{l: first step vanishing thm}
it follows that
$\lambda- \rho_Q \in {\rm Exp}(Q, v\,|\, f_\lambda)$
for $\lambda$ in a full open subset $\Omega_0$ of $\Omega.$
Since $\Omega_0 \cap \Omega'$ is non-empty, we obtain
a contradiction with (\ref{e: hypothesis vanishing thm new}).
Now suppose that $|\deltamap| = k > 1,$ and assume that the result
has already been established for $\deltamap \in {\msy D}Qmaps$ with $|\deltamap| < k.$
Fix $\nu \in {\rm supp}\, (\deltamap)$ and put
$\deltamap' = \deltamap - \deltamap_\nu.$ Then $\deltamap' \in {\msy D}Qmaps;$
moreover, $|\delta_\nu| =1$ and $|\delta'| = k-1.$
Fix any $D \in {\msy D}GH$ and define the family $g$ by
(\ref{e: defi g as D minus gamma f}).
Then $g \in {\cal E}_{Q}({\rm X}p\,:\, \tau\,:\, \Omega\,:\, \delta')$
by Lemma \ref{l: stability family under D new}.
Moreover, it readily follows from
Lemma \ref{l: stability family with globality under D}
that the family $g$ belongs to
${\cal E}_{Q}({\rm X}p\,:\, \tau\,:\,\Omega\,:\, \delta')_\lambdaob.$
For $\lambda \in \Omega$ and $v \in {\msy N}Kaq$ we have
\begin{equation}
\label{e: inclusion exponents F gl}
{\rm Exp}(Q,v\,|\, g_\lambda) \subset {\rm Exp}(Q, v \,|\, f_\lambda) - {\msy N} \Sigma_r(Q),
\end{equation}
in view of Lemma \ref{l: radial component applied to expansion} (b).
Moreover, by hypothesis we have the following inclusion, for every
$\lambda \in \Omega',$
\begin{equation}
\label{e: inclusion exponents f gl}
{\rm Exp}(Q,v\,|\, f_\lambda) \subset [W(\lambda + Y)|_{{\mathfrak a}Qq} - \rho_Q - {\msy N} \Sigma_r(Q)]
\setminus \{\lambda - \rho_Q\}.
\end{equation}
Combining (\ref{e: inclusion exponents F gl}) and
(\ref{e: inclusion exponents f gl}) we infer that ${\rm Exp}(Q,w\,|\, g_\lambda)$
does not contain $\lambda - \rho_Q$ for
$\lambda \in \Omega'$ and every $w \in {\msy N}Kaq.$
Consequently, the family $g$ satisfies the hypotheses
of Theorem \ref{t: vanishing theorem new}.
Since $|\deltamap'| = k-1,$ it follows from the induction hypothesis
that $g =0.$ Since $D$ was arbitrary, we see that $f_\lambda$ is
annihilated by $I_{\deltamap_\nu, \lambda},$ for every $\lambda \in \Omega.$
Hence $f$ belongs to ${\cal E}_{Q}({\rm X}p\,:\, \tau\,:\,\Omega\,:\, \deltamap_\nu)_\lambdaob.$
Since $|\deltamap_\nu| =1 < k,$ it now follows from the induction hypothesis that
$f=0.$
~
$\square$\medbreak\noindent\medbreak
The following result is also based on Lemma
\ref{l: second step vanishing thm}.
\begin{cor}
\label{c: variant of vanishing theorem}
Let $\Omega\subset {\mathfrak a}Qqdc$ be a connected dense open subset,
$Y \subset {}^*\fa_{Q\iq}dc$ a finite subset, and $\delta \in {\msy D}Qmaps.$
Let $f \in {\cal E}_{Q,Y}({\rm X}p\,:\,\tau\,:\,\Omega\,:\, \delta)_\lambdaob$
and $s_1\in W$. If
$$
(s_1\lambda + WY - \rho - {\msy N}\Delta)\cap {\rm Exp}(P, w\,|\, f_\lambda)=\emptyset,
$$
for all $\lambda$ in a non-empty open subset of $\Omega$
and for all $w\in N_K({\mathfrak a}q)$, then $f=0$.
\end{cor}
\par\noindent{\bf Proof:}{\ }{\ } Assume that $f\neq0$. Then there exists an element
$\lambda\in \Omega \cap {\mathfrak a}Qqdczero(P, WY)$ such that $f_\lambda\neq0$,
and then
\begin{equation}
\label{e: not an inclusion}
s\lambda + st \eta - \rho - \mu \in {\rm Exp}(P, w\,|\, f_\lambda)
\end{equation}
for some $s\in W$, $t \in W_Q,\, \eta \in Y$, $\mu \in {\msy N}{\msy D}elta$ and
$w\in N_K({\mathfrak a}q)$.
As remarked in the beginning of the proof of
Lemma \ref{l: second step vanishing thm},
(\ref{e: not an inclusion})
then holds for all $\lambda$ in a full open subset of $\Omega$.
Hence Lemma \ref{l: second step vanishing thm} applies;
its final statement contradicts the present assumption for $s_1$.~
$\square$\medbreak\noindent\medbreak
Finally in this section we will show that for a family in
${\cal E}_{Q}({\rm X}p\,:\,\tau\,:\,\Omega\,:\, \delta)$ that allows a smooth
extension to ${\rm X}$, the hypothesis of asymptotic globality can be
left out in the vanishing theorem. Let
$${\cal E}_{Q}({\rm X}\,:\,\tau\,:\,\Omega\,:\, \delta)=
\{f\in{\cal E}_{Q}({\rm X}p\,:\,\tau\,:\,\Omega\,:\, \delta)\mid
f_\lambda\in C^\infty({\rm X}\,:\,\tau), \lambda\in\Omega\}.$$
\begin{cor}
Let $Q \in \cP_\gs$ and $\deltamap \in {\msy D}Qmaps.$
Let $\Omega \subset {\mathfrak a}Qqdc$
be a $Q$-distinguished open subset and let
$f \in {\cal E}_Q({\rm X}\,:\,\tau\,:\,\Omega\,:\, \delta).$
Assume that there exists a non-empty open subset
$\Omega' \subset \Omega$ such that, for each $v \in {}^Q\cW, $
$$
\lambda - \rho_Q \notin {\rm Exp}(Q,v \,|\, f_\lambda), \quad\quad (\lambda \in \Omega').
$$
Then $f = 0.$
\end{cor}
\par\noindent{\bf Proof:}{\ }{\ } As in the proof of Theorem \ref{t: vanishing theorem new}
we proceed by induction on $|\delta|$. If $|\delta|=0$ the result is trivial.
If $|\delta|=1$ it follows from Proposition
\ref{p: global eigen implies as global} that
${\cal E}_Q({\rm X}\,:\,\tau\,:\,\Omega\,:\, \delta)\subset
{\cal E}_Q({\rm X}p\,:\,\tau\,:\,\Omega\,:\, \delta)_\lambdaob$, and then the result
follows directly from Theorem \ref{t: vanishing theorem new}.
Now suppose that $|\delta|=k>1$, and assume that the result has already been
established for all $\delta\in D_Q$ with $|\delta|<k$. Let $\delta'$ and $g$
be as in the proof of Theorem \ref{t: vanishing theorem new}.
Then it is easily seen that $g\in {\cal E}_Q({\rm X}\,:\,\tau\,:\,\Omega\,:\, \delta')$.
For the rest of the proof we can now proceed exactly as in the proof of
Theorem \ref{t: vanishing theorem new}.~
$\square$\medbreak\noindent\medbreak
\section{Laurent functionals}
\label{s: Laurent functionals}
In this section we define Laurent functionals and describe their
actions on suitable spaces of meromorphic functions.
Throughout this section, $V$ will be a finite dimensional
real linear space, equipped with a (positive definite) inner product $\inp{\,\cdot\,}{\,\cdot\,}.$
Its complexification $V_\C$ is equipped with the complex bilinear
extension of this inner product.
Let $X$ be a (possibly empty) finite set of non-zero elements of $V.$
At this stage we allow proportionality between elements of $X.$
By an $X$-hyperplane in $V_\C,$ we mean an affine hyperplane of the form
$H = a + \alpha_{{\scriptscriptstyle \C}}^\perp,$ with $a \in V_\C,$ $\alpha \in X.$ The hyperplane is
called real if $a$ can be chosen from $V,$ or, equivalently, if it is
the complexification of a real hyperplane from $V.$
A locally finite collection of $X$-hyperplanes in $V_\C$ is called
an $X$-configuration in $V_\C.$ It is called real if all its
hyperplanes are real.
If $a \in V_\C,$ we denote the collection of $X$-hyperplanes in $V_\C$
through $a$ by $\cH(a, X) = \cH(V_\C, a, X).$
If $E$ is a complete locally convex space, then by
$\cM(a, X,E) = \cM(V_\C, a, X,E)$
we denote the ring of germs of $E$-valued meromorphic functions at $a$ whose singular
locus at $a$ is contained in $\cH(a,X).$ Here and in the following we will suppress
the space $E$ in the notation if $E = {\msy C}.$ Thus, $\cM(a,X) = \cM(a,X,{\msy C}).$
Let ${\msy N}^X$ denote the set of maps $X \rightarrow {\msy N}.$ If $d \in {\msy N}^X,$
we define the polynomial function $\pi_{a,d} = \pi_{a,X,d}: V_\C \rightarrow {\msy C}$ by
\begin{equation}
\label{e: defi pi a X d}
\pi_{a,d}(z) ={\rm pr}od_{\xi \in X} \inp{\xi}{z -a}^{d(\xi)}, \quad\quad (z \in V_\C).
\end{equation}
If $X=\emptyset$ then ${\msy N}^X$ has one element which we agree to denote by $0$.
We also agree that $\pi_{a,0}=1$.
Let ${\cal O}_a(E)= {\cal O}_a(V_\C, E)$ denote the ring of germs of $E$-valued holomorphic
functions at $a.$ Then
$$
\cM(a,X,E) = \cup_{d \in {\msy N}^X} \;\; \pi_{a,d}^{-1} {\cal O}_a(E).
$$
In the following we shall identify $S(V)$ with the algebra of constant
coefficient holomorphic differential operators on $V_\C$ in the usual way;
in particular an element
$v \in V$ corresponds to the operator
$\varphi :to v\varphi(z) = \left.\frac{d}{d\tau}\right|_{\tau=0} \varphi(z + \tau v).$
\begin{defi}
\label{d: Laurent functional at a point}
{\rm (Laurent functional at a point)\ }
An $X$-Laurent functional at $a$ is a linear functional $\cL: \cM(a,X) \rightarrow {\msy C}$
such that for every $d \in {\msy N}^X$ there exists an element $u_d \in S(V)$ such that
\begin{equation}
\label{e: Lau and u d}
\cL \varphi = u_d(\pi_{a,d} \varphi)(a),
\end{equation}
for all $\varphi \in \pi_{a,d}^{-1} {\cal O}_a.$
The space of all Laurent functionals at $a$ is denoted by $\cM(a, X)^*_{\rm laur} =
\cM(V_\C, a, X)^*_{\rm laur}.$
\end{defi}
\begin{rem}
Obviously, the string $(u_d)_{d \in {\msy N}^X}$ of elements from $S(V)$ is
uniquely determined by the requirement (\ref{e: Lau and u d}). We shall denote
it by $u_{\cal L}.$
If $E$ is a complete locally convex space, then $X$-Laurent functionals
at $a$ may naturally be viewed as linear maps from $\cM(a,X,E)$ to $E$.
Indeed, let $\cL \in \cM(a, X)^*_{\rm laur}$ and let
$u_\cL = (u_d)_{d \in {\msy N}^X}$ be the associated string of elements from $S(V).$
If $\varphi \in \pi_{a,d}^{-1}{\cal O}_a(E)$ then $\cL \varphi$
is given by formula (\ref{e: Lau and u d}).
\end{rem}
Let $T_a: z :to z + a$ denote translation by $a$ in $V_\C.$
Then $T_a$ maps $\cH(0,X)$ bijectively onto $\cH(a,X).$
Pull-back under $T_a$ induces an isomorphism of rings $T_a^*: \varphi :to \varphi\,{\scriptstyle\circ}\, T_a$
from ${\cal O}_a$ onto ${\cal O}_0.$ Therefore, pull-back under $T_a$ also induces
an isomorphism of rings $T_a^*: \cM(a, X) \rightarrow \cM(0,X).$
By transposition we obtain an isomorphism of linear spaces
$T_{a*}: \cM(0, X)^* \rightarrow \cM(a,X)^*.$ It is readily seen that $T_a^*(\pi_{a,d})
=\pi_{0,d}$ for every $d \in {\msy N}^X.$ {}From the definition of Laurent functionals
it now follows that $T_{a*}$ maps $\cM(0,X)^*_{\rm laur}$ isomorphically
onto $\cM(a,X)^*_{\rm laur}.$ Moreover,
$$
u_{T_{a*}\cL} = u_{\cL}
$$
for all $\cL \in \cM(0,X)^*.$
Let $X'$ be another finite collection of non-zero elements of $V.$ We say
that $X$ and $X'$ are proportional if $\cH(0,X) = \cH(0,X').$
\begin{lemma}
\label{l: Laurent functionals and proportional X}
Let $X,X'$ be proportional finite subsets of $V\setminus\{0\}$
and let $a \in V_\C.$ Then $\cM(a, X) = \cM(a, X')$ and
$\cM(a,X)^*_{\rm laur} = \cM(a,X')^*_{\rm laur}.$
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ } It is obvious that $\cM(a, X) = \cM(a, X')$.
Let $\cL \in \cM(a,X)^*=\cM(a, X')^*$, and assume that
$\cL \in \cM(a,X')^*_{\rm laur}.$ Let $(u_{d'})_{d'\in{\msy N}^{X'}}$
be the associated string. Let $d\in{\msy N}^X$.
Then, by proportionality, there exists $d'\in{\msy N}^{X'}$ and
$c\in{\msy R}\setminus\{0\}$ such that $\pi_{a,X,d}=c\pi_{a,X',d'}.$
Let $u_d=c^{-1}u_{d'}$, then (\ref{e: Lau and u d}) follows immediately.
This shows that $\cL \in \cM(a,X)^*_{\rm laur}$
and establishes the inclusion $\cM(a,X')^*_{\rm laur} \subset \cM(a,X)^*_{\rm laur}.$
The converse inclusion is proved similarly.
~
$\square$\medbreak\noindent\medbreak
Following
the method
of \bib{BSres}, Sect.\ 1.3,
we shall now give
a description of the space of strings
$u_\cL,$ as $\cL \in \cM(a,X)^*_{\rm laur}$.
Put $\varpi_d := \pi_{0,d}$ and equip the space ${\msy N}^X$ with the partial ordering
${\rm pr}eceq$ defined by $d' {\rm pr}eceq d$ if and only if $d'(\xi) \leq d(\xi)$ for
every $\xi \in X.$
If $d'{\rm pr}eceq d$ then we define $d -d'$ componentwise as suggested by the notation.
In \bib{BSres}, Sect.\ 1.3, we defined the linear space
$S_{\leftarrow}(V,X)$ as follows.
Let $d,d' \in {\msy N}^X$ with $d' {\rm pr}eceq d.$ If $u \in S(V),$ then by the Leibniz
rule there exists a unique $u' \in S(V)$ such that
$$
u(\varpi_{d - d'}\varphi)(0) = u'(\varphi)(0), \quad\quad (\varphi \in {\cal O}_0).
$$
We denote the element $u'$ by $j_{d',d}(u).$ The map $j_{d',d}: S(V) \rightarrow S(V)$ thus defined
is linear. Note
that it only depends on $d - d';$ note also that, for $d,d',d'' \in {\msy N}^X$ with
$d'' {\rm pr}eceq d' {\rm pr}eceq d,$
$$
j_{d'',d'}\,{\scriptstyle\circ}\, j_{d', d} = j_{d'', d}.
$$
We now define $S_\leftarrow(V,X)$ as the linear space of strings $(u_d)_{d \in {\msy N}^X}$ in
$S(V)$ such that $j_{d',d}( u_d) = u_{d'}$ for all $d,d \in {\msy N}^X$ with $d' {\rm pr}eceq d.$
Thus, this space is the projective limit:
$$
S_\leftarrow(V,X) = \lim_{\leftarrow} (S(V), j_{\cdot}).
$$
The natural map $S_\leftarrow(V,X) \rightarrow S(V)$ that maps a string to its $d$-component
is denoted by $j_d.$
\begin{lemma}
\label{l: iso Laurent functionals with Sproj}
The map $\cL :to u_\cL$ is a linear isomorphism from $\cM(a,X)^*_{\rm laur}$ onto
$S_\leftarrow(V,X).$
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
See \bib{BSfi}, Appendix B, Lemma B.2.
~
$\square$\medbreak\noindent\medbreak
\begin{lemma}
\label{l: extension to Laurent functional}
Let $a \in V_\C,$ $d \in {\msy N}^X$ and $u \in S(V).$
Then there exists a Laurent functional
$\cL \in \cM(a, X)^*_{\rm laur}$ such that $(u_\cL)_d = u.$
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
See \bib{BSres}, Lemma 1.7.
~
$\square$\medbreak\noindent\medbreak
\begin{rem}
\label{r: evaluation is a Laurent functional}
In particular, it follows that for each $a\in V_{\msy C}$ there
exists a Laurent functional ${\cal L}\in{\cal M}(a,X)^*_{\rm laur}$ such that
${\cal L}\varphi=\varphi(a)$ for all $\varphi\in{\cal O}_a$. Note however, that this
functional is not unique, unless $X=\emptyset$.
\end{rem}
\begin{lemma}
\label{l: annihilator of annihilator}
Let ${\cal M}(a,X)^{*{\cal O}}_{\rm laur}$ denote the annihilator
of ${\cal O}_a$ in ${\cal M}(a,X)^*_{\rm laur}$. Then all functions $\varphi$ in
${\cal M}(a,X)$, that are annihilated by ${\cal M}(a,X)^{*{\cal O}}_{\rm laur}$,
belong to ${\cal O}_a$.
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ } We may assume that $a=0$. Let $\varphi\in{\cal M}(0,X)$
and assume that $\varphi\not\in{\cal O}_0$. Then there exist elements
$d, d'\in{\msy N}^X$ and $\xi\in X$ such that $\pi_{0,d'}=\xi\pi_{0,d}$
and $\pi_{0,d'}\varphi\in{\cal O}_0$ but $\pi_{0,d}\varphi\not\in{\cal O}_0$.
Here we have written $\xi$ also for the function $z:to
\inp{\xi}{z}$ on $V_\biC$.
Since $\pi_{0,d'}\varphi$ is not divisible by $\xi$, its
restriction to $\xi^\perp=\xi^{-1}(0)$ does not vanish.
Hence there exists $u\in S(\xi^\perp)$ such that
$u(\pi_{0,d'}\varphi)(0)\neq 0$.
By Lemma \ref{l: extension to Laurent functional} there exists an element
${\cal L}\in{\cal M}(a,X)^*_{\rm laur}$ such that the $d'$ term of $u_\cL$ is $u$. Then
${\cal L}\varphi=u(\pi_{0,d'}\varphi)(0)\neq0$.
However, for each $\psi\in{\cal O}_0$
we have ${\cal L}\psi=u(\pi_{0,d'}\psi)(0)=[\xi u(\pi_{0,d}\psi)](0)=0$.
Hence ${\cal L}\in{\cal M}(a,X)^{*{\cal O}}_{\rm laur}$.~
$\square$\medbreak\noindent\medbreak
We extend the notion of a Laurent functional as follows.
The disjoint union of the spaces $\cM(a,X)^*_{\rm laur}$ as $a \in V_\C$ is denoted
by $\cM(*,X)^*_{\rm laur} = \cM(V_\C, *, X)^*_{\rm laur}.$
By a section of $\cM(*, X)^*_{\rm laur}$
we mean a map $\cL: V_\C \rightarrow \cM(*,X)^*_{\rm laur}$ with $\cL_a \in \cM(a,X)^*_{\rm laur}$
for all $a \in V_\C.$
The closure of the set $\{a \in V_\C\mid \cL_a \neq 0\}$ is called the support
of $\cL$ and denoted by ${\rm supp}\,(\cL).$
\begin{defi}
{\rm (Laurent functional)}
An $X$-Laurent functional on $V_\C$ is a finitely supported section of
$\cM(*, X)^*_{\rm laur}.$
The set of $X$-Laurent functionals is denoted by $\cM(V_\C, X)^*_{\rm laur}$
and equipped with the obvious structure of a linear space.
Is $S$ is a subset of $V_\C,$ we define the space
$\cM(S, X)^*_{\rm laur} = \cM(V_\C, S, X)^*_{\rm laur}$
by
$$
\cM(S, X)^*_{\rm laur} = \{ \cL \in \cM(V_\C, X)^*_{\rm laur} \mid {\rm supp}\, \cL \subset S \}
$$
and call this the space of $X$-Laurent functionals on $V_\C$ supported in $S.$
\end{defi}
\begin{rem}
\label{r: Laurent functionals at a as subspace}
Note
that, for $a \in V_\C,$ the map $\cM(\{a\}, X)^*_{\rm laur} \rightarrow \cM(a, X)^*_{\rm laur},$
defined by $\cL :to \cL_a,$ is a linear isomorphism. Accordingly we shall
view $\cM(a, X)^*_{\rm laur}$ as a linear subspace of $\cM(V_\C, X)^*_{\rm laur}.$
In this way $\cM(S, X)_{\rm laur}^*$ becomes identified with
the algebraic direct sum of the linear spaces $\cM(a, X)_{\rm laur}^*,$ as $a \in S,$
for $S$ any subset of $V_\C.$ Accordingly,
if $\cL \in \cM(V_\C, X)^*_{\rm laur},$ then $\cL_a \in
\cM(a, X)^*_{\rm laur} \subset \cM(V_\C, X)^*_{\rm laur}$ for $a \in V_\C,$ and
$$
\cL = \sum_{a \in {\rm supp}\, \cL} \cL_a.
$$
\end{rem}
\begin{lemma}
\label{l: global Laurent functionals and proportional X}
Let $X$ and $X'$ be proportional finite subsets of $V\setminus \{0\}.$
Then $$\cM(V_\C, X)^*_{\rm laur} = \cM(V_\C, X')^*_{\rm laur}.$$
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
This is an immediate consequence of
Lemma \ref{l: Laurent functionals and proportional X} and the above definition.
~
$\square$\medbreak\noindent\medbreak
We proceed by discussing the action of a Laurent functional on meromorphic
functions.
Let $E$ be a complete locally convex space and $\Omega\subset V_\C$ an open
subset.
If $a \in \Omega,$
then by $\cM(\Omega, a, X, E)$ we denote the space of meromorphic functions
$\varphi: \Omega \rightarrow E$ whose germ $\varphi_a$ at $a$ belongs to $\cM(a,X,E).$
If $S\subset \Omega,$ we define
$$
\cM(\Omega, S, X, E) := \cap_{a \in S} \cM(\Omega, a, X, E).
$$
Finally, we write $\cM(\Omega, X, E )$ for $\cM(\Omega, \Omega, X, E).$
In particular, $\cM(V_\C, X, E)$ denotes the space of functions
$\varphi \in \cM(V_\C, E)$ with singular locus ${\rm sing}(\varphi)$ contained in an $X$-configuration.
There is a natural pairing $\cM(S,X)^*_{\rm laur} \times \cM(\Omega, S, X, E) \rightarrow E,$ given
by
\begin{equation}
\label{e: pairing Lau and functions}
\cL \varphi = \sum_{a \in {\rm supp}\, \cL} \cL_a \varphi_a.
\end{equation}
\hide{The pairing naturally induces a linear map $\cM(S, X)^*_{\rm laur} \rightarrow \cM(\Omega, S, X, E)^*.$
Although we do not need the following result in the rest of the paper, we give
a proof for the sake of completeness.}
\begin{lemma}
Let $S \subset V_\biC$ be arbitrary, and let $\Omega$ be an open
subset of $V_\biC$ containing $S.$ Then the pairing given by
(\ref{e: pairing Lau and functions}) for $E={\msy C}$
induces a linear embedding
$$
\cM(S,X)^*_{\rm laur} \hookrightarrows \cM(\Omega, S, X)^*.
$$
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
Let $\cL \in \cM(S,X)^*_{\rm laur}$ and assume that $\cL = 0$ on
$\cM(\Omega, S, X).$ We may assume that $S={\rm supp}\, \cL$.
For every $a \in S$ we write $u^a = (u^a_d)_{d \in \N^X}$ for the string
determined by~$\cL_a.$
Select $b \in S.$ Then it suffices to prove
that $\cL_b = 0.$
Fix $d \in \N^X$ and $\phi \in {\cal O}_b.$ Then it suffices to show that
$u^b_d(\phi)(b) = 0.$
For every $a \in S\setminus\{b\}$ we may select
$d(a) \in \N^X$ such that $\pi_{a,d(a)}\pi_{b,d}^{-1}$ is holomorphic at
$a.$ Moreover, we put $d(b) =d.$
For $a \in S$ there exists a unique $v_a \in S(V)$ such that
for all $f \in {\cal O}_a$ we have
$$
v_a(f)(a) = u^a_{d(a)}(\pi_{a,d(a)}\pi_{b,d}^{-1}f)(a).
$$
We note that $v_b = u_{d}^b.$
We may now apply the
lemma below,
with $E_a = {\msy C} v_a,$ for $a \in S,$ and, finally with $\xi_a = 0$ if $a \neq b$ and
with $\xi_b$ defined by $\xi_b(v_b) = v_b(\phi)(b).$
Hence there exists a polynomial function $\psi$ on $V_\biC$ such that
$v_a(\psi)(a) = 0$ for all $a \in S\setminus \{b\},$ and such that
$v_b(\psi)(b) = v_b(\phi)(b).$
Define $\varphi =\pi_{b,d}^{-1}\psi .$ Then $\varphi \in \cM(\Omega, S, X).$
Hence ${\cal L}\varphi = 0.$ On the other hand,
\begin{eqnarray*}
{\cal L}\varphi = \sum_{a \in S} {\cal L}_a\varphi_a &=&
\sum_{a \in S} {\cal L}_a(\pi_{a,d(a)}^{-1} \pi_{a,d(a)}\pi_{b,d}^{-1} \psi)
\\
&=&
\sum_{a \in S} u^a_{d(a)}(\pi_{a,d(a)}\pi_{b,d}^{-1} \psi)(a)
=
\sum_{a \in S} v_a( \psi)(a)
=
v_b(\psi)(b) = u^b_d(\phi)(b).
\end{eqnarray*}
It follows that $u^b_{d}(\phi)(b) =0.$
~
$\square$\medbreak\noindent\medbreak
\begin{lemma}
Let $S \subset V_\biC$ be a finite set. Suppose that for every $a \in S$ a
finite dimensional complex linear subspace $E_a \subset S(V)$ together with
a complex linear functional $\xi_a \in E_a^*$ is given.
Then there exists a polynomial function $\psi$ on $V_\biC$ such
that $u\psi(a) = \xi_a(u)$ for every $a \in S$ and all $u \in E_a.$
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
This result is well known.
~
$\square$\medbreak\noindent\medbreak
We proceed by discussing the push-forward of a Laurent functional by an injective linear mapping.
Let $V_0$ be a real linear space and $\iota: V_0 \rightarrow V$ an injective linear map.
We assume that no element of $X$ is orthogonal to $\iota(V_0)$.
We equip $V_0$ with the pull-back of the inner product of $V$ under $\iota$
and denote the corresponding transpose of $\iota$ by $p.$
Then $X_0 := p(X)$ consists of non-zero elements.
We denote the complex linear extensions of $\iota$ and $p$ by the same symbols.
Then, if $H \subset V_\C$ is an $X$-hyperplane, its preimage
$\iota^{-1}(H)$ is an $X_0$-hyperplane of $V_{0\C}.$
Let $a_0 \in V_{0\C}$ and put $a =\iota(a_0).$
Then pull-back by $\iota$ induces a
natural algebra homomorphism $\iota^*: {\cal O}_a(V_\C) \rightarrow {\cal O}_{a_0}(V_{0\C}).$
On the other hand, pull-back by $p$ induces
a natural algebra homomorphism $p^*: {\cal O}_{a_0}(V_{0\C}) \rightarrow {\cal O}_a(V_\C).$
{}From $p\,{\scriptstyle\circ}\, \iota= I_{V_0}$ it follows that $\iota^* \,{\scriptstyle\circ}\, p^* = I$ on
${\cal O}_{a_0}(V_{0\C}),$ hence $\iota^*$ is surjective.
If $d: X \rightarrow {\msy N}$ is a map, then we write $p_*(d)$ for the map
$X_0 \rightarrow {\msy N}$ defined by
$$
p_*(d) (\xi_0) = \sum_{\xi \in X, p(\xi) = \xi_0} d(\xi).
$$
One readily verifies that for every $d: X \rightarrow {\msy N}$ we have
\begin{equation}
\label{e: pull back of pi}
\iota^* (\pi_{a,X,d}) = \pi_{a_0, X_0, p_*(d)}.
\end{equation}
Let $E$ be a complete locally convex space.
Then it follows that pull-back by $\iota$ induces a linear
map
\begin{equation}
\label{e: pull back by iota}
\iota^*: \cM(V_\C,a, X, E) \rightarrow \cM(V_{0\C},a_0, X_0, E).
\end{equation}
\begin{lemma}
\label{l: injectivity pull back by iota}
The linear map $\iota^*$ in (\ref{e: pull back by iota}) is surjective.
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
Let $d_0 : X_0 \rightarrow {\msy N}$ be a map. Then one readily checks that there
exists a map $d: X \rightarrow {\msy N}$ such that $d_0 = p_*(d).$ {}From this it
follows that
$$
\pi_{a_0,X_0, d_0}^{-1} {\cal O}_{a_0}(V_{0\C},E)
=
\iota^* (\pi_{a,X, d}^{-1})\; \iota^*p^*( {\cal O}_{a_0}(V_{0\C},E) )
\subset
\iota^*( \pi_{a,X,d}^{-1} {\cal O}_a(V_\C,E) ).
$$
where the first equality follows from (\ref{e: pull back of pi}).
~
$\square$\medbreak\noindent\medbreak
The pull-back map $\iota^*$ in (\ref{e: pull back by iota}) with $E = {\msy C}$
has a transpose $\iota_*: \cM(V_{0\C},a_0, X_0)^* \rightarrow \cM(V_\C,a, X)^* $
which is injective by Lemma \ref{l: injectivity pull back by iota}.
\begin{lemma}
\label{l: iota under star}
The map $\iota_*$ maps $\cM(V_{0\C},a_0, X_0)^*_{\rm laur}$ injectively
into $\cM(V_\C,a, X)^*_{\rm laur}.$
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
Let
$\cL \in \cM(V_{0\C},a_0, X_0)^*_{\rm laur}.$ Then it suffices to
show that $\iota_* \cL$ belongs to the space
$\cM(V_\C,a, X)^*_{\rm laur}.$
We first note that
$\iota : V_0 \rightarrow V$ has a unique extension to
an algebra homomorphism $\iota_*: S(V_0) \rightarrow S(V).$ One readily verifies
that $u [\iota^*(\varphi)] = \iota^*(\iota_*(u)\varphi)$ for every
$\varphi \in {\cal O}_a(V_\C)$
and all $u \in S(V_0).$
Let $d$ be a map $X \rightarrow {\msy N}.$ Then there exists a $u_d \in S(V_0)$
such that
$\cL = {\rm ev}_{a_0}\,{\scriptstyle\circ}\, u_d \,{\scriptstyle\circ}\, \pi_{a_0,X_0, p_*(d)}$ on
$\pi_{a_0,X_0, p_*(d)}^{-1}{\cal O}_{a_0}(V_{0\C});$
here
${\rm ev}_{a_0}$ denotes evaluation
at the point $a_0.$
Put $v_d = \iota_*(u_d).$ Then, for $\varphi \in {\cal O}_a(V_\C),$
$$
\iota_*(\cL) [\pi_{a,X,d}^{-1} \varphi]
=
\cL[\iota^*(\pi_{a,X,d})^{-1} \iota^* \varphi)]
=
\cL [\pi_{a_0,X_0,p_*d}^{-1} \iota^* \varphi]
=
\iota^*(v_d \varphi)(a_0) = v_d \varphi(a).
$$
Hence $\iota_*(\cL) = {\rm ev}_a \,{\scriptstyle\circ}\, v_d \,{\scriptstyle\circ}\,\pi_{a,X,d}$ on
$\pi_{a,X,d}^{-1} {\cal O}_a(V_\C)$ and we see that
$\iota_*(\cL)\in\cM(V_\C, a, X)^*_{\rm laur}.$
~
$\square$\medbreak\noindent\medbreak
There exists a unique linear
map $\iota_*: \cM(V_{0\C}, X_0)^*_{\rm laur} \rightarrow \cM(V_\C, X)^*_{\rm laur}$
that restricts to the map $\iota_*$ of
Lemma \ref{l: iota under star}
for every $a_0 \in V_{0\C},$ see Remark
\ref{r: Laurent functionals at a as subspace}.
Clearly,
${\rm supp}\,(\iota_* \cL) = \iota({\rm supp}\,(\cL)),$ for every $\cL \in \cM(V_{0\C}, X_0)^*_{\rm laur}.$
On the other hand, if $E$ is a complete locally convex space,
$\Omega \subset V_\C$ open subset
and $S \subset \iota^{-1}(\Omega)$ a subset, then pull-back by $\iota$
induces a natural map
$\iota^*: \cM(\Omega, \iota(S), X, E) \rightarrow \cM(\iota^{-1}(\Omega), S, X_0, E).$ Moreover,
if $\cL \in \cM(V_{0\C}, S, X_0)^*_{\rm laur}$ and $\varphi \in \cM(\Omega,
\iota(S), X, E),$ then
\begin{equation}
\label{e: iota push forward and pull back}
\iota_*(\cL)\varphi = \cL[\iota^*\varphi].
\end{equation}
We end this section with a discussion of the multiplication
by a meromorphic function and the application of a differential operator
to a Laurent functional.
First, assume that $a \in V_\biC$ and that $\psi\in \cM(a, X).$
Then multiplication by $\psi$ induces a linear endomorphism of
$\cM(a, X),$ which we denote
by $m_\psi.$ The transpose of this linear endomorphism is denoted by
$m_\psi^*: \cM(a, X)^* \rightarrow \cM(a, X)^*.$ It readily follows from the definition
of $X$-Laurent functionals at $a$ that $m_\psi^*$ leaves the space $\cM(a,X)^*_{\rm laur}$
of those functionals invariant.
Let now $S \subset V_\biC$ be a finite subset, let $\Omega \subset V_\biC$ be an open subset
containing $S$ and let $\psi \in \cM(\Omega, S, X).$ If
$\cL \in \cM(V_\biC, S, X)^*_{\rm laur},$
we define the Laurent functional $m_\psi^*(\cL)\in\cM(V_\biC,S,X)^*_{\rm laur}$
by
$$
m_\psi^*(\cL) = \sum_{a \in S} m_{\psi_a}^*(\cL_a).
$$
On the other hand, multiplication
by $\psi$ induces a linear endomorphism of $\cM(\Omega, S, X),$ and it
is immediate from the definitions that
\begin{equation}
\label{e: previous commutative diagram}
m_\psi^*(\cL)(\varphi)=\cL(\psi\varphi)
\end{equation}
for $\varphi\in \cM(\Omega,S,X)$.
\begin{lemma}
\label{l: diff of Laur}
Let $v\in S(V)$, then $v\varphi\in{\cal M}(a,X)$ for all $\varphi\in{\cal M}(a,X)$,
and the transpose $\partial^*_v$ of the endomorphism
$\partial_v\,:\,on \varphi:to v\varphi$ of ${\cal M}(a,X)$
leaves ${\cal M}(a,X)^*_{\rm laur}$ invariant.
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ } We may assume $v\in V$. Let $d\in {\msy N}^X$ and define $d'\in {\msy N}^X$
by $d'(\xi)=d(\xi)+1$ for all $\xi\in X$. Then
$\pi_{a,d}$ divides $v(\pi_{a,d'})$, and hence
$$\pi_{a,d'}v\varphi=v(\pi_{a,d'}\varphi)-v(\pi_{a,d'})\varphi\in{\cal O}_a$$
for all $\varphi\in \pi_{a,d}^{-1}{\cal O}_a$.
Thus $\partial_v\varphi=v\varphi\in\pi_{a,d'}^{-1}{\cal O}_a$ for
$\varphi\in \pi_{a,d}^{-1}{\cal O}_a$.
Let now ${\cal L}\in{\cal M}(a,X)^*_{\rm laur}$, and let $u=u_{\cal L}\inS_\leftarrow(V,X)$.
Then for $d,d'$ and $\varphi$ as above
$$\partial_v^*{\cal L}(\varphi)={\cal L}(v\varphi)=u_{d'}(\pi_{a,d'}v\varphi)(a)
=u_{d'}v(\pi_{a,d'}\varphi)(a)-u_{d'}(v(\pi_{a,d'})\varphi)(a).$$
Each term on the right hand side of this equation has the form
$u'(p\varphi)(a)$ with $u'\in S(V)$ and $p$ a polynomial which is divisible by
$\pi_{a,d}$. Hence, by the Leibniz rule, $\partial_v^*{\cal L}(f)$ has the
required form $u''(\pi_{a,d}\varphi)(a)$, where $u''\in S(V)$.~
$\square$\medbreak\noindent\medbreak
For ${\cal L}\in{\cal M}(V_{\msy C},X)^*_{\rm laur}$ and $v\in S(V)$ we now define
$\partial_v^*{\cal L}\in{\cal M}(V_{\msy C},X)^*_{\rm laur}$ by
$$\partial_v^*{\cal L}=\sum_{a\in{\rm supp}\,{\cal L}} \partial_v^*{\cal L}_a.$$ It
is immediately seen that $\partial_v^*{\cal L}(\varphi)={\cal L}(\partial_v\varphi)$
for each $\varphi\in{\cal M}(\Omega,{\rm supp}\,{\cal L},X)$, where $\Omega$ is an
arbitrary open neighborhood of ${\rm supp}\,{\cal L}$.
\section{Laurent operators}
\label{s: Laurent operators}
In this section we discuss Laurent operators, originally introduced in \bib{BSres}, Section 5,
in the slightly different context of meromorphic functions with values in a complete locally
convex space, whose singular locus is contained in an $X$-configuration.
Let $V$ and $X$ be as in the previous section,
let $\cH$ be an $X$-configuration
and let $E$ be a complete locally convex space.
We define $\cM(V_\C, \cH, E)$ to be the space of meromorphic functions
$\varphi: V_\C \rightarrow E$ whose singular locus is contained in $\cup \cH.$
If $\cH$ is real, we put $\cH_V = \{H \cap V \mid H \in \cH\}.$
Then $\cM(V_\C, \cH) = \cM(V_\C, \cH, {\msy C})$
equals the space $\cM(V, \cH_V)$ introduced in \bib{BSres}.
It is convenient to select a minimal subset $X^0$ of $X$ that is proportional to
$X.$ Then for every $X$-hyperplane $H \subset V_\C$ there exists a unique
$\alpha_H \in X^0$ and a unique first order polynomial $l_H$ of the form
$z :to \inp{\alpha_H}{z} - c,$ with $c \in {\msy C},$ such that $H = l_H^{-1}(0).$
Note that a different choice of $X^0$ causes only a change of $l_H$ by a non-zero factor.
Let ${\msy N}^\cH$ denote the collection of maps $\cH \rightarrow {\msy N}.$
\begin{rem}
\label{r: convention about d}
If $d \in {\msy N}^\cH,$ then for convenience we agree to write $d(H) = 0$ for any $X$-hyperplane
$H$ not contained in $\cH.$
\end{rem}
If $\omega \subset V_\C$ is a bounded subset and $d \in {\msy N}^\cH$
we define the polynomial function $\pi_{\omega, d}: V_\C \rightarrow {\msy C}$
by
\begin{equation}
\label{e: defi pi omega d}
\pi_{\omega, d} = {\rm pr}od_{H \in \cH\atop H \cap \omega \not= \emptyset} l_H^{d(H)}
\end{equation}
Note that a change of $X^0$ only causes this polynomial to
be multiplied by a positive
factor. Let $\cM(V_\C, \cH, d, E)$
be the collection of meromorphic functions $\varphi\in \cM(V_\C, E)$ such that
$\pi_{\omega, d} \varphi\in {\cal O}(\omega, E)$
for every bounded open subset $\omega \subset V_\C.$
We equip the space $\cM(V_\C, \cH, d, E)$ with the weakest locally convex topology
such that for every bounded open subset $\omega \subset V_\C$ the map
map $\varphi :to \pi_{\omega, d}\varphi$ is continuous into ${\cal O}(\omega, E).$
This topology is complete; moreover, it is Fr\'echet if $E$ is Fr\'echet.
We now note that
\begin{equation}
\label{e: mer hyp E as union}
\cM(V_\C, \cH , E) = \cup_{d \in {\msy N}^\cH} \cM(V_\C, \cH , d, E).
\end{equation}
We equip ${\msy N}^\cH$ with the partial ordering ${\rm pr}eceq$ defined by $d' {\rm pr}eceq d$
if and only if $d'(H) \leq d(H)$ for all $H \in \cH.$ If $d,d'$ are elements of ${\msy N}^\cH$
with $d' {\rm pr}eceq d$ then $\cM(V_\C, \cH , d', E) \subset \cM(V_\C, \cH , d, E)$
and the inclusion map $i_{d', d}$ is continuous. Thus, the inclusion maps form a directed
family and from (\ref{e: mer hyp E as union}) we see that the space $\cM(V_\C, \cH, E)$ may
be viewed as the direct limit of the spaces $\cM(V_\C, \cH , d, E).$ Accordingly
we equip $\cM(V_\C, \cH ,E)$ with the direct limit locally convex topology.
By an $X$-subspace of $V_\C$ we mean any non-empty intersection of
$X$-hyperplanes; we agree that $V_\biC$ itself is also an $X$-subspace.
We denote the set of such affine subspaces by ${\cal A} = {\cal A}(V_\C, X).$
For $L \in {\cal A}$ there exists a unique real linear subspace $V_L \subset V$ such that
$L = a + V_{L\C}$ for some $a \in V_\C.$ The intersection
$V_{L\C}^\perp \cap L$ consists of a single point, called the central point of $L;$
it is denoted by $c(L).$ The space $L$ is said to be real if $c(L) \in V;$
this means precisely that $L$ is the complexification of an affine subspace of
$V.$ Translation by $c(L)$
induces an affine isomorphism from
$V_{L\C}$ onto $L.$ Via this isomorphism we equip $L$ with the structure of a complex
linear space together with a real form that is equipped with an inner product.
If $L \in {\cal A},$ the collection of $X$-hyperplanes containing $L$ is finite; we denote
this collection by $\cH(L, X).$ Moreover, we put $X(L):= X \cap V_L^\perp$ and
$X^0(L):= X^0 \cap V_L^\perp.$ {}From the definition of $X^0$ it
follows that the map $H :to \alpha_H$ is a bijection from $\cH(L, X)$ onto
$X^0(L).$ Accordingly we shall identify the sets ${\msy N}^{\cHbLX}$ and ${\msy N}^{X^0(L)}.$
If $\cH$ is any $X$-configuration and $d \in {\msy N}^{\cH},$
we define the polynomial function
$q_{L,d}$ by
$$
q_{L,d}: = {\rm pr}od_{H \in \cH(L,X)} l_H^{d(H)},
$$
see also Remark \ref{r: convention about d}.
Let $X_r$ be the orthogonal projection of $X\setminus X(L)$ onto $V_L;$ then
$X_r$ is a finite set of non-zero elements. Its image in $L$ under translation
by $c(L)$ is denoted by $X_L.$
If $\cH$ is an $X$-configuration in $V_\C,$ then
the collection
$$
\cH_L := \{H\cap L\mid H \in \cH,\; \emptyset \subsetneqq H\cap L \subsetneqq L\}
$$
is an $X_L$-configuration in $L;$ here $L$ is viewed as a complex linear space
in the way described above.
We now assume that $L \in {\cal A}$ and
that $\cH$ is an $X$-configuration in $V_\C.$ In accordance with
\bib{BSres}, Sect. 1.3,
a linear map $R: \cM(V_\C, \cH) \rightarrow \cM(L, \cH_L)$ is called a
Laurent operator
if for every $d \in \cH^{\msy N}$ there exists an element $u_d \in S(V_L^\perp)$
such that
\begin{equation}
\label{e: defi Laurent operator by u d}
R\varphi = u_d(q_{L,d} \varphi)|_L \text{for all} \varphi \in \cM(V_\C, \cH, d).
\end{equation}
The space of such Laurent operators is denoted by $\cLr(V_\C, L, \cH).$
Assume now in addition that $\cH$ contains $\cH(L,X).$
Then as in loc.\ cit.\ it is seen that, for $R \in \cLr(V_\C, L, \cH)$ and
$d \in {\msy N}^\cH$,
the element $u_d \in S(V_L^\perp)$ such that
(\ref{e: defi Laurent operator by u d}) holds, is uniquely determined.
Moreover, it only depends on the restriction of $d$ to $\cH(L,X),$ and
the associated string $u_R:= (u_d\mid d \in \N^{\Hyp(L,X)})$ belongs to
$S_\leftarrow(V_L^\perp, X^0(L)).$
As in to \bib{BSres}, Lemma 1.5,
the map $R :to u_R$ defines a linear isomorphism
\begin{equation}
\label{e: iso Laurent operators with Sproj}
\cLr(V_\C, L, \cH) \simeq S_\leftarrow(V_L^\perp, X^0(L)).
\end{equation}
If $E$ is a complete locally convex space,
and $R \in \cLr(V_\C, L, \cH)$ a Laurent operator,
we may define a linear operator $R_E$ from $\cM(V_\C, \cH, E)$
to $\cM(L, \cH_L, E)$ by the formula (\ref{e: defi Laurent operator by u d}), for
$\varphi \in \cM(V_\C, \cH, d, E)$ and with
$u_d$ equal to the $d$-component of $u_R.$ We shall often denote $R_E$ by $R$ as well.
\begin{rem} Here we note that the algebraic tensor product
$\cM(V_\C, \cH) \otimes E$ naturally embeds onto a subspace of
$\cM(V_\C, \cH, E)$ which is dense. Thus, $R_E$ is the unique continuous
linear extension of $R \otimes I_E.$ However, we shall not need this.
\end{rem}
\begin{lemma}
\label{l: continuity Laurent operator}
Let $L \in {\cal A}$ and let $\cH$ be an $X$-configuration in
$V_\C$ containing $\cH(L,X).$
Let $R \in \cLr(V_\C, L, \cH).$
Then for every $d \in {\msy N}^\cH$ there exists a $d' \in {\msy N}^{\cH_L}$
with the following property. For every complete locally convex space $E$
the operator $R_E$ maps $\cM(V_\C, \cH, d, E)$ continuously into
the space $\cM(L, \cH_L, d', E).$
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
This is proved in a similar fashion as \bib{BSres}, Lemma 1.10.
~
$\square$\medbreak\noindent\medbreak
We shall now relate Laurent operators to the Laurent functionals
introduced in the previous section.
Let $X_r^0$ be a minimal subset of $X_r$
subject to the condition that it be proportional to $X_r.$
Let $X_L^0$ be its image in $L$ under translation by $c(L).$
Thus, with respect to the linear
structure of $L,$ the set $X_L^0$ is an analogue for the pair
$(L, X_L)$
of the set $X^0$ for the pair
$(V,X).$
\begin{lemma}
\label{l: iso Laurent functionals and operators}
Let $L\in \cA$ and let $\cH$ be an $X$-configuration in $V_\C$
containing
$\cH(L,X).$
Let $E$ be a complete locally convex space.
\begin{enumerate}
\item[{\rm (a)}]
If $\varphi \in \cM(V_\C, \cH, E),$ then for
$w \in L\setminus \cup \cH_L$ the function $z :to \varphi(w + z)$
is meromorphic on $V^\perp_{L\biC},$ with a germ at $0$ that belongs to
$\cM(V^\perp_{L\biC},0, X(L),E).$
\minspace\item[{\rm (b)}]
If $\cL \in \cM(V^\perp_{L\biC}, 0, X(L))^*_{\rm laur} $ is an $X(L)$-Laurent functional in
$V^\perp_{L\biC},$ supported at the origin,
then for $\varphi \in \cM(V_\C, \cH,E)$ the function
\begin{equation}
\label{e: defi Lau star}
\cL_*\varphi : w :to \cL(\varphi(w + \,\cdot\, ))
\end{equation}
belongs to the space $\cM(L, \cH_L,E).$
The operator $\cL_*: \cM(V_\C, \cH) \rightarrow \cM(L, \cH_L),$
defined by (\ref{e: defi Lau star})
for $E ={\msy C},$
is a Laurent operator.
\minspace\item[{\rm (c)}]
The map $\cL :to \cL_*,$ defined by (\ref{e: defi Lau star})
for $E = {\msy C},$ is an isomorphism from the space
$\cM(V^\perp_{L\biC}, 0, X(L))^*_{\rm laur}$ onto the space $\cLr(V_\C, L,\cH).$
This isomorphism corresponds with the identity
on $S_\leftarrow(V^\perp_L,X^0(L)),$ via the isomorphisms
of Lemma \ref{l: iso Laurent functionals with Sproj} and
eq.\ (\ref{e: iso Laurent operators with Sproj}).
\end{enumerate}
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
See \bib{BSfi}, Appendix B, Lemma B.3.
~
$\square$\medbreak\noindent\medbreak
\begin{rem}
In the formulation of (c) we have used that the spaces
$\cM(V^\perp_{L\biC}, 0, X(L))^*_{\rm laur}$ and $\cM(V^\perp_{L\biC}, 0, X^0(L))^*_{\rm laur}$
are equal, see Lemma \ref{l: Laurent functionals and proportional X}.
\end{rem}
We now assume that $\cH$ is an $X$-configuration, and that $L \in {\cal A}.$
If $a \in V^\perp_{L\biC},$ then by $\cH_L(a)$ we denote the collection
of hyperplanes $H'$ in $L$ for which there exists a $H \in \cH$
such that $H' = L \cap [(-a) + H].$ Thus, $\cH_L(a) = (T_{-a}\cH)_L$ and we see that
$\cH_L(a)$ is an $X_L$-configuration.
If $S \subset V^\perp_{L\biC}$ is a finite subset,
then
\begin{equation}
\label{e: defi Hyp L S}
\cH_L(S) = \cup_{a \in S} \cH_L(a)
\end{equation}
is an $X_L$-configuration in $L$ as well.
The corresponding set of regular points in $L$
equals
$$
L\setminus \cup \cH_L(S) =
\{w \in L\mid \forall a \in S \,\forall H \in \cH: \;\;
a + w \in H \Rightarrow a + L \subset H \}.
$$
\begin{cor}
\label{c: continuity of Laustar}
Let $L \in {\cal A}$ and let $\cH$ be an $X$-configuration.
Let $S \subset V^\perp_{L\biC}$ be a finite subset and let $E$ be a complete locally
convex space.
\begin{enumerate}
\item[{\rm (a)}]
For every $\varphi \in \cM(V_\C, \cH,E)$ and each $w \in L\setminus\cup \cH_L(S),$
there exists an open neighborhood $\Omega$ of $S$ in $V^\perp_{L\biC}$ such that the
function
$\varphi(w + \,\cdot\,): z :to \varphi(w + z)$ belongs to $\cM(\Omega, X(L),E).$
\minspace\item[{\rm (b)}]
Let $\cL \in \cM(V^\perp_{L\biC}, X(L))^*_{\rm laur}$ be a Laurent functional supported at $S.$
For every $\varphi \in \cM(V_\C, \cH,E)$ the function
$\cL_*\varphi: L\setminus\cup \cH_L(S) \rightarrow E$
defined by
\begin{equation}
\label{e: defi Laustar}
\cL_*\varphi(w) := \cL(\varphi(w + \,\cdot\,))
\end{equation}
belongs to $\cM(L, \cH_L(S),E).$
Finally, $\cL_*$ is a continuous linear map from $\cM(V_\C, \cH,E)$ to
$\cM(L, \cH_L(S), E).$ In fact, for every $d \in {\msy N}^\cH$ there exists
a $d' \in {\msy N}^{\cH_L(S)},$ independent of $E,$ such that
$\cLstar$ maps $\cM(V_\C, \cH,d,E)$ continuously into $\cM(L, \cH_L(S), d', E).$
\end{enumerate}
\end{cor}
\par\noindent{\bf Proof:}{\ }{\ }
It
suffices to prove the result for $S$ consisting of a single point $a.$
Applying a translation by $-a$ if necessary, we may as well assume that $a = 0.$
Then $\cH_L(S) = \cH_L(0) = \cH_L.$
Let $\cH'$ be the union of $\cH$ with $\cH(L, X).$
Then $\cM(V_\C, \cH,E )\subset \cM(V_\C, \cH', E)$
and $(\cH')_L=\cH_L=\cH_L(S)$, hence assertions (a) and (b) of
Lemma \ref{l: iso Laurent functionals and operators}
with $\cH'$ in place of $\cH$ imply assertion (a) and (b),
except for the final statement about the continuity.
For the final statement of (b), we note that by
Lemma \ref{l: iso Laurent functionals and operators}(b),
$\cL_*$ is a Laurent operator
$ \cM(V_\C, \cH') \rightarrow \cM(L, \cH_L(S)).$
Let $d: \cH \rightarrow {\msy N}$ be a map. We extend $d$ to $\cH'$
by triviality on $\cH'\setminus \cH.$ Then according to Lemma
\ref{l: continuity Laurent operator} there exists a map $d': \cH_L(S) \rightarrow {\msy N}$ such that
for any complete locally convex space $E$ the map
$$
\cL_*: \cM(V_\C, \cH', d, E) \rightarrow \cM(L, \cH_L(S), d', E)
$$
is continuous linear. Since $d$ is zero on $\cH'\setminus \cH,$
the first of these spaces equals $\cM(V_\C, \cH, d, E)$ and the asserted
continuity follows.
~
$\square$\medbreak\noindent\medbreak
\begin{lemma}
\label{l: eval of Laustar is a Lau}
Let $L$, ${\cal H}$, $S$ and ${\cal L}$ be as in Cor.\ \ref{c: continuity of Laustar},
and fix $w\in L\setminus\cup{\cal H}_L(S)$. There exists a Laurent
functional (in general not unique) ${\cal L}'\in{\cal M}(V_{\msy C},X)^*_{\rm laur}$,
supported in $w+S$, such that ${\cal L}'\varphi={\cal L}(\varphi(w+\,\cdot\,))$
for all $\varphi\in{\cal M}(V_{\msy C},{\cal H})$.\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ } As in the proof of Cor.\ \ref{c: continuity of Laustar}
we may assume that $S=\{0\}$. Let $\tilde{\cal H}={\cal H}\cup{\cal H}(w,X)$.
Then ${\cal L}_*\,:\,on\varphi:to{\cal L}(\varphi(w+\,\cdot\,))$ is a Laurent operator
in $\cLr(V_{\msy C},L,\tilde{\cal H})$, according to Lemma
\ref{l: iso Laurent functionals and operators} (b).
On the other hand, it follows from
Lemma \ref{l: extension to Laurent functional} (see Remark
\ref{r: evaluation is a Laurent functional}) that there exists a
(in general not unique) $X_L$-Laurent
functional ${\cal L}''$ on $L$ such that $\psi(w)={\cal L}''(\psi_w)$
for each $\psi\in{\cal O}_w(L)$.
The functional $\psi:to{\cal L}''(\psi_w)$ is defined
for $\psi\in\cM(L,\tilde{\cal H}_L)$, and it may be viewed as a
Laurent operator in $\cLr(L,\{w\},\tilde{\cal H}_L)$, which we
denote by the same symbol ${\cal L}''$ (see \bib{BSfi} Appendix, Remark B.4).
It now follows from \bib{BSres}, Lemma 1.8 that the composed map
${\cal L}''\,{\scriptstyle\circ}\,{\cal L}_*$ belongs to $\cLr(V_{\msy C},\{w\},\tilde{\cal H})$
and hence by \bib{BSfi} Appendix, Remark B.4
it is given by an $X$-Laurent functional ${\cal L}'$, supported at $w$.
In particular, for $\varphi\in{\cal M}(V_{\msy C},{\cal H})$ we have from Lemma
\ref{l: iso Laurent functionals and operators} (b) that
$w:to{\cal L}(\varphi(w+\,\cdot\,))$ is holomorphic in a neighborhood of $w$,
hence its evaluation at $w$ is obtained from the application of ${\cal L}''$ to it.
Thus ${\cal L}(\varphi(w+\,\cdot\,))={\cal L}'\varphi$
for $\varphi\in{\cal M}(V_{\msy C},{\cal H})$.~
$\square$\medbreak\noindent\medbreak
Recall from Section \ref{s: Laurent functionals} that
$\cM(V_\C, X, E)$ is the union of the spaces
$\cM(V_\C, \cH, E)$ with $\cH$ an $X$-configuration.
\begin{lemma}
\label{l: Lau under star new}
Let
$L \in {\cal A}$ and let $\cL \in \cM(V^\perp_{L\biC}, X(L))^*_{\rm laur}$ be a Laurent
functional. Then for any complete locally convex space $E$
there exists a unique linear operator
$$
\cL_*: \cM(V_\C, X, E) \rightarrow \cM(L, X_L, E)
$$
that coincides on the subspace $\cM(V_\C, \cH,E)$
with the operator $\cL_*$ defined in
Corollary \ref{c: continuity of Laustar}, for every $X$-configuration $\cH$ in $V_\C.$
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
Let $\cH_1$ and $\cH_2$ be two $X$-configurations. Let $S = {\rm supp}\,(\cL)$
and let, for $j=1,2,$
the continuous linear operator
$\cL^j_*: \cM(V_\C, \cH_j,E) \rightarrow \cM(L, \cH_{jL}(S),E)$ be
defined as in Corollary \ref{c: continuity of Laustar}
with $\cH_j$ in place of $\cH.$
Then it suffices to show that $\cL_*^1$ and $\cL_*^2$ coincide on the
intersection of $\cM(V_\C, \cH_1,E)$ and $\cM(V_\C, \cH_2,E).$
That intersection equals $\cM(V_\C, \cH_1 \cap \cH_2,E ).$
Let $\varphi$ be a function in the latter space,
then from the defining formula (\ref{e: defi Laustar})
it follows that $\cL_{*}^1 \varphi = \cL_*^2\varphi$
on the intersection of the sets $L\setminus \cup \cH_{jL}(S),$ for $j=1,2.$
This implies
that $\cL_{*}^1 \varphi$ and $\cL_*^2\varphi$ coincide as elements of
$\cM(L).$
~
$\square$\medbreak\noindent\medbreak
We end this section with another useful consequence.
\begin{lemma}
\label{l: diagonal action of Laurent functional}
Let $\cL \in \cM(V^\perp_{L\biC}, X(L))^*_{\rm laur}.$ Let the finite subset $\widetilde X$
of $V\times V\setminus \{(0,0)\}$ be defined by
$\widetilde X = (X \times \{0\}) \cup
(\{0\} \times X).$
If $\Phi \in \cM(V_\C \times V_\C, \widetilde X),$ then
$$
{P,s}i: (w_1, w_2) :to \cL(\Phi(\,\cdot\, + w_1, \,\cdot\, + w_2))
$$
defines a function in $\cM(L\times L, \widetilde X_L),$
where
$\widetilde X_L = (X_L \times\{c(L)\}) \cup (\{c(L)\} \times X_L).$
In particular, the pull-back of ${P,s}i$ under the diagonal embedding
$j: L \rightarrow L \times L$ belongs to the space $\cM(L, X_L).$
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
Equip $V^\perp_L \times V^\perp_L$ with one half times the direct sum
inner product.
Then the diagonal embedding $\iota: z :to (z,z)$ is an isometry of
$V^\perp_L$ into $V^\perp_L \times V^\perp_L.$ Its adjoint is the map
$p: (z_1, z_2) :to \frac12 (z_1 + z_2)$ from $V^\perp_L \times V^\perp_L$ onto $V^\perp_L.$
The intersection $\widetilde X(L) := \widetilde X \cap (V^\perp_L\times V^\perp_L)$
equals $(X(L) \times \{0\} )\cup (\{0\} \times X(L)).$ Its image under $p$ is given by
$\widetilde X(L)_0 = \frac12 X(L).$ Thus, according to Lemma
\ref{l: global Laurent functionals and proportional X},
the space of
$\widetilde X(L)_0$-Laurent functionals on $V^\perp_{L\biC}$ is equal to the space of
$X(L)$-Laurent functionals on $V^\perp_{L\biC}.$ Hence,
according to Lemma \ref{l: iota under star} and the remark following
its proof, we have an associated
push-forward map $\iota_*$ from $\cM(V^\perp_{L\biC}, X(L))^*_{\rm laur}$ to
$\cM(V^\perp_{L\biC} \times V^\perp_{L\biC}, \widetilde X(L))^*_{\rm laur}.$
For generic $w_1, w_2 \in L$ we define the meromorphic function $\Phi^{(w_1, w_2)}$
on $V^\perp_{L\biC} \times V^\perp_{L\biC}$ by $\Phi^{(w_1, w_2)}(z_1, z_2) = \Phi(w_1 + z_1 , w_2 + z_2).$
The definition of ${P,s}i$ may now be rewritten as
${P,s}i(w_1, w_2) = \cL [\iota^*(\Phi^{(w_1, w_2)})].$
By (\ref{e: iota push forward and pull back})
it follows that
${P,s}i(w_1, w_2) = \iota_*(\cL)(\Phi^{(w_1, w_2)}),$ or, equivalently,
in the notation
of Lemma \ref{l: Lau under star new},
$$
{P,s}i = [\iota_*(\cL)]_*\Phi.
$$
We now observe that $\widetilde X_L = (\widetilde X)_{L \times L}.$
Hence it follows by application of
Lemma \ref{l: Lau under star new}.
that ${P,s}i\in\cM(L\times L,\tilde X_L)$.
There exists an $\tilde X_L$-configuration $\tilde\cH$ in
$L\times L$ such that ${P,s}i\in\cM(L\times L,\tilde\cH)$.
Any hyperplane $\tilde H\in\tilde\cH$ is of the form
$\tilde H=H\times L$ or $\tilde H=L\times H$, with $H$ an $X_L$-hyperplane
in $L$. In both cases $j^{-1}(\tilde H)=H$. It now follows that
$j^{-1}(\tilde\cH)$ is an $X_L$-configuration in $L$,
and that $j^*{P,s}i\in{\cal M}(L,X_L)$.
~
$\square$\medbreak\noindent\medbreak
\section{Analytic families of a special type}
\label{s: spec fam new}
In
this section we introduce a space ${\cal E}hypQgd$ of analytic families
of ${\msy D}X$-finite $\tau$-spherical functions whose singular locus is
a $\Sigma$-configuration. The definition of this space is motivated
by the fact that
it contains the families obtained from applying
Laurent functionals to Eisenstein integrals related
to a minimal $\sigma$-parabolic subgroup, as we shall see in
the following sections, and by the fact that
the vanishing theorem is applicable, see
Theorem \ref{t: special vanishing theorem}.
In this section we fix a choice $\Sigma^+$ of positive roots
for $\Sigma$ and denote by $P_0$ the associated minimal
standard $\sigma$-parabolic subgroup.
\begin{defi}
\label{d: Cephyp}
Let $Q \in \cP_\gs$ and let $Y \subset {}^*\fa_{Q\iq}dc$ be a finite subset.
We define
\begin{equation}
\label{e: CephypQY}
{\msy C}ephypQY
\end{equation}
to be the space of functions
$f:{\mathfrak a}Qqdc \times {\rm X}p \rightarrow V_\tau,$ meromorphic
in the first variable, for which there exist a constant $k \in {\msy N},$
a $\Sigma_r(Q)$-hyperplane configuration $\cH$ in ${\mathfrak a}Qqdc$ and a function
$d: {\cal H} \rightarrow {\msy N}$ such that
the following conditions are fulfilled.
\begin{enumerate}
\item[{\rm (a)}]
The function $\lambda :to f_\lambda$ belongs to
$\cM({\mathfrak a}Qqdc, \cH, d, {\msy C}i({\rm X}p\,:\, \tau)).$
\minspace\item[{\rm (b)}]
For every $P \in \cP_\gs^{\rm min}$ and $v \in {\msy N}Kaq$ there exist functions
$
q_{s,\xi}(P,v\,|\, f)$ in $ P_k({\mathfrak a}q) \otimes
\cM({\mathfrak a}Qqdc, \cH, d, {\msy C}i({\rm X}zerov\,:\, \tau_{\iM})),$
for $s \in W/W_Q$ and $\xi \in -sW_Q Y + {\msy N} {\msy D}elta(P),$ with the
following property.
For all $\lambda \in {\mathfrak a}Qqdc\setminus \cup \cH,$ $m \in {\rm X}zerov$ and $a \in A_\iq^+(P),$
\begin{equation}
\label{e: expansion Cephyp}
f_\lambda(mav) = \sum_{s \in W/W_Q} a^{s\lambda - \rho_P} \sum_{\xi \in -sW_Q Y + {\msy N} \Delta(P)}
a^{-\xi}\, q_{s,\xi}(P,v\,|\, f, \log a)( \lambda, m),
\end{equation}
where the ${\msy D}P$-exponential polynomial series of each inner sum
converges neatly on $A_\iq^+(P).$
\minspace\item[{\rm (c)}]
For every $P \in \cP_\gs^{\rm min},$ $v \in {\msy N}Kaq$ and $s\in W/W_Q,$ the series
$$
\sum_{\xi \in -sW_Q Y + {\msy N}{\msy D}P} a^{-\xi} q_{s, \xi}(P,v\,|\, f,\log a)
$$
converges neatly on $A_\iq^+(P),$ as an exponential polynomial series
with coefficients in the space
$\cM({\mathfrak a}Qqdc, \cH, d, {\msy C}i({\rm X}zerov \,:\, \tau_{\iM})).$
\end{enumerate}
Finally, we define
\begin{equation}
\label{e: Cephyp}
{\msy C}ephyp: = C^{{\rm ep},{\rm hyp}}_{P_0, \{0\}}({\rm X}p\,:\, \tau).
\end{equation}
\end{defi}
\begin{rem}
\label{r: relation between hypfam and fam}
Note the analogy between the above definition and Definition
\ref{d: anfamQY newer}. In fact, let $\Omega = {\mathfrak a}Qqdc\setminus \cup \cH,$
then it follows immediately from the definitions
that the restriction of $f$ to $\Omega \times {\rm X}p$
belongs to $C^{\rm ep}_{Q, Y}({\rm X}p\,:\, \tau \,:\, \Omega).$
Moreover, it follows from Lemma \ref{l: pointwise expansion of family}
that the functions $q_{s,\mu}(P,v\,|\, f)$ introduced
above are unique, and that the notation used here is consistent
with the notation in Definition
\ref{d: anfamQY newer}.
The precise relation between the definitions is given in
Lemma \ref{l: relation specfam and fam second} below.
\end{rem}
\begin{rem}
\label{r: second on Cephyp}
In analogy with Remark \ref{r: on defi anfamQY new} we note that
the space (\ref{e: CephypQY}) depends on $Q$ through its $\sigma$-split component
$A_{Q\iq}.$
Moreover, it suffices in the above definition to require conditions
(b) and (c) for
a fixed $P \in \cP_\gs^{\rm min}$ and all $v$ in a given set ${\cal W} \subset {\msy N}Kaq$
of representatives
for $W/W_{K \cap H}.$ Alternatively, it suffices to require those conditions
for a fixed given $v\in {\msy N}Kaq$ and each $P \in \cP_\gs^{\rm min}.$
Finally, we note that ${\mathfrak a}_{P_0 {\rm q}} = {\mathfrak a}q,$ hence ${}^*{\mathfrak a}_{P_0} = \{0\}.$
Thus, if $Q = P_0,$ we only need to consider the finite set
$Y = \{0\}.$ This explains the limitation in (\ref{e: Cephyp}).
\end{rem}
It follows from Remark \ref{r: relation between hypfam and fam}
that the following definition of the notion of
asymptotic degree is in accordance
with the definition of the similar notion in Definition \ref{d: anfamQY newer}.
\begin{defi}
\label{d: asymptotic degree Cephyp}
Let $f \in {\msy C}ephypQY.$
We define the asymptotic degree of $f,$ denoted ${\rm deg}_{\rm a}(f),$ to be
the smallest integer $k$ for which there
exist $\cH, d$ such that the conditions of Definition \ref{d: Cephyp}
are fulfilled.
Moreover, we denote by $\cH_f$ the smallest $\Sigma_r(Q)$-configuration
in ${\mathfrak a}Qqdc$
such that the conditions of
Definition \ref{d: Cephyp} are fulfilled with $k = {\rm deg}_{\rm a}(f)$ and for
some $d: \cH_f \rightarrow {\msy N}.$ These choices being fixed,
we denote by $d_f$ the ${\rm pr}eceq$-minimal
map $\cH_f \rightarrow {\msy N}$ for which the conditions of the definition are fulfilled.
Finally, we put ${\rm reg}a (f) := {\mathfrak a}Qqdc\setminus \cup\cH_f.$
\end{defi}
If $Q \in \cP_\gs,$ we denote
by $\Sigma_{r0}(Q)$ the set of indivisible roots in $\Sigma_r(Q),$ i.e., the roots
$\alpha \in \Sigma_r(Q)$ with $]0, 1]\alpha \cap \Sigma_r(Q) = \{\alpha \}.$
Moreover, we put $\Sigma^+_0 = \Sigma_{r0}(P_0).$
Let $\cH$ be a $\Sigma_r(Q)$-configuration in ${\mathfrak a}Qqdc$ and $d: \cH \rightarrow {\msy N}$ a map.
If $\omega \subset {\mathfrak a}Qqdc$
is a bounded subset, we define $\pi_{\omega,d}$
as in (\ref{e: defi pi omega d}) with $V = {\mathfrak a}Qqd,$ $X = \Sigma_r(Q)$ and
$X^0 = \Sigma_{r0}(Q).$
\begin{lemma}
\label{l: relation specfam and fam second}
Let $Q \in \cP_\gs,$ $Y \subset {}^*\fa_{Q\iq}dc$ a finite subset,
$\cH$ a $\Sigma_r(Q)$-configuration
in ${\mathfrak a}Qqdc$ and $d \in {\msy N}^\cH.$
Assume that $f \in \cM({\mathfrak a}Qqdc, {\msy C}i({\rm X}p\,:\, \tau)).$
Then the following two conditions
are equivalent.
\begin{enumerate}
\item[{\rm (a)}]
The function $f$ belongs to ${\msy C}ephypQY$
and satisfies $\cH_f \subset \cH$ and $d_f {\rm pr}eceq d.$
\minspace\item[{\rm (b)}]
For every non-empty bounded open subset $\omega \subset {\mathfrak a}Qqdc,$
the function $ f_{\pi_{\omega, d}}: (\lambda, x) :to \pi_{\omega, d}(\lambda) f(\lambda, x),$
$\omega \times {\rm X}p \rightarrow V_\tau$
belongs to $C_{Q,Y}^{\rm ep}({\rm X}p\,:\, \tau \,:\, \omega).$
\end{enumerate}
Moreover, if one of the above equivalent conditions is fulfilled,
then for every non-empty bounded open subset $\omega \subset {\mathfrak a}qdc$ and
all $P \in \cP_\gs^{\rm min},$ $v \in {\msy N}Kaq,$ $s \in W/W_Q$ and $\xi \in -sW_Q Y + {\msy N}{\msy D}P,$
\begin{equation}
\label{e: relation q of fpi and q}
q_{s, \xi}(P,v\,|\, f_{\pi_{\omega,d}}) = \pi_{\omega, d}\, q_{s, \xi} (P,v\,|\, f),
\end{equation}
where on the right-hand side we have identified $\pi_{\omega, d}$ with the
function $1 \otimes \pi_{\omega, d} \otimes 1$ in $P({\mathfrak a}q) \otimes {\cal O}(\omega) \otimes
{\msy C}i({\rm X}zerov \,:\, \tau).$
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
Assume that (a) holds and that $\omega \subset {\mathfrak a}Qqdc$ is a non-empty bounded
open subset. Put $\pi = \pi_{\omega,d}$ and $f_\pi=f_{\pi_{\omega,d}}.$
It follows from Definition \ref{d: Cephyp} (a) that
$f_\pi: \omega \times {\rm X}p \rightarrow V_\tau$
is smooth and that $f_{\pi\lambda}$ is $\tau$-spherical for every $\lambda \in \omega.$
Thus, it remains to verify conditions (b) and (c) of Definition
\ref{d: anfamQY newer} for $f_\pi.$
Let $P \in \cP_\gs^{\rm min}$ and $v \in {\msy N}Kaq.$ For $s \in W/W_Q$ and
$\xi \in -sW_QY + {\msy N} {\msy D}P$ we define
$$
q_{s,\xi}'(P, v \,|\, f_\pi, X, \lambda, m) := \pi(\lambda) q_{s, \xi} (P, v\,|\, f, X, \lambda, m).
$$
Then conditions (b) and (c) of Definition \ref{d: anfamQY newer},
with $k = {\rm deg}_{\rm a} f$
and with $q_{s,\xi}'$ in place of
$q_{s, \xi},$
follow from the similar conditions of Definition \ref{d: Cephyp}.
Thus, it follows that
$f_\pi \in C^{\rm ep}_{Q,Y}({\rm X}p \,:\, \tau \,:\, \omega)$
and that (\ref{e: relation q of fpi and q}) holds for
all $P \in \cP_\gs^{\rm min},$ $v \in {\msy N}Kaq,$ $s \in W$ and $\xi \in -sW_QY +{\msy N} {\msy D}P.$
Now assume that (b) holds, then it suffices to show that (a) holds.
Let $\omega$ be a bounded non-empty
open subset of ${\mathfrak a}Qqdc.$ Then it follows
from Definition \ref{d: anfamQY newer} that the function
$f_\pi=f_{\pi_{\omega,d}}: \omega \times {\rm X}p \rightarrow V_\tau$ is smooth; moreover,
from condition (a) of the mentioned definition it follows that $f_{\pi,\lambda}$
is $\tau$-spherical for every $\lambda \in \omega.$
Hence the map $\lambda :to f_\pi$ belongs to
${\cal O}(\omega, {\msy C}i({\rm X}p\,:\, \tau)).$
Since $\omega$ was arbitrary, this implies that $\lambda :to
f_\lambda$ belongs to $\cM({\mathfrak a}Qqdc, \cH, d, {\msy C}i({\rm X}p\,:\, \tau)).$
Hence $f$ satisfies condition (a) of Definition
\ref{d: Cephyp}.
Let now $P \in \cP_\gs^{\rm min}$ and $v \in {\msy N}Kaq.$ Then
it remains to establish conditions (b) and (c) of
that definition.
If $\omega$ is a non-empty bounded open subset
of ${\mathfrak a}Qqdc,$ then obviously the restriction to $\omega\setminus \cup \cH $
of the function $f_\pi$ belongs
to $C^{\rm ep}_{Q,Y}({\rm X}p\,:\, \tau\,:\, \omega\setminus \cup\cH).$ Moreover,
since $\pi_{\omega, d}$ is nowhere zero on $\omega \setminus \cup \cH,$ it
follows from division by $\pi_{\omega, d}$
that the restriction $f|_{(\omega\setminus \cup \cH) \times {\rm X}p}$ belongs
to $C^{\rm ep}_{Q,Y}({\rm X}p\,:\, \tau\,:\, \omega\setminus \cup\cH).$
Hence, in view of
Lemma \ref{l: CepQY is sheaf}, the function
$f$ belongs to
$C^{\rm ep}_{Q,Y}({\rm X}p\,:\, \tau\,:\, \Omega),$
where $\Omega:= {\mathfrak a}Qqdc \setminus \cup \cH.$ Let $k = {\rm deg}_{\rm a} f.$
It follows from the division by $\pi_{\omega,d}$, that
for every $s \in W$ and $\xi \in -sW_QY +{\msy N} {\msy D}P,$
$$
\pi_{\omega,d}(\lambda) q_{s, \xi}(P,v\,|\, f, \,\cdot\,, \lambda)
=
q_{s, \xi}(P, v\,|\, f_\pi, \,\cdot\,, \lambda),
\quad\quad (\lambda \in \omega\setminus \cup \cH).
$$
In particular, the function $(X,\lambda) :to
\pi_{\omega,d}(\lambda) q_{s, \xi}(P,v\,|\, f, X, \lambda)$ belongs
to the space $P_k({\mathfrak a}q) \otimes {\cal O}(\omega, {\msy C}i({\rm X}zerov \,:\, \tau_{\iM})).$ Since
$\omega $ is arbitrary, this implies that $f$ satisfies
condition (b) of Definition \ref{d: Cephyp}.
{}From condition (c) of Definition \ref{d: anfamQY newer} with
$f_\pi$
and $\omega$
in place of $f$ and $\Omega,$ respectively, it follows that, for $s \in W,$
the series
$$
\sum_{\xi\in -sW_QY +{\msy N}{\msy D}P} a^{-\xi}
\pi_{\omega, d}(\lambda)\,q_{s, \xi}(P, v\,|\, f ,\log a , \lambda)
$$
converges neatly on $A_\iq^+(P)$ as a ${\msy D}P$-exponential polynomial series
with coefficients in ${\cal O}(\omega, {\msy C}i({\rm X}zerov \,:\, \tau)).$
Since $\omega$ was arbitrary, it follows from the definition of the
topology on $\cM({\mathfrak a}Qqdc, \cH, d, {\msy C}i({\rm X}zerov \,:\, \tau_{\iM}))$
(see Section \ref{s: Laurent operators})
that $f$ satisfies condition (c) of Definition \ref{d: Cephyp}.
~
$\square$\medbreak\noindent\medbreak
\begin{lemma}
Let $f\in {\msy C}ephypQY$ and $D \in {\msy D}X.$ Then $Df \in {\msy C}ephypQY.$
Moreover, $\cH_{Df} \subset \cH_f,$ $d_{Df}{\rm pr}eceq d_f$
and ${\rm deg}_{\rm a} Df \leq {\rm deg}_{\rm a} f.$
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
This follows from a straightforward combination of Lemma
\ref{l: relation specfam and fam second}
with Proposition \ref{p: D on families}.
~
$\square$\medbreak\noindent\medbreak
If $f\in {\msy C}ephypQY,$ then by Remark \ref{r: relation between hypfam and fam}
the function
$f$ belongs to $C^{\rm ep}_{Q,Y}({\rm X}p\,:\, \tau\,:\, \Omega),$
with $\Omega = {\rm reg}a f.$
Let $k = {\rm deg}_{\rm a} f.$
For $P \in \cP_\gs,$ $v \in {\msy N}Kaq,$ $\sigma \in W/\!\sim_{P|Q}$ and
$\xi \in -\sigma \cdot Y + {\msy N}{\msy D}rP,$ let
$q_{\sigma, \xi}(P, v \,|\, f) \in P_k({\mathfrak a}Pq) \otimes {\cal O}(\Omega, {\msy C}i({\rm X}Pvp\,:\, \tau_P))$
be the function defined in Theorem \ref{t: behavior along the walls for families new}.
\begin{lemma}
\label{l: asymptotics along walls for CephypQY}
Let $Q\in \cP_\gs$ and $Y \subset {}^*\fa_{Q\iq}dc$ a finite subset.
Assume that
$f \in {\msy C}ephypQY$ and put $k = {\rm deg}_{\rm a} f.$
Let $P \in \cP_\gs$ and $v \in {\msy N}Kaq.$
Then, for every
$\lambda \in {\rm reg}a f,$
the set ${\rm Exp}(P, v\,|\, f_\lambda)$ is contained in $W(\lambda+ Y)|_{{\mathfrak a}Pq} - \rho_P - {\msy N}{\msy D}rP.$
Moreover, let $\sigma \in W/\!\sim_{P|Q}.$ Then
\begin{enumerate}
\item[{\rm (a)}]
for every $\xi \in -\sigma\cdot Y + {\msy N} {\msy D}rP,$
$$
q_{\sigma, \xi}(P,v \,|\, f)
\in P_k({\mathfrak a}Pq) \otimes \cM({\mathfrak a}Qqdc, \cH_f, d_f, {\msy C}i({\rm X}Pvp\,:\,\tau_P));
$$
\minspace\item[{\rm (b)}]
for every $R > 1,$ the series
$$
\sum_{\xi \in -\sigma \cdot Y + {\msy N} {\msy D}rP} a^{-\xi} q_{\sigma, \xi}(P,v\,|\, f,\log a)
$$
converges neatly on $A_{P\iq}^+(R^{-1})$
as a ${\msy D}rP$-exponential polynomial series with coefficients
in $\cM({\mathfrak a}qdc, \cH_f, d_f, {\msy C}i({\rm X}Pvp[R]\,:\, \tau_P)).$
\end{enumerate}
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
Let $\Omega={\rm reg}a f$.
Then $f\in C^{\rm ep}_{Q, Y}({\rm X}p\,:\, \tau \,:\, \Omega)$.
It follows from Theorem \ref{t: behavior along the walls for families new}
that the assertion about the $(P,v)$-exponents of $f_\lambda$ holds.
That (a) and (b) hold can be seen as in the last part of the proof of
Lemma \ref{l: relation specfam and fam second}, with
the reference to Definition \ref{d: anfamQY newer} replaced
by reference to Theorem \ref{t: behavior along the walls for families new}.
~
$\square$\medbreak\noindent\medbreak
The following definition is the analogue for ${\msy C}ephypQY$ of
Definitions \ref{d: defi cE Q Y revised new}
and \ref{d: family for the vanishing thm newer}.
\begin{defi}
\label{d: defi cEhyp Q Y gd}
Let $Q \in \cP_\gs$ and $\delta \in {\msy D}Qmaps.$ Then for $Y \subset {}^*\fa_{Q\iq}dc$ a finite
subset we define
$$
{\cal E}hypQYgd
$$
to be the space of functions $f \in {\msy C}ephypQY$
(see Definition \ref{d: Cephyp})
such that, for all
$\lambda \in {\rm reg}a(f),$
the function $f_\lambda: x :to f(\lambda, x)$ is annihilated by the cofinite ideal
$I_{\gd, \gl}.$
Moreover, we define
$$
{\cal E}hypQgd: = \bigcup_{Y \subset {}^*\fa_{Q\iq}dc \;{\rm finite}} {\cal E}hypQYgd.
$$
The spaces
$$
{\cal E}hypQYgd_\lambdaob,\quad {\cal E}hypQgd_\lambdaob
$$
are defined to be the spaces of functions $f$ in ${\cal E}hypQYgd$, resp.\
${\cal E}hypQgd$, for which the condition in Definition
\ref{d: family for the vanishing thm newer}
is satisfied by the restriction to $\Omega={\rm reg}a f$.
Finally, we define $${\cal E}^{\rm hyp}_0({\rm X}p\,:\,\tau\,:\,\delta):=
{\cal E}^{\rm hyp}_{P_0}({\rm X}p\,:\,\tau\,:\,\delta),\quad
{\cal E}^{\rm hyp}_0({\rm X}p\,:\,\tau\,:\,\delta)_\lambdaob:=
{\cal E}^{\rm hyp}_{P_0}({\rm X}p\,:\,\tau\,:\,\delta)_\lambdaob$$
for $\delta\in D_{P_0}$.
\end{defi}
\begin{rem}
\label{r: only local annihilation needed}
Combining Lemmas \ref{l: relation specfam and fam second} and
\ref{l: locally killed by I gl gd} we
see that, in the above definition of ${\cal E}hypQYgd$, it suffices to require
that $I_{\gd, \gl}$ annihilates $f_\lambda$ for $\lambda$ in a non-empty open
subset of ${\rm reg}a(f).$
\end{rem}
We now come to a special case of the vanishing theorem that will
be particularly useful in the following.
Let ${}^Q\cW \subset {\msy N}Kaq$ be a complete set of representatives for $W_Q\backslash W/W_{K \cap H}.$
\begin{thm}
{\rm (A special case of the vanishing theorem)\ }
\label{t: special vanishing theorem}
Let $Q \in \cP_\gs$
and let $\delta\in D_Q$.
Let $f \in {\cal E}hypQgd_\lambdaob$ and let $\Omega'$ be a non-empty open subset of
${\rm reg}a f.$ If
$$
\lambda - \rho_Q \notin {\rm Exp}(Q,u\,|\, f_\lambda)
$$
for each $u \in {}^Q\cW$ and all $\lambda \in \Omega',$
then $f = 0.$
\end{thm}
\par\noindent{\bf Proof:}{\ }{\ }
Put $\Omega = {\rm reg}a(f).$
It follows immediately from the definitions that
the restriction $f_\Omega$ of $f$ to $\Omega$ is a family in
${\cal E}_Q({\rm X}p\,:\, \tau \,:\, \Omega\,:\, \delta)_\lambdaob.$
Moreover, being the complement of a locally finite collection
of hyperplanes, $\Omega$ is $Q$-distinguished in ${\mathfrak a}Qqdc.$
It follows that $f_\Omega$ satisfies all hypothesis of
Theorem \ref{t: vanishing theorem new}; hence $f_\Omega = 0$
and hence $f=0$.
~
$\square$\medbreak\noindent\medbreak
\section{Action of Laurent functionals on analytic families}
\label{s: act Laufunc}
Let $Q\in \cP_\gs$ be fixed.
We shall discuss the application of a Laurent functional
$\cL \in \cM({}^*\fa_{Q\iq}dc, \Sigma_Q)^*_{\rm laur},$
to families $f\in{\msy C}ephyp.$ More precisely, we want to set up natural
conditions on $f$ under which the family obtained from applying $\cL$
to $f$ belongs to ${\cal E}hypQgd_\lambdaob$, so that Theorem
\ref{t: special vanishing theorem} is applicable.
Given a $\Sigma$-configuration $\cH$
in ${\mathfrak a}qdc$ and a finite subset $S \subset {}^*\fa_{Q\iq}dc$ we define
the $\Sigma_r(Q)$-configuration
$\cH_Q(S) = \cH_{{\mathfrak a}Qqdc}(S)$ as in (\ref{e: defi Hyp L S}),
with $V = {\mathfrak a}qdc,$ $X = \Sigma,$ and $L = {\mathfrak a}Qqdc.$ Thus, for
$\nu \in {\mathfrak a}Qqdc$ we have
$$
\nu\notin\cup \cH_Q(S) \iff [\; \forall \lambda \in S \,\forall H \in \cH :\;\;\;
\lambda + \nu \in H \Rightarrow \lambda + {\mathfrak a}Qqdc \subset H\;].
$$
We recall from Lemma \ref{l: Lau under star new} that a Laurent functional
$\cL \in \cM({}^*\fa_{Q\iq}dc, \Sigma_Q)_{\rm laur}^*$ induces a
linear operator
\defU{U}
\begin{equation}
\label{e: Laustar in root context}
\cL_*: \cM({\mathfrak a}qdc, \Sigma, U) \rightarrow \cM({\mathfrak a}Qqdc, \Sigma_r(Q), U),
\end{equation}
for any complete locally convex space $U.$
\begin{lemma}
\label{l: Laustar for roots and d prime}
Let $\cL \in \cM({}^*\fa_{Q\iq}dc, \Sigma_Q)_{\rm laur}^*$ and put $Y = {\rm supp}\, \cL.$
Let $\cH$ be a $\Sigma$-configuration in ${\mathfrak a}qdc,$ and let $\cH'=\cH_Q(Y)$.
Then for every
map $d: \cH \rightarrow {\msy N}$ there exists a map $d': \cH' \rightarrow {\msy N}$ such that, for every complete
locally convex space $U,$ the linear map (\ref{e: Laustar in root context})
restricts to a continuous linear operator
$$
\cL_*: \cM({\mathfrak a}qdc, \cH,d, U) \rightarrow \cM({\mathfrak a}Qqdc, \cH', d', U),
$$
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
This follows immediately from Corollary \ref{c: continuity of Laustar}.
~
$\square$\medbreak\noindent\medbreak
For the formulation of the next result it will be convenient
to introduce a particular linear map. Let $\cL \in \cM({}^*\fa_{Q\iq}dc, \Sigma_Q)_{\rm laur}^*$
and let $\lambda_0 \in Y:= {\rm supp}\, \cL.$
Let $\cL_{\lambda_0}\in \cM({}^*\fa_{Q\iq}dc, \Sigma_Q)_{\rm laur}^*$
be the Laurent functional supported at $\lambda_0,$
defined as in Remark \ref{r: Laurent functionals at a as subspace},
and let $U$ be a complete locally convex space.
If $P \in \cP_\gs$ and
$s \in W_P\backslash W,$ then we define the linear operator $\cL_{\lambda_0 *}^{P,s}$ from
$\cM({\mathfrak a}qdc, \Sigma, U) $
into $C({\mathfrak a}Pq, \cM({\mathfrak a}Qqdc, \Sigma_r(Q), U) )$
by the formula
\begin{equation}
\label{e: defi Laustar s}
\cL_{\lambda_0*}^{P,s}\varphi (X, \nu) = e^{-s(\lambda_0 + \nu)(X)}
\cL_{\lambda_0*}[ e^{s(\,\cdot\,)(X)} \varphi(\,\cdot\,) ](\nu),
\end{equation}
for $\varphi \in \cM({\mathfrak a}qdc, \cH, U),$
$X \in {\mathfrak a}Pq$ and
$\nu \in {\mathfrak a}Qqdc\setminus\cup\cH_Q(Y).$
If $f \in {\msy C}ephyp,$ then $f,$ viewed as the function
$\lambda :to f_\lambda,$ belongs to the complete
locally convex space $\cM({\mathfrak a}qdc, \cH_f, d_f, {\msy C}i({\rm X}p\,:\, \tau)).$
Accordingly,
\begin{equation}
\label{e: Laustar f in merhyp}
\cL_* f \in \cM({\mathfrak a}Qqdc, \cH', d', {\msy C}i({\rm X}p\,:\, \tau)),
\end{equation}
where $\cH'=\cH_{fQ}(Y)$ and $d': \cH'\rightarrow {\msy N}$ is associated with $\cL, \cH_f$ and $d_f$ as in Lemma
\ref{l: Laustar for roots and d prime}. We note that by definition
\begin{equation}
\label{e: defi Laustar of family}
\cLstar f(\nu, x) = \cL [ f(\,\cdot\, + \nu, x) ],\quad\quad
(\nu \in {\mathfrak a}Qqdc\setminus \cup \cH', \; x \in {\rm X}p).
\end{equation}
\begin{prop}
\label{p: Lau to hyp family}
Let $Q\in \cP_\gs$ and let $\cL \in \cM({}^*\fa_{Q\iq}, \Sigma_Q)^*_{\rm laur}$
be a Laurent functional with support contained in the finite subset
$Y \subset {}^*\fa_{Q\iq}dc.$
Assume that $f \in {\msy C}ephyp,$ and let $k={\rm deg}_{\rm a} f$.
\begin{enumerate}
\item[{\rm (a)}]
The function $\cLstar f,$ defined as in
(\ref{e: defi Laustar of family}), belongs
to the space ${\msy C}ephypQY.$ Moreover, $\cH_{\cLstar f}\subset\cH'=
\cH_{fQ}(Y)$
and ${\rm deg}_{\rm a} \cLstar f \leq k+ k',$
with $k' \in {\msy N}$ a constant only depending on $\cL, \cH_f$ and $d_f.$
\minspace\item[{\rm (b)}]
Let $P \in \cP_\gs, v \in {\msy N}Kaq.$ Then, for $\sigma \in W/\!\sim_{P|Q}$ and
$\xi \in -\sigma \cdot Y + {\msy N} {\msy D}rP,$
\begin{equation}
\label{e: sum for q of Laustar f}
q_{\sigma, \xi}(P, v\,|\, \cL_*f, X, \nu)
=
\sum_{\lambda \in Y}
\sum_{s \in W_P\backslash W, \, [s] = \sigma \atop s\lambda|_{{\mathfrak a}Pq} + \xi \in {\msy N}{\msy D}rP}
\cL_{\lambda*}^{P,s} \left[
q_{s, s\lambda|_{{\mathfrak a}Pq} + \xi} (P,v\,|\, f)(X,\,\cdot\,)\right] (\nu, X),
\end{equation}
for all $X \in {\mathfrak a}Pq$ and $\nu \in {\mathfrak a}Qqdc\setminus \cup \cH'.$
In particular,
\begin{eqnarray*}
&&{\rm Exp}(P,v\,|\, (\cLstar f)_\nu)\\
&&\quad\subset
\{s(\nu+\lambda)|_{{\mathfrak a}Pq}-\rho_P-\mu\mid s\in W, \lambda\in Y, \mu\in{\msy N}{\msy D}elta_r(P),
q_{s,\mu}(P,v\,|\, f)\neq0\}.
\end{eqnarray*}
\end{enumerate}
\end{prop}
\begin{rem}
\label{r: sum with empty index set}
Note that the index set of the inner sum in (\ref{e: sum for q of Laustar f})
may be empty. We agree that such a sum should be interpreted
as zero.
\end{rem}
The following lemma prepares for the proof of the proposition.
\begin{lemma}
\label{l: continuity Laustar s new}
Let $\cL \in \cM({}^*\fa_{Q\iq}dc, \Sigma_Q)_{\rm laur}^*$ be a Laurent functional with
support contained in the finite set $Y \subset {}^*\fa_{Q\iq}dc.$
Let ${\cal H}$ be a $\Sigma$-configuration in ${\mathfrak a}qdc$ and $d: {\cal H} \rightarrow {\msy N}$ a map.
Let $\cH'=\cH_Q(Y)$ and
$d': \cH' \rightarrow {\msy N}$ be as in Lemma \ref{l: Laustar for roots and d prime}.
There exists a
natural number $k' \in {\msy N}$ with the following property.
For every $\lambda_0 \in Y,$ every $P \in \cP_\gs,$
each $s \in W_P\backslash W$ and
any complete locally convex space
$U,$ the operator $\cL_{\lambda_0*}^{P,s}$ restricts to a continuous linear map
$$
\cL_{\lambda_0*}^{P,s} :\;\; \cM({\mathfrak a}qdc, \cH, d, U) \rightarrow P_{k'}({\mathfrak a}Pq)\otimes
\cM ({\mathfrak a}Qqdc, \cH', d', U).
$$
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
For a fixed $X \in {\mathfrak a}Pq,$ multiplication by the holomorphic
function
$e^{s(\,\cdot\,)(X)}: {\mathfrak a}qdc \rightarrow {\msy C}$ yields a continuous linear endomorphism
of the space $\cM({\mathfrak a}qdc, \cH, d, U);$ similarly, multiplication
by the holomorphic function $e^{-s(\lambda_0 + \,\cdot\,)(X)}: {\mathfrak a}Qqdc \rightarrow {\msy C}$
yields a continuous linear endomorphism of $\cM({\mathfrak a}Qqdc, \cH', d', U).$
It now follows from (\ref{e: defi Laustar s}) that for a fixed
$X \in {\mathfrak a}Pq,$ the function $\cL_{\lambda_0*}^{P,s}\varphi (X)$ belongs
to the space $\cM({\mathfrak a}Qqdc, \cH', d', U)$ and depends continuously
on $\varphi.$
Thus, it remains to establish the
polynomial dependence on $X.$
For any $\Sigma$-hyperplane $H \subset {\mathfrak a}qdc$ we denote by
$\alpha_H$ the root from $\Sigma_0^+$ such that $H$ is a translate
of $\alpha_{H{\scriptscriptstyle \C}}^\perp.$ Let $\Sigma_{Q,0}^+:= \Sigma_Q \cap \Sigma_0^+$
and let $d_0: \Sigma_{Q,0}^+ \rightarrow {\msy N}$ be defined
by $d_0(\alpha) = d(\alpha^\perp + \lambda_0);$ thus
$d_0(\alpha) = 0$ if $\alpha^\perp + \lambda_0 \notin \cH.$
We define $\pi_0 = \pi_{\lambda_0, d_0}$ as in (\ref{e: defi pi a X d}) with
${}^*\fa_{Q\iq}d, \lambda_0, \Sigma_{Q,0}^+$ and $ d_0$ in place of $V, a, X$
and $d,$ respectively.
If $\varphi \in \cM({\mathfrak a}qdc, \cH, d, U),$ then for
$\nu \in {\mathfrak a}Qqdc\setminus \cup\cH',$ the germ
of the function $\varphi^\nu: \lambda :to \varphi(\lambda + \nu)$ at $\lambda_0$
belongs to $\pi_0^{-1} {\cal O}_{\lambda_0}({}^*\fa_{Q\iq}dc, U)$.
Hence there exists
a constant coefficient differential operator $u_0 \in S({}^*\fa_{Q\iq}d),$ independent
of $U,$ such that
\begin{equation}
\label{e: action of Laustar as dif op}
\cL_{\lambda_0 *} \varphi (\nu) = u_0[ \pi_0(\,\cdot\,)\varphi(\,\cdot\, + \nu) ](\lambda_0),
\quad\quad (\nu \in {\mathfrak a}Qqd\setminus \cup \cH'),
\end{equation}
for any $\varphi \in \cM({\mathfrak a}qdc, \cH, d, U).$
Inserting (\ref{e: action of Laustar as dif op}) in (\ref{e: defi Laustar s})
we find that
\begin{eqnarray*}
\cL_{\lambda_0*}^{P,s} \varphi (X,\nu)
&=& e^{-s(\lambda_0 + \nu)(X)} u_0[ e^{s(\,\cdot\, +\nu )(X)}
\pi_0(\,\cdot\,) \varphi(\,\cdot\, + \nu) ](\lambda_0)\\
&=& e^{-s(\lambda_0)(X)} u_0[ e^{s(\,\cdot\,)(X)}
\pi_0(\,\cdot\,) \varphi(\,\cdot\, + \nu) ](\lambda_0).
\end{eqnarray*}
By application of the Leibniz rule it finally follows that
this expression is polynomial in the variable $X$ of degree
at most $k' := {\rm order}(u_0).$
~
$\square$\medbreak\noindent\medbreak
\medbreak\noindent
{\bf Proof of Proposition \ref{p: Lau to hyp family}:\ }
By
linearity we may assume that
${\rm supp}\, \cL$ consists of a single point $\lambda_0 \in {}^*\fa_{Q\iq}dc.$
Let $\cH=\cH_f$ and $d = d_f$, and let
$d':\cH'\rightarrow{\msy N}$ and $k'\in{\msy N}$ be associated as in Lemmas
\ref{l: Laustar for roots and d prime} and
\ref{l: continuity Laustar s new}.
We will establish parts (a), (b) and (c) of Definition \ref{d: Cephyp}
for $\cLstar f$ with $k$, $\cH$ and $d$ replaced by $k+k'$,
$\cH'$ and $d'$.
Note that part (a) was observed already in (\ref{e: Laustar f in merhyp}).
Put $\Omega:= {\mathfrak a}Qqdc\setminus \cup \cH'.$ Then, in particular,
the function $\cLstar f: \Omega \times {\rm X}p \rightarrow V_\tau$ is smooth.
We will establish parts (b) and (c) of Definition \ref{d: Cephyp}
by obtaining an exponential polynomial
expansion for $(\cLstar f)_\nu,$ for $\nu \in \Omega$,
along $P\in\cP_\gs^{\rm min}$. However, having the proof of
(\ref{e: sum for q of Laustar f}) in mind, we assume only
$P \in \cP_\gs$ at present.
Let $v \in {\msy N}Kaq.$
Then $f \in C^{\rm ep}_{P_0, \{0\}}({\rm X}p\,:\, \tau\,:\, {\mathfrak a}qdc\setminus \cup \cH)$
by Remark \ref{r: relation between hypfam and fam}. Hence
by Lemma \ref{l: asymptotics along walls for CephypQY} and
(\ref{e: series for fam along P}) we obtain,
for $\lambda \in {\mathfrak a}qdc\setminus\cup{\cal H},$
\begin{equation}
\label{e: f as sum fs}
f(\lambda, mav) = \sum_{s \in W_P \backslash W}
f_{s}(\lambda, a, m),\quad\quad (m \in {\rm X}Pvp,\; a \in A_{P\iq}^+(R_{P,v}(m)^{-1})),
\end{equation}
where the functions $f_{s}$ on the right-hand side are defined by
\begin{equation}
\label{e: series for fs}
f_{s}(\lambda,a,m ) = a^{s \lambda -\rho_P } \sum_{\mu \in {\msy N}{\msy D}rP}
a^{-\mu} q_{s,\mu} (P,v\,|\, f)( \log a , \lambda , m).
\end{equation}
Here the functions
$q_{s,\mu}(P,v\,|\, f)$ belong to the space
$P_k({\mathfrak a}Pq) \otimes \cM({\mathfrak a}qdc, \cH, d,
{\msy C}i({\rm X}Pvp\,:\, \tau_P)).$
By Lemma \ref{l: asymptotics along walls for CephypQY} (b),
for every $R>1$ the series
in (\ref{e: series for fs}) converges neatly
on $A_{P\iq}^+(R^{-1})$
as a series with coefficients in
$\cM({\mathfrak a}qdc, \cH, d, C^\infty({\rm X}Pvp[R]\,:\, \tau_P)).$
By (\ref{e: defi Laustar s}) we have, for
$\nu \in \Omega,$ $m \in {\rm X}Pvp[R]$ and $a \in A_{P\iq}^+(R^{-1})$
$$\cLstar(f_s)(\nu,a,m)= a^{s(\lambda_0+\nu)-\rho_P}
\cL_{\lambda_0*}^{P,s} [\sum_{\mu \in {\msy N}{\msy D}rP}
a^{-\mu} q_{s,\mu} (P,v\,|\, f) ( \log a , \,\cdot\, , m)](\log a,\nu).$$
It follows from
Lemma \ref{l: continuity Laustar s new} that
$\cL_{\lambda_0*}^{P,s}$ may
be applied term by term to the series. Moreover,
the resulting series is neatly convergent on
$A_{P\iq}^+(R^{-1})$ as a ${\msy D}rP$-exponential polynomial series with coefficients
in $\cM({\mathfrak a}Qqdc, \cHpr, d', {\msy C}i({\rm X}Pvp[R]\,:\,\tau_P)).$
The application of $\cLstar$
thus leads to the following identity,
\begin{equation}
\label{e: series for Laustarfs}
\cLstar(f_s)(\nu, a, m) = a^{s(\lambda_0 + \nu) - \rho_P}
\sum_{\mu \in {\msy N}{\msy D}rP} a^{-\mu} q_{s,\mu}^\cL (P,v\,|\, f)(\log a, \nu, m),
\end{equation}
where the function
$q_{s,\mu}^\cL(P,v\,|\, f): {\mathfrak a}Pq \times \Omega \rightarrow {\msy C}i({\rm X}Pvp\,:\, \tau_P)$ is given
by
\begin{equation}
\label{e: defi q Lau s mu new}
q_{s,\mu}^\cL(P,v\,|\, f)(\log a, \nu)
=
\cL_{\lambda_0*}^{P,s} [q_{s,\mu}(P,v\,|\, f,\log a, \,\cdot\,)](\log a,\nu).
\end{equation}
Using Lemma \ref{l: continuity Laustar s new} we deduce that
$$
q_{s,\mu}^\cL(P,v\,|\, f)
\in
P_{k+ k'}({\mathfrak a}Pq) \otimes
\cM({\mathfrak a}Qqdc, \cHpr, d', {\msy C}i({\rm X}Pvp\,:\, \tau_P)).
$$
Combining (\ref{e: series for Laustarfs}) with
(\ref{e: f as sum fs}) we obtain
an exponential polynomial expansion along $(P,v)$ for the
$\tau$-spherical function $(\cLstar f)_\nu$ as
\begin{equation}
\label{e: series for Laustar f nu}
(\cLstar f)_\nu (mav) = \sum_{s \in W_P\backslash W}
a^{s(\lambda_0 + \nu) -\rho_P}
\sum_{\mu \in {\msy N}{\msy D}rP}
a^{-\mu} q_{s,\mu}^\cL(P,v\,|\, f)(\log a, \nu,m).
\end{equation}
If $s \in W_P\backslash W$ and $\nu \in {\mathfrak a}Qqdc,$ then $s\nu|_{{\mathfrak a}Pq} = [s]\nu|_{{\mathfrak a}Pq},$
where $[s]$ denotes the class of $s$ in $W/\!\sim_{P|Q}.$ It follows that the series
in (\ref{e: series for Laustar f nu}) may be rewritten as
$$
\sum_{\sigma \in W/\!\sim_{P|Q}} a^{\sigma \nu - \rho_P}
\sum_{s \in W_P\backslash W, [s] = \sigma\atop\mu \in {\msy N}{\msy D}rP}
a^{s \lambda_0 -\mu} q_{s,\mu}^\cL(P,v\,|\, f)(\log a, \nu,m).
$$
The exponents $s\lambda_0 - \mu$ as $s \in W_P\backslash W,$ $[s] =\sigma$ and $\mu \in {\msy N} {\msy D}rP,$
are all of the form $-\xi,$ with $\xi \in -\sigma\cdot\{\lambda_0\} + {\msy N}{\msy D}rP.$
Thus, we see that, for $\nu \in \Omega,$ $m\in {\rm X}Pvp[R]$ and
$a \in A_{P\iq}^+(R^{-1}),$
\begin{equation}
\label{e: series for Laustar f along P v}
(\cLstar f)_\nu (mav) = \sum_{\sigma \in W/\!\sim_{P|Q}}
a^{\sigma\nu -\rho_P}
\sum_{\xi \in -\sigma\cdot\{\lambda_0\} + {\msy N}{\msy D}rP} a^{-\xi}\;\widetilde
q_{\sigma, \xi}(\log a, \nu, m)
\end{equation}
with
\begin{eqnarray}
\label{e: q gs xi of Laustar f}
\widetilde q_{\sigma, \xi}
&= &
\sum_{s \in W_P\backslash W,\, [s] = \sigma \atop s\lambda_0|_{{\mathfrak a}Pq} + \xi \in {\msy N}{\msy D}rP}
q_{s,s\lambda_0|_{{\mathfrak a}Pq} + \xi }^\cL(P,v\,|\, f)\\
&\in&
P_{k+ k'}({\mathfrak a}Pq) \otimes
\cM({\mathfrak a}Qqdc, \cHpr, d', {\msy C}i({\rm X}Pvp\,:\, \tau_P)).
\nonumber
\end{eqnarray}
{}From what we said earlier about the convergence
of the series in (\ref{e: series for Laustarfs}), it follows that, for every $R>1,$ the
inner series on the right-hand side of (\ref{e: series for Laustar f along P v})
converges neatly on $A_{P\iq}^+(R^{-1})$ as a
${\msy D}rP$-exponential polynomial series with
coefficients in the space $\cM({\mathfrak a}Qqdc, \cH', d', {\msy C}i({\rm X}Pvp[R] \,:\,\tau_P)).$
If $P$ is minimal, then ${\rm X}Pvp[R] = {\rm X}zerov$
and we see that $\cLstar f$ satisfies conditions (b) and (c)
of Definition \ref{d: Cephyp} with
$q_{\sigma,\xi}(P,v\,|\, \cLstar f) = \tilde q_{\sigma, \xi}$
for $\sigma\inW/\!\sim_{P|Q}\,=W/W_Q$.
This establishes part (a) of the proposition.
For general $P$ we now see that the functions $\tilde q_{\sigma, \xi}$
introduced above coincide with functions $q_{\sigma, \xi}(P,v\,|\, \cLstar f)$
introduced in Theorem \ref{t: behavior along the walls for families new}.
Finally, combining (\ref{e: q gs xi of Laustar f}) and
(\ref{e: defi q Lau s mu new}) we see that we have
established part (b) of the proposition as well.
~
$\square$\medbreak\noindent\medbreak
\begin{lemma}
\label{l: Lau to family of eigenfunctions new}
Let $\delta\in D_{P_0}$ and $f \in {\cal E}_0^{\rm hyp}({\rm X}p\,:\, \tau \,:\, \delta).$
Let $Q\in \cP_\gs$ and $\cL \in \cM({}^*\fa_{Q\iq}dc, \Sigma_Q)^*_{\rm laur},$ and
put $Y = {\rm supp}\, \cL.$
There exists a $\delta' \in {\msy D}Qmaps$ such that
$$
\cL_*f \in {\cal E}_{Q,Y}^{\rm hyp}({\rm X}p\,:\, \tau\,:\, \delta').
$$
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
It follows from Proposition \ref{p: Lau to hyp family}
that $\cLstar f \in C^{{\rm ep}, {\rm hyp}}_{Q,Y}({\rm X}p\,:\,\tau).$
Moreover, ${\rm reg}a{\cLstar f} \supset \Omega = {\mathfrak a}Qqdc\setminus \cH_{fQ}(Y).$
Then in view of Definition \ref{d: defi cEhyp Q Y gd} and Remark
\ref{r: only local annihilation needed}
it suffices to establish the existence of
a $\deltamap' \in {\msy D}Qmaps$ such that, for every $\nu \in \Omega,$ the function
$(\cL_*f)_\nu$ is annihilated by the cofinite ideal
$I_{\delta',\nu}.$
By linearity we may assume that ${\rm supp}\, \cL$ consists of a single
point $\lambda_0 \in {}^*\fa_{Q\iq}dc.$ Then $\cL = \cL_{\lambda_0}.$
Let $\pi_0,u_0$ be as in the proof of Lemma \ref{l: continuity Laustar s new}.
Then from (\ref{e: action of Laustar as dif op}) we see that
$$
(\cLstar f)_\nu( x) = u_0 [\pi_0(\,\cdot\,) f(\,\cdot\, + \nu, x)](\lambda_0) ,
$$
for $x \in {\rm X}p,\, \nu \in \Omega.$
Moreover, since $(\lambda, x) :to \pi_0(\lambda)f_{\lambda + \nu}(x)$
is smooth in a neighborhood of $\{\lambda_0\} \times {\rm X}p,$
it follows that, for $D \in {\msy D}X,$ $\nu \in \Omega$ and $x \in {\rm X}p,$
\begin{equation}
\label{e: D on Laustar f}
D(\cLstar f)_\nu (x) =
u_0[ \pi_0 (\,\cdot\,) D (f_{\,\cdot\, + \nu})(x) ](\lambda_0).
\end{equation}
Put $l = {\rm order}(u_0)$ and define $\delta'\in D_Q$ by
${\rm supp}\,\delta'=\{\lambda_0\}+{\rm supp}\,\delta$ and $\delta'(\lambda_0+\Lambda)=\delta(\Lambda)+l$ for
$\Lambda\in{\rm supp}\,\delta$. It suffices to prove the following.
Let elements $D^\Lambda_i\in{\msy D}X$ be given for $i=1,\dots,\delta(\Lambda)+l$,
for each $\Lambda\in{\rm supp}\,\delta$, and define the differential operator
\begin{equation}
\label{e: defi Dnu}
D_\nu:={\rm pr}od_{\Lambda\in{\rm supp}\,\delta}{\rm pr}od_{i=1}^{\delta(\Lambda)+l}
(D^\Lambda_i-\gamma(D^\Lambda_i,\lambda_0+\Lambda+\nu))\in{\msy D}X
\end{equation}
for $\nu\in{\mathfrak a}Qqdc$.
Then $D_\nu$ annihilates $(\cLstar f)_\nu$ for each $\nu\in\Omega$.
It follows from (\ref{e: defi Laustar of family}) and
(\ref{e: D on Laustar f}) that
\begin{equation}
\label{e: Dnu on Laustar f}
D_\nu(\cLstar f)_\nu(x)= u_0[\pi_0(\,\cdot\,) D_\nu f_{\,\cdot\,+\nu}(x)](\lambda_0),
\end{equation}
where the dots indicate a variable in ${}^*\fa_{Q\iq}dc$.
We write each factor in $D_\nu$ as
\begin{eqnarray*}
&&D^\Lambda_i-\gamma(D^\Lambda_i,\lambda_0+\Lambda+\nu)\\
&&\quad\quad=[D^\Lambda_i-\gamma(D^\Lambda_i,\,\cdot\,+\Lambda+\nu)]+
[\gamma(D^\Lambda_i,\,\cdot\,+\Lambda+\nu)-\gamma(D^\Lambda_i,\lambda_0+\Lambda+\nu)],
\end{eqnarray*}
also with variables in ${}^*\fa_{Q\iq}dc$ indicated by dots.
Inserting this into (\ref{e: defi Dnu}) and
(\ref{e: Dnu on Laustar f}) we obtain an
expression for $D_\nu(\cLstar f)_\nu(x)$ as a sum of terms
each of the form
\begin{equation}
\label{e: summands of Dnu Laustar f}
u_0[\pi_0(\,\cdot\,)
{\rm pr}od_{\Lambda\in{\rm supp}\,\delta}p^\Lambda(\,\cdot\,)D^\Lambda(\,\cdot\,)
f_{\,\cdot\,+\nu}(x)](\lambda_0),
\end{equation}
where
$$D^\Lambda(\lambda)={\rm pr}od_{i\in S_\Lambda}[D^\Lambda_i-\gamma(D^\Lambda_i,\lambda+\Lambda+\nu)]$$
and
$$p^\Lambda(\lambda)={\rm pr}od_{i\in S_\Lambda^c}
[\gamma(D^\Lambda_i,\lambda+\Lambda+\nu)-\gamma(D^\Lambda_i,\lambda_0+\Lambda+\nu)]$$
with $S_\Lambda$ a subset of $\{1,\ldots,\delta(\Lambda)+l\}$
and $S^c_\Lambda$ its complement in this set.
On the one hand, if $S_\Lambda$ has fewer than
$\delta(\Lambda)$ elements for some $\Lambda$, there are at least
$l+1$ factors in the corresponding product $p^\Lambda$. Since each of these
factors vanish at $\lambda_0$, it follows from the Leibniz rule that then
(\ref{e: summands of Dnu Laustar f}) vanishes. On the other hand,
if for each $\Lambda$ the set $S_\Lambda$ has at least $\delta(\Lambda)$ elements,
then the differential operator ${\rm pr}od_{\Lambda}D^\Lambda(\lambda)$
annihilates $f_{\lambda+\nu}$, again causing
(\ref{e: summands of Dnu Laustar f}) to vanish. It follows
that $D_\nu(\cLstar f)_\nu(x)=0$.
~
$\square$\medbreak\noindent\medbreak
In the following definition we introduce a notion of asymptotic
globality that is somewhat stronger than the one
in Definition \ref{d: s globality new}. It is motivated by the fact that
it carries over by the application of Laurent functionals, as we shall see
in Proposition \ref{p: transference of s globality}
\begin{defi}
\label{d: holomorphic s globality new}
Let $Q \in \cP_\gs,$ and let $Y\subset {}^*\fa_{Q\iq}dc$ be finite.
Let $P \in \cP_\gs,$ $v \in{\msy N}Kaq$ and $\sigma \in W/\!\sim_{P|Q}.$
\begin{enumerate}
\item[{\rm (a)}] Let $\Omega \subset {\mathfrak a}Qqdc$ be an open subset.
A family
$f \in C^{\rm ep}_{Q,Y}({\rm X}p\,:\, \tau \,:\,\Omega)$
is called holomorphically $\sigma$-global along
$(P,v)$ if there exists a full open subset $\Omega^*$ of ${\mathfrak a}Qqdc$ such
that, for every $\xi \in -\sigma\cdot Y + {\msy N}{\msy D}rP,$
the function $\lambda :to q_{\sigma,\xi}(P,v\,|\, f,\,\cdot\,)(\lambda)$ is a
holomorphic $P_k({\mathfrak a}Pq) \otimes {\msy C}i({\rm X}Pv \,:\, \tau_P)$-valued function on
$\Omega^*\cap\Omega,$ for some $k \in {\msy N}.$
\minspace\item[{\rm (b)}] A family $f \in C^{{\rm ep},{\rm hyp}}_{Q,Y}({\rm X}p\,:\, \tau)$ is called
holomorphically $\sigma$-global along $(P,v)$ if its restriction
to $\Omega={\rm reg}a f$ is holomorphically $\sigma$-global along $(P,v)$, according to (a).
\end{enumerate}
\end{defi}
It is easily seen that the property of holomorphic globality
according to (a) of the above definition implies the globality
in Definition \ref{d: s globality new}.
We have the following analogue of Lemma \ref{l: transformation of globality},
describing
how the property of holomorphic globality transforms under
the action of ${\msy N}Kaq.$
\begin{lemma}
\label{l: transformation of holomorphic globality}
Let $Q$, $Y$, $P$, $v$ and $\sigma$ be as above,
and let $f \in C^{{\rm ep},{\rm hyp}}_{Q,Y} ({\rm X}p \,:\, \tau).$
If $f$ is holomorphically $\sigma$-global along $(P,v),$ then
$f$ is holomorphically $u\sigma$-global along $(uPu^{-1}, uv),$ for every $u \in {\msy N}Kaq.$
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
The proof is completely analogous to the proof
of Lemma \ref{l: transformation of globality},
involving an application of Lemma
\ref{l: transformation holo coeffs}.
~
$\square$\medbreak\noindent\medbreak
\begin{prop}
\label{p: mero H globality newer}
Let $Q \in \cP_\gs,$
$Y\subset {}^*\fa_{Q\iq}dc$ a finite subset and let
$P \in \cP_\gs,$ $v \in{\msy N}Kaq$ and $\sigma \in W/\!\sim_{P|Q}.$
Let $f \in {\msy C}ephypQY$ and put $\cH=\cH_f$, $d=d_f$ and $k={\rm deg}_{\rm a} f$.
The family $f$ is holomorphically $\sigma$-global along
$(P,v)$ if and only if, for every element $\xi \in - \sigma\cdot Y + {\msy N}{\msy D}rP,$
the function $\lambda :to q_{\sigma, \xi}(P,v\,|\, f,\,\cdot\,)(\lambda)$
belongs to the space $\cM({\mathfrak a}Qqdc, \cH, d, P_k({\mathfrak a}Pq) \otimes
{\msy C}i({\rm X}Pv\,:\, \tau_P)).$
\end{prop}
\par\noindent{\bf Proof:}{\ }{\ }
The `if'-statement is obvious. Assume that $f$ is holomorphically
$\sigma$-global along $(P,v)$, and let $\xi \in -\sigma \cdot Y + {\msy N}{\msy D}rP.$
According to Lemma \ref{l: asymptotics along walls for CephypQY},
the function
\begin{equation}
\label{e: globality of q}
\lambda :to q_{\sigma, \xi}(P,v\,|\, f, \,\cdot\, , \lambda)
\end{equation}
belongs to the space
\begin{equation}
\label{e: space to which q belongs}
\cM({\mathfrak a}Qqdc, \cH,d, P_k({\mathfrak a}Pq)\otimes {\msy C}i({\rm X}Pvp\,:\, \tau_P)).
\end{equation}
Let $\Omega={\rm reg}a(f)$ and let
$\Omega^*$ be a full open subset of ${\mathfrak a}Qqdc$ satisfying the properties of
Definition \ref{d: holomorphic s globality new} (a) for the restriction
of $f$ to $\Omega$.
Then the function (\ref{e: globality of q}) not only belongs to the space
(\ref{e: space to which q belongs}), but also to the space
${\cal O}(\Omega^*\cap\Omega, P_l({\mathfrak a}Pq) \otimes {\msy C}i({\rm X}Pv\,:\, \tau_P)),$ for some $l \in {\msy N}.$
In particular we see that this is true with $l = k.$
Let now $X \in{\mathfrak a}Pq$ be fixed.
Then it suffices to show that the function
(\ref{e: globality of q}), with $X$ substituted for the dot, belongs to
the space
$\cM({\mathfrak a}Qqdc, \cH, d, {\msy C}i({\rm X}Pv\,:\, \tau_P)).$ To prove the latter,
we fix an arbitrary bounded non-empty open set $\omega \subset {\mathfrak a}Qqdc$ and
put $\pi := \pi_{\omega, d},$ see above Lemma
\ref{l: relation specfam and fam second}.
Then the function
$F: \omega \times {\rm X}Pvp \rightarrow V_\tau,$ defined by
$$
F(\lambda, m) = \pi(\lambda)\, q_{\sigma, \xi}(P,v\,|\, f, X, \lambda)(m)
$$
is $C^\infty$ and holomorphic in its first variable.
Moreover, let $\omega_0$ be the full open subset
$\omega \cap \Omega^*\cap\Omega$
of $\omega.$ Then by what we said above,
the restricted function $ F|_{\omega_0 \times {\rm X}Pvp}$
admits a smooth extension to the manifold $\omega_0 \times {\rm X}Pv.$
It now follows from Corollary \ref{c: aux smooth extension} that
$F$ has a unique smooth extension
to $\omega \times {\rm X}Pv;$
this extension is holomorphic in the first variable.
It follows that the function
$\lambda :to \pi(\lambda) q_{\sigma, \xi}(P,v\,|\, f, X, \lambda)$
belongs to ${\cal O}(\omega, {\msy C}i({\rm X}Pv\,:\, \tau_P)).$
Since $\omega$ was arbitrary, this completes the proof.
~
$\square$\medbreak\noindent\medbreak
\begin{prop}
\label{p: transference of s globality}
Let $f \in {\msy C}ephyp,$ let $Q \in \cP_\gs$
and let ${\cal L}$ be a Laurent functional in ${\cal M}({}^*\fa_{Q\iq}dc, \Sigma_Q)^*_{\rm laur}.$
Put $Y = {\rm supp}\, \cL.$
Let $P \in \cP_\gs,$ $ v \in {\msy N}Kaq$ and $\sigma \in W/\!\sim_{P|Q}.$
If $f$ is holomorphically $s$-global along $(P,v)$ for every $s \in W_P\backslash W$
with $[s] = \sigma,$
then ${\cal L}_* f \in {\msy C}ephypQY$ is holomorphically
$\sigma$-global along $(P,v).$
\end{prop}
\par\noindent{\bf Proof:}{\ }{\ }
It follows from Proposition \ref{p: Lau to hyp family} (a)
that ${\cal L}_* f \in {\msy C}ephypQY.$
Assume that $f$ satisfies the globality assumptions. Then
it remains to establish the assertion on $\sigma$-globality for $\cLstar f$.
Let $k = {\rm deg}_{\rm a} f.$
Let $\cH = \cH_f,$ $ d = d_f$ and $\cH' = \cH_Q(Y).$
Moreover, let $d': \cH' \rightarrow {\msy N}$ be associated with these data
as in Lemma \ref{l: Laustar for roots and d prime} and let $k' \in {\msy N}$
be associated as in Proposition \ref{p: Lau to hyp family} (a).
According to the latter proposition, the set
$\Omega' = {\mathfrak a}Qqdc\setminus\cup \cH'$
is contained in ${\rm reg}a(\cLstar f)$.
Let
$\xi \in - \sigma\cdot Y + {\msy N} {\msy D}rP.$ Moreover, let $s \in W_P\backslash W$
be such that $[s] = \sigma$
and let $\lambda_0 \in Y$ be such that $\eta:=s\lambda_0|_{{\mathfrak a}Pq} + \xi $
belongs to ${\msy N} {\msy D}rP.$
Then by Proposition \ref{p: mero H globality newer}, the function
$$
\lambda :to q_{s,\eta}(P,v\,|\, f, \,\cdot\,,\lambda)
$$
belongs to $\cM({\mathfrak a}Qqdc, \cH, d, P_k({\mathfrak a}Pq) \otimes {\msy C}i({\rm X}Pv \,:\, \tau_P)).$
Using Lemma \ref{l: continuity Laustar s new} with $C^\infty({\rm X}Pv\,:\, \tau_P)$
in place of $U,$ we see that, for $X \in {\mathfrak a}Pq,$ the function
$$
\varphi_X := \cL^{P,s}_{\lambda_0 *} [q_{s, \eta}(P,v\,|\, f, X, \,\cdot\,)]
$$
belongs to $\cM({\mathfrak a}Qqdc, \cHpr, d', P_{k'}({\mathfrak a}Pq) \otimes {\msy C}i({\rm X}Pv\,:\, \tau_P)).$
Moreover, it depends on $X \in {\mathfrak a}Pq$ as a polynomial function of degree at most $k.$
It follows that the function $(\nu, X) :to \varphi_X(\nu)(X)$
belongs to the space
\begin{equation}
\label{e: Mer prime pol k kpr}
\cM({\mathfrak a}Qqdc, \cHpr, d', P_{k + k'}({\mathfrak a}Pq) \otimes {\msy C}i({\rm X}Pv\,:\, \tau_P)).
\end{equation}
Each term in the finite sum (\ref{e: sum for q of Laustar f}) is of this form. Hence the function
$$
(\nu, X) :to q_{\sigma, \xi}(P, v\,|\, \cLstar f , X, \nu)
$$
belongs to the space (\ref{e: Mer prime pol k kpr}) as well.
This holds for all $\xi \in - \sigma\cdot Y + {\msy N}{\msy D}rP.$ Therefore the
restriction of $\cLstar f$ to ${\rm reg}a(\cLstar f)$ satisfies Definition
\ref{d: holomorphic s globality new} (a)
with $\Omega^*=\Omega'$.
~
$\square$\medbreak\noindent\medbreak
The following definition is an analogue of the final part of
Definition \ref{d: defi cEhyp Q Y gd},
replacing the globality condition by a condition of
holomorphic globality.
\begin{defi}
\label{d: family for special case of the vanishing thm newer}
Let $Q\in \cP_\gs$ and let $\delta \in {\msy D}Qmaps.$ We define
$$
{\cal E}hypQgdhglob
$$
to be the space of functions $f \in {\cal E}hypQgd$ satisfying the following
condition.
\hbox{\hspace{-12pt}
\vbox{
\begin{enumerate}
\item[]
For each $s \in W$ and every $P \in \cP_\gs^1$ with $s({\mathfrak a}Qq) \not\subset {\mathfrak a}Pq,$
the family $f$ is holomorphically $[s]$-global along $(P,v),$ for all
$v \in {\msy N}Kaq;$
here $[s]$ denotes the image of $s$ in $W/{\sim_{P|Q}}= W_P \backslash W / W_Q.$
\end{enumerate}}}
\noindent
If $Y \subset {}^*\fa_{Q\iq}dc$ is a finite subset, we define
$$
{\cal E}hypQYgdhglob = {\cal E}hypQYgd \cap {\cal E}hypQgdhglob.
$$
\end{defi}
It is easily seen that ${\cal E}hypQgdhglob\subset{\cal E}hypQgd_\lambdaob$.
As in Lemma \ref{l: minimal condition for glob new} the above
condition allows a reduction to a smaller set of $(s,P).$
\begin{lemma}
\label{l: minimal condition for hglob new}
Let $Q\in \cP_\gs$ be standard, let
$\delta \in {\msy D}Qmaps$ and $f \in {\cal E}^{\rm hyp}_Q({\rm X}p\,:\, \tau\,:\, \delta).$
Then $f$ belongs to ${\cal E}^{\rm hyp}_Q({\rm X}p\,:\, \tau \,:\, \delta)_{\rm hglob}$ if and
only if the following condition is fulfilled.
\hbox{\hspace{-12pt}
\vbox{
\begin{enumerate}
\item[]
For each $s \in W$ and every $\alpha \in \Delta$ with $s^{-1} \alpha|_{{\mathfrak a}Qq} \neq 0,$
the family $f$ is holomorphically $[s]$-global along $({}^*Pga,v),$
for all $v \in {\msy N}Kaq.$
\end{enumerate}
}}
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
The proof is similar to the proof of
Lemma \ref{l: minimal condition for glob new},
involving Lemma \ref{l: transformation of holomorphic globality}
instead of Lemma \ref{l: transformation of globality}.
~
$\square$\medbreak\noindent\medbreak
We now come to the main result of this section, which provides
a source of functions to which the vanishing theorem
(Theorem \ref{t: special vanishing theorem}) can be applied.
\begin{thm}
\label{t: source of functions by Lau new}
Let $\delta\in D_{P_0}$ and
$f \in {\cal E}zerohyp({\rm X}p\,:\, \tau\,:\,\delta)_{\rm hglob},$ let $Q \in \cP_\gs$
be a standard
$\sigma$-parabolic subgroup
and let $\cL \in \cM({}^*\fa_{Q\iq}dc, \Sigma_Q)^*_{\rm laur}.$
Put $Y = {\rm supp}\, \cL.$
Then there exists a $\delta' \in {\msy D}Qmaps$ such that
$$
\cLstar f \in {\cal E}^{\rm hyp}_{Q,Y}({\rm X}p\,:\, \tau \,:\, \delta')_{\rm hglob}.
$$
\end{thm}
\par\noindent{\bf Proof:}{\ }{\ }
{}From Lemma \ref{l: Lau to family of eigenfunctions new}
it follows that ${\cal L}_*f$ is
a family in ${\cal E}^{\rm hyp}_{Q,Y}({\rm X}p \,:\, \tau\,:\, \deltamap')$ for some
$\deltamap' \in {\msy D}Qmaps.$
Let $s \in W$ and $\alpha \in \Delta$ be such that
$s^{-1}\alpha |_{{\mathfrak a}Qq} \neq 0.$
Then every $t\in W_\alpha s W_Q$ also satisfies the condition
$t^{-1}\alpha |_{{\mathfrak a}Qq} \neq 0;$ hence $t({\mathfrak a}Qq) \not\subset {\mathfrak a}_{\alpha{\rm q}}.$
Thus, from the hypothesis it follows that
$f$ is holomorphically $W_\alpha t$-global along $(P_\alpha,v)$
for every $t$ in the double coset $W_\alpha s W_Q.$
According to Lemma \ref{l: WPQ as cosets}, see also Remark \ref{r: W Pga Q new},
the latter set equals the class $[s]$ of $s$ for
the equivalence relation $\sim_{{{}^*Pga}|Q}$ in $W.$
It now follows from Proposition
\ref{p: transference of s globality}
that ${\cal L}_*f$ is holomorphically $[s]$-global
along $({{}^*Pga},v).$
We conclude that ${\cal L}_* f$ satisfies the conditions
of Lemma \ref{l: minimal condition for hglob new},
hence belongs
to ${\cal E}^{\rm hyp}_{Q,Y}({\rm X}p \,:\, \tau \,:\,\deltamap')_{\rm hglob}.$
~
$\square$\medbreak\noindent\medbreak
\section{Partial Eisenstein integrals}
\label{s: partial Eisenstein integrals}
In this section we will define partial Eisenstein integrals and
show that they belong to the families of eigenfunctions
introduced in the previous section.
We start by recalling some properties of Eisenstein integrals.
Let $P\in \cP_\gs^{\rm min}$ be a minimal $\sigma$-parabolic subgroup.
Let $(\tau, V_\tau)$ be a finite dimensional unitary representation of $K.$
Let ${\cal W} \subset {\msy N}Kaq$ be a fixed set of representatives
for $W/W_{K \cap H}.$ Following
\bib{BSmc}, eq.\ (5.1), we define the
complex linear space ${}^\circ {\cal C}= {}^\circ\cC(\tau)$ as the following formal direct sum
of finite dimensional linear spaces
\begin{equation}
\label{e: defi oCtau}
{}^\circ {\cal C}:= \oplus_{w \in {\cal W}}\;\; {\msy C}i({\rm X}_{0,w}\,:\, \tau_{\iM}).
\end{equation}
Every summand in the above sum, as $w \in {\cal W},$ is a finite dimensional subspace
of the Hilbert space $L^2({\rm X}_{0,w}, V_\tau);$ here the $L^2$-inner product
is defined relative to the normalized $M$-invariant measure of the compact
space ${\rm X}_{0,w} = M/ M \cap wHw^{-1}$ and the Hilbert structure of $V_\tau.$
Thus, every summand is a finite dimensional Hilbert space of its own right.
The formal direct sum ${}^\circ {\cal C}$ is equipped with the direct sum inner product,
turning (\ref{e: defi oCtau}) into an orthogonal direct sum.
For $\psi \in {}^\circ {\cal C},$ $\lambda \in {\mathfrak a}qdc$ and $x \in {\rm X},$
the Eisenstein integral $E(\psi\,:\,\lambda \,:\, x) = E(P \,:\, \psi \,:\, \lambda \,:\, x)$
and its normalized version
${E^\circ}(\psi\,:\, \lambda \,:\, x)= {E^\circ}(P\,:\, \psi\,:\, \lambda \,:\, x)$ are defined as in
\bib{BSmc}, \S{} 5. The Eisenstein integrals are $\tau$-spherical functions
of $x,$ depend meromorphically on $\lambda$ and linearly on $\psi.$ We
view ${E^\circ}(\lambda \,:\, x) := {E^\circ}(\,\cdot\, \,:\, \lambda \,:\, x)$
(and similarly its unnormalized version)
as an element
of ${\rm Hom}({}^\circ {\cal C}, V_\tau)\simeq V_\tau \otimes {}^\circ {\cal C}^*.$ Thus, for generic $\lambda \in {\mathfrak a}qdc,$
${E^\circ}(\lambda)$ is a $\tau \otimes 1$-spherical function on ${\rm X}.$ The connection between
the unnormalized and the normalized Eisenstein integral is now given by the identity
\begin{equation}
\label{e: relation unnormalized and normalized Eis}
{E^\circ}(\lambda\,:\, x ) = E(\lambda\,:\, x ) \,{\scriptstyle\circ}\, C(1 \,:\, \lambda)^{-1},\quad\quad (x \in{\rm X}),
\end{equation}
for generic $\lambda \in {\mathfrak a}qdc.$
Here $C(1 \,:\, \lambda) := C_{P|P}(1 \,:\, \lambda)$ is a meromorphic ${\rm End}({}^\circ {\cal C})$-valued
function of $\lambda \in {\mathfrak a}qdc;$ see \bib{BSmc}, p.\ 283.
The Eisenstein integral is ${\msy D}GH$-finite. In fact, we recall from \bib{BSmc}, eq.\ (5.11),
that there exists a homomorphism $\mu$ from ${\msy D}GH$ to the algebra of
${\rm End}({}^\circ {\cal C})$-valued polynomial functions on ${\mathfrak a}qdc$ such that
$$
D {E^\circ}(\lambda) = [I \otimes \mu(D\,:\, \lambda)^*] {E^\circ}(\lambda), \quad\quad (D \in {\msy D}GH).
$$
It now follows from Lemma \ref{l: DX finite in exppol} that,
for generic $\lambda \in {\mathfrak a}qdc,$ the Eisenstein integral ${E^\circ}(\lambda)$
belongs to $C^{\rm ep}({\rm X}p \,:\, \tau \otimes 1).$ It therefore has expansions
of the form (\ref{e: expansion f on PqPw with m}).
These expansions have been determined
explicitly in \bib{BSexp}. We
recall some of the results of that
paper.
In \bib{BSexp}, eq.\ (15), we define a function
$\Phi_{P}(\lambda \,:\, \,\cdot\,)$ on $A_\iq^+(P)$ by an exponential polynomial
series with coefficients in ${\rm End}(V_\tau_{\iM}KH)$ of the form
\begin{equation}
\label{e: expansion Phi gl}
\Phi_{P}(\lambda \,:\, a) =
a^{\lambda - \rho_P} \sum_{\nu \in {\msy D}elta(P)} a^{-\nu} \Gamma_{{P},\nu}(\lambda),\quad\quad
(a \in A_\iq^+(P)).
\end{equation}
Note that here ${P}$ replaces the $Q$ of \bib{BSexp}, Sect. 5; also, in
\bib{BSexp} we suppressed the $Q$ in the notation. The coefficients
in the expansion (\ref{e: expansion Phi gl})
are defined by recursive relations (see \bib{BSexp}, eq.\ (18)
and Prop.\ 5.2); it follows from these that the coefficients
depend meromorphically on $\lambda,$ and that the expansion (\ref{e: expansion Phi gl})
converges to a smooth function on $A_\iq^+(P),$
depending meromorphically on $\lambda.$ In fact, we have the following stronger result.
Let $\Pi_{\Sigma, \R}$ be the collection of polynomial
functions ${\mathfrak a}qdc \rightarrow {\msy C}$ that can be written as finite
products of linear factors of the form $\lambda :to \inp{\lambda}{\alpha} - c,$ with
$\alpha \in \Sigma$ and $c \in {\msy R}.$
For $R \in {\msy R},$ we define the set
$$
{\mathfrak a}qd({P},R):= \{ \lambda \in {\mathfrak a}qdc \mid {\msy R}e\inp{\lambda}{\alpha} < R \;\; \forall\alpha \in \Sigma(P)\}.
$$
\begin{lemma}
\label{l: neat convergence of series Phi}
Let $R \in {\msy R}.$ Then there exists a polynomial function $p \in \Pi_{\Sigma, \R}$ such that
the functions $ p \Gamma_{{P},\nu},$ for $\nu \in {\msy N}{\msy D}elta(P),$
are all regular on ${\mathfrak a}q({P},R).$
Moreover, if $p$ is a polynomial function with the above property, then
the series
\begin{equation}
\label{e: series with Gamma}
\sum_{\nu \in {\msy N}{\msy D}elta(P)} a^{-\nu} p(\,\cdot\,) \Gamma_{{P},\nu}(\,\cdot\,)
\end{equation}
converges neatly on $A_\iq^+(P)$ as an exponential series with coefficients
in ${\cal O}({\mathfrak a}qd({P},R)) \otimes {\rm End}(V_\tau_{\iM}KH).$ In particular,
the function $(a,\lambda) :to p(\lambda)\Phi_{P}(\lambda \,:\, a)$ is smooth
on $A_\iq^+(P) \times {\mathfrak a}qd({P},R),$ and in addition holomorphic in its
second variable.
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
Let $p_R$ be the polynomial function
described in \bib{BSexp}, Thm.\ 9.1.
As in the proof of that theorem, it follows from the
estimates in \bib{BSexp}, Thm.\ 7.4, that the power series
$$
{P,s}i(\lambda\,:\, z) = \sum_{\nu \in {\msy N} \Delta(P)} z^{-\nu} p_R(\lambda)\Gamma_{{P},\nu}(\lambda)
$$
converges absolutely locally uniformly in the variables $z \in D^{{\msy D}elta(P)}$ and
$\lambda \in {\mathfrak a}qd({P},R).$ Here we have used the notation of Sect.\ 1 of the present paper.
Since $p_R(\lambda) \Phi_{P}(\lambda\,:\, a) = a^{\lambda - \rho_P} {P,s}i(\lambda\,:\, {\underline z}(a)),$
for $a \in A_\iq^+(P),$
this implies all assertions of the lemma with $p_R$ in place of $p.$
This is not immediately good enough, since $p_R$ is a finite product of
linear factors of the form $\lambda :to \inp{\lambda}{\nu} -c,$ with
$\nu \in {\msy N}\Delta(P)$ and $c \in {\msy R},$ see \bib{BSexp}, the equation preceding Lemma 7.3.
To overcome this, we invoke
\bib{BSexp}, Prop.\ 9.4. It follows from that result
and its proof that there exists a $p \in \Pi_{\Sigma, \R}$ such that
$p\Gamma_{P, \nu}$ is regular on ${\mathfrak a}qd({P},R),$
for every $\nu \in {\msy N} {{\msy D}elta(P)}.$
Let $p$ be any polynomial with this property, and let ${}^{\scriptscriptstyle\backprime} p $ be the least common
multiple of $p$ and $p_R.$ Then all assertions of the lemma hold with ${}^{\scriptscriptstyle\backprime} p$ in
place of $p.$ Let $q$ be the quotient of ${}^{\scriptscriptstyle\backprime} p$ by $p.$
Denote the image of the linear endomorphism $m_q: \varphi :to q \varphi$ of ${\cal O}({\mathfrak a}qd({P},R))$
by ${\cal F},$ and equip this space with the locally convex topology
induced from $ {\cal O}({\mathfrak a}qd({P},R)).$ It follows from an easy application
of the Cauchy integral formula that $m_q$ is a topological linear isomorphism
from ${\cal O}({\mathfrak a}qd({P}, R))$ onto ${\cal F};$ see also \bib{BSmc}, Lemma 20.7.
As said above, all assertions of the lemma hold with ${}^{\scriptscriptstyle\backprime} p$ in place of $p;$
on the other hand, by the hypothesis the series
(\ref{e: series with Gamma}) with ${}^{\scriptscriptstyle\backprime} p$ in place of $p$ has coefficients in ${\cal F}.$
Applying the continuous linear map $m_q^{-1}$ to that series,
we infer that all assertions of the lemma are true
with the polynomial $q^{-1} {}^{\scriptscriptstyle\backprime} p = p.$
~
$\square$\medbreak\noindent\medbreak
Following
\bib{BSexp}, Sect. 11, we define the function
$\Phi_{{P},w}: {\mathfrak a}qdc\times A_\iq^+(P) \rightarrow {\rm End}(V_\tau_{\iM}KwH),$ for $w \in {\cal W},$ by
\begin{equation}
\label{e: defi Phi P w}
\Phi_{{P},w}(\lambda\,:\, a) = \tau(w) \,{\scriptstyle\circ}\, \Phi_{w^{-1}{P} w}(w^{-1} \lambda
\,:\, w^{-1} aw) \,{\scriptstyle\circ}\, \tau(w)^{-1}.
\end{equation}
Following \bib{BSmc}, p.\ 283, we define normalized $C$-functions
$C^\circ(s \,:\, \lambda) = C^\circ_{{P}|{P}}(s\,:\, \lambda),$ for $s \in W,$ by
\begin{equation}
\label{e: defi normalized c function}
C^\circ(s\,:\, \lambda)= C(s\,:\, \lambda) \,{\scriptstyle\circ}\, C(1 \,:\, \lambda)^{-1};
\end{equation}
these are ${\rm End}({}^\circ {\cal C})$-valued meromorphic functions of $\lambda \in {\mathfrak a}qdc.$
{}From (\ref{e: relation unnormalized and normalized Eis}) and \bib{BSexp}, eq.\ (54),
we now obtain the following description of the normalized Eisenstein integral
in terms of the functions $\Phi_{{P},w}.$
Let $\psi \in {}^\circ {\cal C}$ and $w \in {\cal W}.$ Then, for $a \in A_\iq^+(P),$
\begin{equation}
\label{e: deco nE in Phi P w}
{E^\circ}(\lambda \,:\, aw)\psi = \sum_{s \in W} \Phi_{{P},w}(s \lambda \,:\, a)
[C^\circ(s\,:\, \lambda)\psi]_w(e),
\end{equation}
as a meromorphic identity in $\lambda \in {\mathfrak a}qd.$
{}From
(\ref{e: defi Phi P w}) and (\ref{e: expansion Phi gl}) it follows that, for $w \in {\cal W},$ the function
$\Phi_{{P},w}$
is given by the series
\begin{equation}
\label{e: series for Phi P w}
\Phi_{{P},w}(\lambda\,:\, a) = a^{\lambda - \rho_P} \sum_{\nu \in {\msy N}{{\msy D}elta(P)}} a^{-\nu} \Gamma_{{P},w,\nu}(\lambda),
\end{equation}
with coefficients
\begin{equation}
\label{e: defi of Gamma P w}
\Gamma_{{P},w,\nu}(\lambda) =
\tau(w) \,{\scriptstyle\circ}\, \Gamma_{w^{-1}{P} w, w^{-1}\nu}(w^{-1} \lambda)\,{\scriptstyle\circ}\, \tau(w)^{-1}.
\end{equation}
We now have the following result on the convergence of the series
(\ref{e: series for Phi P w}).
\begin{cor}
\label{c: neat convergence of Phi P w}
Let $w \in {\cal W}.$ Then there exists a locally finite real $\Sigma$-hyperplane configuration
$\cH = \cH_w$ in ${\mathfrak a}qdc$ and a map $d = d_w: \cH \rightarrow {\msy N},$ such that
the functions $\Gamma_{{P},w,\nu}$ belong to $\cM({\mathfrak a}qdc, \cH, d, {\rm End}(V_\tau_{\iM}KwH)),$
for every $\nu \in {\msy N}\Delta(P).$ Moreover, the series
\begin{equation}
\label{e: series with Gamma P w}
\sum_{\nu \in {\msy N} \Delta(P)} a^{-\nu} \Gamma_{{P},w,\nu}
\end{equation}
converges neatly on $A_\iq^+(P)$ as an exponential polynomial series
with coefficients in the space $\cM({\mathfrak a}qdc,\cH, d, {\rm End}(V_\tau_{\iM}KwH)).$ In particular,
the function $\lambda :to \Phi_{{P},w}(\lambda\,:\, \,\cdot\,)$ belongs to the space
$\cM({\mathfrak a}qdc,\cH, d, {\msy C}i(A_\iq^+(P)) \otimes {\rm End}(V_\tau_{\iM}KwH)).$
\end{cor}
\par\noindent{\bf Proof:}{\ }{\ }
For $w =1$ the assertion of the corollary follows immediately
from Lemma \ref{l: neat convergence of series Phi}.
For arbitrary $w \in {\cal W}$ it then follows by application of
(\ref{e: defi of Gamma P w}).
~
$\square$\medbreak\noindent\medbreak
For $s \in W$ we define the so called partial Eisenstein integral
$E_{+,s}( \lambda)= E_{+,s}({P}\,:\,\lambda)$ as the $\tau \otimes 1$-spherical
function ${\rm X}p \rightarrow V_\tau \otimes {}^\circ {\cal C}^*$ determined by
\begin{equation}
\label{e: defi partial Eis}
E_{+,s}(\lambda\,:\, a w) \psi = \Phi_{{P},w}(s\lambda \,:\, a) [C^\circ(s\,:\, \lambda)\psi]_w(e),
\end{equation}
for $\psi \in {}^\circ {\cal C},\; w \in {\cal W},\;a \in A_\iq^+(P)$ and generic $\lambda \in {\mathfrak a}qdc$
(use the isomorphism (\ref{e: the iso T down P cW})).
It follows from Corollary \ref{c: neat convergence of Phi P w}
that $E_{+,s}$ is a meromorphic $C^\infty({\rm X}p\,:\,\tau\otimes 1)$-valued
function on ${\mathfrak a}qdc$.
By sphericality it follows from
(\ref{e: deco nE in Phi P w}) and (\ref{e: defi partial Eis}) that
\begin{equation}
\label{e: splitting of Eis}
{E^\circ}( \lambda) = \sum_{s \in W} E_{+,s}( \lambda)\quad\text{on}\quad {\rm X}p.
\end{equation}
It follows from the definitions and the isomorphism (\ref{e: isomorphism of exppol})
that, for generic $\lambda \in {\mathfrak a}qdc,$ the function
$E_{+,s}( \lambda)\psi$ belongs to $C^{\rm ep}({\rm X}p \,:\, \tau \otimes 1)$
for each $\psi\in {}^\circ {\cal C}$. Moreover,
\begin{equation}
\label{e: exponents Eps}
{\rm Exp}(P,v\,|\, E_{+,s}(\lambda)\psi)\subset s\lambda - \rho_P -{\msy N} \Delta(P),
\end{equation}
for every $v \in {\cal W}$ and hence also for every $v \in {\msy N}Kaq.$
Thus, we see that (\ref{e: splitting of Eis}) is
the splitting of Lemma \ref{l: splitting lemma} applied to
the Eisenstein integral.
We abbreviate $E_+(\lambda) = E_{+,1}( \lambda).$ Then from (\ref{e: defi partial Eis}) and (\ref{e: defi normalized c function}) we see that
$$
E_+(\lambda)(aw)\psi = \Phi_{{P},w}(\lambda\,:\, a)\psi_w(e),
$$
for $\psi \in {}^\circ {\cal C},$ $w \in {\cal W},$ $a \in A_\iq^+(P)$ and generic $\lambda \in {\mathfrak a}qdc.$
Moreover, the following holds as a meromorphic identity in $\lambda \in {\mathfrak a}qdc$
\begin{equation}
\label{e: Esp in terms of Ep and C}
E_{+,s} (\lambda\,:\, x) = E_+(s\lambda\,:\, x) \,C^\circ(s\,:\, \lambda).
\end{equation}
In the next lemma we will need the following notation.
If $\Lambda \in {\mathfrak b}kdc,$ we denote by ${}^\circ {\cal C}[\Lambda]$ the subspace of ${}^\circ {\cal C}$
consisting of elements $\psi$
satisfying $\mu(D\,:\, \lambda)\psi = \gamma(D\,:\, \Lambda + \lambda)\psi$ for all $D \in {\msy D}GH,$
$\lambda \in {\mathfrak a}qdc.$
We recall from \bib{BSmc}, eq.\ (5.14), that ${}^\circ {\cal C}$ is a finite direct sum
$$
{}^\circ {\cal C} = \oplus_\Lambda\; {}^\circ {\cal C}[\Lambda],
$$
where $\Lambda$ ranges over a finite
subset $L_\tau$ of ${\mathfrak b}kdc.$
For each $\Lambda\in{\mathfrak b}kdc,$ we denote by $\cEhypgL$
the space ${\cal E}^{\rm hyp}_{0}({\rm X}p \,:\, \tau \,:\,\deltamap)$
(see Definition \ref{d: defi cEhyp Q Y gd})
where $\delta\in D_P$ is the characteristic function of $\{\Lambda\}$.
\begin{lemma}
\label{l: f sub s belongs to specfam}
Let $P \in \cP_\gs^{\rm min},$ $t \in W$ and $\psi \in {}^\circ {\cal C}[\Lambda],$
where $\Lambda\in {\mathfrak b}kdc$. Define the family
$f=f_{\{t\}}:{\mathfrak a}qdc \times {\rm X}p \rightarrow V_\tau,$
by
$$
f(\lambda, x) = E_+t(P\,:\,\lambda\,:\, x)\psi.
$$
Then $f\in{\cal E}^{\rm hyp}_{0}({\rm X}p \,:\, \tau \,:\,\Lambda)$
and ${\rm deg}_{\rm a} f = 0$.
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ } According to Definition \ref{d: defi cEhyp Q Y gd} and
Remark \ref{r: only local annihilation needed},
in order to prove that $f\in{\cal E}^{\rm hyp}_{0}({\rm X}p \,:\, \tau \,:\,\Lambda)$
we must establish that $f\in C^{{\rm ep},{\rm hyp}}_0({\rm X}p\,:\,\tau)$ and that
$f_\lambda$ is annihilated by $I_{\Lambda+\lambda}$ for $\lambda$ in a non-empty
open subset of ${\rm reg}a f$.
We first assume that $t=1.$
Then $f(\lambda,x) = E_{+}(\lambda\,:\, x)\psi.$
It follows immediately from \bib{BSexp} Cor.\ 9.3 and the hypothesis on
$\psi$ that $f_\lambda$ is annihilated by $I_{\Lambda+\lambda}$ for generic $\lambda\in{\mathfrak a}qdc$.
We will now show that
$f \in C^{{\rm ep},{\rm hyp}}_0({\rm X}p\,:\, \tau).$
Let
$\cH$ be the union of the hyperplane configurations
$\cH_w,\, w \in {\cal W},$ of
Corollary \ref{c: neat convergence of Phi P w},
and let $d: \cH \rightarrow {\msy N}$ be defined by
$
d = \max_{w \in {\cal W}} \,d_w
$
(see Remark \ref{r: convention about d}).
Then for every complete locally convex space $U,$ the spaces
$\cM({\mathfrak a}qdc, \cH_w, d_w, U)$ are included in the space $\cM({\mathfrak a}qdc, \cH, d, U),$
with continuous inclusion maps.
Hence for each $w\in{\cal W}$
the series (\ref{e: series with Gamma P w}) converges as
a $\Delta(P)$-exponential polynomial series on $A_\iq^+(P),$ with coefficients in
the space
$\cM({\mathfrak a}qdc, \cH, d, {\rm End}(V_\tau^{K_\iM \cap wHw^{-1}})).$
Moreover,
the function $\lambda :to \Phi_{{P},w}(\lambda \,:\, \,\cdot\,)$
is contained in $\cM({\mathfrak a}qdc,\cH, d, {\msy C}i(A_\iq^+(P)) \otimes {\rm End}(V_\tau_{\iM}KwH))$.
On the other hand, from (\ref{e: defi partial Eis})
and (\ref{e: defi normalized c function}) with $s =1,$
it follows that
\begin{equation}
\label{e: TdownPw f as Phi}
T_P^\downarroww (f_\lambda)(a) = f(\lambda, aw) = \Phi_{P,w}(\lambda \,:\, a) \psi_w(e),
\end{equation}
for all $w \in {\cal W},$ $a \in A_\iq^+(P)$ and $\lambda \in {\mathfrak a}qdc\setminus \cup \cH.$
Hence the function $\lambda :to T_P^\downarroww(f_\lambda)$ belongs
to the space $\cM({\mathfrak a}qdc, \cH, d, {\msy C}i(A_\iq^+(P),V_\tau_{\iM}KwH)).$
In view of the isomorphism (\ref{e: the iso T down P cW}), it now follows
that
the function $\lambda :to f_\lambda$ belongs
to $\cM({\mathfrak a}qdc, \cH,d, {\msy C}i({\rm X}p\,:\, \tau)).$
This establishes condition (a) of
Definition \ref{d: Cephyp}, with $Q = P_0$ and $Y = \{0\}.$
The evaluation map $\psi :to \psi(e)$
is a linear isomorphism
from ${\msy C}i({\rm X}zerow\,:\, \tau_{\iM})$ onto $V_\tau_{\iM}KwH.$ Thus, for $w \in {\cal W}$
and $\nu \in {\msy N} {\msy D}P$ we may define a
function $\tilde q_{1,\nu}(P,w\,|\, f) :
{\mathfrak a}qdc \rightarrow {\msy C}i({\rm X}zerow\,:\, \tau_{\iM})$
by
\begin{equation}
\label{e: tilde q one as Gamma}
\tilde q_{1,\nu}(P,w\,|\, f , \lambda, e) = \Gamma_{P,w,\nu}(\lambda)\psi_w(e),
\end{equation}
for $\lambda \in {\mathfrak a}qdc.$
Then $\tilde q_{1,\nu}(P,w \,|\, f)
\in \cM({\mathfrak a}qd, \cH, d, {\msy C}i({\rm X}zerow\,:\, \tau_{\iM})).$
Moreover, from what we said earlier about the convergence
of the series (\ref{e: series with Gamma P w}), it follows that,
for $w \in {\cal W},$ the series
$$
\sum_{\nu \in {\msy N}{\msy D}P} a^{-\nu} \tilde q_{1, \nu}(P, w\,|\, f)
$$
converges neatly as a ${\msy D}P$-exponential polynomial series on $A_\iq^+(P),$
with coefficients in $\cM({\mathfrak a}qdc, \cH, d, {\msy C}i({\rm X}zerow\,:\, \tau_{\iM})).$
{}From (\ref{e: TdownPw f as Phi}), (\ref{e: series for Phi P w})
and (\ref{e: tilde q one as Gamma})
it follows by sphericality that, for
$w \in {\cal W},$ $\lambda \in {\mathfrak a}qdc \setminus \cup \cH,$ $m \in {\rm X}zerow$
and $a \in A_\iq^+(P),$
$$
f_\lambda(maw) = a^{\lambda -\rho_P} \sum_{\nu \in {\msy N} \Delta(P)}
a^{-\nu} \tilde q_{1, \nu} (P,w\,|\, f)(\lambda, m),
$$
This establishes assertions
(b) and (c) of Definition \ref{d: Cephyp} with a fixed $P,$ arbitrary
$v\in {\cal W},$ and, for $\nu \in {\msy N}\Delta(P),$ $X\in{\mathfrak a}q$,
$$
q_{s, \nu}(P,v \,|\, f,X) = \left\{
\begin{array}{lcl}
\tilde q_{1, \nu}(P, v \,|\, f)& \text{for} &s=1;\\
0& \text{for} &s \in W\setminus \{1\}.
\end{array}
\right.
$$
In view of Remark \ref{r: second on Cephyp} we have shown that
$f \in C^{{\rm ep},{\rm hyp}}_0({\rm X}p\,:\, \tau).$ Moreover,
${\rm deg}_{\rm a} f=0$.
This completes the proof for $t =1.$
Let now $t \in W$ be arbitrary and
let $\tilde t \in W_0({\mathfrak b})$ be such that $\tilde t |_{{\mathfrak a}q} = t;$ see
the text preceding
Lemma \ref{l: restriction on leading exponents}.
{}From (\ref{e: Esp in terms of Ep and C}) we see that
\begin{equation}
\label{e: expression f t in Ep and nC}
f(\lambda, x) = E_+(t\lambda\,:\, x) C^\circ(t\,:\, \lambda) \psi.
\end{equation}
It follows from \bib{BSmc}, Lemma 20.6, that there exists
a $\Sigma$-configuration $\cH'$ in ${\mathfrak a}qdc$ and a map $d': \cH' \rightarrow {\msy N},$
such that
\begin{equation}
\label{e: C-function in MerSigma}
C^\circ(t\,:\, \,\cdot\,)\in\cM({\mathfrak a}qdc, \cH', d', {\rm End}({}^\circ {\cal C})).
\end{equation}
{}From \bib{BSmc}, eq.\ (5.13),
it follows that $C^\circ(t \,:\, \lambda)$
maps ${}^\circ {\cal C}[\Lambda]$ into ${}^\circ {\cal C}[\tilde t \Lambda].$
Fix a basis $\psi_1,\ldots, \psi_s$ for ${}^\circ {\cal C}[\tilde t \Lambda].$
Then there exist unique functions $c_j \in \cM({\mathfrak a}qdc, \cH', d')$
such that
\begin{equation}
\label{e: nC t gl in components}
C^\circ(t\,:\, \lambda) \psi = \sum_{j=1}^r c_j(\lambda) \psi_j.
\end{equation}
For $1\leq j \leq r$ we define the family $g_j: {\mathfrak a}qdc \times {\rm X}p \rightarrow V_\tau$ by
\begin{equation}
\label{e: defi g j}
g_j(\lambda, x) = E_+(\lambda \,:\, x) \psi_j.
\end{equation}
Then by the first part of the proof, each $g_j$ belongs to
${\cal E}starzero({\rm X}p\,:\,\tau\,:\,\tilde t \Lambda).$
Moreover, for every $1 \leq j \leq r,$
the family $g_j$ satisfies the conditions
of Definition \ref{d: Cephyp} with $Q = P_0$ and $Y = \{0\},$ with
$\cH$ and $d$ as in the first part of the proof,
and with $k =0.$
For $1 \leq j \leq r$ we define the family $f_j: {\mathfrak a}qdc \times {\rm X}p \rightarrow V_\tau$ by
$f_j(\lambda, x) = g_j(t\lambda, x).$ Then we readily see
that $f_j$ satisfies the conditions of Definition \ref{d: Cephyp} with
$t^{-1}\cH$ and $d\,{\scriptstyle\circ}\, t$ in place of $\cH$ and $d,$ respectively,
and with $k =0.$ Hence $f_j \in C^{{\rm ep},{\rm hyp}}_0({\rm X}p\,:\, \tau).$
Since $I_{\tilde t \Lambda + t\lambda} = I_{ \Lambda+ \lambda}$ we see that
$f_j \in {\cal E}starzero({\rm X}p\,:\, \tau\,:\, \Lambda)$. Moreover,
${\rm deg}_{\rm a} f_j=0$.
Combining (\ref{e: expression f t in Ep and nC})
and (\ref{e: nC t gl in components}) with
(\ref{e: defi g j}) and the definition of $f_j,$
we find that
$$
f(\lambda, x)
= \sum_{j=1}^r c_j(\lambda) f_j(\lambda, x).
$$
Let $\cH'' = t^{-1}\cH \cup \cH'$
and define $d'': \cH'' \rightarrow {\msy N}$
by $d''(H) = d(tH) + d'(H)$ (see Remark \ref{r: convention about d}).
Then by linearity it readily follows that $f$ satisfies all
conditions of Definition \ref{d: Cephyp}, with $k=0$ and with
$\cH''$ and $d''$ in place of $\cH$ and $d,$
respectively. Hence $f \in C^{{\rm ep}, {\rm hyp}}_0({\rm X}p\,:\, \tau)$
and ${\rm deg}_{\rm a} f = 0.$
Moreover, for generic $\lambda,$
$f_\lambda$ is annihilated by $I_{\Lambda +\lambda},$
and hence $f\in{\cal E}starzero({\rm X}p\,:\, \tau\,:\, \Lambda)$. ~
$\square$\medbreak\noindent\medbreak
\begin{cor}
\label{c: Lau to partial Eis}
Let assumptions be as in Lemma \ref{l: f sub s belongs to specfam}
and let $Q$ be a $\sigma$-parabolic subgroup.
Let $\cL \in \cM({}^*\fa_{Q\iq}dc, \Sigma_Q)^*_{\rm laur}$.
Then $\cLstar f \in {\cal E}^{\rm hyp}_{Q,Y}({\rm X}p\,:\, \tau \,:\, \delta)$
for $Y = {\rm supp}\, \cL$ and $\delta$ a suitable element in ${\msy D}Qmaps$.
Moreover, $${\rm Exp}(P,v\,|\, (\cLstar f)_\nu)\subset t(\nu+Y)-\rho_P-{\msy N}\Delta(P)$$
for $v\in {\msy N}Kaq$ and $\nu\in{\rm reg}a \cLstar f$.
\end{cor}
\par\noindent{\bf Proof:}{\ }{\ } This follows immediately from Lemmas
\ref{l: f sub s belongs to specfam} and
\ref{l: Lau to family of eigenfunctions new},
and from (\ref{e: exponents Eps})
combined with the final statement in Proposition
\ref{p: Lau to hyp family} (b).
~
$\square$\medbreak\noindent\medbreak
\begin{lemma}
\label{l: globality for the full Eisenstein}
Let $\psi \in {}^\circ {\cal C}[\Lambda]$ where $\Lambda\in {\mathfrak b}kdc$.
Then the family $f:{\mathfrak a}qdc \times {\rm X}p \rightarrow V_\tau,$ defined by
$$
f(\lambda, x) = {E^\circ}({}^*Po\,:\, \lambda \,:\, x) \psi
$$
belongs to $\cEhypgL.$ Moreover, ${\rm deg}_{\rm a} f=0$ and
for all $P \in \cP_\gs, v \in {\msy N}Kaq$
and every
$s \in W_P\backslash W,$ the family $f$ is holomorphically $s$-global along $(P,v).$
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
The function $f$ equals the sum, for $t \in W,$ of the functions $f_{\{t\}}$ defined
in Lemma \ref{l: f sub s belongs to specfam}, with ${}^*Po$ in place of $P.$
Hence $f \in \cEhypgL$ and ${\rm deg}_{\rm a} f = 0.$
Moreover, for each $\lambda\in{\rm reg}a f$, the function
$f_\lambda$ is asymptotically global along all pairs $(P,v)$
by Proposition \ref{p: global eigen implies as global}.
Thus, it remains to prove the assertion on holomorphic globality.
In view of Lemma \ref{l: transformation of holomorphic globality},
it suffices to do this for arbitrary
$P \in \cP_\gs$ and the special value $v = e.$
In the rest of this proof we shall use notation of the paper \bib{BSft}.
According to \bib{BSft}, Lemma 14, there exists a locally finite collection $\cH$ of
$\Sigma$-hyperplanes such that $\lambda :to f_\lambda$ is holomorphic
on $\Omega_0: = {\mathfrak a}qdc\setminus \cup\cH,$
with values in ${\msy C}i({\rm X}\,:\, \tau).$
According to the same mentioned lemma it follows that
$f \in {\cal E}_*(G/H, V_\tau, \Omega_0).$
According to \bib{BSft}, p.\ 562, Cor.\ 1, for generic $\lambda \in {\mathfrak a}qdc$
the function $f_\lambda$ has an asymptotic expansion of the form
\begin{equation}
\label{e: expansion Eis in terms of p}
f_\lambda(x\exp tX) \sim \sum_{s \in W_P\backslash W\atop \nu \in {\msy N}{\msy D}rP}
p_{P,\nu}(f_\lambda\,:\, s\,:\, \lambda)(x)\, e^{(s \lambda -\rho_P - \nu)(tX)} \;\; (t \rightarrow \infty)
\end{equation}
for $X \in {\mathfrak a}Pq$ at every $X_0\in {\mathfrak a}_{P{\rm q}}^+.$
Proposition 10 of \bib{BSft} is valid with ${\cal E}_*(G/H, V_\tau, \Omega_0)$
in place of ${\cal E}_*(G/H, \Lambda, \Omega_0),$ by the remarks in the beginning of
\bib{BSft}, Sect.\ 12.
In particular, there exists a full open subset
${}^{\scriptscriptstyle\backprime}{}^{\scriptscriptstyle\backprime}\faqdc$ of ${\mathfrak a}qdc$ such that, for all $s \in W_P\backslash W$ and
$\nu \in {\msy N}{\msy D}rP,$ the coefficient
$p_{P,\nu}(f_\lambda\,:\, s\,:\, \lambda)$ is holomorphic as a $C^\infty(G,V_\tau)$-valued
function of $\lambda$ on the full open set $\Omega_0 \cap {}^{\scriptscriptstyle\backprime}{}^{\scriptscriptstyle\backprime}\faqdc.$
On the other hand, since $f \in {\cal E}^{\rm hyp}_0({\rm X}p\,:\, \tau\,:\,\Lambda),$ and
${\rm deg}_{\rm a} f = 0,$
the expansion (\ref{e: expansion Cephyp})
holds, with $k = 0$ and $Y = \{0\},$
for all $\lambda \in \Omega:= {\rm reg}a f.$
Thus, if $\lambda \in \Omega \cap {\mathfrak a}_{{\rm q}{\scriptscriptstyle \C}}^{*0}(P, \{0\})$ is generic,
then it follows from comparing the expansions (\ref{e: expansion Eis in terms of p})
and (\ref{e: expansion Cephyp}), and
using Lemma \ref{l: exponents disjoint} and uniqueness
of asymptotics (see the proof of
Lemma \ref{l: uniqueness of asymp}), that
\begin{equation}
\label{e: equality of q and p of Eis}
q_{s,\nu}(P,e \,|\, f, X, \lambda)(m) = p_{P,\nu}(f_\lambda\,:\, s\,:\, \lambda)(m),
\end{equation}
for all $s \in W_P\backslash W,$ $\nu \in {\msy N} {\msy D}rP,$
$X \in {\mathfrak a}Pq$ and $m \in M_{P,+};$ here we have
written $M_{P,+}$ for the preimage of ${\rm X}_{P,e,+}$ in $M_P.$
By analytic continuation the equality
(\ref{e: equality of q and p of Eis})
holds for all $\lambda$ in the full, hence connected, open
subset $\Omega':= \Omega \cap \Omega_0 \cap {}^{\scriptscriptstyle\backprime}{}^{\scriptscriptstyle\backprime}\faqdc$ of ${\mathfrak a}qdc.$
In particular it follows
that $\lambda :to q_{s,\nu}(P,e \,|\, f, \lambda)$ is holomorphic
on $\Omega'$ as a function with values in
$P_0({\mathfrak a}Pq) \otimes {\msy C}i({\rm X}_{P,e}\,:\, \tau_P),$ for all
$s \in W_P\backslash W$ and $\nu \in {\msy N} {\msy D}rP.$ This establishes the
assertion on holomorphic globality,
see Definition \ref{d: holomorphic s globality new}.
~
$\square$\medbreak\noindent\medbreak
\begin{lemma}
\label{l: s globality for partial Eisenstein}
Let $\Lambda\in {\mathfrak b}kdc$, $\psi \in {}^\circ {\cal C}[\Lambda],$ $S \subset W$ and define
$ f_S:{\mathfrak a}qdc \times {\rm X}p \rightarrow V_\tau$
by
$$
f_S(\lambda,x) := \sum_{s \in S} E_{+,s}({}^*Po\,:\, \lambda \,:\, x)\psi.
$$
Then the family $f_S$ belongs to $\cEhypgL.$ Moreover, let $t \in W$ and
$\alpha \in {\msy D}elta$, and assume that either $W_\alpha t\subset S$
or $W_\alpha t\cap S=\emptyset$, where $W_\alpha = \{1, s_\alpha\}.$ Then
the family $f_S$ is holomorphically $W_\alpha t$-global along
$({{}^*Pga}, v),$ for every $v \in {\msy N}Kaq.$
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
The first assertion is an immediate consequence of Lemma
\ref{l: f sub s belongs to specfam} with ${}^*Po$ in place of $P.$
Let $v \in {\msy N}Kaq.$
It follows from (\ref{e: exponents Eps}) and Theorem
\ref{t: transitivity of asymptotics} that
$${\rm Exp}(P_\alpha,v\,|\, f_{\{s\}\lambda})\subset s\lambda|_{{\mathfrak a}gaq}-\rho_\alpha-{\msy N}{\msy D}r({}^*Pga)$$
for each $s\in W$.
For $\lambda$ in the full open subset ${\mathfrak a}_{{{}^*Pga} q {\scriptscriptstyle \C}}^{*0}({}^*Po, \{0\})$
of ${\mathfrak a}gaqdc$ the sets $s\lambda|_{{\mathfrak a}gaq} - \rho_\alpha - {\msy N}{\msy D}r({{}^*Pga})$
are mutually disjoint for different $[s]=W_\alpha s$ from $W_\alpha \backslash W,$
see Lemmas \ref{l: exponents disjoint} and \ref{l: WPQ as cosets}.
Hence
\begin{equation}
\label{e: q sub t is zero}
q_{[t], \xi}({{}^*Pga},v \mid f_{\{s\}}) = 0,
\end{equation}
for all $s\in W\setminus W_\alpha t$ and all $\xi \in {\msy D}r({{}^*Pga}).$
First assume that $W_\alpha t \cap S = \emptyset.$
Then
it follows from (\ref{e: q sub t is zero}) that
$q_{[t],\xi}({{}^*Pga}, v \mid f_S) = 0$
for all $\xi \in {\msy D}r({{}^*Pga}).$ Hence $f_S$ is holomorphically $[t]$-global
along $({{}^*Pga}, v)$.
Next assume that $W_\alpha t \subset S.$ Let $S^c=W\setminus S$.
Then $f_S=f_W-f_{S^c}$, and it follows from Lemma
\ref{l: globality for the full Eisenstein}
and what was just proved, that $f_S$ is holomorphically $[t]$-global along
$({{}^*Pga}, v).$
~
$\square$\medbreak\noindent\medbreak
If $Q \in \cP_\gs$ is standard, then we define the subset $W^Q$ of $W$ by
\begin{equation}
\label{e: defi W^Q}
W^Q = \{ s \in W \mid s(\Delta_Q) \subset \Sigma^+ \}
\end{equation}
It is well known, see e.g. \bib{Carter}, Thm.\ 2.5.8, that the multiplication
map $W^Q \times W_Q \rightarrow W$ is bijective. Moreover, if $s\in W^Q$ and
$t \in W_Q,$ then $l(st) = l(s) + l(t);$
here $l: W \rightarrow {\msy N}$ denotes the length function relative to $\Delta.$ In particular
this means that
$W^Q$ consists of the minimal length representatives in $W$ of the cosets in
$W/W_Q.$
\begin{lemma}
\label{l: W Q and left Wga invariance}
Let $s \in W,$ $\alpha \in \Delta$ and assume that
$s^{-1}\alpha |_{{\mathfrak a}Qq} \neq 0.$ Let $t \in W_Q$. Then
$s \in W^Qt$ if and only if $s_\alpha s \in W^Qt$.
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
The hypothesis $s^{-1}\alpha |_{{\mathfrak a}Qq} \neq 0$ is also satisfied by
the elements $s_1 = s t^{-1}$ and $s_2=s_\alpha s t^{-1}$.
Hence we need only prove the implication
$s \in W^Q{\msy R}ightarrow s_\alpha s \in W^Q$.
Assume that $s \in W^Q.$ Then $s(\Delta_Q) \subset \Sigma^+.$
{}From the hypothesis it follows that $s^{-1}\alpha \notin \Delta_Q,$
hence $\alpha \notin s(\Delta_Q).$ Since $\alpha$ is simple, it follows
that $s_\alpha (s(\Delta_Q)) \subset \Sigma^+.$ Hence $s_\alpha s \in W^Q$.
~
$\square$\medbreak\noindent\medbreak
\begin{cor}
\label{c: Lau on partial WQ Eis}
Let $\psi \in {}^\circ {\cal C}[\Lambda]$ where $\Lambda\in{\mathfrak b}kdc$ and let $Q \in \cP_\gs$ be
a standard parabolic subgroup. Fix $t \in W_Q,$ and
let the family $f: {\mathfrak a}qdc \times {\rm X}p \rightarrow V_\tau$ be defined
by
$$
f(\lambda, x) = \sum_{s \in W^Q} E_{+,st}({}^*Po\,:\,\lambda\,:\, x)(\psi).
$$
Then $f \in \cEhypgL_{\rm hglob}.$
If $\cL \in \cM({}^*\fa_{Q\iq}dc, \Sigma_Q)^*_{\rm laur},$ then the
family
${\cal L}_* f$ belongs to the space ${\cal E}hypQYgdhglob,$
where $Y = {\rm supp}\, \cL,$ and where $\deltamap$ is a suitable element of ${\msy D}Qmaps.$
\end{cor}
\par\noindent{\bf Proof:}{\ }{\ }
Let $S = W^Qt.$
Then, $f = f_S,$ where we have used the notation
of Lemma \ref{l: s globality for partial Eisenstein}.
It follows from the mentioned lemma that $f \in \cEhypgL$.
Moreover, let $s \in W$ and $\alpha \in \Delta$ be such that
$s^{-1} \alpha |_{{\mathfrak a}Qq} \neq 0.$
Then it follows from Lemma
\ref{l: W Q and left Wga invariance} that either
$W_\alpha s \subset S$ or $W_\alpha s \cap S$ is empty.
Hence it follows from Lemma \ref{l: s globality for partial Eisenstein}
that $f$ is holomorphically $W_\alpha s$-global along $(P_\alpha, v),$ for every
$v \in {\msy N}Kaq.$ Thus $f \in \cEhypgL_{\rm hglob}$ by Lemma
\ref{l: minimal condition for hglob new}.
The remaining assertion now follows from
Theorem \ref{t: source of functions by Lau new}.
~
$\square$\medbreak\noindent\medbreak
\section{Asymptotics of partial Eisenstein integrals}
\label{s: asymptotics of partial Eisenstein integrals}
Let $P\in \cP_\gs^{\rm min}$ and let
$Q$ be a $\sigma$-parabolic subgroup containing $P.$
For the application of the asymptotic vanishing theorem,
Theorem \ref{t: vanishing theorem new},
in the next section we need
to determine the coefficient of the leading exponent in the $(Q,v)$-expansion
of the Eisenstein integral ${E^\circ}(P\,:\, \lambda),$ for every $v \in {\msy N}Kaq.$
To formulate a result in this direction,
we need some additional notation.
Let $v \in {\msy N}Kaq$ and select
a complete set of representatives ${\cal W}_{Q,v}$
in ${\msy N}KQaq$ for
$W_Q/ W_{K_Q \cap vHv^{-1}}.$
We define ${}^\circ {\cal C}(Q,v) = {}^\circ {\cal C}(Q,v,\tau)$ to be the analogue of the space
${}^\circ {\cal C}$ for the data ${\rm X}oneQv, \tau_Q.$ Thus
\begin{equation}
\label{e: dir sum oC Q v}
{}^\circ {\cal C}(Q,v) = \oplus_{u \in {\cal W}_{Q,v}} {\msy C}i(M/M \cap uv H (uv)^{-1} \,:\, \tau)
\end{equation}
with an orthogonal direct sum. Note that ${}^\circ {\cal C}(Q,v)$ is also the analogue of
${}^\circ {\cal C}$ for the data ${\rm X}Qv, \tau_Q.$
One readily checks that the map ${\cal W}_{Q,v} \rightarrow W/W_{K \cap H}$ given by
$u :to {\rm Ad}(uv)|{\mathfrak a}q$ is injective. Hence we may extend ${\cal W}_{Q,v}v $
to a complete set ${\cal W} \subset {\msy N}Kaq$ of representatives for
$W/W_{K \cap H}.$ If $w \in {\cal W},$ then $w \in {\cal W}_{Q,v}v \iff wv^{-1} \in K_Q.$
With such choices made we have a natural isometric embedding
$
{\rm Q}v : {}^\circ {\cal C}(Q,v) \hookrightarrow {}^\circ {\cal C},
$
defined by
\begin{equation}
\label{e: defi i Q v}
({\rm Q}v\psi)_w = \left\{
\begin{array}{ll}
\psi_{wv^{-1}}&\text{if} w \in {\cal W}_{Q,v} v,\\
0 &\text{otherwise.}
\end{array}
\right.
\end{equation}
The adjoint of the embedding ${\rm Q}v$ is denoted by
$
{\rm pr}_{Q,v} : {}^\circ {\cal C} \rightarrow {}^\circ {\cal C}(Q,v).
$
It is given by the following formula, for $\psi \in {}^\circ {\cal C},$
\begin{equation}
\label{e: formula for pr Q v}
({\rm pr}_{Q,v}\psi)_u = \psi_{uv}, \quad\quad (u \in {\cal W}_{Q,v}).
\end{equation}
The normalized Eisenstein integral associated with the data
${\rm X}oneQv, \tau_Q$ and ${}^*P:= P \cap M_{1Q}$ is denoted
by
${E^\circ}({\rm X}oneQv \,:\, {}^*P \,:\, \nu),$ for $\nu \in{\mathfrak a}qdc.$
Similarly, the partial Eisenstein integrals associated with these data
are denoted by
$E_{+,s}({\rm X}oneQv \,:\, {}^*P\,:\, \nu),$ for $s \in W_Q$ and
$\nu \in {\mathfrak a}qdc.$ Note that all of these are $(\tau_Q \otimes 1)$-spherical
smooth functions on ${\rm X}oneQvp$ with values in
${\rm Hom}({}^\circ {\cal C}Qv, V_\tau)\simeq V_\tau \otimes {}^\circ {\cal C}Qv^*.$
\begin{prop}
\label{p: q Q v of nE}
Let $P \in \cP_\gs^{\rm min},$ $Q \in \cP_\gs$ and assume that $Q\supset P.$
Let $v \in {\msy N}Kaq$, and choose ${\cal W}_{Q,v}$, ${\cal W}$
as above such that ${\cal W}_{Q,v} \subset {\cal W} v^{-1}$.
Let $\psi \in {}^\circ {\cal C}$ and let the family $f:{\mathfrak a}qdc \times {\rm X} \rightarrow V_\tau$
be defined by
$$
f(\lambda, x) = {E^\circ}(P \,:\, \lambda \,:\, x) \psi.
$$
Then, for $\lambda \in {\mathfrak a}qdc$ generic, and for all $X \in {\mathfrak a}Qq$ and $m \in {\rm X}Qvp,$
\begin{equation}
\label{e: q gl for nE}
q_{\lambda|_{{\mathfrak a}Qq} - \rho_Q}(Q,v \,|\, f_\lambda,X,m) =
{E^\circ}({\rm X}oneQv \,:\, {}^*P \,:\, \lambda \,:\, m)\,{\rm pr}Qv \psi.
\end{equation}
\end{prop}
\par\noindent{\bf Proof:}{\ }{\ }
We first assume that
$v = e.$ Then ${\rm X}oneQv = {\rm X}_{1Q,e} = M_{1Q}/ M_{1Q} \cap H.$
Moreover, the set ${\cal W}_Q:= {\cal W}_{Q,e}$ is contained in ${\cal W}.$
{}From \bib{BSft}, p.\ 563, Thm.\ 4, it follows that
$$
q_{\lambda|_{{\mathfrak a}Qq} - \rho_Q}(Q,e\,|\, f_\lambda , X, m) =
{E^\circ}({\rm X}_{1Q,e} \,:\, {}^*P \,:\, \lambda \,:\, m)\, {\rm pr}_Q \psi,
$$
for generic $\lambda \in {\mathfrak a}qdc$ and all $X \in {\mathfrak a}Qq$ and $m \in {\rm X}_{Q,e,+}.$
Here ${\rm pr}_Q$ is the natural projection map from ${}^\circ {\cal C}$ onto
${}^\circ {\cal C}_Q(\tau) = \oplus_{v \in {\cal W}_Q} {\msy C}i(M/M\cap vHv^{-1}\,:\, \tau_{\iM}),$
see \bib{BSft}, pp.\ 544 and 547. Thus, ${\rm pr}_Q$ equals the map ${\rm pr}_{Q,e}$ defined above
and it follows that (\ref{e: q gl for nE}) holds with $v=e.$
To establish the result for arbitrary $v\in {\msy N}Kaq,$ we first need a lemma.
\medbreak
{}From Remark \ref{r: extreme cases subspaces}
we recall that ${\rm X}_v = {\rm X}_{1G,v} = G/vHv^{-1}.$
The set ${\cal W}_{G,v}:= {\cal W} v^{-1}$ is a complete set of representatives
for $W/W_{K \cap vHv^{-1}}.$ Accordingly, the analogue ${}^\circ {\cal C}(G,v) = {}^\circ {\cal C}(G,v,\tau)$
of the space ${}^\circ {\cal C}$ is given by
(\ref{e: dir sum oC Q v}) with $G$ in place of $Q.$
The associated map ${\rm i}_{G,v}: {}^\circ {\cal C}(G,v) \rightarrow {}^\circ {\cal C}$ is now
a bijective isometry; moreover, its adjoint
${\rm pr}_{G,v}$ is its two-sided inverse.
We recall from the end of Section \ref{s: asymp walls} that
right translation by $v$ induces a topological linear
isomorphism $R_v$ from ${\msy C}i({\rm X}\,:\, \tau)$ onto
${\msy C}i({\rm X}_{v} \,:\, \tau).$ In the following lemma
we will relate the right translate of ${E^\circ}(P\,:\, \lambda)$ to the normalized
Eisenstein integral associated with ${\rm X}_{v},$ ${\cal W} v^{-1}$ and $P.$
\begin{lemma}
\label{l: Rv of nE}
Let $\psi \in {}^\circ {\cal C}.$ Then, for generic $\lambda \in {\mathfrak a}qdc,$
\begin{equation}
\label{e: Rv of nE}
R_v ({E^\circ}({\rm X}\,:\, P\,:\, \lambda)\psi) = {E^\circ}({\rm X}_{v} \,:\, P \,:\, \lambda)
[{\rm pr}_{G,v}\psi].
\end{equation}
The formula remains valid if the normalized Eisenstein integrals
are replaced by their unnormalized versions.
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
We first prove the formula for the unnormalized Eisenstein integrals.
Let $\lambda \in {\mathfrak a}qdc$ be such that
$\inp{{\msy R}e \lambda + \rho_P }{\alpha} < 0$ for all $\alpha \in \Sigma(P).$
Define the function $\tilde\psi(\lambda): G \rightarrow V_\tau$ as in \bib{BSft}, Eq.\ (19).
Then $E(P\,:\, \lambda \,:\, x) \psi = \int_K \tau(k)\, \tilde\psi(\lambda \,:\, k^{-1} x) \, dk.$
Hence $E(P\,:\, \lambda \,:\, x v)\psi = \int_K \tau(k) \tilde\psi_{G,v}(\lambda\,:\, k^{-1} x) \, dk,$
where $\tilde\psi_{G,v}(\lambda\,:\, x) = \tilde\psi(\lambda\,:\, xv).$ One now readily checks
that $\tilde \psi_{G,v}(\lambda)$ is the analogue of $\tilde\psi(\lambda),$ associated
with the data ${\rm X}_{v}, {\cal W} v^{-1}$ and with the element
$\psi_{G,v} := {\rm pr}_{G,v}\psi $ of ${}^\circ {\cal C}(G,v).$ {}From
this we obtain the equality (\ref{e: Rv of nE}) for
the present $\lambda.$ For general $\lambda,$ the result follows by
meromorphic continuation.
Let $Q \in \cP_\gs^{\rm min}.$ Then it follows, by application of Lemma \ref{l: q of Rv f}
and
the definition of the $\bf c$-functions (cf.\ \bib{BSft}, \S{} 4), that, for every $s \in W,$ each $u \in {\cal W} v^{-1}$
and generic $\lambda \in {\mathfrak a}qdc,$ we have
$[C_{Q|P}({\rm X}\,:\, s \,:\, \lambda) \psi]_{uv} =
[C_{Q|P}({\rm X}_{v}\,:\, s\,:\, \lambda) {\rm pr}_{G,v} \psi]_u.$ In other words,
$$
{\rm pr}_{G,v} \,{\scriptstyle\circ}\, C_{Q|P}({\rm X}\,:\, s \,:\, \lambda) = C_{Q|P}({\rm X}_{v}\,:\, s\,:\, \lambda)\,{\scriptstyle\circ}\,
{\rm pr}_{G,v}.
$$
The proof is completed by combining this equation, after substitution of $P$ and $1$
for $Q$ and $s,$ respectively,
with the unnormalized version of
(\ref{e: Rv of nE}) and the definitions of the normalized Eisenstein integrals
(cf.\ \bib{BSft}, eq.\ (49)).
~
$\square$\medbreak\noindent\medbreak
\medbreak\noindent
{\bf Completion of the proof of Prop.\ \ref{p: q Q v of nE}.\ }
Let $v \in {\msy N}Kaq$ be arbitrary. Then from Lemmas \ref{l: q of Rv f},
\ref{l: Rv of nE} and equation
(\ref{e: q gl for nE}) with ${\rm X}_{v},$ $e$ and ${\rm pr}_{G,v}\psi$
in place of ${\rm X},$ $v$ and $\psi,$ respectively,
it follows that, for $X \in {\mathfrak a}Qq$ and $m \in {\rm X}Qvp,$
\begin{eqnarray*}
q_{\lambda|_{{\mathfrak a}Qq} - \rho_Q}(Q,v \,|\, f_\lambda , X, m) &=&
q_{\lambda|_{{\mathfrak a}Qq} - \rho_Q}(Q, e \,|\, R_v(f_\lambda) , X, m) \\
&=& {E^\circ}(\tilde {\rm X}_{1Q,e} \,:\, {}^*P \,:\, \lambda \,:\, m) \, \tilde {\rm pr}_{Q,e} {\rm pr}_{G,v}\psi.
\end{eqnarray*}
In the last expression the two tildes over objects indicate that the analogues of
the objects for the symmetric space ${\rm X}_{v}$ are taken.
We now observe that $\tilde {\rm X}_{1Q,e}$ equals the space
$M_{1Q} / M_{1Q} \cap e vHv^{-1} e = {\rm X}_{1Q,v}.$
Hence, to establish (\ref{e: q gl for nE}), it suffices to show that $\tilde {\rm pr}_{Q,e} {\rm pr}_{G,v}\psi
={\rm pr}_{Q,v} \psi.$
For this we note that $\tilde {\rm pr}_{Q,e}$ is the projection
from ${}^\circ {\cal C}(G,v)$ onto the sum of the components
parametrized by the elements $u$ of $M_{1Q} \cap {\cal W} v^{-1} = {\cal W}_{Q,v}.$
Moreover, for $u \in {\cal W}_{Q,v},$
$$
[\tilde {\rm pr}_{Q,e} {\rm pr}_{G,v}\psi]_u = [{\rm pr}_{G,v}\psi]_{u} =
\psi_{uv}
=
[{\rm pr}_{Q,v}\psi]_u.
$$
~
$\square$\medbreak\noindent\medbreak
The result just proved generalizes to partial Eisenstein integrals.
\begin{prop}
\label{p: q Q v of sum partial eis}
Let $P \in \cP_\gs^{\rm min}.$
Let $\psi \in {}^\circ {\cal C},$ let
$S \subset W$ and let the family $f = f_S $
be defined by
$$
f(\lambda, x) = \sum_{s \in S} E_{+,s}(P\,:\, \lambda \,:\, x) \psi,
$$
see Lemma \ref{l: s globality for partial Eisenstein}.
Assume that $Q \in \cP_\gs$ contains $P$ and that $v \in {\msy N}Kaq.$
Then, for generic $\lambda \in {\mathfrak a}qdc,$ and all $X \in {\mathfrak a}Qq$ and $m \in {\rm X}Qvp,$
\begin{equation}
\label{e: q gl for sum partial Eis}
q_{\lambda|_{{\mathfrak a}Qq} - \rho_Q}(Q,v \,|\, f_\lambda,X,m) =
\sum_{s \in S \cap W_Q}
E_{+,s}({\rm X}oneQv \,:\, {}^*P \,:\, \lambda\,:\, m)\,{\rm pr}Qv \psi.
\end{equation}
In particular, if $S\cap W_Q=\emptyset$ then
$\lambda|_{{\mathfrak a}Qq} - \rho_Q\notin {\rm Exp}(Q,v\,|\, f_\lambda)$.
\end{prop}
\par\noindent{\bf Proof:}{\ }{\ }
For $S = W$ this result is precisely
Prop.\ \ref{p: q Q v of nE}.
We shall use transitivity of asymptotics to derive the result for arbitrary $S$
from it.
It suffices to prove the
above identity for $m = bu \in {\rm X}Qvp,$
with $u \in {\msy N}KQaq$ and $b \in {}^*\!A_{Q\iq}p({}^*P)$ arbitrary.
According to Lemma \ref{l: f sub s belongs to specfam}
and Remark \ref{r: relation between hypfam and fam},
the function $f_S$ belongs to
$C^{{\rm ep}}_{0,\{0\}}({\rm X}p\,:\, \tau\,:\, \Omega),$
for the full open subset $\Omega: = {\rm reg}a{f_S}$ of ${\mathfrak a}qdc.$
Hence, according to Theorem \ref{t: transitivity of asymptotics for families new}
with
${}^*Po, Q$ and $P$
in place of $Q, P$ and $P_1,$ respectively,
for $\lambda \in {\mathfrak a}qdc$ generic the following holds, with
$[1]$ the class of $1 \in W$ in $W/\sim_{Q|{}^*Po} = W_Q\backslash W,$
\begin{eqnarray*}
q_{\lambda|_{{\mathfrak a}Qq} - \rho_Q}(Q,v \,|\, f_{S\lambda},X,bu)
&=& q_{[1],0}(Q,v\,|\, f_S, X)(\lambda, bu) \\
& = &
\sum_{s \in W_Q}
\sum_{\mu \in {\msy N}{\msy D}elta_Q(P)} b^{s \lambda - \rho_P - \mu}
q_{s,\mu}(P, uv\,|\, f_S, X + \log b)(\lambda, e).
\end{eqnarray*}
Now, for all $s,t\in W$, $\mu\in{\msy N}\Delta$ and $v\in{\msy N}Kaq$ it follows from
(\ref{e: exponents Eps}) and Lemma \ref{l: exponents disjoint}
that $q_{s,\mu}(P,v\,|\, f_{\{t\}})=0$ if $s\neq t$. Hence
$$
q_{s,\mu}(P, v\,|\, f_S ) =
\left\{
\begin{array}{ll}
q_{s,\mu}(P, v\,|\, f_W)& \text{if} s \in S, \\
0 & \text{otherwise.}
\end{array}
\right.
$$
Thus,
we obtain that
\begin{equation}
\label{e: formula for q one zero of fS}
q_{\lambda|_{{\mathfrak a}Qq} - \rho_Q}(Q,v \,|\, f_{S\lambda},X,bu)
=
\sum_{s \in S \cap W_Q}
\sum_{\mu \in {\msy N}{\msy D}elta_Q(P)} b^{s \lambda - \rho - \mu}
q_{s,\mu}(P, uv\,|\, f_W, X + \log b)(\lambda, e).
\end{equation}
This equation is valid for any subset $S$ of $W;$ in particular,
it
is valid for $S = W.$ Using (\ref{e: q gl for nE}) we now obtain that,
for any $u \in {\msy N}KQaq$ and all $b \in {}^*\!A_{Q\iq}p({}^*P),$
\begin{equation}
\label{e: Eisenstein integral for spXQv}
{E^\circ}({\rm X}oneQv\,:\, {}^*P \,:\, \lambda\,:\, bu)\,{\rm pr}Qv \psi
=
\sum_{s \in W_Q}
\sum_{\mu \in {\msy N}{\msy D}elta_Q(P)} b^{s \lambda - \rho - \mu}
q_{s,\mu}(P, uv\,|\, f_W, X + \log b)(\lambda, e).
\end{equation}
This is the ${\msy D}QP$-exponential polynomial expansion of the Eisenstein integral
along $({}^*P, u).$ In view of (\ref{e: splitting of Eis})
and the remark following (\ref{e: exponents Eps}), with
${\rm X}_{1Q,v}$ in place of ${\rm X}$,
we infer from (\ref{e: Eisenstein integral for spXQv}) that,
for each $s \in W_Q,$ and every $u\in {\msy N}KQaq$ and $b \in {}^*\!A_{Q\iq}p({}^*P),$
\begin{equation}
\label{e: expansion for partial Eisenstein integral along Q}
E_{+,s}({\rm X}oneQv \,:\, {}^*P \,:\, \lambda\,:\, bu){\rm pr}Qv \psi
=
\sum_{\mu \in {\msy N}{\msy D}elta_Q(P)} b^{s \lambda - \rho - \mu}
q_{s,\mu}(P, uv\,|\, f_W, X + \log b)(\lambda, e).
\end{equation}
Finally, (\ref{e: q gl for sum partial Eis}) with $m = bu$ follows
by combining (\ref{e: formula for q one zero of fS})
and (\ref{e: expansion for partial Eisenstein integral along Q}).
~
$\square$\medbreak\noindent\medbreak
We end this section with a
generalization of Proposition \ref{p: q Q v of sum partial eis},
involving the application of a Laurent functional.
\begin{prop}
\label{p: q of Laustar of part Eis}
Let assumptions be as in Prop.\ \ref{p: q Q v of sum partial eis}
and let
${\cal L} \in \cM({}^*\fa_{Q\iq}dc, \Sigma_Q)^*_{\rm laur}.$
Then the family $\cL_*f$ defined by $\cL_*f(\nu,x) = \cL[f(\,\cdot\, + \nu , x)],$
for generic $\nu \in {\mathfrak a}Qqdc$ and $x \in {\rm X}p,$ belongs to
${\cal E}hypQYgd,$
with $Y = {\rm supp}\, \cL$ and for a suitable $\delta \in {\rm D}_Q.$
Moreover, for generic $\nu \in {\mathfrak a}Qqdc$
and all $X \in {\mathfrak a}Qq$ and $m \in {\rm X}Qvp,$
\begin{equation}
\label{e: formula for q of Laustar f}
q_{\nu- \rho_Q}(Q,v\,|\, ({\cal L}_* f)_\nu , X, m) =
{\cal L}[\sum_{s \in S \cap W_Q}
E_{+,s}({\rm X}oneQv \,:\, {}^*P \,:\, \,\cdot\, + \nu \,:\, m)\,{\rm pr}Qv \psi].
\end{equation}
In particular, if $S\cap W_Q=\emptyset$ then
$\nu - \rho_Q\notin {\rm Exp}(Q,v\,|\, ({\cal L}_* f)_\nu)$.
\end{prop}
\par\noindent{\bf Proof:}{\ }{\ }
The first assertion follows from Cor.\ \ref{c: Lau to partial Eis}.
For the
second assertion, we note that $\cL_*f \in {\msy C}epQY({\rm X}p\,:\, \tau \,:\, \Omega),$
where $\Omega$ is the full open subset ${\mathfrak a}Qqdc\setminus\cup \cH_{\cLstar f}$
of ${\mathfrak a}Qqdc,$ see Remark \ref{r: relation between hypfam and fam}.
The set $\Omega_*: = \Omega \cap {\mathfrak a}Qqdczero(P,\{0\})$ is a full open subset
of ${\mathfrak a}Qqdc.$ Moreover, from (\ref{e: q of f in gl versus q of fgl new})
it follows that,
for $\nu \in \Omega_*,$
\begin{equation}
\label{e: q nu min rho as q one zero}
q_{\nu - \rho_Q}(Q,v\,|\, (\cL_* f)_\nu , X) = q_{[1],0}(Q,v\,|\, \cL_* f, X)(\nu),
\quad (X \in {\mathfrak a}Qq);
\end{equation}
here $[1]$ denotes the image of the identity element of $W$ in $W/\sim_{Q|Q}.$
The expression on the right-hand side of the above equation is given
by (\ref{e: sum for q of Laustar f}), with $P=Q, \sigma = [1] \in W/\sim_{Q|Q}$ and $\xi = 0.$
Note that an element $s \in W$ satisfies $[s] = [1]$ if and only if $s \in W_Q.$
It follows from this that $[1]\cdot Y = \{0\}.$
Hence from (\ref{e: sum for q of Laustar f}) and (\ref{e: defi Laustar s})
we conclude, with $\bar 1$ denoting
the image of $1 \in W$ in $W_Q\backslash W,$
\begin{eqnarray}
q_{[1],0}(Q,v\,|\, \cL_* f, X)(\nu)
&=&
\sum_{\lambda \in Y} \cL_{\lambda *}^{Q, \bar 1} [q_{\bar 1, 0}(Q,v\,|\, f)(X,\,\cdot\,)](\nu,X)
\nonumber\\
\label{e: q one zero as Laustar of q one zero}
&=&
\sum_{\lambda \in Y}
e^{-(\lambda + \nu)(X)}\cL_{\lambda*}
[ e^{(\,\cdot\,)(X)}q_{\bar 1, 0}(Q,v\,|\, f)(X, \,\cdot\,)](\nu).
\end{eqnarray}
for $X \in {\mathfrak a}Qq$ and generic $\nu \in {\mathfrak a}Qqdc.$ From
$(\lambda + \nu)(X) = \nu(X)$ we deduce that the last expression
in (\ref{e: q one zero as Laustar of q one zero}) equals
$\sum_{\lambda \in Y} \cL_{\lambda*}
[q_{\bar 1, 0}(Q,v\,|\, f)(X, \,\cdot\,)](\nu).$
Hence from (\ref{e: q nu min rho as q one zero}) and
(\ref{e: q one zero as Laustar of q one zero}) we obtain
\begin{equation}
\label{e: q as Laustar q}
q_{\nu - \rho_Q}(Q,v\,|\, (\cL_* f)_\nu , X)
=
\cL_{*}[q_{\bar 1, 0}(Q,v\,|\, f)(X, \,\cdot\,)](\nu).
\end{equation}
It follows from (\ref{e: q gl for sum partial Eis})
and (\ref{e: q of f in gl versus q of fgl new})
that, for $X \in {\mathfrak a}Qq,$ $m \in {\rm X}Qvp,$
\begin{equation}
\label{e: q bar one zero as Eis}
q_{\bar 1, 0}(Q,v\,|\, f)(X, \lambda, m) =
\sum_{s \in S \cap W_Q}
E_{+,s}({\rm X}oneQv \,:\, {}^*P \,:\, \lambda\,:\, m)\,{\rm pr}Qv \psi,
\end{equation}
as a meromorphic identity in $\lambda \in {\mathfrak a}qdc.$
The equality (\ref{e: formula for q of Laustar f})
now follows by combining
(\ref{e: q as Laustar q})
with
(\ref{e: q bar one zero as Eis}).
~
$\square$\medbreak\noindent\medbreak
\section{Induction of relations}
\label{s: induction of relations}
After the preparations of the previous sections
we are now able to apply the vanishing theorem,
Theorem \ref{t: special vanishing theorem}, to families
obtained from applying Laurent functionals to partial Eisenstein integrals.
This will lead to what we call induction of relations.
We retain the notation of the previous section.
Moreover, we assume that $Q \in \cP_\gs$ is a standard parabolic subgroup.
Thus ${}^*P_0: = M_Q \cap {}^*Po$ is the standard
minimal $\sigma$-parabolic subgroup of $M_Q,$ relative to the positive
system $\Sigma_Q^+:= \Sigma_Q \cap \Sigma.$
We assume that ${}^Q\cW$ is a complete
set of representatives in ${\msy N}Kaq$ for the double coset space $W_Q\backslash W/W_{K \cap H}.$
We also assume that for each $v \in {}^Q\cW$
a set ${\cal W}_{Q,v}$ as above (\ref{e: dir sum oC Q v}) is chosen.
Then one readily verifies that
\begin{equation}
\label{e: cW as disjoint union over QW}
{\cal W} = \cup_{v \in {}^Q\cW} \;\;{\cal W}_{Q,v} v \quad\quad \text{(disjoint union).}
\end{equation}
is a complete set of representatives for $W/W_{K \cap H}$ in ${\msy N}Kaq.$
Combining this with (\ref{e: defi i Q v}) and (\ref{e: formula for pr Q v})
we find that
$$
\sum_{v \in {}^Q\cW} {\rm Q}v\,{\scriptstyle\circ}\, {\rm pr}_{Q,v} = I_{{}^\circ {\cal C}}.
$$
Combining
(\ref{e: cW as disjoint union over QW}) with
(\ref{e: defi i Q v}) and (\ref{e: formula for pr Q v}),
it also follows, for $u,v \in {}^Q\cW,$ that
\begin{equation}
\label{e: prQu after iQv}
{\rm pr}_{Q,u} \,{\scriptstyle\circ}\, {\rm i}_{Q,v} = \left\{
\begin{array}{ll}
I_{{}^\circ {\cal C}(Q,v)} & \text{if} u=v;\\
0 &\text{otherwise.}
\end{array}
\right.
\end{equation}
\begin{thm}
\label{t: induction of relations, new}
{\rm (Induction of relations)\ }
Let
${\cal L}_t\in {\cal M}({}^*\fa_{Q\iq}dc, \Sigma_Q)^*_{\rm laur} \otimes {}^\circ {\cal C}$ be given
for each $t\in W_Q$.
If, for each $v \in {}^Q\cW$,
\begin{equation}
\label{e: hypo ind rels with pr, new}
\sum_{t\in W_Q}{\cal L}_t[E_{+,t}(\spX_{Q,v}\,:\,{}^*P_0\,:\,\,\cdot\,\,:\, m)
\,{\scriptstyle\circ}\, {\rm pr}_{Q,v} ]
=
0,\quad\quad (m \in \spX_{Q,v}p)
\end{equation}
then for each $s\in W^Q$ the following holds
as a meromorphic identity in $\nu \in {\mathfrak a}Qqdc:$
\begin{equation}
\label{e: conclusion ind rels with pr, new}
\sum_{t\in W_Q}{\cal L}_t[\;
E_{+,st}({\rm X}\,:\,{}^*Po\,:\, \,\cdot\, + \nu \,:\, x)\;]
= 0, \quad\quad (x \in {\rm X}p).
\end{equation}
Conversely, if the identity (\ref{e: conclusion ind rels with pr, new})
holds for some $s\in W^Q$ and all
$\nu$ in a non-empty open subset of ${\mathfrak a}Qqdc$, then
(\ref{e: hypo ind rels with pr, new}) holds
for each $v \in {}^Q\cW$.
\end{thm}
\par\noindent{\bf Proof:}{\ }{\ }
Define for each $w\in W$ the family
$g_w\,:\,on {\mathfrak a}Qqdc \times {\rm X}p \rightarrow V_\tau \otimes {}^\circ {\cal C}^*$ by
$$
g_w(\nu,x) = {\cal L}_t[\;E_{+,st}({\rm X}\,:\, P_0\,:\, \,\cdot\,+\nu \,:\, x)\;]
$$
for generic $\nu\in {\mathfrak a}Qqdc$ and every $x \in {\rm X}p;$
the elements $s\in W^Q, t\in W_Q$ are determined by
the unique product decomposition $w=st$ (see below (\ref{e: defi W^Q})).
It follows from Cor.\ \ref{c: Lau to partial Eis}, that there
exist $\delta_w \in {\rm D}_Q$ such that
$g_w \in {\cal E}_{Q,Y_w}^{\rm hyp}({\rm X}p \,:\, \tau\,:\, \delta_w)$;
here $Y_w = {\rm supp}\, \cL_t,$ where $t\in W_Q$ is determined as above.
If we put $Y = \cup Y_w$ and $\delta = \max(\delta_w),$ then
$g_w$ belongs to ${\cal E}QY^{\rm hyp}({\rm X}p\,:\, \tau\,:\, \delta)$
for all $w\in W$. Moreover, for generic $\nu\in{\mathfrak a}Qqdc$,
\begin{equation}
\label{e: 3}
{\rm Exp}(P_0,v\,|\, (g_w)_\nu)\subset w(\nu+Y)-\rho-{\msy N}{\msy D}elta.
\end{equation}
In view of Proposition \ref{p: q of Laustar of part Eis} it also follows
for $X\in{\mathfrak a}Qq$, $m \in \spX_{Q,v}p$ and generic $\nu\in {\mathfrak a}Qqdc$ that
\begin{equation}
\label{e: 4}
q_{\nu - \rho_Q}(Q,v\,|\, (g_t)_\nu, X, m) =
{\cal L}_t[E_{+,t}(\spX_{Q,v}\,:\,{}^*P_0\,:\,\,\cdot\,+\nu\,:\, m)
\,{\scriptstyle\circ}\, {\rm pr}_{Q,v} ] \quad\quad (t\in W_Q),
\end{equation}
and
\begin{equation}
\label{e: 4'}
q_{\nu - \rho_Q}(Q,v\,|\, (g_w)_\nu, X, m) = 0
\quad\quad (w\notin W_Q).
\end{equation}
According to Cor.\ \ref{c: Lau on partial WQ Eis}
the family $\sum_{s\in W^Q} g_{st}$
belongs to the space
${\cal E}_{Q,Y}^{\rm hyp}({\rm X}p\,:\, \tau \,:\, \delta)_\lambdaob$ for each $t\in W_Q$.
Hence so does the family
$g=\sum_{w\in W} g_w=\sum_{t\in W_Q,s\in W^Q} g_{st}$.
Moreover, by (\ref{e: 4}) and (\ref{e: 4'})
$$
q_{\nu - \rho_Q}(Q,v\,|\, (g)_\nu, X, m) =
\sum_{t\in W_Q}{\cal L}_t[E_{+,t}(\spX_{Q,v}\,:\,{}^*P_0\,:\,\,\cdot\,+\nu\,:\, m)
\,{\scriptstyle\circ}\, {\rm pr}_{Q,v} ] \quad\quad (m \in \spX_{Q,v}p).
$$
{}From Theorem \ref{t: special vanishing theorem}
we now see that (\ref{e: hypo ind rels with pr, new})
holds for each $v \in {}^Q\cW$ if and only if $g=0$.
On the other hand, let $g^s=\sum_{t\in W_Q} g_{st}$ for $s\in W^Q$.
It follows from (\ref{e: 3}) that
$${\rm Exp}(P_0,v\,|\, (g^s)_\nu)
\subset s\nu+WY-\rho-{\msy N}{\msy D}elta.$$
Since the latter sets are mutually disjoint as $s$ runs over
$W^Q$, for $\nu$ in a full open subset
(see Lemma \ref{l: exponents disjoint}), we conclude that
for such $\nu$,
$$(s\nu+WY-\rho-{\msy N}{\msy D}elta)\cap{\rm Exp}(P_0,v\,|\, g_\nu)
={\rm Exp}(P_0,v\,|\, (g^s)_\nu).$$
Hence $g=0$ implies that $g^s=0$
for each $s\in W^Q$. Conversely
it follows from Corollary \ref{c: variant of vanishing theorem} that
$g=0$ if $g^s=0$
for some $s\in W^Q$.
The theorem follows immediately.
~
$\square$\medbreak\noindent\medbreak
\begin{cor}
\label{c: ind rels with i, new}
Let $v \in {}^Q\cW$ and let
${\cal L}_t\in{\cal M}({}^*\fa_{Q\iq}dc, \Sigma_Q)^*_{\rm laur} \otimes {}^\circ {\cal C}(Q,v)$ be given
for each $t\in W_Q$. If
\begin{equation}
\label{e: hypo ind rels with i, new}
\sum_{t\in W_Q}
{\cal L}_t[E_{+,t}(\spX_{Q,v}\,:\, {}^*P_0\,:\, \,\cdot\,\,:\, m) ]
=0,\quad\quad (m \in \spX_{Q,v}p)
\end{equation}
then for each $s\in W^Q$ the following holds
as a meromorphic identity in $\nu \in {\mathfrak a}Qqdc:$
\begin{equation}
\label{e: conclusion ind rels with i, new}
\sum_{t\in W_Q}{\cal L}_t[\;
E_{+,st}({\rm X}\,:\,{}^*Po\,:\, \,\cdot\, + \nu \,:\, x)\,{\scriptstyle\circ}\,{\rm Q}v\;]
= 0,\quad\quad (x\in {\rm X}p).
\end{equation}
Conversely, if the identity (\ref{e: conclusion ind rels with i, new})
holds for some $s\in W^Q$ and all
$\nu$ in a non-empty open subset of ${\mathfrak a}Qqdc$, then
(\ref{e: hypo ind rels with i, new}) holds.
\end{cor}
\par\noindent{\bf Proof:}{\ }{\ }
For $t\in W_Q$ we define the functional
$\cL_t^\circ\in \cM({}^*\fa_{Q\iq}dc,\Sigma_Q)^*_{\rm laur} \otimes {}^\circ {\cal C}$
by
$\cL_t^\circ = [ I \otimes {\rm Q}v ] (\cL_t).$
Then for $F \in \cM({}^*\fa_{Q\iq}dc,\Sigma_Q) \otimes {}^\circ {\cal C}^*$
we have
\begin{equation}
\label{e: Lau t circ}
\cL_t^\circ F = \cL_t [ F(\,\cdot\,)\,{\rm Q}v].
\end{equation}
Let $u \in {}^Q\cW.$ Then from
(\ref{e: prQu after iQv}) and (\ref{e: hypo ind rels with i, new})
we deduce
that (\ref{e: hypo ind rels with pr, new}) holds with $u$ and $\cL_t^\circ$
in place of $v$ and $\cL_t,$ respectively.
It follows that (\ref{e: conclusion ind rels with pr, new}) holds with $\cL_t^\circ$
in place of $\cL_t.$ In view of (\ref{e: Lau t circ})
this implies (\ref{e: conclusion ind rels with i, new}).
The converse statement is seen similarly.
~
$\square$\medbreak\noindent\medbreak
Another useful formulation of the principle of induction of relations
is the following.
\begin{cor}
\label{c: ind rels version 3, new}
Let $v \in {}^Q\cW.$ Let $\cL_t\in \cM({}^*\fa_{Q\iq}dc, \Sigma_Q)^*_{\rm laur}$
and $\varphi_t \in \cM({\mathfrak a}qdc,\Sigma) \otimes {}^\circ {\cal C}(Q,v)$ be given
for each $t\in W_Q$.
Assume that
\begin{equation}
\label{e: hypo ind rels with i and nu, new}
\sum_{t\in W_Q}\cL_t[E_{+,t}({\rm X}Qv\,:\, {}^*P_0\,:\, \,\cdot\, \,:\, m)
\varphi_t(\,\cdot\, + \nu)]
=0,\quad\quad(m \in {\rm X}Qvp)
\end{equation}
for generic $\nu \in {\mathfrak a}Qqdc.$ Define
$\psi_t = (I \otimes {\rm Q}v)\varphi_t \in \cM({\mathfrak a}qdc, \Sigma) \otimes {}^\circ {\cal C},
$
for $t\in W_Q.$
Then, for each $s\in W^Q$,
$$
\sum_{t\in W_Q}\cL_t [E_{+,st}({\rm X}\,:\,{}^*Po\,:\, \,\cdot\, + \nu \,:\, x)
\psi_t (\,\cdot\, + \nu )]
=
0,\quad\quad(x\in {\rm X}p)
$$
as an identity
of $V_\tau$-valued meromorphic functions in the variable $\nu \in {\mathfrak a}Qqdc.$
\end{cor}
\par\noindent{\bf Proof:}{\ }{\ }
Let $\cH$ be a $\Sigma$-configuration such that ${\rm sing}(\varphi_t)\subset\cup\cH,$
for each $t\in W_Q.$ Moreover, let
$Y=\cup_{t\in W_Q}{\rm supp}\,\cL_t \subset {}^*\fa_{Q\iq}dc$. Fix $t\in W_Q.$ Let
$\cH':= \cH_{{\mathfrak a}Qqdc}(Y)$ be the $\Sigma_r(Q)$-configuration
in ${\mathfrak a}Qqdc$ defined as in Corollary \ref{c: continuity of Laustar}.
Let $\nu \in {\mathfrak a}Qqdc \setminus \cup \cH';$ then the function
$\varphi_t^\nu: \lambda :to \varphi_t(\lambda + \nu)$ belongs to
$\cM({}^*\fa_{Q\iq}dc, Y, \Sigma_Q).$
It follows from (\ref{e: previous commutative diagram})
that the functional
$\cL_t^\nu \in \cM({}^*\fa_{Q\iq}dc, \Sigma_Q)^* \otimes {}^\circ {\cal C}(Q,v)$ defined by
$$
\cL_t^\nu[F(\,\cdot\,)] := \cL_t[F(\,\cdot\,)\varphi_t(\,\cdot\, + \nu)],
$$
for $F \in \cM({}^*\fa_{Q\iq}dc, \Sigma_Q) \otimes {}^\circ {\cal C}(Q,v)^*,$
is a ${}^\circ {\cal C}(Q,v)$-valued
$\Sigma_Q$-Laurent functional on ${}^*\fa_{Q\iq}dc.$
The hypothesis (\ref{e: hypo ind rels with i and nu, new}) may be rewritten
as (\ref{e: hypo ind rels with i, new}) with $\cL_t^\nu$ in place of
$\cL_t,$ for each $t\in W_Q.$ By application of Corollary
\ref{c: ind rels with i, new} we therefore obtain, for
$\nu \in {\mathfrak a}Qqdc\setminus \cup \cH',$ that
\begin{equation}
\label{e: conclusion ind rels with i nu, new}
\sum_{t\in W_Q}\cL_t [\;
E_{+,st}({\rm X}\,:\, P_0\,:\, \,\cdot\, + \mu\,:\, x)\, \psi_t(\,\cdot\, + \nu) \;]
=
0
\end{equation}
as an identity of $V_\tau$-valued meromorphic functions in the variable
$\mu \in {\mathfrak a}Qqdc.$ According to Lemma
\ref{l: diagonal action of Laurent functional} the expression
in this equation defines a meromorphic $V_\tau$-valued
function on ${\mathfrak a}Qqdc \times {\mathfrak a}Qqdc$ whose restriction to the diagonal
is a meromorphic function on ${\mathfrak a}Qqdc.$ Thus, if we substitute $\nu$ for $\mu$
in (\ref{e: conclusion ind rels with i nu, new}), we obtain an identity of
$V_\tau$-valued meromorphic functions in the variable $\nu \in {\mathfrak a}Qqdc.$
~
$\square$\medbreak\noindent\medbreak
\begin{cor}
\label{c: induction of relations}
Let ${\cal L}_1,{\cal L}_2 \in {\cal M}({}^*\fa_{Q\iq}dc, \Sigma_Q)^*_{\rm laur} \otimes {}^\circ {\cal C}$.
If, for each $v \in {}^Q\cW$,
\begin{equation}
\label{e: hypo ind rels with pr}
{\cal L}_1[ {E^\circ}p(\spX_{Q,v}\,:\, {}^*P_0\,:\,\,\cdot\,\,:\, m) \,{\scriptstyle\circ}\, {\rm pr}_{Q,v} ]
=
{\cal L}_2[
{E^\circ}(\spX_{Q,v}\,:\,{}^*P_0\,:\, \,\cdot\, \,:\, m) \,{\scriptstyle\circ}\, {\rm pr}_{Q,v} ],
\quad\quad(m \in \spX_{Q,v}p)
\end{equation}
then the following holds
as a $V_\tau$-valued meromorphic identity in $\nu \in {\mathfrak a}Qqdc:$
\begin{equation}
\label{e: conclusion ind rels with pr}
{\cal L}_1[\;
\sum_{s \in W^Q}
E_{+,s}({\rm X}\,:\,{}^*Po\,:\, \,\cdot\, + \nu \,:\, x)\;]
= {\cal L}_2 [\;
{E^\circ}({\rm X}\,:\,{}^*Po\,:\, \,\cdot\, + \nu \,:\, x) \;],\quad\quad(x\in{\rm X}p).
\end{equation}
In particular, for regular values of $\nu,$ the expression
on the left extends smoothly in the variable $x$ to all of $X.$
Conversely, if the identity (\ref{e: conclusion ind rels with pr})
holds for $\nu$ in a non-empty open subset of ${\mathfrak a}Qqdc$, then
(\ref{e: hypo ind rels with pr}) holds for each $v\in{}^Q\cW$.
\end{cor}
\par\noindent{\bf Proof:}{\ }{\ } It follows from (\ref{e: splitting of Eis}) that
${E^\circ}(\spX_{Q,v}\,:\,{}^*P_0\,:\, \lambda)=\sum_{t\in W_Q}
E_{+,t}(\spX_{Q,v}\,:\, {}^*P_0\,:\,\lambda).$
Define ${\cal L}_t \in {\cal M}({}^*\fa_{Q\iq}dc, \Sigma_Q)^*_{\rm laur} \otimes {}^\circ {\cal C}$
for $t\in W_Q$ as follows. If $t=e$ then ${\cal L}_t:={\cal L}_2-{\cal L}_1$,
and otherwise ${\cal L}_t:={\cal L}_2$. Then the hypothesis
(\ref{e: hypo ind rels with pr, new}) in Theorem
\ref{t: induction of relations, new}
follows from (\ref{e: hypo ind rels with pr}).
Hence the conclusion (\ref{e: conclusion ind rels with pr, new})
holds for each $s\in W^Q$. By summation over $s$ this implies that
\begin{equation}
\label{e: equivalent of conclusion ind rels with pr}
\sum_{s\in W^Q}\sum_{t\in W_Q}
\cL_t[E_{+,st}({\rm X}\,:\,{}^*Po\,:\, \,\cdot\, + \nu \,:\, x)]=0,
\quad\quad(x\in {\rm X}p)
\end{equation}
which, by the definition of the operators $\cL_t$ is equivalent
to (\ref{e: conclusion ind rels with pr}).
For the converse, let $g^s(\nu,x)$ denote the expression in
(\ref{e: conclusion ind rels with pr, new}), as in the proof of
Theorem \ref{t: induction of relations, new}, with $\cL_t$
specified as above. Then it was seen in the mentioned proof that
if the sum $g$ of the $g^s$ vanishes then so does each $g^s$
separately. Now (\ref{e: conclusion ind rels with pr})
implies (\ref{e: equivalent of conclusion ind rels with pr})
which exactly reads that $g=0$. Thus
(\ref{e: conclusion ind rels with pr, new}) holds for each $s\in W^Q$,
so that the converse statement in Theorem
\ref{t: induction of relations, new} can be applied.
~
$\square$\medbreak\noindent\medbreak
The result just proved allows a straightforward
corollary similar to Corollary \ref{c: ind rels with i, new},
in which the maps ${\rm Q}v$ are used instead of the maps ${\rm pr}_{Q,v}$.
We omit the details. The following result is derived from Corollary
\ref{c: ind rels version 3, new} in exactly the same way as the first
part of Corollary \ref{c: induction of relations} was
derived from Theorem \ref{t: induction of relations, new}.
\begin{cor}
\label{c: ind rels version 3}
Let $v \in {}^Q\cW.$ Let $\cL_1, \cL_2\in \cM({}^*\fa_{Q\iq}dc, \Sigma_Q)^*_{\rm laur}$ be
$\Sigma_Q$-Laurent functionals on ${}^*\fa_{Q\iq}dc,$
and let $\varphi_1,\varphi_2 \in \cM({\mathfrak a}qdc,\Sigma) \otimes {}^\circ {\cal C}(Q,v).$
Assume that
$$
\cL_1(E_+({\rm X}Qv\,:\, {}^*P_0\,:\, \,\cdot\, \,:\, m) \varphi_1(\,\cdot\, + \nu))
=
\cL_2({E^\circ}({\rm X}Qv\,:\, {}^*P_0\,:\, \,\cdot\,\,:\, m)\varphi_2(\,\cdot\, +\nu)),
$$
for all $m \in {\rm X}Qvp$ and generic $\nu \in {\mathfrak a}Qqdc.$ Define
$\psi_j = (I \otimes {\rm Q}v)\varphi_j \in \cM({\mathfrak a}qdc, \Sigma) \otimes {}^\circ {\cal C},
$
for $j=1,2.$
Then, for every $x\in {\rm X}p,$
$$
\cL_1 (\sum_{s \in W^Q} E_{+,s}({\rm X}\,:\,{}^*Po\,:\, \,\cdot\, + \nu \,:\, x) \psi_1 (\,\cdot\, + \nu ))
=
\cL_2({E^\circ}({\rm X}\,:\,{}^*Po\,:\, \,\cdot\, + \nu \,:\, x) \psi_2(\,\cdot\, + \nu )),
$$
as an identity
of $V_\tau$-valued meromorphic functions in the variable $\nu \in {\mathfrak a}Qqdc.$
\end{cor}
\begin{cor} Let $v\in{}^Q\cW$ and let $\psi_t\in
{\cal M}({}^*\fa_{Q\iq}dc, \Sigma_Q)\otimes {}^\circ {\cal C}(Q,v)$ be given
for each $t\in W_Q$. Let $\lambda_0\in {}^*\fa_{Q\iq}dc$.
Assume that for each $m\in\spX_{Q,v}p$, the meromorphic
$V_\tau$-valued function on ${\mathfrak a}Qqdc$, given by
$$
\lambda:to\sum_{t\in W_Q}E_{+,t}(\spX_{Q,v}\,:\,{}^*P_0\,:\,\lambda\,:\, m)
\psi_t(\lambda),
$$
is regular at $\lambda_0$. Then for $s\in W^Q$,
$x\in{\rm X}p$ and generic $\nu \in {\mathfrak a}Qqdc$
the meromorphic function
\begin{equation}
\label{e: function sum of Epst}
\lambda:to\sum_{t\in W_Q}
E_{+,st}({\rm X}\,:\,{}^*Po\,:\, \lambda+ \nu \,:\, x){\rm Q}v\psi_t(\lambda)
\end{equation}
is also regular at $\lambda_0$.
\end{cor}
\par\noindent{\bf Proof:}{\ }{\ } The function in (\ref{e: function sum of Epst}) has a germ
at $\lambda_0$ in $\cM({}^*\fa_{Q\iq}dc,\lambda_0,\Sigma_Q)$. By
Lemma \ref{l: annihilator of annihilator} it suffices to show
that it is annihilated by $\cM({}^*\fa_{Q\iq}dc,\lambda_0,\Sigma_Q)^{*{\cal O}}_{\rm laur}$.
Let ${\cal L}\in\cM({}^*\fa_{Q\iq}dc,\lambda_0,\Sigma_Q)^{*{\cal O}}_{\rm laur}$ and
define ${\cal L}_t\in{\cal M}({}^*\fa_{Q\iq}dc, \Sigma_Q)^*_{\rm laur} \otimes {}^\circ {\cal C}$
for $t\in W_Q$ by $\cL_t=m^*_{\psi_t}\cL$, see
(\ref{e: previous commutative diagram}). The desired conclusion
now follows from Corollary \ref{c: ind rels with i, new}.~
$\square$\medbreak\noindent\medbreak
We shall now give an equivalent formulation of the induction
of relations. We call it the lifting principle. For the group case
a similar principle was formulated by Casselman,
see \bib{Arthur}, Thm.\ II.4.1, however with Eisenstein integrals
that carry a different normalization.
\def\cA_\laur{{\cal A}_{\rm laur}}
\begin{defi}
\label{d: defi Aspace}
The space $\cA_\laur({\rm X}p\,:\,\tau)$ is defined as
the space of functions
$$x:to \cL[E_+({}^*Po\,:\,\,\cdot\,\,:\, x)]\in V_\tau$$
where $\cL\in{\cal M}({\mathfrak a}qdc,\Sigma)^*_{\rm laur}\otimes{}^\circ {\cal C}$.
It is a linear subspace of $C^\infty({\rm X}p\,:\,\tau).$
\end{defi}
It follows from Corollary \ref{c: Lau to partial Eis} with $Q=G$ that
$\cA_\laur({\rm X}p\,:\,\tau)$ consists of ${\msy D}GH$-finite functions
in $C^{\rm ep}({\rm X}p\,:\,\tau)$.
\begin{rem} Let $\cL\in{\cal M}({\mathfrak a}qdc,\Sigma)^*_{\rm laur}\otimes{}^\circ {\cal C}$.
Then $\cL[\varphi(\,\cdot\,)E_+({}^*Po\,:\,\,\cdot\,)]\in\cA_\laur({\rm X}p\,:\,\tau)$
for all $\varphi\in{\cal M}({\mathfrak a}qdc,\Sigma)$ (see (\ref{e: previous commutative diagram})).
In particular, it follows from (\ref{e: C-function in MerSigma}) that
$C^\circ(s\,:\,\,\cdot\,)\in {\cal M}({\mathfrak a}qdc,\Sigma)\otimes{\rm End}({}^\circ {\cal C}).$
Hence it follows from the identity
(\ref{e: Esp in terms of Ep and C})
that $\cL[E_{+,s}({}^*Po\,:\,\,\cdot\,)]\in\cA_\laur({\rm X}p\,:\,\tau)$
for each $s\in W$. Moreover, by similar reasoning it can be seen that
the space $\cA_\laur({\rm X}p\,:\,\tau)$ does not depend on the choice of
$P_0\in\cP_\gs^{\rm min}$.
\end{rem}
\begin{rem} Let $\lambda_0\in{\mathfrak a}qdc$ and
$\varphi\in{\cal M}({\mathfrak a}qdc,\Sigma)\otimes{}^\circ {\cal C}$, and assume that
$\lambda:to E_+({}^*Po\,:\,\lambda)\varphi(\lambda)$ is regular at $\lambda_0$.
Then the function
$x:to u[E_+({}^*Po\,:\,\lambda\,:\, x)\varphi(\lambda)]|_{\lambda=\lambda_0}$
belongs to $\cA_\laur({\rm X}p\,:\,\tau)$ for each $u\in S({\mathfrak a}qd)$
(see the previous remark and Lemma \ref{l: diff of Laur}).
Moreover, it follows easily from the definition of
${\cal M}({\mathfrak a}qdc,\Sigma)^*_{\rm laur}$ that $\cA_\laur({\rm X}p\,:\,\tau)$ is spanned
by functions of this form.
\end{rem}
\begin{thm}
\label{t: lifting principle}
{\rm (Lifting principle)}
Let $Q\in\cP_\gs$ be a standard parabolic subgroup,
and let $s\in W^Q$ be fixed.
\begin{enumerate}
\item[{\rm (a)}] There exists for each $v\in{}^Q\cW$ a unique linear map
$$F_{+,s,v}\,:\,on \cA_\laur({\rm X}_{Q,v,+}\,:\,\tau_Q)
\rightarrow{\cal M}({\mathfrak a}Qqdc,\Sigma_r(Q),C^\infty({\rm X}p\,:\,\tau))$$ with the following
property. If
$\varphi\in\cA_\laur({\rm X}_{Q,v,+}\,:\,\tau_Q)$ is given by
\begin{equation}
\label{e: 5}
\varphi(m)=
\sum_{t\in W_Q}\cL_t[E_{+,t}(\spX_{Q,v}\,:\,{}^*P_0\,:\,\,\cdot\,\,:\, m)]
\quad\quad(m\in \spX_{Q,v}p),
\end{equation}
for some $\cL_t\in{\cal M}({}^*\fa_{Q\iq}dc, \Sigma_Q)^*_{\rm laur} \otimes {}^\circ {\cal C}(Q,v)$,
$t\in W_Q$, then
\begin{equation}
\label{e: 6}
F_{+,s,v}(\varphi)(\nu,x)=\sum_{t\in W_Q}\cL_t[\;
E_{+,st}({\rm X}\,:\,{}^*Po\,:\, \,\cdot\, + \nu \,:\, x)\;{\rm Q}v\;]
\end{equation}
for $x \in {\rm X}p$ and generic $\nu\in{\mathfrak a}Qqdc$.
\minspace\item[{\rm (b)}]
The function $x:to F_{+,s,v}(\varphi,\nu,x)$ belongs to $\cA_\laur({\rm X}p\,:\,\tau)$
for generic $\nu$.
\minspace\item[{\rm (c)}]
The map $$F_{+,s}\,:\,on\quad \oplus_{v\in{}^Q\cW}\cA_\laur({\rm X}_{Q,v,+}\,:\,\tau_Q)
\rightarrow{\cal M}({\mathfrak a}Qqdc,\Sigma_r(Q),C^\infty({\rm X}p\,:\,\tau)),$$ given by
$F_{+,s}(\varphi)=\sum_v F_{+,s,v}\varphi_v$,
is injective.
\end{enumerate}
\end{thm}
\par\noindent{\bf Proof:}{\ }{\ } The uniqueness is clear from Definition \ref{d: defi Aspace}.
We use (\ref{e: 5}) and (\ref{e: 6}) as the definition of $F_{+,s,v}$;
the fact that $F_{+,s,v}(\varphi)$ is well defined for all
$\varphi\in\cA_\laur({\rm X}_{Q,v,+}\,:\,\tau_Q)$ is equivalent with the first
statement in Theorem \ref{t: induction of relations, new}
(see also Corollary \ref{c: ind rels with i, new}). Once the definition makes sense,
it is easily seen that $F_{+,s,v}(\varphi)$ depends linearly on $\varphi$.
That $F_{+,s,v}(\varphi,\nu)\in\cA_\laur({\rm X}p\,:\,\tau)$ for generic $\nu$
is seen from Lemma \ref{l: eval of Laustar is a Lau}.
Finally, the injectivity of $F_{+,s}$ is equivalent with the
final statement of Theorem \ref{t: induction of relations, new}.~
$\square$\medbreak\noindent\medbreak
\begin{rem} Note that
with $\varphi_v={E^\circ}(\spX_{Q,v}\,:\,{}^*P_0\,:\,\lambda)$ for each $v\in{}^Q\cW$ we obtain
$$\sum_{t\in W_Q}E_{+,st}({\rm X}\,:\,{}^*Po\,:\, \lambda + \nu\,:\, x )\;{\rm Q}v=
F_{+,s,v}(\varphi_v,\nu,x),
$$
for $x \in {\rm X}p$, and hence by summation over $v$ and $s$
$$
{E^\circ}({\rm X}\,:\,{}^*Po\,:\, \lambda + \nu \,:\, x)=
\sum_{s\in W^Q} F_{+,s}(\varphi,\nu,x).
$$
\end{rem}
\begin{rem} In \bib{BSfi}, Definition 10.7, we define the generalized
Eisenstein integral $E^\circ_F(\psi\,:\,\nu)\in{\msy C}i({\rm X}\,:\,\tau)$ for
$\psi\in{\cal C}_F$, $\nu\in{\mathfrak a}_{F{\rm q}{\scriptscriptstyle \C}}^*$ (with the notation of {\it loc.\ cit.}).
By comparison with Theorem \ref{t: lifting principle}
for $Q=P_F$ it is easily seen that
$E^\circ_F(\psi\,:\,\nu\,:\, x)=F_{+,1}(\psi,\nu,x)$
for $x\in{\rm X}p$.
\end{rem}
\section{Appendix A: spaces of holomorphic functions}
\label{s: certain function spaces}
If
$\Omega$ is a complex analytic manifold, then by ${\cal O}(\Omega)$ we denote the space
of holomorphic and by $\cM(\Omega)$ the space of meromorphic functions on $\Omega.$
If $V$ is a complete locally convex (Hausdorff) space, we say that a function $\varphi: \Omega \rightarrow V$ is
holomorphic if for every $a \in \Omega$ there exists a holomorphic coordinatisation $z = (z_1, \ldots, z_n)$
at $a$ such that in a neighborhood of $a$ the function $\varphi$ is expressible as
a converging $V$-valued power series in the coordinates $z.$ The space of such holomorphic
functions is denoted by ${\cal O}(\Omega, V).$ We equip this space with a
locally convex topology as follows. Let ${\cal P}$ be a separating collection of
continuous seminorms
for $V.$ For every $p \in {\cal P}$ and every compact set $K \subset \Omega$ we define
the seminorm $\nu_{K,p}$ on ${\cal O}(\Omega, V)$ by $\nu_{K,p}(\varphi) = \sup_{K} p\,{\scriptstyle\circ}\, \varphi.$
This collection of seminorms is separating
hence equips ${\cal O}(\Omega, V)$ with a locally convex topology.
Note that this topology is independent of the choice of ${\cal P}.$ Moreover, it is complete; it is
Fr\'echet if $V$ is a Fr\'echet space.
We recall that ${\cal O}(\Omega, V)$ is a closed subspace of ${\msy C}i(\Omega, V).$
Indeed, if $\bar\partial$ denotes the anti-linear part of exterior differentiation,
then ${\cal O}(\Omega, V)$ is the kernel of $\bar\partial$ in ${\msy C}i(\Omega, V).$
A densely defined function $\varphi: \Omega \rightarrow V$ is called meromorphic
if for every $a \in \Omega$ there exists an open neighborhood
$U$ of $a,$
and a function $\psi \in {\cal O}(U)\setminus\{0\}$ such that $\psi \varphi \in {\cal O}(U,V).$
As usual, meromorphic functions are considered to be equal if
they coincide on a dense open subset. The space of $V$-valued meromorphic
functions on $\Omega$ is denoted by $\cM(\Omega, V).$
If $\varphi$ is an $V$-valued
meromorphic function on $\Omega$ we define ${\rm reg}(\varphi)$ to be the largest open
subset $U$ of $\Omega$ for which $\varphi|_U$ coincides (densely) with an element of ${\cal O}(U, V).$
The complement ${\rm sing}(\varphi) = \Omega\setminus {\rm reg}(\varphi)$ is called the singular locus of $\varphi.$
\begin{lemma}
\label{l: natural isomorphism of function spaces}
Let $X$ be a $C^\infty$ and $\Omega$ a complex analytic manifold.
Let $V$ be a complete locally convex space.
Let ${\cal F}$ be the locally convex space of $C^\infty$-functions
$X \times \Omega \rightarrow V$ that are holomorphic in the second variable.
Given $f \in {\cal F}$ and $x \in X,$ we define the function
${}_1f(x) : \Omega \rightarrow V$ by ${}_1f(x)(z) = f(x, z).$ Given $z \in \Omega$
we define the function ${}_2f(z): X \rightarrow V$ by ${}_2f(z)(x) = f(x, z).$
\begin{enumerate}
\item[{\rm (a)}]
The map $f :to {}_1f$ defines a natural isomorphism
of locally convex spaces from ${\cal F}$ onto ${\msy C}i(X, {\cal O}(\Omega, V)).$
\minspace\item[{\rm (b)}]
The map $f :to {}_2f$ defines a natural isomorphism
of locally convex spaces from ${\cal F}$ onto ${\cal O}(\Omega, {\msy C}i(X,V)).$
\end{enumerate}
In particular, the above maps lead to a natural isomorphism
$$
{\msy C}i(X, {\cal O}(\Omega, V)) \simeq {\cal O}(\Omega, {\msy C}i(X, V)).
$$
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
The above isomorphisms are valid with ${\cal O}$ replaced by ${\msy C}i$ everywhere.
This is a well known fact, and basically a
straightforward consequence of the definitions, though somewhat tedious to check.
The isomorphisms with ${\cal O}$ are seen to be valid by showing that the appropriate
kernels of the operator $\bar\partial$ correspond.
Checking this involves a local application of the multivariable Cauchy integral formula.
~
$\square$\medbreak\noindent\medbreak
\section{Appendix B: removable singularities}
\label{s: appendix b}
We discuss a variation on the idea of removable singularities
for holomorphic functions that is particularly useful for application
in the present paper.
A subset $T$ of a finite dimensional complex analytic manifold $\Omega$ will be called thin
if for every $\lambda\in \Omega$ there exists a connected open neighborhood $U$ and
a non-zero holomorphic function $\varphi \in {\cal O}(U)$
such that $T \cap U \subset \varphi^{-1}(0)$, see
\bib{GuRo}, p.\ 19.
An open subset $U$ of $\Omega$ will be called full if its complement is thin.
It is clear that a full subset of $\Omega$ is dense in $\Omega$.
Note that the union of finitely many thin subsets is thin again; accordingly,
the intersection of finitely many full open subsets of $\Omega$ is again a full open
subset. Obviously any union of full open subsets
is a full open subset.
Note also that if $\Omega$ is connected, then every full open subset of $\Omega$
is connected (\bib{GuRo} p.\ 20).
\begin{lemma}
Let $j: V \rightarrow W$ be an injective continuous linear
map of complete locally convex Hausdorf spaces,
and let $F$ be a $W$-valued holomorphic function
on a complex analytic manifold $\Omega.$
Assume that there exists a full open subset $\Omega_0$ of $\Omega$
and a holomorphic function $G_0: \Omega_0 \rightarrow V$ such that
such that $F = j \,{\scriptstyle\circ}\, G_0$ on $\Omega_0.$
Then there exists a unique holomorphic map $G : \Omega \rightarrow V$
such that $j\,{\scriptstyle\circ}\, G = F.$
\end{lemma}
\par\noindent{\bf Proof:}{\ }{\ }
Clearly the result is of a local nature
in the $\Omega$-variable, so that we may assume that
$\Omega$ is a connected open subset of ${\msy C}^n,$ for some $n \in {\msy N}.$
Moreover, we may as well assume that $\Omega_0 = \Omega\setminus \varphi^{-1}(0),$
with $\varphi \in {\cal O}(\Omega)$ a non-zero holomorphic function.
Fix $\lambda_0 \in \Omega.$ Since $\varphi$ is non-zero, the function
$z:to \varphi(\lambda_0+z\mu)$, defined on a neighborhood of 0 in ${\msy C}$,
is non-zero for some $\mu\in{\msy C}^n\setminus \{0\}$. Being holomorphic,
this function then takes the value 0 in isolated points. Hence we may
choose $\mu$ such that $\lambda_0+z\mu\in\Omega_0$ for
$0<|z|\le 1$. By compactness there exists an open neighborhood $N_0$
of $\lambda_0$ in $\Omega$ such that $\lambda + z \mu \in
\Omega$ for all $\lambda \in N_0$
and $z \in {\msy C}$ with $|z|\le 1,$ and such that
$\lambda + z \mu \in\Omega_0$ for $|z| =1.$
By the Cauchy integral formula we have:
\begin{equation}
\label{e: Cauchy int}
F(\lambda) = \frac{1}{2\pi i} \int_{\partial D} F(\lambda + z \mu) \frac{dz}{z}.
\end{equation}
Here $\partial D$ denotes the boundary of the unit circle
in ${\msy C},$ equipped with the orientation induced by the complex structure,
i.e., the counter clockwise direction.
Note that the $W$-valued (or $V$-valued) integration is well defined, since
$W$ (or $V$) is complete locally convex. In the integrand
of (\ref{e: Cauchy int}) the function
$F(\lambda + z \mu)$ may be replaced by
$ j(G_0(\lambda + z \mu)).$ Using that
$j$ is continuous linear we then obtain that
\begin{equation}
\label{e: F is j F zero}
F(\lambda) = j ( G(\lambda) ),
\end{equation}
where
$$
G(\lambda) := \frac{1}{2 \pi i} \int_{\partial D}
G_0(\lambda + z \mu) \; \frac{dz}{z}\quad\quad (\lambda\in N_0).
$$
Clearly $G: N_0 \rightarrow V$ is a holomorphic function;
moreover, it is uniquely determined by equation
(\ref{e: F is j F zero}), since $j$ is injective. This implies
that the local definition of $G$ is independent of the particular
choice of $\mu.$ Moreover, it also follows
from (\ref{e: F is j F zero}) and the injectivity of
$j$ that all local definitions
match and determine a global holomorphic function $G: \Omega \rightarrow V$
satisfying our requirement.
~
$\square$\medbreak\noindent\medbreak
\begin{cor}
\label{c: aux smooth extension}
Let $\Omega_0$ be a full open subset of a complex analytic manifold $\Omega$
and let $X_0$ be a dense open subset of a $C^\infty$-manifold
$X.$ Moreover, let $F: \Omega \times X_0 \rightarrow {\msy C}$ be
a $C^\infty$ function that is holomorphic in its first variable,
and assume that its restriction to $\Omega_0 \times X_0$ has
a smooth extension to $\Omega_0 \times X.$ Then the
function $F$ has a unique smooth extension to $\Omega \times X.$
Moreover, the extension
is holomorphic in its first variable.
\end{cor}
\par\noindent{\bf Proof:}{\ }{\ }
As in the proof of the above lemma we may as well assume
that $\Omega$ is an open subset of ${\msy C}^n,$ for some $n.$
Let $V = C^\infty(X)$ and $W= C^\infty(X_0)$ be equipped with
the usual Fr\'echet topologies. Restriction to $X_0$
induces an injective continuous linear map $j: V \rightarrow W.$
By
Lemma \ref{l: natural isomorphism of function spaces}(b)
we see that the function
$\widetilde F: \Omega \rightarrow W ,$
defined by $\widetilde F(z) = F(z, \,\cdot\,)$
is holomorphic. Let $G_0$ be the extension
of $(z, x) :to F(z, x)$ to
a smooth map $\Omega_0 \times X \rightarrow {\msy C}.$ Then by density and continuity
the function $G_0$ satisfies the Cauchy-Riemann equations
in its first variable. Hence it is holomorphic in its
first variable, and it follows that the function
$\widetilde G_0: \Omega_0 \rightarrow V$ defined by $\widetilde G_0(z) = G_0(z, \,\cdot\,)$
is holomorphic. {}From the definitions given
we obtain that $ \widetilde F = j \,{\scriptstyle\circ}\, \widetilde G_0$ on $\Omega_0.$
By the above lemma
there exists a unique holomorphic function $\widetilde G: \Omega \rightarrow V$
such that $\widetilde F = j \,{\scriptstyle\circ}\, \widetilde G.$
The function $G: (z, x) :to \widetilde G(z)(x)$
is the desired extension of $F.$
~
$\square$\medbreak\noindent\medbreak
\def{\rm ad}\,ritem#1{\hbox{\small #1}}
\def\hbox{\hspace{3.5cm}}{\hbox{\hspace{3.5cm}}}
\def@{@}
\def{\rm ad}\,derik{\vbox{
{\rm ad}\,ritem{E.P. van den Ban}
{\rm ad}\,ritem{Mathematisch Instituut}
{\rm ad}\,ritem{Universiteit Utrecht}
{\rm ad}\,ritem{PO Box 80 010}
{\rm ad}\,ritem{3508 TA Utrecht}
{\rm ad}\,ritem{Netherlands}
{\rm ad}\,ritem{E-mail: ban{@}math.uu.nl}
}
}
\def{\rm ad}\,dhenrik{\vbox{
{\rm ad}\,ritem{H. Schlichtkrull}
{\rm ad}\,ritem{Matematisk Institut}
{\rm ad}\,ritem{K\o benhavns Universitet}
{\rm ad}\,ritem{Universitetsparken 5}
{\rm ad}\,ritem{2100 K\o benhavn \O}
{\rm ad}\,ritem{Denmark}
{\rm ad}\,ritem{E-mail: [email protected]}
}
}
\hbox{\vbox{{\rm ad}\,derik}\vbox{\hbox{\hspace{3.5cm}}}\vbox{{\rm ad}\,dhenrik}}
\end{document}
|
\begin{document}
\title{Whitney towers and the Kontsevich integral}
\author{Rob Schneiderman\\Peter Teichner}
\address{Courant Institute of Mathematical Sciences,
New York University\\251 Mercer Street, New York, NY 10012-1185, USA}
\address{Department of Mathematics, University of California\\Berkeley,
CA 94720-3840, USA}
\gtemail{\mailto{[email protected]},
\mailto{[email protected]}}
\asciiemail{[email protected], [email protected]}
\begin{abstract}
We continue to develop an obstruction theory for embedding $2$--spheres
into $4$--manifolds in terms of Whitney towers. The proposed intersection
invariants take values in certain graded abelian groups generated
by labelled trivalent trees, and with relations well known from the
3--dimensional theory of finite type invariants. Surprisingly, the
same exact relations arise in 4 dimensions, for example the Jacobi (or
IHX) relation comes in our context from the freedom of choosing Whitney
arcs. We use the finite type theory to show that our invariants agree
with the (leading term of the tree part of the) Kontsevich integral
in the case where the $4$--manifold is obtained from the $4$--ball by
attaching handles along a link in the $3$--sphere.
\epsilonnd{abstract}
\asciiabstract{We continue to develop an obstruction theory for embedding 2-spheres
into 4-manifolds in terms of Whitney towers. The proposed intersection
invariants take values in certain graded abelian groups generated
by labelled trivalent trees, and with relations well known from the
3-dimensional theory of finite type invariants. Surprisingly, the
same exact relations arise in 4 dimensions, for example the Jacobi (or
IHX) relation comes in our context from the freedom of choosing Whitney
arcs. We use the finite type theory to show that our invariants agree
with the (leading term of the tree part of the) Kontsevich integral
in the case where the 4-manifold is obtained from the 4-ball by
attaching handles along a link in the 3-sphere.}
\primaryclass{57M99}
\secondaryclass{57M25}
\keywords{$2$--sphere, $4$--manifold, link concordance,
Kontsevich integral, Milnor invariants, Whitney tower}
\asciikeywords{2-sphere, 4-manifold, link concordance,
Kontsevich integral, Milnor invariants, Whitney tower}
\maketitlepage
\cl {\small\it Dedicated to Andrew Casson on the occasion of his 60th
birthday}
\section{Introduction}
Two of Andrew Casson's wonderful contributions to topology were
his work on {\epsilonm flexible handles} (now called {\epsilonm Casson
towers}) in $4$--manifolds, and his invariant for homology
$3$--spheres, counting representations into $SU(2)$. In this paper
we will describe an obstruction theory for disjointly embedding
collections of 2--spheres (or 2--disks with fixed boundary) into a
4--manifold that provides a connection between these two aspects
of Casson's work. This connection is somewhat indirect, otherwise
our paper would be called {\epsilonm Casson towers and the Casson
invariant}. In other words, we shall switch from Casson towers to
Whitney towers, and from the Casson invariant to the Kontsevich
integral. It would be very satisfying to find a more
straightforward relationship between Casson's two contributions.
To explain the connection, recall that the Casson invariant is the
lowest order (nontrivial) {\epsilonm finite type invariant} of homology
$3$--spheres. These finite type invariants take values in certain
graded abelian groups generated by trivalent graphs. Being {\epsilonm
invariants}, they measure the {\epsilonm uniqueness} of $3$--manifolds
or links in $3$--manifolds. We shall explain how similar graphs,
better, unitrivalent trees, arise in {\epsilonm existence} questions for
$4$--manifolds or surfaces in $4$--manifolds. It is not totally
surprising that raising the dimension by one takes uniqueness to
existence questions, after all an isotopy of, say, a knot in a
$3$--manifold $M^3$ is nothing but a certain annulus in the
$4$--manifold $M \times I$. However, the details of such a
translation from one dimension to the next are not at all obvious.
In the easiest setting one would like to find obstructions for
making the images of maps $A_i\co(D^2,S^1)\to (X^4,\partial X)$ {\epsilonm
disjoint}, without changing the homotopy classes (and without
trying to embed the $A_i$). In fact, Casson's main Embedding
Theorem in \cite[Lecture 1]{Ca} is an example of a special case of
this problem: Casson showed that if $X$ is simply connected, all
intersection numbers between the $A_i$ vanish, and $A_i$ have {\epsilonm
algebraic dual spheres}, then the problem has a positive solution.
He used inverses of the Whitney move, now known as Casson or
finger moves, to introduce many self-intersections, while
trivializing the fundamental group of the complement of one disk
at a time (and hence enabling the other disks to be mapped {\epsilonm
disjointly}). He then went on to construct Casson towers (with
prescribed boundary circles) by iterating the procedure
indefinitely, using the fact that the complement of a finite
height Casson tower can be made simply connected. These ideas
inspired Mike Freedman who proved in \cite{F} that a neighborhood
of a Casson tower actually contains an embedded flat disk.
The presence of algebraic dual spheres in Casson's theorem comes
from the fact that the proposed application was to the s-cobordism
theorem and to the exactness of the surgery sequence in
dimension~4. Indeed, Freedman's theorem implies these results in
the topological category (for {\epsilonm good} fundamental groups).
There is a more general context in which disjoint maps of disks or
spheres can be constructed, namely in the presence of a {\epsilonm
non-repeated Whitney tower} (of sufficiently high order), see
Theorem~\ref{thm:nonrepeating} below and \cite{ST2}. The first
order stage of this Whitney tower is guaranteed by the vanishing
of the intersection numbers whereas the existence of the higher
order stages are obstructed by our new proposed invariants. They
take values in certain graded abelian groups generated by
trivalent trees, which are basically the spines of the Whitney
towers. The difference between a Casson tower and a Whitney tower
is that in the latter, fewer disks are attached at each stage: In
a Casson tower, every intersection point $p$ leads to a new disk
(with boundary an arc leaving on one sheet at $p$ and arriving at
the other sheet), whereas a Whitney tower only has a new disk for
certain {\epsilonm pairs} of intersection points. In particular, it is
usually only possibly to find Casson towers in simply connected
$4$--manifolds, whereas Whitney towers are not restricted by the
fundamental group. In fact, in our theory the fundamental group
leads to a decoration of the trivalent trees in question, thus
giving a much bigger variety of possible obstructions. In
addition, Freedman's reimbedding theorem shows that a Casson tower
of height 3 already contains an embedded flat disk. However, there
are Whitney towers of arbitrary order {\epsilonm not} containing disks,
which explains the use of these ``weaker'' towers in an
obstruction theory.
Our Theorem~\ref{thm:nonrepeating} implies Casson's result because
algebraic dual spheres can be used to construct non-repeating
Whitney towers of arbitrary order. This is already implicit in
\cite{FQ}, so our main contribution is a theory {\epsilonm in the
absence} of algebraic dual spheres. For example, this applies to
concordance questions for links in $3$--space. In this context we
prove in Theorem~\ref{thm:Milnor} below that our invariants agree
rationally with (the leading term of) the tree part of the
Kontsevich integral, which is the universal finite type
concordance invariant \cite{HM}. This relates our obstruction
theory to the finite type theory and, in particular, to the Casson
invariant. It should be mentioned here that Habegger and Masbaum
show in \cite{HM} that (the leading term of) the tree part of the
Kontsevich integral carries exactly the same information as
Milnor's $\overline{\mu}$--invariants which were first observed to
be concordance invariants by Casson in \cite{Ca2}. Reversing the
logic, we have found a $4$--dimensional geometric interpretation
of this part of the Kontsevich integral, in terms of higher order
intersections among Whitney disks. See \cite{CT2} for an
interpretation in terms of {\epsilonm gropes} in $3$--dimensions which
is stronger in the sense that it works for (the leading term of)
the Kontsevich integral, not just of the tree part.
At the time of writing, the setting of Theorem~\ref{thm:Milnor} is
actually the only case where we have a proof that our intersection
invariant is independent of the choice of a Whitney tower, but see
Conjecture~\ref{conj:well defined}. What we do prove in
Theorem~\ref{thm:build-tower} is that the vanishing of our
intersection invariant for a Whitney tower of order~$n$ enables
one to build a Whitney tower of the next order~$(n+1)$. In that
sense, we are producing an obstruction theory since disjointly
embedded sheets $A_i$ allow Whitney towers of arbitrary order.
We close this introduction by pointing out that the Whitney towers
used in this paper are generalizations of the ones in \cite{COT}
in that disks of higher order are here allowed to intersect
previous stages, as long as these intersection points are paired
up by Whitney disks (up to the desired order). In our language, the distinction is made in
terms of saying that these Whitney towers have an {\epsilonm order}
whereas the Whitney towers of \cite{COT} (where different order
Whitney disks don't intersect) have a {\epsilonm height}. This is the
precise analogue of {\epsilonm class} versus {\epsilonm height} in the theory
of gropes, see eg \cite{T}, ultimately coming from the
distinction between the lower central series and the derived
series of a group. The latter explains why Whitney towers with
a height carry more subtle information. In fact, they are {\epsilonm
not} related to the usual finite type theory and hence it is much
more difficult to define an obstruction theory. At present, such a
theory only exists for knot concordance \cite{COT}, \cite{CT} (using von
Neumann signatures to prove nontriviality) and it would be
extremely interesting to develop it more generally, ie in the
context of $2$--spheres in $4$--manifolds.
\section{Statement of results}
We continue to develop the obstruction theory for embedding $2$--spheres
into $4$--manifolds started in \cite{ST}. To fix notation, let $X$ be a
$4$--manifold and $A_1,\dots, A_m$ be generic immersions of $2$--spheres
(or $2$--disks with fixed boundary) into $X$. We shall work in the smooth
setting, even though the techniques of \cite{FQ} allow a generalization
of our work to locally flat surfaces in a topological manifold. The goal
is to construct obstructions for changing the $A_i$, in their regular
homotopy class, to embeddings with disjoint images. This is already a
very interesting problem for $m=1$ but we shall not restrict to this case.
The first, well known, invariants are the Wall intersection ``numbers''
\cite{W}
$$\negativeleftarrowmbda(A_i,A_j) = \sum_{p\in A_i\cap A_j} \epsilon_p\cdot g_p \quad
\in \mathbb{Z}\pi, \quad \pi:=\pi_1X.$$
These count how often $A_i$ and $A_j$ intersect algebraically, including
a group element $g_p\in\pi$ and a sign $\epsilon_p$ for each intersection
point. Similarly, there are self-intersection numbers $\mu(A_i)$ which
are well defined only in a certain quotient of the group ring, see
below. Recall that in higher dimensions (where $A_i$ are $k$--spheres,
$k>2$ and $X$ is $2k$--dimensional) the vanishing of these invariants
implies that after a finite sequence of Whitney moves \cite{Wh} the $A_i$
can be represented by disjoint embeddings. In dimension 4, there are well
known problems to this procedure (since $2+k=2k$ for $k=2$), the most
important one being that, generically, the Whitney disks intersect the
$2$--spheres $A_i$. The first precise statement concerning the failure
of the Whitney trick in dimension~4 was given by Kervaire and Milnor
in \cite{KM}.
In \cite{ST} we assumed that these primary intersection numbers
vanish which means geometrically that all intersections and
self-intersections can be paired by Whitney disks: For each pair
of intersection points between $A_i$ and $A_j$ (if $i=j$ these are
self-intersections), choose one {\epsilonm Whitney arc} on $A_i$ and one
on $A_j$ connecting these two points. Since the fundamental group
is controlled in Wall's invariant, the two Whitney arcs together
form a null homotopic circle in the ambient $4$--manifold, which
hence bounds a disk, the {\epsilonm Whitney disk}. Using a choice for
such disks, one for each pair of intersection points, we
constructed a secondary invariant
$$\tau(A_i,A_j,A_k) \in \mathbb{Z}\pi \times \mathbb{Z}\pi /\dots$$
which measures how the Whitney disks intersect the spheres $A_i$. Here the
indices $i,j,k$ may be repeated, obtaining several slightly distinct
geometrical cases just like for Wall's invariants. We recall that
by standard procedures the Whitney disks can always be assumed to be
disjointly embedded (and framed), and that the only thing which hinders a
successful Whitney move is the fact that they are in general not disjoint
from the original spheres $A_i$.
We will first explain a way to unify the above invariants, then
suggest a vast generalization and finally discuss a relation to
Milnor invariants and the Kontsevich integral (for classical links).
For this purpose, assume that the $A_i$ intersect and self-intersect
generically, and call the collection $A_1,\dots,A_m$ a {\epsilonm Whitney
tower of order~0}. Similarly, if Wall's invariants vanish, and one has
chosen generic Whitney disks $W_I$ which pair all intersections and
self-intersections of the $A_i$ then one obtains a {Whitney tower of
order~1}. If the $\tau$--invariants vanish, then one can chose Whitney
disks for all the intersections of the $A_i$ with the $W_I$ to obtain
a {Whitney tower of order~2}. This procedure can be continued and
we give a precise definition of a {\epsilonm Whitney tower of order~n} in
Section~\ref{W-tower-tau-sec}. This definition includes orientations of
all the surfaces $A_i,W_I,\dots$ in the tower, as well as base points
on these surfaces together with whiskers connecting these base points
to the base point of $X$.
\begin{figure}[ht!]
\centerline{\includegraphics[scale=.4]{W-t5.eps}}
\caption{Part of a Whitney tower (left), and part of the
unitrivalent tree $t_p$ associated to an unpaired intersection point
$p$ in a Whitney tower (right).}
\negativeleftarrowbel{W-tower5-int-point-tree-fig}
\epsilonnd{figure}
\subsection{The intersection tree $\tau_n(\mathcal{W})$}
\negativeleftarrowbel{tau-intro}
Our first observation is that one can canonically associate to
each {\epsilonm unpaired} intersection point $p$ of a Whitney tower
$\mathcal{W}$ a {\epsilonm decorated} unitrivalent tree $t_p$ of {\epsilonm
order}~$n$. The order is the number of trivalent vertices and the
decoration is as follows: the univalent vertices of $t_p$ are
labelled by the $A_i$ or more abstractly, by $i\in\{1,\dots,m\}$,
the edges are labelled by elements from the fundamental group
$\pi$, and the edges and trivalent vertices are oriented. The tree
$t_p$ sits naturally as a subset of $\mathcal{W}$
(Figure~\ref{W-tower5-int-point-tree-fig}, details in
Section~\ref{W-tower-tau-sec}) with each trivalent vertex lying in
a Whitney disk and each univalent vertex lying in some $A_i$. Each
edge of $t_p$ is a sheet-changing path between vertices in
adjacent surfaces, with the group element labelling the edge
determined by the loop formed from the path together with the
whiskers on the adjacent surfaces. For example, in a Whitney tower
of order~0, any intersection point $p$ between $A_i$ and $A_j$ has
order~0 and gives a tree $t_p$ consisting of a single edge whose
univalent vertices (labelled by $i$ and $j$) correspond to
basepoints in $A_i$ and $A_j$. This edge is labelled by the group
element $g_p$ determined by a loop formed from the whiskers on
$A_i$ and $A_j$ together with a path that changes sheets at $p$
where the orientation of the edge corresponds to the direction of
the path. For intersection points of order~1 in an order--1 Whitney
tower, one gets decorated Y--trees with one trivalent vertex and
three univalent vertices labelled by $i,j,k$ (which can repeat).
The central point of this paper is that in an order--$n$ Whitney
tower $\mathcal{W}$ the trees that correspond to the (unpaired) order--$n$
intersection points of $\mathcal{W}$ represent a ``higher order''
obstruction to homotoping (rel boundary) the $A_i$ to disjoint
embeddings. Just like the intersection number $\negativeleftarrowmbda(A_i,A_j)$
is a sum over all intersection points between $A_i$ and $A_j$, we
define the {\epsilonm intersection tree} $\tau_n(\mathcal{W})$ of an order--$n$
Whitney tower $\mathcal{W}$ to be
$$\tau_n(\mathcal{W}):= \sum_p \epsilon_p\cdot t_p \quad \in \mathcal{T}_n(\pi,m).$$
The sum is taken over all order--$n$ intersection points $p$ in
$\mathcal{W}$ and we consider this sum as taking values in the free
abelian group generated by (isomorphism classes of) decorated
trees as above, modulo several relations that are motivated
geometrically (explained briefly below and in detail in
Section~\ref{W-tower-tau-sec}, particularly
Section~\ref{subsec:T}). We denote this quotient by
$$\mathcal{T}(\pi,m)=\bigoplus_{n=0}^\infty \mathcal{T}_n(\pi,m),$$
where the order~$n$ is the number of trivalent vertices and the univalent
labels come from $\{1,\dots,m\}$, possibly repeated. If this index set
is undetermined (or unimportant) we shall just write $\mathcal{T}_n(\pi)$.
The order--0 trees are just single edges and it turns out that
$$\mathcal{T}_0(\pi,1) \cong \mathbb{Z}\pi/ \negativeleftarrowngle \bar g- g \negativerightarrowngle, \quad
\bar g:=w_1(g)\cdot g^{-1},$$
where $w_1\co\pi\to \mathbb{Z}/2$ is the first Stiefel--Whitney class of the
ambient 4--manifold. The quotient comes from the fact that an edge
with two identical labels has an additional symmetry which changes
the orientation of the edge. Moreover, our invariant $\tau_0$
gives exactly Wall's self-intersection invariant $\mu$. To get
Wall's intersection number $ \negativeleftarrowmbda(A_1,A_2)$ we just need to
evaluate $\tau_0$ in order~0 with {\epsilonm exactly} two labels~$1,2$.
The invariants $\tau$ from \cite{ST} are exactly $\tau_1(\mathcal{W})$ in
the various versions of $\mathcal{T}_1(\pi)$, depending on the allowed
labels.
A short discussion of the relations in $\mathcal{T}(\pi)$ is in order.
They reflect the various choices made in the construction of the
Whitney tower, as will be discussed in
Section~\ref{W-tower-tau-sec} (see also Figure~\ref{Relations-fig}
in Section~\ref{W-tower-tau-sec}). As a consequence, working {\epsilonm
modulo} these relations makes our intersection tree $\tau_n$ {\epsilonm
independent} of the choices below.
\begin{itemize}
\item Changing orientations on Whitney disks gives AS,
{\epsilonm antisymmetry} relations; they introduce a sign when the cyclic
ordering of a trivalent vertex is switched.
\item Changing the orientation of an edge changes the label $g$ to $\bar
g$, the OR {\epsilonm orientation} relation.
\item Changing the whiskers gives HOL, {\epsilonm holonomy} relations; they
multiply the labels of 3 edges coming into a
trivalent vertex by a group element.
\item Changing the choice of Whitney arcs, ie of the boundaries of
Whitney disks, gives the IHX relations.
\epsilonnd{itemize}
The last type of relations, well known in dimension~3, is maybe the
most surprising aspect of our $4$--dimensional theory. We feel that
our explanation in terms of the indeterminacy of Whitney arcs is very
satisfying \cite{CST}. It should be pointed out that graded abelian
groups like $\mathcal{T}(\pi)$ arose independently in the $3$--dimensional work
of Garoufalidis, Kricker and Levine \cite{GK}, \cite{GL}. They study
trivalent graphs (instead of unitrivalent trees) and $\pi$ is usually
a $3$--manifold group. In some form, the Kontsevich integral gives
invariants of links (or $3$--manifolds) with values in such graded
abelian groups. So these are invariants for the {\epsilonm uniqueness} of
$3$--dimensional objects, whereas our invariants measure {\epsilonm existence}
of $4$--dimensional things. In that sense, it might not come as a surprise
that there is an overlap between these theories. Note that the restriction
to trees is a well known feature if one wants {\epsilonm concordance invariants}
in the $3$--dimensional context, see \cite{CT2} or \cite{HM}.
To make it possible that the intersection tree $\tau_n(\mathcal{W})$ only
depends on the $A_i$, it is in fact necessary to introduce two
more types of relations which correspond to changing the choices
of Whitney disks (for fixed choices of boundaries:
\begin{itemize}
\item The INT {\epsilonm interior or intersection} relations come from the
choice of the {\epsilonm interiors} of Whitney disks (which can be changed by
summing into any $2$--spheres). More generally, they measure
indeterminacies coming from certain lower order intersection trees for
Whitney towers on subsets of the $A_i$ together with other $2$--spheres. A
special case of these relations will be examined in detail in \cite{ST2}.
\item The FR {\epsilonm framing} relations are generated by certain $2$--torsion
elements which correspond to manipulations of the interiors of Whitney
disks that affect their normal framings. This will be described in
\cite{ST3} but see Figure~\ref{FR-relations-fig}.
\epsilonnd{itemize}
\begin{figure}[ht!]
\centerline{\includegraphics[scale=.4]{FR-rel.eps}}
\caption{FR relations in order one and three (in a simply-connected
$4$--manifold).}
\negativeleftarrowbel{FR-relations-fig}
\epsilonnd{figure}
The INT relations are more subtle in that they actually depend on
the ambient $4$--manifold $X$, rather than just on its fundamental
group. Both, INT and FR relations will not play a role in this
paper, however we will provide evidence supporting the following
conjecture by proving a closely related special case.
\begin{conj} \negativeleftarrowbel{conj:well defined}
The intersection tree $\tau_n(\mathcal{W})\in \mathcal{T}_n(\pi,m)/\mathrm{INT, FR}$
is independent of the choice of the Whitney tower $\mathcal{W}$. In fact, it
only depends on the regular homotopy classes of the original maps $A_i$,
and should be written as $\tau_n(A_1,\dots,A_m)$.
\epsilonnd{conj}
This result is well known in the Wall case, ie for $n=0$, and it was
proven in general for $n=1$ in \cite{ST} (and previously in the simply
connected case for $n=1$ in \cite{Ma} and \cite{FQ}).
The following result reflects the obstruction theoretic nature of the
intersection tree $\tau$.
\begin{thm} \negativeleftarrowbel{thm:build-tower}
Let $A_i$ be properly immersed simply-connected surfaces in a
$4$--manifold, or connected surfaces in a simply-connected
$4$--manifold. If $\mathcal{W}$ is an order--$n$ Whitney tower on the $A_i$ with
vanishing intersection tree $\tau_n(\mathcal{W})\in \mathcal{T}_n(\pi,m)$, then there
is an order--$(n+1)$ Whitney tower on maps $A_i'$ which are regularly
homotopic (rel boundary) to $A_i$.
\epsilonnd{thm}
Theorem~\ref{thm:build-tower} will be proved in
Section~\ref{sec:build-tower-proof}.
\subsection{Immersions with disjoint images} \negativeleftarrowbel{sec:disjoint}
A special case of our invariant only counts those trees $t_p$
whose univalent labels are non-repeating, which means that the
number $m$ of spheres $A_i$ is two more than the order $n$ of the
intersection point $p$, $m=n+2$. Geometrically, one wants to
totally ignore {\epsilonm self}-intersections of the spheres $A_i$ and
in fact none of the (higher order analogues of) self-intersections
in the Whitney tower are paired up. This leads to the notion of a
{\epsilonm non-repeated Whitney tower} $\mathcal{W}$ which has also a {\epsilonm
non-repeated intersection tree} $\negativeleftarrowmbda(\mathcal{W})$ that generalizes
the $\negativeleftarrowmbda$--invariant of Wall's intersection form. We shall
explain these notions in a different paper \cite{ST2} where we
also prove the following beautiful application of the theory.
\begin{thm} \negativeleftarrowbel{thm:nonrepeating}
If the 2--spheres $A_1,\dots, A_{n+2}$ admit a non-repeated Whitney
tower $\mathcal{W}$ of order~$n$, such that $\negativeleftarrowmbda(\mathcal{W})$ vanishes in
$\mathcal{T}_n(\pi,n+2)$, then the homotopy classes (rel boundary) of the
$A_i$ can be represented by immersions with disjoint images.
\epsilonnd{thm}
Again, this result was well known for $n=0$ (see eg \cite{Ko}),
and was proven for $n=1$ in \cite{ST} (and for trivial fundamental
group in \cite{Y}). In the special case discussed in the next
section, this result says that a link $L$ in $S^3$ has vanishing
non-repeating Milnor invariants if and only if it bounds disjoint
immersions of disks in $D^4$. In fact, this singular concordance
can then be improved to a {\epsilonm link homotopy} from $L$ to the
unlink (\cite{Go}, \cite{Gi}). This is Milnor's original theorem
\cite{M1}.
\subsection{Relation to Milnor invariants and the Kontsevich integral}
\negativeleftarrowbel{sec:Milnor}
For a link $L \subset S^3$, there are unique homotopy classes (rel
boundary) $A_i\co D^2 \to D^4$ of immersions extending $L$. Therefore,
the previous discussion should apply to give link invariants via Whitney
towers. The {\epsilonm reduced} Kontsevich integral $Z^t(L)$ is the tree
part of the Kontsevich integral of $L$ and in \cite{HM} Habegger and
Masbaum have shown that the {\epsilonm first non-vanishing term} of $Z^t(L)-1$
carries exactly the same information as the first non-vanishing Milnor
invariants $\mu(L)$. These are the Milnor invariants with repeating
indices, also denoted $\bar\mu$--invariants \cite{M2}. We shall not
make this distinction and we consider only the ``first non-vanishing''
invariants. In the general case one needs to consider {\epsilonm string}
links \cite{HM}.
Denote by $K_n(L)$ the order--$n$ term of $Z^t(L)-1$. Now observe
that $K_n(L)$ takes values exactly in $\mathcal{T}_n(m)\otimes\mathbb{Q}$, where
$m$ is the number of components of $L$ and the {\epsilonm order}~$n$ is
the number of trivalent vertices. Here the relations in $\mathcal{T}(m)$
simplify dramatically because $\pi_1(D^4)=0=\pi_2(D^4)$ and in
fact they reduce to exactly the AS and IHX relations used in the
usual definition of the Kontsevich integral. We note that the most
commonly used degree in papers on the Kontsevich integral is one
half the total number of vertices. For unitrivalent trees, this
degree is one more than the number of trivalent vertices, ie one
{\epsilonm more} than the order that we are using here.
For an oriented link $L\subset S^3$, consider the following four statements.
\begin{enumerate}
\item $L$ bounds a Whitney tower of order~$n$ in $D^4$.
\item $L$ bounds disjointly embedded framed gropes of class~$(n+1)$ in $D^4$.
\item $L$ has vanishing $\mu$--invariants of length~$\leq (n+1)$.
\item All terms in $Z^t(L)-1$ having order~$\leq (n-1)$ vanish.
\epsilonnd{enumerate}
Then $\mathrm{(i)}$ is equivalent to $\mathrm{(ii)}$ by \cite{S},
$\mathrm{(iii)}$ is equivalent to $\mathrm{(iv)}$ by \cite{HM}, and
$\mathrm{(ii)}$ implies $\mathrm{(iii)}$ by \cite{KT}.
The following theorem gives the relation between the Kontsevich integral
and our intersection tree $\tau$ in the context of the above results.
\begin{thm} \negativeleftarrowbel{thm:Milnor}
If $L$ bounds a Whitney tower $\mathcal{W}$ of order~$n$ in $D^4$, then
$$K_n(L)=\tau_n(\mathcal{W}) \quad \in \mathcal{T}_n(m)\otimes\mathbb{Q}$$
which shows that rationally, $\tau_n(L):=\tau_n(\mathcal{W})$ only depends on
(the concordance class of) $L$ and can be used to calculate the first
non-vanishing terms of the reduced Kontsevich integral as well as the
Milnor invariants.
\epsilonnd{thm}
\begin{rem} In \cite{ST3} we shall explain a direct geometric relation between our intersection trees
and Milnor's invariants, completely avoiding the Kontsevich integral.
\epsilonnd{rem}
\begin{rem} \negativeleftarrowbel{rem:torsionfree}
In the nonrepeating case, the groups $\mathcal{T}_n(n+2)$ are torsionfree,
and hence tensoring with $\mathbb{Q}$ does not lose any information. This
implies our above Conjecture~\ref{conj:well defined} for this very
special case (since the FR and INT relations are trivial). By
results in \cite{Ko}, Theorem~\ref{thm:Milnor} also implies the
conjecture for the $2$--spheres in the simply connected
$4$--manifold formed by attaching 0--framed $2$--handles to the
$4$--ball along $L$ in the nonrepeating case (or rationally in the
repeating case). It is not unreasonable to believe that the groups
$\mathcal{T}_{2n}(m)$ are also torsionfree (with repeated labels allowed).
Note that $\mathcal{T}_1(1) \cong \mathbb{Z}/2$ which corresponds exactly to the
Arf invariant of a knot (see \cite{Ma}, \cite{S2}, \cite{ST}) and
hence shows that statement (iv) does {\epsilonm not} imply (i) in the
above theorem. In general, the FR relations are non-trivial for
odd orders as will be explained in \cite{ST3}; see
Figure~\ref{FR-relations-fig} for an example.
\epsilonnd{rem}
\section{Whitney towers and intersection trees}\negativeleftarrowbel{W-tower-tau-sec}
The goal of this section is to define the $n$th-order intersection
tree $\tau_n(\mathcal{W})$ of an order--$n$ Whitney tower $\mathcal{W}$ in an oriented
$4$--manifold $X$. After giving the precise definition of a Whitney
tower $\mathcal{W}$, an indexing of the surfaces in $\mathcal{W}$ is given in terms
of bracketings and rooted trees which are labelled, oriented and then
decorated by elements of the fundamental group $\pi :=\pi_1X$. The
unrooted decorated tree $t_p$ associated to an intersection point $p$
in $\mathcal{W}$ then corresponds to a pairing of the rooted trees associated to
the intersecting surfaces. Finally, $\tau_n(\mathcal{W})$ is defined as a signed
sum of the $t_p$ in the group $\mathcal{T}_n(\pi,m)$, see Section~\ref{subsec:T}.
\subsection{Whitney towers}\negativeleftarrowbel{W-tower-sub-sec}
We assume our $4$--manifolds are oriented and equipped with a
basepoint. The reader is referred to \cite{FQ} for details on immersed
surfaces in $4$--manifolds, including {\epsilonm Whitney moves} and (Casson)
{\epsilonm finger moves}. For more on Whitney towers see \cite{CST}, \cite{S},
\cite{S2}.
\begin{defn}\negativeleftarrowbel{w-tower-defn}\mbox{}
\begin{itemize}
\item A {\epsilonm surface of order 0} in a 4--manifold $X$
is a properly immersed surface (boundary embedded in the boundary
of $X$ and interior immersed in the interior of $X$). A {\epsilonm
Whitney tower of order 0} in $X$ is a collection of order--0
surfaces.
\item The {\epsilonm order of a (transverse) intersection point} between a
surface of order $n_1$ and a surface of order $n_2$ is $n_1+n_2$.
\item The {\epsilonm order of a Whitney disk} is $n+1$ if it pairs intersection
points of order $n$.
\item For $n\geq 0$,
a {\epsilonm Whitney tower $\mathcal{W}$ of order $n+1$} is a Whitney tower of
order $n$ together with Whitney disks pairing all order--$n$
intersection points of $\mathcal{W}$. These top order disks are allowed
to intersect each other as well as lower order surfaces.
\epsilonnd{itemize}
The Whitney disks in a Whitney tower are required to be {\epsilonm
framed} (\cite{FQ}) and have disjointly embedded boundaries.
Intersections in surface interiors are assumed to be transverse. A
Whitney tower is {\epsilonm oriented} if all its surfaces (order--$0$
surfaces {\epsilonm and} Whitney disks) are oriented. A {\epsilonm based}
Whitney tower includes a chosen basepoint on each surface
(including Whitney disks) together with a {\epsilonm whisker} (arc) for
each surface connecting the chosen basepoints to the basepoint of
the ambient $4$--manifold.
\epsilonnd{defn}
Some further terminology: If $\mathcal{W}$ is an order--$n$ Whitney tower
containing $A_i$ as its order--$0$ surfaces then the $A_i$ are said
to {\epsilonm admit} an order--$n$ Whitney tower and we say that $\mathcal{W}$ is
a Whitney tower {\epsilonm on} the $A_i$.
\subsection{Rooted trees and brackets}\negativeleftarrowbel{trees-brackets}
Non-associative ordered {\epsilonm bracketings} of elements from some
index set correspond to {\epsilonm rooted labelled vertex-oriented
unitrivalent trees} as follows. Here {\epsilonm rooted} means ``having a
preferred univalent vertex'' (the {\epsilonm root}), {\epsilonm labelled}
means that each non-root univalent vertex is labelled by an
element from the index set and {\epsilonm vertex-oriented} means that
each trivalent vertex is equipped with a cyclic ordering of its
incident edges. The {\epsilonm order} of a tree is the number of
trivalent vertices.
A bracketing $(i)$ of a singleton element $i$ from the index set
corresponds to the rooted order--0 tree $t(i)$ consisting of a single
edge with one vertex labelled by $i$ and the other vertex designated as
the root. A bracketing $(I,J)$ of brackets $I$ and $J$ corresponds to
the {\epsilonm rooted product} $t(I,J):=t(I) \ast t(J)$ of the trees $t(I)$
and $t(J)$ which identifies together the roots of $t(I)$ and $t(J)$
to a single vertex and ``sprouts'' a new rooted edge at this vertex
(Figure~\ref{bracket-treesA-fig}) with the cyclic order at the new
trivalent vertex given by taking the edges coming from $I$, $J$ and the
root in that order.
\begin{figure}[ht!]
\centerline{\includegraphics[scale=.5]{brack.eps}}
\caption{Rooted trees $t(i)$ and $t(j)$ (upper left) and their rooted product
$t(i,j)=t(i)\ast t(j)$ (lower left);
$t(i_1,i_2)$ and $t(j_1,(j_2,j_3))$ (upper right) and their rooted product
$t((i_1,i_2),(j_1,(j_2,j_3)))=t(i_1,i_2)\ast t(j_1,(j_2,j_3))$
(lower right). In this figure all trivalent orientations are clockwise
in the plane.}
\negativeleftarrowbel{bracket-treesA-fig}
\epsilonnd{figure}
Thus, the non-root univalent vertices of the tree $t(I)$
associated to a bracket $I$ are labelled by elements from the
index set and the trivalent vertices correspond to sub-bracketings
of $I$, with the trivalent vertex adjacent to the root
corresponding to $I$.
\begin{rem}\negativeleftarrowbel{rem:w-tree-f-move}
The rooted product $\ast$ can be ``realized'' geometrically by a
finger-move: Pushing a Whitney disk $W_I$ through another Whitney
disk $W_J$ creates $W_{(I,J)}$ with $t(W_{(I,J)})=t(W_I)\ast
t(W_J)$.
\epsilonnd{rem}
This remark uses the upcoming assignment of a rooted tree $t(W)$
to a Whitney disk $W$ inside a Whitney tower $\mathcal{W}$. In the easiest
version, one starts with a root for $W$ and then introduces one
branching (trivalent vertex) while reading off which two sheets of
$\mathcal{W}$ are paired by $W$. Then one continues with the same
procedure for the two sheets to inductively obtain $t(W)$. In the
next section we shall make this procedure precise, and in fact
explain directly how orientations on the Whitney disks lead to
vertex-orientations of the corresponding trees.
\subsection{Rooted trees for oriented Whitney towers}
Let $\mathcal{W}$ be an oriented Whitney tower on order--$0$ surfaces $A_i$
for $i=1,2,\ldots,m$. The orientations on the surfaces in $\mathcal{W}$
set up an indexing of the surfaces in $\mathcal{W}$ by bracketings $I$
from $\{1,2,\ldots,m\}$ and their corresponding rooted vertex
oriented unitrivalent $m$--labelled trees $t(I)$
(\ref{trees-brackets}) via the following conventions:
A bracketing $(i)$ of a singleton element $i$ from the index set
and the corresponding rooted order--0 tree $t(A_i):=t(i)$ are
associated to each order--$0$ surface $A_i$. The bracket $(I,J)$
and the corresponding tree $t(W_{(I,J)}):=t(I,J)$ are associated
to a Whitney disk $W_{(I,J)}$, pairing intersections between $W_I$
and $W_J$, with the ordering of the components $I$ and $J$ in the
associated bracket $(I,J)$ chosen so that the orientation of
$W_{(I,J)}$ is the same as that given by orienting its boundary
$\partial W_{(I,J)}$ from the negative intersection point to the
positive intersection point {\epsilonm first} along $W_I$ {\epsilonm then}
back along $W_J$ to the negative intersection point, together with
a second inward pointing tangent vector.
We use brackets as subscripts to index surfaces in $\mathcal{W}$, writing
$A_i$ for an order--$0$ surface (dropping the brackets around the
singleton $i$) and $W_{(i,j)}$ for a first-order Whitney disk that
pairs intersections between $A_i$ and $A_j$, etc.. When writing
$W_{(I,J)}$ for a Whitney disk pairing intersections between $W_I$
and $W_J$, the understanding is that if a bracket $I$ is just a
singleton $(i)$ then the surface $W_I=W_{(i)}$ is just the
order--$0$ surface $A_i$. In general, the order of $W_I$ is equal
to the order of (ie the number of trivalent vertices of)
$t(W_I)$.
It will be helpful to consider each tree $t(W_I)$ as a subset of
$\mathcal{W}$: Assuming that $\mathcal{W}$ is based (Definition~\ref{w-tower-defn}),
map the vertices (other than the root) of $t(W_I)$ to the basepoints
of the surfaces whose indices are contained as sub-brackets of $I$
and map the edges (other than the edge adjacent to the root) of
$t(W_I)$ to sheet-changing paths between basepoints, as illustrated in
Figure~\ref{W-disk-treeC-OR-fig} (disregarding, for the moment, the dotted
loop which will be explained in \ref{decorating-tree-subsec}). Then embed
the root and its edge anywhere in the {\epsilonm negative corner} of $W_I$
(see next paragraph).
It can be arranged that this mapping of $t(W_I)$ into $\mathcal{W}$ has the
property that the trivalent orientations of $t(W_I)$ are induced by
the orientations of the corresponding Whitney disks: Note that the pair
of edges which pass from a trivalent vertex down into the lower order
surfaces paired by a Whitney disk determine a ``corner'' of the Whitney
disk which does not contain the other edge of the trivalent vertex. If
this corner contains the {\epsilonm positive} intersection point paired by
the Whitney disk, then the vertex orientation and the Whitney disk
orientation agree. Our figures are drawn to satisfy this convention.
\subsection{Orientation choices on Whitney disks}
\negativeleftarrowbel{subsec:orientations}
Via our bracket-orientation convention, changing the orientation on a
Whitney disk $W_{(I,J)}$ changes its tree from $t(W_{(I,J)})=t(I,J)$ to
$t(W_{(J,I)})=t(J,I)$, ie changes the cyclic orientation of the associated
trivalent vertex. In addition, changing the orientation of a {\epsilonm single}
lower order Whitney disk $W_K$ corresponding to a trivalent vertex of
$t(W_{(I,J)})$ (so $K$ is a sub-bracket of $(I,J)$, with $K\neq (I,J)$)
changes the cyclic orientations at exactly {\epsilonm two} trivalent vertices of
$t(W_{(I,J)})$: the one corresponding to $W_K$ and the adjacent one which
corresponds to a Whitney disk pairing intersections between $W_K$ and some
other surface. This is because changing the orientation of $W_K$ reverses
the signs of the intersection points between $W_K$ and anything else.
\begin{figure}[ht!]
\centerline{\includegraphics[scale=.5]{W-disk.eps}}
\caption{A Whitney disk $W_{((I_1,I_2),I_3)}$ and its
associated tree $t(W_{((I_1,I_2),I_3)})$ shown (left) as a
subset of the Whitney tower and (right) as an abstract
rooted tree. The boundaries of the Whitney disks are
oriented according to our bracket-orientation conventions
using the indicated signs of the intersection points. The
dashed path indicates a sheet-changing loop (based at the
basepoint of the ambient $4$--manifold $X$) which
determines the element $g_e\in\pi_1X$ decorating the
corresponding oriented edge as described in
\ref{decorating-tree-subsec}.} \negativeleftarrowbel{W-disk-treeC-OR-fig}
\epsilonnd{figure}
\subsection{Decorated trees for Whitney towers}
\negativeleftarrowbel{decorating-tree-subsec}
Let $t(W_I)$ be the (oriented labelled rooted) tree associated to
a Whitney disk $W_I$ in an oriented based Whitney tower $\mathcal{W}$ in
a $4$--manifold $X$. Thinking of $t(W_I)$ as a subset of $\mathcal{W}$ as
described above, any edge $e$ of $t(W_I)$, other than the root-edge,
corresponds to a sheet-changing path connecting the basepoints
of adjacent surfaces in $\mathcal{W}$. For a chosen orientation of $e$,
this path together with the whiskers on the adjacent surfaces form
an oriented loop which determines an element $g_e$ of $\pi:=\pi_1X$
(Figure~\ref{W-disk-treeC-OR-fig}). Fixing (arbitrarily) orientations for
all the (non-root) edges in $t(W_I)$ and labelling each oriented edge
with an element of $\pi$ in this way yields the {\epsilonm decorated rooted
tree} associated to $W_I$ (which will still be denoted by $t(W_I)$). Note
that switching the orientation of $e$ changes $g_e$ to $g_e^{-1}$ which
explains the OR orientation reversal relation mentioned in \ref{tau-intro}
and shown in Figure~\ref{Relations-fig}. (Since we are working in an
orientable $4$--manifold, $\omega_1(g_e)$ is trivial.) Also, changing the
choice of whisker on a Whitney disk has the effect of left multiplication
on the group elements associated to the three edges adjacent to and
oriented away from the trivalent vertex corresponding to the Whitney
disk accounting for the HOL relation.
When decorations are understood, we will also denote a decorated tree
by $t(I)$ where the underlying tree corresponds to the bracket $I$.
\subsection{Decorated trees for intersection points}
\negativeleftarrowbel{int-point-trees}
If $p$ is a transverse intersection point between $W_I$ and $W_J$
in $\mathcal{W}$ then the {\epsilonm decorated tree} $t_p$ associated to $p$ is
defined as follows. Identify the roots of the decorated trees $t(W_I)$
and $t(W_J)$ to a single (non-vertex) point. The two edges that were
adjacent to the roots of $t(W_I)$ and $t(W_J)$ now form a single edge
$e_p$. Chose an orientation of $e_p$ and decorate $e_p$ by the element
of $\pi$ determined by the whiskers on $W_I$ and $W_J$ together with a
path connecting the basepoints of $W_I$ and $W_J$ that changes sheets
only at $p$ with the orientation induced by $e_p$.
\begin{figure}[ht!]
\centerline{\includegraphics[scale=.5]{W-disk-i.eps}}
\caption{The punctured tree $t^{\circ}_p$ associated to an
intersection point $p\in W_I\cap W_J$ (for
$I=((I_1,I_2),I_3)$ and $J=(J_1,J_2)$) shown as a subset
of the Whitney tower and as an abstract labelled
(punctured) tree. Decorations other than $g_p$ are
suppressed and the sheet-changing loop that determines
$g_p$ is indicated by the dashed path.}
\negativeleftarrowbel{W-disk-int-point-treeD-OR-fig}
\epsilonnd{figure}
Thus, the decorated tree $t_p$ is unrooted and every edge of $t_p$
is oriented and decorated with an element of $\pi$. Note that the
order of $p$ is equal to the order of $t_p$ (the number of
trivalent vertices).
The mappings of $t(W_I)$ and $t(W_J)$ into $\mathcal{W}$ give rise to a
mapping of $t_p$ into $\mathcal{W}$: Just map the root vertices of $W_I$
and $W_J$ to $p$ and the adjacent edges become a sheet-changing
path between the basepoints of $W_I$ and $W_J$
(Figure~\ref{W-disk-int-point-treeD-OR-fig}). This mapping is an
embedding of $t_p$ into $\mathcal{W}$ if all the Whitney disks ``beneath''
$W_I$ and $W_J$ (corresponding to sub-brackets of $I$ and $J$) are
distinct.
We will sometimes keep track of the edge of $t_p$ that corresponds
to $p$ by marking that edge with a small linking circle as in
Figure~\ref{W-disk-int-point-treeD-OR-fig}; such a {\epsilonm punctured tree}
will be denoted by $t^{\circ}_p$.
It will be convenient to formalize the above description of the
(unrooted) decorated tree $t_p$ as a pairing (over the group
$\pi$) of rooted decorated trees: Given a pair $t(I)$ and $t(J)$
of rooted decorated trees and an element $g\in\pi$, define the
{\epsilonm inner product} $t(I)\cdot_g t(J)$ to be the unrooted
decorated tree gotten by identifying together the root vertices of
$t(I)$ and $t(J)$ to a single (non-vertex) point in an edge
labelled by $g$ as illustrated in
Figure~\ref{inner-product-treesC-fig}. Thus, in this notation we
have $t_p:=t(W_I)\cdot_{g_p} t(W_J)$ for $p\in W_I\cap W_J$ as
just described above.
\begin{figure}[ht!]
\centerline{\includegraphics[scale=.5]{inner-p.eps}}
\caption{A pair of decorated rooted trees $t(I)$ and $t(J)$
corresponding to order--1 Whitney disks $W_{I}$ and
$W_{J}$ with $I=(i_1,i_2)$ and $J=(j_1,j_2)$ (left), and the inner product
$t_p=t(W_I)\cdot_{g_p} t(W_J)=t(I)\cdot_{g_p} t(J)$ associated to
an order--2 intersection point $p\in W_{I}\cap W_{J}$ (right).}
\negativeleftarrowbel{inner-product-treesC-fig}
\epsilonnd{figure}
\subsection{The antisymmetry AS relation}\negativeleftarrowbel{subsec:AS}
If a Whitney tower $\mathcal{W}$ is oriented then there is one more piece
of information that we need to keep track of: the sign $\epsilon_p$ of
an unpaired intersection point
$$p\in W_I\cap W_J\subset \mathcal{W}.$$
$\epsilon_p$ is computed, in the usual way, by comparing the orientation
determined by $W_I$ and $W_J$ at $p$ with the orientation of the
ambient $4$--manifold $X$ at $p$.
Changing the orientation on the Whitney disk $W_I$ changes the
{\epsilonm signed tree} $\epsilonpsilon_p\cdot t_p$ by the AS antisymmetry
relation mentioned in \ref{tau-intro}: The cyclic orientation of
the vertex corresponding to $W_I$ in $t_p$ is switched and so is
the sign $\epsilonpsilon_p$ of the intersection with $W_J$. Moreover,
changing the orientation of a single Whitney disk, other than
$W_I$ or $W_J$, preserves the sign $\epsilonpsilon_p$ and changes the
cyclic orientations at {\epsilonm two} trivalent vertices of $t_p$, as
pointed out above in Section~\ref{subsec:orientations}.
Consequently, working modulo the AS relation makes the signed tree
$\epsilonpsilon_p\cdot t_p$ independent of the choices of orientations
for the Whitney disks in $\mathcal{W}$.
The dependence on orientations for the original sheets $A_i$
remains: changing the orientation of one $A_i$ introduces an
additional sign into $\epsilonpsilon_p\cdot t_p$ if $t_p$ has an odd
number of $i$--labelled vertices.
\subsection{The intersection tree $\tau_n(\mathcal{W})$}\negativeleftarrowbel{subsec:T}
We would next like to add up the unpaired intersection points of a
given Whitney tower in some algebraic structure. For that purpose,
let $\mathcal{T}_n(\pi,m)$ denote the abelian group generated by
(isomorphism classes of) decorated trees of order~$n$ modulo the
relations shown in Figure~\ref{Relations-fig}. That is, each
generator is an (unrooted) unitrivalent tree having
\begin{itemize}
\item $n$ cyclically oriented trivalent vertices,
\item $n+2$ univalent vertices labelled by elements of $\{1,\dots,m\}$, and
\item $2n+1$ oriented edges decorated by elements of $\pi$.
\epsilonnd{itemize}
\begin{figure}[ht!]
\centerline{\includegraphics[scale=0.9]{Relat.eps}}
\caption{The AS, OR, HOL and IHX relations in $\mathcal{T}_n(\pi,m)$ for
$a$, $b$, $c$, $d$, $1$ and $g$ in $\pi$ with $\overline{g}=g^{-1}$. All
trivalent orientations are induced from a fixed orientation of the plane.}
\negativeleftarrowbel{Relations-fig}
\epsilonnd{figure}
\begin{defn}\negativeleftarrowbel{int-tree-defn}
Let $\mathcal{W}$ be an order--$n$ Whitney tower on properly immersed
simply-connected oriented surfaces $A_1,\dots,A_m$ in a
$4$--manifold $X$. (In fact, the $A_i$ only need to be
$\pi_1$--null, see \cite{FQ}.) Define the $n$th-order {\epsilonm
intersection tree} of $\mathcal{W}$ by
$$\tau_n(\mathcal{W}):= \sum_p \epsilon_p\cdot t_p \quad \in \quad \mathcal{T}_n(\pi,m)$$
where the sum is over all order--$n$ intersection points $p$ in $\mathcal{W}$.
\epsilonnd{defn}
As explained above, the AS relations make sure that $\tau_n(\mathcal{W})$
actually does not depend on the choice of orientations for the
Whitney disks. Similarly, the HOL and OR relations make sure that
$\tau_n(\mathcal{W})$ does not depend on the choice of whiskers, or edge
orientations. In other words, $\tau_n(\mathcal{W})$ is defined by first
choosing whiskers and orientations (on edges and Whitney disks)
and then proving independence of these choices.
\begin{rem} Using the HOL relation or, more concretely, by choosing
the whiskers on the Whitney disks appropriately, one can normalize
the trees $t_p$ so that all interior edges and one univalent edge
are decorated with the trivial group element $1\in\pi$. Thus, one
can interpret $\tau_n(\mathcal{W})$ as living in a quotient of the
integral group ring of the $(n+1)$--fold product of $\pi$.
By slightly refining our notation, signs can be associated
formally to all tree edges and the edge decorations can be
extended linearly to elements of the group ring $\mathbb{Z} [\pi ]$
(compare \cite{GK}, \cite{GL}). Similarly, one can extend the
labels on the univalent vertices to the free abelian group on
$\{1,\dots,m\}$.
\epsilonnd{rem}
\section{Proof of Theorem~\ref{thm:build-tower}}\negativeleftarrowbel{sec:build-tower-proof}
Our proof of Theorem~\ref{thm:build-tower} will be constructive in
the sense that we describe how to build the next order Whitney
tower by geometrically realizing all the relations in
$\mathcal{T}_n(\pi,m)$. However, it should be mentioned that since the
groups $\mathcal{T}_n(\pi,m)$ do not in general have a canonical basis we
are sidestepping the ``word problem'' in $\mathcal{T}_n(\pi,m)$. The main
construction (Lemma~\ref{transfer-lemma}) of the proof shows how
to exchange {\epsilonm algebraic cancellation} of pairs of intersection
points for {\epsilonm geometric cancellation} (by Whitney disks) in the
case that the intersection points are {\epsilonm simple} (have certain
standard right- or left-normed trees,
\ref{simple-int-transfer-lem-subsec}). This algebraic cancellation
occurs in the lift $\widehat{\mathcal{T}}$ of $\mathcal{T}$ which forgets the IHX
relation. The general case is then reduced to this case using
geometric IHX constructions from \cite{CST} and \cite{S} to show
that an order--$n$ Whitney tower $\mathcal{W}$ with
$\tau_n(\mathcal{W})=0\in\mathcal{T}_n(\pi,m)$ can be modified so that all
order--$n$ intersections come in simple algebraically-cancelling
pairs.
To simplify the exposition and highlight the combinatorial structure
of Whitney towers, we will emphasize the simply-connected case, often
dropping the group $\pi$ from notation. Refining the constructions to
cover the general case for the most part only requires checking that
whiskers can be (re)-chosen appropriately. At a first reading it doesn't
hurt to ignore group elements entirely and only the simply-connected
version of Theorem~\ref{thm:build-tower} will be used later in the proof
of Theorem~\ref{thm:Milnor}.
We begin with some notation and lemmas. All Whitney towers are assumed
oriented, labelled and based.
\subsection{Geometric intersection trees for Whitney towers}
\negativeleftarrowbel{geo-int-tree-w-tower}
For an (oriented, labelled, based) Whitney tower $\mathcal{W}$ define $t_n(\mathcal{W})$,
the ($n$th-order, oriented) {\epsilonm geometric intersection tree} of $\mathcal{W}$,
to be the disjoint union of signed (decorated) trees
$$t_n(\mathcal{W}):=\amalg_p \epsilon_p\cdot t_p$$
over all unpaired order--$n$ intersection points $p \in \mathcal{W}$. (An unsigned
version of $t_n(\mathcal{W})$ was defined for unoriented Whitney towers in
\cite{S}.) The next two pairs of definitions and lemmas will illustrate
how $t_n(\mathcal{W})$ captures the essential geometric structure of $\mathcal{W}$.
\subsection{Split subtowers}\negativeleftarrowbel{split-subtowers}
The Whitney disks in an arbitrary Whitney tower may have multiple self-intersections and
intersections with other surfaces. However, it is not difficult to modify an arbitrary Whitney
tower so that each Whitney disk is embedded and contains either a single Whitney arc or unpaired
intersection point (Lemma~\ref{split-tower-lem} below). This is best expressed using the notion of
{\epsilonm split subtowers} and splitting a Whitney tower into split subtowers will serve to simplify
geometric constructions and combinatorial arguments.
The purpose of constructing a Whitney tower is to provide
information on the homotopy classes (rel boundary) of its
order--$0$ surfaces. However, when describing and manipulating {\epsilonm
subsets} of a Whitney tower it is natural to consider {\epsilonm
sub}towers on sheets of surfaces which are not {\epsilonm properly}
immersed:
\begin{defn}\negativeleftarrowbel{subtower-defi}
A {\epsilonm subtower} is a Whitney tower except that the boundaries of
the immersed order--$0$ surfaces in a subtower are allowed to lie
in the interior of the $4$--manifold (instead of being required to
lie in the boundary). The boundaries of the order--$0$ surfaces in
a subtower are still required to be embedded. The notions of {\epsilonm
order} for intersection points and Whitney disks are the same as
in Definition~\ref{w-tower-defn}.
\epsilonnd{defn}
In this paper we will only be concerned with subtowers whose order--0
surfaces are sheets in the order--0 surfaces of an actual Whitney tower. In
this case, the surfaces of the subtower inherit the same orientations
and indexing by brackets as the Whitney tower. Thus, the association of
decorated trees to surfaces and intersection points is also the same.
\begin{defn}\negativeleftarrowbel{split-subtower-defi}
A subtower $\mathcal{W}_p$ is {\epsilonm split} if it satisfies all of the following:
\begin{enumerate}
\item $\mathcal{W}_p$ contains a single unpaired intersection point $p$,
\item the order--$0$ surfaces of $\mathcal{W}_p$ are all embedded $2$--disks,
\item the Whitney disks of $\mathcal{W}_p$ are all embedded,
\item the interior of any surface in $\mathcal{W}_p$ either contains $p$ or
contains a single Whitney arc of a Whitney disk in $\mathcal{W}_p$,
\item $\mathcal{W}_p$ is connected (as a $2$--complex in the $4$--manifold).
\epsilonnd{enumerate}
Moreover, a Whitney tower $\mathcal{W}$ is called {\epsilonm split} if all the
unpaired intersection points of $\mathcal{W}$ are contained in disjoint
split subtowers on sheets of the order--$0$ surfaces of $\mathcal{W}$.
\epsilonnd{defn}
Note that a normal thickening of a split subtower $\mathcal{W}_p$ in the
ambient $4$--manifold is just the $4$--disk $D^4$ which is a
regular neighborhood of the embedded tree $t_p$ associated to the
unpaired intersection point $p$.
\begin{figure}[ht!]
\centerline{\includegraphics[scale=.5]{splitt.eps}}
\caption{Part of a Whitney tower before (top) and after (bottom) applying
the splitting procedure described in the proof of Lemma~\ref{split-tower-lem}.}
\negativeleftarrowbel{splitting-and-split-tower-fig}
\epsilonnd{figure}
\subsection{Split Whitney towers}\negativeleftarrowbel{split-w-tower-subsec}
The splitting of a Whitney tower into split subtowers described in the
following lemma is analogous to Krushkal's splitting of a grope into
genus one gropes \cite{Kr}.
\begin{lem}\negativeleftarrowbel{split-tower-lem}
Let $\mathcal{W}$ be a Whitney tower on order--0 surfaces $A_i$. Then there exists
a split Whitney tower $\mathcal{W}_{\mathrm{split}}$ contained in any regular
neighborhood of $\mathcal{W}$ such that:
\begin{enumerate}
\item The order--0 surfaces $A'_i$ of $\mathcal{W}_{\mathrm{split}}$ only differ
from the $A_i$ by finger moves.
\item The geometric intersection trees $t(\mathcal{W})$ and
$t(\mathcal{W}_{\mathrm{split}})$ are isomorphic.
\epsilonnd {enumerate}
\epsilonnd{lem}
The isomorphism in item~(ii) includes decorations and signs.
\begin{proof}
Starting with the highest-order Whitney disks of $\mathcal{W}$, apply finger moves
as indicated in Figure~\ref{splitting-and-split-tower-fig}. Working
down through the lower-order Whitney disks yields the desired
$\mathcal{W}_{\text{split}}$. Choosing whiskers and orientations appropriately
for the new Whitney disks preserves the decorations on the trees
associated to the unpaired intersection points.
\epsilonnd{proof}
An advantage of splitting a Whitney tower is that the geometric
intersection tree sits as an {\epsilonm embedded} subset (\ref{int-point-trees})
and all the singularities of the split Whitney tower are contained in
disjointly embedded $4$--balls, each of which is a regular neighborhood
of an intersection point tree. In this sense the decomposition of a
Whitney tower into split subtowers corresponds to the idea that the trees
associated to the unpaired intersection points capture the essential
structure of a Whitney tower. The next lemma can be interpreted as
justifying that this essential structure is indeed captured by the {\epsilonm
un}-punctured trees rather than the punctured trees in the sense that an
unpaired intersection point (corresponding to a punctured edge) can be
``moved'' to any other edge of its tree.
\begin{figure}[ht!]
\centerline{\includegraphics[scale=.6]{tree-m.eps}}
\caption{A local picture of the tree associated to the split
subtower $\mathcal{W}$
before (left) and $\mathcal{W}'$ after (right) the Whitney move in the proof
of Lemma~\ref{subtower-lemma} illustrated in
Figure~\ref{tree-move1-OR-fig} and Figure~\ref{tree-move2-OR-fig}.}
\negativeleftarrowbel{tree-move-trees-fig}
\epsilonnd{figure}
\begin{lem}\negativeleftarrowbel{subtower-lemma}
Let $\mathcal{W}\subset X$ be a split subtower on order--$0$ sheets $s_i$
with unpaired intersection point $p=W_I\cap W_J\subset \mathcal{W}$.
Denote by $\nu(\mathcal{W})$ a normal thickening of $\mathcal{W}$ in $X$ so that
$\partial s_i\subset\partial\nu(\mathcal{W})\subset\nu(\mathcal{W})\cong D^4$. If
$I'$ and $J'$ are any brackets such that the decorated trees
$t(I')\cdot t(J')=t_p=t(I)\cdot t(J)$, then after a homotopy (rel
$\partial$) of the $s_i$ in $\nu(\mathcal{W})$ the $s_i$ admit a split
subtower $\mathcal{W}'\subset\nu(\mathcal{W})$ with single unpaired intersection
point $p'=W_{I'}\cap W_{J'}\subset \mathcal{W}'$ such that
$\epsilonpsilon_{p'}\cdot t_{p'}=\epsilonpsilon_p\cdot t_p$.
\epsilonnd{lem}
\begin{figure}[ht!]
\centerline{\includegraphics[scale=.55]{tree-m1.eps}}
\caption{The unpaired intersection point $p=W_I\cap W_J$ in the split
subtower $\mathcal{W}$ of Lemma~\ref{subtower-lemma} (left), and the unpaired
intersection point $p'=W_{I'}\cap W_{J'}$ in $\mathcal{W}'$ after the Whitney move (right).
Signs and orientations are indicated for the case $\epsilon_p=+$, with brackets
corresponding to the trivalent orientations in Figure~\ref{tree-move-trees-fig}.}
\negativeleftarrowbel{tree-move1-OR-fig}
\epsilonnd{figure}
\begin{figure}[ht!]
\centerline{\includegraphics[scale=.55]{tree-m2.eps}}
\caption{This figure shows that the oriented punctured trees associated
to $p$ and $p'$ in Figure~\ref{tree-move1-OR-fig} differ as indicated in
Figure~\ref{tree-move-trees-fig}.}
\negativeleftarrowbel{tree-move2-OR-fig}
\epsilonnd{figure}
\begin{proof}(of Lemma~\ref{subtower-lemma})
It is enough to show that the puncture in $t^{\circ}_p$ can be
``moved'' to either {\epsilonm adjacent} edge, since by iterating it can
be moved to any edge of $t_p$. Specifically, it is enough to
consider the case where $J=(J_1,J_2)$, $I'=(I,J_1)$ and $J'=J_2$
so that $I\cdot(J_1,J_2)=(I,J_1)\cdot J_2$ as in
Figure~\ref{tree-move-trees-fig}. (Here we are assuming that $W_J$
is not order--$0$ since if both $W_I$ and $W_J$ are order--$0$ there
is nothing to prove.) The proof is given by the maneuver
illustrated in Figure~\ref{tree-move1-OR-fig}: Use the Whitney
disk $W_J$ to guide a Whitney move on $W_{J_1}$. This eliminates
the intersections between $W_{J_1}$ and $W_{J_2}$ (as well as
eliminating $W_J$ and $p$) at the cost of creating a new
cancelling pair of intersections between $W_{J_1}$ and $W_I$. This
new cancelling pair can be paired by a Whitney disk $W_{(I,J_1)}$
having a single intersection point $p'$ with $W_{J_2}$. That this
achieves the desired effect on the punctured tree can be seen in
Figure~\ref{tree-move2-OR-fig} by referring to the signs and
orientations in Figure~\ref{tree-move1-OR-fig}. See also the
discussion in pages 20--22 of \cite{ST} which includes group
elements.
\epsilonnd{proof}
\subsection{Algebraically- and geometrically-cancelling pairs}
\negativeleftarrowbel{alg-geo-pairs}
Let $\widehat{\mathcal{T}}_n(\pi,m)$ denote the group of order--$n$ decorated trees
modulo all the relations in Figure~\ref{Relations-fig} {\epsilonm except}
the IHX relation. We say that a pair of intersection points $p_+$
and $p_-$ in $\mathcal{W}$ {\epsilonm cancel algebraically} if $\epsilonpsilon _{p_+}\cdot
t_{p_+}=-\epsilonpsilon _{p_-}\cdot t_{p_-}\in\widehat{\mathcal{T}}_n(\pi,m)$. There
is a summation map that sends the disjoint union $t_n(\mathcal{W})=\amalg_p
\epsilon_p\cdot t_p$ to an element $\widehat{\tau}_n(\mathcal{W}):=\sum_p \epsilon_p\cdot
t_p\in\widehat{\mathcal{T}}_n(\pi,m)$ and the vanishing of $\widehat{\tau}_n(\mathcal{W})$
is equivalent to being able to arrange all of the order--$n$ intersection
points of $\mathcal{W}$ into algebraically-cancelling pairs.
Given an algebraically-cancelling pair $p_{\pm}$ in a split Whitney
tower, one can chose orientations and whiskers on the Whitney disks in
the split subtowers containing $p_{\pm}$ so that the trees $t_{p_{\pm}}$
have identical orientations (and decorations) with $\epsilon_{p_+}=-\epsilon_{p_-}$.
(This is because the OR, HOL and AS relations are realized by these
choices, as described in Sections \ref{decorating-tree-subsec} and
\ref{subsec:AS}.)
A pair of intersection points $p_+$ and $p_-$ in $\mathcal{W}$ {\epsilonm cancel
geometrically} if they can be paired by a Whitney disk. Geometric
cancellation implies algebraic cancellation, but the converse is not
true since two algebraically-cancelling intersection points might not
lie on the same Whitney disks.
The next lemma gives sufficient conditions for a sort of converse
involving some additional work.
\subsection{Simple intersection points and the transfer lemma}
\negativeleftarrowbel{simple-int-transfer-lem-subsec}
\begin{figure}[ht!]
\centerline{\includegraphics[scale=.6]{simple-t.eps}}
\caption{From left to right, the non-simple tree of lowest-order
(order--4) and the simple trees of order 4, 5
and $6+n$.}
\negativeleftarrowbel{simple-trees-fig}
\epsilonnd{figure}
Following the terminology of \cite{MKS} for iterated commutators of group
elements, we say that an intersection point $p\in \mathcal{W}$ is {\epsilonm simple}
if its tree $t_p$ is simple (right- or left-normed) as illustrated in
Figure~\ref{simple-trees-fig}. The proof of the next lemma shows how to
exchange simple algebraically-cancelling pairs of intersection points
for geometrically-cancelling pairs.
\begin{lem}\negativeleftarrowbel{transfer-lemma}
Let $\mathcal{W}$ be an order--$n$ Whitney tower on order--0 surfaces $A_i$ such
that all order--$n$ intersection points of $\mathcal{W}$ come in {\epsilonm simple}
algebraically-cancelling pairs. Then the $A_i$ are homotopic (rel
boundary) to $A'_i$ which admit an order--$(n+1)$ Whitney tower.
\epsilonnd{lem}
\begin{figure}[ht!]
\centerline{\includegraphics[scale=.5]{trans-1A.eps}}
\nocolon \caption{}
\negativeleftarrowbel{transfer-move-1A-fig}
\epsilonnd{figure}
\begin{proof}
We will describe a modification of $\mathcal{W}$ which exchanges one
algebraically-cancelling simple pair of order $n$ for another at the
cost of only creating geometrically-cancelling pairs. Iterating this
modification will, at the $n$th iteration, exchange an
algebraically-cancelling pair for {\epsilonm only} geometrically-cancelling
pairs. This modification is described in \cite{Y} for the case $n=1$
in a simply-connected manifold. (See also \cite{ST} for the $n=1$
non-simply-connected case.) Applying this procedure to all
algebraically-cancelling pairs will complete the proof. We will discuss
only the simply-connected case; the reader can easily add group elements
to the figures (as in \cite{ST}).
We may assume that $\mathcal{W}$ is split by Lemma~\ref{split-tower-lem}.
Let $p_+$ and $p_-$ be a simple algebraically-cancelling pair of
order--$n$ intersection points in $\mathcal{W}$. By ``pushing the puncture out
to an end of the simple tree'' using Lemma~\ref{subtower-lemma},
we may further assume that $p_+$ and $p_-$ are intersections
between some order--0 surface $A_{i_0}$ and order--$n$ Whitney
disks $W^+_{I_1}$ and $W^-_{I_1}$ respectively where, for
this proof only, $I_k$ will denote a simple bracket of the form
$I_k:=(i_k,(i_{(k+1)},(\ldots,(i_{n},i_{(n+1)})\dots))\!=\!(i_k,I_{(k+1)})$
for $1\leq k\leq n+1$ and $I_{(n+1)}=i_{(n+1)}$.
\begin{figure}[ht!]
\centerline{\includegraphics[scale=.5]{trans-2A.eps}}
\nocolon \caption{}
\negativeleftarrowbel{transfer-move-2A-fig}
\epsilonnd{figure}
The first step in the modification is illustrated in
Figure~\ref{transfer-move-1A-fig} which shows how to exchange
$p_-\in A_{i_0}\cap W^-_{I_1}$ for $p'_-\in A_{i_0}\cap
W^+_{I_1}$, which cancels geometrically with $p_+$, at the cost of
creating a geometrically-cancelling pair of intersection points
between $A_{i_0}$ and $A_{i_1}$. Note that this first step is
possible because both $A_{i_0}$ and $A_{i_1}$ are {\epsilonm connected}.
The modification is completed by choosing Whitney disks for the
new geometrically-cancelling pairs as illustrated in
Figure~\ref{transfer-move-2A-fig}, which shows that a new
algebraically-cancelling pair $q_{\pm}\in W_{(i_0,i_1)}\cap
W^{\pm}_{I_2}$ has been created (recall that boundaries of Whitney
disks must be disjointly embedded). In the case $n=1$, $q_{\pm}$
would also cancel geometrically since then $I_{(n+1)}=i_{(n+1)}$
means that $W^{+}_{I_2}=W^{-}_{I_2}=A_{i_2}$ which is connected.
Note that $W_{(i_0,i_1)}$ is embedded (in a neighborhood of a
contractible 1--complex) and contains only the pair $q_{\pm}$ in
its interior. The Whitney disk $W_{(i_0,(i_1,I_2))}$ may intersect
anything but we don't care because it is a Whitney disk of order
$n+1$ and hence can only contain intersections of order strictly
greater than $n$.
\begin{figure}[ht!]
\centerline{\includegraphics[scale=.5]{trans-2B.eps}}
\nocolon \caption{}
\negativeleftarrowbel{transfer-move-2B-fig}
\epsilonnd{figure}
Now, assuming $n\geq 2$, apply this modification to $q_{\pm}$ as
illustrated in Figure~\ref{transfer-move-2B-fig}. Note that this
is only possible because we have the {\epsilonm connected} surface
$A_{i_2}$ to ``push along'', since we originally started with the
{\epsilonm simple} pair $p_{\pm}$ so that $W^{\pm}_{I_2}=W_{(i_2,I_3)}$.
\begin{figure}[ht!]
\centerline{\includegraphics[scale=.5]{trans-2C.eps}}
\nocolon \caption{}
\negativeleftarrowbel{transfer-move-2C-fig}
\epsilonnd{figure}
The $k$th iteration of this modification is illustrated in
Figure~\ref{transfer-move-2C-fig} where, for this proof only, we
denote the simple bracket $J_k:=(\ldots((i_0,i_1),i_2),\ldots,i_k)$
for $1\leq k\leq n$. The procedure terminates when $k=n$ meaning that
$W^{\pm}_{I_{(k+1)}}=W_{I_{(n+1)}}=A_{i_{(n+1)}}$ which is connected so
only geometrically-cancelling pairs are created.
This procedure can be applied to all the (simple) algebraically-cancelling
pairs: One can always find disjoint arcs between Whitney arcs
in the $A_{i_k}$ to guide the modification and all new Whitney disks of
order~$\leq n$ are contained in neighborhoods of these arcs so that no
unexpected intersections of order less than or equal to $n$ are created.
\epsilonnd{proof}
\subsection{Geometric IHX and the Proof of Theorem~\ref{thm:build-tower}}
\negativeleftarrowbel{build-thm-proof}
Given $\mathcal{W}$ as in Theorem~\ref{thm:build-tower}, we will reduce the proof
to the case handled by Lemma~\ref{transfer-lemma} by using geometric
constructions and results from \cite{CST} and \cite{S}. Achieving the
hypotheses of Lemma~\ref{transfer-lemma} will involve two steps: First
$\mathcal{W}$ will be modified to have only algebraically-cancelling pairs by
using the ``$4$--dimensional IHX construction'' in \cite{CST}. Then the
algebraically-cancelling pairs will be exchanged for simple
algebraically-cancelling pairs, using a related IHX construction of
\cite{S}. This second step is based on the effect of doing a Whitney move
on a Whitney disk in a split subtower and mimics the usual algebraic proof
that the group of unitrivalent trees modulo the IHX and AS relations is
spanned by simple trees (\cite{B2}, \cite{CT1}).
\subsection{Creating algebraically-cancelling pairs}
The vanishing of $\tau_n(\mathcal{W})\in{\mathcal{T}}^t_n(\pi,m)$
means that $\tau_n(\mathcal{W})$ lifts to
$\widehat{\tau}_n(\mathcal{W})\in\mathrm{span}\{\mathrm{I}-\mathrm{H}+\mathrm{X}\}
<\widehat{\mathcal{T}}_n(\pi,m)$.
To get only algebraically-cancelling pairs we apply the following
corollary of the {\epsilonm $4$--dimensional IHX Theorem} in \cite{CST}:
\begin{prop}\negativeleftarrowbel{prop:IHX} Let $\mathcal{W}$ be any order--$n$ Whitney tower on
order--0 surfaces $A_i$. Then, given any decorated order--$n$ unitrivalent
trees $\mathrm{I}$, $\mathrm{H}$ and $\mathrm{X}$ differing only by the
local $\mathrm{IHX}$ relation of Figure~\ref{Relations-fig}, there exists
an order--$n$ Whitney tower $\mathcal{W}'$ on $A'_i$ homotopic (rel boundary)
to the $A_i$ such that
$$t_n(\mathcal{W}')=t_n(\mathcal{W})+\mathrm{I}-\mathrm{H}+\mathrm{X}.$$
\epsilonnd{prop}
Note that the ``sum'' on the right hand side is really a disjoint
union of signed decorated trees; the summation map takes this
equation to the corresponding equation in
$\widehat{\mathcal{T}}_n(\pi,m)$.
\begin{proof}
As observed in Remark~\ref{rem:w-tree-f-move}, creating a ``clean''
Whitney disk by applying a finger move to surfaces in a Whitney tower
``realizes'' the rooted product $\ast$ on the corresponding rooted
trees. Since finger moves are supported near arcs, one can modify $\mathcal{W}$
to create any number of clean Whitney disks realizing arbitrary rooted
decorated trees without changing $t_n(\mathcal{W})$. Let $W^i$, $i=1,2,3,4$ be
four such Whitney disks which correspond to the four fixed vertices of
the trees I, H and X in the statement. (Of course if any of the fixed
vertices is univalent then the corresponding ``Whitney disk'' is just
an order--0 surface.)
Now the $4$--dimensional IHX Theorem of \cite{CST} says that there
exists an order--2 Whitney tower $\mathcal{W}_{\mathrm{IHX}}$ on oriented
$2$--spheres $A_i$, $i=1,2,3,4$, in a $4$--ball having geometric
intersection tree $t_2(\mathcal{W}_{\mathrm{IHX}})$ equal precisely to the
order--2 IHX relation. So by tubing $A_i$ into $W^i$, for each $i$,
we can get $\mathcal{W}'$ as desired. No unexpected intersections are
created since the entire construction takes place near a
collection of arcs and the (arbitrarily small) $4$--ball. (In the
decorated case the desired group elements are controlled by the
tubes.)
\epsilonnd{proof}
So by applying Proposition~\ref{prop:IHX} as necessary we can assume
that $\widehat{\tau}_n(\mathcal{W})=0\in\widehat{\mathcal{T}}_n(\pi,m)$ which means
that all order--$n$ intersection points can be arranged in
algebraically-cancelling pairs.
\subsection{Simplifying the cancelling pairs}
In case there are algebraically-cancelling pairs which are not simple, we
appeal to results in \cite{S}: Proposition~7.1 of \cite{S} describes an
algorithm for modifying a Whitney tower to have only simple intersection
points. This geometric algorithm, which mimics the algebraic algorithm
described in \cite{B2} and \cite{CT1}, depends on a ``Whitney move''
version of the IHX relation (Lemma~7.2 of \cite{S}) which replaces
a split subtower $\mathcal{W}_p$ by two split subtowers $\mathcal{W}_{p'}$ and
$\mathcal{W}_{p''}$ and has the effect of replacing $\epsilon_p\cdot t_p=\mathrm{I}$
by $\mathrm{H}-\mathrm{X}=\epsilon_{p'}\cdot t_{p'}+\epsilon_{p''}\cdot t_{p''}$
in the geometric intersection tree. The point of the algorithm is
that the trees H and X are ``closer'' to being simple and by iterating
one is eventually left with only simple trees. (The construction is
supported in a neighborhood of $\mathcal{W}_p$ so no unwanted intersections are
created.) Although Proposition~7.1 and Lemma~7.2 of \cite{S} are only
proved in the unoriented undecorated case it is not hard to add signs
to the intersection points in the diagrams in \cite{S} and apply the
conventions of this paper, especially having seen the related proof of
Lemma~\ref{transfer-lemma} above.
So in the present setting we have only algebraically-cancelling
pairs of order--$n$ intersection points in an order--$n$ Whitney
tower $\mathcal{W}$ which we may assume is split by
Lemma~\ref{split-tower-lem}. If any of these cancelling pairs are
not simple, then we apply the just-mentioned IHX algorithm of
\cite{S} {\epsilonm pairwise} (so as to preserve
$\widehat{\tau}_n(\mathcal{W})=0\in\widehat{\mathcal{T}}_n(\pi,m)$) until we are
left with only simple algebraically-cancelling pairs. The proof of
Theorem~\ref{thm:build-tower} is now complete by
Lemma~\ref{transfer-lemma}.
$\square$
\section{Proof of Theorem~\ref{thm:Milnor}}
The proof of Theorem~\ref{thm:Milnor} uses results from \cite{HM},
\cite{KT} and \cite{S} as well Theorem~\ref{thm:build-tower} to compare
an arbitrary link $L$ to certain well-known standard links which generate
the first non-vanishing Milnor and $Z^t$ invariants.
\begin{figure}[ht!]
\centerline{\includegraphics[scale=.35]{Coc-Hab.eps}}
\caption{From left to right: An order--2 (positively signed)
vertex-oriented tree I whose univalent vertices correspond to the
components of an unlink; The Bing--Cochran--Habiro link $L_\mathrm{I}$;
An order--2 Whitney tower $\mathcal{W}$ bounded by $L_\mathrm{I}$ with
$\tau_2(\mathcal{W})=\mathrm{I}$.}
\negativeleftarrowbel{Cochran-Habiro-fig}
\epsilonnd{figure}
\subsection{Bing--Cochran--Habiro links}\negativeleftarrowbel{Cochran-Habiro-subsec}
Given a collection $\sigma$ of signed labelled vertex-oriented
order--$n$ trees, Cochran \cite{C} and Habiro \cite{H} have described,
using Bing doubling and clasper surgery respectively, how to construct
(from the unlink) a link $L_{\sigma}$ such that $K_n(L_{\sigma})$ is
represented by $\sigma$ (considered as a sum). Habiro's construction
applies more generally to unitrivalent {\epsilonm graphs}, but for trees the
two constructions coincide (by applying Kirby calculus to a framed link
surgery description).
Given such a {\epsilonm Bing--Cochran--Habiro link} $L_{\sigma}$, we will use the following two facts:
\begin{enumerate}
\item $L_{\sigma}$ bounds an order--$n$ Whitney tower $\mathcal{W}_{\sigma}$
with $\tau_n(\mathcal{W}_{\sigma})=\sigma\in \mathcal{T}_n(m)$.
\item $K_n(L_{\sigma})=\sigma\in \mathcal{T}_n(m)\otimes\mathbb{Q}$.
\epsilonnd{enumerate}
The Whitney tower $\mathcal{W}$ in statement (i) is easily constructed by
``pulling apart'' a Bing double in Cochran's construction (see
Figure~\ref{Cochran-Habiro-fig}): This creates Whitney disks whose
boundaries are essentially the {\epsilonm derived links} in \cite{C} and each
$t_p\in\sigma$ corresponds to a {\epsilonm derived linking}. Alternatively,
starting with Habiro's clasper surgery description one can apply
the translation to {\epsilonm grope cobordism} of \cite{CT1} and then the
translation to Whitney towers of \cite{S} and \cite{CST}.
For statement (ii), see Section 8 of \cite{HM}. Although \cite{HM} works
with {\epsilonm string} links, the first non-vanishing term of $Z^t(L)-1$
is equal to the first non-vanishing term of $Z^t(SL)-1$ where $SL$
is any string link whose closure is $L$ (see Section~5 of \cite{MV}).
\subsection{Whitney towers and the Kontsevich integral}
Let $L$ and $\mathcal{W}$ be as hypothesized in Theorem~\ref{thm:Milnor}.
Denote by $\sigma$ any disjoint union of signed (labelled
vertex-oriented) trees which represents $\tau_n(\mathcal{W})\in\mathcal{T}_n(m)$,
eg the geometric intersection tree $t(\mathcal{W})$ of $\mathcal{W}$
(\ref{geo-int-tree-w-tower}). Let $L_{\sigma}$ be a
Bing--Cochran--Habiro link formed from the unlink using $\sigma$.
Then, by (i) of \ref{Cochran-Habiro-subsec}, $L_{\sigma}$ bounds
an order--$n$ Whitney tower $\mathcal{W}_{\sigma}$ in $D^4$ with
$\tau_n(\mathcal{W}_{\sigma})=\tau_n(\mathcal{W})\in \mathcal{T}_n(m)$. Now think of $\mathcal{W}$
and $\mathcal{W}_{\sigma}$ as each sitting in a copy of $S^3\times I$
($D^4$ with a neighborhood of a point removed). By gluing together
the two copies of $S^3\times I$ (along the $S^3$ boundary of the
removed neighborhoods) and connecting each order--0 $2$--disk of
$\mathcal{W}$ with the corresponding order--0 $2$--disk of
$\mathcal{W}_{\sigma}$ by a small tube we get properly immersed annuli
$A_i$ in $S^3\times I$ cobounded by the link components. Since the
tubes may be chosen to avoid creating new intersection points, the
$A_i$ admit an order--$n$ Whitney tower $\mathcal{W}'$ with
$$\tau_n(\mathcal{W}')=\tau_n(\mathcal{W})-\tau_n(\mathcal{W}_{\sigma})=0\in \mathcal{T}_n(m)$$
where the minus sign comes from reversing the orientation of one of
the two copies of $S^3\times I$. By Theorem~\ref{thm:build-tower},
the vanishing of $\tau_n(\mathcal{W}')$ implies that (after a homotopy rel
boundary) the $A_i$ admit a Whitney tower of order~$n$, that is, $L$ and
$L_{\sigma}$ are {\epsilonm order--$n$ Whitney equivalent}. By the main theorem
in \cite{S}, order--$n$ Whitney equivalence implies (in fact is equivalent
to) {\epsilonm class~$(n+1)$ grope concordance}, meaning that we can conclude
that the components of $L$ and $L_{\sigma}$ cobound disjoint properly
embedded annulus-like gropes of class~$(n+1)$. This implies, by \cite{KT}
Corollary~4.2, that $L$ and $L_{\sigma}$ have the same $\mu$--invariants
of length less than or equal to $(n+1)$. It follows from \cite{HM} that
$K_n(L)=K_n(L_{\sigma})$ which is equal to $\sigma\in\mathcal{T}_n(m)\otimes\mathbb{Q}$
by (ii) of \ref{Cochran-Habiro-subsec} above.
\epsilonndproof
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|
\begin{document}
\title{Local vanishing mean oscillation}
\begin{abstract}We consider various notions of vanishing mean oscillation on a (possibly unbounded) domain $\Omega \subset {\mathbb R}n$, and prove an analogue of Sarason's theorem, giving sufficient conditions for the density of bounded Lipschitz functions in the nonhomogeneous space ${\rm vmo}(\Omega)$. We also study ${\rm cmo}(\Omega)$, the closure in ${\rm bmo}O$ of the continuous functions with compact support in $\Omega$. Using these approximation results, we prove that there is a bounded extension from ${\rm vmo}(\Omega)$ and ${\rm cmo}(\Omega)$ to the corresponding spaces on ${\mathbb R}n$, if and only if $\Omega$ is a locally uniform domain.
\end{abstract}
\section{Introduction}
In the theory of function spaces, it is useful to define ``local" versions of certain spaces. For function spaces on ${\mathbb R}n$, this can refer to functions defined globally which belong to a certain space when restricted to compact sets (such as ${L^p_\loc}({\mathbb R}n)$) or alternatively to spaces defined on a given domain $\Omega \subset {\mathbb R}n$. In the latter case one can then use ``local" to refer to behavior away from the boundary ${\partial\Omega}$. When considering behavior up to the boundary, the notions of ``vanishing" at the boundary, approximation by smooth functions, and extension to ${\mathbb R}n$ are interrelated and intimately connected with the geometry of the domain (for example, in the case of Sobolev spaces and Triebel-Lizorkin spaces, see \cite{Brudnyi1, KoskelaRajalaZhang, KoskelaZhang, Jones2, Rychkov, SSS}).
In the case of functions of bounded mean oscillation, there are various notions of ``local" and ``vanishing" in the literature. The original definition of bounded mean oscillation by John and Nirenberg \cite{JN} was on a fixed cube $Q_0 \subset {\mathbb R}n$: $f \in {\rm BMO}(Q_0)$ if
$$\sup\limits_{Q\subset Q_0} \fint_Q |f(x) - f_Q| dx < \infty,$$
where the supremum is over all parallel subcubes of $Q_0$, $|Q|$ denotes Lebesgue measure, and $f_Q:= \fint_Q f: = |Q|^{-1} \int_Q f$ is the average of $f$ on $Q$.
This can be extended to give a definition of ${\rm BMO}(\Omega)$ on any domain (or even open set) $\Omega \subset {\mathbb R}n$ by taking the
supremum over all cubes $Q \subset \Omega$ with sides parallel to the axes. When $\Omega$ is connected, this supremum defines a norm modulo constants, and ${\rm BMO}(\Omega)$ is a Banach space.
A more refined measure of the mean oscillation of a function $f \in {L^1}loc({\mathbb R}n)$ is given by the {\em modulus of mean oscillation}
\begin{equation}
\label{eqn-modulus}
\omega(f, t):= \sup\limits_{\substack{\ell(Q)< t\\Q\subset{\mathbb R}n}} \fint_Q |f(x) - f_Q| dx, \quad t > 0,
\end{equation}
where $\ell(Q)$ denotes the sidelength of $Q$. We then have $\|f\|_{{\rm BMO}({\mathbb R}n)}: = \sup_{t > 0}\omega(f,t)$. A local version of BMO can be defined by fixing a finite $T > 0$ and considering locally integrable functions with
$\omega(f,T) < \infty$.
The set of such functions does not depend on the choice of $T$ and is strictly larger than ${\rm BMO}({\mathbb R}n)$ since it contains, for example, all uniformly continuous functions.
Another version of BMO which is called ``local" is the space ${\rm bmo}({\mathbb R}n)$ (not to be confused with what is known as ``little" BMO and has the same notation), introduced by Goldberg \cite{Goldberg} as the dual of the local Hardy space $h^1({\mathbb R}n)$ and consisting of locally integrable functions $f$ satisfying
$$\|f\|_{{\rm bmo}({\mathbb R}n)} : = \omega(f, 1) + \sup_{\ell(Q) \geq 1} |f|_Q < \infty,$$
where once more the supremum is taken over all cubes $Q \subset {\mathbb R}n$ with sides parallel to the axes. Here again the scale $1$ can be replaced by any finite $T$ without changing the collection of functions in ${\rm bmo}({\mathbb R}n)$, only affecting the norm, and we can also restrict the second term to the supremum of the averages of $|f|$ on cubes whose sidelength is exactly equal to $1$. As sets of functions, ${\rm bmo}({\mathbb R}n)$ is strictly smaller than ${\rm BMO}({\mathbb R}n)$ (it does not contain $\log |x|$, for example) and should be considered as a nonhomogeneous version of BMO, not taken modulo constants. Both $h^1$ and ${\rm bmo}$ are part of the scale of nonhomogeneous Triebel-Lizorkin spaces - see \cite[Theorem 1.7.1]{Triebel}.
The notion of vanishing mean oscillation was introduced by Sarason \cite{Sarason}. The space ${\rm VMO}({\mathbb R}n)$ can be defined using either one of the two characterizations in the following theorem, which was proved in \cite{Sarason} for the case $n = 1$.
\begin{theoremA}{A}[Sarason]
\label{thm-VMO}
For $f \in {\rm BMO}({\mathbb R}n)$,
\begin{equation}
\label{eq-VMO}
\displaystyle{\lim_{t \rightarrow 0^+} \omega(f, t) = 0}
\end{equation}
if and only if
$f \in \overline{{\rm UC}({\mathbb R}n) \cap {\rm BMO}({\mathbb R}n)}$, the closure of the uniformly continuous functions in ${\rm BMO}$.
\end{theoremA}
A smaller space which is sometimes also called ${\rm VMO}({\mathbb R}n)$ (see \cite{CoifmanWeiss}), and serves as a predual to the Hardy space $H^1({\mathbb R}n)$, is the closure in ${\rm BMO}({\mathbb R}n)$ of the continuous functions with compact support (or equivalently the ${C^\infty}$ functions with compact support). We will denote this space by ${\rm CMO}({\mathbb R}n)$ for ``continuous mean oscillation", following Neri \cite{Neri}. As stated in \cite{Neri} and proved by Uchiyama in \cite{Uchiyama}, in addition to \eqref{eq-VMO}, functions in ${\rm CMO}({\mathbb R}n)$ also satisfy vanishing mean oscillation conditions as the size of the cube increases to $\infty$ and as the cube itself goes to $\infty$. Recently subspaces between ${\rm CMO}({\mathbb R}n)$ and ${\rm VMO}({\mathbb R}n)$ were considered in \cite{TXYY, TorresXue}.
The nonhomogeneous versions of the spaces ${\rm VMO}({\mathbb R}n)$ and ${\rm CMO}({\mathbb R}n)$, denoted ${\rm vmo}({\mathbb R}n)$ and ${\rm cmo}({\mathbb R}n)$, are the corresponding subspaces of ${\rm bmo}({\mathbb R}n)$, and the vanishing mean oscillation conditions characterizing the latter were given in \cite{Bourdaud, Dafni} (see Proposition~\ref{prop-cmo}). Bourdaud's paper \cite{Bourdaud} contains extensive coverage of the properties of ${\rm BMO}({\mathbb R}n)$ and ${\rm bmo}({\mathbb R}n)$ (treating it modulo constants as a subspace of ${\rm BMO}$) as well as their vanishing subspaces.
The focus of our work are the versions of these spaces on a domain $\Omega \subset {\mathbb R}n$, and the corresponding approximation and extension results. The definition of the modulus of oscillation can be adapted by restricting the cubes to lie inside the domain, namely
\begin{equation}
\label{eqn-modulus-Omega}
{\omega_\Omega}(f, t):= \sup\limits_{\substack{\ell(Q)< t\\Q\subset\Omega}} \fint_Q |f(x) - f_Q| dx, \quad t > 0,
\end{equation}
and ${\rm BMO}(\Omega)$ defined to consist of those $f \in {L^1}loc(\Omega)$ with $\sup_{t > 0}{\omega_\Omega}(f,t) < \infty$. The question of the definition of ${\rm VMO}(\Omega)$ is more delicate: for which domains does a version of Sarason's theorem hold? For a bounded domain $\Omega$, Brezis and Nirenberg \cite{BN2} give many results on ${\rm VMO}(\Omega)$, including a strong version of Sarason's result, which they attribute to Jones, not only identifying the subspace of ${\rm BMO}(\Omega)$ consisting of functions with vanishing mean oscillation with the closure of the uniformly continuous functions, but also with the closure of the continuous, or smooth, functions with compact support in $\Omega$.
Jones \cite{Jones} showed that
there is a bounded linear extension from ${\rm BMO}(\Omega)$ to ${\rm BMO}({\mathbb R}n)$ if and only if $\Omega$ is a uniform domain. In \cite{BD1}, we composed Jones' extension operator with an averaging operator on the complement of ${\overline{\Omega}}$ to obtain an operator simultaneously extending ${\rm BMO}$, ${\rm VMO}$ and ${\rm CMO}$, as well as Lipschitz functions, on a uniform domain $\Omega$, and also characterized such domains in terms of the existence of a bounded extension from ${\rm CMO}(\Omega)$ to ${\rm BMO}({\mathbb R}n)$. As a corollary, we obtained a version of Sarason's theorem for these domains.
Turning to the nonhomogeneous case, in \cite{BD2} we proved a version of Jones' theorem for ${\rm bmo}$, identifying the extension domains with locally uniform domains, which we in turn showed are equivalent to the ${(\epsilon,\delta)}$-domains used in Jones' extension results for Sobolev spaces in \cite{Jones2}.
To define ${\rm bmo}$ on $\Omega$ we need to fix a scale $\lambda$. We say $f \in {\rm bmo}lo$ if $f$ is integrable on every cube $Q \subset \Omega$ and
\begin{equation}
\label{def-bmolo}
\|f\|_{\rm bmo}lo :={\omega_\Omega}(f, \lambda) + \sup_{Q\subset \Omega, \ell(Q) \geq \lambda} |f|_Q < \infty.
\end{equation}
Due to the extension theorem in \cite{BD2}, if $\Omega$ is an ${(\epsilon,\delta)}$-domain, there is a natural scale $\lambda_{\epsilon, \delta}$ such that membership in ${\rm bmo}lo$ is independent of $\lambda$ provided $\lambda \leq \lambda_{\epsilon, \delta}$, and we can define ${\rm bmo}O$ to be ${\rm bmo}_{\lambda_{\epsilon, \delta}}(\Omega)$. As pointed out above, in the case $\Omega = {\mathbb R}n$, one can take $\lambda$ to be any finite positive number.
In the special case when $\Omega$ is a bounded domain, ${\rm BMO}(\Omega)$ and ${\rm bmo}lo$ coincide for $\lambda$ sufficiently small, provided we consider them both modulo constants or fix the average on some large cube in $\Omega$ to be zero, say. In such a case Jones' extension in \cite{Jones} will vanish on all cubes sufficiently far away from $\Omega$.
In \cite{BD0},
we gave an extension operator for ${\rm VMO}(\Omega)$ on a bounded uniform domain which does not use averaging and preserves the property of the Jones extension that the values of the extended function on a Whitney cube of ${\mathbb R}n \setminus {\overline{\Omega}}$ are completely determined by the average of the original function on a matching cube inside $\Omega$.
Turning to unbounded domains, in the present paper we prove an extension theorem for the vanishing mean oscillation subspaces of ${\rm bmo}(\Omega)$, or equivalently a nonhomogeneous version of the VMO extension theorem in \cite{BD1}. Unlike in \cite{BD1}, where we used the extension result to prove the analogue of Sarason's theorem, here we first prove the approximation theorem and then obtain the extension as a corollary. It is natural to state the approximation in terms of Lipschitz functions since, as shown in \cite{BD1}, the averaging process used by Sarason yields this strong form of uniform continuity, and it is used for both the approximation and the extension arguments. As in the setting of metric measure spaces, Lipschitz functions are the ``smooth" functions in this context.
\begin{theorem}
\label{thm-approxdomain}
Suppose $\Omega \subset {\mathbb R}n$ is a domain and $\lambda$ is such that for all $0 < \lambda' \leq \lambda$, ${\rm bmo}_{\lambda'}(\Omega) = {\rm bmo}lo$ as sets. Then
$$\{f \in {\rm bmo}lo: \lim_{t \rightarrow 0^+} {\omega_\Omega}(f, t) = 0\}= \overline{{\rm UC}(\Omega) \cap {L^\infty}(\Omega)} = \overline{{\rm Lip}_b(\Omega)} = \overline{{\rm Lip}_{b,0}(\Omega)},$$
where the closures are in the ${\rm bmo}lo$ norm, ${\rm Lip}_b(\Omega)$ denotes the bounded Lipschitz functions in $\Omega$, equipped with the nonhomogeneous norm
$$\|f\|_{{\rm Lip}_b}:= \|f\|_\infty + \sup_{x \neq y} \frac{|f(x) - f(y)|}{|x - y|},$$
and ${\rm Lip}_{b,0}(\Omega)$ consists of those $f \in {\rm Lip}_b(\Omega)$ with ${\rm dist}({\rm supp}(f), {\partial\Omega}) > 0$.
\end{theorem}
As pointed out above, for an ${(\epsilon,\delta)}$-domain $\Omega$, the hypothesis of the theorem holds for $\lambda = \lambda_{\epsilon, \delta}$, and as a result we can not only unambiguously define ${\rm bmo}O$ but also ${\rm vmo}(\Omega)$. We define ${\rm cmo}(\Omega)$ to be the closure in ${\rm bmo}O$ of $C_c(\Omega)$, the continuous functions with compact support in $\Omega$. This can be identified (see Proposition~\ref{prop-vmo-cmo}) with the space of functions in ${\rm vmo}O$ which vanish at infinity. Note that both ${\rm Lip}_b(\Omega)$ and $C_c(\Omega)$ are continuously embedded in ${\rm bmo}lo$ since $\|f\|_{\rm bmo}lo \leq \|f\|_\infty$.
Using these definitions, we can state our extension result.
\begin{theorem}
\label{thm2}
Let $\Omega \subset {\mathbb R}n$ be an ${(\epsilon,\delta)}$-domain. Then there exists a linear extension operator $T$ such that
\begin{itemize}
\item[\rm{(i)}] $T:{\rm bmo}O \to {\rm bmo}({\mathbb R}n)$ is bounded;
\item[\rm{(ii)}] $T:{\rm vmo}(\Omega) \to {\rm vmo}({\mathbb R}n)$ is bounded;
\item[\rm{(iii)}] $T:{\rm cmo}(\Omega) \to {\rm cmo}({\mathbb R}n)$ is bounded;
\item[\rm{(iv)}] $T:{\rm Lip}_b(\Omega) \to {\rm Lip}_b({\mathbb R}n)$ is bounded.
\end{itemize}
Boundedness in (i)-(iv) refers to the ${\rm bmo}$ norm while in (v) the boundedness is with respect to the norm $\|f\|_{{\rm Lip}_b}$.
\end{theorem}
Note that while Theorem~\ref{thm-approxdomain} holds under weaker assumptions on the domain (see Section~\ref{sec-examples} for examples which are not ${(\epsilon,\delta)}$ domains), it is not possible to weaken the assumptions in Theorem~\ref{thm2}, as the converse holds. This follows from the results in \cite{BD2} by observing that the functions used in the proof of Theorem 3.1 there, constructed from the quasihyperbolic metric, are continuous with compact support in $\Omega$. Thus we can state the converse under the weakest hypotheses.
\begin{theorem}[\cite{BD2}]
\label{thm3}
If $\Omega \subset {\mathbb R}n$ is a domain and for some $\lambda > 0$ there is an extension operator $T: C_c(\Omega)\to {\rm BMO}({\mathbb R}n)$ such that for some $C$ and all $f \in C_c(\Omega)$,
$$\|Tf\|_{{\rm BMO}({\mathbb R}n)}\leq C\|f\|_{\rm bmo}lo,$$
then $\Omega$ is an ${(\epsilon,\delta)}$-domain.
\end{theorem}
We start, in Section~\ref{sec-approximation}, with some results on approximation by smooth and compactly supported functions in ${\rm bmo}({\mathbb R}n)$, and clarify the notion of ``vanishing at infinity" in this context, answering a question posed by Bourdaud in \cite{Bourdaud}. Section~\ref{sec-approximation2} addresses the same questions on a domain $\Omega$, culminating in the proof of Theorem~\ref{thm-approxdomain} in Subsection~\ref{sec-approx-vmo}. The proof of Theorem~\ref{thm2} can be found in Section~\ref{sec-extension1}. Finally, Section~\ref{sec-examples} provides some examples and counterexamples to illustrate the results.
\section{Approximation and vanishing mean oscillation in ${\mathbb R}n$}
\label{sec-approximation}
The approximation of functions in ${\rm VMO}({\mathbb R}n)$ by Lipschitz functions can be proved by the same technique use by Sarason on ${\mathbb R}$, namely first taking the averages of the function on a sufficiently fine grid of cubes, and then smoothing out the resulting step function by convolution with a mollifier. Even when convolving with the normalized characteristic function of a ball, this produces a Lipschitz function (see \cite{BD1}). Averaging functions of vanishing mean oscillation in order to get smooth functions is a standard technique, and can be used to show that if the modulus of oscillation is sufficiently rapidly decreasing, the functions themselves are smooth, as originally shown by Campanato \cite{Campanato}, Meyers \cite{Meyers} and Spanne \cite{Spanne} (see also \cite{Sarason2}). For ${\rm BMO}$ functions without vanishing mean oscillation, on a bounded domain, averaging results in Lipschitz up to a log factor, as shown in \cite[Lemma B.9]{BN1}.
For $f \in {\rm bmo}({\mathbb R}n)$, the functions resulting from this process, for a given grid size, are bounded, as we can control the average of $f$ on a cube $Q$ with $\ell(Q) < 1$ by taking a chain $Q = Q_0 \subset Q_1 \subset \ldots Q_k$ with $\ell(Q_i) = 2\ell(Q_{i-1})$, $1 \leq i \leq k -1$, and $1 = \ell(Q_k) \leq 2\ell(Q_{k-1}) < 2$, and using the standard estimate
\begin{equation}
\label{eq-logbmo}
|f|_Q \leq \sum_{i = 1}^k ||f|_{Q_{i-1}} - |f|_{Q_i}| + |f|_{Q_k} \leq 2^{n+1} k \sup_{1 \leq i \leq k} \fint_{Q_i} |f - f_{Q_i}|+ |f|_{Q_k} \lesssim \log\Big(\frac 2{\ell(Q)}\Big)\|f\|_{\rm bmo}.
\end{equation}
Here and throughout the paper we use $a \lesssim b$ to denote the existence of a constant $C$ (usually only depending on the dimension) such that $a \leq Cb$.
This gives us the result of Bourdaud \cite[Th\'eor\`eme 1]{Bourdaud} identifying ${\rm vmo}({\mathbb R}n)$ with the closure of the bounded uniformly continuous functions in ${\rm bmo}({\mathbb R}n)$.
\begin{theoremA}{B}[\cite{Bourdaud}]
\label{thm-bourdaud}
A function $f \in {\rm bmo}({\mathbb R}n)$ satisfies \eqref{eq-VMO} if and only if it can be approximated in the ${\rm bmo}$ norm by bounded uniformly continuous functions.
\end{theoremA}
Bourdaud also proves (see \cite[Th\'eor\`eme 4]{Bourdaud}) that functions in ${\rm vmo}({\mathbb R}n)$ can be approximated by $C^\infty$ functions, but such functions do not have uniform (on ${\mathbb R}n$) bounds on their derivatives. What is possible, as noted above, is to strengthen uniform continuity to Lipschitz continuity.
One can get more smoothness when there is {\em vanishing at infinity}. The following characterization of ${\rm cmo}({\mathbb R}n)$ captured this notion in the ${\rm bmo}$ sense.
\begin{prop}[\cite{Bourdaud,Dafni}]
\label{prop-cmo}
For $f \in {\rm bmo}({\mathbb R}n)$, the following are equivalent:
\begin{enumerate}
\item $f \in {\rm cmo}({\mathbb R}n)$, the closure in ${\rm bmo}({\mathbb R}n)$ of $C_c({\mathbb R}n)$;
\item $f$ satisfies \eqref{eq-VMO} together with $\displaystyle{\lim_{\beta \rightarrow \infty} \sup\{|f|_Q : {\rm dist}(Q, 0) > \beta, \ell(Q)\geq1\} =0}$;
\item $f$ satisfies \eqref{eq-VMO} together with $$\limsup_{x \rightarrow \infty} \sup\left\{\fint_Q |f - f_Q| : \mbox{center of } Q = x, \ell(Q) \leq 1\right\} = 0=\limsup_{x \rightarrow \infty}|f|_{Q_0 + x},$$
where $Q_0 = [0,1]^n$.
\end{enumerate}
\end{prop}
The equivalence of the first two conditions was proved by the second author in \cite{Dafni}, independently of the work of Bourdaud \cite{Bourdaud}, who proved the equivalence of the first and third condition. Bourdaud defines ${\rm cmo}({\mathbb R}n)$ as the closure in ${\rm bmo}({\mathbb R}n)$ of ${\mathcal D}({\mathbb R}n)$, the set of smooth functions with compact support, so the identifications above include the approximation by Lipschitz functions of compact support. Note that the fact that the vanishing at infinity of the averages over large cubes in the second condition is sufficient to give the vanishing at infinity of the oscillation on small cubes in the third condition can be seen by combining Sarason's VMO condition \eqref{eq-VMO} with the standard log estimate \eqref{eq-logbmo}, but may fail when not working on all of ${\mathbb R}n$ (see Example~\ref{example2}).
Bourdaud, in the same paper \cite[p.\ 1217 (1)]{Bourdaud}, poses as a question for further study the characterization of the notion of vanishing at infinity for functions in ${\rm bmo}({\mathbb R}n)$. We answer this question by giving an analogue of Proposition~\ref{prop-cmo} for functions which do not necessarily satisfy the VMO condition \eqref{eq-VMO} .
\begin{definition}
We denote by ${\rm bmo}c({\mathbb R}n)$ the subspace of functions $f$ in ${\rm bmo}({\mathbb R}n)$ which have compact support. We say $f \in {\rm bmo}({\mathbb R}n)$ {\em vanishes at infinity} if
\begin{equation}
\label{vanishing_at_infinity}
\lim_{R \rightarrow \infty}\|f\|_{{\rm bmo}({\mathbb R}n \setminus \overline{B(0, R)})} = 0,
\end{equation}
where the ${\rm bmo}$ norm is taken here in the sense of Definition~\ref{def-bmolo} with $\lambda = 1$.
\end{definition}
\begin{prop}
\label{prop-bmo_0}
For $f \in {\rm bmo}({\mathbb R}n)$, the following are equivalent:
\begin{enumerate}
\item[(i)] $f$ is in the closure of ${\rm bmo}c({\mathbb R}n)$ in ${\rm bmo}({\mathbb R}n)$;
\item[(ii)] $f$ vanishes at infinity;
\item[(iii)] $\displaystyle{\lim_{\beta \rightarrow \infty} \gamma(f,\beta) = 0}$, where
$$\gamma(f,\beta) := \sup_{{\rm dist}(Q, 0) > \beta, \ell(Q)< 1} \fint_Q |f - f_Q| + \sup_{{\rm dist}(Q, 0) > \beta, \ell(Q)=1}|f|_Q;$$
and
\item[(iv)]
\begin{equation*}
\lim_{\beta \rightarrow \infty} \left(\sup_{{\rm dist}(Q, 0) > \beta, \ell(Q)< 1} \fint_Q |f - f_Q| + \sup_{{\rm dist}(Q, 0) > \beta, \ell(Q)\leq 1}|f|_Q\ell(Q)\right) = 0.
\end{equation*}
\end{enumerate}
\end{prop}
\begin{proof}
Condition \eqref{vanishing_at_infinity} is satisfied for any function of compact support, and is preserved when taking limits in the $\|\cdot\|_{{\rm bmo}({\mathbb R}n)}$ norm, so (i) $\implies$ (ii). Moreover, $\gamma(f,\beta) \leq
\|f\|_{{\rm bmo}({\mathbb R}n \setminus \overline{B(0, \beta)})}$ so (ii) $\implies$ (iii).
To show (iii) $\implies$ (iv), it suffices to bound $|f|_Q\ell(Q)$ for a cube $Q$ with ${\rm dist}(Q, 0) > \beta$, and $\ell(Q) < 1$. Proceeding as in \eqref{eq-logbmo} with $Q \subset Q_k \subset {\mathbb R}n \setminus \overline{B(0,\beta)}$ and $\ell(Q_k)= 1$, we have
$$|f|_Q\lesssim \log(2/\ell(Q))\;\gamma(f, \beta) \lesssim \ell(Q)^{-1} \gamma(f, \beta).$$
This shows the quantity in parenthesis in (iv) is controlled by $\gamma(f, \beta)$.
We will now complete the circle of equivalences by proving that (iv) $\implies$ (i).
Assume $f$ satisfies condition (iv) (and hence also (iii)).
First note that if $P$ is a cube whose sidelength is greater than $1$, then taking a cube $P'\supset P$ with $\ell(P')$ an integer and $\frac{\ell(P')}{\ell(P)} < 2$, and writing $P'$ as the union of cubes $Q_i$ of sidelength $1$
and pairwise disjoint interiors, we have
\begin{eqnarray*}
\fint_P|f| & \leq & \frac{1}{|P|} \left(\sum_{{\rm dist}(Q_i,0) \leq \beta} \int_{P \cap Q_i} |f| + \sum_{{\rm dist}(Q_i,0) > \beta} |Q_i||f|_{Q_i}\right)\\
& \leq & \frac{1}{|P|}\int_{P \cap B(0, \beta + \sqrt{n})} |f| +2^n \sup_{{\rm dist}(Q, 0) > \beta, \ell(Q)=1}|f|_Q.
\end{eqnarray*}
Thus (iii) implies the seemingly stronger vanishing condition for averages over large cubes,
\begin{equation}
\label{eq-largecubes}
\lim_{\beta \rightarrow \infty} \left[ \sup_{{\rm dist}(Q, 0) > \beta, \ell(Q)\geq 1}|f|_Q\ + \sup_{\ell(Q) \geq \beta}|f|_Q\right] =0,
\end{equation}
and in particular (iii) $\implies$ (ii).
Fix $k \in {\mathbb N}$ and let
$f_k = \psi_k f$, where $\psi_k$ is a Lipschitz function which is equal to $1$ on $B(0, k)$ and to $0$ outside $B(0, 2k)$, with $0 \leq \psi_k \leq 1$ and $\|\psi_k\|_{\rm Lip} \leq k^{-1}$. Since the support of $f_k$ is compact, we just need to show $\|f - f_k\|_{{\rm bmo}({\mathbb R}n)} \rightarrow 0$ as $k \rightarrow \infty$.
Write $g_k = 1 - \psi_k$ so $f - f_k = fg_k$. For any cube $Q$, $|fg_k|_Q \leq |f|_Q$,
so by (iii) and \eqref{eq-largecubes} we know the averages of $fg_k$ decay as the size of the cube increases. Hence we may assume that ${\rm diam}(Q) < k/2$;
from this, knowing $f g_k = 0$ on $B(0,k)$, we can restrict to cubes with ${\rm dist}(Q, 0) \geq k/2$, and therefore, again by \eqref{eq-largecubes}, to $\ell(Q) < 1$.
In order to bound the oscillation of $fg_k$ over such cubes $Q$, setting $c_Q = f_Q(g_k)_Q$, we have
\begin{eqnarray}
\label{eq-Leibniz}
\fint_Q |fg_k -c_Q| & \leq &
\|g_k\|_\infty \fint_Q |f- f_Q| + 2|f_Q| \fint_Q |g_k - (g_k)_Q|\\
\nonumber
& \leq &
\fint_Q |f- f_Q| + 2|f_Q|\; {\rm diam}(Q_k)\; \|g_k\|_{\rm Lip}\\
\nonumber & \lesssim & \sup_{{\rm dist}(Q, 0) \geq k/2, \ell(Q)< 1} \left(\fint_Q |f - f_Q| +|f_Q|\ell(Q)k^{-1}\right).
\end{eqnarray}
By (iv), this supremum will go to zero as $k \rightarrow \infty$.
\end{proof}
In \cite[Proposition 3.3]{BonamiFeuto}, estimate \eqref{eq-Leibniz} above and the logarithmic estimate \eqref{eq-logbmo} are used to bound the ${\rm bmo}$ norm of the product of a function in ${\rm bmo}({\mathbb R}n)$ with a bounded function in ${\rm lmo}({\mathbb R}n)$. However, as can be seen in the proof, the logarithmic estimate \eqref{eq-logbmo} itself is not necessary to prove the approximation (i), and may not hold in an arbitrary domain (see Examples~\ref{example1} and \ref{example2}).
\section{Approximation and vanishing mean oscillation on a domain}
\label{sec-approximation2}
Our goal is to study the approximation by Lipschitz functions and functions of compact support and prove Theorem~\ref{thm-approxdomain}, which is the analogue of Theorems~\ref{thm-VMO} and \ref{thm-bourdaud}, as well as analogues of Propositions~\ref{prop-cmo} and \ref{prop-bmo_0} in the nonhomogeneous space ${\rm bmo}$ on a domain $\Omega$. Such results are not only of interest in themselves but will lead to a simple proof of the extension theorem, Theorem~\ref{thm2}.
\subsection{Approximation in ${\rm bmo}lo$}
We start with the case where we do not assume any Sarason-type vanishing mean oscillation condition. We will need the following notation and terminology.
\begin{definition}
\label{def-vanishing_at}
We denote by ${\rm bmo}loc$ the subspace of functions $f$ in ${\rm bmo}lo$ which have compact support in $\Omega$. We say $f \in {\rm bmo}lo$ {\em vanishes at infinity} if
\begin{equation}
\label{vanishing_at_infinity_O}
\lim_{R \rightarrow \infty}\|f\|_{{\rm bmo}l(\Omega \setminus \overline{B(0, R)})} = 0.
\end{equation}
We say $f \in {\rm bmo}lo$ {\em vanishes at the boundary} if
\begin{equation}
\label{vanishing_at_bO}
\lim_{t \rightarrow 0} \|f\|_{{\rm bmo}l(\Omega \setminus \mathring{\Omega}_{t})} = 0
\end{equation}
where we define
$$\mathring{\Omega}_{t}:= \{x\in \Omega: {d_\Omega}(x) \geq t\}, \quad {d_\Omega}(x):= {\rm dist}(x, {\partial\Omega}).$$
\end{definition}
To avoid confusion, it should be pointed out that in \cite{BD2}, $\lambda$ was replaced by $\lambda/4$ in the definition of $\mathring{\Omega}_{\lambda}$, while in \cite{Bourdaud} the notation $\Omega_t$ was used completely differently, for the set ${\mathbb R}n \setminus\overline{B(0,t)}$.
Note that for an arbitrary domain $\Omega$ and $\lambda > 0$, if $f \in {\rm bmo}lo$ with ${\rm dist}({\rm supp}(f),{\partial\Omega})) > 0$ then $\|f\|_{{\rm bmo}l(\Omega \setminus \mathring{\Omega}_{t})} = 0$ for all sufficiently small $t$. Since $\|f\|_{{\rm bmo}l(\Omega \setminus \mathring{\Omega}_{t})} \leq \|f\|_{\rm bmo}lo$, we have that the limit of such functions vanishes at the boundary.
Similarly, if $f \in {\rm bmo}lo$ has compact support in ${\overline{\Omega}}$ (we use the notation ${\rm bmo}locobar$ to denote such functions) then $f$ vanishes outside a bounded set and $\|f\|_{{\rm bmo}l(\Omega \setminus \overline{B(0, R)})} = 0$ for all sufficiently large $R$. This again is dominated by the ${\rm bmo}lo$ norm, so the limit of such functions vanishes at infinity.
In order to get the reverse implications, we need to assume, for functions in ${\rm bmo}lo$, some control of the averages over small cubes in terms of the sidelength of the cubes. This is accomplished by the following two lemmas, stated and proved for general domains.
\begin{lem}
\label{lem-vanishing_at_infinity}
Let $\Omega$ be a domain, $\lambda > 0$ and $f \in {\rm bmo}lo$. Suppose $f$ vanishes at infinity and
\begin{equation}
\label{eq-1/t}
\sup_{Q \subset \Omega, {\rm dist}(Q, 0) > \beta, \ell(Q)< \lambda}|f_Q| \ell(Q) < \infty
\end{equation}
for $\beta$ sufficiently large.
Then $f$ is in the closure of ${\rm bmo}locobar$ in ${\rm bmo}lo$.
\end{lem}
\begin{proof}
We proceed as in the proof of Proposition~\ref{prop-bmo_0}. The vanishing at infinity immediately implies $\displaystyle{\lim_{\beta \rightarrow \infty} \gamma_\Omega(f,\beta) = 0}$, where
$$\gamma_\Omega(f,\beta) := \sup_{Q \subset \Omega, {\rm dist}(Q, 0) > \beta, \ell(Q)< \lambda} \fint_Q |f - f_Q| + \sup_{Q \subset \Omega, {\rm dist}(Q, 0) > \beta, \ell(Q)=\lambda}|f|_Q.$$
By restricting the arguments to cubes lying inside $\Omega$, we get from this an analogue of \eqref{eq-largecubes}, namely
\begin{equation}
\label{eq-largecubes-Omega}
\lim_{\beta \rightarrow \infty} \left[ \sup_{Q \subset \Omega, {\rm dist}(Q, 0) > \beta, \ell(Q)\geq \lambda}|f|_Q\ + \sup_{Q \subset \Omega, \ell(Q) \geq \beta}|f|_Q\right] =0.
\end{equation}
Then we define the functions $f_k = f \psi_k$, noting that the supports of these functions are bounded and are therefore compact subsets of ${\overline{\Omega}}$. We continue with the arguments in the proof of Proposition~\ref{prop-bmo_0}, restricted to cubes $Q \subset \Omega$. In the final step, we apply \eqref{eq-Leibniz} to estimate the oscillation of $f - f_k$ on cubes $Q \subset \Omega$ with ${\rm dist}(Q, 0) \geq k/2$ and $\ell(Q) < \lambda$. As in the final step of that proof, all that is needed for this is the vanishing of $\gamma_\Omega(f,\beta)$ and \eqref{eq-1/t}.
\end{proof}
To obtain the analogous result for the vanishing at the boundary, we need to refine our hypothesis.
\begin{lem}
\label{lem-vanishing_at_boundary}
Let $\Omega$ be a domain, $\lambda > 0$ and $f \in {\rm bmo}lo$. Suppose $f$ vanishes at the boundary and there exists a
function $\varphi_\lambda: (0,\lambda/2) \rightarrow (0,\infty)$ with $\varphi_\lambda(t) = 1$ for $t \geq \lambda/4$, $t\varphi_\lambda(t)$ monotone increasing, $\int_0^{\lambda /4}\frac{dt}{t\varphi_\lambda(t)} = \infty$, and such that
\begin{equation}
\label{eq-varphi}
\sup_{\substack{2Q \subset \Omega\\ {\rm dist}(Q, {\partial\Omega}) < \lambda/4}}|f_Q| \lesssim \varphi_{\lambda}(\ell(Q)).
\end{equation}
Then there exists a sequence $\{f_j\}$ converging to $f$ in ${\rm bmo}lo$, with ${\rm dist}({\rm supp}(f_j),{\partial\Omega})) > 0$ for each $j$.
\end{lem}
\begin{proof} Set $f_j = fh_j$, where $h_j$ are modified versions of the auxiliary functions introduced in the proof of \cite[Theorem 1]{BN2}:
$$h_j(x) := \Big(1 - \frac 1 j \int_{{d_\Omega}(x)}^{\lambda/4} \frac{dt}{t\varphi_\lambda(t)}\Big)_+.$$
Here the notation $F_+$ denotes $\max(F,0)$.
For $x \in \mathring{\Omega}_{\lambda}o$ we have that $h_j(x) = 1$, giving $f_j(x) = f(x)$. Moreover, by the hypothesis on $\varphi_\lambda$, there is a sequence of positive numbers $\alpha_j$ converging to zero such that ${d_\Omega}(x) \leq \alpha_j \iff \int_{{d_\Omega}(x)}^{\lambda/4} \frac{dt}{t\varphi_\lambda(t)} \geq j \iff h_j(x) = 0$, so ${\rm dist}({\rm supp}(f_j),{\partial\Omega}))\geq \alpha_j$. That $f_j \in {\rm bmo}lo$ and $f_j \rightarrow f$ will follow by estimating $\|f - f_j\|_{\rm bmo}lo$.
Let $g = f - f_j$. To estimate $\|g\|_{\rm bmo}lo$, we use a ``local-to-global" property - see \cite[Theorem A1.1]{BN2} applied to cubes (balls in the $\ell^\infty$ norm) and \cite[Lemma 3.5]{BD2}:
$$\|g\|_{\rm bmo}lo \lesssim \sup_{2Q \subset \Omega} \fint_Q |g - g_Q| + \|g\|_{{L^\infty}(\mathring{\Omega}_{\lambda}o)}.$$
Since $g = 0$ on $\mathring{\Omega}_{\lambda}o$, it remains to bound the oscillation of $g$ over cubes $Q$ with $2Q \subset \Omega$ and ${\rm dist}(Q, {\partial\Omega}) < \lambda/4$, which means $\ell(Q) < \lambda/2$.
As in the proof of Proposition~\ref{prop-bmo_0}, we can control the oscillation of $g = f(1-h_j)$ over $Q$ by
\begin{equation}
\label{eqn-split}
\|1 - h_j\|_{{L^\infty}(Q)} \fint_Q |f - f_Q| + |f_Q| \fint_Q |h_j - (h_j)_Q|.
\end{equation}
For a given $\eta > 0$, take $t_0$ sufficiently small so that $\|f\|_{{\rm bmo}l(\Omega \setminus \mathring{\Omega}_{t}o)} < \eta/2$. Then if $Q \cap \mathring{\Omega}_{t}o = \varnothing$ we have $\fint_Q |f(x) - f_Q| dx < \eta/2$ and
$\|1 - h_j\|_{{L^\infty}(Q)} \leq 1$.
When $Q \cap \mathring{\Omega}_{t}o$ is nonempty, the condition $2Q \subset \Omega$ forces ${\rm dist}(Q, {\partial\Omega}) \geq t_0/(1 + \sqrt{n})$. Hence for $j$ sufficiently large so that $\alpha_j < t_0/(1 + \sqrt{n})$, we have
$$|1 - h_j(x)|= \frac 1 j \int_{{d_\Omega}(x)}^{\lambda/4} \frac{dt}{t\varphi_\lambda(t)} \leq \frac 1 j \int_{\frac{t_0}{1 + \sqrt{n}}}^{\lambda/4} \frac{dt}{t\varphi_\lambda(t)}, \quad \forall\; x \in Q.$$
Since the integral on the right depends only on $\lambda$ and $t_0$, bounding the oscillation of $f$ on $Q$ by $\|f\|_{\rm bmo}lo$, we can make the first term in \eqref{eqn-split} smaller than $\eta/2$ by taking $j$ sufficiently large.
For the second term in \eqref{eqn-split}, letting $\Phi_\lambda(x) = \int_{{d_\Omega}(x)}^{\lambda/4} \frac{dt}{t\varphi_\lambda(t)}$ and noting that truncations do not increase oscillation and that ${d_\Omega}$ is a $1$-Lipschitz function, we can proceed as in the proof of \cite[Lemma 4]{BN2}:
\begin{eqnarray}
\nonumber
|f_Q| \fint_Q |h_j(x) - (h_j)_Q| dx & \leq & \frac {|f_Q|} j \fint_Q \fint_Q |\Phi_{\lambda}(x) - \Phi_{\lambda}(y)| dxdy\\
\nonumber
& \lesssim & \frac{\varphi_\lambda(\ell(Q))} j \frac{{\rm diam}(Q)}{\inf_{x \in Q}{d_\Omega}(x)\varphi_\lambda({d_\Omega}(x))} \\
\nonumber
& \lesssim & \frac 1 j \frac{\ell(Q)\varphi_\lambda(\ell(Q))}{{\rm dist}(Q,{\partial\Omega}) \varphi_{\lambda}({\rm dist}(Q,{\partial\Omega}))}\\
\label{eq-lmo}
& \lesssim & \frac {1} j.
\end{eqnarray}
Here we have used \eqref{eq-varphi} and the fact that $2Q \subset \Omega$ implies $\ell(Q) \leq {\rm dist}(Q,{\partial\Omega})$, as well as the fact that $t\varphi_\lambda(t)$ is an increasing function.
Thus the second term in \eqref{eqn-split} can be made small for $j$ sufficiently large.
\end{proof}
The choice of $\varphi_\lambda(t)= \frac 1 t$, corresponding to the bound in \eqref{eq-1/t}, does not satisfy the conditions in Lemma~\ref{lem-vanishing_at_boundary}. At the other extreme, putting $\varphi_\lambda(t) = 1$ in the hypotheses of Lemma~\ref{lem-vanishing_at_boundary} corresponds to $f$ being a bounded function, which is the special case shown in the proof of Theorem 1 in \cite{BN2} for a bounded domain, and in Lemma 3 of \cite{BD1} for a general domain. In these results the vanishing at the boundary follows immediately from assuming the Sarason-type VMO condition restricted to $\Omega$, as will be seen in Section~\ref{sec-approx-vmo}.
We now introduce certain geometric conditions on the domain which will allow us to apply Lemmas~\ref{lem-vanishing_at_infinity} and \ref{lem-vanishing_at_boundary} to all functions in ${\rm bmo}lo$. Let us recall the definition of an ${(\epsilon,\delta)}$-domain by Jones \cite{Jones2}.
\begin{definition}
Given $\epsilon\in (0,1]$ and $\delta>0$, an ${(\epsilon,\delta)}$-domain is a domain $\Omega$ such that every pair of points $x,y$ in $\Omega$ with $|x-y|< \delta$
may be joined by a rectifiable curve $\gamma$ lying in $\Omega$, with
\begin{equation*}
\mbox{arclength}(\gamma) \leq \epsilon^{-1}|x-y|,
\end{equation*}
and such that for any point $z$ on $\gamma$,
\begin{equation*}
{\rm dist}(z, {\partial\Omega}) \geq \epsilon\frac{|z-x||z-y|}{|x-y|} .
\end{equation*}
\end{definition}
For such domains we have the following analogue of \eqref{eq-logbmo}, which can be shown directly by a combination of Proposition 4.9 and Lemma 4.4 in \cite{BD2}, or by extending $f$ to ${\rm bmo}l({\mathbb R}n)$ and then using \eqref{eq-logbmo}.
\begin{lem}[\cite{BD2}]
\label{lem-logbmolo}
Let $\Omega$ be an ${(\epsilon,\delta)}$-domain and $0 < \lambda \leq \lambda_{\epsilon, \delta}:= \frac{\epsilon^2 \delta}{320 n(1 + \sqrt{n} \epsilon)}$. If $f \in {\rm bmo}lo$ and $Q \subset \Omega$ is a cube with sidelength $\ell(Q) < \lambda$, then
\begin{equation}
\label{eq-logbmolo}
|f|_Q \lesssim \left(1 + \log\left(\frac{\lambda}{\ell(Q)}\right)\right) \|f\|_{{\rm bmo}lo}.
\end{equation}
\end{lem}
If $\lambda_1 < \lambda_2 \leq \lambda_{\epsilon, \delta}$, then \eqref{eq-logbmolo} gives
\begin{equation}
\label{eq-normequiv}
\|f\|_{\rm bmo}ltwo \leq 2\|f\|_{\rm bmo}lone \leq C \log \frac{\lambda_2}{\lambda_1}\|f\|_{\rm bmo}ltwo.
\end{equation}
This justifies using the notation ${\rm bmo}O$ for ${\rm bmo}lo$, where we fix $\lambda = \lambda_{\epsilon, \delta}$ for the norm. The same estimates also show that the notions of vanishing at infinity and vanishing at the boundary are independent of the choice of $\lambda$. Moreover, ${\rm bmo}loc$ can be denoted by ${\rm bmo}co$ independently of $\lambda$.
With the help of the three lemmas, we can prove an analogue of Proposition~\ref{prop-bmo_0}.
\begin{prop}
\label{prop-bmolo_0}
Let $\Omega$ be an ${(\epsilon,\delta)}$-domain and $f \in {\rm bmo}O$. Then the following are equivalent:
\begin{enumerate}
\item[(i)] there exists a sequence $\{f_j\}$ converging to $f$ in ${\rm bmo}O$, with ${\rm dist}({\rm supp}(f_j),{\partial\Omega})) > 0$ for each $j$;
\item[(ii)] $f$ vanishes at the boundary;
\end{enumerate}
Moreover, we have the equivalence of the following two conditions:
\begin{enumerate}
\item[(a)] $f$ is in the closure of ${\rm bmo}co$ in ${\rm bmo}O$;
\item[(b)] $f$ vanishes at the boundary and vanishes at infinity.
\end{enumerate}
\end{prop}
\begin{proof}
As was pointed out above, the implication in one direction of each pair holds in any domain, so it remains to show the implications (ii) $\implies$ (i) and (b) $\implies$ (a).
Since $\Omega$ is an ${(\epsilon,\delta)}$ domain and $f \in {\rm bmo}O$, by Lemma~\ref{lem-logbmolo}, both \eqref{eq-1/t} and \eqref{eq-varphi}
with
$$\varphi_\lambda(t) = 1 + \log_+\frac \lambda {4t}, \quad \lambda = \lambda_{\epsilon, \delta},$$
hold for $f$ and also for $|f|$.
Assuming
$f$ vanishes at the boundary, we obtain from Lemma~\ref{lem-vanishing_at_boundary} the approximation by functions $f_j = fh_j$ in ${\rm bmo}O$ which are supported away from the boundary.
If in addition $f$ vanishes at infinity, we apply the proof of Lemma~\ref{lem-vanishing_at_infinity} to $f_j$ and obtain the sequence of functions $f_{j,k} = f_j\psi_k = f h_j \psi_k$ having compact support in $\Omega$. We need to show that by choosing $j$ and $k$ sufficiently large, we can make $\|f_{j,k} - f\|_{\rm bmo}O$ small. This means estimating $\|f_{j,k} - f_j\|_{\rm bmo}O$.
From the proof of Lemma~\ref{lem-vanishing_at_boundary} we know that $0 \leq h_j \leq 1$ so for every $Q \subset \Omega$, $|f_j|_Q \leq |f|_Q$. As $f$ vanishes at infinity, this means that \eqref{eq-largecubes-Omega} holds with $f$ replaced by $f_j$. Thus it suffices to bound the oscillation of $f_{j,k} - f_j$ on cubes $Q \subset \Omega$ with ${\rm dist}(Q, 0) \geq k/2$ and $\ell(Q) < \lambda$. From \eqref{eq-Leibniz}, we can control this oscillation by
$$\sup_{Q \subset \Omega, {\rm dist}(Q, 0) \geq k/2, \ell(Q)< \lambda} \left(\fint_Q |f_j - (f_j)_Q| +|(f_j)_Q|\ell(Q)k^{-1}\right),$$
which in turn, by \eqref{eq-1/t}, \eqref{eq-varphi}, \eqref{eqn-split} and \eqref{eq-lmo}, can be controlled by
$$ \sup_{Q \subset \Omega, {\rm dist}(Q, 0) \geq k/2, \ell(Q)< \lambda} \fint_Q |f - f_Q| + j^{-1} + k^{-1}.
$$
It is then possible to choose $j$ and $k$ sufficient large to make both this quantity and $\|f_{j} - f\|_{\rm bmo}O$ small.
\end{proof}
\subsection{Approximation in ${\rm vmo}lo$}
\label{sec-approx-vmo}
We now consider functions in ${\rm bmo}lo$ which also satisfy a Sarason-type VMO condition, namely
\begin{equation}
\label{eq-VMO-Omega}
\lim_{t \rightarrow 0^+} {\omega_\Omega}(f, t) = 0,
\end{equation}
where ${\omega_\Omega}(f,t)$ is defined in \eqref{eqn-modulus-Omega}.
The following two lemmas will lead to the proof of Theorem~\ref{thm-approxdomain}.
\begin{lem}
\label{lem-bounded}
Let $\Omega$ be a domain, $\lambda > 0$. If $f$ is a bounded function satisfying \eqref{eq-VMO-Omega} then $f$
can be approximated in ${\rm bmo}lo$ by bounded Lipschitz functions supported away from ${\partial\Omega}$.
\end{lem}
\begin{proof} This is the special case discussed after the proof of Lemma~\ref{lem-vanishing_at_boundary}, where the lemma can be applied with $\varphi_\lambda(t) = 1$ and tells us that we may approximate $f$ in ${\rm bmo}lo$ by functions supported away from ${\partial\Omega}$. Since the approximations are products of $f$ with bounded functions $h_j$, they are bounded. To further approximate such functions by Lipschitz functions supported away from ${\partial\Omega}$ can be accomplished by the averaging process described at the beginning of Section~\ref{sec-approximation}, and whose details can be found in Section 2 of \cite{BD1}. As the functions are supported away from ${\partial\Omega}$, the grid can be chosen fine enough so that the resulting Lipschitz functions are also supported away from the boundary. Moreover, while the approximation in \cite{BD1} is in the ${\rm BMO}$ norm, it can be seen from the calculations following \cite[Equation (9)]{BD1} that for large cubes, what is estimated are the averages of the error function, so the approximation is actually in the ${\rm bmo}lo$ norm. This argument is also referred to in the proof of \cite[Proposition 3]{BD1}.
However, unlike in that case where $f$ is assumed to have compact support, or the domain is bounded, as in \cite{BN2}, here we cannot immediately obtain smooth functions with uniform bounds by convolving with a smooth mollifier, since we do not necessarily have control of the $L^1$ norm of $f$.
\end{proof}
Based on the previous lemma, we will be able to prove Theorem~\ref{thm-approxdomain} if we can approximate $f$ in the ${\rm bmo}lo$ norm by bounded functions. The standard way to do this is by truncations, namely setting $f^t = \max(\min(f, t), -t)$ and letting $t \rightarrow \infty$. In the case of a bounded domain, or when $f$ has compact support, the convergence of $f^t$ to $f$ in $L^1$ together with \eqref{eq-VMO-Omega} gives the convergence in ${\rm BMO}(\Omega)$ (see \cite[Lemma A1.4]{BN2}) and also in ${\rm bmo}lo$.
For unbounded domains and general $f \in {\rm bmo}lo$,
we need to find a criterion for approximation by bounded functions. This is accomplished by the following lemma, which assumes some uniform control of the averages of $f$ on cubes of a given size.
Since we are assuming \eqref{eq-VMO-Omega}, we can get by with a weaker assumption than those in Lemmas~\ref{lem-vanishing_at_infinity} and~\ref{lem-vanishing_at_boundary}.
\begin{lem}
\label{lem-vmo_equivalence}
Let $\Omega$ be a domain, $\lambda > 0$ and $f \in {\rm bmo}lo$. Suppose $f$ satisfies \eqref {eq-VMO-Omega} and for every $\ell > 0$,
$$C_\ell = \sup_{2Q \subset \Omega, \ell(Q) \geq \ell} |f|_Q < \infty.$$
Then $f$
can be approximated in ${\rm bmo}lo$ by bounded functions.
\end{lem}
\begin{proof}
For a positive $\ell < \lambda/8\sqrt{n}$ and $x \in \Omega$, let $\ell(x) = \min(\frac{{d_\Omega}(x)}{2\sqrt{n}}, \ell)$, take $Q_x$ to be the cube centered at $x$ with sidelength $\ell(x)$, and set
$$\tilde{f}(x) = \fint_{Q_x} f, \quad g(x) = \max(\min(\tilde{f}(x), C_\ell), -C_\ell).$$
As $g$ is a truncation, $|g| \leq C_\ell$.
Since $2Q_x \subset \Omega$, by the definition of $\ell(x)$ we have
$|\tilde{f}(x)|\leq C_\ell$
whenever ${d_\Omega}(x) \geq 2\ell\sqrt{n}$, i.e.\ when $x \in \mathring{\Omega}_{2\ell\sqrt{n}}$, so $g = \tilde{f}$ on $\mathring{\Omega}_{2\ell\sqrt{n}}$.
To estimate the ${\rm bmo}lo$ norm of $h := f - g$, we again use the local-to-global property (\cite[Lemma 3.5]{BD2}):
\begin{equation}
\label{eq-localglobal}
\|h\|_{\rm bmo}lo \lesssim \sup_{2Q \subset \Omega} \fint_B |h - h_Q| + \sup_{2Q \subset \Omega, \ell(Q) \geq \lambda/2} |h|_Q.
\end{equation}
Note that cubes over which the supremum in the second term on the right is taken satisfy ${\rm dist}(Q, {\partial\Omega}) \geq \ell(Q)/2 \geq \lambda/4$, so to bound this term it suffices to control the averages $|h|_Q$ for $Q \subset \mathring{\Omega}_{\lambda}o$. By the choice of $\ell$, $\mathring{\Omega}_{\lambda}o \subset \mathring{\Omega}_{2\ell\sqrt{n}}$.
So let us first consider a cube $Q\subset \mathring{\Omega}_{2\ell\sqrt{n}}$. For $x \in Q$,
$g(x) = \tilde{f}(x) = f_{Q_x}$ with $\ell(Q_x)= \ell$. Suppose $\ell(Q) \geq \ell$. Then for $z \in Q$,
\begin{equation}
\label{eq-Vdelta}
2^{-n}\ell^n \leq |Q_z \cap Q| \leq \ell^n,
\end{equation}
and therefore
$$\fint_{Q_z \cap Q} |h| \leq \frac{2^n}{\ell^n}\int_{Q_z \cap Q} |f(x) - f_{Q_x}| dx \leq \frac{2^n}{\ell^{2n}}\int_{Q_z} \int_{Q_x} |f(x) - f(y)| dy dx
\leq 2^{1 + 3n} \fint_{2Q_z} |f - f_{2Q_z}|,$$
where we have used the fact that $x \in Q_z \implies Q_x \subset 2Q_z$.
Since $2Q_z\subset \Omega$, the right-hand-side is bounded by $2^{1 + 3n}{\omega_\Omega}(f,2\ell)$.
From this, applying \eqref{eq-Vdelta} again, we get that
$$
\fint_{Q} |h|
\leq \fint_{Q} \left\{\int_{z \in Q_x \cap Q} \frac{2^n dz}{|Q_z \cap Q|}\right\} |h(x)| dx\
= 2^n\fint_{Q}\fint_{x \in Q_z \cap Q} |h(x)| dx dz \lesssim {\omega_\Omega}(f,2\ell).
$$
Thus ${\omega_\Omega}(f,2\ell)$ controls the second term in \eqref{eq-localglobal}, as well as bounding the oscillation of $h$ over cubes in $\mathring{\Omega}_{2\ell\sqrt{n}}$ with $\ell(Q) \geq \ell$.
It now remains to estimate the oscillation of $h$ over cubes $Q$ with $2Q \subset \Omega$ and such that either $Q \not\subset \mathring{\Omega}_{2\ell\sqrt{n}}$ or $\ell(Q) < \ell$. In the first case we also have $\ell(Q) \leq 2{\rm dist}(Q,{\partial\Omega}) < 4\sqrt{n}\ell$ so we will deal with the two simultaneously.
Since $ |h - h_Q| \leq |f - f_Q| + |g - g_Q|$ and truncation reduces oscillation, we have
$$\fint_Q |h - h_Q| \leq {\omega_\Omega}(f,4\sqrt{n}\ell) + \fint_Q |\tilde{f}(x) - \tilde{f}_Q| dx \leq {\omega_\Omega}(f,4\sqrt{n}\ell) + \fint_Q \fint_Q |f_{Q_x} - f_{Q_y}| dy dx.$$
For $x \in Q$, we take $Q'_x$ to be the cube centered at $x$ of sidelength $d = \frac{{\rm dist}(Q,{\partial\Omega})}{2\sqrt{n}}$, noting that
$$d = \min\Big(\frac{{\rm dist}(Q,{\partial\Omega})}{2\sqrt{n}} ,\ell\Big) \leq \ell(x) \leq \frac{{\rm dist}(Q,{\partial\Omega}) + {\rm diam}(Q)}{2\sqrt{n}} \leq c d, \quad c:= 1 + 2\sqrt{n}.$$
Then
$$ |f_{Q'_x} - f_{Q_z}| \leq \fint_{Q'_x}|f - f_{Q_x}| \leq c^n \fint_{Q_x}|f - f_{Q_x}| \leq c^n \omega(f, 4\sqrt{n}\ell).$$
The triangle inequality then gives us
$$\fint_Q \fint_Q |f_{Q_z} - f_{Q_y}| dy dx \leq 2c^n\omega(f, 4\sqrt{n}\ell) + \fint_Q \fint_Q |f_{Q'_x} - f_{Q'_y}| dy dx.$$
Finally, writing the averages as convolution with $d^{-n}\chi_{[-d/2,d/2]^n}$, we can bound the second term on the right-hand-side by
$$ \fint_Q \fint_Q \fint_{[-d/2,d/2]^n} |f(x - z) - f(y - z)| dz dy dx = \fint_{[-d/2,d/2]^n}\fint_{Q-z} \fint_{Q-z} |f(u) - f(v)| du dv dz.$$
This quantity is also bounded by $\omega(f, 4\sqrt{n}\ell)$, since for $z \in [-d/2,d/2]^n$, the choice of $d$ gives ${\rm dist}(Q-z, {\partial\Omega}) \geq {\rm dist}(Q,{\partial\Omega}) - d/2 > 0$ so $Q - z \subset \Omega$.
In summary, for the given $\ell$, we have shown the existence of a bounded function $g$ such that $\|f - g\|_{\rm bmo}lo \lesssim \omega(f, 4\sqrt{n}\ell)$, where the constants depend only on $n$. By choice of $\ell$, we can make this as small as we like.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm-approxdomain}]
On any domain $\Omega$ and for any $\lambda > 0$, if $f$ can be approximated in ${\rm bmo}lo$ by uniformly continuous functions, then
\eqref{eq-VMO-Omega} holds. For the converse inclusions we show the strongest, namely that any function in ${\rm bmo}lo$, with $\lambda$ as in the hypotheses of the theorem, can be approximated by bounded Lipschitz functions supported away from the boundary. This follows from Lemmas~\ref{lem-bounded} and \ref{lem-vmo_equivalence} by noting that for any $\ell \le \lambda$, $C_\ell \leq \|f\|_{{\rm bmo}_\ell(\Omega)} < \infty$.
\end{proof}
As a corollary of Theorem~\ref{thm-approxdomain}, Lemma~\ref{lem-logbmolo} and Proposition~\ref{prop-bmolo_0}, we get the following.
\begin{prop}
\label{prop-vmo-cmo}
Let $\Omega$ be an ${(\epsilon,\delta)}$-domain and $f \in {\rm bmo}O$. Then $f$ can be approximated in ${\rm bmo}O$ by bounded Lipschitz functions supported away from ${\partial\Omega}$ if and only if
\eqref{eq-VMO-Omega} holds.
In addition, the following are equivalent:
\begin{enumerate}
\item[(i)] $f$ is in ${\rm cmo}(\Omega)$, the closure of $C_c(\Omega)$ in ${\rm bmo}O$;
\item[(ii)] $f$ is in the closure of $C_c^\infty(\Omega)$ in ${\rm bmo}O$;
\item[(iii)] $f$ vanishes at infinity and
\eqref{eq-VMO-Omega} holds.
\end{enumerate}
\end{prop}
\begin{proof}
We only need to show the implication (iii) $\implies$ (ii), as convolution with a smooth mollifier gives the approximation, in the $L^\infty$ norm, of a function in $C_c(\Omega)$ by functions in $C_c^\infty(\Omega)$. If $f$ vanishes at infinity and
\eqref{eq-VMO-Omega} holds, then $f$ vanishes at the boundary and we can follow the proof of Proposition~\ref{prop-bmolo_0}, starting with the sequence $\{f_j\}$ of bounded Lipschitz functions supported away from ${\partial\Omega}$ which approximates $f$, and multiplying by the cut-off functions $\psi_k$ to make them of compact support in $\Omega$.
\end{proof}
\section{Proof of Theorem~\ref{thm2}: extension from a locally uniform domain}
\label{sec-extension1}
In this section we assume $\Omega$ is an ${(\epsilon,\delta)}$ domain and fix $\lambda\leq \lambda_{\epsilon, \delta}$, where $\lambda_{\epsilon, \delta}:= \frac{\epsilon^2 \delta}{320 n(1 + \sqrt{n} \epsilon)}$.
As in \cite{BD2}, for $f\in {\rm bmo}lo$ we define a function ${\mathbb T}lam f:{\mathbb R}n \to {\mathbb R}$ by
\begin{equation}\label{Ext_operator_T}
{\mathbb T}lam f (x) = \begin{cases}
f(x) & \textup{if } x\in \Omega; \\
f_{Q^*} &\textup{if } x\in Q\in E': \ell(Q)\leq \lambda; \\
0 & \textup{otherwise} .
\end{cases}
\end{equation}
Here $E'$ denotes the Whitney decomposition of $\Omega'$, the complement of ${\overline{\Omega}}$ (see \cite[Section VI.1]{Stein} for the definition and properties of Whitney cubes), and we have fixed, for each $Q \in E'$ with $\ell(Q)\leq \lambda$, a choice of matching cube $Q^* \in E$ with $\ell(Q) \leq \ell(Q^*) \leq 4\ell(Q)$ and ${\rm dist}(Q,Q^*) \leq C\ell(Q)$. The existence of such a cube is guaranteed by \cite[Lemma 2.4]{Jones2}, since $\lambda_{\epsilon, \delta} \leq \epsilon \delta/(16 n)$.
By \cite[Lemma 2.3]{Jones2}, ${\partial\Omega}$ has measure zero, so it suffices to extend $f$ to $\Omega'$.
We now proceed to average the step function ${\mathbb T}lam f$ on $\Omega'$, as in \cite{BD1}.
Set
$$ \widetilde{\Tlam} f(x) = \left\{ \begin{array}{cc}
f(x), & x\in \Omega,\\
A ({\mathbb T}lam f)(x) & x\in \Omega',
\end{array}
\right.
$$
where the averaging operator $A$, applied applied to $\phi = {\mathbb T}lam f$ on $\Omega'$, is defined by
$$A(\phi)(x) := \fint_{B(x,R(x))} \phi(y) dy, \quad x \in \Omega'.$$
Here
\begin{equation}
\label{eqn-R}
R(x) = c_n {d_\Omega}(x)
\end{equation}
where for $x \in \Omega'$, with an abuse of notation we denote ${d_\Omega}(x) := {\rm dist}(x, {\partial\Omega})$, just like for points inside $\Omega$. As in \cite{BD1}, we choose the constant $c_n$ sufficiently small so that for $x \in \Omega'$, the collection
$${\mathcal N}(x): = \{Q \in E': Q \cap B(x,R(x)) \neq \varnothing\}$$
consists exactly of the Whitney cube containing $x$ and the Whitney cubes adjacent to it, and moreover that at the center $x_Q$ of a cube $Q \in E'$, $A(\phi)(x_Q) = \phi(x_Q)$.
Having proved in \cite{BD2} the boundedness of the map $f \rightarrow {\mathbb T}lam f$ from ${\rm bmo}lo$ to ${\rm bmo}l({\mathbb R}n)$, we now note that it also applies to the map $f \rightarrow \widetilde{\Tlam} f$. As in \cite{BD1}, this follows from the fact that the difference between ${\mathbb T}lam f$ and its averaging $\widetilde{\Tlam} f = A ({\mathbb T}lam f)$ on a Whitney cube in $\Omega'$ is bounded by a constant times $\|{\mathbb T}lam f\|_{{\rm BMO}({\mathbb R}n)}$, since by the choice of $c_n$, the averaging takes places only on adjacent Whitney cubes.
Having part (i) of Theorem~\ref{thm2}, we proceed to part (iv), which will give us the other two parts by approximation.
\subsection{Boundedness on ${\rm Lip}_b(\Omega)$}
The ideas are similar to those in Section 3.3 of \cite{BD1}, with two main differences. First, we are dealing with bounded functions, which makes things easier. On the other hand, we need to deal with the ``jump" in the definition of ${\mathbb T}lam$, namely the separate definitions for cubes near and far away from the boundary.
Starting with a bounded Lipschitz function $f$ on $\Omega$, by definition $\|\widetilde{\Tlam} f\|_\infty = \|f\|_\infty$. If we can show that $\widetilde{\Tlam} f$ is Lipschitz on $\Omega'$ with $\|\widetilde{\Tlam} f\|_{{\rm Lip}_b(\Omega')} \lesssim \|f\|_{{\rm Lip}_b(\Omega)}$,
then the same argument as in Section 3.3 of \cite{BD1}, i.e.\ extending to ${\partial\Omega}$ by uniform continuity, will give us the desired Lipschitz continuity on all of ${\mathbb R}n$ and prove part (iv) of Theorem~\ref{thm2}.
As in \cite[Lemma 1 and Corollary 3]{BD1}, the key is the inherent local Lipschitz nature of the averaging itself, which gives, for $x_1, x_2 \in \Omega'$,
\begin{equation}
\label{eq-Tlam} |\widetilde{\Tlam} f(x_1)-\widetilde{\Tlam} f(x_2)| \le C \sup_{Q,Q' \in {\mathcal N}(x_1) \cup {\mathcal N}(x_2)} \left|({\mathbb T}lam f)_{Q}- ({\mathbb T}lam f)_{Q'}\right| \min\left(\frac{|x_1 - x_2|}{\min_{i=1,2} {d_\Omega}(x_i)}, 1\right).
\end{equation}
Thus if both $x_i$ are sufficiently far from ${\partial\Omega}$, say contained in Whitney cubes of sidelength at least $\lambda/40\sqrt{n}$, then
$$
|\widetilde{\Tlam} f(x_1)-\widetilde{\Tlam} f(x_2)| \leq C_{\lambda,n} \|{\mathbb T}lam f\|_\infty |x_1 - x_2| \leq C_{\lambda,n} \|f\|_\infty |x_1 - x_2|.
$$
On the other hand, if one of the points, say $x_1$, belongs to a Whitney cube of sidelength less than $\lambda/40\sqrt{n}$, while the other, $x_2$, belongs to a cube of sidelength greater than $\lambda/4$, then by the triangle inequality and the properties of Whitney cubes,
$$|x_1 - x_2| \geq {d_\Omega}(x_2) - {d_\Omega}(x_1) \geq \lambda/4 - (5\sqrt{n}\lambda/40\sqrt{n}) = \lambda/8,$$
and the trivial $L^\infty$ bound (not using \eqref{eq-Tlam}) gives
$$
|\widetilde{\Tlam} f(x_1)-\widetilde{\Tlam} f(x_2)| \leq C_{\lambda} \|\widetilde{\Tlam} f\|_\infty |x_1 - x_2| = C_{\lambda} \| f\|_\infty |x_1 - x_2| .
$$
Thus it remains to consider the case when both points lie in Whitney cubes, say $Q_1, Q_2$, respectively, of sidelength at most $\lambda/4$.
Recalling that the Whitney cubes in ${\mathcal N}(x_i)$ are either $Q_i$ or adjacent to it, by the properties of adjacent Whitney cubes, any cubes $Q,Q'$ in ${\mathcal N}(x_1) \cup {\mathcal N}(x_2)$, have sidelength bounded by $\lambda$, so
$$
|\widetilde{\Tlam} f(x_1)-\widetilde{\Tlam} f(x_2)| \le C\sup_{Q,Q' \in {\mathcal N}(x_1) \cup {\mathcal N}(x_2)} \left|f_{Q^*}- f_{(Q')^*}\right| \min\left(\frac{|x_1 - x_2|}{\min_{i=1,2} {d_\Omega}(x_i)}, 1\right).
$$
In the special case when $Q_1$ and $Q_2$ are adjacent (which includes the case $Q_1 = Q_2$), since $\lambda < \epsilon \delta / 16n$, we can apply \cite[Lemma 2.8]{Jones2} to get that the shortest Whitney chain connecting $Q_1^*$ and $Q_2^*$ has length $m$ bounded by a constant.
By transitivity, this also applies to any two cubes $Q^*, (Q')^*$ which are matching to cubes $Q, Q' \in {\mathcal N}(x_1) \cup {\mathcal N}(x_2)$.
Along such a chain, the difference of averages of $f$ on two adjacent cubes is bounded by $\|f\|_{{\rm Lip}(\Omega)}$ times the sum of the diameters of the two cubes. Again by the properties of Whitney cubes, the largest cube along such a chain has diameter at most $4^m$ times the smallest. This gives (see also \cite[Lemma 4]{BD1} for the general case of a function in ${\rm BMO}(\Omega)$ on any domain $\Omega$) that
$$\left|f_{Q^*}- f_{(Q')^*}\right| \lesssim \|f\|_{{\rm Lip}(\Omega)}\ell(Q^*).$$
Since in this special case the sidelengths of $Q\in {\mathcal N}(x_1) \cup {\mathcal N}(x_2)$ and a matching cube $Q^*$ are comparable to $\ell(Q_i)$, which are in turn comparable to ${d_\Omega}(x_i)$, we get that
$$
|\widetilde{\Tlam} f(x_1)-\widetilde{\Tlam} f(x_2)| \lesssim \|f\|_{{\rm Lip}(\Omega)}|x_1 - x_2|.
$$
Finally, when $Q_1$ and $Q_2$ are not adjacent, we follow the argument in Section 3.3 of \cite{BD1}. In this case $|x_1 - x_2|$ must be at least as large as the sidelength of the smallest Whitney cube adjacent to either $Q_1$ or $Q_2$, which means $|x_1 - x_2| \gtrsim \max_{i=1,2}\ell(Q_i)$,
so the estimate is achieved by showing that
$$\left|f_{Q_1^*}- f_{Q_2^*}\right| \lesssim \|f\|_{{\rm Lip}(\Omega)}\max_{i=1,2}\ell(Q_i) \lesssim \|f\|_{{\rm Lip}(\Omega)}|x_1 - x_2|,$$
and noting that the left-hand-side is just the difference of values of $\widetilde{\Tlam} f$ at the center points of the $Q_i$, by the choice of $c_n$ in \eqref{eqn-R}. The comparison of the value of $\widetilde{\Tlam} f(x_i)$ to the value of $\widetilde{\Tlam} f$ at the center of $Q_i$ can be bound by the same quantity using the case of adjacent (in this case identical) cubes above.
It should be noted that while Lipschitz extensions do not need any restriction on the domain, for this particular extension we have had to use some of the geometric properties of ${(\epsilon,\delta)}$ domains, namely the ``reflection" lemmas from \cite{Jones2}, in particular Lemmas 2.5 and 2.8.
\subsection{The ${\rm vmo}$ and ${\rm cmo}$ extensions}
We follow the argument in \cite[Sections 3.4 and 3.5]{BD1}. Here Proposition~\ref{prop-vmo-cmo} gives us, in the case of $f \in {\rm vmo}(\Omega)$, the approximation by bounded Lipschitz functions in ${\rm bmo}O$, and part (ii) of Theorem~\ref{thm2} follows from parts (i), (iv) and Theorem~\ref{thm-bourdaud}.
For ${\rm cmo}(\Omega)$, we just need to add to this argument the fact that the extension maps functions of compact support in $\Omega$ to functions of compact support in ${\mathbb R}n$. This is shown in \cite[Section 3.5]{BD1} in the homogeneous case (where compact support means the function is constant outside a compact set), but the case here is simpler since the extension ${\mathbb T}lam f$ is zero, by definition, on Whitney cubes sufficiently far from ${\partial\Omega}$ and therefore, by the local nature of the averaging process (i.e.\ the choice of $c_n$ in \eqref{eqn-R}), so is $\widetilde{\Tlam} f$. More specifically, it will vanish on any $Q \in E'$ all of whose neighbors have sidelength greater than $\lambda$, which means it is guaranteed to vanish if $\ell(Q) > 4\lambda$.
Suppose $f \in {\rm bmo}O$ is supported in $B(0,R)$. By the argument above, we only need to study $\widetilde{\Tlam} f$ on Whitney cubes $Q \in E'$ with $\ell(Q) \le 4\lambda$. Since $4\lambda < \epsilon \delta/16n$, we can apply \cite[Lemmas 2.4 and 2.5]{Jones2} to such cubes to conclude that their matching cubes $Q^*$ must lie within a distance comparable to $\ell(Q)$. Thus ${\mathbb T}lam f = 0$ on $Q$ whenever ${\rm dist}(Q, 0) \ge R + C_{\epsilon, n, \lambda}$. Again by the local nature of the averaging process, this gives $\widetilde{\Tlam} f = 0$ on any Whitney cube in $\Omega'$ such that all its neighbors lie in the complement of $B(0,R + C_{\epsilon, n, \lambda})$. Since we are only looking at cubes of sidelength bounded by $4\lambda$, this shows that $\widetilde{\Tlam} f$ is supported in $B(0, R')$ for some $R' > R$.
\section{Examples}
\label{sec-examples}
We close the paper with a couple of examples to illustrate Theorem~\ref{thm-approxdomain} and some of the results in Section~\ref{sec-approximation2}.
\begin{example}
\label{example1} \textnormal{In ${\mathbb R}^2$, our domain $\Omega$ consists of the left half-plane $\{(x,y): x < 0\}$ connected to infinitely many disjoint strips lying in the right half-plane, parallel to the positive $x$ axis.
The $n$th strip $S_n$ has vertical width $1/n$ and horizontal length $L_n$. If we take a cube of sidelength $\ell$, there are only finitely many strips $S_n$ which can contain this cube, namely those with $n < \ell^{-1}$. }
\textnormal{If $f$ is in ${\rm bmo}lo$ and $\ell < 1/n < \lambda$, then starting from a cube $Q$ of sidelength $\ell$ in $S_n$, to reach a cube of sidelength $\lambda$ we have to get to the left half-plane, which means going through a chain of Whitney cubes in $S_n$ of sidelength $\gtrsim \ell$ and hence whose length $m \lesssim L_n\ell^{-1}$. The standard telescoping sum argument then gives the bound $|f|_Q \lesssim L_n\ell^{-1}\|f\|_{\rm bmo}lo$. Since $n$ is bounded by $\ell^{-1}$, we get that for $0 < \lambda' < \lambda$,
$$\|f\|_{{\rm bmo}_{\lambda'}(\Omega)}\lesssim \|f\|_{\rm bmo}lo + \sup\{|f|_Q: \lambda' \leq \ell(Q) < \lambda\}
\lesssim \|f\|_{\rm bmo}lo \Big(1 + \max_{n < \frac{1}{\lambda'}}\frac{L_n}{\lambda'}\Big) < \infty.$$
By Theorem~\ref{thm-approxdomain}, any $f \in {\rm bmo}lo$ which satisfies the vanishing mean oscillation condition $\displaystyle{\lim_{t \rightarrow 0^+} {\omega_\Omega}(f, t) = 0}$ can therefore be approximated in ${\rm bmo}lo$ by bounded Lipschitz functions.
}
\textnormal{On the other hand, such a domain $\Omega$ can only be an ${(\epsilon,\delta)}$ domain if the lengths $L_n$ go to zero sufficiently fast as $n \rightarrow \infty$, in order to guarantee that when two points are at some distance $\delta' \leq \delta$, they can belong to $S_n$ only for sufficiently small $n$, depending on $\delta'$, and such $S_n$ are wide enough for them to be joined by the appropriate $\epsilon$ cigar. If $L_n$ do not go to zero, $\Omega$ will not be ${(\epsilon,\delta)}$.}
\textnormal{How fast do $L_n$ have to go to zero for $\Omega$ to be ${(\epsilon,\delta)}$? Consider the function $f(x,y)$ on $\Omega$ which is $0$ on the left half-plane and is equal to $nx$ on $S_n$. Since all cubes contained in the strip $S_n$ are of sidelength at most $1/n$, the mean oscillation of $f$ is bounded by $1$. Moreover, since only finitely many of the strips contain cubes of sidelength at least $\lambda$, and $0 \leq f \leq n L_n$ on $S_n$, we have
$$\sup\{|f|_Q: \ell(Q) \geq \lambda\} \lesssim \max_{n < \frac{1}{\lambda}}n L_n < \infty.$$
If we could extend $f$ to a function in ${\rm bmo}({\mathbb R}n)$ then its averages would have to satisfy the logarithmic estimate \eqref{eq-logbmo}, so for each $n$, taking a cube $Q$ of sidelength $\frac{1}{2n}$ at the rightmost tip of $S_n$, we would need to have
$$n L_n \approx |f|_Q \lesssim \|f\|_{{\rm bmo}lo} \log\Big(\frac{2}{\ell(Q)}\Big) = \|f\|_{{\rm bmo}lo} \log(4n).$$
Thus $L_n$ must be ${\mathcal O}(\frac{\log n}{n})$ as $n \rightarrow \infty$.}
\end{example}
\begin{example}
\label{example2} \textnormal{We now give a variation on Example~\ref{example1} which does not satisfy the hypotheses of Theorem~\ref{thm-approxdomain}. For each integer $n \geq 1$, instead of one strip we attach to the left half-plane $n$ disjoint strips in the right half-plane, $S_{n,j}, 1 \leq j \leq n$, with vertical width $\frac{1}{j}$ and horizontal length $n$. This means that for any $\ell < 1$, there will be cubes of size $\ell$ in infinitely many strips, further and further away from the left half-plane.}
\textnormal{For this domain, fixing any $\lambda > 0$, we want to show there is a function $f$ in ${\rm bmo}lo$ which is not in ${\rm bmo}_{\lambda'}(\Omega)$ for some $\lambda' < \lambda$.
Define $f$ to be $0$ in the left half-plane and on any strip which contains cubes of size $\lambda$. On all the other strips $S_{n,j}$, define $f(x,y) = c_j x$, $c_j \neq 0$. Then $f$ is Lipschitz with constant $c_j$ on $S_{n,j}$, which contains only cubes of sidelength bounded by $\frac{1}{j}$, and therefore has vanishing mean oscillation provided $\frac{c_j}{j} \rightarrow 0$ as $j \rightarrow \infty$. It is also in ${\rm bmo}lo$ because it is zero on cubes of sidelength $\lambda$ or greater contained in $\Omega$. However, for $\lambda' < \frac{1}{j} < \lambda$, there will be cubes $Q$ of size $\lambda'$ in all $S_{n,j}$ with $n \geq j$, for which the averages $|f|_Q \approx n c_j$. Thus $f \not\in {\rm bmo}_{\lambda'}(\Omega)$. }
\textnormal{Note that while $f$ satisfies the vanishing mean oscillation condition $\displaystyle{\lim_{t \rightarrow 0^+} {\omega_\Omega}(f, t) = 0}$, and its averages over all cubes of side $\lambda$ or larger are zero, it does not vanish at infinity in the sense Definition~\ref{def-vanishing_at}, as can be seen by looking at cubes of sidelength approximately $1$ in $S_{n,1}$, for arbitrary large $n$, whose distance from the origin is going to infinity but on which the oscillation of $f$ is approximately $c_1$. Thus the implication $2 \implies 3$ in Proposition~\ref{prop-cmo} does not hold in a general domain.}
\end{example}
\end{document}
|
\begin{document}
\title{Experimental proposal for symmetric minimal two-qubit state tomography}
\author{Amir Kalev}
\affiliation{Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, 117543, Singapore}
\author{Jiangwei Shang}
\affiliation{Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, 117543, Singapore}
\author{Berthold-Georg Englert}
\affiliation{Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, 117543, Singapore}
\affiliation{Department of Physics, National University of Singapore, 2 Science Drive 3, 117542, Singapore}
\date{Finalized on Dec. 13, 11; Posted on Mar. 08, 12}
\begin{abstract}
We propose an experiment that realizes a symmetric informationally complete (SIC) probability-operator measurement (POM) in the four-dimensional Hilbert space of a qubit pair. The qubit pair is carried by a single photon as a polarization qubit and a path qubit. The implementation of the SIC POM is accomplished with the means of linear optics. The experimental scheme exploits a new approach to SIC POMs that uses a two-step process: a measurement with full-rank outcomes, followed by a projective measurement on a basis that is chosen in accordance with the result of the first measurement. The basis of the first measurement and the four bases of the second measurements are pairwise unbiased --- a hint at a possibly profound link between SIC POMs and mutually unbiased bases.
\end{abstract}
\maketitle
Quantum state tomography, the procedure for inferring the state of a quantum system from measurements applied to it, is an important component in most, if not all, quantum computation and quantum communication tasks. The successful execution of such tasks hinges in part on the ability to assess with high efficiency the state of the system at various stages.
A general measurement in quantum mechanics is a probability-operator measurement (POM). A POM is informationally complete (IC) if any state of the system is determined completely by the probabilities for the POM outcomes \cite{ic1,ic2,ic3}. State tomography infers these probabilities from the data acquired with the aid of the POM.
A symmetric IC POM (SIC POM) is an IC POM of a particular kind. In a $d$-dimensional Hilbert space (of kets) it is composed of $d^2$ subnormalized rank-1 projectors, $\{{\cal P}_j\}_{j=1}^{d^2}$, with equal pairwise fidelity of $1/(d+1)$. Their high symmetry and high tomographic efficiency have attracted the attention of many researchers, and a lot of work, both analytical and numerical, has been devoted to the construction of SIC POMs in various dimensions, see e.g. \cite{berge04,renes04,appleby05,scott06,scott10,zauner11}.
In contrast to the major theoretical progress, up to date, all experiments and even proposals for experiments implementing SIC POMs have been limited to the very basic quantum system, the two-level system (qubit) \cite{ling06,pimenta10}, with the exception of \cite{steinberg11} where a SIC POM for a three-level system was approximated. This is, in part, due to the fact that there is no systematic procedure for implementing SIC POMs in higher dimensions, in a simple experimental set-up.
In this contribution we suggest a feasible experiment that implements a SIC POM for four-dimensional Hilbert space of a qubit pair. The qubit pair is carried by a single photon, and the measurement is realized by passive linear optical elements and photodetectors. The experiment is clearly feasible with current technology.
The proposal exploits a new theoretical approach for constructing SIC POMs \cite{our}. In this approach, we `break' the SIC POM into two successive measurements, each with $d$ outcomes, with the intention that each measurement would be relatively easy to implement. Unexpectedly, we find that the two successive measurements that make up the SIC POM in dimension 4, not only have a strikingly simple form, but also hint at a close relation between the structure of mutually unbiased bases (MUB) and that of the SIC POM; Ref.~\cite{durt10} is a recent review on MUB.
Let us begin by setting up nomenclature and notations. A general measurement on a quantum system is composed of a set of outcomes. The latter are mathematically represented by positive operators ${\cal P}_j$ that sum up to the identity operator. The probability of obtaining the outcome ${\cal P}_j$ is given by the Born rule: \mbox{$p_j = \tr{{\cal P}_j\rho}$}, where $\rho$ is the pre-measurement statistical operator of the system. If the $j$th outcome is found, the post-measurement statistical operator of the system is given by
\mbox{$\rho_{j}=\frac{1}{p_j}P_j\rho P^\dagger_j$},
where $P_j$ is the relevant Kraus operator for the $j$th outcome, \mbox{${\cal P}_j=P^\dagger_j P _j$}. Note that the decomposition of the ${\cal P}$s into the corresponding Kraus operators is not unique; for example, \mbox{$P^\dagger_j P_j$} is invariant under the unitary transformation \mbox{$P_j\rightarrow U_jP_j$}, with different $U_j$s corresponding to different implementations of the POM.
Suppose that a given system is subjected to a sequence of two POMs, each with $d$ outcomes, \mbox{$\{{\cal A}_k=A^\dagger_k A_k\}_{k=1}^{d}$}, followed by \mbox{$\{{\cal B}^{\scriptscriptstyle(k)}_j\}_{j=1}^{d}$}, where the superscript $k$ indicates that in general the second measurement depends on the actual outcome of the first measurement. Following Born's rule, the probability of obtaining the $n$th and $m$th outcomes for the first and second measurements is given by \mbox{$\tr{\rho A^\dagger_n {\cal B}^{\scriptscriptstyle(n)}_mA_n}$}. Accordingly, the two successive measurements are equivalent to a single POM with $d^2$ outcomes \mbox{${\cal P}_{n,m}=A^\dagger_n {\cal B}^{\scriptscriptstyle(n)}_mA_n$} with \mbox{$n,m=1,\;\ldots,\;d$}. Indeed, upon finding the over-all outcome ${\cal P}_{n,m}$, we know that the $n$th outcome of the first POM and the $m$th outcome of the second POM are the case. In what follows we will identify the $A$s and the ${\cal B}$s such that the ${\cal P}$s make up the SIC POM for a qubit pair. But before doing so, we discuss such an identification for a single qubit.
All SIC POMs in two-dimensional Hilbert space are unitarily equivalent to the ``tetrahedron measurement'' (TM), whose outcomes correspond to four vectors that define a tetrahedron in the Bloch sphere \cite{berge04,renes04}. The TM could be realized by a sequence of two measurements, as sketched in Fig.~\ref{fig:tetrahedron}. Here, the qubit is encoded in a spatial alternative of a single photon (``path qubit''): traveling on the left or on the right. A unitary transformation on the qubit state amounts to sending the photon through a set of beam splitters (BSs) and phase shifters (PSs) \cite{englert01}.
\begin{figure}
\caption{An optical implementation of the tetrahedron measurement using two successive measurements.}
\label{fig:tetrahedron}
\end{figure}
In this optical setting the TM is implemented as follows: First, two BSs (BS1 and BS2) are used to implement the Kraus operators \mbox{$A_1={\rm diag}(t_1,t_2)$} and \mbox{$A_2={\rm diag}(r_1,r_2)$}, where ${\rm diag}$ stands for a diagonal matrix, and $t_i$ and $r_i$ are the transmission and reflection amplitudes of the $i$th BS. A photon which enters the apparatus with a path statistical operator $\rho$, exits at port $k$ with the statistical operator \mbox{$A_k\rho A_k^\dagger/\tr{A_k\rho A_k^\dagger}$}. For the values \mbox{${\textstyle t_1{=}r_2{=}\sqrt{\frac1{2}-\frac1{\sqrt{12}}}}$} and \mbox{${\textstyle t_2{=}r_1{=}\sqrt{\frac1{2}+\frac1{\sqrt{12}}}}$}, these operators correspond to the measurement outcomes \mbox{${\cal A}_k{=}\frac1{2}(1+\frac1{\sqrt{3}}(-1)^k\sigma_3)$} with \mbox{$k{=}1,2$} (the $\sigma$s are the Pauli operators with $\sigma_3$ diagonal in the left-right basis). Then a photon that exits the first measurement apparatus at port 1 is measured in the $\sigma_1$ basis while a photon that exits at port 2 is a measured in the $\sigma_2$ basis. These measurements could be realized by balanced BSs and appropriate PSs, as indicated in the figure.
Actually, the TM was successfully implemented in an optical system \cite{ling06}, where the qubit was encoded in a photon's polarization (``polarization qubit'') rather than in spatial alternatives. The set-up of \cite{ling06} also consisted of a sequence of two measurements, quite analogous to what is described above. In that set-up, a partially polarizing beam splitter (PPBS) was used to implement the Kraus operators $A_k$ and then, depending on whether the photon was transmitted or reflected, a measurement of $\sigma_1$ or $\sigma_2$ followed.
Before moving on to the qubit-pair case, let us close the present discussion with three remarks: (i) In the above construction, the qubit MUB play a central role; they are used to construct, by means of successive measurements, the SIC POM. We will see below that such a relation appears in dimension 4 as well. (ii) A practical implementation of the scheme presented in Fig.~\ref{fig:tetrahedron} requires the stabilization of the interferometer loop defined by the four BSs. (iii) SIC POMs for a three-level system could be implemented by using a similar set-up, but with allowing the photon to take three different paths \cite{our}.
\begin{figure*}
\caption{A successive-measurement scheme for realizing the SIC POM of a qubit pair. Here the two-qubit state is encoded in the spatial-polarization state of a single photon.}
\label{fig:sicpom_dim4}
\end{figure*}
In dimension 4, there is only one known SIC POM, and all the other known SIC POMs are unitarily equivalent to it \cite{appleby05}. This SIC POM is composed of 16 subnormalized projectors onto 16 (fiducial) kets. The latter are represented in the following matrices as columns with \mbox{$N=\sqrt{5+\sqrt{5}}$} and {$\chi=\sqrt{2+\sqrt{5}}$} \cite{bengtsson10},
\begin{alignat}{2}\label{fiddim4}
&\frac1{N}{\left(
\begin{array}{rrrr}
\chi&\chi&\chi&\chi\\
1&-1&1&-1\\
1&1&-1&-1\\
1&-1&-1&1
\end{array}
\right)}\!,
&\;&\frac1{N}{\left(
\begin{array}{rrrr}
1&1&1&1 \\
1&-1&1&-1\\
i\chi&i\chi&-i\chi&-i\chi\\
-i&i&i&-i
\end{array}
\right)}\!,\nonumber\\
&\frac1{N}{\left(
\begin{array}{rrrr}
1&1&1&1 \\
i\chi&-i\chi&i\chi&-i\chi\\
i&i&-i&-i\\
-1&1&1&-1
\end{array}
\right)}\!,
&\;&\frac1{N}{\left(
\begin{array}{rrrr}
1&1&1&1 \\
i&-i&i&-i\\
1&1&-1&-1\\
-i\chi&i\chi&i\chi&-i\chi
\end{array}
\right)}\!.
\end{alignat}
Each of these matrices could be written as a diagonal matrix times a unitary matrix. The set of bases, corresponding to each unitary matrix, together with the computational basis, form the complete set of MUB in dimension 4. To be more specific, the diagonal matrices are
\begin{align}\label{Adim4}
A_1=&\frac1{N}{\rm diag}(\chi,1,1,1),\;\;A_3=\frac1{N}{\rm diag}(1,1,\chi,1),\nonumber\\
A_2=&\frac1{N}{\rm diag}(1,\chi,1,1),\;\;A_4=\frac1{N}{\rm diag}(1,1,1,\chi),
\end{align}
and the unitary matrices are
\begin{alignat}{2}\label{Bdim4}
{\cal U}_1\!&=\!\frac1{2}{\left(
\begin{array}{rrrr}
1&1&1&1 \\
1&-1&1&-1\\
1&1&-1&-1\\
1&-1&-1&1
\end{array}
\right)}\!,\;
&{\cal U}_3\!&=\!\frac1{2}{\left(
\begin{array}{rrrr}
1&1&1&1 \\
1&-1&1&-1\\
i&i&-i&-i\\
-i&i&i&-i
\end{array}
\right)}\!,\nonumber\\
{\cal U}_2\!&=\!\frac1{2}{\left(
\begin{array}{rrrr}
1&1&1&1 \\
i&-i&i&-i\\
i&i&-i&-i\\
-1&1&1&-1
\end{array}
\right)}\!,\;
&{\cal U}_4\!&=\!\frac1{2}{\left(
\begin{array}{rrrr}
1&1&1&1 \\
i&-i&i&-i\\
1&1&-1&-1\\
-i&i&i&-i
\end{array}
\right)}\!.
\end{alignat}
Noting that \mbox{$\sum_jA^\dagger_jA_j=1$}, we identify the $A$s with the Kraus operators of a measurement.
Actually, the operations of Eq.~\eqref{Bdim4} transform the computational basis into the MUB,
\begin{align}\label{MUBdim4}
{\mathfrak B}_1&=\left\{\begin{array}{r}
\frac1{\sqrt{2}}(\ket{0}+\ket{1})\\
\frac1{\sqrt{2}}(\ket{0}-\ket{1})\\
\end{array}\right\}\otimes\left\{\begin{array}{r}
\frac1{\sqrt{2}}(\ket{0}+\ket{1})\\
\frac1{\sqrt{2}}(\ket{0}-\ket{1})\\
\end{array}\right\},\nonumber\\
{\mathfrak B}_2&=\left\{\begin{array}{r}
\frac1{\sqrt{2}}(\ket{0}+i\ket{1})\\
\frac1{\sqrt{2}}(\ket{0}-i\ket{1})\\
\end{array}\right\}\otimes\left\{\begin{array}{r}
\frac1{\sqrt{2}}(\ket{0}+i\ket{1})\\
\frac1{\sqrt{2}}(\ket{0}-i\ket{1})\\
\end{array}\right\},\nonumber\\
{\mathfrak B}_3&={\rm CZ}\left\{\begin{array}{r}
\frac1{\sqrt{2}}(\ket{0}+i\ket{1})\\
\frac1{\sqrt{2}}(\ket{0}-i\ket{1})\\
\end{array}\right\}\otimes\left\{\begin{array}{r}
\frac1{\sqrt{2}}(\ket{0}+\ket{1})\\
\frac1{\sqrt{2}}(\ket{0}-\ket{1})\\
\end{array}\right\},\nonumber\\
{\mathfrak B}_4&={\rm CZ}\left\{\begin{array}{r}
\frac1{\sqrt{2}}(\ket{0}+\ket{1})\\
\frac1{\sqrt{2}}(\ket{0}-\ket{1})\\
\end{array}\right\}\otimes\left\{\begin{array}{r}
\frac1{\sqrt{2}}(\ket{0}+i\ket{1})\\
\frac1{\sqrt{2}}(\ket{0}-i\ket{1})\\
\end{array}\right\},
\end{align}
where CZ stands for the controlled-Z (phase flip) operation, \mbox{${\rm CZ}={\rm diag}(1,1,1,-1)$}. The bases ${\mathfrak B}_1$ and ${\mathfrak B}_2$ are composed of product states, while the bases ${\mathfrak B}_3$ and ${\mathfrak B}_4$ consist of maximally entangled states.
The structure of fiducial vectors in Eq.~\eqref{fiddim4} (which form the SIC POM in dimension 4) allows us to implement the SIC POM by two successive measurements: A measurement whose Kraus operators are given in Eq.~\eqref{Adim4}, and depending on the measurement outcome, a measurement in one of the MUB of Eq.~\eqref{MUBdim4}. Next, we propose an optical implementation for this scheme.
Our proposal is based on the methods of \cite{englert01} where the two qubits, a polarization qubit and a path qubit, are encoded in a single photon. (We chose here to use a polarization qubit instead of another path qubit in order to avoid as many interferometric loops as possible in the optical set-up.) We consider the vertical (${\sf v}$) and horizontal (${\sf h}$) polarizations as the basic alternative of the polarization qubit, and traveling on the left (${\sf L}$) or on the right (${\sf R}$) as the basic alternative of the path qubit. A unitary transformation on the two-qubit state amounts to sending the photon through a set of passive linear optical elements that unitarily change the state of the path and polarization qubits \cite{englert01}. For our purpose we need the following optical elements:
half-wave plates (HWPs), BSs, polarization and path dependent PSs, and PPBSs. A PPBS is a BS whose reflection and transmission coefficients depend on the polarization. Its action corresponds to a joint unitary transformation on the polarization-path qubits. In the present context, it suffices to consider a PPBS with real amplitude division coefficients $r$ and $t$ that obey the unitarity condition \mbox{$r^2+t^2=1$} for the vertical and horizontal polarizations,
\begin{equation}\label{ppbs}
U_{\!{\textrm{\tiny PPBS}}}=\left(\begin{array}{cccc}
r_v&t_v&0&0\\
-t_v&r_v&0&0\\
0&0&r_h&t_h\\
0&0&t_h&-r_h\\
\end{array}
\right).
\end{equation}
This is a block-diagonal matrix, with the blocks transforming the vertical or horizontal polarization, respectively. Two cases of interest are (i) \mbox{$r_v=t_h=1$}: the polarizing beam splitter (PBS) which totally reflects (transmits) vertically (horizontally) polarized light, and (ii) \mbox{$r_v=r_h=1$}: the CZ gate.
The Kraus operators for the first measurement are listed in Eq.~\eqref{Adim4}. Their realization is schematically drawn in Fig.~\ref{fig:sicpom_dim4} at the `first measurement' part. For each port, we set the parameters of the different optical elements such that a photon which enters the apparatus with a polarization-path statistical operator $\rho$, exits at port $k$ with the two-qubit statistical operator \mbox{$A_k\rho A_k^\dagger/\tr{A_k\rho A_k^\dagger}$}. To be more specific, the apparatus is configured such that the beam splitters BS1a and BS1b have the same properties and so have beam splitters BS2a and BS2b. The PPBSs on the left and right arms also have the same properties. The reflection coefficient of BS1a and BS1b is \mbox{$r_1=1/N$}. The reflection coefficient $r_2$ of BS2a and BS2b satisfies \mbox{$t_1r_2=1/N$}, that is, \mbox{$r_2=1/\sqrt{N^2-1}$}, where $t_1$ is the transmission coefficient of BS1a(b). Setting \mbox{$r_v=t_h=y$} in Eq.~\eqref{ppbs}, the two PPBSs transform vertically polarized incident light $\ket{{\sf v}}$ to the polarizations \mbox{$y\ket{{\sf v}}$} and \mbox{$\sqrt{1-y^2}\ket{{\sf v}}$} in the reflected and transmitted arms, respectively, and horizontally polarized light $\ket{{\sf h}}$ to the polarizations \mbox{$\sqrt{1-y^2}\ket{{\sf h}}$} in reflection and \mbox{$y\ket{{\sf h}}$} in transmission. The amplitude division coefficient $y$ is chosen such that \mbox{$t_1t_2y=1/N$}, and therefore, \mbox{$y=1/\sqrt{N^2-2}$}, where $t_2$ is the transmission coefficient of BS2a(b). These settings ensure that the measurement of Eq.~\eqref{Adim4} is realized.
To complete the measurement scheme, a second measurement is taking place. This measurement depends on the actual outcome of the first measurement, namely, on the output port where the photon exits. For photons emerging from the $k$th port, basis ${\mathfrak B}_k$ of Eq.~\eqref{MUBdim4} is measured. In order to measure in a given basis, ${\mathfrak B}_k$, we first apply a unitary operation ${\cal U}_k$ of Eq.~\eqref{Bdim4} that transforms the basis ${\mathfrak B}_k$ into the computational basis and then measure in the computational basis by using PBSs and photodetectors, as illustrated in Fig.~\ref{fig:sicpom_dim4} at the `second measurement' part.
To implement the unitary transformations of Eq.~\eqref{Bdim4}, one could use either a single, specially designed, birefringent material, or a sequence of wave plates and PPBSs. Considering the latter option, these unitary transformations are
\begin{align}\label{unitaries}
{\cal U}_1&=U_{\!{\textrm{\tiny HWP}}}\otimes U_{\!{\textrm{\tiny BS}}},\nonumber\\
{\cal U}_2&=\left(U_{\!{\textrm{\tiny PS}}}U_{\!{\textrm{\tiny HWP}}}\right) \otimes\left( U_{\!{\textrm{\tiny PS}}}U_{\!{\textrm{\tiny BS}}}\right),\nonumber\\
{\cal U}_3&={\rm CZ}\Big(\left(U_{\!{\textrm{\tiny PS}}}U_{\!{\textrm{\tiny HWP}}}\right)\otimes U_{\!{\textrm{\tiny BS}}}\Big),\nonumber\\
{\cal U}_4&={\rm CZ}\Big(U_{\!{\textrm{\tiny HWP}}}\otimes\left(U_{\!{\textrm{\tiny PS}}}U_{\!{\textrm{\tiny BS}}}\right)\Big),
\end{align}
where \mbox{$U_{\!{\textrm{\tiny PS}}}={\rm diag}(1,i)$} shifts the phase of the path and polarization qubits by $\pi/2$, and $U_{\!{\textrm{\tiny HWP}}}$ and $U_{\!{\textrm{\tiny BS}}}$ implement the Hadamard gate,
\begin{equation}
H=\frac{1}{\sqrt{2}}{\left(\begin{array}{cc}
1&1 \\
1&-1
\end{array}
\right)},
\end{equation}
for the polarization qubit and the path qubit, respectively. For this aim, we use a HWP with its major axis at an angle $\pi/8$ to the optical axis, and a balanced BS.
We see that the unitary transformation ${\cal U}_1$ and ${\cal U}_2$ can be decomposed into a tensor product of two unitary transformations, one for the path qubit and one for the polarization qubit. The unitary transformation ${\cal U}_3$ and ${\cal U}_4$ are not of that kind and could be realized, for example, by using PPBSs together with a Mach-Zehnder interferometer. This closes our proposal.
To conclude, we are proposing a feasible experimental scheme that implements the SIC POM for a two-qubit system. Our scheme uses linear optical elements and photodetectors, and is, therefore, well within the reach of existing technology. The proposal is based on a successive-measurement approach to SIC POMs. We found that the SIC POM for the qubit pair corresponds to a POM diagonal in the computational basis, followed by projections onto bases which are mutually unbiased. We observed that this unique construction is owed to a structural relation between the fiducial vectors and the MUB vectors in dimension 4.
On a more general note, we believe that it would be interesting to learn, if and how this scheme can be generalized to higher dimensions. Such a study could be of a theoretical and a practical use; it might teach us about the SIC POMs' structure in high dimensions and provide new ideas for implementing them. For example, it is possible to show \cite{our} that any SIC POM which is covariant with respect to the Heisenberg-Weyl group can be realized by two successive measurements, each of a rather simple form. This is, in particular, interesting for dimension three, where one has a one-parameter family of non-equivalent group-covariant SIC POMs \cite{appleby05}.
We would like to thank Huangjun Zhu for valuable and stimulating discussions.
Centre for Quantum Technologies is a Research Centre of Excellence funded by Ministry of Education and National Research Foundation of Singapore.
\end{document}
|
\begin{document}
\title[]{On a weighted Trudinger-Moser type inequality on the whole space and related maximizing problem}
\author{Van Hoang Nguyen}
\address{Institut de Math\'ematiques de Toulouse, Universit\'e Paul Sabatier 118 route de Narbonne, 31062 Toulouse C\'edex 09, France}
\email{[email protected]}
\author{Futoshi Takahashi}
\address{Department of Mathematics, Osaka City University \& OCAMI, Sumiyoshi-ku, Osaka, 558-8585, Japan}
\email{[email protected]}
\subjclass[2010]{Primary 35A23; Secondary 26D10.}
\keywords{Trudinger-Moser inequality, weighted Sobolev spaces, maximizing problem.}
\date{\today}
\dedicatory{}
\begin{abstract}
In this paper, we establish a weighted Trudinger-Moser type inequality with the full Sobolev norm constraint on the whole Euclidean space.
Main tool is the singular Trudinger-Moser inequality on the whole space recently established by Adimurthi and Yang, and a transformation of functions.
We also discuss the existence and non-existence of maximizers for the associated variational problem.
\end{abstract}
\maketitle
\section{Introduction}
Let $\Omega \subset \mathbb{R}^N$, $N \ge 2$ be a domain with finite volume.
Then the Sobolev embedding theorem assures that $W^{1,N}_0(\Omega) \hookrightarrow L^q(\Omega)$ for any $q \in [1, +\infty)$,
however, as the function $\log \left( \log (e/|x|) \right) \in W^{1,N}_0(B)$, $B$ the unit ball in $\mathbb{R}^N$, shows,
the embedding $W^{1,N}_0(\Omega) \hookrightarrow L^{\infty}(\Omega)$ does not hold.
Instead, functions in $W^{1,N}_0(\Omega)$ enjoy the exponential summability:
\[
W^{1,N}_0(\Omega) \hookrightarrow \{ u \in L^N(\Omega) \, : \, \int_{\Omega} \exp \left(\alpha |u|^{\frac{N}{N-1}} \right) dx < \infty \quad \text{for any} \, \alpha > 0 \},
\]
see Yudovich \cite{Yudovich}, Pohozaev \cite{Pohozaev}, and Trudinger \cite{Trudinger}.
Moser \cite{Moser} improved the above embedding as follows, now known as the Trudinger-Moser inequality:
Define
\[
TM(N, \Omega, \alpha) = \sup_{u \in W^{1,N}_0(\Omega) \atop \| \nabla u \|_{L^N(\Omega)} \le 1} \frac{1}{|\Omega|} \int_{\Omega} \exp (\alpha |u|^{\frac{N}{N-1}}) dx.
\]
Then we have
\begin{align*}
TM(N, \Omega, \alpha)
\begin{cases}
&< \infty, \quad \alpha \le \alpha_N, \\
&= \infty, \quad \alpha > \alpha_N,
\end{cases}
\end{align*}
here and henceforth $\alpha_N = N \omega_{N-1}^{\frac{1}{N-1}}$ and $\omega_{N-1}$ denotes the area of the unit sphere $S^{N-1}$ in $\mathbb{R}^N$.
On the attainability of the supremum, Carleson-Chang \cite{Carleson-Chang}, Flucher \cite{Flucher}, and Lin \cite{KCLin} proved that
$TM(N, \Omega, \alpha)$ is attained on any bounded domain for all $0 < \alpha \le \alpha_N$.
Later, Adimurthi-Sandeep \cite{Adimurthi-Sandeep} established a weighted (singular) Trudinger-Moser inequality as follows:
Let $0 \le \beta < N$ and put $\alpha_{N, \beta} = \left(\frac{N-\beta}{N}\right) \alpha_N$.
Define
\[
\omegaidetilde{TM}(N, \Omega, \alpha, \beta) = \sup_{u \in W^{1,N}_0(\Omega) \atop \| \nabla u \|_{L^N(\Omega)} \le 1} \frac{1}{|\Omega|} \int_{\Omega} \exp (\alpha |u|^{\frac{N}{N-1}}) \frac{dx}{|x|^{\beta}}.
\]
Then it is proved that
\[
\omegaidetilde{TM}(N, \Omega, \alpha, \beta)
\begin{cases}
&< \infty, \quad \alpha \le \alpha_{N, \beta}, \\
&= \infty, \quad \alpha > \alpha_{N, \beta}.
\end{cases}
\]
On the attainability of the supremum, recently Csat\'o-Roy \cite{Csato-Roy(CVPDE)}, \cite{Csato-Roy(CPDE)} proved that
$\omegaidetilde{TM}(2, \Omega, \alpha, \beta)$ is attained for $0 < \alpha \le \alpha_{2,\beta} = 2\pi(2-\beta)$ for any bounded domain $\Omega \subset \mathbb{R}^2$.
For other types of weighted Trudinger-Moser inequalities, see for example,
\cite{Calanchi}, \cite{Calanchi-Ruf(JDE)}, \cite{Calanchi-Ruf(NA)}, \cite{Furtado-Medeiros-Severo}, \cite{Lam-Lu}, \cite{Souza}, \cite{Souza-O}, \cite{Yang},
to name a few.
On domains with infinite volume, for example on the whole space $\mathbb{R}^N$, the Trudinger-Moser inequality does not hold as it is.
However, several variants are known on the whole space.
In the following, let
\[
\Phi_N(t) = e^t - \sum_{j=0}^{N-2} \frac{t^j}{j!}
\]
denote the truncated exponential function.
First,
Ogawa \cite{Ogawa}, Ogawa-Ozawa \cite{Ogawa-Ozawa}, Cao \cite{Cao}, Ozawa \cite{Ozawa(JFA)}, and Adachi-Tanaka \cite{Adachi-Tanaka}
proved that the following inequality holds true, which we call Adachi-Tanaka type Trudinger-Moser inequality:
Define
\begin{align}
\lambdabel{AT-sup}
A(N, \alpha) = \sup_{u \in W^{1,N}(\mathbb{R}^N) \setminus \{ 0 \} \atop \| \nabla u \|_{L^N(\mathbb{R}^N)} \le 1} \frac{1}{\| u \|^N_{L^N(\mathbb{R}^N)}} \int_{\mathbb{R}^N} \Phi_N (\alpha |u|^{\frac{N}{N-1}}) dx.
\end{align}
Then
\begin{equation}
\lambdabel{AT-TM}
A(N, \alpha)
\begin{cases}
&< \infty, \quad \alpha \, < \, \alpha_N, \\
&= \infty, \quad \alpha \ge \alpha_N.
\end{cases}
\end{equation}
The functional in (\mathbb{R}f{AT-sup})
\[
F(u) = \frac{1}{\| u \|^N_{L^N(\mathbb{R}^N)}} \int_{\mathbb{R}^N} \Phi_N (\alpha |u|^{\frac{N}{N-1}}) dx
\]
enjoys the scale invariance under the scaling $u(x) \mapsto u_{\lambda}(x) = u(\lambda x)$ for $\lambda > 0$, i.e., $F(u_{\lambda}) = F(u)$ for any $u \in W^{1,N}(\mathbb{R}^N) \setminus \{ 0 \}$.
Note that the critical exponent $\alpha = \alpha_N$ is not allowed for the finiteness of the supremum.
On the attainability of the supremum, Ishiwata-Nakamura-Wadade \cite{Ishiwata-Nakamura-Wadade} proved that $A(N, \alpha)$ is attained for any $\alpha \in (0, \alpha_N)$.
In this sense, Adachi-Tanaka type Trudinger-Moser inequality has a subcritical nature of the problem.
On the other hand, Ruf \cite{Ruf} and Li-Ruf \cite{Li-Ruf} proved that the following inequality holds true:
Define
\begin{align}
\lambdabel{LR-sup}
B(N, \alpha) = \sup_{u \in W^{1,N}(\mathbb{R}^N) \atop \| u \|_{W^{1,N}(\mathbb{R}^N)} \le 1} \int_{\mathbb{R}^N} \Phi_N (\alpha |u|^{\frac{N}{N-1}}) dx.
\end{align}
Then
\begin{equation}
\lambdabel{LR-TM}
B(N, \alpha)
\begin{cases}
&< \infty, \quad \alpha \, \le \, \alpha_N, \\
&= \infty, \quad \alpha > \alpha_N.
\end{cases}
\end{equation}
Here $\| u \|_{W^{1,N}(\mathbb{R}^N)} = \left( \| \nabla u \|_{L^N(\mathbb{R}^N)}^N + \| u \|_{L^N(\mathbb{R}^N)}^N \right)^{1/N}$ is the full Sobolev norm.
Note that the scale invariance $(u \mapsto u_{\lambda})$ does not hold for this inequality.
Also the critical exponent $\alpha = \alpha_N$ is permitted to the finiteness of (\mathbb{R}f{LR-sup}).
Concerning the attainability of $B(N, \alpha)$, it is known that $B(N, \alpha)$ is attained for $0 < \alpha \le \alpha_N$ if $N \ge 3$ \cite{Ruf}.
On the other hand when $N = 2$, there exists an explicit constant $\alpha_* > 0$ related to the Gagliardo-Nirenberg inequality in $\mathbb{R}^2$
such that $B(2, \alpha)$ is attained for $\alpha_* < \alpha \le \alpha_2 (= 4\pi)$ \cite{Ruf}, \cite{Ishiwata}.
However, if $\alpha >0$ is sufficiently small, then $B(2,\alpha)$ is not attained \cite{Ishiwata}.
The non-attainability of $B(2,\alpha)$ for $\alpha$ sufficiently small is attributed to the non-compactness of ``vanishing" maximizing sequences, as described in \cite{Ishiwata}.
In the following, we are interested in the weighted version of the Trudinger-Moser inequalities on the whole space.
Let $N \ge 2$, $-\infty < \gamma < N$ and define the weighted Sobolev space $X^{1,N}_{\gamma}(\mathbb{R}^N)$ as
\begin{align*}
&X^{1,N}_{\gamma}(\mathbb{R}^N) = {\dot W}^{1,N}(\mathbb{R}^N) \cap L^N(\mathbb{R}^N, |x|^{-\gamma} dx) \\
&=\{ u \in L^1_{loc}(\mathbb{R}^N) \, : \, \| u \|_{X^{1,N}_{\gamma}(\mathbb{R}^N)} = \left( \| \nabla u \|_N^N + \| u \|_{N, \gamma}^N \right)^{1/N} < \infty \},
\end{align*}
where we use the notation $\| u \|_{N, \gamma}$ for $\left( \int_{\mathbb{R}^N} \frac{|u|^N}{|x|^{\gamma}} dx \right)^{1/N}$.
We also denote by $X^{1,N}_{\gamma,rad}(\mathbb R^N)$ the subspace of $X^{1,N}_\gamma(\mathbb R^N)$ consisting of radial functions.
We note that a special form of the Caffarelli-Kohn-Nirenberg inequality in \cite{Caffarelli-Kohn-Nirenberg}:
\begin{equation}
\lambdabel{CKN}
\| u \|_{N, \beta} \le C \| u \|_{N, \gamma}^{\frac{N-\beta}{N-\gamma}} \| \nabla u \|_{N}^{1 - \frac{N-\beta}{N-\gamma}}
\end{equation}
implies that $X^{1,N}_{\gamma}(\mathbb{R}^N) \subset X^{1,N}_{\beta}(\mathbb{R}^N)$ when $\gamma \le \beta$.
From now on, we assume
\begin{equation}
\lambdabel{assumption:weighted AT}
N \ge 2, \quad -\infty < \gamma \le \beta < N
\end{equation}
and put $\alpha_{N, \beta} = \left(\frac{N-\beta}{N}\right) \alpha_N$.
Recently, Ishiwata-Nakamura-Wadade \cite{Ishiwata-Nakamura-Wadade} proved that the following weighted Adachi-Tanaka type Trudinger-Moser inequality holds true:
Define
\begin{equation}
\lambdabel{weighted AT-sup(radial)}
\tilde{A}_{rad}(N, \alpha, \beta, \gamma)
= \sup_{u \in X^{1,N}_{\gamma, rad}(\mathbb{R}^N) \setminus \{ 0 \} \atop \| \nabla u \|_{L^N(\mathbb{R}^N)} \le 1} \frac{1}{\| u \|_{N, \gamma}^{N(\frac{N-\beta}{N-\gamma})}}
\int_{\mathbb{R}^N} \Phi_N (\alpha |u|^{\frac{N}{N-1}}) \frac{dx}{|x|^{\beta}}.
\end{equation}
Then for $N, \beta, \gamma$ satisfying (\mathbb{R}f{assumption:weighted AT}), we have
\begin{align}
\lambdabel{weighted AT(radial)}
\tilde{A}_{rad}(N, \alpha, \beta, \gamma)
&\begin{cases}
&< \infty, \quad \alpha \, < \, \alpha_{N, \beta}, \\
&= \infty, \quad \alpha \ge \alpha_{N, \beta}.
\end{cases}
\end{align}
Later, Dong-Lu \cite{Dong-Lu} extends the result in the non-radial setting.
Let
\begin{equation}
\lambdabel{weighted AT-sup}
\tilde{A}(N, \alpha, \beta, \gamma)
= \sup_{u \in X^{1,N}_{\gamma}(\mathbb{R}^N) \setminus \{ 0 \} \atop \| \nabla u \|_{L^N(\mathbb{R}^N)} \le 1} \frac{1}{\| u \|_{N, \gamma}^{N(\frac{N-\beta}{N-\gamma})}}
\int_{\mathbb{R}^N} \Phi_N (\alpha |u|^{\frac{N}{N-1}}) \frac{dx}{|x|^{\beta}}.
\end{equation}
Then the corresponding result holds true also for $\tilde{A}(N, \alpha, \beta, \gamma)$.
Attainability of the best constant (\mathbb{R}f{weighted AT-sup(radial)}), (\mathbb{R}f{weighted AT-sup}) is also considered in \cite{Ishiwata-Nakamura-Wadade} and \cite{Dong-Lu}:
$\tilde{A}_{rad}(N, \alpha, \beta, \gamma)$ and $\tilde{A}(N, \alpha, \beta, \gamma)$ are attained for any $0 < \alpha < \alpha_{N, \beta}$.
First purpose of this note is to establish the weighted Li-Ruf type Trudinger-Moser inequality on the weighted Sobolev space $X^{1,N}_{\gamma}(\mathbb{R}^N)$
with $N, \beta, \gamma$ satisfying (\mathbb{R}f{assumption:weighted AT}).
Define
\begin{align}
\lambdabel{weighted LR-sup(radial)}
&\tilde{B}_{rad}(N, \alpha, \beta, \gamma)
= \sup_{u \in X^{1,N}_{\gamma, rad}(\mathbb{R}^N) \atop \| u \|_{X^{1,N}_{\gamma}(\mathbb{R}^N)} \le 1} \int_{\mathbb{R}^N} \Phi_N (\alpha |u|^{\frac{N}{N-1}}) \frac{dx}{|x|^{\beta}}, \\
\lambdabel{weighted LR-sup}
&\tilde{B}(N, \alpha, \beta, \gamma)
= \sup_{u \in X^{1,N}_{\gamma}(\mathbb{R}^N) \atop \| u \|_{X^{1,N}_{\gamma}(\mathbb{R}^N)} \le 1} \int_{\mathbb{R}^N} \Phi_N (\alpha |u|^{\frac{N}{N-1}}) \frac{dx}{|x|^{\beta}}.
\end{align}
\begin{theorem}(Weighted Li-Ruf type inequality)
\lambdabel{Theorem:weighted LR}
Assume (\mathbb{R}f{assumption:weighted AT}) and put $\alpha_{N, \beta} = \left(\frac{N-\beta}{N}\right) \alpha_N$.
Then we have
\begin{align}
\lambdabel{weighted LR(radial)}
&\tilde{B}_{rad}(N, \alpha, \beta, \gamma)
\begin{cases}
&< \infty, \quad \alpha \, \le \, \alpha_{N, \beta}, \\
&= \infty, \quad \alpha > \alpha_{N, \beta}.
\end{cases}
\end{align}
Furthermore if $0 \le \gamma \le \beta < N$, we have
\begin{align}
\lambdabel{weighted LR}
&\tilde{B}(N, \alpha, \beta, \gamma)
\begin{cases}
&< \infty, \quad \alpha \, \le \, \alpha_{N, \beta}, \\
&= \infty, \quad \alpha > \alpha_{N, \beta}.
\end{cases}
\end{align}
\end{theorem}
We also study the existence and non-existence of maximizers for the weighted Trudinger-Moser inequalities \eqref{weighted LR(radial)} and \eqref{weighted LR}.
\begin{theorem}
\lambdabel{Maximizers(radial)}
Assume (\mathbb{R}f{assumption:weighted AT}). Then the following statements hold.
\begin{enumerate}
\item[(i)] If $N \geq 3$ then $\tilde{B}_{rad}(N,\alpha,\beta,\gamma)$ is attained for any $0< \alpha \leq \alpha_{N,\beta}$.
\item[(ii)] If $N=2$ then $\tilde{B}_{rad}(2,\alpha,\beta,\gamma)$ is attained for any $0< \alpha \leq \alpha_{2,\beta}$ if $\beta > \gamma$,
while there exists $\alpha_* > 0$ such that $\tilde{B}_{rad}(2,\alpha,\beta,\beta)$ is attained for any $\alpha_* < \alpha < \alpha_{2,\beta}$.
\item[(iii)] $\tilde{B}_{rad}(2,\alpha,\beta,\beta)$ is not attained for sufficiently small $\alpha > 0$.
\end{enumerate}
\end{theorem}
\begin{theorem}
\lambdabel{Maximizers}
Let $N\geq 2$, $0\leq \gamma \leq \beta < N$.
Then the following statements hold.
\begin{enumerate}
\item[(i)] If $N \geq 3$ then $\tilde{B}(N,\alpha,\beta,\gamma)$ is attained for any $0< \alpha \leq \alpha_{N,\beta}$.
\item[(ii)] If $N = 2$ then $\tilde{B}(2,\alpha,\beta,\gamma)$ is attained for any $0< \alpha \leq \alpha_{2,\beta}$ if $\beta > \gamma$,
while there exists $\alpha_* > 0$ such that $\tilde{B}(2,\alpha,\beta,\beta)$ is attained for any $\alpha_* < \alpha < \alpha_{2,\beta}$.
\item[(iii)] $\tilde{B}(2,\alpha,\beta,\beta)$ is not attained for sufficiently small $\alpha > 0$.
\end{enumerate}
\end{theorem}
Next, we study the relation between the suprema of Adachi-Tanaka type and Li-Ruf type weighted Trudinger-Moser inequalities,
along the line of Lam-Lu-Zhang \cite{Lam-Lu-Zhang}.
Set $\tilde{B}_{rad}(N, \beta, \gamma) = \tilde{B}_{rad}(N, \alpha_{N, \beta},\beta, \gamma)$ in (\mathbb{R}f{weighted LR-sup(radial)}),
and
$\tilde{B}(N, \beta, \gamma) = \tilde{B}(N, \alpha_{N, \beta},\beta, \gamma)$ in (\mathbb{R}f{weighted LR-sup}),
i.e.,
\begin{align}
\lambdabel{weighted LR-sup-critical(radial)}
&\tilde{B}_{rad}(N, \beta, \gamma)
= \sup_{u \in X^{1,N}_{\gamma, rad}(\mathbb{R}^N) \atop \| u \|_{X^{1,N}_{\gamma}} \le 1} \int_{\mathbb{R}^N} \Phi_N (\alpha_{N, \beta} |u|^{\frac{N}{N-1}}) \frac{dx}{|x|^{\beta}}, \\
\lambdabel{weighted LR-sup-critical}
&\tilde{B}(N, \beta, \gamma)
= \sup_{u \in X^{1,N}_{\gamma}(\mathbb{R}^N) \atop \| u \|_{X^{1,N}_{\gamma}} \le 1} \int_{\mathbb{R}^N} \Phi_N (\alpha_{N, \beta} |u|^{\frac{N}{N-1}}) \frac{dx}{|x|^{\beta}},
\end{align}
for $N, \beta, \gamma$ satisfying (\mathbb{R}f{assumption:weighted AT}).
Then $\tilde{B}_{rad}(N, \beta, \gamma) < \infty$, and $\tilde{B}(N, \beta, \gamma) < \infty$ if $\gamma \ge 0$, by Theorem \mathbb{R}f{Theorem:weighted LR}.
\begin{theorem}(Relation)
Assume (\mathbb{R}f{assumption:weighted AT}).
Then we have
\lambdabel{Theorem:relation}
\[
\tilde{B}_{rad}(N, \beta, \gamma) = \sup_{\alpha \in (0, \alpha_{N,\beta})} \left( \frac{1 - \left(\frac{\alpha}{\alpha_{N,\beta}}\right)^{N-1}}{\left(\frac{\alpha}{\alpha_{N,\beta}}\right)^{N-1}} \right)^{\frac{N-\beta}{N-\gamma}}
\tilde{A}_{rad}(N, \alpha, \beta, \gamma).
\]
Furthermore if $\gamma \ge 0$, we have
\[
\tilde{B}(N, \beta, \gamma) = \sup_{\alpha \in (0, \alpha_{N,\beta})} \left( \frac{1 - \left(\frac{\alpha}{\alpha_{N,\beta}}\right)^{N-1}}{\left(\frac{\alpha}{\alpha_{N,\beta}}\right)^{N-1}} \right)^{\frac{N-\beta}{N-\gamma}}
\tilde{A}(N, \alpha, \beta, \gamma).
\]
\end{theorem}
Note that this implies $\tilde{A}_{rad}(N, \alpha, \beta, \gamma) < \infty$ for $N, \beta, \gamma$ satisfying (\mathbb{R}f{assumption:weighted AT}),
and $\tilde{A}(N, \alpha, \beta, \gamma) < \infty$ if $0 \le \gamma \le \beta < N$, by Theorem \mathbb{R}f{Theorem:weighted LR}.
Furthermore, we prove how $\tilde{A}_{rad}(N, \alpha, \beta, \gamma)$ and $\tilde{A}(N, \alpha, \beta, \gamma)$ behaves as $\alpha$ approaches to $\alpha_{N, \beta}$ from the below:
\begin{theorem}(Asymptotic behavior of Adachi-Tanaka supremum)
\lambdabel{Theorem:asymptotic}
Assume (\mathbb{R}f{assumption:weighted AT}).
Then there exist positive constants $C_1, C_2$ (depending on $N$, $\beta$, and $\gamma$) such that for $\alpha$ close enough to $\alpha_{N, \beta}$,
the estimate
\[
\left( \frac{C_1}{1 - \left(\frac{\alpha}{\alpha_{N,\beta}}\right)^{N-1}} \right)^{\frac{N-\beta}{N-\gamma}} \le \tilde{A}_{rad}(N, \alpha, \beta, \gamma)
\le \left( \frac{C_2}{1 - \left(\frac{\alpha}{\alpha_{N,\beta}}\right)^{N-1}} \right)^{\frac{N-\beta}{N-\gamma}}
\]
holds.
Corresponding estimates hold true for $\tilde{A}(N, \alpha, \beta, \gamma)$ if $\gamma \ge 0$.
\end{theorem}
Note that the estimate from the above follows from Theorem \mathbb{R}f{Theorem:relation}.
On the other hand, we will see that the estimate from the below follows from a computation using the Moser sequence.
The organization of the paper is as follows:
In section 2, we prove Theorem \mathbb{R}f{Theorem:weighted LR}.
Main tools are a transformation which relates a function in $X^{1,N}_\gamma(\mathbb{R}^N)$ to a function in $W^{1,N}(\mathbb{R}^N)$,
and the singular Trudinger-Moser type inequality recently proved by Adimurthi and Yang \cite{Adimurthi-Yang}, see also de Souza and de O \cite{Souza-O}.
In section 3, we prove the existence part of Theorems \mathbb{R}f{Maximizers(radial)}, \mathbb{R}f{Maximizers} (i) (ii).
In section 4, we prove the nonexistence part of Theorem \mathbb{R}f{Maximizers(radial)}, \mathbb{R}f{Maximizers} (iii).
Finally in section 5, we prove Theorem \mathbb{R}f{Theorem:relation} and Theorem \mathbb{R}f{Theorem:asymptotic}.
The letter $C$ will denote various positive constant which varies from line to line, but is independent of functions under consideration.
\section{Proof of Theorem \mathbb{R}f{Theorem:weighted LR}.}
In this section, we prove Theorem \mathbb{R}f{Theorem:weighted LR}.
We will use the following singular Trudinger-Moser inequality on the whole space $\mathbb R^N$:
For any $\beta \in [0, N)$, define
\begin{equation}
\lambdabel{singular TM-sup}
\tilde{B}(N,\alpha,\beta,0) = \sup_{u \in W^{1,N}(\mathbb R^N), \atop \|u \|_{W^{1,N}} \leq 1} \int_{\mathbb R^N} \Phi_N(\alpha |u|^{\frac N{N-1}}) \frac{dx}{|x|^{\beta}}.
\end{equation}
Then it holds
\begin{equation}
\lambdabel{singular TM}
\tilde{B}(N, \alpha, \beta, 0)
\begin{cases}
&< \infty, \quad \alpha \, \le \, \alpha_{N, \beta}, \\
&= \infty, \quad \alpha > \alpha_{N, \beta}.
\end{cases}
\end{equation}
Here $\| u \|_{W^{1,N}} = \left( \|\nabla u\|_N^N + \|u\|_N^N \right)^{1/N}$ denotes the full norm of the Sobolev space $W^{1,N}(\mathbb{R}^N)$.
Note that the inequality \eqref{singular TM} was first established by Ruf \cite{Ruf} for the case $N =2$ and $\beta =0$.
It then was extended to the case $N\geq 3$ and $\beta =0$ by Li and Ruf \cite{Li-Ruf}.
The case $N\geq 2$ and $\beta \in (0,N)$ was proved by Adimurthi and Yang \cite{Adimurthi-Yang}, see also de Souza and de O \cite{Souza-O}.
\noindent
{\it Proof of Theorem \mathbb{R}f{Theorem:weighted LR}}:
We define the vector-valued function $F$ by
\[
F(x) = \left(\frac {N-\gamma}N\right)^{\frac{N}{N-\gamma}}|x|^{\frac{\gamma}{N-\gamma}} x.
\]
Its Jacobian matrix is
\begin{align*}
DF(x) &= \left(\frac {N-\gamma}N\right)^{\frac{N}{N-\gamma}}|x|^{\frac{\gamma}{N-\gamma}} \left( Id_{N} + \frac{\gamma}{N-\gamma} \frac{x}{|x|} \, \otimes \, \frac{x}{|x|}\right)\\
&=\frac{N-\gamma}N |F(x)|^{\frac \gamma N}\left( Id_{N} + \frac{\gamma}{N-\gamma} \frac{x}{|x|} \, \otimes \, \frac{x}{|x|}\right).
\end{align*}
where $Id_N$ denotes the $N\times N$ identity matrix and $v\otimes v = (v_i v_j)_{1 \le i, j \le N}$ denotes the matrix
corresponding to the orthogonal projection onto the line generated by the unit vector $v = (v_1, \cdots, v_N) \in \mathbb{R}^N$, i.e., the map $x\mapsto (x\cdot v) v$.
Since a matrix of the form $I + \alpha v \otimes v$, $\alpha \in \mathbb{R}$, has eigenvalues $1$ (with multiplicity $N-1$) and $1 + \alpha$ (with multiplicity $1$),
we see that
\begin{equation}\lambdabel{eq:JacobianF}
det(DF(x)) = \left(\frac{N-\gamma}N\right)^{N-1} |F(x)|^{\gamma}.
\end{equation}
Let $u \in X^{1,N}_\gamma(\mathbb R^N)$ be such that $\|u\|_{X^{1,N}_\gamma} \leq 1$. We introduce a change of functions as follows.
\begin{equation}\lambdabel{eq:changefunct}
v(x) = \left(\frac{N-\gamma}N\right)^{\frac {N-1}N}\, u(F(x)).
\end{equation}
A simple calculation shows that
\begin{align*}
\nabla v(x)&= \left(\frac {N-\gamma}N\right)^{\frac{N-1}N}DF(x) (\nabla u(F(x)))\\
& = \left(\frac{N-\gamma}N\right)^{\frac{2N-1}N}|F(x)|^{\frac{\gamma}{N}} \left(\nabla u(F(x)) + \frac{\gamma}{N-\gamma}\left(\nabla u(F(x)) \cdot \frac x{|x|}\right) \frac x{|x|}\right),
\end{align*}
and hence
\[
|\nabla v(x)|^2 =\left(\frac{N-\gamma}N\right)^{\frac{2(2N-1)}N}|F(x)|^{\frac{2\gamma}{N}} \left(|\nabla u(F(x))|^2 + \frac{\gamma(2N-\gamma)}{(N-\gamma)^2} \left(\nabla u(F(x)) \cdot \frac x{|x|}\right)^2\right).
\]
Since $\left(\nabla u(F(x)) \cdot \frac x{|x|}\right)^2 \leq |\nabla u(F(x))|^2$, we then have
\begin{equation}
\lambdabel{eq:pointwiseineq}
|\nabla v(x)| \leq \left(\frac {N-\gamma}N\right)^{\frac{N-1}N} |F(x)|^{\frac\gamma{N}} |\nabla u(F(x))|= \left(\det (DF(x))\right)^{\frac1N} |\nabla u(F(x))|
\end{equation}
if $\gamma \ge 0$, with equality if and only if $\left(\nabla u(F(x)) \cdot \frac x{|x|}\right)^2 = |\nabla u(F(x))|^2$ when $\gamma >0$.
If $\gamma =0$ the inequality \eqref{eq:pointwiseineq} is an equality.
Note that the inequality \eqref{eq:pointwiseineq} does not hold if $\gamma < 0$ and $u$ is not radial function.
In fact, a reversed inequality occurs in this case. Moreover, \eqref{eq:pointwiseineq} becomes an equality if $u$ is a radial function for any $-\infty < \gamma < N$.
Integrating both sides of \eqref{eq:pointwiseineq} on $\mathbb R^N$, we obtain
\begin{equation}
\lambdabel{eq:comparenorm}
\|\nabla v\|_N \leq \|\nabla u\|_N.
\end{equation}
Moreover, for any function $G$ on $[0,\infty)$, using the change of variables, we get
\begin{multline}
\lambdabel{eq:changeintegral}
\int_{\mathbb R^N} G\left(|u(x)|^{\frac N{N-1}}\right) |x|^{-\delta} dx \\
= \left(\frac {N-\gamma}N\right)^{N-1+ \frac{N(\gamma -\delta)}{N-\gamma}}\int_{\mathbb R^N} G\left(\frac N{N-\gamma} |v(y)|^{\frac N{N-1}}\right) |y|^{\frac{N(\gamma -\delta)}{N-\gamma}} dy.
\end{multline}
Consequently, by choosing $G(t) =t^{N-1}$ and $\delta =\gamma$, we get $\|u\|_{N,\gamma} = \|v\|_N$ and hence
\begin{equation}
\lambdabel{eq:norm}
\|u\|_{X^{1,N}_\gamma}^N = \|\nabla u\|_N^N + \int_{\mathbb R^N} |u(x)|^{N} |x|^{-\gamma} dx \geq \|\nabla v\|_N^N + \|v\|_N^N=\|v\|_{W^{1,N}}^N.
\end{equation}
We remark again that \eqref{eq:comparenorm} and \eqref{eq:norm} become equalities if $u$ is radial function for any $\gamma < N$.
Thus $\|v\|_{W^{1,N}} \leq 1$ if $\|u\|_{X^{1,N}_\gamma} \leq 1$.
By choosing $G(t) = \Phi_N(\alpha t)$ and $\delta =\beta \geq \gamma$, we get
\begin{multline}
\lambdabel{eq:changeintegral*}
\int_{\mathbb R^N} \Phi_N\left(\alpha|u(x)|^{\frac N{N-1}}\right) |x|^{-\beta} dx\\
= \left(\frac {N-\gamma}N\right)^{N-1+ \frac{N(\gamma -\beta)}{N-\gamma}}\int_{\mathbb R^N} \Phi_N\left(\frac N{N-\gamma} \alpha|v(y)|^{\frac N{N-1}}\right) |y|^{-\frac{N(\beta-\gamma)}{N-\gamma}} dy.
\end{multline}
Denote
\begin{equation*}
\lambdabel{btilde}
\tilde{\beta} = \frac{N(\beta-\gamma)}{N-\gamma} \in [0,N).
\end{equation*}
By using \eqref{eq:norm} and \eqref{eq:changeintegral*} and applying the singular Trudinger-Moser inequality \eqref{singular TM},
we get
\begin{align*}
&\sup_{u\in X^{1,N}_\gamma(\mathbb R^N), \|u\|_{X^{1,N}_\gamma} \leq 1} \int_{\mathbb R^N} \Phi_N\left(\alpha|u(x)|^{\frac N{N-1}}\right) |x|^{-\beta} dx\\
&\leq\left(\frac {N-\gamma}N\right)^{N-1+ \frac{N(\gamma -\beta)}{N-\gamma}}\sup_{v\in W^{1,N}(\mathbb R^N), \|v\|_{W^{1,N}} \leq 1}\int_{\mathbb R^N} \Phi_N\left(\frac N{N-\gamma} \alpha|v(y)|^{\frac N{N-1}}\right) |y|^{-\tilde \beta} dy\\
&=\left(\frac {N-\gamma}N\right)^{N-1+ \frac{N(\gamma -\beta)}{N-\gamma}} \tilde{B}\left(N, \frac{N}{N-\gamma} \alpha, \tilde \beta,0\right)\\
&<\infty,
\end{align*}
since $\frac{N}{N-\gamma} \alpha \leq \frac{N}{N-\gamma} \alpha_{N,\beta} = \frac{N-\beta}{N-\gamma} \alpha_N = \left(\frac{N-\tilde{\beta}}{N}\right) \alpha_N = \alpha_{N,\tilde \beta}$.
If $u$ is radial then so is $v$.
In this case, \eqref{eq:pointwiseineq}, \eqref{eq:comparenorm} become equalities, and hence so does \eqref{eq:norm}.
Then the conclusion follows again from the singular Trudinger-Moser inequality \eqref{singular TM}.
We finish the proof of Theorem \mathbb{R}f{Theorem:weighted LR} by showing that $\tilde{B}(N, \alpha, \beta, \gamma) = \infty$ and $\tilde{B}_{rad}(N, \alpha, \beta, \gamma) = \infty$
when $\alpha > \alpha_{N, \beta}$.
Since $\tilde{B}_{rad}(N, \alpha, \beta, \gamma) \leq \tilde{B}(N, \alpha, \beta, \gamma)$, it is enough to prove that $\tilde{B}_{rad}(N, \alpha, \beta, \gamma) = \infty$.
Suppose the contrary that $\tilde{B}_{rad}(N, \alpha, \beta, \gamma) < \infty$ for some $\alpha > \alpha_{N,\beta}$.
Using again the transformation of functions \eqref{eq:changefunct} for radial functions $u \in X_{\gamma}^{1,N}$,
we then have equalities in \eqref{eq:pointwiseineq}, \eqref{eq:comparenorm}, and hence in \eqref{eq:norm}.
Evidently, the transformation of functions \eqref{eq:changefunct} is a bijection between $X^{1,N}_{\gamma,rad}$ and $W^{1,N}_{rad}$ and preserves the equality in \eqref{eq:norm}.
Consequently, we have
\[
\tilde{B}_{rad}(N, \alpha, \beta, \gamma) = \left(\frac {N-\gamma}N\right)^{N-1+ \frac{N(\gamma -\beta)}{N-\gamma}} \tilde{B}_{rad}\left(N, \frac{N}{N-\gamma} \alpha, \tilde \beta,0\right),
\]
with $\tilde{\beta} = \frac{N(\beta-\gamma)}{N-\gamma} \in [0,N)$. Hence $\tilde{B}_{rad}\left(N, \frac{N}{N-\gamma} \alpha, \tilde \beta,0\right) < \infty$.
By rearrangement argument, we have
\[
\tilde{B}\left(N, \frac{N}{N-\gamma} \alpha, \tilde \beta,0\right) = \tilde{B}_{rad}\left(N, \frac{N}{N-\gamma} \alpha, \tilde \beta,0\right) < \infty
\]
which violates the result of Adimurthi and Yang since $\frac{N}{N-\gamma} \alpha > \alpha_{N,\tilde{\beta}}$.
For the later purpose, we also prove here directly $\tilde{B}_{rad}(N, \alpha, \beta, \gamma) = \infty$ when $\alpha > \alpha_{N, \beta}$
by using the weighted Moser sequence as in \cite{Ishiwata-Nakamura-Wadade}, \cite{Lam-Lu-Zhang}:
Let $-\infty < \gamma \le \beta < N$ and for $n \in \mathbb{N}$ set
\[
A_n = \left( \frac{1}{\omega_{N-1}} \right)^{1/N} \left( \frac{n}{N-\beta} \right)^{-1/N}, \quad b_n = \frac{n}{N-\beta},
\]
so that $\left( A_n b_n \right)^{\frac{N}{N-1}} = n/\alpha_{N, \beta}$.
Put
\begin{align}
\lambdabel{Moser sequence}
&u_n =
\begin{cases}
A_n b_n, &\quad \text{if} \, |x| < e^{-b_n}, \\
A_n \log (1/|x|), &\quad \text{if} \, e^{-b_n} < |x| < 1, \\
0, &\quad \text{if} \, 1 \le |x|.
\end{cases}
\end{align}
Then direct calculation shows that
\begin{align}
\lambdabel{Moser_estimates(1)}
&\| \nabla u_n \|_{L^N(\mathbb{R}^N)} = 1, \\
\lambdabel{Moser_estimates(2)}
&\| u_n \|^N_{N, \gamma} = \frac{N-\beta}{(N-\gamma)^{N+1}} \Gamma(N+1) (1/n) + o(1/n)
\end{align}
as $n \to \infty$.
Thus $u_n \in X^{1,N}_{\gamma, rad}(\mathbb{R}^N)$.
In fact for (\mathbb{R}f{Moser_estimates(2)}), we compute
\begin{align*}
\| u_n \|_{N, \gamma}^N &= \omega_{N-1} \int_0^{e^{-b_n}} (A_n b_n)^N r^{N-1-\gamma} dr + \omega_{N-1} \int_{e^{-b_n}}^1 A_n^N (\log (1/r))^N r^{N-1-\gamma} dr \\
&= I + II.
\end{align*}
We see
\[
I = \omega_{N-1} (A_n b_n )^N \left[ \frac{r^{N-\gamma}}{N-\gamma} \right]_{r=0}^{r=e^{-b_n}} = \omega_{N-1} \left( \frac{n}{\alpha_{N, \beta}} \right)^{N-1} \frac{e^{-(\frac{N-\gamma}{N-\beta}) n} }{N-\gamma} = o(1/n)
\]
as $n \to \infty$.
Also
\begin{align*}
II &= \left( \frac{N-\beta}{n} \right) \int_{e^{-b_n}}^1 (\log (1/r))^N r^{N-1-\gamma} dr \\
&= \left( \frac{N-\beta}{n} \right) \int_0^{b_n} \rho^N e^{-(N-\gamma) \rho} d\rho
= \frac{N-\beta}{(N-\gamma)^{N+1}} (1/n) \int_0^{(N-\gamma) b_n} \rho^N e^{-\rho} d\rho \\
&= \frac{N-\beta}{(N-\gamma)^{N+1}} (1/n) \Gamma(N+1) + o(1/n).
\end{align*}
Thus we obtain (\mathbb{R}f{Moser_estimates(2)}).
Now, put $v_n(x) = \lambda_n u_n(x)$ where $u_n$ is the weighted Moser sequence in (\mathbb{R}f{Moser sequence}) and
$\lambda_n > 0$ is chosen so that $\lambda_n^N + \lambda_n^N \| u_n \|_{N, \gamma}^N = 1$.
Thus we have $\| \nabla v_n \|_{L^N}^N + \| v_n \|_{N,\gamma}^N = 1$ for any $n \in \mathbb{N}$.
By (\mathbb{R}f{Moser_estimates(2)}) with $\beta = \gamma$, we see that $\lambda_n^N = 1 - O(1/n)$ as $n \to \infty$.
For $\alpha > \alpha_{N, \beta}$, we calculate
\begin{align*}
&\int_{\mathbb{R}^N} \Phi_N (\alpha |v_n|^{\frac{N}{N-1}}) \frac{dx}{|x|^{\beta}}
\ge \int_{\{ 0 \le |x| \le e^{-b_n} \}} \Phi_N (\alpha |v_n|^{\frac{N}{N-1}}) \frac{dx}{|x|^{\beta}} \\
&= \int_{\{ 0 \le |x| \le e^{-b_n} \}} \left( e^{\alpha |v_n|^{\frac{N}{N-1}}} - \sum_{j=0}^{N-2} \frac{\alpha^j}{j!} |v_n|^{\frac{Nj}{N-1}} \right)
\frac{dx}{|x|^{\beta}} \\
&\ge \left\{ \exp \left( \frac{n \alpha}{\alpha_{N, \beta}} \lambda_n^{\frac{N}{N-1}} \right) - O(n^{N-1}) \right\}
\int_{\{ 0 \le |x| \le e^{-b_n} \}} \frac{dx}{|x|^{\beta}} \\
&\ge \left\{ \exp \left( \frac{n \alpha}{\alpha_{N,\beta}} \left(1 - O\left(\frac{1}{n^{\frac{1}{N-1}}}\right)\right) \right) - O(n^{N-1}) \right\} \left( \frac{\omega_{N-1}}{N-\beta} \right) e^{-n} \to + \infty
\end{align*}
as $n \to \infty$.
Here we have used that for $0 \le |x| \le e^{-b_n}$,
\[
\alpha |v_n|^{\frac{N}{N-1}} = \alpha \lambda_n^{\frac{N}{N-1}} (A_n b_n)^{\frac{N}{N-1}} = \frac{n \alpha}{\alpha_{N, \beta}} \lambda_n^{\frac{N}{N-1}}
\]
by definition of $A_n$ and $b_n$.
Also we used that for $0 \le |x| \le e^{-b_n}$,
\[
|v_n|^{\frac{Nj}{N-1}} = \lambda_n^{\frac{Nj}{N-1}} (A_n b_n)^{\frac{Nj}{N-1}} \le C n^j \le C n^{N-1}
\]
for $0 \le j \le N-2$ and $n$ is large.
This proves Theorem \mathbb{R}f{Theorem:weighted LR} completely.
\qed
\section{Existence of maximizers for the weighted Trudinger-Moser inequality}
As explained in the Introduction, the existence and non-existence of maximizers for (\mathbb{R}f{singular TM-sup}) is well known.
Now, let us recall it here.
\begin{prop}
\lambdabel{maximizersingular}
The following statements hold,
\begin{enumerate}
\item[(i)] If $N \geq 3$ then $\tilde{B}(N, \alpha,0,0)$ is attained for any $0 < \alpha \leq \alpha_N$ (see \cite{Ishiwata, Li-Ruf}).
\item[(ii)] If $N =2$, there exists $0 < \alpha_*< \alpha_2 =4\pi$ such that $\tilde{B}(2,\alpha,0,0)$ is attained for any $\alpha_* < \alpha \leq \alpha_2$ (see \cite{Ishiwata, Ruf}).
\item[(iii)] If $\beta \in (0,N)$ and $N \geq 2$ then $\tilde{B}(N,\alpha,\beta,0)$ is attained for any $0< \alpha \leq \alpha_{N,\beta}$ (see \cite{LiYang}).
\item[(iv)] $\tilde{B}(2,\alpha,0,0)$ is not attained for any sufficiently small $\alpha >0$ (see \cite{Ishiwata}).
\end{enumerate}
\end{prop}
The existence part (iii) of Proposition \mathbb{R}f{maximizersingular} is recently proved by X. Li, and Y. Yang \cite{LiYang} by a blow-up analysis.
\begin{remark}
\lambdabel{remark1}
By a rearrangement argument, the maximizers for \eqref{singular TM-sup}, if exist, must be a decreasing spherical symmetric function if $\beta \in (0,N)$ and up to a translation if $\beta =0$.
\end{remark}
The proofs of the existence part (i) (ii) of Theorem \mathbb{R}f{Maximizers(radial)} and \mathbb{R}f{Maximizers} are completely similar by using the formula of change of functions \eqref{eq:changefunct}
and the results on the existence of maximizers for \eqref{singular TM-sup}.
So we prove Theorem \mathbb{R}f{Maximizers} only here.
As we have seen from the proof of Theorem \mathbb{R}f{Theorem:weighted LR} that
\[
\tilde{B}(N,\alpha,\beta,\gamma) \leq \left(\frac {N-\gamma}N\right)^{N-1+ \frac{N(\gamma -\beta)}{N-\gamma}} \tilde{B}\left(N, \frac{N}{N-\gamma} \alpha, \tilde \beta,0\right)
\]
if $0 \le \gamma \le \beta < N$, where $\tilde{\beta} = N(\beta-\gamma)/(N-\gamma) \in [0,N)$.
If $N, \alpha, \beta$ and $\gamma$ satisfy the condition (i) and (ii) of Theorem \mathbb{R}f{Maximizers},
then $N$, $N \alpha/(N-\gamma)$ and $\tilde{\beta}$ satisfy the condition (i)--(iii) of Proposition \mathbb{R}f{maximizersingular},
hence there exists a maximizer $v \in W^{1,N}(\mathbb{R}^N)$ for $\tilde{B}\left( N, \frac{N}{N-\gamma} \alpha, \tilde{\beta} ,0 \right)$ with $\|v\|_N^N + \|\nabla v\|_N^N =1$ and
\[
\int_{\mathbb R^N} \Phi_N\left(\frac N{N-\gamma} \alpha|v(y)|^{\frac N{N-1}}\right) |y|^{-\tilde \beta} dy = \tilde{B}\left(N, \frac{N}{N-\gamma} \alpha, \tilde \beta,0\right).
\]
As mentioned in Remark \mathbb{R}f{remark1}, we can assume that $v$ is a radial function.
Let $u \in X^{1,N}_{\gamma}$ be a function defined by \eqref{eq:changefunct}. Note that $u$ is also a radial function, hence $\eqref{eq:pointwiseineq}$ becomes an equality. So do \eqref{eq:comparenorm} and \eqref{eq:norm}. Hence, we get
\[
\|u\|_{X^{1,N}_\gamma}^N = \|\nabla v\|_N^N + \|v\|_N^N =1,
\]
and by \eqref{eq:changeintegral*}
\[
\int_{\mathbb R^N} \Phi_N\left(\alpha|u(x)|^{\frac N{N-1}}\right) |x|^{-\beta} dx = \left(\frac {N-\gamma}N\right)^{N-1+ \frac{N(\gamma -\beta)}{N-\gamma}}\tilde{B}\left(N, \frac{N}{N-\gamma} \alpha, \tilde \beta,0\right).
\]
This shows that $u$ is a maximizer for $\tilde{B}(N,\alpha,\beta,\gamma)$.
\qed
\section{Non-existence of maximizers for the weighted Trudinger-Moser inequality}
In this section, we prove the non-existence part (iii) of Theorem \mathbb{R}f{Maximizers}.
The proof of (iii) of Theorem \mathbb{R}f{Maximizers(radial)} is completely similar.
We follow Ishiwata's argument in \cite{Ishiwata}.
Assume $0 \le \beta < 2$, $0 < \alpha \le \alpha_{2, \beta} = 2\pi(2-\beta)$ and recall
\begin{align*}
&\tilde{B}(2, \alpha, \beta, \beta)
= \sup_{u \in X^{1,2}_{\beta}(\mathbb{R}^2) \atop \| u \|_{X^{1,2}_{\beta}(\mathbb{R}^2)} \le 1} \int_{\mathbb{R}^2} \left( e^{\alpha u^2} -1 \right) \frac{dx}{|x|^{\beta}}.
\end{align*}
We will show that $\tilde{B}(2, \alpha, \beta, \beta)$ is not attained if $\alpha > 0$ sufficiently small.
Set
\begin{align*}
M = \left\{ u \in X^{1,2}_{\beta}(\mathbb{R}^2) \, : \, \| u \|_{X^{1,2}_{\beta}} =\left( \| \nabla u \|_2^2 + \| u \|_{2, \beta}^2 \right)^{1/2} = 1 \right\}
\end{align*}
be the unit sphere in the Hilbert space $X^{1,2}_{\beta}(\mathbb{R}^2)$ and
\begin{align*}
J_{\alpha} :M \to \mathbb{R}, \quad J_{\alpha}(u) = \int_{\mathbb{R}^2} \left( e^{\alpha u^2} - 1 \right) \frac{dx}{|x|^{\beta}}
\end{align*}
be the corresponding functional defined on $M$.
Actually, we will prove the stronger claim that $J_{\alpha}$ has no critical point on $M$ when $\alpha > 0$ is sufficiently small.
Assume the contrary that there existed $v \in M$ such that $v$ is a critical point of $J_{\alpha}$ on $M$.
Define an orbit on $M$ through $v$ as
\[
v_{\tau}(x) = \qquad $\blacksquare$rt{\tau} v(\qquad $\blacksquare$rt{\tau} x) \quad \tau \in (0,\infty), \quad w_{\tau} = \frac{v_{\tau}}{\| v_{\tau} \|_{X^{1,2}_{\beta}}} \in M.
\]
Since $w_{\tau}|_{\tau = 1} = v$, we must have
\begin{equation}
\lambdabel{ddt=0}
\frac{d}{d\tau} \Big|_{\tau = 1} J_{\alpha}(w_{\tau}) = 0.
\end{equation}
Note that
\[
\| \nabla v_{\tau} \|_{L^2(\mathbb{R}^2)}^2 = \tau \| \nabla v \|_{L^2(\mathbb{R}^2)}^2, \quad \| v_{\tau} \|_{p, \beta}^p = \tau^{\frac{p+\beta-2}{2}}\| v \|_{p, \beta}^p
\]
for $p > 1$.
Thus,
\begin{align*}
&J_{\alpha}(w_{\tau}) = \int_{\mathbb{R}^2} \left( e^{\alpha w_{\tau}^2} - 1 \right) \frac{dx}{|x|^{\beta}}
= \int_{\mathbb{R}^2} \sum_{j=1}^{\infty} \frac{\alpha^j}{j!} \frac{v_{\tau}^{2j}(x)}{\| v_{\tau} \|_{X^{1,2}_{\beta}}^{2j}} \frac{dx}{|x|^{\beta}} \\
&= \sum_{j=1}^{\infty} \frac{\alpha^j}{j!} \frac{\| v_{\tau} \|_{2j, \beta}^{2j}}{\left(\| \nabla v_{\tau} \|_2^2 + \| v_{\tau}\|_{2, \beta}^2 \right)^j}
= \sum_{j=1}^{\infty} \frac{\alpha^j}{j!} \frac{\tau^{j -1 +\frac{\beta}{2}} \| v \|_{2j, \beta}^{2j}}{\left( \tau \| \nabla v \|_2^2 + \tau^{\frac{\beta}{2}} \| v \|_{2,\beta}^2 \right)^j}.
\end{align*}
By using an elementary computation
\begin{align*}
&f(\tau) = \frac{\tau^{j-1 + \frac{\beta}{2}} c}{(\tau a + \tau^{\frac{\beta}{2}} b)^j}, \quad a = \| \nabla v \|_2^2, \, b = \| v \|_{2,\beta}^2, \, c = \| v \|_{2j,\beta}^{2j}, \\
&f'(\tau) = (1- \frac{\beta}{2}) \frac{\tau^{j-2 + \frac{\beta}{2}} c}{(\tau a + \tau^{\frac{\beta}{2}} b)^{j+1}} \left\{ -\tau a + (j-1)b \right\},
\end{align*}
we estimate $\frac{d}{d\tau} \Big|_{\tau = 1} J_{\alpha}(w_{\tau})$:
\begin{align}
\lambdabel{d_dtau}
&\frac{d}{d\tau} \Big|_{\tau = 1} J_{\alpha}(w_{\tau}) \notag \\
&= \sum_{j=1}^{\infty} \left[ \frac{\alpha^j}{j!} (1-\frac{\beta}{2}) \frac{\tau^{j-2 + \beta/2} \| v \|_{2j, \beta}^{2j}}{\left( \tau \| \nabla v \|_2^2 + \tau^{\beta/2} \| v \|_{2,\beta}^2 \right)^{j+1}}
\left\{ -\tau \| \nabla v \|_2^2 + (j-1) \| v \|_{2,\beta}^2 \right\} \right]_{\tau = 1} \notag \\
&= - \alpha (1-\frac{\beta}{2}) \| \nabla v \|_2^2 \| v \|_{2, \beta}^2
+ \sum_{j=2}^{\infty} \frac{\alpha^j}{j!} (1-\frac{\beta}{2}) \| v \|_{2j, \beta}^{2j} \left\{ -\| \nabla v \|_2^2 + (j-1) \| v \|_{2,\beta}^2 \right\} \notag \\
&\le \alpha (1-\frac{\beta}{2}) \| \nabla v \|_2^2 \| v \|_{2,\beta}^2
\left\{ -1 + \sum_{j=2}^{\infty} \frac{\alpha^{j-1}}{(j-1)!} \frac{\| v \|_{2j, \beta}^{2j}}{\| \nabla v \|_2^2 \| v \|_{2,\beta}^2} \right\},
\end{align}
since $-\| \nabla v \|_2^2 + (j-1) \| v \|_{2,\beta}^2 \le j$.
Now, we state a lemma.
Unweighted version of the next lemma is proved in \cite{Ishiwata}:Lemma 3.1,
and the proof of the next is a simple modification of the one given there
using the weighted Adachi-Tanaka type Trudinger-Moser inequality:
\[
\tilde{A}(2, \alpha, \beta, \beta) = \sup_{u \in X^{1,2}_{\beta}(\mathbb{R}^2) \setminus \{ 0 \} \atop \| \nabla u \|_{L^2(\mathbb{R}^2)} \le 1} \frac{1}{\| u \|^2_{2, \beta}} \int_{\mathbb{R}^2} \left( e^{\alpha u^2} - 1 \right) \frac{dx}{|x|^{\beta}} < \infty
\]
for $\alpha \in (0, \alpha_{2, \beta})$ if $\beta \ge 0$, and the expansion of the exponential function.
\begin{lemma}
\lambdabel{Lemma:Ishiwata:Lemma3.1}
For any $\alpha \in (0, \alpha_{2, \beta})$, there exists $C_{\alpha} > 0$ such that
\[
\| u \|_{2j,\beta}^{2j} \le C_{\alpha} \frac{j!}{\alpha^j} \| \nabla u \|_2^{2j-2} \| u \|_{2, \beta}^2
\]
holds for any $u \in X^{1,2}_{\beta}(\mathbb{R}^2)$ and $j \in \mathbb{N}$, $j \ge 2$.
\end{lemma}
By this lemma,
if we take $\alpha < \tilde{\alpha} < \alpha_{2,\beta}$ and put $C = C_{\tilde{\alpha}}$,
we see
\begin{align*}
\frac{\| v \|_{2j, \beta}^{2j}}{\| \nabla v \|_2^2 \| v \|_{2,\beta}^2} \le C \frac{j!}{\tilde{\alpha}^j} \| \nabla v \|_{2j}^{2j-4}
\le C \frac{j!}{\tilde{\alpha}^j}
\end{align*}
for $j \ge 2$ since $v \in M$.
Thus we have
\[
\sum_{j=2}^{\infty} \frac{\alpha^{j-1}}{(j-1)!} \frac{\| v \|_{2j, \beta}^{2j}}{\| \nabla v \|_2^2 \| v \|_{2,\beta}^2} \le \sum_{j=2}^{\infty} \frac{C \alpha^{j-1}}{(j-1)!} \frac{j!}{\tilde{\alpha}^j}
= (\frac{C \alpha}{\tilde{\alpha}^2}) \sum_{j=2}^{\infty} \left( \frac{\alpha}{\tilde{\alpha}} \right)^{j-2} j \le \alpha C^{\prime}
\]
for some $C^{\prime} > 0$.
Inserting this into the former estimate (\mathbb{R}f{d_dtau}), we obtain
\begin{align*}
\frac{d}{d\tau} \Big|_{\tau = 1} J_{\alpha}(w_{\tau}) \le (1-\frac{\beta}{2}) \alpha \| \nabla v \|_2^2 \| v \|_{2,\beta}^2 (-1 + C^{\prime}\alpha) < 0
\end{align*}
when $\alpha >0$ is sufficiently small.
This contradicts to (\mathbb{R}f{ddt=0}).
\qed
\section{Proof of Theorem \mathbb{R}f{Theorem:relation} and \mathbb{R}f{Theorem:asymptotic}.}
In this section, we prove Theorem \mathbb{R}f{Theorem:relation} and Theorem \mathbb{R}f{Theorem:asymptotic}.
As stated in the Introduction, we follow the argument by Lam-Lu-Zhang \cite{Lam-Lu-Zhang}.
First, we prepare several lemmata.
\begin{lemma}
\lambdabel{Lemma1}
Assume (\mathbb{R}f{assumption:weighted AT}) and set
\begin{equation}
\lambdabel{A_hat}
\omegaidehat{A}(N, \alpha, \beta, \gamma)
= \sup_{{u \in X^{1,N}_{\gamma}(\mathbb{R}^N) \setminus \{ 0 \} \atop \| \nabla u \|_{L^N(\mathbb{R}^N)} \le 1} \atop \| u \|_{N, \gamma} = 1}
\int_{\mathbb{R}^N} \Phi_N (\alpha |u|^{\frac{N}{N-1}}) \frac{dx}{|x|^{\beta}}.
\end{equation}
Let $\tilde{A}(N, \alpha, \beta, \gamma)$ be defined as in (\mathbb{R}f{weighted AT-sup}).
Then $\tilde{A}(N, \alpha, \beta, \gamma) = \omegaidehat{A}(N, \alpha, \beta, \gamma)$ for any $\alpha > 0$.
Similarly, $\tilde{A}_{rad}(N, \alpha, \beta, \gamma) = \omegaidehat{A}_{rad}(N, \alpha, \beta, \gamma)$ for any $\alpha > 0$,
where $\omegaidehat{A}_{rad}(N, \alpha, \beta, \gamma)$ is defined similar to (\mathbb{R}f{A_hat}) and $\omegaidehat{A}_{rad}(N, \alpha, \beta, \gamma)$ is defined in (\mathbb{R}f{weighted AT-sup(radial)}).
\end{lemma}
\begin{proof}
For any $u \in X^{1,N}_{\gamma}(\mathbb{R}^N) \setminus \{ 0 \}$ and $\lambda >0$, we put $u_{\lambda}(x) = u(\lambda x)$ for $x \in \mathbb{R}^N$.
Then it is easy to see that
\begin{equation}
\lambdabel{scaling}
\begin{cases}
&\| \nabla u_{\lambda} \|_{L^N(\mathbb{R}^N)}^N = \| \nabla u \|_{L^N(\mathbb{R}^N)}^N, \\
&\| u_{\lambda} \|^N_{N, \gamma} = \lambda^{-(N-\gamma)} \| u \|^N_{N, \gamma}.
\end{cases}
\end{equation}
Thus for any $u \in X^{1,N}_{\gamma}(\mathbb{R}^N) \setminus \{ 0 \}$ with $\| \nabla u \|_{L^N(\mathbb{R}^N)} \le 1$,
if we choose $\lambda = \| u \|^{N/(N-\gamma)}_{N, \gamma}$, then $u_{\lambda} \in X^{1,N}_{\gamma}(\mathbb{R}^N)$ satisfies
\[
\| \nabla u_{\lambda} \|_{L^N(\mathbb{R}^N)} \le 1 \quad \text{and} \quad \| u_{\lambda} \|^N_{N, \gamma} = 1.
\]
Thus
\[
\omegaidehat{A}(N, \alpha, \beta, \gamma) \ge
\int_{\mathbb{R}^N} \Phi_N (\alpha |u_{\lambda}|^{\frac{N}{N-1}}) \frac{dx}{|x|^{\beta}} =
\frac{1}{\| u \|_{N, \gamma}^{\frac{N(N-\beta)}{N-\gamma}}} \int_{\mathbb{R}^N} \Phi_N (\alpha |u|^{\frac{N}{N-1}}) \frac{dx}{|x|^{\beta}}
\]
which implies $\omegaidehat{A}(N, \alpha, \beta, \gamma) \ge \tilde{A}(N, \alpha, \beta, \gamma)$.
The opposite inequality is trivial.
\end{proof}
\begin{lemma}
\lambdabel{Lemma2}
Assume (\mathbb{R}f{assumption:weighted AT}) and set $\tilde{B}(N, \beta, \gamma)$ as in (\mathbb{R}f{weighted LR-sup-critical}).
Then we have
\[
\tilde{A}(N, \alpha, \beta, \gamma) \le \left( \frac{\left(\frac{\alpha}{\alpha_{N,\beta}}\right)^{N-1}}{1 - \left(\frac{\alpha}{\alpha_{N,\beta}}\right)^{N-1}} \right)^{\frac{N-\beta}{N-\gamma}} \tilde{B}(N, \beta, \gamma)
\]
for any $0 < \alpha < \alpha_{N, \beta}$.
The same relation holds for $\tilde{A}_{rad}(N, \alpha, \beta, \gamma)$ in (\mathbb{R}f{weighted AT-sup(radial)}) and $\tilde{B}_{rad}(N, \beta, \gamma)$ in (\mathbb{R}f{weighted LR-sup-critical(radial)}).
\end{lemma}
\begin{proof}
Choose any $u \in X^{1,N}_{\gamma}$ with $\| \nabla u \|_{L^N(\mathbb{R}^N)} \le 1$ and $\| u \|_{N, \gamma} = 1$.
Put $v(x) = C u(\lambda x)$ where $C \in (0,1)$ and $\lambda > 0$ are defined as
\[
C = \left( \frac{\alpha}{\alpha_{N, \beta}} \right)^{\frac{N-1}{N}} \quad \text{and} \quad
\lambda = \left( \frac{C^N}{1 - C^N} \right)^{1/(N-\gamma)}.
\]
Then by scaling rules (\mathbb{R}f{scaling}), we see
\begin{align*}
\| v \|^N_{X^{1,N}_{\gamma}} &= \| \nabla v \|^N_N + \| v \|^N_{N, \gamma}
= C^N \| \nabla u \|^N_N + \lambda^{-(N-\gamma)} C^N \| u \|^N_{N, \gamma} \\
&\le C^N + \lambda^{-(N-\gamma)} C^N = 1.
\end{align*}
Also we have
\begin{align*}
\int_{\mathbb{R}^N} \Phi_N (\alpha_{N, \beta} |v|^{\frac{N}{N-1}}) \frac{dx}{|x|^{\beta}} &=
\lambda^{-(N-\beta)} \int_{\mathbb{R}^N} \Phi_N \left( \alpha_{N, \beta} C^{\frac{N}{N-1}} |u|^{\frac{N}{N-1}} \right) \frac{dx}{|x|^{\beta}} \\
&= \lambda^{-(N-\beta)} \int_{\mathbb{R}^N} \Phi_N \left( \alpha |u|^{\frac{N}{N-1}} \right) \frac{dx}{|x|^{\beta}}.
\end{align*}
Thus testing $\tilde{B}(N, \beta, \gamma)$ by $v$, we see
\[
\tilde{B}(N, \beta, \gamma) \ge \left( \frac{1 - C^N}{C^N} \right)^{\frac{N-\beta}{N-\gamma}} \int_{\mathbb{R}^N} \Phi_N \left( \alpha |u|^{\frac{N}{N-1}} \right) \frac{dx}{|x|^{\beta}}.
\]
By taking the supremum for $u \in X^{1,N}_{\gamma}$ with $\| \nabla u \|_{L^N(\mathbb{R}^N)} \le 1$ and $\| u \|_{N, \gamma} = 1$,
we have
\[
\tilde{B}(N, \beta, \gamma) \ge \left( \frac{1 - \left(\frac{\alpha}{\alpha_{N,\beta}}\right)^{N-1}}{\left(\frac{\alpha}{\alpha_{N,\beta}}\right)^{N-1}} \right)^{\frac{N-\beta}{N-\gamma}} \omegaidehat{A}(N, \alpha, \beta, \gamma).
\]
Finally, Lemma \mathbb{R}f{Lemma1} implies the result.
The proof of
\[
\tilde{B}_{rad}(N, \beta, \gamma) \ge \left( \frac{1 - \left(\frac{\alpha}{\alpha_{N,\beta}}\right)^{N-1}}{\left(\frac{\alpha}{\alpha_{N,\beta}}\right)^{N-1}} \right)^{\frac{N-\beta}{N-\gamma}} \omegaidehat{A}_{rad}(N, \alpha, \beta, \gamma)
\]
is similar.
\end{proof}
\noindent
{\it Proof of Theorem \mathbb{R}f{Theorem:relation}}:
We prove the relation between $\tilde{B}(N, \beta, \gamma)$ and $\tilde{A}(N, \alpha, \beta, \gamma)$ only.
The assertion that
\[
\tilde{B}(N, \beta, \gamma) \ge \sup_{\alpha \in (0, \alpha_{N, \beta})} \left( \frac{1 - \left(\frac{\alpha}{\alpha_{N,\beta}}\right)^{N-1}}{\left(\frac{\alpha}{\alpha_{N,\beta}}\right)^{N-1}} \right)^{\frac{N-\beta}{N-\gamma}}
\tilde{A}(N, \alpha, \beta, \gamma)
\]
follows from Lemma \mathbb{R}f{Lemma2}.
Note that $\tilde{B}(N, \beta,\gamma) < \infty$ when $0 \le \gamma \le \beta < N$ by Theorem \mathbb{R}f{Theorem:weighted LR}.
Let us prove the opposite inequality.
Let $\{ u_n \} \subset X^{1,N}_{\gamma}(\mathbb{R}^N)$, $u_n \ne 0$, $\| \nabla u_n \|^N_{L^N} + \| u_n \|^N_{N, \gamma} \le 1$,
be a maximizing sequence of $\tilde{B}(N, \beta, \gamma)$:
\[
\int_{\mathbb{R}^N} \Phi_N (\alpha_{N, \beta} |u_n|^{\frac{N}{N-1}}) \frac{dx}{|x|^{\beta}} = \tilde{B}(N, \beta, \gamma) + o(1)
\]
as $n \to \infty$.
We may assume $\| \nabla u_n \|^N_{L^N(\mathbb{R}^N)} < 1$ for any $n \in \mathbb{N}$.
Define
\[
\begin{cases}
&v_n(x) = \frac{u_n(\lambda_n x)}{\| \nabla u_n \|_N}, \quad (x \in \mathbb{R}^N) \\
&\lambda_n = \left( \frac{1 - \| \nabla u_n \|_N^N}{\| \nabla u_n \|^N_N} \right)^{1/(N-\gamma)} > 0.
\end{cases}
\]
Thus by (\mathbb{R}f{scaling}), we see
\begin{align*}
&\| \nabla v_n \|^N_{L^N(\mathbb{R}^N)} = 1, \\
&\| v_n \|^{\frac{N(N-\beta)}{N-\gamma}}_{N, \gamma} = \left( \frac{\lambda_n^{-(N-\gamma)}}{\| \nabla u_n \|^N_N} \| u_n \|^N_{N, \gamma} \right)^{\frac{N-\beta}{N-\gamma}}
= \left( \frac {\| u_n \|^N_{N, \gamma}}{1 - \| \nabla u_n \|^N_N} \right)^{\frac{N-\beta}{N-\gamma}} \le 1,
\end{align*}
since $\| \nabla u_n \|^N_N + \| u_n \|^N_{N, \gamma} \le 1$.
Thus, setting
\[
\alpha_n = \alpha_{N, \beta} \| \nabla u_n \|^{\frac{N}{N-1}}_N < \alpha_{N, \beta}
\]
for any $n \in \mathbb{N}$,
we may test $\tilde{A}(N, \alpha_n, \beta, \gamma)$ by $\{ v_n \}$, which results in
\begin{align*}
\tilde{B}(N, \beta, \gamma) + o(1) &= \int_{\mathbb{R}^N} \Phi_N (\alpha_{N, \beta} |u_n(y)|^{\frac{N}{N-1}}) \frac{dy}{|y|^{\beta}} \\
&= \lambda_n^{N-\beta} \int_{\mathbb{R}^N} \Phi_N (\alpha_{N, \beta} \| \nabla u_n \|_N^{\frac{N}{N-1}} |v_n(x)|^{\frac{N}{N-1}}) \frac{dx}{|x|^{\beta}} \\
&= \lambda_n^{N-\beta} \int_{\mathbb{R}^N} \Phi_N (\alpha_n |v_n(x)|^{\frac{N}{N-1}}) \frac{dx}{|x|^{\beta}} \\
&\le \lambda_n^{N-\beta} \left( \frac{1}{\| v_n \|_{N, \beta}^N} \right)^{\frac{N-\beta}{N-\gamma}} \int_{\mathbb{R}^N} \Phi_N (\alpha_n |v_n(x)|^{\frac{N}{N-1}}) \frac{dx}{|x|^{\beta}} \\
&\le \lambda_n^{N-\beta} \tilde{A}(N, \alpha_n, \beta, \gamma) = \left( \frac{1 - \| \nabla u_n \|_{N}^N}{\| \nabla u_n \|^N_{N}} \right)^{\frac{N-\beta}{N-\gamma}} \tilde{A}(N, \alpha_n, \beta, \gamma) \\
&= \left( \frac{1 - \left(\frac{\alpha_n}{\alpha_{N,\beta}}\right)^{N-1}}{\left(\frac{\alpha_n}{\alpha_{N,\beta}}\right)^{N-1}} \right)^{\frac{N-\beta}{N-\gamma}} \tilde{A}(N, \alpha_n, \beta, \gamma) \\
&\le \sup_{\alpha \in (0, \alpha_{N, \beta})} \left( \frac{1 - \left(\frac{\alpha}{\alpha_{N,\beta}}\right)^{N-1}}{\left(\frac{\alpha}{\alpha_{N,\beta}}\right)^{N-1}} \right)^{\frac{N-\beta}{N-\gamma}} \tilde{A}(N, \alpha, \beta, \gamma).
\end{align*}
Here we have used a change of variables $y = \lambda_n x$ for the second equality, and $\| v_n \|^{\frac{N(N-\beta)}{N-\gamma}}_{N, \gamma} \le 1$ for the first inequality.
Letting $n \to \infty$, we have the desired result.
\qed
\noindent
{\it Proof of Theorem \mathbb{R}f{Theorem:asymptotic}}:
Again, we prove theorem for $\tilde{A}(N, \alpha, \beta, \gamma)$ only.
The assertion that
\[
\tilde{A}(N, \alpha, \beta, \gamma) \le \left( \frac{C_2}{1 - \left(\frac{\alpha}{\alpha_{N,\beta}}\right)^{N-1}} \right)^{\frac{N-\beta}{N-\gamma}}
\]
follows form Theorem \mathbb{R}f{Theorem:relation} and the fact that $\tilde{B}(N, \beta, \gamma) < \infty$ when $0 \le \gamma \le \beta < N$.
For the rest, we need to prove that there exists $C > 0$ such that for any $\alpha < \alpha_{N, \beta}$ sufficiently close to $\alpha_{N, \beta}$,
it holds that
\begin{equation}
\lambdabel{Lower}
\left( \frac{C}{1 - \left(\frac{\alpha}{\alpha_{N,\beta}}\right)^{N-1}} \right)^{\frac{N-\beta}{N-\gamma}} \le \tilde{A}(N, \alpha, \beta, \gamma).
\end{equation}
For that purpose, we use the weighted Moser sequence (\mathbb{R}f{Moser sequence}) again.
By (\mathbb{R}f{Moser_estimates(2)}), we have $N_1 \in \mathbb{N}$ such that
if $n \in \mathbb{N}$ satisfies $n \ge N_1$, then it holds
\begin{equation}
\lambdabel{lower1}
\| u_n \|_{N, \gamma}^N \le \frac{2 (N-\gamma) \Gamma (N+1)}{(N-\beta)^{N+1}} (1/n).
\end{equation}
On the other hand,
\begin{align*}
\int_{\mathbb{R}^N} \Phi_N(\alpha |u_n|^{N/(N-1)}) \frac{dx}{|x|^{\beta}} &\ge \omega_{N-1} \int_0^{e^{-b_n}} \Phi_N \left( \alpha (A_n b_n)^{N/(N-1)} \right) r^{N-1-\beta} dr \\
&= \frac{\omega_{N-1}}{N-\beta} \Phi_N \left( (\alpha/\alpha_{N,\beta}) n \right) \left[ r^{N-\beta} \right]_{r=0}^{r = e^{-b_n}} \\
& = \frac{\omega_{N-1}}{N-\beta} \Phi_N \left( (\alpha/\alpha_{N,\beta}) n \right) e^{-n}.
\end{align*}
Note that there exists $N_2 \in \mathbb{N}$ such that if $n \ge N_2$ then $\Phi_N \left( (\alpha/\alpha_{N,\beta}) n \right) \ge \frac{1}{2} e^{(\alpha/\alpha_{N,\beta}) n}$.
Thus we have
\begin{equation}
\lambdabel{lower2}
\int_{\mathbb{R}^N} \Phi_N(\alpha |u_n|^{N/(N-1)}) \frac{dx}{|x|^{\beta}} \ge \frac{1}{2} \left( \frac{\omega_{N-1}}{N-\beta} \right) e^{-(1 - \frac{\alpha}{\alpha_{N,\beta}}) n}.
\end{equation}
Combining (\mathbb{R}f{lower1}) and (\mathbb{R}f{lower2}), we have $C_1(N, \beta, \gamma) > 0$ such that
\begin{equation}
\lambdabel{lower3}
\frac{1}{ \| u_n \|_{N, \gamma}^{\frac{N(N-\beta)}{N-\gamma}}} \int_{\mathbb{R}^N} \Phi_N (\alpha |u_n|^{N/(N-1)}) \frac{dx}{|x|^{\beta}} \ge C_1(N, \beta, \gamma) n^{\frac{N-\beta}{N-\gamma}} e^{-(1 - \frac{\alpha}{\alpha_{N,\beta}}) n}
\end{equation}
holds when $n \ge \max \{ N_1, N_2 \}$.
Note that $\lim_{x \to 1} \left( \frac{1-x^{N-1}}{1-x} \right) = N-1$, thus
\[
\frac{1 - (\alpha/\alpha_{N, \beta})^{N-1}}{1 - (\alpha/\alpha_{N,\beta})} \ge \frac{N-1}{2}
\]
if $\alpha/\alpha_{N, \beta} < 1$ is very close to $1$.
Now, for any $\alpha > 0$ sufficiently close to $\alpha_{N, \beta}$ so that
\begin{align}
\lambdabel{cond_alpha}
\begin{cases}
&\max \{ N_1, N_2 \} < \left( \frac{2}{1 - \alpha/\alpha_{N,\beta}} \right), \\
&\frac{1 - (\alpha/\alpha_{N, \beta})^{N-1}}{1 - (\alpha/\alpha_{N,\beta})} \ge \frac{N-1}{2},
\end{cases}
\end{align}
we can find $n \in \mathbb{N}$ such that
\begin{align}
\lambdabel{cond_n}
\begin{cases}
&\max \{ N_1, N_2 \} \le n \le \left( \frac{2}{1 - \alpha/\alpha_{N,\beta}} \right), \\
&\left( \frac{1}{1 - \alpha/\alpha_{N,\beta}} \right) \le n.
\end{cases}
\end{align}
We fix $n \in \mathbb{N}$ satisfying (\mathbb{R}f{cond_n}).
Then by $1 \le n (1 - \alpha/\alpha_{N, \beta}) \le 2$, (\mathbb{R}f{lower3}) and (\mathbb{R}f{cond_alpha}), we have
\begin{align*}
&\frac{1}{ \| u_n \|_{N, \beta}^N} \int_{\mathbb{R}^N} \Phi_N(\alpha |u_n|^{N/(N-1)}) \frac{dx}{|x|^{\beta}} \ge C_1(N, \beta, \gamma) n^{\frac{N-\beta}{N-\gamma}} e^{-2} \\
&\ge C_2(N, \beta, \gamma) \left( \frac{1}{1 - (\alpha/\alpha_{N, \beta})} \right)^{\frac{N-\beta}{N-\gamma}} \ge \frac{N-1}{2} C_2(N, \beta, \gamma) \left( \frac{1}{1 - (\alpha/\alpha_{N, \beta})^{N-1}} \right)^{\frac{N-\beta}{N-\gamma}} \\
&= C_3(N, \beta, \gamma) \left( \frac{1}{1 - (\alpha/\alpha_{N, \beta})^{N-1}} \right)^{\frac{N-\beta}{N-\gamma}},
\end{align*}
where $C_2(N, \beta, \gamma) = e^{-2} C_1(N, \beta, \gamma)$ and $C_3(N, \beta, \gamma) = \frac{N-1}{2} C_2(N, \beta, \gamma)$.
Thus we have (\mathbb{R}f{Lower})
for some $C > 0$ independent of $\alpha$ which is sufficiently close to $\alpha_{N, \beta}$.
\qed
\noindent
{\bf Acknowledgments.}
The first author (V. H. N.) was supported by CIMI's postdoctoral research fellowship. The second author (F.T.) was supported by
JSPS Grant-in-Aid for Scientific Research (B), No.15H03631, and
JSPS Grant-in-Aid for Challenging Exploratory Research, No.26610030.
\end{document}
|
\begin{document}
\title{Spectrum and Analytic Functional Calculus in Real and Quaternionic Frameworks}
{\it Keywords: spectrum in real and quaternionic contexts;
holomorphic stem functions; analytic functional calculus for real and quaternionic operators}
{\it AMS Subject Classification:} 47A10; 30G35; 47A60
\begin{abstract}
We present an approach to the spectrum and analytic functional calculus
for quaternionic linear operators, following the corresponding results
concerning the real linear operators. In fact, the construction of the analytic functional calculus for real linear operators can be refined to get a similar construction for quaternionic linear ones, in a classical manner, using a Riesz-Dunford-Gelfand type kernel, and considering spectra
in the complex plane. A quaternionic joint spectrum for
pairs of operators is also discussed, and an analytic functional calculus is constructed, via a Martinelli type kernel in two variables.
\end{abstract}
{\it Keywords:} real and quaternionic operators; spectra;
analytic functional\break calculus.
{\it Mathematics Subject Classification} 2010: 47A10; 47A60; 30G35
{\bf s}ection{Introduction}\label{I}
In this text we consider ${\mathbb R}$-,\,${\mathbb C}$-, and ${\mathbb H}$-linear operators, that is,
real, complex and quaternionic linear operators, respectively.
While the spectrum of a linear operator is traditionally defined
for complex linear operators, it is sometimes useful to have it
also for real linear operators, as well as for quaternionic linear ones.
The definition of the spectrum
for a real linear operator goes seemingly back to Kaplansky (see \cite{Kap}), and it can be stated as follows. If $T$ is a real linear operator on the real vector space ${\mathcal{V}}$, a point $u+iv$
($u,v\in{\mathbb R}$) is in the spectrum of $T$ if the operator $(u-T)^2+v^2$ is not
invertible on ${\mathcal{V}}$, where the scalars are identified with multiples of the identity on ${\mathcal{V}}$. Although this definition involves only operators acting in ${\mathcal{V}}$, the spectrum is, nevertheless, a subset of the complex plane. As a matter of fact, a motivation of this choice can be illustrated via the complexification of the space ${\mathcal{V}}$ (see Section \ref{SC}).
The spectral theory for quaternionic linear operators is largely
discussed in numerous work, in particular in the monographs
\cite{CoSaSt} and \cite{CoGaKi}, and in many of their references as well.
In these works, the construction of an analytic functional
calculus (called $S$-{\it analytic functional calculus}) means to associate to each function from the class of the so-called {\it slice hyperholomorphic} or {\it slice regular functions} a quaternionic linear operator, using a specific noncommutative kernel.
The idea of the present work is to replace the class of slice regular functions by a class holomorphic
functions, using a commutative kernel of type Riesz-
Dunford-Gelfand. These two classes are isomorphic via a Cauchy type
transform (see \cite{Vas5}), and the image of the analytic functional calculus is the same, as one might expect (see Remark \ref{twofc}).
As in the case of real operators, the verbatim extension of the classical definition of the spectrum for quaternionic operators is not appropriate, and so a different definition using the squares of operators and real numbers was given, which can be found in \cite{CoSaSt} (see also \cite{CoGaKi}). We discuss this definition in our framework (see Definition \ref{Q-spectrum}), showing later that its ''complex border`` contains the most significant information, leading to the construction of an analytic functional calculus, equivalent to that obtained via the slice hyperholomorphic functions.
In fact, we first consider the spectrum for real operators on real Banach spaces, and sketch the construction of an analytic
functional calculus for them, using some classical ideas (see Theorem \ref{R_afc}). Then we
extend this framework to a quaternionic one, showing that the
approach from the real case can be easily adapted to the new situation.
As already mentioned, and unlike in \cite{CoSaSt} or \cite{CoGaKi}, our functional calculus is obtained via a
Riesz-Dunford-Gelfand formula, defined in a partially commutatative context, rather than the non-commutative Cauchy type formula used by previous authors. Our analytic functional calculus
holds for a class of analytic operator valued functions, whose definition extends that of stem functions, and it applies, in particular, to a large family of quaternionic linear operators. Moreover, we can show that the analytic functional calculus obtained in this way is equivalent to the analytic functional calculus obtained in \cite{CoSaSt} or \cite{CoGaKi},
in the sense that the images of these functional calculi coincide (see Remark \ref{twofc}).
We finally discuss the case of pairs of commuting real operators,
in the spirit of \cite{Vas4}, showing some connections with the quaternionic case. Specifically, we define a quaternionic spectrum for them and construct an analytic functional calculus using a Martinelli type formula, showing that for such a construction only a sort of ''complex border`` of the
quaternionic spectrum should be used.
This work is just an introductory one. Hopefully, more contributions on this line will be presented in the future.
{\bf s}ection{Spectrum and Conjugation}\label{SC}
Let ${\mathcal{A}}$ be a unital real Banach algebra, not necessarily commutative. As mentioned in the Introduction, the (complex) spectrum of an element $a\in{\mathcal{A}}$ may be defined by the equality
\begin{equation}\label{resp}
{\bf s}igma_{\mathbb C}(a)=\{u+iv;(u-a)^2+v^2\,\, {\rm is\,\,not\,\,invertible}, u,v\in{\mathbb R}\},
\end{equation}
This set is {\it conjugate symmetric}, that is $u+iv\in{\bf s}igma_{\mathbb C}(a)$ if and only if $u-iv\in{\bf s}igma_{\mathbb C}(a)$. A known motivation of this definition comes from the following remark.
Fixing a unital real Banach algebra ${\mathcal{A}}$, we denote by
${\mathcal{A}}_{\mathbb C}$ the complexification of ${\mathcal{A}}$, which is given by
$A_{\mathbb C}={\mathbb C}\otimes_{\mathbb R}{\mathcal{A}}$, written simply as ${\mathcal{A}}+i{\mathcal{A}}$, where the sum is direct, identifying the element $1\otimes a+i\otimes b$ with the element $a+ib$, for all $a,b\in{\mathcal{A}}$.
Then ${\mathcal{A}}_{\mathbb C}$ is a unital complex
algebra, which can be organized as a Banach algebra, with a (not
necessarily unique) convenient norm. To fix the ideas, we recall
that the product of two elements is given by
$(a+ib)(c+id)=ac-bd+i(ad+bc)$ for all $a,b,c,d\in{\mathcal{A}}$, and the norm may be defind by
${\mathcal{V}}ert a+ib{\mathcal{V}}ert={\mathcal{V}}ert a{\mathcal{V}}ert+{\mathcal{V}}ert b{\mathcal{V}}ert$, where ${\mathcal{V}}ert*{\mathcal{V}}ert$
is the norm of ${\mathcal{A}}$.
In the algebra ${\mathcal{A}}_{\mathbb C}$,
the complex numbers commute with all elements of ${\mathcal{A}}$. Moreover,
we have a {\it conjugation} given by
$$
{\mathcal{A}}_{\mathbb C}\ni a+ib\mapsto a-ib\in{\mathcal{A}}_{\mathbb C},\,a,b\in{\mathcal{A}},
$$
which is a unital conjugate-linear automorphism, whose square is the identity. In particular, an arbitrary element $a+ib$ is invertible if and only if $a-ib$ is invertible.
The usual spectrum, defined for each element $a\in{\mathcal{A}}_{\mathbb C}$, will be
denoted by ${\bf s}igma(a)$. Regarding the algebra ${\mathcal{A}}$ as a real
subalgebra of ${\mathcal{A}}_{\mathbb C}$, one has the following.
\begin{Lem}\label{com_sp_re_op} For every $a\in{\mathcal{A}}$ we have the equality
${\bf s}igma_{\mathbb C}(a)={\bf s}igma(a)$.
\end{Lem}
{\it Proof.} The result is well known but we give a short proof, because
a similar idea will be later used.
Let $\lambda=u+iv$ with $u,v\in{\mathbb R}$ arbitrary. Assuming
$\lambda-a$ invertible, we also have $\bar{\lambda}-a$ invertible. From the obvious identity
$$
(u-a)^2+v^2=(u+iv-a)(u-iv-a),
$$
we deduce that the element $(u-a)^2+v^2$ is invertible,
implying the inclusion ${\bf s}igma_{\mathbb C}(a){\bf s}ubset{\bf s}igma(a)$.
Conversely, if $(u-a)^2+v^2$ is invertible, then both
$u+iv-a,u-iv-a$ are invertible via the decomposition from above, showing that we also have ${\bf s}igma_{\mathbb C}(a){\bf s}upset{\bf s}igma(a)$.
\begin{Rem}\rm The spectrum ${\bf s}igma(a)$ with $a\in{\mathcal{A}}$ is always a conjugate symmetric set.
\end{Rem}
We are particularly interested to apply the discussion from above to the context of linear operators. The spectral theory for real linear operators is well known, and it is developed actually in the framework of linear relations (see \cite{BaZa}).
Nevertheless, we present here a different approach, which can
be applied, with minor changes, to the case of some quaternionic operators.
For a real or complex Banach space $\mathcal{V}$, we denote by
$\mathcal{B(V)}$ the algebra of all bounded ${\mathbb R}$-(
respectively ${\mathbb C}$-)linear operators on $\mathcal{V}$.
As before, the multiples of the identity will be identified with the corresponding scalars.
Let ${\mathcal{V}}$ be a real Banach space, and let ${\mathcal{V}}_{\mathbb C}$ be its complexification, which, as above, is identified with the direct sum ${\mathcal{V}}+i{\mathcal{V}}$. Each operator
$T\in\mathcal{B(V)}$ has a natural extension to an operator
$T_{\mathbb C}\in\mathcal{B}({\mathcal{V}}_{\mathbb C})$, given by $T_{\mathbb C}(x+iy)=Tx+iTy,\,
x,y\in{\mathcal{V}}$. Moreover, the map $\mathcal{B(V)}\ni T\mapsto T_{\mathbb C}\in\mathcal{B}({\mathcal{V}}_{\mathbb C})$ is unital, ${\mathbb R}$-linear and multiplicative. In particular, $T\in\mathcal{B(V)}$ is invertible if and only if
$T_{\mathbb C}\in\mathcal{B}({\mathcal{V}}_{\mathbb C})$ is invertible.
Fixing an operator $S\in\mathcal{B}(\mathcal{V}_{\mathbb C})$, we define
the operator $S^\flat\in\mathcal{B}(\mathcal{V}_{\mathbb C})$ to be equal to $CSC$, where $C:{\mathcal{V}}_{\mathbb C}\mapsto{\mathcal{V}}_{\mathbb C}$ is the conjugation
$x+iy\mapsto x-iy,\,x,y\in{\mathcal{V}}$. It is easily seen that the map
$\mathcal{B}(\mathcal{V}_{\mathbb C})\ni S\mapsto S^\flat\in \mathcal{B}(\mathcal{V}_{\mathbb C})$ is a unital conjugate-linear automorphism, whose square is the identity on $\mathcal{B}(\mathcal{V}_{\mathbb C})$.
Because $\mathcal{V}=\{u\in\mathcal{V}_{\mathbb C}; Cu=u\}$, we have
$S^\flat=S$ if and only if $S(\mathcal{V}){\bf s}ubset\mathcal{V}$.
In particular, we have $T_{\mathbb C}^\flat=T_{\mathbb C}$. In fact, because of the representation
$$
S=\frac{1}{2}(S+S^{\flat})+ i\frac{1}{2i}(S-S^{\flat}),\,\,
S\in\mathcal{B}(\mathcal{V}_{\mathbb C}),
$$
where $(S+S^{\flat})({\mathcal{V}}){\bf s}ubset{\mathcal{V}}, i(S-S^{\flat})({\mathcal{V}}){\bf s}ubset{\mathcal{V}}$, the algebras $\mathcal{B}(\mathcal{V}_{\mathbb C})$ and $\mathcal{B(V)}_{\mathbb C}$ are isomorphic and they will be often identified, and $\mathcal{B(V)}$ will be regarded as a (real) subalgebra of $\mathcal{B}(\mathcal{V})_{\mathbb C}$. In particular, if $S=U+iV$, with $U,V\in\mathcal{B(V)}$, we have
$S^\flat=U-iV$, so the map $S\mapsto S^\flat$ is the conjugation
of the complex algebra $\mathcal{B}(\mathcal{V})_{\mathbb C}$ induced
by the conjugation $C$ of ${\mathcal{V}}_{\mathbb C}$.
For every operator $S\in\mathcal{B}(\mathcal{V}_{\mathbb C})$, we denote,
as before, by ${\bf s}igma(S)$ its usual spectrum. As
$\mathcal{B(V)}$ is a real algebra,
the (complex) spectrum of an operator $T\in\mathcal{B(V)}$ is given by the equality (\ref{resp}):
$$
{\bf s}igma_{\mathbb C}(T)=\{u+iv;(u-T)^2+v^2\,\, {\rm is\,\,not\,\,invertible}, u,v\in{\mathbb R}\}.
$$
\begin{Cor} For every $T\in\mathcal{B}({\mathcal{V}})$ we have the equality
${\bf s}igma_{\mathbb C}(T)={\bf s}igma(T_{\mathbb C})$.
\end{Cor}
{\bf s}ection{Analytic Functional Calculus for Real
\\ Operators}
Having a concept of spectrum for real operators, an important step for further development is the construction of an analytic functional calculus. Such a construction has been done actually
in the context of real linear relations in \cite{BaZa}. In
what follows we shall present a similar construction for real
linear operators. Although the case of linear relations looks more general, unlike in \cite{BaZa}, we perform our construction using a class of operator valued analytic functions insted of scalar valued analytic functions. Moreover, our arguments look simpler, and
the construction is a model for a more general one, to get an analytic
functional calculus for quaternionic linear operators.
If ${\mathcal{V}}$ is a real Banach space, and so each operator $T\in\mathcal{B}({\mathcal{V}})$ has a complex spectrum ${\bf s}igma_{\mathbb C}(T)$, which is
compact and nonempty, one can use the classical Riesz-Dunford functional calculus, in a slightly generalized form (that is, replacing the scalar-valued analytic functions by operator-valued analytic ones, which is a well known idea).
The use of vector versions of the Cauchy formula is simplified by adopting the following definition. Let $U{\bf s}ubset{\mathbb C}$ be open. An open subset
$\Delta{\bf s}ubset U$ will be called a {\it Cauchy domain} (in $U$) if
$\Delta{\bf s}ubset\bar{\Delta}{\bf s}ubset U$ and the boundary of $\Delta$ consists of a finite family of closed curves, piecewise smooth, positively oriented. Note that a Cauchy domain is bounded but not necessarily connected.
\begin{Rem}\label{afcro}\rm If $\mathcal{V}$ is a real Banach space, and
$T\in\mathcal{B(V})$, we have the usual analytic functional calculus for the operator $T_{\mathbb C}\in\mathcal{B}(\mathcal{V}_{\mathbb C})$
(see \cite{DuSc}). That is, in a slightly generalized form, and for later use, if $U{\bf s}upset{\bf s}igma(T_{\mathbb C})$ is an open set in ${\mathbb C}$ and
$F:U\mapsto\mathcal{B}(\mathcal{V}_{\mathbb C})$ is analytic, we put
$$
F(T_{\mathbb C})=\frac{1}{2\pi i}\int_\Gamma F({\bf z}eta)({\bf z}eta-T_{\mathbb C})^{-1}
d{\bf z}eta,
$$
where $\Gamma$ is the boundary of a Cauchy domain $\Delta$ containing
${\bf s}igma(T_{\mathbb C})$ in $U$. In fact, because ${\bf s}igma(T_{\mathbb C})$ is conjugate symmetric, we may and shall assume
that both $U$ and $\Gamma$ are conjugate symmetric. Because the
function ${\bf z}eta\mapsto F({\bf z}eta)({\bf z}eta-T_{\mathbb C})^{-1}$ is analytic
in $U{\bf s}etminus{\bf s}igma(T_{\mathbb C})$, the integral does not depend on the
particular choice of the Cauchy domain $\Delta$ containing
${\bf s}igma(T_C)$.
A natural
question is to find an appropriate condition to we have
$F(T_{\mathbb C})^\flat=F(T_{\mathbb C})$, which would
imply the invariance of $\mathcal{V}$ under $F(T_{\mathbb C})$.
\end{Rem}
With the notation of Remark \ref{afcro}, we have the following.
\begin{Thm}\label{afcro1} Let $U{\bf s}ubset{\mathbb C}$ be open and conjugate symmetric. If $F:U\mapsto\mathcal{B}(\mathcal{V}_{\mathbb C})$ is analytic and $F({\bf z}eta)^\flat=F(\bar{{\bf z}eta})$ for all ${\bf z}eta\in U$, then $F(T_{\mathbb C})^\flat=F(T_{\mathbb C})$ for all $T\in\mathcal{B}(\mathcal{V})$ with ${\bf s}igma_{\mathbb C}(T){\bf s}ubset U$.
\end{Thm}
{\it Proof.}\, We use the notation from Remark \ref{afcro}, assuming, in
addition, that $\Gamma$ is conjugate symmetric as well. We put
$\Gamma_\pm:=\Gamma\cap{\mathbb C}_\pm$, where ${\mathbb C}_+$ (resp. ${\mathbb C}_-$) equals to $\{\lambda\in{\mathbb C};\Im\lambda\ge0\}$ (resp. $\{\lambda\in{\mathbb C};\Im\lambda\le0\}$). We write $\Gamma_+=\cup_{j=1}^m\Gamma_{j+}$, where $\Gamma_{j+}$ are the connected components
of $\Gamma_+$. Similarly, we write $\Gamma_-=\cup_{j=1}^m\Gamma_{j-}$, where $\Gamma_{j-}$ are the connected components
of $\Gamma_-$, and $\Gamma_{j-}$ is the reflexion of
$\Gamma_{j+}$ with respect of the real axis.
As $\Gamma$ is a finite union of Jordan piecewise smooth closed curves,
for each index
$j$ we have a parametrization $\phi_j:[0,1]\mapsto{\mathbb C}$, positively oriented, such that
$\phi_j([0,1])=\Gamma_{j+}$. Taking into account that the function $t\mapsto\overline{\phi_j(t)}$ is a
parametrization of $\Gamma_{j-}$ negatively oriented, and setting
$\Gamma_j=\Gamma_{j+}\cup\Gamma_{j-}$, we can write
$$
F_j(T_{\mathbb C}):=\frac{1}{2\pi i}\int_{\Gamma_j} F({\bf z}eta)({\bf z}eta-T_{\mathbb C})^{-1}
d{\bf z}eta=
$$
$$
\frac{1}{2\pi i}\int_0^1 F(\phi_j(t))(\phi_j(t)-T_{\mathbb C})^{-1}
\phi_j'(t)dt
$$
$$
-\frac{1}{2\pi i}\int_0^1 F(\overline{\phi_j(t)})(\overline{\phi_j(t)}-T_{\mathbb C})^{-1}\overline{\phi_j'(t)}dt.
$$
Therefore,
$$
F_j(T_{\mathbb C})^\flat=
-\frac{1}{2\pi i}\int_0^1 F(\phi_j(t))^\flat(\overline{\phi_j(t)}-T_{\mathbb C})^{-1}\overline{\phi_j'(t)}dt
$$
$$
+\frac{1}{2\pi i}\int_0^1 F(\overline{\phi_j(t)})^\flat(\phi_j(t)-T_{\mathbb C})^{-1}\phi_j'(t)dt.
$$
According to our assumption on the function $F$, we obtain
$F_j(T_{\mathbb C})=F_j(T_{\mathbb C})^\flat$ for all $j$, and therefore
$$
F(T_{\mathbb C})^\flat={\bf s}um_{j=1}^mF_j(T_{\mathbb C})^\flat={\bf s}um_{j=1}^mF_j(T_{\mathbb C})=F(T_{\mathbb C}), $$
which concludes the proof.
\begin{Rem}\label{stem_anal}\rm If ${\mathcal{A}}$ is a unital real Banach algebra,
${\mathcal{A}}_{\mathbb C}$ its complexification, and $U{\bf s}ubset{\mathbb C}$ is open, we denote by
$\mathcal{O}(U,{\mathcal{A}}_{\mathbb C})$ the algebra of all analytic ${\mathcal{A}}_{\mathbb C}$-valued functions. If $U$ is conjugate symmetric, and
${\mathcal{A}}_{\mathbb C}\ni a\mapsto \bar{a}\in
{\mathcal{A}}_{\mathbb C}$ is its natural conjugation, we denote by $\mathcal{O}_s(U,{\mathcal{A}}_{\mathbb C})$ the real subalgebra of $\mathcal{O}(U,{\mathcal{A}}_{\mathbb C})$ consisting of those functions $F$ with the property $F(\bar{{\bf z}eta})=\overline{F({\bf z}eta)}$
for all ${\bf z}eta\in U$. Adapting a well known terminology, such
functions will be called (${\mathcal{A}}_{\mathbb C}$-{\it valued $)$ stem functions}.
When ${\mathcal{A}}={\mathbb R}$, so ${\mathcal{A}}_{\mathbb C}={\mathbb C}$, the space $\mathcal{O}_s(U,{\mathbb C})$ will be denoted by
$\mathcal{O}_s(U)$, which is a real algebra. Note that
$\mathcal{O}_s(U,{\mathcal{A}}_{\mathbb C})$ is also a bilateral $\mathcal{O}_s(U)$-module.
In the next result, we identify the algebra
$\mathcal{B}(\mathcal{V})$ with a subalgebra of
$\mathcal{B}(\mathcal{V})_{\mathbb C}$. In ths case, when $F\in \mathcal{O}_s(U,\mathcal{B}(\mathcal{V})_{\mathbb C})$, we shall write
$$
F(T)=\frac{1}{2\pi i}\int_\Gamma F({\bf z}eta)({\bf z}eta-T)^{-1}
d{\bf z}eta,
$$
noting that the right hand side of this formula belongs to
$\mathcal{B}(\mathcal{V})$, by Theorem \ref{afcro1}.
\end{Rem}
The properties of the map $F\mapsto F(T)$, which can be called
the {\it (left) analytic functional calculus of} $T$, are summarized by
the following.
\begin{Thm}\label{R_afc} Let ${\mathcal{V}}$ be a real Banach space, let $U{\bf s}ubset{\mathbb C}$ be a conjugate symmetric open set, and let $T\in\mathcal{B}(\mathcal{V})$, with
${\bf s}igma_{\mathbb C}(T){\bf s}ubset U$. Then the assignment
$$
{\mathcal O}_s(U,\mathcal{B}(\mathcal{V})_{\mathbb C})\ni F\mapsto F(T)\in\mathcal{B}(\mathcal{V})
$$
is an ${\mathbb R}$-linear map, and the map
$$
{\mathcal O}_s(U)\ni f\mapsto f(T)\in\mathcal{B}(\mathcal{V})
$$
is a unital real algebra morphism.
Moreover, the following properties are true:
(1) For all $F\in\mathcal{O}_s(U,\mathcal{B}(\mathcal{V})_{\mathbb C}),\, f\in{\mathcal O}_s(U)$, we have $(Ff)(T)=F(T)f(T)$.
(2) For every polynomial $P({\bf z}eta)={\bf s}um_{n=0}^m A_n{\bf z}eta^n,\,{\bf z}eta\in{\mathbb C}$, with $A_n\in\mathcal{B}(\mathcal{V})$ for all $n=0,1,\ldots,m$, we have $P(T)={\bf s}um_{n=0}^m A_n T^n\in\mathcal{B}(\mathcal{V})$.
\end{Thm}
{\it Proof.\,} The arguments are more or less standard (see
\cite{DuSc}). The ${\mathbb R}$-linearity of the maps
$$
{\mathcal O}_s(U,\mathcal{B}(\mathcal{V})_{\mathbb C})\ni F\mapsto F(T)\in\mathcal{B}(\mathcal{V}),\,
{\mathcal O}_s(U)\ni f\mapsto f(T)\in\mathcal{B}(\mathcal{V}),
$$
is clear. The second one is actually multiplicative, which follows from the multiplicativiry of the usual analytic functional calculus of $T$.
In fact, we have a more general property, specifically
$(Ff)(T)=F(T)f(T)$ for all $F\in\mathcal{O}_s(U,\mathcal{B}(\mathcal{V})_{\mathbb C}),\, f\in{\mathcal O}_s(U)$. This follows from the equalities,
$$
(Ff)(T)=\frac{1}{2\pi i}\int_{\Gamma_0} F({\bf z}eta)f({\bf z}eta)({\bf z}eta-T)^{-1}d{\bf z}eta=
$$
$$
\left(\frac{1}{2\pi i}\int_{\Gamma_0} F({\bf z}eta)({\bf z}eta-T)^{-1}d{\bf z}eta\right)
\left(\frac{1}{2\pi i}\int_{\Gamma} f(\eta)(\eta-T)^{-1}d\eta\right)=F(T)f(T),
$$
obtained as in the classical case (see \cite{DuSc}, Section VII.3), which holds because $f$ is ${\mathbb C}$-valued and commutes with the operators in $\mathcal{B}(\mathcal{V})$. Here $\Gamma,\,\Gamma_0$ are the boundaries of two Cauchy
domains $\Delta,\,\Delta_0$ respectively, such that $\Delta{\bf s}upset
\bar{\Delta}_0$, and $\Delta_0$ contains ${\bf s}igma(T)$.
Note that, in particular, for every polynomial $P({\bf z}eta)={\bf s}um_{n=0}^m A_n{\bf z}eta^n$ with $A_n\in\mathcal{B}(\mathcal{V})$ for all $n=0,1,\ldots,m$, we have $P(T)={\bf s}um_{n=0}^m A_n q^n\in\mathcal{B}(\mathcal{V})$ for all $T\in\mathcal{B}(\mathcal{V})$.
\begin{Exa}\label{ex1}\rm Let $\mathcal{V}={\mathbb R}^2$, so $\mathcal{V}_{\mathbb C}={\mathbb C}^2$, endowed with its natural Hilbert space
structure. Let us first observe that we have
$$
S=\left(\begin{array}{cc} a_1 & a_2 \\ a_3 & a_4
\end{array}\right)\,\,\Longleftrightarrow
S^\flat=\left(\begin{array}{cc} \bar{a}_1 & \bar{a}_2 \\
\bar{a}_3 & \bar{a}_4\end{array}\right),
$$
for all $a_1,a_2,a_3,a_4\in{\mathbb C}$.
Next we consider the operator $T\in\mathcal{B}({\mathbb R}^2)$ given by the matrix
$$
T=\left(\begin{array}{cc} u & v \\ -v & u
\end{array}\right),
$$
where $u,v\in{\mathbb R}, v\neq0$. The extension $T_{\mathbb C}$ of the operator
$T$ to ${\mathbb C}^2$, which is a normal operator, is given by the same formula. Note that
$$
{\bf s}igma_{\mathbb C}(T)=\{\lambda\in{\mathbb C};(\lambda-u)^2+v^2=0\}=
\{u\pm iv\}={\bf s}igma(T_{\mathbb C}).
$$
Note also that the vectors $\nu_\pm=({\bf s}qrt{2})^{-1}(1,\pm i)$
are normalized eigenvectors for $T_{\mathbb C}$ corresponding to the eigenvalues $u\pm iv$, respectively.
The spectral projections of $T_{\mathbb C}$ corresponding to
these eigenvalues are given by
$$
E_\pm(T_{\mathbb C}){\bf w}=\langle {\bf w},\nu_\pm\rangle \nu_\pm=\frac{1}{2}\left(\begin{array}{cc} 1 & \mp i \\ \pm i & 1 \end{array}\right)\left(\begin{array}{c} w_1 \\ w_2 \end{array}\right),
$$
for all ${\bf w}=(w_1,w_2)\in{\mathbb C}^2$.
Let $U{\bf s}ubset{\mathbb C}$ be an open set with $U{\bf s}upset\{u\pm iv\}$,
and let $F:U\mapsto\mathcal{B}({\mathbb C}^2)$ be analytic.
We shall compute explicitly $F(T_{\mathbb C})$. Let $\Delta$ be a Cauchy
domain contained in $U$ with its boundary $\Gamma$, and containing the points $u\pm iv$. Assuming $v>0$, we have
$$
F(T_{\mathbb C})=\frac{1}{2\pi i}\int_\Gamma F({\bf z}eta)({\bf z}eta-T_{\mathbb C})^{-1}
d{\bf z}eta=
$$
$$
F(u+iv)E_+(T_{\mathbb C})+F(u-iv)E_-(T_{\mathbb C})=
$$
$$
\frac{1}{2}F(u+iv)\left(\begin{array}{cc} 1 & -i \\ i & 1 \end{array}\right)+\frac{1}{2}F(u-iv)\left(\begin{array}{cc} 1 & i \\ -i & 1 \end{array}\right).
$$
Assume now that $F(T_{\mathbb C})^\flat=F(T_{\mathbb C})$. Then we must have
$$
(F(u+iv)-F(u-iv)^\flat)\left(\begin{array}{cc} 1 & -i \\ i & 1
\end{array}\right)=
(F(u+iv)^\flat-F(u-iv))\left(\begin{array}{cc} 1 & i \\ -i & 1
\end{array}\right).
$$
We also have the equalities
$$
\left(\begin{array}{cc} 1 & -i \\ i & 1
\end{array}\right)\left(\begin{array}{c} 1 \\ i \end{array} \right)=2\left(\begin{array}{c} 1 \\ i \end{array}\right),\,\,
\left(\begin{array}{cc} 1 & -i \\ i & 1
\end{array}\right)\left(\begin{array}{c} 1 \\ -i \end{array} \right)=0,
$$
$$
\left(\begin{array}{cc} 1 & i \\ -i & 1
\end{array}\right)\left(\begin{array}{c} 1 \\ -i \end{array} \right)=2\left(\begin{array}{c} 1 \\ -i \end{array}\right),\,\,
\left(\begin{array}{cc} 1 & i \\ -i & 1
\end{array}\right)\left(\begin{array}{c} 1 \\ i \end{array} \right)=0,
$$
Using these equalities, we finally deduce that
$$
(F(u+iv)-F(u-iv)^\flat)\left(\begin{array}{c} 1 \\ i \end{array}\right)=0,
$$
and
$$
(F(u-iv)-F(u+iv)^\flat)\left(\begin{array}{c} 1 \\ -i \end{array}\right)=0,
$$
which are necessary conditions for the equality $F(T_{\mathbb C})^\flat=F(T_{\mathbb C})$.
As a matter of fact, this example shows, in particular, that the condition
$F({\bf z}eta)^\flat=F(\bar{{\bf z}eta})$ for all ${\bf z}eta\in U$,
used in Theorem \ref{afcro1}, is sufficient but it might not be always necessary.
\end{Exa}
{\bf s}ection{Analytic Functional Calculus for Quaternionic
\\ Operators}
{\bf s}ubsection{Quaternionic Spectrum}
We now recall some known definitions and elementary
facts (see, for instance, \cite{CoSaSt}, Section 4.6, and/or
\cite{Vas5}).
Let ${\mathbb H}$ be the abstract algebra of quaternions, which is the four-dimensional ${\mathbb R}$-algebra with
unit $1$, generated by the ''imaginary units`` $\{\bf{j,k,l}\}$,
which satisfy
$$
{\bf jk=-kj=l,\,kl=-lk=j,\,lj=-jl=k,\,jj=kk=ll}=-1.
$$
We may assume that ${\mathbb H}{\bf s}upset{\mathbb R}$ identifying every number
$x\in{\mathbb R}$ with the element $x1\in{\mathbb H}$.
The algebra ${\mathbb H}$ has a natural multiplicative norm given by
$$
{\mathcal{V}}ert {\bf x}{\mathcal{V}}ert={\bf s}qrt{x_0^2+x_1^2+x_2^2+x_0^2},\,\,{\bf x}= x_0+x_1{\bf j}+x_2{\bf k}+x_3{\bf l},\,\,x_0,x_1,x_2,x_3\in{\mathbb R},
$$
and a natural involution
$$
{\mathbb H}\ni{\bf x} = x_0+x_1{\bf j}+x_2{\bf k}+x_3{\bf l}\mapsto
{\bf x}^*= x_0-x_1{\bf j}-x_2{\bf k}-x_3{\bf l}\in{\mathbb H}.
$$
Note that ${\bf x}{\bf x}^*={\bf x}^*{\bf x}={\mathcal{V}}ert{\bf x}{\mathcal{V}}ert^2$, implying, in particular, that every element ${\bf x}\in{\mathbb H}{\bf s}etminus\{0\}$ is invertible, and ${\bf x}^{-1}=
{\mathcal{V}}ert {\bf x}{\mathcal{V}}ert^{-2}{\bf x}^*$.
For an arbitrary quaternion ${\bf x}= x_0+x_1{\bf j}+x_2{\bf k}+x_3{\bf l},\,\,x_0,x_1,x_2,x_3\in{\mathbb R}$, we set ${\mathbb R}e{\bf x}=x_0=
({\bf x}+{\bf x}^*)/2$, and $\Im{\bf x}=x_1{\bf j}+x_2{\bf k}+x_3{\bf l}=({\bf x}-{\bf x}^*)/2$, that is, the {\it real} and
{\it imaginary part} of ${\bf x}$, respectively.
We consider the complexification ${\mathbb C}\otimes_{\mathbb R}{\mathbb H}$
of the ${\mathbb R}$-algebra ${\mathbb H}$ (see also \cite{GhMoPe}), which will be
identified with the direct sum ${\mathbb M}={\mathbb H}+i{\mathbb H}$.
Of course, the algebra ${\mathbb M}$ contains the complex field ${\mathbb C}$. Moreover, in the algebra ${\mathbb M}$, the elements of ${\mathbb H}$ commute with all complex numbers. In particular, the ''imaginary units``
$\bf j,k,l$ of the algebra ${\mathbb H}$ are independent of and commute with the imaginary unit $i$ of the complex plane ${\mathbb C}$.
In the algebra ${\mathbb M}$, there also exists a natural conjugation given
by $\bar{\bf a}={\bf b}-i{\bf c}$, where ${\bf a}={\bf b}+i{\bf c}$ is arbitrary in ${\mathbb M}$, with ${\bf b},{\bf c}\in{\mathbb H}$ (see also
\cite{GhMoPe}). Note that $\overline{\bf a+b}=\bar{\bf a}+\bar{\bf b}$, and $\overline{\bf ab}=\bar{\bf a}\bar{\bf b}$, in particular $\overline{r\bf a}=r\bar{\bf a}$ for all ${\bf a},{\bf b}\in{\mathbb M}$, and $r\in{\mathbb R}$. Moreover, $\bar{{\bf a}}={\bf a}$ if and only if ${\bf a}\in{\mathbb H}$, which is a useful characterization of the elements of ${\mathbb H}$ among those of ${\mathbb M}$.
\begin{Rem}\label{Qspectrum}\rm
In the algebra ${\mathbb M}$ we have the identities
$$
(\lambda-{\bf x}^*)(\lambda-{\bf x})=(\lambda-{\bf x})(\lambda-{\bf x}^*)=\lambda^2-
\lambda({\bf x}+{\bf x}^*)+{\mathcal{V}}ert {\bf x}{\mathcal{V}}ert^2\in{\mathbb C},
$$
for all $\lambda\in{\mathbb C}$ and ${\bf x}\in{\mathbb H}$. If the complex number
$\lambda^2-2\lambda{\mathbb R}e{\bf x}+{\mathcal{V}}ert {\bf x}{\mathcal{V}}ert^2$ is
nonnull, then both element $\lambda-{\bf x}^*,\,\lambda-{\bf x}$ are invertible. Conversely, if $\lambda-{\bf x}$ is invertible, we must have
$\lambda^2-2\lambda{\mathbb R}e{\bf x}+{\mathcal{V}}ert {\bf x}{\mathcal{V}}ert^2$ nonnull; otherwise we
would have $\lambda={\bf x}^*\in{\mathbb R}$, so $\lambda={\bf x}\in{\mathbb R}$, which is not possible. Therefore, the element
$\lambda-{\bf x}\in{\mathbb M}$ is invertible if and only if the complex number $\lambda^2-2\lambda{\mathbb R}e{\bf x}+{\mathcal{V}}ert {\bf x}{\mathcal{V}}ert^2$ is
nonnull. Hence, the element
$\lambda-{\bf x}\in{\mathbb M}$ is not invertible if and only if $\lambda=
{\mathbb R}e{\bf x}\pm i{\mathcal{V}}ert\Im{\bf x}{\mathcal{V}}ert$. In this way, the {\it spectrum} of a quaternion ${\bf x}\in{\mathbb H}$ is given by the equality ${\bf s}igma({\bf x})=\{s_\pm(\bf x)\}$, where
$s_\pm(\bf x)={\mathbb R}e{\bf x}\pm i{\mathcal{V}}ert\Im{\bf x}{\mathcal{V}}ert$ are the {\it eigenvalues} of $\bf x$ (see also \cite{Vas4,Vas5}).
The polynomial $P_{\bf x}(\lambda)=\lambda^2-2\lambda{\mathbb R}e{\bf x}+{\mathcal{V}}ert {\bf x}{\mathcal{V}}ert^2$ is the {\it minimal polynomial} of $\bf x$. In fact,
the equality ${\bf s}igma({\bf y})= {\bf s}igma({\bf x})$ for some ${\bf x,y}\in{\mathbb H}$
is an equivalence relation in the algebra ${\mathbb H}$, which holds if and only if $P_{\bf x}=P_{\bf y}$. In fact, setting $\mathbb{S}=\{\mathfrak{\kappa}\in{\mathbb H};{\mathbb R}e\mathfrak{\kappa}=0, {\mathcal{V}}ert \mathfrak{\kappa}{\mathcal{V}}ert=1\}$ (that is the unit sphere of purely imaginary quaternions), representig an arbitrary
quaternion $\bf x$ under the form $x_0+y_0 \mathfrak{\kappa}_0$, with
$x_0,y_0\in{\mathbb R}$ and $\mathfrak{\kappa}_0\in\mathbb{S}$, a quaternion $\bf y$
is equivalent to $\bf x$ if anf only if it is of the form $x_0+y_0 \mathfrak{\kappa}$ for some $\mathfrak{\kappa}\in\mathbb{S}$ (see \cite{Bre} or \cite{Vas5} for some details).
\end{Rem}
\begin{Rem}\label{Hspace}\rm
Following \cite{CoSaSt}, a {\it right ${\mathbb H}$-vector space}
$\mathcal{V}$ is a real vector space
having a right multiplication with the elements of ${\mathbb H}$, such that $(x+y){\bf q}=x{\bf q}+y{\bf q},\,x({\bf q}+{\bf s})=
x{\bf q}+x{\bf s},\, x({\bf q}{\bf s})=(x{\bf q}){\bf s}$
for all $x,y\in\mathcal{V}$ and ${\bf q},{\bf s}\in{\mathbb H}$.
If $\mathcal{V}$ is also a Banach space the operator
$T\in\mathcal{B(V)}$ is {\it right ${\mathbb H}$-linear} if
$T(x{\bf q})=T(x){\bf q}$ for all $x\in\mathcal{V}$ and
${\bf q}\in{\mathbb H}$. The set of right ${\mathbb H}$ linear operators will be
denoted by $\mathcal{B^{\rm r}(V)}$, which is, in particular, a unital
real algebra.
In a similar way, one defines the concept of a {\it left ${\mathbb H}$-vector space}. A real vector space $\mathcal{V}$ will be said to be an {\it ${\mathbb H}$-vector space} if it is simultaneously a right ${\mathbb H}$- and a left ${\mathbb H}$-vector space. As noticed in \cite{CoSaSt},
it is the framework of ${\mathbb H}$-vector spaces an appropriate one
for the study of right ${\mathbb H}$-linear operators.
If ${\mathcal{V}}$ is ${\mathbb H}$-vector space which is also a Banach space, then
${\mathcal{V}}$ is said to be a {\it Banach ${\mathbb H}$-space}. In this case, we also assume that $ R_{\bf q}\in \mathcal{B}({\mathcal{V}})$, and
the map ${\mathbb H}\ni{\bf q}\mapsto R_{\bf q}\in
\mathcal{B}({\mathcal{V}})$ is norm continuous, where $R_{\bf q}$ is the right multiplication of the elements of
$\mathcal{V}$ by a given quaternion ${\bf q}\in{\mathbb H}$. Similarly,
if $L_{\bf q}$ is the left multiplication of the elements of
$\mathcal{V}$ by the quaternion ${\bf q}\in{\mathbb H}$, we assume that
$ L_{\bf q}\in \mathcal{B}({\mathcal{V}})$ for all ${\bf q}\in{\mathbb H}$, and that
the map ${\mathbb H}\ni{\bf q}\mapsto L_{\bf q}\in \mathcal{B}({\mathcal{V}})$ is norm continuous. Note also that
$$
\mathcal{B^{\rm r}(V)}=\{T\in\mathcal{B(V)};TR_{\bf q}=R_{\bf q} T,\,{\bf q}\in{\mathbb H}\}.
$$
To adapt the discussion regarding the real algebras to this case, we first consider the complexification ${\mathcal{V}}_{\mathbb C}$ of ${\mathcal{V}}$. Because ${\mathcal{V}}$ is an ${\mathbb H}$-bimodule, the space ${\mathcal{V}}_{\mathbb C}$ is actually an ${\mathbb M}$-bimodule, via the multiplications
$$
({\bf q}+i{\bf s})(x+iy)={\bf q} x-{\bf s} y+i({\bf q} y+{\bf s} x),
(x+iy)({\bf q}+i{\bf s})=x{\bf q}-y{\bf s}+i(y{\bf q}+x{\bf s}),
$$
for all ${\bf q}+i{\bf s}\in{\mathbb M},\,{\bf q},{\bf s}\in{\mathbb H},\,x+iy\in{\mathcal{V}}_{\mathbb C},\,x,y\in{\mathcal{V}}$. Moreover, the operator
$T_{\mathbb C}$ is right ${\mathbb M}$-linear, that is $T_{\mathbb C}((x+iy)({\bf q}+i{\bf s}))=
T_{\mathbb C}(x+iy)({\bf q}+i{\bf s})$ for all ${\bf q}+i{\bf s}\in{\mathbb M},\,x+iy\in{\mathcal{V}}_{\mathbb C}$, via a direct computation.
Let $C$ be the conjugation of ${\mathcal{V}}_{\mathbb C}$. As in the real case, for every
$S\in \mathcal{B}({\mathcal{V}}_{\mathbb C})$, we put $S^\flat=CSC$. The left and right
multiplication with the quaternion ${\bf q}$ on ${\mathcal{V}}_{\mathbb C}$ will be also denoted by $L_{\bf q},R_{\bf q}$, respectively, as elements of $\mathcal{B}({\mathcal{V}}_{\mathbb C})$. We set
$$
\mathcal{B}^{\rm r}({\mathcal{V}}_{\mathbb C})=\{S\in \mathcal{B}({\mathcal{V}}_{\mathbb C}); SR_{\bf q}=R_{\bf q} S,\,{\bf q}\in{\mathbb H}\},
$$
which is a unital complex algebra containing all operators $L_{\bf q},
{\bf q}\in{\mathbb H}$.
Note that if $S\in\mathcal{B}^{\rm r}({\mathcal{V}}_{\mathbb C})$, then $S^\flat\in\mathcal{B}^{\rm r}({\mathcal{V}}_{\mathbb C})$. Indeed, because $CR_{\bf q}=R_{\bf q} C$, we also have $S^\flat R_{\bf q}=R_{\bf q} S^\flat$. In fact, as we have
$(S+S^\flat)({\mathcal{V}}){\bf s}ubset{\mathcal{V}}$ and $i(S-S^\flat)({\mathcal{V}}){\bf s}ubset{\mathcal{V}}$, it folows that
the algebras $\mathcal{B}^{\rm r}({\mathcal{V}}_{\mathbb C}),\,\mathcal{B}^{\rm r}({\mathcal{V}})_{\mathbb C}$
are isomorphic, and they will be often identified, where
$\mathcal{B^{\rm r}(V)}_{\mathbb C}=\mathcal{B^{\rm r}(V)}+i\mathcal{B^{\rm r}(V)}$ is the complexification of $\mathcal{B^{\rm r}(V)}$, which is also a unital
complex Banach algebra.
\end{Rem}
Looking at the Definition 4.8.1 from \cite{CoSaSt}
(see also \cite{CoGaKi}), we give the folowing.
\begin{Def}\label{Q-spectrum}\rm For a given operator $T\in\mathcal{B^{\rm r}(V)}$, the set
$$
{\bf s}igma_{\mathbb H}(T):=\{{\bf q}\in{\mathbb H}; T^2-2({\mathbb R}e{\bf q})T+{\mathcal{V}}ert{\bf q}{\mathcal{V}}ert^2)\,\,
{\rm not}\,\,{\rm invertible}\}
$$
is called the {\it quaternionic spectrum} (or simply the $Q$-{\it spectrum}) of $T$.
The complement $\rho_{\mathbb H}(T)={\mathbb H}{\bf s}etminus{\bf s}igma_{\mathbb H}(T)$ is called
the {\it quaternionic resolvent} (or simply the $Q$-{\it resolvent}) of $T$.
\end{Def}
Note that, if ${\bf q}\in{\bf s}igma_{\mathbb H}(T)$), then
$\{{\bf s}\in{\mathbb H};{\bf s}igma({\bf s})={\bf s}igma({\bf q})\}{\bf s}ubset{\bf s}igma_{\mathbb H}(T)$.
Assuming that ${\mathcal{V}}$ is a Banach ${\mathbb H}$-space, then
$\mathcal{B^{\rm r}(V)}$ is a unital real Banach ${\mathbb H}$-algebra
(that is, a Banach algebra which also a Banach ${\mathbb H}$-space), via the
algebraic operations $({\bf q} T)(x)={\bf q} T(x)$, and $(T{\bf q})(x)=T({\bf q} x)$ for all
${\bf q}\in{\mathbb H}$ and $x\in{\mathcal{V}}$. Hence the complexification $\mathcal{B^{\rm r}(V)}_{\mathbb C}$ is, in particular, a unital complex Banach algebra. Also note that the complex numbers,
regarded as elements of $\mathcal{B^{\rm r}(V)}_{\mathbb C}$, commute with
the elements of $\mathcal{B^{\rm r}(V)}$. For this reason, for
each $T\in\mathcal{B^{\rm r}(V)}$ we have the resolvent set
$$
\rho_{\mathbb C}(T)=\{\lambda\in{\mathbb C};(T^2-2({\mathbb R}e\lambda)T+\vert\lambda\vert^2)^{-1}\in\mathcal{B^{\rm r}(V)}\}=
$$
$$
\{\lambda\in{\mathbb C};(\lambda-T_{\mathbb C})^{-1}\in \mathcal{B^{\rm r}(V}_{\mathbb C})\}=\rho(T_{\mathbb C}),
$$
and the associated spectrum ${\bf s}igma_{\mathbb C}(T)={\bf s}igma(T_{\mathbb C})$.
Clearly, there exists a strong connexion between ${\bf s}igma_{\mathbb H}(T)$ and
${\bf s}igma_{\mathbb C}(T)$. In fact, the set ${\bf s}igma_{\mathbb C}(T)$ looks like a ''complex border`` of the set ${\bf s}igma_{\mathbb H}(T)$. Specifically, we can prove the following.
\begin{Lem}\label{spec_eg0} For every $T\in\mathcal{B^{\rm r}(V)}$ we have the
equalities
\begin{equation}\label{spec_eg}
{\bf s}igma_{\mathbb H}(T)=\{{\bf q}\in{\mathbb H};{\bf s}igma_{\mathbb C}(T)\cap{\bf s}igma({\bf q})\neq\emptyset\}.
\end{equation}
and
\begin{equation}\label{spec_eg1}
{\bf s}igma_{\mathbb C}(T)=\{\lambda\in{\bf s}igma({\bf q});{\bf q}\in{\bf s}igma_{\mathbb H}(T)\}.
\end{equation}
\end{Lem}
{\it Proof.} Let us prove (\ref{spec_eg}). If ${\bf q}\in {\bf s}igma_{\mathbb H}(T)$, and so the $T^2-2({\mathbb R}e{\bf q})T+{\mathcal{V}}ert{\bf q}{\mathcal{V}}ert^2$ is not invertible, choosing $\lambda\in\{{\mathbb R}e{\bf q}\pm i{\mathcal{V}}ert\Im{\bf q}{\mathcal{V}}ert\}={\bf s}igma({\bf q})$, we clearly have $T^2-2({\mathbb R}e\lambda)T+\vert\lambda\vert^2$ not invertible, implying $\lambda\in{\bf s}igma_{\mathbb C}(T)\cap{\bf s}igma({\bf q})\neq\emptyset$.
Conversely, if for some ${\bf q}\in{\mathbb H}$ there exists $\lambda\in{\bf s}igma_{\mathbb C}(T)\cap{\bf s}igma({\bf q})$, and so $T^2-2({\mathbb R}e\lambda)T+\vert\lambda\vert^2=T^2-2({\mathbb R}e{\bf q})T+{\mathcal{V}}ert{\bf q}{\mathcal{V}}ert^2$ is not invertible, implying ${\bf q}\in{\bf s}igma_{\mathbb H}(T)$.
We now prove (\ref{spec_eg1}). Let $\lambda\in{\bf s}igma_{\mathbb C}(T)$, so the operator $T^2-2({\mathbb R}e\lambda)T+\vert\lambda\vert^2$ is not invertible. Setting
${\bf q}={\mathbb R}e(\lambda)+{\mathcal{V}}ert\Im\lambda{\mathcal{V}}ert\kappa$, with $\kappa\in\mathbb{S}$, we have
$\lambda\in{\bf s}igma({\bf q})$. Moreover, $T^2+2({\mathbb R}e{\bf q})T+{\mathcal{V}}ert{\bf q}{\mathcal{V}}ert^2$ is not
invertible, and so ${\bf q}\in{\bf s}igma_ {\mathbb H}(T)$.
Conversely, if $\lambda\in{\bf s}igma({\bf q})$ for some ${\bf q}\in{\bf s}igma_{\mathbb H}(T)$, then
$\lambda\in\{{\mathbb R}e{\bf q}\pm i{\mathcal{V}}ert\Im({\bf q}){\mathcal{V}}ert\}$, showing that
$T^2-2{\mathbb R}e(\lambda)T+\vert\lambda\vert^2=T^2+2({\mathbb R}e{\bf q})T+{\mathcal{V}}ert{\bf q}{\mathcal{V}}ert^2$ is
not invertible.
{\bf Remark} As expected, the set ${\bf s}igma_{\mathbb H}(T)$ is nonempty and bounded, which follows easily from Lemma \ref{spec_eg0}. It is also compact, as a consequence of Definition \ref{Q-spectrum}, because the set of invertible elements in $\mathcal{B^{\rm r}(V)}$ is open.
We recall that a subset $\Omega{\bf s}ubset{\mathbb H}$ is said to be {\it spectrally saturated}
(see \cite{Vas4},\cite{Vas5}) if whenever ${\bf s}igma({\bf h})={\bf s}igma({\bf q})$ for some ${\bf h}\in{\mathbb H}$ and ${\bf q}\in\Omega$, we also have ${\bf h}\in\Omega$. As noticed in \cite{Vas4} and \cite{Vas5}, this concept coincides with that of {\it axially symmetric set}, introduced in \cite{CoSaSt}.
Note that the subset ${\bf s}igma_{\mathbb H}(T)$ spectrally saturated.
{\bf s}ubsection{Analytic Functional Calculus}
If ${\mathcal{V}}$ is a Banach ${\mathbb H}$-space, because
$\mathcal{B^{\rm r}({\mathcal{V}})}$
is real Banach space, each operator $T\in\mathcal{B^{\rm r}({\mathcal{V}})}$ has a complex spectrum ${\bf s}igma_{\mathbb C}(T)$. Therefore, applying the
corresponding result for real operators, we may construct
an analytic functional calculus using the classical Riesz-Dunford functional calculus, in a slightly generalized form. In this case, our basic complex algebra is $\mathcal{B}^{\rm r}(\mathcal{V})_{\mathbb C}$, endowed with the conjugation
$\mathcal{B}^{\rm r}(\mathcal{V})_{\mathbb C}\ni S\mapsto S^\flat\in\mathcal{B}^{\rm r}(\mathcal{V})_{\mathbb C}$.
\begin{Thm}\label{afcro2} Let $U{\bf s}ubset{\mathbb C}$ be open and conjugate symmetric. If $F:U\mapsto\mathcal{B}^{\rm r}(\mathcal{V}_{\mathbb C})$ is analytic and $F({\bf z}eta)^\flat=F(\bar{{\bf z}eta})$ for all ${\bf z}eta\in U$, then $F(T_{\mathbb C})^\flat=F(T_{\mathbb C})$ for all $T\in\mathcal{B}^{\rm r}(\mathcal{V})$ with
${\bf s}igma_{\mathbb C}(T){\bf s}ubset U$.
\end{Thm}
Both the statement and the proof of Theorem \ref{afcro2} are similar to
those of Theorem \ref{afcro1}, and will be omitted.
As in the real case, we may identify the algebra
$\mathcal{B}^{\rm r}(\mathcal{V})$ with a subalgebra of
$\mathcal{B}^{\rm r}(\mathcal{V})_{\mathbb C}$. In ths case, when $F\in \mathcal{O}_s(U,\mathcal{B}^{\rm r}(\mathcal{V})_{\mathbb C})=
\{F\in{\mathcal O}(U,\mathcal{B}^{\rm r}({\mathcal{V}})_{\mathbb C});F(\bar{{\bf z}eta})=F({\bf z}eta)^{\flat}\,\, \forall {\bf z}eta\in U\}$ (see also Remark \ref{spec_lm}), we can write, via the previous Theorem,
$$
F(T)=\frac{1}{2\pi i}\int_\Gamma F({\bf z}eta)({\bf z}eta-T)^{-1}
d{\bf z}eta\in\mathcal{B}^{\rm r}(\mathcal{V}),
$$
for a suitable choice of $\Gamma$.
The next result provides an {\it analytic functional calculus} for
operators from the real algebra $\mathcal{B}^{\rm r}(\mathcal{V})$.
\begin{Thm}\label{H_afc} Let ${\mathcal{V}}$ be a Banach ${\mathbb H}$-space, let $U{\bf s}ubset{\mathbb C}$ be a conjugate symmetric open set, and let $T\in\mathcal{B}^{\rm r}(\mathcal{V})$, with
${\bf s}igma_{\mathbb C}(T){\bf s}ubset U$. Then the map
$$
{\mathcal O}_s(U,\mathcal{B}^{\rm r}(\mathcal{V})_{\mathbb C})\ni F\mapsto F(T)\in\mathcal{B}^{\rm r}(\mathcal{V})
$$
is ${\mathbb R}$-linear, and the map
$$
{\mathcal O}_s(U)\ni f\mapsto f(T)\in\mathcal{B}^{\rm r}(\mathcal{V})
$$
is a unital real algebra morphism.
Moreover, the following properties are true:
$(1)$ For all $F\in\mathcal{O}_s(U,\mathcal{B}^{\rm r}(\mathcal{V})_{\mathbb C}),\, f\in{\mathcal O}_s(U)$, we have $(Ff)(T)=F(T)f(T)$.
$(2)$ For every polynomial $P({\bf z}eta)={\bf s}um_{n=0}^m A_n{\bf z}eta^n,\,{\bf z}eta\in{\mathbb C}$, with $A_n\in\mathcal{B}^{\rm r}(\mathcal{V})$ for all $n=0,1,\ldots,m$, we have $P(T)={\bf s}um_{n=0}^m A_n T^n\in\mathcal{B}^{\rm r}(\mathcal{V})$.
\end{Thm}
The proof of this result is similar to that of Theorem \ref{R_afc} and will be omitted.
\begin{Rem}\label{leftmult}\rm The algebra ${\mathbb H}$ is, in particular, a Banach
${\mathbb H}$-space. As already noticed, the left multiplications $L_{\bf q},\,{\bf q}\in{\mathbb H},$ are elements of
$\mathcal{B}^{\rm r}({\mathbb H})$. In fact, the map ${\mathbb H}\ni{\bf q}\mapsto L_{\bf q}\in \mathcal{B}^{\rm r}({\mathbb H})$ is a injective morphism of real
algebras allowing the identification of ${\mathbb H}$ with a subalgebra
of $\mathcal{B}^{\rm r}({\mathbb H})$.
\end{Rem}
Let $\Omega{\bf s}ubset{\mathbb H}$ be a spectrally saturated open set, and
let $U=\mathfrak{S}(\Omega):=\{\lambda\in{\mathbb C}, \exists {\bf q}\in\Omega, \lambda\in{\bf s}igma({\bf q})\}$, which is open and conjugate symmetric (see \cite{Vas5}). Denotig by $f_{\mathbb H}$ the function $\Omega\ni{\bf q}\mapsto
f({\bf q}),{\bf q}\in\Omega$, for every $f\in\mathcal{O}_s(U)$, we set
$$
\mathcal{R}(\Omega):=\{f_{\mathbb H}; f\in\mathcal{O}_s(U)\},
$$
which is a commutative real algebra. Defining the function $F_{\mathbb H}$ in a similar way for each $F\in\mathcal{O}_s(U,{\mathbb M})$, we set
$$
\mathcal{R}(\Omega,{\mathbb H}):=\{F_{\mathbb H}; F\in\mathcal{O}_s(U,{\mathbb M})\},
$$
which, according to the next theorem, is a right $\mathcal{R}(\Omega)$-module.
The next result is an analytic functional calculus for quaternions (see \cite{Vas5}, Theorem 5), obtained as a particular case of Theorem \ref{H_afc} (see also its predecessor in \cite{CoSaSt}).
\begin{Thm}\label{H_afc0} Let $\Omega{\bf s}ubset{\mathbb H}$ be a spectrally
saturated open set, and let $U=\mathfrak{S}(\Omega)$.
The space $\mathcal{R}(\Omega)$ is a unital commutative ${\mathbb R}$-algebra, the space $\mathcal{R}(\Omega,{\mathbb H})$ is a right
$\mathcal{R}(\Omega)$-module, the map
$$
{\mathcal O}_s(U,{\mathbb M})\ni F\mapsto F_{\mathbb H}\in\mathcal{R}(\Omega,{\mathbb H})
$$
is a right module isomorphism, and its restriction
$$
{\mathcal O}_s(U)\ni f\mapsto f_{\mathbb H}\in\mathcal{R}(\Omega)
$$
is an ${\mathbb R}$-algebra isomorphism.
Moreover, for every polynomial\,\,$P({\bf z}eta)={\bf s}um_{n=0}^m a_n{\bf z}eta^n,\,{\bf z}eta\in{\mathbb C}$, with $a_n\in{\mathbb H}$ for all $n=0,1,\ldots,m$, we have $P_{\mathbb H}(q)={\bf s}um_{n=0}^m a_n q^n\in{\mathbb H}$ for all $q\in{\mathbb H}$.
\end{Thm}
Most of the assertions of Theorem \ref{H_afc0} can be obtained directly
from Theorem \ref{H_afc}. The injectivity of the map
${\mathcal O}_s(U)\ni f\mapsto f_{\mathbb H}\in\mathcal{R}(\Omega)$, as well as an
alternative complete proof, can be obtained as in the proof of Theorem 5 from \cite{Vas5}.
\begin{Rem}\label{gen_afc}\rm That Theorems \ref{afcro2} and \ref{H_afc}
have practically the same proof as Theorems \ref{afcro1} and \ref{R_afc}
(respectively) is due to the fact that all of them can be obtained as particular
cases of more general results. Indeed, considering a unital real Banach
algebra ${\mathcal{A}}$, and its complexification ${\mathcal{A}}_{\mathbb C}$, identifying ${\mathcal{A}}$ with
a real subalgebra of ${\mathcal{A}}_{\mathbb C}$, for a function $F\in\mathcal{O}_s(U,A_{\mathbb C})$,
where $U{\bf s}ubset{\mathbb C}$ is open and conjugate symmetric,
the element $F(b)\in{\mathcal{A}}$ for each $b\in{\mathcal{A}}$ with ${\bf s}igma_{\mathbb C}(b){\bf s}ubset U$.
The assertion follows as in the proof of Theorem \ref{afcro1}. The other
results also have their counterparts. We omit the details.
\end{Rem}
\begin{Rem}\label{twofc}\rm The space $\mathcal{R}(\Omega,{\mathbb H})$ can be independently defined, and it consists of the set of all
${\mathbb H}$-valued functions, which are {\it slice regular} in the
sense of \cite{CoSaSt}, Definition 4.1.1. They are used in
\cite{CoSaSt} to define a quaternionic functional calculus
for quaternionic linear operators (see also
\cite{CoGaKi}). Roughly speaking, given a quaternionic linear
operator, each regular quaternionic-valued function defined in a neighborhood $\Omega$ of its quaternionic spectrum is associated with
another quaternionic linear operator, replacing formally
the quaternionic variable with that operator. This constraction
is largely explained in the fourth chapter of \cite{CoSaSt}.
Our Theorem \ref{H_afc} constructs an analytic functional calculus with functions from ${\mathcal O}_s(U,\mathcal{B}^{\rm r}(\mathcal{V})_{\mathbb C})$, where $U$ is a a neighborhood of the complex spectrum of a given quaternionic linear operator, leading to another quaternionic linear operator, replacing formally the complex variable with that operator. We can show that those functional calculi are equivalent. This is a consequence of the
fact that the class of regular quaternionic-valued function used by the construction in \cite{CoSaSt} is isomorphic to the class of analytic functions used in our Theorem \ref{H_afc0}. The advantage of our approach is its simplicity and a stronger connection with the classical approach, using spectra defined in the complex plane, and Cauchy type kernels partially commutative.
Let us give a direct argument concerning the equivalence of those analytic functional calculi. For an operator $T\in\mathcal{B}^{\rm r}(\mathcal{V})$, the so-called {\it right $S$-resolvent} is defined via the formula
\begin{equation}\label{kqfc}
S_R^{-1}({\bf s},T)=-(T-{\bf s}^*)(T^2-2{\mathbb R}e({\bf s})T+{\mathcal{V}}ert{\bf s}{\mathcal{V}}ert)^{-1},\,\,{\bf s}\in
\rho_{\mathbb H}(T)
\end{equation}
(see \cite{CoSaSt}, formula (4.27)). Fixing an element $\kappa\in\mathbb{S}$, and a spectrally saturated open set $\Omega{\bf s}ubset{\mathbb H}$, for $\Phi\in\mathcal{R}(\Omega,{\mathbb H})$ one sets
\begin{equation}\label{qfc}
\Phi(T)=\frac{1}{2\pi}\int_{\partial({\mathbb S}igma_\kappa)}\Phi({\bf s})d{\bf s}_\kappa S_R^{-1}({\bf s},T),
\end{equation}
where ${\mathbb S}igma{\bf s}ubset\Omega$ is a spectrally saturated open set containing
${\bf s}igma_{\mathbb H}(T)$, such that ${\mathbb S}igma_\kappa=\{u+v\kappa\in{\mathbb S}igma;u,v\in{\mathbb R}\}$
is a subset whose boundary $\partial({\mathbb S}igma_\kappa)$ consists of a finite family of closed curves, piecewise smooth, positively oriented, and $d{\bf s}_\kappa=-\kappa du\wedge dv$. Formula (\ref{qfc}) is a (right) quaternionic functional calculus, as defined in \cite{CoSaSt}, Section 4.10.
Because the space $\mathcal{V}_{\mathbb C}$ is also an ${\mathbb H}$-space, we may extend these formulas to the
operator $T_{\mathbb C}\in\mathcal{B}^{\rm r}(\mathcal{V}_{\mathbb C})$, extending the operator $T$ to $T_{\mathbb C}$, and replacing $T$ by $T_{\mathbb C}$ in formulas (\ref{kqfc}) and (\ref{qfc}). For the function
$\Phi\in\mathcal{R}(\Omega,{\mathbb H})$ there exists a function $F\in{\mathcal O}_s(U,\mathcal{B}^{\rm r}(\mathcal{V}_{\mathbb C}))$ such that $F_{\mathbb H}=\Phi$. Denoting by
$\Gamma_\kappa$ the boundary of a Cauchy domain in ${\mathbb C}$ containing the compact set
$\cup\{{\bf s}igma({\bf s});{\bf s}\in\overline{{\mathbb S}igma_\kappa}\}$, we can write
$$
\Phi(T_{\mathbb C})=\frac{1}{2\pi}\int_{\partial({\mathbb S}igma_\kappa)}\left(\frac{1}{2\pi i}\int_{\Gamma_\kappa}F({\bf z}eta)({\bf z}eta-{\bf s})^{-1}d{\bf z}eta\right)
d{\bf s}_\kappa S_R^{-1}({\bf s},T_{\mathbb C})=
$$
$$
\frac{1}{2\pi i}\int_{\Gamma_\kappa}F({\bf z}eta)\left(\frac{1}{2\pi}\int_{\partial({\mathbb S}igma_\kappa)}({\bf z}eta-{\bf s})^{-1}d{\bf s}_\kappa S_R^{-1}({\bf s},T_{\mathbb C})\right)d{\bf z}eta.
$$
It follows from the complex linearity of $S_R^{-1}({\bf s},T_{\mathbb C})$, and from
formula (4.49) in \cite{CoSaSt}, that
$$
({\bf z}eta-{\bf s})S_R^{-1}({\bf s},T_{\mathbb C})=S_R^{-1}({\bf s},T_{\mathbb C})({\bf z}eta-T_{\mathbb C})-1,
$$
whence
$$
({\bf z}eta-{\bf s})^{-1}S_R^{-1}({\bf s},T_{\mathbb C})=S_R^{-1}({\bf s},T_{\mathbb C})({\bf z}eta-T_{\mathbb C})^{-1}+
({\bf z}eta-{\bf s})^{-1}({\bf z}eta-T_{\mathbb C})^{-1},
$$
and therefore,
$$
\frac{1}{2\pi}\int_{\partial({\mathbb S}igma_\kappa)}({\bf z}eta-{\bf s})^{-1}d{\bf s}_\kappa S_R^{-1}({\bf s},T_{\mathbb C})=
\frac{1}{2\pi}\int_{\partial({\mathbb S}igma_\kappa)}d{\bf s}_\kappa S_R^{-1}({\bf s},T_{\mathbb C})
({\bf z}eta-T_{\mathbb C})^{-1}+
$$
$$
\frac{1}{2\pi}\int_{\partial({\mathbb S}igma_\kappa)}({\bf z}eta-{\bf s})^{-1}d{\bf s}_\kappa
({\bf z}eta-T_{\mathbb C})^{-1}= ({\bf z}eta-T_{\mathbb C})^{-1},
$$
because
$$
\frac{1}{2\pi}\int_{\partial({\mathbb S}igma_\kappa)}d{\bf s}_\kappa S_R^{-1}({\bf s},T_{\mathbb C})=1\,\,\,{\rm and}
\,\,\,\frac{1}{2\pi}\int_{\partial({\mathbb S}igma_\kappa)}({\bf z}eta-{\bf s})^{-1}d{\bf s}_\kappa =0,
$$
as in Theorem 4.8.11 from \cite{CoSaSt}, since the ${\mathbb M}$-valued function ${\bf s}\mapsto({\bf z}eta-{\bf s})^{-1}$ is analytic in a neighborhood of the
set $\overline{{\mathbb S}igma_\kappa}{\bf s}ubset {\mathbb C}_\kappa$ for each ${\bf z}eta\in\Gamma_\kappa$, respectively. Therefore $\Phi(T_{\mathbb C})=\Phi(T)_{\mathbb C}=F(T_{\mathbb C})=F(T)_{\mathbb C}$, implying $\Phi(T)=F(T)$.
\end{Rem}
{\bf s}ection{Some Examples}
\begin{Exa}\label{ex2}\rm One of the simplest Banach ${\mathbb H}$-space
is the space ${\mathbb H}$ itself. As already noticed (see Remark \ref{leftmult}), taking ${\mathcal{V}}={\mathbb H}$, so ${\mathcal{V}}_{\mathbb C}={\mathbb M}$, and fixing an element ${\bf q}\in{\mathbb H}$, we may consider the operator $L_{\bf q}\in\mathcal{B}^{\rm r}({\mathbb H})$, whose complex spectrum is given by
${\bf s}igma_{\mathbb C}(L_{\bf q})={\bf s}igma({\bf q})=\{{\mathbb R}e{\bf q}\pm i{\mathcal{V}}ert\Im{\bf q}{\mathcal{V}}ert\}$. If
$U{\bf s}ubset{\mathbb C}$ is conjugate symmetric open set containing
${\bf s}igma_{\mathbb C}(L_{\bf q})$, and $F\in\mathcal{O}_s(U,{\mathbb M})$, then we have
\begin{equation}\label{GFC2}
F({L_{\bf q}})=F(s_+({\bf q}))\iota_+(\mathfrak{s}_{\tilde{\bf q}})+F(s_-({\bf q}))\iota_-(\mathfrak{s}_{\tilde{\bf q}}) \in{\mathbb M},
\end{equation}
where $s_\pm({\bf q})={\mathbb R}e{\bf q}\pm i{\mathcal{V}}ert\Im{\bf q}{\mathcal{V}}ert$,
$\tilde{\bf q}=\Im\bf q,\,\mathfrak{s}_{\tilde{\bf q}}=\tilde{\bf q}{\mathcal{V}}ert\tilde{\bf q}{\mathcal{V}}ert^{-1 }$, and
$\iota_\pm(\mathfrak{s}_{\tilde{\bf q}})=2^{-1}
(1\mp i\mathfrak{s}_{\tilde{\bf q}})$ (see \cite{Vas5}, Remark 3).
\end{Exa}
\begin{Exa}\label{spec_lm}\rm Let ${\mathfrak{X}}$ be a topological compact space, and let $C({\mathfrak{X}},{\mathbb M})$ be the space of ${\mathbb M}$-valued continuous
functions on ${\mathfrak{X}}$. Then $C({\mathfrak{X}},{\mathbb H})$ is the real subspace of
$C({\mathfrak{X}},{\mathbb M})$ consisting of ${\mathbb H}$-valued functions, which is also
a Banach ${\mathbb H}$-space with respect to the operations $({\bf q} F)(x)=
{\bf q} F(x)$ and $(F{\bf q})(x)=F(x){\bf q}$ for all $F\in C({\mathfrak{X}},{\mathbb H})$ and $x\in{\mathfrak{X}}$. Moreover, $C({\mathfrak{X}},{\mathbb H})_{\mathbb C}=C({\mathfrak{X}},{\mathbb H}_{\mathbb C})=C({\mathfrak{X}},{\mathbb M})$.
We fix a function $\Theta\in C({\mathfrak{X}},{\mathbb H})$ and define the operator
$T\in\mathcal{B}(C({\mathfrak{X}},{\mathbb H}))$ by the relation $(TF)(x)=\Theta(x)F(x)$ for all $F\in C({\mathfrak{X}},{\mathbb H})$ and $x\in{\mathfrak{X}}$. Note that
$(T(F{\bf q}))(x)=\Theta(x)F(x){\bf q}=((TF){\bf q})(x)$ for all
$F\in C({\mathfrak{X}},{\mathbb H}),{\bf q}\in{\mathbb H}$, and $x\in{\mathfrak{X}}$. In othe words,
$T\in\mathcal{B}^{\rm r}(C({\mathfrak{X}},{\mathbb H}))$. Note also that the operator $T$
is invertible if and only if the function $\Theta$ has no zero in ${\mathfrak{X}}$.
Let us compute the $Q$-spectrum of $T$. According to Definition \ref{Q-spectrum}, we have
$$
\rho_{\mathbb H}(T)=\{{\bf q}\in{\mathbb H}; (T^2-2{\mathbb R}e {\bf q}\,T+{\mathcal{V}}ert {\bf q}{\mathcal{V}}ert^2)^{-1}\in\mathcal{B}^{\rm r}(C({\mathfrak{X}},{\mathbb H}))\}.
$$
Consequently, ${\bf q}\in{\bf s}igma_{\mathbb H}(T)$ if and only if zero is in the range of the function
$$
\tau({\bf q},x):=\Theta(x)^2-2{\mathbb R}e {\bf q}\,\Theta(x)+{\mathcal{V}}ert {\bf q}{\mathcal{V}}ert^2
,\, x\in\mathfrak{X}.
$$
Similarly,
$$
\rho_{\mathbb C}(T)=\{\lambda\in{\mathbb C}; (T^2-2{\mathbb R}e \lambda\,T+{\mathcal{V}}ert \lambda{\mathcal{V}}ert^2)^{-1}\in\mathcal{B}^{\rm r}(C({\mathfrak{X}},{\mathbb H}))\},
$$
and so $\lambda\in {\bf s}igma_{\mathbb C}(T)$ if and only if zero is in the range of the function
$$
\tau(\lambda,x):=\Theta(x)^2-2{\mathbb R}e \lambda\,\Theta(x)+\vert
\lambda\vert^2,\,x\in\mathfrak{X}.
$$
Looking for solutions $u+iv,u,v\in{\mathbb R}$, of the equation
$(u-\Theta(x))^2+v^2=0$, a direct calculation shows that
$u={\mathbb R}e\Theta(x)$ and $v=\pm{\mathcal{V}}ert\Im\Theta(x){\mathcal{V}}ert$. Hence
$$
{\bf s}igma_{\mathbb C}(T)=\{{\mathbb R}e\Theta(x)\pm i{\mathcal{V}}ert\Im\Theta(x){\mathcal{V}}ert;x\in\mathfrak{X}\}=\cup_{x\in\mathfrak{X}}{\bf s}igma(\Theta(x)).
$$
Of course, for every open conjugate symmetric subset
$U{\bf s}ubset{\mathbb C}$ containing ${\bf s}igma_{\mathbb C}(T)$, and for every function
$\Phi\in\mathcal{O}_c(U,\mathcal{B}(C({\mathfrak{X}},{\mathbb M})))$, we may
construct the operator $\Phi(T)\in\mathcal{B}^{\rm r}(C({\mathfrak{X}},{\mathbb H}))$, using
Theorem \ref{H_afc}.
\end{Exa}
{\bf s}ection{Quaternionic Joint Spectrum of Paires}
\label{QJSP}
In many applications, it is more convenient to work
with matrix quaternions rather than with abstract quaternions.
Specifically, one considers the injective unital algebra morphism
$$
{\mathbb H}\ni x_1+y_1{\bf j}+x_2{\bf k}+y_2{\bf l}\mapsto
\left(\begin{array}{cc} x_1+iy_1 & x_2+iy_2 \\ -x_2+iy_2 & x_1-iy_1 \end{array}\right)\in{\mathbb M}_2,
$$
with $x_1,y_1,x_2,y_2\in{\mathbb R},$ where ${\mathbb M}_2$ is the complex algebra of $2\times2$-matrix, whose
image, denoted by ${\mathbb H}_2$ is the real algebra of matrix quaternions. The elements of ${\mathbb H}_2$ can be also written as
matrices of the form
$$
Q({\bf z})=\left(\begin{array}{cc} z_1 & z_2 \\ -\bar{z}_2 & \bar{z_1} \end{array}\right),\,\,{\bf z}=(z_1,z_2)\in{\mathbb C}^2.
$$
A strong connection between the spectral theory of pairs of
commuting operators in a complex Hilbert space and the algebra of quaternions has been firstly noticed in \cite{Vas1}. Another
connection will be presented in this section.
If ${\mathcal{V}}$ is an arbitrary vector space, we denote by ${\mathcal{V}}^2$ the
Cartesian product ${\mathcal{V}}\times{\mathcal{V}}$.
Let $\mathcal{V}$ be a real Banach space, and let
${\bf T}=(T_1,T_2)\in\mathcal{B(V)}^2$ be a pair of commuting
operators. The extended pair
${\bf T}_{\mathbb C}=(T_{1{\mathbb C}},T_{2{\mathbb C}})\in\mathcal{B(V_{\mathbb C})}^2$ also consists of commuting operators. For simplicity, we set
$$
Q({\bf T}_{\mathbb C}):=\left(\begin{array}{cc} T_{1{\mathbb C}} & T_{2{\mathbb C}} \\ -T_{2{\mathbb C}} & T_{1{\mathbb C}}
\end{array}\right)
$$
which acts on the complex Banach space $\mathcal{V}_{\mathbb C}^2$.
We now define the quaternionic resolvent set and spectrum for the case of a pair of operators, inspired by the previous discussion
concerning a single operator.
\begin{Def}\label{Q-jspectrum}\rm Let $\mathcal{V}$ be a real Banach space. For a given pair ${\bf T}=(T_1,T_2)\in\mathcal{B(V)}^2$ of commuting operators, the set of those
$Q({\bf z})\in{\mathbb H}_2,\,{\bf z}=(z_1,z_2)\in{\mathbb C}^2$, such that the operator
$$T_1^2+T_2^2-2{\mathbb R}e{z_1}T_1-2{\mathbb R}e{z_2}T_2+\vert z_1\vert^2+
\vert z_2\vert^2
$$
is invertible in $\mathcal{B(V)}$
is said to be the {\it quaternionic joint resolvent} (or simply the $Q$-{\it joint resolvent}) of ${\bf T}$, and is denoted by
$\rho_{\mathbb H}({\bf T})$.
The complement ${\bf s}igma_{\mathbb H}({\bf T})={\mathbb H}_2{\bf s}etminus\rho_{\mathbb H}({\bf T})$ is called
the {\it quaternionic joint spectrum} (or simply the $Q$-{\it joint spectrum}) of ${\bf T}$.
\end{Def}
For every pair ${\bf T}_{\mathbb C}=(T_{1{\mathbb C}},T_{2{\mathbb C}})\in\mathcal{B(V_{\mathbb C})}^2$ we put
${\bf T}_{\mathbb C}^c=(T_{1{\mathbb C}},-T_{2{\mathbb C}})\in\mathcal{B(V_{\mathbb C})}^2$, and for every pair
${\bf z}=(z_1,z_2)\in{\mathbb C}^2$ we put ${\bf z}^c=(\bar{z}_1,-z_2)\in{\mathbb C}^2$
\begin{Lem}\label{Q-jsp} A matrix quaternion $Q({\bf z})$\,$({\bf z}\in{\mathbb C}^2)$ is in the set $\rho_{\mathbb H}({\bf T})$
if and only if the operators $Q({\bf T}_{\mathbb C})-Q({\bf z}),\,Q({\bf T}_{\mathbb C}^c)-Q({\bf z}^c)$ are invertible in $\mathcal{B}(\mathcal{V}_{\mathbb C}^2)$.
\end{Lem}
{\it Proof}\, The assertion follows from the equalities
$$
\left(\begin{array}{cc} T_{1{\mathbb C}}-z_1 & T_{2{\mathbb C}}-z_2 \\
-T_{2{\mathbb C}}+\bar{z}_2 & T_{1{\mathbb C}}-\bar{z}_1\end{array}\right)
\left(\begin{array}{cc} T_{1{\mathbb C}}-\bar{z}_1 & -T_{2{\mathbb C}}+z_2 \\
T_{2{\mathbb C}}-\bar{z}_2 & T_{1{\mathbb C}}-z_1\end{array}\right)=
$$
$$
\left(\begin{array}{cc} T_{1{\mathbb C}}-\bar{z}_1 & -T_{2{\mathbb C}}+z_2 \\
T_{2{\mathbb C}}-\bar{z}_2 & T_{1{\mathbb C}}-z_1\end{array}\right)
\left(\begin{array}{cc} T_{1{\mathbb C}}-z_1 & T_{2{\mathbb C}}-z_2 \\
-T_{2{\mathbb C}}+\bar{z}_2 & T_{1{\mathbb C}}-\bar{z}_1\end{array}\right)=
$$
$$
[(T_{1{\mathbb C}}-z_1)(T_{1{\mathbb C}}-\bar{z}_1)+
(T_{2{\mathbb C}}-z_2)(T_{2{\mathbb C}}-\bar{z}_2)]{\bf I}.
$$
for all ${\bf z}=(z_1,z_2)\in{\mathbb C}^2$, where $\bf I$ is the identity.
Consequently, the operators $Q({\bf T}_{\mathbb C})-Q({\bf z}),\,Q({\bf T}_{\mathbb C}^c)-Q({\bf z}^c)$ are invertible in
$\mathcal{B}({\mathcal V}_{\mathbb C}^2)$ if and only if the operator
$(T_{1{\mathbb C}}-z_1)(T_{1{\mathbb C}}-\bar{z}_1)+
(T_{2{\mathbb C}}-z_2)(T_{2{\mathbb C}}-\bar{z}_2)$
is invertible in $\mathcal{B}(\mathcal{V}_{\mathbb C})$. Because we have
$$
T_{1{\mathbb C}}^2+T_{2{\mathbb C}}^2-2{\mathbb R}e{z_1}T_{1{\mathbb C}}-2{\mathbb R}e{z_2}T_{2{\mathbb C}}+\vert z_1\vert^2+\vert z_2\vert^2=
$$
$$
[T_1^2+T_1^2-2{\mathbb R}e{z_1}T_1-2{\mathbb R}e{z_2}T_2+\vert z_1\vert^2+
\vert z_2\vert^2]_{\mathbb C},
$$
the operators $Q({\bf T}_{\mathbb C})-Q({\bf z}),\,Q({\bf T}_{\mathbb C}^c)-Q({\bf z}^c)$ are invertible in
$\mathcal{B}({\mathcal V}_{\mathbb C}^2)$ if and only if the operator
$T_1^2+T_1^2-2{\mathbb R}e{z_1}T_1-2{\mathbb R}e{z_2}T_2+\vert z_1\vert^2+
\vert z_2\vert^2$
is invertible in $\mathcal{B(V)}$.
Lemma \ref{Q-jsp} shows that we have the
property $Q({\bf z})\in{\bf s}igma_{\mathbb H}({\bf T})$ if and only if $Q(z^c)\in{\bf s}igma_{\mathbb H}({\bf T}^c)$. Putting
$$
{\bf s}igma_{{\mathbb C}^2}({\bf T}):=\{{\bf z}\in{\mathbb C}^2;Q({\bf z})\in{\bf s}igma_{\mathbb H}({\bf T})\},
$$
the set ${\bf s}igma_{{\mathbb C}^2}({\bf T})$ has a similar property, specifically
$\bf z\in{\bf s}igma_{{\mathbb C}^2}({\bf T})$ if and only if $\bf z^c\in{\bf s}igma_{{\mathbb C}^2}({\bf T}^c)$. As in the quaternionic case, the set ${\bf s}igma_{{\mathbb C}^2}({\bf T})$
looks like a ''complex border`` of the set ${\bf s}igma_{\mathbb H}({\bf T})$.
\begin{Rem}\rm For the extended pair
${\bf T}_{\mathbb C}=(T_{1{\mathbb C}},T_{2{\mathbb C}})\in {B(V_{\mathbb C})}^2$ of the commuting pair ${\bf T}=(T_1,T_2)\in \mathcal{B(V)}$ there is an
interesting connexion with the {\it joint spectral theory} of
J. L. Taylor (see \cite{Tay,Tay2}; see also \cite{Vas3}). Namely, if the operator
$T_{1{\mathbb C}}^2+T_{2{\mathbb C}}^2-2{\mathbb R}e{z_1}T_{1{\mathbb C}}-2{\mathbb R}e{z_2}T_{2{\mathbb C}}+\vert z_1\vert^2+\vert z_2\vert^2$ is invertible, then the point
${\bf z}=(z_1,z_2)$ belongs to the joint resolvent of ${\bf T}_{\mathbb C}$.
Indeed, setting
$$
R_j({\bf T}_{\mathbb C},{\bf z})=(T_{j{\mathbb C}}-\bar{z}_j)(T_{1{\mathbb C}}^2+T_{2{\mathbb C}}^2-2{\mathbb R}e{z_1}T_{1{\mathbb C}}-2{\mathbb R}e{z_2}T_{2{\mathbb C}}+\vert z_1\vert^2+\vert z_2\vert^2)^{-1},
$$ $q=Q({\bf z})$
for $j=1,2$, we clearly have
$$
(T_{1{\mathbb C}}-z_1)R_1({\bf T}_{\mathbb C},{\bf z})+(T_{2{\mathbb C}}-z_2)R_2({\bf T}_{\mathbb C},{\bf z})
={\bf I},
$$
which, according to \cite{Tay}, implies that ${\bf z}$ is in the
joint resolvent of ${\bf T}_{\mathbb C}$. A similar argument shows that,
in this case the point ${\bf z}^c$ belongs to the joint resolvent of ${\bf T}_{\mathbb C}^c$. In addition, if ${\bf s}igma(T_{\mathbb C})$
designates the Taylor spectrum of $T_{\mathbb C}$, we have the
inclusion ${\bf s}igma(T_{\mathbb C}){\bf s}ubset{\bf s}igma_{{\mathbb C}^2}({\bf T})$.
In particular, for every complex-valued function $f$ analytic in a neighborhood of ${\bf s}igma_{{\mathbb C}^2}({\bf T})$, the operator
$f(\bf T_{\mathbb C})$ can be computed via Taylor's analytic functional
calculus. In fact, we have a Martinelli type formula for the
analytic functional calculus:
\end{Rem}
\begin{Thm} Let $\mathcal{V}$ be a real Banach space, let ${\bf T}=(T_1,T_2)\in \mathcal{B(V)}^2$ be a pair of commuting operators, let $U{\bf s}ubset{\mathbb C}^2$ be an open set, let
$D{\bf s}ubset U$ be a bounded domain
containing ${\bf s}igma_{{\mathbb C}^2}({\bf T})$, with piecewise-smooth boundary ${\mathbb S}igma$, and let $f\in\mathcal{O}(U)$. Then we have
$$
f({\bf T}_{\mathbb C})=\frac{1}{(2\pi i)^2}\int_{\mathbb S}igma f({\bf z}))L({\bf z,T_{\mathbb C}})^{-2}(\bar{z}_1-T_{1{\mathbb C}})d\bar{z}_2-(\bar{z}_2-T_{2{\mathbb C}})
d\bar{z}_1]dz_1 dz_2,
$$
where
$$
L({\bf z,T_{\mathbb C}})=T_{1{\mathbb C}}^2+T_{2{\mathbb C}}^2-2{\mathbb R}e{z_1}T_{1{\mathbb C}}-2{\mathbb R}e{z_2}T_{2{\mathbb C}}+\vert z_1\vert^2+\vert z_2\vert^2.
$$
\end{Thm}
{\it Proof.}\, Theorem III.9.9 from \cite{Vas3} implies that the
map $\mathcal{O}(U)\ni f\mapsto f({\bf T}_{\mathbb C})\in\mathcal{B(V_{\mathbb C})}$, defined in terms of Taylor's analytic functional calculus, is unital,
linear, multiplicative, and ordinary complex polynomials in ${\bf z}$
are transformed into polynomials in ${\bf T}_{\mathbb C}$ by simple substitution, where $\mathcal{O}(U)$ is the algebra of all analytic functions in the open set $U{\bf s}ubset{\mathbb C}^2$, provided
$U{\bf s}upset{\bf s}igma({\bf T}_{\mathbb C})$.
The only thing to prove is that, when $U{\bf s}upset{\bf s}igma_{{\mathbb C}^2}({\bf T})$, Taylor's functional calculus is given by the stated (canonical) formula. In order to do that, we use an argument from the proof of Theorem
III.8.1 in \cite{Vas3}, to make explicit the integral III(9.2) from \cite{Vas3} (see also \cite{Lev}).
We consider the exterior algebra
$$
\Lambda[e_1,e_2,\bar{\xi_1},\bar{\xi_2},\mathcal{O}(U)\otimes\mathcal{V}_{\mathbb C}]=
\Lambda[e_1,e_2,\bar{\xi_1},\bar{\xi_2}]\otimes\mathcal{O}(U)\otimes\mathcal{V}_{\mathbb C},
$$
where the indeterminates $e_1,e_2$ are to be associated with the pair ${\bf T}_{\mathbb C}$, we put $\bar{\xi_j}=d\bar{z}_j,\,j=1,2$, and consider the operators
$\delta=(z_1-T_{1{\mathbb C}})\otimes e_1+(z_2-T_{2{\mathbb C}})\otimes e_2,\,\bar{\partial} =
(\partial/\partial\bar{z_1})\otimes\bar{\xi_1}+(\partial/\partial\bar{z_2})\otimes\bar{\xi_2}$, acting naturally on this
exterior algebra, via the calculus with exterior forms.
To simplify the computation, we omit the symbol $\otimes$, and the exterior product will be denoted simply par juxtaposition.
We fix the exterior form $\eta=\eta_2=fye_1e_2$ for some
$f\in\mathcal{O}(U)$ and $y\in\mathcal{X}_{\mathbb C}$, which clearly satisfy the equation
$(\delta+\bar{\partial})\eta=0$, and look for a solution
$\theta$ of the equation $(\delta+\bar{\partial})\theta=\eta$.
We write $\theta=\theta_0+\theta_1$, where $\theta_0,\theta_1$
are of degree $0$ and $1$ in $e_1,e_2$, respectively. Then the equation $(\delta+\bar{\partial})\theta=\eta$ can be written
under the form $\delta\theta_1=\eta,\,\delta\theta_0=-\bar{\partial}\theta_1$, and $\bar{\partial}\theta_0=0$.
Note that
$$
\theta_1=fL({\bf z,T_{\mathbb C}})^{-1}[(\bar{z}_1-T_{1{\mathbb C}})ye_2-
(\bar{z}_2-T_{2{\mathbb C}})]ye_1
$$
is visibly a solution of the equation $\delta\theta_1=\eta$.
Further, we have
$$
\bar{\partial}\theta_1=fL({\bf z,T_{\mathbb C}})^{-2}
[(z_1-T_{1{\mathbb C}})(\bar{z}_2-T_{2{\mathbb C}})y\bar{\xi}_1e_1-
(z_1-T_{1{\mathbb C}})(\bar{z}_1-T_{1{\mathbb C}})y\bar{\xi}_2e_1+
$$
$$
(z_2-T_{2{\mathbb C}})(\bar{z}_2-T_{2{\mathbb C}})y\bar{\xi}_1e_2-
(z_2-T_{2{\mathbb C}})(\bar{z}_1-T_{1{\mathbb C}})y\bar{\xi}_2e_2]=
$$
$$
\delta[fL({\bf z,T_{\mathbb C}})^{-2}(\bar{z}_1-T_{1{\mathbb C}})y\bar{\xi}_2-
fL({\bf z,T_{\mathbb C}})^{-2}(\bar{z}_2-T_{2{\mathbb C}})y\bar{\xi}_1],
$$
so we may define
$$
\theta_0=-fL({\bf z,T_{\mathbb C}})^{-2}(\bar{z}_1-T_{1{\mathbb C}})y\bar{\xi}_2+
fL({\bf z,T_{\mathbb C}})^{-2}(\bar{z}_2-T_{2{\mathbb C}})y\bar{\xi}_1.
$$
Formula III(8.5) from \cite{Vas3} shows that
$$
f({\bf T}_{\mathbb C})y=-\frac{1}{(2\pi i)^2}\int_U\bar{\partial}(\phi\theta_0)dz_1 dz_2=
$$
$$
\frac{1}{(2\pi i)^2}\int_{\mathbb S}igma f({\bf z}))L({\bf z,T_{\mathbb C}})^{-2}[(\bar{z}_1-T_{1{\mathbb C}})yd\bar{z}_2-(\bar{z}_2-T_{2{\mathbb C}})y
d\bar{z}_1]dz_1 dz_2,
$$
for all $y\in\mathcal{X}_{\mathbb C}$, via Stokes's formula, where $\phi$ is a smooth function such
that $\phi=0$ in a neighborhood of ${\bf s}igma_{{\mathbb C}^2}({\bf T})$,
$\phi=1$ on ${\mathbb S}igma$ and the support of $1-\phi$ is compact.
\begin{Rem}\rm (1) We may extend the previous functional calculus to $\mathcal{B(V}_{\mathbb C})$-valued analytic functions, setting, for such a function $F$ and with the notation from above,
$$
F({\bf T}_{\mathbb C})=\frac{1}{(2\pi i)^2}\int_{\mathbb S}igma F({\bf z}))L({\bf z,T_{\mathbb C}})^{-2}(\bar{z}_1-T_{1{\mathbb C}})d\bar{z}_2-(\bar{z}_2-T_{2{\mathbb C}})
d\bar{z}_1]dz_1 dz_2.
$$
In particular, if $F({\bf z})={\bf s}um_{j,k\ge0}A_{jk{\mathbb C}}z_1^jz_2^k$,
with $A_{j,k}\in\mathcal{B(V)}$, where the series is convergent in neighborhood of ${\bf s}igma_{{\mathbb C}^2}({\bf T})$, we obtain
$$F({\bf T}):=F({\bf T}_{\mathbb C})\vert\mathcal{V}={\bf s}um_{j,k\ge0}A_{jk}T_1^jT_2^k\in\mathcal{B(V)}.$$
(2) The connexion of the spectral theory of pairs with the algebra of quaternions is even stronger in the case of complex Hilbert
spaces. Specifically, if $\mathcal{H}$ is a complex Hilbert space and ${\bf V}=(V_1,V_2)$ is a commuting pair of bounded
linear operators on $\mathcal{H}$, a point ${\bf z}=(z_1,z_2)\in{\mathbb C}^2$
is in the joint resolvent of ${\bf V}$ if and only if the
operator $Q({\bf V})-Q({\bf z})$ is invertible in $\mathcal{H}^2$,
where
$$
Q({\bf V})=\left(\begin{array}{cc} V_1 & V_2 \\ -V_2^* & V_1^*
\end{array}\right).
$$
(see \cite{Vas1} for details). In this case, there is also a Martinelli type formula which can be used to construct the associated analytic functional calculus (see \cite{Vas2},\cite{Vas3}). An approach to such a construction in Banach spaces, by using a so-called splitting joint spectrum, can be
found in \cite{MuKo}.
\end{Rem}
\end{document}
|
\begin{document}
\begin{abstract}
Grafakos systematically
proved that $A_\infty$ weights have different characterizations for cubes in Euclidean Spaces in his classical text book. Very recently, Duoandikoetxea, Mart\'{\i}n-Reyes, Ombrosi and Kosz discussed several characterizations of the $A_{\infty}$ weights in the setting of general bases. By conditional expectations, we study $A_\infty$ weights in martingale spaces. Because conditional expectations are Radon-Nikod\'{y}m derivatives with respect to sub$\hbox{-}\sigma\hbox{-}$fields which have no geometric structures, we need new ingredients. Under a regularity assumption on weights, we obtain equivalent characterizations of the $A_{\infty}$ weights. Moreover, using weights modulo conditional expectations, we have one-way implications of different characterizations.
\end{abstract}
\keywords{weight, conditional expectation, maximal operator, median}
\subjclass[2010]{Primary: 60G46; Secondary: 60G48, 60G42}
\maketitle
\section{Introduction }
As is well known, a non-negative function $\omega$ on $\mathbb{R}^n$ is an $A_p$ weight with $p>1,$ if there exists a constant $C$ for all cubes $Q$ such that
$$
\bigg( \frac{1}{|Q|} \int_Q \omega \, d\mu \bigg) \bigg( \frac{1}{|Q|} \int_Q \omega^{-\frac{1}{p-1}} \, d\mu \bigg)^{p-1} \ensuremath{\ell}eq C.
$$
Muckenhoupt \cite{MR293384} observed that the weight has an open property $A_p=\cup_{1<q<p}A_q,$ and shortly after Muckenhoupt \cite{MR350297} defined the $A^M_\infty$ weight as follows: there exist $0<\varepsilon,~\delta<1$ such that for all $E\subseteq Q$ it holds that
$$
|E| < \delta |Q| \ensuremath{\mathbb{R}}ightarrow \omega(E) < \varepsilon \omega(Q).
$$
Then he showed $A^M_{\infty}=\cup_{p>1}A_p.$ Independently,
Coifman and Fefferman \cite{MR358205} introduced an $A^{CF}_\infty$ weight and proved $A^{CF}_{\infty}=\cup_{p>1}A_p,$ where the $A^{CF}_\infty$ weight $\omega$ is defined as follows: there exist $C,~\delta>0$ such that for all $E\subseteq Q$
$$
\frac{\omega(E)}{\omega(Q)} \ensuremath{\ell}eq C \bigg( \frac{|E|}{|Q|} \bigg)^\delta.
$$
Later, a condition $A_{\infty}^{exp}$ defined by a limit of the $A_p$ weight as $p\uparrow\infty$ was studied almost simultaneously in \cite{MR727244} and \cite[p.405]{MR807149} and $\cup_{p>1}A_p=A^{exp}_{\infty}.$
As we have seen, $A^{M}_\infty,$ $A^{CF}_\infty$ and $A_{\infty}^{exp}$ are equivalent. These are geometric characterizations of $\cup_{p>1}A_p$ and systematically studied in Grafakos \cite[Theorem 7.3.3]{MR3243734}. Very recently, Duoandikoetxea, Mart\'{\i}n-Reyes and Ombrosi \cite{MR3473651} compared and discussed different characterizations of $\cup_{p>1}A_p$ in the setting of general bases. Indeed, they studied many other characterizations which are not geometric. Here we list four characterizations for cubes mentioned in \cite{MR3473651}:
\begin{enumerate}
\item [($A^{*}_{\infty}$)]There exists $C > 0$ such that
$$\int_Q M (\omega \chi_Q) dx \ensuremath{\ell}eq C \omega(Q),$$
where $\omega(Q) := \int_Q \omega d\mu$ and $M(\cdot)$ is the Hardy-Littlewood maximal operator(see \cite{MR481968}, \cite{MR883661}). Hyt\"{o}nen and P\'{e}rez \cite{MR3092729} used the weight $A^{*}_{\infty}$ to improve estimates of the bounds in the weighted inequalities.
\item [($A^{log}_{\infty}$)] There exists $C > 0$ such that
$$\int_Q \omega \ensuremath{\ell}og^+ \frac{\omega}{\omega_Q}dx \ensuremath{\ell}eq C \omega(Q),$$
where $\omega_Q:=\omega(Q)/|Q|$(see \cite{MR481968}).
\item [($A^{med}_{\infty}$)] There exists $C > 0$ such that
$$\omega_Q \ensuremath{\ell}eq C m(\omega; Q)$$
where the median of $\omega$ in $Q$ is a number $m(\omega;Q)$ such that $|\{x \in Q : \omega(x) < m(\omega;Q)\}| \ensuremath{\ell}eq |Q| / 2$ and $|\{x \in Q : \omega(x) > m(\omega;Q)\}| \ensuremath{\ell}eq |Q| / 2$ (see \cite{MR529683}). Using the median, Lerner \cite{MR2721744}
obtain a decomposition of an arbitrary measurable function in terms of local mean oscillations.
\item [($A^{\ensuremath{\ell}ambda}_{\infty}$)] There exist $C, \beta > 0$ such that
$$w\big(\{ x \in Q : \omega(x) > \ensuremath{\ell}ambda \}\big) \ensuremath{\ell}eq C \ensuremath{\ell}ambda \, \big|\{x \in Q : \omega(x) > \beta \ensuremath{\ell}ambda \}\big|,$$
where $\ensuremath{\ell}ambda>\omega_Q$. This kind of characterization appeared independently in \cite{MR402038} and \cite{MR358205}.
\end{enumerate}
Although $A^{*}_{\infty},$ $A^{log}_{\infty},$ $A^{med}_{\infty}$ and $A^{\ensuremath{\ell}ambda}_{\infty}$ are not geometric, they are equivalent to $A^{M}_\infty,$ $A^{CF}_\infty$ and $A_{\infty}^{exp}$ for cubes. In
the context of general bases,
the relations between them are more complicated(see \cite{MR3473651} and \cite{MR4446233} for more information).
In this paper, we study $A_\infty$ weights in martingale spaces. Izumisawa and Kazamaki first introduced the
$A_p$ weight for martingales. Under some additional
conditions, they obtained the open property $A_p=\cup_{1<q<p}A_q.$ In martingale setting, it is well known that this property is false in general, because Bonami and L\'{e}pingle \cite{MR544802} showed that for any $p>1,$ there exists a weight $\omega\in A_p,$ but $\omega\notin A_{p-\varepsilon}$ for all $\varepsilon>0$(see also \cite[p.241]{MR1224450}).
Under some additional restrictions, the $A_p$ weight was extensively studied by Dol\'{e}ans-Dade and Meyer \cite{MR544804}. Motivated by the work
of \cite[Theorem 7.3.3]{MR3243734}, \cite{MR3473651} and \cite{MR4446233}, we study
several characterizations of $A_{\infty}$ weights in the setting of martingales. Our first result is the following Theorem \ref{Thm:equa}, which partially depends on a regularity condition $S$(see Definition \ref{regular_p}).
\begin{theorem}\ensuremath{\ell}abel{Thm:equa}Let $\omega$ be a weight. If $\omega\in S,$ then the following are
equivalent.
\begin{enumerate}[\rm (1)]
\item \ensuremath{\ell}abel{Thm:equa_Ap}There exist
$C,~p>1$ such that for all $n\in \mathbb{N}$ we have
\begin{equation}\ensuremath{\ell}abel{Ap}
\mathbb{E}(\omega|\mathcal {F}_n)\mathbb{E}(\omega^{-\frac{1}{p-1}}|\mathcal
{F}_n)^{p-1}\ensuremath{\ell}eq C,
\end{equation}
which is denoted by $\omega\in\bigcup\ensuremath{\ell}imits_{p>1}A_p.$
\item \ensuremath{\ell}abel{Thm:equa_A_exp_infty}There exists a positive
constant $C$ such that for all $n\in
\mathbb{N}$ we have
\begin{equation}\ensuremath{\ell}abel{A_exp_infty}
\mathbb{E}(\omega|\mathcal {F}_n)\ensuremath{\ell}eq C\exp \mathbb{E}(\ensuremath{\ell}og\omega|\mathcal {F}_n),
\end{equation}
which is denoted by $\omega\in A^{exp}_{\infty}.$
\item \ensuremath{\ell}abel{Thm:equa_wAinfty}There exist $0<\gamma,~\delta<1$ such that for all $n\in
\mathbb{N}$ we have
\begin{equation}\ensuremath{\ell}abel{wAinfty}
\mathbb{E}(\chi_{\{\omega\ensuremath{\ell}eq\gamma\omega_n\}}|\mathcal
{F}_n)\ensuremath{\ell}eq\delta<1,
\end{equation}
which is denoted by $\omega\in A^{con}_{\infty}.$
\item \ensuremath{\ell}abel{Thm:equa_R}There exist $0<\alpha,~\beta<1$ such that for all $n\in
\mathbb{N}$ and $A\in\mathcal {F}$ we have
\begin{equation}
\ensuremath{\ell}abel{R}\mathbb{E}(\chi_A|\mathcal {F}_n)\ensuremath{\ell}eq\alpha<1\ensuremath{\mathbb{R}}ightarrow
\mathbb{E}_\omega(\chi_A|\mathcal {F}_n)\ensuremath{\ell}eq\beta<1,\end{equation}
which is denoted by $\omega \in A_{\infty}^{M}.$
\item \ensuremath{\ell}abel{Thm:equa_RH} There exist $C,~q>1$ such that for all $n\in \mathbb{N}$ we have
\begin{equation}\ensuremath{\ell}abel{RH}
\mathbb{E}(\omega^q|\mathcal {F}_n)\ensuremath{\ell}eq
C\mathbb{E}(\omega|\mathcal {F}_n)^q,\end{equation}
which is the reverse H\"{o}lder condition and denoted by $\omega\in \bigcup\ensuremath{\ell}imits_{q>1}RH_{q}$.
\item \ensuremath{\ell}abel{Thm:equa_RR}There exist $0<\varepsilon'<1$ and $C>1$ such that
for all $n\in \mathbb{N}$ we have
\begin{equation}
\ensuremath{\ell}abel{RR}\mathbb{E}_\omega(\chi_A|\mathcal {F}_n)\ensuremath{\ell}eq C\mathbb{E}(\chi_A|\mathcal
{F}_n)^{\varepsilon'}.
\end{equation}
which is denoted by $\omega\in A^{CF}_{\infty}.$
\item \ensuremath{\ell}abel{Thm:equa_rR} There exist $0<\alpha,~\beta<1$ such that for all $n\in
\mathbb{N}$ and $A\in\mathcal {F}$ we have
\begin{equation}
\ensuremath{\ell}abel{rR}\mathbb{E}_\omega(\chi_A|\mathcal {F}_n)\ensuremath{\ell}eq\alpha<1\ensuremath{\mathbb{R}}ightarrow
\mathbb{E}(\chi_A|\mathcal {F}_n)\ensuremath{\ell}eq\beta<1,\end{equation}
which is denoted by $\omega \in \hat{A}_{\infty}^{M}.$
\end{enumerate}
\end{theorem}
We remark that $\omega\in S$ is used only in $\ref{Thm:equa_R}\xRightarrow{S}\ref{Thm:equa_RH}$ and
$\ref{Thm:equa_rR}\xRightarrow{S}\ref{Thm:equa_Ap}$ in Theorem \ref{Thm:equa}.
It is natural to discuss what happens without the condition $S.$ Using weights modulo
conditional expectations $\omega/\omega_n$ instead of weights $\omega,$ we study several characterizations of $A_\infty$ weights.
First we have Theorems \ref{thm:Reverse-test} and \ref{thm:Cond}.
\begin{theorem}\ensuremath{\ell}abel{thm:Reverse-test}The following statements are equivalent.
\begin{enumerate}
\item \ensuremath{\ell}abel{Reverse-test1}$\omega\in \bigcup\ensuremath{\ell}imits_{q>1}RH_{q}$.
\item \ensuremath{\ell}abel{Reverse-test2}$\omega\in A^{CF}_{\infty}.$
\end{enumerate}
\end{theorem}
\begin{theorem}\ensuremath{\ell}abel{thm:Cond}The following statements are equivalent.
\begin{enumerate}
\item \ensuremath{\ell}abel{thm:Cond1}$\omega\in A^{con}_{\infty}.$
\item \ensuremath{\ell}abel{thm:Cond2}$\omega \in A_{\infty}^{M}.$
\end{enumerate}
\end{theorem}
Using a kind of reverse H\"{o}lder condition which appeared in Str\"{o}mberg and Wheeden \cite{MR766221}, we give a characterization of $\omega\in A^{exp}_{\infty},$ which is Theorem \ref{thm:exp-s}.
\begin{theorem}\ensuremath{\ell}abel{thm:exp-s}The following statements are equivalent.
\begin{enumerate}
\item \ensuremath{\ell}abel{thm:exp-s2} $\omega\in A^{exp}_{\infty}.$
\item \ensuremath{\ell}abel{thm:exp-s1}There exists $C>1$ such that for every $s\in(0,1)$ we have
\begin{equation}\ensuremath{\ell}abel{thm:eq-exp-s}
\mathbb{E}(\omega|\mathcal {F}_n)\ensuremath{\ell}eq C\mathbb{E}(\omega^s|\mathcal {F}_n)^{\frac{1}{s}},\end{equation}
which is denoted by $\omega\in A^{SW}_{\infty}.$
\end{enumerate}
\end{theorem}
Modifying $A_{\infty}^{\ensuremath{\ell}ambda}$ and $A_{\infty}^{med}$ in
\cite{MR402038} and \cite{MR529683}, respectively, we have
one-way implications. Indeed,
introducing the quotient $\omega/\omega_n$ into $A_{\infty}^{\ensuremath{\ell}ambda}$ of \cite{MR402038}, we obtain Theorem \ref{thm:imp1}.
\begin{theorem}\ensuremath{\ell}abel{thm:imp1}Let $\omega$ be a weight. We have the sequence of implications
$\eqref{thm:lev}\ensuremath{\mathbb{R}}ightarrow\eqref{thm:rev} \ensuremath{\mathbb{R}}ightarrow\eqref{thm:log}\ensuremath{\mathbb{R}}ightarrow\eqref{thm:imp_Cond2}$
for the following statements.
\begin{enumerate}
\item \ensuremath{\ell}abel{thm:lev}There exist $0<\beta<1$ and $C>1$ such that for all $n\in \mathbb{N}$ and $\ensuremath{\ell}ambda>1$ we have
\begin{equation}\ensuremath{\ell}abel{thm:equ_lev}
\mathbb{E}_\omega(\chi_{\{\frac{\omega}{\omega_n}>\ensuremath{\ell}ambda\}}|\mathcal {F}_n)\ensuremath{\ell}eq C\ensuremath{\ell}ambda\mathbb{E}(\chi_{\{\frac{\omega}{\omega_n}>\beta\ensuremath{\ell}ambda\}}|\mathcal {F}_n),\end{equation}
which is denoted by $\omega\in A_{\infty}^{\ensuremath{\ell}ambda}.$
\item \ensuremath{\ell}abel{thm:rev}
$\omega\in\bigcup\ensuremath{\ell}imits_{q>1}RH_q.$
\item \ensuremath{\ell}abel{thm:log}There exists $C>1$ such that for all $n\in \mathbb{N}$ we have
\begin{equation}\ensuremath{\ell}abel{thm:equ_log}
\mathbb{E}_{\omega}(\ensuremath{\ell}og^+\frac{\omega}{\omega_n}|\mathcal {F}_n)\ensuremath{\ell}eq C,
\end{equation}
which is denoted by $\omega\in A_{\infty}^{log}.$
\item \ensuremath{\ell}abel{thm:imp_Cond2}$\omega \in A_{\infty}^{M}.$
\end{enumerate}
\end{theorem}
As for $A_{\infty}^{med}$ in \cite{MR529683}, we replace the median $m(\omega;Q)$ by the median function $m(\omega,n)$(see Definition \ref{media_f}), which is the key observation in Theorem \ref{thm:imp2}.
\begin{theorem}\ensuremath{\ell}abel{thm:imp2}Let $\omega$ be a weight. We have the sequence of implications
$\eqref{thm:imp2_Ap}\ensuremath{\mathbb{R}}ightarrow\eqref{thm:imp2_Log} \ensuremath{\mathbb{R}}ightarrow\eqref{thm:imp2_Mid}\ensuremath{\mathbb{R}}ightarrow\eqref{thm:imp2_Dou}$
for the following statements.
\begin{enumerate}
\item \ensuremath{\ell}abel{thm:imp2_Ap}$\omega\in\bigcup\ensuremath{\ell}imits_{p>1} A_p.$
\item \ensuremath{\ell}abel{thm:imp2_Log}$\omega\in A^{exp}_{\infty}.$
\item \ensuremath{\ell}abel{thm:imp2_Mid} There exists $C>1$ such that for all $n\in \mathbb{N}$ we have
\begin{equation}\ensuremath{\ell}abel{thm:imp2_EMid}
\omega_n\ensuremath{\ell}eq Cm(\omega,n),
\end{equation}
which is denoted by $\omega\in A_{\infty}^{med}.$
\item \ensuremath{\ell}abel{thm:imp2_Dou}$\omega \in A_{\infty}^{M}.$
\end{enumerate}
\end{theorem}
Now we give Theorem \ref{thm:Wi} which is related to $A^*_{\infty}$ in \cite{MR481968} and \cite{MR883661}.
The main ingredient of Theorem \ref{thm:Wi} is the conditional expectation of tailed maximal operators(see Definition \ref{tailed_o}). It is worth observing that \eqref{tailed_mo} equals $\mathbb{E}(M^*_n(\omega/\omega_n)|\mathcal {F}_n)\ensuremath{\ell}eq C.$
The tailed maximal operators first appeared in \cite{MR1301765} and were used to proved two-weight inequalities for martingales under some additional assumption.
In view of Theorem \ref{thm:Wi}, we have $A^{exp}_{\infty}\subseteq A^*_{\infty}.$
\begin{theorem}\ensuremath{\ell}abel{thm:Wi} Given the following statements.
\begin{enumerate}
\item \ensuremath{\ell}abel{thm:Wi_Ap}$\omega\in \bigcup\ensuremath{\ell}imits_{p>1}A_p.$
\item \ensuremath{\ell}abel{thm:Wi_rev}
$\omega\in\bigcup\ensuremath{\ell}imits_{q>1}RH_q.$
\item \ensuremath{\ell}abel{thm:Wi_log}$\omega\in A^{log}_{\infty}.$
\item \ensuremath{\ell}abel{Thm:Wi_exp}$\omega\in A^{exp}_{\infty}.$
\end{enumerate}
Then each of these statements implies $\omega\in A^*_{\infty},$ i.e., for all $n\in\mathbb{N},$
\begin{equation}\ensuremath{\ell}abel{tailed_mo}
\mathbb{E}(M^*_n(\omega)|\mathcal {F}_n)\ensuremath{\ell}eq C\omega_n.
\end{equation}
\end{theorem}
The paper is organized as follows. Some preliminaries are contained in Sect. \ref{preli}.
In Sect. \ref{regular} we prove Theorem \ref{Thm:equa} for regular weights. Sect. \ref{without} is devoted to
theorems without additional assumptions.
\section{Preliminaries}\ensuremath{\ell}abel{preli}
Let $(\Omega,\mathcal {F},\mu)$ be a complete probability space and
$(\mathcal {F}_n)_{n\geq0}$ an increasing sequence of
sub$\hbox{-}\sigma\hbox{-}$fields of $\mathcal{F}$ with
$\mathcal{F}=\bigvee_{n\geq0}\mathcal{F}_n.$
The conditional expectation with respect
to $(\Omega,\mathcal{F},\mu,\mathcal{F}_n)$ is denoted by $\mathbb{E}(\cdot|\mathcal{F}_n).$
In this paper, for $p\geq1,$ a martingale
$f=(f_n)_{n\geq0}\in L^p(\omega)$ is meant as $f_n=E(f|\mathcal
{F}_n),~f\in L^p(\omega).$ A weight
$\omega$ is a random variable with $\omega>0$ and $
E(\omega)<\infty.$ Without loss of generality, we may assume
$E(\omega)=1$ since otherwise we can replace $\omega$ by $\omega/E(\omega).$
\begin{definition}\ensuremath{\ell}abel{tailed_o}The Doob maximal operator $M$ and the tailed maximal
operator $M_n^*$ for martingale $f=(f_n)$ are defined by
\begin{equation*}Mf=\sup\ensuremath{\ell}imits_{n\geq 0}|f_n|\hbox{ and }M_n^*f=\sup\ensuremath{\ell}imits_{m\geq n}|f_n|,
\end{equation*}
respectively.
\end{definition}
\begin{definition}\ensuremath{\ell}abel{regular_p}The weight $\omega$ is said to satisfy an regularity condition $S$ , if there exists $C>1$
such that for all $n\in \mathbb{N}$ we have
$$\frac{1}{C}\omega_{n-1}\ensuremath{\ell}eq\omega_n\ensuremath{\ell}eq C\omega_{n-1},$$
which is denoted by $\omega\in S.$
\end{definition}
Let $\omega$ be a weight and $d\hat{\mu}=\omega
d\mu.$ We denote the conditional expectation
with respect
to $(\Omega,\mathcal{F},\hat{\mu},\mathcal{F}_n)$ by $\hat{\mathbb{E}}(\cdot|\mathcal{F}_n)$ or $\mathbb{E}_{\omega}(\cdot|\mathcal{F}_n).$ It follows that
$$\mathbb{E}_{\omega}(\cdot|\mathcal{F}_n)=\mathbb{E}(\cdot\omega|\mathcal{F}_n)/\omega_n=\mathbb{E}(\cdot\frac{\omega}{\omega_n}|\mathcal{F}_n).$$ If $A\in \mathcal{F}$,
we denote $\int_A\omega d\mu $ by $|A|_\omega$ and $\int_Ad\mu$ by
$|A|$, respectively.
For
$(\Omega,\mathcal {F},\mu)$ and $(\mathcal {F}_n)_{n\geq0},$ the
family of all stopping times is denoted by $\mathcal {T}.$
\begin{definition}\ensuremath{\ell}abel{media_f}The median function of $\omega$ relative to $\mathcal {F}_n$ is defined as a $\mathcal {F}_n$ measurable function $m(\omega;n)$ such that $\mathbb{E}(\chi_{\{\omega> m(\omega,n)\}}|\mathcal {F}_n)\ensuremath{\ell}eq 1/ 2$
and $\mathbb{E}(\chi_{\{\omega< m(\omega,n)\}}|\mathcal {F}_n)\ensuremath{\ell}eq 1/ 2.$
\end{definition}
We denote the set of non-negative integers by $\mathbb{N}$ and all integers
by $\mathbb{Z},$ respectively. Throughout the paper letter $C$ always denotes a positive constant which may be different in each occurrence.
\section{Equivalent Characterizations of Regular $A_{\infty}$ Weights}\ensuremath{\ell}abel{regular}
The following Lemma \ref{key_lemma_sta} will be used in the proof of Theorem \ref{Thm:equa}.
\begin{lemma}\ensuremath{\ell}abel{key_lemma_sta}Let $v$ be a positive measurable function
and let $0<s_0<+\infty$. If $v\in L^{s_0},$ then
\begin{equation}\ensuremath{\ell}abel{key_lemma}
\mathbb{E}(v^s|\mathcal {F}_n)^{\frac{1}{s}}\downarrow\exp \mathbb{E}(\ensuremath{\ell}og v|\mathcal {F}_n),~\text{~as~}s\downarrow0^+,
\end{equation}\end{lemma}
\begin{proof}[Proof of Lemma \ref{key_lemma_sta}] Because of $v\in L^{s_0},$ we have $v\in L^{s}$ with $0<s<s_0.$
H\"{o}lder's inequality for the conditional expectation(\cite[p.3]{MR1224450}) gives
$\mathbb{E}(v^s|\mathcal {F}_n)^{\frac{1}{s}}\ensuremath{\ell}eq\mathbb{E}(v^t|\mathcal {F}_n)^{\frac{1}{t}}$ with $0<s<t<s_0.$
Following Jensen's inequality for the conditional expectation(\cite[p.5]{MR1224450}), we have
$$
\exp \mathbb{E}(\ensuremath{\ell}og v^s|\mathcal {F}_n)\ensuremath{\ell}eq\mathbb{E}(v^s|\mathcal {F}_n),
$$
which implies \begin{equation}\ensuremath{\ell}abel{eq_left}\exp \mathbb{E}(\ensuremath{\ell}og v|\mathcal {F}_n)\ensuremath{\ell}eq\mathbb{E}(v^s|\mathcal {F}_n)^{\frac{1}{s}}.
\end{equation}
Because of $x\ensuremath{\ell}eq \exp(x-1)$ for $x>0,$ then
$$\mathbb{E}(v^s|\mathcal {F}_n)\ensuremath{\ell}eq\exp \big(\mathbb{E}(v^s|\mathcal {F}_n)-1\big).$$
It follows that
\begin{equation}\ensuremath{\ell}abel{eq_right}\mathbb{E}(v^s|\mathcal {F}_n)^{\frac{1}{s}}\ensuremath{\ell}eq\exp (\frac{\mathbb{E}(v^s|\mathcal {F}_n)-1}{s})=\exp \mathbb{E}(\frac{v^s-1}{s}|\mathcal {F}_n).
\end{equation}
Let $f(x)=\frac{x^s-1}{s}-\frac{x^t-1}{t}$ with $s>t>0$ and $x>0.$ Then $f(1)=0$ is the minimum value of $f$ on $(0,+\infty).$
It follows that for all $x>0$ we have $\frac{x^s-1}{s}\downarrow\ensuremath{\ell}og x,$ as $s\downarrow0^+.$ Using Monotone Convergence Theorem for the conditional expectation(\cite[p.5]{MR1224450}), we obtain that
\begin{eqnarray*}
\mathbb{E}\big(\frac{v^{s_0}-1}{s_0}-\frac{v^s-1}{s}|\mathcal {F}_n\big)&\uparrow&\mathbb{E}\big(\frac{v^{s_0}-1}{s_0}-\ensuremath{\ell}og v|\mathcal {F}_n\big), ~\text{~as~}s\downarrow0^+.
\end{eqnarray*}
Thus \begin{equation}\ensuremath{\ell}abel{eq_further}
\ensuremath{\ell}im\ensuremath{\ell}imits_{s\rightarrow0^+}\mathbb{E}(\frac{\omega^s-1}{s}|\mathcal {F}_n)=\mathbb{E}(\ensuremath{\ell}og\omega|\mathcal {F}_n).
\end{equation}
Combining \eqref{eq_left}, \eqref{eq_right} and\eqref{eq_further}, we deduce
$$\ensuremath{\ell}im\ensuremath{\ell}imits_{s\rightarrow0^+}\mathbb{E}(\omega^s|\mathcal {F}_n)^{\frac{1}{s}}=\exp \mathbb{E}(\ensuremath{\ell}og\omega|\mathcal {F}_n).$$
This completes the proof of \eqref{key_lemma}
\end{proof}
To prove Theorem \ref{Thm:equa}, we use $\omega\in S$ only in $\ref{Thm:equa_R}\xRightarrow{S}\ref{Thm:equa_RH}$ and
$\ref{Thm:equa_rR}\xRightarrow{S}\ref{Thm:equa_Ap}.$
\begin{proof}[Proof of Theorem \ref{Thm:equa}] We shall follow the scheme:
\begin{center}
\end{center}
$\ref{Thm:equa_Ap}\ensuremath{\mathbb{R}}ightarrow\ref{Thm:equa_A_exp_infty}.$ Let $\omega\in A_p.$ Then for all $q\geq p,$ we have $\omega\in A_q.$ In view of Lemma \ref{key_lemma_sta} with $s=\frac{1}{q-1}$ and $v=\frac{1}{\omega},$ we obtain that
$$
\mathbb{E}(\omega^{-\frac{1}{q-1}}|\mathcal {F}_n)^{{q-1}}\downarrow\exp \mathbb{E}(\ensuremath{\ell}og \frac{1}{\omega}|\mathcal {F}_n),~\text{~as~}q\uparrow+\infty.
$$
Thus
$$\mathbb{E}(\omega|\mathcal {F}_n)\exp \mathbb{E}(\ensuremath{\ell}og\frac{1}{\omega}|\mathcal
{F}_n)\ensuremath{\ell}eq C,$$ which implies $\mathbb{E}(\omega|\mathcal {F}_n)\ensuremath{\ell}eq C\exp \mathbb{E}(\ensuremath{\ell}og\omega|\mathcal
{F}_n).$
$\ref{Thm:equa_A_exp_infty}\ensuremath{\mathbb{R}}ightarrow\ref{Thm:equa_wAinfty}.$
Fix $n\in \mathbb{N}.$
Letting $v_n=\exp
\mathbb{E}(\ensuremath{\ell}og\omega|\mathcal {F}_n),$ we have that
\begin{eqnarray*}
1=\frac{1}{v_n}\exp\mathbb{E}(\ensuremath{\ell}og\omega|\mathcal {F}_n)
=\exp\mathbb{E}(\ensuremath{\ell}og\frac{\omega}{v_n}|\mathcal {F}_n).
\end{eqnarray*}
It follows that
\begin{equation}\ensuremath{\ell}abel{equa0}\mathbb{E}(\ensuremath{\ell}og\frac{\omega}{v_n}|\mathcal {F}_n)=0.
\end{equation}
Using \eqref{A_exp_infty}, we obtain that
\begin{equation}\ensuremath{\ell}abel{eqbound}\frac{1}{v_n}\mathbb{E}(\omega|\mathcal {F}_n)
\ensuremath{\ell}eq\frac{C}{v_n}\exp
\mathbb{E}(\ensuremath{\ell}og\omega|\mathcal {F}_n)
=C.\end{equation}
For some $\gamma>0$ to
be chosen later, we observe that
\begin{eqnarray*}
\{\omega\ensuremath{\ell}eq\gamma\omega_n\}
&=&\{\frac{\omega}{v_n}\ensuremath{\ell}eq
\frac{\gamma}{v_n}\mathbb{E}(\omega|\mathcal {F}_n)\}\\
&\subseteq&\{\frac{\omega}{v_n}\ensuremath{\ell}eq\gamma C\}\\
&\subseteq&\{\ensuremath{\ell}og(1+\frac{v_n}{\omega})\geq\ensuremath{\ell}og(1+\frac{1}{\gamma
C})\}.\end{eqnarray*}
Thus
\begin{eqnarray*}
\mathbb{E}(\chi_{\{\omega\ensuremath{\ell}eq\gamma\omega_n\}}|\mathcal{F}_n)
&\ensuremath{\ell}eq&\mathbb{E}(\chi_{\{\ensuremath{\ell}og(1+\frac{v_n}{\omega})\geq\ensuremath{\ell}og(1+\frac{1}{\gamma C})\}}|\mathcal{F}_n)\\
&\ensuremath{\ell}eq&\frac{1}{\ensuremath{\ell}og(1+\frac{1}{\gamma
C})}\mathbb{E}\big(\ensuremath{\ell}og(1+\frac{v_n}{\omega})|\mathcal
{F}_n\big)\\
&=&\frac{1}{\ensuremath{\ell}og(1+\frac{1}{\gamma
C})}\Big(\mathbb{E}\big(\ensuremath{\ell}og(1+\frac{\omega}{v_n})|\mathcal
{F}_n\big)-\mathbb{E}\big(\ensuremath{\ell}og\frac{\omega}{v_n}|\mathcal
{F}_n\big)\Big)\\
&=&\frac{1}{\ensuremath{\ell}og(1+\frac{1}{\gamma
C})}\mathbb{E}\big(\ensuremath{\ell}og(1+\frac{\omega}{v_n})|\mathcal
{F}_n\big),\end{eqnarray*}
where we have used \eqref{equa0}.
It follows from \eqref{eqbound} that
\begin{eqnarray*}
\frac{1}{\ensuremath{\ell}og(1+\frac{1}{\gamma
C})}\mathbb{E}\big(\ensuremath{\ell}og(1+\frac{\omega}{v_n})|\mathcal
{F}_n\big)
&\ensuremath{\ell}eq&\frac{1}{\ensuremath{\ell}og(1+\frac{1}{\gamma
C})}\mathbb{E}(\frac{\omega}{v_n}|\mathcal{F}_n)\\
&\ensuremath{\ell}eq&\frac{C}{\ensuremath{\ell}og(1+\frac{1}{\gamma C})}.
\end{eqnarray*}
Since
$\ensuremath{\ell}im\ensuremath{\ell}imits_{\gamma\rightarrow0}\frac{C}{\ensuremath{\ell}og(1+\frac{1}{\gamma
C})}=0,$ we deduce that \eqref{wAinfty} holds.
$\ref{Thm:equa_wAinfty}\ensuremath{\mathbb{R}}ightarrow\ref{Thm:equa_R}.$ Fix $n\in \mathbb{N}.$ Suppose that $A\in \mathcal {F}$
with $\mathbb{E}_\omega(\chi_A|\mathcal {F}_n)>\beta$ for some $\beta$ to be
chosen later. Denote $A_1^c=A^c\cap\{\omega>\gamma\omega_n\}$ and
$A_2^c=A^c\cap\{\omega\ensuremath{\ell}eq\gamma\omega_n\}.$ Then we have
$$\mathbb{E}(\chi_{A^c}|\mathcal {F}_n)= \mathbb{E}(\chi_{A_1^c}|\mathcal {F}_n)+\mathbb{E}(\chi_{A_2^c}|\mathcal {F}_n)
\ensuremath{\ell}eq \frac{1}{\gamma\omega_n}\mathbb{E}(\omega\chi_{A_1^c}|\mathcal
{F}_n)+\delta.$$ Since $\mathbb{E}(\omega\chi_{A^c}|\mathcal
{F}_n)=\mathbb{E}_\omega(\chi_{A^c}|\mathcal {F}_n)\omega_n,$ it follows that
$$\mathbb{E}(\chi_{A^c}|\mathcal
{F}_n)\ensuremath{\ell}eq\frac{1}{\gamma}\mathbb{E}_\omega(\chi_{A^c}|\mathcal
{F}_n)+\delta<\frac{1-\beta}{\gamma}+\delta.$$ Because of
$\ensuremath{\ell}im\ensuremath{\ell}imits_{\beta\rightarrow1}(\frac{1-\beta}{\gamma}+\delta)=\delta<1,$ it is possible to choose $\beta\in(0,1)$ such that
$\alpha=1-\frac{1-\beta}{\gamma}-\delta\in(0,1).$ Therefore, we
obtain that $$\mathbb{E}(\chi_{A^c}|\mathcal {F}_n)<1-\alpha,$$ which implies $\mathbb{E}(\chi_A|\mathcal {F}_n)>\alpha.$
Thus \ref{Thm:equa_R} is valid.
$\ref{Thm:equa_R}\xRightarrow{S}\ref{Thm:equa_RH}.$ Fix $n\in \mathbb{N}.$ Let
$\tilde{\omega}=:\frac{\omega}{\omega_n}.$ For $m\in \mathbb{N},$ denote
$\tilde{\mathcal {F}}_m=:\mathcal {F}_{n+m}.$ Since $\omega\in
S,$ we obtain a constant $C$ such that
$$\frac{1}{C}\mathbb{E}(\tilde{\omega}|\tilde{\mathcal {F}}_m)\ensuremath{\ell}eq
\mathbb{E}(\tilde{\omega}|\tilde{\mathcal {F}}_{m+1})\ensuremath{\ell}eq C
\mathbb{E}(\tilde{\omega}|\tilde{\mathcal {F}}_m),~\forall~m\in \mathbb{N}.$$ For
$k\in \mathbb{N}$, define
$\tilde{\tau}_k:=\inf\{m:\mathbb{E}(\tilde{\omega}|\tilde{\mathcal
{F}}_m)>\frac{1}{2}2^{kL}\},$ where $L\geq1$ is a large integer to
be chosen momentarily. Trivially, $\tilde{\tau}_0\equiv0$ and
$\tilde{\tau}_k\geq1,~\forall~k\geq1.$ For $k,~m\geq1,$ we have
\begin{eqnarray*}
\mathbb{E}(\chi_{\{\tilde{\tau}_{k+1}<\infty\}}\chi_{\{\tilde{\tau}_k=m\}}|\tilde{\mathcal
{F}}_{\tilde{\tau}_k})
&\ensuremath{\ell}eq&\mathbb{E}\big(\frac{E(\tilde{\omega}|\mathcal
{F}_{\tilde{\tau}_{k+1}})}{\frac{1}{2}2^{(k+1)L}}\chi_{\{\tilde{\tau}_k=m\}}|\tilde{\mathcal{F}}_m\big)\\
&=&2\mathbb{E}\big(\frac{E(\tilde{\omega}|\mathcal
{F}_{\tilde{\tau}_{k+1}})}{2^{(k+1)L}}|\tilde{\mathcal{F}}_{\tilde{\tau}_k}\big)\chi_{\{\tilde{\tau}_k=m\}}\\
&=&\frac{2}{2^{(k+1)L}}\mathbb{E}\big(\mathbb{E}(\tilde{\omega}|\tilde{\mathcal
{F}}_{\tilde{\tau}_{k+1}})|\tilde{\mathcal{F}}_{\tilde{\tau}_k}\big)\chi_{\{\tilde{\tau}_k=m\}}.
\end{eqnarray*} It is clear that $\tilde{\tau}_k\ensuremath{\ell}eq\tilde{\tau}_{k+1},$ then
\begin{eqnarray*}
\mathbb{E}(\chi_{\{\tilde{\tau}_{k+1}<\infty\}}\chi_{\{\tilde{\tau}_k=m\}}|\tilde{\mathcal
{F}}_{\tilde{\tau}_k})
&\ensuremath{\ell}eq&\frac{2}{2^{(k+1)L}}\mathbb{E}(\tilde{\omega}|\tilde{\mathcal
{F}}_{\tilde{\tau}_k})\chi_{\{\tilde{\tau}_k=m\}}\\
&=&\frac{2}{2^{(k+1)L}}\mathbb{E}(\tilde{\omega}|\tilde{\mathcal
{F}}_m)\chi_{\{\tilde{\tau}_k=m\}}\\
&\ensuremath{\ell}eq&\frac{2C}{2^{(k+1)L}}\mathbb{E}(\tilde{\omega}|\tilde{\mathcal
{F}}_{m-1})\chi_{\{\tilde{\tau}_k=m\}}\\
&\ensuremath{\ell}eq&\frac{2^{kL}C}{2^{(k+1)L}}\chi_{\{\tilde{\tau}_k=m\}}.\end{eqnarray*}
Here
we choose $L$ so large that $\frac{2^{kL}C}{2^{(k+1)L}}\ensuremath{\ell}eq \alpha,$
then$$
\mathbb{E}(\chi_{\{\tilde{\tau}_{k+1}<\infty\}}\chi_{\{\tilde{\tau}_k=m\}}|\tilde{\mathcal
{F}}_{\tilde{\tau}_k})
\ensuremath{\ell}eq\alpha\chi_{\{\tilde{\tau}_k=m\}}.$$ Moreover,
\begin{eqnarray*}
\mathbb{E}(\chi_{\{\tilde{\tau}_{k+1}<\infty\}\cap{\{\tilde{\tau}_k=m\}}}|\mathcal
{F}_{n+m})
&=&\mathbb{E}(\chi_{\{\tilde{\tau}_{k+1}<\infty\}}\chi_{\{\tilde{\tau}_k=m\}}|\mathcal
{F}_{n+m})\\
&=&\mathbb{E}(\chi_{\{\tilde{\tau}_{k+1}<\infty\}}\chi_{\{\tilde{\tau}_k=m\}}|\tilde{\mathcal
{F}}_m)\\
&=&\mathbb{E}(\chi_{\{\tilde{\tau}_{k+1}<\infty\}}\chi_{\{\tilde{\tau}_k=m\}}|\tilde{\mathcal
{F}}_{\tilde{\tau}_k})\\
&\ensuremath{\ell}eq&\alpha\chi_{\{\tilde{\tau}_k=m\}}\ensuremath{\ell}eq\alpha.\end{eqnarray*}
Combining with
$\mathbb{E}_{\tilde{\omega}}(\cdot|\mathcal
{F}_{n+m})=\mathbb{E}_\omega(\cdot|\mathcal
{F}_{n+m})$ and\eqref{R}, we have $$
\mathbb{E}_{\tilde{\omega}}(\chi_{\{\tilde{\tau}_{k+1}<\infty\}\cap{\{\tilde{\tau}_k=m\}}}|\mathcal
{F}_{n+m})\ensuremath{\ell}eq\beta.$$
Thus
\begin{eqnarray*}
\mathbb{E}_{\tilde{\omega}}(\chi_{\{\tilde{\tau}_{k+1}<\infty\}}\chi_{{\{\tilde{\tau}_k=m\}}}|\tilde{\mathcal
{F}}_{\tilde{\tau}_k})
&=&\mathbb{E}_{\tilde{\omega}}(\chi_{\{\tilde{\tau}_{k+1}<\infty\}\cap{\{\tilde{\tau}_k=m\}}}|\mathcal
{F}_{n+m})\chi_{\{\tilde{\tau}_k=m\}}\\
&\ensuremath{\ell}eq&\beta\chi_{\{\tilde{\tau}_k=m\}}.\end{eqnarray*} Consequently,
\begin{eqnarray*}
\mathbb{E}_{\tilde{\omega}}(\chi_{\{\tilde{\tau}_{k+1}<\infty\}}|\tilde{\mathcal
{F}}_{\tilde{\tau}_k})
&=&\mathbb{E}_{\tilde{\omega}}(\chi_{\{\tilde{\tau}_{k+1}<\infty\}}\chi_{\{\tilde{\tau}_k<\infty\}}|\tilde{\mathcal
{F}}_{\tilde{\tau}_k})\\
&=&\sum\ensuremath{\ell}imits_{m=1}\ensuremath{\ell}imits^{\infty}\mathbb{E}_{\tilde{\omega}}
(\chi_{\{\tilde{\tau}_{k+1}<\infty\}}\chi_{\{\tilde{\tau}_k=m\}}|\tilde{\mathcal
{F}}_{\tilde{\tau}_k})\\
&=&\sum\ensuremath{\ell}imits_{m=1}\ensuremath{\ell}imits^{\infty}\beta\chi_{\{\tilde{\tau}_k=m\}}\ensuremath{\ell}eq\beta\chi_{\{\tilde{\tau}_k<\infty\}}.\end{eqnarray*}
It follows that \begin{eqnarray*}
\mathbb{E}_{\tilde{\omega}}(\chi_{\{\tilde{\tau}_{k+1}<\infty\}}|\mathcal
{F}_n)
&=&\mathbb{E}_{\tilde{\omega}}(\mathbb{E}_{\tilde{\omega}}(\chi_{\{\tilde{\tau}_{k+1}<\infty\}}|\tilde{\mathcal
{F}}_{\tilde{\tau}_k})|\tilde{\mathcal
{F}}_{\tilde{\tau}_0})\\
&\ensuremath{\ell}eq&\beta
\mathbb{E}_{\tilde{\omega}}(\chi_{\{\tilde{\tau}_k<\infty\}}|\tilde{\mathcal
{F}}_{\tilde{\tau}_0})\\
&\ensuremath{\ell}eq&\beta^k
\mathbb{E}_{\tilde{\omega}}(\chi_{\{\tilde{\tau}_1<\infty\}}|\tilde{\mathcal
{F}}_{\tilde{\tau}_0})\\
&\ensuremath{\ell}eq&\beta^k,\end{eqnarray*} which is also valid for $k=0.$ For some positive
number $\varepsilon$ to be determined later, we have \begin{eqnarray*}
\mathbb{E}\big((\frac{\omega}{\omega_n})^{1+\varepsilon}|\mathcal
{F}_n\big)
&=&\mathbb{E}(\tilde{\omega}^{1+\varepsilon}|\mathcal
{F}_n)\\
&=&\mathbb{E}(\tilde{\omega}^\varepsilon\tilde{\omega}|\mathcal
{F}_n)\\
&\ensuremath{\ell}eq&\mathbb{E}\big(M^*_n(\tilde{\omega})^\varepsilon\tilde{\omega}|\mathcal
{F}_n\big),\end{eqnarray*} where $M^*_n(\cdot)=\sup\ensuremath{\ell}imits_{m\geq0}\mathbb{E}(\cdot|\mathcal
{F}_{m+n})=\sup\ensuremath{\ell}imits_{m\geq0}\mathbb{E}(\cdot|\tilde{\mathcal
{F}}_{m}).$
Because $\bigcap\ensuremath{\ell}imits_{k\in \mathbb{N}}\{\tilde{\tau}_k=\infty\}={\O},$
$\{\tilde{\tau}_0<\infty\}=\Omega$ and $\mathbb{E}_{\tilde{\omega}}\big(
\chi_{\{\tilde{\tau}_0<\infty\}}|\mathcal
{F}_n\big)=1,$
we obtain that \begin{eqnarray*}
\mathbb{E}\big((\frac{\omega}{\omega_n})^{1+\varepsilon}|\mathcal
{F}_n\big)&\ensuremath{\ell}eq&\sum\ensuremath{\ell}imits_{k=0}\ensuremath{\ell}imits^{\infty}
\mathbb{E}\big((M^*_n(\tilde{\omega})^\varepsilon\tilde{\omega}
\chi_{\{\tilde{\tau}_k<\infty\}\cap\{\tilde{\tau}_{k+1}=\infty\}}|\mathcal
{F}_n\big)\\
&\ensuremath{\ell}eq&\frac{1}{2^\varepsilon}\sum\ensuremath{\ell}imits_{k=0}\ensuremath{\ell}imits^{\infty}
2^{(k+1)\varepsilon L}\mathbb{E}\big(\tilde{\omega}
\chi_{\{\tilde{\tau}_k<\infty\}\cap\{\tilde{\tau}_{k+1}=\infty\}}|\mathcal
{F}_n\big)\\
&=&\frac{1}{2^\varepsilon}\sum\ensuremath{\ell}imits_{k=0}\ensuremath{\ell}imits^{\infty}
2^{(k+1)\varepsilon L}\mathbb{E}_{\tilde{\omega}}\big(
\chi_{\{\tilde{\tau}_k<\infty\}\cap\{\tilde{\tau}_{k+1}=\infty\}}|\mathcal
{F}_n\big)\\
&\ensuremath{\ell}eq&\frac{1}{2^\varepsilon}\sum\ensuremath{\ell}imits_{k=0}\ensuremath{\ell}imits^{\infty}
2^{(k+1)\varepsilon L}\mathbb{E}_{\tilde{\omega}}\big(
\chi_{\{\tilde{\tau}_k<\infty\}}|\mathcal
{F}_n\big)\\
&\ensuremath{\ell}eq&\frac{1}{2^\varepsilon}\sum\ensuremath{\ell}imits_{k=0}\ensuremath{\ell}imits^{\infty}
2^{(k+1)\varepsilon L}\beta^{k-1}\\
&=&\frac{2^{\varepsilon
(L-1)}}{\beta}\sum\ensuremath{\ell}imits_{k=0}\ensuremath{\ell}imits^{\infty}
(2^{\varepsilon L}\beta)^k.\end{eqnarray*}
Choosing an $\varepsilon$ small enough, we have
$\sum\ensuremath{\ell}imits_{k=0}\ensuremath{\ell}imits^{\infty}
(2^{\varepsilon L}\beta)^k<\infty.$ Thus, \eqref{RH} is
valid with $C=\frac{2^{\varepsilon
(L-1)}}{\beta}\sum\ensuremath{\ell}imits_{k=0}\ensuremath{\ell}imits^{\infty}
(2^{\varepsilon L}\beta)^k$ and $q=1+\varepsilon.$
$\ref{Thm:equa_RH}\ensuremath{\mathbb{R}}ightarrow\ref{Thm:equa_RR}.$ For $A\in\mathcal {F},$ it is clear that
$\mathbb{E}_\omega(\chi_{A}|\mathcal
{F}_n)\omega_n=E(\omega\chi_{A}|\mathcal {F}_n).$ Applying
H\"{o}lder's inequality, we obtain that
$$\mathbb{E}_\omega(\chi_{A}|\mathcal {F}_n)\omega_n
\ensuremath{\ell}eq \mathbb{E}(\omega^{1+\varepsilon}|\mathcal
{F}_n)^{\frac{1}{1+\varepsilon}}\mathbb{E}(\chi_{A}|\mathcal
{F}_n)^{\frac{\varepsilon}{1+\varepsilon}}.$$ It follows from
\eqref{RH} that $$\mathbb{E}_\omega(\chi_{A}|\mathcal {F}_n)\ensuremath{\ell}eq
C^{\frac{1}{1+\varepsilon}}\mathbb{E}(\chi_{A}|\mathcal
{F}_n)^{\frac{\varepsilon}{1+\varepsilon}},$$ which implies
\eqref{RR} with
$\varepsilon'=\frac{\varepsilon}{1+\varepsilon}.$
$\ref{Thm:equa_RR}\ensuremath{\mathbb{R}}ightarrow\ref{Thm:equa_rR}.$ For $\varepsilon'$ and $C$ in the assumption \ref{Thm:equa_RR}, we fix $\alpha'$ small enough such that
$\beta'=C\alpha'^{\varepsilon'}<1.$ For $B\in\mathcal {F}$ with
$\mathbb{E}(\chi_B|\mathcal {F}_n)\ensuremath{\ell}eq\alpha',$ it follows from \eqref{RR}
that$$\mathbb{E}_\omega(\chi_B|\mathcal {F}_n)\ensuremath{\ell}eq\beta'.$$ Thus for all $A\in \mathcal{F}$ we have
$$\mathbb{E}(\chi_A|\mathcal {F}_n)>1-\alpha'\ensuremath{\mathbb{R}}ightarrow
\mathbb{E}_\omega(\chi_A|\mathcal {F}_n)>1-\beta'.$$
Let $\alpha=1-\beta'$ and $\beta=1-\alpha'.$ Then
$$\mathbb{E}(\chi_A|\mathcal {F}_n)>\beta\ensuremath{\mathbb{R}}ightarrow
\mathbb{E}_\omega(\chi_A|\mathcal {F}_n)>\alpha,$$
which is equivalent to \ref{Thm:equa_rR}.
$\ref{Thm:equa_rR}\xRightarrow{S}\ref{Thm:equa_Ap}.$ Let $\omega_1=\frac{1}{\omega}.$ Then
$\omega_1$ is a weight relative to $\omega d\mu.$ Recalling the definition of $\hat{\mathbb{E}}(\cdot|\mathcal {F}_n),$ we deduce that
$\mathbb{E}(\cdot|\mathcal {F}_n)=\hat{\mathbb{E}}_{\omega_1}(\cdot|\mathcal
{F}_n),~\forall ~n\in \mathbb{N}.$ It follows from \eqref{rR} that
$$\hat{\mathbb{E}}(\chi_A|\mathcal {F}_n)\ensuremath{\ell}eq\alpha<1\ensuremath{\mathbb{R}}ightarrow
\hat{\mathbb{E}}_{\omega_1}(\chi_A|\mathcal {F}_n)\ensuremath{\ell}eq\beta<1,~\forall~A\in
\mathcal{F}.$$ Combining with $\omega_1\in S_{(\omega d\mu)},$ similar to
$\ref{Thm:equa_R}\xRightarrow{S}\ref{Thm:equa_RH},$ we have $\varepsilon$ and $C$ such that
$$\hat{\mathbb{E}}(\omega_1^{1+\varepsilon}|\mathcal {F}_n)\ensuremath{\ell}eq
C\hat{\mathbb{E}}(\omega_1|\mathcal {F}_n)^{1+\varepsilon},~\forall~n\in
\mathbb{N},$$ that is, $$\omega_n\mathbb{E}(\omega^{-\varepsilon}|\mathcal
{F}_n)^{\frac{1}{\varepsilon}}\ensuremath{\ell}eq
C^{\frac{1}{\varepsilon}},~\forall~n\in \mathbb{N}.$$ Thus \eqref{Ap} is
valid with
$\varepsilon=\frac{1}{p-1}.$
\end{proof}
\section{$A_{\infty}$ Weights without Additional Assumptions}\ensuremath{\ell}abel{without}
In this section, we study relations between different characterizations of $A_{\infty}$ weights without additional assumptions. These relations are showed in Figure \ref{figure}.
\begin{center}
\begin{figure}
\caption{Relations without additional assumptions}
\end{figure}
\end{center}
We first prove equivalent characterizations in Theorems \ref{thm:Reverse-test}, \ref{thm:Cond} and \ref{thm:exp-s}.
\begin{proof}[Proof of Theorem \ref{thm:Reverse-test}]
\eqref{Reverse-test1}$\ensuremath{\mathbb{R}}ightarrow$\eqref{Reverse-test2} This is $\ref{Thm:equa_RH}\ensuremath{\mathbb{R}}ightarrow\ref{Thm:equa_RR}$ in the proof of Theorem \ref{Thm:equa}. It holds without additional assumptions $\omega\in S.$
\eqref{Reverse-test2}$\ensuremath{\mathbb{R}}ightarrow$ \eqref{Reverse-test1} Let $B\in \mathcal {F}_n$ and let $E_{\ensuremath{\ell}ambda}=\{\frac{\omega}{\omega_n}>\ensuremath{\ell}ambda\}.$ We have
\begin{eqnarray*}
\ensuremath{\ell}ambda\mu(B\cap E_{\ensuremath{\ell}ambda})
&\ensuremath{\ell}eq&\int_{B\cap E_{\ensuremath{\ell}ambda}}\frac{\omega}{\omega_n}d\ensuremath{\ell}ambda\\
&=&\int_{B}\frac{\omega\chi_{E_{\ensuremath{\ell}ambda}}}{\omega_n}d\mu\\
&=&\int_{B}\frac{\mathbb{E}(\omega\chi_{E_{\ensuremath{\ell}ambda}}|\mathcal
{F}_n)}{\omega_n}d\mu\\
&=&\int_{B}\mathbb{E}_{\omega}(\chi_{E_{\ensuremath{\ell}ambda}}|\mathcal
{F}_n)d\mu.
\end{eqnarray*}
It follows from \eqref{Reverse-test2} and H\"{o}lder's inequality that
\begin{eqnarray*}
\ensuremath{\ell}ambda\mu(B\cap E_{\ensuremath{\ell}ambda})
&\ensuremath{\ell}eq&C\int_{B}\mathbb{E}(\chi_{E_{\ensuremath{\ell}ambda}}|\mathcal{F}_n)^{\varepsilon'}d\mu\\
&\ensuremath{\ell}eq&C\big(\int_{B}\mathbb{E}(\chi_{E_{\ensuremath{\ell}ambda}}|\mathcal{F}_n)d\mu\big)^{\varepsilon'}\mu(B)^{1-\varepsilon'}\\
&=&C\mu(B\cap E_{\ensuremath{\ell}ambda})^{\varepsilon'}\mu(B)^{1-\varepsilon'}.
\end{eqnarray*}
Thus, we have $\ensuremath{\ell}ambda^{\frac{1}{1-\varepsilon'}}\mu(B\cap E_{\ensuremath{\ell}ambda})\ensuremath{\ell}eq C^{\frac{1}{1-\varepsilon'}}\mu(B).$
Let $1<q<\frac{1}{1-\varepsilon'}.$ We obtain that
\begin{eqnarray*}
\int_{B}(\frac{\omega}{\omega_n})^q\mu
&=&q\int_{0}^{+\infty}\ensuremath{\ell}ambda^{q-1}\mu(B\cap E_{\ensuremath{\ell}ambda})d\ensuremath{\ell}ambda\\
&=&q\int_{0}^{1}\ensuremath{\ell}ambda^{q-1}\mu(B\cap E_{\ensuremath{\ell}ambda})d\ensuremath{\ell}ambda
+q\int_{1}^{+\infty}\ensuremath{\ell}ambda^{q-1}\mu(B\cap E_{\ensuremath{\ell}ambda})d\ensuremath{\ell}ambda\\
&\ensuremath{\ell}eq&q(\int_{0}^{1}\ensuremath{\ell}ambda^{q-1}d\ensuremath{\ell}ambda)\mu(B)
+qC^{\frac{1}{1-\varepsilon'}}(\int_{1}^{+\infty}\ensuremath{\ell}ambda^{q-1-\frac{1}{1-\varepsilon'}}d\ensuremath{\ell}ambda)\mu(B)\\
&=&\mu(B)+\frac{C^{\frac{1}{1-\varepsilon'}}q}{\frac{1}{1-\varepsilon'}-q}\mu(B)\\
&=&(1+\frac{C^{\frac{1}{1-\varepsilon'}}q}{\frac{1}{1-\varepsilon'}-q})\mu(B).
\end{eqnarray*}
Since $B$ is arbitrary, we obtain
$$\mathbb{E}(\omega^q|\mathcal {F}_n)(\frac{1}{\omega_n})^q
\ensuremath{\ell}eq 1+\frac{C^{\frac{1}{1-\varepsilon'}}q}{\frac{1}{1-\varepsilon'}-q}.$$
Then $\mathbb{E}(\omega^q|\mathcal {F}_n)\ensuremath{\ell}eq (1+\frac{C^{\frac{1}{1-\varepsilon'}}q}{\frac{1}{1-\varepsilon'}-q})(\omega_n)^q.$
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:Cond}]
Because \eqref{thm:Cond1}$\ensuremath{\mathbb{R}}ightarrow$\eqref{thm:Cond2} is $\ref{Thm:equa_wAinfty}\ensuremath{\mathbb{R}}ightarrow\ref{Thm:equa_R}$ in the proof of Theorem \ref{Thm:equa} without further
assumptions $\omega\in S,$
it suffices to prove \eqref{thm:Cond2}$\ensuremath{\mathbb{R}}ightarrow$
\eqref{thm:Cond1}. Let $0<\gamma <1-\beta.$ Setting $E=\{\omega\ensuremath{\ell}eq\gamma\omega_n\},$ we have
\begin{eqnarray*}
\mathbb{E}_\omega(\chi_{E}|\mathcal {F}_n)&=&\frac{\mathbb{E}(\omega\chi_{E}|\mathcal {F}_n)}{\omega_n}\\
&\ensuremath{\ell}eq&\frac{\mathbb{E}(\gamma\omega_n\chi_{E}|\mathcal {F}_n)}{\omega_n}\\
&\ensuremath{\ell}eq&\gamma\mathbb{E}(\chi_{E}|\mathcal {F}_n)\\
&\ensuremath{\ell}eq&\gamma<1-\beta.
\end{eqnarray*}
It follows that $\mathbb{E}_\omega(\chi_{E^c}|\mathcal {F}_n)>\beta.$ In view of \eqref{thm:Cond2}, we obtain that $$\mathbb{E}(\chi_{E^c}|\mathcal {F}_n)>\alpha,$$
which implies $\mathbb{E}(\chi_{E}|\mathcal {F}_n)\ensuremath{\ell}eq1-\alpha.$
Thus $\eqref{thm:Cond1}$ is valid with $1-\alpha<\delta<1.$
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:exp-s}]
In view of Lemma \ref{key_lemma_sta}, we have that
\begin{equation*}
\mathbb{E}(\omega^s|\mathcal {F}_n)^{\frac{1}{s}}\downarrow\exp \mathbb{E}(\ensuremath{\ell}og\omega|\mathcal {F}_n),~\text{~as~}s\downarrow0^+,
\end{equation*}
which establishes the
equivalence between \eqref{thm:exp-s2} and \eqref{thm:exp-s1}.
\end{proof}
In the rest of this section, we prove Theorems \ref{thm:imp1}, \ref{thm:imp2} and \ref{thm:Wi}.
These are one-way implications.
\begin{proof}[Proof of Theorem \ref{thm:imp1}]
$\eqref{thm:lev}\ensuremath{\mathbb{R}}ightarrow\eqref{thm:rev}$
Let $B\in \mathcal {F}_n.$ We have
\begin{eqnarray*}
\int_B(\frac{\omega}{\omega_n})^{1+\delta}d\mu
&=&\int_B(\frac{\omega}{\omega_n})^{\delta}\frac{\omega}{\omega_n}d\mu\\
&=&\delta\int_0^{+\infty}\ensuremath{\ell}ambda^{\delta-1}
\frac{\omega}{\omega_n}(B\cap\{\frac{\omega}{\omega_n}>\ensuremath{\ell}ambda\})d\ensuremath{\ell}ambda\\
&=&\delta\int_0^{1}\ensuremath{\ell}ambda^{\delta-1}
\frac{\omega}{\omega_n}(B\cap\{\frac{\omega}{\omega_n}>\ensuremath{\ell}ambda\})d\ensuremath{\ell}ambda\\
&&+\delta\int_1^{+\infty}\ensuremath{\ell}ambda^{\delta-1}
\frac{\omega}{\omega_n}(B\cap\{\frac{\omega}{\omega_n}>\ensuremath{\ell}ambda\})d\ensuremath{\ell}ambda.
\end{eqnarray*}
It follows that
\begin{eqnarray*}
\delta\int_0^{1}\ensuremath{\ell}ambda^{\delta-1}
\frac{\omega}{\omega_n}(B\cap\{\frac{\omega}{\omega_n}>\ensuremath{\ell}ambda\})d\ensuremath{\ell}ambda
&\ensuremath{\ell}eq&\frac{\omega}{\omega_n}(B)\delta\int_0^{1}\ensuremath{\ell}ambda^{\delta-1}
d\ensuremath{\ell}ambda\\
&=&\mu(B),
\end{eqnarray*}
where we have used $\frac{\omega}{\omega_n}(B)
=\int_B\frac{\omega}{\omega_n}d\mu
=\int_B\frac{\omega_n}{\omega_n}d\mu=\mu(B).$
Using \eqref{thm:equ_lev}, we obtain the following estimate
\begin{eqnarray*}
&~&\delta\int_{1}^{+\infty}\ensuremath{\ell}ambda^{\delta-1}
\frac{\omega}{\omega_n}(B\cap\{\frac{\omega}{\omega_n}>\ensuremath{\ell}ambda\})d\ensuremath{\ell}ambda\\
&\ensuremath{\ell}eq&C\delta\int_1^{+\infty}\ensuremath{\ell}ambda^{\delta}\mu(B\cap\{\frac{\omega}{\omega_n}>\beta\ensuremath{\ell}ambda\})
d\ensuremath{\ell}ambda\\
&=&\frac{C\delta}{\beta^{1+\delta}}\int_\beta^{+\infty}\ensuremath{\ell}ambda^{\delta}(B\cap\{\frac{\omega}{\omega_n}>\ensuremath{\ell}ambda\})
d\ensuremath{\ell}ambda\\
&\ensuremath{\ell}eq&\frac{C\delta}{(1+\delta)\beta^{1+\delta}}\int_B(\frac{\omega}{\omega_n})^{1+\delta}d\mu.
\end{eqnarray*}
Because of $\ensuremath{\ell}im\ensuremath{\ell}imits_{\delta\rightarrow0}\frac{C\delta}{(1+\delta)\beta^{1+\delta}}=0,$
we can choose $\delta$ such that $\frac{C\delta}{(1+\delta)\beta^{1+\delta}}<\frac{1}{2}.$ Then we have
$$\int_B(\frac{\omega}{\omega_n})^{1+\delta}d\mu\ensuremath{\ell}eq2\mu(B).$$
Since $B\in\mathcal {F}_n$ is arbitrary, it follows that $$\mathbb{E}(\omega^{1+\delta}|\mathcal {F}_n)\ensuremath{\ell}eq2(\omega_n)^{1+\delta}.$$
$\eqref{thm:rev} \ensuremath{\mathbb{R}}ightarrow\eqref{thm:log}$
Let $E_k=\{2^k<\frac{\omega}{\omega_n}\ensuremath{\ell}eq2^{k+1}\}$ for $k\in \mathbb{N}.$ In view of \eqref{thm:rev}, we have
\begin{eqnarray*}
2^{kp}\mathbb{E}(\chi_{E_k}|\mathcal {F}_n)&\ensuremath{\ell}eq&\mathbb{E}\big((\chi_{E_k}\frac{\omega}{\omega_n})^{p}|\mathcal {F}_n\big)\\&\ensuremath{\ell}eq&\mathbb{E}\big((\frac{\omega}{\omega_n})^{p}|\mathcal {F}_n\big)\\&\ensuremath{\ell}eq& C,
\end{eqnarray*}
which implies $\mathbb{E}(\chi_{E_k}|\mathcal {F}_n)\ensuremath{\ell}eq C2^{-kp}.$
It follows that
\begin{eqnarray*}
\mathbb{E}_{\omega}(\ensuremath{\ell}og^+\frac{\omega}{\omega_n}|\mathcal {F}_n)
&=&\mathbb{E}_{\omega}(\sum\ensuremath{\ell}imits_{k=0}^{+\infty}\chi_{E_k}\ensuremath{\ell}og^+\frac{\omega}{\omega_n}|\mathcal {F}_n)\\
&=&\sum\ensuremath{\ell}imits_{k=0}^{+\infty}\mathbb{E}_{\omega}(\chi_{E_k}\ensuremath{\ell}og^+\frac{\omega}{\omega_n}|\mathcal {F}_n)\\
&=&\sum\ensuremath{\ell}imits_{k=0}^{+\infty}\mathbb{E}(\chi_{E_k}\frac{\omega}{\omega_n}\ensuremath{\ell}og^+\frac{\omega}{\omega_n}|\mathcal {F}_n)\\
&\ensuremath{\ell}eq&\sum\ensuremath{\ell}imits_{k=0}^{+\infty}2^{k+1} (k+1)\mathbb{E}(\chi_{E_k}|\mathcal {F}_n)\\
&\ensuremath{\ell}eq&C\sum\ensuremath{\ell}imits_{k=0}^{+\infty}(k+1)2^{k+1}2^{-kq},
\end{eqnarray*}
where the series $\sum\ensuremath{\ell}imits_{k=0}^{+\infty}(k+1)2^{k+1}2^{-kq}$ is convergent. Then we have \eqref{thm:equ_log}.
$\eqref{thm:log}\ensuremath{\mathbb{R}}ightarrow\eqref{thm:imp_Cond2}$
Let $\mathbb{E}(\chi_A|\mathcal {F}_n)\ensuremath{\ell}eq\alpha<1.$ Recall that $ab\ensuremath{\ell}eq a\ensuremath{\ell}og a-a+e^b$ where $a>1$ and $b\geq0.$
Then
\begin{eqnarray*}
\mathbb{E}_\omega(\chi_A|\mathcal {F}_n)
&=& \mathbb{E}_\omega(\chi_{A\cap\{\frac{\omega}{\omega_n}\ensuremath{\ell}eq1\}}|\mathcal {F}_n)+ \mathbb{E}_\omega(\chi_{A\cap\{\frac{\omega}{\omega_n}>1\}}|\mathcal {F}_n)\\
&=& \mathbb{E}(\frac{\omega}{\omega_n}\chi_{A\cap\{\frac{\omega}{\omega_n}\ensuremath{\ell}eq1\}}|\mathcal {F}_n)+ \mathbb{E}(\frac{\omega}{\omega_n}\chi_{A\cap\{\frac{\omega}{\omega_n}>1\}}|\mathcal {F}_n)\\
&\ensuremath{\ell}eq&
\mathbb{E}(\chi_A|\mathcal {F}_n)+\frac{1}{b+1}\mathbb{E}(\frac{\omega}{\omega_n}\ensuremath{\ell}og^+\frac{\omega}{\omega_n}+e^b\chi_A|\mathcal {F}_n)\\
&=&
\mathbb{E}(\chi_A|\mathcal {F}_n)+\frac{1}{b+1}\mathbb{E}(\frac{\omega}{\omega_n}\ensuremath{\ell}og^+\frac{\omega}{\omega_n}|\mathcal{F}_n)+\frac{e^b}{b+1}\mathbb{E}(\chi_A|\mathcal {F}_n)\\
&\ensuremath{\ell}eq&\alpha(1+\frac{e^b}{b+1})+\frac{C}{b+1}.
\end{eqnarray*}
Setting $b=2C-1,$ we can pick an $\alpha$ small enough that
$\alpha(1+\frac{e^b}{b+1})\ensuremath{\ell}eq\frac{1}{4}$ because of $\ensuremath{\ell}im\ensuremath{\ell}imits_{\alpha\rightarrow0}\alpha(1+\frac{e^b}{b+1})=0.$
Thus $\mathbb{E}_\omega(\chi_A|\mathcal {F}_n)\ensuremath{\ell}eq\frac{3}{4}.$
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:imp2}] It suffices to prove $\eqref{thm:imp2_Log} \ensuremath{\mathbb{R}}ightarrow\eqref{thm:imp2_Mid}\ensuremath{\mathbb{R}}ightarrow\eqref{thm:imp2_Dou},$ because $\eqref{thm:imp2_Ap}\ensuremath{\mathbb{R}}ightarrow\eqref{thm:imp2_Log}$ is the one $\ref{Thm:equa_Ap}\ensuremath{\mathbb{R}}ightarrow\ref{Thm:equa_A_exp_infty}$ in Theorem \ref{Thm:equa}.
$\eqref{thm:imp2_Log} \ensuremath{\mathbb{R}}ightarrow\eqref{thm:imp2_Mid}$
Let $E=\{\omega>m(\omega,n)\}.$ Using H\"{o}lder's inequality for the conditional expectation,
we have
\begin{eqnarray*}
\mathbb{E}(\omega^s\chi_E|\mathcal {F}_n)
&\ensuremath{\ell}eq&\mathbb{E}(\omega|\mathcal {F}_n)^s\mathbb{E}(\chi_E|\mathcal {F}_n)^{1-s}\\
&\ensuremath{\ell}eq&2^{s-1}C^s\mathbb{E}(\omega^s|\mathcal {F}_n),
\end{eqnarray*}
where we have used Theorem \ref{thm:exp-s}.
It follows that $\mathbb{E}(\omega^s\chi_E|\mathcal {F}_n)
\ensuremath{\ell}eq\frac{3}{4}\mathbb{E}(\omega^s|\mathcal {F}_n)$ provided $2^{s-1}C^s<\frac{3}{4}.$
Then $\mathbb{E}(\omega^s\chi_{E^c}|\mathcal {F}_n)
\geq\frac{1}{4}\mathbb{E}(\omega^s|\mathcal {F}_n).$ Thus
\begin{eqnarray*}
\frac{1}{4}\mathbb{E}(\omega|\mathcal {F}_n)^s &\ensuremath{\ell}eq&\frac{1}{4}C^s\mathbb{E}(\omega^s|\mathcal {F}_n)\\ &\ensuremath{\ell}eq&C^s\mathbb{E}(\omega^s\chi_{E^c}|\mathcal {F}_n)\\
&\ensuremath{\ell}eq&C^s\mathbb{E}(m(\omega,n)^s\chi_{E^c}|\mathcal {F}_n)\\
&\ensuremath{\ell}eq&C^s\mathbb{E}(m(\omega,n)^s|\mathcal {F}_n)\\
&=&C^sm(\omega,n)^s,
\end{eqnarray*}
which implies $\mathbb{E}(\omega|\mathcal {F}_n)\ensuremath{\ell}eq4^{\frac{1}{s}}Cm(\omega,n).$
$\eqref{thm:imp2_Mid}\ensuremath{\mathbb{R}}ightarrow\eqref{thm:imp2_Dou}$ Let $\alpha<\frac{1}{4}$ and $\mathbb{E}(\chi_E|\mathcal {F}_n)<\frac{1}{4}.$ We claim that $$\mathbb{E}(\chi_{E^c\cap\{\omega\geq m(\omega,n)\}}|\mathcal {F}_n)\geq\frac{1}{4}.$$
Indeed, we have
\begin{eqnarray*}\mathbb{E}(\chi_{E\cup\{\omega< m(\omega,n)\}}|\mathcal {F}_n)
&\ensuremath{\ell}eq&\mathbb{E}(\chi_E|\mathcal {F}_n)+\mathbb{E}(\chi_{\{\omega< m(\omega,n)\}}|\mathcal {F}_n)\\
&<&\frac{1}{4}+\frac{1}{2}=\frac{3}{4}.\end{eqnarray*}
This proves that $$\mathbb{E}(\chi_{E^c\cap\{\omega\geq m(\omega,n)\}}|\mathcal {F}_n)\geq\frac{1}{4}.$$
It follows that
\begin{eqnarray*}
\frac{1}{4}\omega_n
&\ensuremath{\ell}eq& \frac{C}{4}m(\omega,n)\\
&\ensuremath{\ell}eq& Cm(\omega,n)\mathbb{E}(\chi_{E^c\cap\{\omega\geq m(\omega,n)\}}|\mathcal {F}_n)\\
&=& C\mathbb{E}(m(\omega,n)\chi_{E^c\cap\{\omega\geq m(\omega,n)\}}|\mathcal {F}_n)\\
&\ensuremath{\ell}eq& C\mathbb{E}(\omega\chi_{E^c\cap\{\omega\geq m(\omega,n)\}}|\mathcal {F}_n)\\
&\ensuremath{\ell}eq& C\mathbb{E}(\omega\chi_{E^c}|\mathcal {F}_n).
\end{eqnarray*}
Then we have $\omega_n\ensuremath{\ell}eq 4C\mathbb{E}(\omega\chi_{E^c}|\mathcal {F}_n)$ which implies $1\ensuremath{\ell}eq 4C\mathbb{E}_{\omega}(\chi_{E^c}|\mathcal {F}_n).$ Thus $\mathbb{E}_{\omega}(\chi_{E}|\mathcal {F}_n)<\beta$ with $\beta=1-\frac{1}{4C}.$
\end{proof}
Before we prove Theorem \ref{thm:Wi}, we make a couple of observations on the tailed maximal operator which are Lemmas
\ref{doob_con} and \ref{doob_imp}. Lemma \ref{doob_con} is
the conditional version of Doob's inequality which appeared in \cite[p.189]{MR1301765}.
\begin{lemma}\ensuremath{\ell}abel{doob_con} Let $p>1.$ There exists $C>1$ such that for all $n\in \mathbb{N}$ we have
$$\mathbb{E}(M^*_n(f)^{p}|\mathcal {F}_n)\ensuremath{\ell}eq C\mathbb{E}(f^{p}|\mathcal {F}_n).$$
\end{lemma}
Lemma \ref{doob_imp} shows that the tailed operator has the following local property.
\begin{lemma}\ensuremath{\ell}abel{doob_imp} Let $f\in L^1.$ For all $\ensuremath{\ell}ambda>0$ and $n\in\mathbb{N}$ we have
$$\mu(B\cap\{M^*_n(f)>\ensuremath{\ell}ambda\}\ensuremath{\ell}eq\frac{2}{\ensuremath{\ell}ambda}\int_{B\cap\{|f|>\ensuremath{\ell}ambda/2\}}|f|d\mu,$$
where $B\in \mathcal{F}_n.$
\end{lemma}
\begin{proof} [Proof of Lemma \ref{doob_imp}] Because of $B\in \mathcal{F}_n,$ we have
\begin{eqnarray*}
&~&\mu(B\cap\{M^*_n(f)>\ensuremath{\ell}ambda\}\\&=&\mu(\{M^*_n(f\chi_{B})>\ensuremath{\ell}ambda\})\\
&\ensuremath{\ell}eq&\mu(\{M^*_n(f\chi_{B\cap\{|f|>\ensuremath{\ell}ambda/2\}})>\ensuremath{\ell}ambda/2\})+\mu(\{M^*_n(f\chi_{B\cap\{|f|\ensuremath{\ell}eq\ensuremath{\ell}ambda/2\}})>\ensuremath{\ell}ambda/2\})\\
&=&\mu(\{M^*_n(f\chi_{B\cap\{|f|>\ensuremath{\ell}ambda/2\}})>\ensuremath{\ell}ambda/2\}).
\end{eqnarray*}
For $m\in \mathbb{N},$ denote
$\tilde{\mathcal {F}}_m=:\mathcal {F}_{n+m}$ and $\tilde{f}_m=:\mathbb{E}(f\chi_{B\cap\{|f|>\ensuremath{\ell}ambda/2\}}|\mathcal {F}_{n+m}).$
It follows that $(\tilde{f}_m)_{m\geq0}$ is a martingale with respect
to $(\Omega,\mathcal{F},\mu,(\tilde{\mathcal{F}}_m)_{m\geq0}),$ which leads to
$\tilde{M}(\cdot)=M^*_n(\cdot).$ Using the weak $(1,1)$ type inequality for $\tilde{M}(\cdot),$ we have
\begin{eqnarray*}
\mu(\{M^*_n(f\chi_{B\cap\{|f|>\ensuremath{\ell}ambda/2\}})>\ensuremath{\ell}ambda/2\})
&\ensuremath{\ell}eq&\frac{2}{\ensuremath{\ell}ambda}\int_{B\cap\{|f|>\ensuremath{\ell}ambda/2\}}|f|d\mu,
\end{eqnarray*}
which completes the proof.
\end{proof}
Using these Lemmas, we give the proof of Theorem \ref{thm:Wi}.
\begin{proof}[Proof of Theorem \ref{thm:Wi}] In each case, we show that there exists $C>1$ such that
$$\int_{B}M^*_n(\omega)d\mu\ensuremath{\ell}eq C\int_{B}\omega d\mu,$$
for all $B\in \mathcal {F}_n.$ Since B is arbitrary, this proves $\mathbb{E}(M^*_n(\omega)|\mathcal {F}_n)\ensuremath{\ell}eq C\omega_n.$
\eqref{thm:Wi_Ap}
Because $\omega\in A_p,$ we have $\omega^{1-p^\prime}\in A_{p^\prime}$ with $1/p+1/p'=1.$ Recall that $M^*_n$ is bounded on $L^{p^\prime}(\omega^{1-p^\prime})$(\cite[Theorem 6.6.3]{MR1224450}). Then
\begin{eqnarray*}
\int_{B}M^*_n(\omega)d\mu
&=&\int_{B}M^*_n(\omega)\omega^{-\frac{1}{p}}\omega^{\frac{1}{p}}d\mu\\
&\ensuremath{\ell}eq&(\int_{B}M^*_n(\omega)^{p^\prime}\omega^{-\frac{1}{p-1}}d\mu)^{\frac{1}{p^{\prime}}}
(\int_{B}\omega d\mu)^{\frac{1}{p}}\\
&\ensuremath{\ell}eq&C(\int_{B}\omega^{p^\prime}\omega^{-\frac{1}{p-1}}d\mu)^{\frac{1}{p^{\prime}}}
(\int_{B}\omega d\mu)^{\frac{1}{p}}\\
&=&C(\int_{B}\omega d\mu)^{\frac{1}{p^{\prime}}}
(\int_{B}\omega d\mu)^{\frac{1}{p}}\\
&\ensuremath{\ell}eq&C\int_{B}\omega d\mu.
\end{eqnarray*}
\eqref{thm:Wi_rev} By Jensen's inequality for the
conditional expectation, we have
\begin{eqnarray*}
\int_{B}M^*_n(\omega)d\mu
&=&\int_{B}\mathbb{E}(M^*_n(\omega)|\mathcal {F}_n)d\mu\\
&\ensuremath{\ell}eq&\int_{B}\mathbb{E}(M^*_n(\omega)^{p}|\mathcal {F}_n)^{\frac{1}{p}}d\mu.
\end{eqnarray*}
Because of the conditional version of Doob's inequality(Lemma \ref{doob_con}) and $\omega\in\bigcup\ensuremath{\ell}imits_{1<q<\infty} RH_q,$ we obtain that
\begin{eqnarray*}
\int_{B}\mathbb{E}(M^*_n(\omega)^{p}|\mathcal {F}_n)^{\frac{1}{p}}d\mu
&\ensuremath{\ell}eq&C\int_{B}\mathbb{E}(\omega^{p}|\mathcal {F}_n)^{\frac{1}{p}}d\mu\\
&\ensuremath{\ell}eq&C\int_{B}\omega d\mu.
\end{eqnarray*}
Thus
$$\int_{B}M^*_n(\omega)d\mu\ensuremath{\ell}eq C\int_{B}\omega d\mu.$$
\eqref{thm:Wi_log} In view of Lemma \ref{doob_imp}, we have the following estimate
\begin{eqnarray*}
\int_{B}M^*_n(\frac{\omega}{\omega_n})d\mu
&=&\int_0^{+\infty}\mu(B\cap\{M^*_n(\frac{\omega}{\omega_n})>\ensuremath{\ell}ambda\}
)d\ensuremath{\ell}ambda\\
&\ensuremath{\ell}eq&\int_0^{2}\mu(B)d\ensuremath{\ell}ambda+\int_2^{+\infty}\mu(B\cap\{M^*_n(\frac{\omega}{\omega_n})>\ensuremath{\ell}ambda\}
)d\ensuremath{\ell}ambda\\
&\ensuremath{\ell}eq&2\mu(B)+\int_2^{+\infty}\frac{2}{\ensuremath{\ell}ambda}\int_{B\cap\{ \frac{\omega}{\omega_n}>\frac{\ensuremath{\ell}ambda}{2}\}}\frac{\omega}{\omega_n}d\mu d\ensuremath{\ell}ambda\\
&=&2\mu(B)+2\int_1^{+\infty}\frac{1}{\ensuremath{\ell}ambda}\int_{B\cap\{\frac{\omega}{\omega_n}>\ensuremath{\ell}ambda\}}\frac{\omega}{\omega_n}d\mu d\ensuremath{\ell}ambda\\
&=&2\mu(B)+2\int_B\frac{\omega}{\omega_n}\ensuremath{\ell}og^+\frac{\omega}{\omega_n}d\mu\\
&=&2\mu(B)+2\int_B\mathbb{E}_{\omega}(\ensuremath{\ell}og^+\frac{\omega}{\omega_n}|\mathcal {F}_n)d\mu.
\end{eqnarray*}
It follows from \eqref{thm:Wi_log} that
$$\int_{B}M^*_n(\frac{\omega}{\omega_n})d\mu\ensuremath{\ell}eq C\mu(B).$$
Thus $\mathbb{E}\big(M^*_n(\frac{\omega}{\omega_n})|\mathcal {F}_n\big)\ensuremath{\ell}eq C,$ which implies
$\mathbb{E}\big(M^*_n(\omega)|\mathcal {F}_n\big)\ensuremath{\ell}eq C\omega_n.$
\eqref{Thm:Wi_exp} In view of Theorem \ref{thm:exp-s},
there exists $C>1$ such that for every $s\in(0,1)$ we have
$$
\mathbb{E}(\omega|\mathcal {F}_n)\ensuremath{\ell}eq C\mathbb{E}(\omega^s|\mathcal {F}_n)^{\frac{1}{s}}.
$$
Then $M^*_n(\omega)\ensuremath{\ell}eq C\cdot M^*_n(\omega^s)^{\frac{1}{s}}.$ Let $p=:\frac{1}{s}.$ Using Doob's inequality,
we have
\begin{eqnarray*}
\int_{B}M^*_n(\omega)d\mu
&\ensuremath{\ell}eq&C\int_{B}M^*_n(\omega^s)^{\frac{1}{s}}d\mu\\
&\ensuremath{\ell}eq&C(p')^p\int_{B}\omega d\mu,
\end{eqnarray*}
where $\frac{1}{p}+\frac{1}{p'}=1.$
Following from $(p')^p\downarrow e$ as $s\downarrow0,$ we obtain
$$\int_{B}M^*_n(\omega)d\mu
\ensuremath{\ell}eq C e \int_{B}\omega d\mu.$$
\end{proof}
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|
\begin{document}
\title{Sample-Optimal Quantum Process Tomography with Non-Adaptive Incoherent Measurements}
\begin{abstract}
How many copies of a quantum process are necessary and sufficient to construct an approximate classical description of it? We extend the result of Surawy-Stepney, Kahn, Kueng, and Guta (2022)
to show that $\tilde{\mathcal{O}}(\din^3\dout^3/\varepsilon^2)$ copies are sufficient to learn any quantum channel $\mathds{C}^{\din\times \din}\rightarrow\mathds{C}^{\dout\times \dout}$ to within $\varepsilon$ in diamond norm. Moreover, we show that $\Omega(\din^3\dout^3/\varepsilon^2)$ copies are necessary for any strategy using incoherent non-adaptive measurements. This lower bound applies even for ancilla-assisted strategies.
\end{abstract}
We consider the problem of quantum process tomography which consists of approximating an arbitrary quantum channel--any linear map that preserves the axioms of quantum mechanics. This task is an important tool in quantum information processing and quantum control which has been performed in actual experiments (see e.g. \cite{o2004quantum,bialczak2010quantum,yamamoto2010quantum}). Given a quantum channel $\cN: \dC^{\din\times \din} \rightarrow \dC^{\dout\times \dout}$ as a black box, a learner could choose the input state and send it through the unknown quantum channel. Then, it can only extract classical information by performing
a measurement on the output state. It repeats this procedure at different steps. After collecting a sufficient amount of classical data, the goal is to return a quantum channel $\Tilde{\cN}$ satisfying:
\begin{align}\label{diamond}
\forall \rho \in \dC^{\din\times \din} \otimes \dC^{\din\times \din}: \|\id\otimes(\cN-\tilde{\cN})(\rho) \|_1\le \eps \|\rho\|_1
\end{align}
with high probability. In this work, we investigate the optimal complexity of non-adaptive strategies using incoherent measurements. These strategies can only use one copy of the unknown channel at each step and must specify the input states and measurement devices before starting the learning procedure.
\textbf{Contribution} The main contribution of this paper is to show that the optimal complexity of the quantum process tomography with non-adaptive incoherent measurements is $\Tilde{\Theta}(\din^3\dout^3/\eps^2)$.
First, we prove a general lower bound of $\Omega(\din^3\dout^3/\eps^2)$ on the number of incoherent measurements for every non-adaptive process learning algorithm. To do so, we construct an $\Omega(\eps)$-separated family of quantum channels close to the completely depolarizing channel of cardinal $M=\exp(\Omega(\din^2\dout^2))$ by choosing random Choi states of a specific form. This family is used to encode a message from $\{1,\dots, M\}$. A process tomography algorithm can be used to decode this message with the same error probability. Hence, the encoder and decoder should share at least $\Omega(\din^2\dout^2)$ nats of information. On the other hand, we show that the correlation between the encoder and decoder can only increase by at most $\cO(\eps^2/\din\dout)$ nats after each measurement. Note that the naive upper bound on this correlation is $\cO(\eps^2)$, we obtain an improvement by a factor $\din\dout$ by exploiting the randomness in the construction of the quantum channel. This result is stated in Theorem~\ref{thm:lb}. Next, we show that the process tomography algorithm of \cite{surawy2022projected} can be generalized to approximate an unknown quantum channel to within $\eps$ in the diamond norm (\ref{diamond}) using a number of incoherent measurements $\Tilde{\cO}(\din^3\dout^3/\eps^2)$ (Theorem~\ref{upper-bound}). For this, we relate the diamond norm between two quantum channels and the operator norm between their corresponding Choi states which improves on the usual inequality with the $1$-norm: $\|\cM\|_\diamond\le \din\|\cJ_{\cM}\|_1$ (see e.g. \cite{jenvcova2016conditions}).
\textbf{Related work} The first works on process tomography including \cite{chuang1997prescription,poyatos1997complete} follow the strategy of learning the quantum states images of a complete set of basis states
then obtaining the quantum channel by an inversion. The problem of state tomography using incoherent measurements is fully understood even for adaptive strategies \cite{haah2016sample,guctua2020fast,lowe2022lower,chen2022tight}: the optimal complexity is $\Theta(d^3/\eps^2)$. So, learning a quantum channel can be done using $\cO(\din^2\dout^3)$ measurements, but this complexity doesn't take into account the accumulation of errors. The same drawback can be seen in the resource analysis of different strategies by \cite{mohseni2008quantum}.
Another reductive approach is to use the Choi–Jamiołkowski isomorphism \cite{choi1975completely,jamiolkowski1972linear} to reduce the process tomography to state tomography with a higher dimension \cite{leung2000towards,d2001quantum}. However, this requires an ancilla and only implies a sub-optimal upper bound $\cO((\din\dout)^3/(\eps/\din)^2)=\cO(\din^5\dout^3/\eps^2) $ for learning in the diamond norm. \cite{surawy2022projected} propose an algorithm for estimating the Choi state in the $2$ norm that requires only $\tilde{\cO}(d^4/\eps^2)$ ancilla-free incoherent measurements (when $\din=\dout=d$). This article generalizes this result to the diamond norm and general input/output dimensions and shows that this algorithm is optimal up to a logarithmic factor.
\\A special case of quantum process tomography is learning Pauli channels. These channels have weighted Pauli matrices as Kraus operators and can be learned in diamond norm using $\tilde{\cO}(d^3/\eps^2)$ measurements \cite{flammia2020efficient} (here $\din=\dout=d$). Furthermore, it is shown that $\Omega(d^3/\eps^2)$ are necessary for any non-adaptive strategy~\cite{fawzi2023lower}. While the techniques of the lower bound of this article are similar to the one in \cite{fawzi2023lower}, we obtain here a larger lower bound because, in general, we are not restricted to weighted Pauli matrices in the Kraus operators and these latter are implicitly chosen at random.
\section{Preliminaries}
We consider quantum channels
of input dimension $\din$ and output dimension $\dout$. We use the notation $[d] := \{1,\dots,d\}$.
We adopt the bra-ket notation: a column vector is denoted $\ket{\phi}$ and its adjoint is denoted $\bra{\phi}=\ket{\phi}^\dagger$. With this notation, $\spr{\phi}{\psi}$ is the dot product of the vectors $\phi$ and $\psi$ and, for a unit vector $\ket{\phi}\in \mathrm{S}^d$, $\proj{\phi}$ is the rank-$1$ projector on the space spanned by the vector $\phi$. The canonical basis $\{e_i\}_{i\in [d]}$ is denoted $\{\ket{i}\}_{i\in [d]}:=\{\ket{e_i}\}_{i\in [d]}$. A quantum state is a positive semi-definite Hermitian matrix of trace $1$. A $(\din, \dout)$-dimensional quantum channel is a map $\cN: \mathds{C}^{\din\times \din}\rightarrow \mathds{C}^{\dout\times \dout}$ of the form $\cN(\rho)=\sum_{k}A_k \rho A_k^\dagger$ where the Kraus operators $\{A_k\}_{k\in \cK}\in\left( \mathds{C}^{\dout \times\din }\right)^\cK$ satisfy $\sum_{k\in \cK} A_k^\dagger A_k=\mathds{I}_{\din}$. For instance, the identity map $\id(\rho)=\rho$ admits the Kraus operator $\{\mathds{I}\}$ and the completely depolarizing channel $\cD(\rho)=\tr(\rho)\frac{\mathds{I}}{\dout}$ admits the Kraus operators $\left\{\frac{1}{\sqrt{\dout}}\ket{i}\bra{j}\right\}_{j\in[\din], i\in [\dout]}$.
A map $\cN$ is a quantum channel if, and only if, it is:
\begin{itemize}
\item \textbf{completely positive:} for all $\rho\succcurlyeq 0$, $\id\otimes \cN(\rho)\succcurlyeq 0$ and
\item \textbf{trace preserving:} for all $\rho$, $\tr(\cN(\rho))=\tr(\rho)$.
\end{itemize}
We define the diamond distance between two quantum channels $\cN$ and $\cM$ as the diamond norm of their difference:
\begin{align*}
d_\diamond(\cN,\cM):= \max_{\rho }\|\id\otimes(\cN-\cM)(\rho) \|_1
\end{align*}
where the maximization is over quantum states and the Schatten $p$-norm of a matrix $M$ is defined as $\|M\|_p^p=\tr\left(\sqrt{M^\dagger M}^p\right)$. The diamond distance can be thought of as a worst-case distance, while the average case distance is given by the Hilbert-Schmidt or Schatten $2$-norm between the corresponding Choi states. We define the Choi state of the channel $\cN$ as $\cJ_{\cN}= \id\otimes\cN(\proj{\Psi})\in \mathds{C}^{\din \times \din }\otimes \mathds{C}^{\dout \times \dout }$ where $\ket{\Psi}=\frac{1}{\sqrt{\din}}\sum_{i=1}^{\din}\ket{i}\otimes\ket{i}$ is the maximally entangled state. However, to have comparable distances, we will normalize the $2$-norm which is equivalent to unnormalizing the maximally entangled state and we define the $2$-distance as follows:
\begin{align*}
d_2(\cN,\cM):= \din\|\cJ_{\cN}-\cJ_{\cM}\|_2=\|\id\otimes(\cN-\cM)(\din\proj{\Psi}) \|_2.
\end{align*}
This is a valid distance since the map $\cJ : \cN \mapsto \id\otimes\cN(\proj{\Psi}) $ is an isomorphism called the Choi–Jamiołkowski isomorphism \cite{choi1975completely,jamiolkowski1972linear}. Note that $\cJ$ should be positive semi-definite and satisfy $\tr_2(\cJ)=\frac{\dI}{\din}$ to be a valid Choi state (corresponding to a quantum channel).
We consider the channel tomography problem which consists of learning a quantum channel $\cN$ in the diamond distance. Given a precision parameter $\eps>0$, the goal is to construct a quantum channel $\tilde{\cN}$ satisfying with at least a probability $2/3$:
\begin{align*}
d_\diamond(\cN,\tilde{\cN})\le \eps.
\end{align*}
An algorithm $\cA$ is $1/3$-correct for this problem if it outputs a quantum channel $\eps$-close to $\cN$ with a probability of error at most $1/3$.
We choose to learn in the diamond distance because it characterizes the minimal error probability to distinguish between two quantum channels when auxiliary systems are allowed \cite{watrous2018theory}.
The learner can only extract classical information from the unknown quantum channel $\cN$ by performing a measurement on the output state. Throughout the paper, we only consider unentangled or incoherent measurements. That is, the learner can only measure with a $d$ (or $d\times d$)-dimensional measurement device. Precisely, a $d$-dimensional measurement is defined by a POVM (positive operator-valued measure) with a finite number of elements: this is a set of positive semi-definite matrices $\mathcal{M}=\{M_x\}_{x\in \cX}$ acting on the Hilbert space $\mathds{C}^{d}$ and satisfying $\sum_{x\in \cX} M_x=\mathds{I}$. Each element $M_x$ in the POVM $\mathcal{M}$ is associated with the outcome $x\in \cX$. The tuple $\{\tr(\rho M_x)\}_{x\in \cX}$ is non-negative and sums to $1$: it thus defines a probability. Born's rule \cite{1926ZPhy...37..863B} says that the probability that the measurement on a quantum state $\rho$ using the POVM $\cM$ will output $x$ is exactly $\tr(\rho M_x)$. Depending on whether an auxiliary system is allowed to be used, we distinguish two types of strategies.
\begin{figure}
\caption{Illustration of an ancilla-free independent strategy for quantum process tomography.}
\label{Fig: Non-Adap-ancill-free}
\end{figure}
\paragraph{Ancilla-free strategies} At each step $t$, the learner would choose an input $\din$-dimensional state $\rho_t\in \mathds{C}^{\din\times \din}$ and a $\dout$-dimensional measurement device $\cM_t=\{ M_x^t\}_{x\in \cX_t} \in (\mathds{C}^{\dout\times \dout} )^{\cX_t}$. It thus sees the outcome $x_t \in \cX_t$ with a probability $\tr(\cN(\rho_t) M_{x_t}^t)$ (see Fig.~\ref{Fig: Non-Adap-ancill-free}).
\paragraph{Ancilla-assisted strategies} At each step $t$, the learner would choose an input $d\times \din$-dimensional state $\rho_t\in \mathds{C}^{d\times d} \otimes\mathds{C}^{\din\times \din} $ and a $d\times \dout$-dimensional measurement device $\cM_t=\{ M_x^t\}_{x\in \cX_t} \in (\mathds{C}^{d\times d} \otimes\mathds{C}^{\dout\times \dout} )^{\cX_t}$. It thus sees the outcome $x_t \in \cX_t$ with a probability $\tr(\id\otimes\cN(\rho_t) M_{x_t}^t)$ (see Fig.~\ref{Fig: Non-Adap-ancill-ass}).
\begin{figure}
\caption{Illustration of an ancilla-assisted independent strategy for quantum process tomography.}
\label{Fig: Non-Adap-ancill-ass}
\end{figure}
Note that ancilla-assisted strategies were proven to provide an exponential (in the number of qubits $n=\log_2(d)$) advantage over ancilla-free strategies for some problems \cite{chen2022quantum,chen2022exponential}. However, in this work, we show that ancilla-assisted strategies cannot overcome ancilla-free strategies for process tomography. Finally, we only consider non-adaptive strategies: the input states and measurement devices should be chosen before starting the learning procedure and thus cannot depend on the observations.
Given two random variables $X$ and $Y$ taking values in the sets $[d]$ and $[d']$ respectively,
the mutual information between $X$ and $Y$ is the Kullback Leibler divergence between the joint distribution $P_{(X,Y)}$ and the product distribution $P_X\times P_Y$:
\begin{align*}
\cI(X:Y)=\sum_{i=1}^d\sum_{j=1}^{d'} \pr{X=i, Y=j} \log\left(\frac{\pr{X=i, Y=j} }{\pr{X=i}\pr{Y=j} }\right).
\end{align*}
All the logs of this paper are taken in base $e$ and the information is measured in ‘‘nats''.
\section{Lower bound}
In this section, we would like to investigate the intrinsic limitations of learning quantum channels using incoherent measurements. To avoid repetition, we consider only ancilla-assisted strategies since they contain ancilla-free strategies as a special case: one can map every $\din$-dimensional input state $\rho$ to the $d\times \din$-dimensional input state $\tilde{\rho}=\frac{\dI}{d}\otimes \rho$ and every $\dout$-dimensional POVM $\cM= \{M_x\}_{x\in \cX} $ to the $d\times \dout$-dimensional POVM $\tilde{\cM}= \{\dI_{d} \otimes M_x\}_{x\in \cX}$. Mainly, we prove the following theorem:
\begin{theorem}\label{thm:lb} Let $\eps\le 1/16$ and $\dout\ge 4$.
Any non-adaptive ancilla-assisted algorithm for process tomography in diamond distance requires
\begin{align*}
N=\Omega\left( \frac{\din^3\dout^3}{\eps^2} \right)
\end{align*}
incoherent measurements.
\end{theorem}
\begin{proof}
For the proof, we use the construction of the Choi state:
\begin{align*}
\cJ_U=\frac{\dI}{\din \dout}+\frac{\eps}{\din \dout} (U+U^\dagger) - \frac{\eps}{\din \dout} \tr_2(U+U^\dagger)\otimes \frac{\dI}{\dout}
\end{align*}
where $U \sim \Haar(\din \dout)$. $\cJ_U $ is Hermitian and satisfies $\tr_2(\cJ)=\frac{\dI}{\din}$. Moreover, $\cJ_U\succcurlyeq 0$ for $\eps\le 1/4$.
Indeed, $U$ is a unitary so it has an operator norm $1$ thus $\|U+U^\dagger\|_\infty \le 2$. Besides, $\|\tr_2(U+U^\dagger)\otimes \frac{\dI}{\dout}\|_\infty= \frac{1}{\dout}\|\tr_2(U+U^\dagger)\|_\infty \le \max_i \|\dI \otimes \bra{i}(U+U^\dagger)\dI \otimes \ket{i}\|_\infty\le 2$. We claim that:
\begin{lemma}\label{lem:construction}
We can construct an $\eps/2$-separated (according to the diamond distance)
family $\{\cN_x\}_{x\in [M]}$ of cardinal $M=\exp(\Omega(\din^2 \dout^2))$.
\end{lemma}
\begin{proof}
It is sufficient to show that for $U,V\sim \Haar(\din \dout)$:
\begin{align*}
\pr{\|\cJ_U-\cJ_V\|_1\le \eps/2 }\le \exp\left(-\Omega(\din^2 \dout^2)\right).
\end{align*}
because, once this concentration inequality holds, we can choose our family randomly, and by the union bound, it will be $\eps/2$-separated with an overwhelming probability ($1-\exp\left(-\Omega(\din^2 \dout^2)\right)$) using the inequality $d_\diamond(\cN_U,\cN_V)\ge \|\cJ_U-\cJ_V\|_1$.
First, let us lower bound the expected value.
\begin{align*}
\ex{\|\cJ_U-\cJ_V\|_1}&\ge \frac{\eps}{\din \dout} \ex{\|U+U^\dagger-V-V^\dagger\|_1} \\&-\frac{\eps}{\din \dout^2}\ex{\| \tr_2(U+U^\dagger-V-V^\dagger)\otimes \dI \|_1}.
\end{align*}
On one hand, we can upper bound the second expectation using the triangle and the Cauchy-Schwartz inequalities:
\begin{align*}
&\ex{\|\tr_2(U+U^\dagger-V-V^\dagger)\otimes \dI \|_1}
\le 4\ex{\| \tr_2(U)\otimes \dI \|_1}
\\&\le 4\sqrt{\din \dout}\ex{\| \tr_2(U)\otimes \dI \|_2}\le 4\sqrt{\din \dout}
\sqrt{\ex{\tr(\tr_2(U) \tr_2(U^\dagger) \otimes \dI) }}
\\&= 4\sqrt{\din \dout} \sqrt{\dout}
\sqrt{\ex{ \sum_{i} \sum_{k,l} \bra{i} \otimes \bra{k}U \dI \otimes \ket{k}\bra{l}U^\dagger\ket{i} \otimes \ket{l} }}
\\&= 4\sqrt{\din \dout}\sqrt{\dout}
\sqrt{\ex{ \sum_{i=1}^{\din} \sum_{k,l=1}^{\dout} \frac{\din\delta_{k,l}}{\din \dout} }}=4\din \dout.
\end{align*}
On the other hand, we can lower bound the first expectation using Hölder's inequality.
\begin{align*}
\ex{\|U+U^\dagger-V-V^\dagger\|_1}
&\ge \sqrt{\frac{(\ex{\tr(U+U^\dagger-V-V^\dagger)^2})^3}{\ex{\tr(U+U^\dagger-V-V^\dagger)^4}}}
\\&\ge \sqrt{\frac{(4\din \dout)^3}{16\din \dout}}=2\din \dout.
\end{align*}
Therefore:
\begin{align*}
\ex{\|\cJ_U-\cJ_V\|_1}
& \ge \frac{\eps}{\din \dout} \ex{\|U+U^\dagger-V-V^\dagger\|_1} -\frac{4\eps}{\din \dout^2}\ex{\| \tr_2U\otimes \dI \|_1}
\\&\ge 2\eps-\frac{4\eps }{\dout }\ge \eps~~~~~\text{ for }~~~~~ \dout\ge 4.
\end{align*}
Now, we claim that the function $(U,V)\mapsto \|\cJ_U-\cJ_V\|_1$ is $\frac{8\eps}{\sqrt{\din\dout}}$-Lipschitz. Indeed, we have $\|\tr_2(X)\otimes \dI\|_1 \le \sqrt{\din\dout}\|\tr_2(X)\otimes \dI\|_2 = \sqrt{\din}\dout\|\tr_2(X)\|_2\le \sqrt{\din\dout}\dout\|X\|_2 $ where the last inequality can be found in \cite{lidar2008distance}. Therefore, by letting $X=U-U'$ and $Y=V-V'$ and using the triangle inequality we obtain:
\begin{align*}
&|\|\cJ_U-\cJ_V\|_1-\|\cJ_{U'}-\cJ_{V'}\|_1|\\&\le \frac{2\eps}{\din\dout}\Bigg[\| X\|_1+\| Y\|_1 + \left\| \tr_2(X)\otimes \frac{\dI}{\dout}\right\|_1+\left\| \tr_2(Y)\otimes \frac{\dI}{\dout}\right\|_1\Bigg]
\\&\le\frac{2\sqrt{\din\dout}\eps}{\din\dout}\left( \| U-U'\|_2 +\| V-V'\|_2 \right) + \frac{2\sqrt{\din\dout}\dout\eps}{\din\dout^2}\left( \| U-U'\|_2 +\| V-V'\|_2 \right)
\\&\stackrel{\text{}}{\le}
\frac{8\eps}{\sqrt{\din\dout}}\| (U,V)-(U',V')\|_2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
~~~~(\text{Cauchy-Schwartz})
\end{align*}
so by the concentration inequality for Lipschitz functions of $\Haar$ measure \cite{meckes2013spectral}:
\begin{align*}
&\pr{\|\cJ_U-\cJ_V\|_1\le \eps/2 }\\&\le \pr{\|\cJ_U-\cJ_V\|_1-\ex{\|\cJ_U-\cJ_V\|_1}\le -\eps/2 }
\\&\le\exp\left(-\frac{\din\dout\eps^2}{48\times 64\eps^2/\din\dout}\right)= \exp\left(-\Omega(\din^2\dout^2)\right).
\end{align*}
\end{proof}
Now, we use this $\eps/2$-separated family of quantum channels $\{\cN_x\}_{x\in [M]}$ (corresponding to the Choi states $\{\cJ_x\}_{x\in [M]}$ found in Lemma~\ref{lem:construction}) to encode a uniformly random message $X\sim\unif([M])$ by the map $X\mapsto \cN_X$. Using a learning algorithm for process tomography with precision $\eps/4$ and an error probability at most $1/3$, a decoder $Y$ can find $X$ with the same error probability because the family $\{\cN_x\}_{x\in [M]}$ is $\eps/2$-separated. By Fano's inequality, the encoder and decoder should share at least $\Omega(\log(M))$ nats of information.
\begin{lemma} \cite{fano1961transmission}\label{lem:fano} We have
\begin{align*}
\cI(X:Y) \ge 2/3 \log(M)-\log(2)\ge \Omega(d^4).
\end{align*}
\end{lemma}
The remaining part of the proof is to upper bound this mutual information in terms of the number of measurements $N$, the dimensions $\din,\dout$, and the precision parameter $\eps$. Intuitively, the mutual information, after a few measurements, is very small and then it increases when the number of measurements increases.
To make this intuition formal, let $N$ be a number of measurements sufficient for process tomography and let $(I_1,\dots ,I_N)$ be the observations of the learning algorithm, we apply first the data processing inequality to relate the mutual information between the encoder and the decoder with the mutual information between the uniform random variable $X$ and the observations $(I_1,\dots ,I_N)$:
\begin{align*}
\cI(X:Y) \le \cI(X:I_1,\dots, I_N).
\end{align*}
Then we apply the chain rule for the mutual information:
\begin{align*}
\cI(X:I_1,\dots, I_N) &=\sum_{t=1}^N \cI(X:I_t| I_{\le t-1})
\end{align*}
where we use the notation $I_{\le t}= (I_1,\dots, I_t)$ and $\cI(X:I_t| I_{\le t-1})$ is the conditional mutual information between $X$ and $I_t$ given $I_{\le t-1}$. A learning algorithm $\cA$ would choose the input states $\{\rho_t\}_{t\in [N]}$ and measurement devices $\{\cM_t\}_{t\in [N]}$ which can be chosen to have the form $\cM_t= \{\mu^t_i \proj{\phi^t_i}\}_{i\in \cI_t}$ where $\mu^t_i\ge 0$ and $\spr{\phi^t_i}{\phi^t_i}=1$ for all $t,i$.
Using Jensen's inequality, we can prove the following upper bound on the conditional mutual information:
\begin{lemma}\label{lem: cond-mut-inf} For $x\in [M]$, let $\cM_x=\cN_x-\cD$ where $\cD(\rho)=\tr(\rho)\frac{\dI}{\dout}$ is the completely depolarizing channel. We have for all $t\in \{1,\dots, N\}$:
\begin{align*}
& \cI(X:I_t| I_{\le t-1})
\le \frac{3}{M}\sum_{ i\in \cI_t, x\in [M]} \mu^t_{i}\bra{\phi_{i}^t} \id\otimes\cD(\rho_t) \ket{{\phi_{i}^t}}\left(\frac{\bra{\phi_{i}^t} \id\otimes\cM_x(\rho_t) \ket{{\phi_{i}^t}}}{\bra{\phi_{i}^t} \id\otimes\cD(\rho_t) \ket{{\phi_{i}^t}}} \right)^2
\end{align*}
\end{lemma}
\begin{proof}
Let $t \in \{1,\dots, N\}$ and $x\in [M]$.
Let $i=(i_1,\dots, i_t)\in (\cI_1,\dots, \cI_t)$, we can express the joint probability $p$ of $(X,I_1,\dots, I_t)$ as follows:
\begin{align*}
p(x,i_1,\dots, i_t)=\frac{1}{M} \prod_{k=1}^t\mu^k_{i_k}\bra{\phi_{i_k}^k} \id\otimes\cN_x(\rho_k) \ket{{\phi_{i_k}^k}}
\end{align*}
We can remark that, for all $1\le k\le t$:
\begin{align*}
p(x,i_{\le k})&=\mu_{i_k}^k\bra{\phi_{i_k}^k} \id\otimes\cN_x(\rho_k) \ket{{\phi_{i_k}^k}}p(x,i_{\le k-1})
\\&=\mu_{i_k}^k\bra{\phi_{i_k}^k} \id\otimes\cD(\rho_k) \ket{{\phi_{i_k}^k}}(1+ \Phi_{x,i_k}^k) p(x,i_{\le k-1})
\end{align*}
where $\Phi_{x,i_k}^k=\frac{\bra{\phi_{i_k}^k} \id\otimes\cM_x(\rho_k) \ket{{\phi_{i_k}^k}}}{\bra{\phi_{i_k}^k} \id\otimes\cD(\rho_k) \ket{{\phi_{i_k}^k}}} $ because $\cD+\cM_x=\cN_x$.
So, the ratio of conditional probabilities can be written as:
\begin{align*}
& \frac{p(x,i_t| i_{\le t-1})}{p(x|i_{\le t-1})p(i_t|i_{\le t-1})}=\frac{p(x,i_{\le t})p(i_{\le t-1}) }{p(x,i_{\le t-1})p(i_{\le t})}
\\&=\frac{\mu_{i_t}^t\bra{\phi_{i_t}^t} \id\otimes\cD(\rho_t) \ket{{\phi_{i_t}^t}}(1+ \Phi_{x,i_t}^t) p(x,i_{\le t-1})p(i_{\le t-1}) }{p(x,i_{\le t-1})\sum_y p(y,i_{\le t})}
\\&=\frac{\mu_{i_t}^t\bra{\phi_{i_t}^t} \id\otimes\cD(\rho_t) \ket{{\phi_{i_t}^t}}(1+ \Phi_{x,i_t}^t) p(i_{\le t-1}) }{\sum_y p(y,i_{\le t})}
\\&=\frac{\mu_{i_t}^t\bra{\phi_{i_t}^t} \id\otimes\cD(\rho_t) \ket{{\phi_{i_t}^t}}(1+ \Phi_{x,i_t}^t) p(i_{\le t-1}) }{\sum_y \mu_{i_t}^t\bra{\phi_{i_t}^t} \id\otimes\cD(\rho_t) \ket{{\phi_{i_t}^t}}(1+ \Phi_{y,i_t}^t) p(y,i_{\le {t-1}})}
\\&= \frac{(1+ \Phi_{x,i_t}^t) p(i_{\le t-1}) }{\sum_y (1+ \Phi_{y,i_t}^t) p(y,i_{\le {t-1}})}
= \frac{(1+ \Phi_{x,i_t}^t) }{\sum_y (1+ \Phi_{y,i_t}^t) p(y|i_{\le {t-1}})}
\end{align*}
Therefore by Jensen's inequality:
\begin{align*}
& \cI(X:I_t| I_{\le t-1})=\ex{\log\left( \frac{p(x,i_t| i_{\le t-1})}{p(x|i_{\le t-1})p(i_t|i_{\le t-1})}\right) }
\\&= \ex{\log\left(\frac{(1+\Phi_{x,i_t}^t) }{\sum_y p(y|i_{\le t-1})(1+ \Phi_{y,i_t}^t)}\right) }
\\&\le \ex{\log(1+\Phi_{x,i_t}^t) -\sum_y p(y|i_{\le t-1})\log( 1+ \Phi_{y,i_t}^t)}
\\&= \ex{\log(1+\Phi_{x,i_t}^t)} -\sum_y \ex{p(y|i_{\le t-1})\log( 1+ \Phi_{y,i_t}^t)}.
\end{align*}
The first term can be upper bounded using the inequality $\log(1+x)\le x$ verified for all $x\in (-1,\infty)$:
\begin{align*}
& \ex{\log(1+\Phi_{x,i_t}^t)}= \mathds{E}_{x,i\sim p} \log(1+\Phi_{x,i_t}^t)
\\&\le\mathds{E}_{x,i\sim p} \Phi_{x,i_t}^t
=\mathds{E}_{x,i\sim p_{\le t}} \Phi_{x,i_t}^t
\\& = \mathds{E}_{x,i\sim p_{\le t-1}} \sum_{i_t}\mu_{i_t}^t \bra{\phi_{i_t}^t} \id\otimes\cD(\rho_t) \ket{{\phi_{i_t}^t}}(1+\Phi_{x,i_t}^t)\Phi_{x,i_t}^t
\\&= \mathds{E}_{x,i\sim p_{\le t-1}} \sum_{i_t}\mu_{i_t}^t \bra{\phi_{i_t}^t} \id\otimes\cD(\rho_t) \ket{{\phi_{i_t}^t}} (\Phi_{x,i_t}^t)^2
\\&=\frac{1}{M}\sum_{x=1}^M \sum_{i_t}\mu_{i_t}^t \bra{\phi_{i_t}^t} \id\otimes\cD(\rho_t) \ket{{\phi_{i_t}^t}}(\Phi_{x,i_t}^t)^2
\end{align*}
because $\sum_{i_t} \mu_{i_t}^t \bra{\phi_{i_t}^t} \id\otimes\cD(\rho_t) \ket{{\phi_{i_t}^t}}\Phi_{x,i_t}^t = \tr( \id\otimes\cM_x(\rho_t)) = \tr( \id\otimes\cN_x(\rho_t)) -\tr( \id\otimes\cD(\rho_t))= \tr(\rho_t)- \tr(\rho_t)=0 $ and we use the condition that the algorithm is non-adaptive in the last line.
\\On the other hand, the second term can be upper bounded using the inequality $-\log(1+x)\le -x+x^2/2$ verified for all $x\in (-1/2,\infty)$. Let $\lambda_{i_t}^t= \mu_{i_t}^t \bra{\phi_{i_t}^t} \id\otimes\cD(\rho_t) \ket{{\phi_{i_t}^t}} $, we have~:
\begin{align*}
&\ex{-\sum_y p(y|i_{\le t-1})\log( 1+ \Phi_{y,i_t}^t) }
\\&= \sum_y\mathds{E}_{x,i\sim p} p(y|i_{\le t-1})(-\log)( 1+ \Phi_{y,i_t}^t)
\\&=\sum_y\mathds{E}_{x,i\sim p_{\le t}} p(y|i_{\le t-1})(-\log)( 1+ \Phi_{y,i_t}^t)
\\&\le \sum_y\mathds{E}_{x,i\sim p_{\le t}} p(y|i_{\le t-1})( - \Phi_{y,i_t}^t+ (\Phi_{y,i_t}^t)^2/2)
\\&\le\sum_y\mathds{E}_{x,i\sim p_{\le t-1}} p(y|i_{\le t-1})\sum_{i_t}\lambda_{i_t}^t( (\Phi_{x,i_t}^t)^2+ (\Phi_{y,i_t}^t)^2)
\\&=2\sum_y\mathds{E}_{x,i\sim p_{\le t-1}} p(y|i_{\le t-1})\sum_{i_t}\lambda_{i_t}^t (\Phi_{x,i_t}^t)^2
\\&=2\mathds{E}_{x,i\sim p_{\le t-1}} \sum_{i_t}\mu_{i_t}^t \bra{\phi_{i_t}^t} \id\otimes\cD(\rho_t) \ket{{\phi_{i_t}^t}}(\Phi_{x,i_t}^t)^2
\\&=2\mathds{E}_{x,i\sim p_{\le t-1}} \sum_{i_t}\lambda_{i_t}^t (\Phi_{x,i_t}^t)^2
\\&=2\frac{1}{M}\sum_{x=1}^M \sum_{i_t}\mu_{i_t}^t \bra{\phi_{i_t}^t} \id\otimes\cD(\rho_t) \ket{{\phi_{i_t}^t}}(\Phi_{x,i_t}^t)^2
\end{align*}
where we use the condition that the algorithm is non-adaptive in the last line. Since the conditional mutual information is upper bounded by the sum of these two terms, the upper bound on the conditional mutual information follows.
\end{proof}
It remains to approximate every mean $\frac{1}{M}\sum_{x=1}^M$ by the expectation $\mathds{E}_U$.
\begin{lemma}\label{lem:mean_x-meanU}We have with at least a probability $9/10$:
\begin{align*}
& \frac{1}{M}\sum_{t, i, x} \mu^t_{i}\bra{\phi_{i}^t} \id\otimes\cD(\rho_t) \ket{{\phi_{i}^t}}\left(\frac{\bra{\phi_{i}^t} \id\otimes\cM_x(\rho_t) \ket{{\phi_{i}^t}}}{\bra{\phi_{i}^t} \id\otimes\cD(\rho_t) \ket{{\phi_{i}^t}}} \right)^2
\\&\le \sum_{t, i} \mu^t_{i}\bra{\phi_{i}^t} \id\otimes\cD(\rho_t) \ket{{\phi_{i}^t}}\mathds{E}_U\left(\frac{\bra{\phi_{i}^t} \id\otimes\cM_U(\rho_t) \ket{{\phi_{i}^t}}}{\bra{\phi_{i}^t} \id\otimes\cD(\rho_t) \ket{{\phi_{i}^t}}} \right)^2
+16N\eps^2\sqrt{\frac{\log(10)}{M}}.
\end{align*}
\end{lemma}
\begin{proof}
Denote by $f_x^t$
the function $\ket{\phi}\mapsto\frac{ \bra{\phi}\id\otimes\cM_x(\rho_t) \ket{{\phi}}^2}{\bra{\phi} \id\otimes\cD(\rho_t) \ket{{\phi}}^2}$.
We claim that the functions $f^t_x$ are bounded. Indeed, we
can write $\rho_t= \sum_i \lambda_i \proj{\psi_i}$ and every $\ket{\psi_i}$ can be written as $\ket{\psi_i}=A_i \otimes \dI \ket{\Psi}$ so for a unit vector $\ket{\phi}$,
we have:
\begin{align*}
& f^t_x(\ket{\phi})=\frac{ \bra{\phi}\id\otimes\cM_x(\rho_t) \ket{{\phi}}^2}{\bra{\phi} \id\otimes\cD(\rho_t) \ket{{\phi}}^2}
\\&=\frac{4\eps^2\left(\bra{\phi}\sum_i \lambda_i (A_i \otimes \dI)\left(U_x-\tr_2(U_x)\otimes\frac{\dI}{\dout}\right)(A_i^\dagger\otimes \dI)\ket{\phi} \right)^2}{\bra{\phi}\sum_i \lambda_i A_iA_i^\dagger\otimes \dI \ket{\phi}^2}
\\&\le \frac{4\eps^2\bra{\phi}\sum_i \lambda_i (A_i \otimes \dI)(A_i^\dagger\otimes \dI)\ket{\phi}^2\left\|U_x-\tr_2(U_x)\otimes\frac{\dI}{\dout}\right\|_\infty^2}{\bra{\phi}\sum_i \lambda_i A_iA_i^\dagger\otimes \dI \ket{\phi}^2}
\\&\le \frac{16\eps^2\bra{\phi}\sum_i \lambda_i (A_i \otimes \dI)(A_i^\dagger\otimes \dI)\ket{\phi}^2}{\bra{\phi}\sum_i \lambda_i A_iA_i^\dagger\otimes \dI \ket{\phi}^2}=16\eps^2
\end{align*}
where we used that $\|U_x\|_\infty= 1$ and $\|\tr_2(U_x)\|_\infty\le \dout$ for a unitary $U_x$. But we have $\sum_{i} \mu^t_{i}\bra{\phi_{i}^t} \id\otimes\cD(\rho_t) \ket{{\phi_{i}^t}}=\tr(\id\otimes\cD(\rho_t))=1$ so for all $x\in [M]$:
\begin{align*}
\sum_{t, i} \mu^t_{i}\bra{\phi_{i}^t} \id\otimes\cD(\rho_t) \ket{{\phi_{i}^t}}\left(\frac{\bra{\phi_{i}^t} \id\otimes\cM_x(\rho_t) \ket{{\phi_{i}^t}}}{\bra{\phi_{i}^t} \id\otimes\cD(\rho_t) \ket{{\phi_{i}^t}}} \right)^2\le 16N\eps^2.
\end{align*}
Therefore, by Hoeffding's inequality \cite{hoeff} and the union bound, we have with a probability at least $9/10$:
\begin{align*}
& \frac{1}{M} \sum_{x,t, i} \mu^t_{i}\bra{\phi_{i}^t} \id\otimes\cD(\rho_t) \ket{{\phi_{i}^t}}\left(\frac{\bra{\phi_{i}^t} \id\otimes\cM_x(\rho_t) \ket{{\phi_{i}^t}}}{\bra{\phi_{i}^t} \id\otimes\cD(\rho_t) \ket{{\phi_{i}^t}}} \right)^2
\\&\le \sum_{t, i} \mu^t_{i}\bra{\phi_{i}^t} \id\otimes\cD(\rho_t) \ket{{\phi_{i}^t}}\mathds{E}_U\left(\frac{\bra{\phi_{i}^t} \id\otimes\cM_U(\rho_t) \ket{{\phi_{i}^t}}}{\bra{\phi_{i}^t} \id\otimes\cD(\rho_t) \ket{{\phi_{i}^t}}} \right)^2
+16N\eps^2\sqrt{\frac{\log(10)}{M}}.
\end{align*}
\end{proof}
These two Lemmas~\ref{lem: cond-mut-inf},~\ref{lem:mean_x-meanU} imply:
\begin{align}
& \cI(X:I_1,\dots, I_N) =\sum_{t=1}^N \cI(X:I_t| I_{\le t-1})\notag
\\&\le \frac{3}{M}\sum_{ x,t, i} \mu^t_{i}\bra{\phi_{i}^t} \id\otimes\cD(\rho_t) \ket{{\phi_{i}^t}}\mathds{E}\left(\frac{\bra{\phi_{i}^t} \id\otimes\cM_x(\rho_t) \ket{{\phi_{i}^t}}}{\bra{\phi_{i}^t} \id\otimes\cD(\rho_t) \ket{{\phi_{i}^t}}} \right)^2 \notag
\\&\le 3\sum_{ t, i} \mu^t_{i}\bra{\phi_{i}^t} \id\otimes\cD(\rho_t) \ket{{\phi_{i}^t}}\mathds{E}\left(\frac{\bra{\phi_{i}^t} \id\otimes\cM_U(\rho_t) \ket{{\phi_{i}^t}}}{\bra{\phi_{i}^t} \id\otimes\cD(\rho_t) \ket{{\phi_{i}^t}}} \right)^2+48N\eps^2\sqrt{\frac{\log(10)}{M}} \notag
\\&\le 3N\sup_{t,i}\ex{\left(\frac{\bra{\phi_{i}^t} \id\otimes\cM_U(\rho_t) \ket{{\phi_{i}^t}}}{\bra{\phi_{i}^t} \id\otimes\cD(\rho_t) \ket{{\phi_{i}^t}}} \right)^2}+48N\eps^2\sqrt{\frac{\log(10)}{M}} \label{lem:upper bound on mutual info}
\end{align}
where we used that fact that for all $t\in [N]$: $\sum_{i} \mu^t_{i}\bra{\phi_{i}^t} \id\otimes\cD(\rho_t) \ket{{\phi_{i}^t}}= \tr(\id\otimes\cD(\rho_t) )=\tr(\rho_t)=1 $.
The error probability $1/10$ of this approximation can be absorbed in the construction above by asking the unitaries $\{U_x\}_{x\in [M]}$ not only to satisfy the separability condition, but also to satisfy the inequalities in Lemma~\ref{lem:mean_x-meanU}:
\begin{align*}
& \frac{1}{M}\sum_{ t, i, x} \mu^t_{i}\bra{\phi_{i}^t} \id\otimes\cD(\rho_t) \ket{{\phi_{i}^t}}\left(\frac{\bra{\phi_{i}^t} \id\otimes\cM_x(\rho_t) \ket{{\phi_{i}^t}}}{\bra{\phi_{i}^t} \id\otimes\cD(\rho_t) \ket{{\phi_{i}^t}}} \right)^2
\\&\le \sum_{t, i} \mu^t_{i}\bra{\phi_{i}^t} \id\otimes\cD(\rho_t) \ket{{\phi_{i}^t}}\mathds{E}_U\left(\frac{\bra{\phi_{i}^t} \id\otimes\cM_U(\rho_t) \ket{{\phi_{i}^t}}}{\bra{\phi_{i}^t} \id\otimes\cD(\rho_t) \ket{{\phi_{i}^t}}} \right)^2
+48N\eps^2\sqrt{\frac{\log(10)}{M}}.
\end{align*}
Now fix $t\in [N], i_t\in \cI_t$ and $\ket{\phi} = \ket{{\phi_{i_t}^t}}$. Recall that we
can write $\rho_t= \sum_i \lambda_i \proj{\psi_i}$,
the maximally entangled state is denoted $\ket{\Psi}=\frac{1}{\sqrt{\din}}\sum_{i=1}^{\din}\ket{ii}$ and every $\ket{\psi_i}$ can be written as $\ket{\psi_i}=A_i \otimes \dI \ket{\Psi}$ so:
\begin{align}
\id\otimes\cD(\rho_t)&= \sum_i \lambda_i (\id\otimes \cD)(A_i \otimes \dI \proj{\Psi} A_i^\dagger\otimes \dI)\notag
\\&=\sum_i \lambda_i (A_i \otimes \dI) \id\otimes \cD(\proj{\Psi}) (A_i^\dagger\otimes \dI)\notag
\\&=\sum_i \lambda_i (A_i \otimes \dI) \frac{\dI}{\din\dout}(A_i^\dagger\otimes \dI)\notag
\\&=\frac{\sum_i \lambda_i A_iA_i^\dagger}{\din}\otimes \frac{\dI}{\dout}. \label{myeq1}
\end{align}
On the other hand, using the notation $V=U-\tr_2(U)\otimes \frac{\dI}{\dout}$, we can write:
\begin{align*}
&\id\otimes\cM(\rho_t)= \sum_i \lambda_i \id \otimes \cM(A_i \otimes \dI \proj{\Psi} A_i^\dagger\otimes \dI)
\\&=\sum_i \lambda_i (A_i \otimes \dI)\id \otimes (\cN-\cD)( \proj{\Psi}) (A_i^\dagger\otimes \dI)
\\&=\sum_i \lambda_i (A_i \otimes \dI)\left(\cJ_{\cN}-\frac{\dI}{\din\dout}\right) (A_i^\dagger\otimes \dI)
\\&=\frac{\eps}{\din\dout}\sum_i \lambda_i A_i \otimes \dI\left(U+U^\dagger-\tr_2(U+U^\dagger)\otimes \frac{\dI}{\dout}\right) A_i^\dagger\otimes \dI)
\\&=\frac{\eps}{\din\dout}\sum_i \lambda_i\left[ (A_i \otimes \dI)V (A_i^\dagger\otimes \dI)+(A_i \otimes \dI) V^\dagger (A_i^\dagger\otimes \dI)\right].
\end{align*}
By Ineq.~\ref{lem:upper bound on mutual info}, we need to control the expectation $\mathds{E}_U\bra{\phi} \id\otimes\cM_U(\rho_t)\ket{\phi}^2$. First, we replace $\id\otimes\cM(\rho_t)$ with the latter expression, then we apply the inequality $(x+y)^2 \le 2x^2+2y^2$ to separate the terms involving $U$ and the terms involving $\tr_2(U)$. The first term can be computed and bounded as follows.
\begin{align}
&\frac{4\eps^2}{\din^2\dout^2}\ex{\left|\bra{\phi}\left(\sum_{i} \lambda_i (A_i \otimes \dI)U (A_i^\dagger\otimes \dI) \right)\ket{\phi}\right|^2}\notag
\\&= \frac{4\eps^2}{\din^2\dout^2} \sum_{i,j} \frac{\lambda_i \lambda_j }{\din\dout}\left|\tr\left(A_i^\dagger\otimes \dI \ket{\phi} \bra{\phi}A_j\otimes \dI\right)\right|^2\notag
\\&\stackrel{\text{(CS)}}{\le} \frac{4\eps^2}{\din^2\dout^2} \sum_{i,j} \frac{\lambda_i \lambda_j }{\din\dout}\bra{\phi} A_iA_i^\dagger\otimes \dI \ket{\phi} \bra{\phi} A_jA_j^\dagger\otimes \dI \ket{\phi} \notag
\\&= \frac{4\eps^2}{\din^3\dout^3} \left(\bra{\phi}\sum_{i} \lambda_i A_iA_i^\dagger\otimes \dI \ket{\phi} \right)^2. \label{myineq1}
\end{align}
Let's move to the second term which involves the partial trace. Let $M_{ij}=(A_i^\dagger\otimes \dI) \proj{\phi}(A_j\otimes \dI)$.
\begin{align*}
&\frac{4\eps^2}{\din^2\dout^2} \ex{\left|\bra{\phi}\sum_{i} \lambda_i (A_i \otimes \dI)\left(\tr_2(U)\otimes \frac{\dI}{\dout}\right) (A_i^\dagger\otimes \dI) \ket{\phi}\right|^2 }
\\&=\frac{4\eps^2}{\din^2\dout^4} \sum_{i,j} \lambda_i \lambda_j \ex{\tr\big[\left(\tr_2(U)\otimes \dI\right) M_{i,j}\left(\tr_2(U^\dagger)\otimes \dI\right) M_{ij}^\dagger)\big] }
\\&= \frac{4\eps^2}{\din^2\dout^4} \sum_{i,j} \lambda_i \lambda_j \sum_{x,y,z,t=1}^{\din}\sum_{k,l=1}^{\dout}\ex{\bra{xk}U\ket{yk}\bra{zl}U^\dagger\ket{tl}} \tr\Big[\left( \ket{y}\bra{x}\otimes \dI M_{ij}\ket{t}\bra{z}\otimes \dI M_{ij}^\dagger\right) \Big]
\\&= \frac{4\eps^2}{\din^3\dout^5} \sum_{i,j} \lambda_i \lambda_j \sum_{x=t,y=z=1}^{\din}\sum_{k=l=1}^{\dout}\tr\Big[\left( \ket{y}\bra{x}\otimes \dI M_{ij}\ket{x}\bra{y}\otimes \dI M_{ij}^\dagger\right) \Big]
\\&= \frac{4\eps^2}{\din^3\dout^4} \sum_{i,j} \lambda_i \lambda_j \sum_{x,y=1}^{\din}\tr\Big[\left( \ket{y}\bra{x}\otimes \dI M_{ij}\ket{x}\bra{y}\otimes \dI M_{ij}^\dagger\right) \Big].
\end{align*}
To control the latter expression, we write $\ket{\phi}= B^\dagger\otimes \dI \ket{\Psi}$ so that $M_{i,j}= (A_i^\dagger\otimes \dI) \proj{\phi}(A_j\otimes \dI)= (A_i^\dagger B^\dagger\otimes \dI) \proj{\Psi}(BA_j\otimes \dI)$. Using the property of the maximally entangled state $\bra{\Psi}M\otimes \dI\ket{\Psi}=\frac{1}{\din}\tr(M)$ we obtain:
\begin{align*}
& \sum_{x,y=1}^{\din}\tr\Big[\left( \ket{y}\bra{x}\otimes \dI M_{ij}\ket{x}\bra{y}\otimes \dI M_{ij}^\dagger\right) \\&= \sum_{x,y=1}^{\din}\tr\left( \ket{y}\bra{x}\otimes \dI (A_i^\dagger B^\dagger\otimes \dI) \proj{\Psi}(BA_j\otimes \dI) \ket{x}\bra{y}\otimes \dI (A_j^\dagger B^\dagger\otimes \dI) \proj{\Psi}(BA_i\otimes \dI)\right)
\\&= \sum_{x,y=1}^{\din} \bra{\Psi }(BA_j\otimes \dI) \ket{x}\bra{y}\otimes \dI (A_j^\dagger B^\dagger\otimes \dI)\ket{\Psi} \bra{\Psi} (BA_i\otimes \dI)^\dagger\ket{y}\bra{x}\otimes \dI (A_i^\dagger B^\dagger\otimes \dI) \ket{\Psi}
\\&= \frac{1}{\din^2}\sum_{x,y=1}^{\din} \tr(BA_j\ket{x}\bra{y} A_j^\dagger B^\dagger ) \tr( BA_i \ket{y}\bra{x}A_i^\dagger B^\dagger)
\\&= \frac{1}{\din^2}\sum_{x,y=1}^{\din} \bra{y} A_j^\dagger B^\dagger BA_j\ket{x} \bra{x}A_i^\dagger B^\dagger BA_i \ket{y}
=\frac{1}{\din^2}\tr\left( A_j^\dagger B^\dagger BA_jA_i^\dagger B^\dagger BA_i \right).
\end{align*}
On the other hand, we can write
\begin{align*}
\bra{\phi}\sum_{i} \lambda_i A_iA_i^\dagger\otimes \dI \ket{\phi}&= \bra{\Psi}\sum_{i} \lambda_i BA_iA_i^\dagger B^\dagger \otimes \dI \ket{\Psi}
= \frac{1}{\din}\tr\left( \sum_i \lambda_i A_i^\dagger B^\dagger BA_i \right).
\end{align*}
Note that the matrix $\sum_i \lambda_i A_i^\dagger B^\dagger BA_i $ is positive semi-definite so:
\begin{align*}
& \sum_{i,j} \lambda_i \lambda_j\frac{1}{\din^2}\tr\left( A_j^\dagger B^\dagger BA_jA_i^\dagger B^\dagger BA_i \right) = \frac{1}{\din^2}\tr\left( \sum_i \lambda_i A_i^\dagger B^\dagger BA_i \right)^2\notag
\\&\le \Bigg[\frac{1}{\din}\tr\left( \sum_i \lambda_i A_i^\dagger B^\dagger BA_i \right)\Bigg]^2
= \bra{\phi}\sum_{i} \lambda_i A_iA_i^\dagger\otimes \dI \ket{\phi}^2.
\end{align*}
Hence
\begin{align}
&\frac{4\eps^2}{\din^2\dout^2} \ex{\left|\bra{\phi}\sum_{i} \lambda_i (A_i \otimes \dI)\left(\tr_2(U)\otimes \frac{\dI}{\dout}\right) (A_i^\dagger\otimes \dI) \ket{\phi}\right|^2 }\notag
\\&= \frac{4\eps^2}{\din^3\dout^4} \sum_{i,j} \lambda_i \lambda_j \sum_{x,y=1}^{\din}\tr\Big[\left( \ket{y}\bra{x}\otimes \dI M_{ij}\ket{x}\bra{y}\otimes \dI M_{ij}^\dagger\right) \Big]\notag
\\&= \frac{4\eps^2}{\din^3\dout^4} \sum_{i,j} \lambda_i \lambda_j \frac{1}{\din^2}\tr\left( A_j^\dagger B^\dagger BA_jA_i^\dagger B^\dagger BA_i \right)\notag
\\&\le \frac{4\eps^2}{\din^3\dout^4} \bra{\phi}\sum_{i} \lambda_i A_iA_i^\dagger\otimes \dI \ket{\phi}^2\label{myineq2}
\end{align}
Using the equality~\eqref{myeq1} and the two inequalities~\eqref{myineq1} and~\eqref{myineq2}, we deduce:
\begin{align*}
&\ex{\left(\frac{\bra{\phi} \id\otimes\cM_U(\rho_t) \ket{{\phi}}}{\bra{\phi} \id\otimes\cD(\rho_t) \ket{{\phi}}} \right)^2}
\le\frac{\frac{8\eps^2}{\din^3\dout^3} \left(\bra{\phi}\sum_{i} \lambda_i A_iA_i^\dagger\otimes \dI \ket{\phi} \right)^2}{\bra{\phi} \frac{\sum_i \lambda_i A_iA_i^\dagger}{\din}\otimes \frac{\dI}{\dout}\ket{\phi}^2}=\frac{8\eps^2}{\din\dout}.
\end{align*}
Therefore using the inequality~\eqref{lem:upper bound on mutual info}:
\begin{align*}
&\cI(X:I_1,\dots, I_N)=\sum_{t=1}^N \cI(X:I_t| I_{\le t-1})
\\&\le 3 N\sup_{t,i_t}\ex{\left(\frac{\bra{\phi_{i_t}^t} \id\otimes\cM_U(\rho_t) \ket{{\phi_{i_t}^t}}}{\bra{\phi_{i_t}^t} \id\otimes\cD(\rho_t) \ket{{\phi_{i_t}^t}}} \right)^2}+48N\eps^2\sqrt{\frac{\log(10)}{M}}
\\&\le 24N\frac{\eps^2}{\din\dout}+48N\eps^2\sqrt{\frac{\log(10)}{M}}\le \cO\left(N\frac{\eps^2}{\din\dout} \right)
\end{align*}
because $M=\exp(\Omega(\din^2\dout^2))$. But from the data processing inequality and Lemma~\ref{lem:fano}, $\cI(X:I_1,\dots, I_N)\ge \cI(X:Y) \ge \Omega(\din^2\dout^2) $, we deduce that:
\begin{align*}
\cO\left(N\frac{\eps^2}{\din\dout} \right)\ge \cI(X:I_1,\dots, I_N)\ge \Omega(\din^2\dout^2).\end{align*}
Finally, the lower bound follows:
\begin{align*}
N\ge \Omega\left( \frac{\din^3\dout^3}{\eps^2}\right).
\end{align*}
\end{proof}
To assess this lower bound, it is necessary to design an algorithm for quantum process tomography. This will be the object of the following section.
\section{Upper bound}
In this section, we propose an upper bound on the complexity of the quantum process tomography problem. We generalize the algorithm proposed by \cite{surawy2022projected} which is ancilla-free.
\begin{theorem}\cite{surawy2022projected} There is an ancilla-free process tomography algorithm that learns a quantum channel (of $\din=\dout=d$) in the distance $d_2$ using only a number of measurements:
\begin{align*}
N=\cO\left( \frac{d^6\log(d)}{\eps^2}\right).
\end{align*}
\end{theorem}
This algorithm proceeds by providing an unbiased estimator for the Choi state $\cJ_{\cN}$, then projecting this matrix to the space of Choi states (PSD and partial trace $\dI/d$) and finally by invoking the Choi–Jamiołkowski isomorphism we obtain an approximation of the channel. This reduction from learning the Choi state in the operator norm to learning the quantum channel in the $d_2$ distance uses mainly the inequality $d_2(\cN,\cM)=d\|\cJ_{\cN}-\cJ_{\cM}\|_2\le d^2\|\cJ_{\cN}-\cJ_{\cM} \|_\infty$ when $\din= \dout =d$. We generalize this result to the diamond norm and any input/output dimensions.
For this we show the following inequality:
\begin{lemma}\label{lem: diamond infty}
Let $\cN_1$ and $\cN_2$ be two quantum channels. We have:
\begin{align*}
d_\diamond(\cN_1,\cN_2)\le \din \dout\|\cJ_{\cN_1}-\cJ_{\cN_2} \|_\infty.
\end{align*}
\end{lemma}
This inequality can also be obtained by applying the inequality $(3)$ of \cite{nechita2018almost} and the triangle inequality. We provide a simpler proof for completeness.
\begin{proof}
Denote by $\cM=\cN_1-\cN_2$. Let $\ket{\phi}$ be a maximizing unit vector of the diamond norm, i.e., $\|\id \otimes \cM(\proj{\phi})\|_1= d_\diamond(\cN_1,\cN_2)$.
We can write $\ket{\phi}= A\otimes \dI \ket{\Psi} $ where $\ket{\Psi}=\frac{1}{\sqrt{\din}}\sum_{i=1}^{\din} \ket{ii}$ is the maximally entangled state. $\ket{\phi}$ has norm $1$ so $\frac{1}{\din}\tr(A^\dagger A)=\bra{\Psi}A^\dagger A \otimes \dI \ket{\Psi} = \spr{\phi}{\phi}=1. $ On the other hand we can write
\begin{align*}
d_\diamond(\cN_1,\cN_2) &= \|\id \otimes \cM(\proj{\phi})\|_1 \\&= \|\dI \otimes \cM(A\otimes \dI_{\din}\proj{\Psi} A^\dagger \otimes \dI_{\din})\|_1
\\&= \|(A\otimes \dI_{\dout}) \id \otimes \cM(\proj{\Psi})( A^\dagger \otimes \dI_{\dout})\|_1
\\&= \|(A\otimes \dI_{\dout})\cJ_{\cM}( A^\dagger \otimes \dI_{\dout})\|_1.
\end{align*}
$\cJ_{\cM}$ is Hermitian so it can be written as : $\cJ_{\cM}=\sum_i \lambda_i \proj{\psi_i}$. Using the triangle inequality, we obtain:
\begin{align*}
&\|(A\otimes \dI_{\dout})\cJ_{\cM}( A^\dagger \otimes \dI_{\dout})\|_1
\\&= \left\|(A\otimes \dI_{\dout})\sum_i \lambda_i \proj{\psi_i}( A^\dagger \otimes \dI_{\dout})\right\|_1
\\&\le \sum_i |\lambda_i| \|(A\otimes \dI_{\dout}) \proj{\psi_i}( A^\dagger \otimes \dI_{\dout})\|_1
\\&\le \max_i |\lambda_i| \sum_i \|(A\otimes \dI_{\dout}) \proj{\psi_i}( A^\dagger \otimes \dI_{\dout})\|_1
\\&= \|\cJ\|_\infty \sum_i \tr((A\otimes \dI_{\dout}) \proj{\psi_i}( A^\dagger \otimes \dI_{\dout}))
\\&= \|\cJ\|_\infty \tr( AA^\dagger \otimes \dI_{\dout})
= \din \dout \|\cJ\|_\infty.
\end{align*}
\end{proof}
This Lemma shows that the diamond and $2$ distances satisfy the same inequality with respect to the infinity norm between the Choi states when $\din=\dout=d$. Since the algorithm of \cite{surawy2022projected} approximates first the Choi state in the infinity norm, we obtain the same upper bound for the diamond distance. For general dimensions, we obtain the following complexity:
\begin{theorem}\label{upper-bound} There is a non-adaptive ancilla-free process tomography algorithm that learns a quantum channel in the distance $d_\diamond$ using only a number of measurements:
\begin{align*}
N=\cO\left( \frac{\din^3\dout^3\log(\din\dout)}{\eps^2}\right).
\end{align*}
\end{theorem}
This complexity was expected for process tomography with incoherent measurements since the complexity of state tomography with incoherent measurements is $\Theta\left(\frac{d^3}{\eps^2}\right)$ \cite{haah2016sample} and learning $(\din,\dout)$-dimensional channels can be thought of as learning states of dimension $\din\times \dout$. We believe that the $\log(\din\dout)$-factor can be removed from the upper bound in Theorem~\ref{upper-bound} using the techniques of \cite{guctua2020fast}.
The algorithm is formally described in Alg.~\ref{Alg} and is similar to the one in~\cite{surawy2022projected}. By Theorem~\ref{thm:lb}, Alg.~\ref{Alg} is almost optimal.
\begin{algorithm}
\caption{Learning a quantum channel in the diamond distance using ancilla-free independent measurements. }\label{Alg}
\begin{algorithmic}[t!]
\State $N=\cO(\din^3\dout^3\log(\din\dout)/\eps^2)$.
\For{$t =1:N$}
\State Sample two independent copies of $\Haar$ distributed unitaries $V\sim \Haar(\din)$ and $U\sim \Haar(\dout)$ .
\State Let $\ket{v}= V\ket{0}$ be a haar distributed vector.
\State Take the input states $\rho_t=\proj{v}$ and $\sigma_t=\frac{\dI}{\din}$, the output states are respectively $\cN(\proj{v})$ and $\cN\left(\frac{\dI}{\din}\right)$.
\State Perform a measurement on $\cN(\proj{v})$ and $\cN\left(\frac{\dI}{\din}\right)$ using the POVM $\cM_U:=\{U\proj{i}U^\dagger\}_{i\in [\dout]}$ and observe $i_t\sim p_{U,V}:= \{ \bra{i} U^\dagger \cN(\proj{v})U\ket{i} \}_{i\in [\dout]}$ and $j_t\sim q_{U}:= \{ \bra{i} U^\dagger \cN\left(\frac{\dI}{d}\right)U\ket{i} \}_{i\in [\dout]}$.
\State Define $\cJ_t:=(\din+1) \proj{v}^T\otimes ((\dout+1)(U\proj{i_t}U^\dagger )-\dI ) -\dI \otimes((\dout+1)(U\proj{j_t}U^\dagger )-\dI )$
\EndFor
\State Define the estimator $\hat{\cJ}=\frac{1}{N}\sum_{t=1}^{N}\cJ_t$.
\State Find a valid Choi state $\cJ_{\cM}$ such that $\|\cJ_{\cM}-\hat{\cJ}\|_\infty\le \frac{\eps}{2\din\dout}$.\\
\Return the quantum channel $\cM$ corresponding to the Choi state $\cJ_{\cM}$.
\end{algorithmic}
\end{algorithm}
Its analysis is also similar to the one in~\cite{surawy2022projected}.
\paragraph{Correctness} Let us prove that Alg.~\ref{Alg} is $1/3$-correct.
First we show that $\hat{\cJ}=\frac{1}{N}\sum_{t=1}^{N}\cJ_t$ is an unbiased estimator of $\cJ_\cN$. For this, we prove the following lemma relating the Choi state to the average of the tensor product of a random rank-$1$ projector and its image by the quantum channel.
\begin{lemma}\label{lem: exJ}
Let $\ket{\phi}$ be a $\Haar$-distributed random vector. We have the following equality:
\begin{align*}
\cJ_\cN= (\din+1) \ex{\proj{\phi}^T \otimes \cN(\proj{\phi})}-\dI \otimes\cN\left(\frac{\dI}{\din}\right).
\end{align*}
\end{lemma}
\begin{proof}We use the Kraus decomposition of the quantum channel $\cN(\rho)=\sum_k A_k \rho A_K^\dagger$.
We start by writing the following expectation:
\begin{align*}
\ex{\proj{\phi} \otimes \cN(\proj{\phi})}&= \sum_k \ex{\proj{\phi} \otimes A_k\proj{\phi}A_k^\dagger}
\\&= \sum_k \dI \otimes A_k \ex{\proj{\phi} \otimes \proj{\phi}}\dI \otimes A_k^\dagger
\end{align*}
Let $F$ be the flip operator $F=\sum_{i,j=1}^{\din} \ket{ij}\bra{ji}$, if we take the transpose on the fist tensor we obtain the unnormalized maximally entangled state:
\begin{align*}
F^{T_1}= \sum_{i,j=1}^{\din} \ket{i}\bra{j}^T\otimes \ket{j}\bra{i}= \sum_{i,j=1}^{\din} \ket{j}\bra{i}\otimes \ket{j}\bra{i}= \din\proj{\Psi}
\end{align*}
where $\proj{\Psi}=\frac{1}{\din}\sum_{i,j=1}^{\din} \ket{ii}\bra{jj}=\frac{1}{\din}\sum_{i,j=1}^{\din} \ket{i}\bra{j}\otimes\ket{i}\bra{j} $ is the maximally entangled state.
It is known that there is constants $\alpha$ and $\beta$ such that:
\begin{align*}
\ex{\proj{\phi} \otimes \proj{\phi}}= \alpha \dI+\beta F.
\end{align*}
Taking the trace we have the first relation $1=\alpha \din^2+\beta \din$, then taking the trace after multiplying with $F$ we obtain the second relation $\tr(\proj{\phi} \otimes \proj{\phi}F)=\tr(\proj{\phi} \proj{\phi})=1=\alpha \din +\beta \din^2$. These relations imply $\alpha=\beta =\frac{1}{\din(\din+1)}.$ Hence:
\begin{align*}
\ex{\proj{\phi} \otimes \proj{\phi}}= \frac{\dI+ F}{\din(\din+1)}.
\end{align*}
Replacing this expectation on the first expectation yields:
\begin{align*}
&\ex{\proj{\phi}^T \otimes \cN(\proj{\phi})}= \sum_k \ex{\proj{\phi}^T \otimes A_k\proj{\phi}A_k^\dagger}
\\&= \sum_k \dI \otimes A_k \ex{\proj{\phi}^T \otimes \proj{\phi}}\dI \otimes A_k^\dagger
\\&= \frac{1}{\din(\din+1)} \sum_k \dI \otimes A_kA_k^\dagger +\frac{1}{\din(\din+1)} \sum_k \dI \otimes A_k (\din \proj{\Psi}) \dI \otimes A_k^\dagger
\\&= \frac{1}{\din(\din+1)} \dI \otimes \cN(\dI) +\frac{1}{\din+1} \dI \otimes \cN(\proj{\Psi})
\\&= \frac{1}{\din(\din+1)} \dI \otimes \cN(\dI) +\frac{1}{\din+1} \cJ_\cN.
\end{align*}
\end{proof}
Then we compute another expectation:
\begin{lemma} \label{lem: ex}
Let $U\sim \Haar(d)$ and $x\sim p_{U,\rho}:= \{ \bra{i}U^\dagger\rho U\ket{i} \}_{i\in [d]}$, we have
\begin{align*}
\ex{(d+1) U\proj{x}U^\dagger-\dI }= \rho
\end{align*}
\end{lemma}
\begin{proof}
Since the equality is linear in $\rho$ we can without loss of generality restrict ourselves to a pure state $\rho=\proj{\phi}$. Now $x\sim \{ \bra{i}U^\dagger\proj{\phi} U\ket{i} \}_{i\in [d]} $ hence for $k,l \in [d]$, by Weingarten calculus:
\begin{align*}
&\mathds{E}_{U, x\sim p_{U, \phi}}\left(\bra{k}U\proj{x}U^\dagger\ket{l}\right)
\\&= \mathds{E}_{U}\left(\sum_{x=1}^d \bra{x}U^\dagger\proj{\phi} U\ket{x} \bra{k}U\proj{x}U^\dagger\ket{l}\right)
\\&= \mathds{E}_{U}\left(\sum_{x=1}^d \bra{x}U^\dagger\proj{\phi} U\ket{x} \bra{x}U^\dagger\ket{l}\bra{k}U\ket{x}\right)
\\&=\sum_{x=1}^d \frac{1}{d(d+1)}\left(\delta_{l,k}+ \spr{\phi}{l}\spr{k}{\phi}\right)
\\&= \frac{1}{(d+1)}\left(\bra{k}\dI+\proj{\phi}\ket{l})\right)
\end{align*}
Therefore
\begin{align*}
\ex{(d+1) U\proj{x}U^\dagger-\dI }= \proj{\phi}=\rho.
\end{align*}
\end{proof}
Using Lemma~\ref{lem: exJ} and Lemma~\ref{lem: ex} we deduce:
\begin{align*}
\ex{\hat{\cJ}}&= \ex{\cJ_1}
\\&= \mathds{E}_{V,U,i_t,j_t} \big[(\din+1) \proj{v}^T\otimes ((\dout+1)(U\proj{i_t}U^\dagger )-\dI )\big]
\\&-\mathds{E}_{V,U,i_t,j_t} \big[\dI\otimes((\dout+1)(U\proj{j_t}U^\dagger )
- \dI)\big]
\\&= \mathds{E}_{V} \big[(\din + 1) \proj{v}^T\otimes\mathds{E}_{U,i_t} ((\dout + 1)(U\proj{i_t}U^\dagger )-\dI ) \big]
\\&-\big[\dI\otimes\mathds{E}_{U,j_t} ((\dout+1)(U\proj{j_t}U^\dagger )-\dI )\big]
\\&= \mathds{E}_{V} \left((\din+1) \proj{v}^T\otimes \cN(\proj{v})) -\dI\otimes\cN\left(\frac{\dI}{\din}\right)\right)
=\cJ_\cN.
\end{align*}
So the estimator $\hat{\cJ}=\frac{1}{N}\sum_{t=1}^{N}\cJ_t$ is unbiased.
It remains to show a concentration inequality for the random variable $\hat{\cJ}$ so that we can estimate how much steps we need in order to achieve the precision and confidence we aim to. For this, we use the matrix Bernstein inequality \cite{tropp2012user}:
\begin{theorem}\cite{tropp2012user}
Consider a sequence of $n$ independent Hermitian random matrices $A_1,\dots, A_n \in \mathds{C}^{d\times d}$. Assume that each $A_i$ satisfies
\begin{align*}
\ex{A_i}=0 ~~~\text{ and }~~~~ \|A_i\|_\infty \le R \text{ as.}
\end{align*}
Let $\sigma^2= \|\sum_{i=1}^n \ex{A_i^2}\|_\infty$. Then for any $t\ge \frac{\sigma^2}{R}$:
\begin{align*}
\pr{ \left\| \sum_{i=1}^n (A_i-\ex{A_i}) \right\|_\infty \ge t } \le d\exp\left(-\frac{3t}{8R}\right).
\end{align*}
Moreover for any $t\le \frac{\sigma^2}{R}$:
\begin{align*}
\pr{ \left\| \sum_{i=1}^n (A_i-\ex{A_i}) \right\|_\infty \ge t } \le d\exp\left(-\frac{3t^2}{8\sigma^2}\right).
\end{align*}
\end{theorem}
Let $\cJ=\cJ_\cN=\ex{\cJ_t}$. We apply this theorem to the estimator $\hat{\cJ}-\cJ= \frac{1}{N}\sum_{t=1}^{N}(\cJ_t-\cJ)$. Recall that
\begin{align*}
\cJ_t&=(\din+1) \proj{v}^{T}\otimes ((\dout+1)(U\proj{i_t}U^\dagger )-\dI )
-\dI\otimes((\dout+1)(U\proj{j_t}U^\dagger )-\dI ).
\end{align*}
Let $A_t=\frac{\cJ_t-\cJ}{N}$,
we have proven that $\ex{A_t}=\frac{1}{N}\ex{\cJ_t-\cJ}=0$. Moreover
\begin{align*}
\|A_t\|_\infty &=\frac{1}{N} \|\cJ_t-\cJ\|_{\infty}\le \frac{1}{N} (\|\cJ_t\|_{\infty}+\|\cJ\|_{\infty})
\le \frac{8\din\dout}{N}:=R.
\end{align*}
Besides
\begin{align*}
\sigma^2&= \left\|\sum_{t=1}^N \ex{A_t^2}\right\|_\infty= \frac{1}{N}\left\| \ex{(\cJ_1-\cJ)^2}\right\|_\infty
= \frac{1}{N}\left\| \ex{(\cJ_1)^2}\right\|_\infty +\Theta\left(\frac{1}{N}\right).
\end{align*}
Using the identity $\left(a\proj{\phi}-\dI\right)^2= (a^2-2a)\proj{\phi}+\dI$, we have:
\begin{align*}
&\ex{ \big[\dI\otimes((\dout+1)(U\proj{j_t}U^\dagger )-\dI )\big]^2}
\\&=\ex{ (\dI\otimes((\dout^2-1)(U\proj{j_t}U^\dagger )+\dI )}
\\&=\ex{ (\dI\otimes((\dout^2-1)(U\proj{j_t}U^\dagger-\dI/(\dout+1) )+\dout~\dI )}
\\&= (\dout-1)\dI\otimes \cN(\dI/\din) +\dout~\dI\otimes \dI
\end{align*}
has an operator norm at most $\cO(\dout)$ so we can focus on the first term in the definition of $\cJ_1$ which has the main contribution. We have using again the identity $\left(a\proj{\phi}-\dI\right)^2= (a^2-2a)\proj{\phi}+\dI$:
\begin{align*}
&\mathds{E}\big[(\din+1) \proj{v}^{T}\otimes ((\dout+1)(U\proj{i_t}U^\dagger )-\dI )\big]^2
\\&= (\din+1)^2 \ex{\proj{v}^{T}\otimes ((\dout+1)(U\proj{i_t}U^\dagger)-\dI)^2}
\\&= (\din+1)^2 \ex{\proj{v}^{T}\otimes ((\dout^2-1)(U\proj{i_t}U^\dagger)+\dI) }
\\&=(\dout-1) (\din+1)(\cJ+\dI\otimes \cN(\dI/\din)) + \left(\frac{\dout(\din+1)^2}{\din}\right)\dI
\end{align*}
which has an operator norm $\Theta(\din\dout)$. Therefore
\begin{align*}
\sigma^2= \frac{1}{N}\left\| \ex{\cJ_1^2}\right\|_\infty +\Theta\left(\frac{1}{N}\right)=\Theta\left(\frac{\din\dout}{N}\right).
\end{align*}
Since we have $\frac{\sigma^2}{R} \ge \Omega(1)$ we can use the matrix-Bernstein inequality in the regime $t=\frac{\eps}{2\din\dout}\le \cO(1)$:
\begin{align*}
\pr{ \left\| \sum_{t=1}^N (A_t-\ex{A_t}) \right\|_\infty \ge \frac{\eps}{2\din\dout} } &\le \din\dout\exp\left(-\frac{3\eps^2}{8\din^2\dout^2\sigma^2}\right)
\\&\le \din\dout\exp\left(-\frac{CN\eps^2}{\din^3\dout^3}\right)
\end{align*}
where $C>0$ is a universal constant. Hence if $N= \din^3\dout^3\log(3\din\dout)/(C\eps^2)=\cO\left(\din^3\dout^3\log(\din\dout)/\eps^2\right)$ then with a probability at least $2/3$ we have
\begin{align*}
\|\hat{\cJ}-\cJ_\cN\|_\infty = \left\| \sum_{t=1}^N (A_t-\ex{A_t}) \right\|_\infty\le \frac{\eps}{2\din\dout}.
\end{align*}
This implies
that $\|\cJ_\cM-\cJ_\cN\|_\infty\le \frac{\eps}{\din\dout}$ and finally
$\|\cM-\cN\|_\diamond \le \eps$ by Lemma~\ref{lem: diamond infty}. This finishes the proof of the correctness of Alg~\ref{Alg}.
\section{Conclusion and open questions}
In this work, we find the optimal complexity of quantum process tomography using non-adaptive incoherent measurements. Furthermore, we show that ancilla-assisted strategies cannot outperform their ancilla-free counterparts contrary to Pauli channel tomography \cite{chen2022quantum}. Still, many questions remain open. First, it is known that adaptive strategies have the same complexity as non-adaptive ones for state tomography \cite{chen2022tight}, could adaptive strategies overcome non-adaptive ones for quantum process tomography? Secondly, can entangled strategies exploit the symmetry and show a polynomial (in $\din,\dout$) speedup as they do for state tomography \cite{haah2016sample}? Lastly, what would be the potential improvements for simpler problems such as testing identity to a fixed quantum channel or learning the expectations of some given input states and observables?
\printbibliography
\end{document}
|
\begin{document}
\title[Tensor Product Decomposition and Identities]{Tensor Product Decomposition of $\widehat{\mathfrak{sl}}(n)$ Modules and Identities}
\author{Kailash C. Misra \and Evan A. Wilson}
\address{Department of Mathematics, North Carolina State University, Raleigh,
NC 27695-8205}
\email{[email protected]}
\address{
Instituto de Matem\'{a}tica e Estat\'{i}stica\\Universidade de S\~{a}o Paulo}
\email{[email protected]}
\thanks{KCM: partially supported by NSA grant, H98230-12-1-0248,\\
\indent EAW: supported by a postdoctoral fellowship from FAPESP (2011/12079-5).}
\subjclass[2010]{17B67, 17B10, 17B37}
\date{}
\begin{abstract}
We decompose the $\widehat{\mathfrak{sl}}(n)$-module $V(\Lambda_0) \otimes V(\Lambda_i)$ and give generating function identities for the outer multiplicities. In the process we discover some seemingly new partition identities for $n=3, 4$.
\end{abstract}
\maketitle
\section{Introduction}
\par
The affine Lie algebras are the simplest family of infinite dimensional Kac-Moody Lie algebras (cf. \cite{K}). The connection between affine Lie algebra representations and partition identities is well known (for example, see \cite{K1}, \cite{L}, \cite{LM}, \cite{LW}) since 1970's. The affine Lie algebra $\mathfrak{g} = \widehat{\mathfrak{sl}}(n, \mathbb{C})$ is the infinite dimensional analog of the finite dimensional simple Lie algebra $\mathfrak{sl}(n, \mathbb{C})$ of $n \times n$ trace zero matrices. In fact the affine Lie algebra $\widehat{\mathfrak{sl}}(n, \mathbb{C}) = \mathfrak{sl}(n,\mathbb{C}) \otimes \mathbb{C}[t,t^{-1}] \oplus \mathbb{C} c \oplus \mathbb{C} d$ is generated by the degree derivation
$d=1 \otimes t\frac{d}{dt}$, and the Chevalley generators:
$$ e_j=E_{j,j+1} \otimes 1, \qquad f_j=E_{j+1,j} \otimes 1, \qquad h_j= (E_{jj}-E_{j+1,j+1})\otimes 1,\ \ j=1,2,\dots, n-1, $$
$$ e_0=E_{n,1} \otimes t,\qquad f_0=E_{1,n} \otimes t^{-1}, \qquad h_0=(E_{n,n} - E_{1,1}) \otimes 1 + c,$$
where $c=\sum_{j=0}^{n-1} h_j$ spans the one-dimensional center and $E_{i,j}$ denote the $n \times n$ matrix units. With respect to the
\emph{Cartan subalgebra} ${\mathfrak{h}}:=\text{span}_{\mathbb{C}}\{h_0, h_1, \dots, h_{n-1}\} \oplus \mathbb{C}d$, let $\Delta =\{\alpha_0, \alpha_1, \dots, \alpha_{n-1}\}$ be the set of simple roots and
$\Phi$ be the set of roots for $\widehat{\mathfrak{sl}}(n,\mathbb{C})$. Then $\delta = \alpha_0+ \alpha_1+ \cdots + \alpha_{n-1}$ is the null root and $P = \text{span}_{\mathbb{Z}}\{\Lambda_0,\Lambda_1,\dots, \Lambda_{n-1}, \delta\},$ is the \emph{weight lattice} where $\langle \Lambda_j, h_k\rangle=\delta_{jk},$ and $\langle \Lambda_j,d\rangle=0$, for $j=0,1,\dots,n-1$. The \emph{dominant weight lattice} is defined to be $P_+=\text{span}_{\mathbb{Z}_{\geq 0}}\{\Lambda_0,\Lambda_1,\dots,\Lambda_{n-1}\}\oplus \mathbb{Z}\delta.$ By the \emph{level} of $\lambda \in P_+$ we mean the nonnegative integer $\text{level}(\lambda)= \lambda (c)$. For notational convenience we define $\alpha_j = \alpha_{\overline{j}}$ and $\Lambda_j = \Lambda_{\overline{j}}$ for all $j \in \mathbb{Z}$ where $\overline{j}:=j \pmod{n}$.
For $\lambda \in P_+$, let $V(\lambda)$ denote the irreducible integrable $\widehat{\mathfrak{sl}}(n)$-module with highest weight $\lambda$.
For $\lambda , \mu \in P_+$, it is known that the tensor product module $V(\lambda)\otimes V(\mu)$ is completely reducible (cf. \cite[Corollary 10.7 b]{K}), that is:
\begin{equation}
V(\lambda) \otimes V(\mu)=\bigoplus_{\nu \in P_+}c^{\nu}_{\lambda,\mu}V(\nu) \label{gendecomp},
\end{equation}
where $c^{\nu}_{\lambda,\mu}$, called the outer multiplicity, denotes the number of times $V(\nu)$ occurs in this decomposition. In \cite{MW} we studied these outer multiplicities using the crystal base theory for the case $\lambda = \mu = \Lambda_0$ and obtained several identities. In this paper we consider the case $\lambda = \Lambda_0, \mu = \Lambda_i$, where $1 \le i \le n-1$. Using the approach in \cite{MW} we obtain several different identities, some of them seemingly new for the cases $i = 1, n = 3, 4$.
\section{Decomposition of $V(\Lambda_0)\otimes V(\Lambda_i)$}
In order to decompose the module $V(\Lambda_0)\otimes V(\Lambda_i)$ we will us the theory of crystals (\cite{Ka1}, \cite{Ka2}, \cite{Lu}, \cite{HK}) associated with integrable representations of quantum affine algebras. Indeed in this paper we will use the explicit realization of the crystal
$ B(\Lambda_i)$ for the module $V(\Lambda_i)$ in terms of \emph{extended Young diagrams} (or \emph{colored Young diagrams})(\cite{MM}, \cite{JMMO}) which we briefly describe.
Let $I:=\{0,1,\dots n-1\}$ denote the index set for $\widehat{\mathfrak{sl}}(n)$. An \emph{extended Young diagram} is a collection of $I$-colored boxes arranged in left-justified rows and top-justified columns, such that the number of boxes in each row is greater than or equal to the number of boxes in the row below. To every extended Young diagram we associate a \emph{charge}, $i\in I$. In each box, we put a \emph{color} $j\in I$ given by $j \equiv a-b+i \pmod{n}$ where $a$ is the number of columns from the right and $b$ is the number of rows from the top (see figure \ref{pattern}).
\begin{figure}
\caption{Color pattern for an extended Young diagram of charge $i$. All labels are reduced modulo $n$.}
\label{pattern}
\end{figure}
For example, {\scriptsize\young(120,01)} is an extended Young diagram of charge $1$ for $n=3$. The \emph{null diagram} with no boxes---denoted by $\varnothing$---is also considered an extended Young diagram.
A column in an extended Young diagram is $j$-\emph{removable} if the bottom box contains $j$ and can be removed leaving another extended Young diagram. A column is $j$-\emph{admissible} if a box containing $j$ could be added to give another extended Young diagram.
An extended Young diagram is called $n$-\emph{regular} if there are at most $(n-1)$ rows with the same number of boxes. Let $\mathcal{Y}(i)$ denote the collection of all $n$-regular extended Young diagrams of charge $0$. Then $\mathcal{Y}(i)$ can be given the structure of a crystal with the following actions of $\tilde{e}_j$, $\tilde{f}_j$, $\varepsilon_j,$ $\varphi_j$, and wt$(\cdot)$. For each $j\in I$ and $b\in \mathcal{Y}(i)$ we define the \emph{$j$-signature} of $b$ to be the string of $+$'s, and $-$'s in which each $j$-admissible column receives a $+$ and each $j$-removable column receives a $-$ reading from right to left. The \emph{reduced $i$-signature} is the result of recursively canceling all `$+-$' pairs in the $i$-signature leaving a string of the form $(-,\dots, -,+\dots, +)$. The Kashiwara operator $\tilde{e}_j$ acts on $b$ by removing the box corresponding to the rightmost $-$, or maps $b$ to $0$ if there are no minus signs. Similarly, $\tilde{f}_j$ adds a box to the bottom of the column corresponding to the leftmost $+$, or maps $b$ to $0$ if there are no plus signs. The function $\varphi_j(b)$ is the number of $+$ signs in the reduced $j$-signature of $b$ and $\varepsilon_j(b)$ is the number of $-$ signs. We define
wt$:\mathcal{Y}(i)\rightarrow P$ by $b \mapsto \Lambda_i-\sum_{j=0}^{n-1}$\# $\{j$-colored boxes in $b\}\alpha_j$. Then $\mathcal{Y}(i)\cong B(\Lambda_i).$
In order to obtain the decomposition of $V(\Lambda_0)\otimes V(\Lambda_i)$, it suffices to find the set of the \emph{maximal} elements of the crystal base $B(\Lambda_0) \otimes B(\Lambda_i)$, i.e. the set of all $b_1\otimes b_2\in B(\Lambda_0) \otimes B(\Lambda_i)$ for which $\tilde{e}_i (b_1\otimes b_2)=0$ for all $i\in I$. Maximal elements are characterized by the following:
\begin{lemma}[see \cite{JMMO}]
An element $b_1 \otimes b_2 \in B(\Lambda_0)\otimes B(\Lambda_i)$ is maximal if and only if
$\tilde{e}_jb_1=0 $ and $\tilde{e}_j^{\delta_{j0}+1}b_2=0$ for all $j\in I$.
\end{lemma}
\begin{lemma}[\cite{MW}]\label{l2}
An element $b_1 \otimes b_2$ of the $U_q(\widehat{\mathfrak{sl}}(n))$ crystal $B(\Lambda_0) \otimes B(\Lambda_i)$ is maximal if and only if $b_1$ is the null diagram and the following two conditions are satisfied for $b_2$:
\begin{enumerate}
\item the first removable column from the right in $b_2$ is $0$-removable,
\item for all $j \in \{0,1,\dots, n-1\}$, if the $k$th admissible column in $b_2$ is $j$-admissible then the $k+1$st removable column, if it exists, is $j$-removable.
\end{enumerate}
\end{lemma}
As an application of Lemma \ref{l2}, all maximal elements of weight $2\Lambda_0-3\delta$ for $B(\Lambda_0)\otimes B(\Lambda_2)$ with $n=3$ are given in Figure \ref{delta2}.\\
\begin{figure}
\caption{Examples of highest weight elements in $B(\Lambda_0)\otimes B(\Lambda_2)$, for $n=3$.}
\label{delta2}
\end{figure}
We denote a \emph{partition} by a finite sequence $(\lambda_1^{f_1},\lambda_2^{f_2},\dots, \lambda_l^{f_l}),$ where $\lambda_k\in \mathbb{Z}_{> 0},$ $\lambda_k>\lambda_{k+1},$ and $f_k\in \mathbb{Z}_{> 0}$ denotes the multiplicity of
$\lambda_k$. Each $b\in\mathcal{\mathcal{Y}}(i)$ can be uniquely represented as a partition where $\lambda_k$ is the number of boxes in a given row, and $f_k$ is the number of rows having $\lambda_k$ boxes. For example, the two diagrams in Figure \ref{delta2} correspond to the partitions $(5, 4,1^2),$ and $(3^2)$. We can now rephrase Lemma \ref{l2} in terms of partitions as follows.
\begin{lemma}\label{l3}
The highest weight elements of $B(\Lambda_0)\otimes B(\Lambda_i)$ are in a one-to-one correspondence with the set of all partitions $\lambda=(\lambda_1^{f_1},\lambda_2^{f_2},\dots, \lambda_l^{f_l})$ with $f_k< n,$ $k=1,2,\dots, l$, satisfying the conditions:
\begin{enumerate}
\item $ \lambda_1- f_1 +i \equiv 0\pmod{n}$,
\item $f_k+f_{k+1}+\lambda_{k}-\lambda_{k+1}\equiv0 \pmod{n}$, for $k<l$.
\end{enumerate}
\end{lemma}
\begin{proof} The condition that each $f_k< n$ is equivalent to the condition that $b\in\mathcal{\mathcal{Y}}(i)$ is $n$-regular.
The first column from the right is always removable, so by condition (1) of Lemma \ref{l2} the first column must contain a 0-colored box in the bottom row. Since $\lambda_1$ is the number of columns in the diagram and $f_1$ is the number of boxes in the rightmost column, we see that $\lambda_1-f_1 +i\equiv 0 \pmod{n}$.
Now, suppose that the $k$th admissible column is $j$-admissible, and there exists a removable column to the left of that column. The $k$th admissible column is $\lambda_{k}+1$ columns from the left and contains $f_1+f_2+\dots +f_{k-1}$ boxes. Therefore:
\begin{eqnarray}
f_1+f_2+\dots +f_{k-1}-(\lambda_k+1)+i&\equiv&j-1 \pmod{n}\nonumber\\
f_1+f_2+\dots +f_{k-1}-\lambda_k+i&\equiv&j \pmod{n}\label{t1}
\end{eqnarray}
By condition (2) of Lemma \ref{l2} the $k+1$st removable column (the $\lambda_{k+1}$st from the left) is also $j$-removable, and contains $f_1+f_2+\dots +f_k+f_{k+1}$ boxes. Therefore:
\begin{equation}
f_1+f_2+\dots+f_k+f_{k+1}-\lambda_{k+1}+i \equiv j \pmod{n}\label{t2}
\end{equation}
Subtracting equation \eqref{t1} from equation \eqref{t2} we obtain condition (2). Furthermore, if the partition satisfies the conditions (1) and (2), then it is in correspondence with an extended Young diagram as in Lemma \ref{l2}.
\end{proof}
Let $\mathscr{C}_n^i$ be the collection of all partitions satisfying conditions (1) and (2) in Lemma \ref{l3}. Then each such partition in $\mathscr{C}_n^i$ corresponds to a unique maximal element in $B(\Lambda_0)\otimes B(\Lambda_i)$. Let
$\mathscr{C}_{n,\mu}^i$ denote the set of elements in $\mathscr{C}_n^i$ of weight $\mu$. It is known that each connected component of the crystal $B(\Lambda_0)\otimes B(\Lambda_i)$ is the crystal for the corresponding irreducible summand of the module $V(\Lambda_0)\otimes V(\Lambda_i)$. In the following lemma we determine the connected components of the crystal $B(\Lambda_0)\otimes B(\Lambda_i)$.
\begin{lemma}\label{l4}
Each connected component of $B(\Lambda_0)\otimes B(\Lambda_i)$ is isomorphic to \\
$B(\Lambda_t+\Lambda_u-k\delta)$ for some $t,u \in\{0,1,\dots, n-1\}$ such that $t+u \equiv i\pmod{n}$ and $k\in \mathbb{Z}_{\geq 0}$ such that $k\geq t$ if $t\leq i$ and $k\geq t-i$ if $t>i$ .
\end{lemma}
\begin{proof}
Let $b\otimes b'\in B(\Lambda_0)\otimes B(\Lambda_i)$ be maximal. Then $\text{wt}(b)=\Lambda_0$, and $b'$ corresponds to a partition $(\lambda_1^{f_1},\lambda_2^{f_2}\dots,\lambda_l^{f_l})\in \mathscr{C}_n$. We set $\alpha=\Lambda_i-\text{wt}(b')$ and use the weight formula in $B(\Lambda_i)$ to compute $\alpha$:
\begin{eqnarray}
\alpha&=&\sum_{r=1}^{l}\sum_{j_1=1}^{\lambda_{r}}\sum_{j_2=s_{r-1}+1}^{s_{r}}\alpha_{j_1-j_2+i}, \label{trickysum}\\
&=&\sum_{r=1}^{l}\sum_{j_1=1}^{\lambda_{r}}\sum_{j_2=s_{r-1}+1}^{s_{r}}(2\Lambda_{j_1-j_2+i}-\Lambda_{j_1-j_2+i-1}-\Lambda_{j_1-j_2+1}+\delta_{\overline{j_1-j_2+i},0}\delta),\nonumber
\end{eqnarray}
where $s_j:=\sum_{m=1}^j f_m$. The sums telescope, leaving:
\begin{equation}
\alpha=
{\sum_{r=1}^l(\Lambda_{\lambda_r-s_r+i}-\Lambda_{i-s_r}+\Lambda_{i-s_{r-1}}-
\Lambda_{\lambda_r-s_{r-1}+i})}+k\delta,\label{withdelta}
\end{equation}
where $k\in \mathbb{Z}_{\geq 0}$ is the number of $\alpha_0$'s in the sum \eqref{trickysum}. Now consider the sum in \eqref{withdelta}.
\begin{eqnarray*}
\lefteqn{\sum_{r=1}^l(\Lambda_{\lambda_r-s_r+i}-\Lambda_{i-s_r}+\Lambda_{i-s_{r-1}}-
\Lambda_{\lambda_r-s_{r-1}+i})}\\
&=&\Lambda_{\lambda_1-f_1+i}+\sum_{r=2}^l\Lambda_{\lambda_{r-1}+f_{r-1}+f_r-s_{r}+i}-\sum_{r=1}^l\Lambda_{i-s_{r}}
+\sum_{r=1}^l\Lambda_{i-s_{r-1}}-\sum_{r=1}^l\Lambda_{\lambda_r-s_{r-1}+i}\\
&&\qquad \text{since }\lambda_r\equiv \lambda_{r-1}+f_{r-1}+f_{r}\pmod{n} \\
&=&\Lambda_{0}+\sum_{r=2}^l\Lambda_{\lambda_{r-1}-s_{r-2}+i}-\sum_{r=1}^l
\Lambda_{\lambda_r-s_{r-1}+i}+\sum_{r=1}^l\Lambda_{i-s_{r-1}}
-\sum_{r=1}^l\Lambda_{i-s_{r}}\\
&&\qquad \text{since }\lambda_1-f_1+i\equiv 0 \pmod{n}\\
&=&\Lambda_{0}+\sum_{r=1}^{l-1}\Lambda_{\lambda_{r}-s_{r-1}+i}
-\sum_{r=1}^l\Lambda_{\lambda_r-s_{r-1}+i}
+\sum_{r=0}^{l-1}\Lambda_{i-s_{r}}-\sum_{r=1}^l\Lambda_{i-s_{r}}\\
&=&\Lambda_{0}+\Lambda_i-\Lambda_{\lambda_l-s_{l-1}+i}-\Lambda_{i-s_l}.
\end{eqnarray*}
By repeated use of condition (2) of Lemma \ref{l3} we can see that $\lambda_l-s_{l-1}-s_l+2i \equiv \lambda_1-s_0-s_1+2i=\lambda_1-f_1 +2i \equiv i \pmod{n}.$ Therefore, one of the numbers $\overline{\lambda_l-s_{l-1}+i},\overline{i-s_l}$ is in the interval $I=[ i/2 ,(n+i)/2 ]$ and the other is outside this interval. Let $t$ be the one in $I$ and $u$ be the one outside $I$.
There are two cases to consider: $\lceil i/2 \rceil \leq t \leq i$ and $i<t\leq \lfloor (n+i)/2 \rfloor$. Since $\Lambda_t=\Lambda_0+\omega_t$ (\cite{K}, eq. 12.4.3), using well-known formulas for the fundamental dominant weights $\omega_t$ of $\mathfrak{sl}(n)$ (see for example \cite{H}) we have:
\begin {eqnarray*}
\alpha&=&\Lambda_0+\Lambda_{i}-\Lambda_t-\Lambda_{u}+k\delta\\
&=&\Lambda_0+\Lambda_0+\frac{1}{n}\left (\sum_{r=1}^{i}r(n-i)\alpha_r+
\sum_{r=i+1}^{n-1}i(n-r)\alpha_r\right )\\
&&-\:\Lambda_0-\frac{1}{n}\left (\sum_{r=1}^{t}r(n-t)\alpha_r+
\sum_{r=t+1}^{n-1}t(n-r)\alpha_r\right )\\
&&-\:\Lambda_0-\frac{1}{n}\left (\sum_{r=1}^{u}r(n-u)\alpha_r+
\sum_{r=u+1}^{n-1}u(n-r)\alpha_r\right )+k\delta.
\end{eqnarray*}
In the case $t \leq i$ we have $u=i-t$. Therefore:
\begin{eqnarray*}
\alpha&=&-\sum_{r=1}^{u} r\alpha_r-\sum_{r=u+1}^t (i-t)\alpha_r-\sum_{r=t+1}^{i}(i-r)\alpha_r+k\delta\\
&=&\sum_{r=0}^u(k-r)\alpha_r+\sum_{r=u+1}^t (k-i+t)\alpha_r+\sum_{r=t+1}^i (k-i+r)\alpha_r+\sum_{r=i+1}^{n-1}k\alpha_r
\end{eqnarray*}
We must have $k\geq i-t$ in order for the coefficient of $\alpha_{u+1}$ to be $\geq 0$.
If $t > i$ then we have $u=n+i-t$. Therefore:
\begin{eqnarray*}
\alpha&=&-\sum_{r=i+1}^t (r-i)\alpha_r-\sum_{r=t+1}^u (t-i)\alpha_r-\sum_{r=u+1}^{n-1}(n-r)\alpha_r+k\delta\\
&=&\sum_{r=0}^ik\alpha_r+\sum_{r=i+1}^t (k-r+i)\alpha_r+\sum_{r=t+1}^u (k-t+i)\alpha_r+\sum_{r=u+1}^{n-1}(k-n+r)\alpha_r
\end{eqnarray*}
In this case $k\geq t-i$ for the coefficient of $\alpha_{t+1}$ to be $\geq 0$.
Therefore $\text{wt}(b\otimes b')=\Lambda_0+\Lambda_i-\alpha=\Lambda_t+\Lambda_{u}-k\delta$ where $k\geq t-i$. Hence $b\otimes b'$ is a maximal element of weight $\Lambda_t+\Lambda_{n+i-t}-k\delta$, and the component of $B(\Lambda_0)\otimes B(\Lambda_i)$ containing $b\otimes b'$ is isomorphic to $B(\Lambda_t+\Lambda_{u}-k\delta)$ for some $k$ satisfying the given conditions.
\end{proof}
\begin{theorem} \label{partition}
The $\widehat{\mathfrak{sl}}(n)$-module $V(\Lambda_0)\otimes V(\Lambda_i)$ decomposes as the direct sum\\
$\bigoplus_{t=\lceil i/2 \rceil}^{i}\bigoplus_{k=i-t}^{\infty} c^{\Lambda_t+\Lambda_{i-t}-k\delta}_{\Lambda_0,\Lambda_i}V(\Lambda_t+\Lambda_{i-t}-k\delta)\oplus \bigoplus_{t=i+1}^{\lfloor(n+i)/2\rfloor}\bigoplus_{k=t-i}^{\infty} c^{\Lambda_t+\Lambda_{n+i-t}-k\delta}_{\Lambda_0,\Lambda_i} \\ V(\Lambda_t+\Lambda_{n+i-t}-k\delta)$ and the outer multiplicities are given by $$c^{\Lambda_t+\Lambda_{u}-k\delta}_{\Lambda_0,\Lambda_i}=|\mathscr{C}_{n,\Lambda_t+\Lambda_{u}-k\delta}^i|,$$
where the absolute value sign denotes the cardinality.
Here, if $c^{\Lambda_t+\Lambda_{u}-k\delta}_{\Lambda_0,\Lambda_i}=0$ then $V(\Lambda_t+\Lambda_u- k\delta)$ does not occur in the decomposition. Furthermore, if $t<i$ and $k=i-t$ (resp. $t \geq i$ and $k=t-i$) then the unique maximal element is an $(i-t) \times (n-t)$ (resp. $(t-i) \times t$) rectangle.
\end{theorem}
\begin{proof}
By Lemma \ref{l3} $|\mathscr{C}_{n,\mu}^i|$ are the outer multiplicities in the decomposition of $V(\Lambda_0)\otimes V(\Lambda_i)$. Lemma \ref{l4} gives the weights that occur in the decomposition.\\
If two `0' s appeared in the same row or column, then the coefficient of $\alpha_r$ in $\alpha$ would be $>0$ for all $r\in I$, which is ruled out. If $t\leq i$ then we have $$\text{ht}(\alpha)=\frac{(2k-u)(u+1)}{2}+(k-i+t)(t-u)+\frac{(2k-i+t+1)(i-t)}{2}+k(n-i-1).$$
If $k=i-t$ this equals $(i-t)(n-t).$ If $t>i$ we let $k=t-i$ and compute $$\text{ht}(\alpha)=(t-i)(i+1)+\frac{(t-i-1)(t-i)}{2}+\frac{(t-i-1)(t-i)}{2}=t(t-i).$$
\end{proof}
\section{Generating Functions for Outer Multiplicities}
In this section we consider the generating functions for the outer multiplicities in Theorem \ref{partition} and give explicit formulas for these generating functions. First we define the formal series $f$ in the indeterminates $u,v$ as follows:
\begin{equation*}
f(u,v)=\sum_{k=-\infty}^\infty u^{k(k-1)/2}v^{k(k+1)/2}.
\end{equation*}
It is easy to verify that the function $f(u,v)$ satisfies the following properties:
\begin{eqnarray}
f(u,v) &=& f (v,u),\\
f(q^r,q^s)&=& q^r f(q^{2r+s},q^{-r}),\label{shiftpower}
\end{eqnarray}
\noindent for integers $r, s$ not both equal to $0$.
Recall the Euler $\varphi$ function $\varphi(q):=f(-q,-q^2)$.
In what follows we will be using the well known Jacobi triple product identity (cf. \cite{A}):
\begin{equation}\label{tripleprod)}
f(u,v)=\prod_{j=1}^\infty (1-u^k v^k)(1+u^{k-1} v^k)(1+u^k v^{k-1}).
\end{equation}
Recall that the (formal) character of the highest weight irreducible $\mathfrak{g}$-module $V(\lambda),\lambda\in \mathfrak{h}^*$ is defined by the formal power series $\text{ch}(V(\Lambda))=\sum_{\alpha\in Q^+}\dim(V(\lambda)_{\lambda-\alpha})\\e(\lambda-\alpha),$ where $e(\mu),\mu\in \mathfrak{h}^*$ is an element of the group ring of $\mathfrak{h}^*$ satisfying $e(\mu)e(\nu)=e(\mu+\nu), \mu,\nu\in \mathfrak{h}^*.$ The $q$-character (or principally specialized character) $\text{ch}_q(V(\lambda))$ is defined by making the substitution $e(-\alpha_i)\mapsto q,i\in I$ in $e(-\lambda)\text{ch}(V(\lambda)).$ We have the following $q$-character formulas for the $\widehat{\mathfrak{sl}}(n)$-modules $V(\Lambda_i), i \in \{0, 1, \dots , n-1\}$ and $V(\Lambda_0+\Lambda_j)$ for $j\in \{ 0,1, \dots, \lfloor n/2 \rfloor\}$ (cf. \cite{M}):
\begin{equation}
\text{ch}_q(V(\Lambda_i))=\prod_{\substack{j>0 \\ j\not \equiv 0 \pmod{n}}}(1-q^j)^{-1}= \frac{\varphi(q^{n})}{\varphi(q)}.
\label{qdim1}
\end{equation}
\begin{equation}
\text{ch}_q(V(\Lambda_0+\Lambda_j))=\frac{\varphi(q^{n})f(-q^{j+1},-q^{n-j+1})}{\varphi(q)^2}.
\label{qdim2}
\end{equation}
Now we define the generating function
\begin{equation}
B_t^i(q)=\sum_{k=r}^\infty b_{tk}^iq^{k-r},
\end{equation}
where
\begin{equation*}
b_{tk}^i=\begin{cases}
c^{\Lambda_t+\Lambda_{i-t}-k\delta}_{\Lambda_0,\Lambda_i}, \ \ t\in \{\lceil i/2 \rceil ,\dots, i\},\\
c^{\Lambda_t+\Lambda_{n+i-t}-k\delta}_{\Lambda_0,\Lambda_i},t\in \{i+1,\dots, \lfloor (n+i)/2 \rfloor\},
\end{cases}
\end{equation*}
and $r=|i-t|$.
By Theorem \ref{partition} we have:
\begin{multline*}
\text{ch}(V(\Lambda_0)) \text{ch}(V(\Lambda_i))= \sum_{t=\lceil i/2\rceil}^{i}\sum_{k=0}^\infty b_{tk}^i\:\text{ch}(V(\Lambda_t+\Lambda_{i-t}-k\delta))\\
+\:\sum_{t=i+1}^{\lfloor (n+i)/2\rfloor}\sum_{k=0}^\infty b_{tk}^i\:\text{ch}(V(\Lambda_t+\Lambda_{n+i-t}-k\delta))
\end{multline*}
Hence, multiplying both sides by $e(-\Lambda_0-\Lambda_i)$:
\begin{multline*}
e(-\Lambda_0-\Lambda_i)\text{ch}(V(\Lambda_0))\text{ch}(V(\Lambda_i))=\sum_{t=\lceil i/2 \rceil}^{i}\sum_{k=i-t}^{\infty}b_{tk}^i\:\text{ch}(V(\Lambda_t+\Lambda_{i-t}))e(-\Lambda_0-\Lambda_i-k\delta)\\
+\:\sum_{t=i+1}^{\lfloor (n+i)/2\rfloor}\sum_{k=t-i}^{\infty}b_{tk}^i\:\text{ch}(V(\Lambda_t+\Lambda_{n+i-t}))e(-\Lambda_0-\Lambda_i-k\delta).
\end{multline*}
Now, specializing $e(-\alpha_i)=q$, we obtain:
\begin{multline*}
\text{ch}_q(V(\Lambda_0))\text{ch}_q(V(\Lambda_i))=\sum_{t=\lceil i/2 \rceil}^{ i } q^{(i-t)(n-t)}\text{ch}_q(V(\Lambda_t+\Lambda_{i-t}))\sum_{k=i-t}^\infty b_{tk}^i q^{nk}\\
+\:\sum_{t=i+1}^{ \lfloor (n+i)/2\rfloor} q^{t(t-i)}\text{ch}_q(V(\Lambda_t+\Lambda_{n+i-t}))\sum_{k=t-i}^\infty b_{tk}^i q^{nk}
\end{multline*}
which gives
\begin{multline*}
\frac{\varphi(q^{n})^2}{\varphi(q)^2}=\sum_{t=\lceil i/2 \rceil}^{i}q^{(t-i)(t-n)}\frac{\varphi(q^{n})f(-q^{2t-i+1},-q^{n-2t+i+1})}{\varphi(q)^2}B_t^i(q^{n})\\
+\:\sum_{t=i+1}^{\lfloor (n+i)/2\rfloor}q^{t(t-i)}\frac{\varphi(q^{n})f(-q^{2t-i+1},-q^{n-2t+i+1})}{\varphi(q)^2}B_t^i(q^{n})
\end{multline*}
and hence:
\begin{multline}
\varphi(q^{n})=\sum_{t=\lceil i/2 \rceil}^{i}q^{(t-n)(t-i)}f(-q^{2t-i+1},-q^{n-2t+i+1})B_t^i(q^{n})\label{eq2}\\
+\:\sum_{t=i+1}^{\lfloor (n+i)/2\rfloor}q^{t(t-i)}f(-q^{n-2t+i+1},-q^{2t-i+1})B_t^i(q^{n}),
\end{multline}
Note that the series $\varphi(q^{n})=\prod_{j=1}^\infty(1-q^{nj})$ has a zero coefficient in front of $q^j$ whenever $j$ is not a multiple of $n$, and similar is the case for $B_i(q^{n})$. However, this is not the case for $q^{t(t-i)}f(-q^{2t-i+1},-q^{n-2t+i+1})$. So the trick is to rearrange the sum to sort the powers of $q$ carefully as we do below.\\
In the right-hand side of \eqref{eq2} we have:
\begin{eqnarray*}
f(-q^{n-2t+i+1},-q^{2t-i+1})&=&\sum_{j=0}^{n-1} \sum_{\substack{k\in \mathbb{Z}\\ k\equiv j \pmod{n}}}(-1)^k q^{\frac{1}{2}k((n+2)k+4t-2i-n)}\\
&=&\sum_{j=0}^{n-1}\sum_{m \in \mathbb{Z}}(-1)^{nm+j}q^{\frac{1}{2}(nm+j)((n+2)(nm+j)+4t-2i-n))}\nonumber.
\end{eqnarray*}
We separate out terms involving the index $m$ in the exponent of $q$:
\begin{eqnarray*}
\lefteqn{ (nm+j)\left (\frac{(n+2)(nm+j)+4t-2i-n}{2}\right )}\\
&=&(nm+j)\left (\frac{(n+2)nm+(n+2)j+4t-2i-n}{2}\right )\\
&=&nm\left (\frac{(n+2)nm+(n+2)j+4t-2i-n}{2}\right )+j\frac{(n+2)nm}{2}\\
&&+\:j\left (\frac{(n+2)j-n+4t-2i}{2}\right )\\
&=&nm\left (\frac{(n+2)nm+2(n+2)j+4t-2i-n}{2}\right )+j\frac{(n+2)j-n+4t-2i}{2},
\end{eqnarray*}
which gives:
\begin{multline}
f(-q^{n-2t+i+1},-q^{2t-i+1})=\\
\sum_{j=0}^{n-1}(-1)^jq^{\frac{1}{2}j((n+2)j+4t-2i-n)}\left (\sum_{m \in \mathbb{Z}}(-1)^{nm}q^{nm(\frac{1}{2}((n+2)nm+2(n+2)j+4t-2i-n)}\right ).\label{sepeq}
\end{multline}
Taking $q^{1/n}$ in the inner sum in \ref{sepeq} gives a series $\Psi_{tj}^i(q)=f((-1)^nq^r,(-1)^nq^s)$ for $r,s$ satisfying:
\begin{eqnarray*}
s+r&=&n(n+2)\\
s-r&=&2(n+2)j+4t-2i-n.
\end{eqnarray*}
Explicitly:
\begin{equation*}
\Psi_{tj}^i(q)=f\left ((-1)^nq^{\frac{1}{2}n(n+3)-2t+i-(n+2)j },(-1)^nq^{\frac{1}{2}n(n+1)+2t-i+(n+2)j)}\right ).
\end{equation*}
We have:
\begin{equation*}
q^{(t-n)(t-i)}f(-q^{2t+i+1},-q^{n-2t-i+1})=\sum_{j=0}^{n-1} (-1)^j q^{\frac{1}{2}nj(j-1)-n(t-i)+(t+j-i)(t+j)}\Psi_{tj}^i(q^n)
\end{equation*}
and
\begin{equation*}
q^{t(t-i)}f(-q^{2t+i+1},-q^{n-2t-i+1})=\sum_{j=0}^{n-1} (-1)^j q^{ \frac{1}{2}nj(j-1)+(t+j-i)(t+j)}\Psi_{tj}^i(q^n).
\end{equation*}
Thus \eqref{eq2} becomes:
\begin{multline}
\varphi(q^{n})=\sum_{t=\lceil i/2 \rceil}^{i}B_t^i(q^n)\sum_{j=0}^{n-1} (-1)^j q^{\frac{1}{2}nj(j-1)-n(t-i)+(t+j-i)(t+j)}\Psi_{tj}^i(q^n)\label{leq}\\
+\:\sum_{t=i+1}^{\lfloor (n+i)/2\rfloor}B_t^i(q^n)\sum_{j=0}^{n-1} (-1)^j q^{ \frac{1}{2}nj(j-1)+(t+j-i)(t+j)}\Psi_{tj}^i(q^n)\\
\end{multline}
\emph{Example:} If $n=2$ and $i=0$ then $\lceil i/2 \rceil =0,\lfloor (n+i)/2 \rfloor=1,$ and (\ref{leq}) gives:
\begin{equation*}
\varphi(q^2)=B_0^0(q^2)(\Psi_{00}^0(q^2)-q\Psi_{01}^0(q^2))+B_1^0(q^2)(q\Psi_{10}^0(q^2)-q^4\Psi_{11}^0(q^2))
\end{equation*}
where
$$
\Psi_{00}^0(q)=f(q^5,q^3),\qquad\Psi_{01}^0(q)=f(q,q^7),
$$
$$
\Psi_{10}^0(q)=f(q^3,q^5),\qquad\Psi_{11}^0(q)=f(q^{-1},q^9).
$$
One can verify that we have
\begin{eqnarray*}
\Psi_{00}^0(q^2)-q\Psi_{01}^0(q^2)&=&1-q-q^3+q^6+q^{10}-\cdots\\
&=&f(-q,-q^3),
\end{eqnarray*}
and
\begin{eqnarray*}
q\Psi_{10}^0(q^2)-q^4\Psi_{11}^0(q^2)&=&q-q^2-q^4+q^7+q^{11}-\cdots\\
&=&qf(-q,-q^3),
\end{eqnarray*}
as desired.
On the other hand, if $i=1$ then $\lceil i/2 \rceil =\lfloor (n+i)/2 \rfloor=1,$ and (\ref{leq}) gives:
\begin{equation*}
\varphi(q^2)=B_1^1(q^2)(\Psi_{10}^1(q^2)-q^2\Psi_{11}^1(q^2))
\end{equation*}
where
$$
\Psi_{10}^1(q)=f(q^4,q^4),\:\Psi_{11}^1(q)=f(1,q^{8}).
$$
However, this is equivalent to:
\begin{equation*}
\varphi(q^2)=B_1^1(q^2)f(-q^2,-q^2)
\end{equation*}
from which one easily sees:
\begin{equation}\label{n2i1}
B_1^1(q)=\frac{\varphi(q)}{f(-q,-q)}=1+q+q^2+2q^3+2q^4+3q^5+4q^6+5q^7+\cdots.
\end{equation}
\emph{Example:} If $n=3,i=0$ then we have:
\begin{eqnarray*}
\varphi(q^3)&=&B_0^0(q^3)(\Psi_{00}^0(q^3)-q\Psi_{01}^0(q^3)+q^{7}\Psi_{02}^0(q^3))\\
&&+\:B_1^0(q^3)(q\Psi_{10}^0(q^3)-q^4\Psi_{11}^0(q^3)+q^{12}\Psi_{12}^0(q^3))
\end{eqnarray*}
where
$$
\Psi_{00}^0(q)=f(-q^9,-q^6),\Psi_{01}^0(q)=f(-q^4,-q^{11}),\Psi_{02}^0(q)=f(-q^{-1},-q^{16}),
$$
$$
\Psi_{10}^0(q)=f(-q^7,-q^8),\Psi_{11}^0(q)=f(-q^2,-q^{13}),\Psi_{12}^0(q)=f(-q^{-3},-q^{18}).
$$
If $n=3,i=1$ then we have:
\begin{eqnarray*}
\varphi(q^3)&=&B_1^1(q^3)(\Psi_{10}^1(q^3)-q^2\Psi_{11}^1(q^3)+q^{9}\Psi_{12}^1(q^3))\\
&&+\:B_2^1(q^3)(q^2\Psi_{20}^1(q^3)-q^6\Psi_{21}^1(q^3)+q^{15}\Psi_{22}^1(q^3))
\end{eqnarray*}
where
\begin{equation}\label{n3Psi1}
\Psi_{10}^1(q)=f(-q^8,-q^7),\Psi_{11}^1(q)=f(-q^3,-q^{12}),\Psi_{12}^1(q)=f(-q^{-2},-q^{17}),
\end{equation}
\begin{equation}\label{n3Psi2}
\Psi_{20}^1(q)=f(-q^6,-q^9),\Psi_{21}^1(q)=f(-q,-q^{14}),\Psi_{22}^1(q)=f(-q^{-4},-q^{19}).
\end{equation}
The expression $(t+j-i)(t+j)$ appearing in (\ref{leq}) is the only contribution to the exponent of $q$ that possibly has non-zero residue modulo $n$, so we separate the right hand side of (\ref{leq}) into parts having $\overline{(t+j-i)(t+j)}$ equal. We cyclically permute the variable $j$ by $t$ units to $\overline{j-t}$, which transforms $\overline{(t+j-i)(t+j)}$ to $\overline{(j-i)j}$. In the new expression, if $j$ is chosen in the interval $(i/2, (n+i)/2)$ then $(t+\overline{j-t}-i)(t+\overline{j-t})$ gives the same residue modulo $n$ as $(t+\overline{i-j-t}-i)(t+\overline{i-j-t})$ for $j$ in the same interval. Therefore, we have:
\begin{equation}
\varphi(q^{n})=\sum_{t,j=\lceil i/2 \rceil}^{\lfloor(n+i)/2 \rfloor} B_t(q^n)q^{\overline{(j-i)j}}a_{tj}^i(q^n),\label{lineq}
\end{equation}
where
\begin{equation*}
a_{tj}^i(q)=\begin{cases}
(-1)^{\overline{j-t}}q^{\mu(t,i,\overline{j-t})-(t-i)}\Psi_{t,\overline{j-t}}^i(q) \text{ if }t\in [\frac{i}{2},i] \text{ and }j=\frac{i}{2}
\text{ or }\frac{n+i}{2},\\ \\
(-1)^{\overline{j-t}}q^{\mu(t,i,\overline{j-t})-(t-i)}\Psi_{t,\overline{j-t}}^i(q) + (-1)^{\overline{i-j-t}}q^{\mu(t,i,\overline{i-j-t})-(t-i)}\Psi_{t,\overline{i-j-t}}^i(q)\\
\qquad \text{if }t\in [\frac{i}{2},i] \text{ and }j\neq \frac{i}{2},\frac{n+i}{2},\\ \\
(-1)^{\overline{j-t}}q^{\mu(t,i,\overline{j-t})}\Psi_{t,\overline{j-t}}^i(q) \text{ if }t\in [i+1,\frac{n+i}{2}] \text{ and }j=\frac{i}{2} \text{ or }\frac{n+i}{2},\\ \\
(-1)^{\overline{j-t}}q^{\mu(t,i,\overline{j-t})}\Psi_{t,\overline{j-t}}^i(q)+(-1)^{\overline{i-j-t}}q^{\mu(t,i,\overline{i-j-t})}\Psi_{t,\overline{i-j-t}}^i(q) \text{ otherwise,}
\end{cases}
\end{equation*}
and
\begin{equation*}
\mu(t,i,k)=\frac{k(k-1)}{2}+\left \lfloor\frac{(t+k-i)(t+k)}{n}\right \rfloor.
\end{equation*}
Unfortunately, the numbers $\overline{(j-i)j}$--appearing as residues of exponents of $q$ in (\ref{lineq}) may not all be distinct for all integers $j \in [i/2 , (n+i)/2]$, depending on $n$ and $i$. In fact, they are all distinct if and only if $(j-i)j \equiv (j'-i)j' \pmod{n}$ implies $j' \equiv i- j,j \pmod{n}$ for all $j,j'$, i.e. if and only if $(j'-j)(j'+j-i)\equiv 0 \pmod{n}$ has only trivial solutions modulo $n$. The proof of the following is elementary, but we include it for the convenience of the reader.
\begin{proposition}\label{propmod}
The congruence $(j'-j)(j'+j-i)\equiv 0 \pmod{n}$ has only the trivial solutions $j'\equiv j \pmod{n}$ and $j'\equiv i-j \pmod{n}$ if and only if $n$ and $i$ satisfy one of the following conditions:
\begin{enumerate}
\item $i$ is even and $n$ is prime or twice an odd prime,
\item $i$ is odd and $n$ is prime or a power of 2.
\end{enumerate}
\end{proposition}
\begin{proof}
Suppose that $n$ is a composite integer with factorization $n=rs$. If $i$ is even then $j'=r+s+i/2,j=s-r+i/2$ gives a solution that is trivial if and only if $n=2s$ or $n=2r$. Since the factorization of $n$ was chosen arbitrarily, we deduce that $n=2p$ for some prime $p$ (and by examining the cases, we can rule out $n=4$). If $i$ is odd, and $n$ is not a power of $2$, then we can choose a factorization such that $r$ is odd. In such case $j'=s+(r+i)/2, j=-s+(r+i)/2$ gives a non-trivial solution.
Conversely, if $n$ is prime then $n|(j'-j)(j'+j-i)$ implies $n|(j'-j)$ or $n|(j'+j-i)$, i.e. any solution is trivial. If $n=2p$ for an odd prime $p$, and $i$ is even then $2p|(j'-j)(j'+j-i)$ implies that $2$ is a factor of both $j'-j$ and $j'+j-i$, since both have the same parity, and at least one factor is divisible by $p$. These must be the same factor, since $p$ is odd. If $i$ is odd and $n=2^t$, then $2^t|(j'-j)(j'+j-i)$ implies that $2^t|(j'-j)$ or $2^t | (j'+j-i)$ since $j'-j$ and $j'+j-i$ have different parity, which finishes the proof.
\end{proof}
Now assume that $i,n$ satisfy one of the two conditions in Proposition \ref{propmod}. In this case (\ref{lineq}) is equivalent to the following set of linear equations:
\begin{equation*}
q^{\overline{(j-i)j}}\sum_{t=\lceil i/2 \rceil}^{\lfloor (n+i)/2 \rfloor }B_t^i(q^n)a_{tj}^i(q^n)=\delta_{ij}\varphi(q^n), j=\left \lceil\frac{ i}{2} \right \rceil, \dots, \left \lfloor \frac{n+i}{2} \right \rfloor,
\end{equation*}
or,
\begin{equation*}
\sum_{t=\lceil i/2 \rceil}^{\lfloor (n+i)/2 \rfloor }B_t^i(q)a_{tj}^i(q)=\delta_{ij}\varphi(q), j=\left \lceil\frac{i}{2} \right \rceil, \dots, \left \lfloor \frac{n+i}{2} \right \rfloor.
\end{equation*}
which can be written in matrix form as
\begin{equation*}
A^i(q)\mathbf{B}^i(q)=\mathbf{\Phi}^i(q)
\end{equation*}
where
\begin{multline*}
A^i(q)^T=(a_{tj}^i(q))_{t,j\in I}, \mathbf{B}^i(q)=(B_j^i(q))_{j\in I}^T, \mathbf{\Phi}^i(q)=(\delta_{ij}\varphi(q))_{j\in I}^T,\\
I=\left \{ \left \lceil\frac{i}{2} \right \rceil, \dots, \left \lfloor \frac{n+i}{2} \right \rfloor \right \}.
\end{multline*}
Therefore, Cramer's rule yields the following proposition.
\begin{proposition} \label{deteqn} For $0 \le i \le n-1$, $\left \lceil\frac{i}{2} \right \rceil \le t \le \left \lfloor \frac{n+i}{2} \right \rfloor$, we have
\begin{equation*}
B_t^i(q)=\frac{(-1)^{i+t}\varphi(q)\det(\widetilde{A^i(q)}_{it})}{\det (A^i(q))},
\end{equation*}
where $\widetilde{A^i(q)}_{it}$ denotes the matrix $A^i(q)$ with the $i$th row and $t$th column deleted.
\end{proposition}
\section{Examples and Identities}
Comparing the result in Theorem \ref{partition} and Proposition \ref{deteqn} we now have the following theorem which gives generating function identities.
\begin{theorem}\label{th:main}
Let $i$ be as in Proposition \ref{propmod} for $n\geq 2.$ Then, for $t\in \{\lceil i/2 \rceil,\dots,\lfloor (n+i)/2 \rfloor \}$:
\begin{equation}\label{eq:main}
\sum_{k=|i-t|}^\infty|\mathscr{C}^n_{\Lambda_t+\Lambda_u-k\delta}|q^{k-|i-t|}=
\frac{(-1)^{i+t}\varphi(q)\det(\widetilde{A^i(q)}_{it})}{\det (A^i(q))},
\end{equation}
where $u = i-t$ for $t \le i$ and $u = n+i-t$ for $t > i$.
\end{theorem}
\begin{proof}
The left side and the right side of (\ref{eq:main}) both count the outer multiplicity of $V(\Lambda_t+\Lambda_u+k\delta)$ in the decomposition of $V(\Lambda_0)\otimes V(\Lambda_i)$ by Theorem \ref{partition} and Proposition \ref{deteqn} respectively.
\end{proof}
In \cite{MW}, we considered the case $i = 0$ for $n = 2, 3$ and showed that we obtain certain identities in the Slater list \cite{S} and some new identities for $i=0$ and $n=3$.
In this paper we consider the case $i=1$ for $n = 2, 3, 4$ and obtain some seemingly new identities.
In the case $n=2,i=1$, as we have seen in (\ref{n2i1}):
\begin{equation*}
B_1^1(q)=\frac{\varphi(q)}{f(-q,-q)}=\frac{\prod_{j=1}^\infty(1-q^j)}{\prod_{j=1}^\infty(1-q^{2j})(1-q^{2j-1})^2}=\frac{1}{\prod_{j=1}^\infty(1-q^{2j-1})}
\end{equation*}
However, the set $\mathscr{C}_{2, \Lambda_1 + \Lambda_0 -k\delta}^1$ is the set of partitions of $2k$ into distinct even parts which is the same as the number of partitions of $k$ into distinct parts. Thus Theorem \ref{th:main} in this case becomes the Euler's identity (see \cite{A}). This agrees with the corresponding result given in \cite{F}.
We now consider the case $i=1, n=3$. In this case, $\lceil i/2 \rceil = 1, \lfloor(n+i)/2\rfloor=2$. Using (\ref{shiftpower}), (\ref{n3Psi1}), and (\ref{n3Psi2}),
the matrix $A^1(q)$ in Proposition \ref{deteqn} is:
\begin{eqnarray*}
A^1(q) &=& \begin{pmatrix}
\Psi_{10}^1(q)+q^3\Psi_{12}^1(q)&q^5\Psi_{22}^1(q)-q^2\Psi_{21}^1(q)\\
-\Psi_{11}^1(q)&\Psi_{20}^1(q)
\end{pmatrix}\\
&=&
\begin{pmatrix}
f(-q^8,-q^7)+q^3f(-q^{-2},-q^{17})&q^5f(-q^{-4},-q^{19})-q^2f(-q,-q^{14})\\
-f(-q^3,-q^{12})&f(-q^6,-q^9)
\end{pmatrix}\\
&=&
\begin{pmatrix}
f(-q^8,-q^7)+qf(-q^{2},-q^{13})&qf(-q^{4},-q^{11})-q^2f(-q,-q^{14})\\
-f(-q^3,-q^{12})&f(-q^6,-q^9)
\end{pmatrix}\\
\end{eqnarray*}
Now, using Frank Garvan's Maple $q$-series package (\cite{G}) we see that $\det (A^1(q)) = \varphi(q)^2$. Therefore, Proposition \ref{deteqn} gives the following $q$-series for the outer multiplicities (using (\ref{tripleprod)})):
\begin{eqnarray*}
B_1^1(q)&=&\frac{f(-q^6,-q^9)}{\varphi(q)}\\
&=&\frac{\prod_{j=1}^{\infty}(1-q^{15j})(1-q^{15j-6})(1-q^{15j-9})}{\prod_{j=1}^{\infty}(1-q^j)}\\
B_2^1(q)&=&\frac{f(-q^3,-q^{12})}{\varphi(q)}\\
&=&\frac{\prod_{j=1}^{\infty}(1-q^{15j})(1-q^{15j-3})(1-q^{15j-12})}{\prod_{j=1}^{\infty}(1-q^j)}
\end{eqnarray*}
Now Theorem \ref{th:main} gives the following identities:
\begin{equation}
\frac{\prod_{j=1}^{\infty}(1-q^{15j})(1-q^{15j-6})(1-q^{15j-9})}{\prod_{j=1}^{\infty}(1-q^j)} = \sum_{k=0}^\infty a(k) q^k,
\end{equation}
\begin{equation}
\frac{\prod_{j=1}^{\infty}(1-q^{15j})(1-q^{15j-3})(1-q^{15j-12})}{\prod_{j=1}^{\infty}(1-q^j)} = \sum_{k=0}^\infty b(k) q^k,
\end{equation}
where $a(k)$ (respectively $b(k)$) is the number of partitions $(\lambda_1^{f_1},\lambda_2^{f_2},\dots, \lambda_l^{f_l})$ of $3k$ (respectively $3k+2$) with $f_1 - \lambda_1 +1 \equiv 0\pmod{3}$,
$f_j+f_{j+1}+\lambda_{j}-\lambda_{j+1}\equiv0 \pmod{3}$, $f_j <3$ for $1\leq j<l$.
Now we consider the case $i=1$ and $n=4$ which satisfies the conditions in
Proposition \ref{propmod}. In this case we have $\lceil i/2 \rceil = 1, \lfloor(n+i)/2\rfloor=2$ and
\begin{equation*}
\Psi_{10}^1(q)=f(q^{13},q^{11}),\Psi_{11}^1(q)=f(q^{7},q^{17}),\Psi_{12}^1(q)=f(q,q^{23}),\Psi_{13}^1(q)=f(q^{-5},q^{29})
\end{equation*}
\begin{equation*}
\Psi_{20}^1(q)=f(q^{11},q^{13}),\Psi_{21}^1(q)=f(q^{5},q^{19}),\Psi_{22}^1(q)=f(q^{-1},q^{25}),\Psi_{23}^1(q)=f(q^{-7},q^{31}).
\end{equation*}
Hence we have:
\begin{eqnarray*}
A^1(q)&=&
\begin{pmatrix}
\Psi_{10}^1(q)-q^6\Psi_{13}^1(q)&-q^8\Psi_{23}^1(q)+q^4\Psi_{22}^1(q)\\
-\Psi_{11}^1(q)+q^{2}\Psi_{12}^1(q)&\Psi_{20}^1(q)-q\Psi_{21}^1(q)
\end{pmatrix}
\\
&=&
\begin{pmatrix}
f(q^{13},q^{11})-q^6f(q^{-5},q^{29})&-q^8f(q^{-7},q^{31})+q^4f(q^{-1},q^{25})\\
-f(q^7,q^{17})+q^2f(q,q^{23})&f(q^{11},q^{13})-qf(q^5,q^{19})
\end{pmatrix}\\
&=&
\begin{pmatrix}
f(q^{13},q^{11})-qf(q^{5},q^{19})&-qf(q^{7},q^{17})+q^3f(q,q^{23})\\
-f(q^7,q^{17})+q^2f(q,q^{23})&f(q^{11},q^{13})-qf(q^5,q^{19})
\end{pmatrix}
\end{eqnarray*}
Using Frank Garvan's $q$-series package we see that $\det(A^1(q))=\varphi(q)f(-q,-q).$ Therefore:
\begin{eqnarray*}
B_1^1(q)&=&\frac{f(q^{11},q^{13})-qf(q^5,q^{19})}{f(-q,-q)}\\
B_2^1(q)&=&\frac{f(q^7,q^{17})-q^2f(q,q^{23})}{f(-q,-q)}
\end{eqnarray*}
Hence by Theorem \ref{th:main} we have the following identities:
\begin{equation}
\frac{f(q^{11},q^{13})-qf(q^5,q^{19})}{f(-q,-q)} = \sum_{k=0}^\infty c(k) q^k,
\end{equation}
and
\begin{equation}
\frac{f(q^7,q^{17})-q^2f(q,q^{23})}{f(-q,-q)} = \sum_{k=0}^\infty d(k) q^k,
\end{equation}
where $c(k)$ (respectively $d(k)$) is the number of partitions $(\lambda_1^{f_1},\lambda_2^{f_2},\dots, \lambda_l^{f_l})$ of $4k$ (respectively $4k+2$) with $f_1 - \lambda_1 +1 \equiv 0\pmod{4}$, $f_j+f_{j+1}+\lambda_{j}-\lambda_{j+1}\equiv0 \pmod{4}$, $f_j <4$, for $1\leq j<l$.
\end{document}
|
\begin{document}
\author{Yu-Chu Lin,\ Chi-Cheung Poon,\ Dong-Ho Tsai\thanks{Research supported by NCTS
and NSC\ of Taiwan under grant number\ 96-2115-M-007-010-MY3.}}
\title{CONTRACTING CONVEX IMMERSED CLOSED\ PLANE CURVES\ WITH\ SLOW\ SPEED OF
CURVATURE\thanks{AMS\ Subject Classifications:\ 35K15, 35K55.}}
\maketitle
\begin{abstract}
We study the contraction of a convex immersed plane curve with speed $\frac
{1}{\alpha}k^{\alpha},\ $where $\alpha\in(0,1]\ $is a constant and show that,
if the blow-up rate of the curvature is of type one, it will converge to a
\textbf{homothetic self-similar solution}.\ We also discuss a special
symmetric\ case of type two blow-up and show that it converges to a
\textbf{translational self-similar solution}. In the case of curve shortening
flow (i.e., when $\alpha=1$), this translational self-similar solution is the
familiar "\textbf{Grim Reaper}"\ (a terminology due to M. Grayson\ \cite{GR}).
\end{abstract}
\section{Introduction.}
Let $\gamma_{0}\ $be\ a convex immersed smooth\ closed plane curve with
\emph{rotation index} (number of times its tangent vector winds around as one
goes along the curve)$\ m\in\mathbb{N},$ parametrized by a smooth immersion
$X_{0}\left( \varphi\right) :S^{1}\rightarrow\mathbb{R}^{2}.\ $
Here\ "convex" means that $\gamma_{0}$ has no inflection points (i.e.,
curvature is positive everywhere). In general, such a\ curve $\gamma_{0}\ $can
have self-intersections (if $m\geq2$). \
A family of convex immersed closed curves $X\left( \varphi,t\right)
:S^{1}\times\lbrack0,T)\rightarrow\mathbb{R}^{2}\ $(with rotation index $m$
and initial data $\gamma_{0}$)\ is said to evolve (contract)\ under the
$k^{\alpha}\ $flow, where $\alpha>0\ $is a constant, if we have
\[
\left( \bigstar\right) \ \cdot\cdot\cdot\ \left\{
\begin{array}
[c]{l}
\dfrac{\partial X}{\partial t}\left( \varphi,t\right) =\dfrac{1}{\alpha
}k^{\alpha}\left( \varphi,t\right) \mathbf{N}\left( \varphi,t\right)
,\ \ \ \forall\ \ \left( \varphi,t\right) \in S^{1}\times\lbrack0,T)
\\
X\left( \varphi,0\right) =X_{0}\left( \varphi\right) \in C^{\infty}\left(
S^{1}\right) ,\ \ \ \varphi\in S^{1},
\end{array}
\right.
\]
where $k\left( \varphi,t\right) $ is the curvature of the curve $\gamma
_{t}:=X\left( \cdot,t\right) \ $at $\varphi,$ and $\mathbf{N}\left(
\varphi,t\right) \ $is the unit normal vector of the curve\ $\gamma_{t}
.\ $Throughout this paper the constant $\alpha$ is assumed to be $0<\alpha
\leq1\ $(in such a case, we shall call $\left( \bigstar\right) $
a$\ $\textbf{slow speed\ }contraction).$\ $Here we use the convention that for
convex plane curves\ the curvature$\ k>0$ is positive\ everywhere\ and as for
the direction of the normal $\mathbf{N},\mathbf{\ }$we choose $\mathbf{N}
=\left( 0,1\right) $ at a point with minimum $y$-coordinate and extend it
continuously to the whole curve.\
When $\alpha=1\ $(i.e., the well-known \emph{curve-shortening flow}),$\ $our
setting is exactly the same as in the interesting paper by Angenent
\cite{ANG}, where the flow $\left( \bigstar\right) $ contracts $\gamma_{0}$
with singularity forming in finite time.\
Our aim is to study the asymptotic behavior of the contracting flow $\left(
\bigstar\right) $ with $m\geq2\ $(the immersed case), trying to generalize
results in \cite{ANG} to the case $\alpha\in(0,1].\ $The behavior of the
contracting flow$\ \left( \bigstar\right) $\ with \textbf{fast speed}, i.e.,
when$\ 1<\alpha<\infty,$ has been discussed in \cite{PT}. Note that $\alpha
\in(1,\infty)$ corresponds to $p\in\left( 1,2\right) $ in the equation
$\left( \clubsuit\right) $ below.
When $m=1\ $(the embedded case), the initial curve $X_{0}\ $is embedded and
the convergence behavior of the flow $\left( \bigstar\right) $ for general
$\alpha>0$ is well understood due to a series of nice papers by Ben
Andrews\ \cite{AN1},\ \cite{AN3}\ and \cite{AN4}. For the information of the
readers,\ we give a brief summary provided by Andrews\footnote{We thank Ben
Andrews for giving us this summary.}:
\begin{theorem}
(\emph{Ben Andrews}\textsf{\ }\cite{AN1},\ \cite{AN3},\ \cite{AN4}
)\label{thm-ben}\ For $m=1$\ and any $\alpha>0,$ the curve $\gamma_{t}$
contracts to a point in finite time. If $0<\alpha<1/3,$ then for generic
initial data there is no limit of the curves rescaled about the final point
(the isoperimetric ratio approaches infinity); and the exceptional ones where
the isoperimetric ratio remains bounded converge to homothetic solutions,
which have been classified. For $\alpha>1/3,$ the rescaled solutions converge
to circles; and for $\alpha=1/3,$ they converge to ellipses.
\end{theorem}
\begin{remark}
As a consequence of Theorem \ref{thm-ben}, we have the following interesting
elliptic\ result. For $0<\lambda<3\ $(here $\lambda=1/\alpha$)$,$ the only
positive $2\pi$-periodic solution to the equation
\begin{equation}
w^{\lambda}\left( x\right) \left[ w_{xx}\left( x\right) +w\left(
x\right) \right] =1,\ \ \ x\in S^{1} \label{w-lamda}
\end{equation}
is $w\left( x\right) \equiv1.$ But for $\lambda\geq3,$ we begin to have
nonconstant $2\pi$-periodic\ solutions. For example, when $\lambda=3,$ there
is a family of positive $2\pi$-periodic solutions to equation (\ref{w-lamda})
of the form ($b\in\mathbb{R}\ $is a parameter)
\begin{equation}
w(x)=\left( \frac{1}{1+b^{2}}\right) ^{1/4}\sqrt{1+b^{2}\cos^{2}
x},\;\;\ x\in\lbrack0,2\pi] \label{wb}
\end{equation}
where we obtain (\ref{wb}) by computing the curvature (or support
function)\ of an ellipse.\ The function $w\left( x\right) \ $of (\ref{wb})
has maximum value occurred at $x=0,\ $with value$\ \left( 1+b^{2}\right)
^{1/4}\geq1$.
\end{remark}
Let $x\in S_{m}^{1}:=\mathbb{R}/2m\pi\mathbb{Z}\ $be the tangent angle of
$\gamma_{t}$ (a\ function defined on $S_{m}^{1}\ $means that it is periodic
with period $2m\pi$).\ In terms of the variable $\left( x,t\right) $, it is
known that the curvature quantity $v\left( x,t\right) =k^{\alpha}\left(
x,t\right) \ $of $\gamma_{t}\ $in $\left( \bigstar\right) $ will satisfy
the quasilinear parabolic PDE$\ $(a function defined on $S_{m}^{1}\ $means
that it is periodic with period $2m\pi$)$\ $
\[
\left( \clubsuit\right) \ \cdot\cdot\cdot\ \left\{
\begin{array}
[c]{l}
\dfrac{\partial v}{\partial t}=v^{p}\left( v_{xx}+v\right) ,\ \ \ p=1+\dfrac
{1}{\alpha}\in\lbrack2,\infty),\ \ \ 0<\alpha\leq1
\\
v\left( x,0\right) =v_{0}\left( x\right) >0\ \ \ \text{for all\ \ \ }x\in
S_{m}^{1}
\\
v\left( x,t\right) =v\left( x+2m\pi,t\right) \ \ \ \text{for
all\ \ \ }x\in\mathbb{R},\ \ \ t>0,
\end{array}
\right.
\]
where$\ k_{0}\left( x\right) $ is the curvature of $\gamma_{0}$
and$\ v_{0}(x)=k_{0}^{\alpha}\left( x\right) ,\ x\in S_{m}^{1}.\ $Moreover,
it is also known that $\left( \clubsuit\right) \ $is equivalent to $\left(
\bigstar\right) \ $(under the assumption that $v_{0}\left( x\right) >0$
satisfies the integral condition (\ref{integral-cond}) below).$\ $As
$\gamma_{0}\ $is a closed curve, in $\left( \bigstar\right) $\ the initial
data $v_{0}\left( x\right) =k_{0}^{\alpha}\left( x\right) >0$ in $\left(
\clubsuit\right) $ must satisfy$\ $the integral condition
\begin{equation}
\int_{S_{m}^{1}}v_{0}^{1-p}\left( x\right) e^{ix}dx=0,\ \ \ e^{ix}=\cos
x+i\sin x \label{integral-cond}
\end{equation}
where $\int_{S_{m}^{1}}\ $means $\int_{-m\pi}^{m\pi}.\ $Also note that
(\ref{integral-cond})\ is preserved under $\left( \clubsuit\right) $, i.e.,
if initially $v_{0}\left( x\right) \ $satisfies\ (\ref{integral-cond}), so
does $v\left( x,t\right) .$
>From now on we shall focus on $\left( \clubsuit\right) \ $with $p\in
\lbrack2,\infty)\ $and $m\geq2\ $with the smooth\ initial data $v_{0}\left(
x\right) >0\ $in$\ \left( \clubsuit\right) \ $satisfying
(\ref{integral-cond}).\ In Lemma \ref{lem3} we shall discuss a result when the
initial function $v_{0}(x)\ $does not satisfy the integral condition
(\ref{integral-cond}). The overall understanding is that when
(\ref{integral-cond}) is satisfied, then we are talking about the geometric
flow $\left( \bigstar\right) .$ If not, then one can simply view $\left(
\clubsuit\right) $\ as a pure analytical problem.\
Since equation $\left( \clubsuit\right) $\ is parabolic, regularity theory
implies the existence of a unique smooth periodic solution $v\left(
x,t\right) $ on $S_{m}^{1}\times\lbrack0,T)$ for some $T>0.\ $Each $v\left(
\cdot,t\right) ,\ t\in\lbrack0,T),$ remains smooth, positive, and periodic
over $\mathbb{R\ }$with period $2m\pi.$ By the equivalence, the flow $\left(
\bigstar\right) $ also has short time existence of a smooth\ solution.\ Each
$\gamma_{t}$ remains convex, closed, and immersed with rotation index $m$ for
all $t\in\lbrack0,T).\ $
The classical \emph{curve-shortening flow} is when $\alpha=1\ $(or $p=2$); see
Gage-Hamilton \cite{GH} for the embedded case (i.e.,$\ m=1$)$\ $and Angenent
\cite{ANG}, Angenent-Vel\'{a}zquez\ \cite{AV}\ for the immersed case
(i.e.,$\ m\geq2$).\ When $m=1,$ the value $\alpha=1/3$ in Theorem
\ref{thm-ben} corresponds to $p=4$ in $\left( \clubsuit\right) $. For more
information on the evolution\ (expansion or contraction) of convex closed
curves in $\mathbb{R}^{2}$, see Andrews \cite{AN1},\ Chou-Zhu\ \cite{CZ}, and
the references therein.\
\begin{remark}
If in $\left( \clubsuit\right) \ $the constant $\alpha\ $is negative\ (let
$\alpha=-\beta,\ \beta>0$), then the corresponding flow in $\left(
\bigstar\right) $\ is to expand$\ \gamma_{0}\ $along its outward normal
vector direction with speed $1/\left( \beta k^{\beta}\right) $. The
evolution of $v=1/k^{\beta}\ $is\ given by
\begin{equation}
\dfrac{\partial v}{\partial t}=v^{p}(v_{xx}+v),\ \ \ \ \ v\left( x,t\right)
=v\left( x+2m\pi,t\right) ,\ \ \ \ \ p=1+\frac{1}{\alpha}\in\left(
-\infty,1\right) . \label{v-minus}
\end{equation}
Finally if one expands $\gamma_{0}$ along its outward normal vector direction
with the exponential speed $\exp\left( 1/k\right) $, the evolution of
$v=e^{1/k}\ $is
\begin{equation}
\frac{\partial v}{\partial t}=v\left( v_{xx}+v\right) ,\ \ \ \ \ v\left(
x,t\right) =v\left( x+2m\pi,t\right) , \label{v-1}
\end{equation}
which fills in the gap for $p=1.$\ Hence $p=1$ in $\left( \clubsuit\right)
\ $separates the contraction case from the expansion case. \
\end{remark}
When $m\geq2,\ $the behavior of solutions $v\left( x,t\right) \ $to the
equation$\ $
\begin{equation}
\dfrac{\partial v}{\partial t}=v^{p}\left( v_{xx}+v\right)
,\ \ \ \ \ v\left( x,0\right) =v_{0}\left( x\right) >0\in C^{\infty
}\left( S_{m}^{1}\right) ,\ \ \ \ \ p\in\left( -\infty,\infty\right)
\label{v-gen}
\end{equation}
for $p\in(-\infty,0],$\ $p\in\left( 0,1\right) ,\ p=1,\ p\in\left(
1,2\right) ,\ p\in\lbrack2,\infty)$\ are all quite different.\ This means
that equation\ (\ref{v-gen})\ has at least\textbf{ }the following "critical
values".$\ $Each case has its own feature explained below.\
\begin{itemize}
\item $p=0.\ $The case of the linear heat equation for the function$\ e^{-t}
v$, or the case of expanding flow with speed $1/k$.\ It also separates the
sublinear case ($p<0$)\ and the superlinear case\ ($p>0$).
\item $p=1.\ $The case which separates the contraction case from the expansion
case.$\ $In such a case,\ (\ref{integral-cond}) becomes
\begin{equation}
\int_{S_{m}^{1}}\log v_{0}\left( x\right) e^{ix}dx=0. \label{log}
\end{equation}
Since the behavior of $\log x$ is different\ from $x^{1-p}$ for $p\neq1,$ this
case is quite special.
\item $p=2.\ $The case of the classical \emph{curve-shortening flow}.$\ $It is
the gradient flow of the length functional.\ As we shall see below, for
$p\geq2,\ $(\ref{v-gen}) begins to have type-two blow-up (or type-two
singularity in $\left( \bigstar\right) $). Thus $p=2$ separates the type-one
blow-up and the type-two\ blow-up\ \ (see the definition for type-one and
type-two\ blow-up below).
\end{itemize}
\begin{remark}
By Andrews's Theorem \ref{thm-ben}, one can also view $p=4$ as a critical
value although it is for $m=1.$ Also see the discussions before Remark
\ref{rmk1}.\
\end{remark}
The behavior of solutions of (\ref{v-gen})\ for $p\geq2$ is most unknown to
us, especially the blow-up rate of a type-two singularity.\ The case $p=1$ is
also complicated.\ Some proofs valid for $p\neq1$ can not be carried over to
the case $p=1.$ To see their differences, we refer to the papers by
Angenent\ \cite{ANG} ($p=2$), Angenent-Vel\'{a}zquez\ \cite{AV}\ ($p=2$
),\ \cite{PT}\ ($1<p<2$), \cite{T3}\ ($p=1$), \cite{LPT}\ ($0<p<1$)\ and
Urbas\ \cite{U1}\ ($p\leq0$) for details.\
Let$\ v_{\min}\left( t\right) =\min_{x\in S_{m}^{1}}v\left( x,t\right)
\ $and similarly for $v_{\max}\left( t\right) .\ $For all $p\in\left(
-\infty,\infty\right) \ $in equation\ (\ref{v-gen}),\ as long as solution
exists,$\ v_{\min}\left( t\right) \ $is always increasing due to the maximum
principle.\ By the parabolic regularity theory, it is also known that smooth
solution $v\left( x,t\right) $ to equation (\ref{v-gen})\ exists on some
maximal time interval $[0,T_{\max}),$ where $v_{\max}\left( t\right)
\ $blows up at $T_{\max}\ $($v_{\max}\left( t\right) $ will be eventually
increasing for $t$ close to $T_{\max}$).\ For $p\leq0,\ T_{\max}=\infty$\ and
for $p>0,\ T_{\max}<\infty$.
When $T_{\max}<\infty\ $(i.e., when $p>0$), if we let $R\left( t\right) $ be
the unique solution to the ODE
\begin{equation}
\frac{dR}{dt}=R^{p+1}\left( t\right) ,\ \ \ \ \ R\left( T_{\max}\right)
=\infty\label{R-ode}
\end{equation}
then $R\left( t\right) =\left[ p\left( T_{\max}-t\right) \right]
^{-1/p}$ and the comparison principle implies that$\ $
\begin{equation}
0<v_{\min}\left( t\right) \leq R\left( t\right) \leq v_{\max}\left(
t\right) \ \ \ \text{for all}\ \ \ t\in\lbrack0,T_{\max}). \label{vR}
\end{equation}
We define the following terminology:\ if there exists a constant $C,$
independent\ of time, such that
\begin{equation}
0<v_{\max}\left( t\right) \leq CR\left( t\right) \ \ \ \text{for
all}\ \ \ t\in\lbrack0,T_{\max}) \label{t-one}
\end{equation}
then we say $v\left( x,t\right) $ has \textbf{type-one} blow-up. If not,
i.e.,$\ $if$\ v_{\max}\left( t\right) /R\left( t\right) \ $is not
bounded\ on$\ t\in\lbrack0,T_{\max}),\ $then we say $v\left( x,t\right) $
has \textbf{type-two} blow-up. A type-two blow-up is clearly much more complicated.\
It has been shown in p. 158 of \cite{LPT} that for $p\in\left( 0,2\right) $
there is \textbf{no} type-two blow-up for any $m\in\mathbb{N}\ $(include
$m=1$)\ and any positive\ initial data $v_{0}\left( x\right) \in C^{\infty
}\left( S_{m}^{1}\right) \ $(no matter it satisfies (\ref{integral-cond}) or
not), i.e., all blow-ups are of type-one in $\left( \clubsuit\right) \ $for
$p\in\left( 0,2\right) \ $and for any $v_{0}\left( x\right) >0$
.$\ $However,\ for $p\in\left( 0,2\right) ,$ the limit function $w\left(
x\right) \ $($w\left( x\right) $ is the limit of the rescaled
solution\ $v\left( x,t\right) /R\left( t\right) $)$\ $may be either
$w\left( x\right) >0\ $everywhere on\ $S_{m}^{1}\ $(call it
\textbf{nondegenerate}) or $w\left( x\right) \geq0\ $but with $w\left(
x\right) =0$\textsf{ }somewhere on\ $S_{m}^{1}\ $(call it \textbf{degenerate}
). If $w\left( x\right) $ is nondegenerate, it gives rise to a
\textbf{self-similar (homothetic)\ solution}.\ If $w\left( x\right) $ is
degenerate, its behavior (regularity)\ for $p\in\left( 0,1\right)
,\ p=1,\ $and$\ p\in\left( 1,2\right) $ are all different.\
In Angenent \cite{ANG} (the case $p=2$), he employed an elegant unstable
manifold analysis of a shrinking spiral (travelling wave solution) and used it
to prove a\textsf{ }Harnack-type estimate,\ i.e.,\textsf{\ }Lemma 7.1. of
\cite{ANG}. This is the key estimate to ensure convergence to a positive
self-similar solution under type-one blow-up\ (see p. 605,\ Theorem A of his paper).\
We shall see that his proof can be carried to the case $p\in\lbrack
2,\infty),\ $assuming that we have type-one blow-up.\ Hence we can obtain
convergence of equation $\left( \clubsuit\right) $ (or the flow $\left(
\bigstar\right) $)\ to a self-similar solution$\ w\left( x\right) $ under
type-one blow-up. Unlike the case for $p\in\left( 0,2\right) ,\ $now
$w\left( x\right) \ $is positive everywhere on $S_{m}^{1}\ $(for
$p\in\lbrack2,\infty)$) and is an entire periodic solution to the
corresponding steady state\ ODE. See Theorem \ref{thmA} in Section
\ref{type-one}.
In summary, the above says that for $p\in\lbrack2,\infty)\ $there is either
type-one blow-up\ or type-two blow-up.\ Moreover, if we have type-one blow-up,
then the limit function $w\left( x\right) $ is always \emph{nondegenerate},
i.e., $w\left( x\right) >0$ everywhere.\
As for Theorem\ B of \cite{ANG},\ it is rather straightforward to generalize
it to the case $p\geq2.\ $See Theorem\ \ref{thmB}.
Finally we also discuss a special \textbf{symmetric}\ case of type-two blow-up
and obtain a convergence to the cosine function (see Theorem \ref{thmC}\ and
Theorem \ref{thmC-1}). In flow\ $\left( \bigstar\right) $, this convergence
gives rise to a \textbf{translational self-similar solution}. When $\alpha
=1$\ (curve-shortening flow), this translational self-similar solution is the
Grayson's "\textbf{grim reaper}", i.e., the graph of $y=-\log\cos x,$
$x\in\left( -\pi/2,\pi/2\right) .\ $One can view Theorem \ref{thmC}\ and
Theorem \ref{thmC-1}\ as partial generalizations of Theorem\ C of \cite{ANG}
to the case $p\geq2.\ $
In conclusion, we can generalize Theorem A, Theorem\ B,\ and part of Theorem C
in p. 605 of Angenent \cite{ANG} to the case $p\geq2.\ $
To end this introductory section, we point out that solutions of (\ref{v-gen})
for the sublinear case $p<0$ are well-behaved as it is bounded by the
following super-sub solutions
\begin{equation}
\left[ \left( \min_{x\in S_{m}^{1}}v_{0}\left( x\right) \right)
^{-p}-pt\right] ^{-1/p}\leq v\left( x,t\right) \leq\left[ \left(
\max_{x\in S_{m}^{1}}v_{0}\left( x\right) \right) ^{-p}-pt\right] ^{-1/p}
\label{minmax}
\end{equation}
for all$\ t\in\lbrack0,\infty).\ $In particular we have
\[
1\leq\frac{v_{\max}\left( t\right) }{v_{\min}\left( t\right) }\leq
U\left( t\right) :=\frac{\left[ \left( \max_{x\in S_{m}^{1}}v_{0}\left(
x\right) \right) ^{-p}-pt\right] ^{-1/p}}{\left[ \left( \min_{x\in
S_{m}^{1}}v_{0}\left( x\right) \right) ^{-p}-pt\right] ^{-1/p}}
,\ \ \ t\in\lbrack0,\infty)
\]
where$\ U\left( t\right) $ is a \textbf{decreasing} function on$\ [0,\infty
)\ $with $\lim_{t\rightarrow\infty}U\left( t\right) =1.\ $In fact, the
quantity $v_{\max}\left( t\right) /v_{\min}\left( t\right) $ also
decreases to $1$\ as $t\rightarrow\infty$. As a consequence, by regularity
theory, the rescaled solution$\ u\left( x,t\right) :=v\left( x,t\right)
/R\left( t\right) $ will converge as $t\rightarrow\infty\ $to the constant
function $1$ in $C^{k}\left( S_{m}^{1}\right) \ $for any $k\in\mathbb{N}.$
Here $R\left( t\right) $ can be \emph{any}\ solution to the ODE
$dR/dt=R^{1+p}\ $($p<0$)$\ $with $R\left( 0\right) >0.$ The geometric
meaning is that when $\alpha\in\left( -1,0\right) ,$ the expanding flow
$\left( \bigstar\right) $ converges (after rescaling)\ to the $m$-fold cover
of $S^{1}\ $in any $C^{k}$-norm.\ See Urbas \cite{U1} also.\
\section{Some basic estimates.}
>From now on we assume $p\geq2\ $and $m\geq2,\ $with the smooth\ initial data
$v_{0}\left( x\right) >0\ $in$\ \left( \clubsuit\right) \ $satisfying
(\ref{integral-cond}).\ For convenience, denote the maximal space-time domain
$S_{m}^{1}\times\lbrack0,T_{\max})\ $by $\Omega_{m}$. In below, if the proof
of a lemma is omitted, then it is either straightforward or similar to those
established in \cite{GH} or \cite{ANG} for the case $p=2$. Hence we will not
repeat it.\
\begin{lemma}
\label{lem1}(\emph{gradient estimate in integral form}) There exists a
constant $C$ depending only on $v_{0}$ such that
\begin{equation}
\int_{S_{m}^{1}}v_{x}^{2}\left( x,t\right) dx\leq\int_{S_{m}^{1}}
v^{2}\left( x,t\right) dx+C \label{grad-int}
\end{equation}
for all $t\in\lbrack0,T_{\max}),\ $where $v_{x}^{2}$ means $\left( \partial
v/\partial x\right) ^{2}.\ $In particular, for $\varepsilon>0$
sufficiently\ small,\ there exists\ a number\ $\delta>0,\ $depending only on
$\varepsilon$, such that
\begin{equation}
\left( 1-\varepsilon\right) v_{\max}\left( t\right) \leq v\left(
x,t\right) +\sqrt{2m\pi C} \label{local-Har}
\end{equation}
for all\ $x\in\left( x_{t}-\delta^{2},x_{t}+\delta^{2}\right) \ $and all
$t\in\lbrack0,T_{\max}),\ $where$\ v\left( x_{t},t\right) =v_{\max}\left(
t\right) .\ $
\end{lemma}
\begin{lemma}
\label{lem2}(\emph{gradient estimate})\ For solution $v\left( x,t\right)
\ $to equation $\left( \clubsuit\right) $\ on $\Omega_{m}\ $we have, at each
point $(x,t),$\ either
\begin{equation}
\left( v_{xx}+v\right) \left( x,t\right) >0 \label{Ang2}
\end{equation}
or
\begin{equation}
v_{x}^{2}\left( x,t\right) +v^{2}\left( x,t\right) \leq\max_{x\in
S_{m}^{1}}\left[ \left( v_{0}\right) _{x}^{2}\left( x\right) +v_{0}
^{2}\left( x\right) \right] :=\sigma. \label{Ang3}
\end{equation}
In particular we have
\begin{equation}
\left\vert v_{x}\left( x,t\right) \right\vert \leq\max\left\{
\lambda,\ v_{\max}\left( t\right) \right\} \label{vvv}
\end{equation}
for all $\left( x,t\right) \in\Omega_{m},$ where $\lambda>0$ is a constant
depending only on $v_{0}$.\ As a consequence we also know that $v_{\max
}\left( t\right) $ is eventually increasing for $t$ close to $T_{\max}.\ $
\end{lemma}
\begin{lemma}
\label{lem2-1}(\emph{behavior near maximum point})\ Let $v_{\max}\left(
t\right) =v\left( x_{t},t\right) \ $for some $x_{t}\in S_{m}^{1}$.$\ $If at
any time $t\in\lbrack0,T_{\max})$ we have $v_{\max}(t)>\sigma,$\ where
$\sigma$ is from (\ref{Ang3}),\ then
\begin{equation}
v\left( x,t\right) >v_{\max}\left( t\right) \cos\left( x-x_{t}\right)
\label{vvcos}
\end{equation}
for all $x$\ with $0<\left\vert x-x_{t}\right\vert <\arccos\left(
\sigma/v_{\max}\left( t\right) \right) .$
\end{lemma}
With the help of the above basic estimates, we can generalize Theorem B of
\cite{ANG} to the case $p\geq2:$
\begin{theorem}
\label{thmB}(\emph{rough upper bound of}\textsf{ }$v_{\max}\left( t\right)
$)\ If $v_{\max}\left( t\right) $ blows up at time $T_{\max},$ then there
holds the following
\begin{equation}
\lim_{t\rightarrow T_{\max}}\left( T_{\max}-t\right) v_{\max}\left(
t\right) =0. \label{rough-est}
\end{equation}
\end{theorem}
\begin{remark}
For now, by (\ref{vR})\ and (\ref{rough-est}) we have the rough estimate
\[
\left[ p\left( T_{\max}-t\right) \right] ^{-1/p}\leq v_{\max}\left(
t\right) \leq\frac{C}{T_{\max}-t}\ \ \ \text{for all\ \ \ }t\in
\lbrack0,T_{\max}),
\]
where$\ p\geq2$ and $C$ is some constant independent\ of time.\
\end{remark}
\proof
Let
\[
I\left( t\right) =\int_{S_{m}^{1}}v^{1-p}\left( x,t\right)
dx>0,\ \ \ p\geq2.
\]
By (\ref{local-Har})\ in Lemma \ref{lem1}\ we have for $t$ close to $T_{\max}$
the estimate
\[
\int_{S_{m}^{1}}v\left( x,t\right) dx\geq cv_{\max}\left( t\right)
\]
where $c>0$ is a constant independent\ of time.\ Hence there is a time
$t_{\ast}$ close to $T_{\max}\ $such that
\[
-I^{\prime}\left( t\right) =\left( p-1\right) \int_{S_{m}^{1}}v\left(
x,t\right) dx\geq\left( p-1\right) cv_{\max}\left( t\right)
>0\ \ \ \text{for all\ \ \ }t\in\lbrack t_{\ast},T_{\max})
\]
By integration of $I^{\prime}\left( t\right) $ on the interval $[t_{\ast
},T_{\max}),$ we obtain
\[
\left( p-1\right) c\int_{t_{\ast}}^{T_{\max}}v_{\max}\left( t\right)
dt\leq I\left( t_{\ast}\right) -\left( \lim_{t\rightarrow T_{\max}}I\left(
t\right) \right) \leq I\left( t_{\ast}\right) <\infty
\]
and so the integral$\ \int_{0}^{T_{\max}}v_{\max}\left( t\right) dt\ $is
finite. Since by Lemma \ref{lem2} $v_{\max}\left( t\right) $ is eventually
increasing, we may also assume that $v_{\max}\left( t\right) $ is increasing
on $[t_{\ast},T_{\max})$ and conclude
\begin{equation}
\left( T_{\max}-t\right) v_{\max}\left( t\right) \leq\int_{t}^{T_{\max}
}v_{\max}\left( s\right) ds\ \ \ \text{for all\ \ \ }t\in\lbrack t_{\ast
},T_{\max}). \label{ttt}
\end{equation}
Letting $t\rightarrow T_{\max},$ the right hand side of (\ref{ttt}) converges
to zero and the proof is done.$
\square$
\section{Type-one blow-up implies $C^{\infty}$ convergence. \label{type-one}}
Throughout this section we assume the solution $v\left( x,t\right) $ to
equation $\left( \clubsuit\right) $ has type-one blow-up. We shall consider
its asymptotic\ behavior by the obvious rescaling $u\left( x,t\right)
:=v\left( x,t\right) /R\left( t\right) ,\ $where\ $R\left( t\right)
\ $is from\ (\ref{R-ode}),\ and let $\tau\in\lbrack0,\infty)\ $be the new time
given by the relation$\ t=T_{\max}\left( 1-e^{-p\tau}\right) ,\ t\in
\lbrack0,T_{\max}),\ $which is motivated by the requirement $d\tau
/dt=R^{p}\left( t\right) ,\ $then the function
\begin{equation}
u\left( x,\tau\right) =p^{1/p}T_{\max}^{1/p}e^{-\tau}v\left( x,T_{\max
}\left( 1-e^{-p\tau}\right) \right) >0,\ \ \ x\in S_{m}^{1},\ \ \ \tau
\in\lbrack0,\infty) \label{rescale}
\end{equation}
\ will be a positive, \textbf{bounded}, solution of the rescaled equation
\begin{equation}
\left\{
\begin{array}
[c]{l}
\dfrac{\partial u}{\partial\tau}=u^{p}\left( u_{xx}+u-u^{1-p}\right)
,\ \ \ p\geq2
\\
u\left( x,\tau\right) =u(x+2m\pi,\tau)
\end{array}
\right. \label{dudt}
\end{equation}
for all$\ \left( x,\tau\right) \in S_{m}^{1}\times\lbrack0,\infty),\ $with
$u\left( x,0\right) =u_{0}\left( x\right) :=p^{1/p}T_{\max}^{1/p}
v_{0}\left( x\right) >0.$ Moreover, we have
\begin{equation}
0<u_{\min}\left( \tau\right) \leq1\leq u_{\max}\left( \tau\right)
\label{UU}
\end{equation}
for\ all$\ \tau\in\lbrack0,\infty)\ $due to (\ref{vR}). By (\ref{vvv}) we also
have the uniform gradient estimate
\begin{equation}
\left\vert u_{x}\left( x,\tau\right) \right\vert \leq C\ \ \ \text{for
all\ \ \ }\left( x,\tau\right) \in S_{m}^{1}\times\lbrack0,\infty)
\label{grad-est}
\end{equation}
where $C$\ is a constant depending only on $v_{0}.$
We shall generalize Angenent's Lemma 7.1 in\ \cite{ANG} to the following:
\begin{theorem}
\label{thm-type-one}(\emph{gradient estimate for type-one blow-up})\ Let
$v\left( x,t\right) $ be a \emph{type-one solution} to equation $\left(
\clubsuit\right) $\ with $p\geq2$. Then the rescaled
bounded\ positive\ function $u\left( x,\tau\right) $ of (\ref{rescale})
satisfies \
\begin{equation}
\left\vert u_{x}\left( x,\tau\right) \right\vert \leq\lambda u\left(
x,\tau\right) \ \ \ \text{for all\ \ \ }\left( x,\tau\right) \in S_{m}
^{1}\times\lbrack0,\infty) \label{KK}
\end{equation}
where $\lambda$ is a constant depending only on$\ u_{0}$.
\end{theorem}
\begin{remark}
Theorem \ref{thm-type-one}$\ $fails for $p\in\left( 0,2\right) .\ $
\end{remark}
\begin{remark}
By (\ref{KK})\ we have the estimate
\begin{equation}
1\leq u_{\max}\left( \tau\right) \leq e^{2\lambda m\pi}u_{\min}\left(
\tau\right) \label{KK1}
\end{equation}
for all $\tau\in\lbrack0,\infty)\ $and hence $u_{\min}\left( \tau\right) $
has a positive lower bound for $\tau\in\lbrack0,\infty)$ and equation
(\ref{dudt}) is uniformly parabolic\ on $S_{m}^{1}\times\lbrack0,\infty)$.
\end{remark}
\begin{remark}
>From the proof we see that Theorem \ref{thm-type-one} remains valid even the
initial condition $v_{0}\left( x\right) $ does not satisfy the integral
condition (\ref{integral-cond}).\ This observation is important and will be
used in Lemma \ref{lem3} below.
\end{remark}
\subsection{Angenent's method of shrinking spirals.}
Since Theorem \ref{thm-type-one}\ is valid for $p=2$, we assume $p>2.\ $Our
method of proof is similar to that originally used by Angenent in
\cite{ANG}.\ At the same time we also provide some additional details and see
why we need the condition $p>2.\ $Consider a special solution
(\emph{travelling wave solution}) of the form $U\left( x,\tau\right)
=U\left( x-c\tau\right) ,\ c>0\ $($c\ $is a constant), to the equation
\begin{equation}
\dfrac{\partial u}{\partial\tau}=u^{p}\left( u_{xx}+u-u^{1-p}\right) .
\label{eq-ss}
\end{equation}
A positive function $U\left( \xi\right) \ $over some interval $I$ will
generate a solution if and only if
\begin{equation}
U^{p}\left( \xi\right) U^{\prime\prime}\left( \xi\right) +U^{p+1}\left(
\xi\right) -U\left( \xi\right) +cU^{\prime}\left( \xi\right)
=0\ \ \ \text{for all\ \ \ }\xi\in I. \label{Q1}
\end{equation}
For such a$\ U\left( \xi\right) >0$ satisfying equation (\ref{Q1}) on $I$ we
have
\begin{equation}
\frac{d}{d\xi}\left[ \left( U^{\prime}\left( \xi\right) \right)
^{2}+U^{2}\left( \xi\right) -\frac{2}{2-p}U^{2-p}\left( \xi\right)
\right] =-2c\frac{\left( U^{\prime}\left( \xi\right) \right) ^{2}}
{U^{p}\left( \xi\right) }\leq0 \label{Lyapunov}
\end{equation}
and so the function$\ E\left( \xi\right) $ given by
\begin{equation}
E\left( \xi\right) :=\left( U^{\prime}\left( \xi\right) \right)
^{2}+U^{2}\left( \xi\right) -\frac{2}{2-p}U^{2-p}\left( \xi\right)
,\ \ \ \xi\in I,\ \ \ p>2, \label{E}
\end{equation}
is decreasing in $\xi\in I\ $if $c\neq0.\ $For$\ c\neq0$ the only periodic
solution for (\ref{Q1})\ is the constant$\ U\left( \xi\right) \equiv1.$
For $c=0,$ $E\left( \xi\right) $ is a positive constant independent\ of
$\xi\in I.\ $It is obvious that any positive\ solution $U\left( \xi\right) $
satisfying the equation $E\left( \xi\right) =const.>0$ can not become too
small over its domain\ since$\ -2\left( 2-p\right) ^{-1}U^{2-p}\left(
\xi\right) \rightarrow\infty$ as$\ U\left( \xi\right) \rightarrow0^{+}
.\ $Thus any solution $U\left( \xi\right) \ $to the ODE $U^{p}
U^{\prime\prime}+U^{p+1}-U=0\ $is a\ positive periodic function on$\ \xi
\in\left( -\infty,\infty\right) \ $(here we need the condition
$p>2$)\ satisfying
\begin{equation}
\left( U^{\prime}\left( \xi\right) \right) ^{2}+U^{2}\left( \xi\right)
-\frac{2}{2-p}U^{2-p}\left( \xi\right) =b^{2}-\frac{2}{2-p}b^{2-p}
=a^{2}-\frac{2}{2-p}a^{2-p},\ \ \ \xi\in\left( -\infty,\infty\right) ,
\label{ODE}
\end{equation}
where $b\geq1$ ($a\leq1$)\ is the maximum (minimum)\ value of$\ U\left(
\xi\right) $ over$\ \left( -\infty,\infty\right) .$
Similar to Theorem 6.1 of \cite{ANG}, we claim the following:
\begin{theorem}
\label{thm-type-one-1}Assume $p>2.\ $For any small $c>0,$ there is a unique
positive\ solution $U_{c}$ $\in C^{\infty}\left( (-\infty,0]\right) \ $of
(\ref{Q1})$\ $with $\lim_{\xi\rightarrow-\infty}U_{c}\left( \xi\right)
=0\ $and\ the following properties:
\begin{equation}
\left\{
\begin{array}
[c]{l}
\text{(i).}\ \ U_{c}^{\prime}\left( \xi\right) >0\ \ \ \text{for \ \ }\xi
\in(-\infty,0),
\\
\text{(ii).\ \ }U_{c}^{\prime}\left( 0\right) =0
\\
\text{(iii).\ \ }U_{c}^{\prime}\left( \xi\right) \leq\lambda_{c}U\left(
\xi\right) \ \ \ \text{for \ \ }\xi\in(-\infty,0],
\end{array}
\right. \label{Q2}
\end{equation}
where $\lambda_{c}>0\ $is a large constant depending on $c\ $(and
$p\ $also).$\ $As a function of $c>0,$ $U_{c}\left( 0\right) $ is strictly
decreasing and given any $\delta>0$ and $A>0,$ one can choose $c=c\left(
\delta,A\right) >0\ $so small that
\begin{equation}
U_{c}\left( 0\right) >\delta^{-1}\ \ \ \ \ \ \ \text{and\ \ \ \ \ \ \ }
U_{c}^{\prime}\left( \xi\right) >A\ \ \text{\ whenever\ \ \ }\delta\leq
U_{c}\left( \xi\right) \leq\delta^{-1}. \label{Q3}
\end{equation}
\end{theorem}
\proof
It will be convenient to look at $H=U^{p}>0$ instead of $U$ itself. We have
$U=H^{1/p}$ and (\ref{Q1}) is equivalent\ to
\[
HH_{\xi\xi}=-cH_{\xi}+\frac{p-1}{p}\left( H_{\xi}\right) ^{2}-pH^{2}
+pH,\ \ \ H\left( \xi\right) =U^{p}\left( \xi\right) ,
\]
which can be written as the first order system (let $G=H_{\xi}$)
\begin{equation}
\left\{
\begin{array}
[c]{l}
HH_{\xi}=HG
\\
HG_{\xi}=-pH^{2}+pH-cG+\frac{p-1}{p}G^{2}.
\end{array}
\right. \label{ODE-sys}
\end{equation}
Thus \emph{up to a reparametrization} (since there is a factor $H$ in front of
$H_{\xi}\ $and $G_{\xi}$), the positive solutions of (\ref{Q1}) are in
one-to-one correspondence with the orbits of the vector field
\begin{equation}
X_{c}\left( H,G\right) =\left( HG,\ -pH^{2}+pH-cG+\frac{p-1}{p}
G^{2}\right) \label{X}
\end{equation}
lying on the region$\ R^{+}=\left\{ \left( H,G\right) :H>0\right\} .\ $We
shall analyze the phase portrait of the vector field $X_{c}\left( H,G\right)
\ $in $R^{+}.$ Note that $X_{c}$ has three zeros $\left( 0,0\right) ,$
$\left( 1,0\right) ,$ and $\left( 0,pc/\left( p-1\right) \right) \ $in
$R^{+}.\ $For a given small $c>0,\ $our aim is to look at certain special
solution $\left( H\left( \xi\right) ,G\left( \xi\right) \right) $ of the
system (\ref{ODE-sys}) with $H\left( \xi\right) >0\ $everywhere.\
If we compute the linearization of $X_{c}\left( H,G\right) $ at these
equilibrium points, we obtain the three\ matrices
\[
M=
\begin{pmatrix}
G & H\\
p-2pH & -c+\frac{2\left( p-1\right) }{p}G
\end{pmatrix}
=
\begin{pmatrix}
0 & 0\\
p & -c
\end{pmatrix}
,\ \ \
\begin{pmatrix}
0 & 1\\
-p & -c
\end{pmatrix}
,\ \ \
\begin{pmatrix}
\frac{pc}{p-1} & 0\\
p & c
\end{pmatrix}
\]
at $\left( 0,0\right) ,$ $\left( 1,0\right) \ $and $\left( 0,pc/\left(
p-1\right) \right) $ respectively.\ The eigenvalues of them are given
respectively by
\[
\lambda=0,\ -c;\ \ \ \ \ \lambda=\frac{-c\pm\sqrt{c^{2}-4p}}{2}
;\ \ \ \ \ \lambda=\frac{pc}{p-1},\ c
\]
where $c>0$ is a small constant to be chosen later\ on. Therefore, $\left(
0,0\right) $ is a degenerate zero of $X_{c}$ (it is not a hyperbolic fixed
point of $X_{c}$),$\ \left( 1,0\right) $ is a spiraling sink of $X_{c}$ if
$c>0$ is small with $c^{2}-4p<0,$ and $\left( 0,pc/\left( p-1\right)
\right) $ is a source of $X_{c}.$
By definition, the unstable set $W^{u}\left( O\right) $ of the origin
$O=\left( 0,0\right) \ $consists of all orbits of $X_{c},$ which tend to $O$
as $\xi\rightarrow-\infty.$ As the origin is degenerate, one needs to analyze
further to know what$\ W^{u}\left( O\right) $ looks like.\ Note that if
$U_{c}\left( \xi\right) $ is the solution satisfying Theorem
\ref{thm-type-one-1}, then
\[
\left( H_{c}\left( \xi\right) ,G_{c}\left( \xi\right) \right) =\left(
U_{c}^{p}\left( \xi\right) ,\ pU_{c}^{p-1}\left( \xi\right) U_{c}^{\prime
}\left( \xi\right) \right)
\]
parametrizes a trajectory of $X_{c}$ in the unstable set $W^{u}\left(
O\right) \ $of the origin.\ Thus one needs to look at $W^{u}\left( O\right)
.\ $
\ \ \ \
\underline{\textbf{Existence of a trajectory in }$W^{u}\left( O\right) .$
}\ \ Given a constant $\lambda>0$ and let $l_{\lambda}$ be the half line
$G=\lambda H,$ $H\geq0.$ The half line $l_{\lambda}\ $has upward normal
$\left( -\lambda,1\right) $ and along it we have
\[
\left\langle X_{c},\left( -\lambda,1\right) \right\rangle =\left\{
p-c\lambda-\left( p+\frac{\lambda^{2}}{p}\right) H\right\} H,\ \ \ \lambda
>0,\ \ \ H\geq0.
\]
Now choose two positive$\ \lambda_{1},\ \lambda_{2}$ such that $\lambda
_{1}<p/c\ $and$\ \lambda_{2}=p/c.\ $Put$\ h^{\ast}=\left( p-c\lambda
_{1}\right) /\left( p+\lambda_{1}^{2}/p\right) >0,$ and define the points
\[
A=\left( h^{\ast},\lambda_{1}h^{\ast}\right) ,\ \ \ B=\left( h^{\ast
},\lambda_{2}h^{\ast}\right) ,\ \ \ O=\left( 0,0\right) .
\]
Along the segment $OA\ $with $0\leq H\leq h^{\ast},$ we have $\left\langle
X_{c},\left( -\lambda_{1},1\right) \right\rangle \geq0\ $($=0\ $only at
$H=0\ $or$\ H=h^{\ast}$) and so the vector field $X_{c},$ when restricted to
$OA,$ is pointing toward the upper half of segment $OA.\ $Similarly, along the
segment $OB$ we have$\ \left\langle X_{c},\left( -\lambda_{2},1\right)
\right\rangle \leq0\ $($=0\ $only at $H=0$).\ Hence the vector field $X_{c},$
when restricted to $OB,$ is pointing toward the lower half of the segment.
Finally, along the segment $AB,$ we have$\ \left\langle X_{c},\left(
1,0\right) \right\rangle =h^{\ast}G>0$ for all $G$ with$\ \lambda_{1}h^{\ast
}<G\ <\lambda_{2}h^{\ast}.\ $We conclude that the\ trajectories\ of$\ X_{c}
\ $enter\ the\ triangle$\ OAB\ $through\ the\ sides$\ OA\ $and$\ OB,\ $and
they leave\ $OAB\ $through the vertical side $AB.$
For any point $\left( H,G\right) $ inside the triangle $OAB,$ it has the
form $G=\lambda H,$ for some $\lambda_{1}\leq\lambda\leq\lambda_{2}=p/c$ and
hence
\begin{align}
G_{\xi} & =\frac{1}{H}\left( -pH^{2}+pH-c\lambda H+\frac{p-1}{p}\lambda
^{2}H^{2}\right) \nonumber\\
& =\left( p-c\lambda\right) +\left( \frac{p-1}{p}\lambda^{2}-p\right)
H\geq\left( \frac{p-1}{p}\lambda_{1}^{2}-p\right) H. \label{G}
\end{align}
We may choose $c$ small enough and $\lambda_{1}<p/c$ larger than $p/\sqrt
{p-1}$ such that $p/\sqrt{p-1}<\lambda_{1}\leq\lambda\leq\lambda_{2}=p/c\ $and
conclude that
\[
G_{\xi}\geq\left( \frac{p-1}{p}\lambda_{1}^{2}-p\right) H>0
\]
for all $\lambda_{1}\leq\lambda\leq\lambda_{2}$ and $0<H\leq h^{\ast}=\left(
p-c\lambda_{1}\right) /\left( p+\lambda_{1}^{2}/p\right) .$ Therefore
$G_{\xi}>0$ in the interior of the triangle $OAB$ and it follows from the
\emph{Wa\.{z}ewski's Principle} that at least one of the trajectories through
$AB$ tends to the origin as $\xi\rightarrow-\infty.$
In conclusion, we see that as long as $c>0$ is small enough (depending only on
$p$), there exists a trajectory in the unstable set\textbf{ }$W^{u}\left(
O\right) .$
\ \ \ \ \
\underline{\textbf{Uniqueness of the trajectory in }$W^{u}\left( O\right)
.$}\ \ Express the second equation of (\ref{ODE-sys}) as
\[
HG_{\xi}=-p\left( H-\frac{1}{2}\right) ^{2}+\frac{p-1}{p}\left( G-\frac
{pc}{2\left( p-1\right) }\right) ^{2}+\frac{p}{4}\left( 1-\frac{c^{2}
}{p-1}\right) .
\]
We see that if $c<\sqrt{p-1},$ then the set $\left\{ \left( H,G\right)
:H>0,\ G_{\xi}=0\right\} $ is the part of the hyperbola
\[
\Gamma:\ p\left( H-\frac{1}{2}\right) ^{2}-\frac{p-1}{p}\left( G-\frac
{pc}{2\left( p-1\right) }\right) ^{2}=\frac{p}{4}\left( 1-\frac{c^{2}
}{p-1}\right)
\]
lying in $R^{+}=\left\{ \left( H,G\right) :H>0\right\} .\ $Here $\Gamma$
is centered at $\left( 1/2,\ pc/\left( 2\left( p-1\right) \right)
\right) $ and passes through the three equilibrium points $\left(
0,0\right) ,$ $\left( 0,pc/\left( p-1\right) \right) \ $and $\left(
1,0\right) .$
Consider the region $\Omega$ enclosed by the segment $\left\{ \left(
0,G\right) :0\leq G\leq pc/\left( p-1\right) \right\} $ and the left
branch of $\Gamma.\ $For each\ point$\ \left( H,G\right) \ $inside
$\Omega,\ $the vector field $X_{c}\left( H,G\right) =\left( HH_{\xi
},HG_{\xi}\right) \ $satisfies $HH_{\xi}=HG>0\ $and $HG_{\xi}<0.$ This
implies that any trajectory $\left( H,G\right) \ $in\ $W^{u}\left(
O\right) $ will not pass through the region $\Omega$ and thus there exists a
large constant $\lambda_{c}\ $(say$\ \lambda_{c}>p/c,\ $where $p/c$ is the
slope of $\Gamma\ $at the origin) such that any any trajectory $\left(
H\left( \xi\right) ,G\left( \xi\right) \right) $ in $W^{u}\left(
O\right) $ satisfies $G\leq\lambda_{c}H,\ $as $\xi\rightarrow-\infty.$
Let $\left( H_{1},G_{1}\right) $ and $\left( H_{2},G_{2}\right) \ $be two
different orbits in $W^{u}\left( O\right) .\ $From the above observation,
near the origin they can be represented as the graphs $G_{1}=g_{1}\left(
H\right) ,\ G_{2}=g_{2}\left( H\right) ,$ where the $g_{i}$ are solutions
of the equation
\[
g^{\prime}\left( H\right) =\frac{HG_{\xi}}{HH_{\xi}}=\frac{-pH^{2}
+pH-cG+\frac{p-1}{p}G^{2}}{HG}=\frac{\left( p-1\right) g\left( H\right)
-pc}{pH}+\frac{p-pH}{g\left( H\right) }.
\]
Orbits cannot intersect, so we may assume that $g_{1}\left( H\right)
<g_{2}\left( H\right) .$Their difference $w\left( H\right) =g_{2}\left(
H\right) -g_{1}\left( H\right) >0$ satisfies the equation
\[
w^{\prime}\left( H\right) =\left( \frac{p-1}{pH}-\frac{p-pH}{g_{2}\left(
H\right) g_{1}\left( H\right) }\right) w,\ \ \ w>0.
\]
Now for $H>0$ sufficiently small, we have $g_{1}\left( H\right) \leq\lambda
H,$ $g_{2}\left( H\right) \leq\lambda H$ for some $\lambda,$ and so
\[
w^{\prime}\left( H\right) \leq\left( \frac{p-1}{pH}-\frac{p-pH}
{g_{2}\left( H\right) g_{1}\left( H\right) }\right) w=\frac{w}
{p\lambda^{2}H^{2}}\left[ \lambda^{2}\left( p-1\right) H+p^{2}
H-p^{2}\right] <0
\]
for all\ sufficiently small $H>0,\ $which means that $w\left( H\right)
>0\ $is decreasing on some small interval $[0,\varepsilon),$ $\varepsilon>0.$
However, by $\lim_{H\downarrow0}w\left( H\right) =0$ we get a contradiction
and must have $w\left( H\right) \equiv0.$ Thus the two solutions are in fact equal.
\begin{remark}
Since the trajectory $\left( H\left( \xi\right) ,G\left( \xi\right)
\right) \ $in $W^{u}\left( O\right) $ is unique, in the above
existence\ proof we can choose $\lambda_{1}$ as close to $p/c$ as possible. In
particular $\left( H\left( \xi\right) ,G\left( \xi\right) \right) $ in
$W^{u}\left( O\right) \ $must satisfy the following
\begin{equation}
\frac{p}{c}=\lim_{\xi\rightarrow-\infty}\frac{G\left( \xi\right) }{H\left(
\xi\right) }=\lim_{\xi\rightarrow-\infty}\frac{pU^{p-1}\left( \xi\right)
U^{\prime}\left( \xi\right) }{U^{p}\left( \xi\right) }=\lim_{\xi
\rightarrow-\infty}\frac{pU^{\prime}\left( \xi\right) }{U\left( \xi\right)
}, \label{p/c}
\end{equation}
which implies the asymptotic behavior
\begin{equation}
\lim_{\xi\rightarrow-\infty}\frac{U^{\prime}\left( \xi\right) }{U\left(
\xi\right) }=\frac{1}{c}. \label{1/c}
\end{equation}
\end{remark}
\ \ \ \ \ \ \
Let $\left( H_{c}\left( \xi\right) ,G_{c}\left( \xi\right) \right) $
denote the trajectory whose existence and uniqueness have been established and
let $U_{c}\left( \xi\right) $ be the corresponding function of $\xi.$ That
is, $U_{c}\left( \xi\right) =H_{c}^{1/p}\left( \xi\right) .\ $Recall that
along any positive\ solution $U\left( \xi\right) $ of (\ref{Q1}), we
have$\ dE/d\xi=-2c\left( U^{\prime}\right) ^{2}/U^{p}\leq0,\ $
where$\ E\left( \xi\right) $ is given by (\ref{E}). It follows that the
quantity$\ $
\[
E\left( H,G\right) =\frac{G^{2}}{p^{2}}H^{2/p-2}+H^{2/p}-\frac{2}
{2-p}H^{2/p-1},\ \ \ H=U^{p},\ \ \ G=H_{\xi}
\]
is strictly decreasing on orbits$\ $of $X_{c},\ $except$\ c=0$\ (when\ $c=0$
,\ all orbits are closed\ curves).$\ $Thus $H^{2/p}-2\left( 2-p\right)
^{-1}H^{2/p-1},$ and therefore $H,$ are bounded from above on any orbit of
$X_{c}.$ Furthermore, it also implies that $\left\vert G\right\vert $ is bounded.
Using the fact that $\left( 1,0\right) $ is an attracting spiral point, one
can show that any orbit (here we only care about those orbits with positive
$H\ $everywhere)\ converges to $\left( 1,0\right) ,$ and winds around this
point infinitely many times. In particular, any orbit will intersect the
$H$-axis (to see this, just look at the vector field (\ref{X})). For the
function $U_{c}\left( \xi\right) ,$ this means that it will converge to $1$
as $\xi\rightarrow\infty$ and that it will oscillate infinitely often around
its limit value. Its derivative $U_{c}^{\prime}\left( \xi\right) $ must
therefore vanish infinitely often; by replacing $U_{c}\left( \xi\right) $ by
$U_{c}\left( \xi-\xi_{0}\right) $ for some $\xi_{0}\in\mathbb{R}$ if
necessary, we may assume that the first zero of $U_{c}^{\prime}\ $is $\xi=0$
and $U_{c}^{\prime}\left( \xi\right) >0$ for all $\xi\in(-\infty,0).$
So far we have constructed the solution $U_{c}\left( \xi\right) $ satisfying
(i) and (ii) of (\ref{Q2}). Since we also have (\ref{1/c}), one can choose a
large constant$\lambda_{c}>0\ $so that$\,$(iii) of (\ref{Q2}) is also
satisfied for such $U_{c}\left( \xi\right) .$\newline
\ \ \ \ \
This complete our construction of $U_{c}.\ $To finishes the proof, we need to
verify (\ref{Q3}) for $U_{c}\left( \xi\right) .$
$\ \ \ \ \ \ \ $
We observe that the segment of $W^{u}\left( O\right) $ which lies in the
first quadrant is the graph of some function $G=g_{c}\left( H\right) $ for
$0\leq H\leq h_{c},$ where $\left( h_{c},0\right) $ is the first point of
intersection of $W^{u}\left( O\right) $ with the $H$-axis. Since
$U_{c}\left( 0\right) =h_{c}^{1/p},$ we have to show that $h_{c}$\textbf{
}is\emph{ monotone decreasing} in\textbf{ }$c.$\textbf{ }
Let $c^{\prime}<c\ $be given, and suppose that $h_{c^{\prime}}\leq h_{c}.\ $We
want to derive a contradiction. Assume first that $h_{c^{\prime}}<h_{c}
.\ $Compare the two vector fields $X_{c}\ $and $X_{c^{\prime}}\ $in the first
quadrant. If the backward orbit of $X_{c}\ $through $\left( h_{c},0\right)
\ $and the backward orbit of $X_{c^{\prime}}\ $through $\left( h_{c^{\prime}
},0\right) \ $intersect at a first point $\left( H_{0},G_{0}\right)
,$$G_{0}>0,\ $we have the following comparison of the two vector fields at
$\left( H_{0},G_{0}\right) :$
\[
-pH_{0}^{2}+pH_{0}-cG_{0}+\frac{p-1}{p}G_{0}^{2}<-pH_{0}^{2}+pH_{0}-c^{\prime
}G_{0}+\frac{p-1}{p}G_{0}^{2}
\]
which implies that the backward orbit of $X_{c^{\prime}}\ $through $\left(
h_{c^{\prime}},0\right) \ $cannot pass through the graph of $g_{c}\left(
H\right) .\ $As a consequence, the graph of $g_{c^{\prime}}\left( H\right)
\ $on the domain $0<H<h_{c^{\prime}}\ $must be below the graph of
$g_{c}\left( H\right) .\ $On the other hand by (\ref{1/c})\ we know
\begin{equation}
g_{c}^{\prime}\left( 0\right) =\lim_{\xi\rightarrow-\infty}\frac
{g_{c}\left( H\left( \xi\right) \right) }{H\left( \xi\right) }=\lim
_{\xi\rightarrow-\infty}\frac{G\left( \xi\right) }{H\left( \xi\right)
}=\frac{p}{c} \label{gc1}
\end{equation}
and similarly $g_{c^{\prime}}^{\prime}\left( 0\right) =p/c^{\prime}.\ $Hence
$g_{c^{\prime}}^{\prime}\left( 0\right) >g_{c}^{\prime}\left( 0\right)
\ $and this gives a contradiction
\begin{equation}
\lim_{H\rightarrow0^{+}}g_{c^{\prime}}^{\prime}\left( H\right) =\frac
{p}{c^{\prime}}>\lim_{H\rightarrow0^{+}}g_{c}^{\prime}\left( H\right)
=\frac{p}{c}>0. \label{g}
\end{equation}
Thus $h_{c^{\prime}}<h_{c}\ $is impossible.\ \
If $c^{\prime}<c\ $but $h_{c^{\prime}}=h_{c},\ $then by continuity we must
have$\ 0<g_{c^{\prime}}\left( H\right) \leq g_{c}\left( H\right) \ $for
all\ $0<H<h_{c^{\prime}}.\ $But now estimate (\ref{g}) still holds and we
obtain the same contradiction.
\begin{remark}
By (\ref{gc1})\ and (\ref{1/c}), we have
\begin{align*}
\frac{p}{c} & =g_{c}^{\prime}\left( 0\right) =\lim_{H\rightarrow0^{+}
}g_{c}^{\prime}\left( H\right) =\lim_{\xi\rightarrow-\infty}\frac{G_{\xi}
}{H_{\xi}}\\
& =\lim_{\xi\rightarrow-\infty}\frac{pU^{p-1}\left( \xi\right)
U^{\prime\prime}\left( \xi\right) +p\left( p-1\right) U^{p-2}\left(
\xi\right) \left( U^{\prime}\left( \xi\right) \right) ^{2}}
{pU^{p-1}\left( \xi\right) U^{\prime}\left( \xi\right) }=\lim
_{\xi\rightarrow-\infty}\frac{U^{\prime\prime}\left( \xi\right) }{U^{\prime
}\left( \xi\right) }+\frac{p-1}{c}
\end{align*}
and derive the limit
\begin{equation}
\lim_{\xi\rightarrow-\infty}\frac{U^{\prime\prime}\left( \xi\right)
}{U^{\prime}\left( \xi\right) }=\frac{1}{c}. \label{2/c}
\end{equation}
From\ (\ref{1/c})\ and (\ref{2/c}),\ it is not hard to see that asymptotically
$U\left( \xi\right) $ is given by $ae^{\left( 1/c\right) \xi}\ $as
$\xi\rightarrow-\infty\ $for some constant $a>0.\ $
\end{remark}
\ \ \ \
A similar argument also show that $g_{c}\left( H\right) $ is a strictly
decreasing function of $c$ for fixed $H,$ i.e., as $c\downarrow0,$ the
unstable set $W_{c}^{u}\left( O\right) $ moves upwards.
We next claim that $U_{c}\left( 0\right) \rightarrow\infty\ $
as\ $c\downarrow0.\ $Assume that $U_{c}\left( 0\right) $ were bounded, as
$c\downarrow0.$ Then the $h_{c}^{\prime}s$ would converge to some $h_{0}>1.$
The vector field $X_{c}$ is well-defined and smooth for all $c\in\mathbb{R},$
so the unstable set $W_{c}^{u}\left( O\right) ,$ being the orbit of $X_{c}$
through $\left( h_{c},0\right) ,$ must converge to the orbit of $X_{0}$
through $\left( h_{0},0\right) ,$ where
\[
X_{0}\left( H,G\right) =\left( HG,\ -pH^{2}+pH+\frac{p-1}{p}G^{2}\right)
\]
and the quantity
\[
E\left( H,G\right) =\frac{G^{2}}{p^{2}}H^{2/p-2}+H^{2/p}-\frac{2}
{2-p}H^{2/p-1},\ \ \ (H=U^{p},\ G=pU^{p-1}U_{\xi})
\]
is constant on the orbits of $X_{0}.$ By (\ref{ODE})\ we know that all orbits
of $X_{0}$ are periodic (\emph{due to the condition }$p>2$). In particular,
for $c=0,$ the orbit of $X_{0}$ through $\left( h_{0},0\right) $ will
satisfy the equation
\[
\frac{G^{2}}{p^{2}}H^{2/p-2}+H^{2/p}-\frac{2}{2-p}H^{2/p-1}=h_{0}^{2/p}
-\frac{2}{2-p}h_{0}^{2/p-1}>0,\ \ \ \text{ \ }h_{0}>1
\]
and from this equation we see that the orbit of $X_{0}$ through $\left(
h_{0},0\right) $ will intersect the $H$-axis at some point $\left( h_{\ast
},0\right) ,$ $0<h_{\ast}<1,$ when followed backwards in time, where
$h_{\ast}\ $satisfies
\[
h_{\ast}^{2/p}-\frac{2}{2-p}h_{\ast}^{2/p-1}=h_{0}^{2/p}-\frac{2}{2-p}
h_{0}^{2/p-1}.
\]
By continuous dependence on parameters, the same will be true for some small
$c>0,$ a contradiction. Therefore we have $\lim_{c\downarrow0}U_{c}\left(
0\right) =\infty.$
Recall that for fixed $c>0,$ the quantity
\[
E\left( H\left( \xi\right) ,G\left( \xi\right) \right) =\frac
{G^{2}\left( \xi\right) }{p^{2}}H^{2/p-2}\left( \xi\right) +H^{2/p}\left(
\xi\right) -\frac{2}{2-p}H^{2/p-1}\left( \xi\right)
\]
is strictly decreasing along the unstable orbit. We already know that as
$c\downarrow0,$ $h_{c}\left( 0\right) \uparrow\infty.$ In
particular,\ $E\left( H\left( \xi\right) ,G\left( \xi\right) \right) $
is uniformly large on $(-\infty,0]$ since
\[
E\left( H\left( 0\right) ,G\left( 0\right) \right) =h_{c}^{2/p}\left(
0\right) -\frac{2}{2-p}h_{c}{}^{2/p-1}\left( 0\right) \rightarrow
\infty\ \ \ \text{as \ \ }c\downarrow0.
\]
Now when we confine to the region $\delta\leq H=U^{p}\leq\delta^{-1},$ as
$c\downarrow0,$ we must have $g_{c}\left( H\right) =G\uparrow\infty$ as
$c\downarrow0,$ and uniformly so on the interval $\delta\leq H\leq1/\delta.$\
Since$\ g_{c}\left( H\right) =G=pU_{c}^{p-1}U_{c}^{\prime}$ and $\delta\leq
H\leq1/\delta,$ we must have $U_{c}^{\prime}$ sufficiently large as
$c\downarrow0.$ Therefore (\ref{Q3}) also holds.\ The proof of Theorem
\ref{thm-type-one-1}\ is done.$
\square\ $
\subsection{Proof of Theorem \ref{thm-type-one}.}
Assume $v\left( x,t\right) $ is a \emph{type-one solution} to equation
$\left( \clubsuit\right) $\ with $p\geq2$. Then the rescaled
positive\ function $u\left( x,\tau\right) $ is bounded from above.\ Choose a
large constant $A\ $so that
\begin{equation}
\left\{
\begin{array}
[c]{l}
u\left( x,\tau\right) \leq A\ \ \ \text{for all\ \ \ }\left( x,\tau\right)
\in S_{m}^{1}\times\lbrack0,\infty)
\\
\left\vert u_{x}\left( x,0\right) \right\vert \leq A\ \ \ \text{for
all\ \ \ }x\in S_{m}^{1}
\\
u\left( x,0\right) \geq\frac{1}{A}\ \ \ \text{for all\ \ \ }x\in S_{m}^{1}.
\end{array}
\right. \label{Q4}
\end{equation}
Also choose $c>0$ so small that the solution $U_{c}\left( \xi\right) $ of
the last section satisfies$\ U_{c}\left( 0\right) >A\ $and$\ U_{c}^{\prime
}\left( \xi\right) >A$ whenever$\ A^{-1}\leq U_{c}\left( \xi\right) \leq
A.\ $By Theorem\ \ref{thm-type-one-1} such\ a $c$ exists, together with the
existence of a large constant $\lambda_{c}>0$ such that $0<U_{c}^{\prime
}\left( \xi\right) \leq\lambda_{c}U_{c}\left( \xi\right) \ $for
all\ $\xi\in(-\infty,0].\ $Note that here the number $c\ $and $\lambda_{c}$
both depend on the initial data $u\left( x,0\right) .\ $
For any$\ $fixed$\ \left( x_{0},\tau_{0}\right) $ we have $0<u\left(
x_{0},\tau_{0}\right) <U_{c}\left( 0\right) $ and since $U_{c}\left(
\xi\right) $ is strictly increasing on$\ (-\infty,0]$ there exists a unique
$x_{1}\in\mathbb{R}$ for which$\ U_{c}\left( x_{1}-c\tau_{0}\right)
=u\left( x_{0},\tau_{0}\right) .$ Consider the function
\[
u^{\ast}\left( x,\tau\right) =U_{c}\left( x-x_{0}+x_{1}-c\tau\right)
,\ \ \ u^{\ast}\left( x,\tau\right)
\]
then$\ u^{\ast}\left( x,\tau\right) $ is a solution of$\ $the
equation$\ \partial u/\partial\tau=u^{p}\left( u_{xx}+u-u^{1-p}\right) $ on
the region
\[
Q=\left\{ \left( x,\tau\right) :x<x_{0}-x_{1}+c\tau,\ \tau>0\right\}
\]
and the difference
\begin{equation}
w\left( x,\tau\right) =u^{\ast}\left( x,\tau\right) -u\left(
x,\tau\right) =U_{c}\left( x-x_{0}+x_{1}-c\tau\right) -u\left(
x,\tau\right)
\end{equation}
satisfies a linear parabolic PDE of the form (see \cite{ANG}, p. 608)
\begin{equation}
\frac{\partial w}{\partial\tau}=a\left( x,\tau\right) w_{xx}+b\left(
x,\tau\right) w_{x}+c\left( x,\tau\right) w. \label{linear}
\end{equation}
We note that $w\left( x_{0},\tau_{0}\right) =0$ and on the
boundary$\ \partial Q\bigcap\left\{ \tau>0\right\} $ (i.e.,\ when
$x=x_{0}-x_{1}+c\tau$)\ we have
\[
w\left( x,\tau\right) =U_{c}\left( 0\right) -u\left( x,\tau\right) \geq
U_{c}\left( 0\right) -A>0.
\]
On the other part of $\partial Q,\ $i.e.,\ when $x<x_{0}-x_{1}\ $and $\tau=0$
we have
\[
w\left( x_{0}-x_{1},0\right) =U_{c}\left( 0\right) -u\left( x_{0}
-x_{1},0\right) >0
\]
and$\ w\left( x,0\right) =U_{c}\left( x-x_{0}+x_{1}\right) -u\left(
x,0\right) $ becomes negative as $x\rightarrow-\infty$ due to (\ref{Q4}).
Hence $w\left( x,0\right) $ must have at least one zero $y_{0}\ $on $\left(
-\infty,x_{0}-x_{1}\right) .\ $At any$\ $zero$\ y_{0}$ we have$\ $
\[
\frac{1}{A}\leq u\left( y_{0},0\right) =U_{c}\left( y_{0}-x_{0}
+x_{1}\right) \leq A
\]
and so
\[
w_{x}\left( y_{0},0\right) =U_{c}^{\prime}\left( y_{0}-x_{0}+x_{1}\right)
-u_{x}\left( y_{0},0\right) >A-u_{x}\left( y_{0},0\right) \geq0.
\]
Hence$\ w\left( x,0\right) $ cannot have more than one zero on the
interval$\ \left( -\infty,x_{0}-x_{1}\right) .\ $
By the Sturmian theorem, the number of zeros of $x\rightarrow w\left(
x,\tau\right) ,$ counted with multiplicity,\ cannot increase with time.\ Now
by our construction we have$\ w\left( x_{0},\tau_{0}\right) =0\ $and
since\ this is \emph{the only zero} of$\ w\left( \cdot,\tau_{0}\right)
,\ $we must have$\ w_{x}\left( x_{0},\tau_{0}\right) >0\ $(since $w\left(
x_{0}-x_{1}+c\tau_{0},\tau_{0}\right) >0$ and $w\left( -\infty,\tau
_{0}\right) <0$).\ Thus
\begin{equation}
u_{x}\left( x_{0},\tau_{0}\right) <U_{c}^{\prime}\left( x_{1}-c\tau
_{0}\right) \leq\lambda_{c}U_{c}\left( x_{1}-c\tau_{0}\right) =\lambda
_{c}u\left( x_{0},\tau_{0}\right) .
\end{equation}
By applying the same argument to$\ u\left( -x,\tau\right) $ one can also
obtain $-u_{x}\leq\lambda_{c}u,$ so that $\left\vert u_{x}\left(
x,\tau\right) \right\vert \leq\lambda_{c}u\left( x,\tau\right) $ for all
$\left( x,\tau\right) \in\in S_{m}^{1}\times\lbrack0,\infty).\ $The proof of
Theorem \ref{thm-type-one}\ is done.$
\square$
\subsection{Proof of type-one convergence.}
To go further we need to look more closely at the following\ ODE:
\begin{equation}
w^{\prime\prime}\left( x\right) +w\left( x\right) -w^{1-p}\left(
x\right) =0,\text{\ \ }\ x\in\left( -\infty,\infty\right) ,\ \ \ p>2.
\label{W-ODE}
\end{equation}
It is easy to see that any solution $w\left( x\right) $ to it is
\emph{positive everywhere} and\emph{ periodic} over $x\in\left(
-\infty,\infty\right) \ $(this property is valid for $p\geq2;\ $when
$p\in\left( 0,2\right) ,\ w\left( x\right) $ may have different behavior,
see \cite{LPT}\ and \cite{PT})$.\ $Let $a\leq1$ be the minimal value
of$\ w\left( x\right) $ on$\ \left( -\infty,\infty\right) $.\ Without loss
of generality, we may assume that $a=w\left( 0\right) \ $(and so $w^{\prime
}\left( 0\right) =0$)$\ $and by reflection$\ $(if $w\left( x\right) $ is a
solution, so is $w\left( -x\right) $)$\ w\left( x\right) $ must be
symmetric with respect to any local maximum point or minimum point. It also
satisfies the energy identity
\begin{equation}
\left( w^{\prime}\left( x\right) \right) ^{2}+w^{2}\left( x\right)
-\frac{2}{2-p}w^{2-p}\left( x\right) =F\left( a\right)
\ \ \ \text{for\ all}\ \ \ x\in\left( -\infty,\infty\right) \label{wF}
\end{equation}
where$\ F\left( a\right) =a^{2}-2\left( 2-p\right) ^{-1}a^{2-p}>0.\ $For
$p>2,\ $the$\ $convex\ positive\ function $F\left( s\right) =s^{2}-2\left(
2-p\right) ^{-1}s^{2-p}\ $decreases on $s\in\left( 0,1\right)
\ $with$\ \lim_{s\rightarrow0^{+}}F\left( s\right) =+\infty,$\ and increases
to $+\infty\ $on $\left( 1,\infty\right) $. Given $a\in(0,1],\ $there is a
unique $b\geq1\ $so that $F\left( a\right) =F\left( b\right) ,\ $where
$b=\max_{x\in\mathbb{R}}w\left( x\right) ,\ $and the minimal period
$T=2R\left( a\right) \ $of $w\left( x\right) $ is given by
\begin{equation}
T=2R\left( a\right) =2\int_{a}^{b}\frac{ds}{\sqrt{F\left( a\right)
-F\left( s\right) }}=2\int_{a}^{b}\frac{ds}{\sqrt{\left( a^{2}-\frac
{2}{2-p}a^{2-p}\right) -\left( s^{2}-\frac{2}{2-p}s^{2-p}\right) }
},\ \ \ F\left( b\right) =F\left( a\right) . \label{T}
\end{equation}
The above integral is improper near both$\ a$ and $b.\ $
It has been shown in Urbas \cite{U2} that
\begin{equation}
\lim_{a\rightarrow0^{+}}R\left( a\right) =\frac{\pi}{2},\ \ \ \lim
_{a\rightarrow1^{-}}R\left( a\right) =\frac{\pi}{\sqrt{p}},\ \ \ p\in\left(
2,\infty\right) . \label{ur}
\end{equation}
Moreover, by Corollary 5.6 of Andrews\ \cite{AN3},\ we know that $R\left(
a\right) $ is strictly decreasing in $a\in\left( 0,1\right) $ when
$p\in\left( 4,\infty\right) $\ and strictly increasing in $a\in\left(
0,1\right) $ when $p\in\left( 2,4\right) .\ $When $p=4,$ all solutions of
equation (\ref{W-ODE}) are $\pi$-periodic (see (\ref{wb}) also).\
\begin{remark}
\label{rmk1}As a comparison, when $p\in\left( 0,2\right) $ we have\ (see
\cite{U2} and \cite{AN3} again)
\[
\lim_{a\rightarrow0^{+}}R\left( a\right) =\frac{\pi}{p},\ \ \ \lim
_{a\rightarrow1^{-}}R\left( a\right) =\frac{\pi}{\sqrt{p}},\ \ \ p\in\left(
0,2\right)
\]
and $R\left( a\right) $ is strictly decreasing in $a\in\left( 0,1\right) $
when $p\in\left( 0,1\right) $ and strictly increasing in $a\in\left(
0,1\right) $ when $p\in\left( 1,2\right) .\ $When $p=1,\ $all solutions to
the ODE (\ref{W-ODE}) has period $2\pi.\ $
\end{remark}
One can also write the ODE (\ref{W-ODE})\ as a system
\begin{equation}
\dfrac{dw}{dx}=h,\ \ \ \dfrac{dh}{dx}=-w+w^{1-p},\ \ \ p>2.
\end{equation}
Then the vector field $V\left( w,h\right) =\left( h,-w+w^{1-p}\right) $
has only one equilibrium point $\left( 1,0\right) $ on the
half-plane$\ \left\{ w>0\right\} \ $and the eigenvalues of the linearization
at it are $\lambda=\pm\sqrt{p}i\ $(this matches with the second limit of
(\ref{ur})).\ The phase portrait of $V\ $on $\left\{ w>0\right\} \ $is a
family of \emph{closed orbits} $C\left( a\right) \ $centered at$\ \left(
1,0\right) \ $with period $2R\left( a\right) ,\ $where $a=\min
_{x\in\mathbb{R}}w\left( x\right) $.\ Thus the intersections of $C\left(
a\right) $ and the $w$-axis are $\left( a,0\right) \ $and $\left(
b,0\right) \ $with$\ F\left( a\right) =F\left( b\right) ,$ $w\left(
0\right) =a\leq1,\ w\left( R\left( a\right) \right) =b\geq1,\ w^{\prime
}\left( 0\right) =w^{\prime}\left( R\left( a\right) \right) =0.$
We can now state the following convergence theorem, which is a generalization
of Theorem A of \cite{ANG}:
\begin{theorem}
\label{thmA}(\emph{convergence of type-one blow-up for}\textsf{ }$p>2$)\ Let
$p>2$ and let $v\left( x,t\right) >0\ $be a \emph{type-one solution} of
$\left( \clubsuit\right) \ $defined on some maximal time interval
$[0,T_{\max})$.\ Then as $\tau\rightarrow\infty$ the rescaled solution
$u\left( x,\tau\right) ,$ given by (\ref{rescale}),\ converges in
$C^{\infty}\left( S_{m}^{1}\right) $ to a smooth\ positive $2m\pi$-periodic
function $w\left( x\right) ,\ $which is an entire solution of the ODE
\begin{equation}
w^{\prime\prime}\left( x\right) +w\left( x\right) -w^{1-p}\left(
x\right) =0\ \ \ \text{for all\ \ }\ x\in\mathbb{R}. \label{wode}
\end{equation}
\end{theorem}
\proof
Assume type-one blow-up of $v\left( x,t\right) $.\ For any sequence
$\tau_{n}\rightarrow\infty$ by Arzela-Ascoli theorem there is a subsequence,
which\ we also call it $\tau_{n}$, so that $u\left( x,\tau_{n}\right) $
converges uniformly on $S_{m}^{1}$ to a Lipschitz\ function $w\left(
x\right) \geq0$, which is $2m\pi$-periodic. By\ Theorem \ref{thm-type-one}
$,\ u\left( x,\tau\right) \ $has positive lower bound $e^{-2\lambda m\pi}$
for all $\tau\ $(see (\ref{KK1})),$\ $hence $w\left( x\right) \ $is strictly
positive everywhere.\ Now we can apply similar argument as in Proposition 12
and 14 of \cite{LPT} (since $p>2,$\ $u\left( x,\tau\right) \ $has positive
lower bound is essential in (32),\ p.160 of \cite{LPT}) to obtain the
conclusion that$\ w\left( x\right) $ satisfies the ODE (\ref{wode})
everywhere.\ By regularity theory for uniform parabolic equations,$\ w\left(
x\right) \ $is smooth\ and we have $C^{\infty}$ convergence of $u\left(
x,\tau_{n}\right) $ to $w\left( x\right) \ $as $\tau_{n}\rightarrow\infty$.
If we does not have full time convergence of $u\left( x,\tau\right) $
as$\ \tau\rightarrow\infty,\ $then there will exist two sequence of times
$\tau_{n}\rightarrow\infty$ and$\ \tilde{\tau}_{n}\rightarrow\infty$ such that
$u\left( x,\tau_{n}\right) \rightarrow w\left( x\right) $ and $u\left(
x,\tilde{\tau}_{n}\right) \rightarrow\tilde{w}\left( x\right) ,\ $where
$w,$ $\tilde{w}\ $are \emph{different}\ positive $2m\pi$-periodic solutions of
the ODE (\ref{wode}).\ Let $a,\ \tilde{a}\in(0,1]$ be the minimum values of
$w,\ \tilde{w}.\ $We may assume $a\leq\tilde{a}.\ $Note that although
$w\left( x\right) \ $is different from $\tilde{w}\left( x\right) $,
it\ may be possible to have $a=\tilde{a}$.\ By\ the above discussion, we
have$\ w\left( R\left( a\right) \right) =b\geq w\left( R\left( \tilde
{a}\right) \right) =\tilde{b}\in\lbrack1,\infty).$
If we have $a=\tilde{a},$ then $w\left( x\right) $ must be a translation of
$\tilde{w}\left( x\right) $ and we can find some $x_{0}\in\mathbb{R}$ with
$w^{\prime}\left( x_{0}\right) \tilde{w}^{\prime}\left( x_{0}\right)
<0\ $(i.e., they have different\ signs). This would contradict Proposition 23
of \cite{LPT}. Therefore we only have to consider the case $a<\tilde{a}.$
\ \ \ \ \
For $a<\tilde{a}$ there are two cases to discuss.\
\ \
\underline{Case 1}:\ $p\in\left( 2,\infty\right) ,\ p\neq4.$
\ \ \ \ \
By the discussion before Remark \ref{rmk1}, in such case we must have
$R\left( a\right) \neq R\left( \tilde{a}\right) \ $since $R\left(
a\right) $ is a monotone function in $a\in\left( 0,1\right) .\ $Now
$w\left( x\right) $ and $\tilde{w}\left( x\right) $ have
different\ periods and we can find some $x_{0}\in\mathbb{R}$ such that
$w^{\prime}\left( x_{0}\right) \tilde{w}^{\prime}\left( x_{0}\right)
<0.\ $We obtain the same contradiction due to Proposition 23 of \cite{LPT}.\
\ \ \ \
\underline{Case 2}:\ $p=4.$
\ \ \ \
In this case by (\ref{wb}),\ up to a translation, all solutions to the ODE
$w^{\prime\prime}+w-w^{-3}=0$ are $\pi$-periodic (see \cite{AN3}
,\ \cite{U2})\ and are given by (if $w(0)\geq1\ $is the maximum) the
$1$-parameter family of functions in (\ref{wb}). Unfortunately now for any
$x_{0}\in\mathbb{R}$ we have$\ w^{\prime}\left( x_{0}\right) \tilde
{w}^{\prime}\left( x_{0}\right) \geq0\ $and thus Proposition 23 of
\cite{LPT} is not applicable here.\ A different\ method has to be used
here\footnote{D.-H. Tsai would like to thank Prof. Matano\ for teaching him
the zero-number argument several years ago.\ It is now used in the proof of
Theorem \ref{thmA}.}.
$\ $Let $\mathcal{Z}\left[ w-\tilde{w}\right] $ denote the number of
zero\ points $\xi\in S_{m}^{1}\ $(or $\xi\in\lbrack-m\pi,m\pi)$)
with$\ w\left( \xi\right) -\tilde{w}\left( \xi\right) =0.$ Also let $Y$
denote the function space of all solutions of the ODE (\ref{wode}) on
$\mathbb{R\ }$(since now $p=4,\ $all solutions to the ODE (\ref{wode}) has
minimal period $\pi$)$.\ $For any $\xi\in S_{m}^{1}$ with $w\left(
\xi\right) -\tilde{w}\left( \xi\right) =0$, by uniqueness\ we must have
$w^{\prime}\left( \xi\right) \neq\tilde{w}^{\prime}\left( \xi\right) .$
Hence at each intersection point the graphs of the two functions
$w,\ \tilde{w}\ $are transversal.\
For any $z\left( x\right) \in Y,\ $the difference $u\left( x,\tau\right)
-z\left( x\right) $ satisfies a linear parabolic equation of the form
(\ref{linear}) (since $z\left( x\right) $ is also a solution to the PDE
(\ref{dudt})). By Angenent's result in p. 607 of \cite{ANG}\ (Lemma 2.4\ in
p.\ 165 of Chen-Matano \cite{CM} is more applicable here), the number
$\mathcal{Z}\left[ u\left( \cdot,\tau\right) -z\left( \cdot\right)
\right] $ is \emph{non-increasing} in time $\tau\in\left( 0,\infty\right)
.\ $Also note that we have the convergence of $u\left( x,\tau_{n}\right)
\ $to $w\left( x\right) $ in $C^{1},$ which implies
\[
\mathcal{Z}\left[ u\left( \cdot,\tau_{n}\right) -z\left( \cdot\right)
\right] =\mathcal{Z}\left[ w-\tilde{w}\right]
\]
for all large $n$ and all $z\in Y$ that are sufficiently close to $\tilde{w}$
in $C^{1}$ norm on $S_{m}^{1}.\ $In particular, we can conclude the
following:\ there exists a time $T>0\ $and a number $\delta>0\ $such that
\begin{equation}
\mathcal{Z}\left[ u\left( \cdot,\tau\right) -z\left( \cdot\right)
\right] =\mathcal{Z}\left[ w-\tilde{w}\right] \label{co}
\end{equation}
for all $\tau>T$ and all $z\in Y$ satisfying $\left\Vert z-\tilde
{w}\right\Vert _{C^{1}\left( S_{m}^{1}\right) }<\delta.\ $
The number $\mathcal{Z}\left[ u\left( \cdot,\tau\right) -z\left(
\cdot\right) \right] $ remains a constant for large time implies that the
function $x\rightarrow u\left( x,\tau\right) -z\left( x\right) $ does not
have a degenerate zero (i.e., multiple zero)\ in $S_{m}^{1}$ for any fixed
$\tau>T\ $(see \cite{CM}).$\ $But since $u\left( x,\tilde{\tau}_{n}\right) $
converges to $\tilde{w}\left( x\right) \ $in\ $C^{1}\left( S_{m}
^{1}\right) $ norm as $n\rightarrow\infty,$ the graph of the function
$x\rightarrow u\left( x,\tilde{\tau}_{n}\right) $ must be tangential to the
graph of some $z\in Y\ $satisfying $\left\Vert z-\tilde{w}\right\Vert
_{C^{1}\left( S_{m}^{1}\right) }<\delta.\ $For example, for fixed $x_{0}
\ $one can choose $z\left( x\right) $ to be the solution of
\begin{equation}
\left\{
\begin{array}
[c]{l}
z^{\prime\prime}\left( x\right) +z\left( x\right) -z^{-3}\left( x\right)
=0
\\
z\left( x_{0}\right) =u\left( x_{0},\tilde{\tau}_{n}\right)
,\ \ \ z^{\prime}\left( x_{0}\right) =u_{x}\left( x_{0},\tilde{\tau}
_{n}\right)
\end{array}
\right. \label{z}
\end{equation}
then as $n$ large enough, $z\left( x\right) $ will be close to $\tilde
{w}\left( x\right) $ in $C^{1}\left( S_{m}^{1}\right) $ since $u\left(
x_{0},\tilde{\tau}_{n}\right) \ $is close to $\tilde{w}\left( x_{0}\right)
\ $and $u_{x}\left( x_{0},\tilde{\tau}_{n}\right) $ is close to $\tilde
{w}^{\prime}\left( x_{0}\right) .$ Now $u\left( x,\tilde{\tau}_{n}\right)
-z\left( x\right) $ has a degenerate zero at $x_{0},\ $which is a
contradiction. \
\begin{remark}
Since $p=4,$\ $z\left( x\right) $ to the ODE (\ref{z}) has minimal period
$\pi.\ $In particular, it implies that $z\left( x\right) \in C^{1}\left(
S_{m}^{1}\right) $.
\end{remark}
The above contradiction for either Case 1 or Case 2 implies that $w\left(
x\right) \equiv\tilde{w}\left( x\right) $ and the proof is done.$
\square$
\ \
Theorem \ref{thmA}\ implies that for the contracting flow $\left(
\bigstar\right) $, if $k_{\max}\left( t\right) \left( T_{\max}-t\right)
^{1/\left( \alpha+1\right) }$ remains bounded as $t\rightarrow T_{\max},$
then the rescaled curvature
\[
K\left( x,\tau\right) =\left( p^{1/p}T_{\max}^{1/p}e^{-\tau}\right)
^{1/\alpha}k\left( x,T_{\max}\left( 1-e^{-p\tau}\right) \right)
,\ \ \ \alpha\in(0,1],\ \ \ \tau\in\lbrack0,\infty)
\]
converges in $C^{\infty}$ to a positive $K\left( x\right) \in C^{\infty
}\left( S_{m}^{1}\right) ,\ $which satisfies the ODE
\begin{equation}
\left( K^{\alpha}\right) ^{\prime\prime}\left( x\right) +K^{\alpha}\left(
x\right) -\frac{1}{K\left( x\right) }=0\ \ \ \text{for all\ \ \ }
x\in\mathbb{R}. \label{kk-ode}
\end{equation}
Geometrically this says that the evolving convex immersed closed curve
$\gamma_{t}$ shrinks to a point in an asymptotically self-similar way.\
\section{Type-two blow-up.}
We now turn to the much more difficult type-one blow-up. We point out that in
the proof of Theorem \ref{thm-type-one}, the integral condition
(\ref{integral-cond}) does not come into play at all. Hence even it is not
satisfied, Theorem \ref{thm-type-one} still holds.\ In view of this, we have
the following interesting observation:
\begin{lemma}
\label{lem3}(\textsf{existence of type-two blow-up for }$p\geq2$)\ Assume
$v_{0}\left( x\right) >0\in C^{\infty}\left( S_{m}^{1}\right) $ in
$\left( \clubsuit\right) $ does not satisfy (\ref{integral-cond}), i.e.,
\begin{equation}
\int_{S_{m}^{1}}v_{0}^{1-p}\left( x\right) e^{ix}dx\neq0 \label{not-zero}
\end{equation}
then we have\ type-two blow-up\textsf{ }for the solution $v\left( x,t\right)
$ to $\left( \clubsuit\right) $,\textsf{ }which means
\begin{equation}
\limsup_{t\rightarrow T_{\max}}\left( v_{\max}\left( t\right) \left(
T_{\max}-t\right) ^{1/p}\right) =\infty. \label{type-1}
\end{equation}
\end{lemma}
\proof
Without loss of generality we may assume
\[
\int_{S_{m}^{1}}v_{0}^{1-p}\left( x\right) \cos xdx>0.
\]
Since we have
\begin{equation}
\int_{S_{m}^{1}}v^{1-p}\left( x,t\right) e^{ix}dx=\int_{S_{m}^{1}}
v_{0}^{1-p}\left( x\right) e^{ix}dx
\end{equation}
for all $t\in\lbrack0,T_{\max}),$ $u\left( x,\tau\right) $ satisfies$\ $
\[
\lim_{\tau\rightarrow\infty}\int_{S_{m}^{1}}u^{1-p}\left( x,\tau\right) \cos
xdx=\lim_{\tau\rightarrow\infty}\left( p^{1/p}T_{\max}^{1/p}e^{-\tau}\right)
^{1-p}\int_{S_{m}^{1}}v_{0}^{1-p}\left( x\right) \cos xdx=\infty
\]
which means that $\liminf_{\tau\rightarrow\infty}u_{\min}\left( \tau\right)
=0.\ $If we have\ type-one blow-up, then Theorem \ref{thm-type-one} would
imply\ a positive lower bound of $u_{\min}\left( \tau\right) ,$ a
contradiction.$
\square$
\begin{remark}
Thus for $p\geq2,\ $\textsf{type-two blow-up} \emph{in equation}$\ \left(
\clubsuit\right) $\emph{ is generic}.\ Moreover, type-one blow-up occurs only
when the initial data satisfies the integral condition (\ref{integral-cond}).
\end{remark}
\begin{remark}
When (\ref{integral-cond})\ is satisfied, then either type-one or type-two
blow-up can happen.\ For type-one, just take a separable solution of $\left(
\clubsuit\right) $\ of the form $v\left( x,t\right) =h\left( t\right)
g\left( x\right) ,$ where $g\left( x\right) >0$ on $S_{m}^{1}$ satisfies
the ODE\ (\ref{wode}) and $h\left( t\right) $ satisfies$\ dh/dt=h^{1+p}
,\ h\left( 0\right) >0.\ $For type-two, choose a convex immersed plane curve
with one big loop and one tiny loop. Then the corresponding evolution will
become singular without shrinking to a point in an asymptotically self-similar
way. Hence we obtain a type-two blow-up. The difficulty lies in the estimate
of blow-up rate.\
\end{remark}
\subsection{A special symmetric case for type-two blow-up and convergence.}
In this section we assume the initial data $v_{0}\left( x\right) >0$ to
equation $\left( \clubsuit\right) $ satisfies (\ref{integral-cond})\ and the
following \emph{symmetric condition }
\begin{equation}
v_{0}\left( x\right) =v_{0}\left( -x\right) \ \ \ \text{and \ \ }
v_{0}^{\prime}\left( x\right) <0,\ \ \ \ \ \text{for all\ \ \ }x\in\left(
0,m\pi\right) . \label{v0}
\end{equation}
If $v\left( x,t\right) $ is a solution to $\left( \clubsuit\right) $ with
the above\ initial data $v_{0}\left( x\right) $ then $\tilde{v}\left(
x,t\right) :=v\left( -x,t\right) $ is also a solution to $\left(
\clubsuit\right) $ with $\tilde{v}\left( x,0\right) =v_{0}\left(
-x\right) =v_{0}\left( x\right) $ for all $x\in S_{m}^{1}.$ By uniqueness
we must have
\[
\ v\left( x,t\right) =v\left( -x,t\right) \ \ \ \text{for all\ \ \ }
\left( x,t\right) \in\left( 0,m\pi\right) \times\lbrack0,T_{\max})
\]
which also implies
\begin{equation}
v_{x}\left( 0,t\right) =v_{x}\left( m\pi,t\right) =0\ \ \ \text{for
all\ \ \ }t\in\lbrack0,T_{\max}). \label{v1}
\end{equation}
Also the second condition of (\ref{v0}) implies that $v_{0}^{\prime}\left(
x\right) $ has exactly two zeros on $S_{m}^{1}$ and since the number of zeros
for $v_{x}\left( x,t\right) $ is nonincreasing in time, we must have
\begin{equation}
v_{x}\left( x,t\right) <0\ \ \ \text{for all\ \ \ }\left( x,t\right)
\in\left( 0,m\pi\right) \times\lbrack0,T_{\max}). \label{decr}
\end{equation}
Hence the two conditions of (\ref{v0}) are preserved for all time.\ In
particular, we have $v_{\max}\left( t\right) =v\left( 0,t\right) \ $for
$t\in\lbrack0,T_{\max}).\ $
The main result in this section is the following convergence behavior for
type-two blow-up. One can view it as a partial generalization of Theorem C\ of
\cite{ANG} to the case $p\geq2$ since\ here we assume $v_{0}\left( x\right)
\ $is symmetric\ and our convergence is only uniform,\ weaker than Angenent's
$C^{\infty}$ convergence. However, the advantage of focusing on the symmetric
case (\ref{v0})\ is that we always have \emph{type-two} blow-up and the proof
of convergence in Theorem \ref{thmC}\ below\ is very simple and straightforward.
\begin{lemma}
\label{lem4}Assume $v\left( x,t\right) \ $is a positive solution of $\left(
\clubsuit\right) $ in $S_{m}^{1}$ (with $m\geq2$) where\ $v_{0}\left(
x\right) $ satisfies (\ref{integral-cond}) and (\ref{v0}).\ Then$\ v\left(
x,t\right) \ $has \textbf{type-two} blow-up.
\end{lemma}
\proof
Basically, we follow the arguments in p. 630 of \cite{ANG}.\ If $v\left(
x,t\right) \ $has type-one blow-up, then by (\ref{KK1}) we have
\begin{equation}
v\left( x,t\right) \rightarrow\infty\ \text{as\ }t\rightarrow T_{\max
}\ \ \ \text{for all\ \ \ }x\in\left[ -m\pi,m\pi\right] \text{.}
\label{v-go}
\end{equation}
That is, the blow-up set of $v\left( x,t\right) \ $is the whole domain.\
When $m\ $is even,\ $m=2k,\ k\geq1,\ $consider the function
\[
D\left( t\right) =\int_{0}^{\left( 2k-1\right) \pi}v^{1-p}\left(
x,t\right) \cos xdx.
\]
By (\ref{v-go}), we have
\begin{equation}
D\left( t\right) \rightarrow0\ \ \ \text{as}\ \ \ t\rightarrow T_{\max}.
\label{Dt}
\end{equation}
Now by $\left( \clubsuit\right) $ we compute
\begin{align*}
D^{\prime}\left( t\right) & =\left( 1-p\right) \int_{0}^{\left(
2k-1\right) \pi}\left( v_{xx}\left( x,t\right) +v\left( x,t\right)
\right) \cos xdx\\
& =\left( 1-p\right) \left( v\left( x,t\right) \sin x+v_{x}\left(
x,t\right) \cos x\right) \mid_{0}^{\left( 2k-1\right) \pi}=\left(
p-1\right) v_{x}\left( \left( 2k-1\right) \pi,t\right) <0
\end{align*}
due to (\ref{v1})\ and (\ref{decr}). Hence $D\left( t\right) $ is decreasing
and by (\ref{Dt}), it is positive for all $t\in\lbrack0,T_{\max}).\ $Also the
symmetry of $v\left( x,t\right) \ $implies that
\[
\int_{0}^{2k\pi}v^{1-p}\left( x,t\right) \cos xdx=\frac{1}{2}\int_{-2k\pi
}^{2k\pi}v^{1-p}\left( x,t\right) \cos xdx=0.
\]
Thus we have
\[
0=\int_{0}^{\left( 2k-1\right) \pi}v^{1-p}\left( x,t\right) \cos
xdx+\int_{\left( 2k-1\right) \pi}^{2k\pi}v^{1-p}\left( x,t\right) \cos
xdx
\]
and then
\[
\int_{\left( 2k-1\right) \pi}^{2k\pi}v^{1-p}\left( x,t\right) \cos
xdx<0\ \ \ \text{for all\ \ \ }t\in\lbrack0,T_{\max}).
\]
However, by (\ref{decr})$\ $we have$\ $
\[
\int_{\left( 2k-1\right) \pi}^{2k\pi}v^{1-p}\left( x,t\right) \cos
xdx>0\ \ \ \text{for all\ \ \ }t\in\lbrack0,T_{\max}),
\]
which gives a contradiction.
When $m\ $is odd,\ $m=2k+1,\ k\geq1,\ $we consider the function
\[
D\left( t\right) =\int_{0}^{2k\pi}v^{1-p}\left( x,t\right) \cos xdx
\]
and again by (\ref{v-go}), we have (\ref{Dt}). Now
\begin{align*}
D^{\prime}\left( t\right) & =\left( 1-p\right) \int_{0}^{2k\pi}\left(
v_{xx}\left( x,t\right) +v\left( x,t\right) \right) \cos xdx\\
& =\left( 1-p\right) \left( v\left( x,t\right) \sin x+v_{x}\left(
x,t\right) \cos x\right) \mid_{0}^{2k\pi}=\left( 1-p\right) v_{x}\left(
2k\pi,t\right) >0
\end{align*}
and so $D\left( t\right) $ is increasing and therefore negative for all
time. By symmetry again we obtain
\[
\int_{0}^{\left( 2k+1\right) \pi}v^{1-p}\left( x,t\right) \cos
xdx=\frac{1}{2}\int_{-\left( 2k+1\right) \pi}^{\left( 2k+1\right) \pi
}v^{1-p}\left( x,t\right) \cos xdx=0\
\]
and thus
\[
\int_{2k\pi}^{\left( 2k+1\right) \pi}v^{1-p}\left( x,t\right) \cos
xdx>0\ \ \ \text{for all\ \ \ }t\in\lbrack0,T_{\max}).
\]
However, by (\ref{decr})$\ $we have$\ $
\[
\int_{2k\pi}^{\left( 2k+1\right) \pi}v^{1-p}\left( x,t\right) \cos
xdx<0\ \ \ \text{for all\ \ \ }t\in\lbrack0,T_{\max}),
\]
which gives a contradiction.$
\square$
\begin{theorem}
\label{thmC}(\textsf{convergence of type-two blow-up for }$p\geq
2\ $\textsf{with symmetric} $v_{0}\left( x\right) $) Let$\ \Phi\left(
x\right) =\cos x\ $on $\left[ -\pi/2,\pi/2\right] \ $and $\Phi\left(
x\right) =0\ $otherwise.\ Assume $v_{0}\left( x\right) >0\in C^{\infty
}\left( S_{m}^{1}\right) $ satisfies condition\ (\ref{integral-cond})\ and
(\ref{v0}).$\ $Then there exists a sequence of times $t_{n}\nearrow T_{\max}$
such that
\begin{equation}
\lim_{n\rightarrow\infty}\frac{v\left( x,t_{n}\right) }{v\left(
0,t_{n}\right) }=\Phi\left( x\right) \ \ \ \text{uniformly on\ \ \ }
x\in\left[ -m\pi,m\pi\right] . \label{cos}
\end{equation}
\end{theorem}
\begin{remark}
Note that if we have type-one blow-up, then\ we consider the rescaling
$v\left( x,t\right) /R\left( t\right) ,\ $where $R\left( t\right)
\ $is\ comparable\ to $v_{\max}\left( t\right) .$ Hence here for type-two
blow-up, by analogy, it is reasonable to look at the rescaling$\ v\left(
x,t\right) /v_{\max}\left( t\right) ,$ which is (\ref{cos}).
\end{remark}
\proof
By Lemma \ref{lem4}, $v\left( x,t\right) \ $has type-two blow-up and so
$\left( T_{\max}-t\right) ^{1/p}v_{\max}\left( t\right) \ $is not
bounded\ on$\ t\in\lbrack0,T_{\max}).\ $Hence there exists a sequence
$s_{n}\nearrow T_{\max}$ such that
\begin{equation}
\lim_{n\rightarrow\infty}\left( T_{\max}-s_{n}\right) ^{1/p}v_{\max}\left(
s_{n}\right) =\infty. \label{59}
\end{equation}
Let
\begin{equation}
\psi_{n}\left( x\right) =\frac{1}{T_{\max}-s_{n}}\int_{s_{n}}^{T_{\max}
}\frac{v\left( x,s\right) }{v\left( 0,s\right) }ds,\ \ \ x\in\left[
-m\pi,m\pi\right] . \label{60}
\end{equation}
As we shall be interested in the behavior of $\psi_{n}\left( x\right) $ for
$n$ large, without loss of generality, we may assume that $v_{\max}\left(
t\right) =v\left( 0,t\right) \ $is increasing in time for all $t\in
\lbrack0,T_{\max})$ (see Lemma \ref{lem2}) and by (\ref{vvv})\ we have
\begin{equation}
0<\psi_{n}\left( x\right) \leq1\ \ \ \text{and\ \ }\ \left\vert \psi
_{n}^{\prime}\left( x\right) \right\vert \leq1\ \ \ \text{for all\ \ \ }
x\in\left[ -m\pi,m\pi\right] \label{psi}
\end{equation}
for all $n.\ $We also have$\ \psi_{n}\left( x\right) =\psi_{n}\left(
-x\right) \ $for all $x\in\left[ 0,m\pi\right] \ $and $n.\ $Moreover we
have for all $n\ $that
\begin{equation}
\psi_{n}^{\prime}\left( x\right) <0\ \ \ \text{for all\ \ \ }x\in\left(
0,m\pi\right) . \label{PSI}
\end{equation}
Let $0<K<\pi/2\ $be a fixed number but arbitrary.\ By (\ref{vvcos}) we know
that when $t$ is close to $T_{\max},\ $there holds
\begin{equation}
v\left( x,t\right) \geq v_{\max}\left( t\right) \cos x=v\left(
0,t\right) \cos x\ \ \ \text{for\ all}\ \ \ x\in\left[ -K,K\right] .
\label{vvvcos}
\end{equation}
and so$\ $
\begin{equation}
\lim_{t\rightarrow T_{\max}}v\left( x,t\right) =\infty\ \ \ \text{for\ all}
\ \ \ x\in\left[ -K,K\right] . \label{vvinf}
\end{equation}
Moreover, by Lemma \ref{lem2}, we also have
\[
\left( v_{xx}+v\right) \left( x,t\right) >0,\ \ \ x\in\left[ -K,K\right]
\]
when $t$ is close to $T_{\max}.\ $As a consequence, when $n$ is large,$\ $we
have
\begin{align}
0 & <\frac{1}{T_{\max}-s_{n}}\int_{s_{n}}^{T_{\max}}\frac{v_{xx}\left(
x,s\right) +v\left( x,s\right) }{v\left( 0,s\right) }ds=\frac{1}{T_{\max
}-s_{n}}\int_{s_{n}}^{T_{\max}}\frac{v_{s}\left( x,s\right) }{v^{p}\left(
x,s\right) v\left( 0,s\right) }ds\nonumber\\
& \leq\frac{1}{T_{\max}-s_{n}}\int_{s_{n}}^{T_{\max}}\frac{v_{s}\left(
x,s\right) }{v^{p+1}\left( x,s\right) }ds=\frac{1}{p\left( T_{\max}
-s_{n}\right) v^{p}\left( x,s_{n}\right) }\ \ \ \text{for all\ \ \ }
x\in\left[ -K,K\right] . \label{psi-2}
\end{align}
By (\ref{psi}), we may assume that $\psi_{n}\left( x\right) $ converges
uniformly on $S_{m}^{1}\ $to a some$\ w\left( x\right) \in C^{0}\left(
S_{m}^{1}\right) \ $and $w\left( x\right) \geq0\ $in$\ S_{m}^{1}.$ For any
test function $\varphi\in C_{0}^{\infty}\left( -\pi/2,\pi/2\right) $, choose
$0<K<\pi/2\ $so that $\left( -K,K\right) \ $contains the support of
$\varphi.\ $By Fubini theorem and integration by parts we have
\begin{align}
& \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}w\left( x\right) \left[
\varphi_{xx}\left( x\right) +\varphi\left( x\right) \right]
dx=\lim_{n\rightarrow\infty}\int_{-K}^{K}\psi_{n}\left( x\right) \left[
\varphi_{xx}\left( x\right) +\varphi\left( x\right) \right] dx\nonumber\\
& =\lim_{n\rightarrow\infty}\int_{-K}^{K}\left[ \left( \frac{1}{T_{\max
}-s_{n}}\int_{s_{n}}^{T_{\max}}\frac{v\left( x,s\right) }{v\left(
0,s\right) }ds\right) \left[ \varphi_{xx}\left( x\right) +\varphi\left(
x\right) \right] \right] dx\nonumber\\
& =\lim_{n\rightarrow\infty}\frac{1}{T_{\max}-s_{n}}\int_{s_{n}}^{T_{\max}
}\left[ \int_{-K}^{K}\frac{v\left( x,s\right) }{v\left( 0,s\right)
}\left[ \varphi_{xx}\left( x\right) +\varphi\left( x\right) \right]
dx\right] ds\nonumber\\
& =\lim_{n\rightarrow\infty}\frac{1}{T_{\max}-s_{n}}\int_{s_{n}}^{T_{\max}
}\left[ \int_{-K}^{K}\frac{v_{xx}\left( x,s\right) +v\left( x,s\right)
}{v\left( 0,s\right) }\varphi\left( x\right) dx\right] ds\nonumber\\
& =\lim_{n\rightarrow\infty}\int_{-K}^{K}\left[ \frac{1}{T_{\max}-s_{n}}
\int_{s_{n}}^{T_{\max}}\frac{v_{xx}\left( x,s\right) +v\left( x,s\right)
}{v\left( 0,s\right) }ds\right] \varphi\left( x\right) dx=0
\end{align}
due to (\ref{59}),\ (\ref{vvvcos}) and\ (\ref{psi-2}). This implies that
$w\left( x\right) $ is a weak solution of the ODE $w_{xx}+w=0$ in $\left(
-\pi/2,\pi/2\right) \ $(note that since $\left\vert \psi_{n}^{\prime}\left(
x\right) \right\vert $ is uniformly bounded, the function $w\ $is Lipschitz
continuous\ with $w\in W^{1,2}\left( S_{m}^{1}\right) $))$.$ Regularity
theory implies that $w\left( x\right) $ is smooth\ in $x\in\left(
-\pi/2,\pi/2\right) $ with $w_{xx}+w=0.$ By our definition, $\psi_{n}\left(
x\right) $ is decreasing in $x$ for $x\in\left( 0,m\pi\right) $ and has a
maximum at $x=0$ with $\psi_{n}\left( 0\right) =1.$ This implies that
$w\left( x\right) $ is decreasing for $x\in\left( 0,m\pi\right) $ and has
a maximum at $x=0.\ $Hence $w\left( 0\right) =1,$ $w^{\prime}\left(
0\right) =0,$\ and therefore $w\left( x\right) =\cos x$ for $x\in\left(
-\pi/2,\pi/2\right) .$
By Lemma \ref{lem2-1}, we have for large $n$
\begin{align}
0 & \leq\frac{1}{T_{\max}-s_{n}}\int_{s_{n}}^{T_{\max}}\int_{-K}^{K}\left(
\frac{v\left( x,s\right) }{v\left( 0,s\right) }-\cos x\right)
dxds\nonumber\\
& =\int_{-K}^{K}\frac{1}{T_{\max}-s_{n}}\int_{s_{n}}^{T_{\max}}\left(
\frac{v\left( x,s\right) }{v\left( 0,s\right) }-\cos x\right)
dsdx\nonumber\\
& =\int_{-K}^{K}\left( \psi_{n}\left( x\right) -\cos x\right)
dx\rightarrow0\ \ \ \text{as\ \ \ }n\rightarrow\infty. \label{100}
\end{align}
Hence if we let
\[
f\left( s\right) =\int_{-K}^{K}\left( \frac{v\left( x,s\right) }{v\left(
0,s\right) }-\cos x\right) dx,\ \ \ s\in\lbrack0,T_{\max})
\]
we would have
\[
0\leq\frac{1}{T_{\max}-s_{n}}\int_{s_{n}}^{T_{\max}}f\left( s\right)
ds\rightarrow0\ \ \ \text{as\ \ \ }n\rightarrow\infty.
\]
Therefore by mean value theorem we can find a sequence $s_{n}^{\prime}
,\ s_{n}<s_{n}^{\prime}<T_{\max},$ so that
\[
\int_{-K}^{K}\left( \frac{v\left( x,s_{n}^{\prime}\right) }{v\left(
0,s_{n}^{\prime}\right) }-\cos x\right) dx\rightarrow0\ \ \ \text{as\ \ \ }
n\rightarrow\infty.
\]
Note that both $\cos x$ and $v\left( x,s_{n}^{\prime}\right) /v\left(
0,s_{n}^{\prime}\right) $ are bounded functions with bounded derivatives (and
their bounds are independent\ of $n$), and also
\begin{equation}
\frac{v\left( x,s_{n}^{\prime}\right) }{v\left( 0,s_{n}^{\prime}\right)
}-\cos x\geq0\ \ \ \text{on\ \ \ }\left[ -K,K\right] \label{KKK}
\end{equation}
for large $n.$ Thus\ by Arzela-Ascoli theorem we must have $v\left(
x,s_{n}^{\prime}\right) /v\left( 0,s_{n}^{\prime}\right) \rightarrow\cos x$
(passing to a subsequence if necessary)\ uniformly on $\left[ -K,K\right] $
as $n\rightarrow\infty.\ $
Let $K_{j}$ be a sequence with$\ K_{j}\rightarrow\pi/2$ as $j\rightarrow
\infty.$ For each $j,$ there is a sequence $s_{n}^{\left( j\right) }$ so
that $v\left( x,s_{n}^{\left( j\right) }\right) /v\left( 0,s_{n}^{\left(
j\right) }\right) \rightarrow\cos x$ uniformly on $\left[ -K_{j}
,K_{j}\right] $ as $j\rightarrow\infty.$ By a diagonal argument, there is a
sequence $\lambda_{n}\nearrow T_{\max}$ such that $v\left( x,\lambda
_{n}\right) /v\left( 0,\lambda_{n}\right) \ $converges uniformly to $\cos
x\ $on $\left[ -K,K\right] $ for any $0<K<\pi/2.\ $
To obtain the convergence (\ref{cos})\ on $\left[ -\pi/2,\pi/2\right] $, we
argue as follows (for convenience, any further subsequence of $\lambda_{n}$ is
still denoted as $\lambda_{n}$).\ Assume $v\left( x,\lambda_{n}\right)
/v\left( 0,\lambda_{n}\right) \ $does not converge uniformly to $\cos x\ $on
$\left[ -\pi/2,\pi/2\right] $. Then there exist\ $\varepsilon>0,\ $a
sequence of points $x_{n}\in\left[ -\pi/2,\pi/2\right] ,$\ and a time
subsequence $\lambda_{n},$ so that
\begin{equation}
f\left( x_{n},\lambda_{n}\right) :=\frac{v\left( x_{n},\lambda_{n}\right)
}{v\left( 0,\lambda_{n}\right) }-\cos x_{n}\geq\varepsilon\ \ \ \text{for
all\ \ \ }n. \label{f}
\end{equation}
By the above discussion we may assume that $x_{n}\rightarrow\pi/2.\ $Now by
mean value theorem and (\ref{vvv})
\begin{align}
\varepsilon & <\frac{v\left( x_{n},\lambda_{n}\right) }{v\left(
0,\lambda_{n}\right) }\leq\frac{\left\vert v\left( x_{n},\lambda_{n}\right)
-v\left( x_{n}-\varepsilon/100,\lambda_{n}\right) \right\vert }{v\left(
0,\lambda_{n}\right) }+\frac{v\left( x_{n}-\varepsilon/100,\lambda
_{n}\right) }{v\left( 0,\lambda_{n}\right) }\nonumber\\
& \leq\frac{\varepsilon}{100}+\frac{v\left( x_{n}-\varepsilon/100,\lambda
_{n}\right) }{v\left( 0,\lambda_{n}\right) } \label{ff}
\end{align}
where $v\left( x_{n}-\varepsilon/100,\lambda_{n}\right) /v\left(
0,\lambda_{n}\right) \rightarrow\cos\left( \pi/2-\varepsilon/100\right)
\ $as $n\rightarrow\infty.\ $We have got a contradiction.
Since $v\left( x,\lambda_{n}\right) /v\left( 0,\lambda_{n}\right) \ $is
decreasing in $x\in\left( 0,m\pi\right) \ $for each time\ $\lambda_{n}$,\ it
must converge to zero uniformly outside the interval $\left[ -\pi
/2,\pi/2\right] .\ $The proof of Theorem \ref{thmC} is\ done.$
\square\ $
\ \ \ \ \ \ \ \
We next want to improve Theorem \ref{thmC} and show that the convergence in
(\ref{cos})\ is valid for all $t\rightarrow T_{\max},\ $not just along a
sequence of times $t_{n}\nearrow T_{\max}.\ $In below, we basically follow
similar arguments as in Lemmas 4.4, 4.5, and 4.6 of Friedman-McLeod\ \cite{FM}
\ and look more closely at the solution behavior. These estimates are
interesting on their own also.
In the following we still assume that the initial data$\ v_{0}\left(
x\right) $ satisfies the symmetric condition (\ref{v0}).
\begin{lemma}
\label{lem-new-1}If $x\in\left( \pi/2,\pi\right) ,\ $then
\begin{equation}
\frac{d}{dt}\int_{0}^{x}v^{1-p}\left( y,t\right) \cos ydy<0\ \ \ \text{for
all\ \ \ }t\in\lbrack0,T_{\max}). \label{new1}
\end{equation}
\end{lemma}
\proof
We proceed as in \cite{FM}, Lemma 4.5.\ By direct computation, we have
\[
\frac{d}{dt}\int_{0}^{x}v^{1-p}\left( y,t\right) \cos ydy=\left(
1-p\right) \left[ v_{x}\left( x,t\right) \cos x+v\left( x,t\right) \sin
x\right] <0
\]
since for $x\in\left( \pi/2,\pi\right) $ we have $v_{x}\left( x,t\right)
<0,\ \cos x<0,\ \sin x>0.
\square$
\begin{lemma}
\label{lem-new-2}If $x>\pi/2\ $or $x<-\pi/2,$ then there exists a constant $C$
depending on $x$ such that
\begin{equation}
0<v\left( x,t\right) \leq C\ \ \ \text{for all\ \ \ }t\in\lbrack0,T_{\max})
\label{new2}
\end{equation}
i.e.,\ $v\left( x,t\right) $ does not blow up for $\left\vert x\right\vert
>\pi/2.$
\end{lemma}
\proof
We proceed as in \cite{FM}, Lemma 4.6.\ Since$\ v\left( x,t\right) $ is
decreasing in $x\in\left( 0,m\pi\right) \ $for all time,$\ $without loss of
generality, we may just look at the case $x\in\left( \pi/2,\pi\right)
.\ $Suppose$\ v\left( x,t\right) \ $is not bounded, then there exists a
sequence $t_{n}\nearrow T_{\max}$ so that $v\left( x,t_{n}\right)
\rightarrow\infty.$ By Lemma \ref{lem2} we must have $v\left( x,t\right)
\rightarrow\infty$ as $t\rightarrow T_{\max}.\ $In particular, we have (note
that $v\left( y,t\right) $ is decreasing for $y>0$)
\begin{equation}
\int_{0}^{x}v^{1-p}\left( y,t\right) \cos ydy\rightarrow
0\ \ \ \text{as\ \ \ }t\rightarrow T_{\max}. \label{new3}
\end{equation}
On the other hand, we may write for fixed small $\delta>0$
\begin{align}
& \int_{0}^{x}v^{1-p}\left( y,t\right) \cos ydy\nonumber\\
& =\int_{0}^{\left( \pi-\delta\right) /2}v^{1-p}\left( y,t\right) \cos
ydy+\int_{\left( \pi-\delta\right) /2}^{\left( \pi+\delta\right)
/2}v^{1-p}\left( y,t\right) \cos ydy+\int_{\left( \pi+\delta\right)
/2}^{x}v^{1-p}\left( y,t\right) \cos ydy. \label{new4}
\end{align}
As$\ v\left( x,t\right) >0$ is decreasing in $x\in\left( 0,m\pi\right) ,$
the second term in (\ref{new4})\ is negative for all time$.\ $Also by
(\ref{vvcos})\ in Lemma \ref{lem2-1}, there is a constant $c>0\ $such that
$v\left( y,t\right) \geq cv\left( 0,t\right) $ for all $y\in\left[
0,\left( \pi-\delta\right) /2\right] \ $and all time large enough.\ Finally
for $y\in\left[ \left( \pi+\delta\right) /2,x\right] ,\ $by Theorem
\ref{thmC}\ there exists a\ sequence $t_{n}\nearrow T_{\max}\ $such that
$v\left( y,t_{n}\right) \leq\varepsilon_{n}v\left( 0,t_{n}\right) ,$ where
$\varepsilon_{n}\rightarrow0\ $as $n\rightarrow\infty.\ $Hence$\ $for\ $n$
large enough we conclude
\begin{align*}
& \int_{0}^{x}v^{1-p}\left( y,t_{n}\right) \cos ydy\\
& \leq c^{1-p}v^{1-p}\left( 0,t_{n}\right) \int_{0}^{\left( \pi
-\delta\right) /2}\cos ydy+\varepsilon_{n}^{1-p}v^{1-p}\left( 0,t_{n}
\right) \int_{\left( \pi+\delta\right) /2}^{x}\cos ydy<0.
\end{align*}
This gives a contradiction due to (\ref{new1}) and (\ref{new3}).
The proof for the case $x<-\pi/2$ is similar. $
\square$
\begin{lemma}
\label{lem-new-3}Let $w\left( x\right) $ be a nonnegative\ Lipschitz
function defined on $\left[ -\pi/2,\pi/2\right] .\ $Suppose that
$w\ $satisfies the inequality $w_{xx}+w\geq0\ $on$\ \left( -\pi
/2,\pi/2\right) $ in the sense of distribution. If $w\left( -\pi/2\right)
=w\left( \pi/2\right) =0,$ then $w\left( x\right) =a\cos x,\ $where
$a=\max_{x\in\left[ -\pi/2,\pi/2\right] }w\left( x\right) .$
\end{lemma}
\proof
Let $\varphi_{n}$ be a sequence of smooth nonnegative functions with compact
support in $\left( -\pi/2,\pi/2\right) \ $such that it converges to $w$ in
$H_{0}^{1}\left[ -\pi/2,\pi/2\right] \ $(note that$\ 0\leq w\in H_{0}
^{1}\left[ -\pi/2,\pi/2\right] $)$.\ $Since $w\ $satisfies $w_{xx}+w\geq
0\ $on$\ \left( -\pi/2,\pi/2\right) $ in the sense of distribution, we have
\[
\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left( \varphi_{n}^{\prime}\left(
x\right) w^{\prime}\left( x\right) -\varphi_{n}\left( x\right) w\left(
x\right) \right) dx\leq0\ \ \ \text{for all\ \ \ }n.
\]
Letting $n\rightarrow\infty$ we get
\begin{equation}
\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left( \left( w^{\prime}\left(
x\right) \right) ^{2}-w^{2}\left( x\right) \right) dx\leq0. \label{ww1}
\end{equation}
Note that $\lambda=1$ is the principal eigenvalue of the operator
$d^{2}/dx^{2}$ on the interval$\ \left[ -\pi/2,\pi/2\right] \ $with $\cos x$
the principal eigenfunction satisfying Dirichlet boundary condition. Thus we
obtain
\begin{equation}
\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}w^{2}\left( x\right) dx\leq\int
_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left( w^{\prime}\left( x\right) \right)
^{2}dx \label{ww2}
\end{equation}
where equality holds only when $w$ is a constant multiple of the principal
eigenfunction.\ Equations (\ref{ww1}) and (\ref{ww2})\ imply that $w$ is a
principal eigenfunction on the interval $\left[ -\pi/2,\pi/2\right] $ and so
$w_{xx}+w=0\ $on$\ \left( -\pi/2,\pi/2\right) .\ $Let $a=\max_{x\in\left[
-\pi/2,\pi/2\right] }w\left( x\right) .$ Then we conclude that $w=a\cos
x\ $on$\ \left[ -\pi/2,\pi/2\right] .
\square$
\begin{theorem}
\label{thmC-1}Under the same assumption as in Theorem \ref{thmC} we have
\begin{equation}
\lim_{t\rightarrow T_{\max}}\frac{v\left( x,t\right) }{v\left( 0,t\right)
}=\Phi\left( x\right) \ \ \ \text{uniformly on\ \ \ }x\in\left[ -m\pi
,m\pi\right] . \label{cosine}
\end{equation}
\end{theorem}
\proof
It suffices to prove that for any sequence $t_{j}\nearrow T_{\max}$ there is a
subsequence, also denoted as $t_{j},$ so that
\[
\lim_{j\rightarrow\infty}\frac{v\left( x,t_{j}\right) }{v\left(
0,t_{j}\right) }=\Phi\left( x\right) \ \ \ \text{uniformly on\ \ \ }
x\in\left[ -m\pi,m\pi\right] .
\]
This would imply that the convergence is for all time $t\rightarrow T_{\max
}.\ $Let $t_{j}$ be a sequence with $t_{j}\nearrow T_{\max}.$ By Lemma
\ref{lem2} there is a subsequence $t_{j}$ and a nonnegative
Lipschitz\ function $w\left( x\right) $ defined on $\left[ -m\pi
,m\pi\right] $ so that
\[
\lim_{j\rightarrow\infty}\frac{v\left( x,t_{j}\right) }{v\left(
0,t_{j}\right) }=w\left( x\right) \ \ \ \text{uniformly on\ \ \ }
x\in\left[ -m\pi,m\pi\right] .
\]
We clearly have $\max_{x\in\left[ -\pi/2,\pi/2\right] }w=1\ $and by Lemma
\ref{lem2} it satisfies $w_{xx}+w\geq0\ $on$\ \left( -\pi/2,\pi/2\right) $
in the sense of distribution. By Lemma \ref{lem-new-2}, since $v\left(
x,t\right) $ does not blow up for $\left\vert x\right\vert >\pi/2,\ $we must
have $w\left( x\right) =0\ $for $\left\vert x\right\vert >\pi/2.$ By
continuity, we have$\ w\left( -\pi/2\right) =w\left( \pi/2\right) =0.$
Thus Lemma \ref{lem-new-3} implies that $w\left( x\right) =\cos x\ $for
$\left\vert x\right\vert <\pi/2.\ $The proof is done.$
\square$
\subsection{Convergence to a translational self-similar solution.}
Back to the slow-speed curve contracting flow $\left( \bigstar\right) $
with$\ \alpha\in(0,1]$, the initial curve $\gamma_{0}$ has curvature
$k_{0}\left( x\right) >0$ satisfying (\ref{integral-cond}), i.e.,
\begin{equation}
\int_{S_{m}^{1}}\frac{1}{k_{0}\left( x\right) }e^{ix}dx=0.
\end{equation}
If $k_{0}\left( x\right) $ satisfies the symmetric condition
\begin{equation}
k_{0}\left( x\right) =k_{0}\left( -x\right) \ \ \ \text{and\ \ \ }
k_{0}^{\prime}\left( x\right) <0,\text{\ \ \ \ \ }x\in\left( 0,m\pi\right)
\end{equation}
and $k_{\max}\left( t\right) \ $of$\ \gamma_{t}$ has type-two blow-up, then
by Theorem \ref{thmC-1} we have the convergence
\begin{equation}
\lim_{t\rightarrow T_{\max}}\frac{k\left( x,t\right) }{k\left( 0,t\right)
}=\left( \cos x\right) ^{\frac{1}{\alpha}}\ \ \ \text{uniformly
on\ \ \ }x\in\left[ -\pi/2,\pi/2\right] .
\end{equation}
It is well known that under the flow $\left( \bigstar\right) $, there is a
special \textbf{translational self-similar solution} $\Gamma_{t}\ $translating
in the direction $\left( 0,1\right) $ with unit speed\ (see \cite{NT} or the
book\ \cite{CZ}). For each time $t,\ \Gamma_{t}$ is only a translation of
$\Gamma_{0}\ $(this $\Gamma_{0}$ is not a closed curve, but still convex and
embedded). If we use tangent angle $x$ to parametrize $\Gamma_{0},$ its
parametrization$\ $is given by
\[
\Gamma_{0}=\left( \int_{0}^{x}\frac{\cos\xi}{\left( \cos\xi\right)
^{\frac{1}{\alpha}}}d\xi,\ \int_{0}^{x}\frac{\sin\xi}{\left( \cos\xi\right)
^{\frac{1}{\alpha}}}d\xi\right) ,\ \ \ x\in\left( -\pi/2,\pi/2\right)
\]
where
\[
\int_{0}^{x}\frac{\sin\xi}{\left( \cos\xi\right) ^{\frac{1}{\alpha}}}
d\xi=\left\{
\begin{array}
[c]{l}
\frac{\alpha}{\alpha-1}\left[ 1-\left( \cos x\right) ^{1-\frac{1}{\alpha}
}\right] ,\ \ \ \alpha\in\left( 0,1\right)
\\
-\log\cos x,\ \ \ \alpha=1.
\end{array}
\right.
\]
In particular the curve $\Gamma_{0}\ $goes to infinity as $x\rightarrow\pm
\pi/2.$ The curvature of $\Gamma_{0}\ $at angle $x\ $is given by $k\left(
x\right) =\left( \cos x\right) ^{1/\alpha},\ x\in\left( -\pi
/2,\pi/2\right) ,$ with maximum at $x=0.\ $When $\alpha=1,$ we get\ Grayson's
"\textbf{Grim Reaper}",\textsf{\ }which is$\ \Gamma_{0}=\left( x,-\log\cos
x\right) ,\ x\in\left( -\pi/2,\pi/2\right) .$
Evolve the above given symmetric $\gamma_{0}$ according to the flow $\left(
\bigstar\right) $. For any $t\in\lbrack0,T_{\max}),$ choose the point
$x_{t}\in\gamma_{t}$ at which the curvature is $k_{\max}\left( t\right) $
(by the assumption there is only one such point)\ and translate $\gamma_{t}$
so that $x_{t}$ becomes the origin $O=\left( 0,0\right) $. Call this
translational curve $\tilde{\gamma}_{t}$. Next rotate it so that the unit
tangent vector at the origin of $\tilde{\gamma}_{t}\ $becomes $\left(
1,0\right) ,$ and finally dilate the curve so that its maximal curvature
becomes $1\ $and denote this final curve as $\hat{\gamma}_{t}.$ Theorem
\ref{thmC-1} says that if we have type-two blow-up of $k_{\max}\left(
t\right) ,$ then over the region $x\in\left( -\pi/2,\pi/2\right)
,\ \hat{\gamma}_{t}$ converges to the above translational self-similar
solution\ $\Gamma_{0}$ as $t\rightarrow T_{\max}.$ When $\alpha=1,$ this
phenomenon has been observed by Angenent in \cite{ANG}.\
Thus we can summarize the following important observation of the slow speed
flow $\left( \bigstar\right) $:\ \textbf{for type-one blow-up, the
asymptotic behavior is given by a homothetic self-similar solution, while for
type-two blow-up, the asymptotic behavior (in the special symmetric case)\ is
given by a translational self-similar solution.}
To end this paper we point out that most of the lemmas and theorems remain
valid even the initial condition $v_{0}\left( x\right) $ does not satisfy
the integral condition (\ref{integral-cond}), as long as it is positive,
smooth, and $2m\pi$-periodic.\ They include Lemmas \ref{lem1},\ \ref{lem2}
,\ \ref{lem2-1},\ \ref{lem-new-1} and\ Theorems \ref{thmB}
,\ \ref{thm-type-one}, \ref{thmA},
As for Lemma \ref{lem-new-2}\ and Theorems\ \ref{thmC}, \ref{thmC-1}, if we
add the extra assumption that $\left( T_{\max}-t\right) ^{1/p}v_{\max
}\left( t\right) \ $is not bounded\ on$\ t\in\lbrack0,T_{\max}),\ $then they
are all valid even if $v_{0}\left( x\right) $ does not satisfy
(\ref{integral-cond}).
\emph{In particular, we emphasize\ again that for }$p\in\lbrack2,\infty
),\ $\emph{there is either type-one blow-up\ or type-two blow-up.\ Moreover,
type-one blow-up occurs only when}$\ v_{0}\left( x\right) \ $
\emph{satisfies\ the integral condition (\ref{integral-cond}) and if }
$v_{0}\left( x\right) $\emph{ does not satisfy (\ref{integral-cond}), then
the blow-up is always of type-two.\ Thus the generic blow-up behavior for
}$p\in\lbrack2,\infty)$\emph{ is type-two. }
\subsection{What to do next\ ?}
There is still a difficult question of estimating the type-two blow-up rate of
$v\left( x,t\right) =k^{\alpha}\left( x,t\right) .\ $When $\alpha=1$
(i.e., $p=1+1/\alpha=2$)\ and $m=2,\ $Angenent$\ $
and\ Vel\'{a}zquez\ \cite{AV} had given a nontrivial proof of the existence of
some symmetric\ initial\ data $v_{0}\left( x\right) >0,$ satisfying
(\ref{integral-cond}),\ with the type-two blow-up rate$\ $
\[
v_{\max}\left( t\right) =\left( 1+o\left( 1\right) \right) \sqrt
{\frac{\ln\ln\left( \frac{1}{T_{\max}-t}\right) }{T_{\max}-t}}
\ \ \ \text{as\ \ \ }t\rightarrow T_{\max}
\]
and therefore
\begin{equation}
v_{\max}\left( t\right) \sqrt{T_{\max}-t}\sim\sqrt{\ln\ln\left( \frac
{1}{T_{\max}-t}\right) }\rightarrow\infty\;\;\;\text{as\ \ \ }t\rightarrow
T_{\max}. \label{Ve}
\end{equation}
We are wondering if certain similar estimate holds in the case $\alpha
\in(0,1]\ $(i.e., $p>2$).\ At this moment we do not know and we hope to work
on it in the future.\
\section{Some pictures for the ODE\ (\ref{wode})}
In this section we give some pictures relating to the ODE$\ w^{\prime\prime
}+w-w^{1-p}=0.\ $These pictures can help us understand convergence behavior
(for general $p\in\left( -\infty,\infty\right) $) of the PDE $\partial
u/\partial\tau=u^{p}\left( u_{xx}+u-u^{1-p}\right) \ $(with positive initial
data$\ u_{0}\in C^{\infty}\left( S_{m}^{1}\right) $ and periodic boundary
condition).\ This is because that the ODE is a steady state of the PDE.
Let$\ $
\[
F\left( s\right) =\left\{
\begin{array}
[c]{l}
s^{2}-\frac{2}{2-p}s^{2-p},\ \ \ p\neq2,\ \ \ p\in\left( -\infty
,\infty\right)
\\
s^{2}-2\log s,\ \ \ p=2
\end{array}
\right. ,\ \ \ s\in\left( 0,\infty\right) .
\]
The graphs of $F\left( s\right) $ for $p=-1\in\left( -\infty,0\right)
,\ p=1\in\left( 0,2\right) ,\ p=3\in\lbrack2,\infty)$ are given below:
\[
{\includegraphics[
height=1.8282in,
width=2.4284in
]
{figure1.eps}
}
\]
\begin{center}
$
\begin{array}
[c]{cl}
{\includegraphics[
height=1.8862in,
width=2.5054in
]
{figure2.eps}
}
&
{\includegraphics[
height=1.881in,
width=2.4984in
]
{figure3.eps}
}
\end{array}
\ $
\end{center}
The graph of $F\left( s\right) $ for any $p\in\left( -\infty,0\right)
\ $is analogous to the above picture for $p=-1.\ F\left( s\right) $
increases on $s\in\left( 0,1\right) \ $with$\ F\left( 0\right) =0,$
$F\left( 1\right) >0,$\ and decreases to $-\infty\ $on $\left(
1,\infty\right) .\ $The graph of $F\left( s\right) $ for any $p\in\left(
0,2\right) \ $is analogous to the above picture for $p=1.\ F\left( s\right)
$ decreases on $s\in\left( 0,1\right) \ $with$\ F\left( 0\right)
=0,\ F\left( 1\right) <0,$\ and increases to $+\infty\ $on $\left(
1,\infty\right) .\ $Finally the graph of $F\left( s\right) $ for any
$p\in\lbrack2,\infty)\ $is analogous to the above picture for $p=3.\ F\left(
s\right) $ decreases on $s\in\left( 0,1\right) \ $with$\ \lim
_{s\rightarrow0^{+}}F\left( s\right) =+\infty,$ $F\left( 1\right)
>0,$\ and increases to $+\infty\ $on $\left( 1,\infty\right) $. Also note
that when $p=0,$ $F\left( s\right) \equiv0.\ $
Any solution $w\left( x\right) \ $to the ODE $w^{\prime\prime}+w-w^{1-p}=0$
must satisfy the identity$\ w_{x}^{2}\left( x\right) =F\left( M\right)
-F\left( w\left( x\right) \right) \ $for all $x\ $in the domain $I\ $of
$w\left( x\right) ,\ $on which $w\left( x\right) >0.\ $Here we may assume
$0\in I\ $and $w\left( 0\right) =M\geq1$\ is the maximum value of $w$ on
$I.\ $For $p\in\left( -\infty,0\right) ,$ we only have type-one blow up for
$v\left( x,t\right) $ of equation $\left( \clubsuit\right) $.\ If the
rescaled$\ $solution$\ u\left( x,t\right) =v\left( x,t\right) /R\left(
t\right) $ converges to $w\left( x\right) $ on some interval $I,\ $then we
must have $w\left( x\right) \equiv1\ $over $I$.$\ $Otherwise we have$\ M>1$
and use the first picture to get
\begin{equation}
w_{x}^{2}\left( x\right) =F\left( M\right) -F\left( w\left( x\right)
\right) <0
\end{equation}
for all\ $x\in I\ $such that $1\leq w\left( x\right) <M.\ $This gives a
contradiction and so $w\left( x\right) \equiv1\ $over $I.\ $
The main difference between $p\in\left( 0,2\right) \ $and $p\in
\lbrack2,\infty)\ $is that there exist bump solutions (degenerate)\ to the
ODE\ for$\ p\in\left( 0,2\right) ,$ but for $p\in\lbrack2,\infty),\ $all
solutions to the ODE are positive everywhere\ and periodic over $\mathbb{R}$
(nondegenerate). Again, this can also be seen from the second and third pictures.
\ \ \ \
\textbf{Acknowledgments.\ \ \ }While writing this paper, we had discussions
with several mathematicians including Professors Ben Andrews,\ Sigurd
Angenent,\ Hiroshi Matano, Jong-Sheng Guo and Chia-Hsing Nien.\ We are very
grateful to all of them.\ The third author would like to acknowledge the
support of the National Science Council and the National Center for
Theoretical Sciences\ of Taiwan.
\
\ \ \
\ \ \ \
\
\ \ \ \
Chi-Cheung Poon
Department of Mathematics, National Chung Cheng University, Chiayi 621,\ TAIWAN.
Email:\ \textit{[email protected]}
\ \ \ \ \ \
\ \ \ \ \
\ \ \ \
Yu-Chu Lin and\ Dong-Ho Tsai
Department of Mathematics, National Tsing Hua University, Hsinchu 300,\ TAIWAN.
Email:\ \textit{[email protected],\ \ \ [email protected]}
\end{document}
|
\begin{document}
\title{Absolute and Delay-Dependent Stability of Equations with a
Distributed Delay: a Bridge from Nonlinear Differential to Difference
Equations}
\footnotetext[1]{Partially supported by the NSERC Research Grant}
\footnotetext[2]{Corresponding author. E-mail {\em
[email protected]}. Fax (403)-282-5150. Phone (403)-220-3956.}
\begin{abstract}
We study delay-independent stability in nonlinear models with a
distributed delay which have a positive equilibrium.
Such models frequently occur in population dynamics and other
applications. In particular, we construct a relevant difference
equation such that its stability implies stability
of the equation with a distributed delay and a finite memory.
This result is, generally speaking, incorrect for systems with infinite
memory. If the relevant difference equation is unstable, we describe
the general delay-independent attracting set and also demonstrate that
the equation with a distributed delay is stable
for small enough delays.
\end{abstract}
\noindent
{\bf AMS Subject Classification:} 34K20, 92D25, 34K60, 34K23
\noindent
{\bf Keywords:} equations with a distributed delay, global attractivity,
permanent solutions, Nicholson's blowflies equation, Mackey-Glass
equation.
\section{Introduction}
In models of population dynamics which are described by an autonomous
differential equation
\begin{equation}
\label{01}
\frac{dN}{dt}=f(N)-g(N),
\end{equation}
where $f(N)$ and $g(N)$ are reproduction and mortality rates,
respectively, $f(N)>0$, $g(N)>0$ for $N>0$ and $f(N)>g(N)$ for
$0<N<K$, $f(N)<g(N)$ for $N>K$ ($K$ is the carrying capacity of the environment), the
positive equilibrium $K$ is stable: all positive solutions converge to
$K$ and are monotone.
It was argued that the observed data usually oscillates about the carrying
capacity; in order to model this phenomenon, it was suggested to introduce
delay in the production term
\begin{equation}
\label{02}
\frac{dN}{dt}=f(N(t-\tau))-g(N);
\end{equation}
the latter equation can have oscillatory solutions, and the delay
incorporated in the right hand side can be interpreted as maturation,
production or digestion effects. It is usually assumed that the
mortality rate was proportional to the present population level
\begin{equation}
\label{02new}
\frac{dN}{dt}=-\mu N(t)+ f(N(t-\tau)),~~\mu>0.
\end{equation}
The global behavior of solutions of (\ref{02new}) has been extensively
studied in literature, in particular in the cases of negative and
positive feedback
(see, for example, \cite{KWW1999, KW2001} and references therein),
the chaotic behavior is impossible in the case of the monotone feedback
\cite{MPS}. However in the case when $f(x)$ is a unimodal
function, i.e., increases for $x<K$ and decreases for $x>K$, there may be
delay induced instability and complex dynamics \cite{MG1,LW}.
For a detailed overview of the literature on the dynamics of (\ref{02new})
see the recent papers \cite{RostWu,RostLiz}.
It is demonstrated in \cite{RostWu,RostLiz} that if $f$ is a unimodal function
and positive equilibrium $K$ of the equation $x_{n+1}=f(x_n)$ is globally
asymptotically stable, then all solutions of (\ref{02new}) tend to $K$. In
particular, if $f$ has a negative
Schwarzian derivative, then local stability of the equilibrium of the
difference equation implies its global attractivity \cite{Singer}.
To the best of our knowledge, the first
delay-independent stability conditions were obtained in \cite{IvShark}.
In the present paper we will try to answer the general question: what are
intrinsic properties of the reproduction function $f$ which allow us to
conclude
that any solution of the equation with a finite memory converges to the
equilibrium?
Here we consider both general delays (including integral terms) and
continuous functions $f$ which may have multiple extrema, tend to infinity
at infinity etc.
As special cases, (\ref{02new}) includes
the Nicholson's blowflies equation \cite{nichol,GBN} and the Mackey-Glass
equation \cite{MG1,MG2}.
The Nicholson's blowflies equation
\beq{71}
\dot{x}(t)= -\delta x(t)+ px(t-\tau) e^{-a x(t-\tau)}
\end{equation}
was used in \cite{GBN} to describe the periodic
oscillation in Nicholson's classic experiments \cite{nichol} with the
Australian sheep blowfly, {\em Lucila cuprina}.
Equation \rf{71} with a distributed delay was studied in
\cite{CAMQ2006}, where comprehensive results were obtained for the case $\delta < p <
\delta e$.
The Mackey-Glass equation \cite{MG1,MG2}
\beq{72}
\dot{x}(t) = \frac{ax(t-\tau)}{1+x^{\gamma} (t-\tau)}-bx(t)
\end{equation}
models white blood cells production. Local and global stability of
the positive equilibrium for equation (\ref{72}) with variable delays
was studied in \cite{Liz1,Liz2,KubSak,Saker,BB2006,BBDCDIS}; to the best
of our knowledge, there are no publications on (\ref{72}) with a
distributed delay.
To incorporate random environment influence, some
authors included noise in (\ref{71}) and studied attractivity conditions,
see, for example, \cite{Shaikhet}.
However, in applied problems not only the derivative but also the delay
value can be perturbed. We assume that the production delay is not a
constant $\tau$ but some distributed value which leads to the equation
\beq{first}
\dot{x}(t) = r \left[ \int\limits_{-\infty}^t f( x(s)) d_s R(t,s) - x(t)
\right],
\end{equation}
where ${\displaystyle \int_{t-a}^{t-b} d_s R(t,s) }$ is the probability
that at
time $t$ the maturation delay in the production function is between $b$
and $a$, where $0<b<a$. We will assume that very large delays are
improbable, substituting $-\infty$ in the lower bound with $h(t)\leq t$
which tends to infinity as $t \to \infty$.
In the present paper we consider a rather general form of $f$, which
includes unimodal functions, as well as functions with several extrema.
The only requirement is that $f(x)$ has the only positive fixed point.
The main result claims that if this fixed point is a global attractor
for all positive solutions of the difference equation
\beq{first_difference}
x_{n+1}=f(x_n)
\end{equation}
then all solutions of \rf{first} with positive initial conditions tend to
this fixed point as well.
To some extent this establishes a link between stable
differential equation \rf{01} and difference equation
\rf{first_difference} which can undergo a series of bifurcations and even
transition to chaos.
If \rf{first_difference} is globally stable, so is \rf{first}.
If the unique positive equilibrium of \rf{first_difference} is unstable,
\rf{first} can be stable or not, depending on the delay.
The paper is organized as follows.
In Section 2 we prove that all solutions with positive initial conditions
are positive and bounded and establish some estimates for the lower
and the upper bounds. Section 3 presents sufficient conditions under which
all positive solutions converge to the positive equilibrium.
In Section 4 delay-dependent stability is investigated. In particular, it
is demonstrated that equations are globally attractive for delays small
enough; if \rf{first_difference} is unstable, then we can find such delays
that the positive equilibrium of \rf{first} is not a global attractor.
In Section 5 these results are applied to equations of population dynamics
with a unimodal reproduction function and a distributed delay, in
particular, to the Nicholson's blowflies and Mackey-Glass equations;
some open problems are presented.
\section{Boundedness and Estimates of Solutions}
We consider the equation with a distributed delay
\beq{1a}
\dot{x}(t) = r(t) \left[ \int_{h(t)}^t f( x(s)) d_s R(t,s) - x(t)
\right], ~t \geq 0,
\end{equation}
and the initial condition
\beq{2star}
x(t)=\varphi(t), ~ t\leq 0.
\end{equation}
As special cases, \rf{1a} includes
\begin{enumerate}
\item
{\bf The integrodifferential equation}
\beq{int1}
\dot{x}(t)= r(t) \left[ \int_{h(t)}^t K(t,s) f(x(s))\, ds - x(t)
\right]
\end{equation}
corresponding to the absolutely continuous $R(t, \cdot)$ for any $t$.
Here
$$\int_{h(t)}^t K(t,s) ~ds=1 \mbox{~~for any~~~} t, ~~
K(t,s) = \frac{\partial}{\partial s} R(t,s) \geq 0 $$ is defined
almost everywhere.
\item
{\bf The equation with several concentrated delays}
\beq{conc1}
\dot{x}(t)= r(t) \left[ \sum_{k=1}^m a_k(t) f\left( x(h_k(t))
\right) - x(t) \right],
\end{equation}
with $a_k(t) \geq 0$, $k=1, \cdots , m$, where ${\displaystyle
\sum_{k=1}^m
a_k(t) = 1}$ for any $t$.
This corresponds to ${\displaystyle R(t,s)=\sum_{k=1}^m a_k(t)
\chi_{(h_k(t),\infty)} (s) }$, where $\chi_{I}(t)$ is the characteristic
function of interval $I$.
\end{enumerate}
\noindent
{\bf Definition.} An absolutely continuous in $[0, \infty)$
function $x: \hbox{I\kern-.2em\hbox{R}} \rightarrow \hbox{I\kern-.2em\hbox{R}}$ is
called {\em a solution of the problem}
(\ref{1a}),(\ref{2star}) if it satisfies equation (\ref{1a})
for almost all $t\in [0,\infty)$
and conditions (\ref{2star}) for $t\leq 0$.
The integral in the right hand side of \rf{1a}
should exist almost everywhere. In particular, for \rf{int1}
with a locally integrable kernel, $\varphi$
can be any Lebesgue measurable essentially bounded function. For
\rf{conc1} $\varphi$ should be a Borel measurable bounded function. For
any distribution $R$ the integral exists if $\varphi$ is bounded and
continuous (here we assume $f$ is continuous). Besides, as is commonly
set in population dynamics models, $\varphi(t)$ is
nonnegative and the value at the initial point is positive.
Consider (\ref{1a}),(\ref{2star}) under the following assumptions.
\begin{description}
\item{{\bf (a1)}}
$f:[0,\infty) \to [0,\infty)$ is a continuous function satisfying
Lipschitz condition $|f(x)-f(y)| \leq L |x-y|$, $x,y \geq 0$,
$f(0)=0$, $f(x)>x$ for $0<x<K$ and $0<f(x)<x$ for $x>K$;
\item{{\bf (a2)}}
$h:[0,\infty)\rightarrow \hbox{I\kern-.2em\hbox{R}}$, is a Lebesgue measurable
function, $ h(t)\leq t$,
$\lim\limits_{t\rightarrow \infty}h(t)=\infty;$
\item{{\bf (a3)}}
$r(t)$ is a Lebesgue measurable essentially bounded on $[0,\infty)$
function, $r(t) \geq 0$ for any $t \geq 0$, ${\displaystyle
\int_0^{\infty} r(s)~ds = \infty}$;
\item{{\bf (a4)}}
$R(t, \cdot)$ is a left continuous nondecreasing function
for any $t$, $R(\cdot,s)$ is locally integrable for
any $s$, $R(t,s)=0$, $s \leq h(t)$, $R(t,t^+)=1$.
Here $u(t^+)$ is the right side limit of function $u$ at point $t$.
\item{{\bf (a5)}}
$\varphi: (-\infty,0] \to \hbox{I\kern-.2em\hbox{R}}$ is a continuous bounded function,
$\varphi(t) \geq 0$, $\varphi(0)>0$.
\end{description}
First, let us justify that the solution of \rf{1a},\rf{2star} exists and
is unique.
Denote by ${\bf L}^2([t_0,t_1])$ the space of Lebesgue measurable
functions $x(t)$ such that
$$Q=\int_{t_0}^{t_1} (x(t))^2~dt<\infty , ~~\| x
\|_{{\bf L}^2([c,d])}=\sqrt{Q},$$
by ${\bf C}([t_0,t_1])$ the space of continuous in
$[t_0,t_1]$ functions with the $\sup$-norm.
We will use the following result from the book of Corduneanu
\cite{Cordun} (Theorem 4.5, p. 95). We recall that operator $N$ is
{\em causal} (or {\em Volterra}) if for any two functions $x$ and $y$
and each $t$ the fact that $x(s)=y(s)$, $s \leq t$, implies
$(Nx)(s)=(Ny)(s)$, $s \leq t$.
\begin{uess}
\label{lemma1} \cite{Cordun}
Consider the equation
\beq{55}
\dot{y}(t)=({\cal L}y)(t)+({\cal N}y)(t), ~~t\in [t_0,t_1],
\end{equation}
where ${\cal L}$ is a linear bounded causal operator,
$N: {\bf C}([t_0,t_1]) \to {\bf L}^2([t_0,t_1])$ is a nonlinear causal
operator which
satisfies
\beq{56}
\| {\cal N}x - {\cal N}y \|_{{\bf L}^2([t_0,t_1])} \leq \lambda \| x-y
\|_{{\bf
C}([t_0,t_1])}
\end{equation}
for $\lambda$ sufficiently small. Then there exists a unique absolutely
continuous solution of \rf{55} in $[t_0,t_1]$, with the initial function
being equal to zero for $t<t_0$.
\end{uess}
\begin{guess}
\label{theorem0}
Suppose (a1)-(a5) hold. Then there exists a unique solution of
(\ref{1a}),\rf{2star}.
\end{guess}
{\bf Proof.} To reduce \rf{1a} to the equation with the zero initial
function, for any $t_0 \geq 0$ we can present the integral as a sum of two
integrals
\beq{57}
\dot{x}(t)= -r(t) x(t) + r(t) \int_{t_0}^t f(x(s))~d_s R(t,s) +
r(t) \int_{t_0}^t f(\varphi(s))~d_s R(t,s),
\end{equation}
where
$$ x(t)=0,~t<t_0, ~~~\varphi(t)=0,~t \geq t_0.$$
Here $t_0 \geq 0$ is arbitrary, so we begin with $t_0=0$ and proceed to a
neighboring $t_1$ to prove the existence of a local solution.
Then in \rf{55}
$$({\cal L}x)(t)=-r(t) x(t), ~~({\cal N}x)(t)= r(t) \int_{t_0}^t
f(x(s))~d_s R(t,s) + F(t),$$
where
$$
F(t)=r(t) \int_{t_0}^t f(\varphi(s))~d_s R(t,s), ~~
x(t)=0,~t<t_0, ~~~\varphi(t)=0,~t \geq t_0,$$
and for any $\lambda>0$ there is $t_1$, such that
\begin{eqnarray*}
\| {\cal N}x - {\cal N}y \|_{{\bf L}^2([t_0,t_1])} & \leq & |r(t)| \left\|
\int_{t_0}^t
|f(x(s))-f(y(s))|~d_s R(t,s) \right\|_{{\bf L}^2([t_0,t_1])} \\
& \leq & L \, {\rm ess}\sup_{t \geq 0} |r(t)| \max_{s \in [t_0,t_1]}
|x(s)-y(s)| \left\| \int_{t_0}^t d_s
R(t,s) \right\|_{{\bf L}^2([t_0,t_1])} \\
& \leq & L \, {\rm ess} \sup_{t \geq 0} |r(t)| \, \| x(s)-y(s)
\|_{C([t_0,t_1])}
|t_0 - t_1|
\leq \lambda \| x-y \|_{{\bf C}([t_0,t_1])}
\end{eqnarray*}
for ${\displaystyle |t_0-t_1| \leq \lambda/(L \, {\rm ess} \sup_{t \geq 0}
|r(t)| )}$,
where $L$ was defined in (a1), here $\lambda$ can be chosen small enough.
By Lemma \ref{lemma1} this implies existence and uniqueness of a local
solution for \rf{1a}. This solution is either global or there exists $t_2$
such that either
\beq{58}
\liminf_{t \to t_2} x(t) = -\infty
\end{equation}
or
\beq{59}
\limsup_{t \to t_2} x(t) =\infty ~.
\end{equation}
The initial value is positive, so as far as $x(t)>0$, the solution
is not less than the solution of the initial value problem
$\dot{x}+r(t)x=0$, $x(0)=x_0>0$ which is positive and the former case
\rf{58} is impossible.
In addition, $\dot{x}(t)<0$ for any $$ x(t)>K,~x(t) \geq \max\left\{
\max_{0 \leq s \leq t} x(s), \sup_{s \leq 0} \varphi(s) \right\}, $$ which
contradicts
\rf{59}. Thus there exists a unique global solution, which completes
the proof.
\hbox to 0pt{}
$\rlap{$\sqcap$}\sqcup$
\begin{guess}
\label{theorem1}
Suppose (a1)-(a5) hold.
Then the solution of (\ref{1a}),\rf{2star} is
positive for $t \geq 0$.
\end{guess}
{\bf Proof.} After the substitution
\beq{subst}
y(t)=x(t)\exp\left\{ \int_0^t r(\zeta)~d\zeta \right\},
\end{equation}
equation (\ref{1a}) becomes
\beq{3}
\dot{y}(t) = r(t) \exp\left\{ \int_0^t r(s)~ds\right\} \int_{h(t)}^t f
\left( y(s) \exp\left\{ -\int_0^s
r(\zeta)~d\zeta \right\} \right)~d_s R(t,s),
~t \geq 0.
\end{equation}
Thus $y(0)>0$ and $\dot{y}(t) \geq 0$ as far as $y(s) \geq 0$, $s \leq t$,
consequently, $y(t)>0$ for any $t \geq 0$. Since the signs of $y(t)$
and $x(t)$ coincide, then $x(t)>0$ for any $t \geq 0$.
\hbox to 0pt{}
$\rlap{$\sqcap$}\sqcup$
\noindent
{\bf Definition.} The solution $x(t)$ of (\ref{1a}),\rf{2star} is {\em
permanent} if there exist $A$ and $B$, $B\geq A>0$, such that
$$A \leq x(t) \leq B, ~t \geq 0.$$
In the following we prove permanence of all solutions
of (\ref{1a}) with positive initial conditions; moreover, we
establish bounds for solutions.
\begin{guess}
\label{theorem1a}
Suppose (a1)-(a5) hold. Then a solution of \rf{1a},\rf{2star}
is permanent.
\end{guess}
{\bf Proof.} By Theorem \ref{theorem1} the solution is positive for $t
\geq 0$. By (a2) there exists $t_0>0$ such that $h(t)>0$, $t \geq t_0$.
Since the solution is a continuous positive function, then we can define
\beq{defxminmax}
x_{\min} = \min_{t \in [0,t_0]} x(t)>0, ~~
x_{\max} = \max_{t \in [0,t_0]} x(t).
\end{equation}
Without loss of generality we assume $x_{\min}<K$, $x_{\max}>K$;
otherwise, we can choose $\min\{ x_{\min},\lambda K \}$, $\max\{ x_{\max},
K/\lambda\}$, where $0<\lambda<1$, as $x_{\min}$ and $x_{\max}$,
respectively.
By (a1) the following values are positive
\beq{deffminmax}
M = \max_{x\in [x_{\min},x_{\max}]} f(x), ~~
m = \min_{x\in [x_{\min},x_{\max}]} f(x).
\end{equation}
Define
\beq{defAB}
B= \max \left\{ M, x_{\max}, \max_{x \in [0,K]} f(x) \right\},
~~~
A= \min \left\{ m, x_{\min}, \min_{x \in [K,B]} f(x) \right\}.
\end{equation}
Since $f(x)>x$, $0<x \leq A$ and $f(x)<x$, $x \geq B$, then
there exists $\delta>0$ such that
$f(x) \geq A$ for $A-\delta \leq x \leq B$ and $f(x) \leq B$ for $A \leq x
\leq B+ \delta$.
Let us demonstrate
\beq{bounds}
x(t) \in [A,B], ~t \geq 0.
\end{equation}
By the definition of $A,B$ we have $x(t_0) \in [A,B]$.
Suppose the contrary: $x(t)>B$ or $x(t)<A$ for some
$t>t_0$.
First, let $x(t)>B$ for some $t>t_0$. Then $x(t)=B+\varepsilon$ for some
$\varepsilon \leq \delta$.
Denote
$$S_1= \{ t >t_0 | x(t)>B+\varepsilon\}, ~~t^{\ast} = \inf S_1.$$
Since $x(t_0) \leq B$ then the set
$$ S_2= \{ t | t_0 \leq t < t^{\ast}, ~x(t) \leq B\}
$$
is nonempty, denote $t_{\ast} = \sup S_2$. Then $x(t_{\ast})=B$,
$x(t^{\ast})=B+\varepsilon$; we also have $B\leq x(t) \leq B+\delta$ and $f(x) \leq
B$ in the interval
$[t_{\ast},t^{\ast}]$, thus the derivative is nonpositive
$$
\dot{x}(t) = r(t) \left[ \int_{h(t)}^t f(x(s))~d_s R(t,s) -
x(t) \right] \leq r(t) \left[ \int_{h(t)}^t B~d_s R(t,s) - B
\right]=0,
$$
which contradicts the assumption $x(t^{\ast})=B+\varepsilon > B= x(t_{\ast})$.
Similarly, let us assume that $x(t)=A-\varepsilon$ for some
$\varepsilon>0$, $\varepsilon<\delta$ and some $t>t_0$. After introducing
$$S_1= \{ t >t_0 | x(t)<A-\varepsilon\}, ~t^{\ast} = \inf S_1,
~~S_2= \{ t | t_0 \leq t < t^{\ast}, ~x(t) \geq A \}, ~t_{\ast} = \sup
S_2,
$$
we have $x(t_{\ast})=A$,
$x(t^{\ast})=A-\delta$, $A-\delta\leq x(t) \leq A$ and $f(x(t)) \geq A$
for
$t \in [t_{\ast},t^{\ast}]$, hence
$$
\dot{x}(t) = r(t) \left[ \int_{h(t)}^t f(x(s))~d_s R(t,s) -
x(t) \right] \geq r(t) \left[ \int_{h(t)}^t A~d_s R(t,s) - A
\right]=0.
$$
This contradicts the assumption $x(t^{\ast}) < x(t_{\ast})$.
Consequently (\ref{bounds}) is valid for $t\geq t_0$ and thus for any
$t\geq 0$, the bounds are positive, so the solution is permanent, which
completes the proof.
\hbox to 0pt{}
$\rlap{$\sqcap$}\sqcup$
\noindent
{\bf Example 1.} The statement of Theorem \ref{theorem1a} is not valid if
we omit the condition ${\displaystyle \lim_{t \to \infty} h(t)=\infty}$.
Consider the equation
\beq{ex1eq1}
\dot{x}(t) = 5x(h(t)) e^{-x(h(t))}-x(t),~t\geq 0, ~~h(t) \equiv -1,
~~x(t)=t+1,
~t\in [-1,0],
\end{equation}
which is equivalent to the initial value problem
\beq{ex1eq2}
\dot{x}(t)+x(t)=0, ~~x(0)=1,
\end{equation}
its solution ${\displaystyle x(t)=e^{-t}}$ tends to zero as $t \to \infty$
and so is not permanent.
\section{Absolute Global stability for Stable Difference Equations}
One of the main steps in establishing global stability property is the
proof of the fact that all nonoscillatory about the equilibrium solutions
tend to this equilibrium (see, for example, \cite{GL}). For
ordinary differential equations all solutions are nonoscillatory, for
retarded equations it depends on the delay. Below we demonstrate that
convergence of nonoscillatory solutions to the equilibrium is quite a
common property which is valid for any reproduction function with a unique
positive equilibrium. It can be interpreted as: ``if nonoscillatory,
solutions of delay equations behave asymptotically similar to ordinary
differential equations".
\noindent
{\bf Definition.} A solution $x(t)$ of (\ref{1a}),\rf{2star} is {\em
nonoscillatory about $K$} if there exists $\tau>0$ such that
either $x(t)>K$ or $x(t)<K$ for all $t \geq \tau$. Otherwise, $x(t)$ {\em
oscillates about $K$}.
\begin{guess}
\label{nonosciltheor}
Suppose (a1)-(a5) hold.
Any nonoscillatory about $K$ solution of (\ref{1a}),\rf{2star} converges
to $K$.
\end{guess}
{\bf Proof.}
First, let $x(t)<K$, $t \geq \tau$. Without loss of generality we can
assume $\tau=0$. By (a2) there exists $t_0 \geq 0$ such that $h(t) \geq 0$
for $t \geq t_0$. Denote $A$ as in \rf{defAB}.
By Theorem \ref{theorem1a} we obtain that $x(t) \geq
A$ for any $t \geq 0$. Since $f$ is continuous and $f(x)>x$ for $x<K$, then
${\displaystyle m_0=\inf_{x \in [A,K]} f(x) > A }$.
There may be two
cases: $m_0=K$ and $m_0<K$. In the former case, since $x(t)<K$, we have
$$\dot{x}(t)>r(t)\left[ \inf_{x \in [A,K]} f(x) - x(t) \right] > 0,
~t \geq t_0,
$$
as far as $x(t)>A$, thus the solution of the delay differential
equation is not less than the solution of $\dot{x}(t)=r(t)[K-x(t)]$,
$0<x(t_0)<K$, which is increasing and by (a3) (the integral of $r(t)$
diverges) tends to $K$.
\begin{figure}
\caption{For an arbitrary reproduction function with one positive
equilibrium $K$ we construct a series of such points that eventually
a solution is in $[m_j, K]$, if it does not exceed $K$ and
is in $[ K, M_j]$ if a solution is not less than $K$.
}
\label{figure1}
\end{figure}
Consider the latter case $m_0<K$.
By the definition of $m_0$ and $f(x)>x$ for $x \in [A,K]$ we have
${\displaystyle l_0=\sup\{x<K| f(x)\leq m_0\} < m_0 }$. Taking any
$\alpha$, $l_0<\alpha<m_0$ and assuming $x(t) \leq \alpha$ for any $t$, we
obtain
$$ \dot{x}(t)= r(t) \left[ \int_{h(t)}^t f(x(s)) ~d_s R(t,s) - x(t)
\right] \geq r(t)(m_0-\alpha)>0,$$
which leads to a contradiction $x(t)
\to \infty$ as $t\to \infty$ since $\int_0^{\infty} r(s)~ds $ diverges.
Thus, $x(t_*) \geq \alpha$ for some $t_*$;
moreover, since $\dot{x}(t) \geq 0$ as $x(t) \leq m_0$
then $x(t) \geq \alpha$ for any $t \geq t_*$.
Let $h(t) \geq t_*$, $t>t^*$ for some $t^*$.
By the definition of $l_0$ and $\alpha$ we have ${\displaystyle \tilde{m}=
\inf\limits_{x\in [\alpha,K]} f(x)>m_0}$ and
as far as $x(t) \leq m_0$ and $h(t) \geq t^{\ast}$ the following
inequality holds
$$ \dot{x}(t)= r(t) \left[ \int_{h(t)}^t f(x(s)) ~d_s R(t,s) - x(t)
\right] \geq r(t)(\tilde{m}-m_0)>0.$$
Assuming $x(t) \leq m_0$ for any $t$ we again obtain a contradiction.
Thus, there exist $\mu_1$ and $t_1 > t_0$ such that $x(\mu_1) \geq m_0$ and
$h(t) \geq \mu_1$ for $t \geq t_1$. Then $x(t) \geq
m_0$ for any $t \geq \mu_1$ and $x(h(t))\geq m_0$, $t \geq t_1$.
Further, let ${\displaystyle m_1 = \inf_{x \in
[m_0,K]} f(x)<K}$, here $m_0<m_1$ since $f(x)>x$ for $0<x<K$. Similarly,
we obtain $x(t)\geq m_1$ whenever $t>t_2$, for some $t_2>t_1$.
We continue this
process. It can be finite (for example, in Fig. \ref{figure1} we have $m_2=K$, where
the process stops and we deduce $x(t) \to K$ as $t \to \infty$) or
infinite (see the branch $x(t)>K$ of Fig. 1). In the infinite case we have
an increasing sequence $\{ m_j \}$, ${\displaystyle m_{j+1}=\min_{x
\in [m_j,K]} f(x) }$ which does not exceed $K$, so this sequence has a
limit $d$. Since $f(x)$ is continuous then ${\displaystyle
d=\min_{x\in [d,K]} f(x) }$. If $d<K$ then $f(x)-d$ should attain its
minimum in $[d,K]$ but $f(x)>x \geq d$, so this minimum is positive and
the equality ${\displaystyle d=\min_{x\in [d,K]} f(x) }$ leads to a
contradiction.
Further, let $x(t)>K$. Similarly, we define $B$ as in \rf{defAB}
and
${\displaystyle M_0=\max_{x \in [K,B]} f(x)<B }$. There may be two
cases: $M_0=K$ and $M_0>K$. In the former case we obtain $x(t) \to K$ as
$t \to \infty$.
Consider the latter case. By Theorem \ref{theorem1a}
we have $x(t) \leq B$ for any $x \geq 0$.
By the definition of $M_0$ and $f(x)<x$, $x \in [K,B]$ we have
${\displaystyle s_0=\inf\{x>K| f(x)\geq M_0\} > M_0 }$.
Similar to the case $x(t)<K$ we demonstrate that there exists $\nu_1$ such
that $x(\nu_1) \leq M_0$ and $x(t) \leq M_0$ for any $t \geq \nu_1$. Let
$t_1$ be such that $h(t) \geq \nu_1$ for $t \geq t_1$. Further, we define
${\displaystyle M_1=\max_{x \in [K,M_0]} f(x)<M_0 }$.
We continue this process, it can be finite or infinite (Fig.~\ref{figure1}
illustrates an infinite process for $x(t)>K$).
Similar to the case $x(t)<K$ we obtain $x(t) \to K$ as $t \to \infty$,
which completes the proof.
\hbox to 0pt{}
$\rlap{$\sqcap$}\sqcup$
\noindent
{\bf Example 2.} Let us note that in the case of infinite delays
nonoscillatory solutions do not necessarily tend to the positive
equilibrium. For example, the solution of the equation
$$\dot{x}(t) + x(t) = \frac{e}{2}\, x(0) \, e^{-x(0)}, ~~x(0)=2,$$
which is ${\displaystyle x(t)= \frac{1}{e} + \left( 2-\frac{1}{e} \right)
e^{-t}}$, tends to $1/e$ while the positive equilibrium is $1- \ln 2$,
the monotone solution is nonoscillatory.
Thus for any reproduction function with a unique fixed point $f(x)=x$
nonoscillatory solutions tend to the equilibrium; this is not, generally,
true for oscillatory solutions.
\noindent
{\bf Example 3.} Consider the Nicholson's blowflies equation
\beq{ex3eq1}
\dot{x}(t)= -\delta x(t)+ px(t-\tau) e^{-a x(t-\tau)}.
\end{equation}
Denote
\beq{ex3eq2}
\tau_k= \frac{1}{\delta \sqrt{aN^{\ast} (aN^{\ast}-2)}}
\left[ \arcsin
\left( \frac{\sqrt{aN^{\ast}(aN^{\ast}-2)}}{aN^{\ast}-1} \right) + 2\pi k
\right], ~~k=0,1,2, \cdots ~,
\end{equation}
where $N^{\ast}=1/a \ln(p/\delta)$ is a positive equilibrium.
If $p>\delta e^2$ then
the positive equilibrium is locally asymptotically stable
for $\tau \in [0, \tau_0)$ and is unstable (locally, thus it cannot be
globally attractive) for $\tau>\tau_0$, and \rf{ex3eq1} undergoes a Hopf
bifurcation at $N^{\ast}$ when $\tau=\tau_k$ \cite{Li} for $~k=0,1,2,
\cdots$.
Now we prove that absolute (delay-independent) convergence holds in some
special cases.
\begin{uess}
\label{lemmaimp}
Suppose (a1)-(a5) and at least one of the following conditions holds:
\begin{enumerate}
\item
$f(x)<K$ for any $0<x<K$.
\item
Denote by $x_{\max}([a,K])$, $0 \leq a \leq K$, the greatest point in $[a,K]$
where ${\displaystyle \max_{x \in [a,K]} f(x)}$ is attained and assume
\begin{equation}
\label{attractivitycond}
f(x_{\max}([a,K])) > K \mbox{~for some ~} a \in (0,K)
\Rightarrow \!\!
\min_{ x \in [K, f(x_{\max}([a,K]))]} \!\!\!\! f(x) >a.
\end{equation}
\end{enumerate}
Then any solution of (\ref{1a}),\rf{2star} converges
to $K$.
\end{uess}
{\bf Proof.}
1. By Theorem \ref{theorem1a} any solution is permanent: $A \leq x(t) \leq
B$, where $A<K<B$, $t \geq
0$, and for some $t_0 \geq 0$ we have $h(t) \geq 0$, $t\geq t_0$. Denote
${\displaystyle M_0=\max_{x \in
[K,B]}f(x)}$ and ${\displaystyle m_0=\min_{x \in [A,K]}f(x)}$.
If $M_0 \leq K$ then the derivative is negative for any $x(t)>K$ and
the solution either eventually does not exceed $K$ or is
decreasing for any $t \geq t_0$. In the latter case the solution tends to
the equilibrium; if it has a different limit, we obtain that the
derivative is less than a negative number, which is a contradiction.
In the former case, if there exists $t>0$ such that $x(t)<K$ (otherwise,
we have a nonoscillatory case $x(t) \geq K$ where we have already proved
convergence), then $x(t) \leq K$ for any $t$ (assuming $x(t^{\ast})>K$ we
obtain that the derivative of $x(t)$ is negative almost everywhere, while
the function changes from $K$ to $x(t^{\ast})>K$), which is again a
nonoscillatory case, by Theorem \ref{nonosciltheor} solution $x(t)$
converges to $K$.
Thus, we can consider $M_0 >K$ only. Then we introduce $$ M_1=\max_{x \in
[K,M_0]} f(x)<M_0, ~~ m_1 =\min_{x \in [m_0,K]}>m_0, ~~s_1=\min_{x \in
[K,M_0]} f(x)
$$
and similarly define $M_j,m_j$ and $s_j$, $j=2,3,\cdots $.
There exists $\tau_1$ such that $\min\{ m_1,s_1 \} \leq x(t) \leq
M_1$ for $t\geq \tau_1$ and $t_1$ such that $h(t) \geq \tau_1$, $t\geq
t_1$. Similar to the proof of Theorem \ref{nonosciltheor} we obtain that
there exists a sequence $\tau_1 \leq \tau_2 \leq \cdots \leq \tau_j \leq
\cdots $ such that
$$\min\{ m_j,s_j \} \leq x(t) \leq M_j \mbox{~~for~~~} t \geq \tau_j.$$
We assume that all $M_j>K$, otherwise we have an eventually monotone case.
Since for any continuous $f$ all three sequences are monotone
(nonincreasing $M_j$ and nondecreasing $m_j$, $s_j$) and tend to $K$ (see
the end of the proof of Theorem \ref{nonosciltheor}), then
${\displaystyle \lim_{t \to \infty} x(t)=K}$.
2. Now suppose that \rf{attractivitycond} holds for any $a \geq 0$.
We begin with $A \leq x(t) \leq B$ in Theorem \ref{theorem1a},
$t \geq 0$, where $A,B$ are defined in
\rf{defAB}, $h(t)>0$,
$t \geq t_0$. Denote
\beq{addast1}
M_0=\max_{x \in [A,K]}f(x), ~ m_0=\min_{x \in [K,M_0]}f(x).
\end{equation}
The case $M_0 =K$ was considered in Part 1, thus we can restrict ourselves
to the case $M_0>K$, maximum is attained at $$x_0=x_{\max}([A,K]),$$
here $x_0$ is the greatest point where the maximum is attained.
Let us demonstrate that there exists $\mu_0$ such that $x_0 \leq x(t)
\leq M_0$, $t \geq \mu_0$.
If ${\displaystyle x(t) > S_0=\max \{ M_0, \max_{x \in [K,B]} f(x) \}=
\max_{x \in [A,B]} f(x) \geq K }$
for all $t$ then we have a nonoscillatory case and convergence to $K$,
which is a contradiction. Thus, $x(t_{\ast}) \leq S_0$ for some
$t_{\ast}$; assuming there is $t^{\ast}>t_{\ast}$ such that $x(t_{\ast})
>S_0$ we obtain that the value of the function at the end of the
segment is higher than at the beginning point, while the derivative is
nonpositive. So there exists $\tau_0 \geq t_0$ such that $x(t)
\leq S_0$ for $t>\tau_0$. If $S_0 \neq M_0$ then ${\displaystyle S_1=
\max_{x \in [K,S_0]} f(x) < S_0}$. Similarly, we find $\tau_1$ such that
$x(t)\leq S_1$ for $t$ large enough. Since the sequence
${\displaystyle S_{n+1}= \max_{x \in [K,S_n]} f(x) }$ is nonincreasing
and tends to $K$ then there exists $\tau_1$ such that $x(t) \leq M_0$,
$t \geq \tau_1$. Further, we will consider $t \geq t_1$, where
$h(t) \geq \tau_1$ whenever $t>t_1$, only.
If ${\displaystyle x(t) < s_0=\min \{ m_0, \min_{x \in [A,K]} f(x) \} =
\min_{x \in [A,M_0]} f(x) }$
for all $t$, where $m_0$ was defined in \rf{addast1}, then we have a
nonoscillatory case and convergence to $K$,
which is a contradiction. As above, we prove that $x(t)\geq s_0$ for $t
\geq \nu_0$, here $\nu_0 \geq t_0$. If $s_0<m_0$, then we construct a
sequence ${\displaystyle s_{n+1}= \min_{x \in [s_n,K]} f(x) }$ which
tends to $K$. If $m_0<K$ then some $s_j \geq m_0$.
There is $\mu_0 \geq t_1$ such that $x(t)\geq m_0$, $t\geq
\mu_0$. Consequently, we have found $\mu_0$ such that $m_0 \leq x(t)
\leq M_0$, $t \geq \mu_0$. Let $h(t) \geq \mu_0$, $t\geq t_2$.
Now, if $m_0=K$ then for $t\geq t_2$ the solution increases as far as
$x(t)<K$. Thus, either $x(t) < K$ for any $t\geq t_2$ and this monotone
solution converges to $K$, or, if $x(t_{\ast}) \geq K$ for some $t_{\ast}$,
$x(t) \geq K$, $t \geq t_{\ast}$, and again we have a nonoscillatory
solution which converges to $K$.
Denote
$$M_1= \max_{x \in [m_0,M_0]}f(x), ~~m_1=\min_{x \in [m_0,M_1]}f(x).$$
Let us assume $M_1>K$, $m_1<K$ and demonstrate that there is $\mu_1 \geq t_2$
such that
$m_1 \leq x(t) \leq M_1$,
$x(t) \geq \mu_1$, where
$M_1 = f\left( x_{\max}([m_0,K]) \right)$.
In fact, for any $t \geq t_2$ the solution is nonincreasing as far as
$x(t) \geq M_1$, which gives an upper bound. Considering $t$ where the
equation refers only to the values where this bound is valid, we obtain
that the solution is nondecreasing if $x(t) \leq m_1$,
which together with $m_1<K<M_1$ confirms the statement. We continue the
induction process
$$
M_{n+1}=\max_{x \in [m_n,M_n]} f(x),~~
m_{n+1}=\min_{x \in [m_n, M_{n+1}]} f(x),
$$
where $K \geq m_{n+1} > m_n $, $K \leq M_{n+1} < M_n$. This process can
be infinitely continued if all $m_n < K$, $M_n>K$ (otherwise, at certain
stage we have a ``monotone" case which implies convergence), and there
exists $\mu_j$, $j=0,1,2, \cdots$ such that $m_j \leq x(t) \leq M_j$, $t
\geq \mu_j$. Let us assume that there are infinite sequences $\{ m_n \}$
and $\{ M_n \}$, both are monotone and bounded.
Then there exist limits
$$ x=\lim_{j \to \infty} m_j \leq K \leq \lim_{j \to \infty} M_j =X.$$
Since by the assumptions of the theorem $f(M_j)>m_j$, then $f(X) \geq x$.
If $x=f(X)$ then either $x=X=f(X)=X$ and $x(t)$ converges to $K$ or
$\min\limits_{y \in [K,f_{\max}([x,K])]} f(y) \leq x$ which contradicts
the assumptions of the theorem.
Denote by $x_{\max,j}$, $x_{\min,j}$ the
sequences where minimum $m_j$ and maximum $M_j$ are attained, respectively
(we recall that we choose maximal $x_{\max,j}$ and minimal $x_{\min,j}$
if an extremum is attained at several points). Then $f(x_{\max,j})=M_j$,
$f(x_{\min,j})=m_j$, $m_j \leq x_{\max,j} \leq K$, $K \leq x_{\min,j}
\leq m_j$, $m_{j+1}>x_{\max,j}$, $M_{j+1}<x_{\min,j}$, which implies that
$x_{\max,j}$ and $x_{\min,j}$ tend to $x$ and $X$, respectively.
Let us prove that $x=f(X)$.
Assume the contrary: $f(X)> x+ 2\varepsilon>0$.
Since ${\displaystyle \lim_{j \to \infty} x_{\min,j}=X}$ and $f$ is
continuous then there exist $n_0$ such that $f(x_{\min,j})>x+\varepsilon$,
$j \geq n_0$, but $f(x_{\min,j})=m_{j+1}$, so $m_{j+1}>x+\varepsilon$,
which contradicts the equality ${\displaystyle \lim_{j \to \infty} m_j =
\sup_j m_j=x}$. Thus ${\displaystyle \lim_{t \to \infty} x(t)=K}$,
which completes the proof.
\hbox to 0pt{}
$\rlap{$\sqcap$}\sqcup$
Now we can prove the main result of the present paper.
\begin{guess}
\label{maintheorem1}
Suppose (a1)-(a5) hold and all positive solutions of the difference
equation
\begin{equation}
\label{differ}
x_{n+1}=f(x_n)
\end{equation}
tend to $K$.
Then any positive solution of (\ref{1a}),\rf{2star} converges
to $K$.
\end{guess}
{\bf Proof.} If either conditions of Part 1 of Lemma \ref{lemmaimp} or
\rf{attractivitycond} hold then $K$ attracts all positive solutions of
(\ref{1a}),\rf{2star}. Let us assume that \rf{attractivitycond} does not
hold (and we do not have a monotone case as in Part 1 of Lemma
\ref{lemmaimp}), which means that
for some $a \in (0,K)$ we have
$$
\min_{ x \in [K, f(a)]} f(x) \leq a, \mbox{~~or~~~} K < b \leq f(a), ~~
f(b) \leq a, \mbox{~~where~~} b \in [K, f(a)].
$$
We can assume $f(b)=a$, otherwise, since $f(b)<a$, $f(K)=K>a$
and $f$ is continuous, then there is $c \in [K,b]$ such that
$f(c)=a$. Thus, $b>K$ and $f^2(b)=f(f(b))=f(a) \geq b$. Since $f(x)<x$
for any
$x>K$ then for any $x> M:=\max\limits_{x \in [0,K]} f(x)$ we have
$f^2(x)<x$,
then there is a fixed point of $f^2$ in the segment $[b,M]$,
in addition to a fixed point $x=K$. Since the fixed point $K$
of \rf{differ} cannot be a global attractor unless $K$ is the only fixed
point of $f^2$ \cite{Coppel} then $K$ is not a global
attractor of (\ref{differ}), which completes the proof.
\hbox to 0pt{}
$\rlap{$\sqcap$}\sqcup$
Here we have not considered the case when $f$ has no positive equilibria;
however for completeness we will consider this case as well.
\begin{guess}
\label{maintheorem2}
Suppose (a2)-(a5) hold, $f(0)=0$ and $f(x)<x$ for any $x>0$.
Then any positive solution of (\ref{1a}),\rf{2star} converges
to zero.
\end{guess}
{\bf Proof.} Let $t_0$ be such that $h(t) \geq 0$ for $t \geq t_0$.
Denote ${\displaystyle M_0= \sup_{t \in [0,t_0]} x(t) }$.
Since $f(x)<x$ then $x(t) \leq M_0$ for any $t \geq 0$.
The solution is decreasing as far as $x>M_1$, where ${\displaystyle
M_1=\max_{x \in [0,M_0]} f(x)<M_0}$.
\begin{figure}
\caption{If $f(x)<x$ for any $x>0$ then all positive solutions tend to zero.
}
\label{figure2}
\end{figure}
Suppose $S_1$ is the smallest point not exceeding $M_0$ where this maximum
is attained (see Fig.~\ref{figure2}). Since $f(x)<x$, then $S_1>M_1$. Let us
choose $l_1$ such that
$M_1<l_1<S_1$; as far as $x(t) \geq l_1$ we have $\dot{x}(t) \leq
r(t)(M_1-l_1)$, thus eventually any solution is less than $l_1$. Further,
for some $d>0$ we have $M_1-f(x) \geq d$ if $0<x\leq l_1$. Thus the
derivative at any point $x(t)>M_1$ does not exceed (we recall that
$x(t)\leq l_1$) the negative value of $-d\, r(t)$. Consequently, there
exists $t_1$ such that $x(t) \leq M_1$ for $t \geq t_1$.
By induction, we denote ${\displaystyle M_{k+1}=\sup_{x \in [0,M_k]}
f(x)<M_k }$, $k=1,2,
\cdots $, and prove that there exists $t_{k+1} >t_k$ such that $x(t) \leq
M_{k+1}$ for $t>t_{k+1}$. Since ${\displaystyle \lim_{n \to \infty} M_n
=0}$ then
${\displaystyle \lim_{t\to \infty} x(t)=0}$, i.e., any positive solution
tends to zero.
\hbox to 0pt{}
$\rlap{$\sqcap$}\sqcup$
\section{The Case When Stability Is Delay-Dependent}
In this section we consider the case when stability properties
of \rf{1a} depend on the delay. First, we prove that for small delays
all solutions tend to positive equilibrium $K$.
Second, we demonstrate that if \rf{attractivitycond} does not hold then
there exists equation \rf{1a} with parameters satisfying (a1)-(a5) such
that its attracting set is as close to $[m,M]$ as prescribed, where
\beq{mM}
M=\max_{x\in[0,K]} f(x) > K, ~~m=\min_{x\in[K,M]} f(x)
\end{equation}
and ${\displaystyle \max_{x\in[m,K]} f(x)=M}$ as well.
Let us note that the Lipschitz condition in (a1) implies
$| f(x) -K| \leq L |x-K|$.
\begin{guess}
\label{theorem5}
Suppose (a1)-(a5) hold and
\begin{equation}
\label{cup}
\limsup_{t \to \infty} \int_{h(t)}^t r(s)~ds < \frac{1}{L+1}~.
\end{equation}
Then any positive solution of (\ref{1a}),\rf{2star} converges
to $K$.
\end{guess}
{\bf Proof.}
Without loss of generality we can assume that \rf{cup} is satisfied
for $t \geq 0$. We can find a positive $\lambda<1$ such that
${\displaystyle \int_{h(t)}^t
r(s)~ds < \frac{\lambda}{L+1} }$ for $t \geq 0$. By Theorem
\ref{theorem1a} any solution is permanent:
$$ K-a_0 \leq x(t) \leq K+b_0, ~~t \geq 0, ~0<a_0<K, ~b_0>0.$$
Let us denote $\alpha=\max \{ a_0, b_0 \}$ and prove that there exists
$t_1 \geq 0$ such that $x(t) \leq K+\lambda \alpha$ for any $t > t_1$.
If solution $x(t)$ is nonoscillatory about $K$ then by Theorem
\ref{nonosciltheor} it tends to the equilibrium and thus for
$\varepsilon=\lambda \alpha>0$ there exists $t_1$ such that $x(t) \leq
K+\varepsilon$ for $t \geq t_1$. Now assume that $x(t)$ is oscillating,
$t_0$ is such a point that $h(t) \geq 0$ for $t \geq t_0$ and
$x(t_{\ast})=K$, $x(\tau_{\ast})=K$, where $x(t)>K$ for $t \in
(t_{\ast},\tau_{\ast})$. A continuous function $x(t)$ attains its maximum
in $[t_{\ast},\tau_{\ast}]$, let ${\displaystyle x(t^{\ast})
= \max\limits_{s \in [t_{\ast},\tau_{\ast}]} x(s) = M_0 }$. We claim that
$x(t^{\ast}) \leq K+ \lambda \alpha$. Assume the contrary: $x(t^{\ast})
>K+ \lambda \alpha$.
By (a1) we have ${\displaystyle \max_{x \in [K,M_0]} f(x)=M_1<M_0}$.
Denote
$$M_2= \max \{ M_1, K+ \lambda \alpha \}, ~~
\tau = \sup \left\{ t \in [t_{\ast},t^{\ast}] | x(t) \leq M_2
\right\}.$$
Then $x(t)> M_2 \geq K+\lambda \alpha$, $t \in [\tau,t^{\ast}]$.
Since $K-\alpha \leq x(t) \leq K+\alpha$ for any $t \geq 0$ then by (a1)
$$
\dot{x} (t) = r(t) \left[ \int_{h(t)}^t f(x(s)) ~d_s R(t,s) - x(t)
\right] \leq r(t) \left[ \max\limits_{x \in [K-\alpha, K+\alpha], x>0}
f(x) - (K-\alpha) \right]
$$
\beq{star31}
= r(t) \left[ \max\limits_{x \in [K-\alpha, K+\alpha], x>0}
[f(x)-K] +\alpha \right] \leq r(t) [L\alpha+\alpha] = \alpha(L+1)r(t).
\end{equation}
We remark that $\dot{x} (t) <0$ whenever the following conditions hold:
$t \in (\tau, t^{\ast})$, $h(t) \geq t_{\ast}$ (all referred
prehistory of the solution is between $K$ and $M_1<M_0$) and $x(t)>M_2
\geq M_1$. Since $f(\tau)=M_2$, $f(t^{\ast})>M_2$, then there are points in
$(\tau,t^{\ast})$ where the derivative is positive, thus
$h(t)<t_{\ast}$. Let $\bar{t}$ be such a point. As $r(t) \geq 0$ and
$[t_{\ast},\tau] \subset [h(\bar{t}), \bar{t}]$, then
$$ \int_{t_{\ast}}^{\tau} r(s)~ds \leq \int_{h(\bar{t})}^{\bar{t}}
r(s)~ds < \frac{\lambda}{L+1},
$$
consequently,
$$x(\tau)-x(t_{\ast})= \int_{t_{\ast}}^{\tau} \dot{x} (s)~ds
\leq \int_{t_{\ast}}^{\tau} \alpha(L+1)r(s)~ds< \alpha(L+1)
\frac{\lambda}{L+1} = \lambda \alpha.$$
Hence $x(\tau) < x(t_{\ast})+\lambda \alpha=K+\lambda \alpha$, which
contradicts
the assumption $x(\tau) \geq K+\lambda \alpha$. If we denote
$t_1=t_{\ast}$ then $x(t)<K+\lambda \alpha$ for any $t \geq t_1$.
Similar to the previous case, we can prove that there exists $\tau_1$ such
that $x(t)\geq K-\lambda \alpha$ for $t \geq \tau_1$.
Thus, we accept $K-\lambda \alpha \leq x(t) \leq K+\lambda \alpha$ as new
solution bounds and consider $\tau_2$ such that $h(t) \geq \max \{
t_1,\tau_1 \}$ for $t
\geq \tau_2$. We repeat the induction step and obtain that there exists
$t_2 \geq \tau_2$ such that
$$ K-\lambda^2 \alpha \leq x(t) \leq K+\lambda^2 \alpha, ~~t \geq t_2,
$$
and a sequence of points $t_1 \leq t_2 \leq t_3 \leq t_n \leq \cdots$
such that
$$ K-\lambda^3 \alpha \leq x(t)\leq K+\lambda^3 \alpha, ~~t \geq t_3,
\cdots , K-\lambda^n \alpha \leq x(t) \leq K+\lambda^n \alpha,
~~t \geq t_n, \cdots .$$
Since $0<\lambda<1$ then ${\displaystyle \lim_{t\to\infty} x(t)=K}$,
which completes the proof.
\hbox to 0pt{}
$\rlap{$\sqcap$}\sqcup$
\begin{corol}
Suppose (a1)-(a2),(a4)-(a5) hold, $r>0$ and ${\displaystyle
\sup_{t \geq 0} (t-h(t)) < \frac{1}{r(L+1)} }$ (we recall that
$L$ is the Lipschitz constant defined in (a1)). Then all solutions of the
equation \beq{1aadd}
\dot{x}(t) = r \left[ \int_{h(t)}^t f( x(s)) d_s R(t,s) - x(t)
\right]
\end{equation}
tend to $K$.
\end{corol}
\begin{remark}
\label{Lstar}
Let us remark that the Lipschitz constant $L$ in \rf{cup} can be
substituted by, generally, a smaller value of
\begin{equation}
\label{cup1}
L^{\ast} = \sup_{x \geq 0} \left| \frac{f(x)-K}{x-K} \right|,
\end{equation}
since in \rf{star31} only $|f(x)-K|$ is estimated for $|x-K| < \alpha$.
\end{remark}
Next, let us estimate the attracting set when (\ref{1a}) is not absolutely
stable. We introduce $m,M$ as in \rf{mM} and
denote by $x_{\max}$ the greatest point in $[a,K]$
where ${\displaystyle M = \max_{x \in [0,K]} f(x)}$ is attained,
by $x_{\min}$ the minimal point where ${\displaystyle m= \min_{x \in
[K,M]} f(x)}$ is attained.
\begin{guess}
\label{theorem6}
Suppose $f(x)$ satisfies (a1) and
\beq{2cycle}
m=f(x_{\min})<x_{\max}<K.
\end{equation}
Then for any $a\in (m,x_{\max})$, and any $b \in (K,M)$ such that
${\displaystyle \min_{x \in [K,b]} f(x) < a}$
there exists a problem \rf{1a},\rf{2star} with parameters satisfying
(a2)-(a5) such that
\beq{attracting}
\liminf_{t \to \infty} x(t)=a, ~~\limsup_{t \to \infty} x(t)=b.
\end{equation}
\end{guess}
{\bf Proof.}
Let us fix $a\in (m,x_{\max})$, and $b \in (K,M)$ such that
${\displaystyle d= \min_{x \in [K,b]} f(x) < a}$, see Fig.~\ref{figure3}.
\begin{figure}
\caption{Assuming $a$ and $b$ as in the figure exist, we can construct a delay
equation such that $[a,b]$ is its attracting set.
}
\label{figure3}
\end{figure}
Then there exists $x_1$ such that $K<x_1<b$ and $m_1=f(x_1)<a$.
For the initial function
$$\varphi(t)=a+(t+1)(b-a),~~t \in [-1,0],$$
we have $\varphi(-1)=a$, $\varphi(0)=b$ and $\varphi$ changes continuously
from $a$ to $b$. Since $a<K<x_1<b$, then $x(s_0)=x_1$ for
some $s_0 \in (-1,0)$. We consider the equation
\beq{particular}
\dot{x}(t)= r\left[ f(x(h(t)))-x(t) \right], ~~t \geq 0,
\end{equation}
where $r>0$. Obviously parameters of \rf{particular} and the initial
function satisfy (a3)-(a5).
We choose $h(t)=s_0$, $t \in [0, \tau_1]$, where
${\displaystyle \tau_1= \frac{1}{r} \ln \left( \frac{b-m}{a-m} \right)
>0}$. Then
$$x(t)=(b-m)e^{-rt}+m, ~~t \in [0, \tau_1],$$
so $x(\tau_1)=a$ and $x(t)$ changes continuously
from $b$ to $a$ in $[0, \tau_1]$, $a<x_{\max}<b$, thus there exists $s_1
\in (0,\tau_1)$ such that $x(s_1)=x_{\max}$. We assume
$h(t)=s_1$, $t \in (\tau_1,\tau_2]$, where ${\displaystyle \tau_2= \tau_1
+ \frac{1}{r} \ln \left( \frac{M-a}{M-b} \right) > \tau_1 }$.
The solution is
$$x(t)= (a-M) e^{-r(t-\tau_1)} + M, ~~ t \in [\tau_1,\tau_2], $$
hence $x(\tau_2)=b$. We continue periodically with $h(t)$ piecewise
constant such that
$$ x(h(t))=\left\{ \begin{array}{ll} x_1, & t \in (\tau_{2n},
\tau_{2n+1}], \\ x_{\max} & t \in (\tau_{2n+1},\tau_{2n+2} ], \end{array}
\right. ~~
n=0,1,2 \cdots ,
$$
where $\tau_0=0$, $h(t) \in (\tau_{k-1}, \tau_k)$ if $t \in (\tau_k,
\tau_{k+1})$, $x(\tau_{2n})=b$, $x(\tau_{2n+1})=a$, $n=0,1,2 \cdots $ and
$a \leq x(t) \leq b$ for any $t$. Here the delay is bounded and piecewise
continuous, it obviously satisfies (a2).
\hbox to 0pt{}
$\rlap{$\sqcap$}\sqcup$
\begin{corol}
\label{attract}
If (a1)-(a5) and \rf{2cycle} hold then the sharpest attracting interval
for \rf{1a},\rf{2star} is $(m,M)$, where $m$ and $M$ are defined in
\rf{mM}.
\end{corol}
\begin{remark}
\label{remark2}
Suppose the difference equation $x_{n+1}=f(x_n)$ has a stable 2-cycle
$(a,b)$ in such a way that $f(f(x))<x$ for $a<x<K$, $x>b$ and
$f(f(x))>x$ for $x<a$, $K<x<b$. Then for any $\varepsilon>0$ there exist
such distributed delay and initial function that
$[a+\varepsilon,b-\varepsilon]$ is an attracting interval of
\rf{1a},\rf{2star}.
Moreover, if $K \in (a,b)$, function $f$ satisfies (a1), $f(a)>b$
and $f(b)<a$, then, similar to the proof of Theorem \ref{theorem6}
we can demonstrate that there exists problem \rf{1a},\rf{2star} such that
$(a,b)$ is its attracting interval.
\end{remark}
\begin{remark}
\label{remark3}
The claims of Corollary \ref{attract} and Remark \ref{remark2} to some
extent complement
Theorem 6 in \cite{RostLiz} where the sharpest invariant and attracting
interval was
found in the case when absolute stability fails but there exists a unique
globally attractive 2-cycle for the relevant difference equation (for
equations with a unique constant concentrated delay).
\end{remark}
\section{Applications and Discussion}
The global attractivity results can be applied to any
unimodal function $f$ and any type of finite delay.
For example, we can deduce the following result in the stream of
\cite{RostWu}, see also \cite{Singer} and Proposition 2.1 in \cite{Liz}.
\begin{guess}
\label{maintheoremapplic}
Suppose (a1)-(a5) hold, $f$ is three times continuously differentiable and has the
only critical point $x_0>0$ (maximum),
$$ (Sf)(x)= \frac{f^{\prime\prime\prime}(x)}{f^{\prime} (x)} -
\frac{3}{2} \left( \frac{f^{\prime\prime}(x)}{f^{\prime} (x)} \right)^2
<0 \mbox{ ~ for ~~}x \neq x_0 $$
and $|f^{\prime}(K)| \leq 1$.
Then any solution of (\ref{1a}),\rf{2star} tends to $K$ as $t \to \infty$.
\end{guess}
As an application, consider the Nicholson's blowflies equation with a
distributed delay
\beq{71new}
\dot{x}(t)= -\delta x(t)+ p\int_{h(t)}^t x(s) e^{-a x(s)}~d_s R(t,s),
\end{equation}
where (a2)-(a5) hold.
If $p<\delta$ then by Theorem \ref{maintheorem2} all positive solutions
of \rf{71new} go to extinction, i.e., $x(t) \to 0$ as $t \to \infty$.
If ${\displaystyle 1 < \frac{p}{\delta} < e^2}$ then by Theorem
\ref{maintheorem1} all positive
solutions of \rf{71new} tend to the positive equilibrium ${\displaystyle
K=\frac{1}{a} \ln \left( \frac{p}{\delta} \right)}$.
However, since sustainable oscillations were observed in experiments
\cite{nichol}, we speculate that in the real problem the ratio $p/\delta$
was above $e^2$ and delays significant enough.
Particular cases of \rf{71new} are the equation with a variable delay
\beq{71newa}
\dot{x}(t)= -\delta x(t)+ p x(h(t)) e^{-a x(h(t))},
\end{equation}
including the original equation \rf{71} with a constant delay,
the integrodifferential equation
\beq{71newb}
\dot{x}(t)= -\delta x(t)+ p \int_{h(t)}^t k(t-s) x(s) e^{-a x(s)} \, ds,
\end{equation}
where
\beq{kernel}
\int_{h(t)}^t k(t-s) ~ds =1 \mbox{~ for any ~} t \geq 0,
\end{equation}
and the mixed equation
\beq{71newc}
\dot{x}(t)= -\delta x(t)+ \alpha p x(g(t)) e^{-a x(g(t))}
+ (1-\alpha) p \int_{h(t)}^t k(t-s) x(s) e^{-a x(s)} \,ds,
~~0 \leq \alpha \leq 1.
\end{equation}
By Theorems \ref{maintheorem1}, \ref{maintheorem2} and \ref{theorem5} we
conclude the following result.
\begin{guess}
\label{nicholson}
If
$p \leq \delta$ then all positive solutions of \rf{71new} (and thus any
of \rf{71newa},\rf{71newb},\rf{71newc} ) go to
extinction, if $\delta < p < \delta e^2$, all positive solutions
tend to the positive equilibrium.
If $p \geq e^2$ and
\begin{equation}
\label{nick}
\limsup_{t \to \infty} (t-h(t))< \frac{1}{p+\delta},
\end{equation}
then all positive solutions of \rf{71new} tend to the positive
equilibrium.
\end{guess}
{\bf Proof.} The results for $p < \delta e^2$ are immediate corollaries
of Theorems \ref{maintheorem1} and \ref{maintheorem2}. Since the
absolute value of the derivative of ${\displaystyle f(x)=
\frac{p}{\delta}x e^{-ax}}$ does not
exceed $p/\delta$ (which is attained at $x=0$), then \rf{cup} implies
\rf{nick}.
\hbox to 0pt{}
$\rlap{$\sqcap$}\sqcup$
\begin{remark}
\label{Lstar1}
We can apply Remark \ref{Lstar} to improve condition
(\ref{nick}). In fact, $L^{\ast}$ defined in \rf{cup1}
does not exceed $p/(\delta e^2)$ (the absolute value of the
minimum of the derivative which is attained at $x=2/a$)
if $p \geq \delta e^2$ and the stability condition becomes
\begin{equation}
\label{nick1}
\limsup_{t \to \infty} (t-h(t)) < \frac{1}{p/e^2+\delta}.
\end{equation}
We also notice that for unimodal functions such that their equilibrium
(the fixed point $x=f(x)$) exceeds maximum point $x_{\max}$ and such that the
minimum of the derivative is less than -1 (otherwise, under some
additional conditions which are satisfied for the Nicholson's blowflies
and the Mackey-Glass equations, the relevant difference equations are
stable \cite{Liz}), then $L^{\ast}$ defined in \rf{cup1} does not exceed
the absolute value of the minimum of the derivative.
Really, function ${\displaystyle g(x) = \left| \frac{f(x)-K}{x-K}
\right|}$ satisfies $g(0)=1$, ${\displaystyle 0< g(x) \leq -
\!\! \min_{x \in [x_{\max}, \infty )} \!\!\! f^{\prime} (x) }$ in
$[x_{\max}, \infty )$. The value in the right
hand side of the latter inequality exceeds 1 and $g$ is decreasing in $[0,
x^{\ast}]$, where $x^{\ast}<x_{\max}$ is such point that $f(x^{\ast})=K$, while in
$[x^{\ast},x_{\max})$ function $g(x)$ is less than $g(x_{\max})$, which
does not exceed
the absolute value of the minimum of the derivative.
\end{remark}
Let us compare Theorem \ref{nicholson} and Remark \ref{Lstar1} to
some known global stability results
for the Nicholson's blowflies equation. For example, \cite{GL}
contains the following global stability condition
\beq{GLcond}
\left( e^{\delta \tau}-1 \right) \left( \frac{p}{\delta} -1 \right)<1
\end{equation}
for equation \rf{71} with a constant delay.
Condition \rf{GLcond} is slightly sharper for \rf{71} than \rf{nick} but
\rf{nick} is applicable to equations with variable delays and
integrodifferential equations as well. Condition \rf{nick1} is sharper
than \rf{GLcond}, see Fig. \ref{fignick}. It is also applicable to
equations with a variable and/or distributed delay.
\begin{figure}
\caption{The values of delay for which
(\protect{\ref{71}
\label{fignick}
\end{figure}
For completeness of references, we remark that oscillation of \rf{71new}
was studied in \cite{MCM2008}.
All the main results are also relevant for the Mackey-Glass equation with
a distributed delay
\beq{72new}
\dot{x}(t) = \int_{h(t)}^t \frac{ax(s)}{1+x^{\gamma} (s)}~d_s
R(t,s)-bx(t),~a>0,~b>0, ~\gamma>0,
\end{equation}
which also involves the equation
\beq{72newc}
\dot{x}(t)= -b x(t)+ \frac{\alpha a x(g(t))}{1+x^{\gamma} (g(t))}
+ (1-\alpha) a \int_{h(t)}^t k(t-s) \frac{x(s)}{1+x^{\gamma} (s)}~ds,
~~0 \leq \alpha \leq 1,
\end{equation}
where \rf{kernel} is satisfied,
as well as its special cases of the equation with a variable delay
($\alpha=1$) and the integrodifferential equation ($\alpha=0$).
If $a<b$ then all solutions of \rf{72new} go to extinction. If $a>b$ and
\beq{condMackey}
\gamma < \frac{2a}{a-b}
\end{equation}
then all solutions tend to the positive equilibrium ${\displaystyle
K=\left( \frac{a}{b} -1 \right)^{1/\gamma} }$.
Again, Theorems \ref{maintheorem1}, \ref{maintheorem2} and
\ref{theorem5} imply the following result.
\begin{guess}
\label{mackeyglass}
If
$a \leq b$ then all positive solutions of \rf{72new}
(and thus of \rf{72newc} ) go to
extinction. If either $0< \gamma \leq 2$ or $\gamma>2$ and ${\displaystyle
b < a < \frac{\gamma b}{\gamma-2} }$, then all positive solutions
tend to the positive equilibrium $K$.
If $\gamma >2$, ${\displaystyle a \geq \frac{\gamma b}{\gamma-2} }$
and
\begin{equation}
\label{mack}
\limsup_{t \to \infty} (t-h(t))< \frac{4b^2\gamma}{a^2(\gamma-1)^2 +4 a b
\gamma},
\end{equation}
then all positive solutions of \rf{72new} tend to the positive
equilibrium.
\end{guess}
{\bf Proof. }
We remark that for $\gamma \leq 1$ the function ${\displaystyle f(x)
= \frac{ax}{b(1+x^{\gamma})}}$ is monotone increasing, and we have
stability for any delay. For $1 \leq \gamma \leq 2$ the relevant
difference equation $x_{n+1}=f(x_n)$ is locally asymptotically
stable, we apply Theorem \ref{maintheoremapplic}.
It is known that for the relevant difference
equation local asymptotic stability implies global asymptotic stability
(see \cite{Liz}, Example 3.2), thus the difference equation is globally
asymptotically stable whenever
$$
|f^{\prime} (K)|= \left| 1-\gamma+ \frac{b}{a} \gamma \right|<1,
~~\mbox{ where ~~} \gamma > 2,$$
or $a< \gamma b/(\gamma-2)$. According to Remark \ref{Lstar1}, we can
take the absolute value of the minimum of the derivative which is attained at
${\displaystyle \left( \frac{\gamma+1}{\gamma-1}\right)^{1/\gamma} }$
as $L^{\ast}$:
$$
L^{\ast}=\frac{a(\gamma-1)^2}{4b\gamma}.
$$
Application of Theorem \ref{theorem5} and Remark
\ref{Lstar1} completes the proof.
\hbox to 0pt{}
$\rlap{$\sqcap$}\sqcup$
Finally, let us formulate some open problems.
\begin{enumerate}
\item
Generally, absolute stability results are incorrect for systems with
infinite memory. Deduce stability results when (a2) is not satisfied
but measure $d_s R(t,s)$ decays exponentially with memory
$$
\int_{-\infty}^{t-\tau} d_s R(t,s) \leq S e^{-\alpha \tau}.
$$
\item
For arbitrary $f(x)$ satisfying (a1) and the equation with a constant
delay, characterize attractive sets.
\item
Obtain stability results for equations with a distributed delay in the
multistability case: there are several points satisfying $f(x)=x$.
If $K_1$ is the first positive fixed point and $f(x)<x$ for $0<x<K_1$,
describe initial conditions leading to extinction (Alley effect).
\item
Would the results of the present paper remain valid for the equation
$$
\dot{x}(t) = r(t) \left[ f\left( \int_{h(t)}^t x(s) d_s R(t,s) \right) - x(t)
\right], ~t \geq 0,
$$
if (a1)-(a5) hold?
\end{enumerate}
\end{document}
|
\begin{document}
\title{Streaming Algorithms for Planar Convex Hulls}
\author[1]{Mart\'{\i}n Farach-Colton\footnote{
This research was supported in part by NSF CCF 1637458, NIH 1 U01 CA198952-01, a NetAPP Faculty Fellowship and a gift from Dell/EMC.
}}
\author[1]{Meng Li}
\author[2]{Meng-Tsung Tsai\footnote{
This research was supported in part by the Ministry of Science and Technology of Taiwan under contract MOST grant 107-2218-E-009- 026-MY3, and the Higher Education Sprout Project of National Chiao Tung University and Ministry of Education (MOE), Taiwan.
}}
\affil[1]{Rutgers University, Piscataway, USA \protect\\
\texttt{\{farach, ml910\}@cs.rutgers.edu} }
\affil[2]{National Chiao Tung University, Hsinchu, Taiwan \protect\\
\texttt{[email protected]} }
\date{}
\maketitle
\thispagestyle{empty}
\begin{abstract}
Many classical algorithms are known for computing the convex hull of a
set of $n$ point in $\mathbb{R}^2$ using $O(n)$ space. For large
point sets, whose size exceeds the size of the working space, these
algorithms cannot be directly used. The current best streaming
algorithm for computing the convex hull is computationally expensive,
because it needs to solve a set of linear programs.
In this paper, we propose simpler and faster streaming and W-stream
algorithms for computing the convex hull. Our streaming algorithm
has small pass complexity, which is roughly a square root of the
current best bound, and it is simpler in the sense that our algorithm
mainly relies on computing the convex hulls of smaller point sets.
Our W-stream algorithms, one of which
is deterministic and the other of which is randomized, have
nearly-optimal tradeoff between the pass complexity and space usage,
as we established by a new unconditional lower bound.
\end{abstract}
\textbf{Keywords.} Convex Hulls, Streaming Algorithms, Lower Bounds
\pagenumbering{arabic}
\section{Introduction}\label{sec:intro}
The \emph{convex hull} of a set $P$ of points in $\mathbb{R}^2$ is the smallest convex set that contains $P$. We denote the convex hull of $P$ by $\conv{P}$ and denote the set of extreme points in $\conv{P}$ by $\ext{P}$. Let $n = |P|$ and $h = |\ext{P}|$. Note that $h \le n$ because $\ext{P}$ is a subset of $P$. By computing the convex hull of $P$, we mean outputting the points in $\ext{P}$ in clockwise order.
There is a long line of research on computing the convex hull using $O(n)$ space. In the RAM model, Graham~\cite{g72} gave the first algorithm, called the {\it Graham Scan}, with running time $O(n \log n)$. Subsequently, several algorithms were devised with the same running time, but with different approaches~\cite{ph77,a79,k84,bdh96}. In the output-sensitive model, where the running time depends on $n$ and $h$,
Jarvis~\cite{j73} proposed the {\it Gift Wrapping} algorithm, which has running time $O(nh)$. This algorithm was later improved by Kirkpatrick and Seidel~\cite{ks86} and Chan~\cite{c96}, both of which achieve running time of $O(n \log h)$. In the online model, where input points are given one by one and algorithms need to compute the convex hull of points seen so far, Overmars and van Leeuween's algorithm~\cite{ol81} can update the convex hull in $O(\log^2 n)$ time per incoming point. Brodal and Jacob~\cite{bj02} reduced the update time to $O(\log n)$.
\noindent\textbf{Streaming Model.} The algorithms mentioned above all require $s = \Omega(n)$ working space (memory) in the worst case. Therefore, none of these can handle the case when $s \ll n$, that is, when either $n$ is very large (a massive data set) or $s$ is very small (such as in embedded systems). In order to explore the convex hull problem with such a memory restriction, we consider the standard streaming models~\cite{ruhlthesis03,bbdmw02,cia06,muthu06,cbga10}, where the $n$ given points are stored on a read-only or writable tape in an arbitrary order. If the tape is read-only, then the model is simply called the \emph{streaming model}~\cite{bbdmw02,muthu06}. Otherwise the tape is writable, and the model is called the \emph{W-stream model}~\cite{ruhlthesis03,cia06,cbga10}. We refer to algorithms in the streaming model as \emph{streaming algorithms} and algorithms in the W-stream model as \emph{W-stream algorithms}.
In both models, algorithms can manipulate the working space while reading the points sequentially from the beginning of the tape to the end; however, only algorithms in the W-stream model can modify the tape, detailed in Section~\ref{sec:wrstr}. Hence, algorithms in this model cannot access the input randomly, which is different from the model for in-place algorithms~\cite{Bronn04,Bose07}.
The extreme points are written to a write-only stream. The \emph{pass complexity} of an algorithm refers to the number of times the algorithms scans the tape from the beginning to the end. The goal is to devise streaming and W-stream algorithms that have small pass and space complexities.
No single-pass streaming algorithm can compute the convex hull using $o(n)$ space because it is no easier than sorting $n$ positive numbers in $\mathbb{R}$. Since sorting $n$ numbers using $s$ spaces requires $\Omega(n/s)$ passes~\cite{mp78}, computing the convex hull in a single pass requires linear space. However, Chan and Chen~\cite{Chan07} showed that the space requirement can be significantly reduced if multi-pass algorithms are allowed. Specifically, their streaming algorithm uses $O(\delta^{-2})$
passes, $O(\delta^{-2} h n^\delta)$ space, and $O(\delta^{-2} n \log n)$ time for any constant $\delta \in (0, 1)$. On the other hand, to have small space complexity, one can appeal to a general scheme to convert PRAM algorithms to W-stream algorithms established by Demetrescu et al.~\cite{cbga10}, summarized in Section~\ref{sec:wrstr}. Using this technique yields a W-stream algorithm that uses $O((n/s) \log h)$
passes and $O(s)$ space where $s$ can be as small as constant.
\noindent\textbf{Our Contribution.} We devise a new $O(n \log h)$-time RAM algorithm to compute the convex hull (Section~\ref{sec:pre}). Then, we adapt the RAM algorithm to both models.
In the streaming model, the pass complexity of our algorithm is roughly a square root of that of Chan and Chen's algorithm~\cite{Chan07} if both algorithms have the same space usage. Specifically, we have:
\begin{theorem}\label{thm:read-only}
Given a set $P$ of $n$ points in $\mathbb{R}^2$ on a read-only tape where $|\ext{P}| = h$, there exists a deterministic streaming algorithm to compute the convex hull of $P$ in $O(\delta^{-1})$ passes using $O(\min\{\delta^{-1} h n^\delta \log n, n\})$ space and $O(\delta^{-2} n \log n)$ time for every constant $\delta \in (0, 1)$.
\end{theorem}
In the W-stream model, we adapt the RAM algorithm to two W-stream algorithms. One uses $O(s)$ space for any $s = \Omega(\log n)$ and the other uses $O(s)$ space for any $s = \Omega(1)$. The pass complexity of our W-stream algorithms are $O(\ceil{h/s}\log n)$ and $O(h/s + \log n)$, which are smaller than $O((n/s)\log h)$, the best pass complexity among those W-stream algorithms that are converted from PRAM algorithms in algebraic decision tree model~\cite{cbga10}, when $s \le h$.
The first W-stream algorithm is deterministic, and we get:
\begin{theorem}\label{thm:det}
Given a set $P$ of $n$ points in $\mathbb{R}^2$ where $|\ext{P}| = h$, there exists a deterministic W-stream algorithm to compute the convex hull of $P$ in $O(\lceil h/s \rceil \log n)$ passes using $O(s)$ space and $O(n\log^2 n)$ time
for any $s = \Omega(\log n)$.
\end{theorem}
Next, we randomize the above W-stream algorithm. A logarithmic factor can be shaved off from the pass complexity w.h.p.\footnote{w.h.p. means with probability $1-1/n^{\Omega(1)}$.} We have:
\begin{theorem}\label{thm:rand}
Given a set $P$ of $n$ points in $\mathbb{R}^2$ where $|\ext{P}| = h$, there exists a randomized W-stream algorithm to compute the convex hull of $P$ in $p$ passes using $O(s)$ space and $O(n\log^2 n)$ time
for any $s = \Omega(1)$, where $p = O(h/s + \log n)$ w.h.p.
\end{theorem}
We prove that our W-stream algorithms have nearly-optimal tradeoff between pass and space complexities by showing Theorem~\ref{thm:lower}, which generalizes Guha and McGregor's lower bound (Theorem~8 in~\cite{Guha08}). We remark that this lower bound is sharp because it matches the bounds of our randomized W-stream algorithm when $h = \Omega(s \log n)$.
\begin{theorem}\label{thm:lower}
Given a set $P$ of $n$ points in $\mathbb{R}^2$ where $|\ext{P}| = h = \Omega(1)$, any streaming (or W-stream) algorithm that computes the convex hull of $P$ with success rate $\ge 2/3$, and uses $s$ bits requires $\Omega(\ceil{h/s})$ passes.
\end{theorem}
We note here that space is measured in terms of bits for lower bounds and in terms of points for upper bounds. This asymmetry is a common issue for geometric problems because most geometric problems are analyzed under the RealRAM model, where precision of points (or other geometric objects) is unbounded.
\noindent\textbf{Applications.} Our W-stream algorithms can handle the case for $s \le h$ because it outputs extreme points on the fly. This output stream can be used as an input stream for another streaming algorithm, such as for diameter~\cite{shamosthesis78} and minimum enclosing rectangle~\cite{t83},
both of which rely on Shamos' rotating caliper method~\cite{shamosthesis78}. We apply Theorems~\ref{thm:det} and~\ref{thm:rand} to show Corollary~\ref{cor:diam}, detailed in Section~\ref{sec:proof}.
\begin{corollary}\label{cor:diam}
Given a set $P$ of $n$ points in $\mathbb{R}^2$ where $|\ext{P}| = h$, there exists a deterministic W-stream algorithm to compute the diameter and minimum enclosing rectangles of $P$ in $O(\ceil{h/s}\log n)$ passes using $O(s)$ space and $O(n \log^2 n)$ time for every $s = \Omega(\log n)$. Given randomness, the pass complexity can be reduced to $O(h/s+ \log n)$ w.h.p.
\end{corollary}
\noindent\textbf{Approximate Convex Hulls.} Given the hardness result shown in Theorem~\ref{thm:lower}, we know that one cannot have a constant-pass streaming algorithm that uses $o(h)$ space to compute the convex hull. In view of this, to have constant-pass $o(h)$-space streaming algorithms, one may consider computing an approximate convex hulls. There are several results studying on how to efficiently find an approximate convex hull in the streaming model, based on a given error measurement. The error criterion varies from the Euclidean distance~\cite{hs08}, and Hausdorff metric distance~\cite{Lopez00,Lopez05}, to the relative area error~\cite{rr15}. These algorithms use a single pass, $O(s)$ space, and can bound the given error measurement by a function of $s$.
\noindent\textbf{Paper Organization.} In Section~\ref{sec:pre}, we present a new $O(n\log h)$-time RAM algorithm to compute the convex hull. Then, in Section~\ref{sec:algo}, we present a constant-pass streaming algorithm in the streaming model. In Section~\ref{sec:wrstr}, we present two W-stream algorithms, both of which use $O(s)$ space where $s$ can be as small as $O(\log n)$. We generalize the previous lower bound result in Section~\ref{sec:lower}, and prove a higher (but conditional) lower bound in Section~\ref{sec:clower}. We place the proofs of Corollary~\ref{cor:diam} in Section~\ref{sec:proof}.
\newcommand{\UH}[1]{U(#1)}
\section{Yet another $O(n \log h)$-time algorithm in the RAM model}\label{sec:pre}
Our streaming algorithm is based on a RAM algorithm, which we present in this section. This RAM algorithm is a modification of Kirkpatrick and Seidel's ultimate convex hull algorithm in the RAM model~\cite{ks86}. Chan and Chen's streaming algorithm~\cite{Chan07} is also based on Kirkpatrick and Seidel's algorithm, and thus the structure of these two streaming algorithms have some similarities. The changes are made so that our streaming algorithm does not have to rely on solving linear programs, thus reducing the computation cost compared to Chan and Chen's algorithm.
In what follows, we only discuss how to compute the upper hull because the lower hull can be computed analogously. Formally, computing the \emph{upper hull} $\UH{P}$ of a point set $P$ means outputting that part of the extreme points $v_1, v_2, \ldots, v_t \in \ext{P}$ in clockwise order so that $v_1$ is the leftmost point in $P$ and $v_t$ is the rightmost point in $P$, tie-breaking by picking the point with the largest $y$-coordinate, so that all points in $P$ lie below or on the line passing through $v_i, v_{i+1}$ for each $1 \le i < t$. Note that each of $v_1, v_2, \ldots, v_t$ has a unique $x$-coordinate, and each line that passes through $v_i$ and $v_{i+1}$ for $1 \le i < t$ has a finite slope.
Roughly speaking, Kirkpatrick and Seidel's ultimate convex hull algorithm~\cite{ks86} evenly divides the point set into two subsets by a vertical line $\ell: x = \mu$, finds the hull edge in the upper hull that crosses $\ell$, and recurses on the two separated subsets. By appealing to the point-line duality, finding the crossing hull edge is equivalent to solving a linear program. Chan and Chen's streaming algorithm is adapted from this implementation of the ultimate convex hull algorithm. Their algorithm evenly divides the point set into $r+1$ subsets for $r \ge 1$ by $r$ vertical lines, finds the hull edges in the upper hull that cross these vertical lines, and recurses on the $r+1$ separated subsets. Finding these $r$ crossing hull edges is equivalent to solving $r$ linear programs, where the constraint sets for each are the same but the objective functions are different.
\begin{table}[!h]
\centering
\begin{tabular}{|c|c|c|}
\hline
& Find $r$ hull edges, and recurse. & Find $r$ extreme points, and recurse. \\
\hline \hline
$r = 1$ & Kirkpatrick and Seidel 1986~\cite{ks86} & Chan 1995~\cite{chanthesis95} \\
\hline
any $r \ge 1$ & Chan and Chen 2007~\cite{Chan07}& This paper \\
\hline
\end{tabular}
\caption{Categorization of four $O(n \log h)$-time algorithms for convex hull. \label{tab:4algo}}
\end{table}
In~\cite[Section 2]{chanthesis95}, Chan gives another version of Kirkpatrick and Seidel's ultimate convex hull algorithm, that finds a suitable (possibly random) extreme point, divides the point set into two by $x$-coordinate, and recurses. The extreme point can be found by elementary techniques. Our streaming algorithm is adapted from the latter algorithm. It finds $r$ suitable extreme points for $r \ge 1$, divides the point set into $r+1$ subsets by $x$-coordinate, and recurses on each subset. Though this generalization sounds straightforward, finding the $r$ suitable extreme points needs a different approach from that for finding a single suitable extreme point. We reduce finding these $r$ suitable extreme points to computing the upper hulls of $n/(r+1)$ small point sets. This reduction is the key observation of our RAM algorithm and is described in detail in the subsequent paragraphs. These four algorithms are categorized in Table~\ref{tab:4algo}.
Given $r$, our algorithm partitions $P$ arbitrarily into $G_1, G_2, \ldots, G_{n/(r+1)}$ so that each $G_j$ has size in $[1, r+1]$, and then computes the upper hull of each $G_j$. Let $Q$ be the union of the slopes of the hull edges in the upper hull of $G_1$, $G_2$, \ldots, $G_{n/(r+1)}$, which is a multiset. Let $\sigma_k$ be the slope of rank $k|Q|/(r+1)$ in $Q$, for $k \in [1, r]$, in other words, $\sigma_k$ is the $k$th $(r+1)$-quantile in $Q$. To simplify the presentation, let $\sigma_0 = -\infty$ and $\sigma_{r+1} = \infty$. Let $s_k$ be the extreme point in $P$ that \emph{supports} slope $\sigma_k$, for each $k \in [0, r+1]$. That is, for every point $p \in P$ draw a line passing through $p$ with slope $\sigma_k$, and pick $s_k$ as the point whose line has the highest $y$-intercept. We define $s_0 = p_L$, the point with the smallest $x$-coordinate, and $s_{r+1} = p_R$, the point with the largest $x$-coordinate. If any $s_k$ has more than one candidates, pick the point that has the largest $y$-coordinate. Let $x(p)$ denotes the $x$-coordinate of point $p$, and let $\sigma(p, q)$ denote the slope of the line that passes through points $p$ and $q$.
We use these $s_1, s_2, \ldots, s_r$ as the $r$ \emph{suitable} extreme points with which to refine $P$ into $P_1, P_2, \ldots, P_{r+1}$ where we say the $s_i$ are suitable in that each $P_k$ has size bounded by $O(|P|/(r+1))$. Initially, set $P_k = \emptyset$ for all $k \in [1, r+1]$. The refinement applies the \emph{cascade-pruning} described in Lemma~\ref{lem:cascade-pruning} on $G_j$ for each $j \in [1, n/(r+1)]$, which uses the known pruning technique stated in Lemma~\ref{lem:prune} as a building block, and works as follows:
\begin{itemize}[leftmargin=1.8cm]
\item[Step 1.] Compute $\UH{G_j}$, and obtain the extreme points $v_1, v_2, \ldots, v_t \in \UH{G_j}$ in clockwise order.
\item[Step 2.] Set $P_k \leftarrow P_k \cup \{v_i : i \in [\alpha, \beta], x(s_{k-1}) < x(v_i) < x(s_k)\}$ for each $k \in [1, r+1]$, where $v_\alpha$ (resp. $v_\beta$) is the extreme point in $G_j$ that supports $\sigma_{k-1}$ (resp. $\sigma_{k}$).
\end{itemize}
The pruning in Step 2 is two-fold. For any $i < \alpha$, if $x(v_i) \le x(s_{k-1})$, then such a $v_i$ cannot be placed in $P_k$. Otherwise $x(v_i) > x(s_{k-1})$, and Case 2 of Lemma~\ref{lem:cascade-pruning} applies. Again, such a $v_i$ cannot be placed in $P_k$. Similarly, $v_i$ for any $i > \beta$ cannot be placed in $P_k$ either. Finally, remove the points that lie below or on the line passing through $s_{k-1}, s_k$ from $P_k$ for each $k \in [1, r+1]$.
\begin{lemma}[Chan, \cite{chanthesis95}]\label{lem:prune}
Given a point set $P \subset \mathbb{R}^2$ and a slope $\sigma$, let $s$ be the extreme point in $P$ that supports $\sigma$. Then, for any pair of points $p, q \in P$ where $x(p) < x(q)$,
\begin{itemize}[leftmargin=3cm]
\item[Case 1.] If $\sigma(p, q) \le \sigma$ and $x(q) \le x(s)$, then $q \notin \UH{P}$.
\item[Case 2.] If $\sigma(p, q) \ge \sigma$ and $x(p) \ge x(s)$, then $p \notin \UH{P}$.
\end{itemize}
\end{lemma}
{
\renewcommand{RAM Algorithm}{RAM Algorithm}
\renewcommand{\thealgocf}{}
\begin{algorithm}[H]
Let $G_1, G_2, \ldots, G_{n/(r+1)}$ be any partition of $P$ such that each $G_j$ has size in $[1, r+1]$\;
$Q \leftarrow \emptyset$\;
\ForEach{$G_j$ in the partition}{
Compute the upper hull $v_1, v_2, \ldots, v_t$ of $G_j$\;
\For{$i = 1$ \KwTo $t-1$}{
$\sigma \leftarrow $ the slope of the line passing through $v_i, v_{i+1}$\;
$Q \leftarrow Q \cup \{\sigma\}$\;
}
}
\For{$k = 1$ \KwTo $r$}{
$\sigma_k \leftarrow \mbox{the } k|Q|/(r+1)$-th smallest slope in $Q$\;
$s_k \leftarrow \mbox{the extreme point in $P$ that supports } \sigma_k$\;
}
$(s_0, \sigma_0, s_{r+1}, \sigma_{r+1}) \leftarrow (p_L, -\infty, p_R, \infty)$\;
\For{$k = 1$ \KwTo $r+1$}{
$P_k \leftarrow \emptyset$\;
\ForEach{$G_j$ in the partition}{
Compute the upper hull $v_1, v_2, \ldots, v_t$ of $G_j$\;
Find the extreme point $v_{\alpha}$ (resp. $v_\beta$) in $G_j$ that supports $\sigma_{k-1}$ (resp. $\sigma_k$)\;
$P_k \leftarrow P_k \cup \{v_{\alpha}, v_{\alpha+1}, \ldots, v_{\beta}\}$\;
}
Remove the points that lie below or on the line passing through $s_{k-1}$, $s_{k}$ from $P_k$\;
\If{$P_k \ne \emptyset$}{
Recurse on $P_k \cup \{s_{k-1}, s_k\}$\;
}
}
\caption*{Compute the upper hull $\UH{P}$ of $P$. \label{fig:RAMalgo}}
\end{algorithm}
}
\addtocounter{algocf}{1}
\begin{lemma}[Cascade-pruning]\label{lem:cascade-pruning}
Given a point set $P \subset \mathbb{R}^2$ and a slope $\sigma$, let $s$ be the extreme point in $P$ that supports $\sigma$. Then, for any $G \subseteq P$ whose $\UH{G} = \{v_1, v_2, \ldots, v_t\}$, $x(v_1) < x(v_2) < \cdots < x(v_t)$, and where $\delta \in [1, t]$ is such that $v_\delta$ is the extreme point in $G$ that supports $\sigma$, we have:
\begin{itemize}[leftmargin=2cm]
\item[Case 1.] If $x(v_i) \le x(s)$ for some $i \in [\delta+1, t]$, then $v_{\delta+1}, \ldots, v_{i} \notin \UH{P}$.
\item[Case 2.] If $x(v_i) \ge x(s)$ for some $i \in [1, \delta-1]$, then $v_{i}, \ldots, v_{\delta-1} \notin \UH{P}$.
\end{itemize}
\end{lemma}
\begin{proof}
Observe that $\sigma(v_{j}, v_{j+1}) \ge \sigma$ for all $j \in [1, \delta-1]$ and $\sigma(v_{j-1}, v_{j}) \le \sigma$ for all $j \in [\delta+1, t]$ because $v_1, v_2, \ldots, v_t$ are extreme points in $\UH{G}$ in clockwise order and $v_\delta$ is the extreme point in $G$ that supports $\sigma$. Since there is an $i \in [\delta+1, t]$ such that $x(v_i) \le x(s)$, we have $x(v_j) \le x(s)$ for each $j \in [\delta+1, i]$. The above are exactly the conditions of Case 1 in Lemma~\ref{lem:prune} for all point pairs $(v_{j-1}, v_{j})$ whose $j \in [\delta+1, i]$. Thus, $v_{j} \notin \UH{P}$ for all $j \in [\delta+1, i]$. The other case can be proved analogously.
\end{proof}
We get the exact bound for each $P_k$ in Lemma~\ref{lem:balanced}, noting that $|P_k| \le \frac{3}{4}|P|$ for $r=1$.
\begin{lemma}\label{lem:balanced}
$
|P_k| \le (\frac{2}{r+1}-\frac{1}{(r+1)^2}) |P| \le 2|P|/(r+1) \mbox{ for each } k \in [1, r+1].
$
\end{lemma}
\begin{proof}
To ensure that, for every $k \in [1, r+1]$, $P_k$ is a small fraction of $P$, we use the cascade-pruning procedure described in Lemma~\ref{lem:cascade-pruning}. Let $\{v_{1}, v_{2}, \ldots, v_{t}\}$ be $\UH{G_j}$ for some $j \in [n/(r+1)]$ where $x(v_1) < x(v_2) < \cdots < x(v_t)$. Let $v_{\alpha_j}$ (resp. $v_{\beta_j}$) be the extreme point in $G_j$ that supports $\sigma_{k-1}$ (resp. $\sigma_{k}$).
Let $n_j$ be the number of points in $P_k \cap G_j$. Recall that $P_k$ does not contain any $v_i$ for any $i \notin [\alpha_j, \beta_j]$, and hence $n_j \le \beta_j - \alpha_j+1$. Observe that point pair $(v_{i}, v_{i+1})$ has slope in the open interval $(\sigma_{k-1}, \sigma_{k})$ for each $i \in [\alpha_j, \beta_j-1]$. Since $\sigma_{k-1}$ (resp. $\sigma_{k}$) is the $(k-1)|Q|/(r+1)$-th largest slope (resp. the $k|Q|/(r+1)$-th largest slope) in $Q$, $Q$ has at most $|Q|/(r+1)$ slopes in the open interval $(\sigma_{k-1}, \sigma_{k})$. This yields that
$$
\sum_{j = 1}^{n/(r+1)} n_j-1 \le \frac{|Q|}{r+1} \Rightarrow \sum_{j = 1}^{n/(r+1)} n_j \le \frac{|Q|}{r+1} + \frac{n}{r+1} \le \frac{r|P|}{(r+1)^2} + \frac{|P|}{r+1}
$$
The last inequality holds because $|Q| \le r|P|/(r+1)$, and it establishes that the number of points from all $G_j$'s that comprise $P_k$ for each $k \in [1, r+1]$ is at most $2|P|/(r+1)$.
\end{proof}
For each $k \in [1, r+1]$, if $P_k \neq \emptyset$, then our algorithm recurses on $P_k \cup \{s_{k-1}, s_k\}$. This ensures that every subproblem has an input that contains some intermediate extreme point(s), i.e. not the leftmost and rightmost extreme points, and any two subproblems where one is not an ancestor or a descendant of the other have an empty intersection in their intermediate extreme point set. As a result,
\begin{lemma}\label{lem:leaves}
Our algorithm has $O(h)$ leaf subproblems.
\end{lemma}
We need Lemma~\ref{lem:leaves} to analyze the running time.
\subsection*{Running Time}
Here we analyze the running time of the RAM algorithm for the case of $r = O(1)$ and defer the discussion for the case of $r = \omega(1)$ until the section on streaming algorithms. Let $T_C$ be the recursive computation tree of the RAM algorithm. The root of $T_C$ represents the initial problem of the recursive computation. Every node in $T_C$ has at most $r+1$ child nodes, each of which represents a recursive subproblem.
For a computation node with the input point set $P$ whose $|P| < r$, we use any $O(|P|\log r)$-time algorithm to compute the convex hull. Otherwise, we need to compute $|P|/(r+1)$ convex hulls of point sets of size at most $r+1$, which runs in $O(|P|\log r)$ time (Lines 1-9). In addition, the quantile selection in $Q$ has the running time $O(|Q|\log r) = O(|P|\log r)$ (Line 11). The $r$ suitable extreme points can be found in $O(|P|\log r)$ time by Lemma~\ref{lem:getextp} (Line 12). The pruning procedure can be done in $O(|P|\log r)$ time by a simple merge (Lines 15-26). Hence, each computation node needs $O(|P|\log r)$ time.
Since each child subproblem has an input set $P_k \cup \{s_{k-1}, s_k\}$ of size at most $2|P|/(r+1)+2$ (Lemma~\ref{lem:balanced}), the running time of child subproblem is an $(2/(r+1))$-fraction of its parent subproblem. Hence, $T_C$ is an \emph{$(2/(r+1))$-fading} computation tree where Edelsbrunner and Shi~\cite{es91} define a recursive computation tree to be $\alpha$-fading for some $\alpha < 1$ if the running time of a child subproblem is an $\alpha$-fraction of its parent. In~\cite{chanthesis95}, Chan extends Edelsbrunner and Shi's results and obtains that, if an $\alpha$-fading recursive computation tree has $L$ leaf nodes and the total running time of the nodes on each level is at most $F$, then the recursive computation tree has total running time $O(F \log L)$. Our algorithm has $O(h)$ leave nodes (Lemma~\ref{lem:leaves}) and $O(|P|\log r)$ time for the computation nodes on each level because two subproblems on the same level have their inputs only intersected at one of their extreme points. We get:
\begin{theorem}\label{thm:ram}
The RAM algorithm runs in $O(n \log h \log r)$ time, and for $r = O(1)$ it is an $O(n \log h)$-time algorithm.
\end{theorem}
\section{A Simpler and Faster Streaming Algorithm}\label{sec:algo}
In this section, we show how to adapt our RAM algorithm to the streaming model. Our streaming algorithm is the same as our RAM algorithm, but we execute the subproblems on $T_C$ in BFS order. That is, starting from the root of $T_C$, all subproblems on $T_C$ of the same level are solved together in a round, then their invoked subproblems are solved together in the next round, and so on. We will see in a moment that our algorithm needs to scan the input $O(1)$ times for each round. Therefore, to have an $O(1)$-pass streaming algorithm, our approach requires $r = n^\delta$ for some positive constant $\delta < 1$. By setting $r = n^\delta$, we have:
\begin{lemma}\label{lem:internal}
By setting the parameter $r$ to be $n^\delta$ for any constant $\delta \in (0, 1)$, the recursive computation tree $T_C$ has $O\left(\delta^{-1} h \right)$ nodes.
\end{lemma}
\begin{proof}
This lemma holds because $T_C$ has depth $O(\log_r n) = O(\delta^{-1})$ by Lemma~\ref{lem:balanced} and $T_C$ has $O(h)$ leaf nodes by Lemma~\ref{lem:leaves}.
\end{proof}
We assign a unique identifier $z \in [1, |T_C|]$ to each of $|T_C| = O(\delta^{-1}h)$ subproblems. Let $S_z$ be the subproblem on $T_C$ whose identifier is $z$. For each $z \in [1, |T_C|]$, $S_z$ has input point set $P_z$. $P_z$ is a subsequence of $P$ and is given to $S_z$ as an input stream of $|P_z|$ points. Our algorithm will generate $P_z$ more than once for $S_z$ to access, for all $z \in [1, |T_C|]$. The data structures used in $S_z$ also are suffixed with $z$. To compute $S_z$, naively we need $O(|P_z|)$ space. We will see in a moment that given $P_z$, how to solve $S_z$ using $O(r \log r |P_z|)$ space in $O(r \log |P_z|+ |P_z|\log r)$ time. We will also see how to generate the input for all the subproblems on $T_C$ of depth $d > 0$ in $O(1)$ passes.
We now establish all these claims, after which we will be ready to prove Theorem~\ref{thm:read-only}.
We decompose $S_z$ into the following three subtasks and describe the algorithms for the subtasks in the subsequent subsections.
\begin{enumerate}
\item Given $P_z$, obtain the $r$ quantile slopes $\sigma_1, \sigma_2, \ldots, \sigma_r$.
\item Given $P_z$ and $\sigma_1, \sigma_2, \ldots, \sigma_r$, obtain the $r$ suitable extreme points $s_1, s_2, \ldots, s_r$.
\item After the ancestor subproblems of $S_z$ (excluding $S_z$) are all solved, generate $P_z$.
\end{enumerate}
\subsection{Obtaining the $r$ quantile slopes}
To find the $r$ quantile slopes for $S_z$ (Lines 1-11 in the RAM algorithm) using small space, we use a Greenwald and Khanna~\cite{gk01} quantile summary structure, abbreviated as $QS_z$. This summary is a data structure that supports two operations: insert a slope ($QS_z$.insert($\sigma$)) and query for (an estimate of) the $t$-th smallest slope ($QS_z$.query($t$)) in $Q_z$. Given access to $QS_z$, we do not have to store the entire $P_z$ to obtain the $r$ quantile slopes. Instead, we invoke $QS_z$.insert($\sigma$) for each slope $\sigma \in Q_z$. After updating all slopes in $Q_z$, we obtain an estimate of the $(r+1)$-quantile of $Q_z$ by invoking $QS_z$.query($k|Q_z|/(r+1)$) for all $k \in [1, r]$. The detailed implementation of the above adaption to the streaming model is given in Algorithm~\ref{algo:slope}.
\begin{algorithm}[H]
Initialize $QS_z$\;
$B_z \leftarrow \emptyset$\;
$q_z \leftarrow 0$\tcc*{$q_z$ counts $|Q_z|$}
\ForEach{$p$ in $P_z$}{
$B_z \leftarrow B_z \cup \{p\}$\;
\If{ $|B_z|$ equals $r+1$ \textbf{or} $p$ is the last point in $P_z$}{
Compute the upper hull $v_1, v_2, \ldots, v_t$ of $B_z$\;
\ForEach{$i = 1$ \KwTo $t-1$}{
$QS_z$.insert($\sigma(v_i, v_{i+1})$)\;
$q_z \leftarrow q_z+1$\;
}
$B_z \leftarrow \emptyset$\;
}
}
\ForEach{$k = 1$ \KwTo $r$}{
$\hat{\sigma}_k \leftarrow QS_z$.query($kq_z/(r+1)$)\;
}
\caption{Compute the $r$ approximate quantile slopes for the subproblem $S_z$. \label{algo:slope}}
\end{algorithm}
$QS_z$.query($k|Q_z|/(r+1)$) returns an estimate $\hat{\sigma}_k$ that has an additive error $c|Q_z|$ in the rank, where $c$ is a parameter to be determined. We set $c = \varepsilon/(r+1)$ for some constant $\varepsilon > 0$ so that the additive error cannot increase the depth of $T_C$ by more than a constant factor. Precisely, because the obtained $\hat{\sigma}_k$ has the rank in the range
$$
[(k-\varepsilon)|Q_z|/(r+1), (k+\varepsilon)|Q_z|/(r+1)]
$$
for each $k \in [1, r]$, we need to replace Lemma~\ref{lem:balanced} with Corollary~\ref{cor:balanced}. Such a replacement increases the depth of $T_C$ from $O(\log_{r} n) = O(\delta^{-1})$ to $O(\log_{r/(1+\varepsilon)} n) = O(\delta^{-1})+o(1)$.
\begin{corollary}\label{cor:balanced}
$|P_k| \le (\frac{2+2\varepsilon}{r+1}-\frac{1}{(r+1)^2})|P| \le 2(1+\varepsilon)|P|/(r+1) \mbox{ for each } k \in [1, r+1].$
\end{corollary}
The summary $QS_z$ needs $O\left(\frac{1}{c} \log (c|Q_z|)\right)$ space, and therefore the space usage for each subproblem is $O((r/\varepsilon) \log((\varepsilon/r)|Q_z|))$. In~\cite{yccs16}, it shows that Greenwald and Khanna's quantile summary needs $O(\log |Q_z|)$ time for an update and $O(\log r + \log\log(|Q_z|/r))$ for a query. Because $S_z$ conducts $O(r)$ updates and $O(r)$ queries, we get:
\begin{lemma}\label{lem:getslope}
Given $P_z$, there exists a streaming algorithm that can obtain the $r$ approximate quantile slopes in $Q_z$ to within any constant factor using $O(r \log(|P_z|+r))$ time and $O(r \log(|P_z|/r))$ space.
\end{lemma}
\subsection{Obtaining the $r$ suitable extreme points}
To find the $r$ suitable extreme points in $P_z$ (Line 12 in the RAM algorithm), a naive implementation, which would update the supporting points of $\hat{\sigma}_k$ for all $k \in [1, r]$ once for each point $p \in P_z$, needs $O(r|P_z|)$ running time. To reduce the running time to the claimed time complexity $O(r \log |P_z| + |P_z| \log r)$, we need the following observation.
\begin{observation}\label{obs:pred-succ}
For any non-singleton set $G$ whose extreme points in the upper hull $\UH{G}$ from left to right are $v_1, v_2, \ldots, v_t$, the point in $G$ that supports a given slope $\sigma$ is
$$
s = \left\{
\begin{array}{ll}
v_1 & \mbox{ if } \sigma > \sigma(v_1, v_2) \\
v_t & \mbox{ if } \sigma < \sigma(v_{t-1}, v_t) \\
v_i & \mbox{ if } \sigma(v_{i-1}, v_i) \ge \sigma \ge \sigma(v_i, v_{i+1}) \mbox{ for some } i \in [2, t-2]
\end{array}
\right.
$$
\end{observation}
To find the extreme points in $P_z$ that supports $\hat{\sigma}_k$ for all $k \in [1, r]$, we compute the extreme points $v_1, v_2, \ldots, v_t$ in $P_z$ from left to right, generate a (sorted) list $\ell_A$ of slopes $\sigma(v_1, v_2), \sigma(v_2, v_3), \ldots, \sigma(v_{t-1}, v_t)$, and merge $\ell_A$ with another (sorted) list $\ell_B$ of the approximate $(r+1)$-quantile slopes $\hat{\sigma}_1, \hat{\sigma}_2, \ldots, \hat{\sigma}_r$. By Observation~\ref{obs:pred-succ}, the point $\hat{s}_k$ in $P_z$ that supports $\hat{\sigma}_k$ for each $k \in [1, r]$ can be easily determined by the its predecessor and successor in $\ell_A$. Scanning the merged list suffices to get $\hat{s}_1, \hat{s}_2, \ldots, \hat{s}_k$. Though the above reduces the time complexity to $O(r + |P_z| \log |P_z|)$, the space complexity $O(|P_z|)$ is much higher than the claimed space complexity $O(r \log r|P_z|)$ for $r \ll |P_z|$. To remedy, again, we reduce this problem to computing the upper hulls of $|P_z|/(r+1)$ smaller point sets. First, we partition $P_z$ arbitrarily into $G_1, G_2, \ldots, G_{|P_z|/(r+1)}$ so that each group $G_i$ has size $|G_i| \in [1, r+1]$ points. Then, for each $G_i$ we apply the above accordingly, detailed in Algorithm~\ref{algo:support}. We get:
\begin{lemma}\label{lem:getextp}
Given $P_z$ and sorted $\sigma_1, \sigma_2, \ldots, \sigma_r$, there exists a streaming algorithm that can obtain the extreme points in $P_z$ that support $\sigma_i$ for all $i \in [1, r]$ using $O(r+|P_z|\log r)$ time and $O(r)$ space.
\end{lemma}
\begin{algorithm}[H]
$B_z \leftarrow \emptyset$\;
$\hat{s}_k \leftarrow (0, -\infty)$ for each $k \in [1, r]$\;
\ForEach{$p$ in $P_z$}{
$B_z \leftarrow B_z \cup \{p\}$\;
\If{ $|B_z|$ equals $r+1$ \textbf{or} $p$ is the last point in $P_z$}{
Compute $\UH{B_z}$ and obtain its extreme points from left to right, $v_1, v_2, \ldots, v_t$\;
$\ell_A \leftarrow \sigma(v_1, v_2), \sigma(v_2, v_3), \ldots, \sigma(v_{t-1}, v_t)$\;
$\ell_B \leftarrow \hat{\sigma}_1, \hat{\sigma}_2, \ldots, \hat{\sigma}_r$\;
Merge $\ell_A$ and $\ell_B$\;
\ForEach{$k = 1$ \KwTo $r$}{
Find the predecessor and successor of $\hat{\sigma}_k$ in $\ell_A$ by scanning the merged list\;
By which and Observation~\ref{obs:pred-succ}, obtain the point $b_k$ in $B_z$ that supports $\hat{\sigma}_k$\;
$\hat{s}_k \leftarrow$ the point in $\{\hat{s}_k, b_k\}$ that supports $\hat{\sigma}_k$\;
}
}
}
\caption{Compute the $r$ suitable extreme points for the subproblem $S_z$. \label{algo:support}}
\end{algorithm}
\subsection{Generating the input point set $P_z$ for each subproblem $S_z$}
Recall that we execute the subproblems in $T_C$ in BFS order. Upon executing the subproblems of depth $d$ for any $d > 0$, all the subproblems of depth less than $d$ are done and the associated $r$ quantile slopes and $r$ suitable extreme points are memoized in memory. For $d = 0$, we need to generate the input for the initial problem $S_o$. Because its input point set is exactly $P$, scanning over $P$ suffices.
Given the associated $r$ quantile slopes and $r$ suitable extreme points for all the subproblems of depth less than $d$, to generate the input point sets for all the subproblems of depth $d$, we can directly execute Lines 15-26 in the RAM algorithm for all the subproblems of depth less than $ d$ and ignore Lines 1-14 because the intermediate values, the quantile slopes and suitable extreme points, are already computed and kept in memory. Initially, we allocate a buffer $B_z$ of size $r+1$ for each subproblem $S_z$ of depth less than $d$ so as to temporarily store the incoming input points, i.e. points in $P_z$. Then, we scan $P$ on the input tape once and for each input point $p$ in $P$, we place $p$ in the buffer $B_o$ of $S_o$. Once any buffer $B_z$ gets full or the input terminates, we let $B_z$ be some $G_i$, a part in the partition of $P_z$, and apply the pruning procedure stated in Lines 15-26 in the RAM algorithm. Those points that survive the pruning are flushed, one by one, into the buffers of $S_z$'s child subproblems. We apply the above iteratively until we reach the end of the input tape. The space usage counted on each $S_z$ is $O(|B_z|) = O(r)$ and the overall running time to generate the input point set for all the subproblems of depth $d > 0$ is $O(d n \log r)$ because all the subproblems of each depth $i \in [1, d-1]$ computes the upper hull of points sets, disjoint subsets of $P$. Hence, we get:
\begin{lemma}\label{lem:geninput}
There exists a streaming algorithm that can generate the input for all the subproblems on $T_C$ of depth $d$ for each $d \in [0, \depth{T_C}]$ using $O(1)$ passes, $O(hr)$ space, and $O(d n \log r)$ time.
\end{lemma}
\begin{proof}[Proof of Theorem~\ref{thm:read-only}]
For $r = n^\delta$, $T_C$ has $O(\delta^{-1}h)$ nodes and depth $O(\delta^{-1})$ by Lemmas~\ref{lem:internal} and~\ref{lem:balanced}. Hence, the space complexity of our streaming algorithm is the sum of $O(\delta^{-1}h)$ times the space complexity in Lemmas~\ref{lem:getslope} and~\ref{lem:getextp}, and $O(\delta^{-1})$ times the space complexity in Lemma~\ref{lem:geninput}. The overall space complexity is $O(\delta^{-1} h n^\delta\log n)$. One can obtain the space bound $O(\min\{\delta^{-1} hn^\delta\log n, n\})$ by checking whether $\hbar n^\delta \log n > n$ before proceeding to the subproblems on the next depth, where $\hbar$ is the number of subproblems executed so far and thus $\hbar = O(\delta^{-1}h)$. If so, we compute the convex hull by a RAM algorithm.
Analogously, we have that the pass (resp. time) complexity of our streaming algorithm is $O(\delta^{-1})$ (resp. $O(\delta^{-2} n \log n)$)
\end{proof}
\section{A W-Stream Algorithm Of Nearly-Optimal Pass-Space Tradeoff}\label{sec:wrstr}
Demetrescu et al.~\cite{cbga10} establish a general scheme to convert PRAM algorithms to W-stream algorithms. Theorem~\ref{thm:parallel} is an implication of their main result.
\begin{theorem}[Demetrescu et al.~\cite{cbga10}] \label{thm:parallel}
If there exists a PRAM algorithm that uses $m$ processors to compute the convex hull of $n$ given points in $t$ rounds, then there exists an $O(s)$-space $O(mt/s)$-pass W-stream algorithm to compute the convex hull.
\end{theorem}
There is a long line of research that studies how to compute the convex hull of $n$ given points efficiently in parallel~\cite{chow80,ag86,jag86,akl84,gg91,gs97}. In particular, Akl's PRAM algorithm~\cite{akl84} uses $O(n^\varepsilon)$ processors and runs in $O(n^{1-\varepsilon}\log h)$ time for any $\varepsilon \in (0, 1)$. Converting Akl's PRAM algorithm to a W-stream algorithm by Theorem~\ref{thm:parallel}, we have:
\begin{corollary}\label{cor:auto}
There exists an $O((n/s)\log h)$-pass W-stream algorithm that can compute the convex hull of $n$ given points using $O(s)$ space.
\end{corollary}
The optimal work, i.e., the total number of primitive operations that the processors perform, for any parallel algorithm in the algebraic decision tree model\footnote{Roughly speaking, algorithms are decision trees in which each computation node is able to test the sign of the evaluation of a constant-degree polynomial.}
to compute the convex hull is $O(n \log h)$~\cite{ks86,gs97}. Therefore the W-stream algorithm stated in Corollary~\ref{cor:auto} is already the best possible among those W-stream algorithms that are converted from a PRAM algorithm in the algebraic decision tree model by Theorem~\ref{thm:parallel}. However, in this Section, we will show that such a tradeoff between pass complexity and space usage is suboptimal by devising a W-stream algorithm that has a better pass-space tradeoff. Together with the results shown in Section~\ref{sec:lower}, we have that the pass-space tradeoff of our W-stream algorithm is nearly optimal.
\subsection{Deterministic W-stream Algorithm}
Our deterministic W-stream algorithm is the same as our streaming algorithm, except for the following differences:
\begin{itemize}
\item We set $r = 1$ (rather than $r = n^\delta$) for our deterministic W-stream algorithm. Thus, by Corollary~\ref{cor:balanced} $\depth{T_C}$ increases from $O(\delta^{-1})$ to $O(\log n)$, but the space usage of subproblem $S_z$ decreases from $O(n^\delta \log n)$ to $O(\log n)$ for each $z \in [1, |T_C|]$. Moreover, if the extreme point in the input $P$ that supports the approximate median slope is the leftmost point $p_L$ or the rightmost point $p_R$, i.e. the degenerate case, we replace it with the extreme point that supports $\sigma(p_L, p_R)$. In this way, each subproblem on $T_C$ has a unique extreme point and therefore the number of subproblems on $T_C$ is $O(h)$.
\item Our streaming algorithm executes the subproblems on $T_C$ in BFS order, that is, all subproblems of depth $d$ are executed in a round for each $d \in [0, \depth{T_C}]$. In contrast, our deterministic W-stream algorithm refines a single round into subrounds, in each of which it takes care of $O(s/\log n)$ subproblems, so as to bound the working space by $O(s)$.
\item Note that algorithms in the W-stream model are capable of modifying the input tape. Formally, while scanning the input tape in the $i$-th pass, algorithms can write something on a write-only output stream; in the $(i+1)$-th pass, the input tape read by algorithms is the output tape written in the $i$-th pass. Hence, our deterministic W-stream algorithm is able to assign an attribute to each point $p \in P$ to indicate that $p$ is an input of a certain subproblem. Moreover, our deterministic W-stream algorithm can write down the parameters for every subproblem on the output tape. In each subround, our deterministic W-stream algorithm needs to scan the input twice. The first pass is used to load the parameters of subproblems to be solved in the current subround. The second pass is used to scan the input tape and process the points that are the input points for the subproblems to be solved in the current subround.
\end{itemize}
We are ready to prove Theorem~\ref{thm:det}.
\begin{proof}[Proof of Theorem~\ref{thm:det}]
Suppose there are $h_d$ subproblems of depth $d$ on $T_C$ for each $d \in [0, \depth{T_C}]$, then our deterministic W-stream algorithm has to execute
$$
\sum_{d \in [0, \depth{T_C}]} \left\lceil \frac{h_d}{\lfloor s/\Theta(\log n) \rfloor} \right\rceil = O\left(\lceil h/s \rceil \log n\right)
$$
subrounds for any $s = \Omega(\log n)$. Because our deterministic W-stream algorithm scans the input tape twice for each subround, the pass complexity is $O(\lceil h/s \rceil \log n)$.
As shown in Section~\ref{sec:algo}, subproblem $S_z$ needs $O(|P_z| \log |P_z|)$ running time. Since the input of subproblems of depth $d$ on $T_C$ are disjoint subsets of $P$, for each $d \in [0, |T_C|]$. We get that the time complexity is $O(n \log^2 n)$.
\end{proof}
\subsection{Randomized W-stream Algorithm}
Observe that for $r = 1$, finding the $r$ approximate quantile slopes in $Q_z$ is exactly finding the approximate median slope in $Q_z$. Our algorithms mentioned previously all use Greenwald and Khanna quantile summary structure, which needs $O(\log n)$ space. In our randomized W-stream algorithm, we replace the Greenwald and Khanna quantile summary with a random slope in $Q_z$, thereby reducing the space usage to $O(1)$. As noted by Bhattacharya and Sen~\cite{bs97}, such a replacement cannot increase the depth of $T_C$ by more than a constant factor w.h.p. Consequently, we get Theorem~\ref{thm:rand}.
\begin{proof}[Proof of Theorem~\ref{thm:rand}]
Similar to the arguments used in the proof of Theorem~\ref{thm:det}, the pass complexity of our randomized W-stream algorithm is
$$
\sum_{d \in [0, \depth{T_C}]} \left\lceil \frac{h_d}{\lfloor s/\Theta(1) \rfloor} \right\rceil = O\left(h/s+\log n\right)
$$
for any $s = \Omega(1)$ w.h.p. and the time complexity is $O(n \log^2 n)$ w.h.p.
\end{proof}
\newcommand{Kalyanasundaram and Schintger~\cite{ks92}\xspace}{Kalyanasundaram and Schintger~\cite{ks92}\xspace}
\section{Unconditional Lower Bound}\label{sec:lower}
In this section, we will show that any streaming (or W-stream) algorithm that can compute the convex hull with success rate $> 2/3$ using $O(s)$ space requires $\Omega(\ceil{h/s})$ passes (i.e. Theorem~\ref{thm:lower}). This establishes the near-optimality of our proposed algorithms. We note here that the lower bound holds even if the output is the quantity $|\ext{P}|$, rather than the set $\ext{P}$.
\input{fig2}
We construct a point set $U$ so that it is hard to compute the convex hull of point set $P = Q \cup \{(1, 0), (-1, 0)\}$
for all $Q \subseteq U$, as illustrated in Figure~\ref{fig:hard}. Let $C_1, C_2$ be concentric half circles. The radius of $C_1$ equals 1 and that of $C_2$ is any value in $(k,1)$ for some $k$ to be determined later. Let $a_0, a_1, \ldots, a_{n+1}$ be points distributed evenly on $C_1$ so that $a_0 = (1, 0)$ and $a_{n+1} = (-1, 0)$. Define $b_0, b_1, \ldots, b_{n+1}$ on $C_2$ similarly. Let $k$ be the distance between the origin $O$ and the line $\overleftrightarrow{a_i a_{i+2}}$ for any $i \in [0, n-1]$. Let $U$ be the set $\{a_i : i \in [1, n]\} \cup \{b_i : i \in [1, n]\}$.
Before proceeding to the hardness proof, observe the following geometric property of points in $U$.
\begin{lemma}\label{lem:geo}
For every $Q \subseteq U$, let $R = \ext{Q \cup \{(1, 0), (-1, 0)\}}$. We have that
\begin{enumerate}[leftmargin=1.5cm,label={(\arabic*)}]
\item If $a_i \in Q$, then $a_i \in R$.
\item If $b_i \in Q$, then $b_i \in R$ iff $a_i \notin Q$.
\end{enumerate}
\end{lemma}
\begin{proof}
(1) Since $a_i$ is on $C_1$, $a_i$ cannot be expressed as a convex combination of any other points in $C_1 \setminus \{a_i\}$. That implies $a_i$ is an extreme point of $Q$ as long as $a_i \in Q$, and the same argument holds for every $i$.
(2 $\Rightarrow$) If $a_i \in Q$, then $b_i \in \mathrm{\Delta}a_0a_ia_{n+1}$ and thus $b_i \notin R$.
(2 $\Leftarrow$) If $a_i \notin Q$, then $b_i \in R$. To see why, we draw a tangent line $L_{b_i}$ of $C_2$ passing through $b_i$ as illustrated in Figure~\ref{fig:tangentLineFig}. Since $r > k$, all the points in $U \setminus \{a_i\}$ are strictly on one side of $L_{b_i}$, implying that $b_i$ cannot be expressed as a convex combination of any other points in $R \setminus \{a_i\}$ for all $R$. Thus, $b_i$ is an extreme point if $a_i \notin Q$.
\end{proof}
Lemma~\ref{lem:geo} implies the fact that, for every $Q \subseteq U$,
$$
|\ext{Q \cup \{(1, 0), (-1, 0)\}}| = |Q|+2
$$
if and only if $a_i$ and $b_i$ are not both contained in $Q$ for each $i$. Given this fact, we are ready to perform a reduction from the \emph{set disjointness} problem (a two-party communication game) to computing the convex hull in the streaming (and W-stream) model. Set disjointness is defined as follows:
\begin{itemize}[leftmargin=1.5cm]
\item[Given:] Alice has a private $(\alpha n)$-size subset $A$ of $[n]$, and Bob has another private $(\alpha n)$-size subset $B$ of $[n]$ for some constant $\alpha < 1/2$.
\item[Goal:\;\,] Answer whether $A$ and $B$ have an non-empty intersection.
\end{itemize}
Kalyanasundaram and Schintger~\cite{ks92} show that
\begin{theorem}[Kalyanasundaram and Schintger~\cite{ks92}\xspace]\label{lem:disj}
No matter which 2-way, multi-round protocol Alice and Bob use, they must communicate $\Omega(n)$ bits to answer the set disjointness problem with constant success rate greater than $2/3$.
\end{theorem}
We are ready to proceed to the proof of Theorem~\ref{thm:lower}.
\begin{proof}[Proof of Theorem \ref{thm:lower}]
We claim that, if there exists a streaming (or W-stream) algorithm that can compute the convex hull using $s$ bits in $p$ passes with constant probability greater than $2/3$, then the set disjointness problem can be answered using $(ps)$ bits with constant probability greater than $2/3$. Hence, $ps = \Omega(n)$ or $p = \Omega(\ceil{n/s})$, noting that $p$ is an integer and cannot be a sub-constant.
To prove the claim, for every $A, B \in [n]$ we map them to $Q_A, Q_B \subseteq U$, so that $Q_A = \{a_i : i \in A\}$ and $Q_B = \{b_i : i \in B\}$. Then, we use a tape to store the points in $Q_A \cup Q_B \cup \{(1, 0), (-1, 0)\}$, where $Q_A$ occupies the former half of the tape and the rest of points occupies the latter half of the tape. If there exists an algorithm that can compute the convex hull of $R = Q_A \cup Q_B \cup \{(1, 0), (-1, 0)\}$, it must know what $|\ext{R}|$ is. Given the above fact, we have that $A \cap B = \emptyset$ iff $|\ext{R}| = |Q_A| + |Q_B| +2$. This gives us the ability to solve the set disjointness problem.
Then, we generate four sets of the above arrangement. Let the input of these sets be (1) $A$ and $B$, (2) $\overline{A}$ and $B$, (3) $A$ and $\overline{B}$, and (4) $\overline{A}$ and $\overline{B}$. Instead of distributing the generated points among a half circle, we distribute the generated points among an eighth circle for each set. Then, we concatenate these eighth circles so that they evenly partition a half circle. No matter what $A$ and $B$ are, the total number of extreme points in these sets is $3n$. Since any algorithm that computes convex hull needs to output the extreme points in order, as defined, if we observe the outputted extreme points in the first eighth circle, it suffices to answer the above set disjointness problem, while retaining the number of extreme points to be a fixed value $3n$, given $n$.
Hence, we are able to reduce the set disjointness problem of domain set $[h]$ to computing the convex hull of $3h$ extreme points, one can place $n-3h$ dummy points to the locations that are very close to origin, and thus all dummy points are interior points no matter what $Q_A$ and $Q_B$ are. This establishes Theorem~\ref{thm:lower}.
\end{proof}
\section{Conditional Lower Bound}\label{sec:clower}
In this Section, we prove a conditional lower bound, higher than the unconditional one shown in Section~\ref{sec:lower} for small $h$. This conditional lower bound holds for those algorithms in the algebraic decision tree model, that is:
\begin{itemize}
\item What algorithms can store in memory is a subset of input points.
\item The only operations that algorithms can perform is to test the sign of any continuous function evaluated on the points currently stored in memory.
\end{itemize}
Theorem~\ref{thm:constanthard} is our conditional lower bound, which implies that for constant $h$ any deterministic streaming algorithm that can compute the convex hull in the algebraic decision tree model requires $O(n^\delta)$ space, if the pass complexity $< h/2$. This lower bound is tight because the Gift-Wrapping algorithm can compute the convex hull using $O(h) = O(1)$ space and $h/2$ passes.
\begin{theorem}\label{thm:constanthard}
To compute the upper hull of $n$ points on a plane which has $h$ extreme points, any deterministic streaming algorithm in the algebraic decision tree model that uses $1/\delta$ passes for any $1/\delta < \min\{h/2, \log_h n\}$ requires $\Omega(n^\delta)$ storage of points.
\end{theorem}
\begin{proof}
Given two points $p_L$, $p_R$ and an open disk $D$ that lies above the line $p_L p_R$ and between two vertical lines $x = x_{p_L}$ and $x = x_{p_R}$, consider the problem that computes the upper hull of a $n$-point set $P \subseteq D$, where $P$ satisfies that $U(P \cup \{p_L, p_R\}) = U(P) \cup \{p_L, p_R\}$ and $U(P) = h$. We denote this problem as $S(p_L, p_R, D, n, h)$. The following lemma states that after each pass, the problem $S(p_L, p_R, D, n, h)$ remains as difficult as some problem $S(p^\prime_L, p^\prime_R, D^\prime, (n-2)/s, h-2)$. By setting $s \approx n^\delta$ and applying the lemma repeatedly, we have our theorem.
\end{proof}
\begin{lemma}
Given a problem $S(p_L, p_R, D, n, h)$, for any deterministic streaming algorithm that has storage less than $s$ points, there exists a sequence of $n$ points $P$ inside $D$, a subset $X \subseteq P$, two points $p^\prime_L, p^\prime_R \in P$ and an open disk $D^\prime \subseteq D$, such that after we run the first pass of the algorithm, we have:
\begin{itemize}
\item no point of $X$ is in memory but $p^\prime_L$ and $p^\prime_R$;
\item the result of the pass would be identical if we move the points in $X$ to the arbitrary point in $D^\prime$;
\item the upper hull of $X \cup \{p^\prime_L, p^\prime_R\}$ is equal to the upper hull of $P$;
\item $|X| = \ceil{(n - 2)/ S} - 1$.
\end{itemize}
\end{lemma}
\begin{proof}
Consider $s$ points $p_1=(x_1, y_1), \ldots, p_s=(x_s, y_s)$ such that $p_1, \ldots, p_s \in D$, $p_1, \ldots, p_s$ and $p_L, p_R$ form a strictly concave chain, and for each $p_i$, there exists $q_i^L, q_i^R \in D$ such that $p_L, q_i^L, p_i, q_i^R, p_R$ forms the new upper hull above the concave chain $p_L, p_1, \ldots, p_s, p_R$. In other words, these five points are the upper hull of the point set $p_L, p_1, \ldots, p_s, p_R, q_i^L, q_i^R$. Let $U$ be the set containing all tuples $(x_1, y_1, \ldots, x_s, y_s)$ for such choices of $p_1, \ldots, p_s$. Note that $U$ is open and non-empty. To generate the first $n-2$ points in the stream, the adversary would choose the points only from $p_1, \ldots, p_s$. At every step, he picks $p_i$ such that it is currently not stored in the memory. Because the memory can only store $o(s)$ points, such $p_i$ always exists. By pigeonhole principle, there exists some $p_k$ such that it is chosen by the adversary $\ceil{(n-2)/s} - 1$ times. The adversary would stop to pick such $p_k$ when he is about to choose $p_k$ at $\ceil{(n-2)/s}$-th time but to choose any other $p_i$, no matter it is in memory or not. These $\ceil{(n-2)/s} - 1$ copies of $p_k$ are filled into $X$. Therefore $X$ is not stored in the memory and $|X| = \ceil{(n-2)/s} - 1$.
To satisfy the second condition, during the execution of the algorithm, whenever a test is conducted, we consider the sign of the test function over all the possible tuples in $U$; if not all choices result the same sign, we refine $U$ to be a smaller open and non-empty set in which they do. Then, after the algorithm processes $n-2$ points, because $U$ is open, we can fix the choices of $p_1, \ldots, p_s$ and find an open disk $D_0$ around $p_k$, such that if any copy of $p_k$ is replaced by a point in $D_0$, the outcome of the tests is still the same. Finally, $q_k^L$ and $q_k^R$ are added into the stream as $p_L^\prime, p_R^\prime$. We refine $D_0$ further by finding a smaller open disk $D^\prime \subseteq D_0$ such that it is below two lines $p_L p_L^\prime$ and $p_R p_R^\prime$ and above the line $p_L^\prime p_R^\prime$. Therefore the third condition is satisfied and we are done.
\end{proof}
\section{Omitted Proofs}\label{sec:proof}
\begin{proof}[Proof of Corollary \ref{cor:diam}]
For the diameter, we invoke two instantiations of convex hull W-stream algorithms simultaneously. One reports the extreme point starting at $p_\ell$ (leftmost) and the other starts at $p_r$ (rightmost). They both output the points in the clockwise order, i.e., one outputs the upper hull and the other output the lower one. After the execution of the algorithms,the first $s/2$ points starting from $p_\ell$ in the upper hull and first $s/2$ points starting from $p_r$ in the lower hull are loaded into the memory using one pass. Then we simulate Shamos' method by rotating the initial caliper $\overline{p_\ell p_r}$. When the $s/2$ points in the upper hull or the lower hull are processed, we using another pass to load the next $s/2$ points for the upper hull and the lower hull. We stop when all the extreme points on the convex hull are processed. Note that, it only requires $O(h/s)$ passes, $O(s)$ space and $O(h)$ time to do so.
For the minimum enclosing rectangle, four instantiations of our W-stream algorithms are needed as we are rotating two orthogonal calipers. Then similar approach applies to execute the rotating calipers method detailed in~\cite{t83}. The complexities remain the same. This establishes Corollary~\ref{cor:diam}.
\end{proof}
\input{main.ref}
\end{document}
|
\begin{document}
\title{Equivariant maps between Calogero-Moser spaces}
\author{George Wilson}
\address{Mathematical Institute, 24--29 St Giles, Oxford OX1 3LB, UK}
\email{[email protected]}
\begin{abstract}
We add a last refinement to the results of \mathbb{C}ite{BW1} and
\mathbb{C}ite{BW2} relating ideal classes of the Weyl algebra to
the Calogero-Moser varieties: we show that the bijection constructed
in those papers is {\it uniquely determined} by its equivariance with
respect to the automorphism group of the Weyl algebra.
\end{abstract}
\maketitle
\section{Introduction and statement of results}
Let $\, A \,$ be the Weyl algebra
$\, \mathbb{C}\langle x,y \rangle /(xy - yx -1) \,$, and
let $\, \mathcal{R} \,$ be the space of noncyclic right ideal classes of $\, A \,$
(that is, isomorphism classes of noncyclic finitely generated
rank 1 torsion-free right $\,A$-modules). Let
$\, \mathcal{C} \,$ be the disjoint union of the
{\it Calogero-Moser spaces} $\, \mathcal{C}_n \,$ ($\, n \geq 1 \,$):
we recall that $\, \mathcal{C}_n \,$ is the space
of all simultaneous conjugacy classes of pairs of $\, n \times n \,$
matrices $\, (X,Y) \,$ such that $\, [X,Y] + I \,$ has rank $1 \,$.
It is a smooth irreducible affine variety of dimension $\, 2n \,$
(see \mathbb{C}ite{W}).
For simplicity, in what follows we shall use the same notation
$\, (X,Y) \,$ for
a pair of matrices and for the corresponding point of $\, \mathcal{C}_n \,$.
Let $\,G \,$ be the group of $\mathbb{C}$-automorphisms of
$\, A \,$, and let $\, \Gamma \,$ and $\, \Gamma' \,$ be the isotropy
groups of the generators $\, y \,$ and $\, x \,$ of $\, A \,$. Thus
$\, \Gamma \,$ consists of all automorphisms of the form
$$
\mathbb{P}hi_p(x) = x - p(y) \,, \quad \mathbb{P}hi_p(y) = y
$$
where $\, p \,$ is a polynomial; and similarly $\, \Gamma' \,$ consists of
all automorphisms of the form
$$
\mathbb{P}si_q(x) = x \,, \quad \mathbb{P}si_q(y) = y - q(x) \,
$$
where $\, q \,$ is a polynomial. According to Dixmier (see \mathbb{C}ite{D}),
$\, G \,$ is generated by the subgroups $\, \Gamma \,$ and $\, \Gamma' \,$.
There is an obvious action of $\, G \,$ on $\, \mathcal{R} \,$; we let $\, G \,$ act
on $\, \mathcal{C} \,$ by the formulae
\begin{equation}
\mathbb{P}hi_p(X,Y) = (X + p(Y),\, Y) \,, \quad
\mathbb{P}si_q(X,Y) = (X,\, Y + q(X)) \,.
\end{equation}
According to \mathbb{C}ite{BW1} this $ \, G $-action is {\it transitive}
on each space $\, \mathcal{C}_n \,$. The main result of \mathbb{C}ite{BW1} was
the following.
\begin{theorem}
\label{bw}
There is a bijection between the spaces $\, \mathcal{R} \,$ and $\, \mathcal{C} \,$ which
is equivariant with respect to the above actions of $\, G \,$.
\end{theorem}
This bijection constructed in \mathbb{C}ite{BW1} was obtained in a quite different
way in \mathbb{C}ite{BW2}. The proof in \mathbb{C}ite{BW2} that the two constructions
agree used the fact that equivariance was known in both cases; thus
to prove that the bijections coincide, it was enough to check one point
in each $\, G$-orbit, that is, in each space $\, \mathcal{C}_n \,$. The result
to be proved in the present note is that even this (not difficult) check
was unnecessary.
\begin{theorem}
\label{preT}
There is only one $\, G$-equivariant bijection between the spaces
$\, \mathcal{R} \,$ and $\, \mathcal{C} \,$.
\end{theorem}
Clearly, it is equivalent to show that there is no nontrivial
$\, G$-equivariant bijection from $\, \mathcal{C} \,$ to itself. We shall show
a little more, namely, that (apart from the identity)
there is no $\, G$-equivariant map (for short: $G$-map) at all
from $\, \mathcal{C} \,$ to itself. Since a $\, G$-map
must take each orbit onto another orbit, that amounts to the following
assertion.
\begin{theorem}
\label{T}
{\rm (i)} For any $\, n \geq 1 \,$,
let $\, f : \mathcal{C}_n \to \mathcal{C}_n \,$ be a $\, G$-map.
Then $\, f \,$ is the identity. \\
{\rm (ii)} For $\, n \neq m \,$
there is no $\, G$-map from
$\, \mathcal{C}_n \,$ to $\, \mathcal{C}_m \,$.
\end{theorem}
Since $\, \mathcal{C}_n \,$ and
the action of $\, G \,$ on it are defined by simple formulae involving
matrices, the proof of Theorem~\ref{T}
is just an exercise in linear algebra. Quite
possibly there is a simpler solution to the exercise than the one given below.
The first part of Theorem~\ref{T} is equivalent to the statement
that the isotropy group of
any point of $\, \mathcal{C} \,$ (or $\, \mathcal{R} \,$) coincides with its normalizer in
$\, G \,$ (see section~\ref{trivial} below); in particular, these isotropy
groups are not normal in $\, G \,$, confirming a suspicion of Stafford
(see \mathbb{C}ite{St}, p.~636).
Stafford's conjecture seems to have been the motivation for Kouakou's
work \mathbb{C}ite{K}, which contains a result equivalent to ours.
The proof in \mathbb{C}ite{K} looks quite different from the present one, because
Kouakou does not use the spaces $\, \mathcal{C}_n \,$, but rather the alternative
description of $\,\mathcal{R} \,$ (due to Cannings and Holland, see \mathbb{C}ite{CH}) as the
adelic Grassmannian of \mathbb{C}ite{W}. I have not entirely succeeded in following
the details of \mathbb{C}ite{K}; in any case, it seems
worthwhile to make available the independent verification of the result
offered here.
\begin{remark*}
We have excluded from $\, \mathcal{R} \,$ the cyclic ideal class, corresponding to
the Calogero-Moser space $\, \mathcal{C}_0 \,$ (which is a point).
The reason is very trivial: since there is
always a map from any space to a point, part (ii) of Theorem~\ref{T} would
be false if we included $\, \mathcal{C}_0 \,$. However, Theorem~\ref{preT} would
still be true.
\end{remark*}
\section{Proof of Theorem~\ref{T} in the case $\, n < m \,$}
If we accept (cf.\ \mathbb{C}ite{BW1}, section 11)
that the $\, \mathcal{C}_n \,$ are homogeneous spaces for the
(infinite-dimensional) {\it algebraic} group $\, G \,$, then
Theorem~\ref{T} becomes
obvious in the case $\, n < m \,$. Indeed, any $\,G$-map
from $\, \mathcal{C}_n \,$ to $\, \mathcal{C}_m \,$ would have to be a
surjective map of {\it algebraic varieties}, which is clearly impossible
if $\, n < m \,$, because then
$\, \mathcal{C}_m \,$ has greater dimension ($2m$) than $\, \mathcal{C}_n \,$. For
readers who are not convinced by this argument, we offer a more elementary one,
based on the following lemma.
\begin{lemma}
\label{diag}
Let $\, f : \mathcal{C}_n \to \mathcal{C}_m \,$ be a $\, G$-map.
Suppose that $\, f(X,Y) = (P,Q) \,$, and that
$\, P \,$ is diagonalizable. Then every eigenvalue of $\, P \,$
is an eigenvalue of $\, X \,$.
\end{lemma}
\begin{proof}
Let $\, \mathbb{C}hi \,$ be the minimum polynomial of $\, X \,$: then in
$\, \mathcal{C}_m \,$ we have
$$
(P,Q) = f(X,Y) = f(X,Y + \mathbb{C}hi(X)) = (P, Q + \mathbb{C}hi(P))
$$
(where the last step used the fact that $\, f \,$ has to commute
with the action of $\, \mathbb{P}si_{\mathbb{C}hi} \in G \,$).
That means that there is an invertible matrix $\, A \,$ such that
$$
APA^{-1} = P \text{\ \ and \ } AQA^{-1} = Q + \mathbb{C}hi(P) \ .
$$
We may assume that $\, P = \partialiag(p_1, \ldots, p_m) \,$ is diagonal.
Then since the $\, p_i \,$ are distinct (see \mathbb{C}ite{W}, Proposition 1.10),
$\, A \,$ is diagonal too, so
taking the diagonal entries in the last equation gives
$\, q_{ii} = q_{ii} + \mathbb{C}hi({p_i}) \,$, whence
$\, \mathbb{C}hi({p_i}) = 0 \text{\, for all \,} i \,$.
Thus $\, \mathbb{C}hi(P) = 0 \,$, so the
minimum polynomial of $\, P \,$ divides $\,\mathbb{C}hi \,$. The lemma follows.
\end{proof}
\begin{corollary}
If $\, n < m \,$ there is no $\, G$-map
$\, f : \mathcal{C}_n \to \mathcal{C}_m \,$.
\end{corollary}
\begin{proof}
Choose $\, (P,Q) \in \mathcal{C}_m \,$ with $\, P \,$ diagonalizable. Since
$\, \mathcal{C}_m \,$ is just one $\, G$-orbit, $\, f \,$ is surjective, so we
can choose $\, (X,Y) \in \mathcal{C}_n \,$ with $\, f(X,Y) = (P,Q) \,$. But then
Lemma~\ref{diag} says that $\, X \,$ is an $\, n \times n \,$ matrix
with more than $\,n \,$ distinct eigenvalues, which is impossible.
\end{proof}
\section{The base-point}
\label{base}
A useful subgroup of $\, G \,$
is the group $\, R \,$ of {\it scaling transformations}, defined by
$$
R_{\lambda}(x) = \lambda x \,, \ R_{\lambda}(y) = \lambda^{-1} y
\quad (\lambda \in \mathbb{C}^{\times}) \,.
$$
It acts on $\, \mathcal{C}_n \,$ in a similar way:
\begin{equation}
R_{\lambda}(X,Y) = (\lambda^{-1} X,\, \lambda Y) \,.
\end{equation}
\begin{lemma}
\label{scale}
Suppose that the conjugacy class $\, (X,Y) \in \mathcal{C}_n \,$ is fixed by
the group $\, R \,$. Then $\, X \,$ and $\, Y \,$ are both nilpotent.
\end{lemma}
\begin{proof}
Let $\, \mu \,$ be an eigenvalue of (say) $\, Y \,$. Then
for any $\, \lambda \in \mathbb{C}^{\times} \,$,
$\, \lambda \mu \,$ is an eigenvalue of $\, \lambda Y \,$, which is
(by hypothesis) conjugate to $\, Y \,$. Thus
$\, \lambda \mu \,$ is an eigenvalue of $\, Y \,$ for every
$\, \lambda \in \mathbb{C}^{\times} \,$, which is impossible unless $\, \mu = 0 \,$.
Hence all eigenvalues of $\, Y \,$ must be $\, 0 \,$, that is,
$\, Y \,$ must be nilpotent. The same argument applies to $\, X \,$.
\end{proof}
The converse to Lemma~\ref{scale} is also true, but we shall use that
fact only for the pair $\, (X_0, Y_0) \,$ given by
\begin{equation}
X_0 \,=\, \left(
\begin{array}{ccccc}
0 & 0 & 0 & \ldots & 0\\
1 & 0 & 0 & \ldots & 0\\
0 & 2 & 0 & \partialdots & \vdots \\
\vdots & \vdots & \partialdots & \partialdots & 0 \\
0 & 0 & \ldots & n-1 & 0
\end{array}
\right)\ , \quad
Y_0 \, = \, \left(
\begin{array}{ccccc}
0 & 1 & 0 & \ldots & 0\\
0 & 0 & 1 & \ldots & 0\\
0 & 0 & 0 & \partialdots & \vdots\\
\vdots & \vdots & \partialdots & \partialdots & 1 \\
0 & 0 & \ldots & 0 & 0
\end{array}
\right)\ .
\end{equation}
We shall regard $\, (X_0, Y_0) \,$ as the {\it base-point} in $\, \mathcal{C}_n \,$.
In the rather trivial case $\, n=1 \,$, we have $\, \mathcal{C}_1 = \mathbb{C}^2 $, and
we interpret $\, (X_0, Y_0) \,$ as $\, (0,0) \,$.
\begin{lemma}
\label{scal}
The (conjugacy class of) the
pair $\, (X_0, Y_0) \in \mathcal{C}_n \,$ is fixed by the group $\, R \,$.
\end{lemma}
\begin{proof}
For $\, \lambda \in \mathbb{C}^{\times} \,$, let $ \, d(\lambda) \,$ be the diagonal
matrix
$$
d(\lambda) := \partialiag(\lambda, \lambda^2, \ldots, \lambda^n) \,.
$$
Then $\, d(\lambda)^{-1} X d(\lambda) = \lambda^{-1} X \,$
and $\, d(\lambda)^{-1} Y d(\lambda) = \lambda Y \,$.
\end{proof}
\begin{corollary}
\label{cor}
Let $\, f : \mathcal{C}_n \to \mathcal{C}_m \,$ be a $\, G$-map, and let
$\, f(X_0, Y_0) = (P,Q) \,$. Then $\, P \,$ and $\, Q \,$ are nilpotent.
\end{corollary}
\begin{proof}
This follows at once from Lemmas~\ref{scale} and \ref{scal}, since
a $\,G$-map must respect the fixed point set of any subgroup
of $\, G \,$.
\end{proof}
\section{Proof of Theorem~\ref{T} in the case $\, n > m \,$}
The remaining parts of the proof use the following trivial fact.
\begin{lemma}
\label{ppp}
Let $\, (X,Y) \in \mathcal{C}_n \,$, let $\, p \,$ be any polynomial, and let
$\, \mathbb{C}hi \,$ be divisible by the minimum
polynomial of $\, X + p(Y) \,$. Then the
automorphism $\, \mathbb{P}hi_{-p} \mathbb{P}si_{\mathbb{C}hi} \mathbb{P}hi_p \,$ fixes $\, (X,Y) \,$.
\end{lemma}
\begin{proof}
Since $\, \mathbb{C}hi(X + p(Y)) = 0 \,$ we have
\begin{eqnarray}
\mathbb{P}hi_{-p} \mathbb{P}si_{\mathbb{C}hi} \mathbb{P}hi_p(X,\, Y)
&=& \mathbb{P}hi_{-p} \mathbb{P}si_{\mathbb{C}hi}(X + p(Y),\, Y) \nonumber \\
&=& \mathbb{P}hi_{-p}(X + p(Y),\, Y) \nonumber \\
&=& (X, \,Y)\ , \nonumber
\end{eqnarray}
as claimed.
\end{proof}
\begin{proposition}
\label{n>m}
If $\, n > m > 0 \,$ there is no $\, G$-map
$\, f : \mathcal{C}_n \to \mathcal{C}_m \,$.
\end{proposition}
\begin{proof}
We apply Lemma~\ref{ppp} to the base-point $\, (X_0,Y_0) \in \mathcal{C}_n \,$,
with $\, p(t) = t^{n-1} \,$. The minimum ($=$ characteristic)
polynomial of $\, X_0 + Y_0^{n-1} \,$ is
\begin{equation}
\label{char}
\mathbb{C}hi(t) := \partialet(tI - X_0 - Y_0^{n-1}) = t^n - (n-1)! \ .
\end{equation}
Now suppose that $\, f : \mathcal{C}_n \to \mathcal{C}_m \,$ is a $\, G$-map,
and let $\, f(X_0,Y_0) = (P,Q) \,$: according to Corollary~\ref{cor},
$\, P \,$ and $\, Q \,$ are
nilpotent. They are of size less than $\, n \,$,
so we have $\, P^{n-1} = Q^{n-1} = 0 \,$.
Thus
\begin{eqnarray}
\mathbb{P}hi_{-p} \mathbb{P}si_{\mathbb{C}hi} \mathbb{P}hi_p(P,\, Q)
&=& \mathbb{P}hi_{-p} \mathbb{P}si_{\mathbb{C}hi}(P + Q^{n-1},\, Q) \nonumber \\
&=& \mathbb{P}hi_{-p} \mathbb{P}si_{\mathbb{C}hi}(P,\, Q) \nonumber \\
&=& \mathbb{P}hi_{-p}(P,\, Q + P^n - (n-1)! I) \nonumber \\
&=& \mathbb{P}hi_{-p}(P,\, Q - (n-1)! I) \nonumber \\
&=& (\text{something},\, Q - (n-1)! I)\ . \nonumber
\end{eqnarray}
Now, $\, Q - (n-1)!I \,$ is not conjugate
to $\, Q \,$ (because their eigenvalues are different),
hence $\, \mathbb{P}hi_{-p} \mathbb{P}si_{\mathbb{C}hi} \mathbb{P}hi_p \,$
does not fix $\, (P,Q) \,$. So by Lemma~\ref{ppp}, the isotropy
group of $\, (X_0,Y_0) \,$ is not contained in the isotropy
group of $\, f(X_0,Y_0) \,$. This contradiction shows that $\, f \,$
does not exist.
\end{proof}
\section{Proof of Theorem~\ref{T} in the case $\, n = m \,$}
It remains to show that there is no nontrivial $\, G$-map from
$\, \mathcal{C}_n \,$ to itself. Note that because $\, \mathcal{C}_n \,$ is a single orbit,
any such map must be bijective, and must map each point of $\, \mathcal{C}_n \,$
to a point with {\it the same} isotropy group.
In the case $\, n=1 \,$ the result follows (for
example) from Lemma~\ref{diag}, so from now on we shall assume that
$\, n \geq 2 \,$. Let
$\, f : \mathcal{C}_n \to \mathcal{C}_n \,$ be a $\, G$-map, and let
$\, f(X_0, \, Y_0) = (P,Q) \,$. Again, Corollary~\ref{cor} says
that $\, P \,$ and $\, Q \,$ are nilpotent.
We aim to show that $\, (P,Q) \,$
can only be $\, (X_0, \, Y_0) \,$, whence $\, f \,$ is the identity.
We remark first
that if $\, Q^{n-1} = 0 \,$, then the calculation in the proof of
Proposition~\ref{n>m} still gives a contradiction; thus the Jordan
form of $\, Q \,$ consists of just one block, so we may assume that
$\, Q = Y_0 \,$. Now, it is not hard to classify all the points
$\, (X,\, Y_0) \in \mathcal{C}_n \,$ with $\, X \,$ nilpotent (see \mathbb{C}ite{W},
p.26 for the elementary argument): there are exactly $\, n \,$ of them,
and they all have the form $\, (X(\boldsymbol{a}),\, Y_0) \,$, where
$\, \boldsymbol{a} := (a_1, \ldots, a_{n-1}) \,$ and
$\, X(\boldsymbol{a}) \,$ denotes the subdiagonal matrix
\begin{equation}
X(\boldsymbol{a}) \,=\, \left(
\begin{array}{ccccc}
0 & 0 & 0 & \ldots & 0\\
a_1 & 0 & 0 & \ldots & 0\\
0 & a_2 & 0 & \partialdots & \vdots \\
\vdots & \vdots & \partialdots & \partialdots & 0 \\
0 & 0 & \ldots & a_{n-1} & 0
\end{array}
\right)\ .
\end{equation}
The possible vectors $\, \boldsymbol{a} \,$ that
give points of $\, \mathcal{C}_n \,$ are
\begin{equation}
\label{a}
\boldsymbol{a} = (1,2, \ldots, r-1; -(n-r), \ldots, -2, -1)
\quad \text{for} \quad 1 \leq r \leq n \,
\end{equation}
(so $\, r=n \,$ gives $\, X_0 \,$). Thus so far we have shown that
$\, f(X_0, \,Y_0) \,$ must be one of these points
$\, (X(\boldsymbol{a}),\, Y_0) \,$. To finish the argument, we need
the following easy calculations of characteristic polynomials
(the first generalizes \eqref{char}):
\begin{equation}
\label{1}
\partialet(tI - X(\boldsymbol{a}) - Y_0^{n-1}) \,=\,
t^n - \prod_1^{n-1} a_i
\ ;
\end{equation}
\begin{equation}
\label{2}
\partialet(tI - X(\boldsymbol{a}) - Y_0^{n-2}) \,=\,
t^n - (\prod_1^{n-2} a_i + \prod_2^{n-1} a_i)\, t \ ,
\end{equation}
where the last formula holds only for $\, n \geq 3 \,$.
If $\, \boldsymbol{a} \,$ is one of the vectors
\eqref{a} with $\, 1 < r < n \,$, then the right hand side of
\eqref{2} is just $\, t^n \,$; that is,
$\, X(\boldsymbol{a}) + Y_0^{n-2} \, $ is nilpotent. In fact it is easy
to check that the pair $\, (X(\boldsymbol{a}) + Y_0^{n-2}, \,Y_0) \,$
is conjugate to $\, (X(\boldsymbol{a}), \,Y_0) \,$; that is, the
map $\, (X, Y) \mapsto (X + Y^{n-2}, Y) \,$ fixes
$\, (X(\boldsymbol{a}), \,Y_0) \,$. It does not fix $\, (X_0, \, Y_0) \,$,
so $\, f(X_0, \, Y_0) \,$ cannot be any of these points
$\, (X(\boldsymbol{a}), \,Y_0) \,$. It remains only to see that $\, f \,$
cannot map $\, (X_0, \, Y_0) \,$ to the pair
corresponding to $\, r=1 \,$ in \eqref{a}:
let us call it $\, (X_1, \, Y_0) \,$.
If $\, n \,$ is {\it even} we use \eqref{1}: the characteristic
polynomial of $\, X_0 + Y_0^{n-1} \,$ is $\, \mathbb{C}hi(t) := t^n - (n-1)! \,$
while the characteristic
polynomial of $\, X_1 + Y_0^{n-1} \,$ is $\, t^n + (n-1)! \,$, so that
$\, \mathbb{C}hi(X_1 + Y_0^{n-1}) = -2(n-1)! I \,$. We now apply
Lemma~\ref{ppp} with $\, p(t) = t^{n-1} \,$. According to that lemma,
the map
$\, \mathbb{P}hi_{-p} \mathbb{P}si_{\mathbb{C}hi} \mathbb{P}hi_p \,$ fixes $\, (X_0, \, Y_0) \,$;
on the other hand
\begin{eqnarray}
\mathbb{P}hi_{-p} \mathbb{P}si_{\mathbb{C}hi} \mathbb{P}hi_p(X_1,\, Y_0)
&=& \mathbb{P}hi_{-p} \mathbb{P}si_{\mathbb{C}hi}(X_1 + Y_0^{n-1},\, Y_0) \nonumber \\
&=& \mathbb{P}hi_{-p}(X_1 + Y_0^{n-1}, \, Y_0 - 2(n-1)! I) \nonumber \\
&=& (\text{something},\, Y_0 - 2(n-1)! I) \ . \nonumber
\end{eqnarray}
Since $\, Y_0 - 2(n-1)! I \,$ is not conjugate to $\, Y_0 \,$,
this shows that $\, \mathbb{P}hi_{-p} \mathbb{P}si_{\mathbb{C}hi} \mathbb{P}hi_p \,$ does not fix
$\, (X_1, \, Y_0) \,$. Thus in this case $\, f(X_0, \, Y_0) \,$
cannot be equal to $\, (X_1, \, Y_0) \,$
Finally, if $\, n \,$ is {\it odd}, we have a
similar calculation using \eqref{2}. Setting
$\, \alpha := (n-1)! + (n-2)! \,$, the characteristic polynomial
of $\, X_0 + Y_0^{n-2} \,$ is $\, \mathbb{C}hi(t) := t^n - \alpha t \,$
while the characteristic
polynomial of $\, X_1 + Y_0^{n-2} \, $ is $\, t^n + \alpha t \,$,
so that
$\, \mathbb{C}hi(X_1 + Y_0^{n-2}) = -2 \alpha (X_1 + Y_0^{n-2}) \,$.
We now apply
Lemma~\ref{ppp} with $\, p(t) = t^{n-2} \,$. The map
$\, \mathbb{P}hi_{-p} \mathbb{P}si_{\mathbb{C}hi} \mathbb{P}hi_p \,$ fixes $\, (X_0, \, Y_0) \,$;
on the other hand
\begin{eqnarray}
\mathbb{P}hi_{-p} \mathbb{P}si_{\mathbb{C}hi} \mathbb{P}hi_p(X_1,\, Y_0)
&=& \mathbb{P}hi_{-p} \mathbb{P}si_{\mathbb{C}hi}(X_1 + Y_0^{n-2},\, Y_0) \nonumber \\
&=& \mathbb{P}hi_{-p}(X_1 + Y_0^{n-2}, \, Y_0 - 2 \alpha (X_1 + Y_0^{n-2})) \nonumber \\
&=& (\text{something},\, Y_0 - 2 \alpha (X_1 + Y_0^{n-2})) \ . \nonumber
\end{eqnarray}
The matrix $\, Y_0 - 2 \alpha (X_1 + Y_0^{n-2}) \,$ is not
nilpotent, for example because its square does not have trace zero.
Hence $\, \mathbb{P}hi_{-p} \mathbb{P}si_{\mathbb{C}hi} \mathbb{P}hi_p \,$ does not fix
$\, (X_1, \, Y_0) \,$, and the proof is finished.
\section{Other formulations of Theorem~\ref{T}}
\label{trivial}
The remarks in this section are at the level of ``groups acting on sets'':
that is, we may as well suppose that
$\, \mathcal{R} \,$ denotes any set acted
on by a group $\, G \,$. We are interested in the condition
\begin{equation}
\label{simple}
\text{there is no nontrivial $\,G$-map \ } f : \mathcal{R} \to \mathcal{R} \
\end{equation}
(``nontrivial'' means ``not the identity map''). As we observed above, that is
equivalent to the two conditions
\begin{subequations}
\label{simp1}
\begin{equation}
\label{simpi}
\text{each $\,G$-orbit in $\mathcal{R}$ satisfies \eqref{simple}};
\end{equation}
\vspace*{-10mm}
\begin{equation}
\label{simpii}
\text{ if $\, O_1 \,$ and $\, O_2 \,$ are distinct orbits, there is no
$\,G$-map from $\, O_1 \,$ to $\, O_2 \,$.}
\end{equation}
\end{subequations}
Let us reformulate these conditions in terms of the isotropy groups
$\, G_M \,$ of the points $\, M \in \mathcal{R} \,$.
If $\, H \,$ and $\, K \,$ are subgroups of $\, G \,$, then any $\, G$-map
from $\, G/H \,$ to $\, G/K \,$ to must have the form
$\, \varphi(gH) = g(xK) \,$ for some $\, x \in G \,$. This is well-defined
if and only if we have
$$
x^{-1}Hx \subseteq K \ .
$$
In the case $\, H = K \,$, that says that $\, x \in N_G(H) \,$, where
$\, N_G \,$ denotes the normalizer in $\, G \,$: it follows that the
$\, G$-maps from $\, G/H \,$ to itself correspond 1--1 to the points of
$\, N_G(H)/H \,$. Thus the conditions \eqref{simp1} are equivalent to
\begin{subequations}
\label{simp2}
\begin{equation}
\label{simpa}
\text{for any $\,M \in \mathcal{R} \,$}, \text{\, we have \,} G_M = N_G(G_M)\ ;
\end{equation}
\vspace*{-10mm}
\begin{equation}
\label{simpb}
\text{if \,} M \text{\, and \,} N \text{\, are on different orbits,
no conjugate of \,} G_M
\text{ \,is in \,} G_N \ .
\end{equation}
\end{subequations}
Finally, we note that the conditions \eqref{simp2} are equivalent to the
single assertion
\begin{equation}
\label{simp3}
\text{if \,} G_M \subseteq G_N \,, \text{ then \,} M = N\ .
\end{equation}
Indeed, suppose \eqref{simp3} holds, and let $\, x \in N_G(G_M) \,$,
that is, $\, xG_Mx^{-1} \subseteq G_M \,$, or $\, G_{xM} \subseteq G_M \,$.
By \eqref{simp3}, we then have $\, xM = M \,$, that is, $\, x \in G_M \,$.
Thus \eqref{simp3} $\mathcal{R}ightarrow$ \eqref{simpa}. Now, if \eqref{simpb}
is false, we have $\, xG_Mx^{-1} \subseteq G_N \,$, that is,
$\, G_{xM} \subseteq G_N \,$, for some $\, x \in G \,$ and some
$\, M,N \,$ on different orbits. But since they are on different orbits,
$\, xM \neq N \,$, so \eqref{simp3} is false. Thus
\eqref{simp3} $\mathcal{R}ightarrow$ \eqref{simpb}.
Conversely, suppose \eqref{simp2} holds, and let $\, M,N \,$ be such that
$\, G_M \subseteq G_N \,$. By \eqref{simpb}, $\, M \,$ and $\, N \,$ are
on the same orbit, so $\, M = xN \,$ for some $\, x \in G \,$; hence
$\, G_{M} = xG_Nx^{-1} \subseteq G_N \,$. Thus $\, x \in N_G(G_N) \,$,
so by \eqref{simpa}, $\, x \in G_N \,$: hence $\, M=N \,$, as desired.
It is in the form \eqref{simp3} that our result is stated in \mathbb{C}ite{K}.
\scriptsize
\noindent
\textbf{Acknowledgments}. I thank M.\ K.\ Kouakou for kindly allowing me
see his unpublished work \mathbb{C}ite{K}. The main part of this paper was written
in 2006, when I was a participant in the programme on Noncommutative
Geometry at the Newton Institute, Cambridge; the support of EPSRC Grant
531174 is gratefully acknowledged.
\normalsize
\end{document}
|
\begin{document}
\title{On dipolar quantum gases in the unstable regime}
\author[Jacopo Bellazzini]{Jacopo Bellazzini}
\address{Jacopo Bellazzini
\newline\indent
Universit\` a di Sassari
\newline\indent
Via Piandanna 4, 07100 Sassari, Italy}
\email{[email protected]}
\author[Louis Jeanjean]{Louis Jeanjean}
\address{Louis Jeanjean
\newline\indent
Laboratoire de Math\'ematiques (UMR 6623)
\newline\indent
Universit\'{e} de Franche-Comt\'{e}
\newline\indent
16, Route de Gray 25030 Besan\c{c}on Cedex, France}
\email{[email protected]}
\begin{abstract}
We study the nonlinear Schr\"odinger equation arising in dipolar Bose-Einstein condensate in the unstable regime.
Two cases are studied: the first when the system is free, the second when
gradually a trapping potential is added. In both cases we first focus on the existence and stability/ instability properties of standing waves. Our approach leads to the search of critical points of a constrained functional which is unbounded from below on the constraint. In the free case, by showing that the constrained functional has a so-called {\it mountain pass geometry}, we prove the existence of standing states with least energy, the ground states, and show that any ground state is orbitally unstable. Moreover, when the system is free, we show that small data in the energy space scatter in all regimes, stable and unstable. In the second case, if the trapping potential is small, we prove that two different kind of standing waves appears: one corresponds to a {\it topological local minimizer} of the constrained energy functional and it consists in ground states, the other is again of {\it mountain pass type} but now corresponds to excited states. We also prove that any ground state is a {\it topological local minimizer}. Despite the problem is mass supercritical and the functional unbounded from below, the standing waves associated to the set of ground states turn to be orbitally stable. Actually, from the physical point of view, the introduction of the trapping potential stabilizes the system initially unstable. Related to this we observe that it also creates a gap in the ground state energy level of the system. In addition when the trapping potential is active the presence of standing waves with arbitrary small norm does not permit small data scattering. Eventually some asymptotic results are also given.
\end{abstract}
\maketitle
\section{Introduction}
In the recent years the so-called dipolar Bose-Einstein condensate, i.e a condensate made out of particles possessing
a permanent electric or magnetic dipole moment, have attracted much attention, see e.g. \cite{BaCa,BaCaWa,GlMaStHaMa, LaMeSaLePf, NaPeSa, PeSa, SSZL}. At temperature much smaller than the critical temperature it is well described by the wave function $\psi(x,t)$ whose evolution is governed by the three-dimensional (3D) Gross-Pitaevskii equation (GPE), see e.g. \cite{BaCa,BaCaWa, SSZL, YiLo1, YiLo2},
\begin{equation}
\label{eq:evolution}
i h \frac{\partial \psi(x,t)}{\partial t} = - \frac{h^2}{2m}\nabla^2 \psi + W(x) \psi + U_0|\psi|^2 \psi + (V_{dip} \star |\psi|^2) \psi, \quad x \in {\mathbb R}^3, \quad t>0
\end{equation}
where $t$ is time, $x = (x_1,x_2,x_3)^T \in {\mathbb R}^3 $ is the Cartesian coordinates, $\star$ denotes the convolution, $h$ is the Planck constant, $m$ is the mass of a dipolar particle and $W(x)$ is an external trapping potential. In this paper we shall consider a harmonic potential
$$W(x) = \frac{m}{2}\, a^2\, |x|^2$$
where $a$ is the trapping frequency. $U_0 = 4 \pi h^2 a_s /m$ describes the local interaction between dipoles in the condensate with $a_s$ the $s-$wave scattering length (positive for repulsive interaction and negative for attractive interaction).
The long-range dipolar interaction potential between two dipoles is given by
\begin{equation}
\label{eq:dipole}
V_{dip}(x) = \frac{\mu_0 \mu^2_{dip}}{4 \pi} \, \frac{1 - 3 cos^2 (\theta)}{|x|^3}, \quad x \in {\mathbb R}^3
\end{equation}
where $\mu_0$ is the vacuum magnetic permeability, $\mu_{dip}$ is the permanent magnetic dipole moment and $\theta$ is the angle between the dipole axis and the vector $x$. For simplicity we fix the dipole axis as the vector $(0,0,1)$.
The wave function is normalized according to
\begin{equation}
\label{eq:normalized}
\int_{{\mathbb R}^3} |\psi(x,t)|^2 dx = N
\end{equation}
where $N$ is the total number of dipolar particles in the dipolar BEC.
\newline
This aim of this paper is twofold: first to study the existence of stationary states for \eqref{eq:evolution} satisfying \eqref{eq:normalized} and their stability properties, second to understand how the presence of the external trapping potential influences the dynamics of the system. In order to simplify the mathematical analysis
we rescale (\ref{eq:evolution}) into the following dimensionless GPE,
\begin{equation}
\label{eq:evolutionbis}
i \frac{\partial \psi(x,t)}{\partial t} = - \frac{1}{2}\nabla^2 \psi + \frac{a^2}{2} |x|^2 \psi + \lambda_1 |\psi|^2 \psi + \lambda_2 (K \star |\psi|^2) \psi, \quad x \in {\mathbb R}^3, \quad t>0.
\end{equation}
The dimensionless long-range dipolar interaction potential $K(x)$ is given by
$$
K(x) = \frac{1- 3 cos^2(\theta)}{|x|^3}, \quad x \in {\mathbb R}^3.
$$
The corresponding normalization is now
\begin{equation}
\label{eq:normalizedbis}
N(\psi(\cdot, t)):= ||\psi(\cdot, t)||^2_2 = \int_{{\mathbb R}^3}|\psi(x,t)|^2 dx = \int_{{\mathbb R}^3}|\psi(x,0)|^2 dx = 1
\end{equation}
and the physical parameters $(\lambda_1, \lambda_2)$, which describes the strengh of the two nonlinearities, are given in (\ref{eq:parameters}).
Note that despite the kernel $K$ is highly singular it defines a smooth operator. More precisely the operator $ u \rightarrow K \star u$ can be extended as a continuous operator on $L^p({\mathbb R}^3)$ for all $1 < p < \infty$, see \cite[Lemma 2.1]{CMS}. Actually the local existence and uniqueness of solutions to \eqref{eq:evolutionbis} has been proved in \cite{CMS}.
From now on we deal with (\ref{eq:evolutionbis}) under the condition (\ref{eq:normalizedbis}) and we focus on the case when $\lambda_1$ and $\lambda_2$ fulfills the following conditions
\begin{equation}\left\{\begin{matrix} \label{AS}
\lambda_1-\frac 43 \pi \lambda_2<0,\ &\mbox{ if }& \lambda_2>0; \\
\lambda_1+\frac 83 \pi \lambda_2<0,\ &\mbox{ if }& \lambda_2<0. \\
\end{matrix}\right.
\end{equation}
These conditions which, following the terminology introduced in \cite{CMS}, define the {\it unstable regime} corresponds to the Figure \ref{fig2}.
\begin{figure}
\caption{The unstable regime given by \eqref{AS}
\label{fig2}
\end{figure}
To find stationary states we make the ansatz
\begin{equation}\label{ansatz}
\psi(x,t) = e^{-i \mu t}u(x), \quad x \in {\mathbb R}^3
\end{equation}
where $\mu \in {\mathbb R}$ is the chemical potential and $u(x)$ is a time-independent function. Plugging (\ref{ansatz}) into (\ref{eq:evolutionbis}) we obtain the stationary equation
\begin{equation}
\label{eq:maina}
- \frac{1}{2}\Delta u + \frac{a^2}{2} |x|^2 u + \lambda_1 |u|^2 u + \lambda_2 (K \star |u|^2) u + \mu u =0
\end{equation}
and the corresponding constraint $u \in S(1)$ where
\begin{equation}\label{constraint1}
S(1) = \{ u \in H^1({\mathbb R}^3, {\mathbb C}) \ s.t. \ ||u||_2^2 =1\}.
\end{equation}
In the first part of the paper we consider the situation where the trapping potential is not active, namely when $a=0$. The corresponding stationary equation is then just
\begin{equation}\label{eq:main}
-\frac 12 \Delta u +\lambda_1|u|^2u+\lambda_2(K\star u^2)u+\mu u=0, \quad u \in H^1({\mathbb R}^3, {\mathbb C}).
\end{equation}
We recall, see \cite{AS}, that the energy functional associated with \eqref{eq:main} is given by
\begin{equation}
\label{functional}
E(u):= \frac{1}{2}||\nabla u||_2^2 + \frac{\lambda_1}{2}||u||_4^4 + \frac{\lambda_2}{2} \int_{{\mathbb R}^3} (K\star |u|^2)|u|^2 dx.
\end{equation}
Any critical point of $E(u)$ constrained to $S(1)$ corresponds to a solution of \eqref{eq:main} satisfying \eqref{constraint1}. The parameter $\mu \in {\mathbb R}$ being then found as the Lagrange multiplier.
As we shall prove, under assumption \eqref{AS}, the functional $E(u)$ is unbounded from below on $S(1)$. Actually when \eqref{AS} is not satisfied, one speaks of the {\it stable regime}, the functional $E(u)$ is bounded from below on $S(1)$ and coercive, see \cite{BaCaWa, CaHa}. In that case one can prove that no standing waves exists, see Remark \ref{Jacopo}.
The problem of finding solutions to (\ref{eq:main}) was first considered in \cite{AS}. In \cite[Theorem 1.1]{AS}, assuming (\ref{AS}), Antonelli and Sparber obtain, for any $\mu >0$, the existence of a real positive solution of (\ref{eq:main}), along with some symmetry, regularity and decay properties. To overcome the fact that $E(u)$ is unbounded from below on $S(1)$ they developped an approach in the spirit of
Weinstein \cite{W}. Namely their solutions are obtained as minimizers of the following
scaling invariant functional
\begin{equation}\label{def:w}
J(v):=\frac{||\nabla v||^3_2||v||_2}{-\lambda_1||v||_4^4-\lambda_2\int_{{\mathbb R}^3} (K\star |v|^2)|v|^2}.
\end{equation}
In \cite{AS} it is also shown that (\ref{AS}) are necessary and sufficient conditions to obtain a solution of (\ref{eq:main}).
In this paper we propose an alternative approach. We directly work with $E(u)$ restricted to $S(1)$. We obtain our solution as a {\it mountain pass} critical point, Despite the fact the energy is unbounded from below on $S(1)$, if we restrict to states satisfying \eqref{constraint1} that are stationary for the evolution equation \eqref{eq:evolutionbis}, then the energy is bounded from below by a positive constant. We then show that this constant, corresponding to the mountain pass level, is reached and this will prove the existence of least energy states, also called ground states. As a direct consequence of this variational characterization and using a virial approach we manage to show that the associated standing waves are orbitally unstable.
Denoting the Fourier transform of $u$ by $\mathcal{F}(u):=\int_{{\mathbb R}^3} u(x)e^{-i x \cdot \xi}dx$, the Fourier transform of $K$ is given by $$\hat K(\xi)=\frac{4}{3}\pi (\frac{2\xi_3^2-\xi_1^2-\xi_2^2}{|\xi|^2})\in [-\frac{4}{3}\pi, \frac{8}{3}\pi],$$
see \cite[Lemma 2.3]{CMS}. Then, thanks to
the Plancherel identity, see for example \cite[Theorem 1.25]{BaChDa}, one gets
\begin{equation}\label{def:Plancherel}
\lambda_1||v||_4^4+\lambda_2\int_{{\mathbb R}^3} (K\star |v|^2)|v|^2=\frac{1}{(2\pi)^3}\int_{{\mathbb R}^3} (\lambda_1+\lambda_2 \hat K(\xi))|\hat {v^2}|^2d\xi.
\end{equation}
Thus we can rewrite $E(u)$ as
\begin{equation}
E(u)=\frac{1}{2}\int_{{\mathbb R}^3} |\nabla u|^2dx+\frac{1}{2}\frac{1}{(2\pi)^3}\int_{{\mathbb R}^3} (\lambda_1+\lambda_2 \hat K(\xi))|\hat {u^2}|^2d\xi.
\end{equation}
In order to simplify the notation we define
$$A(u):=\int_{{\mathbb R}^3} |\nabla u|^2dx, \ \ B(u):= \frac{1}{(2\pi)^3}\int_{{\mathbb R}^3} (\lambda_1+\lambda_2 \hat K(\xi))|\hat {u^2}|^2d\xi.$$
$$Q(u):=\int_{{\mathbb R}^3} |\nabla u|^2dx+\frac{3}{2}\frac{1}{(2\pi)^3}\int_{{\mathbb R}^3} (\lambda_1+\lambda_2 \hat K(\xi))|\hat {u^2}|^2d\xi.$$
We also set $H:= H^1({\mathbb R}^N, {\mathbb C})$ and denotes by $||\cdot||$ the corresponding usual norm.
Despite the fact that we are primarily interested in solutions satisfying \eqref{constraint1}, for the mathematical treatment of the problem it is convenient to consider $E(u)$ on the set of constraints
$$S(c)=\left\{ u \in H \ s.t. \ ||u||_2^2=c\right\}.$$
Here $c>0$ and the case $c=1$ corresponds to the normalization \eqref{constraint1}. Given $c>0$ we shall prove that $E(u)$ has a mountain pass geometry on $S(c)$, see \cite{Gh} for a definition. More precisely we prove that there exists a $k>0$ such that
\begin{equation}\label{gamma}
\gamma(c) := \inf_{g \in \Gamma(c)} \max_{t\in [0,1]}E(g(t)) > \max \{\max_{g \in \Gamma(c)}E(g(0)), \max_{g \in \Gamma(c)}E(g(1))\}
\end{equation}
holds in the set
\begin{equation}\label{Gamma}
\Gamma(c) =\{g \in C([0,1],S(c)) \ s.t. \ g(0) \in A_{k},E(g(1))<0\},
\end{equation}
where $$A_{k}= \{u \in S(c) \ s.t. \ \left \| \triangledown u \right \|_2^2\leq k\}.$$
It it standard, see for example \cite[Theorem 3.2]{Gh}, that the mountain pass geometry induces the existence of a Palais-Smale sequence at the level $\gamma(c)$. Namely a sequence $(u_n) \subset S(c)$ such that
$$E(u_n)=\gamma(c)+o(1), \ \ \ ||E'|_{S(c)}(u_n)||_{H^{-1}}=o(1).$$
If one can show in addition the compactness of $(u_n)$, namely that up to a subsequence, $u_n \rightarrow u$ in $H$, then a critical point is found at the level $\gamma(c)$. Actually under the assumptions of \cite{AS}, in the unstable regime, we are able to prove the following
\begin{thm}\label{thm:standing}
Let $c>0$ and assume that \eqref{AS} holds. Then $E(u)$ has a {\it mountain pass geometry} on $S(c)$ and there exists a couple $(u_c, \mu_c)\in H \times {\mathbb R}^{+}$ solution of \eqref{eq:main} with
$||u_c||_2^2=c$ and $E(u_c)=\gamma(c)$. In addition $u_c \in S(c)$ is a ground state.
\end{thm}
Since our definition of ground states does not seem to be completely standard we now precise it.
\begin{definition}\label{ground-state}
Let $c>0$ be arbitrary, we say that $u_c \in S(c)$ is a ground state if
$$E(u_c) = \inf \{E(u) \ s.t. \ u \in S(c), E'|_{S(c)}(u) =0\}.$$
\end{definition}
Namely a solution $u_c \in S(c)$ of \eqref{eq:main} is a ground state if it minimize the energy functional $E(u)$ among all the solutions of \eqref{eq:main} which belong to $S(c)$. We point out that with Definition \ref{ground-state} a ground state may exists even if $E(u)$ is unbounded from below on $S(c)$.
\begin{remark}
To prove that a Palais-Smale sequence converges a first step is to show that it is bounded and this is not given for free for $E(u)$. Note also that, due to the dipolar term, our functional is not invariant by rotations. This lack of symmetry also make delicate to prove the compactness of the sequences. To overcome both difficulties we shall prove the existence of one specific Palais-Smale sequence that fulfill $Q(u_n)=o(1).$ This localization property which follows the original ideas of \cite{Gh}, provides a direct proof of the $H$ boundedness
of the sequence and also, after some work, of its compactness.
\end{remark}
\begin{remark}\label{comparaison}
Theorem \ref{thm:standing} is in the spirit of some recent works \cite{BJL,JeLuWa} in which constrained critical points are obtained for functionals unbounded from below on the constraint. We also refer to \cite{NoTaVe} for a closely related problem.
\end{remark}
To prove Theorem \ref{thm:standing} we establish that
$$\gamma(c) = \inf_{u \in V(c)}E(u)$$
where
$$V(c)=\left\{ u \in S(c) \ s.t. \ Q(u)=0 \right\}.$$
As we shall see $V(c)$ contains all the critical points of $E(u)$ restricted to $S(c)$. Actually we also have
\begin{lem}\label{naturalconstraint}
Let $c>0$ be arbitrary, then $V(c)$ is a natural constraint, i.e each critical point of $E_{|_{V(c)}}$ is a critical point of $E_{|_{S(c)}}$.
\end{lem}
Let us denote the set of minimizers of $E(u)$ on $V(c)$ as
\begin{eqnarray}\label{minimizerset}
\mathcal{M}_c := \{u_c\in V(c) \ s.t. \ \ E(u_c)=\inf_{u\in V(c)}E(u)\}.
\end{eqnarray}
\begin{lem}\label{description}
Let $c>0$ be arbitrary, then
\begin{enumerate}
\item [(i)] If $u_c \in \mathcal{M}_c$ then also $|u_c| \in \mathcal{M}_c$ .
\item [(ii)] Any minimizer $u_c \in\mathcal{M}_c$ has the form $e^{i\theta}|u_c|$ for some $\theta \in \mathbb{S}^1$ and $|u_c(x)| >0 $ a.e. on ${\mathbb R}^3$.
\end{enumerate}
\end{lem}
In view of Lemma \ref{description} each element of $\mathcal{M}_c$ is a real positive function multiply by a constant complex factor.
Our next result connects the solutions found in \cite{AS} with the ones of Theorem \ref{thm:standing}.
\begin{thm}\label{thm:AS}
Let $v \in H$ be, for some $\mu >0$, the solution obtained in \cite[Theorem 1.1]{AS}. Then setting $c = ||v||_2^2$ we have that $E(v) = \gamma(c)$.
\end{thm}
\begin{remark}
Since we do not know if nonnegative solutions of \eqref{eq:main} are, up to translations, unique it is not possible to directly identify the solutions of \cite{AS} with the ones at the mountain pass level.
\end{remark}
Concerning the dynamics, under \eqref{AS} the global well posedness for \eqref{eq:evolutionbis} is not guaranteed in unstable regime. The problem is $L^2$ super-critical and energy estimates do not control the $H$ norm. Conditions for blow-up has been discussed in \cite{CMS}. However we are able to prove the following global existence result in an open nonempty set of $H$
that contains not only small initial data.
\begin{thm}\label{thm:global}
Let $u_0 \in H$ be an initial condition associated to \eqref{eq:evolutionbis} with $c=||u_0||_2^2$. If
$$Q(u_0)>0 \text{ and } E(u_0)<\gamma(c),$$
then the solution of \eqref{eq:evolutionbis} with $a=0$ and initial condition $u_0$ exists globally in times.
\end{thm}
For small data in the energy space we now show that scattering occurs independently of the values of $\lambda_1$ and $\lambda_2$. In particular it occurs in all regimes, stable and unstable.
\begin{thm}\label{thm:scat}
Let $\lambda_1, \lambda_2 \in {\mathbb R}\setminus\{0\}$. There exists $\delta>0$ such that if $||\psi_0||<\delta$ then the solution $\psi(t)$ of \eqref{eq:evolutionbis} with $a=0$ scatters in $H.$ More precisely there exist $\psi_{\pm} \in H$
such that
$$\lim_{t \rightarrow \pm \infty}||\psi(t)-e^{i t\frac{\Delta}{2}}\psi_{\pm}||=0.$$
\end{thm}
\begin{remark}
In case of cubic NLS the classical strategy to show small data scattering in $H$ is to prove that some $L^p_tW^{1,q}_x$ Strichartz admissible norm is uniformly bounded in time. In our case we follow the same strategy recalling that the additional nonlocal convolution term $K$ describing the dipolar interaction is a continuous operator in $L^p$
when $1<p<\infty$. This permits to prove the boundedness of $L^{\frac 83}_{[0,\infty]}W^{1,4}_x$ and hence the scattering.
\end{remark}
We now prove that the standing waves associated to elements in $\mathcal{M}_c$ are unstable in the following sense.
\begin{definition}
A standing wave $e^{i\omega t}v(x)$ is strongly unstable if for any $\varepsilon >0$ there exists $u_0 \in H$ such that $\left \| u_0-v \right \|_{H}< \varepsilon$ and the solution $u(t,\cdot)$ of the equation \eqref{eq:evolutionbis} with $u(0, \cdot)=u_0$ blows up in finite time.
\end{definition}
\begin{thm}\label{thm:instability}
For any $u \in \mathcal{M}_c$ the standing wave $e^{-i \mu_c t}u$ where $\mu_c >0$ is the Lagrange multiplier, is strongly unstable.
\end{thm}
In a second part of the paper we analyse what happens to the system when one add, gradually, a confining potential. We are particularly interested in the existence of ground states and their stability but we shall also obtain the existence of excited states.
When $a>0$ the functional associated to (\ref{eq:maina}) becomes
\begin{equation}
\label{functional}
E_a(u):= \frac{1}{2}||\nabla u||_2^2 + \frac{a^2}{2} |||x|u||_2^2 + \frac{1}{2}\frac{1}{(2\pi)^3}\int_{{\mathbb R}^3} (\lambda_1+\lambda_2 \hat K(\xi))|\hat {u^2}|^2d\xi.
\end{equation}
This functional being now defined on the space
\begin{equation}\label{Sigma}
\Sigma=\left\{ u \in H \ s.t. \ \int |x|^2u^2dx<\infty \right\}.
\end{equation}
The associated norm is
$$||u||_{\Sigma}^2:=||u||_{H}^2+|||x|u||_2^2.$$
It is standard, see \cite{CaHa,CMS}, that $E_a(u)$ is of class $C^1$ on $\Sigma.$ Note that $\Sigma$ has strong compactness properties which will be essential in our analysis. In particular the embedding $\Sigma \hookrightarrow L^p({\mathbb R}^3)$ is compact for $p \in [2,6)$, see for example \cite[Lemma 3.1]{Zh}.
For simplicity we keep the notation $S(c)$ for the constraint which is now given by
$$S(c)=\left\{ u \in \Sigma \ s.t. \ ||u||_2^2=c\right\}.$$
\begin{definition}\label{localminimizer}
For $c>0$ being given we say that $v \in S(c)$ is a topological local minimizer for $E_a(u)$ restricted to $S(c)$ if there exist an open subset $A \subset S(c)$ with $v \in A$, such that
\begin{equation}\label{carlocalminimizer}
E_a(v) = \inf_{u \in A}E_a(u) \quad \mbox{and} \quad E_a(v) < \inf_{u \in \partial A}E_a(u).
\end{equation}
Here the boundary is taken relatively to $S(c)$. If this occurs we say that $v$ is a topological local minimizer for $E_a(u)$ on $A$.
\end{definition}
\begin{thm}
\label{thm: mainn}
Let $c>0$ be given and assume that $(\lambda_1, \lambda_2)$ satisfies \eqref{AS}. Then there exists a value $a_0 = a_0(\lambda_1, \lambda_2) >0$ such that for any $a \in (0, a_0],$
\begin{enumerate}
\item $E_a(u)$ restricted to $S(c)$ admits a ground state $u_a^1$ and there exists a $k>0$ such that $u_a^1$ is a topologial local minimizer for $E_a(u)$ on the set
$$B_{k}=\{u \in S(c)\ s.t. \ ||\nabla u||_2^2 < k\}.$$
In addition any ground state for $E_a(u)$ restricted to $S(c)$ is a topological local minimizer for $E_a(u)$ on $B_{k}$.
\item $E_a(u)$ restricted to $S(c)$ admits a second critical point $u_a^2$ obtained at a mountain pass level and it corresponds to an excited state.
\item The following properties hold
\begin{enumerate}
\item $u_a^1$ and $u_a^2$ are real, non negative.
\item For any $a \in (0, a_0]$, $0 < E_a(u_a^1) < E_a(u_a^2).$
\item Any ground state $u_a \in S(c)$ for $E_a(u)$ on $S(c)$ satisfies $A(u_a) \to 0$ and $E_a(u_a) \to 0$ as $a \to 0$. Also $E_a(u_a^2) \to \gamma(c)$, where $\gamma(c)>0$ is the least energy level of $E(u)$, the functional without the trapping potential.
\end{enumerate}
\end{enumerate}
\end{thm}
\begin{remark}
The change of geometry of the constrained energy functional can be viewed as a consequence of the Heisenberg uncertainty principle, see e.g. \cite{LS},
\begin{equation}\label{Heisenberg}
\left(\int_{{\mathbb R}^3} |\nabla u|^2dx\right)^{\frac 12}\left(\int_{{\mathbb R}^3} |x|^2| u|^2dx\right)^{\frac 12}\geq \frac 32 \left(\int_{{\mathbb R}^3} |u|^2 dx\right).
\end{equation}
Using \eqref{Heisenberg} the energy functional $E_a(u)$, thanks to Gagliardo-Nirenberg inequality, fulfills
$$E_a(u) \geq \frac 12 A(u)+\frac{9a^2c^2}{8 A(u)}+ \frac{1}{2}B(u)\geq \frac 12 A(u)+\frac{9a^2c^2}{8 A(u)} -CA(u)^{\frac 32}c^{\frac 12}$$
for some constant $C>0$.
The fact that $E_a(u)$ admits a topological local minimum is closely related to the previous inequality which implies in particular that
\begin{equation}\label{blow}
\lim_{k \rightarrow 0} \inf_{u \in A_k} E_a(u) =+\infty.
\end{equation}
A qualitative picture is given by Figure 2.
\end{remark}
\begin{figure}
\caption{Qualitative behavior of $E(u)$ (left) and $E_a(u)$ (right). In figure (b) the three curves mimic the behavior $E_a(u)$ for three different values of $a$.}
\label{fig}
\end{figure}
As a byproduct of Theorem \ref{thm: mainn} we are able to show that topological local minimizers, taking $a>0$ fixed, fulfills $||u_a^1||_{\Sigma}\rightarrow 0$ when $c \rightarrow 0$. This fact implies
\begin{cor}\label{Cor}
Under the assumption of Theorem \ref{thm: mainn} small data scattering cannot hold.
\end{cor}
\begin{thm}
\label{stability}
Under the assumptions of Theorem \ref{thm: mainn} any ground state of $E_a(u)$ restricted to $S(c)$ is orbitally stable.
\end{thm}
The proof of Theorem \ref{stability} is simple. By Theorem \ref{thm: mainn} we know that any ground state is a topological local minimizer for $E_a(u)$ on $B_{k}$. By conservation of the energy and of the mass, for any initial data in $B_{k}$ the trajectory remains in $B_{k}$ (and in particular we have global existence). As a consequence of this it is possible to directly apply the classical arguments of Cazenave-Lions \cite{CL} which were developed to show the orbital stability of standing waves characterized as global minimizers. Note however that the energy $E_a(u)$ is unbounded from below on $S(c)$ for any
$a\geq 0$.
\begin{remark}\label{Holger}
From the physical point of view, Theorems \ref{thm:standing}, \ref{thm: mainn} and \ref{stability} show that the introduction of a small trapping potential leads to a stabilization of a system which was originally unstable. Up to our knowledge such physical phenomena had not been observed previously in laboratories or numerically. Note that such stabilizing effect is known to hold for lithium quantum gases (with a negative scattering lenght, attractive interactions), see \cite{BrSaToHu}. We conjecture that as the trapping potential increases the system ceases however to be stable.
\end{remark}
\begin{remark}
From Theorem \ref{thm: mainn} (1) we know that the ground state energy level corresponds to the one of the topological local minimizer $u_a^1$. Also from Theorem \ref{thm: mainn} (3) (c) we see that there is a discontinuity at $a=0$ in the energy level of the ground state (which for $a=0$ corresponds to $\gamma(c)>0$). Thus the addition of a trapping potential, however small, create a {\it gap} in the ground state energy level of the system.
\end{remark}
In contrast to the case $a=0$ where the Lagrange parameter $\mu \in {\mathbb R}$ (namely the chemical potential) associated to any solution is strictly positive, see Lemma \ref{lem:poho}, we now have when $a>0$,
\begin{thm}\label{thm:signmu}
Let $a \in (0, a_0]$ and $u$ be a ground state for $E_a(u)$ restricted to $S(c)$. Then if $a>0$ is sufficiently small $\mu \in {\mathbb R}$ as given in \eqref{eq:maina} satisfies $\mu<0.$
\end{thm}
Finally we analyze what happen when $(\lambda_1,\lambda_2)$ moves from {\it unstable region} towards the border of the {\it stable region}.
\begin{thm}\label{asymtotic}
Let $c >0$ and assume that \eqref{AS} holds. Calling
$\lambda_1' =\lambda_1-\frac 43 \pi \lambda_2$ when $\lambda_2>0$ ($\tilde \lambda_1' =\lambda_1+\frac 83 \pi \lambda_2$ when $\lambda_2<0$) we have
when $\lambda_1' \rightarrow 0^-$ ($\tilde \lambda_1' \rightarrow 0^-$ respectively)
\begin{enumerate}
\item The $H$-norm of the mountain pass solution obtained in Theorem \ref{thm:standing} goes to infinity.
\item We can allow any $a_0 >0$ in Theorem \ref{thm: mainn}.
\end{enumerate}
\end{thm}
We have choosen not to consider in this paper the stability of the standing wave corresponding to $u_a^2$. We conjecture that it is strongly unstable. Note that, due to the fact that the geometry of $E_a(u)$ on $S(c)$ is more complex than the one of $E(u)$, in particular the analogue of Lemma \ref{lem:growth} does not hold, the treatment of this question probably requires new ideas.
We end our paper by an Appendix in which we prove a technical result concerning the Palais-Smale sequences associated to $E_a(u)$.
In the sequel we mainly consider the first case of \eqref{AS}, namely $\lambda_2>0, \ \ \lambda_1-\frac{4}{3}\pi \lambda_2<0$, the second case follows by a similar treatment.
\textbf{Acknowledgements.} The second author thanks P. Antonelli and C. Sparber for a discussion on the interest of showing that the solutions of \cite{AS} are orbitally unstable. The first author thanks Nicola Visciglia for fruitful discussion concerning small data scattering. The two authors also thank W. Bao, H. Hajaiej and A. Montaru for stimulating discussions on a first version of this work. The authors thank Giovanni Stegel for Figure . The second author warmly thanks Holger Kadau for sharing with him his physical insight of the problem. In particular Remark \ref{Holger} and Theorem \ref{asymtotic} originate from our interactions. Finally we thanks the two referees whose comments have permit to improve our manuscript and to avoid to include a wrong result.
\\
\section{Derivation of our dimensionless GPE }
In order to obtain a dimensionless GPE from \eqref{eq:evolution} we introduce the new variables
\begin{equation}
\label{eq:newvariable}
\tilde{t} = t, \quad \tilde{x} = \gamma x \, \mbox{ where } \, \gamma = \sqrt{\frac{m}{h}}, \quad \tilde{\psi}(\tilde{x}, \, \tilde{t}) = \frac{1}{\sqrt{N}}\frac{1}{\gamma^{3/2}} \psi(x,t).
\end{equation}
Plugging (\ref{eq:newvariable}) into (\ref{eq:evolution}), dividing by $ h \sqrt{N} \gamma^{3/2}$
and then removing all $\, \tilde{ }\, $ we obtain the dimensionless GPE
$$i \frac{\partial \psi(x,t)}{\partial t} = - \frac{1}{2}\nabla^2 \psi + \frac{a^2}{2} |x|^2 \psi + \lambda_1 |\psi|^2 \psi + \lambda_2 (K \star |\psi|^2) \psi, \quad x \in {\mathbb R}^3, \quad t>0,
$$
under the normalization
$$N(\psi(\cdot, t)):= ||\psi(\cdot, t)||^2 = \int_{{\mathbb R}^3}|\psi(x,t)|^2 dx = \int_{{\mathbb R}^3}|\psi(x,0)|^2 dx = 1.$$
Here
\begin{equation}
\label{eq:parameters}
\lambda_1 = 4 \pi a_s N \gamma, \quad \lambda_2 = \frac{mN \mu_0 \mu_{dip}^2 }{4 \pi h^2}\gamma
\end{equation}
and the dimensionless long-range dipolar interaction potential $K(x)$ is given by
\begin{equation}
\label{eq:dipolar}
K(x) = \frac{1- 3 cos^2(\theta)}{|x|^3}, \quad x \in {\mathbb R}^3.
\end{equation}
\section{Proof of Theorem \ref{thm:standing}}
First we show that any constrained critical point belongs to $V(c)$ and that the associated Lagrange multiplier is strictly positive.
\begin{lem}\label{lem:poho}
If $v$ is a weak solution of
$$
-\frac 12 \Delta u +\lambda_1|u|^2u+\lambda_2(K\star u^2)u+\mu u=0
$$
then $Q(v)=0$. If we assume $v\neq 0$ then $\mu >0$.
\end{lem}
\begin{proof}
The proof is essentially contained in \cite{AS}. It follows from Pohozaev identity that
$$\frac{1}{4}A(u)+\frac{3}{4}B(u)+\frac{3}{2}\mu ||u||_2^2=0.$$
Moreover, multiplying the equation by $u$ and integrating one obtains
$$\frac{1}{2}A(u)+B(u)+\mu ||u||_2^2=0.$$
The two equalities imply that
$$Q(u)= A(u)+\frac 32 B(u)=0 \quad \mbox{and} \quad A(u)=6\mu ||u||_2^2.$$
\end{proof}
\noindent To understand the geometry of $E(u)$ on $S(c)$ we introduce the scaling
\begin{equation}\label{def:sca}
u^t(x)=t^{\frac 32}u(t x), \quad t>0.
\end{equation}
Observing that $\mathcal{F}{(u^t)^2}(\xi)=\mathcal{F} {u^2}(\frac{\xi}{t})$ the energy rescales as
\begin{equation}\label{def:mainscal}
t \to E(u^{t})=\frac{t^2}{2}A(u)+\frac{t^3}{2}B(u).
\end{equation}
\begin{lem}\label{lem:base}
Let $u \in S(c)$ be such that $ \int_{{\mathbb R}^3} (\lambda_1+\lambda_2 \hat K(\xi))|\hat {u^2}|^2d\xi <0$ then:\\
(1)\ $A(u^t) \to \infty$ and $E(u^t) \to -\infty$, as $t \to \infty$.\\
(2)\ There exists $k_0 >0$ such that $Q(u)>0$ if $||\nabla u||_2\leq k_0.$\\
(3)\ If $E(u)<0$ then $Q(u)<0.$
\end{lem}
\begin{proof}
Using \eqref{def:mainscal} and since it always holds that
\begin{equation}\label{star}
E(u)-\frac{1}{3}Q(u)=\frac{1}{6}A(u)
\end{equation}
we get (1) and (3). Now thanks to Gagliardo-Nirenberg inequality and Plancherel identity there exists a constant $C>0$ such that
$$Q(u)> A(u)+\frac{3}{2}\frac{1}{(2\pi)^{\frac 32}}\int_{{\mathbb R}^3} (\lambda_1-\frac 43 \pi \lambda_2) |\hat {u^2}|^2d\xi=A(u)-C||u||_4^4 \geq A(u)- C A(u)^{\frac 32}||u||_2,$$
and this proves (2).
\end{proof}
Our next lemma is inspired by \cite[Lemma 8.2.5]{TC}.
\begin{lem}\label{lem:growth}
Let $u \in S(c)$ be such that $ \int_{{\mathbb R}^3} (\lambda_1+\lambda_2 \hat K(\xi))|\hat {u^2}|^2d\xi <0$ then we have: \\
(1)\ There exists a unique $t^{\star}(u)>0$, such that $u^{t ^{\star}} \in V(c)$;\\
(2)\ The mapping $t \longmapsto E(u^{t})$ is concave on $[t ^{\star}, \infty)$;\\
(3)\ $t ^{\star}(u)<1$ if and only if $Q(u)<0$;\\
(4)\ $t ^{\star}(u)=1$ if and only if $Q(u)=0$;\\
(5)\ $$Q(u^t)\left\{
\begin{matrix}
\ >0,\ \forall\ t &\in& (0,t^*(u));\\
\ <0, \ \forall\ t&\in& (t^*(u),+\infty).
\end{matrix}\right.$$
(6)\ $E(u^{t})<E(u^{t ^{\star}})$, for any $t>0$ and $t \neq t ^{\star}$;\\
(7)\ $\frac{\partial}{\partial t} E(u^{t})=\frac{1}{t}Q(u^{t})$, $\forall t >0$.
\end{lem}
\begin{proof}
Since $$E(u^{t})=\frac{t^2}{2}A(u)+\frac{t^3}{2}B(u)$$
we have that
$$ \frac{\partial}{\partial t} E(u^{t}) = t A(u)+\frac{3}{2}t^{2}B(u)
= \frac{1}{t}Q(u^{t}).$$
Now we denote
$$y(t)= t A(u) + \frac{3}{2}t^2B(u),
$$
and observe that $Q(u^{t})= t \cdot y(t)$ which proves (7). After direct calculations, we see that:
\begin{eqnarray*}
y'(t)&=& A(u) +3tB(u); \\
y''(t)&=& 3B(u).
\end{eqnarray*}
From the expression of $y'(t)$ and the assumption $B(u)<0$ we know that $y'(t)$ has a unique zero that we denote $t_0>0$ such that $t_0$ is the unique maximum point of $y(t)$. Thus in particular the function $y(t)$ satisfies:\\
(i)\ $y(t_0)=\max_{t >0}y(t)$;\\
(ii)\ $\lim_{t\to +\infty}y(t)=-\infty$;\\
(iii)\ $y(t)$ decreases strictly in $[t_0, +\infty)$ and increases strictly in $(0, t_0]$.
By the continuity of $y(t)$, we deduce that $y(t)$ has a unique zero $t^{\star}>0$. Then $Q(u^{t^*}) =0$ and point (1) follows. Points (2)-(4) and (5) are also easy consequences of (i)-(iii). Finally since $y(t) >0$ on $(0, t^*(u))$ and $y(t) <0$ on $(t^*(u), \infty)$ we get (6).
\end{proof}
\begin{prop}\label{prop}
Let $(u_n) \subset S(c)$ be a bounded Palais-Smale sequence for $E(u)$ restricted to $S(c)$ such that $E(u_n) \to \gamma (c)$. Then there is a sequence $(\mu_n)\subset \mathbb{R}$, such that, up to a subsequence:\\
(1)\ $u_n \rightharpoonup \bar{u}$ weakly in $H$;\\
(2)\ $\mu_n \to \mu$ in $\mathbb{R}$;\\
(3) $- \frac{1}{2}\Delta u_n + \lambda_1 |u_n|^2 u_n + \lambda_2 (K \star |u_n|^2) u + \mu u_n \to 0$ in $H^{-1}$;\\
(4) $- \frac{1}{2}\Delta \bar{u} + \lambda_1 |\bar{u}|^2 \bar{u}+ \lambda_2 (K \star |\bar{u}|^2) \bar{u} + \mu \bar{u}= 0$ in $H^{-1}.$
\end{prop}
\begin{proof}
The proof of Proposition \ref{prop} is standard and we refer to \cite[Proposition 4.1]{BJL} for a proof in a similar context.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:standing}]
The proof requires several steps.
\noindent{\bf Step 1: } {\it $E(u)$ has a Mountain-Pass Geometry on $S(c)$.}
First let us show that letting
$$C_k=\left\{ u \in S(c) \ s.t. \ \ A(u)=k\right\}$$
it is possible to choose $0 <k < 2k_0$, where $k_0>0$ is given in Lemma \ref{lem:base}, such that
\begin{equation}\label{def: mp}
0 \leq \inf_{u \in A_{k}}E(u) \leq \sup_{u\in A_{k}} E(u) < \inf_{u \in C_{2k}} E(u).
\end{equation}
Indeed observe that by Gagliardo-Nirenberg inequality and Plancherel identity, for some positive constants $\tilde C_i$, $i=1,\cdots, 4$,
\begin{equation}\label{useful}
E(u)\leq \frac{A(u)}{2}+\tilde C_1||u||_4^4\leq \frac{A(u)}{2}+ \tilde C_2 A(u)^{\frac 32}||u||_2.
\end{equation}
$$E(u)\geq \frac{A(u)}{2}+\frac{1}{2}\frac{1}{(2\pi)^3}\int_{{\mathbb R}^3} (\lambda_1-\frac 43 \pi \lambda_2) |\hat {u^2}|^2d\xi \geq \frac{A(u)}{2}-\tilde C_3||u||_4^4 \geq \frac{A(u)}{2}- \tilde C_4 A(u)^{\frac 32}||u||_2.$$
The proof of \eqref{def: mp} follows directly from these two estimates taking $k >0$ small enough. Now for an arbitrary $v \in S(c)$ consider the scaling given by
\begin{equation}\label{def:scad1}
v^t(x)=t^{\frac 54}v(t x_1, t x_2, t^{\frac 12} x_3), \quad t>0.
\end{equation}
We have $v^t \in S(c)$ for all $t>0$ and the energy rescales as
$$E(v^t)=\frac{t^2}{2}\int_{{\mathbb R}^3} |\nabla_{x_1,x_2} v|^2dx+\frac{t}{2}\int_{{\mathbb R}^3} |\nabla_{x_3} v|^2dx+\frac{t^{\frac{5}{2}}}{2}\frac{1}{(2\pi)^3}\int_{{\mathbb R}^3} (\lambda_1+\frac43 \pi \lambda_2 \frac{2t \xi_3^2-t^2\xi_1^2-t^2\xi_2^2}{t^2\xi_1^2+t^2\xi_2^2+t \xi_3^2})|\hat {v^2}|^2d\xi.$$
This expression of $E(v^t)$ follows observing that $\mathcal{F}{(u^t)^2}(\xi)=\mathcal{F} {u^2}(\frac{\xi_1}{t}, \frac{\xi_2}{t}, \frac{\xi_3}{\sqrt{t}})$
and by a change of variable. Now under \eqref{AS} we have that
$$\lim_{t \rightarrow \infty }\lambda_1+\frac 43 \pi \lambda_2 \frac{2t \xi_3^2-t^2\xi_1^2-t^2\xi_2^2}{t^2\xi_1^2+t^2\xi_2^2+t \xi_3^2}=\lambda_1-\frac{4}{3}\pi \lambda_2<0,$$
which implies that
$\lim_{t \rightarrow \infty}E(v^t)=-\infty$ thanks to Lebesgue's theorem. Just note that in the second case in \eqref{AS}, $\lambda_2<0, \ \lambda_1+\frac{8}{3}\pi \lambda_2<0$, the same conclusion follows choosing the scaling
\begin{equation}\label{def:scad1}
\tilde{v}^t(x)=\lambda^{\frac 54}v(t^{\frac 34} x_1, t^{\frac 34 } x_2, t x_3), \quad t >0.
\end{equation}
Thus in both cases the class of paths $\Gamma(c)$ defined in \eqref{Gamma} is non void. Now if $g \in \Gamma(c)$ there exists a $\overline{t} \in (0,1)$ such that $g(\overline{t}) \in C_{2k}$. Thus
$$\max_{t \in [0,1]}E(g(t)) \geq E(g(\overline{t})) \geq \inf_{u \in C_{2k}}E(u) > \sup_{u \in A_{k}}E(u),$$
and this implies that $\gamma(c) >0$ where $\gamma(c)$ is given in \eqref{gamma}. Thus $E(u)$ admits on $S(c)$ a mountain pass geometry.
{\bf Step 2: } {\it $\gamma(c) = \displaystyle \inf_{V(c)} E$.}
Let $v\in V(c)$. Since $Q(v) =0$ we get that $B(v) <0$ and considering the scaling
$v^t(x)=t^{\frac 32}v(t x), \, t>0$ we deduce from \eqref{def:mainscal} that there exists a $t_1<<1$ and a $t_2>>1$ such that $v^{t_1}\in A_{k}$ and $E(v^{t_2})<0$. Thus if we define
$$g(\lambda)=v^{(1-\lambda)t_1+\lambda t_2} \ \ \ \lambda\in[0,1],$$
we obtain a path in $\Gamma(c)$. By the definition of $\gamma(c)$
$$\gamma(c)\leq \max_{\lambda\in [0,1]}E(g(\lambda))=E(g(\frac{1-t_1}{t_2-t_1}))=E(v).$$
On the other hand any path $g(t)$ in $\Gamma(c)$ by continuity and Lemma \ref{lem:base} crosses $V(c)$. This shows that
$$\max_{t \in [0,1]} E(g(t))\geq \inf_{u \in V(c)} E(u).$$
{\bf Step 3:} {\it Existence of a bounded Palais-Smale sequence $(u_n) \subset S(c)$ at the level $\gamma(c)$.}
As stated in the Introduction the mountain pass geometry implies the existence of a sequence $(u_n) \subset S(c)$ such that
$$E(u_n)=\gamma(c)+o(1), \ \ \ ||E'|_{S(c)}(u_n)||_{H^{-1}}=o(1).$$
By using an argument due to \cite{Gh} we can strenghten this information and select a specific sequence localized around $V(c)$, namely such that
$dist(u_n, V(c))=o(1)$.
To be more precise taking $F$ as $V(c)$ in \cite[Theorem 4.1]{Gh} we obtain the existence of a sequence $(u_n) \subset S(c)$ such that
$$E(u_n)=\gamma(c)+o(1), \ \ \ ||E'|_{S(c)}(u_n)||_{H^{-1}}=o(1), \ dist(u_n, V(c))=o(1).$$ The fact that taking $F= V(c)$ is possible follows from Steps 1 and 2.
Now, for any fixed $c>0$, it follows directly from \eqref{star} that the set
$$L:= \{ u \in V(c), E(u) \leq \gamma(c) +1 \}$$
is bounded.
On the other hand $||dQ(\cdot)||_{H^{-1}}$ is bounded on any bounded set of $H$ and thus in particular in a neighborhood of $L$. Now, for any $n \in {\mathbb N}$ and any $w \in V(c)$ we can write
$$Q(u_n) = Q(w) + dQ(au_n + (1-a)w) (u_n-w) = dQ(au_n + (1-a)w) (u_n-w)$$
where $a \in [0,1]$. Thus choosing $(w_m) \subset V(c)$ such that
$$||u_n - w_m|| \to dist(u_n, V(c)) \mbox{ as } m \to \infty$$
we obtain, since $dist(u_n, V(c)) \to 0$, that $Q(u_n) = o(1).$
At this point, using again \eqref{star}, we deduce that
$$E(u_n) = \frac{1}{6} ||\nabla u_n||_2^2+o(1)$$
which proves the boundedness of $(u_n) \subset S(c)$.
{\bf Step 4:} {\it For all} $c_1\in (0,c)$ \ \ $\gamma(c_1) > \gamma(c).$ \
We use here the characterization
\begin{equation}\label{addc}
\gamma(c)=\inf_{u\in S(c)} \max_{t>0}E(u^t).
\end{equation}
To show \eqref{addc} let us denote the right hand side by $\gamma_1(c)$. On one hand by Lemma \ref{lem:growth} it is clear that for any $u \in S(c)$ such that $\max_{t>0}E(u^t) < \infty$ there exists a unique $t_0>0$ such that $u^{t_0} \in V(c)$ and $\max_{t>0}E(u^t) = E(u^{t_0}).$ Now $E(u^{t_0}) \geq \gamma(c)$ by Step 2 and we thus get that $\gamma_1(c) \geq \gamma(c)$. On the other hand, for any $u \in V(c)$, $\max_{t>0}E(u^t) = E(u)$ and this readily implies that $\gamma_1(c) \leq \gamma(c)$.
Now, recording \eqref{def:mainscal}, we get after a simple calculation, that
\begin{eqnarray}\label{estima}
\max_{t>0} E(u^t)= \frac{2}{27} \, \frac{A(u)^3}{B(u)^2}.
\end{eqnarray}
Next take $u_1\in S(c_1)$, such that
$$
\max_{t>0} E(u^t_1) < \frac{c}{c_1}\gamma(c_1).$$
From the scaling $u_{\theta}(x):=\theta^{-\frac{1}{2}}u_1(\frac{x}{\theta})$ with $ \theta >0$, we have
$$\|u_{\theta}\|_2^2=\theta^2\|u_1\|_2^2, \quad A(u_{\theta})= A(u_1) \quad \mbox{and} \quad B(u_{\theta})=\theta\, B(u_1).$$
Thus taking $ \displaystyle \theta^2 = \frac{c}{c_1}$ we obtain that $u_{\theta} \in S(c)$ and it follows from \eqref{addc} that
$$ \gamma(c) \leq \max_{t>0} E(u_{\theta}^t)= \frac{2}{27} \, \frac{A(u_1)^3}{B(u_1)^2} \frac{c_1}{c} < \gamma(c_1).$$
{\bf Step 5:} {\it Convergence of the Palais-Smale sequence $(u_n) \subset S(c)$.}
From Step 3 we know that there exists a bounded Palais-Smale sequence $(u_n) \subset S(c)$ such that $E(u_n)\rightarrow \gamma(c)$ and $Q(u_n)=o(1)$. Proposition \ref{prop} then implies that $u_n \rightharpoonup \bar{u}$ with $\bar{u}$ a solution of \eqref{eq:main}. Let us first show that we can assume $\bar u \neq 0$. Notice that
$$\int_{{\mathbb R}^3} (\lambda_1-\frac 43 \pi \lambda_2)|\hat {u_n^2}|^2d\xi<\int_{{\mathbb R}^3} (\lambda_1+\lambda_2 \hat K(\xi))|\hat {u_n^2}|^2d\xi=\frac{2}{3}Q(u_n)- \frac{2}{3}A(u_n)=o(1)-4 \gamma(c).$$
This implies by Plancherel identity that $||u_n||_4\geq C>0.$ At this point since $||u_n||_2^2=c$, $||u_n||_6\leq C A(u_n)^{\frac12}<C$, the classical pqr-Lemma \cite{FLL} implies that
there exists a $\eta>0$, such that
\begin{equation}\label{hyp2}
\inf_n \left|\{ |u_n|>\eta \}\right|>0 \,.
\end{equation}
Here $|\cdot |$ denote the Lebesgue measure. This fact, together with Lieb Translation lemma \cite{Lieb}, assures the existence of a sequence $(x_n)\subset{\mathbb R}^3$ such that a subsequence of $u_n(\cdot+ x_n)$ has a weak limit $\bar u \not\equiv 0$ in $H$. Now let us prove the strong convergence. Since $\bar u$ is non trivial and is a solution of \eqref{eq:main} we can assume by
Lemma \ref{lem:poho} that $\bar{u} \in V(c_1)$ for some $0<c_1 \leq c$.
We recall that
\begin{equation}\label{Splittings}
A(u-\bar u)+ A(\bar u)=A(u_n)+o(1), \ \ B(u-\bar u)+B(\bar u)=B(u_n)+o(1).
\end{equation}
For a proof of the splitting property for $B(u)$ we refer to \cite{AS}. Since $E(u_n) \to \gamma(c)$ the splittings give
\begin{equation}\label{101}
\frac 12A(u_n-\bar u)+\frac 12 A(\bar u)+\frac 12B(u_n-\bar u)+\frac 12B(\bar u)=\gamma(c)+o(1)
\end{equation}
and we also have
\begin{equation}\label{1020}
Q(u_n-\bar u)+Q(\bar u)=Q( u_n)+o(1).
\end{equation}
Since $\bar u\in V(c_1)$ we have by Step 2 that $E(\bar{u}) \geq \gamma(c_1)$ and we deduce from \eqref{101} that
\begin{equation}\label{eq:mono}
E(u_n -\bar u)+\gamma(c_1)\leq \gamma(c)+o(1).
\end{equation}
At this point from \eqref{1020}, \eqref{eq:mono}, Step 4 and using the fact that
$$\frac 16 A(u_n-\bar u) = E(u_n-\bar u)-\frac{1}{3}Q(u_n-\bar u)$$
we deduce that necessarily $c_1 = c$ and $A(u_n-\bar u)=o(1)$. This proves the strong convergence of $(u_n) \subset S(c)$ in $H$.
{\bf Step 5:} {\it Conclusion}
Since $(u_n) \subset S(c)$ converges we deduce from Proposition \ref{prop} the existence of a couple $(u_c, \mu_c) \in H \times {\mathbb R}$ which satisfies \eqref{eq:main} and such that $E(u_c) = \gamma(c)$. By Lemma \ref{lem:poho} we see that $\mu_c >0$. Still from Lemma \ref{lem:poho} and using Step 2 we deduce that $u_c$ is a ground state.
\end{proof}
\section{Proofs of Lemmas \ref{naturalconstraint} and \ref{description}}\label{Section51}
In this section we show that $V(c)$ acts as a natural constraint and derive some properties of the set of ground states of $E(u)$ on $S(c)$.
\begin{proof}[Proof of Lemma \ref{naturalconstraint}]
The fact that $V(c)$ is a $C^1$ manifold is standard by the implicit function theorem. Let $u$ be a critical point of $E_{|_{V(c)}}$, then there exist $\mu_1$ and $\mu_2$ such that
$$E'(u) -\mu_1Q'(u)-2\mu_2u=0.$$
We need to show that $\mu_1=0$. Notice that $u$ fulfills the following equation
\begin{equation}\label{100}
(1-2\mu_1)(-\Delta u) +2(1-3\mu_1)\left(\lambda_1|u|^2u+\lambda_2(K\star u^2)u\right)-2\mu_2 u=0.
\end{equation}
Multiplying \eqref{100} by $u$ and integrating we get
\begin{equation}\label{eq:nat1}
(1-2\mu_1)A(u) +2(1-3\mu_1)B(u)-2\mu_2 ||u||_2^2=0.
\end{equation}
Also from Pohozaev identity
\begin{equation}\label{eq:nat2}
\frac 12 (1-2\mu_1)A(u) +\frac{3}{2}(1-3\mu_1)B(u)-3\mu_2 ||u||_2^2=0.
\end{equation}
Combining \eqref{eq:nat1} and \eqref{eq:nat2} we get
\begin{equation}\label{102}
(1-2\mu_1)A(u) +\frac 32 (1-3\mu_1)B(u)=0.
\end{equation}
Now using the fact that $u \in V(c)$, i.e $A(u) +\frac 32 B(u)=0$, it follows from \eqref{102} that $\mu_1 A(u)=0$. Thus necessarily $\mu_1 = 0$.
\end{proof}
\begin{proof}[Proof of Lemma \ref{description}]
Let $u_c \in H$ with $u_c\in V(c)$. Since $\|\nabla |u_c|\|_2\leq \|\nabla u_c \|_2$ we have that $E(|u_c|)\leq E(u_c)$ and $Q(|u_c|)\leq Q(u_c)=0$. In addition, by Lemma \ref{lem:growth}, there exists $t_0\in (0,1]$ such that $Q(|u_c|^{t_0})=0$. Observe that, since $Q(u_c)=Q(|u_c|^{t_0})=0$, we have
$$
E(|u_c|^{t_0}) = \frac 16 A(|u_c|^{t_0})
= t_0^2\cdot \frac 16 A(|u_c|)
= t_0^2 \cdot E(|u_c|) \leq t_0^2 E(u_c).
$$
Thus if $u_c \in H$ is a minimizer of $E(u)$ on $V(c)$ we have
$$E(u_c)=\inf_{u \in V(c)}E(u)\leq E(|u_c|^{t_0}) \leq t_0^2 \cdot E(u_c),$$
which implies that $t_0=1$ since $t_0 \in (0,1]$. Then $Q(|u_c|)=0$ and we conclude that
\begin{eqnarray}
\|\nabla |u_c|\|_2 = \|\nabla u_c \|_2\quad \mbox{and\quad } E(|u_c|)=E(u_c).
\end{eqnarray}
Thus point (i) follows. Now since $|u_c|$ is a minimizer of $F(u)$ on $V(c)$ we know by Lemmas \ref{naturalconstraint} and \ref{lem:poho}
that it satisfies \eqref{eq:main} for some $\mu_c > 0$. By elliptic regularity theory and the maximum principle it follows that
$|u_c| \in C^1({\mathbb R}^3, {\mathbb R})$ and $|u_c|>0$. At this point, using that $\|\nabla |u_c|\|_2 = \|\nabla u_c \|_2$ the rest of the proof of point (ii) is exactly the same as in the proof of Theorem 4.1 of \cite{HAST}.
\end{proof}
\begin{remark}\label{Jacopo}
Clearly in the stable regime, $B(u) \geq 0$, for any $u \in S(c)$. Then one always have that $Q(u) >0$ on $S(c)$ and, in view of Lemma \ref{lem:poho}, $E(u)$ has no constrained critical points on $S(c)$. Thus no solution of \eqref{eq:main} exists. In Step 1 of the Proof of Theorem \ref{thm:standing} we show that \eqref{AS} is a sufficient condition for the existence of a $u \in S(c)$, such that $E(u) <0$ and thus $B(u) <0$. Thus \eqref{AS} is equivalent to the existence of at least one $u \in S(c)$ such that $B(u) <0$.
\end{remark}
\section{Proof of Theorem \ref{thm:AS}}
The aim of this section is to prove that the solutions obtained by \cite{AS} coincide with minimizers of $E(u)$ on $V(c)$. In \cite[Theorem 1.1]{AS} the solutions of \eqref{eq:main} are obtained as minimizer of the functional
$$J(v):=\frac{A(v)^{\frac{3}{2}}||v||_2}{-B(v)}.$$
Let us call $u$ the minimizer of $J(v)$ that solves
for $\mu >0$
\begin{equation}\label{eq:mainbis}
-\frac 12 \Delta u +\lambda_1|u|^2u+\lambda_2(K\star u^2)u+\mu u=0
\end{equation}
and set $||u||_2^2 = c$. Our aim is to show that $E(u) = \gamma(c)$. Note that scaling properties of $J(u)$ allows to find a solution for any $\mu >0$.
Since $u$ satisfies (\ref{eq:mainbis}) then, by Lemma \ref{lem:poho}, $Q(u)=0$ and this implies that
$$E(u) = \frac{1}{6}A(u) \quad \mbox{and} \quad B(u) = - \frac{2}{3}A(u).$$
It then follows by a direct calculation that
\begin{equation}\label{level}
J(u) = \frac{1}{4}\,6^{3/2}c^{1/2}E^{1/2}(u).
\end{equation}
Now assume that $u$ is not a minimizer of $E(u)$ on $V(c)$. Then denoting by $u_0 \in V(c)$ a minimizer of $E(u)$ on $V(c)$ (we know that it exists by Theorem \ref{thm:standing}) we have that $E(u_0) < E(u)$. Since $u_0 \in V(c)$ we also have that
$$ A(u_0) = 6 E(u_0) \quad \mbox{ and } \quad B(u_0) = - 4 E(u_0).$$
Thus
\begin{equation}\label{levelbis}
J(u_0) = \frac{1}{4}\,6^{3/2}c^{1/2}E^{1/2}(u_0).
\end{equation}
Comparing \eqref{level} and \eqref{levelbis} we derive that
$J(u_0) < J(u)$
which provides a contradiction with the fact that $u$ minimizes $J(v)$.
\section{Proof of Theorem \ref{thm:global}}
Let $u(x,t)$ be the solution of \eqref{eq:evolutionbis} with $u(x,0)=u_0$ and $T_{max} \in (0, \infty]$ its maximal time of existence. Then classically we have either
$$T_{max}=+\infty$$
or
\begin{equation}\label{def: blowup}
T_{max} < + \infty \quad \mbox{ and } \lim_{t \rightarrow T_{max}}||\nabla u(x,t)||_2^2=\infty.
\end{equation}
Since
$$
E(u(x,t))-\frac 13 Q(u(x,t))=\frac 16 A(u(x,t))
$$
and $E(u(x,t))=E(u_0)$ for all $t<T_{max}$, if \eqref{def: blowup} happens then,
we get
$$\lim_{t \rightarrow T_{max}}Q(u(x,t))=-\infty.$$
Since $Q(u(x,0))>0$, by continuity it exists $t_0 \in (0, T_{max})$ such that $Q(u(x, t_0))=0$ with
$E(u(x,t_0))=E(u_0)<\gamma(c)$. This contradicts the definition $\gamma(c) = \inf_{u\in V(c)}E(u)$.
\section{Proof of Theorem \ref{thm:scat}}
We recall the Duhamel formula associated to the evolution equation \eqref{eq:evolutionbis} when $a=0$
$$\psi(t)=U(t)\psi_0- i \lambda_1 \int_0^t U(t-s)(|\psi|^2\psi)(s)ds -i \lambda_2 \int_0^t U(t-s)((K\star |\psi|^2)\psi)(s)ds$$
where
$$U(t)=e^{ it\frac{\Delta}{2}}$$
generates the time evolution of the linear Schr\"odinger equation. We also recall the Strichartz estimates in ${\mathbb R}^d$, $d\geq 3$
\begin{eqnarray}
& ||U(\cdot)\varphi||_{L^q_tL^r_x}\leq C||\varphi||_{L^2} \label{eq:se1}\\
& || \int_0^t U(t-s) F(s) ds||_{L^q_tL^r_x}\leq C||F||_{L^{q_1'}L^{r_1'}} \label{eq:se2}
\end{eqnarray}
where the pairs $(q,r)$, $(q_1,r_1)$ are admissible, i.e $2\leq r\leq \frac{2d}{d-2}$ and $\frac{2}{q}=d(\frac{1}{2}-\frac{1}{r})$ (analogous for $(q_1,r_1)$).
The local Cauchy theory for equation \eqref{eq:evolutionbis} is proved in \cite{CMS}.
\begin{thm}[\cite{CMS}]
There exists $T>0$ depending only on $||\psi_0||$ such that \eqref{eq:evolutionbis} with initial data $\psi_0$ has a unique solution $\psi \in X_{T}$, where
$$X_T=\left\{\psi \in C([0,T]; H^1({\mathbb R}^3)); \ \ \psi, \nabla \psi \in C([0,T]; L^2({\mathbb R}^3))\cap L^{\frac{8}{3}}([0,T];L^4({\mathbb R}^3))\right\}.$$
\end{thm}
For the proof of Theorem \ref{thm:scat} we shall need the following
\begin{prop}\label{prop:scat}
There exists $\delta>0$ such that if $||\psi_0||<\delta$ then the solution $\psi(t)$ of \eqref{eq:evolutionbis} is global and $\sup_t ||\psi||<\infty$.
\end{prop}
\begin{proof}
The proof in the stable regime is a direct consequence of the energy conservation since $E(u)$ is then coercive \cite{BaCaWa,CaHa} and the global well-posedness holds for any initial data in $H$. Under conditions \eqref{AS} there exists initial data that blows up in finite time and hence not all initial data have bounded kinetic energy for all times. We consider for simplicity the case $\lambda_2>0$, $\lambda_1-\frac{4}{3}\pi \lambda_2<0$, the other case is identical. According to Theorem \ref{thm:global} we just need to prove that when $||\psi_0||$ is small one has $Q(\psi_0) >0$ and $E(\psi_0) < \gamma(||\psi_0||_2^2)$. Observe that thanks to Plancherel identity we can write $B(u)$ as
$$B(u)=(\lambda_1-\frac 43 \pi \lambda_2)||u||_4^4+\lambda_2 \int_{{\mathbb R}^3} (\tilde K\star |u|^2)|u|^2dx$$
where the fourier transform of $\tilde K$ is $\hat{\tilde K}=4 \pi \frac{|\xi_3|^2}{|\xi|^2}$.
Hence
$$Q(u)\geq A(u)+ \frac 32 (\lambda_1-\frac 43 \pi \lambda_2)||u||_4^4\geq A(u)\left(1 +C(\lambda_1-\frac 43 \pi \lambda_2)A(u)^{\frac 12}||u||_2\right).$$
In particular $Q(u)>0$ when $||u||$ is sufficiently small. Now consider the ground states energy $\gamma(c)$ and let $u_c \in S(c)$ be a groundstate.
We have
$$0=Q(u_c)\geq A(u_c)\left(1 +C(\lambda_1-\frac 43 \pi \lambda_2)A(u_c)^{\frac 12}||u_c||_2\right)$$
which implies that
$$\lim_{c \rightarrow 0} A(u_c)=+\infty.$$
Now since $E(u_c)=\frac 16 A(u_c)$ we deduce that $\lim_{c \rightarrow 0} \gamma(c)=+\infty.$ Thus when $||\psi_0||$ is small we certainly have $E(\psi_0) < \gamma(||\psi_0||_2^2)$ and the proof is completed.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:scat}]
We follow the classical strategy to show that $||\psi||_{L^p_tW^{1,q}_x}$ is globally bounded in time: this implies scattering.
The admissible pair that we use is $(p,q)=(\frac{8}{3},4)$ with conjugated pair $(p',q')=(\frac 85, \frac 43)$. By using the Duhamel formula we have
\begin{eqnarray}||\psi||_{L^{\frac{8}{3}}_tW^{1,4}_x}\leq ||U(t)\psi_0||_{L^{\frac{8}{3}}_tW^{1,4}_x} + \lambda_1 ||\int_0^t U(t-s)(|\psi|^2\psi)(s)ds||_{L^{\frac{8}{3}}_tW^{1,4}_x} + \nonumber \\
+\lambda_2|| \int_0^t U(t-s)((K\star |\psi|^2)\psi)(s)ds||_{L^{\frac{8}{3}}_tW^{1,4}_x}. \nonumber
\end{eqnarray}
Using Strichartz estimates \eqref{eq:se1}, \eqref{eq:se2} we get
\begin{eqnarray}||\psi||_{L^{\frac{8}{3}}_tW^{1,4}_x}\leq c||\psi_0||_{H^1} + c |||\psi|^2\psi||_{L^{\frac{8}{5}}_tW^{1,\frac{4}{3}}_x} + \nonumber \\
+c||(K\star |\psi|^2)\psi||_{L^{\frac{8}{5}}_tW^{1,\frac{4}{3}}_x}. \nonumber
\end{eqnarray}
First we estimate the terms $|||\psi|^2\psi||_{L^{\frac{8}{5}}_tL^{\frac{4}{3}}_x}$ and $||(K\star |\psi|^2)\psi||_{L^{\frac{8}{5}}_tL^{\frac{4}{3}}_x}.$
By H\"{o}lder inequality we have
$$|||\psi|^2\psi||_{L^{\frac{8}{5}}_tL^{\frac{4}{3}}_x}\leq c||\psi||_{L^{\frac{8}{3}}_tL^{4}_x}|||\psi|^2||_{L^{4}_tL^{2}_x}=c||\psi||_{L^{\frac{8}{3}}_tL^{4}_x}||\psi||_{L^{8}_tL^{4}_x}^2$$
and
$$||(K\star |\psi|^2)\psi||_{L^{\frac{8}{5}}_tL^{\frac{4}{3}}_x}\leq c ||\psi||_{L^{\frac{8}{3}}_tL^{4}_x}||K\star |\psi|^2||_{L^{4}_tL^{2}_x}^2\leq c ||\psi||_{L^{\frac{8}{3}}_tL^{\frac{4}{3}}_x}||\psi||_{L^{8}_tL^{4}_x}^2.$$
Notice that in the last step we used that $||K\star f||_p\leq c||f||_p$, namely the $L^p-L^p$ continuity of $K$ established in \cite[Lemma 2.1]{CMS}.
Now using Proposition \ref{prop:scat} and Sobolev embedding $$||\psi||_{L^{8}_tL^{4}_x}^2\leq ||\psi||_{L^{\frac{8}{3}}_tL^{4}_x}^{\frac{2}{3}}||\psi||_{L^{\infty}_tL^{4}_x}^{\frac 43}\leq c ||\psi||_{L^{\frac{8}{3}}_tL^{4}_x}^{\frac{2}{3}}||\psi||_{L^{\infty}_tH^{1}_x}^{\frac 43}\leq c ||\psi||_{L^{\frac{8}{3}}_tL^{4}_x}^{\frac{2}{3}}$$
and we obtain
$$||\psi|^2\psi||_{L^{\frac{8}{5}}_tL^{\frac{4}{3}}_x}\leq c||\psi||_{L^{\frac{8}{3}}_tL^{4}_x}^{\frac 53}.$$
Now we estimate the terms $||\nabla (|\psi|^2\psi)||_{L^{\frac{8}{5}}_tL^{\frac{4}{3}}_x}$ and $||\nabla (K\star |\psi|^2)\psi||_{L^{\frac{8}{5}}_tL^{\frac{4}{3}}_x}$.
By H\"{o}lder inequality again and arguing as before we get
$$||\nabla (|\psi|^2\psi)||_{L^{\frac{8}{5}}_tL^{\frac{4}{3}}_x}\leq c|| \nabla \psi ||_{L^{\frac{8}{3}}_tL^{4}_x}||\psi^2||_{L^{4}_tL^{2}_x}=c||\nabla \psi||_{L^{\frac{8}{3}}_tL^{4}_x}||\psi||_{L^{8}_tL^{4}_x}^2\leq c ||\psi||_{L^{\frac{8}{3}}_tW^{1,4}_x}^{\frac{5}{3}}.$$
The term $||(K\star |\psi|^2)\nabla \psi||_{L^{\frac{8}{5}}_tL^{\frac{4}{3}}_x}$ behaves identically
$$||(K\star |\psi|^2)\nabla \psi||_{L^{\frac{8}{5}}_tL^{\frac{4}{3}}_x}\leq c ||\nabla \psi||_{L^{\frac{8}{3}}_tL^{4}_x}||K\star |\psi|^2||_{L^{4}_tL^{2}_x}\leq c ||\nabla \psi||_{L^{\frac{8}{3}}_tL^{4}_x}||\psi||_{L^{8}_tL^{4}_x}^2\leq c ||\psi||_{L^{\frac{8}{3}}_tW^{1,4}_x}^{\frac{5}{3}}.$$
The last term to compute is $||(K\star \nabla |\psi|^2) \psi||_{L^{\frac{8}{5}}_tL^{\frac{4}{3}}_x}.$ For this term we argue as before
$$||(K\star \nabla |\psi|^2) \psi||_{L^{\frac{8}{5}}_tL^{\frac{4}{3}}_x}\leq c||\psi||_{L^{8}_tL^{4}_x}||K\star \nabla |\psi|^2||_{L^{2}_tL^{2}_x}\leq c||\psi||_{L^{8}_tL^{4}_x}||(\nabla \psi) \psi||_{L^{2}_tL^{2}_x}$$
and therefore by H\"{o}lder inequality
$$||(K\star \nabla |\psi|^2) \psi||_{L^{\frac{8}{5}}_tL^{\frac{4}{3}}_x}\leq ||\nabla \psi||_{L^{\frac 83}_tL^{4}_x}||\psi||_{L^{8}_tL^{4}_x}^2\leq c ||\psi||_{L^{\frac{8}{3}}_tW^{1,4}_x}^{\frac{5}{3}}.$$
Eventually we proved that
\begin{equation}\label{eq:finalscat}
||\psi||_{L^{\frac{8}{3}}_tW^{1,4}_x}\leq c||\psi_0||_{H^1}+c||\psi||_{L^{\frac{8}{3}}_tW^{1,4}_x}^{\frac 53}.
\end{equation}
Now calling $||\psi||_{L^{\frac{8}{3}}_tW^{1,4}_x}=y$ and $||\psi_0||=b$ and looking at the function $f(y)=y-b-y^{\frac 53}$
we notice that if $b$ is sufficiently small then $\left\{ y \ \ s.t. \ \ f(y)\leq 0 \right\}$ has two connected components.
This implies that choosing $||\psi_0||$ sufficiently small we obtain, for some $C>0$,
\begin{equation}\label{eq:stricbound}
||\psi||_{L^{\frac{8}{3}}_{[0, \infty]}W^{1,4}_x}\leq C.
\end{equation}
Scattering follows now from classical arguments, see e.g \cite{TC}. To conclude it is indeed enough to show that $U(-t)\psi(t) \rightarrow \psi_{+}$ in $H^1({\mathbb R}^3)$. Notice that for $0<t<\tau$ and calling $g:= \lambda_1 |\psi|^2 \psi + \lambda_2 (K \star |\psi|^2) \psi$
and $v(t):=U(-t)\psi(t)$, by \eqref{eq:stricbound} one gets
$$||v(t)-v(\tau)||_{H^1}\leq c ||g(\psi)||_{L^{\frac{8}{5}}_{[t, \tau]} W^{1,\frac{4}{3}}_x}\rightarrow_{t, \tau \rightarrow \infty} 0.$$
Therefore it exists $\psi_{+}$ such that $\lim_{t \rightarrow \infty} ||v(t) -\psi_{+}||=0$ and hence
$$\lim_{t \rightarrow \infty}||\psi(t) -U(t)\psi_{+}||=\lim_{t \rightarrow \infty} ||U(-t)\psi(t)- \psi_{+}||=0.$$
\end{proof}
\section{Proof of Theorem \ref{thm:instability}}
The proof of Theorem \ref{thm:instability} is standard and follows the original approach by Glassey \cite{G} and Berestycki-Cazenave \cite{BECA}. We recall the virial identity, see \cite{CMS} and \cite{Lu},
\begin{equation}\label{virial}
\frac{\mathrm{d}^2}{\mathrm{d} t^2} \left \|xv(t)\right\|_2^2=2\int_{{\mathbb R}^3} |\nabla v|^2dx+3 \int_{{\mathbb R}^3} \lambda_1|v|^4+\lambda_2(K\star |v|^2)|v|^2 dx =2 Q(v)
\end{equation}
and the fact that all real positive solutions of \eqref{eq:main} belongs to $\Sigma$ as given in \eqref{Sigma}. This follow from the decay estimates obtained in \cite{AS}, see also \cite{TC}.
For any $c>0$, we define the set
\begin{eqnarray*}
\Theta= \left \{v \in H\setminus \{0\} \ s.t. \ E(v)<E(u_c),\ \left \| v \right \|_2^2=\left \| u_c \right \|_2^2, \ Q(v)<0 \right \}.
\end{eqnarray*}
The set $\Theta$ contains elements arbitrary close to $u_c$ in $H$. Indeed, letting $v_0(x)=u_c^{\lambda}=\lambda^{\frac{3}{2}}u_c(\lambda x)$, with $\lambda <1$, we see from Lemma \ref{lem:growth} that $v_0 \in \Theta$ and that $v_0 \to u_c$ in $H$ as $\lambda \to 1$.
Let $v(t)$ be the maximal solution of \eqref{eq:evolutionbis} with initial datum $v(0)=v_0$ and $T \in (0, \infty]$ the maximal time of existence. Let us show that $v(t) \in \Theta$ for all $ t \in [0,T)$. From the conservation laws
$$\left \| v(t) \right \|_2^2 = \left \| v_0 \right \|_2^2 =\left \| u_c \right \|_2^2,$$ and $$E(v(t))=E(v_0)<E(u_c).$$ Thus it is enough to verify $Q(v(t))<0$. But $Q(v(t)) \neq 0$ for any $t \in (0, T)$. Otherwise, by the definition of $\gamma(c)$, we would get for a $t_0 \in (0,T)$ that $E(v(t_0)) \geq E(u_c)$ in contradiction with $E(v(t)) < E(u_c)$. Now by continuity of $Q$ we get that $Q(v(t))<0$ and thus that $v(t) \in \Theta$ for all $ t \in [0,T)$.
Now we claim that there exists $\delta >0$, such that
\begin{eqnarray}\label{le2.4}
Q(v(t))\leq - \delta , \ \forall t \in [0,T).
\end{eqnarray}
Let $t \in [0,T)$ be arbitrary but fixed and set $v = v(t)$. Since $Q(v) <0$ we know by Lemma \ref{lem:growth} that $\lambda^{\star}(v)<1$ and that $\lambda \longmapsto E(v^{\lambda})$ is concave on $[\lambda ^{\star}, 1)$.
Hence
\begin{eqnarray*}
E(v^{\lambda^{\star}})-E(v) &\leq& (\lambda^{\star}-1) \frac{\partial}{\partial \lambda}E(v^{\lambda})\mid _{\lambda =1} \\
&=& (\lambda^{\star}-1) Q(v).
\end{eqnarray*}
Thus, since $Q(v(t))<0$, we have
$$E(v)-E(v^{\lambda^{\star}})\geq (1- \lambda^{\star})Q(v)\geq Q(v).$$
It follows from $E(v)=E(v_0)$ and $v^{\lambda^{\star}} \in V(c)$ that
$$Q(v) \leq E(v)-E(v^{\lambda^{\star}})\leq E(v_0)-E(u_c)$$
and this proves the claim. Now from the virial identity \eqref{virial} we deduce that $v(t)$ must blow-up in finite time. Recording that $v_0$ has been taken arbitrarily close to $u_c$, this ends the proof of the theorem.
\section{Proof of Theorem \ref{thm: mainn} and Corollary \ref{Cor}}
This section is devoted to the proof of Theorem \ref{thm: mainn}. Under the scaling given by \eqref{def:sca} we have
\begin{equation}\label{def:mainsca}
t \to E_a(u^{t})=\frac{t^2}{2} A(u) + \frac{a^2}{2t^2} D(u)
+\frac{t^3}{2}B(u) \quad \mbox{where we have set} \quad D(u)= \int_{{\mathbb R}^3}|x|^2 u^2 dx.
\end{equation}
Let us also define $$Q_a(u):= A(u) - a^2 D(u)
+\frac{3}{2}B(u).$$
The proof of Theorem \ref{thm: mainn} requires several steps.
{\bf Step 1: } {\it There exist a $a_0 >0$ such that, for any $a \in (0,a_0], \, E_a(u)$ has a topological local minima/mountain-pass geometry on $S(c)$.}
From \eqref{def: mp} we know that there exists a $k>0$ such that
\begin{equation}\label{def: mpp}
0 \leq \inf_{u \in A_{k}}E(u) \leq \sup_{u\in A_{k}} E(u) < \inf_{u \in C_{2k}} E(u).
\end{equation}
Also by Theorem \ref{thm:standing} there exists a $u_c \in V(c)$ such that $E(u_c)= \gamma(c)$. We consider the path
\begin{equation}\label{sparticulier}
t \to v^t(x) := t^{\frac{3}{2}}u_c(tx), \quad t>0.
\end{equation}
First we fix a $t_1 <<1$ such that $v^{t_1}\in A_{k}$. Then, taking $a>0$ sufficiently small so that
$$\frac{a^2}{2t_1^2}D(v^{t_1}) < \inf_{u \in C_{2k}}E(u) - \sup_{u \in A_{k}}E(u),$$
we obtain that
\begin{equation}\label{test}
0< E_a(v^{t_1}) < \inf_{u \in C_{2k}}E(u) \leq \inf_{u \in C_{2k}}E_a(u).
\end{equation}
Thus in view of \eqref{blow} it is reasonable to search for a minima of $E_a(u)$ inside the set $A_{2k}$.
Now, since $D(v^t) \to 0$ as $t \to + \infty$, we still have that $E_a(v^t) \to - \infty$ as $t \to + \infty$. We fix a $t_2 >>1$ such that $E_a(v^{t_2}) <0$ and define
$$\Gamma_a(c) =\{g \in C([0,1],S(c)) \ s.t. \ \ g(0)=v^{t_1}, g(1)=v^{t_2}\}.$$
Clearly $\Gamma_a(c) \neq \emptyset$ and from \eqref{test} it holds that
$$\gamma_a(c) := \inf_{g \in \Gamma_a(c)} \max_{t \in [0,1]}E_a(g(t)) > \max \{E_a(v^{t_1}), E_a(v^{t_2})\}>0.$$
Namely $E_a(u)$ has a {\it mountain pass geometry} on $S(c)$.
{\bf Step 2: } {\it Existence of a topologial local minimizer.}
Let us prove that there exists a $u_a^1 \in A_{2k}$ which satisfies
$$E_a(u_a^1)=\inf_{u \in A_{2k}} E_a(u)>0.$$
Because of \eqref{test} necessarily $u_a^1 \notin C_{2k}$ and thus $u_a^1$ will be a topological local minimizer for $E_a(u)$ restricted to $S(c)$. Let $(u_n) \subset A_{2k}$ be an arbitrary minimizing sequence associated to
$$I_a(c)=\inf_{u \in A_{2k}} E_a(u).$$
This sequence, being in $A_{2k}$, is bounded and we can assume that it converges weakly to some $u_a^1$. To prove the strong convergence, we use the compactness of the embedding $\Sigma \hookrightarrow L^p({\mathbb R}^3)$ for $p \in [2,6)$. This gives directly that $u_a^1 \in S(c)$. Also since, for some $C>0$,
\begin{equation}\label{ControlB}
|B(u_n-u_a^1)|\leq C ||u_n-u_a^1||_4^4=o(1)
\end{equation}
we get that
$$E_a(u_a^1)\leq \liminf E_a(u_n)=I_a(c).$$
This implies that $E_a(u_a^1)=I_a(c)$ and $A(u_n - u_a^1) \to 0.$ Thus $u_n \to u_a^1$ and $u_a^1$ is a minimizer of $I_a(c)$. Note that since $I_a(|u|) \leq I_a(u), \forall u \in S(c)$ we can assume without restriction that $u_a^1$ is real. More generally a description of the set of topological local minimizers as in Lemma \ref{description} is available in a standard way, see for example \cite{CaHa}.
{\bf Step 3: } {\it Existence of a mountain-pass critical point.}
Let us suppose for a moment the existence of a \emph{bounded} PS sequence such that $E_a(u_n) \to \gamma_a(c)$. The proof of such claim requires some work. We posponed it until the Appendix. The strong convergence then follows from the following equivalent of Proposition \ref{prop}.
\begin{prop}\label{propa}
Let $(u_n) \subset S(c)$ be a bounded Palais-Smale in $\Sigma$ for $E_a(u)$ restricted to $S(c)$ such that $E_a(u_n) \to \gamma_a (c)$. Then there is a sequence $(\mu_n)\subset \mathbb{R}$, such that, up to a subsequence:\\
(1)\ $u_n \rightharpoonup u_a^2$ weakly in $\Sigma$;\\
(2)\ $\mu_n \to \mu$ in $\mathbb{R}$;\\
(3) $- \frac{1}{2}\Delta u_n + \frac{a^2}{2}|x|^2u_n + \lambda_1 |u_n|^2 u_n + \lambda_2 (K \star |u_n|^2) u_n + \mu u_n \to 0$ in $\Sigma^{-1}$;\\
(4) $- \frac{1}{2}\Delta u_a^2 + \frac{a^2}{2}|x|^2u_a^2 + \lambda_1 |u_a^2|^2 u+ \lambda_2 (K \star |u_a^2|^2) u+ \mu u_a^2= 0$ in $\Sigma^{-1}.$
\end{prop}
Indeed, letting (3) and (4) act on $u_n$ we get
$$\frac{1}{2}A(u_n)+\frac{a^2}{2}D(u_n)+B(u_n)+c \mu=o(1)$$
$$\frac{1}{2}A(u_a^2)+\frac{a^2}{2}D(u_a^2)+B(u_a^2)+c \mu=0.$$
Thus by substraction and using the splittings \eqref{Splittings} we get that
$$ \frac{1}{2}A(u_n -u_a^1) + \frac{a^2}{2}D(u_n - u_a^2) + B(u_n - u_a^2) = o(1).$$
Thus from \eqref{ControlB} we deduce that $A(u_n-u_a^2)=o(1)$, $D(u_n-u_a^2)=o(1).$ Namely that $u_n \to u_a^2$.
{\bf Step 4: } {\it $E_a(u_a) \to 0$ and $E_a(u_a^2) \to \gamma(c) $ as $a \to 0$. }
To prove that $E_a(u_a) \to 0$ as $a \to 0$ we just need to observe that $k>0$ in \eqref{def: mpp} can be taken arbitrarily small and that from \eqref{useful} we readily have that
$\inf_{u \in C_{2k}}E(u) \to 0$ as $k \to 0$. Then the conclusion follows from \eqref{test} since $E_a(u_a) \leq E_a(v^{t_1}).$
To show that $E_a(u_a^2) \to \gamma(c) $ namely that $\gamma_a(c) \to \gamma(c)$ as $a \to 0$ it suffices to observe that, on one hand, since $\Gamma_a(c) \subset \Gamma(c)$ and $E_a(u) \geq E(u)$ for all $u \in \Sigma$, then $\gamma_a(c) \geq \gamma(c)$. On the other hand considering the path \eqref{sparticulier} we have, as $a \to 0$,
$$ \gamma_a(c) \leq \sup_{t \in [t_1, + \infty]}E_a(v^t) \leq \sup_{t \in [t_1, + \infty]}E(v^t) + \frac{a^2}{2t_1^2}D(u_c) = \gamma(c) + \frac{a^2}{2t_1^2}D(u_c) \to \gamma(c).$$
{\bf Step 5: } {\it $u_a^1$ is a ground state. In addition any ground state $u_a \in S(c)$ for $E_a(u)$ on $S(c)$ is a topological local minimizer for $E_a(u)$ on $A_{2k}$ and it satisfies $A(u_a) \to 0$ and $E_a(u_a) \to 0$ as $a \to 0$. }
Notice that any constrained critical point $v$ fulfills $Q_a(v)=0$. From the definition of $E_a(u)$ and $Q_a(u)$ we get
\begin{equation*}
E_a(v) - \frac{1}{3}Q_a(v) = \frac{1}{6}A(v) + \frac{5}{6}a^2 D(v)
\end{equation*}
which implies that
\begin{equation}\label{eq:ggr}
E_a(v) \geq \frac{1}{6}A(v).
\end{equation}
Now let $u_a$ be a constrained critical point such that $E_a(u_a) \leq E_a(u_a^1)$. From Step 4 we know that $E_a(u_a^1) \to 0$ when $a \to 0$ and together with \eqref{eq:ggr} this implies that $A(u_a)\to 0$. Notice that $k$ does not depend on $a$ and therefore $u_a \in A_{2k}$ when $a$ is sufficiently small. By definition of
$u_a^1$ we obtain the opposite inequality $E_a(u_a) \geq E_a(u_a^1)$.
\begin{proof}[Proof of Corollary \ref{Cor}]
In order to show that small data scattering cannot hold under the assumption of Theorem \ref{thm: mainn} it is sufficient to prove, for $a>0$ fixed, that our topological local minimizers $u_a$ fulfill $\lim_{c\rightarrow 0}||u_a||_{\Sigma}=0$. In turn it is sufficient to show, fixing an arbitrary $\delta >0$, that for any sufficiently small $c>0$, $u_a \in A_{2\delta}$ and $aD(u_a) \leq 2\delta$.
To establish this property we fix an arbitrary $u_0 \in S(1)$ with $u_0 \in A_{\delta}$ and consider again the mapping from $S(1)$ to $S(c^2)$ given by $u^c(x) = c^{-1/2}u(\frac{x}{c})$. After direct calculations we have that
$$||u_0^{c}||_2^2= c^2, \, A(u_0^c) = A(u_0) = \delta, \, B(u_0^c) = c B(u_0) \mbox{ and } D(u_0^c) = c^4 D(u_0).$$
This leads to
$$E_a(u_a^c) = \delta + a^2 c^4 D(u_0) + cB(u_0).$$
Thus on one hand, when $c>0$ is sufficiently small, we have that
\begin{equation}\label{e1}
E_a(u_a^c) < \frac{3 \delta}{2}.
\end{equation}
On the other hand, we have that for $A(u) = 2\delta$ and $c>0$ small enough
\begin{equation}\label{e2}
E_a(u) \geq E(u) \geq \frac{3 A(u)}{2} = \frac{3 \delta}{2}.
\end{equation}
In view of \eqref{e1} and \eqref{e2} we deduce that $u_a \in A_{2\delta}$ for any $c >0$ small enough. Clearly also \eqref{e1} implies that $aD(u_0) \leq 2 \delta$.
\end{proof}
\section{Proof of Theorem \ref{stability} }
In this section we prove Theorem \ref{stability} following the ideas of \cite{CL}. First of all we recall
the definition of orbital stability.
We define
$$S_{a}=\{e^{i\theta }
u(x) \in S(c) \ s.t. \ \theta\in [0,2\pi), \|u\|_{2}^2=c, \
E_a(u)=E_a(u_a^1), \ A(u)\leq 2k\}.$$
Notice that here we are considering only the set of topological local minimizers.
We say that $S_{a}$ is {\sl orbitally stable} if
for every $\varepsilon>0$ there exists $\delta>0$ such that for any $\psi_{0}\in \Sigma$ with
$\inf_{v\in S_{a}}\|v-\psi_{0}\|_{\Sigma}<\delta$ we have
$$\forall \, t>0 \ \ \ \inf_{v\in S_{a}
} \|\psi(t,.)-v\|_{\Sigma}<\varepsilon$$
where $\psi(t,.)$ is the solution of \eqref{eq:evolutionbis} with initial datum $\psi_{0}$. In order to prove Theorem \ref{stability} we argue by contradiction, i.e we assume that
there exists a $\varepsilon>0$ a sequence of initial
data $(\psi_{n,0})\subset \Sigma$ and a sequence $(t_{n})\subset{\mathbb R}$ such that the maximal solution $\psi_{n}$ with $\psi_{n}(0,.)=\psi_{n,0}$ satisfies
\begin{equation*}
\lim_{n\rightarrow +\infty}\inf_{v\in S_{a}}\|\psi_{n,0}-v\|_{\Sigma}=0 \ \ \ \text{ and }\ \ \inf_{v\in S_{a}}\|\psi_{n}(t_{n},.)-v\|_{\Sigma}\ge\varepsilon.
\end{equation*}
Without restriction we can assume that $\psi_{n,0}\in S(c)$ such that $(\psi_{n,0})$ is a minimizing sequence for $E_a(u)$ inside $A_{2k}$. Also since $A(\psi_{n,0})\leq 2k$ and
\begin{equation}\label{conservationE}
E_a(\psi_{n}(.,t_{n}))=E_a(\psi_{n,0}),
\end{equation}
also $(\psi_{n}(.,t_{n}))$ is a minimizing sequence for $E_a(u)$ inside $A_{2k}$. Indeed since
$$\inf_{u \in A_{2k}}E_a(u) < \inf_{u \in C_{2k}}E_a(u)$$
by continuity we have that $\psi_{n}(.,t_{n})$ lies inside $A_{2k}$. This proves in particular that $\psi_n$ is global for $n \in {\mathbb N}$ large enough. Now since we have proved, in Step 2 of the proof of Theorem \ref{thm: mainn}, that every minimizing sequence in $A_{2k}$ has a subsequence converging in $\Sigma$ to a topological local minimum on $A_{2k}$ we reach a contradiction.
\section{Proofs of Theorems \ref{thm:signmu} and \ref{asymtotic}}
\begin{proof}[Proof of Theorem \ref{thm:signmu}]
Let $u \in S(c)$ be a topological local minimizer for $E_a(u)$ on $A_{2k}$. In particular it is a solution of
\begin{equation}\label{eq:solu}
- \frac{1}{2}\Delta u + \frac{a^2}{2} |x|^2 u + \lambda_1 |u|^2 u + \lambda_2 (K \star |u|^2) u+ \mu u =0.
\end{equation}
Notice also that, as any critical point of $E_a(u)$ on $S(c)$, it satisfies since $Q_a(u)=0$,
\begin{equation}\label{eq:pohomu}
\mu||u||_2^2=\frac 16 \int_{{\mathbb R}^3} |\nabla u|^2dx -\frac 56 a^2 \int_{{\mathbb R}^3} |x|^2|u|^2dx.
\end{equation}
Finally observe that, thanks to Plancherel identity we can write $B(u)$ as
$$B(u)=(\lambda_1-\frac 43 \pi \lambda_2)||u||_4^4+\lambda_2 \int_{{\mathbb R}^3} (\tilde K\star |u|^2)|u|^2dx$$
where the fourier transform of $\tilde K$ is $\hat{\tilde K}=4 \pi \frac{|\xi_3|^2}{|\xi|^2}.$
From now on we discuss separately the two cases $B(u)\geq0$ and $B(u)<0$.
{\it Case $B(u)\geq 0.$} Since $Q_a(u)=0$ the fact that $B(u)\geq 0$ implies that
$\int_{{\mathbb R}^3} |\nabla u|^2dx \leq a^2\int_{{\mathbb R}^3} |x|^2|u|^2dx$. Thus thanks to \eqref{eq:pohomu} we conclude that $\mu<0$.
Case $B(u)<0.$
Any constrained critical point is a critical point of the free functional
$$J_a(u):=\frac 12 E_a(u) +\frac 12 \mu ||u||_2^2.$$
Let us first compute $ \langle J''_a(u) \varepsilon, \varepsilon \rangle $ where $\varepsilon \in H$ is real valued.
It is easy to show that
\begin{eqnarray}
\frac 12 \langle J''_a(u) \varepsilon, \varepsilon \rangle =\frac 14 \int_{{\mathbb R}^3} |\nabla \varepsilon|^2dx+\frac 14 a^2\int_{{\mathbb R}^3} |x|^2|\varepsilon|^2dx+\frac{3}{2}(\lambda_1-\frac 43 \pi \lambda_2)\int_{{\mathbb R}^3} |u|^2|\varepsilon|^2dx+ \nonumber \\
+\frac{\lambda_2}{2} \int_{{\mathbb R}^3} \left( \tilde K\star |u|^2\right) |\varepsilon|^2dx+\lambda_2 \int_{{\mathbb R}^3} \left( \tilde K\star |u\varepsilon|\right)|u \varepsilon|dx +\frac 12 \mu ||\varepsilon||_2^2 \label{eq:j22}.
\end{eqnarray}
Using the fact that $u$ solves \eqref{eq:solu} we then get
$$\frac 12 \langle J''_a(u) u, u \rangle=(\lambda_1-\frac 43 \pi \lambda_2)\int_{{\mathbb R}^3} |u|^4 dx +\lambda_2 \int_{{\mathbb R}^3} \left( \tilde K\star |u|^2\right) |u|^2dx=B(u).$$
Now we claim that
$$ \langle E''_a(u) \varepsilon, \varepsilon \rangle \geq c\left (\int_{{\mathbb R}^3} |\nabla \varepsilon|^2dx+a^2\int_{{\mathbb R}^3} |x|^2\varepsilon^2dx\right).$$
The claim clearly implies that $\mu<0$.
To prove the claim we shall use the fact, established in Step 5 of the proof of Theorem \ref{thm: mainn}, that $\int_{{\mathbb R}^3} |\nabla u|^2dx\rightarrow 0$ when $a\rightarrow 0$.
For simplicity we consider only the case $\lambda_2>0$ and
$\lambda_1-\frac 43 \pi \lambda_2<0$, the other one is identical. It suffices to look at the functional
$$\tilde E_a(u):=\frac{1}{2}||\nabla u||_2^2 + \frac{a^2}{2} |||x|u||_2^2 + \frac 12 (\lambda_1-\frac 43 \pi \lambda_2)\int_{{\mathbb R}^3} |u|^{4}dx.$$
Now, by H\"{o}lder and Sobolev inequalities
we have
$$\langle \tilde E''_a(u)\varepsilon, \varepsilon \rangle \geq \int_{{\mathbb R}^3} |\nabla \varepsilon|^2dx +a\int_{{\mathbb R}^3} |x|^2|\varepsilon|^2 dx +6(\lambda_1-\frac{4}{3}\pi \lambda_2)S^2 (\int_{{\mathbb R}^3} |\nabla \varepsilon|^2dx)||u||_{3}^2$$
( S is the Sobolev best constant $||\varepsilon||_6\leq S||\nabla \varepsilon||_2$) and this implies that
$$\langle E''_a(u)\varepsilon, \varepsilon \rangle \geq \left(\int_{{\mathbb R}^3} |\nabla \varepsilon_1|^2dx \right)(1+6(\lambda_1-\frac{4}{3}\pi \lambda_2)S^2||u||_3^2)+a\int_{{\mathbb R}^3} |x|^2|\varepsilon_1|^2 dx$$
Thus if
\begin{equation}\label{control}
||u||_3\leq \frac{1}{S\sqrt{6(-\lambda_1+\frac{4}{3}\pi \lambda_2)}}
\end{equation}
we obtain that $\langle \tilde E''_a(u)\varepsilon, \varepsilon \rangle \geq 0$. But \eqref{control} happens when $a\rightarrow 0$ thanks to Gagliardo-Nirenberg inequality
$$||u||_3^2\leq C||u||_2||\nabla u||_2$$
and the fact that $\int_{{\mathbb R}^3} |\nabla u|^2dx\rightarrow 0$ when $a\rightarrow 0$.
\end{proof}
\begin{lem}[Asymptotics for $\mu$]
Let $a>0$ be sufficiently small and $u$ be a topological local minimizer for the constrained energy, then the following holds
\begin{enumerate}
\item $\lim_{a \rightarrow 0} \mu=0$;
\item $\mu<-\frac 32 a$ \text{ if } $B(u)\geq 0$;
\item $\mu<-3 a \sqrt{\frac 14 +\frac 32 S^2(\lambda_1-\frac{4}{3}\pi \lambda_2)||u||_3^2}$ \text{ if } $B(u)>0$ and $\lambda_2>0$;
\item $\mu<-3 a \sqrt{\frac 14 +\frac 32 S^2(\lambda_1+\frac{8}{3}\pi \lambda_2)||u||_3^2}$ \text{ if } $B(u)>0$ and $\lambda_2<0.$
\end{enumerate}
\end{lem}
\begin{proof}
The fact that $\lim_{a \rightarrow 0} \mu=0$ follows easily from the relations
$$E_a(u) - \frac{1}{3}Q_a(u) = \frac{1}{6}\int_{{\mathbb R}^3} |\nabla u|^2dx + \frac{5}{6}a^2 \int_{{\mathbb R}^3} |x|^2 |u|^2 dx$$
and
$$E_a(u)=\mu||u||_2^2+ \frac{5}{3}a^2 \int_{{\mathbb R}^3} |x|^2 |u|^2 dx.$$
The proof of the last three points follows from Heisenberg uncertainty principle written in the following form
\begin{equation}\label{eq:hp}
||\nabla u||_2^2 + \omega^2 |||x|u||_2^2-3\omega ||u||_2^2\geq 0 \ \ \ \ \forall u \in \Sigma, \omega>0
\end{equation}
\emph{ Case $B(u)\geq 0$:}\\
The fact that $\mu<-\frac 32 a$ follows from \eqref{eq:hp} since $u$ solves \eqref{eq:solu}.\\
\emph{ Case $B(u)< 0$ and $\lambda_2>0$:}\\
Here we use \eqref{eq:j22}. Using the fact that $\lambda_2>0$ we get
\begin{eqnarray}
0>\frac 12 \langle J''_a(u) u, u \rangle >\frac 14 \int_{{\mathbb R}^3} |\nabla u|^2dx+\frac 14 a^2\int_{{\mathbb R}^3} |x|^2|u|^2dx+\\ \nonumber
+\frac{3}{2}(\lambda_1-\frac 43 \pi \lambda_2)\int_{{\mathbb R}^3} |u|^4dx+\frac 12 \mu ||u||_2^2. \nonumber
\end{eqnarray}
Now arguing as in the proof of Theorem \ref{thm:signmu} we obtain
$$0>(\frac 14 +\frac{3}{2}S^2(\lambda_1-\frac{4}{3}\pi \lambda_2)||u||_3^2)\int_{{\mathbb R}^3} |\nabla u|^2dx+\frac 14 a^2\int_{{\mathbb R}^3} |x|^2|u|^2dx+\frac 12 \mu ||u||_2^2.$$
Calling $\beta=(\frac 14 +\frac{3}{2}S^2(\lambda_1-\frac{4}{3}\pi \lambda_2)||u||_3^2)$ we have from Heisenberg uncertainty principle \eqref{eq:hp}
$$\beta \left(\int_{{\mathbb R}^3} |\nabla u|^2dx+\frac{1}{4\beta} a^2\int_{{\mathbb R}^3} |x|^2|u|^2dx+\frac{1}{2\beta} \mu ||u||_2^2\right)\geq \beta\left( \frac{3a}{2\sqrt{\beta}}+\frac{\mu}{2\beta}\right)||u||_2^2.$$
Remembering that $\beta>0$ for $a>0$ small we obtain the required estimate.\\
\emph{ Case $B(u)< 0$ and $\lambda_2<0$:}\\
This case is identical to the previous one just observing that we can write
$$B(u)=(\lambda_1+\frac{8}{3}\pi \lambda_2)||u||_4^4+\lambda_2\int_{{\mathbb R}^3} \left( K_1\star |u|^2\right) |u|^2dx$$
where $\hat{K_1}=-4\pi \frac{(\xi_1^2+\xi_2^2)}{|\xi|^2}.$
\end{proof}
\begin{proof}[Proof of Theorem \ref{asymtotic}]
We consider just the case $\lambda_2>0$. Since $\hat K(\xi)=\frac{4}{3}\pi (\frac{2\xi_3^2-\xi_1^2-\xi_2^2}{|\xi|^2})\in [-\frac{4}{3}\pi, \frac{8}{3}\pi]$ writing $\lambda_1 = \lambda_1' + \frac{4}{3}\pi \lambda_2$ we have, when $\lambda_1' \leq 0$,
\begin{equation}\label{bounds}
\lambda_1' \, \leq \, \lambda_1 + \lambda_2 \hat K(\xi) \, \leq \, 4 \pi \lambda_2.
\end{equation}
Thus
\begin{eqnarray*}
Q(u) &\geq & A(u) +\frac{3}{2}\frac{1}{(2\pi)^{3}}\int_{{\mathbb R}^3} (\lambda_1-\frac 43 \pi \lambda_2) |\hat {u^2}|^2d\xi
\geq A(u) + C \lambda_1' ||u||_4^4 \\
&\geq & A(u) + \lambda_1' C A(u)^{3/2}||u||_2.
\end{eqnarray*}
In particular, for any $k>0$, taking $\lambda_1' <0$ sufficiently close to $0$ it follows that $Q(u) >0$ on $A_k$. Recording that $Q(u)=0$ for any critical point this proves Point (1). To prove Point (2) first observe that,
\begin{equation}
E(u) \leq \frac{A(u)}{2} + 4 \pi \lambda_2 ||u||^4
\leq \frac{A(u)}{2} + \lambda_2 C A(u)^{3/2}||u||_2
\end{equation}
and thus for any $k>0$,
\begin{equation}\label{fix}
\sup_{u \in A_k} E(u) \quad \mbox{ does not depend on } \lambda_1'.
\end{equation}
Also from \eqref{bounds} we have that
\begin{equation}
E(u) \geq \frac{A(u)}{2} + \frac{\lambda_1'}{2} ||u||^4 \geq \frac{A(u)}{2} + \lambda_1' C A(u)^{3/2}||u||_2
\end{equation}
and then a direct calculation shows that
\begin{equation}\label{limit}
\sup_{k>0}\inf_{u \in C_k}E(u) \to + \infty
\end{equation}
as $\lambda_1' \to 0^-$. Now fix a $v \in S(c)$ with $A(v)=1$. We have
$$ E_a(v) \leq \sup_{u \in A_1}E(u) + \frac{a^2}{2}D(v).$$
From \eqref{fix} and \eqref{limit} we deduce that, for any $a>0$,
$$E_a(v) < \sup_{k>0}\inf_{u \in C_k}E(u)$$
if $|\lambda_1'|$ is sufficiently small. Arguing as in Step 1 of the Proof of Theorem \ref{thm: mainn} this proves Point (2).
\end{proof}
\section{Appendix}
In Step 3 of the proof of Theorem \ref{thm: mainn} we have assumed that $E_a(u)$ constrained to $S(c)$ possesses a bounded Palais-Smale sequence at the level $\gamma_a(c)$. We now prove that it is indeed the case and we also derive additional properties of this sequence.
\begin{lem}\label{lm2}
For any fixed $c> 0$ and any $a \in (0,a_0]$ there exists a sequence $(u_n)\subset S(c)$ and a sequence $(v_n)\subset \Sigma$ of real, non negative functions such that
\begin{equation}\label{SPSC}
\left\{
\begin{array}{l}
E_a(u_n)\to \gamma_c(c)>0,\\
\|E'_a|_{S(c)}(u_n)\|_{\Sigma^{-1}}\to 0,\\
Q_a(u_n)\to 0,\\
\|u_n-v_n\|_{\Sigma}\to 0,\\
\end{array}
\right.
\end{equation}
as $n\to \infty$. Here $\Sigma^{-1}$ denotes the dual space of $\Sigma$.
\end{lem}
From the definition of $E_a(u)$ and $Q_a(u)$ we get that
\begin{equation*}
E_a(u) - \frac{1}{3}Q_a(u) = \frac{1}{6}A(u) + \frac{5}{6}a^2 D(u)
\end{equation*}
and thus we immediately deduce that the Palais-Smale sequence given by Lemma \ref{lm2} is bounded. Note also that since $||u_n - v_n||_{\Sigma} \to 0$, if we manage to show that $u_n \to u$ strongly in $\Sigma$ then the limit $u \in S(c)$ will be a real, non negative, function.
Roughly speaking what Lemma \ref{lm2} says is that it is possible to incorporate into the variational problem the information that any critical point must satisfy the constraint $Q_a(u)=0$. For previous works in that direction we refer to \cite{Je,JeLuWa}. Clearly it is possible to prove Step 3 of Theorem \ref{thm:standing} by this approach, but here we have choosen to use the approach of \cite{Gh} for its simplicity. The proof of the lemma is inspired from \cite[Lemma 3.5]{JeLuWa}. Before proving
Lemma \ref{lm2} we need to introduce some notations and to prove some preliminary results. For any fixed $\mu>0$, we introduce the auxiliary functional
$$\widetilde{E}_{a}: S(c) \times {\mathbb R} \to {\mathbb R},\qquad (u, s)\mapsto E_a(H(u,s)),$$
where $H(u,s)(x):= e^{\frac{N}{2}s}u(e^s x)$, and the set of paths
\begin{align*}
\widetilde{\Gamma}_a(c):= \Big\{\widetilde{g} \in C([0,1], S(c) \times {\mathbb R}) : \ \widetilde{g}(0)= (v^{t_1}, 0),\ \widetilde{g}(1)= (v^{t_2}, 0) \Big\},
\end{align*}
where $v^{t_1}, v^{t_2} \in S(c)$ are defined in the proof of Theorem \ref{thm: mainn} (remember that they are real non negative).
Observe that setting
$$\widetilde{\gamma}_{a}(c):= \inf_{\widetilde{g}\in \widetilde{\Gamma}_a(c)}\max\limits_{t\in [0,1]}\widetilde{E}_a(\widetilde{g}(t)),$$
we have that
\begin{eqnarray}\label{ggamma(c)}
\widetilde{\gamma}_{a}(c) = \gamma_a(c).
\end{eqnarray}
Indeed, by the definitions of $\widetilde{\gamma}_{a}(c)$ and $\gamma_a(c)$, \eqref{ggamma(c)} follows immediately from the observation that the maps
$$\varphi: \Gamma_a(c) \longrightarrow \widetilde{\Gamma}_a(c),\ g \longmapsto \varphi(g):=(g,0),$$
and
$$\psi: \widetilde{\Gamma}_a(c) \longrightarrow \Gamma_a(c),\ \widetilde{g} \longmapsto \psi(\widetilde{g}):=H \circ \widetilde{g},$$
satisfy
$$\widetilde{E}_a(\varphi(g)) = E_a(g)\ \mbox{ and }\ E_a(\psi(\widetilde{g})) = \widetilde{E}_a(\widetilde{g}).$$
In the proof of Lemma \ref{lm2}, the lemma below which has been established by the Ekeland variational principle in \cite[Lemma 2.3]{Je} is used. Hereinafter we denote by $X$ the set $ \Sigma \times {\mathbb R}$ equipped with the norm $\|\cdot\|_X^2 = \|\cdot\|_{\Sigma}^2 + |\cdot|_{{\mathbb R}}^2$ and denote by $X^{-1}$ its dual space.
\begin{lem}\label{lm-ekeland}
Let $\varepsilon>0$. Suppose that $\widetilde{g}_0 \in \widetilde{\Gamma}_a(c)$ satisfies
$$\max\limits_{t\in [0,1]}\widetilde{E}_a(\widetilde{g}_0(t))\leq \widetilde{\gamma }_{a}(c)+ \varepsilon.$$
Then there exists a pair of $(u_0, s_0)\in S(c) \times {\mathbb R}$ such that:
\begin{itemize}
\item [(1)] $\widetilde{E}_a(u_0,s_0) \in [\widetilde{\gamma}_{a}(c)- \varepsilon, \widetilde{\gamma}_{a}(c) + \varepsilon]$;
\item [(2)] $\min\limits_{t\in [0,1]} \| (u_0,s_0)-\widetilde{g}_0(t) \|_X \leq \sqrt{\varepsilon}$;
\item [(3)] $\| \widetilde{E}_a'|_{S(c) \times {\mathbb R}}(u_0,s_0) \|_{X^{-1}}\leq 2\sqrt{\varepsilon}$,\ i.e.
$$|\langle \widetilde{E}_a'(u_0, s_0), z \rangle_{X^{-1}\times X} |\leq 2\sqrt{\varepsilon}\left \| z \right \|_X,$$ holds for all $z \in \widetilde{T}_{(u_0,s_0)}:= \{(z_1,z_2) \in X, \langle u_0, z_1\rangle_2=0\}$.
\end{itemize}
\end{lem}
\begin{proof}[Proof of Lemma \ref{lm2}] For each $n\in {\mathbb N}$, by the definition of $\gamma_a(c)$, there exists a $g_n\in \Gamma_a(c)$ such that
$$\max\limits_{t\in [0,1]}E_a(g_n(t))\leq \gamma_a(c) + \frac{1}{n}.$$
Observe that $|g_n| \in \Gamma_a(c)$ and because $E_a(|u|) \leq E_a(u)$ for all $ \in \Sigma$ we have
$$\max_{t\in [0,1]}E_a(|g_n(t)|)\leq \max_{t\in [0,1]}E_a(g_n(t)).$$
Since $\widetilde{\gamma}_{a}(c)=\gamma_a(c)$, then for each $n\in {\mathbb N}$, $\widetilde{g}_n:=(|g_n|, 0)\in \widetilde{\Gamma}_a(c)$ satisfies
$$\max\limits_{t\in [0,1]}\widetilde{E}_a(\widetilde{g}_n(t))\leq \widetilde{\gamma}_{a}(c) + \frac{1}{n}.$$
Thus applying Lemma \ref{lm-ekeland}, we obtain a sequence $\{(w_n,s_n)\}\subset S(c) \times {\mathbb R}$ such that:
\begin{itemize}
\item [(i)] $\widetilde{E}_a(w_n, s_n) \in [\gamma_a(c)-\frac{1}{n}, \gamma_a(c)+\frac{1}{n}]$;
\item [(ii)] $\min\limits_{t\in [0,1]} \| (w_n, s_n)-(|g_n(t)|, 0) \|_X \leq\frac{1}{\sqrt{n}}$;
\item [(iii)] $\| \widetilde{E}_a'|_{S_(c) \times {\mathbb R}}(w_n, s_n) \|_{X^{-1}}\leq \frac{2}{\sqrt{n}}$.
\end{itemize}
For each $n\in {\mathbb N}$, let $t_n\in [0,1]$ be such that the minimum in (ii) is reached. We claim that setting $u_n:=H(w_n, s_n)$ and $v_n:= |g_n(t_n)|$ the corresponding sequences satisfy \eqref{SPSC}. Indeed, first, from (i) we have that $E_a(u_n) \to \gamma_a(c)$, since $E_a(u_n)=E_a(H(w_n, s_n))=\widetilde{E}_a(w_n, s_n).$
Secondly, by simple calculations, we have that
$$Q_a(u_n) = \langle \widetilde{E}_a'(w_n, s_n), (0,1) \rangle_{X^{-1}\times X},$$
and $(0,1)\in \widetilde{T}_{(w_n, s_n)}$. Thus (iii) yields that $Q_a(u_n) \to 0$. To verify that $\|E'_{a}|_{S(c)}(u_n)\|_{\Sigma^{-1}}\to 0$, it suffices to prove, for $n\in {\mathbb N}$ sufficiently large, that
\begin{eqnarray}\label{F'u}
|\langle E_a'(u_n), \phi\rangle_{\Sigma^{-1}\times \Sigma} |\leq \frac{4}{\sqrt{n}}\left \| \phi \right \|_\Sigma,\ \mbox{ for all }\ \phi \in T_{u_n},
\end{eqnarray}
where $T_{u_n}:=\{\phi \in \Sigma,\ \langle u_n,\phi\rangle_{2}=0\}$. To this end, we note that, for each $\phi\in T_{u_n}$, setting $\widetilde{\phi}=H(\phi, -s_n)$, one has by direct calculations that
$$
\langle E'_{a}(u_n), \phi \rangle_{\Sigma^{\ast}\times \Sigma} =\ \langle \widetilde{E}_a'(w_n, s_n), (\widetilde{\phi}, 0)\rangle_{X^{-1}\times X}.
$$
If $(\widetilde{\phi},0)\in \widetilde{T}_{(w_n,s_n)}$ and $\|(\widetilde{\phi},0)\|_X^2\leq 4\|\phi\|_\Sigma^2$ for $n\in {\mathbb N}$ sufficiently large, then from (iii) we conclude that \eqref{F'u} holds. To check this claim one may observe that
$(\widetilde{\phi},0)\in \widetilde{T}_{(w_n,s_n)} \Leftrightarrow \phi \in T_{u_n}$, and that from (ii) we have
\begin{eqnarray}\label{value-s_n}
|s_n|=|s_n-0|\leq \min\limits_{t\in [0,1]}\|(w_n, s_n)-(|g_n(t)|, 0)\|_X\leq \frac{1}{\sqrt{n}},
\end{eqnarray}
by which we deduce that
\begin{eqnarray*}
\|(\widetilde{\phi},0)\|_X^2 &=& \|\widetilde{\phi}\|_\Sigma^2 = ||\phi||_2^2 + e^{-2s_n} ||\nabla \phi||_2^2 + e^{2s_n} |||x| \phi||_2^2\\
& \leq & 2 \ \|\phi\|_{\Sigma}^2,
\end{eqnarray*}
holds for $n\in {\mathbb N}$ large enough. Thus \eqref{F'u} has been proved. Finally, since $\| (w_n, s_n)-(v_n, 0) \|_X \to 0$ we have in particular that
$\| w_n - v_n \|_\Sigma \to 0.$
Thus from \eqref{value-s_n} and since
$$\| u_n - v_n \|_\Sigma = \| H(w_n, s_n) - v_n \|_\Sigma \leq \| H(w_n, s_n) - w_n\|_\Sigma + \|w_n- v_n \|_\Sigma,$$
we conclude that $\| u_n - v_n \|_\Sigma \to 0$ as $n\to \infty$. At this point, the proof of the lemma is complete.
\end{proof}
\end{document}
|
\begin{document}
\title [Asymptotic behavior of LCM of consecutive reducible quadratic progression terms]
{Asymptotic behavior of the least common multiple of consecutive reducible quadratic
progression terms}
\author{Guoyou Qian}
\address{Mathematical College, Sichuan University, Chengdu 610064, P.R. China}
\email{[email protected], [email protected]}
\author{Shaofang Hong$^*$}
\address{Mathematical College, Sichuan University, Chengdu 610064, P.R. China}
\email{[email protected], [email protected], [email protected] }
\thanks{$^*$Hong is the corresponding author and was supported partially
by National Science Foundation of China Grant \#11371260. Qian was supported
partially by Postdoctoral Science Foundation of China Grant \#2013M530109}
\keywords{Least common multiple; Arithmetic progression; Prime number theorem
for arithmetic progressions; $p$-Adic valuation}
\subjclass[2000]{Primary 11B25, 11N37, 11A05}
\date{\today}
\begin{abstract} Let $l$ and $m$ be two integers with $l>m\ge 0$, and let
$f(x)$ be the product of two linear polynomials with integer coefficients.
In this paper, we show that $\log {\rm lcm}_{mn<i\le ln}\{f(i)\}=An+o(n)$,
where $A$ is a constant depending only on $l$, $m$ and $f$.
\end{abstract}
\maketitle
\section{\bf Introduction}
The study of the least common multiple of consecutive positive
integers was first initiated by Chebyshev for a significant attempt
to prove prime number theorem. From Chebyshev's well-known work
\cite{[Ch]}, one can easily get an equivalent of prime number
theorem which states that $\log {\rm lcm}(1, ..., n)\sim n$ as $n$
tends to infinity. Since then, this topic received attentions of
many authors. Hanson \cite{[Ha]} and Nair \cite{[N]} got the upper
and lower bound of ${\rm lcm}_{1\le i\le n}\{i\}$, respectively.
Bateman, Kalb and Stenger \cite{[BKS]} gave an asymptotic formula of
$\log{\rm lcm}_{1\le i\le n}\{b+ai\}$ as $n$ tends to infinity, where
$a$ and $b$ are coprime integers. Farhi \cite{[F]}, Hong and Feng
\cite{[HF]}, Hong and Yang \cite{[HY]}, Hong and Kominers \cite{[HK]},
Wu, Tan and Hong \cite{[WTH]} and Kane and Kominers \cite{[KK]}
obtained lower bounds of the least common multiple of the first $n$
arithmetic progression terms. Farhi and Kane \cite{[FK]} studied the
least common multiple of consecutive integers. Hong and Qian \cite{[HQ]}
obtained some results on the least common multiple of consecutive arithmetic
progression terms which was consequently extended in one direction by Qian, Tan
and Hong \cite{[QTH2]}. Hong, Qian and Tan \cite{[HQT]} got an asymptotic
formula of the least common multiple of a sequence of products of linear
polynomials. On the other hand, Farhi \cite{[F]} obtained a nontrivial
lower bound for the least common multiple of the quadratic sequence $
\{i^2+1\}_{i=1}^\infty $. Oon \cite{[O]} improved some of the Hong-Kominers
result and Farhi's lower bound. Hong, Luo, Qian and Wang \cite{[HLQW]}
extended Nair's and Oon's lower bound by giving a uniform lower bound.
Qian, Tan and Hong \cite{[QTH]} showed that for any given positive
integer $k$, we have $\log{\rm lcm}_{0\le i\le k}\{(n+i)^2+1\}\sim 2(k+1)
\log n$ as $n\rightarrow \infty $. Recently, Hong and Qian \cite{[HQ2]}
got some interesting results on the least common multiple of consecutive
quadratic progression terms.
Qian and Hong \cite{[QH]} investigated the asymptotic
behavior of the least common multiple of any consecutive arithmetic
progression terms. Let $l$ and $m$ be integers
with $l>m\ge 0$ and let $a\ge 1$ and $b$ be integers such that
$a+b\ge 1$ and $\gcd(a, b)=1$. It is proved in \cite{[QH]} that
$$\log {\rm lcm}_{mn<i\le ln}\{ ai+b\}
\sim \frac{an}{\varphi(a)}\sum_{r=1\atop \gcd(r,a)=1}^{a}B_r
$$
as $n\rightarrow \infty$, where
\begin{align}\label{eq: 1.1}
B_r:={\left\{
\begin{array}{rl}
\frac{l}{r}, &\text{if} \ l\ge \frac{(a+r)m}{r},\\
\sum_{i=0}^{\mathcal{K}-1}\frac{l-m}{r+ai}+\frac{l}{r+a\mathcal{K}},
&\text{if} \ l<\frac{(a+r)m}{r}
\end{array}
\right.}
\end{align}
with
$\mathcal{K}:=\big\lfloor\frac{al-(l-m)r}{a(l-m)}\big\rfloor$
and $\lfloor x\rfloor$ being the largest integer no more than $x$.
In this paper, we mainly concentrate on the asymptotic behavior of the least
common multiple of consecutive reducible quadratic progression terms. There are
two cases about the reducible quadratic progressions. The first case
is $f(x)=(ax+b)^2$ with $a\ge 1$ and $b$ being integers such that $a+b\ge 1$
and $\gcd(a, b)=1$. This case is easy to answer. Actually, by the main result
of \cite{[QH]}, we can derive immediately that
$$\log {\rm lcm}_{mn<i\le ln}\{ (ai+b)^2\}
\sim \frac{2an}{\varphi(a)}\sum_{r=1\atop \gcd(r,a)=1}^{a}B_r
$$
as $n\rightarrow \infty$, where $B_r$ is defined as in (1.1).
Our main goal in the present paper is to treat with the second case
that $f(x)=(a_1x+b_1)(a_2x+b_2)$ with $a_i, b_i\in \mathbb{N}^*$
and $\gcd(a_i, b_i)=1$ for $1\le i\le 2$ and $a_1b_2\ne a_2b_1$.
Let $\mathbb{N}$ be the set of nonnegative integers and
$\mathbb{N}^*:=\mathbb{N}\setminus \{0\}$. For any two positive integers $a$ and $b$,
let $\langle b\rangle_{a}$ denote the smallest positive integer congruent to $b$
modulo $a$ between $1$ and $a$. For any integer $t$, we define $S_t$ by
$S_t:=\{i\in \mathbb{N} : 0\le i\le t\}$. Clearly, $S_t$ is empty if $t$
is negative and so we can define $\sum_{i\in S_t}g(i):=0$ for any arithmetic
function $g$ if $t<0$. We define the following three 4-variable arithmetic functions:
\begin{align}\label{eq: 1.2}
g_r(x,y,z,w):=\Big\lfloor \frac{xyl+ym\langle zr\rangle_{x}-xl\langle wr\rangle_{y}}
{xy(l-m)}\Big\rfloor,
\end{align}
\begin{align}\label{eq: 1.3}
h_r(x,y,z,w):=\Big\lfloor \frac{xm\langle wr\rangle_{y}-yl
\langle zr\rangle_{x}}{xy(l-m)}\Big\rfloor
\end{align}
and
\begin{align}\label{eq: 1.4}
\lambda_r(x,y,z,w):=&\sum_{i\in S_{g_r(x,y,z,w)}}\frac{xl}{\langle zr\rangle_{x}+xi}-
\sum_{i\in S_{g_r(x,y,z,w)-1}}\frac{ym}{\langle wr\rangle_{y}+yi}\\
\nonumber&+\sum_{i\in S_{h_r(x,y,z,w)}}
\Big(\frac{yl}{\langle wr\rangle_{y}+yi}- \frac{xm}{\langle zr\rangle_{x}+xi}
\Big).
\end{align}
We can now state the main result of this paper.
\noindent{\bf Theorem 1.1.} {\it Let $l$ and $m$ be fixed integers
with $l>m\ge 0$. Let $f(x)=(a_1x+b_1)(a_2x+b_2)$,
where $a_i, b_i\in \mathbb{N}^*$
and $\gcd(a_i, b_i)=1$ for $1\le i\le 2$ and $a_1b_2\ne a_2b_1$.
Then
$$\log {\rm lcm}_{mn<i\le ln}\{ f(i)\}
=\frac{n}{\varphi(q)}\sum_{r=1\atop \gcd(r,q)=1}^{q}A_r+o(n),
$$
where $q={\rm lcm}(a_1, a_2)$ and}
\begin{align}\label{eq: 1.5}
A_r:=
{\left\{
\begin{array}{rl}
\lambda_r(a_1, a_2, b_1, b_2) \ {\it if} \ a_1\langle b_2r\rangle_{a_2}\ge
a_2\langle b_1r\rangle_{a_1};\\
\lambda_r(a_2, a_1, b_2, b_1) \ {\it if} \ a_1
\langle b_2r\rangle_{a_2}<a_2\langle b_1r\rangle_{a_1}.
\end{array}
\right.}
\end{align}
Note that Theorem 1.1 is still true if at least one of $b_1$ and $b_2$
is a negative integer.
The paper is organized as follows. In Section 2, we prove two lemmas which
are needed for the proof of Theorem 1.1. The final section will devote to
the proof of Theorem 1.1.
\section{\bf Two lemmas}
In this section, we show two lemmas which are needed in the proof of
Theorem 1.1. Throughout, we let
\begin{align}\label{eq: 4.1}
H_1:=\Big\lfloor\frac{a_1l-(l-m)\langle b_1r\rangle_{a_1}}{a_1(l-m)}\Big\rfloor
\end{align}
and
\begin{align}\label{eq: 4.2}
H_2:=\Big\lfloor\frac{a_2l-(l-m)\langle b_2r\rangle_{a_2}}{a_2(l-m)}\Big\rfloor.
\end{align}
As usual, for any prime number $p$, we let $v_{p}$ be the normalized $p$-adic
valuation on the set of positive integers. Namely, one has $v_p(a)=s$ if
$p^{s}\parallel a$. We begin with the following result.
\noindent{\bf Lemma 2.1.} {\it Let $l, m, q$ and $f(x)$ be defined
as in Theorem 1.1. Then
$$\log {\rm lcm}_{mn<i\le ln}\{ f(i)\}
=\sum_{r'=1\atop \gcd(r',q)=1}^{q}\sum_{p\in \mathcal{P}_{r'}}\log p+O\big(\sqrt{n}\big),
$$
where
\begin{align}\label{eq: 4.3}
\mathcal {P}_{r'}:=\Big\{\text{prime}\ p: \ &p\equiv r'\pmod q \
\text{and} \ p\in \Big(0, (l-m)n\Big]\bigcup\\
\nonumber & \bigg(\bigcup_{j=1}^2\bigcup_{i=0}^{H_j}\Big(\frac{a_jmn}{\langle
b_jr\rangle_{a_j}+a_ji}, \frac{a_jln}
{\langle b_jr\rangle_{a_j}+a_ji}\Big]\bigg)\Big\}
\end{align}
with $r$ being the unique integer satisfying $rr'\equiv1\pmod q$ and $1\le r\le q$.}
\begin{proof}
For simplicity, we define $L_{m, l}^{(f)}(n):={\rm lcm}_{mn<i\le ln}\{f(i)\},$
and let $P_{m, l}^{(f)}(n)$ be the set of all the prime factors of
$L_{m,l}^{(f)}(n)$ not dividing ${\rm lcm}(a_1b_2-a_2b_1, q)$.
We claim that if $p\in P_{m, l}^{(f)}(n)$
and $p|f(i)$ for some integer $mn<i\le ln$, then $p$ divides exactly
one of $a_1i+b_1$ and $a_2i+b_2$. Otherwise, we have $p|(a_1i+b_1)$ and $p|(a_2i+b_2)$.
It implies that $p\mid \big(a_1(a_2i+b_2)-a_2(a_1i+b_1)\big)=a_1b_2-a_2b_1$,
which is impossible since $p\nmid {\rm lcm}(a_1b_2-a_2b_1, q)$.
The claim is proved. But the number of prime factors of
${\rm lcm}(a_1b_2-a_2b_1, q)$ is finite. So we have
\begin{align}\label{eq: 4.4}
&\log L_{m, l}^{(f)}(n)=\log \Big(\prod_{p\in P_{m, l}^{(f)}(n)}p^{v_p(L_{m, l}^{(f)}(n))}
\prod_{p\not\in P_{m, l}^{(f)}(n)}p^{v_p(L_{m, l}^{(f)}(n))}\Big)\\
\nonumber&=\sum_{p\in P_{m, l}^{(f)}(n)}v_p(L_{m, l}^{(f)}(n))\log p+O\big(\log\big(f(ln)\big)\big)\\
\nonumber &=\sum_{p\in P_{m, l}^{(f)}(n)}\log p+
\sum_{p\in P_{m, l}^{(f)}(n)\atop v_p(L_{m, l}^{(f)}(n))\ge 2
}\big(v_p\big(L_{m, l}^{(f)}(n)\big)-1\big)\log p+O\big(\log n\big).\quad\quad\
\end{align}
If $p\in P_{m, l}^{(f)}(n)$ and $v_p(L_{m, l}^{(f)}(n))\ge 2$, then $p^2| f(i)$ for
some integer $i$ with $mn<i\le ln$.
Hence by the claim we obtain that $p^2|(a_1i+b_1)$ or $p^2|(a_2i+b_2)$, which implies that
$$p\le M_n:=\max\{\sqrt{a_1ln+b_1)}, \sqrt{a_2ln+b_2}\}\ll \sqrt{n}.$$
On the other hand, since $p^{v_p(L_{m, l}^{(f)}(n))}\le f(ln)$, it follows that
$$v_p(L_{m, l}^{(f)}(n))\le \frac{\log f(ln)}{\log p}\ll \frac{\log n}{\log p}.$$
Hence we get by the prime number theorem that
$$\sum_{p\in P_{m, l}^{(f)}(n)\atop v_p(L_{m, l}^{(f)}(n))\ge 2
}\big(v_p(L_{m, l}^{(f)}(n))-1\big)\log p\ll \sum_{p\le M_n}\frac{\log n}
{\log p}\log p\ll \sum_{p\le M_n}\log n\ll \frac{\sqrt{n}}{\log \sqrt{n}}\log n\ll \sqrt{n}.$$
It then follows from (\ref{eq: 4.4}) that
\begin{align}\label{eq: 4.5}
\log L_{m, l}^{(f)}(n)=\sum_{p\in P_{m, l}^{(f)}(n)}\log p+O(\sqrt{n})+O(\log n)
=\sum_{p\in P_{m, l}^{(f)}(n)}\log p+O(\sqrt{n}).
\end{align}
First, we give a characterization on the primes in the set
$P_{m, l}^{(f)}(n)$. By $T(q)$ we denote the set of all
positive integers no more than $q$ that are relatively prime to
$q$. Then by the definition of $P_{m, l}^{(f)}(n)$, we
know that each prime in $P_{m, l}^{(f)}(n)$ is relatively prime to
$q$. So each prime $p\in P_{m, l}^{(f)}(n)$ is congruent to $r'$
modulo $q$ for some $r'\in T(q)$. For convenience, we let
\begin{align}\label{eq: 4.6}
{\mathcal Q}_{r'}:=\{p\in P_{m, l}^{(f)}(n): p\equiv r'\pmod q\}.
\end{align}
Thus we derive from (\ref{eq: 4.5}) that
\begin{align}\label{eq: 4.7}
\log L_{m, l}^{(f)}(n)=\sum_{r'\in T(q)}\sum_{p\in P_{m,
l}^{(f)}(n)\atop p\equiv r'\pmod q}\log p+O(\sqrt{n})=\sum_{r'\in
T(q)}\sum_{p\in {\mathcal Q}_{r'}}\log p+O(\sqrt{n}).
\end{align}
For any given $r'\in T(q)$, there is exactly one $r\in T(q)$ such
that $rr'\equiv 1\pmod q$. Thus for any given prime $p\equiv r'\pmod q$,
we have $\langle b_jr\rangle_{a_j}p\equiv \langle
b_jr\rangle_{a_j}r'\equiv b_jrr'\equiv b_j\pmod{a_j}$ for each $1\le
j\le 2$. Since $\gcd(p, a_j)=1$ for $j=1, 2$, we can deduce
that all the terms divisible by $p$ in
the arithmetic progression $\{a_ji+b_j\}_{i=1}^{\infty}$ must be of
the form $(a_jk+\langle b_jr\rangle_{a_j})p$,
where $k\in \mathbb{N}$. It follows that for each $1\le j\le 2$ and
any prime $p\in {\mathcal Q}_{r'}$, we have that $p|(a_ji+b_j)$ for some $mn< i\le
ln$ if and only if there is an integer $i_j\ge 0$ so that
$a_jmn+b_j<(a_ji_j+\langle b_jr\rangle_{a_j})p\le a_jln+b_j$.
Therefore, a prime $p$ congruent to $r'$ modulo $q$ is in $P_{m,
l}^{(f)}(n)$ if and only if $ p\nmid (a_1b_2-a_2b_1)$ and either
$$\frac{a_1mn+b_1}{\langle b_1r\rangle_{a_1}+a_1i_1}<p\le \frac{a_1ln+b_1}{\langle b_1r\rangle_{a_1}+a_1i_1}
$$
for some $i_1\in \mathbb{N}$, or
$$\frac{a_2mn+b_2}{\langle b_2r\rangle_{a_2}+a_2i_2}<p\le \frac{a_2ln+b_2}
{\langle b_2r\rangle_{a_2}+a_2i_2}$$
for some $i_2\in \mathbb{N}$.
Thus we have by (\ref{eq: 4.6}) that
\begin{align}\label{eq: 4.8}
{\mathcal Q}_{r'}=\bigcup_{j=1}^2\bigcup_{i=0}^{\infty}\Big\{ \text{prime}\ p\equiv r'\pmod q:
\frac{a_jmn+b_j}{\langle b_jr\rangle_{a_j}+a_ji}<p&\le
\frac{a_jln+b_j} {\langle b_jr\rangle_{a_j}+a_ji}\\
\nonumber&\ \mbox{and}\ p\nmid (a_1b_2-a_2b_1)\Big\}.
\end{align}
To prove Lemma 2.1, we have to treat with the union on the right-hand side of (\ref{eq: 4.8}).
Since $\gcd(p, a_j)=1$ for any prime $p\equiv r'\pmod q$, then by Lemma 3.6 of \cite{[HQ]},
there is exactly one term divisible by $p$ in any $p$ consecutive terms of
the arithmetic progression $\{ a_ji+b_j\}_{i=1}^{\infty}$ for each $1\le j\le 2$.
Therefore, for any prime $p$ with $p\le (l-m)n$ and $p\equiv r'\pmod q$, there is at
least one term divisible by $p$ in the set $\{(a_1i+b_1)(a_2i+b_2)\}_{i=mn+1}^{ln}$.
Hence we have
\begin{align}\label{eq: Q_3}
\big\{\text{prime}\ p\equiv r'\pmod q:\ p\le (l-m)n\ \mbox{and}\ p\nmid (a_1b_2-a_2b_1)\big\}
\subseteq {\mathcal Q}_{r'}.
\end{align}
By (\ref{eq: 4.1}) and (\ref{eq: 4.2}), for $j=1, 2$, we have that
$$\frac{a_jln+b_j}{\langle b_jr\rangle_{a_j}+a_j(H_j+1)}
<(l-m)n < \frac{a_jln+b_j}{\langle b_jr\rangle_{a_j}+a_jH_j}
$$
for any positive integer $n$ with
$$n>n_0:=\Big\lfloor\frac{b_j}{(a_j(H_j+1)
+\langle b_jr\rangle_{a_j})(l-m)-a_jl}\Big\rfloor.$$
It then follows that for $j=1, 2$ and all
integers $i$ with $i>H_j$, we have
$$\frac{a_jmn+b_j}{\langle b_jr\rangle_{a_j}+a_ji}<\frac{a_jln+b_j}
{\langle b_jr\rangle_{a_j}+a_ji}<(l-m)n$$
for any positive integer $n>n_0$. So we can deduce that
\begin{align*}
&\bigcup_{j=1}^2\bigcup_{i=H_j+1}^{\infty}\Big\{\text{prime}\ p\equiv r'\pmod q:
\frac{a_jmn+b_j}{\langle b_jr\rangle_{a_j}+a_ji}<p\le \frac{a_jln+b_j}{\langle b_jr\rangle_{a_j}+a_ji}\
\mbox{and}\\& p\nmid (a_1b_2-a_2b_1)\Big\}
\subseteq \Big\{\text{prime}\ p\equiv r'\pmod q: p\le (l-m)n\ \mbox{and}\ p\nmid (a_1b_2-a_2b_1)\Big\}
\end{align*}
for any positive integer $n>n_0$. It then follows from (\ref{eq: 4.8}) and (\ref{eq: Q_3}) that
\begin{align}\label{eq: 4.9}
{\mathcal Q}_{r'}&=\Big(\bigcup_{j=1}^2\bigcup_{i=0}^{H_j}\Big\{ \text{prime}\ p\equiv r'\pmod q:
\frac{a_jmn+b_j}{\langle b_jr\rangle_{a_j}+a_ji}<p\le \frac{a_jln+b_j}
{\langle b_jr\rangle_{a_j}+a_ji}\Big\}\\
\nonumber&\bigcup \{\text{prime}\ p\equiv r'\pmod q: p\le (l-m)n\}\Big)
\setminus\{\mbox{prime}\ p:\ p\nmid (a_1b_2-a_2b_1)\}
\end{align}
for any positive integer $n>n_0$.
Comparing (\ref{eq: 4.3}) with (\ref{eq: 4.9}) if $n>n_0$ and comparing (\ref{eq: 4.3})
with (\ref{eq: 4.8}) if $n\le n_0$, we know
that there are at most finitely many primes in the union set
$({\mathcal Q}_{r'}\setminus\mathcal{P}_{r'})\cup (\mathcal{P}_{r'}\setminus {\mathcal Q}_{r'})$
for any positive integer $n$.
Therefore
\begin{align}\label{eq: 4.10}
\sum_{p\in {\mathcal Q}_{r'}}\log p=\sum_{p\in \mathcal{P}_{r'}}\log p+O(\log n).
\end{align}
By (\ref{eq: 4.7}) and (\ref{eq: 4.10}), the desired result follows immediately.
This concludes the proof of Lemma 2.1.
\end{proof}
By Lemma 2.1, to estimate $\log {\rm lcm}_{mn<i\le ln}\{f(i)\}$, it suffices to estimate
$\sum_{p\in \mathcal{P}_{r'}}\log p$ for each integer $r'$ satisfying $1\le r'\le q$ and $\gcd(r', q)=1$,
which will be done in the following.
\noindent{\bf Lemma 2.2.} {\it Let $r'$ and $r$ be any given integers such that $1\le r', r\le q$ and
$rr'\equiv1\pmod{q}$. If $a_1\langle b_2r\rangle_{a_2}\ge a_2\langle b_1r\rangle_{a_1}$, then
$$\sum_{p\in \mathcal{P}_{r'}}\log p=\frac{n}{\varphi(q)} \lambda_r(a_1,a_2,b_1,b_2)+o(n),
$$
where $\mathcal{P}_{r'}$ and $\lambda_r(a_1, a_2, b_1,b_2)$ are defined as in (2.3) and (1.4), respectively.}
\begin{proof}
Since $a_1\langle b_2r\rangle_{a_2} \ge a_2\langle b_1r\rangle_{a_1}$,
we have
\begin{align}\label{eq: Q1}
\frac{a_1ln}{\langle b_1r\rangle_{a_1}+a_1i}\ge \frac{a_2ln}
{\langle b_2r\rangle_{a_2}+a_2i}\ \mbox{and}\ \frac{a_1mn}{\langle b_1r\rangle_{a_1}+a_1i}\ge \frac{a_2mn}
{\langle b_2r\rangle_{a_2}+a_2i}
\end{align}
for any integer $i\ge 0$. On the other hand, for any integer $i\ge 0$, we have
\begin{align}\label{eq: Q2}
\frac{a_1ln}{\langle b_1r\rangle_{a_1}+a_1(i+1)}< \frac{a_2ln}
{\langle b_2r\rangle_{a_2}+a_2i}\ \mbox{and}\ \frac{a_1mn}{\langle b_1r\rangle_{a_1}+a_1(i+1)}\le \frac{a_2mn}
{\langle b_2r\rangle_{a_2}+a_2i}
\end{align}
since $0\le a_1\langle b_2r\rangle_{a_2} -a_2\langle b_1r\rangle_{a_1}< a_1a_2$ and $l>m\ge 0$.
Let $K_1=g_r(a_1, a_2, b_1, b_2)$ and $K_2=h_r(a_1, a_2, b_1, b_2)$.
Then by (\ref{eq: 1.2}) and (\ref{eq: 1.3}), we get
\begin{align}\label{eq: 4.13}
K_1=\Big\lfloor \frac{a_1a_2l+a_2\langle b_1r\rangle_{a_1}m-a_1\langle b_2r\rangle_{a_2}l}
{a_1a_2(l-m)}\Big\rfloor
\end{align}
and
\begin{align}\label{eq: 4.14}
K_2=\Big\lfloor \frac{a_1\langle b_2r\rangle_{a_2}m-a_2
\langle b_1r\rangle_{a_1}l}{a_1a_2(l-m)}\Big\rfloor.
\end{align}
Thus by (1.4), in order to show Lemma 2.2, we only need to prove that
\begin{align}\label{eq: 4.15}
\sum_{p\in \mathcal{P}_{r'}}\log p=& \frac{n}{\varphi(q)}\bigg(\sum_{i\in S_{K_1}}
\frac{a_1l}{\langle b_1r\rangle_{a_1}+a_1i}-
\sum_{i\in S_{K_1-1}}\frac{a_2m}{\langle b_2r\rangle_{a_2}+a_2i}+\\
\nonumber \ &\sum_{i\in S_{K_2}}
\Big(\frac{a_2l}{\langle b_2r\rangle_{a_2}+a_2i}- \frac{a_1m}{\langle b_1r\rangle_{a_1}+a_1i}
\Big)\bigg)+o(n).
\end{align}
In the following we show that (\ref{eq: 4.15}) is true. For this purpose,
we need to analyze the following union
\begin{align}\label{eq: 4.16}
\mathcal{T}_r:=\Big(\bigcup_{j=1}^2\bigcup_{i=0}^{H_j}\Big(\frac{a_jmn}{\langle
b_jr\rangle_{a_j}+a_ji}, \frac{a_jln}
{\langle b_jr\rangle_{a_j}+a_ji}\Big]\Big)\bigcup\big(0, (l-m)n\big],
\end{align}
since (\ref{eq: 4.3}) gives that
\begin{align}
\label{eq: 4.26}
\mathcal{P}_{r'}=\{\text{prime}\ p\equiv r'\pmod q:
\ p\in \mathcal{T}_r\}.
\end{align}
Evidently, we have
\begin{align*}
\frac{a_1l-(l-m)\langle b_1r\rangle_{a_1}}{a_1(l-m)}
-\frac{a_2l-(l-m)\langle b_2r\rangle_{a_2}}{a_2(l-m)}
=\frac{a_1\langle b_2r\rangle_{a_2}-a_2\langle b_1r\rangle_{a_1}}{a_1a_2}
\end{align*}
and $$0\le \frac{a_1\langle b_2r\rangle_{a_2}-a_2\langle b_1r\rangle_{a_1}}{a_1a_2}<1.$$
Thus by (\ref{eq: 4.1}) and (\ref{eq: 4.2}) we get that
\begin{align}\label{eq: 4.17}
H_1=H_2\ \mbox{or}\ H_2+1.
\end{align}
Moreover, for each $1\le j\le 2$, it follows from (\ref{eq: 4.1}) and (\ref{eq: 4.2}) that
$$\frac{a_jm-(l-m)\langle b_jr\rangle_{a_j}}{a_j(l-m)}<H_j\le
\frac{a_jl-(l-m)\langle b_jr\rangle_{a_j}}{a_j(l-m)}.$$
Hence for each $1\le j\le 2$,
\begin{align}\label{eq: 4.18}
\frac{a_jmn}{\langle b_jr\rangle_{a_j}+a_jH_j}<(l-m)n\le \frac{a_jln}
{\langle b_jr\rangle_{a_j}+a_jH_j}.
\end{align}
By (\ref{eq: 4.13}), we have $K_1\ge 0$ and
\begin{align*}
K_1-1\le \frac{a_1a_2m+a_2\langle b_1r\rangle_{a_1}m-a_1
\langle b_2r\rangle_{a_2}l}{a_1a_2(l-m)}<K_1.
\end{align*}
It then follows that
\begin{align}\label{eq: 4.19}
\frac{a_1ln}{\langle b_1r\rangle_{a_1}+a_1(i+1)}> \frac{a_2mn}{\langle b_2r\rangle_{a_2}+a_2i}
\end{align}
for any $i\ge K_1$ and
\begin{align}\label{eq: 4.20}
\frac{a_1ln}{\langle b_1r\rangle_{a_1}+a_1(i+1)}\le \frac{a_2mn}{\langle b_2r\rangle_{a_2}+a_2i}
\end{align}
for any $0\le i\le K_1-1$ if $K_1\ge 1$.
From (\ref{eq: 4.14}), we know that $K_2$ may be smaller than 0, and
$$
a_1a_2K_2(l-m)\le a_1\langle b_2r\rangle_{a_2}m-a_2
\langle b_1r\rangle_{a_1}l<a_1a_2(K_2+1)(l-m).
$$
Thus
\begin{align}\label{eq: 4.21}
\frac{a_2ln}{\langle b_2r\rangle_{a_2}+a_2i}
>\frac{a_1mn}{\langle b_1r\rangle_{a_1}+a_1i}.
\end{align}
for any $i\ge \max(0, K_2+1)$, and
\begin{align}\label{eq: 4.22}
\frac{a_2ln}{\langle b_2r\rangle_{a_2}+a_2i}
\le \frac{a_1mn}{\langle b_1r\rangle_{a_1}+a_1i}
\end{align}
for any $0\le i\le K_2$ if $K_2\ge 0$.
For $j=1, 2$, if $H_j\ge 1$, then by (\ref{eq: 4.1}) and (\ref{eq: 4.2}) we infer that
\begin{align*}
\frac{a_jmn}{\langle b_jr\rangle_{a_j}+a_j(i-1)}-\frac{a_jln}{\langle
b_jr\rangle_{a_j}+a_ji}&=
\frac{a_j\big(a_jln-(l-m)\langle b_jr\rangle_{a_j}n-a_ji(l-m)n\big)}
{(\langle b_jr\rangle_{a_j}+a_j(i-1))(\langle b_jr\rangle_{a_j}+ai)}\\
&\ge \frac{a_j(a_jH_j(l-m)n-a_ji(l-m)n)}{(\langle b_jr\rangle_{a_j}+a_j(i-1))(\langle b_jr\rangle_{a_j}+a_ji)}\\
&= \frac{a_j^2(H_j-i)(l-m)n}{(\langle
b_jr\rangle_{a_j}+a_j(i-1))(\langle b_jr\rangle_{a_j}+a_ji)}\ge 0
\end{align*}
for any integer $i$ with $1\le
i\le H_j$, which means that
$$\frac{a_jmn}{\langle b_jr\rangle_{a_j}+a_j(i-1)}\ge \frac{a_jln}{\langle
b_jr\rangle_{a_j}+a_ji}$$
for any integer $i$ with $1\le i\le H_j$.
Hence for $j=1, 2$, the intersection
\begin{align}\label{eq: 4.23}
\Big(\frac{a_jmn}{\langle b_jr\rangle_{a_j}+a_ji_1},
\frac{a_jln}{\langle b_jr\rangle_{a_j}+a_ji_1}\Big] \bigcap \Big( \frac{a_jmn}{\langle b_jr\rangle_{a_j}+a_ji_2},
\frac{a_jln}{\langle b_jr\rangle_{a_j}+a_ji_2}\Big]
\end{align}
is empty for any $0\le i_1\ne i_2\le H_j$ if $H_j\ge 1$. Now we consider the following two cases.
{\sc Case 1.} $K_1\ge K_2+1$. First, it is easy to see from (\ref{eq: 4.2}), (\ref{eq: 4.13})
and (\ref{eq: 4.14}) that $K_1\le H_2$ and $K_2+1\le H_2$. For any integer $i\ge \max(0, K_2+1)$,
we have by (\ref{eq: Q1}) and (\ref{eq: 4.21}) that
\begin{align}\label{eq: 4.24}
\bigcup_{j=1}^2\Big( \frac{a_jmn}{\langle b_jr\rangle_{a_j}+a_ji},
\frac{a_jln}{\langle b_jr\rangle_{a_j}+a_ji}\Big]=\Big(\frac{a_2mn}{\langle b_2r\rangle_{a_2}+a_2i},
\frac{a_1ln}{\langle b_1r\rangle_{a_1}+a_1i}\Big].
\end{align}
It then follows from (\ref{eq: 4.17})-(\ref{eq: 4.19}) and (\ref{eq: 4.24}) that
\begin{align}\label{eq: 4.25}
&\bigg(\bigcup_{j=1}^2\bigcup_{i=K_1}^{H_j}\Big( \frac{a_jmn}{\langle b_jr\rangle_{a_j}+a_ji},
\frac{a_jln}{\langle b_jr\rangle_{a_j}+a_ji}\Big]\bigg)\bigcup \Big( 0, (l-m)n\Big]\\
\nonumber&=\bigg(\bigcup_{i=K_1}^{H_2}\Big( \frac{a_2mn}{\langle b_2r\rangle_{a_2}+a_2i},
\frac{a_1ln}{\langle b_1r\rangle_{a_1}+a_1i}\Big]\bigg) \bigcup \Big( 0, (l-m)n\Big]\\
\nonumber &\quad \ \bigcup\Big( \frac{a_1mn}{\langle b_1r\rangle_{a_1}+a_1H_1},
\frac{a_1ln}{\langle b_1r\rangle_{a_1}+a_1H_1}\Big]\\
\nonumber &= \Big( \frac{a_2mn}{\langle b_2r\rangle_{a_2}+a_2H_2},
\frac{a_1ln}{\langle b_1r\rangle_{a_1}+a_1K_1}\Big] \bigcup \Big( 0,
\frac{a_1ln}{\langle b_1r\rangle_{a_1}+a_1H_1}\Big]\\
\nonumber &=\Big( 0,
\frac{a_1ln}{\langle b_1r\rangle_{a_1}+a_1K_1}\Big].
\end{align}
Thus we can derive from (\ref{eq: 4.16}) and (\ref{eq: 4.25}) that
\begin{align*}
\mathcal{T}_r=&\bigcup_{j=1}^2\bigg(\bigcup_{i\in S_{K_1-1}}\Big( \frac{a_jmn}{\langle b_jr\rangle_{a_j}+a_ji},
\frac{a_jln}{\langle b_jr\rangle_{a_j}+a_ji}\Big]\bigcup \\
\nonumber&\bigcup_{i=K_1}^{H_j}\Big( \frac{a_jmn}{\langle b_jr\rangle_{a_j}+a_ji},
\frac{a_jln}{\langle b_jr\rangle_{a_j}+a_ji}\Big]\bigg)
\bigcup \Big( 0, (l-m)n\Big]\\
\nonumber =&\bigg(\bigcup_{j=1}^2\bigcup_{i\in S_{K_2}}\Big( \frac{a_jmn}{\langle b_jr\rangle_{a_j}+a_ji},
\frac{a_jln}{\langle b_jr\rangle_{a_j}+a_ji}\Big]\bigg)\bigcup\Big(0,
\frac{a_1ln}{\langle b_1r\rangle_{a_1}+a_1K_1}\Big]\\
\nonumber &\bigcup \bigg(\bigcup_{i\in S_{K_1-1}\setminus
S_{K_2}}\bigcup_{j=1}^2\Big( \frac{a_jmn}{\langle b_jr\rangle_{a_j}+a_ji},
\frac{a_jln}{\langle b_jr\rangle_{a_j}+a_ji}\Big]\bigg).
\end{align*}
It then follows from (\ref{eq: 4.24}) that
\begin{align}\label{eq: 4.27}
\mathcal{T}_r=&\bigg(\bigcup_{j=1}^2\bigcup_{i\in S_{K_2}}
\Big( \frac{a_jmn}{\langle b_jr\rangle_{a_j}+a_ji},
\frac{a_jln}{\langle b_jr\rangle_{a_j}+a_ji}\Big]\bigg)\bigcup\\
\nonumber &\bigg(\bigcup_{i\in S_{K_1-1}\setminus S_{K_2}}
\Big(\frac{a_2mn}{\langle b_2r\rangle_{a_2}+a_2i},
\frac{a_1ln}{\langle b_1r\rangle_{a_1}+a_1i}\Big]\bigg)\bigcup\Big(0,
\frac{a_1ln}{\langle b_1r\rangle_{a_1}+a_1K_1}\Big].
\end{align}
Note that $S_{K_2}$ is empty if $K_2<0$, and $S_{K_1-1}\setminus S_{K_2}$ is empty
if $K_1=K_2+1$ or $K_1=0$. By (\ref{eq: 4.20}), we know that the following union
$$
\bigcup_{i\in S_{K_1-1}\setminus S_{K_2}}\Big(\frac{a_2mn}{\langle b_2r\rangle_{a_2}+a_2i},
\frac{a_1ln}{\langle b_1r\rangle_{a_1}+a_1i}\Big]
$$
is a disjoint union. But by (\ref{eq: 4.20}), (\ref{eq: 4.22}) and (\ref{eq: 4.23}), the union
$$
\bigcup_{j=1}^2\bigcup_{i\in S_{K_2}}\Big( \frac{a_jmn}{\langle b_jr\rangle_{a_j}+a_ji},
\frac{a_jln}{\langle b_jr\rangle_{a_j}+a_ji}\Big]
$$
is a disjoint union. Therefore by (\ref{eq: 4.20}), the union on the right-hand side
of (\ref{eq: 4.27}) is disjoint. Thus applying (\ref{eq: 4.26}), (\ref{eq: 4.27})
and prime number theorem for arithmetic progressions (see, for example \cite{[MV]}),
we obtain that
\begin{align*}
\sum_{p\in \mathcal{P}_{r'}}\log p&=\sum_{j=1}^2\sum_{i\in S_{K_2}}
\sum_{\frac{a_jmn}{\langle b_jr\rangle_{a_j}+a_ji}<
p\le \frac{a_jln}{\langle b_jr\rangle_{a_j}+a_ji}\atop p\equiv r'\pmod q}\log p
+\sum_{p\le \frac{a_1ln}{\langle b_1r\rangle_{a_1}+a_1K_1}\atop p\equiv r'\pmod q}\log p\\
&\ \ +\sum_{i\in S_{K_1-1}\setminus S_{K_2}}\sum_{
\frac{a_2mn}{\langle b_2r\rangle_{a_2}+a_2i}<p\le \frac{a_1ln}{\langle b_1r
\rangle_{a_1}+a_1i}\atop p\equiv r'\pmod q}\log p\\
&=\frac{n}{\varphi(q)}\bigg( \sum_{j=1}^2\sum_{i\in S_{K_2}}\Big( \frac{a_jl}{\langle b_jr\rangle_{a_j}+a_ji}-
\frac{a_jm}{\langle b_jr\rangle_{a_j}+a_ji} \Big)+ \frac{a_1l}{\langle b_1r\rangle_{a_1}+a_1K_1} \\
&\quad\quad\quad\quad+\sum_{i\in S_{K_1-1}\setminus S_{K_2}}\Big( \frac{a_1l}{\langle b_1r\rangle_{a_1}+a_1i}-
\frac{a_2m}{\langle b_2r\rangle_{a_2}+a_2i}\Big)\bigg)+o(n).
\end{align*}
Then (\ref{eq: 4.15}) follows immediately. So (\ref{eq: 4.15}) is proved for Case 1.
{\sc Case 2.} $K_1\le K_2$. Then by (\ref {eq: 4.13}) and (\ref {eq: 4.14}), we have $K_2\ge K_1\ge 0$.
If $K_2+1\le H_2$, applying (\ref{eq: 4.17})-(\ref{eq: 4.19}) and (\ref{eq: 4.24}),
one infers that
\begin{align} \label{eq: 4.30}
&\bigcup_{j=1}^2\bigcup_{i=K_2+1}^{H_j}\Big(\frac{a_jmn}{\langle b_jr\rangle_{a_j}+a_ji},
\frac{a_jln}{\langle b_jr\rangle_{a_j}+a_ji}\Big]\bigcup\Big( 0, (l-m)n\Big]\\
\nonumber &=\bigcup_{i=K_2+1}^{H_2}\Big(\frac{a_2mn}{\langle b_2r\rangle_{a_2}+a_2i},
\frac{a_1ln}{\langle b_1r\rangle_{a_1}+a_1i}\Big]\bigcup\Big( 0, (l-m)n\Big]\\
\nonumber &\quad\quad\bigcup \Big(\frac{a_1mn}{\langle b_1r\rangle_{a_1}+a_1H_1},
\frac{a_1ln}{\langle b_1r\rangle_{a_1}+a_1H_1}\Big]\\
\nonumber &=\Big(\frac{a_2mn}{\langle b_2r\rangle_{a_2}+a_2H_2},
\frac{a_1ln}{\langle b_1r\rangle_{a_1}+a_1(K_2+1)}\Big]
\bigcup\Big(0, \frac{a_1ln}{\langle b_1r\rangle_{a_1}+a_1H_1}\Big]\\
\nonumber &=\Big(0, \frac{a_1ln}{\langle b_1r\rangle_{a_1}+a_1(K_2+1)}\Big].
\end{align}
Hence by (\ref{eq: 4.16}) and (\ref{eq: 4.30}),
\begin{align}\label{eq: 4.31}
\mathcal{T}_r&=\bigg(\bigcup_{j=1}^2\bigcup_{i=0}^{K_2}
\Big(\frac{a_jmn}{\langle b_jr\rangle_{a_j}+a_ji},
\frac{a_jln}{\langle b_jr\rangle_{a_j}+a_ji}\Big]\bigg)\bigcup
\Big(0, \frac{a_1ln}{\langle b_1r\rangle_{a_1}+a_1(K_2+1)}\Big].
\end{align}
Moreover, we have
\begin{align}
\label{eq: 4.32}
&\bigcup_{j=1}^2\bigcup_{i=0}^{K_2}\Big(\frac{a_jmn}{\langle b_jr\rangle_{a_j}+a_ji},
\frac{a_jln}{\langle b_jr\rangle_{a_j}+a_ji}\Big]\\
\nonumber &=\bigg(\bigcup_{j=1}^2\bigcup_{i\in S_{K_1-1}}
\Big(\frac{a_jmn}{\langle b_jr\rangle_{a_j}+a_ji},
\frac{a_jln}{\langle b_jr\rangle_{a_j}+a_ji}\Big]\bigg)\bigcup
\Big(\frac{a_1mn}{\langle b_1r\rangle_{a_1}+a_1K_1},
\frac{a_1ln}{\langle b_1r\rangle_{a_1}+a_1K_1}\Big] \bigcup\\
\nonumber &\bigcup_{i\in S_{K_2-1\setminus S_{K_1-1}}}
\bigg( \Big(\frac{a_1mn}{\langle b_1r\rangle_{a_1}+a_1(i+1)},
\frac{a_1ln}{\langle b_1r\rangle_{a_1}+a_1(i+1)}\Big]\bigcup\\
\nonumber & \Big(\frac{a_2mn}{\langle b_2r\rangle_{a_2}+a_2i},
\frac{a_2ln}{\langle b_2r\rangle_{a_2}+a_2i}\Big]\bigg)
\bigcup \Big(\frac{a_2mn}{\langle b_2r\rangle_{a_2}+a_2K_2},
\frac{a_2ln}{\langle b_2r\rangle_{a_2}+a_2K_2}\Big].
\end{align}
But since $K_2\ge 0$ and $K_2\ge K_1$, applying (\ref{eq: Q2}) and (\ref {eq: 4.19}) gives us that
\begin{align}\label{eq: 4.33}
\Big(0, \frac{a_1ln}{\langle b_1r\rangle_{a_1}+a_1(K_2+1)}\Big] \bigcup
\Big(\frac{a_2mn}{\langle b_2r\rangle_{a_2}+a_2K_2},
\frac{a_2ln}{\langle b_2r\rangle_{a_2}+a_2K_2}\Big]
=\Big(0, \frac{a_2ln}{\langle b_2r\rangle_{a_2}+a_2K_2}\Big].
\end{align}
Therefore by (\ref{eq: Q2}), (\ref{eq: 4.19}) and (\ref{eq: 4.31})-(\ref{eq: 4.33}), we have
\begin{align}\label{eq: 4.34}
\mathcal{T}_r&=\bigg(\bigcup_{j=1}^2\bigcup_{i\in S_{K_1-1}}\Big(\frac{a_jmn}{\langle b_jr\rangle_{a_j}+a_ji},
\frac{a_jln}{\langle b_jr\rangle_{a_j}+a_ji}\Big]\bigg)\bigcup \Big(\frac{a_1mn}{\langle b_1r\rangle_{a_1}+a_1K_1},
\frac{a_1ln}{\langle b_1r\rangle_{a_1}+a_1K_1}\Big]\\
\nonumber &\bigg(\bigcup_{i\in S_{K_2-1\setminus S_{K_1-1}}} \Big(\frac{a_1mn}{\langle b_1r\rangle_{a_1}+a_1(i+1)},
\frac{a_2ln}{\langle b_2r\rangle_{a_2}+a_2i}\Big]\bigg)\bigcup \Big(0, \frac{a_2ln}{\langle b_2r\rangle_{a_2}+a_2K_2}\Big].
\end{align}
Note that $S_{K_2-1}\setminus S_{K_1-1}$ is empty if $K_1=K_2$, and $S_{K_1-1}$ is empty if $K_1=0$.
By (\ref{eq: 4.20}), we have that
$$\frac{a_1ln}{\langle b_1r\rangle_{a_1}+a_1K_1}\le \frac{a_2mn}{\langle b_2r\rangle_{a_2}+a_2(K_1-1)}$$
if $K_1\ge 1$.
Using (\ref{eq: 4.22}), one has
$$
\frac{a_2ln}{\langle b_2r\rangle_{a_2}+a_2K_2}\le \frac{a_1mn}{\langle b_1r\rangle_{a_1}+a_1K_2}
\ {\rm and} \
\frac{a_2ln}{\langle b_2r\rangle_{a_2}+a_2K_1}\le \frac{a_1mn}{\langle b_1r\rangle_{a_1}+a_1K_1}.
$$
Then using (\ref{eq: 4.20}), (\ref{eq: 4.22}) and (\ref{eq: 4.23}), we derive that any two intervals in the
union on the right-hand side of (\ref{eq: 4.34}) are disjoint.
Hence we have by (\ref{eq: 4.26}) and (\ref{eq: 4.34}) that
\begin{align*}
&\sum_{p\in \mathcal{P}_{r'}}\log p=\sum_{j=1}^2\sum_{i\in S_{K_1-1}}\sum_{\frac{a_jmn}{\langle b_jr\rangle_{a_j}+a_ji}<
p\le \frac{a_jln}{\langle b_jr\rangle_{a_j}+a_ji}\atop p\equiv r'\pmod q}\log p+\sum_{p\le
\frac{a_2ln}{\langle b_2r\rangle_{a_2}+a_2K_2}\atop p\equiv r'\pmod q}\log p\\
&\ +\sum_{ \frac{a_1mn}{\langle b_1r\rangle_{a_1}+a_1K_1}<p\le
\frac{a_1ln}{\langle b_1r\rangle_{a_1}+a_1K_1}\atop p\equiv r'\pmod q}\log p+\sum_{i\in S_{K_2-1}\setminus S_{K_1-1}}\sum_{
\frac{a_1mn}{\langle b_1r\rangle_{a_1}+a_1(i+1)}<p\le \frac{a_2ln}{\langle b_2r\rangle_{a_2}+a_2i}
\atop p\equiv r'\pmod q}\log p.
\end{align*}
It then follows from the prime number theorem for arithmetic progressions that
\begin{align*}
&\sum_{p\in \mathcal{P}_{r'}}\log p=\frac{n}{\varphi(q)}
\bigg( \sum_{j=1}^2\sum_{i\in S_{K_1-1}}\Big( \frac{a_jl}{\langle b_jr\rangle_{a_j}+a_ji}-
\frac{a_jm}{\langle b_jr\rangle_{a_j}+a_ji} \Big)+ \frac{a_2l}{\langle b_2r\rangle_{a_2}+a_2K_2}\\
&\quad\quad+\frac{a_1(l-m)}{\langle b_1r\rangle_{a_1}+a_1K_1}+\sum_{i\in S_{K_2-1}\setminus S_{K_1-1}}
\Big( \frac{a_2l}{\langle b_2r\rangle_{a_2}+a_2i}-
\frac{a_1m}{\langle b_1r\rangle_{a_1}+a_1(i+1)}\Big)\bigg)+o(n)\\
&=\frac{n}{\varphi(q)} \bigg(\sum_{i\in S_{K_1}}\frac{a_1l}{\langle b_1r\rangle_{a_1}+a_1i}-
\sum_{i\in S_{K_1-1}}\frac{a_2m}{\langle b_2r\rangle_{a_2}+a_2i}\\
&\quad\quad\quad\quad+\sum_{i\in S_{K_2}}
\Big(\frac{a_2l}{\langle b_2r\rangle_{a_2}+a_2i}-
\frac{a_1m}{\langle b_1r\rangle_{a_1}+a_1i} \Big)\bigg)+o(n)
\end{align*}
as required. Thus (\ref{eq: 4.15}) is true for Case 2.
This completes the proof of Lemma 2.2.
\end{proof}
\section{\bf Proof of Theorem 1.1}
In this section, we use the results presented in Section 2 to give
the proof of Theorem 1.1.
{\it Proof of Theorem 1.1.} Let $r'$ and $r$ be any given integers such that $1\le r', r\le q$ and
$rr'\equiv1\pmod{q}$. Exchanging $a_1$ with $a_2$ and $b_1$ with $b_2$ simultaneously,
$f(x)=(a_2x+b_2)(a_1x+b_1)$ is unchanged, meanwhile the condition
$a_1\langle b_2r\rangle_{a_2}\ge a_2\langle b_1r\rangle_{a_1}$ in Lemma 2.2 becomes
$a_2\langle b_1r\rangle_{a_1}\ge a_1\langle b_2r\rangle_{a_2}$, and in the conclusion of Lemma 2.2,
$\lambda_r(a_1,a_2,b_1,b_2)$ becomes $\lambda_r(a_2,a_1,b_2,b_1)$.
Thus by Lemma 2.2, we obtain that
$$
\sum_{p\in \mathcal{P}_{r'}}\log p=\frac{n}{\varphi(q)} \lambda_r(a_2,a_1,b_2,b_1)+o(n)
$$
if $a_2\langle b_1r\rangle_{a_1}\ge a_1\langle b_2r\rangle_{a_2}$.
Note that if $a_1\langle b_2r\rangle_{a_2}=a_2\langle b_1r\rangle_{a_1}$, then
$$\frac{a_1mn}{\langle b_1r\rangle_{a_1}+a_1i}=
\frac{a_2mn}{\langle b_2r\rangle_{a_2}+a_2i}$$
and
$$\frac{a_1ln}{\langle b_1r\rangle_{a_1}+a_1i}= \frac{a_2ln}{\langle
b_2r\rangle_{a_2}+a_2i}$$
for any integer $i\ge 0$. Moreover, one has by (\ref{eq: 1.2})
and (\ref{eq: 1.3}) that
$$g_r(a_1, a_2, b_1, b_2)=g_r(a_2, a_1, b_2, b_1)$$
and
$$
h_r(a_1, a_2, b_1, b_2)=h_r(a_2, a_1, b_2, b_1)
$$
if $a_1\langle b_2r\rangle_{a_2}=a_2\langle b_1r\rangle_{a_1}$.
It then follows from (\ref{eq: 1.4}) that
$$\lambda_r(a_1,a_2,b_1, b_2)=\lambda_r(a_2, a_1, b_2, b_1)$$
if $a_1\langle b_2r\rangle_{a_2}=a_2\langle b_1r\rangle_{a_1}$.
Now by Lemma 2.2 and the above discussion, we get that
\begin{align}\label{eq: 4.12}
\sum_{p\in \mathcal{P}_{r'}}\log p=\frac{n}{\varphi(q)}A_r+o(n),
\end{align}
where $A_r$ is defined as in (\ref{eq: 1.5}). Since
$rr'\equiv 1\pmod q$ and $1\le r', r\le q$, $r$ runs over the set
of all positive integers no more than $q$ that are relatively
prime to $q$ as $r'$ does, it then follows from (\ref{eq: 4.12})
and Lemma 2.1 that
\begin{align*}
&\log {\rm lcm}_{mn<i\le ln}\{ f(i)\}\\
=&\frac{n}{\varphi(q)}
\sum_{r'=1\atop \gcd(r',q)=1}^{q}A_r+o(n)\\
=&\frac{n}{\varphi(q)}\sum_{r=1\atop \gcd(r,q)=1}^{q}A_r+o(n)
\end{align*}
as desired. This finishes the proof of Theorem 1.1.
$\Box$\\
\end{document}
|
\begin{document}
\title{On the Characterization of $1$-sided error Strongly-Testable Graph
Properties for bounded-degree graphs, including an appendix\
}
\begin{abstract}
We study property testing of (di)graph properties in bounded-degree
graph models. The study of graph properties in bounded-degree
models is one of the focal directions of research in property
testing in the last 15 years. However, despite of the many results
and the extensive research effort, there is no characterization of
the properties that are strongly-testable (i.e., testable with constant query
complexity)
even for $1$-sided error tests.
The bounded-degree model can naturally be generalized to
directed graphs resulting in two models that were
considered in the literature. The first contains the directed
graphs in which the outdegree is bounded but the indegree is
not restricted. In the other, both the outdegree and indegree
are bounded.
We give a characterization of the $1$-sided error
strongly-testable {\em monotone} graph properties, and the $1$-sided
error strongly-testable {\em hereditary} graph properties in all the
bounded-degree directed and undirected graphs models.
{\bf comments: this version corrects minor details in the previous: (a)
removed the non-defined term 'non-redundant' from
theorem 3.3. (b) corrected a typo in example 7.2 page 22}
\end{abstract}
\setcounter{page}{1}
\newcommand{\ignore}[1]{}
\def \qed {\hspace*{0pt}
{\quad \vrule height 1ex width 1ex depth 0pt}
}
\newenvironment{proof}{\par\noindent{\bf Proof.}\quad}{ $\qed$}
\newcommand{\ensuremath{\mathbb R}}{\ensuremath{\mathbb R}}
\newcommand{\ensuremath{\mathbb Z}}{\ensuremath{\mathbb Z}}
\newcommand{\ensuremath{\mathbb N}}{\ensuremath{\mathbb N}}
\newcommand{\ensuremath{\mathcal F}}{\ensuremath{\mathcal F}}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}{Definition}[section]
\newtheorem{claim}[definition]{Claim}
\newtheorem{proposition}{Proposition}[section]
\newtheorem{observation}{Observation}[section]
\newtheorem{remark}[definition]{Remark}
\newtheorem{prop}[section]{Proposition}
\newtheorem{lemma}[proposition]{Lemma}
\newtheorem{crl}{Corollary}
\newtheorem{corol}{Corollary}[section]
\newcommand{\noindent{\bf Proof.}\ \ }{\noindent{\bf Proof.}\ \ }
\newtheorem{corollary}{Corollary}[section]
\newtheorem{fact}[definition]{Fact}
\newtheorem{conj}[theorem]{Conjecture}
\newtheorem{thm}[definition]{Theorem}
\newtheorem{thmm}[proposition]{Theorem}
\newtheorem{thmp}[proposition]{Theorem}
\newtheorem{remarkp}[proposition]{Remark}
\newtheorem{definitionp}[proposition]{Definition}
\newtheorem{claimp}[proposition]{Claim}
\newtheorem{factp}[proposition]{Fact}
\section{Introduction}
Testing graph properties has been at the core of
combinatorial property testing since the very beginning with the
important results of Goldreich-Goldwasser-Ron \cite{GGR98}. There are several different models
of interest.
In the dense graph model an
$n$-vertex graph is given by its $n \times n$ boolean adjacency
matrix. For this model there are
characterizations of the properties that can be tested in constant
amount of queries by $1$-sided error tests \cite{AS08}, $2$-sided
error tests \cite{AFNS09}, and the properties that are defined by
forbidden induced subgraphs and are testable by very small query complexity
\cite{AS06}.
In the other model, called the {\em incidence-list} model, an
$n$-vertex graph is represented by its incidence lists. That is, an
array of size $n$ in which every entry is associated with a vertex,
and contains a list of the neighbours of that vertex.
This model contains the important special case of the {\em bounded-degree model} in which the degree of the vertices
is bounded by a universal parameter $d$ (and hence the lists are of size at most $d$).
The bounded-degree model, first considered in the property testing context
by Goldreich and Ron \cite{GR02},
attracts much of the research interest in combinatorial property
testing in the past decade. One reason is the algorithmic sophistication and wealth of
structural results that were developed in the studies of property testing
in this model.
E.g., the use of random walks to test partition properties, starting in
\cite{bounded-degree}, and with the sophisticated recent results in
\cite{CS10, CPS15} for
expander and clustering testing, the ``local-partition'' oracle
\cite{HKNO09, LR15, NO08}, and
others.
The other motivation is the rapidly growing research of very large
networks, e.g., the Internet, and other natural large
networks such as social networks.
These large networks often turn to be represented by bounded-degree
(di)graphs (or very sparse (di)graphs).
Property testing of sparse
graphs can provide a useful filter to discard unwanted instances at a very low cost (in time
and space), as well as algorithmic and structural insights regarding the
tested properties.
Despite of the focus and wealth of results, the
bounded-degree model remains far from being understood. In particular, as of
present, there is no characterization of the
properties that are testable in constant query complexity,
neither by $2$-sided error tests, nor by $1$-sided error tests.
We focus on $1$-sided error testing. Our main result is a
characterization of the {\em monotone} (di)graph properties, and the
{\em hereditary} (di)graph properties, that are $1$-sided-error {\em
strongly-testable}\footnote{For formal
definiton of ``property testing'' see Section \ref{sec:prelim}}. Here ``strongly-testable'' means that the
property can be tested by a constant number of queries that is
independent of the graph size, but may depend on the distance
parameter $\epsilon$.
The characterization essentially states that a monotone graph property
is strongly-testable if and only if it is close (see Definition
\ref{def:dist}) to a property that is defined by a set of forbidden
subgraphs of constant size (Theorem \ref{thm:main-fb}). For
hereditary property we obtain a similar result (Theorem
\ref{thm:main-fb0.5}) except that
forbidden subgraphs are replaced with forbidden as induced subgraphs.
We
believe that our results form a first step towards a characterization
of all $1$-sided error strongly-testable graph properties in the bounded-degree model.
The bounded-degree model extends naturally to directed graphs. There
are two different models that have been studied for directed
graphs: In the first, the access to the graph is via queries to
outgoing neighbours, and correspondingly, only the out-degree of
vertices is bounded. This model
corresponds to the standard representation of directed graphs in
algorithmic computer science. Namely, where an $n$-vertex directed graph (digraph) is
represented by $n$ lists, each being associated with a distinct vertex $v$
in the graph, and contains the list of {\em forward} edges going out
from $v$.
The access to a $d$-outdegree bounded digraph in this model is via queries of the following
type: a query specifies a pair
$(v,i)$ where $v \in V(G)$ and $i \leq d$. As a response, the
algorithm discovers the $i$th outgoing neighbour of the vertex $v$\footnote{if there is
one, or a special symbol otherwise}. In what follows we abbreviate this model
as the $F(d)$-model, where $d$ is the upper bound on the out-degree of
vertices.
In the other model, both the in-degree
and out-degree are bounded by $d$. In this case an $n$-vertex graph is represented
by $2n$ lists; the list of outgoing edges and the list of incoming edges for
each vertex. The query type changes accordingly and allows both
`outgoing' and `incoming' edge queries. We denote this model
as the $FB(d)$-model (`forward' and ` backward' queries). This model contains the model of undirected $d$-bounded
degree graphs (where each undirected edge is replaced by a pair of
anti-parallel edges).
We note that the $F(d)$ model, as a collection
of graphs, strictly contains the $FB(d)$ model, while algorithmically
it is more restricted by the limited access to the graph.
In all models, an $n$-vertex (di)graph $G$ is said to be
$\epsilon$-far from a (di)graph property $P$ if it is
required to change (delete and/or insert) at least $\epsilon \cdot dn$
edges in order to get a $d$-bounded degree graph (in the corresponding
model) that has the property $P$.
The results in this paper are the characterization of
the {\em monotone} digraph properties and {\em hereditary} digraph
properties
that are $1$-{\em sided error} strongly-testable in the $F(d)$-model
(Theorem \ref{thm:main-mon}, and
\ref{thm:main-hered}). The results for the $FB(d)$ model easily follow from
these for the $F(d)$-model. As the $FB(d)$-model
contains the undirected case, an analogous characterization of graph properties
for the $d$-bounded degree undirected graph model is implied.
We note that these are the first results that do not restrict
the family of graphs, nor the family of testers under consideration
(apart of being
$1$-sided-error).
{\bf Related results:}
There are many results for the bounded-degree model on the testability of specific
properties of graphs or digraphs, cf.
\cite{BR02, CS10, GR00, GR02, NS07, OR11, YI10, PR02},
and others.
In \cite{CPS16} the authors
relate (2-sided error) testability in the $FB(d)$ and $F(d)$ models. Other general results fall typically into three
categories. In the first not all $d$-bounded degree graphs are
considered, but rather a restricted family of graphs. It is shown
e.g., in \cite{HKNO09, LR15, NS13} (and citations
therein)\footnote{\cite{NS13} shows that any graph property is $2$-sided error strongly-testable for any hyperfinite family of
graphs.}
that under certain restriction of the input graphs all graph properties are
$2$-sided error strongly-testable.
The other two types of general results are when the graph
properties under study are restricted, or the class of testers is restricted.
Most relevant for this work are the results of Czumaj, Shapira and
Sohler
\cite{CSS09}, and Goldreich-Ron \cite{GR09}.
In \cite{CSS09} it is shown that any hereditary property is $1${\em-sided error}
strongly-testable if the input graph
belongs to a hereditary and non-expanding family of
graphs
In
\cite{GR09} restricted $1$-sided error testers called
{\em proximity oblivious testers} (POT) for graph properties (and
other properties) are studied. The POT
is not being constructed for an explicitly given distance parameter $\epsilon$. Instead, the tester
works for any distance parameter
$\epsilon$, but its success probability deteriorates as
$\epsilon$ tends to $0$. \cite{GR09} give several general results to when graph
properties have a POT in the bounded-degree model (and other models).
{\bf Techniques and description of results: }
Attempting for a characterization result we
should understand what are the limitations that
a 1-sided error test, making $O(1)$ queries, puts on the structure
of the property it tests. It turns out that this is relatively
simple. Using the tools from \cite{GR09} (see also
\cite{GT01}), one can transform any $1$-sided error
tester into a ``canonical'' one that picks (uniformly) $O(1)$
random vertices in $G$, and then scans the balls of radius $O(1)$
around each. Finally, it makes its decision based only on the subgraph $G'$
it discovers {\em and its interface to the rest of the graph}. To
make this latter point clearer consider the $3$-degree bounded model and the
property of not having a vertex of degree two. This
property is $1$-sided error strongly-testable simply by looking at a random
vertex and rejecting if its degree is exactly $2$. Note
that this decision cannot be concluded just by the fact that
the subgraph seen is a {\em subgraph} of $G$. It is important that the
sampled vertex $v$
{\em is not connected} to any other vertex besides the $2$
discovered neighbours of it. Namely, this property is not specified by
a forbidden subgraph (or induced subgraph). This
suggests the notion of {\em configuration} appearing also in
\cite{GR09}, and defined for our setting in Section
\ref{sec:prelim}.
Loosely speaking a configuration specifies an induced subgraph with
an induced ``interface'' to the rest of the graph (see Definition \ref{def:conf}). With this notion it is
fairly easy to see that any $1$-sided error test can essentially test only graph
properties that are close to being defined by a collection of forbidden
configurations (the additional subtleties arise from the fact that the
tester is actually being designed for a distance parameter $\epsilon$,
and for different $\epsilon$'s testers might reject different
configurations).
Is the converse true? Namely, is every property that is defined by a set of
forbidden configurations (let alone, being ''close'' to
such) strongly-testable? This is open at
this point.
Showing that a property that is defined by a forbidden set of
configurations is $1$-sided error strongly-testable
usually amounts to proving what is called ``removal
lemmas''. Namely, a lemma stating that
if a graph is $\epsilon$-far from a property then it has a {\em
large} number of appearances of forbidden configurations
(here ``large'' is $f(\epsilon) \cdot n$, namely linear in $n$).
In the case of monotone properties the notion of a `forbidden
configurations' can be replaced with `forbidden subgraphs'. A removal lemma is true for monotone properties in all models.
For hereditary properties `forbidden configurations' can be
replaced with `forbidden induced subgraphs'. A removal lemma is also
true for hereditary properties for the $FB(d)$-model, and in a slightly
different form for the $F(d)$-model, but is more complicated to prove. We
use a somewhat
different argument and test for hereditary properties in the later case.
Our main results show that for all the
bounded-degree models, for both
monotone properties, and hereditary properties, a property is
$1$-sided error strongly-testable
if and only if it is ``close'' to a property that is defined by an
appropriate set of forbidden graphs (see Section \ref{sec:prelim} for
the exact definition of ``close'' in this context). It could be that
by replacing forbidden graphs with forbidden configurations,
this becomes true for {\em any graph property}. If indeed true, this will settle the
characterization problem of $1$-sided error strongly-testable
properties (see the discussion at the end of Section \ref{sec:F}).
We do not currently know if a generalization of some sort
is true even for undirected $3$-degree bounded graphs.
Finally, the characterization that we present is a structural result
on $1$-sided error strongly-testable properties. It provides a
better understanding of the different models and the difference
between them.
One could further
ask whether the characterization could be used to easily determine
whether
a given property is $1$-sided error strongly-testable using arguments totally outside the area of property
testing. This is indeed demonstrated (Section \ref{sec:55}) by proving (the known results) that 2-colorability is not $1$-sided error
strongly-testable, and that not having a $k$-star as a minor is
strongly-testable (here $k$ is constant).
{\bf Organization: } We start with the essential notations and
preliminaries in Section \ref{sec:prelim}. Section \ref{sec:results}
contains a statement of our main results for the $F(d)$-model, and
Section \ref{sec:F} contains the proofs of the main results. Section
\ref{sec:5} contains further discussion, and examples of properties
that are strongly-testable but not monotone, neither
hereditary. Section \ref{sec:FB} contains the analogous
characterizations for
the $FB(d)$-model. Finally Sections \ref{sec:55} and \ref{sec:concl}
contain the application of our results to simply prove some known
results, and some concluding remarks, respectively.
\section{Preliminaries}\label{sec:prelim}
\subsection{Graph related notations}
Graphs here are mostly directed, can have anti-parallel edges
but no multiple edges. We will describe the results (and corresponding
definitions) mainly for the
$F(d)$-model which is the more interesting technically. Moreover, as we
do not have a bound on the in-degree for this model, better understanding
this model may form a tiny step towards better understanding testing in sparse graphs (of
unbounded degree).
For a directed graph $G = (V,E)$ we denote by $(u,v)$ the {\em
directed} edge $(u \rightarrow v)$. That is, $(u,v)$ is a forward edge from
$u$. In turn, $v$ will be a member in the outgoing list of neighbours of $u$.
\begin{definition} [Neighbourhood]
For a digraph $G=(V,E)$ and a vertex $v \in V$ we denote by
$\Gamma^+(v)$ the set of outgoing neighbours of
$v$. Formally, $\Gamma^+(v) = \{u~\mid ~(v,u) \in E\}$.
Similarly,
$\Gamma^-(v) = \{u~\mid ~(u,v) \in E\}$ and $\Gamma(v) = \Gamma^+(v) \cup \Gamma^-(v)$.
\end{definition}
Note that for undirected graphs $\Gamma^+, \Gamma^-$ and $\Gamma$
coincide.
We generalize the notion of neighbourhood for sets of vertices: For $S \subseteq V$ we denote by $\Gamma^+(S) = \{y \notin S~\mid ~~ \exists x \in S, ~(x,y) \in E \}$. $\Gamma^-(S)$ and $\Gamma(S)$ are defined analogously.
\begin{definition} [Degree Bound]
For an integer $d,$ a digraph $G$ is called
$d$-bounded-out-degree if for every $v \in V(G), ~
|\Gamma^+(v)| \leq d$.
The $F(d)$-model contains all digraphs
that are $d$-bounded-out-degree.
\end{definition}
Note that the in-degree of a vertex can be arbitrary.
For a (di)graph $G=(V,E)$ and $V' \subseteq V$, we denote by
$G\setminus V'$ the (di)graph on $V \setminus V'$ that is obtained form $G$ by deleting
the vertices in $V'$. We denote by $G[V']$ the induced subgraph of $G$ on
$V'$ (that is, $G[V']$ contains all edges in $E(G)$ with both
endpoints in $V'$).
A directed $k$-star is the graph containing $k+1$ vertices $\{u_i, ~ i=0,
\ldots ,k\}$ and the edges $\{(u_0,u_i)| ~ i=1, \ldots ,k \}$. In
this case $u_0$
is called the ``center''.
\subsection{Properties and testers}
\begin{definition}\label{def:f-model} {\bf (The $F(d)$-model; queries)}
Let $G=(V,E)$ be a graph on $n$ vertices in the
$F(d)$-model. The access to $G$ is via the following oracle:
A query specifies a name of a
vertex $v \in [n]$. As a result the oracle provides
$\Gamma^+(v)$ as an answer.
\end{definition}
Note that an algorithm has no direct access to the incoming edges of a specified vertex $v$.
We note that a standard query in the incident list model is
for a pair $(v,i)$, where $v \in V(G)$ and $i$ an index, on which the
oracle's answer is the $i$th vertex in the ordered list $\Gamma^+(v)$. For $d=
O(1)$ the two query-types are asymptotically equivalent (up to multiplying the number
of queries by a factor of
$d$). We use the definition above to
emphasise that algorithms, as well as properties, are invariant to the order of the vertices in
$\Gamma^+$.
The $FB(d)$-model is similar where for a query $v \in V(G)$, the
answer is the pair of sets $\Gamma^+(v)$ and $\Gamma^-(v)$ (both sets
are of size at most $d$). In the undirected case the result is
$\Gamma(v)$ (of size bounded by $d$). In terms of property testing,
the $d$-bounded degree model for undirected graphs can be seen as a
submodel of $FB(d)$-model where each undirected edge is represented as
two anti-parallel edges.
\ignore{
We note that while we consider that the input for a
tester is a {\em labeled} graph in the corresponding model,
it is rather a data-structure for the graph. Namely, in which there
is an extra arbitrary order on $\Gamma^+(v), \Gamma^-(v)$ for every
vertex $v$. Hence, and unlike in the dense graph model, the same
labeled graph has many possible different representations, and hence inputs. When the tester
makes its query, it gets the neighbourhood order information, and it
might base its decision on it. This of course, seems useless, and will
not be important in the testers we design, however, for a
characterization result, as we need to consider all possible testers,
we need to take the above information into account.
}
\begin{definition} [(di)Graph Properties]
A (di)graph property $P$ is a set of (di)graphs that is closed under isomorphism. Namely if $G \in P$ then any isomorphic copy of $G$ is in $P$.
We write $P= \cup_{n \in \mathbb{N}}P_n$, where $P_n$ is the set of $n$-vertex graphs in $P$.
\end{definition}
\begin{definition}[Graph distance, distance to a
property, distance between properties]\label{def:dist}
Let $G$ and $G'$ be (di)graphs on $n$ vertices in any of the
$d$-bounded degree models (that is, the $F(d)$-model, $FB(d)$, or
$d$-bounded degree undirected graph model). The distance,
$dist(G, G')$, is the number of edges that needs to be deleted and /
or inserted from $G$ in order to make it $G'$.
We say that $G, G'$ are $\epsilon$-far (or $G$ is $\epsilon$-far from $G'$) if
$dist(G,G')$ $> \epsilon dn$.
Otherwise $G,G'$ are said to be $\epsilon$-close.
Let $P_n, Q_n$ be properties of $n$-vertex (di)graphs. $G$ is
$\epsilon$-close to $P_n$ if it is $\epsilon$-close to some
$G' \in P_n$.
We say that $P_n$ $\rm{and}$ $Q_n$ are $\epsilon$-close (or
$P_n$ is $\epsilon$-close to $Q_n$) if
every graph in $P_n$ is $\epsilon$-close to $Q_n,$ and every
graph in $Q_n$ is $\epsilon$-close to $P_n$.
\end{definition}
\begin{definition}[Monotone properties and hereditary properties]\label{def:mon-hered-prop}
A (di)graph property
$P$ is monotone (decreasing) if for every $G=(V,E) \in P$, deleting
any edge $e \in E(G)$ results in a (di)graph $G\setminus \{e\}$ that
is in $P$.
A (di)graph property $P$ is hereditary if for every $G=(V,E) \in P$ and $v \in V(G)$, $G \setminus \{v\} \in P$.
\end{definition}
Many natural (di)graph properties are monotone, e.g., being acyclic,
being $3$-colourable etc. Note that if $P = \cup_{n \in \ensuremath{\mathbb N}} P_n$ is a monotone graph
property then for every $n \in \ensuremath{\mathbb N},~$ $P_n$ is by itself monotone.
\begin{definition}[The (di)Graph Properties $\mathcal{P_{\mathcal{H}}}$
and $P^*_{\mathcal{H}}$]\label{def:pH}
Let $\mathcal{H}$ be a set of
digraphs.
A digraph $G$ is $\mathcal{H}$-free if for every $H \in
\mathcal{H}, $ $G$ does not contain any
subgraph that is isomorphic to $H$.
The monotone property $P_{\mathcal{H}}$
contains all digraphs that are $\mathcal{H}$-free, and
$P_{\mathcal{H}_n}$ contains all $n$-vertex (di)graphs in
$P_{\mathcal{H}}$.
Similarly, we denote by $P^*_{\mathcal{H}}$ the $\rm{hereditary}$
property that is defined by being $\mathcal{H}$-free as $\rm{induced
~ subgraphs}$ and $P^*_{\mathcal{H}_n}$ the set of $n$-vertex
(di)graphs in $P^*_{\mathcal{H}}$.
\end{definition}
\begin{definition}
[bounded-size collections]\label{def:size}
Let $\mathcal{H}$ be a set of (di)graphs. We call
$\mathcal{H}$ a $r$-set if every member $H \in \mathcal{H}$ has
at most $r$ vertices.
\end{definition}
\begin{remark}\label{rem:34}
$ $
\begin{itemize}
\item A natural example of monotone decreasing
graph property is a property $P_{\mathcal{H}}$ that is defined by a family of forbidden
subgraphs $\mathcal{H}$. It is immediate from the definition that every monotone
graph property is defined by a family of forbidden subgraphs but this
family may be infinite.
Recall that $P = \cup_{n \in \ensuremath{\mathbb N}} P_n$ is monotone if and
only if $P_n$ is monotone for every $n$. Namely, being monotone is
defined for every $n$ separately. In this respect being monotone is
not a `global' feature of $P$ but rather a feature
of the individual $P_n, ~ n \in \ensuremath{\mathbb N}$. In what follows it will important
to us how the individual monotone properties $P_n, ~ n \in \ensuremath{\mathbb N}$ are
defined. Obviously for any fixed $n$, $P_n$ is defined by an
$r$-set of forbidden subgraphs, but $r$ may depend on $n$.
To make this clearer, consider the property of being acyclic. This
property is defined by forbidding all di-cycles, which is an infinite
family. For the individual slices $P_n, ~n \in \ensuremath{\mathbb N},$ the
corresponding family although finite, it is {\em not} a $r$-set
unless $r \geq n$. An example of slightly different nature is that of
the monotone property that contains the digraphs that are not
Hamiltonian. For every $n \in \ensuremath{\mathbb N},$ $P_n$ is defined by one forbidden
subgraph (the simple directed $n$-cycle). Thus $P_n$ is defined by a $n$-set of forbidden subgraphs but for no fixed $r$,
$P_n$ can be defined by an $r$-set for every $n$.
This
distinction will become important in our characterization results. It
will turn out that the strongly-testable monotone properties are
tightly related
to properties that are defined by $r$-sets of forbidden
subgraphs for $r$ that is independent of $n$.
\item For family $\mathcal{H}$ of forbidden digraphs, the monotone property of
being $\mathcal{H}$-free is determined by the minimal members of
$\mathcal{H}$ (w.r.t edge deletions). That is, if for $H,H' \in
\mathcal{H}$ it holds that $H$ is a subgraph of $H'$, then being
$\mathcal{H}$-free is identical to being $(\mathcal{H} \setminus
\{H'\})$-free.
\item Hereditary (di)graph properties are very natural in graph theory. It
is immediate from the definition that a property is hereditary if and
only if it is defined by a
collection (possibly infinite) of forbidden {\em induced} subgraphs. E.g.,
the property of not containing an induced (di)cycle of length
$4$, and the property of being bipartite (that is expressed in this
case as not containing an odd size cycle). Both these properties are
monotone and
hereditary.
Hereditary properties are not necessarily monotone, and monotone
properties are not necessarily hereditary. Further, the feature of
being hereditary, unlike being monotone, depends on the entire
property $P =\cup_{n \in \ensuremath{\mathbb N}} P_n$ and cannot be defined for a single
$n$-slice $P_n$.
\end{itemize}
\end{remark}
\noindent
{\bf Testers: }
We define here $1$-sided error testers for digraph properties in the
$F(d)$-model.
\begin{definition} [$1$-sided error $\epsilon$-test for a digraph
property $P$, $F(d)$-model]
A $1$-sided error test for a digraph property $P$ is a randomized algorithm
that gets two parameters, $n = |V(G)|$ and a distance
parameter $\epsilon > 0$. It accesses its input graph
via vertex queries (Definition \ref{def:f-model}), and
satisfies the following two conditions.
\begin{itemize}
\item It accepts every $n$-vertex digraph in $F(d)$ that belongs to $P$ with probability $1$.
\item It rejects every $n$-vertex digraph that is $\epsilon$-far from $P$ with probability at least $1/2$.
\end{itemize}
\end{definition}
The query complexity of the test is the maximum number of queries it
makes for any input
graph (in $P$ or not in $P$) and for every run. Hence the query
complexity is a function of $n$ and $\epsilon$.
\noindent
{\bf A note on the definition of testers:}
A test
for a graph property $P$ is formally an infinite set of tests
$\{T(\epsilon,n)\}_{n \in \ensuremath{\mathbb N}, \epsilon \in (0,1)}$, where
$T(\epsilon,n)$ is a test for $P_n$ and distance parameter $\epsilon$. Namely, we deal here with a
non-uniform model of computation. We often use the term
$\epsilon$-test to emphasize that the test is designed for an error
parameter $\epsilon$. This will be of special importance in this
paper, as for different distance parameters, the test will
behave differently. We are
interested, as usual, in the query complexity $q$ as a function of
$\epsilon$ and $n$. Note further that since our models are
parameterized by $d$, the query complexity (or even the fact whether
a property is testable in the corresponding $d$-bounded degree model)
may depend on $d$. We may state the query complexity dependence on $d$ but this
is of no particular importance in this paper.
\begin{definition}[strong-testability]
Let $Q:(0,1) \mapsto \ensuremath{\mathbb N}$. If a property $P$ has an $\epsilon$-test whose query complexity
on every $n$-vertex graph is bounded by $Q(\epsilon)$, we say that $P$
is $\epsilon$-strongly-testable. If $P$ is
$\epsilon$-strongly-testable for every $\epsilon \in (0,1)$ we
say that $P$ is {\em strongly-testable}.
\end{definition}
\subsection{Configurations - the $F(d)$-model}
The following definition of {\em configuration} is of major importance in
this paper. The motivation behind the definition
is that a configuration is what a tester discovers after making some
queries to the graph. It will turn out that the configuration that a
tester discovers contains
{\em all} the information that is
used by the tester in order to form its decision.
\begin{definition} [Configuration, $F(d)$-model]\label{def:conf}
A configuration is a pair $C=(H,L)$, where $H=(W,F)$ is
a $d$-bounded-out-degree graph, and $L$ is a function $L:
W \rightarrow \{\mathrm{developed}, \mathrm{frontier}\}$. The out-degree of
every frontier vertex is $0$.
\end{definition}
Consider a run of a tester on a graph $G$. The tester discovers {\em all} (the at most $d$)
outgoing neighbours of every queried vertex. At the end of the run,
after making $q$ queries, the tester discovers a subgraph $H$ of $G$. $H$
contains the $q$ vertices that are queried; these correspond
to the $developed$ vertices in the configuration it discovers. $H$ may also
contain vertices that are neighbours of queried vertices but that were
not themselves queried. These vertices are the $frontier$ vertices. A frontier vertex that is
discovered by the tester and was not queried may
have outgoing neighbours, but the
corresponding edges (the forward edges from the frontier vertex) will
not be discovered by the tester. Consequently, the out-degree of a
frontier vertex
in the discovered configuration is $0$. In contrast, all forward edges of a
developed vertex are discovered.
We now make the above formal using the defintion below.
\begin{definition} [$C$-Free, $F(d)$-model]\label{def:C-free}
Let $C=(H,L)$ be a configuration, where $H=(W,F)$ a digraph and $L: W \rightarrow
\{developed, frontier\}$. Let $G=(V,E)$ be a
digraph in the $F(d)$-model.
We say that $G$ has a $C$-$\rm{appearance}$ if there is an injective mapping $\phi : W \rightarrow
V$ with the following two properties:
\begin{itemize}
\item $\forall v,u \in W$ and $L(v)=\rm{developed}$,
$(v,u) \in F$ $~\rm{ if ~ and~ only~ if}~
(\phi(v),\phi(u)) \in E$.
\item For every developed $v,$ if $(\phi(v), x) \in E$
then $\exists u\in W, \phi(u)=x$.
\end{itemize}
We say that $G$ is $C$-free if $G$ has no $C$-appearance.
\end{definition}
The notion of configuration (using slightly different terms) appears also in \cite{CSS09,GR09}.
Let $C=(H,L)$ a configuration with $D \subseteq V(H)$ being the
developed vertices. Definition \ref{def:C-free} implies that if $G$ has
a $C$-appearance on a vertex set $V' = \phi(V(H))$, with $\phi$ being
the mapping as in the definition, then $G[\phi(D)]$ is isomorphic to
$H[D]$. Namely
$H$ induces an isomorphic digraph on its developed vertices as
$G$ does on the vertices that are the images of the developed
set of vertices $D$. Further, the 2nd requirement in
Definition \ref{def:C-free} asserts that for every $v \in D$, all
forward edges of $\phi(v)$ in $G$ are the `images of edges' in
$H$. It is not necessarily that
$G[V']$ is isomorphic as an induced subgraph to
$H$. This is since there might be an edge $(x,y) \in
G[V(H')]$ that is not in $H$. This can happen only if
$x$ is an image of a frontier vertex.
To exemplify Defintion \ref{def:C-free} further, consider $C=(H,L)$, where $H$ is the directed
$2$-star and the center is the only developed
vertex in $H$. A digraph $G$ has a $C$-appearance if and only if it has a vertex $v'$ with
{\em exactly} two outgoing neighbours $u_1',u_2'$. There could be
an edge $(u'_1,u_2') \in G$ and hence the subgraph that $G$ induces on
$\{v',u_1',u_2\}$ might not be isomorphic to $H$. There could also be an
edge $(x',v') \in E(G)$. However, there cannot
be an edge $(v',y) \in E(G)$ where $y \notin \{u_1, u_2\}$.
We sum up this discussion with the following obvious fact.
\begin{fact}
\label{fact:f3}
Let $C=(H,L)$ be a configuration and $G$ a digraph (all with
respect to the $F(d)$-model). Then:
\begin{itemize}
\item If $G$ has a $C=(H,L)$-appearance then $G$ contains $H$ as a subgraph.
\item If $G[V']$ is isomorphic to $H$ as an
induced subgraph, then a subgraph of $G$ that is obtained by
deleting $\Gamma^+(V')$ in $G$ has a $C$-appearance (for the
given $L$).
\end{itemize}
\end{fact}
Finally, looking towards a characterization theorem, it would be of use if we
could restrict the behaviour of possible testers to ``canonical'' ones. This proved useful
in the dense graph model in \cite{GT01} and it is of similar flavour (and
simpler) here. It was already done in \cite{GR09} for undirected
$d$-bounded degree graphs and the extension to directed graphs (in
both models) is straightforward. We state it here in order to
be consistent with our notations.
\begin{definition}[$r$-disc around a vertex, $F(d)$-model]
Let $G$ be a digraph and $r \in \ensuremath{\mathbb N}$. The $r$-disc around $v \in
V(G),$ denoted $~D(v,r)$, is the subgraph of $G$ that is induced by all vertices $u$
for which there is a path from $v$ to $u$ of length at most
$r$.
\end{definition}
We note that a tester can discover the $r$-disc around a given
vertex $v \in V(G)$. This is done by making a
`BFS-like' search from $v$, where at each step the tester queries the
next first discovered but not yet queried vertex that is of distance less than
$r$ from $v$. Discovering $D(v,r)$ takes at most $d^{r}$ queries for a
graph in the $F(d)$-model. It is useful to consider such a
procedure as an augmented query, motivating the following
definition.
\begin{definition} [$r$-disc query, $F(d)$-model]
An $r$-disc query is made by specifying a vertex $v \in V(G)$
for which the answer is the $r$-disc
around $v$.
\end{definition}
\begin{definition} [canonical-testers]\label{def:canonical}
A $(r,q)$-canonical tester for a graph property $P$ is a
tester that chooses $q$ vertices uniformly at random
$\{v_1, \ldots, v_q \}$. It then makes an $r$-disc query around
$v_i,$ for $ i=1, \ldots, q$. Then, depending only on the configuration
it sees and possibly on $n$ (but not the order of the queries, or
the internal coins) it makes its decision.
\end{definition}
The following result \cite{GR09}, shows that
strongly-testable properties can be tested by canonical-testers\footnote{In \cite{GR09} it is done only for undirected graphs,
but the generalization to directed graphs in both models is straightforward.}.
\begin{thm} \label{thm:canonical}
Let $T$ be a
$1$-sided error
$\epsilon$-test for a digraph property $P$ in the
$F(d)$-model. If the query complexity of $T$ is bounded by $q$ then there is
a $(q,q)$-canonical tester
that is a $1$-sided error $\epsilon$-test for $P$.
\end{thm}
Note that a $(r,q)$-canonical-tester is a {\em
`non-adaptive'} algorithm with respect to $r$-disc queries.
\ignore{
\begin{proof}[Sketch]
The proof follows the footsteps of \cite{GT01} where it is shown that
if $P$ has a 1-sided error tester of query complexity $q$, then $P$
has a canonical-tester that has 1-sided error and query complexity
${2q \choose 2}$.
The test $T$ makes at most $q$ queries adaptively, and then based on
the answers it gets, and in addition depending possibly on the names of the vertices,
the order of the queries, the internal coins, and $n$- the graph
size, it makes its decision.
When $T$ makes a query to a vertex $v$, it might be that $v$ is
already discovered by previous answers (or queries), in which case we
call $v$ old. Or it might be
that $v$ is a new vertex that did not appear in queries or answers of
previous queries. We call $v$ new in this case.
We first simulate $T$ by an {\em adaptive} tester $T_1$ that acts as
follows: it first picks {\em adaptively} a set of $q$ vertices, and
perform a $q$-disc query on each. Obviously, such $T_1$ can be
constructed from $T$. Each time that $T$ picks a new vertex, $T_1$
picks the same new vertex (with the same probability), and performs a
$q$-disc query around it. Obviously, queries to old vertices can be
simulated having the information of the disc-queries. Finally, the
decision $T_1$ will take is the same that $T$ takes on the specific
run that is simulated.
We now simulate $T_1$ by a test $T_2$ that makes its $q$ disc-queries
to uniformly chosen random vertices. To do this we chose a
permutation on $n$ elements $\pi \in_R
\mathcal{S}_n$ uniformly at random. Then on an input graph $G$ on $n$
vertices, it behaves just like $T_1$, but
on $\pi(G)$. Namely, if $T_1$ makes a disc-query to a new vertex $v$ at a certain stage $T_2$ will
query $\pi(v)$. Finally, the decision that
$T_2$ will take on the resulting run is the same as $T_1$ would on the
information that is discovered. Namely, if one thinks of
$T_1$ as a distribution over deterministic trees, then each tree (with
its corresponding probability) induces $n!$ trees by choosing the extra
randomness, and behaving as described above.
\begin{comment}
The examination of the rejection probability of $T_2$ is as follows:
For $G \in P$, $T_1$ accepts with probability 1, on every run of it. As any permutation of $G$, $G'=\pi(G)$ is isomorphic to $G$, it is also in $P$ and hence to be accepted with probability 1. So all together, $G$ is accepted with probability 1.
For $G$ that is far from $P$, for every $\pi$, $\pi(G)$ is also far and has to be rejected by $T_1$. Consider a certain path in $T_1$. Note that such a path is a sequence of vertices around which a disc query is made. The sequence is $v_1, v_2, ... , v_q$, where $v_i$ may depends on what is seen (including names of neighbours, and type of subgraphs: this is adaptivity), in the discs around $v_1, ... , v_{i-1}$.
Consider one deterministic tree $T$, in the collection of trees defined by $T_1$. Every $G$ follows exactly one path (deterministically) in $T$. We call the value (either "1" for accept, or $0$ for reject) by $path(G,T)$. Then for any fixed $\pi$:
\[Prob(T_1 accepts \pi(G)) = \sum_{T} pr(T) \cdot path(\pi(G),T) < 1/3\] as $G'$ is isomorphic to $G$.
Summing this up for every $\pi$, we get,
\[1/3 > \sum_{\pi} Pr(\pi) \cdot \sum_{T} pr(T) \cdot path(\pi(G),T) = \sum_{T} pr(T) { \sum_{\pi} pr(\pi) \cdot path(\pi(G),T) }\]
But the sum in the curly brackets (\{...\}) on the right hand side, is just the probability that the certain $T$ accept the average of $pi(G)$, over $G$, which is what happens at $T_2$, and the sum over all $T$, is just the probability that $G$ is accepted by $T_2$.
\end{comment}
It is easy to
see that $T_2$ is still a test for $P$, and further, if $T_1$
is a $1$-sided error test, then so is $T_2$. The reason is that
the probability that $T_2$ accepts $G$ is just the average
(according to the uniform distribution on $\pi \in S_n$), of the
acceptance probability of $G_{\pi}$, the permuted graph according to
$\pi$ (as this is true for every branch of $G_1$). However, since the property $P$ is a digraph
property, it is invariant under permuting as above. Hence, $T_2$
will accept any $G \in P$ with probability that is at
least the minimum acceptance probability of $T$ for any $G' \in
P$, and reject $G$ that is $\epsilon$-far from $P$ with the minimum rejection probability
of $T_1$ for any $G'$ that is $\epsilon$-far from
$P$.
We further note that the distribution induced on the vertices that are
chosen by $T_2$ to be queried is the uniform over all $q$-size subsets
of vertices. This can be formally proven by induction on the order of
vertices that are queried, but is also immediate from the fact that at
any step, the probability that a new $v$ is queried, is identical on
all yet non queried vertices.
Hence, $T_2$ already makes its queries as a canonical-tester does. In
particular it is {\em non adaptive}. We
now get rid of the possible dependence of the decision $T_2$ reaches,
on the labels, order, or internal coins. This is argued exactly as
in \cite{GT01}, since the symmetrization implies that
decisions is invariant to order or labels (removing dependence
on coin flips is trivial for $1$-sided error test; $T_2$ accepts on
seeing a configuration only if it accepts w.p $1$ on seeing this
configuration. For two sided error, a slightly more involved argument
is needed, and we should assume that the error probability of $T_2$ is
less than $1/6$ for $2$-sided error, see details in \cite{GT01}).
\end{proof}
}
\section{Our main results}\label{sec:results}
We consider in what
follows the $F(d)$ model (for constant $d$). The $F(d)$-model is the more natural model from the algorithmic point
of view, being consistent with the standard data structures for
directed graphs. It contains a strictly larger set of graphs than the
$FB(d)$-model (as the in-degree is not
bounded).
From the property testing perspective it is more restricted
algorithmically due to the limited
access to the graph.
We prove here that the strongly-testable {\em monotone} graph
properties are these that are
close (in the sense of Definition \ref{def:dist}) to be expressed by
an $r$-set of forbidden
subgraphs that have some additional connectivity
requirements. For hereditary properties the results are essentially the
same where forbidden subgraphs are replaced
with forbidden induced subgraphs.
We need the following definitions.
\begin{definition} [Component]\label{def:component}
Let $H=(V,E)$ be a directed graph. A subset $V' \subset V$
defines a {\rm component} of $H$, if by disregarding the
directions of the edges of $H$, $V'$ induces a
connected component in the resulting undirected graph.
We say in this case that $H[V']$, the directed subgraph of $H$
that is induced by $V'$, is a component of $H$.
\end{definition}
We note that Definition \ref{def:component} is not a standard
graph-theory term, and we warn the reader not to confuse it with strongly
connected components of the digraph. We are concerned with graphs of
multiple components as the forbidden graphs that define a monotone
property might be such. E.g., let $C_k$ be the directed $k$-cycle, and
consider the property $P_1$ of being $C_3$-free, $P_2$ the property of
being $C_4$-free, $P_3$ the property of being $\{C_3,C_4\}$-free, and
$P_4$ the property of being free of the single graph $H$ that is a
vertex disjoint union of $C_3$ and $C_4$. Namely, a graph is not in
$P_4$ if it has a $C_3$ subgraph and a disjoint
$C_4$ subgraph. All properties $P_i, i=1,2,3,4$ are distinct. The properties
$P_1,P_2,P_4$ are defined by one forbidden graph. $P_3$ is defined by
two forbidden graphs. The forbidden graphs defining $P_1,P_2,P_3$ have
one component each, while the single forbidden graph defining $P_4$
has two components.
\begin{definition} [Rooted digraph]
A digraph $H$ is {\rm rooted} if every
component $H'$ of $H$ has a vertex $v$ such that
for every $u \in V(H')$, there is a di-path from $v$ to $u$ in $H'$.
\end{definition}
We note that a digraph can have many roots. In particular, if it is
strongly connected then every vertex of it is a root.
The significance of $v$ being a root in a component of size at most
$r$ is that
making an $r$-disc query around $v$ will discover the whole component that
contains $v$.
Our main theorem, characterizing the strongly-testable monotone
properties is the following.
\begin{thm}
\label{thm:main-mon}
Let $P = \cup_{n \in \ensuremath{\mathbb N}}P_n$ be a monotone digraph property in the
$F(d)$-model. Then $P$ is strongly-testable $\rm{if~ and~ only~ if}$ there is
a function $r: (0,1) \mapsto \ensuremath{\mathbb N}$ such that for any $\epsilon >0$
and $n \in \ensuremath{\mathbb N},~$
there is a
$r(\epsilon)$-set of rooted
digraphs $\mathcal{H}_n $ such that the property
$P_{\mathcal{H}_n}$ that consists of the
$n$-vertex digraphs that are $\mathcal{H}_n$-free, satisfies the
following two conditions:
\noindent
(a) $P_n \subseteq P_{\mathcal{H}_n} $\\
(b) $P_{\mathcal{H}_n} $ is $\epsilon/2$-close to $P_n$.
\end{thm}
We note that the sets $\{\mathcal{H}_n \}_{n \in \ensuremath{\mathbb N}}$ in Theorem
\ref{thm:main-mon} may depend on $\epsilon$ (as the
bound $r(\epsilon)$ depends on
$\epsilon$).
A Similar theorem for hereditary properties is the following.
Let $\mathcal{H}$ be a set of digraphs. Recall the definition of the
property
$P^*_{\mathcal{H}}$ from Definition \ref{def:mon-hered-prop}. We
denote by $P^*_{\mathcal{H}_n}$ the set of $n$-vertex digraphs in $P^*_{\mathcal{H}}$.
\begin{thm}
\label{thm:main-hered}
Let $P$ be an hereditary digraph property in the
$F(d)$-model. Then $P$ is strongly-testable $\rm{if~ and~ only~ if}$
there are functions
$r: (0,1) \mapsto \ensuremath{\mathbb N}$ and $N:(0,1) \mapsto \ensuremath{\mathbb N}$ such that for any $\epsilon >0$
there is a $r(\epsilon)$-set of rooted
digraphs $\mathcal{H}$ such that for every $n \geq N(\epsilon),$
$P^*_{\mathcal{H}_n}$ satisfies the
following two conditions:
\noindent
(a) $P_n \subseteq P^*_{\mathcal{H}_n} $\\
(b) $P^*_{\mathcal{H}_n} $ is $\epsilon/2$-close to $P_n$.
\end{thm}
\noindent
{\bf Some comments on the results:}
\begin{itemize}
\item
The lower bound $n \geq N(\epsilon)$ in Theorem
\ref{thm:main-hered} is essential and not an artifact of the proof.
Consider the $F(1)$-model and let $C_k$ be the directed cycle of
size $k$. Let $P$ be the property that contains an $n$-vertex graph
if it is free of all cycles $C_k$ for $k \leq \sqrt{n}$ (as induced
subgraphs). This is a strongly-testable hereditary (and monotone) property as
asserted by Theorem \ref{thm:main-hered} and the
set $\mathcal{H}$ that contains all cycles up to size
$\frac{1}{2\epsilon}$, for $N(\epsilon) = 4/\epsilon^2$.
However, for any possible $r$-set $\mathcal{H'}$ for which $P
\subseteq P^*_{\mathcal{H'}}$, for $P$ to $\epsilon$-close to $P^*_{\mathcal{H'}}$, $\mathcal{H'}$ should contain all cycles of
size at most
$1/\epsilon$. But then $P_n \subseteq P^*_{\mathcal{H'}}$
only for $n \geq 1/\epsilon^2$.
\item The `only if' direction of Theorem \ref{thm:main-mon} is
restated as Theorem \ref{thm:main-ff2}. In Theorem
\ref{thm:general} we generalize Theorem \ref{thm:main-ff2} by replacing
the forbidden set of digraphs $\mathcal{H}_n$ with a finite set of
forbidden configurations (see Definitions \ref{def:conf} and
\ref{def:C-free} ). In turn, this stronger (and more immediate
theorem) is true for any strongly-testable digraph property (rather
than just for monotone). Thus, Theorem \ref{thm:general} gives a
necessary condition for {\em any} graph property to be $1$-sided
error strongly-testable. For all we know, this could also be a
sufficient condition. This will be further discussed in Section
\ref{sec:concl}.
\item
One may ask whether
the extra restriction that $P_{\mathcal{H}_n}$ (or $P^*_{\mathcal{H}_n}$ in
case of hereditary property) is $\epsilon/2$-close to
$P_n$ rather than just being $P_n$ is a necessity or rather just an
artifact of our proof. The answer is that this is needed.
Indeed, as mentioned in
the introduction, acyclicity is not strongly-testable in
the $F(d)$-model for large enough $d$, even by $2$-sided error testes
\cite{BR02}.
However,
it is easy to see that directed acyclicity is $1$-sided error strongly-testable in the $F(1)$-model.
\ignore{ Indeed a $1$-bounded degree digraph is composed of a
collection of vertex disjoint paths and simple cycles. Hence if $G$
is $\epsilon$-far from acyclic, it must contain at least
$\epsilon n$ disjoint cycles. It follows that there are at least
$\epsilon n/2$ cycles of length at most $2/\epsilon$. Hence,
sampling a vertex at random and scanning the $2/\epsilon$ disc
around it will discover such cycles with high probability. This
immediately implies a $1$-sided error test of query complexity
$O(1/\epsilon^2)$. }
Acyclicity, while monotone, can not be defined by
an $r$-set of forbidden subgraphs in the $F(1)$-model for any fixed
$r$. Rather, it is
$\epsilon$-close (in the $F(1)$-model) to be $\mathcal{H}$-free as induced graphs for the
$\frac{1}{\epsilon}$-set $\mathcal{H}$ that contains all cycles of
size at most $1/\epsilon$.
\end{itemize}
\section{Proofs of the main results}\label{sec:F}
Here we prove Theorem \ref{thm:main-mon} and Theorem \ref{thm:main-hered}.
We will start by proving the `if' directions for both
theorems in Section \ref{sec:if}. Section \ref{sec:mon-inverse} contains the proofs of the
`only-if' parts.
\subsection{Monotone properties and hereditary
properties that are strongly-testable}\label{sec:if}
Theorem \ref{thm:main-mon} states that if $P=\cup_n P_n$ is $\epsilon$-close
to $P_{\mathcal{H}_n}$ for an $r$-set of rooted digraphs $\mathcal{H}_n$
then $P$ is $1$-sided error strongly-testable. We start by proving that
the monotone property
$P_{\mathcal{H}}$ itself is strongly-testable for a fixed $r$-set $\mathcal{H}$.
Let $\mathcal{H}$ be a $r$-set of digraphs and
$P$ the monotone property that contains the
digraphs that are $\mathcal{H}$-free. Remark \ref{rem:34} implies that we may assume in what follows that
$\mathcal{H}$ does not contain two graphs such that one is a subgraph of
the other. We also note that if $\mathcal{H}$ contains a graph that is an isolated vertex (or a set of
isolated vertices) then $P_\mathcal{H}$ becomes trivial (empty for
large enough $n$). We assume in what follows that the above does not happen.
We start with the following preliminary proposition for the subcase of
Theorem \ref{thm:main-mon}, where $P = P_{\mathcal{H}}$.
\begin{proposition} \label{theorem:directed-out:1} Let $\mathcal{H}$ be a
$r$-set of rooted digraphs and $|\mathcal{H}|=t$. Then the monotone
property $P = P_{\mathcal{H}}$ has a $1$-sided error $\epsilon$-test
in the $F(d)$-model, making $O(t r^2 d^{r+1} \ln r/\epsilon)$
neighbourhood queries.
\end{proposition}
\begin{proof}
The top level idea is simple, and a similar idea was used in
\cite{GR02}: Suppose that a digraph $G$
is $\epsilon$-far from being ${\mathcal{H}}$-free.
We will show that there is a
large set of vertices, each being a root in an
$H$-appearance in $G$ for some $H \in \mathcal{H}$. Hence
sampling of a random vertex and scanning the $r$-disc around it will
find a forbidden $H$-appearance in $G$. Some extra care should be taken for
disconnected forbidden subgraphs.
Formally, we prove that the following test $T(\epsilon, n)$ is a test
for $P_{\mathcal{H}}$.
{\bf $T(\epsilon, n)$: }
Repeat for $\ell = (tr^2 d/\epsilon) \cdot 2\ln r$
times independently: Chose a vertex $v \in_R V(G)$ uniformly
at random and
make an $r$-disc query around $v$.
If some $H \in
\mathcal{H}$ is found as a subgraph in the discovered subgraph
of $G$ then reject. Otherwise accept.
Obviously the test accepts with probability $1$ every graph that is
$\mathcal{H}$-free. Further, the claimed complexity is clear.
Assume that $G$ is a digraph on $n$ vertices that is $\epsilon$-far from
$P_{\mathcal{H}}$. We claim that $G$ contains at least $\epsilon n/r$ edge
disjoint subgraphs, each that is isomorphic to some
$H \in \mathcal{H}$. This is so as let $F$ be any maximal edge
disjoint collection of
subgraphs of $G$, each that is isomorphic to some $H \in
\mathcal{H}$. By deleting all outgoing-edges
that are adjacent to vertices in $F$ (at most $|F|\cdot
r \cdot d$) none of the subgraphs in $F$ is a forbidden
subgraph anymore. Further, no new forbidden
subgraph is created (by the assumption that no graph in
$\mathcal{H}$ is a subgraph of another graph in $\mathcal{H}$).
Therefore, $G$ becomes $\mathcal{H}$-free after
deleting these edges. We conclude that
$|F|\cdot r\cdot d \geq \epsilon nd$.
Fix such a collection of subgraphs $F$. We deduce that there is some fixed
graph $H \in \mathcal{H}$ that is isomorphic to at least
$\frac{|F|}{t} \geq \epsilon n/(tr)$ of the digraphs in $F$. Fix such
$\epsilon n/(tr)$ edge disjoint subgraphs in $G$, which we refer to as $F'$.
Assume first that $H$ is composed of one single rooted
component. Since the subgraphs in $F'$ are edge
disjoint, a root vertex $v$ can appear in at most $d$ such
distinct subgraphs (on account that it must have at least one
forward edge in each such appearance). We conclude that there are at least
$\frac{|F'|}{d} \geq \frac{\epsilon n}{t r d}$ distinct
vertices, each being a root in an $H$-appearances in $G$. Hence, with probability
$\frac{\epsilon}{trd}$ a random vertex $v$
will be one of these roots. Assuming that such a
vertex $v$ is chosen by $T(\epsilon,n)$, then making the
$r$-disc query to $v$ will discover
the corresponding $H$-appearance. Thus the failure probability is bounded by
$(1-\frac{\epsilon}{trd})^\ell < 1/2$.
Finally, assume that $H$ is composed of several rooted
components. Since $|H| < r$, $H$ is composed of at most $r$
components $C_1, \ldots C_a,~ a \leq r$. In this case, finding
$a$ vertices $v_1, \ldots v_a$, with the $i$th being the root
of a subgraph isomorphic to $C_i$ will discover an isomorphic
copy of $H$ in $G$. The probability of sampling a root of a
component of type $C_i$ is at least
$\frac{\epsilon}{t r^2 d}$. The union-bound implies that the
probability that there exists some type that we don't sample a
root of is at most
$a\cdot (1-\frac{\epsilon}{t r^2 d})^{\ell} \le 1/2$. This
concludes the proof.
\end{proof}
It is assumed implicitly in
Proposition \ref{theorem:directed-out:1} that $\mathcal{H}$ is a
collection of digraphs in the $F(d)$-model. Therefore, the fact that $\mathcal{H}$
is an $r$-set implies that $t=|\mathcal{H}|$ is bounded in terms of $r$
(exponentially). Although not of prime interest for this paper,
we still give the above tighter dependence on $t$ because $t$ could be much smaller than
the worst case bound.
For hereditary properties a Proposition analogous to Proposition \ref{theorem:directed-out:1} will be stated. In this
case being $\mathcal{H}$-free as subgraphs is replaced by being free
as {\em induced}
subgraphs. However, unlike the easier case of monotone properties, we
can't
assume that if $G$ is $\epsilon$-far from the property, then it
contains many vertices that are roots of
$\mathcal{H}$-appearances. The reason is that deleting
edges in an $\mathcal{H}$-appearance in $G$ may create a new
$\mathcal{H}$-appearance\footnote{It could be true that for every
$\mathcal{H}$, if
$G$ is far from being $\mathcal{H}$-free as induced subgraphs, then there are many
$\mathcal{H}$-appearances in $G$, but we
do not have a proof nor a counter example for this.}. We
use a different argument.
\begin{definition}
Let $\mathcal{H} $ be a set of
digraphs. We say that $H \in \mathcal{H}$ is essential if the
digraph $H$ is $(\mathcal{H} \setminus \{H\})$-free as induced
subgraph. Namely, $H$ does not contain as an induced subgraph any member of
$\mathcal{H}$ except for itself. If every $H \in \mathcal{H}$ is
essential, we say that $\mathcal{H}$ is non-redundant.
\end{definition}
\begin{proposition} \label{thm:main-ff0.5} Let
$\mathcal{H}$ be a non-redundant $r$-set of
rooted digraphs.
Then the
hereditary property of being $\mathcal{H}$-free as induced subgraphs
is $1$-sided error strongly-testable in the $F(d)$-model.
\end{proposition}
The following lemma is folklore. We state it for completeness.
\begin{lemma}[sampling a random edge]
\label{lem:sample}
Let $G = (V;E)$ be a graph in the $F(d)$-model with $|E(G)| \geq \epsilon nd$.
Then, with probability at least $\epsilon/d$, the following randomized algorithm
outputs an edge $e \in E$ that is distributed uniformly in $E$,
and outputs a special failure indication otherwise.
The algorithm sample a vertex $v \in V (G)$ uniformly at random,
queries this vertex to obtain $\Gamma^+(v)$,
and outputs each edge going out of $v$ with probability $1/d$.
In other words, letting $k=|\Gamma^+(v)|$, the algorithm stops
indicating failure with probability $1-\frac{k}{d}$,
and otherwise it samples $u\in\Gamma^+(v)$ uniformly at random and outputs $e = (v,u)$.
\end{lemma}
\begin{proof}
Since $|E(G)| \geq \epsilon nd$ there are at least $\epsilon n$ vertices
each with outdegree at least $1$. Let this set be $V_1$. The algorithm will output an edge in
the case it chooses $v \in V_1$, and that it does not choose to
indicate failure after choosing $v$. This occurs with probability at least $\epsilon /d$.
The algorithm outputs a fixed edge $e=(v,u)$ with probability
$Pr(e) = \Pr(v) \cdot \frac{deg(v)}{d} \cdot \frac{1}{deg(v)}
= \frac{1}{|V_1|d}$. Since this is identical for all edges, the
algorithm induces the uniform distribution on $E(G)$.
\end{proof}
\begin{proof}[of Proposition \ref{thm:main-ff0.5}]
For this proof, we abbreviate ``$H$-appearance'' and
``$\mathcal{H}$-appearance'' for $H$-appearance as {\em induced}
subgraph, and $\mathcal{H}$-appearance as induced subgraphs, respectively.
We may assume that $\mathcal{H}$ does not include an isolated vertex
as a member, as otherwise, being $\mathcal{H}$-free is an empty
property. Further, we may assume that for no $H \in \mathcal {H},$
$H$ contains an isolated vertex. As otherwise, we replace such $H$
with $H'$ that is obtained from $H$ by removing the isolated
vertices. Obviously, for $n$ large enough, $G$ contains $H$ as an
induced subgraph if and only if $G$ contains $H'$ as induced
subgraph.
The test samples some vertices and scans the $r$-disc around each. It
rejects only if it finds a $\mathcal{H}$-appearance in the subgraph of
$G$ that it discovers. The
vertex set that is sampled is a set of endpoints of
$\ell= \frac{8td\ln r}{\epsilon }$ random edges. This is done by
calling the algorithm of Lemma \ref{lem:sample} for $4d\ell/\epsilon$
times. Note that the lemma guarantee a success probability of
$\epsilon/d$ per edge query only for graphs with
$|E(G)| \geq \epsilon d n$ edges. In general, these $4d\ell/\epsilon$
calls could result in some random edges or none at all. If less than
$\ell$ edges are produced by the $4d\ell/\epsilon$ calles to the
algorithm in Lemma \ref{lem:sample}, the algorithm will stop and
accept. Thus the overal query complexity is
$O(d^2t \ln r/\epsilon^2)$ neighbourhood queries in addition to
$O(td \ln r/ \epsilon)$ $r$-disc queries.
It is clear that for $G$ that is $\mathcal{H}$-free the test accepts
with probability $1$.
Let $G$ be a digraph on $n$ vertices that is $\epsilon$-far from
being ${\mathcal{H}}$-free as induced subgraphs. Since $G$ must be
$\epsilon$-far from the empty graph, it follows that $|E(G)| \geq \epsilon dn$.
This implies that
with probability at least $7/8$ the $4d\ell/\epsilon$ calls to the
algorithm in Lemma \ref{lem:sample} will indeed produce at least
$\ell$ random edges. In what follows we condition the analysis on the
assumption that indeed $\ell$ random edges are produced.
For simplicity we first analyze the test for the case that each $H \in
\mathcal{H}$ has only one rooted component (i.e, this does not cover,
e.g., the property of being free of a disjoint pair of a di-triangle
and a $4$-cycle). The argument for the general case will be
somewhat harder.
Let $S$ be a maximal set of subgraphs of $G$, each being
an $\mathcal{H}$-appearance, and in which
the {\em forward-edges
of the roots are
disjoint}. For each subgraph in $S$ fix one root vertex. Let
this set of vertices be $R$.
Assume first that $|S| \ge
\frac{\epsilon n}{2}$. Then for an edge $e=(u,v)$, sampled uniformly at
random from $E(G)$, $u$ is a root of an $\mathcal{H}$-appearance with probability at least
$p_1 = \frac{\epsilon}{2d}$. Hence, choosing $\ell$ random
edges will find a vertex that is a root of an
$\mathcal{H}$-appearance with
probability of at least $3/4$.
Suppose now that $|S|\le \frac{\epsilon n}{2}$. Then $|R| \le
\frac{\epsilon n}{2}$ (as we fixed one root vertex per member
in $S$). Let $E^-(R) = \{(u,v) \in G~ | ~ v \in
R \}$.
Assume first that $|E^-(R)| < \frac{\epsilon nd}{2}$. Let $E(R)$
be the set of all edges adjacent to $R$ (both incoming and outgoing
edges). Then
$|E(R)| \le
d |R| + |E^-(R)| <
\epsilon nd$. Therefore deleting all edges
in $E(R)$ results in a subgraph in which the vertices in $R$
become isolated and
all old $\mathcal{H}$-appearances in $S$ will be
destroyed. We claim that the resulting graph $G'$ becomes
$\mathcal{H}$-free. Indeed if $G'[V']$ is isomorphic to
some $H \in \mathcal{H}$, either $G[V']$ is also so,
or it is created by the absence of some old edges that
are deleted. In the first case,
$G[V']$ must share an edge $(u,v)$ with an
appearance in $S$, and where $u$ is a root in both
appearances. This cannot happen as the edge $(u,v)$ is
deleted. For the second possibility, as we delete {\em
all} edges (forward and backwards edges) adjacent to
roots, deleting an edge $(u,v)$ makes $u$ isolated
in $G'$ and hence, by the discusion in the first
paragraph of the proof, $u$ cannot be part of an
$\mathcal{H}$-appearance.
The fact that $G'$ becomes
$\mathcal{H}$-free is in contradiction with
the assumption that $G$ is $\epsilon$-far from being
such, as we have deleted less than $\epsilon dn$ edges.
Hence $|E^-(R)| \geq \frac{\epsilon nd}{2}$.
But then sampling a random edge $e \in E(G)$ will
result in $e=(u,v)$ for which $ v \in R$
with success probability at least $\epsilon/2$. Thus, choosing $\ell$ random edges
implies that we pick a root of an $\mathcal{H}$-appearance with
probability at least $3/4$.
We conclude that in all cases (of sizes of $S$) we find a vertex that is
a root vertex of an $\mathcal{H}$-appearance with
probability at least $3/4$. If this happens, then scanning the $r$-disc
around the endpoints of the sampled edges will
discover the $\mathcal{H}$-appearance. This concludes the
proof for this simple case (in which each $H \in
\mathcal{H}$ has a single rooted component).
\noindent
{\bf The general case:} For the general case, the same argument does
not
work directly. To realize what is the difficulty, assume that a
forbidden graph $H$ consists of two components: a di-triangle and a disjoint $4$-cycle.
Assume also that $G$ is $\epsilon$-far from being $H$-free and that
there is a small
number of $H$-appearances in $G$. Then, similarly to the second
case above, we conclude that $E^-(R)$ is large, where $R$ is the
set of roots of the $H$-appearances. This would mean that we
can find a root vertex in an $H$-appearance by making only a small
number of queries. But what if most of these edges are going into vertices
in di-triangles, and only very few to vertices in $4$-cycles. In order to discover a forbidden
subgraph we also need to discover a $4$-cycle. In the general case
we need to
combine more carefully the several cases of different sizes of $E^-(R)$. This we do as follows:
Let $G$ be a digraph on $n$ vertices that is $\epsilon$-far from being
$\mathcal{H}$-free as induced subgraphs (where we no longer assume that each forbidden
graph in $\mathcal{H}$ has only one component).
For $\mathcal{H} = \{H_1, \ldots ,H_t\},$ let $H_i$ be composed of
disjoint components $H_{i,j}, j=1, \ldots j_i$.
Let $S$ be a maximal set of subgraphs of $G$, each being
an $H_{i,j}$-appearance for some $i,j$, and in which
the {\em forward-edges
of the roots are
disjoint}.
We can write $S = \cup_{i,j} S_{i,j}$ where $S_{i,j}$ contains
the corresponding appearances of $H_{i,j}$ in $G$. Let
$R_{i,j}$ be the set of the
corresponding roots, one per each appearance in $S_{i,j}$, and $\gamma_{i,j} = |E^- (R_{i,j})|$. Note that $i$ ranges over $\{1,\ldots ,t\}$
and $j$ ranges over all possible components types of $H_i$ which is a
number $j_i, ~ j_i \in \{1, \ldots ,r\}$.
For each $i \in \{1, \ldots , t\}$ let $I_i = \{j \in \{1, \ldots
,j_i\}~ | ~ |S_{i,j}| < \delta n=\frac{\epsilon n}{2t} \}
$.
{\bf case (a): }
Assume that for some $i \in \{1, \ldots ,t\}$, for every
$j \in I_i$, $\gamma_{i,j} \geq \frac{\epsilon d n}{2t}$.
In this case, for every $j \notin I_i$, for a random edge
$(u,v)\in E(G)$, $u$ is
going to be a root of an $H_{i,j}$ appearance (namely in
$R_{i,j}$) with probability at least $\delta /d = \frac{\epsilon}{2td}$. In addition, for
every $j \in I_i$, a random edge $(u,v)$ picked uniformly from
$E(G)$ will have $v \in R_{i,j}$ with probability at least
$\frac{\gamma_{i,j}}{d^2n} = \frac{\epsilon}{2td}$ (as $v$ could be a root of at most $d$
distinct members in $S$).
Hence sampling $\ell > 4 \ln r \cdot \frac{2td}{\epsilon}$ random
edges implies that a root in an appearance of
$H_{i,j},$ for every $j \in \{1, \ldots ,j_i\},$ will be found with
probability at least $7/8$.
Calling the sampling algorithm of
Lemma \ref{lem:sample} for $4d \ell /\epsilon$ times results in at least $\ell$ random edges with
probability at least $7/8$. Therefore, the overall success probability in
this case is at least $3/4$.
{\bf case (b): }
If case (a) does not hold, then for every $i
\in \{1, \ldots , t\}$, there is $j(i)\in I_i$ for which
$\gamma_{i,j(i)} < \frac{\epsilon d n}{2t}$. (It could be that for some $i$ there are more
than one $j(i)$ as above; in that case, choose an arbitrary one.)
But then deleting, for every $i
\in \{1, \ldots , t\}$, all edges incident to every
root in $S_{i,j(i)}$ (forward and backward edges), all
$\mathcal{H}$-occurrences in $S$ will be destroyed (as for each $H_i$
we have destroyed
all appearances of $H_{i,j(i)}$ in $S$). Moreover, no new appearances are created by the same
reasoning as in the simple case. Finally, we have deleted at most
$\sum_{i=1}^t d|S_{i,j(i)}| + \gamma_{i,j(i)} < \epsilon dn$ edges which
contradicts the assumption that $G$ is $\epsilon$-far from being $\mathcal{H}$-free.
\end{proof}
We have proved so far that monotone or hereditary properties that are
defined by an $r$-set of forbidden rooted digraphs are strongly-testable. To
prove the `if-part' of Theorems \ref{thm:main-mon} and \ref{thm:main-hered}, we will also show
that properties that are {\em close} to such properties are
strongly-testable. This is done next.
The following is a restatement of the `if-part' of Theorem \ref{thm:main-mon}.
\begin{thmm}
\label{thm:main-ff1}
Let $\mathcal{H}$ be a $r$-set of rooted
digraphs and for $n \in \ensuremath{\mathbb N}$ let $P_{\mathcal{H}_n} $ the monotone property that contains all
$n$-vertex digraphs that are $\mathcal{H}$-free as subgraphs. Let
$P=\cup_n P_n$ be a digraph
property in the $F(d)$-model for which, (a) $P_n \subseteq P_{\mathcal{H}_n} $, and (b)
$P_{\mathcal{H}_n} $ is $\epsilon/2$-close to $P_n$. Then, $P$ is $1$-sided
error $\epsilon$-strongly-testable in the $F(d)$-model.
\end{thmm}
\begin{proof}
By Proposition \ref{theorem:directed-out:1}, for every $\delta >0$ there
is a $1$-sided error $\delta$-test for $P_{\mathcal{H}}$. Let
$\delta= \epsilon/2$ and $T$ be a corresponding $1$-sided error
$\delta$-test for $P_{\mathcal{H}}$. We run $T$ on $G$, accept if $T$ accepts and reject
otherwise. If $G \in P_n$ then since $P_n \subseteq P_{\mathcal{H}}$
the test will accept $G$ w.p. $1$. On the other hand, if $G$ is
$\epsilon$-far from $P_n$, then it must be $\epsilon/2$-far from
$P_{\mathcal{H}}$ as $P_{\mathcal{H}_n}$ is $\epsilon/2$-close to
$P_n$. Hence, $G$ is rejected with probability at least $1/2$.
\end{proof}
We state below the corresponding restatement of `if-part' of Theorem
\ref{thm:main-hered}. Its proof is identical to that of Theorem
\ref{thm:main-ff1},
where we replace
Proposition \ref{theorem:directed-out:1} with Proposition
\ref{thm:main-ff0.5}.
\begin{thmm}
\label{thm:main-ff1.5}
Let $\mathcal{H}$ be a non-redundant $r$-set of rooted
digraphs and for $n \in \ensuremath{\mathbb N}$ let $P^*_{\mathcal{H}_n} $ the hereditary property that contains all
$n$-vertex digraphs that are $\mathcal{H}$-free as induced subgraphs. Let
$P=\cup_n P_n$ be a digraph
property in the $F(d)$-model for which (a) $P_n \subseteq P^*_{\mathcal{H}_n} $ and (b)
$P^*_{\mathcal{H}_n} $ is $\epsilon/2$-close to $P_n$. Then $P$ is $1$-sided
error $\epsilon$-strongly-testable in the $F(d)$-model. $\qed$
\end{thmm}
\begin{remarkp}\label{rem:r18}
$ $
\begin{itemize}
\item
Theorem \ref{thm:main-ff1} is stated in
terms of a fixed family of forbidden digraphs $\mathcal{H}$. However,
since the conditions (a) and (b) in the theorem are in terms of
the slices $P_n$, namely for $n$-vertex graphs, the
family $\mathcal{H}= \{\mathcal{H}_n \}$ may depend on $n$. The only global requirement
of $\mathcal{H}_n$ is that it is an $r$-set, where $r$ is a function
of $\epsilon$ only.
To make this clearer consider e.g., the
property $P$ in the $F(d)$-model that contains every $n$-vertex graph $G$ if
$n$ is even, and contains the digraphs that
do not have a directed $4$-cycle otherwise. $P$ is monotone but
it is not defined
by a single set of forbidden subgraphs. Rather, for
every $n,$ $P_n$ is a slice of a property that is defined in this way. Hence, $P$ is
$1$-sided error strongly-testable.
\item Note that the digraph property $P$ that is asserted to be
strongly-testable in Theorem \ref{thm:main-ff1} is not necessarily
monotone. It is only required that it is close to a monotone
property. In this sense, Theorem \ref{thm:main-ff1} is slightly
stronger than the `if-part' of Theorem \ref{thm:main-mon}. An
analogous
remark also holds for the property $P$ in Theorem
\ref{thm:main-ff1.5}
\item Note that in the characterization theorem, Theorem
\ref{thm:main-hered}, we did not restrict the family $\mathcal{H}$
to be non-redundant. This is not need as it is clearly the case
that $P_{\mathcal{H}}^* = P_{\mathcal{H}'}^*$ for $\mathcal{H}'$
that is obtained from $\mathcal{H}$ by removing all non-essential graphs.
\end{itemize}
\end{remarkp}
\subsection{The `only-if' parts of Theorems \ref{thm:main-mon} and \ref{thm:main-hered}}\label{sec:mon-inverse}
Theorem \ref{thm:main-mon} requires
that the corresponding family $\mathcal{H}$ contains members that are rooted.
We first show why this
restriction is needed. We say that $H \in \mathcal{H}$ is {\em
minimal} if there is not $H' \in \mathcal{H} \setminus \{H\}$ for
which $H'$ is a subgraph of $H$.
\begin{proposition} \label{tester:directed-out:2} Let
$\mathcal{H} = \{H_1, \ldots ,H_t\}$ be a set of forbidden digraphs
and $P_{\mathcal{H}}$ be the corresponding monotone property of $n$-vertex graphs.
If for some minimal
$H \in \mathcal{H}$, $H$ is not rooted, then any 1-sided error
$\frac{1}{d|H|}$-test for $P_{\mathcal{H}}$ makes $\Omega(\sqrt{n})$
queries in the $F(d)$-model.
\end{proposition}
\begin{proof}
Assume that $H \in \mathcal{H}$ is minimal and not
rooted. Set $\epsilon = \frac{1}{d|H|}$.
An $\epsilon$-test for $P_{\mathcal{H}}$ that is $1$-sided
error must discover some $H \in \mathcal{H}$ on any run that
rejects. Hence it is enough to prove that any test that
discovers a $\mathcal{H}$-appearance and makes $o(\sqrt{n})$
queries must have a success probability that is less than
$1/2$ on some $n$-vertex graphs that are $\epsilon$-far from $P_{\mathcal{H}}$.
We use Yao's principle to prove the lower bound. Namely, we construct a
probability distribution $\mathcal{D}$ that is supported on
$n$-vertex digraphs in $F(d)$ that are
$\epsilon$-far from $P_{\mathcal{H}}$.
We then
show that any {\em deterministic} algorithm making
$q <\sqrt{\frac{n}{3|H|}}$ queries fails to find a copy of $H \in
\mathcal{H}$ for more than $1/2$ of the inputs weighted according to $\mathcal{D}$.
Let
$G= (V,E)$ be an unlabelled directed graph on $n$ vertices that is a union of
$\frac{n}{|H|}$ vertex disjoint copies\footnote{If $|H|$ does not divide $n$, we augment
$G$ with at most $|H|-1$ isolated vertices to get an
$n$-vertex graph.}
of $H$ . The
distribution $\mathcal{D}$ is formed by labelling $V$ according
to a random permutation uniformly chosen from the set of all
permutation on $n$ elements. Obviously $\mathcal{D}$ is supported on
$\epsilon$-far graphs. Moreover, the only forbidden subgraphs
in each graph supported by $\mathcal{D}$ are disjoint copies of
$H$. Hence, any deterministic $1$-sided error test with
respect to $\mathcal{D}$
ends correctly only when it finds a copy of
$H$.
Let $A$ be any deterministic algorithm making $q$ queries,
adaptively. Every query made by $A$ is of the form $v \in [n]$, where $v$
is either one
of the vertices that occurred as answers for some prior
queries, or $v$ is a new vertex that was not yet seen.
We will augment the algorithm so that on query $v$, the algorithm receives the entire
subgraph $H_v$ containing all vertices {\em reachable from} $v$ in the copy
of $H$ where $v$ lies. Note that this gives more information to the
algorithm in the form of possibly $|H|-2$ additional vertices
but with at least one vertex $w$ in the $H$-appearance of $v$
that is excluded by the assumption that
$H$ is not rooted.
Hence, if the augmented algorithm does not discover a copy
of $H$ neither does $A$. Note further that the additional information makes
the queries of the first type -- namely, queries to vertices that are
the answers to prior queries redundant.
Hence the augmented algorithm will end correctly after making
$q$ queries $v_1, \ldots v_q$ only if it for some distinct
$i,j \in \{1, \ldots ,q\}$, the vertices $v_i$ and $v_j$ belong to the same
component of $G$ but none is reachable from the other. This
probability is clearly bounded by
${q \choose 2} \cdot \frac{|H|}{n} < 1/2$, for our choice of
$q$ and $n$ large enough.
\end{proof}
\subsubsection{The `only-if' part of Theorem
\ref{thm:main-mon}}
Proving the `only-if' part of Theorem
\ref{thm:main-mon} naturally brings us back to
configurations in digraphs as this is what a tester discovers in its
run. This motivates the following definition analogous to Definition
\ref{def:pH}.
\begin{definitionp}\label{def:pHp}
For a set of
configurations $\mathcal{C}$,
the property $\mathcal{P_{\mathcal{C}}}$
contains all graphs that are $C$-free for every $C \in
\mathcal{C}$.
\end{definitionp}
We comment that for an unrestricted set of forbidden configurations
$\mathcal{C}$, $P_{\mathcal{C}}$ may happen to be hereditary,
monotone, or neither (in the $FB(d)$-model, $F(d)$-model and the
undirected bounded-degree graph model). E.g., the property of
not having a vertex of out-degree
exactly $2$ in the $F(3)$-model is a property that is defined by one forbidden
configuration that is the directed $2$-star, where the center is
the only developed vertex.
However, the property is not monotone nor
hereditary (and happens to be strongly-testable).
The following is a restatement of the `only if' part of Theorem
\ref{thm:main-mon} followed by its proof. Note that configurations do
not appear in the statement, but will appear in the proof.
\begin{thmp} \label{thm:main-ff2}
Assume that the monotone property $P = \cup_{n \in \ensuremath{\mathbb N}} P_n$ is $1$-sided
error strongly-testable in the $F(d)$-model. Then for any $\epsilon >0$
there is a $r = r(\epsilon)$ such that for any $n$ there is
a $r(\epsilon)$-set of rooted digraphs $\mathcal{H}_n$
such that the corresponding property $P_{\mathcal{H}_n}$
that contains the $n$-vertex digraphs that are
${\mathcal{H}_n}$-free, satisfies the following two conditions:
\noindent
(a) $P_n \subseteq P_{\mathcal{H}_n} $\\
(b) $P_{\mathcal{H}_n} $ is $\epsilon/2$-close to $P_n$.
\end{thmp}
\begin{proof}
Since $P$ is strongly-testable, Theorem
\ref{thm:canonical} implies that for any $\delta$ there is a $(q,q)$-canonical
$1$-sided error $\delta$-test, $T(n,\delta)$
for $P_n$, where $q = q(\delta)$ is independent of $n$. By
definition $T(n,\delta)$ picks $q$ vertices uniformly at random, makes the
$q$-disc queries around each, and accepts or reject based only on the
configuration of size at most $r(\delta)=q\cdot d^{q+1}$ that it
sees.
Let $$\mathcal{C}_n(\delta)=\{C=(H,L) ~\mid ~~ \exists G \text{ with } n
\text{ vertices which is rejected by } T(n,\delta) \text{ upon seeing the configuration }
C\}.$$
$\mathcal{C}_n(\delta)$ is a well defined set of $r(\delta)$-size configurations as the
decision of $T(n,\delta)$ depends only on the configuration it sees.
Let $\mathcal{H}'(\delta) = \mathcal{H'}_n(\delta)=\{H ~ | C=(H,L) \in \mathcal{C}_n
\}$. Obviously $\mathcal{H'}$ is an $r(\delta)$-set.
For fixed $\epsilon$, $\mathcal{H'}(\epsilon/2)$ will nearly be our required set as
asserted in the theorem. We will show in what follows that the
conditions (a) and (b) of the theorem hold for
$\mathcal{H'}(\epsilon/2)$. We will then need to change it
slightly so that every member of it is rooted while keeping
(a) and (b).
\begin{claimp}\label{cl:fb-mon1}
For every $\delta, $ $P_n \subseteq P_{\mathcal{H'}(\delta)}$.
\end{claimp}
\begin{proof}
Assume for the contrary that $G \in P_n$ but it is not
$\mathcal{H'}(\delta)$-free. Then, for some $V' \subseteq
V, |V'|=|V(H)|$, $G[V']$ contains a subgraph $H \in \mathcal{H'}(\delta)$.
Namely, there is a $1-1$ map between $V'$
and $V(H)$ showing the isomorphism. For simplicity we
identify in what follows $V'$ with $V(H)$.
We claim
that $G$, or a subgraph of it that is obtained by removing some
edges, has a $C$-appearance for a configuration $C = (H,L) \in
\mathcal{C}_n$ (there is such $C=(H,L) \in \mathcal{C}_n$ by the
definition of $\mathcal{H'}(\delta)$).
Indeed, we first remove the set of edges from $G$ so that $G[V']$
is isomorphic to $H$ as an {\em induced} subgraph, resulting in a
graph $G_1$. Now that $G_1[V']$ is isomorphic to $H$, what would
prevent $G_1$ to have $C$-appearance with the label $L$ on the
vertices $V'$? The label $L$ restrict the out-degree of some
vertices; frontier vertices must have zero degree, and developed
vertices should have degree in $G_1$ exactly as they do in $H$ (see
Definition \ref{def:C-free}). But, since $G_1[V']$ is isomorphic to
$H$, removing all edges in $G_1$ that go out of $V'$ results in
$G'$ for which the restrictions that $L$ imposes are met. So, $G'$
has a $C$-appearance.
By monotonicity
of $P_n,$ $G' \in P_n$. Hence, (by the definition of $\mathcal{C}$) there is positive
probability that $T(n,\delta)$ will reject $G'$ contradicting the
assumption that $T(n,\delta)$ is $1$-sided error test for
$P_n$.
\end{proof}
We note that we crucially used here the fact that $P$ is monotone.
\begin{claimp}\label{cl:fb-mon2}
For every $\delta,$ $P_{\mathcal{H'}(\delta)}$ is $\delta$-close to $P_n$.
\end{claimp}
\begin{proof}
Let $G \in P_{\mathcal{H'}(\delta)}$. Then $T(n,\delta)$ accepts
$G$ with probability $1$ by the definition of
$\mathcal{H'}(\delta)$. Hence $G$ must be $\delta$-close to
$P_n$ or else $T(n,\delta)$ would have to reject it with probability
at least $1/2$ (being an $\delta$-test for $P_n$). The other
direction is trivial since $P_n \subseteq P_{\mathcal{H'}(\delta)}$.
\end{proof}
Finally, for fixed $\epsilon$ we could choose
$\mathcal{H'}(\epsilon/2)$ to be the set guaranteed in the
theorem, since by Claims \ref{cl:fb-mon1} and \ref{cl:fb-mon2} the
conditions (a) and (b) hold for $\mathcal{H'}(\epsilon/2)$.
However, the theorem requires also that every $H \in
{\mathcal{H}_n}$ is
rooted, which is not guaranteed for the set $\mathcal{H'}(\epsilon/2)$. We show in what follows that
$\mathcal{H'}(\epsilon/2)$ can be changed so that conditions (a) and
$(b)$ of the theorem still hold and so that every member
of it is rooted.
Let $\mathcal{\tilde{H}} = \mathcal{H'}(\epsilon/2) ~ \cup ~ \{H \in
\mathcal{H'}(\delta)~| ~ \delta < \epsilon/2 ~ {and} ~
|H| \leq r(\epsilon/2) \}$.
Note that $\mathcal{\tilde{H}}$ is an $r(\epsilon/2)$-set for $r()$ as
defined above. In addition, since Claim \ref{cl:fb-mon1} is true for every $\delta$, it
follows that $P_n \subseteq
P_{\mathcal{\tilde{H}}}$. Further, the fact that $\mathcal{H'}(\epsilon/2)
\subseteq \mathcal{\tilde{H}}$ implies that $P_{\mathcal{\tilde{H}}}
\subseteq P_{\mathcal{H'}(\epsilon/2)}$, and hence by
Claim \ref{cl:fb-mon2}
it holds that $P_{\mathcal{\tilde{H}}}$ is $\epsilon/2$-close to $P_n$.
It could be that there are two distinct digraphs
$H,H' \in \mathcal{\tilde{H}}$, where $H$ is a subgraph of
$H'$. For every such pair $(H,H')$ we remove $H'$ from
$\mathcal{\tilde{H}}$ so to result in the set
$\mathcal{H}=\mathcal{H}_n$ for which no member is a subgraph of
another. This is our final set as required for the theorem. Indeed
removing $H'$ when such a pair $(H,H')$ exists does not change $P_{\mathcal{\tilde{H}}}$ at all,
and hence conditions (a) and (b) hold for $\mathcal{H}$.
We claim that each $H \in \mathcal{H}$ is
rooted. The argument for this also exhibits
the advantage of $\mathcal{H}$ in
comparison with the initial $\mathcal{H'}(\epsilon/2)$.
Assume for the contrary that
$H \in \mathcal{H}$ is not rooted, and consider the graph $G_H$
that is composed by $n/|H|$ vertex disjoint copies of $H$.
Proposition \ref{tester:directed-out:2} asserts that any $1$-sided
error algorithm that needs to discover a copy of
$H$ with constant probability makes $\Omega(\sqrt{n})$
queries. Now, this is not a contradiction to the fact that $H$
might be a member of $\mathcal{H'}(\epsilon/2)$ if $\epsilon/2
> \frac{1}{d|H|},$ since the test
$T(n, \epsilon/2)$ does not need to reject $G_H$ in this case. However,
this can not happen if $H$ is a member of $\mathcal{H}$:
Indeed, since $G_H$ is
$\frac{1}{d|H|}$-far from $P_n$ the test $T=T(n,\delta)$
rejects $G_H$ for $\delta = \min\{\epsilon/2,
\frac{1}{d|H|}\}$ with probability at least $1/2$. By the
construction of $G_H$ this can be done only by discovering a subgraph
isomorphic to $H$ or by discovering a subgraph $H'$ of $H$.
The later case is ruled out since the existence of such $H'$ implies
that $H' \in \mathcal{H}$ contradicting the fact that $H \in
\mathcal{H}$.
The former case cannot happen as we argued that
to discover $H$ with constant success probability takes
$\Omega(\sqrt{n})$ queries.
\ignore{
Recall that we have fixed $\epsilon$, and accordingly we have
defined the set $\mathcal{H} = \mathcal{H}_n$. We now consider again
the sets
We first remove from $\mathcal{H}_n$ any non-minimal
member. This does not change $P_{\mathcal{H}_n}$ at all, so
(a) and (b) still hold by Claim \ref{cl:fb-mon1} and Claim
\ref{cl:fb-mon2} respectively. In what follows we refer to
$\mathcal{H}=\mathcal{H}_n$ as the family in which every member is minimal.
Assume for the contrary that
$H \in \mathcal{H}$ is not rooted, and consider the graph $G_H$
that is composed by $n/|H|$ vertex disjoint copies of $H$.
Proposition \ref{tester:directed-out:2} asserts that any $1$-sided
error algorithm that needs to discover a copy of
$H$ with constant probability makes $\Omega(\sqrt{n})$
queries. This is not yet a contradiction since $G_H$ is not
$\epsilon/2$-far from $P$ if $\epsilon >
2d/|H|$. Hence, the test $T=T(n,\epsilon/2)$ guaranteed for
$P$
does not have to reject $G_H$. We end the proof as follows.
We will replace $\mathcal{H}$ with $\mathcal{\tilde{H}}$, by
replacing $H$ with a smaller rooted subgraph $H'$ of $H$,
and so that $P_n \subseteq P_{\mathcal{\tilde{H}}}$ still holds. In turn, it
will immediately follow that $P_{\mathcal{\tilde{H}}}$ is
$\epsilon/2$-close to $P_n$, since $P_{\mathcal{\tilde{H}}}
\subseteq P_{\mathcal{H}}$ on account of $H'$ being a
subgraph of $H$.
As a result, we have removed one non-rooted subgraph from
$\mathcal{H}$ while keeping the conditions (a) and (b) of the
theorem. Then after a finite number of iterations the final set
$\mathcal{\tilde{H}}$ will contain only rooted members.
To perform the basic operation
Consider $\epsilon' = \frac{1}{d|H|}$, the corresponding
canonical test
$T'=T(n,\epsilon')$, the corresponding set of configurations, $\mathcal{C}'$, on
which $T'$ rejects, and the corresponding forbidden
digraphs $\mathcal{H}' = \{H'~|~ C'=(H',L') \in \mathcal{C'}\}$.
Since $G_H$ is $\epsilon'$-far from $P$, it should be
rejected by $T'$ with constant probability. Hence, either $H \in
\mathcal{H}'$ or a subgraph $H'$ of $H$ is
in $\mathcal{H}'$.
The former case is ruled
out by the proof of Proposition \ref{tester:directed-out:2}. For the
later case, we replace $H$ by $H'$ in $\mathcal{H}$. Namely we get
$\mathcal{H}_1 = \mathcal{H} \setminus \{H\} \cup \{H'\}$.
Evidently
$P_{{\mathcal{H}_1}} \subseteq P_{\mathcal{H}}$ (as $H'$ is a subgraph
of $H$), and further $P_n \subseteq P_{\mathcal{H}_1}$ since by
assumption, a graph containing $H'$ is not in $P$. We note that
$\mathcal{H}_1$ may not yet meet our goal, as $H'$ might also be
non-rooted. However, $H'$ is strictly smaller than $H$ (in terms of
number of edges), and hence we replace the whole argument for $H'$
replacing $H$. After a finite number of iterations, we will end
with rooted $H''$ that is a subgraph of $H$, and for which the
corresponding $\mathcal{\tilde{H}} =\mathcal{\tilde{H}}_n = \mathcal{H} \setminus \{H\} \cup
\{H''\}$ is such that $P_n \subseteq P_{\mathcal{\tilde{H}}_n}$ as
explained above.
}
\end{proof}
\subsubsection{The `only-if' part of Theorem
\ref{thm:main-hered}}
The following is a restatement of the `only if' part of Theorem
\ref{thm:main-hered}.
\begin{thmp} \label{thm:main-ff2.5}
Assume that the hereditary
property $P = \cup_{n \in \ensuremath{\mathbb N}} P_n$ is $1$-sided error
strongly-testable in the $F(d)$-model. Then, for any $\epsilon >0$
there is a $r(\epsilon)$-set of rooted digraphs
$\mathcal{H} = \mathcal{H}_\epsilon$ and $n^*_{\epsilon} \in \ensuremath{\mathbb N}$
such that for
every $n > n^*_{\epsilon}$ the property $P^*_{\mathcal{H}_n}$ that contains the
$n$-vertex digraphs that are $\mathcal{H}$-free satisfies the
following two conditions:
\noindent
(a) $P_n \subseteq P^*_{\mathcal{H}_n} $\\
(b) $P^*_{\mathcal{H}_n} $ is $\epsilon/2$-close to $P_n$.
\end{thmp}
\begin{proof}
Assume that $P$ is hereditary and is $1$-sided error
strongly-testable. Theorem \ref{thm:canonical} implies
that for any $\delta\in (0,1)$ and $n \in \ensuremath{\mathbb N}$, there is a collection of canonical-tests $\cup_{\delta \in (0,1], n\in \ensuremath{\mathbb N}} T(\delta, n), $
where
$T(\delta,n)$ is a $1$-sided error $(q,q)$-canonical $\delta$-test for
$P_n$, making at most $q = q(\delta)$ $q$-disc queries.
For every $\delta > 0, ~ n \in \ensuremath{\mathbb N}$, let $\mathcal{C} = \mathcal{C}(\delta,n)$ be the set of
forbidden configurations defined by $T(\delta,n)$, namely these
configurations on which $T(\delta,n)$ reject with some positive
probability.
\begin{claimp}\label{cl:121}
For every $\delta > 0$ and $n' > n \geq q,~$ if $G_{n'} \in
P_{n'}$ then $G_{n'}$ is $\mathcal{C}(\delta,n)$-free.
\end{claimp}
\begin{proof}
Suppose that $G'=G_{n'} \in P_{n'}$ for $n ' > n$. If
$G$ has a $C$-appearance for $C \in
\mathcal{C}(\delta,n)$, then fixing such a $C$-appearance,
and
deleting $n'-n$ vertices without
touching the $C$-appearance in $G'$, results in
a graph $G'$ on $n$ vertices that is in $P$ (as $P$ is
hereditary). However, $G'$ has a $C$-appearance
causing $T(\delta,n)$ to reject it with positive
probability. This contradicts the fact that
$T(\delta,n)$ is $1$-sided error for $P_n$.
\end{proof}
Since for every fixed
$\delta$, all tests $T(\delta,n)$ examine only configurations of
size at most $q$ (that may depend on $\delta$ but not on
$n$), $\mathcal{C}^*(\delta) = \cup_{n \in \ensuremath{\mathbb N}} \mathcal{C}(\delta,
n)$ is finite. Namely, there is some $n(\delta) \in \ensuremath{\mathbb N}$ such
that $\mathcal{C}^*(\delta) = \cup_{n \leq n(\delta)} \mathcal{C}(\delta,
n)$. We conclude, by Claim \ref{cl:121}, that for
every $n > n(\delta),$ if $G \in P_n$ then $G$ is
$\mathcal{C}^*(\delta)$-free.
We now proceed with the proof of the Theorem: Fix $\epsilon$ and let
$\delta = \epsilon/2$.
Set
$n^*_\epsilon = n(\delta) +dr +1$, where $r$ is the maximum
size of a configuration in $\mathcal{C}^*(\delta)$.
At this point we have concluded that for every $n \geq n(\delta)$ the test
$T(\delta,n)$ defines the same family of forbidden
configurations $\mathcal{C}^*(\delta) $.
Recall that for a configuration $C=(H,L)$, if $L(v)=frontier$
then the out-degree of
$v \in V(H)$ is $0$. However, $G$ will have
a $C$-appearance even if $G$ contains an {\em induced} subgraph $G'$ that is
isomorphic to $H \cup (v,x)$, where $L(v)=frontier$ (see
Definition \ref{def:C-free}). This motivates the following
definition, capturing the set of possible induced graphs of
$G$ that will cause a
$\mathcal{C}$-appearance in $G$.
\begin{definitionp}
\label{def:closed}
Let $C=(H,L)$ be a configuration in the $F(d)$-model. Then,
$$cl(C) = \{H' =(V(H), E') ~ | ~ E(H) \subseteq E', ~ and ~\forall (v,x) \in E'
\setminus E(H), ~ L(v)=frontier \}$$
\end{definitionp}
Hence $cl(C)$ consists of all digraphs $H'$ such
that if an $n$-vertex graph $G$ has a $C$-appearance on its vertices $A
\subseteq V(G)$, then $G[A]$ induces a subgraph isomorphic to $H'$
(note that
the outdegree of a frontier vertex in $H'$ might not be zero).
Let $\mathcal{H}= \mathcal{H}_\epsilon= \cup_{C \in
\mathcal{C}^*(\delta)} ~ cl(C)$, and let $ P^*_{\mathcal{H}_n}$ contain
the $n$-vertex digraphs that are $\mathcal{H}$-free as
induced subgraphs. By the definition of $r$, $\mathcal{H}$ is an $r$-set.
\begin{claimp}\label{cl:fb-her1}
For $n \geq n^*_{\epsilon}$, $~ ~ P_n \subseteq P^*_{\mathcal{H}_n}$.
\end{claimp}
\begin{proof}
Assume for the contrary that $G \in P_n$ and $G$ is not
$P^*_{\mathcal{H}_n}$. Then for some $H \in \mathcal{H}$, $G$
contains an $H$-appearance as an induced subgraph on some $V_H
\subset V(G)$. Let $C=(H,L) \in
\mathcal{C}^*(\delta)$ be
the corresponding configuration for which $H \in cl(C)$. By Fact
\ref{fact:f3} the digraph $G'$ that is obtained from $G$ by
deleting the outgoing neighbours of $V_H$ in $G$ has a
$C$-appearance. Let $n' = |V(G')|$.
Note that $n' \geq n(\delta)$. Hence $T(\delta,n')$
would reject $G'$ with a positive probability. But $G' \in P$ on
account of $P$ being hereditary. This contradicts the
fact that $T(\delta,n')$ is a $1$-sided error for $P_{n'}$.
\end{proof}
\begin{claimp}\label{cl:fb-her2}
For $n \geq n^*_{\epsilon}, ~ ~ P^*_{\mathcal{H}_n}$ is $\epsilon/2$-close to $P_n$.
\end{claimp}
\begin{proof}
Let $G \in P^*_{\mathcal{H}_n}$. We claim that $T(\epsilon/2,n)$
accepts $G$ with probability $1$. Indeed assume that $T(\epsilon/2,n)$
rejects $G$ on account of a $C$-appearance. Then by the definition of
$\mathcal{H}_\epsilon$, $G$ would have an induced subgraph $H' \in cl(C)$ for
some $C \in \mathcal{C}^*(\delta)$, contradicting
the fact that $G \in P^*_{\mathcal{H}_n}$. Hence $G$ must be $\epsilon/2$-close to
$P_n$ as $T$ is $\epsilon/2$-test for $P_n$. The other
direction is trivial since $P_n \subseteq P_{\mathcal{H}}$.
\end{proof}
We have proved that the requirements (a), (b) of Theorem
\ref{thm:main-ff2.5} hold for the $r$-set $\mathcal{H}_\epsilon$.
Finally, the fact that each $H \in \mathcal{H}_\epsilon$ is rooted
is argued similarly as in the proof of Theorem \ref{thm:main-ff2}
(the monotone case).
\end{proof}
\subsubsection{A few concluding remarks on Theorem \ref{thm:main-ff2} and
monotone properties.}
It is easy to see that if the property $\mathcal{P_{\mathcal{C}}}$ in the $F(d)$ model is
monotone, then $\mathcal{C}$ is {\em upwards closed} in the sense that
is defined below.
\begin{definitionp}\label{def:upward}
A set of configurations ${\mathcal{C}}$ is upwards-closed if for every
$C=(H,L) \in \mathcal{C}$ and $v$
being developed, adding any edge $(v,u)$ to $H$,
while respecting the degree bound, results in a configuration $C' =
(H',L')$ that is
also in $\mathcal{C}$, where if $u \in V(H)$ then $L'
= L$, otherwise $L'(u) =
frontier$ and $L'(x)=L(x)$ for every other vertex $x$.
\end{definitionp}
\begin{factp}\label{fact:f1}
$P_{\mathcal{C}}$ is monotone if and only if
${\mathcal{C}}$ is upwards-closed. $\qed$
\end{factp}
An immediate conclusion from Fact \ref{fact:f1} is that for monotone
$P_{\mathcal{C}}$, $\mathcal{C}$ can be specified by its minimal
configurations. (w.r.t to Definition \ref{def:upward}).
Next, we generalize Theorem \ref{thm:main-ff2}, moving beyond the
scope of monotone properties. Towards this end we use the following.
\begin{definitionp} [Rooted Configuration]
A configuration $C = (H,L)$, where $H$ is a digraph
and $L$ is a label function, is {\rm rooted} if $H$ is
rooted.
\end{definitionp}
A conclusion from the proof of Theorem \ref{thm:main-ff2} is that
$\mathcal{C}_n$ as defined in the proof is upwards closed and every
minimal (with respect to Definition \ref{def:upward}) configuration in
it is rooted.
However, more can be said: The
following theorem follows directly from the arguments above {\em for any
digraph property}, where we say that a set of configuration $\mathcal{C}$ is an $r$-set if for
every $C = (H,L) \in \mathcal{C}, ~ |V(H)| \leq r$.
\begin{thmp}\label{thm:general}
Assume that the digraph property $P = \cup_{n \in \ensuremath{\mathbb N}} P_n$ is $1$-sided
error strongly-testable in the $F(d)$-model. Then, for any $\epsilon >0$
there is a $r = r(\epsilon)$ such that for any $n$ there is
$r(\epsilon)$-set $\mathcal{C}_n$ of
configurations such that every minimal configuration in
$\mathcal{C}_n$ is rooted and,
\noindent
(a) $P_n \subseteq P_{\mathcal{C}_n}$ and
(b) $P_{\mathcal{C}_n} $ is $\epsilon/2$-close to $P_n$.
\end{thmp}
The proof is essentially identical to the proof of Theorem \ref{thm:main-ff2},
in which we replace subgraphs by configurations and leave out the
parts dealing with monotonicity.
\section{Strongly-Testable properties that are non-monotone neither hereditary}\label{sec:5}
There are $1$-sided-error strongly-testable properties in the
$F(d)$-model (and in all other models too) that are not monotone,
neither are hereditary. Consider e.g., the $F(d)$-model and the
property $P$ of not having a vertex of out-degree $d-1$. This property
is not trivial e.g., the graph that contains $n/d$ vertex disjoint
directed $(d-1)$-stars is $\frac{1}{d}$-far from the property. Moreover
$P$ is non-monotone and not hereditary. But $P$ is strongly-testable as
if $G$ is $\epsilon$-far from $P$ then $G$ contains at least
$\epsilon n$ vertices of degree $d-1$. Indeed, it can be defined by
one forbidden rooted configuration, hence consistent with Theorem
\ref{thm:general}.
A more interesting property that is $1$-sided error strongly-testable while not
monotone nor hereditary is the following property {\em
RV} (for a ``reachable vertex''). Differently from not having a degree $d-1$ vertex, the
property $\Sink$ is not expressible by a finite collection of forbidden
configurations at all. Rather, it is close to such (for any $\epsilon$).
For a digraph $G=(V,E)$ a vertex $s \in V$ is
called ``reachable-by-all'' if there is a directed path
from each vertex in $G$ to $s$.
Note that $G$ may have many such vertices, in particular, if $G$ is
strongly connected then every vertex is reachable-by-all. Let $\Sink$ be the digraph property of having a
vertex that is reachable-by-all. The
property
$\Sink$ is
not trivial, as e.g., a directed matching is far from $\Sink$.
\begin{thm} \label{thm:sink}
The property $\Sink$ is $1$-sided error strongly-testable in the $F(d)$-model.
\end{thm}
\begin{proof}
The following test $T$ is a $1$-sided error $\epsilon$-test for $\Sink$ making
$\frac{1}{d^2\epsilon^2} \cdot d^{O(1/(d\epsilon))}$
queries. The basic idea is very similar to the test (and
proof) for testing connectivity in \cite{GR02}.
\noindent
{\bf Test for $\Sink$, $T(\epsilon)$, for $\epsilon < 1/d$:}
\begin{enumerate}
\item Choose a multiset of vertices $B \subseteq V(G)$
by choosing independently a vertex $v \in V(G)$
uniformly at random, for $b= \frac{200}{d^2\epsilon^2}$ times.
Let $B=\{v_1, \ldots ,v_b \}$ the vertices thus chosen.
\item For $i=1$ to $b$: query the disc
$D(v_i,\frac{2}{d\epsilon})$ around $v_i$, and let $S_i$ be the
set of vertices that is discovered (including $v_i$).
\item If there are distinct $i,j$ such that $\Gamma^+(S_i) = \Gamma^+(S_j)
= \emptyset$ and $S_j \cap S_i = \emptyset$ reject, otherwise accept.
\end{enumerate}
\begin{claim}
$T(\epsilon)$ never reject a digraph in $\Sink$.
\end{claim}
\begin{proof}
Let $G$ have a vertex $a$ that is
reachable-by-all. Then for any $v$ that
is queried, there is a path from $v$ to $a$. Therefore,
for every $i$, either $a$ is in
$D_{v_i} = D(v_i,2/(d\epsilon)~)$, or there is a path from $v_i$ to $a$
that stretches outside $D(v_i,\frac{2}{d\epsilon})$
implying that $\Gamma^+(S_i) \neq
\emptyset$. Therefore, for every $v_i$ and
$v_j,$ $\Gamma^+(S_i)=\Gamma^+(S_j)= \emptyset$ holds
only when $a \in S_i \cap S_j$.
\end{proof}
\begin{claim}
Let $\epsilon < 1/d$ and $G$ be $\epsilon$-far from $\Sink$, then $T(\epsilon)$ rejects $G$ with
probability at least $1/2$.
\end{claim}
\begin{proof}
Let $G$ be $\epsilon$-far from $\Sink$ and $SC(G) = (A,F)$ be the DAG
of the strongly
connected components of $G$. We first claim that $SC(G)$ contains at
least $\epsilon
dn$ components $c \in A$ for which $\Gamma^+(c)=\emptyset$. Indeed, let $c_1, \ldots ,c_k$ be the strongly connected
components of $G$ for which $\Gamma^+(c_i) =
\emptyset$. To see that $k \geq \epsilon dn$ note that
by changing at most $k-1$ edges
(one per $c_i$, connecting it to $c_{i+1}$), $G$ will have a
a vertex that is reachable-by-all in $c_k$.
This implies that there are at least $\epsilon dn/2$
components $c\in A$, of size at most $2/(d\epsilon)$,
for which $\Gamma^+(c) = \emptyset$. We denote this
set of components by $A^*$ and the vertices in $A^*$
by $V^*$. It follows that $|V^*| \geq \epsilon dn/2$,
and hence, with high probability sampling
$b= \frac{200}{d^2\epsilon^2}$ vertices finds two vertices in two distinct
components in $A^*$. Scanning the
$\frac{2}{d\epsilon}$-disc around two such vertices
will cause the test to reject. \ignore{
We conclude that picking a random vertex $v$, it
will be in $V^*$ with probability at least
$\epsilon d/2$. In turn, the expected number of
vertices in $B \cap V^*$ is at least
$b \cdot \epsilon d/2$. Hence, by Chernoff, with
extremely small probability
$|B \cap V^*| \leq \epsilon bd/100 =
2/d\epsilon$. In addition with extremely large
probability all the $b$ vertices chosen for $B$ are
distinct. Assuming that the vertices in $B$ are
distinct and that $|B \cap V^*| > 2/(d\epsilon)$,
then there are two distinct vertices $v,v'$ that
reside in distinct components from $A$ (as each
component in $A$ has size bounded by
$2/(d\epsilon)$). Two such vertices $v,v'$ will
cause the test to reject.}
\end{proof}
Finally, the query complexity is clearly $b \cdot \max_i |S_i|$ which is as stated.
\end{proof}
We note that $\Sink$ cannot be defined by any $r$-set of
forbidden configurations, for $r$ that is independent of $n$. To see this consider a digraph that is
composed of two vertex disjoint simple di-cycles of length $n/2$
each. Such a graph is not in $\Sink$ but every configuration of it of
size at most $n/4$ is shared by the digraph that is composed of one
single directed cycle, which is in $\Sink$. The property
$P_{\mathcal{C}_\epsilon}$ that is actually being tested by a
$1$-sided error test for $\Sink$ is defined by the set
$\mathcal{C}_\epsilon$ in which every configuration is a pair of
vertex-disjoint discs, of the appropriate size, with no outgoing
edges. The property $\Sink$ is a subset of $P_{\mathcal{C}_\epsilon}$
for every $\epsilon$, but the size of $\mathcal{C}_\epsilon$ while
finite for every $\epsilon$, is not bounded when $\epsilon$ tends to
$0$.
\section{The $FB(d)$-model and the undirected bounded-degree graph model}\label{sec:FB}
As already mentioned, the undirected bounded-degree graph model can be
viewed as a submodel of the $FB(d)$-model. Hence, we state the results
only for the $FB(d)$-model. The results are very similar to these for the
$F(d)$-model, except that the restriction that the forbidden members
are rooted
is not needed. In addition, the test for being free of a
finite family of forbidden {\em induced} graphs is similar to the
monotone case due to the bound on incoming degree (this will
further explained in the relevant place below). We define here the appropriate notions and state
the appropriate theorems. We give proofs only where they are
significantly different from these for the $F(d)$ model.
We start with the relevant notions, analogous to these seen for the
$F(d)$-model.
The first notion, which is non-standard due to the type of queries that
is available, is that of $r$-disc.
\begin{definition} [$r$-disc, $FB(d)$-model]\label{def:fb-disc}
Let $r$ be an integer and $v \in V(G)$. $\tilde{D}(v,r)$
denotes the ``$r$-disc'' for the $FB(d)$-model, and is defined recursively as follows:
$\tilde{D}(v,1) = \{v\} \cup \Gamma^+(v) \cup \Gamma^-(v)$.
For $r \geq 2,~$ $\tilde{D}(v,r) = \cup_{u \in \tilde{D}(v,1)} \tilde{D}(u,r-1)$
\end{definition}
That is, $\tilde{D}(v,r)$ contains all vertices that are reachable
from $v$ by path of length at most $r$ that is composed of
edges that may be traversed in the wrong direction. The point
being that the $FB(d)$-model allows for such traversal.
With Definition \ref{def:fb-disc}, an $r$-disc query in the $FB(d)$-model
is defined exactly as in the $F(d)$-model, where $r$-disc are
the corresponding one.
$r$-disc queries generalize basic neighbourhood queries as in the $F(d)$
model, and with the same complexity overhead.
For a family of (di)graphs $\mathcal{H}$, the definitions of being
$\mathcal{H}$-free as subgraphs, or as induced subgraphs are
extended naturally with no alterations (as these are model-independent definitions).
But configurations for
the $FB(d)$ model are defined slightly differently; a configuration is
defined as for the $F(d)$ model, with the extra restriction that the
degree bound holds for both in-degree and out-degree. In addition,
frontier vertices may have non-zero out-degree.
Being $C=(H,L)$-free, for a
configuration $C$, is defined as follows.
\begin{definition} [$C$-Free, $FB(d)$-model]
Let $C=(H,L)$ be a configuration and $G=(V,E)$ a
$d$-bounded degree digraph in the $FB(d)$ model. Let $V'
\subseteq V$. We say that $G[V']$ is a $C$-appearance if
there is a bijection $\phi : V(H) \rightarrow
V'$ such that $\forall v,u \in V(H)$ and $L(v)=\rm{developed}$,
\begin{center}
$(v,u) \in E(H) \leftrightarrow
(\phi(v),\phi(u)) \in E~ ~ $
and $~ ~ (u,v) \in E(H) \leftrightarrow (\phi(u), \phi(v)) \in E$.
\end{center}
Further, for every developed $v,$ if $(\phi(v), x) \in E$
or $(x,\phi(v)) \in E$
then $\exists u\in V(H), \phi(u)=x$.
We say that $G$ is $C$-free if $G$ has no $C$-appearance.
\end{definition}
Finally, the fact that every strongly-testable property is testable by a
canonical tester is also identically the same.
We get the following analog of Theorem \ref{thm:main-mon}.
\begin{thm} \label{thm:main-fb}
A monotone digraph property $P = \cup_n P_n$ is $1$-sided error strongly-testable in
the $FB(d)$-model if and only if for every $\epsilon> 0$ there is a $r = r(\epsilon)$ such that for any $n$ there is a $r$-set of
digraphs $\mathcal{H}_n$ for which the following two
conditions hold (a) $P_n \subseteq P_{\mathcal{H}_n} $ and (b):
$P_{\mathcal{H}_n} $ is $\epsilon/2$-close to $P_n$. $\qed$
\end{thm}
The proof is mostly identical to the corresponding proofs for
the $F(d)$ model and is omitted.
Note that we do not require here that the forbidden digraphs are
rooted. This is not needed anymore, due to the stronger query-type.
The analogous theorem for hereditary properties is.
\begin{thm}\label{thm:main-fb0.5}
An hereditary digraph property $P = \cup_n P_n$ is $1$-sided error strongly-testable in
the $FB(d)$-model if and only if for every $\epsilon> 0$ there
is a $r = r(\epsilon)$, $n^*_{\epsilon} \in \ensuremath{\mathbb N}$ and a
$r$- set of digraphs $\mathcal{H}$, for which
the following conditions hold: for
every $n > n^*_{\epsilon}~$
(a) $P_n \subseteq P^*_{\mathcal{H}}$, and (b) $P^*_{\mathcal{H}} $ is $\epsilon/2$-close to $P$.
\end{thm}
\begin{proof}
The proof of the `only-if' part is identical to that of Theorem
\ref{thm:main-ff2.5} without the restriction (and complication) of
being rooted.
For the `if' part, the analog of Theorem \ref{thm:main-ff1.5} holds
with a simpler proof. The proof starts identically, with $S$ being a
maximal set of induced subgraphs of $G$, each being an
$\mathcal{H}$-appearance (with no restrictions on roots). Then,
deleting all edges adjacent to vertices appearing in $S$ results in
$G$ becoming $\mathcal{H}$-free.
(In the $F(d)$-model, we could not afford deleting all edges adjacent to
$S$ as this could be a large set while $S$ is small, and we had to
resort to sampling a random edge. Here, due to the in-degree bound, if
$G$ is $\epsilon$-far from $\mathcal{H}$-free
then $|S| \geq \epsilon n/4$ (as in the first case of Proposition
\ref{thm:main-ff0.5}).)
The rest of the ``only if'' direction follows from the analog of
Theorem \ref{thm:main-ff1.5}, which is identically stated for the
$FB(d)$ model, leaving out the restriction of that members of $\mathcal{H}$ are
rooted.
\end{proof}
\section{ Two application of the characterization}\label{sec:55}
A characterization is more useful when apart of giving some
structural insight to a feature, it also allows to simply conclude the
existence or lack of a property using the characterization and without
going into the theory behind it. Here we show two applications of
our characterization for proving known results. The first is to show
that the monotone (and hereditary) property of
being $2$-colourable is not strongly-testable (proved in
\cite{GR02}). The second is that the monotone (and also hereditary) property of being $k$-star-free as a {\em minor} is strongly
testable (done as a part of proving other results in
\cite{gold-tree-minor}). The discussion below is done with respect to
the undirected $d$-bounded degree model.
\subsection{$k$-colorability}
It is known that $k$ colorability is
not strongly-testable (even by $2$-sided error tests) for bounded-degree graphs for $k \geq 2$ \cite{GR02}.
Here we reprove the fact without getting into property testing at all. We use the analogous theorem of Theorem
\ref{thm:main-fb} for the undirected model.
Indeed, since $2$-colorability is monotone, if it were
strongly-testable, then the analog of of Theorem
\ref{thm:main-fb} for the undirected bounded-degree model
would imply that there is a $r=r(\epsilon)$ and a $r$-set $\mathcal{H}_\epsilon$ such that
the corresponding conditions (a) and (b) hold.
Namely, there should be a $r$-set $\mathcal{H}$ of graphs such that:
(a) $2$-colorability must be a
subset of a property $P_{\mathcal{H}}$, and (b) that $P_{\mathcal{H}}$ should be $\epsilon$-close to being
$2$-colourable.
Assume that $\mathcal{H} = \mathcal{H}_\epsilon$
is such a set.
By (a) every $H \in \mathcal{H}$ is not
$2$-colourable. Further, $\mathcal{H}$ must contain {\em all} non-$2$-colourable
graphs up to size $d/\epsilon$ (otherwise if a non-$2$-colourable
graph $H_0$ of size
smaller than $d/\epsilon$ is not in
$\mathcal{H}$, then the graph that is composed of $nd/|H_0|$ disjoint
copies of $H_0$ is $\epsilon$-far from $2$-colorability but is in
$P_{\mathcal{H}}$).
Let $d$ be large enough, $\epsilon$ small enough, and take any
good $d$-regular Ramanujan expander (or random $d$-bounded degree graph with no
short cycles). Such a graph is locally a tree, and
hence $\mathcal{H}$-free. However, it is $\epsilon$-far from being
$2$-colourable, as by the expander mixing lemma,
any bipartition of the vertex set has many more than
$\epsilon dn$ edges with both ends in one of the parts. We omit further details.
\subsection{Being $k$-star-free}
The property of $d$-bounded degree undirected graphs of being $k$-star
free as minors is a monotone and hereditary property. It is a simple
instance of the more complex property of being $\mathcal{H}$-minor
free, for a fixed given set of graphs $\mathcal{H}$. It is known and
obvious that for arbitrary $\mathcal{H}$, the property of being
$\mathcal{H}$-minor free is not strongly-testable by $1$-sided error
algorithms, as even acyclicity (namely not having a triangle minor) is not $1$-sided error strongly
testable for $d \geq 3$ \cite{GR02}.
However, for $\mathcal{H}$ being a fixed collection of trees, the
property of being $\mathcal{H}$-free is strongly-testable as was shown
in \cite{gold-tree-minor}. A first (and relatively easy step) in the result of
\cite{gold-tree-minor} is when the only member of $\mathcal{H}$ is the $k$-star (for constant fixed
$k$).
The property $P$ of being $k$-star free as a minor is a monotone property.
We show that being $k$-star free as a minor is $1$-sided error
strongly-testable for the undirected
$d$-bounded degree graph model using Theorem \ref{thm:main-fb}.
Indeed, all we need to show
(for any $\epsilon > 0$)
is an $r(\epsilon)$-set $\mathcal{H}$ such that
following holds: (a) $P_n \subseteq
P_{\mathcal{H}_n}$ and (b) that $P_{\mathcal{H}_n}$ is $\epsilon$-close to
$P_n$. Here $P_{\mathcal{H}_n}$ contains the $n$-vertex graphs in
$P_{\mathcal{H}}$.
We set $\mathcal{H}$ to contain all graphs of size at most
$s=\frac{k}{\epsilon} + kd$ that
contain a $k$-star as a minor. It is obvious from the definition
that $P_n \subseteq P_{\mathcal{H}_n}$.
Let $G \in P_{\mathcal{H}_n}$. We note that for any $S \subseteq
V(G)$ such that $G[S]$ is connected and $|S| \leq s-k$, the edge cut $(S,\bar{S}) = \{(u,v) \in E(G)~|~ u
\in S, ~ v \notin S\}$ has size at most $kd$. This is true as
otherwise contracting $G[S]$ to a single point exhibits a $k$-star
in the subgraph $G[S \cup \Gamma(S)]$ that is of size at most $s+kd$.
Hence, it follows that we can decompose $G$ by iteratively choosing
a vertex $v$ in a large enough component, and removing any connected
subgraph of size $s-k$ containing $v$. This will result in
components of size at most $s-k$, while removing at most $kd$ edges
at each iteration. Thus in total, removing at most
$\frac{n}{s-k}\cdot dk$
edges we get a graph $G'$ that is a subgraph of $G$, and in which
every component is of size at most $s-k$. It follows that $G' \in
P_n$ by definition, and since we have removed at most
$\frac{ndk}{s-k} \leq \epsilon dn$ it implies that $G$ is
$\epsilon$-close to $P_n$.
\section{Concluding Discussion}\label{sec:concl}
Let $\mathcal{C}$ be
a finite set of configurations (in any of the models discussed
above). The property of being $\mathcal{C}$-free is very natural in the
context of bounded-degree (di)graphs. In particular, all monotone and
all hereditary
properties are instances of such properties. Hence, being free of
$\mathcal{C}$ is a collection of properties worth studying (and not
only in the context of property testing).
We have characterized the monotone and hereditary (di)graph properties
that are $1$-sided error strongly-testable in all the corresponding bounded-degree (di)graph
models. Theorem \ref{thm:general} states that every property that is
$1$-sided error strongly-testable in the $F(d)$-model (and the analogous statements
for the other models) is defined by a finite
collection of {\em forbidden configurations} with properties (a)
and (b) as in the theorem. It could be that these are exactly the
properties that are $1$-sided error strongly-testable {\em regardless of being
monotone or hereditary}. The
problem with extending it to a characterization arises for the
analog of
Proposition \ref{theorem:directed-out:1}. We do not know that for a finite
set of rooted {\em configurations} $\mathcal{C}$, $P=P_{\mathcal{C}}$ is
strongly-testable. It could be that
for $G$ that is $\epsilon$-far from $P$, $G$ has only a small
number of appearances
of forbidden configurations and any way of ``correcting'' these
appearances creates new appearances. We do know this e.g., for the
$F(d)$-model if the set of forbidden configurations are
degree bounded\footnote{Forbidden configurations of bounded degree
$d-1$ graphs are easy to `correct' by adding edges so to create vertices of
degree $d$. Hence in this case, if a graph is far from the
property, then it has many vertices in forbidden configurations.} by $d-1$, but not for the general case.
Finally, in the very simple case of the $FB(1)$-model, and hence the {\em undirected}
$2$-degree bounded model too, the inverse of
Theorem \ref{thm:general} does work.
We prove, in this case, that if a graph is far from being
$\mathcal{C}$-free then it has many
$\mathcal{C}$-appearances. This conclusion turns out to be not
entirely trivial, although the family of $2$-degree bounded
graphs is very simple\footnote{For these models every
(di)graph property is strongly-testable by
the results of \cite{NS13}. However, not all properties are $1$-sided
error strongly-testable. E.g., consider the property ``having exactly $n/2$
edges'' for which we do not have small witnesses for being far.}. The argument requires some global considerations beyond
these used for monotone properties and appear in the Appendix.
{\bf Acknowledgment:} we thank Oded
Goldreich for the extensive work he has done in order to improve the presentation of this
paper.
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\appendix{\huge{Appendix}}
\section{``Removal Lemma'' - the case of $2$ bounded degree undirected
graphs and the $F(1)$-model }
The standard removal lemma in our context would be that if a graph $G$
has ``small'' number of $\mathcal{C}$-appearances than it can be made
$\mathcal{C}$-free by removing and inserting a ``small'' number of
edges. Here $\mathcal{C}$ is a collection of configurations rather
than just forbidden subgraphs. We do not know if such a lemma is
correct for $d$-bounded degree graphs and $d \geq 3$. We prove the
following for $2$-bounded degree graphs. It is a very simple case, but
already exhibits why simple local considerations might not be enough.
\begin{lemma}\label{lem:d=1}
Let $\mathcal{C}$ be a $k$-set of forbidden configurations in the
$2$-bounded degree model for undirected graphs.
Let $P_n$ be the property that contains the $n$-vertex
$\mathcal{C}$-free $2$-bounded degree graphs. For $\epsilon <
\frac{1}{4k}$ if $P_n \neq
\emptyset$ and
$G$ is $\epsilon$-far from $P_n$ then $G$ contains $\epsilon^2 n/k$
vertices in $\mathcal{C}$-appearances.
\end{lemma}
Before we present the proof we point why local consideration as in
the proof of Proposition \ref{theorem:directed-out:1} are not
sufficient. Let
$\mathcal{C}$ contain two forbidden configurations: a singleton and a
path of length $2$ where the middle vertex is Developed and the two
endpoints are Frontier.
Consider the property $P$ of being $\mathcal{C}$-free.
If $G$ is in $P$ then $G$ is a
perfect matching and hence the property is not trivial. However, for odd $n$ $P_n= \emptyset$
and the existence of a single $C$-appearances can not be corrected
at all.
\begin{proof}
A configuration $C \in \mathcal{C}$ may be disconnected and composed
of several components. For simplicity we prove the lemma for the case
that for every $C=(H,L) \in \mathcal{C}$, $H$ is connected. The proof for the general
case is more complicated but uses the same ideas.
By assumption $\mathcal{C}$ is a collection cycles
and paths. We may assume that all vertices of degree $2$ in any $C \in
\mathcal{C}$ are
Developed as
$d \leq 2$. There are $3$ possible types of paths in $\mathcal{C}$: A path with both ends
Developed, Both ends Frontier, and a Frontier and Developed ends. We
call such paths $DD$, $FF$ and $FD$ paths respectively. We consider
the zero length path containing a
single isolated vertex as a $DD$ path.
Let $P_n$ be the property that contains the $n$-vertex graphs that are
$\mathcal{C}$-free, and assume that $P_n \neq \emptyset$.
Let $G$ on $n$ vertices be $\epsilon$-far from $P_n$.
Let $c$ be a component of $G$. If $|V(c)| > k/\epsilon$, $c$ is
called `large' and otherwise it is called `small'.
{\bf (i)} Assume first that $\mathcal{C}$ contains no $FF$ path.
A large component $c$ of $G$ may have a $C$-appearance for some $C \in \mathcal{C}$
only if $C$ is a $FD$ path and $c$ is a path. Then by adding the edge
between the endpoints of the path $c$ it will not have a
$\mathcal{C}$-appearances.
Since there are at less than $\epsilon n/k$ large $C$'s, all
relevant appearances are corrected by changing at most $\epsilon n/k$
edges. Let $S$ be the set of all small components of $G$ that have a
$\mathcal{C}$-appearance. Let $|S| = \ell.$
We conclude that $\ell \geq \epsilon n/2$ as other wise we can change
$\cup_{c \in S} c$ into a
unique cycle using at most $2\ell$ edge additions (and if $\ell \leq
k$ we can further make this cycle to be of size at least $k+1$ using
some extra $4$ edge changes) and get a graph that is
$\mathcal{C}$-free.
We conclude that there are at least $\ell \geq \epsilon n/2$ vertex
disjoint $\mathcal{C}$-appearances in $G$, which implies the lemma.
{\bf (ii)} Assume that $\mathcal{C}$ contains a $FF$ path and let $r ~ (\leq
k)$ be the
length of the smallest such path.
{\bf (ii).1}
Assume first that $\mathcal{C}$ contain no singleton.
If a component $c$ in $G$ contains an $r$-length path then every
vertex in it is in an $r$-length $FF$ path. Let $S$ be the set of
vertices in an $r$-length path. Then either $|S| \geq \epsilon^2 n/k$
and we are done, or $|S| <
\epsilon^2n/k$ and we
can delete all edges adjacent to vertices in $S$, (at most
$2|S|$ edges) to obtain a graph
that is free of $r$-size $FF$ paths, and we are back in the previous
case.
{\bf (ii).2}
Assume now that $\mathcal{C}$ contains a singleton (which makes the
corrections in the proofs above impossible, and as shown by the example
before the proof, corrections can not be done locally).
Assume first that for some $\ell \in \ensuremath{\mathbb N}$ there are graphs $G_\ell$ and
$G_{\ell+1}$ on $\ell$ and $\ell+1$ vertices respectively, and such
that both graphs are $\mathcal{C}$-free. Let $\ell$ be the
smallest such integer.
By a basic fact in number theory (Frobenius coin problem), there exists $n_0$ such that for
every $m > n_0,$ $~m$ can be written as $m = a\ell + b(\ell+1)$ for
some $a,b, \geq 0$. We conclude that for any $m \geq n_0$ there is a
graph $G_m$ on $m$ vertices that is composed of $a$ copies of $G_\ell$
and $b$ copies of $G_{\ell+1}$ and that is a $\mathcal{C}$-free.
Let $G$ on $n$ vertices ($n >> \ell$) that is $\epsilon$-far from being
$\mathcal{C}$-free.
Let $S$ be the set of
vertices in large components of $G$. Note that every vertex in every
large component is in a $FF$-forbidden path (except possibly two).
Then, if $|S| \geq \e^2 n/k$ we are done.
Otherwise, let $S_1$ be the set of singletons in $G$. Let $A$ be the set
of vertices in small
components that contain a ${\mathcal{C}}$-appearance.
If $|S_1| \geq \epsilon^2 n/k$, or $|A|\geq
\epsilon^2n/k$ then we are done.
Otherwise, let $V_1 = S \cup S_1 \cup A$. Note that $m=|V_1| \leq
3\epsilon^2n/k$. We assume here that $m \geq n_0$ or otherwise we add
some arbitrary $n_0-m$ additional vertices to $V_1$. We now form a
subgraph $G_m$ on $V_1$ that is $\mathcal{C}$-free. Note that such
$G_m$ exists by our assumptions on $m$. Hence by changing at most $2m
< 6 \epsilon^2 n/k < 2 \epsilon n$ edges (for $n$ large enough),
we have made $G'$ be ${\mathcal{C}}$-free in contradiction with the
assumption that $G$ is $\epsilon$-far from $P_n$.
Assume now that there is no $\ell$ for which $G_\ell, ~ G_{\ell+1}$
are $\mathcal{C}$-free. In this case it follows (by the same coin problem of Frobenius)
that all $G_\ell$ that are $\mathcal{C}$-free have number of vertices
that is congruent to $0$ mod some $\alpha >0$. More over, there is
some fixed $\ell\equiv 0 (\alpha) $ and $G_1, G_2$ on
$\ell, \ell+\alpha$ vertices correspondingly, such that both $G_1,
G_2$ are $\mathcal{C}$-free. This implies that for any $m \equiv 0
(\alpha)$ that is large enough, there is a graph $G_m$ on $m$ vertices
that is $\mathcal{C}$-free.
Let $S,S_1, A, V_1$
as before. We proceed in a similar way: Either $S$ or $S_1$ or $A$ is large enough and
we are done. Otherwise, we make a subgraph $G_m$ on $V_1$ to be
$\mathcal{C}$-free, to result in a graph on $V$ that is $\mathcal{C}$-free,
contradicting the assumption that$G$ is $\epsilon$-far from being
$\mathcal{C}$-free.
The important point here is that since $P_n \neq
\emptyset$ it follows by the discussion above that $n\equiv 0
(\alpha)$. And since $G[V \setminus V_1]]$ is $\mathcal{C}$-free, it follows that $m \equiv
0 (\alpha)$. Hence a $G_m$ on $m$ vertices that is $\mathcal{C}$-free
exists.
\end{proof}
\end{document}
|
\betagin{document}
\title[Barotropic compressible flow with free surface]{The $\mathcal{R}-$bounded operator families arising from the study of the barotropic compressible flows with free surface}
\author[Xin Zhang]{Xin Zhang}
\address{Research Institute of Science and Engineering, Waseda University, 3-4-1 Ookubo, Shinjuku-ku, Tokyo, 169-8555, Japan}
\email{[email protected]}
\subjclass[2010]{Primary: 35Q30; Secondary: 76D05.}
\keywords{$\mathcal{R}$-boundedness, barotropic compressible Navier-Stokes equations, resolvent problem, maximal regularity, analytic semigroup}
\date{\today}
\betagin{abstract}
In this paper, we study some model problem associated to
the free boundary value problem of the barotropic compressible Navier-Stokes equations in general smooth domain with taking surface tension into account.
To obtain the maximal $L_p-L_q$ regularity property of the model problem,
we prove the existence of $\mathcal{R}-$bounded operator families of the resolvent problem via Weis' theory on operator valued Fourier multipliers.
\end{abstract}
\maketitle
\section{Introduction}\lambdabel{sec:intro}
\subsection{Model}
In this paper, we study the following boundary value problem in some general (bounded or unbounded) domain $\Omegaega\subset \mathbb{R}^N$ ($N\geq 2$) surrounded by two disjoint sharp surfaces $\Gammamma_0$ and $\Gammamma_1,$
\betagin{equation}\lambdabel{eq:L_CNS_L}
\left\{\betagin{aligned}
&\partial_t \eta + \gammamma_1 \di \boldsymbol{u} = d
&&\quad\hbox{in}\quad \Omegaega \times \mathbb{R}_+, \\
&\gammamma_1 \partial_t \boldsymbol{u} -\Di\big(\mathbb{S}(\boldsymbol{u})-\gammamma_2 \eta \mathbb{I} \big)= \boldsymbol{f}
&&\quad\hbox{in}\quad \Omegaega \times \mathbb{R}_+,\\
&\big( \mathbb{S}(\boldsymbol{u}) - \gammamma_2 \eta \mathbb{I}\big) \boldsymbol{n}_{\Gammamma_0}
+\sigmagma (m-\Deltalta_{\Gammamma_0})h \,\boldsymbol{n}_{\Gammamma_0} = \boldsymbol{g}
&&\quad\hbox{on}\quad \Gammamma_0 \times \mathbb{R}_+, \\
&\partial_t h - \boldsymbol{u} \cdot \boldsymbol{n}_{\Gammamma_0} = k
&&\quad\hbox{on}\quad \Gammamma_0 \times \mathbb{R}_+,\\
&\boldsymbol{u} = \boldsymbol{0} &&\quad\hbox{on}\quad \Gammamma_1 \times \mathbb{R}_+, \\
&(\eta, \boldsymbol{u},h)|_{t=0} = (\eta_0, \boldsymbol{u}_0, h_0) &&\quad\hbox{in}\quad \Omegaega.
\end{aligned}\right.
\end{equation}
Above $\gammamma_1=\gammamma_1(x),$ $\gammamma_2=\gammamma_2(x)>0$ are smooth,
the matrix $\mathbb{S}(\boldsymbol{u})$ is defined by
\betagin{equation*}
\mathbb{S}(\boldsymbol{u}):= \mu \mathbb{D}(\boldsymbol{u})+ (\nu-\mu)\di \boldsymbol{u}\, \mathbb{I}
\end{equation*}
for the viscosity constants $\mu, \nu>0,$
$\mathbb{D}(\boldsymbol{u}):= \betagin{bmatrix}\partial_k u_j + \partial_j u_k\end{bmatrix}_{N\times N}$
is called the (double) deformation tensor of $\boldsymbol{u}$ and $\mathbb{I}:= \betagin{bmatrix}
\deltalta_{jk}
\end{bmatrix}_{N\times N}.$
In addition, for any vector $\boldsymbol{u}$ and any matrix $\mathbb{A}=\betagin{bmatrix} A_{jk}\end{bmatrix}_{N\times N},$ we write $\di \boldsymbol{u} := \sum_{j=1}^N \partial_j u_j$
and $\Di \mathbb{A} := \sum_{k=1}^N \partial_k A_{jk}.$
In $\eqref{eq:L_CNS_L}_3,$ the constants $\sigmagma,m>0,$ $\boldsymbol{n}_{\Gammamma_0}$ stands for the unit normal vector along $\Gammamma_0$ and $\Deltalta_{\Gammamma_0}$ for the Laplace-Beltrami operator of $\Gammamma_0.$
Given the initial states $(\eta_0,\boldsymbol{u}_0, h_0)$ and source terms $d,\boldsymbol{f},\boldsymbol{g},k,$
the aim is to predict the variation of unknowns $(\eta, \boldsymbol{u}, h).$
\smallbreak
In fact, the model problem \eqref{eq:L_CNS_L} arises from the study the motion of the viscous gases governed by the barotropic compressible Navier-Stokes equations (CNS) in some bounded or unbounded domain $\Omegaega_t \subset \mathbb{R}^N$ ($N\geq 2$) with taking the \emph{surface tension} into account.
For the free boundary value problem of (CNS), we need to determine not only the amplitude of the density, the velocity field of the fluid particles, but also the shape of moving domain $\Omegaega_t.$
In fact, the solvability of (CNS) can be reduced to the linearization form \eqref{eq:L_CNS_L}, which will be our forthcoming work. But let us emphasize here that the role of $h$ in \eqref{eq:L_CNS_L} is to handle the variation of the pattern $\Omegaega_t.$
The study of (CNS) attracts the attention of many mathematicians for a long time.
One may refer to the following works and the references therein for a more complete list of previous works.
The study of (CNS) is challenging even for the initial value problem because of the hyperbolicity from the conservation law of the mass.
For instance, the long time issue for the initial value problem of (CNS) in the whole space was investigated by
Matsumura \& Nishida \cite{MN1979,MN1980}, Hoff \cite{Hoff1995}, Hoff \& Zumbrun \cite{HZ1995} and Danchin \cite{Dan2000}.
On the other hand, for the non-slip (Dirichlet) boundary condition, we refer to the works \cite{MN1983} by Matsumura \& Nishida and \cite{KS1999} by Kobayashi \& Shibata in the exterior domain, and \cite{KK2002,KK2005} by Kagei \& Kobayashi in the half space $\mathbb{R}^N_+$ ($N\geq 2$), and \cite{Kagei2008} by Kagei in the layer.
Next, concerning the free boundary value problem for (CNS) in some smooth bounded domain, Tani in \cite{Ta1981} and
Secchi \& Valli in \cite{SV1983} established the short time solutions, and Zajaczkowski in \cite{Zaja1993} found some long time solutions by ignoring the role of surface tension (i.e. $\sigmagma=0$).
The extension to the surface tension case was studied in \cite{SolTa1991,SolTa1992,Zaja1994}.
In particular, the authors of \cite{SolTa1991,SolTa1992,Zaja1994} proved the long time stability with respect to some trivial equilibrium states within the anisotropic Sobolev framework.
However, the aforementioned works on the free boundary value problem are in $L_2$ or H\"older regularity framework.
To obtain the solutions with $L_p$ in time and $L_q$ in space ($L_p-L_q$ for short) regularity
\footnote{The maximal $L_p-L_q$ regularity of (CNS) is verified by Kakizawa in \cite{Kaki2011} with the Navier boundary condition in the bounded domain as well.},
we refer to the recent works \cite{EvBS2014,GS2014} by Shibata and his group for the case $\sigmagma=0.$
Moreover, for the study of the motion of the two-phase compressible flows, one may refer to \cite{JTW2016,KSS2016} and the references therein.
Our purpose here is to tackle \eqref{eq:L_CNS_L} with the surface tension (i.e. $\sigmagma>0$) in maximal $L_p$-$L_q$ regularity framework.
Here let us emphasize that $\Omegaega$ is not necessary bounded, as long as the boundaries of $\Omegaega$ are uniformly smooth.
Of course, $\Gammamma_1=\emptyset$ in \eqref{eq:L_CNS_L} is allowed by refining our later proof.
That is, we may consider the motion of some bounded isolated mass or the gases in some exterior region.
More precisely, we prove that \eqref{eq:L_CNS_L} has a semigroup structure by imposing $(d,\boldsymbol{f},\boldsymbol{g},k)=\boldsymbol{0}$ (see Theorem \ref{thm:semigroup}). In addition, if the initial data vanish, then the solutions of \eqref{eq:L_CNS_L} admits the maximal $L_p$-$L_q$ regularity (see Theorem \ref{thm:maximal}).
The idea to prove Theorem \ref{thm:semigroup} and Theorem \ref{thm:maximal}
are based on the analysis of the resolvent problem of \eqref{eq:L_CNS_L} (i.e. \eqref{eq:GR_CNS} below).
Most importantly, we show that the solution operator families of \eqref{eq:GR_CNS} are $\mathcal{R}$-bounded, which allows us to apply the Weis' theory on operator valued Fourier multipliers in \cite{Weis2001}.
Furthermore, to overcome the main difficulty of the free boundary condition in \eqref{eq:L_CNS_L}, our study is reduced to some model problem in $\mathbb{R}^N_+$ associated to the generalized Lam\'e operator. In order to use the Weis's theory for the model in $\mathbb{R}^N_+,$ we have to treat the explicit solution formula in terms of Fourier transformation.
Especially, tackling the surface equation in $\eqref{eq:L_CNS_L}_4$ is crucial in our work.
\smallbreak
This paper is folded as follows. In the next section, we will state the main theorem
(i.e.Theorem \ref{thm:GR_CNS}) concerning the generalized resolvent problem \eqref{eq:GR_CNS} and then the proofs of Theorem \ref{thm:semigroup} and Theorem \ref{thm:maximal} in view of Theorem \ref{thm:GR_CNS}.
Afterwards we will study resolvent problem over the half space $\mathbb{R}^N_+$ in Section \ref{sec:halfspace} and the bent half space in Section \ref{sec:bh} respectively.
Finally, we combine the estimates to obtain the results in general domain $\Omegaega$ in the last part of the paper.
\subsection{Notations and functional spaces}
Let us fix the notations in this paper.
In what follows, we denote the Fourier transform in $\mathbb{R}^N$ and its inverse by
\betagin{align*}
\mathcal{F}_{x}[f] (\xi):= \int_{\mathbb{R}^N} e^{-i x\cdot \xi} f(x) dx, \quad
\mathcal{F}_{\xi}^{-1}[g] (x):= \frac{1}{(2\pi)^N}\int_{\mathbb{R}^N} e^{i x\cdot \xi} g(\xi) d\xi.
\end{align*}
For $x=(x',x_{_N}), \xi=(\xi',\xi_{_N}) \in \mathbb{R}^N,$ we sometimes use the partial Fourier (inverse) transform with respect to the horizontal variables,
\betagin{align*}
\mathcal{F}_{x'}[f] (\xi',x_{_N})&:= \int_{\mathbb{R}^{N-1}} e^{-i x'\cdot \xi'} f(x',x_{_N}) dx', \\
\mathcal{F}_{\xi'}^{-1}[g] (x',\xi_{_N})&:= \frac{1}{(2\pi)^{N-1}}\int_{\mathbb{R}^{N-1}} e^{i x'\cdot \xi'} g(\xi',\xi_{_N}) d\xi'.
\end{align*}
Besides, the letter $C(a,b,c,\cdots)$ or $C_{a,b,c,\dots}$denotes that the constant $C$ depends on $a,b,c,\dots$
\smallbreak
Hereafter, $L_q(\Omegaega)$ is the standard Lebesgue space in domain $G \subset \mathbb{R}^N,$
and $H^k_p(G)$ with $k\in \mathbb{N}$ and $1<q<\infty$ stands for the Sobolev space.
In addition, the Besov space $B^{s}_{q,p}(G)$
for some $k-1<s\leq k$ and $(p,q) \in ]1,\infty[^2$
is defined by the real interpolation functor
\betagin{equation*}
B_{q,p}^{s}(G):= \big(L_q(G),H^{k}_q(\Omegaega)\big)_{s\slash k,p}.
\end{equation*}
In particular, we write $W^{s}_q(G)=B^s_{q,q}(G)$ for simplicity,
and $W^{-\bar{s}}_q(G)$ is the dual space of $W^{\bar{s}}_{q'}(G)$ for $0<\bar{s}<1$ and the conjugate index $q':=q\slash (q-1).$
For any Banach spaces $X,Y,$ the total of the bounded linear transformations from $X$ to $Y$ is denoted by $\mathcal{L}(X;Y).$ We also write $\mathcal{L}(X)$ for short if $X=Y.$
In addition, ${\rm Hol}\, (\Lambdambda;X)$ denotes the set of $X$ valued mappings defined on some domain $\Lambdambda \subset \mathbb{C}.$
\smallbreak
Now we give the conditions of the (uniformly) smoothness of $\Omegaega$ in the sequel.
\betagin{defi}\lambdabel{def:domain}
We say that a connected open subset $\Omegaega$ in $\mathbb{R}^N$ ($N \geq 2$) is of class uniform $W^{m-1\slash r}_r$ for some integer $m \geq 2$ and $1<r<\infty.$ if and only if the boundary $\partial \Omegaega$ is uniformly characterized by local $W^{m-1\slash r}_r$ graph functions.
That is, for any point $x_0=(x_0',x_{0N}) \in \partial \Omegaega,$ one can choose a Cartesian coordinate system with origin $x_0$ and coordinates $y=(y',y_{_N}):=(y_1,...,y_{_{N-1}},y_{_N}),$ as well as positive constants $\alphapha, \betata, K$ and some $W^{m-1\slash r}_r$ function $h$ with $\|h\|_{W^{m-1\slash r}_r(B_{\alphapha}'(x_0'))} \leq K$ such that
\betagin{gather*}
\{(y',y_{_N}): h(y')-\betata <y_{_N} < h(y'), |y'|<\alphapha \}=\Omegaega \cap U_{\alphapha,\betata, h}(x_0),\\
\{(y',y_{_N}): y_{_N} = h(y'), |y'|<\alphapha \} = \partial\Omegaega \cap U_{\alphapha,\betata, h}(x_0),
\end{gather*}
where $U_{\alphapha,\betata, h}(x_0):= \{(y',y_{_N}): h(y')-\betata <y_{_N} < h(y')+\betata, |y'|<\alphapha \}$ and $B_\alphapha'(x_0'):=\{y'\in \mathbb{R}^{N-1}: |y'-x_0'| < \alphapha\}.$
Moreover, the choices of $\alphapha, \betata, K$ are independent of the location of $x_0,$
Assume that $\Omegaega$ is some domain in $\mathbb{R}^N$ with disjoint boundaries $\Gammamma_0$ and $\Gammamma_1,$ where the case $\Gammamma_0 = \emptyset$ or $\Gammamma_2=\emptyset$ is allowed.
We say $\Omegaega$ is of type $W^{3,2}_r$ for simplicity, if $\Gammamma_k$ is uniformly $W^{3-1\slash r -k}_{r}$ for $k=0,1.$
\end{defi}
At last, we recall \cite[Theorem 2.1]{Shi2016} on the Laplace-Beltrami operator.
For any $\lambdambda_0>0$ and $0<\varepsilon<\pi\slash 2,$ we introduce the sectorial regions
\footnote{One may also refer to the Figure \ref{fig:sect} below for $\Sigmagma_{\varepsilon,\lambdambda_0}.$}
\betagin{gather*}
\Sigmagma_{\varepsilon}:=\big\{z \in \mathbb{C} \backslash \{0\} : |\arg{z}| \leq \pi -\varepsilon \big\},
\quad
\Sigmagma_{\varepsilon,\lambdambda_0}:=\{z \in \Sigmagma_{\varepsilon} : |z|\geq \lambdambda_0\}.
\end{gather*}
\betagin{prop}\lambdabel{prop:resolvent_LB}
Let $0<\varepsilon<\pi\slash 2,$ $1<q,q':=q\slash (q-1)<\infty,$ $N<r<\infty$ and
$r \geq \max\{q,q'\}.$ For any uniform $W^{2-1\slash r}_r$ boundary $\Gammamma \subset \partial \Omegaega,$ there exists a constant $\lambdambda_1=\lambdambda_1(\varepsilon,\Gammamma)>0,$ such that $\Sigmagma_{\varepsilon,\lambdambda_1}$ is contained in the resolvent set $\rho(\Deltalta_{\Gammamma})$ of $\Deltalta_{\Gammamma}.$ That is, for any $\lambdambda \in \Sigmagma_{\varepsilon,\lambdambda_1}$ and $f \in W^{-1\slash q}_q(\Gammamma),$
the resolvent problem
\betagin{equation*}
(\lambdambda -\Deltalta_{\Gammamma}) u =f \,\,\, \thetaxt{on} \,\,\, \Gammamma
\end{equation*}
admits a unique solution $u \in W^{2-1\slash q}_q(\Gammamma)$ possessing the estimates
\betagin{equation*}
\|u\|_{W^{2-1\slash q}_q(\Gammamma)} \leq C_{\varepsilon,q,r,\Gammamma} \|f\|_{W^{-1\slash q}_q(\Gammamma)}.
\end{equation*}
\end{prop}
\section{Main results}
In this section, we first state the results for the resolvent problem of \eqref{eq:L_CNS_L} in Subsection \ref{subsec:RP}.
Then by applying the estimates of the resolvent problem, we can prove the existence of the semigroup of solution operators associated to \eqref{eq:L_CNS_L} in Subsection \ref{subsec:SG} and the property of maximal regularity of \eqref{eq:L_CNS_L} in the last part.
\subsection{Reduced resolvent problem}
\lambdabel{subsec:RP}
Now, let us begin with the following resolvent problem of \eqref{eq:L_CNS_L},
\betagin{equation}\lambdabel{eq:GR_CNS}
\left\{\betagin{aligned}
&\lambdambda \eta + \gammamma_1 \di \boldsymbol{u} = d
&&\quad\hbox{in}\quad \Omegaega, \\
&\gammamma_1 \lambdambda \boldsymbol{u} -\Di\big(\mathbb{S}(\boldsymbol{u})-\gammamma_2 \eta \mathbb{I} \big)= \boldsymbol{F}
&&\quad\hbox{in}\quad \Omegaega,\\
&\big( \mathbb{S}(\boldsymbol{u}) - \gammamma_2 \eta \mathbb{I}\big) \boldsymbol{n}_{\Gammamma_0}
+\sigmagma (m-\Deltalta_{\Gammamma_0})h \,\boldsymbol{n}_{\Gammamma_0} = \boldsymbol{G}
&&\quad\hbox{on}\quad \Gammamma_0, \\
&\lambdambda h - \boldsymbol{u} \cdot \boldsymbol{n}_{\Gammamma_0} = K
&&\quad\hbox{on}\quad \Gammamma_0,\\
&\boldsymbol{u} = \boldsymbol{0} &&\quad\hbox{on}\quad \Gammamma_1,
\end{aligned}\right.
\end{equation}
where $\gammamma_1 =\gammamma_1(x)$ and $\gammamma_2 =\gammamma_2(x)$ are uniformly continuous functions defined on $\overline{\Omegaega}.$ Moreover, there exist constants $\rho_1,\rho_2,\rho_3$ such that
\betagin{gather}\lambdabel{hyp:gamma_GR}
0<\rho_1 \leq \gammamma_1(x) \leq \rho_2,\quad
0 <\gammamma_2(x) \leq \rho_2, \,\,\, \forall \,\, x \in \overline{\Omegaega},\\ \nonumber
\|(\nabla \gammamma_{1},\nabla \gammamma_{2})\|_{L_r(\Omegaega)} \leq \rho_2, \quad
\rho_3:= \max\big\{ \rho_2, \|\gammamma_1\gammamma_2\|_{L_\infty(\Omegaega) \cap \wh H^1_r(\Omegaega)}\big\},
\end{gather}
with $N<r<\infty.$
For any $\nu, \lambdambda_0>0$ and $0<\varepsilon<\pi\slash 2,$ we set that
\betagin{gather}
\Lambdambda_{\varepsilon, \lambdambda_0} := \boldsymbol{i}g\{z \in \Sigmagma_{\varepsilon,\lambdambda_0} :
\big(\Re z + \frac{\rho_3}{\nu} +\varepsilon\big)^2 +(\Im z )^2
\geq \big( \frac{\rho_3}{\nu} +\varepsilon\big)^2 \boldsymbol{i}g\}.\lambdabel{def:Lambda}
\end{gather}
One may refer to the Figure \ref{fig:sect} for the graph of $\Lambdambda_{\varepsilon,\lambdambda_0}.$
\betagin{figure}[h]
\centering
\betagin{subfigure}[b]{0.4\linewidth}
\betagin{tikzpicture}
\partialth [fill=lightgray] (-4,2)--(-4,1)--(0,0)--(0,2) --(-4,2);
\partialth [fill=lightgray] (-4,-2)--(-4,-1)--(0,0)--(0,-2) --(-4,-2);
\partialth [fill=lightgray] (0,2) --(0,-2)--(1,-2)--(1,2)--(0,2);
\partialth [fill=white] (0,0) circle [radius=0.5];
\coordinate (Origin) at (0,0);
\coordinate (XAxisMin) at (-4,0);
\coordinate (XAxisMax) at (1,0);
\coordinate (YAxisMin) at (0,-2);
\coordinate (YAxisMax) at (0,2);
\draw [thick,-latex] (XAxisMin) -- (XAxisMax)
node[right] {$\Re z$};
\draw [thick,-latex] (YAxisMin) -- (YAxisMax)
node[above] {$\Im z$};
\draw [dashed] (0,0) -- (-4,1);
\draw [dashed] (0,0) -- (-4,-1);
\draw (-1,0) to [out=100, in=170] (-0.9,0.225) ;
\draw (-1.2,0.2) node[left]{$\varepsilon$};
\draw [dashed] (0,0) circle [radius=0.5];
\draw [->] (0,0) -- (0.3, 0.4);
\draw (0.3, 0.1) node{\tiny $\lambdambda_0$};
\end{tikzpicture}
\caption{$\Sigmagma_{\varepsilon,\lambdambda_0}$}
\end{subfigure}
\quad
\betagin{subfigure}[b]{0.4\linewidth}
\betagin{tikzpicture}
\partialth [fill=lightgray] (-4,2)--(-4,1)--(0,0)--(0,2) --(-4,2);
\partialth [fill=lightgray] (-4,-2)--(-4,-1)--(0,0)--(0,-2) --(-4,-2);
\partialth[fill=lightgray] (0,2) --(0,-2)--(1,-2)--(1,2)--(0,2);
\partialth [fill=white] (0,0) circle [radius=0.5];
\partialth [fill=white] (-1.8,0) circle [radius=1.8];
\draw[fill] (-1.8,0) circle [radius=0.025];
\draw (-2.5,-0.3) node{\tiny
$\big(-\frac{\rho_3}{\nu}
-\varepsilon ,0\big)$};
\coordinate (Origin) at (0,0);
\coordinate (XAxisMin) at (-4,0);
\coordinate (XAxisMax) at (1,0);
\coordinate (YAxisMin) at (0,-2);
\coordinate (YAxisMax) at (0,2);
\draw [thick,-latex] (XAxisMin) -- (XAxisMax)
node[right] {$\Re z$};
\draw [thick,-latex] (YAxisMin) -- (YAxisMax)
node[above] {$\Im z$};
\draw [dashed] (0,0) -- (-4,1);
\draw [dashed] (0,0) -- (-4,-1);
\draw (-1,0) to [out=100, in=170] (-0.9,0.225) ;
\draw (-1.2,0.2) node[left]{$\varepsilon$};
\draw [dashed] (0,0) circle [radius=0.5];
\draw [dashed] (-1.8,0) circle [radius=1.8];
\draw [->] (0,0) -- (0.3, 0.4);
\draw (0.3, 0.1) node{\tiny $\lambdambda_0$};
\end{tikzpicture}
\caption{$\Lambdambda_{\varepsilon,\lambdambda_0}$}
\end{subfigure}
\caption{Sectorial regions $\Sigmagma_{\varepsilon,\lambdambda_0}$ and $\Lambdambda_{\varepsilon,\lambdambda_0}$}
\lambdabel{fig:sect}
\end{figure}
Next, we recall the basic theory of the $\mathcal{R}-$boundedness of operator families (see \cite{DHP2003,KW2004} for further discussions).
\betagin{defi}\lambdabel{def:R-bounded}
Let $X,Y$ be two Banach spaces and $\mathcal{L}(X;Y)$ be the collection of all bounded linear operators from $X$ to $Y.$
We say that a family of bounded operators $\tau \subset \mathcal{L}(X,Y)$ is $\mathcal{R}$-bounded if for any $N \in {\mathbb{N}},$ $T_j \in \tau,$ $x_j \in X$ and the Rademacher functions $r_j (t):= \sigmagn (\sigman 2^j \pi t )$ defined for $t \in [0,1],$ the following inequality holds,
\betagin{equation*}
\boldsymbol{i}g\|\sum_{j=1}^N r_j T_j x_j\boldsymbol{i}g\|_{L_p([0,1];Y)} \leq C_p \boldsymbol{i}g\|\sum_{j=1}^N r_j x_j\boldsymbol{i}g\|_{L_p([0,1];X)}
\,\,\, \mbox{for some} \,\,p \in [1,\infty[.
\end{equation*}
Above the choice of $C_p$ depends only on $p$ but not on $N,$ $T_j,$ $x_j,$ $r_j$ and $1 \leq j\leq N.$ The smallest $C_p$ is called $\mathcal{R}$-bound of $\tau$, denoted by $R_{\mathcal{L}(X;Y)}(\tau).$
\end{defi}
Some useful comments on Definition \ref{def:R-bounded}:
\betagin{rema}\lambdabel{rmk:R-bounded}
Let $X, Y, Z$ be Banach spaces.
\betagin{enumerate}
\item $\tau \subset \mathcal{L}(X,Y)$ is $\mathcal{R}$-bounded for any $p\in [1,\infty[$ if $\tau$ is $\mathcal{R}$-bounded for some $p_0 \in [1,\infty[.$
\item Suppose that $\mathcal{T}$ and $\mathcal{S}$ are two $\mathcal{R}-$bounded famlies in $\mathcal{L}(X;Y).$ Then the sum set
$$\mathcal{T} + \mathcal{S}:=\{T+S: T \in \mathcal{T}, S\in \mathcal{S}\}$$
is $\mathcal{R}-$bounded as well, and
$\mathcal{R}_{\mathcal{L}(X;Y)} (\mathcal{T} + \mathcal{S}) \leq \mathcal{R}_{\mathcal{L}(X;Y)} (\mathcal{T} )+\mathcal{R}_{\mathcal{L}(X;Y)} (\mathcal{S}).$
\item Assume that the families $\mathcal{T}\subset \mathcal{L}(X;Y)$ and $\mathcal{S} \subset \mathcal{L}(Y;Z)$ are $\mathcal{R}-$bounded. Then the composition set
$$\mathcal{S}\mathcal{T} :=\{S\circ T: T \in \mathcal{T}, S\in \mathcal{S}\} \subset \mathcal{L}(X;Z)$$
is $\mathcal{R}-$bounded as well, and $\mathcal{R}_{\mathcal{L}(X;Z)} (\mathcal{S}\mathcal{T}) \leq \mathcal{R}_{\mathcal{L}(X;Y)} (\mathcal{T} ) \mathcal{R}_{\mathcal{L}(Y;Z)} (\mathcal{S}).$
\item Let $\mathcal{T} \subset \mathcal{L}\big(L_{q}(G)\big)$ be a family of operators for $1 < q<\infty$ and some domain $G \subset \mathbb{R}^N.$ Then $\mathcal{T}$ is $\mathcal{R}-$bounded if and only if there is a constant $C$ such that
\betagin{equation*}
\boldsymbol{i}g\| \big(\sum_{j=1}^{N_0} |T_j f_{j}|^2 \big)^{1\slash 2} \boldsymbol{i}g\|_{L_q(G)}
\leq C\boldsymbol{i}g\| \big(\sum_{j=1}^{N_0} |f_{j}|^2 \big)^{1\slash 2} \boldsymbol{i}g\|_{L_q(G)},
\end{equation*}
for any $N_0 \in \mathbb{N},$ $f_j \in L_q(G)$ and $T_j \in \mathcal{T}.$
In particular, $\{ T_{\lambdambda} : T_{\lambdambda}f := \lambdambda^{-s} f, \lambdambda \in \Sigmagma_{\varepsilon,\lambdambda_0}\}$ for $s,\lambdambda_0>0$ is a $\mathcal{R}$-bounded family in $\mathcal{L}\big(L_q(G)\big).$
\end{enumerate}
\end{rema}
To prove the maximal regularity property of the model problem, we need the following theorem on the Fourier multiplier obtained in \cite{Weis2001}.
\betagin{theo}[Weis] \lambdabel{thm:Weis}
Let $X$ and $Y$ be two UMD Banach spaces
\footnote{One may refer to \cite{KW2004} for the UMD property.}
and $1<p<\infty.$ Let $M(\cdot)$ be a mapping in
$C^{1}\big(\mathbb{R}\backslash \{0\}; \mathcal{L}(X;Y) \big)$ such that
\betagin{equation*}
\mathcal{R}_{\mathcal{L}(X; Y)} \big(\big\{ (\tau \partial_{\tau})^{\ell} M (\tau) : \tau \in \mathbb{R}\backslash \{0\} \big\}\big)
\leq r_b \quad (\ell =0,1),
\end{equation*}
with some constant $r_b>0.$ Then the multiplier operator $T_{M}(\varphi) := \mathcal{F}^{-1} \big[M\mathcal{F}[\varphi] \big]$ for any $\varphi \in \mathcal{S}(\mathbb{R};X)$ can be uniquely extended to a bounded linear operator from $L_p(\mathbb{R};X)$ into $L_p(\mathbb{R};Y)$ with the bound
\betagin{equation*}
\|T_{M}\|_{\mathcal{L}\big(L_p(\mathbb{R};X);L_p(\mathbb{R};Y)\big)} \leq C_{p,X,Y} r_b.
\end{equation*}
\end{theo}
With above definitions and comments, our main result for the model problem \eqref{eq:GR_CNS} is as follows.
\betagin{theo}\lambdabel{thm:GR_CNS}
Let $0<\varepsilon<\pi\slash 2,$ $\sigmagma, \mu, \nu>0,$ $1<q,q':=q\slash (q-1)<\infty,$ $N<r<\infty$ and
$r \geq \max\{q,q'\}.$ Assume that $\Omegaega$ is of type $W^{3,2}_r,$ $m\geq \lambdambda_1(\varepsilon,\Gammamma_0)$ by Proposition \ref{prop:resolvent_LB}, and \eqref{hyp:gamma_GR} is satisfied.
Set that
\betagin{gather*}
X_q(\Omegaega) := H^1_q(\Omegaega) \times L_q(\Omegaega)^N \times H^{1}_q(\Omegaega)^N\times W^{2-1\slash q}_q(\Gammamma_0),\\
\mathcal{X}_q(\Omegaega) :=H^1_q(\Omegaega) \times L_q(\Omegaega)^N \times L_q(\Omegaega)^N \times H^{1}_q(\Omegaega)^N\times W^{2-1\slash q}_q(\Gammamma_0).
\end{gather*}
For any $(d,\boldsymbol{F},\boldsymbol{G},K) \in X_q(\Omegaega),$ there exist constants $\lambdambda_0,r_b \geq 1$ and operator families
\betagin{align*}
\mathcal{P}(\lambdambda,\Omegaega) & \in
{\rm Hol}\,\boldsymbol{i}g( \Lambdambda_{\varepsilon,\lambdambda_0} ; \mathcal{L}\big(\mathcal{X}_q(\Omegaega);H^1_q(\Omegaega) \big) \boldsymbol{i}g),\\
\mathcal{A}(\lambdambda,\Omegaega) & \in
{\rm Hol}\,\boldsymbol{i}g( \Lambdambda_{\varepsilon,\lambdambda_0} ; \mathcal{L}\big(\mathcal{X}_q(\Omegaega);H^2_q(\Omegaega)^N \big) \boldsymbol{i}g),\\
\mathcal{H}(\lambdambda,\Omegaega) & \in
{\rm Hol}\,\boldsymbol{i}g( \Lambdambda_{\varepsilon,\lambdambda_0} ; \mathcal{L}\big(\mathcal{X}_q(\Omegaega);W^{3-1\slash q}_q(\Gammamma_0) \big) \boldsymbol{i}g),
\end{align*}
such that $(\eta,\boldsymbol{u},h):=\big( \mathcal{P}(\lambdambda,\Omegaega), \mathcal{A}(\lambdambda,\Omegaega), \mathcal{H}(\lambdambda,\Omegaega) \big) (d,\boldsymbol{F},\lambdambda^{1\slash 2}\boldsymbol{G},\boldsymbol{G},K)$
is the unique solution of \eqref{eq:GR_CNS}. Moreover, we have
\betagin{gather*}
\mathcal{R}_{\mathcal{L}\big(\mathcal{X}_q(\Omegaega); H^{1}_q(\Omegaega) \big)}
\boldsymbol{i}g( \boldsymbol{i}g\{ (\tau \partial_{\tau})^{\ell}\big( \lambdambda \mathcal{P}(\lambdambda,\Omegaega)\big) :
\lambdambda \in \Lambdambda_{\varepsilon,\lambdambda_0} \boldsymbol{i}g\}\boldsymbol{i}g) \leq r_b,\\
\mathcal{R}_{\mathcal{L}\big(\mathcal{X}_q(\Omegaega); H^{2-j}_q(\Omegaega)^N \big)}
\boldsymbol{i}g( \boldsymbol{i}g\{ (\tau \partial_{\tau})^{\ell}\big( \lambdambda^{j\slash 2}\mathcal{A}(\lambdambda,\Omegaega)\big) :
\lambdambda \in \Lambdambda_{\varepsilon,\lambdambda_0} \boldsymbol{i}g\}\boldsymbol{i}g) \leq r_b,\\
\mathcal{R}_{\mathcal{L}\big(\mathcal{X}_q(\Omegaega); W^{3-1\slash q-j'}_q(\Gammamma_0) \big)}
\boldsymbol{i}g( \boldsymbol{i}g\{ (\tau \partial_{\tau})^{\ell}\big( \lambdambda^{j'}\mathcal{H}(\lambdambda,\Omegaega)\big) :
\lambdambda \in \Lambdambda_{\varepsilon,\lambdambda_0} \boldsymbol{i}g\}\boldsymbol{i}g) \leq r_b,
\end{gather*}
for $\ell, j'=0,1,$ $j=0,1,2,$ and $\tau := \Im \lambdambda.$
Above the choices of $\lambdambda_0$ and $r_b$ depend solely on the parameters
$\varepsilon,$ $\sigmagma,$ $m,$ $\mu,$ $\nu,$ $q,$ $r,$ $N,$ $\rho_1,$ $\rho_2,$ $\rho_3$ and $\Omegaega.$
\end{theo}
In fact, we shall see that Theorem \ref{thm:GR_CNS} is a consequence of
some generalized model problem.
To this end, let us introduce some parameter $\zeta$ fulfilling $|\zeta|\leq \zeta_0$ and either of the following cases
\betagin{equation*}
\hbox{(C1)}\,\, \zeta = \lambdambda^{-1};\quad
\hbox{(C2)}\,\,\zeta \in \Sigmagma_{\varepsilon} \,\, \hbox{and} \,\,\Re \zeta <0;\quad
\hbox{(C3)}\,\,\Re \zeta \geq 0. \hspace*{1cm}
\end{equation*}
Then we define
\betagin{equation}\lambdabel{eq:Gamma}
\Gammamma_{\varepsilon,\lambdambda_0,\zeta} :=
\betagin{cases}
\Lambdambda_{\varepsilon,\lambdambda_0} & \hbox{for (C1)},\\
\big\{\lambdambda \in \mathbb{C} : \Re \lambdambda \geq \big| \frac{\Re \zeta}{\Im \zeta}\big| |\Im \lambdambda|,
\,\, \Re \lambdambda \geq \lambdambda_0 \big\} & \hbox{for (C2)},\\
\{\lambdambda \in \mathbb{C} : \Re \lambdambda \geq \lambdambda_0 \} & \hbox{for (C3)}.
\end{cases}
\end{equation}
For $\lambdambda \in \Gammamma_{\varepsilon, \lambdambda_0,\zeta},$ we consider the model problem
\betagin{equation}\lambdabel{eq:RR_CNS_0}
\left\{\betagin{aligned}
& \lambdambda \boldsymbol{v} -\gammamma_{1}^{-1}\Di \big( \mathbb{S}(\boldsymbol{v}) + \zeta \gammamma_3 \di \boldsymbol{v} \mathbb{I} \big)
= \boldsymbol{f} &&\quad\hbox{in}\quad \Omegaega,\\
& \big( \mathbb{S}(\boldsymbol{v}) +\zeta \gammamma_3 \di \boldsymbol{v} \mathbb{I} \big)\boldsymbol{n}_{\Gammamma_{0}}
+\sigmagma (m-\Deltalta_{\Gammamma_{0}})h \,\boldsymbol{n}_{\Gammamma_{0}} = \boldsymbol{g} &&\quad\hbox{on}\quad \Gammamma_0, \\
&\lambdambda h - \boldsymbol{v} \cdot \boldsymbol{n}_{\Gammamma_0} = k &&\quad\hbox{on}\quad \Gammamma_0,\\
&\boldsymbol{v} = \boldsymbol{0} &&\quad\hbox{on}\quad \Gammamma_1,\\
\end{aligned}\right.
\end{equation}
where $\gammamma_1$ and $\gammamma_3$ are uniformly continuous functions on $\overline{\Omegaega}$ such that
\betagin{gather}\lambdabel{hyp:gamma_GR_2}
0<\rho_1 \leq \gammamma_1(x) \leq \rho_2,\quad
0 <\gammamma_3(x) \leq \rho_3, \,\,\, \forall \,\, x \in \overline{\Omegaega},\quad
\|(\nabla \gammamma_{1},\nabla \gammamma_{3})\|_{L_r(\Omegaega)} \leq \rho_3,
\end{gather}
for some constants $\rho_1,\rho_2,\rho_3 >0$ and $N<r<\infty.$
The following result concerning \eqref{eq:RR_CNS_0} will be established later.
\betagin{theo}\lambdabel{thm:RR_CNS}
Let $0<\varepsilon<\pi\slash 2,$ $\sigmagma, \mu, \nu>0,$ $1<q<\infty,$ $N<r<\infty$ and
$r \geq q.$ Assume that $\Omegaega$ is of type $W^{3,2}_r,$ $m\geq \lambdambda_1(\varepsilon,\Gammamma_0)$ by Proposition \ref{prop:resolvent_LB},
and \eqref{hyp:gamma_GR_2} is satisfied. Set that
\betagin{equation*}
Y_q(\Omegaega) := L_q(\Omegaega)^N \times H^{1}_q(\Omegaega)^N\times H^{2}_q(\Omegaega),\quad
\mathcal{Y}_q(\Omegaega) := L_q(\Omegaega)^N \times Y_q(\Omegaega).
\end{equation*}
For any $(\boldsymbol{f},\boldsymbol{g},k) \in Y_q(\Omegaega),$ there exist constants $\lambdambda_0, r_b \geq 1$ and operator families
\betagin{align*}
\mathcal{A}_0(\lambdambda,\Omegaega) & \in
{\rm Hol}\,\boldsymbol{i}g( \Gammamma_{\varepsilon,\lambdambda_0,\zeta} ; \mathcal{L}\big(\mathcal{Y}_q(\Omegaega);H^2_q(\Omegaega)^N \big) \boldsymbol{i}g),\\
\mathcal{H}_0(\lambdambda,\Omegaega) & \in
{\rm Hol}\,\boldsymbol{i}g( \Gammamma_{\varepsilon,\lambdambda_0,\zeta} ; \mathcal{L}\big(\mathcal{Y}_q(\Omegaega);H^3_q(\Omegaega) \big) \boldsymbol{i}g),
\end{align*}
such that $(\boldsymbol{v}, h):=\big(\mathcal{A}_0(\lambdambda,\Omegaega),\mathcal{H}_0(\lambdambda,\Omegaega) \big)(\boldsymbol{f},\lambdambda^{1\slash 2}\boldsymbol{g},\boldsymbol{g},k)$
is a solutions of \eqref{eq:RR_CNS_0}. Moreover, we have
\betagin{gather*}
\mathcal{R}_{\mathcal{L}\big(\mathcal{Y}_q(\Omegaega); H^{2-j}_q(\Omegaega)^N \big)}
\boldsymbol{i}g( \boldsymbol{i}g\{ (\tau \partial_{\tau})^{\ell}\big( \lambdambda^{j\slash 2}\mathcal{A}_0(\lambdambda,\Omegaega)\big) :
\lambdambda \in \Gammamma_{\varepsilon,\lambdambda_0,\zeta} \boldsymbol{i}g\}\boldsymbol{i}g) \leq r_b,\\
\mathcal{R}_{\mathcal{L}\big(\mathcal{Y}_q(\Omegaega); H^{3-j'}_q(\Omegaega) \big)}
\boldsymbol{i}g( \boldsymbol{i}g\{ (\tau \partial_{\tau})^{\ell}\big( \lambdambda^{j'}\mathcal{H}_0(\lambdambda,\Omegaega)\big) :
\lambdambda \in \Gammamma_{\varepsilon,\lambdambda_0,\zeta} \boldsymbol{i}g\}\boldsymbol{i}g) \leq r_b,
\end{gather*}
for $\ell, j'=0,1,$ $j=0,1,2,$ and $\tau := \Im \lambdambda.$
Above the constants $\lambdambda_0$ and $r_b$ depend solely on
$\varepsilon,$ $\sigmagma,$ $m,$ $\mu,$ $\nu,$ $\zeta_0,$ $q,$ $r,$ $N,$ $\rho_1,$ $\rho_2,$ $\rho_3$ and $\Omegaega.$
\end{theo}
By admitting Theorem \ref{thm:RR_CNS} for a while, it is not hard to prove Theorem \ref{thm:GR_CNS}.
Before that, let us recall a technical lemma from \cite{Shi2013}.
\betagin{lemm}\lambdabel{lemma:ab_BH}
Let $1\leq q\leq r <\infty$ and $N<r<\infty.$
Suppose that $\Omegaega$ is a uniform $W^{2-1\slash r}_r$ domain.
Then there exists a positive constant $C=C_{N,r,q,\Omegaega}$ such that
\betagin{equation*}
\|ab\|_{L_q(\Omegaega)} \leq \sigmagma_0 \|b\|_{H^1_q(\Omegaega)}
+ C \sigmagma_0^{-\frac{N}{r-N}} \|a\|_{L_r(\Omegaega)}^{\frac{r}{r-N}} \|b\|_{L_q(\Omegaega)},
\end{equation*}
for any $\sigmagma_0>0,$ $a\in L_r(\Omegaega)$ and $b \in H^1_q(\Omegaega).$ In particular, one can replace $\|b\|_{H^1_q(\Omegaega)}$ by $\|\nabla b\|_{L_q(\Omegaega)}$ in the inequality above, whenever $\Omegaega$ is $\mathbb{R}^N$ or $\mathbb{R}^N_+.$
\end{lemm}
\betagin{proof}[The proof of Theorem \ref{thm:GR_CNS}]
By our assumptions on $\Omegaega,$ there exist linear mappings
\betagin{equation*}
\mathcal{R}_{\Gammamma_0}: H^{3}_q(\Omegaega) \rightarrow W^{3-1\slash q}_q(\Gammamma_0)
\,\,\,\thetaxt{and}\,\,\,
\mathcal{E}_{\Gammamma_0}: W^{3-1\slash q}_q(\Gammamma_0) \rightarrow H^{3}_q(\Omegaega),
\end{equation*}
such that $\|\mathcal{R}_{\Gammamma_0}f\|_{W^{n-1\slash q}_q(\Gammamma_0)} \leq C_{q,\Omegaega}\|f\|_{H^n_q(\Omegaega)}$
and $\|\mathcal{E}_{\Gammamma_0}g\|_{H^{n}_q(\Omegaega)} \leq C_{q,\Omegaega}\|g\|_{W^{n-1\slash q}_q(\Gammamma_0)}$ for $n=2,3.$
\smallbreak
Next, after eliminating $\eta$ via $\eqref{eq:GR_CNS}_1,$ $(\boldsymbol{u},h)$ satisfies \eqref{eq:RR_CNS_0}
with $\gammamma_3,$ $\boldsymbol{f},$ $\boldsymbol{g}$ and $k$ given by
\betagin{equation*}
\gammamma_3 := \gammamma_1\gammamma_2,\quad
\boldsymbol{f}:= \gammamma_1^{-1} \boldsymbol{F} - \gammamma_1^{-1} \lambdambda^{-1}\nabla (\gammamma_2 d), \quad
\boldsymbol{g}:= \boldsymbol{G} + \lambdambda^{-1}\gammamma_2 d \, \boldsymbol{n}_{\Gammamma_0}, \quad
k := \mathcal{R}_{\Gammamma_0}K.
\end{equation*}
Denote $\mathcal{S}(\lambdambda)(d,\boldsymbol{F},\lambdambda^{1\slash 2}\boldsymbol{G},\boldsymbol{G},K):=(\boldsymbol{f},\lambdambda^{1\slash 2} \boldsymbol{g}, \boldsymbol{g},k)$
for any $(d,\boldsymbol{F},\boldsymbol{G},K) \in X_q(\Omegaega).$
Then Lemma \ref{lemma:ab_BH} yields for $r\geq q,$
\betagin{equation}\lambdabel{es:R-bdd-S}
\mathcal{R}_{\mathcal{L}\big(\mathcal{X}_q(\Omegaega); \mathcal{Y}_q(\Omegaega) \big)}
\boldsymbol{i}g( \boldsymbol{i}g\{ (\tau \partial_{\tau})^{\ell} \mathcal{S}(\lambdambda) :
\lambdambda \in \Sigmagma_{\varepsilon,1} \boldsymbol{i}g\}\boldsymbol{i}g) \leq C_{\rho_1,\rho_2,\rho_3,\Omegaega}.
\end{equation}
Then we define $\mathcal{A}(\lambdambda,\Omegaega):= \mathcal{A}_0(\lambdambda,\Omegaega) \circ \mathcal{S}(\lambdambda),$
$\mathcal{H}(\lambdambda,\Omegaega):= \mathcal{E}_{\Gammamma_0} \circ \mathcal{H}_0(\lambdambda,\Omegaega) \circ \mathcal{S}(\lambdambda)$ and
\betagin{equation*}
\mathcal{P}(\lambdambda,\Omegaega) (d,\boldsymbol{F},\lambdambda^{1\slash 2}\boldsymbol{G},\boldsymbol{G},K)
:= \lambdambda^{-1} \boldsymbol{i}g( d -\gammamma_1 \di \big( \mathcal{A}(\lambdambda,\Omegaega)(d,\boldsymbol{F},\lambdambda^{1\slash 2}\boldsymbol{G},\boldsymbol{G},K)\big) \boldsymbol{i}g),
\end{equation*}
which are the desired operators due to Theorem \ref{thm:RR_CNS} and \eqref{es:R-bdd-S}.
\smallbreak
Finally, let us prove the uniqueness. Suppose that $\boldsymbol{u} \in H^2_q(\Omegaega)^N$ and $h \in W^{3-1\slash q}_q(\Gammamma_0)$ satisfy \eqref{eq:GR_CNS} with $(d,\boldsymbol{F},\boldsymbol{G},k)$ vanishing.
For any $\lambdambda \in \Gammamma_{\varepsilon, \lambdambda_0,\zeta},$ $\boldsymbol{P}hi \in C_0^{\infty}(\Omegaega)^N$ and $\varphii \in C^{\infty}_0(\mathbb{R}^N),$
we consider that
\betagin{equation}\lambdabel{eq:RR_CNS_1}
\left\{\betagin{aligned}
& \gammamma_{1} \lambdambda \boldsymbol{v} -\Di \big( \mathbb{S}(\boldsymbol{v}) + \zeta \gammamma_3 \di \boldsymbol{v} \mathbb{I} \big)
= \boldsymbol{P}hi &&\quad\hbox{in}\quad \Omegaega,\\
& \big( \mathbb{S}(\boldsymbol{v}) +\zeta \gammamma_3 \di \boldsymbol{v} \mathbb{I} \big)\boldsymbol{n}_{\Gammamma_{0}}
+\sigmagma (m-\Deltalta_{\Gammamma_{0}})\theta \,\boldsymbol{n}_{\Gammamma_{0}} = \boldsymbol{0} &&\quad\hbox{on}\quad \Gammamma_0, \\
&\lambdambda \theta - \boldsymbol{v} \cdot \boldsymbol{n}_{\Gammamma_0} = \varphii &&\quad\hbox{on}\quad \Gammamma_0,\\
&\boldsymbol{v} = \boldsymbol{0} &&\quad\hbox{on}\quad \Gammamma_1.\\
\end{aligned}\right.
\end{equation}
Then there exists a solution $(\boldsymbol{v},\theta) \in H^2_{q'}(\Omegaega)^N \times W^{3-1\slash {q'}}_{q'}(\Gammamma_0)$ of \eqref{eq:RR_CNS_1}.
Note the facts that
\betagin{equation*}
\sum_{i,j=1}^N \int_{\Omegaega} \big(\mathbb{S}(\boldsymbol{u}) + \zeta\gammamma_3 \di \boldsymbol{u} \mathbb{I} \big)^i_j \partial_j v^i \,dx
= \sum_{i,j=1}^N \int_{\Omegaega} \big(\mathbb{S}(\boldsymbol{v}) + \zeta\gammamma_3 \di \boldsymbol{v} \mathbb{I} \big)^i_j \partial_j u^i \,dx,
\end{equation*}
and $\big( (m-\Deltalta_{\Gammamma_0})h,\theta \big)_{\Gammamma_0}=\big( h,(m-\Deltalta_{\Gammamma_0}) \theta \big)_{\Gammamma_0}.$
Then it is not hard to see from \eqref{eq:GR_CNS} and \eqref{eq:RR_CNS_1} that
\betagin{equation*}
(\boldsymbol{u},\boldsymbol{P}hi)_{\Omegaega} = \big( \sigmagma (m-\Deltalta_{\Gammamma_0}) h , \varphii \big)_{\Gammamma_0},
\end{equation*}
which yields that $\boldsymbol{u}|_{\Omegaega}=\boldsymbol{0}$ and $(m-\Deltalta_{\Gammamma_0}) h|_{\Gammamma_0}=0$ for the arbitrary choices of $\boldsymbol{P}hi$ and $\varphii.$
As $m-\Deltalta_{\Gammamma_0}$ is invertible by Proposition \ref{prop:resolvent_LB}, $h|_{\Gammamma_0}=0.$
This completes the proof of Theorem \ref{thm:GR_CNS}.
\end{proof}
\subsection{Generation of analytic semigroup}
\lambdabel{subsec:SG}
In this subsection, we study the following system with nonhomogeneous initial data within the semigroup framework,
\betagin{equation}\lambdabel{eq:eta-Lame_0}
\left\{\betagin{aligned}
&\partial_t \eta + \gammamma_1 \di \boldsymbol{u} = 0
&&\quad\hbox{in}\quad \Omegaega \times \mathbb{R}_+, \\
&\gammamma_1 \partial_t \boldsymbol{u} -\Di\big(\mathbb{S}(\boldsymbol{u})-\gammamma_2 \eta \mathbb{I} \big)= \boldsymbol{0}
&&\quad\hbox{in}\quad \Omegaega \times \mathbb{R}_+,\\
&\big( \mathbb{S}(\boldsymbol{u}) - \gammamma_2 \eta \mathbb{I}\big) \boldsymbol{n}_{\Gammamma_0}
+\sigmagma (m-\Deltalta_{\Gammamma_0})h \,\boldsymbol{n}_{\Gammamma_0} = \boldsymbol{0}
&&\quad\hbox{on}\quad \Gammamma_0 \times \mathbb{R}_+, \\
&\partial_t h - \boldsymbol{u} \cdot \boldsymbol{n}_{\Gammamma_0} = 0
&&\quad\hbox{on}\quad \Gammamma_0 \times \mathbb{R}_+,\\
&\boldsymbol{u} = \boldsymbol{0} &&\quad\hbox{on}\quad \Gammamma_1 \times \mathbb{R}_+, \\
&(\eta, \boldsymbol{u},h)|_{t=0} = (\eta_0, \boldsymbol{u}_0,h_0) &&\quad\hbox{in}\quad \Omegaega.
\end{aligned}\right.
\end{equation}
Note that the boundary conditions in \eqref{eq:eta-Lame_0} are equivalent to
\betagin{equation}\lambdabel{cdt:compatible}
\big(\mathbb{S}(\boldsymbol{u}) \boldsymbol{n}_{\Gammamma_0}\big)_{\tau}|_{\Gammamma_0} = \boldsymbol{0},\quad
\mathbb{S}(\boldsymbol{u})\boldsymbol{n}_{\Gammamma_0} \cdot \boldsymbol{n}_{\Gammamma_0} -\gammamma_2\eta+ \sigmagma(m-\Deltalta_{\Gammamma_0})h |_{\Gammamma_0} =0,\quad
\boldsymbol{u}|_{\Gammamma_1}=\boldsymbol{0}.
\end{equation}
Here $\boldsymbol{f}_{\tau}:=\boldsymbol{f} - (\boldsymbol{f}\cdot \boldsymbol{n}_{\Gammamma_0}) \boldsymbol{n}_{\Gammamma_0}$ stands for the tangential component of $\boldsymbol{f}$ along $\Gammamma_0.$
Then we introduce the functional spaces
\betagin{align*}
\mathfrak{X}_q(\Omegaega)&:=H^1_q(\Omegaega) \times L_q(\Omegaega)^{N}\times W^{2-1\slash q}_q(\Gammamma_0),\\
\mathcal{D}_q(\mathcal{A})&:=\boldsymbol{i}g\{ (\eta,\boldsymbol{u},h) \in \mathfrak{X}_q(\Omegaega): \boldsymbol{u} \in H^2_q(\Omegaega)^N,\,\,h\in W^{3-1\slash q}_q(\Gammamma_0), \,\,
\thetaxt{\eqref{cdt:compatible} holds}\boldsymbol{i}g\},
\end{align*}
endowed with the norms
\betagin{align*}
\|(\eta,\boldsymbol{u},h)\|_{\mathfrak{X}_q(\Omegaega)} &:= \|\eta\|_{H^1_q(\Omegaega)}+\|\boldsymbol{u}\|_{L_q(\Omegaega)}+\|h\|_{W^{2-1\slash q}_q(\Gammamma_0)},\\
\|(\eta,\boldsymbol{u},h)\|_{\mathcal{D}_q(\mathcal{A})}& := \|\eta\|_{H^1_q(\Omegaega)}+\|\boldsymbol{u}\|_{H^2_q(\Omegaega)}+\|h\|_{ W^{3-1\slash q}_q(\Gammamma_0)}.
\end{align*}
Furthermore, define the linear operator
\betagin{equation*}
\mathcal{A} \boldsymbol{U}:=
\betagin{bmatrix}
-\gammamma_1 \di \boldsymbol{u} \\
\gammamma_1^{-1}\Di\big(\mathbb{S}(\boldsymbol{u})-\gammamma_2 \eta \mathbb{I} \big) \\
\boldsymbol{u} \cdot \boldsymbol{n}_{\Gammamma_0}
\end{bmatrix}
\,\,\,\thetaxt{for}\,\,\,\, \boldsymbol{U}:=(\eta,\boldsymbol{u},h) \in \mathcal{D}_q(\mathcal{A}),
\end{equation*}
and the following functional space by the real interpolation theory,
\betagin{equation*}
\mathcal{D}_{q,p}(\Omegaega) := \big( \mathfrak{X}_q(\Omegaega), \mathcal{D}_q(\mathcal{A}) \big)_{1-1 \slash p,p}
\subset H^1_q(\Omegaega) \times B^{2(1-1\slash p)}_{q,p}(\Omegaega) \times B^{3-1\slash q-1\slash p}_{q,p}(\Gammamma_0),
\end{equation*}
with $\|(\eta,\boldsymbol{u},h)\|_{\mathcal{D}_{q,p}(\Omegaega)}:=\|\eta\|_{H^1_q(\Omegaega)}
+\|\boldsymbol{u}\|_{B^{2(1-1\slash p)}_{q,p}(\Omegaega)} +\|h\|_{B^{3-1\slash q-1\slash p}_{q,p}(\Gammamma_0)}.$
\smallbreak
Thanks to above settings, \eqref{eq:eta-Lame_0} can be regarded as the abstract Cauchy problem
\betagin{equation*}
\partial_t \boldsymbol{U} - \mathcal{A}\boldsymbol{U} = \boldsymbol{0} \,\,\, \thetaxt{for}\,\,\, t>0, \quad \boldsymbol{U}|_{t=0}=(\eta_0,\boldsymbol{u}_0,h_0),
\end{equation*}
whose resolvent problem is formulated as follows
\betagin{equation*}
\lambdambda \boldsymbol{U} - \mathcal{A}\boldsymbol{U} = \boldsymbol{F} \,\,\, \thetaxt{for}\,\,\, \lambdambda \in \mathbb{C} \,\,\, \thetaxt{and} \,\,\, \boldsymbol{F}=(d,\boldsymbol{f},k) \in \mathfrak{X}_q(\Omegaega).
\end{equation*}
By Theorem \ref{thm:GR_CNS}, there exists $\lambdambda_0>0$ such that $\Lambdambda_{\varepsilon,\lambdambda_0}$ is contained in the resolvent set $\rho(\mathcal{A})$ of $\mathcal{A}.$
Moreover, we have
\betagin{equation*}
|\lambdambda| \|\boldsymbol{U}\|_{\mathfrak{X}_q(\Omegaega)} + \|\boldsymbol{U}\|_{\mathcal{D}_q(\mathcal{A})} \leq C r_b \|\boldsymbol{F}\|_{\mathfrak{X}_q(\Omegaega)}
\quad (\lambdambda \in \Lambdambda_{\varepsilon,\lambdambda_0}),
\end{equation*}
for some constant $C>0.$
Then by the semigroup theory and interpolation arguments in \cite[Theorem 3.9]{ShiShi2008},
we can furnish the following results.
\betagin{theo}\lambdabel{thm:semigroup}
Let $0<\varepsilon<\pi\slash 2,$ $\sigmagma, \mu, \nu,\rho_1,\rho_2,\rho_3>0,$ $1<q,q':=q\slash (q-1)<\infty,$ $N<r<\infty$ and
$r \geq \max\{q,q'\}.$ Assume that $\Omegaega$ is of type $W^{3,2}_r,$ $m\geq \lambdambda_1(\varepsilon,\Gammamma_0)$ by Proposition \ref{prop:resolvent_LB}, and \eqref{hyp:gamma_GR} is satisfied.
Denote that $\boldsymbol{U}_0 := (\eta_0, \boldsymbol{u}_0,h_0).$
Then there exist positive constants $\gammamma_0,$ $C$ such that the following assertions hold true.
\betagin{enumerate}
\item The operator $\mathcal{A}$ generates a $C^0$ semigroup $\{T(t)\}_{t\geq 0}$ in $\mathfrak{X}_q(\Omegaega),$ which is analytic.
Moreover, we have
\betagin{equation*}
\|\boldsymbol{U}\|_{\mathfrak{X}_q(\Omegaega)} + t\big( \|\partial_t \boldsymbol{U}\|_{\mathfrak{X}_q(\Omegaega)} +\| \boldsymbol{U}\|_{\mathcal{D}_q(\mathcal{A})}\big)
\leq Ce^{\gammamma_0 t} \|\boldsymbol{U}_0\|_{\mathfrak{X}_q(\Omegaega)},
\end{equation*}
\betagin{equation*}
\|\partial_t \boldsymbol{U}\|_{\mathfrak{X}_q(\Omegaega)} +\| \boldsymbol{U}\|_{\mathcal{D}_q(\mathcal{A})}
\leq Ce^{\gammamma_0 t} \|\boldsymbol{U}_0\|_{\mathcal{D}_q(\mathcal{A})},
\end{equation*}
with $\boldsymbol{U}:=T(t) \boldsymbol{U}_0.$
\item For any $\boldsymbol{U}_0 \in \mathcal{D}_{q,p}(\Omegaega),$ \eqref{eq:eta-Lame_0} admits a unique solution
\betagin{equation*}
e^{-\gammamma_0 t}(\eta, \boldsymbol{u},h) \in H^1_{p} \big(\mathbb{R}_+;\mathfrak{X}_q(\Omegaega)\big)
\cap L_{p} \big(\mathbb{R}_+;\mathcal{D}_q(\mathcal{A})\big),
\end{equation*}
satisfying the estimates
\betagin{equation*}
\|e^{-\gammamma_0 t}\partial_t(\eta, \boldsymbol{u},h)\|_{L_p(\mathbb{R}_+;\mathfrak{X}_q(\Omegaega))}
+\|e^{-\gammamma_0 t}(\eta, \boldsymbol{u},h)\|_{L_p(\mathbb{R}_+;\mathcal{D}_q(\mathcal{A}))}
\leq C \|(\eta_0, \boldsymbol{u}_0,h_0)\|_{\mathcal{D}_{q,p}(\Omegaega)}.
\end{equation*}
\end{enumerate}
\end{theo}
\subsection{Maximal $L_p-L_q$ regularity}
In this subsection, we consider the following linear evolution equations with trivial initial data,
\betagin{equation}\lambdabel{eq:eta-Lame_1}
\left\{\betagin{aligned}
&\partial_t \eta + \gammamma_1 \di \boldsymbol{u} = d
&&\quad\hbox{in}\quad \Omegaega \times \mathbb{R}_+, \\
&\gammamma_1 \partial_t \boldsymbol{u} -\Di\big(\mathbb{S}(\boldsymbol{u})-\gammamma_2 \eta \mathbb{I} \big)= \boldsymbol{f}
&&\quad\hbox{in}\quad \Omegaega \times \mathbb{R}_+,\\
&\big( \mathbb{S}(\boldsymbol{u}) - \gammamma_2 \eta \mathbb{I}\big) \boldsymbol{n}_{\Gammamma_0}
+\sigmagma (m-\Deltalta_{\Gammamma_0})h \,\boldsymbol{n}_{\Gammamma_0} = \boldsymbol{g}
&&\quad\hbox{on}\quad \Gammamma_0 \times \mathbb{R}_+, \\
&\partial_t h - \boldsymbol{u} \cdot \boldsymbol{n}_{\Gammamma_0} = k
&&\quad\hbox{on}\quad \Gammamma_0 \times \mathbb{R}_+,\\
&\boldsymbol{u} = \boldsymbol{0} &&\quad\hbox{on}\quad \Gammamma_1 \times \mathbb{R}_+, \\
&(\eta, \boldsymbol{u},h)|_{t=0} = (0, \boldsymbol{0},0) &&\quad\hbox{in}\quad \Omegaega.
\end{aligned}\right.
\end{equation}
To describe the main result for \eqref{eq:eta-Lame_1}, we firstly introduce some useful notations.
For $\lambdambda =\gammamma+ i\tau \in \mathbb{C},$ the Laplace transform and its inverse are formulated by
\betagin{equation*}
\mathcal{L} [f] (\lambdambda):= \int_{\mathbb{R}} e^{-\lambdambda t} f(t) dt = \mathcal{F}_t [e^{-\gammamma t}f(t)](\tau), \quad
\mathcal{L}^{-1}[g] (t):= \frac{1}{2\pi}\int_{\mathbb{R}} e^{\lambdambda t} g(\tau) d\tau =e^{\gammamma t}\mathcal{F}^{-1}_{\tau} [g(\tau)](t).
\end{equation*}
For any $X$ valued function $f,$ we set $\Lambdambda^s_{\gammamma} f (t):= \mathcal{L}^{-1} \big[ \lambdambda^s \mathcal{L}[f](\lambdambda) \big]$ for any $s>0.$
Then the Bessel potential spaces are defined as follows,
\betagin{align*}
H^{s}_{p,\gammamma}(\mathbb{R};X) &:= \{f \in L_p(\mathbb{R};X): e^{-\gammamma t} (\Lambdambda^s_\gammamma f)(t) \in L_p(\mathbb{R};X)\},\\
H^{s}_{p,\gammamma,0}(\mathbb{R};X) &:= \{f \in H^{s}_{p,\gammamma}(\mathbb{R};X): f(t)=0 \,\, \thetaxt{for}\,\,t<0\},
\end{align*}
for any $\gammamma>0$ and $1<p<\infty.$ For any $(p,q) \in ]1,\infty[^2,$ we say $(d,\boldsymbol{f},\boldsymbol{g},k) \in \mathscr{F}_{p,q,\gammamma}$ ($\gammamma>0$),
if $d,$ $\boldsymbol{f},$ $\boldsymbol{g}$ and $k$ fulfil that
\betagin{gather*}
d\in L_{p,\gammamma,0}\big(\mathbb{R};H^1_q(\Omegaega)\big), \quad
\boldsymbol{f} \in L_{p,\gammamma,0}\big(\mathbb{R};L_q(\Omegaega)^N\big),\\
\boldsymbol{g} \in L_{p,\gammamma,0}\big(\mathbb{R};H^1_q(\Omegaega)^N\big)
\cap H^{1\slash 2}_{p,\gammamma,0}\big(\mathbb{R};L_q(\Omegaega)^N\big),\quad
k \in L_{p,\gammamma,0}\big(\mathbb{R};W^{2-1\slash q}_q(\Gammamma_0)\big),
\end{gather*}
with the quantity
\betagin{align*}
\|(d,\boldsymbol{f},\boldsymbol{g},k)\|_{\mathscr{F}_{p,q,\gammamma}}
:= &\|e^{-\gammamma t}d\|_{L_p(\mathbb{R};H^1_q(\Omegaega))}
+ \|e^{-\gammamma t}(\boldsymbol{f},\Lambdambda^{1\slash 2}_{\gammamma}\boldsymbol{g}) \|_{L_p(\mathbb{R};L_q(\Omegaega))} \\
&+ \|e^{-\gammamma t}\boldsymbol{g}\|_{L_p(\mathbb{R};H^1_q(\Omegaega))}
+\|e^{-\gammamma t}k\|_{L_p(\mathbb{R};W^{2-1\slash q}_q(\Gammamma_0))} <\infty.
\end{align*}
Thanks to Theorem \ref{thm:GR_CNS}, we can prove the following result.
\betagin{theo}\lambdabel{thm:maximal}
Let $0<\varepsilon<\pi\slash 2,$ $\sigmagma, \mu, \nu,\rho_1,\rho_2,\rho_3>0,$ $1<q,q':=q\slash (q-1)<\infty,$ $N<r<\infty$ and
$r \geq \max\{q,q'\}.$ Assume that $\Omegaega$ is of type $W^{3,2}_r,$ $m\geq \lambdambda_1(\varepsilon,\Gammamma_0)$ by Proposition \ref{prop:resolvent_LB}, and \eqref{hyp:gamma_GR} is satisfied.
Then there exist constants $\gammamma_0,C>0$ such that the following assertions hold true.
For any $(d,\boldsymbol{f},\boldsymbol{g},k) \in \mathscr{F}_{p,q,\gammamma_0},$
\eqref{eq:eta-Lame_1} admits a unique solution
\betagin{gather*}
\eta \in H^1_{p,\gammamma_0,0}\big(\mathbb{R};H^1_q(\Omegaega)\big),\quad
\boldsymbol{u} \in L_{p,\gammamma_0,0}\big(\mathbb{R};H^2_q(\Omegaega)^N\big) \cap H^1_{p,\gammamma_0,0}\big(\mathbb{R};L_q(\Omegaega)^N\big),\\
h \in L_{p,\gammamma_0,0}\big(\mathbb{R};W^{3-1\slash q}_q(\Gammamma_0)\big) \cap H^1_{p,\gammamma_0,0}\big(\mathbb{R};W^{2-1\slash q}_q(\Gammamma_0)\big).
\end{gather*}
Moreover, we have
\betagin{multline*}
\|e^{-\gammamma_0 t}(\partial_t \eta, \eta)\|_{L_p(\mathbb{R};H^1_q(\Omegaega))}
+\|e^{-\gammamma_0 t}(\partial_t \boldsymbol{u},\Lambdambda_{\gammamma_0}^{1\slash 2}\nabla \boldsymbol{u})\|_{L_p(\mathbb{R};L_q(\Omegaega))}
+\|e^{-\gammamma_0 t} \boldsymbol{u} \|_{L_p(\mathbb{R};H^2_q(\Omegaega))} \\
+\|e^{-\gammamma_0 t} \partial_t h \|_{L_p(\mathbb{R};W^{2-1\slash q}_q(\Gammamma_0))}
+\|e^{-\gammamma_0 t} h\|_{L_p(\mathbb{R};W^{3-1\slash q}_q(\Gammamma_0))}
\leq C\|(d,\boldsymbol{f},\boldsymbol{g},k)\|_{\mathscr{F}_{p,q,\gammamma_0}}.
\end{multline*}
\end{theo}
\betagin{proof}
For simplicity, we denote the Laplace transform of $f$ by $\wh f := \mathcal{L}[f](\lambdambda)$ and
$\boldsymbol{Z}_{\lambdambda}:= (d,\boldsymbol{f}, \Lambdambda^{1\slash 2}_{\gammamma} \boldsymbol{g}, \boldsymbol{g}, k).$
Firstly, by applying the Laplace transform to \eqref{eq:eta-Lame_1}, we find
\betagin{equation}\lambdabel{eq:eta-Lame-2}
\left\{\betagin{aligned}
&\lambdambda \wh \eta + \gammamma_1 \di \wh\boldsymbol{u} =\wh d
&&\quad\hbox{in}\quad \Omegaega, \\
&\gammamma_1 \lambdambda \wh\boldsymbol{u} -\Di\big(\mathbb{S}(\wh\boldsymbol{u})-\gammamma_2 \wh\eta \mathbb{I} \big)= \wh \boldsymbol{f}
&&\quad\hbox{in}\quad \Omegaega,\\
&\big( \mathbb{S}(\wh\boldsymbol{u}) - \gammamma_2 \wh\eta \mathbb{I}\big) \boldsymbol{n}_{\Gammamma_0}
+\sigmagma (m-\Deltalta_{\Gammamma_0}) \wh h \,\boldsymbol{n}_{\Gammamma_0} = \wh \boldsymbol{g}
&&\quad\hbox{on}\quad \Gammamma_0, \\
&\lambdambda \wh h -\wh \boldsymbol{u} \cdot \boldsymbol{n}_{\Gammamma_0} = \wh k
&&\quad\hbox{on}\quad \Gammamma_0,\\
&\wh \boldsymbol{u} = \boldsymbol{0} &&\quad\hbox{on}\quad \Gammamma_1.
\end{aligned}\right.
\end{equation}
Then Theorem \ref{thm:GR_CNS} and Theorem \ref{thm:Weis} imply that
\betagin{equation*}
(\eta,\boldsymbol{u},h):= \boldsymbol{i}g( \mathcal{L}^{-1} \big[ \mathcal{P}(\lambdambda,\Omegaega) \wh\boldsymbol{Z}_{\lambdambda} \big],
\mathcal{L}^{-1} \big[ \mathcal{A}(\lambdambda,\Omegaega) \wh\boldsymbol{Z}_{\lambdambda} \big], \mathcal{L}^{-1} \big[ \mathcal{P}(\lambdambda,\Omegaega) \wh\boldsymbol{Z}_{\lambdambda} \big] \boldsymbol{i}g).
\end{equation*}
is a solution of \eqref{eq:eta-Lame_1}. Moreover, there exists $\lambdambda \in \Lambdambda_{\varepsilon,\lambdambda_0}$ with $\Re \lambdambda=\gammamma_0\geq 1$ such that
\betagin{multline*}
\|e^{-\gammamma_0 t}\partial_t \eta\|_{L_p(\mathbb{R};H^1_q(\Omegaega))} + \|e^{-\gammamma_0 t}\partial_t \boldsymbol{u}\|_{L_p(\mathbb{R};L_q(\Omegaega))}
+ \sum_{j=0,1,2} \|e^{-\gammamma_0 t}\Lambdambda^{j\slash 2}_{\gammamma_0} \boldsymbol{u}\|_{L_p(\mathbb{R};H^{2-j}_q(\Omegaega))}\\
+ \|e^{-\gammamma_0 t}\partial_t h \|_{L_p(\mathbb{R};W^{2-1\slash q}_q(\Omegaega))}
+ \|e^{-\gammamma_0 t} h \|_{L_p(\mathbb{R};W^{3-1\slash q}_q(\Gammamma_0))}
\leq C_p r_b \|(d,\boldsymbol{f},\boldsymbol{g},k)\|_{\mathscr{F}_{p,q,\gammamma_0}}.
\end{multline*}
Next, in order to verify $(\eta,\boldsymbol{u},h)(t)=(0,\boldsymbol{0},0)$ for any $t< 0,$ we consider the following dual problem,
\betagin{equation}\lambdabel{eq:eta-Lame-3}
\left\{\betagin{aligned}
&\partial_t \rho - \gammamma_1 \di \boldsymbol{v} = 0
&&\quad\hbox{in}\quad \Omegaega \times ]-\infty,T[, \\
&\gammamma_1 \partial_t \boldsymbol{v} +\Di\big(\mathbb{S}(\boldsymbol{v}) - \gammamma_2 \rho \mathbb{I} \big)= \boldsymbol{0}
&&\quad\hbox{in}\quad \Omegaega \times ]-\infty,T[,\\
&\big( \mathbb{S}(\boldsymbol{v}) - \gammamma_2 \rho \mathbb{I}\big) \boldsymbol{n}_{\Gammamma_0}
+\sigmagma (m-\Deltalta_{\Gammamma_0})\theta \,\boldsymbol{n}_{\Gammamma_0} = \boldsymbol{0}
&&\quad\hbox{on}\quad \Gammamma_0 \times ]-\infty,T[, \\
&\partial_t \theta + \boldsymbol{v} \cdot \boldsymbol{n}_{\Gammamma_0} = 0
&&\quad\hbox{on}\quad \Gammamma_0 \times ]-\infty,T[,\\
&\boldsymbol{v} = \boldsymbol{0} &&\quad\hbox{on}\quad \Gammamma_1 \times ]-\infty,T[, \\
&(\rho, \boldsymbol{v},\theta)|_{t=T} = (\rho_0, \boldsymbol{v}_0, \theta_0) &&\quad\hbox{in}\quad \Omegaega,
\end{aligned}\right.
\end{equation}
with $T\in \mathbb{R}$ and $ (\rho_0, \boldsymbol{v}_0, \theta_0) \in C^{\infty}_0(\mathbb{R}^N)^{2+N}$ satisfying \eqref{cdt:compatible}.
As $(\wt\rho, \wt\boldsymbol{v},\wt\theta)(t):=(\rho, \boldsymbol{v},\theta)(T-t)$ satisfies \eqref{eq:eta-Lame_0},
we infer from Theorem \ref{thm:semigroup} that
\footnote{Here $\gammamma_0$ in Theorem \ref{thm:maximal} can be chosen large enough such that the estimates in Theorem \ref{thm:semigroup} hold true. }
\betagin{multline*}
\|e^{-\gammamma_0 t}(\partial_t \rho,\rho) \|_{L_p(-\infty,T;H^1_q(\Omegaega))} + \|e^{-\gammamma_0 t}\partial_t \boldsymbol{v}\|_{L_p(-\infty,T;L_q(\Omegaega))}
+ \|e^{-\gammamma_0 t} \boldsymbol{v}\|_{L_p(-\infty,T;H^{2}_q(\Omegaega))}\\
+ \|e^{-\gammamma_0 t}\partial_t \theta \|_{L_p(-\infty,T;W^{2-1\slash q}_q(\Gammamma_0))}
+ \|e^{-\gammamma_0 t} \theta \|_{L_p(-\infty,T;W^{3-1\slash q}_q(\Gammamma_0))}
\leq C\|(\rho_0,\boldsymbol{v}_0,h_0)\|_{\mathcal{D}_{q,p}(\Omegaega)}.
\end{multline*}
Note the fact that $m-\Deltalta_{\Gammamma_0}$ is symmetric.
Then by \eqref{eq:eta-Lame_1} and \eqref{eq:eta-Lame-3}, we have
\betagin{multline}\lambdabel{eq:dual-1}
\frac{d}{dt}\boldsymbol{i}g( (\gammamma_1 \boldsymbol{u},\boldsymbol{v})_{\Omegaega} -(\gammamma_2\gammamma_1^{-1} \eta, \rho)_{\Omegaega}
-\big( \sigmagma(m-\Deltalta_{\Gammamma_0}) h, \theta\big)_{\Gammamma_0} \boldsymbol{i}g)(t)
= -(\gammamma_2\gammamma_1^{-1}\rho,d)_{\Omegaega}(t) +(\boldsymbol{v},\boldsymbol{f})_{\Omegaega}(t)\\
+ (\boldsymbol{v},\boldsymbol{g})_{\Gammamma_0}(t) - \big( \sigmagma(m-\Deltalta_{\Gammamma_0}) \theta, k\big)_{\Gammamma_0} (t),
\end{multline}
for any $t \leq T\in \mathbb{R}.$
Then integrating \eqref{eq:dual-1} on $]-\infty,T]$ yields that
\betagin{multline}\lambdabel{eq:dual-2}
\boldsymbol{i}g( (\gammamma_1 \boldsymbol{u},\boldsymbol{v}_0)_{\Omegaega} -(\gammamma_2\gammamma_1^{-1} \eta, \rho_0)_{\Omegaega}
-\big( \sigmagma(m-\Deltalta_{\Gammamma_0}) h, \theta_0\big)_{\Gammamma_0} \boldsymbol{i}g)(T) \\
=- \int_{-\infty}^T \boldsymbol{i}g( (\gammamma_2\gammamma_1^{-1}\rho,d)_{\Omegaega}-(\boldsymbol{v},\boldsymbol{f})_{\Omegaega}
-(\boldsymbol{v},\boldsymbol{g})_{\Gammamma_0} +\big( \sigmagma(m-\Deltalta_{\Gammamma_0}) \theta, k\big)_{\Gammamma_0} \boldsymbol{i}g) (t) \, dt.
\end{multline}
If $d(t),\boldsymbol{f}(t),\boldsymbol{g}(t)$ and $k(t)$ vanish for $t<0,$ then \eqref{eq:dual-2} gives us
$$ \big(\gammamma_1 \boldsymbol{u}(T),\boldsymbol{v}_0\big)_{\Omegaega} -\big(\gammamma_2\gammamma_1^{-1} \eta(T), \rho_0\big)_{\Omegaega}
-\big( \sigmagma(m-\Deltalta_{\Gammamma_0}) h(T), \theta_0\big)_{\Gammamma_0} =0, \,\,\, \forall\,\,T<0. $$
As $\rho_0,\boldsymbol{v}_0$ and $\theta_0$ are arbitrary smooth functions, $\gammamma_1,\gammamma_2>0$ and $m \in \rho(\Deltalta_{\Gammamma_0}),$
we have $(\eta,\boldsymbol{u},h)(T)=(0,\boldsymbol{0},0)$ for any $T< 0.$
\smallbreak
At last, we prove the uniqueness of \eqref{eq:eta-Lame_1}. Suppose that $(\eta_1,\boldsymbol{u}_1, h_1)$ and $(\eta_2,\boldsymbol{u}_2, h_2)$ are two solutions of \eqref{eq:eta-Lame_1}. Then $(\eta,\boldsymbol{u}, h):=(\eta_2-\eta_1,\boldsymbol{u}_2-\boldsymbol{u}_1,h_2- h_1)$ satisfies \eqref{eq:eta-Lame_1}
by imposing $(d,\boldsymbol{f},\boldsymbol{g},k)=(0,\boldsymbol{0},\boldsymbol{0},0).$ Then \eqref{eq:dual-2} yields that
$(\eta,\boldsymbol{u},h)(T)=(0,\boldsymbol{0},0)$ for any $T>0.$
\end{proof}
\section{Generalized model problem in the half space}
\lambdabel{sec:halfspace}
In order to prove Theorem \ref{thm:RR_CNS}, we consider the following model problem in $\mathbb{R}^N_+$ in this section,
\betagin{equation}\lambdabel{eq:RR_CNS_half}
\left\{\betagin{aligned}
& \lambdambda \boldsymbol{u} -\gammamma_{1}^{-1} \Di \big( \mathbb{S}(\boldsymbol{u}) + \zeta \gammamma_3 \di \boldsymbol{u} \mathbb{I} \big)
= \boldsymbol{F} &&\quad\hbox{in}\quad \mathbb{R}^N_+,\\
&\big(\mathbb{S}(\boldsymbol{u}) +\zeta \gammamma_3 \di \boldsymbol{u} \mathbb{I} \big)\boldsymbol{n}_{0}
+\sigmagma (m-\Deltalta')h \,\boldsymbol{n}_{0} = \boldsymbol{G} &&\quad\hbox{on}\quad \mathbb{R}^N_0, \\
&\lambdambda h - \boldsymbol{u} \cdot \boldsymbol{n}_{0} = K &&\quad\hbox{on}\quad \mathbb{R}^N_0,\\
\end{aligned}\right.
\end{equation}
where $\mathbb{R}^N_0:= \{x=(x',x_{_N})\in \mathbb{R}^N: x_{_N}=0\},$ $\boldsymbol{n}_0 := (0,\dots,0,-1)^{\top}$ and $\Deltalta' := \sum_{j=1}^{N-1}\partial_j^2.$
Moreover, the parameter $\zeta$ and the constants $\gammamma_1,$ $\gammamma_3$ fulfil the conditions
\betagin{equation}\lambdabel{hyp:half_space}
|\zeta| \leq \zeta_0, \,\,\,
0<\rho_1 \leq \gammamma_1 \leq \rho_2, \,\,\,
0 < \gammamma_3 \leq \rho_3
\end{equation}
for some $\rho_1,\rho_2,\rho_3>0.$
Then recalling the definition of $\Gammamma_{\varepsilon,\lambdambda_0,\zeta}$ in \eqref{eq:Gamma}, our main result for \eqref{eq:RR_CNS_half} reads:
\betagin{theo}\lambdabel{thm:GR_half_0}
Assume that $0<\varepsilon<\pi\slash 2,$ $\sigmagma, m,\mu, \nu>0,$ $1<q<\infty$ and \eqref{hyp:half_space} is satisfied. Set that
\betagin{gather*}
Y_q(\mathbb{R}^N_+) := L_q(\mathbb{R}^N_+)^N \times H^{1}_q(\mathbb{R}^N_+)^N\times H^{2}_q(\mathbb{R}^N_+),\quad
\mathcal{Y}_q(\mathbb{R}^N_+) := L_q(\mathbb{R}^N_+)^N \times Y_q(\mathbb{R}^N_+).
\end{gather*}
For any $(\boldsymbol{F},\boldsymbol{G},K) \in Y_q(\mathbb{R}^N_+),$ there exist constants $\lambdambda_0, r_b \geq 1$ and operator families
\betagin{align*}
\mathcal{A}_0(\lambdambda,\mathbb{R}^N_+) & \in
{\rm Hol}\,\boldsymbol{i}g( \Gammamma_{\varepsilon,\lambdambda_0,\zeta} ; \mathcal{L}\big(\mathcal{Y}_q(\mathbb{R}^N_+);H^2_q(\mathbb{R}^N_+)^N \big) \boldsymbol{i}g),\\
\mathcal{H}_0(\lambdambda,\mathbb{R}^N_+) & \in
{\rm Hol}\,\boldsymbol{i}g( \Gammamma_{\varepsilon,\lambdambda_0,\zeta} ; \mathcal{L}\big(\mathcal{Y}_q(\mathbb{R}^N_+);H^3_q(\mathbb{R}^N_+) \big) \boldsymbol{i}g),
\end{align*}
such that $(\boldsymbol{u}, h):=\big( \mathcal{A}_0(\lambdambda,\mathbb{R}^N_+), \mathcal{H}_0(\lambdambda,\mathbb{R}^N_+) \big) (\boldsymbol{F},\lambdambda^{1\slash 2}\boldsymbol{G},\boldsymbol{G},K)$
is a solution of \eqref{eq:RR_CNS_half}. Moreover, we have
\betagin{gather*}
\mathcal{R}_{\mathcal{L}\big(\mathcal{Y}_q(\mathbb{R}^N_+); H^{2-j}_q(\mathbb{R}^N_+)^N \big)}
\boldsymbol{i}g( \boldsymbol{i}g\{ (\tau \partial_{\tau})^{\ell}\big( \lambdambda^{j\slash 2}\mathcal{A}_0(\lambdambda,\mathbb{R}^N_+)\big) : \lambdambda \in
\Gammamma_{\varepsilon,\lambdambda_0,\zeta} \boldsymbol{i}g\}\boldsymbol{i}g) \leq r_b,\\
\mathcal{R}_{\mathcal{L}\big(\mathcal{Y}_q(\mathbb{R}^N_+); H^{3-j'}_q(\mathbb{R}^N_+) \big)}
\boldsymbol{i}g( \boldsymbol{i}g\{ (\tau \partial_{\tau})^{\ell}\big( \lambdambda^{j'}\mathcal{H}_0(\lambdambda,\mathbb{R}^N_+)\big) : \lambdambda \in
\Gammamma_{\varepsilon,\lambdambda_0,\zeta} \boldsymbol{i}g\}\boldsymbol{i}g) \leq r_b,
\end{gather*}
for $\ell,j' =0,1,$ $j=0,1,2,$ and $\tau := \Im \lambdambda.$ Above the choices of $\lambdambda_0$ and $r_b$ depend solely on
$\varepsilon,$ $\sigmagma,$ $m,$ $\mu,$ $\nu,$ $q,$ $N,$ $\zeta_0,$ $\rho_1,$ $\rho_2,$ $\rho_3.$
\end{theo}
\subsection{Reduction and the main idea}
For convenience of the later calculations, we introduce the notations
\betagin{equation*}
\alphapha :=\gammamma_1^{-1} \mu,\,\,\,
\betata := \gammamma_1^{-1}(\nu-\mu),\,\,\,
\zeta':= \gammamma_1^{-1}\gammamma_3 \zeta,\,\,\,
\sigmagma':=\gammamma_1^{-1} \sigmagma,\,\,\,
\boldsymbol{G}':=\gammamma_1^{-1}\boldsymbol{G}.
\end{equation*}
Then \eqref{eq:RR_CNS_half} is equivalent to
\betagin{equation}\lambdabel{eq:GR_half_0}
\left\{\betagin{aligned}
&\lambdambda \boldsymbol{u} - \alphapha \Deltalta \boldsymbol{u}
-(\alphapha +\betata +\zeta') \nabla \di \boldsymbol{u}
= \boldsymbol{F}
&&\quad\hbox{in}\quad \mathbb{R}^N_+,\\
&\alphapha (\partial_{_N} u_j+ \partial_j u_{_N})= -G'_j, \,\, j=1,\dots, N-1,
&&\quad\hbox{on}\quad \mathbb{R}^N_0, \\
& 2\alphapha \partial_{_N} u_{_N} +(\betata + \zeta') \di \boldsymbol{u}
+\sigmagma' (m-\Deltalta')h =-G'_{N}
&&\quad\hbox{on}\quad \mathbb{R}^N_0, \\
&\lambdambda h + u_{_N} = K
&&\quad\hbox{on}\quad \mathbb{R}^N_0.\\
\end{aligned}\right.
\end{equation}
Now let us outline the main idea of solving \eqref{eq:GR_half_0}.
Clearly $\boldsymbol{u}:=\boldsymbol{v}+\boldsymbol{w}$ and $h$ satisfy \eqref{eq:GR_half_0}, if $\boldsymbol{v}$ and $(\boldsymbol{w},h)$ satisfy the following linear systems respectively,
\betagin{equation}\lambdabel{eq:GR_half_1}
\left\{\betagin{aligned}
&\lambdambda \boldsymbol{v} - \alphapha \Deltalta \boldsymbol{v}
-(\alphapha +\betata +\zeta') \nabla \di \boldsymbol{v}
= \boldsymbol{F}
&&\quad\hbox{in}\quad \mathbb{R}^N_+,\\
&\alphapha (\partial_{_N} v_j + \partial_j v_{_N})= -G'_j, \,\, j=1,\dots, N-1,
&&\quad\hbox{on}\quad \mathbb{R}^N_0, \\
& 2\alphapha \partial_{_N} v_{_N} +(\betata + \zeta') \di \boldsymbol{v} =-G'_{N}
&&\quad\hbox{on}\quad \mathbb{R}^N_0, \\
\end{aligned}\right.
\end{equation}
\betagin{equation}\lambdabel{eq:GR_half_2}
\left\{\betagin{aligned}
&\lambdambda \boldsymbol{w} - \alphapha \Deltalta \boldsymbol{w}
-(\alphapha +\betata +\zeta') \nabla \di \boldsymbol{w} = \boldsymbol{0}
&&\quad\hbox{in}\quad \mathbb{R}^N_+,\\
&\alphapha (\partial_{_N} w_j+ \partial_j w_{_N})= 0, \,\, j=1,\dots, N-1,
&&\quad\hbox{on}\quad \mathbb{R}^N_0, \\
& 2\alphapha \partial_{_N} w_{_N} +(\betata + \zeta') \di \boldsymbol{w}
+\sigmagma' (m-\Deltalta')h = 0
&&\quad\hbox{on}\quad \mathbb{R}^N_0, \\
&\lambdambda h + w_{_N} = K-v_{_N}
&&\quad\hbox{on}\quad \mathbb{R}^N_0.\\
\end{aligned}\right.
\end{equation}
For the solvability of \eqref{eq:GR_half_1}, we refer to \cite[Theorem 2.3]{GS2014}.
\betagin{prop}\lambdabel{prop:GR_half_1}
Assume that $0<\varepsilon<\pi\slash 2,$ $\mu, \nu>0,$ $1<q<\infty$ and \eqref{hyp:half_space} is satisfied.
\betagin{equation*}
X_q(\mathbb{R}^N_+):= L_q(\mathbb{R}^N_+)^N \times H^1_q(\mathbb{R}^N_+)^N, \quad
\mathcal{X}_q(\mathbb{R}^N_+):= L_q(\mathbb{R}^N_+)^N \times X_q(\mathbb{R}^N_+).
\end{equation*}
For any $(\boldsymbol{F},\boldsymbol{G}') \in X_q(\mathbb{R}^N_+),$
there exist constants $\lambdambda_0,r_b \geq 1$ and a family of operators
\betagin{equation*}
\mathcal{A}_1(\lambdambda, \mathbb{R}^N_+) \in
{\rm Hol}\,\boldsymbol{i}g( \Gammamma_{\varepsilon,\lambdambda_0,\zeta} ; \mathcal{L}\big(\mathcal{X}_q(\mathbb{R}^N_+);H^2_q(\mathbb{R}^N_+)^N \big) \boldsymbol{i}g),
\end{equation*}
such that $\boldsymbol{v} := \mathcal{A}_1(\lambdambda; \mathbb{R}^N_+)\, (\boldsymbol{F},\lambdambda^{1\slash2}\boldsymbol{G}',\boldsymbol{G}')$ solves \eqref{eq:GR_half_1}.
Moreover, we have
\betagin{equation*}
\mathcal{R}_{\mathcal{L}\big(\mathcal{X}_q(\mathbb{R}^N_+); H^{2-j}_q(\mathbb{R}^N_+)^N \big)}
\boldsymbol{i}g( \boldsymbol{i}g\{ (\tau \partial_{\tau})^{\ell}\big( \lambdambda^{j\slash 2}\mathcal{A}_1(\lambdambda,\mathbb{R}^N_+)\big) : \lambdambda \in
\Gammamma_{\varepsilon,\lambdambda_0,\zeta} \boldsymbol{i}g\}\boldsymbol{i}g) \leq r_b,
\end{equation*}
for $\ell =0,1,$ $j=0,1,2,$ $\tau := \Im \lambdambda.$ Above the choices of $\lambdambda_0$ and $r_b$ depend solely on
$\varepsilon,$ $\mu,$ $\nu,$ $q,$ $N,$ $\zeta_0,$ $\rho_1,$ $\rho_2,$ $\rho_3.$
\end{prop}
Hereafter, we replace $(\sigmagma',\zeta')$ by $(\sigmagma,\zeta)$ for simplicity
\footnote{Thanks to \eqref{hyp:half_space}, the new definition of $\zeta$ is harmless to the shape of $\Gammamma_{\varepsilon,\lambdambda_0,\zeta}$ in \eqref{eq:Gamma}.}.
To handle \eqref{eq:GR_half_2}, it suffices to consider
\betagin{equation}\lambdabel{eq:GR_half_3}
\left\{\betagin{aligned}
&\lambdambda \boldsymbol{u} - \alphapha \Deltalta \boldsymbol{u}
-(\alphapha +\betata +\zeta) \nabla \di \boldsymbol{u} = \boldsymbol{0}
&&\quad\hbox{in}\quad \mathbb{R}^N_+,\\
&\alphapha (\partial_{_N} u_j+ \partial_j u_{_N})= 0, \,\, j=1,\dots, N-1,
&&\quad\hbox{on}\quad \mathbb{R}^N_0, \\
& 2\alphapha \partial_{_N} u_{_N} +(\betata + \zeta) \di \boldsymbol{u}
+\sigmagma (m-\Deltalta')h = 0
&&\quad\hbox{on}\quad \mathbb{R}^N_0, \\
&\lambdambda h + u_{_N} = k
&&\quad\hbox{on}\quad \mathbb{R}^N_0.\\
\end{aligned}\right.
\end{equation}
For \eqref{eq:GR_half_3}, we will establish that:
\betagin{prop}\lambdabel{prop:GR_half_2}
Assume that $0<\varepsilon<\pi\slash 2,$ $\sigmagma, m,\mu, \nu>0,$ $1<q<\infty$ and \eqref{hyp:half_space} is satisfied.
For any $k \in H^2_q(\mathbb{R}^N_+),$
there exist constants $\lambdambda_0,r_b \geq 1$ and the operator families
\betagin{align*}
\mathcal{W}(\lambdambda,\mathbb{R}^N_+) \in &
{\rm Hol}\,\boldsymbol{i}g( \Gammamma_{\varepsilon,\lambdambda_0,\zeta} ; \mathcal{L}\big(H^2_q(\mathbb{R}^N_+);H^2_q(\mathbb{R}^N_+)^N \big) \boldsymbol{i}g),\\
\mathcal{H}(\lambdambda,\mathbb{R}^N_+) \in &
{\rm Hol}\,\boldsymbol{i}g( \Gammamma_{\varepsilon,\lambdambda_0,\zeta} ; \mathcal{L}\big(H^2_q(\mathbb{R}^N_+);H^3_q(\mathbb{R}^N_+) \big) \boldsymbol{i}g),
\end{align*}
such that $(\boldsymbol{u},h):=\big( \mathcal{W}(\lambdambda,\mathbb{R}^N_+),\mathcal{H}(\lambdambda,\mathbb{R}^N_+) \big)k$ is a solution of \eqref{eq:GR_half_3}.
Moreover, we have
\betagin{align*}
\mathcal{R}_{\mathcal{L}\big(H^2_q(\mathbb{R}^N_+); H^{2-j}_q(\mathbb{R}^N_+)^N \big)}
\boldsymbol{i}g( \boldsymbol{i}g\{ (\tau \partial_{\tau})^{\ell}\big( \lambdambda^{j\slash 2}\mathcal{W}(\lambdambda,\mathbb{R}^N_+)\big) : \lambdambda \in
\Gammamma_{\varepsilon,\lambdambda_0,\zeta} \boldsymbol{i}g\}\boldsymbol{i}g) \leq r_b,\\
\mathcal{R}_{\mathcal{L}\big(H^2_q(\mathbb{R}^N_+); H^{3-j'}_q(\mathbb{R}^N_+) \big)}
\boldsymbol{i}g( \boldsymbol{i}g\{ (\tau \partial_{\tau})^{\ell}\big( \lambdambda^{j'}\mathcal{H}(\lambdambda,\mathbb{R}^N_+)\big) : \lambdambda \in
\Gammamma_{\varepsilon,\lambdambda_0,\zeta} \boldsymbol{i}g\}\boldsymbol{i}g) \leq r_b,
\end{align*}
for $\ell,j' =0,1,$ $j=0,1,2,$ $\tau := \Im \lambdambda.$
Above the choices of $\lambdambda_0$ and $r_b$ depend solely on
$\varepsilon,$ $\sigmagma,$ $m,$ $\mu,$ $\nu,$ $q,$ $N,$ $\zeta_0,$ $\rho_1,$ $\rho_2,$ $\rho_3.$
\end{prop}
Now let us point out that Theorem \ref{thm:GR_half_0} is an immediate consequence from Proposition \ref{prop:GR_half_1} and Proposition \ref{prop:GR_half_2} above.
\betagin{proof}[Completion of the proof of Theorem \ref{thm:GR_half_0}]
Set that $v_{_N} :=\mathcal{A}_{1N}(\lambdambda; \mathbb{R}^N_+)\, (\boldsymbol{F},\lambdambda^{1\slash2}\boldsymbol{G}',\boldsymbol{G}')$ due to Proposition \ref{prop:GR_half_1} for $(\boldsymbol{F}, \boldsymbol{G}') \in X_q(\mathbb{R}^N_+).$ Then thanks to \eqref{eq:GR_half_2} and Proposition \ref{prop:GR_half_2}, introduce
\betagin{align*}
\boldsymbol{w} :=& \mathcal{A}_2(\lambdambda,\mathbb{R}^N_+) (\boldsymbol{F},\lambdambda^{1\slash 2}\boldsymbol{G}',\boldsymbol{G}',K)
:=\mathcal{W}(\lambdambda,\mathbb{R}^N_+) \big( K - \mathcal{A}_{1N}(\lambdambda; \mathbb{R}^N_+)\, (\boldsymbol{F},\lambdambda^{1\slash2}\boldsymbol{G}',\boldsymbol{G}') \big),\\
h :=&\mathcal{H}_0(\lambdambda,\mathbb{R}^N_+) (\boldsymbol{F},\lambdambda^{1\slash 2}\boldsymbol{G}',\boldsymbol{G}',K)
:= \mathcal{H}(\lambdambda,\mathbb{R}^N_+) \big( K
- \mathcal{A}_{1N}(\lambdambda; \mathbb{R}^N_+)\, (\boldsymbol{F},\lambdambda^{1\slash2}\boldsymbol{G}',\boldsymbol{G}')\big),
\end{align*}
and $(\boldsymbol{w}, h)$ solves \eqref{eq:GR_half_2} clearly.
Finally, we define
\betagin{align*}
\boldsymbol{u}&:= \mathcal{A}_0 (\lambdambda, \mathbb{R}^N_+) (\boldsymbol{F},\lambdambda^{1\slash 2}\boldsymbol{G}',\boldsymbol{G}', K)
:=\mathcal{A}_1(\lambdambda; \mathbb{R}^N_+)\, (\boldsymbol{F},\lambdambda^{1\slash2}\boldsymbol{G}',\boldsymbol{G}')
+ \mathcal{A}_2(\lambdambda,\mathbb{R}^N_+) (\boldsymbol{F},\lambdambda^{1\slash 2}\boldsymbol{G}',\boldsymbol{G}',K).
\end{align*}
Then thanks to Proposition \ref{prop:GR_half_1}, Proposition \ref{prop:GR_half_2} and Remark \ref{rmk:R-bounded}, $\mathcal{A}_0(\lambdambda,\mathbb{R}^N_+)$ and $\mathcal{H}_0(\lambdambda,\mathbb{R}^N_+)$ are the desired operator families.
\end{proof}
To tackle \eqref{eq:GR_half_3}, our main task is to construct $\mathcal{W}(\lambdambda,\mathbb{R}^N_+)$ and $\mathcal{H}(\lambdambda,\mathbb{R}^N_+)$ in Proposition \ref{prop:GR_half_2}.
To this end, we apply the partial fourier transformation $\mathcal{F}_{x'}$ to \eqref{eq:GR_half_3},
\betagin{equation}\lambdabel{eq:GR_half_4}
\left\{\betagin{aligned}
&(\lambdambda + \alphapha |\xi'|^2) \wh{u_j} - \alphapha \partial_{N}^2 \wh{u_j}
-(\alphapha +\betata +\zeta) (i\xi_j)
(i\xi'\cdot \wh{\boldsymbol{u}'} + \partial_{_N} \wh{u_{_N}}) = 0
&&\quad\hbox{for}\quad x_{_N}>0,\\
&(\lambdambda + \alphapha |\xi'|^2) \wh{u_{_N}} - \alphapha \partial_{N}^2 \wh{u_{_N}}
-(\alphapha +\betata +\zeta) \partial_{_N} (i\xi'\cdot \wh{\boldsymbol{u}'}
+ \partial_{_N} \wh{u_{_N}}) = 0
&&\quad\hbox{for}\quad x_{_N}>0,\\
&\alphapha (\partial_{_N} \wh{u_j} + i\xi_j \wh{u_{_N}})= 0, \,\, j=1,\dots, N-1,
&&\quad\hbox{for}\quad x_{_N}=0, \\
& 2\alphapha \partial_{_N} \wh{u_{_N}} +(\betata + \zeta)
(i\xi'\cdot \wh{\boldsymbol{u}'}+\partial_{_N} \wh{u_{_N}})
+\sigmagma (m+|\xi'|^2) \wh{h} = 0
&&\quad\hbox{for}\quad x_{_N}=0, \\
&\lambdambda \wh{h} + \wh{u_{_N}} =\wh{k}
&&\quad\hbox{for}\quad x_{_N}=0.
\end{aligned}\right.
\end{equation}
For convenience, we take advantage of the following notations,
\betagin{gather}\lambdabel{eq:AB}
A:= \sqrt{(2\alphapha +\betata +\zeta)^{-1} \lambdambda +|\xi'|^2},\qquad
B:=\sqrt{\alphapha^{-1}\lambdambda +|\xi'|^2},\\ \nonumber
\eta := \alphapha^{-1}(\alphapha +\betata +\zeta),\qquad
\mathcal{M}(x_{_N}) := (B-A)^{-1} (e^{-Bx_{_N}} - e^{-Ax_{_N}}),
\end{gather}
and the so-called \emph{Lopatinski} matrix $\mathbb{L}=[L_{ij}]_{2\times 2}$ as well, whose entries are defined by
\betagin{gather}\lambdabel{eq:L_ij}
L_{11} := \frac{\alphapha A (B^2 - |\xi'|^2)}{AB-|\xi'|^2} \ccomma \qquad
L_{12}:= \frac{\alphapha |\xi'|^2 (2AB - |\xi'|^2-B^2)}{AB-|\xi'|^2} \ccomma \\ \nonumber
L_{21}:=\frac{2\alphapha A(B-A)-(\betata +\zeta) (A^2-|\xi'|^2)}{AB-|\xi'|^2} \ccomma
\qquad
L_{22}:=\frac{(2\alphapha+\betata+\zeta)B (A^2-|\xi'|^2)}{AB-|\xi'|^2} \cdot
\end{gather}
By direct calculations, we can conclude the formulas (see \cite[Sec.4]{GS2014}),
\betagin{align}\lambdabel{eq:GR_half_uj_1}
\wh{u_j} (\xi', x_{_N})
=&- \frac{\eta (i\xi_j)(L_{12}+B L_{11})}{(A+B) B \deltat{\mathbb{L}}}
\frac{|\xi'|^2 -A^2}{AB- |\xi'|^2}
\boldsymbol{i}g( B\mathcal{M}(x_{_N}) -e^{-Bx_{_N}} \boldsymbol{i}g)
\sigmagma (m + |\xi'|^2) \widehat{h} (\xi',0) \\\nonumber
&+\frac{(i\xi_j)L_{11}}{B\deltat{\mathbb{L}}}e^{-Bx_{_N}}
\sigmagma (m + |\xi'|^2) \widehat{h} (\xi',0),
\quad \forall\,\, j =1, \dots,N-1, \\ \lambdabel{eq:GR_half_uN_1}
\widehat{u_{_N}} (\xi', x_{_N})
=&\boldsymbol{i}g( \frac{\eta A(L_{12}+B L_{11})}{(A+B) \deltat{\mathbb{L}}}
\frac{|\xi'|^2 -A^2}{AB- |\xi'|^2} \mathcal{M}(x_{_N})
+ \frac{L_{11}}{ \deltat{\mathbb{L}} }e^{-Bx_{_N}} \boldsymbol{i}g)
\sigmagma (m + |\xi'|^2) \widehat{h} (\xi',0).
\end{align}
Then \eqref{eq:GR_half_uN_1} and $\eqref{eq:GR_half_4}_5$ imply formally that
\betagin{equation}\lambdabel{eq:GR_half_h_1}
\widehat{h} (\xi',0) =\frac{\deltat \mathbb{L}}{N(A,B)}\widehat{k}(\xi',0)
\,\,\,\hbox{with}\,\,\,
N(A, B) := \lambdambda \deltat{\mathbb{L}} +\sigmagma L_{11} (m + |\xi'|^2).
\end{equation}
Next, insert \eqref{eq:GR_half_h_1} back into \eqref{eq:GR_half_uj_1}, \eqref{eq:GR_half_uN_1},
and take partial Fourier inverse transformation,
\betagin{align}\lambdabel{eq:GR_half_uj_2}
u_{j}(x) =\mathcal{W}_{j} (\lambdambda,\mathbb{R}^N_+)k&:= \mathcal{F}_{\xi'}^{-1} \boldsymbol{i}g[ n_{j1}(\lambdambda,\xi') (m+|\xi'|^2) \big( B\mathcal{M}(x_{_N}) -e^{-Bx_{_N}} \big) \widehat{k} (\xi',0) \boldsymbol{i}g](x')\\ \notag
& \quad +\mathcal{F}_{\xi'}^{-1} \boldsymbol{i}g[ n_{j2}(\lambdambda,\xi') (m + |\xi'|^2) e^{-Bx_{_N}} \widehat{k} (\xi',0) \boldsymbol{i}g](x'),\\
u_{_N}(x) = \mathcal{W}_{N} (\lambdambda,\mathbb{R}^N_+)k&:= \mathcal{F}_{\xi'}^{-1} \boldsymbol{i}g[ n_{N1}(\lambdambda,\xi') (m+|\xi'|^2)B\mathcal{M}(x_{_N}) \widehat{k} (\xi',0) \boldsymbol{i}g](x') \lambdabel{eq:GR_half_uN_2}\\
& \quad +\mathcal{F}_{\xi'}^{-1} \boldsymbol{i}g[ n_{N2}(\lambdambda,\xi') (m+|\xi'|^2) e^{-Bx_{_N}} \widehat{k} (\xi',0) \boldsymbol{i}g](x'), \notag
\end{align}
where the symbols $n_{J1}(\lambdambda,\xi')$ and $n_{J2}(\lambdambda,\xi')$ ($J=1,\dots,N$) are given by
\betagin{align}\lambdabel{eq:n_Jk_half}
n_{j1}(\lambdambda,\xi') :=&- \frac{\sigmagma\eta (i\xi_j) (L_{12}+B L_{11})}{ B (A+B) N(A,B)}
\frac{|\xi'|^2 -A^2}{AB- |\xi'|^2} \ccomma \\
n_{j2}(\lambdambda,\xi') :=& \frac{\sigmagma(i\xi_j)L_{11}}{BN(A,B)},
\quad \forall\,\, j=1,\dots, N-1, \notag \\
n_{N1}(\lambdambda,\xi') :=& \frac{\sigmagma \eta A (L_{12}+B L_{11})}{B(A+B) N(A,B)}
\frac{|\xi'|^2 -A^2}{AB- |\xi'|^2} \ccomma \notag\\
n_{N2}(\lambdambda,\xi') :=& \frac{\sigmagma L_{11}}{N(A,B)} \cdot \notag
\end{align}
On the other hand, according to \eqref{eq:GR_half_h_1} and Lemma \ref{lemma:N_AB}, we set $h$ as follows,
\betagin{equation}\lambdabel{eq:GR_half_h_2}
h (x) =\big(\mathcal{H}(\lambdambda,\mathbb{R}^N_+) k\big) (x):=
\varphi(x_{_N}) \mathcal{F}^{-1}_{\xi'}\boldsymbol{i}g[
\frac{\deltat \mathbb{L}}{N(A,B)} \, e^{-|\xi'|x_{_N}}
\widehat{k}(\xi',0)\boldsymbol{i}g] (x')
\end{equation}
for any $x=(x',x_{_N}) \in \mathbb{R}^N_+,$
where the cut-off function $\varphi(s) \in C^{\infty}_0( ]-2,2[)$ satisfies
$0\leq \varphi(s)\leq 1$ and
$\varphi(s)=1$ for $|s| <1.$
\subsection{Some preliminary results}
In this subsection, we summarize some results to study the $\mathcal{R}$-boundedness properties of the operator families in \eqref{eq:GR_half_uj_2}, \eqref{eq:GR_half_uN_2} and \eqref{eq:GR_half_h_2}. Firstly, recall the definitions of the class of the symbols.
\betagin{defi}
Assume that $\Lambdambda\subset \mathbb{C},$ $s \in \mathbb{R}$ and $\kappappa' \in \mathbb{N}^{N-1}_0.$
Consider some smooth function $m(\lambdambda,\xi')$ defined in $ \wt\Lambdambda := \Lambdambda \times (\mathbb{R}^{N-1} \backslash \{0\}) $ such that
\betagin{equation*}
|m(\lambdambda,\xi')| \leq C_{\Lambdambda} ( |\lambdambda|^{1\slash 2} + |\xi'| )^s
\,\,\,\thetaxt{for any}\,\,(\lambdambda, \xi') \in \wt \Lambdambda.
\end{equation*}
\betagin{itemize}
\item $m(\lambdambda, \xi')$ is called a multiplier of order $s$ with type 1 on $\wt\Lambdambda,$
denoted by $m \in \boldsymbol{M}_{s,1}(\wt\Lambdambda),$ if there exists a constant $C_{\kappappa',s,\Lambdambda}$ such that
\betagin{equation*}
\big| \partial_{\xi'}^{\kappappa'} (\tau \partial_\tau)^{\ell}m(\lambdambda,\xi')\big|
\leq C_{\kappappa',s,\Lambdambda} ( |\lambdambda|^{1\slash 2} + |\xi'| )^{s-|\kappappa'|}
\end{equation*}
for any $(\lambdambda, \xi') \in \wt \Lambdambda,$ $\tau := \Im \lambdambda$ and $\ell =0,1.$
\item $m(\lambdambda, \xi')$ is called a multiplier of order $s$ with type 2 on $\wt\Lambdambda,$
denoted by $m \in \boldsymbol{M}_{s,2}(\wt\Lambdambda),$ if there exists a constant $C_{\kappappa',s,\Lambdambda}$ such that
\betagin{equation*}
\big| \partial_{\xi'}^{\kappappa'} (\tau \partial_\tau)^{\ell}m(\lambdambda,\xi')\big|
\leq C_{\kappappa',s,\Lambdambda} ( |\lambdambda|^{1\slash 2} + |\xi'| )^{s} |\xi'|^{-|\kappappa'|}
\end{equation*}
for $(\lambdambda, \xi') \in \wt \Lambdambda,$ $\tau := \Im \lambdambda$ and $\ell =0,1.$
\end{itemize}
For any $m\in \boldsymbol{M}_{s,i}(\wt\Lambdambda),$ $i=1,2,$ we denote
$M(m,\Lambdambda):= \max_{|\kappappa'|\leq N}C_{\kappappa',s,\Lambdambda} .$
\end{defi}
\betagin{rema} \lambdabel{rmk:M1M2}
Let $s,s_1, s_2 \in \mathbb{R}$ and $i=1,2.$ Denote
$\boldsymbol{M}_{s,i} :=\boldsymbol{M}_{s,i} \big(\Lambdambda \times (\mathbb{R}^{N-1} \backslash \{0\})\big)$
for short. As it was pointed out in \cite[Lemma 5.1]{ShiShi2012}, we have
\betagin{enumerate}
\item $m_1m_2 \in \boldsymbol{M}_{s_1+s_2,1}$
(or $\boldsymbol{M}_{s_1+s_2,2}$) for
$m_i \in \boldsymbol{M}_{s_i,1}$ (or $m_i \in \boldsymbol{M}_{s_i,2}$ resp.);
\item $m_1m_2 \in \boldsymbol{M}_{s_1+s_2,2}$ for
$m_i \in \boldsymbol{M}_{s_i,i}$ and $i=1,2.$
\end{enumerate}
\end{rema}
Now, let us state some useful results proved in \cite{ES2013,GS2014} on $A,$ $B$ and $\mathbb{L}=[L_{jk}]_{2\times 2}$ in \eqref{eq:AB} and \eqref{eq:L_ij}.
In the following,
we write $\wt \Gammamma_{\varepsilon,\lambdambda_0,\zeta}:= \Gammamma_{\varepsilon,\lambdambda_0,\zeta} \times (\mathbb{R}^{N-1}\backslash\{0\})$
\footnote{For the case (C1), $\zeta:=\gammamma_1^{-1}\gammamma_3 \lambdambda^{-1}.$} for simplicity.
\betagin{lemm}\lambdabel{lemma:basic}
Let $0<\varepsilon<\pi \slash 2,$ $\lambdambda_0,\mu, \nu>0,$ $s\geq 1$ and $\xi'\in \mathbb{R}^{N-1}.$
Assume that \eqref{hyp:half_space} is satisfied.
Then the following assertions hold true.
\betagin{enumerate}
\item For any $\lambdambda \in \Sigmagma_\varepsilon$ and $a>0,$ we have
$\big| a \lambdambda + |\xi'|^2 \big|
\geq \sigman (\varepsilon \slash 2) \big( a |\lambdambda|+|\xi'|^2 \big).$
\item There exists a constant $0<\varepsilon'<\pi\slash 2$ such that
\betagin{equation*}
(s \alphapha + \betata +\zeta)^{-1} \lambdambda \in \Sigmagma_{\varepsilon'},
\,\,\, \forall\,\, \lambdambda \in \Gammamma_{\varepsilon,\lambdambda_0,\zeta},
\end{equation*}
where the choice of $\varepsilon'$ depends solely on $\varepsilon,s,\mu,\nu,\lambdambda_0,\zeta_0,\rho_1,\rho_2,\rho_3.$
\item For any $\lambdambda \in \Gammamma_{\varepsilon,\lambdambda_0,\zeta},$ there exist constants $c,C>0$ such that
\betagin{equation*}
c \big( |\lambdambda| + |\xi'|^2 \big)
\leq \big| (s \alphapha + \betata +\zeta)^{-1} \lambdambda + |\xi'|^2 \big|
\leq C\big( |\lambdambda| + |\xi'|^2 \big) ,
\end{equation*}
with $c=c(\varepsilon,s,\mu,\nu,\lambdambda_0,\zeta_0,\rho_1,\rho_2,\rho_3)$
and $C=C(\varepsilon,s, \mu, \nu,\rho_2).$
\end{enumerate}
\end{lemm}
\betagin{lemm}\lambdabel{lemma:ABL}
Let $0<\varepsilon<\pi \slash 2,$ $\lambdambda_0,\mu, \nu>0,$ $s\in \mathbb{R},$ $\kappappa'\in \mathbb{N}_0^{N-1}$ and $\ell=0,1.$ Assume that \eqref{hyp:half_space} is satisfied.
Then the following assertions hold true.
\betagin{enumerate}
\item For any $(\lambdambda,\xi')\in \wt \Gammamma_{\varepsilon,\lambdambda_0,\zeta},$ we have
\betagin{gather*}
c_{\varepsilon,\mu,\nu,\lambdambda_0,\zeta_0,\rho_1,\rho_2,\rho_3}( |\lambdambda|^{1\slash 2}+|\xi'|) \leq |A|
\leq C_{\varepsilon,\mu, \nu,\rho_2} (|\lambdambda|^{1\slash 2} + |\xi'| ),\\
c_{\varepsilon,\mu,\rho_1,\rho_2} (|\lambdambda|^{1\slash 2} + |\xi'| ) \leq |B|
\leq C_{\varepsilon,\mu,\rho_1,\rho_2} ( |\lambdambda|^{1\slash 2} + |\xi'| ).
\end{gather*}
In fact, $A^s,B^s \in \boldsymbol{M}_{s,1}(\wt\Gammamma_{\varepsilon,\lambdambda_0,\zeta})$
for any $s \in \mathbb{R}$ with
\betagin{gather*}
\big| \partial_{\xi'}^{\kappappa'} (\tau \partial_\tau)^{\ell} A^s \big|
\leq C_{\kappappa',s,\varepsilon,\mu,\nu,\lambdambda_0,\zeta_0,\rho_1,\rho_2,\rho_3}
( |\lambdambda|^{1\slash 2} + |\xi'| )^{s-|\kappappa'|},\\
\big| \partial_{\xi'}^{\kappappa'} (\tau \partial_\tau)^{\ell} B^s \big|
\leq C_{\kappappa',s,\varepsilon,\mu,\rho_1,\rho_2} ( |\lambdambda|^{1\slash 2} + |\xi'| )^{s-|\kappappa'|},
\end{gather*}
for any $(\lambdambda,\xi')\in \wt \Gammamma_{\varepsilon,\lambdambda_0,\zeta}.$
Moreover, there exists some constant $c'=c'(\varepsilon,\mu,\rho_1,\rho_2)$ such that
\betagin{equation*}
\big| \partial_{\xi'}^{\kappappa'} (\tau \partial_\tau)^{\ell}e^{-B x_{_N}}\big|
\leq C_{\kappappa',\varepsilon,\mu,\rho_1,\rho_2}
(|\lambdambda|^{1\slash 2} + |\xi'| )^{-|\kappappa'|} e^{-c'(|\lambdambda|^{1\slash 2} +|\xi'|)x_{_N}},
\end{equation*}
for any $(\lambdambda,\xi', x_{_N}) \in \wt\Gammamma_{\varepsilon,\lambdambda_0,\zeta} \times \mathbb{R}_+.$
\item $L_{11}, L_{22} \in \boldsymbol{M}_{1,1}(\wt\Gammamma_{\varepsilon,\lambdambda_0,\zeta}),$
$L_{12} \in \boldsymbol{M}_{2,1}(\wt\Gammamma_{\varepsilon,\lambdambda_0,\zeta}),$
$L_{21} \in \boldsymbol{M}_{0,1}(\wt\Gammamma_{\varepsilon,\lambdambda_0,\zeta})$
and
$$L^\pm :=(\deltat \mathbb{L})^{\pm 1} \in \boldsymbol{M}_{\pm 2,1}(\wt\Gammamma_{\varepsilon,\lambdambda_0,\zeta}).$$
Moreover, $M(L_{jk}, \Gammamma_{\varepsilon,\lambdambda_0,\zeta} ),$ $j,k=1,2,$
$M(L^\pm,\Gammamma_{\varepsilon,\lambdambda_0,\zeta})$ depend solely on $N,\varepsilon,\mu,\nu,\lambdambda_0,\zeta_0,\rho_1,\rho_2,\rho_3.$
\item Set $Q(\lambdambda,\xi'):=(|\xi'|^2 -A^2) \slash (AB-|\xi'|^2)$ and we have
$Q \in \boldsymbol{M}_{0,1}(\wt\Gammamma_{\varepsilon,\lambdambda_0,\zeta})$
with
$M(Q, \Gammamma_{\varepsilon,\lambdambda_0,\zeta} )$ depending solely on
$N,\varepsilon,\mu,\nu,\lambdambda_0,\zeta_0,\rho_1,\rho_2,\rho_3.$
\end{enumerate}
\end{lemm}
Thanks to Lemma \ref{lemma:basic} and Lemma \ref{lemma:ABL} above, it is easy to show that:
\betagin{lemm}[Interaction]\lambdabel{lemma:int_AB}
Under the same assumptions in Lemma \ref{lemma:ABL}, we have
$$Q':=(AB + |\xi'|^2)^{-1} \in \boldsymbol{M}_{-2,1}(\Gammamma_{\varepsilon,\lambdambda_0}),$$
with $M(Q', \Gammamma_{\varepsilon,\lambdambda_0,\zeta} )$ depending solely on $N,\varepsilon,\mu,\nu,\lambdambda_0,\zeta_0,\rho_1,\rho_2,\rho_3.$
\end{lemm}
\betagin{proof}
In fact, it is sufficient to verify that there exists a constant $c=c(\varepsilon,\mu,\nu,\lambdambda_0,\zeta_0,\rho_1,\rho_2,\rho_3)$
such that
\footnote{We omit the details of the reduction here, because the similar arguments will be employed in the proof of Lemma \ref{lemma:N_AB} later.}
\betagin{equation*}
\big| AB + |\xi'|^2 \big| \geq c(|\lambdambda| + |\xi'|^2),
\,\,\,\forall \,\, (\lambdambda,\xi') \in \wt\Gammamma_{\varepsilon,\lambdambda_0,\zeta}.
\end{equation*}
The lower bound of $AB+|\xi'|^2$ above immediately follows from the fact
\betagin{equation}\lambdabel{claim:AB}
AB \in \Sigmagma_{\varepsilon_0}, \,\,\,\thetaxt{for some}\,\,0<\varepsilon_0 <\pi\slash 2.
\end{equation}
Indeed, Lemma \ref{lemma:basic} and Lemma \ref{lemma:ABL} imply that
\betagin{equation*}
\big| AB + |\xi'|^2 \big| \geq \sigman(\varepsilon_0 \slash 2) (|AB| + |\xi'|^2)
\geq \sigman(\varepsilon_0 \slash 2) c^2 (|\lambdambda|^{1\slash 2} + |\xi'|^2)^2.
\end{equation*}
Now, we prove \eqref{claim:AB}. Thanks to Lemma \ref{lemma:basic}, there exists $0<\varepsilon'<\pi\slash 2$ such that
$z_1:=(2\alphapha +\betata +\zeta)^{-1} \lambdambda$ belongs to $\Sigmagma_{\varepsilon'}.$
Moreover, for any $\lambdambda\in \Gammamma_{\varepsilon,\lambdambda_0,\zeta},$ we have
\betagin{align*}
|\arg A | &= \big| \arg (z_1 + |\xi'|^2 )| \slash 2 \leq |\arg z_1|\slash 2 \leq (\pi-\varepsilon') \slash 2,\\
|\arg B | &= \big| \arg (\alphapha^{-1}\lambdambda + |\xi'|^2 )| \slash 2 \leq |\arg \lambdambda| \slash 2 \leq (\pi-\varepsilon) \slash 2.
\end{align*}
Then $|\arg (AB)| = |\arg A+ \arg B| < \pi -\varepsilon_0$ for any $\varepsilon_0 < (\varepsilon+\varepsilon') \slash 2.$
\end{proof}
Next, we pick up several standard results on $\mathcal{R}$-boundedness (see \cite{ES2013,GS2014, Shi2019} for instance).
\betagin{lemm}\lambdabel{lemma:basic_es}
Under the same assumptions in Lemma \ref{lemma:ABL},
we take $n_1(\lambdambda,\xi') \in \boldsymbol{M}_{-2,1} (\wt\Gammamma_{\varepsilon,\lambdambda_0,\zeta}),$
and consider the operators
\betagin{align*}
\Psi_1(\lambdambda) f (x)&:=\int_0^{\infty} \mathcal{F}^{-1}_{\xi'}\boldsymbol{i}g[ n_1(\lambdambda,\xi')
B e^{-B(x_{_N}+y_{_N})}\mathcal{F}_{y'}[f](\xi',y_{_N})\boldsymbol{i}g] (x') \,d y_{_N},\\
\Psi_2(\lambdambda) f (x) &:=\int_0^{\infty} \mathcal{F}^{-1}_{\xi'}\boldsymbol{i}g[ n_1(\lambdambda,\xi')
B^2 \mathcal{M} (x_{_N}+y_{_N}) \mathcal{F}_{y'}[f](\xi',y_{_N})\boldsymbol{i}g] (x') \,d y_{_N}.
\end{align*}
Then we have
\betagin{equation*}
\mathcal{R}_{\mathcal{L} \big(L_q(\mathbb{R}_+^{N});H^{2-j}_q(\mathbb{R}^N_+)\big)}
\big( \{(\tau \partial_{\tau})^{\ell}\ \lambdambda^{j\slash 2}\Psi_k(\lambdambda): \lambdambda \in \Gammamma_{\varepsilon,\lambdambda_0,\zeta} \} \big) \leq r_k,
\end{equation*}
for $j=0,1,2,$ $k=1,2,$ and $\ell =0,1.$ Above the constants $r_1, r_2$ depend on $M(n_1,\Gammamma_{\varepsilon,\lambdambda_0,\zeta}),$ $N,$ $q,$ $\varepsilon,$ $\mu,$ $\lambdambda_0,$ $\rho_1,$ $\rho_2.$
\end{lemm}
\betagin{lemm}\lambdabel{lemma:basic_es_2}
Assume that $n_2(\lambdambda,\xi')\in \boldsymbol{M}_{0,2} \big(\Lambdambda \times (\mathbb{R}^{N-1} \backslash \{0\}) \big)$
for some domain $\Lambdambda \subset \mathbb{C},$
and the cut-off functions $\varphi,$ $\psi \in C_0^{\infty}(]-2,2[).$
Consider the operators
\betagin{align*}
\Psi_3(\lambdambda) f &:=\varphi(x_{_N}) \int_0^{\infty} \mathcal{F}^{-1}_{\xi'}\boldsymbol{i}g[ n_2(\lambdambda,\xi')
e^{-|\xi'|(x_{_N}+y_{_N})} \psi(y_{_N}) \mathcal{F}_{y'}[f](\xi',y_{_N})\boldsymbol{i}g] (x') \,d y_{_N},\\
\Psi_4(\lambdambda) f &:=\varphi(x_{_N}) \int_0^{\infty} \mathcal{F}^{-1}_{\xi'}\boldsymbol{i}g[ n_2(\lambdambda,\xi')
|\xi'|e^{-|\xi'|(x_{_N}+y_{_N})}\psi(y_{_N})\mathcal{F}_{y'}[f](\xi',y_{_N})\boldsymbol{i}g] (x') \,d y_{_N}.
\end{align*}
Then we have
\betagin{equation*}
\mathcal{R}_{\mathcal{L} \big(L_q(\mathbb{R}_+^{N})\big)}
\big( \{(\tau \partial_{\tau})^{\ell} \Psi_k(\lambdambda): \lambdambda \in \Lambdambda \} \big) \leq r_k,
\end{equation*}
for $k=3,4,$ $\ell =0,1$ and some constants $r_3,r_4$ depending on $M(n_2,\Lambdambda),$ $\varphi,\psi,$ $N,q.$
\end{lemm}
Finally, recall $N(A,B)$ in \eqref{eq:GR_half_h_1}, and we shall see $N(A,B)^{-1}$ is well defined by choosing suitable $\Gammamma_{\varepsilon,\lambdambda_0,\zeta}$ in the next lemma.
\betagin{lemm}\lambdabel{lemma:N_AB}
Under the same assumptions in Lemma \ref{lemma:ABL},
there exist some $\lambdambda_0 \geq 1$ and $C_{\kappappa'}>0$ such that
\betagin{equation}\lambdabel{es:N_AB}
\big| \partial_{\xi'}^{\kappappa'} (\tau \partial_\tau)^\ell \big( N(A,B)^{-1}\big) \big|
\leq C_{\kappappa'} (|\lambdambda|+|\xi'|)^{-1} ( |\lambdambda|^{1\slash 2} +|\xi'|)^{-2-|\kappappa'|},
\end{equation}
for all $\lambdambda \in \Gammamma_{\varepsilon,\lambdambda_0,\zeta},$ $\kappappa'\in \mathbb{N}_0^{N-1}$ and $\ell =0,1.$
Here $\lambdambda_0$ and $C_{\kappappa'}$ depend on $\sigmagma,m,\varepsilon,\mu,\nu,\zeta_0,\rho_1, \rho_2, \rho_3.$
\end{lemm}
\betagin{proof}
We will focus on the proof of \eqref{es:N_AB} with $\ell=0,$ from which the case $\ell =1$ will be derived without any difficulty.
Now let us introduce
\betagin{equation*}
P(\lambdambda, \xi') := \frac{\lambdambda}{AB-|\xi'|^2}
= \frac{\alphapha(2\alphapha+\betata+\zeta)}{3\alphapha+\betata+\zeta}\frac{AB+|\xi'|^2}{(3\alphapha+\betata+\zeta)^{-1}\lambdambda +|\xi'|^2}
\cdot
\end{equation*}
By the definition of $P,$ the matrix $\mathbb{L}$ and $\deltat \mathbb{L}$ are formulated by
\betagin{gather}\lambdabel{eq:L_ij_2}
L_{11} := A P, \qquad
L_{12}:= |\xi'|^2 (2\alphapha-P), \\ \nonumber
L_{21}:=\boldsymbol{i}g(\frac{A}{A+B} -\frac{\betata +\zeta}{2\alphapha +\betata +\zeta}\frac{B}{A+B}\boldsymbol{i}g)P,
\quad L_{22}:=BP, \quad \deltat \mathbb{L} = P D(A,B), \\ \nonumber
D(A,B):= AB P - |\xi'|^2 (2\alphapha -P) \big(\frac{A}{A+B} -\frac{\betata +\zeta}{2\alphapha +\betata +\zeta}\frac{B}{A+B}\big).
\end{gather}
\underline{Case: $\ell=0.$}
{\bf Step 1.} In order to show \eqref{es:N_AB} for $\ell =0,$ let us make some reduction.
Firstly, it is easy to see from the definition of $N(A,B)$ that
\betagin{equation*}
N(A,B) = L_{11} E_{\sigmagma,m}(\lambdambda,\xi') -\lambdambda L_{12}L_{21}
\end{equation*}
for $E_{\sigmagma,m}(\lambdambda,\xi') : = \lambdambda L_{22} +\sigmagma (m+|\xi'|^2).$
According to Lemma \ref{lemma:ABL} and Remark \ref{rmk:M1M2}, there exists a constant
$C_{\kappappa'} = C(\kappappa',\varepsilon,\mu,\nu,\zeta_0,\rho_1, \rho_2, \rho_3)>0$ such that
\betagin{align*}
\big| \partial_{\xi'}^{\kappappa'} (\tau \partial_\tau)^{\ell'} (\lambdambda L_{12}L_{21}) \big|
& \leq C_{\kappappa'} |\lambdambda| ( |\lambdambda|^{1\slash 2} +|\xi'|)^{2-|\kappappa'|},\\
\big| \partial_{\xi'}^{\kappappa'} (\tau \partial_\tau)^{\ell'} E_{\sigmagma,m}(\lambdambda,\xi') \big|
& \leq C_{\kappappa'} \max\{1,\sigmagma,\sigmagma m\} (|\lambdambda| +|\xi'|) ( |\lambdambda|^{1\slash 2} +|\xi'|)^{1-|\kappappa'|},
\end{align*}
for any $(\lambdambda,\xi') \in \wt \Gammamma_{\varepsilon,1,\zeta},$ $\kappappa'\in \mathbb{N}_0^{N-1}$ and $\ell'=0,1.$
Then it is easy to see from Lemma \ref{lemma:ABL} that
\betagin{equation}\lambdabel{es:N_AB_upper}
\big| \partial_{\xi'}^{\kappappa'} (\tau \partial_\tau)^{\ell'} N(A,B) \big|
\leq C_{\kappappa'} \max\{1,\sigmagma,\sigmagma m\} (|\lambdambda| +|\xi'|) ( |\lambdambda|^{1\slash 2} +|\xi'|)^{2-|\kappappa'|},
\end{equation}
for any $(\lambdambda,\xi') \in \wt \Gammamma_{\varepsilon,1,\zeta},$ $\kappappa' \in \mathbb{N}_0^{N-1}$ and $\ell'=0,1.$
\smallbreak
On the other hand, by the Bell formula, we have
\betagin{equation*}
\partial_{\xi'}^{\kappappa'} \big( N(A,B)^{-1}\big) = \sum_{j =1}^{\kappappa'} (-1)^j j ! \big(N(A,B)\big)^{-(j+1)}
\sum_{ \substack{\kappappa'_1 +\cdots +\kappappa'_j =\kappappa'\\|\kappappa'_k| \geq 1}} C^j_{\kappappa'_1,\dots \kappappa'_j}
\big( \partial_{\xi'}^{\kappappa'_1} N(A,B)\big) \cdots \big( \partial_{\xi'}^{\kappappa'_j} N(A,B)\big).
\end{equation*}
Then \eqref{es:N_AB} with $\ell =0$ holds true so long as
\betagin{equation}\lambdabel{es:N_AB_low}
|N(A,B)| \geq c (|\lambdambda| +|\xi'|) (|\lambdambda|^{1\slash 2} +|\xi'|)^2
\end{equation}
for some constant $c,$ because the derivatives of $N(A,B)$ is bounded by \eqref{es:N_AB_upper}.
Furthermore, thanks to the rules in \eqref{eq:L_ij_2}, $N(A,B)= P \wt{N}(A,B)$ with
\betagin{equation*}
\wt{N}(A,B) := \lambdambda D(A,B)+ \sigmagma A (m +|\xi'|^2).
\end{equation*}
By Lemma \ref{lemma:basic} and Lemma \ref{lemma:int_AB},
$|P(\lambdambda,\xi')| \geq C_{\varepsilon,\mu,\nu,\zeta_0,\rho_1,\rho_2,\rho_3}$ for any
$(\lambdambda,\xi') \in \wt \Gammamma_{\varepsilon,1,\zeta}.$
Therefore, it is sufficient to show that $\wt N(A,B)$ is bounded below by the r.h.s. of \eqref{es:N_AB_low}.
{\bf Step 2.} In order to study $\wt N(A,B),$ we need the following technical results.
There exist $\lambdambda_0' \geq 1$ and $\varepsilon_0 \in ]0,\pi\slash 2[$ such that
\betagin{equation}\lambdabel{eq:z_N_AB}
z:= (\alphapha+\betata+\zeta)(2\alphapha+\betata+\zeta)^{-1}\lambdambda \in \Sigmagma_{\varepsilon_0},
\,\,\, \forall \,\, (\lambdambda,\xi') \in \wt\Gammamma_{\varepsilon,\lambdambda_0',\zeta},
\end{equation}
with the choices of $\lambdambda_0'$ and $\varepsilon_0$ depending only on $\varepsilon,\mu,\nu,\zeta_0,\rho_1,\rho_2,\rho_3.$
\smallbreak
For the Case (C1) where $|\zeta|=|\gammamma_1^{-1}\gammamma_3 \lambdambda| \leq \min\{ \rho_1^{-1}\rho_3 |\lambdambda|^{-1},\zeta_0\},$
we infer from Lemma \ref{lemma:basic} that
\betagin{equation*}
(2\alphapha + \betata +\zeta)^{-1} \lambdambda \in \Sigmagma_{\varepsilon'},
\,\,\, \forall \,\, \lambdambda \in \Gammamma_{\varepsilon,1,\zeta}
\end{equation*}
for some $\varepsilon'=\varepsilon'(\varepsilon,\mu,\nu,\zeta_0,\rho_1,\rho_2,\rho_3)$ in $]0,\pi\slash 2[.$
On the other hand, there exists $\varepsilon''\in ]0,\varepsilon'\slash 2[$ such that
$$|\arg (\alphapha+\betata+\zeta)| \leq \varepsilon'', \,\,\, \forall \,\, \lambdambda \in \Gammamma_{\varepsilon,\lambdambda_0',\zeta},$$
by taking $\lambdambda_0'$ large enough. Thus $|\arg z| \leq \pi -\varepsilon'-\varepsilon''.$
For the Case (C2) where $|\arg \zeta|\in ]\pi \slash 2, \pi-\varepsilon[,$ we denote $\omegaega:= (\alphapha+\betata+\zeta)(2\alphapha+\betata+\zeta)^{-1}.$ Assume that
\betagin{equation*}
\Re \zeta \leq -(2\alphapha +\betata) \,\,\,
\hbox{or}\,\,\, -\alphapha -\betata \leq \Re \zeta <0,
\end{equation*}
where we have
\betagin{equation*}
|\arg w| \leq \arctan \frac{\alphapha}{(\alphapha + \betata)\tan \varepsilon} = \arctan \frac{\mu}{(\mu + \nu)\tan \varepsilon} \cdot
\end{equation*}
Then $|\arg z| \leq |\arg w| + \pi \slash 2 \leq \pi-\varepsilon_0$ for some $\varepsilon_0=\varepsilon_0(\mu,\nu,\varepsilon).$
For the situation $\Re \zeta \in ]-(2\alphapha+\betata), -(\alphapha+\betata)[,$ we note that $|\arg \lambdambda| \leq \pi -\arg \zeta$ and then conclude that
\betagin{equation*}
|\arg z| = |\arg w + \arg \lambdambda| \leq \pi - \arctan \frac{(\alphapha+\betata)\tan \varepsilon}{\alphapha}
= \pi - \arctan (\mu^{-1}\nu \tan \varepsilon).
\end{equation*}
At last, \eqref{eq:z_N_AB} is valid for the case (C3) as $|\arg w| < \pi \slash 4$ and
$|\arg \lambdambda| < \pi\slash 2.$
{\bf Step 3.} By assuming $|\lambdambda| |\xi'|^{-2} \leq r \ll 1,$ there exists some
$c_1=c_1(\varepsilon,\mu,\nu,\gammamma_1,\gammamma_2,\zeta_0,\sigmagma)>0$ such that
\betagin{equation}\lambdabel{es:N_AB_wt}
\big| \wt N(A,B) \big| \geq c_1 (|\lambdambda| +|\xi'|) (|\lambdambda|^{1\slash 2} +|\xi'|)^2,
\end{equation}
for any $\lambdambda\in \Gammamma_{\varepsilon, \lambdambda_0',\zeta}.$
By Lemma \ref{lemma:basic}, it is not hard to see that
\betagin{equation*}
A, B=|\xi'|\big(1+O(r)\big) \,\,\,\hbox{and}\,\,\,
P = \frac{2\alphapha(2\alphapha +\betata +\zeta)}{3\alphapha +\betata +\zeta} +O(r).
\end{equation*}
Then \eqref{eq:z_N_AB} and Lemma \ref{lemma:basic} furnish that
\betagin{align}\lambdabel{es:N_AB_wt_1}
\big| |\xi'|^{-2}\wt{N}(A,B) \big|
&=\boldsymbol{i}g| \lambdambda \boldsymbol{i}g(\frac{2\alphapha(\alphapha+\betata+\zeta)}{2\alphapha+\betata+\zeta} +O(r)\boldsymbol{i}g)
+ \sigmagma \big(1+O(r) \big) \big( m|\xi'|^{-1} + |\xi'| \big) \boldsymbol{i}g|\\
& \geq \boldsymbol{i}g|\frac{2\alphapha(\alphapha+\betata+\zeta)}{2\alphapha+\betata+\zeta}\lambdambda
+\sigmagma \big( m|\xi'|^{-1} + |\xi'| \big) \boldsymbol{i}g|
- C_1 r \big(|\lambdambda|+\sigmagma m |\xi'|^{-1} +\sigmagma |\xi'| \big) \notag \\ \notag
& \geq \sigman \big(\frac{\varepsilon_0}{2} \big) \boldsymbol{i}g( 2\alphapha\boldsymbol{i}g|\frac{\alphapha+\betata+\zeta}{2\alphapha+\betata+\zeta}\boldsymbol{i}g| |\lambdambda| +\sigmagma (m|\xi'|^{-1} + |\xi'|) \boldsymbol{i}g) \\ \notag
& \quad - C_1 r \big(|\lambdambda|+\sigmagma m |\xi'|^{-1} +\sigmagma |\xi'| \big)
\geq C_2 \big(|\lambdambda|+\sigmagma m |\xi'|^{-1} +\sigmagma |\xi'| \big),
\end{align}
for some $C_2 =C_2(\varepsilon,\mu,\nu,\zeta_0,\rho_1,\rho_2,\rho_3)$ and any
$(\lambdambda,\xi') \in \wt\Gammamma_{\varepsilon,\lambdambda_0',\zeta}.$
Due to smallness of $r$ and $|\lambdambda| \leq r |\xi'|^2,$ we have
\betagin{equation*}
(|\lambdambda| +|\xi'|) (|\lambdambda|^{1\slash 2} +|\xi'|)^2
\leq 4 (|\xi'|^3 + |\lambdambda||\xi'|^2) \leq 4 (\min\{1,\sigmagma\})^{-1} (|\lambdambda| + \sigmagma |\xi'|^3).
\end{equation*}
Thus \eqref{es:N_AB_wt} holds with $c_1:= C_2 \min\{1,\sigmagma\} \slash 4.$
{\bf Step 4.} For the fixed $r=r(\varepsilon,\mu,\nu,\zeta_0,\rho_1,\rho_2,\rho_3)<1$ in the last step,
\eqref{es:N_AB_wt} still holds true for $|\lambdambda| > r |\xi'|^2,$ $\lambdambda \in \Gammamma_{\varepsilon,\lambdambda_0'',\zeta}$ and some $\lambdambda_0'' \geq 1.$
Indeed, thanks to the boundedness of $P$ and Lemma \ref{lemma:ABL}, there exists a constant $c_2$ such that
\betagin{equation*}
|D(A,B)| = |P^{-1}\deltat \mathbb{L}|\geq c_2 (|\lambdambda|^{1\slash2} + |\xi'|)^2,
\,\,\, \forall \,\, (\lambdambda,\xi') \in \wt\Gammamma_{\varepsilon,1,\zeta}.
\end{equation*}
As $|A| \leq C_3 (|\lambdambda|^{1\slash 2} +|\xi'|)$ for some constant $C_3=C_3(\varepsilon,\mu,\nu,\rho_2),$ we have
\betagin{align*}
\big| \wt N(A,B) \big|
\geq c_2 |\lambdambda| ( |\lambdambda|^{1\slash 2} +|\xi'|)^2 -\sigmagma C_3 ( |\lambdambda|^{1\slash 2} +|\xi'|)(m + |\xi'|^2).
\end{align*}
Next, set $C_\sigmagma := \sigmagma (1 + 2C_3 \slash {c_2})$ and we claim that there exists $\lambdambda_0'' \geq 1$ such that
\betagin{equation}\lambdabel{eq:f_N_AB}
f(\lambdambda,|\xi'|):=|\lambdambda| ( |\lambdambda|^{1\slash 2} +|\xi'|) - C_\sigmagma (m+ |\xi'|^2) \geq 0,
\,\,\, \forall \,\, \lambdambda \in \wt\Gammamma_{\varepsilon,\lambdambda_0'',\zeta}.
\end{equation}
Then \eqref{eq:f_N_AB} implies that
\betagin{align*}
|\wt N(A,B)| & \geq \frac{c_2}{2} \big( |\lambdambda| (|\lambdambda|^{1\slash 2} +|\xi'|)^2
+ \sigmagma (m+|\xi'|^2) (|\lambdambda|^{1\slash 2} +|\xi'|) \big) \\
& \geq \frac{c_2}{4} \min\{1,\sigmagma\} (|\lambdambda| +|\xi'|) (|\lambdambda|^{1\slash 2} +|\xi'|)^2,
\,\,\, \forall \,\, \lambdambda \in \wt\Gammamma_{\varepsilon,\lambdambda_0'',\zeta}.
\end{align*}
For \eqref{eq:f_N_AB}, $f(\lambdambda,|\xi'|) \geq (\lambdambda_0'')^{3\slash 2} -C_\sigmagma m + \big((r\lambdambda_0'')^{1\slash 2} -C_\sigmagma \big) |\xi'|^2$ since $|\lambdambda| \geq \max\{r|\xi'|^2, \lambdambda_0'' \}.$
Thus \eqref{eq:f_N_AB} holds true by choosing $\lambdambda_0''$ large.
Finally, we obtain the lower bound of $\wt N(A,B)$ for $\ell=0,$ by taking $\lambdambda_0 := \max\{\lambdambda_0',\lambdambda_0''\}.$
\underline{Case: $\ell=1.$} In fact, the conclusion for $\ell =1$ is a straightforward result from above discussion.
Note that
$(\tau \partial_\tau) \big( N(A,B)^{-1} \big) = - N(A,B)^{-2} \big(\tau \partial_\tau N(A,B)\big).$
Then our results for the case $\ell=0,$ Leibniz formula and \eqref{es:N_AB_upper} yield the desired bound of $\partial_{\xi'}^{\kappappa'}(\tau \partial_\tau) \big( N(A,B)^{-1} \big) $ in \eqref{es:N_AB} for any $\kappappa' \in \mathbb{N}_0^{N-1}.$
\end{proof}
Thanks to Lemma \ref{lemma:N_AB} above, we end up with the following result.
\betagin{coro}\lambdabel{coro:n_Jk}
Under the same assumptions in Lemma \ref{lemma:ABL}, there exists $\lambdambda_0$ depending solely on $\sigmagma,$ $m,$ $\varepsilon,$ $\mu,$
$\nu,$ $\zeta_0,$ $\rho_1,$ $\rho_2,$ $\rho_3,$ such that
the symbols $n_{Jk}(\lambdambda,\xi')$ defined in \eqref{eq:n_Jk_half} belong to
$\boldsymbol{M}_{-2,1}(\wt\Gammamma_{\varepsilon,\lambdambda_0,\zeta})$ for all $J=1,\dots,N$ and $k=1,2.$
\end{coro}
\betagin{proof}
The study of symbols are nothing but applying directly Lemma \ref{lemma:ABL} and Remark \ref{rmk:M1M2}, and we omit the details here.
\end{proof}
\subsection{Proof of Proposition \ref{prop:GR_half_2}}
In this part, we are studying the operators $\mathcal{W}(\lambdambda,\mathbb{R}^N_+)$ and $\mathcal{H}(\lambdambda,\mathbb{R}^N_+)$ respectively according to different technical results in last subsection.
For $Z \in \{B,|\xi'|\},$ let us recall the following equalities, namely the \emph{Volevich trick},
\betagin{align*}
e^{-Zx_{_N}} \wh{k}(\xi',0)
=& \int_0^{\infty}Z e^{-Z(x_{_N}+y_{_N})}\wh{k}(\xi',y_{_N}) \,d y_{_N}
- \int_0^{\infty}e^{-Z(x_{_N}+y_{_N})}\wh{\partial_{N} k}(\xi',y_{_N}) \,d y_{_N},\\
M(x_{_N})\wh{k}(\xi',0)
= &\int_0^{\infty}e^{-B(x_{_N}+y_{_N})}\wh{k}(\xi',y_{_N}) \,d y_{_N}
+\int_0^{\infty}AM(x_{_N}+y_{_N})\wh{k}(\xi',y_{_N}) \,d y_{_N}\\
& - \int_0^{\infty}M(x_{_N}+y_{_N}) \wh{\partial_{N} k}(\xi',y_{_N}) \,d y_{_N},
\end{align*}
where we have used $\partial_{_N} M(z_{_N})=-e^{-Bz_{_N}}-AM(z_{_N})$ for any $z_{_N}>0.$
\subsubsection{The $\mathcal{R}-$boundedness of $\mathcal{W}(\lambdambda,\mathbb{R}^N_+)$}
According to the identity $|\xi'|^2 =-\sum_{\ell=1}^{N-1} (i\xi_{\ell})^2,$
the solution operator
\betagin{equation*}
\boldsymbol{u}= \mathcal{W} (\lambdambda,\mathbb{R}^N_+)k =\big( \mathcal{W}_{1} (\lambdambda,\mathbb{R}^N_+)k, \dots, \mathcal{W}_{N} (\lambdambda,\mathbb{R}^N_+) k\big)^{\top},
\end{equation*}
in \eqref{eq:GR_half_uj_2} is rewritten by
\betagin{equation*}
\mathcal{W}_{J} (\lambdambda,\mathbb{R}^N_+) k := \mathcal{W}_{J1}(\lambdambda,\mathbb{R}^N_+)\big((m-\Deltalta') k\big)
+\mathcal{W}_{J2}(\lambdambda,\mathbb{R}^N_+)(\partial_{_N} k)
+\mathcal{W}_{J3}(\lambdambda,\mathbb{R}^N_+)(\nabla'\partial_{_N} k)
\end{equation*}
for any $J=1,\dots,N.$
The operators $\mathcal{W}_{Jk} \equiv \mathcal{W}_{Jk}(\lambdambda,\mathbb{R}^N_+)$ ($k=1,2,3$) above are given by
\betagin{align*}
\mathcal{W}_{j1}(F_1) :=&\int_0^{\infty} \mathcal{F}_{\xi'}^{-1} \boldsymbol{i}g[ n_{j1}(\lambdambda,\xi')\frac{A}{B} B^2 \mathcal{M}(x_{_N} + y_{_N}) \wh{F_1}(\xi',y_{_N}) \boldsymbol{i}g](x') \,d y_{_N}\\
&+\int_0^{\infty} \mathcal{F}_{\xi'}^{-1} \boldsymbol{i}g[ n_{j2}(\lambdambda,\xi')
B e^{-B(x_{_N}+y_{_N})} \wh{F_1}(\xi',y_{_N}) \boldsymbol{i}g](x') \,d y_{_N},\\
\mathcal{W}_{j2}(F_2) := & - \int_0^{\infty} \mathcal{F}_{\xi'}^{-1} \boldsymbol{i}g[ n_{j1}(\lambdambda,\xi')\frac{m}{B}
\big( B^2 \mathcal{M}(x_{_N} + y_{_N}) - B e^{-B(x_{_N}+y_{_N})} \big)
\wh{F_2}(\xi',y_{_N}) \boldsymbol{i}g](x') \,d y_{_N}\\
&-\int_0^{\infty} \mathcal{F}_{\xi'}^{-1} \boldsymbol{i}g[ n_{j2}(\lambdambda,\xi') \frac{m}{B}
Be^{-B(x_{_N}+y_{_N})} \wh{F_2}(\xi',y_{_N}) \boldsymbol{i}g](x') \,d y_{_N},\\
\mathcal{W}_{j3}(\boldsymbol{F}') :=&\sum_{\ell=1}^{N-1}\int_0^{\infty} \mathcal{F}_{\xi'}^{-1} \boldsymbol{i}g[
n_{j1}(\lambdambda,\xi')\frac{i\xi_{\ell}}{B} \big( B^2 \mathcal{M}(x_{_N} + y_{_N}) - B e^{-B(x_{_N}+y_{_N})} \big)
\wh{F_{\ell}}(\xi',y_{_N}) \boldsymbol{i}g](x') \,d y_{_N}\\
&+\sum_{\ell=1}^{N-1}\int_0^{\infty} \mathcal{F}_{\xi'}^{-1} \boldsymbol{i}g[n_{j2}(\lambdambda,\xi')\frac{i\xi_{\ell}}{B}
Be^{-B(x_{_N}+y_{_N})} \wh{F_{\ell}}(\xi',y_{_N}) \boldsymbol{i}g](x') \,d y_{_N},
\end{align*}
for any $j =1, \dots,N-1,$ and
\betagin{align*}
\mathcal{W}_{N1}(F_1) :=&\int_0^{\infty} \mathcal{F}_{\xi'}^{-1} \boldsymbol{i}g[ (n_{N1}+n_{N2})(\lambdambda,\xi')
Be^{-B(x_{_N}+y_{_N})} \wh{F_1}(\xi',y_{_N}) \boldsymbol{i}g](x') \,d y_{_N}\\
&+\int_0^{\infty} \mathcal{F}_{\xi'}^{-1} \boldsymbol{i}g[ n_{N1}(\lambdambda,\xi')\frac{A}{B} B^2 \mathcal{M}(x_{_N} + y_{_N}) \wh{F_1}(\xi',y_{_N}) \boldsymbol{i}g](x') \,d y_{_N},\\
\mathcal{W}_{N2}(F_2) := & - \int_0^{\infty} \mathcal{F}_{\xi'}^{-1} \boldsymbol{i}g[ n_{N1}(\lambdambda,\xi')\frac{m}{B}
B^2 \mathcal{M}(x_{_N} + y_{_N}) \wh{F_2}(\xi',y_{_N}) \boldsymbol{i}g](x') \,d y_{_N}\\
&-\int_0^{\infty} \mathcal{F}_{\xi'}^{-1} \boldsymbol{i}g[ n_{N2}(\lambdambda,\xi') \frac{m}{B}
Be^{-B(x_{_N}+y_{_N})} \wh{F_2}(\xi',y_{_N}) \boldsymbol{i}g](x') \,d y_{_N} ,\\
\mathcal{W}_{N3}(\boldsymbol{F}') :=&\sum_{\ell=1}^{N-1}\int_0^{\infty} \mathcal{F}_{\xi'}^{-1} \boldsymbol{i}g[
n_{N1}(\lambdambda,\xi')\frac{i\xi_{\ell}}{B} B^2 \mathcal{M}(x_{_N} + y_{_N})
\wh{F_{\ell}}(\xi',y_{_N}) \boldsymbol{i}g](x') \,d y_{_N}\\
&+\sum_{\ell=1}^{N-1}\int_0^{\infty} \mathcal{F}_{\xi'}^{-1} \boldsymbol{i}g[n_{N2}(\lambdambda,\xi')\frac{i\xi_{\ell}}{B}
Be^{-B(x_{_N}+y_{_N})} \wh{F_{\ell}}(\xi',y_{_N}) \boldsymbol{i}g](x') \,d y_{_N}.
\end{align*}
Note that $\boldsymbol{M}_{-3,1}(\wt\Gammamma_{\varepsilon,\lambdambda_0,\zeta}) \subset \boldsymbol{M}_{-2,1}(\wt\Gammamma_{\varepsilon,\lambdambda_0,\zeta})$ and
$A\slash B,$ $i\xi_{\ell}\slash B$ for $\ell =1,\dots N-1,$ are in $\boldsymbol{M}_{0,1}(\wt\Gammamma_{\varepsilon,\lambdambda_0,\zeta}).$
Then Lemma \ref{lemma:basic_es} and Corollary \ref{coro:n_Jk} imply that
\betagin{gather*}
\mathcal{R}_{\mathcal{L}\big(L_q(\mathbb{R}^N_+); H^{2-j}_q(\mathbb{R}^N_+) \big)}
\boldsymbol{i}g( \boldsymbol{i}g\{ (\tau \partial_{\tau})^{\ell}\big( \lambdambda^{j\slash 2}\mathcal{W}_{Jk}(\lambdambda,\mathbb{R}^N_+)\big) : \lambdambda \in
\Gammamma_{\varepsilon,\lambdambda_0,\zeta} \boldsymbol{i}g\}\boldsymbol{i}g) \leq C,\\
\mathcal{R}_{\mathcal{L}\big(L_q(\mathbb{R}^N_+)^{N-1}; H^{2-j}_q(\mathbb{R}^N_+) \big)}
\boldsymbol{i}g( \boldsymbol{i}g\{ (\tau \partial_{\tau})^{\ell}\big( \lambdambda^{j\slash 2}\mathcal{W}_{J3}(\lambdambda,\mathbb{R}^N_+)\big) : \lambdambda \in
\Gammamma_{\varepsilon,\lambdambda_0,\zeta} \boldsymbol{i}g\}\boldsymbol{i}g) \leq C,
\end{gather*}
for $k=1,2,$ and $j=0,1,2.$ Thus Definition \ref{def:R-bounded} gives us
\betagin{equation*}
\mathcal{R}_{\mathcal{L}\big(H_q^2(\mathbb{R}^N_+); H^{2-j}_q(\mathbb{R}^N_+)^N \big)}
\boldsymbol{i}g( \boldsymbol{i}g\{ (\tau \partial_{\tau})^{\ell}\big( \lambdambda^{j\slash 2}\mathcal{W}(\lambdambda,\mathbb{R}^N_+)\big) : \lambdambda \in
\Gammamma_{\varepsilon,\lambdambda_0,\zeta} \boldsymbol{i}g\}\boldsymbol{i}g) \leq C.
\end{equation*}
\subsubsection{The $\mathcal{R}-$boundedness of $\mathcal{H}(\lambdambda,\mathbb{R}^N_+)$}
Let us apply the Volevich trick to \eqref{eq:GR_half_h_2} and obtain
\betagin{equation*}
h(x) =\big(\mathcal{H}(\lambdambda,\mathbb{R}^N_+)k\big)(x)
=\big(\mathcal{H}_{1}(\lambdambda,\mathbb{R}^N_+)k\big)(x) + \big(\mathcal{H}_{2}(\lambdambda,\mathbb{R}^N_+) k\big)(x),
\end{equation*}
where the operators $\mathcal{H}_{\mathfrak{a}} \equiv\mathcal{H}_{\mathfrak{a}}(\lambdambda,\mathbb{R}^N_+)$ with $\mathfrak{a}=1,2,$ are given by
\betagin{align*}
(\mathcal{H}_{1}k)(x)&:= \varphi(x_{_N}) \int_{0}^{\infty} \mathcal{F}^{-1}_{\xi'}\boldsymbol{i}g[
\frac{\deltat \mathbb{L}}{N(A,B)} \,|\xi'| e^{-|\xi'|(x_{_N}+y_{_N})}
\varphi(y_{_N}) \widehat{k}(\xi',y_{_N})\boldsymbol{i}g] (x') \,dy_{_N},\\
(\mathcal{H}_{2}k)(x)&:=-\varphi(x_{_N}) \int_{0}^{\infty} \mathcal{F}^{-1}_{\xi'}\boldsymbol{i}g[
\frac{\deltat \mathbb{L}}{N(A,B)} \,e^{-|\xi'|(x_{_N}+y_{_N})}
\partial_{{_N}} \big( \varphi(y_{_N}) \widehat{k}(\xi',y_{_N})\big) \boldsymbol{i}g] (x') \,dy_{_N}.
\end{align*}
Clearly, we have for $j=0,1,$
\betagin{align*}
(\lambdambda^j \mathcal{H}_{1}k)(x)&:= \varphi(x_{_N}) \int_{0}^{\infty} \mathcal{F}^{-1}_{\xi'}\boldsymbol{i}g[
\frac{\lambdambda^j\deltat \mathbb{L}}{N(A,B)} \,|\xi'| e^{-|\xi'|(x_{_N}+y_{_N})}
\varphi(y_{_N}) \widehat{k}(\xi',y_{_N})\boldsymbol{i}g] (x') \,dy_{_N},\\
(\lambdambda^j\mathcal{H}_{2}k)(x)&:=-\varphi(x_{_N}) \int_{0}^{\infty} \mathcal{F}^{-1}_{\xi'}\boldsymbol{i}g[
\frac{\lambdambda^j \deltat \mathbb{L}}{N(A,B)} \,e^{-|\xi'|(x_{_N}+y_{_N})}
\partial_{{_N}} \big( \varphi(y_{_N}) \widehat{k}(\xi',y_{_N})\big) \boldsymbol{i}g] (x') \,dy_{_N}.
\end{align*}
On the other hand, assume that $1\leq |\alphapha'|+n\leq3-j$ with $j=0,1.$ By the identity
\betagin{equation*}
1 = \frac{1+|\xi'|^2}{1+|\xi'|^2} = \frac{1}{1+|\xi'|^2} - \sum_{\ell=1}^{N-1} \frac{i\xi_{\ell}}{1+|\xi'|^2} i\xi_{\ell},
\end{equation*}
we have
\betagin{align*}
(\lambdambda^j \partial_{x'}^{\alphapha'}\partial_{_N}^{n} \mathcal{H}_1 k)(x)
=& \sum_{n_1+n_2=n} \binom{n}{n_1}(\partial_{_N}^{n_1} \varphi)(x_{_N})
\int_0^{\infty} \mathcal{F}_{\xi'}^{-1} \boldsymbol{i}g[\frac{\deltat \mathbb{L}}{N(A,B)} \frac{\lambdambda^{j}(i\xi')^{\alphapha'}(-|\xi'|)^{n_2}}{1+|\xi'|^2}
\\
&\hspace*{2cm} \times\,|\xi'| e^{-|\xi'|(x_{_N}+y_{_N})}\varphi(y_{_N}) \rwh{(1-\Deltalta')k}(\xi',y_{_N})
\boldsymbol{i}g](x')\,dy_{_N},\\
(\lambdambda^j \partial_{x'}^{\alphapha'}\partial_{_N}^{n} \mathcal{H}_2 k)(x)
=& -\sum_{n_1+n_2=n} \binom{n}{n_1}(\partial_{_N}^{n_1} \varphi)(x_{_N})
\int_0^{\infty} \mathcal{F}_{\xi'}^{-1} \boldsymbol{i}g[\frac{\deltat \mathbb{L}}{N(A,B)} \frac{\lambdambda^{j}(i\xi')^{\alphapha'}(-|\xi'|)^{n_2}}{1+|\xi'|^2}
\\
&\hspace*{2cm} \times\, e^{-|\xi'|(x_{_N}+y_{_N})}
\partial_{{_N}} \big( \varphi(y_{_N}) \widehat{k}(\xi',y_{_N})\big)
\boldsymbol{i}g](x')\,dy_{_N}\\
& +\sum_{n_1+n_2=n} \sum_{\ell=1}^{N-1}\binom{n}{n_1}(\partial_{_N}^{n_1} \varphi)(x_{_N})
\int_0^{\infty} \mathcal{F}_{\xi'}^{-1} \boldsymbol{i}g[\frac{\deltat \mathbb{L}}{N(A,B)}
\frac{\lambdambda^{j}(i\xi')^{\alphapha'}(-|\xi'|)^{n_2}(i\xi_{\ell})}{(1+|\xi'|^2)|\xi'|} \\
&\hspace*{2cm} \times\, |\xi'|e^{-|\xi'|(x_{_N}+y_{_N})}
\partial_{{_N}} \big( \varphi(y_{_N}) \widehat{\partial_{\ell}k}(\xi',y_{_N})\big)
\boldsymbol{i}g](x')\,dy_{_N}.
\end{align*}
Next, Lemma \ref{lemma:ABL} and Lemma \ref{lemma:N_AB} imply that
\betagin{equation*}
\boldsymbol{i}g| \partial_{\xi'}^{\kappappa'} (\tau \partial_\tau)^\ell \boldsymbol{i}g(\frac{\deltat \mathbb{L}}{N(A,B)} \boldsymbol{i}g) \boldsymbol{i}g|
\leq \frac{C_{\kappappa'}}{|\lambdambda| + |\xi'|}(|\lambdambda|^{1\slash 2} + |\xi'|)^{-|\kappappa'|},
\,\,\,\forall\,\, \kappappa' \in \mathbb{N}_0^{N-1} \,\,\,\hbox{and}\,\,\,\ell =0,1.
\end{equation*}
Then it is not hard to see that the symbols
\betagin{equation*}
\frac{\lambdambda^j\deltat \mathbb{L}}{N(A,B)},\,\,
\frac{\deltat \mathbb{L}}{N(A,B)} \frac{\lambdambda^{j}(i\xi')^{\alphapha'}|\xi'|^{n_2}}{1+|\xi'|^2}, \,\,
\frac{\deltat \mathbb{L}}{N(A,B)} \frac{\lambdambda^{j}(i\xi')^{\alphapha'}|\xi'|^{n_2}(i\xi_{\ell})}{(1+|\xi'|^2)|\xi'|}
\end{equation*}
are of the class $\boldsymbol{M}_{0,2}(\wt\Gammamma_{\varepsilon,\lambdambda_0,\zeta})$ for $j+|\alphapha'|+n_1+n_2 \leq 3$ and $j=0,1.$
Thus we infer from Lemma \ref{lemma:basic_es_2} that
\betagin{gather} \lambdabel{eq:H_half_Rbdd}
\mathcal{R}_{\mathcal{L} \big(L_q(\mathbb{R}_+^{N})\big)}
\big( \{ \lambdambda^{j}\mathcal{H}_1(\lambdambda,\mathbb{R}^N_+): \lambdambda \in \Gammamma_{\varepsilon,\lambdambda_0,\zeta} \} \big) \leq C,\\ \notag
\mathcal{R}_{\mathcal{L} \big(H^{1}_q(\mathbb{R}^N_+);L_q(\mathbb{R}_+^{N})\big)}
\big( \{ \lambdambda^{j}\mathcal{H}_2(\lambdambda,\mathbb{R}^N_+): \lambdambda \in \Gammamma_{\varepsilon,\lambdambda_0,\zeta} \} \big) \leq C,\\ \nonumber
\mathcal{R}_{\mathcal{L} \big(H^{2}_q(\mathbb{R}^N_+); L_q(\mathbb{R}_+^{N})\big)}
\big( \{ \lambdambda^{j}\partial_{x'}^{\alphapha'}\partial_{_N}^n\mathcal{H}_{\mathfrak{a}}(\lambdambda,\mathbb{R}^N_+): \lambdambda \in \Gammamma_{\varepsilon,\lambdambda_0,\zeta} \} \big) \leq C,
\end{gather}
with $1\leq |\alphapha'|+n\leq3-j,$ $j=0,1,$ and $\mathfrak{a}=1,2.$
Then $\mathcal{H}(\lambdambda,\mathbb{R}^N_+)$ has the desired property due to \eqref{eq:H_half_Rbdd} and the definition of $\mathcal{R}-$boundedness. This completes the proof of Proposition \ref{prop:GR_half_2}.
\section{Generalized model problem in the bent half space}
\lambdabel{sec:bh}
Let $\boldsymbol{P}hi$ be a $C^1$ diffeomorphism from $\mathbb{R}^N_{\xi}$ onto $\mathbb{R}^N_x$ and $\boldsymbol{P}hi^{-1}$ be the inverse of $\boldsymbol{P}hi.$
Assume that
\betagin{gather*}
\nabla_{\xi} \boldsymbol{P}hi^\top (\xi) := \mathbb{A} + \mathbb{B}(\xi) = \betagin{bmatrix}
a_{ij}
\end{bmatrix}_{N\times N}
+ \betagin{bmatrix}
b_{ij}(\xi)
\end{bmatrix}_{N\times N}, \\
\mathscr{A}_{\Phi} := \nabla_x (\boldsymbol{P}hi^{-1})^\top|_{x=\boldsymbol{P}hi(\xi)} = \mathbb{A}_{-} + \mathbb{B}_-(\xi)
= \betagin{bmatrix}
\bar{a}_{ij}
\end{bmatrix}_{N\times N}
+ \betagin{bmatrix}
\bar{b}_{ij}(\xi)
\end{bmatrix}_{N\times N},
\end{gather*}
for some constant orthogonal matrices $\mathbb{A}$ and $\mathbb{A}_-.$
Moreover, denote $\Omegaega_+ := \boldsymbol{P}hi(\mathbb{R}^N_+)$ and $\Gammamma_+ := \partial \Omegaega_+ = \boldsymbol{P}hi (\mathbb{R}^N_0).$
Then $\Gammamma_+$ is characterized by the equation
$(\Phi^{-1})_N (x) = 0,$
and the unit outer normal $\boldsymbol{n}_{+}$ to $\Gammamma_+$ is given by
\betagin{equation*}
\boldsymbol{n}_{+} \big(\boldsymbol{P}hi(\xi)\big)
=\frac{\mathscr{A}_{\Phi}\boldsymbol{n}_0}{|\mathscr{A}_{\Phi}\boldsymbol{n}_0|}
= -\frac{(\nabla_x \Phi_{N}^{-1})\big(\boldsymbol{P}hi(\xi)\big) }{\big| (\nabla_x \Phi_{N}^{-1})\big(\boldsymbol{P}hi(\xi)\big) \big|}
= -\frac{ \big( \bar{a}_{1N}+\bar{b}_{1N}(\xi),\dots, \bar{a}_{NN}+\bar{b}_{NN}(\xi)\big)^{\top} }{\boldsymbol{i}g( \sum_{j=1}^N \big( \bar{a}_{Nj}+\bar{b}_{Nj}(\xi) \big)^2 \boldsymbol{i}g)^{1\slash 2} },
\end{equation*}
with $\boldsymbol{n}_0 := (0,\dots,0,-1)^{\top}.$
\smallbreak
Now we assume that $\boldsymbol{P}hi \in H^3_r(\mathbb{R}^N)^N$ for some $N<r<\infty,$
and there exist constants $0<M_1<1\leq M_2 \leq M_3<\infty$ such that
\betagin{equation}\lambdabel{hyp:M1M2_BH}
\|(\mathbb{B}, \mathbb{B}_-)\|_{L_{\infty}(\mathbb{R}^N)} \leq M_1, \,\,\,
\|\nabla(\mathbb{B}, \mathbb{B}_-)\|_{L_{r}(\mathbb{R}^N)} \leq M_2, \,\,\,
\|\nabla^2(\mathbb{B}, \mathbb{B}_-)\|_{L_{r}(\mathbb{R}^N)} \leq M_3.
\end{equation}
For such $\Gammamma_+$ characterized by $H^3_r(\mathbb{R}^N)$ mapping, we consider the following model problem,
\betagin{equation}\lambdabel{eq:RR_CNS_BH}
\left\{\betagin{aligned}
& \lambdambda \boldsymbol{v} -\gammamma_{1}^{-1}\Di \big( \mathbb{S}(\boldsymbol{v}) + \zeta \gammamma_3 \di \boldsymbol{v} \mathbb{I} \big)
= \boldsymbol{f} &&\quad\hbox{in}\quad \Omegaega_+,\\
&\mathbb{S}(\boldsymbol{v}) \boldsymbol{n}_{+} +\zeta \gammamma_3 \di \boldsymbol{v} \,\boldsymbol{n}_{+}
+\sigmagma (m-\Deltalta_{\Gammamma_+})h \,\boldsymbol{n}_{+} = \boldsymbol{g} &&\quad\hbox{on}\quad \Gammamma_+, \\
&\lambdambda h - \boldsymbol{v} \cdot \boldsymbol{n}_{+} = k &&\quad\hbox{on}\quad \Gammamma_+.\\
\end{aligned}\right.
\end{equation}
In \eqref{eq:RR_CNS_BH}, $\gammamma_1$ and $\gammamma_3$ are uniformly continuous functions on $\overline{\Omegaega}_+,$ and there exist some constants $\rwh \gammamma_1,\rwh\gammamma_3$ such that
\betagin{gather}\lambdabel{hyp:gamma_RR_BH}
0<\rho_1 \leq \gammamma_1(x), \rwh{\gammamma_1} \leq \rho_2,\quad
0 < \gammamma_3(x), \rwh{\gammamma_3} \leq \rho_3, \,\,\, \forall \,\, x \in \overline{\Omegaega}_+,
\\ \notag
\sum_{\mathfrak{a}=1,3}\|\gammamma_{\mathfrak{a}} -\rwh{\gammamma_{\mathfrak{a}}}\|_{L_{\infty}(\Omegaega_+)} \leq M_1<1,\quad
\sum_{\mathfrak{a}=1,3} \|\nabla \gammamma_{\mathfrak{a}}\|_{L_r(\Omegaega_+)} \leq CM_2,
\end{gather}
for some constants $\rho_1, \rho_2, \rho_3>0.$
The main result in this section for \eqref{eq:RR_CNS_BH} reads:
\betagin{theo}\lambdabel{thm:GR_BH}
Let $0<\varepsilon<\pi\slash 2,$ $\sigmagma,m, \mu, \nu,\zeta_0>0,$ $1<q<\infty,$ $N<r<\infty$ and $r\geq q.$
Assume that \eqref{hyp:gamma_RR_BH} is satisfied.
For $\Omegaega_+$ given above, we set that
\betagin{equation*}
Y_q(\Omegaega_+) := L_q(\Omegaega_+)^N \times H^{1}_q(\Omegaega_+)^N\times H^{2}_q(\Omegaega_+), \quad
\mathcal{Y}_q(\Omegaega_+) := L_q(\Omegaega_+)^N \times Y_q(\Omegaega_+).
\end{equation*}
Then for any $(\boldsymbol{f},\boldsymbol{g},k) \in Y_q(\Omegaega_+),$ there exist constants $\lambdambda_0,r_b \geq 1$ and operator families
\betagin{align*}
\mathcal{A}_0(\lambdambda,\Omegaega_+) & \in
{\rm Hol}\,\boldsymbol{i}g( \Gammamma_{\varepsilon,\lambdambda_0,\zeta} ; \mathcal{L}\big(\mathcal{Y}_q(\Omegaega_+);H^2_q(\Omegaega_+)^N \big) \boldsymbol{i}g),\\
\mathcal{H}_0(\lambdambda,\Omegaega_+) & \in
{\rm Hol}\,\boldsymbol{i}g( \Gammamma_{\varepsilon,\lambdambda_0,\zeta} ; \mathcal{L}\big(\mathcal{Y}_q(\Omegaega_+);H^3_q(\Omegaega_+) \big) \boldsymbol{i}g),
\end{align*}
such that $(\boldsymbol{v},h):=\big(\mathcal{A}_0(\lambdambda,\Omegaega_+), \mathcal{H}_0(\lambdambda,\Omegaega_+) \big)(\boldsymbol{f},\lambdambda^{1\slash 2}\boldsymbol{g},\boldsymbol{g},k)$
is a solution of \eqref{eq:RR_CNS_BH}. Moreover, we have
\betagin{gather*}
\mathcal{R}_{\mathcal{L}\big(\mathcal{Y}_q(\Omegaega_+); H^{2-j}_q(\Omegaega_+)^N \big)}
\boldsymbol{i}g( \boldsymbol{i}g\{ (\tau \partial_{\tau})^{\ell}\big( \lambdambda^{j\slash 2}\mathcal{A}_0(\lambdambda,\Omegaega_+)\big) : \lambdambda \in
\Gammamma_{\varepsilon,\lambdambda_0,\zeta} \boldsymbol{i}g\}\boldsymbol{i}g) \leq r_b,\\
\mathcal{R}_{\mathcal{L}\big(\mathcal{Y}_q(\Omegaega_+); H^{3-j'}_q(\Omegaega_+) \big)}
\boldsymbol{i}g( \boldsymbol{i}g\{ (\tau \partial_{\tau})^{\ell}\big( \lambdambda^{j'}\mathcal{H}_0(\lambdambda,\Omegaega_+)\big) : \lambdambda \in
\Gammamma_{\varepsilon,\lambdambda_0,\zeta} \boldsymbol{i}g\}\boldsymbol{i}g) \leq r_b,
\end{gather*}
for $\ell, j'=0,1,$ $j=0,1,2,$ and $\tau := \Im \lambdambda.$
Above the constants $\lambdambda_0$ and $r_b$ depend solely on
$\varepsilon,$ $\sigmagma,$ $m,$ $\mu,$ $\nu,$ $q,$ $r,$ $N,$ $\zeta_0,$ $\rho_1,$ $\rho_2,$ $\rho_3.$
\end{theo}
\subsection{Reduction of the \eqref{eq:RR_CNS_BH}}
Introduce $\boldsymbol{w} (\xi):=\mathbb{A}^{\top}_- \boldsymbol{v} \big(\boldsymbol{P}hi(\xi)\big)$
and $H(\xi):= h\big(\boldsymbol{P}hi(\xi)\big).$
Let us derive the equations of $\boldsymbol{w}$ and $H$ according to \eqref{eq:RR_CNS_BH}.
Firstly, thanks to the facts that
\betagin{equation*}
\nabla_x \boldsymbol{v}^{\top} \big(\boldsymbol{P}hi(\xi)\big) =\mathscr{A}P \nabla_\xi \boldsymbol{w}^\top\mathbb{A}_-^{\top},\quad
(\di_x \boldsymbol{v}) \big(\boldsymbol{P}hi(\xi)\big) =\mathscr{A}P: (\nabla_\xi \boldsymbol{w}^\top\mathbb{A}_-^{\top}),
\end{equation*}
we obtain that
\betagin{align}
\mathbb{F}_0(\boldsymbol{w}) &:= \big( \mathbb{S}(\boldsymbol{v}) + \zeta \gammamma_3 \di \boldsymbol{v} \mathbb{I} \big) \big(\boldsymbol{P}hi(\xi)\big) \notag \\ \notag
&=\mu (\mathscr{A}P \nabla_{\xi} \boldsymbol{w}^{\top}\mathbb{A}^{\top}_- + \mathbb{A}_- \nabla^{\top}_{\xi} \boldsymbol{w}\mathscr{A}P^{\top})
+\big( \nu-\mu + \zeta (\gammamma_3 \circ \boldsymbol{P}hi) \big) \big( (\mathbb{A}_-^{\top} \mathscr{A}P) : \nabla_{\xi} \boldsymbol{w}^{\top} \big) \mathbb{I},\\
\mathbb{F}_0(\boldsymbol{w})\mathscr{A}P&:=\mathbb{A}_- \big( \mathbb{S}(\boldsymbol{w}) + \zeta \wh\gammamma_3 \di_{\xi} \boldsymbol{w} \, \mathbb{I} \big) + \mathbb{F}(\boldsymbol{w}),\lambdabel{eq:RR_tensor_1}
\end{align}
where $\mathbb{F}(\boldsymbol{w}):=\mathbb{F}_1(\boldsymbol{w})+\mathbb{F}_2(\boldsymbol{w})$ and
\betagin{align*}
\mathbb{F}_1(\boldsymbol{w}) &:= \mathbb{A}_- \mathbb{S}(\boldsymbol{w}) \mathbb{A}_-^{\top} \mathbb{B}_-
+ \mu \big( \mathbb{B}_-\nabla_{\xi} \boldsymbol{w}^{\top}\mathbb{A}_-^{\top}+\mathbb{A}_-\nabla_{\xi}^{\top} \boldsymbol{w}\, \mathbb{B}_-^{\top} \big) \mathscr{A}P \\
& \quad + (\nu-\mu) \big( (\mathbb{A}_-^{\top} \mathbb{B}_-) : \nabla_{\xi} \boldsymbol{w}^{\top} \big) \mathscr{A}P,\\
\mathbb{F}_2(\boldsymbol{w})&:= \zeta \wh\gammamma_3 (\di_{\xi}\boldsymbol{w}) \mathbb{B}_-
+ \zeta \wh\gammamma_3 \big( (\mathbb{A}_-^{\top} \mathbb{B}_-) : \nabla_{\xi} \boldsymbol{w}^{\top} \big) \mathscr{A}P \\
&\quad + \zeta (\gammamma_3\circ \boldsymbol{P}hi -\wh\gammamma_3 )\big( (\mathbb{A}_-^{\top} \mathscr{A}P) : \nabla_{\xi} \boldsymbol{w}^{\top} \big) \mathscr{A}P.
\end{align*}
On the other hand, $\Gammamma_+:= \Phi(\mathbb{R}^N_0)$ and
$\nabla_{\xi} \boldsymbol{P}hi^\top (\xi',0) = \betagin{bmatrix}
a_{ij}
\end{bmatrix}_{N\times N}
+ \betagin{bmatrix}
b_{ij}(\xi',0)
\end{bmatrix}_{N\times N}$ by previous convention.
For simplicity, we write for $i,j=1,\dots,N-1,$
\betagin{equation*}
\boldsymbol{p}hi(\xi'):= \boldsymbol{P}hi(\xi',0),\quad
\partial_{i}:= \partial_{\xi_i}, \quad
\boldsymbol{t}au_i:=\partial_{i} \boldsymbol{p}hi \quad
\hbox{and}\quad
\boldsymbol{t}au_{ij}:=\partial_{j}\partial_{i} \boldsymbol{p}hi.
\end{equation*}
Then the first fundamental form of $\Gammamma_+$ and its inverse are given by
\betagin{equation*}
\mathbb{G}_+ = \betagin{bmatrix} g_{ij}
\end{bmatrix}_{(N-1)\times (N-1)}
:=\betagin{bmatrix}
\boldsymbol{t}au_i \cdot \boldsymbol{t}au_j
\end{bmatrix}_{(N-1)\times (N-1)}, \quad
\mathfrak{g}_+ :=\deltat \mathbb{G}_+, \quad
\mathbb{G}_+^{-1} := \betagin{bmatrix} g^{ij}
\end{bmatrix}_{(N-1)\times (N-1)}.
\end{equation*}
Introduce that
\betagin{equation*}
\wt g_{ij} := g_{ij} - \deltalta_{ij}
=\sum_{k=1}^{N} (b_{ik}a_{jk}+a_{ik}b_{jk} + b_{ik} b_{jk}), \quad
\wt g^{ij} := g^{ij} -\deltalta_{ij} .
\end{equation*}
Then \eqref{hyp:M1M2_BH} implies that
\betagin{gather}\lambdabel{es:geo_1_BH}
\|(\wt g_{ij}, \wt g^{ij})\|_{L_{\infty}(\mathbb{R}^{N-1})} +\|\mathfrak{g}_+ - 1 \|_{L_{\infty}(\mathbb{R}^{N-1})} \leq C M_1, \quad
\|\nabla_{\xi'} (\wt g_{ij}, \wt g^{ij})\|_{L_{r}(\mathbb{R}^{N-1})} \leq C M_2,\\ \notag
\|\nabla_{\xi'}^2 \wt g_{ij}\|_{L_{r}(\mathbb{R}^{N-1})} \leq C M_2 M_3, \quad
\|\nabla_{\xi'}^2 \wt g^{ij}\|_{L_{r}(\mathbb{R}^{N-1})} \leq C M_2^3 M_3.
\end{gather}
\smallbreak
Next, recall the definition of the \emph{Laplace-Beltrami} operator on $\Gammamma_+,$
\betagin{align*}
\big(\Deltalta_{\Gammamma_+} h\big) \big( \boldsymbol{p}hi(\xi') \big)
&:= \frac{1}{ \sqrt{\mathfrak{g}_+}} \partial_i (\sqrt{\mathfrak{g}_+} g^{ij} \partial_j H)
= g^{ij} \partial_{i}\partial_{j} H - g^{ij}\Lambdambda^k_{ij} \partial_k H\\
&= \Deltalta'_{\xi'} H + \mathcal{G}(H),
\end{align*}
where the \emph{Christoffel symbols} $\Lambdambda^k_{ij}:=g^{kr} \boldsymbol{t}au_{ij} \cdot \boldsymbol{t}au_r,$ for $i,j,k,r=1,\dots,N-1,$ and
\betagin{align*}
\mathcal{G}(H) := \wt g^{ij} \partial_{i}\partial_{j} H - g^{ij}\Lambdambda^k_{ij} \partial_k H.
\end{align*}
Furthermore, \eqref{hyp:M1M2_BH} and \eqref{es:geo_1_BH} yield that
\betagin{equation}\lambdabel{es:geo_2_BH}
\|\Lambdambda_{ij}^k\|_{L_\infty(\mathbb{R}^{N-1})} \leq CM_3, \quad
\|\nabla_{\xi'}\Lambdambda_{ij}^k\|_{L_r(\mathbb{R}^{N-1})} \leq CM_2M_3.
\end{equation}
Now it is not hard to see that \eqref{eq:RR_CNS_BH} turns to
\betagin{equation}\lambdabel{eq:RR_CNS_BH_2}
\left\{\betagin{aligned}
& \lambdambda \boldsymbol{w} -\wh\gammamma_{1}^{-1} \Di_{\xi}
\big( \mathbb{S}(\boldsymbol{w}) + \zeta \wh\gammamma_3 \di_{\xi} \boldsymbol{w}\, \mathbb{I}_{N} \big) +\mathcal{F}_1(\boldsymbol{w})
= \boldsymbol{F}_+ &&\quad\hbox{in}\quad \mathbb{R}^N_+,\\
&\big( \mathbb{S}(\boldsymbol{w})+\zeta \wh\gammamma_3 \di_{\xi} \boldsymbol{w} \mathbb{I}_{N} \big)\boldsymbol{n}_{0}
+\sigmagma (m-\Deltalta')H \,\boldsymbol{n}_{0} +\mathcal{F}_2(\boldsymbol{w},H) \boldsymbol{n}_0 = \boldsymbol{G}_+ &&\quad\hbox{on}\quad \mathbb{R}^N_0, \\
&\lambdambda H - \boldsymbol{w} \cdot \boldsymbol{n}_{0} + \mathcal{F}_3(\boldsymbol{w}) \cdot \boldsymbol{n}_0 = K_+ &&\quad\hbox{on}\quad \mathbb{R}^N_0,\\
\end{aligned}\right.
\end{equation}
where $\boldsymbol{F}_+,$ $\boldsymbol{G}_+$ and $K_+$ are defined by
\betagin{equation*}
\boldsymbol{F}_+(\xi):= \mathbb{A}^\top_-(\boldsymbol{f} \circ \boldsymbol{P}hi )(\xi), \quad
\boldsymbol{G}_+(\xi):= |\mathscr{A}_{\Phi}\boldsymbol{n}_0|\mathbb{A}_-^{\top} (\boldsymbol{g} \circ \boldsymbol{P}hi )(\xi), \quad
K_+(\xi):= (k \circ \boldsymbol{P}hi )(\xi),
\end{equation*}
and the operators
\betagin{align*}
\mathcal{F}_1(\boldsymbol{w})&:=(\wh\gammamma_{1}^{-1}-\gammamma_1^{-1}\circ \boldsymbol{P}hi)
\Di_{\xi}\big(\mathbb{S}(\boldsymbol{w}) + \zeta \wh\gammamma_3 \di_{\xi} \boldsymbol{w} \mathbb{I}_{N}\big)
-(\gammamma_1^{-1}\circ\boldsymbol{P}hi) \mathbb{A}_-^{\top} \Di_{\xi} \mathbb{F}(\boldsymbol{w})\\
& \quad +(\gammamma_1^{-1}\circ\boldsymbol{P}hi) \mathbb{A}_-^{\top} \mathbb{F}_0(\boldsymbol{w}) \Di \mathscr{A}P,\\
\mathcal{F}_2(\boldsymbol{w},H)&:= \mathbb{F}_b(\boldsymbol{w}) + \mathbb{G}_b(H),\\
\mathbb{F}_b(\boldsymbol{w})&:=\mathbb{A}_-^{\top} \mathbb{F}(\boldsymbol{w}),\\
\mathbb{G}_b(H)&:= \sigmagma m H \mathbb{A}_-^{\top} \mathbb{B}_-
-\sigmagma \mathcal{G}(H) \mathbb{I}_{N} -\sigmagma (g^{ij} \partial_{i}\partial_{j} H - g^{ij}\Lambdambda^k_{ij} \partial_k H) \mathbb{A}^{\top}_-\mathbb{B}_-,\\
\mathcal{F}_3(\boldsymbol{w})&:= (1 - |\mathscr{A}_{\Phi} \boldsymbol{n}_0|^{-1}) \boldsymbol{w}
- |\mathscr{A}_{\Phi} \boldsymbol{n}_0|^{-1} \mathbb{B}_-^{\top}\mathbb{A}_- \boldsymbol{w}.
\end{align*}
\smallbreak
By \eqref{hyp:M1M2_BH}, $\boldsymbol{Z}_+:=(\boldsymbol{F}_+,\lambdambda^{1\slash 2}\boldsymbol{G}_+,\boldsymbol{G}_+,K_+)$ belongs to
$\mathcal{Y}_q(\mathbb{R}^N_+)$ with
\betagin{equation*}
\|\boldsymbol{Z}_+\|_{\mathcal{Y}_q(\mathbb{R}^N_+)}
=\| (\boldsymbol{F}_+,\lambdambda^{1\slash 2}\boldsymbol{G}_+,\boldsymbol{G}_+,K_+)\|_{\mathcal{Y}_q(\mathbb{R}^N_+)}
\leq C_{N,q} \|(\boldsymbol{f},\lambdambda^{1\slash 2}\boldsymbol{g},\boldsymbol{g},k)\|_{\mathcal{Y}_q(\Omegaega_+)}.
\end{equation*}
Thus, according to Theorem \ref{thm:GR_half_0},
$\boldsymbol{u}:=\mathcal{A}_0(\lambdambda,\mathbb{R}^N_+) \boldsymbol{Z}_+$ and $\mathfrak{h}:=\mathcal{H}_0(\lambdambda,\mathbb{R}^N_+) \boldsymbol{Z}_+$ satisfy
\betagin{equation}\lambdabel{eq:RR_CNS_BH_3}
\left\{\betagin{aligned}
& \lambdambda \boldsymbol{u} -\wh\gammamma_{1}^{-1} \Di_{\xi}
\big( \mathbb{S}(\boldsymbol{u}) + \zeta \wh\gammamma_3 \di_{\xi} \boldsymbol{u}\, \mathbb{I}_{N} \big) +\mathcal{F}_1(\boldsymbol{u})
= \boldsymbol{F}_+ +\mathcal{R}_1(\lambdambda)\boldsymbol{Z}_+
&&\quad\hbox{in}\quad \mathbb{R}^N_+,\\
&\big( \mathbb{S}(\boldsymbol{u})+\zeta \wh\gammamma_3 \di_{\xi} \boldsymbol{u} \mathbb{I}_{N} \big)\boldsymbol{n}_{0}
+\sigmagma (m-\Deltalta')\mathfrak{h} \,\boldsymbol{n}_{0} +\mathcal{F}_2(\boldsymbol{u},\mathfrak{h}) \boldsymbol{n}_0 = \boldsymbol{G}_+ + \mathcal{R}_2(\lambdambda)\boldsymbol{Z}_+
&&\quad\hbox{on}\quad \mathbb{R}^N_0, \\
&\lambdambda \mathfrak{h} - \boldsymbol{u} \cdot \boldsymbol{n}_{0} + \mathcal{F}_3(\boldsymbol{u}) \cdot \boldsymbol{n}_0 = K_+ +\mathcal{R}_3(\lambdambda) \boldsymbol{Z}_+
&&\quad\hbox{on}\quad \mathbb{R}^N_0,\\
\end{aligned}\right.
\end{equation}
where $\mathcal{R}_k(\lambdambda)\boldsymbol{Z}_+,$ for $k=1,2,3,$ are defined by
\betagin{align*}
\mathcal{R}_1(\lambdambda)\boldsymbol{Z}_+ &:= \mathcal{F}_1 \big( \mathcal{A}_0 (\lambdambda,\mathbb{R}^N_+) \boldsymbol{Z}_+\big),\\
\mathcal{R}_2 (\lambdambda) \boldsymbol{Z}_+ &:= \mathcal{F}_2 \big( \mathcal{A}_0 (\lambdambda,\mathbb{R}^N_+)\boldsymbol{Z}_+, \mathcal{H}_0 (\lambdambda,\mathbb{R}^N_+)\boldsymbol{Z}_+\big)\boldsymbol{n}_0,\\
\mathcal{R}_3(\lambdambda)\boldsymbol{Z}_+ & :=\mathcal{F}_3 \big( \mathcal{A}_0 (\lambdambda,\mathbb{R}^N_+)\boldsymbol{Z}_+\big) \cdot \boldsymbol{n}_0.
\end{align*}
\smallbreak
Next, set that $\mathcal{R}(\lambdambda) := \big( \mathcal{R}_1(\lambdambda),\mathcal{R}_2(\lambdambda), \mathcal{R}_3(\lambdambda) \big)^{\top}$ and
$F_{\lambdambda}(\boldsymbol{f},\boldsymbol{g},k):= ( \boldsymbol{f},\lambdambda^{1\slash 2}\boldsymbol{g}, \boldsymbol{g}, k)^{\top}$ for any $(\boldsymbol{f},\boldsymbol{g},k)$ in $Y_q(\mathbb{R}^N_+).$
Then we claim that there exist some constants $C$ and $\lambdambda_0$ depending only on $\varepsilon,$ $\sigmagma,$ $m,$ $\mu,$ $\nu,$ $q,$ $r,$ $N,$ $\zeta_0,$ $\rho_1,$ $\rho_2,$ $\rho_3,$ such that
\betagin{multline}\lambdabel{es:key_Rbd_BH}
\mathcal{R}_{\mathcal{L}\big(\mathcal{Y}_q(\mathbb{R}^N_+)\big)}
\big\{ (\tau \partial_{\tau})^{\ell}F_{\lambdambda}\mathcal{R}(\lambdambda):\lambdambda\in \Gammamma_{\varepsilon,\lambdambda_0,\zeta}\big\}
\leq C (M_1 + \sigmagma_0) (1+\sigmagma+\sigmagma m\lambdambda_0^{-1} ) \\
+ C \lambdambda_0^{-1 \slash 2} (\sigmagma +1)
\big( M_3 + \sigmagma_0^{-\frac{N}{r-N}} M_2^{\frac{r}{r-N}} (M_3^{\frac{r}{r-N}}+m )\big),
\hspace*{1cm}
\end{multline}
for any $0<\sigmagma_0 <1.$
The proof of \eqref{es:key_Rbd_BH} is postponed to the next subsection.
\smallbreak
Let us continue the proof of Theorem \ref{thm:GR_BH} by admitting \eqref{es:key_Rbd_BH} for a while.
By choosing $\sigmagma_0, M_1$ small and $\lambdambda_0$ large enough in \eqref{es:key_Rbd_BH}, it holds that
\betagin{equation}\lambdabel{es:FR_BH_1}
\mathcal{R}_{\mathcal{L}\big(\mathcal{Y}_q(\mathbb{R}^N_+)\big)}
\big\{ (\tau \partial_{\tau})^{\ell}F_{\lambdambda}\mathcal{R}(\lambdambda):\lambdambda\in \Gammamma_{\varepsilon,\lambdambda_0,\zeta}\big\} \leq 1\slash 2.
\end{equation}
In particular, \eqref{es:FR_BH_1} yields that
\betagin{equation}\lambdabel{es:RF_BH_1}
\|F_{\lambdambda} \mathcal{R}(\lambdambda) F_{\lambdambda}(\boldsymbol{F}_+,\boldsymbol{G}_+,K_+)\|_{\mathcal{Y}_q(\mathbb{R}^N_+)}
\leq 1\slash 2 \|F_{\lambdambda}(\boldsymbol{F}_+,\boldsymbol{G}_+,K_+)\|_{\mathcal{Y}_q(\mathbb{R}^N_+)},
\,\,\, \forall \,\, \lambdambda \in \Gammamma_{\varepsilon,\lambdambda_0,\zeta},
\end{equation}
and that
\betagin{equation}\lambdabel{es:FR_BH_2}
\mathcal{R}_{\mathcal{L}\big(\mathcal{Y}_q(\mathbb{R}^N_+)\big)}
\big\{ (\tau \partial_{\tau})^{\ell}\big(Id + F_{\lambdambda}\mathcal{R}(\lambdambda) \big)^{-1}:
\lambdambda\in \Gammamma_{\varepsilon,\lambdambda_0,\zeta}\big\} \leq 1\slash 2.
\end{equation}
Now note that the norm $\|\cdot\|_{Y_q(\mathbb{R}^N_+)}$ is equivalent to $\|\cdot\|_{Y_{q,\lambdambda}(\mathbb{R}^N_+)}$
for any fixed $\lambdambda \in \Gammamma_{\varepsilon,\lambdambda_0,\zeta}$ in $Y_q(\mathbb{R}^N_+)$ with
\betagin{equation*}
\|(\boldsymbol{f},\boldsymbol{g},k)\|_{Y_{q,\lambdambda}(\mathbb{R}^N_+)} := \|F_{\lambdambda} (\boldsymbol{f},\boldsymbol{g},k)\|_{\mathcal{Y}_q(\mathbb{R}^N_+)}, \,\,\,
\forall (\boldsymbol{f},\boldsymbol{g},k) \in Y_q(\mathbb{R}^N_+).
\end{equation*}
Denote $Y_{q,\lambdambda}(\mathbb{R}^N_+):= \big(Y_q(\mathbb{R}^N_+),\|\cdot\|_{Y_{q,\lambdambda}(\mathbb{R}^N_+)}\big)$
and \eqref{es:RF_BH_1} implies that
\betagin{equation*}
\|\mathcal{R}(\lambdambda)F_{\lambdambda}\|_{\mathcal{L}\big(Y_{q,\lambdambda}(\mathbb{R}^N_+)\big)} \leq 1 \slash 2.
\end{equation*}
Thus the inverse operator $(Id + \mathcal{R}(\lambdambda)F_{\lambdambda})^{-1}$ exists for any fixed $\lambdambda\in \Gammamma_{\varepsilon,\lambdambda_0,\zeta}$ with
\betagin{equation*}
\| (Id + \mathcal{R}(\lambdambda)F_{\lambdambda})^{-1}\|_{\mathcal{L}\big(Y_{q,\lambdambda}(\mathbb{R}^N_+)\big)} \leq 2.
\end{equation*}
Then for any $(\boldsymbol{F}_+,\boldsymbol{G}_+,K_+) \in Y_q(\mathbb{R}^N_+),$ we set that
\betagin{align*}
\boldsymbol{w}&:= \mathcal{A}_0(\lambdambda,\mathbb{R}^N_+)F_{\lambdambda} (Id + \mathcal{R}(\lambdambda)F_{\lambdambda})^{-1} (\boldsymbol{F}_+,\boldsymbol{G}_+,K_+),\\
H&:= \mathcal{H}_0(\lambdambda,\mathbb{R}^N_+)F_{\lambdambda} (Id + \mathcal{R}(\lambdambda)F_{\lambdambda})^{-1} (\boldsymbol{F}_+,\boldsymbol{G}_+,K_+),
\end{align*}
which solve \eqref{eq:RR_CNS_BH_2} by keeping \eqref{eq:RR_CNS_BH_3} in mind. As
\betagin{equation*}
F_{\lambdambda} (Id + \mathcal{R}(\lambdambda)F_{\lambdambda})^{-1}
=\sum_{j=0}^{\infty} F_{\lambdambda} \big(-\mathcal{R}(\lambdambda)F_{\lambdambda}\big)^j
= \big(Id + F_{\lambdambda}\mathcal{R}(\lambdambda) \big)^{-1} F_{\lambdambda},
\end{equation*}
the solution $(\boldsymbol{w}, H)$ above can be written by
\betagin{align*}
\boldsymbol{w}= \mathcal{A}_0(\lambdambda,\mathbb{R}^N_+) \big(Id + F_{\lambdambda}\mathcal{R}(\lambdambda) \big)^{-1} \boldsymbol{Z}_+, \quad
H= \mathcal{H}_0(\lambdambda,\mathbb{R}^N_+) \big(Id + F_{\lambdambda}\mathcal{R}(\lambdambda) \big)^{-1} \boldsymbol{Z}_+
\end{align*}
for $\boldsymbol{Z}_+ := (\boldsymbol{F}_+,\lambdambda^{1\slash 2}\boldsymbol{G}_+,\boldsymbol{G}_+,K_+).$
Now we introduce that operators $\mathcal{T}_k$ for $k=1,2,3,$
\betagin{align*}
\mathcal{T}_1 F_\lambdambda (\boldsymbol{f}, \boldsymbol{g}, k)
&:= F_{\lambdambda}\big( \mathbb{A}^\top_-(\boldsymbol{f} \circ \boldsymbol{P}hi ), |\mathscr{A}_{\Phi}\boldsymbol{n}_0|\mathbb{A}_-^{\top} (\boldsymbol{g} \circ \boldsymbol{P}hi ), k \circ \boldsymbol{P}hi \big),\\
\mathcal{T}_2 \,\boldsymbol{w}(x)& := \mathbb{A}^{\top}_- (\boldsymbol{w} \circ \boldsymbol{P}hi^{-1}) \,\,\, \hbox{and}\,\,\,
\mathcal{T}_3 H (x) := H\circ \boldsymbol{P}hi^{-1},
\end{align*}
which, according to assumptions on $\boldsymbol{P}hi$ and Lemma \ref{lemma:ab_BH}, satisfy
\betagin{equation}\lambdabel{eq:T123_BH}
\mathcal{T}_1 \in \mathcal{L}\big( \mathcal{Y}_q(\Omegaega_+); \mathcal{Y}_q(\mathbb{R}^N_+)\big), \,\,\,
\mathcal{T}_2 \in \mathcal{L} \big( H^2_q(\mathbb{R}^N_+)^N; H^2_q(\Omegaega_+)^N \big),\,\,\,
\mathcal{T}_3 \in \mathcal{L} \big( H^3_q(\mathbb{R}^N_+); H^3_q(\Omegaega_+)\big).
\end{equation}
At last, according to \eqref{es:FR_BH_2}, \eqref{eq:T123_BH}, Theorem \ref{thm:GR_half_0}
and Remark \ref{rmk:R-bounded},
\betagin{align*}
\mathcal{A}_0(\lambdambda,\Omegaega_+) := \mathcal{T}_2 \, \mathcal{A}_0(\lambdambda,\mathbb{R}^N_+) \big(Id + F_{\lambdambda}\mathcal{R}(\lambdambda) \big)^{-1} \mathcal{T}_1,\\
\mathcal{H}_0(\lambdambda,\Omegaega_+) := \mathcal{T}_3 \, \mathcal{H}_0(\lambdambda,\mathbb{R}^N_+) \big(Id + F_{\lambdambda}\mathcal{R}(\lambdambda) \big)^{-1} \mathcal{T}_1,
\end{align*}
are the desired solution operators.
\subsection{Proof of \eqref{es:key_Rbd_BH}}
Here we are proving the technical estimate \eqref{es:key_Rbd_BH}. In the rest of this subsection, we always admit that $\tau:=\Im \lambdambda$ for $\lambdambda \in \Gammamma_{\varepsilon,\lambdambda_0,\zeta},$ $\ell,\ell'=0,1,$ and the constants $M_1,M_2,M_3$ given by \eqref{hyp:M1M2_BH}.
For brevity, we say the constant $K>0$ is \emph{admissible} if the choice of $K$ depends only on
the parameters
$\varepsilon,$ $\sigmagma,$ $m,$ $\mu,$ $\nu,$ $q,$ $r,$ $N,$ $\zeta_0,$ $\rho_1,$ $\rho_2,$ $\rho_3.$
\subsubsection*{The study of $\mathcal{R}_1(\lambdambda)$}
Recall $\boldsymbol{w} := \mathcal{A}_0(\lambdambda,\mathbb{R}^N_+) \boldsymbol{Z}_+$ for
$\boldsymbol{Z}_+:=(\boldsymbol{F}_+,\lambdambda^{1\slash 2}\boldsymbol{G}_+,\boldsymbol{G}_+,K_+) \in \mathcal{Y}_q(\mathbb{R}^N_+).$
Firstly, we introduce the operator families $\mathcal{T}_{ij} (\lambdambda),$ $\mathcal{T}_{ijk}(\lambdambda),$ $i,j,k=1,\dots,N,$ such that
\betagin{align*}
\mathbb{F}(\boldsymbol{w})_{ij}= \wt \mathcal{T}_{ij}(\lambdambda) ( \lambdambda^{1\slash 2} \nabla \boldsymbol{w})
= \mathcal{T}_{ij}(\lambdambda) (\boldsymbol{Z}_+), \quad
\partial_{k} \mathbb{F}(\boldsymbol{w})_{ij} = \mathcal{T}_{ijk}(\lambdambda) (\boldsymbol{Z}_+).
\end{align*}
Moreover, it is easy to see from \eqref{hyp:M1M2_BH} and \eqref{hyp:gamma_RR_BH} that
\betagin{equation*}
\mathcal{R}_{\mathcal{L}\big(L_q(\mathbb{R}^N_+)^{N^2}; L_q(\mathbb{R}^N_+)\big)}
\big\{(\tau \partial_{\tau})^{\ell} \lambdambda^{\ell'\slash 2} \wt\mathcal{T}_{ij}(\lambdambda):\lambdambda\in \Sigmagma_{\varepsilon,\lambdambda_0}\big\}
\leq C \lambdambda_0^{(\ell'-1)\slash 2} M_1,
\end{equation*}
which, together with Theorem \ref{thm:GR_half_0} and Remark \ref{rmk:R-bounded}, yields that
\betagin{equation}\lambdabel{es:R_Tij_BH}
\mathcal{R}_{\mathcal{L}\big(\mathcal{Y}_q(\mathbb{R}^N_+); L_q(\mathbb{R}^N_+)\big)}
\big\{(\tau \partial_{\tau})^{\ell} \lambdambda^{\ell'\slash 2} \mathcal{T}_{ij}(\lambdambda):\lambdambda\in \Gammamma_{\varepsilon,\lambdambda_0,\zeta}\big\}
\leq C \lambdambda_0^{(\ell'-1)\slash 2} M_1,
\end{equation}
for some admissible constants $C$ and $\lambdambda_0 \geq 1.$
\smallbreak
To study $\mathcal{T}_{ijk}(\lambdambda),$ we define
$$\wt \mathcal{T}_{ijk}(\lambdambda)(\boldsymbol{Z}_+):=\partial_i w_j \partial_k \mathbb{B}_-
= \partial_i \big( \mathcal{A}_{0j}(\lambdambda, \mathbb{R}^N_+) \boldsymbol{Z}_+ \big) \partial_k \mathbb{B}_- .$$
For any $N \in \mathbb{N},$ $\lambdambda_{\alphapha} \in \Gammamma_{\varepsilon,\lambdambda_0,\zeta},$ $\boldsymbol{Z}_{+\alphapha} \in \mathcal{Y}_q(\mathbb{R}^N_+),$
$w_{\alphapha j}:= \mathcal{A}_{0j}(\lambdambda, \mathbb{R}^N_+) \boldsymbol{Z}_{+\alphapha}$ for all $\alphapha =1,\dots,N,$
Lemma \ref{lemma:ab_BH} and Remark \ref{rmk:R-bounded} imply that
\betagin{align*}
\boldsymbol{i}g\|\sum_{\alphapha=1}^N \varepsilon_\alphapha \wt\mathcal{T}_{ijk}(\lambdambda_\alphapha) \boldsymbol{Z}_{+\alphapha}
\boldsymbol{i}g\|_{L_q \big(\Omegaega;L_q(\mathbb{R}^N_+)^{N^2}\big)}
&\leq \sigmagma_0 \boldsymbol{i}g\| \nabla \partial_i \big( \sum_{\alphapha=1}^N \varepsilon_\alphapha w_{\alphapha j} \big)
\boldsymbol{i}g\|_{L_q \big(\Omegaega;L_q(\mathbb{R}^N_+)^{N}\big)} \\
& \quad + C_r \lambdambda_0^{-1\slash 2} \sigmagma_0^{-\frac{N}{r-N}} M_2^{\frac{r}{r-N}}
\boldsymbol{i}g\|\sum_{\alphapha=1}^N \varepsilon_\alphapha \lambdambda_{\alphapha}^{1\slash 2} \partial_i w_{\alphapha j}
\boldsymbol{i}g\|_{L_q \big(\Omegaega;L_q(\mathbb{R}^N_+)\big)}.
\end{align*}
Then we can conclude from Theorem \ref{thm:GR_half_0} that
\betagin{equation} \lambdabel{es:R_Tijk_BH_1}
\mathcal{R}_{\mathcal{L}\big(\mathcal{Y}_q(\mathbb{R}^N_+); L_q(\mathbb{R}^N_+)\big)}
\big\{(\tau \partial_{\tau})^{\ell} \wt \mathcal{T}_{ijk}(\lambdambda):\lambdambda\in \Gammamma_{\varepsilon,\lambdambda_0,\zeta}\big\}
\leq C \boldsymbol{i}g(\sigmagma_0 + \lambdambda_0^{-1\slash 2}\sigmagma_0^{-\frac{N}{r-N}}M_2^{\frac{r}{r-N}}\boldsymbol{i}g),
\end{equation}
which gives us that
\betagin{equation}\lambdabel{es:R_Tijk_BH_2}
\mathcal{R}_{\mathcal{L}\big(\mathcal{Y}_q(\mathbb{R}^N_+); L_q(\mathbb{R}^N_+)\big)}
\big\{(\tau \partial_{\tau})^{\ell}\mathcal{T}_{ijk}(\lambdambda):\lambdambda\in \Gammamma_{\varepsilon,\lambdambda_0,\zeta}\big\}
\leq C \boldsymbol{i}g( M_1 +\sigmagma_0 + \lambdambda_0^{-1\slash 2}\sigmagma_0^{-\frac{N}{r-N}}M_2^{\frac{r}{r-N}}\boldsymbol{i}g),
\end{equation}
for some admissible constants $C$ and $\lambdambda_0 \geq 1.$
Moreover, the bound of $\mathcal{R}_1(\lambdambda)$ is clear by \eqref{es:R_Tijk_BH_1} and \eqref{es:R_Tijk_BH_2},
\betagin{equation}\lambdabel{es:R1_BH}
\mathcal{R}_{\mathcal{L}\big(\mathcal{Y}_q(\mathbb{R}^N_+); L_q(\mathbb{R}^N_+)^N\big)}
\big\{(\tau \partial_{\tau})^{\ell}\mathcal{R}_1(\lambdambda):\lambdambda\in \Gammamma_{\varepsilon,\lambdambda_0,\zeta}\big\} \leq C
\boldsymbol{i}g( M_1 +\sigmagma_0 + \lambdambda_0^{-1\slash 2}\sigmagma_0^{-\frac{N}{r-N}}M_2^{\frac{r}{r-N}}\boldsymbol{i}g).
\end{equation}
\subsubsection*{The study of $\mathcal{R}_2(\lambdambda)$}
Set that
\betagin{equation*}
\mathcal{R}_{21}(\lambdambda) \boldsymbol{Z}_+ :=\mathbb{F}_b\big( \mathcal{A}_0 (\lambdambda,\mathbb{R}^N_+)\boldsymbol{Z}_+\big)\boldsymbol{n}_0, \quad
\mathcal{R}_{22}(\lambdambda) \boldsymbol{Z}_+ :=\mathbb{G}_b\big( \mathcal{H}_0 (\lambdambda,\mathbb{R}^N_+)\boldsymbol{Z}_+\big)\boldsymbol{n}_0.
\end{equation*}
According to \eqref{es:R_Tij_BH} and \eqref{es:R_Tijk_BH_2}, we have
\betagin{equation*}
\mathcal{R}_{\mathcal{L}\big(\mathcal{Y}_q(\mathbb{R}^N_+); L_q(\mathbb{R}^N_+)^N\big)}
\big\{ (\tau \partial_{\tau})^{\ell} \lambdambda^{\ell'\slash 2}\mathcal{R}_{21}(\lambdambda):\lambdambda\in \Gammamma_{\varepsilon,\lambdambda_0,\zeta}\big\}
\leq C \lambdambda_0^{(\ell'-1)\slash 2} M_1,
\end{equation*}
\betagin{equation*}
\mathcal{R}_{\mathcal{L}\big(\mathcal{Y}_q(\mathbb{R}^N_+); H^1_q(\mathbb{R}^N_+)^N\big)}
\big\{(\tau \partial_{\tau})^{\ell}\mathcal{R}_{21}(\lambdambda):\lambdambda\in \Gammamma_{\varepsilon,\lambdambda_0,\zeta}\big\}
\leq C \boldsymbol{i}g( M_1 +\sigmagma_0 + \lambdambda_0^{-1\slash 2}\sigmagma_0^{-\frac{N}{r-N}}M_2^{\frac{r}{r-N}}\boldsymbol{i}g),
\end{equation*}
for some admissible constants $C$ and $\lambdambda_0 \geq 1.$
\smallbreak
Now let us study $\mathcal{R}_{22}(\lambdambda)$ and recall that
\betagin{equation*}
\mathbb{G}_b(H) = \sigmagma(m -\Deltalta')H \mathbb{A}_-^{\top} \mathbb{B}_-
- \sigmagma \mathcal{G}(H) \mathbb{A}_-^{\top} \mathscr{A}P,
\,\,\, \thetaxt{with}\,\,\, H := \mathcal{H}_0 (\lambdambda,\mathbb{R}^N_+)\boldsymbol{Z}_+.
\end{equation*}
Next, define $\mathcal{T}_{\mathcal{G}}^0(\lambdambda)(\boldsymbol{Z}_+) := \mathcal{G}(H)$ and
$\mathcal{T}_{\mathcal{G},\alphapha}^1(\lambdambda)(\boldsymbol{Z}_+) :=\partial_{\alphapha} \mathcal{G}(H)$ for $\alphapha=1,\dots,N.$
Then arguing as \eqref{es:R_Tij_BH} and \eqref{es:R_Tijk_BH_1},
we infer from \eqref{es:geo_1_BH}, \eqref{es:geo_2_BH} and Lemma \ref{lemma:ab_BH} that
\betagin{equation}\lambdabel{es:G_BH_1}
\mathcal{R}_{\mathcal{L}\big(\mathcal{Y}_q(\mathbb{R}^N_+);L_q(\mathbb{R}^N_+)\big)}
\big\{ (\tau \partial_{\tau})^{\ell}\lambdambda^{\ell'\slash 2}\mathcal{T}_{\mathcal{G}}^{0}(\lambdambda):\lambdambda\in \Gammamma_{\varepsilon,\lambdambda_0,\zeta}\big\}
\leq C \lambdambda_0^{-1+\ell'\slash 2}M_3,
\end{equation}
\betagin{multline}\lambdabel{es:G_BH_2}
\mathcal{R}_{\mathcal{L}\big(\mathcal{Y}_q(\mathbb{R}^N_+);L_q(\mathbb{R}^N_+)\big)}
\big\{ (\tau \partial_{\tau})^{\ell}\mathcal{T}_{\mathcal{G},\alphapha}^{1}(\lambdambda):\lambdambda\in \Gammamma_{\varepsilon,\lambdambda_0,\zeta}\big\} \\
\leq C \boldsymbol{i}g( M_1 +\sigmagma_0 + \lambdambda_0^{-1}
\big( M_3 + \sigmagma_0^{-\frac{N}{r-N}} (M_2M_3)^{\frac{r}{r-N}} \big) \boldsymbol{i}g), \hspace*{2cm}
\end{multline}
for some admissible constants $C$ and $\lambdambda_0.$
Thus \eqref{es:G_BH_1} and \eqref{es:G_BH_2} yield that
\betagin{equation*}
\mathcal{R}_{\mathcal{L}\big(\mathcal{Y}_q(\mathbb{R}^N_+); L_q(\mathbb{R}^N_+)^N\big)}
\big\{ (\tau \partial_{\tau})^{\ell} \lambdambda^{\ell'\slash 2}\mathcal{R}_{22}(\lambdambda):\lambdambda\in \Gammamma_{\varepsilon,\lambdambda_0,\zeta}\big\}
\leq C \sigmagma \lambdambda_0^{-1+\ell'\slash 2} (M_1 \max\{1,m\} + M_3),
\end{equation*}
\betagin{multline*}
\mathcal{R}_{\mathcal{L}\big(\mathcal{Y}_q(\mathbb{R}^N_+); H^1_q(\mathbb{R}^N_+)^N\big)}
\big\{(\tau \partial_{\tau})^{\ell}\mathcal{R}_{22}(\lambdambda):\lambdambda\in \Gammamma_{\varepsilon,\lambdambda_0,\zeta}\big\}
\leq C\sigmagma (M_1 + \sigmagma_0) \max\{1, m\lambdambda_0^{-1}\}\\
+ C \sigmagma \lambdambda_0^{-1}
\big( M_3 + \sigmagma_0^{-\frac{N}{r-N}} M_2^{\frac{r}{r-N}} (M_3^{\frac{r}{r-N}}+\max\{1,m\} )\big).
\hspace*{1cm}
\end{multline*}
Finally, combing the discussions on $\mathcal{R}_{21}(\lambdambda)$ and $\mathcal{R}_{22}(\lambdambda)$ yields that
\betagin{equation}\lambdabel{es:R2_BH_1}
\mathcal{R}_{\mathcal{L}\big(\mathcal{Y}_q(\mathbb{R}^N_+); L_q(\mathbb{R}^N_+)^N\big)}
\big\{ (\tau \partial_{\tau})^{\ell} \lambdambda^{1\slash 2}\mathcal{R}_{2}(\lambdambda):\lambdambda\in \Gammamma_{\varepsilon,\lambdambda_0,\zeta}\big\}
\leq C \big( M_1 + \sigmagma \lambdambda_0^{-1\slash 2}(m+ M_3) \big),
\end{equation}
\betagin{multline}\lambdabel{es:R2_BH_2}
\mathcal{R}_{\mathcal{L}\big(\mathcal{Y}_q(\mathbb{R}^N_+); H^1_q(\mathbb{R}^N_+)^N\big)}
\big\{(\tau \partial_{\tau})^{\ell}\mathcal{R}_{2}(\lambdambda):\lambdambda\in \Gammamma_{\varepsilon,\lambdambda_0,\zeta}\big\}
\leq C (M_1 + \sigmagma_0) (1+\sigmagma+\sigmagma m\lambdambda_0^{-1} ) \\
+ C \lambdambda_0^{-1 \slash 2} (\sigmagma +1)
\big( M_3 + \sigmagma_0^{-\frac{N}{r-N}} M_2^{\frac{r}{r-N}} (M_3^{\frac{r}{r-N}}+m )\big).
\hspace*{1cm}
\end{multline}
\subsubsection*{The study of $\mathcal{R}_3(\lambdambda)$}
By direct calculations, we have
\betagin{equation*}
\big| 1- |\mathscr{A}_{\Phi} \boldsymbol{n}_0|^{-1} \big| \lesssim |\mathbb{B}_-|, \,\,\,
\big| \nabla |\mathscr{A}_{\Phi} \boldsymbol{n}_0|^{-1} \big| \lesssim |\nabla \mathbb{B}_-|, \,\,\,
\big| \nabla^2 |\mathscr{A}_{\Phi} \boldsymbol{n}_0|^{-1} \big| \lesssim |\nabla \mathbb{B}_-|^2 + |\nabla^2\mathbb{B}_-|.
\end{equation*}
Thus \eqref{hyp:M1M2_BH} and $r>N$ imply that
\betagin{equation}\lambdabel{es:An_0_BH}
\big\| 1- |\mathscr{A}_{\Phi} \boldsymbol{n}_0|^{-1} \big\|_{L_{\infty}(\mathbb{R}^N)} \lesssim M_1, \,\,\,
\big\| \nabla |\mathscr{A}_{\Phi} \boldsymbol{n}_0|^{-1} \big\|_{L_r(\mathbb{R}^N)} \lesssim M_2, \,\,\,
\big\| \nabla^2 |\mathscr{A}_{\Phi} \boldsymbol{n}_0|^{-1} \big\|_{L_r(\mathbb{R}^N)} \lesssim M_2M_3.
\end{equation}
For $j,k=1,\dots,N,$ we denote that
\betagin{gather*}
\mathcal{T}_3^0(\lambdambda)(\boldsymbol{Z}_+)=\mathcal{F}_3(\boldsymbol{w}), \quad
\mathcal{T}_{3,j}^{1}(\lambdambda)(\boldsymbol{Z}_+)=\partial_j \mathcal{F}_3(\boldsymbol{w}), \quad
\mathcal{T}_{3,jk}^{2}(\boldsymbol{Z}_+)=\partial_k \partial_j \mathcal{F}_3(\boldsymbol{w}).
\end{gather*}
Then \eqref{es:An_0_BH}, Lemma \ref{lemma:ab_BH} and Theorem \ref{thm:GR_half_0} immediately yield that
\betagin{equation*}
\mathcal{R}_{\mathcal{L}\big( \mathcal{Y}_q(\mathbb{R}^N_+); L_q(\mathbb{R}^N_+)^{N} \big)}
\big\{(\tau \partial_{\tau})^{\ell}\mathcal{T}_{3}^{0}(\lambdambda):\lambdambda\in \Gammamma_{\varepsilon,\lambdambda_0,\zeta}\big\}
\leq C \lambdambda_0^{-1}M_1,
\end{equation*}
\betagin{equation*}
\mathcal{R}_{\mathcal{L}\big(\mathcal{Y}_q(\mathbb{R}^N_+); L_q(\mathbb{R}^N_+)^{N}\big)}
\big\{(\tau \partial_{\tau})^{\ell}\mathcal{T}_{3,j}^{1}(\lambdambda):\lambdambda\in \Gammamma_{\varepsilon,\lambdambda_0,\zeta}\big\}
\leq C \boldsymbol{i}g( \lambdambda_0^{-1\slash 2}(M_1 + \sigmagma_0)
+ \lambdambda_0^{-1} \sigmagma_0^{-\frac{N}{r-N}} M_2^{\frac{r}{r-N}} \boldsymbol{i}g),
\end{equation*}
\betagin{equation*}
\mathcal{R}_{\mathcal{L}\big(\mathcal{Y}_q(\mathbb{R}^N_+) ; L_q(\mathbb{R}^N_+)^{N}\big)}
\big\{(\tau \partial_{\tau})^{\ell}\mathcal{T}_{3,jk}^{2}(\lambdambda):\lambdambda\in \Gammamma_{\varepsilon,\lambdambda_0,\zeta}\big\}
\leq C \boldsymbol{i}g( M_1 + \sigmagma_0
+ \lambdambda_0^{-1\slash 2} \big( \sigmagma_0 M_3 + \sigmagma_0^{-\frac{N}{r-N}} (M_2M_3)^{\frac{r}{r-N}} \big) \boldsymbol{i}g),
\end{equation*}
for any $0<\sigmagma_0<1.$ Therefore we infer from Remark \ref{rmk:R-bounded} and Theorem \ref{thm:GR_half_0} that
\betagin{equation}\lambdabel{es:R3_BH}
\mathcal{R}_{\mathcal{L}\big(\mathcal{Y}_q(\mathbb{R}^N_+);H^2_q(\mathbb{R}^N_+)\big)}
\big\{(\tau \partial_{\tau})^{\ell}\mathcal{R}_{3}(\lambdambda):\lambdambda\in \Gammamma_{\varepsilon,\lambdambda_0,\zeta}\big\}
\leq C \boldsymbol{i}g( M_1 + \sigmagma_0
+ \lambdambda_0^{-1\slash 2} \big( M_3 + \sigmagma_0^{-\frac{N}{r-N}} (M_2M_3)^{\frac{r}{r-N}} \big) \boldsymbol{i}g).
\end{equation}
At last, we can easily conclude \eqref{es:key_Rbd_BH} by combining \eqref{es:R1_BH}, \eqref{es:R2_BH_1}, \eqref{es:R2_BH_2} and \eqref{es:R3_BH}. This completes the proof of \eqref{es:key_Rbd_BH}.
\subsection{Review of other model problems}
To study the model problem in the general domain, let us review some results in \cite{EvBS2014}.
By the notations $\boldsymbol{P}hi,$ $\mathbb{B}$ and $\mathbb{B}_-$ in the beginning of Section \ref{sec:bh},
we assume that $\boldsymbol{P}hi$ is a $H^2_r$ diffeomorphism for some $N<r<\infty,$
and there exist constants $0<M_1<1\leq M_2<\infty$ such that
\betagin{equation*}
\|(\mathbb{B}, \mathbb{B}_-)\|_{L_{\infty}(\mathbb{R}^N)} \leq M_1,\quad
\|\nabla(\mathbb{B}, \mathbb{B}_-)\|_{L_{r}(\mathbb{R}^N)} \leq M_2.
\end{equation*}
For $\Omegaega_+:= \boldsymbol{P}hi(\mathbb{R}^N_+)$ and $\Gammamma_+:= \boldsymbol{P}hi(\mathbb{R}^N_0),$ we consider the following model problem
\betagin{equation}\lambdabel{eq:RR_CNS_BH_D}
\left\{\betagin{aligned}
& \lambdambda \boldsymbol{v} -\gammamma_{1}^{-1}\Di \mathbb{S}(\boldsymbol{v}) - \lambdambda^{-1} \gammamma_1^{-1} \nabla (\gammamma_3 \di \boldsymbol{v})
= \boldsymbol{f} &&\quad\hbox{in}\quad \Omegaega_+,\\
&\boldsymbol{v}=0 &&\quad\hbox{on}\quad \Gammamma_+,
\end{aligned}\right.
\end{equation}
where $\gammamma_1$ and $\gammamma_3$ satisfy the conditions in \eqref{hyp:gamma_RR_BH}.
\betagin{theo}\lambdabel{thm:RR_BH_D}
Let $0<\varepsilon<\pi\slash 2,$ $\mu, \nu,\zeta_0>0,$ $1<q<\infty,$ $N<r<\infty$ and $r\geq q.$
Assume that $\Omegaega_+$ is given as above and \eqref{hyp:gamma_RR_BH} is satisfied.
Then for any $\boldsymbol{f}\in L_q(\Omegaega_+)^N,$ there exist constants $\lambdambda_0, r_b \geq 1$ and a family of operators
\betagin{align*}
\mathcal{A}_1(\lambdambda,\Omegaega_+) \in
{\rm Hol}\,\boldsymbol{i}g(\Gammamma_{\varepsilon,\lambdambda_0,\zeta}; \mathcal{L}\big(L_q(\Omegaega_+)^N;H^2_q(\Omegaega_+)^N \big) \boldsymbol{i}g),
\end{align*}
such that $\boldsymbol{v}:=\mathcal{A}_1(\lambdambda,\Omegaega_+) \boldsymbol{f}$ is a solution of \eqref{eq:RR_CNS_BH_D}. Moreover, we have
\betagin{gather*}
\mathcal{R}_{\mathcal{L}\big(L_q(\Omegaega_+)^N; H^{2-j}_q(\Omegaega_+)^N \big)}
\boldsymbol{i}g( \boldsymbol{i}g\{ (\tau \partial_{\tau})^{\ell}\big( \lambdambda^{j\slash 2}\mathcal{A}_1(\lambdambda,\Omegaega_+)\big) : \lambdambda \in
\Gammamma_{\varepsilon,\lambdambda_0,\zeta} \boldsymbol{i}g\}\boldsymbol{i}g) \leq r_b,
\end{gather*}
for $\ell=0,1,$ $j=0,1,2,$ and $\tau := \Im \lambdambda.$ Above the constants $\lambdambda_0$ and $r_b$ depend solely on
$\varepsilon,$ $\mu,$ $\nu,$ $q,$ $r,$ $N,$ $\zeta_0,$ $\rho_1,$ $\rho_2,$ $\rho_3.$
\end{theo}
Next, let us consider the generalized system in the whole space $\mathbb{R}^N,$
\betagin{equation}\lambdabel{eq:RR_CNS_whole}
\lambdambda \boldsymbol{v} -\gammamma_{1}^{-1}\Di \mathbb{S}(\boldsymbol{v}) - \lambdambda^{-1} \gammamma_1^{-1} \nabla (\gammamma_3 \di \boldsymbol{v})
= \boldsymbol{f} \quad\hbox{in}\quad \mathbb{R}^N.
\end{equation}
For \eqref{eq:RR_CNS_whole}, we recall \cite[Theorem 3.11]{EvBS2014} here.
\betagin{theo}\lambdabel{thm:RR_CNS_whole}
Let $0<\varepsilon<\pi\slash 2,$ $\mu, \nu,\zeta_0>0,$ $1<q<\infty,$ $N<r<\infty$ and $r\geq q.$
Assume that $\gammamma_1$ and $\gammamma_3$ are uniformly continuous functions in $\mathbb{R}^N$ and
satisfy \eqref{hyp:gamma_RR_BH} by changing $\Omegaega_+$ to $\mathbb{R}^N.$
For any $\boldsymbol{f}\in L_q(\mathbb{R}^N)^N,$
there exist constants $\lambdambda_0,r_b \geq 1$ and a family of operators
\betagin{align*}
\mathcal{A}_2(\lambdambda,\mathbb{R}^N) & \in
{\rm Hol}\,\boldsymbol{i}g(\Gammamma_{\varepsilon,\lambdambda_0,\zeta}; \mathcal{L}\big(L_q(\mathbb{R}^N)^N;H^2_q(\mathbb{R}^N)^N \big) \boldsymbol{i}g),
\end{align*}
such that $\boldsymbol{v}:=\mathcal{A}_2(\lambdambda,\mathbb{R}^N) \boldsymbol{f}$ is a solution of \eqref{eq:RR_CNS_whole}. Moreover, we have
\betagin{gather*}
\mathcal{R}_{\mathcal{L}\big(L_q(\mathbb{R}^N)^N; H^{2-j}_q(\mathbb{R}^N)^N \big)}
\boldsymbol{i}g( \boldsymbol{i}g\{ (\tau \partial_{\tau})^{\ell}\big( \lambdambda^{j\slash 2}\mathcal{A}_2(\lambdambda,\mathbb{R}^N)\big) : \lambdambda \in
\Gammamma_{\varepsilon,\lambdambda_0,\zeta} \boldsymbol{i}g\}\boldsymbol{i}g) \leq r_b,
\end{gather*}
for $\ell=0,1,$ $j=0,1,2,$ and $\tau := \Im \lambdambda.$ Above the constants $\lambdambda_0$ and $r_b$ depend solely on
$\varepsilon,$ $\mu,$ $\nu,$ $q,$ $r,$ $N,$ $\zeta_0,$ $\rho_1,$ $\rho_2,$ $\rho_3.$
\end{theo}
\section{Full model problem in the general domain}
This section is dedicated to the study of \eqref{eq:RR_CNS_0}. After the review of some auxiliary results, we will construct the solutions for model problems by the localization procedure due to Section \ref{sec:bh}, and then establish the leading part of the solution of \eqref{eq:RR_CNS_0}. At last, we will see the remainder part from the parametrix is harmless in the sense of $\mathcal{R}-$boundedness.
\subsection{Some auxiliary results}
According to \cite{ES2013}, let us list some properties for the uniformly smooth domain $\Omegaega$ in the class $W^{3,2}_r.$
\betagin{prop}\lambdabel{prop:domain}
Let $\Omegaega$ is of type $W^{3,2}_r$ by Definition \ref{def:domain} for some $N<r<\infty$ and $B_{d}(x):=\{y\in \mathbb{R}^N: |x-y|<d\}$ be the ball in $\mathbb{R}^N$ with radius $d>0.$
Then there exist the constants $0<d_1, d_2, d_3<1,$ $0<M_1<1\leq \min\{M_2,M_3\}<\infty,$ at most countably many mappings $\boldsymbol{P}hi^i_j$ ($i=0,1$) of the class $H^{3-i}_r(\mathbb{R}^N;\mathbb{R}^N)$ and points $x_j^0 \in \Gammamma_0,$ $x_j^1 \in \Gammamma_1,$ $x_j^2 \in \Omegaega$ such that the following assertions hold:
\betagin{enumerate}
\item The mappings $\boldsymbol{P}hi^i_j:\mathbb{R}^N \rightarrow \mathbb{R}^N$ ($i=0,1, j\in \mathbb{N}$) are $H^{3-i}_r$ diffeomorphism.
\item Denote $B^i_j:= B_{d_i}(x^i_j)$ ($i=0,1,2, j\in \mathbb{N}$) for simplicity and we have
\betagin{gather*}
\Omegaega = \bigcup_{i=0,1} \boldsymbol{i}g( \bigcup_{j\in \mathbb{N}}\big( \boldsymbol{P}hi_j^i(\mathbb{R}^N_+) \cap B^i_j\big) \boldsymbol{i}g)
\bigcup \boldsymbol{i}g( \bigcup_{j\in \mathbb{N}} B^2_j \boldsymbol{i}g), \quad
\bigcup_{j\in \mathbb{N}} B^2_j \subset \Omegaega,\\
\boldsymbol{P}hi_j^i(\mathbb{R}^N_+) \cap B^i_j =\Omegaega \cap B^i_j, \quad \boldsymbol{P}hi_j^i(\mathbb{R}^N_0) \cap B^i_j =\Gammamma_i \cap B^i_j, \,\,\, i=0,1, j \in \mathbb{N}.
\end{gather*}
\item There exist $C^{\infty}$ functions $\zeta^i_j, \wt\zeta^i_j$ ($i=0, 1,2, j\in\mathbb{N}$) such that
\betagin{gather*}
0 \leq \zeta^i_j,\widetilde{\zeta}^i_j \leq 1, \,\,\, {\rm supp}\, \zeta^i_j,\, {\rm supp}\, \widetilde{\zeta}^i_j \subset B^i_j,\,\,\, \widetilde{\zeta}^i_j\zeta^i_j=\zeta^i_j,\\
\|(\zeta^0_j,\wt\zeta^0_j)\|_{C^3(\mathbb{R}^N)} +\sum_{i=1,2} \|(\zeta^i_j,\widetilde{\zeta}^i_j)\|_{C^2(\mathbb{R}^N)} \leq c_0,\\
\sum_{i=0,1,2}\sum_{j\in\mathbb{N}} \zeta^i_j =1 \,\,\thetaxt{on}\,\,\,\overline{\Omegaega},\quad
\sum_{j\in \mathbb{N}}\zeta^0_j=1 \,\,\thetaxt{on}\,\,\,\Gammamma_0,\quad
\sum_{j\in \mathbb{N}}\zeta^1_j=1 \,\,\thetaxt{on}\,\,\,\Gammamma_1.
\end{gather*}
Here the choice of $c_0$ is dependent on $N,r, M_1,M_2,M_3,$ but independent on $j.$
\item Denote $\boldsymbol{P}si^i_j:= (\boldsymbol{P}hi^i_j)^{-1}$ for $i=0,1$ and $j\in \mathbb{N}.$ Then
\betagin{gather*}
\nabla_{\xi} (\boldsymbol{P}hi^i_j)^\top (\xi) = \mathbb{A}^i_j + \mathbb{B}^i_j(\xi),\quad
\nabla_x (\boldsymbol{P}si^i_j)^\top|_{x=\boldsymbol{P}hi^i_j(\xi)} = \mathbb{A}^i_{j,-} + \mathbb{B}^i_{j,-}(\xi) ,
\end{gather*}
where $\mathbb{A}^i_j, \mathbb{A}^i_{j,-}$ are orthogonal constant matrices and $\mathbb{B}^i_j, \mathbb{B}^i_{j,-}$ satisfy
\betagin{equation*}
\|(\mathbb{B}^i_j, \mathbb{B}^i_{j,-})\|_{L_{\infty}(\mathbb{R}^N)} \leq M_1,\,\,\,
\|\nabla(\mathbb{B}^i_j, \mathbb{B}^i_{j,-})\|_{L_{r}(\mathbb{R}^N)} \leq M_2,\,\,\,
\|\nabla^2(\mathbb{B}^0_j, \mathbb{B}^0_{j,-})\|_{L_{r}(\mathbb{R}^N)} \leq M_3.
\end{equation*}
\item There exists an integer $L \geq 2$ such that any $L+1$ distinct balls in $\{B^i_j, i=0,1,2,j\in \mathbb{N}\}$ have an empty intersection.
\end{enumerate}
\end{prop}
Let us give some useful comments on Proposition \ref{prop:domain}.
\betagin{itemize}
\item Thanks to \eqref{hyp:gamma_GR_2}, we assume that
\betagin{equation}\lambdabel{hyp:gamma_Om_2}
\sum_{\alphapha=1,3} \|\gammamma_{\alphapha}(\cdot) - \gammamma_{\alphapha}(x^i_j) \|_{L_{\infty} ( \Omegaega \cap B^i_j)} \leq M_1,
\,\,\, \sum_{\alphapha=1,3} \|\nabla \gammamma_\alphapha\|_{L_r(\Omegaega \cap B^i_j)} \leq M_2,
\end{equation}
up the choices of $d_i$ and $M_2.$
\item On the other hand, by the last property of Proposition \ref{prop:domain}, we have
\betagin{equation}\lambdabel{es:fip}
\big( \sum_{i=0,1,2} \sum_{j\in \mathbb{N}} \|f\|^r_{L_r(\Omegaega \cap B^i_j)} \big) ^{1\slash r}\leq C_{r,L} \|f\|_{L_r(\Omegaega)}
\end{equation}
for any $f\in L_r(\Omegaega)$ and $1\leq r<\infty.$
In particular, we infer from \eqref{es:fip} that
\betagin{equation}\lambdabel{es:fip_2}
\big( \sum_{i=0,1,2} \sum_{j\in \mathbb{N}} \|\boldsymbol{F}\|^q_{\mathcal{Y}_q(\Omegaega \cap B^i_j)} \big) ^{1\slash q}\leq C_{q,L} \|\boldsymbol{F}\|_{\mathcal{Y}_q(\Omegaega)},
\end{equation}
for any $\boldsymbol{F} \in \mathcal{Y}_q(\Omegaega)$ and $1<q<\infty.$
\end{itemize}
For $\Omegaega$ given by Proposition \ref{prop:domain}, we adopt the notations
\betagin{equation}\lambdabel{eq:notation_Om}
\Omegaega^i_j := \boldsymbol{P}hi^i_j(\mathbb{R}^N_+), \,\,\,
\Gammamma^i_j := \partial \Omegaega^i_j = \boldsymbol{P}hi^i_j(\mathbb{R}^N_0), \,\,\,
\Omegaega^2_j:= \mathbb{R}^N, \,\,\, \forall \,\, i=0,1, \,\, j \in \mathbb{N},
\end{equation}
and denote $\boldsymbol{n}_j^0$ for the unit normal vectors subject to $\Gammamma^0_j$ in what follows.
Now we recall the following results proved in \cite[Section 9.5.1]{Shi2016b}.
\betagin{prop}\lambdabel{prop:sum}
Assume that $\Omegaega$ satisfies Proposition \ref{prop:domain}, $1 < q <\infty,$ $i=0,1,2,$ and $n \in \mathbb{N}_0.$
For any $j\in \mathbb{N}$ and $i=0,1,2,$
take $\eta^i_j \in C_0^{\infty}(B^i_j)$ and $\{f^i_j : f^i_j \in H^n_q(\Omegaega^i_j)\}_{j\in \mathbb{N}}$ such that
\betagin{equation*}
\|\eta^i_j\|_{C^n(\mathbb{R}^N)} \leq c_1, \quad
\big\| \big\{ \|f^i_j\|_{H^n_q(\Omegaega^i_j)} \big\}_{j\in \mathbb{N}} \big\|_{\ell_q(\mathbb{N})} < \infty,
\end{equation*}
with some constant $c_1$ independent on $i$ and $j.$
Then the infinite sum $f^i:= \sum_{j\in \mathbb{N}} \eta^i_j f^i_j $ exists in $H^n_q(\Omegaega)$ for any $i=0,1,2,$ fulfilling
\betagin{equation*}
\|f^i\|_{H^n_q(\Omegaega)} \leq C_{n,q,L}c_1
\big\| \|f^i_j\|_{H^n_q(\Omegaega^i_j)} \big\|_{\ell_q(\mathbb{N})}.
\end{equation*}
\end{prop}
Let us end up this part with some comment on the unit normal vector $\boldsymbol{n}_{\Gammamma_0}$ to $\Gammamma_0.$
We regard $\boldsymbol{n}_{\Gammamma_0}$ as its natural extension to $\Omegaega$ through
$\boldsymbol{n}_{\Gammamma_0}= \sum_{j\in \mathbb{N}} \zeta^0_j \boldsymbol{n}^0_j.$
In addition, Proposition \ref{prop:domain} yields that
\betagin{equation}\lambdabel{es:normal_0j}
\|\boldsymbol{n}^0_j\|_{L_{\infty}(\mathbb{R}^N)} \leq 1, \quad
\| \nabla \boldsymbol{n}^0_j\|_{L_r(\mathbb{R}^N)} \leq C_{N} M_2, \quad
\|\nabla \boldsymbol{n}^0_j\|_{H^1_r(\mathbb{R}^N)} \leq C_{N} M_2 M_3.
\end{equation}
Thanks to Proposition \ref{prop:sum}, \eqref{es:fip} and \eqref{es:normal_0j}, it is not hard to see that
\betagin{equation}\lambdabel{es:normal_0}
\|f \boldsymbol{n}_{\Gammamma_0}\|_{H^m_q(\Omegaega)} \leq C_{N,q,L} c_0 M_2M_3 \|f\|_{H^m_q(\Omegaega)}, \,\,\, \forall\,\,f \in H^m_q(\Omegaega),
\end{equation}
with $m=0,1,2,$ and $1<q\leq r.$
\subsection{Localization}
Recall the notations in \eqref{eq:notation_Om} and set that
\betagin{equation}\lambdabel{eq:def_gamma_ij}
\gammamma^i_{j\alphapha}(x):= \big( \gammamma_{\alphapha} (x) - \gammamma_{\alphapha} (x^i_j) \big) \widetilde{\zeta}^i_j (x) + \gammamma_{\alphapha} (x^i_j),
\,\,\forall \,\, x\in \Omegaega^i_j, \,\, \alphapha=1,3,\,\, i=0,1,2.
\end{equation}
Then it is not hard to see from \eqref{hyp:gamma_GR_2}, \eqref{hyp:gamma_Om_2} and Proposition \ref{prop:domain} that
\betagin{gather}\lambdabel{hyp:gamma_ij}
0<\rho_1 \leq \gammamma^i_{j1} (x) \leq \rho_2, \,\,\,
0 < \gammamma^i_{j3}(x) \leq \rho_3, \,\,\, \forall \,\,x \in \overline{\Omegaega^i_j}, \\ \nonumber
\sum_{\alphapha=1,3} \|\gammamma^i_{j\alphapha}(\cdot) - \gammamma^i_{j\alphapha}(x^i_j) \|_{L_{\infty} ( \Omegaega^i_j)} \leq M_1,
\,\,\, \sum_{\alphapha=1,3} \|\nabla \gammamma_\alphapha\|_{L_r(\Omegaega^i_j)} \leq M_2+C_{r,N}c_0.
\end{gather}
Then we consider the following model problems for any $j\in \mathbb{N},$
\betagin{equation}\lambdabel{eq:RR_CNS_loc_0}
\left\{\betagin{aligned}
& \lambdambda \boldsymbol{v}_j^0 -(\gammamma_{j1}^0)^{-1}\Di \big( \mathbb{S}(\boldsymbol{v}^0_j) + \zeta \gammamma_{j3}^0 \di \boldsymbol{v}^0_j \mathbb{I}\big) =\wt\zeta^0_j \boldsymbol{f}
&&\quad\hbox{in}\quad \Omegaega^0_j,\\
& \big( \mathbb{S}(\boldsymbol{v}^0_j) +\zeta \gammamma^0_{j3} \di \boldsymbol{v}^0_j \mathbb{I}\big)\boldsymbol{n}^0_{j}
+\sigmagma (m-\Deltalta_{\Gammamma^0_j})h^0_j \,\boldsymbol{n}^0_{j} =\wt\zeta^0_j \boldsymbol{g}
&&\quad\hbox{on}\quad \Gammamma^0_j, \\
&\lambdambda h^0_j - \boldsymbol{v}^0_j \cdot \boldsymbol{n}^0_{j} = \wt\zeta^0_j k
&&\quad\hbox{on}\quad \Gammamma^0_j,
\end{aligned}\right.
\end{equation}
\betagin{equation}\lambdabel{eq:RR_CNS_loc_1}
\left\{\betagin{aligned}
& \lambdambda \boldsymbol{v}_j^1 -(\gammamma_{j1}^1)^{-1}\Di \big( \mathbb{S}(\boldsymbol{v}^1_j) +\zeta \gammamma_{j3}^1 \di \boldsymbol{v}^1_j \mathbb{I} \big) =\wt\zeta^1_j \boldsymbol{f}
&&\quad\hbox{in}\quad \Omegaega^1_j,\\
& \boldsymbol{v}^1_j =0
&&\quad\hbox{on}\quad \Gammamma^1_j,
\end{aligned}\right.
\end{equation}
\betagin{equation}\lambdabel{eq:RR_CNS_loc_2}
\betagin{aligned}
& \lambdambda \boldsymbol{v}_j^2 -(\gammamma_{j1}^2)^{-1} \Di \big( \mathbb{S}(\boldsymbol{v}^2_j) + \zeta \gammamma_{j3}^2 \di \boldsymbol{v}^2_j \mathbb{I} \big)=\wt\zeta^2_j \boldsymbol{f}
&&\quad\hbox{in}\quad \Omegaega^2_j.
\end{aligned}
\end{equation}
Thanks to \eqref{hyp:gamma_ij}, Theorem \ref{thm:GR_BH}, Theorem \ref{thm:RR_BH_D} and Theorem \ref{thm:RR_CNS_whole}, there exist constants $\lambdambda_0,r_b \geq 1$ and the families of operators
\betagin{align*}
\mathcal{A}_0(\lambdambda,\Omegaega^0_j) & \in
{\rm Hol}\,\boldsymbol{i}g(\Gammamma_{\varepsilon,\lambdambda_0,\zeta}; \mathcal{L}\big(\mathcal{Y}_q(\Omegaega^0_j);H^2_q(\Omegaega^0_j)^N \big) \boldsymbol{i}g),\\
\mathcal{H}_0(\lambdambda,\Omegaega^0_j ) & \in
{\rm Hol}\,\boldsymbol{i}g(\Gammamma_{\varepsilon,\lambdambda_0,\zeta}; \mathcal{L}\big(\mathcal{Y}_q(\Omegaega^0_j);H^3_q(\Omegaega^0_j)\big) \boldsymbol{i}g),\\
\mathcal{A}_{i}(\lambdambda,\Omegaega^{i}_j) & \in
{\rm Hol}\,\boldsymbol{i}g(\Gammamma_{\varepsilon,\lambdambda_0,\zeta}; \mathcal{L}\big(L_q(\Omegaega^i_j)^N;H^2_q(\Omegaega^i_j)^N \big) \boldsymbol{i}g)
\quad (i=1,2),
\end{align*}
such that
\betagin{align*}
\boldsymbol{v}^0_j &:= \mathcal{A}_0(\lambdambda,\Omegaega^0_j)
\big( \wt\zeta^0_j\boldsymbol{f},\lambdambda^{1\slash 2}\wt\zeta^0_j\boldsymbol{g},\wt\zeta^0_j \boldsymbol{g},\wt\zeta^0_j k \big), \\
h^0_j &:= \mathcal{H}_0(\lambdambda,\Omegaega^0_j)
\big( \wt\zeta^0_j\boldsymbol{f},\lambdambda^{1\slash 2}\wt\zeta^0_j\boldsymbol{g},\wt\zeta^0_j \boldsymbol{g},\wt\zeta^0_j k \big), \\
\boldsymbol{v}^i_j &:= \mathcal{A}_i(\lambdambda,\Omegaega^i_j) \wt\zeta^i_j\boldsymbol{f} \quad (i=1,2),
\end{align*}
satisfy \eqref{eq:RR_CNS_loc_0}, \eqref{eq:RR_CNS_loc_1} and \eqref{eq:RR_CNS_loc_2} respectively.
Moreover, we have
\betagin{gather}
\mathcal{R}_{\mathcal{L}\big(\mathcal{Y}_q(\Omegaega^0_j); H^{2-k}_q(\Omegaega^0_j)^N \big)}
\boldsymbol{i}g( \boldsymbol{i}g\{ (\tau \partial_{\tau})^{\ell}\big( \lambdambda^{k\slash 2}\mathcal{A}_0(\lambdambda,\Omegaega^0_j)\big) : \lambdambda \in
\Gammamma_{\varepsilon,\lambdambda_0,\zeta} \boldsymbol{i}g\}\boldsymbol{i}g) \leq r_{b}, \nonumber \\
\mathcal{R}_{\mathcal{L}\big(\mathcal{Y}_q(\Omegaega^0_j); H^{3-k'}_q(\Omegaega^0_j) \big)}
\boldsymbol{i}g( \boldsymbol{i}g\{ (\tau \partial_{\tau})^{\ell}\big( \lambdambda^{k'}\mathcal{H}_0(\lambdambda,\Omegaega^0_j)\big) : \lambdambda \in
\Gammamma_{\varepsilon,\lambdambda_0,\zeta} \boldsymbol{i}g\}\boldsymbol{i}g) \leq r_{b}, \lambdabel{es:Rbdd_loc} \\
\mathcal{R}_{\mathcal{L}\big(L_q(\Omegaega^i_j)^N; H^{2-k}_q(\Omegaega^i_j)^N \big)}
\boldsymbol{i}g( \boldsymbol{i}g\{ (\tau \partial_{\tau})^{\ell}\big( \lambdambda^{k\slash 2}\mathcal{A}_i(\lambdambda,\Omegaega^i_j)\big) : \lambdambda \in
\Gammamma_{\varepsilon,\lambdambda_0,\zeta} \boldsymbol{i}g\}\boldsymbol{i}g) \leq r_{b}, \nonumber
\end{gather}
for $i=1,2,$ $k=0,1,2,$ $k',\ell=0,1,$ $\tau := \Im \lambdambda$ and $j\in \mathbb{N}.$
The constants $\lambdambda_0$ and $r_b$ depend solely on
$\varepsilon,$ $\sigmagma,$ $m,$ $\mu,$ $\nu,$ $\zeta_0,$ $q,$ $r,$ $N,$ $\rho_1,$ $\rho_2,$ $\rho_3$ and $\Omegaega.$
In particular, we have
\betagin{gather}
\sum_{k=0,1,2} \|\lambdambda^{k\slash 2} \boldsymbol{v}^0_j \|_{H^{2-k}_q (\Omegaega^0_j)}
+ \sum_{k' =0,1} \| \lambdambda^{k'} h^0_j\|_{H^{3-k'}_q(\Omegaega^0_j)}
\leq 5r_b (1+2c_0)\| (\boldsymbol{f},\lambdambda^{1\slash 2}\boldsymbol{g},\boldsymbol{g},k ) \|_{\mathcal{Y}_q(\Omegaega \cap B^0_j)},
\nonumber \\ \lambdabel{es:RR_loc_1}
\sum_{i=1,2}\sum_{k=0,1,2} \|\lambdambda^{k\slash 2} \boldsymbol{v}^i_j \|_{H^{2-k}_q (\Omegaega^i_j)}
\leq 6r_b \|\boldsymbol{f}\|_{L_q(\Omegaega \cap B^i_j)}, \,\,\,\forall j\in \mathbb{N},
\end{gather}
where we have used the fact that
\betagin{equation*}
\big\| \wt\zeta^i_j \boldsymbol{F} \big\|_{\mathcal{Y}_q(\Omegaega^i_j)}
\leq (1+2c_0) \|\boldsymbol{F}\|_{\mathcal{Y}_q(\Omegaega \cap B^i_j)},
\,\,\, \thetaxt{for all}\,\,\, \boldsymbol{F} \in \mathcal{Y}_q(\Omegaega)
\,\,\, \thetaxt{and}\,\,\,i=0,1,2 .
\end{equation*}
\subsection{Construction of a parametrix}
Now we define $F_{\lambdambda}(\boldsymbol{f},\boldsymbol{g},k):= ( \boldsymbol{f},\lambdambda^{1\slash 2}\boldsymbol{g}, \boldsymbol{g}, k)^{\top}$
for any $(\boldsymbol{f},\boldsymbol{g},k) \in Y_q(\Omegaega)$ and $\lambdambda \in \Gammamma_{\varepsilon,\lambdambda_0,\zeta},$
and then introduce
\betagin{align*}
\boldsymbol{v}&=\mathcal{A}_{\mathfrak{p}}(\lambdambda) F_{\lambdambda}(\boldsymbol{f}, \boldsymbol{g},k) :=\sum_{i\in\{0,1,2\}} \sum_{j\in \mathbb{N}} \zeta^i_j \boldsymbol{v}^i_j \\
&= \sum_{j\in \mathbb{N}} \zeta^0_j \mathcal{A}_0(\lambdambda,\Omegaega^0_j)
\big( \wt\zeta^0_j\boldsymbol{f},\lambdambda^{1\slash 2}\wt\zeta^0_j\boldsymbol{g},\wt\zeta^0_j \boldsymbol{g},\wt\zeta^0_j k \big)
+\sum_{i\in\{1,2\}} \sum_{j\in \mathbb{N}} \zeta^i_j \mathcal{A}_i(\lambdambda,\Omegaega^i_j) \wt\zeta^i_j\boldsymbol{f},\\
h &= \mathcal{H}_{\mathfrak{p}}(\lambdambda) F_{\lambdambda}(\boldsymbol{f}, \boldsymbol{g},k) := \sum_{j\in \mathbb{N}} \zeta^0_j h^0_j
=\sum_{j\in \mathbb{N}} \zeta^0_j \mathcal{H}_0(\lambdambda,\Omegaega^0_j)
\big( \wt\zeta^0_j\boldsymbol{f},\lambdambda^{1\slash 2}\wt\zeta^0_j\boldsymbol{g},\wt\zeta^0_j \boldsymbol{g},\wt\zeta^0_j k \big).
\end{align*}
By Proposition \ref{prop:sum}, \eqref{es:fip_2} and \eqref{es:RR_loc_1}, we have $\boldsymbol{v} \in H^2_q(\Omegaega)^N$ and $h\in H^3_q(\Omegaega).$
Moreover, according to \eqref{eq:def_gamma_ij}, \eqref{eq:RR_CNS_loc_0}, \eqref{eq:RR_CNS_loc_1} and \eqref{eq:RR_CNS_loc_2},
$\boldsymbol{v}$ and $h$ satisfy
\betagin{equation}\lambdabel{eq:RR_CNS_para}
\left\{\betagin{aligned}
& \lambdambda \boldsymbol{v} -\gammamma_{1}^{-1}\Di \big( \mathbb{S}(\boldsymbol{v}) + \zeta \gammamma_3 \di \boldsymbol{v} \mathbb{I} \big)
= \boldsymbol{f} - \mathcal{V}_1 (\lambdambda) F_{\lambdambda}(\boldsymbol{f},\boldsymbol{g},k) &&\quad\hbox{in}\quad \Omegaega,\\
& \big( \mathbb{S}(\boldsymbol{v}) +\zeta \gammamma_3 \di \boldsymbol{v} \mathbb{I} \big)\boldsymbol{n}_{\Gammamma_{0}}
+\sigmagma (m-\Deltalta_{\Gammamma_{0}})h \,\boldsymbol{n}_{\Gammamma_{0}} = \boldsymbol{g}- \mathcal{V}_2(\lambdambda) F_{\lambdambda}(\boldsymbol{f},\boldsymbol{g},k) &&\quad\hbox{on}\quad \Gammamma_0, \\
&\lambdambda h - \boldsymbol{v} \cdot \boldsymbol{n}_{\Gammamma_0} = k &&\quad\hbox{on}\quad \Gammamma_0,\\
&\boldsymbol{v} = \boldsymbol{0} &&\quad\hbox{on}\quad \Gammamma_1,\\
\end{aligned}\right.
\end{equation}
where the operators $\mathcal{V}_1 (\lambdambda)$ and $\mathcal{V}_2(\lambdambda)$ are given by
\betagin{align*}
\mathcal{V}_1(\lambdambda)F_{\lambdambda} (\boldsymbol{f},\boldsymbol{g},k)
:= &\sum_{i\in \{0,1,2\}}\sum_{j\in \mathbb{N}} (\gammamma^i_{j1})^{-1} \boldsymbol{i}g( \Di \mathbb{S}(\zeta^i_j \boldsymbol{v}^i_j) - \zeta^i_j \Di \mathbb{S}(\boldsymbol{v}^i_j) \\
& + \zeta \Di \big( \gammamma^i_{j3} \di (\zeta^i_j \boldsymbol{v}^i_j) \mathbb{I} \big)
- \zeta \zeta^i_j \Di \big(\gammamma^i_{j3}\di \boldsymbol{v}^i_j \mathbb{I} \big) \boldsymbol{i}g),\\
\mathcal{V}_2(\lambdambda) F_{\lambdambda}(\boldsymbol{f},\boldsymbol{g},k)
:=&-\sum_{j\in \mathbb{N}} \boldsymbol{i}g( \mathbb{S}(\zeta^0_j \boldsymbol{v}^0_j) - \zeta^0_j \mathbb{S}(\boldsymbol{v}^0_j)
+ \zeta \gammamma^0_{j3} \big( \di (\zeta^0_j \boldsymbol{v}^0_j) -\zeta^0_j \di \boldsymbol{v}^0_j \big) \mathbb{I}\boldsymbol{i}g) \boldsymbol{n}^0_j \\
& + \sum_{j\in \mathbb{N}} \big( \Deltalta_{\Gammamma^0_j} (\zeta^0_j h^0_j) -\zeta^0_j \Deltalta_{\Gammamma^0_j} h^0_j \big) \boldsymbol{n}^0_j.
\end{align*}
By the definitions of $\mathcal{A}_{\mathfrak{p}}(\lambdambda)$ and $\mathcal{H}_{\mathfrak{p}}(\lambdambda),$ we can infer from Proposition \ref{prop:sum}, \eqref{es:Rbdd_loc} and \eqref{es:fip_2} that,
\betagin{align*}
\mathcal{A}_{\mathfrak{p}}(\lambdambda) & \in
{\rm Hol}\,\boldsymbol{i}g(\Gammamma_{\varepsilon,\lambdambda_0,\zeta}; \mathcal{L}\big(\mathcal{Y}_q(\Omegaega);H^2_q(\Omegaega)^N \big) \boldsymbol{i}g),\\
\mathcal{H}_{\mathfrak{p}}(\lambdambda) & \in
{\rm Hol}\,\boldsymbol{i}g(\Gammamma_{\varepsilon,\lambdambda_0,\zeta}; \mathcal{L}\big(\mathcal{Y}_q(\Omegaega);H^3_q(\Omegaega) \big) \boldsymbol{i}g).
\end{align*}
In addition, there exists some constant $C$ depending solely on $q,L$ and $c_0$ such that
\betagin{gather}\lambdabel{es:R-bdd_para}
\mathcal{R}_{\mathcal{L}\big(\mathcal{Y}_q(\Omegaega); H^{2-k}_q(\Omegaega)^N \big)}
\boldsymbol{i}g( \boldsymbol{i}g\{ (\tau \partial_{\tau})^{\ell}\big( \lambdambda^{k\slash 2}\mathcal{A}_{\mathfrak{p}}(\lambdambda)\big) : \lambdambda \in
\Gammamma_{\varepsilon,\lambdambda_0,\zeta} \boldsymbol{i}g\}\boldsymbol{i}g) \leq C r_b,\\ \nonumber
\mathcal{R}_{\mathcal{L}\big(\mathcal{Y}_q(\Omegaega); H^{3-k'}_q(\Omegaega) \big)}
\boldsymbol{i}g( \boldsymbol{i}g\{ (\tau \partial_{\tau})^{\ell}\big( \lambdambda^{k'}\mathcal{H}_{\mathfrak{p}}(\lambdambda)\big) : \lambdambda \in
\Gammamma_{\varepsilon,\lambdambda_0,\zeta} \boldsymbol{i}g\}\boldsymbol{i}g) \leq C r_b,
\end{gather}
for any $\ell,k'=0,1,$ $k=0,1,2,$ and $\tau := \Im \lambdambda.$
\smallbreak
Here we just prove the estimates of $\mathcal{H}_{\mathfrak{p}}(\lambdambda)$ for instance. Take any $N_0 \in \mathbb{N},$ $\mathfrak{a}=1,\dots,N_0,$ $\boldsymbol{F}_{\mathfrak{a}} \in \mathcal{Y}_q(\Omegaega),$ the Rademacher functions $r_{\mathfrak{a}}.$ Then Proposition \ref{prop:sum} gives us that
\betagin{equation*}
\boldsymbol{i}g\| \sum_{\mathfrak{a}=1}^{N_0} r_{\mathfrak{a}} (\cdot) (\tau \partial_\tau )^{\ell} \big(\lambdambda^{k'}_{\mathfrak{a}} \mathcal{H}_{\mathfrak{p}}(\lambdambda_{\mathfrak{a}})\boldsymbol{F}_{\mathfrak{a}} \big) \boldsymbol{i}g\|_{H^{3-k'}_q(\Omegaega)}^q \leq C \sum_{j\in \mathbb{N}}
\boldsymbol{i}g\| \sum_{\mathfrak{a}=1}^{N_0} r_{\mathfrak{a}} (\cdot) (\tau \partial_\tau )^{\ell}
\big(\lambdambda^{k'}_{\mathfrak{a}} \mathcal{H}_0(\lambdambda_{\mathfrak{a}},\Omegaega^0_j)(\wt\zeta^0_j\boldsymbol{F}_{\mathfrak{a}}) \big),
\boldsymbol{i}g\|_{H^{3-k'}_q(\Omegaega^0_j)}^q
\end{equation*}
for some constant $C =C (q,L,c_0).$
Combining above bound, \eqref{es:Rbdd_loc}, Minkowski inequalities and \eqref{es:fip_2} yields that
\betagin{align*}
\boldsymbol{i}g\| \sum_{\mathfrak{a}=1}^{N_0} r_{\mathfrak{a}} (\cdot) (\tau \partial_\tau )^{\ell}
\big(\lambdambda^{k'}_{\mathfrak{a}} \mathcal{H}_{\mathfrak{p}}(\lambdambda_{\mathfrak{a}})\boldsymbol{F}_{\mathfrak{a}} \big) \boldsymbol{i}g\|_{ L_q([0,1]; H^{3-k'}_q(\Omegaega))}
& \leq C r_b \boldsymbol{i}g( \sum_{j\in \mathbb{N}}
\boldsymbol{i}g\| \sum_{\mathfrak{a}=1}^{N_0} r_{\mathfrak{a}} (\cdot) \boldsymbol{F}_{\mathfrak{a}} \boldsymbol{i}g\|_{L_q([0,1];\mathcal{Y}_q(\Omegaega \cap B^0_j)) }^q \boldsymbol{i}g) ^{1\slash q}\\
& \leq C r_b \boldsymbol{i}g\| \sum_{\mathfrak{a}=1}^{N_0} r_{\mathfrak{a}} (\cdot) \boldsymbol{F}_{\mathfrak{a}} \boldsymbol{i}g\|_{L_q([0,1];\mathcal{Y}_q(\Omegaega))}.
\end{align*}
This completes the study of $\mathcal{H}_{\mathfrak{p}}(\lambdambda).$
Next we denote $\mathcal{V}(\lambdambda):= \big( \mathcal{V}_1(\lambdambda), \mathcal{V}_2(\lambdambda),0 \big)^{\top}$ and claim
\betagin{equation} \lambdabel{es:key_Rbd_FV}
\mathcal{R}_{\mathcal{L}(\mathcal{Y}_q(\Omegaega))}
\big\{ (\tau \partial_{\tau})^{\ell}F_{\lambdambda}\mathcal{V}(\lambdambda):\lambdambda\in \Gammamma_{\varepsilon,\lambdambda_0,\zeta}\big\}
\leq C r_b \lambdambda_0^{-1\slash 2} ,
\end{equation}
whose proof is postponed to the next subsection.
By choosing $\lambdambda_0$ in \eqref{es:key_Rbd_FV} large enough, we have
\betagin{equation}\lambdabel{es:FV_1}
\|F_{\lambdambda} \mathcal{V}(\lambdambda) F_{\lambdambda}(\boldsymbol{f},\boldsymbol{g},k)\|_{\mathcal{Y}_q(\Omegaega)}
\leq 1\slash 2 \|F_{\lambdambda}(\boldsymbol{f},\boldsymbol{g},k)\|_{\mathcal{Y}_q(\Omegaega)},
\,\,\, \forall \,\, \lambdambda \in \Gammamma_{\varepsilon,\lambdambda_0,\zeta}.
\end{equation}
\betagin{equation}\lambdabel{es:FV_2}
\mathcal{R}_{\mathcal{L}(\mathcal{Y}_q(\Omegaega))}
\big\{ (\tau \partial_{\tau})^{\ell}\big(Id - F_{\lambdambda}\mathcal{V}(\lambdambda) \big)^{-1}:
\lambdambda\in \Gammamma_{\varepsilon,\lambdambda_0,\zeta}\big\} \leq 2.
\end{equation}
Thanks to \eqref{es:FV_1}, $Id-\mathcal{V}(\lambdambda)F_{\lambdambda}$ is invertible on the space $Y_{q,\lambdambda}(\Omegaega)$ (cf. $Y_{q,\lambdambda}(\mathbb{R}^N_+)$ in Sect. \ref{sec:bh}). Let us set
\betagin{equation*}
\boldsymbol{v}:= \mathcal{A}_{\mathfrak{p}}(\lambdambda) F_{\lambdambda} \big( Id -\mathcal{V}(\lambdambda)F_{\lambdambda} \big)^{-1}(\boldsymbol{f},\boldsymbol{g},k) , \quad
h:= \mathcal{H}_{\mathfrak{p}}(\lambdambda) F_{\lambdambda} \big( Id -\mathcal{V}(\lambdambda)F_{\lambdambda} \big)^{-1}(\boldsymbol{f},\boldsymbol{g},k).
\end{equation*}
Then $\boldsymbol{v}$ and $h$ satisfy \eqref{eq:RR_CNS_0}. Note the fact that
\betagin{equation*}
F_{\lambdambda} \big( Id -\mathcal{V}(\lambdambda)F_{\lambdambda} \big)^{-1}
=\sum_{j=0}^{\infty} F_{\lambdambda} \big(\mathcal{V}(\lambdambda)F_{\lambdambda}\big)^j
= \big( Id -F_{\lambdambda}\mathcal{V}(\lambdambda) \big)^{-1} F_{\lambdambda}.
\end{equation*}
Then $\mathcal{A}_0(\lambdambda,\Omegaega):= \mathcal{A}_{\mathfrak{p}}(\lambdambda) \big( Id -F_{\lambdambda}\mathcal{V}(\lambdambda) \big)^{-1}$ and
$\mathcal{H}_0(\lambdambda,\Omegaega):= \mathcal{H}_{\mathfrak{p}}(\lambdambda) \big( Id -F_{\lambdambda}\mathcal{V}(\lambdambda) \big)^{-1}$ are desired operators due to
\eqref{es:R-bdd_para} and \eqref{es:FV_2}.
\subsection{Proof of \eqref{es:key_Rbd_FV}}
For any $\boldsymbol{F}= (\boldsymbol{F}^1,\boldsymbol{F}^2,\boldsymbol{F}^3,F^4) \in \mathcal{Y}_q(\Omegaega),$ denote that
\betagin{equation*}
\mathcal{S}^0_j (\lambdambda)\boldsymbol{F}:= \mathcal{A}_0(\lambdambda,\Omegaega^0_j)(\wt \zeta^0_j \boldsymbol{F}), \,\,\,
\mathcal{T}^0_j (\lambdambda)\boldsymbol{F}:= \mathcal{H}_0(\lambdambda,\Omegaega^0_j)(\wt \zeta^0_j \boldsymbol{F}),\,\,\,
\mathcal{S}^i_j (\lambdambda)\boldsymbol{F}:= \mathcal{A}_i(\lambdambda,\Omegaega^i_j)(\wt \zeta^i_j \boldsymbol{F}^1)
\end{equation*}
for $i=1,2.$ Then we have
\betagin{align*}
\mathcal{V}_1(\lambdambda) \boldsymbol{F} =& \sum_{i=0,1,2}\sum_{j\in \mathbb{N}} (\gammamma_{j1}^i)^{-1}
\boldsymbol{i}g( \mathbb{S}\big( \mathcal{S}^i_j (\lambdambda)\boldsymbol{F}\big)
+\zeta \gammamma^i_{j3} \di \big( \mathcal{S}^i_j(\lambdambda)\boldsymbol{F} \big) \mathbb{I} \boldsymbol{i}g)\nabla \zeta^i_j\\
&+ \sum_{i=0,1,2}\sum_{j\in \mathbb{N}} (\gammamma_{j1}^i)^{-1} \Di \boldsymbol{i}g(
\mu \big( \nabla \zeta^i_j \otimes \mathcal{S}^i_j(\lambdambda)\boldsymbol{F} + \mathcal{S}^i_j(\lambdambda)\boldsymbol{F} \otimes \nabla \zeta^i_j \big)
\boldsymbol{i}g) \\
& + \sum_{i=0,1,2}\sum_{j\in \mathbb{N}} (\gammamma_{j1}^i)^{-1} \Di \boldsymbol{i}g(
(\nu-\mu + \zeta \gammamma^i_{j3}) \big( \mathcal{S}^i_j(\lambdambda)\boldsymbol{F} \cdot \nabla \zeta^i_j\big) \mathbb{I} \boldsymbol{i}g),
\end{align*}
\betagin{align*}
\mathcal{V}_2(\lambdambda)\boldsymbol{F}=&- \sum_{j\in \mathbb{N}} \mu
\big( \nabla \zeta^0_j \otimes \mathcal{S}^0_j(\lambdambda)\boldsymbol{F} + \mathcal{S}^0_j(\lambdambda)\boldsymbol{F} \otimes \nabla \zeta^0_j \big) \boldsymbol{n}^0_j\\
&- \sum_{j\in \mathbb{N}} (\nu-\mu + \zeta \gammamma^0_{j3}) \big( \mathcal{S}^0_j(\lambdambda)\boldsymbol{F} \cdot \nabla \zeta^0_j\big) \boldsymbol{n}^0_j\\
& + \sum_{j \in \mathbb{N}} \big( (\wt\Deltalta_{\Gammamma^0_j} \zeta^0_j) \mathcal{T}^0_j (\lambdambda)\boldsymbol{F}
+ 2 \wt\nabla_{\Gammamma^0_j}\zeta^0_j \cdot \wt\nabla_{\Gammamma^0_j}\mathcal{T}^0_j (\lambdambda)\boldsymbol{F} \big) \boldsymbol{n}^0_j.
\end{align*}
where $\wt\nabla_{\Gammamma^0_j} f := \Pi_{\Gammamma^0_j}\nabla f,$ $\Pi_{\Gammamma^0_j} := \mathbb{I} - \boldsymbol{n}^0_j \otimes \boldsymbol{n}^0_j$ and
\betagin{equation*}
\wt \Deltalta_{\Gammamma^0_j} f := \Deltalta f - \tr (\Pi_{\Gammamma^0_j} \nabla (\boldsymbol{n}^0_j)^{\top}) (\boldsymbol{n}^0_j \nabla f) - (\boldsymbol{n}^0_j)^{\top} (\nabla^2 f) \boldsymbol{n}^0_j,
\end{equation*}
for any smooth function $f$ defined near $\Gammamma^0_j.$
In the formula of $\mathcal{V}_2(\lambdambda),$ we have used the fact that
\betagin{equation*}
\nabla_{\Gammamma^0_j} f = \wt \nabla_{\Gammamma^0_j} f, \,\,\,
\Deltalta_{\Gammamma^0_j} f = \wt \Deltalta_{\Gammamma^0_j} f \,\,\, \thetaxt{on} \,\,\, \Gammamma^0_j.
\end{equation*}
Furthermore, \eqref{es:normal_0j} immediately yields that
\betagin{align} \lambdabel{es:cut-off_0j}
\big\|\wt\nabla_{\Gammamma^0_j} \zeta^0_j \big\|_{H^1_{\infty}(\mathbb{R}^N)}
+\big\|\wt\Deltalta_{\Gammamma^0_j} \zeta^0_j \big\|_{L_{\infty}(\mathbb{R}^N)} \leq C_{N} M_2M_3 c_0.
\end{align}
Then thanks to Lemma \ref{lemma:ab_BH}, \eqref{es:Rbdd_loc}, \eqref{es:normal_0j}, \eqref{es:cut-off_0j} and \eqref{es:fip_2}, it is not hard to see that for any $0<\sigmagma_0<1$ and $\ell,\ell'=0,1,$
\betagin{equation} \lambdabel{es:key_Rbd_FV_1}
\mathcal{R}_{\mathcal{L}(\mathcal{Y}_q(\Omegaega);L_q(\Omegaega)^N )}
\big\{ (\tau \partial_{\tau})^{\ell}\mathcal{V}_1(\lambdambda):\lambdambda\in \Gammamma_{\varepsilon,\lambdambda_0,\zeta}\big\}
\leq C r_b\big( \lambdambda_0^{-1\slash 2} + \sigmagma_0^{-N \slash (r-N)} \lambdambda_0^{-1}\big),
\end{equation}
\betagin{equation} \lambdabel{es:key_Rbd_FV_2_1}
\mathcal{R}_{\mathcal{L}(\mathcal{Y}_q(\Omegaega);L_q(\Omegaega)^N )}
\big\{ (\tau \partial_{\tau})^{\ell}\lambdambda^{\ell'\slash 2}\mathcal{V}_2(\lambdambda):\lambdambda\in \Gammamma_{\varepsilon,\lambdambda_0,\zeta}\big\}
\leq C r_b \lambdambda_0^{-1+\ell'\slash 2},
\end{equation}
\betagin{equation} \lambdabel{es:key_Rbd_FV_2_2}
\mathcal{R}_{\mathcal{L}(\mathcal{Y}_q(\Omegaega);H^1_q(\Omegaega)^N )}
\big\{ (\tau \partial_{\tau})^{\ell} \mathcal{V}_2(\lambdambda):\lambdambda\in \Gammamma_{\varepsilon,\lambdambda_0,\zeta}\big\}
\leq C r_b\big( \lambdambda_0^{-1\slash 2} + \sigmagma_0^{-N \slash (r-N)} \lambdambda_0^{-1}\big),
\end{equation}
with some constant $C$ depending on $N,r,q,L,c_0, \mu,\nu,\zeta_0,\rho_1,\rho_2,\rho_3,M_2,M_3.$
Here, for instance, we only consider
$$ \mathcal{V}^{\alphapha}_{2,stk}(\lambdambda) \boldsymbol{F}
:= \sum_{j\in \mathbb{N}} \big( \deltalta_{st} - (\boldsymbol{n}^0_j)_s(\boldsymbol{n}^0_j)_t \big) \partial_{\alphapha} \partial_{t}(\boldsymbol{n}^0_j)_s (\boldsymbol{n}^0_j)_k (\partial_k \zeta^0_j)
\mathcal{T}^0_j (\lambdambda)\boldsymbol{F} \boldsymbol{n}^0_j, $$
with $\alphapha,s,t,k=1,\dots,N,$ arising from the study of
$\sum_{j \in \mathbb{N}} (\partial_{\alphapha} \wt\Deltalta_{\Gammamma^0_j} \zeta^0_j) \mathcal{T}^0_j (\lambdambda)\boldsymbol{F} \boldsymbol{n}^0_j$ in \eqref{es:key_Rbd_FV_2_2}.
For any $N_0\in \mathbb{N},$ $\mathfrak{a}=1,\dots,N_0,$ $\boldsymbol{F}_{\mathfrak{a}} \in \mathcal{Y}_q(\Omegaega)$ and the Rademacher functions $r_{\mathfrak{a}},$
Proposition \ref{prop:sum}, Lemma \ref{lemma:ab_BH} and \eqref{es:normal_0j} imply that
\betagin{align*}
\boldsymbol{i}g\| \sum_{\mathfrak{a}=1}^{N_0} r_{\mathfrak{a}} (\cdot) (\tau \partial_\tau )^{\ell} \big(\mathcal{V}^{\alphapha}_{2,stk}(\lambdambda_{\mathfrak{a}})\boldsymbol{F}_{\mathfrak{a}} \big) \boldsymbol{i}g\|_{L_q(\Omegaega)}
\leq C_{N,q,r,L}c_0 \boldsymbol{i}g(\boldsymbol{i}g\|\sum_{\mathfrak{a}=1}^{N_0} r_{\mathfrak{a}} (\cdot) (\tau \partial_\tau )^{\ell} \nabla \mathcal{T}^0_j (\lambdambda_{\mathfrak{a}})\boldsymbol{F}_{\mathfrak{a}}\boldsymbol{i}g\|_{\ell_q(\mathbb{N};L_{q}(\Omegaega^0_j))}\\
+ (\sigmagma_0^{-1} M_2M_3)^{N\slash (r-N)} \boldsymbol{i}g\|\sum_{\mathfrak{a}=1}^{N_0} r_{\mathfrak{a}} (\cdot) (\tau \partial_\tau )^{\ell}\mathcal{T}^0_j (\lambdambda_{\mathfrak{a}})\boldsymbol{F}_{\mathfrak{a}}\boldsymbol{i}g\|_{\ell_q(\mathbb{N};L_{q}(\Omegaega^0_j))}
\boldsymbol{i}g),
\end{align*}
for any $\sigmagma_0 \in ]0,1[.$
Thus we see from Minkowski's inequalities, \eqref{es:Rbdd_loc} and \eqref{es:fip_2} that,
\betagin{equation*}
\boldsymbol{i}g\| \sum_{\mathfrak{a}=1}^{N_0} r_{\mathfrak{a}} (\cdot) (\tau \partial_\tau )^{\ell} \big(\mathcal{V}^{\alphapha}_{2,stk}(\lambdambda_{\mathfrak{a}})\boldsymbol{F}_{\mathfrak{a}} \big) \boldsymbol{i}g\|_{L_q([0,1];L_q(\Omegaega))}
\leq C r_b \sigmagma_0^{-N \slash (r-N)} \lambdambda_0^{-1}
\boldsymbol{i}g\| \sum_{\mathfrak{a}=1}^{N_0} r_{\mathfrak{a}} (\cdot) \boldsymbol{F}_{\mathfrak{a}} \boldsymbol{i}g\|_{L_q([0,1];\mathcal{Y}_q(\Omegaega))},
\end{equation*}
for some constant $C$ solely depending on $N,r,q,L,c_0,M_2,M_3.$
Finally, put \eqref{es:key_Rbd_FV_1}, \eqref{es:key_Rbd_FV_2_1} and \eqref{es:key_Rbd_FV_2_2} together and we can conclude \eqref{es:key_Rbd_FV} by choosing $\sigmagma_0=1\slash 2.$
\end{document}
|
\begin{document}
\title{Computing quasidegrees of $A$-graded modules}
\author{Roberto Barrera}
\address[]{Department of Mathematics, Texas State University, San Marcos, Texas, 78666 U.S.A.}
\email{[email protected]}
\subjclass{}
\date{}
\begin{abstract}
We describe the main functions of the \texttt{Macaulay2} package \texttt{Quasidegrees}. The purpose of this package is to compute the quasidegree set of a finitely generated $\Z^d$-graded module presented as the cokernel of a monomial matrix. We provide examples with motivation coming from $A$-hypergeometric systems.
\end{abstract}
\maketitle
\section{Introduction}\label{Intro}
Throughout $R=\k[x_1,\ldots,x_n]$ is a $\Z^d$-graded polynomial ring over a field $\k$ and $\mm=\< x_1,\ldots,x_n\>$ denotes the homogeneous maximal ideal in $R$. Let $M=\Oplus_{\beta\in\Z^d} M_\beta$ be a $\Z^d$-graded $R$-module. The \defn{true degree set} of $M$ is
$$\tdeg(M)=\{\beta\in\Z^d\mid M_\beta\neq0\}.$$
The \defn{quasidegree set} of $M$, denoted $\qdeg(M)$, is the Zariski closure in $\C^d$ of $\tdeg(M)$.
The purpose of the \texttt{Macaulay2}\cite{M2} package \texttt{Quasidegrees}\cite{Qd} is to compute the quasidegree set of a finitely generated $\Z^d$-graded module presented as the cokernel of a monomial matrix. By a monomial matrix, we mean a matrix where each entry is either zero or a monomial in $R$. The initial motivation for \texttt{Quasidegrees} was to compute the quasidegree sets of certain local cohomology modules supported at $\mm$ of $\Z^d$-graded $R$-modules so there are some methods in the package specific to local cohomology. Recall that the \defn{ith local cohomology module} of $M$ with support at the ideal $I\subset R$ is the $i$th right derived functor of the left exact $I$-torsion functor
$$\Gamma_I(M)=\{m\in M\mid I^tm=0 \text{ for some }t\in\N\}$$
on the category of $R$-modules.
By the vanishing theorems of local cohomology \cite{CAAG}, the quasidegree sets of the local cohomology modules supported at $\mm$ of $M$ can be seen as measuring how far the module is from being Cohen-Macaulay. From the $A$-hypergeometric systems point of view, the quasidegree set of the non-top local cohomology modules supported at $\mathfrak{m}$ of $R/I_A$, where $I_A$ is the toric ideal associated to $A$ in $R$, determine the parameters $\beta$ where the $A$-hypergeometric system $H_A(\beta)$ has rank higher than expected (see Section \ref{hypergeometricsystems}).
\section{Quasidegrees}\label{Quasidegrees}
The main function of \texttt{Quasidegrees} is \texttt{quasidegrees}, which computes the quasidegree set of a module that is presented by a monomial matrix.
We use the idea of standard pairs of monomial ideals to compute the quasidegree set of a $\Z^d$-graded $R$-module. Given a monomial $x^\bu$ and a subset $Z\subset\{x_1,\ldots,x_n\}$, the pair $(x^\bu,Z)$ indexes the monomials $x^\bu\cdot x^\bv$ where $\supp(x^\bv)\subset Z$. A \defn{standard pair} of a monomial ideal $I\subset R$ is a pair $(x^\bu,Z)$ satisfying:
\begin{enumerate}
\item $\supp(x^\bu)\cap Z=\varempty$,
\item all of the monomials indexed by $(x^\bu,Z)$ are outside of $I$,
\item $(x^\bu,Z)$ is maximal in the sense that $(x^\bu,Z)\nsubseteq(x^\bv,Y)$ for any other pair $(x^\bv,Y)$ satisfying the first two conditions.
\end{enumerate}
To compute the quasidegree set of $M$ we first find a monomial presentation of $M$ so that $M$ is the cokernel of a monomial matrix $\phi$. We then compute the standard pairs of the ideals generated by the rows of $\phi$ and to each standard pair we associate the degrees of the corresponding variables. The following algorithm is implemented in \texttt{Quasidegrees}. The input is an $R$-module presented by a monomial matrix $\phi:R^s\rightarrow R^t$. As in \texttt{Macaulay2}, we write the degree of the $k$th factor of $R^t$ next to the $k$th row of the matrix $\phi$.
\begin{algorithm}[H]
\caption{Compute $\qdeg(M)$}
\label{alg1}
\begin{algorithmic}
\REQUIRE $R$-module $M$ presented by monomial matrix \\
\hskip1cm$\phi=\alpha_i[c_{j,k}\bx^{\bu_{j,k}}]:R^s\rightarrow R^t$
\ENSURE $\qdeg(M)$
\STATE $Q=\varempty$
\FOR{ $1\leq k\leq t$}
\STATE $SP=\{\text{standard pairs of }\<c_{k,1}\bx^{\bu_{k,1}},c_{k,2}\bx^{\bu_{k,2}},\ldots,c_{k,s}\bx^{\bu_{k,s}}\>\}$
\STATE $Q=Q\cup\{\deg(\bx^\bu)+\alpha_k+\sum_{x_i\in F}\C\cdot\deg(x_i)\mid(\bx^\bu,Z)\in SP\}$
\ENDFOR
\RETURN Q
\end{algorithmic}
\end{algorithm}
In the implementation of Algorithm \ref{alg1} in \texttt{Macaulay2}, we represent the output as a list of pairs $(\bu,Z)$ with $\bu\in\Q^d$ and $Z\subset\Q^d$ where the pair $(\bu,Z)$ represents the plane
$$\bu+\sum_{\bv\in Z}\C\cdot\bv.$$
The union of these planes over all such pairs in the output is the quasidegree set of $M$.
The following is an example of \texttt{Quasidegrees} computing the quasidegree set of an $R$-module:
\begin{verbatim}
i1 : R=QQ[x,y,z]
o1 = R
o1 : PolynomialRing
i2 : I=ideal(x*y,y*z)
o2 = ideal (x*y, y*z)
o2 : Ideal of R
i3 : M=R^1/I
o3 = cokernel | xy yz |
1
o3 : R-module, quotient of R
i4 : Q = quasidegrees M
o4 = {{0, {| 1 |}}, {0, {| 1 |, | 1 |}}}
o4 : List
\end{verbatim}
The above example displays a caveat of \texttt{quasidegrees} in that there may be some redundancies in the output. By a redundacy, we mean when one plane in the output is contained in another. The redundancy above is clear:
$$\qdeg(\k[x,y,z]/\langle xy,yz\rangle)=\C=\{z_1+z_2\in\C\mid z_1,z_2\in\C\}.$$
The function \texttt{removeRedundancy} gets rid of redundancies in the list of planes:
\begin{verbatim}
i5 : removeRedundancy Q
o5 = {{0, {| 1 |, | 1 |}}}
o5 : List
\end{verbatim}
\section{Quasidegrees and hypergeometric systems}\label{hypergeometricsystems}
In this section, we discuss the motivation for \texttt{Quasidegrees} and the methods in \texttt{Quasidegrees} that aid us in our studies. Let $A=[a_1~a_2~\cdots~a_n]$ be an integer $(d\times n)$-matrix with $\Z A=\Z^d$ and such that the cone over its columns is pointed. There is a natural $\Z^d$-grading of $R$ by the columns of $A$ given by $\deg(x_j)=a_j$, the $j$th column of $A$. A module that is homogeneous with respect to this grading is said to be {\it A-graded}. By the assumptions on $A$, $R$ is positively graded by $A$, that is, the only polynomials of degree 0 are the constants. Given such a matrix $A$ and a polynomial ring $R$ in $n$ variables, the method \texttt{toGradedRing} gives $R$ an $A$-grading. For example, let $A=\left(\begin{smallmatrix}1&1&1&1&~1\\0&0&1&1&~0\\0&1&1&0&-2\end{smallmatrix}\right)$. We make the $A$-graded polynomial ring $\Q[x_1,x_2,x_3,x_4,x_5]$ :
\begin{verbatim}
i6 : A=matrix{{1,1,1,1,1},{0,0,1,1,0},{0,1,1,0,-2}}
o6 = | 1 1 1 1 1 |
| 0 0 1 1 0 |
| 0 1 1 0 -2 |
3 5
o6 : Matrix ZZ <--- ZZ
i7 : R=QQ[x_1..x_5]
o7 = R
o7 : PolynomialRing
i8 : R=toGradedRing(A,R)
o8 = R
o8 : PolynomialRing
i9 : describe R
o9 = QQ[x , x , x , x , x , Degrees => {{1}, {1}, {1}, {1},
1 2 3 4 5 {0} {0} {1} {1}
{0} {1} {1} {0}
-------------------------------------------------------
{1 }}, Heft => {1, 2:0}, MonomialOrder =>
{0 }
{-2}
-------------------------------------------------------
{MonomialSize => 32}, DegreeRank => 3]
{GRevLex => {5:1} }
{Position => Up }
\end{verbatim}
The \defn{toric ideal associated to} $A$ in $R$ is the binomial ideal
$$I_A=\langle\bx^\bu-\bx^\bv:A\bu=A\bv\rangle.$$
The method \texttt{toricIdeal} computes the toric ideal associated to $A$ in the ring $R$. We continue with the $A$ and $R$ from the above example and compute the toric ideal $I_A$ associated to $A$ in $R$:
\begin{verbatim}
i10 : I=toricIdeal(A,R)
2 2 2 3
o10 = ideal (x x - x x , x x - x x , x x - x x x , x -
1 3 2 4 1 4 3 5 1 4 2 3 5 1
-------------------------------------------------------
2
x x )
2 5
o10 : Ideal of R
\end{verbatim}
We now introduce $A$-hypergeometric systems. Given a matrix $A\in\Z^{d\times n}$ as above and a $\beta\in\C^d$, the \defn{A-hypergeometric system with parameter $\beta\in\C^d$} \cite{SST}, denoted $H_A(\beta)$, is the system of partial differential equations:
\begin{align*}
&\frac{\partial^{|\bv|}}{\partial\bx^\bv}\phi(\bx)=\frac{\partial^{|\bu|}}{\partial\bx^\bu}\phi(\bx)\text{ for all $\bu,\bv$, $A\bu=A\bv$}\\
&\sum_{j=1}^n a_{ij}x_j\frac{\partial}{\partial x_j}\phi(\bx)=\beta_i\phi(\bx),\text{ for }i=1,\ldots, d.
\end{align*}
Such systems are sometimes called \defn{GKZ-hypergeometric systems}. The function \texttt{gkz} in the \texttt{Macaulay2} package \texttt{Dmodules} computes this system as an ideal in the Weyl algebra. The \defn{rank} of $H_A(\beta)$ is
{\small
$$\rank(H_A(\beta))=\dim_\C\left\{\begin{array}{l}\text{germs of holomorphic solutions of $H_A(\beta)$}\\\text{near a generic nonsingular point}\end{array}\right\}.$$}
The function \texttt{holonomicRank} in \texttt{Dmodules} computes the rank of an $A$-hypergeometric system. In general, rank is not a constant function of $\beta$. Denote $\vol(A)$ to be $d!$ times the Euclidean volume of $\conv(A\cup\{0\})$ the convex hull of the columns of $A$ and the origin in $\R^d$. The following theorem gives the parameters $\beta$ for which $\rank(H_A(\beta))$ is higher than expected:
\begin{theorem}\label{rankthm}\cite{MMW}
Let $H_A(\beta)$ be an $A$-hypergeometric system with parameter $\beta$. If $\beta\in\qdeg(\bigoplus_{i=0}^{d-1}H_\mm^i(R/I_A))$ then $\rank(H_A(\beta))>\vol(A)$. Otherwise, $\rank(H_A(\beta))=\vol(A)$.
\end{theorem}
Since Theorem \ref{rankthm} was the initial motivation for \texttt{Quasidegrees}, the package has a method \texttt{quasidegreesLocalCohomology} (abbreviated \texttt{qlc}) to compute the quasidegree set of the local cohomology modules $H_\mm^i(R/I_A)$. If the input is an integer $i$ and the $R$-module $R/I_A$, then the method computes $\qdeg(H^i_\mm(R/I_A))$. If the input is only the module $R/I_A$, the method computes the quasidegree set in Theorem \ref{rankthm}.
We use graded local duality to compute the local cohomology modules of a finitely generated $A$-graded $R$-module supported at the maximal ideal $\mm$:
\begin{theorem}\label{GLC}(Graded local duality \cite{BH98,M01})
Given an $A$-graded $R$-module $M$, there is an $A$-graded vector space isomorphism
$$\Ext_R^{n-i}(M,R)_\alpha\cong \Hom_\k(H_\mm^i(M)_{-\alpha-\veps_A},\k)$$
where $\mm=\langle x_1,\ldots,x_n\rangle$ and $\veps_A=\sum_{j=1}^n a_j$.
\end{theorem}
The algorithm implemented for \texttt{quasidegreesLocalCohomology} is essentially Algorithm \ref{alg1} applied to the $\Ext$-modules of $M$ with the additional twist of $\varepsilon_A$ coming from local duality. For our purposes, we exploit the fact that the higher syzygies of $R/I_A$ are generated by monomials in $R^m$ (see \cite{CCA}, Chapter 9).
Continuing our running example, we use \texttt{quasidegreesLocalCohomology} to compute the quasidegree set of $\bigoplus_{i=0}^{d-1}H_\mm^i(R/I_A)$:
\begin{verbatim}
i11 : M=R^1/I
o11 = cokernel | x_1x_3-x_2x_4 x_1x_4^2-x_3^2x_5
x_1^2x_4-x_2x_3x_5 x_1^3-x_2^2x_5 |
1
o11 : R-module, quotient of R
i12 : quasidegreesLocalCohomology M
o12 = {{| 0 |, {| 1 |}}}
| 0 | | 0 |
| 1 | | -2 |
o12 : List
\end{verbatim}
Thus
\begin{equation}\label{plane}
\qdeg\left(\bigoplus_{i=0}^{d-1}H_\mm^i(R/I_A)\right)=\left[\begin{smallmatrix}0\\0\\1\end{smallmatrix}\right]+\C\cdot\left[\begin{smallmatrix}\phantom{-}1\\\phantom{-}0\\-2\end{smallmatrix}\right].\end{equation}
As a check, we use the methods \texttt{gkz} and \texttt{holonomicRank} from the package \texttt{Dmodules} to compute $\rank(H_A(0))$ and $\rank(H_A(\beta))$ for two different $\beta$ in (\ref{plane}) and demonstrate a rank jump:
\begin{verbatim}
i13 : holonomicRank gkz(A,{0,0,0}) -- vol A in this case
o13 = 4
i14 : holonomicRank gkz(A,{0,0,1})
o14 = 5
i15 : holonomicRank gkz(A,{3/2,0,-2})
o15 = 5
\end{verbatim}
\end{document}
|
\begin{equation}gin{document}
\nablatle{ $\varepsilon$-regularity criteria in anisotropic Lebesgue spaces and Leray's self-similar solutions to
the 3D Navier-Stokes equations }
\author{Yanqing Wang\fracootnote{ Department of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, Henan 450002, P. R. China Email: [email protected]},\;~Gang Wu\fracootnote{School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, P. R. China Email: [email protected]} ~~and ~\,
Daoguo Zhou\fracootnote{
College of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo, Henan 454000, P. R. China Email:
[email protected] }}
\date{}
\title{ $ arepsilon$-regularity criteria in anisotropic Lebesgue spaces and Leray's self-similar solutions to
the 3D Navier-Stokes equations }
\begin{equation}gin{abstract}
In this paper,
we establish some $\varepsilon$-regularity criteria in anisotropic Lebesgue spaces for suitable weak solutions to the 3D Navier-Stokes equations as follows: \begin{equation}\begin{aligned} &\limsup\limits_{\varrho\rightarrow0}
\varrho^{1-\frac{2}{p}-\mathrm{div}um\limits^{3}_{j=1}\frac{1}{q_{j}}}
\|u\|_{L_{t}^{p}L^{\overrightarrow{q}}_{x}(Q(\varrho))} \leq\varepsilon,
~~\frac{2}{p}+\mathrm{div}um\limits^{3}_{j=1}\frac{1}{q_{j}} \wred{\leq2}~~~~~\textxt{with}~q_{j} > 1;\\&
\Delta bel{wwwolf0}
\mathrm{div}up_{-1\leq t\leq0}\|u\|_{L^{\overrightarrow{q}}(B(1))} < \varepsilon,~~\frac{1}{q_{1}}+\frac{1}{q_{2}}+\frac{1}{q_{3}}\wred{<2\quad \textxt{with}\, 1<q_{j}<\infty;}\\&\|u \|_{L_{t}^{p}L^{\overrightarrow{q}}_{x}(Q(1))} +\|\Pi\|_{L^{1}(Q(1))}\leq\varepsilon,
\wred{\quad \frac2p+\mathrm{div}um^{3}_{j=1}\frac{1}{q_{j}} <2 ~~~\textxt{with}~~ 1<q_{j}<\infty,}
\end{aligned}
\end{equation}
which extends the previous results in \cite{[TX],[GKT],[HWZ],[GP],[Wolf1],[CV],[CKN]}.
As an application, in the spirit of \cite{[CW]},
we prove that there
does not exist a nontrivial Leray's backward self-similar solution with profiles in $L^{\overrightarrow{p}}(\mathbb{R}^{3})$ with $\frac{1}{p_{1}}+\frac{1}{p_{2}}+\frac{1}{p_{3}}<2$.
This generalizes the corresponding results of \cite{[NRS],[CW],[Tsai],[GP2]}.
\end{abstract}
\noindent {\bf MSC(2000):}\quad 35B65, 35D30, 76D05 \\\noindent
{\bf Keywords:} Navier-Stokes equations; suitable weak solutions; regularity; self-similar solutions;anisotropic Lebesgue spaces
\mathrm{div}ection{Introduction}
\Delta bel{intro}
\mathrm{div}etcounter{section}{1}\mathrm{div}etcounter{equation}{0}
We study the following incompressible Navier-Stokes equations in three-dimensional space
\begin{equation}\left\{\begin{aligned}\Delta bel{NS}
&u_{t} -\textxt{Div\,}elta u+ u\cdot\nabla
u+\nabla \Pi=0, ~~\mathrm{div}\, u=0,\\
&u|_{t=0}=u_0,
\end{aligned}\right.\end{equation}
where $u $ stands for the flow velocity field, the scalar function $\Pi$ represents the pressure. The
initial velocity $u_0$ satisfies $\textxt{div}\,u_0=0$.
We are concerned with the regularity of suitable weak solutions satisfying local energy inequality to the Navier-Stokes system \eqref{NS}.
A point $(x,t)$ is said to be a regular point if $|u| $ is bounded at some neighbourhood of this point. Otherwise, $(x,t)$ is singular point.
The local energy inequality of \eqref{NS} is due to Scheffer in \cite{[Scheffer1],[Scheffer2]}. In this direction, a milestones result that one dimensional Hausdorff measure of the possible space-time singular \wgr{points set} of suitable weak solutions to the 3D
Navier-Stokes equations is zero was obtained by
Caffarelli, Kohn and Nirenberg in \cite{[CKN]}. This result relies on the following two $\varepsilon$-regularity criteria in \cite{[CKN]} \wgr{for suitable weak solutions to} \eqref{NS}.
One holds at one scale: $(0,0)$ is regular point provided \begin{equation}gin{equation}\Delta bel{CKN} \|u\|_{L^{3}(Q(1))}+\|u\Pi\|_{L^1(Q(1))}+\|\Pi\|_{L_{t}^{5/4}L_{x}^{ 1}(Q(1))}
\leq \varepsilon.
\end{equation}
The other needs infinitely many scales and an alternative assumption of \eqref{CKN} is that \begin{equation}\Delta bel{ckn2}
\limsup_{\varrho\rightarrow0} \varrho^{-\frac{1}{2}} \|\nabla u\|_{L^{2}(Q(\varrho))}
\leq \varepsilon.
\end{equation}
An alternative condition of \eqref{ckn2} is \wred{due to} Tian and Xin \cite{[TX]}
\begin{equation}\Delta bel{tx}
\limsup_{\varrho\rightarrow0} \varrho^{-\frac{2}{3}} \| u\|_{L^{3}(Q(\varrho))}
\leq \varepsilon.
\end{equation}
Gustafson, Kang and Tsai \cite{[GKT]} enhanced \eqref{ckn2} and \eqref{tx} to the following results
\begin{equation}gin{align}\Delta bel{tsai1}
&\limsup_{\varrho\to 0 }\,\, \varrho^{1- \fracrac 2p- \fracrac 3 q}
\|u-\overline{u}_{\varrho}\|_{L_{t}^{p}L_{x}^{q}\wred{(Q(\varrho))}} \leq \varepsilon,\quad
1\leq 2/p +3/q\leq 2,\; 1\leq p, q \leq \infty;\\
&\limsup_{\varrho\to 0 }\,\, \varrho^{2-\fracrac 2p - \fracrac 3 q}
\|\nabla u\|_{L_{t}^{p}L_{x}^{q}} \leq \varepsilon,\quad
2\le 2/p +3/q \le 3, \; 1 \le p,q \wred{\le \infty.}
\Delta bel{gkt2}
\end{align}
Besides suitable weak solutions, there exists other kind of weak solutions equipping energy inequality to the Navier-Stokes equations \eqref{NS}. This kind of weak solutions are called Leray-Hopf weak solutions.
A number of papers have been devoted to the study of regularity of Leray-Hopf weak solutions and many sufficient regularity conditions are established (see for example, \cite{[CMZ1],[CMZ2],[CMZ3],[HLLZ],[CZ],[CZZ],[KZ],[ZP],[NP],[KZ],[Wolf],[Qian],[Zheng],[GCS],[GKS],[CT]}).
In particular, utilizing the anisotropic Lebesgue spaces, Zheng first studied anisotropic regularity criterion in terms of one velocity component in \cite{[Zheng]}. Later, Qian \cite{[Qian]}; Guo, Caggio and Skalak \cite{[GCS]}; Guo, Kucera and Skalak \cite{[GKS]}, further considered regularity condition in anisotropic Lebesgue spaces \wgr{for the Leary-Hopf weak solutions of} system \eqref{NS}. It is worth pointing out that Sobolev-embedding theorem in anisotropic Lebesgue space was established in these works. For the details, see Lemma \ref{zc} in Section \ref{sec2}.
Inspired by recent works \cite{[Zheng],[Qian],[GKS],[GCS]}, we investigate $\varepsilon$-regularity criteria \wgr{for the} 3D Navier-Stokes equations in anisotropic Lebesgue spaces.
Now we formulate our result as follows
\begin{equation}gin{theorem}
\Delta bel{thcw1}Let $(u,\,\Pi)$ be a suitable weak solutions to \eqref{NS} in $Q(\varrho)$. There exists a positive constant $\varepsilon_{1} $ \wred{such that if} \begin{equation}\Delta bel{spe1}\limsup\limits_{\varrho\rightarrow0}
\varrho^{1-\frac{2}{p}-\mathrm{div}um\limits^{3}_{j=1}\frac{1}{q_{j}}}
\|u-\overline{u}_{\varrho}\|_{L_{t}^{p}L^{\overrightarrow{q}}_{x}(Q(\varrho))} \wred{\leq\varepsilon_1},
~~\frac{2}{p}+\mathrm{div}um\limits^{3}_{j=1}\frac{1}{q_{j}} \leq2,~~~~~\textxt{with}~q_{j} > 1;
\end{equation}
\wred{then} $(0,0)$ is a regular point.
\end{theorem}
\begin{equation}gin{remark}
\wred{The valid range of $q$} in \eqref{tsai1} is greater than or equal to $\frac32$. Hence, we extend \eqref{ckn2}-\eqref{tsai1} to anisotropic Lebesgue spaces as well as the range of index of spatial integral.
\end{remark}
\wred{Combining} Theorem
\ref{thcw1} and
the absolute continuity of Lebesgues integral immediately imply the following result, which is of independent interest.
\begin{equation}gin{coro}\Delta bel{coro}
Suppose that $(u,\,\Pi)$ is a suitable weak solution to \eqref{NS}. Then $(0,0)$ is a regular point provided one of the following conditions holds
\begin{equation}gin{align}
(1).~~ &u\in L^{p}_{t}L^{\overrightarrow{q}}_{x}(Q(\varrho)), ~~~~ \textxt{with} ~~~ \frac{2}{p}+\mathrm{div}um^{3}_{j=1}\frac{1}{q_{j}}=1, ~~ 1< q_{j} ; \Delta bel{serin1}\\
(2).~~ & u\in L^{\infty}_{t}L^{\overrightarrow{q}}_{x}(Q(\varrho)),~~
\textxt{and} ~~~~ \|u\|_{\wred{L_t^{\infty}L_x^{\overrightarrow{q}}}(Q(\varrho))}\leq\varepsilon~~~~ \textxt{with} ~~~ \mathrm{div}um^{3}_{j=1}\frac{1}{q_{j}}=1, ~~ \wred{1< q_{j}.} \Delta bel{serin2}
\end{align}
\end{coro}
\begin{equation}gin{remark}
As in \cite{[GKT]}, one can examine that weak solutions satisfying \eqref{serin1} or \eqref{serin2} with $q_{j}>2$ \wred{are suitable weak solutions}. Therefore, this generalizes Serrin's
classical work \cite{[Serrin]}.
\end{remark}
\begin{equation}gin{remark}
After we finished the main part of this paper, we learnt that
an interesting work involving well-posedness \wred{of \eqref{NS} with initial data} in $L^{\overrightarrow{q}}(\mathbb{R}^{3}) $ was established by Phan \cite{[Phan]}, where $\overrightarrow{q}$ meets $\mathrm{div}um\limits_{j=1}^3\frac{1}{q_{j}}=1 $ with $q_{j}>2$.
\end{remark}
Next we \wgr{turn our attention} to the $\varepsilon$-regularity criteria at one scale in the type of \eqref{CKN}. In particular,
Choi and Vasseur \cite{[CV]},
Guevara and Phuc \cite{[GP]} improved \eqref{CKN} to
\begin{equation}gin{equation}\Delta bel{CVGP}
\|u\|_{L_{t}^{ \infty}L_{x}^{2 }(Q(1))} + \|\nabla u\|_{L^{2}(Q(1))} +\|\Pi\|_{L^{1}(Q(1))} \leq \varepsilon.
\end{equation}
Recently,
Guevara and Phuc \cite{[GP]} found that \eqref{CKN} can be replaced by the follows
\begin{equation}\Delta bel{GP}
\|u\|_{L_{t}^{2p}L_{x}^{ 2q} (Q(1))}+\|\Pi\|_{L_{t}^{p}L_{x}^{ q}(Q(1))}\leq \varepsilon,
~~~\frac2p+\frac3q=\frac72
~~~\textxt{with}~1\leq p\leq2.\end{equation}
Authors in \cite{[HWZ]} further extended \eqref{GP} to
\begin{equation}\Delta bel{opt}
\|u\|_{L_{t}^{p}L_{x}^{q}(Q(1))}+\|\Pi\|_{L^{1}(Q(1))}\leq\varepsilon,~~1\leq \frac2p+\frac3q <2, 1\leq p,\,q\leq\infty.
\end{equation}
Very recently, an alternative proof of \eqref{opt} was presented by Dong and Wang \cite{[DW]}. Moreover, for a short summary on $\varepsilon$-regularity criteria at one scale we refer the reader to \cite{[HWZ]} and references therein.
The second result in this paper concerns $\varepsilon$-regularity criteria in anisotropic Lebesgue spaces at one scale, which generalizes the corresponding results in \eqref{opt}.
\begin{equation}gin{theorem}\Delta bel{anatones}
Let the pair $(u, \Pi)$ be a suitable weak solution to the 3D Navier-Stokes system \eqref{NS} in $Q(1)$.
There exists an absolute positive constant $\varepsilon$
such that if the pair $(u,\Pi)$ satisfies \begin{equation}gin{align}\Delta bel{anatonen}\|u &\|_{L_{t}^{p}L^{\overrightarrow{q}}_{x}(Q(1))} +\|\Pi\|_{L^{1}(Q(1))}\leq\varepsilon,\end{align}
\wgr{where}
\begin{equation}gin{align}&\frac2p+\mathrm{div}um^{3}_{j=1}\frac{1}{q_{j}} <2 ~~~\textxt{with}~~ 1<q_{j}<\infty,\Delta bel{partragnge}\end{align}
\wgr{then} $u\in L^{\infty}(Q(1/2)).$
\end{theorem}
\begin{equation}gin{remark}
Theorem \ref{anatones}
extends the recent results obtained by Guevara and Phuc in \cite{[GP]} and \cite{[HWZ]}.
\end{remark}
Very recently, by means of $\varepsilon$-regularity criteria at one scale just in terms of velocity filed,
Chae and Wolf \cite{[CW]} considered Leray's backward self-similar solutions to the Navier-Stokes equations with the \wred{profile in $L^{p}(\mathbb{R}^{3})$} ($p>\frac32$). The
backward self-similar singular solutions to the Navier-Stokes was \wred{introduced by Leray}, who was the first to observe that one can construct singular solutions of the Navier-Stokes equations via
backward self-similar solutions.
The pair $(u,\Pi)$ \wred{is said} to be backward self-similar solutions if
$(u,\Pi)$ satisfy, for $a>0, T\in \mathbb{R},$
\begin{equation}\begin{aligned}
&u(x,t)=\wred{\frac{1}{\mathrm{div}qrt{2a(T-t)}}U\bigg(\frac{x}{\mathrm{div}qrt{2a(T-t)}}\bigg)}, \Delta bel{learysb}\\
&\Pi(x,t)=\wred{\frac{1}{ 2a(T-t)}P\bigg(\frac{x}{\mathrm{div}qrt{2a(T-t)}}\bigg)},
\end{aligned}\end{equation}
where $U=(U_{1},U_{2},U_{3})$ and $P$ are defined in $\mathbb{R}^{3}$
and the pair $(u(x,t),\Pi(x,t))$ is defined in $\mathbb{R}^{3}\nablames(-\infty,T)$.
We obtain a singular solution at $t=T$ if \wred{$U\neq0$ and}
\begin{equation} \Delta bel{SNS}
-\textxt{Div\,}elta U+a U+a(y\cdot \nabla)U+U\cdot\nabla U+ \nabla P=0 , ~~\mathrm{div}\, U=0, ~~y\in \mathbb{R}^{3}.
\end{equation}
In \wred{this direction}, \wred{the breakthrough} was made by Ne\v{c}as, R\r{a}u\v{z}i\v{c}ka and \v{S}ver\'{a}k in \cite{[NRS]}, where they prove that Leary's backward self-similar solutions is trivial under $U\in L^{3}(\mathbb{R}^{3})$.
Subsequently, Tsai \cite{[Tsai]} show that
the solution $U\in L^{p}( \mathbb{R}^{3})$ with $3<p<\infty$ in system \eqref{SNS} is zero
and
the solution $U\in L^{\infty}(\mathbb{R}^{3})$ in system \eqref{SNS} is constant.
Very recently, Guevara and Phuc \cite{[GP2]},Chae and Wolf \cite{[CW]} show that there does not exist a nontrivial solutions of \eqref{SNS} under more general assumptions. To the knowledge of the authors, we summarize the known results concerning Leary's backward self-similar solutions with profiles in
isotropic Lebesgue spaces $L^{p}(\mathbb{R}^{3})$.
\begin{equation}gin{center}\begin{equation}gin{tabular}{|p{2cm}|p{2.5cm}|p{3.5cm}|p{2cm}|p{2cm}|}
\hline
\centering $\frac{3}{2}<p$ & \centering $\frac{12}{5}<p<6$& \centering$ p=3$ & \centering $3<p<\infty$ & $p=\infty$ \\ \hline
\centering$U=0$&\centering$U=0$&\centering$U=0$&\centering$U=0$ &$U=cons$\\ \hline
Chae and Wolf \cite{[CW]}& Guevara and Phuc \cite{[GP2]}& Ne\v{c}as, R\r{a}u\v{z}i\v{c}ka and \v{S}ver\'{a}k \cite{[NRS]} & Tsai \cite{[Tsai]} & Tsai \cite{[Tsai]}
\\ \hline
\end{tabular}\end{center}
We shall investigate Leray's backward self-similar solutions to the Navier-Stokes equations with the profile in anisotropic Lebesgue spaces. More precisely, we have the following statement.
\begin{equation}gin{theorem}\Delta bel{[Leraybackself]}
Let the pair $(U,P) \in C^{\infty}(\mathbb{R}^{3})^{3}\nablames C^{\infty}(\mathbb{R}^{3})$ be a solutions to \eqref{SNS}. Assume that, for $ 1<p_{j}<\infty $,
\begin{equation}\Delta bel{[Leraybackselfr]}
U\in L^{\overrightarrow{p}}(\mathbb{R}^{3})~~
\textxt{with}~~ \frac{1}{p_{1}}+\frac{1}{p_{2}}+\frac{1}{p_{3}}<2,\end{equation}
then $U=0$.
\end{theorem}
\begin{equation}gin{remark}
A special case of (\ref{[Leraybackselfr]}) is $U\in L_{1}^{3/2}L_{2}^{3/2}L_{3}^{p_{3}}(\mathbb{R}^{3})$ \wred{with $p_3>\frac32$}, which is more general than that in \cite{[CW]}.
\end{remark}
\begin{equation}gin{remark}
We would like to mention that Phan recently considered
Liouville type theorems for 3D stationary Navier-Stokes equations in weighted mixed-norm Lebesgue
spaces.
\end{remark}
The proof of Theorem \ref{[Leraybackself]} is based on Chae and Wolf's approach \cite{[CW]}. We present the framework of the argument utilized in \cite{[CW]}.
\begin{equation}gin{enumerate}[-]
\item
First, Chae and Wolf derived the $\varepsilon$-regularity criteria at one scale without the pressure \wred{as below}
\begin{equation} \Delta bel{wolfchae}
\mathrm{div}up_{-1\leq t\leq0}\|u\|_{L^{q}(B(1))} < \varepsilon,~~\frac32<q.
\end{equation}
\item
Second, it follows from \cite[Theorem 1, p.31 ]{[Tsai]} that $U$ belonging
to $L^{\infty}(\mathbb{ R}^{3})\bigcap L^{p}(\mathbb{ R}^{3}) $ with $1\leq p<\infty$ is trivial. Therefore, the key point in this step is to show that $U\in L^{\infty}(\mathbb{ R}^{3})$. It is well-known that $u\in C^{\infty}(\mathbb{R}^{3})$. As a consequence, \wred{it suffices} to prove that the bound of $U(x)$ for $|x|\geq R$ with
suffice large $R$. The decay at infinity of $L^{p}(\mathbb{R}^{3})$ together with the $\varepsilon$-regularity criteria \eqref{wolfchae} yields the desired estimate.
\end{enumerate}
Based on this, we see that it is enough to generalize
\eqref{wolfchae} to extend Leray's self-similar solutions. Therefore, Theorem \ref{[Leraybackself]} turns out to be a consequence of the following result.
\begin{equation}gin{theorem}\Delta bel{the1.5}
Let the pair $(u, \Pi)$ be a suitable weak solution to the 3D Navier-Stokes system \eqref{NS} in $Q(1)$.
There exists an absolute positive constant $\varepsilon$
such that if $u$ satisfies
\begin{equation} \Delta bel{wwwolf}
\mathrm{div}up_{-1\leq t\leq0}\|u\|_{L^{\overrightarrow{q}}(B(1))} < \varepsilon,~~\frac{1}{q_{1}}+\frac{1}{q_{2}}+\frac{1}{q_{3}}<2, 1<q_{j}<\infty.
\end{equation}
then, $u\in L^{\infty}(Q(1/16)).$
\end{theorem}
The proof of \eqref{wolfchae} and \eqref{wwwolf} relies on the local suitable weak solutions via local pressure projection. The local pressure projection and local suitable weak solutions is due to Wolf \cite{[Wolf1]}. The novelty in this concept is that the local energy
inequality
\eqref{wloc1} removed the non-local effect of the pressure term.
It is shown that any suitable weak solutions is local suitable weak solutions in \cite{[CW17]}. We refer the reader to \cite{[JWZ]} and \cite{[WWZ]} for other $\varepsilon$-regularity criteria at one scale without pressure.
This paper is organized as follows. In the second section,
we present the notations and some known fact such as interpolation inequality and Sobolev embedding theorem in anisotropic Lebesgue spaces. In addition, we also recall
the local suitable weak solutions.
Section 3 is devoted to
proving Theorem \ref{thcw1}. In Section 4, we prove Theorems \ref{anatones} involving $\varepsilon$-regularity criteria at one scale. In \wred{final section}, we complete the proof of Theorems \ref{the1.5} concerning $\varepsilon$-regularity criteria at one scale just via velocity filed. This also means Theorem \ref{[Leraybackself]}.
\mathrm{div}ection{Notations and \wred{some known facts}} \Delta bel{sec2}
A function $f$ belongs to the anisotropic Lebesgue spaces $L^{\overrightarrow{q}}_{x}(\Omega)$
if
$$\|f\|_{L^{\overrightarrow{q}}_{x}(\Omega)}=\|f\|_{L_{1}^{q_{1}}L_{2}^{q_{2}}L_{3}^{q_{3}}(\Omega)}=
\Big\|\big\|\|u\|_{L_{1}^{q_{1}}(\{x_1:x\in\Omega\})}\big\|_{L_{2}^{q_{2}}(\{{x_2:x\in\Omega}\})}\Big\|_{L_{3}^{q_{3}} (\{x_3:x\in\Omega\})}<\infty. $$
The study of
anisotropic Lebesgue spaces first appears in
Benedek and Panzone \cite{[BP]}.
\wred{Various topics} on mixed Lebesgue spaces
were established (see e.g. \cite{[Ward],[RRT],[BP]} \wred{and references therein}).
For $p\in [1,\,\infty]$, the notation $L^{p}((0,\,T);X)$ stands for the set of measurable functions on the interval $(0,\,T)$ with values in $X$ and $\|f(t,\cdot)\|_{X}$ belongs to $L^{p}(0,\,T)$.
For simplicity, we write $$\|f\| _{L_{t}^{p}L_{x}^{\overrightarrow{q}}(Q(\varrho))}:=\|f\| _{L^{p}(-\varrho^{2},0;L^{\overrightarrow{q}}(B(\varrho)))}, $$
where $Q(\varrho)=B(\varrho)\nablames ( -\varrho^{2}, 0)$ and $ B(\varrho)$ denotes the ball of center $0$ and radius $\varrho$.
In what follows, for the sake of simplicity of presentation, we define $$\wred{\frac{1}{\overrightarrow{q}}=\fracrac{1}{q_1}+\fracrac{1}{q_{2}}+\fracrac{1}{q_{3}}}. $$
We denote
the average of $f$ on the ball $B(r)$ by
$\overline{f}_{r}$.
Moreover, for the convenience of the reader, we state a fact which will be frequently used below
\begin{equation}\Delta bel{simplefact}
\wgr{\Omega_{1}\mathrm{div}ubseteq\Omega_{2}\mathrm{div}ubseteq\Omega_{3},}
\end{equation}
where
$$\Omega_{1}=\{x:|x|<1\},\Omega_{2}=\{x:|x_1|,|x_2|,|x_3|<1\},
\Omega_{3}=\{x:|x|<\mathrm{div}qrt{3}\}.
$$
The classical Sobolev space $W^{k,p}(\Omega)$ is equipped with the norm $\|f\|_{W^{k,p}(\Omega)}=\mathrm{div}um\limits_{\alpha =0}^{k}\|D^{\alpha}f\|_{L^{p}(\Omega)}$. We denote by $ \dot{H}^{s}$ homogeneous Sobolev spaces with the norm $\|f\|^{2} _{\dot{H}^{s}}= \int_{\mathbb{R}^{3}} |\xi|^{2s}|\hat{f}(\xi)|^{2}d\xi$.
We will also use the summation convention on repeated indices.
$C$ is an absolute
constant which may be different from line to line unless otherwise stated.
Now, for the convenience of readers, we recall the \wred{classical definition of suitable weak solution} to the Navier-Stokes system \eqref{NS}.
\begin{equation}gin{definition}\Delta bel{defi1}
A pair $(u, \,\Pi)$ is called a suitable weak solution to the Navier-Stokes equations \eqref{NS} provided the following conditions are satisfied,
\begin{equation}gin{enumerate}[(1)]
\item $u \in L^{\infty}(-T,\,0;\,L^{2}(\mathbb{R}^{3}))\cap L^{2}(-T,\,0;\,\dot{H}^{1}(\mathbb{R}^{3})),\,\Pi\in
L^{3/2}(-T,\,0;L^{3/2}(\mathbb{R}^{3}));$
\item$(u, ~\Pi)$~solves (\ref{NS}) in $\mathbb{R}^{3}\nablames (-T,\,0) $ in the sense of distributions;
\item$(u, ~\Pi)$ satisfies the following inequality, for a.e. $t\in[-T,0]$,
\begin{equation}gin{align}
&\int_{\mathbb{R}^{3}} |u(x,t)|^{2} \phi(x,t) dx
+2\int^{t}_{-T}\int_{\mathbb{R} ^{3 }}
|\nabla u|^{2}\phi dxds\nonumber\\ \leq& \int^{t}_{-T }\int_{\mathbb{R}^{3}} |u|^{2}
(\partial_{s}\phi+\textxt{Div\,}elta \phi)dxds
+ \int^{t}_{-T }
\int_{\mathbb{R}^{3}}u\cdot\nabla\phi (|u|^{2} +2\Pi)dxds, \Delta bel{loc}
\end{align}
where non-negative function $\phi(x,s)\in C_{0}^{\infty}(\mathbb{R}^{3}\nablames (-T,0) )$.\Delta bel{SWS3}
\end{enumerate}
\end{definition}
\begin{equation}gin{lemma}{\cite{[BP]}}\Delta bel{interi}
Suppose $\Omega\mathrm{div}ubset\mathbb{R}^{3}$ and $1\leq s_{j}\leq r\leq t_{j}\leq\infty$ and
$$
\frac{1}{r}=\frac{\theta}{s_{j}}+\frac{1-\theta}{t_{j}}.
$$
Assume also $f\in L^{\overrightarrow{s}}(\Omega)\cap L^{\overrightarrow{t}}(\Omega)$. Then
$ f\in L^{\overrightarrow{r}}(\Omega)$ and
\begin{equation}\Delta bel{interinequality}
\|f\|_{L^{r}(\Omega) }
\leq \|f\|^{\theta}_{L^{\overrightarrow{s}}(\Omega) }\|f\|^{1-\theta}_{L^{\overrightarrow{t}}(\Omega) }.
\end{equation}
\end{lemma}
\begin{equation}gin{proof}
By the H\"older inequality, we know that
$$\begin{aligned}
\|f\|_{L^{r}(\Omega)}&=\||f|^{r\theta}|f|^{r(1-\theta)}\|^{\frac{1}{r}}_{L^{1}(\Omega)}
\\&\leq \||f|^{r\theta}\|^{\frac{1}{r}}_{L^{\frac{\overrightarrow{s}}{r\theta}}(\Omega)} \||f|^{r(1-\theta)}\|^{\frac{1}{r}}_{L^{\frac{\overrightarrow{t}}{r(1-\theta)}}(\Omega)}
\\&\leq
\|f\|^{\theta}_{L^{\overrightarrow{s}}(\Omega) }\|f\|^{1-\theta}_{L^{\overrightarrow{t}}(\Omega) }.
\end{aligned}$$
\end{proof}
We recall the Sobolev
embedding theorem in anisotropic Lebesgue space in the full three-dimensional space. We refer to \cite{[GCS]} for the proof of the following result.
\begin{equation}gin{lemma}{\cite{[Zheng],[GCS],[Qian],[GKS]}}\Delta bel{zc} Let $q_{1},q_{2},q_{3}\in[2,\infty)$ and \wred{$0\leq \frac{1}{\overrightarrow{q}}-\frac12\leq 1$}. Then there exists a constant $C$ such that
\begin{equation}gin{align}\Delta bel{anis}\|f\|_{\wred{L^{\overrightarrow{q}}(\mathbb{R}^3)}}
&\leq C
\|\partial_{1}f\|^{\frac{q_{1}-2}{2q_{1}}}_{L^{2}(\mathbb{R}^{3})}\|\partial_{2}f\|^{\frac{q_{2}-2}{2q_{2}}}_{L^{2}(\mathbb{R}^{3})}
\|\partial_{3}f\|^{\frac{q_{3}-2}{2q_{3}}}_{L^{2}(\mathbb{R}^{3})}\|f\|^{\frac{1}{\overrightarrow{q}}-\frac12}_{L^{2}(\mathbb{R}^{3})}
\nonumber\\&\leq C\|\nabla f\|_{L^{2}(\mathbb{R}^{3})}^{\frac32-{\frac{1}{\overrightarrow{q}}}}\|f\|^{\frac{1}{\overrightarrow{q}}-\frac12}_{L^{2}(\mathbb{R}^{3})}.
\end{align}
\end{lemma}
\wred{We can state} the local version of the above lemma.
\begin{equation}gin{lemma}\Delta bel{zcl} \wred{Let $q_{1},q_{2},q_{3}\in[2,\infty)$ and $0\leq \frac{1}{\overrightarrow{q}}-\frac12\leq 1$.} Then, for $\varrho>0$ and $0<\xi<\eta$, there exists a constant $C$ such that
\begin{equation}gin{align}
&\|f\|_{L_{1}^{q_{1}} L_{2}^{q_{2}} L_{3}^{q_{3}}(B(\varrho) )} \leq C\|\nabla f\|_{L^{2}(B(\mathrm{div}qrt{2}\varrho ))}^{\frac32-\frac{1}{\overrightarrow{q}}}\|f\|^{\frac{1}{\overrightarrow{q}}-\frac{1}{2}}_{L^{2}(B(\mathrm{div}qrt{2}\varrho))} +C\varrho^{-({\frac32-\frac{1}{\overrightarrow{q}}})}\|f\| _{L^{2}(B(\mathrm{div}qrt{2}\varrho ))},
\Delta bel{locan}\\
&\|f\|_{L_{1}^{q_{1}} L_{2}^{q_{2}} L_{3}^{q_{3}}(B(\frac{\xi+3\eta}{4}) )} \leq C\|\nabla f\|_{L^{2}(B( \eta))}^{ \frac32-\frac{1}{\overrightarrow{q}}} \|f\|^{\frac{1}{\overrightarrow{q}}-\frac{1}{2}}_{L^{2}(B( \eta))} + C(\eta-\xi)^{- ({\frac32-\frac{1}{\overrightarrow{q}}})}\|f\| _{L^{2}(B( \eta))}.\Delta bel{locanlast}
\end{align}
\end{lemma}
\begin{equation}gin{proof}
Let $\phi(x)$ be non-negative smooth function supported in $B(\mathrm{div}qrt{2}\varrho)$ such that
$\phi(x)\equiv1$ on $B(\varrho )$, $0\leq\phi(x)\leq1$ and $|\nabla \phi| \leq C/\varrho $.\\
Making use of\eqref{anis}, we see that
$$\begin{aligned}\|f\|_{L_{1}^{q_{1}} L_{2}^{q_{2}} L_{3}^{q_{3}}(B(\varrho))}\leq&
\|f\phi\|_{L_{1}^{q_{1}} L_{2}^{q_{2}} L_{3}^{q_{3}}(\mathbb{R}^{3})}\\
\leq& C\|\nabla (f\phi)\|_{L^{2}(\mathbb{R}^{3})}^{\frac32-\frac{1}{\overrightarrow{q}}}\|f\phi\|^{\frac{1}{\overrightarrow{q}}-\frac{1}{2}}_{L^{2}(\mathbb{R}^{3})} \\\leq& C\Big(\|\phi\nabla f\|_{L^{2}(\mathbb{R}^{3})}+\|f\nabla \phi\|_{L^{2}(\mathbb{R}^{3})}\Big)^{\frac32-\frac{1}{\overrightarrow{q}}}\|f\phi\|^{\frac{1}{\overrightarrow{q}}-\frac{1}{2}}_{L^{2}(\mathbb{R}^{3})} \\\leq& C\|\nabla f\|_{L^{2}(B(\mathrm{div}qrt{2}\varrho))}^{\frac32-\frac{1}{\overrightarrow{q}}}\|f \|^{\frac{1}{\overrightarrow{q}}-\frac{1}{2}}_{L^{2}(B(\mathrm{div}qrt{2}\varrho))} +C\varrho^{-(\frac32-\frac{1}{\overrightarrow{q}})}\|f\| _{L^{2}(B(\mathrm{div}qrt{2}\varrho))},
\end{aligned}$$
which means \eqref{locan}.\\
Along the exact same lines as the above proof, we have \eqref{locanlast}.
This achieves the proof of the desired estimate.
\end{proof}
By the Poincar\'e inequality, we know that
$$\|f-\wred{\begin{aligned}r{f}_{B(\mathrm{div}qrt{2}\varrho)}}\|_{L^{2}(B(\mathrm{div}qrt{2}\varrho))}\leq
C\rho\|\nabla f\|_{L^{2}(B(\mathrm{div}qrt{2}\varrho))}.
$$
This allows us to derive from \eqref{locan} that, for any $\int_{B(\mathrm{div}qrt{2}\varrho)}fdx=0$, \wred{$0\leq \frac{1}{\overrightarrow{q}}-\frac12\leq 1$,}
\begin{equation}
\wred{\|f\|_{L_{1}^{q_{1} } L_{2}^{q_{2} } L_{3}^{_{q_{3} }}(B(\varrho) )} \leq C\|\nabla f\|_{L^{2}(B(\mathrm{div}qrt{2}\varrho ))}^{\frac32-\frac{1}{\overrightarrow{q}}}\|f\|^{\frac{1}{\overrightarrow{q}}-\frac12}_{L^{2}(B(\mathrm{div}qrt{2}\varrho))},}
\Delta bel{locan0}\end{equation}
which means that
$$\begin{aligned}
\|f\|_{L^{m}_{t}L_{1}^{q_{1} } L_{2}^{q_{2} } L_{3}^{_{q_{3} }}(Q(\varrho) )} &\leq C\|\nabla f\|_{L^{2}(Q(\mathrm{div}qrt{2}\varrho ))}^{\frac32-\frac{1}{\overrightarrow{q}}}\|f\|^{\frac{1}{\overrightarrow{q}}-\frac12}_{L^{\infty}_{t}L^{2}(B(\mathrm{div}qrt{2}\varrho))}
\\&\leq C\|\nabla f\|_{L^{2}(Q(\mathrm{div}qrt{2}\varrho ))}^{\frac{2}{m}}\|f\|^{1-\frac{2}{m}}_{L^{\infty}_{t}L^{2}(B(\mathrm{div}qrt{2}\varrho))},
\end{aligned}$$
where
$$
\frac{2}{m}+\frac{1}{\overrightarrow{q}}=\frac32.
$$
Next, we state another lemma on decomposition of pressure obtained in \cite{[HWZ]} that will be used in the proof of Theorem \ref{anatones}.
\begin{equation}gin{lemma}\cite{[HWZ]}\Delta bel{lem2}
Let $\Phi$ denote the standard normalized fundamental solution of Laplace equation in $\mathbb{R}^{3}$. For $0<\xi<\eta$, we consider smooth cut-off function $\psi\in C^{\infty}_{0}(B(\frac{\xi+3\eta}{4}))$ such that $0\leq\psi\leq1$ in $B(\eta)$, $\psi\equiv1$ in $B(\frac{3\xi+5\eta}{8})$ and $|\nabla^{k}\psi |\leq C/(\eta-\xi)^{k}$ with $k=1,2$ in \wgr{$B(\eta)$}. Then we may split pressure $\Pi$ in \eqref{NS} \wgr{as below}
\begin{equation}\Delta bel{decompose pk}
\Pi(x):=\Pi_{1}(x)+\Pi_{2}(x)+\Pi_{3}(x), \quad x\in B(\frac{\xi+\eta}{2}),
\end{equation}
where
$$\begin{aligned}
\Pi_{1}(x)=&-\partial_{i}\partial_{j}\wgr{\Phi} \ast (\psi (u_{j}u_{i})) ,\\
\Pi_{2}(x)
=&2\partial_{i}\wgr{\Phi} \ast(\partial_{j}\psi(u_{j}u_{i}))- \wgr{\Phi} \ast
(\partial_{i}\partial_{j}\psi u_{j}u_{i}), \\
\Pi_{3}(x)
=&2\partial_{i}\wgr{\Phi} \ast(\partial_{i}\psi \Pi)-\wgr{\Phi} \ast(\partial_{i}\partial_{i}\psi \Pi).
\end{aligned}
$$
Moreover, there holds
\begin{equation}gin{align}
& \| \Pi_1\|_{L^{3/2}(Q(\frac{\xi+\eta}{2}))}
\leq C\| u\|^{2}_{L^{3}(Q(\frac{\xi+3\eta}{4}))};\Delta bel{p1estimate}\\
& \| \Pi_2\|_{L^{3/2}(Q(\frac{\xi+\eta}{2}))}
\leq \wgr{\frac{C\eta^{3}}{(\eta-\xi)^{3}}}\| u\|^{2}_{L^{3}(Q(\frac{\xi+3\eta}{4}))};\Delta bel{p2estimate}\\
& \| \Pi_3\|_{\wgr{L^1L^{2}}(Q(\frac{\xi+\eta}{2}))}
\leq \wgr{\frac{C\eta^{3/2 }}{(\eta-\xi)^{3}}}\|\Pi\|_{L^{1}(Q(\frac{\xi+3\eta}{4}))}.\Delta bel{p3estimate}
\end{align}
\end{lemma}
\wred{Now we introduce Wolf's local} pressure projection $\mathcal{W}_{p,\Omega}:$ $W^{-1,p}(\Omega)\rightarrow W^{-1,p}(\Omega)$ $(1<p<\infty)$.
More precisely, for any $f\in W^{-1,p}(\Omega)$, we define \wred{$\mathcal{W}_{p,\Omega}(f)= \nabla\Pi$}, where $\Pi$ satisfies \eqref{GMS}.
Let $\Omega$ be a bounded domain with $\partial\Omega\in C^{1}$.
According to the $L^p$ theorem of Stokes system in \cite[Theorem 2.1, p149]{[GSS]},
there exists a unique pair $(b,\Pi)\in W^{1,p}(\Omega)\nablames L^{p}(\Omega)$ such that
\begin{equation}\Delta bel{GMS}
-\textxt{Div\,}elta b+\nabla\Pi=f,~~ \textxt{div}\,b=0, ~~b|_{\partial\Omega}=0,~~ \int_{\Omega}\Pi dx=0.
\end{equation}
Moreover, this pair is subject to the inequality
$$
\|b\|_{\wred{W^{1,p}}(\Omega)}+\|\Pi\|_{\wred{L^p}(\Omega)}\leq C\|f\|_{\wred{W^{-1,p}}(\Omega)}.
$$
Let $\nabla\Pi= \mathcal{W}_{p,\Omega}(f)$ $(f\in L^p(\Omega))$, then $\| \Pi\|_{L^p(\Omega)}\leq C\|f\|_{L^p(\Omega)},$ where we used the fact that $L^{p}(\Omega)\hookrightarrow W^{-1,p}(\Omega)$. Moreover, from $\textxt{Div\,}elta \Pi=\textxt{div}\,f$, we see that $\| \nabla\Pi\|_{L^p(\Omega)}\leq C(\|f\|_{L^p(\Omega)}+ \| \Pi\|_{L^p(\Omega)}) \leq C\|f\|_{L^p(\Omega)}.$
\wred{For any} ball $B(R)\mathrm{div}ubseteq \mathbb{R}^{3}$, by the local pressure projection,
Wolf et al. presented the pressure decomposition
$$
- \nabla \Pi = - \partial _ { t } \nabla \Pi _ { h } - \nabla \Pi_ { 1 } - \nabla \Pi_ { 2 },
$$
where
$$\nabla\Pi_{h}=-\mathcal{W}_{p,B(R)}(u),~~ \nabla\Pi_{1}=\mathcal{W}_{p,B(R)}(\textxt{Div\,}elta u),~~\nabla\Pi_{2}=-\mathcal{W}_{p,B(R)}( u\cdot\nabla u).$$ After denoting $v=u+\nabla\Pi_{h}$, one gets the local
energy inequality,
for a.e. $t\in[-T,0]$ and non-negative function $\phi(x,s)\in C_{0}^{\infty}(\mathbb{R}^{3}\nablames (-T,0) )$,
\begin{equation}gin{align}
&\int_{B(r)}|v|^2\phi (x,t) d x+ \int^{t}_{-T }\int_{B(r)}\big|\nabla v\big|^2\wred{\phi (x,s)} d x ds \nonumber\\ \leq& \int^{t}_{-T }\int_{B(r)} | v |^2( \textxt{Div\,}elta \phi + \partial_{t}\phi ) d x d s +\int^{t}_{-T }\int_{B(r)}|v|^{2}u\cdot\nabla \phi \wred{dxds}\nonumber\\
& +\int^{t}_{-T }\int_{B(r)} \phi ( u\otimes v :\nabla^{2}\Pi_{h} ) \wred{dxds} +\int^{t}_{-T }\int_{B(r)} \Pi_{1}v\cdot\nabla \phi dxds+\int^{t}_{-T }\int_{B(r)} \Pi_{2}v\cdot\nabla \phi dxds.\Delta bel{wloc1}
\end{align}
With this in hand, we present the Wolf's new definition of suitable weak solutions of Navier-Stokes equations \eqref{NS}.
\begin{equation}gin{definition}\Delta bel{defi}
A pair $(u, \,\Pi)$ is called a suitable weak solution to the Navier-Stokes equations \eqref{NS} provided the following conditions are satisfied,
\begin{equation}gin{enumerate}[(1)]
\item $u \in L^{\infty}(-T,\,0;\,L^{2}(\mathbb{R}^{3}))\cap L^{2}(-T,\,0;\,\dot{H}^{1}(\mathbb{R}^{3})),\,\Pi\in
L^{3/2}(-T,\,0;L^{3/2}(\mathbb{R}^{3}));$\Delta bel{SWS1}
\item$(u, ~\Pi)$~solves (\ref{NS}) in $\mathbb{R}^{3}\nablames (-T,\,0) $ in the sense of distributions;\Delta bel{SWS2}
\item The local energy inequality \eqref{wloc1} is valid and $\nabla \Pi_{h}$ is a harmonic function. In addition, $ \nabla\Pi_{h}, \nabla\Pi_{1}$ and $\nabla\Pi_{2}$ meet the \wred{following facts}
\begin{equation}gin{align} &\|\nabla\Pi_{h}\|_{L^p(B(R))}\leq \|u\|_{L^p(B(R))}, \Delta bel{ph}\\
&\|\wred{\Pi_{1}}\|_{L^2(B(R))}\leq \|\nabla u\|_{L^2(B(R))},\Delta bel{p1}\\
&\|\wred{\Pi_{2}}\|_{L^{p/2}(B(R))}\leq \| |u|^{2}\|_{L^{p/2}(B(R))}.\Delta bel{p2}
\end{align} \end{enumerate}
\end{definition}
\noindent
We list some
interior estimates
of harmonic functions $\textxt{Div\,}elta h=0$, which will be frequently utilized later. Let $1\leq p,q\leq\infty$ and $0<r<\rho$, then, it holds
\begin{equation}\Delta bel{h1}\|\nabla^{k}h\|_{L^{q}
(B(r))}\leq \frac{Cr^{\frac{n}{q}}}{(\rho-r)^{\frac{n}{p}+k}}\|h\|_{L^{p}(B(\rho))}.\end{equation}
\begin{equation}\Delta bel{h2}
\| h-\overline{h}_{r}\|_{L^{q}
(B(r))}\leq \frac{Cr^{\frac{n}{q}+1}}{(\rho-r)^{\frac{n}{q}+1 }}\|h-\overline{h}_{\rho}\|_{L^{q}(B(\rho))}.\end{equation}
\mathrm{div}ection{Regularity criteria in anisotropic Lebesgue space at infinitely many scales}
Inspired by \cite{[GKT]}, we present the proof of Theorem \ref{thcw1} by Lemma \ref{ineq} and Lemma \ref{presure}.
\wred{By
the natural scaling} property of Navier-Stoke equations \eqref{NS}, we introduce the following dimensionless quantities,
\begin{equation}gin{align}
&E_{\ast}(\varrho)=\fracrac{1}{\varrho}\iint_{\wgr{Q(\varrho)}}|\nabla u|^2dx dt,& E(\varrho)=\mathrm{div}up_{-\varrho^2\leq t<0}\fracrac{1}{\varrho}\int_{B(\varrho)}|u|^2dx,\nonumber\\
&E_{p}(\varrho)=\fracrac{1}{r^{5-p}}\iint_{\wgr{Q(\varrho)}}|u|^{p}dx dt,&D_{3/2}(\varrho)=\fracrac{1}{r^{2}}\iint_{\wgr{Q(\varrho)}}|\Pi-\begin{aligned}r{\Pi}_{\wgr{B(\varrho)}}|^{\fracrac{3}{2}}dx
dt. \nonumber
\end{align}
According to the H\"older inequality, it suffices to prove Theorem \ref{thcw1} for the case
$$
\wred{\frac{2}{p}+\mathrm{div}um\limits_{j=1}\limits^{3}\fracrac{1}{q_{j}}=2.}
$$
Therefore, we introduce the dimensionless quantities below
\begin{equation}gin{align}
\wred{E_{p,\overrightarrow{q}}(u,\,\varrho)=
\varrho^{-1}\|u-\overline{u}_{\varrho}\|_{L^{p}_tL^{\overrightarrow{q}}_{x}(Q(\varrho))}.}
\nonumber\end{align}
To prove Theorem \ref{thcw1}, we need the following crucial lemma.
\begin{equation}gin{lemma}\Delta bel{ineq}
For $0<\mathrm{div}qrt{6}\mu\leq \rho$,~
there is an absolute constant $C$ independent of $\mu$ and $\rho$,~ such that
\begin{equation}gin{align}
\wred{E_{3}(\mu)\leq C\Big(\frac{\rho}{\mu}\Big)^{2}E_{p,\overrightarrow{q}} (u,\,\rho) E_{\ast}(\rho)^{1-\frac{1}{p}} E^{\frac{1}{p}}(\rho) +C\Big(\frac{\mu}{\rho}\Big)E_{3}(\rho).}
\Delta bel{ineq2/2} \end{align}
\end{lemma}
\begin{equation}gin{proof}
We begin by proving the following crucial estimate
\begin{equation}\begin{aligned}\Delta bel{key2.9} \iint_{Q(\varrho)} |v|^{3}dxds&=\iint_{Q(\varrho)} |v| |v|^{2}dxds
\\& \leq C \|v\|_{\wred{L_{t}^{p}L_x^{\overrightarrow{q}}}(Q(\varrho))} \|v\|^{2}_{\wred{L_{t}^{2p^{\ast}}L_x^{2\overrightarrow{q}^{\ast}}}(Q(\varrho))}
\\& \leq C \|v\|_{\wred{L_{t}^{p}L_x^{\overrightarrow{q}}}(Q(\varrho))} \|\nabla v\|_{L^{2}(Q(\mathrm{div}qrt{2}\varrho ))}^{\frac{2}{p^{\ast}}}\|v\|^{2-\frac{2}{p^{\ast}}}_{L^{\infty}_{t}\wred{L_x^{2}(Q(\mathrm{div}qrt{2}\varrho))}},
\\& \leq C \|v\|_{\wred{L_{t}^{p}L_x^{\overrightarrow{q}}}(Q(\varrho))} \|\nabla v\|_{L^{2}(Q(\mathrm{div}qrt{2}\varrho ))}^{2-\frac{2}{p}}\|v\|^{\frac{2}{p}}_{L^{\infty}_{t}\wred{L_x^{2}(Q(\mathrm{div}qrt{2}\varrho))}},
\end{aligned}\end{equation}
where $v=u-\begin{aligned}r{u}_{\mathrm{div}qrt{2}\varrho}$.
\wred{Noticing that} $\frac{2}{p}+\frac{1}{\overrightarrow{q}}=2$, we infer that
\begin{equation}\begin{aligned}\Delta bel{lem2.3.2.7}
\iint_{\wgr{Q(\varrho)}}|v|^{3}dxdt & \leq C\|v\|_{L_{t}^{p}L^{\overrightarrow{q}}_{x}\wgr{(Q(\varrho))}} \|\nabla v\|_{L^{2}(Q(\mathrm{div}qrt{2}\varrho))}^{ \frac{1}{\overrightarrow{q}} } \wred{\|v\|^{2-\frac{1}{\overrightarrow{q}}}_{L_t^{\infty}L_x^{2}\wgr{(Q(\mathrm{div}qrt{2}\varrho))}},}
\end{aligned}\end{equation}
\wred{which entails} that
\begin{equation}\begin{aligned}
&\iint_{Q( \varrho)}|u- \overline{u}_{\mathrm{div}qrt{2}\varrho}|^{3}dxdt \\
\leq& C\|u- \overline{ u}_{\mathrm{div}qrt{2}\varrho }\|_{L_{t}^{p}L^{\overrightarrow{q}}_{x}(Q(\varrho))} \|\nabla u\|_{L^{2}(\wgr{Q(\mathrm{div}qrt{2}\varrho)})}^{ \frac{1}{\overrightarrow{q}} } \|u- \overline{ u}_{\mathrm{div}qrt{2}\varrho} \|^{2-\frac{1}{\overrightarrow{q}}}_{\wgr{L^{\infty}L^{2}}(Q(\mathrm{div}qrt{2}\varrho))}.\Delta bel{eq3.4}
\end{aligned}\end{equation}
By virtue of the triangle inequality, \wred{we obtain}
\begin{equation}gin{align}\nonumber
\iint_{Q(\mu)}|u|^{3}dx\leq& C\iint_{Q(\mu)}|u-\begin{aligned}r{u}_{{\rho}}|^{3}dx
+C\iint_{\wred{Q(\mu)}}|\begin{aligned}r{u}_{{\rho}}|^{3} dx\\
\leq& C\iint_{Q(\rho/\mathrm{div}qrt{6})}|u-\begin{aligned}r{u}_{{\rho_{}}}|^{3}dx
+
C\frac{\mu^{3}}{\rho^{3}}\Big( \iint_{\wred{Q(\rho)}}|u|^{3}dx\Big). \Delta bel{lem2.31}
\end{align}
This \wred{together with \eqref{eq3.4}} implies the desired estimate \eqref{ineq2/2}.
\end{proof}\begin{equation}gin{lemma}\Delta bel{presure}
For $0<4\mathrm{div}qrt{6}\mu\leq \rho$, there exists an absolute constant $C$ independent of $\mu$ and $\rho$ such that
\begin{equation}gin{align}
&D_{3/2}(\mu)\leq
C\left(\frac{\rho}{\mu}\right)
^{2}\wgr{E_{p,\overrightarrow{q} }}(\rho)\wred{E_{\ast}(\rho)^{1-\frac{1}{p}} E^{\frac{1}{p}}(\rho)}
+C\left(\frac{\mu}{\rho}\right)^{\frac{5}{2}}D_{3/2}(\rho).\Delta bel{pe}
\end{align}
\end{lemma}
\begin{equation}gin{proof}
We consider the usual cut-off function $\phi\in C^{\infty}_{0}(B(\frac{\rho}{ \mathrm{div}qrt{6}}))$ such that $\phi\equiv1$ on $B(\frac{3\rho}{4\mathrm{div}qrt{6}})$ with $0\leq\phi\leq1$,
$|\nabla\phi |\leq C\rho^{-1} $ and $ ~|\nabla^{2}\phi |\leq
C\rho^{-2}.$\\
\wgr{Due to} the incompressible condition, the pressure equation can be \wgr{written as}
$$
\partial_{i}\partial_{i}(\Pi\phi)=-\phi \partial_{i}\partial_{j} U_{i,j}
+2\partial_{i}\phi\partial_{i}\Pi+\Pi\partial_{i}\partial_{i}\phi
,$$
where $U_{i,j}=(u_{j}- \begin{aligned}r{u}_{{\rho_{/\mathrm{div}qrt{6}}}})(u_{i}-\begin{aligned}r{u}_{{\rho_{/\mathrm{div}qrt{6}}}})$. \wgr{Thus it follows} that, for $x\in B(\frac{3\rho}{4\mathrm{div}qrt{6}})$
\begin{equation}\begin{aligned}\Delta bel{pp}
\Pi(x)=&\Phi\ast \{-\phi \partial_{i}\partial_{j} U_{i,j}
+2\partial_{i}\phi\partial_{i}\wgr{\Pi}+\wgr{\Pi}\partial_{i}\partial_{i}\phi
\}\\
=&-\partial_{i}\partial_{j}\Phi \ast (\phi U_{i,j} )\\
&+2\partial_{i}\Phi \ast(\partial_{j}\phi U_{i,j} )-\Phi \ast
(\partial_{i}\partial_{j}\phi U_{i,j} )\\
& \wgr{+2}\partial_{i}\Phi \ast(\partial_{i}\phi \Pi) -\Phi \ast(\partial_{i}\partial_{i}\phi \Pi)\\
\wgr{=:} &P_{1}(x)+P_{2}(x)+P_{3}(x),
\end{aligned}\end{equation}
where $\Phi$ stands for the standard normalized fundamental solution of Laplace equation in $\mathbb{R}^{3}$.\\
Since $\phi(x)=1, $ where $x\in B(\mu)$ ($0<\mu\leq\frac{\rho}{2\mathrm{div}qrt{6}} $), we have
\[
\textxt{Div\,}elta(P_{2}(x)+P_{3}(x))=0.
\]
According to the interior estimate of harmonic function
and the H\"older inequality, we thus have, for every
$ x_{0}\in B(\frac{\rho}{4\mathrm{div}qrt{6}} )$,
\begin{equation}\begin{aligned}
|\nabla (P_{2}+P_{3})(x_{0})|&\leq \frac{C}{\rho^{4}}\|(P_{2}+P_{3})\|_{L^{1}(B_{x_{0}}(\frac{\rho}{4\mathrm{div}qrt{6}}))}
\\
&\leq \frac{C}{\rho^{4}}\|(P_{2}+P_{3})\|_{L^{1}(B(\frac{\rho}{2\mathrm{div}qrt{6}}))}\\
&\leq \frac{C}{\rho^{4}}\rho^{3(1-\frac{1}{q})} \|(P_{2}+P_{3})\|_{\wgr{L^{3/2}}(B(\frac{\rho}{2\mathrm{div}qrt{6}}))}.\Delta bel{lem2.4.2.15}
\end{aligned}\end{equation}
We infer from \eqref{lem2.4.2.15} that
\begin{equation}\wgr{\|\nabla (P_{2}+P_3)\|^{3/2}_{L^{\infty}(B(\frac{\rho}{4\mathrm{div}qrt{6}}))}\leq C \rho^{-9/2}\|P_2+P_3\|^{3/2}_{L^{3/2}(B(\frac{\rho}{2\mathrm{div}qrt{6}}))}}.\Delta bel{lem2.4.2.16}\end{equation}
Using the mean value theorem and \eqref{lem2.4.2.16} , for any $\mu\leq \frac{\rho}{4\mathrm{div}qrt{6}}$, we arrive at
\begin{equation}\begin{aligned}\Delta bel{lem2.4.2.17}
\|(P_{2}+P_{3})-\overline{(P_{2}+P_{3})}_{\mu}\|^{3/2}_{L^{3/2}(B(\mu))}\leq&
C\mu^{3} \|(P_{2}+P_{3})-\overline{(P_{2}+P_{3})}_{\mu}\|^{\wgr{3/2}}_{L^{\infty}(B(\mu))}\\
\leq& C
\mu^{3} (2\mu)^{\wgr{3/2}}\|\nabla (P_{2}+P_{3})\|^{\wgr{3/2}}_{L^{\infty}(B(\frac{\rho}{4\mathrm{div}qrt{6}}))}\\
\leq& C\Bigig(\frac{\mu}{\rho}\Bigig)^{\frac{9}{2}}\|(P_{2}+P_{3})\|^{3/2}_{L^{3/2}
(B(\frac{\rho}{2\mathrm{div}qrt{6}}))}.
\end{aligned}\end{equation}
By time integration, we get
$$
\|(P_{2}+P_{3})-\overline{(P_{2}+P_{3})}_{\mu}\|
^{\frac{3}{2}}_{L^{\frac{3}{2}}(Q(\mu))}\leq C\Bigig(\frac{\mu}{\rho}\Bigig)^{ \frac{9}{2}}
\|(P_{2}+P_{3})\|^{\frac{3}{2}}_{L^{\frac{3}{2}}(Q(\frac{\rho}{2\mathrm{div}qrt{6}}))}.
$$
As $(P_{2}+P_{3})-((P_{2}+P_{3}))_{B(\frac{\rho}{2\mathrm{div}qrt{6}})}$ is also a Harmonic function on $B(\frac{\rho}{2\mathrm{div}qrt{6}})$, we deduce taht
$$\begin{aligned}
&\|(P_{2}+P_{3})-\overline{(P_{2}+P_{3})}_{\mu}\|
^{\frac{3}{2}}_{L^{3/2}(Q(\mu))}
\\
\leq & C\Bigig(\frac{\mu}{\rho}\Bigig)^{ \frac{9}{2}}
\|(P_{2}+P_{3})-\overline{(P_{2}+P_{3})}_{\frac{\rho}{2\mathrm{div}qrt{6}}}\|
^{\frac{3}{2}}
_{L^{\frac{3}{2}}(Q(\frac{\rho}{2\mathrm{div}qrt{6}}))}.
\end{aligned}$$
The triangle inequality guarantees that
$$\begin{aligned}
&\|(P_{2}+P_{3})-\overline{(P_{2}+P_{3})}_{(\frac{\rho}{2\mathrm{div}qrt{6}})}\|_{L^{\frac{3}{2}}(Q(\frac{\rho}{2\mathrm{div}qrt{6}}))}\\
\leq& \|\Pi-\overline{\Pi}_{(\frac{\rho}{2\mathrm{div}qrt{6}})}\|_{L^{\frac{3}{2}}(Q(\frac{\rho}{2\mathrm{div}qrt{6}}))}
+\|P_{1}-\overline{P_{1}}_{(\frac{\rho}{2\mathrm{div}qrt{6}})}\|_{L^{\frac{3}{2}}(Q((\frac{\rho}{2\mathrm{div}qrt{6}})))}
\\
\leq& \wgr{C}\|\wred{\Pi-\overline{\Pi}_{\rho}}\|_{L^{\frac{3}{2}}(Q(\frac{\rho}{2\mathrm{div}qrt{6}}))}
+\wgr{C}\|P_{1}\|_{L^{\frac{3}{2}}(Q(\frac{\rho}{2\mathrm{div}qrt{6}}))},
\end{aligned}$$
which leads to that
\begin{equation}\Delta bel{p2rou}\begin{aligned}
&\|(P_{2}+P_{3})-\overline{(P_{2}+P_{3})}_{\mu}\|
^{\frac{3}{2}}_{L^{\frac{3}{3}}(Q(\mu))}\\
\leq& C\Bigig(\frac{\mu}{\rho}\Bigig)^{ \frac{9}{2}}\Bigig(\|\Pi-\overline{\Pi}
_{(\rho)}\|^{\frac{3}{2}}_{L^{\frac{3}{2}}(Q(\frac{\rho}{2\mathrm{div}qrt{6}}))}
+\|P_{1}\|^{\frac{3}{2}}_{L^{\frac{3}{2}}(Q(\frac{\rho}{2\mathrm{div}qrt{6}}))}\Bigig)
\\
\leq& C\Bigig(\frac{\mu}{\rho}\Bigig)^{ \frac{9}{2}}\Bigig(\|\Pi-\overline{\Pi}
_{(\rho)}\|^{\frac{3}{2}}_{L^{\frac{3}{2}}(Q(\rho))}
+\|P_{1}\|^{\frac{3}{2}}_{L^{\frac{3}{2}}(Q(\frac{\rho}{2\mathrm{div}qrt{6}}))}\Bigig).
\end{aligned}
\end{equation}
By virtue of the H\"older inequality and the argument in \eqref{key2.9}, we get
$$\begin{aligned}
&\iint_{Q(\rho/\mathrm{div}qrt{6})}|u-\begin{aligned}r{u}_{{\rho_{/\mathrm{div}qrt{6}}}}|^{3}dxds\\
\leq& C\iint_{Q(\rho/\mathrm{div}qrt{6})}|\wred{u- \overline{u}_{\rho/\mathrm{div}qrt{3}} }|^{3}dxds\\
\leq& C \|u- \overline{ u_{\rho} }\|_{L^{p}L^{\overrightarrow{q}}(Q(\rho))} \|\nabla u\|_{L^{2} (Q(\rho))}^{2-\frac{2}{p} } \|u\|^{\frac{2}{p}}_{L^{\infty}L^{2}(Q(\rho))}.
\end{aligned}$$
The classical Calder\'on-Zygmund Theorem and the latter inequality implies that
\begin{equation}\Delta bel{lem2.4.2}\begin{aligned}
\iint_{Q(\frac{\rho}{2\mathrm{div}qrt{6}})}|P_{1}(x)|^{\frac{3}{2}}dxds
\leq& C \iint_{Q(\frac{\rho}{\mathrm{div}qrt{6}})}|u-\overline{u}_{\rho/\mathrm{div}qrt{6}}|^{3}
dx\\
\leq& C \|u- \overline{ u_{\rho} }\|_{L^{p}L^{\overrightarrow{q}}(Q(\rho))} \|\nabla u\|_{L^{2} (Q(\rho))}^{2-\frac{2}{p} } \|u\|^{\frac{2}{p}}_{L^{\infty}L^{2}(Q(\rho))},
\end{aligned}\end{equation}
and
\begin{equation}\Delta bel{lem2.4.3}\begin{aligned}
\iint_{Q(\mu)}|P_{1}(x)|^{\frac{3}{2}}dx& \leq C \|u- \overline{ u_{\rho} }\|_{L^{p}L^{\overrightarrow{q}}(Q(\rho))} \|\nabla u\|_{L^{2} (Q(\rho))}^{2-\frac{2}{p} } \|u\|^{\frac{2}{p}}_{L^{\infty}L^{2}(Q(\rho))}.
\end{aligned}\end{equation}
The \wgr{inequalities} \eqref{p2rou}-\eqref{lem2.4.3} \wgr{allow} us to deduce that
\begin{equation}\Delta bel{lem2.3}
\begin{aligned}
\iint_{Q(\mu)}|\Pi-\Pi_{\mu}|^{\frac{3}{2}}dxds \leq& C\iint_{Q(\mu)}
|P_{1}-(P_{1})_{\mu}|^{\frac{3}{2}}
+\wgr{\big|P_{2}+P_{3}-(P_{2}+P_{3})_{\mu}\big|^{\frac{3}{2}}}dx \\
\leq& C \|u- \overline{ u_{\rho} }\|_{L^{p}L^{\overrightarrow{q}}(Q(\rho))} \|\nabla u\|_{L^{2} (Q(\rho))}^{2-\frac{2}{p} } \|u\|^{\frac{2}{p}}_{L^{\infty}L^{2}(Q(\rho))}
\\
& +C\left(\frac{\mu}{\rho}\right)
^{\frac{9}{2}}\int_{B(\rho)}|\Pi-\Pi_{\rho}|^{\frac{3}{2}}.
\end{aligned}
\end{equation}
We readily get
\begin{equation}\Delta bel{lem2.42}
\begin{aligned}
\frac{1}{\mu^2}\iint_{Q(\mu)}|\Pi-\Pi_{\mu}|^{\frac{3}{2}}
\leq& \wred{C\frac{1}{\mu^2}}\|u- \overline{ u_{\rho} }\|_{L^{p}L^{\overrightarrow{q}}(Q(\rho))} \|\nabla u\|_{L^{2} (Q(\rho))}^{2-\frac{2}{p} } \|u\|^{\frac{2}{p}}_{L^{\infty}L^{2}(Q(\rho))}\\
& +C\left(\frac{\mu}{\rho}\right)
^{\frac{5}{2}}
\frac{1}{\rho^{2}}\iint_{Q(\rho)}|\Pi-\overline{\Pi}_{\rho}|^{\frac{3}{2}}dx,
\end{aligned}
\end{equation}
which leads to
\begin{equation}gin{align}
D_{3/2}(\mu)\leq &
C\left(\frac{\rho}{\mu}\right)
^{2}\wred{E_{p,\overrightarrow{q}}(\rho)E_{\ast}(\rho)^{1-\frac{1}{p}} E^{\frac{1}{p}}(\rho)}
+C\left(\frac{\mu}{\rho}\right)^{\frac{5}{2}}D_{3/2}(\rho)\wgr{.}\Delta bel{presure4}
\end{align}
The proof of this lemma is completed.
\end{proof}
\begin{equation}gin{proof}[Proof of Theorem \ref{thcw1}]
\wred{From} \eqref{spe1}, we know that
there is a constant $\varrho_0$ such that, for any $\varrho\leq \varrho_{0}$,
$$
\wred{\varrho^{1-\frac{2}{p}-\frac{1}{q_1}-\frac{1}{q_2}-\frac{1}{q_3}}
\|u-\overline{ u_{\varrho} }\|_{L_{t}^{p}L_{1}^{q_1}L_{2}^{q_2}L_{3}^{q_3}(Q(\varrho))}}\leq\varepsilon_{1}.
$$
By the Young inequality and local energy inequality \eqref{loc}, we have
\begin{equation}\Delta bel{eq:88}\begin{aligned}
E(\rho)+E_{\ast}(\rho)\leq& C\Bigig[E^{2/3}_{3}(2\rho)+E_{3}(2\rho)
+\wred{D_{3/2}}(2\rho)\Bigig]\\
\leq& C\Bigig[1+E_{3}(2\rho)
+\wred{D_{3/2}}(2\rho)\Bigig].
\end{aligned}\end{equation}
From \wgr{\eqref{eq:88} and} \eqref{ineq2/2} in Lemma \ref{ineq}, we see that, for
$2\mathrm{div}qrt{6}\mu\leq\rho$,
\begin{equation}\begin{aligned}
E_{3}(\mu)\leq&C \left(\dfracrac{\rho}{\mu}\right)^{2}
\wgr{E_{p,\overrightarrow{q}}}(\rho/2)\wred{E_{\ast}(\rho/2)^{1-\frac{1}{p}} E^{\frac{1}{p}}(\rho/2)}
+C\left(\dfracrac{\mu}{\rho}\right)E_{3}(\rho/2)\\
\leq&C \left(\dfracrac{\rho}{\mu}\right)^{2}
\wgr{E_{p,\overrightarrow{q}}}(\rho/2)\Big( 1+E_{3}(\rho)
+\wred{D_{3/2}}(\rho)\Big)
+C\left(\dfracrac{\mu}{\rho}\right)E_{3}(\rho/2)
\\
\leq&C \left(\dfracrac{\rho}{\mu}\right)^{2}
\wgr{E_{p,\overrightarrow{q}}}(\rho )\Big( 1+E_{3}(\rho)
+\wred{D_{3/2}}(\rho)\Big)
+C\left(\dfracrac{\mu}{\rho}\right)E_{3}(\rho ).\Delta bel{3.2} \end{aligned}\end{equation}
It follows form \eqref{pe} in Lemma \ref{presure} that, for $8\mathrm{div}qrt{6}\mu\leq\rho$,
\begin{equation}\Delta bel{3.3}
D_{3/2}(\mu)\leq
C\left(\frac{\rho}{\mu}\right)
^{2}\wgr{E_{p,\overrightarrow{q}}}(\rho)\Big(1+E_{3}(\rho)
+D_{3/2}(\rho)\Big)
+C\left(\frac{\mu}{\rho}\right)^{\frac{5}{2}}D_{3/2}(\rho).
\end{equation}
Before going further, we set
$$F(\mu)= E_{3}(\mu)+\wred{D_{3/2}(\mu)}.$$
With the help of \eqref{3.2} and \eqref{3.3}, we conclude that
$$\begin{aligned}
F(\mu)\leq&
C
\left(\dfracrac{\rho}{\mu}\right)^{2}
\wgr{E_{p,\overrightarrow{q}}}(\rho)F(\rho)
\wgr{+C}\left(\dfracrac{\rho}{\mu}\right)^{2}
\wred{E_{p,\overrightarrow{q}}}(\rho)
+C
\left(\dfracrac{\mu}{\rho}\right)
F(\rho)\\
\leq & C_{1}\Delta mbda^{-2}\varepsilon_{1} F(\rho)+
C_{2}\Delta mbda^{-2} \varepsilon_{1} +C_{3}\Delta mbda F(\rho),
\end{aligned}$$
\wgr{where $\Delta mbda=\frac{\mu}{\rho}\leq \frac{1}{8\mathrm{div}qrt{6}}$} and $\rho\leq \varrho_{0} $.\\
Choosing $\Delta mbda,~\varepsilon_{1}$ such that $\wgr{\theta}=2C_{3}\Delta mbda<1$ and $\varepsilon_{1}=\min\{ \frac{\wgr{\theta}\Delta mbda^{2}}{2C_{1}} ,
\frac{(1-\wgr{\theta})\Delta mbda^{2}\varepsilon}{2C_{2}\Delta mbda^{-2}}
\}$ \wgr{where $\varepsilon$ is the constant in \eqref{CKN}}, we see that
\begin{equation}\Delta bel{iter}
F(\Delta mbda\rho)\leq \wgr{\theta}F(\rho)+C_{2}\Delta mbda^{-2} \varepsilon_{1}. \end{equation}
We iterate $\eqref{iter}$ to get
\[
F(\Delta mbda^{k}\rho)\leq \wgr{\theta}^{k}F(\rho)+\frac{1}{2}\Delta mbda^{2}\varepsilon. \]
According to the definition of $F(r)$, for a fixed $\varrho_{0}>0$, we know that there exists a positive number $ K_{0}$~such that
$$\wgr{\theta}^{K_{0}}F(\varrho_{0})\wred{\leq}\frac{M(\|u\|_{L^{\infty}L^{2}},\|u\|_{L^{2}W^{1,2}},
\|\Pi\|_{L^{3/2}L^{3/2}})}{\varrho_{0}^{2}}\wgr{\theta}^{K_{0}}
\leq\dfracrac{1}{2}\varepsilon \Delta mbda^{2}.$$
We denote $\varrho_{1}:=\Delta mbda^{K_{0}}\varrho_{0}$. Then, for all $0<\varrho\leq \varrho_{1}$ , $\exists k\geq
K_{0}$,~such that $\Delta mbda^{k+1}\varrho_{0}\leq \varrho\leq \Delta mbda^{k} \varrho_{0}$, there holds
\[
\begin{equation}gin{aligned}
& E_{3}(\varrho)+D_{3/2}(\varrho)\\
=&\fracrac{1}{\varrho^{2}}\iint_{Q(\varrho)}| u|^3dxdt+
\fracrac{1}{\varrho^{2}}\iint_{Q(\varrho)}|\Pi-\overline{\Pi}_{\varrho}|^{\fracrac{3}{2}}dxdt\\
\leq& \fracrac{1}{(\Delta mbda^{k+1}\varrho_{0})^{2}}\iint_{Q(\Delta mbda^{k}\varrho_{0})}| u|^3dxdt
+
\fracrac{1}{(\Delta mbda^{k+1}\varrho_{0})^{2}}\iint_{Q(\Delta mbda^{k}\varrho_{0})}|\Pi-\overline{\Pi}_{\Delta mbda^{k}\varrho_{0}}|^{\fracrac{3}{2}}dxdt
\\
\leq & \frac{1}{\Delta mbda^{2}}F(\Delta mbda^{k}\varrho_{0}) \\
\leq &\frac{1}{\Delta mbda^{2}}(\wgr{\theta}^{k-K_{0}}\wgr{\theta}^{K_{0}}
F(\varrho_{0})+\frac{1}{2}\Delta mbda^{2}\varepsilon )\\
\leq &\varepsilon.
\end{aligned}
\]
This together with \eqref{CKN} completes the \wred{proof of} Theorem \ref{thcw1}.
\end{proof}
\mathrm{div}ection{ Regularity criteria at one scale }\Delta bel{sec5}
Before going further, we write
\begin{equation}
\alpha=\frac{2}{\frac{2}{p}+\frac{1}{\overrightarrow{q}}}.
\Delta bel{aerfa1}\end{equation}
By virtue of the H\"older inequality, we just need consider the case that $\alpha$
is very close to 1 to show Theorem \ref{anatones}. Therefore, for any $1<q_{i}<\infty$, we have
\begin{equation}
q_{i} \leq \frac{2\alpha}{\alpha-1}.
\Delta bel{q>2}\end{equation}
\begin{equation}gin{lemma}\Delta bel{zc2} Let $\alpha$ \wgr{be given} in \eqref{aerfa1}. For $0<\xi<\eta$, there is an absolute constant $C$ such that \wred{for $\varrho=\fracrac{\xi+3\eta}{4}$}
\begin{equation}\begin{aligned}\Delta bel{zw}
\|u\|_{L^{3}(Q(\varrho))}^{3}\leq C
\eta^{\frac{3(\alpha-1)}{2}}\|u\|^{\alpha}_{\wred{L_t^{p}L_x^{\overrightarrow{q}}(Q(\eta))}} \Bigig\{\big[1+
\frac{\eta^{{\frac{7-3\alpha-\frac{4\alpha}{p}}{2}}}}
{(\eta-\varrho)^{\frac{7-3\alpha-\frac{4\alpha}{p}}{2}}}\big]
\|u\|_{\wred{L_t^{\infty}L_x^{2}} (Q(\eta))}^{3-\alpha}+\|\nabla u\|_{L^{2}(Q(\eta))}^{3-\alpha }\Bigig\}.
\end{aligned}\end{equation}
\end{lemma}
\begin{equation}gin{proof}
By the interpolation inequality in Lemma \ref{interi}, we find
\begin{equation}\Delta bel{4.2}
\|u\|_{L^{3}(B(\varrho))}\leq \|u\|^{^{\frac{\alpha}{3}}}_{L^{\overrightarrow{q}}(B(\varrho))}
\|u\|^{\frac{3-\alpha}{3}}_{L^{\overrightarrow{t}}(B(\varrho))}.
\end{equation}
This together with \eqref{q>2} implies that
$$
t_{j}=\frac{3-\alpha}{1-\frac{\alpha}{q_{j}}}\geq2, ~~\textxt{and}~~ \frac{1}{\overrightarrow{t}}=\frac{1+\frac{2\alpha}{p}}{3-\alpha}\geq\frac12.
$$
As a consequece, we can apply \eqref{locanlast} to obtain
$$\begin{aligned}
\|u\|_{L^{\overrightarrow{t}}(B(\varrho))}
&\leq \wred{C
\|\nabla u \|_{L^{2}(B(\eta))}^{\frac{3}{2}-\frac{1}{\overrightarrow{t}}}
\|u\|_{L^{2}(B(\eta))}^{\frac{1}{\overrightarrow{t}}-\frac12}
+C(\eta-\varrho)^{\frac{1}{\overrightarrow{t}}-\frac{3}{2}}
\| u \|_{L^{2}(B(\eta))}.}
\end{aligned}$$
Plugging this into \eqref{4.2}, we know that
$$\begin{aligned}
\|u\|^{3}_{L^{3}(B(\varrho))}\leq & \wred{C}\|u\|^{\alpha}_{L^{\overrightarrow{q}}(B(\eta))}
(\|\nabla u \|_{L^{2}(\eta)}^{\frac{3}{2}-\wred{\frac{1}{\overrightarrow{t}}}}
\|u\|_{L^{2}(B(\eta))}^{\wred{\frac{1}{\overrightarrow{t}}}-\frac12}
+(\eta-\varrho)^{\frac{1}{\overrightarrow{t}}-\frac{3}{2}}
\| u \|_{L^{2}(B(\eta))})^{3-\alpha} \\
\leq& \wred{C}\|u\|^{\alpha}_{L^{\overrightarrow{q}}(\wred{B(\eta)})}
(\|\nabla u \|_{L^{2}(B(\eta))}^{\frac{7-3\alpha-\frac{4\alpha}{p}}{2 }}
\|u\|_{L^{2}(B(\eta))}^{\frac{\alpha-1+\frac{4\alpha}{p}}{2 }}+
(\eta-\varrho)^{\wred{-\frac{7-3\alpha-\frac{4\alpha}{p}}{2 }}}
\| u \|_{L^{2}(B(\eta))}^{3-\alpha})\wred{.}
\end{aligned}$$
According to time integration and the Holder inequality, we see that
$$\begin{aligned}
&\|u\|^{3}_{L^{3}(Q(\varrho))}\\
\leq & \wred{C\eta^{\frac{3(\alpha-1)}{2}}\|u\|^{\alpha}_{L_t^{p}L_x^{\overrightarrow{q}}(Q(\eta))}} \Big[\|\nabla u\|_{L^{2}(Q(\rho))}^{\frac{7-3\alpha-\frac{4\alpha}{p} }{2}}\|u\|^{\frac{\alpha-1+\frac{4\alpha}{p}}{2}}_{\wred{L_t^{\infty}L_x^{2}}(Q(\eta))}
+\frac{\eta^{{\frac{7-3\alpha-\frac{4\alpha}{p}}{2}}}}{(\eta-\varrho)^{\frac{7-3\alpha-\frac{4\alpha}{p}}{2}}}
\|u\|^{3-\alpha}_{\wred{L_t^{\infty}L_x^{2}}(Q(\eta))}\Big]
\\ \leq& \wred{C}\eta^{\frac{3(\alpha-1)}{2}}\|u\|^{\alpha}_{\wred{L_t^{p}L_x^{\overrightarrow{q}}}(Q(\eta))} \Bigig\{\big[1+
\frac{\eta^{{\frac{7-3\alpha-\frac{4\alpha}{p}}{2}}}}
{(\eta-\varrho)^{\frac{7-3\alpha-\frac{4\alpha}{p}}{2}}}\big]
\|u\|_{\wred{L_t^{\infty}L_x^{2}} (Q(\eta))}^{3-\alpha}+\|\nabla u\|_{L^{2}(Q(\eta))}^{3-\alpha }\Bigig\}.
\end{aligned}
$$
\end{proof}
\wred{Now}, \wgr{we prove} Theorem \ref{anatones}. Since the proof is parallel to the one used in \cite{[HWZ]}, we just sketch the proof.
\begin{equation}gin{proof}[Proof of Theorem \ref{anatones}] It suffices to proof the following inequality, for any $R>0$,
\begin{equation}\begin{aligned} \Delta bel{key ineq}
&\|u\|^2_{L_{t}^{\infty}L_{x}^{2}(Q(R/2))}+\|\nabla u\|^2_{L^{2}(Q(R/2))} \\ \leq& \frac{C}{R^{( 4-3\alpha)/\alpha}} \|u\|_{L_{t}^{p}L^{\overrightarrow{q}}_{x}(Q(R))} ^{2} + \frac{C}{R^{( 5-3\alpha)/(\alpha-1)}} \|u\|_{L_{t}^{p}L^{\overrightarrow{q}}_{x}(Q(R))} ^{\frac{2\alpha}{\alpha-1}} + \wgr{\frac{C}{R^{5}}}\|\Pi\|^{2}_{L^{1}(Q(R))}.
\end{aligned}\end{equation}
Indeed, consider $0<R/2\leq \xi<\frac{3\xi+\eta}{4}<\frac{\xi+\eta}{2}<\frac{\xi+3\eta}{4}\wgr{<\eta\leq R}$. Let $\phi(x,t)$ be non-negative smooth function supported in $Q(\frac{\xi+\eta}{2})$ such that
$\phi(x,t)\equiv1$ on $Q(\frac{3\xi+\eta}{4})$,
$|\nabla \phi| \leq C/(\eta-\xi) $ and $
|\nabla^{2}\phi|+|\partial_{t}\phi|\leq C/(\eta-\xi)^{2} .$
The local energy inequality \eqref{loc}, the decomposition of pressure in Lemma \ref{lem2} and the H\"older inequality ensure that
\begin{equation}gin{align}
&\int_{B(\frac{\eta+\xi}{2})} |u(x,t)|^{2} \phi(x,t) dx
+2\iint_{Q(\frac{\eta+\xi}{2})}
|\nabla u|^{2}\phi dxds\nonumber\\\leq & \frac{C\wgr{\eta^{5/3}}}{(\eta-\xi)^{2}}\Big(\iint_{Q(\wgr{\frac{\xi+3\eta}{4}})} |u|^{3}
dxds\Bigig)^{2/3}+\frac{C}{(\eta-\xi)} \iint_{Q(\wred{\frac{\xi+3\eta}{4}})} |u|^{3} dxds\nonumber\\&+ \frac{C\eta^{3}}{(\eta-\xi)^{4}} \|u\|^{3}_{L^{3}(Q(\wgr{\frac{\xi+3\eta}{4}}))}+\frac{C\eta^{3/2}}{(\eta-\xi)^{4}} \| \Pi\|_{L^{1} (Q(\eta))} \|u\|_{\wred{L_t^{\infty}L_x^2}(Q(\eta)))}\nonumber\\
=&:I+II+III\wred{+IV.}\Delta bel{last3}
\end{align}
Combining
\eqref{zw} and the Young inequality, we obtain
\begin{equation}gin{align}\nonumber I\leq & \frac{C\eta^{3+\frac{2}{\alpha} }}{(\eta-\xi)^{6/\alpha}} \|u\|^{2}_{L_{t}^{p}L^{\overrightarrow{q}}(Q(\eta))} +\frac{C\eta^{3+\frac{2}{\alpha} }}{(\eta-\xi)^{6/\alpha}} \big[1+
\frac{\eta^{{\frac{7-3\alpha-\frac{4\alpha}{p}}{2}}}}
{(\eta-\xi)^{\frac{7-3\alpha-\frac{4\alpha}{p}}{2}}}\big]^{\frac{2}{\alpha}}\|u\|^{2}_{\wred{L_{t}^{p}L_x^{\overrightarrow{q}}}(Q(\eta))} \\&+\frac{1}{6}\Big(\|u\|_{\wred{L_t^{\infty}L_x^2} (Q(\eta))}^{2}+\|\nabla u\|_{L^{2}(Q(\eta))}^{2}\Big),\Delta bel{last2}\\
\nonumber II
\leq & \frac{C\eta^{3}}{(\eta-\xi)^{\frac{2}{\alpha-1}}}\|u\|^{\frac{2\alpha}{\alpha-1}}_{\wred{L_{t}^{p}L_x^{\overrightarrow{q}}}(Q(\eta))}
\Big\{1+\big[1+
\frac{\eta^{{\frac{7-3\alpha-\frac{4\alpha}{p}}{2}}}}
{(\eta-\varrho)^{\frac{7-3\alpha-\frac{4\alpha}{p}}{2}}}\big]^{\frac{2}{\alpha-1}}\Big\} \\&
+\frac{1}{6}\Big(\|u\|_{\wred{L_t^{\infty}L_x^2}(Q(\eta))}^{2}+\|\nabla u\|_{L^{2}(Q(\eta))}^{2}\Big),\\ \nonumber
III\leq& \frac{C\eta^{\frac{3(\alpha+1)}{\alpha-1} }}{(\eta-\xi)^{\frac{8}{(\alpha-1)}}}
\|u\|^{\frac{2\alpha}{\alpha-1}}
_{\wred{L_{t}^{p}L_x^{\overrightarrow{q}}}(Q(\eta))}
\Big\{1+\big[1+
\frac{\eta^{{\frac{7-3\alpha-\frac{4\alpha}{p}}{2}}}}
{(\eta-\varrho)^{\frac{7-3\alpha-\frac{4\alpha}{p}}{2}}}\big]^{\frac{2}{\alpha-1}}\Big\} \\&
+\frac{1}{6}\Big(\|u\|_{\wred{L_t^{\infty}L_x^2}(Q(\eta))}^{2}+\|\nabla u\|_{L^{2}(Q(\eta))}^{2}\Big).
\end{align}
Using the Young inequality again, we conclude that
\begin{equation} IV\leq \frac{C\eta^{3 }}{(\eta-\xi)^{8}} \| \Pi\|^{2}_{L^{1} (Q(\eta))} +\frac{1}{6} \|u\|^{2}_{L^{2,\infty}(Q(\eta)))}.\Delta bel{last1}\end{equation}
After plugging \eqref{last2}-\eqref{last1} into \eqref{last3}, we apply
iteration Lemma \cite[Lemma V.3.1, p.161]{[Giaquinta]} to finish the proof.
\end{proof}
\mathrm{div}ection{ Regularity criteria at one scale \wred{without pressure}}\Delta bel{sec4}
As explained \wred{in Section 1}, the proof of Theorem \ref{[Leraybackself]} reduces to the proof of \wred{Theorem \ref{the1.5}.} Thanks to \eqref{wolfchae} or the results in \cite{[Wolf1],[JWZ]}, we just need to prove the following Caccioppoli type inequality \wred{given in Lemma \ref{zc22}}. We write
\begin{equation}\Delta bel{aerfa2}
\wred{\alpha=\frac{2}{\frac{1}{\overrightarrow{q}}}.}
\end{equation}
By the H\"older inequality, we just need consider the case that $\alpha$
is very close to 1. Therefore, for any $1<q_{i}<\infty$, we have
\begin{equation}
\frac{2}{3}\geq (\frac{2}{3}-\frac{1}{q_{i}})\alpha.
\Delta bel{q>24}\end{equation}
\wred{For example, $\overrightarrow{q}=(1000,\frac{1000}{990},\frac{1000}{990})$ do not hold for \eqref{q>24}, by the H\"older inequality, we just consider $(1000,\frac{1000}{999},\frac{1000}{999})$ to ensure that \eqref{q>24} is valid.}
\begin{equation}gin{lemma}\Delta bel{zc22} Let $\alpha$ \wgr{be given} in \eqref{aerfa2}. For \wred{any $R>0$}, there is an absolute constant $C$ such that
\begin{equation}\begin{aligned}\Delta bel{zwl}
&\|u\|^{2}_{\wred{L_t^{3}L_x^{\frac{18}{5}}}(Q(\frac{R}{2}))}+ \|\nabla u\|^{2}_{L^{2}(Q(\frac{R}{2}))}\\
\leq &
CR^{\frac{3\alpha-4}{\alpha}}\| u\|^{2}_{\wred{L_t^{\infty}L_x^{\overrightarrow{q}}}(Q(R))}
+CR^{\frac{5\alpha-8}{2(\alpha-1)}}\| u\|^{\frac{3\alpha}{2(\alpha-1)}}_{\wred{L_t^{\infty}L_x^{\overrightarrow{q}}}(Q(R))}
+CR^{\frac{4\alpha-6}{\alpha}}\| u\|^{3}_{\wred{L_t^{\infty}L_x^{\overrightarrow{q}}}(Q(R))}.
\end{aligned}\end{equation}
\end{lemma}
\begin{equation}gin{proof}
\wred{Consider} $0<R/2\leq r<\frac{3r+\rho}{4}<\frac{r+\rho}{2}<\rho\leq R$. Let $\phi(x,t)$ be non-negative smooth function supported in $Q(\frac{r+\rho}{2})$ such that
$\phi(x,t)\equiv1$ on $Q(\frac{3r+\rho}{4})$,
$|\nabla \phi| \leq C/(\rho-r) $ and $
|\nabla^{2}\phi|+|\partial_{t}\phi|\leq C/(\rho-r)^{2} .$
The interpolation inequality in \cite{[BP]} allows us to arrive at \begin{equation}\Delta bel{4.24}
\|u\|_{L^{3}\wred{(B(\frac{r+3\rho}{4}))}}\leq \|u\|^{^{\frac{\alpha}{4-\alpha}}}_{L^{\overrightarrow{q}}(\wred{B(\frac{r+3\rho}{4})})}
\|u\|^{\frac{2(2-\alpha)}{4-\alpha}}_{L^{\overrightarrow{t}}(\wred{B(\frac{r+3\rho}{4})})},
\end{equation}
Combining this and \eqref{q>24} yields that
$$
\frac{1}{\overrightarrow{t}}=\frac12, ~~\textxt{and}~~~t_{j}=\frac{6q_{i}(2-\alpha)}{(4-\alpha)q_j-3\alpha}\geq2.
$$
Thence, inequality \eqref{locanlast} gives
$$\begin{aligned}
\|u\|_{L^{\overrightarrow{t}}(\wred{B}(\frac{r+3\rho}{4}))}&\leq
\|u-\wred{\overline{u}_{\rho}} \|_{L^{\overrightarrow{t}}(B(\frac{r+3\rho}{4}))}
+\|\wred{\overline{u}_{\rho}}\|_{L^{\overrightarrow{t}}(B(\frac{r+3\rho}{4}))}\\
&\leq\wred{C}
\|\nabla u \|_{L^{2}(\wred{B(\rho)})}
+\wred{C}\rho^{\frac{1}{\overrightarrow{t}}-\frac{1}{\overrightarrow{q}}}
\| u \|_{L^{\overrightarrow{q}}(\wred{B(\rho)})}.
\end{aligned}$$
\wred{Plugging this into \eqref{4.24}}, we infer that \wred{for $\theta=\fracrac{\alpha}{4-\alpha}$,}
$$\begin{aligned}
\|u\|^{3}_{L^{3}B(\frac{r+3\rho}{4})}&\leq \wred{C}\|u\|^{3\theta}_{L^{\overrightarrow{q}}(B(\frac{r+3\rho}{4}))}
\|u\|^{3(1-\theta)}_{L^{\overrightarrow{t}}(B(\frac{r+3\rho}{4}))} \\ &\leq\wred{C}
\|u\|^{3\theta}_{\wred{L^{\overrightarrow{q}}}(B(\frac{r+3\rho}{4}))}\Big[\|\nabla u\|_{L^{2}\wred{(B(\rho))}}+\rho^{\frac{1}{\overrightarrow{t}}-\frac{1}{\overrightarrow{q}}}\|u\|_{L^{\overrightarrow{q}}(\wred{B(\rho)})}
\Big]^{3(1-\theta)}.
\end{aligned}$$
As a consequence, we infer \wred{that}
$$
\|u\|^{3}_{L^{3}(Q(\frac{r+3\rho}{4}))}\leq \wred{C}\rho^{\frac{4(\alpha-1)}{4-\alpha}}\|u\|^{\frac{3\alpha}{4-\alpha}}_{\wred{L_t^{\infty}L_x^{\overrightarrow{q}}}(Q(\rho))} \|\nabla u\|_{L^{2}(Q(\rho))}^{\frac{3(4-2\alpha)}{4-\alpha}}
+\wred{C}\rho^{\frac{5\alpha-6}{\alpha}}\|u\|^{3}_{\wred{L_t^{\infty}L_x^{\overrightarrow{q}}}(Q(\rho))}.
$$
Let $\nabla\Pi_{h}=\mathcal{W}_{3,B(\frac{r+3\rho}{4})}(u)$, then,
from \eqref{ph}-\eqref{p2}, we have
\begin{equation}gin{align}
&\|\nabla \Pi_{h}\|_{L^{3}(Q(\frac{r+3\rho}{4}))}\leq C\|u\|_{L^{3}(Q(\frac{r+3\rho}{4}))},\Delta bel{wp1}\\
&\| \Pi_{1}\|_{L^{2}(Q(\frac{r+3\rho}{4}))}\leq C\|\nabla u\|_{L^{2}(Q(\frac{r+3\rho}{4}))},\Delta bel{wp2}\\
&\| \Pi_{2}\|_{L^{\frac{3}{2}}(Q(\frac{r+3\rho}{4}))}\leq C\| |u|^{2}\|_{L^{\frac{3}{2}}(Q(\frac{r+3\rho}{4}))}.\Delta bel{wp3}
\end{align}
Since $v=u+\nabla\Pi_{h}$, the H\"older inequality and \eqref{wp1} allows us to write
\begin{equation}gin{align}
\iint_{Q(\rho)} \wred{| v|^2}\Bigig| \textxt{Div\,}elta \phi^{4}+ \partial_{t}\phi^{4}\Bigig| \leq& \frac{C}{(\rho-r)^{2}}\iint_{Q(\frac{r+\rho}{2})} |u|^{2}+|\nabla\Pi_{h}|^{2}
\nonumber\\\leq& \frac{C\rho^{5/3}}{(\rho-r)^{2}}\Big(\iint_{Q(\frac{r+\rho}{2})} |u|^{3}+|\nabla\Pi_{h}|^{3}\Big)^{\frac{2}{3}}
\nonumber\\\leq& \frac{C\rho^{5/3}}{(\rho-r)^{2}}\|u\|_{L^{3}(Q(\frac{r+3\rho}{4}))}^{2}.\Delta bel{53.2} \end{align}
It follows from H\"older's inequality, $v=u+\nabla\Pi_{h}$ and \eqref{wp1} that
\begin{equation}gin{align}
\iint_{Q(\rho)}|v|^{2}\phi^{3}u\cdot\nabla \phi
\leq \frac{C}{(\rho-r)}
\| u \|^{3}_{L^{3}(Q(\frac{r+3\rho}{4}))}.
\end{align}
According to interior estimate of harmonic function \eqref{h1} and \eqref{wp1}, we have
$$\begin{aligned}\|\nabla^{2}\Pi_{h} \|_{L^{20/7}(Q(\frac{r+\rho}{2}))}&\leq
\frac{\wred{C} (r+\rho) }{(\rho-r)^{ 2}}
\|\nabla\Pi_{h} \|_{L^{3}(Q( \frac{r+3\rho}{4} ))}\\
&\leq
\frac{ C\rho }{(\rho-r)^{ 2}}
\|u \|_{L^{3}(Q( \frac{r+3\rho}{4}))},
\end{aligned}$$
from which it follows that
\begin{equation}gin{align}
&\iint_{Q(\rho)} \phi^{4}( u\otimes v :\nabla^{2}\Pi_{h} ) \nonumber\\
\leq&
\| v\phi^{2}\|_{L^{3}(Q(\frac{r+\rho}{2}))}
\| u \|_{L^{3}(Q(\frac{r+\rho}{2}))}\|\nabla^{2}\Pi_{h} \|_{L^{3}(Q(\frac{r+\rho}{2}))}
\nonumber\\
\leq &\frac{ C\rho }{(\rho-r)^{ 2}}
\|u \|^{3}_{L^{3}(Q( \frac{r+3\rho}{4} ))}.
\end{align}
Taking advantage of the H\"older inequality, \eqref{wp2} and Young's inequality, we infer that
\begin{equation}gin{align}
\iint_{Q(\rho)} \phi^{3} \Pi_{1}v\cdot\nabla \phi
&\leq \frac{C}{(\rho-r)}\| v\|_{L^{2}(Q(\frac{r+\rho}{2}))}
\| \Pi_{1} \|_{L^{2}(Q(\frac{r+\rho}{2}))} \nonumber\\
&\leq \frac{C}{(\rho-r)^{2}}\| v\|^{2}_{L^{2}(Q(\frac{r+\rho}{2}))}
+\frac{1}{16}\| \Pi_{1} \|^{2}_{L^{2}(Q(\frac{r+3\rho}{4}))} \nonumber\\
&\leq \frac{C\rho^{5/3}}{(\rho-r)^{2}}\|u\|_{L^{3}(Q(\frac{r+3\rho}{4}))}^{2}+\frac{1}{16}\| \nabla u\|^{2}_{L^{2}(Q(\frac{r+3\rho}{4}))}.
\end{align}
We conclude from the H\"older inequality \wred{and \eqref{wp3} that}
\begin{equation}gin{align}
\iint_{Q(\rho)} \phi^{3} \Pi_{2}v\cdot\nabla \phi
\leq \frac{C}{(\rho-r)}\| v\phi^{2}\|_{L^{3}(Q(\frac{r+\rho}{2}))}
\| \Pi_{2} \|_{L^{\frac{3}{2}}(Q(\frac{r+\rho}{2}))} \leq\frac{ C }{(\rho-r)}
\|u \|^{3}_{L^{3}(Q(\frac{r+3\rho}{4} ))}.\Delta bel{locp5}
\end{align}
Plugging \eqref{53.2}-\eqref{locp5} into local energy \wred{inequality \eqref{wloc1}, we infer} that
\begin{equation}gin{align}
\mathrm{div}up_{-\rho^{2}\leq t\leq0}\int_{B(\rho)}|v\phi^{2}|^2 + \iint_{Q(\rho)}\big|\nabla( v\phi^{2})\big|^2 \leq& \frac{C\rho^{5/3}}{(\rho-r)^{2}}\|u\|_{L^{3}(Q(\frac{r+3\rho}{4}))}^{2} +\frac{ C\rho }{(\rho-r)^{ 2}}
\|u \|^{3}_{L^{3}(Q( \frac{r+3\rho}{4} ))}\nonumber\\&+\frac{ C }{(\rho-r)}
\|u \|^{3}_{L^{3}(Q( \frac{r+3\rho}{4} ))}\wred{+\frac{1}{16}\| \nabla u\|^{2}_{L^{2}(Q(\frac{r+3\rho}{4}))}.}\Delta bel{keyl}
\end{align}
Applying the interior estimate of harmonic function
\eqref{h1} and \eqref{wp1} implies that
$$\begin{aligned}\|\nabla\Pi_{h}\|^{2}_{\wred{L_t^{3}L_x^{\frac{18}{5}}}(Q(r))}&\leq \frac{Cr^{\frac{5}{3}}}{(\rho-r)^{2}}\|\nabla\Pi_{h}\|^{2}_{L^{3}Q(\wred{\frac{r+3\rho}{4}})}\leq \frac{Cr^{\frac{5}{3}}}{(\rho-r)^{2}}\wred{\|u\|^{2}_{L^{3}Q(\frac{r+3\rho}{4})},}
\end{aligned}$$
\wred{which together with the triangle inequality and \eqref{keyl} ensures} that
$$\begin{aligned}
\|u\|^{2}_{\wred{L_t^{3}L_x^{\frac{18}{5}}}(Q(r))} \leq& \|v\|^{2}_{\wred{L_t^{3}L_x^{\frac{18}{5}}}(Q(r))}+\|\nabla\Pi_{h}\|^{2}_{\wred{L_t^{3}L_x^{\frac{18}{5}}}(Q(r))}\\
\leq& C\Big\{\|v\|_{\wred{L_t^{\infty}L_x^2}(Q(r))}^{2}+\|\nabla v\|_{L^{2}(Q(r))}^{2}\Big\}+\frac{Cr^{\frac{5}{3}}}{(\rho-r)^{2}}\|u\|^{2}_{L^{3}Q(\frac{r+3\rho}{4})}\\
\leq & \frac{C\rho^{5/3}}{(\rho-r)^{2}}\|u\|_{L^{3}(Q(\frac{r+3\rho}{4}))}^{2}+\frac{ C\rho }{(\rho-r)^{ 2}}
\|u \|^{3}_{L^{3}(Q( \frac{r+3\rho}{4} ))} \\&+\frac{ C }{(\rho-r)}
\|u \|^{3}_{L^{3}(Q( \frac{r+3\rho}{4} ))}+\frac{1}{16}\| \nabla u\|^{2}_{L^{2}(Q(\frac{r+3\rho}{4}))}.
\end{aligned}$$
\wred{Using \eqref{h1} and \eqref{wp1}} once again, we know that
$$
\|\nabla^{2} \Pi_{h}\|^{2}_{L^{2}(Q(r))}\leq \frac{Cr^{3}}{(\rho-r)^{5}} \|\nabla\Pi_{h}\|^{2}_{L^{2}(Q(\frac{r+\rho}{2}))}\leq \frac{Cr^{3}\rho^{5/3}}{(\rho-r)^{5}} \| u \|^{2}_{L^{3}(Q(\frac{r+3\rho}{4}))}.
$$
Combining the triangle \wred{inequality and \eqref{keyl}} yields
\begin{equation}gin{align}
\|\nabla u\|^{2}_{L^{2}(Q(r))}&\leq \|\nabla v\|^{2}_{L^{2}(Q(r))}+
\|\nabla^{2} \Pi_{h}\|^{2}_{L^{2}(Q(r))}\nonumber\\\leq&
\Big\{\frac{C\rho^{5/3}}{(\rho-r)^{2}}+\frac{Cr^{3}\rho^{5/3}}{(\rho-r)^{5}} \Big\} \| u \|^{2}_{L^{3}(Q(\frac{r+3\rho}{4}))}\nonumber\\&
+\Big\{\frac{ C\rho }{(\rho-r)^{ 2}} +\frac{ C }{(\rho-r)}\Big\}
\|u \|^{3}_{L^{3}(Q( \frac{r+3\rho}{4} ))}\wred{+\frac{1}{16}\| \nabla u\|^{2}_{L^{2}(Q(\frac{r+3\rho}{4}))}.}
\end{align}
Then, the following estimate holds
$$\begin{aligned}
&\|u\|^{2}_{\wred{L_t^{3}L_x^{\frac{18}{5}}}(Q(r))}+ \|\nabla u\|^{2}_{L^{2}(Q(r))}
\\\leq&
\Big\{\frac{C\rho^{5/3}}{(\rho-r)^{2}}+\frac{Cr^{3}\rho^{5/3}}{(\rho-r)^{5}} \Big\}^{\frac{4-\alpha}{\alpha}}\rho^{\frac{8(\alpha-1)}{3\alpha}} \| u \|^{2}_{\wred{L_t^{\infty}L_x^{\overrightarrow{q}}}(Q(\rho))}\nonumber\\
&+\Big\{\frac{C\rho^{5/3}}{(\rho-r)^{2}}+\frac{Cr^{3}\rho^{5/3}}{(\rho-r)^{5}} \Big\}
\rho^{\frac{2(5\alpha-6)}{3\alpha}}\|u\|^{2}_{\wred{L_t^{\infty}L_x^{\overrightarrow{q}}}(Q(\rho))}\nonumber\\&
+\Big\{\frac{ C\rho }{(\rho-r)^{ 2}} +\frac{ C }{(\rho-r)}\Big\}^{\frac{4-\alpha}{2(\alpha-1)}}\rho^{2}
\|u \|^{\frac{3\alpha}{2(\alpha-1)}}_{\wred{L_t^{\infty}L_x^{\overrightarrow{q}}}(Q( \rho))}\nonumber\\
&+\Big\{\frac{ C\rho }{(\rho-r)^{ 2}} +\frac{ C }{(\rho-r)}\Big\}\rho^{\frac{5\alpha-6}{\alpha}}\|u\|^{3}_{\wred{L_t^{\infty}L_x^{\overrightarrow{q}}}(Q(\rho))}+\frac{3}{16}\| \nabla u\|^{2}_{L^{2}(Q(\rho))}.
\end{aligned} $$
Now, we are in a position to apply \wred{iteration lemma} \cite[Lemma V.3.1, p.161 ]{[Giaquinta]} to the latter estimate to find that
$$\begin{aligned}
&\|u\|^{2}_{\wred{L_t^{3}L_x^{\frac{18}{5}}}(Q(\frac{R}{2}))}+ \|\nabla u\|^{2}_{L^{2}(Q(\frac{R}{2}))}\\
\leq &
CR^{\frac{3\alpha-4}{\alpha}}\| u\|^{2}_{\wred{L_t^{\infty}L_x^{\overrightarrow{q}}}(Q(R))}
+CR^{\frac{5\alpha-8}{2(\alpha-1)}}\| u\|^{\frac{3\alpha}{2(\alpha-1)}}_{\wred{L_t^{\infty}L_x^{\overrightarrow{q}}}(Q(R))}
+CR^{\frac{4\alpha-6}{\alpha}}\| u\|^{3}_{\wred{L_t^{\infty}L_x^{\overrightarrow{q}}}(Q(R))}.
\end{aligned}$$
This achieves the proof of \wred{this lemma}.
\end{proof}
\mathrm{div}ection*{Acknowledgement}
The authors would like to express their sincere gratitude to Dr. Xiaoxin Zheng at the School
of Mathematics and Systems Science, Beihang University, for calling our attention to the
problem involving $\varepsilon$-regularity criteria in anisotropic Lebesgue spaces.
The research of Wang was partially supported by the National Natural Science Foundation of China under grant No. 11601492 and
the Youth Core Teachers Foundation of Zhengzhou University of
Light Industry.
\wgr{The research of Wu was partially supported by the National
Natural Science Foundation of China under grant No. 11771423 and No. 11671378.}
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|
{\mbox {\bf b}}egin{document}
\title{\Large Graph Sparsification via Refinement Sampling}
\setcounter{page}{0}
\author{Ashish Goel\thanks{
Departments of Management Science and Engineering and (by courtesy)
Computer Science, Stanford University.
Email: {\tt [email protected]}.
Research supported in part by NSF award IIS-0904325.}\\
\and Michael Kapralov\thanks{
Institute for Computational and Mathematical Engineering, Stanford University.
Email: {\tt [email protected]}. Research supported by a Stanford Graduate Fellowship.}\\
\and Sanjeev Khanna\thanks{Department of Computer and Information Science, University of Pennsylvania,
Philadelphia PA. Email: {\tt [email protected]}.
Supported in
part by NSF Awards CCF-0635084 and IIS-0904314.}
}
\maketitle
\thispagestyle{empty}
\pdfbookmark[1]{Abstract}{MyAbstract}
{\mbox {\bf b}}egin{abstract}
A graph $G'(V,E')$ is an {{{\epsilon}ilon}m ${{\epsilon}ilon}ps$-sparsification} of $G$ for some ${{\epsilon}ilon}ps >
0$, if every (weighted) cut in $G'$ is within $(1\pm {{\epsilon}ilon})$ of the
corresponding cut in $G$. A celebrated result of Bencz\'{u}r and Karger
shows that for every undirected graph $G$, an ${{\epsilon}ilon}ps$-sparsification with
$O(n \log n/{{\epsilon}ilon}^2)$ edges can be constructed in $O(m\log^2n)$ time. The
notion of cut-preserving graph sparsification has played an important role
in speeding up algorithms for several fundamental network design and routing
problems. Applications to modern massive data sets often constrain
algorithms to use computation models that restrict random access to the
input. The semi-streaming model, in which the algorithm is constrained to
use $\tilde O(n)$ space, has been shown to be a good abstraction for
analyzing graph algorithms in applications to large data sets. Recently, a
semi-streaming algorithm for graph sparsification was presented by Anh and
Guha; the total running time of their implementation is $\Omega(mn)$, too
large for applications where both space and time are important. In this
paper, we introduce a new technique for graph sparsification, namely {{{\epsilon}ilon}m
refinement sampling}, that gives an $\tilde{O}(m)$ time semi-streaming
algorithm for graph sparsification.
Specifically, we show that refinement sampling can be used to design a
one-pass streaming algorithm for sparsification that takes $O(\log\log n)$
time per edge, uses $O(\log^2 n)$ space per node, and outputs an
${{\epsilon}ilon}ps$-sparsifier with $O(n\log^3 n/{{\epsilon}ilon}^2)$ edges. At a slightly increased
space and time complexity, we can reduce the sparsifier size to $O(n \log
n/{{\epsilon}ilon}^2)$ edges matching the Bencz\'{u}r-Karger result, while improving upon
the Bencz\'{u}r-Karger runtime for $m=\omega(n\log^3 n)$. Finally, we show
that an ${{\epsilon}ilon}ps$-sparsifier with $O(n \log n/{{\epsilon}ilon}^2)$ edges can be constructed
in two passes over the data and $O(m)$ time whenever $m =
\Omega(n^{1+\delta})$ for some constant $\delta > 0$. As a by-product of our approach,
we also obtain an $O(m \log\log n+ n \log n)$ time streaming algorithm to compute
a sparse $k$-connectivity certificate of a graph.
\iffalse
We consider the problem of constructing sparsifiers of undirected graphs in the semi-streaming model. Our main result is an efficient one-pass algorithm for constructing a Bencz\'{u}r-Karger type sparsifier. The algorithm can be implemented using $O(\log\log n)$ work per edge and $O(\log^2 n)$ space per node. The number of edges in the sampled graph is $O(n\log^3 n/{{\epsilon}ilon}^2)$. A two-stage sampling scheme can be used to reduce the size of the sample to $O(n\log n/{{\epsilon}ilon}^2)$ without violating the restrictions of the semi-streaming model or introducing additional passes. The resulting one-pass algorithm take amortized $O(\log\log n+(n/m)\log^5 n)$ work per edge, which improves upon the runtime of the original Bencz\'{u}r-Karger sparsification scheme when $m=\Omega(n\log^3 n)$.
\fi
{{\epsilon}ilon}nd{abstract}
\section{Introduction}
The notion of graph sparsification was introduced in \cite{benczurkarger96},
where the authors gave a near linear time procedure that takes as input an
undirected graph $G$ on $n$ vertices and constructs a weighted subgraph $H$ of
$G$ with $O(n\log n/{{\epsilon}ilon}^2)$ edges such that the value of every cut in $H$ is
within a $1\pm {{\epsilon}ilon}$ factor of the value of the corresponding cut in $G$. This
algorithm has subsequently been used to speed up algorithms for finding
approximately minimum or sparsest cuts in graphs (\cite{benczurkarger96,
krv06}), as well as in a host of other applications (e.g. \cite{kl02}). A
more general class of spectral sparsifiers was recently introduced by Spielman
and Srivastava in \cite{ss:sample2008}. The algorithms developed in
\cite{benczurkarger96} and \cite{ss:sample2008} take near-linear time in the
size of the graph and produce very high quality sparsifiers, but require
random access to the edges of the input graph $G$, which is often
prohibitively expensive in applications to modern massive data sets. The
streaming model of computation, which restricts algorithms to use a small
number of passes over the input and space polylogarithmic in the size of the
input, has been studied extensively in various application domains
(e.g. \cite{b:streaming}), but has proven too restrictive for even the
simplest graph algorithms (even testing $s-t$ connectivity requires
$\Omega(n)$ space). The less restrictive semi-streaming model, in which the
algorithm is restricted to use $\tilde O(n)$ space, is more suited for graph
algorithms~\cite{fkmsz05}. The problem of constructing graph sparsifiers in
the semi-streaming model was recently posed by Anh and Guha~\cite{anh-guha},
who gave a one-pass algorithm for finding Bencz\'{u}r-Karger type sparsifiers
with a slightly larger number of edges than the original Bencz\'{u}r-Karger
algorithm, i.e. $O(n\log n\log\frac{m}{n}/{{\epsilon}ilon}^2)$ as opposed to $O(n\log
n/{{\epsilon}ilon}^2)$. Their algorithm requires only one pass over the data, and their
analysis is quite non-trivial. However, its time complexity is
$\Omega(mn\mbox{\ polylog}(n))$, making it impractical for applications where
both time and space are important constraints\footnote{As is often the case
for semi-streaming algorithms, Anh and Guha do not explicitly compute the
running time of their algorithm; $\Omega(mn\mbox{\ polylog}(n))$ is the best
running time we can come up with for their algorithm.}
Apart from the issue of random access vs disk, the semi-streaming model is
also important for scenarios where edges of the graph are
revealed one at a time by an external process. For example, this application maps
well to online social networks where edges arrive one by one, but efficient
network computations may be required at any time, making it particularly
useful to have a dynamically maintained sparsifier.
\paragraph{Our results:} We introduce the concept of {{\epsilon}ilon}mph{refinement
sampling}. At a high level, the basic idea is to sample edges at
geometrically decreasing rates, using the sampled edges at each rate to refine
the connected components from the previous rate. The sampling rate at which
the two endpoints of an edge get separated into different connected components
is used as an approximate measure of the ``strength'' of that edge. We use
refinement sampling to obtain two algorithms for computing Bencz\'{u}r-Karger
type sparsifiers of undirected graphs in the semi-streaming model
efficiently. The first algorithm requires $O(\log n)$ passes, $O(\log n)$
space per node, $O(\log n\log\log n)$ work per edge and produces sparsifiers
with $O(n\log^2 n/{{\epsilon}ilon}^2)$ edges. The second algorithm requires one pass over
the edges of the graph, $O(\log^2 n)$ space per node, $O(\log\log n)$ work per
edge and produces sparsifiers with $O(n\log^3 n/{{\epsilon}ilon}^2)$ edges. Several
properties of these results are worth noting:
{\mbox {\bf b}}egin{enumerate}
\item In the incremental model, the amortized running time per edge arrival is
$O(\log \log n)$, which is quite practical and much better than the
previously best known running time of $\Omega(n)$.
\item The sample size can be improved for both algorithms by running the original
Benc\'{u}r-Karger algorithm on the sampled graph without violating the
restrictions of the semi-streaming model, yielding $O(\log n\log \log
n+(\frac{n}{m})\log^4 n)$ and $O(\log \log n+(\frac{n}{m})\log^5 n)$ amortized work per edge
respectively.
\item Somewhat surprisingly, this two-stage (but still semi-streaming)
algorithm improves upon the runtime of the original sparsification scheme
when $m=\omega(n\log^2n)$ for the $O(\log n)$-pass version and
$m=\omega(n\log^3 n)$ for the one-pass version.
\item As a by-product of our analysis, we show that refinement sampling can be
regarded as a one-pass algorithm for producing a sparse connectivity
certificate of a weighted undirected graph (see Corollary
\ref{cor:sparse-cert}). Thus we obtaining a streaming analog
of the Nagamochi-Ibaraki result~\cite{nagamochi-ibaraki} for producing sparse
certificates, which is in turn used in the Benc\'{u}r-Karger sampling.
{{\epsilon}ilon}nd{enumerate}
Finally, in Section \ref{sec:two-pass} we give an algorithm for
constructing $O(n\log n/{{\epsilon}ilon}^2)$-size sparsifiers in $O(m)$ time using two
passes over the input when $m=\Omega(n^{1+\delta})$.
\noindent
\paragraph{Related Work:}
In \cite{anh-guha} the authors give an algorithm
for sparsification in the semi-streaming model based on the observation that one can use the constructed sparsification of the currently received part of the graph to estimate of the strong connectivity of a newly received edge. A brief outline of the algorithm is as follows. Denote the edges of $G$ in their order in the stream by $e_1,\ldots, e_m$. Set $H_0=(V, {{\epsilon}ilon}mptyset)$. For every $t>0$ compute the strength $s_t$ of $e_t$ in $H_{t-1}$, and with probability $p_{e_t}=\min\{\rho/s_t, 1\}$ set $H_{t}=(V, E(H_{t-1})\cup \{e_t\})$, giving $e_t$ weight $1/p_{e_t}$ in $H_{t}$ and $H_t=H_{t-1}$ otherwise. For every $t$ the graph $H_t$ is an ${{\epsilon}ilon}$-sparsification of the subgraph received by time $t$.
The authors show that this algorithm yields an ${{\epsilon}ilon}$-sparsifier with $O(n\log n\log\frac{m}{n}/{{\epsilon}ilon}^2)$ edges. However, it is unclear how one can calculate the strengths $s_t$ efficiently. A naive implementation would take $\Omega(n)$ time for each $t$, resulting in $\Omega(mn)$ time overall. One could conceivably use the fact that $H_{t-1}$ is always a subgraph of $H_{t}$, but to the best of our knowledge there are no results on efficiently calculating or approximating {{\epsilon}ilon}mph{strong} connectivities in the incremental model.
It is important to emphasize that our techniques for obtaining an efficient one-pass sparsification algorithm are very different from the approach of \cite{anh-guha}. In particular, the structure of dependencies in the sampling process is quite different. In the algorithm of \cite{anh-guha} edges are not sampled independently since the probability with which an edge is sampled depends on the the coin tosses for edges that came earlier in the stream. Our approach, on the other hand, decouples the process of estimating edge strengths from the process of producing the output sample, thus simplifying analysis and making a direct invocation of the Bencz\'{u}r-Karger sampling theorem possible.
\noindent
\paragraph{Organization:}
Section~\ref{sec:prelim} introduces some notation as well as reviews the
Bencz\'{u}r-Karger sampling algorithm. We then introduce in Section~\ref{sec:refinement}
our {{\epsilon}ilon}mph{refinement sampling} scheme, and show how it can be used to obtain a sparsification algorithm requiring $O(\log n)$ passes and $O(\log n\log\log n)$ work per edge. The size of the sampled graph is $O(n\log^2 n/{{\epsilon}ilon}^2)$, i.e. at most $O(\log n)$ times larger than that produced by Bencz\'{u}r-Karger sampling.
Finally, in Section \ref{sec:onepass} we build on the ideas of Section~\ref{sec:refinement}
to obtain a one-pass algorithm with $O(\log \log n)$ work per edge at the expense of increasing the size of the sample to $O(n\log^3 n/{{\epsilon}ilon}^2)$.
\section{Preliminaries}
\label{sec:prelim}
We will denote by $G(V, E)$ the input undirected graph with vertex set $V$ and edge set $E$ with $|V|=n$ and $|E|=m$.
For any ${{\epsilon}ilon}ps > 0$, we say that a weighted graph $G'(V,E')$ is an {{{\epsilon}ilon}m ${{\epsilon}ilon}ps$-sparsification}
of $G$ if every (weighted) cut in $G'$ is within $(1\pm {{\epsilon}ilon})$ of the corresponding
cut in $G$.
Given any two collections of sets that partition $V$, say $S_1$ and $S_2$,
we say that
$S_2$ is a {{{\epsilon}ilon}m refinement} of $S_1$ if for any $X \in S_1$ and $Y \in S_2$,
either $X \cap Y = {{\epsilon}ilon}mptyset$ or $Y \subset X$. In other words, $S_1 \cup S_2$
form a laminar set system.
\subsection{Bencz\'{u}r-Karger Sampling Scheme}
We say that a graph is {{\epsilon}ilon}mph{$k$-connected} if the value of each cut in $G$ is at least $k$.
The Bencz\'{u}r-Karger sampling scheme uses a more strict notion of connectivity, referred to as
{{\epsilon}ilon}mph{strong connectivity}, defined as follows:
{\mbox {\bf b}}egin{definition}\cite{benczurkarger96}
A {{\epsilon}ilon}mph{$k$-strong component} is a maximal $k$-connected vertex-induced subgraph.
The {{\epsilon}ilon}mph{strong connectivity} of an edge $e$, denoted by $s_e$, is the largest $k$ such that a $k$-strong component contains $e$.
{{\epsilon}ilon}nd{definition}
Note that the set of $k$-strong components form a partition of the vertex set of $G$, and the set of $k+1$-strong components forms a refinement this partition.
We say $e$ is {{\epsilon}ilon}mph{$k$-strong} if its strong connectivity is $k$ or more, and {{\epsilon}ilon}mph{$k$-weak} otherwise.
The following simple lemma will be useful in our analysis.
{\mbox {\bf b}}egin{lemma}\cite{benczurkarger96}\label{lm:k-weak}
The number of $k$-weak edges in a graph on $n$ vertices is bounded by $k(n-1)$.
{{\epsilon}ilon}nd{lemma}
The sampling algorithm relies on the following result:
{\mbox {\bf b}}egin{theorem}\cite{benczurkarger96} \label{thm:bk-sampling}
Let $G'$ be obtained by sampling edges of $G$ with probability $p_e=\min\{\frac{\rho}{{{\epsilon}ilon}^2 s_e}, 1\}$, where $\rho=16(d+2)\ln n$, and giving each sampled edge weight $1/p_e$. Then
$G'$ is an ${{\epsilon}ilon}ps$-sparsification of $G$ with probability at least $1-n^{-d}$.
Moreover, expected number of edges in $G'$ is $O(n \log n)$.
{{\epsilon}ilon}nd{theorem}
It follows easily from the proof of theorem \ref{thm:bk-sampling} in \cite{benczurkarger96} that
if we sample using an {{{\epsilon}ilon}m underestimate} of edge strengths, the resulting graph is
still an ${{\epsilon}ilon}ps$-sparsification.
{\mbox {\bf b}}egin{corollary}\label{cor:oversampling}
Let $G'$ be obtained by sampling each edge of $G$ with probability $\tilde p_e\geq p_e$ and and give every sampled edge $e$ weight $1/\tilde p_e$.
Then
$G'$ is an ${{\epsilon}ilon}ps$-sparsification of $G$ with probability at least $1-n^{-d}$.
{{\epsilon}ilon}nd{corollary}
In \cite{benczurkarger96} the authors give an $O(m\log^2 n)$ time algorithm for calculating estimates of strong connectivities that are sufficient for sampling. The algorithm, however, requires random access to the edges of the graph, which is disallowed in the semi-streaming model. More precisely,
the procedure for estimating edge strengths given in \cite{benczurkarger96} relies on the Nagamochi-Ibaraki algorithm for obtaining sparse certificates for edge-connectivity
in $O(m)$ time (\cite{nagamochi-ibaraki}). The algorithm of \cite{nagamochi-ibaraki} relies on random access to edges of the graph and to the best of our knowledge no streaming implementation is known. In fact we show in Corollary \ref{cor:sparse-cert} that refinement sampling yields a streaming algorithm for producing sparse certificates for edge-connectivity in one pass over the data.
In what follows we will consider unweighted graphs to simplify notation. The results obtained can be easily extended to the polynomially weighted case as outlined in Remark \ref{rmk:weights} at the end of Section \ref{sec:onepass}.
\section{Refinement Sampling}
\label{sec:refinement}
We start by introducing the idea of refinement sampling that gives a simple algorithm
for efficiently computing a BK-sample, and serves as a building block for our streaming algorithms.
To motivate refinement sampling, let us consider the simpler problem of
identifying all edges of strength at least $k$ in the input graph $G(V,E)$. A natural idea to do so
is as follows: (a) generate a graph $G'$ by sampling edges of $G$ with probability $\tilde{O}(1/k)$,
(b) find connected components of $G'$, and (c) output all edges $(u,v) \in E$ as such that
$u$ and $v$ are in the same connected component in $G'$.
The sampling rate of $\tilde{O}(1/k)$ suggests that if an edge $(u,v)$ has
strong connectivity below $k$, the vertices $u$ and $v$ would end up in different components in $G'$,
and conversely, if the strong connectivity of $(u,v)$ is above $k$, they are likely to stay connected
and hence output in step $(c)$.
While this process indeed filters out most $k$-weak edges,
it is easy to construct examples where the output will contain many edges of strength $1$ even
though $k$ is polynomially large (a star graph, for instance).
The idea of refinement sampling is to get around this by successively {{{\epsilon}ilon}m refining} the sample
obtained in the final step $(c)$ above.
In designing our algorithm, we will repeatedly invoke the subroutine
{{\mbox {\bf b}}f \mbox{\sc{Refine}}}$(S, p)$ that essentially implements the simple idea described above.
{\mbox {\bf b}}egin{description}
\item[Function:] {{\mbox {\bf b}}f \mbox{\sc{Refine}}}$(S, p)$
\item[Input:] Partition $S$ of $V$, sampling probability $p$.
\item[Output:] Partition $S'$ of $V$, a refinement of $S$.
\item[1.] Take a uniform sample $E'$ of edges of $E$ with probability $p$.
\item[2.] For each $U\in S, U\subseteq V$ let $C(U)$ be the set of connected components of $U$ induced by $E'$.
\item[3.] Return $S':=\cup_{U\in S} C(U)$.
{{\epsilon}ilon}nd{description}
It is easy to see that {{\mbox {\bf b}}f \mbox{\sc{Refine}}}~can be implemented using $O(n)$ space, a total of $n$ {{\mbox {\bf b}}f \mbox{\sc{Union}}}~ operations with $O(n\log n)$ overall cost and $m$ {{\mbox {\bf b}}f \mbox{\sc{Find}}}~ operations with $O(1)$ cost per operation, for an overall running time of
$O(n \log n + m)$(see, e.g. \cite{b:clr}).
Also, {{\mbox {\bf b}}f \mbox{\sc{Refine}}} ~can be implemented using a single pass over the set of edges.
A scheme of refinement relations between $S_{l,k}$ is given in Fig. \ref{fig:f1}.
The {{\mbox {\bf b}}f refinement sampling} algorithm computes partitions $S_{l,j}$ for $l=1,\ldots,L$ and $j=0, 1, \ldots,K$. Here $L=\log (2n)$ is the number of strength levels (the factor of $2$ is chosen for convenience to ensure that $S_{L, K}$ consists of isolated vertices whp), $K$ is a parameter which we call the {{{\epsilon}ilon}m strengthening} parameter. Also, we choose a parameter $\phi> 0$, which we will refer to as the oversampling parameter. For a partition $S$, let $X(S)$ denote all the edges in $E$ which have endpoints in two different sets in $S$. The partitions are computed as follows:
{\mbox {\bf b}}egin{description}
\item[Algorithm 1 (Refinement Sampling)]
\item[Initialization:] $S_{l,0} = \{V \}$ for $l=1,\ldots,L$.
\item[1.] Set $k:=1$
\item[2.] For each $l$, $1\leq l\leq L$, set $S_{l,k}:={{\mbox {\bf b}}f \mbox{\sc{Refine}}} (S_{l,k-1}, 2^{-l})$.
\item[3.] Set $k:=k+1$. If $k<K$, go to step 1.
\item[4.] For each $e\in E$ define $L(e) = \min\left\lbrace l : e \in X(S_{l,K})\right\rbrace$. Sample edge $e$ with probability $z(e) = \min\{1, \frac{\phi}{{{\epsilon}ilon}^2 2^{L(e)}}\}$ and assign it weight $1/z(e)$. Let $R(\phi,K)$ denote the set of edges sampled during this step; we call this the
refinement sample of $G$.
{{\epsilon}ilon}nd{description}
The following two lemmas relate the probabilities $z(e)$ to the sampling probabilities used in the Bencz\'{u}r-Karger sampling scheme.
{\mbox {\bf b}}egin{lemma}\label{lm:upper}
For any $K>0$, with probability at least $1-K n^{-d}$ every edge $e$ satisfies $z(e) \leq 4\phi \rho/({{\epsilon}ilon}^2 s_e)$.
{{\epsilon}ilon}nd{lemma}
{\mbox {\bf b}}egin{proof}
Consider an edge $e$ with strong connectivity $s_e$, and let $C$ denote the $s_e$-strongly connected component containing $e$. By Theorem \ref{thm:bk-sampling}, sampling with probability $\min\{4\rho/s_e, 1\}$ preserves all cuts up to $1\pm \frac1{2}$ in $C$ with probability at least $1-n^{-d}$. Hence, all $s_e$-strongly connected components stay connected after $K$ passes of {{\mbox {\bf b}}f \mbox{\sc{Refine}}}~for all $l>0$ such that $2^{-l}\geq 4\rho/s_e$, yielding the lemma.
{{\epsilon}ilon}nd{proof}
{\mbox {\bf b}}egin{lemma}\label{lm:lower}
If $K > \log_{4/3} n$, then $2^{-L(e)+1}\geq 1/(2s_e)$ for every $e\in E(G)$ with probability at least $1-Ke^{-(n-1)/100}$.
{{\epsilon}ilon}nd{lemma}
{\mbox {\bf b}}egin{proof}
Consider a level $l$ such that $p=2^{-l} < 1/(2s_e)$. Let $H$ be the graph
obtained by contracting all $(s_e+1)$-strong components in $G$ into supernodes.
Since $H$ contains only $(s_e+1)$-weak edges, the number of edges is at most $s_e(n-1)$ by Lemma \ref{lm:k-weak}. As the expected number of $(s_e+1)$-weak edges in the sample is at most $(n-1)/2$, by
Chernoff bounds, the probability that the
number of $(s_e+1)$-weak edges in the sample exceeds $3(n-1)/4$ is at most $(e^{1/4}(5/4)^{-5/4})^{-(n-1)/2}<e^{-(n-1)/100}$. Thus
at least one quarter of the supernodes get isolated in each iteration. Hence, no $(s_e+1)$-weak edge survives after $K=\log_{4/3} n$ rounds of refinement sampling with probability at least $1-K e^{-(n-1)/100}$. Since $L(e)$ was defined as the least $l$ such that $e\in X(S_{l, K})$, the endpoints of $e$ were connected in $S_{L(e)-1, K}$, so $2^{-L(e)+1}\geq 1/(2s_e)$.
{{\epsilon}ilon}nd{proof}
{\mbox {\bf b}}egin{theorem}
Let $G'$ be the graph obtained by running Algorithm 1 with $\phi:=4\rho$. Then $G'$ has
$O(n\log^2 n/{{\epsilon}ilon}^2)$ edges in expectation, and is an
${{\epsilon}ilon}ps$-sparsification of $G$ with probability at least $1-n^{-d+1}$.
{{\epsilon}ilon}nd{theorem}
{\mbox {\bf b}}egin{proof}
We have from lemma \ref{lm:lower} and the choice of $\phi$ that the sampling probabilities dominate those used in Bencz\'{u}r-Karger sampling with probability at least $1-K e^{-(n-1)/100}$. Hence, by corollary \ref{cor:oversampling}
we have that every cut in $G'$ is within $1\pm {{\epsilon}ilon}ps$ of its value in $G$ with probability at least $1-K e^{-(n-1)/100}-n^{-d}$. The expected size of the sample is $O(n\log^2 n/{{\epsilon}ilon}^2)$ by lemma \ref{lm:upper} together with the fact that $\rho=O(\log n)$. The probability of failure of the estimate in lemma \ref{lm:lower} is at most $Kn^{-d}$, so all bounds hold with probability at least $1-Kn^{-d}+K e^{-(n-1)/100}-n^{-d}>1-n^{-d+1}$ for sufficiently large $n$. The high probability bound on the number of edges follows by an application of the Chernoff bound.
{{\epsilon}ilon}nd{proof}
{\mbox {\bf b}}egin{figure}
{\mbox {\bf b}}egin{picture}(20,10)(0,0)
\put(1, 8){\framebox(1.7, 1){$S_{1,1}$}}
\put(4, 8){\framebox(1.7, 1){$S_{1,2}$}}
\put(7, 8){\makebox(1.7, 1){$\ldots$}}
\put(10, 8){\framebox(1.7, 1){$S_{1,K-1}$}}
\put(13, 8){\framebox(1.7, 1){$S_{1,K}$}}
\put(2.7,8.5){\vector(1, 0){1.3}}
\put(5.7,8.5){\vector(1, 0){1.3}}
\put(8.7,8.5){\vector(1, 0){1.3}}
\put(11.7,8.5){\vector(1, 0){1.3}}
\put(1, 6){\framebox(1.7, 1){$S_{2,1}$}}
\put(4, 6){\framebox(1.7, 1){$S_{2,2}$}}
\put(7, 6){\makebox(1.7, 1){$\ldots$}}
\put(10, 6){\framebox(1.7, 1){$S_{2,K-1}$}}
\put(13, 6){\framebox(1.7, 1){$S_{2,K}$}}
\put(2.7,6.5){\vector(1, 0){1.3}}
\put(5.7,6.5){\vector(1, 0){1.3}}
\put(8.7,6.5){\vector(1, 0){1.3}}
\put(11.7,6.5){\vector(1, 0){1.3}}
\put(1, 5){\makebox(1.7, 1){$\vdots$}}
\put(4, 5){\makebox(1.7, 1){$\vdots$}}
\put(7, 5){\makebox(1.7, 1){$\vdots$}}
\put(10, 5){\makebox(1.7, 1){$\vdots$}}
\put(13, 5){\makebox(1.7, 1){$\vdots$}}
\put(1, 4){\framebox(1.7, 1){$S_{L-1,1}$}}
\put(4, 4){\framebox(1.7, 1){$S_{L-1,2}$}}
\put(7, 4){\makebox(1.7, 1){$\ldots$}}
\put(10, 4){\framebox(1.7, 1){$S_{L-1,K-1}$}}
\put(13, 4){\framebox(1.7, 1){$S_{L-1,K}$}}
\put(2.7,4.5){\vector(1, 0){1.3}}
\put(5.7,4.5){\vector(1, 0){1.3}}
\put(8.7,4.5){\vector(1, 0){1.3}}
\put(11.7,4.5){\vector(1, 0){1.3}}
\put(1, 2){\framebox(1.7, 1){$S_{L,1}$}}
\put(4, 2){\framebox(1.7, 1){$S_{L,2}$}}
\put(7, 2){\makebox(1.7, 1){$\ldots$}}
\put(10, 2){\framebox(1.7, 1){$S_{L,K-1}$}}
\put(13, 2){\framebox(1.7, 1){$S_{L,K}$}}
\put(2.7,2.5){\vector(1, 0){1.3}}
\put(5.7,2.5){\vector(1, 0){1.3}}
\put(8.7,2.5){\vector(1, 0){1.3}}
\put(11.7,2.5){\vector(1, 0){1.3}}
{{\epsilon}ilon}nd{picture}
\caption{Scheme of refinement relations between partitions for Algorithm 1.}
\label{fig:f1}
{{\epsilon}ilon}nd{figure}
The next lemma follows from the discussion above:
{\mbox {\bf b}}egin{lemma}
For any ${{\epsilon}ilon}ps > 0$, an ${{\epsilon}ilon}ps$-sparsification of $G$ with $O(n \log^2 n/{{\epsilon}ilon}ps^2)$ edges
can be constructed in $O(\log n)$ passes of {{\mbox {\bf b}}f \mbox{\sc{Refine}}}\ using
$O(\log n)$ space per node and $O(\log^2 n)$ time per edge.
{{\epsilon}ilon}nd{lemma}
We now note that one $\log n$ factor in the running time comes from the fact that during each pass $k$ Algorithm 1 flips a coin at every level $l$ to decide whether or not to include $e$ into $S_{l, k}$ when $e\in S_{l, k-1}$. If we could guarantee that $S_{l, k}$ is a refinement of $S_{l', k}$ for all $l'<l$ and for all $k$, we would be able to use binary search to find the largest $l$ such that $e\in S_{l, k}$ in $O(\log \log n)$ time.
Algorithm 2 given below uses iterative sampling to ensure a scheme of refinement relations given in Fig. \ref{fig:f2}. For each edge $e$, $1 \le k \le K$, and $1 \le {{\epsilon}ilon}ll \le L$, we
define for convenience independent Bernoulli random variables $A_{l,k,e}$ such
${{\mbox {\bf b}}f \mbox{{\mbox {\bf b}}f Pr}}[A_{l,k,e}=1]=1/2$, even though the algorithm will not always need to flip all these $O(\log^2 n)$ coins. Also define $U_{l,k,e}=\prod_{j\leq l} A_{j,k,e}$.
The algorithm uses connectivity data structures $D_{l,k}$, $1\leq l\leq L, 1\leq k\leq K$.
Adding an edge $e$ to $D_{l,k}$ merges the components that the endpoints of $e$ belong to in $D_{l,k}$.
{\mbox {\bf b}}egin{description}
\item[Algorithm 2 (An $O(\log n)$-Pass Sparsifier)]
\item[Input:] Edges of $G$ streamed in adversarial order: $(e_1,\ldots, e_m)$.
\item[Output:] A sparsification $G'$ of $G$.
\item[Initialization:] Set $E':={{\epsilon}ilon}mptyset$.
\item[1.] For all $k=1,\ldots, K$
\item[2.] Set $t=1$.
\item[3.] For all $l=1,\ldots, L$
\item[4.] Add $e_t=(u_t, v_t)$ to $D_{l, k}$ if $U_{l,k,e}=1$ and $u_t$ and $v_t$ are connected in $D_{(l, k-1)}$.
\item[5.] Set $t:=t+1$. Go to step 1 if $t\leq m$.
\item[6.] For each $e_t$ define $L'(e_t)$ as the minimum $l$ such that $u_t$ and $v_t$ are not connected in $D_{l, K}$. Set $z'(e_t):=\min\left\{1, \frac{4\rho}{{{\epsilon}ilon}^2 2^{L'(e_t)}}\right\}$. Output $e_t$ with probability $z'(e_t)$, giving it weight $1/z'(e_t)$.
{{\epsilon}ilon}nd{description}
{\mbox {\bf b}}egin{theorem}
For any ${{\epsilon}ilon}ps > 0$, there exists an $O(\log n)$-pass streaming algorithm that produces an ${{\epsilon}ilon}ps$-sparsification $G'$ of a graph $G$ with at most $O(n\log^2 n/{{\epsilon}ilon}^2)$ edges using
$O((n/m) \log n + \log n \log \log n)$ time per edge.
{{\epsilon}ilon}nd{theorem}
{\mbox {\bf b}}egin{proof}
The correctness of Algorithm 2 follows in the same way as for Algorithm 1 above, so it remains to determine its runtime. An $O((n/m)\log n+1)$ term per edge comes from amortized $O(n\log n+m)$ complexity of UNION-FIND operations. The $\log n$ factor in the runtime comes from the $\log n$ passes, and we now show that step 3 can be implemented in $O(\log \log n)$ time. First note that since $S_{l',k'}$ is a refinement of $S_{l,k}$ whenever $l'\geq l$ and $k'\geq k$, one can use binary search to determine the largest $l_0$ such that $u_t$ and $v_t$ are connected in $D_{l_0-1,k-1}$. One then keeps flipping a fair coin
and adding $e$ to connectivity data structures $D_{l, k}$ for successive $l\geq l_0$ as long as the coin keeps coming up heads. Since $2$ such steps are performed on average, it takes $O(K)=O(\log n)$ amortized time per edge by the Chernoff bound. Putting these estimates together, we obtain the claimed time complexity.
{{\epsilon}ilon}nd{proof}
The scheme of refinement relations between $S_{l,k}$ is depicted in Fig. \ref{fig:f2}.
{\mbox {\bf b}}egin{corollary}
For any ${{\epsilon}ilon}ps > 0$, there is an $O(\log n)$-pass algorithm
that produces an ${{\epsilon}ilon}ps$-sparsification $G'$ of an input graph $G$ with at
most $O(n\log n/{{\epsilon}ilon}^2)$ edges using $O(\log^2 n)$ space per node,
and performing $O(\log n\log\log n+(n/m)\log^4 n)$ amortized work per edge.
{{\epsilon}ilon}nd{corollary}
{\mbox {\bf b}}egin{proof}
One can obtain a sparsification $G'$ with $O(n \log^2 n/{{\epsilon}ilon}^2)$ edges by running Algorithm 2 on the input graph $G$, and then run the Bencz\'{u}r-Karger algorithm
on $G'$ without violating the restrictions of the semi-streaming model. Note that even though $G'$ is a weighted graph, this will have overhead $O(\log^2 n)$ per edge of $G'$ since the weights are polynomial. Since $G'$ has $O(n\log^2 n)$ edges, the amortized work per edge of $G$ is $O(\log n\log\log n+(n/m)\log^4 n)$.
The Bencz\'{u}r-Karger algorithm can be implemented using space proportional to the size of the graph, which yields $O(\log^2 n)$ space per node.
{{\epsilon}ilon}nd{proof}
{\mbox {\bf b}}egin{remark}
The algorithm improves upon the runtime of the Bencz\'{u}r-Karger sparsification scheme when $m=\omega(n\log^2 n)$.
{{\epsilon}ilon}nd{remark}
\section{A One-pass $\tilde{O}(n+m)$-Time Algorithm for Graph Sparsification} \label{sec:onepass}
In this section we convert Algorithm 2 obtained in the previous section to a one-pass algorithm.
We will design a one-pass algorithm that produces an
${{\epsilon}ilon}ps$-sparsifier with $O(n\log^3 n/{{\epsilon}ilon}ps^2)$ edges using only
$O(\log \log n)$ amortized work per edge. A simple post-processing step at the
end of the algorithm will allow us to reduce the size to $O(n\log n/{{\epsilon}ilon}ps^2)$ edges
with a slightly increased space and time complexity. The main difficulty is that in going
from $O(\log n)$ passes to a one-pass algorithm,
we need to introduce and analyze new dependencies in the sampling process.
As before, the algorithm maintains connectivity data structures $D_{l, k}$, where $1\leq l\leq L$ and $1\leq k\leq K$. In addition to indexing $D_{l,k}$ by pairs $(l,k)$ we shall also write $D_{J}$ for $D_{l,k}$, where $J=K(l-1)+k$, so that $1\leq J \leq LK$.
This induces a natural ordering on $D_{l,k}$, illustrated in Fig. \ref{fig:f3},
that corresponds to the structure of refinement relations. We will assume for simplicity of presentation that $D_{0}=D_{1,0}$ is a connectivity data structure in which all vertices are connected.
For each edge $e$, $1 \le {{\epsilon}ilon}ll \le L$, and $1 \le k \le K$, we define
an independent Bernoulli random variable $A'_{l, k, e}$ with ${{\mbox {\bf b}}f \mbox{{\mbox {\bf b}}f Pr}}[A'_{l, k, e}=1]=2^{-l}$.
The algorithm is as follows:
{\mbox {\bf b}}egin{description}
\item[Algorithm 3 (A One-Pass Sparsifier)]
\item[Input:] Edges of $G$ streamed in adversarial order: $(e_1,\ldots, e_m)$.
\item[Output:] A sparsification $G'$ of $G$.
\item[Initialization:] Set $E':={{\epsilon}ilon}mptyset$.
\item[1.] Set $t=1$.
\item[2.] For all $J=1,\ldots, LK$ ($J=(l,k)$)
\item[3.] Add $e_t=(u_t, v_t)$ to $D_{J}$ if $A'_{l,k,e}=1$ and $u_t$ and $v_t$ are connected in $D_{J-1}$.
\item[4.] Define $L'(e_t)$ as the minimum $l$ such that $u_t$ and $v_t$ are not connected in $D_{l, K}$. Set $z'(e_t):=\min\left\{1, \frac{4\rho}{{{\epsilon}ilon}^2 2^{L'(e_t)}}\right\}$. Output $e_t$ with probability $z'(e_t)$, giving it weight $1/z'(e_t)$.
\item[5.] Set $t:=t+1$. Go to step 2 if $t\leq m$.
{{\epsilon}ilon}nd{description}
Informally, Algorithm 3 underestimates strength of some edges until the data structures $D_{l, k}$ become properly connected but proceeds similarly to Algorithms 1 and 2 after that. Our main goal in the rest of the section is to show that this underestimation of strengths does not lead to a large increase in the size of the sample.
Note that not all $LK=\Theta(\log^2 n)$ coin tosses $A'_{l, k, e}$ per edge are necessary for an implementation of Algorithm 3 (in particular, we will show that Algorithm 3 can be implemented with $O(\log\log n)=o(LK)$ work per edge). However, the random variables $A'_{l,k,e}$ are useful for analysis purposes. We now show that Algorithm 3 outputs a sparsification $G'$ of $G$ with $O(n\log^3 n/{{\epsilon}ilon}^2)$ edges whp.
{\mbox {\bf b}}egin{lemma} \label{lm:sparsification}
For any ${{\epsilon}ilon}ps > 0$, w.h.p. the graph $G'$ is an ${{\epsilon}ilon}ps$-sparsification of $G$.
{{\epsilon}ilon}nd{lemma}
{\mbox {\bf b}}egin{proof}
We can couple behaviors of Algorithms 1 and 3 using
the coin tosses $A'_{l, k, e}$ to show that $L(e)\geq L'(e)$ for every edge $e$, i.e. $z'(e)\geq z(e)$. Hence $G'$ is a sparsification of $G$ by Corollary \ref{cor:oversampling}.
{{\epsilon}ilon}nd{proof}
It remains to upper bound the size of the sample. The following lemma is crucial to our analysis; its proof is deferred to the Appendix \ref{app:xd}
due to space limitations.
{\mbox {\bf b}}egin{lemma} \label{lm:xd}
Let $G(V, E)$ be an undirected graph. Consider the execution of Algorithm 3, and for
$1 \leq J \le LK$ where $J=(l,k)$, let $X^{J}$ denote
the set of edges $e=(u, v)$ such that $u$ and $v$ are connected in $D_{J-1}$ when $e$ arrives. Then $|E\setminus X^{J}|=O(K 2^l n)$ with high probability.
{{\epsilon}ilon}nd{lemma}
{\mbox {\bf b}}egin{lemma} \label{lm:bound}
The number of edges in $G'$ is $O(n \log^3 n/{{\epsilon}ilon}^2)$ with high probability.
{{\epsilon}ilon}nd{lemma}
{\mbox {\bf b}}egin{proof}
Recall that Algorithm 3 samples an edge $e_t=(u_t, v_t)$ with probability $z'(e_t)=\min\left\{1, \frac{4\rho}{{{\epsilon}ilon}^2 2^{L'(e_t)}}\right\}$, where
$L'(e_t)$ is the minimum $l$ such that $u_t$ and $v_t$ are not connected in $D_{l, K}$.
As before, for $J=(l,k)$, we denote by $X^J$ the set of edges $e=(u, v)$ such that $u$ and $v$ are connected in $D_{J-1}$ when $e$ arrives.
Note that w.h.p. $X^{(L, 1)}={{\epsilon}ilon}mptyset$ w.h.p. by our choice of $L=\log(2n)$. For each $1\leq l\leq L$, let $Y_l=X^{(l, 1)}\setminus X^{(l+1, 1)}$. We have by Lemma~\ref{lm:xd} that $\sum_{1\leq j\leq l}|Y_j|=O(K2^ln)$ w.h.p. Also note that edges in $Y_l$ are sampled with probability at most $\frac{4\rho}{{{\epsilon}ilon}^2 2^{l-1}}$.
Hence, we get that the expected number of edges in the sample is at most
{\mbox {\bf b}}egin{equation*}
\sum_{l=1}^{L} |Y_l|\cdot \frac{4\rho}{{{\epsilon}ilon}^2 2^{l-1}}=O\left(\sum_{l=1}^{L} K 2^l n\cdot \frac{4\rho}{{{\epsilon}ilon}^2 2^{l-1}}\right)=O(n\log^3 n/{{\epsilon}ilon}^2).
{{\epsilon}ilon}nd{equation*}
The high probability bound now follows by standard concentration inequalities.
{{\epsilon}ilon}nd{proof}
Finally, we have the following theorem.
{\mbox {\bf b}}egin{theorem}
For any ${{\epsilon}ilon}ps > 0$ and $d > 0$,
there exists a one-pass algorithm that given the edges of an undirected graph $G$ streamed in adversarial order, produces an ${{\epsilon}ilon}ps$-sparsifier $G'$ with $O(n\log^3 n/{{\epsilon}ilon}^2)$ edges with probability at least $1-n^{-d}$. The algorithm takes $O(\log\log n)$ amortized time per edge and uses $O(\log^2 n)$ space per node.
{{\epsilon}ilon}nd{theorem}
{\mbox {\bf b}}egin{proof}
Lemma \ref{lm:sparsification} and Lemma \ref{lm:bound} together establish that
$G'$ is an ${{\epsilon}ilon}ps$-sparsifier $G'$ with $O(n\log^3 n/{{\epsilon}ilon}^2)$ edges.
It remains to prove the stated runtime bounds.
Note that when an edge $e_t=(u_t, v_t)$ is
processed in step 3 of Algorithm 3, it is not necessary to add $e_t$ to any data structure $D_{J}$ in which $u_t$ and $v_t$ are already connected. Also, since $D_{J}$ is a refinement of $D_{J'}$ whenever $J'\leq J$, for every edge $e_t$ there exists $J^*$ such that $u_t$ and $v_t$ are connected in $D_{J}$ for any $J\leq J^*$ and not connected for any $J\geq J^*$. The value of $J^*$ can be found in $O(\log \log n)$ time by binary search. Now we need to keep adding $e_t$ to $D_{J}$, for each $J\geq J^*$ such that $U_{l, k, e_t}=1$. However, we have that ${{\mbox {\bf b}}f \mbox{{\mbox {\bf b}}f E}}\left[\sum_{J\geq J^*} U'_{l, k, e_t}\right]=O(1)$. Amortizing over all edges, we get $O(1)$ per edge using standard concentration inequalities.
{{\epsilon}ilon}nd{proof}
{\mbox {\bf b}}egin{corollary} \label{cor:logn-sample}
For any ${{\epsilon}ilon}ps > 0$ and $d > 0$,
there exists a one-pass algorithm that given the edges of an undirected graph $G$ streamed in adversarial order, produces an ${{\epsilon}ilon}ps$-sparsifier $G'$ with $O(n\log n/{{\epsilon}ilon}^2)$ edges with probability at least $1-n^{-d}$. The algorithm takes amortized
$O( \log\log n+(n/m)\log^5 n)$ time per edge and uses $O(\log^3 n)$ space per node.
{{\epsilon}ilon}nd{corollary}
{\mbox {\bf b}}egin{proof}
One can obtain a sparsification of $G'$ with $O(n \log^3 n/{{\epsilon}ilon}^2)$ edges by running Algorithm 3 on the input graph $G$, and then run the Bencz\'{u}r-Karger algorithm
on $G'$ without violating the restrictions of the semi-streaming model. Note that even though $G'$ is a weighted graph, this will have overhead $O(\log^2 n)$ per edge of $G'$ since the weights are polynomial. Since $G'$ has $O(n\log^3 n)$ edges, the amortized work per edge of $G$ is $O(\log n\log\log n+(n/m)\log^5 n)$. The Bencz\'{u}r-Karger algorithm can be implemented using space proportional to the size of the graph, which yields $O(\log^3 n)$ space per node.
{{\epsilon}ilon}nd{proof}
{\mbox {\bf b}}egin{remark}
The algorithm avove improves upon the runtime of the Bencz\'{u}r-Karger sparsification scheme when $m=\omega(n\log^3 n)$.
{{\epsilon}ilon}nd{remark}
\noindent
{{\mbox {\bf b}}f Sparse $k$-connectivity Certificates:}
Our analysis of the performance of refinement sampling is along broadly similar lines to the analysis of the strength estimation routine
in \cite{benczurkarger96}. To make this analogy more precise, we note that refinement sampling as used in Algorithm 3 in fact produces a {{\epsilon}ilon}mph{sparse connectivity certificate} of $G$, similarly to the algorithm of Nagamochi-Ibaraki\cite{nagamochi-ibaraki}, although with slightly weaker guarantees
on size.
\iffalse
{\mbox {\bf b}}egin{definition}\cite{benczurkarger96}
A sparse $k$-connectivity certificate, or simply a $k$-certificate, for an $n$-vertex graph $G$ is a subgraph $H$ of $G$ such that
{\mbox {\bf b}}egin{enumerate}
\item $H$ has $k(n-1)$ and
\item $H$ contains all edges crossing cuts of value $k$ or less.
{{\epsilon}ilon}nd{enumerate}
{{\epsilon}ilon}nd{definition}
\fi
A {{{\epsilon}ilon}m $k$-connectivity certificate}, or simply a {{{\epsilon}ilon}m $k$-certificate},
for an $n$-vertex graph $G$ is a subgraph $H$ of $G$ such that
contains all edges crossing cuts of size $k$ or less in $G$. Such a certificate
always exists with $O(kn)$ edges, and moreover, there are graphs where
$\Omega(kn)$ edges are necessary.
The algorithm of \cite{nagamochi-ibaraki} depends on random access to edges of $G$ to
produce a $k$-certificate with $O(kn)$ edges in $O(m)$ time.
We now show that refinement sampling gives a one-pass algorithm
to produce a $k$-certificate with $O(kn \log^2 n)$ edges in
time $O(m\log\log n+n\log n)$. The result is summarized in the following corollary:
{\mbox {\bf b}}egin{corollary}\label{cor:sparse-cert}
Whp for each $l\geq 1$ the set $X(D_{l, K})$ is a $2^l$-certificate of $G$
with $O(\log^2n)2^l n$ edges.
{{\epsilon}ilon}nd{corollary}
{\mbox {\bf b}}egin{proof}
Whp $X(D_{l, K})$ contains all $2^l$-weak edges, in particular those that cross cuts of size at most $2^l$. The bound on the size follows by Lemma~\ref{lm:xd}.
{{\epsilon}ilon}nd{proof}
{\mbox {\bf b}}egin{remark}\label{rmk:weights}
Algorithms 1-3 can be easily extended to graphs with polynomially bounded integer weights on edges.
If we denote by $W$ the largest edge weight, then it is sufficient to set the number of levels $L$ to $\log (2nW)$ instead of $\log (2n)$ and the number of passes to $\log_{4/3} nW$ instead of $\log_{4/3} n$.
A weighted edge is then viewed as several parallel edges, and sampling can be performed efficiently for such edges by sampling directly from the corresponding binomial distribution.
{{\epsilon}ilon}nd{remark}
\section{A Linear-time Algorithm for $O(n \log n/{{\epsilon}ilon}^2)$-size Sparsifiers} \label{sec:two-pass}
We now present an algorithm for computing an ${{\epsilon}ilon}ps$-sparsification with $O(n \log n/{{\epsilon}ilon}ps^2)$
edges in $O(m\log \frac1{\delta}+ n^{1+\delta})$ expected time for any $\delta>0$. Thus, the algorithm runs in linear-time whenever $m=\Omega(n^{1 + \Omega(1)})$.
We note that no (randomized) algorithm can output an ${{\epsilon}ilon}ps$-sparsification in sub-linear time even if
there is no restriction on the size of the sparsifier. This is easily seen by considering the
family of graphs formed by disjoint union of two $n$-vertex graphs $G_1$ and $G_2$
with $m$ edges each, and a single edge $e$ connecting the two graphs.
The cut that separates $G_1$ from $G_2$ has a single edge $e$, and hence
any ${{\epsilon}ilon}ps$-sparsifier must include $e$. On the other hand, it is easy to see that $\Omega(m)$
probes are needed in expectation to discover the edge $e$.
Our algorithm can in fact be viewed as a {{{\epsilon}ilon}m two-pass} streaming algorithm, and we present is
as such below.
As before, let $G=(V, E)$ be an undirected unweighted graph. We will use Algorithm 3 as a building block of our construction. We now describe each of the passes.
{\mbox {\bf b}}egin{description}
\item[First pass:] Sample every edge of $G$ uniformly at random with probability $p=4/\log n$. Denote the resulting graph by $G'=(V, E')$. Give the stream of sampled edges to Algorithm 3 as the input stream, and save the state of the connectivity data structures $D_{l, K}$ for all $1\leq l\leq L$ at the end of execution. For $1 \le l \le L$, let $D^*_{l}$ denote these connectivity data structures (we will also refer to $D^*_l$ as partitions in what follows).
{{\epsilon}ilon}nd{description}
Note that the first pass takes $O(m)$ expected time since Algorithm 3 has an overhead $O(\log\log n)$ time per edge and the expected size of $|E'|$ is $|E|/\log n$.
Recall that the partitions $D^*_l$ are used in Algorithm 3 to estimate strength of edges $e\in E'$. We now show that these partitions can also be used to estimate strength of edges in $E$. The following lemma establishes a relationship between the edge strengths in $G'$ and $G$.
For every edge $e \in E$, let $s'_e$ denote the strength of edge $e$ in the graph $G'_e(V, E'\cup \{e\})$.
{\mbox {\bf b}}egin{lemma}
Whp $s'_e \le s_e \le 2 s_e'\log n + \rho\log n$ for all $e \in E$,
where $\rho=16(d+2)\ln n$ is the oversampling parameter in Karger sampling.
{{\epsilon}ilon}nd{lemma}
{\mbox {\bf b}}egin{proof}
The first inequality is trivially true since $G'_e$ is a subgraph of $G$. For the second one, let us first
consider any edge $e\in E$ with $s_e > \rho \log n$. Let $C$ be the $s_e$-strong
component in $G$ that contains the edge $e$. By Karger's theorem, whp the capacity of any cut
defined by a partition of vertices in $C$ decreases by a factor of at most $2\log n$ after sampling edges of $G$ with probability $p=4/\log n=\rho/((1/2)^2\rho\log n)$, i.e. in going
from $G$ to $G'$. So any cut in $C$, restricted to edges in $E'$ has size at least $s_e/(2\log n)$, implying that $s'_e\geq s_e/(2\log n)$. Finally, for any edge $e$ with $s_e\leq \rho\log n$, $s'_e$ is at least $1$, and the inequality thus follows.
{{\epsilon}ilon}nd{proof}
We now discuss the second pass over the data.
Recall that in order to estimate the strength $s'_e$ of an edge $e\in E'$, Algorithm 3 finds the minimum $L(e)$ such that the endpoints of $e$ are not connected in $D^*_{l}$ by doing a binary search over the range $[1..L]$. For an edge $e\in G$ we estimate its strength in $G'_e$ by doing binary
search as before, but stopping the binary search as soon as the size of the interval is smaller than $\delta L$, thus taking $O(\log\frac{1}{\delta})$ time per edge and obtaining an estimate that is away from the true value by a factor of at most $n^{\delta}$. Let $s''_e$ denote this estimate, that is, $s'_e n^{-\delta} \le s''_e \le s'_e n^\delta$. Now sampling every edge with probability $p_e=\min\left\{\frac{\rho n^{\delta}}{{{\epsilon}ilon}^2 s''_e}, 1\right\}$ and giving each sampled edge weight $1/p_e$
yields an ${{\epsilon}ilon}$-sparsification $G''=(V, E'')$ of $G$ whp.
Moreover, we have that w.h.p. $|E''|=\tilde{O}(n^{1+\delta})$.
Finally, we provide the graph $G''$ as input to Algorithm 3 followed by applying Bencz\'{u}r-Karger sampling as outlined in Corollary \ref{cor:logn-sample},
obtaining a sparsifier of size $O(n\log n/{{\epsilon}ilon}^2)$.
We now summarize the second pass.
{\mbox {\bf b}}egin{description}
\item[Second pass:] For each edge $e$ of the input graph $G$:
{\mbox {\bf b}}egin{itemize}
\item Perform $O(\log\frac1{\delta})$ steps of binary search to calculate $s_e''$.
\item
Sample edge $e$ with probability $p_e=\min\{\frac{\rho n^{\delta}}{{{\epsilon}ilon}^2 s''_e}, 1\}$.
\item
If $e$ is sampled, assign it a weight of $1/p_e$, and pass it as an input
to a fresh invocation of Algorithm 3, followed by Bencz\'{u}r-Karger sampling as
outlined in Corollary \ref{cor:logn-sample}, giving the final sparsification.
{{\epsilon}ilon}nd{itemize}
{{\epsilon}ilon}nd{description}
Note that the total time taken in the second pass is $O(m\log\frac1{\delta})+\tilde O(n^{1+\delta})$.
We have proved the following
{\mbox {\bf b}}egin{theorem}
For any ${{\epsilon}ilon}ps > 0$ and $\delta > 0$, there exists a two-pass algorithm that produces
an ${{\epsilon}ilon}$-sparsifier in time $O(m\log\frac1{\delta})+\tilde O(n^{1+\delta})$. Thus the algorithm
runs in linear-time when $m=\Omega(n^{1+\delta})$ and $\delta$ is constant.
{{\epsilon}ilon}nd{theorem}
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\caption{Scheme of refinement relations for Algorithm 2.}
\label{fig:f2}
{{\epsilon}ilon}nd{figure}
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\caption{Scheme of refinement for Algorithm 3.}
\label{fig:f3}
{{\epsilon}ilon}nd{figure}
\pdfbookmark[1]{References}{MyRefs}
\newcommand{{{{\epsilon}ilon}m et al.}char}[1]{$^{#1}$}
{\mbox {\bf b}}egin{thebibliography}{FKM{{{{\epsilon}ilon}m et al.}char{+}}05}
{\mbox {\bf b}}ibitem[AG09]{anh-guha}
K.~Ahn and S.~Guha.
\newblock On graph problems in a semi-streaming model.
\newblock {{{\epsilon}ilon}m Automata, languages and programming: Algorithms and complexity},
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\appendix
\section{Proof of Lemma \ref{lm:xd}}
\label{app:xd}
We denote the edges of $G$ in their order in the stream by $E=(e_1,\ldots, e_m)$. In what follows we shall treat edge sets as ordered sets, and for any
$E_1\subseteq E$ write $E\setminus E_1$ to denote the result of removing edges of $E_1$ from $E$ while preserving the order of the remaining edges. For a stream of edges $E$ we shall write $E_t$ to denote the set of the first $t$ edges in the stream.
For a $\kappa$-connected component $C$ of a graph $G$ we will write $|C|$ to denote the number of vertices in $C$. Also, we will denote the result of sampling the edges of $C$ uniformly at random with probability $p$ by $C'$. The following simple lemma will be useful in our analysis:
{\mbox {\bf b}}egin{lemma} \label{lm:components-weak}
Let $C$ be a $\kappa$-connected component of $G$ for some positive integer $\kappa$. Denote the graph obtained by sampling edges of $C$ with probability $p\geq \lambda/\kappa$ by $C'$. Then the number of connected components in $C'$ is at most $\gamma |C|$ with probability at least $1-e^{-{{\epsilon}ilon}ta|C|}$, where $\gamma=(7/8+e^{-\lambda/2}/8)$ and ${{\epsilon}ilon}ta=1-e^{-\lambda/2}$.
{{\epsilon}ilon}nd{lemma}
{\mbox {\bf b}}egin{proof}
Choose $A, B\subset V(C)$ so that $A\cup B=V(C), A\cap B={{\epsilon}ilon}mptyset$, $|A|\geq |V(C)|/2$ and for every $v\in A$ at least half of its edges that go to vertices in $C$ go to $B$.
Note that such a partition always exists: starting from any arbitrary partition of vertices of $C$,
we can repeatedly move a vertex from one side to the other if it increases the number of edges
going across the partition, and upon termination, the larger side corresponds to the set $A$.
Denote by $Y$ the number of vertices of $A$ that belong to components of size at least $2$.
Note that $Y$ can be expressed as sum of $|A|$ independent $0/1$ Bernoulli random
variables. Let $\mu:={{\mbox {\bf b}}f \mbox{{\mbox {\bf b}}f E}}[Y]$; we
have that $\mu \geq |A|(1-(1-\lambda/\kappa)^{\kappa/2})\geq |A|(1-e^{-\lambda/2})$. We get by the Chernoff bound that ${{\mbox {\bf b}}f \mbox{{\mbox {\bf b}}f Pr}}[Y\leq |A|(1-e^{-\lambda/2})/2]\leq e^{-2\mu}\leq e^{-|C|(1-e^{-\lambda/2})}=e^{-{{\epsilon}ilon}ta |C|}$. Hence, at least a $(1-e^{-\lambda/2})/4$ fraction of the vertices of $C$ are in components of size at least $2$. Hence, the number of connected components is at most a $1-(1-e^{-\lambda/2})/8=7/8+e^{-\lambda/2}/8=\gamma$ fraction of the number of vertices of $C$.
{{\epsilon}ilon}nd{proof}
{\mbox {\bf b}}egin{proofof}{Lemma \ref{lm:xd}}
The proof is by induction on $J$. We prove that w.h.p. for every $J=(l,k)$ one has $|E\setminus X^J|\leq \sum_{1\leq J'=(l',k')\leq J-1} c_1 2^{l'}n$ for a constant $c_1>0$.
{\mbox {\bf b}}egin{description}
\item[Base: $J=1$] Since everything is connected in $D_{0}$ by definition, the claim holds.
\item[Inductive step: $J\to J+1$] The outline of the proof
is as follows. For every $J=(l,k)$ we consider the edges of the stream that the algorithm tries to add to $D_J$, identify a sequence of $2^l$-strongly connected components $C_0,C_1\ldots$ in the partially received graph, and use lemma \ref{lm:components-weak} to show that the number of connected components decreases fast because only a small fraction of vertices in the sampled $2^l$-strongly connected components are isolated. We thus show that, informally, it will take $O(2^ln)$ edges to make the connectivity data structure $D_{J}$ in Algorithm 3 connected. The connected components $C_s$ are defined by induction on $s$. The vertices of $C_s$ are elements of a partition $P_s$ of the vertex set $V$ of the graph $G$. We shall use an auxiliary sequence of graphs which we denote by $H_s^t$.
Let $P_0$ be the partition consisting of isolated vertices of $V$. We treat the base case $s=0$ separately to simplify exposition. We use the definition of $\gamma$ and ${{\epsilon}ilon}ta$ from lemma \ref{lm:components-weak} with $\lambda=1$ since we are considering $2^l$-connected components when $J=(k,l)$.
{\mbox {\bf b}}egin{description}
\item[Base case: $s=0$.] Set $H_0^t=(P_0, \{e_1,\ldots, e_t\})$, i.e. $H_0^t$ is the partially received graph up to time $t$. Let $t_0^*$ be the the first value of $t$ such that $s_{H_0^{t}}(e_t)\geq 2^{l}$. This means that $e_{t_0^*}$ belongs to a $2^{l}$-strongly connected component in $H_0^{t_0^*}$. Note that this component does not contain any $(2^{l}+1)$-strongly connected components. Denote this component by $C_0$ (note that the number of edges in $C_0$ is at most $2^{l}|C_0|$ by lemma \ref{lm:k-weak}). Denote the random variables that correspond to sampling edges of $C_0$ by $R_0$. Let $X_0$ be an indicator variable that equals $1$ if the number of connected components in $C'_0$ is at most $\gamma|C_0|$ and $0$ otherwise. By lemma \ref{lm:components-weak} we have that ${{\mbox {\bf b}}f \mbox{{\mbox {\bf b}}f Pr}}[X_0=1]\geq 1-e^{-{{\epsilon}ilon}ta|C_0|}$.
For a partition $P$ denote $\text{diag}(P)=\{(u, u): u\in P\}$. Define $P_1$ by merging partitions of $P_0$ that belong to connected components in $C'_0$ if $X_0=1$ and as equal to $P_0$ otherwise. Let $E^1=E\setminus (E(C_0)\cup \text{diag}(P_1))$, i.e. we remove edges of $C_0$ and also edges that connect vertices that belong to the same partition in $P_1$.
Note that we can safely remove these edges since their endpoints are connected in $D_J$ when they arrive. Define $H_1^t=(P_1, E^1_t)$, i.e. $H_1^t$ is the partially received graph on the modified stream of edges.
\item[Inductive step: $s\to s+1$.]
As in the base case, let $t_{s}^*$ be the the first value of $t$ such that $s_{H_{s}^{t}}(e_t)\geq 2^{l}$. This means that $e_{t_{s}^*}$ belongs to a $2^{l}$-connected component in $H_{s}^{t_{s}^*}$. Denote this component by $C_{s}$(note that the number of edges in $C_{s}$ is at most $2^{l}|C_{s}|$ by lemma \ref{lm:k-weak}). Denote the random variables that correspond to sampling edges of $C_s$ by $R_s$. Let $X_{s}$ be an indicator variable that equals $1$ if the number of connected components in $C'_{s}$ is at most $\gamma |C_{s}|$ and $0$ otherwise. By lemma \ref{lm:components-weak} we have that ${{\mbox {\bf b}}f \mbox{{\mbox {\bf b}}f Pr}}[X_{s}=1]\geq 1-e^{-{{\epsilon}ilon}ta|C_{s}|}$. Define $P_{s+1}$ by merging together vertices that belong to connected components in $C'_{s}$. Let $E^{s+1}=E^{s}\setminus (E(C_{s})\cup \text{diag}(P_s))$. Denote
$H_s^t=(P_s, E^{s}_t)$.
{{\epsilon}ilon}nd{description}
It is important to note that at each step $s$ we only flip coins $R_s$ that correspond to edges in $E(C_{s})$, and delete only those edges from $E^{s}$.
While there may be edges going across partitions $P_s$ for which we do not perform a coin flip,
there number is bounded by $O(2^ln)$ since these edges do not contain a $2^l$-connected component.
Note that for any $s>0$ the number of connected components in $P_s$ is at most
{\mbox {\bf b}}egin{equation*}
n-\sum_{j=1}^s (1-\gamma) |C_j| X_j.
{{\epsilon}ilon}nd{equation*}
We now show that it is very unlikely that $\sum_{j=1}^s|C_j| X_j$ is more than a constant factor smaller than $\sum_{j=1}^s|C_j|$, thus showing that the number of connected components cannot be more than $1$ when $\sum_{j=1}^s|C_j|\geq \frac{cn}{1-\gamma}$ for an appropriate constant $c>0$.
For any constant $d>0$ define $I^+=\{i\geq 0: |C_i|>((d+2)/{{\epsilon}ilon}ta)\log n\}$ and $I^-=\{i\geq 0: |C_i|\leq ((d+2)/{{\epsilon}ilon}ta)\log n\}$.
Also define $Z_i^+=\sum_{0\leq j\leq i, j\in I^+} X_j |C_j|, Z_i^-=\sum_{0\leq j\leq i, j\in I^-} X_j|C_j|-|C_j|(1-e^{-{{\epsilon}ilon}ta |C_j|})$.
First note that one has ${{\mbox {\bf b}}f \mbox{{\mbox {\bf b}}f Pr}}[X_j=1]\geq 1-n^{-d-2}$ for any $j\in I^+$ by lemma \ref{lm:components-weak}. Hence, it follows by taking the union bound that $i\leq n^2$ one has ${{\mbox {\bf b}}f \mbox{{\mbox {\bf b}}f Pr}}[Z_i^+=\sum_{j\in I^+, j\leq i} |C_j|]\geq 1-n^{-d}$.
We now consider $Z_i^-$. Note that $Z_i^-$'s define a martingale sequence with respect to $R_{i-1},\ldots, R_0$: ${{\mbox {\bf b}}f \mbox{{\mbox {\bf b}}f E}}[Z_i^-|R_{i-1},\ldots, R_0]=Z_{i-1}^-$.
Also, $|Z_{i}^--Z_{i-1}^-|\leq ((d+2)/{{\epsilon}ilon}ta)\log n$ for all $i$. Hence, by Azuma's inequality (see, e.g. \cite{b:alonspencer}) one has
{\mbox {\bf b}}egin{equation*}
{{\mbox {\bf b}}f \mbox{{\mbox {\bf b}}f Pr}}[Z_{i}^-<t]<{{\epsilon}ilon}xp\left(-\frac{t^2}{2 i (((d+2)/{{\epsilon}ilon}ta)\log n)^2}\right).
{{\epsilon}ilon}nd{equation*}
Now consider the smallest value $\tau$ such that $\sum_{j\leq \tau} |C_j|=\sum_{j\leq \tau,j\in I^+} |C_j|+\sum_{j\leq i,j\in I^-} |C_j|=S^++S^-\geq \frac{4n}{(1-e^{-2{{\epsilon}ilon}ta})(1-\gamma)}$.
Note that $\tau<n/(2(1-e^{-2{{\epsilon}ilon}ta})(1-\gamma))$ since $|C_i|\geq 2$. If $S^+\geq \frac{2n}{(1-e^{-2{{\epsilon}ilon}ta})(1-\gamma)}\geq 2n/(1-\gamma)$, then we have that $Z_\tau^+=S^+>2n/(1-\gamma)$ with probability at least $1-n^{-d}$. Thus,
{\mbox {\bf b}}egin{equation*}
n-\sum_{j=1}^\tau (1-\gamma) |C_j| X_j\leq n-(1-\gamma) Z_\tau^+\leq 0.
{{\epsilon}ilon}nd{equation*}
Otherwise $S^-\geq \frac{2n}{(1-e^{-2{{\epsilon}ilon}ta})(1-\gamma)}$ and by Azuma's inequality we have
{\mbox {\bf b}}egin{equation*}
{{\mbox {\bf b}}f \mbox{{\mbox {\bf b}}f Pr}}[Z_{\tau}^-<-n]<{{\epsilon}ilon}xp\left(-\frac{n^2}{2 \tau (((d+2)/{{\epsilon}ilon}ta)\log n)^2}\right)\leq {{\epsilon}ilon}xp\left(-\frac{n}{(((d+2)/{{\epsilon}ilon}ta)\log n)^2}\right)<n^{-d}.
{{\epsilon}ilon}nd{equation*}
Since $|C_i|\geq 2$, we have $|C_i|(1-e^{-{{\epsilon}ilon}ta |C_i|})\geq |C_i|(1-e^{-2{{\epsilon}ilon}ta})$ and thus we get
{\mbox {\bf b}}egin{equation*}
{\mbox {\bf b}}egin{split}
n-\sum_{j=1}^\tau (1-\gamma) |C_j| X_j<n-(1-\gamma) \left[\sum_{1\leq j\leq \tau, j\in I^-} |C_j|(1-e^{-{{\epsilon}ilon}ta |C_j|})+Z_\tau^-\right]\\
<n-(1-\gamma) \left[(1-e^{-2{{\epsilon}ilon}ta})\sum_{1\leq j\leq \tau, j\in I^-} |C_j|+Z_\tau^-\right]\\
<n-(1-\gamma) \left[\frac{2n}{1-\gamma}+n\right]<0\\
{{\epsilon}ilon}nd{split}
{{\epsilon}ilon}nd{equation*}
We have shown that there exists a constant $c'>0$ such that with probability at least $1-n^{-d}$ after $c' 2^l n$ edges are sampled by the algorithm at level $J$ all subsequent edges will have their endpoints connected in $D_J$. Note that we never flipped coins for those edges that did not contain a $2^{l}$-connected component.
Setting $c_1=c'+1$, we have that w.h.p. $|E\setminus X^J|\leq c_1 2^l n+|E\setminus X^{J-1}|$. By the inductive hypothesis we have that $|E\setminus X^{J-1}|\leq \sum_{1\leq J'=(l',k')\leq J-2} c_1 2^{l'}n$, which together with the previous estimate gives us the desired result.
{{\epsilon}ilon}nd{description}
It now follows that $|E\setminus X^J|\leq \sum_{1\leq J'=(l',k')\leq J-1} c_1 2^{l'}n=O(K2^ln)$ w.h.p., finishing the proof of the lemma.
{{\epsilon}ilon}nd{proofof}
{{\epsilon}ilon}nd{document}
|
\begin{document}
\title[K\"AHLER MANIFOLDS ADMITTING A FLAT COMPLEX...]
{K\"AHLER MANIFOLDS ADMITTING A FLAT COMPLEX CONFORMAL CONNECTION}
\author{G. Ganchev and V. Mihova}
\address{Bulgarian Academy of Sciences, Institute of Mathematics and
Informatics, Acad. G. Bonchev Str. bl. 8, 1113 Sofia, Bulgaria}
\email{[email protected]}
\address{Faculty of Mathematics and Informatics, University of Sofia,
J. Bouchier Str. 5, (1164) Sofia, Bulgaria}
\email{[email protected]}
\subjclass{Primary 53B35, Secondary 53C25}
\keywords{Complex conformal connection, Bochner-K\"ahler manifolds with
special scalar distribution, warped product K\"ahler manifolds.}
\begin{abstract}
\vskip 2mm
We prove that any K\"ahler manifold admitting a flat complex conformal
connection is a Bochner-K\"ahler manifold with special scalar
distribution and zero geometric constants. Applying the local structural
theorem for such manifolds we obtain a complete description of
the K\"ahler manifolds under consideration.
\end{abstract}
\maketitle
\section{Introduction}
Let $(M,g,J)\;(\dim \,M=2n)$ be a K\"ahler manifold with complex structure
$J$, metric $g$, Levi-Civita connection $\nabla$, curvature tensor $R$,
Ricci tensor $\rho$ and scalar curvature $\tau$.
The Bochner curvature tensor $B(R)$ is given by
$$\begin{array}{l}
B(R)(X,Y)Z=R(X,Y)Z\\
[2mm]
-Q(Y,Z)X+Q(X,Z)Y-g(Y,Z)Q(X)+g(X,Z)Q(Y)\\
[2mm]
-Q(JY,Z)JX+Q(JX,Z)JY+2Q(JX,Y)JZ\\
[2mm]
-g(JY,Z)JQ(X)+g(JX,Z)JQ(Y)+2g(JX,Y)JQ(Z), \quad X,Y,Z\in {\mathfrak X}M,
\end{array} $$
where $Q(X,Y)=\displaystyle{\frac{1}{2(n+2)}\,\rho(X,Y)
-\frac{\tau}{8(n+1)(n+2)}\,g(X,Y)}$ and $Q(X)$ is the corresponding
tensor of type (1.1).
The manifold is said to be {\it Bochner flat (Bochner-K\"ahler)} if its
Bochner curvature tensor vanishes identically, i.e.
$$\begin{array}{l}
R(X,Y)Z=\\
[2mm]
Q(Y,Z)X-Q(X,Z)Y+g(Y,Z)Q(X)-g(X,Z)Q(Y)\\
[2mm]
+Q(JY,Z)JX-Q(JX,Z)JY-2Q(JX,Y)JZ\\
[2mm]
+g(JY,Z)JQ(X)-g(JX,Z)JQ(Y)-2g(JX,Y)JQ(Z), \quad X,Y,Z\in {\mathfrak X}M,
\end{array} \leqno{(1.1)}$$
For any real $\mathcal{C}^{\infty}$ function $u$ on $M$
we denote $\omega=du$ and $P= grad\,u$.
In \cite{Y} Yano introduced on a K\"ahler manifold a complex conformal
connection and proved
\vskip 2mm
{\bf Theorem A.} {\it If in a $2n$-dimensional\, $(n\geq 2)$ \,K\"ahler
manifold there exists a scalar function $u$ such that the
complex conformal connection
$$\begin{array}{ll}
\mathcal{D}_XY=&\nabla_XY+ \omega(X)Y+\omega(Y)X-g(X,Y)P\\
[2mm]
&-\omega(JX)JY-\omega(JY)JX-g(JX,Y)JP, \quad X,Y\in{\mathfrak X}M,
\end{array}$$
is of zero curvature, then the
Bochner curvature tensor of the manifold vanishes.}
\vskip 2mm
In \cite{S} Seino proved the inverse
\vskip 2mm
{\bf Theorem B.} {\it In a K\"ahlerian space with vanishing Bochner
curvature tensor if there exists a non-constant function $u$ satisfying
the equality
$$(\nabla_X\omega)(Y)+2\omega(JX)\omega(JY)+\omega(P)g(X,Y)=0,$$
then the complex conformal connection is of zero curvature.}
\vskip 2mm
In this paper we prove
\vskip 2mm
{\bf Theorem 3.1.} {\it A K\"ahler manifold $(M,g,J) \;
(\dim M = 2n \geq 6)$ admits a flat complex conformal connection
if and only if it is a Bochner-K\"ahler manifold whose scalar distribution
$D_{\tau}$ is a $B_0$-distribution with function $a+k^2=0$ and
geometric constants ${\mathfrak B}=\frak b_0=0.$}
\vskip 2mm
Applying the local structural theorem \cite{GM2} for Bochner-K\"ahler
manifolds whose scalar distribution is a $B_0$-distribution, we
describe locally all K\"ahler manifolds admitting a flat complex
conformal connection.
\section{Preliminaries}
Let $(M,g,J) \, (\dim M=2n)$ be a K\"ahler manifold with metric $g$,
complex structure $J$ and Levi-Civita connection $\nabla$. We denote by
${\mathfrak X}M$ the Lie algebra of all $\mathcal{C}^{\infty}$ vector fields on $M$.
The fundamental K\"ahler form $\Omega$ is defined as follows
$$\Omega(X,Y)=g(JX,Y), \quad X,Y \in {\mathfrak X}M.$$
For any $\mathcal{C}^{\infty}$ real function $u$ on $M$ we consider
the conformal metric $\bar g= e^{2u}g$. We denote the 1-form
$\omega:= du$ and $P:= grad\, u$ with respect to the metric $g$.
Then $(M,\bar g,J)$ is a locally conformal K\"ahler manifold, or a
$W_4$-manifold in the classification scheme of \cite{GH}. The
fundamental K\"ahler form and the Lee form of the structure
$(\bar g,J)$ are $\bar \Omega (X,Y)=\bar g(JX,Y),
\; X,Y \in {\mathfrak X}M$ and $\bar \omega =2\omega = 2du$, respectively.
The Lee vector $\bar P$ corresponding to $\bar \omega$ with respect to
the metric $\bar g$ is $\bar P= 2e^{-2u}P$.
The unique linear connection $\mathcal{D}$ with torsion $\mathcal{T}$
satisfying the conditions:
$$\begin{array}{l}
1)\, \mathcal{D}J=0;\\
[2mm]
2)\, \mathcal{D}\bar g=0;\\
[2mm]
3)\, \mathcal{T}=-\,\bar\Omega \otimes J\bar P\end{array} \leqno{(2.1)}$$
is said to be a {\it complex conformal connection} \cite{Y}.
In terms of the K\"ahler structure $(g,J)$ $\mathcal{D}$ is given by
$$\begin{array}{ll}
\mathcal{D}_XY=&\nabla_XY+ \omega(X)Y+\omega(Y)X-g(X,Y)P\\
[2mm]
&-\omega(JX)JY-\omega(JY)JX-g(JX,Y)JP, \quad X,Y\in{\mathfrak X}M.
\end{array}\leqno{(2.2)}$$
The conditions (2.1) in terms of the K\"ahler structure $(g,J)$
become
$$\begin{array}{l}
1)\, \mathcal{D}J=0;\\
[2mm]
2)\, \mathcal{D}g=-2\,\omega \otimes g;\\
[2mm]
3)\, \mathcal{T}=-2\,\Omega \otimes JP.\end{array} \leqno{(2.3)}$$
Denote by $\mathcal{R}$ the curvature tensor of the complex conformal
connection $\mathcal {D}$. Taking into account (2.2) we have the
relation between $R$ and $\mathcal{R}$:
$$\begin{array}{l}
\mathcal{R}(X,Y)Z=R(X,Y)Z\\
[1mm]
-\{(\nabla_Y\omega)(Z)-\omega(Y)\omega(Z)+\omega(JY)\omega(JZ)
+\displaystyle{\frac{1}{2}}\,\omega(P)g(Y,Z)\}X\\
[2mm]
+\{(\nabla_X\omega)(Z)-\omega(X)\omega(Z)+\omega(JX)\omega(JZ)
+\displaystyle{\frac{1}{2}}\,\omega(P)g(X,Z)\}Y\\
[2mm]
-g(Y,Z)\{\nabla_XP-\omega(X)P-\omega(JX)JP+\displaystyle{\frac{1}{2}}
\omega(P)X\}\\
[2mm]
+g(X,Z)\{\nabla_YP-\omega(Y)P-\omega(JY)JP+\displaystyle{\frac{1}{2}}
\omega(P)Y\}\\
[2mm]
+\{(\nabla_Y\omega)(JZ)-\omega(Y)\omega(JZ)-\omega(JY)\omega(Z)
+\displaystyle{\frac{1}{2}}\,\omega(P)g(Y,JZ)\}JX\\
[2mm]
-\{(\nabla_X\omega)(JZ)-\omega(X)\omega(JZ)-\omega(JX)\omega(Z)
+\displaystyle{\frac{1}{2}}\,\omega(P)g(X,JZ)\}JY\\
[2mm]
+g(Y,JZ)\{\nabla_XJP-\omega(X)JP+\omega(JX)P+\displaystyle{\frac{1}{2}}
\omega(P)JX\}\\
[2mm]
-g(X,JZ)\{\nabla_YJP-\omega(Y)JP+\omega(JY)P+\displaystyle{\frac{1}{2}}
\omega(P)JY\}\\
[2mm]
-(\nabla_X\omega)(JY)JZ+(\nabla_Y\omega)(JX)JZ+
2g(X,JY)\{\omega(JZ)P+\omega(Z)JP\}
\end{array}\leqno{(2.4)}$$
for all $X,Y,Z\in{\mathfrak X}M.$
From (2.4) it follows that the curvature tensor $\mathcal {R}$ satisfies
the first Bianchi identity (i.e. $\mathcal {R}$ is a K\"ahler tensor) if
and only if \cite{S}:
$$(\nabla_X\omega)(Y)+2\omega(JX)\omega(JY)+\omega(P)g(X,Y)=0,\quad
X,Y\in{\mathfrak X}M,\leqno{(2.5)}$$
which is equivalent to the condition
$$\mathcal{D}_XP=0, \quad X\in{\mathfrak X}M.$$
If the 1-form $\omega$ satisfies (2.5), then (2.4) becomes
$$\begin{array}{l}
\mathcal{R}(X,Y)Z=R(X,Y)Z\\
[2mm]
+L(Y,Z)X-L(X,Z)Y+g(Y,Z)L(X)-g(X,Z)L(Y)\\
[2mm]
+L(JY,Z)JX-L(JX,Z)JY-2L(JX,Y)JZ\\
[2mm]
+g(JY,Z)JL(X)-g(JX,Z)JL(Y)-2g(JX,Y)JL(Z), \quad X,Y,Z\in {\mathfrak X}M,
\end{array} \leqno{(2.6)}$$
where $L(X,Y)=\omega(X)\omega(Y)+\omega(JX)\omega(JY)+
\displaystyle{\frac{1}{2}}\,\omega(P)g(X,Y)$ and $L(X)$ is the
corresponding tensor of type (1,1) with respect to the K\"ahler
metric $g$.
If $(M,g,J)$ admits a flat complex conformal connection (2.2),
then $\mathcal{R}$ satisfies the first Bianchi identity, i.e.
(2.5) holds good. Then (2.6) implies that the K\"ahler manifold is
Bochner flat.
Conversely, if $(M,g,J)$ admits a 1-form $\omega$ satisfying (2.5),
then (2.4) becomes (2.6). The condition $(M,g,J)$ is Bochner
flat implies that ${\mathcal R}=0$, i.e. the complex conformal connection
(2.2) is flat.
\section{A Curvature characterization of K\"ahler manifolds admitting
flat complex conformal connection}
For any Bochner-K\"ahler manifold $(M,g,J)$ in \cite{GM2} we proved that
$$\begin{array}{ll}
(\nabla_X\,\rho)(Y,Z)=&\displaystyle{\frac{1}{4(n+1)}\,\{2d\tau(X)g(Y,Z)
+d\tau(Y)g(X,Z)+d\tau(Z)g(X,Y)}\\
[3mm]
&+d\tau(JY)g(X,JZ)+d\tau(JZ)g(X,JY)\}, \quad X,Y,Z \in {\mathfrak X}M.
\end{array}\leqno (3.1)$$
This equality shows that
the conditions $\tau = const$ and $\nabla\rho = 0$ are equivalent on a
Bochner-K\"ahler manifold. Because of the structural theorem in
\cite {TL} the case $B(R)=0, \; d\tau =0$, can be considered as
well-studied.
We consider Bochner-K\"ahler manifolds satisfying the condition
$d\tau \neq 0$ for all points $p \in M.$
This condition allows us to introduce the frame field
$$\left\{ \xi = \frac{grad \, \tau}{\Vert d\tau \Vert},
\quad J\xi = \frac{Jgrad \, \tau}{\Vert d\tau \Vert} \right\}$$
and the $J$-invariant distributions $D_{\tau}$ and
$D^{\perp}_{\tau}=span\{\xi,J\xi\}$.
Thus our approach to the local theory of Bochner-K\"ahler manifolds is
to treat them as K\"ahler manifolds $(M,g,J,D_{\tau})$ endowed with
a $J$-invariant distribution $D_{\tau}$ generated by the K\"ahler
structure $(g,J)$. We call this distribution
{\it the scalar distribution} of the manifold \cite{GM2}.
A J-invariant distribution
$D_{\tau},\, (D_{\tau}^{\perp} = span \{\xi, J\xi \})$ is said to be a
$B_0$-distribution \cite {GM1} if $\dim M=2n\geqq 6$ and
$$\begin{array}{l}
i) \displaystyle{ \quad \nabla _{x_0} \xi =
\frac{k}{2}\,x_0, \quad k \neq 0,}\quad x_0 \in D_{\tau},\\
[2mm]
ii) \quad \nabla_{J\xi}\xi=-p^*J\xi,\\
[2mm]
iii)\quad \nabla _{\xi} \xi = 0,
\end{array}$$
where $k$ and $p^*$ are functions on $M$.
The above conditions are equivalent to the equalities
$$\begin{array}{l}
\displaystyle{\nabla_X\xi=\frac{k}{2}\{X-\eta(X)\xi+\eta(JX)J\xi\}
+p^*\eta(JX)J\xi, \quad X\in {\mathfrak X}M,}\\
[2mm]
\displaystyle{dk = \xi(k)\,\eta, \quad p^* = -\frac{\xi(k)+k^2}{k}.}
\end{array} \leqno (3.2)$$
In \cite{GM2} we have shown that
$${\mathfrak B}=\Vert \rho \Vert^2 - \frac{\tau^2}{2(n+1)} +
\frac{\mathbb Delta \tau}{n+1}\leqno{(3.3)}$$
is a constant on any Bochner-K\"ahler manifold. We call this constant
{\it the Bochner constant} of the manifold.
Let us denote
$$\begin{array}{ll}
4\pi (X,Y)Z &:= g(Y,Z)X - g(X,Z)Y - 2g(JX,Y)JZ\\
[2mm]
&+ g(JY,Z)JX - g(JX,Z)JY,\\
[2mm]
8\Phi(X,Y)Z&:=g(Y,Z)(\eta(X)\xi-\eta(JX)J\xi)
-g(X,Z)(\eta(Y)\xi-\eta(JY)J\xi)\\
[2mm]
&+g(JY,Z)(\eta(X)J\xi+\eta(JX)\xi)-g(JX,Z)(\eta(Y)J\xi
+\eta(JY)\xi)\\
[2mm]
&-2g(JX,Y)(\eta(Z)J\xi+\eta(JZ)\xi)\\
[2mm]
&+(\eta(Y)\eta(Z)+
\eta(JY)\eta(JZ))X
-(\eta(X)\eta(Z)+\eta(JX)\eta(JZ))Y\\
[2mm]
&-(\eta(Y)\eta(JZ)-\eta(JY)\eta(Z))JX
+(\eta(X)\eta(JZ)-\eta(JX)\eta(Z))JY\\
[2mm]
&+2(\eta(X)\eta(JY)-\eta(JX)\eta(Y))JZ,
\quad X, Y, Z \in {\mathfrak X}M.
\end{array}$$
In \cite{GM2} we have also proved that
\vskip 1mm
{\it If $(M,g,J) \; (\dim M = 2n \geq 6)$ is a Bochner-K\"ahler manifold
whose scalar distribution $D_{\tau}$ is a
$B_0$-distribution, then
$$R = a\pi + b\Phi, \quad b \neq 0,\leqno{(3.4)}$$
where $a, \, b$ are the following functions on $M$
$$a=\frac{\tau}{(n+1)(n+2)}+\frac{2{\mathfrak b}_0}{n+2},\quad
b=\frac{2\tau}{(n+1)(n+2)}-\frac{2n{\mathfrak b}_0}{n+2},\leqno{(3.5)}$$
and
$$b_0= \frac{2a-b}{2}= \, const. \leqno{(3.6)}$$}
In \cite{GM2} we studied three classes of Bochner-K\"ahler manifolds
whose scalar distribution is a $B_0$-distribution according to the function
$a+k^2$:
$$a+k^2>0,\quad a+k^2=0,\quad a+k^2<0.$$
Now we can prove a curvature characterization of K\"ahler manifolds
admitting a flat complex conformal connection.
\begin{thm} \label{T:3.1}
A K\"ahler manifold $(M,g,J) \; (\dim M = 2n \geq 6)$
admits a flat complex conformal connection if and only if
it is a Bochner-K\"ahler manifold whose scalar distribution
$D_{\tau}$ is a $B_0$-distribution with function $a+k^2=0$ and
geometric constants ${\mathfrak B}=\frak b_0=0.$
\end{thm}
{\bf Proof.} Let $u$ be a $\mathcal{C}^{\infty}$ function on $M$,
such that the complex conformal connection $\mathcal{D}$, given
by (2.2) with $\omega = du\neq 0,\; P= grad \,u$, is flat.
Then (2.5) and (2.6) imply that the curvature tensor $R$ of
$M$ has the structure (1.1).
Comparing the tensor $Q$ from (1.1) and the tensor $L$ from (2.6)
we obtain
$$\rho(X,Y)=-2(n+2)\{\omega(X)\omega(Y)+\omega(JX)\omega(JY)+
\omega(P)g(X,Y)\},\; X, Y \in {\mathfrak X}M \leqno(3.7)$$
and
$$\rho(X,P)=-2(n+2)\omega(P)\omega(X), \quad X\in {\mathfrak X}M. \leqno(3.8)$$
After taking a trace in (3.7) we also get
$$\tau=-4(n+1)(n+2)\omega(P).\leqno (3.9)$$
Taking into account (2.5) we calculate from (3.7)
$$\begin{array}{ll}
(\nabla_X\,\rho)(Y,Z)=&2(n+2)\, \omega(P)\{2\omega(X)g(Y,Z)
+\omega(Y)g(X,Z)+\omega(Z)g(X,Y)\\
[3mm]
&+\omega(JY)g(X,JZ)+\omega(JZ)g(X,JY)\}, \quad X,Y,Z \in {\mathfrak X}M.
\end{array}\leqno (3.10)$$
Comparing (3.1) and (3.10) in view of (3.9), we obtain
$$ \omega=-\frac{d\tau}{2\tau},\quad P=-\frac{grad\,\tau}{2\tau},
\quad \Vert d\tau\Vert^2=\frac{-{\tau}^3}{(n+1)(n+2)}.
\leqno (3.11)$$
The unit vector field $\displaystyle{\xi = \frac{grad \,
\tau}{\Vert d\tau \Vert}}$ because of (3.11) gets the form
$$\xi = 2\sqrt{\frac{(n+1)(n+2)}{-\tau}}\,P.$$
From (2.5) and (3.9) we obtain
$$\nabla_X\,\xi=-\frac{1}{2}\,\sqrt{\frac{-\tau}{(n+1)(n+2)}}
\{X-\eta(X)\xi-2\eta(JX)J\xi\}, \quad X \in {\mathfrak X}M.
\leqno (3.12)$$
Now from (3.2) and (3.12) it follows that the scalar distribution
$D_{\tau}$ of the manifold is a $B_0$-distribution with functions
$$k=-\sqrt{\frac{-\tau}{(n+1)(n+2)}}\,, \quad
p^*=\frac{3}{2}\,\sqrt{\frac{-\tau}{(n+1)(n+2)}}\,.\leqno(3.13)$$
Then (2.6) and (3.11) give that the curvature tensor $R$ of the
manifold has the form
$$R=\frac{\tau}{(n+1)(n+2)}\,(\pi + 2\,\Phi)$$
and the functions $a$ and $b$ are
$$a=\frac{\tau}{(n+1)(n+2)}, \quad b=\frac{2\tau}{(n+1)(n+2)}.
\leqno(3.14)$$
From (3.13) and (3.14) we find $a+k^2=0$. The equalities (3.6) and (3.14)
imply that $\frak{b}_0=0$.
Taking into account (2.5), (3.11) and (3.7) we find
$$\mathbb Delta\,\tau=\frac{-\tau^2}{n+1}, \quad \Vert \rho \Vert^2
=\frac{(n+3)\tau^2}{2(n+1)^2}.\leqno (3.15)$$
Replacing $\mathbb Delta\,\tau$ and $\Vert \rho \Vert^2$ in (3.3) we obtain
${\mathfrak B}=0$.
\vskip 2mm
For the inverse, let $(M,g,J)$ be a Bochner-K\"ahler manifold whose
scalar distribution is a $B_0$-distribution. Then it follows
\cite{GM2} that (3.2), (3.4) and (3.5) hold good. Under the condition
$\frak{b}_0=0$ we find that the functions $a$ and $b$ satisfy (3.14).
The condition $a+k^2=0$ implies that $k^2=-a=
\displaystyle{\frac{-\tau}{(n+1)(n+2)}}$\,,\, i.e. \,$\tau<0$. From
Theorem 3.5 in \cite{GM1} it follows that
$$\Vert d\tau \Vert = \xi(\tau)=\frac{(n+1)(n+2)}{2}\,\xi(b)=
\frac{(n+1)(n+2)}{2}\,kb>0,$$
which gives that the function $k$ is negative and
$$k=-\sqrt{\frac{-\tau}{(n+1)(n+2)}},\quad
\Vert d\tau\Vert^2=\frac{-{\tau}^3}{(n+1)(n+2)},\quad
p^* = \frac{3}{2}\,\sqrt{\frac{-\tau}{(n+1)(n+2)}}\,.$$
Then, from the equality (3.2) for any $X,Y \in {\mathfrak X}M$ we have
$$(\nabla_X\,\eta)(Y)=-\frac{1}{2}\,\sqrt{\frac{-\tau}{(n+1)(n+2)}}
\{g(X,Y)-\eta(X)\eta(Y)+2\eta(JX)\eta(JY)\}.$$
Putting $2u:=-\ln {(-\tau)}$ and $\omega:=du=
-\displaystyle{\frac{d\tau}{2\tau}
=-\frac{\Vert d\tau \Vert}{2\tau}\,\eta=-\frac{k}{2}\,\eta}$
we prove that $(\nabla_X\,\omega)(Y)$ satisfies (2.5) and the
complex conformal connection (2.2) is flat.
{\bf QED}
\vskip 2mm
Let $(Q_0,g_0,\varphi,\tilde\xi_0, \tilde\eta_0)$ be an
$\alpha_0$-Sasakian space form \cite{JV} with constant
$\varphi$-holomorphic sectional curvatures $H_0$.
In \cite{GM2} we introduced warped product K\"ahler manifolds,
which are completely determined by the underlying
$\alpha_0$-Sasakian space form $Q_0$ of type
$H_0+3\alpha_0^2 \gtreqqless 0$
and the generating function $p(t), \; t \in I \subset \mathbb{R}$.
In order to obtain a local description of the K\"ahler manifolds
admitting a flat complex conformal connection we apply Theorem 6.1
in \cite {GM2}, which states:
{\it Any Bochner-K\"ahler manifold whose scalar distribution is a
$B_0$-distribution locally has the structure of a warped
product K\"ahler manifold with generating function $p(t)$
$($or $t(p))$ of type $1. - 13.$}
According to Theorem \ref{T:3.1} any K\"ahler manifold $(M,g,J),\;
(\dim M =2n \geqq 6)$ admitting a flat complex conformal connection
is locally a Bochner-K\"ahler manifold whose scalar distribution
is a $B_0$-distribution with function $a+k^2=0$ and constants
${\mathfrak B}=\frak b_0=0$. In terms of \cite{GM2} the conditions
${\mathfrak B}=\frak b_0=0$ are equivalent to the conditions
$\frak K= \frak b_0=0$.
Hence, $(M,g,J)$ is a warped product
Bochner-K\"ahler manifold whose underlying
$\alpha_0$-Sasakian space form is of type
$H_0+3\alpha_0^2 = 0$ with metric
$$g=p^2(t)\displaystyle{\left\{g_0+
\left(\frac{1}{\alpha_0}\frac{dp}{dt}-1\right)\,
\tilde\eta_0\otimes\tilde\eta_0\right\}+
\eta\otimes\eta},$$
generated by the function
$$p(t)=\frac{1}{\sqrt[3]{1-3\alpha_0(t-t_0)}}\,, \quad
t\in \left(-\infty,\,\frac{1+3\alpha_0t_0}{3\alpha_0}\right)$$
of type 9. \cite{GM2}.
This metric is not complete.
Especially in the case $\alpha_0=1$ the underlying manifold is a
Sasakian space form with $H_0=-3$. Sasakian space forms of type
$H_0=-3$ have been studied by Ogiue \cite {O} and Okumura \cite{Ok}.
A classification theorem for Sasakian space forms under the assumption
of completeness has been given by Tanno \cite{T}.
\end{document}
|
\begin{document}
\title{Heat and entropy flows in Carnot groups}
\author[L. Ambrosio]{Luigi Ambrosio}
\address{Scuola Normale Superiore, Piazza Cavalieri 7, 56126 Pisa, Italy}
\email{[email protected]}
\author[G. Stefani]{Giorgio Stefani}
\address{Scuola Normale Superiore, Piazza Cavalieri 7, 56126 Pisa, Italy}
\email{[email protected]}
\date{\today}
\keywords{Carnot group, entropy, gradient flow, sub-elliptic heat equation}
\subjclass[2010]{53C17, 28A33, 35K05}
\begin{abstract}
We prove the correspondence between the solutions of the sub-elliptic heat equation in a Carnot group~$\mathfrak{m}athbb{G}$ and the gradient flows of the relative entropy functional in the Wasserstein space of probability measures on~$\mathfrak{m}athbb{G}$. Our result completely answers a question left open in a previous paper by N.~Juillet, where the same correspondence was proved for $\mathfrak{m}athbb{G}=\mathfrak{m}athbb{H}^n$, the $n$-dimensional Heisenberg group.
\end{abstract}
\mathfrak{m}aketitle
\section{Introduction}
Since the pioneering works~\cites{JKO98,O01} and the monograph~\cite{AGS08}, in the last twenty years there has been an increasing interest in the study of the relation between evolution equations and gradient flows of energy functionals in a large variety of different frameworks, see~\cites{AGS14,AS07,CM14,E10,EM14,FSS10,G10,GKO13,GO12,KL09,M11,O09,OS09,S07,V09}.
The prominent case in the literature is represented by the connection between the heat equation and the relative entropy functional. It is well-known that the heat equation
\begin{equation}\label{intro_eq:heat_eq}\tag{\textbf{HE}}
\begin{cases}
\partial_t u_t=\mathfrak{m}athrm{D}elta u_t & \text{in}\ (0,+\infty)\times\mathfrak{m}athbb{R}^n,\\
u_0=\bar{u}\in L^2(\mathfrak{m}athbb{R}^n) & \text{on}\ \set*{0}\times\mathfrak{m}athbb{R}^n,
\end{cases}
\end{equation}
can be seen as the gradient flow in $L^2(\mathfrak{m}athbb{R}^n)$ of the Dirichlet energy $\mathfrak{m}athsf{D}(u)=\int_{\mathfrak{m}athbb{R}^n}|\nabla u|^2\ dx$ accordingly to the general approach introduced in~\cite{B73}. If the initial datum $\bar u\in L^2(\mathfrak{m}athbb{R}^n)$ is such that $\mathfrak{m}u_0=\bar{u}\,\mathfrak{m}athcal{L}^n\in\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{R}^n)$, where
\begin{equation*}
\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{R}^n)=\set*{\mathfrak{m}u\in\mathfrak{m}athcal{P}(\mathfrak{m}athbb{R}^n) : \int_{\mathfrak{m}athbb{R}^n} |x|^2\ d\mathfrak{m}u(x)<+\infty},
\end{equation*}
then the solution $(u_t)_{t\ge0}$ of~\eqref{intro_eq:heat_eq} induces a curve $(\mathfrak{m}u_t)_{t\ge0}\subset\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{R}^n)$, $\mathfrak{m}u_t=u_t\,\mathfrak{m}athcal{L}^n$. If we endow the set $\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{R}^n)$ with its usual Wasserstein distance~$\mathfrak{m}athsf{W}$, then the curve $(\mathfrak{m}u_t)_{t\ge0}$ is locally absolutely continuous with locally integrable squared $\mathfrak{m}athsf{W}$-derivative in $[0,+\infty)$. On the one hand, since $(u_t)_{t\ge0}$ satisfies~\eqref{intro_eq:heat_eq}, the curve $(\mathfrak{m}u_t)_{t\ge0}$ naturally solves in the weak sense the continuity equation
\begin{equation}\label{intro_eq:CE}\tag{\textbf{CE}}
\begin{cases}
\partial_t\mathfrak{m}u_t+\diverg(v_t\mathfrak{m}u_t)=0 & \text{in}\ (0,+\infty)\times\mathfrak{m}athbb{R}^n,\\
\mathfrak{m}u_0=\bar{u}\,\mathfrak{m}athcal{L}^n & \text{on}\ \set*{0}\times\mathfrak{m}athbb{R}^n,
\end{cases}
\end{equation}
where the velocity vector field $(v_t)_{t\ge0}$ is given by $v_t=-{\nabla u_t}/{u_t}$. On the other hand, the \emph{relative entropy}
\begin{equation*}
\mathfrak{m}athsf{Ent}(\mathfrak{m}u)=\int_{\mathfrak{m}athbb{R}^n} u\log u\ dx,
\qquad
\text{for}\ \mathfrak{m}u=u\,\mathfrak{m}athcal{L}^n\in\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{R}^n),
\end{equation*}
computed along a curve $(\mathfrak{m}u_t)_{t\ge0}\subset\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{R}^n)$ satisfying~\eqref{intro_eq:CE} for a given velocity field $(v_t)_{t\ge0}$ is such that
\begin{equation*}
\frac{d}{dt}\mathfrak{m}athsf{Ent}(\mathfrak{m}u_t)=-\int_{\mathfrak{m}athbb{R}^n} (\log u_t+1)\,\diverg(v_t u_t)\ dx=\int_{\mathfrak{m}athbb{R}^n}\scalar*{v_t,\frac{\nabla u_t}{u_t}}\ d\mathfrak{m}u_t.
\end{equation*}
In analogy with the Hilbertian case, but using Otto's calculus~\cite{O01} in the interpretation of the right hand side, one says that $(\mathfrak{m}u_t)_{t\ge0}$ is a gradient flow of the entropy in $(\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{R}^n),\mathfrak{m}athsf{W})$ if and only if the curve $t\mathfrak{m}apsto\mathfrak{m}athsf{Ent}(\mathfrak{m}u_t)$ has maximal dissipation rate. This happens if and only if $v_t=-{\nabla u_t}/{u_t}$, i.e.\ when $(u_t)_{t\ge0}$ satisfies~\eqref{intro_eq:heat_eq}.
Although not fully rigorous, the argument presented above contains all the key tools needed to establish the correspondence between the heat flow and the entropy flow in a general metric measure space $(X,\mathfrak{m}athsf{d},\mathfrak{m})$. Both the heat equation~\eqref{intro_eq:heat_eq} and the continuity equation~\eqref{intro_eq:CE} have been adequately understood in this general context. For~\eqref{intro_eq:heat_eq}, one relaxes the Dirichlet energy to the so-called \emph{Cheeger energy}
\begin{equation*}
\mathfrak{m}athsf{Ch}(u)=\inf\set*{\liminf_n\int_X |\mathfrak{m}athrm{D} u_n|^2\ d\mathfrak{m} : u_n\to u\ \text{in}\ L^2(X,\mathfrak{m}athsf{d},\mathfrak{m}),\ u_n\in\Lip(X)},
\end{equation*}
where the \emph{local Lipschitz constant} $|\mathfrak{m}athrm{D} u|(x)=\limsup\limits_{y\to x}\frac{|u(y)-u(x)|}{\mathfrak{m}athsf{d}(x,y)}$ of $u\in\Lip(X)$ plays the same role of the absolute value of the gradient in~$\mathfrak{m}athbb{R}^n$. It can be shown that the naturally associated Sobolev space $W^{1,2}(X,\mathfrak{m}athsf{d},\mathfrak{m})$ is a Banach space (not Hilbertian in general) and that the functional~$\mathfrak{m}athsf{Ch}$ is convex, so that~\eqref{intro_eq:heat_eq} can still be interpreted as its gradient flow in the Hilbert space $L^2(X,\mathfrak{m}athsf{d},\mathfrak{m})$, see~\cite{AGS14}. For~\eqref{intro_eq:CE}, one introduces an appropriate space $\textsf{S}^2(X)$ of test functions in $W^{1,2}(X,\mathfrak{m}athsf{d},\mathfrak{m})$ and says that $(\mathfrak{m}u_t)_{t>0}$ satisfies the continuity equation with respect to a family of maps $(L_t)_{t>0}\colon\mathfrak{m}athsf{S}^2(X)\to\mathfrak{m}athbb{R}$ if $t\mathfrak{m}apsto\int_X f\ d\mathfrak{m}u_t$ is absolutely continuous for every $f\in\mathfrak{m}athsf{S}^2(X)$ with $\frac{d}{dt}\int_X f\ d\mathfrak{m}u_t=L_t(f)$ for a.e.\ $t>0$, see~\cite{GH15}.
The notion of gradient flow of the entropy functional
\begin{equation*}
\mathfrak{m}athsf{Ent}_\mathfrak{m}(\mathfrak{m}u)=\int_X \varrho\log\varrho\ d\mathfrak{m},
\qquad
\text{for}\ \mathfrak{m}u=\varrho\mathfrak{m}\in\mathfrak{m}athcal{P}_2(X),
\end{equation*}
in the Wasserstein space $(\mathfrak{m}athcal{P}_2(X),\mathfrak{m}athsf{W})$ can be rigorously defined by requiring the validity of the following
sharp \textit{energy dissipation inequality}
\begin{equation*}
\mathfrak{m}athsf{Ent}_\mathfrak{m}(\mathfrak{m}u_t)+\frac{1}{2}\int_s^t|\dot{\mathfrak{m}u}_r|^2\ dr+\frac{1}{2}\int_s^t|\mathfrak{m}athrm{D}^-\mathfrak{m}athsf{Ent}_\mathfrak{m}|^2(\mathfrak{m}u_r)\ dr\le \mathfrak{m}athsf{Ent}_\mathfrak{m}(\mathfrak{m}u_s)
\end{equation*}
for all $0\le s\le t$, where $t\mathfrak{m}apsto|\dot{\mathfrak{m}u}_t|$ is the $\mathfrak{m}athsf{W}$-derivative of the curve $(\mathfrak{m}u_t)_{t>0}\subset\mathfrak{m}athcal{P}_2(X)$ and
\begin{equation*}
|\mathfrak{m}athrm{D}^-\mathfrak{m}athsf{Ent}_\mathfrak{m}|(\mathfrak{m}u)=\limsup\limits_{\nu\to\mathfrak{m}u}\mathfrak{m}ax\set*{\frac{\mathfrak{m}athsf{Ent}_\mathfrak{m}(\mathfrak{m}u)-\mathfrak{m}athsf{Ent}_\mathfrak{m}(\nu)}{\mathfrak{m}athsf{W}(\mathfrak{m}u,\nu)},0}
\end{equation*}
is the so-called \emph{descending slope} of the entropy. Note that this definition is consistent with the standard one in Hilbert spaces,
since $u'(t)=-\nabla E(u(t))$ is equivalent to
\begin{equation*}
\frac{1}{2}|u'|^2(t)+\frac{1}{2}|\nabla E(u(t))|^2\le-\frac{d}{dt} E(u(t))
\end{equation*}
by combining the chain rule with Cauchy--Schwarz and Young's inequalities.
As pointed out in~\cites{G10,AGS14}, this abstract approach provides a complete equivalence between the two gradient flows if the entropy is \emph{$K$-convex along geodesics} in $(\mathfrak{m}athcal{P}_2(X),\mathfrak{m}athsf{W})$ for some $K\in\mathfrak{m}athbb{R}$, that is, if
\begin{equation}\label{intro_eq:K-convexity}\tag{\textbf{K}}
\mathfrak{m}athsf{Ent}_\mathfrak{m}(\mathfrak{m}u_t)\le(1-t)\mathfrak{m}athsf{Ent}_\mathfrak{m}(\mathfrak{m}u_0)+t\,\mathfrak{m}athsf{Ent}_\mathfrak{m}(\mathfrak{m}u_1)-\frac{K}{2}t(1-t)\mathfrak{m}athsf{W}(\mathfrak{m}u_0,\mathfrak{m}u_1)^2,
\qquad
t\in[0,1],
\end{equation}
holds for a class of constant speed geodesics $(\mathfrak{m}u_t)_{t\in[0,1]}\subset\mathfrak{m}athcal{P}_2(X)$ sufficiently large to join any pair of points
in~$\mathfrak{m}athcal{P}_2(X)$. The $K$-convexity (also known as \emph{displacement convexity}) of the entropy heavily depends on the structure of $(X,\mathfrak{m}athsf{d},\mathfrak{m})$ and encodes a precise information about the ambient space: if $X$ is a Riemannian manifold, then~\eqref{intro_eq:K-convexity} is valid if and only if the Ricci curvature satisfies $\mathfrak{m}athrm{Ric}\ge K$, see~\cite{vRS05}. For this reason, if property~\eqref{intro_eq:K-convexity} holds, then $(X,\mathfrak{m}athsf{d},\mathfrak{m})$ is called a space with \emph{generalized Ricci curvature} bounded from below,
or simply a $CD(K,\infty)$ space.
According to this general framework, the correspondence between heat flow and entropy flow has been proved on Riemannian manifolds with Ricci curvature bounded from below, see~\cite{E10}, and on compact \emph{Alexandrov spaces}, see~\cites{GKO13,GO12,O09}. Alexandrov spaces are considered as metric measure spaces with \emph{generalized sectional curvature} bounded from below (a condition stronger than~\eqref{intro_eq:K-convexity}, see~\cite{P11}).
If $(X,\mathfrak{m}athsf{d},\mathfrak{m})$ is not a $CD(K,\infty)$ space, then the picture is less clear. As stated in~\cite{AGS14}*{Theorem~8.5}, the correspondence between heat flow and entropy flow still holds if the descending slope $|\mathfrak{m}athrm{D}^-\mathfrak{m}athsf{Ent}_\mathfrak{m}|$ of the entropy is an upper gradient of~$\mathfrak{m}athsf{Ent}_\mathfrak{m}$ and satisfies a precise lower semicontinuity property, basically equivalent to the equality between $|\mathfrak{m}athrm{D}^-\mathfrak{m}athsf{Ent}_\mathfrak{m}|$ and the so-called \emph{Fisher information}. These assumptions are weaker than~\eqref{intro_eq:K-convexity} but not easy to check for a given non-$CD(K,\infty)$ space.
In~\cites{J14,J09}, it was proved that the \emph{Heisenberg group}~$\mathfrak{m}athbb{H}^n$ is a non-$CD(K,\infty)$ space in which nevertheless the correspondence between heat flow and gradient flow holds. The Heisenberg group is the simplest non-commutative \emph{Carnot group}. Carnot groups are one of the most studied examples of Carnot--Carathéodory spaces, see~\cites{BLU07,LeD16,M02} and the references therein for an account on this subject. The proof of the correspondence of the two flows in~$\mathfrak{m}athbb{H}^n$ presented in~\cite{J14} essentially splits into two parts. The first part shows that a solution of the sub-elliptic heat equation $\partial_t u_t+\mathfrak{m}athrm{D}elta_{\mathfrak{m}athbb{H}^n} u_t=0$ corresponds to a gradient flow of the entropy in the Wasserstein space $(\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{H}^n),\mathfrak{m}athsf{W}_{\mathfrak{m}athbb{H}^n})$ induced by the Carnot-Carathéodory distance~$\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}}$. The direct computations needed are justified by some precise estimates on the \emph{sub-elliptic heat kernel} in~$\mathfrak{m}athbb{H}^n$ given in~\cites{L06,L07}. The second part proves that a gradient flow of the entropy in $(\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{H}^n),\mathfrak{m}athsf{W}_{\mathfrak{m}athbb{H}^n})$ induces a sub-elliptic heat diffusion in~$\mathfrak{m}athbb{H}^n$. The argument is based on a clever regularization of the gradient flow $(\mathfrak{m}u_t)_{t\ge0}$ based on the particular structure of the Lie algebra of~$\mathfrak{m}athbb{H}^n$.
An open question arisen in~\cite{J14}*{Remark~5.3} was to extend the correspondence of the two flows to any Carnot group. The aim of the present work is to give a positive answer to this problem. To prove that a solution of the sub-elliptic heat equation corresponds to a gradient flow of the entropy, we essentially follow the same strategy of~\cite{J14}. Since the results of~\cites{L06,L07} are not known for a general Carnot group, we instead rely on the weaker estimates given in~\cite{VSC92} valid in any nilpotent Lie group. To show that a gradient flow of the entropy induces a sub-elliptic heat diffusion, we regularize the gradient flow $(\mathfrak{m}u_t)_{t\ge0}$ both in time and space via convolution with smooth kernels. This regularization does not depend on the structure of the Lie algebra of the group, but nevertheless allows us to preserve the key quantities involved, such as the \emph{continuity equation} and the \emph{Fisher information}. In the presentation of the proofs, we also take advantage of a few results taken from the general setting of metric measures spaces developed in~\cite{AGS14} and in the references therein.
The paper is organized as follows. In \cref{sec:preliminaries} we collect the standard definitions and well-known facts that are used throughout the work. The precise statement of our main result is given in \cref{th:main} at the end of this part. In \cref{sec:CE_and_ent_slope} we extend the technical results presented in~\cite{J14}*{Sections~3 and~4} to any Carnot group with minor modifications and we prove that Carnot groups are non-$CD(K,\infty)$ spaces (see \cref{prop:carnot_not_CD}), generalizing the analogous result obtained in~\cite{J09}. Finally, in \cref{sec:proof_of_main_result}, we prove the correspondence of the two flows.
\section{Preliminaries}
\label{sec:preliminaries}
\subsection{AC curves, entropy and gradient flows}
Let $(X,\mathfrak{m}athsf{d})$ be a metric space, let $I\subset\mathfrak{m}athbb{R}$ be a closed interval and let $p\in[1,+\infty]$. We say that a curve $\gamma\colon I\to X$ belongs to $AC^p(I;(X,d))$ if
\begin{equation}\label{eq:def_AC_curve}
\mathfrak{m}athsf{d}(\gamma_s,\gamma_t)\le\int_s^t g(r)\ dr
\qquad
s,t\in I,\ s<t,
\end{equation}
for some $g\in L^p(I)$. The space $AC^p_{\rm loc}(I;(X,d))$ is defined analogously. The case $p=1$ corresponds to \emph{absolutely continuous curves} and is simply denoted by $AC(I;(X,d))$. It turns out that, if $\gamma\in AC^p(I;(X,d))$, there is a minimal function $g\in L^p(I)$ satisfying~\eqref{eq:def_AC_curve}, called \emph{metric derivative} of the curve~$\gamma$, which is given by
\begin{equation*}
|\dot{\gamma}_t|:=\lim_{s\to t}\frac{d(\gamma_s,\gamma_t)}{|s-t|}
\qquad
\text{for a.e.}\ t\in I.
\end{equation*}
See~\cite{AGS08}*{Theorem~1.1.2} for the simple proof. We call $(X,d)$ a \emph{geodesic} metric space if for every $x,y\in X$ there exists a curve $\gamma\colon[0,1]\to X$ such that $\gamma(0)=x$, $\gamma(1)=y$ and
\begin{equation*}
\mathfrak{m}athsf{d}(\gamma_s,\gamma_t)=|s-t|\,\mathfrak{m}athsf{d}(\gamma_0,\gamma_1)
\qquad
\forall s,t\in[0,1].
\end{equation*}
Let $\mathfrak{m}athbb{R}^*=\mathfrak{m}athbb{R}\cup\set{-\infty,+\infty}$ and let $f\colon X\to\mathfrak{m}athbb{R}^*$ be a function. We define the \emph{effective domain} of~$f$ as
\begin{equation*}
\dom(f):=\set*{x\in X : f(x)\in\mathfrak{m}athbb{R}}.
\end{equation*}
Given $x\in\dom(f)$, we define the \emph{local Lipschitz constant} of~$f$ at~$x$ by
\begin{equation*}
|\mathfrak{m}athrm{D} f|(x):=\limsup_{y\to x}\frac{|f(y)-f(x)|}{\mathfrak{m}athsf{d}(x,y)}.
\end{equation*}
The \emph{descending slope} and the \emph{ascending slope} of~$f$ at~$x$ are respectively given by
\begin{equation*}
|\mathfrak{m}athrm{D}^-f|(x):=\limsup_{y\to x}\frac{[f(y)-f(x)]^-}{\mathfrak{m}athsf{d}(x,y)},
\qquad
|\mathfrak{m}athrm{D}^+f|(x):=\limsup_{y\to x}\frac{[f(y)-f(x)]^+}{\mathfrak{m}athsf{d}(x,y)}.
\end{equation*}
Here $a^+$ and $a^-$ denote the positive and negative part of $a\in\mathfrak{m}athbb{R}$ respectively. When $x\in\dom(f)$ is an isolated point of $X$, we set $|\mathfrak{m}athrm{D} f|(x)=|\mathfrak{m}athrm{D}^-f|(x)=|\mathfrak{m}athrm{D}^+f|(x)=0$. By convention, we set $|\mathfrak{m}athrm{D} f|(x)=|\mathfrak{m}athrm{D}^-f|(x)=|\mathfrak{m}athrm{D}^+f|(x)=+\infty$ for all $x\in X\setminus\dom(f)$.
\begin{definition}[Gradient flow]\label{def:metric_GF}
Let $E\colon X\to\mathfrak{m}athbb{R}\cup\set*{+\infty}$ be a function. We say that a curve $\gamma\in AC_{\rm loc}([0,+\infty);(X,\mathfrak{m}athsf{d}))$ is a \emph{(metric) gradient flow} of~$E$ starting from $\gamma_0\in\dom(E)$ if the \emph{energy dissipation inequality (EDI)}
\begin{equation}\label{eq:def_EDI}
E(\gamma_t)+\frac{1}{2}\int_s^t|\dot{\gamma}_r|^2\ dr+\frac{1}{2}\int_s^t|\mathfrak{m}athrm{D}^-E|^2(\gamma_r)\ dr\le E(\gamma_s)
\end{equation}
holds for all $s,t\ge0$ with $s\le t$.
\end{definition}
Note that, if $(\gamma_t)_{t\ge0}$ is a gradient flow of~$E$, then $\gamma_t\in\dom(E)$ for all $t\ge0$ and $\gamma\in AC^2_{\rm loc}([0,+\infty);(X,\mathfrak{m}athsf{d}))$ with $t\mathfrak{m}apsto|\mathfrak{m}athrm{D}^-E|(\gamma_t)\in L^2_{\rm loc}([0,+\infty))$. Moreover, the function $t\mathfrak{m}apsto E(\gamma_t)$ is non-increasing on~$[0,+\infty)$ and thus a.e.\ differentiable and locally integrable.
\begin{remark}\label{rem:GF_AC}
As observed in~\cite{AGS14}*{Section~2.5}, if the function $t\mathfrak{m}apsto E(\gamma_t)$ is locally absolutely continuous on~$(0,+\infty)$, then~\eqref{eq:def_EDI} holds as an equality by the chain rule and Young's inequality. In this case, \eqref{eq:def_EDI} is also equivalent to
\begin{equation*}
\frac{d}{dt}E(\gamma_t)=-|\dot{\gamma}_t|^2=-|\mathfrak{m}athrm{D}^-E|^2(\gamma_t)
\qquad
\text{for a.e.}\ t>0.
\end{equation*}
\end{remark}
\subsection{Wasserstein space}
\label{subsec:wasserstein_space}
We now briefly recall some properties of the Wasserstein space needed for our purposes. For a more detailed introduction to this topic, we refer the interested reader to~\cite{AG13}*{Section~3}.
Let $(X,\mathfrak{m}athsf{d})$ be a Polish space, i.e.\ a complete and separable metric space. We denote by $\mathfrak{m}athcal{P}(X)$ the set of probability Borel measures on~$X$. The \emph{Wasserstein distance} $\mathfrak{m}athsf{W}$ between $\mathfrak{m}u,\nu\in\mathfrak{m}athcal{P}(X)$ is given by
\begin{equation}\label{eq:def_W_2}
\mathfrak{m}athsf{W}(\mathfrak{m}u,\nu)^2=\inf\set*{\int_{X\times X} \mathfrak{m}athsf{d}(x,y)^2\ d\pi : \pi\in\mathfrak{m}athbb{G}amma(\mathfrak{m}u,\nu)},
\end{equation}
where
\begin{equation}\label{eq:def_coupling}
\mathfrak{m}athbb{G}amma(\mathfrak{m}u,\nu):=\set*{\pi\in\mathfrak{m}athcal{P}(X\times X) : (p_1)_\#\pi=\mathfrak{m}u,\ (p_2)_\#\pi=\nu}.
\end{equation}
Here $p_i\colon X\times X\to X$, $i=1,2$, are the the canonical projections on the components. As usual, if $\mathfrak{m}u\in\mathfrak{m}athcal{P}(X)$ and $T\colon X\to Y$ is a $\mathfrak{m}u$-measurable map with values in the topological space~$Y$, the \emph{push-forward measure} $T_\#(\mathfrak{m}u)\in\mathfrak{m}athcal{P}(Y)$ is defined by $T_\#(\mathfrak{m}u)(B):=\mathfrak{m}u(T^{-1}(B))$ for every Borel set $B\subset Y$. The set $\mathfrak{m}athbb{G}amma(\mathfrak{m}u,\nu)$ introduced in~\eqref{eq:def_coupling} is call the set of \emph{admissible plans} or \emph{couplings} for the pair~$(\mathfrak{m}u,\nu)$. For any Polish space $(X,\mathfrak{m}athsf{d})$, there exist optimal couplings where the infimum in~\eqref{eq:def_W_2} is achieved.
The function~$\mathfrak{m}athsf{W}$ is a distance on the so-called \emph{Wasserstein space} $(\mathfrak{m}athcal{P}_2(X),\mathfrak{m}athsf{W})$, where
\begin{equation*}
\mathfrak{m}athcal{P}_2(X):=\set*{\mathfrak{m}u\in\mathfrak{m}athcal{P}(X) : \int_X \mathfrak{m}athsf{d}(x,x_0)^2\ d\mathfrak{m}u(x)<+\infty\
\text{for some, and thus any,}\ x_0\in X}.
\end{equation*}
The space $(\mathfrak{m}athcal{P}_2(X),\mathfrak{m}athsf{W})$ is Polish. If $(X,\mathfrak{m}athsf{d})$ is geodesic, then $(\mathfrak{m}athcal{P}_2(X),\mathfrak{m}athsf{W})$ is geodesic as well. Moreover, $\mathfrak{m}u_n\xrightarrow{\mathfrak{m}athsf{W}}\mathfrak{m}u$ if and only if $\mathfrak{m}u_n\rightharpoonup\mathfrak{m}u$ and
\begin{equation*}
\int_X \mathfrak{m}athsf{d}(x,x_0)^2\ d\mathfrak{m}u_n(x)\to\int_X \mathfrak{m}athsf{d}(x,x_0)^2\ d\mathfrak{m}u(x)\ \qquad\text{for some}\ x_0\in X.
\end{equation*}
As usual, we write $\mathfrak{m}u_n\rightharpoonup\mathfrak{m}u$ if $\int_X\varphi\ d\mathfrak{m}u_n\to\int_X\varphi\ d\mathfrak{m}u$ for all $\varphi\in C_b(X)$.
\subsection{Relative entropy}
Let $(X,\mathfrak{m}athsf{d},\mathfrak{m})$ be a metric measure space, where $(X,\mathfrak{m}athsf{d})$ is a Polish metric space and $\mathfrak{m}$ is a non-negative, Borel and $\sigma$-finite measure. We assume that the space $(X,\mathfrak{m}athsf{d},\mathfrak{m})$ satisfies the following structural assumption: there exist a point $x_0\in X$ and two constants $c_1,c_2>0$ such that
\begin{equation}\label{eq:assumption_on_measure}
\mathfrak{m}\left(\set*{x\in X : \mathfrak{m}athsf{d}(x,x_0)<r}\right)\le c_1 e^{c_2 r^2}.
\end{equation}
The \emph{relative entropy} $\mathfrak{m}athsf{Ent}_\mathfrak{m}\colon\mathfrak{m}athcal{P}_2(X)\to(-\infty,+\infty]$ is defined as
\begin{equation}\label{eq:def_entropy}
\mathfrak{m}athsf{Ent}_\mathfrak{m}(\mathfrak{m}u):=
\begin{cases}
\displaystyle\int_X \varrho\log\varrho\ d\mathfrak{m} & \text{if}\ \mathfrak{m}u=\varrho\mathfrak{m}\in\mathfrak{m}athcal{P}_2(X) ,\\[3mm]
+\infty & \text{otherwise}.
\end{cases}
\end{equation}
According to our definition, $\mathfrak{m}u\in\dom(\mathfrak{m}athsf{Ent}_\mathfrak{m})$ implies that $\mathfrak{m}u\in\mathfrak{m}athcal{P}_2(X)$ and that the effective domain $\dom(\mathfrak{m}athsf{Ent}_\mathfrak{m})$ is convex. As pointed out in~\cite{AGS14}*{Section~7.1}, the structural assumption~\eqref{eq:assumption_on_measure} guarantees that in fact $\mathfrak{m}athsf{Ent}(\mathfrak{m}u)>-\infty$ for all $\mathfrak{m}u\in\mathfrak{m}athcal{P}_2(X)$.
When $\mathfrak{m}\in\mathfrak{m}athcal{P}(X)$, the entropy functional~$\mathfrak{m}athsf{Ent}_\mathfrak{m}$ naturally extends to~$\mathfrak{m}athcal{P}(X)$, is lower semicontinuous with respect to the weak convergence in~$\mathfrak{m}athcal{P}(X)$ and positive by Jensen's inequality. In addition, if $F\colon X\to Y$ is a Borel map, then
\begin{equation}\label{eq:ent_push-forward_formula}
\mathfrak{m}athsf{Ent}_{F_\#\mathfrak{m}}(F_\#\mathfrak{m}u)\le\mathfrak{m}athsf{Ent}_\mathfrak{m}(\mathfrak{m}u)
\qquad
\text{for all}\ \mathfrak{m}u\in\mathfrak{m}athcal{P}(X),
\end{equation}
with equality if~$F$ is injective, see~\cite{AGS08}*{Lemma~9.4.5}.
When $\mathfrak{m}(X)=+\infty$, if we set $\mathfrak{m}athfrak{n}:=e^{-c\,\mathfrak{m}athsf{d}(\cdot,x_0)^2}\mathfrak{m}$ for some $x_0\in X$, where $c>0$ is chosen so that $\mathfrak{m}athfrak{n}(X)<+\infty$ (note that the existence of such $c>0$ is guaranteed by~\eqref{eq:assumption_on_measure}), then we obtain the useful formula
\begin{equation}\label{eq:ent_useful_formula}
\mathfrak{m}athsf{Ent}_\mathfrak{m}(\mathfrak{m}u)=\mathfrak{m}athsf{Ent}_{\mathfrak{m}athfrak{n}}(\mathfrak{m}u)-c\int_X\mathfrak{m}athsf{d}(x,x_0)^2\ d\mathfrak{m}u
\qquad
\text{for all}\ \mathfrak{m}u\in\mathfrak{m}athcal{P}_2(X).
\end{equation}
This shows that $\mathfrak{m}athsf{Ent}_\mathfrak{m}$ is lower semicontinuous in $(\mathfrak{m}athcal{P}_2(X),\mathfrak{m}athsf{W})$.
\subsection{Carnot groups}
\label{subsec:carnot_groups}
Let $\mathfrak{m}athbb{G}$ be a Carnot group, i.e.\ a connected, simply connected and nilpotent Lie group whose Lie algebra $\mathfrak{m}athfrak{g}$ of left-invariant vector fields has dimension $n$ and admits a stratification of step $\kappa$,
\begin{equation*}
\mathfrak{m}athfrak{g}=V_1\oplus V_2\oplus\cdots\oplus V_\kappa
\end{equation*}
with
\begin{equation*}
V_i=[V_1,V_{i-1}]\quad \text{for } i=1,\dots,\kappa, \qquad [V_1,V_\kappa]=\set{0}.
\end{equation*}
We set $m_i=\dim(V_i)$ and $h_i=m_1+\dots+m_i$ for $i=1,\dots,\kappa$, with $h_0=0$ and $h_\kappa=n$. We fix an adapted basis of $\mathfrak{m}athfrak{g}$, i.e.\ a basis $X_1,\dots,X_n$ such that
\begin{equation*}
X_{h_{i-1}+1},\dots,X_{h_i}\ \text{is a basis of}\ V_i,\qquad i=1,\dots,\kappa.
\end{equation*}
Using exponential coordinates, we can identify~$\mathfrak{m}athbb{G}$ with $\mathfrak{m}athbb{R}^n$ endowed with the group law determined by the Campbell--Hausdorff formula (in particular, the identity $e\in\mathfrak{m}athbb{G}$ corresponds to $0\in\mathfrak{m}athbb{R}^n$ and $x^{-1}=-x$ for $x\in\mathfrak{m}athbb{G}$). It is not restrictive to assume that $X_i(0)=\mathfrak{m}athrm{e}_i$ for any $i=1,\dots,n$; therefore, by left-invariance, for any $x\in\mathfrak{m}athbb{G}$ we get
\begin{equation}\label{eq:X_i_dleft_transl}
X_i(x)=dl_x\mathfrak{m}athrm{e}_i, \qquad i=1,\dots,n,
\end{equation}
where $l_x\colon\mathfrak{m}athbb{G}\to\mathfrak{m}athbb{G}$ is the left-translation by $x\in\mathfrak{m}athbb{G}$, i.e.\ $l_x(y)=xy$ for any $y\in\mathfrak{m}athbb{G}$. We endow $\mathfrak{m}athfrak{g}$ with the left-invariant Riemannian metric $\scalar*{\cdot,\cdot}_\mathfrak{m}athbb{G}$ that makes the basis $X_1,\dots,X_n$ orthonormal. For any $i=1,\dots,n$, we define the gradient with respect to the layer $V_i$ as
\begin{equation*}
\nabla_{V_i} f:=\sum_{j=h_{i-1}+1}^{h_i} (X_j f) \, X_j\in V_i.
\end{equation*}
We let $H\mathfrak{m}athbb{G}\subset T\mathfrak{m}athbb{G}$ be the \emph{horizontal tangent bundle} of the group~$\mathfrak{m}athbb{G}$, i.e.\ the left-invariant sub-bundle of the tangent bundle~$T\mathfrak{m}athbb{G}$ such that $H_e\mathfrak{m}athbb{G}=\set*{X(0) : X\in V_1}$. We use the distinguished notation $\nabla_\mathfrak{m}athbb{G}:=\nabla_{V_1}$ for the \emph{horizontal gradient}.
For any $i=1,\dots,n$, we define the degree $d(i)\in\set*{1,\dots,\kappa}$ of the basis vector field $X_i$ as $d(i)=j$ if and only if $X_i\in V_j$. With this notion, the one-parameter family of group dilations $(\partiallta_\lambda)_{\lambda\ge0}\colon\mathfrak{m}athbb{G}\to\mathfrak{m}athbb{G}$ is given by
\begin{equation}\label{eq:def_dilation}
\partiallta_\lambda(x)=\partiallta_\lambda(x_1,\dots,x_n):=(\lambda x_1,\dots,\lambda^{d(i)} x_i,\dots,\lambda^\kappa x_n) \qquad\text{for all}\ x\in\mathfrak{m}athbb{G}.
\end{equation}
The Haar measure of the group $\mathfrak{m}athbb{G}$ coincides with the $n$-dimensional Lebesgue measure~$\mathfrak{m}athcal{L}^n$ and has the homogeneity property $\mathfrak{m}athcal{L}^n(\partiallta_\lambda(E))=\lambda^Q\mathfrak{m}athcal{L}^n(E)$, where the integer $Q=\sum_{i=1}^\kappa i\dim(V_i)$ is the \emph{homogeneous dimension} of the group.
We endow the group $\mathfrak{m}athbb{G}$ with the canonical Carnot--Carathéodory structure induced by~$H\mathfrak{m}athbb{G}$. We say that a Lipschitz curve $\gamma\colon[0,1]\to\mathfrak{m}athbb{G}$ is a \emph{horizontal curve} if $\dot{\gamma}(t)\in H_{\gamma(t)}\mathfrak{m}athbb{G}$ for a.e.\ $t\in[0,1]$. The \emph{Carnot--Carathéodory distance} between $x,y\in\mathfrak{m}athbb{G}$ is then defined as
\begin{equation*}
\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}}(x,y)=\inf\set*{\int_0^1\|\dot{\gamma}(t)\|_\mathfrak{m}athbb{G}\ dt : \gamma\ \text{is horizontal},\ \gamma(0)=x,\ \gamma(1)=y}.
\end{equation*}
By Chow--Rashevskii's Theorem, the function $\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}}$ is in fact a distance, which is also left-invariant and homogeneous with respect to the dilations defined in~\eqref{eq:def_dilation}, precisely $\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}}(zx,zy)=\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}}(x,y)$ and $\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}}(\partiallta_\lambda(x),\partiallta_\lambda(y))=\lambda\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}}(x,y)$ for all $x,y,z\in\mathfrak{m}athbb{G}$ and $\lambda\ge0$. The resulting metric space $(\mathfrak{m}athbb{G},\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}})$ is a Polish geodesic space. We let $\mathfrak{m}athsf{B}_\mathfrak{m}athbb{G}(x,r)$ be the $\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}}$-ball centred at $x\in\mathfrak{m}athbb{G}$ of radius $r>0$. Note that $\mathfrak{m}athcal{L}^n(\mathfrak{m}athsf{B}_\mathfrak{m}athbb{G}(x,r))=c_n r^Q$, where $c_n=\mathfrak{m}athcal{L}^n(\mathfrak{m}athsf{B}_\mathfrak{m}athbb{G}(0,1))$. In particular, the metric measure space $(\mathfrak{m}athbb{G},\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}},\mathfrak{m}athcal{L}^n)$ satisfies the structural assumption~\eqref{eq:assumption_on_measure}.
Let us write $x=(\tilde{x}_1,\dots,\tilde{x}_\kappa)$, where $\tilde{x}_i:=(x_{h_{i-1}+1},\dots,x_{h_i})$ for $i=1,\dots,\kappa$. As proved in~\cite{FSSC03}*{Theorem~5.1}, there exist suitable constants $c_1=1$, $c_2,\dots,c_k\in(0,1)$ depending only on the group structure of~$\mathfrak{m}athbb{G}$ such that
\begin{equation}\label{eq:def_box_norm}
\mathfrak{m}athsf{d}_\infty(x,0):=\mathfrak{m}ax\set*{c_i\abs*{\tilde{x}_i}_{\mathfrak{m}athbb{R}^{m_i}}^{1/i} : i=1,\dots,\kappa}, \qquad x\in\mathfrak{m}athbb{G},
\end{equation}
induces a left-invariant and homogeneous distance $\mathfrak{m}athsf{d}_\infty(x,y):=\mathfrak{m}athsf{d}_\infty(y^{-1}x,0)$, $x,y\in\mathfrak{m}athbb{G}$, which is equivalent to~$\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}}$.
Let $1\le p<+\infty$ and let $\Omega\subset\mathfrak{m}athbb{R}^n$ be an open set. The \emph{horizontal Sobolev space}
\begin{equation}\label{eq:def_horiz_sobolev_space}
W^{1,p}_\mathfrak{m}athbb{G}(\Omega):=\set*{u\in L^p(\Omega) : X_i u\in L^p(\Omega),\ i=1,\dots,m_1}
\end{equation}
endowed with the norm
\begin{equation*}
\|u\|_{W^{1,p}_\mathfrak{m}athbb{G}(\Omega)}:=\|u\|_{L^p(\Omega)}+\sum_{i=1}^{m_1}\|X_i u\|_{L^p(\Omega)}
\end{equation*}
is a reflexive Banach space, see~\cite{FSSC96}*{Proposition~1.1.2}. By~\cite{FSSC96}*{Theorem~1.2.3}, the set $C^\infty(\Omega)\cap W^{1,p}_\mathfrak{m}athbb{G}(\Omega)$ is dense in $W^{1,p}_\mathfrak{m}athbb{G}(\Omega)$. By a standard cut-off argument, we get that $C^\infty_c(\mathfrak{m}athbb{R}^n)$ is dense in $W^{1,p}_\mathfrak{m}athbb{G}(\mathfrak{m}athbb{R}^n)$.
\subsection{Riemannian approximation}
\label{subsec:riemannian_approx}
The metric space $(\mathfrak{m}athbb{G},\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}})$ can be seen as the limit in the \emph{pointed Gromov--Hausdorff sense} as $\varepsilon\to0$ of a family of Riemannian manifolds $\set*{(\mathfrak{m}athbb{G}_\varepsilon,\partialps)}_{\varepsilon>0}$ defined as follows, see~\cite{CDPT07}*{Theorem~2.12}. For any $\varepsilon>0$, we define the Riemannian approximation $(\mathfrak{m}athbb{G}_\varepsilon,\partialps)$ of the Carnot group $(\mathfrak{m}athbb{G},\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}})$ as the manifold $\mathfrak{m}athbb{R}^n$ endowed with the Riemannian metric $g_\varepsilon(\cdot,\cdot)\equiv\scalar*{\cdot,\cdot}_\varepsilon$ that makes orthonormal the vector fields $\varepsilon^{d(i)-1}X_i$, $i=1,\dots,n$, i.e.\ such that
\begin{equation*}
\scalar*{X_i,X_j}_\varepsilon=\varepsilon^{2-d(i)-d(j)}\partiallta_{ij}, \qquad i,j=1,\dots,n.
\end{equation*}
We let $\partialps$ be the Riemannian distance induced by the metric~$g_\varepsilon$. Note that $\partialps$ is left-invariant and satisfies $\partialps\le\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}}$ for all $\varepsilon>0$. For any $\varepsilon>0$, the $\varepsilon$-Riemannian gradient is defined as
\begin{equation*}
\nabla_\varepsilon f=\sum_{i=1}^n\varepsilon^{2(d(i)-1)}(X_i f) \, X_i=\sum_{i=1}^\kappa\varepsilon^{2(i-1)}\nabla_{V_i}f.
\end{equation*}
By~\eqref{eq:X_i_dleft_transl}, we get that
\begin{equation*}
g_\varepsilon(x)=(dl_x)^T D_\varepsilon\,(dl_x), \qquad x\in\mathfrak{m}athbb{G},
\end{equation*}
where $D_\varepsilon$ is the diagonal block matrix given by
\begin{equation*}
D_\varepsilon=\diag(\mathfrak{m}athbf{1}_{m_1},\varepsilon^{-2}\mathfrak{m}athbf{1}_{m_2},\dots,\varepsilon^{-2(i-1)}\mathfrak{m}athbf{1}_{m_i},\dots,\varepsilon^{-2(\kappa-1)}\mathfrak{m}athbf{1}_{m_\kappa}).
\end{equation*}
As a consequence, the Riemannian volume element is given by
\begin{equation*}
\vol_\varepsilon=\sqrt{\partialt g_\varepsilon}\,dx_1\wedge\dots\wedge dx_n=\varepsilon^{n-Q}\mathfrak{m}athcal{L}^n.
\end{equation*}
We remark that, for each $\varepsilon>0$, the $n$-dimensional Riemannian manifold $(\mathfrak{m}athbb{G}_\varepsilon,\partialps)$ has Ricci curvature bounded from below. More precisely, there exists a constant $K>0$, depending only on the Carnot group $\mathfrak{m}athbb{G}$, such that
\begin{equation}\label{eq:ricci_bounded_below}
\ric_\varepsilon\ge-K\varepsilon^{-2}
\qquad
\text{for all}\ \varepsilon>0.
\end{equation}
By scaling invariance, the proof of inequality~\eqref{eq:ricci_bounded_below} can be reduced to the case $\varepsilon=1$, which in turn is a direct consequence of~\cite{M76}*{Lemma~1.1}.
In the sequel, we will consider the metric measure space $(\mathfrak{m}athbb{G}_\varepsilon,\partialps,\mathfrak{m}athcal{L}^n)$, i.e.\ the Riemannian manifold $(\mathfrak{m}athbb{G}_\varepsilon,\partialps,\textrm{vol}_\varepsilon)$ with a rescaled volume measure. Both these two spaces satisfy the structural assumption~\eqref{eq:assumption_on_measure}. Moreover, we have
\begin{equation}\label{eq:entropy_riem_vs_leb}
\mathfrak{m}athsf{Ent}_{\textrm{vol}_\varepsilon}(\mathfrak{m}u)=\mathfrak{m}athsf{Ent}(\mathfrak{m}u)+\log(\varepsilon^{Q-n})
\qquad
\text{for all}\ \varepsilon>0.
\end{equation}
Here and in the following, $\mathfrak{m}athsf{Ent}$ denotes the entropy with respect to the reference measure~$\mathfrak{m}athcal{L}^n$.
\subsection{Sub-elliptic heat equation}
We let $\mathfrak{m}athrm{D}elta_\mathfrak{m}athbb{G}=\sum_{i=1}^{m_1} X_i^2$ be the so-called \emph{sub-Laplacian operator}. Since the horizontal vector fields $X_1,\dots,X_{h_1}$ satisfy H\"ormander's condition, by H\"ormander's theorem the \emph{sub-elliptic heat operator} $\partial_t-\mathfrak{m}athrm{D}elta_\mathfrak{m}athbb{G}$ is \emph{hypoelliptic}, meaning that its fundamental solution $\mathfrak{m}athsf{h}\colon(0,+\infty)\times\mathfrak{m}athbb{G}\to(0,+\infty)$, $\mathfrak{m}athsf{h}_t(x)=\mathfrak{m}athsf{h}(t,x)$, the so-called \emph{heat kernel}, is smooth. In the following result, we collect some properties of the heat kernel that will be used in the sequel. We refer the reader to~\cite{VSC92}*{Chapter~IV} and to the references therein for the proof.
\begin{theorem}[Properties of the heat kernel]\label{th:property_heat_kernel}
The heat kernel $\mathfrak{m}athsf{h}\colon(0,+\infty)\times\mathfrak{m}athbb{G}\to(0,+\infty)$ satisfies the following properties:
\begin{enumerate}[(i)]
\item $\mathfrak{m}athsf{h}_t(x^{-1})=\mathfrak{m}athsf{h}_t(x)$ for any $(t,x)\in(0,+\infty)\times\mathfrak{m}athbb{G}$;
\item $\mathfrak{m}athsf{h}_{\lambda^2 t}(\partiallta_\lambda(x))=\lambda^{-Q}\mathfrak{m}athsf{h}_t(x)$ for any $\lambda>0$ and $(t,x)\in(0,+\infty)\times\mathfrak{m}athbb{G}$;
\item $\int_\mathfrak{m}athbb{G}\mathfrak{m}athsf{h}_t\ dx=1$ for any $t>0$;
\item there exists $C>0$, depending only on $\mathfrak{m}athbb{G}$, such that
\begin{equation}\label{eq:heat_estimate_above}
\mathfrak{m}athsf{h}_t(x)\le Ct^{-Q/2}\exp\left(-\frac{\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}}(x,0)^2}{4t}\right)
\qquad
\forall (t,x)\in(0,+\infty)\times\mathfrak{m}athbb{G};
\end{equation}
\item for any $\varepsilon>0$, there exists $C_\varepsilon>0$ such that
\begin{equation}\label{eq:heat_estimate_below}
\mathfrak{m}athsf{h}_t(x)\ge C_\varepsilon t^{-Q/2}\exp\left(-\frac{\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}}(x,0)^2}{4(1-\varepsilon)t}\right)
\qquad
\forall (t,x)\in(0,+\infty)\times\mathfrak{m}athbb{G};
\end{equation}
\item for every $j,l\in\mathfrak{m}athbb{N}$ and $\varepsilon>0$, there exists $C_\varepsilon(j,l)>0$ such that
\begin{equation}\label{eq:heat_estimate_derivatives}
|(\partial_t)^l X_{i_1}\cdots X_{i_j}\mathfrak{m}athsf{h}_t(x)|\le C_\varepsilon(j,l)t^{-\frac{Q+j+2l}{2}}\exp\left(-\frac{\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}}(x,0)^2}{4(1+\varepsilon)t}\right)
\qquad
\forall (t,x)\in(0,+\infty)\times\mathfrak{m}athbb{G},
\end{equation}
where $X_{i_1}\cdots X_{i_j}\in V_1$.
\end{enumerate}
\end{theorem}
Given $\varrho\in L^1(\mathfrak{m}athbb{G})$, the function
\begin{equation}\label{eq:sol_heat_diff}
\varrho_t(x)=(\varrho\star\mathfrak{m}athsf{h}_t)(x)=\int_\mathfrak{m}athbb{G} \mathfrak{m}athsf{h}_t(y^{-1} x)\, \varrho(y)\ dy,
\qquad
(t,x)\in(0,+\infty)\times\mathfrak{m}athbb{G},
\end{equation}
is smooth and is a solution of the heat diffusion problem
\begin{equation}\label{eq:heat_diffusion}
\begin{cases}
\partial_t\varrho_t=\mathfrak{m}athrm{D}elta_\mathfrak{m}athbb{G}\varrho_t &\text{in}\ (0,+\infty)\times\mathfrak{m}athbb{G},\\
\varrho_0=\varrho, &\text{on}\ \{0\}\times\mathfrak{m}athbb{G}.
\end{cases}
\end{equation}
The initial datum is assumed in the $L^1$-sense, i.e.\ $\lim\limits_{t\to0}\varrho_t=\varrho$ in $L^1(\mathfrak{m}athbb{G})$. As a consequence of the properties of the heat kernel, if $\varrho\ge0$ then the solution $(\varrho_t)_{t\ge0}$ in~\eqref{eq:sol_heat_diff} is everywhere positive and satisfies
\begin{equation*}
\int_\mathfrak{m}athbb{G}\varrho_t(x)\ dx=\|\varrho\|_{L^1(\mathfrak{m}athbb{G})}
\qquad
\forall t>0.
\end{equation*}
In addition, if $\varrho\mathfrak{m}athcal{L}^n\in\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G})$ then $(\varrho_t\mathfrak{m}athcal{L}^n)_{t\ge 0}\subset\mathfrak{m}athcal{P}_2(X)$. Indeed, by~\eqref{eq:heat_estimate_above}, we have
\begin{equation*}
C_t:=\int_\mathfrak{m}athbb{G}\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}}(x,0)^2\,\mathfrak{m}athsf{h}_t(x)\ dx<+\infty
\qquad
\forall t>0.
\end{equation*}
Thus, by triangular inequality, we have
\begin{equation*}
(\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}}(\cdot,0)^2\star\mathfrak{m}athsf{h}_t)(x)
=\int_\mathfrak{m}athbb{G}\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}}(xy^{-1},0)^2\,\mathfrak{m}athsf{h}_t(y)\ dy
\le2\,\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}}(x,0)^2+2C_t,
\end{equation*}
so that, for all $t>0$, we get
\begin{equation}\label{eq:heat_second_moment}
\int_\mathfrak{m}athbb{G}\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}}(x,0)^2\,\varrho_t(x)\ dx
=\int_\mathfrak{m}athbb{G}(\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}}(\cdot,0)^2\star\mathfrak{m}athsf{h}_t)(x)\,\varrho(x)\ dx
\le2\int_\mathfrak{m}athbb{G}\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}}(x,0)^2\,\varrho(x)\ dx+2C_t.
\end{equation}
\subsection{Main result}
We are now ready to state the main result of the paper. The proof is given in \cref{sec:proof_of_main_result} and deals with the two parts of the statement separately.
\begin{theorem}\label{th:main}
Let $(\mathfrak{m}athbb{G},\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}},\mathfrak{m}athcal{L}^n)$ be a Carnot group and let $\varrho_0\in L^1(\mathfrak{m}athbb{G})$ be such that $\mathfrak{m}u_0=\varrho_0\mathfrak{m}athcal{L}^n\in\dom(\mathfrak{m}athsf{Ent})$. If $(\varrho_t)_{t\ge0}$ solves the sub-elliptic heat equation $\partial_t\varrho_t=\mathfrak{m}athrm{D}elta_\mathfrak{m}athbb{G}\varrho_t$ with initial datum $\varrho_0$, then $\mathfrak{m}u_t=\varrho_t\mathfrak{m}athcal{L}^n$ is a gradient flow of $\mathfrak{m}athsf{Ent}$ in $(\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G}),\mathfrak{m}athsf{W}_\mathfrak{m}athbb{G})$ starting from~$\mathfrak{m}u_0$.
Conversely, if $(\mathfrak{m}u_t)_{t\ge0}$ is a gradient flow of $\mathfrak{m}athsf{Ent}$ in $(\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G}),\mathfrak{m}athsf{W}_\mathfrak{m}athbb{G})$ starting from~$\mathfrak{m}u_0$, then $\mathfrak{m}u_t=\varrho_t\mathfrak{m}athcal{L}^n$ for all $t\ge0$ and $(\varrho_t)_{t\ge0}$ solves the sub-elliptic heat equation $\partial_t\varrho_t=\mathfrak{m}athrm{D}elta_\mathfrak{m}athbb{G}\varrho_t$ with initial datum $\varrho_0$.
\end{theorem}
\section{Continuity equation and slope of the entropy}
\label{sec:CE_and_ent_slope}
\subsection{The Wasserstein space on the approximating Riemannian manifold}
Let $(\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G}_\varepsilon),\mathfrak{m}athsf{W}_\varepsilon)$ be the Wasserstein space introduced in \cref{subsec:wasserstein_space} relative to the metric measure space $(\mathfrak{m}athbb{G}_\varepsilon,\partialps,\mathfrak{m}athcal{L}^n)$. As observed in \cref{subsec:riemannian_approx}, $(\mathfrak{m}athbb{G}_\varepsilon,\partialps,\mathfrak{m}athcal{L}^n)$ is an $n$-dimensional Riemannian manifold (with rescaled volume measure) whose Ricci curvature is bounded from below. Here we collect some known results taken from~\cites{E10,V09} concerning the space $(\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G}_\varepsilon),\mathfrak{m}athsf{W}_\varepsilon)$. In the original statements, the canonical reference measure is the Riemannian volume. Keeping in mind that $\textrm{vol}_\varepsilon=\varepsilon^{n-Q}\mathfrak{m}athcal{L}^n$ and the relation~\eqref{eq:entropy_riem_vs_leb}, in our statements each quantity is rescaled accordingly. All time-dependent vector fields appearing in the sequel are tacitly understood to be Borel measurable.
Let $\mathfrak{m}u\in\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G}_\varepsilon)$ be given. We define the space
\begin{equation*}
L^2_\varepsilon(\mathfrak{m}u)=\set*{\xi\in\mathfrak{m}athcal{S}(T\mathfrak{m}athbb{G}_\varepsilon) : \int_{\mathfrak{m}athbb{G}_\varepsilon}\|\xi\|_\varepsilon^2\ d\mathfrak{m}u<+\infty}.
\end{equation*}
Here $\mathfrak{m}athcal{S}(T\mathfrak{m}athbb{G}_\varepsilon)$ denotes the set of sections of the tangent bundle $T\mathfrak{m}athbb{G}_\varepsilon$. Moreover, we define the \emph{`tangent space'} of $(\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G}_\varepsilon),\mathfrak{m}athsf{W}_\varepsilon)$ at~$\mathfrak{m}u$ as
\begin{equation*}
\tang_\varepsilon(\mathfrak{m}u)=\overline{\set*{\nabla_\varepsilon\varphi : \varphi\in C^\infty_c(\mathfrak{m}athbb{R}^n)}}^{L^2_\varepsilon(\mathfrak{m}u)}.
\end{equation*}
The `tangent space' $\tang_\varepsilon(\mathfrak{m}u)$ was first introduced in~\cite{O01}. We refer the reader to~\cite{AGS08}*{Chapter~12} and to~\cite{V09}*{Chapters~13 and~15} for a detailed discussion on this space.
Let $\varepsilon>0$ be fixed. Given $I\subset\mathfrak{m}athbb{R}$ an open interval and a time-dependent vector field $v^\varepsilon\colon I\times\mathfrak{m}athbb{G}_\varepsilon\to T\mathfrak{m}athbb{G}_\varepsilon$, $(t,x)\mathfrak{m}apsto v^\varepsilon_t(x)\in T_x\mathfrak{m}athbb{G}_\varepsilon$, we say that a curve $(\mathfrak{m}u_t)_{t\in I}\subset\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G}_\varepsilon)$ satisfies the \emph{continuity equation}
\begin{equation}\label{eq:CE_eps}
\partial_t\mathfrak{m}u_t+\diverg(v^\varepsilon_t\mathfrak{m}u_t)=0 \qquad\text{in}\ I\times\mathfrak{m}athbb{G}_\varepsilon
\end{equation}
\emph{in the sense of distributions} if
\begin{equation*}
\int_I\int_{\mathfrak{m}athbb{G}_\varepsilon}\|v^\varepsilon_t(x)\|_\varepsilon\,d\mathfrak{m}u_t(x)\,dt<+\infty
\end{equation*}
and
\begin{equation*}
\int_I\int_{\mathfrak{m}athbb{G}_\varepsilon}\partial_t\varphi(t,x)+\scalar*{v^\varepsilon_t(x),\nabla_\varepsilon\varphi(t,x)}_\varepsilon\,d\mathfrak{m}u_t(x)\,dt=0
\qquad
\forall \varphi\in C^\infty_c(I\times\mathfrak{m}athbb{R}^n).
\end{equation*}
We can thus state the following result, see~\cite{E10}*{Proposition~2.5} for the proof. Here and in the sequel, the metric derivative in the Wasserstein space $(\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G}_\varepsilon),\mathfrak{m}athsf{W}_\varepsilon)$ of a curve $(\mathfrak{m}u_t)_{t\in I}\subset\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G}_\varepsilon)$ is denoted by~$|\dot{\mathfrak{m}u}_t|_\varepsilon$.
\begin{proposition}[Continuity equation in $(\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G}_\varepsilon),\mathfrak{m}athsf{W}_\varepsilon)$]\label{prop:CE_eps}
Let $\varepsilon>0$ be fixed and let $I\subset\mathfrak{m}athbb{R}$ be an open interval. If $(\mathfrak{m}u_t)_t\in AC^2_{\rm loc}(I;\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G}_\varepsilon))$, then there exists a time-dependent vector field $v^\varepsilon\colon I\times\mathfrak{m}athbb{G}_\varepsilon\to T\mathfrak{m}athbb{G}_\varepsilon$ with $t\mathfrak{m}apsto\|v^\varepsilon_t\|_{L^2_\varepsilon(\mathfrak{m}u_t)}\in L^2_{\rm loc}(I)$ such that
\begin{equation}\label{eq:v_eps}
v^\varepsilon_t\in\tang_\varepsilon(\mathfrak{m}u_t)
\qquad
\text{for a.e.}\ t\in I
\end{equation}
and the continuity equation~\eqref{eq:CE_eps} holds in the sense of distributions. The vector field~$v^\varepsilon_t$ is uniquely determined in $L^2_\varepsilon(\mathfrak{m}u_t)$ by~\eqref{eq:CE_eps} and~\eqref{eq:v_eps} for a.e.\ $t\in I$ and we have
\begin{equation*}
\|v^\varepsilon_t\|_{L^2_\varepsilon(\mathfrak{m}u_t)}=|\dot{\mathfrak{m}u}_t|_\varepsilon
\qquad
\text{for a.e.}\ t\in I.
\end{equation*}
Conversely, if $(\mathfrak{m}u_t)_{t\in I}\subset\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G}_\varepsilon)$ is a curve satisfying~\eqref{eq:CE_eps} for some $(v^\varepsilon_t)_{t\in I}$ such that $t\mathfrak{m}apsto\|v^\varepsilon_t\|_{L^2_\varepsilon(\mathfrak{m}u_t)}\in L^2_{\rm loc}(I)$, then $(\mathfrak{m}u_t)_t\in AC^2_{\rm loc}(I;(\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G}_\varepsilon),\mathfrak{m}athsf{W}_\varepsilon))$ with
\begin{equation*}
|\dot{\mathfrak{m}u}_t|_\varepsilon\le\|v^\varepsilon_t\|_{L^2_\varepsilon(\mathfrak{m}u_t)}
\qquad
\text{for a.e.}\ t\in I.
\end{equation*}
\end{proposition}
\noindent
We can interpret the time-dependent vector field $(v^\varepsilon_t)_{t\in I}$ given by \cref{prop:CE_eps} as the `tangent vector' of the curve $(\mathfrak{m}u_t)_{t\in I}$ in $(\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G}_\varepsilon),\mathfrak{m}athsf{W}_\varepsilon)$. As remarked in~\cite{E10}*{Section~2}, for a.e.\ $t\in I$ the vector field $v^\varepsilon_t$ has minimal $L^2_\varepsilon(\mathfrak{m}u_t)$-norm among all time-dependent vector fields satisfying~\eqref{eq:CE_eps}. Moreover, this minimality is equivalent to~\eqref{eq:v_eps}.
In the following result and in the sequel, $|\mathfrak{m}athrm{D}_\varepsilon^-\mathfrak{m}athsf{Ent}|(\mathfrak{m}u)$ denotes the descending slope of the entropy~$\mathfrak{m}athsf{Ent}$ at the point $\mathfrak{m}u\in\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G}_\varepsilon)$ in the Wasserstein space $(\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G}_\varepsilon),\mathfrak{m}athsf{W}_\varepsilon)$.
\begin{proposition}\label{prop:slope_ent_eps}
Let $\varepsilon>0$ be fixed and let $\mathfrak{m}u=\varrho\mathfrak{m}athcal{L}^n\in\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G}_\varepsilon)$. The following statements are equivalent:
\begin{enumerate}[(i)]
\item $|\mathfrak{m}athrm{D}_\varepsilon^-\mathfrak{m}athsf{Ent}|(\mathfrak{m}u)<+\infty$;
\item $\varrho\in W^{1,1}_{\rm loc}(\mathfrak{m}athbb{G}_\varepsilon)$ and $\nabla_\varepsilon\varrho=w^\varepsilon\varrho$ for some $w^\varepsilon\in L^2_\varepsilon(\mathfrak{m}u)$.
\end{enumerate}
In this case, $w^\varepsilon\in\tang_\varepsilon(\mathfrak{m}u)$ and $|\mathfrak{m}athrm{D}_\varepsilon^-\mathfrak{m}athsf{Ent}|(\mathfrak{m}u)=\|w^\varepsilon\|_{L^2_\varepsilon(\mathfrak{m}u)}$. Moreover, for any $\nu\in\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G}_\varepsilon)$, we have
\begin{equation}\label{eq:HWI_eps}
\mathfrak{m}athsf{Ent}(\nu)\ge\mathfrak{m}athsf{Ent}(\mathfrak{m}u)-\|w^\varepsilon\|_{L^2_\varepsilon(\mathfrak{m}u)}\,\mathfrak{m}athsf{W}_\varepsilon(\nu,\mathfrak{m}u)-\tfrac{K}{2\varepsilon^2}\,\mathfrak{m}athsf{W}_\varepsilon(\nu,\mathfrak{m}u)^2,
\end{equation}
where $K>0$ is the constant appearing in~\eqref{eq:ricci_bounded_below}.
\end{proposition}
\noindent
The equivalence part in \cref{prop:slope_ent_eps} is proved in~\cite{E10}*{Proposition~4.3}. Inequality~\eqref{eq:HWI_eps} is the so-called \emph{HWI inequality} and follows from~\cite{V09}*{Theorem~23.14}, see~\cite{V09}*{Remark~23.16}.
The quantity
\begin{equation*}
\mathfrak{m}athsf{F}_\varepsilon(\varrho)=\|w^\varepsilon\|_{L^2_\varepsilon(\mathfrak{m}u)}^2=\int_{\mathfrak{m}athbb{G}_\varepsilon\cap\set*{\varrho>0}}\frac{\|\nabla_\varepsilon\varrho\|_\varepsilon^2}{\varrho}\ d\mathfrak{m}athcal{L}^n
\end{equation*}
appearing in \cref{prop:slope_ent_eps} is the so-called \emph{Fisher information} of $\mathfrak{m}u=\varrho\mathfrak{m}athcal{L}^n\in\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G}_\varepsilon)$. The inequality $F_\varepsilon(\varrho)\le|\mathfrak{m}athrm{D}_\varepsilon^-\mathfrak{m}athsf{Ent}|(\mathfrak{m}u)$ holds in the context of metric measure spaces, see~\cite{AGS14}*{Theorem~7.4}. The converse inequality does not hold in such a generality and heavily depends on the lower semicontinuity of the descending slope $|\mathfrak{m}athrm{D}_\varepsilon^-\mathfrak{m}athsf{Ent}|$, see~\cite{AGS14}*{Theorem~7.6}.
\subsection{The Wasserstein space on the Carnot group}
Let $(\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G}),\mathfrak{m}athsf{W}_\mathfrak{m}athbb{G})$ be the Wasserstein space introduced in \cref{subsec:wasserstein_space} relative to the metric measure space $(\mathfrak{m}athbb{G},\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}},\mathfrak{m}athcal{L}^n)$. In this section, we discuss the counterparts of \cref{prop:CE_eps} and \cref{prop:slope_ent_eps} in the space $(\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G}),\mathfrak{m}athsf{W}_\mathfrak{m}athbb{G})$. All time-dependent vector fields appearing in the sequel are tacitly understood to be Borel measurable.
Let $\mathfrak{m}u\in\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G})$ be given. We define the space
\begin{equation*}
L^2_\mathfrak{m}athbb{G}(\mathfrak{m}u)=\set*{\xi\in\mathfrak{m}athcal{S}(H\mathfrak{m}athbb{G}) : \int_{\mathfrak{m}athbb{G}}\|\xi\|_\mathfrak{m}athbb{G}^2\ d\mathfrak{m}u<+\infty}.
\end{equation*}
Here $\mathfrak{m}athcal{S}(H\mathfrak{m}athbb{G})$ denotes the set of sections of the horizontal tangent bundle $H\mathfrak{m}athbb{G}$. Moreover, we define the \emph{`tangent space'} of $(\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G}),\mathfrak{m}athsf{W}_\mathfrak{m}athbb{G})$ at~$\mathfrak{m}u$ as
\begin{equation*}
\tang_\mathfrak{m}athbb{G}(\mathfrak{m}u)=\overline{\set*{\nabla_\mathfrak{m}athbb{G}\varphi : \varphi\in C^\infty_c(\mathfrak{m}athbb{R}^n)}}^{L^2_\mathfrak{m}athbb{G}(\mathfrak{m}u)}.
\end{equation*}
Given $I\subset\mathfrak{m}athbb{R}$ an open interval and a horizontal time-dependent vector field $v^\mathfrak{m}athbb{G}\colon I\times\mathfrak{m}athbb{G}\to H\mathfrak{m}athbb{G}$, $(t,x)\mathfrak{m}apsto v^\mathfrak{m}athbb{G}_t(x)\in H_x\mathfrak{m}athbb{G}$, we say that a curve $(\mathfrak{m}u_t)_{t\in I}\subset\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G})$ satisfies the continuity equation
\begin{equation}\label{eq:CE}
\partial_t\mathfrak{m}u_t+\diverg(v^\mathfrak{m}athbb{G}_t\mathfrak{m}u_t)=0 \qquad\text{in}\ I\times\mathfrak{m}athbb{G}_\varepsilon
\end{equation}
\emph{in the sense of distributions} if
\begin{equation*}
\int_I\int_\mathfrak{m}athbb{G}\|v^\mathfrak{m}athbb{G}_t(x)\|_\mathfrak{m}athbb{G}\,d\mathfrak{m}u_t(x)\,dt<+\infty
\end{equation*}
and
\begin{equation*}
\int_I\int_\mathfrak{m}athbb{G}\partial_t\varphi(t,x)+\scalar*{v^\mathfrak{m}athbb{G}_t(x),\nabla_\mathfrak{m}athbb{G}\varphi(t,x)}_\mathfrak{m}athbb{G}\,d\mathfrak{m}u_t(x)\,dt=0
\qquad
\forall \varphi\in C^\infty_c(I\times\mathfrak{m}athbb{R}^n).
\end{equation*}
The following result is the exact analogue of \cref{prop:CE_eps}. Here and in the sequel, the metric derivative in the Wasserstein space $(\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G}),\mathfrak{m}athsf{W}_\mathfrak{m}athbb{G})$ of a curve $(\mathfrak{m}u_t)_{t\in I}\subset\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G})$ is denoted by~$|\dot{\mathfrak{m}u}_t|_\mathfrak{m}athbb{G}$.
\begin{proposition}[Continuity equation in $(\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G}),\mathfrak{m}athsf{W}_\mathfrak{m}athbb{G})$]\label{prop:CE}
Let $I\subset\mathfrak{m}athbb{R}$ be an open interval. If $(\mathfrak{m}u_t)_t\in AC^2_{\rm loc}(I;(\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G}),\mathfrak{m}athsf{W}_\mathfrak{m}athbb{G}))$, then there exists a horizontal time-dependent vector field $v^\mathfrak{m}athbb{G}\colon I\times\mathfrak{m}athbb{G}\to H\mathfrak{m}athbb{G}$ with $t\mathfrak{m}apsto\|v^\mathfrak{m}athbb{G}_t\|_{L^2_\mathfrak{m}athbb{G}(\mathfrak{m}u_t)}\in L^2_{\rm loc}(I)$ such that
\begin{equation}\label{eq:v_G}
v^\mathfrak{m}athbb{G}_t\in\tang_\mathfrak{m}athbb{G}(\mathfrak{m}u_t)
\qquad
\text{for a.e.}\ t\in I
\end{equation}
and the continuity equation~\eqref{eq:CE} holds in the sense of distributions. The vector field~$v^\mathfrak{m}athbb{G}_t$ is uniquely determined in $L^2_\mathfrak{m}athbb{G}(\mathfrak{m}u_t)$ by~\eqref{eq:CE} and~\eqref{eq:v_G} for a.e.\ $t\in I$ and we have
\begin{equation*}
\|v^\mathfrak{m}athbb{G}_t\|_{L^2_\mathfrak{m}athbb{G}(\mathfrak{m}u_t)}=|\dot{\mathfrak{m}u}_t|_\mathfrak{m}athbb{G}
\qquad
\text{for a.e.}\ t\in I.
\end{equation*}
Conversely, if $(\mathfrak{m}u_t)_{t\in I}\subset\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G})$ is a curve satisfying~\eqref{eq:CE} for some $(v^\mathfrak{m}athbb{G}_t)_{t\in I}$ such that $t\mathfrak{m}apsto\|v^\mathfrak{m}athbb{G}_t\|_{L^2_\mathfrak{m}athbb{G}(\mathfrak{m}u_t)}\in L^2_{\rm loc}(I)$, then $(\mathfrak{m}u_t)_t\in AC^2_{\rm loc}(I;(\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G}),\mathfrak{m}athsf{W}_\mathfrak{m}athbb{G}))$ with
\begin{equation*}
|\dot{\mathfrak{m}u}_t|_\mathfrak{m}athbb{G}\le\|v^\mathfrak{m}athbb{G}_t\|_{L^2_\mathfrak{m}athbb{G}(\mathfrak{m}u_t)}
\qquad
\text{for a.e.}\ t\in I.
\end{equation*}
\end{proposition}
\noindent
As for \cref{prop:CE_eps}, we can interpret the horizontal time-dependent vector field $(v^\mathfrak{m}athbb{G}_t)_{t\in I}$ given by \cref{prop:CE} as the `tangent vector' of the curve $(\mathfrak{m}u_t)_{t\in I}$ in $(\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G}),\mathfrak{m}athsf{W}_\mathfrak{m}athbb{G})$. An easy adaptation of~\cite{E10}*{Lemma~2.4} to the sub-Riemannian manifold $(\mathfrak{m}athbb{G},\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}},\mathfrak{m}athcal{L}^n)$ again shows that for a.e.\ $t\in I$ the vector field $v^\mathfrak{m}athbb{G}_t$ has minimal $L^2_\mathfrak{m}athbb{G}(\mathfrak{m}u_t)$-norm among all time-dependent vector fields satisfying~\eqref{eq:CE} and, moreover, that this minimality is equivalent to~\eqref{eq:v_G}.
\cref{prop:CE} can be obtained applying the general results obtained in~\cite{GH15} to the metric measure space $(\mathfrak{m}athbb{G},\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}},\mathfrak{m}athcal{L}^n)$. Below we give a direct proof exploiting \cref{prop:CE_eps}. The argument is very similar to the one of~\cite{J14}*{Proposition~3.1} and we only sketch it.
\begin{proof}
If $(\mathfrak{m}u_t)_t\in AC^2_{\rm loc}(I;(\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G}),\mathfrak{m}athsf{W}_\mathfrak{m}athbb{G}))$, then also $(\mathfrak{m}u_t)_t\in AC^2_{\rm loc}(I;(\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G}_\varepsilon),\mathfrak{m}athsf{W}_\varepsilon))$ for every $\varepsilon>0$, since $\partialps\le\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}}$. Let $v^\varepsilon\colon I\times\mathfrak{m}athbb{G}_\varepsilon\to T\mathfrak{m}athbb{G}_\varepsilon$ be the time-dependent vector field given by \cref{prop:CE_eps}. Note that
\begin{equation}\label{eq:equilim_v_eps}
\int_\mathfrak{m}athbb{G}\|v^\varepsilon_t\|_\varepsilon^2\ d\mathfrak{m}u_t=|\dot{\mathfrak{m}u}_t|_\varepsilon^2\le|\dot{\mathfrak{m}u}_t|_\mathfrak{m}athbb{G}^2
\qquad
\text{for a.e.}\ t\in I.
\end{equation}
Moreover
\begin{equation}\label{eq:norm_v_eps}
\|v^\varepsilon_t\|_\varepsilon^2=\|v^{\varepsilon,V_1}_t\|_1^2+\sum_{i=2}^\kappa\varepsilon^{2(1-i)}\|v^{\varepsilon,V_i}_t\|_1^2
\qquad
\text{for all}\ \varepsilon>0,
\end{equation}
where $v^{\varepsilon,V_i}_t$ denotes the projection of $v^\varepsilon_t$ on~$V_i$. Combining~\eqref{eq:equilim_v_eps} and~\eqref{eq:norm_v_eps}, we find a sequence $(\varepsilon_k)_{k\in\mathfrak{m}athbb{N}}$, with $\varepsilon_k\to0$, and a horizontal time-dependent vector field $v^\mathfrak{m}athbb{G}\colon I\times\mathfrak{m}athbb{G}\to H\mathfrak{m}athbb{G}$ such that $v^{\varepsilon_k,V_1}\rightharpoonup v^\mathfrak{m}athbb{G}$ and $v^{\varepsilon_k,V_i}\to0$ for all $i=2,\dots,\kappa$ as $k\to+\infty$ locally in time in the $L^2$-norm on $I\times\mathfrak{m}athbb{G}$ naturally induced by the norm $\|\cdot\|_1$ and the measure $d\mathfrak{m}u_tdt$. In particular, $t\mathfrak{m}apsto\|v^\mathfrak{m}athbb{G}_t\|_{L^2_\mathfrak{m}athbb{G}(\mathfrak{m}u_t)}\in L^2_{\rm loc}(I)$ and $\|v^\mathfrak{m}athbb{G}_t\|_{L^2_\mathfrak{m}athbb{G}(\mathfrak{m}u_t)}\le|\dot{\mathfrak{m}u}_t|_\mathfrak{m}athbb{G}$ for a.e.\ $t\in I$. To prove~\eqref{eq:CE}, fix a test function $\varphi\in C^\infty_c(I\times\mathfrak{m}athbb{R}^n)$ and pass to the limit as $\varepsilon\to0^+$ in~\eqref{eq:CE_eps}.
Conversely, if $(\mathfrak{m}u_t)_{t\in I}\subset\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G})$ satisfies~\eqref{eq:CE} for some horizontal time-dependent vector field $(v^\mathfrak{m}athbb{G}_t)_{t\in I}$ such that $t\mathfrak{m}apsto\|v^\mathfrak{m}athbb{G}_t\|_{L^2_\varepsilon(\mathfrak{m}u_t)}\in L^2_{\rm loc}(I)$, then we can apply \cref{prop:CE_eps} for $\varepsilon=1$. By the \emph{superposition principle} stated in~\cite{B08}*{Theorem~5.8} applied to the Riemannian manifold $(\mathfrak{m}athbb{G}_1,\mathfrak{m}athsf{d}_1,\mathfrak{m}athcal{L}^n)$, we find a probability measure $\nu\in\mathfrak{m}athcal{P}(C(I;(\mathfrak{m}athbb{G}_1,\mathfrak{m}athsf{d}_1)))$, concentrated on $AC^2_{\rm loc}(I;(\mathfrak{m}athbb{G}_1,\mathfrak{m}athsf{d}_1))$, such that $\mathfrak{m}u_t=(\mathfrak{m}athsf{e}_t)_\#\nu$ for all $t\in I$ and with the property that $\nu$-a.e.\ curve $\gamma\in C(I;(\mathfrak{m}athbb{G}_1,\mathfrak{m}athsf{d}_1))$ is an absolutely continuous integral curve of the vector field~$v^\mathfrak{m}athbb{G}$. Here $\mathfrak{m}athsf{e}_t\colon C(I;(\mathfrak{m}athbb{G}_1,\mathfrak{m}athsf{d}_1))\to\mathfrak{m}athbb{G}$ denotes the evaluation map at time $t\in I$. Since $v^\mathfrak{m}athbb{G}$ is horizontal, $\nu$-a.e.\ curve $\gamma\in C(I;(\mathfrak{m}athbb{G}_1,\mathfrak{m}athsf{d}_1))$ is horizontal. Therefore, for all $s,t\in I$, $s<t$, we have
\begin{equation*}
\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}}(\gamma(t),\gamma(s))\le\int_s^t\|\dot{\gamma}(r)\|_\mathfrak{m}athbb{G}\ dr
=\int_s^t\|v^\mathfrak{m}athbb{G}_r(\gamma(r))\|_\mathfrak{m}athbb{G}\ dr
\end{equation*}
and we can thus estimate
\begin{align*}
\mathfrak{m}athsf{W}_\mathfrak{m}athbb{G}^2(\mathfrak{m}u_t,\mathfrak{m}u_s)&\le\int_{\mathfrak{m}athbb{G}\times\mathfrak{m}athbb{G}}\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}}^2(x,y)\ d
(\mathfrak{m}athsf{e}_t,\mathfrak{m}athsf{e}_s)_\#\nu(x,y)
=\int_{AC^2_{\rm loc}}\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}}^2(\gamma(t),\gamma(s))\ d\nu(\gamma)\\
&\le(t-s)\int_{AC^2_{\rm loc}}\int_s^t\|v^\mathfrak{m}athbb{G}_r(\gamma(r))\|_\mathfrak{m}athbb{G}^2\ dr d\nu(\gamma)
=(t-s)\int_s^t\int_\mathfrak{m}athbb{G}\|v^\mathfrak{m}athbb{G}_r\|_\mathfrak{m}athbb{G}^2\ d\mathfrak{m}u_r dr.
\end{align*}
This immediately gives $|\dot{\mathfrak{m}u}_t|_\mathfrak{m}athbb{G}\le\|v^\mathfrak{m}athbb{G}_t\|_{L^2_\mathfrak{m}athbb{G}(\mathfrak{m}u_t)}$ for a.e.\ $t\in I$, which in turn proves~\eqref{eq:v_G}.
\end{proof}
To establish an analogue of \cref{prop:slope_ent_eps}, we need to prove the two inequalities separately. For $\mathfrak{m}u=\varrho\mathfrak{m}athcal{L}^n\in\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G})$, the inequality $\mathfrak{m}athsf{F}_\mathfrak{m}athbb{G}(\varrho)\le|\mathfrak{m}athrm{D}^-_\mathfrak{m}athbb{G}\mathfrak{m}athsf{Ent}|(\mathfrak{m}u)$ is stated in \cref{prop:slope_ent_G_general} below. Here and in the sequel, $|\mathfrak{m}athrm{D}_\mathfrak{m}athbb{G}^-\mathfrak{m}athsf{Ent}|(\mathfrak{m}u)$ denotes the descending slope of the entropy~$\mathfrak{m}athsf{Ent}$ at the point $\mathfrak{m}u\in\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G})$ in the Wasserstein space $(\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G}),\mathfrak{m}athsf{W}_\mathfrak{m}athbb{G})$.
\begin{proposition}\label{prop:slope_ent_G_general}
Let $\mathfrak{m}u=\varrho\mathfrak{m}athcal{L}^n\in\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G})$. If $|\mathfrak{m}athrm{D}^-_\mathfrak{m}athbb{G}\mathfrak{m}athsf{Ent}|(\mathfrak{m}u)<+\infty$, then $\varrho\in W^{1,1}_{\mathfrak{m}athbb{G},\,\rm loc}(\mathfrak{m}athbb{G})$ and $\nabla_\mathfrak{m}athbb{G}\varrho=w^\mathfrak{m}athbb{G}\varrho$ for some horizontal vector field $w^\mathfrak{m}athbb{G}\in L^2(\mathfrak{m}u)$ with $\|w^\mathfrak{m}athbb{G}\|_{L^2_\mathfrak{m}athbb{G}(\mathfrak{m}u)}\le|\mathfrak{m}athrm{D}^-_\mathfrak{m}athbb{G}\mathfrak{m}athsf{Ent}|(\mathfrak{m}u)$.
\end{proposition}
\noindent
\cref{prop:slope_ent_G_general} can be obtained by applying~\cite{AGS14}*{Theorem~7.4} to the metric measure space $(\mathfrak{m}athbb{G},\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}},\mathfrak{m}athcal{L}^n)$. Below we give a direct proof of this result which is closer in the spirit to the one in the Riemannian setting, see~\cite{E10}*{Lemma~4.2}. See also~\cite{J14}*{Proposition~3.1}.
\begin{proof}
Let $V\in C^\infty_c(\mathfrak{m}athbb{G};H\mathfrak{m}athbb{G})$ be a smooth horizontal vector field with compact support. Then there exists $\partiallta>0$ such that, for any $t\in(-\partiallta,\partiallta)$, the flow map of the vector field~$V$ at time~$t$, namely
\begin{equation*}
F_t(x):=\exp_x(t V), \qquad x\in\mathfrak{m}athbb{G},
\end{equation*}
is a diffeomorphism and $J_t=\partialt(D F_t)$ is such that $c^{-1}\le J_t\le c$ for some $c\ge1$. By the change of variable formula, the measure $\mathfrak{m}u_t:=(F_t)_\#\mathfrak{m}u$ is such that $\mathfrak{m}u_t=\varrho_t\mathfrak{m}athcal{L}^n$ with $J_t\varrho_t=\varrho\circ F_t^{-1}$ for $t\in(-\partiallta,\partiallta)$. Let us set $H(r)=r\log r$ for $r\ge0$. Then, for $t\in(-\partiallta,\partiallta)$,
\begin{equation*}
\mathfrak{m}athsf{Ent}(\mathfrak{m}u_t)=\int_\mathfrak{m}athbb{G} H(\varrho_t)\ dx=\int_\mathfrak{m}athbb{G} H\left(\frac{\varrho}{J_t}\right) J_t\ dx=\mathfrak{m}athsf{Ent}(\mathfrak{m}u)-\int_\mathfrak{m}athbb{G} \varrho\log(J_t)\ dx<+\infty.
\end{equation*}
Note that $J_0=1$, $\dot{J}_0=\diverg V$ and that $t\mathfrak{m}apsto\dot{J}_t J_t^{-1}$ is uniformly bounded for $t\in(-\partiallta,\partiallta)$. Thus we have
\begin{equation*}
\frac{d}{dt}\mathfrak{m}athsf{Ent}(\mathfrak{m}u_t)\bigg|_{t=0}
=-\frac{d}{dt}\int_\mathfrak{m}athbb{G} \varrho\log(J_t)\ dx\bigg|_{t=0}\ dx
=\int_\mathfrak{m}athbb{G} -\varrho\,\frac{\dot{J}_t}{J_t}\bigg|_{t=0}\ dx
=-\int_\mathfrak{m}athbb{G}\varrho\diverg V\ dx.
\end{equation*}
On the other hand, we have
\begin{equation*}
\mathfrak{m}athsf{W}_\mathfrak{m}athbb{G}^2(\mathfrak{m}u_t,\mathfrak{m}u)=\mathfrak{m}athsf{W}_\mathfrak{m}athbb{G}^2((F_t)_\#\mathfrak{m}u,\mathfrak{m}u)\le\int_\mathfrak{m}athbb{G}\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}}^2(F_t(x),x)\ d\mathfrak{m}u(x)
\end{equation*}
and so
\begin{equation*}
\limsup_{t\to0}\frac{\mathfrak{m}athsf{W}_\mathfrak{m}athbb{G}^2(\mathfrak{m}u_t,\mathfrak{m}u)}{|t|^2}
\le\int_\mathfrak{m}athbb{G}\limsup_{t\to0}\frac{\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}}^2(F_t(x),x)}{|t|^2}\ d\mathfrak{m}u(x)
=\int_\mathfrak{m}athbb{G}\|V\|_\mathfrak{m}athbb{G}^2\ d\mathfrak{m}u.
\end{equation*}
Hence
\begin{align*}
-\frac{d}{dt}\mathfrak{m}athsf{Ent}(\mathfrak{m}u_t)\bigg|_{t=0}\le\limsup_{t\to0}\frac{[\mathfrak{m}athsf{Ent}(\mathfrak{m}u_t)-\mathfrak{m}athsf{Ent}(\mathfrak{m}u)]^-}{\mathfrak{m}athsf{W}_\mathfrak{m}athbb{G}(\mathfrak{m}u_t,\mathfrak{m}u)}\cdot\frac{\mathfrak{m}athsf{W}_\mathfrak{m}athbb{G}(\mathfrak{m}u_t,\mathfrak{m}u)}{|t|}\le|\mathfrak{m}athrm{D}^-_\mathfrak{m}athbb{G}\mathfrak{m}athsf{Ent}|(\mathfrak{m}u)\left(\int_\mathfrak{m}athbb{G}\|V\|_\mathfrak{m}athbb{G}^2\ d\mathfrak{m}u\right)^\frac{1}{2}
\end{align*}
and thus
\begin{equation*}
\left|\int_\mathfrak{m}athbb{G}\varrho\diverg V\ dx\right|\le|\mathfrak{m}athrm{D}^-_\mathfrak{m}athbb{G}\mathfrak{m}athsf{Ent}|(\mathfrak{m}u)\left(\int_\mathfrak{m}athbb{G}\|V\|_\mathfrak{m}athbb{G}^2\ d\mathfrak{m}u\right)^\frac{1}{2}.
\end{equation*}
By Riesz representation theorem, we conclude that there exists a horizontal vector field $w^\mathfrak{m}athbb{G}\in L^2_\mathfrak{m}athbb{G}(\mathfrak{m}u)$ such that $\|w_\mathfrak{m}athbb{G}\|_{L^2_\mathfrak{m}athbb{G}(\mathfrak{m}u)}\le|\mathfrak{m}athrm{D}^-_\mathfrak{m}athbb{G}\mathfrak{m}athsf{Ent}|(\mathfrak{m}u)$ and
\begin{equation*}
-\int_\mathfrak{m}athbb{G}\varrho\diverg V\ dx=\int_\mathfrak{m}athbb{G}\scalar*{w^\mathfrak{m}athbb{G},V}_\mathfrak{m}athbb{G}\ d\mathfrak{m}u \qquad \text{for all}\ V\in C^\infty_c(\mathfrak{m}athbb{G};H\mathfrak{m}athbb{G}).
\end{equation*}
This implies that $\nabla_\mathfrak{m}athbb{G}\varrho=w^\mathfrak{m}athbb{G}\varrho$ and the proof is complete.
\end{proof}
We call the quantity
\begin{equation*}
\mathfrak{m}athsf{F}_\mathfrak{m}athbb{G}(\varrho)=\|w^\mathfrak{m}athbb{G}\|_{L^2_\mathfrak{m}athbb{G}(\mathfrak{m}u)}^2=\int_{\mathfrak{m}athbb{G}\cap\set*{\varrho>0}}\frac{\|\nabla_\mathfrak{m}athbb{G}\varrho\|_\mathfrak{m}athbb{G}^2}{\varrho}\ d\mathfrak{m}athcal{L}^n
\end{equation*}
appearing in \cref{prop:slope_ent_G_general} the \emph{horizontal Fisher information} of $\mathfrak{m}u=\varrho\mathfrak{m}athcal{L}^n\in\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G})$. On its effective domain, $\mathfrak{m}athsf{F}_\mathfrak{m}athbb{G}$ is convex and sequentially lower semicontinuous with respect to the weak topology of $L^1(\mathfrak{m}athbb{G})$, see~\cite{AGS14}*{Lemma~4.10}.
Given $\mathfrak{m}u=\varrho\mathfrak{m}athcal{L}^n\in\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G})$, it is not clear how to prove the inequality $|\mathfrak{m}athrm{D}^-_\mathfrak{m}athbb{G}\mathfrak{m}athsf{Ent}|^2(\mathfrak{m}u)\le\mathfrak{m}athsf{F}_\mathfrak{m}athbb{G}(\varrho)$ under the mere condition $|\mathfrak{m}athrm{D}^-_\mathfrak{m}athbb{G}\mathfrak{m}athsf{Ent}|(\mathfrak{m}u)<+\infty$. Following~\cite{J14}*{Proposition~3.4}, in \cref{prop:slope_ent_G_special} below we show that the condition $|\mathfrak{m}athrm{D}^-_\varepsilon\mathfrak{m}athsf{Ent}|(\mathfrak{m}u)<+\infty$ for some $\varepsilon>0$ (and thus any) implies that $|\mathfrak{m}athrm{D}^-_\mathfrak{m}athbb{G}\mathfrak{m}athsf{Ent}|^2(\mathfrak{m}u)\le\mathfrak{m}athsf{F}_\mathfrak{m}athbb{G}(\varrho)$.
\begin{proposition}\label{prop:slope_ent_G_special}
Let $\mathfrak{m}u=\varrho\mathfrak{m}athcal{L}^n\in\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G})$. If $|\mathfrak{m}athrm{D}_\varepsilon^-\mathfrak{m}athsf{Ent}|(\mathfrak{m}u)<+\infty$ for some $\varepsilon>0$, then also $|\mathfrak{m}athrm{D}^-_\mathfrak{m}athbb{G}\mathfrak{m}athsf{Ent}|(\mathfrak{m}u)<+\infty$ and moreover $\mathfrak{m}athsf{F}_\mathfrak{m}athbb{G}(\varrho)=|\mathfrak{m}athrm{D}^-_\mathfrak{m}athbb{G}\mathfrak{m}athsf{Ent}|^2(\mathfrak{m}u)$.
\end{proposition}
\begin{proof}
Since $|\mathfrak{m}athrm{D}_\varepsilon^-\mathfrak{m}athsf{Ent}|(\mathfrak{m}u)<+\infty$, we have $\mathfrak{m}athsf{Ent}(\mathfrak{m}u)<+\infty$. Since $\partialps\le\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}}$ and so $\mathfrak{m}athsf{W}_\varepsilon\le\mathfrak{m}athsf{W}_\mathfrak{m}athbb{G}$, we also have $|\mathfrak{m}athrm{D}^-_\mathfrak{m}athbb{G}\mathfrak{m}athsf{Ent}|(\mathfrak{m}u)\le|\mathfrak{m}athrm{D}_\varepsilon^-\mathfrak{m}athsf{Ent}|(\mathfrak{m}u)$. By \cref{prop:slope_ent_G_general}, we conclude that $\varrho\in W^{1,1}_{\mathfrak{m}athbb{G},\,\rm loc}(\mathfrak{m}athbb{G})$ and $\nabla_\mathfrak{m}athbb{G}\varrho=w^\mathfrak{m}athbb{G}\varrho$ for some horizontal vector field $w^\mathfrak{m}athbb{G}\in L^2_\mathfrak{m}athbb{G}(\mathfrak{m}u)$ with $\|w^\mathfrak{m}athbb{G}\|_{L^2_\mathfrak{m}athbb{G}(\mathfrak{m}u)}\le|\mathfrak{m}athrm{D}^-_\mathfrak{m}athbb{G}\mathfrak{m}athsf{Ent}|(\mathfrak{m}u)$. We now prove the converse inequality. Since $|\mathfrak{m}athrm{D}_\varepsilon^-\mathfrak{m}athsf{Ent}|(\mathfrak{m}u)<+\infty$, by \cref{prop:slope_ent_eps} we have $\mathfrak{m}athsf{F}_\varepsilon(\varrho)=|\mathfrak{m}athrm{D}_\varepsilon^-\mathfrak{m}athsf{Ent}|^2(\mathfrak{m}u)$ and
\begin{equation}\label{eq:hwi_ineq_eps}
\mathfrak{m}athsf{Ent}(\nu)\ge\mathfrak{m}athsf{Ent}(\mathfrak{m}u)-\mathfrak{m}athsf{F}^{1/2}_\varepsilon(\varrho)\,\mathfrak{m}athsf{W}_\varepsilon(\nu,\mathfrak{m}u)-\tfrac{K}{2\varepsilon^2}\,\mathfrak{m}athsf{W}_\varepsilon^2(\nu,\mathfrak{m}u)
\end{equation}
for any $\nu\in\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G})$. Take $\varepsilon=\mathfrak{m}athsf{W}_\mathfrak{m}athbb{G}(\nu,\mathfrak{m}u)^{1/4}$ and assume $\varepsilon<1$. Since $\mathfrak{m}athsf{W}_\varepsilon\le\mathfrak{m}athsf{W}_\mathfrak{m}athbb{G}$, from~\eqref{eq:hwi_ineq_eps} we get
\begin{align}\label{eq:calo_hwi_eps}
\mathfrak{m}athsf{Ent}(\nu)&\ge\mathfrak{m}athsf{Ent}(\mathfrak{m}u)-\mathfrak{m}athsf{F}^{1/2}_\varepsilon(\varrho)\,\mathfrak{m}athsf{W}_\mathfrak{m}athbb{G}(\nu,\mathfrak{m}u)-\tfrac{K}{2\varepsilon^2}\,\mathfrak{m}athsf{W}_\mathfrak{m}athbb{G}^2(\nu,\mathfrak{m}u)\nonumber\\
&=\mathfrak{m}athsf{Ent}(\mathfrak{m}u)-\mathfrak{m}athsf{F}^{1/2}_\varepsilon(\varrho)\,\mathfrak{m}athsf{W}_\mathfrak{m}athbb{G}(\nu,\mathfrak{m}u)-\tfrac{K}{2}\,\mathfrak{m}athsf{W}_\mathfrak{m}athbb{G}^{3/2}(\nu,\mathfrak{m}u).
\end{align}
We need to bound $\mathfrak{m}athsf{F}_\varepsilon(\varrho)$ from above in terms of $\mathfrak{m}athsf{F}_\mathfrak{m}athbb{G}(\varrho)$. To do so, observe that
\begin{equation*}
\nabla_\varepsilon\varrho=\nabla_\mathfrak{m}athbb{G}\varrho+\sum_{i=2}^k\varepsilon^{2(i-1)}\nabla_{V_i}\varrho,\qquad
\|\nabla_\varepsilon\varrho\|_\varepsilon^2=\|\nabla_\mathfrak{m}athbb{G}\varrho\|_\mathfrak{m}athbb{G}^2+\sum_{i=2}^k\varepsilon^{2(i-1)}\|\nabla_{V_i}\varrho\|_\mathfrak{m}athbb{G}^2.
\end{equation*}
In particular, $\tfrac{\nabla_{V_i}\varrho}{\varrho}\in L^2_\mathfrak{m}athbb{G}(\mathfrak{m}u)$ for all $i=2,\dots,k$. Recalling the inequality $(1+r)\le\left(1+\frac{r}{2}\right)^2$ for $r\ge0$, we can estimate
\begin{align*}
\mathfrak{m}athsf{F}_\varepsilon(\varrho)&=\mathfrak{m}athsf{F}_\mathfrak{m}athbb{G}(\varrho)+\sum_{i=2}^k\varepsilon^{2(i-1)}\left\|\tfrac{\nabla_{V_i}\varrho}{\varrho}\right\|_{L^2_\mathfrak{m}athbb{G}(\mathfrak{m}u)}^2
=\mathfrak{m}athsf{F}_\mathfrak{m}athbb{G}(\varrho)\,\left(1+\frac{1}{\mathfrak{m}athsf{F}_\mathfrak{m}athbb{G}(\varrho)}\sum_{i=2}^k\varepsilon^{2(i-1)}\left\|\tfrac{\nabla_{V_i}\varrho}{\varrho}\right\|_{L^2_\mathfrak{m}athbb{G}(\mathfrak{m}u)}^2\right)\\
&\le\mathfrak{m}athsf{F}_\mathfrak{m}athbb{G}(\varrho)\,\left(1+\frac{1}{2\mathfrak{m}athsf{F}_\mathfrak{m}athbb{G}(\varrho)}\sum_{i=2}^k\varepsilon^{2(i-1)}\left\|\tfrac{\nabla_{V_i}\varrho}{\varrho}\right\|_{L^2_\mathfrak{m}athbb{G}(\mathfrak{m}u)}^2\right)^2
\end{align*}
and thus
\begin{equation}\label{eq:estimate_fisher_fisher}
\mathfrak{m}athsf{F}_\varepsilon^{1/2}(\varrho)\le\mathfrak{m}athsf{F}_\mathfrak{m}athbb{G}^{1/2}(\varrho)\,\left(1+\frac{1}{2\mathfrak{m}athsf{F}_\mathfrak{m}athbb{G}(\varrho)}\sum_{i=2}^k\varepsilon^{2(i-1)}\left\|\tfrac{\nabla_{V_i}\varrho}{\varrho}\right\|_{L^2_\mathfrak{m}athbb{G}(\mathfrak{m}u)}^2\right).
\end{equation}
Inserting~\eqref{eq:estimate_fisher_fisher} into~\eqref{eq:calo_hwi_eps}, we finally get
\begin{equation*}
\mathfrak{m}athsf{Ent}(\nu)\ge\mathfrak{m}athsf{Ent}(\mathfrak{m}u)-\mathfrak{m}athsf{F}^{1/2}_\mathfrak{m}athbb{G}(\varrho)\,\mathfrak{m}athsf{W}_\mathfrak{m}athbb{G}(\nu,\mathfrak{m}u)-C\,\mathfrak{m}athsf{W}_\mathfrak{m}athbb{G}^{3/2}(\nu,\mathfrak{m}u)
\end{equation*}
for some $C>0$ independent of~$\varepsilon$. This immediately leads to $|\mathfrak{m}athrm{D}^-_\mathfrak{m}athbb{G}\mathfrak{m}athsf{Ent}|(\mathfrak{m}u)\le\mathfrak{m}athsf{F}^{1/2}_\mathfrak{m}athbb{G}(\varrho)$.
\end{proof}
\subsection{Carnot groups are non-\texorpdfstring{$CD(K,\infty)$}{CD(K,infty)} spaces}
As stated in~\cite{AGS14}*{Theorem~7.6}, if the metric measure space $(X,\mathfrak{m}athsf{d},\mathfrak{m}athfrak{m})$ is Polish and satisfies~\eqref{eq:assumption_on_measure}, then the properties
\begin{enumerate}[(i)]
\item $|\mathfrak{m}athrm{D}^-\mathfrak{m}athsf{Ent}|^2(\mathfrak{m}u)=\mathfrak{m}athsf{F}(\varrho)$ for all $\mathfrak{m}u=\varrho\mathfrak{m}athfrak{m}\in\dom(\mathfrak{m}athsf{Ent})$;
\item\label{item:lsc_slope} $|\mathfrak{m}athrm{D}^-\mathfrak{m}athsf{Ent}|$ is sequentially lower semicontinuous with respect to convergence with moments in~$\mathfrak{m}athcal{P}(X)$ on sublevels of~$\mathfrak{m}athsf{Ent}$;
\end{enumerate}
are equivalent. We do not know if property~\eqref{item:lsc_slope} is true for the space $(\mathfrak{m}athbb{G},\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}},\mathfrak{m}athcal{L}^n)$ and this is why in \cref{prop:slope_ent_G_special} we needed the additional assumption $|\mathfrak{m}athrm{D}^-_\varepsilon\mathfrak{m}athsf{Ent}|(\mathfrak{m}u)<+\infty$.
By~\cite{AGS14}*{Theorem~9.3}, property~\eqref{item:lsc_slope} holds true if $(X,\mathfrak{m}athsf{d},\mathfrak{m}athfrak{m})$ is $CD(K,\infty)$ for some $K\in\mathfrak{m}athbb{R}$. As the following result shows, (non-commutative) Carnot groups are not $CD(K,\infty)$, so that the validity of property~\eqref{item:lsc_slope} in these metric measure spaces is an open problem. Note that \cref{prop:carnot_not_CD} below was already known for the Heisenberg groups, see~\cite{J09}.
\begin{proposition}\label{prop:carnot_not_CD}
If $(\mathfrak{m}athbb{G},\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}},\mathfrak{m}athcal{L}^n)$ is a non-commutative Carnot group, then the metric measure space $(\mathfrak{m}athbb{G},\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}},\mathfrak{m}athcal{L}^n)$ is not $CD(K,\infty)$ for any $K\in\mathfrak{m}athbb{R}$.
\end{proposition}
\begin{proof}
By contradiction, assume that $(\mathfrak{m}athbb{G},\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}},\mathfrak{m}athcal{L}^n)$ is a $CD(K,\infty)$ space for some $K\in\mathfrak{m}athbb{R}$. Since the Dirichlet--Cheeger energy associate to the horizontal gradient is \emph{quadratic} on $L^2(\mathfrak{m}athbb{G},\mathfrak{m}athcal{L}^n)$ (see~\cite{AGS14-2}*{Section~4.3} for a definition), by~\cite{AGMR15}*{Theorem~6.1} we deduce that $(\mathfrak{m}athbb{G},\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}},\mathfrak{m}athcal{L}^n)$ is a ($\sigma$-finite) $RCD(K,\infty)$ space. By~\cite{AGMR15}*{Theorem~7.2}, we deduce that $(\mathfrak{m}athbb{G},\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}},\mathfrak{m}athcal{L}^n)$ satisfies the $BE(K,\infty)$ property, that is,
\begin{equation}\label{eq:BE}
\|\nabla_\mathfrak{m}athbb{G} (P_t f)\|_\mathfrak{m}athbb{G}^2\le e^{-2Kt} P_t(\|\nabla f\|_\mathfrak{m}athbb{G}^2),
\qquad
\text{for all }
t\ge 0,\ f\in C^\infty_c(\mathfrak{m}athbb{R}^n).
\end{equation}
Here and in the rest of the proof, we set $P_t f:=f\star\mathfrak{m}athsf{h}_t$ for short. Arguing similarly as in the proof of~\cite{W11}*{Theorem~1.1}, it is possible to prove that~\eqref{eq:BE} is equivalent to the following \emph{reverse Poincaré inequality}
\begin{equation}\label{eq:reverse_P}
P_t(f^2)-(P_t f)^2\ge 2I_{2K}(t)\,\|\nabla (P_t f)\|_\mathfrak{m}athbb{G}^2,
\qquad
\text{for all }
t\ge 0,\ f\in C^\infty_c(\mathfrak{m}athbb{R}^n),
\end{equation}
where $I_K(t):=\frac{e^{Kt}-1}{K}$ if $K\ne0$ and $I_0(t):=t$. Now, by~\cite{BB16}*{Propositions~2.5 and~2.6}, there exists a constant $\Lambda\in\left[\frac{Q}{2m_1},\frac{Q}{m_1}\right]$ (where~$Q$ and~$m_1$ are as in \cref{subsec:carnot_groups}) such that the inequality
\begin{equation}\label{eq:reverse_P_sharp}
P_t(f^2)-(P_t f)^2\ge\frac{t}{\Lambda}\,\|\nabla (P_t f)\|_\mathfrak{m}athbb{G}^2,
\qquad
\text{for all }
t\ge 0,\ f\in C^\infty_c(\mathfrak{m}athbb{R}^n),
\end{equation}
holds true and, moreover, is sharp. Comparing~\eqref{eq:reverse_P} and~\eqref{eq:reverse_P_sharp}, we thus must have that $\Lambda\le\frac{t}{2I_{2K}(t)}$ for all $t>0$. Passing to the limit as $t\to0^+$, we get that $\Lambda\le\frac{1}{2}$, so that $Q\le m_1$. This immediately implies that~$\mathfrak{m}athbb{G}$ is commutative, a contradiction.
\end{proof}
\section{Proof of the main result}
\label{sec:proof_of_main_result}
\subsection{Heat diffusions are gradient flows of the entropy}
In this section we prove the first part of \cref{th:main}. The argument follows the strategy outlined in~\cite{J14}*{Section~4.1}.
The following technical lemma will be applied to horizontal vector fields in the proof of \cref{prop:entropy_dissipation} below. The proof is exactly the same of~\cite{J14}*{Lemma~4.1} and we omit it.
\begin{lemma}\label{lemma:divergence}
Let $V\colon\mathfrak{m}athbb{R}^n\to\mathfrak{m}athbb{R}^n$ be a vector field with locally Lipschitz coefficients such that $|V|_{\mathfrak{m}athbb{R}^n}\in L^1(\mathfrak{m}athbb{R}^n)$ and $\diverg V\in L^1(\mathfrak{m}athbb{R}^n)$. Then $\int_{\mathfrak{m}athbb{R}^n}\diverg V\ dx=0$.
\end{lemma}
\cref{prop:entropy_dissipation} below states that the function $t\mathfrak{m}apsto\mathfrak{m}athsf{Ent}(\varrho_t\mathfrak{m}athcal{L}^n)$ is locally absolutely continuous if $(\varrho_t)_{t\ge0}$ solves the sub-elliptic heat equation~\eqref{eq:heat_diffusion} with initial datum $\varrho_0\in L^1(\mathfrak{m}athbb{G})$ such that $\mathfrak{m}u_0=\varrho_0\mathfrak{m}athcal{L}^n\in\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G})$. By~\cite{AGS14}*{Proposition~4.22}, this result is true under the stronger assumption that $\varrho_0\in L^1(\mathfrak{m}athbb{G})\cap L^2(\mathfrak{m}athbb{G})$. Here the point is to remove the $L^2$-integrability condition on the initial datum exploiting the estimates on the heat kernel collected in \cref{th:property_heat_kernel}, see also~\cite{J14}*{Section~4.1.1}.
\begin{proposition}[Entropy dissipation]\label{prop:entropy_dissipation}
Let $\varrho_0\in L^1(\mathfrak{m}athbb{G})$ be such that $\mathfrak{m}u_0=\varrho_0\mathfrak{m}athcal{L}^n\in\dom(\mathfrak{m}athsf{Ent})$. If $(\varrho_t)_{t\ge0}$ solves the sub-elliptic heat equation $\partial_t\varrho_t=\mathfrak{m}athrm{D}elta_\mathfrak{m}athbb{G}\varrho_t$ with initial datum~$\varrho_0$, then the map $t\mathfrak{m}apsto\mathfrak{m}athsf{Ent}(\mathfrak{m}u_t)$, $\mathfrak{m}u_t=\varrho_t\mathfrak{m}athcal{L}^n$, is locally absolutely continuous on $(0,+\infty)$ and it holds
\begin{equation}\label{eq:entropy_dissipation}
\frac{d}{dt}\mathfrak{m}athsf{Ent}(\mathfrak{m}u_t)=-\int_{\mathfrak{m}athbb{G}\cap\set*{\varrho_t>0}}\frac{\|\nabla_\mathfrak{m}athbb{G}\varrho_t\|^2_\mathfrak{m}athbb{G}}{\varrho_t}\ dx
\qquad
\text{for a.e.}\ t>0.
\end{equation}
\end{proposition}
\begin{proof}
Note that $\mathfrak{m}u_t\in\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G})$ for all $t>0$ by~\eqref{eq:heat_second_moment}. Hence $\mathfrak{m}athsf{Ent}(\mathfrak{m}u_t)>-\infty$ for all $t>0$. Since $C_t:=\sup_{x\in\mathfrak{m}athbb{G}}\mathfrak{m}athsf{h}_t(x)<+\infty$ for each fixed $t>0$ by~\eqref{eq:heat_estimate_above}, we get that $\varrho_t\le C_t$ for all $t>0$. Thus $\mathfrak{m}athsf{Ent}(\mathfrak{m}u_t)<+\infty$ for all $t>0$.
For each $m\in\mathfrak{m}athbb{N}$, define
\begin{equation}\label{eq:def_truncated_entropy}
z_m(r):=\mathfrak{m}in\set*{m,\mathfrak{m}ax\set*{1+\log r,-m}}, \qquad H_m(r):=\int_0^r z_m(s)\ ds, \qquad r\ge0.
\end{equation}
Note that $H_m$ is of class $C^1$ on $[0,+\infty)$ with $H_m'$ is globally Lipschitz and bounded. We claim that
\begin{equation}\label{eq:deriv_1}
\frac{d}{dt}\int_\mathfrak{m}athbb{G} H_m(\varrho_t)\ dx=\int_\mathfrak{m}athbb{G} z_m(\varrho_t)\mathfrak{m}athrm{D}elta_\mathfrak{m}athbb{G}\varrho_t\ dx
\qquad
\forall t>0,\ \forall m\in\mathfrak{m}athbb{N}.
\end{equation}
Indeed, we have $|H_m(\varrho_t)|\le m\varrho_t\in L^1(\mathfrak{m}athbb{G})$ and, given $[a,b]\subset(0,+\infty)$, by~\eqref{eq:heat_estimate_derivatives} the function $x\mathfrak{m}apsto\sup_{t\in[a,b]}|\mathfrak{m}athrm{D}elta_\mathfrak{m}athbb{G}\mathfrak{m}athsf{h}_t(x)|$ is bounded. Thus
\begin{equation*}
\sup_{t\in[a,b]}\left|\frac{d}{dt}H_m(\varrho_t)\right|\le m\sup_{t\in[a,b]}\left(\varrho_0\star|\mathfrak{m}athrm{D}elta_\mathfrak{m}athbb{G}\mathfrak{m}athsf{h}_t|\right)\le m\,\varrho_0\star\sup_{t\in[a,b]}|\mathfrak{m}athrm{D}elta_\mathfrak{m}athbb{G}\mathfrak{m}athsf{h}_t|\in L^1(\mathfrak{m}athbb{G}).
\end{equation*}
Therefore~\eqref{eq:deriv_1} follows by differentiation under integral sign. We now claim that
\begin{equation}\label{eq:deriv_2}
\int_\mathfrak{m}athbb{G} z_m(\varrho_t)\mathfrak{m}athrm{D}elta_\mathfrak{m}athbb{G}\varrho_t\ dx=-\int_{\set*{e^{-m-1}<\varrho_t<e^{m-1}}} \frac{\|\nabla_\mathfrak{m}athbb{G}\varrho_t\|_\mathfrak{m}athbb{G}^2}{\varrho_t}\ dx
\qquad
\forall t>0,\ \forall m\in\mathfrak{m}athbb{N}.
\end{equation}
Indeed, by Cauchy--Schwarz inequality, we have
\begin{equation}\label{eq:C-S_for_horizontal_grad}
\begin{split}
\frac{\|\nabla_\mathfrak{m}athbb{G}\varrho_t(x)\|_\mathfrak{m}athbb{G}^2}{\varrho_t(x)}
&\le\frac{\left[(\varrho_0\star\|\nabla_\mathfrak{m}athbb{G}\mathfrak{m}athsf{h}_t\|_\mathfrak{m}athbb{G})(x)\right]^2}{(\varrho_0\star\mathfrak{m}athsf{h}_t)(x)}\\
&=\frac{1}{(\varrho_0\star\mathfrak{m}athsf{h}_t)(x)}\left[\int_\mathfrak{m}athbb{G}\sqrt{\varrho_0(xy^{-1})}\,\frac{\|\nabla_\mathfrak{m}athbb{G}\mathfrak{m}athsf{h}_t(y)\|_\mathfrak{m}athbb{G}}{\sqrt{\mathfrak{m}athsf{h}_t(y)}}\cdot\sqrt{\varrho_0(xy^{-1})}\,\sqrt{\mathfrak{m}athsf{h}_t(y)}\ dy\right]^2\\
&\le\frac{1}{(\varrho_0\star\mathfrak{m}athsf{h}_t)(x)}
\left(\int_\mathfrak{m}athbb{G}\varrho_0(xy^{-1})\,\frac{\|\nabla_\mathfrak{m}athbb{G}\mathfrak{m}athsf{h}_t(y)\|_\mathfrak{m}athbb{G}^2}{\mathfrak{m}athsf{h}_t(y)}\ dy\right)
\left(\int_\mathfrak{m}athbb{G}\varrho_0(xy^{-1})\,\mathfrak{m}athsf{h}_t(y)\ dy\right)\\
&\le\left(\varrho_0\star\frac{\|\nabla_\mathfrak{m}athbb{G}\mathfrak{m}athsf{h}_t\|_\mathfrak{m}athbb{G}^2}{\mathfrak{m}athsf{h}_t}\right)(x)
\qquad
\text{for all}\ x\in\mathfrak{m}athbb{G}.
\end{split}
\end{equation}
Thus, by~\eqref{eq:heat_estimate_below} and~\eqref{eq:heat_estimate_derivatives}, we get
\begin{equation}\label{eq:go_to_h}
\int_{\set*{e^{-m-1}<\varrho_t<e^{m-1}}}\frac{\|\nabla_\mathfrak{m}athbb{G}\varrho_t\|_\mathfrak{m}athbb{G}^2}{\varrho_t}\ dx\le\int_\mathfrak{m}athbb{G}\varrho_0\star\frac{\|\nabla_\mathfrak{m}athbb{G}\mathfrak{m}athsf{h}_t\|_\mathfrak{m}athbb{G}^2}{\mathfrak{m}athsf{h}_t}\ dx=\int_\mathfrak{m}athbb{G}\frac{\|\nabla_\mathfrak{m}athbb{G}\mathfrak{m}athsf{h}_t\|_\mathfrak{m}athbb{G}^2}{\mathfrak{m}athsf{h}_t}\ dx<+\infty.
\end{equation}
This, together with~\eqref{eq:deriv_1}, proves that
\begin{align}\label{eq:diverg_L1}
\diverg(z_m(\varrho_t)\nabla_\mathfrak{m}athbb{G}\varrho_t)=z_m'(\varrho_t)\|\nabla_\mathfrak{m}athbb{G}\varrho_t\|_\mathfrak{m}athbb{G}^2-z_m(\varrho_t)\mathfrak{m}athrm{D}elta_\mathfrak{m}athbb{G}\varrho_t\in L^1(\mathfrak{m}athbb{G}).
\end{align}
Thus~\eqref{eq:deriv_2} follows by integration by parts provided that
\begin{equation}\label{eq:diverg_0}
\int_\mathfrak{m}athbb{G}\diverg(z_m(\varrho_t)\nabla_\mathfrak{m}athbb{G}\varrho_t)\ dx=0.
\end{equation}
To prove~\eqref{eq:diverg_0}, we apply \cref{lemma:divergence} to the vector field $V=z_m(\varrho_t)\nabla_\mathfrak{m}athbb{G}\varrho_t$. By~\eqref{eq:diverg_L1}, we already know that $\diverg V\in L^1(\mathfrak{m}athbb{G})$, so we just need to prove that $|V|_{\mathfrak{m}athbb{R}^n}\in L^1(\mathfrak{m}athbb{G})$. Note that
\begin{equation*}
\int_\mathfrak{m}athbb{G}|V|_{\mathfrak{m}athbb{R}^n}\ dx\le m\int_\mathfrak{m}athbb{G}\varrho_0\star|\nabla_\mathfrak{m}athbb{G}\mathfrak{m}athsf{h}_t|_{\mathfrak{m}athbb{R}^n}\ dx=m\int_\mathfrak{m}athbb{G} |\nabla_\mathfrak{m}athbb{G}\mathfrak{m}athsf{h}_t|_{\mathfrak{m}athbb{R}^n}\ dx,
\end{equation*}
so it is enough to prove that $|\nabla_\mathfrak{m}athbb{G}\mathfrak{m}athsf{h}_t|_{\mathfrak{m}athbb{R}^n}\in L^1(\mathfrak{m}athbb{G})$. But we have
\begin{equation*}
|\nabla_\mathfrak{m}athbb{G}\mathfrak{m}athsf{h}_t(x)|_{\mathfrak{m}athbb{R}^n}\le p(x_1,\dots,x_n)\|\nabla_\mathfrak{m}athbb{G}\mathfrak{m}athsf{h}_t(x)\|_\mathfrak{m}athbb{G}, \qquad x\in\mathfrak{m}athbb{G},
\end{equation*}
where $p\colon\mathfrak{m}athbb{R}^n\to[0,+\infty)$ is a function with polynomial growth, because the horizontal vector fields $X_1,\dots,X_{h_1}$ have polynomial coefficients. Since $\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}}$ is equivalent to $\mathfrak{m}athsf{d}_\infty$, where~$\mathfrak{m}athsf{d}_\infty$ was introduced in~\eqref{eq:def_box_norm}, by~\eqref{eq:heat_estimate_derivatives} we conclude that $|\nabla_\mathfrak{m}athbb{G}\mathfrak{m}athsf{h}_t|_{\mathfrak{m}athbb{R}^n}\in L^1(\mathfrak{m}athbb{G})$. This completes the proof of~\eqref{eq:diverg_0}.
Combining~\eqref{eq:deriv_1} and~\eqref{eq:deriv_2}, we thus get
\begin{equation*}
\frac{d}{dt}\int_\mathfrak{m}athbb{G} H_m(\varrho_t)\ dx=-\int_{\set*{e^{-m-1}<\varrho_t<e^{m-1}}} \frac{\|\nabla_\mathfrak{m}athbb{G}\varrho_t\|_\mathfrak{m}athbb{G}^2}{\varrho_t}\ dx
\qquad
\forall t>0,\ \forall m\in\mathfrak{m}athbb{N}.
\end{equation*}
Note that
\begin{equation}\label{eq:fisher_L1_loc}
t\mathfrak{m}apsto\int_\mathfrak{m}athbb{G} \frac{\|\nabla_\mathfrak{m}athbb{G}\varrho_t\|_\mathfrak{m}athbb{G}^2}{\varrho_t}\ dx\in L^1_{\rm loc}(0,+\infty).
\end{equation}
Indeed, from~\eqref{eq:heat_estimate_below} and~\eqref{eq:heat_estimate_derivatives} we deduce that
\begin{equation*}
t\mathfrak{m}apsto\int_\mathfrak{m}athbb{G} \frac{\|\nabla_\mathfrak{m}athbb{G}\mathfrak{m}athsf{h}_t\|_\mathfrak{m}athbb{G}^2}{\mathfrak{m}athsf{h}_t}\ dx\in L^1_{\rm loc}(0,+\infty).
\end{equation*}
Recalling~\eqref{eq:C-S_for_horizontal_grad} and~\eqref{eq:go_to_h}, this immediately implies~\eqref{eq:fisher_L1_loc}. Therefore
\begin{equation*}
\int_\mathfrak{m}athbb{G} H_m(\varrho_{t_1})\ dx-\int_\mathfrak{m}athbb{G} H_m(\varrho_{t_0})\ dx=-\int_{t_0}^{t_1}\int_{\set*{e^{-m-1}<\varrho_t<e^{m-1}}} \frac{\|\nabla_\mathfrak{m}athbb{G}\varrho_t\|_\mathfrak{m}athbb{G}^2}{\varrho_t}\ dx dt
\end{equation*}
for any $t_0,t_1\in(0,+\infty)$ with $t_0<t_1$ and $m\in\mathfrak{m}athbb{N}$. We now pass to the limit as $m\to+\infty$. Note that $H_m(r)\to r\log r$ as $m\to+\infty$ and that for all $m\in\mathfrak{m}athbb{N}$
\begin{equation}\label{eq:truncated_ent_prop_1}
r\log r\le H_{m+1}(r)\le H_m(r) \qquad\text{for}\ r\in[0,1]
\end{equation}
and
\begin{equation}\label{eq:truncated_ent_prop_2}
0\le H_m(r)\le 1+r\log r \qquad\text{for}\ r\in[1,+\infty).
\end{equation}
Thus
\begin{equation*}
\lim_{m\to+\infty}\int_\mathfrak{m}athbb{G} H_m(\varrho_t)\ dx=\int_\mathfrak{m}athbb{G} \varrho_t\log\varrho_t\ dx
\end{equation*}
by the monotone convergence theorem on $\set*{\varrho_t\le1}$ and by the dominated convergence theorem on $\set*{\varrho_t>1}$. Moreover
\begin{equation*}
\lim_{m\to+\infty}\int_{t_0}^{t_1}\int_{\set*{e^{-m-1}<\varrho_t<e^{m-1}}} \frac{\|\nabla_\mathfrak{m}athbb{G}\varrho_t\|_\mathfrak{m}athbb{G}^2}{\varrho_t}\ dx dt
=\int_{t_0}^{t_1}\int_{\mathfrak{m}athbb{G}\cap\set*{\varrho_t>0}} \frac{\|\nabla_\mathfrak{m}athbb{G}\varrho_t\|_\mathfrak{m}athbb{G}^2}{\varrho_t}\ dx dt
\end{equation*}
by the monotone convergence theorem. This concludes the proof.
\end{proof}
We are now ready to prove the first part of \cref{th:main}. The argument follows the strategy outlined in~\cite{J14}*{Section~4.1}. See also the first part of the proof of~\cite{AGS14}*{Theorem~8.5}.
\begin{theorem}\label{th:heat_diff_implies_GF_ent}
Let $\varrho_0\in L^1(\mathfrak{m}athbb{G})$ be such that $\mathfrak{m}u_0=\varrho_0\mathfrak{m}athcal{L}^n\in\dom(\mathfrak{m}athsf{Ent})$. If $(\varrho_t)_{t\ge0}$ solves the sub-elliptic heat equation $\partial_t\varrho_t=\mathfrak{m}athrm{D}elta_\mathfrak{m}athbb{G}\varrho_t$ with initial datum $\varrho_0$, then $\mathfrak{m}u_t=\varrho_t\mathfrak{m}athcal{L}^n$ is a gradient flow of $\mathfrak{m}athsf{Ent}$ in $(\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G}),\mathfrak{m}athsf{W}_\mathfrak{m}athbb{G})$ starting from~$\mathfrak{m}u_0$.
\end{theorem}
\begin{proof}
Note that $(\mathfrak{m}u_t)_{t>0}\subset\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G})$, see the proof of \cref{prop:entropy_dissipation}. Moreover, $(\mathfrak{m}u_t)_{t>0}$ satisfies~\eqref{eq:CE} with $v^\mathfrak{m}athbb{G}_t=\nabla_\mathfrak{m}athbb{G}\varrho_t/\varrho_t$ for $t>0$. Note that $t\mathfrak{m}apsto\|v^\mathfrak{m}athbb{G}_t\|_{L^2_\mathfrak{m}athbb{G}(\mathfrak{m}u_t)}\in L^2_{\rm loc}(0,+\infty)$ by~\eqref{eq:heat_estimate_below}, \eqref{eq:heat_estimate_derivatives}, \eqref{eq:C-S_for_horizontal_grad} and~\eqref{eq:go_to_h}. By \cref{prop:CE} we conclude that
\begin{equation}\label{eq:estim_deriv_heat_curve}
|\dot{\mathfrak{m}u}_t|^2\le\int_{\mathfrak{m}athbb{G}\cap\set*{\varrho_t>0}}\frac{\|\nabla\varrho_t\|_\mathfrak{m}athbb{G}^2}{\varrho_t}\ dx
\qquad
\text{for a.e.}\ t>0.
\end{equation}
By \cref{prop:entropy_dissipation}, the map $t\mathfrak{m}apsto\mathfrak{m}athsf{Ent}(\mathfrak{m}u_t)$ is locally absolutely continuous on $(0,+\infty)$ and so, by the chain rule, we get
\begin{equation}\label{eq:chain_rule_ent}
-\frac{d}{dt}\mathfrak{m}athsf{Ent}(\mathfrak{m}u_t)\le|\mathfrak{m}athrm{D}^-_\mathfrak{m}athbb{G}\mathfrak{m}athsf{Ent}|(\mathfrak{m}u_t)\cdot|\dot{\mathfrak{m}u}_t|_\mathfrak{m}athbb{G} \qquad
\text{for a.e.}\ t>0.
\end{equation}
Thus, if we prove that
\begin{equation}\label{eq:slope_ent_finite_heat}
|\mathfrak{m}athrm{D}^-_\mathfrak{m}athbb{G}\mathfrak{m}athsf{Ent}|^2(\mathfrak{m}u_t)=\int_{\mathfrak{m}athbb{G}\cap\set*{\varrho_t>0}}\frac{\|\nabla_\mathfrak{m}athbb{G}\varrho_t\|_\mathfrak{m}athbb{G}^2}{\varrho_t}\ dx
\qquad
\text{for a.e.}\ t>0
\end{equation}
then, combining this equality with~\eqref{eq:entropy_dissipation}, \eqref{eq:estim_deriv_heat_curve} and~\eqref{eq:chain_rule_ent}, we find that
\begin{equation*}
|\dot{\mathfrak{m}u}_t|_\mathfrak{m}athbb{G}=|\mathfrak{m}athrm{D}^-_\mathfrak{m}athbb{G}\mathfrak{m}athsf{Ent}|(\mathfrak{m}u_t),
\qquad
\frac{d}{dt}\mathfrak{m}athsf{Ent}(\mathfrak{m}u_t)=-|\mathfrak{m}athrm{D}^-_\mathfrak{m}athbb{G}\mathfrak{m}athsf{Ent}|(\mathfrak{m}u_t)\cdot|\dot{\mathfrak{m}u}_t|_\mathfrak{m}athbb{G}
\qquad
\text{for a.e.}\ t>0,
\end{equation*}
so that $(\mathfrak{m}u_t)_{t\ge0}$ is a gradient flow of~$\mathfrak{m}athsf{Ent}$ starting from~$\mathfrak{m}u_0$ as observed in \cref{rem:GF_AC}.
We now prove~\eqref{eq:slope_ent_finite_heat}. To do so, we apply \cref{prop:slope_ent_G_special}. We need to check that $|\mathfrak{m}athrm{D}^-_\varepsilon\mathfrak{m}athsf{Ent}|(\mathfrak{m}u_t)<+\infty$ for some $\varepsilon>0$. To prove this, we apply \cref{prop:slope_ent_eps}. Since $\partialps\le\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}}$, we have $\mathfrak{m}u_t\in\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G}_\varepsilon)$ for all $\varepsilon>0$. Moreover
\begin{equation*}
\mathfrak{m}athsf{F}_\varepsilon(\varrho_t)=\mathfrak{m}athsf{F}_\mathfrak{m}athbb{G}(\varrho_t)+\sum_{i=2}^k\varepsilon^{2(i-1)}\int_{\mathfrak{m}athbb{G}\cap\set*{\varrho_t>0}}\frac{\|\nabla_{V_i}\varrho_t\|_\mathfrak{m}athbb{G}^2}{\varrho_t}\ dx.
\end{equation*}
Since $\mathfrak{m}athsf{F}_\mathfrak{m}athbb{G}(\varrho_t)<+\infty$, we just need to prove that
\begin{equation*}
\int_{\mathfrak{m}athbb{G}\cap\set*{\varrho_t>0}}\frac{\|\nabla_{V_i}\varrho_t\|_\mathfrak{m}athbb{G}^2}{\varrho_t}\ dx<+\infty
\end{equation*}
for all $i=2,\dots,\kappa$. Indeed, arguing as in~\eqref{eq:C-S_for_horizontal_grad}, by Cauchy--Schwarz inequality we have
\begin{align*}
\frac{\|\nabla_{V_i}\varrho_t\|_\mathfrak{m}athbb{G}^2}{\varrho_t}
\le\frac{\left(\varrho\star\|\nabla_{V_i}\mathfrak{m}athsf{h}_t\|_\mathfrak{m}athbb{G}\right)^2}{\varrho\star\mathfrak{m}athsf{h}_t}
=\frac{1}{\varrho\star\mathfrak{m}athsf{h}_t}\left[\varrho\star\left(\frac{\|\nabla_{V_i}\mathfrak{m}athsf{h}_t\|_\mathfrak{m}athbb{G}}{\sqrt{\mathfrak{m}athsf{h}_t}}\,\sqrt{\mathfrak{m}athsf{h}_t}\right)\right]^2
\le\varrho\star\frac{\|\nabla_{V_i}\mathfrak{m}athsf{h}_t\|_\mathfrak{m}athbb{G}^2}{\mathfrak{m}athsf{h}_t}.
\end{align*}
Therefore, by~\eqref{eq:heat_estimate_below} and~\eqref{eq:heat_estimate_derivatives}, we get
\begin{equation*}
\int_{\mathfrak{m}athbb{G}\cap\set*{\varrho_t>0}}\frac{\|\nabla_{V_i}\varrho_t\|_\mathfrak{m}athbb{G}^2}{\varrho_t}\ dx
\le\int_\mathfrak{m}athbb{G}\varrho\star\frac{\|\nabla_{V_i}\mathfrak{m}athsf{h}_t\|_\mathfrak{m}athbb{G}^2}{\mathfrak{m}athsf{h}_t}\ dx
=\int_\mathfrak{m}athbb{G}\frac{\|\nabla_{V_i}\mathfrak{m}athsf{h}_t\|_\mathfrak{m}athbb{G}^2}{\mathfrak{m}athsf{h}_t}\ dx<+\infty.
\end{equation*}
This concludes the proof.
\end{proof}
\subsection{Gradient flows of the entropy are heat diffusions}
In this section we prove the second part of \cref{th:main}. Our argument is different from the one presented in~\cite{J14}*{Section~4.2}. However, as observed in~\cite{J14}*{Remark~5.3}, the techniques developed in~\cite{J14}*{Section~4.2} can be adapted in order to obtain a proof of \cref{th:GF_ent_implies_heat_diff} below for any Carnot group~$\mathfrak{m}athbb{G}$ of step~$2$.
Let us start with the following remark. If $(\mathfrak{m}u_t)_{t\ge0}$ is a gradient flow of $\mathfrak{m}athsf{Ent}$ in $(\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G}),\mathfrak{m}athsf{W}_\mathfrak{m}athbb{G})$ then, recalling \cref{def:metric_GF}, we have that $\mathfrak{m}u_t\in\dom(\mathfrak{m}athsf{Ent})$ for all $t\ge0$. By~\eqref{eq:def_entropy}, this means that $\mathfrak{m}u_t=\varrho_t\mathfrak{m}athcal{L}^n$ for some probability density $\varrho_t\in L^1(\mathfrak{m}athbb{G})$ for all $t\ge0$. In addition, $t\mathfrak{m}apsto|\mathfrak{m}athrm{D}_\mathfrak{m}athbb{G}^-\mathfrak{m}athsf{Ent}|(\mathfrak{m}u_t)\in L^2_{\rm loc}([0,+\infty))$ and the function $t\mathfrak{m}apsto\mathfrak{m}athsf{Ent}(\mathfrak{m}u_t)$ is non-increasing, therefore a.e.\ differentiable and locally integrable on~$[0,+\infty)$.
\cref{lemma:GF_ent_plus_AC_is_heat_diff} below shows that it is enough to establish~\eqref{eq:GF_ent_plus_AC} in order to prove the second part of \cref{th:main}. For the proof, see also the last paragraph of~\cite{J14}*{Section~4.2}.
\begin{lemma}\label{lemma:GF_ent_plus_AC_is_heat_diff}
Let $\varrho_0\in L^1(\mathfrak{m}athbb{G})$ be such that $\mathfrak{m}u_0=\varrho_0\mathfrak{m}athcal{L}^n\in\dom(\mathfrak{m}athsf{Ent})$. Assume $(\mathfrak{m}u_t)_{t\ge0}$ is a gradient flow of $\mathfrak{m}athsf{Ent}$ in $(\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G}),\mathfrak{m}athsf{W}_\mathfrak{m}athbb{G})$ starting from~$\mathfrak{m}u_0$, with $\mathfrak{m}u_t=\varrho_t\mathfrak{m}athcal{L}^n$ for all $t\ge0$. Let $(v^\mathfrak{m}athbb{G}_t)_{t>0}$ and $(w^\mathfrak{m}athbb{G}_t)_{t>0}$, $w^\mathfrak{m}athbb{G}_t=\nabla_\mathfrak{m}athbb{G}\varrho_t/\varrho_t$, be the horizontal time-dependent vector fields given by \cref{prop:CE} and \cref{prop:slope_ent_G_general} respectively. If it holds
\begin{equation}\label{eq:GF_ent_plus_AC}
-\frac{d}{dt}\mathfrak{m}athsf{Ent}(\mathfrak{m}u_t)\le\int_\mathfrak{m}athbb{G}\scalar*{-w^\mathfrak{m}athbb{G}_t,v^\mathfrak{m}athbb{G}_t}_\mathfrak{m}athbb{G}\ d\mathfrak{m}u_t
\qquad
\text{for a.e.}\ t>0,
\end{equation}
then $(\varrho_t)_{t\ge0}$ solves the sub-elliptic heat equation $\partial_t\varrho_t=\mathfrak{m}athrm{D}elta_\mathfrak{m}athbb{G}\varrho_t$ with initial datum $\varrho_0$.
\end{lemma}
\begin{proof}
From \cref{def:metric_GF} we get that
\begin{equation*}
\mathfrak{m}athsf{Ent}(\mathfrak{m}u_t)+\frac{1}{2}\int_s^t|\dot{\mathfrak{m}u}_r|_\mathfrak{m}athbb{G}^2\ dr+\frac{1}{2}\int_s^t|\mathfrak{m}athrm{D}^-\mathfrak{m}athsf{Ent}|_\mathfrak{m}athbb{G}^2(\mathfrak{m}u_r)\ dr\le\mathfrak{m}athsf{Ent}(\mathfrak{m}u_s)
\end{equation*}
for all $s,t\ge0$ with $s\le t$. Therefore
\begin{equation*}
-\frac{d}{dt}\mathfrak{m}athsf{Ent}(\mathfrak{m}u_t)\ge\frac{1}{2}|\dot{\mathfrak{m}u}_t|_\mathfrak{m}athbb{G}^2+\frac{1}{2}|\mathfrak{m}athrm{D}^-\mathfrak{m}athsf{Ent}|^2_\mathfrak{m}athbb{G}(\mathfrak{m}u_t)
\qquad
\text{for a.e.}\ t>0.
\end{equation*}
By Young's inequality, \cref{prop:CE} and \cref{prop:slope_ent_G_general}, we thus get
\begin{equation}\label{eq:GF_ent_plus_AC_opposite}
-\frac{d}{dt}\mathfrak{m}athsf{Ent}(\mathfrak{m}u_t)\ge|\dot{\mathfrak{m}u}_t|_\mathfrak{m}athbb{G}\cdot|\mathfrak{m}athrm{D}^-_\mathfrak{m}athbb{G}\mathfrak{m}athsf{Ent}|(\mathfrak{m}u_t)\ge\|v^\mathfrak{m}athbb{G}_t\|_{L^2_\mathfrak{m}athbb{G}(\mathfrak{m}u_t)}\cdot\|w^\mathfrak{m}athbb{G}_t\|_{L^2_\mathfrak{m}athbb{G}(\mathfrak{m}u_t)}
\qquad
\text{for a.e.}\ t>0.
\end{equation}
Combining~\eqref{eq:GF_ent_plus_AC} and~\eqref{eq:GF_ent_plus_AC_opposite}, by Cauchy--Schwarz inequality we conclude that $v^\mathfrak{m}athbb{G}_t=-w^\mathfrak{m}athbb{G}_t=-\nabla_\mathfrak{m}athbb{G}\varrho_t/\varrho_t$ in $L^2_\mathfrak{m}athbb{G}(\mathfrak{m}u_t)$ for a.e.\ $t>0$. This immediately implies that $(\varrho_t)_{t\ge0}$ solves the sub-elliptic heat equation $\partial_t\varrho_t=\mathfrak{m}athrm{D}elta_\mathfrak{m}athbb{G}\varrho_t$ with initial datum $\varrho_0$ \emph{in the sense of distributions}, i.e.\
\begin{equation*}
\int_0^{+\infty}\int_\mathfrak{m}athbb{G}\partial_t\varphi_t(x)+\mathfrak{m}athrm{D}elta_\mathfrak{m}athbb{G}\varphi_t(x)\ d\mathfrak{m}u_t\,dt+\int_\mathfrak{m}athbb{G}\varphi_0(x)\ d\mathfrak{m}u_0(x)=0
\qquad
\forall\varphi\in C^\infty_c([0,+\infty)\times\mathfrak{m}athbb{R}^n).
\end{equation*}
By well-known results on hypoelliptic operators, this implies that $(\varrho_t)_{t\ge0}$ solves the sub-elliptic heat equation $\partial_t\varrho_t=\mathfrak{m}athrm{D}elta_\mathfrak{m}athbb{G}\varrho_t$ with initial datum $\varrho_0$.
\end{proof}
To prove \cref{th:GF_ent_implies_heat_diff} below we need some preliminaries. The following two lemmas are natural adaptations of~\cite{AS07}*{Lemma~2.14} to our setting.
\begin{lemma}\label{lemma:AFP_func_meas}
Let $\mathfrak{m}u\in\mathfrak{m}athcal{P}(\mathfrak{m}athbb{G})$ and $\sigma\in L^1(\mathfrak{m}athbb{G})$ with $\sigma\ge0$. Let $\nu\in\mathfrak{m}athcal{M}(\mathfrak{m}athbb{G};\mathfrak{m}athbb{R}^m)$ be a $\mathfrak{m}athbb{R}^m$-valued Borel measure with finite total variation and such that $|\nu|\ll\mathfrak{m}u$. Then
\begin{equation}\label{eq:afp_func_meas}
\int_{\mathfrak{m}athbb{G}}\ \abs*{\frac{\sigma\star\nu}{\sigma\star\mathfrak{m}u}}^2\sigma\star\mathfrak{m}u\ dx
\le\int_{\mathfrak{m}athbb{G}}\ \abs*{\frac{\nu}{\mathfrak{m}u}}^2\, d\mathfrak{m}u.
\end{equation}
In addition, if $(\sigma_k)_{k\in\mathfrak{m}athbb{N}}\subset L^1(\mathfrak{m}athbb{G})$, $\sigma_k\ge0$, weakly converges to the Dirac mass $\partiallta_0$ and $\frac{\nu}{\mathfrak{m}u}\in L^2(\mathfrak{m}athbb{G},\mathfrak{m}u)$, then
\begin{equation}\label{eq:afp_func_meas_limit}
\lim_{k\to+\infty}\int_{\mathfrak{m}athbb{G}}\ \abs*{\frac{\sigma_k\star\nu}{\sigma_k\star\mathfrak{m}u}}^2\sigma_k\star\mathfrak{m}u\ dx
=\int_{\mathfrak{m}athbb{G}}\ \abs*{\frac{\nu}{\mathfrak{m}u}}^2\, d\mathfrak{m}u.
\end{equation}
\end{lemma}
\begin{proof}
Inequality~\eqref{eq:afp_func_meas} follows from Jensen inequality and is proved in~\cite{AS07}*{Lemma~2.14}. We briefly recall the argument for the reader's convenience. Consider the map $\Phi\colon\mathfrak{m}athbb{R}^m\times\mathfrak{m}athbb{R}\to[0,+\infty]$ given by
\begin{equation*}
\Phi(z,t):=
\begin{cases}
\dfrac{|z|^2}{t} & \text{if}\ t>0,\\
0 & \text{if}\ (z,t)=(0,0),\\
+\infty & \text{if either}\ t<0\ \text{or}\ t=0,\ z\ne0.
\end{cases}
\end{equation*}
Then $\Phi$ is convex, lower semicontinuous and positively 1-homogeneous. By Jensen's inequality we have
\begin{equation}\label{eq:afp_Jensen}
\Phi\left(\int_\mathfrak{m}athbb{G}\psi(x)\ d\vartheta(x)\right)
\le\int_\mathfrak{m}athbb{G}\Phi(\psi(x))\ d\vartheta(x)
\end{equation}
for any Borel function $\psi\colon\mathfrak{m}athbb{G}\to\mathfrak{m}athbb{R}^{m+1}$ and any positive and finite measure~$\vartheta$ on~$\mathfrak{m}athbb{G}$. Fix $x\in\mathfrak{m}athbb{G}$ and apply~\eqref{eq:afp_Jensen} with $\psi(y)=\left(\frac{\nu}{\mathfrak{m}u}(y),1\right)$ and $d\vartheta(y)=\sigma(xy^{-1})d\mathfrak{m}u(y)$ to obtain
\begin{align*}
\abs*{\frac{(\sigma\star\nu)(x)}{(\sigma\star\mathfrak{m}u)(x)}}^2(\sigma\star\mathfrak{m}u)(x)
&=\Phi\left(\int_\mathfrak{m}athbb{G}\frac{\nu}{\mathfrak{m}u}(y)\,\sigma(xy^{-1})\ d\mathfrak{m}u(y),\int_\mathfrak{m}athbb{G}\sigma(xy^{-1})\ d\mathfrak{m}u(y)\right)\\
&\le\int_\mathfrak{m}athbb{G}\Phi\left(\frac{\nu}{\mathfrak{m}u}(y),1\right)\sigma(xy^{-1})\ d\mathfrak{m}u(y)
=\int_\mathfrak{m}athbb{G}\abs*{\frac{\nu}{\mathfrak{m}u}}^2(y)\,\sigma(xy^{-1})\ d\mathfrak{m}u(y),
\end{align*}
which immediately gives~\eqref{eq:afp_func_meas}. The limit in~\eqref{eq:afp_func_meas_limit} follows by the joint lower semicontinuity of the functional $(\nu,\mathfrak{m}u)\mathfrak{m}apsto\int_{\mathfrak{m}athbb{G}}\ \abs*{\frac{\nu}{\mathfrak{m}u}}^2\, d\mathfrak{m}u$, see Theorem~2.34 and Example~2.36 in~\cite{AFP00}.
\end{proof}
In \cref{lemma:AFP_func_meas_time} below and in the rest of the paper, we let $f*g$ be the convolution of the two functions $f,g$ with respect to the time variable. We keep the notation $f\star g$ for the convolution of $f,g$ with respect to the space variable.
\begin{lemma}\label{lemma:AFP_func_meas_time}
Let $\mathfrak{m}u_t=\varrho_t\mathfrak{m}athcal{L}^n\in\mathfrak{m}athcal{P}(\mathfrak{m}athbb{G})$ for all $t\in\mathfrak{m}athbb{R}$ and let $\vartheta\in L^1(\mathfrak{m}athbb{R})$, $\vartheta\ge0$. If the horizontal time-dependent vector field $v\colon\mathfrak{m}athbb{R}\times\mathfrak{m}athbb{G}\to H\mathfrak{m}athbb{G}$ satisfies $v_t\in L^2_\mathfrak{m}athbb{G}(\mathfrak{m}u_t)$ for a.e.\ $t\in\mathfrak{m}athbb{R}$, then
\begin{equation}\label{eq:afp_func_meas_time}
\int_\mathfrak{m}athbb{G}\left\|\frac{\vartheta*(\varrho_\cdot v_\cdot)(t)}{\vartheta*\varrho_\cdot(t)}\right\|_\mathfrak{m}athbb{G}^2\,\vartheta*\varrho_\cdot(t)\ dx
\le\vartheta*\left(\int_\mathfrak{m}athbb{G}\|v_\cdot\|_\mathfrak{m}athbb{G}^2\ d\mathfrak{m}u_\cdot\right)(t)
\qquad
\text{for all}\ t\in\mathfrak{m}athbb{R}.
\end{equation}
In addition, if $(\vartheta_j)_{j\in\mathfrak{m}athbb{N}}\subset L^1(\mathfrak{m}athbb{G})$, $\vartheta_j\ge0$, weakly converges to the Dirac mass $\partiallta_0$, then
\begin{equation}\label{eq:afp_func_meas_limit_time}
\lim_{j\to+\infty}\int_\mathfrak{m}athbb{G}\left\|\frac{\vartheta_j*(\varrho_\cdot v_\cdot)(t)}{\vartheta_j*\varrho_\cdot(t)}\right\|_\mathfrak{m}athbb{G}^2\,\vartheta_j*\varrho_\cdot(t)\ dx
=\int_\mathfrak{m}athbb{G}\|v_t\|_\mathfrak{m}athbb{G}^2\ d\mathfrak{m}u_t
\qquad
\text{for a.e.}\ t\in\mathfrak{m}athbb{R}.
\end{equation}
\end{lemma}
\begin{proof}
Inequality~\eqref{eq:afp_func_meas_time} follows from~\eqref{eq:afp_Jensen} in the same way of~\eqref{eq:afp_func_meas}, so we omit the details. For~\eqref{eq:afp_func_meas_limit_time}, set $\mathfrak{m}u^j_t=\vartheta_j*\mathfrak{m}u_\cdot(t)$ and $\nu^j_t=\vartheta_j*(v_\cdot\mathfrak{m}u_\cdot)(t)$ for all $t\in\mathfrak{m}athbb{R}$ and $j\in\mathfrak{m}athbb{N}$. Then $\|\nu^j_t\|_\mathfrak{m}athbb{G}\ll\mathfrak{m}u^j_t$ and $\nu^j_t\rightharpoonup\nu_t=v_t\mathfrak{m}u_t$ for a.e.\ $t\in\mathfrak{m}athbb{R}$, so that
\begin{equation*}
\liminf_{j\to+\infty}\int_\mathfrak{m}athbb{G}\left\|\frac{\vartheta_j*(\varrho_\cdot v_\cdot)(t)}{\vartheta_j*\varrho_\cdot(t)}\right\|_\mathfrak{m}athbb{G}^2\,\vartheta_j*\varrho_\cdot(t)\ dx
=\liminf_{j\to+\infty}\int_\mathfrak{m}athbb{G}\left\|\frac{\nu^j_t}{\mathfrak{m}u^j_t}\right\|_\mathfrak{m}athbb{G}^2 d\mathfrak{m}u^j_t
\ge\int_\mathfrak{m}athbb{G}\left\|\frac{\nu_t}{\mathfrak{m}u_t}\right\|_\mathfrak{m}athbb{G}^2 d\mathfrak{m}u_t
\end{equation*}
for a.e.\ $t\in\mathfrak{m}athbb{R}$ by Theorem~2.34 and Example~2.36 in~\cite{AFP00}.
\end{proof}
The following lemma is an elementary result relating weak convergence and convergence of scalar products of vector fields. We prove it here for the reader's convenience.
\begin{lemma}\label{lemma:polarization}
For $k\in\mathfrak{m}athbb{N}$, let $\mathfrak{m}u_k,\mathfrak{m}u\in\mathfrak{m}athcal{P}(\mathfrak{m}athbb{G})$ and let $v_k,w_k,v,w\colon\mathfrak{m}athbb{G}\to T\mathfrak{m}athbb{G}$ be Borel vector fields. Assume that $\mathfrak{m}u_k\rightharpoonup\mathfrak{m}u$, $v_k\mathfrak{m}u_k\rightharpoonup v\mathfrak{m}u$ and $w_k\mathfrak{m}u_k\rightharpoonup w\mathfrak{m}u$ as $k\to+\infty$. If
\begin{equation*}
\limsup_{k\to+\infty}\int_\mathfrak{m}athbb{G}\|v_k\|_\mathfrak{m}athbb{G}^2\ d\mathfrak{m}u_k\le\int_\mathfrak{m}athbb{G}\|v\|_\mathfrak{m}athbb{G}^2\ d\mathfrak{m}u<+\infty
\qquad\text{and}\qquad
\limsup_{k\to+\infty}\int_\mathfrak{m}athbb{G}\|w_k\|_\mathfrak{m}athbb{G}^2\ d\mathfrak{m}u_k<+\infty,
\end{equation*}
then
\begin{equation}\label{eq:polarization}
\lim_{k\to+\infty}\int_\mathfrak{m}athbb{G}\scalar*{v_k,w_k}_\mathfrak{m}athbb{G}\ d\mathfrak{m}u_k=\int_\mathfrak{m}athbb{G}\scalar*{v,w}_\mathfrak{m}athbb{G}\ d\mathfrak{m}u.
\end{equation}
\end{lemma}
\begin{proof}
By lower semicontinuity, we know that $\lim\limits_{k\to+\infty}\int_\mathfrak{m}athbb{G}\|v_k\|_\mathfrak{m}athbb{G}^2\ d\mathfrak{m}u_k=\int_\mathfrak{m}athbb{G}\|v\|^2_\mathfrak{m}athbb{G}\ d\mathfrak{m}u$ and
\begin{equation*}
\liminf_{k\to+\infty}\int_\mathfrak{m}athbb{G}\|tv_k+w_k\|_\mathfrak{m}athbb{G}^2\ d\mathfrak{m}u_k
\geq
\int_\mathfrak{m}athbb{G}\|tv+w\|_\mathfrak{m}athbb{G}^2\ d\mathfrak{m}u
\qquad
\text{for all}\ t\in\mathfrak{m}athbb{R}.
\end{equation*}
Expanding the squares, we get
\begin{equation*}
\liminf_{k\to+\infty}\left(2t\int_\mathfrak{m}athbb{G}\scalar*{v_k,w_k}_\mathfrak{m}athbb{G}\ d\mathfrak{m}u_k+\int_\mathfrak{m}athbb{G}\|w_k\|_\mathfrak{m}athbb{G}^2\ d\mathfrak{m}u_k\right)
\geq
2t\int_\mathfrak{m}athbb{G}\scalar*{v,w}_\mathfrak{m}athbb{G}\ d\mathfrak{m}u
\qquad
\text{for all}\ t\in\mathfrak{m}athbb{R}.
\end{equation*}
Choosing $t>0$, dividing both sides by~$t$ and letting $t\to+\infty$ gives the $\liminf$ inequality in~\eqref{eq:polarization}. Choosing $t<0$, a similar argument gives the $\limsup$ inequality in~\eqref{eq:polarization}.
\end{proof}
We are now ready to prove the second part of \cref{th:main}.
\begin{theorem}\label{th:GF_ent_implies_heat_diff}
Let $\varrho_0\in L^1(\mathfrak{m}athbb{G})$ be such that $\mathfrak{m}u_0=\varrho_0\mathfrak{m}athcal{L}^n\in\dom(\mathfrak{m}athsf{Ent})$. If $(\mathfrak{m}u_t)_{t\ge0}$ is a gradient flow of $\mathfrak{m}athsf{Ent}$ in $(\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G}),\mathfrak{m}athsf{W}_\mathfrak{m}athbb{G})$ starting from~$\mathfrak{m}u_0$, then $\mathfrak{m}u_t=\varrho_t\mathfrak{m}athcal{L}^n$ for all $t\ge0$ and $(\varrho_t)_{t\ge0}$ solves the sub-elliptic heat equation $\partial_t\varrho_t=\mathfrak{m}athrm{D}elta_\mathfrak{m}athbb{G}\varrho_t$ with initial datum $\varrho_0$. In particular, $t\mathfrak{m}apsto\mathfrak{m}athsf{Ent}(\mathfrak{m}u_t)$ is locally absolutely continuous on $(0,+\infty)$.
\end{theorem}
\begin{proof}
By \cref{lemma:GF_ent_plus_AC_is_heat_diff}, we just need to show that the map $t\mathfrak{m}apsto\mathfrak{m}athsf{Ent}(\mathfrak{m}u_t)$ satisfies~\eqref{eq:GF_ent_plus_AC}. It is not restrictive to extend $(\mathfrak{m}u_t)_{t\ge0}$ in time to the whole~$\mathfrak{m}athbb{R}$ by setting $\mathfrak{m}u_t=\mathfrak{m}u_0$ for all $t\le0$. So from now on we assume $\mathfrak{m}u_t\in AC^2_{\rm loc}(\mathfrak{m}athbb{R};(\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G}),\mathfrak{m}athsf{W}_\mathfrak{m}athbb{G}))$. The time-dependent vector field $(v^\mathfrak{m}athbb{G})_{t>0}$ given by \cref{prop:CE} extends to the whole~$\mathfrak{m}athbb{R}$ accordingly. Note that $(\mathfrak{m}u_t)_{t\in\mathfrak{m}athbb{R}}$ is a gradient flow of~$\mathfrak{m}athsf{Ent}$ in the following sense: for each $h\in\mathfrak{m}athbb{R}$, $(\mathfrak{m}u_{t+h})_{t\ge0}$ is a gradient flow on~$\mathfrak{m}athsf{Ent}$ starting from~$\mathfrak{m}u_h$. By \cref{def:metric_GF}, we get $t\mathfrak{m}apsto|\mathfrak{m}athrm{D}_\mathfrak{m}athbb{G}^-\mathfrak{m}athsf{Ent}|(\mathfrak{m}u_t)\in L^2_{\rm loc}(\mathfrak{m}athbb{R})$ , so that $t\mathfrak{m}apsto\mathfrak{m}athsf{F}_\mathfrak{m}athbb{G}(\varrho_t)\in L^1_{\rm loc}(\mathfrak{m}athbb{R})$ by \cref{prop:slope_ent_G_general}.
We divide the proof in three main steps.
\mathfrak{m}edskip
\textit{Step~1: smoothing in the time variable}. Let $\vartheta\colon\mathfrak{m}athbb{R}\to\mathfrak{m}athbb{R}$ be a symmetric smooth mollifier in~$\mathfrak{m}athbb{R}$, i.e.\
\begin{equation*}
\vartheta\in C^\infty_c(\mathfrak{m}athbb{R}),\quad
\supp\vartheta\subset[-1,1],\quad
0\le\vartheta\le1,\quad
\int_{\mathfrak{m}athbb{R}}\vartheta(t)\ dt=1.
\end{equation*}
We set $\vartheta_j(t):=j\,\vartheta(jt)$ for all $t\in\mathfrak{m}athbb{R}$ and $j\in\mathfrak{m}athbb{N}$. We define
\begin{equation*}
\mathfrak{m}u_t^j:=\varrho^j_t\mathfrak{m}athcal{L}^n,\qquad \varrho^j_t:=(\vartheta_j*\varrho_\cdot)(t)=\int_\mathfrak{m}athbb{R}\vartheta_j(t-s)\,\varrho_s\ ds
\qquad
\forall t\in\mathfrak{m}athbb{R},\ \forall j\in\mathfrak{m}athbb{N}.
\end{equation*}
For any $s,t\in\mathfrak{m}athbb{R}$, let $\pi_{s,t}\in\mathfrak{m}athbb{G}amma_0(\mathfrak{m}u_s,\mathfrak{m}u_t)$ be an optimal coupling between~$\mathfrak{m}u_s$ and~$\mathfrak{m}u_t$. An easy computation shows that $\pi^j_t\in\mathfrak{m}athcal{P}(\mathfrak{m}athbb{G}\times\mathfrak{m}athbb{G})$ given by
\begin{equation*}
\int_{\mathfrak{m}athbb{G}\times\mathfrak{m}athbb{G}}\varphi(x,y)\ d\pi^j_t(x,y)=\int_\mathfrak{m}athbb{R}\vartheta_j(t-s)\int_{\mathfrak{m}athbb{G}\times\mathfrak{m}athbb{G}}\varphi(x,y)\ d\pi_{s,t}(x,y)\,ds,
\end{equation*}
for any $\varphi\colon\mathfrak{m}athbb{G}\times\mathfrak{m}athbb{G}\to[0,+\infty)$ Borel, is a coupling between~$\mathfrak{m}u^j_t$ and~$\mathfrak{m}u_t$. Hence we get
\begin{equation*}
\mathfrak{m}athsf{W}_\mathfrak{m}athbb{G}(\mathfrak{m}u^j_t,\mathfrak{m}u_t)^2\le\int_\mathfrak{m}athbb{R}\vartheta_j(t-s)\,\mathfrak{m}athsf{W}_\mathfrak{m}athbb{G}(\mathfrak{m}u_s,\mathfrak{m}u_t)^2\ ds
\qquad
\forall t\in\mathfrak{m}athbb{R},\ \forall j\in\mathfrak{m}athbb{N}.
\end{equation*}
Therefore $\lim_{j\to+\infty}\mathfrak{m}athsf{W}_\mathfrak{m}athbb{G}(\mathfrak{m}u_t^j,\mathfrak{m}u_t)=0$ for all $t\in\mathfrak{m}athbb{R}$. This implies that $\mathfrak{m}u^j_t\rightharpoonup\mathfrak{m}u_t$ as $j\to+\infty$ and
\begin{equation}\label{eq:2nd_moment_mu_j}
\lim_{j\to+\infty}\int_\mathfrak{m}athbb{G}\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}}(x,0)^2\ d\mathfrak{m}u_t^j(x)=\int_\mathfrak{m}athbb{G}\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}}(x,0)^2\ d\mathfrak{m}u_t(x)
\qquad
\forall t\in\mathfrak{m}athbb{R}.
\end{equation}
In particular, $(\mathfrak{m}u^j_t)_{t\in\mathfrak{m}athbb{R}}\subset\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G})$, $\mathfrak{m}athsf{Ent}(\mathfrak{m}u^j_t)>-\infty$ for all $j\in\mathfrak{m}athbb{N}$ and
\begin{equation}\label{eq:mu_j_liminf}
\liminf_{j\to+\infty}\mathfrak{m}athsf{Ent}(\mathfrak{m}u^j_t)\ge\mathfrak{m}athsf{Ent}(\mathfrak{m}u_t)
\qquad
\forall t\in\mathfrak{m}athbb{R}.
\end{equation}
We claim that
\begin{equation}\label{eq:mu_j_limsup}
\limsup_{j\to+\infty}\mathfrak{m}athsf{Ent}(\mathfrak{m}u^j_t)\le\mathfrak{m}athsf{Ent}(\mathfrak{m}u_t)
\qquad
\text{for a.e.}\ t\in\mathfrak{m}athbb{R}.
\end{equation}
Indeed, define the new reference measure $\nu:=e^{-c\,\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}}^2(\cdot,0)}\mathfrak{m}athcal{L}^n$, where $c>0$ is chosen so that $\nu\in\mathfrak{m}athcal{P}(\mathfrak{m}athbb{G})$. Since the function $\mathfrak{m}athsf{h}at{H}(r):=r\log r+(1-r)$, for $r\ge0$, is convex and non-negative, by Jensen's inequality we have
\begin{align*}
\mathfrak{m}athsf{Ent}_\nu(\mathfrak{m}u^j_t)&
=\int_\mathfrak{m}athbb{G}\mathfrak{m}athsf{h}at{H}\left(e^{c\,\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}}^2(\cdot,0)}\,\vartheta_j*\varrho_\cdot(t)\right)\,d\nu
=\int_\mathfrak{m}athbb{G}\mathfrak{m}athsf{h}at{H}\left(\vartheta_j*\left(e^{c\,\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}}^2(\cdot,0)}\varrho_\cdot\right)(t)\right)\,d\nu\\
&\le\int_\mathfrak{m}athbb{G}\vartheta_j*\mathfrak{m}athsf{h}at{H}\left(e^{c\,\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}}^2(\cdot,0)}\varrho_\cdot\right)(t)\,d\nu
=\vartheta_j*\mathfrak{m}athsf{Ent}_\nu(\mathfrak{m}u_\cdot)(t)
\qquad
\forall t\in\mathfrak{m}athbb{R},\ \forall j\in\mathfrak{m}athbb{N}.
\end{align*}
Therefore $\limsup_{j\to+\infty}\mathfrak{m}athsf{Ent}_\nu(\mathfrak{m}u^j_t)\le\mathfrak{m}athsf{Ent}_\nu(\mathfrak{m}u_t)$ for a.e.\ $t\in\mathfrak{m}athbb{R}$. Thus~\eqref{eq:mu_j_limsup} follows by~\eqref{eq:ent_useful_formula} and~\eqref{eq:2nd_moment_mu_j}. Combining~\eqref{eq:mu_j_liminf} and~\eqref{eq:mu_j_limsup}, we get
\begin{equation}\label{eq:mu_j_lim}
\lim_{j\to+\infty}\mathfrak{m}athsf{Ent}(\mathfrak{m}u^j_t)=\mathfrak{m}athsf{Ent}(\mathfrak{m}u_t)
\qquad
\text{for a.e.}\ t\in\mathfrak{m}athbb{R}.
\end{equation}
Let $(v^\mathfrak{m}athbb{G}_t)_{t\in\mathfrak{m}athbb{R}}$ be the horizontal time-dependent vector field relative to~$(\mathfrak{m}u_t)_{t\in\mathfrak{m}athbb{R}}$ given by \cref{prop:CE}. Let $(v^j_t)_{t\in\mathfrak{m}athbb{R}}$ be the horizontal time-dependent vector field given by
\begin{equation}\label{eq:def_v_j}
v^j_t=\frac{\vartheta_j*(\varrho_\cdot v_\cdot)(t)}{\varrho^j_t}
\qquad
\forall t\in\mathfrak{m}athbb{R}.
\end{equation}
We claim that $v_t^j\in L^2_\mathfrak{m}athbb{G}(\mathfrak{m}u_t^j)$ for all $t\in\mathfrak{m}athbb{R}$. Indeed, applying \cref{lemma:AFP_func_meas_time}, we get
\begin{equation*}
\int_\mathfrak{m}athbb{G}\|v^j_t\|_\mathfrak{m}athbb{G}^2\ d\mathfrak{m}u^j_t
\le\vartheta_j*\left(\int_\mathfrak{m}athbb{G}\|v_\cdot\|_\mathfrak{m}athbb{G}^2\ d\mathfrak{m}u_\cdot\right)(t)
=\vartheta_j*|\dot{\mathfrak{m}u}_\cdot|_\mathfrak{m}athbb{G}^2(t)
\qquad
\forall t\in\mathfrak{m}athbb{R}.
\end{equation*}
We also claim that $(\mathfrak{m}u^j_t)_{t\in\mathfrak{m}athbb{R}}$ solves $\partial_t\mathfrak{m}u^j_t+\diverg(v^j_t\mathfrak{m}u^j_t)=0$ in the sense of distributions for all $j\in\mathfrak{m}athbb{N}$. Indeed, if $\varphi\in C^\infty_c(\mathfrak{m}athbb{R}\times\mathfrak{m}athbb{R}^n)$, then also $\varphi^j:=\vartheta_j*\varphi\in C^\infty_c(\mathfrak{m}athbb{R}\times\mathfrak{m}athbb{R}^n)$, so that
\begin{align*}
&\int_\mathfrak{m}athbb{R}\int_\mathfrak{m}athbb{G}\partial_t\varphi^j(t,x)+\scalar*{\nabla_\mathfrak{m}athbb{G}\varphi^j(t,x),v^j_t(x)}_\mathfrak{m}athbb{G}\,d\mathfrak{m}u^j_t(x)\,dt=\\
&\mathfrak{m}athsf{h}space*{1cm}=\int_\mathfrak{m}athbb{R}\vartheta_j*\left(\int_\mathfrak{m}athbb{G}\partial_t\varphi(\cdot,x)+\scalar*{\nabla_\mathfrak{m}athbb{G}\varphi(\cdot,x),v_\cdot(x)}_\mathfrak{m}athbb{G}\ d\mathfrak{m}u_\cdot(x)\right)(t)\,dt\\
&\mathfrak{m}athsf{h}space*{1cm}=\int_\mathfrak{m}athbb{R}\int_\mathfrak{m}athbb{G}\partial_t\varphi(t,x)+\scalar*{\nabla_\mathfrak{m}athbb{G}\varphi(t,x),v_t(x)}_\mathfrak{m}athbb{G}\,d\mathfrak{m}u_t(x)\,dt=0
\qquad
\forall j\in\mathfrak{m}athbb{N}.
\end{align*}
By \cref{prop:CE}, we conclude that $(\mathfrak{m}u^j_t)_t\in AC^2_{\rm loc}(\mathfrak{m}athbb{R};(\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G}),\mathfrak{m}athsf{W}_\mathfrak{m}athbb{G}))$ with $|\dot{\mathfrak{m}u}^j_t|_\mathfrak{m}athbb{G}^2\le\vartheta_j*|\dot{\mathfrak{m}u}_\cdot|_\mathfrak{m}athbb{G}^2(t)$ for all $t\in\mathfrak{m}athbb{R}$ and $j\in\mathfrak{m}athbb{N}$.
Finally, we claim that $\mathfrak{m}athsf{F}_\mathfrak{m}athbb{G}(\varrho^j_t)\le\vartheta_j*\mathfrak{m}athsf{F}_\mathfrak{m}athbb{G}(\varrho_\cdot)(t)$ for all $t\in\mathfrak{m}athbb{R}$ and $j\in\mathfrak{m}athbb{N}$. Indeed, arguing as in~\eqref{eq:C-S_for_horizontal_grad}, by Cauchy--Schwarz inequality we have
\begin{equation*}
\|\nabla_\mathfrak{m}athbb{G}\varrho^j_t\|_\mathfrak{m}athbb{G}^2
\le\left[\vartheta_j*\left(\mathfrak{m}athsf{Ch}i_{\set*{\varrho_.>0}}\sqrt{\varrho_\cdot}\,\frac{\|\nabla_\mathfrak{m}athbb{G}\varrho_\cdot\|_\mathfrak{m}athbb{G}}{\sqrt{\varrho_\cdot}}\right)\right]^2(t)
\le\varrho^j_t\,\vartheta_j*\left(\frac{\|\nabla_\mathfrak{m}athbb{G}\varrho_\cdot\|_\mathfrak{m}athbb{G}^2}{\varrho_\cdot}\mathfrak{m}athsf{Ch}i_{\set*{\varrho_\cdot>0}}\right)(t),
\end{equation*}
so that
\begin{equation*}
\mathfrak{m}athsf{F}_\mathfrak{m}athbb{G}(\varrho^j_t)\le\vartheta_j*\left(\int_{\set*{\varrho_\cdot>0}}\frac{\|\nabla_\mathfrak{m}athbb{G}\varrho_\cdot\|_\mathfrak{m}athbb{G}^2}{\varrho_\cdot}\ dx\right)(t)=\vartheta_j*\mathfrak{m}athsf{F}_\mathfrak{m}athbb{G}(\varrho_\cdot)(t).
\end{equation*}
\mathfrak{m}edskip
\textit{Step~2: smoothing in the space variable}.
Let $j\in\mathfrak{m}athbb{N}$ be fixed. Let $\eta\colon\mathfrak{m}athbb{G}\to\mathfrak{m}athbb{R}$ be a symmetric smooth mollifier in $\mathfrak{m}athbb{G}$, i.e.\ a function $\eta\in C^\infty_c(\mathfrak{m}athbb{R}^n)$ such that
\begin{equation*}
\supp\eta\subset B_1,\quad
0\le\eta\le1,\quad
\eta(x^{-1})=\eta(x)\ \forall x\in\mathfrak{m}athbb{G},\quad
\int_{\mathfrak{m}athbb{G}}\eta(x)\ dx=1.
\end{equation*}
We set $\eta_k(x):=k^Q\,\eta(\partiallta_k(x))$ for all $x\in\mathfrak{m}athbb{G}$ and $k\in\mathfrak{m}athbb{N}$. We define
\begin{equation*}
\mathfrak{m}u_t^{j,k}:=\varrho^{j,k}_t\mathfrak{m}athcal{L}^n,\qquad \varrho^{j,k}_t(x):=\eta_k\star\varrho^j_t(x)=\int_\mathfrak{m}athbb{G}\eta_k(xy^{-1})\varrho^j_t(y)\ dy, \quad x\in\mathfrak{m}athbb{G},
\end{equation*}
for all $t\in\mathfrak{m}athbb{R}$ and $k\in\mathfrak{m}athbb{N}$. Note that
\begin{equation}\label{eq:mu_k_left_translation}
\mathfrak{m}u^{j,k}_t=\int_\mathfrak{m}athbb{G} (l_y)_\#\mathfrak{m}u_t^j\ \eta_k(y)dy
\qquad
\forall t\in\mathfrak{m}athbb{R},\ \forall k\in\mathfrak{m}athbb{N},
\end{equation}
where $l_y(x)=yx$, $x,y\in\mathfrak{m}athbb{G}$, denotes the left-translation.
Note that $(\mathfrak{m}u_t^{j,k})_{t\in\mathfrak{m}athbb{R}}\subset\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G})$ for all $k\in\mathfrak{m}athbb{N}$. Indeed, arguing as in~\eqref{eq:heat_second_moment}, we have
\begin{equation}\label{eq:2nd_moment_mu_k}
\begin{split}
\int_\mathfrak{m}athbb{G}\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}}(x,0)^2\ d\mathfrak{m}u^{j,k}_t(x)
&=\int_\mathfrak{m}athbb{G}\left(\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}}(\cdot,0)^2\star\eta_k\right)(x)\ d\mathfrak{m}u^j_t(x)\\
&\le 2\int_\mathfrak{m}athbb{G}\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}}(x,0)^2\ d\mathfrak{m}u^j_t(x)+2\int_\mathfrak{m}athbb{G}\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}}(x,0)^2\,\eta_k(x)\,dx
\end{split}
\end{equation}
for all $t\in\mathfrak{m}athbb{R}$. In particular, $\mathfrak{m}athsf{Ent}(\mathfrak{m}u^{j,k}_t)>-\infty$ for all $t\in\mathfrak{m}athbb{R}$ and $k\in\mathfrak{m}athbb{N}$. Clearly $\mathfrak{m}u^{j,k}_t\rightharpoonup\mathfrak{m}u^j_t$ as $k\to+\infty$ for each fixed $t\in\mathfrak{m}athbb{R}$, so that
\begin{equation}\label{eq:mu_k_liminf}
\liminf_{k\to+\infty}\mathfrak{m}athsf{Ent}(\mathfrak{m}u^{j,k}_t)\ge\mathfrak{m}athsf{Ent}(\mathfrak{m}u^j_t)
\qquad
\forall t\in\mathfrak{m}athbb{R}.
\end{equation}
Moreover, we claim that
\begin{equation}\label{eq:mu_k_limsup}
\limsup_{k\to+\infty}\mathfrak{m}athsf{Ent}(\mathfrak{m}u^{j,k}_t)\le\mathfrak{m}athsf{Ent}(\mathfrak{m}u^j_t)
\qquad
\forall t\in\mathfrak{m}athbb{R}.
\end{equation}
Indeed, let $\nu\in\mathfrak{m}athcal{P}(\mathfrak{m}athbb{G})$ and $\mathfrak{m}athsf{h}at{H}$ as in \textit{Step~1}. Recalling~\eqref{eq:mu_k_left_translation}, by Jensen's inequality we get
\begin{equation}\label{eq:mu_k_ent_push-forward}
\mathfrak{m}athsf{Ent}_\nu(\mathfrak{m}u^{j,k}_t)\le\int_\mathfrak{m}athbb{G} \mathfrak{m}athsf{Ent}_\nu((l_y)_\#\mathfrak{m}u_t^j)\ \eta_k(y)dy.
\end{equation}
Define $\nu_y:=(l_y)_\#\nu$ and note that $\nu_y=e^{-c\,\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}}(\cdot,y)}\mathfrak{m}athcal{L}^n$ for all $y\in\mathfrak{m}athbb{G}$. Thus by~\eqref{eq:ent_push-forward_formula} we have
\begin{equation*}
\mathfrak{m}athsf{Ent}_\nu((l_y)_\#\mathfrak{m}u_t^j)=\mathfrak{m}athsf{Ent}_{\nu_{y^{-1}}}(\mathfrak{m}u_t^j)=\mathfrak{m}athsf{Ent}(\mathfrak{m}u^j_t)+c\int_\mathfrak{m}athbb{G}\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}}(yx,0)^2\ d\mathfrak{m}u^j_t(x)
\qquad
\forall y\in\mathfrak{m}athbb{G}.
\end{equation*}
By the dominated convergence theorem we get that $y\mathfrak{m}apsto\mathfrak{m}athsf{Ent}_\nu((l_y)_\#\mathfrak{m}u_t^j)$ is continuous and therefore~\eqref{eq:mu_k_limsup} follows by passing to the limit as $k\to+\infty$ in~\eqref{eq:mu_k_ent_push-forward}. Combining~\eqref{eq:mu_k_liminf} and~\eqref{eq:mu_k_limsup}, we get
\begin{equation}\label{eq:mu_k_lim}
\lim_{k\to+\infty}\mathfrak{m}athsf{Ent}(\mathfrak{m}u^{j,k}_t)=\mathfrak{m}athsf{Ent}(\mathfrak{m}u^j_t)
\qquad
\forall t\in\mathfrak{m}athbb{R}.
\end{equation}
Let $(v^j_t)_{t\in\mathfrak{m}athbb{R}}$ be as in~\eqref{eq:def_v_j} and let $(v^{j,k}_t)_{t\in\mathfrak{m}athbb{R}}$ be the horizontal time-dependent vector field given by
\begin{equation}\label{eq:def_v_k}
v^{j,k}_t=\frac{\eta_k\star(\varrho^j_t v^j_t)}{\varrho^{j,k}_t}
\qquad
\forall t\in\mathfrak{m}athbb{R},\ \forall k\in\mathfrak{m}athbb{N}.
\end{equation}
We claim that $v_t^{j,k}\in L^2_\mathfrak{m}athbb{G}(\mathfrak{m}u_t^{j,k})$ for all $t\in\mathfrak{m}athbb{R}$. Indeed, applying \cref{lemma:AFP_func_meas}, we get
\begin{equation}\label{eq:mu_k_velocity}
\int_\mathfrak{m}athbb{G}\|v^{j,k}_t\|_\mathfrak{m}athbb{G}^2\ d\mathfrak{m}u^{j,k}_t\le\int_\mathfrak{m}athbb{G}\|v^j_t\|_\mathfrak{m}athbb{G}^2\ d\mathfrak{m}u^j_t\le|\dot{\mathfrak{m}u}^j_t|_\mathfrak{m}athbb{G}^2
\qquad
\forall t\in\mathfrak{m}athbb{R},\ \forall k\in\mathfrak{m}athbb{N}.
\end{equation}
We also claim that $(\mathfrak{m}u^{j,k}_t)_{t\in\mathfrak{m}athbb{R}}$ solves $\partial_t\mathfrak{m}u^{j,k}_t+\diverg(v^{j,k}_t\mathfrak{m}u^{j,k}_t)=0$ in the sense of distributions for all $k\in\mathfrak{m}athbb{N}$. Indeed, if $\varphi\in C^\infty_c(\mathfrak{m}athbb{R}\times\mathfrak{m}athbb{R}^n)$, then also $\varphi^k:=\eta_k\star\varphi\in C^\infty_c(\mathfrak{m}athbb{R}\times\mathfrak{m}athbb{R}^n)$, so that
\begin{align*}
&\int_\mathfrak{m}athbb{R}\int_\mathfrak{m}athbb{G}\partial_t\varphi^k(t,x)+\scalar*{\nabla_\mathfrak{m}athbb{G}\varphi^k(t,x),v^{j,k}_t(x)}_\mathfrak{m}athbb{G}\,d\mathfrak{m}u^{j,k}_t(x)\,dt=\\
&\mathfrak{m}athsf{h}space*{1cm}=\int_\mathfrak{m}athbb{G}\eta_k\star\left(\int_\mathfrak{m}athbb{R}\partial_t\varphi(t,\cdot)+\scalar*{\nabla_\mathfrak{m}athbb{G}\varphi(t,\cdot),v^j_t(\cdot)}_\mathfrak{m}athbb{G}\,\varrho^j_t(\cdot)\,dt\right)(x)\ dx\\
&\mathfrak{m}athsf{h}space*{1cm}=\int_\mathfrak{m}athbb{R}\int_\mathfrak{m}athbb{G}\partial_t\varphi(t,x)+\scalar*{\nabla_\mathfrak{m}athbb{G}\varphi(t,x),v^j_t(x)}_\mathfrak{m}athbb{G}\,d\mathfrak{m}u^j_t(x)\,dt=0
\qquad
\forall k\in\mathfrak{m}athbb{N}.
\end{align*}
Here we have exploited a key property of the space $(\mathfrak{m}athbb{G},\mathfrak{m}athsf{d}_{\mathfrak{m}athsf{cc}},\mathfrak{m}athcal{L}^n)$ which cannot be expected in a general metric measure space, that is, the continuity equation in~\eqref{eq:CE} is preserved under regularization in the space variable. By \cref{prop:CE} and~\eqref{eq:mu_k_velocity}, we conclude that $(\mathfrak{m}u^{j,k}_t)_t\in AC^2_{\rm loc}(\mathfrak{m}athbb{R};(\mathfrak{m}athcal{P}_2(\mathfrak{m}athbb{G}),\mathfrak{m}athsf{W}_\mathfrak{m}athbb{G}))$ with $|\dot{\mathfrak{m}u}^{j,k}_t|_\mathfrak{m}athbb{G}\le\|v^{j,k}_t\|_{L^2_\mathfrak{m}athbb{G}(\mathfrak{m}u^{j,k}_t)}\le\|v^j_t\|_{L^2_\mathfrak{m}athbb{G}(\mathfrak{m}u^j_t)}\le|\dot{\mathfrak{m}u}^j_t|_\mathfrak{m}athbb{G}$ for all $t\in\mathfrak{m}athbb{R}$ and $k\in\mathfrak{m}athbb{N}$.
Finally, we claim that $\mathfrak{m}athsf{F}_\mathfrak{m}athbb{G}(\varrho^{j,k}_t)\le\mathfrak{m}athsf{F}_\mathfrak{m}athbb{G}(\varrho^j_t)$ for all $t\in\mathfrak{m}athbb{R}$ and $k\in\mathfrak{m}athbb{N}$. Indeed, arguing as in~\eqref{eq:C-S_for_horizontal_grad}, by Cauchy--Schwarz inequality we have
\begin{equation*}
\|\nabla_\mathfrak{m}athbb{G}\varrho^{j,k}_t\|_\mathfrak{m}athbb{G}^2
\le\left[\eta_k\star\left(\mathfrak{m}athsf{Ch}i_{\set*{\varrho^j_t>0}}\sqrt{\varrho^j_t}\,\frac{\|\nabla_\mathfrak{m}athbb{G}\varrho^j_t\|_\mathfrak{m}athbb{G}}{\sqrt{\varrho^j_t}}\right)\right]^2
\le\varrho^{j,k}_t\,\eta_k\star\left(\frac{\|\nabla_\mathfrak{m}athbb{G}\varrho^j_t\|_\mathfrak{m}athbb{G}^2}{\varrho^j_t}\mathfrak{m}athsf{Ch}i_{\set*{\varrho^j_t>0}}\right),
\end{equation*}
so that
\begin{align*}
\mathfrak{m}athsf{F}_\mathfrak{m}athbb{G}(\varrho^{j,k}_t)
\le\int_\mathfrak{m}athbb{G}\eta_k\star\left(\frac{\|\nabla_\mathfrak{m}athbb{G}\varrho^j_t\|_\mathfrak{m}athbb{G}^2}{\varrho^j_t}\mathfrak{m}athsf{Ch}i_{\set*{\varrho^j_t>0}}\right)\ dx
=\int_{\set*{\varrho^j_t>0}}\frac{\|\nabla_\mathfrak{m}athbb{G}\varrho^j_t\|_\mathfrak{m}athbb{G}^2}{\varrho^j_t}\ dx
=\mathfrak{m}athsf{F}_\mathfrak{m}athbb{G}(\varrho^j_t).
\end{align*}
\mathfrak{m}edskip
\textit{Step~3: truncated entropy}.
Let $j,k\in\mathfrak{m}athbb{N}$ be fixed. For any $m\in\mathfrak{m}athbb{N}$, consider the maps $z_m,H_m\colon[0,+\infty)\to\mathfrak{m}athbb{R}$ defined in~\eqref{eq:def_truncated_entropy}. We set $\tilde{z}_m(r)=z_m(r)+m$ for all $r\ge0$ and $m\in\mathfrak{m}athbb{N}$. Since $\varrho^{j,k}_t\in\mathfrak{m}athcal{P}(\mathfrak{m}athbb{G})$ for all $t\in\mathfrak{m}athbb{R}$, differentiating under the integral sign we get
\begin{equation*}
\frac{d}{dt}\int_\mathfrak{m}athbb{G} H_m(\varrho^{j,k}_t)\ dx
=\int_\mathfrak{m}athbb{G} \tilde{z}_m(\varrho^{j,k}_t)\,\partial_t\varrho^{j,k}_t\ dx
\end{equation*}
for all $t\in\mathfrak{m}athbb{R}$. Fix $t_0,t_1\in\mathfrak{m}athbb{R}$ with $t_0<t_1$. Then
\begin{equation}\label{eq:truncated_ent_before_AC}
\int_\mathfrak{m}athbb{G} H_m(\varrho^{j,k}_{t_1})\ dx-\int_\mathfrak{m}athbb{G} H_m(\varrho^{j,k}_{t_0})\ dx
=\int_{t_0}^{t_1}\int_\mathfrak{m}athbb{G} \tilde{z}_m(\varrho^{j,k}_t)\,\partial_t\varrho^{j,k}_t\,dxdt.
\end{equation}
Let $(\alpha_i)_{i\in\mathfrak{m}athbb{N}}\subset C^\infty_c(t_0,t_1)$ such that $0\le\alpha_i\le1$ and $\alpha_i\to\mathfrak{m}athsf{Ch}i_{(t_0,t_1)}$ in $L^1(\mathfrak{m}athbb{R})$ as $i\to+\infty$. Let $i\in\mathfrak{m}athbb{N}$ be fixed and consider the function $u_t(x)=\tilde{z}_m(\varrho^{j,k}_t(x))\,\alpha_i(t)$ for all $(t,x)\in\mathfrak{m}athbb{R}\times\mathfrak{m}athbb{R}^n$. We claim that there exists $(\psi^h)_{h\in\mathfrak{m}athbb{N}}\subset C^\infty_c(\mathfrak{m}athbb{R}^{n+1})$ such that
\begin{equation}\label{eq:approx_for_AC}
\lim_{h\to+\infty}\int_\mathfrak{m}athbb{R}\int_\mathfrak{m}athbb{G}\,\abs{u_t(x)-\psi^h_t(x)}^2+\|\nabla_\mathfrak{m}athbb{G} u_t(x)-\nabla_\mathfrak{m}athbb{G}\psi^h_t(x)\|_\mathfrak{m}athbb{G}^2\ dx\,dt=0.
\end{equation}
Indeed, consider the direct product $\mathfrak{m}athbb{G}^*=\mathfrak{m}athbb{R}\times\mathfrak{m}athbb{G}$ and note that~$\mathfrak{m}athbb{G}^*$ is a Carnot group. Recalling~\eqref{eq:def_horiz_sobolev_space}, we know that $C^\infty_c(\mathfrak{m}athbb{R}^{n+1})$ is dense in $W^{1,2}_{\mathfrak{m}athbb{G}^*}(\mathfrak{m}athbb{R}^{n+1})$. Thus, to get~\eqref{eq:approx_for_AC} we just need to prove that $u\in W^{1,2}_{\mathfrak{m}athbb{G}^*}(\mathfrak{m}athbb{R}^{n+1})$ (in fact, the $L^2$-integrability of $\partial_t u$ is not strictly necessary in order to achieve~\eqref{eq:approx_for_AC}). We have $\varrho^{j,k},\partial_t\varrho^{j,k}\in L^\infty(\mathfrak{m}athbb{R}^{n+1})$, because
\begin{equation*}
\|\varrho^{j,k}\|_{L^\infty(\mathfrak{m}athbb{R}^{n+1})}\le\|\eta_k\|_{L^\infty(\mathfrak{m}athbb{R}^n)},
\qquad
\|\partial_t\varrho^{j,k}\|_{L^\infty(\mathfrak{m}athbb{R}^{n+1})}\le\|\vartheta_j'\|_{L^1(\mathfrak{m}athbb{R})}\|\eta_k\|_{L^\infty(\mathfrak{m}athbb{R}^n)}
\end{equation*}
by Young's inequality. Moreover, $\varrho^{j,k}\alpha_i,\partial_t\varrho^{j,k}\alpha_i\in L^1(\mathfrak{m}athbb{R}^{n+1})$, because
\begin{equation*}
\|\varrho^{j,k}\alpha_i\|_{L^1(\mathfrak{m}athbb{R}^{n+1})}=\|\alpha_i\|_{L^1(\mathfrak{m}athbb{R})},
\qquad
\|\partial_t\varrho^{j,k}\alpha_i\|_{L^1(\mathfrak{m}athbb{R}^{n+1})}\le\|\vartheta_j'\|_{L^1(\mathfrak{m}athbb{R})}\|\alpha_i\|_{L^1(\mathfrak{m}athbb{R})}.
\end{equation*}
Therefore $\varrho^{j,k}\alpha_i,\partial_t\varrho^{j,k}\alpha_i\in L^2(\mathfrak{m}athbb{R}^{n+1})$, which immediately gives $u\in L^2(\mathfrak{m}athbb{R}^{n+1})$. Now by \textit{Step~2} we have that
\begin{equation*}
\int_\mathfrak{m}athbb{G}\|\nabla_\mathfrak{m}athbb{G}\varrho^{j,k}_t\|_\mathfrak{m}athbb{G}^2\ dx
\le\|\varrho^{j,k}_t\|_{L^\infty(\mathfrak{m}athbb{R}^n)}\mathfrak{m}athsf{F}_\mathfrak{m}athbb{G}(\varrho^{j,k}_t)
\le\|\varrho^{j,k}_t\|_{L^\infty(\mathfrak{m}athbb{R}^n)}\mathfrak{m}athsf{F}_\mathfrak{m}athbb{G}(\varrho^j_t)
\qquad
\forall t\in\mathfrak{m}athbb{R},
\end{equation*}
so that by \textit{Step~1} we get
\begin{equation*}
\|\nabla_\mathfrak{m}athbb{G}\varrho^{j,k}\alpha_i\|_{L^2(\mathfrak{m}athbb{R}^{n+1})}^2
\le\|\eta_k\|_{L^\infty(\mathfrak{m}athbb{R}^n)}
\|\alpha_i^2\cdot\vartheta_j*\mathfrak{m}athsf{F}_\mathfrak{m}athbb{G}(\varrho_\cdot)\|_{L^1(\mathfrak{m}athbb{R})}.
\end{equation*}
This prove that $\|\nabla_\mathfrak{m}athbb{G} u\|_\mathfrak{m}athbb{G}\in L^2(\mathfrak{m}athbb{R}^{n+1})$. The previous estimates easily imply that also $\partial_t u\in L^2(\mathfrak{m}athbb{R}^{n+1})$. This concludes the proof of~\eqref{eq:approx_for_AC}.
Since $\varrho^{j,k},\partial_t\varrho^{j,k}\in L^\infty(\mathfrak{m}athbb{R}^{n+1})$, by~\eqref{eq:approx_for_AC} we get that
\begin{equation}\label{eq:limit_trunc_ent_1}
\lim_{h\to+\infty}\int_\mathfrak{m}athbb{R}\int_\mathfrak{m}athbb{G}\psi^h_t\,\partial_t\varrho^{j,k}_t\,dx dt
=\int_\mathfrak{m}athbb{R}\alpha_i(t)\int_\mathfrak{m}athbb{G} \tilde{z}_m(\varrho^{j,k}_t)\,\partial_t\varrho^{j,k}_t\, dxdt
\end{equation}
and
\begin{equation}\label{eq:limit_trunc_ent_2}
\lim_{h\to+\infty}\int_\mathfrak{m}athbb{R}\int_\mathfrak{m}athbb{G}\scalar*{\nabla_\mathfrak{m}athbb{G}\psi^h_t,v^{j,k}_t}_\mathfrak{m}athbb{G}\ d\mathfrak{m}u^{j,k}_t\,dt
=\int_\mathfrak{m}athbb{R}\alpha_i(t)\int_\mathfrak{m}athbb{G} \tilde{z}_m'(\varrho^{j,k}_t)\,\scalar*{\nabla_\mathfrak{m}athbb{G}\varrho^{j,k}_t,v^{j,k}_t}_\mathfrak{m}athbb{G}\,d\mathfrak{m}u^{j,k}_t\,dt
\end{equation}
for each fixed $i\in\mathfrak{m}athbb{N}$. Since $\partial_t\mathfrak{m}u^{j,k}_t+\diverg(v^{j,k}_t\mathfrak{m}u^{j,k}_t)=0$ in the sense of distributions by \textit{Step~2}, for each $h\in\mathfrak{m}athbb{N}$ we have
\begin{equation*}
\int_\mathfrak{m}athbb{R}\int_\mathfrak{m}athbb{G}\psi^h_t\,\partial_t\varrho^{j,k}_t\,dx dt
=-\int_\mathfrak{m}athbb{R}\int_\mathfrak{m}athbb{G}\partial_t\psi^h_t\ d\mathfrak{m}u^{j,k}_t\,dt
=\int_\mathfrak{m}athbb{R}\int_\mathfrak{m}athbb{G}\scalar*{\nabla_\mathfrak{m}athbb{G}\psi^h_t,v^{j,k}_t}_\mathfrak{m}athbb{G}\,d\mathfrak{m}u^{j,k}_t\,dt.
\end{equation*}
We can thus combine~\eqref{eq:limit_trunc_ent_1} and~\eqref{eq:limit_trunc_ent_2} to get
\begin{equation}\label{eq:trunc_ent_i}
\int_\mathfrak{m}athbb{R}\alpha_i(t)\int_\mathfrak{m}athbb{G} \tilde{z}_m(\varrho^{j,k}_t)\,\partial_t\varrho^{j,k}_t\, dxdt
=\int_\mathfrak{m}athbb{R}\alpha_i(t)\int_\mathfrak{m}athbb{G} \tilde{z}_m'(\varrho^{j,k}_t)\,\scalar*{\nabla_\mathfrak{m}athbb{G}\varrho^{j,k}_t,v^{j,k}_t}_\mathfrak{m}athbb{G}\,d\mathfrak{m}u^{j,k}_t\,dt.
\end{equation}
Passing to the limit as $i\to+\infty$ in~\eqref{eq:trunc_ent_i}, we finally get that
\begin{equation}\label{eq:limit_through_AC}
\int_{t_0}^{t_1}\int_\mathfrak{m}athbb{G} \tilde{z}_m(\varrho^{j,k}_t)\,\partial_t\varrho^{j,k}_t\, dxdt
=\int_{t_0}^{t_1}\int_\mathfrak{m}athbb{G} \tilde{z}_m'(\varrho^{j,k}_t)\,\scalar*{\nabla_\mathfrak{m}athbb{G}\varrho^{j,k}_t,v^{j,k}_t}_\mathfrak{m}athbb{G}\,d\mathfrak{m}u^{j,k}_t\,dt
\end{equation}
by the dominated convergence theorem. Combining~\eqref{eq:truncated_ent_before_AC} and~\eqref{eq:limit_through_AC}, we get
\begin{equation}\label{eq:truncated_ent_after_AC}
\int_\mathfrak{m}athbb{G} H_m(\varrho^{j,k}_{t_1})\ dx-\int_\mathfrak{m}athbb{G} H_m(\varrho^{j,k}_{t_0})\ dx=\int_{t_0}^{t_1}\int_{\set*{e^{-m-1}<\varrho^{j,k}_t<e^{m-1}}}\scalar*{-w^{j,k}_t,v^{j,k}_t}_\mathfrak{m}athbb{G}\,d\mathfrak{m}u^{j,k}_t\,dt
\end{equation}
for all $t_0,t_1\in\mathfrak{m}athbb{R}$ with $t_0<t_1$, where $w^{j,k}_t=\nabla_\mathfrak{m}athbb{G}\varrho^{j,k}_t/\varrho^{j,k}_t$ in $L^2_\mathfrak{m}athbb{G}(\mathfrak{m}u^{j,k}_t)$ for all $t\in\mathfrak{m}athbb{R}$.
We can now conclude the proof. We pass to the limit as $m\to+\infty$ in~\eqref{eq:truncated_ent_before_AC} and we get
\begin{equation}\label{eq:truncated_ent_limit_m}
\mathfrak{m}athsf{Ent}(\mathfrak{m}u^{j,k}_{t_1})-\mathfrak{m}athsf{Ent}(\mathfrak{m}u^{j,k}_{t_0})=\int_{t_0}^{t_1}\int_\mathfrak{m}athbb{G} \scalar*{-w^{j,k}_t,v^{j,k}_t}_\mathfrak{m}athbb{G}\,d\mathfrak{m}u^{j,k}_t\,dt
\end{equation}
for all $t_0,t_1\in\mathfrak{m}athbb{R}$ with $t_0<t_1$. For the left-hand side of~\eqref{eq:truncated_ent_before_AC}, recall~\eqref{eq:truncated_ent_prop_1} and~\eqref{eq:truncated_ent_prop_2} and apply the monotone convergence theorem on $\set*{\varrho^{j,k}_t\le1}$ and the dominated convergence theorem on $\set*{\varrho^{j,k}_t>1}$. For the right-hand side of~\eqref{eq:truncated_ent_before_AC}, recall that $t\mathfrak{m}apsto\|v^{j,k}_t\|_{L^2_\mathfrak{m}athbb{G}(\mathfrak{m}u^{j,k}_t)}\in L^2_{\rm loc}(\mathfrak{m}athbb{R})$ and that $t\mathfrak{m}apsto\|w^{j,k}_t\|_{L^2_\mathfrak{m}athbb{G}(\mathfrak{m}u^{j,k}_t)}=\mathfrak{m}athsf{F}_\mathfrak{m}athbb{G}^{1/2}(\varrho^{j,k}_t)\in L^2_{\rm loc}(\mathfrak{m}athbb{R})$ by \textit{Step~2} and apply Cauchy--Schwarz inequality and the dominated convergence theorem.
We pass to the limit as $k\to+\infty$ in~\eqref{eq:truncated_ent_limit_m} and we get
\begin{equation}\label{eq:truncated_ent_limit_k}
\mathfrak{m}athsf{Ent}(\mathfrak{m}u^j_{t_1})-\mathfrak{m}athsf{Ent}(\mathfrak{m}u^j_{t_0})=\int_{t_0}^{t_1}\int_\mathfrak{m}athbb{G} \scalar*{-w^j_t,v^j_t}_\mathfrak{m}athbb{G}\ d\mathfrak{m}u^j_t\,dt
\end{equation}
for all $t_0,t_1\in\mathfrak{m}athbb{R}$ with $t_0<t_1$, where $w^j_t=\nabla_\mathfrak{m}athbb{G}\varrho^j_t/\varrho^j_t$ in $L^2_\mathfrak{m}athbb{G}(\mathfrak{m}u^j_t)$ for all $t\in\mathfrak{m}athbb{R}$. For the left-hand side of~\eqref{eq:truncated_ent_limit_m}, recall~\eqref{eq:mu_k_lim}. For the right-hand side of~\eqref{eq:truncated_ent_limit_m}, recall that $t\mathfrak{m}apsto\|v^j_t\|_{L^2_\mathfrak{m}athbb{G}(\mathfrak{m}u^j_t)}\in L^2_{\rm loc}(\mathfrak{m}athbb{R})$ and that $t\mathfrak{m}apsto\|w^j_t\|_{L^2_\mathfrak{m}athbb{G}(\mathfrak{m}u^{j,k}_t)}=\mathfrak{m}athsf{F}_\mathfrak{m}athbb{G}^{1/2}(\varrho^j_t)\in L^2_{\rm loc}(\mathfrak{m}athbb{R})$ by \textit{Step~1}, so that the conclusion follows applying \cref{lemma:AFP_func_meas}, \cref{lemma:polarization}, Cauchy--Schwarz inequality and the dominated convergence theorem.
We finally pass to the limit as $j\to+\infty$ in~\eqref{eq:truncated_ent_limit_k} and we get
\begin{equation}\label{eq:truncated_ent_limit}
\mathfrak{m}athsf{Ent}(\mathfrak{m}u_{t_1})-\mathfrak{m}athsf{Ent}(\mathfrak{m}u_{t_0})=\int_{t_0}^{t_1}\int_\mathfrak{m}athbb{G} \scalar*{-w^\mathfrak{m}athbb{G}_t,v^\mathfrak{m}athbb{G}_t}_\mathfrak{m}athbb{G}\,d\mathfrak{m}u_t\,dt
\end{equation}
for all $t_0,t_1\in\mathfrak{m}athbb{R}\setminus\mathfrak{m}athcal{N}$ with $t_0<t_1$, where $\mathfrak{m}athcal{N}\subset\mathfrak{m}athbb{R}$ is the set of discontinuity points of $t\mathfrak{m}apsto\mathfrak{m}athsf{Ent}(\mathfrak{m}u_t)$ and $w^\mathfrak{m}athbb{G}_t=\nabla_\mathfrak{m}athbb{G}\varrho_t/\varrho_t$ in $L^2_\mathfrak{m}athbb{G}(\mathfrak{m}u_t)$ for a.e.\ $t\in\mathfrak{m}athbb{R}$ by \cref{prop:slope_ent_G_general}. For the left-hand side of~\eqref{eq:truncated_ent_limit_k}, recall~\eqref{eq:mu_j_lim}. For the right-hand side of~\eqref{eq:truncated_ent_limit_k}, recall that $t\mathfrak{m}apsto\|v^\mathfrak{m}athbb{G}_t\|_{L^2_\mathfrak{m}athbb{G}(\mathfrak{m}u_t)}\in L^2_{\rm loc}(\mathfrak{m}athbb{R})$ and that $t\mathfrak{m}apsto\|w^\mathfrak{m}athbb{G}_t\|_{L^2_\mathfrak{m}athbb{G}(\mathfrak{m}u^{j,k}_t)}=\mathfrak{m}athsf{F}_\mathfrak{m}athbb{G}^{1/2}(\varrho_t)\in L^2_{\rm loc}(\mathfrak{m}athbb{R})$, so that the conclusion follows applying \cref{lemma:AFP_func_meas_time}, \cref{lemma:polarization}, Cauchy--Schwarz inequality and the dominated convergence theorem. From~\eqref{eq:truncated_ent_limit} we immediately deduce~\eqref{eq:GF_ent_plus_AC} and we can conclude the proof by \cref{lemma:GF_ent_plus_AC_is_heat_diff}. In particular, by \cref{prop:entropy_dissipation} the map $t\mathfrak{m}apsto\mathfrak{m}athsf{Ent}(\mathfrak{m}u_t)$ is locally absolutely continuous on~$(0,+\infty)$.
\end{proof}
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|
\begin{document}
\title{Games and Big Data: A Scalable \\ Multi-Dimensional Churn Prediction
Model}
\author{\IEEEauthorblockN{Paul Bertens, Anna Guitart and \'{A}frica
Peri\'a\~{n}ez}
\IEEEauthorblockA{
Game Data Science Department\\
Silicon Studio\\
1-21-3 Ebisu Shibuya-ku, Tokyo, Japan\\
\{paul.bertens, anna.guitart, africa.perianez\}@siliconstudio.co.jp}
}
\maketitle
\begin{abstract}
The emergence of mobile games has caused a paradigm shift in the video-game
industry. Game developers now have
at their disposal a plethora of information on their players, and thus can take advantage of
reliable models that can accurately predict
player behavior and scale to huge datasets. Churn prediction, a challenge common
to a variety of sectors, is particularly
relevant for the mobile game industry, as player retention is crucial for the
successful monetization of a game. In this
article, we present an approach to predicting game abandon based on survival
ensembles. Our method provides accurate
predictions on both the level at which each player will leave the game and their
accumulated playtime until that moment. Further,
it is robust to different data distributions and applicable to a wide range
of response variables, while also allowing for efficient parallelization
of the algorithm. This makes our model well suited to perform
real-time analyses of churners, even for games with millions of daily active
users.
\end{abstract}
\begin{IEEEkeywords}
social games; churn prediction; ensemble methods; survival analysis; online
games; user behavior; big data
\end{IEEEkeywords}
\section{Introduction}
\label{intro}
In the last few years, the video-game industry has been shaken by the appearance
of mobile games. Currently,
both traditional console games and mobile games are always online and allow game
developers to record every
action of the players. Such a unique source of information opens the door to
achieving a comprehensive analysis
of player behavior and a full understanding of player needs on quantitative
grounds.
Preventing user abandon is a challenge faced by many industries and especially
relevant for the video-game sector. Indeed,
acquisition campaigns to obtain new players are expensive, while retaining
existing users is more cost-effective.
Identifying churners beforehand allows game owners to perform personalized
promotion campaigns to retain the most valuable players
and efficiently increase monetization. Even though there have been some works on
modeling churn in the field of mobile
games \cite{runge2014, hadiji,rothenbuehler2015hidden}, they generally use
techniques that either make binary predictions, rely
on models that are not readily applicable to different data distributions or are
not able to capture the temporal dynamics
intrinsic to churn. They also present some drawbacks regarding scalability.
In this paper we discuss churn prediction beyond the classical binary approach.
Previous works\cite{perianez2016churn} have shown how to predict player abandon
in terms of days, i.e.\ \emph{time-to-event},
using survival analysis embedded into ensemble modeling. The present study
introduces, for the first time in the mobile-game sector,
a model that accurately predicts the level at which a player is expected to
leave the game and their hours of playtime until that moment.
Our methodology allows for a comprehensive solution to the churn prediction
challenge from several perspectives and dimensions, helping to fully understand
and anticipate player attrition.
\begin{figure}
\caption{Predicted survival probability by level and playtime for three players. The
first player is expected to churn at approximately level 100 and play about 500 hours (red), the second around level 200 and will play
5000 hours (yellow), and a loyal player who is not predicted to leave (green).}
\label{level-users}
\end{figure}
\section{Method}
\subsection{Model}
The method presented here is an extension of previous work on churn prediction
in mobile social games \cite{perianez2016churn} using \textit{conditional inference survival ensembles}
\cite{Hothorn06unbiasedrecursive}. Based on survival analysis \cite{clark}, the model is
capable of performing accurate predictions even when the response variable is
censored.
\begin{figure*}
\caption{ Predicted median survival level vs.\ observed level (left) and
relative deviation (right) for churned players, using the survival ensemble and
Cox regression models}
\label{level}
\end{figure*}
A survival ensemble is an ensemble of survival trees. Every tree calculates weighted Kaplan--Meier estimates to distinguish the different survival characteristics of every sample in the tree nodes.
Linear rank statistics are used as splitting criterion of the nodes, in order to maximize the survival difference among the daughter nodes. Because the
partitions of every tree are computed in two steps, the \textit{conditional inference survival ensembles} \cite{Hothorn06unbiasedrecursive} are not
biased towards predictors with many splits and are more robust to overfitting. First, the optimal split variable is selected
based on the relationship between the covariates and response and then the optimal split point is obtained by comparing two sample linear statistics for all possible partitions of the split covariate.
We have implemented a parallelized version that is more practical in a
production setting and can also make predictions on other response variables,
including level and playtime.
The approach taken here is to parallelize computations using multiple cores on a
single machine. Each core trains a subset of trees from the total ensemble, and
at the end of training all trees are merged to obtain the final model. This
method can be easily extended to run on multiple machines, where each machine
takes a subset of the total ensemble and stores the trained partial models on a
shared disk, finally merging them back into a single model.
A similar parallelization can be used to obtain accurate predictions on individual players: each
core focuses on just a partial subset of players, and full-survival probability
curves for each user are efficiently computed across multiple cores.
\begin{figure*}
\caption{Predicted median survival playtime vs.\ observed playtime (left) and
relative deviation (right) for churned players, using the survival ensemble and
Cox regression models.}
\label{playtime2}
\end{figure*}
\subsection{Dataset}
The data consisted of player action logs collected between 2014 and 2017 from a
major mobile social game, \textit{Age of Ishtaria}, developed by Silicon Studio. The predictions were done on
a subset of the most valuable players, who provide at least 50\% of the revenue (in this case 6.136 players).
Since the model should be applicable to multiple types of games, we compute a
set of input features that can easily be generalized to other games and properly
captures the dynamics of the data. The feature calculation is parallelizable over
all players, and the final dataset is small enough to be scalable to
millions of players.
In particular, from a player's action log, the daily logins, purchases,
playtime, and level-ups are extracted. These are commonly found in most
games and provide the essential information on playing behavior. For each of
these data sources, the mean is calculated over several different time
periods, namely over the player's first nine days, last nine days and full
lifetime. Further, the time elapsed until the first and last daily purchase is
calculated together with the amount spent on that day. Finally, to get the
current state of the player, the total purchases, total playtime, total logins
and current level are also added. This way of constructing the
feature set allows for an easy extension to include other data sources (e.g.\
click counts, experience gained, distance moved, etc.) and it is robust enough to describe the different variability of the data among games.
\subsection{Outcome}
Two additional models based on \cite{perianez2016churn} are implemented to
perform predictions on the number of hours a user will play and the level at
which they will quit.
The models are trained on each of the following response variables:
\begin{itemize}
\item Playtime: How many seconds the user played the game.
\item Level: Latest game level reached by the player.
\end{itemize}
In both cases, the censored variable is whether they churned or not (churn
is defined as not having logged in for 9 days).
\subsection{Features}
For the predictors the most relevant variables from the dataset are selected for each of the models.
\begin{itemize}
\item Playtime model: Level, Days since last Purchase, First purchase amount, Last purchase amount, Purchases in the first 9 days, Loyalty index (number of days connected divided by the lifetime), Days since last level up.
\item Level model: Lifetime, Days since last Purchase,
First purchase amount, Last purchase amount, Purchases in the first 9 days, Loyalty index, Days since last level up.
\end{itemize}
\section{Results}
The model described above outputs a different survival probability curve for every player.
Figure \ref{level-users} illustrates the survival probability
for three different users, each having a distinct survival expectation
determined by their characteristics. In this case the first player is expected
to churn at a very early level, while the third player would reach a much higher
level. Similarly in the right figure, the survival prediction for three players is plotted in terms of playtime, distinguishing diffent degrees of playtime expectancy.
The level and playtime prediction results are displayed in Figures~\ref{level}
and \ref{playtime2}, respectively. We perform a comparison between the standard
Cox regression \cite{Cox1972} and the survival ensemble model (an ensemble of 1.500 trees). When predicting
at what level a player will leave the game, the Cox regression model has more
difficulty capturing the temporal nonlinearity inherent to the problem (i.e. the time between levels is not uniform), as evidenced by a much larger spread in
Figure~\ref{level}. The accuracy of the survival ensemble remains better
throughout the entire level range, with points lying tightly along the identity
line, i.e.\ with small differences between the predicted and observed values.
The accuracy is reduced for higher levels, but this is explained by
the fact that there is less data and the censorship increases (as there are
fewer players at those levels). The same effect is observed for playtime in
Figure~\ref{playtime2}: the predictions are most accurate for players with very
little playtime, whereas the spread becomes more significant as playtime
increases, which can be explained again by the fact that very few users have
played so much.
The integrated Brier scores (IBS)\cite{mogensen2012evaluating} calculated using
bootstrap cross-validation with replacement with 1000 samples \cite{alfons2012cvtools} are listed
in Table \ref{BrierTable_playtime}. It can be seen that the survival ensemble
significantly outperforms the Cox regression model both for level and playtime
prediction. Figure \ref{errorMeasures} also shows that the survival ensemble
error is lower than that of Cox regression over the entire range of both
playtime and levels. Figure \ref{errorMeasures} depicts the non-linearity of
the time per level dimension (i.e. the time between levels is not equally distributed), which indeed will be diferent for every game.
\begin{table}
\centering
\caption{Integrated Brier score (IBS) for level and playtime prediction}
\begin{tabular}{ccc} \hline\noalign{
}
Model & IBS level & IBS playtime \\ \hline
Survival Ensemble & 0.025 & 0.026\\
Cox regression & 0.054 & 0.044\\
Kaplan-Meier & 0.127 & 0.134\\ \hline
\noalign{
}\hline
\end{tabular}
\label{BrierTable_playtime}
\end{table}
\begin{figure}
\caption{Playtime
model (top) and level model (bottom) IBS error curves.}
\label{errorMeasures}
\end{figure}
\section{Summary and Conclusions}
The results show that the method based on \textit{conditional inference survival
ensembles} is able to model churn both in terms of playtime and level,
predicting accurately at which level a player will leave and how long they will
play. This indicates that the model is robust to different data distributions,
and applicable to different types of response variables. While Cox regression
did perform relatively well, it requires a lot of manual effort and also suffers
from scalability issues, which makes it
unsuitable for a production environment. On the other hand, the proposed survival
ensembles are easily adaptable to other type of games and uses a parallelized implementation that can be run not
only on multiple cores but also on multiple machines. This gives game developers
the chance to efficiently obtain full survival probability curves for each
player, and to predict in real time not only when a player will leave the game,
but also at what level they will do it and how many hours they will play before
quitting.
\end{document}
|
\begin{document}
\coverpage{On a conjecture of Street and Whitehead on locally maximal product-free sets}{Chimere S. Anabanti and Sarah B. Hart}{12}
\title{On a conjecture of Street and Whitehead on locally maximal product-free sets\subjclass{Primary 20D60; Secondary 20P05}}
\author{Chimere S. Anabanti\thanks{The first author is supported by a Birkbeck PhD Scholarship}\\ [email protected] \and Sarah B. Hart\\ [email protected]}
\date{May 6, 2015}
\maketitle
\begin{abstract}
\noindent Let $S$ be a non-empty subset of a group $G$. We say $S$ is product-free if $S\cap SS=\varnothing$, and $S$ is locally maximal if whenever $T$ is product-free and $S\subseteq T$, then $S=T$. Finally $S$ fills $G$ if $G^*\subseteq S \sqcup SS$ (where $G^*$ is the set of all non-identity elements of $G$), and $G$ is a filled group if every locally maximal product-free set in $G$ fills $G$. Street and Whitehead \cite{SW1974} investigated filled groups and gave a classification of filled abelian groups. In this paper, we obtain some results about filled groups in the non-abelian case, including a classification of filled groups of odd order. Street and Whitehead conjectured that the finite dihedral group of order $2n$ is not filled when $n=6k+1$ ($k\geq 1$). We disprove this conjecture on dihedral groups, and in doing so obtain a classification of locally maximal product-free sets of sizes 3 and 4 in dihedral groups, continuing earlier work in \cite{AH2015} and \cite{GH2009}.
\keywords{Locally maximal, product-free, sum-free, nonabelian}
\end{abstract}
\section{Introduction}
A non-empty subset $S$ of a finite group $G$ is called product-free if $xy=z$ does not hold for any $x,y,z \in S$. Equivalently, writing $SS$ for $\{xy: x, y \in S\}$, we have $S \cap SS = \emptyset$. Product-free sets were originally studied in abelian groups, and therefore they are often referred to in the literature as sum-free (or sumfree) sets. If $S$ is product-free in $G$, and not properly contained in any other product-free subset of $G$, then we call $S$ a locally maximal product-free set (see \cite{AH2015}, \cite{GH2009} and \cite{SW1974}). On the other hand, a product-free set $S$ is called maximal if no product-free set in $G$ has size bigger than $|S|$. In the latter direction, see \cite{DY1969}, \cite{GR2005} and \cite{RS1970}. There has been a good deal of work on maximal product-free sets in abelian groups; for example Green and Ruzsa in \cite{GR2005} were able to determine, for any abelian group $G$, the
cardinality of the maximal product-free sets of $G$. Gowers \cite[Theorem 3.3]{gowers} proved that if the smallest nontrivial
representation of $G$ is of dimension $k$ then $G$ has no product-free sets of size greater than
$k^{−1/3}n$. Much less is known about sizes of locally maximal product-free sets, in particular the minimal size of a locally maximal product-free set. \\
Since every product-free set is contained in a locally maximal product-free set, we can gain information about product-free sets in a group by studying its locally maximal product-free sets. In connection with group Ramsey Theory, Street and Whitehead \cite{SW1974} noted that every partition of a group $G$ (or in fact, of $G^{\ast}$) into product-free sets can be embedded into a covering by locally maximal product-free sets, and hence to find such partitions, it is useful to understand locally maximal product-free sets. They remarked that many examples of these sets have the additional property that $G^{\ast} \subseteq S \cup SS$, and with that in mind gave the following definition. A subset $S$ of a group $G$ is said to {\em fill} $G$ if $G^*\subseteq S \sqcup SS$. The group $G$ is called a {\em filled group} if every locally maximal product-free set in $G$ fills $G$. Street and Whitehead in \cite{SW1974} and \cite{SW1974A} classified the abelian filled groups and conjectured that the dihedral group of order $2n$ is not filled when $n=6k+1$ for $k\geq 1$. One consequence of our results in this paper is that this conjecture is false.\\
\noindent This paper is aimed at throwing more light on locally maximal product-free sets (LMPFS for short) and filled groups in the non-abelian case. In Section 2 we look at filled groups. We show (Theorem \ref{soluble}) that all non-abelian finite filled groups have even order, and that all finite nilpotent filled groups of even order are 2-groups. Using GAP \cite{gap} we have seen that for groups of order up to 31 the only examples of non-abelian filled groups are 2-groups or dihedral (see Table~\ref{table1}). Therefore the dihedral case is of interest. In Section $3$, we study LMPFS in finite dihedral groups and classify all LMPFS of sizes $3$ and $4$ in dihedral groups. (Groups containing a locally maximal product-free set of size 1 or 2 were classified in \cite{GH2009}.) In Section 4 we look at filled dihedral groups, give a counterexample to the conjecture of Street and Whitehead and obtain some restrictions on the possible orders of filled dihedral groups. \\
In the rest of this section we establish the notation we will need and gather together some useful results. All groups in this paper are finite.
Given a positive integer $n$, we write $C_n=\left\langle x|~x^n=1\right\rangle$ for the cyclic group of order $n$ and $D_{2n}=\langle x,y|~x^{n}=y^2=1, xy=yx^{-1} \rangle$ for the dihedral group of order $2n$ (where $n>1$). In $D_{2n}$ the elements of $\langle x\rangle$ are called rotations and the elements of $\langle x\rangle y$ are called reflections. For any subset $S$ of $D_{2n}$, we write Rot($S$) for $S \cap \langle x \rangle$, the set of rotations of $S$, and Ref($S$) for $S \cap \langle x \rangle y$, the set of reflections of $S$. Let $S$ and $V$ be subsets of a finite group $G$. We define $SV:=\{sv|~s\in S,v\in V\}$, $S^{-1}:=\{s^{-1}|s\in S\}$, $T(S):=S \cup SS \cup SS^{-1} \cup S^{-1}S$ and $\sqrt{S}:=\{x \in G:x^2 \in S\}$. The following results will be used repeatedly.
\begin{lem}[Lemma 3.1 of \cite{GH2009}]\label{GH2009L}
Let $S$ be a product-free set in a finite group $G$. Then $S$ is locally maximal if and only if $G=T(S) \cup \sqrt{S}$.
\end{lem}
The following result is well-known but we include a short proof for the reader's convenience.
\begin{lem} \label{lemma1.2} Let $H$ be a subgroup of a group $G$. Any non-trivial coset of $H$ is product-free in $G$. Further, if $H$ is normal and $Q$ is product-free in $G/H$ then the set $S = \{g \in G: gH \in Q\}$ is product-free in $G$. \end{lem}
\begin{proof}
For the first statement, if for some $h_1, h_2, h_3 \in H$ and $g \in G$ we have $(h_1g)(h_2g) = (h_3g)$, then $g = h_1^{-1}h_3h_2^{-1} \in H$. Therefore if $g \notin H$ we have $(Hg)(Hg) \cap Hg = \emptyset$, so $Hg$ is product-free. Now suppose $H$ is normal with $Q$ and $S$ as defined in the statement of the lemma. Then $SS = \{a \in G: aH \in QQ\}$. The fact that $S$ is product-free now follows immediately from the fact that $Q$ is product-free.
\end{proof}
The following is a straightforward consequence of the definitions.
\begin{pro}\label{AH2015A_Pro}
Each product-free set of size $\frac{|G|}{2}$ in a finite group $G$ is the non-trivial coset of a subgroup of index 2. Furthermore such sets are locally maximal and fill $G$.
\end{pro}
\section{Filled groups}\label{sec3}
Street and Whitehead in \cite{SW1974} and \cite{SW1974A} investigated locally maximal product-free sets properties in some groups. They proved the following results.
\begin{lem}\cite[Lemma 1]{SW1974} \label{swl1} Let $G$ be a finite group and $N$ a normal subgroup of $G$. If $Q$ is a locally maximal product-free set in $G/N$ that does not fill $G/N$, then the set $S$ given by $S = \{g: gN \in Q\}$ is a locally maximal product-free in $G$ that does not fill $G$. That is, if $G$ is filled then $G/N$ is filled. \end{lem}
\begin{thm}\cite[Theorem 2]{SW1974} A finite abelian group is filled if and only if it is $C_3, C_5$ or an elementary abelian 2-group.
\end{thm}
They also observed the following.
\begin{lem}\label{co3}
If $G$ is a filled group with a normal subgroup of index 3, then $G \cong C_3$.
\end{lem}
\begin{proof} Let $N$ be a normal subgroup of index 3 and $S$ be a nontrivial coset. Then $S$ is product-free by Lemma \ref{lemma1.2}. Moreover $S \cup SS \cup SS^{-1} = G$. Therefore $S$ is a locally maximal product-free set by Lemma~\ref{GH2009L} and so $S$ must fill $G$, which implies that $G^{\ast} \subseteq S \cup SS$. But $S \cup SS = G - N$. Therefore $N = \{1\}$ and $G$ is cyclic of order 3. \end{proof}
Street and Whitehead also observed that Lemma \ref{swl1} implies that the quotients of any filled non-abelian group $G$ must themselves be filled. In particular, the quotient of $G$ by its derived group $G'$ must be either an elementary abelian 2-group or cyclic of order 5 (it cannot be cyclic of order 3 by Lemma \ref{co3}). These conditions are not sufficient. The counterexamples given in \cite{SW1974} are $D_{14}$ (which in fact {\em is} a filled group, as we shall show), the quaternion group of order 8 and the alternating group of degree 5. \\
Our main aim in this section is to classify filled groups of odd order. We begin with $p$-groups of odd order.
\begin{pro}\label{4.12}
Suppose $G$ is a finite $p$-group, where $p$ is an odd prime. Then $G$ is filled if and only if $G$ is either $C_3$ or $C_5$.
\end{pro}
\begin{proof}
Certainly $C_3$ and $C_5$ are filled. For the reverse implication, let $G$ be a finite $p$-group of order $p^n$. We proceed by induction on $n$. If $G$ is non-abelian, then the quotient of $G$ by its centre $Z(G)$ is a strictly smaller $p$-group so, inductively, is either $C_3$ or $C_5$ (since $p$ is odd). But it is a basic result that if $G/Z(G)$ is cyclic, then $G$ is abelian, giving a contradiction. Therefore $G$ is abelian, and now the result follows immediately from the classification of filled abelian groups.
\end{proof}
The next theorem, Theorem \ref{soluble}, makes use of an observation in \cite{SW1974}. Theorem 3 of that paper asserts that if $G$ is a finite nonabelian filled group, then either $G = G'$ or $G/G'$ is an elementary abelian 2-group, or $G/G' \cong C_5$ and $|G|$ is even. The proof given is that since $G/G'$ must be a filled abelian group, it is either trivial, or elementary abelian 2-group, or $C_3$ or $C_5$. Now $C_3$ is impossible by Lemma \ref{co3}. So if $G$ has odd order, we must have that $G/G'$ is cyclic of order 5. A set is then described, based on an element $a$ of $G - G'$, which the authors claim is locally maximal product free but does not fill $G$. But in fact the given set is only locally maximal if $a$ has order 5. The existence of such an element is not guaranteed when $G/G'$ is cyclic of order 5, even if $G$ has odd order. We are grateful to Robert Guralnick for providing us with an example of a group without such an element --- the group is an extension of an extraspecial group of order $5^{11}$ by the Frobenius group of order $55$, such that the fifth power of each element of order 5 in the Frobenius group is a central element of order 5 in the extraspecial group. In this case the derived group has index 5 and contains all elements of order 5. We resolve that issue in the following lemma and theorem by reducing to a situation where we can be certain of the existence of the required element.
\begin{lem}\label{order5}
Suppose $G$ is a finite group with a normal subgroup $N$ of index five, such that not every element of order five in $G$ is contained in $N$. If $G$ is filled, then $G$ is cyclic of order 5.
\end{lem}
\begin{proof} Our argument is based on the construction given in \cite{SW1974}. If $N$ is trivial then $G \cong C_5$ and $G$ is filled. If $|N| = 2$, then $G \cong C_{10}$ and $G$ is not filled. So we may assume $N$ has order at least three.
Let $h$ be an element of order 5 in $G$ with $h \notin N$. Then let $S = \{h\} \cup h^2N^{\ast}$ (where $N^{\ast}$ is the set of nonidentity elements of $N$). Then $SS = \{h^2\} \cup h^3N^{\ast} \cup h^4N$. (The fact that $(h^2N^{\ast})^2 = h^4N$ follows because $|N| > 2$.) So $S$ is product-free, but does not fill $G$. Now $SS^{-1} = h^4N^{\ast} \cup hN^{\ast} \cup N$. Thus $T(S) = G - \{h^3\}$. Since $h^3 \in \sqrt S$, we can now conclude that $G = T(S) \cup \sqrt S$, which means $S$ is a locally maximal product-free set that does not fill $G$. So $G$ is not filled.
\end{proof}
\begin{thm}\label{soluble}
The only filled groups of odd order are $C_3$ and $C_5$.
\end{thm}
\begin{proof}
Let $G$ be a nontrivial group of odd order. We proceed by induction on the order of $G$. Groups of order 3 and 5 are filled, so assume $|G| > 5$, and, inductively, that if $H$ is a filled group of odd order with $|H| < |G|$, then $H$ is isomorphic to either $C_3$ or $C_5$. \\
If $G$ is abelian, then $G$ is not filled. So we may assume that $G$ is nonabelian. Then, because $G$ is soluble, the derived group $G'$ is a proper nontrivial normal subgroup of $G$. Therefore $G/G'$ is a filled group of order less than $|G|$, and hence is isomorphic to either $C_3$ or $C_5$. However, as we have noted, if $S$ is any non-trivial coset of a normal subgroup of index 3, then $S$ is a locally maximal product-free set that does not fill $G$. Therefore $G/G'$ is cyclic of order 5. If $G''$ is nontrivial, then we can apply the same argument to $G/G''$, which would imply that $G/G''$ is also cyclic of order 5, and thus that $G'' = G'$, contradicting the solubility of $G$. Therefore $G'' = \{1\}$ and $G'$ is abelian.\\
Since $G$ has order greater than 5, Proposition \ref{4.12} implies that $G$ is not a $p$-group. Thus there is at least one prime $p$, with $p \neq 5$, dividing the order of $G$. Any Sylow $p$-subgroup $K$ of $G'$ is also a Sylow $p$-subgroup of $G$. But $G'$ is normal in $G$, and abelian, whence $K$ is normal in $G$. Now $G/K$ is filled, meaning that $G/K$ has order 5, which implies $K = G'$. Therefore $5$ does not divide the order of $|K|$, which means there are elements of order 5 in $G$ that do not lie in $G'$ (in fact of course all elements of order 5 lie outside of $G'$). Therefore, by Lemma \ref{order5}, $G$ is not filled. The result now follows by induction.
\end{proof}
Groups of even order are of course less amenable to analysis. We have the following step in this direction.
\begin{lem}\label{interesting}
If $G$ is a filled nilpotent group, then $G$ is either a 2-group or isomorphic to $C_3$ or $C_5$.
\end{lem}
\begin{proof}
Let $G$ be a filled nilpotent group. If $G$ has odd order then $G$ is either $C_3$ or $C_5$ by Theorem~\ref{soluble}. So assume $G$ has even order. Then $G$ is a direct product of $p$-groups (its Sylow subgroups), and its Sylow 2-subgroup $N$ is nontrivial. The quotient $G/N$ (which must be filled) is isomorphic to the direct product of the remaining Sylow subgroups, which is a group of odd order. If $N \neq G$ then $G/N$ is either $C_3$ or $C_5$. We know no filled group can have a normal subgroup of index 3, so $G/N$ must be cyclic of order 5, and clearly all elements of order 5 in $G$ lie outside $N$. Thus, by Lemma \ref{order5}, $G$ is not filled. Therefore $N = G$. That is, if $G$ is a filled nilpotent group then $G$ is either a 2-group or isomorphic to $C_3$ or $C_5$.
\end{proof}
In the light of Lemma \ref{interesting} it would be interesting to have a classification of filled 2-groups, as this would enable a full classification of filled nilpotent groups. We will show in Section 4 that $D_8$ is the only filled nonabelian dihedral 2-group. We can eliminate generalised quaternion groups from our enquiries now. For a positive integer $n$, with $n>1$, the generalised quaternion group of order $4n$ is the group $Q_{4n} = \langle a, b: a^{2n} = 1, b^2 = a^n, ba = a^{-1}b\rangle$. We have the following.
\begin{pro}
No generalised quaternion group is filled.
\end{pro}
\begin{proof}
Let $G$ be generalised quaternion. Then $G$ has a cyclic subgroup $N$ of index 2, and $G$ contains a unique involution $z$. Let $S$ be any locally maximal product-free set of $N$ containing $z$. Then because every element of $G - N$ is a square root of $z$, we have that $S$ is locally maximal product-free in $G$. But $S$ clearly does not fill $G$.
\end{proof}
The only nonabelian filled groups we know of that are not 2-groups are dihedral (see Table \ref{table1} for a complete list of the filled groups of order less than 32). Therefore it makes sense to study dihedral groups a little more carefully. This is the object of the next section.
\section{Locally maximal product-free sets in dihedral groups}
\begin{thm}\label{CC1}\label{AH2015A_T}
Let $S$ be a locally maximal product-free set in a finite dihedral group $G$ of order $2n$. Then $|G|\leq |S|^2+|S|$.
\end{thm}
\begin{proof}
Suppose $S$ is a locally maximal product-free set of size $m\geq 1$ in a finite dihedral group $G$. Let $A = $Rot($S$) and $B = $Ref($S$). By the relations in the dihedral group, $BA = A^{-1}B$. We also have that $AA^{-1} = A^{-1}A$ and $B^{-1} = B$. Therefore \begin{align}
T(S) &= S \cup SS \cup SS^{-1} \cup S^{-1}S \nonumber\\
&= A \cup B \cup AA \cup AB \cup A^{-1}B \cup BB \cup AA^{-1}. \label{eq1}
\end{align}
Now, since $G = T(S) \cup \sqrt S$, and $\sqrt S$ cannot contain involutions, it must be the case that Ref($G$) is contained in $T(S)$. That is, we must have \begin{equation}
\mathrm{Ref}(G) = B \cup AB \cup A^{-1}B = B \cup (A \cup A^{-1})B.
\label{eq2}
\end{equation}
If $|A| = k$ and $|S| = m$ we see that $|G| \leq 2(2k+1)(m-k) = 2m + 2(2m-1)k - 4k^2$. This is a quadratic expression in $k$ which attains its maximum value over all $k$ when $k = \frac{2m-1}{4}$, so attains its maximum value over integers $k$ at either $k = \frac{m-1}{2}$ or $k = \frac{m}{2}$. Substituting either value for $k$ into $2(2k+1)(m-k)$ gives $m(m+1)$. We conclude that $|G| \leq |S|^2 + |S|$.
\end{proof}
\begin{rem}\label{RSS}
It follows from the proof of Theorem \ref{CC1} that if $S$ is a locally maximal product-free set in a finite dihedral group $G$, then Ref($G$)$=$Ref($T(S)\cup \sqrt{S}$)$=$Ref($S\sqcup SS$). In particular, $S$ must contain at least one reflection.
\end{rem}
We also remark that nearly all other known upper bounds for the order of a finite group $G$ containing a locally maximal product-free set $S$ are in terms of $|\langle S\rangle|$ rather than $|S|$. See \cite{GH2009}, for example. Theorem \ref{CC1} is thus a useful concrete bound for dihedral groups. The only other known upper bounds for the order of a finite group $G$, in terms of the size of a locally maximal product-free set $S$ in $G$, are for the case where $S\cap S^{-1} = \emptyset$, in which case $|G| \leq 4|S|^2 + 1$ \cite[Corollary 3.10]{GH2009}, and the case where $G$ is cyclic, in which case it is easy to see from Lemma \ref{GH2009L} that $|G| \leq \frac{1}{2}\langle 3|S|^2 + 5|S| + 2\rangle$.\\
From Theorem \ref{AH2015A_T}, it is clear that if $n > 1$, there is no locally maximal product-free set of size $1$ in $D_{2n}$, and that any locally maximal product-free set of size $2$ must appear in $D_4$ or $D_6$. A simple check shows that any LMPFS of size $2$ must be automorphic to $\{x,y\}$ in $D_4$ and $D_6$, where $x$ is an element of order $n$, and $y$ is a reflection. We next look at sets of size 3 and 4. Locally maximal product-free sets of size $3$ have been classified in \cite{AH2015}, building on work in \cite{GH2009}, but for completeness we include the result for dihedral groups here with a brief proof. As no full classification has been given for size $4$, the one here is a step in that direction. For the rest of this section, $G$ will be a finite dihedral group of order $2n$ for $n\geq 3$.
\begin{nota}
Where $n$ is even, we denote the non-identity cosets of the maximal subgroups of $G$ by $M_1$, $M_2$ and $M_3$, where $M_1 =$ Ref($G$), $M_2= \{x,x^3,\cdots,x^{n-1},y,x^2y,\cdots,x^{n-2}y\}$ and \linebreak $M_3 = \{x,x^3,\cdots,x^{n-1},xy,x^3y,\cdots,x^{n-1}y\}$
respectively. If $n$ is odd, then $M_1$ is the only such coset.
\end{nota}
\begin{thm}\label{R1}
If $S$ is a LMPFS of size $3$ in a finite dihedral group $G$, then $G=D_6$ or $D_8$. Furthermore, up to automorphisms of $G$, there is only one such set; viz. $\{y,xy,x^2y\}$ or $\{x^2,y,xy\}$ according as $G=D_6$ or $D_8$.
\end{thm}
\begin{proof}
By Theorem \ref{AH2015A_T} and the fact that a LMPFS of size $3$ cannot be contained in a group of order less than $6$, we have that $6 \leq |G| \leq 12$. By Proposition \ref{AH2015A_Pro}, the only LMPFS of size 3 in $D_6$ is $M_1$. So assume $|G| \geq 8$. Now $S$ contains at least one reflection by Remark \ref{RSS}, and $S$ contains at least one rotation, because otherwise it would be properly contained in the product-free set $M_1$. Suppose $S$ consists of a rotation $a$, and reflections $b_1$ and $b_2$. If $G=D_8$ and $a$ has order $4$ in $G$, then no such $S$ exist (since any such $S$ is either contained in $M_2$ or $M_3$, or not product-free); if $a$ is the unique involution, then $S$ must be mapped by an automorphism of $G$ into $\{x^2,y,xy\}$ since $b_1$ and $b_2$ must be in distinct conjugacy classes of $G$ for such $S$ to exist. If $G=D_{10}$, then by adjoining $a^{-1}$ to $S$, we get a bigger product-free set that contains $S$; thus no such $S$ exist.
Finally, suppose $G=D_{12}$.
If $a$ is the unique involution, then by Equation \eqref{eq2} $|\mathrm{Ref}(G)| = 4$, a contradiction. Suppose $\circ(a)=3$. Observe that Rot($T(S)\cup\sqrt{S}$)$\subseteq(\{1,a,a^{-1},\sqrt{a}\} \cup \{b_1b_2,b_2b_1\})$. Since $\sqrt{a}$ consists of $a^{-1}$ and an element of order $6$, and $\circ(b_1b_2)=\circ(b_2b_1)$, such $S$ cannot exist. Now, suppose $\circ(a)=6$. As Rot($T(S)\cup\sqrt{S}$)$\subseteq(\{1,a,a^2\} \cup \{b_1b_2,b_2b_1\})$, we have that $|$Rot$(T(S)\cup\sqrt{S})|\leq 5<6=|$Rot$(D_{12})|$; thus no such $S$ exist. Similar arguments show that there is no locally maximal product-free set made up of exactly two rotations and one reflection in $D_8$, $D_{10}$ and $D_{12}$.
\end{proof}
\begin{pro}\label{MR3}\label{AH2015A_P}
Let $G$ be a dihedral group of order $2p$, $p>3$ and prime. If $S$ is a locally maximal product-free set of size $4$ in $G$, then $p$ is either 5 or 7 and $S$ contains exactly two non-identity rotations and two reflections.
\end{pro}
\begin{proof}
By Theorem \ref{CC1}, either $G$ is at least one of $D_{10}$ or $D_{14}$, or no such $G$ exists. Let $G$ be either $D_{10}$ or $D_{14}$. Suppose for a contradiction that $S$ does not contain exactly two non-identity rotations and two reflections. By Remark \ref{RSS}, $S$ contains at least one reflection. At the other extreme, if every element of $S$ is a reflection then $S$ is properly contained in $M_1$, so $S$ is not locally maximal. Therefore $S$ contains at least one rotation. Suppose $S$ contains $1$ non-identity rotation (say $x^i$) and three reflections. Then a quick check shows that $S \cup \{x^{-i}\}$ is also product-free, contradicting the maximality of $S$. Similarly, if $S$ contains three non-identity rotations and one reflection, then there exists a rotation in $S$ whose inverse is not in $S$, and by adjoining this inverse to $S$, we again obtain a contradiction to the locally maximal condition on $S$. Therefore $S$ contains exactly two reflections and two rotations.
\end{proof}
\begin{cor}\label{MR5}\label{AH2015A_C}
Suppose $G$ is a dihedral group of order $2p$ ($p>3$ and prime). If $S$ is a LMPFS of size $4$ in $G$, then $p$ is either 5 or 7, and $S$ is mapped by an automorphism of $G$ into $\{x^2,x^3,y,x^{-1}y\}$.
\end{cor}
\begin{pro}\label{P1}
If $S$ is a locally maximal product-free set of size $4$ containing exactly four involutions in a dihedral group $G$ such that $10\leq |G|\leq 20$, then $G$ can only be $D_{12}$. Furthermore, $S$ is mapped by an automorphism of $D_{12}$ into $\{x^3,y,xy,x^2y\}$.
\end{pro}
\begin{proof}
As $M_1$ contains all reflections in $G$, in order for $S$ not to be properly contained in $M_1$, we must have that $\frac{|G|}{2}$ is even and that $S$ contains the unique involution $z$ in the rotation subgroup. But now by Equation \eqref{eq2} we obtain $|\mathrm{Ref}(G)| \leq 6$. Therefore the only possibility is $D_{12}$. So assume $G = D_{12}$ and let $S=\{z,b_1,b_2,b_3\}$, where $b_1,b_2$ and $b_3$ are reflections and $z = x^3$. If $S$ is locally maximal product-free in $D_{12}$, then by Remark \ref{RSS}, Ref($D_{12}$)$=$Ref($S\sqcup SS$)$\subseteq$ $\{b_1,b_2,b_3,x^3b_1,x^3b_2, x^3b_3\}$. If any two elements of Ref($S\sqcup SS$) are equal, then $S$ is not locally maximal in $D_{12}$.
For no two elements of Ref($S\sqcup SS$) to be equal, we must have that $S$ is of the form $\{x^3,x^iy,x^{i+1}y,x^{i+2}y\}$ for $i=0,1,2,3,4$ or $5$. Thus, the only possible choices are $S:=\{x^3,y,xy,x^2y\}$, $S_1:=\{x^3,xy,x^2y,x^3y\}$, $S_2:=\{x^3,x^2y,x^3y,x^4y\}$, $S_3:=\{x^3,x^3y,x^4y,x^5y\}$, $S_4:=\{x^3,y,x^4y,x^5y\}$ and $S_5:=\{x^3,y,xy,x^5y\}$. By Lemma~\ref{GH2009L}, $S$ is locally maximal product-free in $D_{12}$. As the automorphism $\phi_i: x \mapsto x, y\mapsto x^iy$ maps $S$ into $S_i$ for each $i\in [0,5]$, we are done.
\end{proof}
\begin{lem}\label{L2}
Suppose $S$ is a locally maximal product-free set of size $4$ consisting of three involutions and one non-involution in a dihedral group $G$ such that $10\leq |G|\leq 20$. If $G=D_{12}$, then $S$ is automorphic to $\{x^2,x^3,y,x^5y\}$. Moreover, no such $S$ exist if $|G|\neq 12$.
\end{lem}
\begin{proof} The argument splits into two cases: Case I where the involutions are all reflections, and Case~II where $\frac{|G|}{2}$ is even and one of the involutions is the central rotation $x^{|G|/4}$.\\
Case I: Let $S=\{a,b_1,b_2,b_3\}$, where $a$ is the non-involution and $b_1,b_2$ and $b_3$ are reflections.
If $a$ has order at least $4$, then $S\cup\{a^{-1}\}$ is product-free, contradicting the local maximality of $S$. Thus $a$ has order $3$ and $G$ is either $D_{12}$ or $D_{18}$. However, no such combination (with any three reflections) gives a locally maximal product-free set of size $4$. For example in $D_{12}$, if $b_1 = x^iy$, $b_2 = x^jy$ and $b_3 = x^ky$ then at least two of $i, j$ and $k$ must have the same parity. But that implies $a \in SS$, a contradiction.
\\
Case II: Let $S=\{a,b_1,b_2,z\}$, where $a$, $b_1$ and $b_2$ are as in Case I, and $z$ is the unique involution in the rotation subgroup. Here $G$ is $D_{12}$, $D_{16}$ or $D_{20}$. A quick calculation using the fact that \linebreak $G = T(S) \cup \sqrt S$ shows that $$\mathrm{Rot}(G) = \sqrt S \cup\{1, a, a^2, z, az, a^{-1}z, b_1b_2, b_2b_1\}.$$
Since $b_1b_2 = (b_2b_1)^{-1}$, and $S$ is product-free, we have that $a^{-1} \in \sqrt S \cup \{a^2, az\}$. If $a^{-1} \in \sqrt z$ or $a^{-1} = az$ then $a^2 = z$, contradicting the fact that $S$ is product-free. Thus $a^{-1} \in \sqrt a \cup a^2$, which implies $\circ(a) = 3$, and so $G = D_{12}$. By Lemma \ref{GH2009L}, the product-free set $\{x^2,x^3,y,x^5y\}$ is locally maximal. A careful check shows that any other arising LMPFS must be mapped by an automorphism of $D_{12}$ into $\{x^2,x^3,y,x^5y\}$.
\end{proof}
\begin{pro}\label{P3}
Let $S$ be a locally maximal product-free set of size $4$ consisting of two involutions and two non-involutions in a dihedral group $G$ such that $10\leq |G|\leq20$. Then, up to automorphism, $S$ and $G$ are given in the table below.
\begin{center}
\begin{tabular}{|p{0.5cm}|p{5.2cm}|}\hline
$G$ & $\qquad \qquad S$\\ \hline
$D_{10}$ & $\{x^2,x^3,y,x^4y\}$\\ \hline
$D_{12}$ & $\{x,x^5,y,x^3y\}, \{x,x^4,y,x^3y\}$ \\ \hline
$D_{14}$ & $\{x^2,x^3,y,x^6y\}$\\ \hline
$D_{16}$ & $\{x^2,x^3,y,x^7y\}, \{x,x^6,y,x^4y\}$\\ \hline
$D_{18}$ & $\{x^2,x^5,y,x^8y\}$\\ \hline
$D_{20}$ & $\{x,x^8,y,x^5y\}$\\ \hline
\end{tabular}
\end{center}
\end{pro}
\begin{proof}
Case I: Let $S=\{a_1,a_2,b_1,b_2\}$, where $a_1,a_2$ are non-involutions, and $b_1,b_2$ are reflections. Assume for the moment that $|G| > 12$. If $a_2 = a_1^{-1}$, then by Equation \eqref{eq2} $|\mathrm{Ref}(G)| \leq 6$, which contradicts our assumption. Thus $a_2 \neq a_1^{-1}$. In the case of $D_{18}$, this means that at least one of $a_1$ and $a_2$ has order 9. If $\circ(a_1)=9=\circ(a_2)$, with $a_2 \neq a_1^{-1}$, then $S$ is automorphic to $\{x^2,x^5,y,x^8y\}$, which is locally maximal product-free. Next suppose $G$ is $D_{16}$ or $D_{20}$. If $\circ(a_1)=3$ and $\circ(a_2) = 9$ then a quick check shows that $S$ is not locally maximal product-free. Suppose $\circ(a_1)=\frac{|G|}{4}=\circ(a_2)$. $S$ is not feasible in $D_{16}$ since then $a_2 = a_1^{-1}$. On the other hand, such $S$ is not also possible in $D_{20}$ as Ref($T(S)\cup \sqrt{S}$)$\subseteq \{b_1,b_2,a_1b_1,a_1b_2,b_1a_1,b_2a_1,a_2b_1,a_2b_2,b_1a_2,b_2a_2\}$, and either $a_2=a_1^{-1}$ or $\{a_1,a_2\}$ is not product-free.
Suppose $\circ(a_1)=\frac{|G|}{2}=\circ(a_2)$. Then Rot($T(S)\cup \sqrt{S})\subseteq $Rot($T(S))\subseteq \{1,a_1,a_2,a_1a_2,a_1^2,a_2^2,b_1b_2,b_2b_1,a_1^{-1}a_2,a_2^{-1}a_1\}$. So the only possible odd powers of a generator of $C_{\frac{|G|}{2}}$ are $a_1,a_2,b_1b_2$ and $b_2b_1$. If $G=D_{20}$, then no such $S$ exists. On the other hand, if $G=D_{16}$, then as $b_2b_1=(b_1b_2)^{-1}$, we must have that $a_2={a_1}^{-1}$ which leads to the conclusion that Ref($T(S)\cup \sqrt{S}$)$\subseteq \{b_1,b_2,a_1b_1,a_1b_2,a_2b_1,a_2b_2\}$, a contradiction. Finally, suppose $\circ(a_1)=\frac{|G|}{4}<\frac{|G|}{2}=\circ(a_2)$. The set $S$ is locally maximal product-free by Lemma \ref{GH2009L}. A careful check shows that any such set must be mapped by an automorphism of the group into $\{x^2,x^3,y,x^7y\}$ or $\{x,x^6,y,x^4y\}$ if $G=D_{16}$, and $\{x,x^8,y,x^5y\}$ if $G=D_{20}$. Now suppose $G=D_{12}$. If $\circ(a_1)=3=\circ(a_2)$, then no such $S$ exist since $a_2=a_1^{2}$. If $\circ(a_1)=6=\circ(a_2)$, then $a_2=a_1^{-1}$. Such $S$ exists, and must be mapped by an automorphism of $D_{12}$ into $\{x,x^5,y,x^3y\}$. If $\circ(a_1) \neq \circ(a_2)$. Any resulting product-free set is locally maximal, and must be mapped by an automorphism of $G$ into $\{x,x^4,y,x^3y\}$. Finally, in the case $G=D_{10}$ or $D_{14}$, the result follows from Corollary \ref{AH2015A_C}.
\\
Case II: Let $S=\{a_1,a_2,b,z\}$, where $a_1,a_2$ are non-involutions, $b$ is a reflection and $z$ is the unique involution in Rot($G$). As $|$Ref($T(S)\cup \sqrt{S}$)$|\leq |\{zb,a_1b,a_2b,a_1^{-1}b,a_2^{-1}b\}|$, $|G|\leq 10$. So $S$ can only exist in $D_{10}$. By Proposition \ref{AH2015A_P}, no such $S$ exist in $D_{10}$.
\end{proof}
\begin{lem}\label{L4}
There is no locally maximal product-free set of size $4$ consisting of at most one involution in a finite dihedral group.
\end{lem}
\begin{proof}
Any locally maximal product-free set $S$ of size $4$ must contain at least one reflection; otherwise $T(S) \cup \sqrt{S} \subseteq \langle S \rangle$ which is cyclic. Now, suppose $S=\{a_1,a_2,a_3,b\}$, where $a_1,a_2$ and $a_3$ are non-involutions, and $b$ is a reflection. As $|$Ref($T(S)\cup \sqrt{S}$)$|\leq |\{a_1b,a_2b,a_3b,$ $a_1^{-1}b,a_2^{-1}b,a_3^{-1}b\}|=6$, and a LMPFS of size $4$ cannot be contained in a group of order less than $8$, we must have that $8\leq |G|\leq 12$. By Propositions \ref{AH2015A_Pro} and \ref{AH2015A_P}, no such $S$ exists in $D_{8}$ and $D_{10}$ respectively. As no three non-involutions can form a locally maximal product-free set in $C_6$, no such $S$ exist in $D_{12}$.
\end{proof}
\noindent We are now in a position to classify all locally maximal product-free sets of size 4 in dihedral groups.
\begin{thm}\label{R2}
Suppose $S$ is a LMPFS of size $4$ in a dihedral group $G$. Then up to automorphisms of $G$, the possible choices are given as follows:
\begin{center}
\begin{tabular}{|p{0.5cm}|p{10.5cm}|}\hline
$|G|$ & $\qquad \qquad \qquad \quad S$\\ \hline
$8$ & $\{y,xy,x^2y,x^3y\}, \{x,x^3,y,x^2y\}$\\ \hline
$10$ & $\{x^2,x^3,y,x^4y\}$\\ \hline
$12$ & $\{x^3,y,xy,x^2y\}, \{x^2,x^3,y,x^5y\}, \{x,x^5,y,x^3y\}, \{x,x^4,y,x^3y\}$ \\ \hline
$14$ & $\{x^2,x^3,y,x^6y\}$\\ \hline
$16$ & $\{x^2,x^3,y,x^7y\}, \{x,x^6,y,x^4y\}$\\ \hline
$18$ & $\{x^2,x^5,y,x^8y\}$\\ \hline
$20$ & $\{x,x^8,y,x^5y\}$\\ \hline
\end{tabular}
\end{center}
\end{thm}
\begin{proof}
By Theorem \ref{AH2015A_T} and the fact that a locally maximal product-free set of size $4$ cannot be contained in a group of order less than $8$, we must have that $8\leq|G|\leq 20$. If $G=D_8$, then by Proposition \ref{AH2015A_Pro}, either $S=\{y,xy,x^2y,x^3y\}$ or it is mapped by an automorphism of $G$ into $\{x,x^3,y,x^2y\}$. Now, suppose $10\leq |G|\leq 20$. The result follows from Proposition \ref{P1}, Lemma \ref{L2}, Proposition \ref{P3} and Lemma \ref{L4}.
\end{proof}
\section{Filled dihedral groups}
In this section we obtain some facts about filled dihedral groups.
In \cite{SW1974} the authors asserted that the dihedral group of order $2n$ is not a filled group for $n=6k+1$. They went further to produce a locally maximal product-free set ($S:=\{x^{2k+1},\dots,x^{4k},x^{2k+1}y,\dots,x^{4k}y\}$) which they claim does not fill $D_{2n}$. However we have the following.
\begin{pro}\label{P5}
Let $G$ be a dihedral group of order $2n$ for $n=6k+1$ and $k\geq 1$. Then the set $S:=\{x^{2k+1},\dots,x^{4k},x^{2k+1}y,\dots,x^{4k}y\}$ is product-free but not locally maximal in $G$.
\end{pro}
\begin{proof}
The fact that $S$ is product-free follows from our proof since every subset of a product-free set is product-free. So, we only show that $S$ is not locally maximal. To do this, we show that the set $V:=\{x^{2k+1},\dots,x^{4k},x^{2k}y,x^{2k+1}y,\dots,x^{4k}y\}$, which properly contains $S$, is product-free. (One may also do same using $U:=\{x^{2k+1},\dots,x^{4k},x^{2k+1}y,$ $\dots,x^{4k}y,x^{4k+1}y\}$.) Let $A = $ Rot($V$) and $B = $ Ref($V$). We note that $V = V^{-1}$ and so $BA = A^{-1}B = AB$. Therefore $VV = AA \cup AB \cup BA \cup BB = AA \cup BB \cup AB$. Thus $$VV = \{1,x,\cdots,x^{2k}\}\sqcup \{x^{4k+1},x^{4k+2},\cdots,x^{6k}\} \sqcup \{y,xy,\cdots,x^{2k-1}y\}\sqcup \{x^{4k+1}y,x^{4k+2}y,\cdots,x^{6k}y\}.$$ As $V\cap VV=\varnothing$, the set $V$ is product-free.
\end{proof}
Incidentally, we note here that our Proposition \ref{MR3} shows that the list of locally maximal product-free sets of size $4$ in $D_{14}$ given in Table $1$ of \cite{SW1974} is not correct. In particular, the authors claimed that $S=\{a,ab,a^3b,a^6b\}$ is locally maximal. However, this is not true as $S$ is contained in a product-free set of size $5$; viz. $\{a,a^{-1},ab,a^3b,a^6b\}$.
\begin{rem}\label{R6}
Observe that $G=V \sqcup VV$ in the proof of Proposition \ref{P5} above. Thus, $V$ fills $G$. By Lemma \ref{GH2009L} therefore, $V$ is a locally maximal product-free subset of $D_{12n+2}$.
\end{rem}
\noindent We give (without proof) Proposition \ref{P7} and Lemma \ref{L8}, whose proofs are similar to those in Section~3.
\begin{pro}\label{P7}
Up to automorphisms of $D_{14}$, the only locally maximal product free set of size $5$ in $D_{14}$ is $V$, where $V$ is as defined in the proof of Proposition \ref{P5}.
\end{pro}
\begin{lem}\label{L8}
There is no locally maximal product-free set of size $6$ in $D_{14}$.
\end{lem}
\noindent The following result together with Proposition \ref{P5} disprove the stated conjecture.
\begin{thm}\label{T9}
$D_{14}$ is a filled group.
\end{thm}
\begin{proof}
From our discussion in Section~3, if $S$ is a locally maximal product-free set in $D_{14}$, then $4\leq |S|\leq 7$. If $|S|=4$, then by Theorem \ref{R2}, $S$ is mapped by an automorphism of $D_{14}$ into $W:=\{x^2,x^3,y,x^6y\}$. As $WW=\{1,x,x^4,x^5,x^6,xy,x^2y,x^3y,x^4y,x^5y\}$, the set $W$ fills $D_{14}$. By Remark \ref{R6} and Proposition \ref{P7}, any locally maximal product-free set of size $5$ in $D_{14}$ fills the group. By Lemma \ref{L8}, there is no locally maximal product-free set of size $6$ in $D_{14}$. By Proposition \ref{AH2015A_Pro}, the only locally maximal product-free set of size $7$ in $D_{14}$ is $M_1$, which by definition, fills $D_{14}$. Since every locally maximal product-free set in $D_{14}$ fills $D_{14}$, therefore $D_{14}$ is a filled group.
\end{proof}
\noindent The disproved conjecture of Street and Whitehead left us with no other known example of a dihedral group which is not a filled group. We show that such examples exist as follows:
\begin{thm}\label{last}
If $S$ is a locally maximal product-free set of size $k\geq 3$ in a finite dihedral group $G$ of order $k(k+1)$, then $S$ does not fill $G$.
\end{thm}
\begin{proof}
Suppose $S$ is a locally maximal product-free set of size $k\geq 3$ in a finite dihedral group $G$ of order $k(k+1)$. As $|SS|\leq k^2$, and $S\cap SS = \varnothing$, the set $S\sqcup SS$ fills $G$ if and only if $|S\sqcup SS|=k^2+k$. As $|S|\geq 3$, either $S$ contains two rotations or two reflections. If $S$ contains two reflections $b_1$ and $b_2$, then as $b_1^2=b_2^2=1$, we must have that $|SS|<k^2$. On the other hand, if $S$ contains two rotations $a_1$ and $a_2$, then as $a_1a_2=a_2a_1$, we must have that $|SS|<k^2$. In either case, $|S\sqcup SS|<k^2+k$; so $S$ does not fill $G$.
\end{proof}
An example of the construction given in Theorem \ref{last} exists in $D_{20}$ as $S=\{x,x^8,y,x^5y\}$ is locally maximal in $D_{20}$ but does not fill the group. Thus, not every dihedral group is a filled group.
\begin{rem}
\noindent Street and Whitehead in \cite{SW1974} and \cite{SW1974A} pointed out that any dihedral group of order less than $14$ is a filled group. We have shown that $D_{14}$ is also filled. In fact, the first example of a non-filled dihedral group is $D_{16}$. By Theorem \ref{R2}, the set $Y:=\{x,x^6,y,x^4y\}$ is locally maximal in $D_{16}$. However, $|Rot(Y\sqcup YY)|=6<8$.
\end{rem}
We make the following more general observation.
\begin{pro}\label{dih1} If $8$ divides $n$, then $D_{2n}$ is not filled. In particular, the only filled dihedral 2-groups are $D_4$ and $D_8$.
\end{pro}
\begin{proof} Suppose $8$ divides $n$. Let $H = \langle x^{8}\rangle $. Then $H$ is a normal subgroup of $D_{2n}$ whose quotient is dihedral of order 16. By Lemma \ref{swl1}, and the fact that $D_{16}$ is not filled, we see that $D_{2n}$ is not filled.
\end{proof}
We finish this paper with a table giving the known filled groups of order up to 31.
\begin{table}[h!]
\begin{center} \begin{tabular}{c|c} Order & Groups\\
\hline 2 & $C_2$\\
3 & $C_3$\\
4 & $C_2 \times C_2$\\
5 & $C_5$\\
6 & $D_6$\\
8 & $C_2^3$, $D_8$\\
10 & $D_{10}$\\
12 & $D_{12}$\\
14 & $D_{14}$\\
16 & $C_2^4$, $D_8 \times C_2$\\
22 & $D_{22}$
\end{tabular}
\caption{Filled Groups of Order less than 32}\label{table1}\end{center}
\end{table}
Table \ref{table1} was calculated using GAP, along with the results obtained in this paper. For example we only needed to check nonabelian groups of even order which have no normal subgroups of index 3. We can see that the known nonabelian filled groups are either 2-groups or dihedral (or both). The same reasoning as in Proposition \ref{dih1}, applied to the dihedral groups in Table \ref{table1} implies that $D_{2n}$ is not filled if $n$ is divisible by $8, 9, 10, 12, 13, 14$ or $15$, but we are not able to fully classify the filled dihedral groups.
\end{document}
|
\begin{document}
\title{PR-CIM: a Variation-Aware Binary-Neural-Network Framework for Process-Resilient Computation-in-memory}
\author{
Minh-Son Le\\
\texttt{[email protected]}
\and
Thi-Nhan Pham\\
\texttt{[email protected]}
\and
Thanh-Dat Nguyen\\
\texttt{[email protected]}
\and
Ik-Joon Chang\\
\texttt{[email protected]}
}
\maketitle
\begin{abstract}
Binary neural networks (BNNs) that use 1-bit weights and activations have garnered interest as extreme quantization provides low power dissipation. By implementing BNNs as computing-in-memory (CIM), which computes multiplication and accumulations on memory arrays in an analog fashion, namely analog CIM, we can further improve the energy efficiency to process neural networks. However, analog CIMs suffer from the potential problem that process variation degrades the accuracy of BNNs. Our Monte-Carlo simulations show that in an SRAM-based analog CIM of VGG-9, the classification accuracy of CIFAR-10 is degraded even below 20\% under process variations of 65nm CMOS. To overcome this problem, we present a variation-aware BNN framework. The proposed framework is developed for SRAM-based BNN CIMs since SRAM is most widely used as on-chip memory, however easily extensible to BNN CIMs based on other memories. Our extensive experimental results show that under process variation of 65nm CMOS, our framework significantly improves the CIFAR-10 accuracies of SRAM-based BNN CIMs, from 10\% and 10.1\% to 87.76\% and 77.74\% for VGG-9 and RESNET-18 respectively.
\end{abstract}
\begin{IEEEkeywords}
BNN, Deep Neural Network, Computation-in-memory, SRAM, Polar Neural Network
\end{IEEEkeywords}
\section{Introduction} \label{introduction}
Deep neural networks (DNNs) have shown outstanding performance to surpass human-level accuracy in many applications such as image processing, voice recognition, and language translation. Many researchers have made hard efforts to deploy the inference of DNNs at resource-constraint edge devices, still challenging due to the following reasons. It is well-known that DNNs accompany myriad parameters, and hence, a large memory size is indispensable to operate DNNs efficiently. At resource-constraint edge devices, it is not easy to increase the size of embedded memories above a certain level, leading to many DRAM footprints. In addition, the computational overhead of DNNs is considerable as well. These result in substantial energy dissipation, the major hurdle of edge devices.
Recently, some researches have shown the possibility to clear the hurdle. One of the most representative researches is a binary neural network (BNN), where all weights and activations are binarized. Despite the extreme 1-bit quantization, BNN delivers reasonably good accuracy \cite{BNN, XNOR, binaryduo}. Due to the aggressive quantization, BNNs have a small model size and can be processed with small-sized embedded memories, ensuring low energy dissipation. The energy efficiency of BNNs is further improved by directly computing BNNs in embedded memories such as SRAM, e-FLASH, and STT-MRAM, namely computation-in-memory (CIM) \cite{BNN/TNN-SRAM, parallelizing_sram, XNOR-free, flash-BNN, MRIMA}. Most BNN CIM hardware computes multiplication and accumulations by using the concept of analog computing, so-called analog CIM, further enhancing energy efficiency. However, in reality, we should consider the potential problem that process variation significantly degrades the accuracy of BNNs to operate on analog CIM platforms. The low-resolution weights of BNNs tend to make BNNs prone to errors due to process variations, strongly motivating techniques to alleviate this problem.
This work presents a variation-aware BNN framework to deliver accurate analog CIM operations even under process variations of scaled technologies, based on the conventional 6T-SRAM. Recently, many emerging non-volatile memories (eNVM) based CIMs have gathered interest because of their high density and low standby power\cite{Device_Variation_crossArray, TernaryWeight-RRAM, flash-BNN, MRIMA, Fully_parallel_RRAM, XNOR-RRAM}. However, the eNVM-based CIMs face many challenges in manufacturing actual hardware, while SRAM has advantages in terms of the actual design perspective and hence, plays a dominant role in the CIM design. Considering such a situation, we develop the variation-aware BNN framework on the SRAM-based CIM. However, we can easily extend the developed framework for the CIMs of other memories.
The contribution of this paper can be summarized as follows.
\begin{itemize}
\item We derive mathematical models related to the effect of process variations on the SRAM-based BNN CIM circuit. In the CIM circuit, an SRAM cell current represents the multiplication results of the weight and activation corresponding to the SRAM cell. Due to parametric process variations, the SRAM cell current is varied, affecting analog computation. Our model regards such a situation as the weight variation of BNNs. We extracted the models of the weight variations by using Monte-Carlo (MC) simulations in 65nm CMOS.
\item Based on the derived model, we present the variation-aware BNN training framework to produce variation-resilient training results. During the training, BNNs are considered bi-polar neural networks due to the weight variations aforementioned. We demonstrate the efficacy of the developed framework through extensive simulations.
\item We optimize biasing voltages of bit-lines (BLs) and word-lines (WLs) of SRAM, further improving the accuracy of the SRAM-based CIM.
\end{itemize}
The remaining part of this paper is organized as follows. In section \ref{sec:pre}, we explain the background regarding BNN, the architecture of SRAM-based CIM, how DNNs can be mapped onto SRAM-based CIM arrays, and in-memory batch normalization. In section \ref{sec:framework}, we present the variation-aware framework and optimization methodology for biasing voltages of BLs and WLs of SRAM. Section \ref{sec:validation} validates the efficacy of our framework. Lastly, we conclude the paper in section \ref{sec:conclusion}.
\section{Preliminaries}\label{sec:pre}
\subsection{Binary Neural Network}\label{sec:bnn}
In BNN, all weights and activations are binarized, significantly enhancing the DNN inference energy efficiency. Many researchers have shown that despite such a low precision format, BNN delivers good inference accuracy \cite{BNN, XNOR, binaryduo}. The firstly introduced BNN \cite{BNN} uses the sign function for the binarization of both weight and activation, where all weights and activations become \textquoteleft+1' or \textquoteleft-1'. However, some state-of-the-art (SOTA) works improved the accuracy of BNN by using the activations of \textquoteleft0' or \textquoteleft1' \cite{binaryduo} while they still employ the sign function for the binarization of weights. Considering such a trend, we use the following activation function.
\begin{equation}
\label{eq_act_bin}
BinAct(X) = \begin{cases}
1, & X \geq thresh \\
0, & X < thresh
\end{cases}
\end{equation}
, where $thresh$ is the activation threshold. Our experiment results, where $thresh$ is assumed as \textquoteleft0.5', are shown in Table \ref{tbl:baseline}, showing the similar trend to SOTA works \cite{binaryduo} as well. Since we use the activation function of (\ref{eq_act_bin}), activations have the values of \textquoteleft0' or \textquoteleft1'.
\begin{table}[t]
\caption{Comparison of two binary activation cases, (+1/-1) and (1/0)}
\label{tbl:baseline}
\centering
\resizebox{1.0\linewidth}{!}{
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
\multicolumn{1}{|c|}{\multirow{2}{*}{Network}} & \multicolumn{1}{c|}{\multirow{2}{*}{Dataset}} & \multicolumn{1}{c|}{\multirow{2}{*}{\shortstack{Full\\precision}}} & \multicolumn{2}{c|}{BNN} & \multicolumn{2}{c|}{{Split BNN}}\\\cline{4-7}
&&&\makecell{(+1/-1)} & \makecell{(1/0)} & \makecell{(+1/-1)} & \makecell{(1/0)}\\
\hline
VGG-9 & CIFAR-10 & 93.71 & 89.77 & 91.36 & 88.51 & 88.70 \\
RESNET-18 & CIFAR-10 & 91.17 & 82.82 & 83.06 & 78.30 & 81.08 \\
\hline
\end{tabular}
}
\end{table}
\subsection{The Architecture of SRAM-based CIM}\label{sec:sram_config}
\begin{figure}
\caption{6T-SRAM based CIM architecture for a BNN and truth table of input neurons and weights.}
\label{fig:cell}
\end{figure}
\begin{figure}
\caption{Sense amplifier circuit}
\label{fig:SA}
\end{figure}
Fig. \ref{fig:cell} shows the most widely used 6T-SRAM based CIM architecture and cell configuration for the BNN computation, which refers to the design of Rui Liu et al. \cite{parallelizing_sram}. We consider such an architecture for our proposed BNN framework, discussed in section \ref{sec:framework}.
In Fig. \ref{fig:cell}, weights of BNNs are stored to 6T-SRAM cells, and the bitwise multiplications between weights and input activations of the network are directly computed with an analog fashion inside the SRAM array. Let us assume that the weight of \textquoteleft+1' is represented by Q = 1, QB = 0, and the weight of \textquoteleft-1' as the inverted cell. When we operate a BNN on the given configuration, the input activations of a certain BNN layer become the digital values of WLs, since the activation function of (\ref{eq_act_bin}) is considered in this work, as mentioned in section \ref{sec:bnn}. When the inference is executed, all WLs are biased upon the input activations and then, the product of the $i^{th}$ weight and the $i^{th}$ activation becomes the difference between the $bl$ and $blb$ cell currents, \textquoteleft$i_{cell\_bl\_i}$ - $i_{cell\_blb\_i}$' in Fig. \ref{fig:cell}. All cell currents are accumulated to the currents of BL and BLB, implying that \textquoteleft$I_{BL}$-$I_{BLB}$' becomes the multiply-and-accumulation (MAC) output. \textquoteleft$I_{BL}$-$I_{BLB}$' is sensed by the differential current sense-amplifier (CSA), producing the binary activation output based on (\ref{eq_act_bin}). Please, note that when $thresh$ of (\ref{eq_act_bin}) is not zero, a certain circuitry is necessary to implement this. Further, we need to implement batch normalization properly. These are embedded to the sense-amplifier of Fig. \ref{fig:SA}, discussed in section \ref{sec:bn}.
\begin{table}[t]
\caption{Number of split groups for VGG-9}
\label{tbl:number_of_groups_vgg9}
\centering
\resizebox{1.0\linewidth}{!}{
\begin{tabular}{|c|c|c|}
\hline
\multicolumn{1}{|c|}{\multirow{2}{*}{Layer}} & \multicolumn{1}{c|}{\multirow{2}{*}{Input count per output}} & \multicolumn{1}{c|}{\multirow{1}{*}{Number of groups}} \\
&&\makecell{(Input size = 256)} \\
\hline
1 & 3x3x3 & - \\
2 & 3x3x128 & 6 \\
3 & 3x3x128 & 6 \\
4 & 3x3x256 & 9 \\
5 & 3x3x256 & 9 \\
6 & 3x3x512 & 18 \\
7 & 8192 & 32 \\
8 & 1024 & 4 \\
9 & 1024 & - \\
\hline
\end{tabular}
}
\end{table}
\subsection{Mapping DNNs onto SRAM-based CIM arrays} \label{sec:mappingdnn}
\begin{table}[t]
\caption{Number of split groups for RESNET-18}
\label{tbl:number_of_groups_resnet18}
\centering
\resizebox{1.0\linewidth}{!}{
\begin{tabular}{|c|c|c|}
\hline
\multicolumn{1}{|c|}{\multirow{2}{*}{Layer}} & \multicolumn{1}{c|}{\multirow{2}{*}{Input count per output}} & \multicolumn{1}{c|}{\multirow{1}{*}{Number of groups}} \\
&&\makecell{(Input size = 256)} \\
\hline
1 & 3x3x3 & - \\
2$\rightarrow$7 & 3x3x16 & 1 \\
8$\rightarrow$13 & 3x3x32 & 2 \\
14$\rightarrow$19 & 3x3x64 & 3 \\
20 & 64 & - \\
\hline
\end{tabular}
}
\end{table}
\subsubsection{Input Splitting}\label{sec:splitting}
In the CIM architecture of Fig. \ref{fig:cell}, we store weights to SRAM and control the potentials of WL upon the input activations. Then, SRAM directly computes the matrix multiplications of convolution and fully-connected (FC) layers by using analog computing techniques. In such a scheme, the maximum matrix size that SRAM can calculate at once is dependent on the SRAM array size, which relies on the physical design constraints. Unfortunately, the computed matrix size often exceeds the SRAM array size. Fig. \ref{fig:mapping} shows such a situation well. Here, some convolution layers have 4-dimensional weights. We can regard a convolution layer with 4-dimensional weights as a 2-dimensional matrix with the size of $($kernel size $\times$ kernel size $\times$ input channel size$)$ $\times$ $($output channel size$)$. For instance, in Fig. \ref{fig:mapping}, the 4-dimensional convolution layer whose kernel, input channel, and output channel sizes are 3, 128, and 256, respectively, is considered as the 1152 $(=3 \times 3 \times 128) \times 256$ matrix. To compute the matrix on the circuit of Fig. \ref{fig:cell}, we need 1152 memory rows. Since it is challenging to implement the SRAM array with 1152 rows, we need to properly split the matrix by considering the SRAM array size. Under such a circumstance, an SRAM CIM circuit can deal with one split part of the matrix and produces the corresponding partial sum. All partial sums delivered by the SRAM CIM circuits should be accumulated to complete the matrix computation. SOTA works showed that the precision of the partial sums significantly affects the accuracy of the computed BNNs \cite{parallelizing_sram, XNOR-RRAM, bnn_hardware_codesign, Device_Variation_crossArray}. To obtain multi-bit partial sums in the SRAM CIM circuits, we need ADCs to produce multi-bit outputs, incurring large area and energy overhead.
To address this problem, the authors of \cite{bnn_hardware_codesign} developed an input splitting technique, which we employed in this work. A large convolution or FC layer is reconstructed into several smaller groups, as shown in Fig. \ref{fig:mapping}, whose input number should be smaller or equal to the number of rows in an SRAM array. Hence, the SRAM array of Fig. \ref{fig:cell} computes the weighted-sums of each group, and the CSAs produce their own 1-bit outputs. Then, the outputs of all groups are merged to fit the input size of the following BNN layer, which is done by digital machines to obtain accurate merging without the effect of process variations.
The accuracy based on the input splitting technique is compared with the accuracy of the baseline as shown in Table \ref{tbl:baseline}. We assume a typical SRAM array size, $256 \times 256$. The first layer of the BNN that processes the input image and the last layer of the BNN that computes the score of each class are excluded from the input splitting, computed by digital machines. Such an approach follows SOTA works \cite{BNN, XNOR, binaryduo, bnn_hardware_codesign} related to BNN. The split BNN accuracies of Table \ref{tbl:baseline} are considered as the baselines when our techniques are evaluated. For reference, we summarized the number of groups per BNN layer after input splitting on Table \ref{tbl:number_of_groups_vgg9} and Table \ref{tbl:number_of_groups_resnet18} for VGG-9 and RESNET-18 networks respectively.
\begin{figure}
\caption{SRAM-based CIM mapping}
\label{fig:mapping}
\end{figure}
\subsubsection{Mapping}\label{sec:mapping}
We make more detailed discussions regarding the mapping between convolution layers of BNNs and SRAM-based CIM arrays, shown in Fig. \ref{fig:mapping}. As aforementioned, convolution layers are split to ensure that their size is equal to or less than the number of rows in the SRAM array. As shown in Table \ref{tbl:number_of_groups_vgg9}, the number of split groups of the layer is six $(=\lceil1152/256\rceil+1)$, and the input channels per group are 21 $(=128/6)$ in VGG-9. Hence, the input size of each group is 189 $(=3\times3\times21)$, which is smaller than the number of rows in the SRAM array. Consequently, each group can be regarded as a 2-dimensional matrix with the size of $(3\times3\times21) \times (256)$. Under this circumstance, we can have the mapping strategy that all weights corresponding to each output channel are stored on one column of the SRAM array. The outputs of each group, which is binary (\textquoteleft0' or \textquoteleft1'), are obtained from the macros. We can manage FC layers with the above mapping strategy as well.
\subsection{In-memory batch normalization} \label{sec:bn}
\begin{table}[t]
\centering
\caption{Software to hardware conversion of $BnBinAct()$}
\label{tbl:bn_converted}
\resizebox{1.0\linewidth}{!}{
\begin{tabular}{|c|c|}
\hline
Software implementation & \makecell{Hardware implementation} \\
\hline
{\(BnBinAct(X) = \begin{cases}1, & X \geq X_{th} \\0, & X < X_{th} \end{cases} \)} & \makecell{\( Output(Y) = \begin{cases}1, & Y \geq X_{th} \times \textcolor{red}{IM} \\0, & Y < X_{th} \times \textcolor{red}{IM} \end{cases} \)} \\
& where $Y = I_{BL} - I_{BLB}$ \\
\hline
\end{tabular}
}
\end{table}
\begin{figure}
\caption{Batch normalization merging in inference phase}
\label{fig:bn_merged}
\end{figure}
Batch normalization (BN) is the technique to stabilize the learning process, significantly reducing the number of training epochs. In BNNs, BN highly affects the training accuracy \cite{bn_binary}, more critical. BN can be described by the following equation.
\begin{equation} \label{eq_bn}
Y = \gamma\frac{X - \mu}{\sqrt{\sigma^2 + \epsilon}} + \beta
\end{equation}
, where X and Y are the input and the output of BN, and $\gamma,\beta,\mu,\sigma$, and $\epsilon$ are a weight, a bias, a mean, a standard deviation, and a sufficiently small constant, respectively. During the back propagation of the training, these four parameters are updated and used to normalize the output of the current batch. In the inference, these parameters become constant, and hence, BN can be regarded as a linear transformation function. As shown in Fig. \ref{fig:bn_merged}, in the inference, the output of the BN layer becomes the input of the activation function (\ref{eq_act_bin}). In this work, we merge (\ref{eq_act_bin}) and (\ref{eq_bn}), whose function is named as $BnBinAct()$. The merged function can be expressed by
\begin{equation}\label{eq_bn_merged}
BnBinAct(X) = \begin{cases}
1, & X \geq X_{th} \\
0, & X < X_{th}
\end{cases}
\end{equation}
, where X is the output of weighted-sum layer, and
\begin{equation}
X_{th} = \frac{(thresh - \beta)}{\gamma}(\sqrt{\sigma^2 + \epsilon}) + \mu.
\end{equation}
Most previous works assume that BN is computed by software \cite{BNN/TNN-SRAM, parallelizing_sram, Fully_parallel_RRAM, XNOR-RRAM}. In such a scenario, ADCs should convert accumulated BL currents to high-precision digital values, which are fed to digital processors, not suitable for edge devices due to large energy overhead. To address this problem, in \cite{batch-merg} BN is implemented as additional cells. In this work, we simply handle the problem by implementing the merged function as the variable current biasing embedded to the differential CSA, shown in Fig. \ref{fig:SA}.
The biasing current values can be derived from the conversion rule of Table \ref{tbl:bn_converted}. Two current biasing, $I_{Thres\_Neg}$ and $I_{Thres\_Pos}$, are necessary to deal with both positive and negative $X_{th}$ cases. Since the major contribution of this work is to present a variation-aware framework, the detailed discussion regarding the operation of the CSA is not discussed.
\iffalse
The output of CSA is given by the following equation.
\begin{equation}\label{xth}
Output(I_{BL} - I_{BLB}) &= \begin{cases}
1, & I_{BL} - I_{BLB} \geq X_{th} \times \textcolor{red}{IM} \\
0, & I_{BL} - I_{BLB} < X_{th} \times \textcolor{red}{IM}
\end{cases}
\end{equation}
For the case $X_{th}>0$, \ref{xth} can be expressed by
\begin{equation}\label{xth}
\begin{split}
Output(I_{BL} - I_{BLB}) &= \begin{cases}
1, & I_{BL} - I_{BLB} - X_{th} \times \textcolor{red}{IM} \geq 0 \\
0, & I_{BL} - I_{BLB} - X_{th} \times \textcolor{red}{IM} < 0
\end{cases}
\end{split}
\end{equation}
\fi
\section{A Variation-aware Binary Neural Network Framework}\label{sec:framework}
\subsection{Variation-aware Models for SRAM-based BNN CIM}\label{sec:model}
\begin{figure}
\caption{Cell current distributions}
\label{fig:cell_dis}
\end{figure}
\begin{figure}
\caption{Weight distribution under process variations}
\label{fig:weight_dist}
\end{figure}
In this section, we present a variation-aware BNN framework to enhance the reliability of CIM under process variations. The framework assumes SRAM-based CIM, whose configuration is discussed in section \ref{sec:sram_config}. To develop such a BNN framework, firstly, variation-aware models are investigated and derived as follows.
\begin{figure}
\caption{Biasing voltages of SRAM optimization methodology}
\label{fig:flowchart}
\end{figure}
In the given configuration (Fig. \ref{fig:cell}), as discussed, the MAC output is defined by \textquoteleft$I_{BL}$-$I_{BLB}$', which described as
\begin{equation}
\label{eq_total_sram_output}
\begin{split}
I_{BL} - I_{BLB} &= \Sigma_{i=0}^{N-1}(i_{cell\_bl\_i} - i_{cell\_blb\_i}) \times WL_{i} \\
&= \Sigma_{i=0}^{N-1}((W_i) \times IM) \times WL_{i}
\end{split}
\end{equation}
, where $W_i$ is the $i^{th}$ weight stored in the SRAM array (i.e., $W_i$ is \textquoteleft+1' or \textquoteleft-1'), $WL_i$ is the $i^{th}$ word-line ($WL$) status (ON or OFF), which corresponds to activation values (`1' or `0'), and $IM$ is the current margin that is absolute value of difference current between BL and BLB for one cell (i.e., one bitwise-multiply operation), where no process variations are assumed. In reality, both \textquoteleft$i_{cell\_bl\_i}$' and \textquoteleft $i_{cell\_blb\_i}$' experiences process variations, which can be regarded as the variation of $W_i$ in (\ref{eq_total_sram_output}). Let us model the weight variation as $\Delta_{(W_i)}$. Consequently, the product of the $i^{th}$ weight and the $i^{th}$ activation, \textquoteleft$i_{cell\_bl\_i}$ - $i_{cell\_blb\_i}$', can be redefined as
\begin{equation}\label{eq_currentvariation}
\begin{split}
i_{cell\_bl\_i} - i_{cell\_blb\_i} = (W_i + \Delta_{(W_i)}) \times IM.
\end{split}
\end{equation}
In this work, the analysis of process variations was performed through 10,000 MC simulations using statistical models from 65nm CMOS. Without the loss of generality, for one cell configuration as in Fig. \ref{fig:cell} we investigate the case that stored weight as \textquoteleft+1' (i.e., Q = 1 and QB = 0). Therefore, when WL is ON (i.e., input neuron is 1), under process variations, the $bl$ and the $blb$ cell currents have the log-normal distributions of $LN(\mu_{bl}$, $\sigma^2_{bl})$ and $LN(\mu_{blb}, \sigma^2_{blb}$) as shown in Fig. \ref{fig:cell_dis}. Then, we can sequentially derive the following equations.
\begin{equation}\label{eq_Deltas}
\Delta_{(W_i)} = \begin{cases}
\Delta_{(+1)} = \frac{1}{IM} \times $($i_{cell\_bl\_i}$ - $i_{cell\_blb\_i}$) - 1$ \\
\Delta_{(-1)} = \frac{1}{IM} \times $($i_{cell\_blb\_i}$ - $i_{cell\_bl\_i}$) + 1$
\end{cases}
\end{equation}
Based on (\ref{eq_Deltas}) and the models that is taken from Fig. \ref{fig:cell_dis}, we obtain the weight distribution under process variations, illustrated in Fig. \ref{fig:weight_dist}. This shows that under process variations of the given CIM configuration, binary weights (-1/+1) are transformed to analog weights ($-1 + \Delta_{(-1)}$/$+1 + \Delta_{(+1)}$) of a BNN, whose distributions are log-normal.
\subsection{Variation-aware Framework for Bi-polar Neural Networks}\label{sec:training}
\begin{algorithm}[t]
\caption{Training a reconstructed L-layer BNN with variation-aware weights and activations. C is the cost function for minibatch, $\lambda$ - the learning rate decay factor and L the number of layers. $\circ$ indicates element-wise multiplication. The function Sign() specifies how to binarize the weights. Polarize() (\ref{eq_polarize}) is used to polarize the binarized one. The activations are clipped to [0, 1] by Clip() function. The function StoQuantize() (\ref{eq_sto_act}) specifies how to binarize the variation-aware activations. BatchNorm() and BackBatchNorm() defines how to batch-normalize and back-propagate the activations, respectively. Update() specifies how to update the parameters when their gradients are known. Straight-Through Estimator (STE) is used for estimating gradients for (\ref{eq_act_bin}) as in \cite{BNN}. Split() and Merge() functions are for input splitting and merging step, as discussed in section \ref{sec:mapping}. $ArraySize$ is the size of SRAM, which is set to 256.}
\label{algo1}
\begin{algorithmic}
\REQUIRE a minibatch of inputs and target $(a_0, a^*)$, previous weights W, previous BatchNorm parameters ($\gamma$, $\beta$), $ArraySize$, weights initialization coefficients from \cite{MSRA_initialization} $\alpha$, and previous learning rate $\eta$.
\ENSURE updated weights $W^{t+1}$, updated BatchNorm parameters ($\gamma^{t+1}$, $\beta^{t+1}$) and updated learning rate $\eta^{t+1}.$
\STATE {1. Computing the parameters gradients:}
\STATE {1.1 Forward propagation:}
\FOR{$k = 1$ to L}
\STATE // Input size per array
\STATE $InputSize = Kernel \times Kernel \times InputChannels$
\STATE // Number of groups
\STATE $nGroups = \lceil InputSize / ArraySize \rceil$
\WHILE{$InputSize \% nGroups \neq 0$}
\STATE $nGroups = nGroups + 1$
\ENDWHILE
\STATE // Input splitting
\STATE $a^b_{k-1} \leftarrow Split(a^b_{k-1},nGroups)$
\STATE $W_k \leftarrow Split(W_k,nGroups) $
\FOR{$i = 1$ to nGroups}
\STATE $W_k^b[i] \leftarrow Sign(W_k[i])$
\STATE $W_k^b[i] \leftarrow Polarize(W_k^b[i])$
\STATE $s_k[i] \leftarrow a^b_{k-1}[i]W^b_k[i]$
\ENDFOR
\STATE $a_k \leftarrow BatchNorm(s_k, \gamma_k, \beta_k)$
\IF{$k < L$}
\STATE $a_k \leftarrow Clip(a_k, 0, 1)$
\STATE $a^b_k \leftarrow StoQuantize(a_k)$
\STATE $a^b_k \leftarrow Merge(a^b_k,nGroups)$
\STATE $a^b_k \leftarrow BinAct(a^b_k)$
\ENDIF
\ENDFOR
\end{algorithmic}
\end{algorithm}
\begin{algorithm}[htpb]
\begin{algorithmic}
\caption*{\textbf{Algorithm} \textbf{\ref{algo1}} [Continued]}
\STATE {1.2 Backward propagation:}
\STATE Compute $g_{a_L} = \frac{\partial C}{\partial a_L}$ knowing $a_L$ and $a^*$
\FOR{$k = L$ to 1}
\IF{$k < L$}
\STATE $g_{a_k} \leftarrow g_{a^b_k} \circ 1_{0 \leq a_k \leq thresh}$ (STE)
\ENDIF
\STATE $(g_{s_k}, g_{\gamma_k}, g_{\beta_k}) \leftarrow BackBatchNorm(g_{a_k}, s_k, \gamma_k, \beta_k)$
\FOR{$i = 1$ to nGroups}
\STATE $g_{a^b_{k-1}[i]} \leftarrow g_{s_k[i]}W^b_k[i]$
\STATE $g_{W^b_k[i]} \leftarrow g^T_{s_k[i]}a^b_{k-1}[i]$
\ENDFOR
\ENDFOR
\STATE {2. Accumulating the parameters gradients:}
\FOR{$k = 1$ to L}
\STATE $\gamma^{t+1}_k \leftarrow Update(\gamma_k, \eta, g_{\gamma_k})$
\STATE $\beta^{t+1}_k \leftarrow Update(\beta_k, \eta, g_{\beta_k})$
\FOR{$i = 1$ to nGroups}
\STATE $W^{t+1}_k[i] \leftarrow Update(W_k[i], \alpha_k[i]\eta, g_{W^b_k[i]})$
\ENDFOR
\STATE $\eta^{t+1} \leftarrow \lambda\eta$
\ENDFOR
\end{algorithmic}
\end{algorithm}
The discussion of section \ref{sec:model} shows that with the effect of process variations, each weight stored in memory array experience process variations with the weight variation as $\Delta_{(W_i)}$. Then, the weight stored in each SRAM cell is not an exact digital value of +1 or -1 but can be redefined as
\begin{equation}\label{eq_polarize}
Polarize(W_i) = \begin{cases}
+1 + \Delta_{(+1)}, & \text{when } W_i = +1 \\
-1 + \Delta_{(-1)}, & \text{when } W_i = -1
\end{cases}
\end{equation}
, where $W_i$ is a binarized weight, and $\Delta_{(+1)}$ and $\Delta_{(-1)}$ are random stochastic parameters to express the effect of process variations, whose distributions are obtained from (\ref{eq_Deltas}). Our training framework is described as Algorithm \ref{algo1}, where the function of (\ref{eq_polarize}) is exploited. In the variation-aware training, we train BNNs based on Algorithm \ref{algo1} from scratch.
Please note that due to process variations of CSA, the activation threshold of (\ref{eq_act_bin}) can be varied. By taking into consideration this, in Algorithm \ref{algo1} we employ the stochastic activation function, whose equation is given by (\ref{eq_sto_act}), instead of the deterministic activation function of (\ref{eq_act_bin}).
\begin{equation}\label{eq_sto_act}
StoQuantize(X) = \begin{cases}
1, & X \geq (thresh + \Delta_{act}) \\
0, & X < (thresh + \Delta_{act})
\end{cases}
\end{equation}
with
\begin{equation}\label{eq_delta_act}
\Delta_{act} \sim N(0, stddev)
\end{equation}
, where the standard deviation of $\Delta_{act}$ is properly assumed.
When the training step is completed, only binarized weights are left for the inference and the SRAM-based CIM. However, in the inference, the quantized weights need to be polarized again to evaluate the effect of process variation. Since the stochastic function of (\ref{eq_polarize}) is used in the variation-aware inference, we executed the inference 100 times, whose distribution is observed.
\subsection{Optimization of Biasing Voltages}\label{optimization}
In this section, we propose the optimization methodology for biasing voltages of WLs and BLs of SRAM, respectively expressed as $V_{WL}$ and $V_{BL}$, which is shown in Fig. \ref{fig:flowchart}. This methodology provides steps to find the optimal biasing voltages, which delivers the best accuracy. Firstly, the $V_{WL}$ and $V_{BL}$ configuration is set to run MC circuit simulations of the SRAM cell. If a cell flipping occurs, the $V_{WL}$ and $V_{BL}$ configuration is discarded to ensure the accuracy for BNNs. If a cell flipping does not happen, mean and variance of $i_{cell\_bl}$ and $i_{cell\_blb}$ distributions are fed to the variation-aware BNN framework (Section \ref{sec:training}). After conducting the variation-aware training and variation-aware inferences, the average accuracy is collected and compared with those of other $V_{WL}$ and $V_{BL}$ configurations.
\begin{figure}
\caption{Average inference accuracy before variation-aware training of RESNET-18 (full-precision shortcut) on CIFAR-10}
\label{fig:before_resnet18}
\end{figure}
\begin{figure}
\caption{Average inference accuracy before variation-aware training of VGG-9 on CIFAR-10}
\label{fig:before_vgg9}
\end{figure}
\begin{figure}
\caption{Average inference accuracy after variation-aware training of RESNET-18 (full-precision shortcut) on CIFAR-10}
\label{fig:after_resnet18}
\end{figure}
\begin{figure}
\caption{Average inference accuracy after variation-aware training of VGG-9 on CIFAR-10}
\label{fig:after_vgg9}
\end{figure}
\subsection{Modeling of IR-drop}\label{sec:ir_drop}
The resistance of the power lines in an SRAM array causes IR-drop, causing the drop of supply voltages. Such an effect is not considered in our experiments. However, we can easily model the drop effect by applying lower supply voltages in our MC simulations.
\section{Validation of Our Framework}\label{sec:validation}
\subsection{Experimental Setting}\label{sec:setting}
We evaluate the efficacy of our proposed framework with different biasing voltages of BLs and WLs in 6T-SRAM. Under process variations of 65nm CMOS, we estimate the average inference accuracies before and after variation-aware training of RESNET-18 and VGG-9 with CIFAR-10.
In training, the loss is minimized with the algorithm of \cite{adam}, the initial learning rate is set to 0.01, and the maximum number of epochs is 150 for VGG-9 and 250 for RESNET-18. We decay the learning rate by 0.31 when scalar statistics of validation accuracy do not change enough. The variation-aware inferences are 100 times executed with the batch size of 1 for 50000 validation images, whose accuracies are averaged. In (\ref{eq_delta_act}), the standard deviation is assumed as 10\% of $thresh$ in (\ref{eq_act_bin}) and (\ref{eq_sto_act}).
For the baseline, we do not consider the effect of process variation while the input splitting technique, mentioned in section \ref{sec:mapping}, is used. For RESNET-18, we assume that short-cuts have the data format of full-precision. Many SOTA works \cite{binaryduo} employed such an approach since the accuracy of RESNETs is sensitive to the quantization errors of short-cuts, which is followed in this work.
\subsection{Results and Discussion}\label{sec:results}
The average inference accuracy before applying variation-aware training of both RESNET-18 and VGG-9, shown in Fig. \ref{fig:before_resnet18} and \ref{fig:before_vgg9}, clearly show that in SRAM-based analog CIMs, the classification accuracy of CIFAR-10 is severely degraded by process variations, even below 20\% for all $V_{WL}$ and $V_{BL}$ voltage configurations under consideration.
Our variation-aware training framework addresses this problem. Fig. \ref{fig:after_resnet18} shows the inference accuracies of RESNET-18 after using the variation-aware training, where the effect of process variations is considered for the inference as well. The results demonstrate that our variation-aware training framework significantly improves the accuracy under process variations. For instance, when $V_{WL}$=0.9V and $V_{BL}$=0.4V, the accuracy was 10\% under process variations of 65nm CMOS (Fig. \ref{fig:before_resnet18}). Our variation-aware training framework provides a quantum leap for the accuracy of this $V_{WL}$ and $V_{BL}$ biasing condition, to 77.74\%.
We have similar results in VGG-9. Our variation-aware training framework supports remarkable accuracy improvement under the effect of process variations. For instance, in Fig. \ref{fig:before_vgg9}, the accuracies are 10\% for two biasing cases of $V_{WL}$=0.4/$V_{BL}$=0.3, and $V_{WL}$=0.9/$V_{BL}$=0.4 while, in Fig. \ref{fig:after_vgg9}, the accuracies corresponding to the above two cases become 74.51\% and 87.76\%. This well validates the efficacy of our variation-aware training framework.
As illustrated in Fig. \ref{fig:after_resnet18} and \ref{fig:after_vgg9}, the accuracy under process variations tends to increase as $V_{WL}$ increases. It is since cell currents are larger with the higher $V_{WL}$, providing better immunity to process variations. However, when $V_{WL}$ is above a certain level, some SRAM cells are flipped due to a negative read static noise margin \cite{static_noise_margin} (In Fig. \ref{fig:before_resnet18}, \ref{fig:before_vgg9}, \ref{fig:after_resnet18} and \ref{fig:after_vgg9}, the biasing case with the SRAM cell flipping is marked with ``Flipped".). It significantly degrades the accuracy. Considering this factor, we decide the optimal biasing point of $V_{WL}$ and $V_{BL}$, which is the case that $V_{WL}$=0.9V and $V_{BL}$=0.4V. At this point, the CIFAR-10 accuracies of RESNET-18 and VGG-9 are 77.74\% and 87.76\%, respectively.
\section{Conclusion}\label{sec:conclusion}
By directly computing BNNs in embedded memories, namely computation-in-memory (CIM), we can obtain the utmost energy efficiency for edge devices. However, such an approach suffers from considerable accuracy degradation due to process variation. We present a variation-aware BNN framework on a configuration of SRAM-based CIM, whose efficacy is validated by extensive simulations.
\end{document}
|
\begin{document}
\mathbf{m}aketitle
\begin{abstract}
Over-parameterized residual networks are amongst the most successful convolutional neural architectures for image processing. Here we study their properties through their Gaussian Process and Neural Tangent kernels. We derive explicit formulas for these kernels, analyze their spectra and provide bounds on their implied condition numbers. Our results indicate that (1) with ReLU activation, the eigenvalues of these residual kernels decay polynomially at a similar rate as the same kernels when skip connections are not used, thus maintaining a similar frequency bias; (2) however, residual kernels are more locally biased. Our analysis further shows that the matrices obtained by these residual kernels yield favorable condition numbers at finite depths than those obtained without the skip connections, enabling therefore faster convergence of training with gradient descent.
\mathbf{m}athbf{e}nd{abstract}
\mathbf{s}ection{Introduction}
In the past decade, deep convolutional neural network (CNN) architectures with hundreds and even thousands of layers have been utilized for various image processing tasks. Theoretical work has indicated that shallow networks may need exponentially more nodes than deep networks to achieve the same expressive power \citep{telgarsky2016benefits, poggio2017and}. A critical contribution to the utilization of deeper networks has been the introduction of Residual Networks \citep{he2016deep}.
To gain an understanding of these networks, we turn to a recent line of work that has made precise the connection between neural networks and kernel ridge regression (KRR) when the width of a network (the number of channels in a CNN) tends to infinity. In particular, for such a network $f(\mathbf{x};\theta)$, KRR with respect to the corresponding Gaussian Process Kernel (GPK) $\mathbf{m}athcal{K}(\mathbf{x},\mathbf{z})=\mathbf{m}athbb{E}_{\theta}[f(\mathbf{x};\theta) \cdot f(\mathbf{z};\theta)]$ (also called Conjugate Kernel or NNGP Kernel) is equivalent to training the final layer while keeping the weights of the other layers at their initial values \citep{lee2017deep}. Furthermore, KRR with respect to the neural tangent kernel $\Theta\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)=\mathbf{m}athbb{E}_\theta\left[\left\langle \mathbf{m}athbf{f}rac{\partial f\left(\mathbf{x};\theta{\textnormal{i}}ght)}{\partial\theta},\mathbf{m}athbf{f}rac{\partial f\left(\mathbf{z};\theta{\textnormal{i}}ght)}{\partial\theta}{\textnormal{i}}ght{\textnormal{a}}ngle {\textnormal{i}}ght]$ is equivalent to training the entire network \citep{jacot2018neural}. Here $\mathbf{x}$ and $\mathbf{z}$ represent input data items, $\theta$ are the network parameters, and expectation is computed with respect to the distribution of the initialization of the network parameters.
We distinguish between four different models; Convolutional Gaussian Process Kernel (CGPK), Convolutional Neural Tangent Kernel (CNTK), and ResCGPK, ResCNTK for the same kernels with additional skip connections.
\citet{yang2020tensor, yang2021tensor} showed that for any architecture made up of convolutions, skip-connections, and ReLUs, in the infinite width limit the network converges almost surely to its NTK. This guarantees that sufficiently over-parameterized ResNets converge to their ResCNTK.
\citet{lee2019wide, lee2020finite} showed that these kernels are highly predictive of finite width networks as well. Therefore, by analyzing the spectrum and behavior of these kernels at various depths, we can better understand the role of skip connections.
Thus the question of what we can learn about skip connections through the use of these kernels begs to be asked. In this work, we aim to do precisely that. By analyzing the relevant kernels, we expect to gain information that is applicable to finite width networks.
Our contributions include:
\begin{enumerate}
\item A precise closed form recursive formula for the Gaussian Process and Neural Tangent Kernels of both equivariant and invariant convolutional ResNet architectures.
\item A spectral decomposition of these kernels with normalized input and ReLU activation, showing that the eigenvalues decay polynomially with the frequency of the eigenfunctions.
\item A comparison of eigenvalues with non-residual CNNs, showing that ResNets resemble a weighted ensemble of CNNs of different depths, and thus place a larger emphasis on nearby pixels than CNNs.
\item An analysis of the condition number associated with the kernels by relating them to the so called double-constant kernels. We use these tools to show that skip connections speed up the training of the GPK.
\mathbf{m}athbf{e}nd{enumerate}
Derivations and proofs are given in the Appendix.
\mathbf{s}ection{Related Work}
The equivalence between over-parameterized neural networks and positive definite kernels was made precise in \citep{lee2017deep, jacot2018neural, allen2019convergence, lee2019wide, chizat2019lazy, yang2020tensor} amongst others.
\citet{arora2019exact} derived NTK and GPK formulas for convolutional architectures and trained these kernels on CIFAR-10. \citet{arora2019harnessing} showed subsequently that CNTKs can outperform standard CNNs on small data tasks.
A number of studies analyzed NTK for fully connected (FC) architectures and their associated Reproducing Kernel Hilbert Spaces (RKHS). These works showed for training data drawn from a uniform distribution over the hypersphere that the eigenvalues of NTK and GPK are the spherical harmonics and with ReLU activation the eigenvalues decay polynomially with frequency \citep{bietti2020deep}.
\citet{bietti2019inductive} further derived explicit feature maps for these kernels. \citet{geifman2020similarity} and \citet{chen2020deep} showed that these kernels share the same functions in their RKHS with the Laplace Kernel, restricted to the hypersphere.
Recent works applied spectral analysis to kernels associated with standard convolutional architectures that include no skip connections. \citep{geifman2022spectral} characterized the eigenfunctions and eigenvalues of CGPK and CNTK. \cite{xiao2022eigenspace,cagnetta2022wide} studied CNTK with non-overlapped filters, while \cite{xiao2022eigenspace} focused on high dimensional inputs.
Formulas for NTK for residual, \tilde{p}h{fully connected} networks were derived and analyzed in \citet{huang2020deep,tirer2022kernel}. They further showed that, in contrast with FC-NTK and with a particular choice of balancing parameter relating the skip and the residual connections, ResNTK does not become degenerate as the depth tends to infinity. As we mention later in this manuscript, this result critically depends on the assumption that the last layer is \tilde{p}h{not} trained. \citet{belfer2021spectral} showed that the eigenvalues of ResNTK for fully connected architectures decay polynomially at the same rate as NTK for networks without skip connections, indicating that residual and conventional FC architectures are subject to the same frequency bias.
In related works, \citep{du2019gradient} proved that training over-parametrized convolutional ResNets converges to a global minimum. \citep{balduzzi2017shattered,philipp2018gradients,orhan2017skip} showed that deep residual networks better address the problems of vanishing and exploding gradients compared to standard networks, as well as singularities that are present in these models. \citet{veit2016residual} made the empirical observation that ResNets behave like an ensemble of networks. This result is echoed in our proofs, which indicate that the eigenvalues of ResCNTK are made of weighted sums of eigenvalues of CNTK for an ensemble of networks of different depths.
Below we derive explicit formulas and analyze kernels corresponding to \tilde{p}h{residual, convolutional} network architectures. We provide lower and upper bounds on the eigenvalues of ResCNTK and ResCGPK. Our results indicate that these residual kernels are subject to the same frequency bias as their standard convolutional counterparts. However, they further indicate that residual kernels are significantly more locally biased than non-residual kernels. Indeed, locality has recently been attributed as a main reason for the success of convolutional networks \citep{shalev2020computational, favero2021locality}. Moreover, we show that with the standard choice of constant balancing parameter used in practical residual networks, ResCGPK attains a better condition number than the standard CGPK, allowing it to train significantly more efficiently. This result is motivated by the work of \citet{lee2019wide, xiao2020disentangling} and \citet{chen2021neural}, who related between the condition number of NTK and the trainability of corresponding finite width networks.
\mathbf{s}ection{Preliminaries}
We consider mutli-channel 1-D input signals $\mathbf{x}\in\mathbf{m}athbb{R}^{C_0\times d}$ of length $d$ with $C_0$ channels. We use 1-D input signals to simplify notations and note that all our results can naturally be extended to 2-D signals. Let $\mathbf{m}athbb{MS}\left( C_{0},d{\textnormal{i}}ght) = \underset{d\text{ times }}{\underbrace{\mathbf{m}athbb{S}^{ C_{0}-1}\times\ldots\times\mathbf{m}athbb{S}^{ C_{0}-1}}}\mathbf{s}ubseteq\mathbf{s}qrt{d}\,\mathbf{m}athbb{S}^{d C_0-1}$ be the multi-sphere, so $\mathbf{x}=(\mathbf{x}_1,...,\mathbf{x}_d)\in\mathbf{m}athbb{MS}\left( C_{0},d{\textnormal{i}}ght)$ iff $\mathbf{m}athbf{f}orall i\in [d],\mathbf{n}orm{\mathbf{x}_i}=1$. \mathbf{m}athbf{f}v{For our analysis, we assume that the input signals are distributed uniformly on the multi-sphere}.
The discrete convolution of a filter $\mathbf{w}\in\mathbf{m}athbb{R}^{q}$ with a vector ${\bm{v}}\in\mathbf{m}athbb{R}^{d}$ is defined as
$[\mathbf{w}*{\bm{v}}]_{i}=\mathbf{s}um_{j=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}[\mathbf{w}]_{j+\mathbf{m}athbf{f}rac{q+1}{2}}[{\bm{v}}]_{ i+j}$, where $1\leq i\leq d$.
We use circular padding, so indices $[{\bm{v}}]_j$ with $j \le 0$ and $j > d$ are well defined.
We use multi-index notation denoted by bold letters, i.e., $\mathbf{n},\mathbf{m}athbf{k}\in\mathbf{m}athbb{N}^d$, where $N$ is the set of natural numbers including zero. $b_{\mathbf{n}},\lambda_{\mathbf{m}athbf{k}}\in\mathbf{m}athbb{R}$ are scalars that depend on $\mathbf{n},\mathbf{m}athbf{k}$, and for $\mathbf{t}\in\mathbf{m}athbb{R}^d$ we let $\mathbf{t}^{\mathbf{n}}=t_1^{n_1}\cdot...\cdot t_d^{n_d}$. As is convention, we say that $\mathbf{n}\mathbf{m}athbf{g}eq\mathbf{m}athbf{k}$ iff $n_i\mathbf{m}athbf{g}eq k_i$ for all $i\in[d]$. Thus, the power series $\mathbf{s}um_{\mathbf{n}\mathbf{m}athbf{g}eq\mathbf{z}ero}b_{\mathbf{n}}\mathbf{t}^{\mathbf{n}}$ should read $\mathbf{s}um_{n_1\mathbf{m}athbf{g}eq0,n_2\mathbf{m}athbf{g}eq0,...}b_{n_1,n_2,...}t_1^{n_1}t_2^{n_2}...$
We further use the following notation to denote sub-vectors and sub-matrices. $\mathbf{m}athbf{f}orall i\in\mathbf{m}athbb{N}$, let $\mathbf{m}athcal{D}_{i}=\left( i+j{\textnormal{i}}ght)_{j=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}$, so that $\left[{\bm{v}}{\textnormal{i}}ght]_{\mathbf{m}athcal{D}_{i}}=\left[{\bm{v}}{\textnormal{i}}ght]_{ i-\mathbf{m}athbf{f}rac{q-1}{2}: i+\mathbf{m}athbf{f}rac{q-1}{2}}$. Additionally, $\mathbf{m}athbf{f}orall i,i^{'}\in\mathbf{m}athbb{N}$, let $\mathbf{m}athcal{D}_{i,i^{'}}=\mathbf{m}athcal{D}_{i}\times D_{i^{'}}$,
so that for a matrix $\bm{M}$ we can write: $\bm{M}_{\mathbf{m}athcal{D}_{i,i^{'}}}=\bm{M}_{i-\mathbf{m}athbf{f}rac{q-1}{2}:i+\mathbf{m}athbf{f}rac{q-1}{2},i^{'}-\mathbf{m}athbf{f}rac{q-1}{2}:i^{'}-\mathbf{m}athbf{f}rac{q-1}{2}}$.
We use $\left(\mathbf{s}_{i} {\bm{v}}{\textnormal{i}}ght)_{j}=v_{j+i}$ to denote the cyclic shift of ${\bm{v}}$ to the left by $i$ pixels.
Finally, for every kernel $\mathbf{m}athcal{K}:\mathbf{m}athbb{R}^d\times\mathbf{m}athbb{R}^d\to\mathbf{m}athbb{R}$ we define the normalized kernel to be $\overline{\mathbf{m}athcal{K}}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)=\mathbf{m}athbf{f}rac{\mathbf{m}athcal{K}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)}{\mathbf{s}qrt{\mathbf{m}athcal{K}\left(\mathbf{x},\mathbf{x}{\textnormal{i}}ght)\mathbf{m}athcal{K}\left(\mathbf{z},\mathbf{z}{\textnormal{i}}ght)}}$. Note that $\overline{\mathbf{m}athcal{K}}\left(\mathbf{x},\mathbf{x}{\textnormal{i}}ght) = 1, \mathbf{m}athbf{f}orall \mathbf{x} \in \mathbf{m}athbb{R}^d$, and $\overline{\mathbf{m}athcal{K}}\in[-1,1]$.
\mathbf{s}ubsection{Convolutional ResNet}
\label{sec:conv_resnet}
We consider a residual, convolutional neural network with $L$ hidden layer (often just called ResNet). Let $\mathbf{x}\in\mathbf{m}athbb{R}^{C_{0} \times d}$ and $q$ be the filter size. We define the hidden layers of the Network as:
\begin{align}
f_i^{\left(0{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)= \mathbf{m}athbf{f}rac{1}{\mathbf{s}qrt{ C_{0}}}\mathbf{s}um_{j=1}^{ C_{0}}\mathbf{V}_{1,j,i}^{\left(0{\textnormal{i}}ght)}*\mathbf{x}_j, ~~~~~~~ g_i^{\left(1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght) = \mathbf{m}athbf{f}rac{1}{\mathbf{s}qrt{ C_{0}}}\mathbf{s}um_{j=1}^{ C_{0}}\mathbf{W}_{1,j,i}^{\left(1{\textnormal{i}}ght)}*\mathbf{x}_j \label{def:network0}
\mathbf{m}athbf{e}nd{align}
\begin{align}
f_{i}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)=f_{i}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)+\alpha\mathbf{s}qrt{\mathbf{m}athbf{f}rac{c_{v}}{qC_{l}}}\mathbf{s}um_{j=1}^{C_{l}}\mathbf{V}_{:,j,i}^{\left(l{\textnormal{i}}ght)}*\mathbf{s}igma\left(g_{j}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght),~~~l=1,...,L,~~i=1,\ldots,C_{l}\label{def:network3}
\mathbf{m}athbf{e}nd{align}
\begin{align}
g_{i}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)=\mathbf{s}qrt{\mathbf{m}athbf{f}rac{c_{w}}{qC_{l-1}}}\mathbf{s}um_{j=1}^{C_{l-1}}\mathbf{W}_{:,j,i}^{\left(l{\textnormal{i}}ght)}*f_{j}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght),~~~l=2,...,L,~~i=1,\ldots,C_{l},
\label{def:network2}
\mathbf{m}athbf{e}nd{align}
where $C_{l}\in\mathbf{m}athbb{N}$ is the
number of channels in the $l$'th layer; $\mathbf{s}igma$ is a nonlinear activation function, which in our analysis below is the ReLU
function; $\mathbf{W}^{\left(l{\textnormal{i}}ght)}\in\mathbf{m}athbb{R}^{q\times C_{l-1}\times C_{l}},\mathbf{V}^{\left(l{\textnormal{i}}ght)}\in\mathbf{m}athbb{R}^{q\times C_{l}\times C_{l}},\mathbf{W}^{\left(1{\textnormal{i}}ght)},\mathbf{V}^{\left(0{\textnormal{i}}ght)}\in\mathbf{m}athbb{R}^{1\times C_{0}\times C_{1}}$ are the network parameters, where $\mathbf{W}^{\left(1{\textnormal{i}}ght)},\mathbf{V}^{\left(0{\textnormal{i}}ght)}$ are convolution filters of size 1, and $\mathbf{V}^{\left(0{\textnormal{i}}ght)}$ is fixed throughout training; $c_v,c_w\in\mathbf{m}athbb{R}$ are normalizing factors set commonly as $c_v=1/\mathbf{m}athbb{E}_{u\mathbf{s}im\mathbf{m}athcal{N}(0,1)}[\mathbf{s}igma(u)]$ (for ReLU $c_v=2$) and $c_w=1$; $\alpha$ is a balancing factor typically set in applications to $\alpha=1$, however previous analyses of non-covolutional kernels also considered $\alpha=L^{-\mathbf{m}athbf{g}amma}$, with $0 \le \mathbf{m}athbf{g}amma \le 1$. We will occasionally omit explicit reference to $c_v$ and $c_w$ and assume in such cases that $c_v=2$ and $c_w=1$.
As in \citet{geifman2022spectral}, we consider three options for the final layer of the network:
\[
f^{\mathbf{m}athbb{E}q}\left(\mathbf{x};\theta{\textnormal{i}}ght):=\mathbf{m}athbf{f}rac{1}{\mathbf{s}qrt{C_{L}}} \mathbf{W}^{\mathbf{m}athbb{E}q}f_{:,1}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)
\]
\[
f^{\text{Tr}}\left(\mathbf{x};\theta{\textnormal{i}}ght):=\mathbf{m}athbf{f}rac{1}{\mathbf{s}qrt{d}\mathbf{s}qrt{C_{L}}}\left\langle \mathbf{W}^{\text{Tr}},f^{\left(L{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght{\textnormal{a}}ngle
\]
\[
f^{\text{GAP}}\left(\mathbf{x};\theta{\textnormal{i}}ght):=\mathbf{m}athbf{f}rac{1}{d\mathbf{s}qrt{C_{L}}}\mathbf{W}^{\text{GAP}}f^{\left(L{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\bm{e}}c{\bm{1}},
\]
where $\mathbf{W}^{\mathbf{m}athbb{E}q},\mathbf{W}^{\text{GAP}}\in\mathbf{m}athbb{R}^{1\times C_{L}},\mathbf{W}^{\text{Tr}}\in\mathbf{m}athbb{R}^{C_{L}\times d }$ and ${\bm{e}}c{\bm{1}} = (1,\ldots,1)^{T}\in\mathbf{m}athbb{R}^{ d }$.
$f^{\mathbf{m}athbb{E}q}$ is fully convolutional. Therefore, applying it to all shifted versions of the input results in a network that is shift-equivariant. $f^{\text{Tr}}$ implements a linear layer in the last layer and $f^{\text{GAP}}$ implements a global average pooling (GAP) layer, resulting in a shift invariant network. \mathbf{m}athbf{f}v{ The three heads allow us to analyze kernels corresponding to (1) shift equivariant networks (e.g., image segmentation networks), (2) a convolutional network followed by a fully connected head, akin to AlexNet \citep{krizhevsky2017imagenet} (but with additional skip connections), and (3) a convnet followed by global average pooling, akin to ResNet \citep{he2016deep}.}
Note that $f^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)\in\mathbf{m}athbb{R}^{C_{l}\times d }$
and $g^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)\in\mathbf{m}athbb{R}^{C_{l}\times d }$.
$\theta$ denote all the network parameters, which we initialize from a standard Gaussian distribution as in \citep{jacot2018neural}.
\mathbf{s}ubsection{Multi-dot product Kernels}
Following \citep{geifman2022spectral}, we call a kernel $\mathbf{m}athcal{K}:\mathbf{m}athbb{MS}\left( C_{0},d{\textnormal{i}}ght)\times\mathbf{m}athbb{MS}\left( C_{0},d{\textnormal{i}}ght)\to\mathbf{m}athbb{R}$ \tilde{p}h{multi-dot product} if $\mathbf{m}athcal{K}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)=\mathbf{m}athcal{K}\left(\mathbf{t}{\textnormal{i}}ght)$ where $\mathbf{t}=\left(\left(\mathbf{x}^T\mathbf{z}{\textnormal{i}}ght)_{1,1},\left(\mathbf{x}^T\mathbf{z}{\textnormal{i}}ght)_{2,2},\ldots,\left(\mathbf{x}^T\mathbf{z}{\textnormal{i}}ght)_{d,d}{\textnormal{i}}ght)\in [-1,1]^d$ (note the overload of notation which should be clear by context.) \mathbf{m}athbf{f}v{Under our uniform distribution assumption on the multi-sphere,} multi-dot product kernels can be decomposed as $\mathbf{m}athcal{K}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)=\mathbf{s}um_{\mathbf{m}athbf{k},\mathbf{m}athbf{j}}\lambda_{\mathbf{m}athbf{k}} \bm{Y}_{\mathbf{m}athbf{k},\mathbf{m}athbf{j}}\left(\mathbf{x}{\textnormal{i}}ght) \bm{Y}_{\mathbf{m}athbf{k},\mathbf{m}athbf{j}} \left(\mathbf{z}{\textnormal{i}}ght)$, where $\mathbf{m}athbf{k},\mathbf{m}athbf{j}\in\mathbf{m}athbb{N}^d$. $\bm{Y}_{\mathbf{m}athbf{k},\mathbf{m}athbf{j}}\left(\mathbf{x}{\textnormal{i}}ght)$ (the eigenfunctions of $\mathbf{m}athcal{K}$) are products of spherical harmonics in $\mathbf{m}athbb{S}^{C_0-1}$, $\bm{Y}_{\mathbf{m}athbf{k},\mathbf{m}athbf{j}}\left(\mathbf{x}{\textnormal{i}}ght)=\prod_{i=1}^d Y_{k_{i},j_{i}}\left(\mathbf{x}_i{\textnormal{i}}ght)$ with $k_i\mathbf{m}athbf{g}eq0$, $j_i\in\left[N(C_0,k_i){\textnormal{i}}ght]$, where $N(C_0,k_i)$ denotes the number of harmonics of frequency $k_i$ in $\mathbf{m}athbb{S}^{C_0-1}$. For $C_0=2$ these are products of Fourier series in a $d$-dimensional torus. Note that the eigenvalues $\lambda_{\mathbf{m}athbf{k}}$ are non-negative and do not depend on $\mathbf{m}athbf{j}$.
Using Mercer's Representation of RKHSs \citep{kanagawa2018gaussian}, we have that the RKHS $\mathbf{m}athcal{H}_{\mathbf{m}athcal{K}}$ of $\mathbf{m}athcal{K}$ is
\[
\mathbf{m}athcal{H}_{\mathbf{m}athcal{K}} := \left\{f=\mathbf{s}um_{\mathbf{m}athbf{k},\mathbf{m}athbf{j}}\alpha_{\mathbf{m}athbf{k},\mathbf{m}athbf{j}}\bm{Y}_{\mathbf{m}athbf{k},\mathbf{m}athbf{j}}~~{\bm{e}}rt~~\mathbf{n}orm{f}^2_{\mathbf{m}athcal{H}_{\mathbf{m}athcal{K}}}=\mathbf{s}um_{\mathbf{m}athbf{k},\mathbf{m}athbf{j}}\mathbf{m}athbf{f}rac{\alpha_{\mathbf{m}athbf{k},\mathbf{m}athbf{j}}^2}{\lambda_{\mathbf{m}athbf{k}}}<\infty{\textnormal{i}}ght\}.
\]
For multi-dot product kernels the normalized kernel simplifies to $\overline{\mathbf{m}athcal{K}}\left(\mathbf{t}{\textnormal{i}}ght)=\mathbf{m}athbf{f}rac{\mathbf{m}athcal{K}\left(\mathbf{t}{\textnormal{i}}ght)}{\mathbf{m}athcal{K}\left({\bm{e}}c{\bm{1}}{\textnormal{i}}ght)}$, where ${\bm{e}}c{\bm{1}}=(1,...,1)^T\in\mathbf{m}athbb{R}^d$.
$\mathbf{m}athcal{K}$ and $\overline{\mathbf{m}athcal{K}}$ thus differ by a constant, and so they share the same eigenfunctions and their eigenvalues differ by a multiplicative constant.
\mathbf{s}ection{Kernel Derivations}
We next provide explicit formulas for ResCGPK and ResCNTK.
\mathbf{s}ubsection{ResCGPK}
Given a network $f(\mathbf{x};\theta)$, the corresponding Gaussian process kernel is defined as $\mathbf{m}athbb{E}_{\theta}\left[f(\mathbf{x};\theta)f(\mathbf{z};\theta){\textnormal{i}}ght]$. Below we consider the network in Sec.~{\textnormal{e}}f{sec:conv_resnet}, which can have either one of three heads, the equivariant head, trace or GAP. We denote the corresponding ResCGPK by $\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}$, $\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\text{Tr}}$ and $\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\text{GAP}}$, where $L$ denotes the number of layers. We proceed with the following definition.
\begin{definition}
Let $\mathbf{x},\mathbf{z}\in\mathbf{m}athbb{R}^{ C_{0}\times d}$ and $f$ be a residual network with $L$ layers. For every $1\leq l\leq L$ (and $i$ is an arbitrary choice of channel) denote by
\begin{align}
\Sigma_{j,j^{'}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght):=\mathbf{m}athbb{E}_{\theta}\left[g_{ij}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)g_{ij^{'}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght],
\mathbf{m}athbf{e}nd{align}
\begin{align}
\Lambda_{j,j^{'}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)=\begin{pmatrix}\Sigma_{j,j}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{x}{\textnormal{i}}ght) & \Sigma_{j,j^{'}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)\\
\Sigma_{j^{'},j}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{z},\mathbf{x}{\textnormal{i}}ght) & \Sigma_{j^{'},j^{'}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{z},\mathbf{z}{\textnormal{i}}ght)
\mathbf{m}athbf{e}nd{pmatrix}
\mathbf{m}athbf{e}nd{align}
\begin{align}
K_{j,j^{'}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)=c_{v}c_{w}\underset{\left(u,v{\textnormal{i}}ght)\mathbf{s}im\mathbf{m}athcal{N}\left(0,\Lambda_{j,j^{'}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght)}{\mathbf{m}athbb{E}}\left[\mathbf{s}igma\left(u{\textnormal{i}}ght)\mathbf{s}igma\left(v{\textnormal{i}}ght){\textnormal{i}}ght]
\label{eq:K}
\mathbf{m}athbf{e}nd{align}
\begin{align}
\dot{K}_{j,j^{'}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)=c_{v}c_{w}\underset{\left(u,v{\textnormal{i}}ght)\mathbf{s}im\mathbf{m}athcal{N}\left(0,\Lambda_{j,j^{'}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght)}{\mathbf{m}athbb{E}}\left[\dot{\mathbf{s}igma}\left(u{\textnormal{i}}ght)\dot{\mathbf{s}igma}\left(v{\textnormal{i}}ght){\textnormal{i}}ght],
\label{eq:Kdot}
\mathbf{m}athbf{e}nd{align}
where $\dot{\mathbf{s}igma}$ is the derivative of the ReLU function expressed by the indicator $\bm{1}_{x\mathbf{m}athbf{g}eq0}$.
\mathbf{m}athbf{e}nd{definition}
Our first contribution is to give an exact formula for the ResCGPK. We refer the reader to the appendix for the precise derivation and note here some of the key ideas. We give precise formulas for $\Sigma, K$ and $\dot{K}$ and prove that
\[\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)
= \Sigma_{1,1}^{\left(1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)+ \mathbf{m}athbf{f}rac{\alpha^{2}}{qc_{w}}\mathbf{s}um_{l=1}^{L}\text{tr}\left(K_{\mathbf{m}athcal{D}_{1,1}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght).\]
This gives us the equivariant kernel, and by showing that $\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\text{Tr}}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)=\mathbf{m}athbf{f}rac{1}{d}\mathbf{s}um_{j=1}^{ d }\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{s}_j\mathbf{x},\mathbf{s}_j\mathbf{z}{\textnormal{i}}ght)$ and $\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\text{GAP}}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)=\mathbf{m}athbf{f}rac{1}{d^2}\mathbf{s}um_{j,j^{'}=1}^{ d }\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{s}_j\mathbf{x},\mathbf{s}_{j^{'}}\mathbf{z}{\textnormal{i}}ght)$ we obtain precise formulas for the trace and GAP kernels.
For clarity, we give here the case of the normalized ResCGPK with multi-sphere inputs, which we prove to simplify significantly. The full derivation for arbitrary inputs is given in Appendix {\textnormal{e}}f{ap:1}.
\begin{theorem} \label{Thm:ResCGPK}
[Multi-Sphere Case]
For any $\mathbf{x},\mathbf{z}\in \mathbf{m}athbb{MS}\left( C_{0},d{\textnormal{i}}ght)$ let $\mathbf{t} = \left(\left(\mathbf{x}^T\mathbf{z}{\textnormal{i}}ght)_{1,1},\ldots,\left(\mathbf{x}^T\mathbf{z}{\textnormal{i}}ght)_{d,d}{\textnormal{i}}ght)$ $\in[-1,1]^d$. Fixing $c_v=2,c_w=1$ and let $\kappa_1(u)=\mathbf{m}athbf{f}rac{\mathbf{s}qrt{1-{\textnormal{h}}o^2}+\left(\pi-\cos^{-1}\left(u{\textnormal{i}}ght){\textnormal{i}}ght)u}{\pi}$. Then,
\[
\overline{\mathbf{m}athcal{K}}^{\left(1{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{t}{\textnormal{i}}ght) = \mathbf{m}athbf{f}rac{1}{1+\alpha^2}\left(t_1 + \mathbf{m}athbf{f}rac{\alpha^2}{q}\mathbf{s}um_{k=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\kappa_1\left(t_{1+k}{\textnormal{i}}ght){\textnormal{i}}ght)
\]
\[
\overline{\mathbf{m}athcal{K}}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{t}{\textnormal{i}}ght) = \mathbf{m}athbf{f}rac{1}{1+\alpha^2}\left(\overline{\mathbf{m}athcal{K}}^{\left(L-1{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{t}{\textnormal{i}}ght) + \mathbf{m}athbf{f}rac{\alpha^{2}}{q^2} \mathbf{s}um_{k=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\mathbf{s}um_{k^{'}=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\kappa_1\left(\overline{\mathbf{m}athcal{K}}^{\left(L-1{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{s}_{k+k^{'}}\mathbf{t}{\textnormal{i}}ght){\textnormal{i}}ght){\textnormal{i}}ght).
\]
\mathbf{m}athbf{e}nd{theorem}
\mathbf{s}ubsection{ResCNTK}
For $\mathbf{x},\mathbf{z}\in\mathbf{m}athbb{R}^{ C_{0} \times d}$ and $f(\mathbf{x};\theta)$ an $L$ layer ResNet, ResCNTK is defined as
$\mathbf{m}athbb{E}\left[\left\langle \mathbf{m}athbf{f}rac{\partial f\left(\mathbf{x};\theta{\textnormal{i}}ght)}{\partial\theta},\mathbf{m}athbf{f}rac{\partial f\left(\mathbf{z};\theta{\textnormal{i}}ght)}{\partial\theta}{\textnormal{i}}ght{\textnormal{a}}ngle {\textnormal{i}}ght]$. Considering the three heads in Sec.~{\textnormal{e}}f{sec:conv_resnet}, we denote the corresponding kernels by $\Theta^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q},\Theta^{\left(L{\textnormal{i}}ght)}_{\text{Tr}}$ and $\Theta^{\left(L{\textnormal{i}}ght)}_{\text{GAP}}$, depending on the choice of last layer. Our second contribution is providing a formula for the ResCNTK for arbitrary inputs.
\begin{theorem}\label{thm:2}
Let $\mathbf{x},\mathbf{z}\in\mathbf{m}athbb{R}^{ C_{0}\times d}$ and $f$ be a residual network with $L$ layers. Then, the ResCNTK for $f$ has the form
\begin{align*}
\Theta^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght) = \mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)
+ \mathbf{m}athbf{f}rac{\alpha^{2}}{q}\mathbf{s}um_{l=1}^{L}\mathbf{s}um_{p=1}^{ d }P_{p}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)
\left(\text{tr}\left(\dot{K}_{\mathbf{m}athcal{D}_{p,p}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)\odot\Sigma_{\mathbf{m}athcal{D}_{p,p}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght) + K_{\mathbf{m}athcal{D}_{p,p}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght) {\textnormal{i}}ght) {\textnormal{i}}ght),
\mathbf{m}athbf{e}nd{align*}
where $\mathbf{m}athbf{f}orall1\leq j\leq d, P_{j}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)=\mathbf{m}athbf{f}rac{\bm{1}_{j=1}}{c_w}$ and for $1\leq l\leq L-1$,
\[
P_{j}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)=P_{j}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght) + \mathbf{m}athbf{f}rac{\alpha^2}{q^2}\mathbf{s}um_{k=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\dot{K}_{j+k, j+k}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)
\mathbf{s}um_{k^{'}=\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}P_{j+k+k^{'}}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght).
\]
The Tr and GAP kernels are given by
$\Theta^{\left(L{\textnormal{i}}ght)}_{\text{Tr}}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)=\mathbf{m}athbf{f}rac{1}{d}\mathbf{s}um_{j=1}^{d}\Theta^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{s}_j\mathbf{x},\mathbf{s}_j\mathbf{z}{\textnormal{i}}ght)$ and $\Theta^{\left(L{\textnormal{i}}ght)}_{\text{GAP}}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)=\mathbf{m}athbf{f}rac{1}{d^2}\mathbf{s}um_{j,j^{'}=1}^{d^2}\Theta^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{s}_j\mathbf{x},\mathbf{s}_{j^{'}}\mathbf{z}{\textnormal{i}}ght)$.
\mathbf{m}athbf{e}nd{theorem}
\mathbf{s}ection{Spectral Decomposition}
$\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}$ and $\Theta^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}$ are multi-dot product kernels, and therefore their eigenfunctions consist of spherical harmonic products. Next, we derive bounds on their eigenvalues. We subsequently use a result due to \citep{geifman2022spectral} to extend these to their trace and GAP versions.
\mathbf{s}ubsection{Asymptotic Bounds}
The next theorem provides upper and lower bounds on the eigenvalues of $\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}$ or $\Theta^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}$.
\begin{theorem}\label{thm:sd}
Let $\mathbf{x},\mathbf{z}\in\mathbf{m}athbb{MS}\left( C_{0},d{\textnormal{i}}ght)$, \mathbf{m}athbf{f}v{where $d$ denotes the number of pixels and $C_0$ denotes the number of input channels for each pixel}. The eigenvalues $\lambda_{\mathbf{m}athbf{k}}$ of either $\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}$ or $\Theta^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}$ satisfy
\[
c_1\prod_{i\in\mathbf{m}athcal{R},k_i>0}k_i^{-(C_0+2\mathbf{n}u_a-3)}\leq\lambda_{\mathbf{m}athbf{k}}\leq c_2 \prod_{i\in\mathbf{m}athcal{R},k_i>0}k_i^{-(C_0+2\mathbf{n}u_b-3)}
\]
where $\mathbf{n}u_a=2.5$ and $\mathbf{n}u_b$ is $1+\mathbf{m}athbf{f}rac{3}{2d}$ for ResCGPK and $1+\mathbf{m}athbf{f}rac{1}{2d}$ for ResCNTK. $c_1,c_2>0$ are constants that depend on $L$. \mathbf{m}athbf{f}v{The set $\mathbf{m}athcal{R}$ denotes the receptive field, defined as the set of indices of input pixels that affect the kernel output.}
\mathbf{m}athbf{e}nd{theorem}
We note that these bounds are identical, up to constants, to those obtained with convolutional networks that do not include skip connections \citep{geifman2022spectral}, although the proof for the case of ResNet is more complex. Overall, the theorem shows that over-parameterized ResNets are biased toward low-frequency functions. In particular, with input distributed uniformly on the multi-sphere, the time required to train such a network to fit an eigenfunction with gradient descent (GD) is inversely proportional to the respective eigenvalue \citep{basri2020frequency}. Consequently, training a network to fit a high frequency function is polynomially slower than training the network to fit a low frequency function. Note, however, that the rate of decay of the eigenvalues depends on the number of pixels over which the target function has high frequencies. Training a target function whose high frequencies are concentrated in a few pixels is much faster than if the same frequencies are spread over many pixels. This can be seen in Figure~{\textnormal{e}}f{fig:eigen}, which shows for a target function of frequency $k$ in $m$ pixels, that the exponent (depicted by the slope of the lines) grows with $m$. The same behaviour is seen when the skip connections are removed. This is different from fully connected architectures, in which the decay rate of the eigenvalues depends on the dimension of the input and is invariant to the pixel spread of frequencies, see \citep{geifman2022spectral} for a comparison of the eigenvalue decay for standard CNNs and FC architectures.
\mathbf{s}ubsection{Locality bias and Ensemble Behavior}
To better understand the difference between ResNets and vanilla CNNs we next turn to a fine-grained analysis of the decay.
Consider an $l$-layer CNN that is identical to our ResNet but with the skip connections removed. Let $p_i^{(l)}$ be the number of paths from input pixel $i$ to the output in \mathbf{m}athbf{f}v{the corresponding CGPK, or equivalently, the number of paths in the same CNN but in which there is only one channel in each node.}.
\begin{theorem} \label{thm:eigen}
For both $\Theta^{(L)}_{\mathbf{m}athbb{E}q}$ or $\mathbf{m}athcal{K}^{(L)}_{\mathbf{m}athbb{E}q}$ there exist scalars $A>1$ and $c_l>0$ \text{ s.t.\ } letting $c_{\mathbf{m}athbf{k},l}=c_l\prod_{i=1}^d A^{\mathbf{m}in(p_i^{(l)}, k_i)}$ for every $1\leq l\leq L$ and $c_{\mathbf{m}athbf{k}}=\mathbf{s}um_{l=1}^L c_{\mathbf{m}athbf{k},l}$, it holds that
\[
\lambda_{\mathbf{m}athbf{k}} \mathbf{m}athbf{g}eq c_{\mathbf{m}athbf{k}}\underset{k_i>0}{\prod_{i=1}^d}k_i^{-C_0-2}.
\]
\mathbf{m}athbf{e}nd{theorem}
The constant $c_{\mathbf{m}athbf{k}}$ differs significantly from that of CNTK (without skip connections) which takes the form $c_{\mathbf{m}athbf{k},L}$ \citep{geifman2022spectral}. In particular, notice that the constants in the ResCNTK are (up to scale factors) the sum of the constants of the CNTK at depths $1,\ldots,L$. Thus, a major contribution of the paper is providing theoretical justification for the following result, observed empirically in \citep{veit2016residual}: \tilde{p}h{over-parameterized ResNets act like a weighted ensemble of CNNs of various depths}. In particular, information from smaller receptive fields is propagated through the skip connections, resulting in larger eigenvalues for frequencies that correspond to smaller receptive fields.
Figure {\textnormal{e}}f{fig:eigen} shows the eigenvalues computed numerically for various frequencies, for both the CGPK and ResCGPK. Consistent with our results, eigenfunctions with high frequencies concentrated in a few pixels, e.g., $\mathbf{m}athbf{k}=(k,0,0,0)$ have larger eigenvalues than those with frequencies spread over more pixels, e.g., $\mathbf{m}athbf{k}=(k,k,k,k)$. See appendix {\textnormal{e}}f{app:exp_eig_decay} for implementation details.
Figure~{\textnormal{e}}f{fig:erfs} shows the effective receptive field (ERF) induced by ResCNTK compared to that of a network and to the kernel and network with the skip connections removed. The ERF is defined to be $\partial f^{\mathbf{m}athbb{E}q}(\mathbf{x};\theta)/\partial \mathbf{x}$ for ResNet \citep{luo2016understanding} \mathbf{m}athbf{f}v{and $\Theta^{\mathbf{m}athbb{E}q}(\mathbf{x},\mathbf{x})/\partial\mathbf{x}$ for ResCNTK. A similar calculation is applied to CNN and CNTK}. We see that residual networks and their kernels give rise to an increased locality bias (more weight at the center of the receptive field (for the equivariant architecture) or to nearby pixels (at the trace and GAP architectures).
\begin{figure}[tb]
\centering
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{images/eigen_decay_new_eq.eps}
\mathbf{m}athbf{e}nd{subfigure}
\mathbf{m}athbf{h}fill
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{images/eigen_compare_rescgpk.eps}
\mathbf{m}athbf{e}nd{subfigure}
\caption{Left: The eigenvalues of ResCGPK (filled circles and solid lines) computed numerically for various eigenfunctions, compared to those of CGPK (empty circles and dashed lines). Here $L=3$, $q=2$, $d=4$ and the output head is the equivariant one. The slopes (respectively, -5.25, -6.8, -8.5, -9.8 for CGPK and -5.3, -6.84, -8.77, -9.85 for ResCGPK) approximate the exponent for each pattern. Notice that the slope increases for eigenfunctions involving more pixels. Right: Additional eigenvalues of ResCGPK. Notice that the eigenvalues for eigenfunction involving nearby pixels (in red) are larger compared to one involving farther pixels (in blue).}
\label{fig:eigen}
\mathbf{m}athbf{e}nd{figure}
\begin{figure}[tb]
\centering
\begin{subfigure}[b]{0.14\textwidth}
\centering
\includegraphics[width=\textwidth]{images/erf/rec_res_cntk_8.eps}
\caption{ResCNTK}
\mathbf{m}athbf{e}nd{subfigure}
\mathbf{m}athbf{h}fill
\begin{subfigure}[b]{0.14\textwidth}
\centering
\includegraphics[width=\textwidth]{images/erf/rec_cntk_8.eps}
\caption{CNTK}
\mathbf{m}athbf{e}nd{subfigure}
\mathbf{m}athbf{h}fill
\begin{subfigure}[b]{0.14\textwidth}
\centering
\includegraphics[width=\textwidth]{images/erf/erf_eq_res_32_w512_L8.eps}
\caption{ResNet}
\mathbf{m}athbf{e}nd{subfigure}
\mathbf{m}athbf{h}fill
\begin{subfigure}[b]{0.14\textwidth}
\centering
\includegraphics[width=\textwidth]{images/erf/erf_eq_cnn_32_w512_L8.eps}
\caption{CNN}
\mathbf{m}athbf{e}nd{subfigure}
\mathbf{m}athbf{h}fill
\begin{subfigure}[b]{0.07\textwidth}
\centering
\includegraphics[width=\textwidth,height=0.1\textheight]{images/color_bar.eps}
\mathbf{m}athbf{e}nd{subfigure}
\caption{The effective receptive field of ResCNTK (left) compared to that of actual ResNet and to CNTK and CNN (i.e., no skip connections). We followed \protect\citep{luo2016understanding} in computing the ERF, where the networks are first trained on CIFAR-10. All values are re-scaled to the [0,1] interval. We used $L=8$ in all cases.}
\label{fig:erfs}
\mathbf{m}athbf{e}nd{figure}
\mathbf{s}ubsection{Extension to $f^{\text{Tr}}$ and $f^{\text{GAP}}$}
Using \citep{geifman2022spectral}[Thm.\ 3.7], we can extend our analysis of equivariant kernels to trace and GAP kernels. In particular, for ResCNTK, the eigenfunctions of the trace kernel are a product of spherical harmonics. In addition, let $\lambda_{\mathbf{m}athbf{k}}$ denote the eigenvalues of $\Theta_{\mathbf{m}athbb{E}q}^{(L)}$, then the eigenvalues of $\Theta_{\text{Tr}}^{(L)}$ are $\lambda_{\mathbf{m}athbf{k}}^{\text{Tr}}=\mathbf{m}athbf{f}rac{1}{d}\mathbf{s}um_{i=0}^{d-1}\lambda_{\mathbf{s}_i\mathbf{m}athbf{k}}$, i.e., average over all shifts of the frequency vector $\mathbf{m}athbf{k}$. This implies that for the trace kernel, the eigenvalues (but not the eigenfunctions) are invariant to shift. For the GAP kernel, the eigenfunctions are $\mathbf{m}athbf{f}rac{1}{\mathbf{s}qrt{d}}\mathbf{s}um_{i=0}^{d-1}\bm{Y}_{\mathbf{s}_i\mathbf{m}athbf{k},\mathbf{s}_i\mathbf{m}athbf{j}}$, i.e., scaled shifted sums of spherical harmonic products. These eigenfunctions are shift invariant and generally span all shift invariant functions. The eigenvalues of the GAP kernel are identical to those of the Trace kernel. The eigenfunctions and eigenvalues of the trace and GAP ResCGPK are determined in a similar way. Finally, we note that the eigenvalues for the trace and GAP kernels decay at the same rate as their equivariant counterparts, and likewise they are biased in frequency and in locality. Moreover, while the equivariant kernel is biased to prefer functions that depend on the center of the receptive field (position biased), the trace and GAP kernels are biased to prefer functions that depend on nearby pixels.
\mathbf{s}ection{Stability at Large Depths}
\mathbf{s}ubsection{Decaying Balancing Parameter $\alpha$}
We next investigate the effects of skip connections in very deep networks. Here the setting of balancing parameter $\alpha$ between the skip and residual connection \mathbf{m}athbf{e}qref{def:network3} plays a critical role. Previous work on residual, non-convolutional kernels \citet{huang2020deep,belfer2021spectral} proposed to use a balancing parameter of the form $\alpha=L^{-\mathbf{m}athbf{g}amma}$ for $\mathbf{m}athbf{g}amma\in(0.5,1]$, arguing that a decaying $\alpha$ contributes to the stability of the kernel for very deep architectures. However, below we prove that in this setting as the depth $L$ tends to infinity, ResCNTK converges to a simple dot-product, $\mathbf{m}athbf{k}(\mathbf{x},\mathbf{z})=\mathbf{x}^T\mathbf{z}$, corresponding to a 1-layer, linear neural network, which may be considered degenerate. We subsequently further elaborate on the connection between this result and previous work and provide a more comprehensive discussion in Appendix {\textnormal{e}}f{app:depth_disc}.
\begin{theorem}\label{thm:decay}
Suppose $\alpha=L^{-\mathbf{m}athbf{g}amma}$ with $\mathbf{m}athbf{g}amma\in (0.5,1]$. Then, for any $\mathbf{t}\in [-1,1]^d$ it holds that $\overline{\Theta}_{\mathbf{m}athbb{E}q}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght)\underset{L\to\infty}{\longrightarrow} \overline{\Sigma}_{1,1}^{\left(1{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght)=t_1$ and likewise $\overline{\mathbf{m}athcal{K}}_{\mathbf{m}athbb{E}q}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght)\underset{L\to\infty}{\longrightarrow} \overline{\Sigma}_{1,1}^{\left(1{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght)=t_1$.
\mathbf{m}athbf{e}nd{theorem}
Clearly, this limit kernel, which corresponds to a linear network with no hierarchical features if undesired. A comparison to the previous work of \citet{huang2020deep,belfer2021spectral}, which addressed residual kernels for fully connected architectures, is due here. This previous work proved that FC-ResNTK converges when $L$ tends to infinity to a two-layer FC-NTK. They however made the additional assumption that the top-most layer is not trained. This assumption turned out to be critical to their result -- training the last layer yields a result analogous to ours, namely, that as $L$ tends to infinity FC-ResNTK converges to a simple dot product. Similarly, if we consider ResCNTK in which we do not train the last layer we will get that the limit kernel is the CNTK corresponding to a two-layer convolutional neural network. However, while a two-layer FC-NTK is universal, the set of functions produced by a two-layer CNTK is very limited; therefore, this limit kernel is also not desired. We conclude that the standard setting of $\alpha=1$ is preferable for convolutional architectures.
\mathbf{s}ubsection{The Condition Number of the ResCGPK Matrix with $\alpha=1$}
Next we investigate the properties of ResCGPK when the balancing factor is set to $\alpha=1$. For ResCGPK and CGPK and any training distribution, we use \tilde{p}h{double-constant matrices} \citep{o2021double} to bound the condition numbers of their kernel matrices. We further show that with any depth, the lower bound for ResCGPK matrices is lower than that of CGPK matrices (and show empirically that these bounds are close to the actual condition numbers). \mathbf{m}athbf{f}v{\citet{lee2019wide, xiao2020disentangling, chen2021neural} argued that a smaller condition number of the NTK matrix implies that training the corresponding neural network with GD convergences faster.} Our analysis therefore indicates that GD with ResCGPK should generally be faster than GD with CGPK. This phenomenon may partly explain the advantage of residual networks over standard convolutional architectures.
\mathbf{m}athbf{f}v{Recall that the condition number of a matrix $\bm{A}\mathbf{s}ucceq 0$ is defined as ${\textnormal{h}}o(\bm{A}):=\lambda_{\mathbf{m}ax}/\lambda_{\mathbf{m}in}$}. Consider an $n \times n$ \tilde{p}h{double-constant matrix} $\bm{B}_{\tilde{b},b}$ that includes $\tilde b$ in the diagonal entries and $b$ in each off-diagonal entry.
The eigenvalues of $\bm{B}_{\tilde{b},b}$ are $\lambda_1=\tilde{b}-b+nb$ and $\lambda_2=...=\lambda_n=\tilde{b}-b$. Suppose $\tilde{b}=1,0<b\leq 1$, then $\bm{B}_{1,b}$ is positive semi-definite and its condition number is ${\textnormal{h}}o(\bm{B}_{1,b})=1+\mathbf{m}athbf{f}rac{nb}{1-b}$. This condition number diverges when either $b=1$ or $n$ tends to infinity.
The following lemma relates the condition numbers of kernel matrices with that of double-constant matrices.
\begin{lemma} \label{lemma:dcb}
Let $\bm{A}\in\mathbf{m}athbb{R}^{n\times n}$ ($n\mathbf{m}athbf{g}eq 2)$ be a normalized kernel matrix with $\mathbf{s}um_{i\mathbf{n}eq j}\bm{A}_{ij}\mathbf{m}athbf{g}eq 0$. Let $\bm{B}(\bm{A})=\bm{B}_{1,b}$ with $b=\mathbf{m}athbf{f}rac{1}{n(n-1)}\mathbf{s}um_{i\mathbf{n}eq j}\bm{A}_{ij}$ and $\mathbf{m}athbf{e}psilon=\mathbf{s}up_{i}\mathbf{s}um_{j\mathbf{n}eq i}\abs{\bm{A}_{ij}-\bm{B}(\bm{A})_{ij}}$. Then,
\begin{enumerate}
\item ${\textnormal{h}}o\left(\bm{B}(\bm{A}){\textnormal{i}}ght) \leq {\textnormal{h}}o(\bm{A})$.
\item If $\mathbf{m}athbf{e}psilon<\lambda_{\mathbf{m}in}(\bm{B}(\bm{A}))$ then ${\textnormal{h}}o(\bm{A}) \leq \mathbf{m}athbf{f}rac{\lambda_{\mathbf{m}ax}(\bm{B}(\bm{A}))+\mathbf{m}athbf{e}psilon}{\lambda_{\mathbf{m}in}(\bm{B}(\bm{A}))-\mathbf{m}athbf{e}psilon} $,
\mathbf{m}athbf{e}nd{enumerate}
where $\lambda_{\mathbf{m}ax}$ and $\lambda_{\mathbf{m}in}$ denote the maximal and minimal eigenvalues of $\bm{B}(\bm{A})$.
\mathbf{m}athbf{e}nd{lemma}
The following theorem uses double-constant matrices to compare kernel matrices produced by ResCGPK and those produced by CGPK with no skip connections.
\begin{theorem}\text{ }\label{thm:cond}
Let $\bar{K}_{\text{ResCGPK}}^{(L)}$ and $\bar{K}_{\text{CGPK}}^{(L)}$ respectively denote kernel matrices for the normalized trace kernels ResCGPK and CGPK of depth $L$. Let $\bm{B}\left(K{\textnormal{i}}ght)$ be a double-constant matrix defined for a matrix $K$ as in Lemma~{\textnormal{e}}f{lemma:dcb}. Then,
\begin{enumerate}
\item $\mathbf{n}orm{\bar{K}_{\text{ResCGPK}}^{(L)} - \bm{B}\left(\bar{K}_{\text{ResCGPK}}^{(L)}{\textnormal{i}}ght)}_1 \underset{L\to\infty}{\longrightarrow}0$ and $\mathbf{n}orm{\bar{K}_{\text{CGPK}}^{(L)} - \bm{B}\left(\bar{K}_{\text{CGPK}}^{(L)}{\textnormal{i}}ght)}_1 \underset{L\to\infty}{\longrightarrow}0$.
\item ${\textnormal{h}}o\left(\bm{B}\left(\bar{K}_{\text{ResCGPK}}^{(L)}{\textnormal{i}}ght){\textnormal{i}}ght) \underset{L\to\infty}{\longrightarrow}\infty$ and ${\textnormal{h}}o\left(\bm{B}\left(\bar{K}_{\text{CGPK}}^{(L)}{\textnormal{i}}ght){\textnormal{i}}ght) \underset{L\to\infty}{\longrightarrow}\infty$.
\item $\mathbf{m}athbf{e}xists L_0\in\mathbf{m}athbb{N} \text{ s.t.\ } \mathbf{m}athbf{f}orall L\mathbf{m}athbf{g}eq L_0$, ${\textnormal{h}}o\left(\bm{B}\left(\bar{K}_{\text{ResCGPK}}^{(L)}{\textnormal{i}}ght) {\textnormal{i}}ght) < {\textnormal{h}}o\left(\bm{B}\left(\bar{K}_{\text{CGPK}}^{(L)}{\textnormal{i}}ght) {\textnormal{i}}ght)$.
\mathbf{m}athbf{e}nd{enumerate}
\mathbf{m}athbf{e}nd{theorem}
\begin{figure}[t]
\centering
\includegraphics[width=0.38\textwidth]{images/cond_compare_cgpk_circular.eps}
\caption{The condition number of ResCGPK-Tr (solid blue) as a function of depth compared to that of CGPK-Tr (solid red) and the corresponding lower and upper bounds (dashed lines) computed with $n=100$.
}
\label{fig:condition_num}
\mathbf{m}athbf{e}nd{figure}
The theorem establishes that, while the condition numbers of both $\bm{B}\left(\bar{K}_{ResCGPK}^{(L)}{\textnormal{i}}ght)$ and $\bm{B}\left(\bar{K}_{\text{CGPK}}^{(L)}{\textnormal{i}}ght)$ diverge as $L \to \infty$, the condition number of $\bm{B}\left(\bar{K}_{ResCGPK}^{(L)}{\textnormal{i}}ght)$ is smaller than that of $\bm{B}\left(\bar{K}_{\text{CGPK}}^{(L)}{\textnormal{i}}ght)$ for all $L>L_0$. ($L_0$ is the minimal $L$ \text{ s.t.\ } the entries of the double constant matrices are non-negative. We notice in practice that $L_0 \approx 2$.) We can therefore use Lemma {\textnormal{e}}f{lemma:dcb} to derive approximate bounds for the condition numbers obtained with ResCGPK and CGPK. Figure~{\textnormal{e}}f{fig:condition_num} indeed shows that the condition number of the CGPK matrix diverges faster than that of ResCGPK and is significantly larger at any finite depth $L$. The approximate bounds, particularly the lower bounds, closely match the actual condition numbers produced by the kernels. (We note that with training sampled from a uniform distribution on the multi-sphere, the upper bound can be somewhat improved. In this case, the constant vector is the eigenvector of maximal eigenvalue for both $\bm{A}$ and $\bm{B}(\bm{A})$, and thus the rows of $\bm{A}$ sum to the same value, yielding ${\textnormal{h}}o(\bm{A}) \leq \mathbf{m}athbf{f}rac{\lambda_{\mathbf{m}ax}(\bm{B}(\bm{A}))}{\lambda_{\mathbf{m}in}(\bm{B}(\bm{A}))-\mathbf{m}athbf{e}psilon}$ with $\mathbf{m}athbf{e}psilon=\mathbf{m}athbf{f}rac{1}{n} \mathbf{n}orm{\bm{A}-\bm{B}(\bm{A})}_1$. We used this upper bound in our plot in Figure~{\textnormal{e}}f{fig:condition_num}.)
To the best of our knowledge, this is the first paper that establishes a relationship between skip connections and the condition number of the kernel matrix.
\mathbf{s}ection{Conclusion}
We derived formulas for the Gaussian process and neural tangent kernels associated with convolutional residual networks, analyzed their spectra, and provided bounds on their implied condition numbers. Our results indicate that over-parameterized residual networks are subject to both frequency and locality bias, and that they can be trained faster than standard convolutional networks. In future work, we hope to gain further insight by tightening our bounds. We further intend to apply our analysis of the condition number of kernel matrices to characterize the speed of training in various other architectures.
\mathbf{s}ection*{Acknowledgement}
This research was partially supported by the Israeli Council for Higher Education (CHE) via the Weizmann Data Science Research Center and by research grants from the Estate of Tully and Michele Plesser and the Anita James Rosen Foundation.
\mathbf{n}ewpage
\appendix
\mathbf{s}ection*{Appendix}
Below we provide derivations and proofs for our paper.
\mathbf{s}ection{Derivation of ResCGPK}
In this section, we derive explicit formulas for ResCGPK. We begin with a few preliminaries. As in \citep{jacot2018neural}, we assume the network parameters $\theta$ are initialized with a standard Gaussian distribution,
$\theta\mathbf{s}im\mathbf{m}athcal{N}\left(0,I{\textnormal{i}}ght)$. Therefore, at initialization, for every pair of parameters, $\theta_{i},\theta_{j}$,
\begin{align}
\mathbf{m}athbb{E}\left[\theta_{i}\cdot\theta_{j}{\textnormal{i}}ght]=\delta_{ij}.\label{eq:exp_par}
\mathbf{m}athbf{e}nd{align}
We note that \citet{lee2019wide} proved the convergence of a network with this initialization to its NTK.
For a vector ${\bm{v}}$, we use the notation $v_{*}$ to denote an entry of ${\bm{v}}$ with arbitrary index.
\mathbf{s}ubsection{A closed formula for $K$ and $\dot{K}$}
For $u\in\left[-1,1{\textnormal{i}}ght]$, let $\kappa_0(u)=\mathbf{m}athbf{f}rac{\pi-\cos^{-1}\left(u{\textnormal{i}}ght)}{\pi}$ and $\kappa_1(u)=\mathbf{m}athbf{f}rac{\mathbf{s}qrt{1-u^2}+\left(\pi-\cos^{-1}\left(u{\textnormal{i}}ght){\textnormal{i}}ght)u}{\pi}$ be the arc-cosine kernels defined in \citet{cho2009kernel}. \citet{daniely2016toward} showed that
\begin{align}
\label{eq:k_form}
K^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)=\mathbf{m}athbf{f}rac{c_vc_w}{2}\mathbf{s}qrt{\Sigma^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{x}{\textnormal{i}}ght)\Sigma^{\left(l{\textnormal{i}}ght)}\left(\mathbf{z},\mathbf{z}{\textnormal{i}}ght)}\kappa_1\left(\overline{\Sigma}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght)
\mathbf{m}athbf{e}nd{align}
and
\begin{align}
\dot{K}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)=\mathbf{m}athbf{f}rac{c_vc_w}{2}\kappa_0\left(\overline{\Sigma}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght),
\mathbf{m}athbf{e}nd{align}
where $K^{(l)}$ and $\dot K^{(l)}$ are defined in \mathbf{m}athbf{e}qref{eq:K} and \mathbf{m}athbf{e}qref{eq:Kdot} and $c_v,c_w$ are defined in Sec.~{\textnormal{e}}f{sec:conv_resnet}.
\mathbf{s}ubsection{ResCGPK Derivation}\label{ap:1}
\begin{theorem}
\label{thm:1}
For an $L$-layer neural network $f$ and $\mathbf{x},\mathbf{z}\in\mathbf{m}athbb{R}^{ C_{0} \times d}$,
\[
\Sigma_{j,j^{'}}^{\left(1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)=\mathbf{m}athbf{f}rac{1}{C_0}\left(\mathbf{x}^T\mathbf{z}{\textnormal{i}}ght)_{j,j^{'}}
\]
\[
\Sigma_{j,j^{'}}^{\left(2{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)=\mathbf{m}athbf{f}rac{c_w}{q}\text{tr}\left(\Sigma_{\mathbf{m}athcal{D}_{j,j^{'}}}^{\left(1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght) + \mathbf{m}athbf{f}rac{\alpha^{2}}{q^{2}}\mathbf{s}um_{k=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\text{tr}\left(K_{\mathbf{m}athcal{D}_{j+k,j^{'}+k}}^{\left(1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght).
\]
For $3\leq l \leq L$,
\[
\Sigma_{j,j^{'}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)=\Sigma_{j,j^{'}}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)+\mathbf{m}athbf{f}rac{\alpha^{2}}{q^{2}}\mathbf{s}um_{k=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\text{tr}\left(K_{\mathbf{m}athcal{D}_{j+k,j^{'}+k}}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght).
\]
Finally, for the output layer
\[
\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)
= \Sigma_{1,1}^{\left(1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)+ \mathbf{m}athbf{f}rac{\alpha^{2}}{qc_{w}}\mathbf{s}um_{l=1}^{L}\text{tr}\left(K_{\mathbf{m}athcal{D}_{1,1}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght)
\]
\[
\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\text{Tr}}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)=\mathbf{m}athbf{f}rac{1}{d}\mathbf{s}um_{j=1}^{ d }\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{s}_j\mathbf{x},\mathbf{s}_j\mathbf{z}{\textnormal{i}}ght)
\]
\[
\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\text{GAP}}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)=\mathbf{m}athbf{f}rac{1}{d^2}\mathbf{s}um_{j,j^{'}=1}^{ d }\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{s}_j\mathbf{x},\mathbf{s}_{j^{'}}\mathbf{z}{\textnormal{i}}ght).
\]
\mathbf{m}athbf{e}nd{theorem}
\begin{proof}
We begin by deriving a formula for $\Sigma^{(l)(\mathbf{x},\mathbf{z})}$.
The case of $l=1$ is shown in Lemma ({\textnormal{e}}f{lem:c0}). For $2<l\leq L$, the strategy is to express $\mathbf{m}athbb{E}\left[g_{ij}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)g_{ij^{'}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]$
using $\mathbf{m}athbb{E}\left[f_{c,l}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)f_{c,l^{'}}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]$
and vice versa (which we can do using Lemma {\textnormal{e}}f{lem:c1}).
This way we derive an expression for $\mathbf{m}athbb{E}\left[f_{c,l}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)f_{c,l^{'}}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]$
in Lemma ({\textnormal{e}}f{lem:c2}) and subsequently get:
\begin{align*}
\mathbf{m}athbb{E}\left[g_{ij}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)g_{ij^{'}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght] &
\underset{\text{Lemma }{\textnormal{e}}f{lem:c1}}{=}
\mathbf{m}athbf{f}rac{c_{w}}{qC_{l-1}}\mathbf{s}um_{k=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\mathbf{s}um_{c=1}^{C_{l-1}}\mathbf{m}athbb{E}\left[f_{c,j+k}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)f_{c,j^{'}+k}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght] \\ &
\underset{\text{Lemma }{\textnormal{e}}f{lem:c2}}{=} \underset{\text{Denote by }A}{\underbrace{\mathbf{m}athbf{f}rac{c_{w}}{qC_{l-1}}\mathbf{s}um_{c=1}^{C_{l-1}}\left(\mathbf{s}um_{k=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\mathbf{m}athbb{E}\left[f_{c,j+k}^{\left(l-2{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)f_{c,j^{'}+k}^{\left(l-2{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]{\textnormal{i}}ght)}}+\\
& ~~~~~~~ +\alpha^{2}\mathbf{m}athbf{f}rac{c_{v}c_{w}}{q^{2}C_{l-1}}\mathbf{s}um_{c=1}^{C_{l-1}}\left(\mathbf{s}um_{k=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\mathbf{s}um_{k^{'}=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\mathbf{m}athbb{E}\left[\mathbf{s}igma\left(g^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght)_{c,j+k+k^{'}}\mathbf{s}igma\left(g^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght)_{c,j^{'}+k+k^{'}}{\textnormal{i}}ght]{\textnormal{i}}ght) \\
& = A +\alpha^{2}\mathbf{m}athbf{f}rac{c_{v}c_{w}}{q^{2}}\mathbf{s}um_{k,k^{'}=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\mathbf{m}athbb{E}\left[\mathbf{s}igma\left(g^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght)_{*,j+k+k^{'}}\mathbf{s}igma\left(g^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght)_{*,j^{'}+k+k^{'}}{\textnormal{i}}ght] \\
& = A + \mathbf{m}athbf{f}rac{\alpha^{2}}{q^{2}}\mathbf{s}um_{k=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\text{tr}\left(K_{\mathbf{m}athcal{D}_{j+k,j^{'}+k}}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght).
\mathbf{m}athbf{e}nd{align*}
If $l>2$ then using Lemma ({\textnormal{e}}f{lem:c1}) we obtain $A=\mathbf{m}athbb{E}\left[g_{ij}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)g_{ij^{'}}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]=\Sigma_{j,j^{'}}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)$.
Otherwise if $l=2$ then using Lemma ({\textnormal{e}}f{lem:c0}) we obtain $A=\mathbf{m}athbf{f}rac{c_w}{q}\text{tr}\left(\Sigma_{\mathbf{m}athcal{D}_{j,j^{'}}}^{\left(1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght)$.
We leave the three output layers to lemma {\textnormal{e}}f{lem:c3}
\mathbf{m}athbf{e}nd{proof}
\begin{lemma} \label{lem:c0}
\[
\mathbf{m}athbb{E}\left[f_{ij}^{\left(0{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)f_{ij^{'}}^{\left(0{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]
=\mathbf{m}athbb{E}\left[g_{ij}^{\left(1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)g_{ij^{'}}^{\left(1{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]=\mathbf{m}athbf{f}rac{1}{ C_{0}}\left(\mathbf{x}^T\mathbf{z}{\textnormal{i}}ght)_{j,j^{'}}.
\]
\mathbf{m}athbf{e}nd{lemma}
\begin{proof}
For $g^{\left(1{\textnormal{i}}ght)}$ we have:
\[
\Sigma^{\left(1{\textnormal{i}}ght)}_{j,j^{'}}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)=\mathbf{m}athbb{E}\left[g_{ij}^{\left(1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)g_{ij^{'}}^{\left(1{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]
=\mathbf{m}athbf{f}rac{1}{ C_{0}}\mathbf{s}um_{l,l^{'}=1}^{ C_{0}}\mathbf{m}athbb{E}\left[\mathbf{W}^{\left(1{\textnormal{i}}ght)}_{1,l,i}\mathbf{x}_{l,j}\mathbf{W}^{\left(1{\textnormal{i}}ght)}_{1,l^{'},i}\mathbf{z}_{l^{'},j^{'}}{\textnormal{i}}ght]
\]
\[
= \mathbf{m}athbf{f}rac{1}{ C_{0}}\mathbf{s}um_{l,l^{'}=1}^{ C_{0}}\mathbf{m}athbb{E}\left[\mathbf{W}^{\left(1{\textnormal{i}}ght)}_{1,l,i}\mathbf{W}^{\left(1{\textnormal{i}}ght)}_{1,l^{'},i}{\textnormal{i}}ght] \mathbf{m}athbb{E}\left[\mathbf{x}_{l,j}\mathbf{z}_{l^{'}j^{'}}{\textnormal{i}}ght]
\underset{\mathbf{m}athbf{e}qref{eq:exp_par}}{=} \mathbf{m}athbf{f}rac{1}{ C_{0}}\mathbf{s}um_{l=1}^{ C_{0}} \mathbf{x}_{l,j}\mathbf{z}_{l,j^{'}}
=\mathbf{m}athbf{f}rac{1}{ C_{0}}\left(\mathbf{x}^T\mathbf{z}{\textnormal{i}}ght)_{j,j^{'}}.
\]
For $f^{\left(1{\textnormal{i}}ght)}$ the proof is analogous, by simply replacing $\mathbf{W}^{\left(1{\textnormal{i}}ght)}$ with $\mathbf{V}^{\left(0{\textnormal{i}}ght)}$.
\mathbf{m}athbf{e}nd{proof}
\begin{lemma}
\label{lem:c1}$\mathbf{m}athbf{f}orall 2\leq l \leq L, 1\leq i,i^{'}\leq C_{l},1\leq j,j^{'}\leq d$,
we have:
\[
\mathbf{m}athbb{E}\left[g_{ij}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)g_{i^{'}j^{'}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]=\delta_{i,i^{'}}\mathbf{m}athbf{f}rac{c_{w}}{qC_{l-1}}\mathbf{s}um_{k=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\mathbf{s}um_{c=1}^{C_{l-1}}\mathbf{m}athbb{E}\left[f_{c,j+k}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)f_{c,j^{'}+k}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght].
\]
\mathbf{m}athbf{e}nd{lemma}
\begin{proof}
\[
\mathbf{m}athbb{E}\left[g_{ij}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)g_{i^{'}j^{'}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]=\mathbf{m}athbf{f}rac{c_{w}}{qC_{l-1}}\mathbf{m}athbb{E}\left[\left[\mathbf{s}um_{c=1}^{C_{l-1}}\mathbf{W}_{:,c,i}^{\left(l{\textnormal{i}}ght)}*f_{c}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght]_{j}\left[\mathbf{s}um_{c^{'}=1}^{C_{l-1}}\mathbf{W}_{:,c^{'},i^{'}}^{\left(l{\textnormal{i}}ght)}*f_{c^{'}}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]_{j^{'}}{\textnormal{i}}ght]
\]
\[
=\mathbf{m}athbf{f}rac{c_{w}}{qC_{l-1}}\mathbf{s}um_{c,c^{'}=1}^{C_{l}}\mathbf{s}um_{k,k^{'}=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\mathbf{m}athbb{E}\left[\mathbf{W}_{k+\mathbf{m}athbf{f}rac{q+1}{2},c,i}^{\left(l{\textnormal{i}}ght)}\mathbf{W}_{k^{'}+\mathbf{m}athbf{f}rac{q+1}{2},c^{'},i^{'}}^{\left(l{\textnormal{i}}ght)}{\textnormal{i}}ght]\mathbf{m}athbb{E}\left[f_{c,j+k}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)f_{c^{'},j^{'}+k^{'}}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]
\]
\begin{align}
\underset{Equation{\textnormal{e}}f{eq:exp_par}}{=}\delta_{i,i^{'}}\mathbf{m}athbf{f}rac{c_{w}}{qC_{l-1}}\mathbf{s}um_{k=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\mathbf{s}um_{c=1}^{C_{l-1}}\mathbf{m}athbb{E}\left[f_{c,j+k}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)f_{c,j^{'}+k}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]
\label{eq:helper1}
\mathbf{m}athbf{e}nd{align}
\mathbf{m}athbf{e}nd{proof}
\begin{lemma}
\label{lem:c2}$\mathbf{m}athbf{f}orall 1\leq l \leq L, 1\leq i,i^{'}\leq C_{l},1\leq j,j^{'}\leq d$, we have:
\[
\mathbf{m}athbb{E}\left[f_{ij}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)f_{ij^{'}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]=\mathbf{m}athbb{E}\left[f_{ij}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)f_{ij^{'}}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]
+\alpha^2\mathbf{m}athbf{f}rac{c_v}{q}\mathbf{s}um_{k=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\mathbf{m}athbb{E}\left[\mathbf{s}igma\left(g^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght)_{*,j+k}\mathbf{s}igma\left(g^{\left(l{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght)_{*,j^{'}+k}{\textnormal{i}}ght].
\]
\mathbf{m}athbf{e}nd{lemma}
\begin{proof}
Using the definition for $f_{ij}^{\left(l{\textnormal{i}}ght)}$
(Def.~{\textnormal{e}}f{def:network3}), we get the expression:
\[
\mathbf{m}athbb{E}\left[f_{ij}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)f_{ij^{'}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]=\mathbf{m}athbb{E}\left[f_{ij}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)f_{ij^{'}}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]+
\]
\[
\underset{\text{Denote by }B_{1}}{\underbrace{\mathbf{m}athbb{E}\left[f_{ij}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)\left[\mathbf{s}um_{c=1}^{C_{l}}\mathbf{V}_{:,c,i}^{\left(l{\textnormal{i}}ght)}*\mathbf{s}igma\left(g_{c}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght){\textnormal{i}}ght]_{j^{'}}{\textnormal{i}}ght]}}+\underset{\text{Denote by }B_{2}}{\underbrace{\mathbf{m}athbb{E}\left[f_{ij^{'}}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght)\left[\mathbf{s}um_{c=1}^{C_{l}}\mathbf{V}_{:,c,i}^{\left(l{\textnormal{i}}ght)}*\mathbf{s}igma\left(g_{c}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght){\textnormal{i}}ght]_{j}{\textnormal{i}}ght]}}+
\]
\[
+\alpha^{2}\mathbf{m}athbf{f}rac{c_{v}}{q}\underset{\text{Denote by }A}{\underbrace{\mathbf{m}athbf{f}rac{1}{C_{l}}\mathbf{m}athbb{E}\left[\left[\mathbf{s}um_{c=1}^{C_{l}}\mathbf{V}_{:,c,i}^{\left(l{\textnormal{i}}ght)}*\mathbf{s}igma\left(g_{c}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght){\textnormal{i}}ght]_{j}\left[\mathbf{s}um_{c^{'}=1}^{C_{l}}\mathbf{V}_{:,c^{'},i}^{\left(l{\textnormal{i}}ght)}*\mathbf{s}igma\left(g_{c^{'}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght){\textnormal{i}}ght]_{j^{'}}{\textnormal{i}}ght]}}.
\]
We will deal with this expression in parts. First, for $B_{1}$ we get:
\[
B_{1}=\mathbf{s}um_{k=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\mathbf{s}um_{c=1}^{C_{l}}\mathbf{m}athbb{E}\left[f_{ij}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)\mathbf{V}_{k+\mathbf{m}athbf{f}rac{q+1}{2},c,i}^{\left(l{\textnormal{i}}ght)}\mathbf{s}igma\left(g_{c}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght)_{c,j^{'}+k}{\textnormal{i}}ght]=0,
\]
where the rightmost equality follows from \mathbf{m}athbf{e}qref{eq:exp_par} and
the fact that $\mathbf{m}athbb{E}\left[\mathbf{V}{\textnormal{i}}ght]=0$ mean in expectation in every index.
Analogously, we also get $B_{2}=0$.
Opening $A$ and using Equation \mathbf{m}athbf{e}qref{eq:exp_par} we get:
\[
A=\mathbf{m}athbf{f}rac{1}{C_{l}}\mathbf{s}um_{k^{'}=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\mathbf{s}um_{k=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\mathbf{s}um_{c=1}^{C_{l}}\mathbf{s}um_{c^{'}=1}^{C_{l}}\mathbf{m}athbb{E}\left[\mathbf{V}_{k+\mathbf{m}athbf{f}rac{q+1}{2},c,i}\mathbf{s}igma\left(g^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght)_{c,j+k}\mathbf{V}_{k^{'}+\mathbf{m}athbf{f}rac{q+1}{2},c^{'},i}\mathbf{s}igma\left(g^{\left(l{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght)_{c^{'},j^{'}+k^{'}}{\textnormal{i}}ght]
\]
\[
\underset{\mathbf{m}athbf{e}qref{eq:exp_par}}{=}\mathbf{s}um_{k=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\mathbf{m}athbf{f}rac{1}{C_{l}}\mathbf{s}um_{c=1}^{C_{l}}\mathbf{m}athbb{E}\left[\mathbf{s}igma\left(g^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght)_{c,j+k}\mathbf{s}igma\left(g^{\left(l{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght)_{c,j^{'}+k}{\textnormal{i}}ght]
\]
\[
=\mathbf{s}um_{k=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\mathbf{m}athbb{E}\left[\mathbf{s}igma\left(g^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght)_{*,j+k}\mathbf{s}igma\left(g^{\left(l{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght)_{*,j^{'}+k}{\textnormal{i}}ght].
\]
Overall, we obtain
\[
\mathbf{m}athbb{E}\left[f_{ij}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)f_{ij^{'}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]=\mathbf{m}athbb{E}\left[f_{ij}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)f_{ij^{'}}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]
+\alpha^2\mathbf{m}athbf{f}rac{c_v}{q}\mathbf{s}um_{k=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\mathbf{m}athbb{E}\left[\mathbf{s}igma\left(g^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght)_{*,j+k}\mathbf{s}igma\left(g^{\left(l{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght)_{*,j^{'}+k}{\textnormal{i}}ght].
\]
\mathbf{m}athbf{e}nd{proof}
\begin{lemma}
\label{lem:c3}
\[
\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)
= \Sigma_{1,1}^{\left(0{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)+ \mathbf{m}athbf{f}rac{\alpha^{2}}{qc_{w}}\mathbf{s}um_{l=1}^{L}\text{tr}\left(K_{\mathbf{m}athcal{D}_{1,1}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght).
\]
\mathbf{m}athbf{e}nd{lemma}
\begin{proof}
We start by proving the case for $\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}$. Observe that:
\[
\mathbf{m}athbb{E}_{\theta}\left[f^{\mathbf{m}athbb{E}q}\left(\mathbf{x};\theta {\textnormal{i}}ght)f^{\mathbf{m}athbb{E}q}\left(\mathbf{z};\theta {\textnormal{i}}ght){\textnormal{i}}ght]
=\mathbf{m}athbf{f}rac{1}{C_{L}}\mathbf{s}um_{i,i^{'}=1}^{C_{L}} \mathbf{m}athbb{E}\left[W_{i}^{\mathbf{m}athbb{E}q}f_{i,1}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)W_{i^{'}}^{\mathbf{m}athbb{E}q}f_{i^{'},1}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]
\]
\[
=\mathbf{m}athbf{f}rac{1}{C_{L}}\mathbf{s}um_{i,i^{'}=1}^{C_{L}} \delta_{ii^{'}}\mathbf{m}athbb{E}\left[f_{i,1}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)f_{i^{'},1}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]
=\mathbf{m}athbf{f}rac{1}{C_{L}}\mathbf{s}um_{i=1}^{C_{L}} \mathbf{m}athbb{E}\left[f_{i,1}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)f_{i,1}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght].
\]
Applying Lemma {\textnormal{e}}f{lem:c2} recursively we obtain
\[
\mathbf{m}athbb{E}_{\theta}\left[f^{\mathbf{m}athbb{E}q}\left(\mathbf{x};\theta {\textnormal{i}}ght)f^{\mathbf{m}athbb{E}q}\left(\mathbf{z};\theta {\textnormal{i}}ght){\textnormal{i}}ght]
\]
\[
=\mathbf{m}athbf{f}rac{1}{C_{L}}\mathbf{s}um_{i=1}^{C_{L}} \left(
\mathbf{m}athbb{E}\left[f_{i,1}^{\left(L-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)f_{i,1}^{\left(L-1{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]+\alpha^{2}\mathbf{m}athbf{f}rac{c_{v}}{q}\mathbf{s}um_{k=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\mathbf{m}athbb{E}\left[\mathbf{s}igma\left(g^{\left(L{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght)_{*,1+k}\mathbf{s}igma\left(g^{\left(L{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght)_{*,1+k}{\textnormal{i}}ght]
{\textnormal{i}}ght)
\]
\[
=\mathbf{m}athbf{f}rac{1}{C_{L}}\mathbf{s}um_{i=1}^{C_{L}} \left( \mathbf{m}athbb{E}\left[f_{i,1}^{\left(L-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)f_{i,1}^{\left(L-1{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]+\mathbf{m}athbf{f}rac{\alpha^{2}}{qc_{w}}\text{tr}\left(K_{\mathbf{m}athcal{D}_{1,1}}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght)
{\textnormal{i}}ght)
\]
\[
\underset{\text{Lemma {\textnormal{e}}f{lem:c2}}}{=}\mathbf{m}athbf{f}rac{1}{C_{L}}\mathbf{s}um_{i=1}^{C_{L}} \left( \mathbf{m}athbb{E}\left[f_{i,1}^{\left(L-2{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)f_{i,1}^{\left(L-2{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]+\mathbf{m}athbf{f}rac{\alpha^{2}}{qc_{w}}\mathbf{s}um_{l=L-1}^{L}\text{tr}\left(K_{\mathbf{m}athcal{D}_{1,1}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght)
{\textnormal{i}}ght).
\]
Applying Lemma {\textnormal{e}}f{lem:c2} recursively for all layers we obtain
\[
\mathbf{m}athbb{E}_{\theta}\left[f^{\mathbf{m}athbb{E}q}\left(\mathbf{x};\theta {\textnormal{i}}ght)f^{\mathbf{m}athbb{E}q}\left(\mathbf{z};\theta {\textnormal{i}}ght){\textnormal{i}}ght]
=\mathbf{m}athbf{f}rac{1}{C_{L}}\mathbf{s}um_{i=1}^{C_{L}} \left( \mathbf{m}athbb{E}\left[f_{i,1}^{\left(0{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)f_{i,1}^{\left(0{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]+\mathbf{m}athbf{f}rac{\alpha^{2}}{qc_{w}}\mathbf{s}um_{l=1}^{L}\text{tr}\left(K_{\mathbf{m}athcal{D}_{1,1^{'}}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght){\textnormal{i}}ght)
\]
\[
=\Sigma_{1,1}^{\left(1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)+\mathbf{m}athbf{f}rac{\alpha^{2}}{qc_{w}}\mathbf{s}um_{l=1}^{L}\text{tr}\left(K_{\mathbf{m}athcal{D}_{1,1}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght).
\]
\mathbf{m}athbf{e}nd{proof}
\begin{lemma}
\[
\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\text{Tr}}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)=\mathbf{m}athbf{f}rac{1}{d}\mathbf{s}um_{j=1}^{ d }\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{s}_j\mathbf{x},\mathbf{s}_j\mathbf{z}{\textnormal{i}}ght)
\]
\[
\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\text{GAP}}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)=\mathbf{m}athbf{f}rac{1}{d^2}\mathbf{s}um_{j,j^{'}=1}^{ d }\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{s}_j\mathbf{x},\mathbf{s}_{j^{'}}\mathbf{z}{\textnormal{i}}ght).
\]
\mathbf{m}athbf{e}nd{lemma}
\begin{proof}
For $f^{\text{Tr}}$ we have:
\[
\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\text{Tr}}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)=\mathbf{m}athbb{E}_{\theta}\left[f^{\text{Tr}}\left(\mathbf{x};\theta {\textnormal{i}}ght)f^{\text{Tr}}\left(\mathbf{z};\theta {\textnormal{i}}ght){\textnormal{i}}ght]
=\mathbf{m}athbf{f}rac{1}{C_{L} d }\mathbf{s}um_{j,j^{'}=1}^{ d }\mathbf{s}um_{i,i^{'}=1}^{C_{L}} \mathbf{m}athbb{E}\left[W_{i,j}^{\text{Tr}}f_{i,j}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)W_{i^{'},j^{'}}^{\text{Tr}}f_{i^{'},j^{'}}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]
\]
\[
=\mathbf{m}athbf{f}rac{1}{d}\mathbf{s}um_{j=1}^{ d }\left(\mathbf{m}athbf{f}rac{1}{C_{L}}\mathbf{s}um_{i=1}^{C_{L}} \mathbf{m}athbb{E}\left[f_{i,j}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)f_{i,j}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]{\textnormal{i}}ght)
=\mathbf{m}athbf{f}rac{1}{d}\mathbf{s}um_{j=1}^{ d }\left(\mathbf{m}athbf{f}rac{1}{C_{L}}\mathbf{s}um_{i=1}^{C_{L}} \mathbf{m}athbb{E}\left[f_{i,1}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{s}_j\mathbf{x}{\textnormal{i}}ght)f_{i,1}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{s}_j\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]{\textnormal{i}}ght),
\]
where the part inside the parentheses was shown in the previous Lemma {\textnormal{e}}f{lem:c3} to equal $\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{s}_j\mathbf{x},\mathbf{s}_j\mathbf{z}{\textnormal{i}}ght)$.
For $f^{\text{GAP}}$ we analogously obtain
\[
\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\text{GAP}}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)=\mathbf{m}athbb{E}_{\theta}\left[f^{\text{GAP}}\left(\mathbf{x};\theta {\textnormal{i}}ght)f^{\text{GAP}}\left(\mathbf{z};\theta {\textnormal{i}}ght){\textnormal{i}}ght]
=\mathbf{m}athbf{f}rac{1}{C_{L}\mathbf{m}athbf{f}rac{d^2}{s^2}}\mathbf{s}um_{j,j^{'}=1}^{ d }\mathbf{s}um_{i,i^{'}=1}^{C_{L}} \mathbf{m}athbb{E}\left[W_{i}^{\text{GAP}}f_{i,j}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)W_{i^{'}}^{\text{GAP}}f_{i^{'},j^{'}}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]
\]
\[
=\mathbf{m}athbf{f}rac{1}{d^2}\mathbf{s}um_{j,j^{'}=1}^{ d }\left(\mathbf{m}athbf{f}rac{1}{C_{L}}\mathbf{s}um_{i=1}^{C_{L}} \mathbf{m}athbb{E}\left[f_{i,j}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)f_{i,j^{'}}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]{\textnormal{i}}ght)
=\mathbf{m}athbf{f}rac{1}{d^2}\mathbf{s}um_{j,j^{'}=1}^{ d }\left(\mathbf{m}athbf{f}rac{1}{C_{L}}\mathbf{s}um_{i=1}^{C_{L}} \mathbf{m}athbb{E}\left[f_{i,1}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{s}_j\mathbf{x}{\textnormal{i}}ght)f_{i,1}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{s}_{j^{'}}\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]{\textnormal{i}}ght),
\]
from which the claim follows.
\mathbf{m}athbf{e}nd{proof}
\mathbf{s}ubsection{Formulas for Multisphere Input: Proof of Theorem {\textnormal{e}}f{Thm:ResCGPK}}
\begin{lemma}
\label{lem:sd1}
For an $L$-layer ResNet $f$ and $\mathbf{x}\in\mathbf{m}athbb{MS}\left( C_{0},d{\textnormal{i}}ght)$, and for every $1\leq l\leq L, 1\leq j,j^{'}\leq d$
\[
\Sigma_{j,j^{'}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{x}{\textnormal{i}}ght)=
\begin{cases}
\mathbf{m}athbf{f}rac{1}{C_0} & l=1\\
\mathbf{m}athbf{f}rac{\left(1+\alpha^2 \mathbf{m}athbf{f}rac{c_vc_w}{2} {\textnormal{i}}ght)^{l-2}\left(2c_w + \alpha^2c_vc_w{\textnormal{i}}ght)}{2C_{0}} & l \mathbf{m}athbf{g}eq 2.
\mathbf{m}athbf{e}nd{cases}
\]
\mathbf{m}athbf{e}nd{lemma}
\begin{proof}
We prove this by induction using the formula in Theorem ({\textnormal{e}}f{thm:1}). For $l=1$, since by assumption $\mathbf{n}orm{\mathbf{x}_i}=1$, for every $i$ we get that $\mathbf{x}^T\mathbf{x}$ is the $d\times d$ matrix with 1 in every entry.
Therefore,
\[
\Sigma_{j,j^{'}}^{\left(1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{x}{\textnormal{i}}ght)=\mathbf{m}athbf{f}rac{1}{C_{0}}\left(\mathbf{x}^T\mathbf{x}{\textnormal{i}}ght)_{j,j^{'}}
=\mathbf{m}athbf{f}rac{1}{C_0}.
\]
Similarly, for $l=2$:
\[
\Sigma_{j,j^{'}}^{\left(2{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{x}{\textnormal{i}}ght)=\mathbf{m}athbf{f}rac{c_w}{q}\text{tr}\left(\Sigma_{\mathbf{m}athcal{D}_{j,j^{'}}}^{\left(1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght) + \mathbf{m}athbf{f}rac{\alpha^{2}}{q^{2}}\mathbf{s}um_{k=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\text{tr}\left(K_{\mathbf{m}athcal{D}_{j+k,j^{'}+k}}^{\left(1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght).
\]
We can plug in the induction hypothesis, and express $K$ as in \mathbf{m}athbf{e}qref{eq:k_form}, obtaining
\[
\Sigma_{j,j^{'}}^{\left(2{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{x}{\textnormal{i}}ght)=\mathbf{m}athbf{f}rac{c_w}{C_0} + \mathbf{m}athbf{f}rac{\alpha^{2}}{q^{2}}\mathbf{s}um_{k,k^{'}=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\mathbf{m}athbf{f}rac{c_vc_w}{2}\kappa_1\left(1{\textnormal{i}}ght)\mathbf{s}qrt{\Sigma_{j+k+k^{'},j^{'}+k+k^{'}}^{\left(1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{x}{\textnormal{i}}ght)\Sigma_{j+k+k^{'},j^{'}+k+k^{'}}^{\left(1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{x}{\textnormal{i}}ght)}
\]
\[
=\mathbf{m}athbf{f}rac{c_w}{C_0} + \alpha^{2}\mathbf{m}athbf{f}rac{c_vc_w}{2}N_1 = \mathbf{m}athbf{f}rac{2c_w + \alpha^2c_vc_w}{2C_0},
\]
where we used the fact that $\kappa_1\left(1{\textnormal{i}}ght)=1$. The proof for $l\mathbf{m}athbf{g}eq3$ is analogous:
\[
\Sigma_{j,j}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{x}{\textnormal{i}}ght)=\Sigma_{j,j}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{x}{\textnormal{i}}ght)+\mathbf{m}athbf{f}rac{\alpha^{2}}{q^{2}}\mathbf{s}um_{k=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\text{tr}\left(K_{\mathbf{m}athcal{D}_{j+k,j+k}}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght)
\]
\[
=N_{L-1}+\mathbf{m}athbf{f}rac{\alpha^{2}}{q^{2}} \mathbf{s}um_{k,k^{'}=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\mathbf{m}athbf{f}rac{c_vc_w}{2}N_{L-1}\kappa_1\left(1{\textnormal{i}}ght)
= \left(1+\alpha ^2 \mathbf{m}athbf{f}rac{c_vc_w}{2}{\textnormal{i}}ght)N_{L-1}
\]
\[
=\mathbf{m}athbf{f}rac{\left(1+\alpha^2 \mathbf{m}athbf{f}rac{c_vc_w}{2} {\textnormal{i}}ght)^{l-2}\left(c_w + \alpha^2c_vc_w{\textnormal{i}}ght)}{2C_{0}}.
\]
\mathbf{m}athbf{e}nd{proof}
\begin{lemma} \label{lem:k_eq_ms}
For any $L\in\mathbf{m}athbb{N}$ let $N_L$ be the value of $\Sigma_{j,j}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{x}{\textnormal{i}}ght)$ from Lemma ${\textnormal{e}}f{lem:sd1}$. Let $\mathbf{x},\mathbf{z}\in\mathbf{m}athbb{MS}\left( C_{0},d{\textnormal{i}}ght)$, then
\[
\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght) = \Sigma_{1,1}^{\left(1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)+ \mathbf{m}athbf{f}rac{\alpha^{2}c_v}{2q}\mathbf{s}um_{l=1}^{L}N_{l} \text{tr}\left(\kappa_1\left(\overline{\Sigma}_{\mathbf{m}athcal{D}_{1,1}}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)
{\textnormal{i}}ght){\textnormal{i}}ght).
\]
\mathbf{m}athbf{e}nd{lemma}
\begin{proof}
Let $L\in\mathbf{m}athbb{N}$. We know from Theorem {\textnormal{e}}f{thm:1} that
\[
\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght) = \Sigma_{1,1}^{\left(1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)+ \mathbf{m}athbf{f}rac{\alpha^{2}}{qc_{w}}\mathbf{s}um_{l=1}^{L}\text{tr}\left(K_{\mathbf{m}athcal{D}_{1,1}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght).
\]
By expressing $K$ as in \mathbf{m}athbf{e}qref{eq:k_form} and using Lemma {\textnormal{e}}f{lem:sd1} we get
\[
\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght) = \Sigma_{1,1}^{\left(1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)+ \mathbf{m}athbf{f}rac{\alpha^{2}c_v}{2q}\mathbf{s}um_{l=1}^{L}N_{l} \text{tr}\left(\kappa_1\left(\overline{\Sigma}_{\mathbf{m}athcal{D}_{1,1}}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)
{\textnormal{i}}ght){\textnormal{i}}ght).
\]
\mathbf{m}athbf{e}nd{proof}
\begin{corollary}
Fix $c_v=2,c_w=1$ then
\[
\overline{\mathbf{m}athcal{K}}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght) = \mathbf{m}athbf{f}rac{C_0}{(1+\alpha^2)^{L}}\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght).
\]
\mathbf{m}athbf{e}nd{corollary}
\begin{proof}
Using the previous lemma and the fact that $\kappa(1)=1$,
\[
\overline{\mathbf{m}athcal{K}}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{x},\mathbf{x}{\textnormal{i}}ght) = \Sigma_{1,1}^{\left(1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{x}{\textnormal{i}}ght)+ \alpha^{2}\mathbf{s}um_{l=1}^{L}N_{l}
=\mathbf{m}athbf{f}rac{1}{C_0}\left(1+\alpha^2\left(1+(1+\alpha^2)\mathbf{s}um_{l=0}^{L-2}(1+\alpha^2)^l{\textnormal{i}}ght){\textnormal{i}}ght)
\]
\[
=\mathbf{m}athbf{f}rac{1}{C_0}\left(1+\alpha^2\left(1+(1+\alpha^2)\mathbf{m}athbf{f}rac{1-(1+\alpha^2)^{L-1}}{1-(1+\alpha^2)}{\textnormal{i}}ght){\textnormal{i}}ght)
=\mathbf{m}athbf{f}rac{1}{C_0}\left(1+\alpha^2\left(1+\mathbf{m}athbf{f}rac{(1+\alpha^2)^{L}-(1+\alpha^2)}{\alpha^2}{\textnormal{i}}ght){\textnormal{i}}ght)
\]
\[
=\mathbf{m}athbf{f}rac{(1+\alpha^2)^{L}}{C_0}
\].
\mathbf{m}athbf{e}nd{proof}
\begin{proposition}\label{prop:cgpk_expr_gen}
For any $L\in\mathbf{m}athbb{N}$, let $\mathbf{x},\mathbf{z}\in\mathbf{m}athbb{MS}\left( C_{0},d{\textnormal{i}}ght)$, and denote $\mathbf{t} = \left(\left(\mathbf{x}^T\mathbf{z}{\textnormal{i}}ght)_{1,1},\left(\mathbf{x}^T\mathbf{z}{\textnormal{i}}ght)_{2,2},\ldots,\left(\mathbf{x}^T\mathbf{z}{\textnormal{i}}ght)_{d,d}{\textnormal{i}}ght)\in[-1,1]^d$. Suppose that $\alpha$ is fixed for all networks of different depths, then:
\[
\mathbf{m}athcal{K}^{\left(1{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{t}{\textnormal{i}}ght) = \mathbf{m}athbf{f}rac{1}{C_0}\mathbf{t}_1 + \mathbf{m}athbf{f}rac{\alpha^2}{qc_wC_0}\mathbf{s}um_{k=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\kappa_1\left(\mathbf{t}_{1+k}{\textnormal{i}}ght)
\]
and
\[
\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{t}{\textnormal{i}}ght) = \mathbf{m}athcal{K}^{\left(L-1{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{t}{\textnormal{i}}ght) + \tilde{N}_{L} \mathbf{s}um_{k,k^{'}=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\kappa_1\left(\mathbf{m}athbf{f}rac{\mathbf{m}athcal{K}^{\left(L-1{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{s}_{k+k^{'}}\mathbf{t}{\textnormal{i}}ght)}{N_{L}}{\textnormal{i}}ght)
\]
where $N_L$ be the value of $\Sigma_{j,j}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{x}{\textnormal{i}}ght)$ from Lemma ${\textnormal{e}}f{lem:sd1}$
\mathbf{m}athbf{e}nd{proposition}
\begin{proof}
If $\alpha$ is fixed, then for all $1\leq l\leq L$ the definition of $K^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)$ does not depend on $L$ (just that $l<L$). Therefore, using Lemma {\textnormal{e}}f{lem:k_eq_ms} we obtain
\[
\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght) = \mathbf{m}athcal{K}^{\left(L-1{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght) + \mathbf{m}athbf{f}rac{\alpha^{2}c_v}{2q}N_{L} \text{tr}\left(\kappa_1\left(\overline{\Sigma}_{\mathbf{m}athcal{D}_{1,1}}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)
{\textnormal{i}}ght){\textnormal{i}}ght).
\]
To simplify this further, observe first that a direct consequence of Theorem {\textnormal{e}}f{thm:1} is that $\mathbf{m}athbf{f}orall L\mathbf{m}athbf{g}eq 2$
\[
\Sigma_{{1,1}}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght) = \mathbf{m}athbf{f}rac{c_w}{q}\mathbf{s}um_{k=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\mathbf{m}athcal{K}^{\left(L-1{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{s}_k\mathbf{x},\mathbf{s}_k \mathbf{z}{\textnormal{i}}ght).
\]
We therefore get
\[
\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght) = \mathbf{m}athcal{K}^{\left(L-1{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght) + \mathbf{m}athbf{f}rac{\alpha^{2}c_vc_w}{2q^2}N_{L} \mathbf{s}um_{k,k^{'}=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\kappa_1\left(\overline{\mathbf{m}athcal{K}}^{\left(L-1{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{s}_{k+k^{'}}\mathbf{x},\mathbf{s}_{k+k^{'}} \mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght).
\]
\mathbf{m}athbf{e}nd{proof}
\begin{corollary}\label{cor:cgpk}
For any $\mathbf{x},\mathbf{z}\in \mathbf{m}athbb{MS}\left( C_{0},d{\textnormal{i}}ght)$, let $\mathbf{t} = \left(\left(\mathbf{x}^T\mathbf{z}{\textnormal{i}}ght)_{1,1},\left(\mathbf{x}^T\mathbf{z}{\textnormal{i}}ght)_{2,2},\ldots,\left(\mathbf{x}^T\mathbf{z}{\textnormal{i}}ght)_{d,d}{\textnormal{i}}ght)\in[-1,1]^d$. Fix $c_v=2,c_w=1$ and some $\alpha$ for all neural networks. Then,
\[
\overline{\mathbf{m}athcal{K}}^{\left(1{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{t}{\textnormal{i}}ght) = \mathbf{m}athbf{f}rac{1}{1+\alpha^2}\left(\mathbf{t}_1 + \mathbf{m}athbf{f}rac{\alpha^2}{q}\mathbf{s}um_{k=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\kappa_1\left(\mathbf{t}_{1+k}{\textnormal{i}}ght){\textnormal{i}}ght)
\]
\[
\overline{\mathbf{m}athcal{K}}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{t}{\textnormal{i}}ght) = \mathbf{m}athbf{f}rac{1}{1+\alpha^2}\left(\overline{\mathbf{m}athcal{K}}^{\left(L-1{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{t}{\textnormal{i}}ght) + \mathbf{m}athbf{f}rac{\alpha^{2}}{q^2} \mathbf{s}um_{k,k^{'}=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\kappa_1\left(\overline{\mathbf{m}athcal{K}}^{\left(L-1{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{s}_{k+k^{'}}\mathbf{t}{\textnormal{i}}ght){\textnormal{i}}ght){\textnormal{i}}ght).
\]
\mathbf{m}athbf{e}nd{corollary}
\mathbf{s}ection{Derivation of ResCNTK}
\label{ap:2}
\mathbf{s}ubsection{Rewriting the Neural Network }
The convolution of $\mathbf{w}\in\mathbf{m}athbb{R}^{q}$
with a vector ${\bm{v}}\in\mathbf{m}athbb{R}^{d}$ can be rewritten as:
\[
[\mathbf{w}*{\bm{v}}]_{i}=\mathbf{s}um_{j=1}^{q}[\mathbf{w}]_{j}[{\bm{v}}]_{i+j-\mathbf{m}athbf{f}rac{q+1}{2}}.
\]
Therefore, let ${\bm{p}}\left({\bm{v}}{\textnormal{i}}ght)\in\mathbf{m}athbb{R}^{q\times d}$ be $\left[{\bm{p}}\left({\bm{v}}{\textnormal{i}}ght){\textnormal{i}}ght]_{ij}:=[{\bm{v}}]_{i+j-\mathbf{m}athbf{f}rac{q+1}{2}}$.
Then we can rewrite the above as:
\[
\mathbf{w}*{\bm{v}}=\left(\mathbf{w}^{T}{\bm{p}}\left({\bm{v}}{\textnormal{i}}ght){\textnormal{i}}ght)^{T}={\bm{p}}\left({\bm{v}}{\textnormal{i}}ght)^{T}\mathbf{w}.
\]
Using this definition, if we instead have $\mathbf{w}\in\mathbf{m}athbb{R}^{q\times c}$ then:
\[
\bm{A}_{ij}:=[\mathbf{w}_{:,i}*{\bm{v}}]_{j}=\left[{\bm{p}}\left({\bm{v}}{\textnormal{i}}ght)^{T}\mathbf{w}_{:,i}{\textnormal{i}}ght]_{j}=\left[{\bm{p}}\left({\bm{v}}{\textnormal{i}}ght)^{T}\mathbf{w}{\textnormal{i}}ght]_{ji}\Longrightarrow\bm{A}=\mathbf{w}^{T}{\bm{p}}\left({\bm{v}}{\textnormal{i}}ght).
\]
Lastly, if we instead have $\mathbf{w}\in\mathbf{m}athbb{R}^{q\times c\times c'}$ and
$v\in\mathbf{m}athbb{R}^{c'\times d}$ then:
\[
\bm{A}_{ij}:=\mathbf{s}um_{k=1}^{c'}[\mathbf{w}_{:,k,i}*{\bm{v}}_{k}]_{j}=\mathbf{s}um_{k=1}^{c^{'}}\mathbf{w}_{:,k,:}^{T}{\bm{p}}\left({\bm{v}}_{k}{\textnormal{i}}ght).
\]
We can now rewrite the network architecture as:
\begin{align}
f^{\left(0{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)=\mathbf{m}athbf{f}rac{1}{\mathbf{s}qrt{C_{0}}}\left(\mathbf{V}^{\left(0{\textnormal{i}}ght)}_1{\textnormal{i}}ght)^T\mathbf{x}\label{def:network0-1}
\mathbf{m}athbf{e}nd{align}
\begin{align}
g^{\left(1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)=\mathbf{m}athbf{f}rac{1}{\mathbf{s}qrt{C_{0}}}\left(\mathbf{W}^{\left(1{\textnormal{i}}ght)}_1{\textnormal{i}}ght)^T\mathbf{x}\label{def:network1-1}
\mathbf{m}athbf{e}nd{align}
\begin{align}
f^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)=f^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)+\alpha\mathbf{s}qrt{\mathbf{m}athbf{f}rac{c_{v}}{qC_{l}}}\mathbf{s}um_{j=1}^{C_{l}}\left(\mathbf{V}_{:,j,:}^{\left(l{\textnormal{i}}ght)}{\textnormal{i}}ght)^{T}{\bm{p}}\left(\mathbf{s}igma\left(g_{j}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght){\textnormal{i}}ght)~~~l=1,\ldots,L\label{def:network3-1}
\mathbf{m}athbf{e}nd{align}
\begin{align}
g^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)=\mathbf{s}qrt{\mathbf{m}athbf{f}rac{c_{w}}{qC_{l-1}}}\mathbf{s}um_{j=1}^{C_{l-1}}\left(\mathbf{W}_{:,j,:}^{\left(l{\textnormal{i}}ght)}{\textnormal{i}}ght)^{T}{\bm{p}}\left(f_{j}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght)~~~l=2,\ldots,L\label{def:network2-1},
\mathbf{m}athbf{e}nd{align}
and as before we have an output layer that corresponds to one of: $f^{\mathbf{m}athbb{E}q},f^{\text{Tr}},$ or $f^{\text{GAP}}$.
\mathbf{s}ubsection{Notations}
We use a numerator layout notation, i.e., for $y\in\mathbf{m}athbb{R},\bm{A}\in\mathbf{m}athbb{R}^{m\times n}$
we denote:
\[
\mathbf{m}athbb{R}^{n\times m}\mathbf{n}i\mathbf{m}athbf{f}rac{\partial y}{\partial\bm{A}}=\begin{bmatrix}\mathbf{m}athbf{f}rac{\partial y}{\partial\bm{A}_{11}} & \mathbf{m}athbf{f}rac{\partial y}{\partial\bm{A}_{21}} & \cdots & \mathbf{m}athbf{f}rac{\partial y}{\partial\bm{A}_{m1}}\\
\mathbf{m}athbf{f}rac{\partial y}{\partial\bm{A}_{12}} & \mathbf{m}athbf{f}rac{\partial y}{\partial\bm{A}_{22}} & \cdots & \mathbf{m}athbf{f}rac{\partial y}{\partial\bm{A}_{m2}}\\
{\bm{d}}ots & {\bm{d}}ots & \ddots & {\bm{d}}ots\\
\mathbf{m}athbf{f}rac{\partial y}{\partial\bm{A}_{1n}} & \mathbf{m}athbf{f}rac{\partial y}{\partial\bm{A}_{2n}} & \cdots & \mathbf{m}athbf{f}rac{\partial y}{\partial\bm{A}_{mn}}
\mathbf{m}athbf{e}nd{bmatrix}.
\]
Also, let $\bm{J}_{mn}^{ij}$ be the $m\times n$ matrix with $1$ in coordinate
$\left(i,j{\textnormal{i}}ght)$ and $0$ elsewhere. we write $\bm{J}^{ij}$ when $m,n$ are clear by
context.
Also, let $\bm{J}^{\mathbf{m}athcal{D}_{i}}=\mathbf{s}um_{m}\delta_{m\in\mathbf{m}athcal{D}_{i}}\bm{J}_{ d ,q}^{m,m-i+\mathbf{m}athbf{f}rac{q+1}{2}}$.
\mathbf{s}ubsection{Chain Rule Reminder}
Recall that by the chain rule we know that we can decompose the Jacobian
of a composition of functions $h\circ\psi\left({\bm{v}}{\textnormal{i}}ght)$ as:
\[
J_{h\circ\psi}\left({\bm{v}}{\textnormal{i}}ght)=J_{h}\left(\psi\left({\bm{v}}{\textnormal{i}}ght){\textnormal{i}}ght)J_{\psi}\left({\bm{v}}{\textnormal{i}}ght).
\]
When $h,\psi$ are scalar functions we can write
\[
\mathbf{m}athbf{f}rac{\partial h}{\partial{\bm{v}}}=\mathbf{m}athbf{f}rac{\partial h}{\partial\psi}\mathbf{m}athbf{f}rac{\partial\psi}{\partial{\bm{v}}}.
\]
However, if $h$ is scalar valued and $\psi\left({\bm{v}}{\textnormal{i}}ght)$ is a matrix we have
\[
\left[\mathbf{m}athbf{f}rac{\partial h\circ\psi}{\partial{\bm{v}}}{\textnormal{i}}ght]_{ij}=\mathbf{m}athbf{f}rac{\partial h\circ\psi}{\partial{\bm{v}}_{ji}}=\mathbf{s}um_{p,q}\mathbf{m}athbf{f}rac{\partial h}{\partial\psi_{pq}}\mathbf{m}athbf{f}rac{\partial\psi_{pq}}{\partial{\bm{v}}_{ji}}=\text{tr}\left(\mathbf{m}athbf{f}rac{\partial h}{\partial\psi}\mathbf{m}athbf{f}rac{\partial\psi}{\partial{\bm{v}}_{ji}}{\textnormal{i}}ght).
\]
As such, the following definitions will come in handy:
\begin{definition}
$\mathbf{m}athbf{f}orall1\leq l\leq L,1\leq j,j^{'}\leq d$, let
\[
b^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght):=\left(\mathbf{m}athbf{f}rac{\partial f^{\mathbf{m}athbb{E}q}\left(\mathbf{x};\theta{\textnormal{i}}ght)}{\partial f^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)}{\textnormal{i}}ght)^{T}, ~~\Pi_{j,j^{'}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght):=\mathbf{m}athbf{f}rac{1}{c_{w}}\mathbf{m}athbb{E}\left[\left(b^{\left(l{\textnormal{i}}ght)^{T}}\left(\mathbf{x}{\textnormal{i}}ght)b^{\left(l{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght)_{j,j^{'}}{\textnormal{i}}ght].
\]
Notice that $f^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)\in\mathbf{m}athbb{R}^{C_{l}\times d }\Longrightarrow b^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)\in\mathbf{m}athbb{R}^{C_{l}\times d }$.
\mathbf{m}athbf{e}nd{definition}
\begin{remark}
There is a slight abuse of notation in the definition of $b^{\left(l{\textnormal{i}}ght)}$.
By Lemma {\textnormal{e}}f{lem:5} $b^{\left(L+1{\textnormal{i}}ght)}$ only depends on the weights of the last layer. Therefore, by the recursive formula for $b^{\left(l{\textnormal{i}}ght)}$ in Lemma {\textnormal{e}}f{lem:4} and plugging in Lemma {\textnormal{e}}f{lem:6} we get that $b^{\left(l{\textnormal{i}}ght)}$ can be written using only $W^{\left(l{\textnormal{i}}ght)},\ldots,W^{(L)},W^{\mathbf{m}athbb{E}q},V^{\left(l{\textnormal{i}}ght)},\ldots,V^{\left(L{\textnormal{i}}ght)}$ and $\dot{\mathbf{s}igma}\left({g^{\left(l+1{\textnormal{i}}ght)}(\mathbf{x})}{\textnormal{i}}ght),\ldots,\dot{\mathbf{s}igma}\left({g^{\left(L{\textnormal{i}}ght)}(\mathbf{x})}{\textnormal{i}}ght)$, where the latter are indicator functions that are always multiplied by some of $W^{\left(l{\textnormal{i}}ght)},\ldots,W^{(L)},W^{\mathbf{m}athbb{E}q},V^{\left(l{\textnormal{i}}ght)},\ldots,V^{\left(L{\textnormal{i}}ght)}$. It is easy to see now that for any $l^{'}\leq l$,
$\mathbf{m}athbb{E}\left[b_{ij}^{\left(l{\textnormal{i}}ght)}(\mathbf{x})f_{i^{'}j^{'}}^{\left(l^{'}{\textnormal{i}}ght)}(\mathbf{x}){\textnormal{i}}ght]=0=
\mathbf{m}athbb{E}\left[b_{ij}^{\left(l{\textnormal{i}}ght)}(\mathbf{x}){\textnormal{i}}ght]\mathbf{m}athbb{E}\left[f_{i^{'}j^{'}}^{\left(l^{'}{\textnormal{i}}ght)}(\mathbf{x}){\textnormal{i}}ght]$
and as a result, $b^{\left(l{\textnormal{i}}ght)}$ is uncorrelated with $f^{\left(0{\textnormal{i}}ght)},\ldots,f^{\left(l{\textnormal{i}}ght)},g^{\left(1{\textnormal{i}}ght)},\ldots,g^{\left(l{\textnormal{i}}ght)},\mathbf{x}$ and $\mathbf{z}$ \\
\mathbf{m}athbf{e}nd{remark}
\mathbf{s}ubsection{Proof of Theorem {\textnormal{e}}f{thm:2} in the main text}
We start with a lemma that relates the trace and GAP ResCNTK to the equivariant kernel.
\begin{lemma}
For an $L$ layer ResNet,
\[
\Theta^{\left(L{\textnormal{i}}ght)}_{\text{Tr}}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)=\mathbf{m}athbf{f}rac{1}{d}\mathbf{s}um_{j=1}^{d}\Theta^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{s}_j\mathbf{x},\mathbf{s}_j\mathbf{z}{\textnormal{i}}ght)
\]
and
\[
\Theta^{\left(L{\textnormal{i}}ght)}_{\text{GAP}}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)=\mathbf{m}athbf{f}rac{1}{d^2}\mathbf{s}um_{j,j^{'}=1}^{d^2}\Theta^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{s}_j\mathbf{x},\mathbf{s}_{j^{'}}\mathbf{z}{\textnormal{i}}ght).
\]
\mathbf{m}athbf{e}nd{lemma}
\begin{proof}
\[
\mathbf{m}athbb{E}\left[\left\langle \mathbf{m}athbf{f}rac{\partial f^{\text{Tr}}\left(\mathbf{x};\theta{\textnormal{i}}ght)}{\partial \theta}, \mathbf{m}athbf{f}rac{\partial f^{\text{Tr}}\left(\mathbf{z};\theta{\textnormal{i}}ght)}{\partial \theta}{\textnormal{i}}ght{\textnormal{a}}ngle{\textnormal{i}}ght]
= \mathbf{m}athbf{f}rac{1}{C_L d }\mathbf{s}um_{i,i^{'}=1}^{C_L} \mathbf{s}um_{j,j^{'}=1}^{ d } \mathbf{m}athbb{E}\left[\left\langle \mathbf{m}athbf{f}rac{\partial \mathbf{W}^{\text{Tr}}_{ij}f^{\left(L{\textnormal{i}}ght)}_{ij}\left(\mathbf{x}{\textnormal{i}}ght)}{\partial \theta}, \mathbf{m}athbf{f}rac{\partial \mathbf{W}^{\text{Tr}}_{i^{'}j^{'}}f^{\left(L{\textnormal{i}}ght)}_{i^{'}j^{'}}\left(\mathbf{z}{\textnormal{i}}ght)}{\partial \theta} {\textnormal{i}}ght{\textnormal{a}}ngle{\textnormal{i}}ght]
\]
\[
\underset{\mathbf{m}athbf{e}qref{eq:exp_par}}{=}\mathbf{m}athbf{f}rac{1}{d}\mathbf{s}um_{j=1}^{ d } \mathbf{m}athbb{E}\left[\left\langle \mathbf{m}athbf{f}rac{\partial f^{\mathbf{m}athbb{E}q}\left(\mathbf{s}_{j-1}\mathbf{x};\theta{\textnormal{i}}ght)}{\partial \theta}, \mathbf{m}athbf{f}rac{\partial f^{\mathbf{m}athbb{E}q}\left(\mathbf{s}_{j-1}\mathbf{z};\theta{\textnormal{i}}ght)}{\partial \theta}{\textnormal{i}}ght{\textnormal{a}}ngle{\textnormal{i}}ght].
\]
Similarly for GAP:
\[
\mathbf{m}athbb{E}\left[\left\langle \mathbf{m}athbf{f}rac{\partial f^{\text{GAP}}\left(\mathbf{x};\theta{\textnormal{i}}ght)}{\partial \theta}, \mathbf{m}athbf{f}rac{\partial f^{\text{GAP}}\left(\mathbf{z};\theta{\textnormal{i}}ght)}{\partial \theta}{\textnormal{i}}ght{\textnormal{a}}ngle{\textnormal{i}}ght]
= \mathbf{m}athbf{f}rac{1}{C_L d^2 }\mathbf{s}um_{i,i^{'}=1}^{C_L} \mathbf{s}um_{j,j^{'}=1}^{d} \mathbf{m}athbb{E}\left[\left\langle \mathbf{m}athbf{f}rac{\partial \mathbf{W}^{\text{GAP}}_{i}f^{\left(L{\textnormal{i}}ght)}_{ij}\left(\mathbf{x}{\textnormal{i}}ght)}{\partial \theta}, \mathbf{m}athbf{f}rac{\partial \mathbf{W}^{\text{GAP}}_{i^{'}}f^{\left(L{\textnormal{i}}ght)}_{i^{'}j^{'}}\left(\mathbf{z}{\textnormal{i}}ght)}{\partial \theta} {\textnormal{i}}ght{\textnormal{a}}ngle{\textnormal{i}}ght]
\]
\[
\underset{\mathbf{m}athbf{e}qref{eq:exp_par}}{=}\mathbf{m}athbf{f}rac{1}{d^2}\mathbf{s}um_{j,j^{'}=1}^{ d } \mathbf{m}athbb{E}\left[\left\langle \mathbf{m}athbf{f}rac{\partial f^{\mathbf{m}athbb{E}q}\left(\mathbf{s}_{j-1}\mathbf{x};\theta{\textnormal{i}}ght)}{\partial \theta}, \mathbf{m}athbf{f}rac{\partial f^{\mathbf{m}athbb{E}q}\left(\mathbf{s}_{j-1}\mathbf{z};\theta{\textnormal{i}}ght)}{\partial \theta}{\textnormal{i}}ght{\textnormal{a}}ngle{\textnormal{i}}ght]
\].
\mathbf{m}athbf{e}nd{proof}
We now return to the main proof. Using the lemma above, it remains to prove the theorem for $f^{\mathbf{m}athbb{E}q}$
\begin{proposition}
Theorem ({\textnormal{e}}f{thm:2}) holds for the case of $f=f^{\mathbf{m}athbb{E}q}$.
\mathbf{m}athbf{e}nd{proposition}
\begin{proof}
By linearity of the derivative operation and expectation we can rewrite:
\begin{align*}
\Theta^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght) &= \mathbf{m}athbb{E}\left[\left\langle \mathbf{m}athbf{f}rac{\partial f\left(\mathbf{x};\theta{\textnormal{i}}ght)}{\partial\theta},\mathbf{m}athbf{f}rac{\partial f\left(\mathbf{z};\theta{\textnormal{i}}ght)}{\partial\theta}{\textnormal{i}}ght{\textnormal{a}}ngle {\textnormal{i}}ght] \\
& = \mathbf{m}athbb{E}\left[\left\langle \mathbf{m}athbf{f}rac{\partial f\left(\mathbf{x};\theta{\textnormal{i}}ght)}{\partial\mathbf{W}^{\left(\mathbf{m}athbb{E}q{\textnormal{i}}ght)}},\mathbf{m}athbf{f}rac{\partial f\left(\mathbf{z};\theta{\textnormal{i}}ght)}{\partial\mathbf{W}^{\left(\mathbf{m}athbb{E}q{\textnormal{i}}ght)}}{\textnormal{i}}ght{\textnormal{a}}ngle {\textnormal{i}}ght]
+\mathbf{s}um_{l=1}^{L}\mathbf{m}athbb{E}\left[\left\langle \mathbf{m}athbf{f}rac{\partial f\left(\mathbf{x};\theta{\textnormal{i}}ght)}{\partial\mathbf{W}^{\left(l{\textnormal{i}}ght)}},\mathbf{m}athbf{f}rac{\partial f\left(\mathbf{z};\theta{\textnormal{i}}ght)}{\partial\mathbf{W}^{\left(l{\textnormal{i}}ght)}}{\textnormal{i}}ght{\textnormal{a}}ngle {\textnormal{i}}ght]+\mathbf{m}athbb{E}\left[\left\langle \mathbf{m}athbf{f}rac{\partial f\left(\mathbf{x};\theta{\textnormal{i}}ght)}{\partial\mathbf{V}^{\left(l{\textnormal{i}}ght)}},\mathbf{m}athbf{f}rac{\partial f\left(\mathbf{z};\theta{\textnormal{i}}ght)}{\partial\mathbf{V}^{\left(l{\textnormal{i}}ght)}}{\textnormal{i}}ght{\textnormal{a}}ngle {\textnormal{i}}ght].
\mathbf{m}athbf{e}nd{align*}
We deal with each term separately, starting with the first term.
\[
\mathbf{m}athbb{E}\left[\left\langle \mathbf{m}athbf{f}rac{\partial f\left(\mathbf{x};\theta{\textnormal{i}}ght)}{\partial\mathbf{W}^{\mathbf{m}athbb{E}q}},\mathbf{m}athbf{f}rac{\partial f\left(\mathbf{z};\theta{\textnormal{i}}ght)}{\partial\mathbf{W}^{\mathbf{m}athbb{E}q}}{\textnormal{i}}ght{\textnormal{a}}ngle {\textnormal{i}}ght]=
\mathbf{m}athbf{f}rac{1}{C_{L}}\mathbf{m}athbb{E}\left[\left\langle f^{\left(L{\textnormal{i}}ght)}_{:,1}\left(\mathbf{x}{\textnormal{i}}ght),f^{\left(L{\textnormal{i}}ght)}_{:,1}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght{\textnormal{a}}ngle{\textnormal{i}}ght]=\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght).
\]
Next, to handle $\mathbf{m}athbb{E}\left[\left\langle \mathbf{m}athbf{f}rac{\partial f\left(\mathbf{x};\theta{\textnormal{i}}ght)}{\partial\mathbf{V}^{\left(l{\textnormal{i}}ght)}},\mathbf{m}athbf{f}rac{\partial f\left(\mathbf{z};\theta{\textnormal{i}}ght)}{\partial\mathbf{V}^{\left(l{\textnormal{i}}ght)}}{\textnormal{i}}ght{\textnormal{a}}ngle {\textnormal{i}}ght]$,
observe that $\mathbf{m}athbf{f}orall1\leq l\leq L,1\leq i\leq C_{l}$ we can express $\mathbf{m}athbf{f}rac{\partial f\left(\mathbf{x};\theta{\textnormal{i}}ght)}{\partial\mathbf{V}_{:,i,:}^{\left(l{\textnormal{i}}ght)}}$
as follows:
\[
\mathbf{m}athbf{f}rac{\partial f\left(\mathbf{x};\theta{\textnormal{i}}ght)}{\partial\mathbf{V}_{:,i,:}^{\left(l{\textnormal{i}}ght)}}\underset{\text{Lemma}{\textnormal{e}}f{lem:3}}{=}\alpha\mathbf{s}qrt{\mathbf{m}athbf{f}rac{c_{v}}{qC_{l}}}\left(\mathbf{m}athbf{f}rac{\partial f\left(\mathbf{x};\theta{\textnormal{i}}ght)}{\partial f^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)}{\textnormal{i}}ght)^{T}{\bm{p}}^{T}\left(\mathbf{s}igma\left(g_{i}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght){\textnormal{i}}ght)
\]
\[
=\alpha\mathbf{s}qrt{\mathbf{m}athbf{f}rac{c_{v}}{qC_{l}}}b^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\bm{p}}^{T}\left(\mathbf{s}igma\left(g_{i}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght){\textnormal{i}}ght).
\]
Notice that Lemma {\textnormal{e}}f{lem:8} implies that the conditions of Lemma {\textnormal{e}}f{lem:7} are satisfied. Therefore,
\[
\mathbf{m}athbb{E}\left[\left\langle \mathbf{m}athbf{f}rac{\partial f\left(\mathbf{x};\theta{\textnormal{i}}ght)}{\partial\mathbf{V}_{:,i,:}^{\left(l{\textnormal{i}}ght)}},\mathbf{m}athbf{f}rac{\partial f\left(\mathbf{z};\theta{\textnormal{i}}ght)}{\partial\mathbf{V}_{:,i:,}^{\left(l{\textnormal{i}}ght)}}{\textnormal{i}}ght{\textnormal{a}}ngle {\textnormal{i}}ght]=\alpha^{2}\mathbf{m}athbf{f}rac{c_{v}}{qC_{l}}\mathbf{s}um_{p=1}^{ d }\Pi_{p,p}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)\mathbf{m}athbb{E}\left[\left({\bm{p}}^{T}\left(\mathbf{s}igma\left(g_{i}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght){\textnormal{i}}ght){\bm{p}}\left(\mathbf{s}igma\left(g_{i}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght){\textnormal{i}}ght){\textnormal{i}}ght)_{pp}{\textnormal{i}}ght]
\]
\[
=\alpha^{2}\mathbf{m}athbf{f}rac{c_{v}c_{w}}{qC_{l}}\mathbf{s}um_{p=1}^{ d }\Pi_{p,p}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)\mathbf{s}um_{k=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\mathbf{m}athbb{E}\left[\left(\mathbf{s}igma\left(g_{i,p+k}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght){\textnormal{i}}ght)\mathbf{s}igma\left(g_{i,p+k}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght){\textnormal{i}}ght]
\]
\[
=\mathbf{m}athbf{f}rac{\alpha^2}{qC_{l}}\mathbf{s}um_{p=1}^{d}\Pi_{p,p}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)\text{tr}\left(K_{\mathbf{m}athcal{D}_{p,p}}^{\left(l{\textnormal{i}}ght)}{\textnormal{i}}ght).
\]
Note that as this does not depend on $i$. We therefore obtain
\[
\mathbf{m}athbb{E}\left[\left\langle \mathbf{m}athbf{f}rac{\partial f\left(\mathbf{x};\theta{\textnormal{i}}ght)}{\partial\mathbf{V}^{\left(l{\textnormal{i}}ght)}},\mathbf{m}athbf{f}rac{\partial f\left(\mathbf{z};\theta{\textnormal{i}}ght)}{\partial\mathbf{V}^{\left(l{\textnormal{i}}ght)}}{\textnormal{i}}ght{\textnormal{a}}ngle {\textnormal{i}}ght]
=\mathbf{s}um_{i=1}^{C_l}\mathbf{m}athbb{E}\left[\left\langle \mathbf{m}athbf{f}rac{\partial f\left(\mathbf{x};\theta{\textnormal{i}}ght)}{\partial\mathbf{V}_{:,i,:}^{\left(l{\textnormal{i}}ght)}},\mathbf{m}athbf{f}rac{\partial f\left(\mathbf{z};\theta{\textnormal{i}}ght)}{\partial\mathbf{V}_{:,i:,}^{\left(l{\textnormal{i}}ght)}}{\textnormal{i}}ght{\textnormal{a}}ngle {\textnormal{i}}ght]
=\mathbf{m}athbf{f}rac{\alpha^2}{q}\mathbf{s}um_{p=1}^{ d }\Pi_{p,p}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)\text{tr}\left(K_{\mathbf{m}athcal{D}_{p,p}}^{\left(l{\textnormal{i}}ght)}{\textnormal{i}}ght).
\]
The next term we deal with is $\mathbf{m}athbb{E}\left[\left\langle \mathbf{m}athbf{f}rac{\partial f\left(\mathbf{x};\theta{\textnormal{i}}ght)}{\partial\mathbf{W}^{\left(l{\textnormal{i}}ght)}},\mathbf{m}athbf{f}rac{\partial f\left(\mathbf{z};\theta{\textnormal{i}}ght)}{\partial\mathbf{W}^{\left(l{\textnormal{i}}ght)}}{\textnormal{i}}ght{\textnormal{a}}ngle {\textnormal{i}}ght]$.
Using Lemma {\textnormal{e}}f{lem:13} we have
\[
\mathbf{m}athbb{E}\left[\left\langle \mathbf{m}athbf{f}rac{\partial f\left(\mathbf{x};\theta{\textnormal{i}}ght)}{\partial\mathbf{W}^{\left(l{\textnormal{i}}ght)}},\mathbf{m}athbf{f}rac{\partial f\left(\mathbf{z};\theta{\textnormal{i}}ght)}{\partial\mathbf{W}^{\left(l{\textnormal{i}}ght)}}{\textnormal{i}}ght{\textnormal{a}}ngle {\textnormal{i}}ght]=\mathbf{m}athbf{f}rac{\alpha^{2}}{q}\mathbf{s}um_{p=1}^{ d }\Pi_{p,p}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)\text{tr}\left(\dot{K}_{\mathbf{m}athcal{D}_{p,p}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)\odot\Sigma_{\mathbf{m}athcal{D}_{p,p}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght).
\]
Putting these together we obtain
\begin{align*}
\Theta^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght) &= \mathbf{m}athbb{E}\left[\left\langle \mathbf{m}athbf{f}rac{\partial f\left(\mathbf{x};\theta{\textnormal{i}}ght)}{\partial\theta},\mathbf{m}athbf{f}rac{\partial f\left(\mathbf{z};\theta{\textnormal{i}}ght)}{\partial\theta}{\textnormal{i}}ght{\textnormal{a}}ngle {\textnormal{i}}ght] \\
& = \mathbf{m}athbb{E}\left[\left\langle \mathbf{m}athbf{f}rac{\partial f\left(\mathbf{x};\theta{\textnormal{i}}ght)}{\partial\mathbf{W}^{\left(\mathbf{m}athbb{E}q{\textnormal{i}}ght)}},\mathbf{m}athbf{f}rac{\partial f\left(\mathbf{z};\theta{\textnormal{i}}ght)}{\partial\mathbf{W}^{\left(\mathbf{m}athbb{E}q{\textnormal{i}}ght)}}{\textnormal{i}}ght{\textnormal{a}}ngle {\textnormal{i}}ght]
+\mathbf{s}um_{l=1}^{L}\mathbf{m}athbb{E}\left[\left\langle \mathbf{m}athbf{f}rac{\partial f\left(\mathbf{x};\theta{\textnormal{i}}ght)}{\partial\mathbf{W}^{\left(l{\textnormal{i}}ght)}},\mathbf{m}athbf{f}rac{\partial f\left(\mathbf{z};\theta{\textnormal{i}}ght)}{\partial\mathbf{W}^{\left(l{\textnormal{i}}ght)}}{\textnormal{i}}ght{\textnormal{a}}ngle {\textnormal{i}}ght]+\mathbf{m}athbb{E}\left[\left\langle \mathbf{m}athbf{f}rac{\partial f\left(\mathbf{x};\theta{\textnormal{i}}ght)}{\partial\mathbf{V}^{\left(l{\textnormal{i}}ght)}},\mathbf{m}athbf{f}rac{\partial f\left(\mathbf{z};\theta{\textnormal{i}}ght)}{\partial\mathbf{V}^{\left(l{\textnormal{i}}ght)}}{\textnormal{i}}ght{\textnormal{a}}ngle {\textnormal{i}}ght] \\
& = \mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)
+ \mathbf{m}athbf{f}rac{\alpha^{2}}{q}\mathbf{s}um_{l=1}^{L}\mathbf{s}um_{p=1}^{ d }\Pi_{p,p}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)
\left(\text{tr}\left(\dot{K}_{\mathbf{m}athcal{D}_{p,p}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)\odot\Sigma_{\mathbf{m}athcal{D}_{p,p}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght) + K_{\mathbf{m}athcal{D}_{p,p}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght) {\textnormal{i}}ght) {\textnormal{i}}ght).
\mathbf{m}athbf{e}nd{align*}
Finally, we provide a formula for of $\Pi$ to Lemma {\textnormal{e}}f{lem:9}, and denoting $P_j=\Pi_{j,j}$ completes the proof.
\mathbf{m}athbf{e}nd{proof}
\begin{lemma}
\label{lem:3}
Let $\psi$ a real valued function and $\bm{X},\bm{A}$ matrices. If $\partial\psi\left(h\left(\bm{X}^{T}\bm{A}{\textnormal{i}}ght){\textnormal{i}}ght)$ is well
defined then
$\mathbf{m}athbf{f}rac{\partial\psi\left(h\left(\bm{X}^{T}\bm{A}{\textnormal{i}}ght){\textnormal{i}}ght)}{\partial\bm{X}}=\left(\mathbf{m}athbf{f}rac{\partial\psi}{\partial\left(\bm{X}^{T}\bm{A}{\textnormal{i}}ght)}{\textnormal{i}}ght)^{T}\bm{A}^{T}=\mathbf{s}um_{s,t}\mathbf{m}athbf{f}rac{\partial\psi}{\partial h\left(\bm{X}^{T}\bm{A}{\textnormal{i}}ght)_{s,t}}\mathbf{m}athbf{f}rac{\partial h\left(\bm{X}^{T}\bm{A}{\textnormal{i}}ght)_{s,t}}{\left(\bm{X}^{T}\bm{A}{\textnormal{i}}ght)}\bm{A}^{T}$.
\mathbf{m}athbf{e}nd{lemma}
\begin{proof}
First, by the linearity of derivatives we get that
\[
\mathbf{m}athbf{f}rac{\partial\left(\bm{X}^{T}\bm{A}{\textnormal{i}}ght)_{ij}}{\partial\bm{X}_{nm}}=\mathbf{s}um_{k}\mathbf{m}athbf{f}rac{\partial\bm{X}_{ki}\bm{A}_{kj}}{\partial\bm{X}_{nm}}=\delta_{im}\bm{A}_{nj}.
\]
Using the chain rule we get
\[
\left[\mathbf{m}athbf{f}rac{\partial\psi}{\partial\bm{X}}{\textnormal{i}}ght]_{mn}=\mathbf{s}um_{i,j}\mathbf{m}athbf{f}rac{\partial\psi}{\partial\left(\bm{X}^{T}\bm{A}{\textnormal{i}}ght)_{ij}}\mathbf{m}athbf{f}rac{\partial\left(\bm{X}^{T}\bm{A}{\textnormal{i}}ght)_{ij}}{\partial\bm{X}_{nm}}=\mathbf{s}um_{j}\mathbf{m}athbf{f}rac{\partial\psi}{\partial\left(\bm{X}^{T}\bm{A}{\textnormal{i}}ght)_{mj}}\bm{A}_{nj}
\]
\[
=\mathbf{s}um_{j}\bm{A}_{nj}\left[\mathbf{m}athbf{f}rac{\partial\psi}{\partial\left(\bm{X}^{T}\bm{A}{\textnormal{i}}ght)}{\textnormal{i}}ght]_{jm}=\left[\left(\bm{A}\mathbf{m}athbf{f}rac{\partial\psi}{\partial\left(\bm{X}^{T}\bm{A}{\textnormal{i}}ght)}{\textnormal{i}}ght)^{T}{\textnormal{i}}ght]_{nm}
\]
\[
\Longrightarrow\mathbf{m}athbf{f}rac{\partial\psi}{\partial\bm{X}}=\left(\mathbf{m}athbf{f}rac{\partial\psi}{\partial\left(\bm{X}^{T}\bm{A}{\textnormal{i}}ght)}{\textnormal{i}}ght)^{T}\bm{A}^{T}=\mathbf{s}um_{s,t}\mathbf{m}athbf{f}rac{\partial\psi}{\partial h\left(\bm{X}^{T}\bm{A}{\textnormal{i}}ght)_{s,t}}\mathbf{m}athbf{f}rac{\partial h\left(\bm{X}^{T}\bm{A}{\textnormal{i}}ght)_{s,t}}{\left(\bm{X}^{T}\bm{A}{\textnormal{i}}ght)}\bm{A}^{T}.
\]
\mathbf{m}athbf{e}nd{proof}
\begin{lemma}
\label{lem:4}$\mathbf{m}athbf{f}orall2\leq l\leq L$,
\[
b^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)=\mathbf{s}um_{m=1}^{C_{l-1}}\mathbf{s}um_{n=1}^{d}b_{mn}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)\left(\mathbf{m}athbf{f}rac{\partial f_{mn}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)}{\partial f^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)}{\textnormal{i}}ght)^{T}.
\]
\mathbf{m}athbf{e}nd{lemma}
\begin{proof}
By the definition of $b^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)$ we have
\[
b^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)=\left(\mathbf{m}athbf{f}rac{\partial f\left(\mathbf{x};\theta{\textnormal{i}}ght)}{\partial f^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)}{\textnormal{i}}ght)^{T}=\left(\mathbf{s}um_{m=1}^{C_{l-1}}\mathbf{s}um_{n=1}^{ d }\mathbf{m}athbf{f}rac{\partial f\left(\mathbf{x};\theta{\textnormal{i}}ght)}{\partial f_{mn}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)}\mathbf{m}athbf{f}rac{\partial f_{mn}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)}{\partial f^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)}{\textnormal{i}}ght)^{T}
\]
\[
=\mathbf{s}um_{m=1}^{C_{l-1}}\mathbf{s}um_{n=1}^{ d }b_{mn}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)\left(\mathbf{m}athbf{f}rac{\partial f_{mn}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)}{\partial f^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)}{\textnormal{i}}ght)^{T}.
\]
\mathbf{m}athbf{e}nd{proof}
\begin{lemma}
\label{lem:5}
\[
\mathbf{m}athbb{E}\left[\left({b^{\left(L{\textnormal{i}}ght)}}^{T}\left(\mathbf{x}{\textnormal{i}}ght)b^{\left(L{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght)_{j,j^{'}}{\textnormal{i}}ght]=\bm{1}_{j=j^{'}=1}.
\]
\mathbf{m}athbf{e}nd{lemma}
\begin{proof}
\[
b_{:,j}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)=\mathbf{m}athbf{f}rac{1}{\mathbf{s}qrt{C_{L}}}\left(\mathbf{m}athbf{f}rac{\partial}{\partial f_{:,j}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)}W^{\mathbf{m}athbb{E}q}f_{:,1}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght) {\textnormal{i}}ght)^{T}=\delta_{j,1}\mathbf{m}athbf{f}rac{1}{\mathbf{s}qrt{C_{L}}}W^{\mathbf{m}athbb{E}q}.
\]
Therefore,
\[
\mathbf{m}athbb{E}\left[\left({b^{\left(L{\textnormal{i}}ght)}}^{T}\left(\mathbf{x}{\textnormal{i}}ght)b^{\left(L{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght)_{j,j^{'}}{\textnormal{i}}ght]=\mathbf{s}um_{k=1}^{C_{L}}\mathbf{m}athbb{E}\left[b_{kj}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)b_{kj^{'}}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]=\mathbf{m}athbf{f}rac{\delta_{j,1}\delta_{j^{'},1}}{C_{L} d }\mathbf{s}um_{k=1}^{C_{L}}\mathbf{m}athbb{E}\left[W_{k}^{\mathbf{m}athbb{E}q}W_{k^{'}}^{\mathbf{m}athbb{E}q}{\textnormal{i}}ght]=\delta_{j,1}\delta_{j^{'},1}.
\]
\mathbf{m}athbf{e}nd{proof}
\begin{lemma}
\label{lem:6}$\mathbf{m}athbf{f}orall2\leq l\leq L$,
\[
\mathbf{m}athbf{f}rac{\partial f_{mn}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)}{\partial f^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)}=\bm{J}^{nm}+\mathbf{m}athbf{f}rac{\alpha}{q}\mathbf{s}qrt{\mathbf{m}athbf{f}rac{c_{w}c_{v}}{C_{l}C_{l-1}}}\mathbf{s}um_{j=1}^{C_{l}}\mathbf{s}um_{k=1}^{q}\mathbf{V}_{k,j,m}^{\left(l{\textnormal{i}}ght)}\dot{\mathbf{s}igma}\left(g_{j,n+k-\mathbf{m}athbf{f}rac{q+1}{2}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght)\bm{J}^{\mathbf{m}athcal{D}_{n+k-\mathbf{m}athbf{f}rac{q+1}{2}}}\mathbf{W}_{:,:,j}^{\left(l{\textnormal{i}}ght)}.
\]
\mathbf{m}athbf{e}nd{lemma}
\begin{proof}
by the definition of $f^{\left(l{\textnormal{i}}ght)}$ we have
\[
f^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)=f^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)+\mathbf{m}athbf{f}rac{\alpha}{q}\mathbf{s}qrt{\mathbf{m}athbf{f}rac{c_{w}c_{v}}{C_{l}C_{l-1}}}\mathbf{s}um_{j=1}^{C_{l}}\mathbf{V}_{:,j,:}^{\left(l{\textnormal{i}}ght)^{T}}{\bm{p}}\left(\mathbf{s}igma\left(g_{j}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght){\textnormal{i}}ght).
\]
Taking a derivative of $f_{mn}^{\left(l{\textnormal{i}}ght)}$ w.r.t.\ $f^{\left(l-1{\textnormal{i}}ght)}$ we obtain
\[
\mathbf{m}athbf{f}rac{\partial f_{mn}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)}{\partial f^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)}=\bm{J}^{nm}+\mathbf{m}athbf{f}rac{\alpha}{q}\mathbf{s}qrt{\mathbf{m}athbf{f}rac{c_{w}c_{v}}{C_{l}C_{l-1}}}\mathbf{s}um_{j=1}^{C_{l}}\mathbf{m}athbf{f}rac{\partial}{\partial f^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)}\underset{\text{denote by }y_{j}}{\underbrace{\left(\left(\mathbf{V}_{:,j,:}^{\left(l{\textnormal{i}}ght)}{\textnormal{i}}ght)^{T}{\bm{p}}\left(\mathbf{s}igma\left(g_{j}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght){\textnormal{i}}ght){\textnormal{i}}ght)_{mn}}}.
\]
To simplify this, notice first that the derivative
can be expressed as
\[
\mathbf{m}athbf{f}rac{\partial}{\partial f^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)}y_{j}=\mathbf{s}um_{k=1}^{q}\mathbf{V}_{k,j,m}^{\left(l{\textnormal{i}}ght)}\mathbf{m}athbf{f}rac{\partial}{\partial f^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)}{\bm{p}}_{kn}\left(\mathbf{s}igma\left(g_{j}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght){\textnormal{i}}ght)=\mathbf{s}um_{k=1}^{q}\mathbf{V}_{k,j,m}^{\left(l{\textnormal{i}}ght)}\mathbf{m}athbf{f}rac{\partial}{\partial f^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)}\mathbf{s}igma\left(g_{j,n+k-\mathbf{m}athbf{f}rac{q+1}{2}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght).
\]
Using the chain rule we can express the derivative of $\mathbf{s}igma\left(g_{j,n+k-\mathbf{m}athbf{f}rac{q+1}{2}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght)$
as follows:
\[
\left[\mathbf{m}athbf{f}rac{\mathbf{s}igma\left(g_{j,n+k-\mathbf{m}athbf{f}rac{q+1}{2}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght)}{\partial f^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)}{\textnormal{i}}ght]_{m^{'}n^{'}}=\dot{\mathbf{s}igma}\left(g_{j,n+k-\mathbf{m}athbf{f}rac{q+1}{2}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght)\cdot\mathbf{m}athbf{f}rac{\partial\mathbf{s}um_{j^{'}=1}^{C_{l-1}}\left\langle\mathbf{W}_{:,j^{'},j}^{\left(l{\textnormal{i}}ght)},f_{j^{'},\mathbf{m}athcal{D}_{n+k-\mathbf{m}athbf{f}rac{q+1}{2}}}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght{\textnormal{a}}ngle}{\partial f_{n^{'}m^{'}}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)}
\]
\[
=\dot{\mathbf{s}igma}\left(g_{j,n+k-\mathbf{m}athbf{f}rac{q+1}{2}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght)\mathbf{s}um_{j^{'}=1}^{C_{l-1}}\delta_{n^{'}j^{'}}\bm{1}_{m^{'}\in\mathbf{m}athcal{D}_{n+k-\mathbf{m}athbf{f}rac{q+1}{2}}}\mathbf{W}_{m^{'}-n-k+q+1,j^{'},j}^{\left(l{\textnormal{i}}ght)}
\]
\[
=\dot{\mathbf{s}igma}\left(g_{j,n+k-\mathbf{m}athbf{f}rac{q+1}{2}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght)\bm{1}_{m^{'}\in\mathbf{m}athcal{D}_{n+k-\mathbf{m}athbf{f}rac{q+1}{2}}}\mathbf{W}_{m^{'}-n-k+q+1,j^{'},j}^{\left(l{\textnormal{i}}ght)}.
\]
\[
=\mathbf{m}athbf{f}rac{\partial g_{j,n+k-\mathbf{m}athbf{f}rac{q+1}{2}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)}{\partial f^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)}=\dot{\mathbf{s}igma}\left(g_{j,n+k-\mathbf{m}athbf{f}rac{q+1}{2}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght)\bm{J}^{\mathbf{m}athcal{D}_{n+k-\mathbf{m}athbf{f}rac{q+1}{2}}}\mathbf{W}_{:,:,j}^{\left(l{\textnormal{i}}ght)}
\]
In summary we obtain
\[
\mathbf{m}athbf{f}rac{\partial f_{mn}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)}{\partial f^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)}=\bm{J}^{nm}+\mathbf{m}athbf{f}rac{\alpha}{q}\mathbf{s}qrt{\mathbf{m}athbf{f}rac{c_{w}c_{v}}{C_{l}C_{l-1}}}\mathbf{s}um_{j=1}^{C_{l}}\mathbf{s}um_{k=1}^{q}\mathbf{V}_{k,j,m}^{\left(l{\textnormal{i}}ght)}\dot{\mathbf{s}igma}\left(g_{j,n+k-\mathbf{m}athbf{f}rac{q+1}{2}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght)\bm{J}^{\mathbf{m}athcal{D}_{n+k-\mathbf{m}athbf{f}rac{q+1}{2}}}\mathbf{W}_{:,:,j}^{\left(l{\textnormal{i}}ght)}.
\]
\mathbf{m}athbf{e}nd{proof}
\begin{lemma}
\label{lem:7}
For any two matrices $\bm{M},\bm{M}^{'}\in\mathbf{m}athbb{R}^{m\times s}$ and $\bm{N},\bm{N}^{'}\in\mathbf{m}athbb{R}^{s\times n}$, if $\bm{M}^{T}\bm{M}^{'}$ is uncorrelated with $\bm{N}\bm{N}^{{'}^{T}}$, and for every
$k\mathbf{n}eq k^{'}$ either $\mathbf{m}athbb{E}\left[\left(\bm{M}^{T}\bm{M}^{'}{\textnormal{i}}ght)_{kk^{'}}{\textnormal{i}}ght]=0$
or $\mathbf{m}athbb{E}\left[\left(\bm{N}\bm{N}^{{'}^{T}}{\textnormal{i}}ght)_{k^{'}k}{\textnormal{i}}ght]=0$, then
\[
\mathbf{m}athbb{E}\left[\left\langle \bm{M}\bm{N},\bm{M}^{'}\bm{N}^{'}{\textnormal{i}}ght{\textnormal{a}}ngle {\textnormal{i}}ght]=\mathbf{s}um_{p=1}^{s}\mathbf{m}athbb{E}\left[\left(\bm{M}^{T}\bm{M}^{'}{\textnormal{i}}ght)_{p,p}{\textnormal{i}}ght]\mathbf{m}athbb{E}\left[\left(\bm{N}^{T}\bm{N}^{'}{\textnormal{i}}ght)_{p,p}{\textnormal{i}}ght]=\mathbf{m}athbb{E}\left[\text{tr}\left(\bm{M}^{T}\bm{M}^{'}\odot \bm{N}^{T}\bm{N}^{'}{\textnormal{i}}ght){\textnormal{i}}ght].
\]
\mathbf{m}athbf{e}nd{lemma}
\begin{proof}
Following the definition of an inner product we have
\[
\mathbf{m}athbb{E}\left[\left\langle \bm{M}\bm{N},\bm{M}^{'}\bm{N}^{'}{\textnormal{i}}ght{\textnormal{a}}ngle {\textnormal{i}}ght]=\mathbf{m}athbb{E}\left[\mathbf{s}um_{i=1}^{m}\mathbf{s}um_{j=1}^{n}\left(\bm{M}\bm{N}{\textnormal{i}}ght)_{ij}\left(\bm{M}^{'}\bm{N}^{'}{\textnormal{i}}ght)_{ij}{\textnormal{i}}ght]=\mathbf{m}athbb{E}\left[\mathbf{s}um_{i=1}^{m}\mathbf{s}um_{j=1}^{n}\left(\mathbf{s}um_{p=1}^{s}\bm{M}_{i,p}\bm{N}_{p,j}{\textnormal{i}}ght)\left(\mathbf{s}um_{p^{'}=1}^{s}\bm{M}_{i,p^{'}}^{'}\bm{N}_{p^{'}j}^{'}{\textnormal{i}}ght){\textnormal{i}}ght]
\]
\[
=\mathbf{m}athbb{E}\left[\mathbf{s}um_{i=1}^{m}\mathbf{s}um_{j=1}^{n}\mathbf{s}um_{p=1}^{s}\mathbf{s}um_{p^{'}=1}^{s}\bm{M}_{i,p}\bm{M}_{ip^{'}}^{'}\bm{N}_{p,j}\bm{N}_{p^{'}j}^{'}{\textnormal{i}}ght]\underset{\text{uncorrelated}}{=}\mathbf{s}um_{p=1}^{s}
\mathbf{s}um_{p^{'}=1}^{s}\mathbf{m}athbb{E}\left[\left(\bm{M}^{T}\bm{M}^{'}{\textnormal{i}}ght)_{p,p^{'}}{\textnormal{i}}ght]\mathbf{m}athbb{E}\left[\left(\bm{N}\bm{N}^{{'}^{T}}{\textnormal{i}}ght)_{p^{'}p}{\textnormal{i}}ght]
\]
\[
\underset{0\text{ when }p\mathbf{n}eq p^{'}}{=}\mathbf{s}um_{p=1}^{s}\mathbf{m}athbb{E}\left[\left(\bm{M}^{T}\bm{M}^{'}{\textnormal{i}}ght)_{p,p}{\textnormal{i}}ght]\mathbf{m}athbb{E}\left[\left(\bm{N}\bm{N}^{{'}^{T}}{\textnormal{i}}ght)_{p,p}{\textnormal{i}}ght]=\mathbf{m}athbb{E}\left[\text{tr}\left(\bm{M}^{T}\bm{M}^{'}\odot \bm{N}^{T}\bm{N}^{'}{\textnormal{i}}ght){\textnormal{i}}ght].
\]
\mathbf{m}athbf{e}nd{proof}
\begin{lemma}
\label{lem:8}
$\mathbf{m}athbf{f}orall1\leq l\leq L-1,1\leq k,k^{'}\leq C_{l},1\leq j,j^{'}\leq d$,
it holds that
\[
\mathbf{m}athbb{E}\left[b_{kj}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)b_{k^{'}j^{'}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]=
\delta_{kk^{'}}\delta_{jj^{'}}\left(\mathbf{m}athbb{E}\left[b_{kj}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)b_{kj}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght] + \mathbf{m}athbf{f}rac{\alpha^2c_w}{q^2C_{l}}\text{tr}\left(
\left(\mathbf{s}um_{p=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\Pi_{\mathbf{m}athcal{D}_{j+p,j+p}}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght)\odot\dot{K}_{\mathbf{m}athcal{D}_{j,j}}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght){\textnormal{i}}ght).
\]
\mathbf{m}athbf{e}nd{lemma}
\begin{proof}
By Lemma {\textnormal{e}}f{lem:4} we have
\[
\mathbf{m}athbb{E}\left[b_{kj}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)b_{k^{'}j^{'}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]=\mathbf{m}athbb{E}\left[\mathbf{s}um_{m=1}^{C_{l}}\mathbf{s}um_{n=1}^{ d }b_{mn}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)\mathbf{m}athbf{f}rac{\partial f_{mn}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)}{\partial f_{kj}^{\left(l{\textnormal{i}}ght)}},\mathbf{s}um_{m^{'}=1}^{C_{l}}\mathbf{s}um_{n^{'}=1}^{ d }b_{m^{'}n^{'}}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght)\mathbf{m}athbf{f}rac{\partial f_{m^{'}n^{'}}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght)}{\partial f_{k^{'}j^{'}}^{\left(l{\textnormal{i}}ght)}}{\textnormal{i}}ght]
\]
\[
=\mathbf{s}um_{m=1}^{C_{l}}\mathbf{s}um_{n=1}^{ d }\mathbf{s}um_{m^{'}=1}^{C_{l}}\mathbf{s}um_{n^{'}=1}^{ d }\mathbf{m}athbb{E}\left[b_{mn}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)b_{m^{'}n^{'}}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght)\mathbf{m}athbf{f}rac{\partial f_{mn}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)}{\partial f_{kj}^{\left(l{\textnormal{i}}ght)}}\mathbf{m}athbf{f}rac{\partial f_{m^{'}n^{'}}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght)}{\partial f_{k^{'}j^{'}}^{\left(l{\textnormal{i}}ght)}}{\textnormal{i}}ght].
\]
Now as $b^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)$ is uncorrelated with
$\mathbf{m}athbf{f}rac{\partial f_{mn}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)}{\partial f_{kj}^{\left(l{\textnormal{i}}ght)}}$
we get
\[
=\mathbf{s}um_{m=1}^{C_{l}}\mathbf{s}um_{n=1}^{ d }\mathbf{s}um_{m^{'}=1}^{C_{l}}\mathbf{s}um_{n^{'}=1}^{ d }\mathbf{m}athbb{E}\left[b_{mn}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)b_{m^{'}n^{'}}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]\mathbf{m}athbb{E}\left[\mathbf{m}athbf{f}rac{\partial f_{mn}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)}{\partial f_{kj}^{\left(l{\textnormal{i}}ght)}}\mathbf{m}athbf{f}rac{\partial f_{m^{'}n^{'}}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght)}{\partial f_{k^{'}j^{'}}^{\left(l{\textnormal{i}}ght)}}{\textnormal{i}}ght].
\]
By induction (where the base case is Lemma {\textnormal{e}}f{lem:5}), we can assume that if $m\mathbf{n}eq m'$ or $n\mathbf{n}eq n'$
then $\mathbf{m}athbb{E}\left[b_{mn}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)b_{m'n'}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght]=0$.
We therefore obtain
\[
\mathbf{m}athbb{E}\left[b_{kj}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)b_{k^{'}j^{'}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght]=\mathbf{s}um_{m=1}^{C_{l}}\mathbf{s}um_{n=1}^{ d }\mathbf{m}athbb{E}\left[b_{mn}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)b_{mn}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]\mathbf{m}athbb{E}\left[\mathbf{m}athbf{f}rac{\partial f_{mn}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)}{\partial f_{kj}^{\left(l{\textnormal{i}}ght)}}\mathbf{m}athbf{f}rac{\partial f_{mn}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght)}{\partial f_{k^{'}j^{'}}^{\left(l{\textnormal{i}}ght)}}{\textnormal{i}}ght].
\]
It remains to calculate $\mathbf{m}athbb{E}\left[\mathbf{m}athbf{f}rac{\partial f_{mn}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)}{\partial f_{kj}^{\left(l{\textnormal{i}}ght)}}\mathbf{m}athbf{f}rac{\partial f_{mn}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght)}{\partial f_{k^{'}j^{'}}^{\left(l{\textnormal{i}}ght)}}{\textnormal{i}}ght]$.
Lemma {\textnormal{e}}f{lem:6} states that
\begin{align*}
\mathbf{m}athbf{f}rac{\partial f_{mn}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)}{\partial f_{kj}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)}
& =\left[\mathbf{m}athbf{f}rac{\partial f_{mn}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)}{\partial f^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)}{\textnormal{i}}ght]_{jk}\\
& = \underset{\text{Denote by }A_{kj}^{mn}}{\underbrace{\delta_{km}\delta_{jn}}}
+\underset{\text{Denote by }C}{\underbrace{\mathbf{m}athbf{f}rac{\alpha}{q}\mathbf{s}qrt{\mathbf{m}athbf{f}rac{c_{w}c_{v}}{C_{l}C_{l+1}}}}}\underset{B_{kj}^{mn}\left(\mathbf{x}{\textnormal{i}}ght)}{\underbrace{\mathbf{s}um_{s=1}^{C_{l+1}}
\mathbf{s}um_{p=1}^{q}\mathbf{V}_{p,s,m}^{\left(l+1{\textnormal{i}}ght)}\dot{\mathbf{s}igma}\left(g_{s,p+n-\mathbf{m}athbf{f}rac{q+1}{2}}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght)\bm{1}_{j\in\mathbf{m}athcal{D}_{n+p-\mathbf{m}athbf{f}rac{q-1}{2}}}\mathbf{W}_{j-n-p+q+1,k,s}^{\left(l+1{\textnormal{i}}ght)}}}.
\mathbf{m}athbf{e}nd{align*}
Using this notation we have:
\[
\mathbf{m}athbb{E}\left[\mathbf{m}athbf{f}rac{\partial f_{mn}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)}{\partial f_{kj}^{\left(l{\textnormal{i}}ght)}}\mathbf{m}athbf{f}rac{\partial f_{mn}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght)}{\partial f_{k^{'}j^{'}}^{\left(l{\textnormal{i}}ght)}}{\textnormal{i}}ght]=A_{kj}^{mn}A_{k^{'}j^{'}}^{mn}+C\left(A_{kj}^{mn}\mathbf{m}athbb{E}\left[B_{k^{'}j^{'}}^{mn}{\textnormal{i}}ght]+A_{k^{'}j^{'}}^{mn}\mathbf{m}athbb{E}\left[B_{kj}^{mn}{\textnormal{i}}ght]{\textnormal{i}}ght)+C^{2}\mathbf{m}athbb{E}\left[B_{kj}^{mn}\left(\mathbf{x}{\textnormal{i}}ght)B_{k^{'}j^{'}}^{mn}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght].
\]
We will deal with each term separately. First, we consider the first term:
\[
A_{kj}^{mn}A_{k^{'}j^{'}}^{mn}=\delta_{km}\delta_{jn}\delta_{k^{'}m}\delta_{j^{'}n}=\delta_{kk^{'}m}\delta_{jj^{'}n},
\]
where we use the notation $\delta_{kk'm}=\begin{cases}
1 & k=k'=m\\
0 & \mathbf{m}athrm{otherwise}
\mathbf{m}athbf{e}nd{cases}$ and likewise for $\delta_{jj'n}$.
For the second term, since $\mathbf{V},\mathbf{W}$ are $0$ meaned i.i.d Gaussians (Equation ({\textnormal{e}}f{eq:exp_par})),
and $\dot{\mathbf{s}igma}$ is the indicator function we have $\mathbf{m}athbb{E}\left[B_{kj}^{mn}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght]=\mathbf{m}athbb{E}\left[B_{k'j'}^{mn}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]=0$, Therefore,
\[
C\left(A_{kj}^{mn}\mathbf{m}athbb{E}\left[B_{k^{'}j^{'}}^{mn}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]+A_{k^{'}j^{'}}^{mn}\mathbf{m}athbb{E}\left[B_{kj}^{mn}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght]{\textnormal{i}}ght)=0.
\]
For the last term, by definition
\[
B_{kj}^{mn}\left(\mathbf{x}{\textnormal{i}}ght)=\mathbf{s}um_{s=1}^{C_{l+1}}\mathbf{s}um_{p=1}^{q}\mathbf{V}_{p,s,m}^{\left(l+1{\textnormal{i}}ght)}\dot{\mathbf{s}igma}\left(g_{s,n+p-\mathbf{m}athbf{f}rac{q+1}{2}}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght)\bm{1}_{j\in\mathbf{m}athcal{D}_{n+p-\mathbf{m}athbf{f}rac{q+1}{2}}}\mathbf{W}_{j-n-p+q+1,k,s}^{\left(l+1{\textnormal{i}}ght)}
\]
and
\[
B_{k^{'}j^{'}}^{mn}\left(\mathbf{z}{\textnormal{i}}ght)=\mathbf{s}um_{s^{'}=1}^{C_{l+1}}\mathbf{s}um_{p^{'}=1}^{q}\mathbf{V}_{p^{'},s^{'},m}^{\left(l+1{\textnormal{i}}ght)}\dot{\mathbf{s}igma}\left(g_{s^{'},n+p^{'}-\mathbf{m}athbf{f}rac{q+1}{2}}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght)\bm{1}_{j^{'}\in\mathbf{m}athcal{D}_{n+p^{'}-\mathbf{m}athbf{f}rac{q+1}{2}}}\mathbf{W}_{j^{'}-n-p^{'}+q+1,k^{'},s^{'}}^{\left(l+1{\textnormal{i}}ght)}.
\]
From \mathbf{m}athbf{e}qref{eq:exp_par}, $\mathbf{m}athbb{E}\left[\mathbf{V}_{p,s,m}^{\left(l+1{\textnormal{i}}ght)}\mathbf{V}_{p',s',m}^{\left(l+1{\textnormal{i}}ght)}{\textnormal{i}}ght]=\delta_{p,p'}\delta_{s,s'}$,
and since they are uncorrelated with $\mathbf{W}^{\left(l+1{\textnormal{i}}ght)}$ and $g^{\left(l+1{\textnormal{i}}ght)}$ then
\[
\mathbf{m}athbb{E}\left[B_{kj}^{mn}\left(\mathbf{x}{\textnormal{i}}ght)B_{k^{'}j^{'}}^{mn}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]=
\]
\\
\[
=C^{2}\mathbf{s}um_{s=1}^{C_{l+1}}\mathbf{s}um_{p=1}^{q}\mathbf{m}athbb{E}\left[\dot{\mathbf{s}igma}\left(g_{s,n+p-\mathbf{m}athbf{f}rac{q+1}{2}}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght)\mathbf{W}_{j-n-p+q+1,k,s}^{\left(l+1{\textnormal{i}}ght)}\dot{\mathbf{s}igma}\left(g_{s,p+n-\mathbf{m}athbf{f}rac{q+1}{2}}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght)\mathbf{W}_{j'-n-p+q+1,k',s}^{\left(l+1{\textnormal{i}}ght)}{\textnormal{i}}ght]\bm{1}_{j,j'\in\mathbf{m}athcal{D}_{n+p-\mathbf{m}athbf{f}rac{q+1}{2}}}
\]
\[
=\delta_{kk'}\delta_{jj'}C^{2}\mathbf{s}um_{s=1}^{C_{l+1}}\mathbf{s}um_{p=1}^{q}\mathbf{m}athbb{E}\left[\dot{\mathbf{s}igma}\left(g_{s,n+p-\mathbf{m}athbf{f}rac{q+1}{2}}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght)\dot{\mathbf{s}igma}\left(g_{s,n+p-\mathbf{m}athbf{f}rac{q+1}{2}}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght){\textnormal{i}}ght]\bm{1}_{j,j'\in\mathbf{m}athcal{D}_{n+p-\mathbf{m}athbf{f}rac{q+1}{2}}}
\]
\[
=\delta_{kk'}\delta_{jj'}\mathbf{m}athbf{f}rac{\alpha^2}{q^2C_{l}}\mathbf{s}um_{p=1}^{q}\dot{K}_{n+p-\mathbf{m}athbf{f}rac{q+1}{2},n+p-\mathbf{m}athbf{f}rac{q+1}{2}}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)\bm{1}_{j,j'\in\mathbf{m}athcal{D}_{n+p-\mathbf{m}athbf{f}rac{q+1}{2}}}
\]
Overall,
\[
\mathbf{m}athbb{E}\left[b_{kj}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)b_{k'j'}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]=
\]
\[
=\mathbf{s}um_{m=1}^{C_{l}}\mathbf{s}um_{n=1}^{d}\mathbf{m}athbb{E}\left[b_{mn}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)b_{mn}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]\left(\delta_{kk'm}\delta_{jj'n}+\delta_{kk'}\delta_{jj'}\mathbf{m}athbf{f}rac{\alpha^2}{q^2C_{l}}\mathbf{s}um_{p=1}^{q}\dot{K}_{n+p-\mathbf{m}athbf{f}rac{q+1}{2},n+p-\mathbf{m}athbf{f}rac{q+1}{2}}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)\bm{1}_{j,j'\in\mathbf{m}athcal{D}_{n+p-\mathbf{m}athbf{f}rac{q+1}{2}}}{\textnormal{i}}ght)
\]
\[
=\delta_{kk'}\delta_{jj'}\left(\mathbf{m}athbb{E}\left[b_{kj}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)b_{kj}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght] + \mathbf{m}athbf{f}rac{\alpha^2}{q^2C_{l}}\mathbf{s}um_{n=1}^{d}\mathbf{s}um_{p=1}^{q}\Pi_{n,n}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)\dot{K}_{n+p-\mathbf{m}athbf{f}rac{q+1}{2},n+p-\mathbf{m}athbf{f}rac{q+1}{2}}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)\bm{1}_{j,j'\in\mathbf{m}athcal{D}_{n+p-\mathbf{m}athbf{f}rac{q+1}{2}}}{\textnormal{i}}ght).
\]
Now observe that
\[
j\in\mathbf{m}athcal{D}_{n+p-\mathbf{m}athbf{f}rac{q+1}{2}}\Longleftrightarrow n+p-q\leq j \leq n+p-1 \Longleftrightarrow j-p+1\leq n \leq j - p + q.
\]
So we can rewrite the above as
\[
\mathbf{m}athbb{E}\left[b_{kj}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)b_{k'j'}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]=
\]
\[
=\delta_{kk'}\delta_{jj'}\left(\mathbf{m}athbb{E}\left[b_{kj}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)b_{kj}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght] + \mathbf{m}athbf{f}rac{\alpha^2c_w}{q^2C_{l}}\mathbf{s}um_{p=1}^{q}\mathbf{s}um_{n=j-p+1}^{j-p+q}\Pi_{n,n}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)\dot{K}_{n+p-\mathbf{m}athbf{f}rac{q+1}{2},n+p-\mathbf{m}athbf{f}rac{q+1}{2}}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght)
\]
\[
=\delta_{kk'}\delta_{jj'}\left(\mathbf{m}athbb{E}\left[b_{kj}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)b_{kj}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght] + \mathbf{m}athbf{f}rac{\alpha^2c_w}{q^2C_{l}}\mathbf{s}um_{p=1}^{q}\text{tr}\left(\Pi_{\mathbf{m}athcal{D}_{j-p+\mathbf{m}athbf{f}rac{q+1}{2},j-p+\mathbf{m}athbf{f}rac{q+1}{2}}}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)\odot\dot{K}_{\mathbf{m}athcal{D}_{j,j}}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght){\textnormal{i}}ght)
\]
\[
=\delta_{kk'}\delta_{jj'}\left(\mathbf{m}athbb{E}\left[b_{kj}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)b_{kj}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght] + \mathbf{m}athbf{f}rac{\alpha^2c_w}{q^2C_{l}}\text{tr}\left(
\left(\mathbf{s}um_{p=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\Pi_{\mathbf{m}athcal{D}_{j+p,j+p}}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght)\odot\dot{K}_{\mathbf{m}athcal{D}_{j,j}}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght){\textnormal{i}}ght).
\]
\mathbf{m}athbf{e}nd{proof}
\begin{lemma}\label{lem:9}
$\mathbf{m}athbf{f}orall1\leq j,j^{'}\leq d$,
\[
\Pi_{j,j^{'}}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)=\mathbf{m}athbf{f}rac{1}{dc_w}\bm{1}_{j=j^{'}=1},
\]
and $\mathbf{m}athbf{f}orall1\leq l\leq L-1,1\leq j,j^{'}\leq d$,
it holds that:
\[
\Pi_{j,j^{'}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)=\delta_{jj^{'}}\left(\Pi_{j,j}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght) + \mathbf{m}athbf{f}rac{\alpha^2}{q^2}\text{tr}\left(
\left(\mathbf{s}um_{p=\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\Pi_{\mathbf{m}athcal{D}_{j+p,j+p}}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght)\odot\dot{K}_{\mathbf{m}athcal{D}_{j,j}}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght){\textnormal{i}}ght).
\]
\mathbf{m}athbf{e}nd{lemma}
\begin{proof}
First, from Lemma {\textnormal{e}}f{lem:5} we know that:
\[
\mathbf{m}athbf{f}rac{1}{c_w}\mathbf{m}athbb{E}\left[\left(b^{\left(L{\textnormal{i}}ght)^{T}}\left(\mathbf{x}{\textnormal{i}}ght)b^{\left(L{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght)_{j,j'}{\textnormal{i}}ght]=\mathbf{m}athbf{f}rac{1}{dc_w}\delta_{j,j'}.
\]
Now let $1\leq l \leq L-1$. Using lemma {\textnormal{e}}f{lem:8} we get that
\[
\mathbf{m}athbf{f}rac{1}{c_{w}}\mathbf{m}athbb{E}\left[\left(b^{\left(l{\textnormal{i}}ght)^{T}}\left(\mathbf{x}{\textnormal{i}}ght)b^{\left(l{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght)_{j,j'}{\textnormal{i}}ght]=\mathbf{m}athbf{f}rac{1}{c_{w}}\mathbf{s}um_{k=1}^{C_{l}}\mathbf{m}athbb{E}\left[b_{kj}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)b_{kj'}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]
\]
\[
=\mathbf{s}um_{k=1}^{C_{l}}\delta_{jj'}\mathbf{m}athbf{f}rac{1}{c_{w}}\left(\mathbf{m}athbb{E}\left[b_{kj}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)b_{kj}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght] + \delta_{kk'}\delta_{jj'}\mathbf{m}athbf{f}rac{\alpha^2c_w}{q^2C_{l}}\text{tr}\left(
\left(\mathbf{s}um_{p=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\Pi_{\mathbf{m}athcal{D}_{j+p,j+p}}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght)\odot\dot{K}_{\mathbf{m}athcal{D}_{j,j}}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght){\textnormal{i}}ght)
\]
\[
=\delta_{jj'}\left(\Pi_{j,j}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght) + \mathbf{m}athbf{f}rac{\alpha^2}{q^2}\text{tr}\left(
\left(\mathbf{s}um_{p=\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\Pi_{\mathbf{m}athcal{D}_{j+p,j+p}}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght)\odot\dot{K}_{\mathbf{m}athcal{D}_{j,j}}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght){\textnormal{i}}ght).
\]
\mathbf{m}athbf{e}nd{proof}
\begin{lemma}
\label{lem:13}
$\mathbf{m}athbf{f}orall1\leq l\leq L$,
\[
\mathbf{m}athbb{E}\left[\left\langle \mathbf{m}athbf{f}rac{\partial f\left(\mathbf{x};\theta{\textnormal{i}}ght)}{\partial\mathbf{W}^{\left(l{\textnormal{i}}ght)}},\mathbf{m}athbf{f}rac{\partial f\left(\mathbf{z};\theta{\textnormal{i}}ght)}{\partial\mathbf{W}^{\left(l{\textnormal{i}}ght)}}{\textnormal{i}}ght{\textnormal{a}}ngle {\textnormal{i}}ght]=\mathbf{m}athbf{f}rac{\alpha^{2}}{q}\mathbf{s}um_{n=1}^{d}\Pi_{nn}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)\text{tr}\left(\dot{K}_{\mathbf{m}athcal{D}_{n,n}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)\odot\Sigma_{\mathbf{m}athcal{D}_{n,n}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght).
\]
\mathbf{m}athbf{e}nd{lemma}
\begin{proof}
We first show the case for $2\leq l\leq L$.
$\mathbf{m}athbf{f}orall2\leq l\leq L,1\leq i\leq C_{l-1},1\leq c\leq C_{l}$, we can express $\mathbf{m}athbf{f}rac{\partial f\left(\mathbf{x};\theta{\textnormal{i}}ght)}{\partial\mathbf{W}_{:,i,c}^{\left(l{\textnormal{i}}ght)}}$ as
\[
\mathbf{m}athbf{f}rac{\partial f_{mn}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)}{\partial\mathbf{W}_{:,i,c}^{\left(l{\textnormal{i}}ght)}}=\mathbf{m}athbf{f}rac{\partial}{\partial\mathbf{W}_{:,i,c}^{\left(l{\textnormal{i}}ght)}}\left[f^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)+\alpha\mathbf{s}qrt{\mathbf{m}athbf{f}rac{c_{v}}{qC_{l}}}\mathbf{s}um_{j=1}^{C_{l}}\left(\mathbf{V}_{:,j,:}^{\left(l{\textnormal{i}}ght)}{\textnormal{i}}ght)^{T}{\bm{p}}\left(\mathbf{s}igma\left(g_{j}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght){\textnormal{i}}ght){\textnormal{i}}ght]_{mn}.
\]
Since $g_{j}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)$ depends on $\mathbf{W}_{:,i,c}^{\left(l{\textnormal{i}}ght)}$
iff $c=j$, we get
\[
\mathbf{m}athbf{f}rac{\partial f_{mn}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)}{\partial\mathbf{W}_{:,i,c}^{\left(l{\textnormal{i}}ght)}}=\alpha\mathbf{s}qrt{\mathbf{m}athbf{f}rac{c_{v}}{qC_{l}}}\mathbf{s}um_{k=1}^{q}\mathbf{V}_{k,c,m}^{\left(l{\textnormal{i}}ght)}\mathbf{m}athbf{f}rac{\partial}{\partial\mathbf{W}_{:,i,c}^{\left(l{\textnormal{i}}ght)}}\mathbf{s}igma\left(g_{c,n+k-\mathbf{m}athbf{f}rac{q+1}{2}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght)
\]
\[
=\mathbf{m}athbf{f}rac{\alpha}{q}\mathbf{s}qrt{\mathbf{m}athbf{f}rac{c_{w}c_{v}}{C_{l}C_{l-1}}}\mathbf{s}um_{k=1}^{q}\mathbf{V}_{k,c,m}^{\left(l{\textnormal{i}}ght)}\dot{\mathbf{s}igma}\left(g_{c,n+k-\mathbf{m}athbf{f}rac{q+1}{2}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght)\mathbf{m}athbf{f}rac{\partial}{\partial\mathbf{W}_{:,i,c}^{\left(l{\textnormal{i}}ght)}}\left[\left(W_{:,i,:}^{\left(l{\textnormal{i}}ght)}{\textnormal{i}}ght)^{T}{\bm{p}}\left(f_{i}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght){\textnormal{i}}ght]_{c,n+k-\mathbf{m}athbf{f}rac{q+1}{2}}
\]
\[
=\mathbf{m}athbf{f}rac{\alpha}{q}\mathbf{s}qrt{\mathbf{m}athbf{f}rac{c_{w}c_{v}}{C_{l}C_{l-1}}}\mathbf{s}um_{k=1}^{q}\mathbf{V}_{k,c,m}^{\left(l{\textnormal{i}}ght)}\dot{\mathbf{s}igma}\left(g_{c,n+k-\mathbf{m}athbf{f}rac{q+1}{2}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght)\left({\bm{p}}^{T}\left(f_{i}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght){\textnormal{i}}ght)_{n+k-\mathbf{m}athbf{f}rac{q+1}{2},:}.
\]
Now we have
\[
\mathbf{m}athbf{f}rac{\partial f\left(\mathbf{x};\theta{\textnormal{i}}ght)}{\partial\mathbf{W}_{:,i,c}^{\left(l{\textnormal{i}}ght)}}=\mathbf{s}um_{m=1}^{C_{l}}\mathbf{s}um_{n=1}^{ d }\mathbf{m}athbf{f}rac{\partial f\left(\mathbf{x};\theta{\textnormal{i}}ght)}{\partial f_{mn}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)}\mathbf{m}athbf{f}rac{\partial f_{mn}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)}{\partial\mathbf{W}_{:,i,c}^{\left(l{\textnormal{i}}ght)}}
\]
\[
=\mathbf{m}athbf{f}rac{\alpha}{q}\mathbf{s}qrt{\mathbf{m}athbf{f}rac{c_{w}c_{v}}{C_{l}C_{l-1}}}\mathbf{s}um_{m=1}^{C_{l}}\mathbf{s}um_{n=1}^{ d }\mathbf{s}um_{k=1}^{q}b_{mn}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)\mathbf{V}_{k,c,m}^{\left(l{\textnormal{i}}ght)}\dot{\mathbf{s}igma}\left(g_{c,n+k-\mathbf{m}athbf{f}rac{q+1}{2}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght)\left({\bm{p}}^{T}\left(f_{i}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght){\textnormal{i}}ght)_{n+k-\mathbf{m}athbf{f}rac{q+1}{2},:}.
\]
Taking the inner product we get the following expression:
\[
\mathbf{m}athbb{E}\left[\mathbf{m}athbf{f}rac{\partial f\left(\mathbf{x};\theta{\textnormal{i}}ght)}{\partial\mathbf{W}_{:,i,c}^{\left(l{\textnormal{i}}ght)}},\mathbf{m}athbf{f}rac{\partial f\left(\mathbf{z};\theta{\textnormal{i}}ght)}{\partial\mathbf{W}_{:,i,c}^{\left(l{\textnormal{i}}ght)}}{\textnormal{i}}ght]=\mathbf{m}athbf{f}rac{\alpha^{2}c_{w}c_{v}}{q^{2}C_{l}C_{l-1}}\mathbf{s}um_{m=1}^{C_{l}}\mathbf{s}um_{m^{'}=1}^{C_{l}}\mathbf{s}um_{n=1}^{ d }\mathbf{s}um_{n^{'}=1}^{ d }\mathbf{s}um_{k=1}^{q}\mathbf{s}um_{k^{'}=1}^{q}\mathbf{m}athbb{E}\left[b_{mn}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)b_{mn^{'}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]\cdot\mathbf{m}athbb{E}\left[\mathbf{V}_{k,c,m}^{\left(l{\textnormal{i}}ght)}\mathbf{V}_{k^{'},c,m^{'}}^{\left(l{\textnormal{i}}ght)}{\textnormal{i}}ght]\cdot
\]
\[
\cdot\mathbf{m}athbb{E}\left[\left\langle \dot{\mathbf{s}igma}\left(g_{c,n+k-\mathbf{m}athbf{f}rac{q+1}{2}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght)\left({\bm{p}}^{T}\left(f_{i}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght){\textnormal{i}}ght)_{n+k-\mathbf{m}athbf{f}rac{q+1}{2},:},\dot{\mathbf{s}igma}\left(g_{c,n^{'}+k^{'}-\mathbf{m}athbf{f}rac{q+1}{2}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght)\left({\bm{p}}^{T}\left(f_{i}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght){\textnormal{i}}ght)_{n^{'}+k^{'}-\mathbf{m}athbf{f}rac{q+1}{2},:}{\textnormal{i}}ght{\textnormal{a}}ngle {\textnormal{i}}ght].
\]
Note however that $\mathbf{m}athbf{e}qref{eq:exp_par}$ implies that $\mathbf{V}_{k,c,m}^{\left(l{\textnormal{i}}ght)}\mathbf{V}_{k',c,m'}^{\left(l{\textnormal{i}}ght)}=\delta_{kk'}\delta_{mm'}$, and Lemma {\textnormal{e}}f{lem:8} implies that $\mathbf{m}athbb{E}\left[b_{mn}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)b_{mn'}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]=0$ when $n\mathbf{n}eq n'$. Therefore,
\begin{align*}
& \mathbf{m}athbb{E}\left[\mathbf{m}athbf{f}rac{\partial f\left(\mathbf{x};\theta{\textnormal{i}}ght)}{\partial\mathbf{W}_{:,i,c}^{\left(l{\textnormal{i}}ght)}},\mathbf{m}athbf{f}rac{\partial f\left(\mathbf{z};\theta{\textnormal{i}}ght)}{\partial\mathbf{W}_{:,i,c}^{\left(l{\textnormal{i}}ght)}}{\textnormal{i}}ght] = \\
&=\mathbf{m}athbf{f}rac{\alpha^{2}c_{w}c_{v}}{q^{2}C_{l}C_{l-1}}\mathbf{s}um_{n=1}^{ d }\mathbf{s}um_{k=1}^{q}\underset{=c_w\Pi_{nn}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)}{\underbrace{\left(\mathbf{s}um_{m=1}^{C_{l}}\mathbf{m}athbb{E}\left[b_{mn}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)b_{mn}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]{\textnormal{i}}ght)}}\cdot\\
& \cdot\mathbf{m}athbb{E}\left[\left\langle \dot{\mathbf{s}igma}\left(g_{c,n+k-\mathbf{m}athbf{f}rac{q+1}{2}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght)\left({\bm{p}}^{T}\left(f_{i}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght){\textnormal{i}}ght)_{n+k-\mathbf{m}athbf{f}rac{q+1}{2},:},\dot{\mathbf{s}igma}\left(g_{c,n+k-\mathbf{m}athbf{f}rac{q+1}{2}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght)\left({\bm{p}}^{T}\left(f_{i}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght){\textnormal{i}}ght)_{n+k-\mathbf{m}athbf{f}rac{q+1}{2},:}{\textnormal{i}}ght{\textnormal{a}}ngle {\textnormal{i}}ght]
\mathbf{m}athbf{e}nd{align*}
\[
=\mathbf{m}athbf{f}rac{\alpha^{2}c_{w}}{q^{2}C_{l}C_{l-1}}\mathbf{s}um_{n=1}^{ d }\Pi_{nn}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)\mathbf{s}um_{k=1}^{q}\left[\dot{K}_{\mathbf{m}athcal{D}_{n,n}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]_{kk}\mathbf{s}um_{k^{'}=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\mathbf{m}athbb{E}\left[f_{i,n+k-\mathbf{m}athbf{f}rac{q+1}{2}+k^{'}}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)f_{i,n+k-\mathbf{m}athbf{f}rac{q+1}{2}+k^{'}}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght].
\]
As a result, by linearity and the fact that this result does not depend on $c$ we obtain
\[
\mathbf{m}athbb{E}\left[\left\langle \mathbf{m}athbf{f}rac{\partial f\left(\mathbf{x};\theta{\textnormal{i}}ght)}{\partial\mathbf{W}^{\left(l{\textnormal{i}}ght)}},\mathbf{m}athbf{f}rac{\partial f\left(\mathbf{z};\theta{\textnormal{i}}ght)}{\partial\mathbf{W}^{\left(l{\textnormal{i}}ght)}}{\textnormal{i}}ght{\textnormal{a}}ngle {\textnormal{i}}ght]=
\]
\[
=\mathbf{m}athbf{f}rac{\alpha^{2}c_{w}}{q^{2}C_{l-1}}\mathbf{s}um_{n=1}^{d }\Pi_{nn}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)\mathbf{s}um_{k=1}^{q}\left[\dot{K}_{\mathbf{m}athcal{D}_{n,n}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]_{kk}\mathbf{s}um_{k'=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\mathbf{s}um_{i=1}^{C_{l-1}}\mathbf{m}athbb{E}\left[f_{i,n+k-\mathbf{m}athbf{f}rac{q+1}{2}+k'}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)f_{i,n+k-\mathbf{m}athbf{f}rac{q+1}{2}+k'}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght].
\]
Using Lemma {\textnormal{e}}f{lem:c1} the last term simplifies as follows:
\[
\mathbf{s}um_{k'=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\mathbf{s}um_{i=1}^{C_{l-1}}\mathbf{m}athbb{E}\left[f_{i,n+k-\mathbf{m}athbf{f}rac{q+1}{2}+k'}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)f_{i,n+k-\mathbf{m}athbf{f}rac{q+1}{2}+k'}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]
\]
\[
=\mathbf{m}athbf{f}rac{qC_{l-1}}{c_{w}}\mathbf{m}athbb{E}\left[g_{*,n+k-\mathbf{m}athbf{f}rac{q+1}{2}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght)g_{*,n+k-\mathbf{m}athbf{f}rac{q+1}{2}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]=\mathbf{m}athbf{f}rac{qC_{l-1}}{c_{w}}\Sigma_{n+k-\mathbf{m}athbf{f}rac{q+1}{2},n+k-\mathbf{m}athbf{f}rac{q+1}{2}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)
\].
Finally we obtain
\[
\mathbf{m}athbb{E}\left[\left\langle \mathbf{m}athbf{f}rac{\partial f\left(\mathbf{x};\theta{\textnormal{i}}ght)}{\partial\mathbf{W}^{\left(l{\textnormal{i}}ght)}},\mathbf{m}athbf{f}rac{\partial f\left(\mathbf{z};\theta{\textnormal{i}}ght)}{\partial\mathbf{W}^{\left(l{\textnormal{i}}ght)}}{\textnormal{i}}ght{\textnormal{a}}ngle {\textnormal{i}}ght]=\mathbf{m}athbf{f}rac{\alpha^{2}}{q}\mathbf{s}um_{n=1}^{ d }\Pi_{nn}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)\mathbf{s}um_{k=1}^{q}\left[\dot{K}_{\mathbf{m}athcal{D}_{n,n}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]_{kk}\left[\Sigma_{\mathbf{m}athcal{D}_{n,n}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]_{kk}
\]
\[
=\mathbf{m}athbf{f}rac{\alpha^{2}}{q}\mathbf{s}um_{n=1}^{ d }\Pi_{nn}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)\text{tr}\left(\dot{K}_{\mathbf{m}athcal{D}_{n,n}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)\odot\Sigma_{\mathbf{m}athcal{D}_{n,n}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght).
\]
The case of $l=1$ is analogous, except that we replace ${\bm{p}}^T\left(f_{i}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{x}{\textnormal{i}}ght){\textnormal{i}}ght)$ with $\mathbf{x}_i^T$ and similarly for $\mathbf{z}$, (making minor adjustments accordingly). We therefore obtain
\[
\mathbf{m}athbb{E}\left[\left\langle \mathbf{m}athbf{f}rac{\partial f\left(\mathbf{x};\theta{\textnormal{i}}ght)}{\partial\mathbf{W}^{\left(1{\textnormal{i}}ght)}},\mathbf{m}athbf{f}rac{\partial f\left(\mathbf{z};\theta{\textnormal{i}}ght)}{\partial\mathbf{W}^{\left(1{\textnormal{i}}ght)}}{\textnormal{i}}ght{\textnormal{a}}ngle {\textnormal{i}}ght]=
\]
\[
=\mathbf{m}athbf{f}rac{\alpha^{2}}{qC_{0}}\mathbf{s}um_{n=1}^{ d }\Pi_{nn}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)\mathbf{s}um_{k=1}^{q}\left[\dot{K}_{\mathbf{m}athcal{D}_{n,n}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght]_{kk}\mathbf{s}um_{i=1}^{C_{0}}\mathbf{m}athbb{E}\left[\mathbf{x}_{i,n+k-\mathbf{m}athbf{f}rac{q+1}{2}}\mathbf{z}_{i,n+k-\mathbf{m}athbf{f}rac{q+1}{2}}{\textnormal{i}}ght]
\]
\[
=\mathbf{m}athbf{f}rac{\alpha^{2}}{q}\mathbf{s}um_{n=1}^{ d }\Pi_{nn}^{\left(1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)\text{tr}\left(\dot{K}_{\mathbf{m}athcal{D}_{n,n}}^{\left(1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)\odot\Sigma_{\mathbf{m}athcal{D}_{n,n}}^{\left(1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght){\textnormal{i}}ght).
\]
\mathbf{m}athbf{e}nd{proof}
\mathbf{s}ection{Spectral Decomposition} \label{ap:3}
\mathbf{s}ubsection{Proof of Theorem {\textnormal{e}}f{thm:sd} in the main text}
\begin{proof}
The strategy is to bound the Taylor expansion of the kernels. We use qualities of the Ordinary Bell Polynomials for the lower bound, and use previous work on singularity analysis \citep{flajolet2009analytic,chen2020deep,bietti2020deep,belfer2021spectral} for the upper bound. The details can be found in the lemmas that follow, resulting in that
$\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}$ can be written as $\mathbf{s}um_{\mathbf{n}\mathbf{m}athbf{g}eq0}b_\mathbf{n}\mathbf{t}^{\mathbf{n}}$ with
\[
c_1\mathbf{n}^{-2.5} \leq b_{\mathbf{n}} \leq c_2 \mathbf{n}^{-\mathbf{m}athbf{f}rac{3}{2d}-1}
\]
and $\Theta^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}$ can similarly be written as $\mathbf{s}um_{\mathbf{n}\mathbf{m}athbf{g}eq0}b_\mathbf{n}\mathbf{t}^{\mathbf{n}}$ with
\[
c_1\mathbf{n}^{-2.5} \leq b_{\mathbf{n}} \leq c_2 \mathbf{n}^{-\mathbf{m}athbf{f}rac{1}{2d}-1}
\]
Thus, applying \citet{geifman2022spectral}[Theorems 3.3,3.4] completes the proof.
\mathbf{m}athbf{e}nd{proof}
\begin{lemma} \label{lem:b1}
$\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}$ can be written as $\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{t}{\textnormal{i}}ght)=\mathbf{s}um_{\mathbf{n}\mathbf{m}athbf{g}eq0}b_\mathbf{n}\mathbf{t}^{\mathbf{n}}$ with
\[
c_1\mathbf{n}^{-\mathbf{n}u} \leq b_{\mathbf{n}},
\]
where $c_1$ is a constant if the receptive field of $\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}$ includes $\mathbf{n}$ and $0$ otherwise that depends on $L$.
\mathbf{m}athbf{e}nd{lemma}
\begin{proof}
Let $\kappa_1(t)=\mathbf{s}um_{n=0}^{\infty}a_nt^n$ be the power series expansion of $\kappa_1(t)$, where $a_n\mathbf{s}im n^{-2.5}$ \citep{chen2020deep}.
We prove this by induction on $L$, starting with $L=1$:
\[
\mathbf{m}athcal{K}^{\left(1{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{t}{\textnormal{i}}ght) =\mathbf{m}athbf{f}rac{1}{C_0}\mathbf{t}_1 + \mathbf{m}athbf{f}rac{\alpha^2}{qc_wC_0} \mathbf{s}um_{k=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\kappa_1\left(\mathbf{t}_{1+k}{\textnormal{i}}ght)
= \mathbf{m}athbf{f}rac{1}{C_0}\mathbf{t}_1 + \mathbf{m}athbf{f}rac{\alpha^2}{qc_wC_0} \mathbf{s}um_{k=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}} \mathbf{s}um_{n=0}^{\infty}a_n\mathbf{t}_{1+k}^n.
\]
By letting $\mathbf{m}athbf{e}_{i}$ be the multi-index with $1$ in the $i$ index and $0$ elsewhere,
it is clear that we can write the above as $\mathbf{s}um_{\mathbf{n}\mathbf{m}athbf{g}eq0}b_{\mathbf{n}}\mathbf{t}^{\mathbf{n}}$ where
\[
b_{\mathbf{n}}= \begin{cases}
\mathbf{m}athbf{f}rac{\alpha^{2}}{c_w C_{0}} a_{0} & \mathbf{n} = 0\\
\mathbf{m}athbf{f}rac{\alpha^{2}}{qc_w C_{0}} a_{n} + \mathbf{m}athbf{f}rac{1}{C_0}\delta_{j,1} & \mathbf{n} = n\mathbf{m}athbf{e}_j \text{ for } -\mathbf{m}athbf{f}rac{q-1}{2}\leq j\leq \mathbf{m}athbf{f}rac{q-1}{2}\text{ and } n\in\mathbf{m}athbb{N}\\
0 & \text{otherwise}.
\mathbf{m}athbf{e}nd{cases}
\]
For $L\mathbf{m}athbf{g}eq2$, by the induction hypothesis $\mathbf{m}athcal{K}^{\left(L-1{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{t}{\textnormal{i}}ght)=\mathbf{s}um_{\mathbf{n}\mathbf{m}athbf{g}eq0}\tilde{b}_\mathbf{n}\mathbf{t}^{\mathbf{n}}\text{ s.t.\ } c_1\mathbf{n}^{-\mathbf{n}u} \leq \tilde{b}_{\mathbf{n}}$. Let $N_L$ be the value of $\Sigma_{j,j}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{x}{\textnormal{i}}ght)$ from Lemma {\textnormal{e}}f{lem:sd1} and $\tilde{N}_L=\mathbf{m}athbf{f}rac{\alpha^2c_vc_w}{2q^2}N_L$.
\[
\kappa_1\left(\mathbf{m}athbf{f}rac{\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{s}_j\mathbf{t}{\textnormal{i}}ght)}{N_{L}}{\textnormal{i}}ght)
=\mathbf{s}um_{n=0}^{\infty}\mathbf{m}athbf{f}rac{a_n}{N_L^n}\left(\mathbf{s}um_{\mathbf{n}\mathbf{m}athbf{g}eq0}\tilde{b}_{\mathbf{n}}\left(\mathbf{s}_j\mathbf{t}{\textnormal{i}}ght)^{\mathbf{n}}{\textnormal{i}}ght)^n
=\mathbf{s}um_{n=0}^{\infty}\mathbf{m}athbf{f}rac{a_n}{N_L^n}\left(\mathbf{s}um_{\mathbf{n}\mathbf{m}athbf{g}eq0}\tilde{b}_{\mathbf{s}_{-j}\mathbf{n}}\left(\mathbf{t}{\textnormal{i}}ght)^{\mathbf{s}_{-j}\mathbf{n}}{\textnormal{i}}ght)^n.
\]
Using the derivations for the multivariate ordinary Bell Polynomials in \citet{withers2010multivariate, schumann2019multivariate} we can rewrite the above as:
\[
\kappa_1\left(\mathbf{m}athbf{f}rac{\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{s}_j\mathbf{t}{\textnormal{i}}ght)}{N_{L}}{\textnormal{i}}ght)=\mathbf{s}um_{\mathbf{n}\mathbf{m}athbf{g}eq 0}\left(\mathbf{s}um_{n=0}^{\infty}\mathbf{m}athbf{f}rac{a_n}{N_L^n} \mathbf{m}athbf{h}at{B}_{\mathbf{n},n}\left(\mathbf{s}_{-j}\tilde{\mathbf{m}athbf{b}}{\textnormal{i}}ght){\textnormal{i}}ght)\mathbf{t}^{\mathbf{n}},
\]
where $\tilde{\mathbf{m}athbf{b}}=\left(\tilde{b}_{\mathbf{n}}{\textnormal{i}}ght)_{\mathbf{n}\mathbf{m}athbf{g}eq0}$ and $\mathbf{m}athbf{h}at{B}_{\mathbf{n},n}$ denotes the ordinary Bell Polynomials.
Plugging this in to our formula for the ResCGPK from Proposition {\textnormal{e}}f{prop:cgpk_expr_gen} we get
\[
\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{t}{\textnormal{i}}ght) = \mathbf{m}athcal{K}^{\left(L-1{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{t}{\textnormal{i}}ght) + \tilde{N}_{L} \mathbf{s}um_{k,k^{'}=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\kappa_1\left(\mathbf{m}athbf{f}rac{\mathbf{m}athcal{K}^{\left(L-1{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{s}_{k+k^{'}}\mathbf{t}{\textnormal{i}}ght)}{N_{L}}{\textnormal{i}}ght)
\]
\[
=\mathbf{s}um_{\mathbf{n}\mathbf{m}athbf{g}eq 0}\left( \tilde{b}_{\mathbf{n}} + \tilde{N}_{L} \mathbf{s}um_{j=-q+1}^{q-1}\abs{q-j}\mathbf{s}um_{n=0}^{\infty}\mathbf{m}athbf{f}rac{a_n}{N_L^n} \mathbf{m}athbf{h}at{B}_{\mathbf{n},n}\left(\mathbf{s}_{-j}\tilde{\mathbf{m}athbf{b}}{\textnormal{i}}ght){\textnormal{i}}ght)\mathbf{t}^{\mathbf{n}}.
\]
Let this be $\mathbf{s}um_{\mathbf{n}\mathbf{m}athbf{g}eq0}b_{\mathbf{n}}\mathbf{t}^{\mathbf{n}}$. All the terms in $b_{\mathbf{n}}$ are positive so for a lower bound, it suffices to sum up only specific $n$'s and $j$'s. We choose $n=1,j=0$, and since \citet{withers2010multivariate} showed that $\mathbf{m}athbf{h}at{B}_{\mathbf{n},1}\left(\tilde{\mathbf{m}athbf{b}}{\textnormal{i}}ght)=\tilde{b}_{\mathbf{n}}$ we get that
\[
b_{\mathbf{n}}\mathbf{m}athbf{g}eq \tilde{b}_{\mathbf{n}} + q\mathbf{m}athbf{f}rac{a_1\tilde{N}_{L}}{N_{L}}\tilde{b}_{\mathbf{n}}
=\left(1+\mathbf{m}athbf{f}rac{a_1 \alpha^2c_vc_w}{2q}{\textnormal{i}}ght)c_1\mathbf{n}^{-\mathbf{n}u},
\]
where the last equality is by the induction hypothesis.
\mathbf{m}athbf{e}nd{proof}
\begin{lemma}\label{lem:bntk}
The bound in Lemma ({\textnormal{e}}f{lem:b1}) holds for $\Theta^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}$.
\mathbf{m}athbf{e}nd{lemma}
\begin{proof}
Denote by $b_{\mathbf{n}}\left(\mathbf{m}athcal{K}{\textnormal{i}}ght)$ the Taylor coefficients for some kernel $\mathbf{m}athcal{K}$.
Theorem {\textnormal{e}}f{thm:2} implies that
\begin{align*}
\Theta^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght) = \mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)
+ \underset{\text{Denote by }\mathbf{m}athcal{K}^{'}}{\underbrace{\mathbf{m}athbf{f}rac{\alpha^{2}}{q}\mathbf{s}um_{l=1}^{L}\mathbf{s}um_{p=1}^{ d }\Pi_{tt}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)
\left(\text{tr}\left(\dot{K}_{\mathbf{m}athcal{D}_{p,p}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)\odot\Sigma_{\mathbf{m}athcal{D}_{p,p}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght) + K_{\mathbf{m}athcal{D}_{p,p}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght) {\textnormal{i}}ght) {\textnormal{i}}ght)}}.
\mathbf{m}athbf{e}nd{align*}
Since for any positive definite kernel the Taylor coefficients are non negative, we get that:
\[
b_{\mathbf{n}}\left(\Theta^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}{\textnormal{i}}ght) = b_{\mathbf{n}}\left(\mathbf{m}athcal{K}^{\mathbf{m}athbb{E}q}{\textnormal{i}}ght) + b_{\mathbf{n}}\left(\mathbf{m}athcal{K}^{'}{\textnormal{i}}ght)\mathbf{m}athbf{g}eq b_{\mathbf{n}}\left(\mathbf{m}athcal{K}^{\mathbf{m}athbb{E}q}{\textnormal{i}}ght).
\]
\mathbf{m}athbf{e}nd{proof}
\begin{definition}
Let $\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\text{FC}}$ be the GPK and $\Theta^{\left(L{\textnormal{i}}ght)}_{\text{FC}}$ the NTK of the bias free fully connected ResNet defined in \citet{huang2020deep}. For $\mathbf{x},\mathbf{z}\in\mathbf{m}athbb{S}^{C_0-1},u=\mathbf{x}^T\mathbf{z}$, following the derivation in \citet{huang2020deep} and \citet{belfer2021spectral} (in particular, Appendix B.1 of the latter), the normalized version of these kernels will be:
\[
\overline{\mathbf{m}athcal{K}}^{\left(0{\textnormal{i}}ght)}_{\text{FC}}\left(u{\textnormal{i}}ght)=u
\]
\[
\overline{\mathbf{m}athcal{K}}^{\left(L{\textnormal{i}}ght)}_{\text{FC}}\left(u{\textnormal{i}}ght)=\mathbf{m}athbf{f}rac{1}{1+\alpha^2}\left(\overline{\mathbf{m}athcal{K}}^{\left(L-1{\textnormal{i}}ght)}_{\text{FC}}\left(u{\textnormal{i}}ght) + \alpha^2 \kappa_1\left(\overline{\mathbf{m}athcal{K}}^{\left(L-1{\textnormal{i}}ght)}_{\text{FC}}\left(u{\textnormal{i}}ght){\textnormal{i}}ght){\textnormal{i}}ght)
\]
\[
\overline{\Theta}^{\left(L{\textnormal{i}}ght)}_{\text{FC}}=\mathbf{m}athbf{f}rac{1}{2Lv_{L-1}}\mathbf{s}um_{l=1}^{L}v_{l-1}\tilde{P}^{\left(l{\textnormal{i}}ght)}\left(u{\textnormal{i}}ght)\left(\kappa_1\left(\overline{\mathbf{m}athcal{K}}^{\left(l-1{\textnormal{i}}ght)}_{\text{FC}}\left(u{\textnormal{i}}ght){\textnormal{i}}ght)+\overline{\mathbf{m}athcal{K}}^{\left(l-1{\textnormal{i}}ght)}_{\text{FC}}\left(u{\textnormal{i}}ght)\kappa_0\left(\overline{\mathbf{m}athcal{K}}^{\left(l-1{\textnormal{i}}ght)}_{\text{FC}}\left(u{\textnormal{i}}ght){\textnormal{i}}ght){\textnormal{i}}ght),
\]
where $v_l=\left(1+\alpha^2{\textnormal{i}}ght)^l$, $\tilde{P}^L=1$ and $\tilde{P}^{l}\left(u{\textnormal{i}}ght)=\tilde{P}^{l+1}\left(u{\textnormal{i}}ght)\left(1+\alpha^2\kappa_0\left(\overline{\mathbf{m}athcal{K}}^{\left(l+1{\textnormal{i}}ght)}_{\text{FC}}\left(u{\textnormal{i}}ght){\textnormal{i}}ght){\textnormal{i}}ght)$.
\mathbf{m}athbf{e}nd{definition}
\begin{lemma}\label{lem:sdfc}
For $u\in\mathbf{m}athbb{R}^d$, letting $\mathbf{t}_1=\mathbf{t}_2=\ldots=u$ we obtain $\overline{\mathbf{m}athcal{K}}^{\left(L{\textnormal{i}}ght)}_{\text{FC}}\left(u{\textnormal{i}}ght)=\overline{\mathbf{m}athcal{K}}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}(\mathbf{t})$ and letting $\boldsymbol{k}^{\left(L{\textnormal{i}}ght)}=\Theta^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}-\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}$ we obtain $\overline{\Theta}^{\left(L{\textnormal{i}}ght)}_{\text{FC}}\left(u{\textnormal{i}}ght)=\overline{\boldsymbol{k}}^{\left(L{\textnormal{i}}ght)}(\mathbf{t})$.
\mathbf{m}athbf{e}nd{lemma}
\begin{proof}
For the GPK, this is immediate from Corollary {\textnormal{e}}f{cor:cgpk}.
For the ResCNTK, first recall that
\[
\Theta^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght) = \mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)
+ \mathbf{m}athbf{f}rac{\alpha^{2}}{q}\mathbf{s}um_{l=1}^{L}\mathbf{s}um_{p=1}^{d }P_{p}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)
\left(\text{tr}\left(\dot{K}_{\mathbf{m}athcal{D}_{p,p}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght)\odot\Sigma_{\mathbf{m}athcal{D}_{p,p}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght) + K_{\mathbf{m}athcal{D}_{p,p}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght) {\textnormal{i}}ght) {\textnormal{i}}ght).
\]
Observe first that a direct consequence of Theorem {\textnormal{e}}f{thm:1} is that $\mathbf{m}athbf{f}orall L\mathbf{m}athbf{g}eq 2$
\[
\Sigma_{{1,1}}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght) = \mathbf{m}athbf{f}rac{c_w}{q}\mathbf{s}um_{k=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\mathbf{m}athcal{K}^{\left(L-1{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{s}_k\mathbf{x},\mathbf{s}_k \mathbf{z}{\textnormal{i}}ght).
\]
and $\Sigma_{{1,1}}^{\left(1{\textnormal{i}}ght)}\left(\mathbf{x},\mathbf{z}{\textnormal{i}}ght) = \mathbf{m}athbf{f}rac{1}{C_0}u$.
Therefore, by choice of $\mathbf{t}$, $\Sigma^{\left(l{\textnormal{i}}ght)}, K^{\left(l{\textnormal{i}}ght)}$ and $\dot{K}^{\left(l{\textnormal{i}}ght)}$ have constant diagonals, and so the terms in the trace do not depend on the index $p$, and we get
\[
\boldsymbol{k}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght) = \alpha^{2}\mathbf{s}um_{l=1}^{L}N_{l+1}\left( \kappa_0\left(\overline{\mathbf{m}athcal{K}}^{\left(l-1{\textnormal{i}}ght)}_{\text{FC}}\left(u{\textnormal{i}}ght){\textnormal{i}}ght) \overline{\mathbf{m}athcal{K}}^{\left(l-1{\textnormal{i}}ght)}_{\text{FC}}\left(u{\textnormal{i}}ght)
+ \kappa_1\left(\overline{\mathbf{m}athcal{K}}^{\left(l-1{\textnormal{i}}ght)}_{\text{FC}}\left(u{\textnormal{i}}ght){\textnormal{i}}ght) {\textnormal{i}}ght) \mathbf{s}um_{p=1}^{d}P_p^{\left(l{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght).
\]
Note that for each $1\leq l\leq L$, $\mathbf{s}um_{p=1}^{d}P_p^{\left(l{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght)=\tilde{P}^{l}\left(u{\textnormal{i}}ght)$. For $l=L$ they are both equal to $1$ by definition. Now by induction we assume for $l+1$ and now show for $l$.
\[
\mathbf{s}um_{p=1}^d P_{p}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght)=\mathbf{s}um_{p=1}^d\left( P_{p}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght) + \mathbf{m}athbf{f}rac{\alpha^2}{q^2}\text{tr}\left(
\left(\mathbf{s}um_{k=\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}P_{\mathbf{m}athcal{D}_{p+k}}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght){\textnormal{i}}ght)\odot\dot{K}_{\mathbf{m}athcal{D}_{p,p}}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght){\textnormal{i}}ght) {\textnormal{i}}ght)
\]
\[
=\mathbf{s}um_{p=1}^d\left( P_p^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght) + \mathbf{m}athbf{f}rac{\alpha^2}{q^2}\kappa_0\left(\overline{K}_{FC}^{\left(l+1{\textnormal{i}}ght)}{\textnormal{i}}ght)\text{tr}\left(\mathbf{s}um_{k=\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}P_{\mathbf{m}athcal{D}_{p+k}}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght){\textnormal{i}}ght) {\textnormal{i}}ght)
\]
\[
=\left(\mathbf{s}um_{p=1}^d P_{p}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght) {\textnormal{i}}ght)
+ \alpha^2 \kappa_0\left(\overline{K}_{FC}^{\left(l+1{\textnormal{i}}ght)}{\textnormal{i}}ght) \left (\mathbf{m}athbf{f}rac{1}{q^2} \mathbf{s}um_{k=\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}} \left(\mathbf{s}um_{p=1}^d P_{p}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght) {\textnormal{i}}ght){\textnormal{i}}ght).
\]
Plugging this in the induction hypothesis we prove the induction. So overall:
\[
\boldsymbol{k}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght) = \alpha^{2}\mathbf{s}um_{l=1}^{L}N_{l+1} \tilde{P}^{l}\left(u{\textnormal{i}}ght) \left( \kappa_0\left(\overline{\mathbf{m}athcal{K}}^{\left(l-1{\textnormal{i}}ght)}_{\text{FC}}\left(u{\textnormal{i}}ght){\textnormal{i}}ght) \overline{\mathbf{m}athcal{K}}^{\left(l-1{\textnormal{i}}ght)}_{\text{FC}}\left(u{\textnormal{i}}ght)
+ \kappa_1\left(\overline{\mathbf{m}athcal{K}}^{\left(l-1{\textnormal{i}}ght)}_{\text{FC}}\left(u{\textnormal{i}}ght){\textnormal{i}}ght) {\textnormal{i}}ght).
\]
Since
\begin{align}\label{lem:norm_pi}
N_{l+1}=N_2v_{l-1},
\mathbf{m}athbf{e}nd{align}
normalizing $\boldsymbol{k}^{\left(L{\textnormal{i}}ght)}$ completes the proof.
\mathbf{m}athbf{e}nd{proof}
\begin{lemma}[\citep{bietti2020deep} Section~3.2]
\label{lem:sdk1}
For a small $t>0$,
\[
\kappa_1\left(1-t{\textnormal{i}}ght)=1-t-\mathbf{m}athbf{f}rac{2\mathbf{s}qrt{2}}{3\pi}t^{\mathbf{m}athbf{f}rac{3}{2}}+\mathbf{m}athcal{O}\left(t^{\mathbf{m}athbf{f}rac{5}{2}}{\textnormal{i}}ght).
\]
\mathbf{m}athbf{e}nd{lemma}
\begin{lemma} \label{lem:sdfu}
For all $L\mathbf{m}athbf{g}eq0$, and for a small $t>0$,
\[
\overline{\mathbf{m}athcal{K}}_{\text{FC}}^{\left(L{\textnormal{i}}ght)}\left(1-t{\textnormal{i}}ght)={1-t} + \Theta\left(t^{\mathbf{m}athbf{f}rac{3}{2}}{\textnormal{i}}ght).
\]
\mathbf{m}athbf{e}nd{lemma}
\begin{remark}
This lemma and its proof are a slightly modified version of Lemma B.4 from \citet{belfer2021spectral} where we tighten the bound.
\mathbf{m}athbf{e}nd{remark}
\begin{proof}
We prove this by induction. For $l=0,\overline{\mathbf{m}athcal{K}}_{\text{FC}}^{\left(0{\textnormal{i}}ght)}\left(1-t{\textnormal{i}}ght)=1-t$, trivially satisfying the lemma. Suppose the lemma holds for $l-1$, then using Lemma {\textnormal{e}}f{lem:sdk1},
\[
\overline{\mathbf{m}athcal{K}}^{\left(L{\textnormal{i}}ght)}_{\text{FC}}\left(1-t{\textnormal{i}}ght)=\mathbf{m}athbf{f}rac{1}{1+\alpha^2}\left(\overline{\mathbf{m}athcal{K}}^{\left(L-1{\textnormal{i}}ght)}_{\text{FC}}\left(1-t{\textnormal{i}}ght) + \alpha^2 \kappa_1\left(\overline{\mathbf{m}athcal{K}}^{\left(L-1{\textnormal{i}}ght)}_{\text{FC}}\left(1-t{\textnormal{i}}ght){\textnormal{i}}ght){\textnormal{i}}ght)
\]
\[
=\mathbf{m}athbf{f}rac{1}{1+\alpha^2}\left(1-t+\Theta\left(t^{\mathbf{m}athbf{f}rac{3}{2}}{\textnormal{i}}ght) + \alpha^2 \kappa_1\left(1-t+\Theta\left(t^{\mathbf{m}athbf{f}rac{3}{2}}{\textnormal{i}}ght){\textnormal{i}}ght){\textnormal{i}}ght)
\]
\[
=\mathbf{m}athbf{f}rac{1}{1+\alpha^2}\left(1-t+\Theta\left(t^{\mathbf{m}athbf{f}rac{3}{2}}{\textnormal{i}}ght) + \alpha^2 \left(1-t+\Theta\left(t^{\mathbf{m}athbf{f}rac{3}{2}}{\textnormal{i}}ght) + \mathbf{m}athbf{f}rac{2\mathbf{s}qrt{2}}{3\pi}\left(t-\Theta\left(t^{\mathbf{m}athbf{f}rac{3}{2}}{\textnormal{i}}ght){\textnormal{i}}ght)^{\mathbf{m}athbf{f}rac{3}{2}} + \mathbf{m}athcal{O}\left(t-\Theta\left(t^{\mathbf{m}athbf{f}rac{3}{2}}{\textnormal{i}}ght){\textnormal{i}}ght)^{\mathbf{m}athbf{f}rac{5}{2}} {\textnormal{i}}ght) {\textnormal{i}}ght)
\]
\[
=\mathbf{m}athbf{f}rac{1}{1+\alpha^2}\left(1-t+\Theta\left(t^{\mathbf{m}athbf{f}rac{3}{2}}{\textnormal{i}}ght) + \alpha^2 \left(1-t+\Theta\left(t^{\mathbf{m}athbf{f}rac{3}{2}}{\textnormal{i}}ght){\textnormal{i}}ght){\textnormal{i}}ght)
\]
\[
={1-t} + \Theta\left(t^{\mathbf{m}athbf{f}rac{3}{2}}{\textnormal{i}}ght).
\]
\mathbf{m}athbf{e}nd{proof}
\begin{lemma} \label{lem:sdtk}
$\overline{\mathbf{m}athcal{K}}_{\text{FC}}^{\left(L{\textnormal{i}}ght)}\left(u{\textnormal{i}}ght)$ and $\overline{\Theta}_{\text{FC}}^{\left(L{\textnormal{i}}ght)}\left(u{\textnormal{i}}ght)$ can be written as $\mathbf{s}um_{n=0}^{\infty}a_nu^n$ with $a_n\mathbf{s}im n^{-2.5}$ for $\overline{\mathbf{m}athcal{K}}_{\text{FC}}^{\left(L{\textnormal{i}}ght)}$ and $a_n\mathbf{s}im n^{-1.5}$ for $\overline{\Theta}_{\text{FC}}^{\left(L{\textnormal{i}}ght)}$.
\mathbf{m}athbf{e}nd{lemma}
\begin{proof}
Using Lemma {\textnormal{e}}f{lem:sdfu} we know that for $t\mathbf{n}earrow 1$, it holds that $\overline{\mathbf{m}athcal{K}}_{\text{FC}}^{\left(L{\textnormal{i}}ght)}\left(t{\textnormal{i}}ght)=t + f(t)$ where $f(t)=\Theta\left(\left(1-t{\textnormal{i}}ght)^{\mathbf{m}athbf{f}rac{3}{2}}{\textnormal{i}}ght)$.
Using \citep{flajolet2009analytic}[page 392 Thm. VI.1] we get that $f$ admits a Taylor expansion $\mathbf{s}um_{n=0}^{\infty}a_nt^n$ around $0$ with $a_n\mathbf{s}im n^{-2.5}$. Therefore, $\overline{\mathbf{m}athcal{K}}_{\text{FC}}^{\left(L{\textnormal{i}}ght)}\left(u{\textnormal{i}}ght)$ has a Taylor expansion with coefficients that exhibit the same decay.
For $\overline{\Theta}_{\text{FC}}^{\left(L{\textnormal{i}}ght)}\left(u{\textnormal{i}}ght)$, using \citep{belfer2021spectral}[Lemma 4.5] we know that for $t\mathbf{n}earrow 1$, it holds that $\overline{\Theta}_{\text{FC}}^{\left(L{\textnormal{i}}ght)}\left(t{\textnormal{i}}ght)=1 + c_1\left(1-t{\textnormal{i}}ght)^{\mathbf{m}athbf{f}rac{1}{2}}+o\left(\left(1-t{\textnormal{i}}ght)^{\mathbf{m}athbf{f}rac{1}{2}}{\textnormal{i}}ght)$ for some constant $c_1<0$.
Similarly to the previous case, using \citep{flajolet2009analytic}[Thm. VI.1, page 392] we get the desired bound.
\mathbf{m}athbf{e}nd{proof}
\begin{lemma}
Both $\overline{\mathbf{m}athcal{K}}_{\mathbf{m}athbb{E}q}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght)$ and $\overline{\Theta}_{\mathbf{m}athbb{E}q}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght)$ can be written as $\mathbf{s}um_{\mathbf{n}\mathbf{m}athbf{g}eq0}b_{\mathbf{n}}\mathbf{t}^{\mathbf{n}}$ with
\[
b_{\mathbf{n}}\leq c_2\mathbf{n}^{-\mathbf{n}u},
\]
where $\mathbf{n}u=\mathbf{m}athbf{f}rac{3}{2d}+1$ for $\overline{\mathbf{m}athcal{K}}_{\mathbf{m}athbb{E}q}^{\left(L{\textnormal{i}}ght)}$ and $\mathbf{n}u=\mathbf{n}u=\mathbf{m}athbf{f}rac{1}{2d}+1$ for $\overline{\Theta}_{\mathbf{m}athbb{E}q}^{\left(L{\textnormal{i}}ght)}$ and $c_2$ depends on $L$.
\mathbf{m}athbf{e}nd{lemma}
\begin{proof}
By Lemma {\textnormal{e}}f{lem:sdtk}, $\overline{\mathbf{m}athcal{K}}_{\text{FC}}^{\left(L{\textnormal{i}}ght)}\left(u{\textnormal{i}}ght)=\mathbf{s}um_{n=0}^{\infty}a_nu^n$ with $a_n\mathbf{s}im n^{-2.5}$.
Moreover, we have that $\overline{\mathbf{m}athcal{K}}_{\mathbf{m}athbb{E}q}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght)=\mathbf{s}um_{\mathbf{n}\mathbf{m}athbf{g}eq 0}b_{\mathbf{n}}\mathbf{t}^{\mathbf{n}}$.
Together with lemma {\textnormal{e}}f{lem:sdfc} we get that
\[
\overline{\mathbf{m}athcal{K}}_{\mathbf{m}athbb{E}q}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght)=\mathbf{s}um_{\mathbf{n}\mathbf{m}athbf{g}eq 0}b_{\mathbf{n}}\mathbf{t}^{\mathbf{n}}
=\mathbf{s}um_{\mathbf{n}\mathbf{m}athbf{g}eq 0}b_{\mathbf{n}}u^{\abs{\mathbf{n}}}
=\mathbf{s}um_{n=0}^{\infty}b_{\mathbf{n}}u^{\abs{\mathbf{n}}}
=\mathbf{s}um_{n=0}^{\infty}u^n\mathbf{s}um_{\abs{\mathbf{n}}=n}b_{\mathbf{n}}.
\]
The uniqueness of the power series implies
\[
\mathbf{s}um_{\abs{\mathbf{n}}=n}b_{\mathbf{n}}=a_n=\Theta\left(n^{-2.5}{\textnormal{i}}ght).
\]
Plugging in Lemma D.8 From \citep{geifman2022spectral} we get that $b_{\mathbf{n}}\leq c_2\mathbf{n}^{-\mathbf{m}athbf{f}rac{3}{2d}-1}$ for some constant $c_2>0$.
For the bound for ResCNTK, since by Lemma {\textnormal{e}}f{lem:sdtk} $\overline{\Theta}_{\text{FC}}^{\left(L{\textnormal{i}}ght)}\left(u{\textnormal{i}}ght)= \mathbf{s}um_{n=0}^{\infty}\tilde{a}_nu^n$ with $\tilde{a}_n\mathbf{s}im n^{-1.5}$
and $\overline{\boldsymbol{k}}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght) = \mathbf{s}um_{\mathbf{n}\mathbf{m}athbf{g}eq 0}\tilde{b}_{\mathbf{n}}\mathbf{t}^{\mathbf{n}}$, we can analogously get that $\tilde{b}_{\mathbf{n}}\leq c^{'}\mathbf{n}^{-\mathbf{m}athbf{f}rac{1}{2d}-1}$ for some constant $c^{'}>0$. (The difference from the different bound comes from the referenced lemma in \citep{geifman2022spectral}.)
Since $\boldsymbol{k}^{\left(L{\textnormal{i}}ght)}=\Theta^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}-\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}$, by combining the two results we get that $\overline{\Theta}_{\mathbf{m}athbb{E}q}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght)$ can be written as $\mathbf{s}um_{\mathbf{n}\mathbf{m}athbf{g}eq 0}b^{'}_{\mathbf{n}}\mathbf{t}^{\mathbf{n}}$ with $b^{'}_{\mathbf{n}}\leq c_2\mathbf{n}^{-\mathbf{m}athbf{f}rac{1}{2d}-1}$.
\mathbf{m}athbf{e}nd{proof}
\mathbf{s}ection{Positional Bias of Eigenvalues} \label{ap:4}
\mathbf{s}ubsection{Proof of Theorem {\textnormal{e}}f{thm:eigen} in the main text}
We define a "stride-q" version of the ResCGPK. Let $Q=\{-\mathbf{m}athbf{f}rac{q-1}{2}\,\ldots,\mathbf{m}athbf{f}rac{q-1}{2}\}, R_0={0}^{2L-1}$ (i.e a set that contains only the tuple $\mathbf{z}ero=(0,\ldots,0)$), and for $l\mathbf{m}athbf{g}eq1$ let $R_l:=Q^{2l-1}\times \{0\}^{2(L-l)}$ (i.e tuples where the first $2l-1$ elements are in $Q$ and the last $2(L-l)$ elements are $0$).
We let $[-1,1]^{R_l}$ be elements of the form $\tilde{\mathbf{t}}_{\mathbf{m}athbf{a}}$ which are $[-1,1]$ valued parameters indexed by tuples $R_l$. Also, for every $k,k^{'}\in Q$ and $1\leq l \leq L$ define $\iota^l_{k,k^{'}}:[-1,1]^{R_L}\to[-1,1]^{R_{L}}$ by $\iota^l_{k,k^{'}}(\tilde{\mathbf{t}})_{\mathbf{m}athbf{a}}=\tilde{\mathbf{t}}_{(a_1,\ldots,a_{2l-3},k,k^{'},a_{2l}\ldots)}$.
We now define the kernel $\boldsymbol{k}^{\left(1{\textnormal{i}}ght)}:[-1,1]^{R_1}\to[-1,1]$ to be
\[
\boldsymbol{k}^{\left(1{\textnormal{i}}ght)}(\tilde{\mathbf{t}}) = \mathbf{m}athbf{f}rac{1}{1+\alpha^2}\left(\underset{:=\boldsymbol{k}_1^{\left(1{\textnormal{i}}ght)}(\tilde{\mathbf{t}})}{\underbrace{\tilde{t}_\mathbf{z}ero}}
+ \alpha^2\underset{:=\boldsymbol{k}_2^{\left(1{\textnormal{i}}ght)}(\tilde{\mathbf{t}})}{\underbrace{\mathbf{m}athbf{f}rac{1}{q}\mathbf{s}um_{k=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\kappa_1\left(\tilde{t}_{(k,0,\ldots,0)}{\textnormal{i}}ght)}}{\textnormal{i}}ght).
\]
Also, for all $2\leq l\leq L$ we let $\boldsymbol{k}^{\left(l{\textnormal{i}}ght)}:[-1,1]^{R_l}\to[-1,1]$ be
\[
\boldsymbol{k}^{\left(l{\textnormal{i}}ght)}(\tilde{\mathbf{t}}) = \mathbf{m}athbf{f}rac{1}{1+\alpha^2}\left(\underset{:=\boldsymbol{k}_1^{\left(l{\textnormal{i}}ght)}(\tilde{\mathbf{t}})}{\underbrace{\boldsymbol{k}^{\left(l-1{\textnormal{i}}ght)}(\iota^l_{0,0}(\tilde{\mathbf{t}}))}}
+ \alpha^2\underset{:=\boldsymbol{k}_2^{\left(l{\textnormal{i}}ght)}(\tilde{\mathbf{t}})}{\underbrace{\mathbf{m}athbf{f}rac{1}{q^2}\mathbf{s}um_{k,k^{'}=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\kappa_1\left(\boldsymbol{k}^{\left(l-1{\textnormal{i}}ght)}\left(\iota^l_{k,k^{'}}(\tilde{\mathbf{t}}){\textnormal{i}}ght){\textnormal{i}}ght)}}{\textnormal{i}}ght).
\]
We also define the change of variables $S_l:R_{l}\to [d]$ by $S_l(\mathbf{m}athbf{a})=\abs{\mathbf{m}athbf{a}}+1$ (reminder: $|\cdot|$ on a multi-index means sum of all entries).
We are now ready to define a correspondence between the stride-q ResCGPK and the standard one. Namely, for every $\mathbf{t}\in [-1,1]^d$ we let $\phi(\mathbf{t})\in[-1,1]^{R_L}$ be $\phi(\mathbf{t})_{\mathbf{m}athbf{a}}:=\mathbf{t}_{S_L(\mathbf{m}athbf{a})}$, I claim that $\boldsymbol{k}^{(L)}(\phi(\mathbf{t}))=\overline{\mathbf{m}athcal{K}}_{\mathbf{m}athbb{E}q}^{\left(L{\textnormal{i}}ght)}(\mathbf{t})$
First observe that for $\mathbf{m}athbf{a}\in R^{l}$ it holds that $\iota^l_{k,k^{'}}(\phi(\mathbf{t}))_{\mathbf{m}athbf{a}}=\mathbf{t}_{\abs{\mathbf{m}athbf{a}}+1+k+k^{'}}=\phi(\mathbf{s}_{k+k^{'}}\mathbf{t})_{\mathbf{m}athbf{a}}$ (because $\mathbf{m}athbf{a}\in R^{l}$ so $\mathbf{m}athbf{a}_{2l-2},\mathbf{m}athbf{a}_{2l-1}=0$). Also notice that $\iota^l_{k,k^{'}}(\phi(\mathbf{t}))\in R^{l+1}$.
Thus, we recursively get that for $\mathbf{m}athbf{a}\in R^1$, $\iota^1_{k_1,k_1^{'}}\circ\ldots \circ\iota^L_{k_L,k_L^{'}}(\phi(\mathbf{t}))_{\mathbf{m}athbf{a}}=\phi(\mathbf{s}_{k_1+k_1^{'}}\circ\ldots\circ\mathbf{s}_{k_L+k_L^{'}}\mathbf{t})_{\mathbf{m}athbf{a}}$.
Now, for $L=1$ we trivially have that $\boldsymbol{k}^{\left(1{\textnormal{i}}ght)}(\phi(\mathbf{t}))=\overline{\mathbf{m}athcal{K}}_{\mathbf{m}athbb{E}q}^{\left(1{\textnormal{i}}ght)}(\mathbf{t})$. So for $\mathbf{m}athbf{a}\in R^1$,
\[
\boldsymbol{k}^{\left(1{\textnormal{i}}ght)}\left(\iota^1_{k_1,k_1^{'}}\circ\ldots \circ\iota^L_{k_L,k_L^{'}}(\phi(\mathbf{t})){\textnormal{i}}ght)_{\mathbf{m}athbf{a}}
=\boldsymbol{k}^{\left(1{\textnormal{i}}ght)}\left(\phi(\mathbf{s}_{k_1+k_1^{'}}\circ\ldots\circ\mathbf{s}_{k_L+k_L^{'}}\mathbf{t}){\textnormal{i}}ght)_{\mathbf{m}athbf{a}}
=\overline{\mathbf{m}athcal{K}}_{\mathbf{m}athbb{E}q}^{\left(1{\textnormal{i}}ght)}(\mathbf{s}_{k_1+k_1^{'}}\circ\ldots\circ\mathbf{s}_{k_L+k_L^{'}}\mathbf{t})
\]
As such, we get that
\[
\boldsymbol{k}^{\left(2{\textnormal{i}}ght)}\left(\iota^2_{k_1,k_1^{'}}\circ\ldots \circ\iota^L_{k_L,k_L^{'}}(\phi(\mathbf{t})){\textnormal{i}}ght)_{\mathbf{m}athbf{a}}
=\overline{\mathbf{m}athcal{K}}_{\mathbf{m}athbb{E}q}^{\left(2{\textnormal{i}}ght)}(\mathbf{s}_{k_2+k_2^{'}}\circ\ldots\circ\mathbf{s}_{k_L+k_L^{'}}\mathbf{t})
\]
And continuing by induction we eventually get $\boldsymbol{k}^{\left(L-1{\textnormal{i}}ght)}\left(\iota^L_{k,k^{'}}(\phi(\mathbf{t})){\textnormal{i}}ght)=\overline{\mathbf{m}athcal{K}}_{\mathbf{m}athbb{E}q}^{\left(L-1{\textnormal{i}}ght)}(\mathbf{s}_{k+k^{'}}\mathbf{t})$ which implies that $\boldsymbol{k}^{\left(L{\textnormal{i}}ght)}(\phi(\mathbf{t}))=\overline{\mathbf{m}athcal{K}}_{\mathbf{m}athbb{E}q}^{\left(L{\textnormal{i}}ght)}(\mathbf{t})$.
We now move towards better understanding their Taylor expansions. For a function $f(\mathbf{t})$ that can be approximated by a Taylor series $\mathbf{s}um_{\mathbf{n}=0}^{\infty}a_{\mathbf{n}}t^{\mathbf{n}}$ let $\left[{\mathbf{t}}^{\mathbf{n}}{\textnormal{i}}ght]f$ denote the coefficient of $\mathbf{t}^{\mathbf{n}}$ in its Taylor series (meaning $a_{\mathbf{n}}$). Let $M_l(\mathbf{n})=\left\{\tilde{\mathbf{n}}\in \mathbf{m}athbb{Z}_{\mathbf{m}athbf{g}eq0}^{R_L} \mathbf{m}id \text{supp}\tilde{\mathbf{n}}\mathbf{s}ubseteq R_l \text{ and }\mathbf{m}athbf{f}orall i\in [d], \mathbf{s}um_{j\in S_L^{-1}(i)}\tilde{\mathbf{n}}_j=\mathbf{n}_i{\textnormal{i}}ght\}$ be the set of multi-indices which are indexed by tuples in $R_L$ with support in $R_l$ that correspond to $\mathbf{n}$. So if $\tilde{\mathbf{n}}\in M_l(\mathbf{n})$ we get that for every $\mathbf{t}\in[-1,1]^d$, $\phi(\mathbf{t})^{\tilde{\mathbf{n}}}=\mathbf{t}^{\mathbf{n}}$.
We get a correspondence between the Taylor expansions of the kernel as follows:
\[
\mathbf{s}um_{\mathbf{n}\mathbf{m}athbf{g}eq 0}b_{\mathbf{n}}\mathbf{t}^{\mathbf{n}}
=\overline{\mathbf{m}athcal{K}}_{\mathbf{m}athbb{E}q}^{\left(L{\textnormal{i}}ght)}(\mathbf{t})
=\boldsymbol{k}^{\left(l{\textnormal{i}}ght)}(\tilde{\mathbf{t}})
=\mathbf{s}um_{\tilde{\mathbf{n}}\mathbf{m}athbf{g}eq 0}\tilde{b}_{\tilde{\mathbf{n}}}\tilde{\mathbf{t}}^{\tilde{\mathbf{n}}}
=\mathbf{s}um_{\mathbf{n}\mathbf{m}athbf{g}eq 0}\left(\mathbf{s}um_{\tilde{\mathbf{n}}\in M_L(\mathbf{n})}\tilde{b}_{\tilde{\mathbf{n}}}{\textnormal{i}}ght)\mathbf{t}^{\mathbf{n}}.
\]
By the uniqueness of the power series,
\[
b_{\mathbf{n}} = \mathbf{s}um_{\tilde{\mathbf{n}}\in M_L(\mathbf{n})}\tilde{b}_{\tilde{\mathbf{n}}}
= \mathbf{m}athbf{f}rac{1}{1+\alpha^2}\left(\mathbf{s}um_{\tilde{\mathbf{n}}\in M_{L-1}(\mathbf{n})}[\tilde{\mathbf{t}}^{\tilde{\mathbf{n}}}]\boldsymbol{k}_1^{\left(L{\textnormal{i}}ght)} + \alpha^2 \mathbf{s}um_{\tilde{\mathbf{n}}\in M_{L}(\mathbf{n})}[\tilde{\mathbf{t}}^{\tilde{\mathbf{n}}}]\boldsymbol{k}_2^{\left(L{\textnormal{i}}ght)}{\textnormal{i}}ght).
\]
Now $\boldsymbol{k}_1^{\left(L{\textnormal{i}}ght)}(\tilde{\mathbf{t}})=\mathbf{m}athbf{f}rac{1}{1+\alpha^2}\left(\boldsymbol{k}_1^{\left(L-1{\textnormal{i}}ght)}(\tilde{\mathbf{t}})+\alpha^2\boldsymbol{k}_2^{\left(L-1{\textnormal{i}}ght)}(\tilde{\mathbf{t}}){\textnormal{i}}ght)$ so we can continue to apply this recursively and eventually get:
\[
= \mathbf{m}athbf{f}rac{1}{(1+\alpha^2)^L}\mathbf{s}um_{\tilde{\mathbf{n}}\in M_0(\mathbf{n})}[\tilde{\mathbf{t}}^{\tilde{\mathbf{n}}}]\boldsymbol{k}_1^{\left(1{\textnormal{i}}ght)} + \mathbf{s}um_{l=1}^{L} \left(\mathbf{m}athbf{f}rac{\alpha^{2}}{1+\alpha^2}{\textnormal{i}}ght)^{L-l+1} \mathbf{s}um_{\tilde{\mathbf{n}}\in M_l(\mathbf{n})}[\tilde{\mathbf{t}}^{\tilde{\mathbf{n}}}]\boldsymbol{k}_2^{\left(l{\textnormal{i}}ght)}
\]
\[
= \mathbf{m}athbf{f}rac{1}{(1+\alpha^2)^L}\bm{1}_{\text{supp}\mathbf{n}\mathbf{s}ubseteq R_0} + \mathbf{s}um_{l=1}^{L} \left(\mathbf{m}athbf{f}rac{\alpha^{2}}{1+\alpha^2}{\textnormal{i}}ght)^{L-l+1} \mathbf{s}um_{\tilde{\mathbf{n}}\in M_l(\mathbf{n})}[\tilde{\mathbf{t}}^{\tilde{\mathbf{n}}}]\boldsymbol{k}_2^{\left(l{\textnormal{i}}ght)}.
\]
Now let $\mathbf{n}u=2.5$, via the proof in Lemma {\textnormal{e}}f{lem:b1} we know that for every $1\leq l\leq L$ there is some $\tilde{c}_l>0 \text{ s.t.\ }$
\[
\left(\mathbf{m}athbf{f}rac{\alpha^{2}}{1+\alpha^2}{\textnormal{i}}ght)^{L-l+1} \mathbf{s}um_{\tilde{\mathbf{n}}\in M_l(\mathbf{n})}[\tilde{\mathbf{t}}^{\tilde{\mathbf{n}}}]\boldsymbol{k}_2^{\left(l{\textnormal{i}}ght)}(\tilde{\mathbf{t}})
\mathbf{m}athbf{g}eq \tilde{c}_l\mathbf{s}um_{\tilde{\mathbf{n}}\in M_l(\mathbf{n})}\tilde{\mathbf{n}}^{-\mathbf{n}u}.
\]
Let $p_i(l)=\abs{S_l^{-1}(i)}$, the number of paths from an input pixel $i$ to the output of an $l$ layer CGPK. Using \citet{geifman2022spectral}[Lemma C.4, C.5] we get that for $A>1$, some $c_l$ constants, and $c_{\mathbf{n},l}=c_l\prod_{i=1}^d A^{\mathbf{m}in(p_i^{(l)}, \mathbf{n}_i)}$ it holds that $c_{\mathbf{n},l}\mathbf{n}^{-\mathbf{n}u} \leq \left(\mathbf{m}athbf{f}rac{\alpha^{2}}{1+\alpha^2}{\textnormal{i}}ght)^{L-l+1}\mathbf{s}um_{\tilde{\mathbf{n}}\in M_l(\mathbf{n})}[\tilde{\mathbf{t}}^{\tilde{\mathbf{n}}}]\boldsymbol{k}^{\left(l{\textnormal{i}}ght)}(\tilde{\mathbf{t}})$.
Overall, we obtain that
\[
b_{\mathbf{n}}\mathbf{m}athbf{g}eq \mathbf{s}um_{l=0}^Lc_{\mathbf{n},l}\mathbf{n}^{-\mathbf{n}u}.
\]
Now consider the kernel $\mathbf{m}athbf{h}at{\boldsymbol{k}}^{(l)}(\mathbf{t})=\mathbf{s}um_{\mathbf{n}\mathbf{m}athbf{g}eq0}c_{\mathbf{n},l}\mathbf{n}^{-\mathbf{n}u}$ then by \citet{geifman2022spectral}[Lemma C.7], the eigenvalues of this kernel satisfy
\[
\lambda_{\mathbf{m}athbf{k}}(\mathbf{m}athbf{h}at{\boldsymbol{k}}^{(l)})\mathbf{m}athbf{g}eq c_{\mathbf{m}athbf{k},l}\underset{n_i>0}{\prod_{i=1}^d}k_i^{-C_0-2},
\]
where $c_{\mathbf{m}athbf{k},l}=\tilde{c}_l\prod_{i=1}^d A^{\mathbf{m}in(p_i^{(L)}, k_i)}$ for some constant $\tilde{c}_l$.
As for every $l$ the eigenvectors that correspond to $\lambda_{\mathbf{m}athbf{k}}(\mathbf{m}athbf{h}at{k}^{(l)})$ are the same (given by the spherical harmonics), we get by linearity that
\[
\lambda_{\mathbf{m}athbf{k}}\left(\overline{\mathbf{m}athcal{K}}^{(L)}_{\mathbf{m}athbb{E}q}{\textnormal{i}}ght) \mathbf{m}athbf{g}eq \mathbf{s}um_{l=1}^L c_{\mathbf{m}athbf{k},l}\underset{n_i>0}{\prod_{i=1}^d}k_i^{-C_0-2}.
\]
As in Lemma {\textnormal{e}}f{lem:bntk}, this also gives a bound on the eigenvalues of $\Theta_{\mathbf{m}athbb{E}q}^{(L)}$.
\mathbf{s}ection{Infinite Depth Limit}
\mathbf{s}ubsection{Proof of Theorem {\textnormal{e}}f{thm:decay} in the main text}\label{ap:6.1}
\begin{lemma}
Suppose that $\alpha=L^{-\mathbf{m}athbf{g}amma}$ for $\mathbf{m}athbf{g}amma\in (0.5,1]$ then for any $\mathbf{t}\in [-1,1]^d$, $\abs{\overline{\Theta}_{\mathbf{m}athbb{E}q}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght) - \overline{\Sigma}_{1,1}^{\left(1{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght)}\leq\mathbf{m}athcal{O}\left(L^{1-2\mathbf{m}athbf{g}amma}{\textnormal{i}}ght)$.
\mathbf{m}athbf{e}nd{lemma}
\begin{proof}
First recall that
\[
\Theta^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{t}{\textnormal{i}}ght) = \mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{t}{\textnormal{i}}ght)
+ \mathbf{m}athbf{f}rac{\alpha^{2}}{q}\mathbf{s}um_{l=1}^{L}\mathbf{s}um_{p=1}^{d }P_{p}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght)
\left(\text{tr}\left(\dot{K}_{\mathbf{m}athcal{D}_{p,p}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght)\odot\Sigma_{\mathbf{m}athcal{D}_{p,p}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght) + K_{\mathbf{m}athcal{D}_{p,p}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght) {\textnormal{i}}ght) {\textnormal{i}}ght).
\]
Let $\boldsymbol{k}^{(L)}(\mathbf{t}) = \mathbf{m}athbf{f}rac{1}{\alpha^2}\left(\Theta^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{t}{\textnormal{i}}ght) - \mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{t}{\textnormal{i}}ght){\textnormal{i}}ght)$ so that $\Theta^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{t}{\textnormal{i}}ght) = \mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{t}{\textnormal{i}}ght)
+ \alpha^2 \boldsymbol{k}^{(L)}(\mathbf{t})$. Using our calculations in \mathbf{m}athbf{e}qref{lem:norm_pi} we know that $\boldsymbol{k}({\bm{e}}c{\bm{1}})=\mathbf{m}athbf{f}rac{2}{C_0}L(1+\alpha^2)^{L-1}$.
Therefore,
\[
\alpha^2\boldsymbol{k}({\bm{e}}c{\bm{1}})
= \mathbf{m}athbf{f}rac{2}{C_0}(\alpha^2 L) (1+\alpha^2)^{L-1}
\leq \mathbf{m}athbf{f}rac{2}{C_0} (\alpha^2 L) (1+\alpha^2)^{L}
= \mathbf{m}athbf{f}rac{2}{C_0}L^{1-2\mathbf{m}athbf{g}amma}\left(1+\mathbf{m}athbf{f}rac{1}{L^{2\mathbf{m}athbf{g}amma}}{\textnormal{i}}ght)^L
\]
\[
\leq \mathbf{m}athbf{f}rac{2}{C_0}L^{1-2\mathbf{m}athbf{g}amma}\left(\left(1+\mathbf{m}athbf{f}rac{1}{L^{2\mathbf{m}athbf{g}amma}}{\textnormal{i}}ght)^{L^{2\mathbf{m}athbf{g}amma}}{\textnormal{i}}ght)^{L^{-2\mathbf{m}athbf{g}amma+1}}
=\mathbf{m}athbf{f}rac{2}{C_0}L^{1-2\mathbf{m}athbf{g}amma}e^{L^{1-2\mathbf{m}athbf{g}amma}} = \mathbf{m}athcal{O}\left(L^{1-2\mathbf{m}athbf{g}amma}{\textnormal{i}}ght).
\]
Consequently, $\abs{\Theta^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{t}{\textnormal{i}}ght)-\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{t}{\textnormal{i}}ght)}\leq \alpha^2\boldsymbol{k}({\bm{e}}c{\bm{1}}) \leq \mathbf{m}athcal{O}\left(L^{1-2\mathbf{m}athbf{g}amma}{\textnormal{i}}ght)$. It therefore suffices to prove that $\abs{\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{t}{\textnormal{i}}ght)-\overline{\Sigma}_{1,1}^{\left(1{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght)}\leq \mathbf{m}athcal{O}\left(L^{1-2\mathbf{m}athbf{g}amma}{\textnormal{i}}ght)$.
To avoid confusion, we denote by $\overline{\mathbf{m}athcal{K}}_{\mathbf{m}athbb{E}q}^{\left(L,\alpha{\textnormal{i}}ght)}(\mathbf{t})$ the ResCGPK with a specific $\alpha$ (that may not necessarily be $L^{-\mathbf{m}athbf{g}amma}$). For all $2 \leq l \leq L$ as a result of Corollary {\textnormal{e}}f{cor:cgpk} we get that
\[
\abs{\overline{\mathbf{m}athcal{K}}_{\mathbf{m}athbb{E}q}^{\left(l,L^{-\mathbf{m}athbf{g}amma}{\textnormal{i}}ght)}(\mathbf{t}) - \overline{\mathbf{m}athcal{K}}_{\mathbf{m}athbb{E}q}^{\left(l-1, L^{-\mathbf{m}athbf{g}amma}{\textnormal{i}}ght)}(\mathbf{t})}
\]
\[
\leq \abs{\mathbf{m}athbf{f}rac{1}{1+\alpha^2}\left(\overline{\mathbf{m}athcal{K}}_{\mathbf{m}athbb{E}q}^{\left(l-1, L^{-\mathbf{m}athbf{g}amma}{\textnormal{i}}ght)}(\mathbf{t}) + \mathbf{m}athbf{f}rac{\alpha^2}{q^2}\mathbf{s}um_{k,k^{'}=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\kappa_1\left(\overline{\mathbf{m}athcal{\mathbf{m}athcal{K}}}^{\left(l-1, L^{-\mathbf{m}athbf{g}amma}{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{s}_{k+k^{'}}\mathbf{t}{\textnormal{i}}ght){\textnormal{i}}ght){\textnormal{i}}ght) - \overline{\mathbf{m}athcal{K}}_{\mathbf{m}athbb{E}q}^{\left(l-1, L^{-\mathbf{m}athbf{g}amma}{\textnormal{i}}ght)}(\mathbf{t})}
\]
\[
=\mathbf{m}athbf{f}rac{\alpha^2}{1+\alpha^2}\abs{\mathbf{m}athbf{f}rac{1}{q^2}\mathbf{s}um_{k,k^{'}=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\kappa_1\left(\overline{\mathbf{m}athcal{K}}^{\left(l-1, L^{-\mathbf{m}athbf{g}amma}{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{s}_{k+k^{'}}\mathbf{t}{\textnormal{i}}ght){\textnormal{i}}ght)-\overline{K}_{\mathbf{m}athbb{E}q}^{\left(l-1, L^{-\mathbf{m}athbf{g}amma}{\textnormal{i}}ght)}(\mathbf{t})}
\leq \mathbf{m}athbf{f}rac{\alpha^2}{1+\alpha^2},
\]
Where the last inequality follows because $\overline{K}_{\mathbf{m}athbb{E}q}^{\left(l-1, L^{-\mathbf{m}athbf{g}amma}{\textnormal{i}}ght)}\in[-1,1]$ and so is $\kappa_1$, This implies that
\[
\abs{\overline{\mathbf{m}athcal{K}}_{\mathbf{m}athbb{E}q}^{\left(L, L^{-\mathbf{m}athbf{g}amma}{\textnormal{i}}ght)}(\mathbf{t}) - \overline{\Sigma}^{\left(1{\textnormal{i}}ght)}_{1,1}(\mathbf{t})} = \abs{\overline{\mathbf{m}athcal{K}}_{\mathbf{m}athbb{E}q}^{\left(L, L^{-\mathbf{m}athbf{g}amma}{\textnormal{i}}ght)}(\mathbf{t}) - \overline{\mathbf{m}athcal{K}}_{\mathbf{m}athbb{E}q}^{\left(1, L^{-\mathbf{m}athbf{g}amma}{\textnormal{i}}ght)}(\mathbf{t})} \leq L\cdot \mathbf{m}athbf{f}rac{\alpha^2}{1+\alpha^2} \leq \mathbf{m}athcal{O}\left(L^{1-2\mathbf{m}athbf{g}amma}{\textnormal{i}}ght),
\]
which completes the proof.
\mathbf{m}athbf{e}nd{proof}
\mathbf{s}ubsection{Proof of Theorem {\textnormal{e}}f{thm:cond} in the main text} \label{ap:6.2}
Our goal of this subsections is to prove the following:
\begin{theorem}\text{ }
Let $\bar{K}_{\text{ResCGPK}}^{(L)}$ and $\bar{K}_{\text{CGPK}}^{(L)}$ respectively denote kernel matrices for the normalized trace kernels ResCGPK and CGPK of depth $L$. Let $\bm{B}\left(K{\textnormal{i}}ght)$ be a double-constant matrix defined for a matrix $K$ as in Lemma~{\textnormal{e}}f{lemma:dcb}. Then,
\begin{enumerate}
\item $\mathbf{n}orm{\bar{K}_{\text{ResCGPK}}^{(L)} - \bm{B}\left(\bar{K}_{\text{ResCGPK}}^{(L)}{\textnormal{i}}ght)}_1 \underset{L\to\infty}{\longrightarrow}0$ and $\mathbf{n}orm{\bar{K}_{\text{CGPK}}^{(L)} - \bm{B}\left(\bar{K}_{\text{CGPK}}^{(L)}{\textnormal{i}}ght)}_1 \underset{L\to\infty}{\longrightarrow}0$.
\item ${\textnormal{h}}o\left(\bm{B}\left(\bar{K}_{\text{ResCGPK}}^{(L)}{\textnormal{i}}ght){\textnormal{i}}ght) \underset{L\to\infty}{\longrightarrow}\infty$ and ${\textnormal{h}}o\left(\bm{B}\left(\bar{K}_{\text{CGPK}}^{(L)}{\textnormal{i}}ght){\textnormal{i}}ght) \underset{L\to\infty}{\longrightarrow}\infty$.
\item $\mathbf{m}athbf{e}xists L_0\in\mathbf{m}athbb{N} \text{ s.t.\ } \mathbf{m}athbf{f}orall L\mathbf{m}athbf{g}eq L_0$, ${\textnormal{h}}o\left(\bm{B}\left(\bar{K}_{\text{ResCGPK}}^{(L)}{\textnormal{i}}ght) {\textnormal{i}}ght) < {\textnormal{h}}o\left(\bm{B}\left(\bar{K}_{\text{CGPK}}^{(L)}{\textnormal{i}}ght) {\textnormal{i}}ght)$.
\mathbf{m}athbf{e}nd{enumerate}
\mathbf{m}athbf{e}nd{theorem}
\begin{proof}
We give here the main ideas and leave the dirty work to the lemmas.
By Lemmas {\textnormal{e}}f{lem:k_conv} and {\textnormal{e}}f{lemma:id1} both $\bar{K}_{\text{ResCGPK}}^{(L)}$ and $\bar{K}_{\text{CGPK}}^{(L)}$ tend towards $\bm{B}_{1,1}$ as $L\to\infty$.
As such, $\bm{B}\left(\bar{K}_{\text{ResCGPK}}^{(L)}{\textnormal{i}}ght)\underset{L\to\infty}{\longrightarrow}\bm{B}_{1,1}$ and so we get (1).
For (2), observe that $\lambda_{\mathbf{m}in}(\bm{B}(\bar{K}_{\text{CGPK}}^{(L)}))=1-\mathbf{m}athbf{f}rac{1}{n(n-1)}\mathbf{s}um_{i\mathbf{n}eq j}\bar{K}_{\text{CGPK}}^{(L)}\underset{L\to\infty}{\longrightarrow}0$.
For (3), let $L_0 \in \mathbf{m}athbb{N}$ be the minimal such that the entries of $\bm{B}\left(\bar{K}_{\text{ResCGPK}}^{(L)}{\textnormal{i}}ght)$ and $\bm{B}\left(\bar{K}_{\text{CGPK}}^{(L)}{\textnormal{i}}ght)$ are non negative and let $L\mathbf{m}athbf{g}eq L_0$. By lemma ${\textnormal{e}}f{lem:k_conv}$ we get that $\mathbf{m}athbf{f}rac{1}{n(n-1)}\mathbf{s}um_{i\mathbf{n}eq j}\bar{K}_{\text{CGPK}}^{(L)} > \mathbf{m}athbf{f}rac{1}{n(n-1)}\mathbf{s}um_{i\mathbf{n}eq j}\bar{K}_{\text{ResCGPK}}^{(L)}$.
Since ${\textnormal{h}}o(\bm{B}_{1,b})=1+n\mathbf{m}athbf{f}rac{b}{1-b}$ we get that ${\textnormal{h}}o\left(\bm{B}\left(\bar{K}_{\text{ResCGPK}}^{(L)}{\textnormal{i}}ght) {\textnormal{i}}ght) < {\textnormal{h}}o\left(\bm{B}\left(\bar{K}_{\text{CGPK}}^{(L)}{\textnormal{i}}ght) {\textnormal{i}}ght)$ as desired.
\mathbf{m}athbf{e}nd{proof}
\begin{lemma} \label{lemma:id1}
Suppose that $\alpha$ is a constant that does not depend on $L$, then for any $\mathbf{t}\in [-1,1]^d, \overline{\mathbf{m}athcal{K}}_{\mathbf{m}athbb{E}q}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght)\to 1$ as $L\to\infty$.
\mathbf{m}athbf{e}nd{lemma}
\begin{proof}
Denote by $\overline{\mathbf{m}athcal{K}}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght)$ the vector that is $\overline{\mathbf{m}athcal{K}}_{\mathbf{m}athbb{E}q}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{s}_{j-1}\mathbf{t}{\textnormal{i}}ght)$ in the $j'th$ coordinate.
Using Corollary {\textnormal{e}}f{cor:cgpk}, let $\mathbf{m}athbb{E}\left[\overline{\mathbf{m}athcal{K}}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght){\textnormal{i}}ght]$ denote the mean of the vector $\overline{\mathbf{m}athcal{K}}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght)$, then by linearity we get:
\[
\mathbf{m}athbb{E}\left[\overline{\mathbf{m}athcal{K}}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght){\textnormal{i}}ght]
= \mathbf{m}athbf{f}rac{1}{1+\alpha^2} \left( \mathbf{m}athbb{E}\left[\overline{\mathbf{m}athcal{K}}^{\left(L-1{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght) {\textnormal{i}}ght]
+ \mathbf{m}athbf{f}rac{\alpha^{2}}{q^2} \mathbf{s}um_{k,k^{'}=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\mathbf{m}athbb{E}\left[\kappa_1\left(\overline{\mathbf{m}athcal{K}}^{\left(L-1{\textnormal{i}}ght)}\left(\mathbf{s}_{k+k^{'}}\mathbf{t}{\textnormal{i}}ght){\textnormal{i}}ght) {\textnormal{i}}ght] {\textnormal{i}}ght),
\]
where we let $\kappa_1$ act point-wise on vectors. Since we can permute $\mathbf{t}$ without chaning the mean (i.e., for any $j$, $\mathbf{m}athbb{E}\left[\overline{\mathbf{m}athcal{K}}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght){\textnormal{i}}ght]=\mathbf{m}athbb{E}\left[\overline{\mathbf{m}athcal{K}}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{s}_j\mathbf{t}{\textnormal{i}}ght){\textnormal{i}}ght]$) we get:
\begin{equation}\label{eq:expec}
\mathbf{m}athbb{E}\left[\overline{\mathbf{m}athcal{K}}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght){\textnormal{i}}ght] = \mathbf{m}athbf{f}rac{1}{1+\alpha^2} \left( \mathbf{m}athbb{E}\left[\overline{\mathbf{m}athcal{K}}^{\left(L-1{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght) {\textnormal{i}}ght]
+ \alpha^{2} \mathbf{m}athbb{E}\left[\kappa_1\left(\overline{\mathbf{m}athcal{K}}^{\left(L-1{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght){\textnormal{i}}ght) {\textnormal{i}}ght] {\textnormal{i}}ght)
\mathbf{m}athbf{e}nd{equation}
\[
= \mathbf{m}athbb{E}\left[\overline{\mathbf{m}athcal{K}}^{\left(L-1{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght) {\textnormal{i}}ght] + \mathbf{m}athbf{f}rac{\alpha^2}{1+\alpha^2} \left(\mathbf{m}athbb{E}\left[\kappa_1\left(\overline{\mathbf{m}athcal{K}}^{\left(L-1{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght){\textnormal{i}}ght) {\textnormal{i}}ght] - \mathbf{m}athbb{E}\left[\overline{\mathbf{m}athcal{K}}^{\left(L-1{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght) {\textnormal{i}}ght] {\textnormal{i}}ght)
\]
\begin{equation} \label{eq:to_al}
\mathbf{m}athbf{g}eq \mathbf{m}athbb{E}\left[\overline{\mathbf{m}athcal{K}}^{\left(L-1{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght) {\textnormal{i}}ght] + \mathbf{m}athbf{f}rac{\alpha^2}{1+\alpha^2} \left(\kappa_1\left(\mathbf{m}athbb{E}\left[\overline{\mathbf{m}athcal{K}}^{\left(L-1{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght){\textnormal{i}}ght]{\textnormal{i}}ght) - \mathbf{m}athbb{E}\left[\overline{\mathbf{m}athcal{K}}^{\left(L-1{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght) {\textnormal{i}}ght] {\textnormal{i}}ght),
\mathbf{m}athbf{e}nd{equation}
where the last inequality is Jensen's inequality (since $\kappa_1$ is convex \citep{daniely2016toward}).
THerefore, let $a_L=\mathbf{m}athbb{E}\left[\overline{\mathbf{m}athcal{K}}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght) {\textnormal{i}}ght]$, we can rewrite \mathbf{m}athbf{e}qref{eq:to_al} as:
\[
a_L \mathbf{m}athbf{g}eq a_{L-1} + \mathbf{m}athbf{f}rac{\alpha^2}{1+\alpha^2} \left(\kappa_1\left(a_{L-1}{\textnormal{i}}ght) - a_{L-1 }{\textnormal{i}}ght).
\]
We therefore need to show that $a_L\to 1$ as $L\to\infty$. Since $a_L$ is monotonically increasing ($\kappa_1(u) > u$ for all $u\in[-1,1]$ \citep{daniely2016toward}) and bounded in $[-1,1]$ it suffices to show that for all $\mathbf{m}athbf{e}psilon > 0$ there exists some $L\in\mathbf{m}athbb{N}\text{ s.t.\ } a_L \mathbf{m}athbf{g}eq 1-\mathbf{m}athbf{e}psilon$. Suppose not, then let $\mathbf{m}athbf{e}psilon>0 \text{ s.t.\ } $ for all $L$, $a_L<1-\mathbf{m}athbf{e}psilon$.
As $\mathbf{m}athbf{f}rac{d}{du}\kappa_1(u)=\kappa_0(u)$ and $\kappa_0(u)\in[0,1]$ \citep{daniely2016toward} then $h(u):=\kappa_1(u)-u$ satisfies $\mathbf{m}athbf{f}rac{d}{du}h(u)=\kappa_0(u)-1\leq0$ with equality iff $u=1$.
Therefore, for any $u\in[-1,1-\mathbf{m}athbf{e}psilon], h(u)\mathbf{m}athbf{g}eq h(1-\mathbf{m}athbf{e}psilon) > 0$ (The $>0$ is because $\kappa_1(u)>u$ for $u\in[-1,1-\mathbf{m}athbf{e}psilon]$). Since we assumed by contradiction that for every $L\in\mathbf{m}athbb{N},a_L<1-\mathbf{m}athbf{e}psilon,$ we get that $h(a_L)\mathbf{m}athbf{g}eq h(1-\mathbf{m}athbf{e}psilon)$ and thus
\[
a_L \mathbf{m}athbf{g}eq a_{L-1} + \mathbf{m}athbf{f}rac{\alpha^2}{1+\alpha^2} h(a_{L-1})
\mathbf{m}athbf{g}eq a_{L-1} + \mathbf{m}athbf{f}rac{\alpha^2}{1+\alpha^2} h(1-\mathbf{m}athbf{e}psilon)
\mathbf{m}athbf{g}eq a_0 + L \mathbf{m}athbf{f}rac{\alpha^2}{1+\alpha^2} h(1-\mathbf{m}athbf{e}psilon)\underset{L\to\infty}{\to}\infty.
\]
However, since $a_L\in[-1,1]$ this leads to a contradiction.
\mathbf{m}athbf{e}nd{proof}
\begin{lemma}\label{lem:k_conv}
Let $\mathbf{m}u^{(i)}(\mathbf{t})=\mathbf{m}athbf{f}rac{1}{d}\mathbf{s}um_{j=1}^{d}\underset{L\text{ times}}{\underbrace{\kappa_1\circ\ldots\circ\kappa_1}}\left(t_j{\textnormal{i}}ght)$, (an average of the entries of $\mathbf{t}$ after $i$ compositions of $\kappa_1$), where $\kappa_1(t)=\mathbf{m}athbf{f}rac{1}{\pi}\left(\mathbf{s}qrt{1-t^2}+\left(\pi-\arccos(t){\textnormal{i}}ght)t{\textnormal{i}}ght)$. Let $\overline{\mathbf{m}athcal{K}}_{\text{CGPK-Tr}}^{(L)}(\mathbf{t})$ be the corresponding CGPK-Tr without skip connections. Then
\[
\overline{\mathbf{m}athcal{K}}_{\text{CGPK-Tr}}^{(L)}(\mathbf{t}) - \overline{\mathbf{m}athcal{K}}_{\text{Tr}}^{(L)}(\mathbf{t})
\mathbf{m}athbf{g}eq \mathbf{s}um_{l=1}^{L}\mathbf{m}athbf{f}rac{\mathbf{m}u^{(l)}(\mathbf{t})-\mathbf{m}u^{(l-1)}(\mathbf{t})}{(1+\alpha^2)^{L-l+1}},
\]
where if $\mathbf{t} \mathbf{n}eq {\bm{e}}c{\bm{1}}$ this quantity is strictly positive.
\mathbf{m}athbf{e}nd{lemma}
\begin{proof}
Let $\boldsymbol{k}_{\mathbf{m}athbb{E}q}^{(L)}(\mathbf{t})$ be the normalized CGPK-EqNet and $\boldsymbol{k}^{(L)}(\mathbf{t})$ be the matrix that is $\boldsymbol{k}_{\mathbf{m}athbb{E}q}^{(L)}(\mathbf{s}_{1+j}\mathbf{t})$ in the $j$ index. Similarly define the matrix $\overline{\mathbf{m}athcal{K}}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght)$ using $\overline{\mathbf{m}athcal{K}}_{\mathbf{m}athbb{E}q}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght)$. For convenience let $\overline{\mathbf{m}athcal{K}}^{\left(0{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght)=\boldsymbol{k}^{\left(0{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght)=\mathbf{t}$.
Note that $\mathbf{m}athbb{E}\left[\boldsymbol{k}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght) {\textnormal{i}}ght]=\overline{\mathbf{m}athcal{K}}_{\text{CGPK-Tr}}^{(L)}(\mathbf{t})$ and $\mathbf{m}athbb{E}\left[\overline{\mathbf{m}athcal{K}}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght) {\textnormal{i}}ght] = \overline{\mathbf{m}athcal{K}}_{\text{Tr}}^{(L)}(\mathbf{t})$.
By equation {\textnormal{e}}f{eq:expec} we have that:
\[
\mathbf{m}athbb{E}\left[\overline{\mathbf{m}athcal{K}}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght) {\textnormal{i}}ght] = \mathbf{m}athbf{f}rac{1}{1+\alpha^2} \left( \mathbf{m}athbb{E}\left[\overline{\mathbf{m}athcal{K}}^{\left(L-1{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght) {\textnormal{i}}ght]
+ \alpha^{2} \mathbf{m}athbb{E}\left[\kappa_1\left(\overline{\mathbf{m}athcal{K}}^{\left(L-1{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght){\textnormal{i}}ght) {\textnormal{i}}ght] {\textnormal{i}}ght),
\]
and similarly it can be readily verified that
\[
\mathbf{m}athbb{E}\left[\boldsymbol{k}^{\left(L{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght) {\textnormal{i}}ght] = \mathbf{m}athbb{E}\left[\kappa_1\left(\boldsymbol{k}^{\left(L-1{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght){\textnormal{i}}ght) {\textnormal{i}}ght].
\]
(Note that the CGPK is naturally normalized so we can omit the bar.) We prove this by induction. For $L=1$ we have:
\[
\mathbf{m}athbb{E}\left[\boldsymbol{k}^{(1)}(\mathbf{t}){\textnormal{i}}ght] - \mathbf{m}athbb{E}\left[\overline{\mathbf{m}athcal{K}}^{(1)}(\mathbf{t}){\textnormal{i}}ght]
= \mathbf{m}athbf{f}rac{1}{1+\alpha^2}\left(\mathbf{m}u^{(1)}-\mathbf{m}u^{(0)}{\textnormal{i}}ght).
\]
Now assume for $L-1\in\mathbf{m}athbb{N}$, then
\[
\mathbf{m}athbb{E}\left[\boldsymbol{k}^{(L)}(\mathbf{t}){\textnormal{i}}ght] - \mathbf{m}athbb{E}\left[\overline{\mathbf{m}athcal{K}}^{(L)}(\mathbf{t}){\textnormal{i}}ght]
= \mathbf{m}athbb{E}\left[\kappa_1\left(\boldsymbol{k}^{\left(L-1{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght){\textnormal{i}}ght) {\textnormal{i}}ght] - \mathbf{m}athbf{f}rac{1}{1+\alpha^2} \left( \mathbf{m}athbb{E}\left[\overline{\mathbf{m}athcal{K}}^{\left(L-1{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght) {\textnormal{i}}ght]
+ \alpha^{2} \mathbf{m}athbb{E}\left[\kappa_1\left(\overline{\mathbf{m}athcal{K}}^{\left(L-1{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght){\textnormal{i}}ght) {\textnormal{i}}ght] {\textnormal{i}}ght).
\]
Since $\kappa_1$ is increasing, using the induction hypothesis we know that $-\alpha^2 \mathbf{m}athbb{E}\left[\kappa_1\left(\overline{\mathbf{m}athcal{K}}^{\left(L-1{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght){\textnormal{i}}ght) {\textnormal{i}}ght] \mathbf{m}athbf{g}eq -\alpha^2 \mathbf{m}athbb{E}\left[\kappa_1\left(\boldsymbol{k}^{\left(L-1{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght){\textnormal{i}}ght) {\textnormal{i}}ght]$, and therefore
\[
\mathbf{m}athbb{E}\left[\boldsymbol{k}^{(L)}(\mathbf{t}){\textnormal{i}}ght] - \mathbf{m}athbb{E}\left[\overline{\mathbf{m}athcal{K}}^{(L)}(\mathbf{t}){\textnormal{i}}ght] \mathbf{m}athbf{g}eq
\]
\[
\mathbf{m}athbf{g}eq \mathbf{m}athbb{E}\left[\kappa_1\left(\boldsymbol{k}^{\left(L-1{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght){\textnormal{i}}ght) {\textnormal{i}}ght] - \mathbf{m}athbf{f}rac{1}{1+\alpha^2} \left( \mathbf{m}athbb{E}\left[\overline{\mathbf{m}athcal{K}}^{\left(L-1{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght) {\textnormal{i}}ght]
+ \alpha^{2} \mathbf{m}athbb{E}\left[\kappa_1\left(\boldsymbol{k}^{\left(L-1{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght){\textnormal{i}}ght) {\textnormal{i}}ght] {\textnormal{i}}ght)
\]
\[
=\mathbf{m}athbf{f}rac{1}{1+\alpha^2}\left(\mathbf{m}athbb{E}\left[\kappa_1\left(\boldsymbol{k}^{\left(L-1{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght){\textnormal{i}}ght) {\textnormal{i}}ght] - \mathbf{m}athbb{E}\left[\overline{\mathbf{m}athcal{K}}^{\left(L-1{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght) {\textnormal{i}}ght] {\textnormal{i}}ght)
\]
\[
= \mathbf{m}athbf{f}rac{1}{1+\alpha^2}\left(\left(\mathbf{m}u^{(L)}-\mathbf{m}u^{(L-1)}{\textnormal{i}}ght)
+ \left(\mathbf{m}athbb{E}\left[\boldsymbol{k}^{(L-1)}(\mathbf{t}){\textnormal{i}}ght]
- \mathbf{m}athbb{E}\left[\overline{\mathbf{m}athcal{K}}^{(L-1)}(\mathbf{t}){\textnormal{i}}ght] {\textnormal{i}}ght) {\textnormal{i}}ght).
\]
Applying the induction hypothesis recursively provides the desired result.
\mathbf{m}athbf{e}nd{proof}
\begin{lemma}[Lemma {\textnormal{e}}f{lemma:dcb} in the main text]
Let $\bm{A}\in\mathbf{m}athbb{R}^{n\times n}$ ($n\mathbf{m}athbf{g}eq 2)$ be a normalized kernel matrix with $\mathbf{s}um_{i\mathbf{n}eq j}\bm{A}_{ij}\mathbf{m}athbf{g}eq 0$. Let $\bm{B}(\bm{A})=\bm{B}_{1,b}$ with $b=\mathbf{m}athbf{f}rac{1}{n(n-1)}\mathbf{s}um_{i\mathbf{n}eq j}\bm{A}_{ij}$ and $\mathbf{m}athbf{e}psilon=\mathbf{s}up_{i}\mathbf{s}um_{j\mathbf{n}eq i}\abs{\bm{A}_{ij}-\bm{B}(\bm{A})_{ij}}$. Then,
\begin{enumerate}
\item ${\textnormal{h}}o\left(\bm{B}(\bm{A}){\textnormal{i}}ght) \leq {\textnormal{h}}o(\bm{A})$.
\item If $\mathbf{m}athbf{e}psilon<\lambda_{\mathbf{m}in}(\bm{B}(\bm{A}))$ then ${\textnormal{h}}o(\bm{A}) \leq \mathbf{m}athbf{f}rac{\lambda_{\mathbf{m}ax}(\bm{B}(\bm{A}))+\mathbf{m}athbf{e}psilon}{\lambda_{\mathbf{m}in}(\bm{B}(\bm{A}))-\mathbf{m}athbf{e}psilon} $,
\mathbf{m}athbf{e}nd{enumerate}
where $\lambda_{\mathbf{m}ax}$ and $\lambda_{\mathbf{m}in}$ denote the maximal and minimal eigenvalues of $\bm{B}(\bm{A})$.
\mathbf{m}athbf{e}nd{lemma}
\begin{proof}
For (1), using \citet{marsli2015bounds}[Theorem 4.4] we have
\[
{\textnormal{h}}o(\bm{A})\mathbf{m}athbf{g}eq \mathbf{m}athbf{f}rac{\mathbf{m}athbf{g}amma_2(\bm{A})}{\mathbf{m}athbf{g}amma_1(\bm{A})},
\]
where
\[
\mathbf{m}athbf{g}amma_1(\bm{A}) = \mathbf{m}in\{\mathbf{m}athbf{f}rac{1}{n}\mathbf{s}um_{i=1}^n \mathbf{s}um_{j=1}^n \bm{A}_{ij} ~~~ , ~~~ \mathbf{m}athbf{f}rac{1}{n}\mathbf{s}um_{i=1}^n \bm{A}_{i,i} - \mathbf{m}athbf{f}rac{1}{n(n-1)}\mathbf{s}um_{i\mathbf{n}eq j} \bm{A}_{ij}\}
\]
\[
\mathbf{m}athbf{g}amma_2(\bm{A}) = \mathbf{m}ax\{\mathbf{m}athbf{f}rac{1}{n}\mathbf{s}um_{i=1}^n \mathbf{s}um_{j=1}^n \bm{A}_{ij} ~~~ , ~~~ \mathbf{m}athbf{f}rac{1}{n}\mathbf{s}um_{i=1}^n \bm{A}_{ii} - \mathbf{m}athbf{f}rac{1}{n(n-1)}\mathbf{s}um_{i\mathbf{n}eq j} \bm{A}_{ij}\}.
\]
By the assumptions that the diagonal entries of $\bm{A}$ are $1$ and that $\mathbf{s}um_{i\mathbf{n}eq j}\bm{A}_{ij}\mathbf{m}athbf{g}eq 0$, we get that $\mathbf{m}athbf{g}amma_1(\bm{A}) = 1 - \mathbf{m}athbf{f}rac{1}{n(n-1)}\mathbf{s}um_{i\mathbf{n}eq j} \bm{A}_{ij}=\lambda_{\mathbf{m}in}(\bm{B}(\bm{A}))$ and $\mathbf{m}athbf{g}amma_2(\bm{A}) = \mathbf{m}athbf{f}rac{1}{n}\mathbf{s}um_{i=1}^n \mathbf{s}um_{j=1}^n \bm{A}_{ij}=\lambda_{\mathbf{m}ax}(\bm{B}(\bm{A}))$. Therefore,
\[
{\textnormal{h}}o(\bm{A})\mathbf{m}athbf{g}eq \mathbf{m}athbf{f}rac{\mathbf{m}athbf{g}amma_2(\bm{A})}{\mathbf{m}athbf{g}amma_1(\bm{A})} = \mathbf{m}athbf{f}rac{\lambda_{\mathbf{m}ax}(\bm{B}(\bm{A}))}{\lambda_{\mathbf{m}in}(\bm{B}(\bm{A}))} = {\textnormal{h}}o(\bm{B}(\bm{A})).
\]
For (2), by the Gershgorin circle theorem, since $\bm{A}-\bm{B}(\bm{A})$ is a matrix with diagonal zero, every eigenvalue $\lambda$ of $\bm{A}-\bm{B}(\bm{A})$ must satisfy $\abs{\lambda}\leq \mathbf{m}athbf{e}psilon$. Since $\bm{A}$ and $\bm{A}-\bm{B}(\bm{A})$ are symmetric, it holds that $\lambda_{\mathbf{m}ax}(\bm{A})\leq \lambda_{\mathbf{m}ax}(\bm{B}(\bm{A})) + \lambda_{\mathbf{m}ax}(\bm{A}-\bm{B}(\bm{A}))$ and $\lambda_{\mathbf{m}in}(\bm{A}) \mathbf{m}athbf{g}eq \lambda_{\mathbf{m}in}(\bm{B}(\bm{A})) + \lambda_{\mathbf{m}in}(\bm{A}-\bm{B}(\bm{A}))$ from which the lemma follows.
\mathbf{m}athbf{e}nd{proof}
\mathbf{s}ection{Infinite Depth Discussion}\label{app:depth_disc}
Lemma {\textnormal{e}}f{lem:sdfc} states that for $u\in\mathbf{m}athbb{R}^d$, letting $\mathbf{t}_1=\mathbf{t}_2=\ldots=u$, we obtain $\overline{\mathbf{m}athcal{K}}^{\left(L{\textnormal{i}}ght)}_{\text{FC}}\left(u{\textnormal{i}}ght)=\overline{\mathbf{m}athcal{K}}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}(\mathbf{t})$, and letting $\boldsymbol{k}^{\left(L{\textnormal{i}}ght)}=\Theta^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}-\mathbf{m}athcal{K}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}$ we obtain $\overline{\Theta}^{\left(L{\textnormal{i}}ght)}_{\text{FC}}\left(u{\textnormal{i}}ght)=\overline{\boldsymbol{k}}^{\left(L{\textnormal{i}}ght)}(\mathbf{t})$, where the fully connected ResNTK and ResGPK are defined in \citet{huang2020deep}.
One may ask why does $\overline{\Theta}^{\left(L{\textnormal{i}}ght)}_{\text{FC}}\left(u{\textnormal{i}}ght)=\overline{\boldsymbol{k}}^{\left(L{\textnormal{i}}ght)}(\mathbf{t})$ and not $\overline{\Theta}^{\left(L{\textnormal{i}}ght)}_{\text{FC}}\left(u{\textnormal{i}}ght)=\overline{\Theta}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{t}{\textnormal{i}}ght)$? This is in fact a consequence of \citet{huang2020deep} not training the last layer (denoted by ${\bm{v}}$ in their paper.) If they were to train the parameters ${\bm{v}}$, the term $\mathbf{m}athbb{E}\left[\left\langle \mathbf{m}athbf{f}rac{\partial f(\mathbf{x};\theta)}{\partial {\bm{v}}} \mathbf{m}athbf{f}rac{\partial f(\mathbf{z};\theta)}{\partial {\bm{v}}} {\textnormal{i}}ght{\textnormal{a}}ngle{\textnormal{i}}ght]$ would be added to their ResNTK expression. But this term is exactly equal to the ResGPK. Therefore, training the last layer adds the ResCGPK to the ResNTK expression. This is indeed confirmed in \citep{tirer2022kernel}, who derived ResNTK when the last layer is trained.
So if the last layer is trained, we would have $\overline{\Theta}^{\left(L{\textnormal{i}}ght)}_{\text{FC}}\left(u{\textnormal{i}}ght)=\overline{\Theta}^{\left(L{\textnormal{i}}ght)}_{\mathbf{m}athbb{E}q}\left(\mathbf{t}{\textnormal{i}}ght)$, and thus Theorem {\textnormal{e}}f{thm:decay} would imply that $\overline{\Theta}^{\left(L{\textnormal{i}}ght)}_{\text{FC}}\left(u{\textnormal{i}}ght)\underset{L\to\infty}{\longrightarrow}u$. Intuitively, this happens because the term $u$ exists in the ResGPK, and is the only term that is not multiplies by $\alpha$. So if $\alpha$ decays quickly enough, $u$ becomes the dominant term.
Instead, by eliminating the ResGPK from the ResNTK expression, all the terms are multiplies by $\alpha$. So after normalizing, the two layer ResNTK becomes equivalent to the two layer FC-NTK \citep{belfer2021spectral}.
If we were to not train the last layer, we would have a similar result, where the resulting kernel would correspond to a 2 layer CNTK. We give here a sketch proof (the details are analogous to \citet{belfer2021spectral}). Theorem {\textnormal{e}}f{thm:1} states that
\[
\Sigma_{j,j^{'}}^{\left(2{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght)=\mathbf{m}athbf{f}rac{c_w}{q}\text{tr}\left(\Sigma_{\mathbf{m}athcal{D}_{j,j^{'}}}^{\left(1{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght){\textnormal{i}}ght) + \mathbf{m}athbf{f}rac{\alpha^{2}}{q^{2}}\mathbf{s}um_{k=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\text{tr}\left(K_{\mathbf{m}athcal{D}_{j+k,j^{'}+k}}^{\left(1{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght){\textnormal{i}}ght).
\]
For $3\leq l \leq L$,
\[
\Sigma_{j,j^{'}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght)=\Sigma_{j,j^{'}}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght)+\mathbf{m}athbf{f}rac{\alpha^{2}}{q^{2}}\mathbf{s}um_{k=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\text{tr}\left(K_{\mathbf{m}athcal{D}_{j+k,j^{'}+k}}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght){\textnormal{i}}ght).
\]
So for $\mathbf{m}athbf{g}amma=L^{-\mathbf{m}athbf{g}amma}$ we have that for all $l\mathbf{m}athbf{g}eq 3$,
\[
\abs{\Sigma^{(l)}_{j,j^{'}}(\mathbf{t}) - \Sigma^{(l-1)}_{j,j^{'}}(\mathbf{t})} \leq \alpha^2 \abs{\mathbf{s}um_{k=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}\text{tr}\left(K_{\mathbf{m}athcal{D}_{j+k,j^{'}+k}}^{\left(l-1{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght){\textnormal{i}}ght)} \leq \alpha^2.
\]
So $\abs{\Sigma^{(l)}_{j,j^{'}}(\mathbf{t}) - \Sigma^{(2)}_{j,j^{'}}(\mathbf{t})} \leq L\cdot \alpha^2=L^{1-2\mathbf{m}athbf{g}amma}$.
In turn we also have $\abs{K^{(l)}_{j,j^{'}}(\mathbf{t}) - K^{(2)}_{j,j^{'}}(\mathbf{t})}=L^{1-2\mathbf{m}athbf{g}amma}$ and $\abs{\dot{K}^{(l)}_{j,j^{'}}(\mathbf{t}) - \dot{K}^{(2)}_{j,j^{'}}(\mathbf{t})}=L^{1-2\mathbf{m}athbf{g}amma}$.
Furthermore, by Theorem {\textnormal{e}}f{thm:2} we have:
\[
\abs{P^{(l+1)}_{j}(\mathbf{t})-P^{(l)}_{j}(\mathbf{t})} \leq \abs{\mathbf{m}athbf{f}rac{\alpha^2}{q^2}\text{tr}\left(
\left(\mathbf{s}um_{p=\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}P_{\mathbf{m}athcal{D}_{j+p}}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght){\textnormal{i}}ght)\odot\dot{K}_{\mathbf{m}athcal{D}_{j,j}}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght){\textnormal{i}}ght)}
\]
\[
\leq \alpha^2 \abs{\mathbf{m}athbf{f}rac{1}{q^2}\mathbf{s}um_{p=\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}P_{\mathbf{m}athcal{D}_{j+p}}^{\left(l+1{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght)}
\leq \alpha^2 \abs{P_{1}^{\left(l+1{\textnormal{i}}ght)}\left({\bm{e}}c{\bm{1}}{\textnormal{i}}ght)}
\leq \alpha^2 (1+\alpha^2)^{L-l}.
\]
Now because it holds that:
\[
\alpha^2 (1+\alpha^2)^{L-l}=(1+\mathbf{m}athbf{f}rac{1}{L^{2\mathbf{m}athbf{g}amma}})^{L-l}\leq (1+\mathbf{m}athbf{f}rac{1}{L^{2\mathbf{m}athbf{g}amma}})^{L^{2\mathbf{m}athbf{g}amma}\cdot{L^{1-2\mathbf{m}athbf{g}amma}}}\leq e^{L^{1-2\mathbf{m}athbf{g}amma}},
\]
Since $P^{(L)}_{j}(\mathbf{t})={\bm{e}}c{\bm{1}}_{j=1}$ we analogously get $\abs{{\bm{e}}c{\bm{1}}_{j=1}-P^{(2)}_{j}(\mathbf{t})}\leq L^{1-2\mathbf{m}athbf{g}amma} \cdot e^{L^{1-2\mathbf{m}athbf{g}amma}} \mathbf{m}athcal{O}(L^{1-2\mathbf{m}athbf{g}amma})$.
Now recall that
\[
\boldsymbol{k}^{(L)}(\mathbf{t}) = \mathbf{m}athbf{f}rac{\alpha^{2}}{q}\mathbf{s}um_{l=1}^{L}\underset{\text{Denote by }\boldsymbol{k}^{(L,l)}(\mathbf{t})}{\underbrace{\mathbf{s}um_{p=1}^{ d }P_{p}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght)
\left(\text{tr}\left(\dot{K}_{\mathbf{m}athcal{D}_{p,p}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght)\odot\Sigma_{\mathbf{m}athcal{D}_{p,p}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght) + K_{\mathbf{m}athcal{D}_{p,p}}^{\left(l{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght) {\textnormal{i}}ght) {\textnormal{i}}ght)}}\]
where by \mathbf{m}athbf{e}qref{lem:norm_pi} we know that $\boldsymbol{k}^{(L)}({\bm{e}}c{\bm{1}})=\alpha^2\mathbf{m}athbf{f}rac{2}{C_0}L(1+\alpha^2)^{L-1}$.
Normalizing the kernel implies
\[\overline{\boldsymbol{k}}^{(L)}(\mathbf{t}) = \mathbf{m}athbf{f}rac{\boldsymbol{k}^{(L)}(\mathbf{t})}{\boldsymbol{k}^{(L)}({\bm{e}}c{\bm{1}})} \approx \boldsymbol{k}^{(L,2)}(\mathbf{t})
\approx \mathbf{m}athbf{f}rac{1}{C}\text{tr}\left(\dot{K}_{\mathbf{m}athcal{D}_{1,1}}^{\left(2{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght)\odot\Sigma_{\mathbf{m}athcal{D}_{1,1}}^{\left(2{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght) + K_{\mathbf{m}athcal{D}_{1,1}}^{\left(2{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght) {\textnormal{i}}ght),
\]
where $C$ is some normalizing constant (Note that after normalizing, $\boldsymbol{k}^{(L,1)}(\mathbf{t})$ becomes negligible.)
For such $\alpha$, $\overline{\Sigma}_{j,j^{'}}^{(2)}(\mathbf{t})\underset{L\to\infty}{\longrightarrow}\overline{\mathbf{m}athbf{f}rac{1}{q}\text{tr}\left(\Sigma_{\mathbf{m}athcal{D}_{1,1}}(\mathbf{t}){\textnormal{i}}ght)}$.
As such, after normalizing, in the infinite depth limit, the expression $\text{tr}\left(\dot{K}_{\mathbf{m}athcal{D}_{1,1}}^{\left(2{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght)\odot\Sigma_{\mathbf{m}athcal{D}_{1,1}}^{\left(2{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght) + K_{\mathbf{m}athcal{D}_{1,1}}^{\left(2{\textnormal{i}}ght)}\left(\mathbf{t}{\textnormal{i}}ght) {\textnormal{i}}ght)$ becomes the two layer CGPK (aka one hidden layer, denoted by $L=1$) with inputs $\mathbf{m}athbf{h}at{\mathbf{t}}$ where $\mathbf{m}athbf{h}at{t}_i= \mathbf{m}athbf{f}rac{1}{q}\mathbf{s}um_{k=-\mathbf{m}athbf{f}rac{q-1}{2}}^{\mathbf{m}athbf{f}rac{q-1}{2}}t_{i+k}$.
\mathbf{s}ection{Eigenvalue Decay Experiment}\label{app:exp_eig_decay}
We use \citep{geifman2022spectral}[Lemma A.6] to numerically compute the eigenvalues. Namely, for each frequency in Figure {\textnormal{e}}f{fig:eigen} we compute the Gegenbaur polynomials and the kernel, and numerically integrate. Note that as this integration requires evaluating the kernel many times, we are limited to $d=4$ and $L=3$. To prevent the receptive field from being much larger than $d$, and in order to better match the CGPK expression from \citep{geifman2022spectral}, we slightly modify the ResCGPK to include one convolution in every layer instead of two, where the layer ends after the ReLU. Thus the kernel computed is with $d=4,q=2,\alpha=1$ is:
\[
\boldsymbol{k}_{0}(\mathbf{t})=\mathbf{m}athbf{f}rac{1}{1+\beta}(\beta t_i+ \kappa_1(t_i))
\]
\[
\boldsymbol{k}_{i}(\mathbf{t})=\mathbf{m}athbf{f}rac{1}{1+\beta}\left(\beta t_i + \kappa_1\left(\mathbf{m}athbf{f}rac{1}{2}(t_{i}+t_{i+1}){\textnormal{i}}ght){\textnormal{i}}ght),
\]
where $\beta=0$ is the CGPK from \citep{geifman2022spectral} and $\beta=1$ is the modified ResCGPK.
\mathbf{m}athbf{e}nd{document}
|
\begin{document}
\date{}
\title
{An analytic derivation of the variance for the Abelian distribution}
\author{Anirban Das, \\
Department of Mathematics, Pennsylvania State University\\
[email protected]
}
\title{Modeling Avalanches of Neurons}
\begin{abstract}{ The Abelian distribution has been studied recently in models for neural avalanches. This paper uncovers new properties about the distribution, ways in which these properties can be useful are indicated.}
\end{abstract}
{\tiny AMS Classification: 60E05, 92C20, 05A10, 62E15. }
{\tiny Key words: Abelian distribution, neural avalanches, discrete distributions.}
\section{Introduction}
The Abelian distribution is a distribution that is in important in models studying neural avalanches (See \cite{eurich2002finite}, \cite{levina2008mathematical} \cite{levina2014abelian}). Neural avalanches were observed by John Beggs and
Dietmar Plenz (\cite{beggs2003neuronal}, \cite{beggs2004neuronal}). In the experiment cultured slices were planted on a multielectrode array and local field potential signals were recorded. The data consisted of short intervals of activity, when one or more electrodes detected LFPs above the threshold, separated by longer periods of silence. A set of such consecutively
active frames was called an avalanche. The size of an avalanche is defined as
the number of electrodes which were active during the avalanche. The data collected showed that avalanche sizes followed the power-law distribution (with exponent $-\frac{3}{2}$) with the exponential cutoff at the
size of the multielectrode array. Neuronal avalanches have also been recently identified in vivo in the normalized LFPs extracted from ongoing activity in awake macaque monkeys(\cite{petermann2009spontaneous}). A model for such neural avalanches was studied in
\cite{eurich2002finite}, in the model the probability distribution of getting avalanches of size $Z_{N,p}$, was derived as $$P(Z_{N,p} = b)= C_{N,p}{{N-1}\choose{b-1}}p^{b-1}(1-bp)^{(N-b-1)}b^{b-2}$$, where $C_{N,p}$ is the normalization constant defined by
$C_{N,p}= \frac{1-Np}{1-(N-1)p} .$ This is the Abelian distribution. The parameter $N$ is the number of neurons, the parameter $p= \frac{\alpha}{N}$, where $\alpha \in (0,1)$ captures the amount of dissipation in the system.
In \cite{denker2014ergodicity} a model for studying avalanches in dynamical systems was constructed, and a similar distribution was derived, we shall call it the Avalanche distribution. In \cite{levina2008mathematical}, the Mean of the Abelian distribution was calculated. This paper calculates the variance of the Abelian distribution analytically. We shall use properties of the Avalanche distribution, for doing this. This underlines the close relationship between the two distributions. At the end of \cite{denker2014ergodicity}, there is a note by Wenbo Li(\cite{WenboVLi}), where Li states some results (without proof) and claims to have found a way for calculating the variance(does not give an explicit formula) of the Abelian distribution. Unfortunately Dr Li died before the results could be published. Here we prove some of Dr Li's claims, and use them to calculate the Variance of the Abelian distribution. Also we show what happens to the variance as the parameter $N$(\ref{def_ab_dist}) goes to $ + \infty$.
The first Section, deals with the calculation of the Variance for a given $N$ and $p$. the second section introduces the Stirling numbers. Some properties of the Stirling numbers are derived. These will be used in Section 3, to study how the variance of the distribution behaves as the parameter $N$ goes to infinity. Finally he Appendix contains some technical proofs that have been excluded from the main text.
\section{The variance of the Abelian distribution}
\begin{defn}{ \textbf{ Abelian Distribution }}{\label{def_ab_dist}}
\\ The Abelian distribution $Z_{N,p}$ is a probability distribution on $\{1,2, \cdots ,N\}$defined by the probability density
$$P(Z_{N,p} = b)= C_{N,p}{{N-1}\choose{b-1}}p^{b-1}(1-bp)^{(N-b-1)}b^{b-2}$$ Where $C_{N,p}$ is the normalization constant defined by
$C_{N,p}= \frac{1-Np}{1-(N-1)p} .$ The parameter $N$ must be an integer, the parameter $p$ lies in $(0,\frac{1}{N})$.
\end{defn}
That this is indeed is a distribution was proved in \cite{levina2008mathematical}, see also \cite{levina2014abelian}. The $p$ in the Abelian distribution is often taken as $\frac{\alpha}{N}$, where $0<\alpha < 1$. It was also proved in \cite{levina2008mathematical}, \cite{levina2014abelian} that :
\begin{lem}\label{ab_mean}
$ E(Z_{N,\frac{\alpha}{N}})= \frac{N}{N-(N-1)\alpha}$.
\end{lem}
In \cite{denker2014ergodicity} we find the following distribution:
\begin{defn}{ \textbf{ Avalanche Distribution }}
The Avalanche distribution $X_{N,p}$ is a probability distribution on $\{0,1,2, \cdots ,N\}$ defined by the probability density
$$P(X_{N,p} = b)= {{N}\choose{b}}p^{b}(1-(b+1)p)^{N-b}(b+1)^{b-1}.$$The parameter $N$ must be an integer, the parameter $p$ lies in $(0,\frac{1}{N})$.
\end{defn}
Wenbo Li in \cite{WenboVLi} states without proof the following result about the mean of the Avalanche distribution. We attach a proof of the statement in the Appendix.
\begin{lem}{\label{av_mean}}
$ E(X_{N,p})= \sum\limits_{i=1}^{N} \frac{N !}{(N-i)!} p^i.$
\end{lem}
Also define $Y_{N,p} \; = \; X_{N,p}+1$.
Thus
\begin{eqnarray*}
P(Y_{N,p} = b) &=& P(X_{N,p} = b-1)\\
& =& {{N}\choose{b-1}}p^{b-1}(1-bp)^{(N-b+1)}b^{b-2}.
\end{eqnarray*}
With these results at hand we are ready to find the Variance of $Z_{N,p}$
\begin{thm}
The second moment of the Abelian distribution is as follows:
\begin{equation}\label{eq2_2}
E({Z_{N,p}}^{2})= \frac{{C_{N,p}}}{p}[\frac{1}{1-Np}-1-\sum\limits_{i=1}^{N-1} {\frac{(N-1)!}{(N-1-i)!}p^i}]
\end{equation}
And the variance of the distribution is
\begin{equation}
V(Z_{N,p})=\frac{{C_{N,p}}}{p}[\frac{1}{1-Np}-1-\sum\limits_{i=1}^{N-1} {\frac{(N-1)!}{(N-1-i)!}p^i}] - {(\frac{N}{N-(N-1)\alpha})}^{2}.
\end{equation}
\end{thm}
\begin{proof}
\begin{eqnarray*}
E(Y_{N,p}) &=& \sum\limits_{b=1}^{N+1} b^{b-1} {{N}\choose{b-1}}p^{b-1}(1-bp)^{(N-b+1)} \\
&=& \sum\limits_{b=1}^{N+1} b^{b-1} {{N}\choose{b-1}}p^{b-1}(1-bp)^{(N-b)} - \sum\limits_{b=1}^{N+1} p \times b^{b} {{N}\choose{b-1}}p^{b-1}(1-bp)^{(N-b)} \\
&=& \frac{1}{C_{N+1,p}} E(Z_{N+1,p})- p \frac{1}{{C_{N+1,p}}}E({Z_{N+1,p}}^2)
\end{eqnarray*}
We know $E(Z_{N+1,p})$ from \ref{ab_mean},
using \ref{av_mean} we can compute $E(Y_{N,p})$.
Using the above two facts one finds $ E({Z_{N,p}}^{2})$. Since $V(Z_{N,p})= E({Z_{N,p}}^{2}) - {E(Z_{N,p})}^{2} $ , the variance too can be found from this.\\
\end{proof}
\section{Stirling numbers}
Our chief goal , for the rest of this paper will be to find how the variance of the Abelian distribution behaves as $N$ goes to infinity. In order to do this we shall use the Stirling number of the first kind. The Stirling numbers were so named by N. Nielson (1906) in honor of James Stirling, who introduced them in his Methodus Differentialis (1730) \cite{tweddle2012james}, without using any notation for them. The notation in this paper is due to J. Riordan \cite{riordan2012introduction}. This section gives some definitions, and results from \cite{charalambides2005combinatorial}. We then proceed to state and prove a few Lemmas of our own \ref{lemma_J_5_1}, \ref{lemma_J_5_2}, \ref{lemma_J_6}. These final three will be used in the section titled Asymptotic behavior of the Variance of the Abelian Distribution.\\
We will use the notation$(x)_{n}$ for the polynomial $x(x-1)(x-2) \cdots (x-(n-1))$. This is called the factorial moment of order $n$.
The coefficients of such polynomials are called the Stirling numbers. Formally we have,
$(x-r)_{i}= \sum\limits_{j=0}^{i} s(i,j;r) x^{j} $. Set $s(0,0;r)=1$. For $i\ge j \ge 0$, $s(i,j;r)$ are called the non- centered Stirling numbers of the first kind.
We will be chiefly interested in r= 1, when
$(x-1)_{i}= \sum\limits_{j=0}^{i} s(i,j;1) x^{j}$. When $i \ge j >0$ denote by ${\tau^{i}}_{j}$ the class of all possible subsets of $\{1,2,3 \cdots i \}$, which are of cardinality $j$.
The following can be found in Chapter 2 of \cite{charalambides2005combinatorial}.
\begin{equation}\label{eq2_4}
| s(i,j;1)| = (-1)^{i-j} s(i,j;1)
\end{equation}
\begin{equation}\label{eq2_5}
(x+i)_{i}= \sum\limits_{j=0}^{i} |s(i,j;1)| x^{j}
\end{equation}
\begin{equation}\label{eq2_7}
s(i,i;1)=1
\end{equation}
\begin{equation}\label{eq2_8}
s(i,i-1;1)=-\frac{i(i+1)}{2}
\end{equation}
\begin{equation}\label{eq2_6}
|s(i,j;1)|= i! \sum _{\{r_1,r_2, \cdots , r_j \} \in {\tau^{i}}_{j}} \frac{1}{r_1 r_2 \cdots r_j} \text{, This holds for }i \ge j >0
\end{equation}
Now we make some definitions, and prove some results of interest to us.
\begin{defn} \label{definition_5_4}
Given positive integer $i$. $P_i$ is a polynomial of degree $i (i\ge 0)$ defined as $P_i(x)= \sum\limits_{j=0}^{i}s(i+2,j;1)x^{j}$.
$h_i$ is a polynomial of degree $i+2$ defined as $h_i(x)= x^{i+1}(\frac{(i+2)(i+3)}{2}-x)$.
\end{defn}
\begin{lem}\label{lemma_J_6}
$i, N$ be positive integers, $N-3 \ge i$. Then
$$
P_i(N)={(N-1)}_{i+2}+h_i(N)
$$
Further when $N-3 \ge i \ge \sqrt{2N}$, $h_i(N) >0$, and also $2N^{i+3}>P_i(N)>{(N-1)}_{i+2} \ge 0 $
\end{lem}
\begin{proof}
The proof is a straightforward calculation.
\end{proof}
\begin{lem}\label{lemma_J_5_1}
There exists a polynomial $f(x)$, of degree $4$, such that for all integers with $i,j$ with $i \ge j \ge 0$, we have $f(i) \ge 0$ and
\begin{equation}
|s(i+2,j;1)| \leq |s(i,j;1)|f(i).
\end{equation}
\end{lem}
\begin{proof}
See Appendix for Proof
\end{proof}
Before ending the section we will state one last technical Lemma that will find use in the next section.
\begin{lem}\label{lemma_J_5_2}
When $0 \le i < \sqrt{2N}$, $\prod_{j=1}^{i}(1+\frac{j}{N}) \le e^{2}$.
\end{lem}
\begin{proof}
It would be enough to show $\sum_{j=1}^{i}\ln (1+\frac{j}{N}) \le 2$. This is exactly what we do.
\begin{align*}
\sum_{j=1}^{i}\ln (1+\frac{j}{N}) &\le \sum_{j=1}^{i} \frac{j}{N} && \text{(Using the fact } \ln (1+x) \le x, \text {when} \;x >-1)\\
&= \frac{1}{2N} \times(i)(i+1) \le 2
\end{align*}
\end{proof}
\section{Asymptotic behavior of the variance of the Abelian Distribution}
We will see how the variance of $Z_{N,p}$ behaves as $N$ tends to infinity, here we take $p=\frac {\alpha}{N}$. Here is the result.
\begin{thm}
$Z_{N,p}$ be the Abelian distribution with parametres $p$ and $N$. For $ 0 <\alpha < 1$, $\lim_{N \to +\infty} V(Z_{N, \frac{\alpha}{N}})= \frac{\alpha}{(1- \alpha)^{3}} $
\end{thm}
\begin{proof}
Let $p= \frac{\alpha}{N}$. Restating \ref{eq2_2} we get
$$
E({Z_{N,p}}^{2})= \frac{{C_{N,p}}}{p}[\frac{1}{1-Np}-1-\sum\limits_{i=1}^{N-1} (N-1)_{i}p^i]
$$
The fact that $s(i,i,1)=1$ is used in the following calculations
\begin{eqnarray*}
E({Z_{N,\frac{\alpha}{N}}}^{2}) &=& \frac{{C_{N,p}}}{p}[\frac{1}{1-Np}-1-\sum\limits_{i=1}^{N-1} p^i\sum\limits_{j=0}^{i}s(i,j;1)N^{j}]\\
&=& \frac{{C_{N,p}}}{p}[\sum\limits_{i=1}^{\infty}(Np)^i-\sum\limits_{i=1}^{N-1} p^i\sum\limits_{j=0}^{i}s(i,j;1)N^{j}] \\
&=& \frac{{C_{N,p}}}{p}[\sum\limits_{i=N}^{\infty}(Np)^i-\sum\limits_{i=1}^{N-1} p^i\sum\limits_{j=0}^{i-1}s(i,j;1)N^{j}] \\
&=& \frac{{C_{N,p}}}{p}[\frac{{\alpha}^N}{1-\alpha}-\sum\limits_{i=1}^{N-1} p^i\sum\limits_{j=0}^{i-1}s(i,j;1)N^{j}]
\end{eqnarray*}
Hence we have
\begin{equation}\label{eq2_10}
E({Z_{N,\frac{\alpha}{N}}}^{2}) = C_{N,p}[J_1- J_2]
\end{equation}
where $J_1= \frac{{\alpha}^N}{\frac{\alpha}{N}(1-\alpha)}$, $J_2 = \frac{1}{p}\sum\limits_{i=1}^{N-1} p^i\sum\limits_{j=0}^{i-1}s(i,j;1)N^{j} $\\
We easily observe
\begin{eqnarray*}
\lim_{N \to +\infty}C_{N,p}=lim_{N \to +\infty}C_{N,\frac{\alpha}{N}} =1, \;
\lim_{N \to +\infty}J_1= 0.
\end{eqnarray*}
Now to focus on $J_2$
\begin{eqnarray*}
J_2 &=& \frac{1}{p}p\sum\limits_{i=0}^{N-2} p^i\sum\limits_{j=0}^{i}s(i+1,j;1)N^{j}\\
&=& \sum\limits_{i=0}^{N-2} p^i\sum\limits_{j=0}^{i}s(i+1,j;1)N^{j} \\
&=& \sum\limits_{i=0}^{N-2}p^{i}N^{i}s(i+1,i;1) \; \;+\; \; \sum\limits_{i=1}^{N-2} p^i\sum\limits_{j=0}^{i-1}s(i+1,j;1)N^{j}
\end{eqnarray*}
So we have
\begin{equation}\label{eq2_11}
J_2=J_3+ J_4
\end{equation}
where $J_3=\sum\limits_{i=0}^{N-2}p^{i}N^{i}s(i+1,i;1) $, and $J_4=\sum\limits_{i=1}^{N-2} p^i\sum\limits_{j=0}^{i-1}s(i+1,j;1)N^{j} $\\
\begin{eqnarray*}
J_3 &=& \sum\limits_{i=0}^{N-2} {\alpha}^{i}s(i+1,i;1)
= - \sum\limits_{i=0}^{N-2} {\alpha}^{i}\frac{(i+1)(i+2)}{2}
\end{eqnarray*}
This yields
$$
lim_{N \to +\infty}J_3= -\frac{1}{(1-\alpha)^{3}}
$$
\begin{eqnarray*}
J_4 &=& \sum\limits_{i=1}^{N-2} p^i\sum\limits_{j=0}^{i-1}s(i+1,j;1)N^{j}\\
&=& p \sum\limits_{i=0}^{N-3} p^i\sum\limits_{j=0}^{i}s(i+2,j;1)N^{j}\\
&=& p \sum\limits_{i=0}^{\sqrt{2N}-1} p^i\sum\limits_{j=0}^{i}s(i+2,j;1)N^{j} +
p \sum\limits_{i=\sqrt{2N}}^{N-3} p^i\sum\limits_{j=0}^{i}s(i+2,j;1)N^{j}\\
&=& J_5+ J_6,
\end{eqnarray*}
where $J_5=\sum\limits_{i=0}^{\sqrt{2N}-1} p^i\sum\limits_{j=0}^{i}s(i+2,j;1)N^{j}$ and $J_6=p \sum\limits_{i=\sqrt{2N}}^{N-3} p^i\sum\limits_{j=0}^{i}s(i+2,j;1)N^{j} $.
\begin{description}
\item[We next show $lim_{N \to +\infty} J_5 =0, f$ below is defined in \ref{lemma_J_5_1}.]
\begin{align*}
|J_5| &= |p(\sum\limits_{i=0}^{\sqrt{2N}-1} p^i\sum\limits_{j=0}^{i}s(i+2,j;1)N^{j})|\\
&\le p(\sum\limits_{i=0}^{\sqrt{2N}-1} p^i\sum\limits_{j=0}^{i}|s(i+2,j;1)|N^{j})\\
&\le p(\sum\limits_{i=0}^{\sqrt{2N}-1} p^i\sum\limits_{j=0}^{i}f(i)|s(i,j;1)|N^{j}) \\
&\le p(\sum\limits_{i=0}^{\sqrt{2N}-1} p^if(i)\prod_{j=1}^{i}(N+j))\\
&\le \frac{\alpha}{N}(\sum\limits_{i=0}^{\sqrt{2N}-1} \alpha^if(i)\prod_{j=1}^{i}(1+\frac{j}{N}))&& \text{(Putting } p =\frac{\alpha}{N})\\
&\le \frac{\alpha}{N}(\sum\limits_{i=0}^{\sqrt{2N}-1} \alpha^if(i)e^{2})
\end{align*}
Since the last expression is an upper bound for $|J_5|$, and it approaches zero as $N$ approaches $\infty$, we are done.
\item[Next we will show $lim_{N \to +\infty} J_6 =0 $, the $P_i$ below is defined in \ref{lemma_J_6}]
\begin{align*}
|J_6| &= |p(\sum\limits_{i=\sqrt{2N}}^{N-3} p^i P_i(N))|\\
&= p(\sum\limits_{i=\sqrt{2N}}^{N-3} p^i P_i(N))\\
&\le p(\sum\limits_{i=\sqrt{2N}}^{N-3} p^i 2 N^{i+3} )\\
&= \frac{\alpha}{N}(\sum\limits_{i=\sqrt{2N}}^{N-3} \alpha^i 2 N^{3} )&& \text{(Putting } p =\frac{\alpha}{N})\\
&= 2N^{2} \alpha (\sum\limits_{i=\sqrt{2N}}^{N-3} \alpha^i )
= 2N^{2} \alpha^{\sqrt{2N}+1} \frac{1-\alpha^{(N-3)-2\sqrt{N}}}{1-\alpha}.
\end{align*}
Since the last expression is an upper bound for $|J_6|$, and it approaches zero as $N$ approaches $\infty$, we are done.
\end{description}
From (\ref{eq2_10}), and the estimations of $J_1,\; J_3, \; J_4$,
we see
$$
lim_{N \to +\infty} E({Z_{N,\frac{\alpha}{N}}}^{2})= (1-\alpha)^{-3}
$$
Also
$$
lim_{N \to +\infty} {E(Z_{N,\frac{\alpha}{N}})}^{2}= lim_{N \to +\infty} \frac{1}{1-\frac{(N-1)\alpha}{N}}= \frac{1}{(1-\alpha)^2}
$$
From this the theorem follows.
\end{proof}
\section{Remarks}
The choice of $\alpha$ which is most interesting for studying the biological phenomenon, are those values close to $1$, these are the values for which the distribution behaves closest to a power law as has been show in \cite{eurich2002finite}. Also Levina \cite{levina2007dynamical} shows these are the values of $\alpha$ the system settles to if one starts with dynamical synapses with a different suitable $\alpha$. This result shows that such systems have very high variance when we deal with a lot of neurons. This is consistent with a power law distribution of exponent $-\frac{3}{2}$ as observed in the experiments of \cite{beggs2004neuronal}, \cite{beggs2003neuronal}. An explicit expression for Variance , as has been found here, often proves useful for inferring details about parameters from available data.
\begin{appendices}
\section{Methods}
Here, we give proofs of certain lemmas, that we omitted from the main text.
\begin{proof}{ of \ref{av_mean}}
Let $(U_{i})_{i=1}^{N}$ be i.i.d uniformly distributed in $[0,1]$, $N$ is a fixed positive integers, $p$ is a fixed number in $(0,\frac{1}{N})$ . From this one may recursively construct the random sequences $(\epsilon_{i,N})_{i=1}^{N}$ as follows
\begin{align*}
\epsilon_{0,N}= 1 , \;
\epsilon_{1,N}=\sum\limits_{j=1}^{N} \mathbbm{1}_{[1-p,1]}(U_j) \\
\epsilon_{k,N}= \sum\limits_{j=1}^{N}\mathbbm{1}_{[1-p\sum\limits_{i=0}^{k-1} \epsilon_{i,N}, \; 1-p\sum\limits_{i=0}^{k-2} \epsilon_{i,N} ]} (U_j),\;
S_{N,p} = \sum\limits_{i=1}^{N} \epsilon_{i,N}
\end{align*}
It was shown in \cite{denker2014ergodicity} that $S_{N,p}$ has the same distribution as the Avalanche distribution.
Thus it is suffice to prove that
$ E(\epsilon_{k,N})= \frac{N !}{(N-k)!} p^k, \; \; \forall k \geq 1$, \\ We do so by induction for $k=1$,
\begin{eqnarray*}
E(\epsilon_{1,N}) &=& \sum\limits_{i=1}^{N} P(U_i > 1-p)
= \sum\limits_{i=1}^{N} p
= Np
\end{eqnarray*}
By Inductive hypothesis the result holds for $k=k-1$, now for $k=k$,
\begin{eqnarray*}
E(\epsilon_{k,N}) &=& \sum\limits_{i=1}^{N} P(p\sum\limits_{m=0}^{k-1} \epsilon_{m,N} \ge 1-U_i \ge p\sum\limits_{m=0}^{k-2}\epsilon_{m,N}) \\
& =& \sum\limits_{i=1}^{N} \sum\limits_{j=1}^{N-1} jp \; P(\epsilon_{k-1,N-1}= j)\\
& =& Np \sum\limits_{j=1}^{N-1} j P(\epsilon_{k-1,N-1}= j)\;\; \; \; \; \; \; \textbf{ [Taking conditions on value of $\epsilon_{k-1,N}$}]\\
& =& Np \frac{N-1 !}{(N-k )!} p^{k-1} \;\; \; \; \; \; \; \; \; \; \;\; \; \; \; \; \;\; \; \; \; \textbf{[By Inductive Hypothesis]}\\
& =& \frac{N!}{(N-k)!} p^{k}.
\end{eqnarray*}
\end{proof}
\begin{proof}{of \ref{lemma_J_5_1}}
For the moment , consider $i \ge j > 0$, the situation where $i \ge j=0$, will be treated at the end separately.
Using Equation ~\ref{eq2_6}, we get
\begin{eqnarray*}
|s(i+2,j;1)| & = & (i+2)! \sum _{\{r_1,r_2, \cdots , r_j \} \in {\tau^{i}}_{j}} \frac{1}{r_1 r_2 \cdots r_j} \\
& &+ (i+2)!\sum _{\{r_1,r_2, \cdots , r_{j-1} \} \in {\tau^{i}}_{j-1}} \frac{1}{(i+1)r_1 r_2 \cdots r_{j-1}} \\
& & + (i+2)! \sum _{\{r_1,r_2, \cdots , r_{j-1} \} \in {\tau^{i}}_{j-1}} \frac{1}{(i+2)r_1 r_2 \cdots r_{j-1}}\\
& & + (i+2)!\sum _{\{r_1,r_2, \cdots , r_{j-2} \} \in {\tau^{i}}_{j-2}} \frac{1}{(i+1)(i+2)r_1 r_2 \cdots r_{j-2}}\label{initial equation}
\end{eqnarray*}
Now for $i \ge j >0$ consider the function $F_{i.j}:\tau^{i}_{j-1}\rightarrow \tau^{i}_{j}$ ($F_{i,j}$ is a function which takes sets to sets)defined as
$$
F_{i,j}(\{ r_1,r_2, \cdots , r_{j-1} \})= \{ l, r_1,r_2, \cdots , r_{j-1} \}
$$
where $l$ is the least number in $\{ 1,2, \cdots , i \}$, which is not in $\{ r_1,r_2, \cdots , r_{j-1} \}$
\begin{equation*}
\forall K \in \ \tau^{i}_{j}, \; \; \; |{F_{i,j}}^{-1}(K)| \le j \le i
\end{equation*}
Also
\begin{equation*}
\forall \{ r_1,r_2, \cdots , r_{j-1} \} \in \tau^{i}_{j-1}, \;\; \; \frac{1}{ (i+1)r_1 r_2 \cdots r_{j-1}} \le
\frac{1}{\prod_{g \in F_{i,j}(\{ r_1,r_2, \cdots , r_{j-1} \})} g}
\end{equation*}
It follows that
\begin{eqnarray*}
\sum _{\{r_1,r_2, \cdots , r_{j-1} \} \in {\tau^{i}}_{j-1}} \frac{1}{(i+1)r_1 r_2 \cdots r_{j-1}}
& \le & \sum _{\{r_1,r_2, \cdots , r_{j-1} \} \in {\tau^{i}}_{j-1}} \frac{1}{\prod_{g \in F_{i,j}(\{ r_1,r_2, \cdots , r_{j-1} \})} g}\\
& \le & \sum _{\{r_1,r_2, \cdots , r_j \} \in {\tau^{i}}_{j}} |{F_{i,j}}^{-1}(\{r_1,r_2, \cdots , r_j \})| \frac{1}{(r_1 r_2 \cdots r_j)}\\
& \le & i \sum _{\{r_1,r_2, \cdots , r_j \} \in {\tau^{i}}_{j}} \frac{1}{r_1 r_2 \cdots r_j}
\end{eqnarray*}
Thus
$$
\frac{i \times (i+2)!}{(i)!}|s(i,j;1)| \; \; \; \geq (i+2)!\sum _{\{r_1,r_2, \cdots , r_{j-1} \} \in {\tau^{i}}_{j-1}} \frac{1}{(i+1)r_1 r_2 \cdots r_{j-1}}
$$.
Similarly
$$
\frac{i \times (i+2)!}{(i)!}|s(i,j;1)|\; \; \geq (i+2)!\sum _{\{r_1,r_2, \cdots , r_{j-1} \} \in {\tau^{i}}_{j-1}} \frac{1}{(i+2)r_1 r_2 \cdots r_{j-1}}
$$
and
$$
\frac{i \times (i-1) \times (i+2)!}{(i)!}|s(i,j;1)| \; \; \; \geq (i+2)!\sum _{\{r_1,r_2, \cdots , r_{j-2} \} \in {\tau^{i}}_{j-2}} \frac{1}{(i+1)(i+2)r_1 r_2 \cdots r_{j-2}}
$$
Using the above three in the initial equation for $|s(i+2,j;1)|$, we get
\begin{eqnarray}
|s(i+2,j;1)| &\le& ((i+1)(i+2)+2(i+1)(i+2)i + (i+1)(i+2)i(i-1))|s(i,j;1)|\\
&\le &((i+1)(i+2)+2(i+1)(i+2)i + (i+1)(i+2)i(i-1)+4)|s(i,j;1)| \label{appendix_eq_1}
\end{eqnarray}
The polynomial $((x+1)(x+2)+2(x+1)(x+2)x + (x+1)(x+2)x(x-1))+4$ is defined as $f$, we have shown above that it satisfies the prescribed properties for $i \ge j >0$.\\
When $i>j=0$, $|s(i+2,0;1)|= (i+1)(i+2)|s(i,0;1)|$, when $i=j=0$, $s(2,0;1) = 4 < f(0)s(2,0;1)$. So \ref{appendix_eq_1} still holds.
\end{proof}
\end{appendices}
\section*{Acknowledgments}
I would like to thank Prof. Manfred Denker for his lucid explanations and guidance.
\end{document}
|
\begin{document}
\title{High validity entanglement verification with finite copies of a quantum state}
\author{Pawe{\l} Cie\'sli\'nski}
\orcid{0000-0001-5975-6265}
\affiliation{Institute of Theoretical Physics and Astrophysics, Faculty of Mathematics, Physics, and Informatics, University of Gdańsk, 80-308 Gdańsk, Poland}
\email{[email protected]}
\author{Jan Dziewior}
\affiliation{Max Planck Institute for Quantum Optics, 85748 Garching, Germany}
\affiliation{Faculty of Physics, Ludwig Maximilian University, 80799 Munich, Germany}
\affiliation{Munich Center for Quantum Science and Technology, 80799 Munich, Germany}
\author{Lukas Knips}
\orcid{0000-0002-7404-1708}
\affiliation{Max Planck Institute for Quantum Optics, 85748 Garching, Germany}
\affiliation{Faculty of Physics, Ludwig Maximilian University, 80799 Munich, Germany}
\affiliation{Munich Center for Quantum Science and Technology, 80799 Munich, Germany}
\author{Waldemar K{\l}obus}
\orcid{0000-0002-9176-4510}
\affiliation{Institute of Theoretical Physics and Astrophysics, Faculty of Mathematics, Physics, and Informatics, University of Gdańsk, 80-308 Gdańsk, Poland}
\author{Jasmin Meinecke}
\affiliation{Max Planck Institute for Quantum Optics, 85748 Garching, Germany}
\affiliation{Faculty of Physics, Ludwig Maximilian University, 80799 Munich, Germany}
\affiliation{Munich Center for Quantum Science and Technology, 80799 Munich, Germany}
\author{Tomasz Paterek}
\affiliation{Institute of Theoretical Physics and Astrophysics, Faculty of Mathematics, Physics, and Informatics, University of Gdańsk, 80-308 Gdańsk, Poland}
\author{Harald Weinfurter}
\affiliation{Max Planck Institute for Quantum Optics, 85748 Garching, Germany}
\affiliation{Faculty of Physics, Ludwig Maximilian University, 80799 Munich, Germany}
\affiliation{Munich Center for Quantum Science and Technology, 80799 Munich, Germany}
\author{Wies{\l}aw Laskowski}
\orcid{0000-0001-5166-0373}
\affiliation{Institute of Theoretical Physics and Astrophysics, Faculty of Mathematics, Physics, and Informatics, University of Gdańsk, 80-308 Gdańsk, Poland}
\affiliation{International Centre for Theory of Quantum Technologies, University of Gdańsk, 80-308 Gdańsk, Poland}
\maketitle
\begin{abstract}
Detecting entanglement of multipartite quantum states is an inherently probabilistic process due to a finite number of measured samples.
The level of confidence of entanglement detection can be used to quantify the probability that the measured signal is coming from a separable state and provides a meaningful figure of merit for big data sets.
Yet, for limited sample sizes, to avoid serious misinterpretations of the experimental results, one should not only consider the probability that a separable state gave rise to the measured signal, but should also include information about the probability that the signal came from an entangled state.
We demonstrate this explicitly and propose a comprehensive method of entanglement detection when only a very limited amount of data is available.
The method is based on a non-linear combination of correlation functions and is independent of system size.
As an example, we derive the optimal number of measurement settings and clicks per setting revealing entanglement with only $20$ copies of a state.
\end{abstract}
\section{Introduction}
Quantum entanglement is long recognised as an important prerequisite of modern quantum technologies.
Its detection is accordingly a well studied topic with a plethora of different methods available.
The field has evolved towards strategies directly applicable to experimental data which inevitably is limited to a finite number of detection events.
If this number is large various forms of entanglement witnesses~\cite{Witness,Review} provide practically deterministic entanglement verification~\cite{finitedata2}.
Interestingly, also the analysis of smaller data sets allows to detect entanglement~\cite{finitedata}, most recently
with quantum state verification methods~\cite{qsv, qsv2} and tailored game-like protocols in which particles are measured one by one~\cite{Bori1,Bori2,Bori3}.
These methods are of high practical relevance in all cases where only a limited amount or only partial data is accessible, e.g.~when an inefficient source has to be characterized quickly or one is interested in entanglement detection in large-scale quantum systems.
\begin{figure}
\caption{\label{FIG1}
\label{FIG1}
\end{figure}
The finiteness of data sets used to derive a conclusion about entanglement inevitably leads to a probabilistic nature of this conclusion. By far the two most prominent ways to quantify the \text{validity} of such a probabilistic statement are the \text{confidence} and the \text{credibility}, respectively related to the Frequentist and Bayesian approaches. Both of these measures capture different important aspects of statistical validity. A certification scheme based on either one aims to ensure, albeit with different figures of merit, a small error if the outcome of the method speaks in favor of entanglement, i.e.~a small error of a false positive.
Yet, in the case of low statistics a small error of a false positive might come at the price of a large probability of a false negative.
This tradeoff is illustrated in Fig.~{\ref{FIG1}}, which shows that for smaller statistics one is required to go to more and more strict criteria to reliably distinguish between results compatible with entangled states and results compatible with separable states.
In consequence, due the increased strictness it becomes also more and more improbable that an entangled state will pass the test.
Thus, especially in the case of small statistics, confidence or credibility is not the only figure of merit and we argue that one also has to ensure the \textit{efficiency} of the method
quantifying what fraction of entangled systems is expected to pass the criterion.
This change of mindset is necessary in order to avoid significant misinterpretations that limited sample sizes can cause if statistical reasoning is not carried out with appropriate care.
The paper is organised as follows. We begin by recalling a non-linear entanglement witness~\cite{Badziag}, which is adapted here to the case of finite sample size.
It is based on estimations of correlation functions and, importantly, is independent of system size.
We then provide a statistical analysis of entanglement verification with limited sample sizes both from a Frequentist as well as from a Bayesian perspective.
The underlying calculations and modelling are for the most part the same for the both approaches.
Based on that we introduce a general entanglement verification scheme with high validity and provide several examples including one that is conclusive with validity (confidence or credibility, respectively) of $97.5\%$ and at the same time admitting between $7\%$ and $27\%$ efficiency using just $20$ state copies.
\section{Entanglement witness for finite data sets}
\subsection{Correlation Length and Entanglement}
Our method is based on the entanglement detection scheme introduced in Ref.~\cite{Badziag} which we extend to the case of a finite set of measured data.
We focus here on $N$ qubits, i.e.~two-level quantum systems, but the method is clearly extendable to higher dimensions.
Any arbitrary state of this system can be decomposed in terms of tensor products of Pauli operators as
\begin{equation}
\rho = \frac{1}{2^N} \sum_{\mu_1, \dots, \mu_N = 0}^3 T_{\mu_1 \dots \mu_N} \sigma_{\mu_1} \otimes \dots \otimes \sigma_{\mu_N},
\end{equation}
where $\sigma_{\mu_k}$ is the $\mu_k$-th local Pauli matrix of the $k$-th party ($\sigma_0$ being identity),
while $T_{\mu_1 \dots \mu_N}$ are the correlation functions defined by
\begin{equation}
T_{\mu_1 \dots \mu_N} = \langle R_1 \ldots R_N \rangle = \Tr(\rho \, \sigma_{\mu_1} \otimes \dots \otimes \sigma_{\mu_N}).
\label{EQ_T}
\end{equation}
The correlation function is the average of the product of local measurement results $R_k = \pm 1$ when suitable local Pauli measurements are executed.
Note that this definition does not only include the \textit{full correlations}, for which all parties perform measurements, but also \textit{marginal correlations}, i.e. values of $T$ to which only a subset of parties contributes, which formally correspond to setting the remaining indices $\mu_i = 0$.
The starting point of our detection scheme is the insight that entanglement is implied if the values of sufficiently many of the full correlations are found to be sufficiently large.
In particular, consider the \textit{correlation length} $S^\infty$ defined by
\begin{align}
S^\infty \equiv \sum_{j_1, \dots, j_N = 1}^3 T_{j_1 \dots j_N}^2,
\label{EQ_CRIT_INF}
\end{align}
which is the sum of all full correlations squared.
We use the superscript ``$\infty$'' here to indicate that Eq.~(\ref{EQ_CRIT_INF}) contains the theoretical predictions, to which the experimental values converge with infinite statistics.
It has been proven~\cite{HJ1,HJ2,HJ3,Minh1,Minh2} that
\begin{equation}
S^\infty>1 \Rightarrow \text{state is entangled}.
\end{equation}
Equivalently, the probability for separable states to achieve $S^\infty>1$ vanishes, i.e.,
\begin{align}
P(S^\infty > 1|\mathrm{sep}) = 0,
\end{align}
where ``$\mathrm{sep}$'' denotes a conditioning on the state being separable.
We will also consider the conditioning on ``$\mathrm{ent}$'', namely that the state is entangled.
Since $S^\infty$ is the sum of non-negative values, it is sufficient to measure only a subset of correlations, i.e. $S^\infty_M$, violating the bound to certify entanglement.
\subsection{Effect of Finite Statistics}
For the usual application of the condition $S^\infty>1$, the actual correlation functions are estimated with a finite number of experimental trials.
We denote such measured correlations by $\tau_{j_1 \dots j_N}$ and to simplify the notation also write them $\tau_i$, where the index $i \in \left[1, 3^N\right]$ indicates the measurement settings.
Thus, instead of the ideal value of $S^\infty$ based on infinite statistics, experimentally we have only access to the modified value $\mathcal{S}$ based on $\tau_i$ from finite statistics, with
\begin{equation}
\mathcal{S} \equiv \sum_{i = 1}^M \tau_{i}^2.
\label{EQ_S_FINITE}
\end{equation}
Additionally, the summation is running over only $M$ indices, introducing the possibility that not all $3^N$ settings are measured.
The set of numbers $\{n_i\}$ shall denote the amount of state copies measured for each setting, which can vary from setting to setting.
To understand how finite statistics modifies the value of $\mathcal{S}$ compared to $S^\infty$, it is necessary to first consider the effect on the values $\tau_i$.
Assuming the system is measured along setting $i$ exactly $n_i$ times, the measured correlation value becomes
\begin{equation}
\tau_i = \frac{n^+_i - n^-_i}{n_i}
\label{EQ_TAU}
\end{equation}
where $n^{\pm}_i$ are the numbers of trials in which the product of local results equals $R_1 \cdots R_N = \pm 1$, with $n^+_i + n^-_i = n_i$.
For a large $n_i$ we obtain $\tau_i \approx T_i$, but for a small $n_i$ the measured $\tau_i$ take a potentially different value from the discrete set $\tau_i\in\{ -1, -1 + \frac{2}{n_i}, \dots, 1 - \frac{2}{n_i}, 1\}$.
Given a certain $N$-qubit state, expressed by a set of $T_i$, it is possible to explicitly calculate the probability distribution $P(\tau_i)$ over these discrete values of $\tau_i$ by employing the binomial distribution as demonstrated in Appendix~\ref{APP_prob}.
The mean and variance of this distribution are given as
\begin{equation}
\langle \tau_i \rangle = T_i, \qquad \mathrm{Var}(\tau_i) = \sigma_i^2/n_i,
\end{equation}
where $\sigma_i^2$ is a function of the ideal quantum prediction equal to $\sigma_i^2 = 1 - T_i^2$, see also \cite{KnipsFiniteStatistics}.
Note that the variance depends on the correlation value itself, e.g., a perfect correlation has no variance.
This already hints that the method is most useful for states with high correlation values.
Using $P(\tau_i)$ to calculate the mean and variance for the squared correlations yields
\begin{equation}
\langle \tau_i^2 \rangle = T_i^2 + \mathrm{Var}(\tau_i), \qquad \mathrm{Var}(\tau_i^2) = \frac{2(n_i-1)(1-T_i^2)[(2n_i-3) T_i^2 + 1]}{n_i^3}.
\end{equation}
The mean value of the square has a consistent shift by exactly the variance of the estimation from the finite data.
This is a consequence of taking the square --- the negative values in the distribution are transformed into positive values leading to the consistent change.
This fact is important in experimental analysis of finite data sets and has already been addressed in Ref.~\cite{Knips}.
Taking into account the particular distributions $P(\tau_i)$ and considering how the values $\tau_i$ combine into effective values of $\mathcal{S}$ (see also Appendix~\ref{APP_prob}), it becomes possible to calculate an explicit probability distribution $P(\mathcal{S})$, given a set of $M$ settings and $n_i$ measurements per setting.
Assuming for example that all $M$ settings are measured with the same amount of measurements $n$, i.e. $n_i = n$, the mean and variance of $P(\mathcal{S})$ can be shown to be
\begin{equation}
\langle \mathcal{S} \rangle = \frac{(n-1)S^\infty_M+M}{n}, \qquad \mathrm{Var}( \mathcal{S} ) = \sum_{i = 1}^M \mathrm{Var}(\tau_i^2),
\label{EQ_Sfin_meanVar}
\end{equation}
where the second equation stems from the independent estimations of each $\tau_i^2$.
\subsection{Violation of Bound as Entanglement Witness}
Two examples of distributions $P(\mathcal{S})$ are shown in Fig.~\ref{FIG1}. The entangled state $\rho$ is chosen as a graph state $\ket{\psi_{\mathrm{Graph}}}$ mixed with white noise,
\begin{equation}
\rho = p \proj{\psi_{\mathrm{Graph}}} + (1-p) \frac{\openone}{2^N} ,
\label{eq_rho}
\end{equation}
where $p = 0.8$.
It becomes clear from the distributions illustrated in Fig.~\ref{FIG1} that in this detection scenario the correlations compatible with separability violate the bound of $1$ with a significant probability and thus the criterion $S^\infty>1$ for entanglement cannot be extended to $\mathcal{S}$.
Given the result (\ref{EQ_Sfin_meanVar}) our approach is thus to formulate a similar criterion $\mathcal{S} > \mathcal{B}$ for entanglement with a new bound $\mathcal{B} > 1$.
Since, however, this new bound is no longer as sharp as the bound of $1$ obtained with infinite statistics, one has to take great care with the probabilistic analysis of the method.
Choosing, e.g., $\mathcal{B} = 2$ and $n=10$ one finds that correlations compatible with separability reach this value with a probability of $P(\mathcal{S} \geq \mathcal{B}|\mathrm{sep}) = 4.2\%$, which according to the usual analysis~\cite{finitedata2} implies a confidence level of $1 - P(\mathcal{S} \geq \mathcal{B}|\mathrm{sep}) = 95.8\%$.
As already pointed out in the introduction we want to show that this confidence level by itself does not capture entirely the quality of the method for entanglement detection.
While it is very improbable to violate the bound $\mathcal{B} = 2$ with separable correlations, at the same time, the probability $P(\mathcal{S} \geq \mathcal{B}|\rho)$ to violate it with the state $\rho$ in (\ref{eq_rho}) is small, too, as $\rho$ achieves a value of $\mathcal{S} \geq 2$ with only $6.9\%$.
Indeed this procedure would wrongly certify entanglement in only $4.2\%$ of the cases (which validates the confidence), but at the same time it would lead to $93\%$ of undetermined cases when in fact the state is entangled.
Furthermore, for any single experiment showing a violation of the bound, we are clearly not justified to conclude (far from a certainty of $95.8\%$) that the state is entangled, as there might exist a separable state that gives rise to the same value of $\mathcal{S}$ with a comparable probability.
These considerations show that in general, apart from the probability $P(\mathcal{S} \geq \mathcal{B}|\mathrm{sep})$, it is essential to also consider the probability $P(\mathcal{S} \geq \mathcal{B}|\rho)$ conditioned on a particular entangled state $\rho$ or the probability that any entangled state gives rise to violation of the bound $P(\mathcal{S} \geq \mathcal{B}|\mathrm{ent})$.
In summary, we propose the following purpose for entanglement certification scheme with very finite data sets.
Using our ability to explicitly calculate the distributions $P(\mathcal{S})$ for various states, we want to find a set of $M$ settings, each measured on $\{n_i\}$ copies of the state, and a bound $\mathcal{B}$ on $\mathcal{S}$, such that we achieve both a high \textit{validity} and a high \textit{efficiency} of entanglement certification.
In the following section we will present two different ways in which the validity of such a certification scheme is defined according to the Frequentist and Bayesian approaches.
The efficiency $r$ is the same in both cases and defined as $r\equiv P(\mathcal{S}\geq \mathcal{B}|\mathrm{ent})$, i.e., it measures the fraction of entangled states which is expected to actually be successfully certified.
Our scheme can be formulated as the following optimization problem:
\begin{quote}
Given a certain minimal degree of validity $\alpha$ and a total number $m$ of state copies, what set of $M$ observables, each estimated on $\{n_i\}$ copies of the state, and what bound $\mathcal{B}$ are to be chosen such that an observation of $\mathcal{S} \geq \mathcal{B}$ certifies entanglement with \textit{validity} $q \geq \alpha$, while keeping the \textit{efficiency} of the method $r = P(\mathcal{S}\geq \mathcal{B}|\mathrm{ent})$ as high as possible.
\end{quote}
\section{Two Approaches to Validity}
\subsection{Degree of Confidence $q_F$}
In the Frequentist approach the validity of the test is captured by the \textit{confidence} $q_F$ of a statistical experiment, which is understood as the minimal rate $q_F$ with which the experiment provides the correct conclusion.
This is a statement about the overall performance of the experiment and does not allow any probabilistic assertions about the correctness of a particular single run of the procedure.
For example, assuming a procedure with $q_F = 95\%$ and a separable test state, it might happen that the conclusion in roughly $5\%$ of the runs will be that the state is entangled.
Although in fact the procedure will be correct $95\%$ of the time, the experimenter would clearly never be justified in believing the correctness of the conclusion that the state is entangled in a particular experimental run.
Conversely, such a method is perfectly suitable to, e.g., certify that a certain device produces entangled states with at least say $50\%$ reliability, since in this case a whole series of repeated experiments is considered.
Furthermore, $q_F$ has the great advantage of being completely independent of any prior assumptions about the underlying system.
To apply our method according to the Frequentist approach, one constructs confidence intervals $I_{\mathcal{S}}$ for each outcome $\mathcal{S}$.
These confidence intervals, being rather confidence sets in our case, can be $\{\mathrm{sep}\}$, $\{\mathrm{ent}\}$, or $\{\mathrm{sep},\mathrm{ent}\}$, where the last case effectively corresponds to undecided.
To achieve an overall degree of confidence $q_F$, the necessary condition is that at least a fraction $q_F$ of the probability mass functions has to be contained within the intervals
\begin{align}
\label{eq_freq_basic}
\mathrm{i)} \quad \sum_{\{\mathcal{S}|\mathrm{sep} \in I_{\mathcal{S}}\}} P(\mathcal{S}|\mathrm{sep}) \qquad &> q_F \quad \land \nonumber \\
\mathrm{ii)} \quad \sum_{\{\mathcal{S}|\mathrm{ent} \in I_{\mathcal{S}}\}} P(\mathcal{S}|\mathrm{ent}) \qquad &> q_F.
\end{align}
We follow the basic idea to use the bound $\mathcal{B}$ for the definition of the confidence intervals $I_\mathcal{S}$. Since our goal is the certification of entanglement from violation of the bound we determine the intervals as $\{I_\mathcal{S} = \{\mathrm{ent}\}|\mathcal{S} \geq \mathcal{B}\}$ and $\{I_\mathcal{S} = \{\mathrm{sep},\mathrm{ent}\}|\mathcal{S} < \mathcal{B}\}$, where the latter corresponds to an undecided outcome, i.e., no certification of entanglement.
This effectively means that only errors of type $I$ (false positive) are of importance, changing condition i) to
\begin{align}
\label{eq_freq_main}
\sum_{\{\mathcal{S}|\mathcal{S}<\mathcal{B}\}} P(\mathcal{S}|\mathrm{sep}) > q_F \quad \Longleftrightarrow \sum_{\{\mathcal{S}|\mathcal{S} \ge \mathcal{B}\}} P(\mathcal{S}|\mathrm{sep}) \le 1-q_F.
\end{align}
Condition ii) on the other hand becomes superfluous as $\{\mathcal{S}|\mathrm{ent} \in I_{\mathcal{S}}\} = \{\mathcal{S}\}$, i.e., the confidence interval of every $\mathcal{S}$ contains the outcome $\mathrm{ent}$, and thus $q_F$ is completely independent of $P(\mathcal{S}|\mathrm{ent})$.
As long as only the degree of confidence, i.e.~the validity of the conclusion is concerned, it is thus sufficient to consider only $P(\mathcal{S}|\mathrm{sep})$ from the point of view of the Frequentist approach.
\begin{figure}
\caption{\label{FIG_CUMUL}
\label{FIG_CUMUL}
\end{figure}
In our scheme, and in fact in most practical applications, one is not only interested in a valid, but also in an efficient method.
In the context of the Frequentist approach, this corresponds to minimizing $e_2$, the error of type II (rate of false negatives), which exactly corresponds to maximizing the efficiency $r$ as defined above.
Repeating the same argumentation as for the confidence $q_F$ one derives the condition for $r$ given by:
\begin{align}
\label{eq_freq_eff}
r = 1-e_2 = \sum_{\{\mathcal{S}|\mathcal{S}\geq\mathcal{B}\}} P(\mathcal{S}|\mathrm{ent}).
\end{align}
Thus, also from a Frequentist point of view it is necessary to keep track of $P(\mathcal{S}|\mathrm{ent})$ to determine and maximize the efficiency of the method.
See Fig.~\ref{FIG_CUMUL} for an illustration.
\subsection{Degree of Belief $q_B$}
In a Bayesian approach, obtaining a particular measurement outcome $\mathcal{S}$ allows the experimenter to come to a certain \textit{degree of belief} $q_B$ that the state is entangled, which corresponds to the probability $P(\mathrm{ent}|\mathcal{S})$.
This directly implies a degree of belief for the opposite assertion, namely that the state is separable, with $P(\mathrm{sep}|\mathcal{S}) = 1 - P(\mathrm{ent}|\mathcal{S})$.
From a Bayesian point of view, a natural way to define a certification method with a validity of $q_B$, is to certify entanglement whenever $P(\mathrm{ent}|\mathcal{S}) \geq q_B$.
In particular, again the bound $\mathcal{B}$ is to be found such that
\begin{align}
P(\mathrm{ent}|\mathcal{S}) \geq q_B, \quad \textrm{ for all } \quad \mathcal{S}\geq\mathcal{B},
\end{align}
i.e., whenever we observe a value of $\mathcal{S} \geq \mathcal{B}$ the certification of entanglement is credible at least with degree $q_B$.
The explicit formula for $P(\mathrm{ent}|\mathcal{S})$ is given according to Bayes rule as
\begin{equation}
P(\mathrm{ent}|\mathcal{S}) = \frac{P(\mathcal{S}|\mathrm{ent})P(\mathrm{ent})}{P(\mathcal{S})}
= \frac{P(\mathcal{S}|\mathrm{ent})P(\mathrm{ent})}{P(\mathcal{S}|\mathrm{ent})P(\mathrm{ent}) + P(\mathcal{S}|\mathrm{sep})P(\mathrm{sep})}
= \frac{R}{1+R},
\label{eq_bayes_main}
\end{equation}
with
\begin{align}
R \equiv \frac{P(\mathcal{S}|\mathrm{ent})}{P(\mathcal{S}|\mathrm{sep})}\frac{P(\mathrm{ent})}{P(\mathrm{sep})} = \frac{P(\mathcal{S} \land \mathrm{ent})}{P(\mathcal{S} \land \mathrm{sep})},
\label{R}
\end{align}
where the ratio $R$ has been introduced.
To calculate the value of $P(\mathrm{ent}|\mathcal{S})$, it is necessary to know the distributions $P(\mathcal{S}|\mathrm{sep})$ and $P(\mathcal{S}|\mathrm{ent})$, just as in the Frequentist approach.
However, in general, Bayesian reasoning also rests on the \textit{prior}, i.e., the best guess about the absolute probability distributions before the experiment is performed.
Here, we express this prior via the ratio $P(\mathrm{ent})/P(\mathrm{sep})$.
In a practical certification scheme this quantity is estimated via a suitable lower bound as we will demonstrate in an example.
The value $r = P(\mathcal{S}\geq\mathcal{B}|\mathrm{ent})$ can be used as the quantifier of the efficiency also in the Bayesian approach.
As $R$ only depends on the quotient $P(\mathcal{S}|\mathrm{ent}) / P(\mathcal{S}|\mathrm{sep})$, $r$ is left as an independent quantity to be maximized.
Let us also briefly mention why the relevant prior is not discussed in the usual detection of quantum entanglement, via witnesses, in the limit of many measurement results ~\cite{finitedata2}.
In such a case, for $n_i \gg 1$, it is practically impossible for a separable state to exceed the bound of the witness, i.e., $P(\mathcal{S} | \mathrm{sep}) \approx 0$.
Therefore, Eq.~(\ref{eq_bayes_main}) simplifies for these values of $\mathcal{S}$ and becomes $P(\mathrm{ent} | \mathcal{S}) \approx 1$, allowing to reveal entanglement independently of the priors.
\subsection{Estimating the Probability Distributions}
Both approaches, Frequentistic and Bayesian, arrive at the task of calculating the two conditioned distributions of $\mathcal{S}$.
However, while so far we demonstrated how to calculate $P(\mathcal{S}|\rho)$ given a specific state $\rho$, we in fact require the quantities $P(\mathcal{S}|\mathrm{sep})$ and $P(\mathcal{S}|\mathrm{ent})$ conditioned on the whole subspaces of \textit{all} separable and entangled states, respectively.
Each particular states $\{\rho^\mathrm{sep}_j\}$ and $\{\rho^\mathrm{ent}_j\}$ imply different distributions of $\mathcal{S}$ such that the relevant quantities have to be decomposed as
\begin{align}
P(\mathcal{S}|\mathrm{sep}) = \sum_j P(S|\rho^\mathrm{sep}_j) P(\rho^\mathrm{sep}_j|\mathrm{sep}), \\
P(\mathcal{S}|\mathrm{ent}) = \sum_j P(S|\rho^\mathrm{ent}_j) P(\rho^\mathrm{ent}_j|\mathrm{ent}).
\end{align}
These expressions illustrate the underlying dependence on the distributions of separable and entangled states in their particular subspaces $P(\rho^\mathrm{sep}_j|\mathrm{sep})$ and $P(\rho^\mathrm{ent}_j|\mathrm{ent})$.
Unfortunately, evaluating those expressions is notoriously daunting~\cite{bergeSCMCsampling}.
For the separable states it is possible to avoid any consideration of state distributions and instead consider a worst-case distribution of $\mathcal{S}$.
Both for the Frequentist as well as the Bayesian approach this can be achieved by introducing an upper bound $F(\mathcal{S})$ on $P(\mathcal{S}|\mathrm{sep})$ satisfying
\begin{align}
\forall_{\rho^\mathrm{sep}_j}\, \forall_{\mathcal{S}>\mathcal{B}} \; F(\mathcal{S}) \geq P(\mathcal{S}|\rho^\mathrm{sep}_j).
\end{align}
The strategy to find the upper bound $F$ is based on finding a set of $M$ correlations $T_i$ corresponding to $\tilde{\rho}$ such that $P(\mathcal{S}|\tilde{\rho}) \geq P(\mathcal{S}|\mathrm{sep})$ for any separable state and any $\mathcal{S} \geq \mathcal{B}$.
We note that $\tilde{\rho}$ might not even be a proper density matrix, this is why we often refer to such correlations as ``compatible with separability''.
This problem is equivalent to maximising $\sum_i T_i^2$ under the constraint that $\sum_i T_i^2 \leq 1$ (recall that 1 is the bound given by ideal quantum predictions for separable states).
In Appendix~\ref{APP_1M} we describe computations which verify, for a chosen value of $n$ and $M$, that the optimal correlations compatible with separability depend on the value of the bound $\mathcal{B}$, but for sufficiently high $\mathcal{B}$ the correlations $T_i^2 = 1/M$ were always universal for uniform distributions of $n$ copies per each setting.
Due to this important fact our method is independent of the number of particles being tested, only the number of measurement settings enters the separability bound.
In analogy to the distribution conditioned on separable states in principle both approaches could proceed given a lower bound on $P(\mathcal{S}|\mathrm{ent})$.
However, this is completely unfeasible for several reasons: (i) entangled states can easily yield very small values of $\mathcal{S}$ due to statistical fluctuations,
(ii) some entangled states have small (or even vanishing) $S^\infty$~\cite{Kaszlikowski2008,Laskowski2012,Schwemmer2015NoCorr,Tran2017},
(iii) bad choice of measurement settings, e.g. ones that give $S^{\infty}$ close to 1.
Thus, $P(\mathcal{S}|\mathrm{ent})$ is obtained by modelling a certain state distribution $P(\rho^\mathrm{ent}_j|\mathrm{ent})$, which fits the currently employed apparatus for state generation, as illustrated in the example in the next section.
Note that, while breaking to some extent the independence from the prior gained in a Frequentist approach, the procedure fits the Bayesian approach very well.
\section{Examples}
Consider a scenario in which, due to some external factors, the apparatus used in an experiment gets slightly misaligned. We want to reliably and efficiently verify if the produced states are still entangled knowing that the visibility of our noisy graph state $\rho$ in Eq.~(\ref{eq_rho}) could drop down to, say, $75\%$. We set the target validity to $q_F=q_B=97.5\%$ and consider the source that sends this state or a separable state with equal \textit{prior} probabilities, i.e., $P(\mathrm{sep}) / P(\mathrm{ent}) = 1$.
According to our optimization scheme, we want to determine the optimal set of measurements, i.e., their amount $M$ and in general also the measurement direction, together with an optimal distributions of measurements $\{n_i\}$.
Here, we simplify the problem by adding the constraint that each setting is measured the same amount of times $n$ with $M n = 20$.
Fig.~\ref{FIG_20OPT} presents the two relevant quantities $P(\mathcal{S}\geq\mathcal{B}|\mathrm{sep})$ and $P(\mathcal{S}< \mathcal{B}|\mathrm{ent})$ which both determine validity and efficiency in the Frequentist and in the Bayesian case.
Independently of the approach used, ideally, both those probabilities should be kept as small as possible.
Fig.~\ref{FIG_20OPT} shows that this is best realised for $n=4$ and $M=5$.
Given all of the information we calculate the bound $\mathcal{B}$ for the Frequentist approach to be $\mathcal{B}_F=4$ and $\mathcal{B}_B=5$ obtained through the Bayesian reasoning. Assume that we have measured $\mathcal{S}=5$. In this case our entanglement detection scheme is conclusive with the given confidence or credibility equal to $97.5\%$.
While this calculation is based on a prior probability $P(ent) = 50\%$, a modification of the prior towards a worse scenario, does not change the conclusion for $S=5$, even up to a $P(sep) \approx 68\%$.
\begin{figure}
\caption{\label{FIG_20OPT}
\label{FIG_20OPT}
\end{figure}
The Bayesian approach helps us conclude that the measured state is entangled, but, similarly to the Frequentist approach, it does not contain information about how often do we detect the entanglement. This is why the efficiency $r$ may be added to our reasoning.
In our example with $p=3/4,n = 4, M=5$ for the measurement outcome $\mathcal{S}=5$ the efficiency is given by $r=P(\mathcal{S}=5 | \mathrm{ent})\approx 7\%$. If the priors were chosen differently, say $75\%/25\%$ the Bayesian bound gets shifted to $4.25$, leading to the probability $P(\mathcal{S}=4.25 | \mathrm{ent})\approx 20\%$ and hence an increased efficiency of $r\approx 27\%$.
Note that this order of magnitude for efficiency will also be found for many noise models different from the white noise.
Extensive examples of optimal certification strategies that allow different distributions of state copies per setting are presented in Appendix~\ref{APP_3}. It also provides further discussion on validity for both the Frequentist and Bayesian frames as well as their comparison. What is interesting is that given some maximal number of state copies that can be used in an experiment different approaches to probability can lead to performing different amount of measurements and different bound choices.
\section{Conclusions}
To certify entanglement, one is usually interested in a measure of validity, i.e., to be convinced that the certification does not falsely conclude entanglement.
We have shown that for small data sets, one should not only consider the probability of a separable state to look rather entangled, but also the probability of an entangled state to look rather separable.
This change of mindset is necessary if we are to make robust conclusions in the presence of finite statistics.
We have discussed entanglement certification schemes for small data samples from the points of view of both Frequentist and Bayesian interpretation.
In order to quantify how often an entangled state is detected as such, we introduced the measure of efficiency that in effect allows comparison between different entanglement detection schemes.
In general a higher validity of the test comes at the price of lowering the efficiency of entanglement detection. For example, in the limited resources entanglement detection scheme proposed in Ref.~\cite{Bori1} the validity is defined according to the Frequentist approach as the confidence level $1-P(\delta)$, where $P(\delta)$ is the probability to observe a measurement result that concludes entanglement for a separable state. The efficiency of that protocol is not discussed, while according to the present findings it has important implications for the practical use of the method. Based on our results the confidence level of $\approx 99\%$, obtained in the discussed paper with the number of state copies as small as $20$, entails a very small probability of actually observing an outcome indicating entanglement.
The methods introduced here are especially helpful for a resource-efficient estimation of the performance of a known apparatus subject to variations of external parameters or in multipartite experiments with rare detection events, e.g., multi-photon setups based on coincidence clicks.
For example our scheme could be employed to quickly certify the quality of a large quantum processor before a time consuming computation task.
\section{Acknowledgments}
We thank Borivoje Daki\'c for useful discussions.
This research was supported by the DFG (Germany) and NCN (Poland) within the joint funding initiative ``Beethoven2'' (2016/23/G/ST2/04273, 381445721),
and by the DFG under Germany’s Excellence Strategy EXC-2111-390814868.
WL acknowledges partial support from the Foundation for Polish Science (IRAP project ICTQT, Contract No. 2018/MAB/5, co-financed by EU via Smart Growth Operational Programme).
TP is supported by the Polish National Agency for Academic Exchange NAWA Project No. PPN/PPO/2018/1/00007/U/00001.
JD acknowledges support from the PhD program IMPRS-QST.
\appendix
\section{Probabilities of Results for $\mathcal{S}$ with Finite Statistics}
\label{APP_prob}
Since the product of measurement results is either equal to $+1$ or $-1$, the probability that in $n$ trials we obtain $n_+$ products equal to $+1$ is given by the binomial distribution:
\begin{equation}
p_i(n_+) = {n \choose n_+} \left( \frac{1 + T_i}{2} \right)^{n_+} \left( \frac{1 - T_i}{2} \right)^{n_-},
\end{equation}
where $(1 \pm T_i)/2$ is the probability to obtain the product equal to $\pm 1$ in a single trial, estimated from the ideal quantum mechanical prediction. All random variables are assumed to be independent and identically distributed.
Using Eq.~(\ref{EQ_TAU}) the correlation $\tau_i$ is also binomially distributed as
\begin{equation}
\mathrm{Prob}(\tau_i) = p_i\left(\frac{n}{2}(1+\tau_i)\right).
\end{equation}
Note that due to finiteness of $n$ also $\tau_i$ can take on a finite set of values $\{ -1, -1 + \frac{2}{n}, \dots, 1 - \frac{2}{n}, 1\}$.
We are interested in statistical properties of this distribution as well as of $\mathcal{S}$.
In order to write the probability distribution of different values of $\mathcal{S}$ we first note that
\begin{equation}
\mathrm{Prob}(\tau_i^2) = \mathrm{Prob}(\tau_i) + \mathrm{Prob}(- \tau_i) \quad \textrm{for} \quad \tau_i \ne 0,
\end{equation}
because both values $\pm \tau_i$ give the same square.
In the special case of $\tau_i = 0$, the probability $\mathrm{Prob}(\tau_i^2)$ is just given by $\mathrm{Prob}(\tau_i = 0)$.
Since $\mathcal{S}$ is defined as the sum of squares, a particular value of $\mathcal{S}$ can be realised in many ways,
e.g. the value $\mathcal{S} = 1$ can be achieved if any one $\tau_i^2 = 1$ and all the other squared correlations are zero, or when all $\tau_i^2 = 1/M$, etc.
Assuming that all the squared correlations $\tau_i^2$ are independently distributed, the values of $\mathcal{S}$ satisfy:
\begin{equation}
\mathrm{Prob}[\mathcal{S}] = \sum_{\tau_1^2 + \dots + \tau_M^2 = \mathcal{S}} \mathrm{Prob}(\tau_1^2) \cdots \mathrm{Prob}(\tau_M^2).
\end{equation}
These distributions were used in the main text to derive mean values and variances of the introduced quantities.
\section{Correlations compatible with separability}
\label{APP_1M}
We have written the following program in order to find the set of correlation functions that give the highest probability of $\mathcal{S} \geq \mathcal{B}$
and at the same time the correlations compatible with separability. The program verifies that for each $n_i=n$ and sufficiently high $\mathcal{B}$ all of the correlations could be taken as $T_i^2=1/M$. Its application for different choices of state copies per setting with ready to use code are presented in Appendix~\ref{APP_3}.
To this end we compute $\mathrm{Prob}[\mathcal{S}]$ and cumulative probability function $\sum_{\mathcal{S} \geq \mathcal{B}} \mathrm{Prob}[\mathcal{S}]$.
They are parameterized with to-be-determined correlations $\lbrace T_1,T_2,...,T_M \rbrace$ as well as the bound $\mathcal{B}$.
The heart of the computation is the numerical maximization of $\sum_{\mathcal{S} \geq \mathcal{B}} \mathrm{Prob}[\mathcal{S}]$, for the fixed value of $\mathcal{B}$, with respect to $\lbrace T_1,T_2,...,T_M \rbrace$.
We constrain the set of possible correlations to those that yield $S^{\infty} \leq 1$, with each $T_i \in [0,1]$ (recall that only squares enter the entanglement condition).
Two methods were used for finding the maximum: simulated annealing and Nelder-Mead method.
Both gave similar results with the Nelder-Mead method being more precise in the considered problem.
The output of the algorithm is a list of $\lbrace T_1,T_2,...,T_M \rbrace$ that maximizes $\sum_{\mathcal{S} \geq \mathcal{B}} \mathrm{Prob}[\mathcal{S}]$ for the given bound $\mathcal{B}$ under the stated constraints.
We have run this program for all combinations of $n$ and $M$ that use $20, 18$ and $16$ state copies in total with uniform $n_i$'s distribution.
The results can be grouped into two cases.
For values of $\mathcal{B}$ which are close to $1$ the optimal correlations are all equal to zero except one $T_i=1$.
For sufficiently high values of $\mathcal{B}$ the optimal correlations are all the same, i.e. $T_i^2=1/M$ for each $i$.
\section{Examples of Optimal Certification Strategies}
\label{APP_3}
We present certification schemes with as high as possible efficiency given a certain minimal target reliability and total number of measurements.
An example will be provided where different certification strategies are found to be optimal, depending on whether a Frequentist or Bayesian approach is chosen.
We estimate the distribution $P(\rho^\mathrm{ent}_j|\mathrm{ent})$ based on the model of a noisy two-qubit graph state (\ref{eq_rho}), with a mean $\bar{p} = 0.85$ and a standard deviation $\Delta p = 0.04$, such that
\begin{align}
P(p) = \begin{cases}
\mathcal{N} e^{-\frac{(p-\bar{p})^2}{2(\Delta p)^2}}& p \in [\frac{1}{\sqrt{2}},1] \\
0 & \text{else},
\end{cases}
\end{align}
with proper normalization $\mathcal{N}$.
Since this state has three non-vanishing correlations we set the number of settings to $M=3$.
At the same time at most $15$ state copies were allowed and a minimal target validity of $70\%$ was required.
The worst case distribution compatible with separable states was adapted to the number of measured settings $M$, where the correlation value was always set to $1/\sqrt{M}$.
\subsection{Optimization with Frequentist Figure of Merit}
If the optimization is based on a Frequentist understanding of validity, i.e., achieving at least $70\%$ confidence, it is optimal to use all of the $15$ measurement runs and distribute them among the three settings as follows: $n_1 = 6$, $n_2 = 5$ and $n_3 = 4$. We shall briefly denote this configuration as $[6 5 4]$.
Table \ref{tab_bestStratFreq} illustrates the resulting probability distributions.
Since given a separable state the value of $\mathcal{S}$ will lie below $\mathcal{B} = 1.8044$ with $70.82\%$ probability, this value is chosen as the bound.
This implies an efficiency of $81.45\%$ corresponding the probability of obtaining a $\mathcal{S} \geq \mathcal{B}$ given an entangled state.
For comparison, we note that
since the credibility $q_B$ for the last four values of $\mathcal{S}$ lies above $70\%$, according to the Bayesian approach the bound $\mathcal{B}$ would be put at $2.25$, resulting in an efficiency of only $68.93\%$.
\begin{table}[ht!]
\centering
\begin{tabular}{|c||c|c||c|c||c|}
\hline
& & $q_F = $ & & $\mathrm{eff} = $ & $q_B = $ \\
$\mathcal{S}$ & $P(\mathcal{S}|\mathrm{sep})$ & $P(\leq \mathcal{S}|\mathrm{sep})$ & $P(\mathcal{S}|\mathrm{ent})$ & $P(\geq \mathcal{S}|\mathrm{ent})$ & $P(\mathrm{ent}|\mathcal{S})$ \\ \hline \hline
0.04 & 0.004287 & 0.004287 & 2.298e-05 & 1 & 0.005332 \\
0.1511 & 0.01286 & 0.01715 & 0.0001619 & 1 & 0.01244 \\
0.29 & 0.01143 & 0.02858 & 0.0001439 & 0.9998 & 0.01244 \\
0.36 & 0.00643 & 0.03501 & 9.647e-05 & 0.9997 & 0.01478 \\
0.4011 & 0.03429 & 0.0693 & 0.001057 & 0.9996 & 0.02989 \\
0.4711 & 0.01929 & 0.08859 & 0.0007115 & 0.9985 & 0.03557 \\
0.4844 & 0.018 & 0.1066 & 0.0006202 & 0.9978 & 0.0333 \\
0.61 & 0.01715 & 0.1237 & 0.0006324 & 0.9972 & 0.03557 \\
0.7211 & 0.05144 & 0.1752 & 0.004882 & 0.9966 & 0.08669 \\
0.7344 & 0.04801 & 0.2232 & 0.004242 & 0.9917 & 0.08119 \\
0.8044 & 0.02701 & 0.2502 & 0.002872 & 0.9874 & 0.09611 \\
1 & 0.004715 & 0.2549 & 0.0001874 & 0.9846 & 0.03823 \\
1.04 & 0.02115 & 0.2761 & 0.001394 & 0.9844 & 0.06185 \\
1.054 & 0.07202 & 0.3481 & 0.02079 & 0.983 & 0.224 \\
1.111 & 0.01415 & 0.3622 & 0.001449 & 0.9622 & 0.09289 \\
1.151 & 0.03001 & 0.3922 & 0.002651 & 0.9607 & 0.08119 \\
1.25 & 0.01257 & 0.4048 & 0.001288 & 0.9581 & 0.09289 \\
1.29 & 0.02972 & 0.4345 & 0.007566 & 0.9568 & 0.2029 \\
1.36 & 0.03172 & 0.4662 & 0.006745 & 0.9492 & 0.1753 \\
1.361 & 0.03772 & 0.504 & 0.01048 & 0.9425 & 0.2174 \\
1.444 & 0.0198 & 0.5238 & 0.006174 & 0.932 & 0.2377 \\
1.471 & 0.04501 & 0.5688 & 0.01299 & 0.9258 & 0.224 \\
1.484 & 0.04201 & 0.6108 & 0.01125 & 0.9128 & 0.2112 \\
1.61 & 0.04458 & 0.6554 & 0.03961 & 0.9016 & 0.4705 \\
1.694 & 0.05281 & \textbf{\color{blue}0.7082} & 0.04749 & 0.862 & 0.4735 \\
\arrayrulecolor{blue} \hline \arrayrulecolor{black}
\textbf{\color{blue}1.804} & 0.06301 & 0.7712 & 0.05894 & \textbf{\color{blue}0.8145} & 0.4833 \\
2 & 0.02326 & 0.7945 & 0.01521 & 0.7556 & 0.3953 \\
2.04 & 0.02601 & 0.8205 & 0.02134 & 0.7403 & 0.4508 \\
2.111 & 0.03301 & 0.8535 & 0.02968 & 0.719 & 0.4735 \\
\arrayrulecolor{red} \hline \arrayrulecolor{black}
\textbf{\color{red}2.25} & 0.03269 & 0.8862 & 0.09734 & \textbf{\color{red}0.6893} &\textbf{\color{red} 0.7486} \\
2.36 & 0.03901 & 0.9252 & 0.1209 & 0.592 & \textbf{\color{red}0.7561} \\
2.444 & 0.04621 & 0.9714 & 0.145 & 0.4711 & \textbf{\color{red}0.7583} \\
3 & 0.02861 & 1 & 0.3261 & 0.3261 & \textbf{\color{red}0.9193} \\
\hline
\end{tabular}
\caption{Best certification strategy according to the Frequentist approach.
The $15$ measurements are distributed among the $3$ settings as $[6 5 4]$. We mark in blue the bound, confidence, and efficiency from a Frequentist perspective and in red the bound, credibilitiy and efficiency from a Bayesian perspective.}
\label{tab_bestStratFreq}
\end{table}
\subsection{Optimization with Bayesian Figure of Merit}
Table \ref{tab_bestStratBayes} shows the probability distributions optimal with respect to the Bayesian approach, i.e., achieving at least $70\%$ credibility.
It turns out that in this case it is better to use only $14$ of the $15$ possible measurements and distribute them as $[6 4 4]$ among the $3$ settings.
It is an interesting property of the Bayesian approach that adding the last data point decreases the probability that an entangled state violates the bound.
Again for the last three values the credibility is above $70\%$, resulting in the same bound of $2.25$ as in the first example.
Now, however, the Bayesian approach achieves the higher efficiency of $71.76\%$.
At the same time the Frequentist procedure fares worse than in the previous scheme, as it achieves an efficiency of $77.92\%$ with the resulting bound of $2$.
\begin{table}[ht!]
\centering
\begin{tabular}{|c||c|c||c|c||c|}
\hline
& & $q_F = $ & & $\mathrm{eff} = $ & $q_B = $ \\
$\mathcal{S}$ & $P(\mathcal{S}|\mathrm{sep})$ & $P(\leq \mathcal{S}|\mathrm{sep})$ & $P(\mathcal{S}|\mathrm{ent})$ & $P(\geq \mathcal{S}|\mathrm{ent})$ & $P(\mathrm{ent}|\mathcal{S})$ \\ \hline \hline
0 & 0.002572 & 0.002572 & 1.379e-05 & 1 & 0.005332\\
0.1111 & 0.007716 & 0.01029 & 9.716e-05 & 1 & 0.01244\\
0.25 & 0.01372 & 0.02401 & 0.0001727 & 0.9999 & 0.01244\\
0.3611 & 0.04115 & 0.06516 & 0.001268 & 0.9997 & 0.02989\\
0.4444 & 0.0108 & 0.07596 & 0.0003721 & 0.9984 & 0.0333\\
0.5 & 0.01829 & 0.09425 & 0.0005635 & 0.9981 & 0.02989\\
0.6111 & 0.05487 & 0.1491 & 0.004329 & 0.9975 & 0.07312\\
0.6944 & 0.05761 & 0.2067 & 0.005091 & 0.9932 & 0.08119\\
0.9444 & 0.07682 & 0.2836 & 0.01833 & 0.9881 & 0.1926\\
1 & 0.01869 & 0.3022 & 0.001043 & 0.9698 & 0.05287\\
1.111 & 0.03601 & 0.3382 & 0.003182 & 0.9687 & 0.08119\\
1.25 & 0.06767 & 0.4059 & 0.01191 & 0.9655 & 0.1496\\
1.361 & 0.09602 & 0.5019 & 0.02291 & 0.9536 & 0.1926\\
1.444 & 0.05041 & 0.5524 & 0.0135 & 0.9307 & 0.2112\\
1.5 & 0.04755 & 0.5999 & 0.03471 & 0.9172 & 0.4219\\
1.694 & 0.1344 & \textbf{\color{blue}0.7343} & 0.1033 & 0.8825 & 0.4345\\
\arrayrulecolor{blue} \hline \arrayrulecolor{black}
\textbf{\color{blue}2} & 0.04521 & 0.7795 & 0.02936 & \textbf{\color{blue}0.7792} & 0.3937\\
2.111 & 0.04201 & 0.8216 & 0.03228 & 0.7498 & 0.4345\\
\arrayrulecolor{red} \hline \arrayrulecolor{black}
\textbf{\color{red}2.25} & 0.08322 & 0.9048 & 0.2105 & \textbf{\color{red}0.7176} & \textbf{\color{red}0.7167}\\
2.444 & 0.05881 & 0.9636 & 0.1568 & 0.507 & \textbf{\color{red}0.7272}\\
3 & 0.03641 & 1 & 0.3502 & 0.3502 & \textbf{\color{red}0.9058}\\
\hline
\end{tabular}
\caption{Best certification strategy according to the Bayesian approach.
Only $14$ of at most $15$ measurements are performed and they are distributed among the $3$ settings as $[6 4 4]$.
The color coding is the same as in Table \ref{tab_bestStratFreq}.}
\label{tab_bestStratBayes}
\end{table}
\subsection{Comparison and Remarks}
Comparing the two strategies used for constructing the Tables \ref{tab_bestStratFreq} and \ref{tab_bestStratBayes}, one notes the following differences: (i) the choice of measurement settings might differ for the Frequentist and the Bayes approach;
(ii) The column showing the confidence $q_F$ is monotonous, whereas the column of the credibility $q_B$ is not.
In Fig.~\ref{FIG_FreqVsBayes}, those two columns are visually compared for both tables.
It is important to note that although we subsume confidence and credibility by the term ``validity'', they measure different quantities. Although in the two tables the efficiency in the Frequentist framework is always larger for a given validity than in the Bayesian framework, this conclusion does not always hold. Namely, the efficiency in the Bayesian case depends on the prior which could have been chosen such that the Bayesian efficiency was larger than Frequentist.
This just emphasises again that confidence and credibility are not answering the same question.
\begin{figure}
\caption{
Comparison of two choices of assigning different number of measurement runs to different settings. The left panel is for the choice optimal within the Frequentist framework and the right panel within the Bayesian framework, for a targeted validity of $0.7$.
By construction, the confidence ($q_F$) is monotonous, whereas the credibility ($q_B$) is not.
A distribution of measurement settings is called optimal (within one of the frameworks) if it maximizes the efficiency, i.e., more entangled states are detected than for all other distributions of measurement settings.
}
\label{FIG_FreqVsBayes}
\end{figure}
\end{document}
|
\begin{document}
\begin{frontmatter}
\title{Law of the absorption time of some positive self-similar Markov processes\thanksref{T1}}
\runtitle{Absorption time of some Markov processes}
\thankstext{T1}{Supported in part by Swiss National Fund Grant
2000021--121901.}
\begin{aug}
\author[A]{\fnms{P.} \snm{Patie}\corref{}\ead[label=e1]{[email protected]}}
\runauthor{P. Patie}
\affiliation{Universit\'{e} Libre de Bruxelles}
\address[A]{D\'{e}partement de Math\'{e}matiques\\
Universit\'{e} Libre de Bruxelles\\
Boulevard du Triomphe\\
B-1050, Bruxelles\\
Belgique\\
\printead{e1}}
\end{aug}
\received{\smonth{4} \syear{2010}}
\revised{\smonth{12} \syear{2010}}
\begin{abstract}
Let $X$ be a spectrally negative self-similar Markov process with $0$
as an absorbing state. In this paper, we show that the distribution of
the absorption time is absolutely continuous with an infinitely
continuously differentiable density. We provide a power series and a
contour integral representation of this density. Then, by means of
probabilistic arguments, we deduce some interesting analytical
properties satisfied by these functions, which include, for instance,
several types of hypergeometric functions. We also give several
characterizations of the Kesten's constant appearing in the study of
the asymptotic tail distribution of the absorbtion time. We end the
paper by detailing some known and new examples. In particular, we
offer an alternative proof of the recent result obtained by Bernyk,
Dalang and Peskir [\textit{Ann. Probab.} \textbf{36} (2008) 1777--1789]
regarding the law of the maximum of spectrally positive L\'{e}vy stable
processes.
\end{abstract}
\begin{keyword}[class=AMS]
\kwd{60E07}
\kwd{60G18}
\kwd{60G51}
\kwd{33E30}.
\end{keyword}
\begin{keyword}
\kwd{Self-similar processes}
\kwd{absorption time}
\kwd{L\'{e}vy processes}
\kwd{exponential functional}
\kwd{generalized hypergeometric
functions}.
\end{keyword}
\pdfkeywords{60E07, 60G18, 60G51, 33E30, Self-similar processes,
absorption time, Levy processes, exponential functional,
generalized hypergeometric functions}
\end{frontmatter}
\section{Introduction}
Let $X=((X_t)_{t\geq0}, (\mathbb{Q}_x)_{x>0})$ be a self-similar Hunt process
with values in $ [0,\infty)$. It means that $X$ is a right-continuous
strong Markov process with quasi-left continuous trajectories and there
exists $\alpha>0$ such that $X$ enjoys the following self-similarity
property: for each $c>0$ and $x\geq0$,
\[
\mbox{the law of the process } (c^{-1}X_{c^{\alpha}t})_{t\geq0},
\mbox{under } \mathbb{Q}_x, \mbox{ is } \mathbb{Q}_{x/c}.
\]
$1/\alpha$ is called the index of self-similarity. The purpose of the
paper is to describe the law of the stopping time
\[
T_0=\inf\{s>0; X_s=0\}
\]
with the usual convention that $\inf\{\varnothing\}=\infty$. The class of
positive self-similar Markov processes (for short pssMp) has been
introduced and studied by Lamperti~\cite{Lamperti-72}. In particular,
he showed that for each fixed $\alpha> 0$, there is a~bijective correspondence
between pssMp with index $\alpha$ and (possibly killed) real-valued L\'{e}vy
processes, that is, processes with stationary and independent increments.
More specifically, by introducing the additive functional
\[
\Sigma_t=\inf\biggl\{s>0; A_s =\int_0^sX_r^{-\alpha}\,dr>t\biggr\},
\]
Lamperti \cite{Lamperti-72} showed that the process $\xi=(\xi
_t)_{t\geq
0}$, defined by
\begin{equation} \label{eq:ss}
\xi_t = \log(X_{\Sigma_t}),\qquad 0\leq t<T_0,
\end{equation}
is a (possibly killed) L\'{e}vy process. We denote the law of the
process $\xi$ when starting at $0$ by $\mathrm{P}$.
It is plain that
\[
\Sigma_t = \int_0^t e^{\alpha\xi_s} \,ds
\]
and writing $q\geq0$ for the killing rate of the L\'{e}vy process, one
gets the identity in distribution
\[
(T_0,
\mathbb{Q}_x)\stackrel{(d)}{=}(x^{\alpha}\Sigma_{{\mathbf{e}}_{q}},\mathrm{P}),
\]
where ${\mathbf{e}}_q$ is an independent exponential random variable of parameter
$q$ (we have ${\mathbf{e}}_0=\infty$). Lamperti \cite{Lamperti-72} explained
that, either $q>0$ and $X$ reaches $0$ by a jump, that is,
\[
\mathbb{Q}_x(X_{T_{0-}}>0, T_0 <\infty)=1\qquad \forall x>0,
\]
or $\xi$ drifts to $-\infty$ and $X$ reaches $0$, that is,
\[
\mathbb{Q}_x(X_{T_{0-}}=0, T_0 <\infty)=1\qquad \forall x>0.
\]
We gather these two possibilities in the following hypothesis.
\begin{longlist}[H:]
\item[H:]\hypertarget{HypoH}\mbox{}
\textit{Either} $q>0$ \textit{or}
$\lim_{t\rightarrow\infty} \xi_t = -\infty$ \textit{a.s.}
\textit{and} $q=0$.
\end{longlist}
The law of $T_0$ or equivalently of $\Sigma_{{\mathbf{e}}_{q}}$ turns out
to be a
key object in various settings. It appears, for instance, in the study
of coagulation-fragmentation processes \cite{Bertoin-02-f} and
continuous state branching processes with immigration \cite
{Patie-CBI-09}. We also mention that, recently, in the SLE context,
Alberts and Sheffield~\cite{Alberts-Sheffield-08} describe a
measure-valued function supported on the intersection of a~chordal
SLE$(\kappa)$ curve with $\mathbb{R}$, $4<\kappa<8$, in terms of the law of the
absorption time~$T_0$ of some Bessel processes which form the class of
pssMp having continuous trajectories.
The law of $\Sigma_{{\mathbf{e}}_{q}}$ is also critical for the pricing
of Asian
options in mathematical finance (see, e.g., \cite
{Patie-Asian-09}), but also for computing perpetuities in insurance
mathematics (see, e.g., \cite{Dufresne-90}).
Unfortunately, beside some isolated cases the distribution of $T_0$ is
not attainable. We mention the papers
\cite{Carmona-Petit-Yor-97,Gjessing-Paulsen-97} and \cite{Patie-CBI-09} where such
examples can be found and refer to the survey paper \cite{Bertoin-Yor-05} for a
description of these cases. Besides, two notable exceptions might be
worth mentioning: when $X$ is a Bessel process of negative\vadjust{\goodbreak} index and
when $X$ is a regular spectrally negative stable L\'{e}vy process
killed upon entering the negative half-line. In the former case,
several proofs can be found in the literature, see, for instance, the
excellent monograph of Yor \cite{Yor-01} and the more recent survey
papers of Matsumoto and Yor \cite{Matsumoto-Yor-05-1} and \cite
{Matsumoto-Yor-05-2}. However, most of the proofs rely on the knowledge
of the semigroup of Bessel processes. For the second case,
Bernyk, Dalang and Peskir \cite{Bernyk-Dalang-Peskir-08} derive a representation of
the distribution of $T_0$ by inverting, in a nontrivial way, the known
expression of the Wiener--Hopf factorization of stable one-sided L\'
{e}vy processes. Our approach will differ from these two cases since we
do not have, in general, access neither to the semigroup of $X$ nor to
the Laplace transform of $T_0$.
The remaining part of the paper is organized as follows. In the next
section, we state our main results including the smoothness and the
representation as an absolutely convergent power series of the
distribution of $T_0$.
The proof of these results is presented in Section \ref{sec3}. Finally, in the
last section, we present a few consequences of the main result and we
detail some known and new examples. We also mention that some of the
results stated in Theorem~\ref{thm:2} below were announced without
proofs in the note \cite{Patie-09-cras}.
\section{Main results}\label{sec2}
Henceforth, we assume that $X$ is a pssMp of index $1/\alpha>0$ and of
the spectrally negative type. It means that it is associated via the
Lamperti mapping to a possibly killed L\'{e}vy process $\xi$ which is
spectrally negative. We exclude the cases when $\xi$ is degenerate,
that is, when $\xi$ is the negative of a subordinator or a pure drift
process. We recall that $\mathrm{P}$ (resp., $\mathbb{E}l$) stands for the law
(resp., the expectation operator) of $\xi$ with $\xi_0=0$. The law of
$\xi$ is determined by its Laplace exponent $\bar{\psi}(u)=\psi
(u)-q$, where $q\geq0$ is the killing rate and $\psi$ admits the
following L\'{e}vy--Khintchine representation: for any $u\geq0$,
\[
\psi(u) = \bar{b} u + \frac{\sigma}{2} u^2 + \int_{-\infty}^0
\bigl(e^{u r} -1
-ur{{\mathbb{I}}}_{\{|r|<1\}} \bigr)\nu(dr),
\]
where $ \bar{b}\in\mathbb{R}, \sigma\geq0$ and the measure
$\nu$ is such that $\int_{-\infty}^0 (1
\wedge r^2 ) \nu(dr) <+ \infty$.
We shall refer to $\xi$ (resp., $\bar{\psi}$) as the underlying
L\'
{e}vy process (resp., Laplace exponent) of $X$.
Let us now proceed by recalling some basic properties of the Laplace
exponent $\psi$, which can be found, for instance, in Bertoin \cite
{Bertoin-96}. First, it is plain that $\lim_{u \rightarrow
\infty}\psi(u)=+\infty$ and by monotone convergence, one gets
$\mathbb{E}l[\xi_1]= \bar{b} +\int_{-\infty}^{-1}r\nu(dr) \in[-\infty
,\infty)$.
We shall also need the value of the constant $\Lambda=\lim
_{u\rightarrow\infty} \frac{\psi(\alpha u)}{u}$ which is given (see
\cite{Bertoin-96}, Corollary VII.5) by
\[
\Lambda= \cases{
\displaystyle \alpha b =\alpha\biggl(\bar{b}-\int_{-1}^0 r \nu(dr)\biggr), &\quad if
$\sigma=0$ and $\displaystyle \int_{-\infty}^0 (1 \wedge
r)\nu(dr)<\infty$,\vspace*{2pt}\cr
+\infty, &\quad otherwise.}
\]
Since we have excluded the degenerate cases, we easily check that
$b>0$. Next, we recall that the\vadjust{\goodbreak} mapping $u\mapsto\psi(u)$ is
continuous and increasing on $[\phi(0),\infty)$, where $\phi(0)$ stands
for the largest solution to the equation $\psi(u)=0$. Thus, $\psi$
has a
well-defined inverse function $\phi\dvtx[0,\infty)\rightarrow
[\phi(0),\infty)$ which is also continuous and increasing. In order to
simplify the notation we write, for any $q\geq0$, $\gamma=\phi(q)>0$.
Then, it is easily seen that
\[
\mathbb{E}l[e^{ \gamma\xi_1}] = 1.
\]
We also note that the condition \hyperlink{HypoH}{H} is equivalent to the
requirement $\phi(q)>0$.
Next, we set $\psi_{\gamma}(u)=\psi(u+\gamma)-\psi(\gamma)$ and
observing that $\psi_{\gamma}(0)=0$, we deduce that $\psi_{\gamma}$ is
the Laplace exponent of a conservative spectrally negative L\'{e}vy
process. We also point out that $\psi_{\gamma}'(0^+)=\psi'(\gamma)>0$
and $\lim_{u\rightarrow\infty} \frac{\psi_{\gamma}(u)}{u}=\lim
_{u\rightarrow\infty} \frac{\psi(u)}{u}$.
We proceed by introducing more notation taken from Patie
\cite{Patie-OU-06} and \cite{Patie-06c}. First, for a function $f$ and for
any $\alpha>0$, we write
\[
a_{s}(f;\alpha) = \prod_{k=1}^{\infty}\frac{f(\alpha(k
+s)
)}{f(\alpha k)}, \qquad s\in\mathbb{C},
\]
whenever the infinite product exists. Note that, for instance, $a_0(\psi
;\alpha)=1$ and for any $n=1,2,\ldots,$
\begin{equation} \label{eq:coef}
a_n(\psi;\alpha) =\Biggl( \prod_{k=1}^n\psi(\alpha k)\Biggr)^{-1}.
\end{equation}
Next, we introduce, for any $\rho\in\mathbb{C}$ such that $\mathfrak
{Re}(\rho)>0$, the
power series
\begin{equation} \label{eq:f1}
\mathcal{I}p(\rho;z)= \frac{1}{\Gamma(\rho)}\sum_{n=0}^{\infty}
a_n(\psi;\alpha)
\Gamma(\rho+n) z^{n},
\end{equation}
where $\Gamma$ stands for the Gamma function. By means of classical
criteria, it is easily seen that the function $z \mapsto\mathcal{I}p(\rho;z)$
is analytic in the disc $ \{z \in\mathbb{C};\break |z|<\Lambda\}$. In particular,
in the case $\Lambda=+\infty$, that is, when the process $\xi$ has paths
of unbounded variations, $\mathcal{I}p(\rho;z)$ is an entire function in $z$.
Moreover, for any $|z|<\Lambda$, the mapping $\rho\mapsto\mathcal{I}p(\rho;z)$
is a meromorphic function defined for all complex numbers $\rho$ except
at the poles of the Gamma function, which are the points $\rho
=0,-1,\ldots$ However, they are removable singularities. Indeed, for
any $|z|<\Lambda$ and any integer $N\in\mathbb{N}$, one has, by means of the
recurrence relation $\Gamma(z+1)=z\Gamma(z)$,
\[
\mathcal{I}p(0;z)=1
\]
and
\[
\mathcal{I}p(-N;z) = \sum_{n=0}^{N}(-1)^n \frac{\Gamma(N+1)}{\Gamma
(N+1-n)}a_n(\psi;\alpha) z^{ n}.
\]
Thus, by uniqueness of the analytic continuation, for any $|z|<\Lambda
$, $\mathcal{I}p(\rho;z)$ is an entire function in $\rho$. Before stating our
main result,\vadjust{\goodbreak} we show that in the case $\Lambda=\alpha b$, the power
series (\ref{eq:f1}) can be represented, in the left half-plane, as
another convergent power series which corresponds to an analytic
continuation in this domain. To this end, we aim to use the co-called
Euler transformation; see, for example, \cite{Norlund-55}, page 294.
However, this transformation can be performed if and only if the
singularity of the function $\mathcal{I}pg(\rho;z)$ on the circle $|z|=\Lambda$
is located at the point $z=\Lambda$. In order to show that our family
of functions satisfies this property, we first provide a contour
integral representation of $\mathcal{I}pg(\rho;z)$ which turns out to be an
analytic continuation in the entire complex plane cut along the
positive real axis. Then, we are able to apply the Euler
transformation to derive a series representation.\vspace*{-2pt}
\begin{prop} \label{prop:ac}
Let $\Lambda=\alpha b$, then $\mathcal{I}pg(\rho;z)$ is analytic in the disc
$|z|<\alpha b$ and for any fixed $\rho=0,-1,\ldots,$ the mapping
$z\mapsto\mathcal{I}pg(\rho;z)$, as a~polynomial, is an entire function.
Moreover, for any $\rho\neq0,-1,\ldots, \mathcal{I}pg(\rho;z)$ admits an
analytic continuation in the entire complex plane cut along the
positive real axis given by
\begin{eqnarray} \label{eq:ci}
\mathcal{I}pg(\rho;z) = \frac{1}{2i \pi\Gamma(\rho)}\int_{-i\infty
}^{i\infty}
a_s(\varphi_{\gamma};\alpha) \Gamma(s+\rho)\Gamma(-s)
\biggl(-\frac
{z}{\alpha}\biggr)^{s} \,ds,\nonumber\\[-8pt]\\[-8pt]
&&\eqntext{|{\arg}(-z)|<\pi,}
\end{eqnarray}
where the contour is indented to ensure that all poles
(resp., nonnegative poles) of $\Gamma(\rho+s)$ [resp., $\Gamma(-s)$] lie
to the left (resp., right) of the intended imaginary axis.
Consequently, for any $\rho\in\mathbb{C}$, $\mathcal{I}pg(\rho;z)$ admits, in the
half-plane $\mathfrak{Re}(z)<\frac{\alpha b}{2}$, the following power series
representation
\begin{equation}\label{eq:h}\qquad
\mathcal{I}pg(\rho;z) = \biggl(1-\frac{z}{\alpha b}\biggr)^{-\rho} \sum
_{n=0}^{\infty} \mathcal{I}pg(-n;\alpha b)\frac{\Gamma(\rho+n)}{n!\Gamma
(\rho
)}\biggl(\frac{z}{z-\alpha b}\biggr)^n.
\end{equation}
Finally, for any fixed $\mathfrak{Re}(z)<\frac{\alpha b}{2}$, $\mathcal{I}pg
(\rho;z)$ is
an entire function in the argument~$\rho$.\vspace*{-2pt}
\end{prop}
\begin{remark}
A specific instance of the mapping $\mathcal{I}_{\psi_{\gamma}}(\rho;x)$ when
$\Lambda=\alpha b$, is the hypergeometric function ${}_2F_1$. In this
case, the representation (\ref{eq:h}) is known as the Euler
transformation which has the remarkable feature that the power series
on the right-hand side of (\ref{eq:h}) is still an hypergeometric
function ${}_2F_1$. We refer to the Section \ref{ex:3} below for more
details on this example.\vspace*{-2pt}
\end{remark}
We are now ready to state our main result.\vspace*{-2pt}
\begin{theorem} \label{thm:2}
Let $q\geq0$, assume that $\phi(q)>0$ and set $\gamma=\phi(q)$ and
$\gamma_{\alpha}=\gamma/\alpha$.
Then, there exists a constant $C_{\gamma}>0$ such that
\begin{equation} \label{eq:const1}
\mathcal{I}_{\psi_{\gamma}}(\gamma_{\alpha};-t) \sim \frac{t^{-\gamma
_{\alpha
}}}{C_{\gamma}} \qquad\mbox{as } t\rightarrow\infty\vadjust{\goodbreak}
\end{equation}
($f(t)\sim g(t) $ as $t\rightarrow a$ means that $\lim_{t\rightarrow
a}\frac{f(t)}{g(t)}=1$ for any $a \in[0,\infty]$) and
\begin{equation}\label{eq:pd}
S(t) = C_{\gamma}t^{-\gamma_{\alpha}} \mathcal{I}_{\psi_{\gamma}}(\gamma
_{\alpha
};-t^{-1}),\qquad t>0,
\end{equation}
where, by self-similarity, we have set $S(tx^{-\alpha})=\mathbb{Q}
_{x}(T_0\geq
t), x,t>0$.
Finally, the law of $T_0$ under $\mathbb{Q}_1$ is absolutely continuous with an
infinitely continuously differentiable density denoted by $s$ and given by
\[
s(t) = \gamma_{\alpha} C_{\gamma} t^{-\gamma_{\alpha}-1} \mathcal{I}
_{\psi
_{\gamma}}(1+\gamma_{\alpha};-t^{-1}), \qquad t>0.
\]
\end{theorem}
\begin{remark}
In the case $\Lambda=\infty$, we easily check that, for any
$\mathfrak{Re}
(\rho
)>0$, the mapping $x\mapsto\mathcal{I}_{\psi_{\gamma}}(\rho;x)$ is increasing
on $[0,\infty)$. Hence, we deduce from the above theorem that the
entire function $z\mapsto\mathcal{I}_{\psi_{\gamma}}(\gamma_{\alpha};z)$
has no
real zeros.
\end{remark}
In the above theorem, the constant $C_{\gamma}$ is characterized by the
behavior of the function $\mathcal{I}_{\psi_{\gamma}}(\gamma_{\alpha};-t)$ for
large values of $t$. In what follows, we provide some representations
of this constant in terms of the Laplace exponent $\psi_{\gamma}$.
\begin{prop} \label{prop:ck}
\begin{longlist}[(1)]
\item[(1)] If $\Lambda=\alpha b$, then
\[
C_{\gamma} = \alpha^{\gamma_{\alpha}} a_{-\gamma_{\alpha
}}(\varphi
_{\gamma};\alpha),
\]
where $\varphi_{\gamma}(u)= b- \int_0^{\infty}e^{-ur}\int_{-\infty
}^{-r}e^{\gamma v}\nu(dv)\,dr$.
\item[(2)] Otherwise, we have
\[
C_{\gamma} =\cases{\psi'_{\gamma}(0^+), &\quad if $\gamma_{\alpha} =
1$,\cr
\displaystyle \alpha^n\psi'_{\gamma}(0^+)\Biggl(\prod_{k=1}^n\varphi_{\gamma
}(\alpha
k)\Biggr)^{-1}, &\quad if $\gamma_{\alpha} = n+1, n=1,2\ldots,$
\cr
\displaystyle \frac{ \alpha^{2\gamma_{\alpha}}}{\Gamma(1-\gamma_{\alpha
})}a_{-\gamma
_{\alpha}}(\bar{\varphi}_{\gamma};\alpha), &\quad otherwise,}
\]
where $\varphi_{\gamma}(\alpha u)= \psi_{\gamma}(\alpha u) / \alpha
u$ and
\[
\bar{\varphi}_{\gamma}( u)=\frac{\hat{b}}{ u} + \frac{\sigma}{2}
+ \int
^{\infty}_0 e^{- u r}\int_{-\infty}^{-r}\int_{-\infty
}^{-s}e^{\gamma
v}\nu(dv) \,ds \,dr
\]
with $\hat{b}=\bar{b}+\sigma\gamma+\int_{-\infty}^0
(e^{\gamma
r}-{{\mathbb{I}}}_{\{|r|<1\}})r\nu(dr)$.
\item[(3)] Finally, if $q=0$ and $0<\gamma_{\alpha}<1$, then
\[
\mathcal{I}p(r)\sim C_{\gamma} \Gamma(1-\gamma_{\alpha} )
r^{\gamma_{\alpha}}\mathcal{I}_{\psi_{\gamma}}(r)
\qquad\mbox{as } r\rightarrow
\infty,
\]
where $\mathcal{I}p(r)= \sum_{n=0}^{\infty}
a_n(\psi;\alpha) r^{n}$ is an entire function.
\end{longlist}
\end{prop}
\section{Proofs}\label{sec3}
\subsection{A useful analytic continuation} \label{sec:pt1}
The first claim of Proposition \ref{prop:ac} follows from the
discussion preceding the proposition. Thus, let us assume that\vadjust{\goodbreak}
$\rho\neq0,-1,\ldots$ Since $\psi_{\gamma}'(0^+)>0$, $\psi
_{\gamma}$
is well defined and analytic in the positive right half-plane and $\psi
_{\gamma}(u)>0$ for any $u>0$. Our next aim is to extend the
coefficients $a_n(\psi_{\gamma},\alpha)$ to a function of the complex
variable. Since the paths of the L\'{e}vy process $\xi$ are of bounded
variation, its Laplace exponent $\psi_{\gamma}$ admits the following
representation (see \cite{Bertoin-96}, Section VII.3):
\[
\psi_{\gamma}(u)= u\bigl(b-\hat{v}_{\gamma}(u)\bigr),
\]
where $\hat{v}_{\gamma}(u)=\int_0^{\infty}e^{-ur}\int_{-\infty
}^{-r}e^{\gamma v}\nu(dv)\,dr$. Thus, for any $n\geq0$, we have
\[
a_n(\psi_{\gamma},\alpha)=\frac{1}{\Gamma(n+1)\alpha
^n}a_n(\varphi
_{\gamma};\alpha)
\]
with $a_n(\varphi_{\gamma};\alpha)^{-1}=\prod_{k=1}^n \varphi
_{\gamma
}(\alpha k)$ and $a_0(\varphi_{\gamma};\alpha)=1$.
It is plain that the mapping $\hat{v}_{\gamma}$ is analytic in
$F_{-\gamma}=\{s\in\mathbb{C}; \mathfrak{Re}(s)> -\gamma\}$ and $\hat
{v}_{\gamma
}(u)$ is decreasing on~$\mathbb{R}^+$ with $0<\hat{v}_{\gamma}(0)< b$ since
$\psi_{\gamma}(0^+)>0$. Then, we may write
\begin{eqnarray*}
a_{s}(\psi_{\gamma};\alpha)&=& \frac{1}{\Gamma(s+1)\alpha
^s}a_s(\varphi
_{\gamma};\alpha)\\
&=&\frac{1}{\Gamma(s+1)\alpha^s}\prod
_{k=1}^{\infty
}\frac{\varphi_{\gamma}(\alpha(k +s))}{\varphi_{\gamma}(\alpha k)},
\end{eqnarray*}
where the infinite product is easily seen to be absolutely convergent
for any $\mathfrak{Re}(s)>0$ by taking the logarithm and noting that
$|\hat
{v}_{\gamma}(s)|\leq\hat{v}_{\gamma}(\mathfrak{Re}(s))$; see, for
example, \cite{Titchmarsh-39}, Section 1.41. Moreover, $a_s(\varphi
_{\gamma};\alpha)$ satisfies the functional equation
\[
a_{s+1}(\varphi_{\gamma};\alpha)=\frac{1}{\varphi_{\gamma}(\alpha
(s+1))}a_{s}(\varphi_{\gamma};\alpha),
\]
which shows that $a_s(\varphi_{\gamma};\alpha)$ is analytic in the
half-plane $F_{-\gamma-1}=\{s\in\mathbb{C};\break \mathfrak{Re}(s)> -1-\gamma\}$.
Consequently, $a_{s}(\varphi_{\gamma};\alpha)$ is bounded on any closed
subset of $F_{-\gamma-1}$.
Then, we set $G(s)=\Gamma(s+\rho)\Gamma(-s) a_{s}(\varphi_{\gamma
};\alpha)$ and define
\[
\mathfrak{I}_{\mathfrak{L}_R} =- \frac{1}{2i \pi\Gamma(\rho)}\int
_{\mathfrak{L}_R}G(s) \biggl(-\frac{z}{\alpha}\biggr)^s \,ds,
\]
where the integral is taken in a clockwise direction round the contour
$\mathfrak{L}_R$, consisting of a large semi-circle, of center the
origin and radius $R$, lying to the right of the imaginary axis. This
contour is intended to ensure that all poles (resp., nonnegative poles)
of $\Gamma(\rho+s)$ [resp., $\Gamma(- s)$] lie to the left (resp., right)
of the intended imaginary axis. This contour is always possible since
we have assumed that $\rho\neq0,-1,\ldots.$ We can split $\mathfrak
{I}_{\mathfrak{L}_R}$ up into two integrals,~$\mathfrak{I}_{\mathfrak
{A}_{iR}}$ along the imaginary axis and, writing $s=Re^{i\theta}$,
\[
\mathfrak{I}_{\mathfrak{C}_R}=-\frac{1}{2\pi i}\int_{-\pi/2}^{\pi
/2}G(Re^{i\theta}) \biggl(-\frac{z}{\alpha}\biggr)^{Re^{i\theta
}}Re^{i\theta} \,d\theta.
\]
Recalling the following well-known
asymptotic formulae (see,
e.g., \cite{Paris-Kaminski-01}, Section~2.4), as $|s|\rightarrow
\infty$,
\begin{eqnarray*}
\Gamma(s+ \rho) &\sim&\sqrt{2\pi}
e^{-Re^{i\theta}}R^{Re^{i\theta}+\rho-{1}/{2}}e^{i\theta
(Re^{i\theta
}+\rho-{1}/{2})}, \qquad |\theta| < \pi,\\
\Gamma(-s) &\sim&e^{-\pi R|{\sin\theta}|}
e^{Re^{i\theta}}R^{-Re^{i\theta}-{1}/{2}}e^{i\theta
(-Re^{i\theta
}-{1}/{2})},\qquad |\theta| < \pi,
\end{eqnarray*}
and
\[
\biggl|\biggl(-\frac{z}{\alpha}\biggr)^{s}\biggr|\sim|\alpha
z|^{R\cos
\theta}e^{-R\sin\theta\arg(-z)},
\]
we deduce that as $|s|\rightarrow\infty$
\begin{eqnarray}\label{eq:ae}
\biggl|G(s)\biggl(-\frac{z}{\alpha}\biggr)^{s}\biggr| &\sim&
a R^{\mathfrak{Re}(\rho)-1}|\alpha z|^{R\cos\theta}\nonumber\\[-8pt]\\[-8pt]
&&{}\times\cases{ e^{-R
|{\sin
\theta}|
(\pi+\arg(-z))}, &\quad $0<\theta\leq
\pi/2$,\cr
e^{-R |{\sin\theta}| (\pi-\arg(-z))}, &\quad
$-\pi/2\leq\theta<0$,}\nonumber
\end{eqnarray}
where $a$ is a positive constant.
On the one hand, along the path ${\mathfrak{A}_{iR}}$ we have $\theta
=\pm\frac{\pi}{2}$ and thus as $|z|\rightarrow\infty$
\[
\biggl|G(s) \biggl(-\frac{z}{\alpha}\biggr)^{s}\biggr| \sim
a R^{\mathfrak{Re}(\rho)-1}e^{\pm({\pi}/{2}) \mathcal{I}m(\rho)} \cases
{ e^{-R
(\pi
+\arg(-z))}, &\quad
$\theta=\pi/2$,\cr
e^{-R (\pi-\arg(-z))}, &\quad $\theta=-\pi/2$.}
\]
For the integral (\ref{eq:ci}) to converge absolutely, it is therefore
required that $|{\arg}(-z)|<\pi$. On the other hand,
the asymptotic estimate (\ref{eq:ae}) gives, as $R\rightarrow\infty$,
\[
\mathfrak{I}_{\mathfrak{C}_R}\rightarrow0 \qquad\mbox{if } |z|<1
\mbox{ and } |{\arg}(-z)|<\pi.
\]
Thus, as $R\rightarrow\infty$,
\[
\mathfrak{I}_{\mathfrak{L}_R} \rightarrow-\frac{1}{2i \pi}\int
_{-i\infty}^{i\infty}G(s) \biggl(-\frac{z}{\alpha}\biggr)^{s} \,ds.
\]
Finally, evaluating $\mathfrak{I}_{\mathfrak{L}_R}$ by the Cauchy
integral theorem and letting $R\rightarrow\infty$, we get
\begin{eqnarray}
\frac{1}{2i\pi}\int_{-i\infty}^{i\infty}G(s) \biggl(-\frac
{z}{\alpha
}\biggr)^{s} \,ds=\frac{1}{\Gamma(\rho)}\sum_{n=0}^{\infty}
a_n(\psi_{\gamma},\alpha)
\Gamma(\rho+n) z^{n}, \nonumber\\
&&\eqntext{|z|<1 \mbox{ and } |{\arg}(-z)|<\pi.}
\end{eqnarray}
Therefore, the integral (\ref{eq:ci}) offers an analytic continuation
of the mapping $z\mapsto\mathcal{I}pg(\rho;z)$ in the entire complex plane cut
along the positive real axis.
Moreover, we deduce from such an analytic continuation that the power
series (\ref{eq:f1}) has an unique singularity on the circle
$|z|=\alpha b$ located at the point $z=\alpha b>0$. Now, following a
device developed for hypergeometric series (see N{\o}rlund\vadjust{\goodbreak}
\cite{Norlund-55}, pages 294 and 295), we introduce the function $\mathcal{H}$
defined for some $a\in\mathbb{C}$ by
\[
\mathcal{H}_{\psi_{\gamma},a}(\rho;z) = (1-z)^{-\rho}
\mathcal{I}pg
\biggl(\rho;\frac{a \alpha b z}{z-1} \biggr).\vspace*{-1pt}
\]
Note that
\begin{equation}\label{eq:hir}
\mathcal{I}pg(\rho;az) = \biggl(1-\frac{z}{\alpha b}\biggr)^{-\rho}
\mathcal
{H}_{\psi_{\gamma},a}\biggl(\rho;\frac{z}{z-\alpha b} \biggr).\vspace*{-1pt}
\end{equation}
Thus, denoting by $(b_n)_{n\geq0}$ the coefficients of the power series
$\mathcal{H}_{\psi_{\gamma},a}(\rho;z)$, we have $b_0=a_0$ and by means
of residues calculus, with $\mathfrak{C}$ a circle around $0$ of small
radius and with positive orientation, we have for $n\geq1$,
\begin{eqnarray*}
b_n &=& \frac{1}{2\pi i } \int_{\mathfrak{C}}
\frac
{\mathcal{H}_{\psi_{\gamma},a}(\rho;z)}{z^{n+1}}\,dz
\\[-2pt]
&=& (-1)^n\frac{1}{2\pi i } \int_{\mathfrak{C}}(1-z
)^{-\rho} \mathcal{I}pg\biggl(\rho;\frac{a \alpha b z}{z-1} \biggr) \frac
{dz}{z^{n+1}}
\,dv\\[-2pt]
&=& \frac{1}{\Gamma(\rho)}\sum_{k=0}^{n}(-a \alpha b)^{k}a_k(\psi
_{\gamma};\alpha)
\frac{\Gamma(\rho+n)}{\Gamma(n-k+1)}.\vspace*{-1pt}
\end{eqnarray*}
Thus, one gets
\[
(1-z)^{-\rho} \mathcal{I}pg\biggl(\rho;\frac{a \alpha b z}{z-1} \biggr)
=\sum
_{n=0}^{\infty}\mathcal{I}pg(-n;a \alpha b) \frac{\Gamma(\rho+n)}{\Gamma
(\rho
)n!} z^n.\vspace*{-1pt}
\]
From Weierstrass's double series theorem, the above identity is true if
$|z|<\frac{1}{1+|a|}$. Moreover, the function on the left-hand side has
a singularity at $z=1$ and $z=\frac{1}{1-a}$. Thus, the series on the
right-hand side is convergent if $|z|<1$ and $|z(1-a)|<1$. By choosing
$a=1$, we conclude by observing that the series on the right-hand side
of (\ref{eq:hir}) is convergent for $\mathfrak{Re}(z)<\frac{\alpha
b}{2}$.\vspace*{-2pt}
\subsection{The distribution of $T_0$} \label{sec:pt}
We proceed by introducing the Ornstein--Uhlenbeck process
$U=(U_t)_{t\geq
0}$ defined by
\[
U_t=e^{\tilde{\alpha}t}X_{\tilde{\alpha}u(t)},\qquad t\geq0,\vspace*{-1pt}
\]
where $\tilde{\alpha}=\alpha^{-1}$ and $\tilde{\alpha}u(t)=1-e^{-t}$. Next, we put
\[
H_0=\inf\{s>0; U_s=0\}\vspace*{-1pt}
\]
and set
\[
1-K(x) = \mathbb{Q}_x(H_0<\infty), \qquad x>0.\vspace*{-1pt}
\]
We are now ready to state the following.\vspace*{-2pt}
\begin{prop} \label{thm:1}
Assume that the condition \textup{\hyperlink{HypoH}{H}} holds. Then, for any $x>0$ and
$t>0$, we have
\begin{equation} \label{eq:ii}
K(xt^{-\tilde{\alpha}}) = \mathbb{Q}_x( T_0\geq t)\vspace*{-1pt}
\end{equation}
and $P$ is increasing on $\mathbb{R}^+$ with\vadjust{\goodbreak} $\lim_{x\rightarrow\infty
}K(x)=1$ and $K(0)=0$.
\end{prop}
\begin{pf}
First, a simple time change yields the following identity in
distribution:
\[
H_0\stackrel{(d)}{=}-\log(1-{T_0}\wedge1).
\]
Thus, we deduce that
\begin{eqnarray*}
1-K(x) &=&\mathbb{Q}_x(H_0<\infty)\\
&=&\mathbb{Q}_x( T_0<1).
\end{eqnarray*}
Then, invoking the self-similarity property of $X$ we obtain the identity
\[
\mathbb{Q}_x( T_0\geq t)=\mathbb{Q}_{xt^{-\tilde{\alpha}}}( T_0\geq1)
\]
from which we deduce the identity (\ref{eq:ii}) and the properties
stated on $P$.
\end{pf}
According to Proposition \ref{thm:1}, our goal now is to derive an
expression of the function $K(x)=1-\mathbb{Q}_{x}(H_0< \infty), x>0$.
Relying on the following identity:
\[
K(x)= \lim_{a\rightarrow\infty} \lim_{q\rightarrow0} \mathbb{E}_{x}
\bigl[e^{-qH_a}\mathbb{I}_{\{H_a<H_0\}}\bigr],
\]
where $H_a = \inf\{s>0; U_s\geq a\}$, the problem reduces to the
computation of the functional $\mathbb{E}_{x}[e^{-qH_a}\mathbb{I}_{\{
H_a<H_0\}}]$. Actually, for technical reasons, we must deal first
with the functional $\mathbb{E}^{(\gamma)}_{x}[e^{-qH_a}]$ which is
the Laplace transform of the first passage time above for the
Ornstein--Uhlenbeck process associated to the pssMp $X$ with underlying
Laplace exponent $\psi_{\gamma}$. Finally, by means of Doob h-transform
arguments, we will be able to relate the latter functional to the
former one.
We use the notation introduced in Theorem \ref{thm:2} and take first
$X$ with underlying Laplace exponent $\psi_{\gamma}$. We denote its law
(resp., its expectation operator) by $\mathbb{Q}^{(\gamma)}$ (resp., $\mathbb{E}
^{(\gamma
)}$). In order to simplify the notation we set, without loss of
generality, $\alpha=1$. We recall that $\psi_{\gamma}(0)=0$ and
$\psi
_{\gamma}(0^+)>0$ and hence the condition \hyperlink{HypoH}{H} does not hold.
Next, we simply write $Q^{(\gamma)}=(Q^{(\gamma)}_t)_{t\geq0}$ for the
semigroup of $X$, that is, for any bounded Borelian function~$g$ and~$t$, $x>0$, one has
\[
Q^{(\gamma)}_tg(x)=\mathbb{E}^{(\gamma)}_x[g(X_t)].
\]
From \cite{Bertoin-Yor-02-b}, we have that $Q^{(\gamma)}$ is a Feller
semigroup on $[0,\infty)$. Next, we say, for any $r\in\mathbb{R}$, that a
function $I$ is $r$-invariant for $Q^{(\gamma)}$ if
\[
e^{-rt}Q^{(\gamma)}_tI(x)=I(x), \qquad x>0.
\]
We start with the following lemma which is obtained readily from
\cite{Patie-06c}, Theorem~1.
\begin{lemma} \label{lem:1}
For any $r>0$, the mapping $x\mapsto \mathcal{I}pg(-rx)$ is $-r$-invariant for
$Q^{(\gamma)}$.\vadjust{\goodbreak}
\end{lemma}
Following a device developed by the author in \cite{Patie-08a}, we show
how to construct some specific time--space invariant functions for the
semigroup $Q^{(\gamma)}$ in terms of its $r$-invariant functions. We
now state the following result which is a slight generalization of
\cite{Patie-08a}, Theorem 1 and Corollary 3.2.
\begin{lemma} \label{prop:iou}
For any $\mathfrak{Re}(\rho)>0$, the mapping $x\mapsto\mathcal{I}pg(\rho
;-x)$ satisfies
the identity, for any $0\leq t< 1$,
\begin{equation} \label{eq:ts}
(1- t)^{-\rho}Q^{(\gamma)}_t \bigl(d_{(1-t)^{-1}}\mathcal{I}pg\bigr)
(\rho
;-x) = \mathcal{I}pg(\rho;-x), \qquad x>0,
\end{equation}
where $d_cf(x)=f(cx), c>0$.
\end{lemma}
Next, we introduce the stopping time $ D_a$ defined, for any $a>0$, by
\[
D_a = \inf\{0<s\leq1; X_s=a(1- s) \}.
\]
Writing $(a)_+=\max(a,0)$, we have
\begin{equation} \label{eq:hi}
e^{-H_a}\stackrel{(d)}{=}(1- D_a)_+
\end{equation}
and, in particular, for $a=0$, since $D_0\stackrel{(d)}{=}T_0\wedge1$,
we obtain
\[
e^{-H_0}\stackrel{(d)}{=}(1- T_0)_+.
\]
For any $a>0$, we set
\[
\kappa(a) = \inf\{\kappa\in\mathbb{R}^{+}; \mathcal{I}pg(\kappa;-a)=0\}
\]
with the usual convention that $\inf\{\varnothing\}=\infty$, for the
smallest positive real zero of the function $\mathcal{I}pg(\cdot;-a)$. We are now
ready to state the following.
\begin{cor} \label{cor:1}
Let $0\leq x\leq a$. Then, for any $\rho\in\mathbb{C}$ with $\mathfrak
{Re}(\rho
)<\kappa(a)$, we have
\[
\mathbb{E}^{(\gamma)}_x[(1 - D_a)_+^{-\rho}]=
\frac{\mathcal{I}pg(\rho; -x)}{\mathcal{I}pg(\rho;-a)}.
\]
Consequently, for any real $\kappa$ such that $\kappa<\kappa(a)$, the mapping
$x\mapsto\mathcal{I}pg(\kappa;-x)$ is positive on $\mathbb{R}^+$.
\end{cor}
\begin{pf}
Since $X$ under $\mathbb{Q}^{(\gamma)}$ is a Feller process on $[0,\infty)$,
we can start by fixing $x=0$ and $a>0$. Then, recalling that $\mathcal{I}pg
(0,-a)=1$, we observe that $\mathcal{I}pg(\kappa;-a )$ is positive for any
$0\leq\kappa< \kappa(a)$ reals. The existence of such an interval follows
from the fact that the zeros of a nonconstant holomorphic function are
isolated. Thus, by combining the identity (\ref{eq:ts}) with the Dynkin
formula (see, e.g., \cite{Dynkin-65}, Theorem 12.4), applied to
the bounded stopping time~$D_a$, we deduce, for any $0\leq\kappa< \kappa(a)
$, that
\begin{equation} \label{eq:ma}
\mathbb{E}^{(\gamma)}_0[(1- D_a)_+^{-\kappa}] = \frac{1}{\mathcal{I}pg
(\kappa;-a )}.\vadjust{\goodbreak}
\end{equation}
Next, we recall, from identity (\ref{eq:hi}), that
\[
e^{\kappa H_a}\stackrel{(d)}{=}(1- D_a)^{-\kappa}_+.
\]
Since $H_a$ is a positive random variable, as a Laplace transform, the
left-hand side on identity (\ref{eq:ma}) is analytic in the
half-plane $\{\rho\in\mathbb{C}; \mathfrak{Re}(\rho)<\kappa(a)\}$ and positive
on $\mathbb{R}
^+$; see,
for example, \cite{Widder-41}, Chapter II.
Then, let us assume that there exists a complex number $\rho(a)$ in the
strip $0\leq\mathfrak{Re}(\rho(a))<\kappa(a)$ such that $\mathcal{I}pg(\rho(a)
;-a )=0$.
However, as the left-hand side of (\ref{eq:ma}) is analytic with
respect to the argument $\kappa$ in this strip, we deduce, by the
principle of analytic continuation, that this is not possible.
Moreover, we get that $\mathcal{I}pg(\rho;-a )$ has no zeros on $\{\rho\in\mathbb{C};
\mathfrak{Re}(\rho)<\kappa(a)\}$ and is positive on $\{\kappa\in\mathbb{R};\break
\kappa
<\kappa(a)\}
$. Finally, let us consider a real number $a_1$ such that $0< a_1\leq
a$. Clearly, $\mathbb{Q}^{(\gamma)}_0$-a.s. $(1- D_{a_1})_+^{-\kappa}\leq(1-
D_a)_+^{-\kappa}$, for any $0\leq\kappa< \kappa(a)\wedge\kappa(a_1)$. Then
we deduce from~(\ref{eq:ma}), for any $0\leq\kappa< \kappa(a)\wedge
\kappa(a_1)$, that
\[
0<\frac{1}{\mathcal{I}pg(\kappa;-a_1 )} \leq\frac{1}{\mathcal{I}pg(\kappa;-a )}.
\]
Thus, it is not difficult to see that $ \kappa(a_1) \geq\kappa(a)$.
Therefore, since $ \kappa(x) \geq\kappa(a)$, for any $0\leq x\leq a$, the
strong Markov property and the absence of positive jumps of $X$
complete the proof.
\end{pf}
The choice of starting our computation under the law $\mathbb{Q}^{(\gamma)}$
was motivated by the previous proof where it was necessary to start $X$
at $0$ in order to get some information about the sign of the function
$\mathcal{I}pg(\kappa,-a)$. This device would not have been possible under $\mathbb{Q}$.
We proceed to the proof of Theorem~\ref{thm:2} which we now split into
two parts: the case when $X$ reaches $0$ continuously, that is, $q=0$
and $\mathbb{E}l[\xi_1]<0$ and the case when $X$ reaches $0$ by a jump, that
is, $q>0$.
\subsubsection{Continuous killing}
Here, we assume that $q=0$ and $\mathbb{E}l[\xi_1]<0$. Thus, in this case,
$\gamma=\phi(0)$ and $\psi_{\gamma}(u)=\psi(\gamma+u)$ with $\psi
_{\gamma}'(0^+)>0$.
\begin{lemma} \label{lem:2}
Writing $\kappa'(a)=\kappa(a)-\gamma>0$, we have, for any $\kappa
<\kappa'( a)$ and $0<x\leq a$,
\[
\mathbb{E}_x\bigl[(1- D_a)^{-\kappa}{{\mathbb{I}}}_{\{D_a<T_0\wedge1\}
}\bigr]=\frac
{x^{\gamma}}{a^{\gamma}}\frac{\mathcal{I}pg(\kappa+ \gamma; -x )}{\mathcal{I}pg
(\kappa
+\gamma;- a )}.
\]
In particular, for any $0<x\leq a$, we have
\[
\mathbb{Q}_x[ D_a<T_0\wedge1] = \frac{x^{\gamma}}{a^{\gamma
}}\frac
{\mathcal{I}pg(\gamma; -x )}{\mathcal{I}pg(\gamma;- a )}.
\]
\end{lemma}
\begin{pf}
We start by using the fact that the function $x\mapsto x^{-\gamma}$ is
excessive for $Q^{(\gamma)}_t$; see, for example, \cite{Rivero-05}. In
particular, one\vadjust{\goodbreak} has, for any $t>0$ and for any $F$ a $\mathcal{F}_t$-measurable
and bounded random variable,
\[
\mathbb{E}^{(\gamma)}_{x}[F] =\mathbb{E}_x[X_t^{\gamma} F, t<T_0
],\qquad x>0.
\]
Note that this relation also holds for any $\mathcal{F}_{\infty}$-stopping time.
Moreover, proceeding as in the proof of Corollary \ref{cor:1}, one gets
that the Mellin transform of the positive random variable $(1- D_a)_+$
is well defined for any real $\kappa$ such that $\kappa\leq0$. Thus,
since $X$ has no positive jumps, one obtains by means of both Corollary
\ref{cor:1} and the optional stopping theorem, for any $\kappa\leq0$,
\begin{eqnarray*}
\mathbb{E}_x\bigl[(1 - D_a)_+^{-\kappa}{{\mathbb{I}}}_{\{D_a<T_0\}}
\bigr]&=&\frac
{x^{\gamma
}}{a^{\gamma}}\mathbb{E}^{(\gamma)}_x\bigl[(1 - D_a)_+^{-(\kappa+ \gamma
)}
\bigr] \\
&=& \frac{x^{\gamma}}{a^{\gamma}}\frac{\mathcal{I}pg(\kappa+ \gamma; -x
)}{\mathcal{I}pg
(\kappa+ \gamma; -a )}.
\end{eqnarray*}
We deduce that $\kappa'(a)>0$ and the proof is completed by letting
$\kappa\rightarrow0$.
\end{pf}
We are now ready to complete the proof of Theorem \ref{thm:2} in the
case $\gamma=\phi(0)$. One gets that
\begin{eqnarray*}
\mathbb{Q}_x[ D_a<T_0\wedge1]&=&\mathbb{Q}_x[\tilde{\alpha}u(H_a)<\tilde{\alpha}u
(H_0)\wedge
1]\\
&=& \mathbb{Q}_x[H_a<H_0]
\end{eqnarray*}
since $\tilde{\alpha}u$ is increasing and $\tilde{\alpha}u^{-1}(1)=\infty$. Thus, as $X$ has
no positive jumps, one deduces that
\begin{eqnarray*}
\lim_{a\rightarrow\infty}\mathbb{Q}_x[ D_a<T_0\wedge1]&=& \mathbb{Q}
_x
[H_0=\infty]\\
&=& K(x).
\end{eqnarray*}
As we have learnt from Corollary \ref{cor:1} and Lemma \ref{lem:2} that
the mapping $x\mapsto\mathcal{I}pg( \gamma;-x )$ is positive on $\mathbb{R}^+$, it
means that there exists a constant $C_{\gamma}>0$ such that
\[
\mathcal{I}pg( \gamma;-x ) \sim C_{\gamma}^{-1} x^{-\gamma} \qquad \mbox
{as } x\rightarrow\infty.
\]
Then, recalling that $\lim_{x\rightarrow\infty}K(x)=1$, we obtain
\[
K(x)=C_{\gamma}x^{\gamma}\mathcal{I}pg( \gamma; -x ).
\]
Hence, we deduce the expression of $S$ from the identity
$S(t)=K(t^{-1})$. Finally, the series $\mathcal{I}pg( \gamma;-x )$ being
absolutely continuous, the expression of the density $s$ is obtained by
differentiating terms by terms. Indeed, one has
\begin{eqnarray*}
s(t)&=&-\frac{d}{dt}S(t)\\
&=& C_{\gamma}t^{-\gamma-1}\frac{1}{\Gamma( \gamma)}\sum
_{n=0}^{\infty}
(-1)^n a_n(\psi)(\gamma+n) \Gamma( \gamma+n)t^{-n}\\
&=&\frac{\Gamma(\gamma+1)}{\Gamma(\gamma)} C_{\gamma}t^{-\gamma
-1} \mathcal{I}pg
(1+ \gamma;-t ).
\end{eqnarray*}
The expression of the successive derivatives are obtained by means of
an induction argument.
\subsubsection{$X$ reaches $0$ by a jump}
Throughout this part, we assume that $\xi$ is a spectrally negative L\'
{e}vy process killed at some independent exponential time of parameter
$q>0$. Recall that, for any $u\geq0$, ${\bar{\psi}}(u)=\psi(u)-q$, $\phi$ is
such that $\psi\circ\phi(u)=u$ and with $\gamma=\phi(q)$, we
easily see
that $\psi_{\gamma}(u)={\bar{\psi}}(u+\gamma)$ and \mbox{$\psi'_{\gamma}(0^+)>0$}.
\begin{lemma}\label{lemmaa}
Writing $\kappa'(a)=\kappa(a)-\gamma>0$, we have, for any $\kappa
<\kappa'( a)$ and $0<x\leq a$,
\[
\mathbb{E}_x\bigl[(1- D_a)_+^{-\kappa}{{\mathbb{I}}}_{\{D_a<T_0\}}
\bigr]=\frac
{x^{\gamma
}}{a^{\gamma}}\frac{\mathcal{I}pg(\kappa+ \gamma;- x )}{\mathcal{I}pg(\kappa
+\gamma;- a )}.
\]
In particular,
\[
\mathbb{Q}_x[ D_a<T_0\wedge1 ]=\frac{x^{\gamma}}{a^{\gamma
}}\frac
{\mathcal{I}pg(\gamma;-x )}{\mathcal{I}pg(\gamma; -a )}.
\]
\end{lemma}
\begin{remark}\label{rem:1}
$\!\!$Writing $D_a^+=\inf\{s>0; X_s=a(1+s)\}$ and $K(x;a)=\mathbb{Q}_x[
D_a<T_0\wedge1]$, we deduce from \cite{Patie-08a}, Corollary 3.2,
the following identity:
\[
\mathbb{Q}_x[ D^+_a<T_0]=K(-x,-a), \qquad 0<x\leq a<\Lambda.
\]
It would be interesting to prove such a formula directly from the
definition of $D_a$ and $D_a^+$.
\end{remark}
\begin{pf*}{Proof of Lemma \ref{lemmaa}}
Let us observe from the Lamperti mapping~(\ref{eq:ss}) that the
semigroup $(Q_t)_{t\geq0}$ of $X$ is given for a function $f$
positive and measurable on $\mathbb{R}^+$ by
\[
Q_t f(x) = \mathbb{E}^{q}_x[e^{-q A_t}f(X_t)], \qquad t\geq0,x>0,
\]
where $\mathbb{E}^{q}$ stands for the expectation operator associated to the
law of $X$ with underlying Laplace exponent $\psi$. Thus, for any $\mathcal{F}
_{\infty}$-stopping time $T$, one has
\[
\mathbb{E}_{x}[f(X_T)]=\mathbb{E}^{q}_x[e^{-q A_{T}}f(X_T)].
\]
Moreover, as $\xi$ has independent increments, it is plain that the
process $(e^{-qt+\gamma\xi_t})_{t\geq0}$ is a $\mathrm{P}^{q}$-martingale,
where $\mathrm{P}^{q}$ stands for the law of the L\'{e}vy process with Laplace
exponent $\psi$. By time change, one deduces that the process
$(X_t^{\gamma}e^{-qA_t})_{t\geq0}$ is a $\mathbb{Q}_1$-martingale. Thus, one
can define a new probability measure, which we denote by $\mathbb{Q}^{(\gamma
)}$, as follows, for any $t>0$ and for any $F$ a $\mathcal{F}_t$-measurable and
bounded random variable,
\[
\mathbb{E}^{(\gamma)}_x[F]=\mathbb{E}_x^{q}[X_t^{\gamma
}e^{-qA_t}F],\qquad x>0.
\]
It is easily seen that the underlying Laplace exponent of $X$, under
$\mathbb{Q}
^{(\gamma)}$, is~$\psi_{\gamma}$.
Hence, one gets by the absence of positive jumps for $X$ and an
application of the optional stopping theorem, that, for any $0<x\leq a$
and $\kappa\leq0$,
\begin{eqnarray*}
\mathbb{E}_x\bigl[(1-D_a)_+^{-\kappa}{{\mathbb{I}}}_{\{D_a<T_0\}}
\bigr]&=& \mathbb{E}
^{q}_x
\bigl[e^{-q A_{D_a}}(1- D_a)_+^{-\kappa}{{\mathbb{I}}}_{\{D_a<T_0\}}
\bigr]\\
&=& \biggl(\frac{x}{a}\biggr)^{\gamma}\mathbb{E}^{(\gamma)}_x\bigl[(1-
D_a)_+^{-(\kappa+\gamma)}\bigr]\\
&=&\biggl(\frac{x}{a}\biggr)^{\gamma}\frac{\mathcal{I}pg(\kappa+\gamma;-x
)}{\mathcal{I}pg
(\kappa+\gamma;-a )},
\end{eqnarray*}
where the last line follows from Corollary \ref{cor:1} since $\psi
_{\gamma}'(0^+)>0$. The proof of the lemma is complete.
\end{pf*}
The proof of the theorem is completed by following a line of reasoning
similar to the previous case.
\subsection{\texorpdfstring{Proof of Proposition \protect\ref{prop:ck}}
{Proof of Proposition 2.5}}
Let us start by pointing out that it is not difficult to check that we
have, in all cases, $C_{\gamma}>0$. Moreover, let us first assume that
$\lim_{u\rightarrow\infty} \frac{\psi(u)}{u}=b$. From Proposition
\ref
{prop:ac}, we have
\begin{eqnarray}
\mathcal{I}pg( \gamma_{\alpha} ;-z) =
\frac{1}{2i \pi\Gamma( \gamma_{\alpha} )}\int_{-i\infty
}^{i\infty}
a_s(\varphi_{\gamma};\alpha) \Gamma(s+ \gamma_{\alpha} )\Gamma
(-s)
\biggl(\frac{z}{\alpha}\biggr)^{s} \,ds, \nonumber\\
&&\eqntext{|{\arg}(z)|<\pi.}
\end{eqnarray}
Hence, upon displacement of the path to the left in order to include
the first pole of $\Gamma(s+ \gamma_{\alpha} )$ we obtain, from Theorem
\ref{thm:2} and a residue computation, that
\[
\mathcal{I}pg( \gamma_{\alpha} ;-z) = \alpha^{ \gamma_{\alpha} }a_{-
\gamma
_{\alpha} }(\varphi_{\gamma};\alpha) z^{- \gamma_{\alpha} }+ o(z^{-
\gamma_{\alpha} }),
\]
which gives the characterization of $C_{\gamma}$ in this case.
For the other case, that is, when $\Lambda=+\infty$, one may follow a
line of reasoning similar to the proof of Proposition \ref{prop:ac}.
Indeed, as $0<\psi_{\gamma}'(0^+)<\infty$, we have, for any $u>0$,
\begin{eqnarray*}
\psi_{\gamma}(\alpha u) &=& \hat{b} \alpha u + \frac{\sigma}{2}
(\alpha
u)^2 + \int_{-\infty}^0 (e^{\alpha u r} -1
-\alpha ur)e^{\gamma r}\nu(dr) \\
&=& (\alpha u)^{2}\bar{\varphi}_{\gamma}(\alpha u),
\end{eqnarray*}
where $\hat{b}=\bar{b}+\sigma\gamma+\int_{-\infty}^0
(e^{\gamma
r}-{{\mathbb{I}}}_{\{|r|<1\}})r\nu(dr)$ and
\[
\bar{\varphi}_{\gamma}(\alpha u)=\frac{\hat{b}}{\alpha u} + \frac
{\sigma
}{2} + \int^{\infty}_0 e^{-\alpha u r}\int_{-\infty}^{-r}\int
_{-\infty
}^{-s}e^{\gamma v}\nu(dv) \,ds \,dr.
\]
Thus, as above, one may define the function
\begin{eqnarray*}
a_{s}(\psi_{\gamma};\alpha)&=& \frac{1}{\alpha^2\Gamma
^2(s+1)}a_s(\bar
{\varphi}_{\gamma};\alpha)\\[-2pt]
&=&\frac{1}{\alpha^2 \Gamma^2(s+1)}\prod_{k=1}^{\infty}\frac{\bar
{\varphi}_{\gamma}(\alpha(k +s+1))}{\bar{\varphi}_{\gamma}(\alpha k)}
\end{eqnarray*}
and observe the identity
\[
a_{s+1}(\bar{\varphi}_{\gamma};\alpha)=\frac{1}{\bar{\varphi
}_{\gamma
}(\alpha(s+1))}a_{s}(\bar{\varphi}_{\gamma};\alpha)
\]
with $a_0(\bar{\varphi}_{\gamma};\alpha)=1$. Hence, $a_{s}(\bar
{\varphi
}_{\gamma};\alpha)$ is a meromorphic function in $F_{-\gamma}=\{s\in
\mathbb{C}
; \mathfrak{Re}(s)>-\gamma-1\}$ with simple poles at the points
$s_k=-k-1$ for
$k=0,1,\ldots$ and $s_k>-\gamma-1$. We obtain, writing $\bar
{G}(s)=\frac{a_s(\bar{\varphi}_{\gamma};\alpha)}{\Gamma(s+1)}
\Gamma(s+
\gamma_{\alpha} )\Gamma(-s)$, the following identity:
\[
\mathcal{I}pg( \gamma_{\alpha} ;-z) = \frac{1}{2i \pi\Gamma( \gamma
_{\alpha}
)}\int_{-i\infty}^{i\infty} \bar{G}(s)\biggl(\frac{z}{\alpha
^2}\biggr)^{s} \,ds,
\]
which is now valid in the sector $ |{\arg}(z)|<\pi/2$. As above, after a
displacement of the path to the left in order to include the first pole
of $\Gamma(s+ \gamma_{\alpha} )$ we obtain, from Theorem \ref{thm:2}
and a residue computation, that
\begin{eqnarray*}
\mathcal{I}pg( \gamma_{\alpha} ;-z) &=&
\frac{1}{\Gamma( \gamma_{\alpha} )}\Biggl(\sum_{k=1}^{[ \gamma
_{\alpha}
]}\frac{\Gamma(k)\operatorname{Res}_{j=-k}a_{j}(\bar{\varphi}_{\gamma
};\alpha
)}{\Gamma(1-k)} \biggl(\frac{z}{\alpha^2}\biggr)^{-k} \\[-2pt]
&&\hspace*{83.6pt}{}+
\operatorname{Res}_{s=- \gamma_{\alpha} }\bar{G}(s) \biggl(\frac{z}{\alpha
^2}\biggr)^{-s}\Biggr)\\[-2pt]
&&{}+ o(z^{- \gamma_{\alpha} }),
\end{eqnarray*}
where the sum is $0$ if $[ \gamma_{\alpha} ]$, the integer part of $
\gamma_{\alpha} $, is lower than $1$.
Since $a_{s}(\bar{\varphi}_{\gamma},\alpha)$ has a simple pole at
$j=-1,\ldots,-[ \gamma_{\alpha} ]$, the terms in the sum vanish. Hence,
if $ \gamma_{\alpha} $ is not an integer $\bar{G}(s)$ has a simple
pole at $- \gamma_{\alpha} $ and the expression of $C_{\gamma}$ follows
readily in this case. If $ \gamma_{\alpha} =n+1$, then $\bar
{G}(s)$ has a double pole at $-(n+1)$ and using the recurrence
relations of both the gamma function and $a_{s}(\bar{\varphi}_{\gamma
};\alpha)$, we deduce that
\begin{eqnarray*}
&&\operatorname{Res}_{s=-(n+1)}\bar{G}(s)\\[-2pt]
&&\qquad= \lim_{s\rightarrow
-n-1}\frac{d}{ds}\bigl( (s+n+1)^2 \bar{G}(s)\bigr)\\[-2pt]
&&\qquad= \lim_{s\rightarrow-n-1}\frac{d}{ds}\Biggl( \alpha^{-n-2} \prod
_{k=1}^{n}\varphi_{\gamma}\bigl(\alpha(s+k)\bigr)\psi_{\gamma}\bigl(\alpha
(s+n+1)\bigr)a_{s+n+1}(\bar{\varphi}_{\gamma};\alpha)\Biggr)\\[-2pt]
&&\qquad= \alpha^{-n-2}\Gamma(n+1)\psi'_{\gamma}(0^+)\prod
_{k=1}^{n}\varphi
(\alpha k)
\end{eqnarray*}
and the result follows.
The second part of the proposition is proved as follows. Let us recall
that in \cite{Patie-06c}, the expression of the Laplace transform of
$T_0$, in the case $\mathbb{E}l[\xi_1]<0$, $q=0$ and $\gamma<\alpha$ is given
for any $r,x \geq0$ as follows:
\begin{equation} \label{eq:lt_exp}
\mathbb{E}_x[e^{-r T_0} ] = \mathcal{N}_{\psi,\gamma}(r x ),
\end{equation}
where
\[
\mathcal{N}_{\psi,\gamma}(r) =\mathcal{I}p(r)-
C(\gamma)r^{ \gamma_{\alpha}}\mathcal{I}t(r)
\]
and the positive constant $C(\gamma)$ is characterized by
\[
\mathcal{I}p(r)\sim C(\gamma)
r^{\gamma_{\alpha}}\mathcal{I}_{\psi_{\gamma}}(r)
\qquad\mbox{as } r\rightarrow
\infty.
\]
Next, let us write $\hat{F}(r)=\mathbb{E}_1[1-e^{-r T_0} ]$. Then,
from (\ref{eq:lt_exp}), one deduces easily that
\[
\hat{F}(r) \sim C(\gamma)r^{\gamma_{\alpha}} \qquad\mbox{as }
r\rightarrow0,
\]
which is equivalent, according to Bingham, Goldie and Teugels \cite{Bingham-Goldie-Teugels-89},
Corollary~8.1.7, to
\[
S(t)\sim\frac{ C(\gamma)}{\Gamma(1-\gamma_{\alpha})} t^{-\gamma
_{\alpha
}} \qquad\mbox{as } t\rightarrow\infty,
\]
which completes the proof.
\section{Some final remarks and illustrative examples} \label{sec:exa}
We start by offering a~few consequences of Theorem \ref{thm:2}.
\begin{cor}\label{cor:dd}
With the notation used and introduced in Theorem~\ref{thm:2}, we have,
writing $s^{(m)}=\frac{d^m }{dt^m}s$,
\[
s^{(m)}(t)= (-1)^m \frac{\Gamma(m+1+ \gamma_{\alpha} )}{\Gamma(
\gamma
_{\alpha} )} C_{\gamma}t^{- \gamma_{\alpha} -1-m} \mathcal{I}pg(m+1+ \gamma
_{\alpha} ;-t^{-1}), \qquad t>0.
\]
Moreover,
\[
S(t)\sim C_{\gamma}t^{- \gamma_{\alpha} } \qquad\mbox{as }
t\rightarrow\infty
\]
and, for any $m=0,1\ldots,$
\[
s^{(m)}(t) \sim (-1)^m C_{\gamma}\frac{\Gamma(m+1+ \gamma_{\alpha}
)}{\Gamma( \gamma_{\alpha} )} t^{- \gamma_{\alpha} -1-m}
\qquad\mbox{as } t\rightarrow\infty.
\]
\end{cor}
As pointed out by several authors (see Carmona, Petit and Yor~\cite
{Carmona-Petit-Yor-97}, Rive\-ro~\cite{Rivero-05} and Maulik and Zwart
\cite{Maulik2006}) the study of the exponential functional is also
motivated by its connection to some interesting random affine equations
which have been deeply studied by Kesten \cite{Kesten-73}. Relying on a
result of Kesten, Rivero (\cite{Rivero-05}, Lemma 4) shows that there
exists a constant $C>0$ such that one has the following asymptotic behavior
\[
S(t) \sim Ct^{-\tilde{\alpha}\gamma} \qquad\mbox{as } t\rightarrow\infty,
\]
whenever the L\'{e}vy process satisfies a set of conditions. As we have
excluded the case when $-\xi$ is a subordinator, it is not difficult to
verify that the L\'{e}vy processes we consider in this paper satisfy
Rivero's conditions. Hence, Theorem \ref{thm:2} and Proposition \ref
{prop:ck} offers several characterizations of the Kesten's constant. We
also point out that the asymptotic behavior of the density in Corollary
\ref{cor:dd} could not be deduced directly from Rivero's result since
we do not know whether or not the density is ultimately monotone.
\subsection{The Bessel processes} \label{sub:bess}
We consider $\xi$ to be a $2$-scaled Brownian motion with drift $2b
\in
\mathbb{R}$ and killed at some independent exponential time of parameter $q>0$,
that is, ${\bar{\psi}}(u)=2u^2+2b u-q$ and $2\phi(q)=\sqrt{2q+b^2}-b$. Note that
$\psi_{\phi(q)}(u)=2u^2+(2b+\phi(q)) u$. Its associated
self-similar process~$X$ is well known to be a Bessel process of index
$b$ killed at a rate $q\int_0^{t}X_s^{-2}\,ds$. Moreover, we obtain,
setting $\varrho=b+2\phi(q)$,
\begin{eqnarray*}
\mathcal{I}_{\psi_{\phi(q)}}(\rho;-x)&=&\frac{\Gamma(\varrho
+1)}{\Gamma(\rho)}\sum_{n=0}^{\infty} (-1)^n\frac{\Gamma(\rho
+n)}{n!\Gamma(n+\varrho+1)}(x/2)^n\\
&=&\Phi(\rho, \varrho+1;-x/2),
\end{eqnarray*}
where $\Phi$ stands for the confluent hypergeometric function. We refer
to Lebedev (\cite{Lebedev-72}, Section 9) for useful properties of this
function. Next, using the following asymptotic:
\[
\Phi(\rho, \varrho+1;-x) \sim\frac{\Gamma(\varrho
+1)}{\Gamma
(\varrho+1-\rho)} x^{-\rho} \qquad\mbox{as } x\rightarrow
\infty,
\]
we get that $C_{\phi(q)}=\frac{\Gamma(\varrho+1-\phi(q))}{2^{\phi
(q)}\Gamma(\varrho+1)}$.
Thus, we obtain, recalling that, for any $q>0$, $\varrho-\phi
(q)=b+\phi(q)>0$,
\begin{eqnarray*}
s_{\phi(q)}(t)&=&\phi(q) \frac{\Gamma(\varrho+1-\phi(q))}{2^{\phi
(q)}\Gamma(\varrho+1)}t^{-\phi(q)-1}\Phi\bigl(1+\phi(q), \varrho
+1;-(2t)^{-1}\bigr)\\
&=&\frac{b+\phi(q)}{2^{\phi(q)}\Gamma(\phi(q))}t^{-\phi(q)-1}
\int
_0^{1}e^{-{u}/({2t})}(1-u)^{\varrho-\phi(q)-1}u^{\phi(q)}\,du,
\end{eqnarray*}
which is expression (5.a) in \cite{Yor-01}, page 105.
Considering now the case $q=0$ and $b<0$, we obtain readily that $\phi
(0)=-b$ and
\begin{eqnarray*}
s_{\phi(0)}(t)&=&\frac{2^b}{\Gamma(-b)}t^{b-1}\Phi\bigl(1-b,
1-b;-(2t)^{-1}\bigr)\\
&=&\frac{2^b}{\Gamma(-b)}t^{b-1}e^{-{1}/({2t})}.
\end{eqnarray*}
Hence, we deduce the well-known identity $(T_0,\mathbb{Q}_1)\stackrel{(d)}{=}
\frac{1}{2G_{-b}}$ where we recall that $G_{-b}$ stands for a Gamma
random variable of parameter $-b>0$.
\subsection{Law of the maximum of spectrally positive stable L\'{e}vy
processes} \label{sec:ms}
Let~$Z$ be an $\alpha$-stable spectrally negative L\'{e}vy process,
with $1<\alpha<2$. Let us denote by~$X$ the process $Z$ killed upon
entering\vspace*{1pt} into the negative half-line.~$X$ is then a pssMp. Next, we
denote by $\hat{Z}$ the dual of $Z$, that is, $\hat{Z}=-Z$ which is a
$\alpha$-stable spectrally positive L\'{e}vy process. Then, by means of
the translation invariance of L\'{e}vy processes, we deduce readily the
following identities:
\begin{eqnarray*}
\mathbb{Q}_x(T_0 \leq t)&=& \mathbb{P}_x\Bigl(\inf_{0<s\leq t}
Z_s\leq
0\Bigr)\\
&=& \mathbb{P}\Bigl(\max_{0<s\leq t} \hat{Z}_s\geq x \Bigr),
\end{eqnarray*}
which can be written as follows:
\begin{equation} \label{eq:sa}
\mathbb{P}\Bigl(\max_{0\leq s \leq t}\hat{Z_s}\geq x
\Bigr)=K(xt^{-\alpha
}), \qquad x,t>0.
\end{equation}
The Laplace exponent of the underlying L\'{e}vy process of $X$ has been
computed Patie \cite{Patie-CBI-09} in terms of the Pochhammer symbol.
Instead of using this expression, we follow an alternative route.
Indeed, in \cite{Patie-OU-06}, the author computed the unique
increasing invariant function, say $P_{+}$, of the Ornstein--Uhlenbeck
process defined by
\[
\tilde{U}_t=e^{- t/\alpha}X_{e^{t}-1}, \qquad t\geq0.
\]
The function $P_+$, is given, with $C$ a constant to be determined and
writing $\tilde{\alpha}=1/\alpha$, by
\begin{eqnarray*} \label{eq:lin}
P_+(x) &=& C x^{\alpha-1}\sum_{n=0}^{\infty} \frac{\Gamma
(n+1-\tilde
{\alpha})}{\Gamma(\alpha n +\alpha)}\alpha^n x^{\alpha n} \\&=& C
x^{\alpha-1}{}_2\Psi_1
\left(\matrix{
(1,1),(1,1-\tilde{\alpha}) \cr
(\alpha,\alpha)}\bigg|
\alpha x^{\alpha}\right)
, \qquad x \geq0,
\end{eqnarray*}
where ${}_2\Psi_1$ stands for the Wright hypergeometric function. From
Remark~\ref{rem:1}, we have $K(x)= K_{+}(e^{i\pi/\alpha} x)$. Note that
$K(0)=0$ and using the large asymptotic of the function ${}_2\Psi_1$
(details can be found in \cite{Patie-08-Asym}), we get as $x\rightarrow
\infty$,
\[
{}_2\Psi_1
\left(\matrix{(1,1),(1,1-\tilde{\alpha}) \cr
(\alpha,\alpha)}
\bigg|-x^{\alpha}\right) \sim\biggl(\frac{\sin( \tilde{\alpha}\pi
)}{\pi
}\biggr)^{-1} x^{1-\alpha}.
\]
Hence, by setting $C=\frac{\sin( \tilde{\alpha}\pi)}{\pi}$, we obtain
the required condition\break $\lim_{x\rightarrow\infty} K(\infty)=1$ and
\[
K(x)=\frac{\sin( \tilde{\alpha}\pi)}{\pi}x^{\alpha-1}{}_2\Psi
_1\left(\matrix{
(1,1),(1,1-\tilde{\alpha}) \cr
(\alpha,\alpha)}\bigg|
-x^{\alpha}\right).
\]
Next, from identity (\ref{eq:sa}), we find that
\[
\mathbb{P}\Bigl(\max_{0\leq s\leq1}\hat{Z}_s\geq x\Bigr)= P(x),
\]
where $\hat{Z} $ is a spectrally positive stable process of index
$\alpha$. Thus, by differentiating, one gets the following expression
for the density:
\[
k(x)
=\frac{\sin( \tilde{\alpha}\pi)}{\pi}x^{\alpha-2}{}_2\Psi
_1
\left(\matrix{
(1,1),(1,1-\tilde{\alpha}) \cr
(\alpha,\alpha-1)}
\bigg| -x^{\alpha}\right),
\]
which is the expression found by Bernyk, Dalang and Peskir \cite{Bernyk-Dalang-Peskir-08},
Theorem 1.
\subsection{The self-similar saw-tooth processes} \label{ex:3}
Finally, we consider the so-called saw-tooth process introduced and
deeply studied by Carmona, Petit and Yor \cite{Carmona-Petit-Yor-98}. It is a
self-similar positive Markov process of index $\alpha=1$ with
underlying L\'{e}vy process the sum of a drift of parameter $b=1$ and
the negative of a compound Poisson process of parameter
$\beta>0$ whose jumps are exponentially distributed with parameter
$\delta+\beta-1>0$, that is,
\[
\psi(u) =u\frac{u+\delta-1 }{u+\delta+ \beta-1}, \qquad u\geq0.
\]
Moreover, in \cite{Carmona-Petit-Yor-98}, the authors show that
\[
\phi(q) =\tfrac{1}{2}\bigl(q-(\delta-1)+\bar{\phi}(q)
\bigr), \qquad q\geq0,
\]
where $\bar{\phi}(q)=\sqrt{(q-(\delta-1))^{2}+4(\delta+\beta-1) q}$.
Let us proceed with the case \mbox{$q=0$}. Note, for $1-\beta<\delta<1$, that
$\gamma=1-\delta$ and
\[
\psi_{1-\delta}(u)=u\frac{u+1-\delta}{u+\beta}.
\]
Thus,
\[
a_n(\psi_{1-\delta},1)= \frac{\Gamma(n+1+\beta)\Gamma(2-\delta
)}{\Gamma
(1+\beta)\Gamma(n+1)\Gamma(n+2-\delta)},\qquad
a_0=1,
\]
and for $|z|<1$
\begin{eqnarray*}
\mathcal{I}_{\psi_{1-\delta}}(\rho;-z)&=&\frac{\Gamma
(2-\delta
)}{\Gamma(\rho)\Gamma(1+\beta)}\sum_{n=0}^{\infty} (-1)^n\frac
{\Gamma
(\rho+n)\Gamma(n+1+\beta)}{\Gamma(n+2-\delta)n!} z^n\\
&=&{}_2F_1(\rho,1+\beta,2-\delta;-z),
\end{eqnarray*}
where ${}_2F_1(a,b;x)$ stands for the hypergeometric function; see
Lebedev \cite{Lebedev-72}, Section 9, for a detailed account on this
function. Next, recalling the identity
\[
{}_2F_1(-n,1+\beta,\delta;1) = \frac{\Gamma(2-\delta)\Gamma
(n+1-\delta
-\beta)}{\Gamma(2-\delta+n)\Gamma(1-\delta-\beta)},
\]
we recover from (\ref{eq:h}) the well-known identity
\[
{}_2F_1(\rho,1+\beta,2-\delta;z) = (1-z)^{-\rho}{}_2F_1\biggl(\rho
,1-\delta-\beta,\delta;\frac{z}{z-1}\biggr),
\]
which provides an analytic continuation of the hypergeometric function
into the half-plane $\mathfrak{Re}(z)<\frac{1}{2}$. Finally,
using the asymptotic
\[
{}_2F_1(\rho,1+\beta,2-\delta;-x) \sim\frac{\Gamma(2-\delta
)\Gamma
(1+\beta-\rho)}{\Gamma(2-\delta-\rho)\Gamma(1+\beta)} x^{-\rho}
\qquad \mbox{as } x\rightarrow\infty,
\]
one obtains
\[
S(t)=\frac{\Gamma(1+\beta)}{\Gamma(2-\delta)\Gamma(\beta
+\delta)}
t^{\delta-1}{}_2F_1(1-\delta,1+\beta,\delta;-t^{-1}).
\]
Moreover, after some easy computations, one gets for
$\gamma=\phi(q),q>0$,
\[
\psi_{\phi(q)}(u)=u\frac{u+\phi(q)}{u+\beta+\delta+\phi(q)-1}.
\]
Thus, proceeding as above, we obtain
\[
\mathcal{I}_{\psi_{\phi(q)}}(\rho;-z)={}_2F_1\bigl(\rho
,\beta+\delta
+\phi(q),1+\phi(q);-z\bigr)
\]
and
\begin{eqnarray*}
S(t)&=&\frac{\Gamma(\beta+\delta+\phi(q))\Gamma(1+\bar{\phi
}(q)-\phi
(q))}{\Gamma(1+\bar{\phi}(q))\Gamma(\beta+\delta)}\\
&&{}\times t^{-\phi
(q)}{}_2F_1\bigl(\phi(q),\beta+\delta+\phi(q),1+\bar{\phi
}(q);-t^{-1}\bigr).
\end{eqnarray*}
\section*{Acknowledgment}
I am grateful to M. Savov and an anonymous referee for their comments which
significantly helped in improving the presentation of the paper.
\printaddresses
\end{document}
|
\begin{document}
\title{Relating maximum entropy, resilient behavior and game-theoretic equilibrium feedback operators in multi-channel systems}
\author{Getachew~K.~Befekadu~\IEEEmembership{}
and~Panos~J.~Antsaklis
\thanks{This work was supported in part by the National Science Foundation under Grant No. CNS-1035655. The first author acknowledges support from the College of Engineering, University of Notre Dame.}
\IEEEcompsocitemizethanks{\IEEEcompsocthanksitem G. K. Befekadu is with the Department
of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA.\protect\\
E-mail: [email protected]
\IEEEcompsocthanksitem P. J. Antsaklis is with the Department
of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA.\protect\\
E-mail: [email protected]\protect\\
Version - December 19, 2013.}}
\markboth{}
{Shell \MakeLowercase{\textit{et al.}}: Bare Advanced Demo of IEEEtran.cls for Journals}
\IEEEcompsoctitleabstractindextext{
\begin{abstract}
In this paper, we first draw a connection between the existence of a stationary density function (which corresponds to an equilibrium state in the sense of statistical mechanics) and a set of feedback operators in a multi-channel system that strategically interacts in a game-theoretic framework. In particular, we show that there exists a set of (game-theoretic) equilibrium feedback operators such that the composition of the multi-channel system with this set of equilibrium feedback operators, when described by density functions, will evolve towards an equilibrium state in such a way that the entropy of the whole system is maximized. As a result of this, we are led to study, by a means of a stationary density function (i.e., a common fixed-point) for a family of Frobenius-Perron operators, how the dynamics of the system together with the equilibrium feedback operators determine the evolution of the density functions, and how this information translates into the maximum entropy behavior of the system. Later, we use such results to examine the resilient behavior of this set of equilibrium feedback operators, when there is a small random perturbation in the system.
\end{abstract}
\begin{IEEEkeywords}
Equilibrium feedback operators; Frobenius-Perron operators; game theory; maximum entropy; multi-channel system; resilient behavior; small random perturbation.
\end{IEEEkeywords}}
\maketitle
\IEEEdisplaynotcompsoctitleabstractindextext
\IEEEpeerreviewmaketitle
\section{Introduction} \label{S1}
The main purpose of this paper is to draw a connection between the existence of a stationary density (that corresponds to an equilibrium state in the sense of statistical mechanics) and a set of feedback operators in a multi-channel system that interacts strategically in a game-theoretic framework. We first specify a game in a strategic form over an infinite-horizon -- where, in the course of the game, each feedback operator generates automatically a feedback control in response to the action of other feedback operators through the system (i.e., using the current state-information of the system) and, similarly, any number of feedback operators can decide on to play their feedback strategies simultaneously. However, each of these feedback operators are expected to respond in some sense of best-response correspondence to the strategies of the other feedback operators in the system. In such a scenario, it is well known that the notion of (game-theoretic) equilibrium will offer a suitable framework to study or characterize the robust property of all equilibrium solutions under a family of information structures -- since no one can improve his payoff by deviating unilaterally from this strategy once the equilibrium strategy is attained (e.g., see \cite{Aub93}, \cite{Nas51} or \cite{Ros65} on the notions of optimums and strategic equilibria in games).\footnote{In this paper, we consider this set of feedback operators as noncooperative agents (or players), but fully-rational entities, {\em over an infinite-horizon}, in a game-theoretic sense. Further, {\em at each instant-time}, each feedback operator knows that the others will look for feedback strategies, but they are not necessarily informed about each others strategies.}
In view of the above arguments, we present in this paper an extension of game-theoretic formalism for multi-channel systems that tend to move towards an equilibrium or ``maximum entropy" state in the sense of statistical mechanics (e.g., see Lanford \cite[pp\,77--95]{Lan73} or Ruelle \cite{Rue78}) -- when the criterion is to minimize the relative entropy between any two density functions, for large-time, with respect to the control channels or the class of admissible control functions (i.e., the set of feedback operators). This further allows us to establish a connection between the existence of a stationary density function (which corresponds to a unique equilibrium state) and a set of feedback operators that strategically interacts in the system.
Moreover, based on a common fixed-point for a family of Frobenius-Perron operators, we provide a sufficient condition on the existence of a set of (game-theoretic) equilibrium feedback operators such that when the composition of the multi-channel system with this set of equilibrium feedback operators, described by density functions, evolves towards an equilibrium state in such a way that the entropy of the whole system is maximized. As a result of this, we are led to study, how the dynamics of the system together with these equilibrium feedback operators determine the evolution of the density functions, and how this information translates into the maximum entropy behavior of the system. Later, we use such results to examine the resilient behavior of this set of equilibrium feedback operators, when there is a random perturbation in the system. We, in particular, establish sufficient conditions, based on the convergence of invariant measures (i.e., stochastic stability -- in the sense of deterministic limit (e.g., see \cite{You86}, \cite{BalYo93}, \cite{Kif88} or \cite{FreWe12} for related discussions), that will guarantee the resilient behavior for the set of equilibrium feedback operators with respect to random perturbations in the system.
Here, we hasten to add that such a study, which involves evidence of systems exhibiting resilient behavior, would undoubtedly provide a better understanding of reliability or prescribing an optimal (sub-optimal) degree of redundancy in decentralized control systems. Finally, in the information theoretic-games, we also note that the notions of entropy, game-theoretical equilibrium and complexity, based on the {\em Maximum Entropy Principle} (MaxEnt) of Jaynes \cite{Jay57a}, \cite{Jay57b} and the {\em $I$-\,divergence metric} of Csisz\'{a}r \cite{Csi67}, \cite{Csi75}, have been investigated in the context of zero-sum games by Tops\o e (e.g., see \cite{Top93} or \cite{Top04}), Gr\"{u}nwald and Dawid \cite{GruDa04} and, similarly, by Haussler \cite{Hau97}. Moreover, we observe that the notion of entropy (and its variants) has been well discussed in systems theory literature in the context of robustness analysis and/or synthesis for systems with uncertainties (e.g., see \cite{PetJaDu00} and \cite{ChaKyRe06}).
The remainder of the paper is organized as follows. In Section~\ref{S2}, we recall the necessary background and present some preliminary results that are relevant to our paper. Section~\ref{S3} introduces a family of mappings for multi-channel systems that will be used for our main results. In Section~\ref{S4} we present our main results -- where we establish a three way connection between the existence of an equilibrium state (i.e., the maximum entropy in the sense of statistical mechanics), a common stationary density function for the family of Frobenius-Perron operators, and a set of (game-theoretic) equilibrium feedback operators. This section also discusses an extension of the resilient behavior (to these equilibrium feedback operators), when there is a small random perturbation in the system.
\section{Background, Definitions, and Notations} \label{S2}
In the following, we provide the necessary background and recall some known results from measure theory that will be useful in the sequel. The results are standard (and will be stated without proof); and they can be found in standard graduate books (e.g., see \cite{Hal74}, \cite{GikSko75} and \cite{Yos95} on the measure theory; and see also \cite{LasMac94} or \cite{Kif88} on the stochastic aspects of dynamical systems).
\begin{definition} \label{DFN1}
Let $(X, \mathscr{A}, \mu)$ be a measure space and $L^1(X, \mathscr{A}, \mu)$ be the space of all possible real-valued measurable functions $\vartheta\colon X \rightarrow \mathbb{R}$ satisfying
\begin{align}
\int_{X} \vert \vartheta(x)\vert \mu(dx) < \infty. \label{EQ1}
\end{align}
If $S\colon X \rightarrow X$ is a measurable nonsingular transformation, i.e., $\mu(S^{-1}(A))=0$ for all $A \in \mathscr{A}$ such that $\mu(A)=0$, then the operator $P \colon L^1(X, \mathscr{A}, \mu) \rightarrow L^1(X, \mathscr{A}, \mu)$ defined by
\begin{align}
\int_{A} P \vartheta(x) \mu(dx) = \int_{S^{-1}(A)} \vartheta(x) \mu(dx), ~~ \forall A \in \mathscr{A}, \label{EQ2}
\end{align}
is called the Frobenius-Perron operator with respect to $S$.
\end{definition}
\begin{definition}\label{DFN2}
Let $(X, \mathscr{A}, \mu)$ be a measure space. Define
\begin{align}
D(X, \mathscr{A}, \mu) = \Bigl \{ \vartheta(x) \in L^1(X, \mathscr{A}, \mu) \, \Bigl \vert \,\vartheta(x) \ge 0 ~~ \text{and} ~~ \bigl\lVert \vartheta(x) \bigr\rVert_{L^1(X, \mathscr{A}, \mu)}=1 \Bigr\}. \label{EQ3}
\end{align}
Then, any continuous function $\vartheta(x) \in D(X, \mathscr{A}, \mu)$ is called a density function.
\end{definition}
\begin{definition} \label{DFN3}
Let $(X, \mathscr{A}, \mu)$ be a measure space. If $S\colon X \rightarrow X$ is a nonsingular transformation and $\zeta(x) \in L^{\infty}(X, \mathscr{A}, \mu)$. Then, the operator $U \colon L^{\infty}(X, \mathscr{A}, \mu) \rightarrow L^{\infty}(X, \mathscr{A}, \mu)$ defined by
\begin{align}
U \zeta(x) = \zeta\bigl(S(x)\bigr), \label{EQ4}
\end{align}
is called the Koopman operator with respect to $S$.
\end{definition}
Note that for every $\zeta(x) \in L^{\infty}(X, \mathscr{A}, \mu)$
\begin{align}
\bigl \lVert U \zeta(x) \bigr \lVert_{L^{\infty}(X, \mathscr{A}, \mu)} \le \bigl \lVert \zeta(x) \bigr \lVert_{L^{\infty}(X, \mathscr{A}, \mu)}. \label{EQ5}
\end{align}
Moreover, for every $\vartheta(x) \in L^1(X, \mathscr{A}, \mu)$ and $\zeta(x) \in L^{\infty}(X, \mathscr{A}, \mu)$, then we have
\begin{align}
\langle P \vartheta,\, \zeta \rangle = \langle \vartheta,\, U \zeta \rangle, \label{EQ6}
\end{align}
so that the operator $U$ is an adjoint to the Frobenius-Perron operator $P$.\footnote{$\langle P \vartheta,\, \zeta \rangle \triangleq \int_{X} P\vartheta(x)\zeta(x) \mu(dx)$.}
\begin{remark}
The transformation $S$ is said to be measure preserving if $\mu\bigl(S^{-1}(A)\bigr)=\mu\bigl(A\bigr)$ for all $A \in \mathscr{A}$. Note that the property of measure preserving depends both on $S$ and $\mu$.
\end{remark}
\begin{definition} \label{DFN4}
Let $\vartheta(x) \in L^1(X, \mathscr{A}, \mu)$ and $\vartheta(x) \ge 0$. If the measure
\begin{align}
\mu_{\vartheta}(A) = \int_{A} \vartheta(x) \mu\bigr(dx\bigl), \label{EQ7}
\end{align}
is absolutely continuous with respect to the measure $\mu$, then $\vartheta(x)$ is called the Radon-Nikodym derivative of $\mu_{\vartheta}$ with respect to $\mu$.
\end{definition}
\begin{theorem} \label{TH1}
Let $(X, \mathscr{A}, \mu)$ be a measure space, $S\colon X \rightarrow X$ be a nonsingular transformation, and let $P$ be the Frobenius-Perron operator with respect to $S$. Consider a nonnegative function $\vartheta(x) \in L^1(X, \mathscr{A}, \mu)$, i.e., $\vartheta(x) > 0$, $\forall x \in X$. Then, a measure $\mu_{\vartheta}$ given by
\begin{align}
\mu_{\vartheta}(A) = \int_{A} \vartheta(x) \mu(dx), \quad \forall A \in \mathscr{A}, \label{EQ8}
\end{align}
is invariant, if and only if, $\vartheta(x)$ is a stationary density function (i.e., a fixed-point) of $P$.
\end{theorem}
\begin{theorem} \label{TH2}
Let $(X, \mathscr{A}, \mu)$ be a measure space and $S\colon X \rightarrow X$ be a nonsingular transformation. $S$ is ergodic, if and only if, for every measurable function $\vartheta \colon X \rightarrow \mathbb{R}$
\begin{align}
\vartheta(S(x)) = \vartheta(x), \label{EQ9}
\end{align}
for almost all $x \in X$, implies that $\vartheta(x)$ is constant almost everywhere.
\end{theorem}
\begin{definition} \label{DFN5} Convergence of sequences of functions (e.g., see \cite{Kre85} or \cite{DunSch58}).
\begin{enumerate} [(i)]
\item A sequence of functions $\bigl\{\vartheta_n(x)\bigl\}$, $\vartheta_n(x) \in L^1(X, \mathscr{A}, \mu)$, is {\em weakly Ces\`{a}ro convergent} to $\vartheta(x) \in L^1(X, \mathscr{A}, \mu)$ if
\begin{align}
\lim_{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^n \langle \vartheta_n, \,\zeta \rangle = \langle \vartheta, \,\zeta \rangle, \quad \forall \zeta \in L^{\infty}(X, \mathscr{A}, \mu). \label{EQ10}
\end{align}
\item A sequence of functions $\bigl\{\vartheta_n(x)\bigl\}$, $\vartheta_n(x) \in L^1(X, \mathscr{A}, \mu)$, is {\em weakly convergent} to $\vartheta(x) \in L^1(X, \mathscr{A}, \mu)$ if
\begin{align}
\lim_{n \rightarrow \infty} \langle \vartheta_n, \,\zeta \rangle = \langle \vartheta, \,\zeta \rangle, \quad \forall \zeta \in L^{\infty}(X, \mathscr{A}, \mu). \label{EQ11}
\end{align}
\item A sequence of functions $\bigl\{\vartheta_n(x)\bigl\}$, $\vartheta_n(x) \in L^1(X, \mathscr{A}, \mu)$, is {\em strongly convergent} to $\vartheta(x) \in L^1(X, \mathscr{A}, \mu)$ if
\begin{align}
\lim_{n \rightarrow \infty} \bigl\Vert \vartheta_n(x) - \vartheta(x) \bigr\Vert_{L^1(X, \mathscr{A}, \mu)} = 0. \label{EQ12}
\end{align}
\end{enumerate}
\end{definition}
\begin{theorem} [Chebyshev's inequality] \label{TH3}
Let $(X, \mathscr{A}, \mu)$ be a measure space and let $V \colon X \rightarrow \mathbb{R}_{+}$ be an arbitrary nonnegative measurable function. Define
\begin{align}
\mathit{E}\bigl(V(x) \bigl\vert \vartheta(x)\bigr) = \int_{X} V(x) \vartheta(x) \mu(dx), ~~ \forall \vartheta(x) \in D(X, \mathscr{A}, \mu). \label{EQ13}
\end{align}
If \, $G_{\alpha} = \bigl\{ x \in X \,\bigl\vert\, V(x) < \alpha \bigr\}$, then
\begin{align}
\int_{G_{\alpha}} V(x) \mu(dx) = 1 - \mathit{E}\bigl(V(x) \bigl\vert \vartheta(x)\bigr). \label{EQ14}
\end{align}
\end{theorem}
\begin{theorem} \label{TH4}
Let $(X, \mathscr{A}, \mu)$ be a finite measure space (i.e., $\mu(X) < \infty$) and let $S\colon X \rightarrow X$ be a measure preserving and ergodic. Then, for any integrable function $\vartheta_{\ast}(x)$, the average of $\vartheta(x)$ along the trajectory of $S$ is equal to almost everywhere to the average of $\vartheta(x)$ over the space $X$, i.e.,\footnote{{\bf Theorem} {\bf (Birkhoff's individual ergodic theorem)} Let $(X, \mathscr{A}, \mu)$ be a measure space, $S \colon X \rightarrow X$ be a measurable transformation, and let $\vartheta \colon X \rightarrow \mathbb{R}$ be an integrable function. If the measure $\mu$ is invariant, then there exists an integrable function $\vartheta_{\ast}(x)$ such that
\begin{align*}
\vartheta_{\ast}(x) = \lim_{n \rightarrow \infty} \frac{1}{n}\sum_{k=1}^{n-1} \vartheta(S^{k}(x)),
\end{align*}
for almost all $x \in X$.}
\begin{align}
\lim_{n \rightarrow \infty} \frac{1}{n}\sum_{k=1}^{n-1} \vartheta(S^{k}(x)) = \frac{1}{\mu(X)} \int_{X} \vartheta(x)\mu(dx). \label{EQ15}
\end{align}
\end{theorem}
\begin{corollary} \label{CR1}
Let $(X, \mathscr{A}, \mu)$ be a finite measure space and $S\colon X \rightarrow X$ be measure preserving and ergodic. Then, for any set $A \in \mathscr{A}$, $\mu(A) > 0$, and for almost all $x \in X$, the fraction of the points $\bigl\{S^k(x) \bigr\}$ in $A \in \mathscr{A}$ as $k \rightarrow \infty$ is given by $\mu(A)/\mu(X)$.
\end{corollary}
\begin{corollary} \label{CR2}
Let $(X, \mathscr{A}, \mu)$ be a normalized measure space (i.e., $\mu(X)=1$) and let $S \colon X \rightarrow X$ be measure preserving. Suppose that $P$ is the Frobenius-Perron with respect to $S$. Then, $S$ is ergodic if and only if
\begin{align}
\lim_{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n-1} P^n \vartheta(x) = 1, \label{EQ16}
\end{align}
for every $\vartheta(x) \in D(X, \mathscr{A}, \mu)$.
\end{corollary}
\begin{theorem} \label{TH5}
Let $(X, \mathscr{A}, \mu)$ be a measure space, $S\colon X \rightarrow X$ be a nonsingular transformation, and let $P$ be the Frobenius-Perron operator with respect to $S$. If $S$ is ergodic, then there exist at most one stationary density function $\vartheta_{\ast}(x)$ of $P$ (i.e., a fixed-point of $P\vartheta_{\ast}(x)=\vartheta_{\ast}(x)$). Furthermore, if there is a unique stationary density function $\vartheta_{\ast}(x)$ of $P$ and $\vartheta_{\ast}(x)>0$ almost everywhere, then $S$ is ergodic.
\end{theorem}
\begin{corollary} \label{CR3}
Let $(X, \mathscr{A}, \mu)$ be a measure space, $S \colon X \rightarrow X$ be a nonsingular transformation, and let $P \colon L^1(X, \mathscr{A}, \mu) \rightarrow L^1(X, \mathscr{A}, \mu)$ be the Frobenius-Perron operator $P$ with respect to $S$. If, for some $\vartheta(x) \in D(X, \mathscr{A}, \mu)$, there is an $\ell(x) \in D(X, \mathscr{A}, \mu)$ such that
\begin{align}
P^n\vartheta(x) \le \ell(x), ~~ \forall n \ge 0. \label{EQ17}
\end{align}
Then, there is a stationary density function $\vartheta_{\ast}(x) \in D(X, \mathscr{A}, \mu)$ such that $P \vartheta_{\ast}(x) = \vartheta_{\ast}(x)$.
\end{corollary}
\begin{definition}\label{DFN6}
If $\vartheta(x) \in D(X, \mathscr{A}, \mu)$, then the entropy of $\vartheta(x)$ is defined by
\begin{align}
\mathcal{H}(\vartheta(x)) = - \int_{X} \vartheta(x) \ln \vartheta(x) \mu(dx). \label{EQ18}
\end{align}
\end{definition}
\begin{remark}
The following integral inequality (which is useful for verifying the extreme properties of $\mathcal{H}(\vartheta(x))$) holds for any $\vartheta(x), \xi(x) \in D(X, \mathscr{A}, \mu)$
\begin{align}
-\int_{X} \vartheta(x) \ln \vartheta(x) \mu(dx) \le -\int_{X} \vartheta(x) \ln \xi(x) \mu(dx). \label{EQ19}
\end{align}
Note that, in general, we have the following Gibbs inequality
\begin{align*}
\vartheta(x) - \vartheta(x) \ln \vartheta(x) \le \xi(x) - \vartheta(x) \ln \xi(x),
\end{align*}
for any two nonnegative measurable functions $\vartheta(x), \xi(x) \in L^1(X, \mathscr{A}, \mu)$.
\end{remark}
\begin{proposition} \label{PR1}
Let $(X, \mathscr{A}, \mu)$ be a finite measure space. Consider all (nonnegative) possible density functions $\vartheta(x)$ defined on $X$. Then, for such a family of density functions, the maximum entropy occurs for a constant density function
\begin{align}
\vartheta_0(x) = 1/\mu(X), \label{EQ20}
\end{align}
and for any other density function $\vartheta(x)$, the entropy is strictly less than $\mathcal{H}(\vartheta_0(x))$, i.e.,
\begin{align*}
-\int_{X} \vartheta(x) \ln \vartheta(x) \mu(dx) \le -\ln\left(\frac{1}{\mu(X)}\right).
\end{align*}
\end{proposition}
\begin{proof}
Take any $\vartheta(x) \in D(X, \mathscr{A}, \mu)$. Then, the entropy of $\vartheta(x)$ is given by
\begin{align*}
\mathcal{H}(\vartheta(x)) = -\int_{X} \vartheta(x) \ln \vartheta(x) \mu(dx).
\end{align*}
Using Equation~\eqref{EQ19}, we have the following
\begin{align*}
\mathcal{H}(\vartheta(x)) & \le -\int_{X} \vartheta(x) \ln \vartheta_0(x) \mu(dx), \\
&= -\ln \left(\frac{1}{\mu(X)}\right) \int_{X} \vartheta(x) \mu(dx),
\end{align*}
and the equality is satisfied only, when $\vartheta(x)=\vartheta_0(x)$. Note that the entropy for $\vartheta_0(x)$ is also given by
\begin{align*}
\mathcal{H}(\vartheta_0(x)) &= - \int_{X} \frac{1}{\mu(X)} \ln \left(\frac{1}{\mu(X)}\right) \mu(dx),\\
&= - \ln \left(\frac{1}{\mu(X)}\right).
\end{align*}
Hence, $\mathcal{H}(\vartheta(x)) \le \mathcal{H}(\vartheta_0(x))$ for all $\vartheta(x) \in D(X, \mathscr{A}, \mu)$.
$\Box$
\end{proof}
\begin{definition}\label{DFN7}
Let $\vartheta(x), \xi(x) \in L^1(X, \mathscr{A}, \mu)$ be two nonnegative measurable functions such that $\operatorname{supp}\xi(x) \subset \operatorname{supp}\vartheta(x)$. Then, the relative entropy of $\xi(x)$ with respect to $\vartheta(x)$ is defined by
\footnote{
{\bf Lemma} (\cite[Voigt (1981)]{Voi81}) Suppose that $P$ is a Markov operator, then
\begin{align*}
\mathcal{H}_{\rm r}(P^n\xi(x)\, \vert\, P^n \vartheta(x)) \ge \mathcal{H}_{\rm r}(\xi(x)\, \vert\, \vartheta(x)), ~~ \forall \vartheta(x) \in D(X, \mathscr{A}, \mu),
\end{align*}
for any nonnegative measurable function $\xi(x)$.
\begin{remark}
Notice that any linear operator $P \colon L^1(X, \mathscr{A}, \mu) \rightarrow L^1(X, \mathscr{A}, \mu)$ satisfying
\begin{enumerate} [(i)]
\item $P \vartheta(x) \ge 0$ and
\item $\Vert P \vartheta(x)\Vert_{L^1(X, \mathscr{A}, \mu)} = \Vert \vartheta(x)\Vert_{L^1(X, \mathscr{A}, \mu)}$
\end{enumerate}
for any nonnegative measurable function $\vartheta(x) \in L^1(X, \mathscr{A}, \mu)$ is called a Markov operator.
\end{remark}}
\begin{align}
\mathcal{H}_{\rm r}(\xi(x)\,\vert\,\vartheta(x)) & = \int_{X} \xi(x) \ln \left (\frac{\xi(x)}{\vartheta(x)}\right) \mu(dx),\notag \\
&= \int_{X} \bigl(\xi(x) \ln \xi(x) - \xi(x) \ln \vartheta(x) \bigr) \mu(dx). \label{EQ21}
\end{align}
\end{definition}
\begin{remark}
Note that the relative entropy $\mathcal{H}_{\rm r}(\xi(x)\,\vert\,\vartheta(x))$, which measures the deviation of $\xi(x)$ from the density function $\vartheta(x)$, has the following properties.
\begin{enumerate} [(i)]
\item If $\xi(x), \vartheta(x) \in D(X, \mathscr{A}, \mu)$, then $\mathcal{H}_{\rm r}(\xi(x)\,\vert\,\vartheta(x)) \ge 0$ and $\mathcal{H}_{\rm r}(\xi(x)\,\vert\,\vartheta(x)) = 0$ if and only if $\xi(x) = \vartheta(x)$.
\item If $\vartheta(x)$ is constant density and $\vartheta(x) = 1$, then $\mathcal{H}_{\rm r}(\xi(x)\,\vert\,1) = \mathcal{H}(\xi(x))$. Thus, the relative entropy is a generalization of entropy.
\end{enumerate}
\end{remark}
\begin{remark}
For any $\vartheta(x) \in L^1(X, \mathscr{A}, \mu)$, the support of $\vartheta(x)$ is defined by $\operatorname{supp} \vartheta(x) = \bigl\{x \in X \,\vert \, \vartheta(x) \neq 0 \bigr\}$.
\end{remark}
\section{A family of mappings for multi-channel systems} \label{S3}
Consider the following continuous-time multi-channel system
\begin{align}
\dot{x}(t) &= A(t) x(t) + \sum\nolimits_{j \in \mathcal{N}} B_j(t) u_j(t), ~~ x(t_0)=x_0, ~~ t \in [t_0, +\infty ), \label{EQ22}
\end{align}
where $A(\cdot) \in \mathbb{R}^{d \times d}$, $B_j(\cdot) \in \mathbb{R}^{d \times r_j}$, $x(t) \in X$ is the state of the system, $u_j(t) \in U_j$ is the control input to the $j$th\,-\,channel and $\mathcal{N} \triangleq \{1, 2, \ldots, N\}$ represents the set of control input channels (or the set of feedback operators) in the system.
Moreover, we consider the following class of admissible control strategies that will be useful in Section~\ref{S4} (i.e., in a game-theoretic formalism)
\begin{align}
U_{\mathcal{L}} \subseteq \biggm\{u(t) \in \prod\nolimits_{j \in \mathcal{N}} \underbrace{L^{2}(\mathbb{R}_{+},\mathbb{R}^{r_j}) \cap L^{\infty}(\mathbb{R}_{+},\mathbb{R}^{r_j})}_{\triangleq U_j}\biggm\}, \label{EQ23}
\end{align}
where $u(t)$ is given by $u(t)=\bigl(u_1(t), \, u_{2}(t), \, \ldots, \, u_N(t)\bigr)$.
In what follows, suppose there exists a set of feedback operators $\bigl(\mathcal{L}_1^{\ast},\, \mathcal{L}_2^{\ast},\, \ldots,\, \mathcal{L}_N^{\ast}\bigr)$ from a class of linear operators \,$\mathscr{L} \colon X \rightarrow U_{\mathcal{L}}$ (i.e., $(\mathcal{L}_jx)(t) \in U_j$ for $j \in \mathcal{N}$) with strategies $\bigl(\mathcal{L}_j^{\ast}x\bigr)(t) \in U_j$ for $t \ge t_0$ and for $j \in \mathcal{N}$. Further, let $\phi_j\bigl(t; t_0, x_0, \bigl(\widehat{u_j(t), u^{\ast}_{\neg j}(t)}\bigr)\bigr) \in X$ be the unique solution of the $j$th\,-\,subsystem
\begin{align}
\dot{x}^j(t) &= \Bigl(A(t)+ \sum\nolimits_{i \in \mathcal{N}_{\neg j}} B_i(t) \mathcal{L}_i^{\ast}(t) \Bigr) x^j(t) + B_j(t) u_j(t), \label{EQ24}
\end{align}
with an initial condition $x_0 \in X$ and control inputs given by
\begin{align}
\bigl(\widehat{u_j(t), u^{\ast}_{\neg j}(t)}\bigr)\triangleq\bigl(u^{\ast}_1(t),\, u^{\ast}_{2}(t),\,\ldots,\,u^{\ast}_{j-1}(t),\, u_{j}(t),\, u^{\ast}_{j+1}(t),\,\ldots,\, u^{\ast}_N(t)\bigr)\in U_{\mathcal{L}}, \label{EQ25}
\end{align}
where $u^{\ast}_i(t) = \mathcal{L}_i^{\ast}(t) x^j(t)$ for $i \in \mathcal{N}_{\neg j} \triangleq \mathcal{N} \backslash \{j\}$ and $j \in \mathcal{N}$.
Furthermore, we may require that the control input for the $j$th\,-\,channel to be $u_j(t)=\bigl(\mathcal{L}_jx^j\bigr)(t) \in U_j$ and with this set of linear feedback operators
\begin{align*}
\underbrace{\bigl(\mathcal{L}_1^{\ast},\,\ldots,\, \mathcal{L}_{j-1}^{\ast},\,\mathcal{L}_{j},\,\mathcal{L}_{j+1}^{\ast},\, \ldots,\,\mathcal{L}_N^{\ast}\bigr)}_{ \triangleq \bigl(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast}\bigr)} \in \mathscr{L}.
\end{align*}
Then, the unique solution $\phi_j\bigl(t; t_0, x_0, \bigl(\widehat{u_j(t), u^{\ast}_{\neg j}(t)}\bigr)\bigr)$ will take the form
\begin{align}
\phi_j\bigl(t; t_0, x_0, \bigl(\widehat{u_j(t), u^{\ast}_{\neg j}(t)}\bigr)\bigr) = \underbrace{\Phi^{\mathcal{L}_{\neg j}^{\ast}}(t, t_0)\, \Phi^{\mathcal{L}_j}(t, t_0)}_{\triangleq \Phi_t^{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast})}} x_0, ~~ \forall t \in [t_0,\,+\infty), \label{EQ26}
\end{align}
where
\begin{align}
\frac{\partial\Phi^{\mathcal{L}_{\neg j}^{\ast}}(t, \tau)}{\partial t} &= \Bigl(A(t) + \sum\nolimits_{i \in \mathcal{N}_{\neg j}} B_i(t) \mathcal{L}_i^{\ast}(t) \Bigr)\Phi^{\mathcal{L}_{\neg j}^{\ast}}(t, \tau), \label{EQ27} \\
\frac{\partial\Phi^{\mathcal{L}_j}(t, \tau)}{\partial t} &= B_j^{\ast}(t)\Phi^{\mathcal{L}_j}(t, \tau), \label{EQ28}
\end{align}
with both $\Phi^{\mathcal{L}_{\neg j}^{\ast}}(\tau, \tau)$ and $\Phi^{\mathcal{L}_j}(\tau, \tau)$ are identity matrices; and $B^{\ast}(t)$ is given by
\begin{align}
B_j^{\ast}(t) = \Bigl(\Phi^{\mathcal{L}_{\neg j}^{\ast}}(t, \tau)\Bigr)^{-1} B_j(t)\mathcal{L}_j(t) \Phi^{\mathcal{L}_{\neg j}^{\ast}}(t, \tau), \label{EQ29}
\end{align}
for each $j \in \mathcal{N}$.\footnote{Note that, with $t_0 = \tau$, if we take the partial derivative of $\Phi_t^{(\mathcal{L}_j,\,\mathcal{L}_{\neg j}^{\ast})}$ with respect to $t$ and make use of Equations~\eqref{EQ6} and \eqref{EQ7} together with Equation~\eqref{EQ8}, then we have
\begin{align*}
\frac{\partial}{\partial t}\Bigl(\Phi_t^{(\mathcal{L}_j,\,\mathcal{L}_{\neg j}^{\ast})}\Bigr) &= \frac{\partial}{\partial t}\Bigl(\Phi^{\mathcal{L}_{\neg j}^{\ast}}(t, \tau)\Bigr)\,\Phi^{\mathcal{L}_j}(t, \tau) + \Phi^{\mathcal{L}_{\neg j}^{\ast}}(t, \tau)\, \frac{\partial}{\partial t}\Bigl(\Phi^{\mathcal{L}_j}(t, \tau))\Bigr), \\
&= \Bigl(A(t) + \sum\nolimits_{i \in \mathcal{N}_{\neg j}} B_i(t) \mathcal{L}_i^{\ast}(t) \Bigr)\Phi^{\mathcal{L}_{\neg j}^{\ast}}(t, \tau)\, \Phi^{\mathcal{L}_j}(t, \tau) \\
& \quad\quad\quad \quad + \Phi^{\mathcal{L}_{\neg j}^{\ast}}(t, \tau) \Bigl(\Phi^{\mathcal{L}_{\neg j}^{\ast}}(t, \tau)\Bigr)^{-1} B_j(t)\mathcal{L}_j(t)\,\Phi^{\mathcal{L}_{\neg j}^{\ast}}(t, \tau) \,\Phi^{\mathcal{L}_j}(t, \tau)), \\
&= \left(\Bigl(A(t) + \sum\nolimits_{i \in \mathcal{N}_{\neg j}} B_i(t) \mathcal{L}_i^{\ast}(t) \Bigr) + B_j(t)\mathcal{L}_j(t) \right) \,\Phi_t^{(\mathcal{L}_j,\,\mathcal{L}_{\neg j}^{\ast})}.
\end{align*}
Moreover, we note that $\Phi^{\mathcal{L}_{\neg j}^{\ast}}(t, \tau)$ satisfies the following
\begin{align*}
\Phi^{\mathcal{L}_{\neg j}^{\ast}}(t_2, t_1)\, \Phi^{\mathcal{L}_{\neg j}^{\ast}}(t_1, \tau) = \Phi^{\mathcal{L}_{\neg j}^{\ast}}(t_2, \tau), \quad \forall t_1, t_2 \in [\tau,\,+\infty),
\end{align*}
(e.g., see \cite{Che62} for such a decomposition that arises in differential equations).}
In the following, we assume that $X$ is a topological Hausdorff space and $\mathscr{A}$ is a $\sigma$\,-\,algebra of Borel set, i.e., the smallest $\sigma$\,-\,algebra which contains all open, and thus closed, subsets of $X$. With this, for any $t \ge 0$ (assuming that $t_0=0)$ and $(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast})\in \mathscr{L}$, we can consider a family of continuous mappings (or transformations) $\Bigl\{\Phi_t^{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast})} \Bigr\}_{t \ge 0}$
on $X$ satisfying
\begin{align}
(\mathbb{R}_{+} \times X) \ni (t,\,x) \mapsto \Phi_t^{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast})} x \in X. \label{EQ30}
\end{align}
Note that, for each fixed $t \ge 0$, the transformation $\Phi_t^{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast})}$ is measurable, i.e., we have $\Bigl(\Phi_t^{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast})}\Bigr)^{-1}(A) \in \mathscr{A}$ for all $A \in \mathcal{A}$, where $\Bigl(\Phi_t^{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast})}\Bigr)^{-1}(A)$ denotes the set of all $x$ such that $\Phi_t^{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast})} x \in A$ (e.g., see \cite{LasPia77} for invariant measures on topological spaces; see also \cite{BauSig75} for topological properties of measure spaces).
Then, we can introduce the following definitions.
\begin{definition}\label{DFN8}
A measure $\mu$ is called invariant under the family of measurable transformations $\Bigl\{\Phi_t^{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast})} \Bigr\}_{t \ge 0}$ if
\begin{align}
\mu\Bigl(\Phi_t^{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast})} \bigr)^{-1}(A) = \mu\bigl(A\bigr), ~~ \forall A \in \mathcal{A}. \label{EQ31}
\end{align}
\end{definition}
Next, we assume that the family of transformations $\Bigl\{\Phi_t^{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast})} \Bigr\}_{t \ge 0}$ are nonsingular and, for each fixed $t \ge 0$, the unique Frobenius-Perron operator $\mathit{P}_t^{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast})} \colon L^1(X, \mathscr{A}, \mu)\rightarrow L^1(X, \mathscr{A}, \mu)$ is then defined by
\begin{align}
\int_{A} \mathit{P}_t^{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast})} \vartheta(x) \mu(dx) = \int_{\Bigl(\Phi_t^{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast})} \bigr)^{-1}(A) } \vartheta(x) \mu(dx), ~~ \forall A \in \mathscr{A}. \label{EQ32}
\end{align}
\begin{definition}\label{DFN9}
Let $(X, \mathscr{A}, \mu)$ be a measure space, then the family of operators $\Bigl\{\mathit{P}_t^{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast})}\Bigr\}_{t \ge 0}$, $\forall (\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast}) \in \mathscr{L}$, $\forall t \ge 0$ and $\forall j \in \mathcal{N}$, satisfies the following properties.
\begin{enumerate}[(P1)]
\item For all $\vartheta_1(x), \vartheta_2(x) \in L^1(X, \mathscr{A}, \mu)$ and $\lambda_1, \lambda_2 \in \mathbb{R}$
\begin{align*}
\mathit{P}_t^{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast})} \Bigl\{\lambda_1 \vartheta_1(x) + \lambda_2 \vartheta_2(x) \Bigr\}= \lambda_1 \mathit{P}_t^{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast})} \vartheta_1(x) + \lambda_2 \mathit{P}_t^{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast})} \vartheta_2(x).
\end{align*}
\item $\mathit{P}_t^{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast})} \vartheta(x) \ge 0$ if $\vartheta(x) \ge 0$.
\item For all $\vartheta(x) \in L^1(X, \mathscr{A}, \mu)$,
\begin{align*}
\int_{X} \mathit{P}_t^{(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast})} \vartheta(x) \mu(dx) = \int_{X} \vartheta(x)\mu(dx),
\end{align*}
\end{enumerate}
is called a semigroup; and it is uniformly continuous, if
\begin{align}
\lim_{t \rightarrow t_0} \Bigl \Vert \mathit{P}_t^{(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast})} \vartheta(x) - \mathit{P}_{t_0}^{(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast})} \vartheta(x) \Bigr\Vert_{L^1(X, \mathscr{A}, \mu)} = 0, ~~ \forall t_0 \ge 0, \label{EQ33}
\end{align}
for each $\vartheta(x) \in L^1(X, \mathscr{A}, \mu)$ with $(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast}) \in \mathscr{L}$.
\end{definition}
\section{Main Results} \label{S4}
\subsection{Game-theoretic formalism} \label{S4(1)}
In the following, we specify a game in a feedback strategic form -- where, in the course of the game, each feedback operator generates automatically a feedback control in response to the action of other feedback operators via the system state $x(t)$ for $t \in [t_0,\,+\infty)$. For example, the $j$th\,-\,feedback operator can generate a feedback control $u_j(t)=\bigl(\mathcal{L}_jx^j\bigr)(t)$ in response to the actions of other feedback operators $u^{\ast}_i(t)= \bigl(\mathcal{L}_i^{\ast} x^j\bigr)(t)$ for $i \in \mathcal{N}_{\neg j}$, where $\bigl(\widehat{u_j(t), u^{\ast}_{\neg j}(t)}\bigr) \in U_{\mathcal{L}}$, and, similarly, any number of feedback operators can decide on to play feedback strategies simultaneously. Hence, for such a game to have a set of stable (game-theoretic) equilibrium feedback operators (which is also robust to small perturbations in the system or strategies played by others), then each feedback operator is required to respond (in some sense of best-response correspondence) to the others strategies.
To this end, it will be useful to consider the following criterion functions
\begin{align}
\mathscr{L} \ni (\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast}) \mapsto \mathcal{H}_{\rm r}\Bigr(\mathit{P}_t^{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast})}\vartheta(x)\,\bigl\lvert\,\vartheta(x) \Bigl) \in \mathbb{R}_{-} \cup \{-\infty\}, ~~ \forall t\ge 0, ~~ \forall j \in \mathcal{N}, \label{EQ34}
\end{align}
over the class of admissible control functions $U_{\mathcal{L}}$ (or the set of linear feedback operators $\mathscr{L}$) and for any $\vartheta(x) \in D(X, \mathscr{A}, \mu)$.
Note that, under the game-theoretic framework, if there exists a set of equilibrium feedback operators $\bigl(\mathcal{L}_1^{\ast},\, \mathcal{L}_2^{\ast},\, \ldots,\, \mathcal{L}_N^{\ast}\bigr)$ (from the class of linear feedback operators $\mathscr{L}$). Then, this set of equilibrium feedback operators decreases the relative entropy between any two density functions from $D(X, \mathscr{A}, \mu)$ for all $t \ge 0$. On the other hand, if there exists a unique stationary density function (i.e., a common fixed-point) $\vartheta_{\ast}(x) \in D(X, \mathscr{A}, \mu)$ for the family of Frobenius-Perron operators $\mathit{P}_t^{(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast})}$ for each fixed $t \ge 0$. Then, the composition of the multi-channel system with this set of equilibrium feedback operators, when described by density functions, will evolve towards the unique equilibrium state, i.e., the entropy of the whole system will be maximized (see Lanford \cite[pp\,1--113]{Lan73} for an exposition of equilibrium states and entropy in statistical mechanics).
Therefore, more formally, we have the following definition for the set of equilibrium feedback operators $(\mathcal{L}_1^{\ast},\, \mathcal{L}_2^{\ast}, \, \ldots, \, \mathcal{L}_N^{\ast}) \in \mathscr{L}$.
\begin{definition}\label{DFN10}
We shall say that a set of linear system operators $(\mathcal{L}_1^{\ast},\, \mathcal{L}_2^{\ast}, \, \ldots, \, \mathcal{L}_N^{\ast}) \in \mathscr{L}$ is a set of (game-theoretic) equilibrium feedback operators, if they produce control responses given by
\begin{align}
u_j^{\ast}(t) = \bigl(\mathcal{L}_j^{\ast}x\bigr)(t) \in U_j, ~~ \forall j \in \mathcal{N}, \label{EQ35}
\end{align}
for $t \in [0, \infty]$ and satisfy further the following conditions
\begin{align}
\left.\begin{array}{r}
\mathcal{H}_{\rm r}\Bigl(\mathit{P}_t^{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast})}\vartheta(x)\,\bigl\lvert\,\vartheta(x) \Bigr) \ge \mathcal{H}_{\rm r}\Bigl(\mathit{P}_t^{(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast})}\vartheta(x)\,\bigl\lvert\,\vartheta(x) \Bigr), \quad \forall t \ge 0, ~~ \\
~~~~ \forall (\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast}) \in \mathscr{L}, ~ \forall j \in \mathcal{N}, ~~ \\ \\
D(X, \mathscr{A}, \mu)\ni \mathit{P}_t^{(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast})}\vartheta(x) \rightarrow \vartheta_{\ast}(x) \in D(X, \mathscr{A}, \mu) \quad \text{as} \quad t \rightarrow \infty, ~~ \\ \\
\mathcal{H}\Bigl(\mathit{P}_t^{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast})}\vartheta(x)\Bigr) \le \mathcal{H}\Bigl(\mathit{P}_t^{(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast})}\vartheta_{\ast}(x)\Bigr), \quad \forall t \ge 0, \quad \forall j \in \mathcal{N}, ~~
\end{array} \right \} \label{EQ36}
\end{align}
for each $\vartheta(x) \in D(X, \mathscr{A}, \mu)$.
\end{definition}
\begin{remark}
We remark that the relative entropy $\mathcal{H}_{\rm r}(\cdot \lvert \cdot)$ in Equation~\eqref{EQ36} is determined with respect to $\mathcal{L}_j$ for each $j \in \mathcal{N}$; while the others $L_{\neg j}^{\ast}$ remain fixed.
\end{remark}
Then, we formally state the main objective of this paper.
\begin{problem}
Provide a sufficient condition for the existence of a set of equilibrium feedback operators in the multi-channel system (that interacts strategically in a game-theoretic framework) such that when the composition of the multi-channel system with this set of equilibrium feedback operators, described by density functions, will evolve towards an equilibrium state in such a way that the entropy of the whole system is maximized.
\end{problem}
\subsection{Existence of a set of (game-theoretic) equilibrium feedback operators}\label{S4(2)}
In the following, we provide a sufficient condition for the existence of a unique equilibrium state that is associated with a stationary density function (i.e., a common fixed-point) for the family of Frobenius-Perron operators $\Bigl\{\mathit{P}_t^{(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast})}\Bigr\}_{t \ge 0}$.
\begin{proposition}\label{PR2}
Let $B(X, \mathscr{A}, \mu)$ be an open ball in $D(X, \mathscr{A}, \mu)$ of center $\vartheta_0(x) \in D(X, \mathscr{A}, \mu)$ and radius $\beta$, i.e,,
\begin{align}
B(X, \mathscr{A}, \mu) = \Bigl \{ \vartheta(x) \in D(X, \mathscr{A}, \mu) \, \Bigl \vert \, \bigl\lVert \vartheta(x) - \vartheta_0(x) \bigr\rVert_{L^1(X, \mathscr{A}, \mu)} \le \beta \Bigr\}. \label{EQ37}
\end{align}
Suppose that there exists a set of feedback operators $\bigr(\mathcal{L}_1^{\ast}, \mathcal{L}_2^{\ast}, \ldots, \mathcal{L}_N^{\ast} \bigl)\,\in\negthinspace\mathscr{L}$ such that the family of Frobenius-Perron operators $\Bigr\{\mathit{P}_t^{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast})} \Bigl\}_{t \ge 0}$ with respect to $\Phi_t^{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast})}$ satisfies
\begin{align}
\sup_{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast}) \in \mathscr{L}} \Bigl \Vert \mathit{P}_t^{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast})} \vartheta_2(x) &- \mathit{P}_t^{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast})} \vartheta_1(x) \Bigr\Vert_{L^1(X, \mathscr{A}, \mu)} \notag \\
&\le \kappa \bigl \Vert \vartheta_2(x) - \vartheta_1(x) \bigr\Vert_{L^1(X, \mathscr{A}, \mu)}, ~ \forall t \ge 0, ~ \forall j \in \mathcal{N}, \label{EQ38}
\end{align}
for any two $\vartheta_1(x), \vartheta_2(x) \in B(X, \mathscr{A}, \mu)$, where $\kappa$ is a positive constant which is less than one.
Then, if
\begin{align}
\sup_{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast}) \in \mathscr{L}} \Bigl \Vert \mathit{P}_t^{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast})} \vartheta_0(x) - \vartheta_0(x) \Bigr\Vert_{L^1(X, \mathscr{A}, \mu)} \le \beta\bigl(1 - \kappa \bigr), ~~ \forall t \ge 0, ~~ \forall j \in \mathcal{N}, \label{EQ39}
\end{align}
there is at least one stationary density function (i.e., a common fixed-point) ~ $\vartheta_{\ast}(x) \in B(X, \mathscr{A}, \mu)$ such that
\begin{align}
\mathit{P}_t^{(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast})} \vartheta_{\ast}(x) = \vartheta_{\ast}(x), ~~ \forall t \ge 0, ~ \forall j \in \mathcal{N}. \label{EQ40}
\end{align}
Furthermore, there exists a unique equilibrium state, which corresponds with $\vartheta_{\ast}(x)$, if the measure $\mu_{\ast}$
\begin{align}
\mu_{\ast}(A) = \int_{A} \vartheta_{\ast}(x) \mu(dx), ~~ \forall A \in \mathscr{A}, \label{EQ41}
\end{align}
is invariant with respect to $\Phi_t^{(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast})}$ for each fixed $t \ge 0$.\footnote{Note that the supremum in Equation~\eqref{EQ38} (and also in Equation~\eqref{EQ39}) is computed with respect to $\mathcal{L}_j$ with $(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast}) \in \mathscr{L}$ for each $j \in \mathcal{N}$, while $\mathcal{L}_{\neg j}^{\ast}$ remains fixed.}
\end{proposition}
\begin{proof}
Observe that $\mathit{P}_t^{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast})}$ is continuous for each $t \ge 0$ and for any $\vartheta(x) \in D(X, \mathscr{A}, \mu)$ (cf. Equation~\eqref{EQ33}). For $\vartheta_{\ast}(x) \in B(X, \mathscr{A}, \mu)$, we will show that there exists a convergent sequence of functions $\bigl\{ \vartheta_n(x)\bigr\}$ such that
\begin{align*}
B(X, \mathscr{A}, \mu) \ni \vartheta_n(x) = \mathit{P}_t^{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast})}\vartheta_{n-1}(x) - \vartheta_0(x), ~ \forall n \ge 0, ~ \forall j \in \mathcal{N}, ~ \forall (\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast}) \in \mathscr{L},
\end{align*}
We see that if $\vartheta_p(x)$ is defined in $B(X, \mathscr{A}, \mu)$ for $1 \le p \le n$, i.e., $\mathit{P}_t^{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast})}\vartheta_{p-1}(x) -\vartheta_0(x) \in B(X, \mathscr{A}, \mu)$, $\forall p \in [1, n]$, then we have followings
\begin{align*}
\vartheta_p(x) - \vartheta_{p-1}(x) = \mathit{P}_t^{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast})}\vartheta_{p-1}(x) - \mathit{P}_t^{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast})}\vartheta_{p-2}(x).
\end{align*}
and
\begin{align*}
&\bigl \Vert \vartheta_p(x) - \vartheta_{p-1}(x) \bigr\Vert_{L^1(X, \mathscr{A}, \mu)} \notag \\
&\quad\quad\quad \le \kappa \sup_{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast}) \in \mathscr{L}}\bigl\Vert \mathit{P}_t^{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast})}\vartheta_{p-1}(x) - \mathit{P}_t^{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast})}\vartheta_{p-2}(x)\bigr \Vert_{L^1(X, \mathscr{A}, \mu)}, \notag \\
&\quad\quad\quad\quad\quad\quad\quad\quad ~~ ~ \forall t \ge 0, ~ \forall j \in \mathcal{N}.
\end{align*}
With $(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast}) \in \mathscr{L}$, we conclude that
\begin{align*}
\bigl \Vert \vartheta_p(x) - \vartheta_{p-1}(x) \bigr\Vert_{L^1(X, \mathscr{A}, \mu)} \le \kappa^{p-1}\bigl\Vert\vartheta_1(x)\bigr \Vert_{L^1(X, \mathscr{A}, \mu)},
\end{align*}
which further gives us
\begin{align*}
\bigl \Vert \vartheta_p(x)\bigr\Vert_{L^1(X, \mathscr{A}, \mu)} \le (1+ \kappa + \kappa^2+ \cdots + \kappa^{p-1})\bigl\Vert\vartheta_1(x)\bigr \Vert_{L^1(X, \mathscr{A}, \mu)},
\end{align*}
and
\begin{align*}
\bigl \Vert \vartheta_p(x)\bigr\Vert_{L^1(X, \mathscr{A}, \mu)} \le \frac{1}{1- \kappa} \bigl\Vert\vartheta_1(x)\bigr \Vert_{L^1(X, \mathscr{A}, \mu)} < \beta.
\end{align*}
Hence, this agrees with our claim, i.e.,
\begin{align*}
\sup_{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast}) \in \mathscr{L}}\bigl\Vert \mathit{P}_t^{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast})}\vartheta_0(x) - \vartheta_0(x)\bigr \Vert_{L^1(X, \mathscr{A}, \mu)} < \beta(1- \kappa), ~~ \forall t \ge 0, ~ \forall j \in \mathcal{N}.
\end{align*}
Note that, for any $n \ge 0$, we have
\begin{align*}
\bigl \Vert \vartheta_n(x) - \vartheta_{n-1}(x) \bigr\Vert_{L^1(X, \mathscr{A}, \mu)} \le \kappa^{n-1}\bigl\Vert\vartheta_1(x)\bigr \Vert_{L^1(X, \mathscr{A}, \mu)},
\end{align*}
which is strongly convergent (i.e., $\lim_{n \rightarrow \infty} \bigl \Vert \vartheta_n(x) - \vartheta_{n-1}(x) \bigr\Vert_{L^1(X, \mathscr{A}, \mu)} = 0$). Then, by passing to a limit, we conclude that there exists a common fixed-point (or a stationary density function) $\vartheta_{\ast}(x) \in B(X, \mathscr{A}, \mu)$ for the family of Frobenius-Perron operators $\Bigl\{\mathit{P}_t^{(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast})} \Bigr\}_{t \ge 0}$ that satisfies
\begin{align*}
\mathit{P}_t^{(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast})} \vartheta_{\ast}(x) = \vartheta_{\ast}(x), ~~ \forall t \ge 0,
\end{align*}
which also corresponds to the unique equilibrium state, in the sense of statistical mechanics, for the multi-channel system together with $(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast}) \in \mathscr{L}$.\footnote{\label{FT1}Note that the following also holds true
\begin{align*}
\lim_{t \rightarrow \infty} \mathit{P}_t^{(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast})} \vartheta(x) = \vartheta_{\ast}(x),
\end{align*}
for any $\vartheta(x) \in B(X, \mathscr{A}, \mu)$.}
Moreover, from Theorem~\ref{TH1}, we see that the measure $\mu_{\ast}$, i.e.,
\begin{align*}
\mu_{\ast}(A) = \int_{A} \vartheta_{\ast}(x) \mu(dx), ~~ \forall A \in \mathscr{A},
\end{align*}
is invariant with respect to $\Phi_t^{(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast})}$ for each fixed $t \ge 0$.
$\Box$
\end{proof}
The above proposition (i.e., Proposition~\ref{PR2}) is important because of the three way connection it draws between the existence of a common stationary density function $\vartheta_{\ast}(x) \in D(X, \mathscr{A}, \mu)$ for $\Bigl\{\mathit{P}_t^{(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast})}\Bigr\}_{t \ge 0}$ (i.e., the unique equilibrium state), the invariant measure $\mu_{\ast}$ (i.e., the measure preserving property of $\Phi_t^{(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast})}$ for all $t \ge 0$) and the set of equilibrium feedback operators $\bigr(\mathcal{L}_1^{\ast}, \mathcal{L}_2^{\ast}, \ldots, \mathcal{L}_N^{\ast} \bigl) \in \mathscr{L}$. Moreover, the corresponding maximum entropy $\mathcal{H}_{max} \bigl(\vartheta_{\ast}(x) \bigr)$ is given by
\begin{align*}
\mathcal{H}_{\rm max} \bigl(\vartheta_{\ast}(x) \bigr) = -\int_{X} \vartheta_{\ast}(x) \ln \vartheta_{\ast}(x) \mu(dx).
\end{align*}
\begin{remark}
We remark that, in the above proposition, a fixed-point theorem is implicitly used for deriving a sufficient condition for the existence of a common stationary density function for the family of Frobenius-Perron operators (e.g., see Dunford and Schwartz \cite[pp\,456]{DunSch58} or Dieudonn\'{e} \cite[pp\,261]{Dieu60}).
\end{remark}
\subsection{Asymptotic stability of the family of Frobenius-Perron operators} \label{S4(3)}
Here, we provide a connection between the stationary density function $\vartheta_{\ast}(x) \in D(X, \mathscr{A}, \mu)$ (which corresponds to the equilibrium state) and the asymptotic stability of the family of Frobenius-Perron operators $\Bigl\{\mathit{P}_t^{(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast})} \Bigr\}_{t \ge 0}$. Note that, from Proposition~\ref{PR2}, any initial density function $\vartheta(x) \in D(X, \mathscr{A}, \mu)$ under the action of the family of Frobenius-Perron operators $\Bigl\{\mathit{P}_t^{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast})} \Bigr\}_{t \ge 0}$ will only converge to a unique stationary density function $\vartheta_{\ast}(x) \in D(X, \mathscr{A}, \mu)$, if the relative entropy
\begin{align}
\sup_{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast}) \in \mathscr{L}} \mathcal{H}_{\rm r}\Bigr(\mathit{P}_t^{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast})}\vartheta(x)\,\bigl\lvert\,\vartheta_{\ast}(x) \Bigl), ~~ \forall j \in \mathcal{N}, \label{EQ42}
\end{align}
tends zero as $t \rightarrow \infty$, and when the set of feedback operators attains a (game-theoretic) equilibrium.
Then, we have the following corollary that exactly establishes the connection between the relative entropy and the stationary density function (where the latter corresponds to the unique equilibrium state).
\begin{corollary}\label{CR4}
Suppose that the set of equilibrium feedback operators $\bigl(\mathcal{L}_1^{\ast}, \mathcal{L}_2^{\ast}, \ldots, \mathcal{L}_N^{\ast} \bigr) \in \mathscr{L}$ satisfies Proposition~\ref{PR2}. Then,
\begin{align}
\lim_{t \rightarrow \infty} \mathcal{H}_{\rm r}\Bigl(\mathit{P}_t^{(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast})}\vartheta(x)\,\bigl\lvert\,\vartheta_{\ast}(x) \Bigr) = 0, ~~ \forall j \in \mathcal{N}, \label{EQ43}
\end{align}
for each $\vartheta(x) \in D(X, \mathscr{A}, \mu)$ such that $\mathcal{H}_{\rm r}\bigr(\vartheta(x)\,\bigl\lvert\,\vartheta_{\ast}(x) \bigl)$ is finite.
\end{corollary}
\begin{proof}
From Proposition~\ref{PR2}, if $\bigl(\mathcal{L}_1^{\ast}, \mathcal{L}_2^{\ast}, \ldots, \mathcal{L}_N^{\ast} \bigr) \in \mathscr{L}$ is a set of equilibrium feedback operators. Then, there is a common fixed-point density function $\vartheta_{\ast}(x) \in D(X, \mathscr{A}, \mu)$ such that
\begin{align*}
\vartheta_{\ast}(x) \in \bigcap_{t \ge 0} \overline{\mathit{P}_t^{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast})}\vartheta(x)} \neq \varnothing, ~~ \forall (\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast}) \in \mathscr{L}, ~~ \forall j \in \mathcal{N}, \\
\forall \vartheta(x) \in D(X, \mathscr{A}, \mu),
\end{align*}
and
\begin{align*}
\mathit{P}_t^{(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast})}\vartheta_{\ast}(x) = \vartheta_{\ast}(x), ~~ \forall t \ge 0,
\end{align*}
with (cf. Equation~\eqref{EQ36} or Footnote~\ref{FT1})
\begin{align*}
\lim_{t \rightarrow \infty} \mathit{P}_t^{(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast})}\vartheta(x) = \vartheta_{\ast}(x), ~~ \forall \vartheta(x) \in D(X, \mathscr{A}, \mu).
\end{align*}
Further, if $\mathcal{H}_{\rm r}\bigr(\vartheta(x)\,\bigl\lvert\,\vartheta_{\ast}(x) \bigl)$ is finite, then we have
\begin{align*}
\mathcal{H}_{\rm r}\Bigl(\mathit{P}_t^{(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast})}\vartheta(x)\,\bigl\lvert\,\vartheta_{\ast}(x) \Bigr) \rightarrow 0 \quad \text{as} \quad t \rightarrow \infty,
\end{align*}
for any $ \vartheta(x) \in D(X, \mathscr{A}, \mu)$.
$\Box$
\end{proof}
\subsection{Resilient behavior of a set of (game-theoretic) equilibrium feedback operators} \label{S4(4)}
Here, we consider the following systems with a small random perturbation term
\begin{align}
d Z_{\epsilon}^j(t) = \Bigl(A(t) + \sum\nolimits_{i \in \mathcal{N}_{\neg j}} B_i(t) \mathcal{L}_i^{\ast}(t) \Bigr)Z_{\epsilon}^j(t) dt + B_j(t)\mathcal{L}_i(t) Z_{\epsilon}^j(t) dt \notag \\
\quad + \sqrt{\epsilon} \,\sigma(t, Z_{\epsilon}^j(t))\,d W(t), ~~ Z_{\epsilon}^j(0) = x_0, \notag \\
~(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast}) \in \mathscr{L}, ~ j \in \mathcal{N}, \label{EQ44}
\end{align}
where $\sigma(t, Z_{\epsilon}^j(t)) \in \mathbb{R}^{d \times d}$ is a diffusion term, $W(t)$ is a $d$-dimensional Wiener process and $\epsilon$ is a small positive number, which represents the level of perturbation in the system. Note that we assume here there exists a set of equilibrium feedback operators $\bigl(\mathcal{L}_1^{\ast}, \mathcal{L}_2^{\ast}, \ldots, \mathcal{L}_N^{\ast} \bigr) \in \mathscr{L}$, when $\epsilon = 0$ (which corresponds to the unperturbed multi-channel system). Then, we investigate, as $\epsilon \rightarrow 0$, the asymptotic stability behavior of an invariant measure for the family of Frobenius-Perron operators $\Bigl\{\mathit{P}_{\epsilon, t}^{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast})} \Bigr\}_{t \ge 0}$, with $(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast}) \in \mathscr{L}$, which corresponds to the multi-channel system with a small random perturbation.\footnote{We remark that such a solution for Equation~\eqref{EQ44} is assumed to have continuous sample paths with probability one (see Kunita \cite{Kun90} for additional information).}
\begin{remark}
Note that, in general, the evolution of the density function is given by
\begin{align*}
\mathit{P}_{\epsilon, t}^{(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast})}\vartheta(x) = \int_{X} \Gamma_{\epsilon, t}(x, y)\vartheta(y)\mu(dy), \quad \forall t \ge 0,
\end{align*}
where $\Gamma_{\epsilon, t}(\cdot,\cdot)$ is the kernel (i.e., the fundamental solution), which is independent of the initial density function $\vartheta(x) \in D(X, \mathscr{A}, \mu)$. Moreover, it is well known that the solution, which is associated with Cauchy problem, satisfies the Fokker-Planck (or Kolmogorov forward) equation that is completely specified, with some additional regularity conditions, by $\Bigl(A(t) + \sum\nolimits_{j \in \mathcal{N}} B_j(t) \mathcal{L}_j^{\ast}(t)\Bigr)Z_{\epsilon}(t)$ and $\sqrt{\epsilon}\,\sigma(t, Z_{\epsilon}(t))$ (e.g., see also \cite{GikSko75} or \cite{Kun90}).
\end{remark}
In what follows, we provide additional results, based on the asymptotic stability of an invariant measure, that partly establish the resilient behavior for the set of equilibrium feedback operators with respect to the random perturbation in the system.
\begin{proposition}\label{PR3}
For any continuous density function $\vartheta(x) \in D(X, \mathscr{A}, \mu)$, suppose that
\begin{align}
\sup_{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast}) \in \mathscr{L}}\,\Bigl \Vert \mathit{P}_{\epsilon, t}^{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast})} \vartheta(x) - \mathit{P}_t^{(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast})} \vartheta(x) \Bigr\Vert_{L^1(X, \mathscr{A}, \mu)}, ~~ \forall t \ge 0, \forall j \in \mathcal{N}, \label{EQ45}
\end{align}
tends to zero in a weak* topology on $X$ as $\epsilon \rightarrow 0$. Then, the weak limit of invariant measure $\mu_{\ast}^{\epsilon}$ of $\Big\{\mathit{P}_{\epsilon,t}^{(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast})} \Bigr\}_{t \ge 0}$ is absolutely continuous with respect to the invariant measure $\mu_{\ast}$, where the latter corresponds to the family of Frobenius-Perron operators $\Bigl\{\mathit{P}_t^{(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast})} \Bigr\}_{t \ge 0}$.
\end{proposition}
\begin{proof}
Note that, from the standard perturbation arguments for linear operators, if the set of equilibrium feedback operators $\bigl(\mathcal{L}_1^{\ast}, \mathcal{L}_2^{\ast}, \ldots, \mathcal{L}_N^{\ast} \bigr) \in \mathscr{L}$ and the fixed-point density function $\vartheta_{\ast}(x) \in D(X, \mathscr{A}, \mu)$ (i.e., $\mathit{P}_t^{(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast})}\vartheta_{\ast}(x)=\vartheta_{\ast}(x)$, $\forall t \ge 0$) satisfy Proposition~\ref{PR2}. Then, the following holds
\begin{align*}
\lim_{\epsilon \rightarrow 0} \left(\sup_{x \in X} \,\int_{X} \left \vert \mathit{P}_{\epsilon, t}^{(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast})} \vartheta(x) - \mathit{P}_t^{(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast})} \vartheta(x)\right\vert \mu(dx) \right)=0, ~~ \forall \vartheta(x) \in D(X, \mathscr{A}, \mu),
\end{align*}
for any fixed $t \ge 0$. This further implies the following
\begin{align*}
\lim_{\epsilon \rightarrow 0} \,\Bigl \Vert \vartheta_{\ast}^{\epsilon}(x) - \vartheta_{\ast}(x) \Bigr\Vert_{L^1(X, \mathscr{A}, \mu)} = 0,
\end{align*}
where $\vartheta_{\ast}^{\epsilon}(x)$ is invariant of $\mathit{P}_{\epsilon, t}^{(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast})}$ for each fixed $t \ge 0$.
In order for $\mu_{\ast}^{\epsilon}$ to be absolutely continuous with respect to $\mu_{\ast}$, i.e., $\mu_{\ast}^{\epsilon} \ll \mu_{\ast}$ and $\mu_{\ast}^{\epsilon} (A) = \int_{A} \vartheta_{\ast}^{\epsilon}(x) \mu_{\ast}^{\epsilon}(dx)$, $\forall A \in \mathscr{A}$, it is suffice to show that, for any fixed $t \ge 0$, the family of Frobenius-Perron operators $\mathit{P}_{\epsilon, t}^{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast})}$, with respect to $(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast}) \in \mathscr{L}$ for all $j \in \mathcal{N}$, should not be too different from $\mathit{P}_t^{(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast})}$ for small $\epsilon \ge 0$ (cf. Remark~\ref{RM8} below).
On the other hand, under the game-theoretic framework (cf. Proposition~\ref{PR2}), each of these feedback operators are required to respond in some sense of best-response correspondence to the others feedback strategies in the system. As a result of this, the following will hold true
\begin{align*}
\sup_{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast}) \in \mathscr{L}}\,\Bigl \Vert \mathit{P}_{\epsilon, t}^{(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast})} \vartheta_{\ast}(x) - \underbrace{\mathit{P}_t^{(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast})}\vartheta_{\ast}(x)}_{=\vartheta_{\ast}(x), ~\forall t \ge 0} \Bigr\Vert_{L^1(X, \mathscr{A}, \mu)} \rightarrow 0 \quad \text{as} \quad \epsilon \rightarrow 0,
\end{align*}
for each $j \in \mathcal{N}$, when only the set of feedback operators attains a robust/stable (game-theoretic) equilibrium solution $(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast}) \in \mathscr{L}$. Note that, in the above equation, the supremum is computed with respect to $\mathcal{L}_j$ with $(\mathcal{L}_j, \mathcal{L}_{\neg j}^{\ast}) \in \mathscr{L}$ for each $j \in \mathcal{N}$, while others $\mathcal{L}_{\neg j}^{\ast}$ remain fixed, and when there is also a small random perturbation in the system.
Then, we see that $\mu_{\ast}^{\epsilon}$ tends to $\mu_{\ast}$ weakly as $\epsilon \rightarrow 0$. This completes the proof.
$\Box$
\end{proof}
\begin{remark}\label{RM8}
We remark that, in general, the relation between $\mathit{P}_{\epsilon, t}^{(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast})}$ and $\mathit{P}_t^{(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast})}$ depends on the family of transformations $\Bigl\{\Phi_t^{(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast})} \Bigr\}_{t \ge 0}$ (with respect to the set of equilibrium feedback operators $(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast}) \in \mathscr{L}$) as well as on the measure space $L^1(X, \mathscr{A}, \mu)$ (see also \cite{BalYo93} and \cite{Kel82}).\end{remark}
We conclude this subsection with the following corollary, which is concerned with the resilient behavior of the set of equilibrium feedback operators, when there is a small random perturbation in the system. The proof follows similar arguments as in the proofs of Proposition~\ref{PR3} and Corollary~\ref{CR4}, and therefore will be omitted.
\begin{corollary}\label{CR5}
For $\epsilon > 0$ and $\operatorname{supp}\vartheta_{\ast}^{\epsilon}(x) \subset \operatorname{supp}\vartheta_{\ast}(x)$, if the relative entropy of the multi-channel system, with a random perturbation term, satisfies the following condition
\begin{align}
\mathcal{H}_{\rm r}\Bigr(\mathit{P}_{\epsilon, t}^{(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast})}\vartheta(x)\,\bigl\lvert\,\mathit{P}_t^{(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast})}\vartheta(x)\Bigl)\le \theta_{\epsilon}^{(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast})}, ~~ \forall t \ge 0, ~ \forall \vartheta(x) \in D(X, \mathscr{A}, \mu), \label{EQ46}
\end{align}
where $\theta_{\epsilon}^{(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast})}$ is a small positive number that depends on $\epsilon$ (and also tends to zero as $\epsilon \rightarrow 0$). Then, the set of equilibrium feedback operators $\bigl(\mathcal{L}_1^{\ast}, \mathcal{L}_2^{\ast}, \ldots, \mathcal{L}_N^{\ast} \bigr) \in \mathscr{L}$ exhibits a resilient behavior.
\end{corollary}
The above corollary states that the set of equilibrium feedback operators exhibits a resilient behavior, when the contribution of the perturbation term, to move away the system from the invariant measure $\mu_{\ast}$, is bounded from above for all $t \ge 0$.
We also note that the following holds true (see Equation~\eqref{EQ45})
\begin{align}
\lim_{t \rightarrow \infty} \, \Bigl \Vert \mathit{P}_{\epsilon, t}^{(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast})} \vartheta(x) - \vartheta_{\ast}(x) \Bigr \Vert_{L^1(X, \mathscr{A}, \mu)} \rightarrow 0 \quad \text{as} \quad \epsilon \rightarrow 0, \label{EQ47}
\end{align}
for any $\vartheta(x) \in D(X, \mathscr{A}, \mu)$. Therefore, such a bound in Equation~\eqref{EQ46} is an immediate consequence of this fact.\footnote{For small $\epsilon \ge 0$, notice that
\begin{align*}
\lim_{t \rightarrow \infty} \int_{X} \mathit{P}_{\epsilon, t}^{(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast})} \vartheta(x) \mu(dx) = \int_{X} \vartheta_{\ast}^{\epsilon}(x)\mu(dx), \quad \forall \vartheta(x) \in D(X, \mathscr{A}, \mu),
\end{align*}
when the stochastic semigroup $\mathit{P}_{\epsilon, t}^{(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast})}$ is asymptotically stable for each fixed $t \ge 0$ (cf. \cite[Sec.~11.9]{LasMac94}).}
\begin{remark}
Finally, we note that although we have not discussed the limiting behavior, as $\epsilon \rightarrow 0$, of the family of measures $\bigl\{\mu_{\ast}^{\epsilon}\bigr\}$ on the space $L^1(X, \mathscr{A}, \mu)$. It appears that the {\em theory of large deviations} can be used to estimate explicitly the rate at which this family of measures converges to the limit measure $\mu_{\ast}$, where the latter is invariant with respect to $\Phi_t^{(\mathcal{L}_j^{\ast}, \mathcal{L}_{\neg j}^{\ast})}$ for each fixed $t \ge 0$ (e.g., see \cite{Tou09}, \cite{Eli95} or \cite{DemZe98} for a detailed exposition of this theory).
\end{remark}
\end{document}
|
\begin{document}
\title{An Explicit Presentation of the Grothendieck Ring of Finitely
Generated $\mathbb{F}_{q}[SL_{2}(\mathbb{F}_{q})]$-Modules}
\author{Davide A. Reduzzi \\
University of California at Los Angeles\\
{\scriptsize [email protected]}}
\maketitle
\begin{abstract}
Let $p$ be a prime and $q=p^{g}$. We show that the Grothendieck ring of
finitely generated $\mathbb{F}_{q}[SL_{2}(\mathbb{F}_{q})]$-modules is
naturally isomorphic to the quotient of the polynomial algebra $
\mathbb{Z}
\lbrack x]$ by the ideal generated by $f^{[g]}(x)-x$, where $
f(x)=\tsum\nolimits_{j=0}^{\left\lfloor p/2\right\rfloor }\left( -1\right)
^{j}\tfrac{p}{p-j}\tbinom{p-j}{j}x^{p-2j}$, and the superscript $[g]$
denotes $g$-fold composition of polynomials. We conjecture that a similar
result holds for simply connected semisimple algebraic groups defined and
split over a finite field.
\end{abstract}
\section{Introduction}
In \cite{Se01}, J-P. Serre discovered a puzzling identity involving
characteristic $p$ symmetric powers representations of the group $GL_{2}(
\mathbb{F}_{q})$, viewed as elements of the Grothendieck ring $K_{0}\left(
GL_{2}(\mathbb{F}_{q})\right) $ of finitely generated $\mathbb{F}_{q}[GL_{2}(
\mathbb{F}_{q})]$-modules.
More precisely, fix a rational prime $p$, a positive integer $g$, and set $
q=p^{g}$. Denote by $\mathbb{F}_{q}$ a field with $q$ elements and by $
\mathfrak{G}$ the group $SL_{2}(\mathbb{F}_{q})\subset GL_{2}(\mathbb{F}
_{q}) $. For any non-negative integer $k$, denote by $\mathfrak{M}_{k}$ the $
(k+1)$-dimensional representation $\limfunc{Sym}\nolimits^{k}\mathbb{F}
_{q}^{2}$ of $\mathfrak{G}$. Motivated by the computation of the
Euler-Poincar\'{e} characteristic of the twisted sheaf $\mathcal{O}\left(
k\right) $ on $\mathbb{P}_{\mathbb{F}_{q}}^{1}$, in \cite{Se01} Serre
extended the definition of the modules $\mathfrak{M}_{k}$'s for negative
values of $k$, and showed that for any integer $k$ the following relation
holds in the ring $K_{0}\left( \mathfrak{G}\right) $:
\begin{equation}
\mathfrak{M}_{k}-\mathfrak{M}_{k-(q+1)}=\mathfrak{M}_{k-\left( q-1\right) }-
\mathfrak{M}_{k-2q}. \tag{$\Sigma $}
\end{equation}
The dimensional shiftings by $q+1$ and $q-1$ occurring in Serre's relation
can be obtained by applying opportune intertwining operators $\Theta _{q}$
and $D$ to the symmetric powers modules. This has been exploited in \cite{Re}
\ for the study of cohomological weight shiftings for elliptic modular forms
modulo $p$.
Motivated by generalizations of the above considerations to Hilbert modular
forms, families of generalized $\Theta _{q}$ and $D$ operators are defined
in \cite{Re2}, and the following identity in $K_{0}\left( \mathfrak{G}
\right) $ is proved for any integers $k,h$ and $i$:
\begin{equation}
\mathfrak{M}_{k}^{[i]}\mathfrak{M}_{h}^{[i+1]}-\mathfrak{M}_{k-p}^{[i]}
\mathfrak{M}_{h-1}^{[i+1]}=\mathfrak{M}_{k-p}^{[i]}\mathfrak{M}
_{h+1}^{[i+1]}-\mathfrak{M}_{k-2p}^{[i]}\mathfrak{M}_{h}^{[i+1]}.
\tag{$\Phi $}
\end{equation}
\noindent Here the superscript $[i]$ denote the $i$th Frobenius twisting on
the corresponding virtual representation.
Using Glover's product identity, one sees that $\left( \Phi \right) $ is
equivalent to $\left( \Sigma \right) $ in case $g=1$, but it is stronger for
$g>1.$
In this paper we apply formula $(\Phi )$ to determine an explicit
presentation of the Grothendieck ring $K_{0}\left( \mathfrak{G}\right) $. We
treat the case of $\mathfrak{G}=SL_{2}(\mathbb{F}_{q})$ instead of $GL_{2}(
\mathbb{F}_{q})$, so we will not need to consider determinant twists that
would make the set of relations more complicated; following the same methods
we describe below, one could easily work with $GL_{2}(\mathbb{F}_{q})$
instead.
Our main result is the following (cf. Theorem \ref{presentation}):
\begin{theorem}
Denote by $\mathfrak{X}$ the standard representation of $\mathfrak{G}$ on $
\mathbb{F}_{q}^{2}$ and let $\mathfrak{x}$ be an indeterminate over $
\mathbb{Z}
$. The assignment $\mathfrak{X\longmapsto x}$\ induces an isomorphism of
rings:
\begin{equation*}
K_{0}\left( \mathfrak{G}\right) \simeq \frac{
\mathbb{Z}
\lbrack \mathfrak{x}]}{\left( \mathfrak{f}^{[g]}(\mathfrak{x})-\mathfrak{x}
\right)
\mathbb{Z}
\lbrack \mathfrak{x}]},
\end{equation*}
\noindent where $\mathfrak{f}^{[g]}(\mathfrak{x})=\left( \mathfrak{f}\circ
...\circ \mathfrak{f}\right) \left( \mathfrak{x}\right) $ is the polynomial
of $
\mathbb{Z}
\lbrack \mathfrak{x}]$ having degree $p^{g}$ obtained by $g$-fold
composition of the monic degree $p$ polynomial:
\begin{equation*}
\mathfrak{f}(\mathfrak{x})=\dsum\nolimits_{j=0}^{\left\lfloor
p/2\right\rfloor }\left( -1\right) ^{j}\dfrac{p}{p-j}\dbinom{p-j}{j}
\mathfrak{x}^{p-2j}.
\end{equation*}
\end{theorem}
Proposition \ref{gth compo} gives an explicit closed formula for $\mathfrak{f
}^{[g]}(\mathfrak{x})$. Notice that, since $\mathfrak{f}(\mathfrak{x})\equiv
\mathfrak{x}^{p}(\func{mod}p
\mathbb{Z}
\lbrack \mathfrak{x}])$, the structure of the generic and special fibers of
the ring $K_{0}\left( \mathfrak{G}\right) \otimes _{
\mathbb{Z}
}
\mathbb{Z}
_{p}$ are easily determined (Corollary \ref{cor_presentation}). On the other
side, the arithmetic properties of the polynomial $\mathfrak{f}(\mathfrak{x}
) $ over $
\mathbb{Q}
$ seem to be more complicated.
In the last paragraph of the paper we prove the following fact (Proposition
\ref{tensorp}): assume $\mathbb{G}$ is a simply connected, semisimple
algebraic group defined and split over $\mathbb{F}_{q}$. If $\mathfrak{M}$
is an $\mathbb{F}_{q}[\mathbb{G]}$-rational module of finite $\mathbb{F}_{q}$
-dimension, then the multiplicity of an irreducible $\mathbb{F}_{q}[\mathbb{
G]}$-rational module $\mathfrak{V}$ as a Jordan-H\"{o}lder constituent of $
\mathfrak{M}^{[i]}$ is congruent modulo $p$ to the multiplicity of $
\mathfrak{V}$ as a Jordan-H\"{o}lder constituent of $\mathfrak{M}^{\otimes
p^{i}}$, for any positive integer $i$.
Motivated by this result, we are led to conjecture that the Grothendieck
ring of a Chevalley group arising from a rank $\ell $ algebraic group $
\mathbb{G}$ as above is isomorphic to the algebra
\begin{equation*}
\frac{
\mathbb{Z}
\lbrack \mathfrak{x}_{1},...,\mathfrak{x}_{\ell }]}{\left( \mathfrak{f}
_{1}^{[g]}(\mathfrak{x}_{1})-\mathfrak{x}_{1}\mathfrak{,...,f}_{\ell }^{[g]}(
\mathfrak{x}_{\ell })-\mathfrak{x}_{\ell }\right)
\mathbb{Z}
\lbrack \mathfrak{x}_{1},...,\mathfrak{x}_{\ell }]},
\end{equation*}
\noindent where for any $i$, $\mathfrak{f}_{i}^{[g]}(\mathfrak{x}_{i})$ is
the $g$-fold composition of the degree $p$ monic polynomial $\mathfrak{f}
_{i}(\mathfrak{x}_{i})\in
\mathbb{Z}
\lbrack \mathfrak{x}_{i}]$. We conjecture that $\mathfrak{f}_{i}(\mathfrak{x}
_{i})\equiv \mathfrak{x}_{i}^{p}(\func{mod}p
\mathbb{Z}
\lbrack \mathfrak{x}_{i}])$ for any value of $i$. Some of the evidence for
this conjecture is presented at the end of paragraph 4.
\textbf{Conventions} All the group representations in this paper are left
representations on a module of finite length over a fixed ring. If $R$ is an
algebra over a ring $A$, and $\mathcal{S}$ is a subset of $R$, the symbol $A[
\mathcal{S}]$ denotes the $A$-subalgebra of $R$ generated by $\mathcal{S}$.
\section{Weight shiftings identities in $K_{0}\left( \mathfrak{G}\right) $}
Fix a rational prime $p$, a positive integer $g$, and set $q=p^{g}$. Denote
by $\mathbb{F}_{q}$ a finite field with $q$ elements and fix an algebraic
closure $\overline{\mathbb{F}}_{q}$ of $\mathbb{F}_{q}$; let $\sigma \in
\limfunc{Gal}\left( \mathbb{F}_{q}/\mathbb{F}_{p}\right) $ be the arithmetic
absolute Frobenius element. Denote by $\mathfrak{G}$ the group $SL_{2}(
\mathbb{F}_{q})$.
For any $i\in
\mathbb{Z}
$, the Frobenius power $\sigma ^{i}$ induces a function $\mathfrak{G}
\longrightarrow \mathfrak{G}$ obtained by applying $\sigma ^{i}$ to each
entry of the matrices in $\mathfrak{G}$: composing this map with the action
of $\mathfrak{G}$ on a given $\mathbb{F}_{q}[\mathfrak{G}]$-module $
\mathfrak{M}$ gives to the latter a new structure of $\mathfrak{G}$-module,
that is denoted $\mathfrak{M}^{[i]}$ and called the $i$th Frobenius twist of
$\mathfrak{M}.$
If $f:\mathfrak{M}\longrightarrow \mathfrak{N}$ is a homomorphism of $
\mathbb{F}_{q}[\mathfrak{G}]$-modules and $i\in
\mathbb{Z}
$, we denote by $f^{[i]}:\mathfrak{M}^{[i]}\longrightarrow \mathfrak{N}
^{[i]} $ the $\mathfrak{G}$-homomorphism defined by $f^{[i]}(x)=f(x)$ for
all $x\in \mathfrak{M}^{[i]}$.
Let $\mathfrak{X}$ denote the standard representation of $\mathfrak{G}$ on $
\mathbb{F}_{q}^{2}$ and, for any positive integer $k$, define
\begin{equation*}
\mathfrak{M}_{k}=\limfunc{Sym}\nolimits^{k}\mathfrak{X}
\end{equation*}
to be the $k$th symmetric power of $\mathfrak{X}$, so that in particular $
\mathfrak{X=M}_{1}$. Let $\mathfrak{M}_{0}$ be the trivial representation of
$\mathfrak{G}$ on $\mathbb{F}_{q}$.
Observe that we can identify $\mathfrak{M}_{k}$ with the $\mathbb{F}_{q}$
-vector space of homogeneous degree $k$ polynomials over $\mathbb{F}_{q}$ in
two variables $X$ and $Y$, endowed with the action of $\mathfrak{G}$ induced
by:
\begin{center}
\begin{equation*}
\left(
\begin{array}{cc}
a & b \\
c & d
\end{array}
\right) \cdot X=aX+cY,\ \ \ \left(
\begin{array}{cc}
a & b \\
c & d
\end{array}
\right) \cdot Y=bX+dY.
\end{equation*}
\end{center}
As a consequence of Steinberg's restriction and tensor product theorems (
\cite{St}) we have:
\begin{proposition}
\label{stei}All and only the irreducible representations of $\mathfrak{G}$
over $\mathbb{F}_{q}$ are of the form:
\begin{equation*}
\dbigotimes\nolimits_{i=0}^{g-1}\mathfrak{M}_{k_{i}}^{[i]},
\end{equation*}
where $k_{0},...,k_{g-1}$ are integers such that $0\leq k_{i}\leq p-1$ for $
i=0,...,g-1$, and the tensor products are over $\mathbb{F}_{q}$.
Furthermore, the above representations are pairwise non-isomorphic.
\end{proposition}
Denote by $K_{0}(\mathfrak{G})$ the Grothendieck group of finitely generated
$\mathbb{F}_{q}[\mathfrak{G}]$-modules. $K_{0}(\mathfrak{G})$ is isomorphic
to the free abelian group generated by the isomorphism classes of
irreducible representations of $\mathfrak{G}$ over $\mathbb{F}_{q}$, so that
it has rank $q$ over $
\mathbb{Z}
$. If $\mathfrak{M}$ is an $\mathbb{F}_{q}[\mathfrak{G}]$-module, we will
denote by the same symbol $\mathfrak{M}$ its class in $K_{0}(\mathfrak{G})$,
if no confusion arises.
Tensor product over $\mathbb{F}_{q}$ induces on $K_{0}(\mathfrak{G})$ a
structure of commutative ring with identity; we denote the product in $K_{0}(
\mathfrak{G})$ by $\cdot $ or by simple juxtaposition. All the tensor
products we will consider in the sequel are over $\mathbb{F}_{q}$, unless
otherwise specified.
We can extend the definition of the virtual representation $\mathfrak{M}_{k}$
for $k<0$ in a way that is coherent with Brauer character computations. In
\cite{Se01}, the determination the Euler-Poincar\'{e} characteristic of the
twisted sheaf $\mathcal{O}\left( k\right) $ on $\mathbb{P}_{\mathbb{F}
_{q}}^{1}$ suggests the following:
\begin{definition}
Let $k$ be a negative integer. Define the element $\mathfrak{M}_{k}$ of the
Grothendieck group $K_{0}\left( \mathfrak{G}\right) $ of $\mathfrak{G}$ over
$\mathbb{F}_{q}$ by:
\begin{equation*}
\mathfrak{M}_{k}=\left\{
\begin{array}{cc}
0 & \text{if }k=-1 \\
\mathfrak{M}_{-k-2} & \text{if }k\leq -2.
\end{array}
\right.
\end{equation*}
\end{definition}
The following result summarizes some non-trivial identities that hold in the
ring $K_{0}\left( \mathfrak{G}\right) $:
\begin{theorem}
Let $k$ and $h$ be any integers. The following identities hold in $
K_{0}\left( \mathfrak{G}\right) $:
\begin{equation}
\mathfrak{M}_{k}=-\mathfrak{M}_{-k-2} \tag{$\Delta _{g}$}
\end{equation}
\begin{equation}
\mathfrak{M}_{k}-\mathfrak{M}_{k-(q+1)}=\mathfrak{M}_{k-\left( q-1\right) }-
\mathfrak{M}_{k-2q} \tag{$\Sigma _{g}$}
\end{equation}
\begin{equation}
\mathfrak{M}_{k}\mathfrak{M}_{h}=\mathfrak{M}_{k+h}+\mathfrak{M}_{k-1}
\mathfrak{M}_{h-1} \tag{$\Pi _{g}$}
\end{equation}
\begin{equation}
\mathfrak{M}_{k}=\mathfrak{M}_{k-p}\mathfrak{M}_{1}^{[1]}-\mathfrak{M}
_{k-2p}. \tag{$\Phi _{g}$}
\end{equation}
\end{theorem}
\textbf{Proof }Formulae $(\Delta _{g})$ and $(\Sigma _{g})$ are proved in
\cite{Se01} via a Brauer characters computation. Formula $(\Pi _{g})$ comes
from an exact sequence of $\mathfrak{G}$-modules constructed in \cite{Glo}.
Formula $(\Phi _{g})$ is proved in section 3 of \cite{Re2}. $\blacksquare $
We remark that formula $(\Phi _{g})$ appeared in \cite{Re2} also in the form:
\begin{equation*}
\mathfrak{M}_{k}^{[i]}\mathfrak{M}_{h}^{[i+1]}-\mathfrak{M}_{k-p}^{[i]}
\mathfrak{M}_{h-1}^{[i+1]}=\mathfrak{M}_{k-p}^{[i]}\mathfrak{M}
_{h+1}^{[i+1]}-\mathfrak{M}_{k-2p}^{[i]}\mathfrak{M}_{h}^{[i+1]},
\end{equation*}
\noindent where $k,h$ and $i$ are any integers.
The product formula $(\Pi _{1})$ implies that $(\Phi _{1})$ and $\left(
\Sigma _{1}\right) $ are equivalent. If $g>1$, $(\Phi _{g})$ cannot be
deduced from $(\Sigma _{g})$ and $(\Pi _{g})$: the proof of this fact,
contained in \cite{Re2}, is indirect and throughout the paper the knowledge
of Serre's relation $(\Sigma _{g})$ will allow sometimes to bypass long
computations involving Frobenius twists, when $g>1$.
In \cite{Re2} it is also proved that for $g\geq 1$, we can use the relations
$(\Delta _{g}),(\Phi _{g}),(\Pi _{g})$ to explicitly compute the Jordan-H
\"{o}lder factors of any virtual representations of the form $
\tprod\nolimits_{i=0}^{g-1}\mathfrak{M}_{k_{i}}^{[i]}$, where $
k_{0},...,k_{g-1}\in
\mathbb{Z}
$.
\section{Presentation of $K_{0}\left( \mathfrak{G}\right) $}
We keep the notation introduced in the previous paragraph.
\begin{lemma}
\label{1generator}The ring $K_{0}\left( \mathfrak{G}\right) $ is generated
by $\mathfrak{X}$ as a $
\mathbb{Z}
$-algebra.
\end{lemma}
\textbf{Proof }By Proposition \ref{stei}, $K_{0}\left( \mathfrak{G}\right) $
is freely\ generated as a $
\mathbb{Z}
$-module by the $q$ elements $\tprod\nolimits_{i=0}^{g-1}\mathfrak{M}
_{k_{i}}^{[i]}$, where $0\leq k_{i}\leq p-1$ for any $i$. It is therefore
enough to show that for all integers $i,k$ such that $0\leq i\leq g-1$ and $
0\leq k\leq p-1$ we have $\mathfrak{M}_{k}^{[i]}\in
\mathbb{Z}
\lbrack \mathfrak{X}]$.
Applying $(\Pi _{g})$ we obtain the recursive relations:
\begin{equation}
\mathfrak{M}_{2}=\mathfrak{X}^{2}-1,\text{ }\mathfrak{M}_{n}=\mathfrak{
X\cdot M}_{n-1}-\mathfrak{M}_{n-2}\text{ \ (}n>2\text{),} \label{m}
\end{equation}
so that $\mathfrak{M}_{k}\in
\mathbb{Z}
\lbrack \mathfrak{X}]$ for all $k\geq 0$. Twisting (\ref{m})\ by powers of
Frobenius, we obtain:
\begin{equation*}
\mathfrak{M}_{2}^{[i]}=\left( \mathfrak{X}^{[i]}\right) ^{2}-1,\text{ }
\mathfrak{M}_{n}^{[i]}=\mathfrak{X}^{[i]}\cdot \mathfrak{M}_{n-1}^{[i]}-
\mathfrak{M}_{n-2}^{[i]}\text{ \ (}n>2\text{),}
\end{equation*}
\noindent for all $0\leq i\leq g-1$, so that $\mathfrak{M}_{k}^{[i]}\in
\mathbb{Z}
\lbrack \mathfrak{X,X}^{[1]},...,\mathfrak{X}^{[g-1]}]$ for all $k\geq 0$
and:
\begin{equation*}
K_{0}\left( \mathfrak{G}\right) =
\mathbb{Z}
\lbrack \mathfrak{X,X}^{[1]},...,\mathfrak{X}^{[g-1]}].
\end{equation*}
By $(\Phi _{g})$, we have $\mathfrak{M}_{p}=\mathfrak{M}_{1}^{[1]}-\mathfrak{
M}_{-p}$, and applying $(\Delta _{g})$ we obtain $\mathfrak{X}^{[1]}=
\mathfrak{M}_{p}-\mathfrak{M}_{p-2}$, so that $\mathfrak{X}^{[1]}\in
\mathbb{Z}
\lbrack \mathfrak{X}]$, as $\mathfrak{M}_{k}\in
\mathbb{Z}
\lbrack \mathfrak{X}]$ for all $k\geq 0$. We also obtain that, for any $
0\leq i\leq g-1$, we have:
\begin{equation}
\mathfrak{X}^{[i+1]}=\mathfrak{M}_{p}^{[i]}-\mathfrak{M}_{p-2}^{[i]},
\label{ff}
\end{equation}
and we conclude $\mathfrak{X}^{[1]},...,\mathfrak{X}^{[g-1]}\in
\mathbb{Z}
\lbrack \mathfrak{X}]$, implying $K_{0}\left( \mathfrak{G}\right) =
\mathbb{Z}
\lbrack \mathfrak{X}]$. $\blacksquare $
Let $\mathfrak{x}$ be an indeterminate over $
\mathbb{Z}
$ and define the following two families of polynomials of $
\mathbb{Z}
\lbrack \mathfrak{x}]$:
\begin{eqnarray*}
&&\left\{
\begin{array}{l}
\mathfrak{m}_{0}(\mathfrak{x})=1 \\
\mathfrak{m}_{1}(\mathfrak{x})=\mathfrak{x} \\
\mathfrak{m}_{2}(\mathfrak{x})=\mathfrak{x}^{2}-1 \\
\mathfrak{m}_{n}(\mathfrak{x})=\mathfrak{x\cdot m}_{n-1}(\mathfrak{x})-
\mathfrak{m}_{n-2}(\mathfrak{x})\ \text{\ \ }(n>2);
\end{array}
\right. \\
&& \\
&&\left\{
\begin{array}{l}
\mathfrak{f}^{[0]}(\mathfrak{x})=\mathfrak{x} \\
\mathfrak{f}^{[1]}(\mathfrak{x})=\mathfrak{m}_{p}(\mathfrak{x})-\mathfrak{m}
_{p-2}(\mathfrak{x}) \\
\mathfrak{f}^{[i]}(\mathfrak{x})=\text{ }\left( \mathfrak{f}^{[1]}\circ
\mathfrak{f}^{[i-1]}\right) (\mathfrak{x})\ =\mathfrak{m}_{p}(\mathfrak{f}
^{[i-1]}(\mathfrak{x}))-\mathfrak{m}_{p-2}(\mathfrak{f}^{[i-1]}(\mathfrak{x}
))\text{\ \ \ }(i>1).
\end{array}
\right.
\end{eqnarray*}
Observe that for any non-negative integer $n$, $\mathfrak{m}_{n}(\mathfrak{x}
)$ is a monic polynomial of degree $n$, so that for any non-negative integer
$i$, $\mathfrak{f}^{[i]}(\mathfrak{x})$ is a monic polynomial of degree $
p^{i}$.
\begin{lemma}
For any non-negative integer $i$, we have $\mathfrak{f}^{[i]}(\mathfrak{X})=
\mathfrak{X}^{[i]}$ in $K_{0}\left( \mathfrak{G}\right) $.
\end{lemma}
\textbf{Proof }Notice first that, by definition of $\mathfrak{m}_{n}(
\mathfrak{x})$ and by formula (\ref{m}), one has:
\begin{equation}
\mathfrak{m}_{n}(\mathfrak{X})=\mathfrak{M}_{n} \label{meaningful1}
\end{equation}
\noindent in $K_{0}\left( \mathfrak{G}\right) $ ($n\geq 0$). To prove the
lemma, we use induction on $i$. If $i=0$, the statement is clear; if $i=1$
it follows from formulae (\ref{meaningful1}) and (\ref{ff}). Assume $i\geq 1$
fixed and suppose $\mathfrak{f}^{[i]}(\mathfrak{X})=\mathfrak{X}^{[i]}$. We
have:
\begin{eqnarray*}
\mathfrak{f}^{[i+1]}(\mathfrak{X}) &=&\mathfrak{m}_{p}(\mathfrak{f}^{[i]}(
\mathfrak{X}))-\mathfrak{m}_{p-2}(\mathfrak{f}^{[i]}(\mathfrak{X})) \\
&=&\mathfrak{m}_{p}(\mathfrak{X}^{[i]})-\mathfrak{m}_{p-2}(\mathfrak{X}
^{[i]}).
\end{eqnarray*}
\noindent Observe that Frobenius twists do not act on the coefficients of
virtual representations in $K_{0}\left( \mathfrak{G}\right) $, so that the
last term above is equal to $\mathfrak{m}_{p}(\mathfrak{X})^{[i]}-\mathfrak{m
}_{p-2}(\mathfrak{X})^{[i]}$. By formula (\ref{meaningful1}), the latter is $
\mathfrak{M}_{p}^{[i]}-\mathfrak{M}_{p-2}^{[i]}$. By formula (\ref{ff}),
this is $\mathfrak{M}_{p}^{[i]}-\mathfrak{M}_{p-2}^{[i]}=\mathfrak{X}
^{[i+1]} $. $\blacksquare $
\begin{proposition}
There is an isomorphism of rings:
\begin{equation*}
\frac{
\mathbb{Z}
\lbrack \mathfrak{x}]}{\left( \mathfrak{f}^{[g]}(\mathfrak{x})-\mathfrak{x}
\right)
\mathbb{Z}
\lbrack \mathfrak{x}]}\simeq K_{0}\left( \mathfrak{G}\right) ,
\end{equation*}
\noindent induced by mapping the indeterminate $\mathfrak{x}$ of the
polynomial ring $
\mathbb{Z}
\lbrack \mathfrak{x}]$ into the class of the representation $\mathfrak{X}$
of $\mathfrak{G}$.
\end{proposition}
\textbf{Proof }By Proposition \ref{1generator}, the ring homomorphism $
\mathbb{Z}
\lbrack \mathfrak{x}]\longrightarrow K_{0}\left( \mathfrak{G}\right) $
induced by $\mathfrak{x}\mapsto \mathfrak{X}$ is surjective. Since $
\mathfrak{X}^{[g]}=\mathfrak{X}$ in $K_{0}\left( \mathfrak{G}\right) $, and
since by the above lemma we have $\mathfrak{f}^{[g]}(\mathfrak{X})=\mathfrak{
X}^{[g]}$, the above assignment induces an epimorphism
\begin{equation*}
\pi :\frac{
\mathbb{Z}
\lbrack \mathfrak{x}]}{\left( \mathfrak{f}^{[g]}(\mathfrak{x})-\mathfrak{x}
\right)
\mathbb{Z}
\lbrack \mathfrak{x}]}\longrightarrow K_{0}\left( \mathfrak{G}\right) .
\end{equation*}
Since $\mathfrak{f}^{[g]}(\mathfrak{x})-\mathfrak{x}$ is a polynomial of
degree $p^{g}$ and since $K_{0}\left( \mathfrak{G}\right) $ is $
\mathbb{Z}
$-free of rank $p^{g}$, after tensoring with $
\mathbb{Q}
$ the map $\pi $ defines an isomorphism of $
\mathbb{Q}
$-vector spaces. This implies that $\ker \pi $ is a finitely generated
torsion $
\mathbb{Z}
$-submodule of $\frac{
\mathbb{Z}
\lbrack \mathfrak{x}]}{\left( \mathfrak{f}^{[g]}(\mathfrak{x})-\mathfrak{x}
\right)
\mathbb{Z}
\lbrack \mathfrak{x}]}$, and hence it is trivial since $\mathfrak{f}^{[g]}(
\mathfrak{x})-\mathfrak{x}$ is monic. We conclude that $\pi $ is an
isomorphism of rings. $\blacksquare $
We are now left with determining an explicit formula for the polynomial $
\mathfrak{f}^{[g]}(\mathfrak{x})\in
\mathbb{Z}
\lbrack \mathfrak{x}]$.
\begin{lemma}
For any non-negative integer $n$ we have:
\begin{equation*}
\mathfrak{m}_{n}(\mathfrak{x})=\dsum\nolimits_{j=0}^{\left\lfloor
n/2\right\rfloor }\left( -1\right) ^{j}\dbinom{n-j}{j}\mathfrak{x}^{n-2j}.
\end{equation*}
\noindent \noindent (Where, for any integer $h$, $\left\lfloor
h\right\rfloor $ denotes the largest integer not greater than $h$).
\end{lemma}
\textbf{Proof }We use induction on $n\geq 0$; denote by $\mathfrak{m}
_{n}^{\prime }(\mathfrak{x})$ the right hand side of the above formula. We
have $\mathfrak{m}_{0}^{\prime }(\mathfrak{x})=1=\mathfrak{m}_{0}(\mathfrak{x
})$, $\mathfrak{m}_{1}^{\prime }(\mathfrak{x})=\mathfrak{x}=\mathfrak{m}_{1}(
\mathfrak{x})$ and $\mathfrak{m}_{2}^{\prime }(\mathfrak{x})=\mathfrak{x}
^{2}-1=\mathfrak{m}_{2}(\mathfrak{x})$. If $n>2$ we have by induction:
\begin{eqnarray*}
\mathfrak{m}_{n}(\mathfrak{x}) &=&\mathfrak{xm}_{n-1}(\mathfrak{x})-
\mathfrak{m}_{n-2}(\mathfrak{x})=\mathfrak{xm}_{n-1}^{\prime }(\mathfrak{x})-
\mathfrak{m}_{n-2}^{\prime }(\mathfrak{x}) \\
&=&\tsum\nolimits_{j=0}^{\left\lfloor (n-1)/2\right\rfloor }\left( -1\right)
^{j}\tbinom{n-1-j}{j}\mathfrak{x}^{n-2j}-\tsum\nolimits_{j=0}^{\left\lfloor
(n-2)/2\right\rfloor }\left( -1\right) ^{j}\tbinom{n-2-j}{j}\mathfrak{x}
^{n-2(j+1)}.
\end{eqnarray*}
\noindent If $n>2$ is even, $\left\lfloor (n-1)/2\right\rfloor =\left\lfloor
(n-2)/2\right\rfloor =(n/2)-1$ and:
\begin{eqnarray*}
\mathfrak{m}_{n}(\mathfrak{x}) &=&\tsum\nolimits_{j=0}^{(n/2)-1}\left(
-1\right) ^{j}\tbinom{n-1-j}{j}\mathfrak{x}^{n-2j}+\tsum
\nolimits_{j=1}^{n/2}\left( -1\right) ^{j}\tbinom{n-1-j}{j-1}\mathfrak{x}
^{n-2j} \\
&=&\mathfrak{x}^{n}+\left( \tsum\nolimits_{j=1}^{(n/2)-1}\left( -1\right)
^{j}\left[ \tbinom{n-1-j}{j}+\tbinom{n-1-j}{j-1}\right] \mathfrak{x}
^{n-2j}\right) +(-1)^{n/2} \\
&=&\mathfrak{x}^{n}+\left( \tsum\nolimits_{j=1}^{(n/2)-1}\left( -1\right)
^{j}\tbinom{n-j}{j}\mathfrak{x}^{n-2j}\right) +(-1)^{n/2} \\
&=&\tsum\nolimits_{j=0}^{\left\lfloor n/2\right\rfloor }\left( -1\right) ^{j}
\tbinom{n-j}{j}\mathfrak{x}^{n-2j} \\
&=&\mathfrak{m}_{n}^{\prime }(\mathfrak{x}).
\end{eqnarray*}
\noindent If $n>2$ is odd, $\left\lfloor (n-1)/2\right\rfloor =(n-1)/2$, $
\left\lfloor (n-2)/2\right\rfloor =(n-3)/2$ and:
\begin{eqnarray*}
\mathfrak{m}_{n}(\mathfrak{x}) &=&\tsum\nolimits_{j=0}^{(n-1)/2}\left(
-1\right) ^{j}\tbinom{n-1-j}{j}\mathfrak{x}^{n-2j}+\tsum
\nolimits_{j=1}^{(n-1)/2}\left( -1\right) ^{j}\tbinom{n-1-j}{j-1}\mathfrak{x}
^{n-2j} \\
&=&\mathfrak{x}^{n}+\left( \tsum\nolimits_{j=1}^{(n-1)/2}\left( -1\right)
^{j}\left[ \tbinom{n-1-j}{j}+\tbinom{n-1-j}{j-1}\right] \mathfrak{x}
^{n-2j}\right) \\
&=&\mathfrak{x}^{n}+\left( \tsum\nolimits_{j=1}^{(n-1)/2}\left( -1\right)
^{j}\tbinom{n-j}{j}\mathfrak{x}^{n-2j}\right) \\
&=&\tsum\nolimits_{j=0}^{\left\lfloor n/2\right\rfloor }\left( -1\right) ^{j}
\tbinom{n-j}{j}\mathfrak{x}^{n-2j} \\
&=&\mathfrak{m}_{n}^{\prime }(\mathfrak{x}).\text{ \ }\blacksquare
\end{eqnarray*}
\begin{corollary}
\label{n-n2}Let $n\geq 2$ be an integer. Then:
\begin{equation*}
\mathfrak{m}_{n}(\mathfrak{x})-\mathfrak{m}_{n-2}(\mathfrak{x}
)=\dsum\nolimits_{j=0}^{\left\lfloor n/2\right\rfloor }\left( -1\right) ^{j}
\dfrac{n}{n-j}\dbinom{n-j}{j}\mathfrak{x}^{n-2j}.
\end{equation*}
\end{corollary}
\textbf{Proof }This is a computation using the previous lemma. We
distinguish two cases: if $n\geq 2$ is even we have:
\begin{eqnarray*}
\mathfrak{m}_{n}(\mathfrak{x})-\mathfrak{m}_{n-2}(\mathfrak{x})
&=&\tsum\nolimits_{j=0}^{n/2}\left( -1\right) ^{j}\tbinom{n-j}{j}\mathfrak{x}
^{n-2j}-\tsum\nolimits_{j=0}^{(n/2)-1}\left( -1\right) ^{j}\tbinom{n-2-j}{j}
\mathfrak{x}^{n-2(j+1)} \\
&=&\tsum\nolimits_{j=0}^{n/2}\left( -1\right) ^{j}\tbinom{n-j}{j}\mathfrak{x}
^{n-2j}+\tsum\nolimits_{j=1}^{n/2}\left( -1\right) ^{j}\tbinom{n-1-j}{j-1}
\mathfrak{x}^{n-2j} \\
&=&\mathfrak{x}^{n}+\tsum\nolimits_{j=1}^{n/2}\left( -1\right) ^{j}\left[
\tbinom{n-j}{j}+\tbinom{n-1-j}{j-1}\right] \mathfrak{x}^{n-2j} \\
&=&\mathfrak{x}^{n}+\tsum\nolimits_{j=1}^{n/2}\left( -1\right) ^{j}\tfrac{n}{
n-j}\tbinom{n-j}{j}\mathfrak{x}^{n-2j}.
\end{eqnarray*}
\noindent If $n\geq 3$ is odd we have:
\begin{eqnarray*}
\mathfrak{m}_{n}(\mathfrak{x})-\mathfrak{m}_{n-2}(\mathfrak{x})
&=&\tsum\nolimits_{j=0}^{(n-1)/2}\left( -1\right) ^{j}\tbinom{n-j}{j}
\mathfrak{x}^{n-2j}-\tsum\nolimits_{j=0}^{(n-3)/2}\left( -1\right) ^{j}
\tbinom{n-2-j}{j}\mathfrak{x}^{n-2(j+1)} \\
&=&\tsum\nolimits_{j=0}^{(n-1)/2}\left( -1\right) ^{j}\tbinom{n-j}{j}
\mathfrak{x}^{n-2j}+\tsum\nolimits_{j=1}^{(n-1)/2}\left( -1\right) ^{j}
\tbinom{n-1-j}{j-1}\mathfrak{x}^{n-2j} \\
&=&\mathfrak{x}^{n}+\tsum\nolimits_{j=1}^{(n-1)/2}\left( -1\right) ^{j}\left[
\tbinom{n-j}{j}+\tbinom{n-1-j}{j-1}\right] \mathfrak{x}^{n-2j} \\
&=&\mathfrak{x}^{n}+\tsum\nolimits_{j=1}^{(n-1)/2}\left( -1\right) ^{j}
\tfrac{n}{n-j}\tbinom{n-j}{j}\mathfrak{x}^{n-2j}.\text{ \ }\blacksquare
\end{eqnarray*}
We have proved:
\begin{theorem}
\label{presentation}Let $g$ be a positive integer, $p$ a prime, $q=p^{g}$
and set $\mathfrak{G}=SL_{2}(\mathbb{F}_{q})$. Denote by $\mathfrak{X}$ the
standard representation of $\mathfrak{G}$ on $\mathbb{F}_{q}^{2}$ and let $
\mathfrak{x}$ be an indeterminate over $
\mathbb{Z}
$. The assignment $\mathfrak{X\longmapsto x}$\ induces an isomorphism of
rings:
\begin{equation*}
K_{0}\left( \mathfrak{G}\right) \simeq \frac{
\mathbb{Z}
\lbrack \mathfrak{x}]}{\left( \mathfrak{f}^{[g]}(\mathfrak{x})-\mathfrak{x}
\right)
\mathbb{Z}
\lbrack \mathfrak{x}]},
\end{equation*}
\noindent where $\mathfrak{f}^{[g]}(\mathfrak{x})=\left( \mathfrak{f}\circ
\mathfrak{f}\circ ...\circ \mathfrak{f}\right) \left( \mathfrak{x}\right) $
is the monic polynomial of $
\mathbb{Z}
\lbrack \mathfrak{x}]$ having degree $p^{g}$ that is obtained by composing $
g $-times wit itself the monic degree $p$ polynomial:
\begin{equation*}
\mathfrak{f}(\mathfrak{x}):=\dsum\nolimits_{j=0}^{\left\lfloor
p/2\right\rfloor }\left( -1\right) ^{j}\dfrac{p}{p-j}\dbinom{p-j}{j}
\mathfrak{x}^{p-2j}.
\end{equation*}
\end{theorem}
At the time of writing of this paper, we do not know much about the
properties of the polynomial $\mathfrak{f}^{[g]}(\mathfrak{x})-\mathfrak{x}$
when viewed over $
\mathbb{Z}
$. Notice that if $p>2$, $\mathfrak{f}^{[1]}(\mathfrak{x})-\mathfrak{x}$ is
an odd polynomial; using the easy to check facts that for any integer $n\geq
0$ we have $\mathfrak{m}_{n}\left( 2\right) =n+1,$ and that:
\begin{equation*}
\mathfrak{m}_{n}\left( 1\right) =\left\{
\begin{array}{c}
-1\text{, if }n\equiv 3,4(\func{mod}6) \\
0\text{, \ if }n\equiv 2,5(\func{mod}6) \\
1\text{, if }n\equiv 0,1(\func{mod}6),
\end{array}
\right.
\end{equation*}
\noindent one deduce that $\mathfrak{f}^{[1]}(\mathfrak{x})-\mathfrak{x}$
always admits $0,\pm 1,\pm 2$ as roots, as long as $p>3$ (roots $0$ and $\pm
2$ also occur for $p=3$). Furthermore, from computer elaborations, $
\mathfrak{f}^{[1]}(\mathfrak{x})-\mathfrak{x}$ seems to have only real roots.
In general, it is natural to ask what we can say about the irreducible
factors over $
\mathbb{Q}
$ of $\mathfrak{f}^{[g]}(\mathfrak{x})-\mathfrak{x}$. We do not have an
answer for this. Nevertheless, after tensoring $K_{0}\left( \mathfrak{G}
\right) $ with $
\mathbb{Z}
_{p}$, we can prove:
\begin{corollary}
\noindent \label{cor_presentation}Let $K_{0}\left( \mathfrak{G}\right)
_{p}=K_{0}\left( \mathfrak{G}\right) \otimes _{
\mathbb{Z}
}
\mathbb{Z}
_{p}$. For any positive divisor $d$ of $g$, let $\psi (d)$ be the number of
monic irreducible polynomials of degree $d$ in $\mathbb{F}_{p}[\mathfrak{x}]$
. Then:
\begin{description}
\item[(a)] The special fiber of $K_{0}\left( \mathfrak{G}\right) _{p}$ is a
split$\mathbb{\ F}_{p}$-algebra isomorphic to $\dprod\nolimits_{d|g}\mathbb{F
}_{p^{d}}^{\psi (d)};$
\item[(b)] The generic fiber of $K_{0}\left( \mathfrak{G}\right) _{p}$ is a
split $
\mathbb{Q}
_{p}$-algebra isomorphic to $\dprod\nolimits_{d|g}\mathbb{
\mathbb{Q}
}_{p^{d}}^{\psi (d)}.$
\end{description}
\noindent (Here we denoted by $\mathbb{
\mathbb{Q}
}_{p^{d}}$ the degree $d$ unramified extension of $\mathbb{
\mathbb{Q}
}_{p}$ contained in a fixed algebraic closure of $\mathbb{
\mathbb{Q}
}_{p}$).
\end{corollary}
\textbf{Proof }By the explicit formula given above for $\mathfrak{f}(
\mathfrak{x})$, we see that $\mathfrak{f}(\mathfrak{x})\equiv \mathfrak{x}
^{p}(\func{mod}p
\mathbb{Z}
\lbrack \mathfrak{x}])$: this is clear if $p=2$, otherwise notice that $
\tfrac{p}{p-j}\tbinom{p-j}{j}=p\cdot \tfrac{(p-j-1)!}{j!\cdot (p-2j)!}$ and
the last denominator is prime to $p$ if $1\leq j\leq \tfrac{p-1}{2}$,
implying that $\tfrac{(p-j-1)!}{j!\cdot (p-2j)!}\in
\mathbb{Z}
$. We conclude that
\begin{equation*}
\mathfrak{f}^{[g]}(\mathfrak{x})-\mathfrak{x\equiv x}^{q}-\mathfrak{x}(\func{
mod}p
\mathbb{Z}
\lbrack \mathfrak{x}])
\end{equation*}
and statement (a) follows. Part (b)\ follows from (a) and Hensel's lemma. $
\blacksquare $
\begin{remark}
We also have isomorphisms of algebras: $K_{0}\left( \mathfrak{G}\right)
_{p}\otimes _{
\mathbb{Z}
_{p}}\mathbb{F}_{q}\simeq \left( \mathbb{F}_{q}\right) ^{q}$ and $
K_{0}\left( \mathfrak{G}\right) _{p}\otimes _{
\mathbb{Z}
_{p}}\mathbb{
\mathbb{Q}
}_{q}\simeq \left( \mathbb{
\mathbb{Q}
}_{q}\right) ^{q}$.
\end{remark}
It is interesting to notice that we can give an explicit formula also for $
\mathfrak{f}^{[g]}(\mathfrak{x})$. As the following proposition uses Serre's
relation $(\Sigma _{g})$, it seems that an explicit formula for $\mathfrak{f}
^{[i]}(\mathfrak{x})$ when $i\neq 1,g$ would probably require more work.
\begin{proposition}
\label{gth compo}We have $\mathfrak{f}^{[g]}(\mathfrak{x})=\mathfrak{m}_{q}(
\mathfrak{x})-\mathfrak{m}_{q-2}(\mathfrak{x})$, so that:
\begin{equation*}
\mathfrak{f}^{[g]}(\mathfrak{x})=\dsum\nolimits_{j=0}^{\left\lfloor
q/2\right\rfloor }\left( -1\right) ^{j}\dfrac{q}{q-j}\dbinom{q-j}{j}
\mathfrak{x}^{q-2j}.
\end{equation*}
\end{proposition}
\textbf{Proof }Let $\tilde{\pi}:
\mathbb{Z}
\lbrack \mathfrak{x}]\rightarrow K_{0}\left( \mathfrak{G}\right) $ be the
epimorphism of rings obtained by sending $\mathfrak{x}$ to $\mathfrak{X}$.
Relation $(\Sigma _{g})$ implies that $\mathfrak{M}_{1}=\mathfrak{M}_{q}-
\mathfrak{M}_{q-2}$ in $K_{0}(\mathfrak{G})$, that is $\mathfrak{M}_{q}-
\mathfrak{M}_{q-2}-\mathfrak{X}=0$. This means, by formula (\ref{meaningful1}
), that $\mathfrak{X}$ satisfies the polynomial $\mathfrak{m}_{q}(\mathfrak{x
})-\mathfrak{m}_{q-2}(\mathfrak{x})-\mathfrak{x\in
\mathbb{Z}
}[\mathfrak{x}]$, so that $\mathfrak{m}_{q}(\mathfrak{x})-\mathfrak{m}_{q-2}(
\mathfrak{x})-\mathfrak{x\in }\ker \tilde{\pi}=\left( \mathfrak{f}^{[g]}(
\mathfrak{x})-\mathfrak{x}\right)
\mathbb{Z}
\lbrack \mathfrak{x}]$. Since $\mathfrak{m}_{q}(\mathfrak{x})-\mathfrak{m}
_{q-2}(\mathfrak{x})-\mathfrak{x}$ and $\mathfrak{f}^{[g]}(\mathfrak{x})-
\mathfrak{x}$ are both monic of degree $q$, the last relation implies that
they have to be equal and $\mathfrak{f}^{[g]}(\mathfrak{x})=\mathfrak{m}_{q}(
\mathfrak{x})-\mathfrak{m}_{q-2}(\mathfrak{x})$. The proposition now follows
from Corollary \ref{n-n2}. $\blacksquare $
\section{A conjecture}
The following fact was pointed out to us by G. Savin:
\begin{proposition}
\label{tensorp}Let $p$ be a prime and $q=p^{g}>1$ be an integral power of $p$
. Let $\mathbb{G}$ be a simply connected semisimple algebraic group defined
and split over $\mathbb{F}_{q}$, and denote by $K_{0}(\mathbb{G)}$ the
Grothendieck ring of $\mathbb{F}_{q}[\mathbb{G]}$-rational modules of finite
$\mathbb{F}_{q}$-dimension. If $\mathfrak{M}$ is an element of $K_{0}(
\mathbb{G)}$ and $i$ is any non-negative integer, we have:
\begin{equation*}
\mathfrak{M}^{[i]}\equiv \mathfrak{M}^{p^{i}}(\func{mod}pK_{0}(\mathbb{G))}.
\end{equation*}
\end{proposition}
\textbf{Proof }Let $\mathbb{T}$\ be a maximal torus of $\mathbb{G}$ defined
and split over $\mathbb{F}_{q}$,\ and denote by $X=X(\mathbb{T})$ its
character group. For any $\lambda \in X$, denote by $e(\lambda )$ the
corresponding basis element of the group ring $
\mathbb{Z}
\lbrack X]$, so that $e(\lambda +\lambda ^{\prime })=e(\lambda )e(\lambda
^{\prime })$ for any characters $\lambda $ and $\lambda ^{\prime }$.\
Fix a $\mathbb{G}$-module $\mathfrak{M}$ and write its formal character as:
\begin{equation*}
\limfunc{ch}\mathfrak{M}=\dsum\nolimits_{\lambda \in X}m_{\lambda }\cdot
e(\lambda ),
\end{equation*}
\noindent where $m_{\lambda }$ is the dimension of the $\lambda $-isotypic
submodule of $\mathfrak{M}$. For a positive integer $i$, the $p^{i}$th power
automorphism of $\mathbb{\bar{F}}_{q}$ induces an action on $
\mathbb{Z}
\lbrack X]$ by sending a basis element $e(\lambda )$ to $e(p^{i}\lambda )$,
so that:
\begin{eqnarray*}
\limfunc{ch}(\mathfrak{M}^{[i]}) &=&\dsum\nolimits_{\lambda \in X}m_{\lambda
}\cdot e(\lambda )^{p^{i}} \\
&\equiv &\left( \dsum\nolimits_{\lambda \in X}m_{\lambda }\cdot e(\lambda
)\right) ^{p^{i}}(\func{mod}p
\mathbb{Z}
\lbrack X]\mathbb{)}\text{.}
\end{eqnarray*}
The formal character $\left( \tsum\nolimits_{\lambda \in X}m_{\lambda }\cdot
e(\lambda )\right) ^{p^{i}}$ is the element associated to $\mathfrak{M}
^{p^{i}}$ by the map $\limfunc{ch}:K_{0}(\mathbb{G})\longrightarrow
\mathbb{Z}
\lbrack X]$. We have therefore:
\begin{equation}
\limfunc{ch}(\mathfrak{M}^{[i]})\equiv \limfunc{ch}(\mathfrak{M}^{p^{i}})
\text{ \ }(\func{mod}p
\mathbb{Z}
\lbrack X]\mathbb{)}\text{.} \tag{$1$} \label{11}
\end{equation}
Let $\mathcal{W}$ denotes the Weyl group of the pair $(\mathbb{G},\mathbb{T}
) $. By \cite{Jan}\ II.5.8, the map $\limfunc{ch}$ induces an isomorphism of
commutative rings:
\begin{equation*}
\limfunc{ch}:K_{0}(\mathbb{G})\overset{\sim }{\longrightarrow }
\mathbb{Z}
\lbrack X]^{\mathcal{W}}.
\end{equation*}
Write $\limfunc{ch}(\mathfrak{M}^{[i]})=\tsum\nolimits_{\lambda \in
X}a_{\lambda }\cdot e(\lambda )$ and $\limfunc{ch}(\mathfrak{M}
^{p^{i}})=\tsum\nolimits_{\lambda \in X}b_{\lambda }\cdot e(\lambda )$, so
that $\limfunc{ch}(\mathfrak{M}^{[i]}-\mathfrak{M}^{p^{i}})=\tsum\nolimits_{
\lambda \in X}\left( a_{\lambda }-b_{\lambda }\right) \cdot e(\lambda )$ is
such that:
\begin{equation}
\dsum\nolimits_{\lambda \in X}\left( a_{\lambda }-b_{\lambda }\right) \cdot
e(\lambda )=\dsum\nolimits_{\lambda \in X}\left( a_{\lambda }-b_{\lambda
}\right) \cdot e(w\lambda ) \tag{$2$} \label{22}
\end{equation}
for all $w\in \mathcal{W}$.
By (\ref{11}), there are integers $c_{\lambda }$ such that $a_{\lambda
}-b_{\lambda }=pc_{\lambda }$ for all $\lambda \in X$. Since $
\mathbb{Z}
\lbrack X]$ is $
\mathbb{Z}
$-flat, we can view it as a subring of $
\mathbb{Q}
[X]$, in which we have, for any $w\in \mathcal{W}$:
\begin{eqnarray*}
w\cdot \left( \dsum\nolimits_{\lambda \in X}c_{\lambda }\cdot e(\lambda
)\right) &=&\frac{1}{p}\dsum\nolimits_{\lambda \in X}\left( a_{\lambda
}-b_{\lambda }\right) \cdot e(w\lambda ) \\
&=&\frac{1}{p}\dsum\nolimits_{\lambda \in X}\left( a_{\lambda }-b_{\lambda
}\right) \cdot e(\lambda ),
\end{eqnarray*}
\noindent where the last equality follows from (\ref{22}). Therefore $w\cdot
\left( \tsum\nolimits_{\lambda \in X}c_{\lambda }\cdot e(\lambda )\right)
=\tsum\nolimits_{\lambda \in X}c_{\lambda }\cdot e(\lambda )$ in $
\mathbb{Z}
\lbrack X]$\ for all $w\in \mathcal{W}$ and
\begin{equation*}
\limfunc{ch}(\mathfrak{M}^{[i]}-\mathfrak{M}^{p^{i}})\in p
\mathbb{Z}
\lbrack X]^{\mathcal{W}}.
\end{equation*}
This\ implies that $\mathfrak{M}^{[i]}\ $is congruent to $\mathfrak{M}
^{p^{i}}$ modulo the ideal generated by $p$ in $K_{0}(\mathbb{G)}$. $
\blacksquare $
Motivated by Theorem \ref{presentation}, Corollary \ref{cor_presentation}
and Proposition \ref{tensorp}, we are led to the following:
\begin{conjecture}
\label{conJ}Let $p$ be a prime and $q=p^{g}>1$ be an integral power of $p$.
Let $\mathbb{G}$ be a simply connected semisimple algebraic group defined
and split over $\mathbb{F}_{q}$, whose rank is $\ell >0$. Denote by $K_{0}(
\mathbb{G}(\mathbb{F}_{q}))$ the Grothendieck ring of finitely generated $
\mathbb{F}_{q}[\mathbb{G}(\mathbb{F}_{q})]$-modules. Then there exist $\ell $
monic polynomials $\mathfrak{f}_{1}(\mathfrak{x}_{1})\in
\mathbb{Z}
\lbrack \mathfrak{x}_{1}],...,\mathfrak{f}_{\ell }(\mathfrak{x}_{\ell })\in
\mathbb{Z}
\lbrack \mathfrak{x}_{\ell }]$ having degree $p$\ such that:
\begin{equation*}
K_{0}\left( \mathbb{G}(\mathbb{F}_{q})\right) \simeq \frac{
\mathbb{Z}
\lbrack \mathfrak{x}_{1},...,\mathfrak{x}_{\ell }]}{\left( \mathfrak{f}
_{1}^{[g]}(\mathfrak{x}_{1})-\mathfrak{x}_{1}\mathfrak{,...,f}_{\ell }^{[g]}(
\mathfrak{x}_{\ell })-\mathfrak{x}_{\ell }\right)
\mathbb{Z}
\lbrack \mathfrak{x}_{1},...,\mathfrak{x}_{\ell }]},
\end{equation*}
\noindent where for any $i$, $1\leq i\leq \ell $, $\mathfrak{f}_{i}^{[g]}(
\mathfrak{x}_{i})$ is the polynomial obtained by composing $\mathfrak{f}_{i}(
\mathfrak{x}_{i})$ with itself $g$ times.
Furthermore, $\mathfrak{f}_{i}(\mathfrak{x}_{i})\equiv \mathfrak{x}_{i}^{p}(
\func{mod}p
\mathbb{Z}
\lbrack \mathfrak{x}_{i}])$ for any $i$, $1\leq i\leq \ell $.
\end{conjecture}
The idea behind the above statement is that if $\pi $ is an isomorphism of $
\mathbb{Z}
\lbrack \mathfrak{x}_{1},...,\mathfrak{x}_{\ell }]/(\mathfrak{f}_{1}^{[g]}(
\mathfrak{x}_{1})-\mathfrak{x}_{1}\mathfrak{,...,f}_{\ell }^{[g]}(\mathfrak{x
}_{\ell })-\mathfrak{x}_{\ell })$ onto $K_{0}\left( \mathbb{G}(\mathbb{F}
_{q})\right) $, and if we set $\mathfrak{X}_{i}:=\pi (\mathfrak{x}_{i})$ for
$1\leq i\leq \ell $, then $\mathfrak{f}_{i}(\mathfrak{X}_{i})\in K_{0}\left(
\mathbb{G}(\mathbb{F}_{q})\right) $ should be the Frobenius twist $\mathfrak{
X}_{i}^{[1]}$. This means that the relations imposed above in the algebra $
\mathbb{Z}
\lbrack \mathfrak{x}_{1},...,\mathfrak{x}_{\ell }]$ are the obvious ones
that translate into $\mathfrak{X}_{i}^{[g]}=\mathfrak{X}_{i}$ for all $i$.
Here is some evidence for the conjecture:
\begin{description}
\item[(a)] As proved in the previous paragraph, the conjecture is true for $
\mathbb{G}=SL_{2}$ over $\mathbb{F}_{q}$ ($\ell =1$), in which case we can
also give an explicit formula for the polynomial $\mathfrak{f}(\mathfrak{x})=
\mathfrak{f}_{1}(\mathfrak{x}_{1})$ (Theorem \ref{presentation}).
\item[(b)] A theorem of Steinberg (\cite{St}) states that if $\mathbb{G}$ is
a simply connected semisimple algebraic group over $\mathbb{F}_{q}$, the
number of semisimple conjugacy classes of $\mathbb{G}(\mathbb{F}_{q})$ is
equal to $q^{\ell }$, where $\ell $ is the rank of $\mathbb{G}$. Therefore $
K_{0}\left( \mathbb{G}(\mathbb{F}_{q})\right) \simeq
\mathbb{Z}
^{q^{\ell }}$ as $
\mathbb{Z}
$-modules, which follows from the conjecture.
\item[(c)] Since $\mathfrak{h}(\mathfrak{x}_{1},...,\mathfrak{x}_{\ell
})^{q}=\mathfrak{h(}\mathfrak{x}_{1}^{q},...,\mathfrak{x}_{\ell }^{q}
\mathfrak{)}$ for any polynomial $\mathfrak{h}(\mathfrak{x}_{1},...,
\mathfrak{x}_{\ell })\in \mathbb{F}_{q}[\mathfrak{x}_{1},...,\mathfrak{x}
_{\ell }]$, the conjecture implies that $\overline{\mathfrak{M}}^{q}=
\overline{\mathfrak{M}}$ for any $\overline{\mathfrak{M}}\in K_{0}(\mathbb{G}
(\mathbb{F}_{q}))\otimes _{
\mathbb{Z}
}\mathbb{F}_{q}$. This fact is also a consequence of Proposition \ref
{tensorp}.
\item[(d)] Assume we are given a surjective homomorphism of $\mathbb{F}_{q}$
-algebras:
\begin{equation*}
\gamma :\mathbb{F}_{q}[\mathfrak{x}_{1},...,\mathfrak{x}_{\ell
}]\longrightarrow K_{0}(\mathbb{G}(\mathbb{F}_{q}))\otimes _{
\mathbb{Z}
}\mathbb{F}_{q}\text{.}
\end{equation*}
Proposition \ref{tensorp} implies that $\gamma (\mathfrak{x}_{i})^{q}=\gamma
(\mathfrak{x}_{i})$ for any integer $i$, $1\leq i\leq \ell $; in particular
the kernel of $\gamma $ contains the ideal generated by the polynomials $
\mathfrak{x}_{1}^{q}-\mathfrak{x}_{1}\mathfrak{,...,x}_{\ell }^{q}-\mathfrak{
x}_{\ell }$. By dimension reasons we must have an isomorphism of $\mathbb{F}
_{q}$-algebras:
\begin{equation*}
\frac{\mathbb{F}_{q}[\mathfrak{x}_{1},...,\mathfrak{x}_{\ell }]}{\left(
\mathfrak{x}_{1}^{q}-\mathfrak{x}_{1}\mathfrak{,...,x}_{\ell }^{q}-\mathfrak{
x}_{\ell }\right) \mathbb{F}_{q}[\mathfrak{x}_{1},...,\mathfrak{x}_{\ell }]}
\overset{\sim }{\longrightarrow }K_{0}(\mathbb{G}(\mathbb{F}_{q}))\otimes _{
\mathbb{Z}
}\mathbb{F}_{q}.
\end{equation*}
This is predicted by Conjecture \ref{conJ}.
\end{description}
If Conjecture \ref{conJ} is correct, one would like to determine explicit
formulae for the polynomials $\mathfrak{f}_{1}(\mathfrak{x}_{1}),...,
\mathfrak{f}_{\ell }(\mathfrak{x}_{\ell })$ and to relate factorization
properties of these polynomials in $
\mathbb{Z}
\lbrack \mathfrak{x}_{i}]$ to algebraic properties of the group $\mathbb{G}(
\mathbb{F}_{q})$.
\end{document}
|
\begin{document}
\renewcommand{1.2}{1.2}
\markboth{
{\footnotesize\rm Xinyi Li, Li Wang and Dan Nettleton}
}
{
{\footnotesize\rm Sparse Model Identification and Learning for Ultra-high-dimensional Additive Partially Linear Models}
}
\renewcommand{\thefootnote}{}
$\ $\par \fontsize{10.95}{14pt plus.8pt minus .6pt}\selectfont
\centerline{\Large\bf Sparse Model Identification and Learning for }
\centerline{\Large\bf Ultra-high-dimensional Additive Partially Linear Models}
\centerline{Xinyi Li$^{a}$, Li Wang$^{b}$ and Dan Nettleton$^{b}$
\footnote{\emph{Address for correspondence}: Li Wang, Department of Statistics and the Statistical Laboratory, Iowa State University, Ames, IA, USA. Email: [email protected]}}
\centerline{\it $^{a}$SAMSI / University of North Carolina at Chapel Hill and $^{b}$Iowa State University}
\fontsize{9}{11.5pt plus.8pt minus .6pt}\selectfont
\begin{quotation}
\noindent \textit{Abstract:} The additive partially linear model (APLM) combines the flexibility of nonparametric regression with the parsimony of regression models, and has been widely used as a popular tool in multivariate nonparametric regression to alleviate the ``curse of dimensionality''. A natural question raised in practice is the choice of structure in the nonparametric part, that is, whether the continuous covariates enter into the model in linear or nonparametric form. In this paper, we present a comprehensive framework for simultaneous sparse model identification and learning for ultra-high-dimensional APLMs where both the linear and nonparametric components are possibly larger than the sample size. We propose a fast and efficient two-stage procedure. In the first stage, we decompose the nonparametric functions into a linear part and a nonlinear part. The nonlinear functions are approximated by constant spline bases, and a triple penalization procedure is proposed to select nonzero components using adaptive group LASSO. In the second stage, we refit data with selected covariates using higher order polynomial splines, and apply spline-backfitted local-linear smoothing to obtain asymptotic normality for the estimators. The procedure is shown to be consistent for model structure identification. It can identify zero, linear, and nonlinear components correctly and efficiently. Inference can be made on both linear coefficients and nonparametric functions. We conduct simulation studies to evaluate the performance of the method and apply the proposed method to a dataset on the Shoot Apical Meristem (SAM) of maize genotypes for illustration.
\noindent \textit{Key words and phrases:} Dimension reduction, inference for ultra-high-dimensional data, semiparametric regression, spline-backfitted local polynomial, structure identification, variable selection.
\end{quotation}
\fontsize{10.95}{14pt plus.8pt minus .6pt}\selectfont
\thispagestyle{empty}
\setcounter{chapter}{1}
\setcounter{equation}{0}
\renewcommand{E.\arabic{equation}}{\arabic{equation}}
\renewcommand{0}{0}
\section{Introduction} \label{SEC:introduction} \vskip 0.1in
In the past three decades, flexible and parsimonious additive partially linear models (APLMs) have been extensively studied and widely used in many statistical applications, including biology, econometrics, engineering, and social science. Examples of recent work on APLMs include \cite{liang2008additive}, \cite{liu2011estimation}, \cite{ma2011spline}, \cite{wang2011estimation}, \cite{ma2013simultaneous}, \cite{wang2014estimation} and \cite{lian2014generalized}. APLMs are natural extensions of classical parametric models with good interpretability and are becoming more and more popular in data analysis.
Suppose we observe $\{(Y_i,\mathbf{Z}_{(i)},$ $\mathbf{X}_{(i)})\}_{i=1}^{n}$. For subject $i =1,\ldots, n$, $Y_i$ is a univariate response, $\mathbf{Z}_{(i)}=( Z_{i1},\ldots ,$ $Z_{ip_1})^{\top}$ is a $p_1$-dimensional vector of covariates that may be linearly associated with the response, and $\mathbf{X}_{(i)}=(X_{i1},\ldots ,X_{i p_2})^{\top}$ is a $p_2$-dimensional vector of continuous covariates that may have nonlinear associations with the response. We assume $\{(Y_i,\mathbf{Z}_{(i)},$ $\mathbf{X}_{(i)})\}_{i=1}^{n}$ is an i.i.d sample from the distribution of $\left(Y,\boldsymbol{Z},\boldsymbol{X}\right) $, satisfying the following model:
\begin{align}
Y_i &= \mu + \mathbf{Z}_{(i)}^{\top}\boldsymbol{\alpha} + \sum_{\ell=1}^{p_2}\phi_{\ell}(X_{i\ell}) + \varepsilon_i
= \mu + \sum_{k=1}^{p_1}Z_{ik}\alpha_k + \sum_{\ell=1}^{p_2}\phi_{\ell}(X_{i\ell}) + \varepsilon_i,
\label{model:aplm}
\end{align}
where $\mu$ is the intercept, $\alpha_k$, $k=1,\ldots,p_1$, are unknown regression coefficients, $\left\{\phi _{\ell}\left( \cdot \right) \right\} _{\ell=1}^{p_2}$ are unknown smooth functions, and each $\phi_{\ell}\left( \cdot \right)$ is centered with ${\rm E} \phi _{\ell}\left(X_{i\ell}\right)=0$ to make model (\ref{model:aplm}) identifiable. The $\mathbf{X}_{(i)}$ is a $p_2$-dimensional vector of zero-mean covariates having density with a compact support. Without loss of generality, we assume that each covariate $\left\{X_{i\ell}\right\}_{\ell=1}^{p_2}$ can be rescaled into an interval $\chi=[a,b]$. The $\varepsilon_i$ terms are iid random errors with mean zero and variance $\sigma^2$.
The APLM is particularly convenient when $\boldsymbol{Z}$ is a vector of categorical or discrete variables, and in this case, the components of $\boldsymbol{Z}$ enter the linear part of model (\ref{model:aplm}) automatically, and the continuous variables usually enter the model nonparametrically. In practice, we might have reasons to believe that some of the continuous variables should enter the model linearly rather than nonparametrically. A natural question is how to determine which continuous covariates have a linear effect and which continuous covariates have a nonlinear effect. If the choice of linear components is correctly specified, then the biases in the estimation of these components are eliminated and root-$n$ convergence rates can be obtained for the linear coefficients. However, such prior knowledge is rarely available, especially when the number of covariates is large. Thus, structure identification, or linear and nonlinear detection, is an important step in the process of building an APLM from high-dimensional data.
When the number of covariates in the model is fixed, structure identification in additive models (AMs) has been studied in the literature. \citet{zhang2011linear} proposed a penalization procedure to identify the linear components in AMs in the context of smoothing splines ANOVA. They demonstrated the consistency of the model structure identification and established the convergence rate of the proposed method specifically under the tensor product design. \citet{huang2012semiparametric} proposed another penalized semiparametric regression approach using a group minimax concave penalty to identify the covariates with linear effects. They showed consistency in determining the linear and nonlinear structure in covariates, and obtained the convergence rate of nonlinear function estimators and asymptotic properties of linear coefficient estimators; but they did not perform variable selection at the same time.
For high-dimensional AMs, \citet{lian2015separation} proposed a double penalization procedure to distinguish covariates that enter the nonparametric and parametric parts and to identify significant covariates simultaneously.
They demonstrated the consistency of the model structure identification, and established the convergence rate of nonlinear function estimators and asymptotic normality of linear coefficient estimators. Despite the nice theoretical properties, their method heavily relies on the local quadratic approximation in \cite{fan2001variable}, which is incapable of producing naturally sparse estimates. In addition, employing the local quadratic approximation can be extremely expensive because it requires the repeated factorization of large matrices, which becomes infeasible when the number of covariates is very large.
Note that all the aforementioned papers \citep{zhang2011linear,huang2012semiparametric,lian2015separation} about structure identification focus on the AM with continuous explanatory variables. However, in many applications, a canonical partitioning of the variables exists. In particular, if there are categorical or discrete explanatory variables, as in the case of the SAM data studies (see the details in Section \ref{sec:application}) and in many genome-wide association studies, we may want to keep discrete explanatory variables separate from the other design variables and let discrete variables enter the linear part of the model directly. In addition, if there is some prior knowledge of certain parametric forms for some specific covariates, such as a linear form, we may lose efficiency if we simply model all the covariates nonparametrically.
The above practical and theoretical concerns motivate our further investigation of the simultaneous variable selection and structure selection problem for flexible and parsimonious APLMs, in which the features of the data suitable for parametric modeling are modeled parametrically and nonparametric components are used only where needed. We consider the setting where both the dimension of the linear components and the dimension of nonlinear components is ultra-high. We propose an efficient and stable penalization procedure for simultaneously identifying linear and nonlinear components, removing insignificant predictors, and estimating the remaining linear and nonlinear components. We prove the proposed \textit{S}parse \textit{M}odel \textit{I}dentification, \textit{L}earning and \textit{E}stimation (referred to as SMILE) procedure is consistent. We propose an iterative group coordinate descent approach to solve the penalized minimization problem efficiently. Our algorithm is very easy to implement because it only involves simple arithmetic operations with no complicated numerical optimization steps, matrix factorizations, or inversions. In one simulation example with $n=500$ and $p_1=p_2=5000$, it takes less than one minute to complete the entire model identification and variable selection process on a regular PC.
After variable selection and structure detection, we would like to provide an inferential tool for the linear and nonparametric components. The spline method is fast and easy to implement; however, the rate of convergence is only established in mean squares sense, and there is no asymptotic distribution or uniform convergence, so no measures of confidence can be assigned to the estimators. In this paper, we propose a two-step spline-backfitted local-linear smoothing (SBLL) procedure for APLM estimation, model selection and simultaneous inference for all the components. In the first stage, we approximate the nonparametric functions $\phi_{\ell} (\cdot)$, $\ell= 1,\ldots,p_2$, with undersmoothed constant spline functions. We perform model selection for the APLM using a triple penalized procedure to select important variables and identify the linear vs. nonlinear structure for the continuous covariates, which is crucial to obtain efficient estimators for the non-zero components. We show that the proposed model selection and structure identification for both parametric and nonparametric terms are consistent, and the estimators of the nonzero linear coefficients and nonzero nonparametric functions are both $L_2$-norm consistent. In the second stage, we refit the data with covariates selected in the first step using higher-order polynomial splines to achieve root-$n$ consistency of the coefficient estimators in the linear part, and apply a one-step local-linear backfitting to the projected nonparametric components obtained from the refitting. Asymptotic normality for both linear coefficient estimators and nonlinear component estimators, as well as simultaneous confidence bands (SCBs) for all nonparametric components, are provided.
The rest of the paper is organized as follows. In Section \ref{sec:method}, we describe the first-stage spline smoothing and propose a triple penalized regularization method for simultaneous model identification and variable selection. The theoretical properties of selection consistency and rates of convergence for the coefficient estimators and nonparametric estimators are developed. Section \ref{sec:nonparmetric} introduces the spline-backfitted local-linear estimators and SCBs for the nonparametric components. The performance of the estimators is assessed by simulations in Section \ref{sec:simulation} and illustrated by application to the SAM data in Section \ref{sec:application}. Some concluding remarks are given in Section \ref{sec:conclusion}. Section A of the online Supplemental Materials evaluates the effect of different smoothing parameters on the performance of the proposed method. Technical details are provided in Section B of the Supplemental Materials.
\setcounter{chapter}{2} \renewcommand{E.\arabic{proposition}}{{2.
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\section{Methodology}
\label{sec:method}
\subsection{Model Setup}
In the following, the functional form (linear vs. nonlinear) for each continuous covariate in model (\ref{model:aplm}) is assumed to be unknown.
In order to decide the form of $\phi_{\ell}$, for each $\ell=1, \ldots p_2$, we can decompose $\phi_{\ell}$ into a linear part and a nonlinear part: $\phi_{\ell}(x) = \beta_{\ell}x + g_{\ell}(x)$,
where $g_{\ell}(x)$ is some unknown smooth nonlinear function (see Assumption (A1) in Appendix \ref{SUBSEC:assump}). For model identifiability, we assume that $\mathrm{E}(X_{i\ell}) = 0$, $\mathrm{E}\{g_{\ell}(X_{i\ell})\} = 0$ and ${\rm E}\{g_{\ell}^{\prime}(X_{i\ell})\} = 0$. The first two constraints $\mathrm{E}(X_{i\ell}) = 0$ and ${\rm E}\{g_{\ell}(X_{i\ell})\} = 0$, are required to guarantee identifiability for the APLM, that is, ${\rm E}\{\phi_{\ell}(X_{i\ell})\} = 0$. The constraint $\mathrm{E}\{g_{\ell}^{\prime}(X_{i\ell})\} = 0$ ensures there is no linear form in nonlinear function $g_{\ell}(x)$. Note that these constraints are also in accordance with the definition of nonlinear contrast space in \cite{zhang2011linear}, which is a subspace of the orthogonal decomposition of RKHS. In the following, we assume $Y_i$ values are centered so that we can express the APLM in (\ref{model:aplm}) without an intercept parameter as
\begin{align}
Y_i &= \sum_{k=1}^{p_1}Z_{ik}\alpha_k + \sum_{\ell=1}^{p_2}X_{i\ell}\beta_{\ell} + \sum_{\ell=1}^{p_2}g_{\ell}(X_{i\ell}) + \varepsilon_i.
\label{EQ:aplm}
\end{align}
In the following, we define predictor variable $Z_k$ as irrelevant in model (\ref{EQ:aplm}), if and only if $\alpha_k=0$, and $X_{\ell}$ as irrelevant if and only if $\beta_{\ell}=0$ and $g_{\ell}(x_{\ell})=0$ for all $x_{\ell}$ on its support. A predictor variable is defined as relevant if and only if it is not irrelevant. Suppose that only an unknown subset of predictor variables is relevant. We are interested in identifying such subsets of relevant predictors consistently while simultaneously estimating their coefficients and/or functions.
For covariates $\boldsymbol{Z}$, we define
\begin{align*}
\mbox{Active index set for $\boldsymbol{Z}$}:& ~\mathcal{S}_z=\{k = 1,\ldots, p_1: \alpha_k \neq 0\}, \\
\mbox{Inactive index set for $\boldsymbol{Z}$}:& ~\mathcal{N}_z=\{k = 1,\ldots, p_1: \alpha_k = 0\}.
\end{align*}
For continuous covariate $X_{\ell}$, we say it is a linear covariate if $\beta_{\ell} \neq 0$ and $g_{\ell}(x_{\ell}) = 0$ for all $x_{\ell}$ on its support, and $X_{\ell}$ is a nonlinear covariate if $g_{\ell}(x_{\ell})\neq 0$. Explicitly, we define the following index sets for $\boldsymbol{X}$:
\begin{align*}
\mbox{Active pure linear index set for $\boldsymbol{X}$}:& ~\mathcal{S}_{x, PL}=\{\ell=1,\ldots, p_2: \beta_{\ell} \neq 0,~ g_{\ell} \equiv 0\}, \\
\mbox{Active nonlinear index set for $\boldsymbol{X}$}:& ~\mathcal{S}_{x, N}=\{\ell=1,\ldots, p_2: g_{\ell} \neq 0\}, \\
\mbox{Inactive index set for $\boldsymbol{X}$}:& ~\mathcal{N}_x=\{\ell=1,\ldots, p_2: \beta_{\ell} = 0,~ g_{\ell} \equiv 0\}.
\end{align*}
Note that the active nonlinear index set for $\boldsymbol{X}$, $\mathcal{S}_{x, N}$, can be decomposed as $\mathcal{S}_{x, N} = \mathcal{S}_{x, LN} \cup \mathcal{S}_{x, PN}$, where $\mathcal{S}_{x, LN}=\{\ell=1,\ldots, p_2: \beta_{\ell} \neq 0, ~g_{\ell} \neq 0\}$ is the index set for covariates whose linear and nonlinear terms in (\ref{EQ:aplm}) are both nonzero, and $\mathcal{S}_{x, PN}=\{\ell=1,\ldots, p_2: \beta_{\ell} = 0, ~g_{\ell} \neq 0\}$ is the index set for active pure nonlinear index set for $\boldsymbol{X}$.
Therefore, the model selection problem for model (\ref{EQ:aplm}) is equivalent to the problem of identifying $\mathcal{S}_z$, $\mathcal{N}_z$, $\mathcal{S}_{x, PL}$, $\mathcal{S}_{x, LN}$, $\mathcal{S}_{x, PN}$ and $\mathcal{N}_x$. To achieve this, we propose to minimize
\begin{equation}
\sum_{i=1}^n \Bigg\{ Y_i - \sum_{k=1}^{p_1}Z_{ik}\alpha_k - \sum_{\ell=1}^{p_2}X_{i\ell}\beta_{\ell} - \sum_{\ell=1}^{p_2}g_{\ell}(X_{i\ell}) \Bigg\}^2 + \sum_{k=1}^{p_1} p_{\lambda_{n1}}(|\alpha_k|) + \sum_{\ell=1}^{p_2} p_{\lambda_{n2}}(|\beta_{\ell}|) + \sum_{\ell=1}^{p_2} p_{\lambda_{n3}}(\|g_{\ell}\|_2),
\label{EQ:loss_penalty}
\end{equation}
where $\Vert g_{\ell}\Vert _{2}^2=\mathrm{E}\{g_{\ell}^{2}(X_{\ell})\}$, and $p_{\lambda _{n1}}\left( \cdot \right)$, $p_{\lambda _{n2}}\left( \cdot \right)$ and $p_{\lambda _{n3}}\left( \cdot \right)$ are penalty functions explained in detail in Section \ref{subsec:selection}. The tuning parameters $\lambda_{n1}$, $\lambda_{n2}$ and $\lambda_{n3}$ decide the complexity of the selected model. The smoothness of predicted nonlinear functions is controlled by $\lambda_{n3}$, and $\lambda_{n1}$, $\lambda_{n2}$ and $\lambda_{n3}$ go to $\infty$ as $n$ increases to $\infty$.
\subsection{Spline Basis Approximation} \label{SUBSEC:spline}
We approximate the smooth functions $\left\{g _{\ell}\left( \cdot \right): \ell=1, \ldots, p_2 \right\}$ in (\ref{EQ:aplm}) by polynomial splines for their simplicity in computation. For example, for each $\ell=1, \ldots, p_2$, let $\upsilon _{0,\ell}, \ldots, \upsilon _{N_n+1,\ell}$ be knots that partition $[a,b]$ with
$a=\upsilon _{0,\ell}<\upsilon _{1,\ell}<\ldots <\upsilon_{N_n,\ell}<\upsilon _{N_n+1,\ell}=b$.
The space of polynomial splines of order $d\geq 1$, $\mathcal{B}^{(d)}_{\ell}[a,b]$, consisting of functions $s(\cdot)$ satisfying (i) the restriction of $s(\cdot)$ to subintervals $[\upsilon _{J,\ell},\upsilon_{J+1,\ell})$, $J = 1, \ldots, N_n + d$, and $\left[\upsilon _{N_n,\ell},\upsilon_{N_n+1,\ell}\right]$, is a polynomial of $(d-1)$-degree (or less); (ii) for $d\geq 2$ and $0 \leq d^{\prime} \leq d-2$, $s(\cdot)$ is $d^{\prime}$ times continuously differentiable on $[a,b]$. Below we denote $b_{J,\ell}^{(d)}(\cdot)$, $J=1,\ldots, N_n+d$, the basis functions of $\mathbb{B}^{(d)}_{\ell}[a,b]$.
To ensure $\mathrm{E}\{g_{\ell}(X_{i\ell})\} = 0$ and $\mathrm{E}\{g_{\ell}^{\prime}(X_{i\ell})\} = 0$, we consider the following normalized first-order B-splines, referred to as piecewise constant splines.
We define for any $\ell =1, \ldots, p_2$ the piecewise constant B-spline function as the indicator
function $I_{J,\ell}\left( x_{\ell}\right) $ of the $\left( N_n+1\right)
$ equally-spaced subintervals of $[a,b]$
with length $H=H_{n}=(b-a) / \left( N_n+1\right)$, that is,
\begin{align*}
I_{J,\ell}\left( x_{\ell}\right) &=\left\{
\begin{array}{ll}
1 & a + JH\leq x_{\ell}<a + \left( J+1\right) H, \\
0 & \mbox{otherwise},
\end{array}
\right. \, J=0,1, \ldots, N_n-1, ~~\\
I_{N_n,\ell}\left( x_{\ell}\right) &=\left\{
\begin{array}{ll}
1 & a + N_n H\leq x_{\ell}\leq b, \\
0 & \mbox{otherwise}.
\end{array}
\right.
\end{align*}
Define the following centered spline basis
\[
b_{J,\ell}^{(1)}\left(x_{\ell}\right) =I_{J,\ell}\left( x_{\ell}\right) -(\| I_{J,\ell}\|_2 /
\|I_{J-1,\ell}\|_2)I_{J-1,\ell}\left(x_{\ell}\right) , \, \forall ~ J =1, \ldots, N_n, \, \ell= 1, \ldots, p_2,
\]
with the standardized version given for any $\ell= 1, \ldots, p_2$,
\begin{equation}
B_{J,\ell}^{(1)}\left( x_{\ell}\right) =b_{J,\ell}^{(1)}\left( x_{\ell}\right)/\| b_{J,\ell}^{(1)}\| _{2}, \, \forall ~ J=1, \ldots, N_n.
\label{DEF:BJalpha}
\end{equation}
So $\mathrm{E} \{B_{J,\ell}^{(1)}(X_{i\ell})\} = 0$, $\mathrm{E} \{B_{J,\ell}^{(1)}(X_{i\ell})\}^2 = 1$.
In practice, we use the empirical distribution of $X_{1\ell},\ldots,X_{n\ell}$ to perform the centering and scaling in the definitions of $b_{J,\ell}^{(1)}(x_{\ell})$ and $B_{J,\ell}^{(1)}( x_{\ell})$.
We approximate the nonparametric function $g_{\ell}(x_{\ell})$, $\ell= 1,\ldots, p_2$, using the above normalized piecewise constant splines
\begin{equation}
g_{\ell}(x_{\ell}) \approx g_{\ell s}(x_{\ell})= \sum_{J=1}^{N_n}\gamma _{J,\ell}B_{J,\ell}^{(1)}(x_{\ell})=\mathbf{B}_{\ell}^{(1)\top}(x_{\ell}) \boldsymbol{\gamma}_{\ell},
\label{EQ:approx}
\end{equation}
where $\mathbf{B}_{\ell}^{(1)}(x_{\ell})=(B_{1,\ell}^{(1)}(x_{\ell}),\ldots ,B_{N_n,\ell}^{(1)}(x_{\ell}))^{\top}$, and $\boldsymbol{\gamma}_{\ell}=\left( \gamma_{1,l},\ldots ,\gamma _{N_n,\ell}\right)^{\top}$ is a vector of the spline coefficients. By using the centered constant spline basis functions, we can guarantee that $n^{-1}\sum_{i=1}^{n} g_{\ell s}(X_{i\ell}) = 0$, and $n^{-1}\sum_{i=1}^{n}g_{\ell s}^{\prime}(X_{i\ell}) = 0$ except at the location of the knots.
Denote a length $N_n$ vector $\mathbf{B}_{i\ell}^{(1)} = (B_{1,\ell}^{(1)}(X_{i\ell}), \ldots, B_{N_n, \ell}^{(1)}(X_{i\ell}))^{\top}$.
For any vector $\boldsymbol{a} \in \mathbb{R}^p$, denote $\Vert \boldsymbol{a} \Vert = (\sum_{\ell = 1}^p a_{\ell}^2)^{1/2}$ as the $L_2$ norm of $\boldsymbol{a}$. Following from (\ref{EQ:approx}), to minimize (\ref{EQ:loss_penalty}), it is approximately equivalent to consider the problem of minimizing
\[
\sum_{i=1}^n \Bigg\{ Y_i - \sum_{k=1}^{p_1}Z_{ik}\alpha_k - \sum_{\ell=1}^{p_2}X_{i\ell}\beta_{\ell} - \sum_{\ell=1}^{p_2}\mathbf{B}_{i\ell}^{(1)}\boldsymbol{\gamma}_{\ell} \Bigg\}^2 + \sum_{k=1}^{p_1} p_{\lambda_{n1}}(|\alpha_k|) + \sum_{\ell=1}^{p_2} p_{\lambda_{n2}}(|\beta_{\ell}|) + \sum_{\ell=1}^{p_2} p_{\lambda_{n3}}(\|\boldsymbol{\gamma}_{\ell}\|).
\]
\subsection{Adaptive Group LASSO Regularization} \label{subsec:selection}
We use adaptive LASSO \citep{zou2006adaptive} and adaptive group LASSO \citep{huang2010variable} for variable selection and estimation. Other popular choices include methods based on the Smoothly Clipped Absolute Deviation penalty \citep{fan2001variable} or the minimax concave penalty \citep{zhang2010nearly}.
Specifically, we start with group LASSO estimators obtained from the following minimization:
\begin{align}
(\widetilde{\boldsymbol{\alpha}}, \widetilde{\boldsymbol{\beta}}, \widetilde{\boldsymbol{\gamma}}) = \underset{{\boldsymbol{\alpha},\boldsymbol{\beta}, \boldsymbol{\gamma}}}{\arg\min} \sum_{i=1}^n \left\{ Y_i - \sum_{k=1}^{p_1}Z_{ik}\alpha_k - \sum_{\ell=1}^{p_2}X_{i\ell}\beta_{\ell} - \sum_{\ell=1}^{p_2}\mathbf{B}_{i\ell}^{(1)}\boldsymbol{\gamma}_{\ell} \right\}^2\notag\\
+ \widetilde{\lambda}_{n1} \sum_{k=1}^{p_1} |\alpha_k| + \widetilde{\lambda}_{n2} \sum_{\ell=1}^{p_2} |\beta_{\ell}| + \widetilde{\lambda}_{n3} \sum_{\ell=1}^{p_2} \|\boldsymbol{\gamma}_{\ell}\|.
\label{EQ:loss_glasso_spline}
\end{align}
Then, let $w_k^{\alpha} = |\widetilde{\alpha}_k|^{-1} I\{|\widetilde{\alpha}_k| > 0\} + \infty \times I\{|\widetilde{\alpha}_k| = 0\}$, $w_{\ell}^{\beta} = |\widetilde{\beta}_{\ell}|^{-1} I\{|\widetilde{\beta}_{\ell}| > 0\} + \infty \times I\{|\widetilde{\beta}_{\ell}| = 0\}$, $w_{\ell}^{\boldsymbol{\gamma}} = \|\widetilde{\boldsymbol{\gamma}}_{\ell}\|^{-1} I\{\|\widetilde{\boldsymbol{\gamma}}_{\ell}\| > 0\} + \infty \times I\{\|\widetilde{\boldsymbol{\gamma}}_{\ell}\| = 0\}$, where by convention, $\infty \times 0 = 0$. The adaptive group LASSO objective function is defined as
\begin{align}
L(\boldsymbol{\alpha}, \boldsymbol{\beta}, \boldsymbol{\gamma}; \lambda_{n1}, \lambda_{n2}, \lambda_{n3}) = \sum_{i=1}^n \left\{ Y_i - \sum_{k=1}^{p_1}Z_{ik}\alpha_k - \sum_{\ell=1}^{p_2}X_{i\ell}\beta_{\ell} - \sum_{\ell=1}^{p_2}\mathbf{B}_{i\ell}^{(1)}\boldsymbol{\gamma}_{\ell} \right\}^2 \nonumber\\
+ \lambda_{n1} \sum_{k=1}^{p_1} w_k^{\alpha} |\alpha_k| + \lambda_{n2} \sum_{\ell=1}^{p_2} w_{\ell}^{\beta} |\beta_{\ell}| + \lambda_{n3}\sum_{\ell=1}^{p_2} w_{\ell}^{\boldsymbol{\gamma}} \|\boldsymbol{\gamma}_{\ell}\|.
\label{EQ:loss_aglasso_spline}
\end{align}
The adaptive group LASSO estimators are minimizers of (\ref{EQ:loss_aglasso_spline}), denoted by
\[
(\widehat{\boldsymbol{\alpha}}, \widehat{\boldsymbol{\beta}}, \widehat{\boldsymbol{\gamma}}) = \underset{{\boldsymbol{\alpha},\boldsymbol{\beta}, \boldsymbol{\gamma}}}{\arg\min} ~ L(\boldsymbol{\alpha}, \boldsymbol{\beta}, \boldsymbol{\gamma}; \lambda_{n1}, \lambda_{n2}, \lambda_{n3}).
\]
The model structure selected is defined by
\begin{align*}
\widehat{\mathcal{S}}_z &= \left\{1 \leq k \leq p_1: |\widehat{\alpha}_k| > 0 \right\}, ~
\widehat{\mathcal{S}}_{x, PL} = \left\{\ell: |\widehat{\beta}_{\ell}| > 0, \|\widehat{\boldsymbol{\gamma}}_{\ell}\| = 0, 1 \leq \ell \leq p_2 \right\} , \nonumber \\
\widehat{\mathcal{S}}_{x, LN} &= \left\{\ell: |\widehat{\beta}_{\ell}| > 0, \|\widehat{\boldsymbol{\gamma}}_{\ell}\| > 0, 1 \leq \ell \leq p_2 \right\} , ~
\widehat{\mathcal{S}}_{x, PN} = \left\{\ell: |\widehat{\beta}_{\ell}| = 0, \|\widehat{\boldsymbol{\gamma}}_{\ell}\| > 0, 1 \leq \ell \leq p_2 \right\}.
\end{align*}
The spline estimators of each component function are
\begin{equation*}
\widehat{g}_{\ell}\left( x_{\ell}\right) =\sum_{J=1}^{N_n}\widehat{\gamma}
_{J,\ell}B_{J,\ell}^{(1)}\left( x_{\ell}\right)
-n^{-1}\sum_{i=1}^{n}\sum_{J=1}^{N_n}\widehat{\gamma}_{J,\ell}B_{J,\ell}^{(1)}\left( X_{i\ell}\right) .
\end{equation*}
Accordingly, the spline estimators for the original component functions $\phi_{\ell}$'s are $\widehat{\phi}_{\ell}\left( x_{\ell}\right) = \widehat{\beta}_{\ell} x_{\ell} + \widehat{g}_{\ell}\left( x_{\ell}\right)$.
The following theorems establish the asymptotic properties of the adaptive group LASSO estimators. Theorem \ref{THM:selection} shows the proposed method can consistently distinguish nonzero components from zero components. Theorem \ref{THM:consistency} gives the convergence rates of the estimators. We only state the main results here. To facilitate the development of the asymptotic properties, we assume the following sparsity condition:
\begin{enumerate}
\item[(A1)] (\textit{Sparsity}) The numbers of nonzero components $|\mathcal{S}_z|$, $|\mathcal{S}_{x,PL}|$ and $|\mathcal{S}_{x,N}|$ are fixed, and there exist positive constants $c_{\alpha}$, $c_{\beta}$ and $c_g$ such that $\min_{k \in \mathcal{S}_z} |{\alpha}_{0k} | \geq c_{\alpha}$, $\min_{\ell \in \mathcal{S}_{x, PL}} | {\beta}_{0\ell} | \geq c_{\beta}$, and $\min_{\ell \in \mathcal{S}_{x, N}} \Vert g_{0\ell} \Vert_2 \geq c_g$.
\end{enumerate}
\noindent Other regularity conditions and proofs are provided in Appendix \ref{SUBSEC:assump}-- \ref{SUBSEC:KKT}.
\begin{theorem}
\label{THM:selection}
Suppose that Assumptions (A1), (A2)--(A6) in Appendix \ref{SUBSEC:assump} hold. As $n\rightarrow \infty$, we have $\widehat{\mathcal{S}}_z=\mathcal{S}_z$, $\widehat{\mathcal{S}}_{x, PL}=\mathcal{S}_{x, PL}$, $\widehat{\mathcal{S}}_{x, LN}=\mathcal{S}_{x, LN}$ and $\widehat{\mathcal{S}}_{x, PN}=\mathcal{S}_{x, PN}$
with probability approaching one.
\end{theorem}
In the following, to avoid confusion, we use $\boldsymbol{\alpha}_{0}=(\alpha_{01},\ldots, \alpha_{0p_1})^{\top}$, $\boldsymbol{\beta}_{0}=(\beta _{01},\ldots ,$ $\beta _{0p_2})^{\top}$ to denote the true parameters in model (\ref{EQ:aplm}), and $\boldsymbol{g}_{0}=(g_{01},\ldots, g_{0p_2})^{\top}$ to denote the nonlinear functions in model (\ref{EQ:aplm}). Let $\boldsymbol{\alpha}_{0}=(\boldsymbol{\alpha}_{0,\mathcal{S}_z}^{\top},\boldsymbol{\alpha}_{0,\mathcal{N}_z}^{\top})^{\top}$, where $\boldsymbol{\alpha}_{0, \mathcal{S}_z}$ consists of all nonzero components of $\boldsymbol{\alpha }_{0}$, and $\boldsymbol{\alpha}_{0, \mathcal{N}_z}=\boldsymbol{0}$ without loss of generality; similarly, let $\boldsymbol{\beta}_{0}=(\boldsymbol{\beta}_{0,\mathcal{S}_{x, L}}^{\top},\boldsymbol{\beta}_{0,\mathcal{N}_x}^{\top})^{\top}$, where $\boldsymbol{\beta}_{0, \mathcal{S}_{x, L}}$ consists of all nonzero components of $\boldsymbol{\beta}_{0}$, and $\boldsymbol{\beta}_{0, \mathcal{N}_x}=\boldsymbol{0}$ without loss of generality.
\begin{theorem}
\label{THM:consistency}
Suppose that Assumptions (A1), (A2)--(A6) in Appendix \ref{SUBSEC:assump} hold. Then
\begin{eqnarray*}
\sum\limits_{k \in \mathcal{S}_z} \, | \widehat{\alpha}_k - \alpha_{0k} |^2 = O_P\left(n^{-1} N_n \right) + O\left(N_n^{-2}\right) + O_P\left(n^{-2}\sum_{j=1}^{3} \lambda_{nj}^2\right), \\
\sum\limits_{\ell \in \mathcal{S}_{x, L}} |\widehat{\beta}_{\ell} - \beta_{0\ell}|^2 = O_P\left( n^{-1} N_n\right) + O\left(N_n^{-2}\right) + O_P\left(n^{-2}\sum_{j=1}^{3} \lambda_{nj}^2\right), \\
\sum\limits_{\ell \in \mathcal{S}_{x, N}} \, \Vert \widehat{g}_{\ell} - g_{0\ell} \Vert_2^2 = O_P\left( n^{-1} N_n\right) + O\left(N_n^{-2}\right) + O_P\left(n^{-2}\sum_{j=1}^{3} \lambda_{nj}^2\right).
\end{eqnarray*}
\end{theorem}
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\section{Two-stage SBLL Estimator and Inference} \label{sec:nonparmetric}
After model selection, our next step is to conduct statistical inference for the nonparametric component functions of those important variables. Although the one-step penalized estimation in Section \ref{subsec:selection} can quickly identify the nonzero nonlinear components, the asymptotic distribution is not available for the resulting estimators.
To obtain estimators whose asymptotic distribution can be used for inference, we first refit the data using selected model,
\begin{equation}
Y_i = \sum_{k \in \widehat{\mathcal{S}}_z} Z_{ik}{\alpha}_k +
\sum_{j \in \widehat{\mathcal{S}}_{x, PL}} X_{ij}{\beta}_j +
\sum_{\ell \in {\widehat{\mathcal{S}}}_{x, N}} {\phi}_{\ell}\left(
X_{i\ell}\right) + \epsilon_i.
\label{model:aplm_refit}
\end{equation}
We approximate the smooth functions $\left\{\phi _{\ell}\left( \cdot \right): \ell \in {\widehat{\mathcal{S}}}_{x, N} \right\}$ in (\ref{model:aplm_refit}) by polynomial splines introduced in Section \ref{SUBSEC:spline}.
Let $\mathcal{B} _{\ell}^{(d)}$ be the space of polynomial splines of order $d$, and $\mathcal{B} _{\ell}^{0}=\{b\in \mathcal{B}_{\ell}^{(d)}: \mathrm{E}\{b(X_{\ell})\} = 0, ~\mathrm{E}\{b^2(X_{\ell})\} < \infty\}$. Working with $\mathcal{B} _{\ell}^{0}$ ensures that the spline functions are centered, see for example \citet[][]{xue2006additive, wang2007spline, wang2014estimation}. Let $\left\{B_{J,\ell}^{(d)}\left( \cdot \right) \right\} _{j=1}^{M_n}$ be a set of standardized spline basis functions for $\mathcal{B} _{\ell}^{0}$ with dimension $M_n=N_{n}+d-1$, where $B_{J,\ell}^{(d)}(x_{\ell}) = b_{J,\ell}^{(d)}(x_{\ell}) / \|b_{J,\ell}^{(d)}\|_2$, $J = 1, \ldots, M_n$, so that $\mathrm{E} \{B_{J,\ell}^{(d)}(X_{\ell})\} \equiv 0$, $\mathrm{E} \{B_{J,\ell}^{(d)}(X_{\ell})\}^2 \equiv 1$. Specifically, if $d=1$, $M_n = N_n$ and $B_{J,\ell}^{(1)}(\cdot)$ is the standardized piecewise constant spline function defined in (\ref{DEF:BJalpha}).
We propose a one-step backfitting using refitted pilot spline estimators in the first stage followed by local-linear estimators. The refitted coefficients are defined as
\begin{align}
(\widehat{\boldsymbol{\alpha}}^{\ast}, \widehat{\boldsymbol{\beta}}^{\ast}, \widehat{\boldsymbol{\gamma}}^{\ast}) = \underset{{\boldsymbol{\alpha},\boldsymbol{\beta}, \boldsymbol{\gamma}}}{\arg\min}
\sum_{i=1}^n \left(Y_i - \sum_{k \in \widehat{\mathcal{S}}_z}Z_{ik}\alpha_k - \sum_{j \in \widehat{\mathcal{S}}_{x, PL}} X_{ij}\beta_j - \sum_{\ell \in {\widehat{\mathcal{S}}}_{x, N}}\mathbf{B}_{i\ell}^{(d)}\boldsymbol{\gamma}_{\ell} \right)^2.
\label{EQ:refit}
\end{align}
Then the refitted spline estimator for nonlinear functions $\phi_{\ell}(\cdot)$ is
\begin{equation}
\widehat{\phi}^{\ast}_{\ell}\left( x_{\ell}\right) = \mathbf{B}_{\ell}^{(d)}\left( x_{\ell}\right)\widehat{\boldsymbol{\gamma}}^{\ast}_{\ell}, \quad \ell \in {\widehat{\mathcal{S}}}_{x, N}.
\label{DEF:phihatstar}
\end{equation}
Next we establish the asymptotic normal distribution for the parametric estimators. To make $\boldsymbol{\beta}_{0, \mathcal{S}_Z}$ estimable at the $\sqrt{n}$ rate, we need a condition to ensure $\boldsymbol{X}$ and $\boldsymbol{Z}$ are not functionally related. Define $\mathcal{F}_{+}=\left\{f(\boldsymbol{x}) =\sum_{\ell \in \mathcal{S}_{x,N}} f_{\ell}(x_{\ell}),~ \mathrm{E}\{f_{\ell}(X_{\ell})\} =0,~\left \| f_{\ell}\right\| _{2}<\infty \right\} $ as the Hilbert space of theoretically centered $L_{2}$ additive functions. For any $k \in \mathcal{S}_z$, let $z_{k}$ be the coordinate mapping that maps $\boldsymbol{Z}$ to its $k$-th component so that $z_{k}(\boldsymbol{Z})=Z_{k}$, and let $\psi_k^z=\mathrm{argmin}_{\psi\in \mathcal{F}_{+}}\|z_{k}-\psi\|_{2}^{2}=\mathrm{argmin}_{\psi\in \mathcal{F}_{+}}\mathrm{E}\{Z_{k}-\psi(\boldsymbol{X})\}^{2}$ be the orthogonal projection of $z_{k}$ onto $\mathcal{F}_{+}$. Let $\widetilde{\boldsymbol{Z}}_{\mathcal{S}_z}=\left\{\psi_k^z(\boldsymbol{X}), k \in \mathcal{S}_z \right\}^{\top}$. Similarly, for any $\ell \in \mathcal{S}_{x,PL}$, let $x_{\ell}$ be the coordinate mapping that maps $\boldsymbol{X}$ to its $\ell$-th component so that $x_{\ell}(\boldsymbol{X})=X_{\ell}$, and let
\begin{equation}
\psi_{\ell}^x=\mathrm{argmin}_{\psi\in \mathcal{F}_{+}}\|x_{\ell}-\psi\|_{2}^{2}=\mathrm{argmin}_{\psi\in \mathcal{F}_{+}}\mathrm{E}\{X_{\ell}-\psi(\boldsymbol{X})\}^{2}
\label{DEF:psi}
\end{equation}
be the orthogonal projection of $x_{\ell}$ onto $\mathcal{F}_{+}$. Let $\widetilde{\boldsymbol{X}}_{\mathcal{S}_{x, PL}}=\left\{\psi_{\ell}^x(\boldsymbol{X}), \ell \in \mathcal{S}_{x,PL} \right\}^{\top}$. Define $\boldsymbol{Z}_{\mathcal{S}_z} = \left( \boldsymbol{Z}_k, k \in \mathcal{S}_z \right)^{\top}$ and $\boldsymbol{X}_{\mathcal{S}_{x, PL}} = \left( \boldsymbol{X}_{\ell}, \ell \in \mathcal{S}_{x, PL} \right)^{\top}$. Denote vector $\boldsymbol{T}$ and $\widetilde{\boldsymbol{T}}$ as $\boldsymbol{T} = (\boldsymbol{Z}_{\mathcal{S}_z}, \boldsymbol{X}_{\mathcal{S}_{x, PL}})^{\top}$, $\widetilde{\boldsymbol{T}} = \left(\widetilde{\boldsymbol{Z}}_{\mathcal{S}_z}, \widetilde{\boldsymbol{X}}_{\mathcal{S}_{x, PL}} \right)$.
\begin{theorem}
\label{THM:normality}
Under the Assumptions (A1), (A2)--(A6), (A3$^{\prime}$) and (A6$^{\prime}$) in Appendix \ref{SUBSEC:assump},
\[
(n \boldsymbol{\Sigma})^{1/2}
\begin{pmatrix}
\widehat{\boldsymbol{\alpha}}_{\mathcal{S}_z}^{\ast} - \boldsymbol{\alpha}_{0, \mathcal{S}_z} \\
\widehat{\boldsymbol{\beta}}_{\mathcal{S}_{x, PL}}^{\ast} - \boldsymbol{\beta}_{0, \mathcal{S}_{x, PL}}
\end{pmatrix}
\stackrel{D}{
\longrightarrow } \mathcal{N}(\boldsymbol{0}, \mathbf{I}),
\]
where $\mathbf{I}$ is an identity matrix and $\boldsymbol{\Sigma}=\sigma^{-2}{\rm E}[(\boldsymbol{T}-\widetilde{\boldsymbol{T}}) (\boldsymbol{T}-\widetilde{\boldsymbol{T}})^{\top}]$.
\end{theorem}
The proof of Theorem \ref{THM:normality} is similar to the proof of \cite{liu2011estimation} and \cite{li2017ultra} and thus omitted. {Let $\mathbf{Z}_{\mathcal{S}_z}=(Z_{ik},k\in \mathcal{S}_z)_{i=1}^{n}$ and $\mathbf{B}_{\mathcal{S}}^{(d)} = (B_{J,\ell}^{(d)}(X_{i\ell}), 1 \leq \ell \leq p_2$, $\ell \in \mathcal{S}_{x,N}, J = 1, \ldots, N_n)_{i=1}^{n}$.
If $\mathcal{S}_z$ and $\mathcal{S}_x$ are given, $\boldsymbol{\Sigma}$ can be consistently estimated by
$\widehat{\boldsymbol{\Sigma}}_{n}=(n\widehat{\sigma}^{2})^{-1}(\mathbf{Z}_{\mathcal{S}_z}-\widehat{\mathbf{Z}}_{\mathcal{S}_z})^{\top}(\mathbf{Z}_{\mathcal{S}_z}-\widehat{\mathbf{Z}}_{\mathcal{S}_z})$,
where $\widehat{\mathbf{Z}}_{\mathcal{S}_z}^{\top}=\mathbf{Z}_{\mathcal{S}_z}^{\top}\mathbf{B}_{\mathcal{S}}^{(d)}\mathbf{U}_{22}^{-1} \mathbf{B}_{\mathcal{S}}^{(d)\top}$ with $\mathbf{U}_{22}$ given in (\ref{EQN:us}) in the Supplemental Materials and $\widehat{\sigma}^{2}=(n-|\mathcal{S}_z|-|\mathcal{S}_x|)^{-1}\|\mathbf{Y}-\widehat{\mathbf{Y}}\|^{2}$.
In practice, we replace $\mathcal{S}_z$ and $\mathcal{S}_x$ with $\widehat{\mathcal{S}}_z$ and $\widehat{\mathcal{S}}_x$, respectively, to obtain the corresponding estimate.}
Let $\Omega_n = \{\widehat{\mathcal{S}}_z = \mathcal{S}_z, \widehat{\mathcal{S}}_{x, PL} = \mathcal{S}_{x, PL} \}$. In the selection step, we estimate $\mathcal{S}_z$ and $\mathcal{S}_{x, PL}$ consistently, that is, $P\left(\Omega_n\right) \rightarrow 1$. Within the event $\Omega_n$, that is, $\widehat{\mathcal{S}}_z = \mathcal{S}_z$ and $\widehat{\mathcal{S}}_{x, PL} = \mathcal{S}_{x, PL}$,
the estimator $(\widehat{\boldsymbol{\alpha}}_{\mathcal{S}_z}^{\ast\top}, \widehat{\boldsymbol{\beta}}_{\mathcal{S}_{x, PL}}^{\ast\top})^{\top}$ is root-$n$ consistent according to Theorem \ref{THM:normality}.
Since $\Omega_n$ is shown to have probability tending to one, we can conclude that $(\widehat{\boldsymbol{\alpha}}_{\widehat{\mathcal{S}}_z}^{\ast\top}, \widehat{\boldsymbol{\beta}}_{\widehat{\mathcal{S}}_{x, PL}}^{\ast\top})^{\top}$ is also root-$n$ consistent.
These refitted pilot estimators defined in (\ref{EQ:refit}) and (\ref{DEF:phihatstar}) are then used to define new pseudo-responses $\widehat{Y}
_{i\ell}$, which are estimates of the unobservable ``oracle'' responses $Y_{i\ell}$
. Specifically,
\begin{align}
\widehat{Y}_{i\ell}=Y_{i}-\left\{
\sum_{k \in \widehat{\mathcal{S}}_z} Z_{ik}\widehat{\alpha}^{\ast}_k +
\sum_{\ell^{\prime} \in \widehat{\mathcal{S}}_{x, PL}} X_{i\ell^{\prime}}\widehat{\beta}_{\ell^{\prime}}^{\ast} +
\sum_{\ell^{\prime\prime} \in \widehat{\mathcal{S}}_{x, N} \setminus \{\ell\}} \widehat{\phi}_{\ell^{\prime\prime}}^{\ast}\left(
X_{i\ell^{\prime\prime}}\right)\right\} , \nonumber \\
Y_{i\ell}=Y_{i}-\left\{
\sum_{k \in \mathcal{S}_z} Z_{ik}\alpha_{0k} +
\sum_{\ell^{\prime} \in \mathcal{S}_{x, PL}} X_{i\ell^{\prime}}\beta_{0\ell^{\prime}} +
\sum_{\ell^{\prime\prime} \in \mathcal{S}_{x, N} \setminus \{\ell\}} \phi_{0\ell^{\prime\prime}}\left(
X_{i\ell^{\prime\prime}}\right)\right\} .
\label{DEF:YilhatYil}
\end{align}
Denote $K(\cdot)$ a continuous kernel function, and let $K_{h_{\ell}}(t)=K(t/h)/h$ be a rescaling of $K$, where $h$ is usually called the bandwidth.
Next, we define the spline-backfitted local-linear (SBLL) estimator of $\phi_{\ell}\left(
x_{\ell}\right) $ as $\widehat{\phi}_{\ell}^{\mathrm{SBLL}}\left( x_{\ell}\right) $ based on $\left\{
X_{i\ell},\widehat{Y}_{i\ell}\right\} _{i=1}^{n}$, which attempts to mimic the
would-be SBLL estimator $\widehat{\phi}_{\ell}^o\left( x_{\ell}\right) $
of $\phi_{\ell}\left( x_{\ell}\right) $ based on $\left\{X_{i\ell},Y_{i\ell}\right\}
_{i=1}^{n}$ if the unobservable ``oracle'' responses $\left\{ Y_{i\ell}\right\}
_{i=1}^{n}$ were available:
\begin{equation}
\left(\widehat{\phi}_{\ell}^o\left( x_{\ell}\right), \widehat{\phi}_{\ell}^{\mathrm{SBLL}}\left( x_{\ell}\right) \right)= \left(1 ~~ 0\right) \left(\mathbf{X}_{\ell}^{\ast\top}\mathbf{W}_{\ell}\mathbf{X}_{\ell}^{\ast}\right)^{-1} \mathbf{X}_{\ell}^{\ast\top}\mathbf{W}_{\ell} (\mathbf{Y}_{\ell}, \widehat{\mathbf{Y}}_{\ell}),
\label{DEF:alphahat-SBLL}
\end{equation}
where $\mathbf{Y}_{\ell} = \left(Y_{1\ell}, \ldots, Y_{n\ell}\right)^{\top}$ and $\widehat{\mathbf{Y}}_{\ell} = (\widehat{Y}_{1\ell}, \ldots, $ $\widehat{Y}_{n\ell})^{\top}$, with $\widehat{Y}_{i\ell}$ and $Y_{i\ell}$ as defined in (\ref{DEF:YilhatYil}), respectively; and the weight and ``design'' matrices are
\[
\mathbf{W}_{\ell} = n^{-1}\text{diag} \{K_{h_{\ell}}(X_{i\ell} - x_{\ell})\}_{i = 1}^n, \quad \mathbf{X}_{\ell}^{\ast\top} =
\begin{pmatrix}
1 & ,\ldots , & 1 \\
X_{1\ell} - x_{\ell} & , \ldots , & X_{n\ell} - x_{\ell}
\end{pmatrix}.
\]
Asymptotic properties of smoothers of $\widehat{\phi}_{\ell}^o\left( x_{\ell}\right), ~ \ell\in \mathcal{S}_{x, N}$, can be easily established. Specifically, let $\mu_{2}(K)=\int u^{2}K\left( u\right) du$, and let $f_{\ell}$ be the probability density function of $X_{\ell}$, then under Assumptions (B1) and (B2) in Appendix \ref{SUBSEC:assump},
\begin{equation}
\sqrt{nh_{\ell}}\left\{\widehat{\phi}_{\ell}^o\left( x_{\ell}\right) -\phi_{0\ell}(x_{\ell}) -b_{\ell}(x_{\ell}) h_{\ell}^{2}\right\} \stackrel{D}{
\longrightarrow }N\left\{0,v_{\ell}^{2}(x_{\ell}) \right\} , ~ \ell\in \mathcal{S}_{x,N},
\label{EQN:alphao_normal}
\end{equation}
where
\begin{equation}
b_{\ell}(x_{\ell}) = \mu_{2}(K)
\phi_{0\ell}^{\prime \prime }(x_{\ell})/2 , \quad
v_{\ell}^{2}(x_{\ell}) = \|K\|_2^2
f_{\ell}^{-1}(x_{\ell}) \sigma^{2}.
\label{EQ:oracle}
\end{equation}
The following theorem states that the asymptotic uniform magnitude of the difference between $\widehat{\phi}_{\ell}^{\mathrm{SBLL}}\left( x_{\ell}\right) $ and $\widehat{\phi}_{\ell}^o\left( x_{\ell}\right) $ is of order $o_{P}\{(nh_{\ell})^{-1/2}\} $, which is dominated by the asymptotic uniform size of $\widehat{\phi}_{\ell}^o\left( x_{\ell}\right) -\phi_{0\ell}(x_{\ell})$. As a result, $\widehat{\phi}_{\ell}^{\mathrm{SBLL}}\left( x_{\ell}\right) $ will have the same asymptotic distribution as $\widehat{\phi}_{\ell}^o\left( x_{\ell}\right) $. We say $x_{\ell}\in\chi_{\ell}$ is a boundary point if and only if $x_{\ell}=a+ch_{\ell}$ or $x_{\ell}=b-ch_{\ell}$ for some $0\leq c<1$ and an interior point otherwise. Let $\chi_{h_{\ell}}$ be the interior of the support $\chi$.
\begin{theorem}
\label{THM:nonlinear-normality}
Suppose the assumptions in Theorem \ref{THM:normality} hold. In addition, if Assumptions (B1) and (B2) in Appendix \ref{SUBSEC:assump} are satisfied, then the SBLL estimator $\widehat{\phi}_{\ell}^{\mathrm{SBLL}}\left( x_{\ell}\right)$ given in (\ref{DEF:alphahat-SBLL}) satisfies
\begin{equation}
\label{EQN:SBLL_Order}
\sup_{x_{\ell}\in \chi_{h_{\ell}}}\left| \widehat{\phi}_{\ell}^{\mathrm{SBLL}}\left( x_{\ell}\right) -\widehat{\phi}_{\ell}^o\left( x_{\ell}\right) \right| =o_{P}\{(nh_{\ell})^{-1/2}\}, ~ \ell\in \mathcal{S}_{x,N}.
\end{equation}
Hence with $b_{\ell}(x_{\ell}) $ and $v_{\ell}^{2}(x_{\ell})$ as defined in (\ref{EQ:oracle}), for any $x_{\ell}$ in its interior support $x_{\ell} \in \chi_{h_{\ell}}$,
\begin{equation}
\sqrt{nh_{\ell}}\left\{\widehat{\phi}_{\ell}^{\mathrm{SBLL}}\left( x_{\ell}\right) -\phi_{0\ell}(x_{\ell}) -b_{\ell}(x_{\ell}) h_{\ell}^{2}\right\} \stackrel{D}{\longrightarrow }\mathcal{N} \left\{0,v_{\ell}^{2}(x_{\ell}) \right\} , ~ \ell\in \mathcal{S}_{x,N}.
\label{EQN:sbll_normal}
\end{equation}
In addition, the estimator $\widehat{\phi}_{\ell}^{\mathrm{SBLL}}\left( x_{\ell}\right)$ satisfies, for any $t$ and $ \ell\in \mathcal{S}_{x,N}$,
\begin{align}
\lim_{n \rightarrow \infty} & \Pr\left\{\sqrt{\ln(h_{\ell}^{-2})} \left(\sup_{x_{\ell} \in \chi_{h_{\ell}}} \frac{\sqrt{nh_{\ell}}}{v_{\ell}(x_{\ell})} |\widehat{\phi}_{\ell}^{\mathrm{SBLL}}\left( x_{\ell}\right) -\phi_{0\ell}(x_{\ell})| - \tau_n\right) < t\right\}
= e^{-2 e^{-t}},
\label{EQN:scb_form}
\end{align}
where $\tau_n = \sqrt{\ln(h_{\ell}^{-2})} + \ln\{\|K^{\prime}\|_2 / (2\pi \|K\|_2)\} / \sqrt{\ln(h_{\ell}^{-2})}$.
\end{theorem}
Theorem \ref{THM:nonlinear-normality} provides analytical expressions for constructing asymptotic confidence intervals and SCBs under certain conditions. Under Assumptions (A1)--(A6), (A3$^{\prime}$), (A6$^{\prime}$), (B1) and (B2) in Appendix \ref{SUBSEC:assump}, for any $\alpha\in(0,1)$, an asymptotic $100(1-\alpha)\%$ pointwise confidence interval for $\phi_{0\ell}(x_{\ell})$ over the interval $\chi_{h_{\ell}}$ is
\[
\widehat{\phi}_{\ell}^{\mathrm{SBLL}}\left( x_{\ell}\right) -\widehat{b}_{\ell}(x_{\ell})h_{\ell}^{2}\pm \widehat{v}_{\ell}(x_{\ell})(nh_{\ell})^{-1/2}, ~ \ell\in \mathcal{S}_{x,N}.
\]
Under Assumptions (A1)--(A6), (A2$^\prime$) (A3$^{\prime}$), (A6$^{\prime}$), (B1) and (B2) in the Appendix, for any $\alpha\in(0,1)$, an asymptotic $100(1-\alpha)\%$ SCB for $\phi_{0\ell}(x_{\ell})$ over the interval $\chi_{h_{\ell}}$ is
\[
\widehat{\phi}_{\ell}^{\mathrm{SBLL}}\left( x_{\ell}\right) \pm \widehat{v}_{\ell}(x_{\ell})(nh_{\ell})^{-1/2}\left[\tau_n - \{\ln(h_{\ell}^{-2})\}^{-1/2} \ln\left\{-\frac{1}{2} \ln(1 - \alpha)\right\} \right], ~ \ell\in \mathcal{S}_{x,N}.
\]
\setcounter{chapter}{4}
\renewcommand{E.\arabic{section}}{\arabic{section}}
\renewcommand{0}{0}
\renewcommand{0}{0}
\renewcommand{E.\arabic{subsection}}{4.\arabic{subsection}}
\setcounter{section}{3}
\section{Implementation and Simulation} \label{sec:simulation}
In this section we discuss practical implementations for the SMILE procedure.
To meet the zero mean requirement specified in Assumption (A4),
we use the centralized $X_{i\ell}^{\ast}$ instead of $X_{i\ell}$ directly, for each $\ell=1, \ldots, p_2$.
At the risk of abusing the notation, we still use symbol $X$ instead of $X^{\ast}$ to avoid creating too many new symbols. To implement the proposed procedure, one needs to select the penalty parameters, the knots for a spline at the selection stage and refitting stage, and the bandwidth for a kernel at the backfitting stage.
\textit{Knot selection.} For spline smoothing involved in both selection and refitting, we suggest placing knots on a grid of evenly spaced sample quantiles. Based on extensive simulation experiments in Section A of the Supplementary Materials, we find that the number of knots often has little effect on the model selection results. Therefore, we recommend using a small number of knots at the model selection stage to reduce the computing cost, especially when the sample size is too small compared to the number of covariates. In practice, $2\sim 5$ interior knots is usually adequate to identify the model structure.
At the refitting stage, Assumption (A6$^\prime$) in the Supplementary Materials suggests the number of interior knots $M_n$ for a refitting spline needs to satisfy: $\{n^{1/(2d)} \vee n^{4/(10d-5)}\} \ll M_n \ll n^{1/3}$, where $d$ is the degree of the polynomial spline basis functions used in the refitting. The widely used quadratic/cubic splines and any polynomial splines of degree $d\geq 2$ all satisfy this condition. Therefore, in practice we suggest take the following rule-of-thumb number of interior knots
\begin{equation*}
\min\{\lfloor n^{1/(2d) \vee 4/(10d-5)}\ln(n)\rfloor, \lfloor n/(4s)\rfloor\}+1,
\end{equation*}
where $s$ is the number of nonlinear components selected at the first stage, and the term $\lfloor n/(4s)\rfloor$ is to guarantee that we have at least four observations in each subinterval between two adjacent knots to avoid getting (near) singular design matrices in the spline refitting.
\textit{Bandwidth selection.} Note that Condition (B2) in the Supplementary Materials requires that the bandwidths in the backfitting are of order $n^{-1/5}$. Thus, the bandwidth selection can be done using a standard routine in the literature. In our numerical studies, we find that the rule-of-thumb bandwidth selector \citep{Fan:Gijbels:96} often works very well in both estimation and SCB construction.
Section A in the Supplementary Materials provides detailed investigations on how the smoothing parameters affect the proposed SMILE method and evaluates the practical performance in finite-sample simulation studies. Next we present our algorithm and discuss how to choose the penalty parameters.
\subsection{Algorithm} \label{subsec:algorithm}
In this section we discuss practical implementations for the SMILE procedure.
To meet the zero mean requirement specified in Assumption (A4),
we use the centralized $X_{i\ell}^{\ast}$ instead of $X_{i\ell}$ directly, for each $\ell=1, \ldots, p_2$.
At the risk of abusing the notation, we still use symbol $X$ instead of $X^{\ast}$ to avoid creating too many new symbols. To implement the proposed procedure, one needs to select the penalty parameters, the knots for a spline at the selection stage and refitting stage, and the bandwidth for a kernel at the backfitting stage.
\textit{Knot selection.} For spline smoothing involved in both selection and refitting, we suggest placing knots on a grid of evenly spaced sample quantiles. Based on extensive simulation experiments in Section A of the Appendix, we find that the number of knots often has little effect on the model selection results. Therefore, we recommend using a small number of knots at the model selection stage to reduce the computing cost, especially when the sample size is small compared to the number of covariates. In practice, $2\sim 5$ interior knots is usually adequate to identify the model structure.
At the refitting stage, Assumption (A6$^\prime$) in the Appendix suggests the number of interior knots $M_n$ for a refitting spline needs to satisfy: $\{n^{1/(2d)} \vee n^{4/(10d-5)}\} \ll M_n \ll n^{1/3}$, where $d$ is the degree of the polynomial spline basis functions used in the refitting. The widely used quadratic/cubic splines and any polynomial splines of degree $d\geq 2$ all satisfy this condition. Therefore, in practice we suggest take the following rule-of-thumb number of interior knots
\begin{equation*}
\min\{\lfloor n^{1/(2d) \vee 4/(10d-5)}\ln(n)\rfloor, \lfloor n/(4s)\rfloor\}+1,
\end{equation*}
where $s$ is the number of nonlinear components selected at the first stage, and the term $\lfloor n/(4s)\rfloor$ is to guarantee that we have at least four observations in each subinterval between two adjacent knots to avoid (near) singular design matrices in the spline refitting.
\textit{Bandwidth selection.} Note that Condition (B2) in the Appendix requires that the bandwidths in the backfitting are of order $n^{-1/5}$. Thus, the bandwidth selection can be done using a standard routine in the literature. In our numerical studies, we find that the rule-of-thumb bandwidth selector \citep{Fan:Gijbels:96} often works very well in both estimation and SCB construction.
Section A in the Appendix provides detailed investigations on how the smoothing parameters affect the proposed SMILE method and evaluates the practical performance in finite-sample simulation studies. Next we present our algorithm and discuss how to choose the penalty parameters.
\subsection{Algorithm} \label{subsec:algorithm}
The minimization of (\ref{EQ:loss_aglasso_spline}) can be solved by the group coordinate descent algorithm \citep{huang2012selective}, implemented using \textsf{R} package \texttt{grpreg} \citep{grpreg}. As for the selection of penalty parameters, we consider two criteria widely used in high-dimensional settings, modified Bayesian information criteria \citep[BIC; see][]{lee2014model} and the extended BIC \citep[EBIC; see][]{chen2008extended, chen2009tournament}:
\begin{align*}
\text{BIC}(\boldsymbol{\lambda}) &= \ln(RSS_{\boldsymbol{\lambda}}) + df_{\boldsymbol{\lambda}} \times \frac{\ln (p_1 + p_2 + p_2N_n) \times \ln(n)}{2n}, \\
\text{EBIC}(\boldsymbol{\lambda}) &= \ln(RSS_{\boldsymbol{\lambda}}) + df_{\boldsymbol{\lambda}} \times \frac{\ln(n)}{n} + df_{\boldsymbol{\lambda}} \times \frac{\ln (p_1 + p_2 + p_2N_n)}{n},
\end{align*}
where $RSS_{\boldsymbol{\lambda}}$ is the residual sum of squares associated with penalty parameters $\boldsymbol{\lambda} = (\lambda_1, \lambda_2,$ $\lambda_3)^{\top}$ and $df_{\boldsymbol{\lambda}}$ is the number of estimated nonzero coefficients for the given $\boldsymbol{\lambda}$. The simulation results are similar based on these two criteria, so in the following, we choose $\lambda_1$ and $\lambda_2$ by modified BIC and $\lambda_3$ by EBIC for illustration using an approach described below.
The classical coordinate descent algorithm deals with the optimization problem with one tuning parameter, and there are several ways to address the triple-penalization or multiple-penalization issue. A natural idea is to solve the optimization problem by searching over a three-dimensional grid for tuning parameters, which can be computationally expensive.
To pose a balance between computational efficiency and precision, we propose to solve the triple-penalization problem in two steps. In the first step, BIC is minimized with a common smoothing parameter $\lambda$, i.e., we set $\lambda_1=\lambda_2=\lambda_3=\lambda$, and we choose $\lambda$ by minimizing BIC($\lambda$) over a grid of $\lambda$ values. Using the selected common smoothing parameter, we obtain the initial estimators $\widehat{\boldsymbol{\alpha}}^{(0)}$, $\widehat{\boldsymbol{\beta}}^{(0)}$ and $\widehat{\boldsymbol{\gamma}}^{(0)}$. In Step 2, $\boldsymbol{\alpha}$, $\boldsymbol{\beta}$ and $\boldsymbol{\gamma}$ estimates are obtained one at a time by minimizing (\ref{EQ:loss_aglasso_spline}). More precisely, an $\boldsymbol{\alpha}$ estimate is obtained with $\boldsymbol{\beta}$, $\boldsymbol{\gamma}$ fixed at current estimates, where $\lambda_1$ is set equal to its minimum BIC value and $\lambda_2=\lambda_3=0$. One cycles in this way through $\boldsymbol{\alpha}$, $\boldsymbol{\beta}$ and $\boldsymbol{\gamma}$ estimation steps for a fixed number of iterations. Three iterations generally works well in practice. Algorithm 1 outlines the iterative group coordinate descent algorithm.
\normalem
\begin{algorithm}
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\caption{Iterative group coordinate descent algorithm}
\Input{Data $\left\{(Y_i, Z_{i1},\ldots,Z_{ip_1}, X_{i1},\ldots,X_{ip_2},\mathbf{B}_{i1}^{(1)},\ldots,\mathbf{B}_{ip_2}^{(1)})\right\}_{i=1}^n$\\
$\widehat{\boldsymbol{\alpha}}^{(0)}$, $\widehat{\boldsymbol{\beta}}^{(0)}$ and $\widehat{\boldsymbol{\gamma}}^{(0)}$: initial parameters of interest \\
$\delta_0$: convergence criterion\\}
\Output{$\widehat{\boldsymbol{\alpha}}$, $\widehat{\boldsymbol{\beta}}$ and $\widehat{\boldsymbol{\gamma}}$: Estimates of $\boldsymbol{\alpha}$, $\boldsymbol{\beta}$ and $\boldsymbol{\gamma}$}
\BlankLine
\While{$\left\|\left(\widehat{\boldsymbol{\alpha}}^{(m+1)\top}, \widehat{\boldsymbol{\beta}}^{(m+1)\top}, \widehat{\boldsymbol{\gamma}}^{(m+1)\top}\right)^{\top}-\left(\widehat{\boldsymbol{\alpha}}^{(m)\top}, \widehat{\boldsymbol{\beta}}^{(m)\top}, \widehat{\boldsymbol{\gamma}}^{(m)\top}\right)^{\top}\right\|^2 > \delta_0$}{
(i) Given $\widehat{\boldsymbol{\beta}}^{(m)}$ and $\widehat{\boldsymbol{\gamma}}^{(m)}$, obtain $w_1^{\alpha\,(m+1)},\ldots, w_{p_1}^{\alpha\,(m+1)}$ by minimizing objective function (\ref{EQ:loss_glasso_spline}) with $\widetilde\lambda_1$ selected via the modified BIC;\\
(ii) Given $\widehat{\boldsymbol{\beta}}^{(m)}$, $\widehat{\boldsymbol{\gamma}}^{(m)}$ and $w_1^{\alpha\,(m+1)},\ldots, w_{p_1}^{\alpha\,(m+1)}$, obtain $\widehat{\boldsymbol{\alpha}}^{(m+1)}$ by minimizing objective function (\ref{EQ:loss_aglasso_spline}) with $\lambda_1$ selected via the modified BIC;\\
(iii) Given $\widehat{\boldsymbol{\alpha}}^{(m+1)}$ and $\widehat{\boldsymbol{\gamma}}^{(m)}$, obtain $w_1^{\beta\,(m+1)},\ldots, w_{p_2}^{\beta\,(m+1)}$ by minimizing objective function (\ref{EQ:loss_glasso_spline}) with $\widetilde\lambda_2$ selected via the modified BIC;\\
(iv) Given $\widehat{\boldsymbol{\alpha}}^{(m+1)}$, $\widehat{\boldsymbol{\gamma}}^{(m)}$ and $w_1^{\beta\,(m+1)},\ldots, w_{p_2}^{\beta\,(m+1)}$, obtain $\widehat{\boldsymbol{\beta}}^{(m+1)}$ by minimizing objective function (\ref{EQ:loss_aglasso_spline}) with $\lambda_2$ selected via the modified BIC;\\
(v) Given $\widehat{\boldsymbol{\alpha}}^{(m+1)}$ and $\widehat{\boldsymbol{\beta}}^{(m+1)}$, obtain $w_1^{\gamma\,(m+1)},\ldots, w_{p_2}^{\gamma\,(m+1)}$ by minimizing objective function (\ref{EQ:loss_glasso_spline}) with $\widetilde\lambda_3$ selected via EBIC;\\
(vi) Given $\widehat{\boldsymbol{\alpha}}^{(m+1)}$, $\widehat{\boldsymbol{\beta}}^{(m+1)}$ and $w_1^{\gamma\,(m+1)},\ldots, w_{p_2}^{\gamma\,(m+1)}$, obtain $\widehat{\boldsymbol{\gamma}}^{(m+1)}$ by minimizing objective function (\ref{EQ:loss_aglasso_spline}) with $\lambda_3$ selected via EBIC.}
\vskip -.1in
Set $\widehat{\boldsymbol{\alpha}}=\widehat{\boldsymbol{\alpha}}^{(m+1)}$, $\widehat{\boldsymbol{\beta}}=\widehat{\boldsymbol{\beta}}^{(m+1)}$ and $\widehat{\boldsymbol{\gamma}}=\widehat{\boldsymbol{\gamma}}^{(m+1)}$.
\label{ALGO:igcd}
\end{algorithm}
\ULforem
\subsection{Simulation Studies}
In this section, we investigate the performance of the proposed sparse model identification and learning estimator, abbreviated as SMILE, in terms of model selection, estimation accuracy and inference performance in a simulation study. We compare SMILE with the sparse APLM estimator with adaptive group LASSO penalty (SAPLM) proposed in \cite{li2017ultra}, the ordinary linear least squares estimator with the adaptive LASSO penalty (SLM), and the oracle estimator (ORACLE), which uses the same estimation techniques as SMILE except that no penalization or data-driven variable selection is used because all active and inactive index sets are treated as known. Note that SAPLM ignores the potential linear structure in covariate $\boldsymbol{X}$, and estimates the effects of each component of $\boldsymbol{X}$ with all nonparametric forms; in contrast, SLM ignores the potential nonlinear structure in covariate $\boldsymbol{X}$ and requires selected components of covariates $\boldsymbol{Z}$ and $\boldsymbol{X}$ to enter the model in a linear form. In terms of the performances of SCBs, we compare SMILE with SAPLM and ORACLE. In our simulation, ORACLE works as a benchmark for estimation comparison. It is worth pointing out that the ORACLE estimator is only computable in simulations, not real examples.
We generate simulated datasets using the APLM structure
\begin{align*}
Y_i = \sum_{k=1}^{p_1}Z_{ik}\alpha_k + \sum_{\ell=1}^{p_2} \phi_{\ell}(X_{i\ell}) + \varepsilon_i,
\end{align*}
where $\alpha_1 = 3$, $\alpha_2 = 4$, $\alpha_3 = -2$, $\alpha_4 = \ldots = \alpha_{p_1} = 0$,
$\phi_1(x) = 9x$, $\phi_2(x) = -1.5 \cos^2(\pi x) + 3 \sin^2(\pi x) - \mathrm{E}\{-1.5 \cos^2(\pi X_2 + 3 \sin^2(\pi X_2)\}$, $\phi_3(x) = 6x + 18x^{2} - \mathrm{E}(6X_3 + 18X_3^{2})$, and $\phi_4(x) = \ldots = \phi_{p_2}(x) = 0$.
Notice that $\phi_1(x)$ is actually a linear function. So there are three variables in the active index set for $\boldsymbol{Z}$, one variable in the active pure linear index set for $\boldsymbol{X}$, one variable in the active pure nonlinear index set for $\boldsymbol{X}$, and one variable in the active linear \& nonlinear index set for $\boldsymbol{X}$.
We simulate $Z_{ik}^{\ast}$ independently from the $\mathrm{Unif}[0, 1]$ and $X_{i\ell}$ independently from the $\mathrm{Unif}[-.5,$ $.5]$, and set $Z_{ik} = I(Z_{ik}^{\ast} > 0.75)$, for $i = 1, \ldots, n$, $k = 1, \ldots, p_1$, $\ell=1, \ldots, p_2$.
To make an ultra-high-dimensional scenario, we let the sample size $n = 300$ and $n = 500$, and consider three different dimensions: $p_1= p_2= p$, where $p$ is taken to be $1000$, $2000$ and $5000$. The error term $\varepsilon_i$ is simulated from $\mathcal{N}(0, \sigma^{2})$ with $\sigma= 0.5$ and $1.0$.
To approximate the nonlinear functions, we use the constant B-spline ($d=1$) with four interior knots for selection and use the cubic B-spline ($d=4$) with four interior knots in the refitting step. For both selection and refitting, the knots are on a grid of evenly spaced sample quantiles. To construct the SCBs, in our simulation studies below, we choose the Epanechnikov kernel function with the rule-of-thumb bandwidth described in Section 4.2 in \cite{Fan:Gijbels:96}, which usually works well in our experimental investigation. More simulation studies have been conducted with different choices for spline knots and kernel bandwidth selectors; see Section A of the Appendix.
We evaluate the methods on the accuracy of variable selection, prediction and inference. In detail, we adopt the following criteria for evaluation:
\begin{enumerate} [{\normalfont (B-i)}]
\item Percent of covariates in $\boldsymbol{Z}$ with nonzero linear coefficients that are correctly identified (``CorrZ''); \\[-18pt]
\item Percent of covariates in $\boldsymbol{Z}$ with zero linear coefficients that are correctly identified (``CorrZ0'');\\[-18pt]
\item Percent of covariates in $\boldsymbol{X}$ with nonzero purely linear functions that are correctly identified (``CorrL'');\\[-18pt]
\item Percent of covariates in $\boldsymbol{X}$ with nonzero purely nonlinear functions that are correctly identified (``CorrN''); \\[-18pt]
\item Percent of covariates in $\boldsymbol{X}$ with nonzero linear and nonlinear functions that are correctly identified (``CorrLN'); \\[-18pt]
\item Percent of covariates in $\boldsymbol{X}$ with zero functions that are correctly identified (``CorrX0'');
\end{enumerate}
\begin{enumerate} [{\normalfont (C-i)}]
\item Percent of covariates in $\boldsymbol{Z}$ with nonzero linear coefficients incorrectly identified as having zero linear coefficients (``Zto0'');\\[-18pt]
\item Percent of covariates in $\boldsymbol{X}$ with nonzero purely linear functions incorrectly identified as having nonlinear functions (``LtoN''); \\[-18pt]
\item Percent of covariates in $\boldsymbol{X}$ with nonzero purely nonlinear functions incorrectly identified as having linear functions (``NtoL''); \\[-18pt]
\item Percent of covariates in $\boldsymbol{X}$ with nonzero linear or nonzero nonlinear functions incorrectly identified as having both zero linear and zero nonlinear functions (``Xto0'');
\end{enumerate}
\begin{enumerate} [{\normalfont (D-i)}]
\item Mean squared errors (MSE) for linear coefficients $\alpha_1$, $\alpha_2$, $\alpha_3$ and $\beta_1$; \\[-18pt]
\item Average MSE (AMSE) for $\phi_1$, $\phi_2$ and $\phi_3$, defined as $n^{-1} \sum_{i = 1}^n \{\widehat{\phi}_{\ell}^{\mathrm{SBLL}}(x_{i\ell}) - \phi_{\ell}(x_{i\ell})\}^2$;\\[-18pt]
\item 10-fold cross-validation mean squared prediction error (CV-MSPE) for the response variable, defined as $10^{-1}\sum_{m = 1}^{10} |\kappa_m|^{-1}\sum_{i \in \kappa_{m}} (\widehat{Y}_i - Y_i)^2$, where $\kappa_1, \ldots, \kappa_{10}$ comprise a random partition of the dataset into $10$ disjoint subsets of approximately equal size, and $\widehat{Y}_i$ is the prediction obtained from all data aside from the subset containing the $i$th observation;\\[-18pt]
\item The coverage rates of the proposed 95\% SCB for functions $\phi_2$ and $\phi_3$ (Coverage).
\end{enumerate}
All these performance measures are computed based on 1000 replicates. Note that Criteria (B-i)--(B-vi) measure the frequency of getting the correct model structure; Criteria (C-i)--(C-iv) measure the frequency of getting an incorrect model structure; Criteria (D-i)--(D-iii) focus on the estimation and prediction accuracy for the model components; and Criterion (D-iv) measures the inferential performance.
The model selection results are provided in Tables \ref{TAB:SIM1-selection1} and \ref{TAB:SIM1-selection2}, respectively. SMILE can effectively identify informative linear and nonlinear components as well as correctly discover the linear and nonlinear structure in covariate $\boldsymbol{X}$, while SAPLM neglects linear structure in $\boldsymbol{X}$ and SLM fails in representing the nonlinear part of covariate $\boldsymbol{X}$. For SMILE, the numbers of correctly selected nonzero covariates in $\boldsymbol{Z}$, linear, nonlinear, linear-and-nonlinear components in $\boldsymbol{X}$, nonzero covariates are very close to ORACLE (100\% for corrZ, corrL, corrN, corrLN, corrZ0 and corrX0, respectively); and the numbers of incorrectly identified components approach to $0$ as the sample size $n$ increases, as shown in Table \ref{TAB:SIM1-selection2}. SMILE is close in the selection of covariates $\boldsymbol{Z}$ to the SAPLM estimator, and it far outperforms SAPLM in identifying the linear-and-nonlinear structure of covariate $\boldsymbol{X}$. From the results in Tables \ref{TAB:SIM1-selection1} and \ref{TAB:SIM1-selection2}, it is also evident that model misspecification leads to poor variable selection performance for SLM. Especially for the selection of covariates in $\boldsymbol{X}$, which is our main focus for real data analysis, SLM fails to select the right nonlinear components in each simulation.
\begin{table}[htbp]
\caption{Statistics (B-i)--(B-vi) comparing the SMILE, SAPLM and SLM.}
\label{TAB:SIM1-selection1}
\renewcommand{0.92}{0.9}
\begin{center}
\begin{tabular}{ccclrrrrrr}
\hline \hline
Size & Noise & & & \multicolumn{2}{c}{Z Part} & \multicolumn{ 4}{c}{X Part} \\
\cmidrule(r){5-6} \cmidrule(l){7-10}
$n$ & sig & $p$ & Method & corrZ & corrZ0 & corrL & corrN & corrLN & corrX0 \\
\hline
300 & 0.5 & 1000 & SMILE & 100 & 99.99960 & 100 & 100 & 100 & 99.99940 \\
& & & SAPLM & 100 & 100 & 0 & 100 & 0 & 100 \\
& & & SLM & 98.6 & 99.99920 & 100 & 0 & 0 & 99.99850 \\
& & 2000 & SMILE & 100 & 99.99995 & 100 & 100 & 100 & 99.99985 \\
& & & SAPLM & 100 & 100 & 0 & 100 & 0 & 100 \\
& & & SLM & 97.3 & 99.99950 & 100 & 0 & 0 & 99.99915 \\
& & 5000 & SMILE & 100 & 99.99996 & 100 & 100 & 100 & 100 \\
& & & SAPLM & 100 & 100 & 0 & 100 & 0 & 100 \\
& & & SLM & 96.63333 & 99.99988 & 100 & 0 & 0 & 99.99974 \\
& 1.0 & 1000 & SMILE & 100 & 99.99920 & 100 & 100 & 100 & 99.99990 \\
& & & SAPLM & 100 & 99.99920 & 0 & 100 & 0 & 100 \\
& & & SLM & 96.56667 & 99.99799 & 100 & 0 & 0 & 99.99719 \\
& & 2000 & SMILE & 99.93333 & 99.99995 & 100 & 99.8 & 99.8 & 99.99975 \\
& & & SAPLM & 100 & 99.99970 & 0 & 100 & 0 & 100 \\
& & & SLM & 95.7 & 99.99975 & 100 & 0 & 0 & 99.99905 \\
& & 5000 & SMILE & 99.86667 & 99.99996 & 100 & 99.5 & 99.5 & 99.99996 \\
& & & SAPLM & 100 & 99.99990 & 0 & 100 & 0 & 100 \\
& & & SLM & 93.73333 & 99.99982 & 100 & 0 & 0 & 99.99978 \\
500 & 0.5 & 1000 & SMILE & 100 & 99.99990 & 100 & 100 & 100 & 99.99980 \\
& & & SAPLM & 100 & 100 & 0 & 100 & 0 & 100 \\
& & & SLM & 100 & 99.99990 & 100 & 0 & 0 & 99.99960 \\
& & 2000 & SMILE & 100 & 99.99995 & 100 & 100 & 100 & 100 \\
& & & SAPLM & 100 & 100 & 0 & 100 & 0 & 100 \\
& & & SLM & 100 & 99.99985 & 100 & 0 & 0 & 99.99985 \\
& & 5000 & SMILE & 100 & 99.99996 & 100 & 100 & 100 & 100 \\
& & & SAPLM & 100 & 100 & 0 & 100 & 0 & 100 \\
& & & SLM & 99.96667 & 99.99994 & 100 & 0 & 0 & 99.99994 \\
& 1.0 & 1000 & SMILE & 100 & 99.99950 & 100 & 100 & 100 & 99.99970 \\
& & & SAPLM & 100 & 100 & 0 & 100 & 0 & 100 \\
& & & SLM & 99.96667 & 99.99940 & 100 & 0 & 0 & 99.99930 \\
& & 2000 & SMILE & 100 & 99.99980 & 100 & 100 & 100 & 99.99990 \\
& & & SAPLM & 100 & 99.99990 & 0 & 100 & 0 & 100 \\
& & & SLM & 99.93333 & 99.99990 & 100 & 0 & 0 & 99.99960 \\
& & 5000 & SMILE & 100 & 99.99994 & 100 & 100 & 100 & 100 \\
& & & SAPLM & 100 & 100 & 0 & 100 & 0 & 100 \\
& & & SLM & 99.76667 & 100 & 100 & 0 & 0 & 99.99994 \\
\hline \hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[htbp]
\caption{Statistics (C-i)--(C-iv) comparing the SMILE, SAPLM and SLM.}
\label{TAB:SIM1-selection2}
\renewcommand{0.92}{0.9}
\begin{center}
\begin{tabular}{ccclrrrr}
\hline \hline
Size & Noise & & & Z Part & \multicolumn{3}{c}{X Part} \\
\cmidrule(r){5-5} \cmidrule(l){6-8}
$n$ & sig & $p$ & Method & Zto0 & LtoN & NtoL & Xto0 \\
\hline
300 & 0.5 & 1000 & SMILE & 0 & 0 & 0 & 0 \\
& & & SAPLM & 0 & 100 & 0 & 0 \\
& & & SLM & 1.4 & 0 & 100 & 33.33333 \\
& & 2000 & SMILE & 0 & 0 & 0 & 0 \\
& & & SAPLM & 0 & 100 & 0 & 0 \\
& & & SLM & 2.7 & 0 & 100 & 33.33333 \\
& & 5000 & SMILE & 0 & 0 & 0 & 0 \\
& & & SAPLM & 0 & 100 & 0 & 0 \\
& & & SLM & 3.36667 & 0 & 100 & 33.33333 \\
& 1.0 & 1000 & SMILE & 0 & 0 & 0 & 0 \\
& & & SAPLM & 0 & 100 & 0 & 0 \\
& & & SLM & 3.43333 & 0 & 100 & 33.33333 \\
& & 2000 & SMILE & 0.06667 & 0 & 0 & 0.06667 \\
& & & SAPLM & 0 & 100 & 0 & 0 \\
& & & SLM & 4.3 & 0 & 100 & 33.33333 \\
& & 5000 & SMILE & 0.13333 & 0 & 0 & 0.16667 \\
& & & SAPLM & 0 & 100 & 0 & 0 \\
& & & SLM & 6.26667 & 0 & 100 & 33.33333 \\
500 & 0.5 & 1000 & SMILE & 0 & 0 & 0 & 0 \\
& & & SAPLM & 0 & 100 & 0 & 0 \\
& & & SLM & 0 & 0 & 100 & 33.33333 \\
& & 2000 & SMILE & 0 & 0 & 0 & 0 \\
& & & SAPLM & 0 & 100 & 0 & 0 \\
& & & SLM & 0 & 0 & 100 & 33.33333 \\
& & 5000 & SMILE & 0 & 0 & 0 & 0 \\
& & & SAPLM & 0 & 100 & 0 & 0 \\
& & & SLM & 0.03333 & 0 & 100 & 33.33333 \\
& 1.0 & 1000 & SMILE & 0 & 0 & 0 & 0 \\
& & & SAPLM & 0 & 100 & 0 & 0 \\
& & & SLM & 0.03333 & 0 & 100 & 33.33333 \\
& & 2000 & SMILE & 0 & 0 & 0 & 0 \\
& & & SAPLM & 0 & 100 & 0 & 0 \\
& & & SLM & 0.06667 & 0 & 100 & 33.33333 \\
& & 5000 & SMILE & 0 & 0 & 0 & 0 \\
& & & SAPLM & 0 & 100 & 0 & 0 \\
& & & SLM & 0.23333 & 0 & 100 & 33.33333 \\
\hline \hline
\end{tabular}
\end{center}
\end{table}
The estimation and prediction results are displayed in Table \ref{TAB:SIM1-estimation}. Specifically, we present the MSEs for linear coefficients $\alpha_1$, $\alpha_2$, $\alpha_3$ and $\beta_1$ and AMSEs for functions $\phi_1$, $\phi_2$ and $\phi_3$ and the CV-MSPEs for predicting $Y$. The case with known active covariates (ORACLE) is also reported in each setting and serves as a gold standard. SMILE performs the best in predicting $Y$ and estimating the coefficients of covariates $\boldsymbol{Z}$, as indicated by CV-MSPE and MSEs for $\alpha_1$, $\alpha_2$ and $\alpha_3$ that are closest to ORACLE in most simulation settings, while SLM is much higher (around 2 $\sim$ 18 times higher). As for the linear structure in $\boldsymbol{X}$, as shown in MSE for $\beta_1$ and AMSE for $\phi_1$, the performance of SMILE is comparable to SAPLM and SLM, even though restricted to the selection bias; as the sample size $n$ increases, the performance of SMILE is perfect and matches with ORACLE. Note that the SAPLM estimator is incapable in estimating $\beta_1$ in this case. The estimation of nonlinear functions $\phi_2$ and $\phi_3$ is also good for SMILE, and matches with ORACLE as sample size $n$ increases. The inferior performance of SAPLM and the poor performance of SLM, in both estimation and prediction, illustrates the importance and necessity of identifying correct model structure.
\begin{table}[htbp]
\small
\caption{Estimation results comparing the ORACLE, SMILE, SAPLM and SLM.}
\label{TAB:SIM1-estimation}
\begin{center}
\renewcommand{0.92}{0.725}
\begin{tabular}{ccclrrrrrrrr}
\hline \hline
& & & & \multicolumn{4}{c}{MSE $(\times 10^{-2})$} & \multicolumn{3}{c}{AMSE $(\times 10^{-2})$} & CV- \\
\cmidrule(r){5-8} \cmidrule(l){9-11}
$n$ & $\sigma$ & $p$ & Method & $\alpha_1$ & $\alpha_2$ & $\alpha_3$ & $\beta_1$ & $\phi_1$ & $\phi_2$ & $\phi_3$ & MSPE \\
\hline
300 &\!\!\! 0.5 &\!\!\! 1000 &\!\!\! ORACLE &\!\!\! 0.47 &\!\!\! 0.48 &\!\!\! 0.50 &\!\!\! 1.06 &\!\!\! 0.09 &\!\!\! 0.98 &\!\!\! 0.83 &\!\!\! 0.28 \\
&\!\!\! &\!\!\! &\!\!\! SMILE &\!\!\! 0.47 &\!\!\! 0.48 &\!\!\! 0.50 &\!\!\! 1.06 &\!\!\! 0.11 &\!\!\! 0.94 &\!\!\! 0.77 &\!\!\! 0.28 \\
&\!\!\! &\!\!\! &\!\!\! SAPLM &\!\!\! 0.49 &\!\!\! 0.48 &\!\!\! 0.54 &\!\!\! - &\!\!\! 0.41 &\!\!\! 0.99 &\!\!\! 0.83 &\!\!\! 0.28 \\
&\!\!\! &\!\!\! &\!\!\! SLM &\!\!\! 9.13 &\!\!\! 8.86 &\!\!\! 25.99 &\!\!\! 18.65 &\!\!\! 1.63 &\!\!\! 253.82 &\!\!\! 178.85 &\!\!\! 4.79 \\
&\!\!\! &\!\!\! 2000 &\!\!\! ORACLE &\!\!\! 0.47 &\!\!\! 0.46 &\!\!\! 0.47 &\!\!\! 1.08 &\!\!\! 0.09 &\!\!\! 0.99 &\!\!\! 0.85 &\!\!\! 0.27 \\
&\!\!\! &\!\!\! &\!\!\! SMILE &\!\!\! 0.47 &\!\!\! 0.45 &\!\!\! 0.47 &\!\!\! 1.08 &\!\!\! 0.19 &\!\!\! 0.95 &\!\!\! 0.79 &\!\!\! 0.27 \\
&\!\!\! &\!\!\! &\!\!\! SAPLM &\!\!\! 0.49 &\!\!\! 0.47 &\!\!\! 0.53 &\!\!\! - &\!\!\! 0.43 &\!\!\! 1.01 &\!\!\! 0.86 &\!\!\! 0.28 \\
&\!\!\! &\!\!\! &\!\!\! SLM &\!\!\! 10.51 &\!\!\! 8.70 &\!\!\! 41.05 &\!\!\! 21.29 &\!\!\! 1.84 &\!\!\! 252.75 &\!\!\! 180.40 &\!\!\! 4.80 \\
&\!\!\! &\!\!\! 5000 &\!\!\! ORACLE &\!\!\! 0.45 &\!\!\! 0.44 &\!\!\! 0.53 &\!\!\! 1.06 &\!\!\! 0.09 &\!\!\! 0.97 &\!\!\! 0.81 &\!\!\! 0.27 \\
&\!\!\! &\!\!\! &\!\!\! SMILE &\!\!\! 0.45 &\!\!\! 0.44 &\!\!\! 0.53 &\!\!\! 1.06 &\!\!\! 0.16 &\!\!\! 0.94 &\!\!\! 0.75 &\!\!\! 0.27 \\
&\!\!\! &\!\!\! &\!\!\! SAPLM &\!\!\! 0.47 &\!\!\! 0.47 &\!\!\! 0.57 &\!\!\! - &\!\!\! 0.42 &\!\!\! 0.98 &\!\!\! 0.81 &\!\!\! 0.28 \\
&\!\!\! &\!\!\! &\!\!\! SLM &\!\!\! 9.29 &\!\!\! 8.66 &\!\!\! 49.64 &\!\!\! 19.90 &\!\!\! 1.73 &\!\!\! 252.44 &\!\!\! 179.41 &\!\!\! 4.83 \\
&\!\!\! 1.0 &\!\!\! 1000 &\!\!\! ORACLE &\!\!\! 1.94 &\!\!\! 1.98 &\!\!\! 1.82 &\!\!\! 4.48 &\!\!\! 0.37 &\!\!\! 2.97 &\!\!\! 2.63 &\!\!\! 1.08 \\
&\!\!\! &\!\!\! &\!\!\! SMILE &\!\!\! 1.94 &\!\!\! 1.98 &\!\!\! 1.82 &\!\!\! 4.48 &\!\!\! 0.56 &\!\!\! 2.80 &\!\!\! 2.29 &\!\!\! 1.09 \\
&\!\!\! &\!\!\! &\!\!\! SAPLM &\!\!\! 1.98 &\!\!\! 2.01 &\!\!\! 1.96 &\!\!\! - &\!\!\! 1.44 &\!\!\! 2.98 &\!\!\! 2.53 &\!\!\! 1.09 \\
&\!\!\! &\!\!\! &\!\!\! SLM &\!\!\! 11.33 &\!\!\! 10.55 &\!\!\! 51.08 &\!\!\! 22.51 &\!\!\! 1.95 &\!\!\! 253.34 &\!\!\! 180.31 &\!\!\! 5.59 \\
&\!\!\! &\!\!\! 2000 &\!\!\! ORACLE &\!\!\! 1.90 &\!\!\! 1.77 &\!\!\! 1.82 &\!\!\! 4.16 &\!\!\! 0.35 &\!\!\! 3.04 &\!\!\! 2.57 &\!\!\! 1.08 \\
&\!\!\! &\!\!\! &\!\!\! SMILE &\!\!\! 1.91 &\!\!\! 1.84 &\!\!\! 2.62 &\!\!\! 4.23 &\!\!\! 0.73 &\!\!\! 3.30 &\!\!\! 3.20 &\!\!\! 1.11 \\
&\!\!\! &\!\!\! &\!\!\! SAPLM &\!\!\! 1.98 &\!\!\! 1.85 &\!\!\! 1.98 &\!\!\! - &\!\!\! 1.40 &\!\!\! 3.07 &\!\!\! 2.49 &\!\!\! 1.09 \\
&\!\!\! &\!\!\! &\!\!\! SLM &\!\!\! 11.04 &\!\!\! 10.19 &\!\!\! 60.67 &\!\!\! 23.02 &\!\!\! 1.99 &\!\!\! 252.71 &\!\!\! 179.98 &\!\!\! 5.61 \\
&\!\!\! &\!\!\! 5000 &\!\!\! ORACLE &\!\!\! 1.71 &\!\!\! 1.89 &\!\!\! 1.93 &\!\!\! 4.03 &\!\!\! 0.33 &\!\!\! 2.93 &\!\!\! 2.54 &\!\!\! 1.08 \\
&\!\!\! &\!\!\! &\!\!\! SMILE &\!\!\! 1.82 &\!\!\! 1.92 &\!\!\! 3.52 &\!\!\! 4.05 &\!\!\! 0.39 &\!\!\! 3.97 &\!\!\! 4.67 &\!\!\! 1.28 \\
&\!\!\! &\!\!\! &\!\!\! SAPLM &\!\!\! 1.77 &\!\!\! 1.97 &\!\!\! 2.09 &\!\!\! - &\!\!\! 1.43 &\!\!\! 2.96 &\!\!\! 2.44 &\!\!\! 1.25 \\
&\!\!\! &\!\!\! &\!\!\! SLM &\!\!\! 16.78 &\!\!\! 10.55 &\!\!\! 80.30 &\!\!\! 23.28 &\!\!\! 2.01 &\!\!\! 252.41 &\!\!\! 180.60 &\!\!\! 5.72 \\
500 &\!\!\! 0.5 &\!\!\! 1000 &\!\!\! ORACLE &\!\!\! 0.27 &\!\!\! 0.28 &\!\!\! 0.28 &\!\!\! 0.67 &\!\!\! 0.06 &\!\!\! 0.67 &\!\!\! 0.58 &\!\!\! 0.27 \\
&\!\!\! &\!\!\! &\!\!\! SMILE &\!\!\! 0.27 &\!\!\! 0.28 &\!\!\! 0.28 &\!\!\! 0.67 &\!\!\! 0.07 &\!\!\! 0.65 &\!\!\! 0.55 &\!\!\! 0.27 \\
&\!\!\! &\!\!\! &\!\!\! SAPLM &\!\!\! 0.29 &\!\!\! 0.29 &\!\!\! 0.31 &\!\!\! - &\!\!\! 0.27 &\!\!\! 0.67 &\!\!\! 0.58 &\!\!\! 0.27 \\
&\!\!\! &\!\!\! &\!\!\! SLM &\!\!\! 5.08 &\!\!\! 4.91 &\!\!\! 5.88 &\!\!\! 10.70 &\!\!\! 0.97 &\!\!\! 253.20 &\!\!\! 180.13 &\!\!\! 4.66 \\
&\!\!\! &\!\!\! 2000 &\!\!\! ORACLE &\!\!\! 0.27 &\!\!\! 0.27 &\!\!\! 0.31 &\!\!\! 0.65 &\!\!\! 0.05 &\!\!\! 0.65 &\!\!\! 0.55 &\!\!\! 0.26 \\
&\!\!\! &\!\!\! &\!\!\! SMILE &\!\!\! 0.27 &\!\!\! 0.27 &\!\!\! 0.31 &\!\!\! 0.65 &\!\!\! 0.06 &\!\!\! 0.63 &\!\!\! 0.52 &\!\!\! 0.26 \\
&\!\!\! &\!\!\! &\!\!\! SAPLM &\!\!\! 0.28 &\!\!\! 0.28 &\!\!\! 0.34 &\!\!\! - &\!\!\! 0.27 &\!\!\! 0.66 &\!\!\! 0.55 &\!\!\! 0.27 \\
&\!\!\! &\!\!\! &\!\!\! SLM &\!\!\! 5.25 &\!\!\! 4.99 &\!\!\! 5.90 &\!\!\! 11.93 &\!\!\! 1.07 &\!\!\! 252.96 &\!\!\! 179.22 &\!\!\! 4.66 \\
&\!\!\! &\!\!\! 5000 &\!\!\! ORACLE &\!\!\! 0.29 &\!\!\! 0.25 &\!\!\! 0.29 &\!\!\! 0.62 &\!\!\! 0.05 &\!\!\! 0.67 &\!\!\! 0.57 &\!\!\! 0.26 \\
&\!\!\! &\!\!\! &\!\!\! SMILE &\!\!\! 0.29 &\!\!\! 0.25 &\!\!\! 0.29 &\!\!\! 0.62 &\!\!\! 0.17 &\!\!\! 0.64 &\!\!\! 0.54 &\!\!\! 0.26 \\
&\!\!\! &\!\!\! &\!\!\! SAPLM &\!\!\! 0.30 &\!\!\! 0.26 &\!\!\! 0.32 &\!\!\! - &\!\!\! 0.28 &\!\!\! 0.67 &\!\!\! 0.57 &\!\!\! 0.27 \\
&\!\!\! &\!\!\! &\!\!\! SLM &\!\!\! 5.30 &\!\!\! 4.87 &\!\!\! 6.35 &\!\!\! 11.96 &\!\!\! 1.07 &\!\!\! 252.99 &\!\!\! 179.99 &\!\!\! 4.66 \\
&\!\!\! 1.0 &\!\!\! 1000 &\!\!\! ORACLE &\!\!\! 1.18 &\!\!\! 1.08 &\!\!\! 1.09 &\!\!\! 2.43 &\!\!\! 0.20 &\!\!\! 1.90 &\!\!\! 1.62 &\!\!\! 1.05 \\
&\!\!\! &\!\!\! &\!\!\! SMILE &\!\!\! 1.18 &\!\!\! 1.08 &\!\!\! 1.09 &\!\!\! 2.43 &\!\!\! 0.56 &\!\!\! 1.83 &\!\!\! 1.47 &\!\!\! 1.05 \\
&\!\!\! &\!\!\! &\!\!\! SAPLM &\!\!\! 1.21 &\!\!\! 1.12 &\!\!\! 1.15 &\!\!\! - &\!\!\! 0.87 &\!\!\! 1.92 &\!\!\! 1.60 &\!\!\! 1.06 \\
&\!\!\! &\!\!\! &\!\!\! SLM &\!\!\! 6.45 &\!\!\! 5.26 &\!\!\! 7.42 &\!\!\! 12.11 &\!\!\! 1.09 &\!\!\! 253.05 &\!\!\! 180.33 &\!\!\! 5.41 \\
&\!\!\! &\!\!\! 2000 &\!\!\! ORACLE &\!\!\! 1.12 &\!\!\! 1.02 &\!\!\! 1.12 &\!\!\! 2.45 &\!\!\! 0.20 &\!\!\! 1.94 &\!\!\! 1.66 &\!\!\! 1.04 \\
&\!\!\! &\!\!\! &\!\!\! SMILE &\!\!\! 1.12 &\!\!\! 1.02 &\!\!\! 1.12 &\!\!\! 2.45 &\!\!\! 0.22 &\!\!\! 1.84 &\!\!\! 1.49 &\!\!\! 1.04 \\
&\!\!\! &\!\!\! &\!\!\! SAPLM &\!\!\! 1.15 &\!\!\! 1.05 &\!\!\! 1.21 &\!\!\! - &\!\!\! 0.85 &\!\!\! 1.94 &\!\!\! 1.63 &\!\!\! 1.05 \\
&\!\!\! &\!\!\! &\!\!\! SLM &\!\!\! 6.12 &\!\!\! 5.99 &\!\!\! 7.62 &\!\!\! 13.76 &\!\!\! 1.22 &\!\!\! 252.81 &\!\!\! 180.10 &\!\!\! 5.43 \\
&\!\!\! &\!\!\! 5000 &\!\!\! ORACLE &\!\!\! 1.12 &\!\!\! 1.05 &\!\!\! 1.16 &\!\!\! 2.46 &\!\!\! 0.20 &\!\!\! 1.96 &\!\!\! 1.67 &\!\!\! 1.05 \\
&\!\!\! &\!\!\! &\!\!\! SMILE &\!\!\! 1.12 &\!\!\! 1.05 &\!\!\! 1.16 &\!\!\! 2.46 &\!\!\! 0.22 &\!\!\! 1.87 &\!\!\! 1.48 &\!\!\! 1.05 \\
&\!\!\! &\!\!\! &\!\!\! SAPLM &\!\!\! 1.14 &\!\!\! 1.08 &\!\!\! 1.22 &\!\!\! - &\!\!\! 0.87 &\!\!\! 1.97 &\!\!\! 1.64 &\!\!\! 1.06 \\
&\!\!\! &\!\!\! &\!\!\! SLM &\!\!\! 6.16 &\!\!\! 5.64 &\!\!\! 9.37 &\!\!\! 12.28 &\!\!\! 1.10 &\!\!\! 252.69 &\!\!\! 180.26 &\!\!\! 5.43 \\
\hline \hline
\end{tabular}
\end{center}
\end{table}
Next we investigate the coverage rates of the proposed SCB. For each replication, we test whether the true functions are covered by the SCB at the simulated values of the covariate in the interval $[-0.5+h,0.5-h]$, where $h$ is the bandwidth. Table \ref{TAB:SIM1-scb} shows the empirical coverage probabilities for a nominal 95\% confidence level out of 500 replications. For comparison, we also provide the SCBs from the SAPLM and ORACLE estimators. From Table \ref{TAB:SIM1-scb}, we observe that coverage probabilities for the SMILE, SAPLM and ORACLE SCBs all approach the nominal levels as $n$ increases, which provides positive confirmation of Theorem \ref{THM:nonlinear-normality}. In most cases, SMILE performs as well as or better than SAPLM, and arrives at about the nominal coverage when $n = 500$ and $\sigma=1.0$. Figure \ref{FIG:SIM-band} depicts the true function $\phi_{\ell}$, the corresponding SMILE $\widehat{\phi}_{\ell}^{\mathrm{SBLL}}$ and the $95\%$ SCB for $\phi_{\ell}$ based on $\widehat{\phi}_{\ell}^{\mathrm{SBLL}}$, for $\ell = 2, 3$, which are based on a typical run with $n = 500$, $p = 1000$ and $\sigma = 1.0$.
\begin{table}[htbp]
\caption{Coverage rates comparing the ORACLE, SMILE and SAPLM.} \vspace*{-.3cm}
\label{TAB:SIM1-scb}
\begin{center}
\renewcommand{0.92}{0.75}
\begin{tabular}{ccccccccc}
\hline \hline
Size & Noise & & \multicolumn{3}{c}{$\phi_2$ Coverage (\%)} & \multicolumn{3}{c}{$\phi_3$ Coverage (\%)} \\
\cmidrule(r){4-6} \cmidrule(l){7-9}
$n$ & $\sigma$ & $p$ & ORACLE & SMILE & SAPLM & ORACLE & SMILE & SAPLM \\
\hline
300 & 0.5 & 1000 & 93.7 & 94.5 & 93.9 & 92.4 & 92.6 & 91.7 \\
& & 2000 & 92.6 & 93.3 & 92.6 & 92.3 & 93.8 & 92.5 \\
& & 5000 & 92.3 & 93.0 & 92.7 & 93.3 & 92.3 & 91.7 \\
& 1 & 1000 & 96.0 & 95.6 & 94.7 & 96.1 & 96.4 & 95.3 \\
& & 2000 & 95.4 & 95.7 & 94.9 & 96.1 & 96.2 & 95.5 \\
& & 5000 & 95.1 & 95.6 & 94.2 & 95.9 & 96.4 & 94.8 \\
500 & 0.5 & 1000 & 92.9 & 93.8 & 93.5 & 92.7 & 90.6 & 92.0 \\
& & 2000 & 92.5 & 92.7 & 92.3 & 92.0 & 92.0 & 92.3 \\
& & 5000 & 92.5 & 92.6 & 91.8 & 91.5 & 89.9 & 90.4 \\
& 1 & 1000 & 97.1 & 96.7 & 96.3 & 96.0 & 96.0 & 95.2 \\
& & 2000 & 95.2 & 95.0 & 94.5 & 95.2 & 94.6 & 94.3 \\
& & 5000 & 94.7 & 95.1 & 95.0 & 96.2 & 96.0 & 95.5 \\
\hline \hline
\end{tabular}
\end{center}
\end{table}
\begin{figure}
\caption{Plots of the SMILE (dashed curve) and the 95\% SCB (shaded area) of the nonparametric component $\phi_{\ell}
\label{FIG:cb_phi2}
\label{FIG:cb_phi3}
\label{FIG:SIM-band}
\end{figure}
Appendices B--D contain the results of additional simulations which show that our proposed SMILE procedure performs well relative to competing methods under a wider range of conditions.
\setcounter{chapter}{5}
\renewcommand{D.\arabic{table}}{{\arabic{table}}}
\renewcommand{D.\arabic{figure}}{{\arabic{figure}}}
\renewcommand{E.\arabic{section}}{\arabic{section}}
\renewcommand{E.\arabic{subsection}}{5.\arabic{subsection}}
\setcounter{section}{4}
\section{Application} \label{sec:application}
We illustrate the application of our proposed method in the ultra-high-dimensional setting by using the SAM data generated by \citet[][]{leiboff2015genetic}. The maize SAM is a small pool of stem cells located in the plant shoot that generate all the above-ground tissues of maize plants. \citet[][]{leiboff2015genetic} showed that SAM volume is correlated with a variety of agronomically important traits in adult plants. The goal of our analysis is to model and predict SAM volume as a function of single nucleotide polymorphism (SNP) genotypes and messenger RNA transcript abundance levels using data from maize inbred lines. Following the preprocessing steps described in Section B.5 in the Supplementary Materials in \cite{li2017ultra}, linear sure independent screening \citep{fan2008sure} for SNP genotypes, and nonlinear independent screening \citep{fan2011nonparametric} for RNA transcripts, the dataset we analyze c
onsists of log-scale SAM volume measurements, binary SNP genotypes at $p_1=5203$ markers, and log-scale measures of abundance for $p_2=1020$ transcripts for each of $n=368$ maize inbred lines.
\citet{li2017ultra} used the APLM to model the relationship between the log SAM volume response and predictors determined by SNP genotypes and RNA transcript abundance levels. Because the SNP genotypes are binary, they naturally entered the linear part of the APLM, and for convenience all the RNA transcripts were included in the nonlinear part of the APLM in \cite{li2017ultra}. As discussed before, failing to account for exactly linear features makes the APLM less efficient statistically and computationally. In the following we apply our proposed SMILE method to distinguish among RNA transcripts entering the nonparametric and parametric parts of the APLM and to identify significant SNP genotypes and RNA transcripts simultaneously.
To compare the results of SMILE to the sparse APLM and the sparse linear regression model, we also analyze the data using the SAPLM and SLM estimators presented in \cite{li2017ultra}. Parallel to the settings in Section \ref{sec:simulation}, we use constant B-splines with four quantile knots for model structure identification, and use cubic B-splines with one quantile knot for nonlinear function approximation. We use the iterative algorithm proposed in Section \ref{subsec:algorithm} for penalty parameter selection and estimation.
As shown in Table \ref{TAB: RNA_sel}, SMILE identified 169 SNPs, 10 RNA transcripts linearly associated with log SAM size and 2 RNA transcripts that have nonlinear association with log SAM size. In contrast, SAPLM selected 177 SNPs and 3 RNA transcripts, and SLM selected 167 SNPs and 32 RNA transcripts. To evaluate the predictive performance of the two methods, we computed 10-fold cross-validation mean squared prediction error (CV-MSPE) for each method. The SMILE-estimated nonlinear function for the selected nonlinear RNA transcript is plotted, along with 95\% SCBs, in Figure \ref{FIG:SAM-band}.
\begin{table}[htbp]
\caption{Selected SNPs and Transcripts by SMILE, SAPLM and SLM.}
\label{TAB: RNA_sel}
\begin{center}
\begin{tabular}{lccc}
\hline\hline
RNA Transcripts Selected& SMILE & SAPLM & SLM \\
\hline
$X_{725}$ & \checkmark & \checkmark & \checkmark \\
$X_{127}$, $X_{136}$, $X_{141}$, $X_{208}$, $X_{289}$, $X_{312}$, $X_{493}$,$X_{749}$,$X_{855}$& \checkmark & & \checkmark \\
$X_{153}^{\ast}$, $X_{677}^{\ast}$ & \checkmark & & \\
$X_{157}$,$X_{701}$ & & \checkmark & \\
$X_{209}$,$X_{314}$, $X_{320}$, $X_{321}$, $X_{342}$, $X_{419}$,$X_{472}$,$X_{489}$,$X_{553}$, & & & \checkmark \\
$X_{589}$,$X_{601}$,$X_{615}$, $X_{783}$,$X_{785}$,$X_{793}$,$X_{846}$,$X_{863}$, $X_{940}$,& & & \checkmark \\
$X_{946}$,$X_{978}$,$X_{1002}$,$X_{1018}$& & & \checkmark \\
\hline
Number of SNP Genotypes & 169 & 177 & 167 \\
Number of Linear RNA Transcripts & 10 & 0 & 32 \\
Number of Functional RNA Transcripts & 2 & 3 & 0 \\
\hline
CV MSPE& 0.060 & 0.102 & 0.132 \\
CV Mean Number of SNPs & 153.9 & 175.9 & 83.1 \\
CV Mean Number of Linear Transcripts & 8.7 & 0 & 17.7 \\
CV Mean Number of Nonlinear Transcripts & 1.9 & 3.8 & 0 \\
\hline\hline
\multicolumn{4}{l}{$\ast$ nonlinear association identified by SMILE for $X_{153}$ and $X_{677}$}
\end{tabular}
\end{center}
\end{table}
\begin{figure}
\caption{Plot of the SMILE (solid curve) and the 95\% confidence band (shaded area) for the selected RNA transcript.}
\label{X153}
\label{X677}
\label{FIG:SAM-band}
\end{figure}
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\section{Discussion} \label{sec:conclusion}
This paper focuses on the simultaneous sparse model identification and learning for ultra-high-dimensional APLMs which strikes a delicate balance between the simplicity of the standard linear regression models and the flexibility of the additive regression models. We proposed a two-stage penalization method, called SMILE, which can efficiently select nonzero components and identify the linear-and-nonlinear structure in the functional terms, as well as simultaneously estimate and make inference for both linear coefficients and nonlinear functions. First, we have devised a groupwise penalization method in the APLM for simultaneous variable selection and structure identification. After identifying important covariates and the functional forms for the selected covariates, we have further constructed SCBs for the nonzero nonparametric functions based on refined spline-backfitted local-linear estimators. Our simulation studies and applications demonstrate the proposed SMILE procedure can be more efficient than penalized linear regression and the penalized APLM without model identification, and can improve predictions.
Our work differs from previous works in practical, theoretical and computational aspects: (i) We perform variable selection and model structure identification simultaneously, for both the linear components in $\boldsymbol{Z}$, and the linear and nonlinear forms for the components of $\boldsymbol{X}$. In contrast, existing works either performs only model structure identification or performs variable selection only for components in $\boldsymbol{X}$.
(ii) Besides the consistency of model structure identification, we also provide inference tools for both the regression coefficients and the component functions.
(iii) Compared to the local quadratic approximation approach used in \cite{lian2015separation}, which cannot provide exactly zero solutions and is inefficient for fitting large regression problems, our proposed iterative group coordinate descent algorithm takes advantage of sparsity in computation and is able to deal with the triple penalization problem very efficiently. (See \cite{breheny2015group} for a detailed comparison of these two algorithms.) Our algorithm is easy to implement and can provide analysis results for large data sets with thousands of dimensions within seconds.
Our work deals with independent observations but can be extended to longitudinal data settings through marginal models or mixed-effects models. In addition, although we consider continuous response variables in our work, or approach can be readily extended to generalized additive partially linear models, to deal with different types of responses. Currently, the APLM assumes that the effects of all covariates are additive, which may overlook the potential interaction between covariates. Our method can be extended to models that can accommodate interactions between covariates, for example, APLMs with interaction terms. We leave such extensions to future work. Another limitation of our work is a reliance on the assumption of constant error variance. However, heteroscedasticity may be encountered in the analysis of genomic data sets. It is of interest to develop a new methodology that allows non-constant error variance for high-dimensional estimation and model selection, and this is another challenge we leave for future work.
\section*{Acknowledgment}
This work was supported by the Iowa State University Plant Sciences Institute Scholars Program. In addition, Wang's research was supported by NSF grant DMS-1542332, and Nettleton's research was supported by NSF grant IOS-1238142. We sincerely thank the Editor, the Associate Editor and the anonymous reviewers for their insightful comments that have lead to significant improvements on the paper.
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\vskip .4in \noindent \textbf{\Large Appendices}
\section*{A. Effect of Smoothing Parameters on Performance of SMILE}
To implement the proposed SMILE procedure, one needs to select the knots for a spline at the selection stage and refitting stage, and the bandwidth for a kernel at the backfitting stage. In this section, we study how these smoothing parameters affect the proposed SMILE method and evaluate the practical performance in the finite-sample simulation studies described in Section 4.2 of the main paper. In the literature of polynomial spline smoothing, the knots for a spline are generally put on a grid of equally spaced sample quantiles \citep{ruppert2002selecting}. Therefore, we only need to investigate the effect of the number of knots on the performance of SMILE.
At the first stage (model selection), we use piecewise constant splines with the number of interior knots $N=2,3,\ldots,8$ in the simulation. Figure \ref{FIG:knots} shows the effect of $N$ on the accuracy of model selection based on the criteria defined in the main paper: (B-i)--(B-vi) and (C-i)--(C-iv). From Figure \ref{FIG:knots}, it appears that the value $N$ has little effect on the selection results. For all combinations of $n$, $p$ and $\sigma$, no matter which $N$ is used, the ``corrZ0", ``corrL", ``corrX0" are all $100\%$, and the ``LtoN" and ``Nto0" are all $0\%$. The values of ``corrZ", ``corrN", ``corrLN" and ``Zto0" and ``Xto0" are not exactly the same when using different values of $N$, but they are almost constant for $N=2,3,\ldots,8$. Especially when the sample size $n=500$, the proposed SMILE is able to identify the true model structure regardless of $p=1000,2000$ or $5000$. When $n=300$ and $p=5000$, the selection results become slightly worse when we increase to $N\geq 6$.
\begin{figure}
\caption{First stage selection results using different number of knots.}
\label{FIG:knots}
\end{figure}
In summary, the values of $N$ often have little effect on the model selection results. Choosing small values of $N$ can also help to reduce computational burden. So we recommend using fewer knots at the model selection stage, especially when the sample size is small compared to the number of predictors. In practice, $N=2\sim 5$ usually would be adequate to identify the model structure.
Next, we study the effect of the smoothing parameters at the refitting stage. For the selected model, we approximate the nonlinear functional components using higher order polynomial splines to obtain more accurate pilot estimators. Then we apply spline backfitted local-linear smoothing to obtain the final SBLL estimators and the corresponding SCBs. According to Assumption (A6$^\prime$), to obtain the SCB with the desired confidence level, the number of interior knots $M_n$ for a refitting spline needs to satisfy: $\{n^{1/(2d)} \vee n^{4/(10d-5)}\} \ll M_n \ll n^{1/3}$, where $d$ is the degree of the polynomial spline basis functions used in the refitting. The widely used quadratic/ cubic splines and any polynomial splines of degree $d\geq 2$ all satisfy this condition. Therefore, in practice we suggest choosing
\begin{equation*}
M_n=\min\{\lfloor n^{1/(2d) \vee 4/(10d-5)}\ln(n)\rfloor, \lfloor n/(4s)\rfloor\}+1,
\end{equation*}
where $s$ is the number of nonlinear components selected at the first stage and the term $\lfloor n/(4s)\rfloor$ is to guarantee that we have at least four observations in each subinterval between two adjacent knots to avoid getting (near) singular design matrices in the spline smoothing.
A researcher with some knowledge of the shape of the nonlinear component may be able to select a more suitable number of knots. In our simulation studies, we try $4$, $6$ and $8$ interior knots to test the sensitivity of the SBLL estimators and the corresponding SCBs.
For the local-linear smoothing in the backfitting, Condition (B2) requires that the bandwidths are of order $n^{-1/5}$. Any bandwidths with this rate lead to the same limiting distribution for $\widehat{\phi}_{\ell}^{\mathrm{SBLL}}$, so the user can consider any standard routine for bandwidth selection. There have been many proposals for bandwidth selection in the literature. In our simulation, we consider three popular bandwidth selectors described in \cite{Fan:Gijbels:96} and \cite{Wand:Jones:95}: rule-of-thumb bandwidth (``thumbBw''), plug-in bandwidth selector (``pluginBw'') and leave-one-out cross-validation bandwidth selector (``regCVBwSelC''). Below we present simulation results to compare the performance of three bandwidth selectors. The kernel that we use here is the Epanechnikov kernel: $K(u)=3/4(1-u^{2})I(|u|\leq 1)$.
To see how the refitting smoothing parameters affect estimation accuracy, we report the average mean square errors (AMSEs) of the SBLL estimators based on $4$, $6$ and $8$ interior knots in the spline refitting and three different bandwidth selectors in the kernel backfitting. Figure \ref{FIG:AMSE-knots-BW} presents the AMSEs of the resulting SBLL estimators based on different combinations of the refitting smoothing parameters. For both $\phi_1$ and $\phi_2$, the AMSEs are very similar across the different combinations of knots and bandwidth selectors.
\begin{figure}
\caption{Average mean squared errors (AMSEs) of the SBLL estimators of $\phi_2$ and $\phi_3$.}
\label{FIG:AMSE-knots-BW}
\end{figure}
Figure \ref{FIG:SCB-knots-BW} shows the coverage rates of the SCBs based on different combinations of knots and bandwidth selectors. From Figure \ref{FIG:SCB-knots-BW}, it is clear that the number of knots for a spline in the refitting has very little effect on the coverage of the SCBs. One also observes that the performances of the SCBs based on different smoothing parameters become more similar with increasing sample size, whereas the coverage rates of the SCBs using the ``thumbBw'' selector are the closest to the nominal level in all the simulation settings.
Thus we recommend the ``thumbBw'' selector, especially when the sample size is small.
\begin{figure}
\caption{Coverage rates of the SCBs for $\phi_2$ and $\phi_3$.}
\label{FIG:SCB-knots-BW}
\end{figure}
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\section*{B. Simulation Studies Using Purely Additive Models or Purely Linear Models}
In this section, we examine the performance the proposed method when the underlying model is either a purely additive model (AM) or a purely linear model (LM). We evaluated the selection, estimation and prediction accuracy, and inference performance of the proposed SMILE method. We also compared the performance of SMILE with the sparse APLM estimator with adaptive group LASSO penalty (SAPLM), the ordinary linear least squares estimator with the adaptive LASSO penalty (SLM), and the oracle estimator (ORACLE), which uses the same estimation techniques as the SMILE except that no penalization or data-driven variable selection is used because all active and inactive index sets are treated as known. All the performance measures were computed based on 200 replicates.
\vskip .1in
\noindent{\bf Case I. A Purely Additive Model.} We generate simulated datasets using the AM structure
\begin{align*}
Y_i = \sum_{\ell=1}^{p} \phi_{\ell}(X_{i\ell}) + \varepsilon_i,
\end{align*}
where
\begin{align*}
\phi_1(x) &= \frac{8\sin(2\pi x)}{2-\sin(2\pi x)} - \mathrm{E}\left\{\frac{8\sin(2\pi X_1)}{2-\sin(2\pi X_1)}\right\}, \\
\phi_2(x) & = -3 \cos^2(\pi x) + 6 \sin^2(\pi x) - \mathrm{E}\{-3 \cos^2(\pi X_2 + 6 \sin^2(\pi X_2)\}, \\
\phi_3(x) & = 6x + 18x^{2} - \mathrm{E}(6X_3 + 18X_3^{2} ),
\end{align*}
and $\phi_4(x) = \ldots = \phi_{p}(x) = 0$.
\vskip .1in
\noindent{\bf Case II. A Purely Linear Model.} We generate simulated datasets using the LM structure:
\begin{align*}
Y_i = \sum_{\ell=1}^{p} \beta_{\ell} X_{i\ell} + \varepsilon_i,
\end{align*}
where $\beta_1=3$, $\beta_2=4$, $\beta_3=-2$, and $\beta_4 = \ldots = \beta_{p} = 0$.
We use the criteria mentioned in Section \ref{sec:simulation} to evaluate the methods on the accuracy of variable selection and prediction. The model selection results are provided in Tables \ref{TAB:AM1} and \ref{TAB:LM1}, respectively. The SMILE can correctly discover the linear or nonlinear structure in covariates $\boldsymbol{X}$, while the SAPLM neglects linear structure in $\boldsymbol{X}$ and SLM fails in presenting the nonlinear part of covariates $\boldsymbol{X}$. For the SMILE, regardless of the underlying models, the percents of nonzero covariates correctly selected are very close to ORACLE (100 for corrX and corrX0, respectively), as shown in Table \ref{TAB:AM1}; and the percents of components incorrectly identified approach to 0 as the sample size $n$ increases, as shown in Table \ref{TAB:AM1}. The SMILE is close in the selection of nonlinear covariates to the SAPLM estimator, and it overwhelms the SAPLM in identifying the linear structure of covariates $\boldsymbol{X}$. SMILE is close in the selection of linear covariates to the SLM estimator, and it overwhelms the SLM in identifying the nonlinear structure of covariates $\boldsymbol{X}$. From the results in Tables \ref{TAB:AM1} and \ref{TAB:LM1}, it is also evident that model misspecification leads to poor variable selection performance for the SLM, as the SLM fails to select the right nonlinear components in each simulation.
The estimation and prediction results are displayed in Tables \ref{TAB:AM2} and \ref{TAB:LM2}. Specifically, we present the AMSEs for functions $\phi_1$, $\phi_2$ and $\phi_3$ and the CV-MSPEs for predicting $Y$. The case with known active covariates (ORACLE) is also reported in each setting and serves as a benchmark. The SMILE performs well in predicting $Y$ regardless of the model structure for the underlying model, as indicated by results closest to ORACLE in CV-MSPE for base cases. The SLM is around 18$\sim$36 times higher than the SMILE in the AM case, and the SAPLM is around 1$\sim$3 times higher than the SMILE in the LM case. The estimation of functions $\phi_1$, $\phi_2$ and $\phi_3$ is also good for the SMILE, and matches with ORACLE as sample size $n$ increases. The inferior performance of the SAPLM in the LM case and the poor performance of SLM in the AM case, in both estimation and prediction, illustrates the importance and necessity of identifying correct model structure.
\begin{table}[htbp]
\caption{AM Case: Selection statistics comparing the SMILE, SAPLM and SLM.}
\label{TAB:AM1}
\begin{center}
\renewcommand{0.92}{0.9}
\begin{tabular}{ccclrrrr}
\hline\hline
Size & Noise & & & \multicolumn{2}{c}{True Selection} & \multicolumn{2}{c}{False Selection} \\
\cmidrule(r){5-6} \cmidrule(l){7-8}
$n$ & $\sigma$ & $p$ & Method & corrX & corrX0 & NtoL & Nto0 \\
\hline
300 & 0.5 & 1000 & SMILE & 100 & 99.9995 & 2.3333 & 0 \\
& & & SAPLM & 100 & 100 & 0 & 0 \\
& & & SLM & 65.1667 & 99.9985 & 65.1667 & 34.8333 \\
& & 2000 & SMILE & 100 & 99.9998 & 3.1667 & 0 \\
& & & SAPLM & 100 & 100 & 0 & 0 \\
& & & SLM & 64 & 99.9995 & 64.0003 & 36.0000 \\
& & 5000 & SMILE & 99.6667 & 99.9994 & 7 & 0.3333 \\
& & & SAPLM & 100 & 100 & 0 & 0 \\
& & & SLM & 63.6667 & 99.9998 & 63.6667 & 36.3333 \\
& 1.0 & 1000 & SMILE & 100 & 100 & 6.5000 & 0 \\
& & & SAPLM & 100 & 100 & 0 & 0 \\
& & & SLM & 64.6667 & 99.9980 & 64.6667 & 35.3333 \\
& & 2000 & SMILE & 99.8333 & 99.9995 & 7.8333 & 0.16667 \\
& & & SAPLM & 100 & 100 & 0 & 0 \\
& & & SLM & 63.5 & 99.9995 & 63.5003 & 36.5000 \\
& & 5000 & SMILE & 99.1667 & 99.9994 & 13.8333 & 0.8333 \\
& & & SAPLM & 100 & 100 & 0 & 0 \\
& & & SLM & 62.1667 & 99.9999 & 62.1667 & 37.8333 \\
500 & 0.5 & 1000 & SMILE & 100 & 99.9995 & 0 & 0 \\
& & & SAPLM & 100 & 100 & 0 & 0 \\
& & & SLM & 66.66667 & 99.999 & 66.6667 & 33.3333 \\
& & 2000 & SMILE & 100 & 99.99975 & 0 & 0 \\
& & & SAPLM & 100 & 100 & 0 & 0 \\
& & & SLM & 66.6667 & 99.99975 & 66.6667 & 33.3333 \\
& & 5000 & SMILE & 100 & 99.9999 & 0 & 0 \\
& & & SAPLM & 100 & 100 & 0 & 0 \\
& & & SLM & 66.6667 & 99.9999 & 66.6667 & 33.3333 \\
& 1.0 & 1000 & SMILE & 100 & 100 & 0 & 0 \\
& & & SAPLM & 100 & 100 & 0 & 0 \\
& & & SLM & 66.6667 & 99.9980 & 66.6667 & 33.3333 \\
& & 2000 & SMILE & 100 & 100 & 0.16667 & 0 \\
& & & SAPLM & 100 & 100 & 0 & 0 \\
& & & SLM & 66.6667 & 99.9995 & 66.6667 & 33.3333 \\
& & 5000 & SMILE & 100 & 99.9998 & 0 & 0 \\
& & & SAPLM & 100 & 100 & 0 & 0 \\
& & & SLM & 66.6667 & 99.9999 & 66.6667 & 33.3333 \\
\hline \hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[htbp]
\caption{AM Case: Estimation statistics comparing the SMILE, SAPLM and SLM.}
\label{TAB:AM2}
\begin{center}
\renewcommand{0.92}{0.72}
\begin{tabular}{ccclrrrr}
\hline\hline
\multicolumn{1}{l}{Size} & \multicolumn{1}{l}{Noise} & & & \multicolumn{3}{c}{AMSE} & \multicolumn{1}{c}{CV-} \\
\cmidrule(lr){5-7}
$n$ & $\sigma$ & $p$ & Method & $\phi_1(\cdot)$ & $\phi_2(\cdot)$ & $\phi_3(\cdot)$ & MSPE \\
\hline
300 & 0.5 & 1000 & ORACLE & 0.0151 & 0.0136 & 0.0094 & 0.2924 \\
& & & SMILE & 0.0236 & 0.0196 & 0.2484 & 0.5400 \\
& & & SAPLM & 0.0152 & 0.0136 & 0.0094 & 0.2924 \\
& & & SLM & 6.0801 & 10.1935 & 1.9317 & 18.7746 \\
& & 2000 & ORACLE & 0.0156 & 0.0136 & 0.0093 & 0.2928 \\
& & & SMILE & 0.0329 & 0.0282 & 0.5159 & 0.5893 \\
& & & SAPLM & 0.0156 & 0.0137 & 0.0092 & 0.2928 \\
& & & SLM & 6.0919 & 10.1303 & 2.0038 & 19.1599 \\
& & 5000 & ORACLE & 0.0154 & 0.0131 & 0.0087 & 0.2922 \\
& & & SMILE & 0.1042 & 0.0841 & 0.7177 & 1.1427 \\
& & & SAPLM & 0.0154 & 0.0131 & 0.0086 & 0.2921 \\
& & & SLM & 6.1067 & 10.0619 & 2.1242 & 19.1425 \\
& 1.0 & 1000 & ORACLE & 0.0446 & 0.0373 & 0.0256 & 1.1294 \\
& & & SMILE & 0.0719 & 0.0535 & 0.7567 & 1.6505 \\
& & & SAPLM & 0.0445 & 0.0373 & 0.0244 & 1.1272 \\
& & & SLM & 6.0704 & 10.1433 & 1.9960 & 19.5683 \\
& & 2000 & ORACLE & 0.0453 & 0.0357 & 0.0272 & 1.1313 \\
& & & SMILE & 0.1605 & 0.0717 & 1.2918 & 2.0046 \\
& & & SAPLM & 0.0452 & 0.0358 & 0.0259 & 1.1291 \\
& & & SLM & 6.2667 & 10.1503 & 2.1672 & 20.0240 \\
& & 5000 & ORACLE & 0.0459 & 0.0367 & 0.0250 & 1.1275 \\
& & & SMILE & 0.4953 & 0.2275 & 1.5863 & 3.6949 \\
& & & SAPLM & 0.0456 & 0.0367 & 0.0235 & 1.1255 \\
& & & SLM & 6.1453 & 10.0619 & 2.2305 & 20.0080 \\
500 & 0.5 & 1000 & ORACLE & 0.0103 & 0.0087 & 0.0064 & 0.2762 \\
& & & SMILE & 0.0103 & 0.0087 & 0.0064 & 0.2762 \\
& & & SAPLM & 0.0103 & 0.0087 & 0.0063 & 0.2762 \\
& & & SLM & 6.0317 & 10.1296 & 1.8271 & 18.3818 \\
& & 2000 & ORACLE & 0.0103 & 0.0088 & 0.0060 & 0.2755 \\
& & & SMILE & 0.0103 & 0.0088 & 0.0060 & 0.2755 \\
& & & SAPLM & 0.0103 & 0.0088 & 0.0061 & 0.2756 \\
& & & SLM & 6.0997 & 10.1132 & 1.8425 & 18.2931 \\
& & 5000 & ORACLE & 0.0105 & 0.0092 & 0.0065 & 0.2775 \\
& & & SMILE & 0.0105 & 0.0092 & 0.0064 & 0.2845 \\
& & & SAPLM & 0.0105 & 0.0092 & 0.0065 & 0.2775 \\
& & & SLM & 6.0894 & 10.1655 & 1.8348 & 18.5435 \\
& 1.0 & 1000 & ORACLE & 0.0297 & 0.0240 & 0.0179 & 1.0784 \\
& & & SMILE & 0.0296 & 0.0240 & 0.0174 & 1.0778 \\
& & & SAPLM & 0.0296 & 0.0240 & 0.0174 & 1.0778 \\
& & & SLM & 6.0328 & 10.1296 & 1.8276 & 19.1402 \\
& & 2000 & ORACLE & 0.0305 & 0.0245 & 0.0169 & 1.0770 \\
& & & SMILE & 0.0306 & 0.0247 & 0.0396 & 1.0846 \\
& & & SAPLM & 0.0304 & 0.0246 & 0.0164 & 1.0765 \\
& & & SLM & 6.1016 & 10.1132 & 1.8431 & 19.0421 \\
& & 5000 & ORACLE & 0.0297 & 0.0255 & 0.0173 & 1.0833 \\
& & & SMILE & 0.0297 & 0.0255 & 0.0170 & 1.1096 \\
& & & SAPLM & 0.0296 & 0.0255 & 0.0170 & 1.0826 \\
& & & SLM & 6.0649 & 10.1493 & 1.8395 & 19.3058 \\
\hline\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[htbp]
\caption{LM Case: Selection Statistics comparing the SMILE, SAPLM and SLM.}
\label{TAB:LM1}
\begin{center}
\renewcommand{0.92}{0.9}
\begin{tabular}{ccclrrrr}
\hline\hline
Size & Noise & & & \multicolumn{2}{c}{True Selection} & \multicolumn{2}{c}{False Selection} \\
\cmidrule(r){5-6} \cmidrule(l){7-8}
$n$ & $\sigma$ & $p$ & Method & corrX & corrX0 & NtoL & Nto0 \\
\hline
300 & 0.5 & 1000 & SMILE & 100 & 99.9975 & 0 & 0 \\
& & & SAPLM & 100 & 100 & 100 & 0 \\
& & & SLM & 100 & 100 & 0 & 0 \\
& & 2000 & SMILE & 100 & 99.9998 & 0 & 0 \\
& & & SAPLM & 100 & 100 & 100 & 0 \\
& & & SLM & 100 & 100 & 0 & 0 \\
& & 5000 & SMILE & 100 & 99.9998 & 0 & 0 \\
& & & SAPLM & 100 & 100 & 100 & 0 \\
& & & SLM & 100 & 100 & 0 & 0 \\
& 1.0 & 1000 & SMILE & 100 & 99.9890 & 0 & 0 \\
& & & SAPLM & 32.8333 & 100 & 32.8450 & 67.1667 \\
& & & SLM & 100 & 99.9995 & 0 & 0 \\
& & 2000 & SMILE & 100 & 99.9940 & 0 & 0 \\
& & & SAPLM & 17.3333 & 100 & 17.3550 & 82.6667 \\
& & & SLM & 100 & 99.9998 & 0 & 0 \\
& & 5000 & SMILE & 100 & 99.9982 & 0 & 0 \\
& & & SAPLM & 4.5000 & 100 & 4.5050 & 95.5000 \\
& & & SLM & 100 & 100 & 0 & 0 \\
500 & 0.5 & 1000 & SMILE & 100 & 100 & 0 & 0 \\
& & & SAPLM & 100 & 100 & 100 & 0 \\
& & & SLM & 100 & 100 & 0 & 0 \\
& & 2000 & SMILE & 100 & 100 & 0 & 0 \\
& & & SAPLM & 100 & 100 & 100 & 0 \\
& & & SLM & 100 & 100 & 0 & 0 \\
& & 5000 & SMILE & 100 & 100 & 0 & 0 \\
& & & SAPLM & 100 & 100 & 100 & 0 \\
& & & SLM & 100 & 100 & 0 & 0 \\
& 1.0 & 1000 & SMILE & 100 & 99.9945 & 0 & 0 \\
& & & SAPLM & 100 & 100 & 100 & 0 \\
& & & SLM & 100 & 100 & 0 & 0 \\
& & 2000 & SMILE & 100 & 99.9983 & 0 & 0 \\
& & & SAPLM & 100 & 100 & 100 & 0 \\
& & & SLM & 100 & 100 & 0 & 0 \\
& & 5000 & SMILE & 100 & 99.9993 & 0 & 0 \\
& & & SAPLM & 99.8333 & 100 & 99.8350 & 0.1667 \\
& & & SLM & 100 & 100 & 0 & 0 \\
\hline\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[htbp]
\caption{LM Case: Estimation Statistics comparing the SMILE, SAPLM and SLM.}
\label{TAB:LM2}
\begin{center}
\renewcommand{0.92}{0.725}
\begin{tabular}{ccclrrrr}
\hline\hline
Size & Noise & & & \multicolumn{3}{c}{AMSE $\times 10^{-3}$} & \multicolumn{1}{c}{CV-} \\
\cmidrule(lr){5-7}
$n$ & $\sigma$ & $p$ & Method & $\phi_1(\cdot)$ & $\phi_2(\cdot)$ & $\phi_3(\cdot)$ & MSPE \\
\hline
300 & 0.5 & 1000 & ORACLE & 0.7675 & 0.9103 & 0.9152 & 0.2548 \\
& & & SMILE & 0.7737 & 0.9264 & 0.9243 & 0.2551 \\
& & & SAPLM & 3.9953 & 4.1778 & 3.9635 & 0.2636 \\
& & & SLM & 1.4344 & 1.6371 & 1.4934 & 0.2549 \\
& & 2000 & ORACLE & 0.8341 & 0.6486 & 0.8838 & 0.2532 \\
& & & SMILE & 0.8335 & 0.6481 & 0.8837 & 0.2534 \\
& & & SAPLM & 3.9852 & 3.9137 & 3.9935 & 0.2619 \\
& & & SLM & 1.4163 & 1.3344 & 1.5203 & 0.2533 \\
& & 5000 & ORACLE & 0.8666 & 0.8955 & 0.6963 & 0.2526 \\
& & & SMILE & 0.8685 & 0.9001 & 0.6955 & 0.2529 \\
& & & SAPLM & 4.2351 & 4.1994 & 3.8121 & 0.2609 \\
& & & SLM & 1.6461 & 1.5967 & 1.3410 & 0.2528 \\
& 1.0 & 1000 & ORACLE & 3.1740 & 3.5473 & 3.5171 & 1.0206 \\
& & & SMILE & 3.1932 & 3.5094 & 3.5415 & 1.0299 \\
& & & SAPLM & 496.6754 & 881.6171 & 234.2496 & 3.2926 \\
& & & SLM & 3.8947 & 4.2478 & 4.1987 & 1.0215 \\
& & 2000 & ORACLE & 2.9228 & 2.7934 & 3.5526 & 1.0079 \\
& & & SMILE & 2.9376 & 2.7424 & 3.5271 & 1.0186 \\
& & & SAPLM & 606.3631 & 1071.5918 & 291.8006 & 3.3790 \\
& & & SLM & 3.5838 & 3.4188 & 4.4221 & 1.0090 \\
& & 5000 & ORACLE & 3.0392 & 3.4082 & 2.9891 & 1.0067 \\
& & & SMILE & 3.0248 & 3.4140 & 3.0178 & 1.0137 \\
& & & SAPLM & 711.5465 & 1265.1620 & 323.5512 & 3.4189 \\
& & & SLM & 3.8804 & 4.1732 & 4.0380 & 1.0076 \\
500 & 0.5 & 1000 & ORACLE & 0.5143 & 0.6210 & 0.4891 & 0.2532 \\
& & & SMILE & 0.5143 & 0.6210 & 0.4892 & 0.2532 \\
& & & SAPLM & 2.5860 & 2.7880 & 2.3860 & 0.2582 \\
& & & SLM & 1.2596 & 1.2759 & 1.0586 & 0.2533 \\
& & 2000 & ORACLE & 0.4745 & 0.5541 & 0.5393 & 0.2518 \\
& & & SMILE & 0.4745 & 0.5541 & 0.5393 & 0.2518 \\
& & & SAPLM & 2.7168 & 2.5863 & 2.3419 & 0.2569 \\
& & & SLM & 1.1549 & 1.2357 & 1.0850 & 0.2519 \\
& & 5000 & ORACLE & 0.5280 & 0.7104 & 0.6061 & 0.2514 \\
& & & SMILE & 0.5280 & 0.7104 & 0.6061 & 0.2514 \\
& & & SAPLM & 2.6505 & 2.8623 & 2.4300 & 0.2563 \\
& & & SLM & 1.2083 & 1.3043 & 1.0216 & 0.2514 \\
& 1.0 & 1000 & ORACLE & 1.9255 & 2.2321 & 2.0733 & 1.0141 \\
& & & SMILE & 1.9323 & 2.2494 & 2.0626 & 1.0170 \\
& & & SAPLM & 8.2207 & 7.9738 & 8.1074 & 1.0418 \\
& & & SLM & 2.7106 & 2.8192 & 2.7374 & 1.0145 \\
& & 2000 & ORACLE & 1.8588 & 2.2051 & 1.9679 & 1.0110 \\
& & & SMILE & 1.8588 & 2.2136 & 1.9783 & 1.0138 \\
& & & SAPLM & 8.6430 & 8.1768 & 7.9037 & 1.0356 \\
& & & SLM & 2.5655 & 2.7929 & 2.5935 & 1.0116 \\
& & 5000 & ORACLE & 2.0815 & 2.1319 & 1.9007 & 1.0055 \\
& & & SMILE & 2.0905 & 2.1545 & 1.9073 & 1.0076 \\
& & & SAPLM & 8.6299 & 8.9538 & 9.5719 & 1.0561 \\
& & & SLM & 2.8374 & 2.7330 & 2.4121 & 1.0058 \\
\hline\hline
\end{tabular}
\end{center}
\end{table}
Next we investigated the coverage rates of the proposed SCB. For each replication, we tested if the true functions can be covered by the SCB at the simulated values of the covariate in $[-0.5+h, 0.5-h]$, where $h$ is the bandwidth. Table \ref{TAB:AM3} shows the empirical coverage probabilities for a nominal 95\% confidence level out of 200 replications. For comparison, we also provided the SCBs from the SAPLM and ORACLE estimator. From Table \ref{TAB:AM3}, one observes that coverage probabilities for the SMILE, SAPLM and ORACLE SCBs all approach the nominal levels as the sample size $n$ increases. In most cases, the SMILE performs as well as or better than the SAPLM and arrives at about the nominal coverage when $n = 500$ and $\sigma = 1.0$.
\begin{table}[htbp]
\caption{AM Case: Coverage rates comparing the SMILE, SAPLM and SLM.}
\label{TAB:AM3}
\renewcommand{0.92}{0.9}
\begin{center}
\begin{tabular}{cccccccccccc}
\hline\hline
Size & Noise & & \multicolumn{3}{c}{ORACLE (\%)} & \multicolumn{3}{c}{SMILE (\%)} & \multicolumn{3}{c}{SAPLM (\%)} \\
\cmidrule(lr){4-6} \cmidrule(lr){7-9}\cmidrule(lr){10-12}
$n$ & $\sigma$ & $p$ & $\phi_1$ & $\phi_2$ & $\phi_3$ & $\phi_1$ & $\phi_2$ & $\phi_3$ & $\phi_1$ & $\phi_2$ & $\phi_3$ \\
\hline
300 & 0.5 & 1000 & 81.0 & 86.0 & 91.0 & 84.5 & 91.5 & 90.5 & 84.5 & 92.0 & 90.5 \\
& & 2000 & 80.5 & 84.5 & 88.0 & 85.5 & 92.0 & 92.2 & 83.0 & 92.0 & 91.5 \\
& & 5000 & 81.5 & 86.5 & 94.5 & 80.9 & 94.0 & 97.1 & 80.0 & 92.5 & 96.0 \\
& 1 & 1000 & 92.0 & 94.0 & 96.0 & 91.0 & 95.5 & 95.3 & 92.5 & 95.5 & 95.5 \\
& & 2000 & 93.0 & 96.0 & 95.5 & 91.0 & 95.0 & 95.9 & 93.5 & 97.0 & 95.0 \\
& & 5000 & 89.0 & 93.5 & 99.0 & 92.3 & 94.4 & 97.8 & 89.0 & 94.5 & 98.0 \\
500 & 0.5 & 1000 & 84.5 & 92.5 & 89.0 & 85.5 & 96.5 & 93.0 & 83.5 & 96.0 & 93.0 \\
& & 2000 & 85.0 & 86.5 & 85.5 & 86.0 & 93.5 & 88.5 & 85.5 & 93.5 & 88.5 \\
& & 5000 & 75.0 & 88.5 & 87.0 & 76.5 & 91.5 & 92.5 & 77.0 & 91.5 & 92.5 \\
& 1 & 1000 & 88.5 & 96.5 & 97.0 & 88.5 & 97.0 & 96.0 & 88.0 & 98.0 & 96.0 \\
& & 2000 & 92.0 & 97.5 & 96.0 & 91.5 & 97.5 & 95.5 & 91.5 & 97.5 & 95.5 \\
& & 5000 & 91.0 & 90.5 & 94.5 & 94.5 & 94.0 & 95.0 & 94.0 & 94.0 & 95.0 \\
\hline\hline
\end{tabular}
\end{center}
\end{table}
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\section*{C. A Simulation Study to Explore the Impacts of Covariate Interactions}
Our model considers the APLM, which focuses on variable selection, estimation, model identification and inference for main effects. There might be scenarios where the responses (measurement of SAM tissues, or phenotypes) are affected by interactions between SNP genotypes and RNA sequences.
To explore the robustness of our method in the behavior of selection and model identification for main terms, we conduct a simulation study under an underlying model that includes interaction terms. To be specific, we simulate datasets using the model:
\[
Y_i = \sum_{k=1}^{p_1}Z_{ik}\alpha_k + \sum_{\ell=1}^{p_2} \phi_{\ell}(X_{i\ell}) + \sum_{m=4}^5 Z_{im}\psi_m(X_{im}) + \varepsilon_i,
\]
where
\begin{align*}
&\alpha_1 = 3, ~ \alpha_2 = 4, ~ \alpha_3 = -2, ~ \alpha_4 = \ldots = \alpha_{p_1} = 0; \\
&\phi_1(x) = 9x,
~~~\phi_2(x) = -1.5 \cos^2(\pi x) + 3 \sin^2(\pi x) - \mathrm{E}\{-1.5 \cos^2(\pi X_2 + 3 \sin^2(\pi X_2)\}, \\
&\phi_3(x) = 6x + 18x^{2} - \mathrm{E}(6X_3 + 18X_3^{2}),
~~~\phi_4(x) = \ldots = \phi_{p_2}(x) = 0; \\
&\psi_4(x)= 6x, ~~\psi_5(x)=\frac{6}{1+\exp(-20x)}.
\end{align*}
We adopt the following similar criteria used in Section \ref{sec:simulation}:
\begin{enumerate}
\item[(B-i')] Percent of covariates in $\boldsymbol{Z}$ with nonzero linear coefficients (i.e., $Z_1$, $Z_2$ and $Z_3$) that are correctly identified (``CorrZ'');
\item[(B-ii')] Percent of covariates in $\boldsymbol{Z}$ with zero linear coefficients (all except $Z_1,\ldots,Z_5$) that are correctly identified (``CorrZ0'');
\item[(B-iii')] Percent of covariates in $\boldsymbol{X}$ with nonzero purely linear functions (i.e., $X_1$) that are correctly identified (``CorrL'');
\item[(B-iv')] Percent of covariates in $\boldsymbol{X}$ with nonzero purely nonlinear functions (i.e., $X_2$) that are correctly identified (``CorrN'');
\item[(B-v')] Percent of covariates in $\boldsymbol{X}$ with nonzero linear and nonlinear functions (i.e., $X_3$) that are correctly identified (``CorrLN');
\item[(B-vi')] Percent of in $\boldsymbol{X}$ with zero functions (all except $X_1,\ldots,X_5$) that are correctly identified covariates (``CorrX0'');
\item[(C-i')] Percent of covariates in $\boldsymbol{Z}$ with nonzero linear coefficients (i.e., $Z_1$, $Z_2$ and $Z_3$) incorrectly identified as having zero linear coefficients (``Zto0'');
\item[(C-ii')] Percent of covariates in $\boldsymbol{X}$ with nonzero purely linear functions (i.e., $X_1$) incorrectly identified as having nonlinear functions (``LtoN'');
\item[(C-iii')] Percent of covariates in $\boldsymbol{X}$ with nonzero purely nonlinear functions (i.e., $X_2$) incorrectly identified as having linear functions (``NtoL'');
\item[(C-iv')] Percent of covariates in $\boldsymbol{X}$ with nonzero linear or nonzero nonlinear functions (i.e., $X_1$, $X_2$ and $X_3$) incorrectly identified as having both zero linear and zero nonlinear functions (``Xto0'').
\end{enumerate}
Note that Criteria (B-i')--(B-vi') measure the frequency of getting the correct model structure; Criteria (C-i')--(C-v') measure the frequency of getting an incorrect model structure. All the above performance measures were computed based on 200 replicates.
The model selection results are provided in Tables \ref{TAB:Interaction1} and \ref{TAB:Interaction2}, respectively. The SMILE can effectively identify informative linear and nonlinear components as well as correctly discover the linear and nonlinear structure in covariates $\boldsymbol{X}$, while the SAPLM neglects the linear structure in $\boldsymbol{X}$ and SLM fails in presenting the nonlinear part of covariates $\boldsymbol{X}$. For the SMILE, the numbers of correctly selected nonzero covariates in $\boldsymbol{Z}$, linear, nonlinear, linear-and-nonlinear components in $\boldsymbol{X}$, nonzero covariates are very close to ORACLE (100\% for corrZ, corrL, corrN, corrLN, corrZ0 and corrX0, respectively); and the numbers of incorrectly identified components approach 0 as the sample size $n$ increases, as shown in Table \ref{TAB:Interaction2}. From the results in Tables \ref{TAB:Interaction1} and \ref{TAB:Interaction2}, it is evident that our method is robust in the sense that main effects are correctly identified in the presence of interaction effects; in contrast, neither SAPLM nor SLM performs well in this scenario. Especially for the selection of nonlinear and linear-nonlinear covariates in $\boldsymbol{X}$, which is our main focus for real data analysis, both SAPLM and SLM fail to select the right nonlinear and linear-nonlinear components in each simulation. Because SMILE, SAPLM, and SLM are based on additive models, none of these approaches are appropriate for detecting interactions.
Table \ref{TAB:Interaction3} reports the percentage of those covariates involved in the interaction ($Z_4$, $Z_5$, $X_4$ and $X_5$) selected out of 200 replications.
As shown in Table \ref{TAB:Interaction3}, SMILE can detect $Z_5$ and $X_5$ in most cases; and the percentages of selection approach 100 as the sample size $n$ increases. In contrast, SAPLM completely fails to select $Z_5$ and $X_5$; while SLM is only slightly worse than SMILE in the detection of $Z_5$, it has poor performance in the detection of $X_5$. For the interaction terms with smaller main-effect signal, i.e., $Z_4$ and $X_5$, SMILE outperforms in the detection of $X_4$ compared to SAPLM and SLM, and the detection power increases when the sample size $n$ increases. In addition, all three methods fail to detect the relevance of $Z_4$, due to the weak main-effect signal and interaction with $X_4$.
\begin{table}[htbp]
\caption{Statistics of true selection comparing the SMILE, SAPLM and SLM.}
\label{TAB:Interaction1}
\begin{center}
\renewcommand{0.92}{0.92}
\begin{tabular}{ccclrrrrrr}
\hline \hline
Size & Noise & & & \multicolumn{2}{c}{Z Part} & \multicolumn{ 4}{c}{X Part} \\
\cmidrule(r){5-6} \cmidrule(l){7-10}
$n$ & sig & $p$ & Method & corrZ & corrZ0 & corrL & corrN & corrLN & corrX0 \\
\hline
300 & 0.5 & 1000 & SMILE & 97.8333 & 99.9985 & 100 & 82.0000 & 82.0000 & 99.9970 \\
& & & SAPLM & 10.5000 & 100 & 0 & 0 & 0 & 100 \\
& & & SLM & 90.0000 & 99.9980 & 100 & 0 & 0 & 99.9965 \\
& & 2000 & SMILE & 95.6667 & 99.9995 & 100 & 55.5000 & 55.5000 & 99.9992 \\
& & & SAPLM & 9.1667 & 100 & 0 & 0 & 0 & 100 \\
& & & SLM & 85.5000 & 99.9995 & 100 & 0 & 0 & 99.9990 \\
& & 5000 & SMILE & 88.6667 & 99.9994 & 100 & 29.0000 & 28.5000 & 99.9997 \\
& & & SAPLM & 6.6667 & 100 & 0 & 0 & 0 & 100 \\
& & & SLM & 80.1667 & 99.9998 & 100 & 0 & 0 & 99.9995 \\
& 1.0 & 1000 & SMILE & 93.8333 & 99.9950 & 100 & 49.0000 & 49.0000 & 99.9990 \\
& & & SAPLM & 7.3333 & 100 & 0 & 0 & 0 & 100 \\
& & & SLM & 86.0000 & 99.9990 & 99.5000 & 0 & 0 & 99.9975 \\
& & 2000 & SMILE & 90.3333 & 99.9967 & 100 & 19.5000 & 19.0000 & 99.9987 \\
& & & SAPLM & 8.5000 & 100 & 0 & 0 & 0 & 100 \\
& & & SLM & 80.5000 & 99.9990 & 99.0000 & 0 & 0 & 99.9992 \\
& & 5000 & SMILE & 86.0000 & 99.9977 & 100 & 7.5000 & 7.0000 & 99.9999 \\
& & & SAPLM & 7.1667 & 100 & 0 & 0 & 0 & 100 \\
& & & SLM & 74.1667 & 99.9999 & 99.0000 & 0 & 0 & 100 \\
500 & 0.5 & 1000 & SMILE & 100 & 100 & 100 & 100 & 100 & 99.9995 \\
& & & SAPLM & 64.0000 & 99.9980 & 0 & 56.0000 & 0 & 100 \\
& & & SLM & 99.6667 & 100 & 100 & 0 & 0 & 99.9985 \\
& & 2000 & SMILE & 100 & 100 & 100 & 100 & 100 & 99.9992 \\
& & & SAPLM & 22.1667 & 100 & 0 & 11.0000 & 0 & 100 \\
& & & SLM & 98.3333 & 100 & 100 & 0 & 0 & 99.9992 \\
& & 5000 & SMILE & 100 & 100 & 100 & 100 & 100 & 99.9999 \\
& & & SAPLM & 14.1667 & 100 & 0 & 0 & 0 & 100 \\
& & & SLM & 98.5000 & 99.9998 & 100 & 0 & 0 & 99.9999 \\
& 1.0 & 1000 & SMILE & 100 & 100 & 100 & 100 & 100 & 99.9990 \\
& & & SAPLM & 24.1667 & 99.9995 & 0 & 9.5000 & 0 & 100 \\
& & & SLM & 98.8333 & 100 & 100 & 0 & 0 & 99.9985 \\
& & 2000 & SMILE & 100 & 99.9992 & 100 & 100 & 100 & 99.9990 \\
& & & SAPLM & 11.8333 & 100 & 0 & 0.5000 & 0 & 100 \\
& & & SLM & 97.1667 & 99.9992 & 100 & 0 & 0 & 99.9987 \\
& & 5000 & SMILE & 100 & 100 & 100 & 99.0000 & 99.0000 & 100 \\
& & & SAPLM & 13.5000 & 100 & 0 & 0 & 0 & 100 \\
& & & SLM & 97.1667 & 100 & 100 & 0 & 0 & 99.9998 \\
\hline \hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[htbp]
\caption{Statistics of false selection comparing the SMILE, SAPLM and SLM.}
\label{TAB:Interaction2}
\begin{center}
\renewcommand{0.92}{0.92}
\begin{tabular}{ccclrrrr}
\hline \hline
Size & Noise & & & Z Part & \multicolumn{3}{c}{X Part} \\
\cmidrule(r){5-5} \cmidrule(l){6-8}
$n$ & sig & $p$ & Method & Zto0 & LtoN & NtoL & Xto0 \\
\hline
300 & 0.5 & 1000 & SMILE & 2.1667 & 0 & 0 & 6.0000 \\
& & & SAPLM & 89.5000 & 0 & 0 & 100 \\
& & & SLM & 10.0000 & 0 & 0 & 33.3333 \\
& & 2000 & SMILE & 4.3333 & 0 & 0 & 14.8333 \\
& & & SAPLM & 90.8333 & 0 & 0 & 100 \\
& & & SLM & 14.5000 & 0 & 0 & 33.6667 \\
& & 5000 & SMILE & 11.3333 & 0 & 0 & 23.6667 \\
& & & SAPLM & 93.3333 & 0 & 0 & 100 \\
& & & SLM & 19.8333 & 0 & 0 & 33.8333 \\
& 1.0 & 1000 & SMILE & 6.1667 & 0 & 0 & 17.0000 \\
& & & SAPLM & 92.6667 & 0 & 0 & 100 \\
& & & SLM & 14.0000 & 0 & 0 & 33.6667 \\
& & 2000 & SMILE & 9.6667 & 0 & 0 & 26.8333 \\
& & & SAPLM & 91.5000 & 0 & 0 & 100 \\
& & & SLM & 19.5000 & 0 & 0 & 34.8333 \\
& & 5000 & SMILE & 14.0000 & 0 & 0 & 30.8333 \\
& & & SAPLM & 92.8333 & 0 & 0 & 100 \\
& & & SLM & 25.8333 & 0 & 0 & 35.0000 \\
500 & 0.5 & 1000 & SMILE & 0 & 0 & 0 & 0 \\
& & & SAPLM & 36.0000 & 56.0000 & 0 & 44.0000 \\
& & & SLM & 0.3333 & 0 & 0 & 33.3333 \\
& & 2000 & SMILE & 0 & 0 & 0 & 0 \\
& & & SAPLM & 77.8333 & 11.0000 & 0 & 89.0000 \\
& & & SLM & 1.6667 & 0 & 0 & 33.3333 \\
& & 5000 & SMILE & 0 & 0 & 0 & 0 \\
& & & SAPLM & 85.8333 & 0 & 0 & 100 \\
& & & SLM & 1.5000 & 0 & 0 & 33.3333 \\
& 1.0 & 1000 & SMILE & 0 & 0 & 0 & 0 \\
& & & SAPLM & 75.8333 & 9.5000 & 0 & 90.5000 \\
& & & SLM & 1.1667 & 0 & 0 & 33.3333 \\
& & 2000 & SMILE & 0 & 0 & 0 & 0 \\
& & & SAPLM & 88.1667 & 0.5000 & 0 & 99.5000 \\
& & & SLM & 2.8333 & 0 & 0 & 33.3333 \\
& & 5000 & SMILE & 0 & 0 & 0 & 0.3333 \\
& & & SAPLM & 86.5000 & 0 & 0 & 100 \\
& & & SLM & 2.8333 & 0 & 0 & 33.3333 \\
\hline \hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[htbp]
\caption{Percents of $Z_4$, $Z_5$, $X_4$ and $X_5$ are selected comparing the SMILE, SAPLM and SLM.}
\label{TAB:Interaction3}
\begin{center}
\renewcommand{0.92}{0.92}
\begin{tabular}{ccclrrrr}
\hline \hline
Size & Noise & & & \multicolumn{2}{c}{Z Part} & \multicolumn{2}{c}{X Part} \\
\cmidrule(r){5-6} \cmidrule(l){7-8}
$n$ & sig & $p$ & Method & \multicolumn{1}{c}{Z4} & \multicolumn{1}{c}{Z5} & \multicolumn{1}{c}{X4} & \multicolumn{1}{c}{X5} \\
\hline
300 & 0.5 & 1000 & SMILE & 0 & 100 & 33 & 79.5 \\
& & & SAPLM & 0 & 8.5 & 0 & 0 \\
& & & SLM & 0 & 98.5 & 5.5 & 21 \\
& & 2000 & SMILE & 0 & 99 & 15.5 & 52 \\
& & & SAPLM & 0 & 6 & 0 & 0 \\
& & & SLM & 0 & 98 & 3 & 20 \\
& & 5000 & SMILE & 0 & 96.5 & 4.5 & 26.5 \\
& & & SAPLM & 0 & 3 & 0 & 0 \\
& & & SLM & 0 & 92 & 2 & 13 \\
& 1.0 & 1000 & SMILE & 0 & 99.5 & 11 & 46 \\
& & & SAPLM & 0 & 7 & 0 & 0 \\
& & & SLM & 0 & 94.5 & 4 & 15 \\
& & 2000 & SMILE & 0 & 98.5 & 4 & 20.5 \\
& & & SAPLM & 0 & 2.5 & 0 & 0 \\
& & & SLM & 0 & 95.5 & 1.5 & 13 \\
& & 5000 & SMILE & 0 & 97 & 1 & 13 \\
& & & SAPLM & 0 & 2.5 & 0 & 0 \\
& & & SLM & 0 & 86.5 & 1 & 7 \\
500 & 0.5 & 1000 & SMILE & 0 & 100 & 85 & 100 \\
& & & SAPLM & 0 & 61.5 & 0 & 0 \\
& & & SLM & 0 & 100 & 21 & 70 \\
& & 2000 & SMILE & 0 & 100 & 75.5 & 100 \\
& & & SAPLM & 0 & 21 & 0 & 0 \\
& & & SLM & 0 & 100 & 12.5 & 60 \\
& & 5000 & SMILE & 0 & 100 & 66 & 99 \\
& & & SAPLM & 0 & 12.5 & 0 & 0 \\
& & & SLM & 0 & 100 & 6.5 & 50 \\
& 1.0 & 1000 & SMILE & 0 & 100 & 73.5 & 98 \\
& & & SAPLM & 0 & 22 & 0 & 0 \\
& & & SLM & 0 & 100 & 14.5 & 58 \\
& & 2000 & SMILE & 0 & 100 & 55 & 94 \\
& & & SAPLM & 0 & 11 & 0 & 0 \\
& & & SLM & 0 & 100 & 7 & 49 \\
& & 5000 & SMILE & 0 & 100 & 41.5 & 91.5 \\
& & & SAPLM & 0 & 11 & 0 & 0 \\
& & & SLM & 0 & 99.5 & 4 & 42 \\
\hline \hline
\end{tabular}
\end{center}
\end{table}
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\section*{D. A Simulation Study Based on the SAM Data}
In this section, we conduct a simulation study using the SNPs and RNA transcripts selected in real data analysis as the active covariates in our data-generating model. This demonstrates the performance of our method when there are many true nonzero components in both linear and nonlinear parts.
With the true linear coefficients and nonlinear functions set to be the same as the estimates obtained in real data analysis in Section 6, we choose the noise level $\sigma$ as 0.01, 0.02, 0.04 and 0.05, in accordance with the errors in real data analysis ($\widehat{\sigma}\approx 0.04$). We compare SMILE with SAPLM and SLM. We still summarize the simulation results by using the statistics described in Section \ref{sec:simulation}. All the performance measures were computed based on 200 replicates.
The model selection results are provided in Tables \ref{SAM:SIM:1} and \ref{SAM:SIM:2}, respectively. The SMILE can effectively identify informative linear and nonlinear components as well as correctly discover the linear and nonlinear structure in covariate $\boldsymbol{X}$, while the SAPLM neglects linear structure in $\boldsymbol{X}$ and SLM fails in presenting the nonlinear part of covariate $\boldsymbol{X}$. For the SMILE, the selection performance in $\boldsymbol{Z}$, including the numbers of correctly selected nonzero and zero covariates in $\boldsymbol{Z}$ (corrZ and corrZ0) and the numbers of incorrectly identified components in $\boldsymbol{Z}$ (Zto0), is very close to the performance of SLM; while SMILE outperforms SLM in the selection of important components of $\boldsymbol{X}$, indicated by the much higher percents of correctly selected components in $\boldsymbol{X}$ (corrL, corrN, and corrX0) and much lower percents in the incorrectly selected components in $\boldsymbol{X}$ (Xto0). The SMILE outperforms SAPLM in almost all statistics. From the results in Tables \ref{SAM:SIM:1} and \ref{SAM:SIM:2}, it is evident that model misspecification leads to poor variable selection performance for the SAPLM and SLM.
\begin{table}[htbp]
\caption{True selection statistics comparing the SMILE, SAPLM and SLM.}
\label{SAM:SIM:1}
\begin{center}
\begin{tabular}{clrrrrr}
\hline\hline
Noise & & \multicolumn{2}{c}{Z Part} & \multicolumn{3}{c}{X Part} \\
\cmidrule(r){3-4}\cmidrule(l){5-7}
$\sigma$ & Method & \multicolumn{1}{l}{corrZ} & \multicolumn{1}{l}{corrZ0} & \multicolumn{1}{l}{corrL} & \multicolumn{1}{l}{corrN} & \multicolumn{1}{l}{corrX0} \\
\hline
0.01 & SMILE & 71.62 & 99.81 & 81.9 & 99.5 & 98.61 \\
& SAPLM & 38.57 & 99.66 & 0 & 0 & 99.43 \\
& SLM & 78.83 & 99.72 & 25.8 & 0 & 99.59 \\
0.02 & SMILE & 69.31 & 99.80 & 75.55 & 96.75 & 98.50 \\
& SAPLM & 38.36 & 99.62 & 0 & 0 & 99.37 \\
& SLM & 77.60 & 99.69 & 26.3 & 0 & 99.37 \\
0.04 & SMILE & 65.80 & 99.74 & 65.45 & 87.5 & 98.07 \\
& SAPLM & 36.95 & 99.56 & 0 & 0 & 99.25 \\
& SLM & 74.19 & 99.57 & 23.35 & 0 & 98.89 \\
0.05 & SMILE & 64.35 & 99.72 & 57.75 & 85.5 & 97.79 \\
& SAPLM & 36.06 & 99.53 & 0 & 0 & 99.20 \\
& SLM & 72.49 & 99.49 & 22.35 & 0 & 98.65 \\
\hline\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[htbp]
\caption{False selection statistics comparing the SMILE, SAPLM and SLM.}
\label{SAM:SIM:2}
\begin{center}
\begin{tabular}{clrrrr}
\hline\hline
Noise & & \multicolumn{1}{c}{Z Part} & \multicolumn{3}{c}{X Part} \\
\cmidrule(l){4-6}
$\sigma$ & Method & Zto0 & LtoN & NtoL & Xto0 \\
\hline
0.01 & SMILE & 28.38 & 3.6 & 0 & 0.08 \\
& SAPLM & 61.43 & 25.8 & 0 & 78.50 \\
& SLM & 21.17 & 0 & 0 & 16.67 \\
0.02 & SMILE & 30.69 & 7.45 & 0 & 0.71 \\
& SAPLM & 61.64 & 26.3 & 0 & 78.08 \\
& SLM & 22.40 & 0 & 0 & 16.67 \\
0.04 & SMILE & 34.20 & 13.95 & 0 & 3.25 \\
& SAPLM & 63.05 & 23.35 & 0 & 80.54 \\
& SLM & 25.81 & 0 & 0 & 16.79 \\
0.05 & SMILE & 35.65 & 17.9 & 0 & 4.58 \\
& SAPLM & 63.94 & 22.35 & 0 & 81.37 \\
& SLM & 27.51 & 0 & 0 & 17.38 \\
\hline\hline
\end{tabular}
\end{center}
\end{table}
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\section*{E. Technical Details}
This section contains some technical assumptions, lemmas and proofs. For any real numbers $a$ and $b$, let $a \vee b$ and $a \wedge b$ denote the maximum and minimum of $a$ and $b$, respectively. For any two sequences $\{a_n\}$, $\{b_n\}$, $n = 1, 2, \ldots$, we use $a_n \asymp b_n$ if there are constants $0 < c_1 < c_2 < \infty$ such that $c_1 < a_n / b_n < c_2$ for all $n$ sufficiently large. On any fixed interval $[a,b]$, we denote the space of the second order smooth functions as $C^{(d)}[a,b]=\left\{f\left|f^{(d)}\in C[a,b]\right\}\right.$ and the class of Lipschitz continuous functions for any fixed constant $C>0$ as Lip $([a,b],C)=\{f| \left|f(x)-f(x^\prime)\right|\leq C\left|x-x^\prime\right|,\forall x, x^\prime \in [a,b]\}$.
Furthermore, let $\mathbf{Y} = (Y_1, \ldots, Y_n)^{\top}$ be an $n$-dimensional vector, $\mathbf{Z} =(\mathbf{Z}_1, \ldots, \mathbf{Z}_{p_1})$ be an $n \times p_1$ matrix, where $\mathbf{Z}_k = (Z_{1k}, \ldots, Z_{nk})^{\top}$, $k = 1, \ldots, p_1$, and $\mathbf{X} =(\mathbf{X}_1, \ldots, \mathbf{X}_{p_2})$ be an $n \times p_2$ matrix, where $\mathbf{X}_{\ell} = (X_{1\ell}, \ldots, X_{n\ell})^{\top}$, $\ell=1, \ldots, p_2$.
Let $\mathbf{B}^{(d)} = (\mathbf{B}_1^{(d)}, \ldots, \mathbf{B}_{p_2}^{(d)})$ be a dimension $n \times (p_2 M_n)$ matrix, where $\mathbf{B}_{\ell}^{(d)} = (\mathbf{B}_{\ell}^{(d)}(X_{1\ell}), \ldots, \mathbf{B}_{\ell}^{(d)}(X_{n\ell}))^{\top}$ is a dimension $n \times M_n$ matrix of spline basis functions of order $d$, for $\ell= 1,\ldots, p_2$. Let $\mathcal{A} \subseteq \{1, \ldots, p_1 + 2p_2\}$ be an index set, and let $|\mathcal{A}|$ denote the cardinality of set $\mathcal{A}$.
\subsection{Technical Assumptions} \label{SUBSEC:assump}
In addition to the sparsity condition (A1) stated in Section \ref{sec:method}, we need the following additional regularity conditions to establish the theoretical results in this paper.
\begin{enumerate}
\item[(A2)] (\textit{Conditions on errors}) The errors $\varepsilon_1, \ldots, \varepsilon_n$ are independent and identically distributed with $\mathrm{E}(\varepsilon_i )= 0$, $\mathrm{Var}(\varepsilon_i) = \sigma^2$, $\mathrm{E} |\varepsilon_i|^{2 + \delta} \leq M_{\delta}$ for some positive constant $M_{\delta}$ ($\delta > 0.5$), and have $b$-sub-gaussian tails, i.e., $\mathrm{E} \{\exp({t\varepsilon})\} \leq \exp({b^2t^2 / 2})$, for any $t \geq 0$ and some $b > 0$.
\item[(A3)] (\textit{Conditions on nonlinear functions}) The additive component function $g_{\ell}(\cdot) \in C^{(2)}[a,$ $b]$, $\ell=1, \ldots, p_2$.
\item[(A4)] (\textit{Conditions on covariates}) Each covariate in the parametric part of the model
is bounded,
that is, there is a positive constant $C_3$ such that $
|Z_{k}|
\leq C_3, \, 1 \leq k \leq p_1$;
also, $\mathrm{E}(X_{\ell})=0$, and there is a positive constant $C_4$ such that $
|X_{\ell}|
\leq C_4, \, 1 \leq \ell \leq p_2$. The joint density function of active pure linear $\boldsymbol{X}$ is continuous and bounded below and above.
Each covariate in the nonparametric part of the model has a continuous density and there exist constants $C_1$ and $C_2$ such that the marginal density function $f_{\ell}$ of $X_{\ell}$ has continuous derivatives on its support, and satisfies $0 < C_1 \leq f_{\ell}(x_{\ell}) \leq C_2 < \infty$ on its support for every $1 \leq \ell \leq p_{2}$. In addition, the eigenvalues of $\mathrm{E} \{(\boldsymbol{Z} \boldsymbol{Z}^{\top}) | \boldsymbol{X}\}$ are bounded away from 0.
\item[(A5)] (\textit{Conditions on the initial estimators}) The initial estimators satisfy $r_{n1}\max\limits_{k \in \mathcal{N}_z} |\widetilde{\alpha}_k| = O_P(1)$, $r_{n2}\max\limits_{\ell \in \mathcal{N}_x} |\widetilde{\beta}_{\ell}| = O_P(1)$, $r_{n3}\max\limits_{\ell \in \mathcal{N}_x} \Vert\widetilde{\boldsymbol{\gamma}}_{\ell} \Vert_2 = O_P(1)$, $r_{n1}$, $r_{n2}$, $r_{n3} \to \infty$, and there exist positive constants $c_{b1}$, $c_{b2}$ and $c_{b3}$ such that $\Pr \left(\min\limits_{k \in \mathcal{S}_z} |\widetilde{\alpha}_{k}| \geq c_{b1}b_{n1}\right) \to 1$, $\Pr \left(\min\limits_{\ell \in \mathcal{S}_{x, L}} |\widetilde{\beta}_{\ell}| \geq c_{b2}b_{n2}\right) \to 1$, $\Pr \left(\min\limits_{\ell \in \mathcal{S}_{x, N}} \Vert \widetilde{\boldsymbol{\gamma}}_{\ell}\Vert_2 \geq c_{b3}b_{n3}\right) \to 1$, where $b_{n1} = \min_{k \in \mathcal{S}_z} |\alpha_{0k}|$, $b_{n2} = \min_{\ell \in \mathcal{S}_{x, L}} |\beta_{0\ell}|$, and $b_{n3} = \min_{\ell \in \mathcal{S}_{x, N}} \Vert g_{0\ell} \Vert_2$.
\item[(A6)] (\textit{Conditions on parameters and spline basis functions}) Let $p_1$ and $p_2$ be the number of linear and nonlinear components, respectively. Suppose that $n^{-1}N_n + n^{-2}\sum_{j=1}^{3}\lambda_{nj}^2 = o(1)$, and
\[
\frac{\sqrt{n\ln(p_1)}}{\lambda_{n1}r_{n1}} + \frac{\sqrt{n\ln(p_2)}}{\lambda_{n2}r_{n2}} + \frac{\sqrt{n N_n\ln(p_2 N_n)}}{\lambda_{n3}r_{n3}} + \sum_{j=1}^{3}\frac{n}{\lambda_{nj}r_{nj}N_n}= o(1).
\]
\end{enumerate}
Assumptions (A1)--(A4) are regularity conditions that are commonly used in the APLM literature. To obtain the selection consistency of the SBLL-AGLASSO, we need an order requirement for a general initial estimator; see Assumption (A5).
Theorem \ref{THM:LASSO-1} below demonstrates that the group LASSO estimator defined in (\ref{EQ:loss_glasso_spline}) satisfies Assumption (A5) under some weak conditions, specifically if $\sum_{j=1}^{3}\widetilde{\lambda}_{nj}^2 \asymp {n \{\ln(p_1) \vee N_n \ln(p_2N_n)\}}$ and
$N_n \asymp n^{1/3}$,
then the consistent rates for the group LASSO estimator in (A5) have order {$r_{n1} \asymp r_{n2} \asymp r_{n3} = O\{n^{1/2} / \sqrt{\ln (p_1) \vee N_n \ln(p_2 N_n)} \}$}. Consequently, Assumption (A6) is equivalent to:
\begin{equation}
\frac{\sum_{j=1}^{3}\lambda_{nj}^2}{n^2}
+ \frac{\ln (p_1) \vee N_n \ln(p_2 N_n)}{(\lambda_{n1} \wedge \lambda_{n2} \wedge \lambda_{n3})}
+ \frac{n^{1/6}\sqrt{\ln (p_1) \vee N_n \ln(p_2 N_n)}}{(\lambda_{n1} \wedge \lambda_{n2} \wedge \lambda_{n3})} = o(1), \label{EQN:order}
\end{equation}
If we take $\lambda_{n1} \asymp \lambda_{n2} \asymp \lambda_{n3} = O(n^{1/2})$, then (\ref{EQN:order}) indicates $p_1 = \exp\{o(n^{1/2})\}$ and $p_2 = \exp\{o(n^{1/6})\}$.
We need the following additional assumptions in order to develop the asymptotic SCBs for the nonparametric components.
\begin{enumerate}
{\renewcommand{-1pt}{-1pt}
\item[(A3$^{\prime}$)] (\textit{Conditions on nonlinear functions}) For any $\ell \in \mathcal{S}_{x,N}$, $\phi_{0\ell} \in {C}^{(d)}[a,b]$, for some integer $d\geq 2$. In addition, $\psi_{\ell}^x$ defined in (\ref{DEF:psi}) satisfies $\psi_{\ell}^x \in C^{(d)}[a,b]$.
\item[(A6$^{\prime}$)](\textit{Conditions on spline basis functions}) The order of the spline basis functions is at least $d$, and the number of interior knots $M_n$ satisfies: $\left\{ n^{1/\left( 2d\right)} \vee n^{4/(10d-5)} \right\} \ll M_{n}\ll n^{1/3}$.
\item[(B1)] (\textit{Conditions on the kernel function}) The kernel function $K \in \mbox{Lip\ }([-1,1],C_{K})$ for some constant $C_{K}>0$, and is bounded, nonnegative, symmetric, and supported on $\left[
-1,1\right] $ with the second moment $\mu_{2}(K)=\int u^{2}K\left( u\right) du$.
\item[(B2)] (\textit{Conditions on bandwidth}) For each $\ell\in \mathcal{S}_{x,N}$, the bandwidth of the kernel $K$ is $h_{\ell}^{-1}=O(n^{1/5}\ln^{\delta}n)$ for some constant $\delta>1/5$.}
\end{enumerate}
Assumptions (A3$^{\prime}$), (B1) and (B2) are typical in the local polynomial smoothing literature; see, for instance, \cite{zheng2016statistical}. Assumption (A6$^{\prime}$) imposes the condition of the number of knots for spline smoothing. For example, if $d=2$, we can
take $M_{n}\sim n^{4/15}\ln n$.
\subsection{Selection and estimation properties of the group LASSO estimators}
In this section, we consider the selection and estimation properties of the group LASSO estimator $\boldsymbol{\widetilde{\boldsymbol{\theta}}} = (\widetilde{\boldsymbol{\alpha}}^{\top}, \widetilde{\boldsymbol{\beta}}^{\top}, \widetilde{\boldsymbol{\gamma}}^{\top})^{\top}$ in (\ref{EQ:loss_glasso_spline}). In the following, denote $\boldsymbol{\alpha} = (\alpha_1, \ldots, \alpha_{p_1})^{\top}$ with length $p_1$, $\boldsymbol{\beta} = (\beta_1, \ldots, \beta_{p_2})^{\top}$ with length $p_2$, and $\boldsymbol{\gamma} = (\boldsymbol{\gamma}_1^{\top}, \ldots, \boldsymbol{\gamma}_{p_2}^{\top})^{\top} $ with length ($p_2 N_n$).
Let
\[
\boldsymbol{\theta}^{\top}\!\! = \!(\boldsymbol{\alpha}^{\top}\!, \boldsymbol{\beta}^{\top}\!, \boldsymbol{\gamma}^{\top}) \!= \!\left(\alpha_1, \ldots, \alpha_{p_1}, \beta_1, \ldots, \beta_{p_2}, \boldsymbol{\gamma}_1^{\top}\!, \ldots, \boldsymbol{\gamma}_{p_2}^{\top} \right)\!=\! \left(\boldsymbol{\theta}_1^{\top}\!, \ldots, \boldsymbol{\theta}_m^{\top}, \ldots, \boldsymbol{\theta}_{p_1 + 2p_2}^{\top}\right),
\]
where $\boldsymbol{\theta}_m = \alpha_m I\{1 \leq m \leq p_1\} + \beta_{m-p_1} I\{p_1+1 \leq m \leq p_1+p_2\} + \boldsymbol{\gamma}_{m - p_1-p_2} I\{p_1+p_2+1 \leq m \leq p_1 + 2p_2 \}$, with $I(\cdot)$ being an indicator function.
Let
\[
\mathbf{D} = (\mathbf{Z}_1, \ldots, \mathbf{Z}_{p_1}, \mathbf{X}_1, \ldots, \mathbf{X}_{p_2}, \mathbf{B}_1^{(1)},\ldots, \mathbf{B}_{p_2}^{(1)}) \equiv \left(\mathbf{D}_1, \ldots, \mathbf{D}_m, \ldots, \mathbf{D}_{p_1 + 2p_2}\right)
\]
be an $n \times (p_1 + p_2 + p_2N_n)$ matrix, where
\[
\mathbf{D}_m = \mathbf{Z}_m I\{1 \leq m \leq p_1\} + \mathbf{X}_m I\{1 \leq m \leq p_1\} + \mathbf{B}_{m - p_1-p_2}^{(1)} I\{p_1+p_2 +1 \leq m \leq p_1 + 2p_2\},
\]
an $n \times d_m$ submatrix of $\mathbf{D}$ with $d_m= I(1 \leq m \leq p_1) + I(p_1 + 1 \leq m \leq p_1+p_2) + N_n I(p_1+p_2 + 1 \leq m \leq p_1 + 2p_2)$. Define
\begin{equation}
\widetilde{\mathcal{S}} = \{m: \Vert \boldsymbol{\widetilde{\boldsymbol{\theta}}}_m\Vert \neq 0, 1 \leq m \leq p_1 + 2p_2\}.
\label{DEF:S_tilde}
\end{equation}
Next we define the active linear index set for $\boldsymbol{X}$ as $\mathcal{S}_{x,L}=\mathcal{S}_{x, PL}\cup\mathcal{S}_{x, LN}$, the inactive linear index set for $\boldsymbol{X}$ as $\mathcal{N}_{x, L}$, and the inactive nonlinear index set for $\boldsymbol{X}$ as $\mathcal{N}_{x, N}$. Note that $\mathcal{N}_{x}=\mathcal{N}_{x, L}\cap\mathcal{N}_{x, N}$. Further, let
\begin{align}
&\mathcal{S} =\mathcal{S}_z \, \cup \, \{\ell + p_1: \ell \in\mathcal{S}_{x,L}\} \, \cup \, \{\ell + p_1 + p_2: \ell \in\mathcal{S}_{x,N}\}, \nonumber \\
&\mathcal{N} =\mathcal{N}_z \, \cup \, \{\ell + p_1: \ell \in\mathcal{N}_{x, L}\} \, \cup \, \{\ell + p_1 + p_2: \ell \in\mathcal{N}_{x, N}\}.
\label{DEF:SN}
\end{align}
For any index set $\mathcal{A} \subseteq \{1, \ldots, p_1 + 2p_2\}$, define $\mathbf{D}_{\mathcal{A}} = \{\mathbf{D}_{m}: m \in \mathcal{A}\}$.
Next denote $\mathbf{C}_\mathcal{A} = n^{-1}\mathbf{D}_\mathcal{A}^{\top}\mathbf{D}_\mathcal{A}$, and let $\pi_{\min}(\mathbf{C}_\mathcal{A})$ and $\pi_{\max}(\mathbf{C}_\mathcal{A})$ represent the minimum and maximum eigenvalues of $\mathbf{C}_\mathcal{A}$, respectively.
\begin{lemma}\label{LEM:eigen}
Let $N_n = O(n^{\gamma})$, where $0 < \gamma < 0.5$. Suppose that $|\mathcal{A}|$ is bounded by a fixed constant independent of n, $p_{1}$ and $p_{2}$. Then under Assumption (A4), with probability approaching one as $n \to \infty$, $c_1 \leq \pi_{\min}(\mathbf{C}_\mathcal{A}) \leq \pi_{\max}(\mathbf{C}_\mathcal{A}) \leq c_2$, where $c_1$ and $c_2$ are two positive constants.
\end{lemma}
\begin{proof}
Similar to the proof of Lemma A.1 in \cite{li2017ultra}.
\end{proof}
\begin{lemma}
\label{LEM:spline-approx}
Under Assumption (A3), there exists a vector $\boldsymbol{\gamma}_0=(\boldsymbol{\gamma}_{0 1}^{\top}, \ldots, \boldsymbol{\gamma}_{0 p_2}^{\top})^{\top}$, such that
$\|\boldsymbol{\gamma}_{0\ell}\|\neq 0$, for $\ell \in \mathcal{S}_{x, N}$, $\|\boldsymbol{\gamma}_{0\ell}\|=0$, $\ell \in \mathcal{N}_{x,N}$
and $\Vert g_{0\ell}-\mathbf{B}_{\ell}^{(d)\top} \boldsymbol{\gamma}_{0\ell}\Vert_2=O(M_n^{-d})$.
\end{lemma}
\begin{proof}
Similar to the proof of Lemma A.2 in \cite{li2017ultra}.
\end{proof}
In the following, we denote $g_{n\ell}(\cdot) = \sum_{J = 1}^{N_n} \gamma_{0\ell J} {B}_{J,\ell}^{(1)}\left(\cdot\right)$ the best constant spline approximation of $g_{0\ell}(\cdot)$ such that $\Vert g_{0\ell} - g_{n\ell} \Vert_{\infty} = \sup_{x \in [a, b]} |g_{0\ell}(x) - g_{n\ell}(x)| = O(N_n^{-1})$. Let $\boldsymbol{\gamma}_{0\ell}=(\gamma_{0\ell J}, J=1,\ldots,N_n)^{\top}$ be the vector of the coefficients of the best spline approximation in Lemma \ref{LEM:spline-approx}. Denote $\boldsymbol{\theta}_{0}^{\top} = (\boldsymbol{\theta}_{01}^{\top}, \ldots, \boldsymbol{\theta}_{0m}^{\top}, \ldots, \boldsymbol{\theta}_{0,p_1 + 2p_2}^{\top}) = (\boldsymbol{\alpha}_{0}^{\top}, \boldsymbol{\beta}_{0}^{\top}, \boldsymbol{\gamma}_{0}^{\top})
= (\alpha_{01}, \ldots, \alpha_{0p_1},$ $\beta_{01},\ldots, \beta_{0p_2}, \boldsymbol{\gamma}_{01}^{\top}, \ldots, \boldsymbol{\gamma}_{0p_2}^{\top} )$. Define $\boldsymbol{\theta}_{\mathcal{A}} = (\boldsymbol{\theta}_{m}^{\top}: m \in \mathcal{A})^{\top}$, $\boldsymbol{\theta}_{0,\mathcal{A}} = (\boldsymbol{\theta}_{0m}^{\top}: m \in \mathcal{A})^{\top}$ and $\widetilde{\boldsymbol{\theta}}_{\mathcal{A}} = (\widetilde{\boldsymbol{\theta}}_m^{\top}: m \in \mathcal{A})^{\top}$.
\begin{theorem}
\label{THM:LASSO-1}
Suppose that Assumptions (A1)--(A4) hold.
\begin{enumerate} [{\normalfont (i)}]
\item
If $\{\ln(p_1) \vee N_n \ln(p_2N_n)\} / {n} \rightarrow 0$ and $n^{-2}\sum_{j=1}^{3} \widetilde{\lambda}_{nj}^2 \rightarrow 0$ as $n \to \infty$, then with probability converging to one, all the nonzero linear parameters $\alpha_{0k}$ and $\beta_{0\ell}$, $k \in \mathcal{S}_z, \ell \in \mathcal{S}_{x,L}$, and nonzero additive components $g_{0\ell}$, $\ell \in \mathcal{S}_{x, N}$, are selected.
\item
In addition,
\begin{align*}
\sum_{k= 1}^{p_1} |\widetilde{\alpha}_k - {\alpha}_{0k}|_2^2 &= O_P\left\{\frac{\ln(p_1) \vee N_n \ln(p_2N_n)}{n}\right\} + O\left(N_n^{-2}\right) + O\left(n^{-2}\sum_{j=1}^{3} \widetilde{\lambda}_{nj}^2\right), \\
\sum_{\ell = 1}^{p_2}| \widetilde{\beta}_{\ell} - {\beta}_{0\ell} |^2 &= O_P\left\{\frac{\ln(p_1) \vee N_n \ln(p_2N_n)}{n}\right\} + O\left(N_n^{-2}\right) + O\left(n^{-2}\sum_{j=1}^{3} \widetilde{\lambda}_{nj}^2\right), \\
\sum_{\ell = 1}^{p_2} \Vert \widetilde{g}_{\ell} - g_{0\ell} \Vert_2^2 &= O_P\left\{\frac{\ln(p_1) \vee N_n \ln(p_2N_n)}{n}\right\} + O\left(N_n^{-2}\right) + O\left(n^{-2}\sum_{j=1}^{3} \widetilde{\lambda}_{nj}^2\right).
\end{align*}
\end{enumerate}
\end{theorem}
\begin{proof}
We prove part (ii) first. Let $\widetilde{\boldsymbol{\theta}}^{\top} \!\!\!\equiv ({\widetilde{\boldsymbol{\theta}}}_1^{\top}, \ldots, \widetilde{\boldsymbol{\theta}}_{p_1 + 2p_2}^{\top}) \!=\! (\widetilde{\alpha}_1, \!\ldots, \!\widetilde{\alpha}_{p_1}, \widetilde{\beta}_1, \ldots, \widetilde{\beta}_{p_2},$ $ \widetilde{\boldsymbol{\gamma}}_{1}^{\top} \ldots, \widetilde{\boldsymbol{\gamma}}_{p_2}^{\top})$. For $\mathcal{S}$ defined in (\ref{DEF:SN}) and $\widetilde{\mathcal{S}}$ defined in (\ref{DEF:S_tilde}), denote $\mathcal{S}^{\prime} = \mathcal{S} \bigcup \widetilde{\mathcal{S}} = \{m: \Vert \boldsymbol{{\theta}}_{0m}\Vert_2 \neq 0 \, \text{or} \, \Vert \boldsymbol{\widetilde{\boldsymbol{\theta}}}_m\Vert_2 \neq 0 \}$ and $d^{\prime} = |\mathcal{S}^{\prime}|$. By Lemma \ref{LEM:gLASSO_select},
$d^{\prime} = O(|\mathcal{S}|)$. Notice that $\mathbf{D}\widetilde{\boldsymbol{\theta}} = \mathbf{D}_{\mathcal{S}^{\prime}}\sigsy{\widetilde{\boldsymbol{\theta}}}$ and $\mathbf{D}\boldsymbol{\theta}_0 = \mathbf{D}_{\mathcal{S}^{\prime}}\boldsymbol{\theta}_{0,\mathcal{S}^{\prime}}$, by the definition of $\widetilde{\boldsymbol{\theta}}$ and $\mathcal{S}^{\prime}$,
\begin{align*}
\Vert \mathbf{Y} - \mathbf{D}_{\mathcal{S}^{\prime}}\sigsy{\widetilde{\boldsymbol{\theta}}} \Vert^2
& - \Vert \mathbf{Y} - \mathbf{D}_{\mathcal{S}^{\prime}}\boldsymbol{\theta}_{0,\mathcal{S}^{\prime}} \Vert^2\\
& \leq \sum_{m\in \mathcal{S}^{\prime}}\{\widetilde{\lambda}_{n1} I(m \leq p_1) + \widetilde{\lambda}_{n2} I(p_1 < m \leq p_1 + p_2) + \widetilde{\lambda}_{n3} I(m > p_1 + p_2)\} \Vert \boldsymbol{\theta}_{0m} \Vert \\
& \quad - \sum_{m\in \mathcal{S}^{\prime}}\{\widetilde{\lambda}_{n1} I(m \leq p_1) + \widetilde{\lambda}_{n2} I(p_1 < m \leq p_1 + p_2) + \widetilde{\lambda}_{n3} I(m > p_1 + p_2)\} \Vert \widetilde{\boldsymbol{\theta}}_m \Vert.
\end{align*}
Let $\boldsymbol{\eta} = \mathbf{Y} - \mathbf{D}\boldsymbol{\theta}_0$
and $\boldsymbol{\nu} = \mathbf{D}_{\mathcal{S}^{\prime}}(\sigsy{\widetilde{\boldsymbol{\theta}}} - {\boldsymbol{\theta}_{0, \mathcal{S}^{\prime}}})$, so $\boldsymbol{\eta} - \boldsymbol{\nu} = \mathbf{Y} - \mathbf{D}_{\mathcal{S}^{\prime}}\sigsy{\widetilde{\boldsymbol{\theta}}}$,
and we have $\Vert \mathbf{Y} - \mathbf{D}_{\mathcal{S}^{\prime}}\sigsy{\widetilde{\boldsymbol{\theta}}} \Vert^2 - \Vert \mathbf{Y} - \mathbf{D}_{\mathcal{S}^{\prime}}\boldsymbol{\theta}_{0,\mathcal{S}^{\prime}} \Vert^2 = \boldsymbol{\nu}^{\top}\boldsymbol{\nu} - 2\boldsymbol{\eta}^{\top}\boldsymbol{\nu}$. Thus, from the triangle inequality and the Cauchy-Schwartz inequality,
\begin{align}
\Vert \boldsymbol{\nu} \Vert^2 \!-\! 2 \boldsymbol{\eta}^{\top}\boldsymbol{\nu}
&\!\leq\! \sum_{m\in \mathcal{S}^{\prime}}\{\widetilde{\lambda}_{n1} I(m \!\leq\! p_1) \!+\! \widetilde{\lambda}_{n2} I(p_1 \!<\! m \leq p_1 + p_2) \!+\! \widetilde{\lambda}_{n3} I(m \!>\! p_1 + p_2)\} (\Vert \boldsymbol{\theta}_{0m} \Vert \!-\! \Vert \widetilde{\boldsymbol{\theta}}_m \Vert) \nonumber \\
&\leq \sqrt{d^{\prime}\sum_{j=1}^3 \widetilde\lambda_{nj}^2} \Vert \sigsy{\widetilde{\boldsymbol{\theta}}} - \boldsymbol{\theta}_{0,\mathcal{S}^{\prime}} \Vert
\leq \frac{ d^{\prime} \sum_{j=1}^3 \widetilde\lambda_{nj}^2}{nc_{*}} + \frac{1}{4} nc_{*}\Vert \sigsy{\widetilde{\boldsymbol{\theta}}} - \boldsymbol{\theta}_{0,\mathcal{S}^{\prime}} \Vert^2,
\label{EQ:nu2_etanu}
\end{align}
where $c_{*}$ is the lower bound of eigenvalues of $n^{-1}\mathbf{D}_{\mathcal{S}^{\prime}}^{\top}\mathbf{D}_{\mathcal{S}^{\prime}}$. By Lemma\ref{LEM:eigen} and Lemma \ref{LEM:gLASSO_select}, $c_{*}\asymp 1$ with probability approaching one. Apparently,
\begin{equation}
\label{EQ:vu2_geq}
\Vert \boldsymbol{\nu} \Vert^2 \geq nc_{*} \Vert \sigsy{\widetilde{\boldsymbol{\theta}}} - \boldsymbol{\theta}_{0,\mathcal{S}^{\prime}} \Vert^2.
\end{equation}
Define $\boldsymbol{\eta}^{\ast} \equiv
\mathbf{D}_{\mathcal{S}^{\prime}}(\mathbf{D}_{\mathcal{S}^{\prime}}^{\top}\mathbf{D}_{\mathcal{S}^{\prime}})^{-1}\mathbf{D}_{\mathcal{S}^{\prime}}^{\top}\boldsymbol{\eta}$ to be the projection of $\boldsymbol{\eta}$ onto the column space of $\mathbf{D}_{\mathcal{S}^{\prime}}$. Obviously, $\boldsymbol{\eta}^{\top}\boldsymbol{\nu} = \boldsymbol{\eta}^{\ast\top}\boldsymbol{\nu}$. By the Cauchy-Schwartz inequality, we have
\begin{equation}
\label{EQ:eta_nu}
2|\boldsymbol{\eta}^{\top}\boldsymbol{\nu}| \leq 2 \Vert \boldsymbol{\eta}^{\ast} \Vert \Vert \boldsymbol{\nu} \Vert \leq 2 \Vert \boldsymbol{\eta}^{\ast} \Vert^2 + \frac{1}{2} \Vert \boldsymbol{\nu} \Vert^2.
\end{equation}
Combining (\ref{EQ:nu2_etanu}), (\ref{EQ:vu2_geq}) and (\ref{EQ:eta_nu}), we obtain
\begin{eqnarray}
\Vert \sigsy{\widetilde{\boldsymbol{\theta}}} - \boldsymbol{\theta}_{0,\mathcal{S}^{\prime}} \Vert^2 &\leq& \frac{8\Vert \boldsymbol{\eta}^{\ast} \Vert^2}{nc_{*}} + \frac{4 d^{\prime}\sum_{j=1}^3 \widetilde\lambda_{nj}^2}{n^2c_{*}^2} .
\label{EQN:the_diffs1}
\end{eqnarray}
With $\eta_i$ defined to be the $i$th element of $\boldsymbol{\eta}$, we have the following decomposition:
\begin{align}
\eta_i = Y_i &- \sum_{k = 1}^{p_1} \, Z_{ik}\alpha_{0k} - \sum_{\ell = 1}^{p_2} \, X_{i\ell}\beta_{0\ell} - \sum_{\ell = 1}^{p_2}\sum_{J = 1}^{N_n} \, \gamma_{0J,l}B_{J,\ell}^{(1)}(X_{i\ell}) \nonumber \\
= Y_i &\!-\! \sum_{k \in \mathcal{S}_z} \, Z_{ik}\alpha_{0k} \!-\! \sum_{\ell \in \mathcal{S}_{x,L}} \, X_{i\ell}\beta_{0\ell} \!-\! \sum_{\ell \in \mathcal{S}_{x,N}}\, g_{0\ell} (X_{i\ell})
\!-\! \sum_{\ell \in \mathcal{S}_{x,N}}\, g_{0\ell} (X_{i\ell}) \!-\! \sum_{\ell \in \mathcal{S}_{x,N}}\sum_{J = 1}^{N_n} \gamma_{0J,l}B_{J,\ell}^{(1)} (X_{i\ell}) \nonumber \\
= \varepsilon_i &+ \sum_{\ell \in \mathcal{S}_{x,N}} \, \delta_{i\ell}, \label{EQN:eta_form}
\end{align}
where $\delta_{i\ell} = g_{0\ell} (X_{i\ell}) - \sum_{J = 1}^{N_n} \, \gamma_{0J,l}B_{J,\ell}^{(1)} (X_{i\ell})$.
Let $\delta_i = \sum_{\ell \in \mathcal{S}_{x,N}} \delta_{i\ell}$,
$\boldsymbol{\delta} = (\delta_1, \ldots, \delta_n)^{\top}$, and $\boldsymbol{\varepsilon} = (\varepsilon_1, \ldots, \varepsilon_n)^{\top}$. Then $\boldsymbol{\eta = \varepsilon + \delta}$. Define $\sigsy{\delta} = (\sum_{\ell + p_1 + p_2 \in \mathcal{S}^{\prime}, 1\leq \ell \leq p_2} \delta_{i\ell}, i = 1, \dots, n)^{\top}$. By (\ref{EQN:eta_form}) and the fact that ${|\delta_{i\ell}| = O_P(N_n^{-1})}$,
\begin{eqnarray}
\Vert \boldsymbol{\eta}^{\ast} \Vert^2 = \Vert \boldsymbol{\varepsilon}^{\ast} + \sigsy{\delta}^{\ast} \Vert^2 \leq 2 \Vert \boldsymbol{\varepsilon}^{\ast} \Vert^2 + 2 \Vert \sigsy{\delta} \Vert^2 \leq 2 \Vert \boldsymbol{\varepsilon}^{\ast} \Vert^2 + O_P (nd^{\prime 2}N_n^{-2}),
\label{EQN:eta_ord}
\end{eqnarray}
where $\boldsymbol{\varepsilon}^{\ast} \equiv \mathbf{P}_{\mathbf{D}_{\mathcal{S}^{\prime}}}\boldsymbol{\varepsilon}$ and $\sigsy{\delta}^{\ast} \equiv \mathbf{P}_{\mathbf{D}_{\mathcal{S}^{\prime}}}\sigsy{\delta}$ are the projections of $\boldsymbol{\varepsilon}$ and $\sigsy{\delta}$ onto the column space of $\mathbf{D}_{\mathcal{S}^{\prime}}$, respectively.
Define $T_1 = \max_{1 \leq k \leq p_1}$ $|n^{-1/2} \sum_{i = 1}^{n} Z_{ik} \varepsilon_i|$, $T_2 = \max_{1 \leq \ell \leq p_2}$ $|n^{-1/2} \sum_{i = 1}^{n} X_{i\ell} \varepsilon_i|$, and $T_3 = \max_{1 \leq \ell \leq p_2, 1 \leq J \leq N_n}$ $|n^{-1/2} \sum_{i = 1}^{n} B_{J,\ell}^{(1)}(X_{i\ell}) \varepsilon_i|$.
Then,
$\Vert \boldsymbol{\varepsilon}^{\ast} \Vert^2 = \Vert \left(\mathbf{D}_{\mathcal{S}^{\prime}}^{\top}\mathbf{D}_{\mathcal{S}^{\prime}}\right)^{-1/2} \mathbf{D}_{\mathcal{S}^{\prime}}^{\top}\boldsymbol{\varepsilon} \Vert^2 \leq (nc_{*})^{-1}\Vert \mathbf{D}_{\mathcal{S}^{\prime}}^{\top}\boldsymbol{\varepsilon} \Vert^2$,
and
\[
\max_{\mathcal{A}: |\mathcal{A}| \leq d^{\prime}} \Vert \mathbf{D}_\mathcal{A}^{\top}\boldsymbol{\varepsilon} \Vert^2 = \max_{\mathcal{A}: |\mathcal{A}| \leq d^{\prime}} \sum_{m \in \mathcal{A}} \Vert \mathbf{D}_m^{\top}\boldsymbol{\varepsilon} \Vert^2 \leq nd^{\prime}(T_1^2 \vee T_2^2 \vee N_nT_3^2).
\]
By Lemma \ref{LEM:2}, $\max_{\mathcal{A}: |\mathcal{A}| \leq d^{\prime}} \Vert \mathbf{D}_\mathcal{A}^{\top}\boldsymbol{\varepsilon} \Vert^2 = O_P[n d^{\prime}\{\ln(p_1) \vee N_n \ln(p_2N_n)\}]$. Therefore,
\begin{eqnarray}
\Vert \boldsymbol{\varepsilon}^{\ast} \Vert^2 = O_P[d^{\prime}c_{*}^{-1}\{\ln(p_1) \vee N_n \ln(p_2N_n)\}].
\label{EQN:eps_ord}
\end{eqnarray}
Combing (\ref{EQN:the_diffs1}), (\ref{EQN:eta_ord}) and (\ref{EQN:eps_ord}), we conclude that
\begin{align*}
\Vert \sigsy{\widetilde{\boldsymbol{\theta}}} - \boldsymbol{\theta}_{0,\mathcal{S}^{\prime}} \Vert^2
& = O_P\left[\frac{d^{\prime}\left\{\ln(p_1) \vee N_n \ln(p_2N_n)\right\}}{nc_{*}}\right] + O\left(\frac{d^{\prime}}{N^2 c_{*}}\right) + \frac{4 d^{\prime} \sum_{j=1}^3 \widetilde\lambda_{nj}^2}{n^{2}c_{*}^{2}} \\
& = O_P\left[n^{-1}\{\ln(p_1) \vee N_n \ln(p_2N_n)\}\right] + O\left(N_n^{-2}\right) + O\left(n^{-2}\sum_{j=1}^{3}\widetilde{\lambda}_{nj}^2\right),
\end{align*}
where the last inequality follows by $d^{\prime} = O(|\mathcal{S}_z| + |\mathcal{S}_{x,L}| + |\mathcal{S}_{x,N}|)$ and $c_{*} \asymp 1$ with probability approaching one.
By the properties of splines \citep{de2001practical},
$\Vert \widetilde{g}_{\ell} - g_{n\ell} \Vert_2^2 \asymp \Vert \widetilde{\boldsymbol{\gamma}}_{\ell} - \boldsymbol{\gamma}_{0\ell} \Vert^2$,
where $g_{n\ell}, \, \ell=1, \ldots, p_2$, is the best approximation for function $g_{\ell}$.
Hence, part (ii) follows from $\sum_{k = 1}^{p_1}| \widetilde{\alpha}_k - {\alpha}_{0k} |^2 = O(\Vert \sigsy{\widetilde{\boldsymbol{\theta}}} - \boldsymbol{\theta}_{0,\mathcal{S}^{\prime}} \Vert^2)$, $\sum_{\ell = 1}^{p_2}| \widetilde{{\beta}}_{\ell} - {\beta}_{0\ell}|^2 = O(\Vert \sigsy{\widetilde{\boldsymbol{\theta}}} - \boldsymbol{\theta}_{0,\mathcal{S}^{\prime}} \Vert^2)$ and $\sum_{\ell = 1}^{p_2}\| \widetilde{\boldsymbol{\gamma}}_{\ell} - {\boldsymbol{\gamma}}_{0\ell} \|^2 = O(\Vert \sigsy{\widetilde{\boldsymbol{\theta}}} - \boldsymbol{\theta}_{0,\mathcal{S}^{\prime}} \Vert^2)$.
We now prove part (i). Under Assumption (A1), if $\Vert \boldsymbol{\theta}_{0m} \Vert \neq 0$ but $\Vert \widetilde{\boldsymbol{\theta}}_{m} \Vert = 0$, then $\Vert \boldsymbol{\theta}_{0m} - \widetilde{\boldsymbol{\theta}}_{m} \Vert \geq c_{\alpha} \vee c_{\beta} \vee c_g$, which contradicts part (ii) when ${\ln(p_1) \vee N_n \ln(p_2N_n)} / {n} \rightarrow 0$, ${\widetilde{\lambda}_{n1}^2} / {n^2} \rightarrow 0$, ${\widetilde{\lambda}_{n2}^2} / {n^2} \rightarrow 0$ and ${\widetilde{\lambda}_{n3}^2} / {n^2} \rightarrow 0$.
The results follow by
\begin{align*}
\widetilde{\boldsymbol{\alpha}} - \boldsymbol{\alpha}_{0} &= \left(\mathbf{I}_{|\mathcal{S}_z|} ~~ \boldsymbol{0}_{|\mathcal{S}_z| \times |\mathcal{S}_{x,L}|} ~~ \boldsymbol{0}_{|\mathcal{S}_z| \times (|\mathcal{S}_{x,N}|N_n)}\right)(\widetilde{\boldsymbol{\theta}} - \boldsymbol{\theta}_{0}), \\
\widetilde{\boldsymbol{\beta}} - \boldsymbol{\beta}_{0} &= \left(\boldsymbol{0}_{|\mathcal{S}_{x,L}| \times |\mathcal{S}_z|} ~~ \mathbf{I}_{|\mathcal{S}_{x,L}|} ~~ \boldsymbol{0}_{|\mathcal{S}_{x,L}|\times (|\mathcal{S}_{x,N}|N_n)}\right)(\widetilde{\boldsymbol{\theta}} - \boldsymbol{\theta}_{0}),\\
\widetilde{\boldsymbol{\gamma}} - \boldsymbol{\gamma}_{0} &= \left(\boldsymbol{0}_{(|\mathcal{S}_{x,N}| N_n) \times |\mathcal{S}_z|} ~~ \boldsymbol{0}_{(|\mathcal{S}_{x,N}| N_n) \times |\mathcal{S}_{x,L}|} ~~ \mathbf{I}_{|\mathcal{S}_{x,N}| N_n}\right)(\widetilde{\boldsymbol{\theta}} - \boldsymbol{\theta}_{0})
\end{align*}
and the definition of $\widetilde{g}_{\ell}$, $1 \leq \ell \leq p_2$.
\end{proof}
\subsection{Selection and estimation properties of the adaptive group LASSO estimators} \label{SUBSEC:KKT}
In this section, we establish the selection and estimation properties of the adaptive group LASSO estimators as stated in Theorems \ref{THM:selection} and \ref{THM:consistency}.
\begin{proof}[Proof of Theorem \ref{THM:selection}]
By the Karush-Kuhn-Tucker (KKT) condition \citep{boyd2004convex}, if $(\widehat{\boldsymbol{\alpha}}, \widehat{\boldsymbol{\beta}}, \widehat{\boldsymbol{\gamma}})$ is the unique minimizer of $L(\boldsymbol{\alpha}, \boldsymbol{\beta}, \boldsymbol{\gamma}; \lambda_1, \lambda_2, \lambda_3)$, it is equivalent to satisfy
\begin{enumerate}
{\renewcommand{-1pt}{-1pt}
\item[(C1-1)] $\mathbf{Z}_k^{\top} \left(\mathbf{Y}-\mathbf{Z}\boldsymbol{\alpha} - \mathbf{X}\boldsymbol{\beta}-\sum_{{\ell^{\prime}} = 1}^{p_2}\mathbf{B}_{\ell^{\prime}}^{(1)}\boldsymbol{\gamma}_{\ell^{\prime}}\right) = \lambda_{n1} w_k^{\alpha}\alpha_k / |\alpha_k|$, for any $k\in \mathcal{S}_z$,
\item[(C1-2)] $\mathbf{X}_{\ell}^{\top} \left(\mathbf{Y}-\mathbf{Z}\boldsymbol{\alpha} - \mathbf{X}\boldsymbol{\beta}-\sum_{{\ell^{\prime}} = 1}^{p_2}\mathbf{B}_{\ell^{\prime}}^{(1)}\boldsymbol{\gamma}_{\ell^{\prime}}\right) = \lambda_{n2} w_{\ell}^{\beta} \beta_{\ell} / |\beta_{\ell}|$, for any $\ell \in \mathcal{S}_{x, L}$,
\item[(C1-3)] $\mathbf{B}_{\ell}^{(1)\top} \left(\mathbf{Y}-\mathbf{Z}\boldsymbol{\alpha} - \mathbf{X}\boldsymbol{\beta}-\sum_{{\ell^{\prime}} = 1}^{p_2}\mathbf{B}_{\ell^{\prime}}^{(1)}\boldsymbol{\gamma}_{\ell^{\prime}}\right) = \lambda_{n3} w_{\ell}^{\gamma} \boldsymbol{\gamma}_{\ell} / \Vert\boldsymbol{\gamma}_{\ell}\Vert$, for any $\ell \in \mathcal{S}_{x,N}$,
\item[(C2)] $\left|\mathbf{Z}_k^{\top} \left(\mathbf{Y}-\mathbf{Z}\boldsymbol{\alpha} - \mathbf{X}\boldsymbol{\beta}-\sum_{{\ell^{\prime}} = 1}^{p_2}\mathbf{B}_{\ell^{\prime}}^{(1)}\boldsymbol{\gamma}_{\ell^{\prime}}\right)\right| \leq\lambda_{n1} w_k^{\alpha}$, for any $k\in \mathcal{N}_z$,
\item[(C3)] $\left|\mathbf{X}_{\ell}^{\top} \left(\mathbf{Y}-\mathbf{Z}\boldsymbol{\alpha} - \mathbf{X}\boldsymbol{\beta}-\sum_{{\ell^{\prime}} = 1}^{p_2}\mathbf{B}_{\ell^{\prime}}^{(1)}\boldsymbol{\gamma}_{\ell^{\prime}}\right)\right|\leq \lambda_{n2} w_{\ell}^{\beta}$, for any $\ell \in \mathcal{N}_x \cup \mathcal{S}_{x, PN}$,
\item[(C4)] $\Vert\mathbf{B}_{\ell}^{(1)\top} \left(\mathbf{Y}-\mathbf{Z}\boldsymbol{\alpha} - \mathbf{X}\boldsymbol{\beta}-\sum_{{\ell^{\prime}} = 1}^{p_2}\mathbf{B}_{\ell^{\prime}}^{(1)}\boldsymbol{\gamma}_{\ell^{\prime}}\right)\Vert \leq \lambda_{n3} w_{\ell}^{\gamma}$, for any $\ell \in \mathcal{N}_x\cup \mathcal{S}_{x, PL}$,
}
\end{enumerate}
Define $ \overline{\boldsymbol{\theta}}^o = \left(\mathbf{D}_{\mathcal{S}}^{\top} \, \mathbf{D}_{\mathcal{S}}\right)^{-1} \mathbf{D}_{\mathcal{S}}^{\top}\mathbf{Y}$, a vector with length $|\mathcal{S}_z| + |\mathcal{S}_{x,L}| + |\mathcal{S}_{x,N}|N_n$.
Denote three vectors, $\boldsymbol{v}_1$, $\boldsymbol{v}_2$ and $\boldsymbol{v}_3$, whose elements are in the form:
\begin{align}
& \boldsymbol{v}_{1m} = \frac{\omega_m^{\alpha}\overline{\theta}_{0m}}{|\overline{\theta}_{0m}|} I\{m \in \mathcal{S}_z\} + \boldsymbol{0}_{N} I\{m - |\mathcal{S}_z| - |\mathcal{S}_{x,LN}| \in \mathcal{S}_{x,N}\}, \label{DEF:vj}\\
& \boldsymbol{v}_{2m} = \frac{\omega_{m - |\mathcal{S}_z|}^{\beta}\overline{\theta}_{0m}}{|\overline{\theta}_{0m}|} I\{m - |\mathcal{S}_z| \in \mathcal{S}_{x, L}\} + \boldsymbol{0}_{N} I\{m - |\mathcal{S}_z| - |\mathcal{S}_{x,LN}| \in \mathcal{S}_{x,N}\}, \notag \\
& \boldsymbol{v}_{3m} = \frac{\omega_{m - |\mathcal{S}_z| - |\mathcal{S}_{x,L}|}^{\gamma}\overline{\boldsymbol{\theta}}_{0m}}{\Vert \overline{\boldsymbol{\theta}}_{0m} \Vert} I\{m - |\mathcal{S}_z| - |\mathcal{S}_{x,LN}| \in \mathcal{S}_{x,N}\},
~ \forall m \in \mathcal{S}. \notag
\end{align}
Next define $\widehat{\boldsymbol{\theta}}^o = (\widehat{\boldsymbol{\theta}}_m^o, 1 \leq m \leq p_1 + 2p_2)^{\top}$,
where $\widehat{\boldsymbol{\theta}}_{\mathcal{S}}^o \equiv (\widehat{\boldsymbol{\theta}}_m^o, m \in \mathcal{S})^{\top} = \left(\mathbf{D}_{\mathcal{S}}^{\top} \, \mathbf{D}_{\mathcal{S}}\right)^{-1} (\mathbf{D}_{\mathcal{S}}^{\top}\mathbf{Y} - \sum_{j=1}^{3}\lambda_{nj} \boldsymbol{v}_j)$,
$\widehat{\theta}_m^o = 0$ for $m \in \mathcal{N}_z$ and $m - p_1 \in \mathcal{N}_{x, L}$, and $\widehat{\boldsymbol{\theta}}_m^o = \boldsymbol{0}_N$ for $m - p_1 - p_2 \in \mathcal{N}_{x,N}$.
So we can represent $\widehat{\boldsymbol{\theta}}^o \equiv (\widehat{\boldsymbol{\theta}}_{\mathcal{S}_z}^{o\top}, \widehat{\boldsymbol{\theta}}_{\mathcal{N}_z}^{o\top}, \widehat{\boldsymbol{\theta}}_{\mathcal{S}_{x, L}}^{o\top}, \widehat{\boldsymbol{\theta}}_{\mathcal{N}_{x, L}}^{o\top}, \widehat{\boldsymbol{\theta}}_{\mathcal{S}_{x, N}}^{o\top}, \widehat{\boldsymbol{\theta}}_{\mathcal{N}_{x, N}}^{o\top})^{\top}$,
and $\widehat{\boldsymbol{\theta}}_{\mathcal{S}}^o \equiv (\widehat{\boldsymbol{\theta}}_{\mathcal{S}_z}^{o\top}, \widehat{\boldsymbol{\theta}}_{\mathcal{S}_{x, L}}^{o\top}, \widehat{\boldsymbol{\theta}}_{\mathcal{S}_{x, N}}^{o\top})^{\top}$.
Denote $\widehat{\mathcal{S}}^o =\{1 \leq m \leq p_1 + 2p_2: \|\widehat{\boldsymbol{\theta}}_m^o\| > 0\}$. Apparently, $\widehat{\mathcal{S}}^o \subseteq \mathcal{S}$.
Notice that $\mathbf{D}\widehat{\boldsymbol{\theta}}^o = \mathbf{D}_{\mathcal{S}}\widehat{\boldsymbol{\theta}}_{\mathcal{S}}^o$ and $\{\mathbf{D}_m, \, m \in \mathcal{S} \}$ are linearly independent,
so by the definition of $\widehat{\boldsymbol{\theta}}^o$, (C1-1), (C1-2) and (C1-3) hold for $\widehat{\boldsymbol{\theta}}^{o}$ if $\widehat{\mathcal{S}}^o \supseteq \mathcal{S}$.
Therefore, if $\widehat{\boldsymbol{\theta}}^o$ satisfies
\begin{enumerate}
{\renewcommand{-1pt}{-1pt}
\item[(C1$^{\prime}$)] $\widehat{\mathcal{S}}^o \supseteq \mathcal{S}$,
\item[(C2$^{\prime}$)] $\left|\mathbf{Z}_k^{\top} \left(\mathbf{Y} - \mathbf{D}\widehat{\boldsymbol{\theta}}^o\right)\right| \leq\lambda_{n1}\omega_k^{\alpha}$, for any $k\in \mathcal{N}_z$,
\item[(C3$^{\prime}$)] $\Vert\mathbf{X}_{\ell}^{\top} \left(\mathbf{Y} - \mathbf{D}\widehat{\boldsymbol{\theta}}^o\right)\Vert \leq \lambda_{n2} \omega_{\ell}^{\beta}$, for any $\ell \in \mathcal{N}_{x, L}$,
\item[(C4$^{\prime}$)] $\Vert\mathbf{B}_{\ell}^{(1)\top} \left(\mathbf{Y} - \mathbf{D}\widehat{\boldsymbol{\theta}}^o\right)\Vert \leq \lambda_{n3} \omega_{\ell}^{\gamma}$, for any $\ell \in \mathcal{N}_{x, N}$,
}
\end{enumerate}
then $\widehat{\boldsymbol{\theta}}^o$ is the unique minimizer of $L_n(\boldsymbol{\theta};\lambda_{n1},\lambda_{n2},\lambda_{n3})$, in other words, $\widehat{\boldsymbol{\theta}}^o = \widehat{\boldsymbol{\theta}}$ with probability approaching one.
Therefore, in order to show $\Pr(\widehat{\mathcal{S}}=\mathcal{S}) \rightarrow 1$, it is equivalent to show $\widehat{\boldsymbol{\theta}}^o$ satisfies (C1$^{\prime}$)--(C3$^{\prime}$) with probability approaching one, as $n \rightarrow \infty$.
Further notice that
\begin{enumerate}
{\renewcommand{-1pt}{-1pt}
\item[(C1$^{\prime\prime}$)] $\Vert \boldsymbol{\theta}_{0m} \Vert - \Vert \widehat{\boldsymbol{\theta}}_m^o \Vert < \Vert \boldsymbol{\theta}_{0m} \Vert$, $\forall \, m \in \mathcal{S}$
}
\end{enumerate}
implies Condition (C1$^{\prime}$).
Therefore, to show $\widehat{\boldsymbol{\theta}}^o$ is the unique minimizer of $L_n(\boldsymbol{\theta};\lambda_{n1},\lambda_{n2},$ $\lambda_{n3})$, and consequently, $\Pr(\widehat{\mathcal{S}}=\mathcal{S}) \rightarrow 1$, it suffices to show that $\widehat{\boldsymbol{\theta}}^o$ satisfies Conditions (C1$^{\prime\prime}$), (C2$^{\prime}$) and (C3$^{\prime}$) with probability approaching one, as $n \rightarrow \infty$.
According to Lemma \ref{LEM:sel_cons_S} and Lemma \ref{LEM:sel_cons_N} below, we obtain that
\begin{align*}
\Pr(\widehat{\mathcal{S}} \neq \mathcal{S}) \leq &\Pr(\Vert \boldsymbol{\theta}_{0m} - \widehat{\boldsymbol{\theta}}_m^o \Vert \geq \Vert \boldsymbol{\theta}_{0m} \Vert, \, \exists \, m \in \mathcal{S}) + \Pr(|\mathbf{Z}_k^{\top} (\mathbf{Y} - \mathbf{D}\widehat{\boldsymbol{\theta}}^o)| > \lambda_{n1}\omega_k^{\alpha} , \, \exists \, k \in \mathcal{N}_z) \\
& + \Pr(\Vert\mathbf{X}_{\ell}^{\top} (\mathbf{Y} - \mathbf{D}\widehat{\boldsymbol{\theta}}^o)\Vert > \lambda_{n2} \omega_{\ell}^{\beta}, \, \exists \, \ell \in \mathcal{N}_{x, L})\\
& + \Pr(\Vert\mathbf{B}_{\ell}^{(1)\top} (\mathbf{Y} - \mathbf{D}\widehat{\boldsymbol{\theta}}^o)\Vert > \lambda_{n3} \omega_{\ell}^{\gamma} , \, \exists \, \ell \in \mathcal{N}_{x, N}) \rightarrow 0,
\end{align*}
as $n \rightarrow \infty$. This completes the proof.
\end{proof}
The following Lemma \ref{LEM:sel_cons_S} and Lemma \ref{LEM:sel_cons_N} are used in the proof of Theorem \ref{THM:selection}.
\begin{lemma}
\label{LEM:sel_cons_S}
Under Assumptions (A3)--(A6), as $n \rightarrow \infty$,
\[
\Pr \left(\Vert \boldsymbol{\theta}_{0m} - \widehat{\boldsymbol{\theta}}_m^o \Vert \geq \Vert \boldsymbol{\theta}_{0m} \Vert, \, \exists \, m \in \mathcal{S}\right) \to 0.
\]
\end{lemma}
\begin{proof}
Let $\mathbf{Q}_{m}$ be an $d_m \times (|\mathcal{S}_z|+|\mathcal{S}_{x,L}|+|\mathcal{S}_{x,N}|N_n\}$ matrix, $d_m = 1$ for $m \in \mathcal{S}_z$ or $m\in\mathcal{S}_{x,L}$, and $d_m = N_n$ for $m\in\mathcal{S}_{x,N}$, with the form
\begin{align*}
\mathbf{Q}_m = \begin{pmatrix}
\mathbf{Q}_{1m} & \boldsymbol{0}_{(|\mathcal{S}_{x,N}|N_n)\times (|\mathcal{S}_{x,N}|N_n)}
\end{pmatrix} & I(m \in \mathcal{S}_z \cup \mathcal{S}_{x,L})\\
& + \begin{pmatrix}
\boldsymbol{0}_{N_n\times (|\mathcal{S}_z|+|\mathcal{S}_{x,L}|)} & \mathbf{Q}_{2, m - |\mathcal{S}_z| - |\mathcal{S}_{x,L}|}
\end{pmatrix} I(m \in \mathcal{S}_{x,N})
\end{align*}
with $\mathbf{Q}_{1m} = (0, \ldots, 0, 1, 0, \ldots, 0)$ and $\mathbf{Q}_{2m} = (\boldsymbol{0}_{N_n\times N_n}, \ldots, \boldsymbol{0}_{N_n\times N_n}, \mathbf{I}_{N_n}, \boldsymbol{0}_{N_n\times N_n}, \ldots,$ $\boldsymbol{0}_{N_n\times N_n})$,
where scalar 1 is the $m$-th element of vector $\mathbf{Q}_{1m} $ with length $|\mathcal{S}_z|$, and an $N_n \times N_n$ identity matrix $\mathbf{I}_{N_n}$ is at the $m$-th block of the $N_n \times (|\mathcal{S}_{x,N}|N_n)$ matrix $\mathbf{Q}_{2m}$ with rest $N_n \times N_n$ matrices of zeros $\boldsymbol{0}_{N_n \times N_n}$.
Then from (\ref{EQN:theta_dfc}), $\widehat{\boldsymbol{\theta}}_m^o - \boldsymbol{\theta}_{0m} = n^{-1} \mathbf{Q}_{m} \, \mathbf{C}_{\mathcal{S}}^{-1} \left(\mathbf{D}_{\mathcal{S}}^{\top} \boldsymbol{\varepsilon} + \mathbf{D}_{\mathcal{S}}^{\top} \boldsymbol{\delta} - \sum_{j=1}^3 \lambda_{nj}\boldsymbol{v}_j\right )$.
By the triangle inequality,
\[
\left\Vert \widehat{\boldsymbol{\theta}}_m^o - \boldsymbol{\theta}_{0m} \right\Vert \leq
n^{-1} \left\Vert \mathbf{Q}_{m} \mathbf{C}_{\mathcal{S}}^{-1} \mathbf{D}_{\mathcal{S}}^{\top} \boldsymbol{\varepsilon} \right\Vert
+ \, n^{-1} \left\Vert \mathbf{Q}_{m} \mathbf{C}_{\mathcal{S}}^{-1} \mathbf{D}_{\mathcal{S}}^{\top} \boldsymbol{\delta} \right\Vert
+ \, n^{-1} \left\Vert \mathbf{Q}_{m} \mathbf{C}_{\mathcal{S}}^{-1} \left(\sum_{j=1}^3 \lambda_{nj}\boldsymbol{v}_j\right)\right\Vert .
\]
Recall that $\pi_1$ and $\pi_2$ are the minimum and maximum eigenvalues of $\mathbf{C}_{\mathcal{S}}$, respectively. By Lemmas \ref{LEM:2} and \ref{LEM:eigen}, the first term on the right-hand side
\begin{align*}
\max\limits_{m \in \mathcal{S}} \, & n^{-1} \Vert \mathbf{Q}_{m} \mathbf{C}_{\mathcal{S}}^{-1} \mathbf{D}_{\mathcal{S}}^{\top} \boldsymbol{\varepsilon} \Vert_2
\leq \max\limits_{m \in \mathcal{S}} \, n^{-1} \pi_1^{-1} \Vert \, \mathbf{Q}_{m} \mathbf{D}_{\mathcal{S}}^{\top} \boldsymbol{\varepsilon} \Vert_2 \\
& = n^{-1} \pi_1^{-1} \max\limits_{k,l \in \mathcal{S}} \, \left\{\sum_{i=1}^n |Z_{ik} \varepsilon_i|, \sum_{i=1}^n |X_{i\ell} \varepsilon_i|, \sum_{i=1}^n \|\mathbf{B}_{i\ell} \varepsilon_i\|\right\}
= {O_P \left(n^{-1/2}N_n^{1/2} \right) }.
\end{align*}
By Lemma \ref{LEM:eigen}, the second term
\begin{align*}
\max\limits_{m \in \mathcal{S}} \, & n^{-1} \Vert \mathbf{Q}_{m} \mathbf{C}_{\mathcal{S}}^{-1} \mathbf{D}_{\mathcal{S}}^{\top} \boldsymbol{\delta} \, \Vert_2
\leq \max\limits_{m \in \mathcal{S}} \, n^{-1} \pi_1^{-1} \Vert \, \mathbf{Q}_{m} \mathbf{D}_{\mathcal{S}}^{\top} \boldsymbol{\delta} \Vert_2 \\
& = n^{-1} \pi_1^{-1} \max\limits_{k,l \in \mathcal{S}} \, \left\{\sum_{i=1}^n |Z_{ik} \delta_i|, \sum_{i=1}^n |X_{i\ell} \delta_i|, \sum_{i=1}^n \|\mathbf{B}_{i\ell} \delta_i\|\right\}
= O_P (N_n^{-1}).
\end{align*}
By Lemma \ref{LEM:eigen} and Lemma \ref{LEM:4}, the third term
\begin{align*}
\max\limits_{m \in \mathcal{S}} n^{-1} & \Vert \mathbf{Q}_{m} \mathbf{C}_{\mathcal{S}}^{-1} \left(\sum_{j=1}^3 \lambda_{nj}\boldsymbol{v}_j\right)\Vert_2
\leq n^{-1} \pi_1^{-1} \left\Vert \sum_{j=1}^3 \lambda_{nj}\boldsymbol{v}_j\right\Vert
= O_P \left\{n^{-1} \left(\sum_{j=1}^3 \lambda_{nj}h_{nj}\right) \right\}.
\end{align*}
Thus, the claim follows by Assumption (A6).
\end{proof}
\begin{lemma}
\label{LEM:sel_cons_N}
Under Assumptions (A3)--(A6), as $n \rightarrow \infty$,
\begin{eqnarray*}
&&\Pr \left(|\mathbf{Z}_k^{\top} (\mathbf{Y} - \mathbf{D}\widehat{\boldsymbol{\theta}}^o)| > \lambda_{n1}\omega_k^{\alpha} , \, \exists \, k \in \mathcal{N}_z \right) \to 0, \\
&&\Pr \left(\Vert\mathbf{X}_{\ell}^{\top} (\mathbf{Y} - \mathbf{D}\widehat{\boldsymbol{\theta}}^o)\Vert > \lambda_{n2} \omega_{\ell}^{\beta}, \, \exists \, \ell \in \mathcal{N}_{x, L} \right) \to 0, \\
&&\Pr \left(\Vert\mathbf{B}_{\ell}^{(1)\top} (\mathbf{Y} - \mathbf{D}\widehat{\boldsymbol{\theta}}^o)\Vert > \lambda_{n3} \omega_{\ell}^{\gamma} , \, \exists \, \ell \in \mathcal{N}_{x, N} \right) \to 0.
\end{eqnarray*}
\end{lemma}
\begin{proof}
Note that
\begin{align}
\mathbf{Y} - \mathbf{D}_{\mathcal{S}}\sigs{\widehat{\boldsymbol{\theta}}}^o
&= \mathbf{Z}_{\mathcal{S}_z} \boldsymbol{\alpha}_{0,\mathcal{S}_z} + \mathbf{X}_{\mathcal{S}_{x,L}} \boldsymbol{\beta}_{0,\mathcal{S}_{x,L}} \!+\! \sum_{\ell \in \mathcal{S}_{x,N}} \, {\alpha}_{0\ell}\left(\mathbf{X}_{\ell}\right) \!+\! \boldsymbol{\varepsilon} - \mathbf{D}_{\mathcal{S}}\sigs{\widehat{\boldsymbol{\theta}}} = \mathbf{D}_{\mathcal{S}} \boldsymbol{\theta}_{0,\mathcal{S}} + \boldsymbol{\delta} + \boldsymbol{\varepsilon}\!-\! \mathbf{D}_{\mathcal{S}}\sigs{\widehat{\boldsymbol{\theta}}} \nonumber \\
&= \left\{\mathbf{I} - \mathbf{D}_{\mathcal{S}}\left(\mathbf{D}_{\mathcal{S}}^{\top} \mathbf{D}_{\mathcal{S}}\right)^{-1}\mathbf{D}_{\mathcal{S}}^{\top} \right\} \left(\boldsymbol{\delta}+\boldsymbol{\varepsilon}\right) + \mathbf{D}_{\mathcal{S}}\left(\mathbf{D}_{\mathcal{S}}^{\top} \mathbf{D}_{\mathcal{S}}\right)^{-1}\left(\sum_{j=1}^3 \lambda_{nj}\boldsymbol{v}_j\right) \nonumber \\
&= \mathbf{H}\boldsymbol{\varepsilon} + \mathbf{H}\boldsymbol{\delta} + n^{-1}\mathbf{D}_{\mathcal{S}}\mathbf{C}_{\mathcal{S}}^{-1}\left(\sum_{j=1}^3 \lambda_{nj}\boldsymbol{v}_j\right). \label{EQN:yds}
\end{align}
For $1 \leq m \leq p_{1} + 2p_{2}$, by (\ref{EQN:yds}), we have
\[
\mathbf{D}_m^{\top} (\mathbf{Y} - \mathbf{D}_{\mathcal{S}}\sigs{\widehat{\boldsymbol{\theta}}}^o) = \mathbf{D}_m^{\top} \mathbf{H}\boldsymbol{\varepsilon} + \mathbf{D}_m^{\top} \mathbf{H}\boldsymbol{\delta} + n^{-1}\mathbf{D}_m^{\top} \mathbf{D}_{\mathcal{S}}\mathbf{C}_{\mathcal{S}}^{-1}\left(\sum_{j=1}^3 \lambda_{nj}\boldsymbol{v}_j\right).
\]
By Lemma \ref{LEM:2},
\begin{align*}
\mathrm{E}\left( \max\limits_{m \in \mathcal{N}_z} \Vert n^{-1/2} \mathbf{D}_m^{\top} \mathbf{H}\boldsymbol{\varepsilon}\Vert_2 \right)
&= \mathrm{E}\left( \max\limits_{k \in \mathcal{N}_z} \Vert n^{-1/2} \mathbf{Z}_k^{\top} \mathbf{H}\boldsymbol{\varepsilon}\Vert_2 \right)
= O \{\sqrt{\ln (p_{1})}\}, \\
\mathrm{E}\left( \max\limits_{m - p_1 \in \mathcal{N}_{x,L}} \Vert n^{-1/2} \mathbf{D}_m^{\top} \mathbf{H}\boldsymbol{\varepsilon}\Vert_2 \right)
&= \mathrm{E}\left( \max\limits_{\ell \in \mathcal{N}_{x,L}} \Vert n^{-1/2} \mathbf{X}_{\ell}^{\top} \mathbf{H}\boldsymbol{\varepsilon}\Vert_2 \right)
= O \{\sqrt{\ln\left(p_{2}\right)} \},, \\
\mathrm{E}\left( \max\limits_{m - p_1 - p_2 \in \mathcal{N}_{x,N}} \Vert n^{-1/2} \mathbf{D}_m^{\top} \mathbf{H}\boldsymbol{\varepsilon}\Vert_2 \right)
&= \mathrm{E}\left( \max\limits_{\ell \in \mathcal{N}_{x,N}} \Vert n^{-1/2} \mathbf{B}_{\ell}^{(1)\top} \mathbf{H}\boldsymbol{\varepsilon}\Vert_2 \right)
= O \{\sqrt{\ln\left(p_{2} N_n \right)} \},
\end{align*}
then for the $\mathbf{D}_m^{\top} \mathbf{H}\boldsymbol{\varepsilon}$ part, from Condition (A5), for all $k \in \mathcal{N}_z, \, \omega_k^{\alpha} = |\widetilde{\alpha}_k|^{-1} = O_P(r_{n1})$, there exists a positive constant $c_{1}$, such that
\begin{eqnarray*}
&& \Pr\left(|\mathbf{Z}_k^{\top} \mathbf{H}\boldsymbol{\varepsilon} | > \lambda_{n1}\omega_k^{\alpha} / 3, \, \exists \, k \in \mathcal{N}_z \right)
\leq \Pr\left(|\mathbf{Z}_k^{\top} \mathbf{H}\boldsymbol{\varepsilon} | > c_1\lambda_{n1}r_{n1}, \, \exists \, k \in \mathcal{N}_z\right) + o(1) \\
&=& \Pr \left( \max \limits_{k \in \mathcal{N}_z} | n^{-1/2} \mathbf{Z}_k^{\top} \mathbf{H}\boldsymbol{\varepsilon}| > c_1 n^{-1 / 2} \lambda_{n3}r_{n1} \right) + o(1) \\
&\leq& n^{1 / 2}(c_1 \lambda_{n3}r_{n1})^{-1}\mathrm{E}\left( \max\limits_{k \in \mathcal{N}_z} | n^{-1/2} \mathbf{Z}_k^{\top} \mathbf{H}\boldsymbol{\varepsilon}| \right) + o(1)
\leq n^{1 / 2} O\{\ln (p_{1})^{1 / 2} \}{c_1^{-1} \lambda_{n3}^{-1}r_{n1}^{-1} } + o(1) \\
&=& O\{n^{1 / 2}\sqrt{\ln (p_{1})}\lambda_{n3}^{-1}r_{n1}^{-1}\} + o(1) .
\end{eqnarray*}
Similarly,
\[
\Pr\left(\Vert \mathbf{X}_{\ell}^{\top} \mathbf{H}\boldsymbol{\varepsilon} \Vert_2 > \lambda_{n2}\omega_{\ell}^{\beta} / 3, \, \exists \, \ell \in \mathcal{N}_{x,L} \right)= O\left(n^{1 / 2}\sqrt{\ln(p_2)}{\lambda_{n2}^{-1} r_{n2}^{-1}}\right) + o(1),
\]
and
\[
\Pr\left(\Vert \mathbf{B}_{\ell}^{(1)\top} \mathbf{H}\boldsymbol{\varepsilon} \Vert_2 > \lambda_{n3}\omega_{\ell}^{\gamma} / 3, \, \exists \,l \in \mathcal{N}_{x,N} \right)= O\left(n^{1 / 2}\sqrt{N_n\ln(p_{2} N_n) }{\lambda_{n3}^{-1} r_{n3}^{-1}}\right) + o(1).
\]
Recall the definition of $\mathcal{N}$ in (\ref{DEF:SN}) and $\pi_3 = \max_{m \notin \mathcal{S}} \Vert n^{-1}\mathbf{D}_m^{\top}\mathbf{D}_m \Vert_2$, by the properties of spline \citep{de2001practical}, the $\mathbf{D}_m^{\top} \mathbf{H}\boldsymbol{\delta}$ term has
\begin{align*}
\max\limits_{m \in \mathcal{N}} \Vert \mathbf{D}_m^{\top} \mathbf{H}\boldsymbol{\delta} \Vert_2
&\leq n^{1/2} \max\limits_{m \notin \mathcal{S}} \Vert \frac{1}{n} \mathbf{D}_m^{\top} \mathbf{D}_m \Vert_2^{1 / 2} \Vert \mathbf{H} \Vert_2 \Vert \boldsymbol{\delta} \Vert_2 \!
= O_{P}\left(n\pi_3^{1 / 2}N_n^{-1}|\mathcal{S}_{x,N}|^{1 / 2}\right) =O_{P}\left(nN_n^{-1}\right).
\end{align*}
and the last term follows by Lemma \ref{LEM:eigen} and (\ref{EQN:v1_order}) and (\ref{EQN:v2_order}) in Lemma \ref{LEM:4} that
\begin{align*}
\max\limits_{m \in \mathcal{N}} &\Vert n^{-1}\mathbf{D}_m^{\top} \mathbf{D}_{\mathcal{S}}\mathbf{C}_{\mathcal{S}}^{-1}\left(\sum_{j=1}^3 \lambda_{nj}\boldsymbol{v}_j\right) \Vert_2
\leq \max\limits_{m \notin \mathcal{S}} \Vert n^{-1 / 2} \mathbf{D}_m \Vert_2 \times \Vert n^{- 1 / 2} \mathbf{D}_{\mathcal{S}}\mathbf{C}_{\mathcal{S}}^{-1 / 2} \Vert_2 \times \Vert \mathbf{C}_{\mathcal{S}}^{-1 / 2} \Vert_2 \\
&\times \Vert \lambda_{n1}\boldsymbol{v}_1 + \lambda_{n2}\boldsymbol{v}_2 + \lambda_{n3}\boldsymbol{v}_3\Vert_2 \leq \pi_3^{1 / 2} \pi_1^{-1 / 2} {O_P\left(\sum_{j=1}^3 \lambda_{nj}h_{nj}\right)}
= {O_P\left(\sum_{j=1}^3 \lambda_{nj}h_{nj}\right)}.
\end{align*}
\end{proof}
\begin{proof} [Proof of Theorem \ref{THM:consistency}]
The idea of the proof is similar to the proof of part (ii) in Theorem \ref{THM:LASSO-1}, but we look at index set $\mathcal{S}$ instead of $\mathcal{S}^{\prime}$. Let $\pi_1$ and $\pi_2$ be the minimum and maximum eigenvalues of $\mathbf{C}_{\mathcal{S}}$, respectively, and let $\pi_3 = \max_{m \notin \mathcal{S}} \Vert n^{-1}\mathbf{D}_m^{\top}\mathbf{D}_m \Vert$. By Lemma \ref{LEM:eigen}, $\pi_1 \asymp 1$, $\pi_2 \asymp 1$ and $\pi_3 \asymp 1$.
For any $\ell\in\mathcal{S}_{x,N}$, let $g_{0\ell} (\mathbf{X}_{\ell})=\left(g_{0\ell} (X_{1\ell}),\ldots, g_{0\ell} (X_{n\ell})\right)^{\top}$, $\boldsymbol{\delta}_{\ell}=g_{0\ell} (\mathbf{X}_{\ell})- \mathbf{B}_{\ell}^{(1)} \boldsymbol{\gamma}_{\ell}$ and $\boldsymbol{\delta}=\sum_{\ell \in \mathcal{S}_{x,N}} \boldsymbol{\delta}_{\ell}$. According to the proof of Theorem \ref{THM:selection}, with probability approaching one, we have
\begin{align*}
{\widehat{\boldsymbol{\theta}}}_{\mathcal{S}} &= \widehat{\boldsymbol{\theta}}_{\mathcal{S}}^o
= \left(\mathbf{D}_{\mathcal{S}}^{\top}\mathbf{D}_{\mathcal{S}}\right)^{-1} \left(\mathbf{D}_{\mathcal{S}}^{\top}\mathbf{Y} - \sum_{j=1}^{3}\lambda_{nj} \boldsymbol{v}_j \right)
= \left(\mathbf{D}_{\mathcal{S}}^{\top} \mathbf{D}_{\mathcal{S}}\right)^{-1} \times\\
&\left[\mathbf{D}_{\mathcal{S}}^{\top} \left\{\mathbf{Z}_{\mathcal{S}_z}\boldsymbol{\alpha}_{0, \mathcal{S}_z} + \mathbf{X}_{\mathcal{S}_{x, L}}\boldsymbol{\beta}_{0, \mathcal{S}_{x, L}} + \sum_{\ell \in \mathcal{S}_{x,N}} \left(\mathbf{B}_{\ell}^{(1)} \boldsymbol{\gamma}_{0\ell} + \boldsymbol{\delta}_{\ell}\right) + \boldsymbol{\varepsilon}\right\} - \sum_{j=1}^{3}\lambda_{nj} \boldsymbol{v}_j \right] \\
&= \boldsymbol{\theta}_{0, \mathcal{S}} + \left(\mathbf{D}_{\mathcal{S}}^{\top} \mathbf{D}_{\mathcal{S}}\right)^{-1} \left\{\mathbf{D}_{\mathcal{S}}^{\top} \left(\boldsymbol{\delta}+\boldsymbol{\varepsilon}\right) - \sum_{j=1}^{3}\lambda_{nj} \boldsymbol{v}_j \right \}.
\end{align*}
Let $\mathbf{C}_{\mathcal{S}} = n^{-1} \mathbf{D}_{\mathcal{S}}^{\top} \mathbf{D}_{\mathcal{S}}$ be an $(|\mathcal{S}_z| + |\mathcal{S}_{x,L}| + |\mathcal{S}_{x,N}|N_n)\times(|\mathcal{S}_z| + |\mathcal{S}_{x,L}| + |\mathcal{S}_{x,N}|N_n)$ matrix, and let $\mathbf{H = I} - \mathbf{D}_{\mathcal{S}} \left(\mathbf{D}_{\mathcal{S}}^{\top} \mathbf{D}_{\mathcal{S}}\right)^{-1}\mathbf{D}_{\mathcal{S}}^{\top}$ be an $n \times n$ matrix, then
\begin{equation}
\widehat{\boldsymbol{\theta}}_{\mathcal{S}} - \boldsymbol{\theta}_{\mathcal{S}}^o = n^{-1} \mathbf{C}_{\mathcal{S}}^{-1} \left\{\mathbf{D}_{\mathcal{S}}^{\top} \left(\boldsymbol{\delta} + \boldsymbol{\varepsilon}\right) - \sum_{j=1}^{3}\lambda_{nj} \boldsymbol{v}_j \right\}.
\label{EQN:theta_dfc}
\end{equation}
For $\boldsymbol{\eta} = \mathbf{Y} - \mathbf{D}\boldsymbol{\theta}$, define $\boldsymbol{\eta}_{*}$ as the projection of $\boldsymbol{\eta}$ to the column space of $\mathbf{D}_{\mathcal{S}}$, that is, $\boldsymbol{\eta}_{*} \equiv \mathbf{P}_{\mathbf{D}_{\mathcal{S}}}\boldsymbol{\eta} = \mathbf{D}_{\mathcal{S}}(\mathbf{D}_{\mathcal{S}}^{\top}\mathbf{D}_{\mathcal{S}})^{-1}\mathbf{D}_{\mathcal{S}}^{\top}\boldsymbol{\eta}$. Then for $\boldsymbol{\varepsilon}_{*} \equiv \mathbf{P}_{\mathbf{D}_{\mathcal{S}}}\boldsymbol{\varepsilon}$, similar to (\ref{EQN:the_diffs1}), (\ref{EQN:eta_ord}) and (\ref{EQN:eps_ord}), and by Lemma \ref{LEM:eigen},
\begin{align*}
\Vert \boldsymbol{\varepsilon}_{*} \Vert^2
= \Vert \left(\mathbf{D}_{\mathcal{S}}^{\top}\mathbf{D}_{\mathcal{S}}\right)^{-1/2} \mathbf{D}_{\mathcal{S}}^{\top}\boldsymbol{\varepsilon} \Vert^2
\leq (n\pi_1)^{-1} \Vert \mathbf{D}_{\mathcal{S}}^{\top}\boldsymbol{\varepsilon} \Vert^2
= O_P\{\pi_1^{-1}(|\mathcal{S}_z|+|\mathcal{S}_{x,L}|+|\mathcal{S}_{x,N}|N_n)\},
\end{align*}
\begin{align*}
\Vert \boldsymbol{\eta}_{*} \Vert^2
\leq 2 \Vert \boldsymbol{\varepsilon}_{*} \Vert^2 + O_P(n|\mathcal{S}_{x,N}|N_n^{-2})
= O_P\{\pi_1^{-1}(|\mathcal{S}_z|+|\mathcal{S}_{x,L}|+|\mathcal{S}_{x,N}|N_n)\}
+ O_P(nN_n^{-2}),
\end{align*}
\begin{align*}
\Vert \widehat{\boldsymbol{\theta}}_{\mathcal{S}} - \boldsymbol{\theta}_{0, \mathcal{S}} \Vert^2
& \leq \frac{8\Vert \boldsymbol{\eta}_{*} \Vert^2}{n\pi_{1}} + \frac{4 \{\lambda_{n1}^2|\mathcal{S}_z| + \lambda_{n2}^2|\mathcal{S}_{x,L}|+ \lambda_{n3}^2|\mathcal{S}_{x,N}|\}}{n^2\pi_{1}^2} \\
& = O_P\left\{ \frac{|\mathcal{S}_z|+|\mathcal{S}_{x,L}|+|\mathcal{S}_{x,N}|N_n}{n \pi_1^2}\right\} + O\left(\frac{|\mathcal{S}_{x,N}|}{\pi_1 N_n^2}\right) \\
& \quad + O_P\left\{\frac{\lambda_{n1}^2|\mathcal{S}_z| + \lambda_{n2}^2|\mathcal{S}_{x,L}|+ \lambda_{n3}^2|\mathcal{S}_{x,N}|}{n^2 \pi_1^2}\right\}.
\end{align*}
Therefore, the results follow by the facts that
\begin{align}
\widehat{\boldsymbol{\alpha}}_{\mathcal{S}_z} - \boldsymbol{\alpha}_{0,\mathcal{S}_z} &= \left(\mathbf{I}_{|\mathcal{S}_z|} ~~ \boldsymbol{0}_{|\mathcal{S}_z| \times |\mathcal{S}_{x,L}|} ~~ \boldsymbol{0}_{|\mathcal{S}_z| \times (|\mathcal{S}_{x,N}|N_n)}\right)(\sigs{\widehat{\boldsymbol{\theta}}} - \boldsymbol{\theta}_{0,\mathcal{S}} ), \nonumber \\
\widehat{\boldsymbol{\beta}}_{\mathcal{S}_{x, L}} - \boldsymbol{\beta}_{0,\mathcal{S}_{x, L}} &= \left(\boldsymbol{0}_{|\mathcal{S}_{x,L}| \times |\mathcal{S}_z|} ~~ \mathbf{I}_{|\mathcal{S}_{x,L}|} ~~ \boldsymbol{0}_{|\mathcal{S}_{x,L}|\times (|\mathcal{S}_{x,N}|N_n)}\right)(\sigs{\widehat{\boldsymbol{\theta}}} - \boldsymbol{\theta}_{0,\mathcal{S}} ), \nonumber \\
\widehat{\boldsymbol{\gamma}}_{\mathcal{S}_{x, N}} - \boldsymbol{\gamma}_{0,\mathcal{S}_{x, N}} &=\left(\boldsymbol{0}_{(|\mathcal{S}_{x,N}| N_n) \times |\mathcal{S}_z|} ~~ \boldsymbol{0}_{(|\mathcal{S}_{x,N}| N_n) \times |\mathcal{S}_{x,L}|} ~~ \mathbf{I}_{|\mathcal{S}_{x,N}| N_n}\right)(\sigs{\widehat{\boldsymbol{\theta}}} - \boldsymbol{\theta}_{0,\mathcal{S}}),
\label{EQN:diffs_betagamma}
\end{align}
and $\Vert \widehat{g}_{\ell} - g_{n\ell} \Vert_2^2 \asymp \Vert \widehat{\boldsymbol{\gamma}}_{\ell} - \boldsymbol{\gamma}_{0\ell} \Vert^2$,
where $\widehat{\boldsymbol{\beta}}_{\mathcal{S}_z} = (\widehat{\beta}_k, k \in \mathcal{S}_z)^{\top}$, $\boldsymbol{\beta}_{0,\mathcal{S}_z} = ({\beta}_{0k}, k \in \mathcal{S}_z)^{\top}$, $\widehat{\boldsymbol{\gamma}}_{\mathcal{S}_{x,N}} = (\widehat{\boldsymbol{\gamma}}_{\ell}, \ell \in \mathcal{S}_{x,N})^{\top}$ and $\boldsymbol{\gamma}_{0,\mathcal{S}_{x,N}} = (\boldsymbol{\gamma}_{\ell}, \ell \in \mathcal{S}_{x,N})^{\top}$.
\end{proof}
\subsection{Proof of Theorem \ref{THM:nonlinear-normality}}
In this section, the spline basis functions considered are of order $d$.
For any index set $\mathcal{A} \subseteq \{1, \ldots, p_1+p_2\}$, denote $\boldsymbol{\beta}_{\mathcal{A}} = (\beta_k, 1 \leq k \leq p_1, k \in \mathcal{A})^{\top}$, $\widehat{\boldsymbol{\beta}}_{\mathcal{A}} = (\widehat{\beta}_k, 1 \leq k \leq p_1, k \in \mathcal{A})^{\top}$, $\boldsymbol{\gamma}_{\mathcal{A}} = (\boldsymbol{\gamma}_{\ell}, 1 \leq \ell \leq p_2$, $\ell + p_1 \in \mathcal{A})^{\top}$ and $\widehat{\boldsymbol{\gamma}}_{\mathcal{A}} = (\widehat{\boldsymbol{\gamma}}_{\ell}, 1 \leq \ell \leq p_2$, $\ell + p_1 \in \mathcal{A})^{\top}$.
Next, denote $\mathbf{Z}_{\mathcal{A}} = (\mathbf{Z}_{i, \mathcal{A}} ^{\top}, i = 1, \ldots, n)^{\top}$, where $\mathbf{Z}_{i, \mathcal{A}} = (Z_{ik}, 1 \leq k \leq p_1, k \in \mathcal{A})^{\top}$, $\mathbf{X}_{i, \mathcal{A}} = (X_{i\ell}, 1 \leq \ell \leq p_2, \ell+p_1 \in \mathcal{A})^{\top}$. Similarly, denote $\mathbf{B}_{\mathcal{A}}^{(d)} = (\mathbf{B}_{i, \mathcal{A}}^{(d)\top}, i = 1, \ldots, n)^{\top}$, where $\mathbf{B}_{i, \mathcal{A}}^{(d)} = (B_{J,\ell}^{(d)}(X_{i\ell}), 1 \leq \ell \leq p_2$, $\ell + p_1 + p_2 \in \mathcal{A}, J = 1, \ldots, N_n)^{\top}$.
Define $\mathbf{T}_{\mathcal{S}} = (\mathbf{Z}_{\mathcal{S}_z}, \mathbf{X}_{\mathcal{S}_{x,PL}})$.
By an abuse of notation, let $\mathbf{D}_{\mathcal{S}} = \left(\mathbf{T}_{\mathcal{S}}, \mathbf{B}_{\mathcal{S}}^{(d)}\right)$,
and we define
\begin{equation}
\mathbf{C}_{\mathcal{S}} = n^{-1}\mathbf{D}_{\mathcal{S}}^{\top} \mathbf{D}_{\mathcal{S}} =
\begin{pmatrix}
n^{-1} \mathbf{T}_{\mathcal{S}}^{\top}\mathbf{T}_{\mathcal{S}} & n^{-1}\mathbf{T}_{\mathcal{S}}^{\top}\mathbf{B}_{\mathcal{S}}^{(d)} \\
n^{-1}\mathbf{B}_{\mathcal{S}}^{(d)\top}\mathbf{T}_{\mathcal{S}} & n^{-1}\mathbf{B}_{\mathcal{S}}^{(d)\top}\mathbf{B}_{\mathcal{S}}^{(d)}
\end{pmatrix} =
\begin{pmatrix}
\mathbf{C}_{11} & \mathbf{C}_{12} \\
\mathbf{C}_{21} & \mathbf{C}_{22}
\end{pmatrix} ,
\label{EQN:cs}
\end{equation}
\begin{equation}
\mathbf{U}_{\mathcal{S}} = \mathbf{C}_{\mathcal{S}} ^{-1} =
\begin{pmatrix}
\mathbf{U}_{11} & - \mathbf{U}_{11} \mathbf{C}_{12} \mathbf{C}_{22}^{-1} \\
- \mathbf{U}_{22} \mathbf{C}_{21} \mathbf{C}_{11}^{-1} & \mathbf{U}_{22}
\end{pmatrix} =
\begin{pmatrix}
\mathbf{U}_{11} & \mathbf{U}_{12} \\
\mathbf{U}_{21} & \mathbf{U}_{22}
\end{pmatrix},
\label{EQN:us}
\end{equation}
where $\mathbf{U}_{11}^{-1} = \mathbf{C}_{11} - \mathbf{C}_{12} \mathbf{C}_{22}^{-1} \mathbf{C}_{21} = n^{-1} \mathbf{T}_{\mathcal{S}}^{\top} \left(\mathbf{I}_n - \mathbf{P}_{\mathbf{B}_{\mathcal{S}}^{(d)}}\right) \mathbf{T}_{\mathcal{S}}$ and $\mathbf{U}_{22}^{-1} = \mathbf{C}_{22} - \mathbf{C}_{21} \mathbf{C}_{11}^{-1} \mathbf{C}_{12}$ $= n^{-1} \mathbf{B}_{\mathcal{S}}^{(d)\top} \left(\mathbf{I}_n - \mathbf{P}_{\mathbf{T}_{\mathcal{S}}}\right) \mathbf{B}_{\mathcal{S}}^{(d)}$,
with $\mathbf{P}_{\mathbf{B}_{\mathcal{S}}^{(d)}}$ and $\mathbf{P}_{\mathbf{T}_{\mathcal{S}}}$ being projection matrices for $\mathbf{B}_{\mathcal{S}}^{(d)}$ and $\mathbf{T}_{\mathcal{S}}$, respectively.
In the following, we give the proof of Theorem \ref{THM:nonlinear-normality}.
\begin{proof}[Proof of Theorem \ref{THM:nonlinear-normality}]
The structure of the proof is consisted of two parts: (i) we show the oracle efficiency of $\widehat{\phi}_{\ell}^{\mathrm{SBLL}}$; (ii) we show the uniform asymptotic normality for the ``oracle'' estimator $\widehat{\phi}_{\ell}^o$.
For part (i), note that for $\ell \in \mathcal{S}_{x,N}$
\[
\widehat{\phi}_{\ell}^{\mathrm{SBLL}}\left( x_{\ell}\right)-\widehat{\phi}_{\ell}^o\left( x_{\ell}\right) = \left(1, \, 0\right) (\mathbf{X}_{\ell}^{\ast\top}\mathbf{W}_{\ell}\mathbf{X}_{\ell}^{\ast})^{-1} \mathbf{X}_{\ell}^{\ast\top}\mathbf{W}_{\ell} (\widehat{\mathbf{Y}}_{\ell} - \mathbf{Y}_{\ell}), ~~\mathrm{where}
\]
\begin{align*}
\widehat{\mathbf{Y}}_{\ell} - \mathbf{Y}_{\ell}
&= \mathbf{Z}_{\mathcal{S}_z} (\boldsymbol{\alpha}_{0, \mathcal{S}_z} - \widehat{\boldsymbol{\alpha}}_{\mathcal{S}_z}^{\ast})
\!\!\!\!+ \mathbf{X}_{\mathcal{S}_{x,PL}} (\boldsymbol{\beta}_{0, \mathcal{S}_{x,PL}} - \widehat{\boldsymbol{\beta}}_{\mathcal{S}_{x,PL}}^{\ast})
+ \sum_{\ell^{\prime} \in \mathcal{S}_{x,N}\setminus \{\ell\} } \left\{\phi_{0\ell^{\prime}} (\mathbf{X}_{\ell^{\prime}}) - \widehat{\phi}_{\ell^{\prime}}^{\ast}(\mathbf{X}_{\ell^{\prime}}) \right\} \\
&=\mathbf{Z}_{\mathcal{S}_z} (\boldsymbol{\alpha}_{0, \mathcal{S}_z} - \widehat{\boldsymbol{\alpha}}_{\mathcal{S}_z}^{\ast})
+ \mathbf{X}_{\mathcal{S}_{x,PL}} (\boldsymbol{\beta}_{0, \mathcal{S}_{x,PL}} - \widehat{\boldsymbol{\beta}}_{\mathcal{S}_{x,PL}}^{\ast}) \\
& \quad \quad + \mathbf{B}_{\mathcal{S}_{x,N} \setminus \{\ell\}}^{(d)} (\boldsymbol{\gamma}_{0, \mathcal{S} _{x,N}\setminus \{\ell\}} - {\widehat{\boldsymbol{\gamma}}}_{\mathcal{S}_{x,N} \setminus \{\ell\}}^{\ast})
+ \sum_{\ell^{\prime} \in \mathcal{S}_{x,N}\setminus \{\ell\} } \left\{\phi_{0\ell^{\prime}} (\mathbf{X}_{\ell^{\prime}}) - \phi_{nl^{\prime}} (\mathbf{X}_{\ell^{\prime}}) \right\} ,
\end{align*}
and
\begin{align*}
\!\!\!\text{diag} (1, h_{\ell}^{-1}) \, &\mathbf{X}_{\ell}^{\ast\top}\mathbf{W}_{\ell}\mathbf{X}_{\ell}^{\ast} \, \text{diag} (1, h_{\ell}^{-1}) \\
= &
\begin{pmatrix}
n^{-1}\sum_{i = 1}^n K_{h_{\ell}}(X_{i\ell} - x_{\ell}) & n^{-1} \sum_{i = 1}^n \left(\frac{X_{i\ell} - x_{\ell}}{h_{\ell}}\right)K_{h_{\ell}}(X_{i\ell} - x_{\ell}) \\
n^{-1} \sum_{i = 1}^n \left(\frac{X_{i\ell} - x_{\ell}}{h_{\ell}}\right) K_{h_{\ell}}(X_{i\ell} - x_{\ell}) & n^{-1}\sum_{i = 1}^n \left(\frac{X_{i\ell} - x_{\ell}}{h_{\ell}}\right)^{2} K_{h_{\ell}}(X_{i\ell} - x_{\ell})
\end{pmatrix} \\
= & f_{\ell}(x_{\ell})
\begin{pmatrix}
1 & 0 \\
0 & \mu_2(K)
\end{pmatrix}
+ u_P(1),
\end{align*}
with $u_P(\cdot) = o_P(\cdot)$ uniformly for all $x_{\ell} \in [a, b]$. So
\[
\left(\mathbf{X}_{\ell}^{\ast\top}\mathbf{W}_{\ell}\mathbf{X}_{\ell}^{\ast}\right)^{-1} = \text{diag} (1, h_{\ell}^{-1}) \, f_{\ell}^{-1}(x_{\ell}) \left\{
\begin{pmatrix}
1 & 0 \\
0 & \mu_2(K)
\end{pmatrix}
+ u_P(1) \right\} \, \text{diag} (1, h_{\ell}^{-1}),
\]
\[
\text{diag} (1, h_{\ell}^{-1}) \, \mathbf{X}_{\ell}^{\ast\top}\mathbf{W}_{\ell} = \frac{1}{n} \times
\begin{pmatrix}
K_{h_{\ell}} (X_{1\ell} - x_{\ell}) & ,\ldots, & K_{h_{\ell}} (X_{n\ell} - x_{\ell}) \\
\left(\frac{X_{1\ell} - x_{\ell}}{h_{\ell}}\right) K_{h_{\ell}} (X_{1\ell} - x_{\ell}) & ,\ldots, & \left(\frac{X_{n\ell} - x_{\ell}}{h_{\ell}}\right) K_{h_{\ell}} (X_{n\ell} - x_{\ell})
\end{pmatrix}.
\]
Thus,
\begin{align}
\widehat{\phi}_{\ell}^{\mathrm{SBLL}}\left( x_{\ell}\right) & -\widehat{\phi}_{\ell}^o\left( x_{\ell}\right) = f_{\ell}^{-1} (x_{\ell}) \left[
\frac{1}{n} \sum_{i = 1}^n K_{h_{\ell}}(X_{i\ell} - x_{\ell}) \mathbf{Z}_{i, \mathcal{S}_z}^{\top} (\boldsymbol{\alpha}_{0, \mathcal{S}_z} - \widehat{\boldsymbol{\alpha}}_{\mathcal{S}_z}^{\ast}) \right . \nonumber \\
& + \frac{1}{n} \sum_{i = 1}^n K_{h_{\ell}}(X_{i\ell} - x_{\ell}) \mathbf{X}_{i, \mathcal{S}_{x,PL}}^{\top} (\boldsymbol{\beta}_{0, \mathcal{S}_{x,PL}} - \widehat{\boldsymbol{\beta}}_{\mathcal{S}_{x,PL}}^{\ast}) \nonumber \\
& + \frac{1}{n} \sum_{i = 1}^n K_{h_{\ell}}(X_{i\ell} - x_{\ell}) \mathbf{B}_{i, \mathcal{S}_{x,N} \setminus \{\ell\}}^{(d)\top} (\boldsymbol{\gamma}_{0, \mathcal{S}_{x,N} \setminus \{\ell\} } - {\widehat{\boldsymbol{\gamma}}}_{\mathcal{S}_{x,N} \setminus \{\ell\} }^{\ast}) \nonumber \\
& \left . + \, \frac{1}{n} \sum_{i = 1}^n \sum_{\ell^{\prime} \in \mathcal{S}_{x,N} \setminus \{\ell\} } K_{h_{\ell}}(X_{i\ell} - x_{\ell}) \left\{\phi_{0\ell^{\prime}} ({X}_{i\ell^{\prime}}) - \phi_{nl^{\prime}} ({X}_{i\ell^{\prime}}) \right\} + u_P(1) \right].
\label{eqn:alpha_diff}
\end{align}
For the first and second summation terms in the right hand side of (\ref{eqn:alpha_diff}), by Theorem \ref{THM:normality}, we have $ n^{-1}\sum_{i = 1}^n K_{h_{\ell}}(X_{i\ell} - x_{\ell}) \mathbf{Z}_{i, \mathcal{S}_z}^{\top} (\boldsymbol{\alpha}_{0, \mathcal{S}_z} - \widehat{\boldsymbol{\alpha}}_{\mathcal{S}_z}^{\ast}) = u_P\left( n^{-1/2} \right) $ and $ n^{-1}\sum_{i = 1}^n K_{h_{\ell}}(X_{i\ell} - x_{\ell}) \mathbf{X}_{i, \mathcal{S}_{x,PL}}^{\top} (\boldsymbol{\beta}_{0, \mathcal{S}_{x,PL}} \!- \widehat{\boldsymbol{\beta}}_{\mathcal{S}_{x,PL}}^{\ast}) \!=\! u_P\left( n^{-1/2} \right)$;
and by Lemma \ref{LEM:spline-approx}, $n^{-1} \!\sum_{i = 1}^n \!\sum_{\ell^{\prime} \in \mathcal{S}_{x,N} \setminus \{\ell\}}K_{h_{\ell}}(X_{i\ell} - x_{\ell}) \left\{\phi_{0\ell^{\prime}} ({X}_{i\ell^{\prime}}) - \phi_{nl^{\prime}} ({X}_{i\ell^{\prime}}) \right\} = u_P\left\{|\mathcal{S}_{x,N}|M_n^{-d}\right\}$.
As for the third terms, define $\zeta_{i\ell} = \phi_{0\ell}(X_{i\ell}) - \sum_{J=1}^{M_n} \gamma_{0J,l}^{\ast} B_{J,\ell}^{(d)}(X_{i\ell})$, $\zeta_{i} = \sum_{\ell \in \mathcal{S}_{x,N}}\zeta_{i\ell}$, and $\boldsymbol{\zeta} = (\zeta_1, \ldots, \zeta_n)^{\top}$, similar to the induction with (\ref{EQN:theta_dfc}), we have
\begin{equation}
\widehat{\boldsymbol{\theta}}_{\mathcal{S}}^{\ast} - \boldsymbol{\theta}_{\mathcal{S}}^o
= n^{-1} \mathbf{C}_{\mathcal{S}}^{-1} \{\mathbf{D}_{\mathcal{S}}^{\top} \left(\boldsymbol{\zeta} + \boldsymbol{\epsilon}\right)\},
\label{EQN:theta_dfc_star}
\end{equation}
then $\widehat{\boldsymbol{\gamma}}_{\mathcal{S}_{x,N}}^{\ast} - \boldsymbol{\gamma}_{0,\mathcal{S}_{x,N}}
= \left(\boldsymbol{0}_{\{|\mathcal{S}_{x,N}|M_n\} \times (|\mathcal{S}_z|+|\mathcal{S}_{x,L}|)} ~ \mathbf{I}_{(|\mathcal{S}_{x,N}|M_n)}\right) \mathbf{C}_{\mathcal{S}}^{-1} n^{-1} \left\{\mathbf{D}_{\mathcal{S}}^{\top} \left(\boldsymbol{\zeta} +\boldsymbol{\epsilon}\right)\right \}$.
Define a diagonal matrix $\mathbf{I}_{\ell}^0 = \mathrm{diag}\{\boldsymbol{1}_{(l-1)M_n}, \boldsymbol{0}_{M_n}, \boldsymbol{1}_{(|\mathcal{S}_{x,N}| - l)M_n}\}$, $\ell \in \mathcal{S}_{x,N}$. Then
\begin{align*}
\mathbf{B}_{i, \mathcal{S}_{x,N} \setminus \{\ell\}}^{(d)\top} (\boldsymbol{\gamma}_{0, \mathcal{S}_{x,N} \setminus \{\ell\} } - {\widehat{\boldsymbol{\gamma}}}_{\mathcal{S}_{x,N}\setminus \{\ell\} }^{\ast} )
= \, \mathbf{B}_{i, \mathcal{S}}^{(d)\top} \mathbf{I}_{\ell}^0
\left(\widehat{\boldsymbol{\gamma}}_{\mathcal{S}_{x,N}}^{\ast} - \boldsymbol{\gamma}_{0,\mathcal{S}_{x,N}}\right).
\end{align*}
Next by Lemma \ref{LEM:spline-approx}, (\ref{EQN:cs}) and (\ref{EQN:us}), for any $\ell \in \mathcal{S}_{x,N}$, we have
\begin{align*}
\frac{1}{n} \sum_{i = 1}^{n} & K_{h_{\ell}}(X_{i\ell} - x_{\ell}) \, \mathbf{B}_{i, \mathcal{S}}^{(d)\top} \mathbf{I}_{\ell}^0
\left(\widehat{\boldsymbol{\gamma}}_{\mathcal{S}_{x,N}}^{\ast} - \boldsymbol{\gamma}_{0,\mathcal{S}_{x,N}}\right)\\
& = \frac{1}{n} \sum_{i = 1}^{n} K_{h_{\ell}}(X_{i\ell} - x_{\ell}) \mathbf{B}_{i, \mathcal{S}}^{(d)\top} \mathbf{I}_{\ell}^0 \mathbf{U}_{22} \left(- \mathbf{C}_{21} \mathbf{C}_{11}^{-1} ~~ \mathbf{I}_{(|\mathcal{S}_{x,N}|M_n)} \right)
\frac{1}{n}\mathbf{B}_{\mathcal{S}}^{(d)\top} \left(\boldsymbol{\zeta}+\boldsymbol{\epsilon}\right) \\
= & \frac{1}{n} \sum_{i = 1}^{n} K_{h_{\ell}}(X_{i\ell} - x_{\ell}) \mathbf{B}_{i, \mathcal{S}}^{(d)\top} \mathbf{I}_{\ell}^0 \mathbf{U}_{22} \frac{1}{n}\mathbf{B}_{\mathcal{S}}^{(d)\top} \left(\mathbf{I}_n - \mathbf{P}_{\mathbf{T}_{\mathcal{S}}}\right) \left(\boldsymbol{\zeta}+\boldsymbol{\epsilon}\right) .
\end{align*}
Following the same idea in the proof of Lemma \ref{LEM:eigen}, we have that
there exist constants $0<c_{U_{2}}<C_{U_{2}}<\infty $, such that with
probability approaching one, $c_{U_{2}}\mathbf{I}
_{|\mathcal{S}_{x,N}|M_n}\leq \mathbf{U}_{22}\leq C_{U_{2}}\mathbf{I}_{|\mathcal{S}_{x,N}|M_n}$.
Similar to Lemma A.4 in \citet[][]{wang2007spline}, for any $\ell,\ell^{\prime} \in \mathcal{S}_{x,N}$ and $\ell\neq \ell^{\prime}$, we have
\begin{align*}
\sup_{x_{\ell} \in \chi_{h_{\ell}}}\max_{1\leq J\leq M_n}&
\left|\frac{1}{n}\!\sum_{i=1}^{n}\!\left[ K_{h_{\ell}}\left(
X_{i\ell}-x_{\ell}\right) B_{J\ell^{\prime}}^{(d)}\left( X_{i\ell^{\prime}}\right)
\!-\!E\{K_{h_{\ell}}\left(
X_{i\ell}-x_{\ell}\right) B_{J\ell^{\prime}}^{(d)}\left( X_{i\ell^{\prime}}\right)\} \right] \right| \!=\! O_P\!\left(\sqrt{\frac{\ln n}{nh_{\ell}}}\right),
\end{align*}
\begin{equation*}
\sup_{x_{\ell} \in \chi_{h_{\ell}}}\max_{1\leq J\leq M_n}\left|
n^{-1}\sum_{i=1}^{n}\mathrm{E}\{K_{h_{\ell}}\left(
X_{i\ell}-x_{\ell}\right) B_{J\ell}^{(d)}\left( X_{i\ell}\right)\} \right|
=O_{P}\left(M_n^{-1/2}\right). \label{EQ:supxi}
\end{equation*}
We can show that
\begin{align*}
\sup_{x_{\ell} \in \chi_{h_{\ell}}}\!\frac{1}{n} \!\sum_{i = 1}^{n} & K_{h_{\ell}}(X_{i\ell} - x_{\ell}) \mathbf{B}_{i, \mathcal{S}}^{(d)\top} \mathbf{I}_{\ell}^0 \mathbf{U}_{22} \frac{1}{n}\mathbf{B}_{\mathcal{S}}^{(d)\top} \left(\mathbf{I}_n - \mathbf{P}_{\mathbf{T}_{\mathcal{S}}}\right) \boldsymbol{\delta}
\!=\!O_{P}\{M_n^{-d + 1} (\sqrt{\ln n / (nh_{\ell})}+M_n^{-1/2})\},
\end{align*}
and by Proposition 2 in \citet[][]{wang2009efficient},
\[
\sup_{x_{\ell} \in \chi_{h_{\ell}}} \frac{1}{n} \sum_{i = 1}^{n} K_{h_{\ell}}(X_{i\ell} - x_{\ell}) \mathbf{B}_{i, \mathcal{S}}^{(d)\top} \mathbf{I}_{\ell}^0 \mathbf{U}_{22} \frac{1}{n}\mathbf{B}_{\mathcal{S}}^{(d)\top} \left(\mathbf{I}_n - \mathbf{P}_{\mathbf{T}_{\mathcal{S}}}\right) \boldsymbol{\epsilon} =O_{P}(\sqrt{\ln(n) / n}).
\]
Therefore,
\[
\sup_{x_{\ell} \in \chi_{h_{\ell}}}|\widehat{\phi}_{\ell}^{\mathrm{SBLL}}\left( x_{\ell}\right)-\widehat{\phi}_{\ell}^o\left( x_{\ell}\right) |
= O_{P}\left\{\sqrt{\frac{\ln n} {n}} + M_n^{-d+1} \left(\sqrt{\frac{\ln n} {nh_{\ell}}}+\sqrt{\frac{1}{M_n}}\right)\right\}.
\]
For part (ii), below we show that for any $t$ and $ \ell\in \mathcal{S}_{x,N}$,
\[
\lim_{n \rightarrow \infty} \Pr\left\{\sqrt{\ln(h_{\ell}^{-2})} \left(\sup_{x_{\ell} \in \chi_{h_{\ell}}} \frac{\sqrt{nh_{\ell}}}{v_{\ell}(x_{\ell})} |\widehat{\phi}_{\ell}^{o}\left( x_{\ell}\right) -\phi_{0\ell}(x_{\ell})| - \tau_n\right) < t\right\}
= e^{-2 e^{-t}},
\]
where
$v_{\ell}^{2}(x_{\ell}) = \|K\|_2^2 f_{\ell}^{-1}(x_{\ell}) \sigma^{2}$,
$\tau_n = \sqrt{\ln(h_{\ell}^{-2})} + \ln\{\|K^{\prime}\|_2 / (2\pi \|K\|_2)\} / \sqrt{\ln(h_{\ell}^{-2})}$.
Define $M_{h}(x)=h_{\ell}^{-1/2}\int K\{(x^{\prime}-x)/h_{\ell}\}dW(x^{\prime})$, where $W(x)$ is a Wiener process defined on $(0,\infty)$. By the Lemma 1 in \cite{zheng2016statistical}, one has
\begin{equation}
\label{EQU:Wiener}
\lim_{n \rightarrow \infty} \Pr\left[ \sqrt{\ln(h_{\ell}^{-2})}
\bigg\{\sup_{x \in \chi_{h_{\ell}}}|M_{h_{\ell}}(x)|/\|K\|_{L_2}^2-\tau_n\bigg\} < t \right] = e^{-2e^{-t}}.
\end{equation}
Recall the definition of $\widehat{\phi}_{\ell}^{o}(x_{\ell})$ in (15), we have
\begin{align*}
\widehat{\phi}_{\ell}^{o}(x_{\ell}) -\phi_{0\ell}(x_{\ell})
=&\left(1 ~~ 0\right) \left(\mathbf{X}_{\ell}^{\ast\top}\mathbf{W}_{\ell}\mathbf{X}_{\ell}^{\ast}\right)^{-1} \mathbf{X}_{\ell}^{\ast\top}\mathbf{W}_{\ell} \mathbf{Y}_{\ell} - \phi_{0\ell}(x_{\ell}) \\
=&f_{\ell}^{-1} (x_{\ell})\frac{1}{n} \sum_{i = 1}^n K_{h_{\ell}}(X_{i\ell} - x_{\ell}) \varepsilon_i + O_P(h_{\ell}^2).
\end{align*}
According to the proof of Theorem 1 in \cite{zheng2016statistical}, we have
\begin{equation}
\label{SCB2}
\sup_{x_{\ell} \in \chi_{h_{\ell}}}\left| \frac{\sqrt{nh_{\ell}}}{v_{\ell}(x_{\ell})} \{\widehat{\phi}_{\ell}^{o}( x_{\ell}) -\phi_{0\ell}(x_{\ell})\}-M_{h_{\ell}}(x_{\ell})/\|K\|_{L_2}^2\right|=o_P(\ln^{-1/2} n).
\end{equation}
Consequently, we have
\[
\sup_{x_{\ell} \in \chi_{h_{\ell}}}\sqrt{\ln(h_{\ell}^{-2})}\bigg| \frac{\sqrt{nh_{\ell}}}{v_{\ell}(x_{\ell})} \{\widehat{\phi}_{\ell}^{o}( x_{\ell}) -\phi_{0\ell}(x_{\ell})\}-M_{h_{\ell}}(x_{\ell})/\|K\|_{L_2}^2\bigg|=o_P(1),
\]
as $\sqrt{\ln(h_{\ell}^{-2})}/\sqrt{\ln(n)}=O(1)$.
The uniformly asymptotic normality of the ``oracle" estimator $\widehat{\phi}_{\ell}^{o}(x_{\ell})$ follows from (\ref{EQU:Wiener}) and Slusky's Theorem.
Hence, the result in (\ref{EQN:SBLL_Order}) is established. Consequently, the result in (\ref{EQN:sbll_normal}) follows from (\ref{EQN:alphao_normal}), and the result in (\ref{EQN:scb_form}) follows from \cite{claeskens2003bootstrap}.
\end{proof}
\subsection{Technical Lemmas}
The following lemmas are used in the proofs of Theorem \ref{THM:LASSO-1} and Theorem \ref{THM:selection}.
\begin{lemma}
\label{LEM:gLASSO_select}
Suppose that Assumptions (A1)--(A4) hold. Recall the definition of $\mathcal{S}$ and $\widetilde{\mathcal{S}}$ in (\ref{DEF:SN}) and (\ref{DEF:S_tilde}),
with probability approaching one, $|\widetilde{\mathcal{S}}| \leq M_1|{\mathcal{S}}| = M_1(|\mathcal{S}_z| + |\mathcal{S}_{x,L}| + |\mathcal{S}_{x,N}|)$ for a finite constant $M_1 > 1$.
\end{lemma}
\begin{proof} [Proof]
The basic idea of the proof is similar to the proofs of Theorem 1 of \cite{zhang2008sparsity} and Theorem 2.1 of \cite{wei2010consistent}. The main differences are the error term shown in (\ref{EQN:eta_form}) and we have a more complex data structure. By Lemma \ref{LEM:spline-approx}, for some constant $C_{4} > 0$, we have $\Vert \boldsymbol{\delta} \Vert_2 \leq C_{4} \sqrt{n |\mathcal{S}_{x,N}| N_n^{-2}} = C_{4} |\mathcal{S}_{x,N}|^{1/2} n^{1 / 2} N_n^{-1}$.
For any positive integers $s_1$, $s_2$ and $s_3$, pick some index sets $\mathcal{A}_1 \subseteq \{1, \ldots, p_{1}\}$, $\mathcal{A}_2 \subseteq \{1, \ldots, p_{2}\}$ and $\mathcal{A}_3 \subseteq \{1, \ldots, p_{2}\}$ such that the cardinalities of $\mathcal{A}_1$, $\mathcal{A}_2$ and $\mathcal{A}_3$ are $|\mathcal{A}_1| = s_1$, $|\mathcal{A}_2| = s_2$ and $|\mathcal{A}_3| = s_3$, respectively. Denote $\mathcal{A} = \mathcal{A}_1 \cup \mathcal{A}_2 \cup \mathcal{A}_3$. Define an $(s_1 + s_2 + s_3 N_n) \times 1$ vector $\mathbf{S}_\mathcal{A} = \left(\widetilde{\lambda}_{n1}\mathbf{u}_{s_1}^{\top}, \widetilde{\lambda}_{n2}\mathbf{u}_{s_2}^{\top}, \widetilde{\lambda}_{n3}\sqrt{N_n}\mathbf{U}_{1}^{\top}, \ldots, \widetilde{\lambda}_{n3}\sqrt{N_n}\mathbf{U}_{s_3}^{\top}\right)^{\top}$, where $\mathbf{u}_{s_1} \in \{\pm 1\}^{s_1}$, $\mathbf{u}_{s_2} \in \{\pm 1\}^{s_2}$ and $\mathbf{U}_j$ is in a unit ball with dimension $N_n$, that is, $\mathbf{U}_j \in \mathbb{R}^N$ and $\Vert \mathbf{U}_j \Vert_2 = 1$, $j = 1, \ldots, s_3$. Let $\mathbf{P}_\mathcal{A} = \mathbf{D}_\mathcal{A} (\mathbf{D}_\mathcal{A}^{\top}\mathbf{D}_\mathcal{A})^{-1}\mathbf{D}_\mathcal{A}^{\top}$ be the projection matrix of $\mathbf{D}_\mathcal{A}$. Define
\[
\chi_{s_1, s_2, s_3} = \max_{\substack{|\mathcal{A}_1| = s_1, |\mathcal{A}_2| = s_2, \\ |\mathcal{A}_3| = s_3 \\ \mathcal{A} = \mathcal{A}_1 \cup \mathcal{A}_2 \cup \mathcal{A}_3}}
\max\limits_{\substack{\mathbf{u}_{s_1} \in \{\pm 1\}^{s_1} \\ \mathbf{u}_{s_2} \in \{\pm 1\}^{s_2} \\ \Vert \mathbf{U}_j \Vert_2 = 1, \, 1 \leq j \leq s_3}}
\frac{|\boldsymbol{\eta}^{\top} \{\mathbf{D}_\mathcal{A}\left(\mathbf{D}_\mathcal{A}^{\top}\mathbf{D}_\mathcal{A}\right)^{-1}\mathbf{S}_\mathcal{A} - (\mathbf{I} - \mathbf{P}_\mathcal{A})\mathbf{D}\boldsymbol{\theta}_0\}|}
{\Vert \mathbf{D}_\mathcal{A}\left(\mathbf{D}_\mathcal{A}^{\top}\mathbf{D}_\mathcal{A}\right)^{-1}\mathbf{S}_\mathcal{A} - (\mathbf{I} - \mathbf{P}_\mathcal{A})\mathbf{D}\boldsymbol{\theta}_0 \Vert_2},
\]
\begin{eqnarray*}
\Omega_{|\mathcal{S}_z|, |\mathcal{S}_{x,L}|, |\mathcal{S}_{x,N}|} = \left\{(\mathbf{D}, \boldsymbol{\eta}): {\chi_{s_1, s_2, s_3} \leq \sigma C_{2} \sqrt{s_1\ln(p_1) \vee s_2\ln(p_2) \vee s_3 N_n \ln(p_3N_n)}}, \right. \\
\left. \forall \, s_1 \geq |\mathcal{S}_z|, \, s_2 \geq |\mathcal{S}_{x,L}|, \, s_3 \geq |\mathcal{S}_{x,N}| \right\},
\end{eqnarray*}
where $C_{2} > 0$ is some sufficiently large constant.
As shown in the proof of Theorem 1 of \citet[][]{zhang2008sparsity} and Theorem 2.1 of \citet[][]{wei2010consistent}, there exists a constant $M_1 > 1$, such that if $\left(\mathbf{D}, \boldsymbol{\eta}\right) \in \Omega_{|\mathcal{S}_z|, |\mathcal{S}_{x,L}|, |\mathcal{S}_{x,N}|}$, then $|\widetilde{\mathcal{S}}| \leq M_1|{\mathcal{S}}| = M_1(|\mathcal{S}_z| + |\mathcal{S}_{x,L}| + |\mathcal{S}_{x,N}|)$.
So it suffices to show that $\left(\mathbf{D}, \boldsymbol{\eta} \right) \in \Omega_{|\mathcal{S}_z|, |\mathcal{S}_{x,L}|, |\mathcal{S}_{x,N}|}$. Denote $\mathbf{V}_\mathcal{A} = \mathbf{D}_\mathcal{A}\left(\mathbf{D}_\mathcal{A}^{\top}\mathbf{D}_\mathcal{A}\right)^{-1}\mathbf{S}_\mathcal{A} - (\mathbf{I} - \mathbf{P}_\mathcal{A})\mathbf{D}\boldsymbol{\theta}_0$, then by the triangle and Cauchy-Schwarz inequalities,
\[
\frac{|\boldsymbol{\eta}^{\top} \, \mathbf{V}_\mathcal{A}|}{\Vert \mathbf{V}_\mathcal{A} \Vert_2} = \frac{|\boldsymbol{\varepsilon}^{\top} \, \mathbf{V}_\mathcal{A} + \boldsymbol{\delta}^{\top} \, \mathbf{V}_\mathcal{A}|}{\Vert \mathbf{V}_\mathcal{A} \Vert_2} \leq \frac{|\boldsymbol{\varepsilon}^{\top} \, \mathbf{V}_\mathcal{A}|}{\Vert \mathbf{V}_\mathcal{A}\Vert_2} + \Vert \boldsymbol{\delta} \Vert_2 .
\]
For the ${|\boldsymbol{\varepsilon}^{\top} \, \mathbf{V}_\mathcal{A}|} / {\Vert \mathbf{V}_\mathcal{A} \Vert_2}$ part, define
\[
\chi_{s_1, s_2, s_3} = \max_{\substack{|\mathcal{A}_1| = s_1, |\mathcal{A}_2| = s_2, \\ |\mathcal{A}_3| = s_3 \\ \mathcal{A} = \mathcal{A}_1 \cup \mathcal{A}_2 \cup \mathcal{A}_3}}
\max\limits_{\substack{\mathbf{u}_{s_1} \in \{\pm 1\}^{s_1} \\ \mathbf{u}_{s_2} \in \{\pm 1\}^{s_2} \\ \Vert \mathbf{U}_j \Vert_2 = 1, \, 1 \leq j \leq s_3}}
\frac{|\boldsymbol{\varepsilon}^{\top} \mathbf{V}_\mathcal{A}|}{\Vert \mathbf{V}_\mathcal{A} \Vert_2},
\]
\begin{eqnarray*}
\Omega_{|\mathcal{S}_z|, |\mathcal{S}_{x,L}|, |\mathcal{S}_{x,N}|}^{\ast} = \left\{(\mathbf{D}, \boldsymbol{\varepsilon}): \chi_{s_1, s_2, s_3}^{\ast} \leq \sigma C_{3} \sqrt{s_1\ln(p_1) \vee s_2\ln(p_2) \vee s_3 N_n \ln(p_3N_n)}, \right. \\
\left. \forall \, s_1 \geq |\mathcal{S}_z|, \, s_2 \geq |\mathcal{S}_{x,L}|, \, s_3 \geq |\mathcal{S}_{x,N}| \right\},
\end{eqnarray*}
where $C_3 > 0$ is some sufficiently large constant. As shown in the proof of Theorem 1 of \citet[][]{zhang2008sparsity} and Theorem 2.1 of \citet[][]{wei2010consistent},
$\Pr(\Omega_{|\mathcal{S}_z|, |\mathcal{S}_{x,L}|, |\mathcal{S}_{x,N}|}^{\ast}) \rightarrow 1$. And for $\Vert \boldsymbol{\delta} \Vert_2 $ part, for $n$ sufficiently large and $N_n \asymp n^{1/3}$,
\[
\Vert \boldsymbol{\delta} \Vert_2 \leq C_{4}|\mathcal{S}_{x,N}|^{1/2} n^{1 / 2} N_n^{-1}
\leq \sigma C_{5} \sqrt{s_1\ln(p_1) \vee s_2\ln(p_2) \vee s_3 N_n \ln(p_2N_n)}.
\]
It follows that $\Pr(\Omega_{|\mathcal{S}_z|, |\mathcal{S}_{x,L}|, |\mathcal{S}_{x,N}|}) \rightarrow 1$. This completes the proof.
\end{proof}
For any random variable $X$, denote $\Vert X \Vert_p = (\mathrm{E}|X|^p)^{1 / p}$ as the $L_p$ norm for random variable $X$; and denote $\Vert X \Vert_{\varphi} = \inf \left\{C > 0: \mathrm{E}\{\varphi\left({|X|}/{C}\right)\} \leq 1 \right\}$ as the \textit{Orlicz} norm for random variable $X$, where $\varphi$ is required as a non-decreasing, convex function with $\varphi(0) = 0$.
\begin{lemma}
\label{LEM:2}
Suppose that Assumptions (A2) and (A4) hold. Let
\begin{align*}
T_{1k} & = n^{-1/2} \sum_{i = 1}^{n} Z_{ik} \varepsilon_i , ~1 \leq k \leq p_1, \quad
T_{2\ell} = n^{-1/2} \sum_{i = 1}^{n} X_{i\ell} \varepsilon_i , ~1 \leq \ell \leq p_2, \\
T_{3J\ell} & = n^{-1/2} \sum_{i = 1}^{n} B_{J,\ell}^{(d)}(X_{i\ell}) \varepsilon_i, ~1 \leq \ell \leq p_2, ~ 1 \leq J \leq N_n,
\end{align*}
and $T_1 = \max\limits_{1 \leq k \leq p_1} |T_{1k}|$, $T_2 = \max\limits_{1 \leq \ell \leq p_2} |T_{2\ell}|$ and $T_3 = \max\limits_{1 \leq \ell \leq p_2, 1 \leq J \leq N_n} |T_{3J\ell}|$. Then we have
\begin{eqnarray*}
&\mathrm{E}(T_1) \leq {C_1\sqrt{\ln(p_1)}}, \quad \mathrm{E}(T_2) \leq {C_2\sqrt{\ln(p_2)}}, \\
&\mathrm{E}(T_3) \leq {C_3 n^{-1/2} \sqrt{\ln(p_2 N_n)} \left(\sqrt{2 C_4\,n N_n \ln(2p_2N_n)} + C_5N_n^{1/2}\ln(2p_2N_n) + n\right)^{1/2} },
\end{eqnarray*}
where $C_1$, $C_2$, $C_3$, $C_4$ and $C_5$ are positive constants.
In particular, when $ {N_n \ln(p_{2}N_n) / n \rightarrow 0}$, we have
\[
\mathrm{E}(T_1) = O\{\sqrt{\ln(p_1)}\}, \quad
\mathrm{E}(T_2) = O\{\sqrt{\ln(p_2)}\}, \quad
\mathrm{E}(T_3) = O\{\sqrt{\ln(p_2N_n)}\}.
\]
\end{lemma}
\begin{proof}
Denote $s_{1nk}^2 = \sum_{i = 1}^n Z_{ik}^2$, $1 \leq k \leq p_1$, $s_{2nl}^2 = \sum_{i = 1}^n X_{i\ell}^2$, $1 \leq \ell \leq p_2$, $s_{3nJ\ell}^2 = \sum_{i = 1}^n \{B_{J,\ell}^{(d)}(X_{i\ell})\}^2$, $1 \leq \ell \leq p_2$, $1 \leq J \leq N_n$.
Next let $s_{1n}^2 = \max_{1 \leq k \leq p_1} s_{1nk}^2$, $s_{2n}^2 = \max_{1 \leq \ell \leq p_2} s_{2nl}^2$ and $s_{3n}^2 = \max_{1 \leq \ell \leq p_2, 1 \leq J \leq N_n} s_{3nJ\ell}^2$.
By Assumption (A2), conditional on $\mathbb{Z}=\{Z_{ik}, \, 1 \leq i \leq n, \, 1 \leq k \leq p_1\}$, $\sqrt{n} \, T_{1k}$ is $b{\left(\sum_{i = 1}^n Z_{ik}^2\right)}^{1/2}$--subgaussian; and conditional on $\mathbb{X}=\left\{X_{i\ell}, \, 1 \leq i \leq n, \, 1 \leq \ell \leq p_2\right\}$, $\sqrt{n} \, T_{2\ell}$ is $b{\left(\sum_{i = 1}^n X_{i\ell}^2\right)}^{1/2}$-subgaussian, and $\sqrt{n} \, T_{3J\ell}$ is $b\left[\sum_{i = 1}^n \{B_{J,\ell}^{(d)} (X_{i\ell})\}^2\right]^{1 / 2}$-subgaussian.
Define $\varphi_p(x) = \exp({x^p}) - 1, \, p \geq 1$. Then $\varphi_p^{-1}(m) = {\left\{\ln(1 + m)\right\}}^{1/p}$. By Assumption (A2) and the maximal inequality for sub-Gaussian random variables (as stated in Lemmas 2.2.1 and 2.2.2 of \citet[][]{van1996weak}),
\begin{align*}
\mathrm{E}\left(T_{1} | \mathbb{Z}\right) & \!=\!
\mathrm{E}\left(\max\limits_{1 \leq k \leq p_1} \big|T_{1k}| | \mathbb{Z}\right)
\!=\! \left\Vert \max\limits_{1 \leq k \leq p_1} |T_{1k}| \big| \mathbb{Z} \right\Vert_1
\!\!\!\leq\! \left\Vert \max\limits_{1 \leq k \leq p_1} |T_{1k}| \big| \mathbb{Z} \right\Vert_{\varphi_1} \!\!\!\!\!\!\leq\! \sqrt{\ln (2)} \left\Vert \max\limits_{1 \leq k \leq p_1} |T_{1k}| \big| \mathbb{Z} \right\Vert_{\varphi_2} \\
& \leq K_1 \sqrt{\ln 2} \sqrt{\ln(1 + p_1)} \, {n}^{-1/2} \max\limits_{1 \leq k \leq p_1} \left\Vert \sqrt{n} \, T_{1k} \big| \left\{Z_{ik}, 1 \leq i \leq n, \, 1 \leq k \leq p_1 \right\} \right\Vert_{\varphi_2} \\
& \leq K_1 \sqrt{\ln 2} \sqrt{\ln(1 + p_1)} \, {n}^{-1/2} \max\limits_{1 \leq k \leq p_1} \left(6b^2 \sum_{i = 1}^n Z_{ik}^2\right)^{1/2} \\
&\leq {C_{5} {n}^{-1/2}s_{1n} \sqrt{\ln(p_1)}}.
\end{align*}
Next,
\begin{align*}
\mathrm{E}\left(T_{2} | \mathbb{X}\right) & =
\mathrm{E}\left(\max\limits_{1 \leq \ell \leq p_2} \big|T_{2\ell}| | \mathbb{X}\right)
= \left\Vert \max\limits_{1 \leq \ell \leq p_2} |T_{2\ell}| \big| \mathbb{X} \right\Vert_1
\leq \sqrt{\ln (2)} \left\Vert \max\limits_{1 \leq \ell \leq p_2} |T_{2\ell}| \big| \mathbb{X} \right\Vert_{\varphi_2} \\
& \leq K_2 \sqrt{\ln 2} \sqrt{\ln(1 + p_2)} \, {n}^{-1/2} \max\limits_{1 \leq \ell \leq p_2} \left\Vert \sqrt{n} \, T_{2\ell} \big| \left\{X_{i\ell}, 1 \leq i \leq n, \, 1 \leq \ell \leq p_2 \right\} \right\Vert_{\varphi_2} \\
& \leq K_2 \sqrt{\ln 2} \sqrt{\ln(1 + p_2)} \, {n}^{-1/2} \max\limits_{1 \leq \ell \leq p_2} \left(6b^2 \sum_{i = 1}^n X_{i\ell}^2\right)^{1/2} \\
&\leq {C_{5} {n}^{-1/2}s_{2n} \sqrt{\ln(p_2)}}.
\end{align*}
\begin{align*}
\mathrm{E}\left(T_{3} | \mathbb{X}\right) & = \mathrm{E}\left(\max\limits_{1 \leq \ell \leq p_2, \, 1 \leq J \leq N_n} |T_{3J\ell}| \big| \mathbb{X}\right) = \left\Vert \max\limits_{1 \leq \ell \leq p_2, \, 1 \leq J \leq N_n} |T_{3J\ell}| \big| \mathbb{X} \right\Vert_1\\
&\leq \sqrt{\ln (2)} \Vert \max\limits_{1 \leq \ell \leq p_2, \, 1 \leq J \leq N_n} |T_{3J\ell}| \big| \mathbb{X} \Vert_{\varphi_2} \\
& \leq K_3 \sqrt{\ln 2} \sqrt{\ln(1 + p_2 N_n)} \, {n}^{-1/2} \max\limits_{1 \leq \ell \leq p_2, \, 1 \leq J \leq N_n} \left[6b^2 \sum_{i = 1}^n \{B_{J,\ell}^{(d)}(X_{i\ell})\}^2 \right]^{1/2} \\
&= {C_{21}{n}^{-1/2} s_{3n} \sqrt{\ln(p_2 N_n)}}.
\end{align*}
Thus,
\begin{align*}
\mathrm{E}\left(T_{1} \right) &\leq {C_{11} {n}^{-1/2} \sqrt{\ln(p_1)} \mathrm{E} (s_{1n})}, ~
\mathrm{E}\left(T_{2} \right) \leq {C_{21} {n}^{-1/2} \sqrt{\ln(p_2)} \mathrm{E} (s_{2n})}, \\
\mathrm{E}\left(T_{3} \right) &\leq {C_{31} {n}^{-1/2} \sqrt{\ln(p_2 N_n)} \, \mathrm{E}(s_{3n})},
\end{align*}
where $K_{1}$, $K_{2}$, $K_3$, $C_{11}$, $C_{21}$ and $C_{31}$ are positive constants. By Assumption (A4), we have $\mathrm{E} (Z_{ik})^2 \leq C_{13}^2 $ and $\mathrm{E} (X_{i\ell})^2 \leq C_{23}^2 $. The properties of normalized B-splines imply that, for every $l, J$, there exist positive constants $C_{13}$, and $C_{4}$, such that $|B_{J,\ell}^{(d)}(X_{i\ell})| \leq C_4 N_n^{1 / 2}$ and $\mathrm{E} \left(B_{J,\ell}^{(d)}(X_{i\ell})\right)^2 = 1$. Therefore,
$\mathrm{E}(s_{1n}^2) = \max\limits_{1 \leq k \leq p_1} \mathrm{E}(s_{1nk}^2 )= \max\limits_{1 \leq k \leq p_1} \sum\limits_{i = 1}^n \mathrm{E} (Z_{ik}^2) \leq n C_{13}^2$, $\mathrm{E}(s_{2n}^2) = \max\limits_{1 \leq \ell \leq p_2} \mathrm{E}(s_{2nl}^2 )= \max\limits_{1 \leq \ell \leq p_2} \sum\limits_{i = 1}^n \mathrm{E} (X_{i\ell}^2)\leq n C_{23}^2$, and $\sum\limits_{i = 1}^n \mathrm{E} \left[\{B_{J,\ell}^{(d)}(X_{i\ell})\}^2 - \mathrm{E}^2 \{B_{J,\ell}^{(d)}(X_{i\ell})\}\right]^2$ $\leq n N_n C_3$.
Thus, by Lemma A.1 of \citet[][]{van2008high}, we have
\[
\mathrm{E} \Bigg[\max_{\substack{1 \leq \ell \leq p_2, \\ 1 \leq J \leq N_n}} \Bigg| \sum\limits_{i = 1}^n \Bigg\{B_{J,\ell}^{(d)}(X_{i\ell})\Bigg\} ^2 - \mathrm{E}^2 \left\{B_{J,\ell}^{(d)}(X_{i\ell})\right\} \Bigg| \Bigg] \leq \sqrt{2C_4 n N_n\ln(2p_2N_n)} + C_5 N_n^{1 / 2} \ln(2p_2N_n).
\]
Therefore, by triangle inequality, $\mathrm{E}(s_{3n}^2) \leq \sqrt{2C_4\,n N_n \ln(2p_2N_n)} + C_5 N_n^{1 / 2} \ln(2p_2N_n) + n$.
Thus, $\mathrm{E}(s_{1n})\leq (\mathrm{E}s_{1n}^2)^{1/2} \leq {\left(C_{13}^2n\right)^{1/2}}$, $\mathrm{E}(s_{2n})\leq (\mathrm{E}s_{2n}^2)^{1/2} \leq {\left(C_{23}^2n\right)^{1/2}}$, and
\[
\mathrm{E}(s_{3n}) \leq (\mathrm{E}s_{3n}^2)^{1/2} \leq {\left\{\sqrt{2C_4 n N_n\ln(2p_2N_n)} + C_5 N_n^{1 / 2} \ln(2p_2N_n) + n\right\}^{1/2}}.
\]
The lemma follows.
\end{proof}
\begin{lemma}
\label{LEM:4}
For
\begin{align*}
\boldsymbol{v}_1 &= \left\{\left(\frac{\omega_k^{\alpha} \overline{\theta}_{0,k}}{|\overline{\theta}_{0,k}|}, k\in\mathcal{S}_z\right)^{\top}, \boldsymbol{0}_{|\mathcal{S}_{x,L}|}^{\top} , \boldsymbol{0}_{|\mathcal{S}_{x,N}|N_n}^{\top} \right\}^{\top},\\
\boldsymbol{v}_2 &= \left\{\boldsymbol{0}_{|\mathcal{S}_z|}^{\top}, \left(\frac{\omega_{\ell}^{\beta} \overline{\theta}_{0,|\mathcal{S}_z|+\ell}}{|\overline{\theta}_{0,|\mathcal{S}_z|+\ell}|}, \ell\in\mathcal{S}_{x,L}\right)^{\top}, \boldsymbol{0}_{|\mathcal{S}_{x,N}|N_n}^{\top} \right\}^{\top}, \\
\boldsymbol{v}_3 &= \left\{\boldsymbol{0}_{|\mathcal{S}_z|}^{\top}, \boldsymbol{0}_{|\mathcal{S}_{x,L}|}^{\top}, \left(\frac{\omega_{\ell}^{\gamma}\overline{\boldsymbol{\theta}}_{0,|\mathcal{S}_z|+|\mathcal{S}_{x,L}|+\ell}^{\top}}{\Vert \overline{\boldsymbol{\theta}}_{0,|\mathcal{S}_z|+|\mathcal{S}_{x,L}|+\ell}\Vert_2}, \ell\in\mathcal{S}_{x,N}\right)^{\top} \right\}^{\top},
\end{align*}
under Assumption (A5),
\begin{align}
\Vert \boldsymbol{v}_1 \Vert_2^2 &= O_P \left( h_{n1}^2 \right) = O_P \left(b_{n1}^{-4}c_{b1}^{-2}r_{n1}^{-2} + |\mathcal{S}_z|b_{n1}^{-2}\right), \label{EQN:v1_order} \\
\Vert \boldsymbol{v}_2 \Vert_2^2 &= O_P \left( h_{n2}^2 \right) = O_P \left\{b_{n2}^{-4}c_{b2}^{-2}r_{n2}^{-2} + |\mathcal{S}_{x,L}|b_{n2}^{-2}\right\}, \label{EQN:v2_order} \\
\Vert \boldsymbol{v}_3 \Vert_2^2 &= O_P \left( h_{n3}^2 \right) = O_P \left\{b_{n3}^{-4}c_{b3}^{-2}r_{n3}^{-2} + |\mathcal{S}_{x,N}|b_{n3}^{-2}\right\}. \label{EQN:v3_order}
\end{align}
\end{lemma}
\begin{proof}
Write
\begin{align*}
\Vert \boldsymbol{v}_1 \Vert _2^2 &= \sum\limits_{k \in \mathcal{S}_z} \left(\omega_k^{\alpha}\right)^2
= \sum\limits_{k \in \mathcal{S}_z} \, |\widetilde{\alpha}_k |^{-2}
= \sum\limits_{k \in \mathcal{S}_z} \, \frac{\alpha_{0k}^2 - \widetilde{\alpha}_k^2}{\alpha_{0k}^2 \widetilde{\alpha}_k^2}
+ \sum\limits_{k \in \mathcal{S}_z} \, |\alpha_{0k}|^{-2} , \\
\Vert \boldsymbol{v}_2 \Vert _2^2 &= \sum\limits_{\ell \in \mathcal{S}_{x,L}} \left(\omega_{\ell}^{\beta}\right)^2
= \sum\limits_{\ell \in \mathcal{S}_{x,L}} \, |\widetilde{\beta}_{\ell} |^{-2}
= \sum\limits_{\ell \in \mathcal{S}_{x,L}} \, \frac{\beta_{0\ell}^2 - \widetilde{\beta}_{\ell}^2}{\beta_{0\ell}^2 \widetilde{\beta}_{\ell}^2}
+ \sum\limits_{\ell \in \mathcal{S}_{x,L}} \, |\beta_{0\ell}|^{-2} , \\
\Vert \boldsymbol{v}_3 \Vert _2^2 &= \sum\limits_{\ell \in \mathcal{S}_{x,N}} \left(\omega_{\ell}^{\gamma}\right)^2
= \sum\limits_{\ell \in \mathcal{S}_{x,N}} \Vert \widetilde{\boldsymbol{\gamma}}_{\ell} \Vert_2^{-2}
= \sum\limits_{\ell \in \mathcal{S}_{x,N}} \, \frac{\Vert\boldsymbol{\gamma}_{0\ell}\Vert_2^2 - \Vert\widetilde{\boldsymbol{\gamma}}_{\ell}\Vert^2}{\Vert\boldsymbol{\gamma}_{0\ell}\Vert_2^2 \Vert\widetilde{\boldsymbol{\gamma}}_{\ell}\Vert_2^2}
+ \sum\limits_{\ell \in \mathcal{S}_{x,N}} \Vert \boldsymbol{\gamma}_{0\ell} \Vert_2^{-2} .
\end{align*}
Under (A5), there exist positive constants $M_1$, $M_2$ and $M_3$, such that
\begin{align*}
\sum_{k \in \mathcal{S}_z} \frac{\left| \alpha_{0k}^2 - \widetilde{\alpha}_k^2 \right|}{\alpha_{0k}^2 \widetilde{\alpha}^2}
&\leq M_1c_{b1}^{-2}b_{n1}^{-4} \|\widetilde{\boldsymbol{\alpha}} - \boldsymbol{\alpha}_0 \|^2
= O_P \left( b_{n1}^{-4}c_{b1}^{-2} r_{n1}^{-2}\right), \\
\sum_{\ell \in \mathcal{S}_{x,L}} \frac{\left| \beta_{0\ell}^2 - \widetilde{\beta}_{\ell}^2 \right|}{\beta_{0\ell}^2 \widetilde{\beta}^2}
&\leq M_2c_{b2}^{-2}b_{n2}^{-4} \|\widetilde{\boldsymbol{\beta}} - \boldsymbol{\beta}_0 \|^2
= O_P \left( b_{n2}^{-4}c_{b2}^{-2} r_{n2}^{-2}\right), \\
\sum_{\ell \in \mathcal{S}_{x,N}} \, \frac{\left|\|\boldsymbol{\gamma}_{0\ell}\|^2 - \Vert\widetilde{\boldsymbol{\gamma}}_{\ell}\Vert^2 \right|}{\Vert\boldsymbol{\gamma}_{0\ell}\Vert^2 \Vert\widetilde{\boldsymbol{\gamma}}\Vert^2}
&\leq M_3 c_{b3}^{-2}b_{n3}^{-4} \Vert \widetilde{\boldsymbol{\gamma}} - \boldsymbol{\gamma}_0 \Vert^2
= O_P \left( b_{n3}^{-4}c_{b3}^{-2} r_{n3}^{-2}\right),
\end{align*}
and the results follow from that $\sum_{k \in \mathcal{S}_z} |\alpha_k |^{-2} \leq |\mathcal{S}_z| b_{n1}^{-2}$, $\sum_{\ell \in \mathcal{S}_{x,L}} |\beta_{\ell} |^{-2} \leq |\mathcal{S}_{x,L}| b_{n2}^{-2}$ and $\sum_{\ell \in \mathcal{S}_{x,N}}\Vert \boldsymbol{\gamma}_{\ell} \Vert^{-2} \leq |\mathcal{S}_{x,N}|b_{n3}^{-2}$.
\end{proof}
{\baselineskip=12pt
}
\end{document}
|
\begin{document}
\title{f Hankel Transform of the First Form $(q, r)$-Dowling Numbers}
\begin{abstract}
In this paper, the Hankel transform of the generalized $q$-exponential polynomial of the first form $(q,r)$-Whitney numbers of the second kind is established using the method of Cigler. Consequently, the Hankel transform of the first form $(q,r)$-Dowling numbers is obtained as special case.
\noindent{\bf Keywords}: $r$-Whitney numbers, $r$-Dowling numbers, generating function, $q$-analogue, $q$-exponential function, $A$-tableau, convolution formula, Hankel transform, Hankel matrix, $k$-binomial transform.
\end{abstract}
\section{Introduction}
The $r$-Dowling numbers $D_{m,r}(n)$ are defined in \cite{CHEON} as the sum of $r$-Whitney numbers of the second $W_{m,r}(n,k)$ \cite{corcino, Mezo2}. More precisely,
$${D}_{m,r}(n):=\sum_{k=0}^nW_{m,r}(n,k).$$
These numbers are certain generalization of ordinary Bell numbers $B_n$ \cite{Com}, $r$-Bell numbers $B_{r}(n)$ \cite{Mezo3}, and noncentral Bell numbers $B_{n,a}$ \cite{Kout}. That is, when $m=1$, the $r$-Dowling numbers reduce to $r$-Bell numbers and noncentral Bell numbers. Furthermore, when $m=1, r=0$, these yield the ordinary Bell numbers.
J. Layman \cite{Lay} defined the Hankel transform of an integers sequence $(a_n)$ as a sequence of the following determinants $d_n$ of Hankel matrix of order $n$
\begin{equation}
d_n=\left|
\begin{matrix}
a_0 & a_1 & a_2 & \dots & a_n \\
a_1 & a_2 & a_3 & \dots & a_{n+1} \\
a_2 & a_3 & a_4 & \dots & a_{n+2} \\
\hdotsfor{5} \\
a_n & a_{n+1} & a_{n+2} & \dots & a_{2n}
\end{matrix}
\right|.
\end{equation}
Aigner \cite{Aig} derived the Hankel transform of the ordinary Bell numbers to be
\begin{equation}\label{det1}
det(B_{i+j})_{0\le i,j\le n}=\prod_{k=0}^nk!
\end{equation}
which is exactly the Hankel transform obtained by Mezo \cite{Mezo3} for $r$-Bell numbers using Layman's Theorem \cite{Lay} on the invariance of Hankel transform.
Using the method of Aiger \cite{Aig} and Layman's Theorem \cite{Lay}, the sequence of $(r,\beta)$-Bell numbers in \cite{Cor2, Cor3}, denoted by $\{G_{n,r,\beta}\}$, has been shown to possess the following Hankel transform (see \cite{Cor1})
$$H(G_{n,r,\beta})=\prod_{j=0}^n\beta^jj!.$$
It is worth mentioning that the $(r,\beta)$-Bell numbers are equivalent to the $r$-Dowling numbers $D_{m,r}(n)$, which are defined in \cite{CHEON} as
$$D_{m,r}(n)=\sum_{k=0}^nW_{m,r}(n,k)$$
with $W_{m,r}(n,k)$ denotes the $r$-Whitney numbers of the second kind introduced by Mezo in \cite{Mezo2}. In \cite{Cor1}, the authors have also tried to derive the Hankel transform of the sequence of $q$-analogue of $(r,\beta)$-Bell numbers. In this attempt, they used the $q$-analogue defined in \cite{Cor4}. But they failed to derive it.
Just recently, another definition of $q$-analogue of $r$-Whitney numbers of the second $W_{m,r}[n,k]_q$ was introduced in \cite{Cor5, Cor6} by means of the following triangular recurrence relation
\begin{equation}\label{whitrec}
W_{m,r}[n,k]_{q}= q^{m(k-1)+r}W_{m,r}[n-1,k-1]_q+[mk+r]_{q} W_{m,r}[n-1,k]_{q}.
\end{equation}
From this definition, two more forms of the $q$-analogue were defined in \cite{Cor5, Cor6} as
\begin{align}
W^*_{m,r}[n,k]_q&:=q^{-kr-m\binom{k}{2}}W_{m,r}[n,k]_q\label{2ndform}\\
\widetilde{W}_{m,r}[n,k]_q&:=q^{kr}W^*_{m,r}[n,k]_q=q^{-m\binom{k}{2}}W_{m,r}[n,k]_q,\label{3rdform}
\end{align}
where $W^*_{m,r}[n,k]_q$ and $\widetilde{W}_{m,r}[n,k]_q$ denote the second and third forms of the $q$-analogue, respectively. Corresponding to these, three forms of $q$-analogues for $r$-Dowling numbers (or $(q,r)$-Dowling numbers) may be defined as follows:
\begin{align}
{D}_{m,r}[n]_q&:=\sum_{k=0}^nW_{m,r}[n,k]_q\label{1stformqDow}\\
{D}^*_{m,r}[n]_q&:=\sum_{k=0}^nW^*_{m,r}[n,k]_q\label{2ndformqDow}\\
\widetilde{D}_{m,r}[n]_q&:=\sum_{k=0}^n\widetilde{W}_{m,r}[n,k]_q.\label{3rdformqDow}
\end{align}
However, among the three forms of $(q,r)$-Dowling numbers, only the first form has not been given a Hankel transform. The third form was thoroughly studied in \cite{Cor6} and was given the Hankel transform as follows
\begin{equation}\label{HankelqRDN}
H(\widetilde{D}_{m,r}[n]_q)=q^{m\binom{n+1}{3}-rn(n+1)}[0]_{q^m}![1]_{q^m}!\ldots [n]_{q^m}![m]_q^{\binom{n+1}{2}}.
\end{equation}
This Hankel transform was derived using the Hankel transform of $q$-exponential polynomials in \cite{Ehr1}, the Layman's Theorem in \cite{Lay} and the Spivey-Steil Theorem in \cite{Spiv}. This method cannot be used to derive the Hankel transform of the first and second forms of $q$-analogues for $r$-Dowling numbers. But the method used by Cigler in \cite{Cig2} was found to be useful to derive the Hankel transforms for the second form of the $(q,r)$-Dowling numbers. The said Hankel transform was derived in \cite{Cor5}, which is given by
\begin{equation*}
H({D}^*_{m,r}[n]_q)=[m]_q^{\binom{n}{2}}q^{\binom{n}{3}+r\binom{n}{2}}\prod_{k=0}^{n-1}[k]_{q^m} !
\end{equation*}
Corcino et al. \cite{Cor7} have made a preliminary investigation for the first form $(q,r)$-Dowling numbers $D_{m,r}[n]_q$ by establishing an explicit formula expressed in terms of the first form $(q,r)$-Whitney numbers of the second kind and $(q,r)$-Whitney-Lah numbers. In this paper, the Hankel transform for the sequence $\left(D_{m,r}[n]_q\right)_{n=0}^{\infty}$ will be established using Cigler's method \cite{Cig2}. However, a more general form of $D_{m,r}[n]_q$, denoted by $\Phi_n[x,r,m]_q$, is considered, which is defined in polynomial form as follows:
\begin{equation}\label{genq1}
\Phi_{n}\left[ x,r,m\right]_q=\sum\limits_{k=0}^{n}W_{m,r}[n,k]_{q}x^{k}
\end{equation}
such that, when $x=1$, $\Phi_n[1,r,m]_q=D_{m,r}[n]_q$.
\section{Generalized $q$-Exponential Polynomials}
We may call $\Phi_{n}\left[ x,r,m\right]_q $ to be the generalized $q$-exponential polynomial of $q$-analogue of $r$-Whitney numbers of the second kind. Note that we can rewrite \eqref{genq1} as
\begin{align}
\Phi_{n-1}\left[ x,r,m\right]_q&=\sum\limits_{k=0}^{n-1}W_{m,r}[n-1,k]_{q}x^{k}\nonumber \\
\Phi_{n-1}\left[ qx,r,m\right]_q&=\sum\limits_{k=0}^{n-1}W_{m,r}[n-1,k]_{q}q^{rk+m\binom{k}{2}+k}x^{k}.\label{eqn_trans}
\end{align}
The following theorem contains a recursive relation for $\Phi_{n}\left[ x,r,m\right]_q $.
\begin{thm}
The generalized $q$-exponential polynomials $\Phi_{n}\left[ x,r,m\right]_q $ of $q$-analogue of $r$-Whitney numbers of the second kind satisfy the following relation
\begin{align}
\Phi_n[x,r,m]_q &=\left[q^{r}x+(q^m-1)q^{r}x^2D_{q^m}+[r]_q+q^r[m]_qxD_{q^m}\right]\Phi_{n-1}[x,r,m]_q.\label{phi}
\end{align}
\end{thm}
\begin{proof}
Using \eqref{whitrec}, equation \eqref{genq1} can be written as
\begin{align*}
&\Phi_{n}\left[ x,r,m\right]_q =\sum\limits_{k=0}^{n}W_{m,r}[n,k]_{q}x^{k}\\
&\;\;\;\;=\sum\limits_{k=0}^{n}q^{mk-m+r}W_{m,r}[n-1,k-1]_qx^{k}+\sum\limits_{k=0}^{n}[mk+r]_qW_{m,r}[n-1,k]_qx^{k}\\
&\;\;\;\;=\sum\limits_{k=0}^{n-1}q^{m(k+1)-m+r}W_{m,r}[n-1,k]_qx^{k+1}+\left([r]_q+q^r[m]_qxD_{q^m}\right)\Phi_{n-1}[x,r,m]_q\\
&\;\;\;\;=x\sum\limits_{k=0}^{n-1}q^{mk+r}W_{m,r}[n-1,k]_qx^k+\left([r]_q+q^r[m]_qxD_{q^m}\right)\Phi_{n-1}[x,r,m]_q\\
&\;\;\;\;=xq^{r}\sum\limits_{k=0}^{n-1}q^{mk}W_{m,r}[n-1,k]_qx^k+\left([r]_q+q^r[m]_qxD_{q^m}\right)\Phi_{n-1}[x,r,m]_q,
\end{align*}
where $D_q$ denotes the $q$-derivative operator defined by
\begin{equation}\label{qDer}
D_qf(x)=\frac{f(x)-f(qx)}{(1-q)x}.
\end{equation}
Hence, using \eqref{eqn_trans}, we have
\begin{align}
\Phi_n[x,r,m]_q &=xq^{r}\Phi_{n-1}[q^mx,r,m]_q+\left([r]_q+q^r[m]_qxD_{q^m}\right)\Phi_{n-1}[x,r,m]_q.\label{genq2}
\end{align}
Note that \eqref{qDer} can be expressed as
\begin{align*}
f(qx)&=(q-1)xD_qf(x)+f(x)\\
f(q^mx)&=(q^m-1)xD_{q^m}f(x)+f(x)\\
f(q^mx)&=((q^m-1)xD_{q^m}+1)f(x).
\end{align*}
This implies that
$$\Phi_{n-1}[q^mx,r,m]_q=\left(1+(q^m-1)xD_{q^m}\right)\Phi_{n-1}[x,r,m]_q.$$
Thus, equation \eqref{genq2} can further be written as
\begin{align*}
\Phi_n[x,r,m]_q &=xq^{r}\left(1+(q^m-1)xD_{q^m}\right)\Phi_{n-1}[x,r,m]_q\\
&\;\;\;\;\;+\left([r]_q+q^r[m]_qxD_{q^m}\right)\Phi_{n-1}[x,r,m]_q\\
&=\left[q^{r}x+(q^m-1)q^{r}x^2D_{q^m}\right.\nonumber\\
&\left.\;\;\;\;\;+[r]_q+q^r[m]_qxD_{q^m}\right]\Phi_{n-1}[x,r,m]_q,
\end{align*}
which is exactly the desired relation.
\end{proof}
\begin{rmk}\rm
Let
\begin{align*}
\hat{D}_{qx}&=\left[q^{r}x+(q^m-1)q^{r}x^2D_{q^m}+[r]_q+q^r[m]_qxD_{q^m}\right],\\
\tilde{D}_{qx}&=\left[q^{r}+(q^m-1)q^{r}xD_{q^m}+[r]_q+q^r[m]_qD_{q^m}\right],\\
\hat{D}_{qx}&=x\tilde{D}_{qx}.
\end{align*}
Then \eqref{phi} can be written as
\begin{align}
\Phi_{n}[x,r,m]_{q}&=\hat{D}_{qx}\Phi_{n-1}[x,r,m]_{q}.\label{Phi2}\\
\Phi_{n}[x,r,m]_{q}&=x\tilde{D}_{qx}\Phi_{n-1}[x,r,m]_{q}.\label{Phi3}
\end{align}
By repeated application of \eqref{Phi2},
\begin{align*}
\Phi_{n}[x,r,m]_{q}&=\hat{D}_{qx}\Phi_{n-1}[x,r,m]_{q}\\
&=\hat{D}_{qx}\left( \hat{D}_{qx}\Phi_{n-2}[x,r,m]_{q}\right)\\
&=\hat{D}_{qx}^{2}\Phi_{n-2}[x,r,m]_{q}\\
&\vdots\\
&=\hat{D}_{qx}^{n}\Phi_{0}[x,r,m]_{q}\\
&=\hat{D}_{qx}^{n}.
\end{align*}
\end{rmk}
\section{Hankel Transform of ${D}_{m,r}[n]_q$}
Let $\langle\langle x\rangle\rangle_{r,m,k}=\prod\limits_{j=0}^{k-1}\frac{\left(x-[r+jm]_q \right) }{q^{r+jm}}=q
^{-rk-m\binom{k}{2}}\langle x\rangle_{r,m,k}$.\\
The horizontal generating function of $W_{m,r}[n,k]_q$ is given by:
\begin{equation*}
\sum\limits_{k=0}^{n}W_{m,r}[n,k]_q\langle x\rangle_{r,m,k}=x^n
\end{equation*}
Define a linear functional $G_{r,q}$ by
\begin{equation*}
G_{r,q}\left( \langle\langle x\rangle\rangle_{r,m,n}\right)=a^n
\end{equation*}
and a linear operator $V_{r,q}$ by
\begin{equation*}
V_{r,q}\left( \langle\langle x\rangle\rangle_{r,m,n}\right)=x^n.
\end{equation*}
Then
\begin{align*}
V_{r,q}\left( x^n\right) &=\sum\limits_{k=0}^{n}W_{m,r}[n,k]_q V_{r,q}\left(\langle x\rangle_{r,m,k}\right)\\
&=\sum\limits_{k=0}^{n}W_{m,r}[n,k]_q q^{rk+m\binom{k}{2}}V_{r,q}\left(\langle\langle x\rangle\rangle_{r,m,k}\right)\\
&=\sum\limits_{k=0}^{n}W_{m,r}[n,k]_q q^{rk+m\binom{k}{2}}x^k\\
&=\Phi_n[x,r,m]_q
\end{align*}
It can be easily verified from equation \eqref{Phi3} that
\begin{equation*}
V_{r,q}xV_{r,q}^{-1}=x\tilde{D}_{qx}.
\end{equation*}
Consider the polynomial
\begin{equation*}
g_{n,q}(x,a,r,m)=\sum\limits_{k=0}^{n}(-a)^{k}q^{\binom{k}{2}}\bracketed{n}{k}_{q}\langle\langle x\rangle\rangle_{r,m,n-k}.
\end{equation*}
Then
\begin{align*}
V_{r,q}\left( g_{n,q}(x,a,r,m)\right)&=\sum\limits_{k=0}^{n}(-a)^{k}q^{\binom{k}{2}}\bracketed{n}{k}_{q}V_{r,q}\left( \langle\langle x\rangle\rangle_{r,m,n-k}\right) \\
&=\sum\limits_{k=0}^{n}(-a)^{k}q^{\binom{k}{2}}\bracketed{n}{k}_{q}x^{n-k}\\
&=p_{n,q}(x,a).
\end{align*}
This implies that $V_{r,q}^{-1}p_{n,q}(x,a)=g_{n,q}(x,a,r,m)$. Now,
\begin{align*}
V_{r,q}xg_{n,q}(x,a,r,m)&=V_{r,q}xV_{r,q}^{-1}p_{n,q}(x,a)\\
&=x\left[q^{r}+(q^m-1)q^{r}xD_{q^m}+[r]_q+q^r[m]_qD_{q^m}\right]p_{n,q}(x,a).
\end{align*}
Applying the operator to $p_{n,q}(x,a)$, we get
\begin{align*}
V_{r,q}xg_{n,q}(x,a,r,m)&=V_{r,q}xV_{r,q}^{-1}p_{n,q}(x,a)\\
&=x\left[q^{r}+(q^m-1)q^{r}xD_{q^m}+[r]_q+q^r[m]_qxD_{q^m}\right]p_{n,q}(x,a)\\
&=q^{r}xp_{n,q}(x,a)+(q^m-1)q^{r}x^2D_{q^m}p_{n,q}(x,a)+[r]_{q}p_{n,q}(x,a)\\
&\;\;\;\;+q^r[m]_qxD_{q^m}p_{n,q}(x,a).
\end{align*}
Note that
\begin{align*}
xp_{n,q}(x,a)&=\sum_{k=0}^{n}(-a)^kq^{\binom{k}{2}}\bracketed{n}{k}x^{n+1-k}\\
&=\sum_{k=0}^{n}(-a)^kq^{\binom{k}{2}}\left(\bracketed{n+1}{k}-q^{n+1-k}\bracketed{n}{k-1}\right)x^{n+1-k}\\
&=\sum_{k=0}^{n}(-a)^kq^{\binom{k}{2}}\bracketed{n+1}{k}x^{n+1-k}-\sum_{k=0}^{n}(-a)^kq^{\binom{k}{2}}q^{n+1-k}\bracketed{n}{k-1}x^{n+1-k}\\
&\;\;\;\;\;\;+(-a)^{n+1}q^{\binom{n+1}{2}}-(-a)^{n+1}q^{\binom{n+1}{2}}.
\end{align*}
\begin{align*}
xp_{n,q}(x,a)&=\sum_{k=0}^{n+1}(-a)^kq^{\binom{k}{2}}\bracketed{n+1}{k}x^{n+1-k}-\sum_{k=-1}^{n-1}(-a)^{k+1}q^{\binom{k+1}{2}}q^{n-k}\bracketed{n}{k}x^{n-k}\\
&\;\;\;\;\;\;-(-a)^{n+1}q^{\binom{n+1}{2}}\\
&=p_{n+1,q}(x,a)-(-a)q^n\sum_{k=0}^{n-1}(-a)^{k+1}q^{\binom{k+1}{2}-k}\bracketed{n}{k}x^{n-k}\\
&\;\;\;\;\;\;-(-a)^{n+1}q^{\binom{n+1}{2}}\\
&=p_{n+1,q}(x,a)+aq^n\sum_{k=0}^{n}(-a)^{k+1}q^{\binom{k}{2}}\bracketed{n}{k}x^{n-k}\\
&=p_{n+1,q}(x,a)+aq^np_{n,q}(x,a).
\end{align*}
So, $q^rxp_{n,q}(x,a)=q^rp_{n+1,q}(x,a)+aq^{n+r}p_{n,q}(x,a)$. With $D_qp_{n,q}(x,a)=[n]_qp_{n-1,q}(x,a)$, we have
$$D_{q^m}p_{n,q}(x,a)=[n]_{q^m}p_{n-1,q}(x,a).$$
Hence,
$$(q^m-1)q^{r}x^2D_{q^m}p_{n,q}(x,a)=(q^m-1)q^{r}x^2[n]_{q^m}p_{n-1,q}(x,a)=(q^{mn}-1)q^{r}x^2p_{n-1,q}(x,a)$$
and
$$q^r[m]_qxD_{q^m}p_{n,q}(x,a)=q^r[m]_qx[n]_{q^m}p_{n-1,q}(x,a)$$
Thus,
\begin{align*}
&V_{r,q}xg_{n,q}(x,a,r,m)=V_{r,q}xV_{r,q}^{-1}p_{n,q}(x,a)\\
&\;\;\;\;\;=q^rxp_{n,q}(x,a)+q^rp_{n+1,q}(x,a)+aq^{n+r}p_{n,q}(x,a)+[r]_{q}p_{n,q}(x,a)\\
&\;\;\;\;\;\;\;\;+(q^{mn}-1)q^{r}x^2p_{n-1,q}(x,a)+q^r[m]_qx[n]_{q^m}p_{n-1,q}(x,a)\\
&\;\;\;\;\;=q^r(p_{n+1,q}(x,a)+q^nap_{n,q}(x,a))+q^rp_{n+1,q}(x,a)+aq^{n+r}p_{n,q}(x,a)+[r]_{q}p_{n,q}(x,a)\\
&\;\;\;\;\;\;\;\;+(q^{mn}-1)q^{r}[p_{n+1,q}(x,a)+(q^na+q^{n-1}a)p_{n,q}(x,a)+q^{2n-2}a^2p_{n-1,a}(x,a)]\\
&\;\;\;\;\;\;\;\;+q^r[m]_q[n]_{q^m}(p_{n,q}(x,a)+q^{n-1}ap_{n-1,q}(x,a)\\
&\;\;\;\;\;=q^r(2+(q^{mn}-1))p_{n+1,q}(x,a)\\
&\;\;\;\;\;\;\;\;+(q^{n+r}a+aq^{n+r}+[r]_{q}+(q^{mn}-1)q^{r}(q^na+q^{n-1}a)+q^r[m]_q[n]_{q^m}) p_{n,q}(x,a)\\
&\;\;\;\;\;\;\;\;+((q^{mn}-1)q^{r}q^{2n-2}a^2+q^r[m]_q[n]_{q^m}q^{n-1}a)p_{n-1,q}(x,a)
\end{align*}
Applying the operator $V_{r,q}^{-1}:p_{n,q}(x,a)\mapsto g_{n,q}(x,a,r,m)$, then
\begin{align*}
&xg_{n,q}(x,a,r,m)=q^r(2+(q^{mn}-1))g_{n+1,q}(x,a,r,m)\\
&\;\;\;\;\;\;\;\;+(2q^{n+r}a+[r]_{q}+(q^{mn}-1)q^{r}(q^na+q^{n-1}a)+q^r[m]_q[n]_{q^m}) g_{n,q}(x,a,r,m)\\
&\;\;\;\;\;\;\;\;+((q^{mn}-1)q^{r}q^{2n-2}a^2+q^r[m]_q[n]_{q^m}q^{n-1}a)g_{n-1,q}(x,a,r,m)
\end{align*}
We set
\begin{equation*}
h_{n,q}(x,a,r,m)=q^{m\binom{n}{2}+rn}g_{n,q}(x,a,r,m).
\end{equation*}
That is,
\begin{equation*}
g_{n,q}(x,a,r,m)=q^{-m\binom{n}{2}-rn}h_{n,q}(x,a,r,m).
\end{equation*}
Then,
\begin{align*}
&xq^{-m\binom{n}{2}-rn}h_{n,q}(x,a,r,m)=q^r(2+(q^{mn}-1))q^{-m\binom{n+1}{2}-r(n+1)}h_{n+1,q}(x,a,r,m)\\
&\;\;\;\;+(2q^{n+r}a+[r]_{q}+(q^{mn}-1)q^{r}(q^na+q^{n-1}a)+q^r[m]_q[n]_{q^m}) q^{-m\binom{n}{2}-rn}h_{n,q}(x,a,r,m)\\
&\;\;\;\;\;+((q^{mn}-1)q^{r}q^{2n-2}a^2+q^r[m]_q[n]_{q^m}q^{n-1}a)q^{-m\binom{n-1}{2}-r(n-1)}h_{n-1,q}(x,a,r,m)
\end{align*}
\begin{align*}
&xh_{n,q}(x,a,r,m)=q^r(2+(q^{mn}-1))q^{-mn-r}h_{n+1,q}(x,a,r,m)\\
&\;\;\;\;+(2q^{n+r}a+[r]_{q}+(q^{mn}-1)q^{r}(q^na+q^{n-1}a)+q^r[m]_q[n]_{q^m}) h_{n,q}(x,a,r,m)\\
&\;\;\;\;\;+((q^{mn}-1)q^{r}q^{2n-2}a^2+q^r[m]_q[n]_{q^m}q^{n-1}a)q^{m(n-1)+r}h_{n-1,q}(x,a,r,m)
\end{align*}
\begin{align}
&xh_{n,q}(x,a,r,m)=(2+(q^{mn}-1))q^{-mn}h_{n+1,q}(x,a,r,m)\nonumber\\
&\;\;\;\;+(2q^{n+r}a+[r]_{q}+(q^{mn}-1)q^{r}(q^na+q^{n-1}a)+q^r[m]_q[n]_{q^m}) h_{n,q}(x,a,r,m)\nonumber\\
&\;\;\;\;\;+((q^{mn}-1)q^{r}q^{2n-2}a^2+q^r[m]_q[n]_{q^m}q^{n-1}a)q^{m(n-1)+r}h_{n-1,q}(x,a,r,m)\label{orthpol}
\end{align}
It is clear that
\begin{align*}
G_{r,q}(h_{n,q}(x,a,r,m))&=G_{r,q}\left( q^{m\binom{n}{2}+rn}g_{n,q}(x,a,r,m)\right)\\
&=q^{m\binom{n}{2}+rn}\sum\limits_{k=0}^{n}(-a)^{k}q^{\binom{k}{2}}\bracketed{n}{k}_{q}G_{r,q}\left(\langle\langle x\rangle\rangle_{r,m,n-k} \right)\\
&=q^{m\binom{n}{2}+rn}\sum\limits_{k=0}^{n}(-a)^{k}q^{\binom{k}{2}}\bracketed{n}{k}_{q}a^{n-k}\\
&=q^{m\binom{n}{2}+rn}p_{n,q}(a,a)\\
&=0,
\end{align*}
\begin{equation*}
G_{r,q}\left( \langle\langle x\rangle\rangle_{rm,o}\right) =a^{0}=1
\end{equation*}
which implies
\begin{equation*}
G_{r,q}(1)=1.
\end{equation*}
and
\begin{align*}
g_{0,q}(x,a,r,m)&=\sum\limits_{k=0}^{0}(-a)^{k}q^{\binom{k}{2}}\bracketed{0}{k}_{q}\langle\langle x\rangle\rangle_{r,m,0-k}\\
&=(-a)^{0}q^{\binom{0}{2}}\bracketed{0}{0}_{q}\langle\langle x\rangle\rangle_{r,m,0}\\
&=1.
\end{align*}
It follows that
\begin{equation*}
h_{0,q}(x,a,r,m)=q^{m\binom{0}{2}+0}g_{0,q}(x,a,r,m)=1
\end{equation*}
and
\begin{equation*}
G_{r,q}\left( h_{0,q}(x,a,r,m)\right)=G_{r,q}(1)=1.
\end{equation*}
Clearly, $G_{r,q}\left( [x]_{q}h_{n,q}(x,a,r,m)\right)=0 $ and from \eqref{orthpol},
\begin{equation*}
xh_{n,q}(x,a,r,m)=g(n)h_{n+1,q}(x,a,r,m)+f(n)h_{n,q}(x,a,r,m)+c(n)h_{n-1,q}(x,a,r,m)
\end{equation*}
where
\begin{align*}
g(n)&=(2+(q^{mn}-1))q^{-mn}\\
f(n)&=(2q^{n+r}a+[r]_{q}+(q^{mn}-1)q^{r}(q^na+q^{n-1}a)+q^r[m]_q[n]_{q^m})\\
c(n)&=((q^{mn}-1)q^{r}q^{2n-2}a^2+q^r[m]_q[n]_{q^m}q^{n-1}a)q^{m(n-1)+r}.
\end{align*}
Then,
\begin{align*}
x^{2}h_{n,q}(x,a,r,m)&=xxh_{n,q}(x,a,r,m)\\
&=x\left[ g(n)h_{n+1,q}(x,a,r,m)+f(n)h_{n,q}(x,a,r,m)+c(n)h_{n-1,q}(x,a,r,m)\right]\\
&=g(n)xh_{n+1,q}(x,a,r,m)+f(n)xh_{n,q}(x,a,r,m)\\
&\;\;\;\;\;+c(n)xh_{n-1,q}(x,a,r,m)\\
&=g(n)g(n+1)h_{n+2,q}(x,a,r,m)+g(n)f(n+1)h_{n+1,q}(x,a,r,m)\\
&\;\;\;\;\;+g(n)c(n+1)h_{n,q}(x,a,r,m)+g(n)f(n)h_{n+1,q}(x,a,r,m)\\
&\;\;\;\;\;+f^{2}(n)h_{n,q}(x,a,r,m)+f(n)c(n)h_{n-1,q}(x,a,r,m)\\
&\;\;\;\;\;+c(n)g(n-1)h_{n,q}(x,a,r,m)+c(n)f(n-1)h_{n-1,q}(x,a,r,m)\\
&\;\;\;\;\;+c(n)c(n-1)h_{n-2,q}(x,a,r,m)\\
&=g(n)g(n+1)h_{n+2,q}(x,a,r,m)\\
&\;\;\;\;\;+\left[ g(n)f(n+1)+g(n)f(n)\right] h_{n+1,q}(x,a,r,m)\\
&\;\;\;\;\;+\left[ g(n)c(n+1)+f^{2}(n)+c(n)g(n-1)\right] h_{n,q}(x,a,r,m)\\
&\;\;\;\;\;+\left[ f(n)c(n)+c(n)f(n-1)\right] h_{n-1,q}(x,a,r,m)\\
&\;\;\;\;\;+c(n)c(n-1)h_{n-2,q}(x,a,r,m).
\end{align*}
Applying the linear functional $G_{r,q}$ to $[x]_{q}^{2}h_{n,q}(x,a,r,m)$ gives,
\begin{align*}
G_{r,q}\left( x^{2}h_{n,q}(x,a,r,m)\right)&= 0\\
&\vdots\\
G_{r,q}\left( x^{k}h_{n,q}(x,a,r,m)\right)&= 0
\end{align*}
for $k<n$. For $k=n$,
\begin{align*}
x^{n}h_{n,q}(x,a,r,m)=&g(n)x^{n-1}h_{n+1,q}(x,a,r,m)+f(n)x^{n-1}h_{n,q}(x,a,r,m)\\
\;\;\;\;\;&+c(n)x^{n-1}h_{n-1,q}(x,a,r,m).
\end{align*}
Then,
\begin{align*}
G_{r,q}\left( x^{n}h_{n,q}(x,a,r,m)\right)=&g(n)G_{r,q}\left( x^{n-1}h_{n+1,q}(x,a,r,m)\right) +f(n)G_{r,q}\left( x^{n-1}h_{n,q}(x,a,r,m)\right) \\
\;\;\;\;\;&+c(n)G_{r,q}\left( x^{n-1}h_{n-1,q}(x,a,r,m)\right)\\
=&c(n)G_{r,q}\left( x^{n-1}h_{n-1,q}(x,a,r,m)\right)\\
=&c(n)c(n-1)G_{r,q}\left( x^{n-2}h_{n-2,q}(x,a,r,m)\right)\\
=&c(n)c(n-1)c(n-2)G_{r,q}\left( x^{n-3}h_{n-3,q}(x,a,r,m)\right)\\
&\vdots\\
=&c(n)c(n-1)c(n-2)\dots c(1)G_{r,q}\left( x^{0}h_{0,q}(x,a,r,m)\right)\\
=&\left[ \prod\limits_{i=1}^{n}c(i)\right] (1)\\
=&\prod\limits_{i=1}^{n}c(i)
\end{align*}
Since $x^{n}h_{n,q}(x,a,r,m)$ is a sequence of orthogonal polynomials with respect to linear functional $G_{r,q}$,
\begin{equation*}
d_{n,q}=G_{r,q}\left( x^{n}h_{n,q}(x,a,r,m)\right) =\prod\limits_{i=1}^{n}c(i)
\end{equation*}
where
\begin{equation*}
c(i)=((q^{mi}-1)q^{r}q^{2i-2}a^2+q^r[m]_q[i]_{q^m}q^{i-1}a)q^{m(i-1)+r}
\end{equation*}
Then
\begin{align*}
d_{n,q}(n,0)&=G_{r,q}\left[ [x]_{q}^{n}h_{n,q}(x,a,r,m)\right] \\
&=\prod\limits_{i=0}^{n-1}d_{i,q}\\
&=\prod\limits_{i=0}^{n-1}\left\lbrace \prod\limits_{j=1}^{i}\left[ ((q^{mj}-1)q^{r}q^{2j-2}a^2+q^r[m]_q[j]_{q^m}q^{j-1}a)q^{m(j-1)+r}\right] \right\rbrace \\
&=\prod\limits_{i=0}^{n-1}\left\lbrace \prod\limits_{j=1}^{i}\left[ aq^{m(j-1)+2r}((q^{mj}-1)q^{2j-2}a+[m]_q[j]_{q^m}q^{j-1})\right] \right\rbrace \\
&=\prod\limits_{i=0}^{n-1}q^{2ri}q^{m(1+2+3+\dots+(i-1))} a^i\prod\limits_{j=1}^{i}\left[ ((q^{mj}-1)q^{2j-2}a+[m]_q[j]_{q^m}q^{j-1})\right] \\
&=\prod\limits_{i=0}^{n-1}q^{2ri}q^{m\binom{i}{2}} a^i\prod\limits_{j=1}^{i}\left[ ((q^{mj}-1)q^{2j-2}a+[m]_q[j]_{q^m}q^{j-1})\right] \\
&=q^{2r(0+1+2+3+\dots+(n-1))+m\left[ \binom{2}{2}+\binom{3}{2}+\dots+\binom{n-1}{2}\right]}a^{0+1+2+\dots+(n-1)}\\
&\;\;\;\prod\limits_{i=0}^{n-1}\prod\limits_{j=1}^{i}\left[ ((q^{mj}-1)q^{2j-2}a+[m]_q[j]_{q^m}q^{j-1})\right] \\
&=q^{2r\binom{n}{2}+(m+1)\binom{n}{3}}a^{\binom{n}{2}}\prod\limits_{i=0}^{n-1}\prod\limits_{j=1}^{i}\left[ ((q^{mj}-1)q^{2j-2}a+[m]_q[j]_{q^m}q^{j-1})\right]\\
&=q^{2r\binom{n}{2}+m\binom{n}{3}}a^{\binom{n}{2}}\prod\limits_{i=0}^{n-1}q^{\binom{i}{2}}\prod\limits_{j=1}^{i}\left[ [mj]_q\left(1-q^j\left(\frac{1-q}{q}\right)a\right)\right]\\
&=q^{2r\binom{n}{2}+(m+1)\binom{n}{3}}a^{\binom{n}{2}}\prod\limits_{i=0}^{n-1}\prod\limits_{j=1}^{i}\left[ [mj]_q\left(1-q^{j-1}(1-q)a\right)\right].
\end{align*}
This result is stated formally in the following theorem.
\begin{thm}\label{thm1}
The Hankel transform of $\Phi_{n}[x,r,m]_{q}$ corresponding to the $0th$ Hankel determinant is given by
\begin{equation}\label{res1}
H\left( \Phi_{n}[x,r,m]_{q}\right) =q^{2r\binom{n}{2}+(m+1)\binom{n}{3}}a^{\binom{n}{2}}\prod\limits_{i=0}^{n-1}\prod\limits_{j=1}^{i}\left[ [mj]_q\left(1-q^{j-1}(1-q)a\right)\right].
\end{equation}
\end{thm}
Note that when $m=1$, \eqref{res1} yields
$$H\left( \Phi_{n}[x,r,1]_{q}\right) =q^{2r\binom{n}{2}+2\binom{n}{3}}a^{\binom{n}{2}}\prod\limits_{i=0}^{n-1}[i]_q!((1-q)a;q)_i$$
where
\begin{equation*}
(x;q)_{i}=\prod\limits_{j=0}^{i-1}\left( 1-q^{j}x\right).
\end{equation*}
This is exactly the result obtained by Cigler \cite{Cig2}.
As a direct consequence of Theorem \ref{thm1}, we have the following corollary, which contains the main result of this paper.
\begin{cor}
The Hankel transform of the sequence $\left(D_{m,r}[n]_q\right)_{n=0}^{\infty}$ is given by
\begin{equation*}
H\left(D_{m,r}[n]_q\right) =q^{2r\binom{n}{2}+(m+1)\binom{n}{3}}\prod\limits_{i=0}^{n-1}((1-q)a;q)_i\prod\limits_{j=1}^{i}[mj]_q.
\end{equation*}
\end{cor}
\begin{thm}
The Hankel transform of $\Phi_{n}[x,r,m]_{q}$ corresponding to the $1st$ Hankel determinant is given by
\begin{align*}
d_{n,q}(n,1)&=q^{2r\binom{n}{2}+(m+1)\binom{n}{3}}a^{\binom{n}{2}}\prod\limits_{i=0}^{n-1}((1-q)a;q)_i\prod\limits_{j=1}^{i}[mj]_q\\
&\;\;\;\;\;\;\;\sum\limits_{k=0}^{n}(-1)^{n}[x]_{q}^{k}q^{\binom{k}{2}}\bracketed{n}{k}_{q}\prod\limits_{j=0}^{k-1}\dfrac{[r+jm]_{q}}{q^{r+jm}}.
\end{align*}
\end{thm}
\begin{proof}
From Gram-Schmidt orthogonalization process, we obtain
\begin{equation*}
d_{n,q}(n,1)=d_{n,q}(n,0)(-1)^{n}p_{n,q}(0)
\end{equation*}
where
$p_{n,q}(0)$ is a sequence of orthogonal polynomials i.e.,
\begin{equation*}
g_{n,q}(x,a,r,m)=\sum\limits_{k=0}^{n}\left( -a\right)^{k}q^{\binom{k}{2}}\bracketed{n}{k}_{q}\langle\langle x\rangle\rangle_{r,m,k}=p_{n,q}(x)
\end{equation*}
which implies
\begin{equation*}
p_{n,q}(0)=\sum\limits_{k=0}^{n}\left( -a\right)^{k}q^{\binom{k}{2}}\bracketed{n}{k}_{q}\langle\langle 0\rangle\rangle_{r,m,k}.
\end{equation*}
Since,
\begin{align*}
\langle\langle 0\rangle\rangle_{r,m,k}&=\prod\limits_{j=0}^{k-1}\dfrac{\left( [0]_{q}-[r+jm]_{q}\right) }{q^{r+jm}}\\
&=\prod\limits_{j=0}^{k-1}\dfrac{-[r+jm]_{q}}{q^{r+jm}}\\
&=\left( \dfrac{-[r]_{q}}{q^r}\right) \left(\dfrac{-[r+j]_q}{q^{r+m}} \right) \left( \dfrac{-[r+(k-1)m]_{q}}{q^{r+(k-1)m}}\right)\\
&=(-1)^{k}\prod\limits_{j=0}^{k-1}\dfrac{[r+jm]_{q}}{q^{r+jm}}.
\end{align*}
Then,
\begin{align*}
p_{n,q}(0)&=\sum\limits_{k=0}^{n}\left( -a\right)^{k}q^{\binom{k}{2}}\bracketed{n}{k}_{q}(-1)^{k}\prod\limits_{j=0}^{k-1}\dfrac{[r+jm]_{q}}{q^{r+jm}}\\
&=\sum\limits_{k=0}^{n}(-1)^{k}a^{k}q^{\binom{k}{2}}\bracketed{n}{k}_{q}(-1)^{k}\prod\limits_{j=0}^{k-1}\dfrac{[r+jm]_{q}}{q^{r+jm}}\\
&=\sum\limits_{k=0}^{n}a^{k}q^{\binom{k}{2}}\bracketed{n}{k}_{q}\prod\limits_{j=0}^{k-1}\dfrac{[r+jm]_{q}}{q^{r+jm}}
\end{align*}
which implies
\begin{equation*}
(-1)^{n}p_{n,q}(0)=\sum\limits_{k=0}^{n}(-1)^{n}[x]_{q}^{k}q^{\binom{k}{2}}\bracketed{n}{k}_{q}\prod\limits_{j=0}^{k-1}\dfrac{[r+jm]_{q}}{q^{r+jm}}.
\end{equation*}
Hence,
\begin{align*}
d_{n,q}(n,1)&=d_{n,q}(n,0)(-1)^{n}p_{n,q}(0)\\
&=q^{2r\binom{n}{2}+(m+1)\binom{n}{3}}a^{\binom{n}{2}}\prod\limits_{i=0}^{n-1}((1-q)a;q)_i\prod\limits_{j=1}^{i}[mj]_q\\
&\;\;\;\;\;\;\;\sum\limits_{k=0}^{n}(-1)^{n}[x]_{q}^{k}q^{\binom{k}{2}}\bracketed{n}{k}_{q}\prod\limits_{j=0}^{k-1}\dfrac{[r+jm]_{q}}{q^{r+jm}}
\end{align*}
\end{proof}
\noindent{\bf Acknowledgement}. This research has been funded by Cebu Normal University (CNU) and the Commission on Higher Education - Grants-in-Aid for Research (CHED-GIA).
\end{document}
|
\begin{document}
\title{f A short note on the Liouville problem for the steady-state Navier-Stokes equations}
\begin{abstract}
Uniqueness of the trivial solution (the zero solution) for the steady-state Navier-Stokes equations is an interesting problem who has known several recent contributions. These results are also known as the Liouville type problem for the steady-state Navier-Stokes equations. In the setting of the $L^p-$ spaces, when $3\leq p \leq 9/2$ it is known that the trivial solution of these equations is the unique one. In this note, we extend this previous result to other values of the parameter $p$. More precisely, we prove that the velocity field must be zero provided that it belongs to the $L^p -$ space with $3/2<p<3$. Moreover, for the large interval of values $9/2<p<+\infty$, we also obtain a partial result on the vanishing of the velocity under an additional hypothesis in terms of the Sobolev space of negative order $\dot{H}^{-1}$. This last result has an interesting corollary when studying the Liouville problem in the natural energy space of these solutions $\dot{H}^{1}$. \\[3mm]
\textbf{Keywords:} Steady-state Navier-Stokes equations; Liouville type problem; Cacciopoli type estimates. \\[3mm]
\textbf{AMS Classification:} 35Q30, 35B53
\end{abstract}
\section{Introduction}
This short note deals with the homogeneous and incompressible steady-state (time-independent) Navier-Stokes equations in the whole space $\Rt:$
\begin{equation*}
(NS) \quad -\Delta \U+(\U \cdot \vec{\nabla})\U +\vec{\nabla}P = 0, \qquad div(\U)=0.
\end{equation*}
Here $\U=(U_1, U_2, U_3): \Rt\longrightarrow \Rt$ is the velocity of the fluid while $P:\Rt\longrightarrow \mathbb{R}$ is the pressure. The equation $\ds{div(\U)=0}$ represents the fluid incompressibility. We recall that $(\U,P)$ is a smooth solution for the (NS) equations if $\U \in \mathcal{C}^{2}(\Rt)$, $P \in \mathcal{C}^{1}(\Rt)$ and if the couple $(\U,P)$ verify these equations in the classical sense.
In the (NS) equations, we may observe that $\U=0$ and $P=0$ is always a smooth solution, also known as the \emph{trivial solution}, and then it is quite natural to ask if the trivial solution is the unique one. In the general setting of the space $\mathcal{C}^{2}(\Rt) \times \mathcal{C}^{1}(\Rt)$, the answer to this question is negative and we are able to give a simple counterexample. Let $f \in \mathcal{C}^{3}(\Rt)$ be a scalar field such that $\Delta f =0$. Then, by setting the velocity $\U= \vec{\nabla} f$ and the pressure $P= - \frac{1}{2} \vert \vec{\nabla} f \vert^2$, and by using some well-known rules of the vector calculus, we have that $(\U,P)$ is also a smooth solution of the (NS) equations. See the Appendix \ref{AppendixA} for all the details. We thus look for some \emph{additional hypothesis} on smooth solutions of the (NS) equations to insure the uniqueness of the trivial solution. This type of problem is also known as a the \emph{Liouville problem} for the (NS) equations. We emphasize this problem has attired a lot of attention in the community of researchers. See, for instance, \cite{ChaeWolf,ChaeWeng,OscarPaper,Ser2016,Seregin3,SereWang} and the references therein.
In the example above, we remark that as the scalar field $f$ is a harmonic function then it is a polynomial and thus $\U = \vec{\nabla} f$ also has a polynomial growth at infinity. This fact strongly suggests that, in order to study the Liouville problem for the (NS) equations, we must seek for \emph{decaying properties} at infinity on the velocity $\U$. Precisely, it was pointed out in the celebrated work of G. Galdi who showed in \cite{Galdi} (Chapter X, Remark X.9.4 and Theorem X.9.5, p. 729) that if $\U$ is a smooth solution of the (NS) equations, and moreover, if $\U \in L^{9/2}(\Rt)$ then we necessarily have $\U=0$. This result follows from the following \emph{Cacciopoli} type estimate: for $R>1$, we denote $B_R=\{ x \in \Rt: \vert x \vert <R \}$ and $C(R/2,R)=\{ x \in \Rt: R/2 < \vert x \vert <R \}$, and for a constant $C>0$ which does not depend on $R$ we have:
\begin{equation*}
\int_{B_R} \vert \vec{\nabla} \otimes \U (x) \vert^2 dx \leq \, C \Vert \U \Vert^{3}_{L^{9/2}(C(R/2,R))} + C R^{-1/3} \Vert \U \Vert^{2}_{L^{9/2}(C(R/2,R))} + C \Vert \U \Vert_{L^{9/2}(C(R/2,R))} \Vert P \Vert_{L^{9/4}(C(R/2,R))}.
\end{equation*}
This estimate yields the identity $\U=0$, provided that $\U \in L^{9/2}(\Rt)$ and $P \in L^{9/4}(\Rt)$. Moreover, it is worth mentioning the value of the integration parameter $9/2$ naturally appears by the well-known scaling properties of (NS) equations.
Galdi's result was recently extended in \cite{Liouville1} for other values of the parameter $p$ in the Lebesgue spaces. More precisely, in Theorem $1$ of \cite{Liouville1}, D. Chamorro, P.G. Lemarié-Rieusset and the author of this note proved that if $\U$ is a smooth solution of the (NS) equations such that $\U \in L^{p}(\Rt)$, with $3 \leq p < 9/2$, then we have $\U=0$ and $P=0$. For this, we performed the next \emph{Cacciopoli} type estimate:
\begin{equation*}
\int_{B_R} \vert \vec{\nabla} \otimes \U (x) \vert^2 dx \leq C\, R^{1-6/p} \Vert \U \Vert_{L^p(B_R)}+ C_p\, R^{2-9/p} \Vert \U \Vert^{3}_{L^p(C(R/2,R))} + C_p \, \Vert \U \Vert_{L^p(C(R/2,R))}\, \Vert P \Vert_{L^{p/2}(C(R/2,R))}.
\end{equation*}
Here, the constant $C_p>0$ is the norm of a certain test function in the space $L^{\frac{p}{p-3}}(\Rt)$ and we thus need condition $ 3 \leq p$. On the other hand, in order to get a uniform control on $R>1$, due to the expression $R^{2-9/p}$ in the second term on the right-hand side, we also need the condition $p\leq 9/2$. Consequently, the Liouville problem is solved in the $L^p-$ space for $3 \leq p \leq 9/2$.
The aim of this short note is to continue with the study of the Liouville problem for the (NS) equations in the setting of the Lebesgue spaces. Specifically, we study this problem for the values of the parameter $p$ outside the interval $3 \leq p \leq 9/2$. For the values of the parameter $p$ lower than $3$, our first result states as follows.
\begin{TheoremeP}\label{Th1} Let $(\U, P)$ be a smooth solution of the (NS) equations. If $\U \in L^{p}(\Rt)$, with $\frac{3}{2}<p<3$, then we have $\U=0$ and $P=0$.
\end{TheoremeP}
The proof is based on a different and more technical \emph{Cacciopoli} type estimate, which is stated in Proposition \ref{lemaTehc-Ser}. This estimate was originally established in \cite{Seregin3}. It is worth mentioning our computations are not longer valid when $1\leq p \leq 3/3$ and, to the best of our knowledge, the Liouville problem for the (NS) equations is still an open question these values of the parameter $p$.
It is also worth emphasizing some recent preprints study this problem when $\U \in L^p(\Rt)$, whit $1 \leq p < 3$, provided that $\vert \U(x) \vert \to 0$ when $\vert x \vert \to +\infty$. However, as $\U$ is assumed a smooth solution, the last hypothesis implies that $\U \in L^{\infty}(\Rt)$. Then, by the interpolations inequalities we have $\U \in L^p \cap L^{\infty}(\Rt)$, hence $\U \in L^{9/2}(\Rt)$. Consequently, the additional vanishing assumption at infinity makes the problem trivial. In the sense, one the main interests on the result above is the study of the Liouville problem in the interval $3/2<p<3$ without any additional assumption.
When $9/2 <p <6$, the Liouville problem in the $L^p -$ spaces was study studied by using some supplementary hypothesis. In \cite{Liouville1}, this problem is solved in the space $L^{p} \cap B^{3/p-3/2}_{\infty, \infty} (\Rt)$, where $B^{3/p-3/2 }_{\infty,\infty} (\Rt)$ is a homogeneous Besov space (\ref{Besov-neg}). Moreover, for the value $p=6$, an interesting result of G. Seregin given in \cite{Ser2016} shows that this problem is solved in the space $L^{6}\cap BMO^{-1}(\Rt)$. On the other hand, to the best of our knowledge, there are not previous results for the values $6<p$. In our next result, we study the Liouville problem in the large interval $9/2<p<+\infty$. As the previous results, we shall need here an additional assumption on the velocity field.
\begin{TheoremeP}\label{Th2} Let $(\U, P)$ be a smooth solution of the (NS) equations. If $\U \in L^{p}(\Rt)\cap \dot{H}^{-1}(\Rt)$, for $9/2 < p <+\infty$, then we have $\U=0$ and $P=0$.
\end{TheoremeP}
One the main interests of this result is the use of the space $\dot{H}^{-1}(\Rt)$, which is the dual space of functions having a finite Dirichlet integral; and which provides us a different condition to solve the Liouville type problem. On the other hand, this result suggests that the value $p=9/2$, found by G. Galdi in \cite{Galdi}, seems to be the upper limit to solve the Liouville problem without any additional assumptions. This fact also suggests to look for non trivial solutions for the (NS) equations in the space $L^p(\Rt)$ with $9/2<p$. However, we think that this is still a very challenging open question.
In this result, the value $p=6$ is of particular interest. Let us recall that the problem Liouville for the (NS) equations was originally stated in the homogeneous Sobolev space $\dot{H}^{1}(\Rt)$ (see for instance the Chapter $X$ in \cite{Galdi} and the Chapter $4$ in \cite{OscarTesis}). This is the natural \emph{energy space} for this equations, and moreover, not rigorous computations strongly suggest that all the solutions $\U \in \dot{H}^{1}(\Rt)$ must be identical to zero.
However, the information $\U \in \dot{H}^{1}(\Rt)$ is not enough to justify all the computations, in particular those estimates involving the nonlinear term $\ds{(\U \cdot \vec{\nabla}) \U}$. Consequently, the Liouville problem for the (NS) equations in the space $\dot{H}^{1}(\Rt)$ is still an outstanding open question far from obvious.
By the Sobolev embeddings we have $\dot{H}^{1}(\Rt)\subset L^6(\Rt)$, and then, we can use the Theorem \ref{Th2} to prove the following corollary. This is a partial result on the Liouville problem in the space $\dot{H}^{1}(\Rt)$.
\begin{CorollaireP}\label{Col} Let be $\U \in \dot{H}^{1}(\Rt)$ be a weak solution of the (NS) equations. If $\ds{\widehat{\U} \in L^{r}_{loc}(\Rt)}$, with $r>6$, then we have $\U =0$.
\end{CorollaireP}
This corollary does not assume any smoothness of the velocity $\U$ and we understand here $\U \in \dot{H}^{1}(\Rt)$ as a weak solution, \emph{i.e.}, $\U$ verifies the (NS) equations in the distributional sense. On the other hand, we recall that the space $\dot{H}^{1}(\Rt)$ is defined as the space of temperate distributions $g$ such that $\widehat{g} \in L^{1}_{loc}(\Rt)$ and $\ds{\int_{\Rt} \vert \xi \vert^2 \vert \widehat{g}(\xi) \vert^2 \, d \xi <+\infty}$. We thus observe that a stronger locally-integrability condition on $\widehat{\U}$ yields the desired identity $\U=0$.
\section{The key tools}\label{Sec2:Previous-Results}
\subsection{Homogeneous Besov spaces.}
The first key tool deals with the homogeneous Besov spaces. Let $0<s<1$, and $1\leq p,q\leq +\infty$. The homogeneous Besov space of positive order: $\dot{B}^{s}_{p, q}(\Rt)$, is defined as the set of $f \in \mathcal{S}'(\Rt)$ such that
\begin{equation}\label{besov-pos1}
\Vert f \Vert_{\dot{B}^{s}_{p, q}}= \left( \int_{\Rt} \frac{\Vert f(\cdot+x)-f(\cdot) \Vert_{L^p}}{\vert x \vert^{3+sq}}\, dx \right)^{1/q} <+\infty, \quad \text{with}, \quad 1\leq p,q<+\infty,
\end{equation}
and
\begin{equation}\label{besov-pos}
\Vert f \Vert_{\dot{B}^{s}_{\infty, \infty}}= \sup_{x \in \Rt} \frac{\Vert f(\cdot+x)-f(\cdot)\Vert_{L^{\infty}}}{\vert x \vert^{s}}<+\infty.
\end{equation}
Moreover, the Besov space of negative order $\dot{B}^{-s}_{\infty, \infty}(\Rt)$ can be characterized by means of the heat kernel $h_t$ as the set of $f \in \mathcal{S}'(\Rt)$ such that
\begin{equation}\label{Besov-neg}
\Vert f \Vert_{\dot{B}^{-s}_{\infty, \infty}} = \sup_{t>0} \, t^{\frac{s}{2}} \Vert h_t \ast f \Vert_{L^{\infty}}<+\infty.
\end{equation}
For more details on the Besov spaces and their application to the theoretical study of the Navier-Stokes equations (stationary or time-dependent) see the Chapter $8$ in the book \cite{PGLR1}.
\subsection{A \emph{Cacciopoli} type estimate}
The second key tool is the following \emph{Cacciopoli} type estimate. This inequality is inspired by the estimate $(2.2)$, page $9$ in \cite{Seregin3} and its proof essentially follows similar ideas. However, for the sake of completeness, we include the main steps of the proof.
\begin{Proposition}\label{lemaTehc-Ser} Let $(\U, P)$ be a smooth solution of the (NS) equation. Let $\V_1$ and $\V_2$ be smooth vector fields such that $\U = \vec{\nabla} \wedge \V_1$ and $\U= \vec{\nabla} \wedge \V_2$. Then, for all $3<q<+\infty $, there exists a constant $C_q>0$ such that such that for all $R>1$ we have:
\begin{equation*}
\int_{B_{R/2}} \vert \vec{\nabla} \otimes \U(x) \vert^2 dx \leq \frac{C_q}{R} \left( \frac{1}{R^3} \int_{\mathcal{C}(R/2,R)} \vert \V_1(x) \vert^2 dx \right) \left( 1+ \left(\frac{1}{R^3} \int_{B_R} \vert \V_2(x) \vert^{q}dx \right)^{\frac{4}{q-3}} \right).
\end{equation*}
\end{Proposition}
\pv We introduce the following cut-off functions, which were considered in \cite{OscarPaper}. For $R>1$ fixed, we define first the function $\varphi_R\in \mathcal{C}^{\infty}_{0}(B_R)$ such that: for $R/2 < \rho < r< R$ it verifies: $\varphi_R(x)=1$ when $\vert x \vert < \rho$, $\varphi_R(x)=0$ when $\vert x \vert >r$ and, for all multi-indice $\vert \alpha \vert \leq 4$, $\Vert \partial^{\alpha} \varphi_R \Vert_{L^\infty} \leq \frac{c}{(r-\rho)^{\vert \alpha \vert}}$. Next, we define the function $\W_R$ as the solution of the problem:
\begin{equation*}
div(\W_R)=\vec{\nabla}\varphi_R\cdot \U, \quad \text{in}\,\, B_r, \quad \text{and}\quad \W_R=0 \,\, \text{on}\,\, \partial B_r.
\end{equation*}
In Lemma $III. 3.1$, page 162 of \cite{Galdi}, it is proven that for $1<q<+\infty$ there exists a solution $\W_R\in W^{1,q}(B_R)$, which verifies $supp\,(\W_R)\subset \mathcal{C}(R/2, R)$ and $\Vert \vec{\nabla}\otimes \W_R\Vert_{L^q(\mathcal{C}(R/2, R))}\leq c \Vert \vec{\nabla}\varphi_R\cdot \U \Vert_{L^q(\mathcal{C}(R/2, R))}$.
We consider now the function $\varphi_R \U-\W_R$, which has a support in the ball $B_r$, and we write
\begin{equation}\label{eq01}
\int_{B_r} \left( -\Delta \U +(\U \cdot \vec{\nabla})\U +\vec{\nabla}P\right)\cdot \left( \varphi_R \U-\W_R \right)dx=0.
\end{equation}
From this identity, by performing some integration by parts, and moreover, for a function $\V_2$ such that $\U = \vec{\nabla} \wedge \V$ on $B_R$, for $1=1/p+1/q$ we have the following estimate (see the page $6$ of \cite{Seregin3} or the appendix of \cite{Liouville1}):
\begin{equation*}
\begin{split}
\int_{B_\rho} \vert \vec{\nabla} \otimes \U \vert^2 dx \leq & \, C \left( \int_{B_r} \vert \vec{\nabla} \otimes \U \vert^2 dx\right)^{1/2} \left( \int_{B_r} \vert \vec{\nabla} \varphi_R \vert^2 \vert \U \vert^2 dx\right)^{1/2}\\
&\, + C \left(\int_{B_r} \vert \vec{\nabla} \otimes \U \vert^2 dx\right)^{1/2} \left( \int_{B_r}\vert \vec{\nabla} \varphi_R \vert \vert \U \vert \vert \V_2 \vert dx \right)^{1/2}\\
\leq & \, C \left( \int_{B_r} \vert \vec{\nabla} \otimes \U \vert^2 dx\right)^{1/2} \left( \int_{B_r} \vert \vec{\nabla} \varphi_R \vert^2 \vert \U \vert^2 dx\right)^{1/2}\\
&\, + C \left(\int_{B_r} \vert \vec{\nabla} \otimes \U \vert^2 dx\right)^{1/2} \left( \int_{B_r}\vert \vec{\nabla} \varphi_R \vert^p \vert \U \vert^p dx \right)^{1/p} \left( \int_{B_r} \vert \V_2 \vert^q dx \right)^{1/q}.
\end{split}
\end{equation*}
Then, we apply the discrete Young inequalities (with $1=1/2+1/2$) in both terms on the right-hand side to get:
\begin{equation}\label{estim-01}
\int_{B_\rho} \vert \vec{\nabla} \otimes \U \vert^2 dx \leq \frac{1}{8} \int_{B_r} \vert \vec{\nabla} \otimes \U \vert^2 dx + C \int_{B_r} \vert \vec{\nabla} \varphi_R \vert^2 \vert \U \vert^2 dx + C \left( \int_{B_r}\vert \vec{\nabla} \varphi_R \cdot \U \vert^p dx \right)^{2/p} \left( \int_{B_r} \vert \V_2 \vert^q dx \right)^{2/q}.
\end{equation}
We must study the third term on the right-hand side. We set $2<p<6$. For $\theta= \frac{3(p-2)}{2p}$ (which verifies $0<\theta<1$) by the interpolation inequalities (with $2/p=\theta/3 + (1-\theta)/1$) and the Sobolev embedding $ \left(\int_{B_r} \vert f \vert^6 dx \right)^{1/6} \leq c \left( \int_{B_r} \vert \vec{\nabla} f \vert^2 dx\right)^{1/2}$, we write:
\begin{equation*}
\begin{split}
&\left( \int_{B_r}\vert \vec{\nabla} \varphi_R \cdot \U \vert^p dx \right)^{2/p} = \, \left( \int_{B_r} \left(\vert \vec{\nabla} \varphi_R \cdot \U \vert^2 \right)^{p/2} dx \right)^{2/p} \\
\leq & \, C \left( \int_{B_r}\vert \vec{\nabla} \varphi_R \vert^2\, \vert \U \vert^2 \, dx \right)^{1-\theta}\left( \int_{B_r} \left( \vert \vec{\nabla} \varphi_R \cdot \U \vert^2 \right)^3 dx \right)^{\theta} \\
\leq & \, C \left( \int_{B_r}\vert \vec{\nabla} \varphi_R \vert^2\, \vert \U \vert^2 \, dx \right)^{1-\theta} \left( \int_{B_r} \left\vert \vec{\nabla} \left( \vec{\nabla} \varphi_R \cdot \U \right) \right\vert^2 dx \right)^{\theta} \\
\leq & \, C \left( \int_{B_r}\vert \vec{\nabla} \varphi_R \vert^2\, \vert \U \vert^2 \, dx \right)^{1-\theta} \left[ \left( \int_{B_r} \left\vert \vec{\nabla} ( \vec{\nabla} \ \varphi_R ) \right\vert^2\, \vert \U \vert^2 dx \right)^{\theta} + \left( \int_{B_r} \vert \vec{\nabla} \varphi_R \vert^2 \vert \vec{\nabla} \otimes \U \vert^2 dx\right)^{\theta} \right]\\
\leq & \, C \left( \int_{C(\rho, r)}\vert \vec{\nabla} \varphi_R \vert^2\, \vert \U \vert^2 \, dx \right)^{1-\theta} \left[ \left( \int_{C(\rho, r)} \left\vert \vec{\nabla} ( \vec{\nabla} \ \varphi_R ) \right\vert^2\, \vert \U \vert^2 dx \right)^{\theta} + \left( \int_{B_r} \vert \vec{\nabla} \varphi_R \vert^2 \vert \vec{\nabla} \otimes \U \vert^2 dx\right)^{\theta} \right]
\end{split}
\end{equation*}
With this estimate, we get back to (\ref{estim-01}). We applying again the discrete Young inequalities (with $1=\theta + (1-\theta)$) and moreover, by the estimate $\Vert \Delta \varphi_R \Vert_{L^\infty} \leq \frac{c}{(r-\rho)^2}$, we write:
\begin{equation}\label{estim-02}
\begin{split}
\int_{B_\rho} \vert \vec{\nabla} \otimes \U \vert^2 dx \leq & \, \frac{1}{4} \int_{B_r} \vert \vec{\nabla} \otimes \U \vert^2 \, dx + C \underbrace{\int_{C(\rho,r)}\vert \vec{\nabla} \varphi_R \vert^2\, \vert \U \vert^2 \, dx}_{(\mathcal{A})} \\
& \, + C \left( \int_{C(\rho,r)} \vert \vec{\nabla} \varphi_R \vert^2 \vert \U \vert^2 dx\right)^{1-\theta} \underbrace{ \left( \int_{C(\rho,r)} \left\vert\vec{\nabla} \left(\vec{\nabla} \varphi_R\right) \right\vert^2 \vert \U \vert^2 dx \right)^{\theta}}_{(\mathcal{B})} \left( \int_{B_r} \vert \V_2 \vert^q dx\right)^{2/q}\\
&\, + C \left(\frac{1}{(r-\rho)^2}\right)^{\frac{\theta}{1-\theta}}\, \left( \int_{C(\rho,r)} \vert \vec{\nabla} \varphi_R \vert^2\, \vert \U \vert^2 dx\right)\left( \int_{B_r} \vert \V_2 \vert^{q} dx\right)^{\frac{2}{q(1-\theta)}}.
\end{split}
\end{equation}
We must estimate now the terms $(\mathcal{A})$ and $(\mathcal{B})$. By Lemma $2.1$ of \cite{Seregin3}, for a smooth function $\V_1$ such that $\U = \vec{\nabla} \wedge \V_1$ we have:
\[ (\mathcal{A}) \leq \frac{C}{(r-\rho)^2} \underbrace{ \left[ \left( \int_{B_r} \vert \vec{\nabla} \otimes \U \vert^2 dx \right)^{1/2} \left( \int_{C(\rho,r)} \vert \V_1 \vert^2 dx\right)^{1/2} + \frac{1}{(r-\rho)^2} \int_{C(\rho,r)} \vert \V_1 \vert^2 dx \right]}_{(\mathcal{C})}, \]
\[ (\mathcal{B}) \leq \frac{C}{(r-\rho)^4} \underbrace{\left[ \left( \int_{B_r} \vert \vec{\nabla} \otimes \U \vert^2 dx \right)^{1/2} \left( \int_{C(\rho,r)} \vert \V_1 \vert^2 dx\right)^{1/2} + \frac{1}{(r-\rho)^2} \int_{C(\rho,r)} \vert \V_1 \vert^2 dx \right]}_{(\mathcal{C})}. \]
We get back to the estimate (\ref{estim-02}) to write:
\begin{equation*}
\begin{split}
\int_{B_\rho} \vert \vec{\nabla} \otimes \U \vert^2 dx \leq & \, \frac{1}{4} \int_{B_r} \vert \vec{\nabla} \otimes \U \vert^2 dx + \frac{C}{(r-\rho)^2} (\mathcal{C}) \\
& \, + \left( \frac{C}{(r-\rho)^2}\, (\mathcal{C}) \right)^{1-\theta} \left(\frac{C}{(r-\rho)^4}\, (\mathcal{C}) \right)^{\theta}\, \left( \int_{B_r} \vert \V_2 \vert^{q} dx\right)^{2/q} \\
&\, + \left( \frac{C}{(r-\rho)^2} \right)^{\frac{\theta}{1-\theta}}\, \frac{C}{(r-\rho)^2}(\mathcal{C}) \left( \int_{B_r} \vert \V \vert^q\, dx\right)^{\frac{2}{q(1-\theta)}}\\
= &\, \frac{1}{4} \int_{B_r} \vert \vec{\nabla} \otimes \U \vert^2 dx + \frac{C}{(r-\rho)^2} (\mathcal{C}) + \frac{C}{(r-\rho)^{2 \theta}}\, \frac{1}{(r-\rho)^2} (\mathcal{C}) \left( \int_{B_r} \vert \V_2 \vert^q \, dx\right)^{2/q} \\
&\, + \left( \frac{C}{(r-\rho)^2} \right)^{\frac{\theta}{1-\theta} +1} \, (\mathcal{C})\, \left( \int_{B_r} \vert \V_2 \vert^q\, dx\right)^{\frac{2}{q(1-\theta)}}.
\end{split}
\end{equation*}
In the last term above, we remark that we have $\frac{\theta}{1-\theta} + 1 = \frac{1}{1-\theta}$. Moreover, by definition of the parameter $\theta= \frac{3(p-2)}{2p}$ and by the relation $1/2 = 1/p +1/q$ we have $q\theta =3$. Thus, we can write:
\begin{equation*}
\begin{split}
&\int_{B_\rho} \vert \vec{\nabla} \otimes \U \vert^2 dx \\
\leq & \, \frac{1}{4} \int_{B_r} \vert \vec{\nabla} \otimes \U \vert^2 dx + \frac{C}{(r-\rho)^2} (\mathcal{C})\, \left[ 1 + \left( \frac{1}{(r-\rho)^{q \theta}} \int_{B_r} \vert \V_2 \vert^q\, dx\right)^{2/q} + \left( \frac{1}{(r-\rho)^{q \theta}} \int_{B_r} \vert \V_2 \vert^q\, dx\right)^{\frac{2}{q(1-\theta)}}\right]\\
=&\, \frac{1}{4} \int_{B_r} \vert \vec{\nabla} \otimes \U \vert^2 dx + \frac{C}{(r-\rho)^2} (\mathcal{C})\, \underbrace{\left[ 1 + \left( \frac{1}{(r-\rho)^{3}} \int_{B_r} \vert \V_2 \vert^q\, dx\right)^{2/q} + \left( \frac{1}{(r-\rho)^{3}} \int_{B_r} \vert \V_2 \vert^q\, dx\right)^{\frac{2}{q(1-\theta)}}\right]}_{(\mathcal{D})}.
\end{split}
\end{equation*}
Now, we write down the whole term $(\mathcal{C})$, and we have
\begin{equation*}
\begin{split}
&\int_{B_\rho} \vert \vec{\nabla} \otimes \U \vert^2 dx \\
\leq &\, \frac{1}{4} \int_{B_r} \vert \vec{\nabla} \otimes \U \vert^2 dx + \frac{C}{(r-\rho)^2} \left[ \left( \int_{B_r} \vert \vec{\nabla} \otimes \U \vert^2 dx \right)^{\frac{1}{2}} \left( \int_{C(\rho,r)} \vert \V_1 \vert^2 dx\right)^{\frac{1}{2}} + \frac{1}{(r-\rho)^2} \int_{C(\rho,r)} \vert \V_1 \vert^2 dx \right](\mathcal{D}) \\
\leq &\, \frac{1}{4} \int_{B_r} \vert \vec{\nabla} \otimes \U \vert^2 dx+ \frac{C}{(r-\rho)^2} \left[ \frac{(r-\rho)^2}{4C} \int_{B_r} \vert \vec{\nabla} \otimes \U \vert^2 dx + \frac{C}{(r-\rho)^2} \int_{C(\rho,r)} \vert \V_1 \vert^2 \, dx \, (\mathcal{D})^2 \right]\\
\leq &\, \frac{1}{2} \int_{B_r} \vert \vec{\nabla}\otimes \U \vert^2 dx + \frac{C}{(r-\rho)^4} \int_{C(\rho,r)} \vert \V_1 \vert^2 dx \, (\mathcal{D})^2
\end{split}
\end{equation*}
Now, we write down the whole term $(\mathcal{D})$ to write:
\begin{equation*}
\begin{split}
\int_{B_\rho} \vert \vec{\nabla} \otimes \U \vert^2 dx
\leq &\,\frac{1}{2} \int_{B_r} \vert \vec{\nabla} \otimes \U \vert^2 dx + \frac{C}{(r-\rho)} \left[ \frac{1}{(r-\rho)^3} \int_{C(\rho,r)}\vert \V_1 \vert^2 dx \right] \\
&\, \times \left[1 + \left( \frac{1}{(r-\rho)^3} \int_{B_r} \vert \V_2 \vert^q\, dx\right)^{4/q} + \left( \frac{1}{(r-\rho)^3} \int_{B_r} \vert \V_2 \vert^q dx\right)^{\frac{4}{q(1-\theta)}} \right].
\end{split}
\end{equation*}
We remark that as $0<1-\theta <1$ and as $q\theta=3$, we have $\frac{4}{q} \leq \frac{4}{q(1-\theta)}= \frac{4}{q-3}$. Then, from the last estimate we obtain:
\begin{equation*}
\begin{split}
\int_{B_\rho} \vert \vec{\nabla} \otimes \U \vert^2 dx \leq\, \frac{1}{2} \int_{B_r} \vert \vec{\nabla} \otimes \U \vert^2 dx+\frac{C}{(r-\rho)} \left[ \frac{1}{(r-\rho)^3} \int_{C(\rho,r)}\vert \V_1 \vert^2 dx \right] \, \left[ 1 + \left( \frac{1}{(r-\rho)^3} \int_{B_r} \V_2 \vert^q \, dx\right)^{\frac{4}{q-3}}\right].
\end{split}
\end{equation*}
Finally, we use a well-known iterative argument, see for instance \cite{Liouville1} and \cite{Giaquinta}, to obtain the Cacciopoli type estimate stated in Proposition \ref{lemaTehc-Ser}, which is now proven. \finpv
\section{Proofs of the results}\label{Sec3:Proofs}
\subsection{Proof of Theorem \ref{Th1}}
Let $(\U, P)$ be a smooth solution of the (NS) equations. We assume that $\U \in L^{p}(\Rt)$ with $3/2<p<3$. In the framework of Proposition \ref{lemaTehc-Ser}, we will set the vector fields $\V_1$ and $\V_2$ as follows: we define first the vector field $\V$ by means of the velocity $\U$ as $\ds{\V= \frac{1}{-\Delta} (\vec{\nabla} \wedge \U),}$ where we have $\U= \vec{\nabla} \wedge \V$. Indeed, as we have $div(\U)=0$, then we can write
$$ \vec{\nabla}\wedge \V=\vec{\nabla}\wedge \left( \vec{\nabla}\wedge \left( \frac{1}{-\Delta} \U \right) \right)=\vec{\nabla} \left( div \left( \frac{1}{-\Delta} \U \right)\right) - \Delta \left( \frac{1}{-\Delta} \U \right) =\U.$$
Now, for all $x \in \Rt$ we set the vector fields $\V_1(x)= \V(x)$ and $\V_2(x)= \V(x)$. Then, for all $3<q<+\infty$ and for all $R>1$, by Proposition \ref{lemaTehc-Ser} we have the estimate:
\begin{equation}\label{eq03}
\int_{B_{R/2}} \vert \vec{\nabla} \otimes \U(x) \vert^2 dx \leq \frac{C_q}{R} \left( \frac{1}{R^3} \int_{\mathcal{C}(R/2,R)} \vert \V(x) \vert^2 dx \right) \left( 1+ \left(\frac{1}{R^3} \int_{B_R} \vert \V(x) \vert^{q}dx \right)^{\frac{4}{q-3}} \right).
\end{equation}
We must study now the term on the right-hand side. We recall that we have $\ds{\V= \frac{1}{-\Delta} (\vec{\nabla} \wedge \U),}$ and moreover, we have $\U \in L^{p}(\Rt)$ with $3/2<p<3$. Then, we get $\V \in L^{3p/(3-p)}(\Rt)$ with $3p/(3-p)>3$. Indeed, we write $\ds{\V= \frac{1}{\sqrt{-\Delta}} (\frac{1}{\sqrt{-\Delta}} (\vec{\nabla} \wedge \U))}$, and then, by the properties of the Riesz potential $\frac{1}{\sqrt{-\Delta}}$, as well by the properties of the Riesz transforms $\frac{\partial_i}{\sqrt{-\Delta}}$, we can write:
$$ \Vert \V \Vert_{L^{3p/(3-p)}} \leq c \left\Vert \frac{1}{\sqrt{-\Delta}} (\frac{1}{\sqrt{-\Delta}} (\vec{\nabla} \wedge \U)) \right\Vert_{L^{3p/(3-p)}} \leq c \left\Vert \frac{1}{\sqrt{-\Delta}} (\vec{\nabla} \wedge \U) \right\Vert_{L^p} \leq c \Vert \U \Vert_{L^p}. $$
Moreover, as $3/2<p<3$ we have $3p/(3-p)>3$. Thus, we set the parameter $q=3p/(3-p)$ and we have $\Vert \V \Vert_{L^q}<+\infty$.
We get back to the estimate (\ref{eq03}) to write:
\begin{equation*}
\begin{split}
\int_{B_{R/2}} \vert \vec{\nabla} \otimes \U(x) \vert^2 dx \leq & \, \frac{C_q}{R^4} \left( \int_{\mathcal{C}(R/2,R)} \vert \V(x) \vert^2 dx \right) \left( 1+ \left(\frac{1}{R^3} \int_{B_R} \vert \V(x) \vert^{q}dx \right)^{\frac{4}{q-3}} \right) \\
\leq & \, \frac{C_q}{R^4} \, R^{6(1/2-1/q)} \left( \int_{\mathcal{C}(R/2,R)} \vert \V(x)\vert^q dx \right)^{2/q} \left( 1+ \left(\frac{1}{R^3} \int_{B_R} \vert \V(x) \vert^{q}dx \right)^{\frac{4}{q-3}} \right)\\
\leq & \, C_q R^{-1-6/q} \left( \int_{\mathcal{C}(R/2,R)} \vert \V(x)\vert^q dx \right)^{2/q} \left( 1+ \left(\frac{1}{R^3} \int_{B_R} \vert \V(x) \vert^{q}dx \right)^{\frac{4}{q-3}} \right).
\end{split}
\end{equation*}
Now, we let $R\to +\infty$ and we obtain $\ds{\int_{\Rt} \vert \vec{\nabla} \otimes \U(x) \vert^2 dx =0}$. But, by the Sobolev embeddings we write $\Vert \U \Vert_{L^6}\leq c \Vert \vec{\nabla} \otimes \U \Vert_{L^2}$ and then we have the identity $\U=0$. Finally, by splitting the pressure $P$ as the well-known expression $\ds{P =\sum_{1 \leq i,j \leq 3} \mathcal{R}_{i}\mathcal{R}_{j}(U_i U_j)}$, we conclude that $P=0$. Theorem \ref{Th1} is now proven. \finpv
\subsection{Proof of Theorem \ref{Th2}}
We assume now that the smooth solution $(\U,P)$ verifies $\U \in L^{p} \cap \dot{H}^{-1}(\Rt)$ with $9/2 \leq p < +\infty$. As before, we define the vector field $\ds{\V= \frac{1}{-\Delta} (\vec{\nabla} \wedge \U)}$, and we set now the vector fields $\V_1$ and $\V_2$ as follows:
\begin{equation}\label{def-V1-V2}
\V_1(x)= \V(x) \quad \text{and} \quad \V_2(x)= \V(x)-\V(0).
\end{equation}
We remark that we have $\U = \vec{\nabla} \wedge \V_1$ and $\U= \vec{\nabla} \wedge \V_2$, and then, for $q=p$ and for all $R>1$, by Proposition \ref{lemaTehc-Ser} we can write:
\begin{equation}\label{Caccioppoli2}
\begin{split}
\int_{B_{R/2}} \vert \vec{\nabla} \otimes \U(x) \vert^2 dx
\leq & \, \frac{C_p}{R} \left(\frac{1}{R^3}\int_{\mathcal{C}(R/2,R) } \vert \V_1(x) \vert^2 dx \right) \left( 1+ \left(\frac{1}{R^3} \int_{B_R} \vert \V_2(x) \vert^{p}dx
\right)^{\frac{4}{p-3}} \right) \\
\leq & \, frac{C_p}{R^4} \left(\int_{\mathcal{C}(R/2,R)} \vert \V_1(x) \vert^2 dx \right) \left( 1+ \left(\frac{1}{R^3} \int_{B_R} \vert \V_2(x) \vert^{p}dx
\right)^{\frac{4}{p-3}} \right) \\
\leq & \, C_p \left( \int_{\mathcal{C}(R/2,R)} \vert \V_1(x) \vert^2 dx \right)\, \left(\frac{1}{R^4}+ \underbrace{\frac{1}{R^4} \left(\frac{1}{R^3} \int_{B_R} \vert \V_2(x) \vert^{p}dx
\right)^{\frac{4}{p-3}}}_{(I(R))} \right).
\end{split}
\end{equation}
We must estimate the term $I(R)$. As we have $\U \in L^p(\Rt)$, and moreover, by the continuous embedding $L^p(\Rt) \subset \dot{B}^{-3/p}_{\infty, \infty}(\Rt)$, we obtain $\U \in \dot{B}^{-3/p}_{\infty, \infty}(\Rt)$. Then, as $\ds{\V= \frac{1}{-\Delta} (\vec{\nabla} \wedge \U)}$ then we get $\V \in \dot{B}^{1-3/p}_{\infty, \infty}(\Rt)$. Moreover, as we have $9/2 \leq p < +\infty$, then we get $\frac{1}{3}\leq 1-\frac{3}{p}<1$, where the Besov space $\dot{B}^{1-3/p}_{\infty, \infty}(\Rt) $ is defined in the formula (\ref{besov-pos}). By this formula, for all $R>1$, we can write $$\ds{\sup_{x \in B_R} \frac{\vert \V(x)-\V(0)\vert}{\vert x \vert^{1-\frac{3}{p}}}\leq \Vert \V \Vert_{\dot{B}^{1-\frac{3}{p}}_{\infty,\infty}}},$$ and by recalling that the vector field $\V_2$ is defined in the second identity in (\ref{def-V1-V2}), then we obtain:
\[\sup_{\vert x \vert <R} \frac{\vert \V_2(x)\vert}{\vert x \vert^{1-\frac{3}{p}}}\leq \Vert \V \Vert_{\dot{B}^{1-\frac{3}{p}}_{\infty,\infty}}. \]
Thereafter, for all $x \in B_R$ we have:
\[ \vert \V_2(x)\vert \leq \Vert \V \Vert_{\dot{B}^{1-\frac{3}{p}}_{\infty,\infty}} \, \vert x \vert^{1-\frac{3}{p}} \leq \Vert \V \Vert_{\dot{B}^{1-\frac{3}{p}}_{\infty,\infty}} \,R^{1-\frac{3}{p}}. \]
We thus have:
\begin{equation*}
\begin{split}
I(R) \leq & \, C\, \frac{1}{R^4} \left(\frac{1}{R^3} \int_{\vert x \vert <R} \vert \V_2(x) \vert^{p}dx \right)^{\frac{4}{p-3}} \leq \Vert \V \Vert_{\dot{B}^{1-\frac{3}{p}}_{\infty,\infty}} \, \frac{1}{R^4} \left( \left( \frac{1}{R^3} \int_{\vert x \vert <R} dx \right) R^{p(1-\frac{3}{p})} \right)^{\frac{4}{p-3}} \\
\leq & \, C÷, \Vert \V \Vert_{\dot{B}^{1-\frac{3}{p}}_{\infty,\infty}} \, \frac{1}{R^4}\left( R^{p-3} \right)^{\frac{4}{p-3}} \leq c \Vert \V \Vert_{\dot{B}^{1-\frac{3}{p}}_{\infty,\infty}} \leq c \Vert \U \Vert_{\dot{B}^{-3/p}_{\infty, \infty}} \leq c \Vert \U \Vert_{L^p}.
\end{split}
\end{equation*}
We get back to (\ref{Caccioppoli2}) to write
$$ \int_{B_{R/2}} \vert \vec{\nabla} \otimes \U(x) \vert^2 dx \leq C_p \left( \int_{\mathcal{C}(R/2,R)} \vert \V_1(x) \vert^2 dx \right) \Vert \U \Vert_{L^p}.$$
We recall that by the first identity in formula (\ref{def-V1-V2}) we have $\V_1=\V$ where $\ds{\V= \frac{1}{-\Delta} (\vec{\nabla} \wedge \U)}$. Moreover, as we have $\U \in \dot{H}^{-1}(\Rt)$ then we obtain $\V \in L^{2}(\Rt)$. Consequently, by letting $R \to +\infty$ we have $\Vert \vec{\nabla} \otimes \U \Vert^{2}_{L^2}=0$. By proceeding as in the end of the proof of Theorem \ref{Th1} we have $\U=0$ and $P=0$. Theorem \ref{Th2} is proven. \finpv
\subsection{Proof of Corollary \ref{Col}}
Let $\U \in \dot{H}^{1}(\Rt)$ be a weak solution of the (NS) equations. First we remark that as $\U \in \dot{H}^{1}(\Rt)$, by the Sobolev embeddings we have $\U \in L^6(\Rt)$, and consequently, $\U \in L^{3}_{loc}(\Rt)$. Then, by Theorem $X.1,1$ at the page 658 in \cite{Galdi} we have $\U \in \mathcal{C}^{\infty}(\Rt)$ and $P \in \mathcal{C}^{\infty}(\Rt)$.
Now, we shall prove that $\U \in \dot{H}^{-1}(\Rt)$. For $\rho>0$ fixed, we write
\[\Vert \U \Vert^{2}_{\dot{H}^{-1}}= \int_{\Rt} \frac{1}{\vert \xi \vert^2} \left\vert \widehat{\U} (\xi)\right\vert^2 \, d\xi = \int_{\vert \xi \vert < \rho} \frac{1}{\vert \xi \vert^2} \left\vert \widehat{\U} (\xi)\right\vert^2 \, d\xi + \int_{\vert \xi \vert \geq \rho} \frac{1}{\vert \xi \vert^2} \left\vert \widehat{\U} (\xi)\right\vert^2 \, d\xi.
\]
In order to control the first term on the right-hand side, by the H\"older inequalities (with $1=2/p+2/r$) we write
\[ \int_{\vert \xi \vert < \rho} \frac{1}{\vert \xi \vert^2} \left\vert \widehat{\U} (\xi)\right\vert^2 \, d\xi \leq \left( \int_{\vert \xi \vert < \rho} \frac{1}{\vert \xi \vert^{p}}\, d\xi \right)^{2/p} \left( \int_{\vert \xi \vert < \rho} \left\vert \widehat{\U} (\xi)\right\vert^{r}\, d \xi\right)^{2/r}.\]
We set $p<3$ and the first term in the right ride converges. Moreover, the hypothesis $\widehat{\U} \in L^{r}_{loc}(\Rt)$, with $r>6$, allows us to conclude that the second term on the right-hand side also converges.
To control the second term on the right-hand side of the last identity, we just write
\[ \int_{\vert \xi \vert \geq \rho} \frac{1}{\vert \xi \vert^2} \left\vert \widehat{\U} (\xi)\right\vert^2 \, d\xi = \int_{\vert \xi \vert \geq \rho} \frac{1}{\vert \xi \vert^4} \vert \xi \vert^2 \left\vert \widehat{\U} (\xi)\right\vert^2 \, d\xi \leq \frac{1}{\rho^4} \Vert \U \Vert^{2}_{\dot{H}^1}<+\infty.\]
We this get $\U \in L^6(\Rt) \cap \dot{H}^{-1}(\Rt)$ and by Theorem \ref{Th2} we have the identities $\U=0$ and $P=0$. \finpv
\begin{appendices}
\section{Appendix}\label{AppendixA}
Let $f \in \mathcal{C}^3(\Rt)$ be a harmonic function. We define $\U= \vec{\nabla} f $ and $\ds{P = - \frac{1}{2} \vert \vec{\nabla} f \vert^2}$, and we will prove that $(\U,P)$ is a solution of the (NS) equations. Indeed, as $\Delta f =0$ we directly have $-\Delta \U=0$. On the other hand, by well-known rules of the vector calculus we have the identity
\[(\U \cdot \vec{\nabla}) \U= \vec{\nabla} \left( \frac{1}{2}\vert \U \vert^2 \right)+ \left(\vec{\nabla} \wedge \U\right)\wedge \U, \]
but, as $\U= \vec{\nabla} f $ then we have $\ds{\vec{\nabla} \wedge \U=0}$, and we can write
\[ (\U \cdot \vec{\nabla}) \U= \frac{1}{2} \vec{\nabla}\left(\vert \U \vert^2\right) =-\vec{\nabla} P.\]
Hence, $(\U,P)$ solves the equation $\ds{-\Delta \U + (\U \cdot \vec{\nabla}) \U +\vec{\nabla} P = 0}$. Moreover, always as by the fact that $f$ is an harmonic function we have $\ds{div(\U)= div (\vec{\nabla} f)=\Delta f =0}$.
\end{appendices}
\section*{Availability of data and materials}
Data sharing not applicable to this article as no datasets were generated
or analyzed during the current study.
\end{document}
|
\begin{document}
\begin{frontmatter}
\title{Conditional positive definiteness as a bridge \\ between $k$--hyponormality and $n$--contractivity}
\author{Chafiq Benhida}
\address{UFR de Math\'{e}matiques, Universit\'{e} des Sciences et
Technologies de Lille, F-59655 \newline Villeneuve-d'Ascq Cedex, France}
\ead{[email protected]}
\author{Ra\'ul E. Curto}
\address{Department of Mathematics, University of Iowa, Iowa City, Iowa 52242-1419, USA}
\ead{[email protected]}
\ead[url]{http://www.math.uiowa.edu/\symbol{126}rcurto/}
\author{George R. Exner}
\address{Department of Mathematics, Bucknell University, Lewisburg, Pennsylvania 17837, USA}
\ead{[email protected]}
\begin{abstract}
For sequences $\alpha \equiv \{\alpha_n\}_{n=0}^{\infty}$ of positive real numbers, called weights, we study the weighted shift operators $W_{\alpha}$ having the property of moment infinite divisibility ($\mathcal{MID}$); that is, for any $p > 0$, the Schur power $W_{\alpha}^p$ is subnormal. \ We first prove that $W_{\alpha}$ is $\mathcal{MID}$ if and only if certain infinite matrices $\log M_{\gamma}(0)$ and $\log M_{\gamma}(1)$ are conditionally positive definite (CPD). \ Here $\gamma$ is the sequence of moments associated with $\alpha$, $M_{\gamma}(0),M_{\gamma}(1)$ are the canonical Hankel matrices whose positive semi-definiteness determines the subnormality of $W_{\alpha}$, and $\log$ is calculated entry-wise (i.e., in the sense of Schur or Hadamard). \ Next, we use conditional positive definiteness to establish a new bridge between $k$--hyponormality and $n$--contractivity, which sheds significant new light on how the two well known staircases from hyponormality to subnormality interact. \ As a consequence, we prove that a contractive weighted shift $W_{\alpha}$ is $\mathcal{MID}$ if and only if for all $p>0$, $M_{\gamma}^p(0)$ and
$M_{\gamma}^p(1)$ are CPD. \
\end{abstract}
\begin{keyword}
Weighted shift, Subnormal, Moment infinitely divisible, Conditionally positive definite, Completely monotone
\textit{2010 Mathematics Subject Classification} \ Primary 47B20, 47B37; Secondary 44A60.
\end{keyword}
\end{frontmatter}
\tableofcontents
\setcounter{tocdepth}{2}
\section{Introduction and Statement of Main Results} \label{Intro}
Let $\mathcal{H}$ denote a separable, complex Hilbert space and $\mathcal{L}(\mathcal{H})$ be the algebra of bounded linear operators on $\mathcal{H}$. \ Recall that an operator $T$ is \textit{subnormal} if it is the restriction to a (closed) invariant subspace of a normal operator, and \textit{hyponormal} if $T^* T \geq T T^*$. \ A unilateral weighted shift $W_{\alpha}$ acting on the classical sequence space $\ell^2(\mathbb{N}_0)$ is called \textit{moment infinitely divisible} (in symbols, $W_{\alpha} \in \mathcal{MID}$) if all Schur powers $W_{\alpha}^p$ are subnormal. \ Thus, the class $\mathcal{MID}$ consists of all subnormal weighted shifts $W_{\alpha}$ with moment sequence $\gamma$ such that $\gamma^p$ is interpolated by a positive Borel probability measure $\mu^{(p)}$, for every $p>0$; $\mu^{(p)}$ is the so-called \textit{Berger measure} of the subnormal weighted shift $W_{\alpha}^p$. \ This is equivalent to the (Schur) infinite divisibility of the two Hankel \textit{moment matrices} $M_{\gamma}(0):=\left(\gamma_{i+j}\right)_{i,j=0}^{\infty}$ and $M_{\gamma}(1):=\left(\gamma_{i+j+1}\right)_{i,j=0}^{\infty}$, where $\gamma$ is the sequence of moments associated with the weight sequence $\alpha$. \ Since all entries in these two matrices are positive, and since the matrices are Hermitian, their infinite divisibility is equivalent to the \textit{conditional positive definiteness} of their (Schur) logarithms $\log M_{\gamma}(0)$ and $\log M_{\gamma}(1)$.
As a consequence, we can prove that $W_{\alpha}$ is $\mathcal{MID}$ if and only if both $\log M_{\gamma}(0)$ and $\log M_{\gamma}(1)$ are CPD. \ This leads to a new characterization of $W_{\alpha} \in \mathcal{MID}$ in terms of the sequence $\delta_n:=\log \left( \ddfrac{\gamma_n \gamma_{n+2}}{\gamma_{n+1}^2}\right) \quad (n \ge 0)$. \
It is well known that subnormality is a much stronger condition than hyponormality. \ For a contractive weighted shift $W_{\alpha}$, the former requires a Berger measure $\mu$ supported in $[0,1]$ that interpolates $\gamma$; the latter requires $\alpha_0 \le \alpha_1 \le \alpha_2 \le \ldots$. \ There are two well-known staircases connecting hyponormality and subnormality, and they correspond to two well-known tests for subnormality. \ The first test involves the Bram-Halmos criterion for subnormality, which requires that $W_{\alpha}$ be $k$--hyponormal for every $k \ge 1$; the second test has to do with the Agler-Embry approach to subnormality, which requires the $n$--contractivity of $W_{\alpha}$ for every $n \ge 1$. \
In this paper we consider the role of conditional positive definiteness in establishing a bridge between the two above mentioned staircases. \ Concretely, as a first step we will see that $k$--hyponormality implies $2k$--contractivity.
In \cite{EJP}, G.R. Exner, I.B. Jung and S.S. Park proved for general operators that $k$--hyponormality implies $2k$--contractivity. \ Implicit in their work was a significant identity, which we use as a point of departure. \ Given a moment sequence $\gamma$, a positive integer $k$, a nonnegative integer $\ell$, and the $(k+1) \times(k+1)$ (Hankel) compressed moment submatrix $M_{\gamma}(\ell,k):=\left( \gamma_{\ell+i+j} \right)_{i,j=0}^{k}$, we consider the expression
\begin{equation} \label{identity1}
Q_{\gamma}(\ell,k):=\bm{v}^{\ast} M_{\gamma}(\ell,k)\bm{v},
\end{equation}
where $\bm{v}$ is the column vector of length $k+1$ with $i$--th coordinate $(-1)^i \binom{k}{i}$; that is, $\bm{v}:=(1, -\binom{k}{1},
\binom{k}{2},-\binom{k}{3}, \ldots, -\binom{k}{k-1}, 1)$. \ Observe that $\sum_{i=0}^{k} v_i=(1-1)^{k}=0$. \ Experimental calculations using {\it Mathematica} \cite{Wol} easily reveal that (\ref{identity1}) becomes
\begin{eqnarray} \label{identity2}
Q_{\gamma}(\ell,k)&=&\gamma_{\ell}-\binom{2k}{1} \gamma_{\ell+1}+\binom{2k}{2} \gamma_{\ell+2}-\binom{2k}{3} \gamma_{\ell+3} + \ldots \nonumber \\
&&-\binom{2k}{2k-1}\gamma_{\ell+2k-1}+\gamma_{\ell+2k} \nonumber \\
&&= \sum_{i=0}^{2k} (-1)^i \binom{2k}{i}\gamma_{\ell+i}.
\end{eqnarray}
Once the proposed form of $Q_{\gamma}(\ell,k)$ is discovered, a proof of (\ref{identity2}) by induction is straightforward.
Positivity of the expression on the right-hand side of (\ref{identity2}) for all $\ell$ is exactly what we need for $2k$--contractivity of $W_{\alpha}$. \ On the other hand, (\ref{identity1}) is one of the expressions used in the determination of CPD for the matrix $M_{\gamma}(\ell,k)$. \ Concretely, if $W_{\alpha}$ is $k$--hyponormal, then $M_{\gamma}(\ell,k) \ge 0$ for all $\ell \ge 0$. \ Then $M_{\gamma}(\ell,k)$ is CPD for all $\ell \ge 0$, and therefore $Q_{\gamma}(\ell,k) \ge 0$ for all $\ell \ge 0$. \ It follows that $\sum_{i=0}^{2k} (-1)^i \binom{2k}{i}\gamma_{\ell+i} \ge 0$ for all $\ell \ge 0$, and this means that $W_{\alpha}$ is $2k$--contractive. \ This is a special case of a more general result, involving $(k,2m)$--CPD matrices, which represents a very useful version of CPD, appropriately localized to keep track of the size of the matrix and the initial moment.
The above calculation is part of a much broader theoretical setting, involving properties of sequences such as complete monotonicity and hypercontractivity, along with their $\log$ analogs, e.g., $\log$ complete monotonicity. \ We discuss aspects of this theory in Section \ref{main}.
We are now ready to state the main results of this paper. \ The proofs will be given in Section \ref{main}. \ For notation and terminology, we refer the reader to Section \ref{prelim}, where we also briefly review some topics from unilateral weighted shifts and matrix theory that will be needed for Section \ref{main}. \
\begin{theorem} \label{thm:equivcondtIDcontraction}
Suppose $W_\alpha$ is a contractive weighted shift. \ Then the following statements are equivalent.
\begin{enumerate}
\item $W_\alpha$ is moment infinitely divisible ($\mathcal{MID}$).
\item $\log M_{\gamma}(0)$ and $\log M_{\gamma}(1)$ are CPD.
\item For every $p>0$, $M_{\gamma}^p(0)$ and $M_{\gamma}^p(1)$ are positive definite.
\item For every $p>0$, $M_{\gamma}^p(0)$ and $M_{\gamma}^p(1)$ are CPD.
\item The moment sequence $\gamma$ is log completely monotone.
\item The weight sequence $\alpha$ is log completely alternating.
\end{enumerate}
\end{theorem}
\begin{proposition} \label{prop:getAglerfrom1}
Let $W_{\alpha}$ be a unilateral weighted shift, $k \ge 1$, and $0 \le m \le k$. \ Assume that $W_{\alpha}$ is $(k,2m)$--CPD. \ Then $W_{\alpha}$ is $2m$--contractive, $(2m+2)$--contractive, \ldots, $2k$--contractive.
\end{proposition}
\begin{theorem}
Let $W_{\alpha}$ be a contractive weighted shift whose weight sequence $\alpha$ has a limit (for example, if $W_{\alpha}$ is hyponormal), and fix $m \in \mathbb{N}$. \ Assume that for all $k \geq m$, $W$ is $(k,2m)$--PD or $(k,2m)$--CPD. \ Then $W$ is subnormal.
\end{theorem}
As a consequence, we obtain a result including what we believe is a new sufficient condition for subnormality.
\begin{proposition} \label{propnew2}
Let $W_{\alpha}$ be a weighted shift with moment sequence $\gamma = (\gamma_n)_{n=0}^\infty$. \ The following statements are equivalent. \newline
(i) $W_{\alpha}$ is $\mathcal{MID}$. \newline
(ii) The sequence $(\delta_n)_{n=0}^\infty$ with $\delta_n = \ln\left(\frac{\gamma_n \gamma_{n+2}}{\gamma_{n+1}^2}\right)$ is a Stieltjes moment sequence. \newline
(iii) The weighted shift with moments $\left(\frac{\delta_n}{\delta_0}\right)$ is subnormal.
\end{proposition}
We now focus on rank-one perturbations of the Agler shifts (see Subsection \ref{Aglershifts} for the definition); our study reveals some remarkable cutoffs for $(k,2m)$--positive definiteness, and for the related weights-squared sequence to be $m$--alternating (for the terminology, see Definition \ref{def:kmpositive}, and page \pageref{defn}, resp.; in particular, $(k,0)$--positive definiteness corresponds to $k$--hyponormality).
\begin{theorem} \label{thm14A}
Let $A_j(x)$ be the perturbation of the Agler shift $A_j$ in which the zeroth weight $(\alpha^{(j)})_0:= \sqrt{\frac{1}{j}}$ is replaced by $(\alpha^{(j)})_0(x) := \sqrt{\frac{x}{j}}$, and let $k \in \mathbb{N}$ and $m \in \{0,1,\ldots,k\}$. \newline
(i) $A_j(x)$ is $(k,2m)$--PD if and only if $x \leq p(j,k,m)$, where
\begin{equation}
p(j,k,m) \!=\! \ddfrac{(k+1-m)(j+k+m-1)}{k^2 + j k + 2m - j m - m^2}\!=\!\ddfrac{(j+k+m)(k-m)+2m+j-1}{(j+k+m)(k-m)+2m}.
\end{equation}
(ii) \ $A_j(x)$ is $(k,2m)$--PD if and only if $A_j(x)$ is $((j+k+m)(k-m)+2m)$--contractive.
\end{theorem}
\begin{proposition}
Let $A_j(x)$ be the zeroth weight perturbation of the Agler shift $A_j$ with weight sequence $\alpha^{(j)}$ as in Theorem \ref{thm14A}. \ Then the weights-squared of $A_j(x)$ are $m$--alternating if and only if
\begin{equation} \label{eq:naltAjofx1}
x \leq 1 + \frac{(j-1) m!}{\prod_{i=1}^{m} (j+i)}.
\end{equation}
If we take $j=2$, which corresponds to the Bergman shift, the weights-squared of $A_2(x)$ are $m$--alternating if and only if $A_2(x)$ is $\frac{(m+1)(m+2)}{2}$--contractive.
\end{proposition}
\section{Notation and Preliminaries} \label{prelim}
\subsection{Unilateral weighted shifts}
To set the notation for weighted shifts, let $\mathbb{N}_0 := \{0, 1, \ldots\}$ and let $\ell^2$ denote the classical Hilbert space $\ell^2(\mathbb{N}_0)$ with canonical orthonormal basis $e_0, e_1, \ldots$ (note that we begin indexing at zero). \ Let $\alpha: \alpha_0, \alpha_1, \ldots$ be a (bounded) positive \textit{weight sequence} and $W_\alpha$ the weighted shift, defined by linearity and $W_\alpha e_j := \alpha_j e_{j+1} \;\; (j \ge 0)$. \ (Weighted shifts can be defined for any bounded sequence $\alpha$; however, for all questions of interest to us, without loss of generality we can assume (as we do) that $\alpha$ is positive.) \ When $\alpha$ is the constant weight sequence $1,1,\ldots$, the resulting (un-weighted) shift is the classical unilateral shift $U_+e_{j}:=e_{j+1} \; (j \ge 0)$. \ The \textit{moments} $\gamma = (\gamma_n)_{n=0}^\infty$ of the shift are defined by $\gamma_0 := 1$ and $\gamma_n := \prod_{j=0}^{n-1} \alpha_j^2$ for $n \geq 1$. \ From \cite[III.8.16]{Con} and \cite{GW}, a weighted shift $W_\alpha$ is subnormal if and only if it has a \textit{Berger measure}, meaning a probability measure $\mu$ supported on $[0, \|W_\alpha\|^2]$ such that
$$
\gamma_n = \int_0^{\|W_\alpha\|^2} t^n d \mu(t), \hspace{.2in} n = 0, 1, \ldots .
$$
Applying the Cauchy-Schwarz inequality in $L^2(\mu)$ to the monomials $t^{n/2}$ and $t^{(n+2)/2}$ yields $\gamma_{n+1}^2 \le \gamma_n \gamma_{n+2} \; (n \ge 0)$, and hence $\alpha_n^2 \le \alpha_{n+1}^2 \; (n \ge 0)$. \ Recall that an operator $T$ is hyponormal if $T^*T-TT^* \ge 0$, and one computes easily that for a weighted shift $W_{\alpha}$ this is exactly $\alpha_n^2 \le \alpha_{n+1}^2$ for all $n \ge 0$.
The canonical polar decomposition of $W_{\alpha}$ is $U_+ P_{\alpha}$, where $P_{\alpha}$ denotes the diagonal operators with diagonal entries $\alpha_0,\alpha_1, \ldots$ . \ The {\it Aluthge transform} \cite{Alu} of $W_{\alpha}$ is given by $AT(W_{\alpha}):=\sqrt{P_{\alpha}}U_+ \sqrt{P_{\alpha}}$, and this is the weighted shift with weight sequence $\sqrt{\alpha_0 \alpha_1},\sqrt{\alpha_1 \alpha_2},\sqrt{\alpha_2 \alpha_3},\ldots$ . \ It is easy to see that $AT$ preserves hyponormality, but whether it preserves subnormality is a nontrivial problem, addressed in detail in \cite{Ex2, CuEx, BCE,BCESC}. \ A sufficient condition for subnormality, pointed out in \cite{CuEx}, is the subnormality of $W_{\sqrt{\alpha}}$.
\subsection{Agler shifts} \label{Aglershifts}
We briefly recall a class of subnormal unilateral weighted shifts with Berger measures that can be easily computed. \ The {\it Agler shifts} $A_j$, $j = 1, 2, \ldots$, are those with weight sequence $\sqrt{\frac{n+1}{n+j}}$, $n = 0, 1, \ldots$ . \ (These were used in \cite{Ag} as part of a model theory for hypercontractions.) \ Since $A_1$ is the unilateral shift, its Berger measure is $\delta_{1}$ (the Dirac point mass at $\{1\}$); for $j \ge 2$ the Berger measure of $A_j$ is $d \mu(t)=(j-1)(1-t)^{j-2}dt$ on $[0,1]$. \ The Agler shifts appear naturally as the weighted shifts associated with the rows (and columns) of the weight diagram for the Drury-Arveson $2$--variable weighted shift \cite[Pages 29--30]{CRC}.
In \cite{Ex2}, G.R. Exner proved that for $j = 2, 3, \ldots$, and $p > 0$, the (Schur) $p$--th power of $A_j$ is subnormal, as is any $m$--th iterated Aluthge transform of $A_j$. \ The proof (which uses monotone function theory) offers no information about the Berger measure of the resulting shift; however, it brings to the fore the significance of complete monotonicity in the study of $\mathcal{MID}$ shifts, discussed in the next subsection.
\subsection{$\mathcal{MID}$ shifts}
It is useful to note, in considering $\mathcal{MID}$ weighted shifts, that raising every weight to the $p$--th power is equivalent to raising every moment to the $p$--th power. \ We refer the reader to \cite{BCE,BCESC} for an initial study of moment infinite divisibility; this paper constitutes a continuation of that study.
A companion to the class of Agler shifts is the class of {\it homographic shifts}, that is, those weighted shifts denoted $S(a,b,c,d)$ (where $a, b, c, d >0$ and with $a d > b c$), with weights $\sqrt{\frac{a n + b}{c n + d}}$. \ These shifts were defined and studied in \cite{CPY}, together with certain subshifts of such shifts, and their subnormality established. \ Observe as well that if we throw away a finite number of terms at the beginning of a completely alternating (or log completely alternating) sequence, what remains is still in the original class. \ Therefore, some of the results to follow may be generalized easily to restrictions of shifts to the canonical invariant subspaces of finite co-dimension.
\begin{lemma}(\cite{Ex2}) \! \! The Agler shifts are all $\mathcal{MID}$, as are the contractive shifts $S(a,b,c,d)$ (i.e., $a, b, c, d >0$, $a d > b c$ and $a \le c$), and their subshifts.
\end{lemma}
By definition, a weighted shift $W_{\alpha}$ is completely hyperexpansive when the inequalities for $n$--contractivity in (\ref{cond100}) are reversed; that is,
$$
\sum_{i=0}^n (-1)^i \binom{n}{i} \gamma_{k+i} \le 0, \qquad k = 0, 1, \ldots ,
$$
for all $n \ge 1$. \ For instance, the Dirichlet shift is completely hyperexpansive. \ We recall that a completely hyperexpansive weighted shift $W_{\alpha}$ gives rise to a subnormal weighted shift by forming a new weight sequence $\delta$ where $\delta_j := \frac{1}{\alpha_j}$ for all $j \ge 0$; further, one cannot necessarily begin with a subnormal shift, and, by taking reciprocals of the weights, generate a completely hyperexpansive shift. \ (See the discussion after \cite[Proposition 6]{At}.) \ Nevertheless, the following result holds.
\begin{lemma} (\cite[Corollary 4.1]{BCE}) \label{lem22} \ Let $W_\alpha$ be a completely hyperexpansive \linebreak weighted shift with positive weight sequence $(\alpha_n)_{n=0}^\infty$. \ Then the weighted shift with weight sequence $(\frac{1}{\alpha_n})_{n=0}^\infty$ is not only subnormal but is $\mathcal{MID}$.
\end{lemma}
Using the definition of $\mathcal{MID}$, and Schur products, one may show that if $W_\alpha$ is $\mathcal{MID}$ then so is $AT(W_\alpha)$. \ We record this and related results in the following.
\begin{lemma}
(i) (\cite[Corollary 3.6]{BCE}) \ If a contractive weighted shift $W_\alpha$ is $\mathcal{MID}$ then so is $AT(W_\alpha)$. \newline
(ii) (\cite[Theorem 4.13]{BCESC}) \ $AT$ maps the class $\mathcal{MID}$ bijectively onto itself. \newline
(iii) (\cite[Theorem 4.4]{BCESC}) \ Suppose that $W_{\alpha}$ is a contractive weighted shift whose weights $\alpha_j$ approach a limit (as $j \rightarrow \infty$). \ Then $AT(W_{\alpha})$ is $\mathcal{MID}$ if and only if $W_{\alpha}$ is.
\end{lemma}
\subsection{$k$--hyponormality} \label{khypon}
For the reader's convenience, we sketch the $k$--hyponormality approach to subnormality, and give a brief version of this background (see \cite{Cu1} and \cite{CF1} for a full discussion and some of the beginnings of this substantial study). \ It is the Bram-Halmos characterization of subnormality (see \cite{Br}) that an operator $T$ is subnormal if and only if, for every $k = 1, 2, \ldots$, a certain $(k+1) \times (k+1)$ operator matrix $A_n(T)$ is positive. \ For $k \ge 1$, an operator is $k$--hyponormal if this positivity condition holds for $k$. \ For weighted shifts, it is well-known from \cite[Theorem 4]{Cu} that $k$--hyponormality reduces to the positivity, for each $n$, of the $(k+1) \times (k+1)$ Hankel moment matrix $M_{\gamma}(\ell,k)$, where
$$
M_{\gamma}(\ell,k) = \left(
\begin{array}{cccc}
\gamma _{n} & \gamma _{n+1} & \cdots & \gamma _{n+k} \\
\gamma _{n+1} & \gamma _{n+2} & \cdots & \gamma _{n+k+1} \\
\vdots & \vdots & \ddots & \vdots \\
\gamma _{n+k} & \gamma _{n+k+1} & \cdots & \gamma _{n+2k}
\end{array}
\right).
$$
\subsection{$n$--contractivity}
Another approach to subnormality, this time for a contractive operator $T$ (that is, $\|T \| \leq 1$), is the Agler-Embry characterization based on the notion of $n$--contractivity. \ For $n \ge 1$, an operator is $n$\textit{--contractive} if
\begin{equation} \label{eq11}
A_n(T):=\sum_{i=0}^n (-1)^i \binom{n}{i} {T^*}^i T^i \geq 0 \quad \; \textrm{(cf. \cite{Ag})}.
\end{equation}
A contractive operator is subnormal if and only if it is $n$--contractive for all positive integers $n$ (cf. \cite{Ag}). \ It is well-known, and follows easily from ${W_\alpha^*}^i W_\alpha^i$ being diagonal, that for a weighted shift it suffices to test this condition on basis vectors and that a weighed shift is $n$--contractive if and only if
\begin{equation} \label{cond100}
\sum_{i=0}^n (-1)^i \binom{n}{i} \gamma_{k+i} \geq 0, \qquad k = 0, 1, \ldots.
\end{equation}
Given a sequence $a = (a_j)_{j=0}^\infty$, let $\nabla$ (the forward difference operator) be defined by
$$
(\nabla a)_j := a_j - a_{j+1},
$$
and the iterated forward difference operators $\nabla^{n}$ by
$$
\nabla^{0} a := a \; \; \textrm{ and } \; \; \nabla^{n} := \nabla (\nabla^{n-1}),
$$
for $n \geq 1$. \ For instance,
\begin{equation} \label{nabla2}
(\nabla^2a)_j=a_j-2a_{j+1}+a_{j+2} \; \; (j \ge 0).
\end{equation}
To ease the notation in some settings, set, for any $n \geq 1$ and $k \geq 0$,
$$
T_a(n,k) := (\nabla^{n} a)_k = \sum_{i=0}^n (-1)^i \binom{n}{i} a_{i+k}
$$
(where we write simply $T(n,k)$ if no confusion as to the sequence $a$ will arise). \ With a slight abuse of language say that a sequence $a$ is $n$--contractive if $T_a(n,k) \geq 0$ for all $k = 0, 1, \ldots$ . \
There is alternative language for these and related notions: a sequence $a$ is $n$\textit{--monotone} if $T_a(n,k) = (\nabla^{n} a)_k \geq 0$ for all $k = 0, 1, \ldots$, $n$\textit{--hypermonotone} if it is $j$--monotone for all $j = 1, \ldots, n$, and \textit{completely monotone} if it is $n$--monotone for all $n = 1, 2, \ldots$. \ \label{defn} Similarly, a sequence is $n$\textit{--alternating} if $T_a(n,k) = (\nabla^{n} a)_k \leq 0$ for all $k = 0, 1, \ldots$, $n$\textit{--hyperalternating} if it is $j$--alternating for all $j = 1, \ldots, n$, and \textit{completely alternating} if it is $n$--alternating for all $n = 1, 2, \ldots$. \ A sequence is $n$\textit{--log monotone} (respectively, \textit{completely log monotone}, $n$\textit{--log alternating}, \textit{completely log alternating}) if the sequence $(\ln a_j)$ is $n$--monotone (respectively, completely monotone, $n$--alternating, completely alternating). (Notice that we define log monotonicity using $n$--monotonicity, so we allow for the sequence to possibly be negative, as in the case of $\ln a$, where $a$ is the sequence of moments of a contractive weighted shift. \ Thus, we allow slightly more generality than in the definition given in \cite[Definition 6.1]{BCR}.)
\subsection{$n$--hypermonotone and $n$--hyperalternating weighted shifts}
\begin{proposition}\label{nXvsnhyperX}
Suppose $n \in \mathbb{N}$ and $(a_k)_{k=0}^\infty$ is a sequence of real numbers, and in addition, $\lim_{k \rightarrow \infty} a_k$ exists.
\begin{enumerate}
\item If $(a_k)_{k=0}^\infty$ is $n$--monotone, it is $n$--hypermonotone.
\item If $(a_k)_{k=0}^\infty$ is $n$--log monotone, it is $n$--log hypermonotone.
\item If $(a_k)_{k=0}^\infty$ is $n$--alternating, it is $n$--hyperalternating.
\item If $(a_k)_{k=0}^\infty$ is $n$--log alternating, it is $n$--log hyperalternating.
\end{enumerate}
\end{proposition}
\begin{proof}
We prove only the first assertion as the other proofs are similar. \ If $n = 1$ there is nothing to prove, so assume $n > 1$. \ Suppose for a contradiction that $(a_k)_{k=0}^\infty$ is $n$--monotone but not $n$--hypermonotone, and let $m$ be the least integer, $1 < m \leq n$, such that the sequence is $m$--monotone but not $(m-1)$--monotone. \ For each $j$ and $k$, recall that $T(j,k)$ is defined by
$$
T(j,k) = \sum_{i=0}^j (-1)^i \binom{j}{i} a_{k+i}.
$$
The positivity of $T(j,k)$ for all $k$ yields $j$--monotonicity. \ Further, it is routine that, for any $j$ and $k$,
\begin{equation} \label{eq:recAnk}
T(j+1,k) = T(j,k) - T(j,k+1).
\end{equation}
Suppose $k_0$ to be the least $k$ such that
$$T(m-1,k_0) = \delta < 0.$$
Then
$$0 \leq T(m,k_0) = T(m-1,k_0) - T(m-1,k_0+1)$$
forces
$$T(m-1,k_0+1) \leq \delta.$$
Repeating the argument, we have
\begin{equation} \label{ineq:negative}
T(m-1,k) \leq \delta < 0, \quad k \geq k_0.
\end{equation}
If $\lim_{k \rightarrow \infty} a_k = L$, then it is elementary that
$$
\lim_{k \rightarrow \infty} T(m-1,k) = \sum_{i=0}^{m-1} (-1)^i \binom{m-1}{i} L = 0,
$$
which is a contradiction of \eqref{ineq:negative}. \qed
\end{proof}
Observe that in light of the Dirichlet shift $D$ (that is, the standard $2$--isometry whose moment sequence is therefore $2$--monotone but not $1$--monotone), some assumption like existence of the limit of the weight sequence is required.
\begin{corollary} \label{contnmonoimpnhypermono}
Suppose $(\gamma_k)_{k=0}^\infty$ is the sequence of moments of a weighted shift. \ If the weighted shift is contractive, and $(\gamma_k)_{k=0}^\infty$ is $n$--monotone for some $n \in \mathbb{N}$, then $(\gamma_k)_{k=0}^\infty$ is $n$--hypermonotone. \ Alternatively, if the sequence $(\gamma_k)_{k=0}^\infty$ is both $1$--monotone and $n$--monotone for some $n \in \mathbb{N}$, then the sequence is $n$--hypermonotone.
\end{corollary}
\begin{proof}
In either case the sequence $(\gamma_k)$ is decreasing and bounded below by zero and so has a limit. \qed
\end{proof}
This transfers to the $n$--log-monotone and $n$--log-hypermonotone case, at least in situations of interest to us.
\begin{corollary} \label{cor:nmonoimpnhypermonoshifts}
Suppose $(\gamma_k)_{k=0}^\infty$ is the sequence of moments of a (bounded) \linebreak weighted shift $W_{\alpha}$. \ If $W_{\alpha}$ is contractive, and its weights have a non-zero limit (in particular, if $W_{\alpha}$ is hyponormal), then for any $n \in \mathbb{N}$, if $(\gamma_k)_{k=0}^\infty$ is $n$--log-monotone, then it is $n$--log-hypermonotone. \ Alternatively, if the shift is hyponormal or its weights have a non-zero limit, and if the sequence $(\gamma_k)_{k=0}^\infty$ is both $1$--log-monotone and $n$--log-monotone for some $n \in \mathbb{N}$, then the sequence is $n$--log-hypermonotone.
\end{corollary}
\begin{proof}
One applies the previous corollary to the sequence $(\log \gamma_k)$, and the only thing to check is that the given conditions yield that the resulting expressions $T(n,k)$ for the sequence of logs actually have limits, which is straightforward combining the terms using rules of logs. \qed
\end{proof}
We leave to the interested reader the appropriate versions of the corollaries for the $n$--alternating and $n$--log alternating case.
Let us record some results on product sequences. \ It is trivial that the Schur (term-wise) product of two $n$--log monotone sequences is again $n$--log monotone (and, in fact, the Schur (term-wise) product of two $n$--log alternating sequences is again $n$--log alternating). \ The good thing holds for $n$--monotone sequences as well.
\begin{lemma}
Suppose $(a_k)_{k=0}^\infty$ and $(b_k)_{k=0}^\infty$ are $n$--hypermonotone sequences. \linebreak \ Then $(a_k b_k)_{k=0}^\infty$ is $n$--hypermonotone. \ In particular, if they are moment sequences arising from a contractive weighted shift, and if each is both $1$--monotone and $n$--monotone, it follows that $(a_k b_k)_{k=0}^\infty$ is $n$--hypermonotone.
\end{lemma}
\begin{proof}
The first claim follows immediately from Leibniz's rule of differences: if $\nabla$ is the usual forward difference operator on sequences, then
\begin{equation} \label{eq:LeibnizRule}
\nabla^{n} (a_k b_k) = \sum_{j=0}^n \binom{n}{j} (\nabla^{n-j}a_{k+j})(\nabla^{j}b_k).
\end{equation}
Since the coefficients are positive and the differences appearing are non-negative, we have the result. \ The second claim follows because we have shown that, in the contractive case, $n$--hypermonotonicity and $n$--monotonicity coincide. \qed
\end{proof}
We end this section with an example where the complete alternating property is crucial in the proof of $\mathcal{MID}$.
\begin{example} (\cite[Example 3.5]{BCE}) \ The weighted shift with weights the sequence $(\alpha_n)$ defined by
$$\alpha_n = 1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n+1} - \ln (n+2), \hspace{.2in}n=0,1, \ldots,$$
or the shift with weights the square roots of these, is $\mathcal{MID}$ and subnormal. \ This is because this sequence is completely alternating (and it increases to the Euler constant $\gamma \approx 0.577\ldots $).
\end{example}
\subsection{Conditionally positive definite matrices}
Throughout this paper, and as is common in the literature, the expression positive definite is to be understood as positive semi-definite; that is, a $k \times k$ matrix $A$ is \textit{positive definite} (in symbols, $A \ge 0$) if $A$ is Hermitian and $\bm{v}^{\ast}A\bm{v} \ge 0$, for all vectors $\bm{v} \in \mathbb{C}^k$. \ Since we are mainly interested in sequences of real numbers, and matrices built from those sequences, we will focus our discussion on matrices over $\mathbb{R}$. \ Amongst canonical examples of $k \times k$ positive matrices we mention the Gram matrices $\bm{v} \bm{v}^{\ast}$, the Hilbert matrix $\left(\ddfrac{1}{i+j-1} \right)_{i,j=1}^k$, the Pascal matrix $\left(\binom{i+j}{i} \right)_{i,j=1}^k$, the Lehmer matrix $\left(\ddfrac{\min \{i,j\}}{\operatorname{max} \{i,j\}} \right)_{i,j=1}^k$, and the GCD matrix $\left(g.c.d.(i,j) \right)_{i,j=1}^k$. \ These matrices have the added property of remaining positive after taking the $p$--th Schur power, for all $p>0$; that is, they are infinitely divisible matrices (cf. \cite{Bh}).
\begin{remark}
\rm{
It is easy to see that all positive definite $2 \times 2$ matrices are infinitely divisible. \ However, the same is not true of $3 \times 3$ matrices, witness the matrix
$$
R(a):=\left(
\begin{array}{ccc}
1 & 1 & a \\
1 & 2 & 1 \\
a & 1 & 1
\end{array}
\right) .
$$
$R(a)$ is positive definite for all $a \in [0,1]$, but $\det R(a)^{1/2}<0$ for small positive values of $a$.
}
\end{remark}
We briefly pause to give a precise definition of conditional positive definiteness; cf. \cite[p. 66]{BCR}.
\begin{definition}
A Hermitian $k \times k$ complex matrix $A \equiv (a_{ij})_{i,j=0}^k$ (i.e., $a_{ji}=\overline{a}_{ij}$ for all $i,j=0,\ldots,k$) is \textit{conditionally positive definite} (abbreviated CPD) whenever $\bm{c}^{\ast}A\bm{c}:=\sum_{i,j=0}^k a_{ij}\bar{c}_ic_j \ge 0$ for all complex vectors $\bm{c}\equiv(c_0,c_1,\ldots,c_k)$ such that $c_0+c_1+\ldots+c_k=0$. \ An \textit{infinite} (scalar) matrix is CPD if all of its principal minors of finite size are CPD.
\end{definition}
Clearly, a positive definite matrix is CPD. \ When $k=2$, $A$ is CPD if and only if $a_{11}-a_{12}-a_{21}+a_{22} \ge 0$. \ When $k=3$, a straightforward calculation reveals that $A$ is CPD if and only if the $2 \times 2$ matrix
$$
\left(
\begin{array}{cc}
a_{11}-a_{12}-a_{21}+a_{22} \; \; & a_{12}-a_{13}-a_{22}+a_{23} \\
a_{21}-a_{22}-a_{31}+a_{32} \; \; & a_{22}-a_{23}-a_{32}+a_{33}
\end{array}
\right).
$$
is positive definite. \ A much more general result holds, as we will see below. \ First, we present two elementary observations.
\begin{remark}
\rm{
Observe that a real Hermitian matrix $A$ is CPD if and only if it satisfies the CPD positivity condition for vectors with real coordinates adding up to zero. \ Here is the argument for the non-trivial direction: given a complex vector $\bm{v}$ whose components sum to zero, we may write $\bm{v} = \bm{x} + {\mathrm{i}}\bm{y}$ where $\bm{x}$ and $\bm{y}$ are real vectors each of whose coordinates must sum to zero. \ Then
\begin{eqnarray} \label{xiy}
\bm{v}^{\ast} A \bm{v} &=& (\bm{x} + {\mathrm{i}}\bm{y})^{\ast} A (\bm{x} + {\mathrm{i}}\bm{y}) \nonumber \\
&=& \bm{x}^T A \bm{x} + \bm{y}^T A \bm{y} - {\mathrm{i}}\bm{y}^T A\bm{x} +{\mathrm{i}}\bm{x}^T A \bm{y} \nonumber \\
&=& \bm{x}^T A \bm{x} + \bm{y}^T A \bm{y} - {\mathrm{i}}(\bm{y}^T A\bm{x} - \bm{x}^T A \bm{y}) \\
&=& \bm{x}^T A \bm{x} + \bm{y}^T A \bm{y}, \nonumber
\end{eqnarray}
using the fact that $\bm{v}^{\ast} A \bm{v}$ must be real (since $A$ is Hermitian) and therefore the imaginary part of (\ref{xiy}) must be zero. \ If we assume that $A$ satisfies conditional positive definiteness for real vectors, we must have $\bm{x}^T A \bm{x} + \bm{y}^T A \bm{y} \ge 0$, and it follows that $\bm{v}^{\ast} A \bm{v} \ge 0$, showing that $A$ satisfies conditional positive definiteness for complex vectors.
}
\end{remark}
\begin{remark}
\rm{
(i) \ (cf. \cite[Lemma 4.1.4]{BaRa}) \ If $A$ is CPD, then $A$ has at most one negative eigenvalue. \newline
(ii) \ Assume that a $k \times k$ matrix $A$ is nonzero, has non-positive entries, and is CPD. \ Then $A$ has exactly one negative eigenvalue. \ For, since $A$ is Hermitian, all eigenvalues are real, and not all eigenvalues can be zero. \ Since the trace of $A$ is non-positive, at least one eigenvalue must be negative. \ By (i), at most one eigenvalue can be negative, and therefore $A$ has exactly one negative eigenvalue. \newline
(iii) \ Consider a contractive unilateral weighted shift $W_{\alpha}$, and its moment matrices $M_{\gamma}(0)$ and $M_{\gamma}(1)$, whose entries are real numbers in the interval $(0,1]$ (recall that we have assumed that the weights are positive). \ Then $\log M_{\gamma}(0,k)$ and $\log M_{\gamma}(1,k)$ are nonzero matrices, have non-positive entries, and are CPD. \ It follows from (ii) that each of these matrices has exactly one negative eigenvalue.
}
\end{remark}
Since we wish to study CPD matrices and weighted shifts, the equivalence of (i) and (ii) in the following result is of special interest to us.
\begin{theorem} \label{thm114}
(cf. \cite[Theorem 4.1.3]{BaRa}, \cite{Bh}, and \cite[Problem 6.3.8]{HJ}) \ For $k \ge 2$, let $A \equiv \left(a_{ij}\right)_{i,j=1}^k$ be a Hermitian $k \times k$ matrix. \ The following statements are equivalent. \newline
(i) \ $A$ is CPD. \newline
(ii) \ $B := \left(a_{ij}-a_{i,j+1}-a_{i+1,j}+a_{i+1,j+1}\right)_{i,j=1}^{k-1}$ is positive definite. \newline
(iii) \ There exist real numbers $\omega_i \; (i=1,\ldots,k)$ such that $C:=\left(a_{ij}-\omega_i-\omega_j \right)_{i,j=1}^k$ is positive definite. \newline
(iv) \ For all $p>0$, the $k \times k$ matrix $e^{pA}$ is positive definite.
\end{theorem}
\begin{remark} \label{rem115}
\rm{
Assume that the matrix $A$ in Theorem \ref{thm114} is Hankel, so the entries are of the form $a_{ij}=w_{i+j}$. \ Then the entries of the matrix $B$ in {\it (ii)} are of the form $w_{i+j}-2w_{i+j+1}+w_{i+j+2}$. \ This sum is precisely what one obtains by letting the square of the forward difference operator $\nabla$ act on $w$ (cf. (\ref{nabla2})). \ We'll revisit this viewpoint in Subsection \ref{condition}.
}
\end{remark}
We now establish a link between conditional positive definiteness and unilateral weighted shifts; cf. Lemma \ref{lem22}.
\begin{remark}
\rm{
Let $A$ be a $k \times k$ matrix with positive entries, and assume that -A is CPD. \ Then the $k \times k$ matrix $R$ of reciprocals, defined as $r_{ij}:=\ddfrac{1}{a_{ij}}$, is infinitely divisible \cite[Subsection 6.3, Problem 11]{HJ}.
}
\end{remark}
We now present a proposition crucial for our purposes, which R. Bhatia \cite{Bh} attributes to C. Loewner and which may be found in \cite{HJ}.
\begin{proposition} (\cite[Theorem 6.3.13]{HJ}) \ \label{prop215}
Suppose $M$ is a symmetric matrix with positive entries. \ Then $M$ is infinitely divisible if and only if $\log M$ (meaning the Hadamard logarithm matrix) is CPD.
\end{proposition}
\begin{remark}
\rm{
CPD matrices are intrinsically associated with \textit{distance} matrices, as follows. \ In $\mathbb{R}^d$, consider $k$ points $\bm{p}^{(1)},\ldots,\bm{p}^{(k)}$ and their Euclidean distances $\left\|\bm{p}^{(i)}-\bm{p}^{(j)}\right\|^2 \; (i,j=1,\ldots,k)$. \ Now form the $k \times k$ matrix $D(\bm{p}^{(1)},\ldots,\bm{p}^{(k)}) \equiv \left(d_{ij}\right)_{i,j=1}^k$ where $d_{ij}(\bm{p}^{(1)},\ldots,\bm{p}^{(k)}):=\left\|\bm{p}^{(i)}-\bm{p}^{(j)}\right\|^2$, for $i,j=1,\ldots,k$; $D \equiv D(\bm{p}^{(1)},\linebreak \ldots,\bm{p}^{(k)})$ is called the \textit{distance} matrix associated with the $k$--tuple $(\bm{p}^{(1)},\ldots,\bm{p}^{(k)})$. \ Clearly, $d_{ii}=0$ for $i=1,\ldots,k$, so the diagonal entries of $D$ are all zero. \ Moreover, $-D$ is CPD, as a calculation reveals. \ In fact, fix $i$ and $j$ and write $\left\|\bm{p}^{(i)}-\bm{p}^{(j)}\right\|^2=\left\|\bm{p}^{(i)}\right\|^2+\left\|\bm{p}^{(j)}\right\|^2-2 \left\langle \bm{p}^{i},\bm{p}^{j}\right\rangle$. \ Now let $\omega_i:=\left\|\bm{p}^{(i)}\right\|^2$, $m_{ij}:=\omega_i+\omega_j$, $M:=(m_{ij})_{i,j=1}^k$, and $g_{ij}:=\left\langle \bm{p}^{i},\bm{p}^{j}\right\rangle$. \
For $k \ge 2$, consider the map from Hermitian $k \times k$ matrices to Hermitian $(k-1) \times (k-1)$ matrices, given by $A \longmapsto \Phi(A):=(a_{ij}-a_{i,j+1}-a_{i+1,j}+a_{i+1,j+1})_{i,j=1}^{k-1}$. \ The above mentioned matrix $M$ is a typical element of the linear subspace of $k \times k$ matrices $A$ such that $\Phi(A)=0$; that is, $M \in \operatorname{ker} \Phi$. \ It follows that $D$ can be written as the difference between a matrix in $\operatorname{ker} \Phi$ and twice the Gramian matrix $G:=(g_{ij})_{i,j=1}^k$. \
}
\end{remark}
Now consider an arbitrary CPD Hermitian $k \times k$ matrix $A \equiv (a_{ij})_{i,j=1}^k$ and let $H_A:=(h_{ij})_{i,j=1}^k$, with $h_{ij}:=(a_{ii}+a_{jj})/2 \quad (i,j=1,\ldots,k)$. \ We already know that $\Phi(H_A) = 0$, so $H_A \in \operatorname{ker} \Phi$. \ Moreover, $A-H_A$ is CPD and has diagonal identically equal to zero; then, by \cite[Theorem 4.1.7]{BaRa}, $-(A-H_A)$ is a distance matrix. \ From the discussion above, we see that
$$
-(A-H_A)=M-2G,
$$
for some $M \in \operatorname{ker} \Phi$ and $G$ Gramian. \
As a result, we can prove that, given a CPD Hermitian $k \times k$ matrix $A$, one can find a positive integer $d$, and $k$ points $\bm{p}^{(i)} \in \mathbb{R}^d \; (i=1,\ldots,k)$, such that $A-2G \in \operatorname{ker} \Phi$, where $G$ is the Gramian of the $k$--tuple $(\bm{p}^{(1)},\ldots,\bm{p}^{(k)})$. \ In particular, $\Phi(A) = 2 \Phi(G) \ge 0$, since it is well-known that Gramians are PD (see, for instance, \cite[Theorem 7.2.10]{HJ2}).
We close this section by mentioning a very recent paper of Z.J. Jab\l o\'nski, I.B. Jung and J. Stochel \cite{JJS}, in which they study the CPD condition for operators on Hilbert space. \ The focus is on CPD sequences of exponential growth, with emphasis given to obtaining criteria for their positive definiteness.
\section{A Bridge Between $k$--hyponormality and $n$--contractivity} \label{main}
Proposition \ref{prop215} establishes an equivalence between the infinite divisibility of a moment matrix $M$ and the conditional positive definiteness of its Schur logarithm $\log M$. \ This allows us to view $k$--hyponormality under a new lens, and describe moment infinite divisibility of a weighted shift $W_{\alpha}$ as follows; for each $k \ge 1$, $i \ge 0$, and $p >0$, the matrix
$$M_{\gamma}^p(i,k) = \begin{pmatrix}
\gamma_i^p & \gamma_{i+1}^p &\gamma_{i+2}^p &\ldots & \gamma_{i+k}^p\\
\gamma_{i+1}^p & \gamma_{i+2}^p & \ldots & \ldots & \gamma_{i+k+1}^p \\
\gamma_{i+2}^p & \ldots & \ldots & \ldots & \gamma_{i+k+2}^p \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
\gamma_{i+k}^p & \gamma_{i+k+1}^p & \gamma_{i+k+2}^p & \ldots & \gamma_{i+2k}^p \\
\end{pmatrix}$$
is positive definite, where the $\gamma_i$ are the moments of the shift. \ (When $p=1$, we will simply write $M_{\gamma}(i,k)$.)
The following two corollaries are immediate. \ Note, however, that in this context we do not need to assume that $W_\alpha$ is a contraction: while the Agler conditions (equivalently, the approach through monotone moment sequences) yield subnormality for a contraction (but not in general, viz. the Dirichlet shift), the $k$--hyponormality conditions do yield subnormality for any shift.
Observe also that if we wish to operate at the level of infinite matrices of moments, we must take care that we examine two such matrices. \ Recall that for subnormality we are in pursuit of \textit{Stieltjes} moment sequences, that is, moment sequences $(\gamma_n)_{n=0}^\infty$ for which the following two matrices are simultaneously positive definite \cite{Ak}:
\begin{equation} \label{def:MofW}
M_{\gamma}(0) \equiv M_{\gamma}(0,\infty):= \left(
\begin{array}{ccccc}
\gamma _{0} & \gamma _{1} & \cdots & \gamma _{k} & \ldots \\
\gamma _{1} & \gamma _{2} & \cdots & \gamma _{k+1} & \ldots\\
\vdots & \vdots & \ddots & \vdots & \vdots\\
\gamma _{k} & \gamma _{k+1} & \cdots & \gamma _{k+2n} & \ldots \\
\vdots & \vdots & \ddots & \vdots & \ldots
\end{array}
\right)
\end{equation}
and
\begin{equation} \label{def:M1ofW}
M_{\gamma}(1) \equiv M_{\gamma}(1,\infty): = \left(
\begin{array}{ccccc}
\gamma _{1} & \gamma _{2} & \cdots & \gamma _{k+1} & \ldots \\
\gamma _{2} & \gamma _{3} & \cdots & \gamma _{k+2} & \ldots\\
\vdots & \vdots & \ddots & \vdots & \vdots\\
\gamma _{k+1} & \gamma _{k+2} & \cdots & \gamma _{k+2n+1} & \ldots \\
\vdots & \vdots & \ddots & \vdots & \ldots
\end{array}
\right).
\end{equation}
\begin{corollary}
Let $W_\alpha$ be a weighted shift. \ Then $W_\alpha$ is $\mathcal{MID}$ if and only if for each $i \geq 0$ and each $k \geq 1$,
$\log M_{\gamma}(i,k)$ is CPD (equivalently, both $\log M_{\gamma}(0)$ and $\log M_{\gamma}(1)$ are CPD).
\end{corollary}
\begin{corollary} \label{cor:infDivkHN}
Let $W_\alpha$ be a weighted shift and let $k \in \mathbb{N}$. \ Then $W_\alpha^p$ is $k$--hyponormal for all $p>0$ if and only if
$\log M_{\gamma}(i,k)$ is CPD for each $i \geq 0$.
\end{corollary}
We leave to the interested reader to show that the corollaries may be improved to add the equivalent condition ``subnormal for each $p$ in some interval $[0, \delta)$ with $\delta > 0$'' (respectively, ``$k$--hyponormal for each $p$ in some interval $[0, \delta)$ with $\delta > 0$''). \ This may be accomplished by considering positivity of Schur products of positive matrices.
We may apply this to a weighted shift to yield the following result. \ This yields a test for subnormality which is, as far as we know, new.
\begin{proposition} \label{propnew1}
Let $W_{\alpha}$ be a weighted shift with moment sequence $\gamma = (\gamma_n)_{n=0}^\infty$. \ The following statements are equivalent. \newline
(i) $W_{\alpha}$ is $\mathcal{MID}$. \newline
(ii) The sequence $(\delta_n)_{n=0}^\infty$ with $\delta_n = \ln\left(\frac{\gamma_n \gamma_{n+2}}{\gamma_{n+1}^2}\right)$ is a Stieltjes moment sequence. \newline
(iii) The weighted shift with moments $\left(\frac{\delta_n}{\delta_0}\right)$ is subnormal.
\end{proposition}
\begin{proof}
This is immediate from the fact that $\log M_{\gamma}(0)$ and $\log M_{\gamma}(1)$ must be CPD, Theorem \ref{thm114} (i) $\Leftrightarrow$ (ii), and the standard two-matrix requirement that a sequence be a Stieltjes moment sequence. \qed
\end{proof}
When the weighted shift is a contraction we may add to the list, in Proposition \ref{propnew1}, of conditions equivalent to moment infinite divisibility, as we do in the following theorem. \ Note that, however, as in the observation following Corollary \ref{cor:infDivkHN}, we may instead add a condition only on some Schur powers $p$ for $p$ in a small interval $[0, \delta)$. \ To extend this to all $p>0$, one needs to resort to Schur multiplication to go from say, $p$ to $2p$, and so on.
We now prove our first main result, which we restate for the reader's convenience.
\begin{theorem} \label{thm:equivcondtIDcontraction2}
Let $W_\alpha$ be a contractive weighted shift, and let $M_{\gamma}(i,k)$, $M_{\gamma}(0)$ and $M_{\gamma}(1)$ be as above. \ The following statements are equivalent.
\begin{enumerate}
\item $W_\alpha$ is $\mathcal{MID}$.
\item $\log M_{\gamma}(0)$ and $\log M_{\gamma}(1)$ are CPD.
\item For every $i \ge 0$ and every $k \ge 1$, $\log M_{\gamma}(i,k)$ is CPD.
\item For every $p>0$, $M_{\gamma}^p(0)$ and $M_{\gamma}^p(1)$ are positive definite.
\item For every $p>0$, $M_{\gamma}^p(0)$ and $M_{\gamma}^p(1)$ are CPD.
\item The moment sequence $\gamma$ is log completely monotone.
\item The weight sequence $\alpha$ is log completely alternating.
\end{enumerate}
\end{theorem}
\begin{proof}
The only assertion not immediate is the equivalence of $M_{\gamma}^p(0)$ and $M_{\gamma}^p(1)$ being CPD (for all $p>0$) to the others. \ Fix some power $p>0$ for the moment. \ Using the observation just before the theorem, the condition in question yields all of the $2k$--monotonicity conditions, and, using Corollary \ref{cor:nmonoimpnhypermonoshifts} we obtain $2k$--hypermonotonicity for all $k$, yielding that the sequence of moments $\gamma^{p}$ is completely monotone. \ Since $p$ was arbitrary, this yields the result. \qed
\end{proof}
In view of the results in this section, one can try and extend the notion of moment infinite divisibility for weighted shifts to arbitrary contractions on Hilbert space, using the polar decomposition factor, as in Problem \ref{problem1} below. \ We plan to pursue this idea in a forthcoming paper.
\begin{problem} \label{problem1}
Starting with the canonical polar decomposition of a subnormal operator, $T \equiv V\left|T\right|$, should one consider the subnormality of $V\left|T\right|^p$ for all $p>0$ as the proper analog of moment infinite divisibility?
\end{problem}
\subsection{A unifying condition} \label{condition}
It is well-known that $k$--hyponormality for all $k = 1, 2, \ldots$ is necessary and sufficient for subnormality of an operator; it is true as well that $n$--contractivity for all $n = 1, 2, \ldots$ is sufficient for subnormality of an operator, and necessary if the operator is a contraction. \ Of course these, in their equivalents as conditions on moments of the shift, hold as well. \ With the exception of \cite[Theorem 1.2]{EJP}, which shows in general that $k$--hyponormality implies $2k$--contractivity, and work in \cite{AE} which gives, in a very special case, a situation in which each $k$--hyponormality coincides exactly with a particular $n$--contractivity, the two sets of conditions have remained resistant to a common point of view. \ It is the goal of this subsection to give a more general condition exhibiting a unified point of view yielding the sets of conditions.
We will have occasion to operate mainly upon a sequence likely in the sequel to be the moment sequence of a weighted shift (often contractive), so we will employ $\gamma = (\gamma_n)_{n=0}^\infty$ for this sequence. \ Recall the forward difference operator $\nabla$ acting on sequences by $\nabla(\gamma)_n = \gamma_n - \gamma_{n+1}$ for all $n = 0, 1, \ldots$, and with its powers defined iteratively as $\nabla^{m+1}(\gamma) := \nabla(\nabla^{m}(\gamma))$ for all $m \geq 0$; we let $\nabla^{0}(\gamma) := \gamma$. \ Note that $m$--contractivity means that $\nabla^{m}(\gamma)$ is a non-negative sequence.
For a sequence $a = (a_n)_{n=0}^\infty$, $k \geq 1$, and $i \geq 0$, we let $M_a(i,k)$ denote the Hankel matrix of size $k+1$ by $k+1$ given by
$$M_a(i,k) := \begin{pmatrix}
a_i & a_{i+1} &a_{i+2} &\ldots & a_{i+k}\\
a_{i+1}& a_{i+2} & \ldots & \ldots & a_{i+k+1} \\
a_{i+2} & \ldots & \ldots & \ldots & a_{i+k+2} \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
a_{i+k} & a_{i+k+1} & a_{i+k+2} & \ldots & a_{i+2k} \\
\end{pmatrix}.$$
(Observe that we use the positive integer $k$ to keep track of $k$--hyponormality, but the actual size of the related matrices is $k+1$, as in Subsection \ref{khypon}.) \ In the case of a weighted shift, $k$--hyponormality is exactly that $M_\gamma(i,k)$ is positive definite for each $i = 0, 1, \ldots$. \ Observe that by Theorem \ref{thm114} (i) $\Leftrightarrow$ (ii) applied to Hankel matrices, it follows that $M_\gamma(i,k)$ is a CPD matrix if and only if $M_{\nabla^{2}(\gamma)}(i,k-1)$ is positive definite; see also Remark \ref{rem115}.
\begin{definition} \label{def:kmpositive}
Let $k \ge 1$ and $0 \le m \le k$. \ We say a sequence $\gamma = (\gamma_n)_{n=0}^\infty$ is $(k,2m)$--PD if, for every $i \ge 0$, the matrix $M_{\nabla^{2m}(\gamma)}(i, k-m)$ is positive definite. \ Similarly, we say the sequence is $(k,2m)$--CPD if, for every $i \ge 0$, the matrix $M_{\nabla^{2m}(\gamma)}(i, k-m)$ is CPD.
\end{definition}
We will abuse language slightly to say a weighted shift $W_\alpha$ with moment sequence $\gamma$ is $(k,2m)$--PD (respectively, $(k,2m)$--CPD) if its moment sequence is. \ Observe that a weighted shift is $k$--hyponormal exactly when it is $(k,0)$--PD; a computation shows that a shift is $(k,2k)$--PD if and only if it is $2k$--contractive. \ This is the sense in which the $k$--hyponormality condition and the $2k$--contractivity condition are simply instances (in fact, extremes) of the single more general condition. \ It is productive to think of the condition as beginning with the moment matrix of size $k+1$, employing repeatedly the matrix transformation implicit in Theorem \ref{thm114}\,(ii), and insisting that the matrix obtained after $m$ such steps be positive definite (respectively, CPD).
Observe also that if the shift $W_\alpha$ is a contraction, with some other mild condition (such as hyponormality), and if $W_\alpha$ is $(k,2k)$--PD, then its moment sequence is $2k$--hypermonotone (see Corollary \ref{contnmonoimpnhypermono}).
We now record the following proposition.
\begin{proposition} \label{prop:getAglerfrom}
Let $W_{\alpha}$ be a unilateral weighted shift, $k \ge 1$, and $0 \le m \le k$. \ Assume that $W_{\alpha}$ is $(k,2m)$--CPD. \ Then $W_{\alpha}$ is $2m$--contractive, $(2m+2)$--contractive, \ldots, $2k$--contractive.
\end{proposition}
\begin{proof}
This is just a computation: use each matrix $M_{\nabla^{2m}(\gamma)}(i, k-m)$ as a quadratic form against an appropriately sized ``negative binomial'' vector, such as $(0,0,1,-1,0)$ or $(0, 1, -3, 3, -1, 0, 0)$. \ To show one significant instance of this calculation, let $k=4$, $m=2$ and $i=\ell$. \ Then $M_{\nabla^2 \gamma}(\ell,k-m)$ becomes
$$
M_{\nabla^2 \gamma}(\ell,2) \!=\!\! \left(
\begin{array}{ccc}
\gamma_{\ell}-2\gamma_{\ell+1}+\gamma_{\ell+2} \; & \gamma_{\ell+1}-2\gamma_{\ell+2}+\gamma_{\ell+3} \; & \gamma_{\ell+2}-2\gamma_{\ell+3}+\gamma_{\ell+4} \\
\gamma_{\ell+1}-2\gamma_{\ell+2}+\gamma_{\ell+3} \; & \gamma_{\ell+2}-2\gamma_{\ell+3}+\gamma_{\ell+4} \; & \gamma_{\ell+3}-2\gamma_{\ell+4}+\gamma_{\ell+5} \\
\gamma_{\ell+2}-2\gamma_{\ell+3}+\gamma_{\ell+4} \; & \gamma_{\ell+3}-2\gamma_{\ell+4}+\gamma_{\ell+5} \; & \gamma_{\ell+4}-2\gamma_{\ell+5}+\gamma_{\ell+6}
\end{array}
\right)\!\!,
$$
and therefore
$$
\left(
\begin{array}{ccc}
1 & -1 & 0
\end{array}
\right) M_{\nabla^2 \gamma}(\ell,2) \left(
\begin{array}{c}
1 \\
-1 \\
0
\end{array}
\right) = \gamma_{\ell} - 4 \gamma_{\ell+1} + 6 \gamma_{\ell+2} - 4 \gamma_{\ell+3} + \gamma_{\ell+4} ,
$$
which is one of the Agler expressions needed to verify $4$--contractivity. \qed
\end{proof}
\subsection{A family of intermediate ladders}
It is well-known that the $k$--hyponormality conditions ($k \in \mathbb{N}$) provide a \linebreak ``ladder'' with subnormality at the top, and that the $n$--contractivity conditions $n=1, 2, \ldots$ serve similarly if the shift is a contraction. \ If $W$ is a contraction whose weights have a limit (for example, it is hyponormal), and if we fix some $m$, the $(k,2m)$--PD or $(k,2m)$--CPD conditions (for $k \geq m$) form a similar ladder. \ The proof is immediate from Proposition \ref{prop:getAglerfrom}, the fact that (in the presence of contractivity) $n$--monotonicity implies $n$--hypermonotonicity, and the Agler-Embry characterization of subnormality as applied to a shift.
\begin{theorem}
Let $W$ be a contractive weighted shift whose weights have a limit (for example, if $W$ is hyponormal), and fix $m \in \mathbb{N}$. \ If for all $k \geq m$, $W$ is $(k,2m)$--PD or $(k,2m)$--CPD, $W$ is subnormal.
\end{theorem}
We turn next to some results showing that these ladders are ``different'' by considering zeroth-weight perturbations $A_j(x)$ of the Agler shifts $A_j$. \ Recall that the moment sequence $\gamma^{(j)} = (\gamma_n^{(j)})_{n=0}^\infty$ of the $j$--th Agler shift satisfies
\begin{equation} \label{eq:Aglerjmoments}
\gamma_n^{(j)} = \frac{(j-1)!\!\;n!}{(n+j-1)!} \; \; \; (j \ge 2, n \ge 0).
\end{equation}
It is a computation to show that
$$\nabla(\gamma^{(j)}) = \frac{j-1}{j}\cdot\gamma^{(j+1)}$$
and that therefore
\begin{equation} \label{eq:nabla2mgammajIS}
\nabla^{2m}(\gamma^{(j)}) = \frac{j-1}{j+2m - 1}\cdot \gamma^{(j+2m)}.
\end{equation}
Observe that this is in general not a moment sequence, because $(\nabla^{2m}(\gamma^{j}))_0$ may not be equal to $1$. \ However, since $A_j$ is subnormal for all $j$, $\nabla^{2m}(\gamma^{j})$ is, up to a normalizing constant, a Stieltjes moment sequence.
\subsection{Cutoffs for rank-one perturbations of the Agler shifts} \label{cutoffs}
Recall that if we form the standard Hankel matrices of moments of a weighted shift, $n$--hyponormality amounts to positivity of all those matrices of size $n+1$ by $n+1$. We have seen as well that instead of insisting these matrices be positive, we might ask only that the matrices of size $n+1$ by $n+1$ be CPD, which is equivalent to the matrices of size $n$ by $n$, and consisting of second differences $(\nabla^{2})$, be positive definite. And of course we could insist only that these matrices be CPD, which is equivalent to matrices of size $n-1$ by $n-1$, and consisting of fourth differences of the original moments, be positive definite. We may continue in this way, terminating by insisting that matrices of size $1$ by $1$, and having the entry consisting of $2n$--th order differences, be positive definite, and this turns out just to be the $2n$--th Agler condition.
We set some notation, so if the original moment sequence is $\gamma = (\gamma_n)$, we write $M_{\nabla^{2m}\gamma}(k,n-1)$ for the matrix of $2m$--th order differences of $\gamma$, of size $n$ by $n$, and with the upper-left-hand entry consisting of the difference beginning at $\gamma_k$. \ However, in what follows we need to consider only those matrices for which $k = 0$, and we abbreviate $M_{\nabla^{2m}\gamma}(0,n-1)$
to $(\nabla^{2m} \gamma)_{n}$. \ Thus, for example,
$$(\nabla^{4} \gamma)_{3} = \left(\begin{array}{ccc}
\gamma_0 - 4 \gamma_1 + \ldots & \gamma_1 - 4 \gamma_2 + \ldots & \gamma_2 - 4 \gamma_3 + \ldots \\
\gamma_1 - 4 \gamma_2 + \ldots & \gamma_2 - 4 \gamma_3 + \ldots & \gamma_3 - 4 \gamma_4 + \ldots \\
\gamma_2 - 4 \gamma_3 + \ldots & \gamma_3 - 4 \gamma_4 + \ldots & \gamma_4 - 4 \gamma_5 + \ldots \\
\end{array}\right).$$
We shall consider perturbations $A_j(x)$ of the $j$--th Agler shift $A_j$ in which the zeroth weight $\alpha_0 = \sqrt{\frac{1}{j}}$ is replaced by $\alpha_0(x) = \sqrt{\frac{x}{j}}$. \ It is well-known that this results in the moment sequence $1, x \gamma^{(j)}_1, x \gamma^{(j)}_2, x \gamma^{(j)}_3, \ldots$. \ The task is to consider the cutoffs in $x$ for which $A_j(x)$ is various $(k,2m)$--PD; as usual with zeroth-weight perturbations, we need to consider only the tests of matrices whose $(1,1)$ entry is $\gamma_0^{(j)} + \ldots$ since other matrices relevant to some $(k,2m)$--PD are simply $x$ times a matrix relevant to $A_{j+2m}$, which is subnormal and so the matrix is positive definite (and hence CPD). \ To detect positive definiteness, we will use the nested determinant test. \ Perturbations $A_2(x)$ of the Bergman shift were studied initially in \cite[Proposition 7]{Cu}, where the specific cutoff for $2$--hyponormality was calculated. \ Here we will follow the approach described in \cite[Section 2]{AE}, both for $k$--hyponormality and $n$--contractivity of Agler-type shifts. \ We will therefore merely sketch the computations here. \ For the reader's convenience, below we state one of the main results in \cite{AE}.
\begin{theorem} \label{AdamsExner}
(\cite[Theorem 2.6]{AE}) \ For $j \ge 2$, let $A_j(x)$ be the rank-one perturbation of the $j$--th Agler shift, as described above. \newline
(i) \ $A_j(x)$ is $n$--contractive if and only if $x \le c(j,n):=\ddfrac{n+j-1}{n}$. \newline
(ii) \ $A_j(x)$ is $k$--hyponormal if and only if $x \le h(j,k):=\ddfrac{k(k+j)+j-1}{k(k+j)}$. \newline
As a consequence, $A_j(x)$ is $k$--hyponormal if and only if $A_j(x)$ is $k(k+j)$--contractive.
\end{theorem}
\begin{remark}
\rm{
(i) \ The quantities on the right-hand side of the inequalities in Theorem \ref{AdamsExner}(i) and (ii) are consistent with previous partial results on rank-one perturbations of weighted shifts (see \cite[Proposition 7]{Cu} and \cite[Problem 3.2 and Proposition 3.4]{CF1}); however, while those results define the $0$--th weight as $x$, for our purposes it is better to write it as $\sqrt \ddfrac{x}{j}$. \newline
(ii) \ In the specific case of the Bergman shift ($j=2$), we get $c(2,n)=\ddfrac{n+1}{n}$ and $h(2,k)=\ddfrac{k^2+2k+1}{k^2+2k}$, so it is immediate that $A_2(x)$ is $2$--hyponormal if and only if it is $8$--contractive.
}
\end{remark}
Our aim is to obtain a comparable result for $(k,2m)$--PD, which will enable us to relate this notion to $n$--contractivity. \ To ease the notation, view $j$ as fixed for the moment (which we then suppress as much as possible) and let $N(k,m)$ denote the matrix $M_{\nabla^{2m}(\gamma^{j})}(0, k-m)$ and let $\hat{N}(k,m)$ denote the matrix obtained from $N(k,m)$ by deleting the first row and first column. \ To detect $(k,2m)$--PD we wish to test, for positivity, the determinant $\det M_{\nabla^{2m}(\gamma^{j}(x))}(0, k-m)$; it is a computation to show that this positivity is equivalent to
$$
x \leq \frac{1}{1 - \frac{\det N(k,m)}{\det \hat{N}(k,m)}}.
$$
In light of \eqref{eq:nabla2mgammajIS} and the fact that $N(k,m)$ is of size $k+1-m$, we have
$$\det N(k,m) = \left(\frac{j-1}{j+2m - 1}\right)^{k+1-m} \det M_{\gamma^{(j+2m)}}(0,k-m)$$
and using again \eqref{eq:nabla2mgammajIS} and that $\hat{N}(k,m)$ is of size $k+1-m-1 = k-m$, we obtain
$$\det \hat{N}(k,m) = \left(\frac{j-1}{j+2m - 1}\right)^{k-m} \det M_{\gamma^{(j+2m)}}(2,k-m-1).$$
The ratio $\frac{\det M_{\gamma^{(j+2m)}}(0,k-m)}{\det M_{\gamma^{(j+2m)}}(2,k-m-1)}$ was computed in \cite[Lemma 2.5]{AE}; transferred to our setting, it is
$$\frac{j + 2m - 1}{(k-m + j + 2m-1)(k-m+1)}.$$
Putting this all together yields the following result.
\begin{theorem} \label{thm14}
Let $A_j(x)$ be the perturbation of the Agler shift $A_j$ in which the zeroth weight $\alpha_0 = \sqrt{\frac{1}{j}}$ is replaced by $\alpha_0(x) = \sqrt{\frac{x}{j}}$, and let $k \in \mathbb{N}$ and \linebreak $m \in \{0,1,\ldots,k\}$. \newline
(i) $A_j(x)$ is $(k,2m)$--PD if and only if $x \leq p(j,k,m)$, where
\begin{equation}
p(j,k,m) = \ddfrac{(k+1-m)(j+k+m-1)}{k^2 + j k + 2m - j m - m^2}=\ddfrac{(j+k+m)(k-m)+2m+j-1}{(j+k+m)(k-m)+2m}.
\end{equation}
(ii) \ $A_j(x)$ is $(k,2m)$--PD if and only if $A_j(x)$ is $((j+k+m)(k-m)+2m)$--contractive.
\end{theorem}
We now present some examples of this result for the case of the Bergman shift ($j=2$). \ Recall that the size of the relevant matrix is $k+1$ (in accordance with the size of the matrix to be tested for $k$--hyponormality). \ In this case, $p(j,k,m)=p(2,k,m)=\ddfrac{k(k+2)-m^2+1}{k(k+2)-m^2}$; observe that the numerator is 1 plus the denominator.
$$
\begin{array}{ccccccccc}
& & & & & m & &\\
& & | & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline
& 1 & | & 4/3 & 3/2 & - & - & - & - \\
& 2 & | & 9/8 & 8/7 & 5/4 & - & - & - \\
k & 3 & | & 16/15 & 15/14 & 12/11 & 7/6 & - & - \\
& 4 & | & 25/24 & 24/23 & 21/20 & 16/15 & 9/8 & - \\
& 5 & | & 36/35 & 35/34 & 32/31 & 27/26 & 20/19 & 11/10
\end{array}.
$$
As described in Theorem \ref{thm14}, each entry in this table corresponds with an exact cutoff for $n$--contractivity, where $n=k(k+2)-m^2$; that is, the perturbation is $n$--contractive if and only if $x \leq \frac{n+1}{n}$, so, for example, $(4,3)$--PD corresponds with the shift being $15$--contractive. \ In general there is not an exact correspondence with any $k$--hyponormal condition: for instance, the $(4,2)$--PD cutoff ($21/20$), falls between the cutoffs for $3$-- and $4$--hyponormality ($16/15$ and $25/24$, respectively). \ (As noted above, the cutoff for $k$--hyponormality is exactly that for $(k,0)$--PD.)
It is worth noting that the cutoffs for $m=1$ and $m=0$ are close (the difference amounts to $\ddfrac{1}{k(k+2)(k(k+2)-1)}$); recall that the latter is exactly the threshold for $k$--hyponormality, while, by Theorem \ref{thm114}(ii), the former is the cutoff for the standard matrix for $k$--hyponormality to be CPD. \ Observe also that the cutoff for $(k,2k)$--PD, which is the same as for $2k$--contractivity, is considerably bigger than that for the standard matrix for $k$--hyponormality to be CPD. \
This shows in particular that, for that matrix to be non-negative as a quadratic form against all the ``negative binomial'' vectors of appropriate size (e.g., \linebreak $(0,1,-1, 0, 0, \ldots), (0,0,1,-3,3,-1,0, \ldots)$ is a weaker property than for it to be CPD. \ This is because if the operator is $2k$--contractive it is, being a contraction, $2k$--hypercontractive, and the quadratic form results just mentioned are all necessarily non-negative. \
It is therefore reasonable to raise the following question.
\begin{question}
Given integers $k\ge 1$ and $m$, with $0 \le m \le k$, what is the set of vectors upon which the original matrix of size $k+1$ is non-negative as a quadratic form if it is $(k,2m)$--PD? \ If $m=0$ it is the whole space; if $m=1$ it is the subspace of dimension $k$ consisting of those vectors whose components sum to zero; if $m=k$ it contains at least the ``negative binomial'' vectors.
\end{question}
\begin{remark}
\rm{
From the above, it is easy to show that the various conditions in $k$ and $m$ are distinct from one another (even just using $j=2$, the Bergman shift). \ It is also true in this setting that there is an easy expression in $x$ equivalent to the shift $A_j(x)$ $p$--contractive:
$$
x \leq \frac{p+j-1}{p}
$$
(cf. Theorem \ref{AdamsExner}). \ It results from this that the condition in Theorem \ref{AdamsExner}(i) is satisfied if and only if $A_j(x)$ is $((j+k+m)(k-m)+2m)$--contractive. This allows us to use contractivity as a ``scale'' to measure and compare the strengths of the various positivities of matrices of certain sizes and certain order of differences.
}
\end{remark}
The following table for $A_2$ illustrates this (we suppress $j=2$ to ease the notation slightly). Recall that if a weighted shift is $n$--hyponormal then it is $2n$--contractive, and that if we begin with $n$--hyponormality, the process terminates in a $1$ by $1$ matrix whose positivity is exactly this $2n$--th Agler expression. Further, each of the intermediate conditions is at least no stronger than the previous one. (In fact, the result above shows that at least in this case the previous condition is strictly stronger than the subsequent one obtained by introducing another $\nabla^{2}$ and decreasing the size of the matrix.) Note also that positive definiteness of an entry in one column is equivalent to conditional positive definiteness of the entry one column to the left. Recall finally that for a contraction, if all the even order contractivity tests are positive, the contraction is subnormal, so each of these ladders has, for contractions, subnormality at the top.
\vspace*{-.5cm}
$$
\hspace*{-.5cm}
\begin{array}{c|c|c|c|c|c}
&n-\mbox{\rm hyponormal}&&&&\\
n& (\gamma^{(x)})_{n+1} \geq 0 & (\nabla^{2} \gamma^{(x)})_{n} \geq 0 & (\nabla^{4} \gamma^{(x)})_{n-1} \geq 0&(\nabla^{6} \gamma^{(x)})_{n-2} \geq 0&(\nabla^{8} \gamma^{(x)})_{n-3} \geq 0 \\ \hline
1& \mbox{3-C} & \mbox{2-C}& -& - & -\\
2& \mbox{8-C}& \mbox{7-C}& \mbox{4-C}& - & - \\
3& \mbox{15-C}& \mbox{14-C}& \mbox{11-C}& \mbox{6-C}& - \\
4& \mbox{24-C}& \mbox{23-C}& \mbox{20-C} & \mbox{15-C}& \mbox{8-C} \\
\end{array}
$$
Put informally, every regularity you see in this display is correct (for example, in any row the ``decreases'' in contractivity have gaps 1, 3, 5, 7, \ldots). The corresponding results for $A_3$ and $A_4$ are similarly orderly.
We conclude this section with a brief digression, to report another fact about these zeroth weight perturbations $A_j(x)$ of the Agler shifts $A_j$.
\begin{proposition}
Let $A_j(x)$ be the zeroth weight perturbation of the Agler shift $A_j$ with weight sequence $\alpha^{(j)}$ as in Theorem \ref{thm14}. \ Then the weights-squared of $A_j(x)$ are $m$--alternating if and only if
\begin{equation} \label{eq:naltAjofx}
x \leq 1 + \frac{(j-1) m!}{\prod_{i=1}^{m} (j+i)}.
\end{equation}
If we take $j=2$, that is, the case of the Bergman shift, the weights-squared \linebreak of $A_2(x)$ are $m$--alternating if and only if $A_2(x)$ is $\frac{(m+1)(m+2)}{2}$--contractive.
\end{proposition}
\begin{proof}
As usual with zeroth weight perturbations, the only condition to check for $m$--alternating is that beginning with $\gamma_0(x)$, since we know that $A_j$ has weights-squared that are completely alternating (\cite[Remark 2.9]{BCE}). \ The relevant expression is
\begin{eqnarray*}
\frac{x}{j} - \binom{m}{1} \frac{2}{j+1} & + & \binom{m}{2} \frac{3}{j+2} - \ldots \pm \binom{m}{m} \frac{m+1}{m + j} \\
&=\!\!&\frac{x}{j} - \frac{1}{j} \!+\! \left(\frac{1}{j} - \binom{m}{1} \frac{2}{j+1} + \binom{m}{2} \frac{3}{j+2}- \ldots \pm \binom{m}{m} \frac{m+1}{m + j}\right) \\
&=& \frac{x}{j} - \frac{1}{j} + (\nabla^{m}\alpha^{(j)})(0).
\end{eqnarray*}
One may compute using an induction argument that
$$
\nabla^{m}(\alpha^{(j)})(n) = \frac{(1-j)m!}{\prod_{i=n}^{n+m}(j+i)},
$$
and this plus an easy calculation yields the result. \ The assertion for the Bergman shift is simply another calculation using \eqref{eq:naltAjofx}. \qed
\end{proof}
One may check that for $A_j(x)$ with $j > 2$ the cutoffs for $m$--alternating do not (necessarily) correspond to some $n$--contractivity; the $n$--contractivity cutoff is $\frac{n+1}{n}$, and what comes out of the expression above need not correspond to an integer $n$. \ When it does, the correspondence is exact. \ In some sense, the cutoffs correspond approximately with $n$--contractivity where $n = \mathcal{O}(m^2)$, as is to be expected from the formula in \eqref{eq:naltAjofx}.
\noindent \textbf{Acknowledgments.} \ The authors are deeply grateful to the referee for a detailed reading of the paper, and for detecting an inconsistency in the use of the phrase ``infinite divisibility." \ The authors also wish to express their appreciation for support and warm hospitality during various visits (which materially aided this work) to Bucknell University, the University of Iowa, and the Universit\'{e} des Sciences et Technologies de Lille, and particularly the Mathematics Departments of these institutions. \ Several examples in this paper were obtained using calculations with the software tool \textit{Mathematica} \cite{Wol}.
\end{document}
|
\begin{document}
\pagestyle{plain}
\title{Classification of vertex-transitive digraphs via automorphism group}
\maketitle
\begin{abstract}
In the mid-1990s, two groups of authors independently obtained classifications of vertex-transitive graphs whose order is a product of two distinct primes. In the intervening years it has become clear that there is additional information concerning these graphs that would be useful, as well as making explicit the extensions of these results to digraphs. Additionally, there are several small errors in some of the papers that were involved in this classification. The purpose of this paper is to fill in the missing information as well as correct all known errors.
\end{abstract}
\section{Introduction}
The initial motivation for this paper came from some work \cite{DobsonHKM2017} done by the first four authors that used a well-known classification of vertex-transitive graphs of order $pq$, where $p$ and $q$ are distinct primes.
The original classification had been obtained by two different groups of authors, each with their own perspective on what properties of these graphs were important.
One group (consisting of Maru\v si\v c and Scapellato) \cite{MarusicS1994} was primarily concerned with determining a minimal transitive subgroup of the automorphism group, while the other (consisting of Praeger, Xu, and several others) \cite{PraegerWX1993,PraegerX1993} was primarily concerned with determining the full automorphism groups of these graphs, and in particular determining all primitive or symmetric permutation groups that can act as the automorphism group of such a graph. Although the results in the classifications are stated for graphs, the proofs as written apply equally to digraphs.
Over the years, it has become apparent that there are ``gaps" in the information about vertex-transitive digraphs of order $pq$ that are not addressed by either approach but would be useful to fill. Specifically, the classification of vertex-transitive digraphs of order $pq$ that have imprimitive almost simple automorphism groups was incomplete, and the full automorphism group of the Maru\v si\v c-Scapellato (di)graphs was unknown. Additionally, there are several errors in this classification, and these have propagated themselves in the literature. The most significant of these errors, at least from the point of view of the difficulty in correcting the error, is with Praeger, Wang, and Xu's classification of symmetric Maru\v si\v c-Scapellato graphs \cite{PraegerWX1993}.
It is the purpose of this paper to fill the ``gaps" described in the preceding paragraph, and to correct the known errors. Finally, widespread reliance on results that contained errors has left a body of results that may or may not be correct; at best, the proofs need to be revised. We have not attempted to address all of these, but we provide a list of those that we are aware of.
\section{Preliminaries}
Throughout, $p$ and $q$ are distinct primes with $q<p$. We begin with basic definitions. In particular, we define the classes of graphs and digraphs that will appear in what follows (with the exception of the Maru\v si\v c-Scapellato digraphs, whose definition is best presented in a group-theoretic context and is therefore postponed to Definition \ref{MSdigraphs}). We denote the arc-set of a digraph $\Gamma$ by $A(\Gamma)$. The most commonly studied class of vertex-transitive digraphs are Cayley digraphs.
\begin{defin}
Let $G$ be a group and $S\subseteq G$. Define the {\bf \mathversion{bold} Cayley digraph of
$G$ with connection set $S$}, denoted ${\rm Z}ay(G,S)$, to be the digraph with $V({\rm Z}ay(G,S)) = G$ and $A({\rm Z}ay(G,S)) = \{(g,gs):g\in G, s\in S\}$.
\end{defin}
Note that we use the term digraphs to include graphs. If $\Gamma$ is a digraph satisfying $(x,y)\in A(\Gamma)$ if and only if $(y,x)\in A(\Gamma)$, then we will say that $\Gamma$ is a {\bf graph}, and replace each pair $(x,y)$ and $(y,x)$ of symmetric ordered pairs in $A(\Gamma)$ by the unordered pair $\{x,y\}$ in the \textbf{edge set} $E(\Gamma)$, which takes the place of the arc set. The next-most-commonly-encountered class of vertex-transitive digraphs are metacirculant digraphs, first defined by Alspach and Parsons \cite{AlspachP1982} (although they only defined metacirculant graphs).
\begin{defin}{\langle}bel{metacirculantdefin}
Let $V = {\mathbb Z}_m\times{\mathbb Z}_n$, $\alpha\in{\mathbb Z}_n^*$, and $S_0,\ldots,S_{m-1}\subseteq{\mathbb Z}_n$ such that $\alpha^mS_i = S_i$, $i\in{\mathbb Z}_n$. Define an {\bf \mathversion{bold}$(m,n,\alpha,S_0,\ldots,S_{m-1})$-metacirculant digraph} $\Gamma = \Gamma(m,n,\alpha,S_0,\ldots,S_{m-1})$ by $V(\Gamma) = {\mathbb Z}_m\times{\mathbb Z}_n$ and $A(\Gamma) = \{(\ell,j),(\ell + i,k)):k-j\in \alpha^\ell S_i\}$.
We also define an {\bf \mathversion{bold} $(m,n)$-metacirculant digraph} to be a digraph that is an $(m,n,\alpha,S_0,\ldots,S_{m-1})$-metacirculant digraph for some $\alpha$ and some $S_0, \ldots, S_{m-1}$ as above.
\end{defin}
Many Cayley digraphs and metacirculant digraphs have the important property of imprimitivity that assists in any effort to understand their automorphisms.
\begin{defin}{\langle}bel{imprimitive}
Let $G\le S_X$ be transitive. A subset $B\subseteq X$ is a {\bf block} of $G$ if whenever $g\in G$, then $g(B)\cap B = \emptyset$ or $B$. For a block $B$ of $G$, the set $\mathcal{B}=\{g(B)\mid g \in G\}$ is called a {\bf $G$-invariant partition}. If $B = \{x\}$ for some $x\in X$ or $B = X$, then $B$ is a {\bf trivial block}. Any other block is nontrivial. If $G$ has a nontrivial block, then $G$ is {\bf imprimitive}. If $G$ is not imprimitive, we say $G$ is {\bf primitive}.
If $\Gamma$ is a digraph, then we say that $\Gamma$ {\bf admits an imprimitive action} if there is some transitive group $G \le {\rm Aut}(\Gamma)$ that is imprimitive. We say that $\Gamma$ {\bf admits no imprimitive action} if every transitive group $G \le {\rm Aut}(\Gamma)$ is primitive. We say that $\Gamma$ is
{\bf primitive} if ${\rm Aut}(\Gamma)$ is primitive, and $\Gamma$ is {\bf imprimitive} if ${\rm Aut}(\Gamma)$ is imprimitive. We refer to any block of ${\rm Aut}(\Gamma)$ as a block of $\Gamma$ also.
\end{defin}
It is important for us to make these definitions about $\Gamma$. One of the sources of confusion in the literature is that Maru\v si\v c and Scapellato referred to a digraph as $m$-imprimitive whenever it admits an imprimitive action with blocks of size $m$, even if the full automorphism group is primitive.
We observe that a digraph $\Gamma$ of order $pq$ must lie in one of three families: $\Gamma$ is primitive; $\Gamma$ is imprimitive with blocks of size $p$; or $\Gamma$ is imprimitive with blocks of size $q$. Note that the second and third families are not mutually exclusive.
Maru\v si\v c provided some of the early analysis of vertex-transitive graphs of order $pq$.
\begin{prop}[Proposition 3.3, \cite{Marusic1988}] {\langle}bel{prop3.3Dragan}
The graphs of order $pq$ that admit an imprimitive action with blocks of size $p$ are precisely the $(q,p)$-metacirculant graphs.
\end{prop}
\begin{thrm}[Theorem 3.4, \cite{Marusic1988}]{\langle}bel{thm3.4Dragan}
Let $\Gamma$ be a graph of order $pq$ that admits an imprimitive action of the group $G$ with a $G$-invariant partition ${\mathcal B}$. Suppose that $\Gamma$ is not a metacirculant graph. Then the kernel of the action of $G$ on $\mathcal B$ is trivial, and $G$ is nonsolvable.
\end{thrm}
The proofs of both of these results as written apply equally to digraphs.
In later work with Scapellato, he extended these results to show the following.
\begin{thrm}[Theorem, \cite{MarusicS1992}]{\langle}bel{thm:MS1992}
Let $\Gamma$ be a vertex-transitive digraph of order $pq$ that admits an imprimitive action but is not metacirculant. Then every (transitive) imprimitive subgroup of ${\rm Aut}(\Gamma)$ admits blocks of size $q$; $p=2^{2^a}+1$ is a Fermat prime, $q$ divides $p-2$, and $\Gamma$ is a Maru\v si\v c-Scapellato graph (see Definition~\ref{MSdigraphs}).
\end{thrm}
The classification of groups of automorphisms as ``primitive" or ``imprimitive" is a natural one. Observe that a primitive group $G$ cannot contain an intransitive normal subgroup, because the orbits of such a group would give rise to a $G$-inviariant partition \cite[Proposition 7.1]{Wielandt1964}. However, $G$-invariant partitions can also arise even if $G$ has no intransitive normal subgroup.
The following definition was first introduced by Praeger.
\begin{defin}
A transitive group is called {\bf quasiprimitive} if every nontrivial normal subgroup is transitive.
\end{defin}
As we have just observed, every primitive group is quasiprimitive, and quasiprimitive groups are a generalization of primitive groups.
Vertex-transitive digraphs with quasimprimitive automorphism groups are usually studied via their orbital digraphs, which we now define.
\begin{defin}
Let $G$ act on $X\times X$ in the canonical way, that is $g(x,y) = (g(x),g(y))$. The orbits of this action are called {\bf orbitals}. One orbital is the diagonal, or $\{(x,x):x\in X\}$, and is called the {\bf trivial orbital}. We assume here that ${\mathcal O}_1,\ldots{\mathcal O}_r$ are the nontrivial orbitals. Define digraphs $\Gamma_1,\ldots,\Gamma_r$ by $V(\Gamma_i) = X$ and $E(\Gamma_i) = {\mathcal O}_i$. The set $\{\Gamma_i:1 \le i \le r\}$ is the set of {\bf \mathversion{bold} orbital digraphs of $G$}. A {\bf \mathversion{bold} generalized orbital digraph of $G$} is an arc-disjoint union of some orbital digraphs of $G$ (that is, identify vertices in the natural way amongst a set of orbital digraphs, and take the new arc set to be the union of the arcs that are in any of the orbital digraphs). We say an orbital is {\bf self-paired} if the corresponding orbital digraph is a graph.
\end{defin}
Orbital digraphs of a group $G$ are often given in terms of their suborbits.
\begin{defin}
Let $G\le S_n$ be transitive and $x$ a point. The orbits of ${\rm Stab}_G(x)$ are the {\bf \mathversion{bold} suborbits of $G$ with respect to $x$}.
\end{defin}
Notice that in an orbital digraph of $G$, the outneighbors of $x$ and inneighbors of $x$ are both suborbits of $G$ with respect to $x$. We finish this section with group- and graph-theoretic terms that relate to graph quotients.
\begin{defin}
Suppose $G\le S_n$ is a transitive group that has a $G$-invariant partition ${\mathcal B}$ consisting of $m$ blocks of size $k$. Then $G$ has an {\bf \mathversion{bold} induced action on ${\mathcal B}$}, denoted $G/{\mathcal B}$. Namely, for $g\in G$, define $g/{\mathcal B}:{\mathcal B}\mapsto{\mathcal B}$ by $g/{\mathcal B}(B) = B'$ if and only if $g(B) = B'$, and set $G/{\mathcal B} = \{g/{\mathcal B}:g\in G\}$. We also define the {\bf \mathversion{bold} fixer of ${\mathcal B}$ in $G$}, denoted ${\rm fix}_G({\mathcal B})$, to be $\{g\in G:g/{\mathcal B} = 1\}$. That is, ${\rm fix}_G({\mathcal B})$ is the subgroup of $G$ which fixes each block of ${\mathcal B}$ set-wise.
\end{defin}
Observe that ${\rm fix}_G({\mathcal B})$ is the kernel of the induced homomorphism $G\mapsto S_{\mathcal B}$ that arose previously in the statement of Theorem~\ref{thm3.4Dragan}, and as such is normal in $G$.
\begin{defin}{\langle}bel{block quotient def}
Let $\Gamma$ be a vertex-transitive digraph that admits an imprimitive action of the group $G$ with a $G$-invariant partition ${\mathcal B}$. Define the {\bf \mathversion{bold} block quotient digraph of $\Gamma$ with respect to ${\mathcal B}$}, denoted $\Gamma/{\mathcal B}$, to be the digraph with vertex set ${\mathcal B}$ and arc set $$\{(B,B'):B\not = B'\in{\mathcal B}\textit{ and }(u,v)\in A(\Gamma)\textit{ for some }u\in B\textit{ and } v \in B'\}.$$
\end{defin}
Note that ${\rm Aut}(\Gamma)/{\mathcal B}\le{\rm Aut}(\Gamma/{\mathcal B})$.
\section{Automorphism groups of $(q,p)$-metacirculant digraphs whose full automorphism group admits only blocks of size $q$}
Our original interest in this problem arose when we were studying a particular Cayley digraph of the nonabelian group of order $21$ whose automorphism group is a nonabelian simple group but is imprimitive. This digraph is included in the Maru\v si\v c-Scapellato characterization as a metacirculant digraph as its automorphism group contains the nonabelian group of order $21$. It does not appear elsewhere in that characterization as Maru\v si\v c and Scapellato were interesteed in finding a minimal transitive subgroup (indeed, they define a primitive graph to be one in which {\it every} transitive subgroup of the automorphism group is primitive), and so they were not concerned with its full automorphism group. This digraph does not occur in the Praeger-Xu characterization, as they were interested in graphs (and occasionally digraphs) whose full automorphism group is primitive (indeed, they define a primitive graph to be one in which the full automorphism group is primitive). So in neither characterization of vertex-transitive graphs of order $pq$ were such digraphs looked for. Finally, this digraph does not arise in \cite[Theorem 3.2(1)]{Dobson2006a} since that result only holds for graphs, not digraphs. Thus there is a small gap in the literature here.
The aim of this section of our paper is to fill in this gap. Fortunately, the work by Maru\v si\v c and Scapellato \cite{MarusicS1992} can be easily modified to help in this goal. Indeed, Maru\v si\v c and Scapellato's work is actually stronger than advertised through the statement of their results, and an additional goal of this section is to make this stronger work more apparent, as from our work on this paper we believe that such stronger statements may be useful. We note that when writing a wreath product, we use the convention that the first group written is acting on the partition, and the second is acting within each block. Some authors, including Praeger et al, use the opposite order.
\begin{thrm}{\langle}bel{gendigraphauto}
Let $\Gamma$ be a vertex-transitive digraph of order $pq$, where $q < p$ are distinct primes such that $G\le {\rm Aut}(\Gamma)$ is quasiprimitive and has a $G$-invariant partition ${\mathcal B}$ with blocks of size $q$. Additionally, suppose that ${\mathcal B}$ is also an ${\rm Aut}(\Gamma)$-invariant partition. Then $G$ is an almost simple group and one of the following is true:
\begin{enumerate}
\item $\Gamma$ is a nontrivial wreath product and ${\rm Aut}(\Gamma)$ contains $G/{\mathcal B}\wr({\rm Stab}_G(B)\vert_B)$ which contains a regular cyclic subgroup $R$, where $B\in {\mathcal B}$, or {\langle}bel{wreath}
\item $\Gamma$ is isomorphic to a generalized orbital digraph of ${\rm PSL}(2,11)$ that is not a generalized orbital digraph of ${\rm PGL}(2,11)$ of order $55$. Moreover, $\Gamma$ is a Cayley digraph of the nonabelian group of order $55$, and its full automorphism group is ${\rm PSL}(2,11)$, or{\langle}bel{case:2-11}
\item $\Gamma$ is isomorphic to a generalized orbital digraph of ${\rm PSL}(3,2)$ of order $21$ that is not a generalized orbital digraph of ${\rm P\Gamma L}(3,2)$. Moreover, $\Gamma$ is a Cayley digraph of the nonabelian group of order $21$, and its full automorphism group is ${\rm PSL}(3,2)$, or {\langle}bel{case:3-2}
\item $\Gamma$ is not metacirculant; $p=2^{2^s}+1$ is a Fermat prime, and $q$ divides $p-2$. Further, the minimal transitive subgroup $G$ of ${\rm Aut}(\Gamma)$ that admits only a $G$-invariant system of $p$ blocks of size $q$ is isomorphic to ${\rm SL}(2,2^s)$, and ${\rm Aut}(\Gamma)$ is isomorphic to a subgroup of ${\rm Aut}({\rm SL}(2,2^s))$. {\langle}bel{MS-graphs}
\end{enumerate}
\end{thrm}
\begin{proof}
Almost all of the proof is contained in \cite{MarusicS1992}. We analyze the digraph structures essentially as they do in the proof of their main theorem.
Since $G$ is quasiprimitive, it has no nontrivial intransitive normal subgroups. So ${\rm fix}_G({\mathcal B}) = 1$ and $G/{\mathcal B}\cong G$ is of prime degree $p$. As $G$ does not have a normal Sylow $p$-subgroup, neither does $G/{\mathcal B}$, and so by Burnside's Theorem \cite[Corollary 3.5B]{DixonM1996} $G/{\mathcal B}$ is doubly-transitive, and by another theorem of Burnside \cite[Theorem 4.1B]{DixonM1996}, $G/{\mathcal B}$ has nonabelian simple socle. Consequently, $G$ is nonsolvable and $G/{\mathcal B}\cong G$ is almost simple.
The $2$-transitive groups of prime degree are known (they are given for example in \cite[Proposition 2.4]{MarusicS1992}). The various cases, with the one exception of ${\rm PSL}(2,2^k)$, are then analyzed in \cite{MarusicS1992}. They are almost all either rejected as impossible using group theoretic arguments or \cite[Proposition 2.1]{MarusicS1992} (which is purely about the permutation group structure and also applies to our situation), or determined to be metacirculants using \cite[Proposition 2.2]{MarusicS1992}, which is almost sufficient for our purposes. Maru\v{s}i\v{c} and Scapellato in fact showed that whenever $G$ is a group satisfying the hypothesis of \cite[Proposition 2.2]{MarusicS1992}, and $\Gamma$ is a digraph (they only considered graphs but their proof works for digraphs) with $G\le{\rm Aut}(\Gamma)$, then either $\Gamma$ or its complement is disconnected. This implies that ${\rm Aut}(\Gamma)$ is a wreath product, and ${\rm Aut}(\Gamma)$ contains $G/{\mathcal B}\wr({\rm Stab}_G(B)\vert_B)$ which contains a regular cyclic subgroup $R$, where $B\in {\mathcal B}$, which is what we need here. There are two possible group structures for $G$ that do not succumb to this general approach, and \cite{MarusicS1992} use direct arguments to show that the corresponding (di)graphs are metacirculant. We need to address these exceptional possibilities separately.
The first exception occurs in the proof of \cite[Proposition 2.7]{MarusicS1992} when handling the case $G = {\rm PSL}(2,11)$ of degree $55$. In this case it is argued that ${\rm PSL}(2,11)$ contains a regular metacyclic subgroup that has blocks of size $11$. This is a contradiction to the hypothesis of \cite[Proposition 2.7]{MarusicS1992}, so finishes the argument for them; for us, it shows that these digraphs are Cayley digraphs (as claimed), and $(q,p)$-metacirculants. It can be verified in \texttt{magma} \cite{magma} that the only regular subgroup of ${\rm PSL}(2,11)$ in its action on $55$ points is the nonabelian group of order $55$. Since ${\rm PGL}(2,11)$ is primitive, the digraphs that arise in this case are precisely those whose full automorphism group is ${\rm PSL}(2,11)$.
The second exception occurs at the beginning of \cite[Proposition 3.5]{MarusicS1992}, namely when $G = {\rm PSL}(3,2)$ and $\Gamma$ is of order $21$. Here, Maru\v si\v c and Scapellato note that ${\rm PSL}(3,2)$ in its action on $21$ points has a (transitive) nonabelian subgroup of order $21$, and so $\Gamma$ is a metacirculant (which is enough for their purposes, and for us again shows that $\Gamma$ is a Cayley graph on the nonabelian group of order $21$). By the Atlas of Finite Simple Groups the group ${\rm PSL}(3,2)$ in its representation on $21$ points has suborbits of length $1$, $2^2$, $4^2$, and $8$, with the suborbits of lengths $4$ being non self-paired. The action of ${\rm P\Gamma L}(3,2)$ is primitive, so again orbital digraphs of that group do not meet our hypotheses and we are interested only in those digraphs whose full automorphism group is ${\rm PSL}(3,2)$.
Finally, the case where $G$ has socle ${\rm PSL}(2,2^k) = {\rm SL}(2,2^k)$ is mainly analyzed in \cite{MarusicS1994}, where, for example, the orbital digraphs of the groups are determined. In \cite[Theorem]{MarusicS1992}, they show if an imprimitive representation of ${\rm SL}(2,2^k)$ has order $qp$ and is contained in the automorphism group of a metacirculant digraph $\Gamma$ of order $qp$, then either it either contains the complete $p$-partite graph where each partition has size $q$ (and are the blocks of ${\mathcal B}$), or is contained in the complement of this graph. These digraphs are easily seen to be circulant as either $\Gamma$ or its complement is again disconnected. The arithmetic conditions are also derived there.
\end{proof}
From a closer analysis of the suborbits of ${\rm PSL}(2,11)$ and of ${\rm PSL}(3,2)$, we can derive additional information about the digraphs that arise in this analysis. For ${\rm PSL}(3,2)$, we use \texttt{magma} for this analysis.
The orbital digraphs of ${\rm PSL}(2,11)$ are examined in \cite[Example 2.1]{LuX2003}.
The suborbits are of length $1,4,4,4,6,12,12,$ and $12$.
Two suborbits of length $12$ are the only ones that are not self-paired, and the corresponding orbital digraphs have automorphism group ${\rm PSL}(2,11)$ which is imprimitive (as ${\rm PSL}(2,11)$ has disconnected orbital digraphs).
Thus, a generalised orbital digraph that is not a graph must use exactly one of these.
Two suborbits of length $4$ have disconnected orbital graphs and their union is an orbital graph of ${\rm PGL}(2,11)$, while all of the other suborbits are also suborbits of ${\rm PGL}(2,11)$.
Thus, in order to avoid ${\rm PGL}(2,11)$ in the automorphism group of an orbital graph, we must include exactly one of these. We summarize this extra information in the following remark.
\begin{hey}
If $\Gamma$ arises in Theorem~\ref{gendigraphauto}(\ref{case:2-11}) and is a graph, then it has a subgraph of valency $4$ that is a disconnected orbital graph of ${\rm PSL}(2,11)$, and the other disconnected orbital graph of ${\rm PSL}(2,11)$ (which is the image of this one under the action of ${\rm PGL}(2,11)$) is not a subgraph of $\Gamma$ (but $\Gamma$ itself is connected).
If $\Gamma$ arises in Theorem~\ref{gendigraphauto}(\ref{case:2-11}) and is not a graph, then it has a subdigraph of valency $12$ that is a non-self-paired orbital digraph of ${\rm PSL}(2,11)$, and whose paired orbital digraph of ${\rm PSL}(2,11)$ is not a subdigraph of $\Gamma$.
If $\Gamma$ arises in Theorem~\ref{gendigraphauto}(\ref{case:3-2}) then \texttt{magma} \cite{magma} has been used to verify that $\Gamma$ cannot be a graph. It has a subdigraph of valency $4$ that is a non-self-paired orbital digraph of ${\rm PSL}(3,2)$, and whose paired orbital digraph of ${\rm PSL}(3,2)$ is not a subdigraph of $\Gamma$.
\end{hey}
\begin{hey}
There are several instances, other than the complete graph and its complement, where a quasiprimitive group $G$ with nontrivial $G$-invariant partition ${\mathcal B}$, is contained in the full automorphism group of a digraph $\Gamma$ of order $qp$, but the automorphism group ${\rm Aut}(\Gamma)$ is primitive. We list the exceptions or not in the same order as in Theorem \ref{gendigraphauto}:
\begin{enumerate}
\item There are no such cases if Theorem \ref{gendigraphauto} (\ref{wreath}) holds as ${\mathbb Z}_{qp}$ is a Burnside group \cite[Corollary 3.5A]{DixonM1996}. This implies ${\rm Aut}(\Gamma)$ is doubly-transitive and so ${\rm Aut}(\Gamma)=S_{qp}$.
\item If Theorem \ref{gendigraphauto} (\ref{case:2-11}) holds then there are graphs whose automorphism group is primitive and equal to ${\rm PGL}(2,11)$ on $55$ points that contain the quasiprimitive and imprimitive representation of ${\rm PSL}(2,11)$ on $55$ points. These graphs are explicitly described in \cite[Lemma 4.3]{PraegerX1993}.
\item If Theorem \ref{gendigraphauto} (\ref{case:3-2}) holds, then there are graphs whose automorphism group is ${\rm P\Gamma L}(3,2)$ in its primitive representation on $21$ points that contains the quasiprimitive and imprimitive representation of ${\rm PSL}(3,2)$ on $21$ points. These graphs are explicitly described in \cite[Example 2.3]{WangX1993}.
\item If Theorem \ref{gendigraphauto} (\ref{MS-graphs}) holds, then there are graphs whose automorphism group is primitive but contains the quasiprimitive and imprimitive representation of ${\rm SL}(2,2^{2^s})$ on $qp$ points. These graphs are explicitly described in the proof of \cite[Theorem 2.1]{MarusicS1994}, starting in the last paragraph on page 192.
\end{enumerate}
\end{hey}
\section{Automorphism groups of Maru\v si\v c-Scapellato digraphs}
We turn now to the next ``gap" in information about vertex-transitive digraphs of order $pq$, where there is also an error. The gap is that there is not an algorithm to calculate the full automorphism group of every vertex-transitive digraph of order $pq$.
The automorphism groups of circulant digraphs of order $pq$ are found in \cite{KlinP1981}.
One of the authors of this paper determined the automorphism groups of metacirculant graphs of order $pq$ that are not circulant \cite{Dobson2006a} and that argument works for digraphs as well provided that the full automorphism group is not an almost simple group (we dealt with this last possibility in the previous section). Praeger and Xu \cite{PraegerX1993} determined the full automorphism group of graphs of order $pq$ in every case where that group is acting primitively. In light of Theorem~\ref{thm:MS1992}, this means that the gap in the problem of determining the full automorphism group of vertex-transitive digraphs of order a product of two distinct primes reduces to determining the automorphism groups of imprimitive Maru\v si\v c-Scapellato digraphs of order $pq$ that are not metacirculant graphs, where $p$ is a Fermat prime, and $q$ divides $p-2$. In the process of filling this gap we will fix an error in \cite{PraegerX1993}. Unfortunately, there is also an error of omission in \cite{PraegerX1993} that we will need to correct, but we leave this for Section~\ref{sec:prim-errors}.
For the remainder of this section, we may therefore assume that $p=2^{2^t}+1$ is a Fermat prime, and $q$ is a divisor of $p-2$ (so $t\ge1$). For convenience, we write $s=2^t$. This means we are considering a restricted subclass of Maru\v{s}i\v{c}-Scapellato digraphs, since the original definition allowed $p=2^s$ without any conditions on $s$.
The Maru\v si\v c-Scapellato digraphs are vertex-transitive digraphs that are generalized orbital digraphs of ${\rm SL}(2,2^s)$. They were first studied by Maru\v si\v c and Scapellato in \cite{MarusicS1992,MarusicS1993}. Praeger, Wang, and Xu \cite{PraegerWX1993} determined the automorphism groups of Maru\v si\v c-Scapellato graphs of order $pq$ that are also symmetric (i.e.\ arc-transitive), partially filling the gap we are addressing here. One of the authors of this paper studied the full automorphism groups of Maru\v si\v c-Scapellato graphs in \cite{Dobson2016} and was able to say a great deal about them, but left their complete determination as an open problem \cite[Problem 1]{Dobson2016}.
We now discuss the construction of Maru\v si\v c-Scapellato graphs, using a combination of the approaches followed in \cite{Dobson2016} and \cite{MarusicS1993}.
Let $I_2$ be the $2\times 2$ identity matrix, and set $Z = \{aI_2:a\in{\mathbb F}_{2^{s}}^*\}$, the set of all {\bf scalar matrices}. The name $Z$ is chosen as $Z = {\rm Z}({\rm GL}(2,2^s))$, the center of ${\rm GL}(2,2^s)$. Let $\mathbb F_{2^s}^2$ denote the set of all $2$-dimensional vectors whose entries lie in $\mathbb F_{2^s}$. Clearly ${\rm SL}(2,2^s)$ is transitive on ${\mathbb F}_{2^s}^2 - \{(0,0)\}$. It is also clear that ${\rm SL}(2,2^s)$ permutes the {\bf projective points} ${\rm PG}(1,2^s)$, where a projective point is the set of all vectors other than $(0,0)$ that lie on a line. Notice that there are $2^s+1$ projective points, and ${\rm PG}(1,2^s)$ is an invariant partition of ${\rm SL}(2,2^s)$ in its action on ${\mathbb F}_{2^s}^2 - \{(0,0)\}$ with $2^s + 1$ blocks of size $2^s - 1$. This action is faithful. That is, ${\rm SL}(2,2^s)/{\rm PG}(1,2^s)\cong{\rm SL}(2,2^s)$, or equivalently, ${\rm fix}_{{\rm SL}(2,2^s)}({\rm PG}(1,2^s)) = 1$.
It is traditional to identify the projective points with elements of ${\mathbb F}_{2^s}\cup\{\infty\}$ in the following way: The nonzero vectors in the one-dimensional subspace generated by $(1,0)$, will be identified with $\infty$. Any other one-dimensional subspace is generated by a vector of the form $(c,1)$, where $c\in{\mathbb F}_{2^s}$. The nonzero vectors in the one-dimensional subspace generated by $(c,1)$ will be identified with $c$.
For $a\in{\mathbb F}_{2^s}^*$, let $\sqrt{a}$ be the unique element of ${\mathbb F}_{2^s}^*$ whose square is $a$, and
$$k_a = \left[\begin{array}{ll}
\sqrt{a} & 0\\
0 & \sqrt{a}^{-1}
\end{array}\right].$$
\noindent Set $K = \{k_a:a\in{\mathbb F}_{2^s}^*\}$. It is clear that $k_a$ stabilizes the projective point $\infty$ and that for any generator $\omega$ of ${\mathbb F}^*_{2^s}$, ${\langle}ngle k_\omega{\rangle}ngle=K$ is cyclic (of order $2^s-1$), since $\sqrt{\omega}$ also generates $\mathbb F^*_{2^s}$.
Additionally, it is clear that every element of the set-wise stabilizer of $\infty$ in ${\rm SL}(2,2^k)$ has the same action on $\infty$ as some element of $K\vert_{\infty}$ (the entry in the top-right position is irrelevant to the action on $\infty$).
Let $J\le K$ be the unique subgroup of order $\ell$, where $\ell$ is a fixed divisor of $2^s - 1$ (under our assumptions, we will take $\ell=(2^s-1)/q$). By \cite[Exercise 1.5.10]{DixonM1996}, every orbit of $J\vert_\infty$ is a block of ${\rm SL}(2,2^s)$, and so ${\rm SL}(2,2^s)$ has an invariant partition ${\mathcal D}_\ell$ with blocks of size $\ell$ (the blocks whose points lie ``within" the projective point $\infty$ of ${\rm PG}(1,2^s)$ -- that is, those blocks consisting of points whose second entry is $0$ -- are the orbits of $J\vert_\infty$, and the other blocks are the images of these orbits under ${\rm SL}(2,2^s)$). These blocks of $\mathcal D_\ell$ will be the vertices of the generalised orbital digraphs of ${\rm SL}(2,2^s)$, and it is the action of ${\rm SL}(2,2^s)$ on these blocks that produces the Maru\v si\v c-Scapellato digraphs. Under our assumptions, there are $pq$ blocks in $\mathcal D_\ell$.
As mentioned above, the blocks of $\mathcal D_\ell$ are the images of the orbits of $J\vert_{\infty}$ under the action of ${\rm SL}(2,2^s)$, so each lies within a point of ${\rm PG}(1,2^s)$; that is, ${\mathcal D}_\ell\preceq{\rm PG}(1,2^s)$.
Now ${\rm SL}(2,2^s)/{\mathcal D}_\ell$ is a faithful representation of ${\rm SL}(2,2^s)$ (as ${\rm fix}_{{\rm SL}(2,2^s)}({\rm PG}(1,2^s)) = 1$ and ${\mathcal D}_\ell\preceq{\rm PG}(1,2^s)$). Additionally, the ${\rm SL}(2,2^s)$-invariant partition ${\rm PG}(1,2^s))$ induces the invariant partition ${\mathcal B} = {\rm PG}(1,2^s)/{\mathcal D}_\ell$ of ${\rm SL}(2,2^s)/{\mathcal D}_\ell$, and $\mathcal B$ consists of $2^s + 1$ blocks whose size in general is $m = (2^s - 1)/\ell$ (under our assumptions, $m=q$). We will use the notation ${\mathcal B} = {\rm PG}(1,2^s)/{\mathcal D}_\ell$ throughout this section. The elements of $\mathcal B$ will be the blocks of our digraphs of order $pq$, and will have size $q$ (and there are $p$ of them), so for our purposes and henceforth in this section, we have $q=m=(2^s-1)/\ell$. It is shown in \cite{MarusicS1993} that ${\mathcal B}$ is the unique complete block system of ${\rm SL}(2,2^s)/{\mathcal D}_\ell$ with blocks of size $q$. The following result is \cite[Lemma 2.3]{MarusicS1993}.
\begin{lem}{\langle}bel{orbitalint}
${\rm SL}(2,2^s)/{\mathcal D}_\ell$ has $q$ suborbits of length $1$ and $q$ suborbits of length $2^s$. Additionally, for a suborbit $S$ of length $2^s$, $\vert S\cap(c/{\mathcal D}_\ell)\vert = 1$ for every projective point $c\in{\rm PG}(1,2^s)$.
\end{lem}
Note that this implies that the valency of an orbital digraph of ${\rm SL}(2,2^s)$ is either $1$ or $2^s$. Additionally, as ${\rm SL}(2,2^s)/{\rm PG}(1,2^s) = {\rm PSL}(2,2^s)$ is doubly-transitive, the previous result also implies that the orbital digraphs of of ${\rm SL}(2,2^s)/{\mathcal D}_\ell$ having valency $2^s$ are graphs. We now define Maru\v si\v c-Scapellato digraphs, and the fact that some orbital digraphs are graphs and some are not will cause us to naturally define these digraphs in terms of the edges which are not arcs as well as arcs that need not be edges.
\begin{defin}{\langle}bel{MSdigraphs}
Let $s$ be a positive integer, and $q$ a divisor of $2^s - 1$, $S\subset{\mathbb Z}_q^*$, $\emptyset \subseteq T\subseteq{\mathbb Z}_q$, and $\omega$ a primitive element of ${\mathbb F}_{2^s}$. The digraph $X(2^s,q,S,T)$ has vertex set ${\rm PG}(1,2^s)\times{\mathbb Z}_q$. The out-neighbors of $(\infty,r)$ are $\{(\infty,r + a):a\in S\}$ while the neighbors of $(\infty,r)$ are $\{(y,r+b):y\in{\mathbb F}_{2^s},b \in T\}$. The out-neighbors of $(x,r)$, $x\in{\mathbb F}_{2^s}$, are given by $\{(x,r + a):a\in S\}$ while the neighbors of $(x,r)$ are
$$\{(\infty,r-b):b \in T\}\cup\{(x + \omega^i,-r+b+2i):i\in{\mathbb Z}_{2^s - 1},b \in T\}.$$
\noindent The digraph $X(2^s,q,S,T)$ is a {\bf Maru\v si\v c-Scapellato digraph}.
\end{defin}
In \cite{MarusicS1993} Maru\v si\v c and Scapellato only defined graphs, but their definition, with the obvious modifications, also define digraphs as above - see \cite{MarusicS1994a}. Additionally, they required that $\emptyset\subset T\subset{\mathbb Z}_q$ as they wished their family to be disjoint from other already known families of graphs. If $\emptyset = T$ or $T = {\mathbb Z}_q$ then the resulting digraphs are either disconnected or complements of disconnected digraphs, and so have automorphism group either a nontrivial wreath product or a symmetric group. They also showed that with their definition, Maru\v si\v c-Scapellato digraphs are isomorphic to some, but not all, generalized orbital digraphs of ${\rm SL}(2,2^s)/{\mathcal D}_\ell$. We prefer the more general definition that includes all generalized orbital digraphs of ${\rm SL}(2,2^s)/{\mathcal D}_\ell$.
However, the distinction Maru\v si\v c and Scapellato made is also important, so if $T = \emptyset$ or ${\mathbb Z}_q$, we will call such a Maru\v si\v c-Scapellato digraph a {\bf degenerate Maru\v si\v c-Scapellato digraph}.
We have said that the vertices of the Maru\v si\v c-Scapellato digraphs are the blocks of $\mathcal D_\ell$. The blocks of $\mathcal D_\ell$ are two-dimensional vectors that are subsets of projective points; in fact, it may be useful to the reader if we describe the blocks of $\mathcal D_\ell$ more precisely here. We assume that a primitive element $\omega$ of ${\mathbb F}_{2^s}$ has been chosen and is fixed. Then each block $D \in \mathcal D_\ell$ has one of the following forms:
\begin{eqnarray*}
\{(\sqrt{\omega}^{qj+r},0): 0 \le j \le \ell-1\} & &\text{(in the projective point $\infty$)}\\
\{(\sqrt{\omega}^{qj+c+r},\sqrt{\omega}^{qj+m}):0 \le j \le \ell-1\}&&\text{(in the projective point $\sqrt{\omega}^c$)}\\
\{(0,\sqrt{\omega}^{qj+r}): 0 \le j \le \ell-1\}&&\text{(in the projective point $0$),}
\end{eqnarray*}
for some fixed $0 \le r \le q-1$. The action of any element of ${\rm SL}(2,2^s)$ on any one of these sets is easy to calculate. Clearly, the definition that we have given for the Maru\v si\v c-Scapellato graphs does not have these sets as vertices; its vertices are the elements of ${\rm PG}(1,2^s) \times {\mathbb Z}_q$.
In \cite[Theorem 3.1]{MarusicS1993}, Maru\v si\v c and Scapellato show that the imprimitive orbital digraphs of ${\rm SL}(2,2^s)$ whose block systems come from the projective points, are precisely the Maru\v si\v c-Scapellato digraphs with the correct correspondence chosen between the blocks of $\mathcal D_\ell$ and the elements of ${\rm PG}(1,2^s)\times {\mathbb Z}_q$. They describe explicitly how certain matrices act on elements of ${\rm PG}(1,2^s)\times {\mathbb Z}_q$.
The action on the first coordinate is straightforward; the set of blocks of $\mathcal D_\ell$ that lie in a particular projective point will correspond to the set of vertices of the digraph whose label has that first coordinate. Thus, any matrix will map a vertex whose first coordinate is some projective point, to a vertex whose first coordinate is the image of that projective point under that matrix. However, the action on the second coordinate is less clear, and this is what they explain in more detail.
In Equations~(10) and~(12) of \cite{MarusicS1993}, they explain that the labeling of the vertices is chosen so that $k_\omega(\infty,r)=(\infty,r+1)$ (where $r \in {\mathbb Z}_q$), and $k_\omega(c,r)=(c\omega,r+1)$ for any projective point $c$ other than $\infty$. They also introduce a family of matrices $$h_b=\left[\begin{matrix}1&b\\0&1\end{matrix}\right],$$ where $b \in {\mathbb F}_{2^s}$. Observe that $H=\{h_b: b \in {\mathbb F}_{2^s}\}$ is a group, and in fact since ${\mathbb F}_{2^s}$ has characteristic $2$, $H$ is an elementary abelian $2$-group.
They note in the paper that the stabilizer of $\infty$ in ${\rm SL}(2,2^s)$ is the set of upper triangular matrices, and this is generated by $k_\omega$ together with $H$. In Equation~(14), they observe that under their labeling, $h_b(\infty,r)=(\infty,r)$, and $h_b(c,r)=(c+b,r)$ when $c$ is any projective point other than $\infty$.
From this point on, our assumptions that $s=2^t$, $p=2^s+1$ is a Fermat prime, and $q$ divides $p-2$ become important.
With the information we now have in hand, we are ready to understand how diagonal matrices act on the vertex labels from ${\rm PG}(1,2^s)\times {\mathbb Z}_q$; this will be valuable to us.
\begin{lem}{\langle}bel{scalar in MS-labeling}
Let $\omega$ be a primitive root of ${\mathbb F}_{2^s}$, so that $\sqrt{\omega}$ is also a primitive root of ${\mathbb F}_{2^s}$. Then the permutation $\sqrt{\omega}I_2$ acts on vertices labeled with elements of ${\rm PG}(1,2^s) \times {\mathbb Z}_q$ by satisfying
$$\sqrt{\omega} I_2(\infty,r) = (\infty, r + 1)\qquad{\text and}\qquad\sqrt{\omega}I_2(c,r) = (c,r - 1),\text{ when } c\neq \infty.$$
\end{lem}
\begin{proof}
Observe that a point of ${\mathbb F}_{2^s}^2$ that lies in the projective point $\infty$ has $0$ as its second entry, so the action of $k_\omega$ on such a point must be identical to the action of $\sqrt{\omega}I_2$. Thus, any set $D \in \mathcal D_\ell$ of points lying in the projective point $\infty$ must have the same image under $\sqrt{\omega}I_2$ as under $k_{\omega}$. If $D$ corresponds to the vertex labelled $(\infty,r)$, then since Maru\v si\v c and Scapellato have told us that $k_\omega(\infty,r)=(\infty,r+1)$, it must also be the case that $\sqrt{\omega}I_2(\infty,r)=(\infty,r+1)$.
Similarly, a point of ${\mathbb F}_{2^s}^2$ that lies in the projective point $0$ has $0$ as its first entry, so the action of $k_{\omega^{-1}}=(k_\omega)^{-1}$ on such a point must be identical to the action of $\sqrt{\omega}I_2$. Again,
Maru\v si\v c and Scapellato have told us that $k_\omega(0,r-1)=(0,r)$, it must be the case that $\sqrt{\omega}I_2(0,r)=(0,r-1)$.
Finally, consider any point of ${\mathbb F}_{2^s}^2$ that lies in the projective point $c$ where $c \neq 0, \infty$, so $c=\sqrt{\omega}^t$ for some $t$. Then the point of ${\mathbb F}_{2^s}^2$ has the form $(\sqrt{\omega}^{qi+t+m},\sqrt{\omega}^{qi+m})$ for some $0 \le i \le \ell-1$ and $0 \le m \le q-1$. Straightforward calculations using the field's characteristic of $2$ show that the action of the matrix $\sqrt{\omega}^{-1}I_2$ has the same effect on such a point as the action of the matrix
$$h_ck_\omega h_c=\left[\begin{matrix}\sqrt{\omega} & \sqrt{\omega}^t(\sqrt{\omega}+\sqrt{\omega}^{-1})\\ 0 & \sqrt{\omega}^{-1}\end{matrix}\right].$$
Using the information from Maru\v si\v c and Scapellato, we know that $h_c(c,r)=(c+c,r)=(0,r)$, $k_\omega(0,r)=(0,r+1)$, and $h_c(0,r+1)=(c,r+1)$. Thus, we must also have $\sqrt{\omega}^{-1}I_2(c,r)=(c,r+1)$, and hence $\sqrt{\omega}I_2(c,r)=(c,r-1)$.
\end{proof}
Let $F:{\mathbb F}_{2^s}\mapsto{\mathbb F}_{2^s}$ be the Frobenius automorphism, and so be given by $F(x) = x^2$. The Frobenius automorphism induces an automorphism $f$ of ${\rm GL}(2,2^s)$ in the natural way - by applying $F$ to the entries of the standard matrix of an element of ${\rm GL}(2,2^s)$.
Observe that since the Frobenius automorphism is an automorphism, we have $Z \cap {\rm SL}(2, 2^s)=\{I_2\}$. Furthermore, every element of $\mathbb F_{2^s}$ is a square, and so every element of $\mathbb F_{2^s}$ arises as the determinant of some matrix in $Z$. Therefore ${\langle}ngle {\rm SL}(2,2^s),Z{\rangle}ngle={\rm GL}(2,2^s)$. Since we know that ${\rm SL}(2,2^s),Z \triangleleft {\rm GL}(2,2^s)$, this implies that ${\rm GL}(2,2^s)={\rm SL}(2,2^s)\times Z$.
We need to introduce some additional notation that will be used throughout the remainder of this section. We use ${\rm \Gamma L}(2,2^s)$ to denote the group ${\rm GL}(2,2^s)\rtimes{\langle}ngle f{\rangle}ngle$. We also use ${\rm \Sigma L}(2,2^s)$ to denote ${\rm SL}(2,2^s)\rtimes{\langle}ngle f{\rangle}ngle$. We know that ${\rm GL}(2,2^s)={\rm SL}(2,2^s)\times Z$, so ${\rm \Gamma L}(2,2^s)=({\rm SL}(2,2^s)\times Z)\rtimes{\langle}ngle f{\rangle}ngle$ where the action of $f$ leaves ${\rm SL}(2,2^s)$ and $Z$ invariant.
\begin{lem}
{\langle}bel{diagonals don't normalize2}
Let $p = 2^s + 1$ be a Fermat prime and $q\vert(2^s - 1)$ a prime, with $q\ell=(2^s-1)$. Let $a$ be the order of $2$ modulo $q$, let $b$ be a divisor of ${\rm gcd}(a,s)$ with $b \neq a$, and let $1\neq L = {\langle} f^b {\rangle}$ (where $f$ is the automorphism of ${\rm GL}(2,2^s)$ induced by the Frobenius automorphism, as described above).
If $1\not = z/{\mathcal D}_\ell\in Z/{\mathcal D}_\ell$ , then $z^{-1} {\langle} {\rm SL}(2,2^s),L {\rangle} z/{\mathcal D}_\ell \not = {\langle} {\rm SL}(2,2^s),L {\rangle}/{\mathcal D}_{ \ell}$.
\end{lem}
\begin{proof}
Let $1\neq L ={\langle} f^b {\rangle}\leq {\langle} f {\rangle}$ and let $G= {\langle} {\rm SL}(2,2^s),L {\rangle}$.
Towards a contradiction, suppose that $1 \not = z/{\mathcal D}_\ell\in Z/{\mathcal D}_\ell$, and $z^{-1} G z/{\mathcal D}_\ell = G/{\mathcal D}_{ \ell}$. Let $Y = {\langle} z/{\mathcal D}_\ell{\rangle}$. As $Z{\triangleleft} {\rm \Gamma L}(2,2^s)$ is cyclic and $Y$ is the unique subgroup of $Z/{\mathcal D}_\ell$ of order $\vert Y\vert$, $Y{\triangleleft} {\rm \Gamma L}(2,2^s)/{\mathcal D}_\ell$.
Since $z^{-1} G z/{\mathcal D}_\ell = G/{\mathcal D}_{ \ell}$, it follows that $G/{\mathcal D}_\ell{\triangleleft} {\langle} Y,G/{\mathcal D}_\ell{\rangle}$. Moreover, since $Y\cap G/{\mathcal D}_\ell = 1$ (this follows from ${\rm GL}(2,2^s)={\rm SL}(2,2^s) \times Z$), we see ${\langle} Y,G/{\mathcal D}_\ell{\rangle}\cong Y\times G/{\mathcal D}_\ell$.
In particular, $z/{\mathcal D}_\ell$ commutes with $f^b/{\mathcal D}_\ell$.
Choose $i$ such that $z=\sqrt{\omega} ^i I_2\in Z$, for some fixed generator $\omega$ of $\mathbb F_{2^s}^*$ ($\sqrt{\omega}$ also generates $\mathbb F_{2^s}^*$). It is straightforward to verify that $z^{-1}f^bzf^{-b} =z^{2^b-1}=\sqrt{\omega}^{i(2^b-1)} I_2$.
On the other hand, since $z/{\mathcal D}_\ell$ commutes with $f/{\mathcal D}_\ell$,
it follows that $(z^{-1}f^bzf^{-b})/{\mathcal D}_\ell=1$, implying $\sqrt{\omega}^{i(2^b-1)} I_2/{\mathcal D}_\ell=1$.
Observe that each block of $\mathcal D_\ell$ has the form $\{(x\sqrt{\omega}^{qj},y\sqrt{\omega}^{qj}): 0 \le j <\ell\},$ for some $x,y \in \mathbb F_{2^s}$. This implies that ${\rm fix}_{Z}({\mathcal D}_\ell)={\langle} \sqrt{\omega}^qI_2 {\rangle} $. Therefore $\sqrt{\omega}^{i(2^b-1)} I_2/{\mathcal D}_\ell=1$ if and only if $\sqrt{\omega}^{i(2^b-1)}\in {\langle} \sqrt{\omega}^q {\rangle}$.
We conclude that $i(2^b-1)\equiv q \pmod{2^s-1}$. Since $q$ divides $2^s-1$, it follows that $q$ divides $i(2^b-1)$. Recall that $a$ is the order of $2$ modulo $q$ and $b<a$. This implies that $2^b\not \equiv 1 \pmod{q}$ and therefore $q$ does not divide $2^b-1$.
Since $q$ is a prime, and $q$ divides $i(2^b-1)$, this implies that $q$ divides $i$. However, this means that $z\in {\langle} \sqrt{\omega}^qI_2 {\rangle}= {\rm fix}_{Z}({\mathcal D}_\ell)$ contradicting the assumption that $ z/{\mathcal D}_\ell \not =1$.
This contradiction establishes that $z^{-1}Gz/{\mathcal D}_\ell\not = G/{\mathcal D}_\ell$, as claimed.
\end{proof}
The first error that we will correct concerns the classification of symmetric Maru\v si\v c-Scapellato graphs given in \cite[Theorem, as it relates to (3.8)]{PraegerWX1993}. In that paper, Lemma 4.9(a) states that the $q$ connected orbital graphs $X(2^s,q,\emptyset,\{t\})$ (where $t \in {\mathbb Z}_{q}$) of ${\rm SL}(2,2^s)$ all have automorphism group ${\rm \Sigma L}(2,2^s)$ (this group is written in \cite{PraegerWX1993} as ${\rm \Gamma L}(2,2^s)$, but it is clear from the proof of \cite[Theorem 3.7]{PraegerWX1993} that they mean the group we are denoting by ${\rm \Sigma L}(2,2^s)$).
Using Lemma~\ref{scalar in MS-labeling} to understand the action of $Z$ on these graphs, we see that for any $z \in Z$ with $z \neq 1$, there exists some $t' \in {\mathbb Z}_q^*$ such that $X(2^s,q,\emptyset,\{t\})^z=X(2^s,q,\emptyset,\{t-t'\})$ (more precisely, if $z=\sqrt{\omega}^iI_2$, then $t'=2i$). Thus, every such $z$ acts as a cyclic permutation on this set of $q$ graphs.
Suppose that $\Gamma$ and $\Gamma^z$ are two of these orbital digraphs with $z/\mathcal D_\ell \neq 1$ (so that the graphs are distinct), and ${\rm Aut}(\Gamma)={\rm \Sigma L}(2,2^s)$ as claimed in \cite{PraegerWX1993}.
Then ${\rm Aut}(\Gamma^z)=z^{-1}({\rm Aut}(\Gamma))z,$ and by taking $b=1$ in Lemma~\ref{diagonals don't normalize2} we see that
${\rm \Sigma L}(2,2^s)/\mathcal D_\ell \neq z^{-1}{\rm \Sigma L}(2,2^s)z/\mathcal D_\ell$,
contradicting their claim that ${\rm Aut}(\Gamma^z)={\rm Aut}(\Gamma)$.
The mathematical error leading to the incorrect statement of \cite[Lemma 4.9]{PraegerWX1993} actually arises in \cite[Lemma 4.8]{PraegerWX1993} where it is concluded that the automorphism group $G$ of any Maru\v si\v c-Scapellato graph satisfies ${\rm SL}(2,2^s)/{\mathcal D}_\ell\le G\le {\rm \Sigma L}(2,2^s)/{\mathcal D}_\ell$ (using our notation). The proof of \cite[Lemma 4.8]{PraegerWX1993} only gives that $G/{\mathcal B} = {\rm \Sigma L}(2,2^s)/{\rm PG}(1,2^s) = {\rm P\Sigma L}(2,2^s)$. If we consider any of the groups that are conjugate to ${\rm \Sigma L}(2,2^s)$ by a scalar matrix, which we have shown in Lemma \ref{diagonals don't normalize2} are distinct modulo $\mathcal D_\ell$, the fact that scalar matrixes fix every point of ${\rm PG}(1,2^s)$ shows that every such group satisfies this equation. With that said, the proof of \cite[Lemma 4.9 (b)]{PraegerWX1993} is correct if we strengthen the hypothesis to assume that ${\rm SL}(2,2^s)/{\mathcal D}_\ell\le G\le{\rm \Sigma L}(2,2^s)/{\mathcal D}_\ell$. So we can restate their result correctly as follows, to identify the symmetric Maru\v si\v c-Scapellato digraphs whose automorphism group is contained in ${\rm \Sigma L}(2,2^s)/\mathcal D_\ell$.
Note that when $G={\rm Aut}(\Gamma)$ where $\Gamma$ is one of these Maru\v si\v c-Scapellato graphs, and ${\rm SL}(2,2^s)/{\mathcal D}_\ell\le G\le{\rm \Sigma L}(2,2^s)/{\mathcal D}_\ell$, all of these actions on $\mathcal D_\ell$ are faithful, so that ${\rm SL}(2,2^s)/{\mathcal D}_\ell\cong {\rm SL}(2,2^s)$, $G/\mathcal D_\ell \cong G$, and ${\rm \Sigma L}(2,2^s)/\mathcal D_\ell \cong {\rm \Sigma L}(2,2^s)$. In \cite{PraegerWX1993}, they were to some extent studying the abstract structure of these groups, and did not make this distinction, which may have contributed to the confusion and does lead to our statement looking somewhat different from theirs.
\begin{thrm}[see \cite{PraegerWX1993}, Lemma 4.9]{\langle}bel{PraegerWXsymmetric}
Let $p = 2^s + 1$ be a Fermat prime and $q\vert(2^s - 1)$ be prime. Let $\Gamma = X(2^s,q,S,T)$ be a symmetric Maru\v si\v c-Scapellato digraph and assume that ${\rm SL}(2,2^s)/\mathcal D_\ell\le {\rm Aut}(\Gamma) \le {\rm \Sigma L}(2,2^s)/\mathcal D_\ell$. Let $a$ be the order of $2$ modulo $q$. Then $S = \emptyset$ and one of the following is true:
\begin{enumerate}
\item $T = \{0\}$, $\Gamma$ has valency $q$, and automorphism group ${\rm \Sigma L}(2,2^s)/\mathcal D_\ell$.
\item There is a divisor $b$ of ${\rm gcd}(a,s)$ and $1 < a/b < q - 1$ such that $T = U_{b,i} = \{i2^{bj}:0\le j < a/b\}$. There are exactly $(q-1)/a$ distinct graphs of this type for a given $b$, each of valency $qa/b$, and the automorphism group of each is ${\langle} {\rm SL}(2,2^s),L{\rangle} /{\mathcal D}_\ell$ where $L\le {\langle} f{\rangle}$ is of order $s/b$. Up to isomorphism, there are exactly $(q-1)/b$ such graphs.
\end{enumerate}
\end{thrm}
Before turning to the characterization of symmetric Maru\v si\v c-Scapellato graphs of order $qp$, we will need a solution to the isomorphism problem for these graphs. This problem has been solved in \cite{Dobson2016}, but the solution there is not suited to our needs. The solution given in \cite{Dobson2016} is also perhaps not optimal in the sense that it requires one check $\vert{\rm \Sigma L}(2,2^s)\vert = \vert{\rm Aut}({\rm SL}(2,2^s))\vert$ maps to determine isomorphism, while we show in the next result that one only needs to check $qs$ maps.
\begin{thrm}{\langle}bel{New MS iso}
Let $p = 2^s + 1$ be a Fermat prime, $q\vert(2^s - 1)$ a prime, and $\Gamma,\Gamma'$ be non-degenerate Maru\v si\v c-Scapellato digraphs. Then $\Gamma$ and $\Gamma'$ are isomorphic if and only if $\delta(\Gamma) = \Gamma'$, where $\delta\in{\langle} Z,f{\rangle}/{\mathcal D}_\ell$.
\end{thrm}
\begin{proof}
It is shown in \cite[Theorem 1]{Dobson2016} that $\Gamma' = \delta(\Gamma)$ for some $\delta$ if and only if this occurs for a $\delta$ that normalizes ${\rm SL}(2,2^s)$. This normalizer is ${\rm \Gamma L}(2,2^s)$ as every element of $S_{qp}$ that normalizes ${\rm SL}(2,2^s)$ can be written in the form $abc$, where $c\in{\rm SL}(2,2^s)$, $b\in{\rm Aut}({\rm SL}(2,2^s))$, and $a$ is contained in the centralizer of ${\rm SL}(2,2^s)$ in $S_{qp}$. As ${\rm SL}(2,2^s) = {\rm PSL}(2,2^s)$ and ${\rm Aut}({\rm PSL}(2,2^s)) = {\rm P\Sigma L}(2,2^s)$, we may take $b\in{\langle} f{\rangle}$. As the centralizer in $S_{qp}$ of ${\rm SL}(2,2^s)$ has order $q$ by \cite[Theorem 4.2A (i)]{DixonM1996} and \cite[Lemma 2.1]{MarusicS1993}, we see that $a\in Z/{\mathcal D}_\ell$.
\end{proof}
We are now ready to determine the symmetric Maru\v si\v c-Scapellato digraphs of order a product of two distinct primes with imprimitive automorphism group.
\begin{thrm}{\langle}bel{MS symmetric}
Let $s = 2^t$, $p = 2^s + 1$ be a Fermat prime, and $q\vert(2^s - 1)$ be prime. Let $\Gamma = X(2^s,q,S,T)$ be a nondegenerate symmetric Maru\v si\v c-Scapellato digraph constructed with the primitive root $w$ of ${\mathbb F}_{2^s}$ with an imprimitive automorphism group. Let $a$ be the order of $2$ modulo $q$, and $d = \sqrt{w}I$. Then $S = \emptyset$ and one of the following is true:
\begin{enumerate}
\item $T = \{-2k\}$, $\Gamma$ has valency $q$, and automorphism group $d^{-k}{\rm \Sigma L}(2,2^s)d^k/{\mathcal D}_\ell$, $k\in{\mathbb Z}_q$.
\item There is a divisor $b$ of ${\rm gcd}(a,2^s)$, $1 < d/e < q - 1$, and $k\in{\mathbb Z}_q$ such that $T = U_{b,i,k} = \{i2^{bj} - 2k:0\le j < a/b\}$. There are exactly $(q-1)b/a$ distinct graphs of this type for a given $b$ and $k$, each of valency $qa/b$, and the automorphism group of each is $d^{-k}{\langle} SL(2,2^s),L{\rangle} d^k/{\mathcal D}_{\ell}$ where $L\le {\langle} f{\rangle}$ is of order $2^s/b$. Up to isomorphism, there are exactly $(q-1)/b$ such graphs.
\end{enumerate}
\end{thrm}
\begin{proof}
For the proof of this result, we will abuse notation by writing $H$ instead of $H/{\mathcal D}_\ell$ where $H/{\mathcal D}_\ell\le d^{-k}{\rm \Sigma L}(2,2^s)d^k/{\mathcal D}_\ell$, and will similarly abuse notation for elements of $d^{-k}{\rm \Sigma L}(2,2^s)d^k/{\mathcal D}_\ell$. This should cause no confusion. The result follows by Theorem \ref{PraegerWXsymmetric} if ${\rm Aut}(\Gamma)\le {\rm \Sigma L}(2,2^s)/$, in which case $k = 0$. Suppose that ${\rm Aut}(\Gamma)$ is not contained in ${\rm \Sigma L}(2,2^s)/$. As ${\rm Aut}(\Gamma)$ is imprimitive and contains ${\rm SL}(2,2^s)/$, by \cite[Theorem]{MarusicS1992} either $\Gamma$ is metacirculant or the only invariant partition of ${\rm Aut}(\Gamma)$ is ${\mathcal B} = {\rm PG}(1,2^s)$ which is also the only invariant partition of ${\rm SL}(2,2^s)$. If $\Gamma$ is metacirculant, then it is degenerate by \cite[Theorem 2.1]{MarusicS1994}. Hence $\Gamma$ is not metacirculant and so ${\rm fix}_{{\rm Aut}(\Gamma)}({\mathcal B}) = 1$ by \cite[Theorem 3.4]{Marusic1988}. Then ${\rm Aut}(\Gamma)/{\mathcal B}\cong{\rm Aut}(\Gamma)$ is a group of prime degree $p$. By \cite[Corollary 3.5B]{DixonM1996} we have ${\rm Aut}(\Gamma)/{\mathcal B}\le{\rm AGL}(1,p)$ or is a doubly-transitive group. By \cite[Theorem 4.1B]{DixonM1996} we see either ${\rm Aut}(\Gamma)/{\mathcal B}\le{\rm AGL}(1,p)$ or is a doubly-transitive group with nonabelian simple socle. If ${\rm Aut}(\Gamma)/{\mathcal B}\le{\rm AGL}(1,p)$ then ${\rm Aut}(\Gamma)$ contains a normal subgroup of order $p$, and so has blocks of size $p$, a contradiction. Thus ${\rm Aut}(\Gamma)/{\mathcal B}$ is a doubly-transitive group with nonabelian simple socle. By \cite[Lemmas 4.5, 4.6, and 4.7]{PraegerWX1993} we have ${\rm SL}(2,2^s){\triangleleft}{\rm Aut}(\Gamma)$.
Now, $N_{S_V}({\rm SL}(2,2^s)) = {\langle} {\rm SL}(2,2^s),f,Z{\rangle}={\rm \Gamma L}(2,2^s)= ({\rm SL}(2,2^s)\times Z)\rtimes {\langle} f{\rangle}$. Thus every element of $N_{S_V}({\rm SL}(2,2^s))$, and hence $\gamma\in{\rm Aut}(\Gamma)$, can be written as $\gamma = f^iz\omega$, where $\omega\in {\rm SL}(2,2^s)$, $z\in Z$, and $i$ is a positive integer. Of course, as ${\rm SL}(2,2^s)\le{\rm Aut}(\Gamma)$, $f^iz\in{\rm Aut}(\Gamma)$ if and only if $f^iz\omega\in{\rm Aut}(\Gamma)$ for some $\omega\in{\rm SL}(2,2^s)$. Then $H = {\rm Aut}(\Gamma)\cap\{f^iz:i\in{\mathbb Z},z\in Z\}$ is a subgroup of ${\rm Aut}(\Gamma)$, and ${\rm Aut}(\Gamma)/{\rm SL}(2,2^s)\cong H$ by the First Isomorphism Theorem. As ${\rm Aut}(\Gamma)\cap Z = 1$, $H$ is isomorphic to a subgroup of ${\langle} f{\rangle}$, and hence ${\rm Aut}(\Gamma)/{\rm SL}(2,2^s)$ is isomorphic to a cyclic $2$-subgroup. Then ${\rm Aut}(\Gamma)/{\rm SL}(2,2^s)$ is conjugate by an element $z/{\rm SL}(2,2^s)\in Z/{\rm SL}(2,2^s)$ to a subgroup of ${\langle} f{\rangle}/{\rm SL}(2,2^s)$, and so $z^{-1}{\rm Aut}(\Gamma)z\le {\rm \Sigma L}(2,2^s)$. Then ${\rm Aut}(\Gamma)\le z{\rm \Sigma L}(2,2^s)z^{-1}$ and $z^{-1}(\Gamma)$ is a nondegenerate symmetric Maru\v si\v c-Scapellato digraph with ${\rm Aut}(z^{-1}(\Gamma))\le{\rm \Sigma L}(2,2^s)$, and so is given by Theorem \ref{PraegerWXsymmetric}.
In order to verify the numbers of symmetric Maru\v si\v c-Scapellato digraphs are as in the result, we need only to see different scalar matrices do indeed give different symmetric Maru\v si\v c-Scapellato digraphs. Suppose that there exist two non-isomorphic symmetric Maru\v si\v c-Scapellato graphs $\Gamma_1$ and $\Gamma_2$ with automorphism groups contained in ${\rm \Sigma L}(2,2^s)$, such that $z_1(\Gamma_1)=z_2(\Gamma_2)$, for $z_1,z_2\in Z$.
Then $\Gamma_2=z_2^{-1}z_1(\Gamma_1)$. This implies that $\Gamma_1$ and $\Gamma_2$ are of the same valency and since by Theorem~\ref{PraegerWXsymmetric} all Maru\v si\v c-Scapellato graphs with the same valency have the same automorphism groups, it follows that ${\rm Aut}(\Gamma_1)={\rm Aut}(\Gamma_2)={\langle} SL(2,2^s), L {\rangle}$, where $L={\langle} f^b {\rangle}$. By Lemma~\ref{diagonals don't normalize2} it follows that $z^{-1}{\langle} SL(2,2^s), L {\rangle} z= {\langle} SL(2,2^s), L {\rangle}$ holds only when $z=1$.
On the other hand, since $\Gamma_2=z_2^{-1}z_1(\Gamma_1)$ it follows that ${\rm Aut}(\Gamma_2)=(z_2^{-1}z_1) {\rm Aut}(\Gamma_1)(z_2^{-1}z_1)^{-1}$, and hence $z_1=z_2$, which implies that different scalar matrices do indeed give different symmetric Maru\v si\v c-Scapellato graphs.
Let $z^{-1}(\Gamma) = X(2^s,q,\emptyset, T)$, where $T = \{0\}$ or $T = U_{b,i} = \{i2^{bj}:0\le j < a/b\}$. Let $z = d^k$ for some positive integer $k$. We need only verify that $T = \{2k\}$ or $U_{b,i,k} = \{i2^{bj} + 2k:0\le j < a/b\}$. Now, in $z^{-1}(\Gamma)$, the neighbors of $(\infty,r)$ are $\{(y,r + u):y\in{\mathbb F}_{2^s},u\in U\}$. Considering $z(z^{-1}(\Gamma)) = \Gamma$ and applying Lemma \ref{scalar in MS-labeling}, we see the neighbors of $(\infty,r + k)$ in $\Gamma$ are $\{(y,r + u - k):y\in{\mathbb F}_{2^s},u\in U\}$. Equivalently, the neighbors of $(\infty,r)$ in $\Gamma$ are $\{(y,r +u - 2k):y\in{\mathbb F}_{2^s},u\in U\}$ and the result follows.
\end{proof}
We now determine the full automorphism group of any Maru\v si\v c-Scapellato digraph.
\begin{thrm}
Let $p = 2^s + 1$ be a Fermat prime, $q\vert(2^s - 1)$ be prime, and $\Gamma$ be a Maru\v si\v c-Scapellato digraph of order $qp$. Then $\Gamma$ or its complement is $X(2^s,q,S,T)$ and one of the following is true.
\begin{enumerate}
\item ${\rm Aut}(\Gamma)$ is primitive and
\begin{enumerate}
\item $s=2$, $qp = 15$, $S = {\mathbb Z}_3^*$ and $T = \{0\},\{1\}$, or $\{2\}$. Then $\Gamma$ is isomorphic to the line graph of $K_6$ and has automorphism group $d^{-1}{\rm \Sigma L}(2,4)d\cong S_6$ for some $d\in Z$.
\item $p = k^2 + 1$, $q = k + 1$, $S = {\mathbb Z}_q^*$ and $\vert T\vert = 1$. Then there exists $d\in Z/{\mathcal D}_\ell$ such that ${\rm Aut}(\Gamma) = d^{-1}{\rm P}\Gamma{\rm Sp}(4,k)d$.
\item $S = {\mathbb Z}_q^*$, $T = {\mathbb Z}_q$, and $\Gamma$ is a complete graph with automorphism group $S_{qp}$.
\end{enumerate}
\item ${\rm Aut}(\Gamma)$ is imprimitive and
\begin{enumerate}
\item $S < {\mathbb Z}_q^*$, $T = {\mathbb Z}_q$, $\Gamma$ is degenerate, and ${\rm Aut}(\Gamma)\cong S_p\wr {\rm Aut}({\rm Z}ay({\mathbb Z}_q,S))$.
\item In all other cases there exists $L\le {\langle} f/{\mathcal D}_\ell{\rangle}$ and $d\in Z/{\mathcal D}_\ell$ such that
$${\rm Aut}(\Gamma) = d^{-1}{\langle}{\rm SL}(2,2^s), L{\rangle} d/{\mathcal D}_\ell$$
\noindent which is isomorphic to a subgroup of ${\rm \Sigma L}(2,2^s)/{\mathcal D}_\ell$ that contains ${\rm SL}(2,2^s)/{\mathcal D}_\ell$.
\end{enumerate}
\end{enumerate}
\end{thrm}
\begin{proof}
As in the previous result, we will abuse notation and drop the ${\mathcal D}_\ell$'s from our notation. The case when ${\rm Aut}(\Gamma) = S_{qp}$ is trivial. The other Maru\v si\v c-Scapellato graphs of order $qp$ with primitive automorphism group were calculated in \cite{MarusicS1994} and their automorphism groups computed in \cite{PraegerX1993}. This gives the information in the result with $d = 1$. We observe that ${\rm \Sigma L}(2,2^s)$ is contained in ${\rm Aut}(\Gamma)$, and so the only possible isomorphisms with other Maru\v si\v c-Scapellato graphs are with elements of $Z$ by Lemma \ref{New MS iso}. That the elements of $Z$ give different graphs follows as they normalize ${\rm SL}(2,2^s)$ but are not contained in ${\rm Aut}(\Gamma)$.
If ${\rm Aut}(\Gamma)$ is imprimitive and $\Gamma$ is degenerate, then $T = {\mathbb Z}_q$ and as ${\rm Aut}(\Gamma)$ is imprimitive, $S\not = {\mathbb Z}_q^*$ as otherwise $\Gamma = K_{qp}$ has a primitive automorphism group. It is then not difficult to see that $\Gamma\cong K_p\wr{\rm Z}ay({\mathbb Z}_q,S)$ and by \cite[Theorem 5.7]{DobsonM2009} ${\rm Aut}(\Gamma)\cong S_p\wr{\rm Aut}({\rm Z}ay({\mathbb Z}_q,S))$.
If ${\rm Aut}(\Gamma)$ is imprimitive and $\Gamma$ is non-degenerate, then $\Gamma$ is a generalized orbital digraph of ${\rm Aut}(\Gamma)$. We write $\Gamma = \Gamma_1\cup\dotsm \cup \Gamma_r$ where each $\Gamma_i$ is an orbital digraph of ${\rm Aut}(\Gamma)$. Note that as ${\rm Aut}(\Gamma)$ is imprimitive, some orbital digraph of ${\rm Aut}(\Gamma)$ is disconnected. Also, each connected orbital digraph of ${\rm Aut}(\Gamma)$ is either symmetric or $1/2$-transitive, and as each orbital digraph of $\Gamma$ is a generalized orbital digraph of ${\rm SL}(2,2^s)$, we see each connected orbital digraph of ${\rm Aut}(\Gamma)$ is symmetric as each connected orbital digraph of ${\rm SL}(2,2^s)$ is symmetric.
If there exist connected orbital digraphs of ${\rm Aut}(\Gamma)$ that are subdigraphs of $\Gamma$ whose automorphism groups are contained in $d^{-1}{\rm \Sigma L}(2,2^s)d$ and $e^{-1}{\rm \Sigma L}(2,2^s)e$ for $d\not = e$ both in $Z$, then
$${\rm SL}(2,2^s)\le{\rm Aut}(\Gamma)\le d^{-1}{\rm \Sigma L}(2,2^s)d\cap e^{-1}{\rm \Sigma L}(2,2^s)e\le d^{-1}{\rm \Sigma L}(2,2^s)d.$$
Thus ${\rm Aut}(\Gamma)$ is a subgroup of $d^{-1}{\rm \Sigma L}(2,2^s)d$ that contains ${\rm SL}(2,2^s)$. Now let ${\rm SL}(2,2^s)\le K\le{\rm \Sigma L}(2,2^s)$. As ${\rm SL}(2,2^s){\triangleleft} {\rm \Sigma L}(2,2^s)$, every element of ${\rm \Sigma L}(2,2^s)$, and consequently every element of $K$, can be written as $gf^c$ for some $g\in{\rm SL}(2,2^s)$ and integer $c$. As ${\rm SL}(2,2^s)\le K$, we have $gf^c\in K$ if and only if $f^c$ in $K$. We conclude $K = {\langle}{\rm SL}(2,2^s),L{\rangle}$, where $L\le {\langle} f{\rangle}$ consists of all powers of $f$ contained in $K$. Then $d^{-1}{\rm SL}(2,2^s)d\le{\rm Aut}(\Gamma)\le d^{-1}{\rm \Sigma L}(2,2^s)d$ and ${\rm Aut}(\Gamma) = d^{-1}{\langle} {\rm SL}(2,2^s),L{\rangle} d$ for some $L\le {\langle} f{\rangle}$ as required. We thus assume that every connected orbital digraph of ${\rm Aut}(\Gamma)$ that is a subdigraph of $\Gamma$ has automorphism group contained in $d^{-1}{\rm \Sigma L}(2,2^s)d$ for some scalar matrix $d\in Z$. Suppose that $\Gamma_1,\ldots,\Gamma_t$, $t\le r$ are the connected orbital digraphs of ${\rm Aut}(\Gamma)$ that are subdigraphs of $\Gamma$. Then by Theorem \ref{MS symmetric}, $\Gamma_i$ has automorphism group $d^{-1}{\langle} {\rm SL}(2,2^s),L_i{\rangle} d$ where $L_i\le {\langle} f{\rangle}$. Then ${\rm Aut}(\Gamma)\le d^{-1}{\langle} {\rm SL}(2,2^s),L'{\rangle} d$ where $L' = \cap_{i=1}^rL_i$. Finally, let $L$ be the subgroup of $L'$ consisting of automorphisms of the subdigraph of $\Gamma$ obtained by removing all edges between elements of ${\rm PG}(1,2^s)/{\mathcal D}_\ell$. Then ${\rm Aut}(\Gamma) = d^{-1}{\langle}{\rm SL}(2,2^s),L{\rangle} d$ and the result follows.
\end{proof}
We remark that the automorphism group of a Maru\v si\v c-Scapellato digraph $\Gamma$ can be calculated quite quickly:
If $\Gamma$ is nondegenerate and ${\rm Aut}(\Gamma)$ imprimitive, then one only needs to determine the subgroup of $d^{-1}{\rm \Sigma L}(2,2^s)d/{\mathcal D}_\ell = {\langle} {\rm SL}(2,2^s),d^{-1}fd{\rangle}/{\mathcal D}_\ell$ which is ${\rm Aut}(\Gamma)$. In particular, one only needs to determine the maximal subgroup of $d^{-1}{\langle} f{\rangle} d$ contained in ${\rm Aut}(\Gamma)$. This can easily be accomplished as all such subgroups can be computed quickly. Let $s = 2^t$, $t\ge 1$. As $F(x) = x^2$, $f$ has order $2^t$, so there are $t+1$ subgroups of ${\langle} f{\rangle}$ each determined by a generator of the form $f^{2^r}$, $0\le r\le t$. As $Z/{\mathcal D}_\ell$ has order at most $2^s$, there are at most $(t+1)2^s$ maps which need to be tested as elements of ${\rm Aut}(\Gamma)$ in order to determine ${\rm Aut}(\Gamma)$.
If $\Gamma$ is degenerate or ${\rm Aut}(\Gamma)$ is primitive, then this can be determined easily as the sets $S$ and $T$ are given explicitly. Again, one only needs to determine $d$, and this can be done as above by checking which $d^{-1}gd$ is contained in ${\rm Aut}(\Gamma)$.
\section{Missing digraphs whose automorphism group is primitive}{\langle}bel{sec:prim-errors}
The first error in the literature is most probably simply an unfortunate typographical error. The misprint occurs in \cite[Table 3]{LiebeckS1985a} for the groups ${\rm PSL}(2,q)$ of degree $q(q^2-1)/24$ with point stabilizer $A_4$. In the ``Comment" column, the paper literally lists ``$q\equiv +3\ ({\rm mod\ } 8), q\le 19$". Of course, as written the ``$+$" is entirely superfluous, but in reality it should be a ``$\pm$". Indeed, without the $\pm$ the group ${\rm PSL}(2,13)$ which has $A_4$ as a maximal subgroup is not listed. The action of ${\rm PSL}(2,13)$ on right cosets of $A_4$ is primitive of degree $\vert{\rm PSL}(2,13)\vert/\vert A_4\vert = 7\cdot 13$. The authors thank Primo\v z Poto\v cnik for pointing out this error.
\begin{lem}{\langle}bel{PSL213lem}
Let ${\rm PSL}(2,13)$ act transitively on $7\cdot 13$ points with point-stabilizer $A_4$.
Then there are $3$ self-paired orbitals of size $4$ all of which are $2$-arc-transitive. No other orbital digraphs are $2$-arc-transitive.
Two of the graphs corresponding to these self-paired orbitals are isomorphic with automorphism group ${\rm PSL}(2,13)$. The graph corresponding to the union of these orbitals is symmetric and has automorphism group ${\rm PGL}(2,13)$.
The graph corresponding to the remaining orbital has automorphism group ${\rm PGL}(2,13)$ and is symmetric.
There is $1$ self-paired orbital of size $6$ whose corresponding graph is symmetric and has automorphism group ${\rm PGL}(2,13)$.
There are $2$ non self-paired orbitals of size $12$ whose corresponding digraphs have automorphism group ${\rm PSL}(2,13)$, and whose union corresponds to a symmetric graph with automorphism group ${\rm PGL}(2,13)$.
There are $4$ self-paired orbitals of size $12$ that are all symmetric, two of which correspond to graphs that are isomorphic with automorphism group ${\rm PSL}(2,13)$. Their union corresponds to a graph that has automorphism group ${\rm PGL}(2,13)$ and is symmetric.
The remaining two self-paired orbitals correspond to graphs that are non-isomorphic and have automorphism group ${\rm PGL}(2,13)$ and are symmetric.
Any other digraph of order $91$ that contains ${\rm PSL}(2,13)$ as a transitive subgroup and is not complete or the complement of a complete graph is a union of the above digraphs and is not symmetric. It will have automorphism group either ${\rm PSL}(2,13)$ or ${\rm PGL}(2,13)$, depending upon whether or not it can be written as a union of graphs all of whose automorphism groups are ${\rm PGL}(2,13)$. If this is possible, then it has automorphism group ${\rm PGL}(2,13)$; otherwise, its automorphism group will be ${\rm PSL}(2,13)$.
\end{lem}
\begin{proof}
The information about the orbital digraphs of ${\rm PSL}(2,13)$, including whether or not they are self-paired and their automorphism groups and whether they are $2$-arc-transitive or symmetric, was obtained using MAGMA. So was information about the automorphism groups of unions of exactly two orbital digraphs. It thus remains to determine the automorphism group of any other digraph of order $91$ that contains ${\rm PSL}(2,13)$ and is not complete or its complement.
Let $\Gamma$ be such a digraph. Then ${\rm Aut}(\Gamma)\not = S_{91}$, and ${\rm Aut}(\Gamma)$ is $2$-closed. There is only one other socle of a primitive but not $2$-transitive subgroup of $S_{91}$, namely ${\rm PSL}(3,9)$ by \cite[Table B.2]{DixonM1996}. However, ${\rm PSL}(3,9)$ contains no subgroup isomorphic to ${\rm PSL}(2,13)$ by \cite{Bloom1967}. Thus ${\rm soc}({\rm Aut}(\Gamma)) = {\rm PSL}(2,13)$ and so ${\rm Aut}(\Gamma) = {\rm PSL}(2,13)$ or ${\rm PGL}(2,13)$. Clearly, if $\Gamma$ can be written as a union of graphs whose automorphism group is ${\rm PGL}(2,13)$, then ${\rm Aut}(\Gamma) = {\rm PGL}(2,13)$. Otherwise, by the first part of this lemma, $\Gamma$ is a union of digraphs one of which has automorphism group ${\rm PSL}(2,13)$ but is not invariant under ${\rm PGL}(2,13)$ and its different image under ${\rm PGL}(2,13)$ is not a subdigraph of $\Gamma$. Hence ${\rm Aut}(\Gamma) = {\rm PSL}(2,13)$.
\end{proof}
This leads to the next error in the literature, which is also mainly typographical. Namely, in \cite[Table 2]{PraegerX1993} the entries for ${\rm PSL}(2,p)$ require $p\ge 11$. For $p = 5$, ${\rm PSL}(2,5)\cong A_5$ is $2$-transitive in its representation of degree $6$, and so any digraph of order $6$ whose automorphism group contains ${\rm PSL}(2,5)$ is necessarily complete or has no arcs and has automorphism group $S_6$. For ${\rm PSL}(2,7)\cong{\rm PSL}(3,2)$, we see from Theorem \ref{gendigraphauto} that there are other digraphs that are not graphs that are not listed in \cite[Table 2]{PraegerX1993}. The error here is more one of omission than a mistake in the proof - in \cite{PraegerX1993} the proofs are for vertex-transitive digraphs and graphs of order at least $5p$ (see for example \cite[Table IV]{PraegerX1993}), $p\ge 7$, as the case when $p = 3$ was already considered in \cite{WangX1993} - but \cite{WangX1993} did not consider digraphs that were not graphs.
The next error involves $M_{23}$ in its actions on $11\cdot 23$ points. There are two actions of $M_{23}$ on $253 = 11\cdot 23$ points. One is on pairs taken from a set of $23$ elements, while the other is on the septads (sets of size $7$) in the Steiner system $S(4,7,23)$ \cite{Conway1985}. The action on pairs gives $M_{23}$ as a transitive subgroup of ${\rm Aut}(T_{23})$, the triangle graph, whose automorphism group is $S_{23}$, and this graph is listed in the row corresponding to $A_{23}$. The action of $M_{23}$ on septads was not considered in \cite{PraegerX1993}.
\begin{lem}{\langle}bel{M23lem}
The action of $M_{23}$ on septads (sets of size $7$) in the Steiner system $S(4,7,23)$ of degree $11\cdot 23$ has two orbital digraphs which are graphs of valency $112$ and $140$. Both of these graphs are Cayley graphs of the nonabelian group of order $11\cdot 23$ and so are also isomorphic to metacirculant graphs. Both graphs have automorphism group $M_{23}$ and neither is $2$-arc-transitive.
\end{lem}
\begin{proof}
The action on septads gives $M_{23}$ as a transitive subgroup of ${\rm Aut}(M_{23})$. By \cite{Atlas} the suborbits are of length $112$ and $140$, and by \cite{Conway1985} there is a maximal subgroup $H$ of $M_{23}$ of order $253$, and $H$ is isomorphic to the Frobenius group of order $253$. By order arguments no element of $H$ is contained in the stabilizer of a septad, and so $H$ must be semiregular. By the Orbit-Stabilizer Theorem we see that $H$ is regular. As Sabidussi showed \cite{Sabidussi1958} that a graph is isomorphic to a Cayley graph of the group $G$ if and only if it contains a regular subgroup isomorphic to $G$, each of the two orbital graphs of $M_{23}$ are Cayley graphs. As every Cayley graph of order $qp$ is a metacirculant graph, these two graphs are also metacirculant.
Turning to the automorphism groups of the two orbital digraphs $\Gamma_1$ and $\Gamma_2$ of $M_{23}$, they are complements of each other and so ${\rm Aut}(\Gamma_1) = {\rm Aut}(\Gamma_2)$. Also, with respect to the $2$-closure of this action of $M_{23}$ (which can be defined as the intersection of the automorphism groups of its orbital digraphs), we have $M_{23}^{(2)} = {\rm Aut}(\Gamma_1)\cap{\rm Aut}(\Gamma_2) = {\rm Aut}(\Gamma_1)$. By \cite[Theorem 1]{LiebeckPS1988a} we have $M_{23}{\triangleleft}{\rm Aut}(\Gamma_1)$. By \cite[Table B.2]{DixonM1996} we have ${\rm Aut}(\Gamma_1) = M_{23}$.
Finally, in order to be $2$-arc-transitive, $d(d-1)$ must divide the order of the stabilizer of a point in $M_{23}$ where $d$ is the valency of $\Gamma_1$ or $\Gamma_2$, and this stabilizer has order $2^7\cdot 3^2\cdot 5\cdot 7$. So neither $\Gamma_1$ nor $\Gamma_2$ is $2$-arc-transitive.
\end{proof}
\begin{table}
\begin{center}
\begin{tabular}{| c | c | c | c | c |}
\hline
${\rm soc}(G)$ & $qp$ & Valency & Cayley & Reference\\
\hline
$A_{qp}$ & $qp$ & $0,qp-1$ & Y & \\
\hline
$A_p$ & $\frac{p(p-1)}{2}$ & $2(p-2),\frac{(p-2)(p-3)}{2}$ & Y* & \cite[3.1]{PraegerX1993}\\
\hline
$A_{p+1}$ & $\frac{p(p+1)}{2}$ & $2(p-1),\frac{(p-1)(p-2)}{2}$ & ${\rm N}^\dagger$ & \cite[3.1]{PraegerX1993} \\
\hline
$A_7$ & $5\cdot 7$ & ${\bf 4},12,18$ & N & \cite[3.2]{PraegerX1993} \\
\hline
${\rm PSL}(4,2)$ & $5\cdot 7$ & $16,18$ & N & \cite[3.3]{PraegerX1993} \\
\hline
${\rm PSL}(5,2)$ & $5\cdot 31$ & $42,112$ & Y & \cite[3.3]{PraegerX1993} \\
\hline
$\Omega^\pm(2d,2)$ & $(2^d\mp 1)(2^d\pm 1)$ & $2^{2d - 2},2(2^{d-1}\mp 1)(2^{d-2}\pm 1)$ & N & \cite[3.4]{PraegerX1993}\\
\hline
${\rm PSp}(4,k)$ & $(k^2 + 1)(k + 1)$ & $k^2 + k$, $k^3$, $k$ even & ${\rm N}^\dagger$ & \cite[3.5]{PraegerX1993}\\
\hline
${\rm PSL}(2,k^2)$ & $k(k^2 + 1)/2$ & $k^2 - 1,\frac{k^2 - k}{2}, k^2 \pm k$, $k \equiv 1\ ({\rm mod\ } 4)$ & N & \cite[4.1]{PraegerX1993}\\
\hline
${\rm PSL}(2,k^2)$ & $k(k^2 + 1)/2$ & $k^2 - 1,\frac{k^2 + k}{2}, k^2 \pm k$, $k\equiv 3\ ({\rm mod\ } 4)$ & N & \cite[4.1]{PraegerX1993}\\
\hline
${\rm PSL}(2,p)$ & $\frac{p(p \mp 1)}{2}$ & $\frac{p\pm 1}{2}, p\pm 1$, or & Y** & \cite[4.4]{PraegerX1993} \\
& & $\frac{p\pm 1}{4}$ or $2(p-1)$ & & \\
\hline
$G = {\rm PGL}(2,7)$ & $3\cdot 7$ & $4,8$ & Y & \cite[Example 2.3]{WangX1993} \\
\hline
$G = {\rm PGL}(2,11)$ & $5\cdot 11$ & ${\bf 4},6,8,12,24$ & Y & \cite[4.3]{PraegerWX1993} \\
\hline
${\rm PSL}(2,13)$ & $7\cdot 13$ & ${\bf 4},6,12,24$ & N & Lemma \ref{PSL213lem} \\
\hline
${\rm PSL}(2,19)$ & $3\cdot 19$ & ${\bf 6},20,30$ & Y & \cite[4.2]{PraegerWX1993} \\
\hline
${\rm PSL}(2,23)$ & $11\cdot 23$ & ${\bf 4},6,8,12,24$ & Y & \cite[4.3]{PraegerWX1993} \\
\hline
${\rm PSL}(2,29)$ & $7\cdot 29$ & $12,20,30,60$ & ${\rm N}^\#$ & \cite[4.2]{PraegerWX1993} \\
\hline
${\rm PSL}(2,59)$ & $29\cdot 59$ & ${\bf 6},10,12,20,30,60$ & Y & \cite[4.2]{PraegerWX1993} \\
\hline
${\rm PSL}(2,61)$ & $31\cdot 61$ & ${\bf 6},10,12,20,30,60$ & N & \cite[4.2]{PraegerWX1993} \\
\hline
$M_{22}$ & $7\cdot 11$ & ${\bf 16},60$ & N & \cite[3.6]{PraegerWX1993} \\
\hline
$M_{23}$ & $11\cdot 23$ & $112,140$ & Y & Lemma \ref{M23lem} \\
\hline
\end{tabular}
\caption{The graphs of order $pq$ with primitive automorphism groups.}
{\langle}bel{table}
\end{center}
\end{table}
\begin{thrm}
Let $\Gamma$ be a vertex-transitive graph of order $qp$, where $q$ and $p$ are distinct primes, whose automorphism group $G$ is simply primitive. Then ${\rm soc}(G)$ is given in Table \ref{table}. There is a boldface entry in the column ``Valency" if and only if there is a $2$-arc-transitive graph of that valency. The superscipt symbols in the table have the following meanings:
\begin{itemize}
\item $*$ means $p\ge 7$,
\item $\dagger$ means that these graphs are also Maru\v si\v c-Scapellato graphs but in the case of $A_{p+1}$ this is only true for $A_6$,
\item $**$ means these graphs are Cayley if and only if $p\equiv 3\ ({\rm mod\ } 4)$,
\item $\#$ means that these graphs are metacirculant graphs which are not Cayley graphs.
\end{itemize}
\end{thrm}
\begin{proof}
Most of the information in the Table \ref{table} is taken directly from the sources in the column ``Reference", with the following exceptions. First, information about $2$-arc-transitive graphs not given in Lemma \ref{PSL213lem} or \ref{M23lem} can be found in \cite{MarusicP2002}. That the generalized orbital digraphs of ${\rm PSL}(2,29)$ are metacirculants is proven in \cite[pg. 192, paragraph 3]{MarusicS1994}. The vertex-transitive graphs of order $pq$ with primitive automorphism group that are also isomorphic to nontrivial Maru\v si\v c-Scapellato graphs are determine in \cite{MarusicS1994} starting at the bottom of page 192.
\end{proof}
\section{Other errors in the literature}{\langle}bel{sec:other-errors}
To end this paper, we list the errors that we are aware in the literature that follow from the errors above and that are not in the original papers where the error was made.
\begin{itemize}
\item The statement of \cite[Theorem 2.5]{DobsonHKM2017} is missing the graphs given in Theorem \ref{gendigraphauto} with imprimitive automorphism group ${\rm PSL}(2,11)$. This result is only used to discuss graphs of order $21$, and so this error does not affect any results proven in the paper.
\item The result \cite[Proposition 2.5]{WangFZWM2016} does not list the symmetric graphs of valency $4$ given by Lemma \ref{PSL213lem}. Consequently, \cite[Lemma 3.4]{WangFZWM2016} has a small gap which can be filled using GAP or MAGMA.
\item The result \cite[Proposition 4.2]{MarusicP2002} is missing the $2$-arc-transitive graphs of valency $4$ given by Lemma \ref{PSL213lem}.
\item The result \cite[Corollary 3.3, Table 1]{Dobson2006a} is missing the graphs given by Lemmas \ref{PSL213lem} and \ref{M23lem}. Additionally, \cite[Theorem 3.2]{Dobson2006a} and \cite[Corollary 3.3]{Dobson2006a} are missing the group ${\rm PSL}(2,11)$ in its imprimitive action action on 55 points. Finally, \cite[Theorem 4.1(3)]{Dobson2006a} is missing these same graphs.
The result from \cite[Theorem 3.2(1)]{Dobson2006a} could be strengthened to digraphs by including the digraphs with simple and imprimitive automorphism groups.
\item The result \cite[Theorem]{LiWWX1994} does not consider the action of ${\rm PSL}(2,13)$ given in Lemma \ref{PSL213lem} nor the action of $M_{23}$ given in Lemma \ref{M23lem}.
\end{itemize}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
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\end{document}
|
\begin{document}
\title{Supplementary Material "Room temperature entanglement storage
using decoherence free subspace in a solid-state spin system"}
\author{F. Wang$^1$, Y.-Y. Huang$^1$, Z.-Y. Zhang$^{1,2}$, C. Zu$^1$, P.-Y. Hou$^1$,
X.-X. Yuan$^1$, W.-B. Wang$^1$, W.-G. Zhang$^1$, L. He$^1$, X.-Y. Chang$^1$,
L.-M. Duan}
\affiliation{Center for Quantum Information, IIIS, Tsinghua University, Beijing 100084,
PR China}
\affiliation{Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA}
\date{\today }
\begin{abstract}
In this supplementary material, we include experimental details and
numerical simulation for parameter calibration, quantum gates, and state
detection in weakly coupled nuclear spin systems. We also describe
experimental technique to control environment and realize general collective
noise model.
\end{abstract}
\date{\today }
\maketitle
\section{Experimental Setup}
The optical setup is similar to what was described in Ref \cite{1}. A major
difference is that another acoustic optical modulator (AOM) in double pass
configuration is added into the optical path right after the first AOM
double pass to constrain the leakage of green laser to a higher order.
The microwave signal is delivered into the system with a waveguide
transmission line fabricated on a cover glass, to which the diamond sample
is attached. The microwave field is generated by a carrier signal modulated
at an IQ mixer by two analog outputs of an Arbitrary Waveform Generator
(AWG) to control the relative phase of the signal. We add a switch
controlled by the digital output of the AWG after the IQ mixer to reduce the
influence of leakage of the carrier signal. The microwaves are then sent to
a high power amplifier and subsequently get delivered to the sample.
The radio frequency signal is generated by the analog channel of the AWG and
amplified through a high power amplifier. We fabricated a coplanar coil with
matching impedance on a PCB board to deliver radio frequency signal. To
achieve a reasonable nuclear spin Rabi frequency, the coil is detached from
the diamond surface with a distance about $2$ mm (The other surface is
attached to the cover glass).
A magnetic field of $B_{z}=480$ Gauss is applied along the NV symmetry axis
using a permanent magnet. The strong magnetic field is used to polarize the
intrinsic nitrogen spin and provide a relative strong Larmor frequency to
the nuclear spins compared to the nuclear spin hyperfine parameters. All the
experiments are performed at room temperature with $10^{6}$ repetitions for
measurement of each data point.
\begin{figure}
\caption{Calibration of nuclear spin environment by the CPMG pulse sequence. What is shown is the measured coherence signal after
the CPMG decoupling sequence as a function of free evolution time $\protect\tau$. The
signal (grey) is taken with a magnetic field $B=490$ Gauss by preparing the
electronic spin in $(|0\rangle-i|1\rangle)/\protect\sqrt{2}
\end{figure}
\begin{figure}
\caption{Nuclear spin ODMR results for spin 1 and spin 2. The experimental scheme
is illustrated in Fig. 1 in the main text. (a,b) Resonant frequency with
electronic spin at $m_s=+1,0,-1$ state for nuclear spin 1 and 2.}
\end{figure}
\section{Calibration of nuclear-spin hyperfine interaction parameters}
Due to the anisotropic property of the hyperfine interaction, nuclear spins
undergo different evolutions with electronic spin at different eigenstates $
m_{s}=+1,0,-1$. We consider the two-spin system composed of an electronic
spin with $m_{s}=0$ and $m_{s}=-1$, denoted as $|0\rangle $ and $|1\rangle $
, and a nuclear spin with components of the spin operator denoted as $
I_{x},I_{y},I_{z}$. With the electronic spin at $|0\rangle $ and $|1\rangle $
state, the nuclear spin Hamiltonian is denoted as $H_{0}$ and $H_{1}$,
respectively, which takes the form
\begin{equation}
H_{0}=\omega _{L}I_{z},
\end{equation}
\begin{equation}
H_{1}=(\omega _{L}+A_{\parallel })I_{z}+A_{\perp }I_{x},
\end{equation}
where $\omega _{L}$ is the nuclear Larmor frequency, $A_{\parallel }$ and $
A_{\perp }$ are the parallel and transverse components of the hyperfine
parameters. Consider a simple equally-spaced sequence of $\pi $ rotations ($
\tau -\pi -2\tau -\pi -\tau $) with pulse number $N=2$ and pulse interval
denoted by $2\tau $, the net result of this specific decoupling sequences is
that the nuclear spin rotates by an angle of $\phi $ around axis $\hat{n}_{0}
$ ($\hat{n}_{1}$) with the electronic spin at $m_{s}=0$ ($m_{s}=-1$) state.
When $\hat{n}_{0}\cdot \hat{n}_{1}=-1$, a resonance condition is satisfied
and the electronic spin gets entangled with the nuclear spin, thus the
electronic coherence collapses after the CPMG sequence (Fig. 1 grey). With $
\omega _{L}\gg A_{\parallel },A_{\perp }$, the condition of resonance is
given by \cite{2}
\begin{equation}
\tau \approx \frac{(2k-1)\pi }{2\omega _{L}+A_{\parallel }}
\end{equation}
where $k$ is the order of resonance. The probability that the coherence is
preserved is given by \cite{2}
\begin{eqnarray}
P &=&(M+1)/2, \\
M &=&1-(1-\hat{n}_{0}\cdot \hat{n}_{1})sin^{2}\frac{N\phi }{2}
\end{eqnarray}
At resonance, with $\omega _{L}\gg A_{\parallel },A_{\perp }$, this equation
becomes \cite{2}
\begin{eqnarray}
P &=&(cos(Nm_{x})+1)/2, \\
m_{x} &=&\frac{A_{\perp }}{\sqrt{(A_{\parallel }+\omega _{L})^{2}+A_{\perp
}^{2}}}
\end{eqnarray}
By fitting the experimental data on the measured electronic spin coherence
after the CPMG sequence to the numerical simulation of the corresponding
dynamics with the fitting parameters $A_{\parallel }$ and $A_{\perp }$,
single nuclear spins can be resolved to a resolution of about $10$ $kHz$
\cite{2}. As illustrated in the main text, to calibrate the nuclear spin
hyperfine parameters more precisely, we run nuclear spin ODMR experiments
with the electronic spin set at $m_{s}=+1,0,-1$ states (Fig. 2). With the
nucelar spin ODMR technique, the hyperfine parameters for the two nuclear
spins used in our experiment are determined with a high precision as
follows:
\begin{equation}
A_{\parallel 1}=-77.02(3)\text{ kHz},\text{ \ }A_{\perp 1}=114.5(1)\text{ kHz
},
\end{equation}
\begin{equation}
A_{\parallel 2}=71.03(3)\text{ kHz},\text{ \ }A_{\perp 2}=58.7(3)\text{ kHz},
\end{equation}
where the number in the bracket denotes the standard deviation on the last
digit.
\begin{figure}
\caption{Characterization of the conditional X gate for nuclear spin 1 and
2. (a) Experimental scheme. The nuclear spin is polarized by swapping
the electronic spin polarization onto the nuclear spin. An optional $\protect
\pi$ rotation is applied to set the electronic spin to $m_s=-1$ state. After
that, the desired gate is applied on the nuclear spin for $N=1,...10$ times
with electronic spin at $m_s=0$ or $m_s=-1$ separately before measuring the
nuclear spin on the Y basis. (b,c) Characterization of conditional X gate
for nuclear spin 1 and 2. Solid lines are fit by the formula $sin(2\protect\pi N/4)*(1-bN)$ with $b=0.012$ and a standard deviation $\sigma=0.011$ in (b),
and with $b=0.025$ and a standard deviation $\sigma=0.014$ in (c).}
\end{figure}
\begin{figure}
\caption{Characterization of the unconditional X gate for nuclear spin 1. (a)
Experimental scheme. The nuclear spin is polarized by swapping the
electronic spin polarization onto the nuclear spin. An optional $\protect\pi$
rotation is applied to set the electronic spin to $m_s=-1$ state. After
that, the desired gate is applied on the nuclear spin for $N=1,...10$ times
with electronic spin at $m_s=0$ or $m_s=-1$ separately before measuring the
nuclear spin on the Y basis. (b) Characterization of unconditional X gate
for nuclear spin 1. Solid lines are fits by the formula $sin(2\protect\pi
N/4)*(1-bN)$ with $b=0$ and a standard deviation $\sigma=0.013$.}
\end{figure}
\begin{figure}
\caption{Characterization of the unconditional Z gate for nuclear spin 1 and
spin 2. (a) Experimental scheme. The nuclear spin is polarized by swapping
the electronic spin polarization onto the nuclear spin. A $\protect\pi/2$
rotation is applied to set the nuclear spin to $(|0\rangle-i|1\rangle)/
\protect\sqrt{2}
\end{figure}
\section{Control of the two weakly coupled nuclear spins}
The conditional and unconditional gates are accomplished by the same pulse
sequence used to calibrate the nuclear spin hyperfine interaction
parameters. The parameters $\tau $ and $N$ (Table I) in each gate are
calculated using the calibrated hyperfine parameters. In Fig. 3, Fig. 4, and
Fig. 5, we apply the same gates (conditional X gate, unconditional X gate,
or unconditional Z gate) $N$ times (with $N=1,...10$) on the polarized
nuclear spin with the electronic spin initialized at different states.
\begin{table}[tbp]
\caption{Spin1, Spin2: gate parameters in the experiment}
\begin{center}
\begin{tabular}{l|c|c|c|c}
& $U$ & $\tau(\mu s)$ & $N$ & total time($\mu s$) \\ \hline
& $R_X^e(\pi/2)$ & $2.579$ & $7$ & 36.1 \\
spin1 & $R_X(\pi/2)$ & $4.123$ & $8$ & 66.0 \\
& $R_Z(\pi/2)$ & $0.047$ & $4$ & 0.4 \\ \hline
& $R_X^e(\pi/2)$ & $2.253$ & $19$ & 85.6 \\
spin2 & $R_Z(\pi/2)$ & $0.039$ & $4$ & 0.3 \\ \hline
\end{tabular}
\end{center}
\end{table}
The gate fidelity is mainly restricted by three factors: (i) The precision
of the hyperfine interaction parameters. (ii) The fluctuation of magnetic
field. (iii) The decoherence of the electronic spin during the pulse
sequence. To suppress the influence from the latter one, we choose two
nuclear spins with hyperfine parameters $A_{\parallel }$ around $\pm 50kHz$
so that they can be isolated from each other and the spin bath even at a
small pulse interval $\tau $. Therefore, the gate time can be controlled
within $100$ $\mu s$ and decoherence contribution by the electronic spin to
the gate infidelity is mitigated. From the slow decay of the oscillations in
Fig. 3, Fig. 4 and Fig. 5, we estimate a gate fidelity of $F\sim 0.988$ ($0.975
$) for the conditional X-gate on the nuclear spin 1 (2). The conditional
gate on the nuclear spin 2 has a lower fidelity as its pulse sequence has a
longer time and thus a larger contribution from the electronic spin
decoherence. The unconditional gate has a higher intrinsic fidelity: we do
not see noticeable fidelity decay after $10$ gates under experimental uncertainty, suggesting its
intrinsic fidelity $F>99\%$.
\section{Initialization and readout of single nuclear spins}
The single nuclear spin initialization and readout is obtained by measuring
the free evolution contrast of the nuclear spin coherence. In Fig. 6, the
nuclear spin is prepared to $(|0\rangle -i|1\rangle )/\sqrt{2}$ with the
electronic spin at $m_{s}=0$ ($m_{s}=-1$) for nuclear spin 1 (spin 2). We
characterize the nuclear spin coherence by projecting the nuclear spin phase
to the electronic spin population. We repeat the initialization process
twice and extract an initialization and readout fidelity of $F_{1}=0.896(6)$
and $F_{2}=0.873(9)$ for nuclear spin 1 and 2.
In experiments the initialization and readout fidelity of nuclear spins are always combined together as there is no direct ways to polarize or measure the state of nuclear spins separately. From the supplementary information of Ref. \cite{6}, the high initialization and readout fidelity indicates that the charge state initialization fidelity is either high ($>0.8$ from our experiment result) or not sensitive to the measurements. We briefly summarize the arguments here respectively under the following two assumptions (i) the re-initialization has no memory for the charge state or (ii) the re-initialization does not change the charge state.
Consider green laser initialization involves both spin states and charge states, the initial state of the electronic spin takes the form:
\begin{equation}
\rho_e=p_1\rho_0+p_2\rho_m+p_3\rho_s+p_4\rho_c
\end{equation}
where $\rho_0$ is the desired $m_s=0$ state, $\rho_m$ denotes the completely mixed state of $m_s=0$ and $m_s=-1$ states, $\rho_s$ denotes the other spin state $m_s=+1$, and $\rho_c$ denotes the $NV^0$ state. The probabilities satisfy the normalization $p_1+p_2+p_3+p_4=1$. In our normalized fluorescence contrast measurement, the $m_s=0$ state gives a signal of $1$, $m_s=\pm 1$ states and $NV^0$ give a signal of $0$.
Assume the unpolarized nuclear spin is in a completely mixed state $\rho_m$, the state of the initialized electronic spin and a single nuclear spin is:
\begin{equation}
\rho=\rho_e\otimes\rho_n=p_1(\rho_0\otimes\rho_m)+p_2(\rho_m\otimes\rho_m)+p_3(\rho_s\otimes\rho_m)+p_4(\rho_c\otimes\rho_m)
\end{equation}
After swapping the nuclear spin with the electronic spin
\begin{equation}
\rho=p_1(\rho_m\otimes\rho_0)+p_2(\rho_m\otimes\rho_m)+p_3(\rho_s\otimes\rho_m)+p_4(\rho_c\otimes\rho_m)
\end{equation}
Under the scenario that re-initialization of the electronic spin has no memory for the charge state, the state becomes:
\begin{equation}
\rho=(p_1\rho_0+p_2\rho_m+p_3\rho_s+p_4\rho_c)\otimes(p_1\rho_0+(1-p_1)\rho_m)
\end{equation}
Reading out the nuclear spin involves another swap gate between the nuclear and the electronic spin:
\begin{equation}
\rho=p_1(p_1\rho_0+(1-p_1)\rho_m)\otimes\rho_0+p_2(p_1\rho_0+(1-p_1)\rho_m)\otimes\rho_m+p_3\rho_s\otimes(p_1\rho_0+(1-p_1)\rho_m)+p_4\rho_c\otimes(p_1\rho_0+(1-p_1)\rho_m)
\end{equation}
Reading out electronic spin using green laser only yields non-zero signal for the electronic spin in the pure state $\rho_0$. Thus the maximum signal contrast is:
\begin{equation}
C_{max}=\frac{p_1^2+p_1p_2}{p_1}=p_1+p_2
\end{equation}
so that the maximum initialization and readout fidelity is calculated by:
\begin{equation}
F_{max}=\frac{1}{2}+\frac{C_{max}}{2}=\frac{1}{2}+\frac{p_1+p_2}{2}
\end{equation}
From the above equation and the initialization and readout fidelity measured in the experiment ($\sim 0.9$), we find $p_1+p_2\sim 0.8$, thus the charge state initialization fidelity $p_1+p_2+p_3>0.8$ under this scenario.
Alternatively, if the electron re-initialization does not change the charge state, the state after re-initialization becomes:
\begin{equation}
\rho=p_1(\frac{p_1\rho_0+p_2\rho_m+p_3\rho_s}{p_1+p_2+p_3}\otimes\rho_0)+(p_2+p_3)(\frac{p_1\rho_0+p_2\rho_m+p_3\rho_s}{p_1+p_2+p_3}\otimes\rho_m)+p_4\rho_c\otimes\rho_m
\end{equation}
After another swap gate,
\begin{equation}
\rho=p_1(\rho_0\otimes\frac{p_1\rho_0+p_2\rho_m}{p_1+p_2+p_3})+p_1(\frac{p_3\rho_s}{p_1+p_2+p_3}\otimes\rho_0)+(p_2+p_3)(\rho_m\otimes\frac{p_1\rho_0+p_2\rho_m}{p_1+p_2+p_3})+(p_2+p_3)(\frac{p_3\rho_s}{p_1+p_2+p_3}\otimes\rho_m)+p_4\rho_c\otimes\rho_m
\end{equation}
So that the maximum initialization and readout fidelity becomes:
\begin{equation}
F_{max}=\frac{1}{2}+\frac{p_1+p_2}{2(p_1+p_2+p_3)}
\end{equation}
This indicates that the initialization and readout fidelity is independent of the charge state initialization fidelity.
\begin{figure}
\caption{Characterization of the initialization and readout fidelity for
nuclear spin 1 and spin 2. (a) Experimental scheme. The nuclear spin is
prepared to $(|0\rangle-i|1\rangle)/\protect\sqrt{2}
\end{figure}
\section{Two-bit quantum state tomography}
The two-bit tomography consists of three-basis (X, Y, Z) single-bit
measurements on the two nuclear spins separately and nine two-bit
correlation measurements. All the measurements are performed by mapping the
nuclear spin information onto the electronic spin population. Figure 7 shows
our experimental scheme for the single-bit and two-bit measurements. The
nuclear spin density matrix is extracted from the two-bit tomography result
with a maximum likelihood calculation \cite{3}.
\begin{figure}
\caption{Quantum state tomography scheme. (a) Single-bit tomography scheme of the three
bases on the two nuclear spins separately. (b) Correlation measurements of
the $9$ bases on the two nuclear spins cooperatively. }
\end{figure}
\section{Numerical simulation}
The numerical simulation is performed in the rotating frame in the
three-qubit system composed of electronic spin and the two target nuclear
spins. We assume a noiseless environment without decoherence and relaxation.
The $350$ ns green laser pumping is simulated by an instant reset of the
electronic spin. After the reset, the nuclear spin state is given by the
partial trace over the electronic spin state. All the parameters in the
simulation is the same as those calibrated by the experiments.
\section{Crosstalk between the two nuclear spins}
As the nuclear hyperfine interactions are always on, gates on one nuclear
spin will affect other nuclear spins and lead to unwanted operations. There
are two types of possible operations on other nuclear spins: (i) Conditional
or unconditional X operations. (ii) Z rotations.
Because conditional and unconditional X operations only happen at specific
time $\tau $ with a very narrow bandwidth, the first type of influence can
be avoided by choosing $\tau $ to bypass X rotations on other nuclear spins.
To reduce the influence from the unwanted Z rotation, we track the phase of
each nuclear spin in experiment through simulation and compensate
accumulated phase at proper time in experimental sequence using the right
parameters fixed from the simulation.
\section{Realization of general collective noise model}
By injecting radio frequency (rf) noise into the system to drive the nuclear
spin transitions, we can realize any collective noise model. The rf signal
is centered at the nuclear spin Larmor frequency with a bandwidth of $10$
kHz. To model a noisy environment with time correlation function of the
shape $exp(-R|\tau |)$, we add up all the frequency components weighted with
function $\sqrt{\frac{2\delta \omega R}{(2\pi n\Delta \omega )^{2}+R^{2}}}$,
where $\Delta \omega =1$ kHz is the discretization step. The noise is turned
on $5$ $\mu $s after the entanglement preparation step and turned off $5$ $
\mu $s before the state tomography measurement to avoid the ac Stark shift
on the electronic spin caused by the rf signal.
\section{Influence of the magnetic field fluctuation}
The magnetic field is calibrated by measuring electronic ODMR signal every
two hours during the experiments. Due to the fluctuation in the lab
temperature, the magnetic field fluctuates on the order of $0.2$ G. The
fluctuation leads to gate errors accumulated on nuclear spin 1 as well as
unwanted phase evolution on nuclear spin 2 in the entangling process. The
induced phase fluctuation over the $10^{6}$ repetitions of measurements of
each experimental density matrix element leads to a drop of the measured
entanglement fidelity. In our numerical simulation, we find that a magnetic
field fluctuation with a Gaussian shape and a standard deviation of $0.15$ G
leads to dropping of the entanglement fidelity from $1$ to $0.92$.
\end{document}
|
\begin{document}
\begin{frontmatter}
\centerline{Theoretical Computer Science, 328/1-2(2004), 151-160}
\title{Reducing the time complexity of testing for local threshold
testability}
\author{A.N. Trahtman}
\date{}
\ead{[email protected]}
\address{Bar-Ilan University, Dep. of Math., 52900, Ramat Gan, Israel}
\begin{abstract}
A locally threshold testable language $L$ is a language with
the property that for some nonnegative integers $k$ and $l$
and for some word $u$ from $L$, a word $v$ belongs to $L$ iff
\\
(1) the prefixes [suffixes] of length $k-1$ of words $u$
and $v$ coincide,
\\
(2) the numbers of occurrences of every factor
of length $k$ in both words $u$ and $v$ are either the same or
greater than $l-1$.
\\
A deterministic finite automaton is called locally threshold testable
if the automaton accepts a locally threshold testable language for some
$l$ and $k$.\\
New necessary and sufficient conditions
for a deterministic finite automaton
to be locally threshold testable are found.
On the basis of these conditions,
we modify the algorithm to verify local
threshold testability of the automaton and to reduce the time
complexity of the algorithm.
The algorithm is implemented as a part of the
$C/C ^{++}$ package TESTAS.
\texttt{http://www.cs.biu.ac.il/$\sim$trakht/Testas.html}.
\end{abstract}
\begin{keyword}
automaton, threshold locally testable, graph, algorithm
\end{keyword}
\end{frontmatter}
\section{Introduction}
The locally threshold testable languages introduced
by Beauquier and
Pin \cite {BP} now have various applications \cite {REG}, \cite {VCG},
\cite {W}. In particular,
stochastic locally threshold testable languages, also known as
$\it n-grams$, are used
in pattern recognition and in speech recognition,
both in acoustic-phonetics decoding and in language modelling
\cite {VCG}.
These languages generalize the concept of
local testability \cite {BS}, \cite {MP},
which can be considered as a special case of
local $l$-threshold testability for $l=1$. \\
Necessary and sufficient conditions of local testability
\cite {K91} form a basis of polynomial-time algorithms for
the local testability problem \cite{K91}, \cite {TC}.
The algorithms were implemented \cite {C}, \cite {C2}, \cite{TC}.\\
Necessary and sufficient conditions of local threshold testability
for deterministic finite automata (DFA) found in \cite {BP} are
based on a syntactic characterization of locally threshold testable
languages \cite {TW}.
A polynomial-time algorithm of order $O(|\Gamma|^5|\Sigma|)$ for the
local threshold testability problem based on some other kind
of necessary and sufficient conditions was described in \cite{TC}
and implemented. We modify the last necessary and sufficient conditions
and reduce in that way the order of the algorithm for
local threshold testability to $O(|\Gamma|^4|\Sigma|)$.
The algorithm was successfully implemented.
\\
\section{Notation and definitions}
Let $\Sigma^+$ [$\Sigma^*$] denote the free semigroup [monoid]
over an alphabet $\Sigma$.
\\
If $w \in \Sigma^+$, let $|w|$ denote the length of $w$.
Let $k$ be a positive integer. Let $i_k(w)$ $[t_k(w)]$ denote
the prefix
[suffix] of $w$ of length $k$ or $w$ if $|w| < k$. Let
$F_{k,j}(w)$ denote
the set of factors of $w$ of length $k$ with at least $j$
occurrences.
A language $L$
is called {\it l-threshold k-testable} if there is an
alphabet $\Sigma$ such that for all
$u$, $v \in \Sigma^+$, if $i_{k-1}(u)=i_{k-1}(v)$, $t_{k-
1}(u)=t_{k-1}(v)$
and $F_{k,j}(u)=F_{k,j}(v)$ for all $j \le l$, then either
both $u$ and $v$
are in $L$ or neither is in $L$. \\
An automaton is {\it $l$-threshold $k$-testable} if the
automaton accepts a
$l$-threshold $k$-testable language.
A language $L$ [an automaton] is
{\it locally threshold} {\it testable} if it is
$l$-threshold $k$-testable for
some $k$ and $l$.\\
Let us now consider the transition graph of a DFA.\\
The action of a word $v \in \Sigma^*$ on a state $\bf q$ is denoted by ${\bf q}v$.
Thus ${\bf q}v$ is the state reached by the unique path of label $v$
starting at $\bf q$.
\\
A state $\bf p$ is a $\it cycle$ $\it state$
if, for some $e \in \Sigma^+$, ${\bf p}e=\bf p$.
\\
A maximal strongly connected component of a directed graph will be
denoted for brevity by $\it SCC$.
\\
We shall write $\bf p \succeq \bf q$ if $\bf q$ is reachable from $\bf p$
(that is, if ${\bf p}v=\bf q$ for some word $v \in \Sigma^*$)
and $\bf p \sim q$ if $\bf p \succeq q$ \& $\bf q \succeq p$
(that is, if $\bf p$ and $\bf q$ are in the same $SCC$).
\\
The number of vertices of a graph $\Gamma$ is denoted by $|\Gamma|$.\\
An oriented labelled graph is {\it complete} if any of its vertex
has outgoing edge with any label from the alphabet of
labels. A non-complete graph can be completed by adding a sink
state and then adding lacking edges from corresponding
vertices to the sink state.\\
The direct product $\Gamma^k$ of $k$ copies of a directed labelled
graph $\Gamma$ over an alphabet $\Sigma$ consists of vertices $({\bf
p}_1, ..., {\bf p}_k)$ and edges (${\bf p}_1, ..., {\bf p}_k) \to
({\bf p}_1\sigma, ..., {\bf p}_k\sigma)$ labelled by $\sigma$.
Here ${\bf p}_i$ are vertices from $\Gamma$, $\sigma \in \Sigma$.
\section{The necessary and sufficient conditions of local
threshold testability}
Let us formulate the result of Beauquier and Pin \cite
{BP} in the
following form:
\begin{thm} $\label{1}$ \cite {BP}
A language $L$ is locally threshold
testable if and only if the syntactic semigroup $S$ of $L$
is aperiodic
and for any two idempotents $e$, $f$ and elements $a$, $u$,
$b$ of $S$, we have
\begin{equation}
eafuebf=ebfueaf.
\label {e1}
\end{equation}
\end{thm}
We now consider a fixed locally threshold testable DFA
with state transition graph $\Gamma$ and transition semigroup $S$.
\begin{lem} $\label{2}$ \cite {K91} \cite {TC}
Let ($\bf p, q$) be a cycle state of $\Gamma^2$.
If ${\bf p} \sim \bf q$, then ${\bf p} = \bf q$.
\end{lem}
\begin{lem} $\label{4}$
Let (${\bf q, t}_1$) and (${\bf q, t}_2$) be cycle states
of $\Gamma^2$.
If $({\bf q, t}_1) \succeq ({\bf q, t}_2)$
and ${\bf q} \succeq {\bf t}_1$ then ${\bf t}_1 \sim {\bf t}_2$.
\end{lem}
\begin{picture}(170,44)
\put(35,18){\circle{4}}
\put(17,20){${\bf t}_1$}
\put(55,8){$a$}
\put(55,43){$a$}
\put(45,26){$b$}
\put(37,42){\vector(1,0){63}}
\put(37,18){\vector(1,0){63}}
\put(102,42){\circle{4}}
\put(112,40){$\bf q$}
\put(35,40){\vector(0,-1){20}}
\put(35,42){\circle{4}}
\put(18,41){$\bf q$}
\put(102,18){\circle{4}}
\put(112,20){${\bf t}_2$}
\put(35,32){\oval(14,38)}
\put(30,4){$e$}
\put(101,32){\oval(14,38)}
\put(105,5){$i$}
\put(145,30){\vector(1,0){20}}
\put(185,27){${\bf t}_1 \sim {\bf t}_2$}
\end{picture}\\
Proof.
One has (${\bf q, t}_1)e = ({\bf q, t}_1$),
(${\bf q, t}_2)i= ({\bf q, t}_2$),
(${\bf q, t}_1)a = ({\bf q, t}_2$),
${\bf q}b = {\bf t}_1$ for some idempotents $e$, $i$
and elements $a$, $b$ from $S$.
The substitution $ai$ in place of $a$ and $e$ in place of $f$ and $u$
in (\ref{e1}) implies $eaiebe=ebeaie$.
Therefore ${\bf t}_2e = {\bf t}_2ie = {\bf t}_1eaie =
{\bf q}ebeaie ={\bf q}eaiebe$.
Thus ${\bf t}_2e = {\bf q}eaiebe={\bf q}ebe = {\bf t}_1e ={\bf t}_1$.
So ${\bf t}_2 \succeq {\bf t}_1$.
We have ${\bf t}_1a = {\bf t}_2$, whence ${\bf t}_1 \sim {\bf t}_2$.
\begin{lem} $\label{5}$
Let ${\bf p, q, t, r, s}$ be states
such that $({\bf p, s})$ and (${\bf r, t}$)
are cycle states of $\Gamma^2$. If $({\bf p,s}) \succeq ({\bf q, t})$
and ${\bf p} \succeq {\bf r} \succeq {\bf s}$,
then ${\bf q} \succeq {\bf t}$.
\end{lem}
\begin{picture}(170,54)
\put(35,18){\circle{4}}
\put(19,20){${\bf s}$}
\put(112,34){$\bf r$}
\put(76,5){$b$}
\put(75,53){$b$}
\put(100,35){\vector(-4,-1){63}}
\put(55,26){$u$}
\put(51,40){$a$}
\put(102,35){\circle{4}}
\put(37,52){\vector(1,0){63}}
\put(37,18){\vector(1,0){63}}
\put(102,52){\circle{4}}
\put(112,50){$\bf q$}
\put(37,52){\vector(4,-1){63}}
\put(35,52){\circle{4}}
\put(20,51){$\bf p$}
\put(102,18){\circle{4}}
\put(112,20){$\bf t$}
\put(37,37){\oval(14,48)}
\put(30,5){$e$}
\put(100,26){\oval(16,28)}
\put(106,6){$i$}
\put(146,35){\vector(1,0){20}}
\put(188,32){${\bf q} \succeq {\bf t}$}
\end{picture}
\\
Proof. One has $({\bf p, s})e=({\bf p, s})$ and
(${\bf r},{\bf t})i=({\bf r},{\bf t})$
for some idempotents $e, i \in S$.
Furthermore, $({\bf p,s})b=({\bf q, t})$,
${\bf p}a={\bf r}$ and ${\bf r}u={\bf s}$
for some elements $a$, $u$, $b \in S$.
In view of (\ref{e1}), ${\bf t}={\bf p}eaiuebi = {\bf p}ebiueai$.
Thus ${\bf t} = {\bf p}ebiueai = {\bf q}iuebi$,
whence ${\bf q} \succeq {\bf t}$.
\begin{lem} $\label{6}$
Let ($\bf q, r$), (${\bf p, s}$), (${\bf q, t}_1$) and (${\bf q, t}_2$)
be cycle states of the graph $\Gamma^2$ such that
$({\bf p, s}) \succeq ({\bf q, t}_i)$,
${\bf q} \succeq {\bf t}_i$ for $i=1,2$
and ${\bf p} \succeq {\bf r} \succeq {\bf s}$.
Then ${\bf t}_1 \sim {\bf t}_2$.
\end{lem}
\begin{picture}(150,70)
\put(25,30){\circle{4}}
\put(9,30){${\bf s}$}
\put(97,27){$\bf t_2$}
\put(109,66){$f_1$}
\put(27,30){\vector(1,0){63}}
\put(90,30){\circle{4}}
\put(47,17){$b_2$}
\put(90,62){\circle{4}}
\put(96,61){$\bf q$}
\put(28,62){\vector(1,0){61}}
\put(90,60){\vector(0,-1){29}}
\put(93,62){\vector(2,-3){9}}
\put(25,62){\circle{4}}
\put(10,61){$\bf p$}
\put(46,65){$b_1, b_2$}
\put(25,30){\vector(4,1){73}}
\put(58,44){$b_1$}
\put(100,48){\circle{4}}
\put(110,46){${\bf t}_1$}
\put(96,56){\oval(24,30)}
\put(25,46){\oval(10,46)}
\put(90,46){\oval(10,46)}
\put(20,15){$e$}
\put(72,16){$f_2$}
\end{picture}
\begin{picture}(180,70)
\put(5,30){\circle{4}}
\put(-12,30){${\bf s}$}
\put(77,40){$\bf r$}
\put(70,45){\vector(-4,-1){63}}
\put(70,45){\circle{4}}
\put(70,62){\circle{4}}
\put(77,60){$\bf q$}
\put(7,62){\vector(1,0){63}}
\put(7,62){\vector(4,-1){63}}
\put(5,62){\circle{4}}
\put(-10,61){$\bf p$}
\put(21,36){$u$}
\put(20,50){$a$}
\put(5,46){\oval(10,46)}
\put(69,53){\oval(10,30)}
\put(0,15){$e$}
\put(66,25){$f$}
\put(95,37){\vector(1,0){20}}
\put(136,35){${\bf t}_1 \sim {\bf t}_2$}
\end{picture}\\
Proof. One has
$({\bf p, s})e=({\bf p, s})$,
$({\bf q, r})f=({\bf q, r})$,
(${\bf q},{\bf t}_1)f_1=({\bf q},{\bf t}_1)$,
(${\bf q},{\bf t}_2)f_2=({\bf q},{\bf t}_2)$
for some idempotents $e$, $f$, $f_1$, $f_2 \in S$,
Furthermore, $({\bf p,s})b_1=({\bf q, t}_1)$,
$({\bf p,s})b_2=({\bf q, t}_2)$,
${\bf p}a={\bf r}$ and ${\bf r}u={\bf s}$
for some elements $a$, $u$, $b_1$, $b_2 \in S$.\\
Let us consider the state ${\bf t}_if$ ($i=1,2$) where the
idempotent $f$ is right unit for $({\bf q, r})$.
The states $({\bf q, t}_i)$ and $({\bf q, t}_if)$
are cycle states,
${\bf q} \succeq {\bf t}_i$,
$({\bf q,t}_i) \succeq ({\bf q, t}_if)$,
whence by Lemma \ref{4}, ${\bf t}_if \sim {\bf t}_i$ for any such $f$.
So ${\bf t}_1f \sim {\bf t}_1$ and ${\bf t}_2f \sim {\bf t}_2$.
\\
The equality of local
threshold testability (\ref{e1}) implies ${\bf t}_1f ={\bf s}b_1f ={\bf
r}ueb_1f = {\bf p}eafueb_1f= {\bf p}eb_1fueaf$. Furthermore, ${\bf
p}b_1 = {\bf p}b_2= {\bf q}$, whence ${\bf t}_1f = {\bf p}eb_1fueaf
= {\bf p}eb_2fueaf = {\bf p}eafueb_2f= {\bf t}_2f$. So ${\bf t}_2f =
{\bf t}_1f$. We have ${\bf t}_1f \sim {\bf t}_1$ and ${\bf t}_2f \sim {\bf t}_2$,
whence ${\bf t}_2 \sim {\bf t}_1$.\qed
If ${\bf p, q, s}$ are states of $\Gamma$ and there exists
some state ${\bf r}$
such that $({\bf q, r})$ and $({\bf p, s})$
are cycle states of $\Gamma^2$,
${\bf p} \succeq {\bf q}$,
and ${\bf p} \succeq {\bf r} \succeq {\bf s}$, then the
non-empty set\\
\centerline{
$ T= \{ t$ $| ({\bf p,s}) \succeq ({\bf q, t})$, ${\bf q} \succeq {\bf t}$
and (${\bf q, t}$) is a cycle state$\}$
}\\
by the previous Lemma is a subset of some $SCC$ from transition graph $\Gamma$
of locally threshold testable automaton.
This $SCC$ will be denoted by $SCC({\bf p, q, s})$. In the case
that $\bf r$ does not exist or $T$ is empty,
let $SCC({\bf p, q, s})$ be empty.
\\
\begin{picture}(180,85)
\put(45,29){\circle{4}}
\put(28,30){${\bf s}$}
\put(104,51){$\bf r$}
\put(88,16){$b$}
\put(85,63){$b$}
\put(108,45){\vector(-4,-1){61}}
\put(109,46){\circle{4}}
\put(47,62){\vector(1,0){68}}
\put(47,29){\vector(1,0){68}}
\put(117,62){\circle{4}}
\put(126,60){$\bf q$}
\put(117,60){\vector(0,-1){30}}
\put(47,62){\vector(4,-1){62}}
\put(45,62){\circle{4}} \put(30,61){$\bf p$}
\put(117,29){\circle{4}}\put(126,30){$\bf t$}
\put(46,46){\oval(10,46)}
\put(112,54){\oval(20,30)}
\put(40,15){$e$}
\put(100,70){$f$}
\put(118,46){\oval(10,46)}
\put(121,14){$i$}
\put(155,45){\vector(1,0){20}}
\put(200,42){${\bf t} \in SCC({\bf p, q, s})$}
\end{picture}\\
By Lemma \ref{6}, $SCC({\bf p, q, s})$ is well-defined for transition graphs of
locally threshold testable automata.
\begin{lem} \label{7}
Let $({\bf p, r}_1)$ and $({\bf p, r}_2)$ be cycle states
of the graph $\Gamma^2$.
Suppose that ${\bf r}_1 \sim {\bf r}_2$,
${\bf q} \succeq {\bf t}_i$,
${\bf p} \succeq {\bf r} \succeq {\bf r}_i$ ($i=1,2$) for some ${\bf r}$
such that $({\bf q, r})$ is a cycle state.
Then ${\bf t}_1 \sim {\bf t}_2$
and $SCC({\bf p,q, r}_1) = SCC({\bf p,q, r}_2)$.
\end{lem}
\begin{picture}(100,70)
\put(16,30){\circle{4}}
\put(0,30){${\bf r}_i$}
\put(98,23){$\bf t_i$}
\put(18,30){\vector(1,0){73}}
\put(91,30){\circle{4}}
\put(51,17){$b_i$}
\put(91,62){\circle{4}}
\put(98,60){$\bf q$}
\put(91,60){\vector(0,-1){29}}
\put(18,62){\vector(1,0){73}}
\put(16,62){\circle{4}}
\put(1,61){$\bf p$}
\put(51,66){$b_i$}
\put(16,46){\oval(10,46)}
\put(90,46){\oval(10,46)}
\put(85,55){\oval(20,32)}
\put(11,15){$e_i$}
\put(76,15){$f_i$}
\put(71,71){$f$}
\put(81,45){\circle{4}}
\put(78,50){${\bf r}$}
\put(81,45){\vector(-4,-1){63}}
\put(18,62){\vector(4,-1){63}}
\put(39,40){$u_i$}
\put(24,1){$i=1,2 $}
\put(105,35){$\&$}
\put(124,35){$\bf r_1 \sim r_2$}
\put(176,36){\vector(1,0){15}}
\put(215,42){$SCC({\bf p,q, r}_1) = SCC({\bf p,q, r}_2)$}
\put(257,23){$\bf t_1 \sim t_2$}
\end{picture}\\
Proof.
One has $({\bf p, r}_1)e_1=({\bf p, r}_1)$,
$({\bf p, r}_2)e_2=({\bf p, r}_2)$, $({\bf q, r})f=({\bf q, r})$
for some idempotents $e_1$, $e_2$, $f \in S$,
${\bf p}\succeq {\bf r}$
and $({\bf p,r}_1)b_1=({\bf q, t}_1)$,
$({\bf p,r}_2)b_2=({\bf q, t}_2)$,
${\bf r}u_i={\bf r}_i$
for some elements $u_i$, $b_1$, $b_2 \in S$.
\\
From $({\bf p,r}_1)e_2=({\bf p, r}_1e_2)$,
by Lemma \ref{4},
it follows that ${\bf r}_1 \sim {\bf r}_1e_2$.
Notice that ${\bf r}_2e_2 = {\bf r}_2 \sim {\bf r}_1$,
whence ${\bf r}_2 \sim {\bf r}_1e_2$.
Therefore, by Lemma \ref{2}, ${\bf r}_2 = {\bf r}_1e_2$.
Furthermore, $({\bf p,r}_1)e_2b_2=({\bf q, r}_1e_2b_2)=
({\bf q, r}_2b_2)= ({\bf q, t}_2)$.
Thus $({\bf p,r}_1) \succeq ({\bf q, t}_2)$.
Then $({\bf p,r}_1) \succeq ({\bf q, t}_1)$
in view of $({\bf p,r}_1)b_1=({\bf q, t}_1)$.
Now by Lemma \ref{6}, the states ${\bf t}_1, {\bf t}_2$
belong to $SCC({\bf p,q, r}_1)$ and
${\bf t}_1 \sim {\bf t}_2$.
Let us notice that
the state ${\bf t}_2$ belongs to
$SCC({\bf p,q, r}_2)$ too. Hence by Lemma \ref{6},
$SCC({\bf p,q, r}_1) = SCC({\bf p,q, r}_2)$.
\begin{lem} \label{12}
If \\
${\bf p}\succeq \bf q$, ${\bf p}\succeq \bf r$ and ${\bf p}e={\bf p}$
for an idempotent $e \in S$,\\
the state $({\bf q, r}$) is a cycle state of
the graph $\Gamma^2$,\\
there exists a state ${\bf r}_1$ such that
(${\bf p, r}_1)$ is a cycle state and ${\bf r} \succeq {\bf r}_1$,\\
then $SCC({\bf p,q,r}_1e)=SCC({\bf p,q,r}_1)$.
\end{lem}
\begin{picture}(170,41)
\put(25,10){\circle{4}}
\put(7,8){${\bf r}_1$}
\put(95,13){$\bf r$}
\put(89,24){\vector(-4,-1){62}}
\put(91,25){\circle{4}}
\put(27,42){\vector(1,0){63}}
\put(91,42){\circle{4}} \put(99,40){$\bf q$}
\put(27,42){\vector(4,-1){63}}
\put(8,41){$\bf p$}
\put(25,42){\circle{4}}
\put(25,26){\oval(14,46)}
\put(91,34){\oval(12,30)}
\put(107,22){$\&$ ${\bf p}e={\bf p}$}
\put(168,24){\vector(1,0){12}}
\put(184,22){$SCC({\bf p, q, r}_1) = SCC({\bf p, q, r}_1e)$}
\end{picture}\\
Proof.
One has ${\bf p}\succeq {\bf r} \succeq {\bf r}_1$.
Then (${\bf p, r}_1) \succeq ({\bf p, r}_1)e = ({\bf p, r}_1e)$ and both these
states are cycle states. Therefore, by Lemma \ref{4},
${\bf r}_1e \sim {\bf r}_1$.
Lemma \ref{7} for ${\bf r}_2 = {\bf r}_1e$
implies now $SCC({\bf p,q,r}_1e)=SCC({\bf p,q, r}_1)$.
\begin{thm} $\label{14}$
$DFA$ {\bf A} with state transition complete graph $\Gamma$
(or completed by a sink state) is locally threshold testable iff
\\
\begin{itemize}
\item 1) for every cycle state ($\bf p, q$) of $\Gamma^2$, ${\bf p}
\sim \bf q$ implies ${\bf p} = \bf q$,
\\
\item 2) for every states ${\bf p, q, t, s}$ of
$\Gamma$ such that
\begin{itemize}
\item (${\bf p, s})$ is a cycle state,
\item $({\bf p,s}) \succeq ({\bf q, t})$,
\item ${\bf p} \succeq {\bf r} \succeq {\bf s}$ and
(${\bf r, t}$) is a cycle state for some ${\bf r}$,
\end{itemize}
it holds ${\bf q} \succeq {\bf t}$. (see figure to Lemma \ref{5})
\\
\item 3) for every states ${\bf p, q, r}$, $SCC({\bf p, q, r})$
is well defined,
\\
\item 4) for every four states ${\bf p, q, r, q}_1$ such that
\begin{itemize}
\item (${\bf p, q}_1)$ and $({\bf q, r}$) are cycle states of
the graph $\Gamma^2$,
\item ${\bf p}\succeq \bf q$ and ${\bf p}\succeq \bf r$,
\item for some state ${\bf r}_1$ such that
(${\bf p, r}_1)$ is a cycle state and
$({\bf q,r})\succeq ({\bf q}_1,{\bf r}_1)$,
\end{itemize}
it holds $SCC({\bf p,q,r}_1)=SCC({\bf p,r,q}_1)$.
\end{itemize}
\end {thm}
\begin{picture}(170,79)
\put(25,30){\circle{4}}
\put(7,28){${\bf r}_1$}
\put(102,44){$\bf r$}
\put(89,44){\vector(-4,-1){62}}
\put(90,60){\vector(-4,-1){54}}
\put(52,44){$u$}
\put(52,28){$u$}
\put(90,45){\circle{4}}
\put(27,62){\vector(1,0){63}}
\put(92,62){\circle{4}} \put(100,60){$\bf q$}
\put(27,62){\vector(4,-1){63}}
\put(25,62){\circle{4}} \put(8,61){$\bf p$}
\put(36,45){\circle{4}}\put(19,45){${\bf q}_1$}
\put(25,46){\oval(14,46)}
\put(29,53){\oval(24,30)}
\put(20,14){$i$}
\put(35,69){$e$}
\put(91,54){\oval(12,30)}
\put(93,28){$f$}
\put(122,45){\vector(1,0){15}}
\put(155,42){$SCC({\bf p, r, q}_1) = SCC({\bf p, q, r}_1)$}
\end{picture}\\
Proof. Let $\bf A$ be a locally threshold testable DFA.
Condition 1 follows in this case from Lemma \ref{2}.
Condition 2 follows from Lemma \ref{5}.
Condition 3 follows from Lemma \ref{6}.
\\
Let us check the last condition.
For some idempotent $e$, it holds (${\bf p, q}_1)e = ({\bf p, q}_1)$.
By Lemma \ref{12},
$SCC({\bf p,q,r}_1e)=SCC({\bf p,q, r}_1)$.
Therefore let us compare $SCC({\bf p,q,r}_1e)$
and $SCC({\bf p,r,q}_1)$.
\begin{picture}(200,84)
\put(25,10){\circle{4}} \put(30,77){$e$}
\put(27,9){\vector(1,0){63}}
\put(3,-2){${\bf r_1}e$}
\put(90,10){\circle{4}}
\put(97,8){${\bf t}_1 \in SCC({\bf p, q, r}_1e)$}
\put(73,76){$f_1$}
\put(58,12){b}
\put(25,30){\circle{4}}
\put(27,30){\vector(1,0){74}}
\put(101,30){\circle{4}}
\put(105,31){${\bf t} \in SCC({\bf p, r, q}_1)$}
\put(114,77){$f$}
\put(43,31){a}
\put(6,30){$\bf q_1$}
\put(88,50){\vector(-3,-2){60}} \put(97,50){$\bf r$}
\put(88,49){\vector(-3,-2){54}}
\put(90,50){\circle{4}}
\put(34,20){u}
\put(90,70){\circle{4}} \put(97,68){$\bf q$}
\put(27,71){\vector(1,0){62}}
\put(25,70){\circle{4}} \put(10,71){$\bf p$}
\put(58,74){b}
\put(88,70){\vector(-3,-2){60}}
\put(88,69){\vector(-3,-2){54}}
\put(48,49){u}
\put(27,70){\vector(3,-1){61}}
\put(54,62){a}
\put(25,40){\oval(10,70)}
\put(89,40){\oval(10,70)}
\put(99,53){\oval(30,56)}
\put(155,50){\vector(1,0){20}}
\put(183,46){$SCC({\bf p, q, r}_1e)=SCC({\bf p, r, q}_1)$}
\end{picture}
One has ${\bf t}_1f = {\bf r}_1ebf = {\bf p}eafuebf$.
Then by (\ref{e1}) ${\bf p}eafuebf = {\bf p}ebfueaf = {\bf q}ueaf =
{\bf q}_1af = \bf t$. So ${\bf t}_1 \succeq \bf t$. Analogously,
${\bf t} \succeq {\bf t}_1$. Therefore ${\bf t}_1 \sim \bf t$, whence
$SCC({\bf p, r, q}_1) = SCC({\bf p, q, r}_1e)$.
Consequently, $SCC({\bf p, q, r}_1)=SCC({\bf p,r, q}_1)$.
\\
Conversely, suppose that all four conditions of the theorem hold.
Our aim is to prove the local threshold
testability of DFA. For this aim, let us consider an arbitrary state
$\bf v$, arbitrary elements $a, u, b$ and idempotents $e, f$ from
the syntactic semigroup $S$ of the automaton.
We must to prove that ${\bf v}eafuebf = {\bf v}ebfueaf$ (Theorem \ref{1}).
\\
Let us denote ${\bf p} = {\bf v}e$, ${\bf q} = {\bf v}ebf$,
${\bf q}_1 = {\bf v}ebfue$, ${\bf t} = {\bf v}ebfueaf$,
${\bf r} = {\bf v}eaf$, ${\bf r}_1 = {\bf v}eafue$,
${\bf t}_1 = {\bf v}eafuebf$.
\\
We have $({\bf p, r}_1) \succeq ({\bf q, t}_1)$, the states
$({\bf p, r}_1)$, $({\bf q, t}_1)$ and $({\bf r, t}_1)$ are cycle states,
${\bf p} \succeq {\bf r} \succeq {\bf r}_1$.
Therefore by condition 2, for ${\bf r}_1 = \bf s$, ${\bf q} \succeq {\bf t}_1$.
Now ${\bf t}_1 \in SCC({\bf p, q, r}_1)$. Analogously
${\bf t} \in SCC({\bf p,r, q}_1)$.
The state
(${\bf p, q}_1)$ is a cycle state and $({\bf q,r})ue = ({\bf q}_1,{\bf r}_1)$.
Hence condition 4 implies
$SCC({\bf p, q, r}_1)=SCC({\bf p,r, q}_1)$.
These sets are well-defined, whence by condition 3, ${\bf t}_1 \sim {\bf t}$.
Both these states have common right unit $f$.
Consequently, $({\bf t, t}_1)$ is a cycle state. Now by condition 1,
${\bf t}_1 = {\bf t}$. Thus ${\bf v}eafuebf = {\bf v}ebfueaf$
is true for an arbitrary state $\bf v$
and the identity $eafuebf = ebfueaf$
of local threshold testability holds.
\\
It remains now to prove the aperiodicity of $S$.
Let $\bf p$ be an arbitrary state and let
$s$ be an arbitrary element of $S$. The semigroup $S$
is finite, whence for some integers $k$ and $m$, it
holds $s^k=s^{k+m}$. Let us consider the states ${\bf p}s^k$ and
${\bf p}s^{k+1}$. We have ${\bf p}s^k \succeq {\bf p}s^{k+1}$ and, in view
$s^k=s^{k+m}= s^{k+1}s^{m-1}$, it holds ${\bf p}s^{k+1} \succeq {\bf
p}s^k$. Thus ${\bf p}s^{k+1} \sim {\bf p}s^k$. Some power of $s$ is
an idempotent and a right unit for both these states. Then by
condition 1, ${\bf p}s^k = {\bf p}s^{k+1}$. Therefore
$S$ is aperiodic,
and thus the automaton is locally threshold testable.
\begin{lem} \label{15}
Let $P({\bf q, r})$ be a non-empty set of cycle states
of a locally threshold testable $DFA$
such that
${\bf p} \succeq {\bf q}$ and ${\bf p}\succeq \bf r$
for a cycle state $({\bf q, r})$.
\\
By ${\bf r}_2 \rho_r {\bf r}_1$ we denote the case that
for a pair of cycle states $({\bf p},{\bf r}_1)$ and $({\bf p},{\bf r}_2)$,
it holds $({\bf q, r}) \succeq ({\bf q}_1,{\bf r}_1)$ and
$({\bf q, r}) \succeq ({\bf q}_1,{\bf r}_2)$.
\\
Then ${\bf r}_1 \rho_r {\bf r}_2$ implies
$SCC({\bf p,q, r}_1) = SCC({\bf p,q, r}_2)$
for any ${\bf p} \in P({\bf q, r})$.
\end{lem}
\begin{picture}(200,86)
\put(2,18){$\bf r_1$}
\put(26,15){$\bf r_2$}
\put(17,30){\circle{4}}
\put(35,30){\circle{4}}
\put(73,80){$f$}
\put(35,8){\circle{4}}
\put(47,34){$u_2$}
\put(53,17){$u_1, u_2$}
\put(31,50){$u_1$}
\put(50,61){$a$}
\put(49,72){$b$}
\put(19,5){$\bf q_1$}
\put(80,50){\circle{4}}
\put(88,46){$\bf q$}
\put(80,70){\circle{4}}
\put(88,68){$\bf r$}
\put(27,71){\vector(1,0){52}}
\put(25,70){\circle{4}}
\put(22,81){$\bf p$}
\put(79,71){\vector(-3,-2){61}}
\put(79,71){\vector(-1,-1){42}}
\put(79,51){\vector(-1,-1){42}}
\put(26,68){\vector(3,-1){53}}
\put(19,50){\oval(22,54)}
\put(32,50){\oval(22,54)}
\put(79,60){\oval(12,32)}
\put(130,50){\vector(1,0){20}}
\put(172,46){$SCC({\bf p, q, r}_1)=SCC({\bf p, q, r}_2)$}
\end{picture}
\\
Proof. One has $({\bf q, r})f = ({\bf q, r})$,
$({\bf q},{\bf r})u_1 = ({\bf q}_1,{\bf r}_1)$,
$({\bf q},{\bf r})u_2 = ({\bf q}_1,{\bf r}_2)$,
${\bf p}a = {\bf q}$, ${\bf p}b = {\bf r}$,
${\bf p}e=\bf p$ for some idempotents $e$, $f$ and elements $u_i$,
$a$, $b$ from $S$.
So ${\bf q}_1 = {\bf p}eafu_2 = {\bf p}eafu_1$, ${\bf p}ebfu_1 = {\bf r}_1$,
${\bf p}ebfu_2 = {\bf r}_2$.
For the state ${\bf r}_1eaf$ from $SCC({\bf p,q, r}_1e)$, it holds
${\bf r}_1eaf = {\bf p}ebfu_1eaf = {\bf p}eafu_1ebf =
{\bf p}eafu_2ebf = {\bf p}ebfu_2eaf = {\bf r}_2eaf \in SCC({\bf p,q, r}_2e)$.
So $SCC({\bf p,q, r}_1e) = SCC({\bf p,q, r}_2e)$.
Thus ${\bf r}_2 \rho_r {\bf r}_1$ implies
$SCC({\bf p,q, r}_1e) = SCC({\bf p,q, r}_2e)$.
By Lemma \ref{12}, $SCC({\bf p,q, r}_ie) = SCC({\bf p,q, r}_i)$,
whence $SCC({\bf p,q, r}_1) = SCC({\bf p,q, r}_2)$.
\begin{cor} \label{l6}
Let $P({\bf q, r})$ be a non-empty set of cycle states $\bf p$
of a locally threshold testable $DFA$ such that
${\bf p} \succeq {\bf q}$ and ${\bf p}\succeq \bf r$
for cycle state $({\bf q, r})$.
\\
Then non-empty $SCC({\bf p,q, r}_1)$ does not depend on ${\bf r}_1$
for any ${\bf p} \in P({\bf q, r})$.
\end{cor}
\section{An algorithm for local threshold
testability}
A linear depth-first search algorithm which finds
all $SCC$ (see \cite {Ta}) will be used.
The algorithm is based on Theorem \ref{14}
for a complete transition graph $\Gamma$
(or $\Gamma$ which is completed by sink state).
The measures of complexity of the transition graph
$\Gamma$ are here $|\Gamma|$ (state complexity),
the sum of the numbers of the states and the transitions
$\it k$ and the size of the alphabet $\it g$ of the labels.
Let us notice that $|\Gamma|(g+1) \ge k$.
\\
Let us find all $SCC$ of the graphs $\Gamma$
and $\Gamma^2$ and all their cycle states.
Furthermore we recognize the reachability on the graph $\Gamma$
and form the table of reachability for all pairs
of states. The step uses $O(|\Gamma|^2g)$ time and space.
\\
{\it The first condition of Theorem \ref{14}.}
For every cycle state ($\bf p, q$) (${\bf p} \neq \bf q$)
from $\Gamma^2$, let us check
the condition ${\bf p} \sim \bf q$. A negative answer for
any considered
cycle state ($\bf p, q$) implies the validity of the condition. In
the opposite case, the automaton is not locally threshold testable.
The time of the step is $O(|\Gamma|^2)$.
\\
{\it The second condition of Theorem \ref{14}.}
For every cycle state $({\bf p,s})$, we form the set $T$
of states ${\bf t} \in \Gamma$ such that ${\bf s} \succeq {\bf t}$
and for some state $\bf r$ holds: (${\bf r, t}$)
is a cycle state and ${\bf p} \succeq {\bf r} \succeq {\bf s}$.
If there exists a state ${\bf q}$ such that
$({\bf p, s}) \succeq ({\bf q, t}$) for ${\bf t} \in T$ and
${\bf q} \not\succeq {\bf t}$, then
the automaton is not threshold locally testable.
It is a step of worst case asymptotic cost $O(|\Gamma|^4g)$
with space complexity $O(|\Gamma|^3)$.
\\
{\it The condition 3 of Theorem \ref{14}.}
For every three states ${\bf p, q, s}$ of the automaton
such that $({\bf p,s})$ is a cycle state,
${\bf p} \succeq {\bf s}$
and ${\bf p}\succeq {\bf q}$,
let us find a state $\bf r$ such that
${\bf p} \succeq {\bf r} \succeq {\bf s}$
and then let us find $SCC({\bf p, q, s})$.
In the case that this set is not well-defined
(for ${\bf t}_1$, ${\bf t}_2$ from $SCC({\bf p, q, s})$
${\bf t}_1 \not\sim {\bf t}_2$), the
automaton is not threshold locally testable (Lemma \ref{6}).
The time required for this step in the worst case
is $O(|\Gamma|^4g)$. The space complexity is $O(|\Gamma|^3)$.
\\
Before checking condition 4, let us check the assertion of Lemma \ref{15}.
For every cycle state $({\bf q, r})$ of the graph $\Gamma^2$, let us form the
set $P({\bf q, r})$ of cycle states $\bf p$ such that ${\bf p}
\succeq {\bf q}$ and ${\bf p}\succeq \bf r$. We continue for non-empty set
$P({\bf q, r})$. For every state ${\bf q}_1$, let us
form the set $R(q_1)$ of states ${\bf r}_1$ such that $({\bf q, r})
\succeq ({\bf q}_1,{\bf r}_1)$ and the state $({\bf q}_1,{\bf r}_1)$
is a cycle state.
Let us consider two states ${\bf r}_1$, ${\bf r}_2$ from the set $R(q_1)$
such that the states
$({\bf p, r}_1)$ and $({\bf p},{\bf r}_2)$ are cycle states
for some $\bf p$ from $P({\bf q, r})$ .
If $SCC({\bf p, q,r}_1) \ne SCC({\bf p, q, r}_2)$, then
the automaton is not locally threshold testable.
\\
{\it The condition 4 of Theorem \ref{14}.}
For every cycle state $({\bf q, r})$ of $\Gamma^2$, let us form the
set $P({\bf q, r})$ of cycle states $\bf p$ such that ${\bf p}
\succeq {\bf q}$ and ${\bf p}\succeq \bf r$. We continue for non-empty set
$P({\bf q, r})$. By Corollary \ref{l6},
$SCC({\bf p, q, r}_1)$ for given $\bf r$ depends
only on the states $\bf p, q$, and $SCC({\bf p, r, q}_1)$
for given $\bf q$ depends only on $\bf p, r$.
If $SCC({\bf p, q, r}_1)$ and $SCC({\bf p, r, q}_1)$ exist and
are not equal, then the automaton is not locally threshold
testable according to condition 4.
The time required for these last two steps in the worst case
is $O(|\Gamma|^4g)$ with $O(|\Gamma|^3)$ space.
\\
A positive answer for all the cases implies
the local threshold testability of the automaton.
The time complexity of the algorithm
is no more than $O(|\Gamma|^4g)$. The space complexity
is $max(O(|\Gamma|^2g),O(|\Gamma|^3))$.
In more conventional formulation, we have $O(k^4)$ time and
$O(k^3)$ space.
\section{Conclusion. The package TESTAS}
The considered algorithm is now implemented as a part of the $C/C
^{++}$ package TESTAS replacing the old version of the algorithm
and reducing the time of execution. The program worked essentially
faster in many cases we have studied because of the structure of the
algorithm. A part of branches of the algorithm have only
$O(|\Gamma|^2g)$ or $O(|\Gamma|^3g)$ time and space complexity.\\
The maximal size of the considered graphs on an ordinary PC
was about several hundreds states with an alphabet of several dozen
letters. The program in such case used memory on hard disc
and works slower.
\\
The package realizes, besides the considered algorithm for local
threshold testability, a set of algorithms for checking local
testability, left local testability, right local testability,
piecewise testability and some other programs.
The package checks also the synchronizeability of the automaton and
finds synchronizing words.
The programs of the package TESTAS analyze:
\\
1) an automaton of the language presented by oriented labelled
graph. The automaton is given by the matrix:
\\
\centerline{ states X labels}
\\
The non-empty (i,j) cell contains the state
from the end of the transition with label
from the j-th column and beginning in the i-th state.
\\
2) An automaton of the language presented by its syntactic semigroup.
The semigroup is presented by the matrix (Cayley graph)
\\
\centerline{ elements X generators}
\\
where the i-th row of the matrix is a list of products of
the i-th element on all generators.
The set of generators is not necessarily minimal, therefore
the multiplication table of the semigroup (Cayley table) is acceptable, too.
\\
Some auxiliary programs of the package find direct products of the
objects and build the syntactic semigroup of the automaton on the base
of the transition graph.
\\
The space complexity of the algorithms which consider the
transition graph of an automaton is not less than $|\Gamma|g$
because of the structure of the input.
The graph programs use usually a table
of reachability defined on the states of the graph.
The table of reachability is a square table
and so we have $|\Gamma|^2$ space complexity.
\\
The number of the states of $\Gamma^n$ is $|\Gamma|^n$,
the alphabet is the same as in $\Gamma$.
So the sum of the numbers of the states
and the transitions of the graph $\Gamma^n$
is not greater than $(g+1)|\Gamma|^n$.
Some algorithms of the package use the powers $\Gamma^2$,
and $\Gamma^3$. So the space complexity of the algorithms
reaches in these cases $|\Gamma|^2\it g$ or $|\Gamma|^3\it g$.
\\
An algorithm for the local testability problem
for the transition graph (\cite {TC})
of $O(k^2)$ (or $O((|\Gamma|^2g)$ ) time and
space is implemented in the package.
An algorithm of $O(|\Gamma|^2g)$ time
and of $O(|\Gamma|^2g)$ space is used for
finding the bounds on the order of local testability for a given
transition graph of the automaton \cite {Tp}.
An algorithm of worst case $O(|\Gamma|^3g)$ time complexity
and of $O(|\Gamma|^2g)$ space complexity checked the $2$-testability \cite {Tp}.
The $1$-testability is verified using an algorithm \cite {K94} of
order $O(|\Gamma|g^2)$.
For checking the $n$-testability \cite {Tp}, we use
an algorithm of worst case asymptotic cost $O(|\Gamma|^3g^{n-1})$
of time complexity with $O(|\Gamma|^2g)$ space.
The time complexity of the last algorithm grows with $n$
and in this way we obtain a non-polynomial algorithm for finding
the order of local testability. However, $n \le
log_gM$ where $M$ is the maximal size of the integer in the
computer memory.
\\
The time complexity of the algorithm to verify piecewise testability of
DFA is $O(|\Gamma|^2g)$. The space complexity of the algorithm is $O(k)$
\cite {TC}.
\\
The algorithms for right and left local testability
for the transition graph are essentially distinct,
moreover, the time complexity of the algorithms differs.
The graph algorithm for
the left local testability problem needs in the worst case
$O(|\Gamma|^3g)$ time and $O(|\Gamma|^3g)$ space and the algorithm
for the right local testability problem for transition
graph of the deterministic finite automaton needs
$O(|\Gamma|^2g)$ time and space \cite {TL}.
\\
The main measure of complexity for semigroup $S$ is the size
of the semigroup $|S|$. We use also the number of generators
(size of alphabet) $\it g$ and the number of idempotents $\it i$.
\\
Algorithms of the package dealing with the transition semigroup
of an automaton use the multiplication table of the semigroup of
$O(|S|^2)$ space.
Other arrays used by the package present subsemigroups or
subsets of the transition semigroup.
So we usually have $O(|S|^2)$ space complexity.
\\
We implement in the package TESTAS
a polynomial-time algorithms of $O(|S|^2)$ time complexity for the
local testability problem and for finding the order of local testability
for a given semigroup \cite{Ts}.\\
The time complexity of the semigroup algorithm
for both left and right local testability is $O(|S|i)$ \cite{TL}.
The time complexity of the semigroup algorithm
for local threshold testability is $O(|S|^3)$.
Piecewise testability is verified
in $O(|S|^2)$ time \cite{TC}.
\section*{Acknowledgments}
I am grateful to the anonymous referees for helpful and detailed comments
that proved very useful in improving the presentation and style of the paper.
\end{document}
|
\begin{document}
\title{Stability in affine optimal control problems constrained by semilinear elliptic partial differential equations hanks{This research was supported by
the Austrian Science Foundation (FWF) under grant I 4571-N.}
\begin{abstract}
This paper investigates stability properties of affine optimal control problems constrained
by semilinear elliptic partial differential equations. This is done by studying the so called metric subregularity
of the set-valued mapping associated with the system of first order necessary optimality conditions.
Preliminary results concerning the differentiability of the functions involved are established,
especially the so-called switching function. Using this ansatz, more general nonlinear perturbations are encompassed,
and under weaker assumptions than the ones previously considered in the literature on
control constrained elliptic problems.
Finally, the applicability of the results is illustrated with some error estimates for the Tikhonov regularization.
\varepsilonnd{abstract}
\section{Introduction}
We consider the following optimal control problem
\begin{align}\label{cost}
\min_{u\in\mathcal U}\left\lbrace\int_{\Omega}\Big[w(x,y)+ s(x,y) u\Big]\,dx\right\rbrace ,
\varepsilonnd{align}
subject to
\begin{align}\label{system}
\left\{ \begin{array}{cclcc}
-\dive\big(A(x)\nabla y\big)+d(x,y)&=&\beta(x)u& \text{in}& \Omega \\
\\ A(x)\nabla y\cdot \nu+b(x)y&=&0 &\text{on}& \partial\Omega.
\varepsilonnd{array} \right.
\varepsilonnd{align}
The set $\Omega\subset\mathbb R^n$ is a bounded domain with Lipschitz boundary, where
$n\in\left\lbrace 2,3\right\rbrace $. The unit outward normal vector field on the boundary $\partial\Omega$, which is single valued a.e. in $\partial\Omega$, is denoted by $\nu$. The control set is given by
\begin{align*}
\mathcal U:=\left\lbrace u:\Omega\to\mathbb R\hspace*{0.2cm}\text{measurable}: \hspace{0.20cm}b_1(x)\le u(x)\le b_2(x) \hspace{0.20cm}\text{for a.e. $x\in\Omega$}\right\rbrace,
\varepsilonnd{align*}
where $b_{1}$ and $b_{2}$ are bounded measurable
functions satisfying $b_1(x)\le b_2(x)$ for a.e. $x\in\Omega$. The functions $w: \Omega\times\mathbb R \to \mathbb R$,
$s:\Omega\times\mathbb R\to\mathbb R$, $d:\Omega\times\mathbb R\to\mathbb R$, $\beta : \Omega \to \mathbb R$ and
$b:\partial\Omega\to\mathbb R$
are real-valued and measurable, and $A:\Omega\to \mathbb R^{n\times n}$ is a measurable matrix-valued function.
There are many motivations for studying stability of solutions, in particular for error analysis of numerical methods,
see e.g., \cite{Pornerelliptic2,Pornerelliptic1}.
Most of the stability results for elliptic control problems are obtained under a second order growth condition
(analogous to the classical Legendre-Clebsch condition). For literature concerning this type of problems, the reader is referred
to \cite{Grielliptic,Hinzeelliptic,Kiebelliptic,Malaelliptic,Morduelliptic,Quielliptic} and the references therein.
In optimal control problems like (\ref{cost})--(\ref{system}), where the control appears linearly
(hence, called affine problems) this growth condition does not hold. The so-called
bang-bang solutions are ubiquitous in this case, see \cite{Casas93,Casasbang,Casanum}.
To give an account of the state of art in stability of bang-bang problems, we mention the works
\cite{Sey2,SubregOsm,Mayersubreg,Prei,Sey1} on optimal control of ordinary differential equations.
Associated results for optimization problems constrained by partial differential equations have been gaining
relevance in recent years, see \cite{Casascone,CT,Casasbang,Casanum,Hinzestruct,Wachelliptic}.
However, its stability has been only investigated in a handful of papers, see e.g.,
\cite{Hinzestruct,MR3810878,Wachelliptic}.
From these works, we mention here particularly \cite{Wachelliptic},
where the authors consider linear perturbations in the state and adjoint equations for a similar problem with
Dirichlet boundary condition. They use the so-called structural assumption (a growth assumption satisfied
near the jumps of the control) on the adjoint variable. This assumption has been widely used in
the literature on bang-bang control of ordinary differential equations in a somewhat different form, see, e.g., \cite{Sey2,SubregOsm,Prei,Sey1}.
The investigations of stability properties of optimization problems, in general, are usually based on
the study of similar properties of the corresponding system of necessary optimality conditions.
The first order necessary optimality conditions for problem (\ref{cost})--(\ref{system}) can be
recast as a system of two elliptic equations (primal and adjoint) and one variational inequality
(representing the minimization condition of the associated Hamiltonian), forming together a {\varepsilonm generalized
equation}, that is, an inclusion involving a set-valued mapping called {\varepsilonm optimality mapping}.
The concept of {\varepsilonm strong metric subregularity}, see \cite{Subreg,Dontchevbook},
of set-valued mappings has shown to be efficient in many applications especially ones related
to error analysis, see \cite{Bonnans}.
This also applies to optimal control problems of ordinary differentials equations, see e.g., \cite{MpcDont,SubregOsm}.
In the present paper we investigate the strong metric subregularity property of the optimality mapping
associated with problem (\ref{cost})--(\ref{system}). We present sufficient conditions for strong subregularity
of this mapping on weaker assumptions than the ones used in literature, see Section \ref{Section5} for precise details.
The structural assumption in \cite{Wachelliptic} is weakened and more general perturbations are considered.
Namely, perturbations in the variational inequality, appearing as a part of the first
order necessary optimality conditions, are considered; which are important in the numerical analysis of ODE and
PDE constrained optimization problems. Moreover, nonlinear perturbations are investigated,
which provides a framework for applications, as illustrated with an estimate related to the Tikhonov regularization.
The concept of linearization is employed in a functional frame in order to deal with nonlinearities.
The needed differentiability of the control-to-adjoint mapping and the switching function (see Section \ref{Section diff})
is proved, and the derivatives are used to obtain adequate estimates needed in the stability results.
Finally, we consider nonlinear perturbations in a general framework. We propose the use of the compact-open
topology to have a notion of “closeness to zero" of the perturbations. In our particular case this topology can me
metrized, providing a more “quantitative" notion. Estimates in this metric are obtained in
Section \ref{Section nonlin}.
\section{Preliminaries}
The euclidean space $\mathbb R^s$ is considered with its usual norm, denoted by $|\cdot|$.
As usual, for $p\in[1,\infty)$, we denote by $L^p(\Omega)$ the space of all measurable $p$-integrable functions
$\psi:\Omega\to\mathbb R^s$ with the norm
\begin{align*}
|\psi|_{L^p(\Omega)}:=\Big(\sum_{i=1}^{s}\int_{\Omega}|\psi_i(x)|^p\,dx\Big)^{\frac{1}{p}}.
\varepsilonnd{align*}
The space $L^\infty(\Omega)$ consists of all measurable essentially bounded functions $\psi:\Omega\to\mathbb R^s$ with the norm
\begin{align*}
|\psi|_{L^\infty(\Omega)}:=\varepsilonsssup_{x\in\Omega}|\psi(x)|.
\varepsilonnd{align*}
We denote by $C(\bar\Omega)$ the space of continuous functions on $\Omega$ that can be extended
continuously to $\bar\Omega$ equipped with the $L^\infty$-norm. We denote by $H^1(\Omega)$ the space
of functions $\psi\in L^2(\Omega)$ having all first order weak derivatives in $L^2(\Omega)$ endowed with its usual norm.
The space $H^1(\Omega)\cap C(\bar \Omega)$ is endowed with the norm
\begin{align*}
|\psi|_{H^1(\Omega)\cap C(\bar \Omega)}:= |\psi|_{H^1(\Omega)}+|\psi|_{C(\bar \Omega)}.
\varepsilonnd{align*}
A function $\psi:\Omega\times\mathbb R\to\mathbb R$ is said to be Carath\'eodory if $\psi(\cdot,y)$ is measurable
for every $y\in\mathbb R$, and $\psi(x,\cdot)$ is continuous for a.e. $x\in\Omega$.
A function $\psi:\Omega\times\mathbb R\to\mathbb R$ is said to be locally Lipschitz, uniformly in the first variable, if
for each $M>0$ there exists $L>0$ such that
\begin{align*}
|\psi(x,y_2)-\psi(x,y_1)|\le L|y_2-y_1|
\varepsilonnd{align*}
for a.e. $x\in\Omega$ and all $y_1,y_2\in [-M,M]$.
In order to abbreviate notation, we define $f,g:\Omega\times\mathbb R\times\mathbb R\to\mathbb R$ by
\begin{align*}
f(x,y,u) := \beta(x) u-d(x,y) \quad \text{and} \quad g(x,y,u) := w(x,y) + s(x,y) u.
\varepsilonnd{align*}
The following assumption is supposed to hold throughout the remainder of the paper. It ensures that
the mathematical objects related to problem (\ref{cost})--(\ref{system}) that we consider are well defined.
Assumption \ref{A1} is quite standard in the literature, see the book \cite{TroeltzschPde}.
\begin{ssmptn}\label{A1}
The following statements are assumed to hold.
\begin{itemize}
\item[(i)] The set $\Omega\subset\mathbb R^n$ is a bounded Lipschitz domain.
The matrix $A(x)$ is symmetric for a.e. $x$ in $\Omega$, and there exists $\alphapha>0$ such that
$\xi \cdot A(x)\xi\ge \alphapha|\xi|^2$ for a.e. $x$ in $\Omega$ and all $\xi\in\mathbb R^n$.
\item[(ii)] The functions $w,s$ and $d$ are Carath\'eodory, twice differentiable with respect to
the {second} variable, and their second derivatives are locally Lipschitz, uniformly in the first variable.
\item[(iii)] The functions $A,\beta,b,d(\cdot,0),d_{y}(\cdot,0),w_y(\cdot,0)$ and $s_y(\cdot,0)$ are measurable and bounded.
\item[(iv)] The function $d_y(\cdot,y)$ is nonnegative a.e. in $\Omega$ for all $y\in\mathbb R$.
The function $b$ is nonnegative a.e. in $\partial\Omega$ and $|b|_{L^\infty(\partial\Omega)}>0$.
\varepsilonnd{itemize}
\varepsilonnd{ssmptn}
Items $(i)$ and $(iv)$ of Assumption \ref{A1} ensure that the partial differential equations appearing in this paper have
unique solutions in the space $H^1(\Omega)\cap L^\infty(\Omega)$.
\subsection{The elliptic operator}
We consider the set $D(\mathcal L)$ of all functions $y\in H^1(\Omega)\cap L^\infty(\Omega)$ for which there exists
$h\in L^2(\Omega)$ such that
\begin{align}\label{weakformulation}
\int_{\Omega}A(x)\nabla y\cdot\nabla\varphi\,dx+\int_{\partial\Omega}b(x)y\varphi\,ds(x)=\int_{\Omega}h\varphi\,dx
\quad\forall \varphi\in H^1(\Omega).
\varepsilonnd{align}
As usual, $ds$ denotes the Lebesgue surface measure. It is easy to see that for each $y\in D(\mathcal L)$ there
exists a unique element $h\in L^2(\Omega)$ such that (\ref{weakformulation}) holds. We define the operator
$\mathcal L:D(\mathcal L)\to L^2(\Omega)$ by assigning each $y\in D(\mathcal L)$ to the function $h\in L^2(\Omega)$
satisfying (\ref{weakformulation}).
By definition, a function $y\in H^1(\Omega)\cap L^\infty(\Omega)$ belongs to $D(\mathcal L)$ if, and only if, it is the weak solution of the linear elliptic partial differential equation
\begin{align*}
\left\{ \begin{array}{cclcc}
-\dive\big(A(x)\nabla y\big)&=&h& \text{in}& \Omega, \\
\\ A(x)\nabla y\cdot \nu+b(x)y&=&0&\text{on}& \partial\Omega
\varepsilonnd{array} \right.
\varepsilonnd{align*}
for some $h\in L^2(\Omega)$. The following lemma is of trivial nature.
\begin{lmm}
The set $D(\mathcal L)$ is a linear subspace of $H^1(\Omega)\cap L^\infty(\Omega)$. Moreover,
the operator $\mathcal L:D(\mathcal L)\to L^2(\Omega)$ is a well defined linear mapping.
\varepsilonnd{lmm}
\iffalse
\begin{proof}{}
We first prove that $\mathcal L:D(\mathcal L)\to L^2(\Omega)$ is well defined. If $h_1,h_2\in L^2(\Omega)$
satisfy $\mathcal Ly=h_1$ and $\mathcal Ly=h_2$ for some $y\in D(\mathcal L)$, then by (\ref{weakformulation}) we have
\begin{align*}
\int_{\Omega}\big(h_1-h_2\big)\varphi=0\quad \forall\varphi\in H^1(\Omega).
\varepsilonnd{align*}
By the fundamental lemma of calculus of variations for locally integrable functions, we obtain $h_1=h_2$.
Now, let $y_1,y_2\in D(\mathcal L)$ and $\alphapha\in\mathbb R$. There exist $h_1,h_2\in L^2(\Omega)$ such that
for $i\in\{1,2\}$,
\begin{align*}
\int_{\Omega}A(x)\nabla y_i\cdot\nabla\varphi\,dx+\int_{\partial\Omega}b(x)y_i\varphi\,ds(x)=\int_{\Omega}h_i\varphi\,dx
\quad\forall \varphi\in H^1(\Omega).
\varepsilonnd{align*}
Consequently,
\begin{align*}
\int_{\Omega}A(x)\nabla \big(y_1+\alphapha y_2\big)\cdot\nabla\varphi\,dx +
int_{\partial\Omega}b(x)\big(y_1+\alphapha y_2\big)\varphi\,ds(x)=\int_{\Omega}\big(h_1+\alphapha h_2\big)\varphi\,dx
\varepsilonnd{align*}
for all $\varphi\in H^1(\Omega)$. Since $h_1+\alphapha h_2$ belongs to $L^2(\Omega)$,
we conclude that $y_1+\alphapha y_2$ belongs to $D(\mathcal L)$ and that
$\mathcal L\big(y_1+\alphapha y_2\big)=h_1+\alphapha h_2=\mathcal Ly_1+\alphapha \mathcal Ly_2$.
\varepsilonnd{proof}
\fi
If $D(\mathcal L)$ is endowed with the norm of $L^2(\Omega)$, then $\mathcal L$ is an unbounded operator
from $D(\mathcal L)$ to $L^2(\Omega)$. Since $A(x)$ is symmetric for a.e. $x\in\Omega$, by (\ref{weakformulation})
we have
\begin{align}\label{intbyparts}
\int_{\Omega} \mathcal Ly\varphi\, dx=\int_\Omega y\mathcal L \varphi\, dx
\varepsilonnd{align}
for all $y,\varphi\in D(\mathcal L)$, the so-called integration by parts formula.
\iffalse
Let $H^1(\Omega)^*$ be the dual of $H^1(\Omega)$ endowed with its dual norm.
We the have the continuous embeddings
\begin{align*}
H^1(\Omega)\hookrightarrow L^2(\Omega)\hookrightarrow H^1(\Omega)^*,
\varepsilonnd{align*}
the so-called Gelfand triple. Hence, to say that $\mathcal H\in H^1(\Omega)^*$ belongs to $L^2(\Omega)$
is to say that $\mathcal H$ can be identified with some element in $L^2(\Omega)$, i.e., there exists
$h\in L^2(\Omega)$ such that
\begin{align*}
\mathcal H(\varphi)=\int_{\Omega} h\varphi \quad \forall\varphi.
\varepsilonnd{align*}
Thus, if we consider the operator $\mathcal A: H^1(\Omega)\to H^1(\Omega)^*$ given by
\begin{align*}
(\mathcal Ay)\varphi:= \int_{\Omega}A(x)\nabla y\cdot\nabla\varphi\,dx+\int_{\partial\Omega}b(x)y\varphi\,ds(x),
\varepsilonnd{align*}
then $D(\mathcal L)= \{y\in H^1(\Omega)\cap L^\infty(\Omega): \mathcal Ay\in L^2(\Omega) \}$ and
$\mathcal L=\mathcal A|_{D(\mathcal L)}$.
\fi
\begin{rmrk}
If $\partial\Omega$ is of class $C^{1,1}$, $A$ is Lipschitz in $\bar\Omega$, and $b$ is Lipschitz and
positive in $\partial\Omega$, then
\begin{align*}
D(\mathcal L)= \{y\in H^2(\Omega):A(\cdot)\nabla y\cdot \nu+b(\cdot)y=0 \},
\varepsilonnd{align*}
and $\mathcal Ly=-\dive\big(A(\cdot)\nabla y\big)$ for all $y\in D(\mathcal L)$, see \cite[Theorem 2.4.2.6]{Grisvard}.
\varepsilonnd{rmrk}
The following lemma shows the inclusion $D(\mathcal L)\subset C(\bar\Omega)$. Its proof can be found in
\cite[Theorem 4.7]{TroeltzschPde} and follows the arguments in \cite{Casas93,Stampacchiaelliptiques}.
\begin{lmm}\label{L1e}
Let $\alphapha\in L^\infty(\Omega)$ be nonnegative and $h\in L^2(\Omega)$. There exists a unique
function $y\in D(\mathcal L)$ such that
\begin{align}\label{linequ}
\mathcal Ly+\alphapha(\cdot)y=h
\varepsilonnd{align}
and this function belongs to $C(\bar\Omega)$. Moreover, for each $r>n/2$ there exists a positive number $c$ such that
\begin{align*}
|y|_{H^1(\Omega)\cap C(\bar\Omega)}\le c|h|_{L^r(\Omega)}
\varepsilonnd{align*}
for all $\alphapha\in L^\infty(\Omega)$ nonnegative, $y\in D(\mathcal L)$, and $h\in L^2(\Omega)\cap L^r(\Omega)$
satisfying (\ref{linequ}).
\varepsilonnd{lmm}
\iffalse
\begin{proof}
See \cite[Theorem 4.7]{TroeltzschPde} or \cite[Theorem 1.25]{Hinzepde}
\varepsilonnd{proof}
\fi
The following technical lemma can be deduced from Lemma \ref{L1e}, see the proof of \cite[Lemma 3.4]{Casanum}. Its use in optimal control of elliptic
partial differential equations dates from the paper \cite[Lemma 2.6]{Casasbang}. It has shown to be useful
for diverse estimates, see \cite{Casasbang,Wachelliptic}.
\begin{lmm}\label{L2e}
There exists a positive number $c$ such that
\begin{align*}
|y|_{L^2(\Omega)}\le c|h|_{L^1(\Omega)}
\varepsilonnd{align*}
for all nonnegative $\alphapha\in L^\infty(\Omega)$, $y\in D(\mathcal L)$ and $h\in L^2(\Omega)$ satisfying (\ref{linequ}).
\varepsilonnd{lmm}
\iffalse
\begin{proof}
Let $\alphapha\in L^\infty(\Omega)$ {be} nonnegative, $y\in D(\mathcal L)$, and $h\in L^2(\Omega)$ {be} such that
$\mathcal Ly+\alphapha(\cdot)y=h$.
By Lemma \ref{L1e}, there exists a continuous function $\varphi\in D(\mathcal L)$ such that
$\mathcal L\varphi+\alphapha(\cdot)\varphi=y$ and $c>0$ such that $|\varphi|_{C(\bar\Omega)}\le c|y|_{L^2(\Omega)}$.
We have then by (\ref{intbyparts})
\begin{align*}
\int_{\Omega} y^2\, dx=\int_{\Omega} \big(\mathcal L\varphi+\alphapha(x)\varphi\big)y\, dx=\int_{\Omega} \big(\mathcal Ly+\alphapha(x)y\big)\varphi\, dx=\int_{\Omega}h\varphi\, dx.
\varepsilonnd{align*}
Hence,
\begin{align*}
|y|^2_{L^2(\Omega)}\le |h|_{L^1(\Omega)}|\varphi|_{L^\infty(\Omega)}\le c|y|_{L^2(\Omega)}|h|_{L^1(\Omega)}.
\varepsilonnd{align*}
The result follows since by Lemma \ref{L1e}, $c$ is independent of $\alphapha,y$ and $h$.
\varepsilonnd{proof}
\fi
{The proof of the next result can be found in
\cite[Theorem 2.11]{Casasnonmonotone} in the case of a Dirichlet problem, see also \cite[Lemma 6.8]{Dintelmann}. Here we adapt the
argument below Theorem 2.1 in \cite[p. 618]{Delosreyes}}.
\begin{lmm}\label{weakconv}
Let $\alphapha\in L^\infty(\Omega)$ be nonnegative, $\{h_m\}_{m=1}^\infty$ be a sequence in $L^2(\Omega)$
and $h\in L^2(\Omega)$. For each $m\in\mathbb N$, let $y_m\in C(\bar\Omega)$ be the unique function
satisfying $\mathcal L y_m+\alphapha(\cdot)y_m=h_m$, and let $y\in C(\bar\Omega)$ be the unique function satisfying
of $\mathcal Ly+\alphapha(\cdot)y=h$. If $h_m\rightharpoonup h$ weakly in $L^2(\Omega)$, then $y_m\to y$ in $C(\bar\Omega)$.
\varepsilonnd{lmm}
\begin{proof}
Let $p\in(2n/(n+2),n/(n-1))$.
Then $W^{1,p}(\Omega)$ is compactly embedded in $L^2(\Omega)$ and consequently, by Schauder's Theorem,
$L^2(\Omega)$ is compactly embedded in $W^{1,p}(\Omega)^*$. By the latter compact embedding, every weakly
convergent sequence in $L^2(\Omega)$ converges also in $W^{1,p}(\Omega)^*$ to the same limit.
Define $\mathcal K:L^2(\Omega)\to C(\bar\Omega)$ by $\mathcal Kh:= y$, where $y\in C(\bar\Omega)$ is the unique
function satisfying $\mathcal Ly+\alphapha(\cdot)=h$.
The result follows from \cite[Theorem 3.14]{Robin}, since that theorem asserts that the linear operator $\mathcal K$
is continuous from $L^2(\Omega)$ endowed with the norm of $W^{1,p}(\Omega)^*$ to $C(\bar\Omega)$.
\varepsilonnd{proof}
\begin{rmrk}\label{remark1}
Using the definitions of the set $D(\mathcal L)$ and the operator $\mathcal L$, we can write in a shorter way
the partial differential equations involved in this paper. For example, given $u\in\mathcal U$, to say that $y$
belongs to $D(\mathcal L)$ and satisfies $\mathcal Ly+d(\cdot,y)=\beta(\cdot)u$
is equivalent to say that $y$ belongs to $ H^1(\Omega)\cap L^\infty(\Omega)$ and satisfies the weak formulation of (\ref{system}), that is
\begin{align*}
\int_{\Omega}A(x)\nabla y\cdot\nabla\varphi\,dx+\int_{\Omega}d(x,y)\varphi\,dx+
\int_{\partial\Omega}b(x)y\varphi\,ds(x)=\int_{\Omega}\beta(x)u \varphi\,dx
\varepsilonnd{align*}
for all $\varphi\in H^1(\Omega)$. This weak formulation makes sense since, by $(ii)$ and $(iii)$ of Assumption \ref{A1},
for any $y\in L^\infty(\Omega)$, the function $d(\cdot,y)$ belongs to $L^\infty(\Omega)$.
\varepsilonnd{rmrk}
\subsection{The control model}
Having in mind Remark \ref{remark1}, given a function $u\in\mathcal U$ we say that $y_u\in D(\mathcal L)$
is the associated state to $u\in\mathcal U$ if
\begin{align}\label{stateeq}
\mathcal Ly_u=f(\cdot,y_u,u).
\varepsilonnd{align}
The following proposition shows that the {mapping} $u\to y_u$ from $ \mathcal{U}$ to $D(\mathcal L)$
is well defined. Its proof can be found in the standard literature; it follows from
\cite[Theorem 4.8]{TroeltzschPde}, see also \cite[p. 212]{TroeltzschPde}.
\begin{prpstn}\label{contstate}
For each $u\in\mathcal U$ there exists a unique state $y_u\in D(\mathcal L)$ associated with $u\in\mathcal U$.
Moreover, $\{y_u:u\in\mathcal U\}$ is a bounded subset of $H^1(\Omega)\cap C(\bar\Omega)$ and for each $r>n/2$
there exists $c>0$ such that
\begin{align*}
|y_{u_2}-y_{u_1}|_{H^1(\Omega)\cap C(\bar\Omega)}\le c|u_2-u_1|_{L^r(\Omega)}
\varepsilonnd{align*}
for all $u_1,u_2\in\mathcal U$.
\varepsilonnd{prpstn}
We call the function $\mathcal G:\mathcal U\to H^1(\Omega)\cap C(\bar\Omega)$ given by $\mathcal G(u):= y_u$
the control-to-state mapping. \iffalse Note that one can easily extend the definition of $\mathcal G$ to all $L^2(\Omega)$
defining $G(u)$ as a function satisfying (\ref{stateeq}) with $u\in L^2(\Omega)$, the existence and uniqueness also follows
from \cite[Theorem 4.8]{TroeltzschPde}. For any $r>n/2$, the control-to-state mapping $\mathcal G$ is Lipschitz
continuous from $\mathcal U$ endowed with the inherited metric from $L^r(\Omega)$ to $ H^1(\Omega)\cap C(\bar\Omega)$.\fi
The functional $\mathcal J:\mathcal U\to\mathbb R$ given by
\begin{align*}
\mathcal J(u):=\int_{\Omega}g(x,y_{u},u)\,dx
\varepsilonnd{align*}
is called the objective functional of problem (\ref{cost})--(\ref{system}).
\begin{dfntn}
Let $\bar u$ belong to $\mathcal U$.
\begin{itemize}
\item[(i)] We say that $\bar u$ is a global solution of problem (\ref{cost})--(\ref{system}) if
$\mathcal J(\bar u)\le\mathcal J(u)$ for all $u\in\mathcal U$.
\item[(ii)] We say that $\bar u$ is a local solution of problem (\ref{cost})--(\ref{system}) if there exists
$\varepsilon_0>0$ such that $\mathcal J(\bar u)\le\mathcal J(u)$ for all $u\in\mathcal U$
with $|u-\bar u|_{L^1(\Omega)}\le\varepsilon_0$.
\item[(iii)] We say that $\bar u$ is a strict local solution of problem (\ref{cost})--(\ref{system}) if
there exists $\varepsilon_0>0$ such that $\mathcal J(\bar u)<\mathcal J(u)$ for all $u\in\mathcal
U$ with $u\neq \bar u$ and $|u-\bar u|_{L^1(\Omega)}\le\varepsilon_0$.
\varepsilonnd{itemize}
\varepsilonnd{dfntn}
Under Assumption \ref{A1}, problem (\ref{cost})--(\ref{system}) has at least one global solution.
The proof is routine and can be obtained by standard arguments; namely, taking a minimizing sequence
and using the weak compactness of $\mathcal U$ in $L^2(\Omega)$.
\begin{lmm}\label{globalsol}
Problem (\ref{cost})--(\ref{system}) has at least one global solution.
\varepsilonnd{lmm}
\iffalse
\begin{proof}
By definition of infimum, there exists a sequence $\{u_n\}_{n=1}^\infty\subset\mathcal U$ such that
$\mathcal J(u_n)\to \inf_{u\in\mathcal U} \mathcal J(u)$. Since $\mathcal U$ is a closed bounded convex
subset of $L^2(\Omega)$, by the Eberlein–\^Smulian Theorem, it is weakly sequentially compact.
There exists a subsequence $\{u_{n_k}\}_{k=1}^\infty$ of $\{u_n\}_{n=1}^\infty$ converging weakly in
$L^2(\Omega)$ to some $\bar u\in\mathcal U$. By Proposition \ref{weakconvergencestates},
$y_{u_{n_k}}\to y_{\bar u}$ converges in $C(\Omega)$. Now,
\begin{align*}
\mathcal J(u_{n_k})=\int_{\Omega}w(x,y_{u_{n_k}})\, dx + \int_{\Omega} s(x,y_{u_{n_k}}) u_{n_k}\, dx.
\varepsilonnd{align*}
By the Lebesgue's dominated convergence Theorem,
\begin{align*}
\int_{\Omega}w(x,y_{u_{n_k}})\, dx\to \int_{\Omega}w(x,y_{u})\, dx.
\varepsilonnd{align*}
By the Lebesgue's dominated convergence Theorem, and since $u_{n_k}\rightharpoonup \bar u$ weakly in $L^2(\Omega)$,
\begin{align*}
\int_{\Omega} s(x,y_{u_{n_k}}) u_{n_k}\, dx=\int_{\Omega} \big[s(x,y_{u_{n_k}})-s(x,y_{\bar u})\big] u_{n_k}\, dx +\int_{\Omega} s(x,y_{\bar u}) u_{u_{n_k}}\, dx \to \int_{\Omega} s(x,y_{\bar u}) \bar u\, dx.
\varepsilonnd{align*}
Therefore, $\mathcal J(u_{n_k})\to \mathcal J(\bar u)$, and $\min_{u\in\mathcal U}\mathcal J(u)=\mathcal J(\bar u)$.
\varepsilonnd{proof}
\fi
In order to make notation simpler, from now on we fix a local solution $\bar u\in\mathcal U$ of problem
(\ref{cost})--(\ref{system}). We call the function $H:\Omega\times\mathbb R\times\mathbb R\times\mathbb R\to\mathbb R$, given by
\begin{align*}
H(x,y,p,u) := g(x,y,u)+ p f(x,y,u),
\varepsilonnd{align*}
the Hamiltonian of problem (\ref{cost})--(\ref{system}).
Given $u\in\mathcal U$, we say that $p_u\in D(\mathcal L)$ is the costate associated with $u\in\mathcal
U$ if
$$
\mathcal Lp_u=H_y(\cdot,y_u,p_u,u).
$$
The following proposition shows that the {mapping} $u\to p_u$ from
$\mathcal{U}$ to $D(\mathcal L)$ is well defined.
We {give} the proof of this elementary result because it {seems not to be} explicitly stated in the literature.
\begin{prpstn}\label{concost}
For each $u\in\mathcal U$ there exists a unique costate $p_u\in D(\mathcal L)$ associated with $u\in\mathcal U$. Moreover, $\{p_u:u\in\mathcal U\}$ is a bounded subset of $H^1(\Omega)\cap C(\bar\Omega)$ and for each $r>n/2$ there exist $c>0$ such that
\begin{align*}
|p_{u_2}-p_{u_1}|_{H^1(\Omega)\cap C(\bar\Omega)}\le c|u_2-u_1|_{L^r(\Omega)}
\varepsilonnd{align*}
for all $u_1,u_2\in\mathcal U$.
\varepsilonnd{prpstn}
\begin{proof}
The existence and uniqueness follows from \iffalse\cite[Theorem 4.10]{TroeltzschPde}\fi Lemma \ref{L1e}.
Given $u\in\mathcal U$, the function $p_u$ satisfies
\begin{align*}
\mathcal{L}p_{u}+d_{y}(\cdot,y_u)p_u=g_{y}(\cdot,y_u,u).
\varepsilonnd{align*}
By $(ii),(iii)$ and $(iv)$ of Assumption \ref{A1}, for each $u\in\mathcal U$, the function $d_y(\cdot,y_{u})$
is nonnegative and belongs to $L^\infty(\Omega)$. By $(ii)$ and $(iii)$ of Assumption \ref{A1}, for each $u\in\mathcal U$
the function $g_{y}(\cdot,y_u,u)$ belongs to $L^\infty(\Omega)$. Furthermore, since by Proposition \ref{contstate}
the set $\{y_{u}: u\in\mathcal U\}$ is bounded in $C(\bar\Omega)$, there exists $M_1>0$ such that
\begin{align*}
|g_{y}(\cdot,y_u,u)|_{L^\infty(\Omega)}\le M_1
\varepsilonnd{align*}
for all $u\in\mathcal U$. By Lemma \ref{L1e}, there exists a positive number $c_1$ such that for all $u\in\mathcal U$
\begin{align*}
|p_{u}|_{H^1(\Omega)\cap C(\bar\Omega)}\le c_1|g_{y}(\cdot,y_u,u)|_{L^\infty(\Omega)}.
\varepsilonnd{align*}
Thus, $M_2:=c_1M_1$ is a bound for the set $\{p_u:u\in\mathcal U\}$ in $H^1(\Omega)\cap C(\bar\Omega)$.
Let $u_1,u_2\in\mathcal U$ and $r>n/2$. We have then
\begin{align*}
\mathcal L(p_{u_2}-p_{u_1})+d_{y}(\cdot,y_{u_2})(p_{u_2}-p_{u_1})=
H_{y}(\cdot,y_{u_2},p_{u_1},u_2)-H_{y}(\cdot,y_{u_1},p_{u_1},u_1).
\varepsilonnd{align*}
By Lemma \ref{L1e}, there exists a positive number $c_2$ (independent of $u_1$ and $u_2$) such that
\begin{align*}
|p_{u_2}-p_{u_1}|_{H^1(\Omega)\cap C(\bar\Omega)}\le c_2|H_{y}(\cdot,y_{u_2},p_{u_1},u_2)-
H_{y}(\cdot,y_{u_1},p_{u_1},u_1)|_{L^r(\Omega)}.
\varepsilonnd{align*}
By $(ii)$ of Assumption \ref{A1} and the boundedness of the set $\{p_u:u\in\mathcal U\}$ in $C(\bar\Omega)$,
there exists $L>0$ such that
\begin{align*}
|H_{y}(\cdot,y_{u_2},p_{u_1},u_2)-H_{y}(\cdot,y_{u_1},p_{u_1},u_1)|
\le L\Big(|y_{u_2}-y_{u_1}|+|u_2-u_1|\Big)\quad\text{a.e. in $\Omega$}.
\varepsilonnd{align*}
Consequently,
\begin{align*}
|p_{u_2}-p_{u_1}|_{H^1(\Omega)\cap C(\bar\Omega)} &\le
{ c_2 L \big(| y_{u_1} - y_{u_2} |_{L^r(\Omega)} + | u_1 - u_2 |_{L^r(\Omega)}\big)}
\\ & \le c_2L \Big((\text{meas}
\hspace*{0.05cm}\Omega) ^\frac{1}{r}|y_{u_2}-y_{u_1}|_{L^\infty(\Omega)}+|u_2-u_1|_{L^r(\Omega)}\Big).
\varepsilonnd{align*}
By Proposition \ref{contstate}, there exists a constant $c_3>0$ (independent of $u_1$ and $u_2$) such that
\begin{align*}
|y_{u_2}-y_{u_1}|_{C(\bar\Omega)}\le c_3|u_2-u_1|_{L^r(\Omega)}.
\varepsilonnd{align*}
Thus,
\begin{align*}
|p_{u_2}-p_{u_1}|_{H^1(\Omega)\cap C(\bar\Omega)}\le c_2L\big(1+c_3 (\text{meas}
\hspace*{0.05cm}\Omega)^\frac{1}{r} \big)|u_2-u_1|_{L^r(\Omega)}.
\varepsilonnd{align*}
The estimate follows defining $c:=c_2L\big(1+c_3 (\text{meas}\hspace*{0.05cm}\Omega)^\frac{1}{r} \big)$.
\varepsilonnd{proof}
We call the function $\mathcal S:\mathcal U\to H^1(\Omega)\cap C(\bar\Omega)$ given by $\mathcal S(u):= p_u$
the control-to-adjoint mapping. The following proposition gives us another useful estimate; it can be easily proved employing Lemma \ref{L2e} and the argument in the proof of \cite[Theorem 4.16]{TroeltzschPde}.
\begin{prpstn}\label{contstatel1}
There exists $c>0$ such that
\begin{align*}
|y_{u_2}-y_{u_1}|_{L^2(\Omega)}+|p_{u_2}-p_{u_1}|_{L^2(\Omega)}\le c|u_2-u_1|_{L^1(\Omega)}
\varepsilonnd{align*}
for all $u_1,u_2\in\mathcal U$.
\varepsilonnd{prpstn}
\iffalse
\begin{proof}
Let $u_1$ and $u_2$ belong to $\mathcal U$. Define $\alphapha:\Omega\to\mathbb R$ by
\begin{align*}
\alphapha(x):= \left\{ \begin{array}{lcc}
\displaystyle\frac{d(x,y_{u_2}(x))-d(x,y_{u_1}(x))}{y_{u_{2}}(x)-y_{u_1}(x)}& if & y_{u_2}(x)\neq y_{u_1}(x) \\
\\ 0& if & y_{u_2}(x)= y_{u_1}(x).
\varepsilonnd{array} \right.
\varepsilonnd{align*}
By $(ii)$ and $(iv)$ of Assumption \ref{A1}, $\alphapha$ belongs to $L^\infty(\Omega)$ and is nonnegative. We have then
\begin{align*}
\mathcal L(y_{u_2}-y_{u_1})+\alphapha(\cdot)(y_{u_2}-y_{u_1})=\beta(\cdot)(u_2-u_1).
\varepsilonnd{align*}
By Lemma \ref{L2e}, there exists a positive number $c_0$ (independent of $u_1$ and $u_2$) such that
\begin{align*}
|y_{u_2}-y_{u_1}|_{L^2(\Omega)}\le c_0|\beta(\cdot)(u_2-u_1)|_{L^1(\Omega)}.
\varepsilonnd{align*}
The estimate follows defining $c:=\ c_0|\beta|_{L^\infty(\Omega)}$.
\varepsilonnd{proof}
\fi
We close this subsection with the following result.
\iffalse
The proof of the following proposition can be found in \cite[Theorem 2.11]{Casasnonmonotone} in the case
of a Dirichlet problem, and it can be easily adapted. It can also be deduced from Lemma \ref{weakconv} with
the same argument given in the proof of Proposition \ref{weakconvergencecostates}.
\fi
\begin{prpstn}\label{weakconvergence}
Let $\{u_m\}_{m=1}^\infty$ be a sequence in $\mathcal U$ and $u\in \mathcal U$. If $u_m\rightharpoonup u$
weakly in $L^2(\Omega)$, then $y_{u_m}\to y_{u}$ and $p_{u_m}\to p_{u}$ in $C(\bar\Omega)$.
\varepsilonnd{prpstn}
\begin{proof}
We prove only the convergence $p_{u_m}\to p_u$ in $C(\bar\Omega)$, the convergence
$y_{u_m}\to y_u$ in $C(\bar\Omega)$ is analogous.
Let $\{p_{u_{m_k}}\}_{k=1}^\infty$ be an arbitrary subsequence of $\{p_{u_m}\}_{m=1}^\infty$.
By the compact embedding $H^1(\Omega)\hookrightarrow L^2(\Omega)$, there exists a subsequence
of $\{p_{u_{m_k}}\}_{k=1}^\infty$, denoted in the same way, and $p\in L^2(\Omega)$ such that
$p_{u_{m_k}}\to p$ in $L^2(\Omega)$. Since $y_{u_{m_k}}\to y_{u}$ in $C(\bar\Omega)$, one can deduce that
\begin{align*}
H_{y}(\cdot,y_{u_{m_k}}, p_{u_{m_k}}, {u_{m_k}})\rightharpoonup H_{y}(\cdot,y_u,p,u) \quad\text{weakly in $L^2(\Omega)$.}
\varepsilonnd{align*}
\iffalse
\begin{align*}
g_{y}(\cdot,y_{u_{m_k}},{u_{m_k}})-d_{y}(\cdot,y_{u_{m_k}})p_{u_{m_k}}\rightharpoonup g_{y}(\cdot,y_u,u)-
d_{y}(\cdot,y_u)p \quad\text{weakly in $L^2(\Omega)$.}
\varepsilonnd{align*}
\fi
By Lemma \ref{weakconv}, we have $p_{u_{m_{k}}}\to p_u$ in $C(\bar\Omega)$. The result follows, since every
subsequence of $\{p_{u_{m}}\}_{m=1}^\infty$ has a {further} subsequence that converges to $p_{u}$ in $C(\bar\Omega)$.
\varepsilonnd{proof}
\section{Differentiability of the mappings involved}\label{Section diff}
\iffalse
Under Assumption \ref{A1}, the Minimum Principle holds. That means, namely, that if $\bar u\in\mathcal U$
is a local solution of of problem (\ref{cost})--(\ref{system}), then the triple $(y_{\bar u},p_{\bar u},\bar u)$ satisfies
the optimality system
\begin{align*}
0&=-Ly+f(\cdot,y,u),\\
0&=-Lp+H_{y}(\cdot,y,p,u),\\
0&\in H_{u}(\cdot,y,p,u)+ n_{\mathcal U}(u),
\varepsilonnd{align*}
where the normal cone $n_{\mathcal U}:\mathcal U\rightrightarrows L^\infty(\Omega)$ to the set $\mathcal U$ is given by
\begin{align}
N_{\mathcal U}(u) := \left\lbrace w\in L^\infty(\Omega) : \int_{\Omega} w(v-u)\,dx\le 0 \quad
\forall v\in L^1(\Omega) \right\rbrace.
\varepsilonnd{align}
\fi
In this section, we prove some preliminary results concerning the differentiability of the control-to-state mapping,
the control-to-adjoint mapping and the switching mapping (to be defined later). Some of these properties
are well known for the control-to-state mapping; see, e.g.,
\cite{Casascone,Casasbang,Casanum,Wachelliptic,TroeltzschPde}. Nevertheless, we require more specific
estimates than the ones in the literature. The differentiability of the control-to-adjoint mapping and the switching mapping
has not been studied before in the literature {on} elliptic control-constrained problems, therefore we devote this section
to obtain appropriate estimates needed in the study of stability in the next section.
\subsection{The state and adjoint mappings}
We begin this subsection recalling the definition of directional derivative, see
\cite[pp.2-4]{Flemming} or \cite[p.171]{Lue}. Let $Y$ be a normed space and ${\mathcal F}: {\mathcal U} \to Y$ a mapping. Given $u \in {\mathcal U}$ and $v\in\mathcal U-u$, if the limit
\begin{align*}
d {\mathcal F}(u;v) := \lim_{\varepsilon \to 0^+} \frac{{\mathcal F}(u + \varepsilon v) - {\mathcal F}(u)}{\varepsilon}
\varepsilonnd{align*}
exists in $Y$, we say that $\mathcal F(u;v)$ is the (G\^ateaux) differential of $\mathcal F$ at $u$ in the direction $v$. Note that by convexity of $\mathcal U$, $u + \varepsilon v$ belongs to $\mathcal U$ for every $u\in\mathcal U$, $v\in\mathcal U-u$ and $\varepsilon \in [0,1]$. We will restrict ourselves to this simple definition of directional derivative, as further differentiability properties are not needed in our analysis of stability.
Recall that $\bar u\in \mathcal U$ is a fixed solution of problem (\ref{cost})--(\ref{system}). As it is well-known,
the differential of the control-to-state mapping at $\bar u$ is related to the linearization of the system equation
around $\bar u$. Bearing this in mind, given $v\in L^2(\Omega)$, we denote by $z_{v}$
the unique\footnote{ The uniqueness follows from Lemma \ref{L1e}, and the fact that equation (\ref{linsta})
can be rewritten as $$\mathcal Lz_{v}+d_{y}(\cdot,y_{\bar u})z_{v}=\beta(\cdot)v.$$} solution of the equation
\begin{align}\label{linsta}
\mathcal L z_{v}=f_y(\cdot, y_{\bar u},\bar u)z_{v}+f_u(\cdot,y_{\bar u},\bar u)v.
\varepsilonnd{align}
The proof of the following estimate can be found in the standard literature, see the proof of
\cite[Theorem 4.17]{TroeltzschPde} for the case of a Neumann boundary problem (the proof is the same for
Robin or Dirichlet boundary). It can also be deduced by the same arguments given in the proof
of Proposition \ref{Ecr}.
\begin{prpstn}\label{Esr}
For each $r>n/2$ there exists $c>0$ such that
\begin{align*}
|y_u-y_{\bar u}-z_{u-\bar u}|_{H^1(\Omega)\cap C(\bar\Omega)}\le c|u-\bar u|_{L^r(\Omega)}^2\quad\quad
\forall u\in\mathcal U.
\varepsilonnd{align*}
\varepsilonnd{prpstn}
One of the first things that can be deduced from Proposition \ref{Esr} is the differentiability of the control-to-state
mapping $\mathcal G$. Given $v\in L^2(\Omega)$ satisfying $\bar u+v\in\mathcal U$, the
differential of the control-to-state mapping $\mathcal G$ at $\bar u$ in the direction $v$ exists and
is given by $d{\mathcal G}(\bar u;v)=z_{v}$. \iffalse Moreover, one can prove that $\mathcal G$ is of class $C^2$. This is a standard application of the Implicit Function Theorem to the function $\mathcal F: D(\mathcal L)\times L^r(\Omega)\to L^r(\Omega)$ given by $\mathcal F(y,u):=\mathcal Ly+d(\cdot,y)-\beta(\cdot)u$, where $r>n/2$; see \cite[Theorem 2.12]{Casasnonmonotone} for details in the Dirichlet boundary case.\fi
For further differentiability properties of the control-to-state mapping, we refer the reader to \cite[Theorem 2.12]{Casasnonmonotone}.
In order to study the differential of the control-to-adjoint
mapping{ we introduce the following notations}.
Given $v\in L^2(\Omega)$, we denote by $q_{v}$ the unique\footnote{ The uniqueness follows from Lemma \ref{L1e},
and the fact that equation (\ref{lincos}) can be rewritten as
$$
\mathcal Lq_{v}+d_{y}(\cdot,y_{\bar u})q_{v}=H_{yy}(\cdot,y_{\bar u}, p_{\bar u},\bar u)z_v+
H_{yu}(\cdot,y_{\bar u}, p_{\bar u},\bar u)v.
$$}
solution of the equation
\begin{align}\label{lincos}
\mathcal Lq_{v}=H_{yy}(\cdot,y_{\bar u}, p_{\bar u},\bar u)z_v+H_{yp}(\cdot,y_{\bar u}, p_{\bar u},\bar u)q_{v}+
H_{yu}(\cdot,y_{\bar u}, p_{\bar u},\bar u)v.
\varepsilonnd{align}
The following estimate is concerned with the differentiability of the control-to-adjoint mapping.
To the best of our knowledge, this result does not appear in the literature; therefore we present its proof, although it is standard.
\begin{prpstn}\label{Ecr}
For each $r>n/2$ there exists $c>0$ such that
\begin{align*}
|p_u-p_{\bar u}-q_{u-\bar u}|_{H^1(\Omega)\cap C(\bar\Omega)}\le c|u-\bar u|_{L^r(\Omega)}^2\quad
\quad \forall u\in\mathcal U.
\varepsilonnd{align*}
\varepsilonnd{prpstn}
\begin{proof}
Given $u\in\mathcal U$,
we define $\psi_{u}:\Omega\to\mathbb R^4$ by $\psi_{u}(x):=(x,y_{u}(x),p_{u}(x),u(x))$.
For each $u\in \mathcal U$, we denote by $\tilde q_{u-\bar u}$ the unique solution of the equation
\begin{align*}
\mathcal L\tilde q_{u-\bar u}=H_{yy}(\psi_{\bar u})(y_{u}-y_{\bar u})+H_{yp}(\psi_{\bar u})\tilde q_{u-\bar u}+
H_{yu}(\psi_{\bar u})(u-\bar u).
\varepsilonnd{align*}
Let $u\in\mathcal U$ and $r>n/2$ be arbitrary. Using the Taylor Theorem (integral form of the remainder) and $(ii)$-$(iii)$ of Assumption \ref{A1},
one can find $\alphapha_1,\alphapha_2,\alphapha_3\in L^\infty(\Omega)$ such that
\begin{align*}
H_{y}(\psi_{u})=&H_{y}(\psi_{\bar u})+H_{yy}(\psi_{\bar u})(y_{u}-y_{\bar u})+H_{yp}(\psi_{\bar u})(p_{u}-p_{\bar u})+H_{yu}(\psi_{\bar u})v\\
&+\alphapha_1(\cdot)(y_{u}-y_{\bar u})^2+\alphapha_2(\cdot)(y_{u}-y_{\bar u})(p_{u}-p_{\bar u})+\alphapha_3(\cdot)(y_{u}-y_{\bar u})v,
\varepsilonnd{align*}
where $v=u-\bar u$. Hence
\begin{align*}
\mathcal L(p_u-p_{\bar u}-\tilde q_{v})=H_{yp}(\psi_{\bar u})(p_{u}-p_{\bar u}-\tilde q_{v})+
\Big[ \alphapha_1(\cdot)(y_{u}-y_{\bar u})+\alphapha_2(\cdot)(p_{u}-p_{\bar u})+\alphapha_3(\cdot)v\Big](y_{u}-y_{\bar u}).
\varepsilonnd{align*}
By Lemma \ref{L1e}, Proposition \ref{contstate} and Proposition \ref{concost}, there exists $c_1>0$ such that
\begin{align*}
|p_u-p_{\bar u}-\tilde q_{v}|_{H^1(\Omega)\cap C(\bar\Omega)}\le c_1|v|_{L^r(\Omega)}^2.
\varepsilonnd{align*}
Now,
\begin{align*}
\mathcal L(\tilde q_{v}- q_{v})=H_{yy}(\psi_{\bar u})(y_{u}-y_{\bar u}-z_{v})+H_{yp}(\psi_{\bar u})(\tilde q_{v}-q_{v}).
\varepsilonnd{align*}
By Lemma \ref{L1e} and Proposition \ref{Esr}, there exists $c_2>0$ such that
\begin{align*}
|\tilde q_{v}- q_{v}|_{H^1(\Omega)\cap C(\bar\Omega)}\le c_2|v|_{L^r(\Omega)}^2.
\varepsilonnd{align*}
Finally, by the triangle inequality
\begin{align*}
|p_u-p_{\bar u}- q_{v}|_{H^1(\Omega)\cap C(\bar\Omega)}\le|p_u-p_{\bar u}-
\tilde q_{v}|_{H^1(\Omega)\cap C(\bar\Omega)}+ |\tilde q_{v}- q_{v}|_{H^1(\Omega)\cap C(\bar\Omega)}.
\varepsilonnd{align*}
The result follows taking $c:=c_1+c_2$.
\varepsilonnd{proof}
Given $v\in L^\infty(\Omega)$ satisfying $\bar u+v\in\mathcal U$, the differential of the control-to-adjoint
mapping $\mathcal S$ at $\bar u$ in the direction $v$ exists and is given by $d{\mathcal S}(\bar u;v)=q_{v}.$ \iffalse It is worth mentioning that the map $\mathcal S$ is of class $C^2$, this can be seen applying the Implicit Function Theorem to the function $\mathcal H:D(\mathcal L)\times L^r(\Omega)\to L^r(\Omega)$ given by $\mathcal H(p,u):=\mathcal Lp-H_y(\cdot,y_u,p,u),$ where $r>n/2$.
\begin{Remark}
It is worth mentioning that the map $\mathcal S$ is of class $C^2$, this can be seen applying the Implicit Function Theorem to the function $\mathcal H:D(\mathcal L)\times L^r(\Omega)\to L^r(\Omega)$ by $\mathcal H(p,u):=\mathcal Lp-H_y(\cdot,y_u,p,u)$, where $r>n/2$. Let $(p,u)\in D(\mathcal L)\times L^r(\Omega)$. Clearly, we have
\begin{align*}
D\mathcal H(p,u)(q,v)=\mathcal Lq+d_y(\cdot,y_u)q-H_{yy}(\cdot,y_u,p,u)z_v-H_{yu}(\cdot,y_u,p,u)v
\varepsilonnd{align*}
for all $(q,v)\in D(\mathcal L)$, where $D\mathcal H(p,u):D(\mathcal L)\times L^r(\Omega)\to L^r(\Omega)$ is the Fr\'echet derivative of $\mathcal H$ at $(p,u)$. Let $\phi:D(\mathcal L)\to L^r(\Omega)$ be given by $\phi(q):=D\mathcal H(p,u)(q,0)=\mathcal Lq+d_y(\cdot,y_u)q$. It is easy to see that $\phi$ is a continuous isomorphism using Lemma \ref{L1e} and the Bounded Inverse Theorem.
\varepsilonnd{Remark}
\fi
We now state further properties concerning the mappings $v\to z_v$ and $v\to q_v$.
\begin{prpstn}\label{furtherprop0}
The following statements hold.
\begin{itemize}
\item[(i)] For each $r>n/2$ there exists a positive number $c$ such that
\begin{align*}
|z_{v}|_{H^1(\Omega)\cap C(\bar\Omega)}+|q_{v}|_{H^1(\Omega)\cap C(\bar\Omega)}\le c|v|_{L^r(\Omega)}\quad\forall v\in L^2(\Omega)\cap L^r(\Omega).
\varepsilonnd{align*}
\item[(ii)] There exists a positive number $c$ such that
\begin{align*}
|z_{v}|_{L^2(\Omega)}+|q_{v}|_{L^2{(\Omega)}}\le c|v|_{L^1(\Omega)} \quad \forall v\in L^2(\Omega).
\varepsilonnd{align*}
\item[(iii)] Let $\{v_k\}_{k=1}^\infty$ be a sequence in $L^2(\Omega)$ and $v\in L^2(\Omega)$. If $v_k\rightharpoonup v$ weakly in $L^2(\Omega)$, then $z_{v_{k}}\to z_{v}$ and $q_{v_{k}}\to q_{v}$ in $C(\bar\Omega)$.
\varepsilonnd{itemize}
\varepsilonnd{prpstn}
\begin{proof}
Items $(i)$ and $(ii)$ follow from Lemma \ref{L1e} and \ref{L2e}, respectively. Item $(iii)$ follows from
Lemma \ref{weakconv}.
\varepsilonnd{proof}
\subsection{The switching mapping}
Let us begin this subsection recalling the first order necessary condition (Pontryagin principle {in integral form}) for problem
(\ref{cost})--(\ref{system}). If $u\in\mathcal U$ is a local solution of problem (\ref{cost})--(\ref{system}), then
\begin{align}\label{varin}
\int_{\Omega}\Big[s(x,y_u)+\beta(x)p_u\Big](w-u)\, dx\ge0\quad\forall w\in\mathcal U.
\varepsilonnd{align}
The variational inequality (\ref{varin}) motivates the following definition. For each $u\in\mathcal U$, define
\begin{align*}
\sigma_{u}:= s(\cdot,y_u)+\beta(\cdot)p_u.
\varepsilonnd{align*}
Observe that $\sigma_u=H_{u}(\cdot,y_u,p_u)$. The mapping $\mathcal Q:\mathcal U\to L^\infty(\Omega)$ given by $\mathcal Q(u):=\sigma_u$ is called
the switching mapping. Given $v\in L^2(\Omega)$, we define
\begin{align*}
\pi_{v}:=H_{uy}(\cdot,y_{\bar u},p_{\bar u})z_{v}+H_{up}(\cdot,y_{\bar u},p_{\bar u})q_v.
\varepsilonnd{align*}
{This definition is justified by the following estimate.}
\begin{prpstn}\label{sigmatilde}
For each $r>n/2$ there exists $c>0$ such that
\begin{align*}
|\sigma_u-\sigma_{\bar u}-\pi_{u-\bar u}|_{L^\infty(\Omega)}\le c|u-\bar u|_{L^r(\Omega)}^2\quad\quad \forall u\in\mathcal U.
\varepsilonnd{align*}
\varepsilonnd{prpstn}
\begin{proof}
Given $u\in\mathcal U$, we define $\psi_{u}:\Omega\to\mathbb R^3$ by $\psi_{u}(x):=(x,y_{u}(x),p_{u}(x))$.
For each $u\in \mathcal U$, we {denote}
\begin{align*}
\tilde\pi_{u-\bar u}:=H_{uy}(\psi_{\bar u})(y_{u}-y_{\bar u})+H_{up}(\psi_{\bar u})(p_u-p_{\bar u}).
\varepsilonnd{align*}
Let $u\in\mathcal U$ and $r>n/2$ be arbitrary, and abbreviate $v=u-\bar u$. Using the Taylor Theorem (integral form of the remainder) and $(ii)$-$(iii)$ of Assumption \ref{A1}, one can find $\alphapha\in L^\infty(\Omega)$ such that
\begin{align*}
H_{u}(\psi_{u})=&H_{u}(\psi_{\bar u})+H_{uy}(\psi_{\bar u})(y_{u}-y_{\bar u})+H_{up}(\psi_{\bar u})(p_{u}-p_{\bar u})+\alphapha(\cdot)(y_{u}-y_{\bar u})^2.
\varepsilonnd{align*}
Therefore, by Proposition \ref{contstate}, there exists $c_1>0$ such that
\begin{align*}
|\sigma_u-\sigma_{\bar u}-\tilde \pi_{v}|_{L^\infty(\Omega)}\le c_1|v|_{L^r(\Omega)}^2.
\varepsilonnd{align*}
Now,
\begin{align*}
|\tilde \pi_{v}-\pi_v|_{L^\infty(\Omega)}\le |H_{uy}(\cdot,y_{\bar u},p_{\bar u})(y_{u}-y_{\bar u}-z_{v})+
H_{up}(\cdot,y_{\bar u},p_{\bar u})(q_u-q_{\bar u}-q_v)|_{L^\infty(\Omega)}.
\varepsilonnd{align*}
Hence, by Proposition \ref{Esr} and \ref{Ecr}, there exists $c_2>0$ such that
\begin{align*}
|\tilde \pi_{v}-\pi_v|_{L^\infty(\Omega)}\le c_2|v|_{L^r(\Omega)}^2.
\varepsilonnd{align*}
Finally, by the triangle inequality,
\begin{align*}
|\sigma_u-\sigma_{\bar u}- \pi_{v}|_{L^\infty(\Omega)}\le|\sigma_u-\sigma_{\bar u}-\tilde \pi_{v}|_{L^\infty(\Omega)}
+|\tilde \pi_{v}-\pi_v|_{L^\infty(\Omega)}.
\varepsilonnd{align*}
The result follows defining $c:=c_1+c_2$.
\varepsilonnd{proof}
Proposition \ref{sigmatilde} yields immediately that the differential of the switching mapping
$\mathcal Q$ at $\bar u$ in any direction $v\in\mathcal U-\bar u$ exists and is given by $d{\mathcal Q}(\bar u;v)=\pi_{v}.$
One of the important features of the mapping $v\to\pi_v$ is the following.
\begin{prpstn}\label{opQ}
For all $v\in L^2(\Omega)$, we have
\begin{align*}
\int_{\Omega} \pi_{v}v\, dx=\int_{\Omega}\Big[ H_{yy}(x,y_{\bar u},p_{\bar u},\bar u)z_{v}^2+
2H_{uy}(x,y_{\bar u},p_{\bar u},\bar u)z_{v} v \Big]\,dx.
\varepsilonnd{align*}
\varepsilonnd{prpstn}
\begin{proof}
In order to simplify notation, we write $\psi_{\bar u}(x):=(x,y_{\bar u}(x),p_{\bar u}(x),\bar u(x))$ for
each $x\in\Omega$. Let $v\in L^2(\Omega)$ be arbitrary. By the integration by parts formula (\ref{intbyparts}) and the concrete form of the Hamiltonian, we get
\begin{align*}
\int_{\Omega} H_{up}(\psi_{\bar u})q_{v}v\, dx&=\int_{\Omega} \big(\mathcal L z_v+d_y(x,y_{\bar u})z_v\big)q_v\,dx
=\int_{\Omega}\big(\mathcal L q_v+d_y(x,y_{\bar u})q_v\big)z_v\, dx\\
&=\int_{\Omega}\big(H_{yy}(\psi_{\bar u})z_v+H_{uy}(\psi_{\bar u})v\big)z_v=
\int_{\Omega}\Big[H_{yy}(\psi_{\bar u})z_v^2+H_{uy}(\psi_{\bar u})z_v v\Big]\, dx.
\varepsilonnd{align*}
The result follows since
\begin{align*}
\int_{\Omega} \pi_{v}v\, dx=\int_{\Omega}H_{uy}(\psi_{\bar u})z_v v\, dx+
\int_{\Omega}H_{up}(\psi_{\bar u})q_v v\, dx.
\varepsilonnd{align*}
\varepsilonnd{proof}
We give further properties of the mapping $v\to\pi_v$ in the next proposition, its proof follows trivially from Proposition
\ref{furtherprop0}.
\begin{prpstn}\label{furprop3}
The following statements hold.
\begin{itemize}
\item[(i)] For each $r>n/2$ there exists a positive number $c$ such that
\begin{align*}
|\pi_{v}|_{L^\infty(\Omega)}\le c|v|_{L^r(\Omega)}\quad\forall v\in L^2(\Omega)\cap L^r(\Omega).
\varepsilonnd{align*}
\item[(ii)] There exists a positive number $c$ such that
\begin{align*}
|\pi_{v}|_{L^2(\Omega)}\le c|v|_{L^1(\Omega)} \quad \forall v\in L^2(\Omega).
\varepsilonnd{align*}
\item[(iii)] Let $\{v_k\}_{k=1}^\infty$ be a sequence in $L^2(\Omega)$ and $v\in L^2(\Omega)$.
If $v_k\rightharpoonup v$ weakly in $L^2(\Omega)$, then $\pi_{v_{k}}\to \pi_{v}$ in $L^\infty(\Omega)$.
\varepsilonnd{itemize}
\varepsilonnd{prpstn}
Proposition \ref{opQ} motivates the following definition.
For each $v\in L^2(\Omega)$, define
\begin{align}\label{quadraticform}
\Lambda(v):=\int_{\Omega}\Big[ H_{yy}(x,y_{\bar u},p_{\bar u},\bar u)z_{v}^2+
2H_{uy}(x,y_{\bar u},p_{\bar u},\bar u)z_v v \Big]\,dx.
\varepsilonnd{align}
\begin{rmrk}
We mention that the quadratic form $\Lambda:L^2(\Omega)\to\mathbb R$ is the second variation of the objective
functional $\mathcal J:\mathcal U\to\mathbb R$ at $\bar u$. By Proposition \ref{opQ}, we also have
the following representation
\begin{align*}
\Lambda(v)=\int_\Omega \pi_{v}v\,dx\quad\forall v\in L^2(\Omega).
\varepsilonnd{align*}
\varepsilonnd{rmrk}
We close this section with a result concerning the quadratic form (\ref{quadraticform}).
\begin{prpstn}\label{lemcas}
Let $\{v_k\}_{k=1}^\infty\subset L^2(\Omega)$ and $v\in L^2(\Omega)$. If $v_k\rightharpoonup v$
weakly in $L^2(\Omega)$, then $\Lambda(v_k)\to \Lambda(v)$.
\varepsilonnd{prpstn}
\begin{proof}
By Proposition \ref{furprop3}, $\pi_{v_k}\to\pi_v$ in $L^\infty(\Omega)$, therefore
\begin{align*}
\Lambda(v_k)=\int_\Omega (\pi_{v_k}-\pi_v)v_k\,dx+\int_\Omega \pi_{v}v_k\,dx \to \int_\Omega \pi_{v}v\,dx.
\varepsilonnd{align*}
\varepsilonnd{proof}
\iffalse
To say that $\bar u\in\mathcal U$ satisfies (\ref{optvar}) is equivalent to say that it satisfies the variational inequality
\begin{align}\label{varine}
0\in \sigma_u+ N_{\mathcal U}( u),
\varepsilonnd{align}
where the normal cone $N_{\mathcal U}:\mathcal U\rightrightarrows L^\infty(\Omega)$ to the set $\mathcal U$ is given by
\begin{align}
N_{\mathcal U}(u) := \left\lbrace w\in L^\infty(\Omega) : \int_{\Omega} w(v-u)\,dx
\le 0 \quad \forall v\in L^1(\Omega) \right\rbrace.
\varepsilonnd{align}
\begin{Corollary}[Minimum principle]\label{mp}
The triple $(y_{\bar u},p_{\bar u},\bar u)$ satisfies
\begin{align*}
\left\{ \begin{array}{cll}
\mathcal Ly_{\bar u}&=&f(\cdot,y_{\bar u},\bar u),\\
\mathcal Lp_{\bar u}&=&H_{y}(\cdot,y_{\bar u},p_{\bar u},\bar u),\\
0&\in& H_{u}(\cdot,y_{\bar u},p_{\bar u}) + N_{\mathcal U}(\bar u).
\varepsilonnd{array} \right.
\varepsilonnd{align*}
\varepsilonnd{Corollary}
\fi
\section{Stability}\label{Section Stab}
In this section, we study the stability of the optimal solution
of problem (\ref{cost})--(\ref{system}) with respect to perturbations.
As usual in optimization, the stability of the solution is derived
from stability of the system of necessary optimality conditions.
The investigated stability property of the latter is the so-called
strong metric Hölder subregularity (SMHSr), see e.g., \cite[Section 3I]{Dontchevbook} or \cite[Section 4]{Subreg}.
After introducing the assumptions we study the SMHSr property
of the variational inequality (9). Then the result is used to obtain
this property for the whole system of necessary optimality conditions
\subsection{The main assumption}
We begin the section recalling that
$\bar u\in\mathcal U$ is a local minimizer of problem (\ref{cost})--(\ref{system}), and the definition of the quadratic form
$\Lambda: L^2(\Omega)\to\mathbb R$ in (\ref{quadraticform}).
\begin{ssmptn}\label{A2}
There exist positive numbers $\alphapha_0,\gamma_0$ and $k^*\in[1,4/n)$ such that
\begin{align}\label{asscond}
\int_{\Omega}\sigma_{\bar u}(u-\bar u)\,dx+\Lambda(u-\bar u)\ge\gamma_0|u-\bar u|_{L^1(\Omega)}^{{k^*}+1},
\varepsilonnd{align}
for all $u\in\mathcal U$ with $|u-\bar u|_{L^1(\Omega)}\le\alphapha_0$.
\varepsilonnd{ssmptn}
Assumption \ref{A2} resembles the well-known $L^2$-coercivity condition in optimal control, with two substantial differences: $(i)$ the left-hand side of (\ref{asscond}) involves a linear term (not only the quadratic form in the $L^2$-coercivity condition); $(ii)$ the $L^1$-norm appears in the right-hand side of (\ref{asscond}). We mention that the standard $L^2$-coercivity condition cannot hold in affine problems.
Assumption \ref{A2} in the particular case ${k^*}=1$ has been used before in the literature on optimal control
problems constrained by ordinary differential equations, see \cite[Assumption A2']{SubregOsm} or
\cite[Assumption A2]{Mayersubreg}. A similar assumption was used in \cite[Assumption 2]{Hoeldersubreg}.\iffalse The upper bound on $k^*$ appears in the proof of Proposition \ref{propofequiv}. This proposition is essential for our analysis of stability.\fi
We first point out that if $\bar u$ satisfies Assumption \ref{A2}, then it must be bang-bang. A control $u\in\mathcal U$ is bang-bang
if $ u(x)\in\{b_1(x),b_2(x)\}$ for a.e. $x$ in $\Omega$. The proof of this result
follows the arguments given in the proof of \cite[Theorem 2.1]{Casanum}.
\begin{prpstn}\label{bangbang}
If $\bar u\in \mathcal U$ satisfies Assumption \ref{A2}, then $\bar u$ is bang-bang.
\varepsilonnd{prpstn}
\begin{proof}
Let $\alphapha_0$ and $\gamma_0$ be the positive numbers in Assumption \ref{A2}.
Suppose that there exists $\varepsilon>0$ and a measurable set $E\subset\Omega$ of positive measure such that
\begin{align*}
\bar u(x)\in [b_1(x)+\varepsilon,b_2(x)-\varepsilon]\quad \text{for a.e. $x\in E$.}
\varepsilonnd{align*}
Define $\varepsilon^*:=\min\{\alphapha_0 (\text{meas}\hspace*{0.05cm} E)^{-1}, \varepsilon\}$.
Let $\{v_m\}_{m=1}^\infty\subset L^2(\Omega)$ be a sequence converging to zero weakly in $L^2(\Omega)$
such that for each $m\in\mathbb N$, $v_{m}(x)\in\{-\varepsilon^*,\varepsilon^*\}$ for a.e. $x\in\Omega$.
For each $m\in \mathbb N$, define
\begin{align*}
u_{m}(x):=\left\{ \begin{array}{lcc}
\bar u(x) & if & x\notin E \\
\\ \bar u(x)+v_{m}(x)& if & x\in E.
\varepsilonnd{array} \right.
\varepsilonnd{align*}
Clearly, for each $m\in\mathbb N$, $u_m$ belongs to $\mathcal U$ and
\begin{align*}
|u_{m}-\bar u|_{L^1(\Omega)}=\varepsilon^* \hspace*{0.05cm}\text{meas}\hspace*{0.05cm} E.
\varepsilonnd{align*}
Hence, by Assumption \ref{A2}
\begin{align}\label{lhs}
\int_{\Omega}\sigma_{\bar u}(u_m-\bar u)\,dx+\Lambda(u_m-\bar u)\ge\gamma_0\Big(\varepsilon^*
\hspace*{0.05cm}\text{meas}\hspace*{0.05cm} E\Big)^{{k^*}+1}
\varepsilonnd{align}
for all $m\in\mathbb N$. Since $u_m\rightharpoonup \bar u$ weakly in $L^2(\Omega)$, we have
by {Proposition} \ref{lemcas} that the left hand side of $(\ref{lhs})$ converges to $0$; a contradiction.
\varepsilonnd{proof}
Proposition \ref{bangbang} makes the following lemma relevant. The proof follows the argument used in the proof of \cite[Theorem 4.4]{BangBangconvergence}. Alternatively, as argued in the proof of \cite[Theorem 4.3]{Wachelliptic}, one can also use \cite[Theorem 1]{Visin} and the fact that for a.e. $x\in\Omega$, $u(x)$ is an extremal point of $\overline{\text{conv}}(\{u_{k}(x)\}_{k=1}^\infty\cup u(x))$ if $u\in\mathcal U$ is bang-bang.
\begin{lmm}\label{weakimpliesstrong}
Let $u\in\mathcal U$ be bang-bang, and $\{u_{k}\}_{k=1}^\infty\subset \mathcal U$ be a sequence.
If $u_k\rightharpoonup u$ weakly in $L^1(\Omega)$, then $u_k\to u$ in $L^1(\Omega)$.
\varepsilonnd{lmm}
\begin{proof}
Let $\Omega_i:=\{x\in\Omega:u(x)=b_i(x)\}$, $i=1,2$. Let $\chi_{\Omega_i}:\Omega\to\{0,1\}$ denote the characteristic function of the set $\Omega_i$, $i=1,2$. Now, by definition of weak convergence
\begin{align*}
\int_{\Omega}|u_k-u|\, dx=\int_{\Omega}\chi_{\Omega_1}(u_n-\bar u)\, dx-\int_{\Omega}\chi_{\Omega_2}(u_n-\bar u)\, dx\to 0.
\varepsilonnd{align*}
\varepsilonnd{proof}
\iffalse
To prove the latter lemma, one can also use \cite[Theorem 1]{Visin} and the fact that if $u\in\mathcal U$ is bang-bang, then $u(x)$ is an extremal point of $\overline{\text{conv}}(\{u_{k}(x)\}_{k=1}^\infty\cup u(x))$ for a.e. $x\in\Omega$.
\fi
The next proposition shows that the switching mapping satisfies a growth condition.
The proof consists of two steps. The first one is to show that Assumption \ref{A2} implies this growth
condition for the linearization of the switching mapping. The second step is to adequately use the linearization
as an approximation of the switching mapping.
\begin{prpstn}\label{propofequiv}
Let Assumption 2 be fulfilled. Then there exist positive numbers $\alphapha$ and $\gamma$ such that
\begin{align*}
\int_{\Omega} \sigma_{u}(u-\bar u)\,dx\ge \gamma |u-\bar u|_{L^1(\Omega)}^{k^*+1}
\varepsilonnd{align*}
for all $ u\in\mathcal U$ with $|u-\bar u|_{L^1(\Omega)}\le\alphapha$.
\varepsilonnd{prpstn}
\begin{proof}{}
Let $\alphapha_0,\gamma_0$ and $k^*$ be the positive numbers in Assumption \ref{A2}. Fix $r\in(n/2,2/{k^*})$.
Using Proposition \ref{sigmatilde}, a constant $c>0$ can be found such that
\begin{align}\label{inevarlr}
|\sigma_u-\sigma_{\bar u}-\pi_{u-\bar u}|_{L^\infty(\Omega)}\le c|u-\bar u|_{L^1(\Omega)}^{2/r}\quad\quad
\forall u\in\mathcal U.
\varepsilonnd{align}
From Proposition \ref{opQ} and Assumption \ref{A2}, we have
\begin{align}\label{intpr}
\int_{\Omega}\Big[\sigma_{\bar u}+\pi_{u-\bar u}\Big](u-\bar u)\, dx=
\int_{\Omega}\sigma_{\bar u}(u-\bar u)\,dx+\Lambda(u-\bar u)\ge \gamma_0 |u-\bar u|_{L^1(\Omega)}^{{k^*}+1}
\varepsilonnd{align}
for all $u\in\mathcal U$ with $|u-\bar u|_{L^1(\Omega)}\le\alphapha_0$. Define $\gamma:=\gamma_0/2$ and
\begin{align*}
\alphapha:=\min\left\lbrace \alphapha_0,\gamma^{\frac{r}{2-{k^*}r}} c^{-{\frac{r}{2-{k^*}r}}}\right\rbrace.
\varepsilonnd{align*}
Then, by (\ref{inevarlr})
\begin{align}\label{inevarlr1}
|\sigma_u-\sigma_{\bar u}-\pi_{u-\bar u}|_{L^\infty(\Omega)}\le c|u-\bar u|_{L^1(\Omega)}^{\frac{2}{r}}=
c|u-\bar u|_{L^1(\Omega)}^{\frac{2}{r}-{k^*}}|u-\bar u|_{L^1(\Omega)}^{k^*}\le
\gamma|u-\bar u|_{L^1(\Omega)}^{k^*}
\varepsilonnd{align}
for all $u\in\mathcal U$ with $|u-\bar u|_{L^1(\Omega)}\le\alphapha$. We have for all $u\in\mathcal U$
\begin{align*}
\int_{\Omega}\sigma_{u}(u-\bar u)\,dx&=\int_{\Omega} \Big[\sigma_{\bar u}+\pi_{u-\bar u}\Big](u-\bar u)\,dx+\int_{\Omega} \Big[\sigma_{u}-\sigma_{\bar u}-\pi_{u-\bar u}\Big](u-\bar u)\,dx.
\varepsilonnd{align*}
Consequently, by (\ref{intpr}) and (\ref{inevarlr1}),
\begin{align*}
\int_{\Omega}\sigma_{u}(u-\bar u)\,dx&\ge\gamma_0 |u-\bar u|_{L^1(\Omega)}^{{k^*}+1}-|
\sigma_{u}-\sigma_{\bar u}-\pi_{u-\bar u}|_{L^\infty(\Omega)}|u-\bar u|_{L^1(\Omega)}\\
&\ge(\gamma_0-\gamma)|u-\bar u|_{L^1(\Omega)}^{{k^*}+1}=\gamma|u-\bar u|_{L^1(\Omega)}^{{k^*}+1}
\varepsilonnd{align*}
for all $u\in\mathcal U$ with $|u-\bar u|_{L^1(\Omega)}\le\alphapha$.
\varepsilonnd{proof}
\subsection{Some existence and stability results}
We now pass to some preparatory lemmas concerning the existence of solutions of inclusions (also called generalized equations, see \cite{Robinsongenequ1}) related to
the first order necessary condition of problem (\ref{cost})--(\ref{system}). Given $r\in[1,\infty]$, we denote by
$\mathbb B_{L^r}(c;\alphapha)$ the closed ball in $L^r(\Omega)$ with center $c\in L^r(\Omega)$ and radius $\alphapha>0$.
The variational inequality (\ref{varin}) can be written as the inclusion
\begin{align}\label{Inclusion}
0\in\sigma_u+ N_{\mathcal U}(u),
\varepsilonnd{align}
where the normal cone at $u$ to the set $\mathcal U$ is given by
\begin{align*}
N_{\mathcal U}(u)=\left\lbrace \sigma\in L^\infty(\Omega): \int_{\Omega}\sigma(w-u)\, dx\le 0\quad \forall w\in\mathcal U\right\rbrace.
\varepsilonnd{align*}
\begin{lmm}\label{Lemainclusion0}
For all $\rho\in {L^\infty(\Omega)}$ and $\varepsilon>0$ there exists
$u\in \mathcal U\cap \mathbb B_{L^1}(\bar u;\varepsilon)$ satisfying
$$
\rho\in\sigma_u+ N_{\mathcal U\cap \hspace*{0.02cm}\mathbb B_{L^1}(\bar u;\varepsilon)}(u).
$$
\varepsilonnd{lmm}
\begin{proof}
Let $\rho\in {L^\infty(\Omega)}$ and $\varepsilon>0$. Consider the functional $\mathcal J_{\rho}:\mathcal U\cap \mathbb B_{L^1}(\bar u;\varepsilon)\to\mathbb R$ given by
\begin{align*}
\mathcal J_{\rho}(u):=\int_{\Omega} \big[g(y_u,u)-\rho u\big]\, dx=\mathcal J(u)-\int_{\Omega}\rho u\,dx.
\varepsilonnd{align*}
The functional $\mathcal J_\rho$ has at least one global minimizer $ u_\rho\in\mathcal U\cap \mathbb B_{L^1}(\bar u;\varepsilon)$ since
$\mathcal U\cap \mathbb B_{L_1}(\bar u;\varepsilon)$ is a weakly sequentially compact subset of $L^2(\Omega)$
and $\mathcal J_{\rho}$ is weakly sequentially continuous. By the Pontryagin principle,
\begin{align*}
\int_{\Omega}\big[\sigma_{u_\rho}-\rho\big](u-u_\rho)\, dx\ge0\quad \forall u\in\mathcal U\cap \mathbb B_{L^1}(\bar u;\varepsilon).
\varepsilonnd{align*}
We have then that $u_\rho$ satisfies
$\rho\in\sigma_{u_\rho}+ N_{\mathcal U\cap \hspace*{0.02cm}\mathbb B_{L^1}(\bar u;\varepsilon)}(u_\rho)$.
\varepsilonnd{proof}
\begin{lmm}\label{coneint}
Let $\mathcal V_1$ and $\mathcal V_2$ be closed and convex subsets of $L^1(\Omega)$
such that $\mathcal V_1\cap\text{int}\hspace*{0.07cm}\mathcal V_2\neq\varepsilonmptyset$. Then
\begin{align}\label{EdecomN}
N_{\mathcal V_1\cap\mathcal V_2}(u)= N_{\mathcal V_1}(u)+N_{\mathcal V_2}(u)
\varepsilonnd{align}
for all $u\in\mathcal V_1\cap\mathcal V_2$.
\varepsilonnd{lmm}
\begin{proof}
Given a set $\mathcal W \subset L^1(\Omega)$, let $s_\mathcal W : L^\infty(\Omega) \to \mathbb R\cup\{+\infty\}$
denote the support function to $\mathcal W$, that is
\begin{align*}
s_\mathcal W(h) := \sup_{w \in \mathcal W} \int_{\Omega}hw\, dx.
\varepsilonnd{align*}
By \cite[Proposition 3.1]{Coneintersection}, the set
$\text{Epi}\hspace{0.07cm} s_{\mathcal V_1} + \text{Epi}\hspace{0.07cm} s_{\mathcal V_2}$ is a weakly${}^*$
closed subset of $L^\infty(\Omega)$.
{Then the representation (\ref{EdecomN}) holds according
to \cite[Theorem 3.1]{Coneintersection}.}
\varepsilonnd{proof}
We can now prove existence of solutions of the inclusion $\rho\in\sigma_u+N_{\mathcal U}(u)$
that are close (in the $L^1$-norm) to $\bar u$ whenever $\rho$ is close to zero
(in the norm $L^\infty$-norm). The proof follows the arguments in \cite[p. 1127]{Bim19}.
\begin{lmm}\label{exisrho}
Let Assumption \ref{A2} hold. Then for each $\varepsilon>0$ there exists $\delta>0$ such that for each
$\rho\in \mathbb B_{L^\infty}(0;\delta)$ there exists $u\in \mathcal U\cap \mathbb B_{L^1}(\bar u;\varepsilon)$
satisfying $\rho\in\sigma_u+ N_{\mathcal U}(u)$.
\varepsilonnd{lmm}
\begin{proof}
Let $\alphapha$ and $\gamma$ be the numbers in Proposition \ref{propofequiv}.
Define $\varepsilon_0:=\min\{\varepsilon,\alphapha\}$ and $\delta:=\varepsilon_0^{k^*}\gamma/2$.
Let $\rho\in L^\infty(\Omega)$ with $|\rho|_{L^\infty(\Omega)}\le \delta$. By Lemma \ref{Lemainclusion0},
there exists $u\in\mathcal U\cap \mathbb B_{L^1}(\bar u;\varepsilon_0)$ such that
\begin{align*}
\rho\in\sigma_u+ N_{\mathcal U\cap \mathbb B_{L^1}(\bar u;\varepsilon_0)}(u).
\varepsilonnd{align*}
Since trivially $\bar u\in \mathcal U\cap \text{int}\hspace*{0.04cm}\mathbb B_{L^1}(\bar u,\varepsilon_0)$,
by Lemma \ref{coneint} we have
\begin{align}\label{intcone}
N_{\mathcal U\cap \mathbb B_{L^1}(\bar u;\varepsilon_0)}(u)= N_{\mathcal U}(u)+
N_{\mathbb B_{L^1}(\bar u;\varepsilon_0)}(u).
\varepsilonnd{align}
Thus there exists $\nu\in N_{\mathbb B_{L^1}(\bar u;\varepsilon_0)}(u)$ such that
\begin{align*}
\rho-\sigma_{u}-\nu\in N_{\mathcal U}(u).
\varepsilonnd{align*}
By definition of the normal cone,
\begin{align}\label{nu0}
0\ge\int_{\Omega}\big(\rho-\sigma_u\big)(\bar u-u)\,dx-\int_{\Omega}\nu(\bar u- u)\, dx.
\varepsilonnd{align}
As $\bar u\in \mathbb B_{L^1}(\bar u;\varepsilon_0)$ and $\nu\in N_{\mathbb B_{L^1}(\bar u;\varepsilon_0)}(u)$, we have
\begin{align*}
\int_{\Omega}\nu(\bar u- u)\, dx\le 0.
\varepsilonnd{align*}
Consequently, by (\ref{nu0}) and Proposition \ref{propofequiv}
\begin{align*}
0\ge\int_{\Omega}\big(\rho-\sigma_u\big)(\bar u-u)\,dx\ge -|\rho|_{L^\infty(\Omega)}|u-
\bar u|_{L^1(\Omega)}+\gamma|u-\bar u|_{L^1(\Omega)}^{{k^*}+1},
\varepsilonnd{align*}
which implies
\begin{align*}
|u-\bar u|_{L^1(\Omega)}\le \gamma^{-\frac{1}{{k^*}}}|\rho|_{L^\infty(\Omega)}^{\frac{1}{{k^*}}}
\le2^{-\frac{1}{{k^*}}}\varepsilon_0<\varepsilon_0.
\varepsilonnd{align*}
As $u\in \text{int}\hspace*{0.07cm} \mathbb B_{L^1}(\bar u;\varepsilon_0)$, we have
$N_{\mathbb B_{L^1}(\bar u;\varepsilon_0)}(u)=\left\lbrace 0\right\rbrace $. Thus by (\ref{intcone}),
\begin{align}
\rho\in\sigma_u+ N_{\mathcal U\cap \mathbb B_{L^1}(\bar u;\varepsilon_0)}(u)=\sigma_u+ N_{\mathcal U}(u).
\varepsilonnd{align}
\varepsilonnd{proof}
The following lemma shows how Proposition \ref{propofequiv} (and consequently Assumption \ref{A2}) is related to H\"{o}lder- stability.
\begin{lmm}\label{Esslem}
Let Assumption \ref{A2} hold. There exist positive numbers $\alphapha,\delta$ and $c$ such that for every $\rho\in \mathbb B_{L^\infty}(0;\delta) $ there exists $u\in\mathbb B_{L^1}(\bar u,\alphapha)$ satisfying $\rho\in\sigma_u+ N_{\mathcal U}(u)$. Moreover, \begin{align}\label{holderine}
|u-\bar u|_{L^1(\Omega)}\le c|\rho|_{L^\infty(\Omega)}^{\frac{1}{{k^*}}}
\varepsilonnd{align}
for all $\rho\in L^\infty(\Omega)$ and $u\in\mathbb B_{L^1}(\bar u;\alphapha)$ satisfying $\rho\in\sigma_u+ N_{\mathcal U}(u)$.
\varepsilonnd{lmm}
\begin{proof}
The existence part follows from Lemma \ref{exisrho}.
Let $\alphapha$ and $\gamma$ be the positive numbers in Proposition \ref{propofequiv}.
Since $\rho-\sigma_u\in N_{\mathcal U}(u)$, we have
\begin{align*}
\int_{\Omega}(\rho-\sigma_{u})(\bar u-u)\,dx\le0.
\varepsilonnd{align*}
By Proposition \ref{propofequiv},
\begin{align*}
0&\ge\int_{\Omega}(\rho-\sigma_{u})(\bar u- u)
\,dx=\int_{\Omega} \sigma_{u}(u-\bar u)\,dx+
\int_{\Omega}\rho(\bar u- u)\,dx\\
&\ge \gamma\left( \int_{\Omega}|u-\bar u|\,dx\right)^{{k^*}+1}-|\rho|_{L^\infty(\Omega)}\int_{\Omega}|u-\bar u|\,dx.
\varepsilonnd{align*}
Hence
\begin{align*}
\displaystyle\int_{\Omega}|u-\bar u|\,dx\le\Big(\frac{1}{\gamma}|\rho|_{L^\infty(\Omega)}\Big)^{1/{k^*}}=
\gamma^{-\frac{1}{{k^*}}}|\rho|_{L^\infty(\Omega)}^{\frac{1}{{k^*}}}.
\varepsilonnd{align*}
The result follows defining $c=\gamma^{-\frac{1}{{k^*}}}$.
\varepsilonnd{proof}
\iffalse
\begin{Lemma}\label{contcone}
Let $\{\theta_m\}_{m=1}^\infty\subset L^\infty(\Omega)$ and $\{u_{m}\}_{m=1}^\infty\subset\mathcal U$ be
sequences such that $\theta_m\in N_{\mathcal U}(u_m)$ for all $m\in\mathbb N$. Let $u\in\mathcal U$ and
$\theta\in L^\infty(\Omega)$. If $u_m\rightharpoonup u$ weakly in $L^{1}(\Omega)$ and $\theta_m\to\theta$
in $L^{\infty}(\Omega)$, then $\theta\in N_{\mathcal U}(u)$.
\varepsilonnd{Lemma}
\begin{proof}
We have for each $w\in \mathcal U$
\begin{align*}
0\ge\int_{\Omega}\theta_m(w-u_m)\, dx=\int_{\Omega}(\theta_m-\theta)(w-u_m)\, dx+\int_{\Omega}\theta(w-u_m)\, dx.
\varepsilonnd{align*}
Taking limit we obtain
\begin{align*}
0\ge\int_{\Omega}\theta(w-u)\, dx,
\varepsilonnd{align*}
which can be rewritten as $\theta\in N_{\mathcal U}(u)$.
\varepsilonnd{proof}
\fi
For inequality (\ref{holderine}) to hold, Lemma \ref{Esslem} requires that the controls are close in the $L^1$-norm to the reference solution (by Lemma \ref{exisrho}, the existence of such controls is guaranteed). This closeness assumption on the controls can be removed if the solution of inclusion (\ref{Inclusion}) is unique. In particular, if (\ref{Inclusion}) has a unique solution, then problem (\ref{cost})-(\ref{system}) has unique optimal control (minimizer).
\begin{lmm}\label{Esslemuni}
Let Assumption \ref{A2} hold, and suppose additionally $0\in \sigma_{ u}+N_{\mathcal U}(u)$ has a unique solution $\bar u\in\mathcal U$. There exist positive numbers $\delta$ and $c$ such that
\begin{align*}
|u-\bar u|_{L^1(\Omega)}\le c|\rho|_{L^\infty(\Omega)}^{\frac{1}{{k^*}}}.
\varepsilonnd{align*}
for all $\rho\in \mathbb B_{L^\infty}(0;\delta)$ and $u\in\mathcal U$ satisfying $\rho\in\sigma_u+ N_{\mathcal U}(u)$.
\varepsilonnd{lmm}
\begin{proof}
Let $\alphapha$ and $c$ be the positive numbers in Lemma \ref{Esslem}.
First we prove that there exists $\delta>0$ such that if $u\in\mathcal U$ and $\rho\in L^\infty(\Omega)$
satisfy $\rho\in\sigma_u+N_{\mathcal U}(u)$ and $|\rho|_{L^\infty(\Omega)}\le\delta$, then
$u\in \mathbb B_{L^1}(\bar u;\alphapha)$. Suppose not, then there exist sequences
$\{\rho_k\}_{k=1}^\infty\subset L^\infty(\Omega)$ and $\{u_{k}\}_{k=1}^\infty\subset \mathcal U$
such that $\rho_k\in \sigma_{u_k}+ N_{\mathcal U}(u_k)$, $\rho_k\to 0$ in $L^\infty(\Omega)$, and $|u_k-\bar u|_{L^1(\Omega)}> \alphapha$.
Since $\mathcal U$ is weakly sequentially compact in $L^2(\Omega)$, there exists a subsequence of
$\{u_k\}_{k=1}^\infty$, denoted in the same way, and $u^*\in\mathcal U$ such that $u_k\rightharpoonup u^*$
weakly in $L^2(\Omega)$. Using Proposition \ref{weakconvergence}, one can see that
$\rho_k-\sigma_{u_k}\to \sigma_{u^*}$ in $L^\infty(\Omega)$. Consequently,
as $\rho_k\in\sigma_{u_k}+ N_{\mathcal U}(u_k)$ for all $n\in\mathbb N$,
we obtain $0\in\sigma_{u^*}+N_{\mathcal U}(u^*)$. Then, by assumption, $u^*=\bar u$, so $u^*$
is bang-bang. By Lemma \ref{weakimpliesstrong}, we have $u_k\to u^*$ in $L^1(\Omega)$; a contradiction.
The result follows from Lemma \ref{Esslem}.
\varepsilonnd{proof}
\subsection{Strong metric subregularity}
Let us begin considering the following system representing the necessary optimality conditions (Pontryagin principle) for problem (\ref{cost})--(\ref{system}):
\begin{align}\label{s1}
\left\{ \begin{array}{cll}
0&=&\mathcal Ly-f(\cdot,y,u),\\
0&=&\mathcal Lp-H_{y}(\cdot,y,p,u),\\
0&\in& H_{u}(\cdot,y,p) + N_{\mathcal U}(u),
\varepsilonnd{array} \right.
\varepsilonnd{align}
If $ u\in\mathcal U$ is a local solution of problem (\ref{cost})--(\ref{system}), then the triple $(y_{u}, p_{u},u)$ is a solution of (\ref{s1}).
Therefore, the mapping that defines the right-hand side is referred to as the {\varepsilonm optimality mapping}. In order to give a strict definition and
recast system (\ref{s1}) in a functional frame, we introduce the metric spaces
\begin{align*}
\mathcal Y:=D(\mathcal L)\times D(\mathcal L)\times\mathcal U\quad\text{and}\quad \mathcal Z:=L^2(\Omega)\times
L^2(\Omega)\times L^\infty(\Omega),
\varepsilonnd{align*}
endowed with the following metrics. For $\psi_i=(y_i,p_i,u_i) \in {\mathcal Y}$ and
$\zeta_i=(\xi_i,\varepsilonta_i,\rho_i) \in {\mathcal Z}$, $i\in\{1,2\}$,
\begin{align*}
& d_{\mathcal Y}(\psi_1,\psi_2):=|y_1-y_2|_{L^2(\Omega)}+|p_1-p_2|_{L^2(\Omega)}+|u_1-u_2|_{L^1(\Omega)},\\
& d_{\mathcal Z}(\zeta_1,\zeta_2):=|\xi_1-\xi_2|_{L^2(\Omega)}+|\varepsilonta_1-\varepsilonta_2|_{L^2(\Omega)}+
|\rho_1-\rho_2|_{L^\infty(\Omega)}.
\varepsilonnd{align*}
Both metrics are shift-invariant. We denote by $\mathbb B_{\mathcal Y}(\psi;\alphapha)$ the closed ball in ${\mathcal Y}$, centered
at $\psi$ and with radius $\alphapha$. The notation for the ball $\mathbb B_{\mathcal Z}(\zeta;\alphapha)$ is analogous.
{Then the optimality mapping is defined as the set-valued mapping $\Phi:\mathcal Y\twoheadrightarrow\mathcal Z$ } given by
\begin{align}\label{optmapping}
\Phi(y,p,u) =\left( \begin{array}{c}
\mathcal Ly - f(\cdot,y,u) \\
\mathcal Lp- H_y(\cdot,y,p,u) \\
H_u(\cdot,y,p,u)+ N_{\mathcal U}(u)
\varepsilonnd{array} \right) .
\varepsilonnd{align}
{Then the optimality system (\ref{s1}) can be recast as the inclusion }
\begin{align}\label{inc}
0\in\Phi(y,p,u).
\varepsilonnd{align}
Our purpose is to study the stability of system (\ref{s1}), or equivalently of inclusion (\ref{inc}), {with respect to perturbations on
the left-hand side}.
{From now on, we denote $\bar \psi := (\bar y, \bar p, \bar u) = (y_{\bar u},p_{\bar u},\bar u)$ , where $\bar u$ is the fixed local solution of problem (\ref{cost})--(\ref{system}).}
\begin{dfntn}\label{Dsmsr}
{The} optimality mapping $\Phi:\mathcal Y\twoheadrightarrow \mathcal Z$ is {called} strongly
{H\"older subregular} with exponent $\lambda>0$ at $(\bar\psi,0)$ if there exist positive numbers
$\alphapha_1,\alphapha_2$ {and $\kappa$} such that
\begin{align}\label{Essr}
d_{\mathcal Y}(\psi,\bar\psi)\le\kappa d_{\mathcal Z}(\zeta,0)^\lambda
\varepsilonnd{align}
for all $\psi\in \mathbb B_{\mathcal Y}(\bar \psi;\alphapha_1)$ and $\zeta\in \mathbb B_{\mathcal Z}(0;\alphapha_2)$
satisfying $\zeta\in\Phi(\psi)$.
\varepsilonnd{dfntn}
{More explicitly, the inequality (\ref{Essr}) reads as
\begin{align} \label{EHe}
|y-y_{\bar u}|_{L^2(\Omega)}+|p-p_{\bar u}|_{L^2(\Omega)}+|u-\bar u|_{L^1(\Omega)}\le
\kappa\Big( |\xi|_{L^2(\Omega)}+|\varepsilonta|_{L^2(\Omega)}+|\rho|_{L^\infty(\Omega)}\Big)^\lambda.
\varepsilonnd{align}}
Hence, if the optimality mapping is {strongly H\"older subregular}, all solutions of the system
\begin{align}\label{s1per}
\left\{ \begin{array}{cll}
\xi&=&\mathcal Ly-f(\cdot,y,u),\\
\varepsilonta&=&\mathcal Lp-H_{y}(\cdot,y,p,u),\\
\rho&\in& H_{u}(\cdot,y,p) + N_{\mathcal U}(u).
\varepsilonnd{array} \right.
\varepsilonnd{align}
that are near $(y_{\bar u},p_{\bar u},\bar u)$ satisfy the H\"older estimate {(\ref{EHe})}
with respect to the perturbations {$\zeta = (\xi,\varepsilonta,\rho)$}, provided they are small enough. The subregularity property is weaker than the well known strong regularity (see \cite[pp. 178-179]{Dontchevbook}); this allows to relax the assumptions to prove stability.
\begin{rmrk}\label{strictlocal}
If
$\Phi$ is strongly H\"older subregular at $(\bar \psi, 0)$, then from (\ref{Essr}) applied with $\zeta = 0$ we obtain that $\bar \psi$ is the unique solution of (\ref{inc})
in $B_{\mathcal Y}(\bar \psi;\alphapha_1)$, hence $\bar u$ is the unique local solution of problem (\ref{cost})--(\ref{system})
in this ball. In particular, $\bar u$ is a strict local minimizer.
\varepsilonnd{rmrk}
\iffalse
\begin{lmm}\label{lem2}
There exists $c>0$ such that
\begin{align*}
|y-y_{u}|_{L^\infty(\Omega)}+|p-p_{u}|_{L^\infty(\Omega)}\le c\Big(|\xi|_{L^2(\Omega)}+
|\varepsilonta|_{L^2(\Omega)}\Big)
\varepsilonnd{align*}
for all $\xi,\varepsilonta\in L^2(\Omega)$, $y,p\in H^1(\Omega)$ and $u\in\mathcal U$ satisfying
\begin{align}\label{s1pers}
\left\{ \begin{array}{cll}
\mathcal Ly&=&f(\cdot,y,u)+\xi,\\
\mathcal Lp&=&H_{y}(\cdot,y,p,u)+\varepsilonta.
\varepsilonnd{array} \right.
\varepsilonnd{align}
\varepsilonnd{lmm}
\begin{proof}
Let $\xi,\varepsilonta\in L^2(\Omega)$, $y,p\in H^1(\Omega)$ and $u\in\mathcal U$ satisfying (\ref{s1pers}).
Using the argument given in the proof of Proposition \ref{contstatel1} and Lemma \ref{L1e}, one can find
a constant $c_1>0$ (independent of $y,\xi$ and $u$) such that
\begin{align*}
|y-y_{u}|_{L^\infty(\Omega)}\le c_1|\xi|_{L^2(\Omega)}.
\varepsilonnd{align*}
The function $p-p_u$ satisfies
\begin{align*}
\mathcal L(p-p_{u})+d_{y}(\cdot,y)(p-p_{u})=H_{y}(\cdot,y,p_{u},u)-H_{y}(\cdot,y_{u},p_{u},u)+\varepsilonta.
\varepsilonnd{align*}
By Lemma \ref{L1e}, there exists $c_2>0$ (independent of $\xi,\varepsilonta,y,p$ and $u$) such that
\begin{align*}
|p-p_{\bar u}|_{L^\infty(\Omega)}\le c_2 |H_{y}(\cdot,y,p_{u},u)-H_{y}(\cdot,y_{u},p_{u},u)+\varepsilonta|_{L^2(\Omega)}.
\varepsilonnd{align*}
Since $H_y$ is locally Lipschitz uniformly in $(y,p,u)$, and the sets $\{y_u:u\in\mathcal U\}$, $\{p_u:u\in\mathcal U\}$
are bounded in $C(\bar\Omega)$, there exists $c_3>0$ (independent of $\xi,\varepsilonta,y,p$ and $u$) such that
\begin{align*}
|H_{y}(\cdot,y,p_{u},u)-H_{y}(\cdot,y_{u},p_{u},u)|_{L^2(\Omega)}\le c_3|y-y_{u}|_{L^\infty(\Omega)}.
\varepsilonnd{align*}
Putting all together,
\begin{align*}
|y-y_{u}|_{L^\infty(\Omega)}+|p-p_{u}|_{L^\infty(\Omega)}\le c_1|\xi|_{L^2(\Omega)}+
c_2c_3c_1|\xi|_{L^2(\Omega)}+c_2|\varepsilonta|_{L^2(\Omega)}.
\varepsilonnd{align*}
The results follows choosing $c:=(c_1+c_2c_3c_1)+c_2$.
\varepsilonnd{proof}
\fi
We are now ready to state our main result.
\begin{thrm}\label{Ssr}
{Let Assumption \ref{A2} hold. Then the optimality mapping $\Phi$ is strongly H\"older subregular
at $(\bar\psi,0)$ with exponent $\lambda = 1/k^*$.}
\varepsilonnd{thrm}
\begin{proof}
Let $\alphapha$ and $c$ be the positive numbers in Lemma \ref{Esslem}.
Let $\zeta=(\xi,\varepsilonta,\rho)\in\mathcal \mathbb B_{\mathcal Z}(0;1)$ and
$\psi=(y,p,u)\in \mathbb B_{\mathcal Y}(\bar\psi;\alphapha)$ such that $\zeta\in\Phi(\psi)$.
By a standard argument, we can find $c_1>0$ (independent of $\psi$ and $\zeta$) such that
\begin{align}\label{mt1}
|y-y_{u}|_{L^\infty(\Omega)}+|p-p_{u}|_{L^\infty(\Omega)}\le c_1\Big(|\xi|_{L^2(\Omega)}+|\varepsilonta|_{L^2(\Omega)}\Big).
\varepsilonnd{align}
Since $H_u$ is locally Lipschitz uniformly in the first variable, and the sets $\{y_u:u\in\mathcal U\}$, $\{p_u:u\in\mathcal U\}$
are bounded in $C(\bar\Omega)$, there exists $c_2>0$ (independent of $\psi$) such that
\begin{align}\label{mt2}
|H_{u}(\cdot,y,p)-H_{u}(\cdot,y_u,p_u)|_{L^\infty(\Omega)}\le c_2\Big
( |y-y_{u}|_{L^\infty(\Omega)}+|p-p_{u}|_{L^\infty(\Omega)}\Big)
\varepsilonnd{align}
Define $\nu:=\rho+H_{u}(\cdot,y_u,p_u) - H_{u}(\cdot,y,p).$ By (\ref{mt1}) and (\ref{mt2}),
there exists $c_3>0$ (independent of $\psi$ and $\zeta$) such that
\begin{align*}
|\nu|_{L^\infty(\Omega)}\le c_3\Big(|\xi|_{L^2(\Omega)}+|\varepsilonta|_{L^2(\Omega)}+
|\rho|_{L^\infty(\Omega)}\Big)=c_3|\zeta|_{\mathcal Z}.
\varepsilonnd{align*}
As $\rho\in H_{u}(\cdot,y,p)+ N_{\mathcal U}(u)$, we have $\nu\in H_{u}(\cdot, y_u,p_u)+ N_{\mathcal U}(u)$.
Then by Lemma \ref{Esslem},
\begin{align}\label{mt3}
|u-\bar u|_{L^1(\Omega)}\le c|\nu|_{L^\infty(\Omega)}^{\frac{1}{{k^*}}}\le
cc_3^{\frac{1}{{k^*}}}|\zeta|^{\frac{1}{{k^*}}}_{\mathcal Z}:=c_4|\zeta|^{\frac{1}{{k^*}}}_{\mathcal Z}.
\varepsilonnd{align}
Now, by Proposition \ref{contstatel1}, there exists $c_5>0$ (independent of $\psi$) such that
$|y_{u}-y_{\bar u}|_{L^2(\Omega)}\le c_5|u-\bar u|_{L^1(\Omega)}$. Consequently, by (\ref{mt3})
\begin{align*}
|y-y_{\bar u}|_{L^2(\Omega)}&\le |y-y_{ u}|_{L^2(\Omega)}+|y_{u}-y_{\bar u}|_{L^2(\Omega)}\\
&\le c_1\text{meas}\hspace{0.08cm}\Omega^{\frac{1}{2}}\Big(|\xi|_{L^2(\Omega)}+|\varepsilonta|_{L^2(\Omega)}\Big)+
c_5|u-\bar u|_{L^1(\Omega)}\\
&\le (c_1\text{meas}\hspace{0.08cm}\Omega^{\frac{1}{2}}+c_5c_4)|\zeta|^{\frac{1}{{k^*}}}_{\mathcal Z}
=:c_6|\zeta|^{\frac{1}{{k^*}}}_{\mathcal Z}.
\varepsilonnd{align*}
Analogously, there exists $c_7>0$ (independent of $\psi$ and $\zeta$) such that
\begin{align*}
|p-p_{\bar u}|_{L^2(\Omega)}\le c_7 |\zeta|^{\frac{1}{{k^*}}}_{\mathcal Z}.
\varepsilonnd{align*}
Putting all together,
\begin{align*}
|y-y_{\bar u}|_{L^2(\Omega)}+|p-p_{\bar u}|_{L^2(\Omega)}+|u-\bar u|_{L^1(\Omega)}\le
(c_4+c_6+c_7)|\zeta|^{\frac{1}{{k^*}}}_{\mathcal Z}.
\varepsilonnd{align*}
Finally, let $\alphapha_1:=\alphapha$, $\alphapha_2:=1$ and $\kappa:=c_4+c_6+c_7$.
Since the constants $c_4, c_6$ and $c_7$ are independent of $\psi$ and $\zeta$, so is $\kappa$.
Thus we have (\ref{Essr})
for all $\psi\in \mathbb B_{\mathcal Y}(\bar \psi;\alphapha_1)$ and $\zeta\in
\mathbb B_{\mathcal Z}(0;\alphapha_2)$ satisfying $\zeta\in\Phi(\psi)$.
\varepsilonnd{proof}
\iffalse
\begin{Lemma}\label{lem3}
Let $\xi,\varepsilonta\in L^2(\Omega)$ be arbitrary. Let $\{u_n\}_{n=1}^\infty$ be a sequence in $\mathcal U$ and $u\in\mathcal U$.
For each $n\in\mathbb N$, let $y_{n},p_n\in C(\bar\Omega)$ be the unique functions satisfying
\begin{align}
\left\{ \begin{array}{cll}
\mathcal Ly_n&=&f(\cdot,y_n,u_n)-\xi,\\
\mathcal Lp_n&=&H_{y}(\cdot,y,p_n,u_n)-\varepsilonta.
\varepsilonnd{array} \right.
\varepsilonnd{align}
If $u_{n}\rightharpoonup u$ weakly in $L^2(\Omega)$, then $y_{n}\to y$ and
$p_n\to p$ in $C(\bar\Omega)$, where $y,p\in C(\bar\Omega)$ satisfy
\begin{align}
\left\{ \begin{array}{cll}
\mathcal Ly&=&f(\cdot,y,u)-\xi,\\
\mathcal Lp&=&H_{y}(\cdot,y,p,u)-\varepsilonta.
\varepsilonnd{array} \right.
\varepsilonnd{align}
\varepsilonnd{Lemma}
\fi
{The strong subregularity property defined above does not require existence of solutions of the perturbed
inclusion (\ref{s1per}) in a neighborhood of the reference solution $\bar \psi$.
The next theorem answers the existence question.}
\begin{thrm}\label{Ssrexist}
Let Assumption \ref{A2} hold. For each $\varepsilon>0$ there exists $\delta>0$ such that for every
$\zeta\in \mathbb B_{\mathcal Z}(0;\delta)$ there exists $\psi\in \mathbb B_{\mathcal Y}(\bar \psi;\varepsilon)$
satisfying the inclusion $\zeta\in \Phi(\psi)$.
\varepsilonnd{thrm}
\begin{proof}
For each $u\in\mathcal U$ and $\zeta=(\xi,\varepsilonta,\rho)\in \mathcal Z$, define
$\nu_{u,\zeta}:=\rho+H_{u}(\cdot,y_u,p_u)-H_{u}(\cdot,y_{u,\zeta},p_{u,\zeta})$,
where $y_{u,\zeta}$ and $p_{u,\zeta}$ are the unique solutions of
\begin{align}\label{s1pers1}
\left\{ \begin{array}{cll}
\mathcal Ly&=&f(\cdot,y,u)+\xi,\\
\mathcal Lp&=&H_{y}(\cdot,y,p,u)+\varepsilonta.
\varepsilonnd{array} \right.
\varepsilonnd{align}
By a standard argument, one can find positive numbers $c_1$ and $c_2$ such that
\begin{align}\label{dl2}
|y_{u,\zeta}-y_{u}|_{L^2(\Omega)}+|p_{u,\zeta}-p_{u}|_{L^2(\Omega)}\le
c_1\Big(|\xi|_{L^2(\Omega)}+|\varepsilonta|_{L^2(\Omega)}\Big),
\varepsilonnd{align}
and $|\nu_{u,\zeta}|_{L^\infty(\Omega)}\le c_2|\zeta|_{\mathcal Z}$ for all $u\in\mathcal U$
and $\zeta\in\mathcal Z$. Let $\varepsilon>0$ be arbitrary.
By Lemma \ref{exisrho}, the exists $\delta_0>0$ such that for each $\nu\in \mathbb B_{L^\infty}(0;\delta_0)$
there exists $u\in \mathcal U\cap \mathbb B_{L^1}(\bar u;\varepsilon/2)$ satisfying $\nu\in\sigma_u+ N_{\mathcal U}(u)$.
Define $\delta:=\min\{c_2^{-1}\delta_0,(2c_1)^{-1}\varepsilon\}$ and let $\zeta^*\in \mathbb B_{\mathcal Z}(0;\delta)$
be arbitrary; we will prove that there exists $u^*\in\mathcal U\cap \mathbb B_{L^1}(\bar u;\varepsilon/2)$
such that $\nu_{u^*,\zeta^*}\in\sigma_{u^*}+N_{\mathcal U}(u^*)$. First, observe that
\begin{align*}
|\nu_{u,\zeta^*}|_{L^\infty(\Omega)}\le c_2|\zeta^*|_{\mathcal Z}\le \delta_0\quad\forall u\in\mathcal U.
\varepsilonnd{align*}
Therefore, by Lemma \ref{exisrho}, we can {inductively define}
a sequence $\{u_{k}\}_{k=1}^\infty\subset \mathcal U$
such that $\nu_{u_{k},\zeta^*}\in\sigma_{u_{k+1}}+N_{\mathcal U}(u_{k+1})$ and
$|u_k-\bar u|_{L^1(\Omega)}\le \varepsilon/2$ for all $k\in\mathbb N$. Since $\mathcal U$ is weakly
compact in $L^2(\Omega)$, we may assume that $u_{k}\rightharpoonup u^*$
weakly in $L^2(\Omega)$ for some $u^*\in\mathcal U$. Weak convergence in $L^2(\Omega)$
implies weak convergence in $L^1(\Omega)$ and $\mathbb B_{L^1}(\bar u;\varepsilon/2)$ is weakly sequentially
closed in $L^1(\Omega)$, therefore $u^*\in \mathbb B_{L^1}(\bar u; \varepsilon/2)$. Using Proposition \ref{weakconvergence}, one can see that
$\nu_{u_k,\zeta^*}-\sigma_{u_{k+1}}\to\nu_{u^*,\zeta^*}-\sigma_{u^*}$ in $L^\infty(\Omega)$,
and consequently that $\nu_{u^*,\zeta^*}\in\sigma_{u^*}+N_{\mathcal U}(u^*)$.
We conclude then that $\zeta^*\in\Phi(\psi^*)$, where $\psi^*:=(y_{u^*,\zeta^*}, p_{u^*,\zeta^*},u^*)$.
Finally, by definition of $\delta$ and (\ref{dl2})
\begin{align*}
|\psi^*-\bar\psi|_{\mathcal Y}\le c_1|\zeta|_{\mathcal Z}+\varepsilon/2\le\varepsilon.
\varepsilonnd{align*}
Thus, $\zeta^*\in\Phi(\psi^*)$ and $\psi^*\in \mathbb B_{\mathcal Y}(\bar\psi;\varepsilon)${, which completes the proof}.
\varepsilonnd{proof}
{The next theorem claims that \varepsilonmph{all} solutions of the perturbed optimality system (\ref{s1per}) are arbitrarily close to the solution
of the unperturbed optimality system, provided that the solution of the latter is globally unique,
Assumption \ref{A2} holds, and the perturbation is sufficiently small.}
\begin{thrm}\label{Ssrexisuni}
Let Assumption \ref{A2} hold and suppose additionally that $\bar\psi$ is the unique element
of $\mathcal Y$ that satisfies $0\in\Phi(\bar\psi)$. For each $\varepsilon>0$ there exists $\delta>0$
such that if $\zeta\in \mathbb B_{\mathcal Z}(0;\delta)$ and $\psi\in\mathcal Y$ satisfy $\zeta\in \Phi(\psi)$,
then $\psi\in \mathbb B_{\mathcal Y}(\bar \psi;\varepsilon)$.
\varepsilonnd{thrm}
\begin{proof}
Let $\delta_0$ and $c_0$ be the positive numbers in Lemma \ref{Esslemuni}.
Let $\zeta=(\xi,\varepsilonta,\rho)\in{\mathcal Z}$ and $\psi=(y,p,u)\in\mathcal Y$ {be} such that $\zeta\in\Phi(\psi)$.
Define $\nu:=\rho+H_{u}(\cdot,y_u,p_u) - H_{u}(\cdot,y,p).$ Arguing as in the proof of Theorem \ref{Ssr},
we can find positive numbers $c_1$ and $c_2$ (independent of $\psi$ and $\zeta$) such that
$|\nu|_{L^\infty(\Omega)}\le c_1|\zeta|_{\mathcal Z}$ and
\begin{align*}
|y-y_{\bar u}|_{L^2(\Omega)}+|p-p_{\bar u}|_{L^2(\Omega)}\le
c_2\Big(|\zeta|_{\mathcal Z}+|u-\bar u|_{L^1(\Omega)}\Big).
\varepsilonnd{align*}
Let $\delta:=\min\{c_1^{-1}\delta_0,(2c_0c_2)^{-k^*}c_1^{-1}\varepsilon^{k^*},(2c_2)^{-1}\varepsilon\}$
and suppose that $\zeta\in \mathbb B_{\mathcal Z}(0;\delta)$.
As $\rho\in H_{u}(\cdot,y,p)+ N_{\mathcal U}(u)$, we have $\nu\in H_{u}(\cdot, y_u,p_u)+ N_{\mathcal U}(u)$.
By Lemma \ref{Esslemuni},
\begin{align*}
|u-\bar u|_{L^1(\Omega)}\le c_0|\nu|_{L^\infty(\Omega)}^{\frac{1}{{k^*}}}\le
c_0c_1^\frac{1}{{k^*}}|\zeta|^{\frac{1}{{k^*}}}_{\mathcal Z}\le c^{-1}_2\varepsilon/2.
\varepsilonnd{align*}
Thus,
\begin{align*}
|y-y_{\bar u}|_{L^2(\Omega)}+|p-p_{\bar u}|_{L^2(\Omega)}+|u-\bar u|_{L^1(\Omega)}\le
c_2\Big(\delta+c^{-1}_2\varepsilon/2\Big)\le\varepsilon.
\varepsilonnd{align*}
\varepsilonnd{proof}
\iffalse
\subsection{Linear Perturbations}\label{linearperturbations}
The study of linear perturbations in the optimal control problem is an easy consequence of the subregularity of the optimality mapping. Set $\Theta:=L^2(\Omega)\times L^2(\Omega)\times L^\infty(\Omega)$, and for each $\zeta=(\xi,\varepsilonta,\rho)\in\Theta$ consider the optimal control problem $\mathcal P_{\zeta}$ given by
\begin{align}\label{costppp}
\quad\min_{u\in\mathcal U}\left\lbrace\int_{\Omega}\Big[g(x,y,u)+\varepsilonta(x) y+\rho(x) u\Big]\,dx\right\rbrace,
\varepsilonnd{align}
subject to
\begin{align}\label{systemppp}
\left\{ \begin{array}{cclcc}
-\dive\big(A(x)\nabla y\big)+d(x,y)&=&\beta(x)u+\xi(x)& \text{in}& \Omega \\
\\ A(x)\nabla y\cdot \nu+b(x)y&=&0 &\text{on}& \partial\Omega.
\varepsilonnd{array} \right.
\varepsilonnd{align}
\fi
\section{Nonlinear Perturbations}\label{Section nonlin}
In this section we apply the subregularity results in Section \ref{Section Stab} for studying the effect of certain nonlinear perturbations on the optimal solution.
We consider the following family of problems
\begin{align}\label{costpp}
\quad\min_{u\in\mathcal U}\left\lbrace\int_{\Omega}\Big[g(x,y,u)+\varepsilonta(x,y,u)\Big]\,dx\right\rbrace,
\varepsilonnd{align}
subject to
\begin{align}\label{systempp}
\left\{ \begin{array}{cclcc}
-\dive\big(A(x)\nabla y\big)+d(x,y)+\xi(x,y)&=&\beta(x)u& \text{in}& \Omega \\
\\ A(x)\nabla y\cdot \nu+b(x)y&=&0 &\text{on}& \partial\Omega.
\varepsilonnd{array} \right.
\varepsilonnd{align}
In order to specify the perturbations $\xi$ and $\varepsilonta$ under consideration and their topology, we begin the section recalling some elementary notions of functional analysis.
As usual, $C(\mathbb R^s)$ denotes the space of all continuous functions $\omega:\mathbb R^s\to\mathbb R$. For each $m\in\mathbb N$, let $K_m$ denote the closed ball in $\mathbb R^s$ centered at zero with radius $m$. Consider the metric on $C(\mathbb R^s)$ given by
\begin{align*}
d_{C}(\omega_1,\omega_2):=\sum_{m=1}^{\infty}\frac{1}{2^m}\frac{|\omega_1-\omega_2|_{L^\infty(K_m)}}{1+|\omega_1-\omega_2|_{L^\infty(K_m)}}.
\varepsilonnd{align*}
This metric induces the compact-convergence topology on $C(\mathbb R^s)$. In this topology, a sequence $\{\omega_m\}_{m=1}^\infty\subset C(\mathbb R^s)$ converges to $\omega\in C(\mathbb R^s)$ if and only if
$|\omega-\omega_m|_{L^\infty(K)}\to0$ for every compact set $K\subset\mathbb R^s$. This topology is also known as the compact-open topology, see \cite[Chapter 7]{Munkres}. The following lemma is straightforward and follows from the definition of $d_{C}$.
\begin{lmm}\label{metlem}
For each compact set $K\subset\mathbb R^s$ there exists $m\in\mathbb N$ such that
\begin{align*}
|\omega_1-\omega_2|_{L^\infty(K)}\le 2^{m}d_{C}(\omega_1,\omega_2)
\varepsilonnd{align*}
for all $\omega_1,\omega_2\in C(\mathbb R^s)$ such that $d_{C}(\omega_1,\omega_2)\le {2^{{-m}}}$.
\varepsilonnd{lmm}
\begin{proof}
Let $K$ be a compact subset of $\mathbb R^s$. There exists $i\in\mathbb N$ such that $K\subset K_i$, where $K_i$ denotes the closed ball in $\mathbb R^s$ centered at zero with radius $i$. Now, by definition of the metric $d_{C}$,
\begin{align*}
\frac{|\omega|_{L^\infty(K_i)}}{1+|\omega|_{L^\infty(K_i)}}\le 2^i d_{C}(\omega,0)\quad\forall \omega\in C(\mathbb R^s).
\varepsilonnd{align*}
Hence,
\begin{align*}
|\omega|_{L^\infty(K_i)}&\le \frac{2^{i}d_{C}(\omega,0)}{1-2^{i}d_{C}(\omega,0)}\le 2^{i+1}d_{C}(\omega,0)
\varepsilonnd{align*}
for all $\omega\in C(\mathbb R^s)$ with $d_{C}(\omega,0)\le 2^{-(i+1)}$. Let $m=i+1$. Then
\begin{align*}
|\omega_2-\omega_1|_{L^\infty(K)}\le|\omega_2-\omega_1|_{L^\infty(K_i)}\le 2^m d_{C}(\omega_2-\omega_1,0)=2^m d_{C}(\omega_2,\omega_1)
\varepsilonnd{align*}
for all $\omega_1,\omega_2\in C(\mathbb R^s)$ with $d_{C}(\omega_1,\omega_2)\le {2^{{-m}}}$.
\varepsilonnd{proof}
\subsection{The perturbations}
We begin describing the space of perturbations appearing in equation (\ref{systempp}). Let $\Upsilon_{s}$ be the set of all continuously differentiable functions $\xi: \mathbb R^n\times \mathbb R\to\mathbb R$ such that $ d_y(x,y)+\xi_{y}(x,y)\ge 0$
for all $x\in\Omega$ and $y\in\mathbb R$. The set $\Upsilon_{s}$ does not constitute a linear space, but it allows to have well-defined states for each perturbation.
\begin{prpstn}\label{contstateper}
For each $u\in\mathcal U$ and $\xi\in \Upsilon_{s}$ there exists a unique function $y^{\xi}_u\in D(\mathcal L)$ satisfying
\begin{align*}
\mathcal Ly^\xi_u+d(\cdot,y^\xi_u)+\xi(\cdot,y^\xi_u)=\beta(\cdot)u.
\varepsilonnd{align*}
Moreover, there exist positive numbers $M$ and $\delta$ such that $|y_{u}^\xi|_{L^\infty(\Omega)}\le M$
for all $u\in\mathcal U$ and $\xi\in\Upsilon_{s}$ with $d_{C}(\xi,0)\le\delta$.
\varepsilonnd{prpstn}
\begin{proof}
The existence follows from \cite[Theorem 4.8]{TroeltzschPde}. Moreover, also from this theorem, there exists $c>0$ such that
\begin{align*}
|y^\xi_u|_{L^\infty(\Omega)}\le c\big|\beta(\cdot)u-d(\cdot,0)-\xi(\cdot,0)\big|_{L^\infty(\Omega)}
\varepsilonnd{align*}
for all $u\in\mathcal U$ and $\xi\in\Upsilon_{s}$. Let $K:=\bar\Omega\times\{0\}$, then by Lemma \ref{metlem} there exists $m\in\mathbb N$ such that
\begin{align*}
|y^\xi_u|_{L^\infty(\Omega)}&\le c\Big( |\beta|_{L^\infty(\Omega)}|u|_{L^\infty(\Omega)}+|d(\cdot,0)|_{L^\infty(\Omega)}+|\xi|_{L^\infty(K)}\Big)\\
&\le c\Big( |\beta|_{L^\infty(\Omega)}\sup_{u \in \mathcal U}|u|_{L^\infty(\Omega)}+|d(\cdot,0)|_{L^\infty(\Omega)}+2^{m}d_{C}(\xi,0)\Big)\\
& \le c\Big( |\beta|_{L^\infty(\Omega)}\sup_{u \in \mathcal U}|u|_{L^\infty(\Omega)}+|d(\cdot,0)|_{L^\infty(\Omega)}+1\Big)
\varepsilonnd{align*}
for all $u\in\mathcal U$ and $\xi\in\Upsilon_{s}$ with $d_{C}(\xi,0)\le 2^{{-m}}$. The result follows defining $\delta:=2^{{-m}}$ and
\begin{align*}
M:=c\Big( |\beta|_{L^\infty(\Omega)}\sup_{u \in \mathcal U}|u|_{L^\infty(\Omega)}+|d(\cdot,0)|_{L^\infty(\Omega)}+1\Big).
\varepsilonnd{align*}
\varepsilonnd{proof}
We now proceed to describe the perturbations appearing in the cost functional (\ref{costpp}). Consider the set $\Upsilon_{c}$ of all continuously differentiable functions $\varepsilonta:\mathbb R^n\times\mathbb R\times\mathbb R\to\mathbb R$ such that $\varepsilonta(x,y,\cdot)$ is convex for all $x\in\Omega$ and $y\in\mathbb R$. We have the following result concerning the adjoint variable of the perturbed problem. Its proof is similar to the one of Proposition \ref{contstateper}.
\begin{prpstn}\label{contcostateper}
For each $u\in\mathcal U$, $\xi\in \Upsilon_{s}$ and $\varepsilonta\in\Upsilon_c$ there exists a unique function $p^{\xi,\varepsilonta}_u\in D(\mathcal L)$ satisfying
\begin{align*}
\mathcal Lp^{\xi,\varepsilonta}_u+\big[d_y(\cdot,y_u^\xi)+\xi_y(\cdot,y^\xi_u)\big]p^{\xi,\varepsilonta}_u=g_y(\cdot,y_{u}^\xi,u)+\varepsilonta_y(\cdot,y_{u}^\xi,u).
\varepsilonnd{align*}
Moreover, there exist positive numbers $M$ and $\delta$ such that $|p_{u}^{\xi,\varepsilonta}|_{L^\infty(\Omega)}\le M$
for all $u\in\mathcal U$, $\xi\in\Upsilon_{s}$ and $\varepsilonta\in\Upsilon_{c}$ with $d_{C}(\xi,0)+d_{C}(\xi_y,0)+d_{C}(\varepsilonta_y,0)\le\delta$.
\varepsilonnd{prpstn}
We denote $\Upsilon:=\Upsilon_{s}\times\Upsilon_{c}$, and write $\zeta:=(\xi,\varepsilonta)$ for a generic element of $\Upsilon$. We endow $\Upsilon$ with the pseudometric $d_\Upsilon:\Upsilon\times\Upsilon\to[0,\infty)$ given by
\begin{align*}
d_{\Upsilon}(\zeta,\zeta'):=d_{C}(\xi,\xi')+d_{C}(\xi_y,\xi_y')+d_{C}(\varepsilonta_y,\varepsilonta_y')+d_{C}(\varepsilonta_u,\varepsilonta_u').
\varepsilonnd{align*}
\subsection{The stability result}
We are now ready to state problem (\ref{costpp})-(\ref{systempp}) in a precise way. Given $\zeta\in\Upsilon$, problem $\mathcal P_\zeta$ is given by
\begin{align}\label{Perprob}
\quad\min_{u\in\mathcal U}\left\lbrace\mathcal J_\zeta(u):=\int_{\Omega}\Big[g(x,y_u^\xi,u)+\varepsilonta(x,y_u^\xi,u)\Big]\,dx\right\rbrace.
\varepsilonnd{align}
Due to the convexity of the cost in the control variable, each problem $\mathcal P_\zeta$ has at least one global solution. For each $\zeta\in\Upsilon$, we fix a local minimizer $\hat u_{\zeta}\in\mathcal U$ of problem
$\mathcal P_{\zeta}$. By the local minimum principle, for each $\zeta=(\xi,\varepsilonta)\in\Upsilon$, the triple $(\hat y_\zeta,\hat p_\zeta,\hat u_\zeta):=({y_{\hat u_\zeta}^\xi},{p_{\hat u_\zeta}^{\xi,\varepsilonta}},\hat u_{\zeta})$ satisfies the system
\begin{align}\label{s1perper}
\left\{ \begin{array}{cll}
0&=&\mathcal Ly-f(\cdot,y,u)-\xi(\cdot,y),\\
0&=&\mathcal Lp-H_{y}(\cdot,y,p,u)+\varepsilonta_y(\cdot,y,u)-\xi_y(\cdot,y)p,\\
0&\in& H_{u}(\cdot,y,p)+ \varepsilonta_u(\cdot,y,u) + N_{\mathcal U}(u).
\varepsilonnd{array} \right.
\varepsilonnd{align}
As a consequence of Theorem \ref{Ssr}, we have the following result.
\begin{thrm}\label{Thmreg0}
Let Assumption \ref{A2} hold. There exist positive numbers $\alphapha,\alphapha'$ and $c$ such that
\begin{align*}
|\hat y_{\zeta}-y_{\bar u}|_{L^2(\Omega)}+|\hat p_{\zeta}-p_{\bar u}|_{L^2(\Omega)}+|\hat u_\zeta-\bar u|_{L^1(\Omega)}\le c d_\Upsilon(\zeta,0)^{1/k^*}
\varepsilonnd{align*}
for all $\zeta\in \Upsilon$ such that $|\hat u_{\zeta}-\bar u|_{L^1(\Omega)}\le\alphapha$ and $d_{\Upsilon}(\zeta,0)\le\alphapha'$.
\varepsilonnd{thrm}
\begin{proof}
By Theorem \ref{Ssr}, the mapping $\Phi$ is strongly H\"older subregular at $(\bar\psi,0)$ with exponent $1/k^*$.
Let $\alphapha_1,\alphapha_2$ and $\kappa$ be the positive numbers in the definition of strong subregularity.
By Proposition \ref{contstateper} and \ref{contcostateper} there exist positive numbers $M$ and $\delta_0$ such that
\begin{align*}
|y_u^\xi|_{L^\infty(\Omega)}+|p^{\xi,\varepsilonta}_u|_{L^\infty(\Omega)}\le M
\varepsilonnd{align*}
for all $u\in\mathcal U$ and $\zeta\in\Upsilon$ with $d_{\Upsilon}(\zeta,0)\le\delta_0$. Let $K:=\bar\Omega\times[-M,M]$. By Lemma \ref{metlem}, there exists $m\in\mathbb N$ such that
\begin{align*}
|\xi(\cdot, y_{u}^\xi)|_{L^2(\Omega)}\le \text{meas}\hspace*{0.06cm}\Omega^{\frac{1}{2}} |\xi|_{L^\infty(K)}\le 2^{m} \text{meas}\hspace*{0.06cm}\Omega^{\frac{1}{2}}d_{C}(\xi,0)\le 2^m \text{meas}\hspace*{0.06cm}\Omega^{\frac{1}{2}} d_{\Upsilon}(\zeta,0)
\varepsilonnd{align*}
for all $u\in\mathcal U$ and $\zeta\in\Upsilon$ with $d_{\Upsilon}(\zeta,0)\le \min\{2^{{-m}},\delta_0\}$. Repeating this argument, we can find positive numbers $\delta$ and $c_0$ such that
\begin{align}\label{oflin}
|\xi(\cdot, y_{u}^\xi)|_{L^2(\Omega)}+|\xi_{y}(\cdot,y^{\xi}_u)p_{u}^{\xi,\varepsilonta}|_{L^2(\Omega)}+|\varepsilonta_y(\cdot,y_{u}^\xi,u)|_{L^2{\Omega}}+|\varepsilonta_{u}(\cdot,y_{u}^\xi,u)|_{L^\infty}\le c_0 d_{\Upsilon}(\zeta,0)
\varepsilonnd{align}
for all $u\in\mathcal U$ and $\zeta\in\Upsilon$ with $d_{\Upsilon}(\zeta,0)\le \delta$. Using Proposition \ref{contstatel1} and Lemma \ref{metlem}, one can find positive numbers $\alphapha$ and $\delta'$ such that
\begin{align*}
|\hat y_{\zeta}-y_{\bar u}|_{L^2(\Omega)}+|\hat p_{\zeta}-p_{\bar u}|_{L^2(\Omega)}+|\hat u_\zeta-\bar u|_{L^1(\Omega)}\le \alphapha_1
\varepsilonnd{align*}
for all $\zeta\in \Upsilon$ with $|\hat u_\zeta-\bar u|_{L^1(\Omega)}\le\alphapha$ and $d_{\Upsilon}(\zeta,0)\le\delta'$. Observe that by (\ref{s1perper}), we have
\begin{align*}
\left( \begin{array}{c}
\xi(\cdot,\hat y_\zeta) \\
-\varepsilonta_y(\cdot,\hat y_\zeta,\hat u_\zeta)+\xi_y(\cdot,\hat y_\zeta)\hat p_\zeta \\
-\varepsilonta_u(\cdot,\hat y_\zeta,\hat u_\zeta)
\varepsilonnd{array} \right) \in \Phi(\hat y_\zeta,\hat p_\zeta,\hat u_{\zeta})
\varepsilonnd{align*}
for all $\zeta\in\Upsilon$. Let $\alphapha':=\min\{c_0^{-1}\alphapha_2,\delta,\delta'\}$. Then by H\"older subregularity of $\Phi$ and (\ref{oflin}),
\begin{align*}
|\hat y_{\zeta}-y_{\bar u}|_{L^2(\Omega)}+|\hat p_{\zeta}-p_{\bar u}|_{L^2(\Omega)}+|\hat u_\zeta-\bar u|_{L^1(\Omega)}\le \kappa c_0^{\frac{1}{k^*}} d_{\Upsilon}(\zeta,0)^{\frac{1}{k^*}}
\varepsilonnd{align*}
for all $\zeta\in\Upsilon$ such that $|\hat u_\zeta-\bar u|_{L^1(\Omega)}\le\alphapha$ and $d_{\Upsilon}(\zeta,0)\le \alphapha'$. The result follows defining $c:=\kappa c_0^{\frac{1}{k^*}}$.
\varepsilonnd{proof}
\iffalse
If the family $\{\hat u_{\zeta}\}_{\zeta\in\Upsilon}$ consists of global solutions, the result in Theorem \ref{Thmreg0}
can be improved.
\begin{Theorem}
Let Assumption \ref{A2} be fulfilled and suppose additionally that each $\hat u_{\zeta}$ is a global solution.
There exist positive numbers $c$ and $\delta$ such that
\begin{align*}
|\hat y_{\zeta}-y_{\bar u}|_{L^2(\Omega)}+|\hat p_{\zeta}-p_{\bar u}|_{L^2(\Omega)}+|\hat u_\zeta-\bar u|_{L^1(\Omega)}\le c d_\Upsilon(\zeta,0)
\varepsilonnd{align*}
for all $\zeta\in \Upsilon$ with $d_\Upsilon(\zeta,0)\le\delta$.
\varepsilonnd{Theorem}
\begin{proof}
Let $\alphapha,\alphapha'$ and $c$ be the positive numbers in Theorem \ref{Thmreg0}. First we prove that there exists $\delta_0>0$
such that $|\hat u_{\varepsilon}-\bar u|_{L^1(\Omega)}\le\alphapha$ for all $\varepsilon\in(0,\delta_0)$.
Suppose not, then there exists a sequence $\{\zeta_n\}_{n=1}^\infty$ converging to zero such that
$|\hat u_{\varepsilon_n}-\bar u|_{L^1(\Omega)}>\alphapha$ for all $n\in\mathbb N$. Since $\mathcal U$ is weakly sequentially
compact in $L^2(\Omega)$, we may assume without loss of generality that $u_{\varepsilon_{n}}\to u^*$ for some
$u^*\in\mathcal U$. Now, since $y_{u_{\varepsilon_n}}\to y_{u^*}$ in $C(\bar\Omega)$, one can deduce
\begin{align*}
J(u^*) \leq \mathop{\mbox{\rm liminf}}\limits_{k\to\infty}\Big[J(u_{\zeta_k})+\int_\Omega \varepsilonta(x,y_{})\Big]
\le \mathop{\mbox{\rm liminf}}\limits_{k\to\infty}\Big[J(\bar u)+\frac{\varepsilon_{k}}{2}|\bar u|_{L^2(\Omega)}\Big]=J(\bar u).
\varepsilonnd{align*}
By Remark \ref{strictlocal}, $\bar u$ is a strict local solution, therefore $u^*=\bar u$.
By Proposition \ref{bangbang}, $u^*=\bar u$ is bang-bang. Weak convergence in $L^2(\Omega)$
implies that of $L^1(\Omega)$; consequently, by Lemma \ref{weakimpliesstrong},
$u_{\varepsilon_{n}}\to u^*$ in $L^1(\Omega)$; a contradiction. The result follows from Theorem \ref{Thmreg}.
\varepsilonnd{proof}
\fi
\subsection{An application: Tikhonov regularization}
In what follows we present an application of the theory derived in the previous chapters, namely the so-called Tikhonov regularization.
For a more detailed description and an account of the state of art, the reader
is referred to \cite{MR3810878,MR2986517,MR3780469}. We derive estimates on the convergence rate of the solution of the regularized problem when the regularization parameter tends to zero.
The results that appear in the literature require the so-called structural assumption and positive-definiteness
(in some sense) of the second derivative of the objective functional. Using Theorem \ref{Ssr}, we can obtain
this results under weaker assumptions than used in the literature so far. One can compare this results with
\cite[Theorem 4.4]{MR3810878} (where a tracking problem with semilinear elliptic equation is considered)
when it comes to stability of the controls. In Section \ref{Section5}, we give more details on how the assumptions
in the literature interplay with Assumption \ref{A2}.
We consider the following family of problems $\{\mathcal P_{\varepsilon}\}_{\varepsilon\ge0}$.
\begin{align}\label{costp}
\quad\min_{u\in\mathcal U}\left\lbrace\int_{\Omega}g(x,y,u)\,dx+\frac{\varepsilon}{2}\int_{\Omega} u^2\,dx\right\rbrace,
\varepsilonnd{align}
subject to
\begin{align}\label{systemp}
\left\{ \begin{array}{cclcc}
-\dive\big(A(x)\nabla y\big)+d(x,y)&=&\beta(x)u& \text{in}& \Omega \\
\\ A(x)\nabla y\cdot \nu+b(x)y&=&0 &\text{on}& \partial\Omega.
\varepsilonnd{array} \right.
\varepsilonnd{align}
\begin{lmm}\label{lemfin}
Let Assumption \ref{A2} be fulfilled. For every $\alphapha>0$ there exists $\varepsilon_\alphapha>0$ such that for every $\varepsilon\in(0,\varepsilon_\alphapha)$ problem $\mathcal P_{\varepsilon}$ has a local solution $\hat u_{\varepsilon}\in\mathcal U\cap \mathbb B_{L^1}(\bar u;\alphapha)$.
\varepsilonnd{lmm}
\begin{proof}
Let $\alphapha>0$ be arbitrary. By Remark \ref{strictlocal}, $\bar u$ is a strict local minimizer, hence there exists $\alphapha^*\le \alphapha$ such that $\mathcal J(\bar u)<\mathcal J(u)$ for all $\bar u\neq u\in\mathcal U\cap \mathbb B_{L^1(\Omega)}(\bar u;\alphapha^*)$.
Consider the family of problems $\mathcal P^*_{\varepsilon}$ given by
\begin{align}\label{label2}
\displaystyle\min_{\mathcal U\cap \mathbb B_{L^1}(\bar u;\alphapha^*)}\left\lbrace \mathcal J(u)+\frac{\varepsilon}{2}|u|_{L^2(\Omega)}^2\right\rbrace .
\varepsilonnd{align}
Each problem $\mathcal P^*_{\varepsilon}$ has a global solution $\hat u_\varepsilon$. There exists $\varepsilon^*>0$
such that $|\hat u_{\varepsilon}-\bar u|_{L^1(\Omega)}\le \alphapha^*/2$ for all $\varepsilon\in(0,\varepsilon^*)$. Suppose the opposite. Then there exists a sequence $\{\varepsilon_{k}\}_{k=1}^\infty$ converging to zero such that
$|\hat u_{\varepsilon_k}-\bar u|_{L^1(\Omega)}>\alphapha^*/2$ for all $k\in\mathbb N$. Since $\mathcal U\cap \mathbb B_{L^1}(\bar u;\alphapha^*)$ is weakly
compact in $L^2(\Omega)$, we may assume without loss of generality that $u_{\varepsilon_{k}}\rightharpoonup u^*$ for some
$u^*\in\mathcal U\cap \mathbb B_{L^1}(\bar u;\alphapha^*)$. Since $y_{\hat u_{\varepsilon_k}}\to y_{u^*}$ in $C(\bar\Omega)$, we obtain that
\begin{align*}
\mathcal J(u^*) \leq \mathop{\mbox{\rm liminf}}\limits_{k\to\infty}\Big[\mathcal J(\hat u_{\varepsilon_k})+\frac{\varepsilon_{k}}{2}|\hat u_{\varepsilon_{k}}|_{L^2(\Omega)}^2\Big]
\le \mathop{\mbox{\rm liminf}}\limits_{k\to\infty}\Big[\mathcal J(\bar u)+\frac{\varepsilon_{k}}{2}|\bar u|_{L^2(\Omega)}^2\Big]=\mathcal J(\bar u).
\varepsilonnd{align*}
Therefore $u^*=\bar u$ since $u^*\in\mathcal U\cap\mathbb B_{L^1(\Omega)}(\bar u;\alphapha^*)$ and $\bar u$ is strict local minimum.
By Proposition \ref{bangbang}, $u^*=\bar u$ is bang-bang. Weak convergence in $L^2(\Omega)$
implies weak convergence in $L^1(\Omega)$; consequently, by Lemma \ref{weakimpliesstrong},
$\hat u_{\varepsilon_{k}}\to u^*$ in $L^1(\Omega)$, which is a contradiction. We can see that for all $\varepsilon\le\varepsilon^*$, $\hat u_{\varepsilon}$ is a local solution of problem $\mathcal P_{\varepsilon}$. Indeed, if $u\in\mathcal U\cap B_{L^1(\Omega)}(\hat u_{\varepsilon}; \alphapha^*/2)$, then
\begin{align*}
|u-\bar u|_{L^1(\Omega)}\le |u-\hat u_{\varepsilon}|_{L^1(\Omega)}+|\hat u_{\varepsilon}-\bar u|_{L^1(\Omega)}\le \alphapha^*,
\varepsilonnd{align*}
and consequently, as $\hat u_{\varepsilon}$ is a global solution of problem $\mathcal P^*_{\varepsilon}$,
\begin{align*}
\mathcal J(\hat u_{\varepsilon})+\frac{\varepsilon}{2}|\hat u_{\varepsilon}|_{L^2(\Omega)}\le \mathcal J(u)+\frac{\varepsilon}{2}|u|_{L^2(\Omega)}.
\varepsilonnd{align*}
The result follows defining $\varepsilon_\alphapha:=\varepsilon^*$.
\varepsilonnd{proof}
\begin{thrm}\label{Treg}
Let Assumption \ref{A2} be fulfilled. Then there exist positive numbers $\alphapha,\kappa$ and $\varepsilon_0$ such that for every $\varepsilon\in(0,\varepsilon_0)$ problem $\mathcal P_{\varepsilon}$ has a local solution $\hat u_{\varepsilon}\in\mathbb B_{L^1}(\bar u;\alphapha)$. Moreover,
\begin{align} \label{Ereg}
|\hat u_{\varepsilon}-\bar u|_{L^1(\Omega)}\le \kappa\hspace*{0.03cm}\varepsilon^{1/k^*}
\varepsilonnd{align}
for every local solution $\hat u_{\varepsilon}$ of problem $\mathcal P_{\varepsilon}$ such that $\varepsilon\in(0,\varepsilon_0)$ and $|\hat u_{\varepsilon}-\bar u|_{L^1(\Omega)} \le \alphapha$.
\varepsilonnd{thrm}
\begin{proof}
The first claim follows from Lemma \ref{lemfin}.
Let $\alphapha,\alphapha'$and $c$ be the positive numbers in Theorem \ref{Thmreg0}. Define $\varepsilonta_\varepsilon:\mathbb R\to\mathbb R$ by $\varepsilonta_\varepsilon(u):=\varepsilon u^2/2$ and $\zeta_\varepsilon:=(0,\varepsilonta_\varepsilon)\in\Upsilon$ for each $\varepsilon>0$. Note that
\begin{align*}
d_C(\varepsilonta_\varepsilon,0):=\sum_{m=1}^\infty \frac{1}{2^m}\frac{\varepsilon m^2/2}{1+\varepsilon m^2/2}=\varepsilon\sum_{m=1}^\infty \frac{1}{2^m}\frac{m^2}{2+\varepsilon m^2}\le\varepsilon\sum_{m=1}^\infty\frac{m^2}{2^{m+1}}=3\varepsilon
\varepsilonnd{align*}
for all $\varepsilon>0$. Analogously,
\begin{align*}
d_C(\frac{\partial \varepsilonta_\varepsilon}{\partial u},0):=\sum_{m=1}^\infty \frac{1}{2^m}\frac{\varepsilon m}{1+\varepsilon m}\le\varepsilon\sum_{m=1}^\infty\frac{m}{2^m}=2\varepsilon
\varepsilonnd{align*}
for all $\varepsilon>0$. We conclude that $d_{\Upsilon}(\zeta_\varepsilon,0)\le5\varepsilon\le \alphapha'$
for all $\varepsilon\in (0,\varepsilon_0)$, where $\varepsilon_0:=\alphapha'/5$. By Theorem \ref{Thmreg0},
\begin{align*}
|\hat u_\varepsilon-\bar u|_{L^1(\Omega)}\le 5^{\frac{1}{k^*}}c\varepsilon^\frac{1}{k^*}
\varepsilonnd{align*}
for all $\varepsilon\in(0,\varepsilon_0)$ such that $|\hat u_\varepsilon-\bar u|_{L^1(\Omega)}\le\alphapha$. \iffalse Let $M > 0$ be a bound for $\mathcal U$ in $L^\infty(\Omega)$. We also have
we have
\begin{align*}
|\hat u_\varepsilon-\bar u|_{L^1(\Omega)}&\le 2M\le 2M\varepsilon^{-\frac{1}{k^*}}\varepsilon^{\frac{1}{k^*}}\le 2M \varepsilon_0^{-\frac{1}{k^*}}\varepsilon^{\frac{1}{k^*}}
\varepsilonnd{align*}
for all $\varepsilon\ge\varepsilon_0$.
Hence, defining
$$
\kappa:= \max \left\{5^{\frac{1}{k^*}}c, \, 2 M^{} \varepsilon_0^{-1/k^*} \right\},
$$
completes the proof.\fi
\varepsilonnd{proof}
\section{Assumptions related {to} subregularity}\label{Section5}
In this section, we gather some results concerning Assumption \ref{A2}, in order to provide sufficient
conditions under which it is fulfilled. Furthermore, we analyze related assumptions and their relation between themselves.
Recall that $\bar u\in\mathcal U$ is a local solution of problem (\ref{cost})--(\ref{system}).
Since $\bar u\in\mathcal U$ satisfies the variational inequality (\ref{varin}), we have
\begin{align*}
\bar u(x)=\left\{ \begin{array}{lcc}
b_{1}(x) & if & \sigma_{\bar u}(x)>0 \\
\\ b_2(x)& if & \sigma_{\bar u}(x)<0.
\varepsilonnd{array} \right.
\varepsilonnd{align*}
We introduce the following extended cone suggested in \cite{Casascone}. For a fixed $\tau>0$ define
\begin{align*}
C_{\bar u}^\tau=\left\lbrace v\in L^2(\Omega): v(x)\left\{ \begin{array}{cll}
=0& \text{if} & |\sigma_{\bar u}(x)|>\tau \mbox{ or } \bar u(x) \in (b_1(x), b_2(x)) \\
\ge 0 & \text{if} & |\sigma_{\bar u}(x)| \leq \tau \mbox{ and } \bar u(x) = b_1(x) \\
\le 0& \text{if} & |\sigma_{\bar u}(x)| \leq \tau \mbox{ and } \bar u(x) = b_2(x)
\varepsilonnd{array} \right. \right\rbrace .
\varepsilonnd{align*}
We introduce the following modification of Assumption \ref{A2}.
\noindent
{\bf Assumption 2${}'$.} {\varepsilonm
There exist positive numbers $\alphapha_0$ and $\gamma_0$ such that
\begin{align*}
\int_{\Omega}\sigma_{\bar u}(u-\bar u)\,dx+\Lambda(u-\bar u)\ge\gamma_0|u-\bar u|_{L^1(\Omega)}^{{k^*}+1},
\varepsilonnd{align*}
for all $u\in\mathcal U$ with $u - \bar u \in C_{\bar u}^\tau \cap {\mathbb B}_{L^1(\Omega)}(\bar u; \alpha_0)$.
}
\noindent
This assumption is seemingly weaker than Assumption \ref{A2}. However, we will prove that the two assumptions are equivalent.
Before that, for technical purposes, we introduce the bilinear form $\Gamma:L^{2}(\Omega)\times L^{2}(\Omega)\to\mathbb R$ given by
\begin{align}\label{bilfor}
\Gamma(v_1,v_2):=\frac{1}{2}\int_{\Omega}\Big[\pi_{v_1}v_2+\pi_{v_2}v_1\Big]\,dx.
\varepsilonnd{align}
The bilinear form is particularly useful because of the following property.
\begin{align}\label{remarkgamma}
\Lambda(v_1+v_2)=\Gamma(v_1,v_1)+2\hspace*{0.02cm}\Gamma(v_1,v_2)+\Gamma(v_2,v_2)\quad\forall v_1,v_2\in L^2(\Omega).
\varepsilonnd{align}
We will require the following technical lemma.
\begin{lmm}\label{Lemgamma}
For every positive number $M$, there exists a positive number $c$ such that
\begin{align*}
|\Gamma(v_1,v_2)|\le c|v_1|_{L^1(\Omega)}^{1/2}|v_2|_{{L^1(\Omega)}}
\varepsilonnd{align*}
for all $v_1,v_2\in \mathbb B_{L^\infty}(0;M)$.
\varepsilonnd{lmm}
\begin{proof}
By Proposition \ref{furprop3}, there exist $c_1,c_2>0$ such that $|\pi_{v}|_{L^\infty(\Omega)}\le c_1|v|_{L^2(\Omega)}$ and $|\pi_{v}|_{L^2(\Omega)}\le c_2|v|_{L^1(\Omega)}$ for all $v\in L^2(\Omega)$. Let $M>0$ be arbitrary. Observe that
\begin{align*}
\Big|\int_{\Omega}\pi_{v_1}v_2\, dx\Big|\le |\pi_{v_1}|_{L^\infty(\Omega)}|v_2|_{L^1(\Omega)}\le c_1M^{\frac{1}{2}}|v_1|_{L^1(\Omega)}^{\frac{1}{2}}|v_2|_{L^1(\Omega)},
\varepsilonnd{align*}
and that
\begin{align*}
\Big|\int_{\Omega}\pi_{v_2}v_1\, dx\Big|\le |\pi_{v_2}|_{L^2(\Omega)}|v_1|_{L^2(\Omega)}\le c_2M^{\frac{1}{2}}|v_1|_{L^1(\Omega)}^{\frac{1}{2}}|v_2|_{L^1(\Omega)}
\varepsilonnd{align*}
for all $v_1,v_2\in \mathbb B_{L^\infty}(0;M)$. There result follows defining $c:=2^{-1}(c_1+c_2)M^{\frac{1}{2}}$.
\varepsilonnd{proof}
\begin{prpstn} \label{PA2}
Assumptions \ref{A2} and 2${\,}'$ are equivalent.
\varepsilonnd{prpstn}
\begin{proof}{}
Clearly Assumption \ref{A2} implies 2${\,}'$.
Let $\alphapha_0$ and $\gamma_0$ be the numbers in Assumption 2${\,}'$. Let $u\in\mathcal U$ and define
\begin{align*}
v_1(x):=\left\{ \begin{array}{lcc}
u(x)-\bar u(x) & if & |\sigma_{\bar u}(x)|\le\tau \\
\\ 0& if & |\sigma_{\bar u}(x)|>\tau,
\varepsilonnd{array} \right.
\varepsilonnd{align*}
and
\begin{align*}
v_2(x):=\left\{ \begin{array}{lcc}
0& if & |\sigma_{\bar u}(x)|\le\tau \\
\\ u(x)-\bar u(x)& if & |\sigma_{\bar u}(x)|>\tau.
\varepsilonnd{array} \right.
\varepsilonnd{align*}
Clearly $v_1\in C_{\bar u}^\tau$ and $v_1+v_2=u-\bar u$. Let $M$ be a bound for $\mathcal U$ in $L^\infty(\Omega)$, and let $c$ be the positive number in Lemma \ref{Lemgamma} corresponding to $2M$.
By Assumption 2${\,}'$,
\begin{align*}
\int_{\Omega}\sigma_{\bar u}(u-\bar u)\,dx&=\int_{\Omega}\sigma_{\bar u}v_1\,dx+\int_{|\sigma_{\bar u}|>\tau}\sigma_{\bar u}v_2\,dx\\
&=\int_{\Omega}\sigma_{\bar u}v_1\,dx+\Lambda(v_1)-\Lambda(v_1)+\int_{|\sigma_{\bar u}|>\tau}\sigma_{\bar u}v_2\,dx\\
&\ge \gamma_0|v_1|^{k+1}+\tau|v_2|_{L^1(\Omega)}-\Lambda(v_1),
\varepsilonnd{align*}
and
\begin{align*}
\Lambda(u-\bar u)&=\Lambda(v_1)+2\Gamma(v_1,v_2)+\Lambda(v_2)\\
&\ge\Lambda(v_1)-2c|v_1|_{L^1(\Omega)}^{1/2}|v_2|_{{L^1(\Omega)}}-c|v_2|_{L^1(\Omega)}^{1/2}|v_2|_{{L^1(\Omega)}}\\
&\ge\Lambda(v_1)-3c|v_2|_{L^1(\Omega)}|u-\bar u|_{L^1(\Omega)}^{1/2}
\varepsilonnd{align*}
for $u\in\mathcal U$ with $|u-\bar u|_{{L^1(\Omega)}}\le\alphapha_0$. Thus
\begin{align*}
\int_{\Omega}\sigma_{\bar u}(u-\bar u)\,dx+\Lambda(u-\bar u)&\ge\gamma_0|v_1|^{k+1}+\tau|v_2|_{L^1(\Omega)}-3c|v_2|_{L^1(\Omega)}|u-\bar u|_{L^1(\Omega)}^{1/2}\\
&= \gamma_0|v_1|^{k+1}+|v_2|_{L^1(\Omega)}\Big(\tau-3c|u-\bar u|_{L^1(\Omega)}^{1/2}\Big)
\varepsilonnd{align*}
for $u\in\mathcal U$ with $|u-\bar u|_{{L^1(\Omega)}}\le\alphapha_0$. Now, by the reverse triangle inequality and Bernoulli's inequality (consider without loss of generality $u\neq\bar u$)
\begin{align*}
|v_1|_{L^1(\Omega)}^{k+1}&=|(u-\bar u)-v_2|_{L^1(\Omega)}^{k+1}\ge \Big(|u-\bar u|_{L^1(\Omega)}-|v_2|_{L^1(\Omega)}\Big)^{k+1}\\
&=|u-\bar u|_{L^1(\Omega)}^{k+1}\Big(1-\frac{|v_2|_{L^1(\Omega)}}{|u-\bar u|_{L^1(\Omega)}}\Big)^{k+1}\ge|u-\bar u|_{L^1(\Omega)}^{k+1}\Big(1-(k+1)\frac{|v_2|_{L^1(\Omega)}}{|u-\bar u|_{L^1(\Omega)}}\Big)\\
&=|u-\bar u|_{L^1(\Omega)}^{k+1}-(k+1)|u-\bar u|_{L^1(\Omega)}^k|v_2|_{L^1(\Omega)}.
\varepsilonnd{align*}
Consequently,
\begin{align*}
\int_{\Omega}\sigma_{\bar u}(u-\bar u)\,dx+\Lambda(u-\bar u)&\ge\gamma_0|v_1|^{k+1}+|v_2|_{L^1(\Omega)}\Big(\tau-3c|u-\bar u|_{L^1(\Omega)}^{1/2}\Big)\\
&\ge\gamma_0|u-\bar u|_{L^1(\Omega)}^{k+1}-\gamma_0(k+1)|u-\bar u|_{L^1(\Omega)}^k|v_2|_{L^1(\Omega)}+|v_2|_{L^1(\Omega)}\Big(\tau-3c|u-\bar u|_{L^1(\Omega)}^{1/2}\Big)\\
&\ge\gamma_0|u-\bar u|_{L^1(\Omega)}^{k+1}+|v_2|_{L^1(\Omega)}\Big(\tau-\gamma_0(k+1)|u-\bar u|_{L^1(\Omega)}^k-3c|u-\bar u|_{L^1(\Omega)}^{1/2}\Big).
\varepsilonnd{align*}
Choosing $\alphapha$ small enough, one can ensure
\begin{align*}
\int_{\Omega}\sigma_{\bar u}(u-\bar u)\,dx+\Lambda(u-\bar u)&\ge \gamma_0|u-\bar u|_{L^1(\Omega)}^{k+1}+|v_2|_{L^1(\Omega)}\Big(\tau-\gamma_0(k+1)|u-\bar u|_{L^1(\Omega)}^k-3c|u-\bar u|_{L^1(\Omega)}^{1/2}\Big)\\
&\ge \gamma_0|u-\bar u|_{L^1(\Omega)}^{k+1}+\frac{\tau}{2}|v_2|_{L^1(\Omega)}\ge \gamma_0|u-\bar u|_{L^1(\Omega)}^{k+1}
\varepsilonnd{align*}
for all $u\in\mathcal U$ with $|u-\bar u|_{L^1(\Omega)}\le\alphapha$.
\varepsilonnd{proof}
Proposition \ref{PA2} allows to split Assumption \ref{A2} in two parts, as it follows in the next theorem.
\begin{thrm} \label{TSsR_2}
Let there exist numbers $\mu_1,\mu_2\in\mathbb R$ and $\alphapha>0$ such that
\begin{align} \label{EH01}
\int_{\Omega}\sigma_{\bar u} v \,dx \ge \mu_1 |v|_{L^1(\Omega)}^{{k^*}+1}
\varepsilonnd{align}
and
\begin{align} \label{EH02}
\Lambda(v)\ge \mu_2|v|_{L^1(\Omega)}^{{k^*}+1}
\varepsilonnd{align}
for every $v \in (\mathcal U - \bar u) \cap C_{\bar u}^\tau \cap {\mathbb B}_{L^1(\Omega)}(\bar u; \alpha)$.
If $\mu_1 + \mu_2 > 0$, then Assumption 2 is fulfilled, hence the optimality mapping $\Phi$
(see (\ref{optmapping})) of problem (\ref{cost})--(\ref{system}) is strongly H\"older subregular with exponent
$\lambda = 1/k^*$ at the reference point $(\bar y, \bar p, \bar u)$.
\varepsilonnd{thrm}
The proof consists of summation of (\ref{EH01}) and (\ref{EH02}) and utilization of Proposition \ref{PA2} and
Theorem \ref{Ssr}.
\noindent
The splitting of Assumption 2 has the advantage that the inequalities in (\ref{EH01}) and (\ref{EH02}) can be
analyzed separately. The next proposition is related to (\ref{EH01}).
The following assumption has become standard in the literature on PDE optimal control problems with
bang-bang controls, see, e.g., \cite{Casasbang,Hinzestruct,Wachelliptic,Wachstruct}.
\begin{ssmptn}\label{A3}
There exists a positive number $\mu_0$ such that
\begin{align*}
\text{meas}\left\lbrace x\in\Omega: |\sigma_{\bar u}(x)|\le\varepsilon\right\rbrace
\le \mu_0\varepsilon^{\frac{1}{k^*}}\quad\forall \varepsilon>0.
\varepsilonnd{align*}
\varepsilonnd{ssmptn}
\begin{prpstn} \label{PA3_1}
The following statements hold.
\begin{itemize}
\item[(i)] If Assumption \ref{A3} is fulfilled then there exists $\mu_1 > 0$
such that (\ref{EH01}) holds for every $v \in \mathcal U -\bar u$.
\item[(ii)] Suppose there exists $\nu > 0$ such that $b_2(x) - b_1(x) \geq \nu$ for a.e. $x \in \Omega$.
If (\ref{EH01}) holds for every $v \in \mathcal U -\bar u$ then Assumption \ref{A3} is fulfilled.
\varepsilonnd{itemize}
\varepsilonnd{prpstn}
\begin{proof}
The proof of the first claim follows \cite[Proposition 3.1]{Wachelliptic}, see also \cite[Proposition 2.7]{Casasbang}.
It has been also proved several times in the literature on ordinary differential equations
in a somewhat stronger form; see, e.g., \cite{Sey2,SubregOsm,Prei,Sey1}.
Let us prove the second claim.
For each $\varepsilon>0$, define
\begin{align*}
u_\varepsilon(x):=\left\{ \begin{array}{lcc}
\bar u(x) & \text{if} & |\sigma_{\bar u}(x)|>\varepsilon \\
\\ b_1(x)& \text{if} & |\sigma_{\bar u}(x)|\le\varepsilon\quad\text{and}\quad
\bar u(x)\in\Big[\displaystyle\frac{b_1(x)+b_2(x)}{2},b_2(x)\Big]\\
\\ b_2(x)& \text{if} & |\sigma_{\bar u}(x)|\le\varepsilon\quad\text{and}\quad
\bar u(x)\in\Big[b_1(x),\displaystyle\frac{b_1(x)+b_2(x)}{2}\Big).
\varepsilonnd{array} \right.
\varepsilonnd{align*}
Clearly each $u_\varepsilon$ belongs to $\mathcal U$, and
\begin{align}\label{onehalfeine}
|u_\varepsilon(x)-\bar u(x)|\ge\frac{1}{2}|b_2(x)-b_1(x)|
\varepsilonnd{align}
for a.e $x\in\left\lbrace s\in\Omega: |\sigma_{\bar u}(s)|\le\varepsilonpsilon\right\rbrace $. From (\ref{EH01}) we have
\begin{align*}
\mu_1\Big(\int_{|\sigma_{\bar u}|\le\varepsilon} |u_{\varepsilon}-\bar u|\,dx\Big)^{k+1}
\le\int_{|\sigma_{\bar u}|\le\varepsilon} \sigma_{\bar u}(u_{\varepsilon}-\bar u)\,dx
\le\varepsilon\int_{|\sigma_{\bar u}|\le\varepsilon} |u_{\varepsilon}-\bar u|\,dx.
\varepsilonnd{align*}
This implies
\begin{align}\label{wsfas}
\int_{|\sigma_{\bar u}|\le\varepsilon} |u_{\varepsilon}-\bar u|\,dx\le\mu_1^{-\frac{1}{k}}\varepsilon^\frac{1}{k}.
\varepsilonnd{align}
Using (\ref{onehalfeine}) and (\ref{wsfas}) we obtain that
\begin{align*}
\text{meas} \left\lbrace x \in \Omega: |\sigma_{\bar u}(x)| \,\le \, \varepsilon \right\rbrace&
=\frac{1}{\nu} \int_{|\sigma_{\bar u}|\le\varepsilon} \nu \,dx
\,\le\,\frac{1}{\nu} \int_{|\sigma_{\bar u}|\le\varepsilon} |b_2-b_1|\,dx
\, \le \, \frac{2}{\nu} \int_{|\sigma_{\bar u}| \le \varepsilon} |u_{\varepsilon}-\bar u| \,dx\\
&\le 2(\mu_1)^{-\frac{1}{k}} \nu^{-1} \, \varepsilon^\frac{1}{k}.
\varepsilonnd{align*}
Thus Assumption \ref{A3} is fulfilled with $\mu_0 :=2 (\mu_1)^{-\frac{1}{k}} \nu^{-1}$.
\varepsilonnd{proof}
{}
\varepsilonnd{document}
|
\begin{equation}gin{document}
\makeatletter
\renewcommand{{\thesection}.{\arabic{equation}}}{{\thesection}.{\arabic{equation}}}
\@addtoreset{equation}{section}
\makeatother
\title{\rightline{\normalsize\it To the memory of A.M. Vinogradov}
Quantum vector fields via quantum doubles and their applications}
\author{\rule{0pt}{7mm} Dimitry Gurevich\thanks{[email protected]}\\
{\small\it Institute for Information Transmission Problems}\\
{\small\it Bolshoy Karetny per.~19, Moscow 127051, Russian Federation}\\
\rule{0pt}{7mm} Pavel Saponov\thanks{[email protected]}\\
{\small\it
National Research University Higher School of Economics,}\\
{\small\it 20 Myasnitskaya Ulitsa, Moscow 101000, Russian Federation}\\
{\small \it and}\\
{\small \it
Institute for High Energy Physics, NRC "Kurchatov Institute"}\\
{\small \it Protvino 142281, Russian Federation }}
\maketitle
\begin{equation}gin{abstract}
By treating generators of the reflection equation algebra corresponding to a Hecke symmetry as quantum analogs of vector fields, we exhibit the corresponding Leibniz rule
via the so-called quantum doubles. The role of the function algebra in such a double is attributed to another copy of the reflection equation algebra. We consider two types
of quantum doubles: these giving rise to the quantum analogs of left vector fields acting on the function algebra and those giving rise to quantum analogs of the adjoint vector
fields acting on the same algebra. Also, we introduce quantum partial derivatives in the generators of the reflection equation algebra and then at the limit $q\rightarrow 1$ we get
quantum partial derivatives on the enveloping algebra $U(gl_N)$ as well as on a certain its extension.
\end{abstract}
\maketitle
\section{Introduction}
The central problem of any attempt to generalize the notion of a vector field on a manifold is the definition of the corresponding generalized Leibniz rule. Thus, the Leibniz rule for
su\-per-fields acting on a su\-per-va\-ri\-ety should take into account the parity of all elements involved. Such su\-per-fields generate super-Lie algebras. In the late 80's one of the
authors introduced analogs of Lie algebras, associated with involutive symmetries, that is braidings\footnote{Recall that by a braiding we mean a linear operator $R:V^{\otimes 2}\to V^{\otimes 2}$
subject to the so-called braid relation
$$
(R\otimes I)(I\otimes R)(R\otimes I)=(I\otimes R)(R\otimes I)(I\otimes R).
$$ }
$R:V^{\otimes 2}\toV^{\otimes 2}$, such that $R^2=I\otimesimes I$ (see \cite{G} and references therein). Hereafter, $V$ is a vector space of dimension $N$ over the field ${\Bbb C}$ and $I$ is the identity
operator in $V$ or the $N\times N$ unit matrix. The Leibniz rule for the corresponding vector fields was also introduced via such symmetries.
Latter it became clear that many aspects of this approach can be formulated in terms of the so-called reflection equation (RE) algebras, associated with Hecke symmetries,
that is braidings $R$ such that
\begin{equation}
R^2 = I\otimesimes I +(q-q^{-1})R,q^{-1}uad q\in {\Bbb C}\setminus \{\pm 1,0\}.
{\lambda}bel{H-cond}
\end{equation}
With a given Hecke symmetry $R$ we associate two forms of the RE algebra: the non-modified ${\cal{L}}(R)$ and modified $hLL(R)$ ones. They are respectively generated by
entries of the matrices $L=(l_i^j)_{1\leq i,j \leq N}$ and $hL=(hl_i^j)_{1\leq i,j \leq N}$, which are subject to the systems of relations
\begin{equation}
R_{12}L_1R_{12} L_1-L_1R_{12} L_1R_{12}=0,
{\lambda}bel{RE}
\end{equation}
\begin{equation}
R_{12}hL_1R_{12} hL_1-hL_1R_{12}hL_1R_{12}=R_{12}hL_1-hL_1 R_{12},
{\lambda}bel{mRE}
\end{equation}
where $L_1 = L\otimesimes I$ and similarly for all other matrices.
Note that these algebras are isomorphic to each other. Their isomorphism can be established by the following relation on the {\em generating matrices} $L$ and $hL$:
\begin{equation}
L=I-(q-q^{-1})\, hL.
{\lambda}bel{iso}
\end{equation}
Thus, in fact, we are dealing with one algebra realized in two different sets of generators: $\{l_i^j\}$ and $\{hl_i^j\}$. Observe that the isomorphism (\ref{iso}) fails at $q=\pm 1$. If a Hecke
symmetry $R$ is a deformation of the usual flip $P$, then at the limit $q\to 1$ the algebra ${\cal{L}}(R)$ tends to $\mathrm{Sym}(gl_N)$, whereas $hLL(R)$ tends to the algebra $U(gl_N)$.
The generators of the RE algebra (modified or not) are usually considered as quantum (or braided) analogs of the classical vector fields. Habitually, the role of the function algebra
is played by the corresponding RTT algebra (see \cite{IP} and references therein). In \cite{GPS2, GPS3} we suggested a different version of the quantum calculus, in which the role of the
function algebra was played by another copy of the RE algebra or by an algebra, generated by the space $V$ or its dual $V^*$.
All these examples can be included into the family of the so-called quantum doubles. By a quantum double we mean a couple of associative algebras $(A,B)$ endowed with a permutation map ${\sigma}:A\otimes B\to B\otimes A$, satisfying a set of certain axioms (see the next section). On the base of this map one can introduce the structure of a unital associative algebra on the linear space $B\otimesimes A$. If the algebra $A$ is endowed with an algebra homomorphism $\varepsilon_A: A\to {\Bbb C}$, then it is possible to define an action of the algebra $A$ onto $B$ (see
formula (\ref{act}) below) and this action will be compatible with the algebraic structures of both components.
The objective of this article is two-folded. First, we compare two quantum doubles, giving rise to quantum analogs of the left vector fields and to quantum analogs of the adjoint vector
fields. These quantum vector fields act on the RE algebra, which plays the role of the function algebra $\mathrm{Sym}(gl_N)$. This construction is exhibited in Section 2 in terms of quantum doubles.
Second, we discuss some applications of these two classes of quantum vector fields. Namely, in Section 3 we consider invariant differential operators, corresponding to the central elements of the RE algebra and represented by the quantum left vector fields, and formulate a conjecture on their spectrum.
In Section 4 we establish a series of matrix Capelli identities which are quantum generalizations of results by A.Okounkov (see \cite{Ok1,Ok2}). These generalizations are based on the quantum analogs of the partial derivatives in the generators of an RE algebra, defined in Section 2. Moreover, in Section 6 we get analogs of the partial derivatives on the enveloping algebra
$U(gl_N)$ by passing to the limit $q\rightarrow 1$ in the corresponding structures of the RE algebra. As for the quantum adjoint vector fields, we show that they can be reduced to
the so-called ``quantum orbits", which are some quotients of the RE algebras. This is done in Section 5.
\section{Quantum doubles and partial derivatives}
\begin{equation}gin{definition}
A couple $(A, B)$ of unital associative algebras $A$ and $B$ is called {\it an associative double} if it is equipped with a linear invertible {\it permutation map}
$$
{\sigma}gma :A \otimes B\to B\otimes A,
$$
which meets the following requirements:
$$
{\sigma}gma\circ(\mu_{A}\otimesimes\mathrm{id}_B) =(\mathrm{id}_B\otimesimes\mu_{A})\circ{\sigma}gma_{12}\circ{\sigma}gma_{23} \quad\mathrm{on}\quad A\otimes A\otimes B,
$$
$$
{\sigma}gma\circ(\mathrm{id}_A\otimesimes\mu_{B}) =(\mu_{B}\otimesimes \mathrm{id}_A)\circ{\sigma}gma_{23}\circ{\sigma}gma_{12} \quad\mathrm{on}\quad A\otimes B\otimes B,
$$
$$
{\sigma}gma(1_A\otimes b)=b\otimes 1_A,\quad {\sigma}gma(a\otimes 1_B)=1_B\otimes a q^{-1}uad\forall\, a\in A,\, \forall\,b\in B.
$$
Here $\mu_A: A\otimes A\to A $ is the multiplication in the algebra $A$, $1_A$ is its unit element, and similarly for $B$.
\end{definition}
In general, if the algebras $A$ and $B$ are introduced via relations on their generators, the verification of the above properties reduces to checking that the map ${\sigma}$ preserves
the ideals defining these algebras. In this sense we say that the defining relations of both algebras are compatible with the permutation map ${\sigma}$.
Note, that an obvious and well-known example of such a permutation map is the flip ${\sigma}gma = P$:
$$
P(a\otimesimes b) = b\otimesimes a\quad \forall\, a\in A,\,b\in B.
$$
If the permutation map ${\sigma}gma$ is not a flip (or a super flip), we call the corresponding double {\it a quantum double} (QD).
By means of the permutation map ${\sigma}$ we can introduce a unital associative algebra which is isomorphic as a vector space to the tensor product $B\otimes A$.
For this we consider a free tensor algebra $\mathrm{T}(A\oplus B)$ and take its quotient over an ideal
generated by the multiplication rules in $A$, $B$ and by elements $a\otimesimes b -{\sigma}gma(a\otimesimes b)$, $\forall \, a\in A$, $\forall\,b\in B$. So, in the quotient algebra the
following relations take place:
\begin{equation}
a\otimesimes b = {\sigma}gma(a\otimesimes b).
{\lambda}bel{perm-rel}
\end{equation}
Below the equalities (\ref{perm-rel}) will be referred to as {\it the permutation relations}.
Moreover, if the algebra $A$ is equipped with an algebra homomorphism $\varepsilon_A :A\to {\Bbb C}$, then it becomes possible to define a linear action
$\triangleright: A\otimesimes B\rightarrow B$ of the algebra $A$ onto $B$:
\begin{equation}
a\triangleright b:=(\mathrm{id}_B\otimes \varepsilon_A) {\sigma}(a\otimes b),\quad \forall\, \,a\in A,\, b\in B.
{\lambda}bel{act}
\end{equation}
Note, that due to requirements imposed on the map ${\sigma}gma$ this action defines a representation of the algebra $A$ in the algebra $B$.
Consider now two examples of the QD which are of the main interest for us. As the first example we consider the QD $(A,B)$ of two RE algebras $A=hLL(R)$ and
$B={\cal{M}}(R)$ equipped with the following permutation map
\begin{equation}
{\sigma}gma:\,R_{12}hL_1 R_{12}\otimesimes M_1\to M_1 \otimesimes R_{12}hL_1 R^{-1}_{12}+R_{12}M_1\otimesimes 1_A.
{\lambda}bel{sii}
\end{equation}
Consequently, the corresponding permutation relations are (from now on we omit the tensor product signs and the unit elements):
\begin{equation}
R_{12}hL_1 R_{12} M_1= M_1R_{12} hL_1 R^{-1}_{12}+R_{12}M_1.
{\lambda}bel{dvaa}
\end{equation}
This QD was introduced in \cite{GPS2, GPS3}. The modified RE algebra $A= hLL(R)$ admits a homomorphism $\varepsilon: hLL(R)\rightarrow {\Bbb C}$ which on generators
of $hLL(R)$ is defined as $\varepsilon(hL)=0$, $\varepsilon(1_A) = 1$. Then, in accordance with (\ref{act}) we get the action of the algebra $hLL(R)$
onto ${\cal{M}}(R)$:
$$
hL_1 R_{12}\triangleright M_1=M_1.
$$
Note, that if we replace the Hecke symmetry $R$ to the flip $P$ in the permutation map (\ref{sii}), then the above action acquires the meaning of the left action of the enveloping
algebra\footnote{We assume the realization of $U(gl_N)$ as an algebra of right invariant differential opereators on $\mathrm{Sym}(gl_N)$.} $U(gl_N)$ onto the commutative algebra
$\mathrm{Sym}(gl_N)$.
Using the change (\ref{iso}) of the generating matrices, we can pass from the QD $(hLL(R), {\cal{M}}(R))$ to that $({\cal{L}}(R), {\cal{M}}(R))$. The permutation relations in
the latter QD take the form
\begin{equation}
R_{12}L_1R_{12} M_1 = M_1 R_{12} L_1R^{-1}_{12}.
{\lambda}bel{dva}
\end{equation}
Since $\varepsilon(L)=I$ we get the action
\begin{equation}
L_1R_{12}\triangleright M_1 = R^{-1}_{12}M_1.
{\lambda}bel{new-act}
\end{equation}
From the technical point of view this form of the QD is easier to deal with, so we use it below for a study of invariant operators.
As the second example consider the QD $(hLL(R), {\cal{M}}(R))$ composed of the same algebras but equipped with another permutation map $\tilde{\sigma}gma$ leading
to the following permutation relations:
\begin{equation}
R_{12}hL_1R_{12} M_1-M_1 R_{12}hL_1R_{12}= R_{12} M_1-M_1R_{12}.
{\lambda}bel{py}
\end{equation}
With the homomorhism $\varepsilon(hL)=0$ we get the action of the form:
$$
hL_1R_{12}\triangleright M_1 = M_1-R^{-1}_{12}M_1R_{12}.
$$
Now, if we pass to the classical limit, assuming that $R$ tends to $P$, the above action again turns into the action of the enveloping algebra $U(gl_N)$ onto commutative
algebra $\mathrm{Sym}(gl_N)$ but the generators of enveloping algebra should be realized as the adjoint vector fields.
Here, we can also express $hL$ via $L$ by means of the relation (\ref{iso}). Then we get a QD $({\cal{L}}(R), {\cal{M}}(R))$, equipped with the permutation relations of the form:
\begin{equation}
R_{12}L_1R_{12} M_1= M_1R_{12} L_1 R_{12}.
{\lambda}bel{pyat}
\end{equation}
As for the homomorphism $\varepsilon$, we always put $\varepsilon(L)=I$ for the generating matrix of the RE algebra ${\cal{L}}(R)$ and $\varepsilon(hL)=0$ for that of the modified
RE algebra $hLL(R)$.
Now, we go back to the QD $(hLL(R), {\cal{M}}(R))$ equipped with the permutation relations (\ref{dvaa}). As was shown in \cite{GPS3}, the matrix $D=\|{\partial}_i^j\|$, defined by
$D= M^{-1}hL$ satisfies the matrix relations:
$$
R^{-1}_{12}D_1 R^{-1}_{12}D_1=D_1 R^{-1}_{12}D_1 R^{-1}_{12}
$$
and
\begin{equation}
D_1R_{12}M_1R_{12}= R_{12}M_1R^{-1}_{12}D_1+ R_{12}.
{\lambda}bel{raz}
\end{equation}
The entries of the matrix $D$ generate an RE algebra\footnote{Observe that $R^{-1}$ is a braiding, subject to the Hecke condition with $q$ replaced by $q^{-1}$.} ${\cal D}(R^{-1})$
which forms a QD with the algebra ${\cal{M}}(R)$, equipped with the permutation relations (\ref{raz}). To define the action of ${\cal D}(R^{-1})$ onto ${\cal{M}}(R)$ we
put $\varepsilon(D)=0$.
We treat the entries of the matrix $D$ as quantum analogs of the usual partial derivatives and call them quantum partial derivatives (QPD). This treatment is motivated by the fact
that if $R=P$ the generators $m_i^j$ become commutative and the entries of the matrix $D$ become the partial derivatives ${\partial}_i^j={\partial}rtial/{\partial}rtial m_j^i$ while the permutation
relations (\ref{raz}) turn into the classical Leibniz rules. The QD $({\cal D}(R^{-1}), {\cal{M}}(R))$ becomes the Heisenberg-Weyl algebra with the coordinates $(m_i^j)_{1\le i,j\le N}$ and the
corresponding partial derivatives $({\partial}rtial_i^j)_{1\le i,j\le N}$.
In the last section the QD $({\cal D}(R^{-1}), {\cal{M}}(R))$ will be used for a definition of analogs of the partial derivatives on the enveloping algebra $U(gl_N)$.
We complete this section with one more example of a QD. Let us take $A={\cal{L}}(R)$ again. As the algebra $B$ we choose the free tensor algebra $T(V)$ of the space $V$. Define
the permutation relations as follows:
\begin{equation}
R_{12} L_1 R_{12} \,x_1=x_1L_2.
{\lambda}bel{perr}
\end{equation}
On taking $\varepsilon(L) = I$, we get the action $L_1R_{12}\triangleright x_1=R_{12}^{-1}\,x_1$. Note that the permutation relations (\ref{perr}) can be reduced
to the $R$-symmetric and $R$-skew-symmetric algebras of $V$ defined respectively by
$$
\mathrm{Sym}_R(V)=T(V)/{\lambda}ngle \mathrm{Im}(q I-R)\rangle,q^{-1}uad \Lambda_R(V)=T(V)/{\lambda}ngle \mathrm{Im}(q^{-1} I+R)\rangle.
$$
\section{Quantum left vector fields and the invariant operators}
Let $hL$ be the generating matrix of the enveloping algebra $U(gl_N)$, that is the matrix $hL$ meets the relation (\ref{mRE}) with $R=P$. It is well known that for all integers $k\ge 1$
the {\it power sums} $\mathrm{Tr} hL^k$ belong to the center of $U(gl_N)$. Therefore, by the Schur lemma in any irreducible finite dimensional $U(gl_N)$-module $V_{\lambda}mbda$ the
images of $\mathrm{Tr} hL^k$ are scalar operators. Note that any such a module $V_{\lambda}mbda$ is labelled by a partition ${\lambda}=({\lambda}_1\geq {\lambda}_2\geq\dots \geq{\lambda}_N)$. The eigenvalues
of the mentioned scalar operators were computed in \cite{PP}.
Let us recall a way of constructing the $U(gl_N)$-modules $V_{\lambda}mbda$. We start from the finite dimensional complex vector space $V$, $\dim_{\,\Bbb C}V = N$. In this space the
standard irreducible fundamental vector representation is realized.
Then for any integer $k\ge 1$ we consider the tensor power $V^{\otimesimes k}$. This space is decomposed into the direct sum of irreducible $U(gl_N)$-modules:
$$
V^{\otimesimes k} = \bigoplus_{{\lambda}mbda\vdash k}\bigoplus_{T_{\lambda}mbda}P_{T_{\lambda}mbda}V^{\otimesimes k}.
$$
Here $T_{\lambda}mbda$ denotes a standard Young table of the Young diagram, corresponding to a partition ${\lambda}mbda\vdash k$. The orthonormal projection operators $P_{T_{\lambda}mbda}$
are images of primitive idempotents $E_{T_{\lambda}mbda}$ of the group algebra ${\Bbb C}[S_k]$ of the symmetric group $S_k$ under the representation of $S_k$ in $V^{\otimesimes k}$
where elements of the group $S_k$ act by permutations of factors of the tensor product $V^{\otimesimes k}$.
The subspaces $V_{T_{\lambda}mbda} = P_{T_{\lambda}mbda}V^{\otimesimes k}$ are irreducible $U(gl_N)$-modules. For different tables $T_{\lambda}mbda$ and $T'_{\lambda}mbda$ of the same diagram ${\lambda}mbda$
these modules are isomorphic to each other and can be treated as different embeddings of the module $V_{\lambda}mbda$ into the space $V^{\otimesimes k}$:
$$
V_{T_{\lambda}mbda}{\sigma}meq V_{T'_{\lambda}mbda}{\sigma}meq V_{\lambda}mbda.
$$
In what follows we assume the Hecke symmetry $R$ to be a deformation of the usual flip. In this case the corresponding modified RE algebra $hLL(R)$ is a deformation of the
enveloping algebra $U(gl_N)$. Moreover, the category of finite dimensional $U(gl_N)$-modules can be also deformed into that of $hLL(R)$-modules. Irreducible objects of this category can be constructed in the same way with the use of the Hecke algebras $H_k(q)$ instead of ${\Bbb C}[S_k]$. Namely, for a generic $q$ in the Hecke algebra $H_k(q)$ there are analogous idempotents $E_{T_{\lambda}mbda}$ (we keep the same notation for them) such that $P_{T_{\lambda}}(R)V^{\otimes k}$ are irreducible $hLL(R)$-modules. Here
$P_{T_{\lambda}mbda}(R) = r_\hho_R(E_{T_{\lambda}mbda})$ and $r_\hho_R:H_k(q)\rightarrow \mathrm{End}(V^{\otimesimes k})$ is an $R$-matrix representation of the Hecke algebra in the space
$V^{\otimesimes k}$. This representation sends an Artin generator $\tau_i$ of the Hecke algebra to $R_i=I^{\otimesimes (i-1)}\otimesimes R\otimesimes I^{\otimesimes(k-i+1)}$. We refer the reader to
(\cite{GPS1}) for detail and to \cite{OP} for an explicit construction of the primitive idempotents $E_{T_{\lambda}mbda}$ in the Hecke algebra.
An important role in the theory of the Hecke algebras belongs to the so-called Jusys-Murphy elements which generates the maximal commutative subagebra in the Hecke algebra
$H_k(q)$. In the $R$-matrix representation they form a set of mutually commutative linear operators $J_i$, $1\le i\le k$, in the space $V^{\otimesimes k}$:
$$
J_1=\mathrm{Id}_{V^{\otimesimes k}},\quad J_i=R_{i-1}J_{i-1}R_{i-1},\quad 2\leq i\leq k.
$$
Introduce also the notation for some ``matrix copies'' of the generating matrix $L$:
$$
L_{\overline 1}= L_{\underline 1}=L_{1},\quad L_{\overline k}=R_{k-1}L_{\overline {k-1}}\,R^{-1}_{k-1},\quad L_{\underline k}=R^{-1}_{k-1}L_{\underline {k-1}}R_{k-1}.
$$
With the above notation we can write the action (\ref{new-act}) in the form:
\begin{equation}
L_{\underline 2}\triangleright M_1=J_2^{-1} M_1.
{\lambda}bel{JM}
\end{equation}
On applying the permutation relations (\ref{dva}) one can generalize this formula on arbitraty ``monomials'' in generators of ${\cal{M}}(R)$:
\begin{equation}
L_{\underline{k+1}}\triangleright M_1\dots M_{\overline k}=J_{k+1}^{-1} M_1\dots M_{\overline k}.
{\lambda}bel{dvaaa}
\end{equation}
Calculating the $R$-trace at the $(k+1)$-th space and taking into account the formula
\begin{equation}
\mathrm{Tr}_{R(k+1)} X_{\overline {k+1}}=\mathrm{Tr}_{R(k+1)} X_{\underline {k+1}}= I_{1\dots k} \mathrm{Tr}_{R} \,X,
{\lambda}bel{syst}
\end{equation}
where $X$ is an arbitrary $N\times N$ matrix, we get the following claim (see \cite{S}).
\begin{equation}gin{proposition} {\lambda}bel{prop:2}
Let ${\lambda}mbda\vdash k$ be a partition and $T_{\lambda}mbda$ be a standard Young table corresponding to the Young diagram of the partition ${\lambda}mbda$.
Then the following relation holds
\begin{equation}
\mathrm{Tr}_R L\triangleright P_{T_{\lambda}mbda}(R) M_1M_{\overline 2}\dots M_{\overline k} =\chi_{{\lambda}} (\mathrm{Tr}_R L)P_{T_{\lambda}mbda}(R)M_1M_{\overline 2}\dots M_{\overline k}, {\lambda}bel{spec}
\end{equation}
where
\begin{equation}
\chi_{{\lambda}} (\mathrm{Tr}r L)=\frac{N_q}{q^{N}}-\frac{\nu}{q^{2N}} \sum_{i=1}^k \, q^{-2c(i)},\quad \nu = q-q^{-1},
{\lambda}bel{dec}
\end{equation}
and the sum is taken over all boxes of the table $T_{\lambda}mbda$. Here $c(i)=n-m$ is the content of the box in which the integer $i$ is located, that is $n$ (respectively $m$) is the
number of the corresponding column (row).
\end{proposition}
Note that the symbol of the table $T$ in the notation $\chi_{\lambda}$ would be irrelevant since the right hand side of (\ref{dec}) actually depens only on the {\it diagram} ${\lambda}mbda$ but not
on the table.
\begin{equation}gin{remark}
Let us point out that formula (\ref{dvaaa}) is the matrix form of the action of the left quantum vector fields on the algebra ${\cal{M}}(R)$. In fact, it is similar to the action of the algebra
${\cal{L}}(R)$ onto the free tensor algebra $T(V)$ considered at the end of the previous section. Namely, using the permutation relations (\ref{perr}) and the homomorphism
$\varepsilon(L) = I$ we can get the action
\begin{equation}
L_{\underline{k+1}}\triangleright x_1\otimes\dots \otimes x_{ k}=J_{k+1}^{-1}x_1\otimes\dots \otimes x_{ k}.
{\lambda}bel{tri}
\end{equation}
Evidently, a formula similar to (\ref{spec}) is also valid in this case.
\end{remark}
We are interested in the spectral analysis of the invariant operators, that is these corresponding to the central elements of the algebra ${\cal{L}}(R)$ acting onto ${\cal{M}}(R)$. The proposition
\ref{prop:2} describes the spectrum of the operator $\mathrm{Tr}r L$, corresponding to the lowest power sum.
Below, we consider some other examles of central elements and describe the spectrum of the corresponding operators in invariant subspaces $\mathrm{Im}(P_{T_{\lambda}mbda})$ extracted
by the projection operators $P_{T_{\lambda}mbda}(R)$. Besides the higher power sums $\mathrm{Tr}_RL^k$, $k\ge 2$, we are interested in the so-called quantum elementary symmetric polynomials $e_k(L)$, which are defined by the formula:
$$
e_k(L)=\mathrm{Tr}_{R(1\dots k)} (A^{(k)} L_1 L_{\overline 2}\dots L_{\overline k}),\quad k \ge 1.
$$
Here $A^{(k)}$ are the $R$-skew-symmetrizers introduced by the following recursion formula:
$$
A^{(1)}=I,\quad A^{(k)}=\frac{1}{k_q}A^{(k-1)}\left(q^{k-1} I-(k-1)_q \, R_{k-1}\right)A^{(k-1)},\quad k\ge 2,
$$
where $k_q=(q^k-q^{-k})/(q-q^{-1})$ is a $q$-integer. Below we assume $q^{2k}\not=1$ for all integer $2\le k\le N$, and, therefore, $k_q\not=0$. Any $R$-skew-symmetrizer $A^{(k)}(R)$ is a particular case of the above idempotents $P_{T_{\lambda}mbda}(R)$, corresponding to one-column diagram ${\lambda}=(1^k)$.
The elementary symmetric polynomials possess a natural and convenient parameterization in terms of the {\it quantum eigenvalues} $\{\mu_i\}_{1\le i\le N}$ of the generating
matrix $L$. These eigenvalues belong to an algebraic extension of the center of the RE algebra ${\cal{L}}(R)$. They are defined via the quantum Cay\-ley-Ha\-mil\-ton
identity on the matrix $L$:
\begin{equation}
L^N-q e_1(L) L^{N-1}+ q^2e_2(L) L^{N-2}+\dots +(-q)^{N} e_{N}(L) I=0.
{\lambda}bel{CH}
\end{equation}
Let us introduce $N$ elements $\mu_i$, $1\le i\le N$, which satisfy a system of $N$ polynomial equations with central elements on the left hand side:
\begin{equation}
q^k e_k(L)=\sum_{1\leq i_1<\dots <i_k\leq N} \mu_{i_1}\dots \mu_{i_k}, \quad 1\le k\le N.
{\lambda}bel{elem-mu}
\end{equation}
The elements $\mu_i$ are also assumed to be central. They are interpreted as quantum analogs of eigenvalues of the matrix $L$ since in virtue of (\ref{elem-mu}) the
Cayley-Hamilton identity (\ref{CH}) can be rewritten in the following factorized form:
\begin{equation}
\prod_{i=1}^N(L-\mu_iI) = 0.
{\lambda}bel{factor-CH}
\end{equation}
In terms of the quantum eigenvalues $\mu_i$ one can express all central symmetric polynomials of the RE algebra --- the power sums, the quantum Schur functions and so on
(see \cite{GPS1,GPS4} for more detail). For example, the power sums in terms of quantum spectrum read:
\begin{equation}
\mathrm{Tr}r L^k=\sum \mu_i^k \, d_i,q^{-1}uad d_i=q^{-1} \prod_{j\not=i}^N \frac{\mu_i-q^{-2}\mu_j}{\mu_i-\mu_j}.
{\lambda}bel{powe}
\end{equation}
Since the elements $\mu_i$ are central, then in any irreducible ${\cal{L}}(R)$-module they are represented by scalar operators. These operators must be compatible with relations
(\ref{elem-mu}). In the paper \cite{GPeS1} we suggested the following conjecture\footnote{In the cited paper the conjecture was formulated for a wider class of Hecke symmetries,
namely those of the bi-rank $(m|0)$. (For the notion of the bi-rank the reader is referred to \cite{GPS1}.) The Hecke symmetries we are dealing with in the present paper is of the
bi-rank $(N|0)$.}.
\begin{equation}gin{conjecture}
In ${\cal{L}}(R)$-invariant subspaces $V_{T_{\lambda}mbda} = \mathrm{Im}P_{T_{\lambda}mbda}(R)$ the central elements $\mu_i$ $1\le i\le N$ are represented (after a proper ordering) by the
following scalar operators:
\begin{equation}
\mu_i\mapsto \chi_{\lambda}mbda(\mu_i)\,\mathrm{Id}_{V_{T_{\lambda}mbda}},q^{-1}uad \chi_{\lambda}(\mu_i)=q^{-2({\lambda}_i+N-i)},
{\lambda}bel{mu-char}
\end{equation}
where ${\lambda}mbda_i$ are elements of a partition ${\lambda}mbda = ({\lambda}mbda_1,{\lambda}mbda_2,\dots ,{\lambda}mbda_N)$.
The representation (\ref{mu-char}) is compatible with the scalar operators corresponding to the quantum elementary symmetric polynomials
$e_k(L)\mapsto \chi_{\lambda}mbda(e_k)\,\mathrm{Id}_{V_{T_{\lambda}mbda}}$, that is:
$$
q^k\,\chi_{{\lambda}}(e_k)=\sum_{1\leq i_{1}<\dots < i_k \leq N} \chi_{{\lambda}}(\mu_{i_1})\dots \chi_{{\lambda}}(\mu_{i_k}).
$$
\end{conjecture}
For the lowest elementary symmetric polynomial $e_1(L) = \mathrm{Tr}_R(L) = q^{-1}\sum_i\mu_i$ this conjecture is easily verified with the use of (\ref{dec}).
We have also checked it for $e_2(L)$ by a direct computation.
Turn now to the modified RE algebra $hLL(R)$ whose generators $hat l_i^j$ are treated as quantum analogs of the right-invariant vector fields. The corresponding Casimir
operators are $\mathrm{Tr}r hL^k$. The most convenient way to perform their spectral analysis is to introduce the quantum eigenvalues $hat\mu_i$ of the generating matrix $hat L$.
Taking into account relation (\ref{iso}), we can rewrite the factorized Cayley-Hamilton identity (\ref{factor-CH}) in the form of the matrix identity for $hat L$:
$$
\prod_{i=1}^N(L-\mu_i I) = 0q^{-1}uad \Rightarrowq^{-1}uad \prod_{i=1}^N(hat L-hat \mu_i I) = 0,
$$
where the quantum eigenvalues $hat \mu_i$ of the generating matrix $hat L$ are connected with those of $L$ by a linear shift:
\begin{equation}
\mu_i=1-\nu hat \mu_i,q^{-1}uad \nu = q-q^{-1}.
{\lambda}bel{mu-shift}
\end{equation}
Here the symbol $1$ stands for the unit element of the RE algebra.
As a direct consequence of (\ref{mu-shift}) we obtain:
$$
\chi_{\lambda}({hat{\mu}_k})=\frac{1- q^{-2\, ({\lambda}_k+N-k)}}{q-q^{-1}} = q^{-({\lambda}mbda_k+N-k)}({\lambda}mbda_k+N-k)_q,\quad 1\le k \le N.
$$
By taking the limit $q\rightarrow1$, we get the spectral values of $U(gl_N)$ generating matrix $hL$ in the irreducible module $V_{\lambda}mbda$:
$$
\chi_{\lambda}({hat{\mu}_k})={\lambda}_k+N-k,\quad 1\le k\le N.
$$
Note that this is in the full agreement with the results of \cite{PP}.
\section{Different forms of the Capelli identity}
Consider the QD $({\cal D}(R^{-1}), {\cal{M}}(R))$ of two RE algebras generated by the enties of matrices $D$ and $M$ and equipped with the permutation relations (\ref{raz}).
Introduce the matrix $hat L=MD$. As was argued in Section 2, its entries generate the modified RE algebra $hLL(R)$ and the permutation relations with entries of the matrix $M$
are given by (\ref{dvaa}).
One of the remarkable properties of the QD under consideration are the following quantum {\it matrix Capelli identities}\footnote{If $R$ is a deformation of the flip $P$ then at the
classical limit $q\rightarrow 1$ these identities transform into those presented in \cite{Ok2}.}.
\begin{equation}gin{theorem} {\rm (\cite{GPeS2}) }
{\lambda}bel{th:1}
For $\forall\,k\ge 1$ the following matrix identity takes place:
\begin{equation}
A^{(k)}hat L_{\overlineerline 1}\,(hat L_{\overline 2}\,+q I)\dots (hat L_{\overline k}\,+q^{k-1}(k-1)_qI\,) \,A^{(k)}=
q^{k(k-1)}A^{(k)}M_{\overlineerline 1}\dots M_{\overline k}\, D_{\overline k}\dots D_{\overlineerline 1}.
{\lambda}bel{th}
\end{equation}
\end{theorem}
Let us point out that identities (\ref{th}) are valid for a wide class of the Hecke symmetries: to prove (\ref{th}) one needs only the braid relation and the Hecke condition (\ref{H-cond})
on $R$. If we impose some additional restrictions on $R$ we can obtain further consequences of (\ref{th}), namely, the quantum version of the determinant Capelli identity.
Thus, assume $R$ to be a deformation of the usual flip $P$, then the $R$-skew-symmetrizer $A^{(N)}$ is a unit rank projector
\begin{equation}
\mathrm{dim}\,\mathrm{Im}\, A^{(N)}=1
{\lambda}bel{unit-rank}
\end{equation}
and $A^{(N+1)}\equiv 0$ (recall that $N=\dim\, V$). A direct consequence of (\ref{unit-rank}) is the existence of two tensors $|u\rangle= \|u_{i_1i_2\dots \,i_N}\|$
and ${\lambda}ngle v| = \|v^{i_1i_2\dots\,i_N}\|$ such that
$$
A^{(N)} = |u\rangle\otimesimes{\lambda}ngle v|, q^{-1}uad {\lambda}ngle v| u\rangle = \sum_{\{i\}} v^{i_1i_2\dots\,i_N}u_{i_1i_2\dots\,i_N} = 1.
$$
The structure tensors $|u\rangle$ and ${\lambda}ngle v|$ allow one to define a quntum analog of the determinant for the generatig matrix of the RE algebra. Namely, we set
$$
{\deltat}_R\,M ={\lambda}ngle v| M_1M_{\overlineerline 2}\dots M_{\overlineerline N} |u\rangle,q^{-1}uad {\deltat}_{R^{-1}}\,D = {\lambda}ngle v| D_{\overline N}\dots D_{\overlineerline 2}D_1)|u \rangle .
$$
With these definitions we can formulate the following corollary of Theorem \ref{th:1}.
\begin{equation}gin{corollary} {\lambda}bel{cor:6}
If $R$ is a deformation of the flip $P$, then the following quantum Capelli identity takes place:
\begin{equation}
\mathrm{Tr}_{R(1\dots N)}A^{(N)}hat L_{\overlineerline 1}\,(hat L_{\overline 2}\,+q I)\dots (hat L_{\overline N}\,+q^{N-1}(N-1)_qI\,) =
q^{-N}{\deltat}_R\, M\, {\deltat}_{R^{-1}}\, D.
{\lambda}bel{dettt}
\end{equation}
\end{corollary}
\begin{equation}gin{remark}
Note that this corollary remains valid mutatis mutandis if the initial Hecke symmetry $R$ is of bi-rank $(m|0)$, $m\leq N$, the corresponding examples were constructed in \cite{G}.
This condition means that $\mathrm{rank}\,A^{(m)}=1$ and $A^{(m+1)} \equiv 0$. In this case we only have to replace $N$ by $m$ in formula (\ref{dettt}), while the structure tensors
$|u\rangle$ and ${\lambda}ngle v|$ should be extracted from the projector $A^{(m)}$.
\end{remark}
In conclusion of the section we note that another version of the quantum Capelli identity was suggested in \cite{NUW}. This version is related to the RTT algebra and is valid for $R$
coming from the QG $U_q(sl_N)$.
\section{Quantum adjoint vector fields and quantum orbits}
In this section we deal with the QD $(hLL(R), {\cal{M}}(R))$, endowed with the permutation relations (\ref{py}). Our main objective here is to show that the action of the modified RE algebra
$hLL(R)$ on ${\cal{M}}(R)$ can be reduced onto some quotients of the algebra ${\cal{M}}(R)$. These quotient algebras are treated as quantum analogs of the coordinate rings of the
$GL(N)$-orbits in $gl_N^*$. Observe that we employ the term ``orbit" in a loose sense since even in the classical case these quotients are ``orbits" for generic matrices\footnote{This
means that the eigenvalues of the matrix are pairwise distinct.} generating the orbit. The question which quantum orbits can be considered as ``generic" is discussed below. The
generators $hat l_i^j$ are treated as quantum adjoint vector fields.
\begin{equation}gin{proposition}{\lambda}bel{prop:8}
In the QD $(hLL(R), {\cal{M}}(R))$ the following relations hold for any integer $k\ge 1$
$$
hL\, \mathrm{Tr}r( M^k)=\mathrm{Tr}r( M^k)hL
$$
and consequently
\begin{equation}
hL\triangleright \mathrm{Tr}r M^k=0.
{\lambda}bel{hL-act}
\end{equation}
\end{proposition}
\noindent
{\bf Proof.}
Successively multiplying the both sides of matrix equality (\ref{py}) by the matrix $M_1$ from the right and transforming the term $M_1R_{12}hat L_1 R_{12}M_1$
with the use of (\ref{py}) we come to the result:
$$
R_{12}hat L_1R_{12} M^k_1 - M_1^kR_{12}hat L_1 R_{12} = R_{12}M_1^k - M_1^kR_{12}.
$$
Then we multiply this equality by $R^{-1}_{12}$ from the left and right and apply the $R$-trace in the second space. These operations lead to the desired result
in virtue of (\ref{syst}):
$$
hL_1 \mathrm{Tr}r M^k - \mathrm{Tr}r M^khL_1 = 0.
$$
To get the action (\ref{hL-act}) we apply the homomorphism $\varepsilon(hat L) = 0$ to the second term.hfill\rule{6.5pt}{6.5pt}
So, the quantum adjoint vector fields kill all elements $\mathrm{Tr}r M^k$. Note that these elements are central in the RE algebra ${\cal{M}}(R)$.
Now, consider the following quotient algebra:
$$
\mathcal{O}({\alpha}_1,\dots ,{\alpha}_N)={\cal{M}}(R)/{\lambda}ngle \mathrm{Tr}r M-{\alpha}_1, \, \mathrm{Tr}r M^2-{\alpha}_2,\dots , \mathrm{Tr}r M^N-{\alpha}_N\rangle,
$$
where ${\alpha}_1,\dots ,{\alpha}_N$ are some complex constants and the notation ${\lambda}ngle J\rangle$ stands for the ideal, generated by a subset $J$ of a given algebra.
We call these quotients the ``quantum orbits".
The quantities ${\alpha}_i$ can be expressed in terms of the eigenvalues $\mu_i$ of the matrix $M$ since for this matrix formula (\ref{powe}) is also valid. In the paper \cite{GS1} we
showed (up to a conjecture on the quantum de Rham complex) that the quantum orbits are generic if
\begin{equation}
\mu_i\not=q^2\mu_j,\quad\forall \, i,j.
{\lambda}bel{orb}
\end{equation}
Observe that by definition a quantum orbit is generic if its cotangent module is projective. Thus, as was shown in \cite{GS1}, under the condition (\ref{orb}) and under the assumption
that the mentioned conjecture is true the cotangent module on the quantum orbit $\mathcal{O}({\alpha}_1,\dots ,{\alpha}_N)$ is projective.
\section{Quantum partial derivatives on $U(gl_N)$ background}
In this section we consider one more application of the quantum left vector fields. Namely, we show how to get analogs of the partial derivatives on the enveloping algebra $U(gl_N)$
via the QD introduced at the end of Section 2.
To this purpose we return to the QD $({\cal D}(R^{-1}), {\cal{M}}(R))$, endowed with the permutation relations (\ref{raz}). Let us introduce a new generating matrix $N$ of the RE algebra ${\cal{M}}(R)$ by the shift analogous to (\ref{iso}):
$$
M=h\, I-(q-q^{-1})\, N,q^{-1}uad h\in{\Bbb C}.
$$
The full list of the relations in the QD $({\cal D}(R^{-1}), hat{\cal N}_h(R))$, generated by the entries of matrices $D$ and $N$, is as follows
$$
R_{12} N_1R_{12} N_1-N_1R_{12} N_1R_{12}=h(R_{12} N_1-N_1 R_{12}),
$$
$$
R^{-1}_{12} D_1 R^{-1}_{12} D_1=D_1R^{-1}_{12} D_1 R^{-1}_{12},
$$
\begin{equation}
D_1R_{12}N_1R_{12}- R_{12}N_1R^{-1}_{12}D_1=R_{12}+h D_1R_{12}.
{\lambda}bel{q-Leib-rule}
\end{equation}
We introduce the deformation parameter $h$ in order to present the modified RE algebra $hat{\cal N}_h(R)$ as a deformation of the RE algebra ${\cal{M}}(R)$, which corresponds to the
value $h=0$.
The entries of the matrix $D=\|{\partial}_i^j\|_{1\leq i,j \leq N}$ can be treated as analogs of partial derivatives in $n_i^j$ and we call them the quantum partial derivatives (QPD). Using the
homomorphism $\varepsilon(D) = 0$, we get the action of the QPD on the generators of $hat{\cal N}_h(R)$:
\begin{equation}
D_1\triangleright N_{\overlineerline 2} = R_{12}^{-1}.
{\lambda}bel{lin-act}
\end{equation}
The relation (\ref{q-Leib-rule}) is a substitution of the Leibniz rule allowing one to extend the action of QPD (\ref{lin-act}) from generators onto the whole algebra $hat{\cal N}_h(R)$.
If $R$ is a deformation of the flip $P$ then at the limit $q\rightarrow 1$ relations (\ref{lin-act}) turn into the classical action of the partial derivatives on the generators of the commutative
algebra\footnote{However, the action of the QPD on higher polynomials in $n_k^l$ differs from the classical formulae and contains corrections depending on $h$.
The full classical picture restores after specializing $h=0$.} $\mathrm{Sym}(gl_N)$:
$$
{\partial}rtial_i^j\triangleright n_k^s = \deltalta_i^s\,\deltalta_k^j.
$$
This is the reason for treating the QPD ${\partial}rtial_i^j$ as ``derivatives'' in generators $n_j^i$.
In general, if the symmetry $R$ is involutive, it is useful to introduce the matrix $hD=D+h^{-1}\, I$. In terms of this matrix the permutation relations (\ref{q-Leib-rule}) take the form:
$$
hD_1R_{12}N_1R_{12} - R_{12}N_1R_{12}hD_1= h hD_1R_{12}
$$
We call the entries of the matrix $hD=\|hpa_i^j\|$ the shifted QPD\footnote{ In fact, only the diagonal elements of the matrix $D$ are shifted.}.
From now on we assume that $R=P$. In this case the permutation relations for the generators of ${\cal N}_h(R)$ turn into those of the universal enveloping algebra $U(gl(N)_h)$:
$$
N_2N_1-N_1N_2 = h(P_{12}N_1 - N_1P_{12}).
$$
The permutation relations (\ref{q-Leib-rule}) give rise to the Leibnitz rule for the QPD, expressed via the coproduct defined on the shifted QPD
$$
\Delta(hpa_i^j)=hpa_k^j\otimes hpa_i^k,
$$
and completed with $\varepsilon(hpa_i^j)=h^{-1}\, \delta_i^j$.
Let us consider the example $N=2$ in more detail. It is convenient to pass to the compact form $u(2)_h$ of the algebra $gl(2)_h$. The Lie algebra $u(2)_h$ is generated by 4 elements
$x$, $y$, $z$ and $t$ with the following Lie brackets:
$$
[x, \, y]=h z,q^{-1}uad [y, \, z]=h x,q^{-1}uad[z, \, x]=h y,q^{-1}uad [t, \, x]=[t, \, y]=[t, \, z]=0.
$$
The corresponding QPD ${\partial}_x$, ${\partial}_y$, ${\partial}_z$ and $hat{\partial}_t$ (we use the ``shifted" derivative in $t$: $hpa_t = {\partial}_t + 2/h$) commute with each other,
while the permutation relations with the $u(2)_h$ generators read:
\begin{equation}
\begin{equation}gin{array}{l@{\quad}l@{\quad}l@{\quad}l}
hpa_t\,t - t\,hpa_t = \frac{h}{2}\,hpa_t & hpa_t\, x - x\,hpa_t
=-\frac{h}{2}\,{\partial}_x &
hpa_t\, y - y\, hpa_t=-\frac{h}{2}\,{\partial}_y &hpa_t\, z - z\,hpa_t=- \frac{h}{2}\,{\partial}_z\\
\rule{0pt}{7mm}
{\partial}_x\, t - t\,{\partial}_x = \frac{h}{2}\,{\partial}_x &{\partial}_x \,x - x\,{\partial}_x = \frac{h}{2}\,hpa_t &
{\partial}_x \, y- y\,{\partial}_x = \frac{h}{2}\,{\partial}_z & {\partial}_x \,z - z\, {\partial}_x = - \frac{h}{2}\,{\partial}_y \\
\rule{0pt}{7mm}
{\partial}_y \,t - t \, {\partial}_y = \frac{h}{2}\,{\partial}_y & {\partial}_y \,x - x\, {\partial}_y = -\frac{h}{2}\,{\partial}_z &
{\partial}_y \,y - y \, {\partial}_y = \frac{h}{2}\,hpa_t & {\partial}_y \,z - z \, {\partial}_ y= \frac{h}{2}\,{\partial}_x\\
\rule{0pt}{7mm}
{\partial}_z \,t - t \,{\partial}_z = \frac{h}{2}\,{\partial}_z & {\partial}_z \,x - x \,{\partial}_z = \frac{h}{2}\,{\partial}_y&
{\partial}_z \,y - y\,{\partial}_z = -\frac{h}{2}\,{\partial}_x & {\partial}_z \,z - z \,{\partial}_z = \frac{h}{2}\,hpa_t.
\end{array}
{\lambda}bel{leib}
\end{equation}
Introduce the matrix $hatt$, composed of the partial derivatives:
\begin{equation}
{hatt}=ihh \left(\begin{equation}gin{array}{rrrr}
hpa_t&{\partial}_x&{\partial}_y&{\partial}_z\\
-{\partial}_x&hpa_t&-{\partial}_z&{\partial}_y\\
-{\partial}_y&{\partial}_z&hpa_t&-{\partial}_x\\
-{\partial}_z&-{\partial}_y&{\partial}_x&hpa_t
\end{array} \right),
{\lambda}bel{seven}
\end{equation}
where $hh=h/2i$.
By $hatt(a)$ we denote the matrix whose entries result from applying the corresponding partial derivatives to an element $a\in U(u(2)_h)$. Thus, we get a linear map of associative
algebras:
$$
U(u(2)_h)\to \mathrm{Mat}_4(U(u(2)_h)) = \mathrm{End}({\Bbb C}^{\,4})\otimesimes U(u(2)_h)
$$
which will be also denoted by the symbol $hatt$:
\begin{equation}
hatt: \, a\mapsto hatt(a)\quad \, \forall\, a\in U(u(2)_h).
{\lambda}bel{mapp}
\end{equation}
We state (see \cite{GS3}) that the map $hatt$ is actually a homomorphism of associative algebras, that is
\begin{equation}
hatt(ab)=hatt(a) \cdot hatt(b),\quad \forall\,a,b\in U(u(2)_h).
{\lambda}bel{mult}
\end{equation}
Consequently, the map $hatt$ defines a representation of the algebra $U(u(2)_h)$:
$$
hatt(x)\cdothatt(y)-hatt(y)\cdothatt(x)=hhatt(z)
$$
and so on. In virtue of the above properties the map $hatt$ is treated as the Leibniz rule for the QPD on the algebra $U(u(2)_h)$.
We want to stress the usefulness of this form of the Leibniz rule. So far, the QPD were defined on the algebra of polynomials in noncommutative generators. Our next aim is to
extend their action onto some elements of a central extension of this algebra. Namely, consider the so-called {\em quantum radius} $r_\hh$:
$$
r_\hh^2=x^2+y^2+z^2+hh^2,
$$
which is an element of a central extension of the algebra $U(u(2)_h)$. We want to extend the action of the QPD onto $r_\hh$. As a reasonable criterium of such an extension we require
that the property (\ref{mult}) would be preserved on the extended algebra. Namely, we require the relation
$$
hatt(r_\hh)^2=hatt(r_\hh^2)=hatt(x^2+y^2+z^2+hh^2)
$$
to be fulfilled.
As follows from computation of \cite{GS3}, the result of applying the map $hatt$ to the quantum radius $r_\hh$ is
$$
{hatt}(r_\hh)=\frac{r_\hh^2+hh^2}{r_\hh}\, I+\frac{ihh}{r_\hh}\, M,
$$
where the matrix $M$ is of the form:
$$
M=
\left(\!\!
\begin{equation}gin{array}{cccc}
0 & x&y&z\\
-x&0&-z&y\\
-y&z& 0 & -x\\
-z&-y& x & 0
\end{array}\right).
$$
Now we are able to compute the action of the QPD on the quantum radius:
$$
{\partial}_t r_\hh=-\frac{ihh}{r_\hh},q^{-1}uad {\partial}_xr_\hh=\frac{x}{r_\hh},q^{-1}uad {\partial}_yr_\hh=\frac{y}{r_\hh},q^{-1}uad {\partial}_zr_\hh=\frac{z}{r_\hh}.
$$
Observe that these formulae possess the usual classical limit as $hh\to 0$ and, consequently, $r_\hh\to r$.
Also, in \cite{GS4} the following problem was discussed: how to extend the action of the QPD on the skew-field $B[B^{-1}]$, where $B$ is a central extension of the algebra
$U(u(2)_h)$. Assuming that the above Leibniz rule is still valid on this skew-field, we have:
$$
hatt (a)hatt (a^{-1})=hatt (e)=I.
$$
Thus, in order to extend the action of the partial derivatives on the skew-field $B[B^{-1}]$ it suffices to invert the matrix $hatt (a)$. In \cite{GS4} we exhibited an example,
in which such inverting was performed with the use of the Cayley-Hamilton identity for the matrix $hatt (a)$.
In conclusion, we note that defining QPD on the extended algebra $U(u(2)_h)$ enables us to introduce some dynamical models on this algebra. Certain of them were considered in
\cite{GS2, GS3}.
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\end{document}
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