set
stringclasses 1
value | id
stringlengths 5
9
| chunk_text
stringlengths 1
115k
| chunk_num_tokens
int64 1
106k
| document_num_tokens
int64 58
521k
| document_language
stringclasses 2
values |
|---|---|---|---|---|---|
train
|
0.47.73
|
\insubsubhead Dunford\,--\,Pettis property of $\,\mLrs42^1(\kern0.37mm\mu\kern0.15mm)$ \label{Ss Dun-Pet}
When treating the reflexive case in the proof of
Theorem \nfss A\,\ref{main Th}\kern0.15mm, we need to know that \mathss03{
\mvLrs42^1(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm) } has the Dunford\,--\,Pettis property,
or in our terminology introduced below, is a {\eightsl DP}{\sl\,--\,space\kern0.15mm}.
This is equivalent to \math{\mLrs42^1(\kern0.37mm\mu\kern0.15mm) } being a
\erm{DP}\,--\,space, and for this reason we here consider this matter to some
extent. Although we shall need the result only for positive measures \math{\mu}
with \mathss36{\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\bigcup\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\mu < \lower1.05mm\hbox{$^+$}\infty}, \,we anyhow
consider the situation for general positive measures.
In what follows, note
that \math{\pi} is said to be a {\it probability measure\kern0.37mm} if{}f \math{
\pi} is a positive measure with \mathss34{\pi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\bigcup\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\pi=1}.
\begin{lemma}\label{Le L^p(m)=L^p(p)}
Let $\,1\le p\le\lower1.05mm\hbox{$^+$}\infty${\,\rm, }and let $\,\mu$ be a \rsigma0finite
positive measure with
$\,{}^{}{\rm rng}\,{}_{{}^{}}\mu\not=\{\kern0.37mm 0\kern0.37mm\}\KPt8$. Then
there is a probability measure $\,\pi$ with
$\,\mLrs03^p(\kern0.37mm\mu\kern0.15mm)$ and $\,\mLrs03^p(\kern0.15mm\pi\kern0.15mm)$
linearly homeomorphic.
\end{lemma}
\begin{proof} Letting \math{\scrmt A\subseteq\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+} be a
finite or countably infinite partition of \mathss36{{}^{}\Cal Omega}, \,let \math{
\bosy a:\scrmt A\to\rbb R^+} be any function with \math{\sum\,\bosy a=1} and
take \vskip.5mm\centerline{$
\pi =
\kern0.37mm\big\langle\,\sum_{\KPt8 B\kern0.37mm\in\kern0.37mm\scrm7 A\,}
(\kern0.37mm\bosy a\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} B\KP1(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} B\kern0.37mm)^{\kern0.37mm\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\,
(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm A\capss31 B\kern0.37mm))) : A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\KP1\rangle \KP1 $.} \inskipline{.5}0
Then \math{\pi} is a probability measure with \math{{{}^{}{\rm dom}\,{}_{{}^{}}}\pi={{}^{}{\rm dom}\,{}_{{}^{}}}\mu} and we
define
$\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm:\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\mLrs03^p(\kern0.37mm\mu\kern0.15mm)
\to\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\mLrs03^p(\kern0.15mm\pi\kern0.15mm)$
by $\smb\Phii\mapsto\smb\Psii$ when $x\in\smb\Phii$ and
$\bigcup\KPt8\{\KPt9\roman b\,B\KP1(\kern0.37mm x\KP1|\KP1 B\kern0.37mm)
:B\in\scrmt A\KP1\}\in\smb\Psii$
where $\roman b\,B=
(\kern0.37mm\bosy a\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} B\kern0.37mm)\kern0.37mm^{\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} p^{-1}}
(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} B\kern0.37mm)\KP1^{p^{-1}}$
for $1\le p < \lower1.05mm\hbox{$^+$}\infty$
and \math{\roman b\,B=1} for
\mathss36{p = \lower1.05mm\hbox{$^+$}\infty}.
\end{proof}
\begin{definitions}\label{df D-P}
(1) \ Say that \math{E} is a {\it\eit{DP\,}--\,space over\kern0.37mm} \math{\bosy K}
if{}f \math{\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}} and \mathss30{E \in \kern-0.3mm} \mathss03{
\roman{LCS}\kern0.4mmps0(K)} and for all \math{F\in\roman{BaS}\kern0.4mmps0(K) } and \math{ \smb U \in
\Cal L\,(\kern0.15mm E\kern0.37mm,\kern0.07mm F\kern0.37mm) } and for \inskipline07
$\scrmt K=\{\,A:A\kern0.37mm\text{ is absolutely \mathss37{\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E}--\,convex and \mathss37{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm E\subsigrs04) }--\,compact }\}
$ from \KP3 \inskipline0{11}
$\smb U\kern0.37mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\bouSet E\subseteq\{\,B:B\kern0.37mm\text{ is relatively \mathss37{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm F\subsigrs00) }--\,compact }\}
$ it follows that \KP3 \inskipline0{14.8}
$\smb U\kern0.37mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\scrmt K\subseteq\{\,B:B\kern0.37mm\text{ is relatively \mathss37{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F}--\,compact }\} \KP1 $ holds, \inskipline{.5}2
(2) \ Say that \math{E} is a {\it\eit{DP\,}--\,space\kern0.37mm} if{}f there is \math{
\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}} \inskipline0{7.7}
such that \math{E} is a \erm{DP\,}--\,space over \mathss32{\bosy K}.
\end{definitions}
Instead of saying that \math{E} is a \erm{DP}\,--\,space, we may also say that
it is a {\it Dunford\,--\,Pettis\kern0.37mm} space. For application of the
Dunford\,--\,Pettis property one should note that by
\cite[Remarks 9.4.1\,(3)\,, p.\ 634]{Edw} for \math{E} complete in
Definitions \ref{df D-P}\,(1) we get an equivalent condition if we instead
take \vskip.2mm\centerline{$
\scrmt K = \{\,A:A\kern0.37mm\text{ is relatively \mathss37{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm E\subsigrs04) }--\,compact }\} \KP1 $.} \inskipline{.4}0
In particular, this holds if with \math{\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}} and \math{\mu} a
positive measure we take \mathss08{E=\mvLrs42^1(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm) }.
For the proof in the general case one possibly uses {\sl Krein's theorem\kern0.15mm}
\cite[9.8.5\kern0.37mm, p.\ 192]{Jr}\,. The Lebesgue case with \math{
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm\bigcup\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\mu < \lower1.05mm\hbox{$^+$}\infty } also follows from Lemma \ref{Le L^1_si-compa}
on page \pageref{Le L^1_si-compa} above, and only this will be needed in the
sequel.
\begin{proposition}\label{Pro DP-seq}
Let \ú$\,E\in\roman{BaS}\kern0.4mmps0(K)$ with \ú$\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}\KP1$. Then $\,E$ is a
Dunford\,--\,Pettis space if and only if \ú$\,
\roman{ev}\circ\kern0.07mm[\KP1\bosy x\kern0.37mm,\kern0.07mm\bosy y\kern.9mm]\kern.2mm\lower.8mm\hbox{\font\SweD =cmr5\SweD f}\kern.2mm\to 0$ holds for all $\,
\bosy x\kern0.37mm,\kern0.15mm\bosy y$ with $\,\bosy x\to\Bnull_E$ in top $\,\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm
E\subsigrs04)$ and $\,\bosy y\to\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E\times\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}$ in top $\,
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm((\kern0.15mm E\dlbetss22\kern0.15mm)\subsigrs03) \KP1 $.
\end{proposition}
\begin{proof} See \cite[Proposition 20.7.1\kern0.15mm, p.\ 473]{Jr} or
\cite[p.\ 636]{Edw}\,.
\end{proof}
| 3,845 | 362,273 |
en
|
train
|
0.47.74
|
\begin{lemma}\label{Le E_{/S} DP imp ...}
With \ú$\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let \ú$\,E\in\roman{BaS}\kern0.4mmps0(K)$ be such that for every $\,
\bosy x$ with \ú$\,\bosy x\to\Bnull_E$ in top $\,\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm E\subsigrs04)$
there is a closed linear subspace $\,S$ in $\,E$ with \ú$\,{}^{}{\rm rng}\,{}_{{}^{}}\bosy x\subseteq S$
and such that $\,E_{\,/\,S}$ is a \eit{DP}\,--\,space. Then $\,E$ is a
\eit{DP}\,--\,space.
\end{lemma}
\begin{proof} Given \math{\bosy x\kern0.37mm,\kern0.15mm\bosy y } with \math{\bosy x\to
\Bnull_E } in top \math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm E\subsigrs04) } and \math{\bosy y\to
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E\times\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } in top \mathss08{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm((\kern0.15mm
E\dlbetss22\kern0.15mm)\subsigrs03) }, \,by Proposition \ref{Pro DP-seq} it suffices
to show that \math{\roman{ev}\circ\kern0.07mm[\KP1\bosy x\kern0.37mm,\kern0.07mm\bosy y\kern.9mm]\kern.2mm\lower.8mm\hbox{\font\SweD =cmr5\SweD f}\kern.2mm\to 0 }
holds. To get this, putting \math{F=E_{\,/\,S} } and \math{\bosy z=\seqss30{
\bosy y\fvalss01 i\KP1|\KP1 S:i\in\mathbb No} } we now have \mathss30{
\roman{ev}\circ\kern0.07mm[\KP1\bosy x\kern0.37mm,\kern0.07mm\bosy y\kern.9mm]\kern.2mm\lower.8mm\hbox{\font\SweD =cmr5\SweD f}\kern.2mm = \kern-0.3mm} \mathss03{
\roman{ev}\circ\kern0.07mm[\KP1\bosy x\kern0.37mm,\kern0.07mm\bosy z\kern.9mm]\kern.2mm\lower.8mm\hbox{\font\SweD =cmr5\SweD f}\kern.2mm } and hence we are done
{\sl if\kern0.15mm} we can show (\kern0.15mm a\kern0.15mm) that \math{\bosy x\to\Bnull_E } in top \mathss30{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm F\subsigrs00) } holds, and (\kern0.15mm b\kern0.15mm) that \math{\bosy z\to
S\times\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } in top \math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm((\kern0.15mm
F\dlbetss20\kern0.15mm)\subsigrs03) } holds. Now (\kern0.15mm a\kern0.15mm) follows trivially from
Hahn\,--\,Banach since given \math{v\in\Cal L\,(\kern0.15mm F\kern0.07mm,\kern0.07mm\bosy K\kern0.37mm) }
there is \math{u} with \mathss30{v\subseteq u\in\Cal L\,(\kern0.15mm E\kern0.37mm,\kern0.07mm\bosy K\kern0.37mm) }
and hence \math{ v\circss00\bosy x = u\circss00\bosy x\to 0 } holds. For
(\kern0.15mm b\kern0.15mm) taking the annihilators \inskipline{.2}{16.5}
$N\aar 0 = \Cal L\,(\kern0.15mm E\kern0.37mm,\kern0.07mm\bosy K\kern0.37mm)\capss31\{\,u :
u\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.3mm S\subseteq\{\kern0.37mm 0\kern0.37mm\}\kern0.15mm\} \KP{13.5} $ and \inskipline0{17}
$S\ar 1 = \Cal L\,(\kern0.15mm E\dlbetss22\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm)\capss31\{\,w :
w\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.3mm N\aar 0\subseteq\{\kern0.37mm 0\kern0.37mm\}\kern0.15mm\} \KP9 $ and putting \inskipline{.2}0
\ú$F\aar 1=E\dlbetss22\kern0.37mm/\tvsquotient N\aar 0 \KPt7 $, \,for \math{ w \ar 1
\in \Cal L\,(\kern0.15mm F\dlbetss20,\kern0.07mm\bosy K\kern0.37mm) } from
\cite[3.13\kern0.37mm, pp.\ 261\,--\,263]{Ho} we first get ex- istence of \math{
w\ar 2 \in \Cal L\,(\kern0.15mm F\aar 1\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm) } with \math{
w\ar 2\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm\smb U = w\ar 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm u\KP1|\KP1 S\kern0.37mm) } for \mathss34{
u\in\smb U\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1 }. Then we get existence of \math{w\in S\ar 1} with \math{
w\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} u=w\ar 2\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm\smb U=w\ar 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm u\KP1|\KP1 S\kern0.37mm) }
for \math{u\kern0.37mm,\kern0.15mm\smb U} as above. Hence we finally get \math{
w\ar 1\kern-0.3mm\circ\kern0.07mm\bosy z=w\circss00\bosy y\to 0 } as required.
\end{proof}
\begin{proposition}\label{Pro L^(p) is DP}
If $\,\pi$ is any probability measure{\kern0.15mm\rm, }then $\,\mLrs42^1(\kern0.15mm\pi\kern0.15mm)$
is a \eit{DP}\,--\,space.
\end{proposition}
\begin{proof} See \cite{Brg} and \cite{Sch}\,, and take \math{T=1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}} in
the latter.
\end{proof}
| 2,612 | 362,273 |
en
|
train
|
0.47.75
|
\begin{proposition}\label{Pro L^(p) is DP}
If $\,\pi$ is any probability measure{\kern0.15mm\rm, }then $\,\mLrs42^1(\kern0.15mm\pi\kern0.15mm)$
is a \eit{DP}\,--\,space.
\end{proposition}
\begin{proof} See \cite{Brg} and \cite{Sch}\,, and take \math{T=1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}} in
the latter.
\end{proof}
\begin{corollary}\label{Cor L^1 is D-S}
For $\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ and for any \inskipline0{21.5}
positive measure $\,\mu$ it holds that $\,
\mvLrs42^1(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm)$ is a \eit{DP}\,--\,space.
\end{corollary}
\begin{proof} From Proposition \ref{Pro L^(p) is DP} we first see that also \math{
\mvLrs42^1(\kern0.37mm\pi\kern0.15mm,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\kern0.37mm) } is a \erm{DP\,}--\,space for any
probability measure \mathss32{\pi}. Indeed, for \math{ G =
\mLrs42^1(\kern0.15mm\pi\kern0.15mm)\kern0.37mm\sqcap\kern0.15mm\mLrs42^1(\kern0.15mm\pi\kern0.15mm) } from
\cite[9.4.3\,(a)\,, p.\ 635]{Edw} we first see that \math{G} is a
\erm{DP\,}--\,space, and since \math{\mLrs42^1(\kern0.15mm\pi\kern0.15mm)\flbb_C} and \math{G}
are naturally linearly homeomorphic, also \math{\mLrs42^1(\kern0.15mm\pi\kern0.15mm)\flbb_C}
is a \erm{DP\,}--\,space. If now \math{\smb U} is a continuous linear map \mathss30{
\mvLrs42^1(\kern0.37mm\pi\kern0.15mm,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\kern0.37mm)\to F}, \,it is also a continuous real
linear map \mathss30{\mLrs42^1(\kern0.15mm\pi\kern0.15mm)\flbb_C\to F\Reit0} whence the
assertion follows by noting that the equality \math{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm E\RHB{.15}{\subsigma}\kern0.07mm) =
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm E\Reit1\kern-0.2mm\RHB{.15}{\subsigma}\kern0.07mm) } holds for every \mathss38{
E\in\roman{LCS}\kern0.4mmps5(\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C) }.
By Lemma \ref{Le L^p(m)=L^p(p)} from the above we know that \math{
\mvLrs42^1(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm) } is a \erm{DP\,}--\,space for any
\rsigma5finite positive measure \mathss36{\mu}. Then by Lemma \ref{Le E_{/S} DP imp ...}
we get the general case as follows. Putting \math{ E =
\mvLrs42^1(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm) } and letting \math{
\bmii8\Phii\to\Bnull_E} in top \math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm E\subsigrs04) } we first find
some countable \math{\scrmt A\subseteq\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\lbb R_+} such that \math{
\|\KP1\varphi\KP1|\KP1 A\KP1\|\Lnorss33^1_\mu=0 } holds for all \mathss30{A\in
\scrmt A} and \math{\varphi\in\bigcup\kern0.37mm{}^{}{\rm rng}\,{}_{{}^{}}\kern0.15mm\bmii8\Phii}. Then taking \math{
\mu\ar 1=\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm\kern-0.3mm\sbig(1\bigcup\,\scrmt A\kern0.37mm) } we have \math{
\mvLrs42^1(\kern0.37mm\mu\ar 1\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm) } a \erm{DP\,}--\,space. Moreover,
we have an obvious strict morphism \math{ \kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm :
\mvLrs42^1(\kern0.37mm\mu\ar 1\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm)\to E } with \mathss03{
{}^{}{\rm rng}\,{}_{{}^{}}\kern0.15mm\bmii8\Phii\subseteq{}^{}{\rm rng}\,{}_{{}^{}}\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm} whence Lemma \ref{Le E_{/S} DP imp ...} gives
the conclusion.
\end{proof}
To fill the gap \q{exists $\ldots\,\eightroman M_{\kern0.37mm\roman t}\,\ldots$} in
\cite[p.\ 3]{Sch} we give the following
\begin{lemma}
With $\,\pi$ a probability measure let \ú$\,E=\mLrs42^1(\kern0.15mm\pi\kern0.15mm)$ and also
let \ú$\,\bmii8\Phii\to\Bnull_E$ in top $\, \tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm E\subsigrs04)\KP1$.
Further let $\,\varepsilon\in\rbb R^+$ and $\, \bosy\varphi \in
\prod{_{_{\kern-.3mm\bold c\kern.15mm}}}\kern0.37mm\bmii8\Phii\kern0.37mm$ with \vskip.3mm\centerline{$
{}^{}{\rm rng}\,{}_{{}^{}}\bosy\varphi\subseteq\{\,\varphi:(\KPt5\varphi\KPt8;\kern0.07mm\pi\kern0.15mm,\kern-0.63mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\kern0.37mm)\kern0.37mm\text{
is measurable }\} \KP1 $.} \inskipline{.5}0
Then there is some \ú$\,\smb M\in\rbb R^+$ such that \ú$\,\|\KP1\varphi\KP1|\KP1
((\kern0.37mm\Abrs03^1\kern-0.2mm\circ\kern0.15mm\varphi\kern0.37mm)\invss24\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm[\KP1\smb M\kern0.15mm,
\lower1.05mm\hbox{$^+$}\infty\KP1{[\KPt8}\sbig)3\,
\big\|\LHB{.4}{\Lnorss33^1_{\emath\char'031}\def\rhoo{\char'032}\def\sigmaa{\char'033}} < \varepsilon$ holds for all $\,
\varphi \in {}^{}{\rm rng}\,{}_{{}^{}}\bosy\varphi \, $.
\end{lemma}
\begin{proof} If the assertion is false, by {\sl dependent choice\kern0.15mm} there is
a stricly increasing \mathss03{\bosy n:\mathbb No\to\mathbb No } such that \math{\varepsilon\le
\|\KP1\varphi\KP1|\KP1((\kern0.37mm\Abrs03^1\kern-0.2mm\circ\kern0.15mm\varphi\kern0.37mm)\invss24\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm
[\KP1 i\kern0.37mm\lower1mm\hbox{$^{^+}$}\kern0.15mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056},\lower1.05mm\hbox{$^+$}\infty\KP1{[\KPt8}\sbig)3\,
\big\|\LHB{.4}{\Lnorss33^1_{\emath\char'031}\def\rhoo{\char'032}\def\sigmaa{\char'033}} } holds for \mathss30{
(\kern0.37mm i\kern0.37mm,\kern0.07mm\varphi\kern0.37mm) \in \kern-0.3mm} \mathss06{\bosy\varphi\circss00\bosy n
}. Noting that \math{{}^{}{\rm rng}\,{}_{{}^{}}\kern0.37mm\bmii8\Phii } is relatively \mathss37{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm E\subsigrs04)}--\,compact, then from Lemma \ref{Le L^1_si-compa}
on page \pageref{Le L^1_si-compa} above it follows indirectly that there is \math{
\delta\in\rbb R^+ } with the property that for \math{
(\kern0.37mm i\kern0.37mm,\kern0.07mm\varphi\kern0.37mm)\in\bosy\varphi\circss00\bosy n } we have \mathss38{
\delta\le\pi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}((\kern0.37mm\Abrs03^1\kern-0.2mm\circ\kern0.15mm\varphi\kern0.37mm)\invss24\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm[\KP1
i\kern0.37mm\lower1mm\hbox{$^{^+}$}\kern0.15mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056},\lower1.05mm\hbox{$^+$}\infty\KP1{[\KPt8}\sbig)0 }. This implies that \mathss03{
i\kern0.37mm\lower1mm\hbox{$^{^+}$}\,\delta\le\|\,\varphi\,\|\Lnorss33^1_{\emath\char'031}\def\rhoo{\char'032}\def\sigmaa{\char'033} } holds, giving
a {\sl contradiction\kern0.15mm} with \mathss36{{}^{}{\rm rng}\,{}_{{}^{}}\kern0.37mm\bmii8\Phii\in\bouSet E }.
\end{proof}
| 3,574 | 362,273 |
en
|
train
|
0.47.76
|
\insubsubhead Absolutely continuous vector measures \label{Ss abs conti}
We here give some basic definitions for vector measures in order to be able to
present a decent proof for Proposition \ref{Pro Phi5.4} below that is needed
as an auxiliary result for the proof of Theorem \nfss A\,\ref{main Th} above.
\begin{definitions}\label{df vec mea}
(1) \ Say that \math{E} is a {\it topologized conoid\,} if{}f there are \math{
a\kern0.37mm,\kern0.15mm c\,,\kern0.15mm o\kern0.37mm,\kern0.15mm R\,,\kern0.15mm S\kern0.37mm,\kern0.15mm\scrmt T} with \math{
\lbb R_+\subseteq R\subseteq\mathbb C } and \math{o\in S} and \math{
(\kern0.15mm S\kern0.37mm,\kern0.07mm\scrmt T\,) } a Hausdorff topological space and \mathss30{ E = {\kern-0.63mm}} \mathss03{
(\kern0.37mm a\kern0.37mm,\kern0.07mm c\,,\kern0.07mm\scrmt T\,) } and \math{a} a function \math{S\times S
\to S} and \math{c} a function \math{R\times S\to S} and such that for all \math{
x\kern0.37mm,\kern0.15mm y\kern0.37mm,\kern0.15mm z\in S } and for all \math{s\kern0.37mm,\kern0.15mm t\in R} it holds that \math{
a\,(\kern0.37mm x\kern0.37mm,\kern0.15mm\cdot\,) } and \math{c\KPt8(\kern0.37mm t\kern0.37mm,\kern0.15mm\cdot\,) } are
continuous \math{\scrmt T\to\scrmt T } and in addition \inskipline09
$a\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm a\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm)\,,\kern0.15mm z\kern0.37mm)=
a\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm x\kern0.37mm,\kern0.07mm a\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm y\kern0.37mm,\kern0.07mm z\kern0.37mm))\,$ and
$\,a\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm)=a\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm y\kern0.37mm,\kern0.07mm x\kern0.37mm)\,$ and \inskipline09
$a\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm x\kern0.37mm,\kern0.07mm o\kern0.37mm)=c\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm 1\kern0.15mm,\kern0.07mm x\kern0.37mm)=x\,$ and
$\,c\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm s\KPt8 t\kern0.37mm,\kern0.07mm x\kern0.37mm)=
c\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm s\kern0.37mm,\kern0.15mm c\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm t\kern0.37mm,\kern0.07mm x\kern0.37mm ))\,$ and \inskipline09
$c\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm s + t\kern0.37mm,\kern0.07mm x\kern0.37mm)=
a\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\,c\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm s\kern0.37mm,\kern0.07mm x\kern0.37mm)\,,\kern0.15mm
c\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm t\kern0.37mm,\kern0.07mm x\kern0.37mm)) \,$ and \inskipline09
$c\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm s\kern0.37mm,\kern0.07mm a\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm))=
a\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\,c\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm s\kern0.37mm,\kern0.07mm x\kern0.37mm)\,,\kern0.07mm
c\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm s\kern0.37mm,\kern0.07mm y\kern0.37mm)) \KP1 $, \inskipline{.5}2
(2) \ Say that \math{m} is an \mathss35{E}{\it--\,measure\kern0.37mm} if{}f \math{E}
is a topologized conoid and \math{(\kern0.37mm\emptyset\,,\kern0.07mm\Bnull_E) \in {\kern-0.63mm}} \inskipline09
$m\in\kern0.15mm^{\roman{dom}\KPt8 m}\,\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E\kern0.37mm$ and for all \math{ A\,,\kern0.15mm B \in
{{}^{}{\rm dom}\,{}_{{}^{}}} m} it holds that \math{\{\,A\cupss31 B\kern0.37mm,\kern0.15mm A\setminus B\,\}\subseteq {\kern-0.63mm}} \inskipline0{8.7}
${{}^{}{\rm dom}\,{}_{{}^{}}} m \kern0.37mm $ and \mathss36{A\capss31 B=\emptyset\impss33
m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm A\cupss31 B\kern0.37mm) = (\kern0.37mm m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A + m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} B\kern0.37mm)\svs E}, \inskipline{.5}2
(3) \ Say that \math{m} is {\it countably \mathss37{E}--\,additive\kern0.37mm} if{}f \math{
m} is an \mathss37{E}--\,measure and \inskipline09
for all countable disjoint \math{\scrmt A\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}} m} with \math{
\bigcup\,\scrmt A\in{{}^{}{\rm dom}\,{}_{{}^{}}} m} \inskipline0{36.5}
it holds that \mathss38{m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm\bigcup\,\scrmt A =
E\vtopsum3\KP1(\kern0.37mm m\KP1|\KP1\scrmt A\kern0.37mm) }, \inskipline{.5}2
(4) \ Say that \math{m} is {\it absolutely \mathss57{\mu}--\,continous\kern0.37mm} in
\math{E} if{}f \math{\mu} is a positive measure \inskipline09
and \math{m} is an \mathss37{E}--\,measure and for every \math{ U\kern-0.3mm \in
\neiBoo E} there is some \math{\delta\in\rbb R^+} \inskipline0{76}
with \mathss30{\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm[\KPp1.1 0\,,\kern0.07mm\delta\KP1{[\kern0.15mm} \subseteq
m\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022} U}, \inskipline{.5}2
(5) \ Say that \math{m} has {\it bounded \mathss57{\mu}--$\KP2^p\,
$variation\kern0.37mm} in \math{E} if{}f \math{1\le p < \lower1.05mm\hbox{$^+$}\infty} and \math{\mu}
is a positive measure and \math{m} is an \mathss35{E}--\,measure with \math{
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}} m} and for ev- ery \math{ \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm \in
\Bqnorm E} there is \math{\smb M\in\lbb R_+} such that \mathss30{
\sum_{\,A\kern0.37mm\in\kern0.37mm\scrm7 A\,}( (\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)
\KP1^{1\kern0.37mm-\,p}\,(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 m\kern0.07mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm)
\RHB{.3}{\KP1^p}\kern0.37mm\sbig)0\le\smb M } holds for all finite disjoint \mathss30{
\scrmt A\subseteq\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+}.
\end{definitions}
In Example \ref{Exa sign mea} on page \pageref{Exa sign mea} we demonstrate
how also the concepts of positive measure and of {\sl signed measure\kern0.15mm} in
the sense of \cite[5.6\kern0.37mm, p.\ 137]{Du} can be subsumed in Definitions \ref{df vec mea}
above. By a {\it real measure\kern0.37mm} we mean any countably \mathss37{\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R
}--\,additive \linebreak
$m\kern0.37mm$ such that \math{{{}^{}{\rm dom}\,{}_{{}^{}}} m} is a \rsigma3algebra. The
definition of {\it complex measure\kern0.37mm} is obtained by taking here \math{
\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C} in place of \mathss36{\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R}. \vskip.3mm
The essential content of \cite[Lemma 5.3\kern0.37mm, p.\ 133]{Phil} is reformulated
in the next
\begin{lemma}\label{Le Phi5.3}
| 3,799 | 362,273 |
en
|
train
|
0.47.77
|
(3) \ Say that \math{m} is {\it countably \mathss37{E}--\,additive\kern0.37mm} if{}f \math{
m} is an \mathss37{E}--\,measure and \inskipline09
for all countable disjoint \math{\scrmt A\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}} m} with \math{
\bigcup\,\scrmt A\in{{}^{}{\rm dom}\,{}_{{}^{}}} m} \inskipline0{36.5}
it holds that \mathss38{m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm\bigcup\,\scrmt A =
E\vtopsum3\KP1(\kern0.37mm m\KP1|\KP1\scrmt A\kern0.37mm) }, \inskipline{.5}2
(4) \ Say that \math{m} is {\it absolutely \mathss57{\mu}--\,continous\kern0.37mm} in
\math{E} if{}f \math{\mu} is a positive measure \inskipline09
and \math{m} is an \mathss37{E}--\,measure and for every \math{ U\kern-0.3mm \in
\neiBoo E} there is some \math{\delta\in\rbb R^+} \inskipline0{76}
with \mathss30{\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm[\KPp1.1 0\,,\kern0.07mm\delta\KP1{[\kern0.15mm} \subseteq
m\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022} U}, \inskipline{.5}2
(5) \ Say that \math{m} has {\it bounded \mathss57{\mu}--$\KP2^p\,
$variation\kern0.37mm} in \math{E} if{}f \math{1\le p < \lower1.05mm\hbox{$^+$}\infty} and \math{\mu}
is a positive measure and \math{m} is an \mathss35{E}--\,measure with \math{
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}} m} and for ev- ery \math{ \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm \in
\Bqnorm E} there is \math{\smb M\in\lbb R_+} such that \mathss30{
\sum_{\,A\kern0.37mm\in\kern0.37mm\scrm7 A\,}( (\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)
\KP1^{1\kern0.37mm-\,p}\,(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 m\kern0.07mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm)
\RHB{.3}{\KP1^p}\kern0.37mm\sbig)0\le\smb M } holds for all finite disjoint \mathss30{
\scrmt A\subseteq\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+}.
\end{definitions}
In Example \ref{Exa sign mea} on page \pageref{Exa sign mea} we demonstrate
how also the concepts of positive measure and of {\sl signed measure\kern0.15mm} in
the sense of \cite[5.6\kern0.37mm, p.\ 137]{Du} can be subsumed in Definitions \ref{df vec mea}
above. By a {\it real measure\kern0.37mm} we mean any countably \mathss37{\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R
}--\,additive \linebreak
$m\kern0.37mm$ such that \math{{{}^{}{\rm dom}\,{}_{{}^{}}} m} is a \rsigma3algebra. The
definition of {\it complex measure\kern0.37mm} is obtained by taking here \math{
\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C} in place of \mathss36{\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R}. \vskip.3mm
The essential content of \cite[Lemma 5.3\kern0.37mm, p.\ 133]{Phil} is reformulated
in the next
\begin{lemma}\label{Le Phi5.3}
Let \ú$\,E\in\roman{LCS}\kern0.4mmps0(K)$ be normable with \ú$\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ and $\,\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm$ a
compatible norm for $\,E${\,\rm, }and let $\,m$ be absolutely $\,\mu\,$--\,
continuous in $\,E$ with \ú$\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}} m$ and \ú$\,\mu\fvalss01{}^{}\Cal Omega <
\infty$ for the set \ú$\,{}^{}\Cal Omega=\bigcup\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\,$. Then for every \ú$\,
\smb M\in\rbb R^+$ there exist some \ú$A\ar 0\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu$ and a countable set \ú$\,
\scrmt A\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}}\mu$ with \ú$\,\scrmt A\cupss31\{\,A\ar 0\kern0.15mm\}$ disjoint and
also with \ú${}^{}\Cal Omega = \bigcup\,\scrmt A\cupss31 A\ar 0$ and such that for all $\,
A\,,\kern0.15mm B$ {\rm\inskipline{.7}2
(1) \ }$A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\capss22\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\ar 0 \impss33
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 m\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\le\smb M\KP1(\kern0.37mm\mu\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm) \KP1
${\rm,\inskipline{.4}2
(2) \ }$A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu$ and $\,A\subseteq B\in\scrmt A \impss33
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 m\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\ge\smb M\KP1(\kern0.37mm\mu\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm) \KP1
$.
\end{lemma}
\begin{proof} We first note that in the case \math{\bosy K=\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C} we have \math{
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm} also a compatible norm for the realification \math{E\Reit4} of \math{E}
and hence we may without loss of generality assume that \math{\bosy K=\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R}
holds. Now we let \math{\roman P\,B\,u} mean that \math{B\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} and \ú$\kern0.37mm
u\in\Cal L\,(\kern0.15mm E\kern0.37mm,\kern-0.2mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\kern0.37mm) $ \linebreak
with \math{
\sup\kern0.37mm\big\{\KPt8|\KP1 u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x\KP1|:x\in\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb24 I\,\}
\le 1 } and that for all \math{A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\capss22\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm B } we have \linebreak
\ú$
u\circss01 m\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\ge\smb M\KP1(\kern0.37mm\mu\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm) \kern0.37mm $
and that \math{ u\circss01 m\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A \le
\smb M\KP1(\kern0.37mm\mu\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm) } holds for all \math{A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu}
with \linebreak
\ú$A\capss31 B=\emptyset \,$. Also let \mathss38{ \scrmt A\kern0.15mm\ar 0 =
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.3mm\cap\kern0.15mm\{\,A:\eexi{B\kern0.37mm,\kern0.15mm u}\,A\subseteq B\kern0.37mm
\text{ and }\kern0.37mm\roman P\,B\,u\KPt8\} }. By con- sidering the set \math{
\scrmt P} of disjoint subsets \math{\scrmt A} of \math{\scrmt A\kern0.15mm\ar 0}
partially ordered by inclusion, from \linebreak
{\sl Zorn's lemma\kern0.15mm} we get existence of some
maximal \math{\scrmt A} of \mathss30{\scrmt P}. Then by \math{
\mu\fvalss02{}^{}\Cal Omega < \lower1.05mm\hbox{$^+$}\infty } we \linebreak
see that \math{\scrmt A} is countable,
and we take \mathss36{ A\ar 0 = {}^{}\Cal Omega\kern0.07mm\setminus\bigcup\,\scrmt A }.
| 3,645 | 362,273 |
en
|
train
|
0.47.78
|
(2) \ }$A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu$ and $\,A\subseteq B\in\scrmt A \impss33
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 m\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\ge\smb M\KP1(\kern0.37mm\mu\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm) \KP1
$.
\end{lemma}
\begin{proof} We first note that in the case \math{\bosy K=\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C} we have \math{
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm} also a compatible norm for the realification \math{E\Reit4} of \math{E}
and hence we may without loss of generality assume that \math{\bosy K=\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R}
holds. Now we let \math{\roman P\,B\,u} mean that \math{B\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} and \ú$\kern0.37mm
u\in\Cal L\,(\kern0.15mm E\kern0.37mm,\kern-0.2mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\kern0.37mm) $ \linebreak
with \math{
\sup\kern0.37mm\big\{\KPt8|\KP1 u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x\KP1|:x\in\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb24 I\,\}
\le 1 } and that for all \math{A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\capss22\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm B } we have \linebreak
\ú$
u\circss01 m\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\ge\smb M\KP1(\kern0.37mm\mu\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm) \kern0.37mm $
and that \math{ u\circss01 m\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A \le
\smb M\KP1(\kern0.37mm\mu\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm) } holds for all \math{A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu}
with \linebreak
\ú$A\capss31 B=\emptyset \,$. Also let \mathss38{ \scrmt A\kern0.15mm\ar 0 =
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.3mm\cap\kern0.15mm\{\,A:\eexi{B\kern0.37mm,\kern0.15mm u}\,A\subseteq B\kern0.37mm
\text{ and }\kern0.37mm\roman P\,B\,u\KPt8\} }. By con- sidering the set \math{
\scrmt P} of disjoint subsets \math{\scrmt A} of \math{\scrmt A\kern0.15mm\ar 0}
partially ordered by inclusion, from \linebreak
{\sl Zorn's lemma\kern0.15mm} we get existence of some
maximal \math{\scrmt A} of \mathss30{\scrmt P}. Then by \math{
\mu\fvalss02{}^{}\Cal Omega < \lower1.05mm\hbox{$^+$}\infty } we \linebreak
see that \math{\scrmt A} is countable,
and we take \mathss36{ A\ar 0 = {}^{}\Cal Omega\kern0.07mm\setminus\bigcup\,\scrmt A }.
Now, for the proof (1) letting \math{A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\capss22\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\ar 0 } we
first note (\kern0.15mm$*$\kern0.15mm) that by \linebreak
maximality of \math{\scrmt A} there cannot
exist \math{B\kern0.37mm,\kern0.15mm u} with \math{\roman P\,B\,u} and \mathss31{A\capss31 B
\in \mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+ }. Fur- \linebreak
thermore, by Hahn\,--\,Banach it
suffices for arbitrarily fixed \math{u\in\Cal L\,(\kern0.15mm E\kern0.37mm,\kern-0.2mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\kern0.37mm) }
with norm \math{ \sup \kern0.37mm \big\{ \KPt8 |\KP1 u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x\KP1| : x \in
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb24 I\,\} \le 1 } to verify that \math{
u\circss01 m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A \le \smb M\KP1(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm) } holds. \linebreak
Since \math{u\circss01 m\KP1|\KP1{{}^{}{\rm dom}\,{}_{{}^{}}}\mu } is absolutely \mathss37{\mu
}--\,continuous, there is a Radon\,--\,Nikodym deri- vative of it, and by
considering one such we see existence of \math{B} with \mathss36{
\roman P\,B\,u}. Then by (\kern0.15mm$*$\kern0.15mm) we have \math{ A\capss31 B \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } whence with \math{ A\ar 1 =
A\kern0.07mm\setminus B } we finally get \vskip.3mm\centerline{$
u\circss01 m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A = u\circss01 m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\ar 1 \le
\smb M\KP1(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\ar 1) =
\smb M\KP1(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm) \KP1 $.} \vskip.3mm
For the proof of (2) letting \math{A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} with \mathss36{ A \subseteq B \in
\scrmt A\subseteq\scrmt A\kern0.15mm\ar 0 }, \,there are some \linebreak
\ú$B\ar 1\kern0.37mm$ and \math{u} with \math{
\roman P\,B\ar 1\kern0.37mm u} and \mathss34{B\subseteq B\ar 1}. Then we also have \math{
A\subseteq B\ar 1} and consequently \linebreak
$u\circ m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A \ge
\smb M\KP1(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm) \kern0.37mm $ whence further \math{
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 m\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\ge\smb M\KP1(\kern0.37mm\mu\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm) }
trivially follows.
\end{proof}
| 3,036 | 362,273 |
en
|
train
|
0.47.79
|
The essential content of \cite[Lemma 5.4\kern0.37mm, p.\ 133]{Phil} is reformulated
in the next
\begin{proposition}\label{Pro Phi5.4}
Let \ú$\,E\in\roman{LCS}\kern0.4mmps0(K)$ be normable with \ú$\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ and $\,\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm$ a
compatible norm for $\,E${\,\rm, }and let $\,m$ be absolutely $\,\mu\,$--\,
continuous in $\,E$ with \ú$\,\mu\fvalss01{}^{}\Cal Omega < \infty$ for \ú$\, {}^{}\Cal Omega =
\bigcup\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\,$. Also let \ú$\,1\le p < \lower1.05mm\hbox{$^+$}\infty$ and let $\,m$ have
bounded $\,\mu\,$--$\RHB{.3}{\KP{1.5} ^p \kern0.37mm}$variation in $\,E\,$. Then
there is a decreasing \ú$\,\bmii8 A\in\kern0.15mm^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu$ with \ú$\,
\lim\,(\kern0.37mm\mu\kern0.07mm\circ\bmii8 A\kern0.37mm)=0$ and such that $\,
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\le i\kern0.37mm\lower1mm\hbox{$^{^+}$}\,(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm)$ holds
for $\,i\in\mathbb No$ and $\,A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu$ with $\,
A\capss32(\kern0.37mm\bmii8 A\fvalss01 i\kern0.37mm)=\emptyset\,$.
\end{proposition}
\begin{proof} We first note that the requirement \math{{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}} m} in
Lemma \ref{Le Phi5.3} holds since from (4) and (5) in Definitions \ref{df vec mea}
we get \math{\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}} m } and \mathss30{
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+} \mathss04{\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}} m }. Now, for each fixed \math{
i\in\mathbb No} taking \math{ \smb M = i\kern0.37mm\lower1mm\hbox{$^{^+}$}\kern0.15mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056} } in Lemma \ref{Le Phi5.3}
above, by {\sl countable choice\kern0.15mm} we get existence of \math{ \scrb8 A \in \kern0.15mm
^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm{{}^{}{\rm dom}\,{}_{{}^{}}}\mu } with the property that for every \math{i\in\mathbb No} we
have \math{\scrb8 A\fvalss11 i} countable and disjoint and such that for all \math{
A\,,\kern0.15mm B} and for \math{ A\kern0.15mm\ar 0 =
{}^{}\Cal Omega\kern0.07mm\setminus\bigcup\KP1(\kern0.15mm\scrb8 A\fvalss11 i\kern0.37mm) } we have \inskipline{.7}3
(a) \ $A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\capss22\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\ar 0 \impss33
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 m\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A \le
i\kern0.37mm\lower1mm\hbox{$^{^+}$}\kern0.15mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056}\,(\kern0.37mm\mu\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm) \KP1 $, \inskipline{.4}3
(b) \ $A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu$ and $\,A\subseteq B\in\scrb8 A\fvalss11 i \impss33
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 m\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A \ge
i\kern0.37mm\lower1mm\hbox{$^{^+}$}\kern0.15mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056}\,(\kern0.37mm\mu\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm) \KP1 $. \inskipline{.7}0
Then we take \ $\bmii8 B \kern0.37mm = \kern0.37mm
\big\langle\KPt8\bigcup\KP1(\kern0.15mm\scrb8 A\fvalss11 i\kern0.37mm) : i\in\mathbb No\,\rangle \,$
and \inskipline{.5}{23}
$ \bmii8 A \kern0.37mm = \kern0.37mm
\big\langle\KPt8\bigcup\KP1(\kern0.37mm\bmii8 B\KP1|\KP1(\kern0.37mm
\mathbb No\kern-0.3mm\setminus i\kern0.37mm)) : i\in\mathbb No\,\rangle \KP1 $. \inskipline{.4}0
It is now clear that \math{\bmii8 A} is decreasing with \mathss36{\bmii8 A \in
\kern0.15mm^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu}, \,and for the proof of the remaining required
properties we proceed as follows.
By the bounded variation property there is \math{\smb M\aR 1\in\rbb R^+ } such
that for all \math{i\in\mathbb No} and for all finite \math{\scrmt A \subseteq
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.3mm\cap\kern0.15mm(\kern0.15mm\scrb8 A\fvalss11 i\kern0.37mm) } in view
of (b) above we have \vskip.5mm\centerline{$
i\kern0.37mm\lower1mm\hbox{$^{^+}$}\kern0.15mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056}\RHB{.3}{\,^p}\,(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\bigcup\,\scrmt A\kern0.37mm)
\le \sum_{\,A\kern0.37mm\in\kern0.37mm\scrm7 A\,}((\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm
)\KP1^{1\kern0.37mm-\kern0.37mm p}\,(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm
)\RHB{.3}{\KP1^p}\kern0.15mm\big)
\le \smb M\aR 1 \, $,} \inskipline{.5}0
and hence \mathss36{\mu\circss11\bmii8 B\fvalss21 i \le
\smb M\aR 1\,i\kern0.37mm\lower1mm\hbox{$^{^+}$}\kern0.15mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056}\RHB{.3}{\,^{\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} p}} }, \,whence further \mathss36{
\lim\,(\kern0.37mm\mu\kern0.07mm\circ\bmii8 B\kern0.37mm) = 0 }. Next considering \mathss03{B\ar 1=
\bmii8 B\fvalss21 i\kern0.37mm\lower1mm\hbox{$^{^+}$}\kern-0.2mm\setminus(\kern0.37mm\bmii8 B\fvalss21 i\kern0.37mm) } for
all \math{A\in\scrb8 A\fvalss11 i\kern0.37mm\lower1mm\hbox{$^{^+}$} } by both (a) and (b) above we
have \vskip.5mm\centerline{$
(\kern0.37mm i\kern0.37mm\lower1mm\hbox{$^{^+}$}\kern0.15mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern-0.2mm + 1\kern0.37mm)\KP1(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm
A\capss31 B\ar 1))
\le \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm A\capss31 B\ar 1)
\le i\kern0.37mm\lower1mm\hbox{$^{^+}$}\,(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm A\capss31 B\ar 1)) $} \inskipline{.5}0
| 3,516 | 362,273 |
en
|
train
|
0.47.80
|
$ \bmii8 A \kern0.37mm = \kern0.37mm
\big\langle\KPt8\bigcup\KP1(\kern0.37mm\bmii8 B\KP1|\KP1(\kern0.37mm
\mathbb No\kern-0.3mm\setminus i\kern0.37mm)) : i\in\mathbb No\,\rangle \KP1 $. \inskipline{.4}0
It is now clear that \math{\bmii8 A} is decreasing with \mathss36{\bmii8 A \in
\kern0.15mm^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu}, \,and for the proof of the remaining required
properties we proceed as follows.
By the bounded variation property there is \math{\smb M\aR 1\in\rbb R^+ } such
that for all \math{i\in\mathbb No} and for all finite \math{\scrmt A \subseteq
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.3mm\cap\kern0.15mm(\kern0.15mm\scrb8 A\fvalss11 i\kern0.37mm) } in view
of (b) above we have \vskip.5mm\centerline{$
i\kern0.37mm\lower1mm\hbox{$^{^+}$}\kern0.15mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056}\RHB{.3}{\,^p}\,(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\bigcup\,\scrmt A\kern0.37mm)
\le \sum_{\,A\kern0.37mm\in\kern0.37mm\scrm7 A\,}((\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm
)\KP1^{1\kern0.37mm-\kern0.37mm p}\,(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm
)\RHB{.3}{\KP1^p}\kern0.15mm\big)
\le \smb M\aR 1 \, $,} \inskipline{.5}0
and hence \mathss36{\mu\circss11\bmii8 B\fvalss21 i \le
\smb M\aR 1\,i\kern0.37mm\lower1mm\hbox{$^{^+}$}\kern0.15mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056}\RHB{.3}{\,^{\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} p}} }, \,whence further \mathss36{
\lim\,(\kern0.37mm\mu\kern0.07mm\circ\bmii8 B\kern0.37mm) = 0 }. Next considering \mathss03{B\ar 1=
\bmii8 B\fvalss21 i\kern0.37mm\lower1mm\hbox{$^{^+}$}\kern-0.2mm\setminus(\kern0.37mm\bmii8 B\fvalss21 i\kern0.37mm) } for
all \math{A\in\scrb8 A\fvalss11 i\kern0.37mm\lower1mm\hbox{$^{^+}$} } by both (a) and (b) above we
have \vskip.5mm\centerline{$
(\kern0.37mm i\kern0.37mm\lower1mm\hbox{$^{^+}$}\kern0.15mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern-0.2mm + 1\kern0.37mm)\KP1(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm
A\capss31 B\ar 1))
\le \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm A\capss31 B\ar 1)
\le i\kern0.37mm\lower1mm\hbox{$^{^+}$}\,(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm A\capss31 B\ar 1)) $} \inskipline{.5}0
and hence \math{\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm A\capss31 B\ar 1) = 0 } whence further \mathss36{
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} B\ar 1 = 0 }. Now for every \math{i\in\mathbb No} we have \math{
\mu\kern0.07mm\circ\bmii8 A\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm i
\le \mu\kern0.07mm\circ\bmii8 B\fvalss21 i +
\sum_{\,j\kern0.37mm\in\KPt5{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\kern0.15mm\setminus\kern0.37mm i\,}(\,\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm
\bmii8 B\fvalss20 j\kern0.37mm\lower1mm\hbox{$^{^+}$}\kern-0.2mm\setminus(\kern0.37mm\bmii8 B\fvalss20 j\kern0.37mm)
))
= \mu\kern0.07mm\circ\bmii8 B\fvalss21 i} and hence we obtain \mathss36{
\lim\,(\kern0.37mm\mu\kern0.07mm\circ\bmii8 A\kern0.37mm) = 0 }. For the remaining property letting \math{
i\in\mathbb No} and \math{A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} with \mathss38{ \emptyset =
A\capss32(\kern0.37mm\bmii8 A\fvalss01 i\kern0.37mm) =
A\capss34\bigcup\KP1(\kern0.37mm\bmii8 B\KP1|\KP1(\kern0.37mm\mathbb No\kern-0.3mm\setminus i\kern0.37mm))
}, \,we hence also have \mathss30{ \emptyset =
A\capss32(\kern0.37mm\bmii8 B\fvalss21 i\kern0.37mm) } \mathss03{{\KN{.7}} =
A\capss34\bigcup\KP1(\kern0.15mm\scrb8 A\fvalss11 i\kern0.37mm) } and consequently by (a) we
obtain \mathss38{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A \le
i\kern0.37mm\lower1mm\hbox{$^{^+}$}\,(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm) }.
\end{proof}
| 2,413 | 362,273 |
en
|
train
|
0.47.81
|
\begin{proposition}\label{Pro mA=int ev_x c mu}
Let $\,\mu$ be a positive measure with \ú$\,\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm\bigcup\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\mu
< \lower1.05mm\hbox{$^+$}\infty${\,\rm, }and with \ú$\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let \ú$\, \vPi \in
\roman{BaS}\kern0.4mmps0(K)$ be such that either $\,\vPi$ is reflexive or $\,\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi$ is a
separable topology. Also let \ú$\,
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1=\seqss44{
\sup\KPt8(\kern0.37mm\Abrs00^1\circ\kern0.15mm u\circss01\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb15 I)
:u\in\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm)}$ where $\,\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm$ is a compatible norm
for $\,\vPi${\kern0.15mm\rm, }and let $\,m$ be a $\,
\vPi\dlbetss01\,$--\,measure with \ú$\,{{}^{}{\rm dom}\,{}_{{}^{}}} m={{}^{}{\rm dom}\,{}_{{}^{}}}\mu$ and such that
\ú$\,\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.2mm\circ\kern0.15mm m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A \le \mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A$ holds
for all \ú$\,A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\,$. Then there are $\,y\kern0.37mm,\kern0.15mm S$ such that
$\,S$ is a separable closed linear subspace in $\,\vPi\dlbetss01$ and
\ú$\,
(\KPt5 y\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm)$ is
simply measurable and Pettis with
\ú$\, m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\fvalss12\xi =
\int_{\,A}\kern0.37mm\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\rmdss11\mu$ for all
\ú$\, A
\in{{}^{}{\rm dom}\,{}_{{}^{}}} m$ and \ú$\,\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi${\kern0.15mm\rm, }and in addition such that also
\ú$\,{}^{}{\rm rng}\,{}_{{}^{}} y\subseteq S$ holds and
\ú$\,(\KPt5 y\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\dlbetss01\kern0.15mm)$ is simply
measurable in the case where $\,\vPi$ is reflexive.
\end{proposition}
\begin{proof} Let \math{{}^{}\Cal Omega=\bigcup\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} and \mathss38{ E =
\mvLrs42^1(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm) }. Putting
\newline
\math{\roman x\,A=
(\kern0.37mm{}^{}\Cal Omega\kern0.07mm\setminus A\kern0.37mm)\times\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\cupss22
(\kern0.15mm A\times\kern-0.2mm\{\kern0.37mm 1\kern0.37mm\}\kern0.15mm\sbig)0} and
\math{\eightroman X\,A=\uniqset\smb\Phii:
\roman x\,A\in\smb\Phii\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E} let \math{D}
be the linear \mathss37{\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E}--\,span of
\mathss38{\{\KPt8\eightroman X\,A:
A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern0.15mm\big\} }. Thus
\math{D} is the set of all \math{\smb\Phii\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E} such that
there is \math{\varphi\in\smb\Phii} with
\math{{}^{}{\rm rng}\,{}_{{}^{}}\varphi} finite. We know that
\math{D} is \mathss37{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E}--\,dense. Then
we let \math{\smb V\aR 0} be the unique linear extension
\math{\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E_{\KPt8|\,D}\to\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm\vPi\dlbetss01\kern0.15mm) } of
\newline
\mathss38{\{\,(\kern0.37mm\eightroman X\,A
\,,\kern0.07mm m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm):
A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern0.15mm\big\} }, \,noting
that by the assumptions on \math{m} we indeed get a linear map.
Now for finite functions \math{\bosy s\subseteq
{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\times\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\bosy K} with \math{{{}^{}{\rm dom}\,{}_{{}^{}}}\bosy s} disjoint and
\newline
\math{\varphi=
\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\bosy K\expnota^\kern0.37mm{}^{}\Cal Omega\kern0.15mm]_{vs}\text{\KPt8-}
\sum_{\,A\kern0.37mm\in\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.3mm\bmii8 s\,}
(\kern0.37mm\bosy s\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)\KP1\roman x\,A
\in\smb\Phii\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E } we obtain \vskip.5mm\centerline{$
|\KP1\smb V\aR 0\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Phii\fvalss01\xi\KP1|=
\big|\,
\sum_{\,A\kern0.37mm\in\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.3mm\bmii8 s\,}
(\kern0.37mm\bosy s\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)\KP1
(\kern0.37mm m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\fvalss01\xi\kern0.37mm)\KP1|\le
\sum_{\,A\kern0.37mm\in\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.3mm\bmii8 s\,}|\KP1
\bosy s\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\KP1|\KP1|\KPp1.2
m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\fvalss01\xi\KP1| $} \inskipline1{21.6}
${}\le
\sum_{\,A\kern0.37mm\in\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.3mm\bmii8 s\,}|\KP1
\bosy s\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\KP1|\KP1
(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)\KP1(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss01\xi\kern0.37mm)=
(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss01\xi\kern0.37mm)\KP1\|\,\varphi\,\|\Lnorss33^1_\mu \, $. \inskipline{.7}0
Consequently \math{\smb V\aR 0} has a unique continuous extension \mathss38{
\smb V\in\Cal L\,(\kern0.15mm E\kern0.37mm,\kern0.07mm\vPi\dlbetss01\kern0.15mm) }.
| 3,405 | 362,273 |
en
|
train
|
0.47.82
|
Now for finite functions \math{\bosy s\subseteq
{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\times\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\bosy K} with \math{{{}^{}{\rm dom}\,{}_{{}^{}}}\bosy s} disjoint and
\newline
\math{\varphi=
\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\bosy K\expnota^\kern0.37mm{}^{}\Cal Omega\kern0.15mm]_{vs}\text{\KPt8-}
\sum_{\,A\kern0.37mm\in\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.3mm\bmii8 s\,}
(\kern0.37mm\bosy s\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)\KP1\roman x\,A
\in\smb\Phii\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E } we obtain \vskip.5mm\centerline{$
|\KP1\smb V\aR 0\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Phii\fvalss01\xi\KP1|=
\big|\,
\sum_{\,A\kern0.37mm\in\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.3mm\bmii8 s\,}
(\kern0.37mm\bosy s\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)\KP1
(\kern0.37mm m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\fvalss01\xi\kern0.37mm)\KP1|\le
\sum_{\,A\kern0.37mm\in\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.3mm\bmii8 s\,}|\KP1
\bosy s\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\KP1|\KP1|\KPp1.2
m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\fvalss01\xi\KP1| $} \inskipline1{21.6}
${}\le
\sum_{\,A\kern0.37mm\in\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.3mm\bmii8 s\,}|\KP1
\bosy s\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\KP1|\KP1
(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)\KP1(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss01\xi\kern0.37mm)=
(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss01\xi\kern0.37mm)\KP1\|\,\varphi\,\|\Lnorss33^1_\mu \, $. \inskipline{.7}0
Consequently \math{\smb V\aR 0} has a unique continuous extension \mathss38{
\smb V\in\Cal L\,(\kern0.15mm E\kern0.37mm,\kern0.07mm\vPi\dlbetss01\kern0.15mm) }.
Now assuming that \math{\vPi} is reflexive and taking
\mathss38{K=\{\KPt8\eightroman X\,A:A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\KPt8\} }, \,from
Lemma \ref{Le L^1_si-compa} on page \pageref{Le L^1_si-compa} above
we see that
\math{K}
is relatively \mathss37{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm E\subsigrs04)}--\,compact. Since by
Corollary \ref{Cor L^1 is D-S} on page \pageref{Cor L^1 is D-S} above
\math{E} is a \erm{DP\,}--\,space, noting that by reflexivity
of \math{\vPi} all bounded sets in \math{\vPi\dlbetss01} are relatively
\mathss37{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm((\kern0.15mm\vPi\dlbetss01\kern0.15mm)\subsigma)}--\,compact, we
see that \math{\smb V\KPt8\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm K} is
relatively \mathss37{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm\vPi\dlbetss01\kern0.15mm)}--\,compact.
Since the linear \mathss37{\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E}--\,span of
\math{K} is \mathss37{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E}--\,dense, it follows that
\math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.07mm\vPi\dlbetss01\kern0.15mm)\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss32{}^{}{\rm rng}\,{}_{{}^{}}\smb V } is a
separable topology. Taking \math{S=
\CltaurdvPidualbeta{}^{}{\rm rng}\,{}_{{}^{}}\smb V} hence by
Proposition \ref{Pro Edw 8.17.8} on page \pageref{Pro Edw 8.17.8} above
there is $y\ar 1 \in \bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm
\mvsLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm)$ with \math{
{}^{}{\rm rng}\,{}_{{}^{}} y\ar 1\subseteq S} and (\kern0.15mm$*$\kern0.15mm) that \math{
\smb V\fvalss60\smb\Phii\fvalss00\xi=
\int_{\KP{1.1}{}^{}\Cal Omega\,}
\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\ar 1\kern-0.63mm\cdot\kern0.07mm\varphi\rmdss21\mu
} holds for \math{
\varphi\in\smb\Phii\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E} and \mathss31{\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi }. By Pettis'
theorem and reflexivity of \math{\vPi} in fact \math{
y\ar 1 \in \bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm
\mvLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlbetss01\kern0.15mm)
} holds. Hence
there is some \math{N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } such that
for \math{B={}^{}\Cal Omega\kern0.07mm\setminus N} we have \math{
(\kern0.37mm y\ar 1\kern0.37mm|\KP1 B\,;\kern0.07mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm B\kern0.37mm,\kern0.07mm\vPi\dlbetss01\kern0.15mm) }
simply measurable, and so taking
\newline
\math{y=N\kern-.2mm\times\kern-.2mm\{\KPt8\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\kern-.2mm\times\kern-.2mm\{\kern0.37mm 0\kern0.37mm\}\kern0.15mm\}\cupss22
(\kern0.37mm y\ar 1\kern0.37mm|\KP1 B\kern0.37mm) } we get
\math{(\KPt5 y\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\dlbetss01\kern0.15mm) } simply measurable. To
conclude the proof in the reflexive case, it suffices to take
\math{\varphi=\roman x\,A} in (\kern0.15mm$*$\kern0.15mm) above.
| 3,053 | 362,273 |
en
|
train
|
0.47.83
|
In the separable case we instead apply Proposition \ref{Pro Edw 8.17.6} on
page \pageref{Pro Edw 8.17.6} above to get existence of \math{y\ar 1} with
(\kern0.15mm$*$\kern0.15mm) above. To see that \math{y\ar 1} can be modified on a set of
measure zero to get some \math{y} with \math{
(\KPt5 y\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) } simply measurable, we proceed as
follows. We take a countable \math{D\ar 1} such that \math{D\ar 1} is \mathss37{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi}--\,dense. For every fixed \linebreak
\mathss03{\xi\in D\ar 1} we now know
that \math{
(\,\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\ar 1\kern0.37mm;\kern0.07mm\mu\,,\kern0.07mm\bosy K\kern0.37mm) }
is almost measurable, and hence there is some \math{ N\aar 1 \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } such that \math{
(\,\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\ar 1\,|\KP1 B\,;\kern0.07mm
\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm B\kern0.37mm,\kern0.07mm\bosy K\kern0.37mm) } for \math{ B =
{}^{}\Cal Omega\kern0.07mm\setminus N\aar 1 } is measurable. Since \math{D\ar 1} is
countable, by {\sl countable choice\kern0.15mm} we then find \mathss30{ N \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } \linebreak
such that with \math{ B =
{}^{}\Cal Omega\kern0.07mm\setminus N } we have \math{
(\,\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\ar 1\,|\KP1 B\,;\kern0.07mm
\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm B\kern0.37mm,\kern0.07mm\bosy K\kern0.37mm) } measurable for all \linebreak \mathss04{
\xi\in D\ar 1 }. By density, we can extend this to hold for all \mathss31{
\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi}. By Proposition \ref{pro-mea-equ} on page \pageref{pro-mea-equ}
above, this gives that \math{
(\kern0.37mm y\ar 1\,|\KP1 B\,;\kern0.07mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm B\kern0.37mm,\kern0.07mm\vPi\dlsigss00\kern0.07mm) }
is simply measurable, and so it suffices to take \math{y} as in the reflexive
case above.
\end{proof}
| 1,128 | 362,273 |
en
|
train
|
0.47.84
|
From the logical point of view, note that in the nonreflexive case in
Proposition \ref{Pro mA=int ev_x c mu} we may trivially take for example \mathss38{
S=\{\KPt8\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\kern-.2mm\times\kern-.2mm\{\kern0.37mm 0\kern0.37mm\}\kern0.15mm\} }. We give below an alternative
proof for the existence of \math{y} above. It has the drawback of not giving
existence of the separable \math{S} that allowed us to deduce the stronger
measurability in the reflexive case. The underlying argument of applying
Alaoglu's theorem is already shortly sketched in \cite[p.\ 131]{Phil}\,, and
in a more explicit manner it is also utilized in
\cite[pp.\ 594\,--\,595]{Edw}\,. This alternative in fact was our first
approach but then we noticed that using Propositions \ref{Pro Edw 8.17.8} and
\ref{Pro Edw 8.17.6} offers a more uniform way to treating the cases (5) and
(6) in Theorem \nfss A\,\ref{main Th} together.
\vskip1mm
{\it$\null
$
Let $\,\mu$ be a positive measure on
$\,{}^{}\Cal Omega$ with $\,\mu\fvalss02{}^{}\Cal Omega < \lower1.05mm\hbox{$^+$}\infty${\,\rm, }and
with
\linebreak
$\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let \ú$\,E\in\roman{BaS}\kern0.4mmps0(K)$
with $\,\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1$ a compatible dual norm for
$\,F=E\dlbetss12\,$.
Also let $\,m$ be an $\,F\,$--\,measure with $\,{{}^{}{\rm dom}\,{}_{{}^{}}} m={{}^{}{\rm dom}\,{}_{{}^{}}}\mu$ and
such that $\,\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.2mm\circ\kern0.15mm m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A \le \mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A$
holds for all $\,A \in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\,$.
Then there is $\, c \in \kern0.15mm ^{}^{}\Cal Omega\,\Cal L\,(\kern0.15mm E\kern0.37mm,\kern0.07mm\bosy K\kern0.37mm)$ such
that $\,(\,c\KPt8;\kern0.15mm\mu\,,\kern0.07mm E\dlsigss12\kern0.07mm)$ is Pettis and such that
$\, m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x =
\int_{\,A}\kern0.37mm\roman{ev}\kern0.15mm\sbi{\emath x}\kern-0.2mm
\circ\kern0.15mm c\rmdss11\mu$ holds for all \ú$\, A
\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu$ and $\,x\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E\,$.
}
\begin{proof} The assertion being trivial if \math{\mu\fvalss02{}^{}\Cal Omega=0}
holds, assuming \math{\mu\fvalss02{}^{}\Cal Omega > 0} we consider the net \math{
(\kern0.15mm\varDelta\,,\kern0.07mm\bosy c\kern0.37mm) } obtained as follows.
Let $\varDelta$ be the set of all pairs
$(\kern0.15mm\scrmt A\,,\kern0.07mm\scrmt B\kern0.37mm)$ where
\math{\scrmt A\,,\kern0.15mm\scrmt B\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}} m\setminus\kern0.15mm
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } are finite partitions of
${}^{}\Cal Omega$ such that for every $B\in\scrmt B$ there is some $A\in\scrmt A$
with $B\subseteq A \,$. Then $\varDelta$ is a direction, and we take \vskip.3mm\centerline{$
\bosy c=\langle\KP1
{}^{}\Cal Omega\times\hbox{\font\SweD =cmssbx10\SweD U}{}\capss31\{\,(\kern0.37mm\eta\kern0.37mm,\kern0.07mm u\kern0.37mm):
\all A\,\eta\in A\in\scrmt A\impss33
u=(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\,
(\kern0.37mm m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)\KPt9\}:\scrmt A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\varDelta
\KP{1.3}\rangle $} \inskipline{.3}0
thus obtaining a function ${{}^{}{\rm dom}\,{}_{{}^{}}}\varDelta\to
\kern0.15mm^{}^{}\Cal Omega\,\Cal L\,(\kern0.15mm E\kern0.37mm,\kern0.15mm\bosy K\kern0.37mm)$
such that for every $\scrmt A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\varDelta$ we have
$\bosy c\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm\scrmt A$ the function ${}^{}\Cal Omega\owns\eta\mapsto
(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\,(\kern0.37mm m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm) $
when $\eta\in A\in\scrmt A$ holds.
We further let \math{\varLambda} be the set of all pairs \math{
(\kern0.37mm\eta\kern0.37mm,\kern0.07mm u\kern0.37mm)\in{}^{}\Cal Omega\times\Cal L\,(\kern0.15mm E\kern0.37mm,\kern0.07mm\bosy K\kern0.37mm) }
such that \math{u} is a \mathss37{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm E\dlsigss12\kern0.07mm) }--\,limit point
of the net \mathss38{ (\kern0.15mm \varDelta \, , \kern0.07mm
\roman{ev}\sbi\eta\kern-0.2mm\circ\kern0.07mm\bosy c\kern0.37mm) }. Then by {\sl Alaoglu's theorem\kern0.15mm}
we have \mathss36{{}^{}\Cal Omega\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}}\varLambda }, \,and hence by the {\sl axiom of
choice\kern0.15mm} there is a function \math{c\subseteq\varLambda} with \mathss06{ {{}^{}{\rm dom}\,{}_{{}^{}}} c =
{}^{}\Cal Omega}. Arbitrarily fixing \mathss34{x\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E}, \,it remains to show that
\math{\roman{ev}\kern0.15mm\sbi{\emath x}\kern-0.2mm\circ\kern0.15mm c } is inte- grable over every \mathss36{
A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu}, \,and that \math{m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x =
\int_{\,A}\kern0.37mm\roman{ev}\kern0.15mm\sbi{\emath x}\kern-0.2mm\circ\kern0.15mm c\rmdss11\mu } holds.
To see this, we let \math{\varphi} be a Radon\,--\,Nikodym derivative
with respect to $\mu$ of
${{}^{}{\rm dom}\,{}_{{}^{}}}\mu\owns A\mapsto
m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\fvalss10 x \,$,
noting that some such exist since by our assumption
for some $\smb M\in\rbb R^+$ we have
$|\KP1 m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\fvalss10 x\KP1|
\le\smb M\KP1(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm)$ for all
$A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\,$. For the same reason we may assume that
$|\KP1\varphi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\KP1|\le\smb M$ holds for all $
\eta\in {}^{}\Cal Omega\,$. We now have
$m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\fvalss10 x=
\int_{\,A}\kern0.37mm\varphi\rmdss11\mu$
for $A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\, $, and it suffices to show
existence of some $N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}$ such that
$\roman{ev}\kern0.15mm\sbi{\emath x}\kern-0.2mm\circ\kern0.15mm c\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta=
\varphi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta$ holds for all
$\eta\in {}^{}\Cal Omega\kern0.07mm\setminus N\kern0.37mm$.
By taking inverse images under $\varphi$ of partitions of
$\mathbb C\capss31\{\,z:|\,z\,|\le\smb M\KPt8\}$ into sets of diameter
${}<i\kern0.37mm\lower1mm\hbox{$^{^+}$}\kern0.15mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056}\,^{\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}$, we obtain a sequence
$\bosy s\ar 1$ of simple functions such that
$|\KP{1.1}\bosy s\ar 1\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm i\fvalss10\eta - \varphi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\KP{1.2}|
< i\kern0.37mm\lower1mm\hbox{$^{^+}$}\kern0.15mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056}\,^{\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}$ holds for all
$i\in\mathbb No$ and $\eta\in {}^{}\Cal Omega\,$.
| 3,900 | 362,273 |
en
|
train
|
0.47.85
|
To see this, we let \math{\varphi} be a Radon\,--\,Nikodym derivative
with respect to $\mu$ of
${{}^{}{\rm dom}\,{}_{{}^{}}}\mu\owns A\mapsto
m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\fvalss10 x \,$,
noting that some such exist since by our assumption
for some $\smb M\in\rbb R^+$ we have
$|\KP1 m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\fvalss10 x\KP1|
\le\smb M\KP1(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm)$ for all
$A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\,$. For the same reason we may assume that
$|\KP1\varphi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\KP1|\le\smb M$ holds for all $
\eta\in {}^{}\Cal Omega\,$. We now have
$m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\fvalss10 x=
\int_{\,A}\kern0.37mm\varphi\rmdss11\mu$
for $A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\, $, and it suffices to show
existence of some $N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}$ such that
$\roman{ev}\kern0.15mm\sbi{\emath x}\kern-0.2mm\circ\kern0.15mm c\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta=
\varphi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta$ holds for all
$\eta\in {}^{}\Cal Omega\kern0.07mm\setminus N\kern0.37mm$.
By taking inverse images under $\varphi$ of partitions of
$\mathbb C\capss31\{\,z:|\,z\,|\le\smb M\KPt8\}$ into sets of diameter
${}<i\kern0.37mm\lower1mm\hbox{$^{^+}$}\kern0.15mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056}\,^{\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}$, we obtain a sequence
$\bosy s\ar 1$ of simple functions such that
$|\KP{1.1}\bosy s\ar 1\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm i\fvalss10\eta - \varphi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\KP{1.2}|
< i\kern0.37mm\lower1mm\hbox{$^{^+}$}\kern0.15mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056}\,^{\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}$ holds for all
$i\in\mathbb No$ and $\eta\in {}^{}\Cal Omega\,$.
If $\sigma\ar 1\in{}^{}{\rm rng}\,{}_{{}^{}}\bosy s\ar 1$ is such that $
\sigma\ar 1\kern-0.3mm\inve\kern0.37mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm s\kern0.37mm\}
\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}$ holds for
some $s\in{}^{}{\rm rng}\,{}_{{}^{}}\sigma\ar 1\,$, on a set of measure zero we can modify
$\sigma\ar 1$ to get another simple function $\sigma$
such that for every
$s\in{}^{}{\rm rng}\,{}_{{}^{}}\sigma$ we have $\sigma\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm s\kern0.37mm\}
\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\,$.
Using this observation in conjunction with
{\sl countable choice\kern0.15mm} we obtain another sequence $\bosy s$ of
simple functions and some $N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}
$ such that for all $\eta\in {}^{}\Cal Omega\kern0.07mm\setminus N$ and $i\in\mathbb No$ we have
$\bosy s\fvalss01 i\fvalss10\eta=
\bosy s\ar 1\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm i\fvalss10\eta\,$.
Now arbitrarily given \math{\eta\in {}^{}\Cal Omega\kern0.07mm\setminus N} and \math{\varepsilon\in
\rbb R^+} we pick some $\sigma\in{}^{}{\rm rng}\,{}_{{}^{}}\bosy s$ such that for all $\eta\ar 1\in
{}^{}\Cal Omega\kern0.07mm\setminus N$ we have
$|\KP1\sigma\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\ar 1\kern-0.2mm - \varphi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\ar 1\,| < \varepsilon \,$. Then
with $A=\sigma\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\,\sigma\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\,\}$ we take either
$\scrmt A=\{\,A\,,\kern0.07mm {}^{}\Cal Omega\kern0.07mm\setminus A\KPt9\}$
or $\scrmt A=\{\kern0.37mm A\,\}$
according to whether
$A\not={}^{}\Cal Omega$ or $A={}^{}\Cal Omega$ holds, getting then
$\scrmt A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\varDelta$ by construction.
If now $\eta\in B\in\scrmt B\in\varDelta\kern0.07mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\,\scrmt A\,\}$
holds, we have $B\subseteq A$ and hence \vskip.5mm\centerline{$
\roman{ev}\kern0.15mm\sbi{\emath x}\kern-0.2mm\circ\kern0.15mm
\roman{ev}\sbi\eta\kern-0.2mm\circ\kern0.07mm\bosy c\fvalss02\scrmt B
=\bosy c\fvalss02\scrmt B\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\fvalss02\xi=
(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} B\kern0.37mm)^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\,
(\kern0.37mm m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} B\fvalss11\xi\kern0.37mm)=
(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} B\kern0.37mm)^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\int_{\KP1 B}\kern0.37mm\varphi\rmdss11\mu $} \inskipline{.5}0
further giving
$|\KP{1.2}
\roman{ev}\kern0.15mm\sbi{\emath x}\kern-0.2mm\circ\kern0.15mm
\roman{ev}\sbi\eta\kern-0.2mm\circ\kern0.07mm\bosy c\fvalss02\scrmt B
- \varphi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\KP{1.2}| < 2\KP1\varepsilon \,$.
Since $c\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x$ is a $\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\bosy K\,$--\,limit point of
the net
$(\kern0.15mm\varDelta\,,\kern0.07mm
\roman{ev}\kern0.15mm\sbi{\emath x}\kern-0.2mm\circ\kern0.15mm
\roman{ev}\sbi\eta\kern-0.2mm\circ\kern0.07mm\bosy c\kern0.37mm) \KPt8 $, this gives
$c\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x=\varphi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\,$, and having
here $\eta\in {}^{}\Cal Omega\kern0.07mm\setminus N$ arbitrarily fixed, we see that
$\roman{ev}\kern0.15mm\sbi{\emath x}\kern-0.2mm\circ\kern0.15mm c\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta=
\varphi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta$ holds for all
$\eta\in {}^{}\Cal Omega\kern0.07mm\setminus N\kern0.37mm$.
\end{proof}
| 3,360 | 362,273 |
en
|
train
|
0.47.86
|
\begin{corollary}\label{Coro q-var}
Let \ú$\,1\le q<\lower1.05mm\hbox{$^+$}\infty$ and let $\,\mu$ be a positive measure on $\,{}^{}\Cal Omega
${\,\rm, }and with \ú$\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let \ú$\,\vPi\in\roman{BaS}\kern0.4mmps0(K)$ be such
that either \ú$\,\vPi$ is reflexive or \ú$\,\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi$ is a separable
topology. Also let $\,m$ be a $\,\vPi\dlbetss01\,$--\,measure with \ú$\,{{}^{}{\rm dom}\,{}_{{}^{}}} m
= \mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+$ and such that $\,m$ is absolutely $\,\mu\,
$--\,continuous in $\,\vPi\dlbetss01$ with $\,m$ having bounded $\,\mu\,$--$
\RHB{.3}{\KP{1.5} ^q \kern0.37mm}$variation in $\,\vPi\dlbetss01\,$. Then there are
some countable disjoint $\,\scrmt A$ and $\,y$ with \ú$\, \scrmt A \subseteq
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+$ and \ú$
(\KPt5 y\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm)$ simply measurable and such that
{\,\rm(1)} and {\,\rm(2)} and {\,\rm(3)} and {\,\rm(4)} below hold for all $\,
A\in{{}^{}{\rm dom}\,{}_{{}^{}}} m$ and $\,A\kern0.07mm\ar 1\in\scrmt A$ and $\,\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi$ and $\,\eta
\in{}^{}\Cal Omega\,$. {\rm\inskipline14
(1)} \ $\eta\not\in\bigcup\,\scrmt A\impss33
y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta=\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\times\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} \KP1 ${\rm, \inskipline{.5}4
(2)} \ $\bigcup\,\scrmt A\capss31 A=\emptyset\impss33 m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A =
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\times\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} \KP1 ${\rm, \inskipline{.5}4
(3)} \ $A\subseteq A\kern0.07mm\ar 1\impss33 m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\fvalss12\xi =
\int_{\,A}\kern0.37mm\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\rmdss11\mu \KPt6 ${\rm, \inskipline{.5}4
(4)} \ $\vPi$ is reflexive $\impss33
(\KPt5 y\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\dlbetss01\kern0.15mm)$ is simply measurable. \vskip1mm
\end{corollary}
\begin{proof} Let \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1=\seqss44{
\sup\KPt8(\kern0.37mm\Abrs00^1\circ\kern0.15mm u\circss01\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb15 I)
:u\in\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm)} } where
\math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm} is some fix- ed compatible norm
for \mathss31{\vPi}. We first show that there is a countable disjoint
\math{\scrmt C\subseteq\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+ } such that
\math{m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A=\vPi\times\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } for all
\math{A\in{{}^{}{\rm dom}\,{}_{{}^{}}} m} with
\mathss36{\bigcup\,\scrmt C\capss31 A=\emptyset}. To see
this, with \math{
\scrmt A\ar 1=
{{}^{}{\rm dom}\,{}_{{}^{}}} m\capss21\{\,A:\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm
m\kern0.07mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\not=0\KPt9\} }
\newline
we let \mathss38{\Cal P=\{\,(\kern0.15mm\scrmt A\,,\kern0.07mm\scrmt B\kern0.37mm)
:\scrmt A\,,\kern0.15mm\scrmt B\kern0.37mm\text{ are disjoint and }\kern0.37mm
\scrmt A\subseteq\scrmt B\subseteq\scrmt A\ar 1\,\} }.
Then $\Cal P$ is a nonempty partial order, and if
$\Cal C$ is a $\Cal P\,$--\,chain, then
$\bigcup\KP1\Cal C$ is an upper $\Cal P\,$--\,bound. Hence by
{\sl Zorn's lemma\kern0.15mm} there exists some $\Cal P\,$--\,maximal $\scrmt C\kern0.37mm$.
Clearly $\scrmt C$ is as required {\sl if it is countable\kern0.15mm}. To verify this,
we note that $\scrmt C
=\{\KP1\roman C\,n:n\in\rbb Z^+\kern0.15mm\big\}$ when
$\roman C\,n$ is the set of all $A\in\scrmt C$
with \mathss31{n^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1} <
(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)\KP1^{1\kern0.37mm-\,q}\,(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm
m\kern0.07mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm)\RHB{.3}{\KPt8^q} }. If $\roman C\,n$
is finite for every $n\in\rbb Z^+$, then $\scrmt C$ is countable.
If $\roman C\,n$ is
infinite for some $n\in\rbb Z^+$, we
get a contradiction with the assumption that
\math{m} has bounded \mathss37{\mu}--$
\RHB{.3}{\KP{1.5} ^q \kern0.37mm}$variation in \mathss34{\vPi\dlbetss01}.
Next, using Proposition \ref{Pro Phi5.4} on page \pageref{Pro Phi5.4}
above we find a countable disjoint
$\scrmt A\subseteq\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+$ with
$\bigcup\,\scrmt A=\bigcup\,\scrmt C$ and such that
$\sup\KPt8\{\KPt8(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\,
(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm m\kern0.07mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm) :
A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.3mm\cap\kern0.37mm\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\ar 1\kern0.37mm\} < \lower1.05mm\hbox{$^+$}\infty$
holds for every fixed \mathss36{A\ar 1\in\scrmt A}.
Indeed, we just apply Proposition \ref{Pro Phi5.4} separately to
\math{\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm C}
for every fixed \math{C\in\scrmt C} and then take
the union of the thus obtained partitions.
Finally we let \math{\scrmt Y} be the set of all
pairs \math{
(\kern0.15mm A\ar 1\kern0.15mm,\kern0.07mm y\ar 1) } with \mathss03{ A\ar 1 \in
\scrmt A} and such that for \mathss03{
\mu\ar 1=\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\ar 1} and
\math{m\ar 1 = m\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\ar 1} we have \math{
(\kern0.37mm y\ar 1\KPt2;\kern0.07mm\mu\ar 1\kern0.15mm,\kern0.07mm\vPi\dlsigss00\kern0.07mm) }
simply measurable and
Pettis with \math{m\ar 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\fvalss12\xi =
\int_{\,A}\kern0.37mm\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\ar 1\rmdss01\mu }
for all \math{
A\in{{}^{}{\rm dom}\,{}_{{}^{}}} m\ar 1} and \mathss31{\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi }, \,and such that also
\math{(\kern0.37mm y\ar 1\KPt2;\kern0.07mm\mu\ar 1\kern0.15mm,\kern0.07mm\vPi\dlbetss01\kern0.15mm) } is simply
measurable if \math{\vPi} is reflexive. Then
considering arbitrarily fixed \math{
A\ar 1\in\scrmt A} and with \vskip.3mm\centerline{$
| 3,816 | 362,273 |
en
|
train
|
0.47.87
|
Next, using Proposition \ref{Pro Phi5.4} on page \pageref{Pro Phi5.4}
above we find a countable disjoint
$\scrmt A\subseteq\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+$ with
$\bigcup\,\scrmt A=\bigcup\,\scrmt C$ and such that
$\sup\KPt8\{\KPt8(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\,
(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm m\kern0.07mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm) :
A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.3mm\cap\kern0.37mm\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\ar 1\kern0.37mm\} < \lower1.05mm\hbox{$^+$}\infty$
holds for every fixed \mathss36{A\ar 1\in\scrmt A}.
Indeed, we just apply Proposition \ref{Pro Phi5.4} separately to
\math{\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm C}
for every fixed \math{C\in\scrmt C} and then take
the union of the thus obtained partitions.
Finally we let \math{\scrmt Y} be the set of all
pairs \math{
(\kern0.15mm A\ar 1\kern0.15mm,\kern0.07mm y\ar 1) } with \mathss03{ A\ar 1 \in
\scrmt A} and such that for \mathss03{
\mu\ar 1=\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\ar 1} and
\math{m\ar 1 = m\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\ar 1} we have \math{
(\kern0.37mm y\ar 1\KPt2;\kern0.07mm\mu\ar 1\kern0.15mm,\kern0.07mm\vPi\dlsigss00\kern0.07mm) }
simply measurable and
Pettis with \math{m\ar 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\fvalss12\xi =
\int_{\,A}\kern0.37mm\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\ar 1\rmdss01\mu }
for all \math{
A\in{{}^{}{\rm dom}\,{}_{{}^{}}} m\ar 1} and \mathss31{\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi }, \,and such that also
\math{(\kern0.37mm y\ar 1\KPt2;\kern0.07mm\mu\ar 1\kern0.15mm,\kern0.07mm\vPi\dlbetss01\kern0.15mm) } is simply
measurable if \math{\vPi} is reflexive. Then
considering arbitrarily fixed \math{
A\ar 1\in\scrmt A} and with \vskip.3mm\centerline{$
\smb M=\sup\KPt8\{\KPt8(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\,
(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm m\kern0.07mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm) :
A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.3mm\cap\kern0.37mm\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\ar 1\kern0.37mm\} $} \inskipline{.3}0
taking \mathss03{ \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\kern0.15mmrim2 =
\{\,(\kern0.37mm\xi\,,\kern0.07mm\smb M\KPt8 t\kern0.37mm) :
(\kern0.37mm\xi\,,\kern0.07mm t\kern0.37mm)\in\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\KPt8\} } in place of
\math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm} in
Proposition \ref{Pro mA=int ev_x c mu}
on page \pageref{Pro mA=int ev_x c mu} above,
we see that \mathss03{ \scrmt A
\subseteq {{}^{}{\rm dom}\,{}_{{}^{}}}\scrmt Y } holds, and hence by {\sl countable choice\kern0.15mm} there
is
a function \mathss30{\scrmt Y\ar 1\subseteq\scrmt Y} with \mathss34{ \scrmt A \subseteq
{{}^{}{\rm dom}\,{}_{{}^{}}}\scrmt Y\ar 1}. Now taking \vskip.3mm\centerline{$
y = (\kern0.37mm{}^{}\Cal Omega\setminus\bigcup\,\scrmt A\,)
\times\kern-0.2mm\{\KPt8\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\times\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\kern0.15mm\}\cupss24
\bigcup\kern0.37mm{}^{}{\rm rng}\,{}_{{}^{}}\scrmt Y\ar 1 \kern0.37mm $,} \inskipline{.3}0
it is clear that all the asserted properties hold.
\end{proof}
| 2,120 | 362,273 |
en
|
train
|
0.47.88
|
Although we shall not need below the result, as an application of the
Dunford\,--\,Pettis property of \math{\mLrs42^1(\kern0.37mm\mu\kern0.15mm) } we reformulate
the a bit mysterious looking assertion \vskip.5mm\centerline{
\q{if $|\tau_0|\tau<\infty$, then $[x(\tau)|\tau\subset\tau_0]$ is compact
valued}} \inskipline{.5}0
from \cite[p.\ 131]{Phil} in the following
\begin{proposition}
Let $\,\mu$ be a positive measure with \ú$\,\sup{}^{}{\rm rng}\,{}_{{}^{}}\mu < \lower1.05mm\hbox{$^+$}\infty${\,\rm, }
and with \ú$\,\bosy K\in{\kern-0.63mm}$ $\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let \ú$\,F\in\roman{BaS}\kern0.4mmps0(K)$ be reflexive
with $\,\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm$ a compatible norm. Also let \ú$\,m \in{\kern-0.63mm}$ \ú$
^{\roman{dom\,}\mu}\,\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E$ be such that \ú$\,m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm A\cupss31 B\kern0.37mm)
= (\kern0.37mm m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A + m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} B\kern0.37mm)\svs E$ and \ú$\,
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A \le \mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A$ hold for all \ú$\,A\,,\kern0.15mm B
\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu$ with \ú$\,A\capss31 B=\emptyset\,$. Then $\,{}^{}{\rm rng}\,{}_{{}^{}} m$ is relatively $\,
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F\,$--\,compact.
\end{proposition}
\begin{proof} Let \math{{}^{}\Cal Omega=\kern0.15mm\bigcup\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\mu } and \mathss38{ E =
\mvLrs42^1(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm) }. Also
putting \math{\roman x\,A=
(\kern0.37mm{}^{}\Cal Omega\kern0.07mm\setminus A\kern0.37mm)\times\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\cupss22
(\kern0.15mm A\times\kern-0.2mm\{\kern0.37mm 1\kern0.37mm\}\kern0.15mm\sbig)0}
and \math{\eightroman X\,A=\uniqset\smb\Phii:
\roman x\,A\in\smb\Phii\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E} let \math{S}
be the linear \mathss37{\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E}--\,span
of
\mathss38{\{\KPt8\eightroman X\,A:
A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern0.15mm\big\} }. Thus
\math{S} is the set of all \math{\smb\Phii\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E} such that
there is \math{\varphi\in\smb\Phii} with
\math{{}^{}{\rm rng}\,{}_{{}^{}}\varphi} finite. We know that
\math{S} is \mathss37{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E}--\,dense. Then
we let \math{\smb V\aR 0} be the unique linear extension
\math{\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E_{\KPt8|\,S}\to\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F } of
\mathss38{\{\,(\kern0.37mm\eightroman X\,A
\,,\kern0.07mm m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm):
A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern0.15mm\big\} }, \,noting that by the assumptions
on \math{m} we indeed get a linear map.
For finite functions \math{\bosy s\subseteq
{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\ar 1\kern-0.3mm\times\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\bosy K} with \math{{{}^{}{\rm dom}\,{}_{{}^{}}}\bosy s} disjoint
and
\newline
\math{\varphi=
\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\bosy K\expnota^\kern0.15mm\aars A_1]_{vs}\text{\KPt8-}
\sum_{\,A\kern0.37mm\in\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.3mm\bmii8 s\,}
(\kern0.37mm\bosy s\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)\KP1\roman x\,A
\in\smb\Phii\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E }
and for \math{u\in\Cal L\,(\kern0.15mm F\kern0.07mm,\kern0.07mm\bosy K\kern0.37mm) } with \math{
\sup\KPt8(\kern0.37mm\Abrs00^1\circ\kern0.15mm u\circss01\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb15 I)
\le 1 } we obtain \vskip.5mm
$|\KP1 u\circss00 \smb V\aR 0\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Phii\KP1|=
\big|\,
\sum_{\,A\kern0.37mm\in\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.3mm\bmii8 s\,}
(\kern0.37mm\bosy s\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)\KP1
(\kern0.37mm u\circss00 m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)\KP1|\le
\sum_{\,A\kern0.37mm\in\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.3mm\bmii8 s\,}|\KP1
\bosy s\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\KP1|\KP1|\KP1
u\circss00 m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\KP1| $ \inskipline1{11}
${}\le
\sum_{\,A\kern0.37mm\in\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.3mm\bmii8 s\,}|\KP1
\bosy s\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\KP1|\KP1
(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)\le
\sum_{\,A\kern0.37mm\in\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.3mm\bmii8 s\,}|\KP1
\bosy s\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\KP1|\KP1
(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)=
\|\,\varphi\,\|\Lnorss33^1_{\aars\mu_1} \, $. \inskipline{.7}0
Consequently \math{\smb V\aR 0} has a unique continuous extension \mathss38{
\smb V\in\Cal L\,(\kern0.15mm E\kern0.37mm,\kern0.07mm F\kern0.37mm) }.
Taking \mathss38{K=\{\KPt8\eightroman X\,A:A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\KPt8\} }, \,from Lemma \ref{Le L^1_si-compa}
on page \pageref{Le L^1_si-compa} above we see that \math{K} is relatively \mathss37{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm E\subsigrs04)}--\,compact. Since by
Corollary \ref{Cor L^1 is D-S} on page \pageref{Cor L^1 is D-S} above
\math{E} is a \erm{DP\,}--\,space, noting that by reflexivity of
\math{F} all bounded sets in \math{F} are relatively
\mathss37{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(F\!\RHB{.25}{\subsigma})}--\,compact, we
see that \math{\smb V\KPt8\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.3mm K} is
relatively \mathss37{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F}--\,compact. Noting that also
\math{{}^{}{\rm rng}\,{}_{{}^{}} m\subseteq\smb V\KPt8\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.3mm K} holds,
we are done.
\end{proof}
| 3,809 | 362,273 |
en
|
train
|
0.47.89
|
\Ssubhead D Duality of Bochner spaces \label{Sec D}
Proceeding by a sequence of lemmas, we here give the proof of
Theorem \nfss A\,\ref{main Th} on page \pageref{main Th} above. From now on
untill the end of the proof of Lemma \nfss A\,\ref{final lemma} on
page \pageref{endmpf} below, without further mention we let \math{p\,,\kern0.15mm
\bosy K\kern0.37mm,\kern0.15mm\vPi\kern0.15mm,\kern0.15mm\mu\,,\kern0.15mm{}^{}\Cal Omega
\,,\kern0.15mm F\kern0.15mm,\kern0.15mm F\aar 1\kern0.37mm,\kern0.15mm\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm } be as in
Theorem \nfss A\,\ref{main Th}\kern0.15mm. For short, we call this assumption
together with the temporary shorthands below
{\it Assumptions
\hbox{\font\≈=cmssi9\≈A}\kern0.15mm}. From Corollary \ref{Coro Io inj etc}
on page \pageref{Coro Io inj etc} above, we see that \math{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm} is an
injective continuous linear map \mathss34{F\aar 1\to F\dlbetss10 }. Since by
Theorem \ref{Th L_s^p Ba} and Corollary \ref{Cor L^p Ban} the spaces \math{
F\aar 1} and \math{F\dlbetss10} are \erm Banachable, by the open mapping
theorem we only need to verify the surjectivity \mathss34{
\Cal L\,(\kern0.15mm F\kern0.07mm,\kern0.07mm\bosy K\kern0.37mm)\subseteq{}^{}{\rm rng}\,{}_{{}^{}}\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm \label{page surj}
}. This we shall do separately for \mathss30{p=1} under (1) or (2) or (3) or
(4) and for \math{1 < p < \lower1.05mm\hbox{$^+$}\infty} under (5) or (6)\,.
Fixing a compatible norm \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm}
for \math{\vPi} and letting \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1} be the dual norm, we introduce the
following shorthands \vskip.6mm
$\|\,\smb X\kern0.37mm\|\sNorF = \inf\kern0.15mm\big\{\KPt8\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 x\KP1
\|\Lnorss33^p_\mu\kern-0.2mm:x\in\smb X\,\}
$ and \KP7 \vskip.4mm
$\|\,\smb Y\KPt8\|\sNorFp = \inf\kern0.15mm\big\{\KPt8\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\ar 1\kern-0.2mm\circ\kern0.15mm y\KP1
\|\Lnorss50^{p^*}_\mu\kern-0.2mm:y\in\smb Y\KPp1.1\}
$ and \KP7 \vskip.4mm
$\|\KPt8\smb U\,\| = \sup\kern0.37mm\big\{\KP1|\KP{1.1}\smb U\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb X\KPt9| :
\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\kern0.37mm\text{ and }\kern0.37mm \|\,\smb X\kern0.37mm\|\sNorF\le 1\KPt9\}
$
and \KP7 \vskip.4mm
$\roman f\,u\,\xi=\uniqset\smb X:{}$ \inskipline0{18.5}
$(\kern0.37mm{}^{}\Cal Omega\setminus\kern-0.2mm{{}^{}{\rm dom}\,{}_{{}^{}}} u\kern0.37mm)\times\kern-0.2mm\{\,\Bnull_\vPi\}\cupss22
\seqss33{((\kern0.37mm u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)\KP1\xi\kern0.37mm)\svs\vPi\kern-0.3mm:\eta\in{{}^{}{\rm dom}\,{}_{{}^{}}} u}
\in\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F \kern0.15mm $. \inskipline{.6}0
Note that by the discussion after the proof of Lemma \ref{Le 0_{L^p}} on page \pageref{discus inf N = N}
above we in fact have \math{\|\,\smb X\kern0.37mm\|\sNorF =
\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 x\KP1 \|\Lnorss33^p_\mu } for \mathss30{x\in\smb X\in
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F}.
\begin{Alemma}\label{LeA(1)}
If under {\,\rm Assumptions \nfss A} also {\,\rm(1)} holds{\kern0.37mm\rm, }then $\,
\Cal L\,(\kern0.15mm F\kern0.07mm,\kern0.07mm\bosy K\kern0.37mm)\subseteq{}^{}{\rm rng}\,{}_{{}^{}}\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\kern0.37mm$.
\end{Alemma}
\begin{proof} Arbitrarily fix \math{ \smb U \in
\Cal L\,(\kern0.15mm F\kern0.07mm,\kern0.07mm\bosy K\kern0.37mm) } and let \math{\scrmt A} and \math{
N\kern0.15mmrim1} be as in Definitions \ref{df decomp}\,(2) on page \pageref{decos A}
above. Then let \math{\scrmt Y} be the set of all pairs \math{
(\kern0.15mm A\ar 1\KPt2;\kern0.15mm y\ar 1\kern0.15mm,\kern0.07mm S\ar 1) } with \mathss03{ A\ar 1 \in
\scrmt A} and \math{{}^{}{\rm rng}\,{}_{{}^{}} y\ar 1\subseteq S\ar 1} and \math{S\ar 1} a separable
closed linear subspace in \math{\vPi\dlbetss01} and such that for \math{
\mu\ar 1=\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\ar 1} and \mathss30{m = \langle\kern0.37mm\seqss33{
\smb U\fvalss11\lfloor\,^{1\kern0.15mm,\kern0.37mm\aars\mu_1\kern0.07mm,\kern0.37mm\vPi}\kern0.37mm\xi\kern0.15mm\sbi A \kern-0.2mm
: \xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} : A \in \mu\ar 1\kern-0.3mm\inve\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+\kern0.37mm
\big\rangle } we have \math{
(\kern0.37mm y\ar 1\KPt2;\kern0.07mm\mu\ar 1\kern0.15mm,\kern0.07mm\vPi\dlbetss01\kern0.15mm) } measurable and
Pettis with \math{m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\fvalss12\xi =
\int_{\,A}\kern0.37mm\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\ar 1\rmdss01\mu } for all \linebreak \mathss03{
A\in{{}^{}{\rm dom}\,{}_{{}^{}}} m} and \mathss31{\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi }. Now considering arbitrarily fixed \math{
A\ar 1\in\scrmt A} and choosing \mathss03{ \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\kern0.15mmrim2 =
\{\,(\kern0.37mm\xi\,,\kern0.15mm t\KPt8\|\KPt8\smb U\,\|\kern0.15mm\sbig)0 :
(\kern0.37mm\xi\,,\kern0.07mm t\kern0.37mm)\in\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\KPt8\} } by Proposition \ref{Pro mA=int ev_x c mu}
on page \pageref{Pro mA=int ev_x c mu} above we see that \mathss03{ \scrmt A
\subseteq {{}^{}{\rm dom}\,{}_{{}^{}}}\scrmt Y } holds, and hence by the {\sl axiom of choice\kern0.15mm} there is
a function \mathss30{\scrmt Y\ar 1\subseteq\scrmt Y} with \mathss34{ \scrmt A \subseteq
{{}^{}{\rm dom}\,{}_{{}^{}}}\scrmt Y\ar 1}. Let \mathss34{ y = (\kern0.37mm{}^{}\Cal Omega\setminus N\kern0.15mmrim1\kern0.15mm)
\times\kern-0.2mm\{\KPt8\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\times\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\kern0.15mm\}\cupss24
\bigcup\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.3mm{}^{}{\rm rng}\,{}_{{}^{}}\scrmt Y\ar 1}.
To verify that \math{y\in\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1} holds, it suffices to get \mathss38{
\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.2mm\circ\kern0.15mm y\KP1\|\Lnorss33^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}_\mu\le
\|\KPt8\smb U\,\| }. This in turn follows if for every fixed \math{A\ar 1\in
\scrmt A} we show existence of some \math{ N \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } such that \math{
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.2mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\le\|\KPt8\smb U\,\| } holds for \mathss30{
\eta\in A\ar 1\kern-0.63mm\setminus N}. Now for \math{A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\capss22\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\ar 1 }
and \math{\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} we have \math{
\smb U\fvalss11\lfloor\,^{1\kern0.15mm,\kern0.37mm\mu\kern0.15mm,\kern0.37mm\vPi}\kern0.37mm\xi\kern0.15mm\sbi A
= m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\fvalss01\xi
= \int_{\,A}\kern0.37mm\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\rmdss11\mu } and hence \vskip.2mm\centerline{$
| 3,978 | 362,273 |
en
|
train
|
0.47.90
|
To verify that \math{y\in\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1} holds, it suffices to get \mathss38{
\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.2mm\circ\kern0.15mm y\KP1\|\Lnorss33^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}_\mu\le
\|\KPt8\smb U\,\| }. This in turn follows if for every fixed \math{A\ar 1\in
\scrmt A} we show existence of some \math{ N \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } such that \math{
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.2mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\le\|\KPt8\smb U\,\| } holds for \mathss30{
\eta\in A\ar 1\kern-0.63mm\setminus N}. Now for \math{A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\capss22\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\ar 1 }
and \math{\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} we have \math{
\smb U\fvalss11\lfloor\,^{1\kern0.15mm,\kern0.37mm\mu\kern0.15mm,\kern0.37mm\vPi}\kern0.37mm\xi\kern0.15mm\sbi A
= m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\fvalss01\xi
= \int_{\,A}\kern0.37mm\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\rmdss11\mu } and hence \vskip.2mm\centerline{$
\big|\kern0.15mm\int_{\,A}\kern0.37mm\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\rmdss11\mu\KP1|
\le \|\KPt8\smb U\,\|\KP1(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss02\xi\kern0.37mm)\KP1
(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm) \KP1 $.} \inskipline{.6}0
Then by Corollary \ref{Coro |f|<M} on page \pageref{Coro |f|<M}
above for every \math{\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} there is \math{ N\aar 1 \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } such that \math{
|\KPp1.1 y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\fvalss01\xi\KP1| \le
\|\KPt8\smb U\,\|\KP1(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss02\xi\kern0.37mm) } holds for \mathss34{ \eta
\in A\ar 1\kern-0.63mm\setminus N\aar 1 }.
Now taking \math{S\ar 1=
\roman{pr}\ar 2\circ\scrmt Y\ar 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.07mm\ar 1 } in place of \math{
S} in Lemma \ref{Le Nu_1 = sup ...} on page \pageref{Le Nu_1 = sup ...} above,
let \math{D} be as given there. Then considering fixed \math{\xi\in D} we find \math{
N\aar 1\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } with \math{
|\KPp1.1 y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\fvalss01\xi\KP1| \le
\|\KPt8\smb U\,\|\KP1(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss02\xi\kern0.37mm) } for all \mathss34{ \eta \in
A\ar 1\kern-0.63mm\setminus N\aar 1}. By {\sl countable choice\kern0.15mm} taking as \math{N}
the union of these \math{N\aar 1} we get \math{
|\KPp1.1 y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\fvalss01\xi\KP1| \le
\|\KPt8\smb U\,\|\KP1(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss02\xi\kern0.37mm) } for all \math{ \eta \in
A\ar 1\kern-0.63mm\setminus N } and \mathss34{\xi\in D}. Now having \mathss38{
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta =
\sup \KPt8(\kern0.37mm\Abrs00^1\circ\kern0.15mm(\kern0.37mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)\kern0.07mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.3mm D\kern0.37mm)
\le\|\KPt8\smb U\,\|} for all \mathss30{\eta\in A\ar 1\kern-0.63mm\setminus N}, \,the
assertion follows.
Thus having \math{y\in\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1} there is \math{\smb Y} with \mathss34{
y\in\smb Y\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1}. To proceed, we first note that we now have \math{
m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\fvalss12\xi = \label{Le A2 final ded}
\int_{\,A}\kern0.37mm\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\rmdss11\mu } for all \math{A\in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+} and \mathss31{\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi}. To see this,
let \math{\scrmt C=\scrmt A\capss31\{\,A\kern0.07mm\ar 1\kern-0.63mm:A\kern0.07mm\ar 1\kern-0.2mm\cap\kern0.15mm A
\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern0.15mm\big\} } and \mathss36{N = N\kern0.15mmrim1\cup\kern0.37mm
\bigcup\KP1(\kern0.15mm\scrmt A\kern0.15mm\setminus\scrmt C\kern0.37mm)\capss21 A }. Then \math{
\scrmt C} is countable since otherwise \math{A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+}
would be contradicted. In addition \math{ N \in \bigcup\KPt8\{\KPt8
\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm N\aar 1\kern-0.3mm:N\aar 1\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} \kern0.15mm\} } holds
with \mathss30{ A = \bigcup\KP1(\kern0.15mm\scrmt C\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss31 A\kern0.37mm)\cupss21 N}.
Now by do- \linebreak minated convergence we obtain \inskipline{.6}{20.2}
$ m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\fvalss12\xi
= \smb U\fvalss11\lfloor\,^{1\kern0.15mm,\kern0.37mm\mu\kern0.15mm,\kern0.37mm\vPi}\kern0.37mm\xi\kern0.15mm\sbi A
= \sum_{\,\aars A_1\kern0.15mm\in\kern0.37mm\scrm7 A\,}(\kern0.37mm
\smb U\fvalss11\lfloor\,^{1\kern0.15mm,\kern0.37mm\mu\kern0.15mm,\kern0.37mm\vPi}\kern0.37mm\xi\kern0.15mm\sbi
{\aars A_1\capss25 A}\kern0.15mm\sbig)0 $ \inskipline{.6}{31}
${}
= \sum_{\,\aars A_1\kern0.15mm\in\kern0.37mm\scrm7 A\,}
m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm A\kern0.07mm\ar 1\kern-0.3mm\cap\kern0.15mm A\kern0.37mm)\fvalss01\xi $ \inskipline{.4}{31}
${}
= \sum_{\,\aars A_1\kern0.15mm\in\kern0.37mm\scrm7 A\kern0.37mm}\int_{\,\aars A_1\capss25 A}\kern0.37mm
\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\rmdss11\mu
= \int_{\,A}\kern0.37mm\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\rmdss11\mu \,$. \inskipline{.8}0
Then by Lemma \ref{Le-first} on page \pageref{Le-first} above we have \vskip.3mm\centerline{$
U=
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\kern-0.2mm\times\mathbb C\capss31\{\,(\kern0.37mm\smb X\kern0.15mm,\kern0.07mm t\kern0.37mm) :
\aall{x\in\smb X}\,
t = \int_{\KPp1.1{}^{}\Cal Omega\,}y\,.\KPt8 x\rmdss11\mu\KPt9\} $} \inskipline{.5}0
and hence \math{ \smb U = \kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\kern-0.2mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm\smb Y} holds and so \math{
\smb U\in{}^{}{\rm rng}\,{}_{{}^{}}\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm} is established.
\end{proof}
| 4,068 | 362,273 |
en
|
train
|
0.47.91
|
\begin{Alemma}\label{LeA(2)}
If under {\,\rm Assumptions \nfss A} also {\,\rm(2)} holds{\kern0.37mm\rm, }then $\,
\Cal L\,(\kern0.15mm F\kern0.07mm,\kern0.07mm\bosy K\kern0.37mm)\subseteq{}^{}{\rm rng}\,{}_{{}^{}}\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\kern0.37mm$.
\end{Alemma}
\begin{proof} Given \mathss38{\smb U
\in\Cal L\,(\kern0.15mm F\kern0.07mm,\kern0.07mm\bosy K\kern0.37mm) }, \,letting \vskip.5mm\centerline{$
\smb V=
\kern0.37mm\big\langle\KPt8
\{\,(\kern0.37mm\xi\,,\kern0.07mm\smb U\fvalss22\roman f\KPt8\varphi\KP1\xi
\kern0.37mm):\vPi:\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\kern0.37mm\text{ and }\kern0.37mm\varphi\in\smb\Phii\,\}
:
\smb\Phii \in
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm\mvLrs42^1(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm)
\KP1\big\rangle \KP1 $,} \inskipline{.5}0
we easily see \math{\smb V \in
\Cal L\,(\kern0.37mm\mvLrs42^1(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm)\,,\kern0.07mm\vPi\dlbetss01\kern0.15mm) }
to hold. Hence by
Proposition \ref{Pro Edw 8.17.6} on page \pageref{Pro Edw 8.17.6} above there
exists some \math{y \in \bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm
\mvsLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) } such that \vskip.5mm\centerline{$
\smb U\fvalss22\roman f\KPt8\varphi\KP1\xi
=\smb V\fvalss60\smb\Phii\fvalss00\xi=
\int_{\KP{1.1}{}^{}\Cal Omega\,}
\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\cdot\varphi\rmdss21\mu $} \inskipline{.5}0
holds for \math{\varphi\in\smb\Phii\in{{}^{}{\rm dom}\,{}_{{}^{}}}\smb V} and \mathss31{ \xi \in
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi }. Noting that from \mathss30{ y \in \bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm
\mvsLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) } we directly get \math{
\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.2mm\circ\kern0.15mm y\KP1\|\Lnorss33^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}_\mu < \lower1.05mm\hbox{$^+$}\infty } now
Lemma \ref{Le-first} gives the conclusion similarly as in the proof of Lemma \nfss A\,\ref{LeA(1)}
above.
\end{proof}
\begin{Alemma}\label{LeA(3)}
If under {\,\rm Assumptions \nfss A} also {\,\rm(3)} holds{\kern0.37mm\rm, }then $\,
\Cal L\,(\kern0.15mm F\kern0.07mm,\kern0.07mm\bosy K\kern0.37mm)\subseteq{}^{}{\rm rng}\,{}_{{}^{}}\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\kern0.37mm$.
\end{Alemma}
\begin{proof} Given \mathss38{\smb U
\in\Cal L\,(\kern0.15mm F\kern0.07mm,\kern0.07mm\bosy K\kern0.37mm) }, \,define $\smb V:
\mvLrs42^1(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm)\to\vPi\dlbetss01$ by
$\smb V\fvalss60\smb\Phii\fvalss00\xi=
\smb U\fvalss22\roman f\KPt8\varphi\KP1\xi$ for
$\varphi \in \smb\Phii \in
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm\mvLrs42^1(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm)$ and $\,\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\kern0.15mm$.
Then by
Proposition \ref{Pro Edw 8.17.8} on page \pageref{Pro Edw 8.17.8} above there
is $y \in \bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm
\mvsLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm)$ with
$\smb U\fvalss22\roman f\KPt8\varphi\KP1\xi
=\smb V\fvalss60\smb\Phii\fvalss00\xi=
\int_{\KP{1.1}{}^{}\Cal Omega\,}
\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\cdot\varphi\rmdss21\mu$ .
The rest proceeds as in the proof of Lemma \nfss A\,\ref{LeA(1)} above.
\end{proof}
| 2,075 | 362,273 |
en
|
train
|
0.47.92
|
\begin{Alemma}\label{LeA(4)}
If under {\,\rm Assumptions \nfss A} also {\,\rm(4)} holds{\kern0.37mm\rm, }then $\,
\Cal L\,(\kern0.15mm F\kern0.07mm,\kern0.07mm\bosy K\kern0.37mm)\subseteq{}^{}{\rm rng}\,{}_{{}^{}}\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\kern0.37mm$.
\end{Alemma}
\begin{proof} \newcommand\erU{\hbox{\font\≈=cmr8\≈U}\kern.7mm}
Arbitrarily fix \mathss38{\smb U \in
\Cal L\,(\kern0.15mm F\kern0.07mm,\kern0.07mm\bosy K\kern0.37mm) }. Putting \math{ G =
\mvLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm) } and letting \math{c} be as
given by (4) in Theorem \nfss A\,\ref{main Th} let \math{c\ar 1 = c } if \math{
\bosy K=\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R} holds and in the complex case let \math{ c\ar 1 =
\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 3\circ\kern0.15mm(\kern0.37mm c\hbox{${}\times\kern-2.7mm\lower.9mm\hbox{\font\SweD =cmr5\SweD f}\kern1.93mm$} c\kern0.37mm)\circ\kern0.15mm\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 2} where with \math{
S = \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\mLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\kern0.15mm) \times
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\mLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\kern0.15mm) } we have \inskipline{.5}{10}
$\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 3 = \{\,(\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm,\kern0.07mm x + \imag\KPt8 y\kern0.37mm) :
x\kern0.37mm,\kern0.15mm y\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\lll^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}\kern0.15mm(\kern0.15mm{}^{}\Cal Omega\kern0.15mm)\KPt8\} \KP{33} $ and \inskipline{.2}{10}
$\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 2 = \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm G\times S\capss31\{\,(\kern0.37mm\smb\Psii\,;\kern0.15mm
\smb\Psii\ar 1\kern0.15mm,\kern0.07mm\smb\Psii\aR 2\kern0.07mm) : \aall{\psi\ar 1\in\smb\Psii\ar 1\KPt2
,\kern0.15mm\psi\ar 2\in\smb\Psii\ar 2}\,
\psi\ar 1\kern-0.2mm + \imag\KPt8\psi\ar 2\in\smb\Psii\,\} \KP1 $. \inskipline{.5}0
Then \math{c\ar 1} is a continuous linear choice function \math{G\to
\lll^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}\kern0.15mm(\kern0.37mm{}^{}\Cal Omega\,,\kern0.07mm\bosy K\kern0.37mm) } and hence there is some \math{
\smb A\in\lbb R_+} with the property that \math{
\|\KP1 c\fvalss01\smb\Psii\KP1\|\lllnor_{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}
\le \smb A\KP1\|\,\psi\,\|\Lnorss33^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}_\mu } holds for \mathss30{
\psi\in\kern-0.63mm} \mathss02{\smb\Psii\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm G}. Further let \math{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1} be
as in Proposition \ref{Pro L^1'=L^i} on page \pageref{Pro L^1'=L^i} above.
Then for \mathss30{ E = \kern-0.63mm } \mathss03{\mvLrs42^1(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm) }
with \math{\erU\xi = \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E\times\ssbb00 C\capss41\{\,
(\kern0.37mm\smb\Phii,\kern0.07mm t\kern0.37mm) : \aall{\varphi\in\smb\Phii}\, t =
\smb U\fvalss11\roman f\,\varphi\KP1\xi\KPt9\} } we obtain \math{ \smb V \in \kern-0.63mm } \mathss03{
\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.15mm\lll^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}\kern0.15mm(\kern0.37mm{}^{}\Cal Omega\,,\kern0.07mm\bosy K\kern0.37mm)) }
by taking \mathss38{\smb V = c\ar 1\kern-0.3mm\circ\kern0.15mm\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1\kern-0.3mm\inve\circ\kern0.15mm
\seqss33{\erU\xi:\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} }.
Now for \math{\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} and \math{\varphi\in\smb\Phii\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E} we
have \math{\smb V\fvalss50\xi\in\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1\kern-0.3mm\inve\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm\erU\xi } and
hence \mathss30{\smb U\fvalss11\roman f\,\varphi\KP1\xi } \mathss03{ =
\erU\xi\kern0.15mm\fvalss10\smb\Phii =
\int_{\KP{1.1}{}^{}\Cal Omega}\kern0.15mm\smb V\fvalss50\xi\cdot\varphi\rmdss11\mu }. Taking \inskipline{.5}{12}
$\smb B = \inf\,\{\KPt8\sup\kern0.37mm\big\{\,\big|\kern0.15mm\int_{\KPp1.1{}^{}\Cal Omega\,}
\psi\cdot\varphi\rmdss21\mu\KP1| : \varphi\in\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E\kern0.37mm\text{
and }\kern0.37mm \|\,\varphi\,\|\Lnorss33^1_\mu\le 1\KPt8\}$ \inskipline{.5}{58.6}
${} : \psi\in\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm G\kern0.37mm\text{ and }\kern0.37mm
\|\,\psi\,\|\Lnorss33^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}_\mu = 1\KPt8\} \KP1 $, \inskipline{.5}0
we have \math{\smb B\in\rbb R^+} unless \math{{}^{}\Cal Omega} is \mathss37{\mu
}--\,negligible in which case the assertion of the lemma to be proved
trivially holds. Then for \math{\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} we get \inskipline{.4}4
(\kern0.15mm$*$\kern0.15mm) \ $\|\KP1\smb V\fvalss50\xi\KP1\|\lllnor_{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty} \le
\smb A\KP1\|\KP1\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1\kern-0.3mm\inve\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm\erU\xi
\KP1\|\Lnorss33^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}_\mu \le \smb A\,\smb B^{\kern0.37mm\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\,
\|\KPt8\smb U\,\|\KP1(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss12\xi\kern0.37mm) \KP1 $. \vskip.5mm
Now taking \mathss38{ y =
\seqss33{\roman{ev}\kern0.15mm\sbi{\eta}\kern-0.2mm\circ
\smb V
:\eta\in{}^{}\Cal Omega} }, \,trivially \math{
(\KPt5 y\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) } is finitely almost sca- larly
measurable, and by (\kern0.15mm$*$\kern0.15mm) above having \math{
\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.2mm\circ\kern0.15mm y\KP1\|\Lnorss33^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}_\mu \le
\smb A\,\smb B^{\kern0.37mm\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\,\|\KPt8\smb U\,\| } we get \mathss02{ y \in
\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1 }. Then the conclusion follows from Lemma \ref{Le-first}
similarly as above.
\end{proof}
| 3,730 | 362,273 |
en
|
train
|
0.47.93
|
As opposed to the case (1) in Lemma \nfss A\,\ref{LeA(1)} above, note that in
the cases (2) and (3) and (4) in Lemmas \nfss A\,\ref{LeA(2)} and
\nfss A\,\ref{LeA(3)} and \nfss A\,\ref{LeA(4)} we only got \math{
\|\,\smb Y\KPt8\|\sNorFp\le\smb A\KP1\|\KP1\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\kern-0.2mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm\smb Y\KPp1.2\| }
for all \mathss03{\smb Y\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1} for some \math{\smb A} with \math{
1\le\smb A<\lower1.05mm\hbox{$^+$}\infty} and possibly \math{1<\smb A}.
\begin{Alemma}\label{final lemma}
If under {\,\rm Assumptions \nfss A} also
{\,\rm(5)} or {\,\rm(6)} holds{\kern0.37mm\rm, } \inskipline{.2}{54.3}
then $\,\Cal L\,(\kern0.15mm F\kern0.07mm,\kern0.07mm\bosy K\kern0.37mm)\subseteq{}^{}{\rm rng}\,{}_{{}^{}}\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\kern0.37mm$.
\end{Alemma}
\begin{proof} Since the verification is quite long ending on page \pageref{endmpf}
below, we devide it into Steps 1\kern0.37mm$,\ldots\,$4\kern0.37mm. Now, arbitrarily fixing \math{
\smb U \in \Cal L\,(\kern0.15mm F\kern0.07mm,\kern0.07mm\bosy K\kern0.37mm) } let \vskip.5mm\centerline{$
m = \langle\kern0.37mm\seqss33{ \smb U\fvalss11
\lfloor\,^{p\kern0.15mm,\kern0.37mm\mu\kern0.15mm,\kern0.37mm\vPi}\kern0.37mm\xi\kern0.15mm\sbi A\kern-0.2mm : \xi \in
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} : A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+\kern0.37mm\big\rangle \KP1 $.}
\Step 1.0 We first show that \math{m} has bounded \mathss37{
\mu}--$\KP{1.5} ^{p\sast}\kern0.37mm$variation in \mathss34{\vPi\dlbetss01
}. Indeed, we show that \math{
\sum_{\,A\kern0.37mm\in\kern0.37mm\scrm7 A\,}( (\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)
\KP1^{1\kern0.37mm-\,p\sast}\kern0.37mm(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm m\kern0.07mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm)
\KP1^{p\sast}\kern0.15mm\sbig)0
\le\|\KPt8\smb U\,\|\KP1^{p\sast} } holds for arbitrarily given finite
disjoint \mathss30{\scrmt A\subseteq\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+}. In order to get
this, we first note that for arbitrarily given \math{
\bosy\xi\in\kern0.15mm^{\scrm7 A}\,(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb15 I) } it suffices to
show that \vskip.0mm\centerline{$
\big(\kern0.15mm
\sum_{\,A\kern0.37mm\in\kern0.37mm\scrm7 A\,}( (\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)
\KP1^{1\kern0.37mm-\,p\sast}\kern0.37mm
|\KPp1.1 m\,.\KPt9\bosy\xi\KPt4\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\KN{.4}A
\KP1|\KP1^{p\sast}\kern0.37mm\sbig)0\sbig)0\KP1^{p\sast\kern0.15mm^{-1}}\kern-0.3mm\le\kern0.15mm
\|\KPt8\smb U\,\| $} \inskipline{.5}0
holds since otherwise we could easily get a contradiction.
$\null
$
In order to get this, taking
\math{s=p^{\,*\kern-0.3mm}-1} and with the short-
\linebreak
hand \math{
\roman v\kern0.37mm A=(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)
\KP1^{p\sast\kern0.15mm^{-1}\kern0.07mm -\kern0.37mm 1}\,(\kern0.37mm
m\,.\KPt9\bosy\xi\KPt4\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\KN{.4}A\kern0.37mm) } putting \math{v=
\seqss33{\roman v\kern0.37mm A:A\in\scrmt A} } we have
\math{v\in\kern0.15mm^{\scrm7 A}\,\ssbb10 C } and we need to show that \math{
\|\,v\,\|\lllnor_{p\sast}\le\|\KPt8\smb U\,\| } holds. We may assume that
\math{\|\,v\,\|\lllnor_{p\sast}\not=0 } holds, and then taking \math{u=
\seqss33{\roman u\,A:A\in\scrmt A} } where
\math{\roman u\,A=
\|\,v\,\|\lllnor_{p\sast}\kern-0.2mm^{\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000}\emath s}\,
|\KP1\roman v\kern0.37mm A\KP1|\KPt8^{\emath s\kern0.37mm - \kern0.37mm 1}\,
\overline{\roman v\kern0.37mm A\RHB{.0}{\KN{.99}\phantom{'}}}
} if \math{\roman v\kern0.37mm A\not=0} holds, otherwise having
\mathss36{\roman u\,A=0}, \,we now have \math{
\|\,u\,\|\lllnor_p=1 } and \vskip.5mm\centerline{$
\big|\kern0.15mm\sum\KP1(\kern0.37mm u\cdot v\kern0.37mm)\KP1|=
\sum\KP1(\kern0.37mm u\cdot v\kern0.37mm)=
\sum_{\,A\kern0.37mm\in\kern0.37mm\scrm7 A\,}(\kern0.37mm\roman u\,A\KP1\roman v\kern0.37mm A\kern0.37mm)=
\|\,v\,\|\lllnor_{p\sast} \KP1 $.} \inskipline{.3}0
Furthermore, with the shorthand \math{ \roman t\,A = \roman u\,A\KP1(\kern0.37mm
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)\,^{\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} p}\LHB{.2}{^{^{-1}}} } we have \inskipline1{13}
$ \big|\kern0.15mm\sum\KP1(\kern0.37mm u\cdot v\kern0.37mm)\KP1|
= \big|\kern0.15mm\sum_{\,A\kern0.37mm\in\kern0.37mm\scrm7 A\,}(\kern0.37mm
\roman u\,A\KP1\roman v\kern0.37mm A\kern0.37mm) \KP1 | $ \inskipline{.6}{31.5}
${} = \big|\kern0.15mm\sum_{\,A\kern0.37mm\in\kern0.37mm\scrm7 A\,}(\,\roman t\,A\KP1(\kern0.37mm
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)\KP1^p\LHB{.2}{^{^{-1}}} (\kern0.37mm
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)\KP1^{p\sast\kern0.15mm^{-1}\kern0.07mm -\kern0.37mm 1}\,(\kern0.37mm
m\,.\KPt9\bosy\xi\KPt4\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\KN{.4}A\kern0.37mm )\kern0.37mm )\KP1 | $ \inskipline{.6}{31.5}
${} = \big|\kern0.15mm\sum_{\,A\kern0.37mm\in\kern0.37mm\scrm7 A\,}(\,\roman t\,A\KP1(\kern0.37mm
m\,.\KPt9\bosy\xi\KPt4\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\KN{.4}A\kern0.37mm )\kern0.37mm )\KP1 |$ \inskipline{.6}{31.5}
${} = \big|\kern0.15mm\sum_{\,A\kern0.37mm\in\kern0.37mm\scrm7 A\,}(\,\roman t\,A\KP1(\kern0.37mm
\smb U\fvalss20
\lfloor\,^{p\kern0.15mm,\kern0.37mm\mu\kern0.15mm,\kern0.37mm\vPi}\kern0.37mm(\kern0.37mm
\bosy\xi\KPt4\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\KN{.4}A\kern0.37mm)\kern0.15mm\sbi A\kern0.37mm))\KP1|$ \inskipline{.6}{31.5}
${} = |\KP1\smb U\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F\text{\KPt8-}
\sum_{\,A\kern0.37mm\in\kern0.37mm\scrm7 A\,}\roman t\,A\KP1
\lfloor\,^{p\kern0.15mm,\kern0.37mm\mu\kern0.15mm,\kern0.37mm\vPi}\kern0.37mm(\kern0.37mm
\bosy\xi\KPt4\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\KN{.4}A\kern0.37mm)\kern0.15mm\sbi A\,|$ \inskipline{.6}{31.5}
${}\le\|\KPt8\smb U\,\|\KP1\|\KP1\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F\text{\KPt8-}
\sum_{\,A\kern0.37mm\in\kern0.37mm\scrm7 A\,}\roman t\,A\KP1
\lfloor\,^{p\kern0.15mm,\kern0.37mm\mu\kern0.15mm,\kern0.37mm\vPi}\kern0.37mm(\kern0.37mm
\bosy\xi\KPt4\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\KN{.4}A\kern0.37mm)\kern0.15mm\sbi A\,\|\sNorF$ \inskipline{.6}{31.5}
| 4,053 | 362,273 |
en
|
train
|
0.47.94
|
$ \big|\kern0.15mm\sum\KP1(\kern0.37mm u\cdot v\kern0.37mm)\KP1|
= \big|\kern0.15mm\sum_{\,A\kern0.37mm\in\kern0.37mm\scrm7 A\,}(\kern0.37mm
\roman u\,A\KP1\roman v\kern0.37mm A\kern0.37mm) \KP1 | $ \inskipline{.6}{31.5}
${} = \big|\kern0.15mm\sum_{\,A\kern0.37mm\in\kern0.37mm\scrm7 A\,}(\,\roman t\,A\KP1(\kern0.37mm
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)\KP1^p\LHB{.2}{^{^{-1}}} (\kern0.37mm
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)\KP1^{p\sast\kern0.15mm^{-1}\kern0.07mm -\kern0.37mm 1}\,(\kern0.37mm
m\,.\KPt9\bosy\xi\KPt4\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\KN{.4}A\kern0.37mm )\kern0.37mm )\KP1 | $ \inskipline{.6}{31.5}
${} = \big|\kern0.15mm\sum_{\,A\kern0.37mm\in\kern0.37mm\scrm7 A\,}(\,\roman t\,A\KP1(\kern0.37mm
m\,.\KPt9\bosy\xi\KPt4\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\KN{.4}A\kern0.37mm )\kern0.37mm )\KP1 |$ \inskipline{.6}{31.5}
${} = \big|\kern0.15mm\sum_{\,A\kern0.37mm\in\kern0.37mm\scrm7 A\,}(\,\roman t\,A\KP1(\kern0.37mm
\smb U\fvalss20
\lfloor\,^{p\kern0.15mm,\kern0.37mm\mu\kern0.15mm,\kern0.37mm\vPi}\kern0.37mm(\kern0.37mm
\bosy\xi\KPt4\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\KN{.4}A\kern0.37mm)\kern0.15mm\sbi A\kern0.37mm))\KP1|$ \inskipline{.6}{31.5}
${} = |\KP1\smb U\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F\text{\KPt8-}
\sum_{\,A\kern0.37mm\in\kern0.37mm\scrm7 A\,}\roman t\,A\KP1
\lfloor\,^{p\kern0.15mm,\kern0.37mm\mu\kern0.15mm,\kern0.37mm\vPi}\kern0.37mm(\kern0.37mm
\bosy\xi\KPt4\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\KN{.4}A\kern0.37mm)\kern0.15mm\sbi A\,|$ \inskipline{.6}{31.5}
${}\le\|\KPt8\smb U\,\|\KP1\|\KP1\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F\text{\KPt8-}
\sum_{\,A\kern0.37mm\in\kern0.37mm\scrm7 A\,}\roman t\,A\KP1
\lfloor\,^{p\kern0.15mm,\kern0.37mm\mu\kern0.15mm,\kern0.37mm\vPi}\kern0.37mm(\kern0.37mm
\bosy\xi\KPt4\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\KN{.4}A\kern0.37mm)\kern0.15mm\sbi A\,\|\sNorF$ \inskipline{.6}{31.5}
${}\le\|\KPt8\smb U\,\|\KP1\big(\kern0.15mm\sum_{\,A\kern0.37mm\in\kern0.37mm\scrm7 A}\kern0.15mm\sbig(3
|\KP1\roman t\,A\KP1|\RHB{.3}{\KP1^p}\,(\kern0.37mm
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)))\KP1^p\LHB{.2}{^{^{-1}}} $ \inskipline{.6}{31.5}
${} = \|\KPt8\smb U\,\|\KP1\|\,u\,\|\lllnor_p=\|\KPt8\smb U\,\| \KP1
$, \,giving the assertion.
\Step 2.0 Noting that the requirement of absolute continuity holds since we
trivially have \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm m\kern0.07mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A \le
\|\KPt8\smb U\,\|\KP1(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)\KPt8\RHB{.2}{^p}{^{^{\kern0.15mm-1}}} }
for any \math{A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+}, \,now let \math{\scrmt A}
and \math{y} be as given by Corollary \ref{Coro q-var} on page \pageref{Coro q-var}
above. Then we have \math{(\KPt5 y\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) } simply
measurable and such that \math{y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta=\vPi\times\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} }
holds for \math{\eta\in{}^{}\Cal Omega\kern0.15mm\setminus\bigcup\,\scrmt A } and such that we
also have \math{m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\fvalss12\xi =
\int_{\,A}\kern0.37mm\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\rmdss11\mu } for \math{
A\kern0.07mm\ar 1 \in\scrmt A} and \math{A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\capss33\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\ar 1} and \mathss31{
\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi}. In addition \math{
(\KPt5 y\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\dlbetss01\kern0.15mm) } is simply measurable if \math{
\vPi} is reflexive.
\Step 3.0 Under (5) or (6) to prove that \math{
\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\KP1\|\Lnorss40^{p\sast}_\mu\le\|\KPt8\smb U\,\| }
holds, noting that in the reflexive case now \math{
(\kern0.37mm\Abrs33^{p\sast}\KN1\circ\KPt2\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\,;\kern0.07mm
\mu\,,\kern-0.3mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\kern0.37mm) } is trivially measurable, and that by Lemma \ref{Le Nu_1 ci y meas}
on page \pageref{Le Nu_1 ci y meas} above the same holds also in the separable
case, it suffices to show that \math{\int_{\KPp1.1{}^{}\Cal Omega\,}\Abrs33^{p\sast}\KN1
\circ\KPt2\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\rmdss11\mu \le
\|\KPt8\smb U\,\|\RHB{.2}{\KP1^{p\sast}} } holds. For this in turn for every
fixed $A\kern0.15mm\ar 0\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern0.15mm$ it suffices to show that \math{
\int_{\,\aars A_0\kern0.15mm}(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)
\RHB{.2}{\KP1^{p\sast}}\kern-0.63mm\rmdss01\mu\,(\kern0.15mm\eta\kern0.15mm)
\le\|\KPt8\smb U\,\|\RHB{.2}{\KP1^{p\sast}} } holds.
Now we can express \math{A\kern0.15mm\ar 0} as the union of an increasing sequence of
$A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+$ such that
$\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y$ is bounded on every $A\,$,
say \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.63mm A\subseteq
[\KP{1.1} 0\,,\kern0.07mm\smb M\KPt9] } with \mathss30{\smb M\in\rbb R^+}, \,and
it further suffices to show that for every such $A$ with
$0 < \mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A$ we have
$
\int_{\,A\KPt8}(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)
\RHB{.2}{\KP1^{p\sast}}\kern-0.63mm\rmdss01\mu\,(\kern0.15mm\eta\kern0.15mm)
\le\|\KPt8\smb U\,\|\RHB{.2}{\KP1^{p\sast}} \kern0.15mm$.
To proceed indirectly, supposing that
$\|\KPt8\smb U\,\|\RHB{.2}{\KP1^{p\sast}}
<\int_{\,A\KPt8}(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)
\RHB{.2}{\KP1^{p\sast}}\kern-0.63mm\rmdss01\mu\,(\kern0.15mm\eta\kern0.15mm)$ holds, we let
| 4,071 | 362,273 |
en
|
train
|
0.47.95
|
Now we can express \math{A\kern0.15mm\ar 0} as the union of an increasing sequence of
$A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+$ such that
$\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y$ is bounded on every $A\,$,
say \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.63mm A\subseteq
[\KP{1.1} 0\,,\kern0.07mm\smb M\KPt9] } with \mathss30{\smb M\in\rbb R^+}, \,and
it further suffices to show that for every such $A$ with
$0 < \mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A$ we have
$
\int_{\,A\KPt8}(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)
\RHB{.2}{\KP1^{p\sast}}\kern-0.63mm\rmdss01\mu\,(\kern0.15mm\eta\kern0.15mm)
\le\|\KPt8\smb U\,\|\RHB{.2}{\KP1^{p\sast}} \kern0.15mm$.
To proceed indirectly, supposing that
$\|\KPt8\smb U\,\|\RHB{.2}{\KP1^{p\sast}}
<\int_{\,A\KPt8}(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)
\RHB{.2}{\KP1^{p\sast}}\kern-0.63mm\rmdss01\mu\,(\kern0.15mm\eta\kern0.15mm)$ holds, we let
$\varepsilon=
\frac 14\KP1
(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)
\RHB{.2}{\,^{\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} p^{\kern0.15mm-1}}}
\big(\kern0.15mm
\int_{\,A\KPt8}(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)
\RHB{.2}{\KP1^{p\sast}}\kern-0.63mm\rmdss01\mu\,(\kern0.15mm\eta\kern0.15mm)
)
\RHB{.2}{\KP1^{p\sast\kern0.37mm^{-1}}}
\kern-0.2mm - \|\KPt8\smb U\,\|\kern0.37mm\sbig)0 \KP1 $. \vskip1mm
Since \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\KP1|\KPt9 A } is positive \mathss37{\mu
}--\,measurable with \mathss35{
\sup\KPt8(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\kern0.37mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.63mm A\kern0.37mm) < \lower1.05mm\hbox{$^+$}\infty}, \,we
can find a finite partion \math{\scrmt A\kern0.15mm\ar 0\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} of \math{A} such
that \math{ |\KP{1.2}\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta -
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\ar 1\,| < \varepsilon } holds
for all \mathss36{\eta\kern0.37mm,\kern0.15mm\eta\ar 1\in A\ar 1\in\scrmt A\kern0.15mm\ar 0 }. Taking \math{
S=\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 1\kern0.37mm\} } and \vskip.4mm\centerline{$
P =
A\times S\capss31\{\,(\kern0.37mm\eta\kern0.37mm,\kern0.07mm\xi\kern0.37mm):
0 \le y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\fvalss01\xi \kern0.37mm \text{ and }\kern0.37mm
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta
< y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\fvalss01\xi + \varepsilon \KPt9\} \KP1 $,} \inskipline{.4}0
we first see that \math{A\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}} P} holds. In the reflexive case letting \math{
S\ar 0} be the closed linear span in \math{\vPi\dlbetss01} of \math{{}^{}{\rm rng}\,{}_{{}^{}} y} we
take \math{\scrmt T\aR 1=\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm\vPi\dlbetss01\kern0.15mm)\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss42(\kern0.37mm
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\inve\kern0.37mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm[\KPp1.1 0\,,\kern0.07mm\smb M\KPt9]\capss42
S\ar 0\kern0.07mm) } whereas in the separable case we put \mathss38{\scrmt T\aR 1 =
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm\vPi\dlsigss00\kern0.07mm)
\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss42(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\inve\kern0.37mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm[
\KP{1.1} 0\,,\kern0.07mm\smb M\KPt9]\kern0.15mm
\sbig)0 }. Noting that in both cases now
\math{\scrmt T\aR 1} is a separable and metrizable and hence
second countable topology, we find some
$\bmii8 U\in\kern0.15mm^{{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}}\,\scrmt T\aR 1$ with
$y\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.63mm A\subseteq\bigcup\,{}^{}{\rm rng}\,{}_{{}^{}}\kern0.37mm\bmii8 U$ and
such that for every
$U\in{}^{}{\rm rng}\,{}_{{}^{}}\bmii8 U$ there are $\xi\,,\kern0.15mm\eta$ with
$(\kern0.37mm\eta\kern0.37mm,\kern0.07mm\xi\kern0.37mm)\in P$ and
\mathss38{U\subseteq\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm)
\capss31\{\,\zeta:
|\KP{1.1}(\kern0.37mm\zeta - y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)\fvalss01\xi\KP{1.1}|
< \varepsilon\KPt8\} }.
| 3,240 | 362,273 |
en
|
train
|
0.47.96
|
P =
A\times S\capss31\{\,(\kern0.37mm\eta\kern0.37mm,\kern0.07mm\xi\kern0.37mm):
0 \le y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\fvalss01\xi \kern0.37mm \text{ and }\kern0.37mm
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta
< y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\fvalss01\xi + \varepsilon \KPt9\} \KP1 $,} \inskipline{.4}0
we first see that \math{A\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}} P} holds. In the reflexive case letting \math{
S\ar 0} be the closed linear span in \math{\vPi\dlbetss01} of \math{{}^{}{\rm rng}\,{}_{{}^{}} y} we
take \math{\scrmt T\aR 1=\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm\vPi\dlbetss01\kern0.15mm)\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss42(\kern0.37mm
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\inve\kern0.37mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm[\KPp1.1 0\,,\kern0.07mm\smb M\KPt9]\capss42
S\ar 0\kern0.07mm) } whereas in the separable case we put \mathss38{\scrmt T\aR 1 =
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm\vPi\dlsigss00\kern0.07mm)
\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss42(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\inve\kern0.37mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm[
\KP{1.1} 0\,,\kern0.07mm\smb M\KPt9]\kern0.15mm
\sbig)0 }. Noting that in both cases now
\math{\scrmt T\aR 1} is a separable and metrizable and hence
second countable topology, we find some
$\bmii8 U\in\kern0.15mm^{{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}}\,\scrmt T\aR 1$ with
$y\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.63mm A\subseteq\bigcup\,{}^{}{\rm rng}\,{}_{{}^{}}\kern0.37mm\bmii8 U$ and
such that for every
$U\in{}^{}{\rm rng}\,{}_{{}^{}}\bmii8 U$ there are $\xi\,,\kern0.15mm\eta$ with
$(\kern0.37mm\eta\kern0.37mm,\kern0.07mm\xi\kern0.37mm)\in P$ and
\mathss38{U\subseteq\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm)
\capss31\{\,\zeta:
|\KP{1.1}(\kern0.37mm\zeta - y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)\fvalss01\xi\KP{1.1}|
< \varepsilon\KPt8\} }.
We next fix some bijection $
\bmii8 A\kern0.15mm\ar 0:k\to\scrmt A\kern0.15mm\ar 0$ with $k\in\mathbb N\,$ and construct the
countable finite or infinite sequence $\bmii8A$ as follows. Indeed, we first
let $\bmii8 A\ar 1$ be the infinite
sequence of possibly empty finite sequences obtained as follows. For
every fixed $i\in\mathbb No$ with
$B=y\invss46[\KP{1.2}
\bmii8 U\fvalss51 i\kern0.15mm\setminus\bigcup\KP1(\kern0.37mm\bmii8 U\KPt8\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm i\kern0.37mm)
\KP{1.1}]$ let
$\bmii8 A\ar 1\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm i$ with
$l\in\mathbb No$ be the unique bijection
$l\to\scrmt A\kern0.15mm\ar 0\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss02 B\kern0.15mm\setminus 1\kern0.15mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}$ ordered by
$\bmii8 A\ar 0 \,$. Then let
$\bmii8 A$ be the infinite concatenation of
$\bmii8 A\ar 1 \kern0.37mm $. Now
$\bmii8 A$ is injective with ${}^{}{\rm rng}\,{}_{{}^{}}\bmii8 A\subseteq
{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\setminus 1\kern0.15mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}$ and such that
${}^{}{\rm rng}\,{}_{{}^{}}\bmii8 A$ is a partition of $A$ refining $\scrmt A\kern0.15mm\ar 0\,$, i.e.\ for
every $i\in{{}^{}{\rm dom}\,{}_{{}^{}}}\bmii8 A$ there is $A\ar 1\in\scrmt A\kern0.15mm\ar 0$ with
$\bmii8 A\fvalss51 i\subseteq A\ar 1 \kern0.37mm$. Possibly by
{\sl countable choice\kern0.15mm} we
take any
$\bosy\eta\in\prod{_{_{\kern-.3mm\bold c\kern.15mm}}}\bmii8 A$ and any $\bosy\xi\in\kern0.15mm^{{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.3mm\bmii6 A}\,S$
such that
$(\kern0.37mm\bosy\eta\fvalss01 i\kern0.37mm,\kern0.07mm\bosy\xi\fvalss01 i\kern0.37mm)
\in P$ holds for all $i\in{{}^{}{\rm dom}\,{}_{{}^{}}}\bmii8 A \,$. Now by construction
$|\KP{1.1}(\kern0.37mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta - y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\ar 1)\fvalss01\xi\KP{1.1}|
< \varepsilon \,$ and $\,
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\ar 1
< y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\ar 1\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm\xi + \varepsilon
\,$
and $0\le y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\ar 1\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm\xi\,$ hold
whenever we have
$(\kern0.37mm i\kern0.37mm,\kern0.07mm A\ar 1)\in\bmii8 A$ and $\eta\in A\ar 1$ and
$(\kern0.37mm i\kern0.37mm,\kern0.07mm\eta\ar 1)\in\bosy\eta$ and
$(\kern0.37mm i\kern0.37mm,\kern0.07mm\xi\kern0.37mm)\in\bosy\xi \,$.
With \math{N\aar 0={{}^{}{\rm dom}\,{}_{{}^{}}}\bmii8 A} we next compute \vskip1mm
$
\int_{\,A\KPt8}(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)
\RHB{.2}{\KP1^{p\sast}}\kern-0.63mm\rmdss01\mu\,(\kern0.15mm\eta\kern0.15mm)
=\sum_{\KPt8 i\kern0.37mm\in\kern0.37mm\aars N_0}\kern0.07mm
\int_{\,\bmii6 A\kern-0.2mm\hbox{\kern.2mm\font\SweD =cmr7\SweD \char'022\kern-.2mm} i\KPt8}(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)
\RHB{.2}{\KP1^{p\sast}}\kern-0.63mm\rmdss01\mu\,(\kern0.15mm\eta\kern0.15mm) $ \inskipline{.7}{27}
${}\le
\sum_{\KPt8 i\kern0.37mm\in\kern0.37mm\aars N_0}\kern0.37mm
(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\circss00\bosy\eta\fvalss01 i + \varepsilon\kern0.37mm)
\RHB{.2}{\KP1^{p\sast}}\kern0.37mm
(\kern0.37mm\mu\circss10\bmii8 A\fvalss01 i\kern0.37mm)$ \inskipline{.7}{20}
${}\le
\sum_{\KPt8 i\kern0.37mm\in\kern0.37mm\aars N_0}\kern0.37mm
(\kern0.37mm y\circss00\bosy\eta\fvalss01 i
\fvalss20(\kern0.37mm\bosy\xi\fvalss11 i\kern0.37mm) + 2\KPt8\varepsilon\kern0.37mm)
\RHB{.2}{\KP1^{p\sast}}\kern0.37mm
(\kern0.37mm\mu\circss10\bmii8 A\fvalss01 i\kern0.37mm)$ \vskip.7mm\centerline{$
{}=\lim\sbi{\kern0.15mm\ssmb N\kern0.37mm\to\kern0.37mm\infty\kern0.37mm}
\sum_{\KPt8 i\kern0.37mm\in\kern0.37mm\ssmb N}\,
(\kern0.37mm y\circss00\bosy\eta\fvalss01 i
\fvalss20(\kern0.37mm\bosy\xi\fvalss11 i\kern0.37mm) + 2\KPt8\varepsilon\kern0.37mm)
\RHB{.2}{\KP1^{p\sast}}\kern0.37mm
(\kern0.37mm\mu\circss10\bmii8 A\fvalss01 i\kern0.37mm) \KP1 $,} \inskipline10
where the last limit expression is valid and needed only in the case where
\math{\bmii8 A} is infinite. According to whether
\math{\bmii8 A} is finite or infinite, with
\math{\smb N=N\aar 0} or
for arbitrarily fixed \math{\smb N\in\mathbb N} considering
\math{u\in\kern0.15mm^{\ssmb N}\KPt8\lbb R_+} given by \vskip.5mm\centerline{$
| 4,020 | 362,273 |
en
|
train
|
0.47.97
|
whenever we have
$(\kern0.37mm i\kern0.37mm,\kern0.07mm A\ar 1)\in\bmii8 A$ and $\eta\in A\ar 1$ and
$(\kern0.37mm i\kern0.37mm,\kern0.07mm\eta\ar 1)\in\bosy\eta$ and
$(\kern0.37mm i\kern0.37mm,\kern0.07mm\xi\kern0.37mm)\in\bosy\xi \,$.
With \math{N\aar 0={{}^{}{\rm dom}\,{}_{{}^{}}}\bmii8 A} we next compute \vskip1mm
$
\int_{\,A\KPt8}(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)
\RHB{.2}{\KP1^{p\sast}}\kern-0.63mm\rmdss01\mu\,(\kern0.15mm\eta\kern0.15mm)
=\sum_{\KPt8 i\kern0.37mm\in\kern0.37mm\aars N_0}\kern0.07mm
\int_{\,\bmii6 A\kern-0.2mm\hbox{\kern.2mm\font\SweD =cmr7\SweD \char'022\kern-.2mm} i\KPt8}(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)
\RHB{.2}{\KP1^{p\sast}}\kern-0.63mm\rmdss01\mu\,(\kern0.15mm\eta\kern0.15mm) $ \inskipline{.7}{27}
${}\le
\sum_{\KPt8 i\kern0.37mm\in\kern0.37mm\aars N_0}\kern0.37mm
(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\circss00\bosy\eta\fvalss01 i + \varepsilon\kern0.37mm)
\RHB{.2}{\KP1^{p\sast}}\kern0.37mm
(\kern0.37mm\mu\circss10\bmii8 A\fvalss01 i\kern0.37mm)$ \inskipline{.7}{20}
${}\le
\sum_{\KPt8 i\kern0.37mm\in\kern0.37mm\aars N_0}\kern0.37mm
(\kern0.37mm y\circss00\bosy\eta\fvalss01 i
\fvalss20(\kern0.37mm\bosy\xi\fvalss11 i\kern0.37mm) + 2\KPt8\varepsilon\kern0.37mm)
\RHB{.2}{\KP1^{p\sast}}\kern0.37mm
(\kern0.37mm\mu\circss10\bmii8 A\fvalss01 i\kern0.37mm)$ \vskip.7mm\centerline{$
{}=\lim\sbi{\kern0.15mm\ssmb N\kern0.37mm\to\kern0.37mm\infty\kern0.37mm}
\sum_{\KPt8 i\kern0.37mm\in\kern0.37mm\ssmb N}\,
(\kern0.37mm y\circss00\bosy\eta\fvalss01 i
\fvalss20(\kern0.37mm\bosy\xi\fvalss11 i\kern0.37mm) + 2\KPt8\varepsilon\kern0.37mm)
\RHB{.2}{\KP1^{p\sast}}\kern0.37mm
(\kern0.37mm\mu\circss10\bmii8 A\fvalss01 i\kern0.37mm) \KP1 $,} \inskipline10
where the last limit expression is valid and needed only in the case where
\math{\bmii8 A} is infinite. According to whether
\math{\bmii8 A} is finite or infinite, with
\math{\smb N=N\aar 0} or
for arbitrarily fixed \math{\smb N\in\mathbb N} considering
\math{u\in\kern0.15mm^{\ssmb N}\KPt8\lbb R_+} given by \vskip.5mm\centerline{$
u=\seqss33{
(\kern0.37mm y\circss00\bosy\eta\fvalss01 i
\fvalss20(\kern0.37mm\bosy\xi\fvalss11 i\kern0.37mm) + 2\KPt8\varepsilon\kern0.37mm)\KP1
(\kern0.37mm\mu\circss10\bmii8 A\fvalss01 i\kern0.37mm)
\RHB{.2}{\KP1^{p\sast\kern0.37mm^{-1}}}\kern-0.63mm:i\in\smb N} \KP1 $,} \inskipline{.5}0
we know that for some \math{v\in\kern0.15mm^{\ssmb N}\KPt8\lbb R_+} with \math{
\|\,v\,\|\lllnor_p=1} we have \mathss30{ \|\,u\,\|\lllnor_{p\sast} = \label{express p-norm}
\sum\KP1(\kern0.37mm u\cdot v\kern0.37mm) } where \math{u\cdot v} is the pointwise product \mathss39{
\smb N\owns i\mapsto u\fvalss01 i\cdot(\kern0.37mm v\fvalss01 i\kern0.37mm) }. Using this,
we get \vskip1mm
$
\sum_{\KPt8 i\kern0.37mm\in\kern0.37mm\ssmb N}\,
(\kern0.37mm y\circss00\bosy\eta\fvalss01 i
\fvalss20(\kern0.37mm\bosy\xi\fvalss11 i\kern0.37mm) + 2\KPt8\varepsilon\kern0.37mm)
\RHB{.2}{\KP1^{p\sast}}\kern0.37mm
(\kern0.37mm\mu\circss10\bmii8 A\fvalss01 i\kern0.37mm)
= \big(\kern0.15mm\sum\KP1
(\kern0.37mm u\cdot v\kern0.37mm))\,^{\,p\sast}$ \vskip.7mm
${}=\big(\kern0.15mm\sum_{\KPt8 i\kern0.37mm\in\kern0.37mm\ssmb N}\,(
(\kern0.37mm y\circss00\bosy\eta\fvalss01 i
\fvalss20(\kern0.37mm\bosy\xi\fvalss11 i\kern0.37mm) + 2\KPt8\varepsilon\kern0.37mm)\KP1
(\kern0.37mm\mu\circss10\bmii8 A\fvalss01 i\kern0.37mm)
\RHB{.2}{\KP1^{p\sast\kern0.37mm^{-1}}}
(\kern0.37mm v\fvalss01 i\kern0.37mm)))\RHB{.2}{\KP1^{p\sast}}$ \vskip.7mm
${}=\big(\kern0.15mm\sum_{\KPt8 i\kern0.37mm\in\kern0.37mm\ssmb N}\kern0.07mm
\int_{\,\bmii6 A\kern-0.2mm\hbox{\kern.2mm\font\SweD =cmr7\SweD \char'022\kern-.2mm} i\KPt8}(
(\kern0.37mm y\circss00\bosy\eta\fvalss01 i
\fvalss20(\kern0.37mm\bosy\xi\fvalss11 i\kern0.37mm) + 2\KPt8\varepsilon\kern0.37mm)\KP1
(\kern0.37mm\mu\circss10\bmii8 A\fvalss01 i\kern0.37mm)
\RHB{.2}{\KP1^{p\sast\kern0.37mm^{-1} - \kern0.37mm 1\,}}
(\kern0.37mm v\fvalss01 i\kern0.37mm))
\rmdss11\mu\,(\kern0.15mm\eta\kern0.15mm))\RHB{.2}{\KP1^{p\sast}}$ \vskip.7mm
${}=(\kern0.37mm\smb I\aR 1\kern-0.2mm + 2\KP1\varepsilon\KP1\smb I\aR 2\kern0.07mm
)\RHB{.2}{\KP1^{p\sast}}$ where we have \vskip.7mm
$\smb I\aR 1 = \sum_{\KPt8 i\kern0.37mm\in\kern0.37mm\ssmb N}\kern0.07mm
\int_{\,\bmii6 A\kern-0.2mm\hbox{\kern.2mm\font\SweD =cmr7\SweD \char'022\kern-.2mm} i\KPt8}y\circss00\bosy\eta\fvalss01 i
\fvalss20(\kern0.37mm\bosy\xi\fvalss11 i\kern0.37mm)\KP1
(\kern0.37mm\mu\circss10\bmii8 A\fvalss01 i\kern0.37mm)
\RHB{.2}{\KP1^{p\sast\kern0.37mm^{-1} - \kern0.37mm 1\,}}(\kern0.37mm v\fvalss01 i\kern0.37mm)
\rmdss11\mu\,(\kern0.15mm\eta\kern0.15mm) \,$ and
$\smb I\aR 2 = \sum_{\KPt8 i\kern0.37mm\in\kern0.37mm\ssmb N}\kern0.07mm
\int_{\,\bmii6 A\kern-0.2mm\hbox{\kern.2mm\font\SweD =cmr7\SweD \char'022\kern-.2mm} i\KPt8}(
\kern0.37mm\mu\circss10\bmii8 A\fvalss01 i\kern0.37mm)
\RHB{.2}{\KP1^{p\sast\kern0.37mm^{-1} - \kern0.37mm 1\,}}
(\kern0.37mm v\fvalss01 i\kern0.37mm)\rmdss11\mu\,(\kern0.15mm\eta\kern0.15mm) \KP1 $. \inskipline10
Now with \math{A\kern0.15mm\ar 1=\bigcup\KP1(\kern0.15mm\bmii8 A\kern0.07mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\smb N\kern0.37mm) }
a direct computation using H\"older's inequality
gives
\math{\smb I\aR 2\le
(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.15mm\ar 1)
\RHB{.2}{\KP1^{p^{\kern0.15mm-1}}}\le
(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)
\RHB{.2}{\KP1^{p^{\kern0.15mm-1}}} }, \,and to estimate \math{\smb I\aR 1}, \,taking
$\bosy\xi\ar 1=
\seqss33{((\kern0.37mm\mu\circss10\bmii8 A\fvalss01 i\kern0.37mm)
\RHB{.2}{\KP1^{p\sast\kern0.37mm^{-1} - \kern0.37mm 1\,}}(\kern0.37mm v\fvalss01 i\kern0.37mm)\KP1
(\kern0.37mm\bosy\xi\fvalss11 i\kern0.37mm))\svs\vPi\kern-0.3mm
:i\in\smb N} \,$ and
$\smb X
=\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F\text{\KPt8-}\sum_{\KPt8 i\kern0.37mm\in\kern0.37mm\ssmb N}\,
\lfloor\,^{p\kern0.15mm,\kern0.37mm\mu\kern0.15mm,\kern0.37mm\vPi}\kern0.37mm
(\kern0.37mm\bosy\xi\fvalss11 i\kern0.37mm)\sbi{\,\bmii6 A\kern-0.2mm\hbox{\kern.2mm\font\SweD =cmr7\SweD \char'022\kern-.2mm} i} \KP1 $, we get \vskip1mm
| 3,859 | 362,273 |
en
|
train
|
0.47.98
|
${}=(\kern0.37mm\smb I\aR 1\kern-0.2mm + 2\KP1\varepsilon\KP1\smb I\aR 2\kern0.07mm
)\RHB{.2}{\KP1^{p\sast}}$ where we have \vskip.7mm
$\smb I\aR 1 = \sum_{\KPt8 i\kern0.37mm\in\kern0.37mm\ssmb N}\kern0.07mm
\int_{\,\bmii6 A\kern-0.2mm\hbox{\kern.2mm\font\SweD =cmr7\SweD \char'022\kern-.2mm} i\KPt8}y\circss00\bosy\eta\fvalss01 i
\fvalss20(\kern0.37mm\bosy\xi\fvalss11 i\kern0.37mm)\KP1
(\kern0.37mm\mu\circss10\bmii8 A\fvalss01 i\kern0.37mm)
\RHB{.2}{\KP1^{p\sast\kern0.37mm^{-1} - \kern0.37mm 1\,}}(\kern0.37mm v\fvalss01 i\kern0.37mm)
\rmdss11\mu\,(\kern0.15mm\eta\kern0.15mm) \,$ and
$\smb I\aR 2 = \sum_{\KPt8 i\kern0.37mm\in\kern0.37mm\ssmb N}\kern0.07mm
\int_{\,\bmii6 A\kern-0.2mm\hbox{\kern.2mm\font\SweD =cmr7\SweD \char'022\kern-.2mm} i\KPt8}(
\kern0.37mm\mu\circss10\bmii8 A\fvalss01 i\kern0.37mm)
\RHB{.2}{\KP1^{p\sast\kern0.37mm^{-1} - \kern0.37mm 1\,}}
(\kern0.37mm v\fvalss01 i\kern0.37mm)\rmdss11\mu\,(\kern0.15mm\eta\kern0.15mm) \KP1 $. \inskipline10
Now with \math{A\kern0.15mm\ar 1=\bigcup\KP1(\kern0.15mm\bmii8 A\kern0.07mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\smb N\kern0.37mm) }
a direct computation using H\"older's inequality
gives
\math{\smb I\aR 2\le
(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.15mm\ar 1)
\RHB{.2}{\KP1^{p^{\kern0.15mm-1}}}\le
(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)
\RHB{.2}{\KP1^{p^{\kern0.15mm-1}}} }, \,and to estimate \math{\smb I\aR 1}, \,taking
$\bosy\xi\ar 1=
\seqss33{((\kern0.37mm\mu\circss10\bmii8 A\fvalss01 i\kern0.37mm)
\RHB{.2}{\KP1^{p\sast\kern0.37mm^{-1} - \kern0.37mm 1\,}}(\kern0.37mm v\fvalss01 i\kern0.37mm)\KP1
(\kern0.37mm\bosy\xi\fvalss11 i\kern0.37mm))\svs\vPi\kern-0.3mm
:i\in\smb N} \,$ and
$\smb X
=\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F\text{\KPt8-}\sum_{\KPt8 i\kern0.37mm\in\kern0.37mm\ssmb N}\,
\lfloor\,^{p\kern0.15mm,\kern0.37mm\mu\kern0.15mm,\kern0.37mm\vPi}\kern0.37mm
(\kern0.37mm\bosy\xi\fvalss11 i\kern0.37mm)\sbi{\,\bmii6 A\kern-0.2mm\hbox{\kern.2mm\font\SweD =cmr7\SweD \char'022\kern-.2mm} i} \KP1 $, we get \vskip1mm
$\smb I\aR 1 = \sum_{\KPt8 i\kern0.37mm\in\kern0.37mm\ssmb N}\kern0.07mm
\int_{\,\bmii6 A\kern-0.2mm\hbox{\kern.2mm\font\SweD =cmr7\SweD \char'022\kern-.2mm} i\KPt8}y\circss00\bosy\eta\fvalss01 i
\fvalss20(\kern0.37mm\bosy\xi\ar 1\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm i\kern0.37mm)
\rmdss11\mu\,(\kern0.15mm\eta\kern0.15mm)
=\smb U\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm\smb X + \smb I\aR 3 $ where \vskip.7mm
$\smb I\aR 3 =
\sum_{\KPt8 i\kern0.37mm\in\kern0.37mm\ssmb N}\kern0.07mm
\int_{\,\bmii6 A\kern-0.2mm\hbox{\kern.2mm\font\SweD =cmr7\SweD \char'022\kern-.2mm} i\KPt8}(\kern0.37mm y\circss00\bosy\eta\fvalss01 i
- y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm\bosy\xi\ar 1\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm i\kern0.37mm)
\rmdss11\mu\,(\kern0.15mm\eta\kern0.15mm) \KP1 $. \inskipline10
A direct computation gives \math{\|\,\smb X\kern0.37mm\|\sNorF \le 1} whence we get \mathss38{
|\KP1\smb U\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm\smb X\,|\le\|\KPt8\smb U\,\|}, \,and further \vskip1mm
$|\KP1\smb I\aR 3\,| \le \varepsilon\,
\sum_{\KPt8 i\kern0.37mm\in\kern0.37mm\ssmb N}\kern0.07mm
\int_{\,\bmii6 A\kern-0.2mm\hbox{\kern.2mm\font\SweD =cmr7\SweD \char'022\kern-.2mm} i\KPt8}
(\kern0.37mm\mu\circss10\bmii8 A\fvalss01 i\kern0.37mm)
\RHB{.2}{\KP1^{p\sast\kern0.37mm^{-1} - \kern0.37mm 1\,}}(\kern0.37mm v\fvalss01 i\kern0.37mm)
\rmdss11\mu\,(\kern0.15mm\eta\kern0.15mm)=
\varepsilon\KP1\smb I\aR 2
\le\varepsilon\KP1(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)
\RHB{.2}{\KP1^{p^{\kern0.15mm-1}}} $. \inskipline10
Putting these results together, and letting
\math{\smb N\to\infty} or taking \math{\smb N={{}^{}{\rm dom}\,{}_{{}^{}}}\bmii8 A} if
\math{\bmii8 A} is finite, \,we finally obtain \inskipline1{11}
$
\int_{\,A\KPt8}(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)
\RHB{.2}{\KP1^{p\sast}}\kern-0.63mm\rmdss01\mu\,(\kern0.15mm\eta\kern0.15mm)
\le
\big(\kern0.37mm \|\KPt8\smb U\,\| + 3\KP1
\varepsilon\KP1(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)
\RHB{.2}{\KP1^{p^{\kern0.15mm-1}}}
\kern0.15mm\big)\RHB{.2}{\KP1^{p\sast}} $ \inskipline{.7}{47}
${}<
\big(\kern0.37mm \|\KPt8\smb U\,\| + 4\KP1
\varepsilon\KP1(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)
\RHB{.2}{\KP1^{p^{\kern0.15mm-1}}}
\kern0.15mm\big)\RHB{.2}{\KP1^{p\sast}} $ \inskipline{.7}{47}
${}=
\int_{\,A\KPt8}(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)
\RHB{.2}{\KP1^{p\sast}}\kern-0.63mm\rmdss01\mu\,(\kern0.15mm\eta\kern0.15mm)
\KP1 $, \,a {\sl contradiction\kern0.15mm}. \vskip1mm
\Step 4.0 Now having obtained \math{
\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\KP1\|\Lnorss40^{p\sast}_\mu\le\|\KPt8\smb U\,\| }
we know that \math{y\in\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1} holds, and hence there is
some \math{\smb Y} with \math{y\in\smb Y\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1}. Then we get \math{
\smb U \in {}^{}{\rm rng}\,{}_{{}^{}}\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm} from Lemma \ref{Le-first} similarly as in the proof of
Lemma \nfss A\,\ref{LeA(1)} on page \pageref{Le A2 final ded} above. \label{endmpf}
\end{proof}
| 3,562 | 362,273 |
en
|
train
|
0.47.99
|
We have now established Theorem \nfss A\,\ref{main Th} since as noted at the
beginning of this section on page \pageref{page surj} above, only the
surjectivity \mathss34{\Cal L\,(\kern0.15mm F\kern0.07mm,\kern0.07mm\bosy K\kern0.37mm)\subseteq{}^{}{\rm rng}\,{}_{{}^{}}\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm }
remained to be verified, and this is done in the various cases in
Lemmas \nfss A\,\ref{LeA(1)}$\,,\ldots\KPt8$\nfss A\,\ref{final lemma} above.
Note also that as opposed to the treatments in \cite{Phil} and \cite{Edw}\,,
we succeeded to handle the cases (5) and (6) simultaneously. In \cite{Phil}
only the case (5) is considered, and the text also contains some quite obscure
passages. In \cite{Edw} the case (6) is treated under the additional
assumption that \math{\mu} be at least positive \erm Radonian.
\Ssubhead E Examples and open problems \label{Sec E}
Below, we have collected some examples in order to make more concrete some
points of the abstract theory given above. We also point out some related open
problems. In the first example we demonstrate that in Theorem \nfss A\,\ref{main Th}
the case (3) {\sl does not\kern0.15mm} cover (1) and (2) even when \math{\mu} is a
probability measure.
\begin{example}\label{Exa big compact} \renewcommand\sNorF{\sNor{\fivemath F}}\renewcommand\erm[1]{\hbox{\font\≈=cmr8\≈#1}}
For \mathss37{{}^{}\Cal Omega=\kern0.15mm^\bbI\ssbb70 I}, \,we construct a probability measure \math{
\mu} on \math{{}^{}\Cal Omega} such that for the space \math{ F =
\mLrs42^1(\kern0.37mm\mu\kern0.37mm) } the topology \math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F} is not separable.
Indeed, for details referring to \cite[199\,--\,203]{Du} let \math{ \mu =
\otimes_{\fiveroman{mea}\,}(\ssbb60 I\times\kern-0.2mm\{\,\LeBmef^{}\,|\KP1
\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm\bbI\KP1\}\kern0.15mm\sbig)0 } be the uncountable product measure of the
Borel\,--\,Lebesgue measure on the closed unit interval.
Now with
$\roman A\,s=
{}^{}\Cal Omega\capss41\{\,\eta:\frac 12 \le \eta\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} s \le 1\KPt9\} \, $ let
$\roman x\,s=
(\kern0.37mm{}^{}\Cal Omega\kern0.15mm\setminus\roman A\,s\kern0.37mm)\times\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}
\cupss22(\kern0.15mm\roman A\,s\times\kern-0.2mm\{\kern0.37mm 2\kern0.37mm\}\kern0.15mm\sbig)0 \, $ and
$\, \erm X\,s=\uniqset\smb X:\roman x\,s\in\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F \kern0.15mm $.
For $\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F$ letting
$\|\,\smb X\kern0.37mm\|\sNorF = \uniqset s:\aall{x\in\smb X}\, s =
\int_{\KP{1.1}{}^{}\Cal Omega\,}|\KP1 x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\KP1|\suba
\rmdss01\mu\,(\kern0.07mm\eta\kern0.07mm)$
and
$\|\KPt9\smb X - \smb Y\KP{1.1}\|\sNorF=
\|\KP1(\kern0.15mm\smb X - \smb Y\,)\svs F\,\|\sNorF$ ,
\noindent
then $\{\KPt8\erm X\,s:s\in\bbI\KP1\}$ is uncountable, and
for $s\kern0.37mm,\kern0.15mm t\in\bbI$ with $s\not=t$ by a simple computation we get
$\|\KP1\erm X\,s - \erm X\,t\KP1\|\sNorF=1 \,$, \,giving the
assertion on nonseparability.
\end{example}
Equally well in Example \ref{Exa big compact} above we could have taken the
uncountable \q{coin tossing} measure \math{ \mu =
\otimea3(\kern0.37mm I\times\kern-0.2mm\{\kern0.37mm\pi\kern0.37mm\}\kern0.15mm\sbig)0 } for any uncountable set \math{
I} when \vskip.3mm\centerline{$
\pi = \{\,1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern0.37mm,\kern0.15mm\{\,1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern0.07mm\}\kern0.15mm\}\kern-.2mm\times\kern-.2mm\big\{\kern0.37mm
\frac 12\kern0.37mm\big\}\cupss22\{\,(\kern0.37mm\emptyset\,,\kern0.07mm 0\kern0.37mm)\,,\kern0.15mm
(\kern0.37mm 2\kern0.15mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern0.37mm,\kern0.07mm 1\kern0.37mm)\,\} \KP1 $.}
\begin{problem}
{\it Does {\,\rm(4)} hold\,} in Theorem \nfss A\,\ref{main Th} when \math{\mu}
is the probability measure constructed in Example \ref{Exa big compact} above?
Observe that \cite[Lemma 8.17.1\,(\kern0.15mm b\kern0.07mm)\,, p.\ 580]{Edw} would give a
positive answer only if \math{
(\kern0.37mm\nsTbb_R\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss02\ssbb05 I)\expnota^\ssbb44 I]_{ti} } were a metrizable
topology, thus requiring the set \math{\mathbb I=[\KPp1.1 0\,,\kern0.07mm 1\KPt9] }
to be countable.
\end{problem}
\begin{example}\label{Exa not trul deco}
For \math{{}^{}\Cal Omega=\mathbb R\times\mathbb R} we construct a {\sl decomposable\kern0.15mm} positive
measure
\math{\mu} on \math{{}^{}\Cal Omega} that is {\sl not truly decomposable\kern0.15mm}. We also
get a
function \math{u:{}^{}\Cal Omega\to\{\KPt8 0\,,\kern0.07mm 1\KPt5\} } with \math{
\upint u\rmdss11\mu=
\lower1.05mm\hbox{$^+$}\infty}
but \math{
\int_{\,A}\kern0.37mm u\rmdss11\mu=
0} for all \mathss34{A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+}. Indeed,
let $\mu$ be the set of all pairs
$(\kern0.15mm A\,,\kern0.07mm s\kern0.37mm)$
with $A\subseteq{}^{}\Cal Omega$ and such that there are
$B\in\{\,A\,,\kern0.07mm{}^{}\Cal Omega\kern0.15mm\setminus A\KPt8\}$ and a
countable $C\subseteq\mathbb R$ such that
$B\,\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm t\kern0.37mm\}\in{{}^{}{\rm dom}\,{}_{{}^{}}}\Lebmef^{}$ holds
for all $t\in C$, and that
$B\,\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm t\kern0.37mm\}=\emptyset$ for
$t\in\ssbb02 R\setminus C$, and that $s=\sum\KP1
\seqss33{\Lebmef^{}\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm
A\kern0.37mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm t\kern0.37mm\}\kern0.37mm\big):t\in\mathbb R} \KP1 $. For $N=
\mathbb R\times\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}$ then $N\not\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu$ but
$A \capss31 N \in \mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} $
for all $A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\lbb R_+$.
It follows that $\mu$ cannot be truly decomposable. To see that $\mu$ is
decomposable, just take \mathss39{\scrmt A=
\big\{\kern0.37mm\{\kern0.37mm t\kern0.37mm\}\kern-0.2mm\times
{]}\KP{1.2} n\kern0.37mm,\kern0.07mm n + 1\KP1]:
t\in\mathbb R\kern0.37mm$ and $\kern0.37mm n\in\mathbb Z\KP1 \} }. One also
observes that for \math{u\ar 0=N\kern-.2mm\times\kern-.2mm\{\kern0.37mm 1\kern0.37mm\} } and \math{u
=(\kern0.37mm{}^{}\Cal Omega\kern0.15mm\setminus N\kern0.37mm)\times\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\cupss22 u\ar 0 }
we have $\upint u\rmdss11\mu=\upint u\ar 0\rmdss01\mu=
\lower1.05mm\hbox{$^+$}\infty$
but $\int_{\,A}\kern0.37mm u\rmdss11\mu=\int_{\,A}\kern0.37mm u\ar 0\rmdss01\mu=
0$ for all $A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\lbb R_+$.
\end{example}
Decomposable but not \rsigma6finite positive measures are given in the next
\begin{example}\label{Exa Haar}
Let \math{g\in\kern0.15mm^{S\kern0.37mm\times\kern0.37mm S}\,S } be a group operation with \math{S}
uncountable. Then with \mathss03{{}^{}\Cal Omega=S\kern-0.2mm\times\mathbb R } and \math{ \scrmt T
= \Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm S\kern0.15mm\hbox{\kern-.2mm${}\times\kern-2.5mm\lower.8mm\hbox{\font\SweD =cmr5\SweD t}\kern1.8mm$}\nsTbb_R } and \inskipline{.2}{4.4}
$a=\{\,(\kern0.37mm s\ar 1\kern0.15mm,\kern0.07mm t\ar 1\KPt2;\kern0.07mm s\ar 2\kern0.37mm,\kern0.07mm t\ar 2\,;\kern0.07mm
s\ar 3\kern0.37mm,\kern0.07mm t\ar 3\kern0.07mm) :
(\kern0.37mm s\ar 1\kern0.15mm,\kern0.07mm s\ar 2\kern0.37mm,\kern0.07mm s\ar 3\kern0.07mm) \in g\kern0.37mm\text{ and }\kern0.37mm
(\kern0.37mm t\ar 1\kern0.15mm,\kern0.07mm t\ar 2\kern0.37mm,\kern0.07mm t\ar 3\kern0.07mm) \in\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\kern-0.2mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD f}\kern.3mm\mathbb R\KP1\}
$
putting \inskipline{.5}{4}
| 4,020 | 362,273 |
en
|
train
|
0.47.100
|
\end{example}
Decomposable but not \rsigma6finite positive measures are given in the next
\begin{example}\label{Exa Haar}
Let \math{g\in\kern0.15mm^{S\kern0.37mm\times\kern0.37mm S}\,S } be a group operation with \math{S}
uncountable. Then with \mathss03{{}^{}\Cal Omega=S\kern-0.2mm\times\mathbb R } and \math{ \scrmt T
= \Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm S\kern0.15mm\hbox{\kern-.2mm${}\times\kern-2.5mm\lower.8mm\hbox{\font\SweD =cmr5\SweD t}\kern1.8mm$}\nsTbb_R } and \inskipline{.2}{4.4}
$a=\{\,(\kern0.37mm s\ar 1\kern0.15mm,\kern0.07mm t\ar 1\KPt2;\kern0.07mm s\ar 2\kern0.37mm,\kern0.07mm t\ar 2\,;\kern0.07mm
s\ar 3\kern0.37mm,\kern0.07mm t\ar 3\kern0.07mm) :
(\kern0.37mm s\ar 1\kern0.15mm,\kern0.07mm s\ar 2\kern0.37mm,\kern0.07mm s\ar 3\kern0.07mm) \in g\kern0.37mm\text{ and }\kern0.37mm
(\kern0.37mm t\ar 1\kern0.15mm,\kern0.07mm t\ar 2\kern0.37mm,\kern0.07mm t\ar 3\kern0.07mm) \in\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\kern-0.2mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD f}\kern.3mm\mathbb R\KP1\}
$
putting \inskipline{.5}{4}
$\mu \kern0.15mm = \kern0.37mm \big\langle\kern0.37mm\sum_{\KPt8\emath s\kern0.37mm\in\kern0.37mm S\KPt8}
(\kern0.37mm\Lebmef^{}\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm A\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm s\kern0.37mm\}\kern0.15mm\sbig)0\big) :
A\subseteq{}^{}\Cal Omega\kern0.37mm\text{ and }\kern0.37mm\aall{s\in S}\,A\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm s\kern0.37mm\}
\in {{}^{}{\rm dom}\,{}_{{}^{}}}\Lebmef^{} \KPt8 \rangle $ \inskipline{.7}0
and \mathss38{\mu\ar 1=\mu\KP1|\KP1\{\,A:{{}^{}{\rm dom}\,{}_{{}^{}}} A\kern0.37mm\text{ is countable or
\math{{{}^{}{\rm dom}\,{}_{{}^{}}}(\kern0.37mm{}^{}\Cal Omega\kern0.07mm\setminus A\kern0.37mm) } is countable }\} },
we
have \linebreak \mathss03{
(\kern0.37mm a\,,\kern0.07mm\scrmt T\,) } a locally compact Hausdorff topological group
with \math{\mu} a modified {\sl Haar me- asure\kern0.15mm} for it and \mathss38{
\mu\ar 1 = \mu\KP1|\KP1\sigmAlg3\{\,K:K\kern0.37mm\text{ is \mathss37{\scrmt T
}--\,compact }\} }. With \vskip.2mm\centerline{$
\scrmt A = \{\kern0.15mm\{\kern0.37mm s\kern0.37mm\}\kern-0.2mm\times[\KP1 n\kern0.37mm,\kern0.07mm n + 1 \KP1 {[\kern0.15mm} :
s\in S\kern0.37mm\text{ and }\kern0.37mm n\in\ssbb04 Z\,\} $} \inskipline{.2}0
one checks that \math{\mu} is {\sl truly decomposable\kern0.15mm} and that \math{
\mu\ar 1} is {\sl decomposable\kern0.15mm}. Note that for \math{ g = \sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\kern-0.2mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD f}\kern.3mm\mathbb R =
\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern-1.7mm\raise1.25mm\hbox{\font\SweD =cmr6\SweD 2}\kern1mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R } we have \math{\mu\ar 1} precisely the \math{\mu} given in
Example \ref{Exa not trul deco} above.
\end{example}
\begin{problem}\label{Prblm z-z mea}
{\it Is $\,\mu$ almost decomposable\kern0.37mm} in the following situation\kern0.15mm? Let \math{
{}^{}\Cal Omega=\kern0.15mm^{2.}\ssbb60 R } and with \math{\roman S\,a\,b=\{\,a + t\KP1(\kern0.37mm
b - a\kern0.37mm):0\le t\le 1\KPt8\} } let \mathss30{\mu\ar 0=\{\,(\KPt5
\roman S\,a\,b\,,\kern0.07mm\|\KPt8 a - b\KP1\|\lllnor_2\kern0.15mm):a\kern0.37mm,\kern0.15mm b\in{}^{}\Cal Omega
\KP1\} } and \ $\mu\kern0.15mm = \kern0.15mm \upCth\kern0.37mm \seqss43{ \inf \kern0.15mm \big \{\kern0.37mm \sum\KP1
(\kern0.37mm\mu\ar 0\circ\bmii8 A\kern0.37mm) : \mu\ar 0\kern-0.2mm : \ebit A \in \kern0.15mm
^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\ar 0\kern0.37mm\text{ and }$ \inskipline0{39}
$ A\subseteq\bigcup\kern0.37mm{}^{}{\rm rng}\,{}_{{}^{}}\ebit A\,\} :
A\subseteq{}^{}\Cal Omega } \KP1|\KP1\sigmAlg3{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\ar 0 \KPt8 $. \inskipline{0.2}0
Since from \cite[Proposition 3.2.4\kern0.37mm, p.\ 72]{Du} we know that \math{\mu} is
a positive measure, the problem is whether there exist \math{\scrmt A\,,\kern0.15mm
N\kern0.15mmrim1 } as required in Definitions \ref{df decomp}\,(2) on page \pageref{decos A}
above. An appeal to intuition suggests that \math{\mu} is {\sl not\kern0.15mm} almost
decomposable, but a possible proof does not seem to be simple.
Note that if we above instead had written \vskip.25mm\centerline{$
\mu=\uniqset m:m\kern0.37mm$ is a positive measure and
\math{{{}^{}{\rm dom}\,{}_{{}^{}}} m=\sigmAlg3{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\ar 0 } and \mathss36{\mu\ar 0\subseteq m },} \inskipline{.25}0
then it might have happened that \math{\mu=\hbox{\font\SweD =cmssbx10\SweD U}{}} holds, and hence the answer
to the above question would trivially have been \q{no}, noting that by the
lacking \rsigma5finiteness the uniqueness in \cite[Theorem 3.1.10\kern0.37mm, p.\ 68]{Du}
is not applicable in this situation.
\end{problem}
{ \newcommand\elvrB{\lower.2mm\hbox{\font\≈=cmr11\≈B\kern.7mm}}
Similarly as in Problem \ref{Prblm z-z mea} above we might ask whether with \math{
k\kern0.37mm,\smb N\in\mathbb N } and \mathss30{ {}^{}\Cal Omega = \kern-0.3mm} \mathss30{\kern0.15mm
^{k\kern0.37mm+\KPt4\ssmb N}\ssbb80 R } and suitably fixed \math{\lambda\in\rbb R^+ }
and \vskip.25mm\centerline{$
\elvrB r=\{\KPt8{}^{}\Cal Omega\capss31\{\,\eta:\|\KP1\eta - \eta\ar 0\,\|\lllnor_2 <
\smb R\,\}:\eta\ar 0\in{}^{}\Cal Omega\kern0.37mm\text{ and }\kern0.37mm
0 < \smb R \le r \, \} $} \inskipline{.25}0
and \math{ \alpha = \seqss30{ t\KPt8^{\ssmb N\,(\kern0.15mm k \kern0.37mm + \kern0.37mm\ssmb N\kern0.37mm)
\kern.2mm\raise1.35mm\hbox{\font\SweD =cmb7\SweD \char'056}\kern.2mm\kern0.15mm^{-1}}\KN{.8}:t\in\lbb R_ +\kern-0.63mm} } the \mathss36{\smb N
}--\,dimensional {\sl Hausdorff measure\kern0.15mm} \inskipline{.3}{24}
$\upCth\kern0.37mm\seqss43{\lim_{\KP1\emath r\kern0.37mm\to\,0^+}\kern0.15mm\inf\kern0.15mm\big\{\,\lambda\kern0.15mm
\sum\,(\kern0.37mm\alpha\circss00\Lebmef^{\kern0.37mm k\kern0.37mm+\kern0.37mm\ssmb N}\kern-0.3mm\circ\ebit B\kern0.37mm
) : {} $ \inskipline0{38.3}
$ \ebit B\in\kern0.15mm^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,\elvrB r\kern0.37mm\text{ and }\kern0.37mm A \subseteq \bigcup \kern0.15mm {}^{}{\rm rng}\,{}_{{}^{}}
\ebit B\,\} : A\subseteq{}^{}\Cal Omega}$ \inskipline{.3}0
is almost decomposable. }
In search for an example of a positive measure that would not be almost
decomposable we noticed the positive measure \math{\mu} in the following
\begin{example}\label{Exa non-Rad?}
Let \math{\mu\ar 1=\scrmt N\times\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\cupss22\{\,(\kern0.37mm
\varOmega\aar 1\kern-0.63mm\setminus N\kern0.15mm,\kern0.07mm 1\kern0.37mm):N\in\scrmt N\KP1\} } where \math{
\varOmega\aar 1} is an uncountable set and \math{\scrmt N} is a \rsigma7ideal
in \math{\varOmega\aar 1} with \math{\varOmega\aar 1\not\in\scrmt N } and \mathss34{
\{\kern0.15mm\{\kern0.37mm\eta\kern0.37mm\}:\eta\in\varOmega\aar 1\kern0.37mm \}\subseteq\scrmt N }. For
example, we might have \math{\varOmega\aar 1=\bbI } and \mathss36{ \scrmt N =
\Lebmef^{}\KN1\inve\kern0.37mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss22\bbI }, \,or \vskip.2mm\centerline{$
\scrmt N=\kern0.15mm\big\{\kern0.37mm\bigcup\,\scrmt A:\scrmt A\kern0.37mm\text{ is countable and }\kern0.37mm
\scrmt A\subseteq\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm\varOmega\aar 1\cap\kern0.15mm\{\,A:\roman{Int\,}\sbi{\scrm7 T\KP1}
\roman{Cl\,}\sbi{\scrm7 T\KPt8}A = \emptyset \KPt9 \} \kern0.15mm \} $} \inskipline{.2}0
where \math{\scrmt T} is a regular locally compact or completely metrizable
topology for \mathss34{\varOmega\aar 1}.
| 3,988 | 362,273 |
en
|
train
|
0.47.101
|
\end{problem}
{ \newcommand\elvrB{\lower.2mm\hbox{\font\≈=cmr11\≈B\kern.7mm}}
Similarly as in Problem \ref{Prblm z-z mea} above we might ask whether with \math{
k\kern0.37mm,\smb N\in\mathbb N } and \mathss30{ {}^{}\Cal Omega = \kern-0.3mm} \mathss30{\kern0.15mm
^{k\kern0.37mm+\KPt4\ssmb N}\ssbb80 R } and suitably fixed \math{\lambda\in\rbb R^+ }
and \vskip.25mm\centerline{$
\elvrB r=\{\KPt8{}^{}\Cal Omega\capss31\{\,\eta:\|\KP1\eta - \eta\ar 0\,\|\lllnor_2 <
\smb R\,\}:\eta\ar 0\in{}^{}\Cal Omega\kern0.37mm\text{ and }\kern0.37mm
0 < \smb R \le r \, \} $} \inskipline{.25}0
and \math{ \alpha = \seqss30{ t\KPt8^{\ssmb N\,(\kern0.15mm k \kern0.37mm + \kern0.37mm\ssmb N\kern0.37mm)
\kern.2mm\raise1.35mm\hbox{\font\SweD =cmb7\SweD \char'056}\kern.2mm\kern0.15mm^{-1}}\KN{.8}:t\in\lbb R_ +\kern-0.63mm} } the \mathss36{\smb N
}--\,dimensional {\sl Hausdorff measure\kern0.15mm} \inskipline{.3}{24}
$\upCth\kern0.37mm\seqss43{\lim_{\KP1\emath r\kern0.37mm\to\,0^+}\kern0.15mm\inf\kern0.15mm\big\{\,\lambda\kern0.15mm
\sum\,(\kern0.37mm\alpha\circss00\Lebmef^{\kern0.37mm k\kern0.37mm+\kern0.37mm\ssmb N}\kern-0.3mm\circ\ebit B\kern0.37mm
) : {} $ \inskipline0{38.3}
$ \ebit B\in\kern0.15mm^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,\elvrB r\kern0.37mm\text{ and }\kern0.37mm A \subseteq \bigcup \kern0.15mm {}^{}{\rm rng}\,{}_{{}^{}}
\ebit B\,\} : A\subseteq{}^{}\Cal Omega}$ \inskipline{.3}0
is almost decomposable. }
In search for an example of a positive measure that would not be almost
decomposable we noticed the positive measure \math{\mu} in the following
\begin{example}\label{Exa non-Rad?}
Let \math{\mu\ar 1=\scrmt N\times\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\cupss22\{\,(\kern0.37mm
\varOmega\aar 1\kern-0.63mm\setminus N\kern0.15mm,\kern0.07mm 1\kern0.37mm):N\in\scrmt N\KP1\} } where \math{
\varOmega\aar 1} is an uncountable set and \math{\scrmt N} is a \rsigma7ideal
in \math{\varOmega\aar 1} with \math{\varOmega\aar 1\not\in\scrmt N } and \mathss34{
\{\kern0.15mm\{\kern0.37mm\eta\kern0.37mm\}:\eta\in\varOmega\aar 1\kern0.37mm \}\subseteq\scrmt N }. For
example, we might have \math{\varOmega\aar 1=\bbI } and \mathss36{ \scrmt N =
\Lebmef^{}\KN1\inve\kern0.37mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss22\bbI }, \,or \vskip.2mm\centerline{$
\scrmt N=\kern0.15mm\big\{\kern0.37mm\bigcup\,\scrmt A:\scrmt A\kern0.37mm\text{ is countable and }\kern0.37mm
\scrmt A\subseteq\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm\varOmega\aar 1\cap\kern0.15mm\{\,A:\roman{Int\,}\sbi{\scrm7 T\KP1}
\roman{Cl\,}\sbi{\scrm7 T\KPt8}A = \emptyset \KPt9 \} \kern0.15mm \} $} \inskipline{.2}0
where \math{\scrmt T} is a regular locally compact or completely metrizable
topology for \mathss34{\varOmega\aar 1}.
With a fixed \math{s\ar 0\in\varOmega\aar 1 } for \math{ \eta\ar 0 =
(\kern0.37mm s\ar 0\KPt5,\kern0.07mm s\ar 0) } and for \math{ {}^{}\Cal Omega =
\varOmega\aar 1\kern-0.3mm\times\varOmega\aar 1 } we then construct a positive
measure \math{\mu} on \math{{}^{}\Cal Omega} such that \math{\mu} is {\sl decomposable\kern0.15mm}
but not \rsigma5finite and such that \math{ \bigcap\,\scrmt A\kern0.15mm\ar 0 =
\{\,\eta\ar 0\kern0.07mm\}\not\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu } and \math{ \bigcap\,\scrmt A \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 1\kern0.37mm\} } hold for \vskip.2mm\centerline{$
\scrmt A\kern0.15mm\ar 0 = \mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 1\kern0.37mm\}\capss22\{\, A :
\eta\ar 0 \in A \KP1 \} $} \inskipline{.2}0
and for all nonempty countable \mathss36{\scrmt A\subseteq\scrmt A\kern0.15mm\ar 0 }.
Indeed, with \math{ \roman P\kern0.37mm\ebit A = \bigcup\KPt8\{\kern0.15mm\{\kern0.37mm s\kern0.37mm\}
\kern-.2mm\times\kern-.2mm A\kern0.07mm\ar 1\kern-0.63mm:(\kern0.37mm s\kern0.37mm,\kern0.07mm A\kern0.07mm\ar 1)\in\ebit A\,\} } and \math{
\scrmt P} the set of all countable functions \math{ \ebit A \subseteq
\varOmega\aar 1\kern-0.3mm\times{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\ar 1 } with \math{[\KPp1.4 s\ar 0 \in
{{}^{}{\rm dom}\,{}_{{}^{}}}\ebit A\kern0.37mm\text{ and }\kern0.37mm\ebit A\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} s\ar 0\not\in\scrmt N\impss33
s\ar 0\in\ebit A\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} s\ar 0 \KP1 ] } we let \math{ \mu =
\{\KPt8(\kern0.15mm A\,,\kern0.07mm t\kern0.37mm):\eexi{\ebit A\in\scrmt P}\,[\KPp1.4 A =
\roman P\kern0.37mm\ebit A\kern0.37mm\text{ and }\kern0.37mm t =
\sum\KP1(((\kern0.37mm\mu\ar 1\kern-0.3mm\circ\kern0.07mm\ebit A\kern0.37mm
)\invss24\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 1\kern0.37mm\}\kern0.15mm\sbig)0\times\kern-0.2mm
\{\kern0.37mm 1\kern0.37mm\}\kern0.15mm\sbig)0\KPp1.4\big] } \inskipline0{71.4}
or $\kern0.37mm[\KPp1.4 A = {}^{}\Cal Omega\kern0.07mm\setminus\roman P\kern0.37mm\ebit A\kern0.37mm\text{ and }\kern0.37mm
t = \lower1.05mm\hbox{$^+$}\infty\KPp1.4]\KP1
\lower.1mm\hbox{\font\≈=cmsy11\≈\char'147}
\KP1 $. \inskipline{.25}0
Note that \math{\{\KPt9\roman P\kern0.37mm\ebit A:\ebit A\in\scrmt P\KP1\} } is a
\rsigma3ring and hence that \math{{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} is a \rsigma3algebra.
One sees that for \math{0 < p\le\lower1.05mm\hbox{$^+$}\infty } and for \math{\vPi\in\roman{LCS}\kern0.4mmps0(K) }
with \math{\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\} } the spaces \math{
\mvLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } and \math{
\lll^p\kern0.07mm(\kern0.15mm\varOmega\aar 1\kern0.15mm,\kern0.07mm\vPi\kern0.37mm) } are linearly homeomorphic
under \math{\smb X\mapsto y } when \mathss03{ y \in \kern0.15mm
^{\aars\varOmega_1}\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi } is such that for \math{x\in\smb X} and \math{
u\in\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm) } and for finite \math{ \ebit A \in
\scrmt P } with \mathss03{ {}^{}{\rm rng}\,{}_{{}^{}}\ebit A \subseteq
\mu\ar 1\kern-0.63mm\inve\kern0.37mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 1\kern0.37mm\} } we have \mathss38{
\int_{\KPp1.1\roman P\KPt2\bmii6 A}\kern0.15mm u\circss00 x\rmdss11\mu
= \sum\KP1(\kern0.37mm u\circss00 y\KP1|\KP1{{}^{}{\rm dom}\,{}_{{}^{}}}\ebit A\kern0.37mm) }.
\end{example}
From Example \ref{Exa non-Rad?} above we arrive at the following
\begin{problem}
| 3,475 | 362,273 |
en
|
train
|
0.47.102
|
or $\kern0.37mm[\KPp1.4 A = {}^{}\Cal Omega\kern0.07mm\setminus\roman P\kern0.37mm\ebit A\kern0.37mm\text{ and }\kern0.37mm
t = \lower1.05mm\hbox{$^+$}\infty\KPp1.4]\KP1
\lower.1mm\hbox{\font\≈=cmsy11\≈\char'147}
\KP1 $. \inskipline{.25}0
Note that \math{\{\KPt9\roman P\kern0.37mm\ebit A:\ebit A\in\scrmt P\KP1\} } is a
\rsigma3ring and hence that \math{{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} is a \rsigma3algebra.
One sees that for \math{0 < p\le\lower1.05mm\hbox{$^+$}\infty } and for \math{\vPi\in\roman{LCS}\kern0.4mmps0(K) }
with \math{\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\} } the spaces \math{
\mvLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } and \math{
\lll^p\kern0.07mm(\kern0.15mm\varOmega\aar 1\kern0.15mm,\kern0.07mm\vPi\kern0.37mm) } are linearly homeomorphic
under \math{\smb X\mapsto y } when \mathss03{ y \in \kern0.15mm
^{\aars\varOmega_1}\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi } is such that for \math{x\in\smb X} and \math{
u\in\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm) } and for finite \math{ \ebit A \in
\scrmt P } with \mathss03{ {}^{}{\rm rng}\,{}_{{}^{}}\ebit A \subseteq
\mu\ar 1\kern-0.63mm\inve\kern0.37mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 1\kern0.37mm\} } we have \mathss38{
\int_{\KPp1.1\roman P\KPt2\bmii6 A}\kern0.15mm u\circss00 x\rmdss11\mu
= \sum\KP1(\kern0.37mm u\circss00 y\KP1|\KP1{{}^{}{\rm dom}\,{}_{{}^{}}}\ebit A\kern0.37mm) }.
\end{example}
From Example \ref{Exa non-Rad?} above we arrive at the following
\begin{problem}
{\it Is $\,\mu$ positive \eit Radonian\kern0.37mm} when \math{\mu =
\Lebmef^{}\,|\KP1(\kern0.37mm\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm\bbI\capss42(\kern0.37mm
\Lebmef^{}\KN1\inve\kern0.37mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\KPt8 0\,,\kern0.07mm 1\,\}\kern0.15mm\sbig)0\sbig)0 }
holds\kern0.37mm? Note that if there is \math{\scrmt T} that
positively \erm Radonizes \math{\mu} above, then necessarily \mathss30{
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm K\in\kern-0.3mm} \mathss03{
\{\KPt8 0\,,\kern0.07mm 1\,\} } holds when \math{K} is \mathss37{
\scrmt T}--\,compact. Furthermore, there is some \mathss30{ N \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } such that \math{
\scrmt T\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss32(\ssbb62 I\setminus N\kern0.37mm) } is a compact topology. Then we
get \math{\mu\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}(\kern0.37mm\scrmt T\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss32(\ssbb62 I\setminus N\kern0.37mm)) \subseteq \kern-0.3mm} \mathss03{
\{\KPt8 0\,,\kern0.07mm 1\,\} } and hence \mathss38{ \mu\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.37mm\scrmt T \subseteq
\{\KPt8 0\,,\kern0.07mm 1\,\} }.
\end{problem}
Observe that if we take the trivially positive \erm Radonian \mathss38{
\mu\ar 0 = \{\KPt8(\kern0.37mm\emptyset\,,\kern0.07mm 0\kern0.37mm)\,,\kern0.15mm(\kern0.37mm 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern0.37mm,\kern0.07mm
1\kern0.37mm)\KPt8 \} }, then for \math{ q = \bbI\times 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056} } and \math{
\mu\ar 2 = \{\,(\,q\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.63mm A\,,\kern0.07mm t\kern0.37mm):(\kern0.15mm A\,,\kern0.07mm t\kern0.37mm)\in
\mu\ar 0\,\} } and for \math{0 \le p \le \lower1.05mm\hbox{$^+$}\infty } and \mathss03{ E =
\mLrs03^p(\kern0.37mm\mu\kern0.15mm) } and \math{F = \mLrs03^p(\kern0.37mm\mu\ar 2\kern0.07mm) } and \math{
\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm = \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E\times\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\capss21\{\,(\kern0.37mm\smb\Phii\kern0.07mm,\kern0.07mm\smb\Psii\kern0.15mm) :
\smb\Phii\capss12\smb\Psii\not=\emptyset \KPt9 \} } we \linebreak
have \math{
\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm:E\to F} a linear homeomorphism. This leads us to the following
\begin{definitions}\label{Df Leb equ}
(1) \ $q\meastss33\mu = \{\KPt7(\,q\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.63mm A\,,\kern0.07mm t\kern0.37mm) :
(\kern0.15mm A\,,\kern0.07mm t\kern0.37mm)\in\mu\KPt8\} \KP1 $, \inskipline{.5}2
(2) \ Say that \math{N\kern0.15mmrim1} is {\it finitely \mathss37{\mu}--\,negligible\kern0.37mm}
if{}f \math{N\kern0.15mmrim1} is \mathss37{\mu}--\,negligible \inskipline0{37.5}
and \math{\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.15mm\lower1.05mm\hbox{$^+$}\infty\,\}\capss13\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm N\kern0.15mmrim1 =
\emptyset} holds, \inskipline{.5}2
(3) \ Say that \math{\mu\ar 1\kern0.15mm,\kern0.15mm\mu\ar 2} are {\it Lebesgue equal\,} if{}f \math{
\mu\sbi{\sixmath\nuu} } for \math{\sbi{ \sixmath\nuu \sixroman{\kern0.37mm =\kern0.37mm
1\kern0.07mm,\kern0.37mm 2}}} is a positive measure and there are \math{N\aar 1\kern0.15mm,\kern0.15mm
N\aar 2\kern0.37mm,\kern0.15mm q\ar 1\kern0.15mm,\kern0.15mm q\ar 2\kern0.37mm,\kern0.15mm Q } with \math{
N\kern-0.3mm\sbi{\sixmath\nuu} } finitely \mathss37{\mu\sbi{\sixmath\nuu}
}--\,negligible and \math{q\sbi{\sixmath\nuu} } is a surjection \mathss03{Q\to
\bigcup\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\sbi{\sixmath\nuu}\kern-0.3mm\setminus N\kern-0.3mm\sbi{\sixmath\nuu} } for \math{\sbi{
\sixmath\nuu \sixroman{\kern0.37mm =\kern0.37mm 1\kern0.07mm,\kern0.37mm 2}} } and \math{ \kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm \in
\Lis(\kern0.15mm E\ar 1\kern0.15mm,\kern0.07mm E\ar 2\kern0.07mm) } holds for \mathss30{E\sbi{\sixmath\nuu}
= \mLrs42^1(\,q\sbi{\sixmath\nuu}\kern-0.63mm\meastss03\mu\sbi{\sixmath\nuu}) } and \mathss38{
\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm = \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E\ar 1\kern-0.3mm\times\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E\ar 2\capss01\{\,(\kern0.37mm\smb\Phii\kern0.07mm , \kern0.07mm
\smb\Psii\kern0.15mm) : \smb\Phii\capss12\smb\Psii\not=\emptyset \KPt9 \} }, \inskipline{.5}2
(4) \ Say that \math{\mu} is {\it essentially positive \eit Radonian\kern0.37mm}
if{}f \inskipline0{9}
there is a positive \erm Radonian \math{\mu\ar 0} such that \math{
\mu\,,\kern0.15mm\mu\ar 0} are Lebesgue equal.
\end{definitions}
| 3,575 | 362,273 |
en
|
train
|
0.47.103
|
\begin{definitions}\label{Df Leb equ}
(1) \ $q\meastss33\mu = \{\KPt7(\,q\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.63mm A\,,\kern0.07mm t\kern0.37mm) :
(\kern0.15mm A\,,\kern0.07mm t\kern0.37mm)\in\mu\KPt8\} \KP1 $, \inskipline{.5}2
(2) \ Say that \math{N\kern0.15mmrim1} is {\it finitely \mathss37{\mu}--\,negligible\kern0.37mm}
if{}f \math{N\kern0.15mmrim1} is \mathss37{\mu}--\,negligible \inskipline0{37.5}
and \math{\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.15mm\lower1.05mm\hbox{$^+$}\infty\,\}\capss13\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm N\kern0.15mmrim1 =
\emptyset} holds, \inskipline{.5}2
(3) \ Say that \math{\mu\ar 1\kern0.15mm,\kern0.15mm\mu\ar 2} are {\it Lebesgue equal\,} if{}f \math{
\mu\sbi{\sixmath\nuu} } for \math{\sbi{ \sixmath\nuu \sixroman{\kern0.37mm =\kern0.37mm
1\kern0.07mm,\kern0.37mm 2}}} is a positive measure and there are \math{N\aar 1\kern0.15mm,\kern0.15mm
N\aar 2\kern0.37mm,\kern0.15mm q\ar 1\kern0.15mm,\kern0.15mm q\ar 2\kern0.37mm,\kern0.15mm Q } with \math{
N\kern-0.3mm\sbi{\sixmath\nuu} } finitely \mathss37{\mu\sbi{\sixmath\nuu}
}--\,negligible and \math{q\sbi{\sixmath\nuu} } is a surjection \mathss03{Q\to
\bigcup\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\sbi{\sixmath\nuu}\kern-0.3mm\setminus N\kern-0.3mm\sbi{\sixmath\nuu} } for \math{\sbi{
\sixmath\nuu \sixroman{\kern0.37mm =\kern0.37mm 1\kern0.07mm,\kern0.37mm 2}} } and \math{ \kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm \in
\Lis(\kern0.15mm E\ar 1\kern0.15mm,\kern0.07mm E\ar 2\kern0.07mm) } holds for \mathss30{E\sbi{\sixmath\nuu}
= \mLrs42^1(\,q\sbi{\sixmath\nuu}\kern-0.63mm\meastss03\mu\sbi{\sixmath\nuu}) } and \mathss38{
\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm = \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E\ar 1\kern-0.3mm\times\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E\ar 2\capss01\{\,(\kern0.37mm\smb\Phii\kern0.07mm , \kern0.07mm
\smb\Psii\kern0.15mm) : \smb\Phii\capss12\smb\Psii\not=\emptyset \KPt9 \} }, \inskipline{.5}2
(4) \ Say that \math{\mu} is {\it essentially positive \eit Radonian\kern0.37mm}
if{}f \inskipline0{9}
there is a positive \erm Radonian \math{\mu\ar 0} such that \math{
\mu\,,\kern0.15mm\mu\ar 0} are Lebesgue equal.
\end{definitions}
Note that if \math{\mu} is a positive measure and \math{q} is a small function
such that for the set \math{N\kern0.15mmrim1=\bigcup\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\setminus{}^{}{\rm rng}\,{}_{{}^{}} q } it
holds that \math{N\kern0.15mmrim1} is \mathss37{\mu}--\,negligible, then \math{
q\meastss33\mu } {\sl need not\kern0.15mm} be a positive measure since it may even
fail to be a function. For example, taking \mathss03{q=\emptyset} and \math{
\mu = \{\KPt8(\kern0.37mm\emptyset\,,\kern0.07mm 0\kern0.37mm)\,,\kern0.15mm(\kern0.37mm 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern0.37mm ,
\lower1.05mm\hbox{$^+$}\infty\kern0.37mm)\KPt8\} } we get \mathss38{ q\meastss33\mu =
1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern-.2mm\times\kern-.2mm\{\KPt7 0\,,\lower1.05mm\hbox{$^+$}\infty\,\} }. However, if we know that \math{
q\meastss33\mu } is a function, then it is also a positive measure as one
quickly verifies. A sufficient condition to guarantee that \math{
q\meastss33\mu } be a function is that \mathss30{N\kern0.15mmrim1} be finitely \mathss37{
\mu}--\,negligible as we have required in Definitions \ref{Df Leb equ}\,(3)
above.
Note also that our Definition \ref{Df Leb equ}\,(3) is not entirely
satisfactory since for example taking \math{ \mu\ar 0 =
\{\,(\kern0.37mm\emptyset\,,\kern0.07mm 0\kern0.37mm)\,\} } we have both \math{
\mLrs42^1(\kern0.37mm\mu\kern0.15mm) } and \math{\mLrs42^1(\kern0.37mm\mu\ar 0\kern0.07mm) } linearly
homeomorphic to the trivial space \math{
(\kern0.37mm 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern-0.2mm\times 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern-0.2mm\times 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern0.37mm , \kern0.07mm
\mathbb R\times 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern-0.2mm\times 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern0.37mm , \kern0.07mm\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern0.07mm) }
but \math{\mu\,,\kern0.15mm\mu\ar 0} are not Lebesgue equal by the above. We further
remark that the relation of being Lebesgue equal is not an equivalence since
for example taking the positive \erm Radonian \math{ \mu\ar 1 =
2\kern0.15mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern-.2mm\times\kern-.2mm\{\kern0.37mm 0\kern0.37mm\} } that is positively \erm Radonized by \math{
\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}} we have both \math{\mu\,,\kern0.15mm\mu\ar 1} Lebesgue equal and \mathss30{
\mu\ar 0\,,\kern0.15mm\mu\ar 1} Lebesgue equal. Now we can pose the following
\begin{problem}\label{Prblm all Rado?}
{\sl Is every positive measure essentially positive \eit Radonian\kern0.37mm}?
\end{problem}
If the answer to the question in Problem \ref{Prblm all Rado?} is positive,
then one might be able to remove from Theorem \nfss A\,\ref{main Th} the
assumption on \math{\mu} being almost decomposable in the case where \math{p=1}
holds. It seems that possibly by using {\sl Kakutani's theorem\kern0.15mm}
\cite[4.23.2\kern0.37mm, p.\ 287]{Edw} one might be able to prove that this indeed is
the case. However, we leave these matters open here.
For example the {\sl Wiener\kern0.15mm} probability {\sl measure\kern0.15mm} in \cite{Pi-Po}
on a non\kern0.37mm-\kern0.37mm locally compact separably metrizable topological space
{\sl is essentially positive \esl Radonian\kern0.15mm} directly by its construction
since it is obtained by restricting a \erm Radonian probability measure on a
compact topological space to a subset of measure unity. More specifically, one
first constructs a probability measure \math{\pi} that is positively
\erm Radonized by a compact topology \mathss30{\scrmt T}. Then for a certain
separably metrizable topological space \math{(\kern0.37mm{}^{}\Cal Omega\,,\kern0.07mm\scrmt U\kern0.37mm) }
one shows that \math{\sigmAlg3\scrmt U={{}^{}{\rm dom}\,{}_{{}^{}}}\pi\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss23{}^{}\Cal Omega } and \math{
{}^{}\Cal Omega\in\pi\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 1\kern0.37mm\} } hold, and finally one defines \linebreak
$\kern.3mm\roman{id}\kern.7mm\kern0.15mm{}^{}\Cal Omega\meastss33\pi\kern0.37mm$ to be the Wiener measure.
We remark that there is some confusion in \cite[pp.\ 12\,--\,25]{Pi-Po} and
that the above is not a review but rather an interpretation of how it could
have been done.
\begin{example}\label{Exa pos Rad C(I)}
It holds that \math{\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm{}^{}\Cal Omega} {\sl positively almost \esl Radonizes\kern0.15mm} \math{
\pi} in the following situation. Let \math{(\kern0.37mm{}^{}\Cal Omega\kern0.37mm,\kern0.07mm\scrmt T\,) }
be a separably metrizable and not locally compact topological space with \math{
{}^{}\Cal Omega} uncountable, and let \math{D} countable and \mathss37{\scrmt T
}--\,dense with \math{\bosy a\in\kern0.15mm^D\KPt8\rbb R^+ } and \mathss04{
\sum\,\bosy a=1}. Then let \mathss38{\pi = \kern0.15mm \big\langle\kern0.37mm\sum\KP1(\kern0.37mm
\bosy a\KPt9|\KPt8 A\kern0.37mm) : A \in \sigmAlg3\scrmt T\KPp1.2\rangle }. For
example, we might have \mathss03{\scrmt T=\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm C\,(\ssbb55 I) } and \math{D}
the set of all polynomial functions with rational coefficients, or the set of
all piecewise affine functions with rational \q{break} points.
\end{example}
\begin{example}\label{Exa Sum Lebm^N}
For \math{{}^{}\Cal Omega = \kern0.15mm^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,\bbI } we obtain a
{\sl decomposable\kern0.15mm}
non\kern0.37mm-\kern0.15mm\rsigma5finite positive mea- \linebreak
sure \math{\mu} on \math{{}^{}\Cal Omega} by
taking \math{\mu=\sum\KP1\seqss30{\roman m\KPt8\alpha:\alpha\subseteq\mathbb No} } where
with \math{m\ar 1 = \Lebmef^{}\,|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm\bbI } and \mathss03{\delta\ar 0 =
{{}^{}{\rm dom}\,{}_{{}^{}}} m\ar 1\kern-0.2mm\times\{\KPt7 0\,,\kern0.07mm 1\KPt6\}\capss22\{\,
(\kern0.15mm A\,,\kern0.07mm t\kern0.37mm) : t = 1 \equivss33 0 \in A \KP1 \} } we have \vskip.3mm\centerline{$
| 4,042 | 362,273 |
en
|
train
|
0.47.104
|
\begin{problem}\label{Prblm all Rado?}
{\sl Is every positive measure essentially positive \eit Radonian\kern0.37mm}?
\end{problem}
If the answer to the question in Problem \ref{Prblm all Rado?} is positive,
then one might be able to remove from Theorem \nfss A\,\ref{main Th} the
assumption on \math{\mu} being almost decomposable in the case where \math{p=1}
holds. It seems that possibly by using {\sl Kakutani's theorem\kern0.15mm}
\cite[4.23.2\kern0.37mm, p.\ 287]{Edw} one might be able to prove that this indeed is
the case. However, we leave these matters open here.
For example the {\sl Wiener\kern0.15mm} probability {\sl measure\kern0.15mm} in \cite{Pi-Po}
on a non\kern0.37mm-\kern0.37mm locally compact separably metrizable topological space
{\sl is essentially positive \esl Radonian\kern0.15mm} directly by its construction
since it is obtained by restricting a \erm Radonian probability measure on a
compact topological space to a subset of measure unity. More specifically, one
first constructs a probability measure \math{\pi} that is positively
\erm Radonized by a compact topology \mathss30{\scrmt T}. Then for a certain
separably metrizable topological space \math{(\kern0.37mm{}^{}\Cal Omega\,,\kern0.07mm\scrmt U\kern0.37mm) }
one shows that \math{\sigmAlg3\scrmt U={{}^{}{\rm dom}\,{}_{{}^{}}}\pi\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss23{}^{}\Cal Omega } and \math{
{}^{}\Cal Omega\in\pi\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 1\kern0.37mm\} } hold, and finally one defines \linebreak
$\kern.3mm\roman{id}\kern.7mm\kern0.15mm{}^{}\Cal Omega\meastss33\pi\kern0.37mm$ to be the Wiener measure.
We remark that there is some confusion in \cite[pp.\ 12\,--\,25]{Pi-Po} and
that the above is not a review but rather an interpretation of how it could
have been done.
\begin{example}\label{Exa pos Rad C(I)}
It holds that \math{\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm{}^{}\Cal Omega} {\sl positively almost \esl Radonizes\kern0.15mm} \math{
\pi} in the following situation. Let \math{(\kern0.37mm{}^{}\Cal Omega\kern0.37mm,\kern0.07mm\scrmt T\,) }
be a separably metrizable and not locally compact topological space with \math{
{}^{}\Cal Omega} uncountable, and let \math{D} countable and \mathss37{\scrmt T
}--\,dense with \math{\bosy a\in\kern0.15mm^D\KPt8\rbb R^+ } and \mathss04{
\sum\,\bosy a=1}. Then let \mathss38{\pi = \kern0.15mm \big\langle\kern0.37mm\sum\KP1(\kern0.37mm
\bosy a\KPt9|\KPt8 A\kern0.37mm) : A \in \sigmAlg3\scrmt T\KPp1.2\rangle }. For
example, we might have \mathss03{\scrmt T=\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm C\,(\ssbb55 I) } and \math{D}
the set of all polynomial functions with rational coefficients, or the set of
all piecewise affine functions with rational \q{break} points.
\end{example}
\begin{example}\label{Exa Sum Lebm^N}
For \math{{}^{}\Cal Omega = \kern0.15mm^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,\bbI } we obtain a
{\sl decomposable\kern0.15mm}
non\kern0.37mm-\kern0.15mm\rsigma5finite positive mea- \linebreak
sure \math{\mu} on \math{{}^{}\Cal Omega} by
taking \math{\mu=\sum\KP1\seqss30{\roman m\KPt8\alpha:\alpha\subseteq\mathbb No} } where
with \math{m\ar 1 = \Lebmef^{}\,|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm\bbI } and \mathss03{\delta\ar 0 =
{{}^{}{\rm dom}\,{}_{{}^{}}} m\ar 1\kern-0.2mm\times\{\KPt7 0\,,\kern0.07mm 1\KPt6\}\capss22\{\,
(\kern0.15mm A\,,\kern0.07mm t\kern0.37mm) : t = 1 \equivss33 0 \in A \KP1 \} } we have \vskip.3mm\centerline{$
\roman m\KPt8\alpha=
\otimea0((\kern0.37mm\mathbb No\kern-0.3mm\setminus\alpha\kern0.37mm)\times\kern-0.2mm\{\KPt8 m\ar 1\}\cupss22
(\kern0.37mm\alpha\times\kern-0.2mm\{\,\delta\ar 0\kern0.07mm\}\kern0.15mm\sbig)0\sbig)0 \KP1 $.} \inskipline{.3}0
Indeed, with \math{\roman A\KPt8\alpha = {}^{}\Cal Omega\capss41\{\,\eta :
\eta\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}=\alpha\KPt8\} } taking \math{ \scrmt A =
\{\,\roman A\KPt8\alpha:\alpha\subseteq\mathbb No\,\} } we have \mathss03{\scrmt A}
uncountable and disjoint with \math{{}^{}\Cal Omega=\bigcup\,\scrmt A} and \mathss38{
\scrmt A\subseteq\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 1\kern0.37mm\} }. Furthermore, for \mathss03{
\alpha\KPt5,\kern0.15mm\kappa\in\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm\mathbb No} we have \math{
\roman m\KPt8\alpha\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm\roman A\KPt8\alpha = 1 } and \mathss36{
\alpha\not=\kappa\impss33\roman m\KPt8\kappa\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm\roman A\KPt8\alpha=0}.
To see these, by straightforward inspection one first verifies the last
assertion which then directly implies that \math{ \scrmt A \subseteq
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 1\kern0.37mm\} } holds. Consequently \math{\mu} is not
\rsigma5finite.
To show that \math{\mu} is decomposable, let \math{A\in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\rbb R^+} and \math{N\kern0.15mmrim1\subseteq{}^{}\Cal Omega} with \mathss30{
\scrmt A\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss42 N\kern0.15mmrim1 \subseteq \kern-0.3mm} \mathss08{
\bigcup\KPt8\{\KPt8\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm N \kern-0.3mm : N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\kern0.15mm\}
}. Then taking \mathss30{ \varLambda\kern0.15mm\ar 0 = \Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm\mathbb No\capss01\{\,\alpha :
A\capss32\roman A\KPt8\alpha\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\rbb R^+\kern0.15mm\big\} } we
have \math{\varLambda\kern0.15mm\ar 0} countable, and also putting \math{ N\aar 1 =
A\kern0.07mm\setminus\bigcup\KPt8\{\,\roman A\KPt8\alpha:\alpha\in\varLambda\kern0.15mm\ar 0
\,\} } we get \mathss30{N\aar 1 \in \kern-0.3mm } \mathss08{
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} }. Indeed, trivially \math{ N\aar 1 \in
{{}^{}{\rm dom}\,{}_{{}^{}}}\mu } holds, and if we have \mathss38{ N\aar 1 \not \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} }, \,then \math{0 <
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm N\aar 1 = \sum\KP1\seqss30{
\roman m\KPt8\alpha\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm N\aar 1\kern-0.3mm : \alpha\in\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm\mathbb No} =
\sum\KP1\seqss30{\roman m\KPt8\alpha\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm N\aar 1\kern-0.3mm : \alpha \in
\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm\mathbb No\kern-0.3mm\setminus\varLambda\kern0.15mm\ar 0} } when- ce there is some \math{
\alpha\in\mathbb No\kern-0.3mm\setminus\varLambda\kern0.15mm\ar 0 } with \math{ 0 <
\roman m\KPt8\alpha\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm N\aar 1 \le \roman m\KPt8\alpha\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A }
and hence \mathss36{\alpha\in\varLambda\kern0.15mm\ar 0}, \,a {\sl contradiction\kern0.15mm}.
Now by {\sl countable choice\kern0.15mm} there is some countable \mathss30{ \scrmt N
\subseteq\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } with \mathss34{
\bigcup\KPt8\{\,N\kern0.15mmrim1\cap\kern0.15mm\roman A\KPt8\alpha : \alpha \in
\varLambda\kern0.15mm\ar 0\,\}\subseteq\bigcup\,\scrmt N }. Then taking \math{ N =
\bigcup\,\scrmt N\cupss42 N\aar 1} we finally get \mathss08{
A\capss31 N\kern0.15mmrim1\subseteq N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} }.
\end{example}
Observe that if in Example \ref{Exa Sum Lebm^N} in place of \math{\mathbb No} we
take any uncountable set, then we obtain a trivial measure \math{\mu} in the
sense that \math{{}^{}{\rm rng}\,{}_{{}^{}}\mu=\{\KPt7 0\,,\lower1.05mm\hbox{$^+$}\infty\,\} } holds.
\begin{problem}\label{Prblm Sum Lebm^N}
| 3,921 | 362,273 |
en
|
train
|
0.47.105
|
\end{example}
Observe that if in Example \ref{Exa Sum Lebm^N} in place of \math{\mathbb No} we
take any uncountable set, then we obtain a trivial measure \math{\mu} in the
sense that \math{{}^{}{\rm rng}\,{}_{{}^{}}\mu=\{\KPt7 0\,,\lower1.05mm\hbox{$^+$}\infty\,\} } holds.
\begin{problem}\label{Prblm Sum Lebm^N}
{\it Is $\,\mu$ positive \eit Radonian\kern0.37mm} in Example \ref{Exa Sum Lebm^N}
above\kern0.15mm? Note that at least we cannot take \math{ \scrmt T =
(\kern0.37mm\nsTbb_R\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}\ssbb25 I)\expnota^\kern0.15mm{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\kern0.15mm]_{ti} } in order to
positively \erm Radonize \math{\mu} since \math{{}^{}\Cal Omega} is \mathss37{\scrmt T
}--\,compact with \mathss35{\mu\fvalss01{}^{}\Cal Omega=\lower1.05mm\hbox{$^+$}\infty}.
\end{problem}
\begin{example}\label{Exa sign mea}
We say that \math{\mu} is a {\it signed measure\kern0.37mm} if{}f \math{ \mu \in \kern0.15mm
^{{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.3mm\mu}\,\ovbbR} with \math{{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} being a \rsigma3algebra and \math{
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm\bigcup\,\scrmt A = \sum\KP1(\kern0.37mm\mu\KP1|\KP1\scrmt A\kern0.37mm) } for
all countable disjoint \mathss35{\scrmt A\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}}\mu}. Then \linebreak
we cannot have \math{
\{\kern0.15mm\kern.2mm\lower1.05mm\hbox{$^-$}infty\,,\lower1.05mm\hbox{$^+$}\infty\KPt8\}\subseteq{}^{}{\rm rng}\,{}_{{}^{}}\mu } since otherwise we could
find \math{A\,,\kern0.15mm B} with \mathss03{
\{\KPt8(\kern0.15mm A\,,\kern.2mm\lower1.05mm\hbox{$^-$}infty\kern0.37mm)\,,\kern0.15mm(\kern0.15mm B\kern0.37mm,\lower1.05mm\hbox{$^+$}\infty\kern0.37mm)\KPt8\} \subseteq
\mu} and \math{A\capss32 B=\emptyset} whence we would get \vskip.3mm\centerline{$
\hbox{\font\SweD =cmssbx10\SweD U}{} = \sum\KP1(\,\mu\KP1|\KPt8\{\,A\,,\kern0.15mm B\KPt8\}\kern0.15mm\sbig)0
= \mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm A\cupss32 B\kern0.37mm) \in \ovbbR \KP1 $,} \inskipline{.3}0
a contradiction following from our sum conventions in \cite{Hif}\,. Now the
positive measures are precisely the signed measures \math{\mu} with \mathss38{
{}^{}{\rm rng}\,{}_{{}^{}}\mu\subseteq[\KPp1.1 0\,,\lower1.05mm\hbox{$^+$}\infty\KPt9] }, \,and a signed measure \math{\mu}
we say to be {\it positively signed\,} if{}f \math{\kern.2mm\lower1.05mm\hbox{$^-$}infty\not\in{}^{}{\rm rng}\,{}_{{}^{}}\mu}
holds. Similarly the con- dition \math{\lower1.05mm\hbox{$^+$}\infty\not\in{}^{}{\rm rng}\,{}_{{}^{}}\mu} defines being
{\it negatively signed\,}. Real measures are now those that are both
positively and negatively signed\kern0.15mm.
We next construct the topologized conoid \math{ \tcbbR_+ =
(\kern0.37mm a\kern0.37mm,\kern0.15mm c\,,\kern0.07mm\scrmt T\,) } so that \math{m} is positively signed
if{}f \math{m} is countably \mathss37{\tcbbR_+}--\,additive and such that \math{
{{}^{}{\rm dom}\,{}_{{}^{}}} m} is a \rsigma3algebra. Indeed, taking \math{R=\lbb R_+} and \math{ S =
{\kern0.37mm]}\,\kern.2mm\lower1.05mm\hbox{$^-$}infty\kern0.37mm,\lower1.05mm\hbox{$^+$}\infty\KP1] } let \math{ \scrmt T =
\barscTbb_R\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss02 S} and \inskipline{.3}{10.3}
$a = \sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern-1.7mm\raise1.25mm\hbox{\font\SweD =cmr6\SweD 2}\kern1mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\cupss31\{\,(\kern0.37mm s\kern0.37mm,\kern0.07mm t\kern0.37mm,\lower1.05mm\hbox{$^+$}\infty\kern0.37mm) :
\lower1.05mm\hbox{$^+$}\infty\in\{\,s\kern0.37mm,\kern0.15mm t\,\}\subseteq S\KP1\} \KP{26.5} $ and \inskipline{.3}{10.7}
$c = \tau\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\KP1|\KP1(\kern0.15mm R\times\hbox{\font\SweD =cmssbx10\SweD U}{}\kern0.37mm)\cupss21\{\,
(\kern0.37mm 0\,,\lower1.05mm\hbox{$^+$}\infty\kern0.37mm,\kern0.07mm 0\kern0.37mm)\,\}\cupss22\{\,
(\kern0.37mm s\kern0.37mm,\lower1.05mm\hbox{$^+$}\infty\kern0.37mm,\lower1.05mm\hbox{$^+$}\infty\kern0.37mm):s\in\rbb R^+\kern0.15mm\big\} \KP1 $. \inskipline{.3}0
Making the obvious modifications we similarly get the topologized conoid \math{
\tcbbR_-} so that \math{m} is negatively signed if{}f \math{m} is countably \mathss37{
\tcbbR_-}--\,additive and such that \mathss30{{{}^{}{\rm dom}\,{}_{{}^{}}} m} is a \rsigma3algebra.
Likewise, we can construct the topologized conoid \math{\tcovbbRplus}
characterizing the positive measures.
\end{example}
\begin{problem}\label{Prblm Io sur}
In Theorem \nfss A\,\ref{main Th} taking for example \math{ \mu =
\LeBmef^{}\,|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm\mathbb I } and \vskip.5mm\centerline{$
\vPi \in \{\KP1\hbox{\font\≈=cmmi12\≈c}\lower.8mm\hbox{\font\≈=cmr6\≈o}\kern.4mm(\ssbb44 I) \, , \kern0.15mm \ell\KPt8^1\kern0.15mm(\ssbb44 I) \, , \kern0.15mm
\LLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\ssbb44 I) \KPt9 \} \KP1 $,} \inskipline{.3}0
hence \math{\vPi} being nonreflexive with \math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi} a nonseparable
topology, for \math{p=1} we see \linebreak
that (3) holds, and then with \math{ F\aar 1 =
\mvsLrs23^{p\sast}\kern-0.63mm(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) } we obtain that \math{
\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm} is a linear ho- \linebreak
meomorphism \mathss30{F\aar 1\to F\dlbetss10}, \,and
hence in particular that \math{\Cal L\,(\kern0.15mm F\kern0.07mm,\kern-0.3mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\kern0.37mm)\subseteq{}^{}{\rm rng}\,{}_{{}^{}}\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm}
holds. However, if we instead take \math{1 < p < \lower1.05mm\hbox{$^+$}\infty} for example with \mathss35{
p=2}, \,then we {\sl do not\kern0.15mm} know whether \math{
\Cal L\,(\kern0.15mm F\kern0.07mm,\kern-0.3mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\kern0.37mm)\subseteq{}^{}{\rm rng}\,{}_{{}^{}}\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm } holds. So under these
circumstances we may ask\kern0.15mm: {\it Is there\kern0.37mm} \math{\smb U} such that \math{
\smb U\in\Cal L\,(\kern0.15mm F\kern0.07mm,\kern-0.3mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\kern0.37mm)\setminus{}^{}{\rm rng}\,{}_{{}^{}}\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm} holds\,{\bf?}
We remark that by suitably adapting the proof of Corollary \ref{Coro q-var}
above it seems \linebreak
to be possible to deduce existence of some \math{y} and a
countable disjoint \mathss30{\scrmt A\subseteq\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+ \kern-0.63mm}
with \mathss30{\bigcup\,\scrmt A = \mathbb I } and such that \math{
(\KPt5 y\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) } is scalarly measurable and such
that we have \math{\smb U\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm\smb X=
\int_{\,A}\kern0.37mm y\,.\KPt8 x\rmdss11\mu } for all \math{x\kern0.37mm,\kern0.15mm\smb X} with \math{
x\in\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F} and \math{{}^{}{\rm rng}\,{}_{{}^{}} x} finite and such that for some \math{
A\in\scrmt A} we have \mathss36{
x\invss46[\KPp1.1\hbox{\font\SweD =cmssbx10\SweD U}{}\kern0.15mm\setminus\{\,\Bnull_\vPi\}\KP1]\subseteq A }. However, we
do not know whether \math{
\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\KP1\|\Lnorss40^{p\sast}_\mu < \lower1.05mm\hbox{$^+$}\infty } holds.
If we could get \math{y} with these properties together with \mathss36{
\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\KP1\|\Lnorss40^{p\sast}_\mu < \lower1.05mm\hbox{$^+$}\infty }, \,then
we would also get \mathss34{\smb U\in{}^{}{\rm rng}\,{}_{{}^{}}\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm}.
\end{problem}
\end{document}
| 3,991 | 362,273 |
en
|
train
|
0.48.0
|
\begin{document}
\begin{abstract}
This paper is devoted to investigate the singular directions of
meromorphic functions in some angular domains. We will confirm the
existence of Hayman $T$ directions in some angular domains. This is
a continuous work of Yang [Yang L., Borel directions of meromorphic
functions in an angular domain, Science in China, Math.
Series(I)(1979), 149-163.] and Zheng [Zheng, J.H., Value
Distribution of Meromorphic Functions, preprint.].
\end{abstract}
\maketitle \keywords{ Keywords and phases: Hayman $T$ direction,
Angular domain, P\'{o}lya peaks, Order}
\maketitle
\section{Introduction and Main Results}
\setcounter{equation}{0} Let $f(z)$ be a meromorphic function on the
whole complex plane. We will use the standard notation of the
Nevanlinna theory of meromorphic functions, such as $T(r,f), N(r,f),
m(r,f), \delta(a,f)$. For the detail, see \cite{Yang}. The order and
lower order of it are defined as follows
$$\lambda(f)=\limsup\limits_{r\rightarrow\infty}\frac{\log T(r,f)}{\log r}$$
and
$$\mu(f)=\liminf\limits_{r\rightarrow\infty}\frac{\log T(r,f)}{\log r}.$$
In view of the second fundamental theorem of Nevanlinna, Zheng
\cite{Zheng} introduced a new singular direction, which is named $T$
direction.\\
\begin{defi}\label{def2} A direction $L: \operatorname{arg} z=\theta$ is called a
$T$ direction of $f(z)$ if for any $\varepsilon>0$, we have
\begin{equation*}
\limsup\limits_{r\rightarrow\infty}\frac{N(r,Z_\varepsilon(\theta),f=a)}{T(r,f)}>0
\end{equation*}
for all but at most two values of $a$ in the extended complex plane
$\widehat{\mathbb{C}}$. Here
$$N(r,\Omega,f=a)=\int_1^r{n(t,\Omega,f=a)\over t}dt,$$
where $n(t,\Omega,f=a)$ is the number of the roots of $f(z)=a$ in
$\Omega\cap\{1<|z|<t\},$ counted according to multiplicity. And
through out this paper, we denote
$Z_\varepsilon(\theta)=\{z:\theta-\varepsilon<\operatorname{arg}
z<\theta+\varepsilon\}$ and $\Omega(\alpha,\beta)=\{z:\alpha<\operatorname{arg}
z<\beta\}$.
\end{defi}
The reason about the name is that we use the Nevanlinna's
characteristic $T(r,f)$ as comparison body. Under the growth
condition
\begin{equation}\limsup\limits_{r\rightarrow\infty}\frac{T(r,f)}{(\log
r)^2}=+\infty. \end{equation} Guo, Zheng and Ng \cite{GuoZhengNg}
confirmed the existence of this type direction and they pointed out
the growth condition (1.1) is sharp. Later, Zhang \cite{zhang}
showed that $T$ directions are different from Borel directions whose
definition can be found in \cite{Hayman}.
In 1979, Yang \cite{Yang01} showed the following theorem, which says that the condition for an angular domain to contain at least one Borel direction.\\
\textbf{Theorem A.} \emph{Let $f(z)$ be a meromorphic function on
the whole complex plane, with $\mu<\infty,0<\lambda\leq\infty$. Let
$\rho$ be a finite number such that $\lambda\geq\rho\geq\mu$ and
$\rho>1/2$. If $f^{(k)}(z)(k\geq0)$ has $p$ distinct deficient
values $a_1,a_2,\cdots,a_p$, then in any angular domain
$\Omega(\alpha,\beta)$ such that
$$\beta-\alpha>\max\{\frac{\pi}{\rho},2\pi-\frac{4}{\rho}\sum\limits_{i=1}^p\arcsin\sqrt{\frac{\delta(a_i,f^{(k)})}{2}}\},$$
$f(z)$ has a Borel direction with order $\geq\rho$.}
Recently, Zheng \cite{JH01} discussed the problem of $T$ directions
of a meromorphic function in one angular domain by proving.\\
\textbf{Theorem B.}\emph{ Let $f(z)$ be a transcendental meromorphic
function with finite lower order $\mu$ and non-zero order $\lambda$
and $f$ has a Nevanlinna deficient value $a\in\widehat{\mathbb{C}}$
with $\delta=\delta(a,f)>0$. For any positive and finite $\tau$ with
$\mu\leq\tau\leq\lambda$, consider the angular domain
$\Omega(\alpha,\beta)$ with
$$\beta-\alpha>\max\{\frac{\pi}{\tau},2\pi-\frac{4}{\tau}\arcsin\sqrt{\frac{\delta}{2}}\}.$$
Then $f(z)$ has a T direction in $\Omega=\Omega(\alpha,\beta)$.}
Following Yang \cite{Yang01} and Zheng \cite{JH01}, we will
continue the discussion of singular directions of $f(z)$ in some
angular domains. The following three questions will be mainly
investigated in this paper.
\begin{que}Can we extend Theorem B to some angular domains
$$X=\bigcup\limits_{j=1}^q\{z:\alpha_j\leq\operatorname{arg} z\leq\beta_j\},$$
where the $q$ pair of real numbers $\{\alpha_j,\beta_j\}$ satisfy
\begin{equation}\label{05}
-\pi\leq\alpha_1<\beta_1\leq\alpha_2<\beta_2\leq\cdots\leq\alpha_q<\beta_q\leq\pi?
\end{equation}
\end{que}
\begin{que} Can $f(z)$ in Theorem B be replaced by any derivative
$f^{(p)}(z)(p\geq0)$?
\end{que}
\begin{que}What can we do if $f(z)$ has many deficient values $a_1, a_2, a_3, \cdots,
a_l$ in Theorem B?
\end{que}
According to the Hayman inequality (see \cite{Hayman}) on the
estimation of $T(r,f)$ in terms of only two integrated counting
functions for the roots of $f(z)=a$ and $f^{(k)}(z)=b$ with
$b\not=0$, Guo, Zheng and Ng proposed in \cite{GuoZhengNg} a
singular direction named Hayman $T$ direction as follows.
\begin{defi}\label{def2}\ Let $f(z)$ be a transcendental meromorphic
function. A direction $L: \operatorname{arg} z=\theta$ is called a Hayman $T$
direction of $f(z)$ if for any small $\varepsilon>0$, any positive
integer $k$ and any complex numbers $a$ and $b\not=0$, we have
\begin{equation*}\label{4}
\limsup\limits_{r\longrightarrow\infty}\frac{N(r,Z_{\varepsilon}(\theta),f=a)
+N(r,Z_{\varepsilon}(\theta),f^{(k)}=b)}{ T(r,f)}>0.
\end{equation*}\end{defi}
Recently, Zheng and the first author \cite{Zheng02} confirmed the
existence of Hayman $T$ direction under the condition that
\begin{equation}\limsup\limits_{r\rightarrow+\infty}\frac{T(r,f)}{(\log r)^3}=+\infty
\end{equation}\\
In the same paper, the authors pointed out the Hayman $T$ direction
is different from the $T$ direction and they gave an example to
show the growth condition (1.3) is sharp. Can we discuss the
problem in some angular domains in the viewpoint of Question 1.1-1.3
? Though out this paper, we define
$$\omega=\max\{\frac{\pi}{\beta_1-\alpha_1},\cdots,\frac{\pi}{\beta_q-\alpha_q}\}.$$
Now, we state our theorems as follows.
\begin{thm}\label{thm1.1}
Let $f(z)$ be a transcendental meromorphic function with finite
lower order $\mu<\infty$, $0<\lambda\leq\infty$. There is an integer
$p\geq0$, such that $f^{(p)}$ has a Nevanlinna deficient value
$a\in\widehat{\mathbb{C}}$ with $\delta(a,f^{(p)})>0$. For $q$ pairs
of real numbers satisfies \eqref{05}. $f$ has at least one Hayman
$T$ direction in $X$ if
\begin{equation}\label{02}
\sum\limits_{j=1}^q(\alpha_{j+1}-\beta_j)<\frac{4}{\sigma}\arcsin\sqrt{\frac{\delta(a,f^{(p)})}{2}},
\end{equation}
where $\mu\leq\sigma\leq\lambda$, and $\omega<\sigma$.
\end{thm}
\begin{thm}\label{thm1.2}
Let $f(z)$ be a transcendental meromorphic function with finite
lower order $\mu<\infty$, $0<\lambda\leq\infty$. There is an integer
$p\geq0$, such that $f^{(p)}$ has $l\geq1$ distinct deficient values
$a_1, a_2, \cdots, a_l$ with the corresponding deficiency
$\delta(a_1,f^{(p)})$, $\delta(a_2,f^{(p)}),
\cdots,\delta(a_l,f^{(p)})$. For $q$ pair of real numbers
$\{\alpha_j,\beta_j\}$ satisfying \eqref{05} and
\begin{equation}\label{02}
\sum\limits_{j=1}^q(\alpha_{j+1}-\beta_j)<\sum\limits_{j=1}^l\frac{4}{\sigma}\arcsin\sqrt{\frac{\delta(a_j,f^{(p)})}{2}},
\end{equation}
where $\mu\leq\sigma\leq\lambda$. If $\omega<\sigma$, then $f$ has
at least one Hayman $T$ direction in $X$.
\end{thm}
We will only prove Theorem \ref{thm1.2}, and Theorem \ref{thm1.1}
is a special case of Theorem \ref{thm1.2}.
| 2,613 | 9,044 |
en
|
train
|
0.48.1
|
\section{Primary knowledge and some lemmas}
In order to prove the theorems, we give some lemmas. The following
result is from \cite{Zheng}.
\begin{lem}
Let $f(z)$ be a transcendental meromorphic function with lower order
$\mu<\infty$ and order $0<\lambda\leq\infty$, then for any positive
number $\mu\leq\sigma\leq\lambda$ and a set $E$ with finite measure,
there exist a sequence $\{r_n\}$, such that
(1) $r_n\notin E$,
$\lim\limits_{n\rightarrow\infty}\frac{r_n}{n}=\infty$;
(2) $\liminf\limits_{n\rightarrow\infty}\frac{\log T(r_n,f)}{\log
r_n}\geq\sigma$;
(3) $T(t,f)<(1+o(1))(\frac{2t}{r_n})^\sigma
T(r_n/2,f),t\in[r_n/n,nr_n]$;
(4)$T(t,f)/t^{\sigma-\varepsilon_n}\leq2^{\sigma+1}T(r_n,f)/r_n^{\sigma-\varepsilon_n},1\leq
t\leq nr_n, \varepsilon_n=[\log n]^{-2}.$
\end{lem}
We recall that $\{r_n\}$ is called the P\'{o}lya peaks of order
$\sigma$ outside $E$. Given a positive function $\Lambda(r)$
satisfying $\lim_{r\rightarrow\infty}\Lambda(r)=0$. For $r>0$ and
$a\in\mathbb{C}$, define
$$D_\Lambda(r,a)=\{\theta\in[-\pi,\pi):\log^+\frac{1}{|f(re^{i\theta})-a|}>\Lambda(r)T(r,f)\},$$
and
$$D_\Lambda(r,\infty)=\{\theta\in[-\pi,\pi):\log^+|f(re^{i\theta})|>\Lambda(r)T(r,f)\}.$$
The following result is called the generalized spread relation, and
Wang in \cite{Wang} proved this.
\begin{lem}\label{lem03}
Let $f(z)$ be transcendental and meromorphic in $\mathbb{C}$ with
the finite lower order $\mu<\infty$ and the positive order
$0<\lambda\leq\infty$ and has $l\geq1$ distinct deficient values
$a_1, a_2, \cdots, a_l$. Then for any sequence of P\'{o}lya peaks
$\{r_n\}$ of order $\sigma>0,\mu\leq\sigma\leq\lambda$ and any
positive function $\Lambda(r)\rightarrow0$ as $r\rightarrow+\infty$,
we have
$$\liminf\limits_{n\rightarrow\infty}\sum\limits_{j=1}^l \operatorname{meas} D_\Lambda(r_n,a_j)\geq\min\{2\pi, \frac{4}{\sigma}\sum\limits_{j=1}^l \arcsin\sqrt{\frac{\delta(a_j,f^{(p)})}{2}}\}.$$
\end{lem}
From \cite{Yang01}, we know that for $a\neq b$ are two deficient
values of $f$, then we have $D_\Lambda(r,a)\bigcap
D_\Lambda(r,b)=\emptyset$.
Nevanlinna theory on the angular domain plays an important role in
this paper. Let us recall the following terms:
\begin{equation*}
\begin{split}
A_{\alpha,\beta}(r,f)&=\frac{\omega}{\pi}\int_1^r(\frac{1}
{t^\omega}-\frac{t^\omega}{r^{2\omega}})\{\log^+|f(te^{i\alpha})|+\log^+|f(te^{i\beta})|\}\frac{dt}{t},\\
B_{\alpha,\beta}(r,f)&=\frac{2\omega}{\pi
r^\omega}\int_{\alpha}^\beta\log^+|f(re^{i\theta})|\sin\omega(\theta-\alpha)d\theta,\\
C_{\alpha,\beta}(r,f)&=2\sum\limits_{1<|b_n|<r}(\frac{1}{|b_n|^\omega}-
\frac{|b_n|^\omega}{r^{2\omega}})\sin\omega(\theta_n-\alpha),
\end{split}
\end{equation*}
where $\omega=\frac{\pi}{\beta-\alpha}$, and
$b_n=|b_n|e^{i\theta_n}$ is a pole of $f(z)$ in the angular domain
$\Omega(\alpha,\beta)$, appeared according to the multiplicities.
The Nevanlinna's angular characteristic is defined as follows:
\begin{equation*}
S_{\alpha,\beta}(r,f)=A_{\alpha,\beta}(r,f)+B_{\alpha,\beta}(r,f)+C_{\alpha,\beta}(r,f).
\end{equation*}
From the definition of $B_{\alpha,\beta}(r,f)$, we have the
following inequality, which will be used in the next.
\begin{equation}\label{06}
B_{\alpha,\beta}(r,f)\geq\frac{2\omega\sin(\omega\varepsilon)}{\pi
r^\omega}\int_{\alpha+\varepsilon}^{\beta-\varepsilon}\log^+|f(re^{i\theta})|d\theta
\end{equation}
The following is the Nevanlinna first and second fundamental
theorem on the angular domains.
\begin{lem}\label{notation01}
Let $f$ be a nonconstant meromorphic function on the angular domain
$\Omega(\alpha,\beta)$. Then for any complex number $a$,
\begin{equation*}
S_{\alpha,\beta}(r,f)=S_{\alpha,\beta}(r,\frac{1}{f-a})+O(1),
r\rightarrow\infty,
\end{equation*}
and for any $q(\geq3)$ distinct points $a_j\in\widehat{\mathbb{C}}\
(j=1,2,\ldots,q)$,
\begin{equation*}\label{nevan01}
\begin{split}
(q-2)S_{\alpha,\beta}(r,f)&\leq\sum\limits_{j=1}^q\overline{C}_{\alpha,\beta}(r,\frac{1}{f-a_j})+Q_{\alpha,\beta}(r,f),
\end{split}
\end{equation*}
where
$$Q_{\alpha,\beta}(r,f)=(A+B)_{\alpha,\beta}(r,\frac{f'}{f})
+\sum\limits_{j=1}^q(A+B)_{\alpha,\beta}(r,\frac{f'}{f-a_j})+O(1).$$
\end{lem}
The key point is the estimation of error term
$Q_{\alpha,\beta}(r,f)$, which can be obtained for our purpose of
this paper as follows. And the following is true(see
\cite{Goldberg}). Write
$$Q(r,f)=A_{\alpha,\beta}(r,\frac{f^{(p)}}{f})+B_{\alpha,\beta}(r,\frac{f^{(p)}}{f}).$$
Then
(1)$Q(r,f)=O(\log r)$ as $r\rightarrow\infty$, when
$\lambda(f)<\infty$.
(2)$Q(r,f)=O(\log r+\log T(r,f))$ as $r\rightarrow\infty$ and
$r\notin E$ when $\lambda(f)=\infty$, where $E$ is a set with finite
linear measure.
The following result is useful for our study, the proof of which is
similar to the case of the characteristic function $T(r,f)$ and
$T(r,f^{(k)})$ on the whole complex plane. For the completeness, we
give out the proof.
\begin{lem}\label{lem01}
Let $f(z)$ be a meromorphic function on the whole complex plane.
Then for any angular domain $\Omega(\alpha,\beta)$, we have
$$S_{\alpha,\beta}(r,f^{(p)})\leq(p+1)S_{\alpha,\beta}(r,f)+O(\log r+\log T(r,f)),$$
possibly outside a set of $r$ with finite measure.
\end{lem}
\begin{proof}
In view of the definition of $S_{\alpha,\beta}(r,f)$ and Lemma
\ref{notation01}, we get the following
\begin{equation*}
\begin{split}
S_{\alpha,\beta}(r,f^{(p)})&\leq
C_{\alpha,\beta}(r,f^{(p)})+(A+B)_{\alpha,\beta}(r,f)+(A+B)_{\alpha,\beta}(r,\frac{f^{(p)}}{f})\\
&=p\overline{C}_{\alpha,\beta}(r,f)+S_{\alpha,\beta}(r,f)+(A+B)_{\alpha,\beta}(r,\frac{f^{(p)}}{f})\\
&\leq(p+1)S_{\alpha,\beta}(r,f)+Q(r,f).
\end{split}
\end{equation*}
\end{proof}
Recall the definition of Ahlfors-Shimizu characteristic in an angle
(see \cite{Tsuji}). Let $f(z)$ be a meromorphic function on an angle
$\Omega=\{z:\alpha\leq\operatorname{arg} z\leq \beta\}$. Set
$\Omega(r)=\Omega\cap\{z:1<|z|<r\}$. Define
$$\mathcal{S}(r,\Omega,f)=\frac{1}{\pi}\int\int_{\Omega(r)}{\left(|f'(z)|\over
1+|f(z)|^2\right)^2}d\sigma$$ and
$$\mathcal{T}(r,\Omega,f)=\int_1^r{\mathcal{S}(t,\Omega,f)\over
t}dt.$$
The following lemma is a theorem in \cite{Zheng02}, which is to
controll the term $\mathcal {T}(r,\Omega_\varepsilon)$ using the
counting functions $N(r,\Omega,f=a)$ and $N(r,\Omega,f^{(k)}=b)$.
\begin{lem}\label{04}
Let $f(z)$ be meromorphic in an angle $\Omega=\{z:\alpha\leq\operatorname{arg} z
\leq\beta\}$. Then for any small $\varepsilon>0$, any positive
integer $k$ and any two complex numbers $a$ and $b\not=0$, we have
\begin{equation}\label{9}
\mathcal{T}(r,\Omega_\varepsilon,f) \leq K\{N(2r,\Omega,f=a)+
N(2r,\Omega,f^{(k)}=b)\}+O(\log^3 r)
\end{equation} for a positive constant $K$ depending only on $k$, where $\Omega_\varepsilon=\{z:\alpha+\varepsilon<\operatorname{arg}
z<\beta-\varepsilon\}$.
\end{lem} In order to prove our theorem, we have to use the following
lemma, which is a consequent result of Theorem 3.1.6 in \cite{JH01}.
\begin{lem}
Let $f(z)$ be a transcendental meromorphic function in the whole
plane, and satisfies the conditions of Theorem \ref{thm1.2} or
Theorem \ref{thm1.1}. Take a sequence of P\'{o}lya peak $\{r_n\}$ of
$f(z)$ of order $\sigma>\omega=\frac{\pi}{\beta-\alpha}$. If $f(z)$
has no Hayman T direction in the angular domain
$\Omega(\alpha,\beta)$, then the following real function satisfy
$\lim\limits_{r\rightarrow\infty}\Lambda(r)=0$, which $\Lambda(r)$
is defined as follows
$$\Lambda(r)^2=\max\{\frac{\mathcal
{T}(r_n,\Omega_\varepsilon,f)}{T(r_n,f)},
\frac{r_n^{\omega}}{T(r_n,f)}\int_1^{r_n}\frac{\mathcal
{T}(t,\Omega_\varepsilon,f)}{t^{\omega+1}}dt,\frac{r_n^\omega[\log
r_n+\log T(r_n,f)]}{T(r_n,f)}\},$$ for $r_n\leq r<r_{n+1}.$
\end{lem}
\begin{proof} We should treat two cases.\\
Case (I). If there is no Hayman $T$ direction on $\Omega$, then from
Lemma \ref{04}, we have
$$\mathcal {T}(r,\Omega_\varepsilon,f)=o(T(2r,f))+O(\log^3r), \ as\ r\rightarrow\infty.$$
Combining Lemma 2.1 and $\sigma>\omega$, we have
\begin{equation*}
\begin{split}
\int_1^{r_n}\frac{\mathcal
{T}(t,\Omega_\varepsilon,f)}{t^{\omega+1}}dt&=o(\int_1^{r_n}\frac{T(2t,f)}{t^{\omega+1}}dt)+\int_1^{r_n}\frac{O(\log^3
t)}{t^{\omega+1}}dt\\
&\leq
o(\int_1^{r_n}\frac{T(r_n,f)}{t^{\omega+1}}(\frac{2t}{r_n})^\sigma
dt)+O(\log^3r_n)\\
&=o(\frac{T(r_n,f)}{r_n^\omega})+O(\log^3r_n)\\
\end{split}
\end{equation*}
Then
$$\frac{r_n^\omega}{T(r_n,f)}\int_1^{r_n}\frac{\mathcal {T}(t,\Omega_\varepsilon)}{t^{\omega+1}}dt\rightarrow0, \ as \ n\rightarrow\infty. $$
Case (II). If $$\limsup\limits_{n\rightarrow\infty}\frac{\mathcal
{T}(r_n,\Omega_\varepsilon,f)}{T(r_n,f)}>0,$$ then by \eqref{9}, we
have
$$\limsup\limits_{n\rightarrow\infty}\frac{N(2r_n,\Omega,f=a)+N(2r_n,\Omega,f^{(k)}=b)}{T(r_n,f)}>0.$$
Since $\{r_n\}$ is a sequence of P\'{o}lya peaks of order $\sigma$,
then we have $$T(2r_n,f)\leq2^\sigma T(r_n,f).$$ Then $\Omega$ must
contain a Hayman $T$ direction of $f(z)$. This is contradict to the
hypothesis.
From Case (I) and Case (II) and notice that $r_n^\omega[\log
r_n+\log T(r_n,f)]/T(r_n,f)\rightarrow0,(n\rightarrow\infty)$, we
have proved that $\limsup_{r\rightarrow\infty}\Lambda(r)=0$.
\end{proof}
The following result was firstly established by Zheng
\cite{JH01}(Theorem 2.4.7), it is crucial for our study.
\begin{lem}
Let $f(z)$ be a function meromorphic on
$\Omega=\Omega(\alpha,\beta)$. Then
$$S_{\alpha,\beta}(r,f)\leq2\omega^2\frac{\mathcal {T}(r,\Omega,f)}{r^\omega}+\omega^3\int_1^r\frac{\mathcal {T}(t,\Omega,f)}{t^{\omega+1}}dt+O(1),\ \ \omega=\frac{\pi}{\beta-\alpha}.$$
\end{lem}
We also have to use the following lemma, which is due to Hayman and
Miles \cite{Miles}.
\begin{lem}\label{lem02}
Let $f(z)$ be meromorphic in the complex plane. Then for a given
$K>1$, there exists a set $M(K)$ with $\overline{\log
dens}M(K)\leq\delta(K)$,
$\delta(K)=\min\{(2e^{K-1}-1)^{-1},(1+e(K-1)exp(e(1-K)))\}$, such
that
$$\limsup\limits_{r\rightarrow+\infty,r\notin M(K)}\frac{T(r,f)}{T(r,f^{(p)})}\leq3eK.$$
\end{lem}
| 3,979 | 9,044 |
en
|
train
|
0.48.2
|
\section{Proof of theorem \ref{thm1.2}}
\begin{proof}
Case(I). $\lambda(f)>\mu$. Then we choose $\sigma$ such that
$\lambda(f^{(p)})=\lambda(f)>\sigma\geq\mu=\mu(f^{(p)}),
\sigma>\omega$. From the inequality \eqref{02}, we can take a real
number $\varepsilon>0$ such that
\begin{equation}\label{01}
\sum\limits_{j=1}^q(\alpha_{j+1}-\beta_j+4\varepsilon)+\varepsilon<\sum\limits_{j=1}^l\frac{4}{\sigma+2\varepsilon}\arcsin\sqrt{\frac{\delta(a_j,f^{(p)})}{2}},
\end{equation}
and $$\lambda(f^{(p)})>\sigma+2\varepsilon>\mu.$$ Then there exists
a sequence of P\'{o}lya peaks $\{r_n\}$ of order
$\sigma+2\varepsilon$ of $f^{(p)}$ such that $\{r_n\}$ are not in
the set of Lemma \ref{lem01} and Lemma \ref{lem02}.
We define $q$ real functions $\Lambda_j(r)(j=1,2,\cdots,q)$ as
follows.
\begin{equation*}
\begin{split}
\Lambda_j(r)^2=\max\{&\frac{\mathcal
{T}(r_n,\Omega(\alpha_j+\varepsilon,\beta_j-\varepsilon),f)}{T(r_n,f)},\\
&\frac{r_n^{\omega_j}}{T(r_n,f)}
\int_1^{r_n}\frac{\mathcal
{T}(t,\Omega(\alpha_j+\varepsilon,\beta_j-\varepsilon),f)}{t^{\omega_j+1}}dt,\frac{r_n^{\omega_j}[\log
r_n+\log T(r_n,f)]}{T(r_n,f)}\},
\end{split}
\end{equation*}
for $r_n\leq r<r_{n+1},
\omega_{j}=\frac{\pi}{\beta_{j}-\alpha_{j}}$. By using Lemma 2.5, we
have $\Lambda_j(r)\rightarrow0$, as $r\rightarrow\infty$, if $f(z)$
has no Hayman $T$ directions on $X$. Set $\Lambda(r)=\max_{1\leq
j\leq q}\{\Lambda_j(r)\}$, we have
$\lim_{r\rightarrow\infty}\Lambda(r)=0$. Therefore for large enough
$n$, by Lemma \ref{lem03} we have
\begin{equation}\label{03}
\sum\limits_{j=1}^l \operatorname{meas} D_\Lambda(r_n,a_j)>\min\{2\pi,
\frac{4}{\sigma+2\varepsilon}\sum\limits_{j=1}^l
\arcsin\sqrt{\frac{\delta(a_j,f^{(p)})}{2}}\}-\varepsilon.
\end{equation}
We note that $\sigma+2\varepsilon>1/2$, we suppose for any $n$
\eqref{03} holds. Set
$$K_n=\operatorname{meas}
((\bigcup\limits_{j=1}^lD_\Lambda(r_n,a_j))\bigcap(\bigcup\limits_{j=1}^q(\alpha_j+2\varepsilon,\beta_j-2\varepsilon))).$$
Combining \eqref{01} with \eqref{03}, we obtain
\begin{equation*}
\begin{split}
K_n&\geq \sum\limits_{j=1}^l\operatorname{meas} (D_\Lambda(r_n,
a_j))-\operatorname{meas}([-\pi,\pi)\backslash\bigcup\limits_{j=1}^q(\alpha_j+2\varepsilon,\beta_j-2\varepsilon))\\
&= \sum\limits_{j=1}^l\operatorname{meas} (D_\Lambda(r_n,
a_j))-\operatorname{meas}(\bigcup\limits_{j=1}^q(\beta_j-2\varepsilon,\alpha_{j+1}+2\varepsilon))\\
&=\sum\limits_{j=1}^l\operatorname{meas} (D_\Lambda(r_n,
a_j))-\sum\limits_{j=1}^q(\alpha_{j+1}-\beta_j+4\varepsilon)>\varepsilon>0.
\end{split}
\end{equation*} It is easy to see that, there exists a $j_0$ such that for
infinitely many $n$, we have
\begin{equation*}
\operatorname{meas}
(\bigcup\limits_{j=1}^lD_\Lambda(r_n,a_j)\bigcap(\alpha_{j_0}+2\varepsilon,\beta_{j_0}-2\varepsilon))>\frac{K_n}{q}>\frac{\varepsilon}{q}.
\end{equation*}
We can assume that the above holds for all the $n$.
Set
$E_{nj}=D(r_n,a_j)\bigcap(\alpha_{j_0}+2\varepsilon,\beta_{j_0}-2\varepsilon)$.
Thus we have
\begin{equation}\label{07}
\begin{split}
\sum\limits_{j=1}^l\int_{\alpha_{j_0}+2\varepsilon}^{\beta_{j_0}-2\varepsilon}\log^+\frac{1}{|f^{(p)}(r_ne^{i\theta})-a_j|}d\theta&\geq
\sum\limits_{j=1}^l\int_{E_{nj}}\log^+\frac{1}{|f^{(p)}(r_ne^{i\theta})-a_j|}d\theta\\
&\geq\sum\limits_{j=1}^l\operatorname{meas}(E_{nj})\Lambda(r_n)T(r_n,f^{(p)})\\&>\frac{\varepsilon}{q}\Lambda(r_n)T(r_n,f^{(p)})\\
&>\frac{\varepsilon}{3eqK}\Lambda(r_n)T(r_n,f).
\end{split}
\end{equation}
The last inequality uses Lemma 2.8.
On the other hand, we have
\begin{equation}\label{08}
\begin{split}
&\sum\limits_{j=1}^l\int_{\alpha_{j_0}+2\varepsilon}^{\beta_{j_0}-2\varepsilon}\log^+\frac{1}{|f^{(p)}(r_ne^{i\theta})-a_j|}d\theta\leq\sum\limits_{j=1}^l
\frac{\pi}{2\omega_{j_0}\sin(\varepsilon\omega_{j_0})}r_n^{\omega_{j_0}}B_{\alpha_{j_0}+\varepsilon,\beta_{j_0}-\varepsilon}(r_n,\frac{1}{f^{(p)}-a_j})\\
&<\sum\limits_{j=1}^l\frac{\pi}{2\omega_{j_0}\sin(\varepsilon\omega_{j_0})}r_n^{\omega_{j_0}}S_{\alpha_{j_0}+\varepsilon,\beta_{j_0}-\varepsilon}(r_n,\frac{1}{f^{(p)}-a_j})\\
&=\frac{l\pi}{2\omega_{j_0}\sin(\varepsilon\omega_{j_0})}r_n^{\omega_{j_0}}S_{\alpha_{j_0}+\varepsilon,\beta_{j_0}-\varepsilon}(r_n,f^{(p)})+O(r_n^{\omega_{j_0}})\\
&\leq\frac{l\pi}{2\omega_{j_0}\sin(\varepsilon\omega_{j_0})}r_n^{\omega_{j_0}}[(p+1)S_{\alpha_{j_0}+\varepsilon,\beta_{j_0}-\varepsilon}(r_n,f)+\log r_n+\log T(r_n,f)]+O(r_n^{\omega_{j_0}})\\
&\leq\frac{l\pi}{2\omega_{j_0}\sin(\varepsilon\omega_{j_0})}(p+1)[2\omega_{j_0}^2\mathcal
{T}(r_n,\Omega(\alpha_{j_0}+\varepsilon,\beta_{j_0}-\varepsilon),f)\\&+\omega_{j_0}^3r_n^{\omega_{j_0}}\int_1^{r_n}\frac{\mathcal
{T}(t,\Omega(\alpha_{j_0}+\varepsilon,\beta_{j_0}-\varepsilon),f)}{t^{\omega_{j_0}+1}}dt]\\
&+\frac{l\pi}{2\omega_{j_0}\sin(\varepsilon\omega_{j_0})}r_n^{\omega_{j_0}}[\log r_n+\log T(r_n,f)]+O(r_n^{\omega_{j_0}})\\
&\leq\frac{l\pi}{2\omega_{j_0}\sin(\varepsilon\omega_{j_0})}(p+1)[2\omega_{j_0}^2\Lambda(r_n)^2T(r_n,f)
+\omega_{j_0}^3\Lambda(r_n)^2T(r_n,f)]\\
&+\frac{l\pi}{2\omega_{j_0}\sin(\varepsilon\omega_{j_0})}r_n^{\omega_{j_0}}[\log
r_n+\log T(r_n,f)]+O(r_n^{\omega_{j_0}}),\ \ \ \
\omega_{j_0}=\frac{\pi}{\beta_{j_0}-\alpha_{j_0}-2\varepsilon}. \
\end{split}
\end{equation}
\eqref{07} and \eqref{08} imply that
$$\Lambda(r_n)\leq O(\Lambda(r_n)^2).$$
A contradiction is derived because $\Lambda(r_n)\rightarrow0$ as
$n\rightarrow\infty$.
Case (II). $\lambda(f)=\mu$. By the same argument as in Case1 with
all the $\sigma+2\varepsilon$ replaced by $\sigma=\mu$, we can
derive the same contradiction.
\end{proof}
\end{document}
\end{document}
| 2,452 | 9,044 |
en
|
train
|
0.49.0
|
\begin{document}
\title{Introducing a new concept of distance on a topological space by generalizing the definition of quasi-pseudo-metric
}
\author{Hamid Shobeiri\footnote{Email address: [email protected] (A.H. Shobeiri)}\\
\footnotesize{{\it Department of Mathematics, K.N. Toosi
University of Technology,}} \\
\footnotesize{{\it P.O.Box 16315-1618, Tehran, Iran}}\\
}
\date{}
\maketitle
\begin{abstract}
In this paper, a new structure is defined on a topological space that equips the space with a concept of distance in order to do that firstly, a generalization of quasi-pseudo-metric space named R.O-metric space is introduced, and some of its basic properties is studied. Afterwards the concept of generalized R.O-metric space is defined .Finally, we establish that every topological space is generalized R.O-metrizable.
\end{abstract}
\textbf{Keywords:} Topological space, Quasi-pseudo-metric space, R.O-metric space, Generalized R.O-metric space. \\
\textbf{AMS Subject Classifications:} 54D80, 54D35, 54E35, 54E99, 54D65 .
\section{Introduction}
Topological spaces are extension of metric spaces. It is well known that each arbitrary topological space is not necessary metrizable (see \cite{munkres1975topology} or \cite{simmons1963introduction}). Therefore despite of the beauty and simplicity of such extension, it involves some limitations. For example, size of neighborhoods of two distinct points are not comparable in topological spaces. In addition, uniform continuity, Cauchy sequence and complete space are no more definable in arbitrary topological spaces. These limitations may raise the idea of defining topological spaces through a new concept of distance, in order to simplify working in these spaces.Defining a new concept of distance, will be useful. In this direction, some mathematicians introduced some structures weaker than metric spaces.
A metric on a set $X$ is a function $d:X\times X\to [0,\infty)$ such that for all $x,y,z\in X$, the following conditions are satisfied:
\begin{align}
d(x,y)=0\Leftrightarrow x=y,\label{I.1}\\
d(x,y)=d(y,x)\label{I.2},\\
d(x,z)\leq d(x,y)+d(y,z). \label{I.3}
\end{align}
One of the generalized metric spaces is semi-metric space that is introduced by Frechet and Menger which satisfies conditions \eqref{I.1} and \eqref{I.2} of definition of metric space (see \cite{cicchese1976questioni}, \cite{wilson1931semi}, \cite{frechet1906quelques} and \cite{menger1928untersuchungen}). In the last few years, the study of non-symmetric topology has received a new derive as a consequence of it's applications to the study of several problems in theoretical computer science and applied physics.
One of such structures is quasi-metric space that is introduced by W.A.Wilson (see \cite{wilson1931quasi}) which has conditions \eqref{I.1} and \eqref{I.3}. One other generalization of metric spaces is called pseudo-metric space, which satisfies conditions $(\displaystyle d(x,x)=0)$, (2) , (3) (see \cite{simmons1963introduction}). Quasi-pseudo-metric space is introduced by Kelly (see \cite{kelly1963bitopological}) which satisfies conditions $( d(x,x)=0)$ and (3). $T_0$-quasi-metric space is quasi-pseudo-metric space that satisfies condition $(d(x,y)=0=d(y,x)\Rightarrow x=y)$ that is presented in paper \cite{kemajou2012isbell}.
Multi-metric space is defined by Smarandache $($see \cite{smarandache2000mixed}, \cite{smarandache2001unifying}$)$, which is a union $\tilde{M} =\bigcup_{i=1}^{n}M_i$, such that each $M_i$ is a space
with metric $d_i$ for all $1\le i\le n$.
The above mentioned structures can not describe all topological spaces. In this paper, it is aimed to present a new structure to be able to describe all topological spaces. It is started by definition of structure that is called R.O-metric space (Right-Oriented-metric space: this terminology comes from non-symmetric meter) which is a generalization of quasi-pseudo-metric space. Then generalized R.O-metric is defined which reforms the definition of topological space. In the first section, the concept of R.O-metric space is defined. In the second section, R.O-metric space is generalized and improved by adding some conditions.
\section{Preliminaries}
\begin{defn}
Let $X$ be a non empty set; a function $\overrightarrow{d}:X\times X\to [0,\infty)$ is called a R.O-metric on $X$ iff for every $x,y,z\in X$, the following conditions hold:
\begin{enumerate}
\item$\overrightarrow{d}(x,x)=0$,
\item $\overrightarrow{d}(x,z)+\overrightarrow{d}(z,y)\neq 0 \Rightarrow \overrightarrow{d}(x,y)\le \overrightarrow{d}(x,z)+\overrightarrow{d}(z,y),$
\end{enumerate}
and then $(X,\overrightarrow{d})$ is called a R.O-metric space.
\end{defn}
\begin{ex}
Every metric space is a R.O-metric space.
\end{ex}
\begin{ex}
As another example , for $X=\{a,b,c\}$ consider $$\overrightarrow{d}(a,b)=\overrightarrow{d}(b,c)=1\; , \overrightarrow{d}(a,c)=2\; ,$$$$\overrightarrow{d}(a,a)=\overrightarrow{d}(b,b)=\overrightarrow{d}(c,c)=\overrightarrow{d}(b,a)=\overrightarrow{d}(c,a)=\overrightarrow{d}(c,b)=0,$$ then $(X,\overrightarrow{d})$ is a R.O-metric space.\ref{fig:55}
\begin{figure}\label{fig:55}
\end{figure}
| 1,683 | 12,035 |
en
|
train
|
0.49.1
|
\section{Preliminaries}
\begin{defn}
Let $X$ be a non empty set; a function $\overrightarrow{d}:X\times X\to [0,\infty)$ is called a R.O-metric on $X$ iff for every $x,y,z\in X$, the following conditions hold:
\begin{enumerate}
\item$\overrightarrow{d}(x,x)=0$,
\item $\overrightarrow{d}(x,z)+\overrightarrow{d}(z,y)\neq 0 \Rightarrow \overrightarrow{d}(x,y)\le \overrightarrow{d}(x,z)+\overrightarrow{d}(z,y),$
\end{enumerate}
and then $(X,\overrightarrow{d})$ is called a R.O-metric space.
\end{defn}
\begin{ex}
Every metric space is a R.O-metric space.
\end{ex}
\begin{ex}
As another example , for $X=\{a,b,c\}$ consider $$\overrightarrow{d}(a,b)=\overrightarrow{d}(b,c)=1\; , \overrightarrow{d}(a,c)=2\; ,$$$$\overrightarrow{d}(a,a)=\overrightarrow{d}(b,b)=\overrightarrow{d}(c,c)=\overrightarrow{d}(b,a)=\overrightarrow{d}(c,a)=\overrightarrow{d}(c,b)=0,$$ then $(X,\overrightarrow{d})$ is a R.O-metric space.\ref{fig:55}
\begin{figure}\label{fig:55}
\end{figure}
\end{ex}
\begin{defn}
In a R.O-metric space $(X,\overrightarrow{d})$,
the set, $V_{r}(p)=\{x\;|\;\overrightarrow{d}(p,x)<r\}$ is called a $r$-ball of a point $p$ with radius $r>0$.
\end{defn}
\begin{n}
Let $(X,\overrightarrow{d})$ be a R.O-metric space ,
then the set $S_{\overrightarrow{d}}=\{V_{r}(p)\;|\;r>0,p\in X\}$ is a subbasis for a topology on X, which is called the generated topology by $\overrightarrow{d}$ and is shown by $\tau_{\overrightarrow{d}}.$
\end{n}
\begin{defn}
Topological space $(X,\tau)$ is called R.O-metrizable iff there exists a R.O-metric $\overrightarrow{d}$ such that $\tau_{\overrightarrow{d}}=\tau$.
\end{defn}
It can be shown that many of the most familiar topological spaces are R.O-metric spaces, here are some examples of non metrizable topological spaces which are R.O-metric spaces:
\begin{ex}
Let $X$ be a set and $\phi \neq A\subseteq X$, then $\tau_{A}=\{B\subseteq X \;|\; A\subseteq B\}\cup \{\emptyset\}$ is a topology on $X$; we define R.O-metric $\overrightarrow{d}$ on $X$ as follows:
\begin{enumerate}
\item $\forall x\in X;\;\overrightarrow{d}(x,x)=0,$ \item $\forall a \in X , \forall b\in A, \; \overrightarrow{d}(a,b)=0,$\item $\forall a\in X , \forall b\in A^{c} ;\; \overrightarrow{d}(a,b)=1.$
\end{enumerate}
The topology $\tau_{\overrightarrow{d}}$ is generated by subbasis $$S_{\overrightarrow{d}}=\{V_{r}(p)\;|\;r>0,p\in X\}=\{V_{1}(a)\;|\;a\in A\}\cup{\{V_{1}(b)\;|\;b\in A^{c}\}}\cup{\{V_{2}(x)\;|\;x\in X\}}$$
$$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\{A\}\cup{\{\{b\}\cup{A}\;|\;b\in A^{c}\}}\cup{\{X\}}.$$
Thus $\tau_{\overrightarrow{d}}=\tau_{A}$.
\end{ex}
\begin{ex}\label{13}
Let $\tau$ be cofinite topology on infinite set $X$, that means topology in which the open sets are the subset of $X$ with finite complements. It is known that (see \cite{hrbacek1999introduction}) the set $X$ can be written as $X=\bigcup_{\alpha\in I}{A_{\alpha}}$ such that for all $\alpha\in I$, $A_{\alpha}$ is countable and $A_{\alpha}=\{x_{\alpha,1},x_{\alpha,2},...\}$. Now define R.O-metric $\overrightarrow{d}$ on $X$ as follows:
\begin{enumerate}
\item$\forall x\in X;\;\overrightarrow{d}(x,x):=0,$\item$\forall\alpha\in I,\;\forall n\in O;(O\;is\;the\;odd\;natural\;numbers)\\\;\overrightarrow{d}(x_{\alpha,n},x_{\alpha,n+1}):=1=\overrightarrow{d}(x_{\alpha,n+1},x_{\alpha,n})$,\item$\overrightarrow{d}(x_{\alpha,n},y):=0=\overrightarrow{d}(x_{\alpha,n+1},z),\;\forall y\ne x_{\alpha,n+1},\;\forall z\ne x_{\alpha,n}$
\end{enumerate}
the induced topology by $\overrightarrow{d}$ which is generated by subbasis $$S_{\overrightarrow{d}}=\{V_{1}(x_{\alpha,n})\;|\;\alpha\in I\;,\;n\in O\}\cup \{V_{1}(x_{\alpha,n})\;|\;\alpha\in I\;,\;n\in E\}\cup\{V_{2}(p)\;|\;p\in X\}$$ $$\;\;\;\;\;\;=\{X-\{x_{\alpha,n+1}\}\;|\;n\in O\;,\;\alpha\in I\}\cup \{X-\{x_{\alpha,n-1}\}\;|\;n\in E\;,\;\alpha\in I\}\cup\{X\},$$
in which E is the even natural numbers. Thus $\tau=\tau_{\overrightarrow{d}}$.
\end{ex}
\begin{ex}
Let $\tau$ be K-topology on $\Bbb{R}$,that means the topology generated by the basis $\{(a,b)\;|\;a,b\in \Bbb{R}\}\cup \{(a,b)-\{\frac{1}{n}\}_{n\in \Bbb{N}}\;|\;a,b\in \Bbb{R}\}$, then we define R.O-metric $\overrightarrow{d}$ as follows:
\begin{enumerate}
\item$\forall x\in X;\;\overrightarrow{d}(x,x):=0,$\item$\forall x\notin \{\frac{1}{n}\}_{n\in {\Bbb{N}}},\;\overrightarrow{d}(x,\frac{1}{n}):=\mid{x-\frac{1}{n}}\mid+1,$\item$\; Otherwise\; \overrightarrow{d}(x,y):=|x-y|.$
\end{enumerate}
Then it is easy to check that
$$S_{\overrightarrow{d}}=\{V_{r}(x)\;|\;x\notin \{\frac{1}{n}\}_{n\in {\Bbb{N}}}\;,\;0< r\le 1\}\cup \{V_{r}(x)\;|\;x\notin \{\frac{1}{n}\}_{n\in {\Bbb{N}}}\;,\;r>1\}$$$$\cup\{V_r(\frac{1}{n})\;|\;n\in\Bbb{N}\;,\;r>0\}=\{(x-r,x+r)-\{\frac{1}{n}\}_{n\in {\Bbb{N}}}\;|\;0<r\le 1\}$$$$\cup\{(x-r,x+r)\;|\;r>1\}\cup\{(\frac{1}{n}-r,\frac{1}{n}+r)\;|\;n\in\Bbb{N}\;,\;r>0\},$$
generates topology of K-topology on $\Bbb{R}$.
\end{ex}
\begin{ex}
For lower limit topology $\Bbb{R}_{l}$, which is a topology on $\Bbb{R}$ that has a basis as $\{[a,b)\;|\;a,b\in \Bbb{R}\}$, define R.O-metric $\overrightarrow{d}$ as follows:
\begin{enumerate}
\item$\forall a\in\Bbb{R};\;\overrightarrow{d}(a,a)=0,$\item$\overrightarrow{d}(a,b)=\begin{cases}a-b+1&b<a\\b-a&a\le b\end{cases}.$
\end{enumerate}
Then $$S_{\overrightarrow{d}}=\{V_r(a)\;|\;0<r\le 1\;,\;a\in\Bbb{R}\}\cup\{V_r(a)\;|\;r>1\;,\;a\in\Bbb{R}\}$$$$=\{[a,a+r)\;|\;0<r\le 1\;,\;a\in\Bbb{R}\}\cup\{(a-r,a+r)\;|\;r>1\;,\;a\in\Bbb{R}\}.$$
Simply we can see $\tau_{\overrightarrow{d}}$ is the lower limit topology.
\end{ex}
Now, we give an example which shows that there can be fined R.O-metric spaces that are NOT qusi-metrizable, pseudo-metrizable and NOT quasi-pseudo-metrizable.
\begin{ex}
Suppose $(X,\tau_c)$ be a cofinite topological space and $Card(X)\ge Card(\Bbb{R})$. By Example \ref{13}, $(X,\tau_c)$ is R.O-metrizable. Now we prove that $(X,\tau_c)$ is not quasi-pseudo-metrizable. If it is quasi-pseudo-metrizable, then there exists a quasi-pseudo-metric $d$, such that $\tau_d=\tau_c$, thus $B=\{V_r(x)\;|\;r>0\;,\;x\in X\}$ is a basis, and for each $x\in X$ and $U$ open set containing $x$, there exists $t>0$ such that $V_t(x)\subseteq U$. Thus $B_x=\{V_r(x)\;|\;r>0\}$ is a local base at $x$, since $V_r(x)^{c}$ is finite, thus $B_x$ is at most countable. Therefore $(X,\tau_c)$ is first countable and it is a contradiction, because $Card(X)\ge Card(\Bbb{R})$ and cofinite topological spaces like $(X,\tau)$ with $Card(X)>Card(\Bbb{N})$ are not first countable $($see \cite{simmons1963introduction}$)$. Since quasi-metrizable space is quasi-pseudo-metrizable, so $(X,\tau_c)$ is not quasi-metrizable. Also if $(X,\tau_c)$ is pseudo-metrizable, then $B$ is a basis for this topology. In addition, for each $x\in X$ and $U$ open set containing $x$, there exist $t>0$ such that $V_t(x)\subseteq U$ and by the same procedure as above, it causes a contradiction.
\end{ex}
\begin{p}
Suppose $(X,\overrightarrow{d})$ is a R.O-metric space and for each $x,y\in X$, define $\overrightarrow{\bar{d}}(x,y)=\frac{\overrightarrow{d}(x,y)}{\overrightarrow{d}(x,y)+1}$. Then $\tau_{\overrightarrow{d}}=\tau_{\overrightarrow{\bar{d}}}$.
\end{p}
\begin{proof}
Obviously condition (1) in the definition of R.O-metric holds. Now for all $x,y\in X$, if $\overrightarrow{d}(x,y)\le \overrightarrow{d}(x,z)+\overrightarrow{d}(z,y)$, then $$\frac{1}{\overrightarrow{d}(x,z)+\overrightarrow{d}(z,y)+1}\le \frac{1}{\overrightarrow{d}(x,y)+1}$$ $$\Rightarrow 1-\frac{1}{\overrightarrow{d}(x,y)+1}\le 1-\frac{1}{\overrightarrow{d}(x,z)+\overrightarrow{d}(z,y)+1}=\frac{\overrightarrow{d}(x,z)+\overrightarrow{d}(z,y)}{\overrightarrow{d}(x,z)+\overrightarrow{d}(z,y)+1}$$ $$\Rightarrow \frac{\overrightarrow{d}(x,y)}{\overrightarrow{d}(x,y)+1}\le \frac{\overrightarrow{d}(x,z)}{\overrightarrow{d}(x,z)+1}+\frac{\overrightarrow{d}(z,y)}{\overrightarrow{d}(z,y)+1}.$$
Therefore $\overrightarrow{\bar{d}}$ is R.O-metric on $X$. To prove $\tau_{\overrightarrow{d}}=\tau_{\overrightarrow{\bar{d}}}$, assume $V_r(x)\in S_{\overrightarrow{d}}$ and $z\in V_r(x)$, so by definition $\overrightarrow{d}(x,z)<r$. If $\overrightarrow{d}(x,z)\ne 0$, then $$\overrightarrow{\bar{d}}(x,z)<\frac{r}{r+1}\Rightarrow z\in \bar{V}_{\frac{r}{r+1}}(x)\in S_{\overrightarrow{\bar{d}}},$$in which $\bar{V}_{\frac{r}{r+1}}(x)$ is a $r$-ball with respect to $\overrightarrow{\bar{d}}$, and if $$\overrightarrow{d}(x,z)=0,\;then\; \overrightarrow{\bar{d}}(x,z)=0<\frac{r}{r+1},$$thus $z\in\bar{V}_{\frac{r}{r+1}}(x)$ and we have $V_r(x)\subseteq \bar{V}_{\frac{r}{r+1}}(x).$ It is easy to check that $\bar{V}_{\frac{r}{r+1}}(x)\subseteq V_r(x),$ for all $x\in X$ and all non negative real numbers. Thus $$S_{\overrightarrow{d}}\subseteq S_{\overrightarrow{\bar{d}}}\;\;\;(*).$$
Now let $\bar{V}_r(x)\in S_{\overrightarrow{\bar{d}}}$ and $z\in \bar{V}_r(x),\;r<1$, then $\overrightarrow{\bar{d}}(x,z)<r$. If $\overrightarrow{\bar{d}}(x,z)\ne 0$, then $\overrightarrow{d}(x,z)<\frac{r}{1-r}$, and if $\overrightarrow{\bar{d}}(x,z)=0$, then $\overrightarrow{d}(x,z)=0<\frac{r}{1-r}$, thus $z\in V_{\frac{r}{1-r}}(x)\in S_{\overrightarrow{d}}$, that implies $\bar{V}_r(x)\subseteq V_{\frac{r}{1-r}}(x)$. It is easy to check that $V_{\frac{r}{1-r}}(x)\subseteq \bar{V}_r(x)$, thus we get $ S_{\overrightarrow{\bar{d}}}\subseteq S_{\overrightarrow{d}}$, and by virtue of $(*)$, $S_{\overrightarrow{\bar{d}}}= S_{\overrightarrow{d}}$, which implies $\tau_{\overrightarrow{d}}=\tau_{\overrightarrow{\bar{d}}}.$
\end{proof}
| 3,984 | 12,035 |
en
|
train
|
0.49.2
|
\begin{n}\label{1}
It is well-known that every finite topological space $(X,\tau)$ has a subbasis $S$ such that $Card(S)\le Card(X)$, since for every point $p$ in $X$ there is the smallest open set with respect to $(\subseteq)$ containing $p$ and the set of these open sets is a subbasis $S$ for $(X,\tau)$ , obviously $Card(S)\le Card(X)$. In the following example we show that this property does not necessarily hold for infinite topological spaces.
\end{n}
\begin{ex} \label{4}
Let $Y=\Bbb{N}\times\Bbb{N}$ , $p$ be a point not in $Y$ and $X=\{p\}\cup Y$. For each function $ f:\Bbb{N}\to\Bbb{N}$, let
$$B_f:=\{p\}\cup\{(k,\ell)\in Y\;|\;\ell\ge f(k)\}\;.$$
Topologize $X$ by making each point of $Y$ isolated and taking $\left\{B_f:f\in{\Bbb{N}^{\Bbb{N}}}\right\}$ as a local subbasis at $p$.
We show that $X$ has no countable subbase.
Let $S=\left\{B_f:f\in{\Bbb{N}^{\Bbb{N}}}\right\}\cup\{(n,m)\;|\;n,m\in\Bbb{N}\}$ and
$$\mathscr{B}=\left\{\bigcap\mathscr{F}:\mathscr{F}\subseteq S\text{ and }\mathscr{F}\text{ is finite}\right\}\;.$$
Thus $\mathscr{B}$ is the base generated by the subbasis $S$. $S$ is infinite and has $|S|$ finite subsets, and therefore $|\mathscr{B}|=|S|$. If $S$ is countable, $\mathscr{B}$ is also countable, and $X$ is second countable and hence first countable. But we show that there is no countable local base at $p$.
Suppose that $\mathscr{U}=\{U_n:n\in\Bbb N\}$ is a countable family of open neighbourhoods of $p$. For each $n\in\Bbb{N}$ there are $f_{n_1},f_{n_2},\ldots,f_{n_{m_n}}\in \Bbb {N}^{\Bbb{ N}}$ such that $\bigcap_{n_1\le i\le n_{m_n}} B_{f_i}\subseteq U_n$. Define
$$g:\Bbb N\to\Bbb N:k\mapsto 1+\max\{f_{k_i}(\ell):\ell\le k,\;1\le i\le m_k\}\;;$$
then $B_g\nsupseteq B_{f_{n_i}}$, because it is evident that $(k,f_{k_i}(k))\in B_{f_{k_i}}$, but $(k,f_{k_i}(k))\notin B_g$ . Therefore for all $n\in \Bbb{N}$, $B_g\nsupseteq U_n$, so $\mathscr{U}$ is not a local base at $p$.
\end{ex}
\begin{n}
If $X$ is infinite and $\overrightarrow{d}$ is a R.O-metric on $X$, by definition of R.O-metric and $S_{\overrightarrow{d}}$, we can see that $Card(S_{\overrightarrow{d}})\le Card(X)$.
Since for every $p$ in $X$, $Card(\{V_{r}(p)\;|\;r\in \Bbb{R}^{+}\})\le Card(X)$ ,
hence $Card(S_{\overrightarrow{d}})=Card(\bigcup_{p\in X}\{V_{r}(p)\;|\;r\in \Bbb{R}^{+}\})\le Card{X}.$ This shows that the topological space in Example \ref{4} is not R.O-metrizable.
\end{n}
\begin{p}
Let $(X,\tau)$ be a topological space. If it has a subbasis $S$ such that $|S|\le |X|$, and there is a function $f:X \to S$ such that $x\in f(x)$, for every $x\in X$, then $(X,\tau)$ is R.O-metrizable.
\end{p}
\begin{proof}
By the hypothesis $S=\{f(x)\;|\;x\in X\}$. For every $x,y\in X$ define $$\overrightarrow{d}(x,y):=\begin{cases} 0&y\in f(x) \\1&otherwise \end{cases}$$
It is easy to check that $S_{\overrightarrow{d}}=S$, hence $\tau_{\overrightarrow{d}}=\tau.$
\end{proof}
\begin{co}
Every finite topological space is R.O-metrizable since by Note \ref{1} we can define $f:X\to S$ such that $f(x)$ is the smallest open set containing $x$.
\end{co}
\begin{cun}
Let $(X,\tau)$ and $S$ be a subbasis of it, such that $Card\;(S)\;\le Card\;(X)$, then $(X,\tau)$ is a R.O-metrizable.
\end{cun}
Now we mention three lemmas that will be useful for the last section.
\begin{lem}\label{2}
Let $(X,\tau)$ be finite $T_0$-topological space, then there exists $a\in X$ such that $\{a\}\in\tau$.
\end{lem}
\begin{proof}
Suppose that $U\in\tau$ be a minimal open set by relation $\subseteq$. If $Card\;(U)>1$, then there exists at least two points $a,b\in U$ and since $(X,\tau)$ is $T_0$, there exists $V\in\tau$ such that $a\in V,\;b\notin V$. Therefore $\emptyset\ne U\cap V\in\tau$ and $Card\;(U\cap V)<Card\;(U)$, this is a contradiction with minimality of $U$.
\end{proof}
\begin{lem}\label{15}
Let $(X,\tau)$ be a topological space and $(\sim)$ be a relation on $X$ by :$$x\sim y\Leftrightarrow \forall U\in\tau,\;(x\in U\Leftrightarrow y\in U).$$
Then $(\sim)$ is an equivalence relation on $X$ and $\tau^{'}=\{\;[U]\;|\;U\in \tau\}$ is a $T_0$-topology on $\mathcal{A}_{\tau}=\{\;[x]\;|\;x\in X\}$.
\end{lem}
\begin{proof}
Obviuosely $(\sim)$ is an equivalence relation on $X$. Suppose $[z]\in [U]\cap [V]$, in which $[U],[V]\in \tau^{'}$, then iff $[z]\in [U\cap V]$. Thus $[U]\cap [V]=[U\cap V]$. Also if $[z]\in\bigcup_{\alpha\in I}[U_{\alpha}]$, then iff $[z]\in [\bigcup_{\alpha\in I}U_{\alpha}]$. Thus $\bigcup_{\alpha\in I}[U_{\alpha}]=[\bigcup_{\alpha\in I}U_{\alpha}]$. Therefore $\tau^{'}$ is a topology on $\mathcal{A}_{\tau}$. Assume $[x]\ne [y]$, thus without losing the quality, there exist $U\in \tau$ such that $x\in U$ and $y\notin U$. Thus $[x]\in [U]$ and $\bar{[y]}\subseteq [U]^c$, so $\bar{[x]}\ne \bar{[y]}$. Therefore $\tau^{'}$ is $T_0$-topology on $\mathcal{A}_{\tau}$.
\end{proof}
\begin{lem}\label{3}
Let $(X,\tau)$ be a topological space. If $(\mathcal{A}_{\tau},\tau^{'})$ is a R.O-metrizable, then $(X,\tau)$ is R.O-metrizable.
\end{lem}
\begin{proof}
There exist R.O-metric $\overrightarrow{d}$ such that $\tau_{\overrightarrow{d}}=\tau^{'}$. Now define for all $x,y\in X$, $$\overrightarrow{D}(x,x)=0,\;\overrightarrow{D}(x,y)=\begin{cases}0&[x]=[y]\\\overrightarrow{d}([x],[y])&[x]\ne [y]\end{cases}.$$ Obviuosly $\overrightarrow{D}$ is a metric on $X$. Assume that $[U]=\bigcup_{\alpha\in I}(\cap_{i=1}^{n_{\alpha}}U_{r_i}([x_i]))$. We clame that $U=\bigcup_{\alpha\in I}(\cap_{i=1}^{n_{\alpha}}U_{r_i}(x_i))$. So suppose that $z\in U$. Since $U=\bigcup_{[x]\in [U]}[x]$, then there exist $\alpha_0\in I$ such that $[z]\in \bigcap_{i=1}^{n_{\alpha_0}}U_{r_i}([x_i])$. Hence for all $1\le i\le n_{\alpha_0}$, $[z]\in U_{r_i}([x_i])$ which means $\overrightarrow{d}([x_i],[z])<r_i$. By definition of $\overrightarrow{D}$, we have $\overrightarrow{D}(x_i,z)<r_i$, thus for all $1\le i\le n_{\alpha_0}$, $z\in U_{r_i}(x_i)$. Therefore $U\subseteq \bigcup_{\alpha\in I}(\cap_{i=1}^{n_{\alpha}}U_{r_i}(x_i))$. Checking $\bigcup_{\alpha\in I}(\cap_{i=1}^{n_{\alpha}}U_{r_i}(x_i))\subseteq U$ is easy. So $U=\bigcup_{\alpha\in I}(\cap_{i=1}^{n_{\alpha}}U_{r_i}(x_i))$, thus $\tau_{\overrightarrow{D}}=\tau$.
\end{proof}
| 2,552 | 12,035 |
en
|
train
|
0.49.3
|
\begin{co}
Every finite topological space is R.O-metrizable since by Note \ref{1} we can define $f:X\to S$ such that $f(x)$ is the smallest open set containing $x$.
\end{co}
\begin{cun}
Let $(X,\tau)$ and $S$ be a subbasis of it, such that $Card\;(S)\;\le Card\;(X)$, then $(X,\tau)$ is a R.O-metrizable.
\end{cun}
Now we mention three lemmas that will be useful for the last section.
\begin{lem}\label{2}
Let $(X,\tau)$ be finite $T_0$-topological space, then there exists $a\in X$ such that $\{a\}\in\tau$.
\end{lem}
\begin{proof}
Suppose that $U\in\tau$ be a minimal open set by relation $\subseteq$. If $Card\;(U)>1$, then there exists at least two points $a,b\in U$ and since $(X,\tau)$ is $T_0$, there exists $V\in\tau$ such that $a\in V,\;b\notin V$. Therefore $\emptyset\ne U\cap V\in\tau$ and $Card\;(U\cap V)<Card\;(U)$, this is a contradiction with minimality of $U$.
\end{proof}
\begin{lem}\label{15}
Let $(X,\tau)$ be a topological space and $(\sim)$ be a relation on $X$ by :$$x\sim y\Leftrightarrow \forall U\in\tau,\;(x\in U\Leftrightarrow y\in U).$$
Then $(\sim)$ is an equivalence relation on $X$ and $\tau^{'}=\{\;[U]\;|\;U\in \tau\}$ is a $T_0$-topology on $\mathcal{A}_{\tau}=\{\;[x]\;|\;x\in X\}$.
\end{lem}
\begin{proof}
Obviuosely $(\sim)$ is an equivalence relation on $X$. Suppose $[z]\in [U]\cap [V]$, in which $[U],[V]\in \tau^{'}$, then iff $[z]\in [U\cap V]$. Thus $[U]\cap [V]=[U\cap V]$. Also if $[z]\in\bigcup_{\alpha\in I}[U_{\alpha}]$, then iff $[z]\in [\bigcup_{\alpha\in I}U_{\alpha}]$. Thus $\bigcup_{\alpha\in I}[U_{\alpha}]=[\bigcup_{\alpha\in I}U_{\alpha}]$. Therefore $\tau^{'}$ is a topology on $\mathcal{A}_{\tau}$. Assume $[x]\ne [y]$, thus without losing the quality, there exist $U\in \tau$ such that $x\in U$ and $y\notin U$. Thus $[x]\in [U]$ and $\bar{[y]}\subseteq [U]^c$, so $\bar{[x]}\ne \bar{[y]}$. Therefore $\tau^{'}$ is $T_0$-topology on $\mathcal{A}_{\tau}$.
\end{proof}
\begin{lem}\label{3}
Let $(X,\tau)$ be a topological space. If $(\mathcal{A}_{\tau},\tau^{'})$ is a R.O-metrizable, then $(X,\tau)$ is R.O-metrizable.
\end{lem}
\begin{proof}
There exist R.O-metric $\overrightarrow{d}$ such that $\tau_{\overrightarrow{d}}=\tau^{'}$. Now define for all $x,y\in X$, $$\overrightarrow{D}(x,x)=0,\;\overrightarrow{D}(x,y)=\begin{cases}0&[x]=[y]\\\overrightarrow{d}([x],[y])&[x]\ne [y]\end{cases}.$$ Obviuosly $\overrightarrow{D}$ is a metric on $X$. Assume that $[U]=\bigcup_{\alpha\in I}(\cap_{i=1}^{n_{\alpha}}U_{r_i}([x_i]))$. We clame that $U=\bigcup_{\alpha\in I}(\cap_{i=1}^{n_{\alpha}}U_{r_i}(x_i))$. So suppose that $z\in U$. Since $U=\bigcup_{[x]\in [U]}[x]$, then there exist $\alpha_0\in I$ such that $[z]\in \bigcap_{i=1}^{n_{\alpha_0}}U_{r_i}([x_i])$. Hence for all $1\le i\le n_{\alpha_0}$, $[z]\in U_{r_i}([x_i])$ which means $\overrightarrow{d}([x_i],[z])<r_i$. By definition of $\overrightarrow{D}$, we have $\overrightarrow{D}(x_i,z)<r_i$, thus for all $1\le i\le n_{\alpha_0}$, $z\in U_{r_i}(x_i)$. Therefore $U\subseteq \bigcup_{\alpha\in I}(\cap_{i=1}^{n_{\alpha}}U_{r_i}(x_i))$. Checking $\bigcup_{\alpha\in I}(\cap_{i=1}^{n_{\alpha}}U_{r_i}(x_i))\subseteq U$ is easy. So $U=\bigcup_{\alpha\in I}(\cap_{i=1}^{n_{\alpha}}U_{r_i}(x_i))$, thus $\tau_{\overrightarrow{D}}=\tau$.
\end{proof}
\section{Generalized R.O-metric space}
\begin{defn}
Suppose $(X,\overrightarrow{d})$ is a R.O-metric space. $(X,\overrightarrow{d},\beta)$ is called a generalized R.O-metric space if and only if there exists a collection $\beta=\{f_{\alpha}:X\to X\;|\;\alpha \in I\}$ such that $Id_X \in \beta$ and $$\forall y\in V_{r,\alpha}(x),\;\exists t>0\;,\;\exists \eta\in I \;s.t\; y\in f_{\eta}(X)\;,\;V_{t,\eta}(y)\subseteq V_{r,\alpha}(x)$$
where $V_{r,\alpha}(x)=\{f_{\alpha}(y)\in X\;|\;d(x,f_{\alpha}(y))<r\; ,\; y\in X\; ,\; x\in f_{\alpha}(X)\}$.
\end{defn}
\begin{n}
If $(X,\overrightarrow{d},\beta)$ is a generalized R.O-metric space the set $S_{\overrightarrow{d,\beta}}=\{V_{r,\alpha}(x)\;|\;x\in X\; ,\; r>0\; ,\; x\in f_{\alpha}(X)\}$ is a basis of a topology on $X$.
Topology generated by $S_{\overrightarrow{d},\beta}$ is denoted by $\tau_{(\overrightarrow{d},\beta)}$.
\end{n}
\begin{defn}
Topological space $(X,\tau)$ is called generalized R.O-metrizable iff there exists $\overrightarrow{d}$ such that $\tau_{(\overrightarrow{d},\beta)}=\tau$.
\end{defn}
\begin{ex} \label{5}
Let $X$ be the set from Example \ref{4} and $\tau$ be the topology on $X$ from the same example,for every $(m,n)$ and $(s,r)$ in $\Bbb{N}^{2}$ such that $(m,n)\ne (s,r)$ define $$\overrightarrow{d}((m,n),(m,n))=0=\overrightarrow{d}(p,(m,n))=\overrightarrow{d}(p,p)$$
$$\overrightarrow{d}((m,n),(s,r))=1=\overrightarrow{d}((m,n),p)$$
And for every $f\in \Bbb{N}^{\Bbb{N}}$ define $F_{f}:X\to X$ as follows : $$F_{f}(p)=p \;,\; F_{f}((m,n))=G((m,n))$$ Where $G:\Bbb{N}^{2} \to B_f$ is a surjective function, let $\beta=\{F_{f}\;|\;f\in \Bbb{N}^{\Bbb{N}}\} \cup \{Id_X\} $.
Now $$V_{(\frac{1}{2} ,f)}(p)$$ $$=\{F_{f}(a)\in X\;|\;\overrightarrow{d}(p,F_{f}(a))<\frac{1}{2} \}$$ $$=\{F_{f}(a) \;|\;a\in X \}=B_f $$ and $$V_{(\frac{1}{2} ,f)}((m,n))$$ $$=\{F_{f}(a)\in X\;|\;\overrightarrow{d}((m,n),F_{f}(a))<\frac{1}{2} \}=\{(m,n)\}$$ But if $r>1$ , $x\in X$ and $f\in \beta$ then $V_{(r,f)}(x)=X$ therefore $\tau_{(\overrightarrow{d},\beta)}=\tau$. So $(X,\tau)$ is generalize R.O-metric space which is NOT R.O-metrizable space.
\end{ex}
\begin{ex} \label{6}
Let $X$ be a infinite set, $p\in X$ be fixed and $B\subset X$ which is $Card\;(B)=Card\;(B^{c})=Card\;(X)$ containing $p$. Suppose $\tau=\{A\subset X\;|\;B\subset A\;,\;A\ne B \}\cup \{\{x\}\;|\;x\in X\;,\;x\ne p\}$ then $\tau$ is a topology on $X$. Let $f_{A}:X\to X$ be surjective function such that $f_{A}(p)=p$ and $f_{A}(x)\ne p \; \forall x\ne p $ and let $\beta=\{f_{A}\;|\;A\in \tau \}\cup \{Id_X\}$. Now For distinct $x$ and $y$ in $X$ such that $x,y\ne p$ define $$\overrightarrow{d}(x,y)=0=\overrightarrow{d}(p,x)=\overrightarrow{d}(p,p)$$ $$\overrightarrow{d}(x,y)=1=\overrightarrow{d}(x,p).$$ It is easy to check that $\tau_{(\overrightarrow{d},\beta)}=\tau$ thus $(X,\tau)$ is a generalized R.O-metrizable space.
\end{ex}
\begin{theorem}
Every topological space $(X,\tau)$ is generalized R.O metrizable
\end{theorem}
\begin{proof}
Suppose $(X,\tau^{'})$ is an arbitrary $T_0$ topological space. Now let $(Y,\tau^{''})$ be a topological space where $Y=\{0,1\}$ and $\tau^{''}=\{\emptyset ,\;\{1\},\;Y\}$ and $\overrightarrow{d}(0,1)=0=\overrightarrow{d}(0,0)=\overrightarrow{d}(1,1),\;\overrightarrow{d}(1,0)=1$ (obviously $\tau_{\overrightarrow{d}}=\tau^{''}$). Assume that $C$ is a proper closed subset of $(X,\tau)$ , Define $f_{C}:X\to Y$ as follows :
$f_{C}(C)=0$ and $f_{C}(X-C)=1$
$f$ is obviously continuous, Let $J=\{f_{C}\;|\;X-C\in \tau\}$. $J$ is a family of continuous maps that separates points from closed sets. Define $$F:X\to Y^J\;\;\;,F(x)=(f_{\alpha}(x))_{\alpha\in J}$$ obviously $F$ is an embedding when $Y^J$ is equipped with the product topology $(F(X)\cong X)$.
Let $ (<)$ be a well-ordering on $J$ define $$\overrightarrow{D}((x_{\alpha}),(y_{\alpha}))=\overrightarrow{d}(1,y_{\beta})$$ where $\beta$ is the smallest index in $(J,<)$ which $x_{\beta}=1$. $\overrightarrow{D}$ is a R.O metric that generates the product topology of $Y^J$ therefore $(F(X),\overrightarrow{D}|_{F(X)})$ is a R.O-metric space.We know that topology induced of $ (Y^J,\tau_{\overrightarrow{D}})$ on $F(X)$ is equal to $\{U\cap F(X)\;|\; U\in \tau_{\overrightarrow{D}}\}$ which is equal to the topology generated by $\{V_{r}(y)\cap F(X)\;|\;r>0\;,\;y\in Y^J\}$. Now let $A=\{V_{r}(y)\cap F(X)\;|\;r>0\;,\;y\in Y^J-F(X)\}$, consider $B=\{f_{r,y}:X\to X\;|\; y\in Y^J-F(X) \;,\; r>0\;,f_{r,y}(X)=V_{r}(y)\cap F(X)\}\cup\{Id_X\}$ we claim that $(F(X),\overrightarrow{D}|_{F(X)},B)$ is a generalized R.O-metric space, it suffices to prove that $\tau_{\overrightarrow{D}|_{F(X)},B}$ generates the induced topology of the product topology of $Y^J$ on $F(X)$. For $y\in Y^J$ and $r>0$ if $y\in F(X)$ then $V_{r}(y)\cap F(X) \in \tau_{\overrightarrow{D}|_{F(X)}}$. Also suppose $y\notin F(X)$, $r>0$ and $z\in V_{r,y}(y)=f_{r,y}(X)=V_{r}(y)\cap F(X)$, then $V_{r,y}(z)\subseteq V_{r,y}(y)$.Thus $\tau_{\overrightarrow{D}|_{F(X)},B}$ generates the induced topology of the product topology of $Y^J$ on $F(X)$. Since $F(X)\cong X$, therefore $(X,\tau^{'})$ is a generalized R.O-metrizable. By Lemma \ref{3} $(A,\tau^{'})$ is a $T_0$-topological space, thus there exist $\overrightarrow{d}$ and $\beta=\{f_{\alpha}:A\to A\}\cup \{Id_X\}$ such that $(A,\overrightarrow{d},\beta)$ is a generalized R.O-metric space. Define $$\overrightarrow{D}(x,y)=\begin{cases}0&[x]=[y]\\\overrightarrow{d}(x,y)&[x]\ne [y]\end{cases}$$ and $$F_{\alpha}:X\to X\;\;,\;\;F_{\alpha}(x)=g_{[x]}(x)$$where $g_{[x]}$ is a map between $[x]$ and $f_{\alpha}([x])$. Therefore $(X,\tau)$ is a generalized R.O-metrizable.
\end{proof}
\noindent
At the end, it is useful to see the figure \ref{fig:16} for understanding the subject.
\begin{figure}
\caption{diagram for topological spaces}
\label{fig:16}
\end{figure}
\end{document}
| 3,816 | 12,035 |
en
|
train
|
0.50.0
|
\betagin{document}
\title{A relative mass cocycle and the mass of asymptotically hyperbolic manifolds}
\author{Andreas \v Cap, and A.\ Rod Gover}
\circperatorname{ad}dress{A.\v C.: Faculty of Mathematics\\
University of Vienna\\
Oskar--Morgenstern--Platz 1\\
1090 Wien\\
Austria\\
A.R.G.:Department of Mathematics\\
The University of Auckland\\
Private Bag 92019\\
Auckland 1142\\
New Zealand}
\email{[email protected]}
\email{[email protected]}
\betagin{abstract}
We construct a cocycle that, for a given $n$-manifold, maps pairs of
asymptotically locally hyperbolic (ALH) metrics to a tractor-valued
$(n-1)$-form field on the conformal infinity. This describes, locally
on the boundary, a relative mass difference between the pairs of ALH
metrics, where the latter are required to be asymptotically related to
a given order that depends on the dimension. It is distinguished as a
geometric object by its property of being invariant under suitable
diffeomorphisms fixing the boundary, and that act on
(either) one of the argument metrics.
Specialising to the case of an ALH metric $h$ that is suitably
asymptotically related to a locally hyperbolic conformally compact
metric, we show that the cocycle detemines an absolute invariant
$c(h)$. This tractor-valued $(n-1)$-form field on the conformal
infinity is canonically associated to $h$ (i.e. is not dependent on
other choices) and is equivariant under the appropriate
diffeomorphisms.
Finally specialising further to the case that the boundary is a sphere and
that a metric $h$ is asymptotically related to a hyperbolic metric on
the interior, we show that the invariant $c(h)$ can be integrated
over the boundary. The result pairs with solutions of the KID (Killing
initial data) equation to recover the known description of hyperbolic
mass integrals of Wang, and Chru\'{s}ciel--Herzlich.
\end{abstract}
\subjclass[2020]{primary: 53A55; secondary: 53C18, 53C25, 53C80, 83C30, 83C60}
\keywords{mass in GR, mass aspect, asymptotically hyperbolic manifolds, geometric
invariants, tractor calculus, conformally compact manifolds}
\thetaanks{A.\v C.\ gratefully acknowledges support by the Austrian Science Fund (FWF):
P33559-N and the hospitality of the University of Auckland. A.R.G.\ gratefully
acknowledges support from the Royal Society of New Zealand via Marsden Grants
16-UOA-051 and 19-UOA-008. We thank P.\ Chru\'sciel for very helpful discussions.}
\maketitle
\pagestyle{myheadings} \markboth{\v Cap, Gover}{Mass and Tractors}
\section{Introduction}\lambdabel{1}
In general relativity, and a number of related mathematical studies,
the notion of a ``mass'' invariant for relevant geometric manifolds is
extremely important and heavily studied \cite{Bartnik,LP,S-Yau}. In
general defining and interpreting a suitable notion of mass is not
straightforward. For so-called asymptotically flat manifolds the
Arnowitt-Deser-Misner (ADM) energy-momentum, is well established and
is usually accepted as the correct definition. Motivated by the
desire to define and study mass in other settings X.\ Wang \cite{Wang}
and P.T.\ Chru\'sciel and M.\ Herzlich \cite{Chrusciel-Herzlich}
introduced a notion of mass integrals and ``energy-momentum'' for
Riemannian manifolds that, in a suitable sense, are asymptotically
hyperbolic. These have immediate applications for a class of static
spacetimes.
The aim of this article is to construct new invariants that capture a notion of mass
density, in the setting of asymptotically hyperbolic metrics. These invariants are
local as quantities on the boundary at infinity and they specialise to recover (by an
integration procedure) the mass as introduced by Wang and Chru\'sciel--Herzlich. The
setting we work in is rather restrictive in some aspects but very general in other
aspects. The main restriction is that we are working in a conformally compact
setting, so we need strong assumptions on the order of asymptotics. On the other
hand, underlying this is an arbitrary manifold with boundary with no restrictions on
the topology of the boundary. We also allow a fairly general ``background metric'';
the core of our results only require a background metric that is asymptotically
locally hyperbolic (ALH), as in Definition \ref{def2.2}, part (1). The main
invariant we construct is associated to a pair of ALH metrics (that are asymptotic to
sufficient order), so it should be thought of as a \textit{relative local mass} or as
a \textit{local mass difference}.
Our basic setting looks as follows. We start with an arbitrary
manifold $\circverline{M}$ of dimension $n\geq 3$, with boundary $\partial M$
and interior $M$. Given two conformally compact metrics $g$ and $h$ on
$M$, there is a well defined notion of $g$ and $h$ approaching each
other asymptotically to certain orders towards the boundary. The
actual order we need depends on the dimension and specializes to the
order required in \cite{Wang} on hyperbolic space. This defines an
equivalence relation on conformally compact metrics and we consider
one equivalence class $\mathcal G$ of such metrics. The only additional
requirement at this point is that $\mathcal G$ consists of
ALH-metrics. Since $\mathcal G$ consists of conformally compact metrics,
each $g\in\mathcal G$ gives rise to a conformal structure on $\partial M$,
called the conformal infinity of $g$. Moreover, the asymptotic
condition used to define $\mathcal G$ is strong enough to ensure that all
metrics in $\mathcal G$ lead to the same conformal infinity. Thus $\mathcal G$
canonically determines a conformal structure $[\mathcal G]$ on $\partial
M$.
This last fact is crucial for the
further development, since the invariants we construct are geometric
objects for this conformal structure on $\partial M$ (and a slightly
stronger structure in case that $\dim(\circverline{M})=3$). Indeed, the conformal
structure $[\mathcal G]$ canonically determines the so-called
\textit{standard tractor bundle} $\mathcal T\partial M\to\partial M$. This
is a vector bundle of rank $n+1$ endowed with a Lorentzian bundle
metric and a metric linear connection \cite{BEG}. We construct
\textit{cocycles} $c$ that associate to each pair of metrics
$g,h\in\mathcal G$ a tractor valued $(n-1)$-form
$c(g,h)\in\Omega^{n-1}(\partial M,\mathcal T\partial M)$, with the property
that $c(h,g)=-c(g,h)$ and $c(g,k)=c(g,h)+c(h,k)$ for all $g,h,k\in\mathcal
G$. These cocycles are the basic ``relative local masses'' we
consider.
The idea of the construction is that any metric $g\in\mathcal G$ determines a conformal
class on $\circverline{M}$ (since the conformal class of $g$ on $M$ extends to the boundary by
conformal compactness) and this restricts to $[\mathcal G]$ at the boundary. Working with
this conformal structure, we can use standard techniques of tractor calculus, and an
instance of what is called a BGG splitting operator, to associate to any $h\in\mathcal G$
a one-form with values in the tactor bundle $\mathcal T\circverline{M}$. The Hodge dual (with
respect to $g$) of this one-form is then shown to admit a smooth extension of the
boundary whose boundary value is defined to be $c(g,h)$, see Propositions
\ref{prop3.1} and \ref{prop3.2}. The proof that this actually defines a cocycle
needs some care because of the use of different conformal structures on $\circverline{M}$, but
is otherwise straigthforward.
Initially, this leads to a two parameter family of cocyles since there
are two constructions of the above type, one depending on the trace of
the difference $g-h$, the other on its trace-free part. Our
constructions do not require charts or coordinates, so, in that sense,
are automatically geometric in nature. The constructions also
readily imply equivariancy with respect to the appropriate diffeomorphisms. If
$\Phi$ is a diffeomorphism of $\circverline{M}$ which preserves the class $\mathcal
G$ (in an obvious sense) then by definition the restriction
$\Phi|_{\partial M}$ is a conformal isometry for $[\mathcal G]$. Therefore,
it naturally acts on sections of $\mathcal T\partial M$ (and of course on
forms on $\partial M$) and for any of the cocycles
$c(\Phii^*g,\Phii^*h)=(\Phii|_{\partial M})^*c(g,h)$, see Proposition
\ref{prop3.3}.
There is a much more subtle compatibility condition with
diffeomorphisms, however. Indeed, consider a diffeomorphism $\Psii$
that is compatible with $[\mathcal G]$ and suppose that $\Psii|_{\partial
M}=\circperatorname{id}_{\partial M}$. Then it turns out that $\Psii$ is asymptotic to
the identity of order $n+1$ in a well-defined sense, see Section
\ref{3.4} and Theorem \ref{thm3.5}. The main technical result of our
article then is that there is a unique ratio of the two parameters for
our cocyles, which ensures that $c(g,\Psii^*h)=c(g,h)$ for any $\Psii$
which is asymptotic to the identity of order $n+1$. Hence, up to an
overall normalization we obtain a unique cocylce $c$ which has this
invariance property in addition to the equivariancy property mentioned
above. This proof is based on an idea in \cite{CDG} that shows that
the action of diffeomorphisms that are asymptotic to the identity can
be absorbed into a geometric condition relating the two metrics (and
an adapted defining function for one of them). The key feature of this
property is that it holds in the general setting of a class of ALH
metrics on a manifold with boundary. In the special cases for which
we obtain invariants of single metrics, this property enables us to
prove equivariancy of such invariants under diffeomorphisms preserving
$\mathcal G$, see below.
To pass from our cocycles to invariants of a single metric, one has to
go to specific situations in which $\mathcal G$ constains particularly
nice metrics. We only discuss the case that $\mathcal G$ locally contains
metrics that are hyperbolic, i.e.\ have constant sectional curvature
$-1$. This of course implies that $(\partial M,[\mathcal G])$ is
conformally flat, but it does not impose further restrictions on the
topology of $\partial M$, see Section \ref{3.8}. Under this
assumption, we show that, for a cocycle from our one-parameter family
and a metric $h\in\mathcal G$, all local hyperbolic metrics $g\in\mathcal G$
locally lead to the same tractor-valued form $c(g,h)$. These local
forms then piece together to define an object
$c(h)\in\Omega^{n-1}(\partial M,\mathcal T\partial M)$ that is canoncially
associated to $h$. We prove that this is equivariant under
diffeomorphisms preserving the class $\mathcal G$, see Theorem
\ref{thm3.8}.
As a last step, we prove that, in the conformally compact setting, the AH mass as
introduced in \cite{Chrusciel-Herzlich} can be obtained by integrating our
invariant. Thus we have to specialize to the case that $\circverline{M}$ is an open
neighborhood of the boundary sphere in the closed unit ball and $\mathcal G$ contains the
restriction of the Poincar\'e metric. This implies that $\partial M=S^{n-1}$ and
$[\mathcal G]$ is the round conformal structure and thus the standard tractor bundle
$\mathcal TS^{n-1}$ can be globally trivialized by parallel sections. Hence $\mathcal
TS^{n-1}$--valued $(n-1)$-forms can be integrated to global parallel sections of
$\mathcal TS^{n-1}$ (say by expanding in any globally parallel frame and then integrating
the coefficient forms). Now it is well known how to make the trivialization of $\mathcal
TS^{n-1}$ explicit, and we show that there is a particularly nice way to do this
using the conformal class $[g]$ on $\circverline{M}$ determined by the Poincar\'e metric. Via
boundary values of parallel tractors in the interior, the parallel sections of $\mathcal
TS^{n-1}$ turn out to be parametrised by the solutions of the KID (Killing initial
data) equation (\ref{KID}) on $M$. But these solutions exactly parametrise the mass
integrals used to define the AH mass in the style of \cite{Chrusciel-Herzlich}, see
\cite{Michel}. This last fact was our original motivation to look for a tractor
interpretation of the AH mass.
Now a solution $V$ of the KID equation determines a parallel section
$s_V$ of $\mathcal TS^{n-1}$ and we can proceed as follows. Given
$h\in\mathcal G$, we can form the invariant $c(h)\in
\Omega^{n-1}(S^{n-1},\mathcal TS^{n-1})$ and integrate it to a parallel
section $\int_{S^{n-1}}c(h)$ of $\mathcal TS^{n-1}$. This can then be
paired, via the tractor metric, with $s_V$. Analyzing the boundary
behavior of the mass integral determined by $V$, we show in Theorem
\ref{thm3.9} that, after appropriate normalization, this pairing
exactly recovers the mass integral, of \cite{Chrusciel-Herzlich},
determined by $V$.
Throughout all manifolds, tensor fields, and related objects, will be
taken to be smooth in the sense of $C^\infty$. For most results lower
regularity would be sufficient, but we do not take that up here.
| 3,636 | 45,932 |
en
|
train
|
0.50.1
|
\section{Setup and tractors}\lambdabel{2}
\subsection{Conformally compact metrics}\lambdabel{2.1}
Let $\circverline{M}$ be a smooth manifold with boundary $\partial M$ and
interior $M$. Recall that a \textit{local defining function} for
$\partial M$ is a smooth function $\rho:U\to\mathbb R_{\geq 0}$ defined
on some open subset $U\subset\circverline{M}$ such that $U\cap\partial
M=\rho^{-1}(\{0\})$ and such that $d\rho|_{U\cap\partial M}$ is
nowhere vanishing. For two local defining functions $\rho$ and
$\hat\rho$ defined on the same open set $U$, there is a smooth
function $f:U\to\mathbb R_{>0}$ such that $\hat\rho=f\rho$. It is often
convenient to write such positive function as $f=e^{\tilde f}$ for
some smooth function $\tilde f$. This notion of definining functions
extends, without problem, from functions to smooth sections of line
bundles. One just has to replace $d\rho$ by the covariant derivative
with respect to any linear connection, which is independent of the
connection along the zero set of the section. In particular, taking the line bundle
concerned to be a density bundle (as discussed above Proposition \ref{prop2.1} below)
leads to \textit{defining densities} \cite{Proj-comp}.
A pseudo-Riemannian metric $g$ on $M$ is called \textit{conformally
compact} if, for any point $x\in\partial M$, there is a local defining
function $\rho$ for $\partial M$ defined on some neighborhood $U$ of
$x$ such that $\rho^2g$ admits a (sufficiently) smooth extension from
$U\cap M$ to all of $U$. It is easy to see that this property is
independent of the choice of defining function, so if $g$ is
conformally compact, then for any local defining function $\hat\rho$
for $\partial M$, the metric $\hat\rho^2 g$ admits a smooth extension
to the boundary. While the metric on the boundary, induced by such an
extension, depends on the chosen defining function, such extensions
are always conformally related. Hence a conformally compact metric on
$M$ gives rise to a well-defined conformal class on $\partial M$,
which is called the \textit{conformal infinity} of $g$. The model
example of a conformally compact metric is the Poincar\'e ball model
of hyperbolic space. Here $\circverline{M}$ is the closed unit ball in $\mathbb
R^{n}$ and one defines a metric $g$ on the open unit ball as
$\tfrac{4}{(1-|x|^2)^2}$ times the Euclidean metric, see
\cite{Graham:Srni}. The resulting conformal infinity is the conformal class of
the round metric on $S^{n-1}$.
There is a conceptual description of conformally compact metrics in
the language of conformal geometry: Since $\rho^2g$ is conformal to
$g$ away from the boundary, it provides a (sufficiently) smooth
extension of the conformal structure on $M$ defined by $g$ to all of
$\circverline{M}$. Conformally compact metrics can be neatly characterized in
this picture via their volume densities. Recall that the volume
density of $g$ is nowhere vanishing, so one can form powers with
arbitrary real exponents to obtain nowehere vanishing densities of all
(non-zero) weights. Each of these densities is parallel for the
connection on the appropriate density bundle induced by the
Levi-Civita connection, and up to constant multiples, this is the only
parallel section.
In the usual conventions of conformal geometry, see \cite{BEG}, the
square of the top exterior power ot the tangent bundle, that is
$\circtimes^2(\Lambdambda^nTM)$, is identified with a line bundle of {\em
densities} of weight $2n$ that we denote $\mathcal{E}[2n]$. This is
oriented and thus there is a standard notion of its roots. With these
conventions, on an $n$-manifold a metric $g$ determines a volume
density $\circperatorname{vol}_g$ that has conformal weight $-n$,
meaning $\circperatorname{vol}_g\in\Gamma(\mathcal E[-n])$. Rescaling $g$ to $\rho^2g$, we get
$\circperatorname{vol}_{\rho^2g}=\rho^n\circperatorname{vol}_g$ and thus
$\circperatorname{vol}_{g}^{-1/n}=\rho\circperatorname{vol}_{\rho^2g}^{-1/n}\in\Gamma(\mathcal E[1])$. Assuming
that $g$ is conformally compact, $\circperatorname{vol}_{\rho^2g}^{-1/n}$ is smooth up
to the boundary and nowhere vanishing, so this equation shows that
$\circperatorname{vol}_{g}^{-1/n}\in\Gamma(\mathcal E[1])$ is a defining density for $\partial
M$. Similar arguments prove the converse, which leads to the following
result.
\betagin{prop}\lambdabel{prop2.1}
Let $\circverline{M}$ be a smooth manifold with boundary $\partial M$ and interior $M$, which
is endowed with a conformal structure $c$. Then a metric $g$ on $M$ which lies in
$c|_M$ is conformally compact if an only if any non-zero section of $\sigma\in\Gamma(\mathcal
E[1]|_M)$ which is parallel for the Levi-Civita connection of $g$ extends by $0$ to a
defining density for $\partial M$.
\end{prop}
\subsection{ALH-metrics and adapted defining functions}\lambdabel{2.2}
Consider a conformally compact metric $g$ and for a local defining
function $\rho$ put $\circverline{g}:=\rho^2g$, and note that this is
smooth and non-degenerate up to the boundary. Hence also the inverse metric
$\circverline{g}^{-1}$ is smooth up to the boundary, so
$\circverline{g}^{-1}(d\rho,d\rho)$ a smooth function on the domain of
definition of $\rho$. Moreover, since $d\rho$ is nowhere vanishing
along the boundary $\circverline{g}^{-1}(d\rho,d\rho)$ has the same
property. Replacing $\rho$ by $\hat\rho:=e^f\rho$, we obtain
$d\hat\rho=\hat\rho df+e^fd\rho$, which easily implies that the
restriction of $\circverline{g}^{-1}(d\rho,d\rho)$ to $\partial M$ is
independent of the defining function $\rho$.
So $\circverline{g}^{-1}(d\rho,d\rho)$ is an invariant of the metric $g$,
and this is related to the asymptotic behavior of the curvature of
$g$, see e.g.\ \cite{Graham:Srni}. In particular, if
$(\rho^2g)^{-1}(d\rho,d\rho)|_{\partial M}\equiv 1$, then the
sectional curvature of $g$ is asymptotically constant $-1$. This
justifies the terminology in the first part of the following
definition and leads to a subclass of defining functions:
\betagin{definition}\lambdabel{def2.2}
Consider a smooth manifold $\circverline{M}=M\cup\partial M$ with boundary and a conformally
compact metric $g$ on $M$.
(1) The metric $g$ is called \textit{asymptotically locally hyperbolic} (or an
ALH-metric) if $(\rho^2g)^{-1}(d\rho,d\rho)|_{\partial M}$ is identically one.
(2) Assume that $g$ is an ALH-metric on $M$ and that $U\subset\circverline{M}$ (with
$U\cap\partial M\neq\emptyset$ to be of interest). Then a local defining function
$\rho$ for $\partial M$ defined on $U$ is called \textit{adpated to $g$} if the
function $(\rho^2 g)^{-1}(d\rho,d\rho)$ is identically one on some open
neighborhood of $U\cap\partial M$.
\end{definition}
\betagin{remark} Note that in the literature the terminology
``asymptotically locally hyperbolic'' is sometimes used for (various) more
restrictive classes of geometry. Also the condition we have (1) is
sometimes referred to as simply ``asymptotically
hyperbolic''. However the latter is also used for rather special
settings where in particular the boundary is necessarily a sphere,
hence our use here.
\end{remark}
The existence of adpated defining functions, as in part (2) of
Definition \ref{def2.2}, can be established by solving an appropriate
non-characteristic first order PDE. The following precise description
of adapted defining functions is given in Lemma 2.1 of
\cite{Graham:Srni}.
\betagin{prop}\lambdabel{prop2.2}
Consider $\circverline{M}=M\cup\partial M$ and an ALH metric $g$ on $M$. Then
for any choice of a representative metric $h$ in the conformal
infinity of $g$, there exists an adapted defining function $\rho$ for
$g$, defined on an open neighborhood of $\partial M$ in $\circverline{M}$,
such that $\rho^2g$ induces the metric $h$ on $\partial
M$. Moreover, for fixed $h$, the germ of $\rho$ along $\partial M$
is uniquely determined.
\end{prop}
| 2,386 | 45,932 |
en
|
train
|
0.50.2
|
\subsection{ALH-metrics and adapted defining functions}\lambdabel{2.2}
Consider a conformally compact metric $g$ and for a local defining
function $\rho$ put $\circverline{g}:=\rho^2g$, and note that this is
smooth and non-degenerate up to the boundary. Hence also the inverse metric
$\circverline{g}^{-1}$ is smooth up to the boundary, so
$\circverline{g}^{-1}(d\rho,d\rho)$ a smooth function on the domain of
definition of $\rho$. Moreover, since $d\rho$ is nowhere vanishing
along the boundary $\circverline{g}^{-1}(d\rho,d\rho)$ has the same
property. Replacing $\rho$ by $\hat\rho:=e^f\rho$, we obtain
$d\hat\rho=\hat\rho df+e^fd\rho$, which easily implies that the
restriction of $\circverline{g}^{-1}(d\rho,d\rho)$ to $\partial M$ is
independent of the defining function $\rho$.
So $\circverline{g}^{-1}(d\rho,d\rho)$ is an invariant of the metric $g$,
and this is related to the asymptotic behavior of the curvature of
$g$, see e.g.\ \cite{Graham:Srni}. In particular, if
$(\rho^2g)^{-1}(d\rho,d\rho)|_{\partial M}\equiv 1$, then the
sectional curvature of $g$ is asymptotically constant $-1$. This
justifies the terminology in the first part of the following
definition and leads to a subclass of defining functions:
\betagin{definition}\lambdabel{def2.2}
Consider a smooth manifold $\circverline{M}=M\cup\partial M$ with boundary and a conformally
compact metric $g$ on $M$.
(1) The metric $g$ is called \textit{asymptotically locally hyperbolic} (or an
ALH-metric) if $(\rho^2g)^{-1}(d\rho,d\rho)|_{\partial M}$ is identically one.
(2) Assume that $g$ is an ALH-metric on $M$ and that $U\subset\circverline{M}$ (with
$U\cap\partial M\neq\emptyset$ to be of interest). Then a local defining function
$\rho$ for $\partial M$ defined on $U$ is called \textit{adpated to $g$} if the
function $(\rho^2 g)^{-1}(d\rho,d\rho)$ is identically one on some open
neighborhood of $U\cap\partial M$.
\end{definition}
\betagin{remark} Note that in the literature the terminology
``asymptotically locally hyperbolic'' is sometimes used for (various) more
restrictive classes of geometry. Also the condition we have (1) is
sometimes referred to as simply ``asymptotically
hyperbolic''. However the latter is also used for rather special
settings where in particular the boundary is necessarily a sphere,
hence our use here.
\end{remark}
The existence of adpated defining functions, as in part (2) of
Definition \ref{def2.2}, can be established by solving an appropriate
non-characteristic first order PDE. The following precise description
of adapted defining functions is given in Lemma 2.1 of
\cite{Graham:Srni}.
\betagin{prop}\lambdabel{prop2.2}
Consider $\circverline{M}=M\cup\partial M$ and an ALH metric $g$ on $M$. Then
for any choice of a representative metric $h$ in the conformal
infinity of $g$, there exists an adapted defining function $\rho$ for
$g$, defined on an open neighborhood of $\partial M$ in $\circverline{M}$,
such that $\rho^2g$ induces the metric $h$ on $\partial
M$. Moreover, for fixed $h$, the germ of $\rho$ along $\partial M$
is uniquely determined.
\end{prop}
\subsection{The basic setup}\lambdabel{2.3}
Defining functions can be used to measure the asymptotic growth (or fall-off) of
functions and more general geometric objects on the interior of a manifold with
boundary. A fundamental property of defining functions is that for a function $f$
that is smooth up to the boundary $f|_{\partial M}\equiv 0$ if and only if for any
local defining function $\rho$ for $\partial M$, we obtain, on the domain of
definition of $\rho$, $f=\rho f_1$ for a function $f_1$ that is smooth up to the
boundary. We say that $f$ is $\mathcal O(\rho)$ in this case, observing that this notion
is actually independent of the specific defining function $\rho$. Similarly if, in
such an expansion, $f_1$ also vanishes along the boundary then this fact does not
depend on the choice of $\rho$, and in that case we say that $f$ is $\mathcal
O(\rho^2)$. Inductively, one obtains the notion that $f$ is $\mathcal O(\rho^N)$ for any
integer $N>0$, which again does not depend on the specific choice of defining
function.
Given a smooth function $f:M\to\mathbb R$ and an integer $N>0$, we then say that $f$ is
${\mathcal O} (\rho^{-N})$ if locally around each $x\in \partial M$ we find a defining
function $\rho$ such that $\rho^Nf$ admits a smooth extension to the boundary. Again,
the fact that such an extension exists is independent of the choice of defining
function, as is vanishing of the boundary value of $\rho^N f$ in some point. In
points where the boundary value of $\rho^Nf$ is nonzero, the actual value does depend
on the choice of $\rho$, however.
For two functions $f_1,f_2:\circverline{M}\to\mathbb R$ and $N>0$, we write $f_1\sigmam_N f_2$ if
$f_1-f_2$ is $\mathcal O(\rho^N)$. By definition, this means that, on the domain of a
local defining function $\rho$, we can write $f_2=f_1+\rho^Nf$ for some function $f$
that is smooth up to the boundary. Observe that this defines an equivalence relation.
All this extends without problems to tensor fields of arbitrary
(fixed) type. This can be seen immediately from looking at coordinate
functions in (boundary) charts. So for $N>0$, a tensor field $t$ on
$M$ is $\mathcal O(\rho^N)$ if can be written as $\rho^N\tilde t$ for a
tensor field $\tilde t$ that is smooth up to the boundary. Likewise,
$t$ is $\mathcal O(\rho^{-N})$ if $\rho^Nt$ admits a smooth extension to
the boundary. In the obvious way we extend the notation $t_1\sigmam_N
t_2$ with $N>0$ to tensor fields $t_1,t_2$ that are smooth up to the
boundary. Observe also that these concepts are compatible in an
obvious sense with tensorial operations, like inserting vector fields
into metrics, etc.
In this langauge, a conformally compact metric $g$ is $\mathcal
O(\rho^{-2})$ and, writing $g=\rho^{-2}\circverline g$, it satisfies that
$\circverline g$ is nowhere vanishing along the boundary. Oberserve that
this implies that the inverse metric $g^{-1}$ is $\mathcal O(\rho^2)$ and,
in particular, vanishes along $\partial M$. For conformally compact
metrics $g$ and $h$ we can consider the metrics $\rho^2g$ and
$\rho^2h$ that are smooth up to the boundary and require that
$\rho^2g\sigmam_N\rho^2h$, which again is independent of the choice of
defining function $\rho$. This defines an equivalence relation on the
set of conformally compact metrics, and to simplify notation, we
formally write this as $g\sigmam_{N-2}h$. In the current article we will
mainly be interested in the case that $N=n=\dim(M)$ but we carry out
most computations for general integers $N>0$, since this does not lead
to difficulties and as a preparation for later extensions.
We will start with an equivalence class $\mathcal G$ of conformally compact metrics on
$M$ with respect to the equivalence relation $\sigmam_{N-2}$ for some $N>0$. Observe
that for two metrics $g,h\in\mathcal G$ and a local defining fuction $\rho$, the metrics
$\rho^2g$ and $\rho^2h$, by definition, admit a smooth extension to the boundary with
the same boundary value. In particular, they induce the same conformal infinity on
$\partial M$. Thus the class $\mathcal G$ of metrics gives rise to a well defined
conformal structure on $\partial M$ that we will denote by $[\mathcal G]$. As we shall
see below, if one metric $g\in\mathcal G$ is ALH, then the same holds for all metrics in
$\mathcal G$. We shall always assume that this is the case from now on. Let us also
remark here, that in the case that $\dim(\circverline{M})=3$, $[\mathcal G]$ induces a stronger
structure that just a conformal structure on $\partial M$ which will be needed in
what follows. This will be discussed in more detail below.
From now on, we will sometimes use abstract index notation. In that
notation we write $g_{ij}$ for the metric $g$, $g^{ij}$ for its
inverse and so on (even though no coordinates or frame field is
chosen). Given $g_{ij},h_{ij}\in\mathcal G$, and fixing a local defining
function $\rho$ for $\partial M$, by definition there is a section
$\mu_{ij}$ that is smooth up to the boundary such that
\betagin{equation}\lambdabel{h-g}
h_{ij}=g_{ij}+\rho^{N-2}\mu_{ij}.
\end{equation}
Since $g^{ij}$ is $\mathcal O(\rho^2)$, we see that $g^{ij}\mu_{ij}=\rho^2\mu$ for some
function $\mu$ that is smooth up to the boundary, whence
$g^{ij}(h_{ij}-g_{ij})=\rho^N\mu$. Using this, we next compute the relation between
the defining densities $\sigma,\tau\in\Gamma(\mathcal E[1])$ determined by $g_{ij}$ and
$h_{ij}$, respectively. Writing
\betagin{equation}\lambdabel{h-g2}
h_{ij}=g_{ik}(\deltalta^k_j+\rho^{N-2}g^{k\ell}\mu_{\ell j}),
\end{equation}
we can take determinants to find that
$\deltat(h_{ij})=\deltat(g_{ik})(1+\rho^N\mu+O(\rho^{N+1}))\in\Gamma(\mathcal
E[-2n])$. (Formally, the determinants are formed by using two
copies of the canonical section of $\Lambda^nTM[-n]$
that expresses the
isomorphism between volume forms and
densities of conformal weight $-n$ on an oriented manifold. Since two
copies of the forms are used, this is well defined even in the
non-orientable case, but this will not be needed here.) To obtain
$\sigma$ and $\tau$, we have to take $\tfrac{-1}{2n}$th powers, and
taking into account that $\sigma$ is $\mathcal O(\rho)$, we get
\betagin{equation}\lambdabel{tau-si}
\tau-\sigma=-\sigma\tfrac{\rho^N}{2n}\mu+\mathcal O(\rho^{N+2}),
\end{equation}
so this is $\mathcal O(\rho^{N+1})$. Moreover, contracting \eqref{h-g2} with $g^{ai}$, we
get $g^{ai}h_{ij}=\deltalta^a_j+\rho^{N-2}g^{ai}\mu_{ij}$, which in turn easily
implies that
\betagin{equation}\lambdabel{hinv}
h^{ij}=g^{ij}-\rho^{N-2}g^{ik}\mu_{k\ell}g^{\ell j}+\mathcal O(\rho^{N+3}).
\end{equation}
Hence $h^{ij}-g^{ij}$ is $\mathcal O(\rho^{N+2})$, which in particular shows that, as
claimed above, $h$ is ALH provided that $g$ has this property.
| 3,066 | 45,932 |
en
|
train
|
0.50.3
|
\subsection{Tractors}\lambdabel{2.4}
For a manifold $K$ of dimension $n\geq 3$ which is endowed with a conformal
structure, the \textit{standard tractor bundle} \cite{BEG,CG-TAMS} is a vector bundle
$\mathcal TK=\mathcal E^A\to K$ of rank $n+2$ endowed with the following data.
\betagin{itemize}
\item A Lorentzian bundle metric, called the \textit{tractor metric}, which we
denote by $\lambdangle\ ,\ \rangle$.
\item A distinguished isotropic line subbundle $\mathcal T^1K\subset \mathcal TK$ that is
isomorphic to the density bundle $\mathcal E[-1]$.
\item A canonical linear connection, called the \textit{tractor connection}, that
preserves the tractor metric and satisfies a non-degeneracy condition.
\end{itemize}
Since $\mathcal T^1K$ is isotropic it is contained in $(\mathcal T^1K)^\perp$, and
$\lambdangle\ ,\ \rangle$ induces a positive definite bundle metric on $(\mathcal
T^1K)^\perp/\mathcal T^1K$. Via the tractor connection, this quotient gets identified
with $\mathcal E^a[-1]$, so the tractor metric gives rise to a section of $\mathcal
E_{ab}[2]$, which is exactly the \textit{conformal metric} $\mathbf{g}_{ab}$ that
defines the conformal structure on $K$. These properties together determine the data
uniquely up to isomorphism. Observe that the inclusion of $\mathcal T^1K\cong\mathcal E[-1]$
into $\mathcal TK=\mathcal E^A$ can be viewed as defining a canonical section
$\mathbf{X}^A\in\Gamma(\mathcal E^A[1])$. Moreover, pairing with $\mathbf{X}^A$, with
respect to the tractor metric, defines an isomorphism $\mathcal TK/(\mathcal
T^1K)^\perp\to\mathcal E[1]$.
As mentioned in \ref{2.3}, we will use these tools in a slightly unusual setting,
since we will deal with different conformal structures on $\circverline{M}$ at the same time,
and only the induced conformal structure on the boundary $\partial M$ will be the
same for all structures in question. Hence we have to carefully discuss how to form
boundary values of sections of the standard tractor bundles.
For our current purposes the ``naive'' approach to tractors (which avoids the
explicit use of Cartan connections or similar tools) is most appropriate and we'll
describe this next. A crucial feature in all approaches to tractors is that the
standard tractor bundle admits a simple description depending on the choice of a
metric $g$ in the conformal class. Such a choice gives an isomorphism $\mathcal E^A\cong
\mathcal E[1]\circplus\mathcal E_a[1]\circplus \mathcal E[-1]$, with the last summand corresponding to
$\mathcal T^1K$ and the last two summands corresponding to $(\mathcal T^1K)^\perp$. The
resulting elements are usually written as column vectors, with the first
component in the top, and there are simple explicit formulae for the tractor metric
and the tractor connection in these terms, namely:
\betagin{equation}\lambdabel{trac-met}
\left\lambdangle\betagin{pmatrix}\sigma \\ \mu_a\\ \nu
\end{pmatrix},\betagin{pmatrix}\tilde\sigma \\ \tilde\mu_a\\ \tilde\nu
\end{pmatrix}\right\rangle=\sigma\tilde\nu+\nu\tilde\sigma+\mathbf{g}^{ab}\mu_a\tilde\mu_b
\end{equation}
with $\mathbf{g}^{ab}$ denoting the inverse of the conformal metric, and
\betagin{equation}\lambdabel{trac-conn}
\qquad \nabla_a \betagin{pmatrix}\sigma\\ \mu_b\\ \nu\end{pmatrix}=
\betagin{pmatrix}\nabla_a\sigma-\mu_a\\ \nabla_a\mu_b+\mathbf{g}_{ab}\nu+
\mbox{\textsf{P}}_{ab}\sigma\\
\nabla_a\nu-\mathbf{g}^{ij}\mbox{\textsf{P}}_{ai}\mu_j\end{pmatrix}.
\end{equation}
In the right hand side of this, we the Levi-Civita connection and the Schouten tensor
$\mbox{\textsf{P}}_{ab}$ of $g$. This is a trace-modification of the Ricci tensor
$\text{Ric}_{ab}$ of $g$ characterized by $\text{Ric}_{ab}=(n-2)\mbox{\textsf{P}}_{ab}+\mbox{\textsf{P}}
\mathbf{g}_{ab}$, where $\mbox{\textsf{P}}=\mathbf{g}^{ij}\mbox{\textsf{P}}_{ij}$.
Changing from $g$ to a metric $\widehat{g}=e^{2f}g$ for
$f\in C^\infty(K,\mathbb R)$ there is an explicit formula for the change of the
identification in terms of $\Upsilon_a=df$, namely
\betagin{equation}\lambdabel{trac-transf}
\betagin{pmatrix}\widehat{\sigma}\\ \widehat{\mu_a}\\ \widehat{\nu}
\end{pmatrix}=\betagin{pmatrix} \sigma\\ \mu_a+\Upsilon_a\sigma
\\ \nu-\mathbf{g}^{ij}(\Upsilon_i\mu_j+\frac{1}{2}\Upsilon_i\Upsilon_j\sigma) \end{pmatrix}
\end{equation}
Now one may turn around the line of argument and define sections of
the tractor bundle as equivalence classes of quadruples consisting of
a metric in the conformal class and sections of $\mathcal E[1]$, $\mathcal
E_a[1]$ and $\mathcal E[-1]$ with respect to the equivalence relation
defined in \eqref{trac-transf}. Recall that the behavior of
the Schouten tensor under a conformal change is given by
\betagin{equation}\lambdabel{Rho-transf}
\widehat{\mbox{\textsf{P}}}_{ab}=\mbox{\textsf{P}}_{ab}-\nabla_a\Upsilon_b+\Upsilon_a\Upsilon_b-\tfrac12\Upsilon_i\Upsilon^ig_{ab}.
\end{equation}
Using this, direct computations show that the definitions in \eqref{trac-met} and
\eqref{trac-conn} are independent of the choice of metric, so they give rise to a
well defined bundle metric and linear connection on the resulting bundle. That this
cannot directly done in a point-wise manner is a consequence of the fact that
tractors are more complicated geometric objects than tensors, since the action of
conformal isometries in a point depends on the two-jet of the isometry in that
point. To come to a point-wise construction, one would have to use 1-jets of metrics
in the conformal class instead.
In the above discussion we have assumed that $n\geq 3$. Indeed, it is well known that
conformal structures in dimension two behave quite differently from higher
dimensions. In particular, they do not allow an equivalent description in term of a
normal Cartan geometry or of tractors. Still we can obtain boundary tractors as
follows. Note first, that associating to a conformal structure a tractor bundle and a
tractor metric via formulae \eqref{trac-transf} and \eqref{trac-met} works without
problems in dimension two. This observation is already sufficient for most of our
results, where we just need a vector bundle canonically associated to some structure
on the boundary. Now one view into the different behavior in dimension two is seen by
the fact that the definition of the Schouten tensor $\mbox{\textsf{P}}_{ab}$ via the Ricci
curvature breaks down. While there are other ways to understand the Schouten tensor
it is nevertheless true that on a 2 dimensional Riemannian manifold there is no
natural tensor that transforms conformally according to (\ref{Rho-transf}). Thus one
cannot use \eqref{trac-conn} to associate to a conformal structure a canonical
connection on the tractor bundle.
However, in the computations needed to verify that \eqref{trac-conn} leads to a well
defined connection only the transformation law \eqref{Rho-transf} for the Schouten
tensor under conformal changes is needed, the relation to the Riemann curvature does
not play a role. (In fact this computation only involves single covariant
derivatives, so there is no chance for curvature terms to arise.) Consequently, the
construction of a canonical connection on the tractor bundle extends to dimension
two, provided that in addition to a conformal class one associates to each metric in
that class a symmetric tensor $\mbox{\textsf{P}}_{ab}$, such that the tensors associated to
conformally related metrics satisfy the transformation law \eqref{Rho-transf}.
The observation just made, for constructing a tractor bundle in
dimension 2, is close to the idea of a M\"obius structure, but
actually it is a slight generalization of the concept of a M\"obius
structure in the sense of \cite{Calderbank:Moebius}; compare in
particular with the MR review \cite{Eastwood:Moebius} of that
article. To define a M\"obius structure, one requires, in addition,
that the trace of the tensor $\mbox{\textsf{P}}_{ab}$ associated to $g_{ab}$ is one
half times the scalar curvature of $g_{ab}$. This can be expressed as
a normalization condition on the curvature of the tractor connection,
but we will not need this. Having said that in the cases in which we will need the
tractor connection in dimension two, we actually will deal with M\"obius
structures, since we get flat tractor connections.
| 2,383 | 45,932 |
en
|
train
|
0.50.4
|
\subsection{Boundary values of tractors}\lambdabel{2.5}
In our usual setting of $\circverline{M}=M\cup \partial M$, equipped with the
conformal class defined by a conformally compact metric, we will deal
with standard-tractor-valued differential forms on $\circverline{M}$. The
strategy is to associate to suitable forms, of this type, a boundary
value, and interpret this as a form taking values in the standard
tractor bundle $\mathcal T\partial M$ of the conformal infinity. As
discussed at the end of Section \ref{2.4} above, such a bundle is
available in all relevant dimensions $\dim(\circverline{M})=n\geq 3$. Moreover,
since this infinity is the same for all the metrics in a class $\mathcal
G$, as discussed in Section \ref{2.2}, this allows us to relate the
boundary values obtained from different metrics in $\mathcal G$. However,
even for $n\geq 4$, it is not obvious how to relate $\mathcal T\circverline{M}$ and
$\mathcal T\partial M$, so we discuss this next. This discussion will also
show how to canonically obtain the ``abstract Schouten tensors''
needed to define a tractor connection on $\mathcal T\partial M$ for
$n=3$. Hence we obtain a uniform description of boundary values in all
dimensions.
Provided that one works with metrics that are smooth up to the
boundary, the discussion of boundary values of tractors can be reduced
to the case of hypersurfaces in conformal manifolds. Observe first
that in our setting, it is no problem to relate density bundles on
$\circverline{M}$ and on $\partial M$ of any conformal weight. This is based on
the fact that $\mathcal E[2]$ can always be viewed as the line subbundle
spanned by the conformal class and one can form boundary values for
metrics in the conformal class (that are smooth up to the
boundary). So the densities of weight $w$ on $\partial M$ are simply
the restriction of the ambient densities of weight $w$ i.e.\ sections
of $\mathcal{E}[w]|_{\partial M}$.
Now the conformal metric and its inverse define inner products on
$\mathcal E^a[-1]$ and its dual $\mathcal E_a[1]$. In the case of a boundary,
there thus is a unique inward pointing unit normal normal
$n^i\in\Gamma(T\circverline{M}[-1]|_{\partial M})$ to the subbundle $T\partial M$,
and we put $n_i:=\mathbf{g}_{ij}n^j\in \Gamma(T^*\circverline{M}[1]|_{\partial
M})$. We will assume that $n^i$ and $n_i$ are (arbitrarily) extended
off the boundary, if needed. For a choice of metric $\bar{g}$ (which
is smooth up to the boundary) in the conformal class, we observe that
the restriction of $\nabla^{\bar g}_in_j$ to $T\partial M\times T\partial
M$ is independent of the chosen extension. This is the (weighted)
second fundamental form of $\partial M$ in $\circverline{M}$ with respect to
$\bar g$, and we can decompose it into a trace-free part and a
trace-part with respect to $\mathbf{g}_{ab}$. A short computation
shows that, along $\partial M$, the tracefree part is conformally
invariant. The trace part can be encoded into the mean curvature
$H^{\bar g}\in \Gammamma(\mathcal{E} [-1])$ of $\partial M$ in $\circverline{M}$
with respect to $\bar g$. Its behaviour under a conformal change
corresponding to $\Upsilon_a$ is given by $H^{\widehat{\bar g}}=H^{\bar
g}+\Upsilon_in^i$.
It is a classical fact, see Section 2.7 of \cite{BEG}, that these
ingredients can be encoded into a conformally invariant normal tractor
$N^A\in\Gamma(\mathcal E^A|_{\partial M})$. In the splitting corresponding to
$\bar g$, the tractor $N^A$ corresponds to the triple $(0,n_i,-H^{\bar
g})$, which shows that $N^A$ is orthogonal to $\mathbf{X}^B$ and has
norm $1$. Conformal invariance of $N^A$ follows immediately from the
change of $H^{\bar g}$ described above. It then turns out that, for
$n\geq 4$, there is a conformally invariant isomorphism between the
orthocomplement ${N^A}^\perp\subset \mathcal T\circverline{M}|_{\partial M}$ and the
intrinsic tractor bundle $\mathcal T\partial M$ of the conformal infinity
\cite{Gover:P-E}. In a scale $\bar g$ that is smooth up to the
boundary, a triple $(\sigma,\mu_a,\nu)$ is orthogonal to $N^A$ if and
only if $\mu_jn^j=H^{\bar g}\sigma $. Such a triple then gets mapped to
$$(\sigma,\mu_a-H^{\bar g} n_a\sigma,\nu+\tfrac12(H^{\bar g})^2\sigma)$$ in the splitting
corresponding to $\bar g|_{\partial M}$, see Section 6.1 of \cite{Curry-Gover}. In
particular, in the case that $\partial M$ is minimal with respect to $\bar g$
(i.e.\ that $H^{\bar g}$ vanishes identically), we simply get the na\"{\i}ve
identification of triples.
As mentioned above, these considerations also show how to obtain a
tractor connection on $\mathcal T\partial M$ in the case that $n=3$. We
can do this in the setting of hypersurfaces and as discussed in
Section \ref{2.4}, we have to associate an ``abstract Schouten
tensor'' to the metrics in the conformal class $\partial M$. The idea
here is simply that for metrics $\bar g$ such that $H^{\bar g}=0$, we
associate the restriction of the Schouten tensor of $\bar g$ to
$\partial M$ as an ``abstract Schouten tensor'' for the metric $\bar
g|_{\partial M}$. If $\bar g$ and $\widehat{\bar g}$ are two such
metrics, then for the change $\Upsilon_a$, we get $\Upsilon_in^i=0$, which
implies that the restriction of $\Upsilon_i\Upsilon^i$ to $\partial M$ coincides
with the trace of the restriction of $\Upsilon_a$ to $T\partial M$. From
this and the Gauss formula we conclude that restrictions of the
Schouten tensors to $\partial M$ satisfy the correct transformation
law \eqref{Rho-transf}. This is already sufficient to obtain a tractor
connection on $\mathcal T\partial M$. Since we know the behavior of all
objects under conformal rescalings, one can deduce a description of
$\mbox{\textsf{P}}_{ab}$ for general metrics, but we won't need that here. A
different approach to induced M\"obius structures on hypersurfaces and
more general submanifolds in conformal manifolds can be found in
\cite{Belgun}.
In any case, it is clear from this description that the above discussion of boundary
values now extends to $n=3$. Finally, consider a class $\mathcal G$, as discussed in
Section \ref{2.2}, and metrics $g,h\in\mathcal G$. Then for the conformal classes $[g]$
and $[h]$ the metrics $\rho^2g$ and $\rho^2h$, for a local defining function $\rho$,
admit a smooth extension to the boundary. Now by definition $\rho^2g\sigmam_N\rho^2h$
and $N\geq 3$. In particular, the difference of their curvatures is $\mathcal
O(\rho^{N-2})$ and $N-2\geq 1$, which implies that the restrictions of their Schouten
tensors to the boundary agree. This implies that, in dimension $n=3$, all metrics in
$\mathcal G$ lead to the same tractor connection on $\mathcal T\partial M$. In higher
dimensions the tractor connection on $\mathcal T\partial M$ is determined by the conformal
structure $[\mathcal G]$, and so the equivalent result holds trivially.
\subsection{The scale tractor}\lambdabel{2.6}
We now combine the ideas about boundary tractors with the conformally
compact situation. This needs one more basic tool of tractor calculus,
the so-called tractor $D$-operator (also called the Thomas
$D$-operator). In the simplest situation, which is all that we need
here, this is an operator $D^A:\mathcal E[w]\to\mathcal E^A[w-1]$, which in
triples is defined by
\betagin{equation}\lambdabel{D-def}
D^A\tau=\left(w(n+2w-2)\tau, (n+2w-2)\nabla_a\tau,
-\mathbf{g}^{ij}(\nabla_i\nabla_j+\mbox{\textsf{P}}_{ij})\tau\right).
\end{equation}
Again, a direct computation shows that this is conformally invariant. We will mainly
need the case that $w=1$, so that $D^A$ maps sections of the quotient bundle $\mathcal
E[1]$ of $\mathcal E^A$ to sections of $\mathcal E^A$. Since the first component of
$\tfrac1nD^A\sigma$ is $\sigma$, this operator is referred to as a \textit{splitting
operator}. In particular, given a metric in a conformal class, we can apply this
splitting operator to the canonical section of $\mathcal E[1]$ obtained from the volume
density of the metric. The resulting section of $\mathcal E^A$ is called the
\textit{scale tractor} associated to the metric.
Computing in the splitting determined by the metric itself, the associated section
$\sigma$ of $\mathcal E[1]$ satisfies $\nabla_a\sigma=0$. Hence in this splitting,
$\tfrac1nD^A\sigma$ corresponds to $(\sigma,0,-\tfrac1n\mbox{\textsf{P}}\sigma)$, where
$\mbox{\textsf{P}}:=\mathbf{g}^{ij}\mbox{\textsf{P}}_{ij}$. Applying the tractor connection to this, we get
$(0,(\mbox{\textsf{P}}_{ab}-\frac{1}{n}\textbf{g}_{ab}\mbox{\textsf{P}})\sigma,-\frac{1}{n}\sigma\nabla_c\mbox{\textsf{P}})$. Observe
that the middle slot of this vanishes iff $\mbox{\textsf{P}}_{ab}$ is pure trace, i.e.\ iff the
metric is Einstein. In that case, $\mbox{\textsf{P}}$ which is just a multiple of the scalar
curvature, is constant, hence a metric is Einstein iff its scale tractor is parallel.
Now we move to the case of $\circverline{M}=M\cup\partial M$ and a conformally compact metric
$g$ on $M$. By Proposition \ref{prop2.1}, the canonical section $\sigma\in\Gamma(\mathcal
E[1])$ which is parallel for $\nabla^g$ admits a smooth extension to the boundary (as
a defining density). Since we have a conformal structure on all of $\circverline{M}$ also the
scale tractor $I^A:=\tfrac{1}{n}D^A\sigma$ and its covariant derivative $\nabla_aI^A$
are smooth up to the boundary. On $M$, we can compute in the splitting determined by
$g$, which shows that $\lambdangle
I,I\rangle=-\tfrac2n\sigma^2\mbox{\textsf{P}}=-\tfrac2ng^{ij}\mbox{\textsf{P}}_{ij}$. Now the relation between the
Schouten tensor $\mbox{\textsf{P}}_{ij}$ and the Ricci-tensor shows that the scalar curvature
$R=g^{ij}\text{Ric}_{ij}$ of $g$ can be written as $2(n-1)g^{ij}\mbox{\textsf{P}}_{ij}$ and thus
$\lambdangle I,I\rangle=-\tfrac1{n(n-1)}R$. In particular, if $g$ is ALH, then this is
identically $1$ along the boundary.
Under slightly stronger assumptions, the restriction of $I^A$ to the boundary is
nicely related to the objects discussed in Section \ref{2.5}. To formulate this,
recall from Section \ref{2.5} that the trace-free part of the second fundamental form
is conformally invariant along $\partial M$. This vanishes identically if and only
if $\partial M$ is totally umbilic in $\circverline{M}$, which thus is a conformally invariant
condition.
\betagin{prop}\lambdabel{prop2.6}
Consider a manifold $\circverline{M}$ with boundary $\partial M$ and interior $M$ and a
conformally compact metric $g$ on $M$; let $\sigma\in\Gamma(\mathcal E[1])$ be the
corresponding density and $I^A:=\tfrac1nD^A\sigma$ the scale tractor.
(1) If $\lambdangle I,I\rangle=1+\mathcal O(\rho^2)$ near to $\partial M$, then the
restriction of $I^A$ to $\partial M$ coincides with the normal tractor $N^A$.
(2) If in addition $g$ is asymptotically Einstein in the sense that
$\nabla_aI^A|_{\partial M}$ vanishes in tangential directions, then the boundary
$\partial M$ is totally umbilic.
\end{prop}
\betagin{proof}
For (1), see Proposition 7.1 of \cite{Curry-Gover} (or Proposition 6
\cite{G-sigma}).
(2) There is a nice characterization for a hypersurface to be totally
umbilic, see \cite{BEG} and e.g.\ Lemma 6.2 of \cite{Curry-Gover}:
Extending the normal tractor $N_A$ arbitrarily off $\partial M$ one
can apply the tractor connection to obtain a tractor valued one
form. The restriction of this derivative to tangential directions
along $\partial M$ is independent of the chosen extension and
$\partial M$ is totally umbilic in $\circverline{M}$ if and only if this
restrictions vanishes. This implies the claim.
\end{proof}
| 3,509 | 45,932 |
en
|
train
|
0.50.5
|
\section{The tractor mass cocycle}\lambdabel{3}
We consider the tractor version of the classical almost hyperbolic
mass here, so the order of asymptotics we need corresponds to
$N=n=\dim(M)$ in the notation of Section \ref{2}.
\subsection{The contribution from the trace}\lambdabel{3.1}
Most of the theory we develop applies in the
general setting of an oriented manifold $\circverline{M}$ with boundary
$\partial M$, interior $M$, and equipped with a class $\mathcal G$ of
metrics on $M$, as introduced in Section \ref{2.3} with $N=n$. The
only additional assumption is that the metrics in $\mathcal G$ are ALH in
the sense of Definition \ref{def2.2}. Given two metrics $g,h\in\mathcal
G$, we denote by $\sigma,\tau\in \Gamma(\mathcal E[1])$ the corresponding powers
of the volume densities of $g$ and $h$. Recall that the class $\mathcal G$
gives rise to a well defined standard tractor bundle $\mathcal T\partial
M$ over $\partial M$. Our aim is to associate to $g$ and $h$ a form
$c(g,h)\in\Omega^{n-1}(\partial M,\mathcal T\partial M)$, i.e.~a top-degree
form on $\partial M$ with values in $\mathcal T\partial M$. Further, we
want this to satisfy a cocyle property, i.e.~that $c(h,g)=-c(g,h)$ and
that $c(g,k)=c(g,h)+c(h,k)$ for $g,h,k\in\mathcal G$.
The first ingredient for this is rather simple: Given $g,h\in\mathcal G$, we use the
conformal structure $[g]$ on $\circverline{M}$, and consider the $\mathcal T\circverline{M}$-valued one-form
$\tfrac{1}n\nabla_bD^A(\tau-\sigma)$. We already know that $\tau$ and $\sigma$ admit a smooth
extension to the boundary, so this is well defined and smooth up to the boundary. Now
on $M$, we can apply the Hodge-$*$-operator determined by $g$ to convert this into a
$\mathcal T\circverline{M}$-valued $(n-1)$-form. The following result shows that this is smooth up to
the boundary and that its boundary value is orthogonal to the normal tractor $N^A$
(and non-zero in general). Thus this boundary value defines a form in
$\Omega^{n-1}(\partial M,\mathcal T\partial M)$.
\betagin{prop}\lambdabel{prop3.1}
In the setting $\circverline{M}=M\cup\partial M$ and $g,h\in\mathcal G$ as described above, let
$\rho$ be a local defining function for the boundary. Put $\circverline{g}:=\rho^2g$,
let $\circverline{\sigma}\in\Gamma(\mathcal E[1])$ be the corresponding density and let
$\circverline{g}_{\infty}$ be the boundary value of $\circverline{g}$.
(1) In terms of the canonical section $\textbf{X}^A\in\mathcal T\circverline{M}[1]$ from Section
\ref{2.4} and the function $\mu$ from \eqref{tau-si} and writing $\rho_a$ for
$d\rho$, we get
\betagin{equation}\lambdabel{c11}
\nabla_bD^A(\tau-\sigma)=
\tfrac{n^2-1}{2}\rho^{n-2}\rho_b\mu\circverline{\sigma}^{-1}\mathbf{X}^A+ \mathcal
O(\rho^{n-1}).
\end{equation}
(2) The form $\star_g\nabla_bD^A(\tau-\sigma)$ is smooth up to the boundary and
its boundary value is given by
\betagin{equation}\lambdabel{c12}
\tfrac{n^2-1}{2}\circperatorname{vol}_{\circverline{g}_{\infty}}\mu_\infty\circverline{\sigma}_{\infty}^{-1}\mathbf{X}^A.
\end{equation}
Here $\circperatorname{vol}_{\circverline{g}_{\infty}}$ is the volume form of $\circverline{g}_\infty$,
$\circverline{\sigma}_{\infty}$ is the corresponding $1$-density on $\partial M$, and
$\mu_{\infty}$ is the boundary value of $\mu$. In particular, this is perpendicular
to $N^A|_{\partial M}$ and thus defines an $(n-1)$-form with values in $\mathcal T\partial
M$.
\end{prop}
\betagin{proof}
Observe first that for a connection $\nabla$, a section $s$ that both are smooth
up to the boundary, and an integer $k>0$, we get $\nabla_a(\rho^k
s)=k\rho^{k-1}\rho_as+\mathcal O(\rho^k)$. This can be applied both to the tractor
connection and to the Levi-Civita connection of $\circverline{g}$, which will both be written
as $\nabla$ in this proof.
Using that $\circverline{\sigma}=\tfrac{\sigma}{\rho}$, we can write formula \eqref{tau-si}
(still for $N=n$) as $\tau-\sigma=-\circverline{\sigma}\tfrac{\rho^{n+1}}{2n}\mu+\mathcal
O(\rho^{n+2})$. Now the defining formula \eqref{D-def} for $D^A$ shows that the first
two slots of $D^A(\tau-\sigma)$ in the splitting determined by $\circverline{g}$ are $\mathcal
O(\rho^{n+1})$ and $\mathcal O(\rho^n)$, respectively, while in the last slot the only
contribution which is not $\mathcal O(\rho^n)$ comes from the double derivative. This
shows that
$$
D^A(\tau-\sigma)=-\tfrac{n(n+1)}{2n}\rho^{n-1}\circverline{\sigma}\mu(-
\textbf{g}^{ij}\rho_i\rho_j)\mathbf{X}^A+\mathcal O(\rho^n).
$$ Using that $\circverline{g}^{ij}=\circverline{\sigma}^2\textbf{g}^{ij}$ and that $g$ is
ALH, we conclude that $\circverline{\sigma}
\textbf{g}^{ij}\rho_i\rho_j=\circverline{\sigma}^{-1}+\mathcal O(\rho)$, so we get
$$
D^A(\tau-\sigma)=\tfrac{n+1}{2}\rho^{n-1}\mu\circverline{\sigma}^{-1}\mathbf{X}^A
+\mathcal O(\rho^n).
$$
From this \eqref{c11} and hence part (1) follows immediately.
(2) Since $\circverline{M}$ is oriented we have an isomorphism
$ \mathcal E[-n]\stackrel{\cong}{\longrightarrow} \Lambda^nT^*\circverline{M} $
that can be interpreted as a canonical section $\betap_{a_1\dots a_n}\in
\Gammamma (\Lambda^nT^*\circverline{M} [n])$. In terms of this, the volume form of $g$ is given by
$\sigma^{-n}\betap_{a_1\dots a_n}$. Now by definition,
$\star_g\rho^{n-2}\rho_a$ is given by contracting $\rho^{n-2}\rho_a$ into the volume
form of $g$ via $g^{-1}$. So this is given by
\betagin{equation}\lambdabel{sigma-tech}
\sigma^{-n}\rho^{n-2}g^{ij}\rho_i\betap_{ja_1\dots a_{n-1}}.
\end{equation}
Now $\sigma^{-2}g^{ij}=\textbf{g}^{ij}$, while
$\sigma^{2-n}\rho^{n-2}=\circverline{\sigma}^{2-n}$. Together with part (1), this shows that
$$
\star_g\nabla_aD^A(\tau-\sigma)=\tfrac{n^2-1}2\circverline{\sigma}^{2-n}\mathbf{g}^{ij}\rho_i\betap_{ja_1\dots
a_{n-1}}\mu\circverline{\sigma}^{-1}\mathbf{X}^A+O(\rho).
$$
This is evidently smooth up to the boundary and its boundary value is a multiple
of $\mathbf{X}^A$ and thus perpendicular to $N^A$. To obtain the interpretation of
the boundary value, we can rewrite \eqref{sigma-tech} as
$\circverline{\sigma}^{-n}\circverline{g}^{ij}\rho_i\betap_{ja_1\dots a_{n-1}}$. Since the first and
last terms combine to give the volume form of $\circverline{g}$ and, along the boundary,
$\circverline{g}^{ib}\rho_i$ gives the unit normal with respect to $\circverline{g}$, we
conclude that the boundary value of \eqref{sigma-tech} is the volume form of
$\circverline{g}_\infty$. From this, \eqref{c12} and thus part (2) follows immediately.
\end{proof}
There actually is a simpler way to write out the boundary value of
$\star_g\nabla_bD^A(\tau-\sigma)$ than \eqref{c12} that needs less choices. The function
$\mu$ defined in \eqref{tau-si} of course depends on the choice of the defining
function $\rho$, and there is no canonical choice of defining function. However,
fixing the metric $g\in\mathcal G$, we of course get the distinguished defining density
$\sigma$, and we can get a more natural version of \eqref{tau-si} by phrasing things
in terms of densities. Namely, for the current setting with $N=n$, we can define
$\nu\in\Gamma(\mathcal E[-n])$ to be the unique density such that
\betagin{equation}\lambdabel{tau-si-dens}
\tau-\sigma=-\tfrac{1}{2n}\sigma^{n+1}\nu+\mathcal O(\rho^{n+2}).
\end{equation}
Then of course $\nu$ is uniquely determined by $g$ and $h$. In terms of a defining
function $\rho$ and the corresponding function $\mu$, we get
$\nu=(\tfrac{\rho}{\sigma})^n\mu$, which shows that $\nu$ is smooth up to the boundary
and non-zero wherever $\mu$ is non-zero. Let us write the boundary value of $\nu$ as
$\nu_\infty$, which by Section \ref{2.5} can be interpreted as a density of weight
$-n$ on $\partial M$. In the setting of Proposition \ref{prop3.1}, we then have
$\circverline{g}=\rho^2g$, so $\circverline{\sigma}=\tfrac{\sigma}{\rho}$. The latter is smooth
up to the boundary and from Section \ref{2.5} we know that its boundary value is the
$1$-density $\circverline{\sigma}_\infty$ on $\partial M$ corresponding to
$\circverline{g}_\infty$. Hence our construction implies that $\circperatorname{vol}_{\circverline{g}_\infty}$
corresponds to the density $\circverline\sigma_{\infty}^{1-n}$, so \eqref{c12} simplifies
$\tfrac{n^2-1}2\circverline{\sigma}_\infty^{-n}\mu_\infty\mathbf{X}^A=
\tfrac{n^2-1}2\nu_\infty\mathbf{X}^A$. Using this we can easily prove that we have
constructed a cocyle.
\betagin{cor}\lambdabel{cor3.1}
Let us denote the section of $\mathcal T\partial M[-n+1]$ associated to $g,h\in\mathcal G$
via formula \eqref{c12} by $c_1(g,h)$. Then $c_1$ is a cocycle in the sense that
$c_1(h,g)=-c_1(g,h)$ and that $c_1(g,k)=c_1(g,h)+c_1(h,k)$ for $g,h,k\in\mathcal G$.
\end{cor}
\betagin{proof}
From $\tau-\sigma=-\tfrac{1}{2n}\sigma^{n+1}\nu+\mathcal O(\rho^{n+2})$, we get
$\tau=\sigma+\mathcal O(\rho^{n+1})$ and hence
$\sigma-\tau=-\tfrac{1}{2n}\tau^{n+1}(-\nu)+\mathcal O(\rho^{n+2})$, so
$c_1(h,g)=-c_1(g,h)$. The second claim follows similarly.
\end{proof}
| 3,104 | 45,932 |
en
|
train
|
0.50.6
|
\subsection{The contribution from the trace-free part}\lambdabel{3.2}
We next need another element of tractor calculus that, again, concerns one-forms with
values in the standard tractor bundle. Returning to the setting of a general
conformal manifold $K$, for $k=1,\dots,n$, there is a natural bundle map
$\partial^*:\Lambda^kT^*K\circtimes \mathcal TK\to\Lambda^{k-1}T^*K\circtimes\mathcal TK$, which is
traditionally called the \textit{Kostant codifferential}. This has the crucial
feature that $\partial^*\circ\partial^*=0$, so for each $k$, one obtains nested natural
subbundles $\circperatorname{im}(\partial^*)\subset\ker(\partial^*)$ and hence there is the
subquotient $\mathcal H_k=\ker(\partial^*)/\circperatorname{im}(\partial^*)$. To make this explicit in low
degrees let us write the spaces $\Lambdambda^kT^*K\circtimes\mathcal TK$ for $k=0,1,2$ in the
obvious extension of the vector notation for standard tractors:
$$
\betagin{pmatrix} \mathcal E[1] \\ \mathcal E_c[1] \\ \mathcal E[-1]\end{pmatrix}
\circverset{\partial^*}{\longleftarrow}
\betagin{pmatrix} \mathcal E_a[1] \\ \mathcal E_{ac}[1] \\ \mathcal E_a[-1]\end{pmatrix}
\circverset{\partial^*}{\longleftarrow}
\betagin{pmatrix} \mathcal E_{[ab]}[1] \\ \mathcal E_{[ab]c}[1] \\ \mathcal E_{[ab]}[-1]\end{pmatrix} .
$$
Here $\mathcal E_{[ab]}$ is the abstract index notation for $\Lambda^2T^*K$.
By general facts, $\partial^*$ maps each row to the row below, so in particular the
bottom row is contained in the kernel of $\partial^*$. Moreover, Kostant's version of
the Bott-Borel-Weil Theorem implies that $\mathcal H_0\cong \mathcal E[1]$ and $\mathcal H_1\cong
\mathcal E_{(ab)_0}[1]$ (symmetric trace-free part). In particular, for $k=0$, we
conclude that $\partial^*$ has to map onto the two bottom rows, so
$\circperatorname{im}(\partial^*)=(\mathcal T^1K)^\perp$. Hence $\mathcal H_0$ coincides with the natural
quotient bundle $\mathcal E[1]$ of $\mathcal TK$ considered above.
In degree $k=1$, we conclude that $\partial^*$ maps the top
slot isomorphically onto the middle slot of $\mathcal TK$, while its restriction to the
middle slot must be a non-zero multiple of the trace. Hence $\ker(\partial^*)$
consists exactly of those elements for which the top slot vanishes and the middle
slot is trace-free. Similarly, we conclude that $\partial^*$ has to map the top slot
of the degree-two-component injectively into the middle slot of the degree-one part,
while the middle slot has to map onto the bottom slot of the degree-one
component. This shows how $\mathcal H_1\cong \mathcal E_{(ab)_0}[1]$ naturally arises as the
subquotient $\ker(\partial^*)/\circperatorname{im}(\partial^*)$ of $T^*K\circtimes\mathcal TK$.
Now similarly to the tractor-D operator, the machinery of BGG sequences constructs a
conformally invariant \textit{splitting operator}
$$
S:\Gamma(\mathcal H_1K)\to \Gamma(\ker(\partial^*))\subset \Omega^1(K,\mathcal TK).
$$
Apart from the fact that for the projection $\pi_H:\ker(\partial^*)\to\mathcal H_1$, one
obtains $\pi_H(S(\alpha))=\alpha$, this operator is characterized by the single property
that, for the covariant exterior derivative $d^\nabla$ induced by the tractor
connection, one gets $\partial^*\circ d^\nabla(S(\alpha))=0$ for any $\alpha\in\Gamma(\mathcal
H_1)$.
To compute the explicit expression for $S$, we again use the notation of triples.
\betagin{lemma}\lambdabel{lem3.2}
Let $K$ be a conformal manifold and let $g$ be a metric in the conformal
class. Then for $\phi=\phi_{ab}\in\Gamma(\mathcal E_{(ab)_0}[1])$, the section
$S(\phi)\in\Omega^1(K,\mathcal TK)$ is, in the splitting determined by $g$, given by
\betagin{equation}\lambdabel{S-formula}
(0,\phi_{ab}, \tfrac{-1}{n-1}\mathbf{g}^{ij}\nabla_i\phi_{aj}).
\end{equation}
\end{lemma}
\betagin{proof}
From above, we know that $S$ has values in $\ker(\partial^*)$ and that this implies
that the first component of $S(\phi)$ has to be zero. The fact that $\pi_H\circ S$ is the
identity map then shows that the middle component has to coincide with
$\phi_{ab}$. Thus it remains to determine the last component, which we temporarily
denote by $\psi=\psi_a\in\Gamma(\mathcal E_a[-1])$. This can be determined by exploiting the
fact that $\partial^*\circ d^\nabla(S(\phi))=0$. To compute $d^\nabla(0,\phi_{bc},\psi_b)$,
we first have to use formula \eqref{trac-conn} to compute
$\nabla_a(0,\phi_{bc},\psi_b)$ viewing the form index $b$ as a mere ``passenger
index''. This leads to $(-\phi_{ba},\nabla_a\phi_{bc}+\mathbf{g}_{ac}\psi_b,*)$ where we
don't compute the last component, which will not be needed in what follows. Then we
have to apply twice the alternation in $a$ and $b$, which kills the first component
by symmetry of $\phi$ and leads to $2(\nabla_{[a}\phi_{b]c}-\psi_{[a}\mathbf{g}_{b]c})$
in the middle component. From the description of $\partial^*$ above we know that
$\partial^*\circ d^\nabla(S(\phi))=0$ is equivalent to the fact that this middle
component lies in the kernel of a surjective natural bundle map to $\mathcal E_a[-1]$.
By naturality, this map has to be a nonzero multiple of the contraction by
$\mathbf{g}^{bc}$. Using trace-freeness of $\phi$, we conclude that $\partial^*\circ
d^\nabla(S(\phi))=0$ is equivalent to
$$
0=\mathbf{g}^{ij}(-\nabla_i\phi_{aj})-(n-1)\psi_a,
$$
which gives the claimed formula.
\end{proof}
Now we return to our setting $\circverline{M}=M\cup\partial M$, and a class $\mathcal G$ of metrics
with $N=n$ as before. Given two metrics $g,h\in\mathcal G$, we now consider the
trace-free part $(h_{ij}-g_{ij})^0$ of $h_{ij}-g_{ij}$ with respect to $g$, which
defines a smooth section of $\mathcal E_{(ab)_0}$. Thus for the density $\sigma\in\Gamma(\mathcal
E[1])$ determined by $g$, we can apply the splitting operator $S$ to
$\sigma(h_{ij}-g_{ij})^0$, to obtain a $\mathcal T\circverline{M}$-valued one-form $\phi_a^B$. We next
prove that this has the right asymptotic behavior to apply $\star_g$ and construct a
boundary value which lies in $\Omega^{n-1}(\partial M,\mathcal T\partial M)$, as we did for
the trace part in Section \ref{3.1} above.
Choosing a local defining function $\rho$ for the boundary, we get the function
$\mu_{ij}$ defined in \eqref{h-g}, and then
\betagin{equation}\lambdabel{h-g0}
(h_{ij}-g_{ij})^0=\rho^{n-2}(\mu_{ij}-\tfrac1n\rho^2\mu g_{ij})+\mathcal O(\rho^{n-1}),
\end{equation}
and clearly $\mu^0_{ij}:=\mu_{ij}-\tfrac1n\rho^2\mu g_{ij}$ defines a section of
$\mathcal E_{(ab)_0}$ that is smooth up to the boundary.
\betagin{prop}\lambdabel{prop3.2}
In the setting and notation of Proposition \ref{prop3.1} and using $\mu_{ij}^0$ as
defined above, we get:
(1) The form $S(\sigma(h_{ij}-g_{ij})^0)\in\mathcal E_a^A$ is given by
\betagin{equation}\lambdabel{c21}
-\rho^{n-2}\circverline{g}^{ij}\rho_i\mu^0_{aj}\circverline{\sigma}^{-1}\mathbf{X}^A+\mathcal
O(\rho^{n-1}).
\end{equation}
(2) The form $\star_gS(\sigma(h_{ij}-g_{ij})^0)$ is smooth up to the boundary and its
boundary value is given by
\betagin{equation}\lambdabel{c22}
-\circverline{g}^{ij}\circverline{g}^{k\ell}\rho_i\rho_k\mu^0_{j\ell}\circperatorname{vol}_{\circverline{g}_\infty}
\circverline{\sigma}_{\infty}^{-1}\mathbf{X}^A.
\end{equation}
This is perpendicular
to $N^A|_{\partial M}$ and thus defines an $(n-1)$-form with values in $\mathcal T\partial
M$.
\end{prop}
\betagin{proof}
(1) As before, we will work in the splitting determined by $\circverline{g}_{ij}$
throughout the proof. By construction
$\sigma(h_{ij}-g_{ij})^0=\sigma\rho^{n-2}\mu_{ij}^0=\circverline{\sigma}\rho^{n-1}\mu_{ij}^0$
is $\mathcal O(\rho^{n-1})$. Using Lemma \ref{lem3.2}, we see that the first slot of
$S(\sigma(h_{ij}-g_{ij})^0)$ vanishes and its middle slot is $\mathcal
O(\rho^{n-1})$. Using the observation from the beginning of the proof of
Proposition \ref{3.1} we see that the covariant derivative of
$\circverline{\sigma}\rho^{n-1}\mu_{ij}^0$, with respect to the Levi-Civita connection of $\circverline{g}_{ij}$, is given
by $(n-1)\circverline{\sigma}\rho^{n-2}\rho_k\mu_{ij}^0+\mathcal O(\rho^{n-1})$. Using this,
the claimed formula follows immediately from Lemma \ref{lem3.2} and the fact that
$\circverline{g}^{ij}=\circverline{\sigma}^2\mathbf{g}^{ij}$.
\noindent (2) Proceeding as in the proof of Proposition \ref{prop3.1}, we now show that
\mbox{$\star_gS((h_{ij}-g_{ij})^0)$} is given by
$$
-\circverline{g}^{ij}\rho_i\mu^0_{kj}\circverline{g}^{k\ell}\betap_{\ell a_1\dots
a_{n-1}}\circverline{\sigma}^{-n-1}\mathbf{X}^A.
$$
This is evidently smooth up to the boundary and, as observed there,
$\circverline{\sigma}^{-n}\betap_{a_1\dots a_n}$ is the volume form of
$\circverline{g}_{ij}$. Writing $\circperatorname{vol}_{\circverline{g}}|_{\partial M}$ as
$d\rho\wedge\circperatorname{vol}_{\circverline{g}_\infty}$ and using that the image of $d\rho$ in
$\Omega^1(\partial M)$ vanishes, we directly get the claimed formula for the boundary
value. The final statement follows the same argument as in Proposition \ref{prop3.1}.
\end{proof}
Similarly as in Section \ref{3.1} above, this admits a more natural interpretation
when working with densities. Again fixing $N=n$, instead of \eqref{h-g} we can start
from
\betagin{equation}\lambdabel{h-g-dens}
h_{ij}=g_{ij}+\sigma^{n-2}\nu_{ij},
\end{equation}
where $\nu_{ij}\in\Gamma(\mathcal E_{(ij)}[-n+2])$ now is a weighted symmetric two-tensor
that is smooth up to the boundary. For a choice of local defining function $\rho$,
the relation to \eqref{h-g} is described by
$\nu_{ij}=(\frac{\rho}{\sigma})^{n-2}\mu_{ij}$. This immediately implies that
$$
\mathbf{g}^{ij}\nu_{ij}=(\tfrac{\rho}{\sigma})^{n-2}\tfrac1{\sigma^2}g^{ij}\mu_{ij}=(\tfrac{\rho}{\sigma})^n\mu,
$$ so this coincides with the density $\nu\in\Gamma(\mathcal E[-n])$ used in Section
\ref{3.1}. The tracefree part $\nu^0_{ij}$, then of course is $\nu_{ij}-\tfrac1n
\mathbf{g}_{ij}\nu=(\frac{\rho}{\sigma})^{n-2}\mu^0_{ij}$. On the other hand, the fact
that $g_{ij}$ is ALH shows that $\tfrac{1}{\rho^2}g^{ij}\rho_i\rho_j$ is identically
one along the boundary. Using $g^{ij}=\sigma^2\mathbf{g}^{ij}$, we conclude that
$\tfrac{\sigma}{\rho}\mathbf{g}^{ij}\rho_i\in\mathcal E^a[-1]$ coincides, along $\partial M$,
with the conformal unit normal $n^j$ from Section \ref{2.5}. Hence
$\circverline{g}^{ij}\rho_j=\circverline{\sigma}n^i$ and hence we can rewrite formula
\eqref{c21} as $-\sigma^{n-2}\nu^0_{ij}n^j\mathbf{X}_A+\mathcal O(\rho^{n-1})$, where we
have extended $n^j$ arbitrarily off the boundary.
To rewrite the formula \eqref{c22} for the boundary value in a similar
way, we use the observation that $\circperatorname{vol}_{\circverline{g}_\infty}$
corresponds to $(\tfrac{\rho}{\sigma})^{n-1}$, as discussed in Section
\ref{3.1}. Using this and the above, see that \eqref{c22} equals
$$
-n^in^j(\nu^0_{ij})_\infty\mathbf{X}_A,
$$
where $(\nu^0_{ij})_\infty$ indicates the boundary value of $\nu^0_{ij}$. Using this
formulation, it is easy to prove that we obtain another cocyle.
\betagin{cor}\lambdabel{cor3.2}
Let us denote the section of $\mathcal T\partial M[-n+1]$ associated to $g,h\in\mathcal G$
via formula \eqref{c22} by $c_2(g,h)$. Then $c_2$ is a cocycle in the sense that
$c_2(h,g)=-c_2(g,h)$ and that $c_2(g,k)=c_2(g,h)+c_2(h,k)$ for $g,h,k\in\mathcal G$.
\end{cor}
\betagin{proof}
Suppose that the tracefree part of $h_{ij}-g_{ij}$ with respect to $g$ is given by
$\sigma^{n-2}\nu^0_{ij}$ for $\nu^0_{ij}\in\Gamma(\mathcal E_{ij}[2-n])$ and similarly the
tracefree part of $(g_{ij}-h_{ij})^0$ with respect to $h$ corresponds to
$\tilde\nu^0_{ij}$. Then one immediately verifies that
$\tilde\nu^0_{ij}=-\nu^0_{ij}+\mathcal O(\rho^{n-2})$. Together with the above formula,
this readily implies that $c_2(h,g)=-c_2(g,h)$. The second claim follows similarly.
\end{proof}
| 4,092 | 45,932 |
en
|
train
|
0.50.7
|
\subsection{Diffeomorphisms}\lambdabel{3.3}
We next start to study the compatibility of the cocycles we have
constructed above with diffeomorphisms. There are various concepts of
compatibility here, and we have to discuss some background
first. Recall that a diffeomorphism $\Psii:\circverline{M}\to\circverline{M}$ of a manifold
with boundary maps $M$ to $M$ and $\partial M$ to $\partial M$. This
also shows that for $x\in\partial M$, the linear isomorphism
$T_x\Psii:T_xM\to T_{\Psii(x)}M$ maps the subspace $T_x\partial M$ to
$T_{\Psii(x)}\partial M$. This readily implies that for a defining
function $\rho$ for $\partial M$, also $\Psii^*\rho=\rho\circ\Psii$ is a
defining function for $\partial M$. This also works for a local
defining function defined on a neighborhood of $\Psii(x)$, which gives
rise to a local defining function defined on a neighborhood of $x$.
This in turn readily implies that the relation $\sigmam_N$ on tensor
fields defined in Section \ref{2.3} is compatible with diffeomorphisms
in the sense that $t\sigmam_N\tilde t$ implies $\Psii^*t\sigmam_N\Psii^*\tilde
t$ for each $N>0$. In particular, given an equivalence class $\mathcal G$
of conformally compact metrics, also $\Psii^*\mathcal G$ is such an
equivalence class. We are particularly interested in the case that
$\Psii^*\mathcal G=\mathcal G$, in which we say that \textit{$\Psii$ preserves
$\mathcal G$}. The diffeomorphisms with this property clearly form a
subgroup of the diffeomorphism group $\circperatorname{Diff}(\circverline{M})$ which we denote by
$\circperatorname{Diff}_{\mathcal G}(\circverline{M})$. From our considerations it follows
immediately that this is equivalent to the fact that there is one
metric $g\in\mathcal G$ such that $\Psii^*g\in\mathcal G$ or equivalently
$\Psii^*g\sigmam_{N-2}g$. (In \cite{CDG} an analogous property is phrased
by saying that $\Psii$ is an ``asymptotic isometry'' of $g$. We don't
use this terminology since $\Psii$ is not more compatible with $g$
than with any other metric in $\mathcal G$.)
Recall from Section \ref{2.3} that all metrics in $\mathcal G$ give rise to the same
conformal infinity on $\partial M$. This implies that for $\Psii\in\circperatorname{Diff}_{\mathcal
G}(\circverline{M})$ the restriction $\Psii_\infty:=\Psii|_{\partial M}$ is not only a
diffeomorphism, but actually a conformal isometry of the conformal infinity of $\mathcal
G$. In particular, it induces a well defined bundle automorphism on the standard
tractor bundle $\mathcal T\partial M$ and hence we can pullback sections of $\mathcal
T\partial M$ along $\Psii_\infty$. This also works for $n=3$ without
problems. Using this, we can prove the first and
simpler compatibility condition of our cocylces with diffeomorphisms.
\betagin{prop}\lambdabel{prop3.3}
Consider a manifold $\circverline{M}=M\cup\partial M$ with boundary and a class
$\mathcal G$ of metrics, in the case $N=n$. Then the cocycles constructed
in Propositions \ref{prop3.1} and \ref{prop3.2} are compatible with
the action of a diffeomorphism $\Psii\in\circperatorname{Diff}_{\mathcal G}(\circverline{M})$ in the
sense that for each such cocycle
$c(\Psii^*g,\Psii^*h)=(\Psii_\infty)^*c(g,h)$. Here
$\Psii_\infty=\Psii|_{\partial M}$, and on the right hand side we have
the action of a conformal isometry of the conformal infinity of $\mathcal
G$ (on $\partial M$) on tractor valued forms.
\end{prop}
\betagin{proof}
This basically is a direct consequence of the invariance properties of the
constructions we use. If $g$ corresponds to $\sigma\in\Gamma(\mathcal E[1])$, then of course
$\Psii^*g$ corresponds to $\Psii^*\sigma$. Moreover, $\Psii_\infty$ defines an conformal
isometry between the conformal structures on $\partial M$ induced by $\Psii^*g$ and
$g$, respectively. Similarly, $\Psii^*h$ corresponds to $\Psii^*\tau$ and naturality of
the tractor constructions
implies that
$\nabla_aD_B(\Psii^*\tau-\Psii^*\sigma)$ (computed in the conformal structure $[\Psii^*g]$)
equals $\Psii^*(\nabla_aD_B(\tau-\sigma))$. Since $\Psii|_M$ is an isometry from $\Psii^*g$
to $g$, we get $\Psii^*\circperatorname{vol}_g=\circperatorname{vol}_{\Psii^*g}$, which implies compatibility with the
Hodge-star. Hence on $M$, we get
$$
\star_{\Psii^*g}\nabla_aD_B(\Psii^*\tau-\Psii^*\sigma)=\Psii^*(\star_g\nabla_aD_B(\tau-\sigma))
$$
and since both sides admit a smooth extension to the boundary, the boundary values
have to coincide, too. But these than are exactly $c_1(\Psii^*g,\Psii^*h)$ and the
pullback induced by the conformal isometry $\Psii_\infty$ of $c_1(g,h)$. This
completes the proof for $c_1$.
For $c_2$, we readily get that $(\Psii^*h-\Psii^*g)^0$ (tracefree part with respect to
$\Psii^*g$) coincides with $\Psii^*(h-g)^0$ (tracefree part with respect to $g$). Using
naturality of the splitting operator $S$, the proof is completed in the
same way as for $c_1$.
\end{proof}
| 1,526 | 45,932 |
en
|
train
|
0.50.8
|
\subsection{Diffeomorphisms asymptotic to the identity}\lambdabel{3.4}
To move towards a more subtle form of compatibility of our cocycles with
diffeomorphisms, we need a concept of asymptotic relation between diffeomorphisms.
\betagin{definition}\lambdabel{def3.4}
Let $\circverline{M}=M\cup\partial M$ be a manifold with boundary and let
$\Psi,\tilde\Psi:\circverline{M}\to\circverline{M}$ be diffeomorphisms.
(1) We say that $\Psi$ and $\widetilde{\Psi}$ are asymptotic of order $N>0$ and write
$\Psii\sigmam_N\widetilde{\Psi}$ if and only if for any function $f\in C^\infty(\circverline{M}, \mathbb R)$
we get $f\circ\Psii\sigmam_Nf\circ\widetilde{\Psi}$ in the sense of Section \ref{2.3}.
(2) For $N>0$, we define $\circperatorname{Diff}_0^N(\circverline{M})$ to be the set of diffeomorphisms which are
asymptotic to the identity $\circperatorname{id}_{\circverline{M}}$ of order $N$.
\end{definition}
Since $\sigmam_N$ clearly defines an equivalence relation on functions,
we readily see that it is an equivalence relation on
diffeomorphisms. Moreover, since the pullback of a local defining
function for $\partial M$ along a diffeomorphism of $\circverline{M}$ again is a
local defining function, we conclude that $\Psii\sigmam_N\widetilde{\Psi}$ implies
$\Psii\circ\Phi\sigmam_N\widetilde{\Psi}\circ\Phi$ and $\Phi\circ\Psii\sigmam_N\Phii\circ\widetilde{\Psi}$ for any
diffeomorphism $\Phii$ of $\circverline{M}$. In particular, this shows either of
$\widetilde{\Psi}^{-1}\circ\Psii\sigmam_N\circperatorname{id}$ and $\widetilde{\Psi}\circ\Psii^{-1}\sigmam_N\circperatorname{id}$ is
equivalent to $\Psii\sigmam_N\widetilde{\Psi}$.
On the other hand, we need some observations on charts. Given a manifold
$\circverline{M}=M\cup\partial M$ with boundary, take a point $x\in\partial M$. Then by
definition, there is a chart $(U,u)$ around $x$, so $U$ is an open neighborhood of
$x$ in $\circverline{M}$ and $u:U\to u(U)$ is a diffeomorphism onto an open subset of an
$n$-dimensional half space. Then $u$ restricts to a diffeomorphism between the open
neighborhood $U\cap\partial M$ of $x$ in $\partial M$ and the open subspace
$u(U)\cap\mathbb R^{n-1}\times\{0\}$ of $\mathbb R^{n-1}$. Then by definition, the last
coordinate function $u^n$ is a local defining function of $\partial M$. Conversely,
any local defining function can locally be used as such a coordinate function in a
chart.
If $\Psii\in\circperatorname{Diff}(\circverline{M})$ is a diffeomorphism, then for a chart $(U,u)$ also
$(\Psii^{-1}(U),u\circ\Psii)$ is a chart. If $\widetilde{\Psi}$ is another diffeomorphism such that
$\widetilde{\Psi}|_{\partial M}=\Psii|_{\partial M}$, then $V:=\Psii^{-1}(U)\cap\widetilde{\Psi}^{-1}(U)$ is
an open subset in $\circverline{M}$ which contains $\Psii^{-1}(U\cap\partial M)$. For any tensor
field $t$ defined on $U$, both $\Psii^*t$ and $\widetilde{\Psi}^*t$ are defined on $V$, and can
be compared asymptotically there. Using these observations, we start by proving a
technical lemma.
\betagin{lemma}\lambdabel{lem3.4}
Let $\circverline{M}=M\cup\partial M$ be a smooth manifold with boundary,
let $\Psii,\widetilde{\Psi}\in\circperatorname{Diff}(\circverline{M})$ be diffeomorphisms, and fix
$N>0$. Then the following conditions are equivalent:
\betagin{itemize}
\item[(i)] $\Psii\sigmam_{N+1}\widetilde{\Psi}$
\item[(ii)] $\Psii|_{\partial M}=\widetilde{\Psi}|_{\partial M}$ and for any tensor field $t$
on $\circverline{M}$, we get $\Psii^*t\sigmam_N\widetilde{\Psi}^*t$.
\item[(iii)] $\Psii|_{\partial M}=\widetilde{\Psi}|_{\partial M}$ and for each $x\in\partial M$, there
is an chart $(U,u)$ for $\circverline{M}$ with $x\in U$, whose coordinate functions $u^i$
satisfy $\Psii^*u^i\sigmam_{N+1}\widetilde{\Psi}^*u^i$ locally around $\Psii^{-1}(x)$.
\end{itemize}
\end{lemma}
\betagin{proof}
Replacing $\Psii$ by $\widetilde{\Psi}^{-1}\circ\Psii$ we may without loss of generality assume
that $\widetilde{\Psi}=\circperatorname{id}_{\circverline{M}}$, which we do throughout the proof.
(i)$\Rightarrow$(iii): We first claim that $\Psii|_{\partial M}=\circperatorname{id}_{\partial
M}$. For $x\in\partial M$, take an open neighborhood $W$ of $x$ in $\partial
M$. Then there is a bump function $f\in C^\infty(\circverline{M},\mathbb R)$ with values in
$[0,1]$ such that $f(x)=1$ and such that $f^{-1}(\{1\})\cap\partial M\subset W$. By
assumption $f\circ\Psii\sigmam_{N+1}f$, so in particular, these functions have to agree on
$\partial M$ and hence at $x$. Since $\Psii(x)\in \partial M$, by construction, we
get $\Psii(x)\in W$. Since $W$ was arbitrary, this implies that $\Psii(x)=x$ and
hence the claim. Having this at hand, we take any chart $(U,u)$ with $x\in U$,
extend the coordinate functions $u^i$ to globally defined functions on $M$ without
changing them locally around $x$ and then (i) immediately implies that
$\Psii^*u^i\sigmam_{N+1}u^i$ locally around $x$.
(iii)$\Rightarrow$(ii): For any tensor field $t$, it suffices to verify
$\Psii^*t\sigmam_N t$ locally around each boundary point $x\in\partial M$. Fixing $x$,
we take a chart $(U,u)$ as in (iii) and its coordinate functions $u^i$ and we work
on $V=\Psii^{-1}(U)\cap U$. Taking a vector field $\timesi\in\mathfrak X(U)$ we can compare
$\Psii^*\timesi$ and $\timesi$ on $V$. We can do this via coordinate expressions with
respect to the chart $(U,u)$ and we denote by $\timesi^i$ and $(\Psii^*\timesi)^i$ the
component functions. By assumption, $u^i\circ\Psii=u^i+\mathcal O(\rho^{N+1})$ and
differentiating this with $\Psii^*\timesi$, we obtain $(\Psi^*\timesi)(u^i)+\mathcal
O(\rho^N)$. Thus we conclude that $(\Psii^*\timesi)(u^i\circ\Psii)\sigmam_N(\Psii^*\timesi)^i$. But
by definition of the pullback, we get
$(\Psii^*\timesi)(u^i\circ\Psii)=\timesi(u^i)\circ\Psii\sigmam_{N+1}\timesi^i$. Overall, we conclude that
$(\Psii^*\timesi)^i\sigmam_N\timesi^i$ on $V$, which implies that $\Psii^*\timesi\sigmam_N\timesi$ on $V$
and hence we get condition (ii) for vector fields.
In particular, this implies that the coordinate vector fields $\partial_i$ of the
chart $(U,u)$ satisfy $\Psii^*\partial_i\sigmam_N\partial_i$ on $V$. On the other hand,
applying the exterior derivative to $u^i\circ\Psii\sigmam_{N+1}u^i$, we conclude that
$\Psii^*du^i=d(u^i\circ\Psii)\sigmam_Ndu^i$. Of course, on $V$ the $du^i$ coincide with the
coordinate one-forms of the chart $(U,u)$. Now given a tensor field $t$ of any
type, we can take $\Psii^*t$ and hook in vector fields $\Psii^*\partial_{i_a}$ and
one-forms $\Psii^*du^{j_b}$. On $V$ this by construction produces one of the
component functions of $t$ up to $\mathcal O(\rho^N)$. On the other hand, by definition
of the pullback, this coincides with the composition of the corresponding coordinate
function of $t$ with $\Psii$, and hence with that coordinate function up to $\mathcal
O(\rho^{N+1})$, so (ii) is satisfied in general.
(ii)$\Rightarrow$(i): Let $f\in C^\infty(\circverline{M},\mathbb R)$ be a smooth function. Then
by assumption we know that $f\circ\Psii\sigmam_Nf$, so choosing a local defining function
$\rho$ for $\partial M$, we get $f\circ\Psii=f+\rho^N\tilde f$ for some smooth function
$\tilde f\in C^\infty(\circverline{M},\mathbb R)$. But then
$\Psii^*df=d(f\circ\Psii)=df+N\rho^{N-1}\tilde fd\rho+\mathcal O(\rho^N)$. However,
condition (ii) also says that $\Psii^*df\sigmam_Ndf$ and since $d\rho|_{\partial M}$ is
nowhere vanishing, this implies that $\tilde f|_{\partial M}=0$. But this implies
that $f\circ\Psii\sigmam_{N+1}f$ and hence condition (i) follows.
\end{proof}
| 2,626 | 45,932 |
en
|
train
|
0.50.9
|
\subsection{The relation to adapted defining functions}\lambdabel{3.5}
In a special case and in quite different language, it has been observed in \cite{CDG}
that there is a close relation between diffeomorphisms asymptotic to the identity and
adapted defining functions. We start discussing this with the following lemma.
\betagin{lemma}\lambdabel{lem3.5}
In our usual setting, of $\circverline{M}=M\cup\partial M$, let $\mathcal G$ be an
equivalence class of ALH metrics on $M$ for the relation $\sigmam_{N-2}$
for some $N\geq 3$. Take two metrics $g,h\in\mathcal G$, and let $\rho$
and $r$ be local defining functions for $\partial M$ defined on the
same open subset $U\subset\circverline{M}$. If $\rho$ is adapted to $g$ and $r$
is adapted to $h$ in the sense of Definition \ref{def2.2} and if
$\rho^2g_{ij}$ and $r^2h_{ij}$ induce the same metric on the boundary,
then $\rho\sigmam_{N+1}r$.
\end{lemma}
\betagin{proof}
We have to analyze the asymptotics of solutions to the PDE that governs the change to
an adapted defining function. Replacing $\circverline{M}$ by an appropriate open subset, we
may assume that $\rho$ and $r$ are defined on all of $\circverline{M}$. Then we can write
$r=\rho e^v$ for some smooth function $v\in C^\infty(\circverline{M},\mathbb R)$ which gives
$dr=rdv+e^vd\rho$. In abstract index notation, this reads as
$r_i=rv_i+e^v\rho_i$. The fact that $r$ is adapted to $h_{ij}$ says that
$r^{-2}h^{ij}r_ir_j$ is identically $1$ on a neighborhood of the boundary, and
inserting we conclude that
\betagin{equation}\lambdabel{adap-PDE}
1\equiv \rho^{-2}h^{ij}\rho_i\rho_j+2\rho^{-1}h^{ij}\rho_iv_j+h^{ij}v_iv_j.
\end{equation}
Observe that $\rho^{-2}h^{ij}$, $\rho_i$ and $v_i$ are all smooth up to the boundary,
so the terms in the right hand side are $\mathcal O(1)$, $\mathcal O(\rho)$, and $\mathcal
O(\rho^2)$, respectively.
Now on the one hand, since $g,h\in\mathcal G$, we know from \eqref{hinv} that
$\rho^{-2}h^{ij}=\rho^{-2}g^{ij}+\mathcal O(\rho^N)$. Since $\rho$ is adapted to $g$,
this means that the first term in the right hand side of \eqref{adap-PDE} is $1+\mathcal
O(\rho^N)$. Inserting into \eqref{adap-PDE}, we conclude that
\betagin{equation}\lambdabel{adap-PDE2}
2\rho^{-1}h^{ij}\rho_iv_j+h^{ij}v_iv_j=\mathcal O(\rho^N).
\end{equation}
On the other hand, $r^2h_{ij}=e^{2v}\rho^2h_{ij}$, and $r^2h_{ij}$, by assumption, is
smooth up to the boundary with the same boundary value as $\rho^2g_{ij}$. Hence our
assumption on the induced metrics on the boundary imply that $e^{2v}|_{\partial
M}=1$, so $v$ has to vanish identically along the boundary and hence $v=\mathcal
O(\rho)$. Inductively, putting $v=\rho^\ell\tilde v$ for $\ell\geq 1$, we get
$v_i=\ell\rho^{\ell-1}\tilde v\rho_i+\mathcal O(\rho^\ell)$, which implies that the left
hand side of \eqref{adap-PDE2} becomes $2\ell\rho^\ell\tilde
v\rho^{-2}h^{ij}\rho_i\rho_j+\mathcal O(\rho^{\ell+1})$. As long as $\ell<N$, this shows
that $\tilde v= \mathcal O (\rho)$, so we conclude that we can write $v=\rho^N\tilde v$
where $\tilde v$ is smooth up to the boundary. But then
$$r=\rho e^v=\rho(1+\rho^N\tilde v+\mathcal O(\rho^{N+1}))=\rho+\mathcal O(\rho^{N+1}),$$
which completes the proof.
\end{proof}
Using this, we can now establish several important properties of
diffeomorphisms that are asymptotic to the identity.
\betagin{thm}\lambdabel{thm3.5}
Let $\circverline{M}=M\cup\partial M$ be a smooth manifold with boundary and, for some $N\geq
3$, let $\mathcal G$ be an equivalence class of ALH metrics on $M$ for the relation
$\sigmam_{N-2}$. Let us denote by $[\mathcal G]$ the conformal structure on $\partial M$
defined by the conformal infinity of $\mathcal G$ and by $\circperatorname{Conf}(\partial M,[\mathcal G])$
the group of its conformal isometries. Then we have
(1) $\circperatorname{Diff}^{N+1}_0(\circverline{M})$ is a normal subgroup in $\circperatorname{Diff}(\circverline{M})$ and
is contained in $\circperatorname{Diff}_{\mathcal G}(\circverline{M})$.
(2) Restriction of diffeomorphisms to the boundary induces a homomorphism
$\circperatorname{Diff}_{\mathcal G}(\circverline{M})\to\circperatorname{Conf}(\partial M,[\mathcal G])$ with kernel
$\circperatorname{Diff}^{N+1}_0(\circverline{M})$.
\end{thm}
\betagin{proof}
(1) The observations on the relation $\sigmam_N$ for diffeomorphisms we have made after
Definition \ref{def3.4} readily imply that $\circperatorname{Diff}^{N+1}_0(\circverline{M})$ is stable under
inversions as well as compositions, and conjugations by arbitrary elements of
$\circperatorname{Diff}(\circverline{M})$. Hence $\circperatorname{Diff}^{N+1}_0(\circverline{M})$ is a normal subgroup of
$\circperatorname{Diff}(\circverline{M})$. Taking $g\in\mathcal G$ and a local defining function $\rho$ for
$\partial M$, we know that $\rho^2g_{ij}$ admits a smooth extension to the
boundary. Thus, given $\Psii\in \circperatorname{Diff}^{N+1}_0(\circverline{M})$, we may apply part (2) of
Lemma \ref{lem3.4} to conclude that $\Psii^*(\rho^2g_{ij})\sigmam_N\rho^2g_{ij}$. Now
$\Psii^*(\rho^2)=(\rho\circ\Psii)^2$ and $\rho\circ\Psii\sigmam_{N+1}\rho$. Hence
$\rho^2\Psii^*g_{ij}\sigmam_N \Psii^*(\rho^2g_{ij})\sigmam_N\rho^2g_{ij}$ and restricting
to $M$ we conclude that $\Psii^*g_{ij}\in\mathcal G$. This completes the proof of (1).
(2) It follows readily from the definitions that, for
$\Psii\in\circperatorname{Diff}_{\mathcal G}(\circverline{M})$, the diffeomorphism $\Psii|_{\partial
M}$ of $\partial M$ preserves the conformal structure $[\mathcal
G]$. Thus we get a homomorphism $\circperatorname{Diff}_{\mathcal
G}(\circverline{M})\to\circperatorname{Conf}(\partial M,[\mathcal G])$ as claimed and it remains
to prove the claim about the kernel. So let us take a diffeomorphism
$\Psii\in\circperatorname{Diff}_{\mathcal G}(\circverline{M})$ such that $\Psii|_{\partial
M}=\circperatorname{id}_{\partial M}$ and we want to show that
$\Psii\sigmam_{N+1}\circperatorname{id}$. To prove this, we can apply condition (iii) of
Lemma \ref{lem3.4} and work locally around a boundary point $x$. Let
us choose $g\in\mathcal G$ and a local defining function $\rho$ for
$\partial M$ which is adapted to $g$ and defined on some open
neighborhood $U$ of $x$ in $\circverline{M}$. Now we consider the normal field
(to $\rho $ level sets) determined by $g$ and $\rho$, i.e.~we put
$\timesi^i:=\rho^{-2}g^{ij}\rho_j$. This admits a smooth
extension to the boundary, and the fact that $\rho$ is adapted to $g$
exactly says that $\rho^2g_{ij}\timesi^i\timesi^j$ is identically one on a
neighborhood of the boundary.
For $x\in\partial M$, we now work on an open neighborhood $W$ of $x$
in $\circverline{M}$ such that $W\subset\Psii^{-1}(U)\cap U$. On $W$, we can
pull back all our data by $\Psii$, thus obtaining
$h_{ij}:=\Psii^*g_{ij}$, $r:=\rho\circ\Psii$, and $\eta:=\Psii^*\timesi$. By
assumption, $h_{ij}\in\mathcal G$ and pulling back the defining equation
for $\timesi^i$ we get $\eta^i=r^{-2}h^{ij}r_j$. Also by pulling back,
we readily see that the $r^2h_{ij}\eta^i\eta^j$ is identically one
on a neighborhood of the boundary. Hence we conclude that the local
defining function $r$ is adapted to $h_{ij}$ and hence Lemma
\ref{lem3.5} shows that $r\sigmam_{N+1}\rho$. Observe that this implies
that $r_j\sigmam_N\rho_j$ and together with $g^{ij}\sigmam_{N+2}h^{ij}$
the defining equations for $\timesi$ and $\eta$ show that
$\timesi\sigmam_N\eta$.
Next, we pass to an appropriate collar of the boundary. We can
choose an open neighborhood $V$ of $x$ in $\partial M$ and a
positive number $\epsilonsilon$ such that the flow map
$(y,t)\mapsto\circperatorname{Fl}^\eta_t(y)$ defines a diffeomorphism $\phi$ from $V\times
[0,\epsilonsilon)$ onto an open subset contained in $\Psii^{-1}(W)\cap W$,
and on which $\rho$ satisfies $(\rho^2 g)^{-1}(d\rho,d\rho)=1$.
Now let us define $\partial_t:=\phi^*\eta$. Note that this is the
coordinate vector field for any product chart on $V\times
[0,\epsilonsilon)$ induced by some chart on $V$. Since
$\eta=\Psii^*\timesi$, the fact the $\Psii$-related vector fields have
$\Psii$-related flows together with $\Psii|_{\partial M}=\circperatorname{id}$
readily implies that $(\Psii\circ\phi)(y,t)=\circperatorname{Fl}^\timesi_t(y)$. Using
Section \ref{3.4}, we see that we can complete the proof by
showing that $\Psii\circ\phi\sigmam_{N+1}\phi$.
By Lemma \ref{lem3.4} it suffices to consider the pullbacks of coordinate functions
of local charts along these diffeomorphisms. We apply this to charts which are
obtained by composing a product chart for $V\times [0,\epsilonsilon)$ with
$(\Psii\circ\phi)^{-1}$. Now the fact that $d\rho(\timesi)\equiv 1$ (and that $\Psii\circ\phi$
maps $V\times \{0\}$ into $\partial M$) shows that applying this construction to
the coordinate $t$ on $[0,\epsilonsilon)$, we obtain $\rho$. As observed above,
$\rho\circ\Psii\sigmam_{N+1}\rho$ and thus $\rho\circ\Psii\circ\phi\sigmam_{N+1}\rho\circ\phi$. Thus
it remains to consider functions $f$ such that $f\circ(\Psii\circ\phi)$ is one of the
boundary coordinate functions, and hence $\partial_t\cdot (f\circ \Psi\circ\phi)\equiv
0$ or, equivalently, $\timesi\cdot f\equiv 0$ on an appropriate neighborhood of the
boundary. We have to compare $f\circ(\Psii\circ\phi)$ to $f\circ\phi$, and of course
$\phi^*\timesi\cdot (f\circ\phii)\equiv 0$.
As we have observed above, we get $\eta\sigmam_N\timesi$
and hence
$\phi^*\timesi\sigmam_N\phi^*\eta=\partial_t$. Thus we get
$\phi^*\timesi=\partial_t+t^N\tilde\timesi$ for some vector field
$\tilde\timesi\in\mathfrak(V\times [0,\epsilonsilon))$. By construction $f\circ\phi$ and
$f\circ(\Psi\circ\phi)$ agree on $V\times\{0\}$, so
$f\circ\phi\sigmam_1f\circ(\Psi\circ\phi)$. Assuming that $f\circ\phi\sigmam_kf\circ(\Psi\circ\phi)$ for
some $k\geq 1$, we get $f\circ\phi=f\circ(\Psi\circ\phi)+t^k\tilde f$ for some $\tilde
f\in C^\infty(V\times[0,\epsilon),\mathbb R)$. Then we compute
$$
0=(\partial_t+t^N\tilde\timesi)\cdot (f\circ(\Psi\circ\phi)+t^k\tilde
f)=0+kt^{k-1}\tilde f+\mathcal O(t^{\text{min}(k,N)})
$$
If $k\leq N$, then this equation shows that $\tilde f$ vanishes along $V\times\{0\}$
and hence $(f\circ\phi)\sigmam_{k+1}f\circ(\Psi\circ\phi)$. Inductively, this gives
$(f\circ\phi)\sigmam_{N+1}f\circ(\Psi\circ\phi)$, which completes the proof.
\end{proof}
| 3,675 | 45,932 |
en
|
train
|
0.50.10
|
\subsection{Aligned metrics}\lambdabel{3.6}
Following an idea in \cite{CDG} we next show that the freedom under diffeomorphisms
asymptotic to the identity can be absorbed in a geometric relation between the
metrics. The analogous condition in \cite{CDG} is phrased as ``transversality''.
\betagin{definition}\lambdabel{def3.6}
Let $\circverline{M}=M\cup\partial M$ be a manifold with boundary and let $\mathcal G$ be an
equivalence class of ALH metrics for the relation $\sigmam_{N-2}$ for some $N\geq
3$. Consider two metrics $g,h\in\mathcal G$ and a local defining function $\rho$ for
$\partial M$ defined on some open subset $U\subset\circverline{M}$. Then we say that $h$
\textit{is aligned with $g$ with respect to $\rho$} if
$$
\rho_ig^{ij}(h_{jk}-g_{jk})\equiv 0
$$
on some open neighborhood of $U\cap\partial M$ in $U$.
\end{definition}
Observe that the condition in Definition \ref{def3.6} can be rewritten
as $\rho_ig^{ij}h_{jk}=\rho_k$. This in turn implies that
$\rho_ih^{ij}=\rho_ig^{ij}$ on a neighborhood of the boundary. This
shows that in the case that $\rho$ is adapted to $g_{ij}$ and $h_{ij}$ is
aligned to $g_{ij}$ with respect to $\rho$, then $\rho$ is also
adapted to $h_{ij}$.
\betagin{thm}\lambdabel{thm3.6}
In our usual setting, of $\circverline{M}=M\cup\partial M$ and a class $\mathcal G$ of metrics,
assume that $g\in\mathcal G$ and $\rho$ is a local defining function for $\partial M$
that is adapted to $g$. Then for any $h\in\mathcal G$ and locally around any point
$x\in\partial M$ in the domain of definition of $\rho$, there exist a diffeomorphism
$\Psii$ such that $\Psii\sigmam_{N+1}\circperatorname{id}$ and such that $\Psii^*h$ is aligned to
$g$ with respect to $\rho$. Moreover, the germ of $\Psii$ along the intersection of
its domain of definition with $\partial M$ is uniquely determined by this condition.
\end{thm}
\betagin{proof}
Since $\rho$ is adapted to $g$ the function
$\rho^{-2}g^{ij}\rho_i\rho_j$ is identically one on some neighborhood
of the boundary. Now given $h_{ij}\in\mathcal G$, we can use Proposition
\ref{prop2.2} to modify $\rho$ to a local defining function $r$
adapted to $h_{ij}$ in such a way that $\rho^2g_{ij}$ and $r^2h_{ij}$
induce the same metric on the boundary, compare also to the proof of
Lemma \ref{lem3.5}. Now we define vector fields $\timesi=\timesi^i$ and
$\eta=\eta^i$ by $\timesi^i:=\rho^{-2}g^{ij}\rho_j$ and
$\eta^i:=r^{-2}h^{ij}r_j$ where as usual we write $\rho_j$ for $d\rho$
and similarly for $r$. Recall from the proof of Theorem \ref{thm3.5} that this implies
that $\timesi\sigmam_N\eta$.
Now, via the construction of collars from the proof of Theorem \ref{thm3.5}, we
obtain a diffeomorphism $\Psii$ which has the property that for any $y$ in an
appropriate neighborhood of $x$ in $\partial M$, we get
$\Psii\circ\circperatorname{Fl}^\timesi_t(y)=\circperatorname{Fl}^\eta_t(y)$. Differentiating this equation shows that
$\timesi=\Psii^*\eta$. By construction the derivative of the function $t\mapsto
r(\circperatorname{Fl}^\eta_t(y))$ is given by
$$
dr(\eta)=r^{-2}r_ih^{ij}r_j\equiv 1,
$$ so $r(y)=0$ shows that $r(\circperatorname{Fl}^\eta_t(y))=t$. In the same way
$\rho(\circperatorname{Fl}^\timesi_t(y))=t$, which shows that $\Psii^*r=\rho$.
Knowing this and $\timesi\sigmam_N\eta$, the last part of the proof of
Theorem \ref{thm3.5} shows that $\Psii\sigmam_{N+1}\circperatorname{id}$.
To show that $\Psii$ has the required property, observe that by
construction $r^2h_{ij}\eta^j=r_i$. That is $i_\eta r^2h=dr$ and in the
same way $i_\timesi \rho^2g=d\rho$. Using this and the above, we now
obtain
$$
i_\timesi\rho^2\Psii^*h=i_{\Psii^*\eta}\Psii^*r^2h=\Psii^*(i_\eta r^2h)=\Psii^*(dr)=d\rho=i_\timesi
\rho^2g.
$$
This shows that inserting $\timesi$ into the bilinear form
$\rho^2(\Psii^*h-g)$, which by construction is smooth up to the
boundary, the result vanishes identically on a neighborhood of the
boundary. This exactly shows that $\Psii^*h$ is aligned with $g$ with
respect to $\rho$, so the proof of existence is complete.
To prove uniqueness, assume that $h$ is aligned to $g$ with respect to $\rho$ and
that a diffeomorphism $\Psii$ such that $\Psii\sigmam_{N+1}\circperatorname{id}$ has the property that also
$\Psii^*h$ is aligned to $g$ with respect to $\rho$. Observe that $\Psii\sigmam_{N+1}\circperatorname{id}$
implies that $\Psii^*h\in\mathcal G$. As observed above, the fact that both $h$ and
$\Psii^*h$ are aligned with $g$ with respect to $\rho$ implies that $\rho$ is adapted
both to $h$ and to $\Psii^*h$. But on the other hand, the fact that $\rho$ is adapted
to $h$ of course implies that $\Psii^*\rho$ is adapted to $\Psii^*h$. Now by
construction $\rho\circ\Psii\sigmam_{N+1}\rho$ and hence $(\rho\circ\Psii)\Psii^*h$ and
$\rho\Psii^*h$ induce the same metric on the boundary. Hence the uniqueness part in
Proposition \ref{prop2.2} implies that $\rho\circ\Psii=\rho$ and hence
$\Psii^*\rho_i=\rho_i$ on a neighborhood of the boundary.
Now of course the inverse metric to $\Psii^*h_{ij}$ is $\Psii^*(h^{ij})$
and $\Psii^*\rho_i\Psii^*h^{ij}=\rho_i\Psii^*h^{ij}$. Since
$\Psii^*h_{ij}$ is aligned with $g_{ij}$ with respect to $\rho$, we get
$ \rho_i\Psii^*h^{ij}=\rho_ig^{ij}$ and since also $h_{ij}$ is aligned
with $g$ with respect to $\rho$, it equals $\rho_ih^{ij}$.
Hence putting $\timesi^j:=\rho_ih^{ij}$ we conclude that $\Psii^*\timesi=\timesi$.
Since $\Psii$ is the identity on the boundary, this implies that
$\Psii(\circperatorname{Fl}^\timesi_t(y))=\circperatorname{Fl}^\timesi_t(y)$ for $y$ in the boundary and $t$
sufficiently small, so $\Psii=\circperatorname{id}$ locally around the boundary.
\end{proof}
| 1,974 | 45,932 |
en
|
train
|
0.50.11
|
\subsection{The action of the aligning diffeomorphism}\lambdabel{3.7}
Fix $g\in\mathcal G$ and a local defining function $\rho$ for the boundary which is
adapted to $g$. Consider another metric $h\in\mathcal G$ and the corresponding tensor
field $\mu_{ij}$ defined by \eqref{h-g}. From Theorem \ref{thm3.6} we then know that
locally around each boundary point, we find an essentially unique diffeomorphism
$\Psii$ such that $\Psii\sigmam_{N+1}\circperatorname{id}$ and such that $\Psii^*h$ is aligned with $g$ with
respect to $\rho$. Since $\Psii^*h\in\mathcal G$, the analog of \eqref{h-g} defines a
tensor field $\tilde\mu_{ij}$ that describes the difference between $\Psii^*h$ and
$g$. We now prove that the boundary value of $\tilde\mu_{ij}$ can be explicitly
computed from the boundary value of $\mu_{ij}$. This will be the crucial step towards
finding combinations of the two cocycles constructed in Sections \ref{3.1} and
\ref{3.2} which are invariant under diffeomorphisms asymptotic to the
identity. Observe that this formally looks like the coordinate formula in Proposition
2.16 of \cite{CDG}, but the actual meaning is different: Our description
does not involve any choice of local coordinates, but uses only abstract indices.
\betagin{thm}\lambdabel{thm3.7}
In the setting of Theorem \ref{thm3.6} for some fixed order $N$, let
$\mu_{ij}$ be the tensor field relating $h_{ij}$ and $g_{ij}$
according to \eqref{h-g}, and let $\tilde\mu_{ij}$ be the
corresponding tensor field relating $\Psii^*(h_{ij})$ and
$g_{ij}$. Then putting $\timesi^i:=\rho^{-2}g^{ij}\rho_j$, we obtain
\betagin{equation}\lambdabel{alignment-diffeo}
\tilde\mu_{ij}=\mu_{ij}-\rho_i\mu_{j\ell}\timesi^\ell-\rho_j\mu_{i\ell}\timesi^\ell+
\tfrac{\timesi^k\mu_{k\ell}\timesi^\ell}{N}(\rho^2g_{ij}+(N-1)\rho_i\rho_j)+\mathcal O(\rho).
\end{equation}
\end{thm}
\betagin{proof}
We use the quantities introduced in the proof of Theorem \ref{thm3.6}:
We denote by $r$ the defining function adapted to $h_{ij}$ such that
$r^2h_{ij}$ and $\rho^2g_{ij}$ induce the same metric on the
boundary. Further we put $\timesi^i:=\rho^{-2}g^{ij}\rho_j$ and
$\eta^i:=r^{-2}h^{ij}r_j$. In terms of these, we know from the proof
of Theorem \ref{thm3.6} that the alignment diffeomorphism $\Psii$
satisfies $r\circ\Psii=\rho$ and $\Psii^*\eta=\timesi$, and hence
$\Psii\circ\circperatorname{Fl}^\timesi_t=\circperatorname{Fl}^\eta_t$ wherever the flows are defined. Moreover,
writing $r=e^v\rho$, we know from the proof of Lemma \ref{lem3.5} that
$v=\rho^N\tilde v$ for a function $\tilde v$ that admits a smooth
extension to the boundary. Moreover, we can compute the boundary value
of $\tilde v$ from that proof: In equation \eqref{adap-PDE} we can
bring the first term on the right hand side to the left hand side and
then use use fact that $\rho$ is adapted to $g$ to rewrite
\eqref{adap-PDE} as
$$
(\rho^{-2}g^{ij}-\rho^{-2}h^{ij})\rho_i\rho_j=2\rho\rho^{-2}h^{ij}\rho_iv_j+\rho^2\rho^{-2}h^{ij}v_iv_j.
$$ In the left hand side, we can insert \eqref{hinv} and use the definition of $\timesi$
to obtain $\rho^N\timesi^k\mu_{k\ell}\timesi^\ell$. On the other hand $v_j=N\rho^{N-1}\tilde
v\rho_j+\mathcal O(\rho^N)$. Inserting this in the right hand side and using that
$\rho^{-2}h^{ij}\rho_i\rho_j=1+\mathcal O(\rho)$, we obtain $2N\rho^N\tilde v+\mathcal
O(\rho^{N+1})$, which shows that
\betagin{equation}\lambdabel{tildev}
\tilde v=\tfrac{1}{2N}\timesi^k\mu_{k\ell}\timesi^\ell+\mathcal O(\rho).
\end{equation}
The basis for the further computation will be the fact that for vector fields
$\zeta_1,\zeta_2$ (which we assume to be smooth up to the boundary), we get
$\Psii^*h(\Psii^*\zeta_1,\Psii^*\zeta_2)=h(\zeta_1,\zeta_2)\circ\Psii$. Multiplying by $\rho^2$, we
obtain
\betagin{equation}\lambdabel{PB}
\rho^2\Psii^*h(\Psii^*\zeta_1,\Psii^*\zeta_2)=(r^2h(\zeta_1,\zeta_2))\circ\Psii,
\end{equation}
and both sides admit a smooth extension to the boundary. Hence the right hand side
equals $r^2h(\zeta_1,\zeta_2)+\mathcal O(\rho^{N+1})$. Inserting $r=\rho e^v$ and
\eqref{h-g}, we conclude that this equals
$$
e^{2v}\rho^2g(\zeta_1,\zeta_2)+e^{2v}\rho^N\mu(\zeta_1,\zeta_2)+\mathcal O(\rho^{N+1}).
$$
Of course, $e^{2v}=1+2\rho^N\tilde v+\mathcal O(\rho^{N+1})$ and we conclude that the
right hand side of \eqref{PB} equals
\betagin{equation}\lambdabel{PB-RHS}
\rho^2g(\zeta_1,\zeta_2)+\rho^N\big(2\tilde v\rho^2g(\zeta_1,\zeta_2)+
\mu(\zeta_1,\zeta_2)\big)+\mathcal O(\rho^{N+1}).
\end{equation}
The left hand side of \eqref{PB}, by definition, can be written as
$\rho^2g(\Psii^*\zeta_1,\Psii^*\zeta_2)+\rho^N\tilde\mu(\Psii^*\zeta_1,\Psii^*\zeta_2)$. Now we
know that $\Psii^*\zeta_1\sigmam_N\zeta_1$ and hence $\Psii^*\zeta_1=\zeta_1+\rho^N\tilde\zeta_1$,
where $\tilde\zeta_1$ admits a smooth extension to the boundary, and similarly for
$\zeta_2$. Inserting this, we obtain
\betagin{equation}\lambdabel{PB-LHS}
\rho^2g(\zeta_1,\zeta_2)+\rho^N\big(\rho^2g(\zeta_1,\tilde\zeta_2)+\rho^2g(\tilde\zeta_1,\zeta_2)+
\tilde\mu(\zeta_1,\zeta_2)\big)+\mathcal O(\rho^{N+1}).
\end{equation}
Since this has to equal \eqref{PB-RHS}, we conclude that
\betagin{equation}\lambdabel{tilde-mu-main}
\tilde\mu(\zeta_1,\zeta_2)=\mu(\zeta_1,\zeta_2)-\rho^2g(\zeta_1,\tilde\zeta_2)-
\rho^2g(\tilde\zeta_1,\zeta_2)+2\tilde v\rho^2g(\zeta_1,\zeta_2)+\mathcal O(\rho).
\end{equation}
The key observation now is that the computation of $\tilde\zeta_1$ and $\tilde\zeta_2$
essentially reduces to the computation of the vector field $\tilde\eta$ which has the
property that $\eta=\timesi+\rho^N\tilde\eta$. As a first step, we claim that for a
vector field $\zetata$ which is tangent to the boundary along the boundary, we have
$\Psii^*\zetata\sigmam_{N+1}\zetata$. This can of course be proved locally, so we can use
local charts obtained from a collar construction as in the proof of Theorem
\ref{thm3.5}. These have $r$ as one coordinate and $\eta$ as the corresponding
coordinate vector field. We first consider the case that $\zeta$ is the coordinate
vector field $\partial_i$ associated to one of the boundary coordinates. Of course,
$0=[\eta,\partial_i]$ and pulling back along $\Psii$, we conclude that
$0=[\timesi,\Psii^*\partial_i]$. Now by Lemma \ref{lem3.4}, we know that $\timesi\sigmam_N\eta$
and $\Psii^*\partial_i\sigmam_N\partial_i$, and we express this via
$\timesi=\eta+r^N\tilde\eta$ and $\Psii^*\partial_i=\partial_i+r^N\tilde\zetata$, where
$\tilde\eta$ and $\tilde\zeta$ admit a smooth extension to the boundary. Plugging these
expressions into the Lie bracket and using that $\partial_i\cdot r=0$ and $\eta\cdot
r=1$, we conclude that
$$
0=[\timesi,\Psii^*\partial_i]=[\eta,\partial_i]+Nr^{N-1}\tilde\zeta+\mathcal O(r^N).
$$
This shows that $\tilde\zeta$ vanishes along the boundary and hence
$\Psii^*\partial_i\sigmam_{N+1}\partial_i$. Now a general vector field $\zetata$ that is
tangent to the boundary along the boundary can be be written as $f\eta+\sum
f_i\partial_i$ for arbitrary smooth functions $f_i$ and a smooth function $f$ which
is $\mathcal O(r)$. Thus the general version of our claim follows readily since
$\Psii^*\eta\sigmam_N\eta$, $f_i\circ\Psii\sigmam_{N+1}f_i$ and $f\circ\Psii\sigmam_{N+1}f$.
Now for any vector field $\zeta$ that is smooth up to the boundary, the difference
$\zeta-d\rho(\zeta)\eta$ is smooth up to the boundary and tangent to the boundary along
the boundary. Of course $\Psii^*(d\rho(\zeta)\eta)=(d\rho(\zeta)\circ\Psii)\timesi$ and so this
equals $d\rho(\zeta)\timesi+\mathcal O(\rho^{N+1})$. Thus, writing
$\zeta=d\rho(\zeta)\eta+(\zeta-d\rho(\zeta)\eta)$ and pulling back, we get
\betagin{equation}\lambdabel{tilde-zeta}
\rho^N\tilde\zeta=\Psii^*\zeta-\zeta=d\rho(\zeta)(\timesi-\eta)+\mathcal O(\rho^{N+1}).
\end{equation}
To compute the difference $\timesi-\eta$, we first use $r=e^v\rho$ to conclude that
$r_j=e^v\rho_j+e^v\rho v_j$ and $v_j=N\rho^{N-1}\tilde v\rho_j+\mathcal O(\rho^N)$. This
shows that $r_j=\rho_j(1+(N+1)\rho^N\tilde v)+\mathcal O(\rho^{N+1})$. Next, by
definition $\eta^i=e^{-2v}\rho^{-2}h^{ij}r_j$ and
$$
e^{-2v}(1+(N+1)\rho^N\tilde v)=1+(N-1)\rho^N\tilde v+\mathcal O(\rho^{N+1}).
$$
Now \eqref{hinv} shows that
$$
\rho^{-2}h^{ij}=\rho^{-2}g^{ij}-\rho^N(\rho^{-2}g^{ik}\mu_{k\ell}\rho^{-2}g^{\ell
j})+\mathcal O(\rho^{N+1}).
$$
Putting all this together, we see that
\betagin{equation}\lambdabel{eta-xi}
\eta^j-\timesi^j=\rho^N((N-1)\tilde v\timesi^j-\rho^{-2}g^{jk}\mu_{k\ell}\timesi^\ell)+\mathcal
O(\rho^{N+1}).
\end{equation}
Dividing the negative of the right hand side by $\rho^N$ and contracting with
$\rho^2g_{ij}$, we obtain $-(N-1)\tilde v\rho_j+\mu_{j\ell}\timesi^{\ell}$. Using this
and \eqref{tilde-zeta}, we can write \eqref{tilde-mu-main} in abstract index notation
as
\betagin{equation}\lambdabel{tilde-mu-main2}
\tilde\mu_{ij}=\mu_{ij}+2(N-1)\tilde v\rho_i\rho_j-\rho_i\mu_{j\ell}\timesi^\ell
-\rho_j\mu_{i\ell}\timesi^\ell +2\tilde v\rho^2g_{ij}+\mathcal O(\rho),
\end{equation}
which together with \eqref{tildev} exactly gives the claimed formula.
\end{proof}
| 3,362 | 45,932 |
en
|
train
|
0.50.12
|
Using this, we can easily deduce that appropriate combinations of the cocycles
constructed in Sections \ref{3.1} and \ref{3.2} remain unchanged if one of the two
metrics involved is pulled back by a diffeomorphisms that is asymptotic to the
identity. Since we are dealing with the situation of the classical mass here, we have
to specialize to the case that $N=n$.
\betagin{cor}\lambdabel{cor3.7}
In our usual setting, of $\circverline{M}=M\cup\partial M$, let $\mathcal G$ be an equivalence
class of ALH metrics on $M$ for the relation $\sigmam_{n-2}$. Let $c_1$ and $c_2$ be the
cocycles constructed in Sections \ref{3.1} and \ref{3.2}, respectively, and let $c$
be a constant multiple of $\tfrac1nc_1+\tfrac12c_2$. Then $c$ defines a cocyle on
$\mathcal G$ that has the property that for metrics $g,h\in\mathcal G$ and any diffeomorphism
$\Phii\in \circperatorname{Diff}_0^{n+1}(\circverline{M})$, we get $c(g,h)=c(g,\Phii^*h)$.
\end{cor}
\betagin{proof}
We fix $g\in\mathcal G$ and a local defining function $\rho$ that is adapted to
$g$. Then we show that for $c=\tfrac1nc_1+\tfrac12c_2$ and the alignment
diffeomorphism $\Psii$ obtained from Theorem \ref{thm3.7}, we get
$c(g,\Psii^*h)=c(g,h)$. The last part of Theorem \ref{thm3.6} shows that $\Psii^*h$
is the unique metric in the the orbit of $h$ under $\circperatorname{Diff}_0^{n+1}(\circverline{M})$ which is
aligned to $g$ with respect to $\rho$. Hence applying the construction of Theorem
\ref{thm3.7} to $\Phii^*h$ for arbitrary $\Phii\in \circperatorname{Diff}_0^{n+1}(\circverline{M})$, we also
have to arrive at $\Psii^*h$, which then implies the result.
Using the formulae in parts (2) of Propositions \ref{prop3.1} and \ref{prop3.2}, we
see that to prove our claim it suffices to show that the boundary value of
$$
\tfrac{n^2-1}{2n}\rho^{-2}g^{ij}\mu_{ij}-\tfrac12\timesi^i\mu^0_{ij}\timesi^j
$$
coincides with the boundary value of the analogous expression formed from
$\tilde\mu_{ij}$. Inserting
$\mu^0_{ij}=\mu_{ij}-\tfrac{1}{n}\rho^{-2}g^{k\ell}\mu_{k\ell}\rho^2g_{ij}$ and
using that $\timesi^i\rho^2g_{ij}\timesi^j=1$ on a neighborhood of $\partial M$, we see that
our expression equals
$$
\tfrac{n}2\rho^{-2}g^{ij}\mu_{ij}-\tfrac12\timesi^i\mu_{ij}\timesi^j.
$$
Contracting $\rho^{-2}g^{ij}$ into formula \eqref{alignment-diffeo} (for the case
$N=n$) and multiplying by $\tfrac{n}2$, we obtain
$$
\tfrac{n}2\rho^{-2}g^{ij}\tilde\mu_{ij}=\tfrac{n}2\rho^{-2}g^{ij}\mu_{ij}-
\tfrac12\timesi^i\mu_{ij}\timesi^j.
$$
By alignment, $\timesi^i\tilde\mu_{ij}=0$, so this proves our claim.
\end{proof}
| 981 | 45,932 |
en
|
train
|
0.50.13
|
\subsection{From relative to absolute invariants}\lambdabel{3.8}
So far, we have not imposed any restriction on the equivalence class
$\mathcal G$ of metrics beyond the fact that it consists of
ALH-metrics. We next show that assuming that $\mathcal G$ locally contains
metrics that are hyperbolic (i.e.~have constant sectional curvature
$-1$), one can use our construction to obtain an invariant for
(single) metrics in $\mathcal G$. This assumption of course implies that
the conformal infinity $[\mathcal G]$ on $\partial M$ is conformally flat,
but as we shall see below, it does not impose further restrictions on
the topology of $\circverline{M}$.
The key step toward this are results on the uniqueness of hyperbolic
metrics with prescribed infinity that are discussed in Chapter 7 of
\cite{FeffGr}. These build on results in \cite{SkenSol} and are
related to the work in \cite{Epstein}.
\betagin{thm}\lambdabel{thm3.8}
Consider our usual setting, of $\circverline{M}=M\cup\partial M$, and an equivalence class
$\mathcal G$ of ALH metrics on $M$ for the relation $\sigmam_{n-2}$. Assume that for each
$x\in\partial M$ there is an open neighborhood $U$ of $x$ in $\circverline{M}$ and a metric $g$
in $\mathcal G$ that is hyperbolic (i.e.\ has constant sectional curvature $-1$) on
$U$. Let $c$ be any constant multiple of the cocycle $\tfrac1n c_1+\tfrac12 c_2$ from
Corollary \ref{cor3.7}.
Then for an open subset $U$ as above, and two metrics $g_1,g_2\in\mathcal
G$ that are hyperbolic on $U$, we get $c(g_1,h)|_{U\cap\partial
M}=c(g_2,h)|_{U\cap\partial M}$ for any $h\in\mathcal G$. Hence these
quantities fit together to a well defined section
$c(h)\in\Omega^{n-1}(\partial M,\mathcal T\partial M)$, thus defining a map
$c$ from $\mathcal G$ to tractor valued differential forms. This is equivariant under
diffeomorphisms preserving $\mathcal G$ in the sense that for
$\Phii\in\circperatorname{Diff}_{\mathcal G}(\circverline{M})$, we obtain
$$
c(\Phii^*h)=(\Phii_\infty)^*c(h).
$$
Here $\Phii_{\infty}:=\Phii|_{\partial M}\in\circperatorname{Conf}(\partial M)$ and in the right hand
side we use the standard action of conformal isometries on tractor valued differential forms.
\end{thm}
\betagin{proof}
Suppose that $g_1,g_2\in\mathcal G$ are hyperbolic on $U$. Then we can apply
Proposition 7.4 of \cite{FeffGr} (see also the discussion of the proof of this
result in \cite{FeffGr}) to their restrictions to $U$. This implies that there is a
neighborhood $V$ of $U\cap\partial M$ in $U$ and a diffeomorphism $\Psii:V\to V$
which restricts to the identity on $U\cap\partial M$ such that
$g_1|_{V}=\Psii^*(g_2|_V)$. Theorem \ref{thm3.5} and Corollary \ref{cor3.7} then
immediately imply that $c(g_1,h)$ and $c(g_2,h)$ coincide on $U\cap\partial M$. It
is then clear that we obtain the map $c$ as claimed.
The equivariancy of $c$ can be proved locally. So we take $\Phii\in\circperatorname{Diff}_{\mathcal
G}(\circverline{M})$ and let $\Phii_\infty$ be its restriction to the boundary. Given
$x\in\partial M$ we find an open neighborhood $U$ of $x$ in $\circverline{M}$ and a metric
$g\in\mathcal G$ such that $g|_U$ is hyperbolic. Now $\Phii^{-1}(U)$ is an open
neighborhood of $\Phii^{-1}(x)$ in $\circverline{M}$ and $\Phii^*g|_{\Phii^{-1}(U)}$ is
hyperbolic on $\Phii^{-1}(U)$. Thus we can compute $c(\Phii^*h)$ as
$c(\Phii^*g,\Phii^*h)$ on $\Phii^{-1}(U)$, and by Proposition \ref{prop3.3} this
coincides with $(\Phii_\infty)^*(c(g,h)|_U)$. Since $c(g,h)|_U=c(h)|_U$, this implies the
claim.
\end{proof}
Suppose that $x\in\partial M$ and $U$ is an open neighborhood of $x$
in $\circverline{M}$ such that $\mathcal G$ contains a metric $g$ which is
hyperbolic on $U$. Then of course the conformal class $[\mathcal G]$ has
to be flat on $U\cap\partial M$. In particular, the assumptions of
Theorem \ref{thm3.8} imply that $(\partial M,[\mathcal G])$ is conformally
flat, which in turn imposes restrictions on $\partial M$. However, if
we are given a manifold $\circverline{M}$ with boundary $\partial M$ and a flat
conformal structure on $\partial M$, then there always is a class $\mathcal G$
of conformally compact metrics on $M$, for which the assumptions of
Theorem \ref{thm3.8} are satisfied, and hence we obtain an invariant for
single metrics in $\mathcal G$.
Indeed, Proposition 7.2 of \cite{FeffGr} (see also the discussion on
p.\ 72 of that reference) shows that there is a hyperbolic metric $g$
on some open neighborhood of $\partial M$ in $\circverline{M}$ which induces the
given boundary structure. Then of course $g$ determines an equivalence
class $\mathcal G$ of conformally compact ALH-metrics on $M$ for which all
the assumptions of Theorem \ref{thm3.8} are satisfied.
We want to point out that it is not clear whether the condition of
conformal flatness in Theorem \ref{thm3.8} is of a fundamental
nature. What one would need in more general situations is a class of
``model metrics'' in $\mathcal G$ which can be characterized well enough
to obtain ``uniqueness up to diffeomorphism'' in a form as used in the
proof of Theorem \ref{thm3.8}. An obvious idea is to assume that $\mathcal
G$ contains at least one Einstein metric (which is a condition that is
stable under diffeomorphism) and then look at appropriate classes of
Einstein metrics in $\mathcal G$. In general, the Einstein condition is
certainly not enough to pin down a metric up to diffeomorphism,
compare with the non-uniqueness issues for the ambient
metric. However, it is well possible that there are situations in
which additional (geometric) conditions can be imposed to ensure
uniqueness.
| 1,758 | 45,932 |
en
|
train
|
0.50.14
|
\subsection{Recovering mass}\lambdabel{3.9}
We now show that in the special case of hyperbolic space that, by an
integration process of our cocycles, we can recover the mass for
asymptotically hyperbolic metrics as introduced by Wang \cite{Wang}
and Chru\'{s}iel-Herzlich \cite{Chrusciel-Herzlich}. To do this, we
specialize to the case that $\circverline{M}$ is an open neighbourhood of the
boundary $S^{n-1}$ in the closed unit ball and that $\mathcal G$ is the
equivalence class of (the restriction to $M$ of) the Poincar\'e metric
which we denote by $g$ here. This of course implies that $[\mathcal G]$ is
the round conformal structure on $S^{n-1}$ and that $\mathcal G$ satisfies
the conditions of Theorem \ref{thm3.8}. Hence we get a map $c:\mathcal
G\to \Omega^{n-1}(S^{n-1},\mathcal TS^{n-1})$ as described there.
If $n\geq
4$, conformal flatness of the round metric on $S^{n-1}$ implies that
the tractor connection $\nabla^{\mathcal T}$ is flat. Moreover, since
$(\partial M,[\mathcal G])$ is the homogeneous model of conformal
structures, the tractor bundle $\mathcal T\partial M$ admits a global
trivialisation by parallel sections. This extends to the case $n=3$
with the tractor connection on $S^2$ constructed as discussed in
Section \ref{2.5}. Indeed, since $g$ is conformally flat and Einstein,
the ambient tractor connection is flat and the scale tractor $I^A$ is
parallel on all of $\circverline{M}$. By Proposition \ref{2.5}~ $\partial M$ is
totally umbilic in $\circverline{M}$, and so the second fundamental form with
respect to any metric conformal to $g$ is pure trace. Using a scale
with vanishing mean curvature, as in Section \ref{2.5}, the second
fundamental form actually vanishes. Hence the ambient Levi Civita
connection restricts to the Levi Civita connection on the boundary and
by definition we use the restriction of the ambient Schouten tensor in
the construction of the tractor connection on the boundary in Section
\ref{2.5}. Hence formula \eqref{trac-conn} directly implies that the
boundary tractor connection coincides with the restriction of the
ambient tractor connection, so it is flat since the normal tractor is
parallel. The trivialisation by parallel sections then works exactly
as in higher dimensions.
This easily implies that, fixing an orientation on $\partial M$, there
is a well defined integral that associates to each form
$\circmega\in\Omega^{n-1}(\partial M, \mathcal T \partial M)$ a parallel section of
$\mathcal T\partial M$. Indeed, on $\partial M$ we can take a global frame
$\{s_i\}$ of $\mathcal T \partial M$ consisting of parallel sections,
expand $\circmega$ as $\sum_i\circmega_is_i$ with $\circmega_i\in\Omega^{n-1}(\partial M)$
and then define $\int_{\partial M}\circmega:=\sum_i(\int_{\partial M}\circmega_i)
s_i$. Of course, any other parallel frame consists of linear
combinations of the $s_i$ with constant coefficients, so the result is
independent of the choice of parallel frame.
Now the boundary tractor metric induces a tensorial map $\Omega^{n-1}(\partial
M,\mathcal T \partial M)\times\Gamma(\mathcal T \partial M )|\to\Omega^{n-1}(\partial M)$,
which we write as $(\circmega,s)\mapsto\lambdangle \circmega,s\rangle$. Observe that
the definition of the integral readily implies that for any parallel
section $s\in\Gamma(\mathcal T \partial M)$ on $\partial M$, we obtain $\lambdangle
\int_{\partial M}\circmega,s\rangle=\int_{\partial
M}\lambdangle\circmega,s\rangle$. In particular, the coefficients of
$\int_{\partial M}\circmega$ with respect to a parallel frame can be
computed as an ordinary integral over an $(n-1)$-form. Given a metric
$h\in\mathcal G$, we can in particular apply this to $c(h)$ and we will to
show that, after appropriate normalization, $\int_{\partial M}c(h)$
recovers the mass of $h$.
We will work on $\circverline{M}$ with the extension of the conformal class of
$g$, which we again denote by $[g]$. Since $g$ is conformally flat,
this leads to a flat tractor connection and $[g]$ restricts to $[\mathcal
G]$ on $\partial M$. Hence any parallel section $s\in\Gamma(\mathcal
T\partial M)$ can be extended to a parallel section of $\mathcal T\circverline{M}$
on a neighborhood of $\partial M$. (In fact, also $\mathcal T\circverline{M}$ is
globally trivialized by parallel sections, but we don't really need
this here.) It is well known that parallel sections of the standard
tractor bundle always are in the image of the tractor
$D$-operator. Denoting by $\sigma\in\Gamma(\mathcal E[1])$ the density
determined by $g$, $D^A(\sigma)$ is parallel since $g$ is Einstein. We
know from Proposition \ref{prop2.6}, that this parallel tractor spans
the tractor normal bundle along the boundary.
On the other hand, on $M$, $\sigma$ is nowhere vanishing. Thus on $M$, we can write any
parallel section of $\mathcal T\circverline{M}$ as $D^A(V\sigma)$ for some smooth function $V:M\to\mathbb
R$. Now $\nabla^{\mathcal T}_aD^A(V\sigma)=0$ is a differential equation on $V$ that can be
easily computed from formulae \eqref{D-def} and \eqref{trac-conn} in the scale $g$,
and using that $\mbox{\textsf{P}}_{ab}$ is a multiple of $g_{ab}$. This shows that
$\nabla_a\nabla_b V$ must have vanishing trace-free part, and it is well known that,
since $D^A(\sigma)$ is parallel, this is also sufficient for $D^A(V\sigma)$ being
parallel (see e.g. \cite{BEG}).
To obtain parallel sections that lead to boundary tractors along
$\partial M$, we can require in addition that $D^A(V\sigma)$ is orthogonal to
$D^A(\sigma)$. One immediately verifies that this condition is equivalent to requiring
that $V$, in addition, satisfies $\Delta V=-2\mbox{\textsf{P}} V=nV$. The two required properties
then can be equivalently encoded as a single equation, the KID (``Killing initial
data'') equation
\betagin{equation}\lambdabel{KID}
\nabla_a\nabla_b V-g_{ab}\Delta V+(n-1)g_{ab}V=0.
\end{equation}
Hence we see that, via $\tfrac1n D^A(V\sigma)$ on $M$, solutions to this
equations parametrise those parallel tractors on $\circverline{M}$, which lie in
$\mathcal T\partial M$ along $\partial M$. But on the other hand,
solutions to this equation parametrise the mass integrals in the
classical approach to the AH version of mass, which was our original
motivation for looking for a tractor description.
\betagin{thm}\lambdabel{thm3.9}
Let $\circverline{M}$ be an open neighborhood of $\partial M=S^{n-1}$ in the closed unit ball,
let $\mathcal G$ be the equivalence class of the restriction of the hyperbolic
(Poincar\'e) metric $g$ to $M$. For a metric $h\in\mathcal G$ consider
$c(h)\in\Omega^{n-1}(\partial M,\mathcal T\partial M)$ as in Theorem \ref{thm3.8}. For a
solution $V$ of the KID equation \eqref{KID}, let $s_V\in\Gamma(\mathcal T\partial M)$ be
the parallel section obtained as the boundary value of $\frac{1}{n}
D^A(V\sigma)\in\Gamma(\mathcal T\circverline{M})$ (with respect to $[g]$).
Then $\lambdangle\int_{\partial M} c(h),s_V\rangle$ coincides with the mass integral
associated to $V$ in \cite{Chrusciel-Herzlich}.
\end{thm}
\betagin{proof}
Using the conformal class $[g]$ on $\circverline{M}$ we have constructed $c(h)=c(g,h)$ as the
boundary value of $\star_g\alpha$ for a certain $\mathcal T\circverline{M}$-valued one-form $\alpha$ on
$M$. Likewise, for a solution $V$ of \eqref{KID}, the parallel boundary tractor $s_V$
is the restriction to $\partial M$ of a parallel section $\tilde s_V$ of $\mathcal
T\circverline{M}$. As we have noted already, $\lambdangle\int_{\partial M}
c(h),s_V\rangle=\int_{\partial M}\lambdangle c(h),s_V\rangle$. This integrand is the
boundary value of $\lambdangle\star_g\alpha,\tilde s_V\rangle$, which equals
$\star_g\lambdangle\alpha,\tilde s_V\rangle$ by definition. Since this form is smooth up to
the boundary, its integral over $\partial M$ equals the limit as $\epsilon\to 0$ of the
integrals over the level sets $S_{\epsilon}=\{x:\rho(x)=\epsilon\}$. Since the mass integral
associated to $V$ is also expressed via a one-form, it suffices to compare that
one-form to $\lambdangle\alpha,\tilde s_V\rangle$. In this comparison, we may work up to
terms that vanish along the boundary after application of $\star_g$ and hence up to
$\mathcal O(\rho^{n-1})$, cf.\ the proof of Proposition \ref{prop3.1}.
We use the description of the mass integral associated to $V$ from
\cite{Michel}, it is shown in that reference that this agrees with the
original mass integral introduces in \cite{Chrusciel-Herzlich}. The
mass integrand associated to a solution $V$ of the KID equation
\eqref{KID} is given by
$$
V(\nabla^i\lambdambda_{ia}-\nabla_a\circperatorname{tr}(\lambdambda))-g^{ij}\lambdambda_{ia}\nabla_jV+
\circperatorname{tr}(\lambdambda)\nabla_aV,
$$
where $\lambda_{ij}=h_{ij}-g_{ij}$, the hyperbolic metric $g$ is used to raise and lower
indices and to form traces, and $\nabla$ is the Levi-Civita connection of $g$.
Decomposing $\lambdambda_{ij}=\lambdambda^0_{ij}+\frac{1}{n}\circperatorname{tr}(\lambdambda)g_{ij}$, this becomes
\betagin{equation}\lambdabel{Michel}
(V\nabla^i\lambdambda^0_{ia}-\lambdambda^0_{ia}\nabla^iV)+
\tfrac{n-1}n(\circperatorname{tr}(\lambdambda)\nabla_aV-V\nabla_a\circperatorname{tr}(\lambdambda)).
\end{equation}
Choosing a defining function $\rho$ adapted to $g_{ij}$, we then obtain
$\lambda^0_{ij}=\rho^{n-2}\mu^0_{ij}$ and $\circperatorname{tr}(\lambda)=\rho^n\mu$ for the quantities
introduced in and below equation \eqref{h-g} (for $N=n$). Note that $\mu^0_{ij}$ and
$\mu$ are smooth up to the boundary. We also know from above that $\sigma V$ is the
projection of the parallel tractor $\tilde s_V=\frac{1}{n}D(\sigma V)$, so this is
smooth up to the boundary. Since $\sigma$ is a defining density for $\partial M$, it
follows that $\rho V$ is smooth up to the boundary.
Now we analyse the two parts of \eqref{Michel} separately, starting with the part
involving $\circperatorname{tr}(\lambda)$. Here we have the advantage that covariant derivatives are only
applied to smooth functions (and not to tensor fields), so the fact that $\nabla$ is
not smooth up to the boundary does not matter. Since $\rho V$ is smooth up to the
boundary,
$\rho\nabla_a\rho V$ is $\mathcal
O(\rho)$. Writing $\rho_a$ for $d\rho$ as before, we compute this as $\rho_a\rho
V+\rho^2\nabla_aV$. Thus $\rho^2\nabla_aV=-\rho_a\rho V+\mathcal O(\rho)$ and in
particular is smooth up to the boundary. Using this, we obtain
$$
\circperatorname{tr}(\lambdambda)\nabla_aV=\rho^n\mu\nabla_aV=-\rho^{n-2}\rho_a\rho V\mu +\mathcal
O(\rho^{n-1}).
$$
Similarly, $V\nabla_a\circperatorname{tr}(\lambda)=V\nabla_a\rho^n\mu=n\rho^{n-2}(\rho
V)\rho_a\mu+O(\rho^{n-1})$. Hence the second part in \eqref{Michel} simply gives
$-\frac{n^2-1}{n}\rho^{n-2}(V\rho)\rho_a\mu+O(\rho^{n-1})$.
Analyzing the second summand in the first part of \eqref{Michel} is similarly
easy. This writes as
$$ -\rho^{-2} g^{ij}\rho^n\mu^0_{ia}\nabla_jV=\rho^{n-2} \rho^{-2}
g^{ij}\rho_j\mu^0_{ia}(\rho V)+\mathcal O(\rho^{n-1}).
$$
| 3,509 | 45,932 |
en
|
train
|
0.50.15
|
Then $\lambdangle\int_{\partial M} c(h),s_V\rangle$ coincides with the mass integral
associated to $V$ in \cite{Chrusciel-Herzlich}.
\end{thm}
\betagin{proof}
Using the conformal class $[g]$ on $\circverline{M}$ we have constructed $c(h)=c(g,h)$ as the
boundary value of $\star_g\alpha$ for a certain $\mathcal T\circverline{M}$-valued one-form $\alpha$ on
$M$. Likewise, for a solution $V$ of \eqref{KID}, the parallel boundary tractor $s_V$
is the restriction to $\partial M$ of a parallel section $\tilde s_V$ of $\mathcal
T\circverline{M}$. As we have noted already, $\lambdangle\int_{\partial M}
c(h),s_V\rangle=\int_{\partial M}\lambdangle c(h),s_V\rangle$. This integrand is the
boundary value of $\lambdangle\star_g\alpha,\tilde s_V\rangle$, which equals
$\star_g\lambdangle\alpha,\tilde s_V\rangle$ by definition. Since this form is smooth up to
the boundary, its integral over $\partial M$ equals the limit as $\epsilon\to 0$ of the
integrals over the level sets $S_{\epsilon}=\{x:\rho(x)=\epsilon\}$. Since the mass integral
associated to $V$ is also expressed via a one-form, it suffices to compare that
one-form to $\lambdangle\alpha,\tilde s_V\rangle$. In this comparison, we may work up to
terms that vanish along the boundary after application of $\star_g$ and hence up to
$\mathcal O(\rho^{n-1})$, cf.\ the proof of Proposition \ref{prop3.1}.
We use the description of the mass integral associated to $V$ from
\cite{Michel}, it is shown in that reference that this agrees with the
original mass integral introduces in \cite{Chrusciel-Herzlich}. The
mass integrand associated to a solution $V$ of the KID equation
\eqref{KID} is given by
$$
V(\nabla^i\lambdambda_{ia}-\nabla_a\circperatorname{tr}(\lambdambda))-g^{ij}\lambdambda_{ia}\nabla_jV+
\circperatorname{tr}(\lambdambda)\nabla_aV,
$$
where $\lambda_{ij}=h_{ij}-g_{ij}$, the hyperbolic metric $g$ is used to raise and lower
indices and to form traces, and $\nabla$ is the Levi-Civita connection of $g$.
Decomposing $\lambdambda_{ij}=\lambdambda^0_{ij}+\frac{1}{n}\circperatorname{tr}(\lambdambda)g_{ij}$, this becomes
\betagin{equation}\lambdabel{Michel}
(V\nabla^i\lambdambda^0_{ia}-\lambdambda^0_{ia}\nabla^iV)+
\tfrac{n-1}n(\circperatorname{tr}(\lambdambda)\nabla_aV-V\nabla_a\circperatorname{tr}(\lambdambda)).
\end{equation}
Choosing a defining function $\rho$ adapted to $g_{ij}$, we then obtain
$\lambda^0_{ij}=\rho^{n-2}\mu^0_{ij}$ and $\circperatorname{tr}(\lambda)=\rho^n\mu$ for the quantities
introduced in and below equation \eqref{h-g} (for $N=n$). Note that $\mu^0_{ij}$ and
$\mu$ are smooth up to the boundary. We also know from above that $\sigma V$ is the
projection of the parallel tractor $\tilde s_V=\frac{1}{n}D(\sigma V)$, so this is
smooth up to the boundary. Since $\sigma$ is a defining density for $\partial M$, it
follows that $\rho V$ is smooth up to the boundary.
Now we analyse the two parts of \eqref{Michel} separately, starting with the part
involving $\circperatorname{tr}(\lambda)$. Here we have the advantage that covariant derivatives are only
applied to smooth functions (and not to tensor fields), so the fact that $\nabla$ is
not smooth up to the boundary does not matter. Since $\rho V$ is smooth up to the
boundary,
$\rho\nabla_a\rho V$ is $\mathcal
O(\rho)$. Writing $\rho_a$ for $d\rho$ as before, we compute this as $\rho_a\rho
V+\rho^2\nabla_aV$. Thus $\rho^2\nabla_aV=-\rho_a\rho V+\mathcal O(\rho)$ and in
particular is smooth up to the boundary. Using this, we obtain
$$
\circperatorname{tr}(\lambdambda)\nabla_aV=\rho^n\mu\nabla_aV=-\rho^{n-2}\rho_a\rho V\mu +\mathcal
O(\rho^{n-1}).
$$
Similarly, $V\nabla_a\circperatorname{tr}(\lambda)=V\nabla_a\rho^n\mu=n\rho^{n-2}(\rho
V)\rho_a\mu+O(\rho^{n-1})$. Hence the second part in \eqref{Michel} simply gives
$-\frac{n^2-1}{n}\rho^{n-2}(V\rho)\rho_a\mu+O(\rho^{n-1})$.
Analyzing the second summand in the first part of \eqref{Michel} is similarly
easy. This writes as
$$ -\rho^{-2} g^{ij}\rho^n\mu^0_{ia}\nabla_jV=\rho^{n-2} \rho^{-2}
g^{ij}\rho_j\mu^0_{ia}(\rho V)+\mathcal O(\rho^{n-1}).
$$
For the first summand in \eqref{Michel}, the analysis is slightly more
complicated. This can be written
$V\rho\rho^{-2}g^{ij}\rho\nabla_i\rho^{n-2}\mu^0_{ja}$ and hence equals
\betagin{equation}\lambdabel{tech}
V\rho \rho^{-2}g^{ij}\rho^{n-2}((n-2)\rho_i\mu^0_{ja}+\rho\nabla_i\mu^0_{ja}).
\end{equation}
Since in the last summand, we apply a covariant derivative to a tensor
field, we have to change to a connection that admits a smooth
extension to the boundary in order to analyse the boundary
behavior. Hence we change from the Levi-Civita connection $\nabla$ of
$g_{ij}$ to the Levi-Civita connection $\bar\nabla$ of $\bar
g_{ij}:=\rho^2g_{ij}$, which has this property. For the usual
conventions, as used in \cite{BEG}, the one form $\Upsilon_a$ associated to
this conformal change is given by $\Upsilon_a=\tfrac{\rho_a}{\rho}$. The
relevant formula for the change of connection is then given by
$$
\nabla_i\mu^0_{ja}=\bar\nabla_i\mu^0_{ja}+2\Upsilon_i\mu^0_{ja}+\Upsilon_j\mu^0_{ia}+
\Upsilon_a\mu^0_{ij}-\Upsilon_k\bar g^{k\ell}\mu^0_{\ell a}\bar g_{ij}-\Upsilon_k\bar
g^{k\ell}\mu^0_{j\ell}\bar g_{ia},
$$
This immediately shows that $\rho\nabla_i\mu^0_{ja}$ admits a
smooth extension to the boundary and its boundary value can be
obtained by dropping the first summand in the right hand side of this
formula and replacing each occurrence of $\Upsilon$ in the remaining terms
by $d\rho$, so $\Upsilon_i$ becomes $\rho_i$ and so on. Inserting this back
into \eqref{tech}, we get a contraction with $\bar g^{ij}$. This kills
the term involving $\mu^0_{ij}$ by trace-freeness, while all other
terms become multiples of $\bar g^{ij}\rho_i\mu^0_{ja}$. The factors
of the individual terms are $2$, $1$, $-n$, and $-1$, respectively, so
we'll get a total contribution of $(2-n)\bar
g^{ij}\rho_i\mu^0_{ja}$. This actually implies that \eqref{tech} is
$\mathcal O(\rho^{n-1})$. Hence we finally conclude that \eqref{Michel}
equals
\betagin{equation}\lambdabel{Michel-asymp}
\rho^{n-2}(\rho V)\left(\rho^{-2}g^{ij}\rho_j\mu^0_{ia}-
\tfrac{n^2-1}n\rho_a\mu\right)+O(\rho^{n-1}).
\end{equation}
Now recall the formula for $\nabla_bD^A(\tau-\sigma)$ from part (1) of Proposition
\ref{prop3.1}, taking into account the definition of $\mathbf{X}^A$. This shows that,
up to $\mathcal O(\rho^{n-1})$, $\nabla_bD^A(\tau-\sigma)$ is given by inserting
$\tfrac{n^2-1}2\rho^{n-2}\rho_b\mu\rho\sigma^{-1}$ into the bottom slot of a
tractor. Pairing this with $\frac{1}{n}D^B(\sigma V)$ using the tractor metric, we
simply simply obtain the product of $\sigma V$ with this bottom slot, i.e.\
$$
\tfrac{n^2-1}2\rho^{n-2}\rho_b\mu(\rho V)+\mathcal O(\rho^{n-1}).
$$
Analyzing the formula for $S(\sigma(h_{ij}-g_{ij})^0)$ from part (1) of Proposition
\ref{prop3.2} we similarly see that the pairing of this with $\frac{1}{n}D^B(\sigma V)$
gives
$$
-\rho^{n-2}\rho^{-2}g^{ij}\rho_j\mu^0_{ia}(\rho V)+\mathcal O(\rho^{n-1}).
$$
But this exactly tells us that, up to $\mathcal O(\rho^{n-1})$, \eqref{Michel-asymp}
equals $\lambdangle\alpha,\tilde s_V\rangle$, where $\alpha$ corresponds to
$c=-\tfrac{2}nc_1-c_2$, which is one of the cocycles identified in Theorem
\ref{thm3.8}.
\end{proof}
| 2,602 | 45,932 |
en
|
train
|
0.50.16
|
\betagin{thebibliography}{10}
\bibitem{BEG} T.N. Bailey, M.G. Eastwood, and A.R. Gover.
\newblock Thomas's structure bundle for conformal, projective and related
structures.
\newblock Rocky Mountain J. \textbf{24} (1994), 1191--1217.
\bibitem{Bartnik} R.\ Bartnik.
\newblock The mass of an asymptotically flat manifold.
\newblock Comm. Pure Appl. Math. \textbf{39} (1986), no.\ 5, 661--693.
\bibitem{Belgun} F.\ Belgun
\newblock Geodesics and submanifold structures in conformal geometry.
\newblock J.\ Geom.\ Phys.\ \textbf{91} (2015) 172--191.
\bibitem{Calderbank:Moebius}
D.M.J.\ Calderbank
\newblock M\"obius structures and two-dimensional Einstein-Weyl geometry.
\newblock J.\ Reine Angew.\ Math.\ \textbf{504} (1998), 37--53.
\bibitem{CG-TAMS} A.\ \v Cap, and A.R.\ Gover.
\newblock Tractor calculi for parabolic geometries.
\newblock Trans.\ Amer.\ Math.\ Soc., {\bf 354} (2002), 1511-1548.
\bibitem{Proj-comp} A.\ \v Cap, and A.R.\ Gover.
\newblock Projective compactifications and Einstein metrics.
\newblock J.\ Reine Angew.\ Math. \textbf{717} (2016) 47-75.
\bibitem{book}
A.~{\v{C}}ap, and J.~Slov{\'{a}}k.
\newblock {\em Parabolic Geometries I: Background and General Theory}.
\newblock Mathematical Surveys and Monographs 154. American Mathematical Society,
Providence, RI, 2009.
\bibitem{Chrusciel-Herzlich}
P.T.\ Chru\'sciel, and M.\ Herzlich.
\newblock The mass of asymptotically hyperbolic Riemannian manifolds.
\newblock Pacific J.\ Math.\ \textbf{212}, no.\ 2 (2003) 231--264.
\bibitem{CDG}
J.\ Cortier, M.\ Dahl, and R.\ Gicquad,
\newblock Mass-like invariants for asymptotically hyperbolic metrics.
\newblock preprint arXiv:1603.07952
\bibitem{Curry-Gover} S.N.\ Curry, and A.R.~Gover.
\newblock An introduction to conformal geometry and tractor calculus, with a view to
applications in general relativity.
\newblock in T.\ Daud\'e, D.\ H\"afner, J.-P.\ Nicolas (eds.), ``Asymptotic analysis
in general relativity'', Cambridge Univ. Press, 2018, 86--170.
\bibitem{Eastwood:Moebius} M.G.\ Eastwood
\newblock MR review of \cite{Calderbank:Moebius}.
\newblock available online at https://mathscinet.ams.org as MR1656822.
\bibitem{Epstein} C. Epstein
\newblock An asymptotic volume formula for convex cocompact hyperbolic manifolds.
\newblock Appendix A in S.J.\ Patterson, P.A.\ Perry: \textit{The divisor of
Selberg's zeta function for Kleinian groups},
\newblock Duke Math.\ J.\ \textbf{106}, no.\ 2 (2001), 321--390.
\bibitem{FeffGr} C.\ Fefferman, and C.R.\ Graham.
\newblock {\em The Ambient Metric.}
\newblock Annals of Mathematics Studies, \textbf{178}. Princeton University Press 2012.
\bibitem{G-sigma} A.R.~Gover.
\newblock Conformal Dirichlet-Neumann Maps and Poincar\'e-Einstein Manifolds.
\newblock SIGMA (Symmetry, Integrability
and Geometry: Methods and Applications) {\bf 3} 100, (2007) 21 pages
\bibitem{Gover:P-E} A.R.~Gover.
\newblock Almost Einstein and Poincar\'e-{E}instein manifolds in
Riemannian signature.
\newblock J.\ Geom.\ Phys. \textbf{60} (2010) 182--204.
\bibitem{Graham:Srni} C.R.\ Graham.
\newblock Volume and area renormalizations for conformally compact Einstein
metrics.
\newblock Rend.\ Circ.\ Mat.\ Palermo Suppl.\ No.\ \textbf{63}
(2000), 31--42.
\bibitem{LP} J.\ M.\ Lee, and T.\ H.\ Parker.
\newblock The Yamabe problem.
\newblock Bull.\ Amer.\ Math.\ Soc.\ (N.S.) \textbf{17}, no.\ 1 (1987) 37--91.
\bibitem{Michel} B.\ Michel.
\newblock Geometric invariance of mass-like asymptotic invariants.
\newblock J.\ Math.\ Phys.\ \textbf{52}, no.\ 5 (2011) 052504, 14 pp.
\bibitem{S-Yau} R.\ Schoen, and S.\ T.\ Yau.
\newblock On the proof of the positive mass conjecture in general relativity.
\newblock Comm.\ Math.\ Phys.\ \textbf{65}, no.\ 1 (1979), 45--76.
\bibitem{SkenSol} K.\ Skenderis, and S.N.\ Solodukin
\newblock Quantum effective action from the AdS/CFT correspondence.
\newblock Phys.\ Lett.\ \textbf{B 472} (2000), 316--322.
\bibitem{Wang} X. Wang.
\newblock Mass for asymptotically hyperbolic manifolds.
\newblock J.\ Differential Geom., \textbf{57} (2001), 273--299.
\end{thebibliography}
\end{document}
| 1,743 | 45,932 |
en
|
train
|
0.51.0
|
\textcolor{blue}egin{eqnarray}gin{document}
\title{Density analysis of BSDEs}
\textcolor{blue}egin{eqnarray}gin{abstract}
In this paper, we study the existence of densities (with respect to the Lebesgue measure) for marginal laws of the solution $(Y,Z)$ to a quadratic growth BSDE. Using the (by now) well-established connection between these equations and their associated semi-linear PDEs, together with the Nourdin-Viens formula, we provide estimates on these densities.
\end{abstract}
{\noindentindent \textit{Key words:} BSDEs; Malliavin Calculus; Density analysis; Nourdin-Viens' Formula; PDEs.
}
\noindentindent
{\noindentindent \textit{AMS 2010 subject classification:} Primary: 60H10; Secondary: 60H07.
\noindentrmalsize
}
\tableofcontents
\section{Introduction}
In recent years the field of Backward Stochastic Differential Equations (BSDEs) has been a subject of growing interest in stochastic calculus, as these equations naturally arise in stochastic control problems in Finance, and as they provide Feynman-Kac type formulae for semi-linear PDEs (\cite{PardouxPeng_92}). Before going further let us recall that a solution to a BSDE is a pair of \textit{regular enough} (in a sense to be made precise) predictable processes $(Y,Z)$ such that
\textcolor{blue}egin{eqnarray}gin{equation}
\label{eq:BSDEintro}
Y_t=\timesi+\int_t^T h(s,Y_s,Z_s)ds -\int_t^T Z_s dW_s, \quad t\in [0,T],
\end{equation}
where $W$ is a one-dimensional Brownian motion, $h$ is a predictable process and $\timesi$ is a $\mathcal{F}_T$-measurable random variable (with $(\mathcal{F}_t)_{t \in [0,T]}$ the natural completed and right-continuous filtration generated by $W$). Since it is generally not possible to provide an explicit solution to \eqref{eq:BSDEintro}, except for instance when $h$ is a linear mapping of $(y,z)$, one of the main issues, especially regarding the applications is to provide a numerical analysis for the solution of a BSDE. This calls for a deep understanding of the regularity of the solution processes $Y$ and $Z$. The classical regularity related to obtaining a numerical scheme for the solution $(Y,Z)$ is the so-called \textit{path regularity} for the $Z$ component originally studied in \cite{Ma_Zhang_PTRF02}. In this paper, we aim at studying another type of regularity namely, we focus on the law of the marginals of the random variables $Y_t$, $Z_t$ at a given time $t$ in $(0,T)$. More precisely, we are interested in providing sufficient conditions which ensure the existence of a density (with respect to the Lebesgue measure) for these marginals on the one hand, and in deriving some estimates on these densities on the other hand. This type of information on the solution is of theoretical and of practical interest since the description of the tails of the (possible) density of $Z_t$ would provide more accurate estimates on the convergence rates of numerical schemes for quadratic growth BSDEs (qgBSDEs in short), that is when $h$ in \eqref{eq:BSDEintro} has quadratic growth in the $z$-variable, as noted in \cite{DosReisPhD}.
Before reviewing the results available in the literature and the one we derive in this paper, we would like to illustrate with the two following simple examples that the existence and the estimate of densities issues for BSDEs are very different from the one concerning the classical (forward) SDEs. For instance consider the following very particular case of \eqref{eq:BSDEintro} given by:
\textcolor{blue}egin{eqnarray}gin{equation}
\label{eq:BSDEintrobis}
Y_t=W_1+\int_t^T (s-W_s) ds -\int_t^T Z_s dW_s, \quad t\in [0,1], \; (T=1).
\end{equation}
This equation should be extremely simple in the sense that the driver $h$ does not depend on $(Y,Z)$, and indeed it can be solved explicitly to get that:
$$ Y_t = W_t \left(-\frac12 + 2t -\frac{t^2}{2}\right), \quad t\in [0,1]. $$
Hence $Y_t$ is a Gaussian random variable for every time $t$ in $(0,2-\sqrt{3})$, then $Y_{2-\sqrt{3}} = 0$, and for $t$ in $(2-\sqrt{3},1]$, $Y_t$ is Gaussian distributed once again. This illustrates the difficulty of the problem and somehow shows how it is different from the study of forward SDEs. This example, even though it is very simple is pretty insightful and will be studied as Example \ref{exemple} in Section \ref{section:lip}.
Concerning the density estimates, the backward case brings here also, significant differences with the forward case as the following example illustrates. Consider the following equation:
\textcolor{blue}egin{eqnarray}gin{equation}
\label{eq:BSDEintroter}
Y_t=W_1^3+\int_t^T 3 W_s ds -\int_t^T Z_s dW_s, \quad t\in [0,1], \; (T=1),
\end{equation}
which can be solved explicitly:
$$ Y_t = W_t^3 +6 W_t(1-t), \quad Z_t = 3 W_t^2 + 6(1-t), \; t\in [0,1],$$
from which we deduce that both $Y_t$ and $Z_t$ admits a density with respect to the Lebesgue's measure for $t$ in $(0,1]$. However, it is clear that neither the law of $Y_t$ nor the one of $Z_t$ admits Gaussian tails. This example will be considered in Section \ref{section:densY} as Example \ref{rem.toostringent}.
Coming back to the
general problem of existence of densities for the marginal laws of $Y$ and $Z$, it is worth mentioning that this issue has been pretty few studied in the literature, since up to our knowledge only references \cite{AntonelliKohatsu,AbouraBourguin} address this question. The first results about this problem have been derived in \cite{AntonelliKohatsu}, where the authors provide existence and smoothness properties of densities for the marginals of the $Y$ component only and when the driver $h$ is Lipschitz continuous in $(y,z)$. Note that two kinds of sufficient conditions for the existence of a density for $Y$ are derived in \cite{AntonelliKohatsu}: the so-called \textit{first-order} (cf. \cite[Theorem 3.1]{AntonelliKohatsu}) and \textit{second-order} (see \cite[Theorem 3.6]{AntonelliKohatsu}) conditions. Concerning the $Z$ component, much less is known since existence of a density for $Z$ has been established in \cite{AbouraBourguin} only under the condition that the driver is linear in $z$. This constitutes, to our point of view, a major restriction since up to a Girsanov transformation this case basically reduces to the situation where the driver does not depend on $z$. Nonetheless, in \cite{AbouraBourguin}, estimates on the densities of the laws of $Y_t$ and $Z_t$ are given using the Nourdin-Viens formula.
In this paper we revisit and extend the results of \cite{AntonelliKohatsu,AbouraBourguin} by providing sufficient conditions for the existence of densities for the marginal laws of the solution $Y_t,Z_t$ (with $t$ an arbitrary time in $(0,T)$) of a qgBSDE with a terminal condition $\timesi$ in \eqref{eq:BSDEintro} given as a deterministic mapping of the value at time $T$ of the solution to a one-dimensional SDE, together with some estimates on these densities. The results concerning the Lipschitz case, \textit{i.e.} when the generator $h$ is Lipschitz, are presented in Section \ref{section:lip}. As recalled above, the case where $h$ is Lipschitz continuous in $(y,z)$ has been investigated in \cite{AntonelliKohatsu} for the $Y$ component only, where the authors have derived two types of sufficient conditions. However, we provide as Example \ref{exemple} a counter-example to \cite[Theorem 3.6]{AntonelliKohatsu} which is devoted to the second-order conditions. This is due to an inefficiency in the proof that can be easily fixed by making a small change in a key quantity in the statement of the result. Hence, we propose a new version of this result as Theorem \ref{AKmodifie}. Then, we gather in Section \ref{section:lip:z} the first existence results of a density for the $Z$ component for Lipschitz BSDEs. Concerning the quadratic case, studied in Section \ref{section:quadratic}, we propose sufficient conditions for the existence of a density first for the $Y$ component of qgBSDEs (in Section \ref{section:quadratic:y}), then for the $Z$ component of qgBSDEs (in Section \ref{section:quadratic:z}). We would like to stress once more at this stage that concerning the existence of a density for the $Y$ component, only the Lipschitz case was known and concerning the control variable $Z$, only the case of linear drivers in $z$ was studied (see \cite[Theorem 4.3]{AbouraBourguin}) up to now, which makes our result a major improvement on the existing literature. Finally, we derive in Section \ref{section:densY}, density estimates for the marginal laws of $Y$ and $Z$ using the Nourdin-Viens formula, and taking advantage of the connection between the solution to a Markovian BSDE and the solution to its associated semi-linear PDE. Note that contrary to \cite{AbouraBourguin}, we do not assume that the Malliavin derivative of $Y$ (or $Z$) to be bounded which is, from our point of view, a too stringent assumption (as illustrated in Example \ref{rem.toostringent}) both from the theoretical and practical point of view. Indeed, such an assumption leads to Gaussian tails for the densities of $Y$ or $Z$.
However, even in seemingly benign situations, we will see that it is not generally the case for BSDEs, and unlike most of the literature, we have obtained tail estimates which are {\it not} Gaussian. This might be seen as a significant difference between BSDEs and diffusive equations (i.e. with an initial condition) like SDEs or SPDEs for instance \cite{Kohatsu_03,Kohatsu_2003b,Nualart_Quer}.
\noindentindent
Before going further, we would like to explain why our results are quite relevant for financial applications and some stochastic control problems. Most of problems in portfolio management, utility maximization or risk sensitive control (see \textit{e.g.} \cite[Section 4.2]{elk_hamadene_matoussi}) can be essentially reduced to study a qgBSDE. Let us present two examples.
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[1.] Assume that a financial agent wants to maximize her utility under constraints, \textit{i.e.} her investment strategies are restricted to a specific closed set $C$, it was proved in \cite{elk_rouge} and \cite{HIM} that her optimal strategies are essentially given through the $Z$ component of a qgBSDE of the form
$$Y_t=\timesi +\int_t^T h(s,Z_s)ds -\int_t^T Z_s dW_s, \; \forall t\in [0,T]\; \mathbb{P}-a.s.$$
with
$$h(s,z):= -z\theta_s - \frac{|\theta_s|^2}{2\alpha}+\frac\alpha2 \text{dist}_{C}^2\left(z+\frac{\theta_s}{\alpha}\right),$$ where $\alpha$ denotes the risk aversion of the investor and $\theta$ is the market price of risk, and where $\text{dist}_C(x)$ denotes the Euclidean distance between $x$ and $C$. Hence, if one obtains a criterion providing density existence for the $Z$ component solution to a qgBSDE with estimates on its tails, then one gets crucial information to study the behaviors of optimal strategies for utility maximization problems. For example, since $Z$ essentially gives the optimal quantity of money which should be invested in the risky asset, being able to estimate the probability that $Z$ becomes large is particularly meaningful in risk management. Besides, the control of the tails of the density of $Z$ could give important information concerning the rate of convergence for numerical schemes to solve numerically BSDEs, so as to compute optimal strategies (see \cite{ImkellerDosreis, ChassagneuxRichou}). For instance, one can check directly that if $\theta$ above is deterministic, $C$ is smooth (that is its boundary is a $C^2$ Jordan arc), and $\timesi=g(W_T)$, where $g$ is any bounded function such that its second-order derivative is non-negative almost everywhere and positive on a set of positive Lebesgue measure (for instance a smoothed butterfly spread), then Theorem \ref{thm_density_z_quadra} below applies and $Z_t$ admits a density for all $t\in(0,T]$.
\item[2.] Assume now that a controller, sensitive to risk, wants to maximize on the control set $ \mathcal{U}$
\textcolor{blue}egin{eqnarray}gin{equation}\label{RSP} J(u):= \mathbb{E}^u\left[ \exp\left(\theta \int_0^T H(s,X_{\cdot},u_s)ds + g(X_T)\right)\right], \; u \in \mathcal{U},\end{equation}
where $\theta$ denotes the sensitiveness of the controller with respect to risk and $X$ denotes a solution to a classical SDE. This is the classical risk sensitive control problem introduced in \cite{jacobson}. Hence, this risk sensitive control problem can be rewritten in term of the well-known risk entropic measure (see \cite{barrieu_elk} for more details). Then, according to \cite[Theorem 4.3]{elk_hamadene_matoussi}, one can find a maximizer $u^\star$ of \eqref{RSP} which is essentially given by a process $Z^\star$ which is the second component of the solution to the following qgBSDE
$$ Y_t^\star= g(X_T)+\int_t^T h(s,x_{\cdot}, Z_s^\star, u^\star_s)+\frac12 |Z^\star_s|^2ds -\int_t^T Z_s^\star dW_s, \;\forall t\in [0,T], \; \mathbb{P}-a.s.,$$
where $h$ is the Hamiltonian process (which is given explicitly in terms of $H$), which is such that $z\longmapsto h(s,x_{\cdot}, z, u_s)+\frac12 |z|^2$ has a quadratic growth for every $s\in [0,T]$ and $u\in \mathcal{U}$. Again, our results give information on the density of $Z^\star$ and thus on the law of the optimal control which is important for obtaining qualitative properties of this optimal control as well as for numerical approximations.
\end{itemize}
| 3,839 | 50,703 |
en
|
train
|
0.51.1
|
\section{Preliminaries}
\subsection{General notations}
In this paper we fix $T \in (0,\infty)$. Let $W:=(W_t)_{t\in [0,T]}$ be a standard one-dimensional Brownian motion on a probability space $(\Omega,\mathcal{F},\mathbb{P})$, and we denote by $\mathbb{F}:=(\mathcal{F}_t)_{t\in [0,T]}$ the natural (completed and right-continuous) filtration generated by $W$. We denote by $\lambda$ the Lebesgue measure on $\mathbb{R}$ and we set for any $p\in [1,+\infty]$, $L^p(\mathbb{P}):=L^p(\Omega,\mathcal{F}_T,\mathbb{P})$ and denote by $\No{\cdot}_p$ the associated norm.
We denote by $\mathcal{C}_b(\mathbb{R}^n)$ ($n \geq 1$) the set of functions from $\mathbb{R}^n$ to $\mathbb{R}$ which are infinitely differentiable with bounded partial derivatives. Similarly, for any $n\geq 1$ and any $p\in\N^*$, we denote by $\mathcal C^p(\R^n)$ the set of functions $f:\R^n\rightarrow \R$ which are $p$-times continuously differentiable. For $f$ in $\mathcal{C}_b(\mathbb{R}^n)$, we set $f_{x_{i_1}\cdots x_{i_n}}$ the $n$-th partial derivative with respect to the variables $x_{i_1},\ldots,x_{i_k}$ with $i_1+\ldots+i_k=n$. For a differentiable mapping $f:\mathbb{R} \longrightarrow \mathbb{R}$, we denote $f'$ its derivative in place of $f_x$. Let us denote, for any $(p,q)\in\N^2$, by ${\cal C}^{p,q}$ the space of functions $f:[0,T]\times\R\rightarrow \R$ which are $p$-times differentiable in $t$ and $q$-times differentiable in space with partial derivatives continuous (in $(t,x)$).
Finally, we introduce the following norms and spaces for any $p\geq 1$. $\mathbb S^p$ is the space of $\mathbb R$-valued, continuous and $\mathbb F$-progressively measurable processes $Y$ s.t.
$$\No{Y}^p_{\mathbb S^p}:=\mathbb E\left[\underset{0\leq t\leq T}{\sup} |Y_t|^p\right]<+\infty.$$
$\mathbb S^\infty$ is the space of $\mathbb R$-valued, continuous and $\F$-progressively measurable processes $Y$ s.t.
$$\No{Y}_{\mathbb S^\infty}:=\underset{0\leq t\leq T}{\sup}\No{Y_t}_\infty<+\infty.$$
$\mathbb H^p$ is the space of $\mathbb R$-valued and $\F$-predictable processes $Z$ such that
$$\No{Z}^p_{\mathbb H^p}:=\mathbb E\left[\left(\int_0^T\abs{Z_t}^2dt\right)^{\frac p2}\right]<+\infty.$$
$\rm{BMO}$ is the space of square integrable, continuous, $\mathbb R$-valued martingales $M$ such that
$$\No{M}_{\rm{BMO}}:=\underset{\tau\in\mathcal T_0^T}{{\rm ess \, sup}}\No{\mathbb E_\tau\left[\left(M_T-M_{\tau}\right)^2\right]}_{\infty}<+\infty,$$
where for any $t\in[0,T]$, $\mathcal T_t^T$ is the set of $\F$-stopping times taking their values in $[t,T]$. Accordingly, $\mathbb H^2_{\rm{BMO}}$ is the space of $\mathbb R$-valued and $\F$-predictable processes $Z$ such that
$$\No{Z}^2_{\mathbb H^2_{\rm{BMO}}}:=\No{\int_0^.Z_sdB_s}_{\rm{BMO}}<+\infty.$$
\subsection{Elements of Malliavin calculus and density analysis}
In this section we introduce the basic material on the Malliavin calculus that we will use in this paper. Set $\mathfrak{H}:=L^2([0,T],\mathcal B([0,T]),\lambda)$, where $\mathcal B([0,T])$ is the Borel $\sigma$-algebra on $[0,T]$, and let us consider the following inner product on $\mathfrak{H}$
$$\langle f,g\rangle :=\int_0^Tf(t)g(t)dt, \quad \forall (f,g) \in \mathfrak{H}^2,$$
with associated norm $\No{\cdot}_{\mathfrak{H}}$. Let $\mathcal{S}$ be the set of cylindrical functionals, that is the set of random variables $F$ in $L^2(\mathbb{P})$ of the form
\textcolor{blue}egin{eqnarray}gin{equation}
\label{eq:cylindrical}
F=f(W_{t_1},\ldots,W_{t_n}), \quad (t_1,\ldots,t_n) \in [0,T]^n, \; f \in \mathcal{C}_b(\mathbb{R}^n), \; n\geq 1.
\end{equation}
For any $F$ in $\mathcal{S}$ of the form \eqref{eq:cylindrical}, the Malliavin derivative $D F$ of $F$ is defined as the following $\mathfrak{H}$-valued random variable:
\textcolor{blue}egin{eqnarray}gin{equation}
\label{eq:DF}
D F:=\sum_{i=1}^n f_{x_i}(W_{t_1},\ldots,W_{t_n}) \textbf{1}_{[0,t_i]}.
\end{equation}
It is then customary to identify $DF$ with the stochastic process $(D_t F)_{t\in [0,T]}$. Denote then by $\mathbb{D}^{1,2}$ the closure of $\mathcal{S}$ with respect to the Sobolev norm $\|\cdot\|_{1,2}$, defined as:
$$ \|F\|_{1,2}:=\mathbb{E}\left[|F|^2\right] + \mathbb{E}\left[\int_0^T |D_t F|^2 dt\right]. $$
In an iterative way, one may define $D^n F$ (for $n\geq 1$) as the following $\mathfrak{H}^{\odot n}$-valued random variable:
$$ D^nF:=D (D^{n-1} F), $$
where $\mathfrak{H}^{\odot n}$ denotes the $n$-times symmetric tensor product of $\mathfrak{H}$. We refer to \cite{Nualartbook} for more details.
We recall the following criterion for absolute continuity of the law of a random variable $F$ with respect to the Lebesgue measure.
\textcolor{blue}egin{eqnarray}gin{Theorem}[Bouleau-Hirsch, see e.g. Theorem 2.1.2 in \cite{Nualartbook}]\label{BH}
Let $F$ be in $\mathbb{D}^{1,2}$. Assume that $\|DF\|_{\mathfrak{H}} >0$, $\mathbb{P}-$a.s. Then $F$ has a probability distribution which is absolutely continuous with respect to the Lebesgue measure on $\mathbb{R}$.
\end{Theorem}
Let $F$ such that $\|DF\|_{\mathfrak{H}} >0$, $\mathbb{P}-$a.s., then the previous criterion implies that $F$ admits a density $\rho_F$ with respect to the Lebesgue measure. Assume there exists in addition a measurable mapping ${\mathfrak{H}at P}i_F$ with ${\mathfrak{H}at P}i_F : \mathbb{R}^{\mathfrak{H}} \rightarrow \mathfrak{H}$, such that $DF={\mathfrak{H}at P}i_F(W)$, then we set:
\textcolor{blue}egin{eqnarray}gin{equation}\label{gzt}
g_F(x):=\int_0^\infty e^{-u} \mathbb{E}\left[\mathbb{E}^*[\langle {\mathfrak{H}at P}i_F(W),\widetilde{{\mathfrak{H}at P}i_F^u}(W)\rangle_{\mathfrak{H}}] \Big{|} F-\mathbb{E}(F)=x\right] du, \ x\in \mathbb{R},
\end{equation}
where $\widetilde{{\mathfrak{H}at P}i_F^u}(W):={\mathfrak{H}at P}i_F(e^{-u}W+\sqrt{1-e^{-2u}}W^*)$ with $W^*$ an independent copy of $W$ defined on a probability space $(\Omega^*,\mathcal{F}^*,\mathbb{P}^*)$, and $\mathbb{E}^*$ denotes the expectation under $\mathbb{P}^*$ (${\mathfrak{H}at P}i_F$ being extended on $\Omega\times \Omega^*$). We recall the following result from \cite{NourdinViens}.
\textcolor{blue}egin{eqnarray}gin{Theorem}[Nourdin-Viens' formula]\label{thm_NourdinViens}
$F$ has a density $\rho$ with the respect to the Lebesgue measure if and only if the random variable $g_F(F-\mathbb{E}[F])$ is positive a.s.. In this case, the support of $\rho$, denoted by $\text{supp}(\rho)$, is a closed interval of $\mathbb{R}$ and for all $x \in \text{supp}(\rho)$:
\textcolor{blue}egin{eqnarray}gin{equation*}
\rho(x)=\frac{\mathbb{E}(|F-\mathbb{E}[F]|)}{2g_{F}(x-\mathbb{E}[F])}\exp{\left( -\int_0^{x-\mathbb{E}[F]} \frac{udu}{g_F(u)} \right)}.
\end{equation*}
\end{Theorem}
\subsection{The FBSDE under consideration}
\label{sub:X}
In this paper, we consider a FBSDE of the form:
\textcolor{blue}egin{eqnarray}gin{equation}\label{edsr}
\textcolor{blue}egin{eqnarray}gin{cases}
\displaystyle X_t= X_0+\int_0^t b(s,X_s) ds +\int_0^t \sigma(s,X_s)dW_s,\ t\in[0,T],\ \mathbb{P}-a.s.\\
\displaystyle Y_t = g(X_T) +\int_t^T h (s,X_s,Y_s,Z_s) ds -\int_t^T Z_s dW_s, \ t\in [0,T],\ \mathbb{P}-a.s.,
\end{cases}
\end{equation} with $X_0$ a given real constant. We denote by $\mathfrak S(X_t)$ the support of the law of $X_t$ under $\mathbb P$, that is to say the smallest closed subset $A$ of $\mathbb{R}$ such that $\mathbb{P}(X_t\in A)=1$.
Throughout this paper we will make the following standing assumption on the process $X$ in \eqref{edsr}.
\textbf{Standing assumptions on $X$:}
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[(X)]
$b,\sigma : [0,T]\times \mathbb{R} \longrightarrow \mathbb{R}$ are continuous in time and continuously differentiable in space for any fixed time $t$ and such that there exist $k_b,k_\sigma >0$ with
$$|b_x(t,x)|\leq k_b,\ |\sigma_x(t,x)|\leq k_\sigma, \text{ for all $x\in\R$}.$$
Besides $b(t,0), \sigma(t,0)$ are bounded functions of $t$ and there exists $c>0$ such that for all $t\in [0,T]$ $$0<c\leq |\sigma(t,\cdot)|, \ \lambda(dx)-a.e.$$
\end{itemize}
\textcolor{blue}egin{eqnarray}gin{Remark}\label{densite_x}According to Theorem 2.1 in \cite{FournierPrintems}, $(X)$ implies that for all $t\in(0,T]$, the law of $X_t$, denoted by $\mathcal{L}(X_t)$, has a density with respect to the Lebesgue measure.
\end{Remark}
Our results will obviously need conditions on the parameters $g$, $h$ which appear in the backward component of \eqref{edsr}. More precisely, one can distinguish between two regimes which call for two different analyses: the case where $h$ exhibits Lipschitz growth in its variables (developed in Section \ref{section:lip}), and the case where $h$ has quadratic growth in the $z$ variable (studied in Section \ref{section:quadratic}). We start with the Lipschitz situation.
| 3,160 | 50,703 |
en
|
train
|
0.51.2
|
\section{The Lipschitz case}
\label{section:lip}
In this section, we focus on the solution $(Y,Z)$ of FBSDE \eqref{edsr} under a Lipschitz-type assumption on the driver $h$. The problem of existence of a density for the marginal laws of $Y$ has been first studied in \cite{AntonelliKohatsu}, when the generator $h$ is assumed to be uniformly Lipschitz continuous in $y$ and $z$. We first recall in Section \ref{sub:lipprel} some general results on Lipschitz FBSDEs, then we review in Section \ref{section:lip:y} the results from \cite{AntonelliKohatsu}. Next, we point out an inefficiency in \cite[Theorem 3.6]{AntonelliKohatsu} by providing a counter example to this result, and we make precise how this small flaw can be corrected, and propose a precised version of it as Theorem \ref{AKmodifie}. Finally, in Section \ref{section:lip:z}, we study the existence of a density for the marginal laws of $Z$ when the generator $h$ of the BSDE satisfies Assumption $($L$)$.
\subsection{Generalities on Lipschitz FBSDEs}
\label{sub:lipprel}
We start by making precise as Assumption $($L$)$ the Lipschitz condition on $h$ and the associated condition on the terminal condition $g$. We set:
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[(L)]
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[(i)] $g : \mathbb{R} \longrightarrow \mathbb{R}$ is such that $\mathbb{E}[g(X_T)^2]<+\infty$.
\item[(ii)] $h : [0,T]\times \mathbb{R}^3 \longrightarrow \mathbb{R}$ is such that there exist $(k_x,k_y,k_z)\in(\R_+^*)^3$ such that for all $(t,x_1,x_2,y_1,y_2,z_1,z_2) \in [0,T]\times \mathbb{R}^6$,
$$ |h(t,x_1,y_1,z_1)-h(t,x_2,y_2,z_2)|\leq k_x|x_1-x_2|+k_y|y_1-y_2|+k_z|z_1-z_2|.$$
\item[(iii)] $\int_0^T |h(s,0,0,0)|^2ds<+\infty$.
\end{itemize}
\end{itemize}
Before going to the density analysis of the $Y$ and $Z$ components we recall briefly well-known facts about existence, uniqueness and Malliavin differentiability for the system \eqref{edsr} which can be found in \cite{PardouxPeng,ElkarouiPengQuenez}.
\textcolor{blue}egin{eqnarray}gin{Proposition}[\cite{PardouxPeng,ElkarouiPengQuenez}]$($Existence and uniqueness$)$\label{propex}
Under Assumptions $(X)$ $($that we recall is given in Section \ref{sub:X}$)$ and $(L)$, there exists a unique solution $(X,Y,Z)$ in $\mathbb{S}^2 \times \mathbb{S}^2 \times \mathbb{H}^2$ to the FBSDE \eqref{edsr}.
\end{Proposition}
Concerning the Malliavin differentiability of $(X,Y,Z)$, it can obtained (see \cite{PardouxPeng} and \cite[Remark of Proposition 5.3]{ElkarouiPengQuenez}) under the following assumptions:
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[(D1)]
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[(i)] $g$ is differentiable, $\mathcal{L}(X_T)-$a.e., $g$ and $g'$ have polynomial growth.
\item[(ii)] $(x,y,z)\mapsto h(t,x,y,z)$ is continuously differentiable for every $t$ in $[0,T]$.
\end{itemize}
\item[(D2)]
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[(i)] $g$ is twice differentiable, $\mathcal{L}(X_T)-$a.e., $g$, $g'$ and $g''$ have polynomial growth.
\item[(ii)] $(x,y,z)\mapsto h(t,x,y,z)$ is twice continuously differentiable for every $t$ in $[0,T]$.
\end{itemize}
\end{itemize}
Note that (D1) ensures that $Y$ is Malliavin differentiable, whereas $(D2)$ ensures it is twice Malliavin differentiable. As it will be made more clear below, since $Z$ can be represented as a Malliavin trace of $Y$, the fact that $Y$ is twice Malliavin differentiable entails that $Z$ is Malliavin differentiable.
\textcolor{blue}egin{eqnarray}gin{Proposition}$($Malliavin differentiabiliy$)$ \label{MD}
Under $(X)$, $(L)$ and $(D1)$, we have for any $t\in[0,T]$ that $(X_t,Y_t) \in (\mathbb{D}^{1,2})^2$, $Z_t \in \mathbb{D}^{1,2}$ for almost every $t$, and for all $0<r\leq t \leq T$:
\textcolor{blue}egin{eqnarray}gin{equation}\label{edsr_derive}
\textcolor{blue}egin{eqnarray}gin{cases}
\displaystyle D_r X_t=\sigma(r,X_r) + \int_r^t b_x(s,X_s) D_r X_s ds + \int_r^t \sigma_x(s,X_s) D_r X_s dW_s\\
\displaystyle D_r Y_t=g'(X_T)D_rX_T+\int_t^T H(s,D_r X_s,D_r Y_s, D_r Z_s)ds -\int_t^T D_r Z_s dW_s,
\end{cases}
\end{equation}
where $H(s,x,y,z):=h_x(s,X_s,Y_s,Z_s)x+h_y(s,X_s,Y_s,Z_s)y+h_z(s,X_s,Y_s,Z_s)z.$
\end{Proposition}
Notice that BSDE \eqref{edsr_derive} is a linear BSDE, whose solution can be computed using the linearization method (see \cite{ElkarouiPengQuenez}).
We will need extra properties on the Malliavin derivative of $Y$ and $Z$ for which the following result will be crucial. These results rely heavily on the Markovian framework we are working with.
\textcolor{blue}egin{eqnarray}gin{Proposition}[\cite{MaZhang,IRR}]\label{prop:Markov}
Let Assumptions $(X)$, $(L)$ and $(D1)$ hold, then there exists a map $u:[0,T] \times \mathbb{R} \longrightarrow \mathbb{R}$ in ${\cal C}^{1,2}$ such that
$$Y_t =u(t,X_t), \quad t\in [0,T], \; \mathbb{P}-a.s.$$
In addition, $Z$ admits a continuous version given by
\textcolor{blue}egin{eqnarray}gin{equation}
\label{eq:Zu'}
Z_t = u_x(t,X_t) \sigma(t,X_t), \quad t\in [0,T], \; \mathbb{P}-a.s.
\end{equation}
\end{Proposition}
In view of Proposition \ref{prop:Markov}, the chain rule formula implies that $Y_t$ belongs to $\mathbb{D}^{2,2}$ and
\textcolor{blue}egin{eqnarray}gin{equation}
\label{eq:D2Y}
D^2 Y_t = u_x(t,X_t) D^2 X_t + u_{xx}(t,X_t) (D X_t)^{\otimes 2}, \quad \mathbb{P}-a.s.
\end{equation}
Note that by definition, $Z$ is an element of $\mathbb{H}^2$. As a consequence, for any fixed element $t$ in $[0,T]$, the random variable $Z_t$ is not uniquely defined, which makes the density analysis ill-posed. However, by the previous proposition, $Z$ admits in our framework a continuous version. From now on, we will always consider this version.
The following Lemma is due to Ma and Zhang in \cite[Lemma 2.4]{MaZhang} and to Pardoux and Peng \cite{PardouxPeng_92} for the representation of $Z$ as a Malliavin trace of $Y$ (see \eqref{prop:PP92} below).
\textcolor{blue}egin{eqnarray}gin{Lemma}\label{lemma_gradient_malliavin}
Let Assumptions $(X)$, $(L)$, $(D1)$ and $(D2)$ hold. Then, there exists a version of $(D_r X_t, D_r Y_t, D_r Z_t)$ for all $0<r\leq t\leq T$ which satisfies:
\textcolor{blue}egin{eqnarray}gin{align*}
D_r X_t&=\nabla X_t (\nabla X_r)^{-1}\sigma(r,X_r),\ D_r Y_t&=\nabla Y_t (\nabla X_r)^{-1}\sigma(r,X_r),\ D_r Z_t&=\nabla Z_t (\nabla X_r)^{-1}\sigma(r,X_r),
\end{align*}
\textcolor{blue}egin{eqnarray}gin{equation}
\label{prop:PP92}
Z_t=D_t Y_t:=\lim_{s \nearrow t} D_s Y_t,\ \mathbb{P}-a.s., \textrm{ for } a.e. \; t \in [0,T],
\end{equation}
where $(\nabla X,\nabla Y,\nabla Z)$ is the solution to the following FBSDE:
\textcolor{blue}egin{eqnarray}gin{equation}\label{edsr_gradient}
\textcolor{blue}egin{eqnarray}gin{cases}
\displaystyle \nabla X_t= \int_0^t b_x(s,X_s) \nabla X_s ds +\int_0^t \sigma_x(s,X_s)\nabla X_sdW_s,\\
\displaystyle \nabla Y_t = g'(X_T)\nabla X_T +\int_t^T \nabla h(s,{\mathfrak{H}at T}eta_s)\cdot \nabla {\mathfrak{H}at T}eta_s ds -\int_t^T \nabla Z_s dW_s.
\end{cases}
\end{equation}
\end{Lemma}
\textcolor{blue}egin{eqnarray}gin{Remark}
Assumptions $(D1)$ and $(D2)$ are linked to the existence of first and second-order Malliavin derivatives for the $Y$ component of the solution of \reff{edsr}. We would like to point out to the reader that we only require the differentiability of $g$, $\mathcal{L}(X_T)-$a.e. Such a relaxation will be particularly useful in the quadratic case $($\textit{i.e.} in Section \ref{section:quadratic}$)$. We emphasize that when we work under Assumption $(X)$, the law of $X_T$ is absolutely continuous with respect to the Lebesgue measure and $X_T$ has finite moments of any order. Thus, thanks to standard approximation arguments, we can show that the usual chain rule formula of Malliavin calculus $($see Proposition 1.2.3. in \cite{Nualartbook}$)$ still holds for the random variable $g(X_T)$, under Assumptions $(D1)$ or $(D2)$.
\end{Remark}
Finally, set the following assumption
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[(M)] There exists a function $f\in \mathcal{C}^2(\mathbb{R})$ such that for all $t\in [0,T]$: $X_t=f(t,W_t)$.
\end{itemize}
We obtain the following proposition
\textcolor{blue}egin{eqnarray}gin{Proposition}\label{prop_dy_dz_r}
Under Assumptions $(M)$, $(L)$ and $(D2)$, for all $0<r,s\leq t\leq T$ we have $D_r Y_t=D_s Y_t=Z_t$ and $D_r Z_t=D_s Z_t$, $\mathbb{P}-$a.s.
\end{Proposition}
\textcolor{blue}egin{eqnarray}gin{proof}
Once again we set ${\mathfrak{H}at T}eta_s:=(X_s,Y_s,Z_s)$. We know that for all $0<r\leq t\leq T$:
\textcolor{blue}egin{eqnarray}gin{align*}
D_r Y_t&=g'(X_T) f'(T,W_T)+\int_t^T (h_x(s,{\mathfrak{H}at T}eta_s)f'(s,W_s)+h_y(s,{\mathfrak{H}at T}eta_s)D_r Y_s+h_z(s,{\mathfrak{H}at T}eta_s)D_r Z_s)ds\\
&\mathfrak{H}space{1cm}- \int_t^T D_r Z_s dW_s. \end{align*}
Then $(D_r Y, D_r Z)$ satisfies a linear BSDE which does not depend on $r$ and by the uniqueness of the solution we deduce that for all $0<r,s\leq t\leq T$ we have $D_r Y_t=D_s Y_t$ and $D_r Z_t=D_s Z_t$, $\mathbb{P}-$a.s. Finally, $D_r Y_t=Z_t$ by \eqref{prop:PP92}.
\textcolor{blue}egin{eqnarray}gin{flushright}
$\qed$
\end{flushright}
\end{proof}
| 3,572 | 50,703 |
en
|
train
|
0.51.3
|
\subsection{Existence of a density for the $Y$ component}
\label{section:lip:y}
We focus in this section on the existence of a density for the marginal laws of the process $Y$ in the Lipschitz case, pursuing the study started in \cite{AntonelliKohatsu}. Towards this goal, we recall first the so-called \textit{first order conditions} introduced in \cite{AntonelliKohatsu}, which are only sufficient, as illustrated in Example \ref{exemple}. We then turn our attention to the \textit{second-order conditions} of Theorem 3.6 in \cite{AntonelliKohatsu}. We point out a (small) inefficiency in the proof of \cite[Theorem 3.6]{AntonelliKohatsu} and provide a corrected version of this result as Theorem \ref{AKmodifie}.
As in \cite{AntonelliKohatsu}, we set for any $A\in{\cal B}(\R)$ (i.e. the Borel $\sigma$-algebra on $\R$), and $t$ in $[0,T]$ such that $\mathbb{P}(X_T \in A | \mathcal{F}_t)>0$:
\textcolor{blue}egin{eqnarray}gin{equation}
\label{eq:barg}
\underline{g}:= \inf\limits_{x \in \mathbb{R}} g'(x), \quad \underline{g}^A:=\inf\limits_{x\in A} g'(x),\quad \overline{g}:= \sup\limits_{x \in \mathbb{R}} g'(x), \quad \overline{g}^A:=\sup\limits_{x\in A} g'(x),
\end{equation}
\textcolor{blue}egin{eqnarray}gin{equation}
\label{eq:barh}
\underline{h}(t):=\inf\limits_{ s\in [t,T], (x,y,z) \in \mathbb{R}^3} h_x(s,x,y,z),\quad \quad \overline{h}(t):=\sup\limits_{ s\in [t,T], (x,y,z) \in \mathbb{R}^3} h_x(s,x,y,z).
\end{equation}
\textcolor{blue}egin{eqnarray}gin{Theorem}$($First-order conditions \cite[Theorem 3.1]{AntonelliKohatsu}$)$\label{thm_H+H-}
Assume that $(X)$, $(L)$ and $(D1)$ hold. Fix some $t\in(0,T]$ and set $K:=k_b+k_y+k_{\sigma}k_z$. If there exists $A\in\mathcal B(\R)$ such that $\mathbb{P}(X_T \in A | \mathcal{F}_t)>0$ and one of the two following assumptions holds
\textcolor{blue}egin{eqnarray}gin{align*}
&(H+)\quad \textcolor{blue}egin{eqnarray}gin{cases}
\displaystyle \underline{g}e^{-\text{sgn}(\underline{g})KT}+\underline{h}(t)\int_t^T e^{-\text{sgn}(\underline{h}(s))Ks}ds\geq0 \\
\displaystyle \underline{g}^Ae^{-\text{sgn}(\underline{g}^A)KT}+\underline{h}(t)\int_t^T e^{-\text{sgn}(\underline{h}(s))Ks}ds>0
\end{cases}\\[0.3em]
&(H-)\quad \textcolor{blue}egin{eqnarray}gin{cases}
\displaystyle \overline{g}e^{-\text{sgn}(\overline{g})KT}+\overline{h}(t)\int_t^T e^{-\text{sgn}(\overline{h}(s))Ks}ds\leq 0 \\
\displaystyle \overline{g}^Ae^{-\text{sgn}(\overline{g}^A)KT}+\overline{h}(t)\int_t^T e^{-\text{sgn}(\overline{h}(s))Ks}ds<0,
\end{cases}
\end{align*}
then $Y_t$ has a law absolutely continuous with respect to the Lebesgue measure.
\end{Theorem}
\textcolor{blue}egin{eqnarray}gin{Remark}
Notice that $\underline{g}$ $($resp. $\overline{g})$ could be equal to $-\infty$ $($resp. $+\infty)$. Then Assumption $(H+)$ $($resp. $(H-))$ cannot be satisfied. Therefore, there is no problem if we allow the extrema of $g$ to take the values $\pm \infty$.
\end{Remark}
\textcolor{blue}egin{eqnarray}gin{Remark}\label{positivite_dy}
In view of the proof of \cite[Theorem 3.1]{AntonelliKohatsu}, one can show that under $(X)$, $(L)$, and $(D1)$ and if $g'\geq 0$ and $\underline{h}(t)\geq 0$ $($resp. $g'\leq 0$ and $\overline{h}(t)\leq 0)$ for $t \in [0,T]$, then for all $0<r\leq t \leq T$, $D_r Y_t \geq 0$ $($resp. $D_r Y_t \leq 0)$ and the inequality is strict if there exists $A\in{\cal B}(\R)$ such that $\mathbb{P}(X_T \in A|\mathcal{F}_t)>0$ and $g'_{|A}>0$ $($resp. $g'_{|A}<0)$.
\end{Remark}
Note that neither Condition $(H+)$ nor Condition $(H-)$ are necessary for getting existence of a density as illustrated in the following example.
\textcolor{blue}egin{eqnarray}gin{Example}\label{exemple}
Let $T=1$, $g(x)=x$, $X=W$, $h(s,x,y,z)=(s-2)x$. In this case, $K=0$ and $h_x(s,x,y,z)=s-2$ for all $(x,y,z)\in \mathbb{R}^3$. For any $t$ in $(0,1]$, we have:
\[ \overline{g}=\underline{g}=1,\ \underline{h}(t)=t-2,\ \overline{h}(t)=-1, \]
so that Assumption $(H-)$ is not satisfied. Indeed,
$$ \overline{g}e^{-\text{sgn}(\overline{g})KT}+\overline{h}(t)\int_t^T e^{-\text{sgn}(\overline{h}(s))Ks}ds=1-(1-t)=t>0.$$
Similarly, $(H+)$ is not satisfied for any $t \in \left(0, (3-\sqrt{5})/2\right)$ since:
$$ \underline{g}e^{-\text{sgn}(\underline{g})KT}+\underline{h}(t)\int_t^T e^{-\text{sgn}(\underline{h}(s))Ks}ds=1+(t-2)(1-t)=-t^2+3t-1,$$ which is negative for $t \in \left(0, (3-\sqrt{5})/2\right)$. We deduce that for $t\in \left(0, (3-\sqrt{5})/2\right)$ neither Assumption $(H+)$ nor Assumption $(H-)$ is satisfied. However, we know that:
\textcolor{blue}egin{eqnarray}gin{align}
\label{eq:counterex}
Y_t &= \mathbb{E}\left[\left.W_1+\int_t^1 (s-2)W_s ds \right| \mathcal{F}_t\right]\noindentnumber\\
&=W_t\left(1+\int_t^1 (s-2)ds\right)=W_t\left(-\frac12+2t-\frac{t^2}{2}\right), \ \forall t\in [0,1],\ \mathbb{P}-a.s.,
\end{align}
which admits a density with respect to the Lebesgue measure except when $t=0$ and $t=2-\sqrt{3}$.
\end{Example}
Notice that in the previous example, the generator does not depend on $z$. In that setting, another result is derived \cite{AntonelliKohatsu}, involving so-called \textit{second order conditions}. There, the authors of \cite{AntonelliKohatsu} benefit from the absence of $z$ in the driver to make a higher order expansion of the Malliavin norm $\int_0^T |D_r Y_t|^2 dr$. The price to pay is that the condition involves a mapping $\tilde{h}$ (see \eqref{eq:htilde} below), which is essentially a sum of derivatives of the driver $h$, which goes beyond the simple derivative $h_x$. However, Example \ref{exemple} provides a counter-example to \cite[Theorem 3.6]{AntonelliKohatsu}. Indeed, the second-order conditions proposed in \cite[Theorem 3.6]{AntonelliKohatsu} entails that $Y_t$ admits a density, when $t\neq \frac12$, so in particular at $t=2-\sqrt{3}$. However from \eqref{eq:counterex}, $Y_{2-\sqrt{3}}=0$. This example proves that \cite[Theorem 3.6]{AntonelliKohatsu} has to be modified. The proof of \cite[Theorem 3.6]{AntonelliKohatsu} is essentially correct, except that in their proof the original Brownian motion $W$ is not a Brownian motion any more under the new measure $\mathbb Q$ defined in \cite[page 275]{AntonelliKohatsu} and need to be replaced by the process $W'_\cdot:=W_\cdot-\int_0^\cdot \sigma_x(s,X_s)ds$ which is a $\mathbb Q$-Brownian motion. This leads to the two extra terms $-(\sigma \sigma_xh_{xx}+z\sigma_xh_{xy})$ in the expression of the mapping \eqref{eq:htilde} below, compareLd to the original expression of $\tilde{h}$ in the statement of \cite[Theorem 3.6]{AntonelliKohatsu}. We refer the reader to Example \ref{counterexample} below and we propose a corrected version of \cite[Theorem 3.6]{AntonelliKohatsu} as Theorem \ref{AKmodifie} (whose proof exactly follows the original one up to the introduction of $W'$), in which the modified second-order conditions are sufficient, and necessary in the special situation of Example \ref{exemple}.
Consider the FBSDE \eqref{edsr} when $h$ does not depend on $z$ and define:
\textcolor{blue}egin{eqnarray}gin{align}
\label{eq:htilde}
\tilde{h}(s,x,y,z):=& -\left( h_{xt}+b h_{xx}-hh_{xy}+\frac12(\sigma^2 h_{xxx}+2z\sigma h_{xxy}+z^2h_{xxy})\right)(s,x,y)\noindentnumber\\
&-\left((h_y+b_x)h_x+\sigma \sigma_xh_{xx}+z\sigma_xh_{xy}\right)
(s,x,y).
\end{align}
$$ \tilde{g}(x):=g'(x)+(T-t)h_x(T,x,g(x)),$$
$$ \underline{\tilde{g}}:=\min\limits_{x\in \mathbb{R}} \tilde{g}(x), \quad \overline{\tilde{g}}:=\max\limits_{x\in \mathbb{R}} \tilde{g}(x),\quad \underline{\tilde{g}}^A:=\min\limits_{x\in A} \tilde{g}(x), \quad \overline{\tilde{g}}^A:=\max\limits_{x\in A} \tilde{g}(x),$$
$$\underline{\tilde{h}}(t):=\min\limits_{[t,T]\times \mathbb{R}^3} \tilde{h}(s,x,y,z), \ \overline{\tilde{h}}(t):=\max\limits_{[t,T]\times \mathbb{R}^3} \tilde{h}(s,x,y,z).$$
The following theorem corrects Theorem 3.6 in \cite{AntonelliKohatsu}.
\textcolor{blue}egin{eqnarray}gin{Theorem}$($Second-order conditions \cite[Theorem 3.6]{AntonelliKohatsu}$)$\label{AKmodifie}
Fix some $t\in(0,T]$, assume that $h$ does not depend on $z$, that Assumptions $(X)$, $(L)$ and $(D1)$ hold and set $K:=k_y+k_b$. If there exists $A\in\mathcal B(\R)$ such that $\mathbb{P}(X_T \in A | \mathcal{F}_t)>0$ and one of the two following assumptions holds
\textcolor{blue}egin{eqnarray}gin{align*}
&\widetilde{(H+)} \quad \textcolor{blue}egin{eqnarray}gin{cases}
\displaystyle \underline{\tilde{g}}e^{-\text{sgn}(\underline{\tilde{g}})KT}+\underline{\tilde{h}}(t)\int_t^T e^{-\text{sgn}(\underline{\tilde{h}}(s))Ks}(T-s)ds\geq0 \\
\displaystyle \underline{\tilde{g}}^Ae^{-\text{sgn}(\underline{\tilde{g}}^A)KT}+\underline{\tilde{h}}(t)\int_t^T e^{-\text{sgn}(\underline{\tilde{h}}(s))Ks}(T-s)ds>0,
\end{cases}\\[0.3em]
&\widetilde{(H-)} \quad \textcolor{blue}egin{eqnarray}gin{cases}
\displaystyle \overline{\tilde{g}}e^{-\text{sgn}(\overline{\tilde{g}})KT}+\overline{\tilde{h}}(t)\int_t^T e^{-\text{sgn}(\overline{\tilde{h}}(s))Ks}(T-s)ds\leq 0 \\
\displaystyle \overline{\tilde{g}}^Ae^{-\text{sgn}(\overline{\tilde{g}}^A)KT}+\overline{\tilde{h}}(t)\int_t^T e^{-\text{sgn}(\overline{\tilde{h}}(s))Ks}(T-s)ds<0,
\end{cases}
\end{align*}
then the first component $Y_t$ of the solution of BSDE \eqref{edsr} has a law which is absolutely continuous with respect to the Lebesgue measure.
\end{Theorem}
\textcolor{blue}egin{eqnarray}gin{Example}\label{counterexample}
We go back to Example \ref{exemple} with $g\equiv Id.$ and $h(s,x,y,z)=(s-2)x$ which does not depend on $z$. On the one hand, we know from \eqref{eq:counterex} that for all $t\in (0,1]$, the law of $Y_t$ has a density except when $t=0$ or $t=2-\sqrt{3}$. On the other hand, our conditions in Theorem \ref{AKmodifie} read:
\[ \overline{\tilde{g}}=\tilde{\underline{g}}=\tilde{g}(x)=t, \quad \tilde{h}(t,x,y)=\overline{\tilde{h}}(t)=\underline{\tilde{h}}(t)=-1, \quad K=0,
\]
from which $\widetilde{(H+)}$ becomes:
\[ t-\int_t^1 (1-s) ds = t-(1-t)+\frac12-\frac{t^2}{2}=-\frac{t^2}{2}+2t-\frac12 >0,\]
and $\widetilde{(H-)}$ becomes:
\[ t-\int_t^1 (1-s) ds = t-(1-t)+\frac12-\frac{t^2}{2}=-\frac{t^2}{2}+2t-\frac12 <0.\]
We hence conclude, in view of Theorem \ref{AKmodifie}, that the law of $Y_t$ has a density with respect to the Lebesgue measure for every $t\in (0,1] \textcolor{blue}ackslash \{2-\sqrt{3}\}$.
| 3,948 | 50,703 |
en
|
train
|
0.51.4
|
Consider the FBSDE \eqref{edsr} when $h$ does not depend on $z$ and define:
\textcolor{blue}egin{eqnarray}gin{align}
\label{eq:htilde}
\tilde{h}(s,x,y,z):=& -\left( h_{xt}+b h_{xx}-hh_{xy}+\frac12(\sigma^2 h_{xxx}+2z\sigma h_{xxy}+z^2h_{xxy})\right)(s,x,y)\noindentnumber\\
&-\left((h_y+b_x)h_x+\sigma \sigma_xh_{xx}+z\sigma_xh_{xy}\right)
(s,x,y).
\end{align}
$$ \tilde{g}(x):=g'(x)+(T-t)h_x(T,x,g(x)),$$
$$ \underline{\tilde{g}}:=\min\limits_{x\in \mathbb{R}} \tilde{g}(x), \quad \overline{\tilde{g}}:=\max\limits_{x\in \mathbb{R}} \tilde{g}(x),\quad \underline{\tilde{g}}^A:=\min\limits_{x\in A} \tilde{g}(x), \quad \overline{\tilde{g}}^A:=\max\limits_{x\in A} \tilde{g}(x),$$
$$\underline{\tilde{h}}(t):=\min\limits_{[t,T]\times \mathbb{R}^3} \tilde{h}(s,x,y,z), \ \overline{\tilde{h}}(t):=\max\limits_{[t,T]\times \mathbb{R}^3} \tilde{h}(s,x,y,z).$$
The following theorem corrects Theorem 3.6 in \cite{AntonelliKohatsu}.
\textcolor{blue}egin{eqnarray}gin{Theorem}$($Second-order conditions \cite[Theorem 3.6]{AntonelliKohatsu}$)$\label{AKmodifie}
Fix some $t\in(0,T]$, assume that $h$ does not depend on $z$, that Assumptions $(X)$, $(L)$ and $(D1)$ hold and set $K:=k_y+k_b$. If there exists $A\in\mathcal B(\R)$ such that $\mathbb{P}(X_T \in A | \mathcal{F}_t)>0$ and one of the two following assumptions holds
\textcolor{blue}egin{eqnarray}gin{align*}
&\widetilde{(H+)} \quad \textcolor{blue}egin{eqnarray}gin{cases}
\displaystyle \underline{\tilde{g}}e^{-\text{sgn}(\underline{\tilde{g}})KT}+\underline{\tilde{h}}(t)\int_t^T e^{-\text{sgn}(\underline{\tilde{h}}(s))Ks}(T-s)ds\geq0 \\
\displaystyle \underline{\tilde{g}}^Ae^{-\text{sgn}(\underline{\tilde{g}}^A)KT}+\underline{\tilde{h}}(t)\int_t^T e^{-\text{sgn}(\underline{\tilde{h}}(s))Ks}(T-s)ds>0,
\end{cases}\\[0.3em]
&\widetilde{(H-)} \quad \textcolor{blue}egin{eqnarray}gin{cases}
\displaystyle \overline{\tilde{g}}e^{-\text{sgn}(\overline{\tilde{g}})KT}+\overline{\tilde{h}}(t)\int_t^T e^{-\text{sgn}(\overline{\tilde{h}}(s))Ks}(T-s)ds\leq 0 \\
\displaystyle \overline{\tilde{g}}^Ae^{-\text{sgn}(\overline{\tilde{g}}^A)KT}+\overline{\tilde{h}}(t)\int_t^T e^{-\text{sgn}(\overline{\tilde{h}}(s))Ks}(T-s)ds<0,
\end{cases}
\end{align*}
then the first component $Y_t$ of the solution of BSDE \eqref{edsr} has a law which is absolutely continuous with respect to the Lebesgue measure.
\end{Theorem}
\textcolor{blue}egin{eqnarray}gin{Example}\label{counterexample}
We go back to Example \ref{exemple} with $g\equiv Id.$ and $h(s,x,y,z)=(s-2)x$ which does not depend on $z$. On the one hand, we know from \eqref{eq:counterex} that for all $t\in (0,1]$, the law of $Y_t$ has a density except when $t=0$ or $t=2-\sqrt{3}$. On the other hand, our conditions in Theorem \ref{AKmodifie} read:
\[ \overline{\tilde{g}}=\tilde{\underline{g}}=\tilde{g}(x)=t, \quad \tilde{h}(t,x,y)=\overline{\tilde{h}}(t)=\underline{\tilde{h}}(t)=-1, \quad K=0,
\]
from which $\widetilde{(H+)}$ becomes:
\[ t-\int_t^1 (1-s) ds = t-(1-t)+\frac12-\frac{t^2}{2}=-\frac{t^2}{2}+2t-\frac12 >0,\]
and $\widetilde{(H-)}$ becomes:
\[ t-\int_t^1 (1-s) ds = t-(1-t)+\frac12-\frac{t^2}{2}=-\frac{t^2}{2}+2t-\frac12 <0.\]
We hence conclude, in view of Theorem \ref{AKmodifie}, that the law of $Y_t$ has a density with respect to the Lebesgue measure for every $t\in (0,1] \textcolor{blue}ackslash \{2-\sqrt{3}\}$.
In this particular example, notice that Theorem \ref{AKmodifie} is more accurate than Theorem \ref{thm_H+H-} since Condition $\widetilde{(H+)}$ and Condition $\widetilde{(H-)}$ are sufficient \underline{and} necessary to obtain the existence of a density for $Y$. Finally, we emphasize once more that the counterpart of Condition $\widetilde{(H-)}$ in \cite[Theorem 3.6]{AntonelliKohatsu} gives that whenever $2t-1<0$, $Y_t$ admits a density, which is clearly satisfied for $t=2-\sqrt{3}$. However we know that $Y_{2-\sqrt{3}}=0$.
\end{Example}
| 1,534 | 50,703 |
en
|
train
|
0.51.5
|
\subsection{Existence of a density for the control variable $Z$}
\label{section:lip:z}
We now turn to the problem of existence of a density for the marginal laws of $Z$. This question was studied in \cite{AbouraBourguin} when the generator is linear in $z$, that is to say $h(t,x,y,z)=\tilde{h}(t,x,y)+\alpha z$, which is from our point of view a too stringent assumption since by a Girsanov transformation this equation basically reduces to a BSDE with a generator which does not depend on $z$. We focus here on a general function $h$ satisfying Assumption (L). Consider the two following assumptions
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[(C+)] $h_x,h_{xx},h_{yy},h_{zz},h_{xy}\geq 0$ and $h_{xz}= h_{yz}= 0$,
\item[(C-)] $h_x,h_{xx},h_{yy},h_{zz},h_{xy}\leq 0$ and $h_{xz}= h_{yz}= 0$.
\end{itemize}
Let $t\in (0,T]$ and $A\in{\cal B}(\R)$. We set:
$$ \underline{g''}:= \min\limits_{x \in \mathfrak{S}(X_T)} g''(x), \quad \underline{g''}^A:=\min\limits_{x\in \mathfrak{S}(X_T) \cap A} g''(x),\quad \underline{g'}:= \min\limits_{x \in \mathfrak{S}(X_T)} g'(x), \quad \underline{g'}^A:=\min\limits_{x\in \mathfrak{S}(X_T) \cap A} g'(x),$$
$$ \underline{h_{xx}}(t):=\min\limits_{ s\in [t,T], (x,y,z) \in \mathbb{R}^3} h_{xx}(s,x,y,z).$$
\textcolor{blue}egin{eqnarray}gin{Theorem}\label{thm_density_z_lip}
Let Assumption $(X)$, $(L)$ and $(D2)$ hold. Let $0< t \leq T$ and assume moreover
\textcolor{blue}egin{eqnarray}gin{itemize}
\item There exist $(\underline a,\overline a)\in(0,+\infty)$, such that $\underline{a}\leq D_r X_u \leq \overline{a}$, for all $0<r<u\leq T$,
\item There exists $\overline b\geq 0$, such that $0\leq D_{r,t}^2 X_u\leq \overline{b}$, for all $0<r,t<u\leq T$,
\item $(C+)$ holds
\item $h_{xy}= 0$ or $(h_{xy}\geq 0$ and $g'\geq 0$, $\mathcal{L}(X_T)-a.e.)$.
\end{itemize}
If there exists a set $A\in \mathcal B(\mathbb{R})$ such that $\mathbb{P}(X_T\in A | \mathcal{F}_t)>0$ and such that
$$ \mathbf{1}_{\{\underline{g''}<0\}}\underline{g''}\overline{a}^2+\underline{g'}\mathbf{1}_{\{\underline{g'}<0\}}\overline{b}+(\mathbf{1}_{\{\underline{g''}\geq 0\}}\underline{g''}+\underline{h_{xx}}(t)(T-t))\underline{a}^2\geq 0,$$
and
$$ (\mathbf{1}_{\{\underline{g''}^A<0\}}\underline{g''}^A\overline{a}^2+\underline{g'}^A\mathbf{1}_{\{\underline{g'}<0\}}\overline{b}) +(\mathbf{1}_{\{\underline{g''}^A\geq 0\}}\underline{g''}^A+\underline{h_{xx}}(t)(T-t))\underline{a}^2>0, $$
then, the law of $Z_t$ has a density with respect to the Lebesgue measure.
\end{Theorem}
\textcolor{blue}egin{eqnarray}gin{proof} Under the assumptions of Theorem \ref{thm_density_z_lip}, we obtain for $0<r,s< t\leq T$:
\textcolor{blue}egin{eqnarray}gin{align*}
D_{r,s}^2 Y_t=&\ g''(X_T)D_r X_T D_s X_T+g'(X_T)D_{r,s}^2 X_T -\int_t^T D_{r,s}^2 Z_u dW_u\\
&+\int_t^T[h_x(u,{\mathfrak{H}at T}eta_u)D_{r,s}^2 X_u+h_{xx}(u,{\mathfrak{H}at T}eta_u)D_r X_u D_s X_u+h_{xy}(u,{\mathfrak{H}at T}eta_u) D_s X_u D_r Y_u\\
&+h_y(u,{\mathfrak{H}at T}eta_u) D_{r,s}^2 Y_u +h_{xy}(u,{\mathfrak{H}at T}eta_u) D_r X_u D_s Y_u+h_{yy}(u,{\mathfrak{H}at T}eta_u) D_r Y_u D_s Y_u\\
&+h_z(u,{\mathfrak{H}at T}eta_u) D_{r,s}^2 Z_u+h_{zz}(u,{\mathfrak{H}at T}eta_u)D_r Z_u D_s Z_u] du.
\end{align*}
Let $\tilde{\mathbb{P}}$ be the probability equivalent to $\mathbb{P}$ such that
\textcolor{blue}egin{eqnarray}gin{equation} \label{tilde_p}
\frac{d\tilde{\mathbb{P}}}{d\mathbb{P}}=\exp\left(\int_0^T h_z(s,{\mathfrak{H}at T}eta_s) dW_s-\frac12\int_0^T\abs{h_z(s,{\mathfrak{H}at T}eta_s)}^2ds\right),
\end{equation}
where $h_z$ is bounded thanks to Assumption (L).
Under $\tilde{\mathbb{P}}$ defined by \eqref{tilde_p}, we obtain:
\textcolor{blue}egin{eqnarray}gin{align*}D_{r,s}^2 Y_t&=\mathbb{E}^{\tilde{\mathbb{P}}}\Big[g''(X_T)D_r X_T D_s X_T+g'(X_T)D_{r,s}^2 X_T\\
& \mathfrak{H}space{1cm}+\int_t^T[h_x(u,{\mathfrak{H}at T}eta_u)D_{r,s}^2 X_u+h_{xx}(u,{\mathfrak{H}at T}eta_u)D_r X_u D_s X_u+h_{xy}(u,{\mathfrak{H}at T}eta_u) D_s X_u D_rY_u\\
&\mathfrak{H}space{2cm}+h_y(u,{\mathfrak{H}at T}eta_u) D_{r,s}^2 Y_u +h_{xy}(u,{\mathfrak{H}at T}eta_u) D_r X_u D_s Y_u+h_{yy}(u,{\mathfrak{H}at T}eta_u) D_r Y_u D_s Y_u\\
&\mathfrak{H}space{2cm}+h_{zz}(u,{\mathfrak{H}at T}eta_u)D_r Z_u D_s Z_u] du\Big|{\cal F}_t\Big].
\end{align*}
By standard linearization techniques, we obtain:
\textcolor{blue}egin{eqnarray}gin{align*}D_{r,s}^2 Y_t&=\mathbb{E}^{\tilde{\mathbb{P}}}\Big[e^{\int_t^T h_y(u,{\mathfrak{H}at T}eta_u)du}(g''(X_T)D_r X_T D_s X_T+g'(X_T)D_{r,s}^2 X_T) \\
&\mathfrak{H}space{1cm}+\int_t^T e^{\int_t^u h_y(v,{\mathfrak{H}at T}eta_v)dv}[h_x(u,{\mathfrak{H}at T}eta_u)D_{r,s}^2 X_u+h_{xx}(u,{\mathfrak{H}at T}eta_u)D_r X_u D_s X_u\\
&\mathfrak{H}space{1cm}\mathfrak{H}space{1cm} + h_{xy}(u,{\mathfrak{H}at T}eta_y) (D_r X_u D_s Y_u +D_s X_u D_r Y_u)\\
&\mathfrak{H}space{1cm}\mathfrak{H}space{1cm}+h_{yy}(u,{\mathfrak{H}at T}eta_u) D_r Y_u D_s Y_u+h_{zz}(u,{\mathfrak{H}at T}eta_u)D_r Z_u D_s Z_u] du\Big|\mathcal F_t\Big]. \end{align*}
Then, using Remark \ref{positivite_dy}, Lemma \ref{lemma_gradient_malliavin} and our assumptions we obtain:
\textcolor{blue}egin{eqnarray}gin{align*}
&e^{\int_t^T h_y(u,{\mathfrak{H}at T}eta_u)du}(g''(X_T)D_r X_T D_s X_T+g'(X_T)D_{r,s}^2 X_T)\\
&\quad +\int_t^T e^{\int_t^u h_y(v,{\mathfrak{H}at T}eta_v)dv}[h_x(u,{\mathfrak{H}at T}eta_u)D_{r,s}^2 X_u+h_{xx}(u,{\mathfrak{H}at T}eta_u)D_r X_u D_s X_u\\
&\quad+ h_{xy}(u,{\mathfrak{H}at T}eta_y) (D_r X_u D_s Y_u +D_s X_u D_r Y_u)\\
&\quad +h_{yy}(u,{\mathfrak{H}at T}eta_u) D_r Y_u D_s Y_u+h_{zz}(u,{\mathfrak{H}at T}eta_u)D_r Z_u D_s Z_u] du\\
&\geq e^{\int_t^T h_y(u,{\mathfrak{H}at T}eta_u)du} \left(\mathbf{1}_{\{\underline{g''}<0\}}\underline{g''}\overline{a}^2+\underline{g'}\mathbf{1}_{\{\underline{g'}<0\}}\overline{b}+(\mathbf{1}_{\{\underline{g''}\geq 0\}}\underline{g''}+\underline{h_{xx}}(t)(T-t))\underline{a}^2\right)\geq 0.
\end{align*}
We deduce that:
\textcolor{blue}egin{eqnarray}gin{align*}D_{r,s}^2 Y_t\geq&\ \mathbb{E}^{\tilde{\mathbb{P}}}\Big[e^{\int_t^T h_y(u,{\mathfrak{H}at T}eta_u)du}\mathbf{1}_{X_T\in A}(g''(X_T)D_r X_T D_s X_T+g'(X_T)D_{r,s}^2 X_T) \\
&+\mathbf{1}_{X_T\in A}\int_t^T e^{-K(u-t)}[h_{xx}(u,{\mathfrak{H}at T}eta_u)\underline{a}^2] du\Big|\mathcal{F}_t\Big]\\
\geq &\ e^{-KT}\left(\mathbf{1}_{\{\underline{g''}^A<0\}}\underline{g''}^A\overline{a}^2+\underline{g'}^A\mathbf{1}_{\{\underline{g'}<0\}}\overline{b}\right)\mathbb{\tilde{\mathbb{P}}}(X_T\in A | \mathcal{F}_t)\\
& +e^{-KT}\left(\mathbf{1}_{\{\underline{g''}^A\geq 0\}}\underline{g''}^A+\underline{h_{xx}}(t)(T-t)\right)\underline{a}^2\mathbb{\tilde{\mathbb{P}}}(X_T\in A | \mathcal{F}_t).
\end{align*}
Using the fact that $D^2 Y_t$ is symmetric, the chain rule formula, \eqref{eq:Zu'} and \eqref{eq:D2Y} and the fact that $\lim_{s \nearrow t} D_{r,s}^2 X_t = \sigma'(t,X_t) D_r X_t$, we have that $ \lim_{s \nearrow t} D^2_{r,s} Y_t =D_r Z_t,$ from which we deduce that $D_r Z_t>0$, $\mathbb{P}-a.s.$ Then according to Bouleau and Hirsch's Theorem, we conclude that the law of $Z_t$ has a density with respect to the Lebesgue measure.
\textcolor{blue}egin{eqnarray}gin{flushright}
$\qed$
\end{flushright}
\end{proof}
| 3,292 | 50,703 |
en
|
train
|
0.51.6
|
\textcolor{blue}egin{eqnarray}gin{Remark}\label{ab}
Notice that the sign assumption on $D^2X$ can be obtained under the following sufficient conditions.
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[$(X+)$] For any $t\in[0,T]$, the maps $x\longmapsto b(t,x)$ and $x\longmapsto \sigma(t,x)$ are respectively in $\mathcal C^2(\R)$ and $\mathcal C^3(\R)$, and there exists $c>0$ such that $$\sigma\geq c>0,\quad \sigma' \geq 0,\quad \sigma'',\sigma'''\leq 0 \; \text{ and } \; [\sigma,[\sigma,b]]\geq 0,$$ where $[b,\sigma]$ denotes the Lie bracket between $b$ and $\sigma$ defined by $[b,\sigma]:=b'\sigma+\sigma' b$.
\item[$(X-)$] For any $t\in[0,T]$, the maps $x\longmapsto b(t,x)$ and $x\longmapsto \sigma(t,x)$ are respectively in $\mathcal C^2(\R)$ and $\mathcal C^3(\R)$, and there exists $c<0$ such that
$$\sigma\leq c<0,\ \sigma' \leq 0,\ \sigma'',\sigma'''\geq 0\text{ and }[\sigma,[\sigma,b]]\leq 0.$$
\end{itemize}
Indeed, according to the first step of the proof of Theorem 4.3 in \cite{AbouraBourguin}, Condition $(X+)$ $($resp. $(X-))$ ensures that $D^2 X$ is non-negative $($resp. non-positive$)$.\end{Remark}
\textcolor{blue}egin{eqnarray}gin{Remark}
\label{rem:X+-}
One can provide an alternative version of the previous result, whose proof follows the same lines as the one of Theorem \ref{thm_density_z_lip}. Fix $t$ in $(0,T]$, let Assumptions $(L)$, $(X)$ and $(D2)$ hold and assume that there exists $A\in{\cal B}(\mathbb{R})$ such that $\mathbb{P}(\left.X_T\in A \right| \mathcal{F}_t)>0$, and such that one of the two following conditions is satisfied:
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[$a)$] $(X+)$ and $(C+)$ hold true and $g''\geq0, \; g''_{\vert A}>0 \text{ and } g' \geq0, \quad \mathcal{L}(X_T)\text{-a.e.}$
\item[$b)$] $(X-)$ and $(C-)$ hold true and $g''\leq0, \; g''_{\vert A}<0 \text{ and } g' \leq0, \quad \mathcal{L}(X_T)\text{-a.e.},$
\end{itemize}
then, for all $t \in (0,T]$, the law of $Z_t$ has a density with the respect to Lebesgue measure.
\end{Remark}
When Assumption (M) holds, Theorem \ref{thm_density_z_lip} takes a different form as shown below in Theorem \ref{densite_z_f}, mainly because of Proposition \ref{prop_dy_dz_r}. Indeed, consider the following assumptions:
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[($\tilde{C}+$)] $h_{zz}\geq 0$ and $h_{xz}=h_{yz}\equiv 0$.
\item[($\tilde{C}-)$] $h_{zz}\leq 0$ and $h_{xz}=h_{yz}\equiv 0$.
\end{itemize}
Under Assumption ($\tilde{C}+)$ or ($\tilde{C}-)$, we recall that:
\textcolor{blue}egin{eqnarray}gin{align*}\label{drz}
D_r Z_t=&\ g''(X_T)|f'(T,W_T)|^2+g'(X_t)f''(T,W_T)\\
&+\int_t^T \Big[h_x(u,{\mathfrak{H}at T}eta_u)f''(u,W_u)+h_y(u,{\mathfrak{H}at T}eta_u) D_{r,t}^2 Y_u \Big]du\\
&+\int_t^T\Big[|f'(u,W_u)|^2 h_{xx}(u,{\mathfrak{H}at T}eta_u)+\left(h_{xy}(u,{\mathfrak{H}at T}eta_u)D_r Y_u+ D_t Y_uh_{xy}(u,{\mathfrak{H}at T}eta_u) \right) f'(u,W_u)\Big]du\\
&+\int_t^T\Big[h_{yy}(u,{\mathfrak{H}at T}eta_u) \underbrace{D_t Y_uD_r Y_u}_{=|Z_u|^2}+\underbrace{D_t Z_u D_r Z_u}_{=|D_r Z_u|^2} h_{zz}(u,{\mathfrak{H}at T}eta_u) \Big]du - \int_t^T D_{r,t}^2 Z_u d\tilde{W}_u,
\end{align*}
with $\tilde{W}:=W-\int_0^\cdot h_z(s,{\mathfrak{H}at T}eta_s) ds$.
We set $\theta=(x,y,z),$ and
\textcolor{blue}egin{eqnarray}gin{align*}
\tilde{h}(t,w,x,y,z,\tilde{z}):= &\ h_{xx}(t,\theta)|f'(t,w)|^2+h_x(t,\theta)f''(t,w)+(h_{yy}(t,\theta)z+2h_{xy}(t,\theta)f'(t,w))z\\
&+h_y(t,\theta)\tilde{z},
\end{align*}
$$ \underline{\tilde{h}}(t)=\min\limits_{(s,w,x,y,z,\tilde{z})\in [t,T]\times \mathbb{R}^5} \tilde{h}(s,w,x,y,z,\tilde{z}), \quad \overline{\tilde{h}}(t)=\max\limits_{(s,w,x,y,z,\tilde{z})\in [t,T]\times \mathbb{R}^5} \tilde{h}(s,w,x,y,z,\tilde{z}).$$
\textcolor{blue}egin{eqnarray}gin{Theorem}\label{densite_z_f}
Assume that $(M)$, $(L)$ and $(D2)$ are satisfied and that there exists $A\in{\cal B}(\R)$ such that $\mathbb{P}(X_T\in A | \mathcal{F}_t)>0$ and one of the two following assumptions holds:
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[$a)$] Assumption $(\tilde{C}+)$, $ \underline{((g'\circ f) f')'}+(T-t)\underline{\tilde{h}}(t) \geq 0$ and
$\underline{((g'\circ f) f')'_A} +(T-t)\underline{\tilde{h}}(t)>0.$
\item[$b)$] Assumption $(\tilde{C}-)$, $\overline{((g'\circ f) f')'} +(T-t)\overline{\tilde{h}}(t)\leq 0$ and $\overline{((g'\circ f) f')'_A}+(T-t)\overline{\tilde{h}}(t) <0.$
\end{itemize}
Then, the law of $Z_t$ is absolutely continuous with respect to the Lebesgue measure on $\mathbb{R}$.
\end{Theorem}
\textcolor{blue}egin{eqnarray}gin{proof}
Using Proposition \ref{prop_dy_dz_r}, we recall that:
\textcolor{blue}egin{eqnarray}gin{align*}
D_r Z_t=&\ g''(X_T)|f'(T,W_T)|^2+g'(X_t)f''(T,W_T)\\
&+\int_t^T \tilde{h}(u,W_u,X_u,Y_u,Z_u,D_r Z_u)+|D_r Z_u|^2 h_{zz}(u) du - \int_t^T D_{r,t}^2 Z_u d\tilde{W}_u,
\end{align*}
where $\tilde{W}:=W-\int_0^\cdot h_z(u,{\mathfrak{H}at T}eta_u) du$.
Then the proof follows exactly the same line as the one of Theorem \ref{thm_density_z_lip}.
\textcolor{blue}egin{eqnarray}gin{flushright}
$\qed$
\end{flushright}
\end{proof}
| 2,169 | 50,703 |
en
|
train
|
0.51.7
|
\section{The quadratic case}
\label{section:quadratic}
We now turn to the quadratic case and provide an extension of both Theorem \ref{thm_H+H-} and Theorem \ref{thm_density_z_lip}. Note however that the assumptions of these theorems do not find immediate counterparts in the quadratic setup since the latter involves the Lipschitz constant of $h$ with respect to the $z$ variable (see Remark \ref{rk:condi}). We also emphasize that existence of densities for the $Y$ and $Z$ components in the quadratic case that we consider here was open until now. We first make precise the quadratic growth setting together with existence, uniqueness and Malliavin differentiability results for these equations in the next section. Then, we investigate respectively in Sections \ref{section:quadratic:y} and \ref{section:quadratic:z} the existence of density for respectively $Y$ and $Z$.
\subsection{Generalities on quadratic FBSDEs}
\label{sub:quadprel}
In contadistinction to the previous section, we will now assume that $h$ exhibits quadratic growth in the $z$ variable. As noted in the introduction, this case is particularly useful for applications, especially in Finance where any pricing and hedging problem on an incomplete market which can be translated into a BSDE analysis will lead to a quadratic BSDE. The precise assumption for dealing with quadratic BSDEs is given as:
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[(Q)]
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[(i)] $g : \mathbb{R} \longrightarrow \mathbb{R}$ is bounded.
\item[(ii)] $h : [0,T]\times \mathbb{R}^3 \longrightarrow \mathbb{R}$ is such that:
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[$\triangleright$] There exists $(K,K_z,K_y)\in(\R_+^*)^3$ such that for all $(t,x,y,z) \in [0,T]\times \mathbb{R}^3$
$$\mathfrak{H}space{-3em}|h(t,x,y,z)|\leq K(1+|y|+|z|^2),\ \abs{h_z}(t,x,y,z)\leq K_z(1+|z|),\ \abs{h_y}(t,x,y,z)\leq K_y.$$
\item[$\triangleright$] There exists $C>0$ such that for all $(t,x,y,z_1,z_2) \in [0,T] \times \mathbb{R}^4$
\textcolor{blue}egin{eqnarray}gin{equation}\label{condition_h} |h(t,x,y,z_1)-h(t,x,y,z_2)| \leq C(1+|z_1|+|z_2|) |z_1-z_2|.\end{equation}
\end{itemize}
\item[(iii)] $\int_0^T |h(s,0,0,0)|^2ds<+\infty$.
\end{itemize}
\end{itemize}
Existence and uniqueness of a solution triplet $(X,Y,Z)$ under Assumption $(Q)$ has been obtained in \cite{Kobylanski}. More precisely:
\textcolor{blue}egin{eqnarray}gin{Proposition}[\cite{Kobylanski}]$($Existence and uniqueness of BSDEs$)$\label{propexq}
Under Assumptions $(X)$ and $(Q)$, there exists a unique solution $(X,Y,Z)$ in $\mathbb{S}^2 \times \mathbb S^\infty\times\mathbb H^2_{\rm BMO}$.
\end{Proposition}
Note that Condition \eqref{condition_h} on the generator $h$ in Assumption $(Q)$ in the one that ensures uniqueness of the solution. Hence, it can be dropped and one can then consider the maximal solution $Y$ of the BSDE, for which our proofs still apply.
Concerning the Malliavin differentiability of the processes $(X,Y,Z)$ it has been obtained in the quadratic case in \cite{AIDR} under the Assumptions $(D1)$ and $(D2)$ (that are defined in Section \ref{sub:lipprel}). Note that Proposition \ref{prop:Markov} still holds true if Assumption $(L)$ is replaced by Assumption $(Q)$. However, although the above proposition is completely proved in \cite{MaZhang} in the Lipschitz case, we did not find a proper reference in the quadratic case, except for \cite{IRR} which proves the result under Assumption $(Q)$, with the exception that $u$ is only shown to be in ${\cal C}^{1,1}$. Nonetheless, one can still obtain the required result by proving that Theorem $3.1$ of \cite{MaZhang} still holds for a BSDE with a driver which is uniformly Lipschitz in $y$ and stochastic Lipschitz in $z$ with a Lipschitz process in $\mathbb H^2_{\rm BMO}$ $($which is exactly the case of the BSDE satisfied by the Malliavin derivative of $Y)$. This can be achieved by following exactly the steps of the proof of Theorem 3.1 in \cite{MaZhang}, where the a priori estimates of their Lemma $2.2$ have to be replaced by those given in Lemma $A.1$ of \cite{IRR}. As in the Lipschitz case, Relation \eqref{eq:D2Y} still holds true under $(Q)$. In addition, as for Proposition \ref{prop:Markov}, the proof of Lemma \ref{lemma_gradient_malliavin} can be extended to the quadratic setting. Finally, Propositions \ref{MD} and \ref{prop_dy_dz_r} are valid if one replaces Assumption $(L)$ by Assumption $(Q)$.
\textcolor{blue}egin{eqnarray}gin{Proposition}$($Malliavin differentiabiliy$)$ \label{MDq}
Under $(X)$, $(Q)$ and $(D1)$, we have for any $t\in[0,T]$ that $(X_t,Y_t) \in (\mathbb{D}^{1,2})^2$, $Z_t \in \mathbb{D}^{1,2}$ for almost every $t$, and for all $0<r\leq t \leq T$:
\textcolor{blue}egin{eqnarray}gin{equation}\label{edsr_derive1}
\textcolor{blue}egin{eqnarray}gin{cases}
\displaystyle D_r X_t=\sigma(r,X_r) + \int_r^t b_x(s,X_s) D_r X_s ds + \int_r^t \sigma_x(s,X_s) D_r X_s dW_s\\
\displaystyle D_r Y_t=g'(X_T)D_rX_T+\int_t^T H(s,D_r X_s,D_r Y_s, D_r Z_s)ds -\int_t^T D_r Z_s dW_s,
\end{cases}
\end{equation}
where $H(s,x,y,z):=h_x(s,X_s,Y_s,Z_s)x+h_y(s,X_s,Y_s,Z_s)y+h_z(s,X_s,Y_s,Z_s)z.$
\end{Proposition}
| 1,829 | 50,703 |
en
|
train
|
0.51.8
|
\subsection{Existence of a density for the $Y$ component}
\label{section:quadratic:y}
\textcolor{blue}egin{eqnarray}gin{Theorem}\label{thm_H+H-_quadra}
Fix $t\in (0,T]$ and assume that $(X)$, $(Q)$ and $(D1)$ hold. If there is $A\in\mathcal B(\R)$ such that $\mathbb{P}(X_T\in A \ | \ \mathcal{F}_t)>0$ and one of the following assumptions holds $($see Definitions \eqref{eq:barg}-\eqref{eq:barh}$)$
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[$(Q+)$] $g'\geq 0$ and $g'_{\vert A} >0$, $\mathcal{L}(X_T)-$a.e. and $\underline{h}(t)\geq 0$,
\item[$(Q-)$] $g'\leq 0,\ g'_{\vert A}<0$, $\mathcal{L}(X_T)-$a.e. and $\overline{h}(t)\leq 0$,
\end{itemize}
then $Y_t$ has a law absolutely continuous with respect to the Lebesgue measure.
\end{Theorem}
\textcolor{blue}egin{eqnarray}gin{proof}
To simplify the notations for any $s$ in $[0,T]$, we set ${\mathfrak{H}at T}eta_s:=(X_s,Y_s,Z_s)$. We set $K:=k_b\vee k_y\vee k_\sigma$. We assume that (Q+) is satisfied (the proof with (Q-) follows the same lines, so we omit it). According to Bouleau-Hirsch's criterion, it is enough to show that $\gamma_{Y_t}:=\int_0^T |D_r Y_t|^2 dr >0$, $\mathbb{P}$-a.s. As in the proof of \cite[Theorem 3.6]{AntonelliKohatsu}, we have for $0\leq r \leq t \leq T$, that $D_r Y_t$ writes down as:
\textcolor{blue}egin{eqnarray}gin{equation}\label{eq:y}
D_r Y_t=g'(X_T)D_r X_T+\int_t^T
h_x(s,{\mathfrak{H}at T}eta_s)D_r X_s+h_y(s,{\mathfrak{H}at T}eta_s)D_r Y_s ds +\int_t^T D_r Z_s dW_s.
\end{equation}
From \reff{eq:y}, and following the expression of $\gamma_{Y_t}$ given in \cite[page 271]{AntonelliKohatsu}, we deduce that
$$ \gamma_{Y_t}=\left(\mathbb{E}\left[ g'(X_T)\zeta_T \psi_T +\int_t^T \psi_s h_x(s,{\mathfrak{H}at T}eta_s)\zeta_s ds | \mathcal{F}_t \right]\right)^2 (\psi_t^{-1})^2 \int_0^t (\zeta_r^{-1}\sigma(r,X_r))^2 dr,$$
with
$$ \psi_t\zeta_t=\underbrace{e^{\int_0^t (b_x(s,X_s)+h_y(s,{\mathfrak{H}at T}eta_s)+\sigma_x(s,X_s)h_z(s,{\mathfrak{H}at T}eta_s))ds}}_{=:E_t} \underbrace{e^{\int_0^t (\sigma_x(s,X_s)+h_z(s,{\mathfrak{H}at T}eta_s)) dW_s - \frac12 \int_0^t (\sigma_x(s,X_s)+h_z(s,{\mathfrak{H}at T}eta_s))^2 ds}}_{=:M_t}.$$
Let $\mathbb{Q}$ the probability measure equivalent to $\mathbb{P}$ with density $\frac{d\mathbb{Q}}{d\mathbb{P}}:=M_T$. Indeed, $M$ is a martingale as $\int_0^\cdot (\sigma_x(s,X_s)+h_z(s,{\mathfrak{H}at T}eta_s)) dW_s$ is a BMO martingale due to the boundedness of $\sigma_x$ (by (X)) and the fact that $|h_z(s,{\mathfrak{H}at T}eta_s)|\leq C(1+|Z_s|)$ (by (Q)) and from the BMO property of $\int_0^\cdot Z_s dW_s$ (by Proposition \ref{propex}). We therefore have:
$$ \mathbb{E}\left[ g'(X_T)\psi_T\zeta_T+\int_t^T \psi_s h_x(s,{\mathfrak{H}at T}eta_s)\zeta_s ds \Big|\mathcal{F}_t\right]=M_t\mathbb{E}^{\mathbb{Q}}\left[ g'(X_T) E_T +\int_t^T h_x(s,{\mathfrak{H}at T}eta_s) E_s ds \Big| \mathcal{F}_t\right].$$
Using (Q+), we know that:
$$g'(X_T)E_T+\int_t^T h_x(s,{\mathfrak{H}at T}eta_s) E_s ds \geq \underline{g}E_T+\underline{h}(t)\int_t^T E_sds\geq 0. $$
Thus,
\textcolor{blue}egin{eqnarray}gin{align*}
&\mathbb{E}\left[ g'(X_T)\psi_T\zeta_T+\int_t^T \psi_s h_x(s,{\mathfrak{H}at T}eta_s)\zeta_s ds \Big|\mathcal{F}_t\right]\\
&\geq M_t \mathbb{E}^{\mathbb{Q}}\left[ \mathbf{1}_{X_T \in A}\left(g'(X_T) E_T +\int_t^T h_x(s,{\mathfrak{H}at T}eta_s) E_s ds\right) \Big| \mathcal{F}_t\right]\\
&\geq M_t\Big(\underline{g}^Ae^{-2KT}\mathbb{E}^\mathbb{Q}\left[ \mathbf{1}_{X_T\in A} e^{-K\int_0^T |h_z(s,{\mathfrak{H}at T}eta_s)|ds}\textcolor{blue}ig| \mathcal{F}_t\right]\\
&\mathfrak{H}space{0.9em}+\underline{h}(t)e^{-2KT}(T-t)\mathbb{E}^\mathbb{Q}\left[ \mathbf{1}_{X_T\in A} e^{-K\int_0^T|h_z(s,{\mathfrak{H}at T}eta_s)|ds} \textcolor{blue}ig| \mathcal{F}_t \right]\Big)\\
&\geq M_t\Big(\underline{g}^Ae^{-2KT}\mathbb{E}^\mathbb{Q}\Big[ \mathbf{1}_{X_T\in A} e^{-K\sqrt{T}\sqrt{\int_0^T |h_z(s,{\mathfrak{H}at T}eta_s)|^2ds}}\textcolor{blue}ig| \mathcal{F}_t\Big]\\
&\mathfrak{H}space{0.9em}+\underline{h}(t)e^{-2KT}(T-t)\mathbb{E}^\mathbb{Q}\Big[ \mathbf{1}_{X_T\in A} e^{-K\sqrt{T}\sqrt{\int_0^T|h_z(s,{\mathfrak{H}at T}eta_s)|^2ds} }\textcolor{blue}ig| \mathcal{F}_t \Big]\Big),
\end{align*}
where the last inequality is due to Cauchy-Schwarz inequality. Besides, according to Assumption (Q), $|h_z(s,{\mathfrak{H}at T}eta_s)|\leq C(1+|Z_s|).$ Then, we deduce that $\int_0^T |h_z(s,{\mathfrak{H}at T}eta_s)|^2ds<+\infty, \ \mathbb{P}-$a.s., since $Z\in \mathbb{H}^2$. Hence, $M_t>0$, $\mathbb{P}-$a.s.
Given that the law of $X_T$ is absolutely continuous with respect to the Lebesgue measure, we deduce that $\mathbb{E}\left[ g'(X_T)\psi_T\zeta_T+\int_t^T \psi_s h_x(s,{\mathfrak{H}at T}eta_s)\zeta_s ds \Big|\mathcal{F}_t\right]>0, \ \mathbb{P}-\text{a.s.}$
We conclude using Theorem \ref{BH}.
\textcolor{blue}egin{eqnarray}gin{flushright}
$\qed$
\end{flushright}
\end{proof}
\textcolor{blue}egin{eqnarray}gin{Remark}
Similarly to Remark \ref{positivite_dy}, the proof of Theorem \ref{thm_H+H-_quadra} shows that under $(X)$, $(Q)$, $(D1)$ and if $g'\geq 0$ and $\underline{h}(t)\geq 0$ $($resp. $g'\leq 0$ and $\overline{h}(t)\leq 0)$ for $t \in [0,T]$, then for all $0<r\leq t \leq T$, $D_r Y_t \geq 0$ $($resp. $D_r Y_t \leq 0)$ and the inequality is strict if there exists $A\in{\cal B}(\R)$ such that $\mathbb{P}(X_T \in A|\mathcal{F}_t)>0$ and $g'_{|A}>0$ $($resp. $g'_{|A}<0)$.
\end{Remark}
\textcolor{blue}egin{eqnarray}gin{Remark}\label{rk:condi}
Conditions $(Q+)$ and $(Q-)$ are stronger than $(H+)$ and $(H-)$, due to the unboundedness of $h_z$, which prevents us from reproducing the same proof than in \cite{AntonelliKohatsu}. Indeed, in this framework the quantity appearing for instance in $(H+)$ becomes:
$$ \underline{g}e^{-2K\text{sgn}(\underline{g})T}e^{-K\text{sgn}(\underline{g})\int_0^T |h_z(s)|ds}+\underline{h}(t)e^{-2K\text{sgn}(\underline{h}(t))T}\int_t^T e^{-K\text{sgn}(\underline{h}(t))\int_0^s |h_z(s)|ds},$$
whose sign for every $K\geq 0$ depends strongly on those of $g'$ and $h_x$. This is why we must use the stronger conditions $(Q+)$ and $(Q-)$.
\end{Remark}
\textcolor{blue}egin{eqnarray}gin{Remark}
In \cite[Corollary 3.5]{dosdos} comonotonicty conditions on the data of a BSDE under Assumption $(Q)$ are given so that $Z_t \geq 0$, $\mathbb{P}-$a.s., $\forall t \in [0,T]$. In addition, the authors claim that strict comonotonicity entails that $Z_t>0$, which implies by Bouleau-Hirsch criterion that the law of $Y_t$ has a density with respect to the Lebesgue measure. However, we do not understand their proof and it is not true that an increasing mapping which is differentiable has a positive derivative everywhere $($even if one relaxes it by asking for a positive derivative Lebesgue-almost everywhere$)$ and one needs an extra assumption to prove that the derivative does not vanish. Indeed, take any closed set of positive Lebesgue measure with empty interior $($for instance the Smith-Volterra-Cantor set on $\R)$. By Whitney's extension Theorem, there exists a differentiable increasing map whose derivative vanishes on this set.
\end{Remark}
| 2,980 | 50,703 |
en
|
train
|
0.51.9
|
\subsection{Existence of a density for the control variable $Z$}
\label{section:quadratic:z}
In this section, we obtain existence results for the density of $Z$ under Assumption (Q). We actually have exactly the same type of results as in the Lipschitz case with similar proofs, which highlights the robustness and flexibility of our approach. Let us detail first the changes that we have to make.
Under (Q), using the fact that for all $s\in [0,T]$ $|h_z(s,{\mathfrak{H}at T}eta_s)|\leq C(1+|Z_s|)$ and according to Proposition \ref{propex} we deduce that $\int_0^\cdot h_z(s,{\mathfrak{H}at T}eta_s) dW_s$ is a BMO-martingale. Then, according to Theorem 2.3 in \cite{Kazamaki}, the stochastic exponential of $\int_0^\cdot h_z(s,{\mathfrak{H}at T}eta_s) dW_s$ is a uniformly integrable martingale and we can apply Girsanov's Theorem. We also emphasize that in (Q), $g$ is not assumed to be twice continuously differentiable. Indeed, to recover the BMO properties linked to quadratic BSDEs (and thus in order to be able to apply the above reasoning), $g$ needs to be bounded, which is incompatible with g convex (or concave). Nevertheless, there exist terminal conditions $g$ which are twice differentiable almost everywhere on the support of the law of $X_T$ (which is some closed subset of $\R$), such that their second-order derivative have a given sign there. As an example, take $X=W$ and $g(x):= f(x)\mathbf{1}_{x\in[a,b]}+f(a)\mathbf{1}_{x\leq a} + f(b) \mathbf{1}_{x\leq b}$ with $f$ a twice differentiable convex function and $a,b \in \mathbb{R}$.
\textcolor{blue}egin{eqnarray}gin{Theorem}\label{thm_density_z_quadra}
Let Assumptions $(X)$, $(Q)$ and $(D2)$ hold. Let $0< t \leq T$ and assume moreover
\textcolor{blue}egin{eqnarray}gin{itemize}
\item There exist $(\underline a,\overline a)$ s.t., $0<\underline{a}\leq D_r X_u \leq \overline{a}$, for all $0<r<u\leq T$.
\item There exists $\overline b$ s.t., $0\leq D_{r,s}^2 X_u\leq \overline{b}$, for all $0<r,s<u\leq T$.
\item $(C+)$ holds and $h_y\geq 0$.
\item $h_{xy}= 0$ or $(h_{xy}\geq 0$ and $g'\geq 0$, $\mathcal{L}(X_T)$-a.e.$)$.
\end{itemize}
If there exists $A\in{\cal B}(\R)$ such that $\mathbb{P}(X_T\in A | \mathcal{F}_t)>0$ and such that:
$$ \mathbf{1}_{\{\underline{g''}<0\}}\underline{g''}\overline{a}^2+\underline{g'}\mathbf{1}_{\{\underline{g'}<0\}}\overline{b}+(\mathbf{1}_{\{\underline{g''}\geq 0\}}\underline{g''}+\underline{h_{xx}}(t)(T-t))\underline{a}^2\geq 0,$$
and
$$ (\mathbf{1}_{\{\underline{g''}^A<0\}}\underline{g''}^A\overline{a}^2+\underline{g'}^A\mathbf{1}_{\{\underline{g'}<0\}}\overline{b}) +(\mathbf{1}_{\{\underline{g''}^A\geq 0\}}\underline{g''}^A+\underline{h_{xx}}(t)(T-t))\underline{a}^2>0,$$
then, the law of $Z_t$ has a density with respect to the Lebesgue measure.
\end{Theorem}
\textcolor{blue}egin{eqnarray}gin{proof} As in the proof of Theorem \ref{thm_density_z_lip}, we notice that for all $0<r,t\leq s\leq T$:
\textcolor{blue}egin{eqnarray}gin{align*}D_{r,s}^2 Y_t&=\mathbb{E}^{\tilde{\mathbb{P}}}\Big[g''(X_T)D_r X_T D_s X_T+g'(X_T)D_{r,s}^2 X_T\\
& \mathfrak{H}space{1cm}+\int_t^T[h_x(u,{\mathfrak{H}at T}eta_u)D_{r,s}^2 X_u+h_{xx}(u,{\mathfrak{H}at T}eta_u)D_r X_u D_s X_u+h_y(u,{\mathfrak{H}at T}eta_u)D_r Y_u D_s Y_u \\
&\mathfrak{H}space{1cm}\mathfrak{H}space{1cm}+h_{yy}(u,{\mathfrak{H}at T}eta_u)D^2_{r,s}Y_u+h_{zz}(u,{\mathfrak{H}at T}eta_u)D_r Z_u D_s Z_u] du\Big|{\cal F}_t\Big], \end{align*}
where $\tilde{\mathbb{P}}$ is the equivalent probability measure to $\mathbb{P}$ with density $$\frac{d\tilde{\mathbb{P}}}{d\mathbb{P}}:=\exp\left(\int_0^T h_z(u,{\mathfrak{H}at T}eta_u)dW_u-\frac12\int_0^T \abs{h_z(u,{\mathfrak{H}at T}eta_u)}^2du\right),$$ given that $\int_0^\cdot h_z(u,{\mathfrak{H}at T}eta_u)dW_u$ is a BMO-martingale and using Theorem 2.3 in \cite{Kazamaki}. Then the proof is similar to that of Theorem \ref{thm_density_z_lip}.
\textcolor{blue}egin{eqnarray}gin{flushright}
$\qed$
\end{flushright}
\end{proof}
\textcolor{blue}egin{eqnarray}gin{Remark} In order to satisfy the condition in Theorem \ref{thm_density_z_quadra}, there are basically two types of sufficient conditions
\textcolor{blue}egin{eqnarray}gin{itemize}
\item First of all, if the support of the law of $X_T$ is bounded from above, then one can take $g$ to continuously differentiable everywhere, non-decreasing, convex and bounded on this support. Then it suffices to take $h$ to be convex in $x$ as well.
\item However, when the support of the law of $X_T$ is no longer bounded from above, then it is no longer possible to find $g$ which is non-decreasing, bounded and convex on this support. We must therefore allow $g''$ to become non-positive, and the role of $h_{xx}$ becomes then crucial, as it has to be sufficiently positive in order to balance $g''$. As an example, take $X:=W$. Then $\overline a=\underline a =1$ and $\overline b=0$. One can choose $g(x):= \frac{1}{1+x^2}$. Then, there exists a positive constant $M$ such that $-2\leq g''(x)\leq M$ and by choosing $h$ such that $h$ satisfies the assumptions in Theorem \ref{thm_density_z_quadra} and $t\in(0,T)$ such that $\underline{h_{xx}}(t)(T-t)\geq 2$, we deduce that $Z_t$ admits a density.
\end{itemize}
\end{Remark}
We give also a theorem under Assumption (M):
\textcolor{blue}egin{eqnarray}gin{Theorem}\label{densite_z_f_quadra}
Assume that $(M)$, $(Q)$ and $(D2)$ are satisfied and that there exists $A\in{\cal B}(\R)$ such that $\mathbb{P}(X_T\in A | \mathcal{F}_t)>0$ and one of the two following assumptions holds:
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[$a)$] Assumption $(\tilde{C}+)$, $\underline{((g'\circ f) f')'}+(T-t)\underline{\tilde{h}}(t) \geq 0$ and
$ \underline{((g'\circ f) f')'_A} +(T-t)\underline{\tilde{h}}(t)>0.$
\item[$b)$] Assumption $(\tilde{C}-)$, $\overline{((g'\circ f) f')'} +(T-t)\overline{\tilde{h}}(t)\leq 0$ and $\overline{((g'\circ f) f')'_A}+(T-t)\overline{\tilde{h}}(t) <0.$
\end{itemize}
Then, the law of $Z_t$ is absolutely continuous with respect to the Lebesgue measure.
\end{Theorem}
The proof is the same as the proof of Theorem \ref{densite_z_f} using the BMO property of $\int_0^\cdot Z_s dW_s$, we therefore omit it. We now turn to the simplest case of quadratic growth BSDE and verify that it is covered by our result.
\textcolor{blue}egin{eqnarray}gin{Example}
Let us consider the following BSDE
\[ Y_t=g(W_T)+\int_t^T \frac12 |Z_s|^2ds -\int_t^T Z_s dW_s,\]
where $g$ is bounded. According to Theorem \ref{thm_density_z_quadra} with $\overline{a}=\underline{a}=1$, $\overline{b}=0$ and $h_{xx}=0$, we deduce that for all $t\in (0,T]$, the law of $Z_t$ has a density with respect to the Lebesgue measure if $g''\geq 0$, $\lambda(dx)$-a.e. and if there exists $A\in{\cal B}(\R)$ with positive Lebesgue measure such that $g''_{|A}>0$.
We emphasize that, as a sanity check, this can be verified by direct calculations. Indeed, using the fact that if $F \in \mathbb{D}^{1,2}$ then $D_r (\mathbb{E}[F|\mathcal{F}_t])=\mathbb{E}[D_r F|\mathcal{F}_t] \mathbf{1}_{[0,t)} (r)$ $($see \cite[Proposition 1.2.4]{Nualartbook}$)$ we deduce that if $0\leq r<t\leq T$ then:
$$D_r Y_t = \frac{\mathbb{E}[g'(W_T)e^{g(W_T)}| \mathcal{F}_t]}{\mathbb{E}[e^{g(W_T)}| \mathcal{F}_t]},$$which does not depend on $r$. Then according to Proposition \ref{prop:PP92},
$Z_t= \frac{\mathbb{E}[g'(W_T)e^{g(W_T)}| \mathcal{F}_t]}{\mathbb{E}[e^{g(W_T)}| \mathcal{F}_t]}.$
Take $0<r<t\leq T$, then:
$$D_r Z_t= \frac{\mathbb{E}[g''(W_T)e^{g(W_T)}+ |g'(W_T)|^2 e^{g(W_T)} | \mathcal{F}_t]\mathbb{E}[e^{g(W_T)}| \mathcal{F}_t]-|\mathbb{E}[g'(W_T)e^{g(W_T)}| \mathcal{F}_t]|^2}{\mathbb{E}[e^{g(W_T)}| \mathcal{F}_t]}.$$
Using Cauchy-Schwarz inequality, if $g''\geq 0$, $\lambda(dx)$-a.e. and if there exists $A\in{\cal B}(\R)$ with positive Lebesgue measure such that $g''_{|A}>0$, we deduce that for all $t\in (0,T]$, $Z_t$ has a density with respect to the Lebesgue measure by Theorem \ref{BH}.
\end{Example}
| 3,022 | 50,703 |
en
|
train
|
0.51.10
|
\section{Density estimates for the marginal laws of $Y$ and $Z$}
\label{section:densY}
Up to now, the density estimates obtained in the literature relied mainly on the fact that the framework considered implied that the Malliavin derivative of $Y$ was bounded. Hence, using the Nourdin-Viens' formula (or more precisely their Corollary 3.5 in \cite{NourdinViens}), it could be showed that the law of $Y$ has Gaussian tails. Although such an approach is perfectly legitimate from the theoretical point of view, let us start by explaining why, as pointed out in the introduction, we think that this is not the natural framework to work with when dealing with BSDEs. Consider indeed the following example.
\textcolor{blue}egin{eqnarray}gin{Example}\label{rem.toostringent}
Let us consider the FBSDE \reff{edsr}, with $T=1$, $g(x):=x^3$, $h(t,x,y,z):=3x$, $b(t,x)=0$, $\sigma(t,x)=1$ and $X_0=0$. Then, simple computations show that the unique solution is given by
$$X_t=W_t,\quad Y_t=W_t^3+6W_t(1-t),\quad Z_t=3W_t^2+6(1-t).$$
Then, both $Y_t$ and $Z_t$ have a law which is absolutely continuous with respect to the Lebesgue measure, for every $t\in(0,1]$, but neither $Y_t$ nor $Z_t$ has Gaussian tails.
\end{Example}
Moreover, when it comes to applications dealing with generators with quadratic growth, assuming that the Malliavin derivative of $Y$ is bounded implies that the process $Z$ itself is bounded as $Z_t=D_t Y_t$, which is seldom satisfied in applications, since in general, one only knows that $Z\in\mathbb{H}^2_{\rm BMO}$.
One of the main applications of the results we obtain in this section is the precise analysis of the error in the truncation method in numerical schemes for quadratic BSDEs, introduced in \cite{ImkellerDosreis} and studied in \cite{ChassagneuxRichou}. We recall that according to Proposition \ref{prop:Markov} there exists a function $v : [0,T]\times \mathbb{R}\longmapsto \mathbb{R}$ in ${\cal C}^{1,2}$ such that $Y_t=v(t,X_t)$ and $Z_t=v_x(t,X_t)\sigma(t,X_t)$. Since we want to study the tails of the laws of $Y$ and $Z$, we will assume from now on that the support of these laws is $\mathbb{R}$, which implies that neither $v$ nor $v'$ is bounded from below or above. Moreover, we emphasize that throughout this section, we will assume that $Y_t$ and $Z_t$ do have a law which is absolutely continuous, so as to highlight the conditions needed to obtain the estimates. Throughout this section we assume that $X_t=W_t$ in \eqref{edsr} (that is $X_0=0, \; \sigma\equiv 1, \; b \equiv 0$).
\subsection{Preliminary results}
We will have to study the asymptotic growth of $v$ and $v_x$ in the neighborhood of $\pm \infty$. To this end, we introduce for any measurable function $f:\R\longrightarrow \R$ the following two kinds of growth rates:
\[ \overline{\alpha_f}:=\inf\left\{\alpha>0,\ \limsup\limits_{|x| \to +\infty} \abs{\frac{f(x)}{x^\alpha}}<+\infty\right\}, \quad \underline{\alpha_f}:=\inf\left\{\alpha>0,\ \liminf\limits_{|x| \to +\infty} \abs{\frac{f(x)}{x^\alpha}}<+\infty\right\}. \]
\textcolor{blue}egin{eqnarray}gin{Lemma}\label{lemme_v-1}
Let $ f \in \mathcal{C}^1(\mathbb{R})$. Assume that for all $x \in \mathbb{R}$, $f'(x)>0$. If $0<\underline{\alpha_f}<+\infty$ then for all positive constant $0<\eta<\underline{\alpha_f}$:
$$\overline{\alpha_{f^{(-1)}}}\leq \frac{1}{\underline{\alpha_f}-\eta},$$
where $f^{(-1)}$ is the inverse function of $f$.
\end{Lemma}
\textcolor{blue}egin{eqnarray}gin{proof}
Using the definition of $\underline{\alpha_f}$, we deduce that for all $\eta>0$,
\textcolor{blue}egin{eqnarray}gin{equation*}
\liminf\limits_{|x| \to + \infty} \abs{\dfrac{f(x)}{x^{\underline{\alpha_f}-\eta}}}=\lim\limits_{|x| \to + \infty} \abs{\dfrac{f(x)}{x^{\underline{\alpha_f}-\eta}}}=+\infty.
\end{equation*}
Since $f$ and $f^{(-1)}$ are increasing and unbounded from above and below, we deduce that there exists $\overline{x}>0$ such that for all $x\geq \overline{x}$, $f(x)$ and $f^{(-1)}$ are positive. Then, for all $M>0$, there exists $x_0 \geq \overline{x}$ such that for all $x\geq x_0>0$ and for all $y \geq M x_0^{\underline{\alpha_f}-\eta}\vee \overline x$
\textcolor{blue}egin{eqnarray}gin{align*}
f(x)\geq M x^{\underline{\alpha_f}-\eta}\Longleftrightarrow &\; f\left((y M^{-1})^\frac{1}{\underline{\alpha_v}-\eta}\right)\geq y\Longleftrightarrow(y M^{-1})^{\frac{1}{\underline{\alpha_f}-\eta}} \geq f^{(-1)}(y).
\end{align*}
This implies directly that $\limsup\limits_{y \to +\infty} \abs{\frac{f^{(-1)}(y)}{y^{\frac{1}{\underline{\alpha_f}-\eta}}}} <+\infty.$ The proof is similar when $y$ goes to $ -\infty$.
\textcolor{blue}egin{eqnarray}gin{flushright}
$\qed$
\end{flushright}
\end{proof}
It is rather natural to expect that for well-behaved functions $f \in \mathcal{C}^1(\mathbb{R})$, $\overline{\alpha_f}=\underline{\alpha_f}$ and $\overline{\alpha_f}=\overline{\alpha_{f'}}+1$. However, the situation is unfortunately not that clear. First of all, this may not be true if $f$ is not monotone. Indeed, let $f(x):=x^2\sin(x)$, then $\overline{\alpha_f}=\underline{\alpha_f}=2$. Furthermore, the strict monotonicity of $f$ is not sufficient either. Without being completely rigorous, let us describe a counterexample. Consider a function $f$ defined on $\R_+$, equal to the identity on $[0,1]$, which then increases as $x^4$ until it crosses $x\longmapsto x^2$ for the first time, which then increases as $x^{1/2}$ until it crosses $x\longmapsto x$ for the first time and so on. Finally, extend it by symmetry to $\R_-$. Then, it can be checked that $\overline{\alpha_f}=2$, $\underline{\alpha_f}=1$, $\overline{\alpha_{f'}}=3$, $\underline{\alpha_{f'}}=0$.
A nice sufficient condition for the aforementioned result to hold is that $f'$ is a \textit{regularly varying function} (see \cite{BinghamGoldieTeugels} and \cite{Seneta}).
\textcolor{blue}egin{eqnarray}gin{Lemma}\label{prop_regularly}
Assume that $f'$ is equivalent in $+ \infty$ $($resp. in $-\infty)$ to a regularly varying function with Karamata's decomposition $x^\textcolor{blue}egin{eqnarray}ta L_1(x)$ where $L_1$ is slowly varying $($resp. $x^\textcolor{blue}egin{eqnarray}ta L_2(x)$ where $L_2$ is slowly varying$)$ and where $\textcolor{blue}egin{eqnarray}ta>0$. Then
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[$\rm{(i)}$] $f$ is equivalent in $+ \infty$ $($resp. in $-\infty)$ to a regularly varying function with Karamata's decomposition $x^{\textcolor{blue}egin{eqnarray}ta+1} \widetilde{L_1}(x)$ where $\widetilde{L_1}$ is slowly varying $($resp. $x^{\textcolor{blue}egin{eqnarray}ta+1} \widetilde{L_2}(x)$ where $\widetilde{L_2}$ is slowly varying$)$.
\item[$\rm{(ii)}$] $ \overline{\alpha_f}=\underline{\alpha_f}=\underline{\alpha_{f'}}+1=\overline{\alpha_{f'}}+1.$
\end{itemize}
\end{Lemma}
\textcolor{blue}egin{eqnarray}gin{proof} By Karamata's Theorem (see Theorem 1.5.11 in \cite{BinghamGoldieTeugels} with $\sigma=1$), for any $x_0 \in \mathbb{R}$:
\textcolor{blue}egin{eqnarray}gin{equation}
\label{eq:cvreg}
\frac{xf'(x)}{f(x)-f(x_0)} \longrightarrow \textcolor{blue}egin{eqnarray}ta +1, \ \text{ when } x\longrightarrow +\infty.
\end{equation}
In addition, $f'$ is equivalent to a regularly varying function with Karamata's decomposition $x^\textcolor{blue}egin{eqnarray}ta L_1(x)$ when $x\longrightarrow +\infty$, hence in view of \eqref{eq:cvreg}, there exists a function $\widetilde{L_1}$ (equivalent to a constant times $L_1$ at $+\infty$) slowly varying such that $f$ is equivalent when $x\longrightarrow +\infty$ to a regularly varying function with Karamata's decomposition $x^{\textcolor{blue}egin{eqnarray}ta+1}\widetilde{L_1}(x)$. The same result holds when $x\longrightarrow -\infty$.
We now show (ii). According to Proposition 1.3.6 (v) in \cite{BinghamGoldieTeugels} and (i), we deduce that:
$$ \overline{\alpha_{f}}=\textcolor{blue}egin{eqnarray}ta+1=\underline{\alpha_{f}} \ \text{ and } \ \overline{\alpha_{f'}}=\textcolor{blue}egin{eqnarray}ta=\underline{\alpha_{f'}} .$$
\textcolor{blue}egin{eqnarray}gin{flushright}
$\qed$
\end{flushright}
\end{proof}
| 2,667 | 50,703 |
en
|
train
|
0.51.11
|
\subsection{A general estimate}
From now on, for a map $(t,x)\longmapsto v(t,x)$, $v'(t,x)$ will denote for simplicity the derivative of $v$ with respect to the space variable. Before enonciating a general theorem which gives us density estimates for the tails of the law of random variables of the form $v(t,W_t)$ and will be used to obtain estimates for the laws of $Y_t$ and $Z_t$, we set some
constants in order to simplify the notations in Theorem \ref{thm_estime_y} below.
\paragraph*{List of constants} Let $\alpha\in (0,+\infty)$, $\alpha' \in\R_+$ and $\tilde\alpha>0$. For $\varepsilon >0$, we set
$$C_{\varepsilon,v,\alpha}:=\sup\limits_{x\in \R,\; t\in [0,T]} \frac{\abs{v(t,x)}}{1+|x|^{\alpha+\varepsilon}},\ \delta_{\alpha'}:= \max(1,2^{\alpha'}), \ \Xi_{\alpha'}:=\frac{\alpha'\Gamma\left(\frac{1+\alpha'}{2}\right)}{2\sqrt{\pi}},\ \mu(\tilde\alpha):=\int_\R\frac{\phi(z)}{1+|z|^{\tilde{\alpha}}} dz,$$
$$ D_{\alpha'}:=\max\left(1+\delta_{\alpha'}\Xi_{\alpha'} +\frac{\delta_{\alpha'}^2}{2}\left( \Xi_{\alpha'} + (1+\alpha')^{-1}\right)^2, \frac12 + \frac{\delta_{\alpha'}}{1+\alpha'}\right),$$
where $\Gamma$ is the usual Euler function and $\phi$ the distribution function of the normal law, defined by
$$\Gamma(x):=\int_0^{+\infty} e^{-t}t^{x-1}dt,\ x> 0, \ \text{and }\phi(x):=\frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}},\ x\in\mathbb R.$$
We emphasize that the following theorem can be applied in much more general cases, and it is clearly not limited to the context of BSDEs. It could for instance be used to provide non-Gaussian tail estimates for the law of solutions to some SDEs. Therefore, it has an interest of its own.
\textcolor{blue}egin{eqnarray}gin{Theorem}\label{thm_estime_y}
Fix $t\in (0, T]$. Let $v: [0,T]\times \mathbb{R} \longrightarrow \mathbb{R}$ in $\mathcal{C}^{1,1}$ and let $P_t:=v(t,W_t)$. Assume furthermore that $P_t\in L^1(\mathbb{P})$, that $v$ is unbounded in $x$ both from above and from below, that $v'>0$, $\underline{\alpha_v}\in (0,+\infty)$, $\overline{\alpha_{v'}}<+\infty$ and that there exist $\tilde{\alpha}> 0$ and $K>0$ such that:
\textcolor{blue}egin{eqnarray}gin{equation}\label{ineg_v'}
\frac{1}{v'(t,x)}\leq K(1+|x|^{\tilde{\alpha}}),\text{ for all $x\in\R$}.
\end{equation}
Then, the law of $P_t$ has a density with respect to the Lebesgue measure, denoted by $\rho_t$, and for all $\varepsilon, \varepsilon' >0$ and for every $y\in\R$
\textcolor{blue}egin{eqnarray}gin{equation}\label{estime_y_alpha}
\rho_t(y)\leq \frac{\mathbb{E}[|P_t-\mathbb{E}[P_t]|]}{2M(\varepsilon')t}\left(1+|y|^{2\tilde{\alpha}{(\overline{\alpha_{v^{(-1)}}}+\varepsilon')}}\right)\exp \left(-\int_0^{y-\mathbb{E}[P_t]} \frac{(M'(\varepsilon,\varepsilon')t)^{-1}xdx}{1+|x+\mathbb{E}[P_t]|^{2(\overline{\alpha_{v'}}+\varepsilon)(\overline{\alpha_{v^{(-1)}}}+\varepsilon')}}\right),
\end{equation} and
\textcolor{blue}egin{eqnarray}gin{equation}\label{estime_y_alpha_bis}
\rho_t(y)\geq \frac{(2M'(\varepsilon,\varepsilon')t)^{-1}\mathbb{E}[|P_t-\mathbb{E}[P_t]|]}{1+|y|^{2(\overline{\alpha_{v'}}+\varepsilon)(\overline{\alpha_{v^{(-1)}}}+\varepsilon')}}\exp \left(-\int_0^{y-\mathbb{E}[P_t]} \frac{x\left(1+|x+\mathbb{E}[P_t]|^{2\tilde{\alpha}(\overline{\alpha_{v^{(-1)}}+\varepsilon')}}\right)dx}{M(\varepsilon')t}\right),
\end{equation}
with $$M'(\varepsilon, \varepsilon'):=C_{\varepsilon,v',\overline{\alpha_{v'}}}^2 D_{\overline{\alpha_{v'}}+\varepsilon}\left(1+C_{\varepsilon',v^{(-1)}, \overline{\alpha_{v^{(-1)}}}}^{2(\overline{\alpha_{v'}}+\varepsilon)}\right) \delta_{2(\overline{\alpha_{v'}}+\varepsilon)} ,$$
and $$ M(\varepsilon'):= \frac{\mu(\tilde\alpha)}{K^2 \left(1+C^{2\tilde \alpha}_{\varepsilon', v^{(-1)}, \overline{\alpha_{v^{(-1)}}}}\delta_{2\tilde\alpha}\right)} ,$$ using the aforementioned definitions of the constants.
\end{Theorem}
\textbf{Proof.} Notice immediately that since the map $x\longmapsto v(t,x)$ is in $\mathcal C^1(\R)$ and increasing, the law of $P_t$ clearly has a density. We prove inequalities \eqref{estime_y_alpha} and \eqref{estime_y_alpha_bis} using Nourdin and Viens' formula (see Theorem \ref{thm_NourdinViens}).The rest of the proof is divided into three steps.
{\textcolor{blue}f Step $1$:} Given that for all $0<r\leq t \leq T$, $D_r P_t=v'(t,W_t)$, the function $g_{P_t}$ defined by \eqref{gzt} becomes
$$ g_{P_t}(y):= \int_0^\infty e^{-a} \mathbb{E}\left[\mathbb{E}^*[\langle {\mathfrak{H}at P}i_{P_t}(W),\widetilde{{\mathfrak{H}at P}i_{P_t}^a}(W)\rangle_{\mathfrak{H}}] | P_t-\mathbb{E}[P_t]=y\right] da, \quad y\in \mathbb{R}, $$
with\footnote{Knowing that $D_r P_t$ does not depend on $r$, ${\mathfrak{H}at P}i_{P_t}(W):[0,T]\longrightarrow L^2(\Omega,{\cal F},\mathbb{P})$ is a random process which is actually constant on $[0,t]$.} ${\mathfrak{H}at P}i_{P_t}(W):=v'(t,W_t)$ and where $\widetilde{{\mathfrak{H}at P}i_{P_t}^a}(W):={\mathfrak{H}at P}i_{P_t}(e^{-a}W+\sqrt{1-e^{-2a}}W^*)$ with $W^*$ an independent copy of $W$ defined on a probability space $(\Omega^*,\mathcal{F}^*,\mathbb{P}^*)$ where $\mathbb{E}^*$ is the expectation under $\mathbb{P}^*$ (${\mathfrak{H}at P}i_{P_t}$ being extended on $\Omega\times \Omega^*$). Letting $\phi(z):=\frac{1}{\sqrt{2\pi t}} e^{-\frac{z^2}{2t}}$, we get that
\textcolor{blue}egin{eqnarray}gin{align} \label{expression_gy}
\noindentnumber g_{P_t}(y)&= \int_0^\infty e^{-a} \mathbb{E}\left[\mathbb{E}^*[\langle {\mathfrak{H}at P}i_{P_t}(W),\widetilde{{\mathfrak{H}at P}i_{P_t}^a}(W)\rangle_{\mathfrak{H}}] | W_t=v^{(-1)}(t,y+\mathbb{E}[P_t])\right] da, \quad y\in \mathbb{R},\\
&=tv'(t,v^{(-1)}(t,y+\mathbb{E}[P_t]))\int_0^\infty e^{-a} \int_\mathbb{R} v'\Big(t,e^{-a} v^{(-1)}(t,y+\mathbb{E}[P_t])+\sqrt{1-e^{-2a}}z\Big)\phi(z) dz da.
\end{align}
{\textcolor{blue}f Step $2$: Upper bound for $g_{P_t}$}
Recall that for all $\varepsilon >0$:
$$0<v'(t,x)\leq C_{\varepsilon,v',\overline{\alpha_{v'}}}\left(1+|x|^{\overline{\alpha_{v'}}+\varepsilon}\right), \quad \forall x \in \mathbb{R}.$$
Then, using \eqref{expression_gy} we get:
\textcolor{blue}egin{eqnarray}gin{align*}
g_{P_t}(y)\leq &\ C_{\varepsilon,v',\overline{\alpha_{v'}}}^2 t \left(1+\abs{v^{(-1)}\left(y+\mathbb{E}[P_t]\right)}^{\overline{\alpha_{v'}}+\varepsilon}\right)\\
&\times \int_0^{+\infty} e^{-a}\int_\mathbb{R}\left(1+ \abs{e^{-a} v^{(-1)}(t,y+\mathbb{E}[P_t])+\sqrt{1-e^{-2a}}z}^{\overline{\alpha_{v'}}+\varepsilon}\right)\phi(z) dz da\\
\leq &\ C_{\varepsilon,v',\overline{\alpha_{v'}}}^2 t \left(1+|v^{(-1)}(y+\mathbb{E}[P_t])|^{(\overline{\alpha_{v'}}+\varepsilon)}\right)\\
&\times \int_0^{+\infty} e^{-a}\int_\mathbb{R}\left(1+\delta_{\overline{\alpha_{v'}}+\varepsilon}\left(e^{-a(\overline{\alpha_{v'}}+\varepsilon)} \abs{ v^{(-1)}(t,y+\mathbb{E}[P_t])}^{\overline{\alpha_{v'}}+\varepsilon}+\abs{z}^{\overline{\alpha_{v'}}+\varepsilon}\right)\right)\phi(z) dz da\\
\leq & \ C_{\varepsilon,v',\overline{\alpha_{v'}}}^2 t \left(1+|v^{(-1)}(y+\mathbb{E}[P_t])|^{\overline{\alpha_{v'}}+\varepsilon}\right)\\
&\times \left(1+ \frac{\delta_{\overline{\alpha_{v'}}+\varepsilon}}{1+\overline{\alpha_{v'}}+\varepsilon} |v^{(-1)}(y+\mathbb{E}[P_t])|^{\overline{\alpha_{v'}}+\varepsilon} +\delta_{\overline{\alpha_{v'}}+\varepsilon} \Xi _{\overline{\alpha_{v'}}+\varepsilon}\right)\\
\leq & \ C_{\varepsilon,v',\overline{\alpha_{v'}}}^2 t D_{\overline{\alpha_{v'}} +\varepsilon}\left( 1+ |v^{(-1)}(y+\mathbb{E}[P_t])|^{2(\overline{\alpha_{v'}}+\varepsilon)}\right).
\end{align*}
By Lemma \ref{lemme_v-1}, $\overline{\alpha_{v^{(-1)}}}$ belongs to $(0,+\infty)$, hence by the definition of $\overline{\alpha_{v^{(-1)}}}$ it holds for all $\varepsilon'>0$ that
\textcolor{blue}egin{eqnarray}gin{equation}\label{upper_bound_gy}g_{P_t}(y)\leq M'(\varepsilon,\varepsilon')t\left(1+|y+\mathbb{E}[P_t]|^{2(\overline{\alpha_{v'}}+\varepsilon)(\overline{\alpha_{(v)^{-1}}}+\varepsilon')}\right).
\end{equation}
{\textcolor{blue}f Step $3$: Lower bound for $g_{P_t}$}
Using Assumption \eqref{ineg_v'} and \eqref{expression_gy} we have that
\textcolor{blue}egin{eqnarray}gin{align*}
g_{P_t}(y)\geq &\ \frac{t}{K^2(1+|v^{(-1)}(t,y+\mathbb{E}[P_t])|^{\tilde{\alpha}})}\\
&\times\int_0^{+\infty} e^{-a}\int_\mathbb{R} \frac{1}{1+|e^{-a} (v)^{-1}(t,y+\mathbb{E}[P_t])|^{\tilde{\alpha}}+|\sqrt{1-e^{-2a}}z|^{\tilde{\alpha}}}\phi(z) dz da.
\end{align*}
Noticing that $|\sqrt{1-e^{-2a}}z|^{\tilde{\alpha}}\leq |z|^{\tilde{\alpha}}$, and that
$$\int_\mathbb{R} \frac{(1+|x|^{\tilde{\alpha}})\phi(z)}{1+|x|^{\tilde{\alpha}}+|z|^{\tilde{\alpha}}} dz\geq \mu(\tilde\alpha), \quad \forall x\in \mathbb{R}$$
we deduce that:
\textcolor{blue}egin{eqnarray}gin{align*}
g_{P_t}(y)&\geq \frac{\mu(\tilde\alpha)t}{K^2(1+|v^{(-1)}(t,y+\mathbb{E}[P_t])|^{\tilde{\alpha}})}\int_0^{+\infty} e^{-a}\frac{1}{1+e^{-a\tilde{\alpha}}|v^{(-1)}(t,y+\mathbb{E}[P_t])|^{\tilde{\alpha}}}da.
\end{align*}
Hence:
$
g_{P_t}(y)\geq \frac{\mu(\tilde\alpha)t}{K^2(1+|v^{(-1)}(t,y+\mathbb{E}[P_t])|^{2\tilde{\alpha}})}.
$
We finally get Relation \eqref{estime_y_alpha_bis} for
$$M(\varepsilon'):= \frac{\mu(\tilde\alpha)}{K^2 \left(1+C^{2\tilde \alpha}_{\varepsilon', v^{(-1)}, \overline{\alpha_{v^{(-1)}}}}\delta_{2\tilde\alpha}\right)}.$$
We conclude using Nourdin and Viens' formula.
\textcolor{blue}egin{eqnarray}gin{flushright}
$\qed
| 3,590 | 50,703 |
en
|
train
|
0.51.12
|
$
\end{flushright}
\textcolor{blue}egin{eqnarray}gin{Corollary}\label{cor_estime_y}
Let the assumptions in Theorem \ref{thm_estime_y} hold, with the same notations. Assume moreover that $0\leq \overline{\alpha_{v'}}<\underline{\alpha_v}<+\infty $. Then there exist $\varepsilon_0,\varepsilon'_0>0$, $y_0>0$ and $\gamma\in(0,1)$ such that for any $|y|> y_0$:
\textcolor{blue}egin{eqnarray}gin{equation}\label{estime_y_alpha_sansint}
\rho_t(y)\leq \frac{\mathbb{E}[|P_t-\mathbb{E}[P_t]|]}{2M(\varepsilon'_0)t}\left(1+|y|^{2\tilde{\alpha}{(\overline{\alpha_{v^{(-1)}}}+\varepsilon'_0)}}\right)\exp\left(-\frac{\abs{y-\mathbb{E}[P_t]}^{2(1-\gamma)}-\abs{y_0-\mathbb{E}[P_t]}^{2(1-\gamma)}}{4(1-\gamma)tM'(\varepsilon_0,\varepsilon'_0)}\right),
\end{equation}
and
\textcolor{blue}egin{eqnarray}gin{align}\label{estime_y_alpha_bis_sansint}
\noindentnumber\rho_t(y)\geq &\ \frac{\mathbb{E}[|P_t-\mathbb{E}[P_t]|]}{2M'(\varepsilon_0,\varepsilon'_0)t\left(1+|y|^{\gamma}\right)}\exp\left(-\frac{\abs{y-\mathbb{E}[P_t]}^{2(\tilde\alpha(\overline{\alpha_{v^{(-1)}}}+\varepsilon'_0)+1)}-\abs{y_0-\mathbb{E}[P_t]}^{2(\tilde\alpha(\overline{\alpha_{v^{(-1)}}}+\varepsilon'_0)+1)}}{M(\varepsilon_0')t(\tilde\alpha(\overline{\alpha_{v^{(-1)}}}+\varepsilon'_0)+1)}\right)\\
&\times \exp\left(-\frac{\abs{y_0-\mathbb{E}[P_t]}^2}{M(\varepsilon_0')t}\left(1+y_0^{2\tilde\alpha\left(\overline{\alpha_{v^{(-1)}}}+\varepsilon'_0\right)}\right)\right).
\end{align}
\end{Corollary}
\textcolor{blue}egin{eqnarray}gin{proof}
Let us define for any $\varepsilon,\varepsilon'>0$
$$\gamma(\varepsilon,\varepsilon'):=(\overline{\alpha_{v'}}+\varepsilon) (\overline{\alpha_{v^{(-1)}}}+\varepsilon').$$
Since we assumed that $0\leq \overline{\alpha_{v'}}<\underline{\alpha_v}<+\infty $, we can deduce using Lemma \ref{lemme_v-1} that there exist some $\varepsilon_0,\varepsilon_0'>0$ such that
$$\gamma:=\gamma(\varepsilon_0,\varepsilon_0')<1.$$
We start with \reff{estime_y_alpha_sansint}. We have from Theorem \ref{thm_estime_y}
$$ \rho_t(y)\leq \frac{\mathbb{E}[|P_t-\mathbb{E}[P_t]|]}{2M(\varepsilon'_0)t}\left(1+|y|^{2\tilde{\alpha}{(\overline{\alpha_{v^{(-1)}}}+\varepsilon'_0)}}\right)\exp \left(-\int_0^{y-\mathbb{E}[P_t]} \frac{xdx}{M'(\varepsilon_0,\varepsilon'_0)t\left(1+|x+\mathbb{E}[P_t]|^{2\gamma}\right)}\right).$$
We notice that $$\lim\limits_{\abs{x}\to +\infty}\frac{x}{M'(\varepsilon_0,\varepsilon_0')t(1+|x+\mathbb{E}[P_t]|^{2\gamma})} \times \frac{1}{\frac{x}{M'(\varepsilon_0,\varepsilon_0')t|x|^{2\gamma}}}=1,$$ so that there exists $x_0$ large enough such that $\frac{x}{M'(\varepsilon_0,\varepsilon_0')t(1+|x+\mathbb{E}[P_t]|^{2\gamma})} \geq \frac{x}{2M'(\varepsilon_0,\varepsilon_0')t|x|^{2\gamma}}$ when $\abs{x}\geq x_0$. Hence, since $\gamma\in(0,1)$, we know that we can find some $y_0>0$ large enough such that if $\abs{y}>y_0$
\textcolor{blue}egin{eqnarray}gin{align*}
& \int_{y_0-\mathbb{E}[P_t]}^{y-\mathbb{E}[P_t]} \frac{xdx}{M'(\varepsilon_0,\varepsilon_0')t(1+|x+\mathbb{E}[P_t]|^{2\gamma})}\\&\geq \int_{y_0-\mathbb{E}[P_t]}^{y-\mathbb{E}[P_t]} \frac{xdx}{2M'(\varepsilon_0,\varepsilon_0')t|x|^{2\gamma}}\\
&=\frac{1}{4(1-\gamma)tM'(\varepsilon_0,\varepsilon'_0)}\left(\abs{y-\mathbb{E}[P_t]}^{2(1-\gamma)}-\abs{y_0-\mathbb{E}[P_t]}^{2(1-\gamma)}\right),
\end{align*}
from which \reff{estime_y_alpha_sansint} follows directly. Similarly, increasing $y_0$ if necessary, we have that for $\abs{y}>y_0$
\textcolor{blue}egin{eqnarray}gin{align*}
&\int_0^{y-\mathbb{E}[P_t]}x\left(1+|x+\mathbb{E}[P_t]|^{2\tilde{\alpha}(\overline{\alpha_{v^{(-1)}}}+\varepsilon'_0)}\right)dx\\
&=\underbrace{\int_0^{y_0-\mathbb{E}[P_t]} x\left(1+|x+\mathbb{E}[P_t]|^{2\tilde{\alpha}(\overline{\alpha_{v^{(-1)}}}+\varepsilon'_0)}\right)dx}_{:=I_1}+\underbrace{\int_{y_0-\mathbb{E}[P_t]}^{y-\mathbb{E}[P_t]} x\left(1+|x+\mathbb{E}[P_t]|^{2\tilde{\alpha}(\overline{\alpha_{v^{(-1)}}}+\varepsilon'_0)}\right)dx}_{:=I_2}.\\
\end{align*}
Using the fact that the function $x\longmapsto 1+|x+\mathbb{E}[P_t]|^{2\tilde{\alpha}(\overline{\alpha_{v^{(-1)}}}+\varepsilon'_0)} $ is convex, we deduce that for $y_0$ large enough
$$I_1\leq \abs{y_0-\mathbb{E}[P_t]}^2\left(1+y_0^{2\tilde\alpha\left(\overline{\alpha_{v^{(-1)}}}+\varepsilon'_0\right)}\right). $$
Moreover, since $\lim\limits_{x\to +\infty} x\left(1+|x+\mathbb{E}[P_t]|^{2\tilde{\alpha}(\overline{\alpha_{v^{(-1)}}+\varepsilon'_0)} +\varepsilon'_0)}\right) \times \frac{1}{x^{2{\tilde{\alpha}(\overline{\alpha_{v^{(-1)}}}+\varepsilon'_0)+1}}}=1 $, we obtain for $x$ large enough
$$ x\left(1+|x+\mathbb{E}[P_t]|^{2\tilde{\alpha}(\overline{\alpha_{v^{(-1)}}+\varepsilon'_0)} +\varepsilon'_0)}\right)\leq 2x^{2{\tilde{\alpha}(\overline{\alpha_{v^{(-1)}}}+\varepsilon'_0)+1}}. $$
Then, we have that for $|y|\geq y_0$
$$ I_2\leq \frac{\abs{y-\mathbb{E}[P_t]}^{2(\tilde\alpha(\overline{\alpha_{v^{(-1)}}}+\varepsilon'_0)+1)}-\abs{y_0-\mathbb{E}[P_t]}^{2(\tilde\alpha(\overline{\alpha_{v^{(-1)}}}+\varepsilon'_0)+1)}}{\tilde\alpha(\overline{\alpha_{v^{(-1)}}}+\varepsilon'_0)+1}.$$
Hence,
\textcolor{blue}egin{eqnarray}gin{align*}
&\int_0^{y-\mathbb{E}[P_t]}x\left(1+|x+\mathbb{E}[P_t]|^{2\tilde{\alpha}(\overline{\alpha_{v^{(-1)}}}+\varepsilon'_0)}\right)dx\\
&\leq \abs{y_0-\mathbb{E}[P_t]}^2\left(1+y_0^{2\tilde\alpha\left(\overline{\alpha_{v^{(-1)}}}+\varepsilon'_0\right)}\right) +\frac{\abs{y-\mathbb{E}[P_t]}^{2(\tilde\alpha(\overline{\alpha_{v^{(-1)}}}+\varepsilon'_0)+1)}-\abs{y_0-\mathbb{E}[P_t]}^{2(\tilde\alpha(\overline{\alpha_{v^{(-1)}}}+\varepsilon'_0)+1)}}{\tilde\alpha(\overline{\alpha_{v^{(-1)}}}+\varepsilon'_0)+1},
\end{align*}
from which the second inequality \reff{estime_y_alpha_bis_sansint} follows directly using \reff{estime_y_alpha_bis}.
\textcolor{blue}egin{eqnarray}gin{flushright}
$\qed$
\end{flushright}
\end{proof}
| 2,353 | 50,703 |
en
|
train
|
0.51.13
|
Finally, we have the following theorem, which is a simple application of the results obtained above in the special cases where we take the random variables $(Y_t,Z_t)$ solutions to the BSDE \reff{edsr} when they can be written $Y_t=v(t,W_t)$ and $Z_t=v'(t,W_t)$.
\textcolor{blue}egin{eqnarray}gin{Theorem}\label{estim.dens}
Let $(Y,Z)$ be the solution to the BSDE \reff{edsr} $($which is assumed to exist and to be unique$)$. Assume that there exists a map $v\in\mathcal C^{1,2}$ such that $Y_t=v(t,W_t)$.
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[$\rm{(i)}$] If in addition, $v'>0$, $0\leq \overline{\alpha_{v'}}<\underline{\alpha_v}<+\infty $ and there exist $K>0$, $\tilde\alpha>0$ such that $v'(t,x)\geq 1/(K(1+\abs{x}^{\tilde\alpha}))$ then, denoting $\rho_{Y_t}$ the density of the law of $Y_t$, there exist $y_0>0$, $C_1,C_2>0$, $p_1\in(0,2)$ and $p_2>0$ $($which are given explicitly in Theorem \ref{thm_estime_y}$)$ such that for any $\abs{y}>y_0$
\textcolor{blue}egin{eqnarray}gin{align*}
\rho_{Y_t}(y)\geq&\ \frac{\mathbb{E}[\abs{Y_t-\mathbb{E}[Y_t]}}{C_2t\left(1+\abs{y}^{1-p_1/2}\right)}\exp\left(-\frac{\abs{y-\mathbb{E}[Y_t]}^{2(p_2+1)}-\abs{y_0-\mathbb{E}[Y_t]}^{2(p_2+1)}}{(p_2+1)C_2t}\right)\\
\rho_{Y_t}(y)\leq&\ \frac{\mathbb{E}[\abs{Y_t-\mathbb{E}[Y_t]}}{C_1t}\left(1+\abs{y}^{2p_2}\right)\exp\left(-\frac{2\abs{y_0-\mathbb{E}[Y_t]}^2}{C_2t}\left(1+y_0^{2p_2}\right)\right)\\
&\times\exp\left(-\frac{\abs{y-\mathbb{E}[Y_t]}^{p_1}-\abs{y_0-\mathbb{E}[Y_t]}^{p_1}}{p_1C_2t}\right).
\end{align*}
\item[$\rm{(ii)}$] If in addition, $v''>0$, $0\leq \overline{\alpha_{v''}}<\underline{\alpha_{v'}}<+\infty $ and there exist $K>0$, $\tilde\alpha>0$ such that $v''(t,x)\geq 1/(K(1+\abs{x}^{\tilde\alpha}))$ then, denoting $\rho_{Z_t}$ the density of the law of $Z_t$, there exists $Z_0>0$, $C_1,C_2>0$, $p_1\in(0,2)$ and $p_2>0$ $($which are given explicitly in Theorem \ref{thm_estime_y}$)$ such that for any $\abs{z}>z_0$
\textcolor{blue}egin{eqnarray}gin{align*}
\rho_{Z_t}(z)\geq&\ \frac{\mathbb{E}[\abs{Z_t-\mathbb{E}[Z_t]}}{C_2t\left(1+\abs{z}^{1-p_1/2}\right)}\exp\left(-\frac{\abs{z-\mathbb{E}[Z_t]}^{2(p_2+1)}-\abs{z_0-\mathbb{E}[Z_t]}^{2(p_2+1)}}{(p_2+1)C_2t}\right)\\
\rho_{Z_t}(y)\leq&\ \frac{\mathbb{E}[\abs{Z_t-\mathbb{E}[Z_t]}}{C_1t}\left(1+\abs{z}^{2p_2}\right)\exp\left(-\frac{2\abs{z_0-\mathbb{E}[Z_t]}^2}{C_2t}\left(1+z_0^{2p_2}\right)\right)\\
&\times\exp\left(-\frac{\abs{z-\mathbb{E}[Z_t]}^{p_1}-\abs{z_0-\mathbb{E}[Z_t]}^{p_1}}{p_1C_2t}\right).
\end{align*}
\end{itemize}
\end{Theorem}
| 1,167 | 50,703 |
en
|
train
|
0.51.14
|
\subsection{Verifying the assumptions of Theorem \ref{estim.dens}}
In this subsection, we give some conditions which ensure that the assumptions in Corollary \ref{cor_estime_y} hold. We recall that under Assumptions (X), (L) or (Q), (D1) and according to Proposition \ref{prop:Markov}, there exists a map $u:[0,T] \times \mathbb{R} \longrightarrow \mathbb{R}$ in $\mathcal{C}^{1,2}$ such that $Y_t =u(t,W_t), \ t\in [0,T], \ \mathbb{P}-$a.s., and $Z$ admits a continuous version given by $ Z_t = u'(t,W_t), \ t\in [0,T], \ \mathbb{P}-$a.s., assuming that $\sigma\equiv1$ and $b\equiv 0$ in the studied FBSDE \eqref{edsr}. Moreover we suppose for simplicity that the generator $h$ of BSDE \eqref{edsr} depends only on $z$, and that $u'$ and $u''$ are\footnote{This assumption is satisfied if $g$ and $h$ are smooth enough.} in $\mathcal{C}^{1,2}$. By a simple application of the non-linear Feynman-Kac formula (see for instance \cite{PardouxPeng_92}), and by differentiating it repeatedly, it can be shown that $u$, $u'$ and $u''$ are respectively classical solutions of the following PDEs:
\textcolor{blue}egin{eqnarray}gin{align}\label{PDE}
&\textcolor{blue}egin{eqnarray}gin{cases}
\displaystyle -u_t(t,x)-\frac12 u_{xx}(t,x)-h(t, u_x(t,x))=0,\ (t,x)\in [0,T)\times \mathbb{R} \\
\displaystyle u(T,x)=g(x),\ x\in \mathbb{R},
\end{cases}\\
\label{PDE'}
& \textcolor{blue}egin{eqnarray}gin{cases}
\displaystyle - u'_t(t,x)-\frac12 u'_{xx}(t,x)-h_z(t, u'(t,x)) u'_x(t,x)=0,\ (t,x)\in [0,T)\times \mathbb{R} \\
\displaystyle u'(T,x)=g'(x),\ x\in \mathbb{R},\end{cases}\\
\label{PDE''}
& \textcolor{blue}egin{eqnarray}gin{cases}
\displaystyle - u''_t(t,x)-\frac12 u''_{xx}(t,x)-h_z(t, u'(t,x)) u''_x(t,x)-h_{zz}(t,u'(t,x))|u''(t,x)|^2=0,\; \mathfrak{H}space{-0.2em}(t,x)\in [0,T)\times \mathbb{R} \\
\displaystyle u''(T,x)=g''(x),\ x\in \mathbb{R}.
\end{cases}
\end{align}
We show in the following proposition and its corollary that under some conditions on $g,g',g''$ and $h, h_z$, the assumptions in Theorem \ref{estim.dens} are satisfied. We emphasize that this is only one possible set of assumptions, and that the required properties of $u$ and its derivatives can be checked on a case by case analysis.
\textcolor{blue}egin{eqnarray}gin{Proposition}\label{prop_illustration_lipy}
Let $u$, $u'$ and $u''$ be respectively the solution to \reff{PDE}, \reff{PDE'} and \reff{PDE''} and assume that a comparison theorem holds for classical super and sub-solutions of these PDEs, in the class of functions with polynomial growth. Assume that there exist $(\varepsilon,\underline C,\overline C)\in(0,1)\times(0,+\infty)^3$, such that for all $x\in \mathbb{R}$
$$\underline{C}(1+|x|^{1-\varepsilon})\leq g(x)\leq \overline{C}(1+|x|^{1+\varepsilon}).$$
Assume moreover that $h$ is non-positive and that there exist $(\varepsilon',\underline{D},\overline{D})\in(0,\varepsilon)\times(0,+\infty)^2$ s.t.
$$\underline{D}(1+|x|^{\varepsilon'})\leq g'(x)\leq \overline{D}(1+|x|^{\varepsilon}).$$
Assume that there exist $(\underline{B},\overline{B}) \in (0,+\infty)^2$ such that for all $x\in \mathbb{R}$
$$\underline{B}\leq g''(x)\leq \overline{B}, \text{ and } 0\leq h_{zz}(t,x)<\frac{1}{4\overline{B}T}.$$
Assume finally that there exist $\lambda\in(0,\varepsilon^{-1}-1]$ and $C>0$ such that $|h_z(t,z)|\leq C(1+|z|^\lambda)$, then for all $(t,x)\in[0,T]\times\R$,
$$\underline{\alpha_u}\in [1-\varepsilon,1+\varepsilon], \ \overline{\alpha_{u'}},\underline{\alpha_{u'}}\in [\varepsilon',\varepsilon],\ \overline{\alpha_{u''}}=0, \ u'(t,x)\geq \underline D \text{ and } u''(t,x)\geq \underline{B}.$$
\end{Proposition}
\textcolor{blue}egin{eqnarray}gin{proof}
Let $\varphi(t,x):=\tilde{C}(T-t)+\overline{C}k_\varepsilon(x)$, where $k_\varepsilon(x)$ is in $\mathcal C^\infty(\R)$, coincides with the function $(1+\abs{x}^{1+\varepsilon})$ outside some closed interval centered at $0$ and is always greater than $(1+\abs{x}^{1+\varepsilon})$. We show that $\varphi$ is a (classical) super-solution to \reff{PDE} for some positive constant $\tilde{C}$ large enough. Indeed we can choose $\tilde{C}>0$ such that for any $(t,x)\in[0,T)\times\R$
$$ -\varphi_t(t,x)-\frac12 \varphi_{xx}(t,x)-h(t, \varphi_x(t,x))=\tilde{C}-\frac12 \overline{C}k_\varepsilon''(x)-h(t,\varphi_x(t,x))\geq 0,$$
since $h\leq 0$ and $\lim\limits_{|x|\to \infty }\frac12 k_\varepsilon''(x)=0$.
Moreover, by the assumption made on $g$, we clearly have for all $x\in\R$, $g(x)\leq \overline{C}k_\varepsilon(x)$, so that we deduce by comparison that for all $(t,x)\in [0,T] \times \mathbb{R}$:
$$ u(t,x)\leq \overline{C}k_\varepsilon(x)+\tilde{C}(T-t).$$
Now, we let $\phi(t,x):=-\tilde{C}_1(T-t)+\underline{C}\kappa_\varepsilon(x)$ for $(t,x)\in [0,T)\times \mathbb{R}$, where $\kappa_\varepsilon(x)$ is in $\mathcal C^\infty(\R)$, coincides with the function $(1+\abs{x}^{1-\varepsilon})$ outside some closed interval centered at $0$ and is always smaller than $(1+\abs{x}^{1-\varepsilon})$. We show that $\phi$ is a classical subsolution to \reff{PDE} for some positive constant $\tilde{C}_1$ large enough. We have
\textcolor{blue}egin{eqnarray}gin{equation}\label{truc} - \phi_t(t,x)-\frac12 \phi_{xx}(t,x)-h(t,\phi_x(t,x))=-\tilde{C}_1+\frac12\underline{C}\kappa_\varepsilon''(x)-h(t, \phi_x(t,x)).
\end{equation}
Given that the quantity $h(t,\phi_x(t,x))=h(t,\underline C\kappa_\varepsilon'(x))$ is bounded because $\lim\limits_{|x|\to \infty}\kappa_\varepsilon'(x)=0$ and $h$ is continuous, we can always choose $\tilde C_1$ so that \reff{truc} is non-positive. Then, since we clearly have for all $x\in\R$, $g(x)\geq \underline{C}\kappa_\varepsilon(x)$, we deduce by comparison that for all $(t,x)\in [0,T] \times \mathbb{R}$:
$$ u(t,x)\geq \overline{C}\kappa_\varepsilon(x)+\tilde{C}_1(T-t).$$
To sum up, we have showed that for all $(t,x)\in [0,T]\times \mathbb{R}$:
$$ \underline{C}\kappa_\varepsilon(x)-\tilde{C}_1(T-t)\leq u(t,x)\leq \overline{C}k_\varepsilon(x)+\tilde{C}(T-t).$$ In other words $[\underline{\alpha_u},\overline{\alpha_u}]\subset [1-\varepsilon,1+\varepsilon]. $
We now study \reff{PDE'}. Define for some constant $\tilde C_2>0$ to be fixed later
\textcolor{blue}egin{eqnarray}gin{align*}
\psi(t,x):=\tilde C_2(T-t)+\overline D\Upsilon_\varepsilon(x),
\end{align*}
where $\Upsilon_\varepsilon(x)$ is in $\mathcal C^\infty(\R)$, coincides with the function $(1+\abs{x}^{\varepsilon})$ outside some closed interval centered at $0$ and is always greater than $(1+\abs{x}^{\varepsilon})$. We then have
\textcolor{blue}egin{eqnarray}gin{align*}
-\psi_t(t,x)-\frac12 \psi_{xx}(t,x)-h_z(t,\psi(t,x)) \psi_x(t,x)=\tilde{C}_2-\frac12\overline{D}\Upsilon_\varepsilon''(x)-h_z(t,\psi(t,x))\overline D\Upsilon_\varepsilon'(x).
\end{align*}
Next, for some constant $C>0$ which may vary from line to line
$$|h_z(t,\psi(t,x))|\leq C(1+\abs{\psi(t,x)}^{\lambda})\leq C(1+\abs{x}^{\lambda\varepsilon}),$$ and since $\lambda \leq\frac1\varepsilon-1$ we deduce that:
$$\abs{h_z(t,\psi(t,x))\overline D\Upsilon_\varepsilon'(x)}\leq C(1+|x|^{\lambda \varepsilon+\varepsilon-1}) \text{, which is bounded. }$$
Since in addition we have $\Upsilon_\varepsilon''(x)\longrightarrow 0$ as $\abs{x}$ goes to $+\infty$, we can always choose $\tilde C_2$ large enough so that
$$- \psi_t(t,x)-\frac12 \psi_{xx}(t,x)-h_z(t,\psi(t,x))\psi_x(t,x)\geq 0.$$
By the assumption we made on $g$, we can use once more the comparison theorem to obtain
$$u'(t,x)\leq \psi(t,x).$$ Similarly, we show that $\underline{D}\Upsilon_{\varepsilon'}(x)-\tilde{C}_3(T-t)$ is a sub-solution of \reff{PDE'} for some positive constant $\tilde{C}_3$, since $\lambda\leq \varepsilon^{-1}-1\leq \varepsilon'^{-1}-1$. Then, by comparison, we deduce that $\overline{\alpha_{u'}}, \underline{\alpha_{u'}} \in [\varepsilon', \varepsilon]$. Moreover, we notice that $\underline{D}\leq g'(x)$ for all $x\in \mathbb{R}$, so $\underline{D}$ is a sub-solution of \reff{PDE'}. Thus, using once more the comparison theorem $u'(t,x)\geq \underline{D}$ for all $(t,x)\in [0,T]\times \mathbb{R}$.
We now study \reff{PDE''}. Given that $h_{zz}$ is non negative and $\underline{B}\leq g''(x)$ for all $(t,x)\in [0,T]\times \mathbb{R}$, we deduce directly that $\underline{B}$ is a sub-solution of \reff{PDE''}. Next, let $\varpi(t,x)=\overline{B}+\frac{\overline{B}}{T^{1-\eta}}(T-t)^{1-\eta}$ where $\eta \in (0,1)$ is chosen small enough so that $h_{zz}(t,x)\leq \frac{1-\eta}{4T\overline{B}}$. Thus,
\textcolor{blue}egin{eqnarray}gin{align*}
& - \varpi_t(t,x)-\frac12 \varpi_{xx}(t,x)-h_z(t, u'(t,x)) \varpi_x(t,x)-h_{zz}(t,u'(t,x))|\varpi(t,x)|^2\\
&=(1-\eta)\frac{\overline{B}}{T^{1-\eta}}(T-t)^{-\eta}-h_{zz}(t,u'(t,x))\overline{B}^2\left(1+\frac{(T-t)^{1-\eta}}{T^{1-\eta}} \right)^2\\
&\geq (1-\eta)\frac{\overline{B}}{T^{1-\eta}}(T-t)^{-\eta}-\frac{1-\eta}{4T}\overline{B}\left(1+\frac{(T-t)^{1-\eta}}{T^{1-\eta}} \right)^2\\
&\geq 0.
\end{align*}
We deduce that $\varpi$ is a super solution of \reff{PDE''}, which by comparison, implies that $u''$ is bounded, so $\overline{\alpha_{u''}}=0$.
\end{proof}
\textcolor{blue}egin{eqnarray}gin{Corollary}
Consider the FBSDE \eqref{edsr} and assume that for all $t\in [0,T]$ $X_t=W_t$ and $h$ depends only on $z$. Let $u(t,X_t):=Y_t$ and assume that $u\in {\cal C}^{1,2}$, $u'\in {\cal C}^{1,2}$ and $u''\in {\cal C}^{1,2}$. Let the assumptions of Proposition \ref{prop_illustration_lipy} hold, and assume moreover that $\varepsilon \in (0,\frac12)$. Then, the assumptions of Theorem \ref{estim.dens} hold.
\end{Corollary}
\textcolor{blue}egin{eqnarray}gin{proof}
According to Proposition \ref{prop_illustration_lipy}, $\underline{\alpha_u}\geq 1-\varepsilon$, $\overline{\alpha_{u'}}\leq \varepsilon$ and $u'(t,x)\geq \underline D,\ (t,x)\in[0,T]\times\R$. From the fact that $\varepsilon$ is smaller than $1/2$, we deduce that $0\leq\overline{\alpha_{u'}}<\underline{\alpha_{u}}<+\infty$. Moreover, $0=\overline{\alpha_{u''}}<\varepsilon'\leq\underline{\alpha_{u'}}$.
\end{proof}
\textcolor{blue}egin{eqnarray}gin{flushright}
$\qed$
\end{flushright}
\section*{Acknowledgments}
Thibaut Mastrolia is grateful to R\'egion Ile-De-France for financial support. The authors thank an Associate Editor and two anonymous Referees for their careful reading of this paper and for insightful suggestions which have greatly improve its presentation.
| 3,856 | 50,703 |
en
|
train
|
0.51.15
|
\section{Table of assumptions-results}
In this appendix we recall the different assumptions made within this paper and we give a summary table of some most significant results on BSDEs including ours.
\paragraph*{Assumption for $X$:}
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[(X)]$b,\sigma : [0,T]\times \mathbb{R} \longrightarrow \mathbb{R}$ are continuous in time and continuously differentiable in space for any fixed time $t$ and such that there exist $k_b,k_\sigma >0$ with
$$|b_x(t,x)|\leq k_b,\ |\sigma_x(t,x)|\leq k_\sigma, \text{ for all $x\in\R$}.$$
Besides $b(t,0), \sigma(t,0)$ are bounded functions of $t$ and there exists $c>0$ such that for all $t\in [0,T]$ $$0<c\leq |\sigma(t,\cdot)|, \ \lambda(dx)-a.e.$$
\end{itemize}
\textbf{List of assumptions for BSDEs:}
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[(L)]
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[(i)] $g : \mathbb{R} \longrightarrow \mathbb{R}$ is such that $\mathbb{E}[g(X_T)^2]<+\infty$.
\item[(ii)] $h : [0,T]\times \mathbb{R}^3 \longrightarrow \mathbb{R}$ is such that there exist $(k_x,k_y,k_z)\in(\R_+^*)^3$ such that for all $(t,x_1,x_2,y_1,y_2,z_1,z_2) \in [0,T]\times \mathbb{R}^6$,
$$ |h(t,x_1,y_1,z_1)-h(t,x_2,y_2,z_2)|\leq k_x|x_1-x_2|+k_y|y_1-y_2|+k_z|z_1-z_2|.$$
\item[(iii)] $\int_0^T |h(s,0,0,0)|^2ds<+\infty$.
\end{itemize}
\item[(Q)]
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[(i)] $g : \mathbb{R} \longrightarrow \mathbb{R}$ is bounded.
\item[(ii)] $h : [0,T]\times \mathbb{R}^3 \longrightarrow \mathbb{R}$ is such that:
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[$\triangleright$] There exists $(K,K_z,K_y)\in(\R_+^*)^3$ such that for all $(t,x,y,z) \in [0,T]\times \mathbb{R}^3$
$$\mathfrak{H}space{-3em}|h(t,x,y,z)|\leq K(1+|y|+|z|^2),\ \abs{h_z}(t,x,y,z)\leq K_z(1+|z|),\ \abs{h_y}(t,x,y,z)\leq K_y.$$
\item[$\triangleright$] There exists $C>0$ such that for all $(t,x,y,z_1,z_2) \in [0,T] \times \mathbb{R}^4$
\textcolor{blue}egin{eqnarray}gin{equation*} |h(t,x,y,z_1)-h(t,x,y,z_2)| \leq C(1+|z_1|+|z_2|) |z_1-z_2|.\end{equation*}
\end{itemize}
\item[(iii)] $\int_0^T |h(s,0,0,0)|^2ds<+\infty$.
\end{itemize}
\end{itemize}
\paragraph*{List of assumptions for Malliavin differentiability of $(X,Y,Z)$:}
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[(D1)]
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[(i)] $g$ is differentiable, $\mathcal{L}(X_T)-$a.e., $g$ and $g'$ have polynomial growth.
\item[(ii)] $(x,y,z)\mapsto h(t,x,y,z)$ is continuously differentiable for every $t$ in $[0,T]$.
\end{itemize}
\item[(D2)]
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[(i)] $g$ is twice differentiable, $\mathcal{L}(X_T)-$a.e., $g$, $g'$ and $g''$ have polynomial growth.
\item[(ii)] $(x,y,z)\mapsto h(t,x,y,z)$ is twice continuously differentiable for every $t$ in $[0,T]$.
\end{itemize}
\end{itemize}
\paragraph*{List of assumptions for the existence of densities for $Y$ and $Z$:}
$$\underline{g}:= \inf\limits_{x \in \mathbb{R}} g'(x), \quad \underline{g}^A:=\inf\limits_{x\in A} g'(x),\quad \overline{g}:= \sup\limits_{x \in \mathbb{R}} g'(x), \quad \overline{g}^A:=\sup\limits_{x\in A} g'(x),$$
$$\underline{h}(t):=\inf\limits_{ s\in [t,T], (x,y,z) \in \mathbb{R}^3} h_x(s,x,y,z),\quad \quad \overline{h}(t):=\sup\limits_{ s\in [t,T], (x,y,z) \in \mathbb{R}^3} h_x(s,x,y,z),$$
and $K:=k_b+k_y+k_{\sigma}k_z$. There exists $A\in\mathcal B(\R)$ such that $\mathbb{P}(X_T \in A | \mathcal{F}_t)>0$ and such that:
\textcolor{blue}egin{eqnarray}gin{align*}
&(H+)\quad \textcolor{blue}egin{eqnarray}gin{cases}
\displaystyle \underline{g}e^{-\text{sgn}(\underline{g})KT}+\underline{h}(t)\int_t^T e^{-\text{sgn}(\underline{h}(s))Ks}ds\geq0 \\
\displaystyle \underline{g}^Ae^{-\text{sgn}(\underline{g}^A)KT}+\underline{h}(t)\int_t^T e^{-\text{sgn}(\underline{h}(s))Ks}ds>0
\end{cases}\\[0.3em]
&(H-)\quad \textcolor{blue}egin{eqnarray}gin{cases}
\displaystyle \overline{g}e^{-\text{sgn}(\overline{g})KT}+\overline{h}(t)\int_t^T e^{-\text{sgn}(\overline{h}(s))Ks}ds\leq 0 \\
\displaystyle \overline{g}^Ae^{-\text{sgn}(\overline{g}^A)KT}+\overline{h}(t)\int_t^T e^{-\text{sgn}(\overline{h}(s))Ks}ds<0,
\end{cases}
\end{align*}
Set
\textcolor{blue}egin{eqnarray}gin{align*}
\tilde{h}(s,x,y,z):=& -\left( h_{xt}+b h_{xx}-hh_{xy}+\frac12(\sigma^2 h_{xxx}+2z\sigma h_{xxy}+z^2h_{xxy})\right)(s,x,y)\noindentnumber\\
&-\left((h_y+b_x)h_x+\sigma \sigma_xh_{xx}+z\sigma_xh_{xy}\right)
(s,x,y).\\
\tilde{g}(x):=&\ g'(x)+(T-t)h_x(T,x,g(x)),
\end{align*}
and
$$ \underline{\tilde{g}}:=\min\limits_{x\in \mathbb{R}} \tilde{g}(x), \quad \overline{\tilde{g}}:=\max\limits_{x\in \mathbb{R}} \tilde{g}(x),\quad \underline{\tilde{g}}^A:=\min\limits_{x\in A} \tilde{g}(x), \quad \overline{\tilde{g}}^A:=\max\limits_{x\in A} \tilde{g}(x),$$
$$\underline{\tilde{h}}(t):=\min\limits_{[t,T]\times \mathbb{R}^3} \tilde{h}(s,x,y,z), \ \overline{\tilde{h}}(t):=\max\limits_{[t,T]\times \mathbb{R}^3} \tilde{h}(s,x,y,z),$$
and set $K:=k_y+k_b$. There exists $A\in\mathcal B(\R)$ such that $\mathbb{P}(X_T \in A | \mathcal{F}_t)>0$ \textcolor{blue}egin{eqnarray}gin{align*}
&\widetilde{(H+)} \quad \textcolor{blue}egin{eqnarray}gin{cases}
\displaystyle \underline{\tilde{g}}e^{-\text{sgn}(\underline{\tilde{g}})KT}+\underline{\tilde{h}}(t)\int_t^T e^{-\text{sgn}(\underline{\tilde{h}}(s))Ks}(T-s)ds\geq0 \\
\displaystyle \underline{\tilde{g}}^Ae^{-\text{sgn}(\underline{\tilde{g}}^A)KT}+\underline{\tilde{h}}(t)\int_t^T e^{-\text{sgn}(\underline{\tilde{h}}(s))Ks}(T-s)ds>0,
\end{cases}\\[0.3em]
&\widetilde{(H-)} \quad \textcolor{blue}egin{eqnarray}gin{cases}
\displaystyle \overline{\tilde{g}}e^{-\text{sgn}(\overline{\tilde{g}})KT}+\overline{\tilde{h}}(t)\int_t^T e^{-\text{sgn}(\overline{\tilde{h}}(s))Ks}(T-s)ds\leq 0 \\
\displaystyle \overline{\tilde{g}}^Ae^{-\text{sgn}(\overline{\tilde{g}}^A)KT}+\overline{\tilde{h}}(t)\int_t^T e^{-\text{sgn}(\overline{\tilde{h}}(s))Ks}(T-s)ds<0.
\end{cases}
\end{align*}
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[$(Q+)$] $g'\geq 0$ and $g'_{\vert A} >0$, $\mathcal{L}(X_T)-$a.e. and $\underline{h}(t)\geq 0$,
\item[$(Q-)$] $g'\leq 0,\ g'_{\vert A}<0$, $\mathcal{L}(X_T)-$a.e. and $\overline{h}(t)\leq 0$,
\item[(Z+)]
\textcolor{blue}egin{eqnarray}gin{itemize}
\item There exist $(\underline a,\overline a)$ s.t., $0<\underline{a}\leq D_r X_u \leq \overline{a}$, for all $0<r<u\leq T$.
\item There exists $\overline b$ s.t., $0\leq D_{r,s}^2 X_u\leq \overline{b}$, for all $0<r,s<u\leq T$.
\item $h_x,h_{xx},h_{yy},h_{zz},h_{xy}\geq 0$ and $h_{xz}= h_{yz}= 0$ (and $h_y\geq 0$ under $(Q)$)
\item $h_{xy}= 0$ or $(h_{xy}\geq 0$ and $g'\geq 0$, $\mathcal{L}(X_T)$-a.e.$)$.
\item We have $$ \mathbf{1}_{\{\underline{g''}<0\}}\underline{g''}\overline{a}^2+\underline{g'}\mathbf{1}_{\{\underline{g'}<0\}}\overline{b}+(\mathbf{1}_{\{\underline{g''}\geq 0\}}\underline{g''}+\underline{h_{xx}}(t)(T-t))\underline{a}^2\geq 0,$$
and
$$ (\mathbf{1}_{\{\underline{g''}^A<0\}}\underline{g''}^A\overline{a}^2+\underline{g'}^A\mathbf{1}_{\{\underline{g'}<0\}}\overline{b}) +(\mathbf{1}_{\{\underline{g''}^A\geq 0\}}\underline{g''}^A+\underline{h_{xx}}(t)(T-t))\underline{a}^2>0,$$
\end{itemize}
\end{itemize}
We give the following summary table which sums up significant results for BSDEs in both the Lipschitz case and the quadratic case with assumptions made and references.
\small
\textcolor{blue}egin{eqnarray}gin{center}
\textcolor{blue}egin{eqnarray}gin{tabular}{|c|c|c|}
\mathfrak{H}line \diagbox{Results}{Cases} & Lipschitz case (L) & Quadratic case (Q) \\
\mathfrak{H}line Existence and uniqueness &\multirow{2}{*}{Prop. \ref{propex} (X)} & \multirow{2}{*}{Prop. \ref{propexq} (X) }\\
of solutions of BSDEs& & \\
\mathfrak{H}line Malliavin differentiability & \multirow{2}{*}{Prop. \ref{MD} (X) and (D1) } & \multirow{2}{*}{Prop. \ref{MDq} (X) and (D1)} \\
of $(X,Y,Z)$ & &\\
\mathfrak{H}line \multirow{2}{*}{Density existence for $Y$} & Th. \ref{thm_H+H-} (X), (D1) and (H+) or (H-)& \multirow{2}{*}{Th. \ref{thm_H+H-_quadra} (X), (D2) and (Q+) or (Q-)} \\
& Th. \ref{AKmodifie} (X), (D1) and ($\widetilde{H+}$) or ($\widetilde{H-}$) &\\
\mathfrak{H}line \multirow{2}{*}{Density existence for $Z$} & \multirow{2}{*}{Th. \ref{thm_density_z_lip} (X), (D2) and (Z+)} & \multirow{2}{*}{Th. \ref{thm_density_z_quadra} (X), (D2) and (Z+)}\\
& &\\
\mathfrak{H}line
\end{tabular}
\end{center}
\noindentrmalsize
\textcolor{blue}egin{eqnarray}gin{thebibliography}{10}
\textcolor{blue}ibitem{AbouraBourguin}
O.~Aboura and S.~Bourguin.
\newblock Density {E}stimates for {S}olutions to {O}ne {D}imensional {B}ackward
{SDE}'s.
\newblock {\em Potential Anal.}, 38(2):573--587, 2013.
\textcolor{blue}ibitem{AIDR}
S.~Ankirchner, P.~Imkeller, and G.~Dos~Reis.
\newblock Classical and variational differentiability of {BSDE}s with quadratic
growth.
\newblock {\em Electron. J. Probab.}, 12:no. 53, 1418--1453 (electronic), 2007.
\textcolor{blue}ibitem{AntonelliKohatsu}
F.~Antonelli and A.~Kohatsu-Higa.
\newblock Densities of one-dimensional backward {SDE}s.
\newblock {\em Potential Anal.}, 22(3):263--287, 2005.
| 3,998 | 50,703 |
en
|
train
|
0.51.16
|
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[$(Q+)$] $g'\geq 0$ and $g'_{\vert A} >0$, $\mathcal{L}(X_T)-$a.e. and $\underline{h}(t)\geq 0$,
\item[$(Q-)$] $g'\leq 0,\ g'_{\vert A}<0$, $\mathcal{L}(X_T)-$a.e. and $\overline{h}(t)\leq 0$,
\item[(Z+)]
\textcolor{blue}egin{eqnarray}gin{itemize}
\item There exist $(\underline a,\overline a)$ s.t., $0<\underline{a}\leq D_r X_u \leq \overline{a}$, for all $0<r<u\leq T$.
\item There exists $\overline b$ s.t., $0\leq D_{r,s}^2 X_u\leq \overline{b}$, for all $0<r,s<u\leq T$.
\item $h_x,h_{xx},h_{yy},h_{zz},h_{xy}\geq 0$ and $h_{xz}= h_{yz}= 0$ (and $h_y\geq 0$ under $(Q)$)
\item $h_{xy}= 0$ or $(h_{xy}\geq 0$ and $g'\geq 0$, $\mathcal{L}(X_T)$-a.e.$)$.
\item We have $$ \mathbf{1}_{\{\underline{g''}<0\}}\underline{g''}\overline{a}^2+\underline{g'}\mathbf{1}_{\{\underline{g'}<0\}}\overline{b}+(\mathbf{1}_{\{\underline{g''}\geq 0\}}\underline{g''}+\underline{h_{xx}}(t)(T-t))\underline{a}^2\geq 0,$$
and
$$ (\mathbf{1}_{\{\underline{g''}^A<0\}}\underline{g''}^A\overline{a}^2+\underline{g'}^A\mathbf{1}_{\{\underline{g'}<0\}}\overline{b}) +(\mathbf{1}_{\{\underline{g''}^A\geq 0\}}\underline{g''}^A+\underline{h_{xx}}(t)(T-t))\underline{a}^2>0,$$
\end{itemize}
\end{itemize}
We give the following summary table which sums up significant results for BSDEs in both the Lipschitz case and the quadratic case with assumptions made and references.
\small
\textcolor{blue}egin{eqnarray}gin{center}
\textcolor{blue}egin{eqnarray}gin{tabular}{|c|c|c|}
\mathfrak{H}line \diagbox{Results}{Cases} & Lipschitz case (L) & Quadratic case (Q) \\
\mathfrak{H}line Existence and uniqueness &\multirow{2}{*}{Prop. \ref{propex} (X)} & \multirow{2}{*}{Prop. \ref{propexq} (X) }\\
of solutions of BSDEs& & \\
\mathfrak{H}line Malliavin differentiability & \multirow{2}{*}{Prop. \ref{MD} (X) and (D1) } & \multirow{2}{*}{Prop. \ref{MDq} (X) and (D1)} \\
of $(X,Y,Z)$ & &\\
\mathfrak{H}line \multirow{2}{*}{Density existence for $Y$} & Th. \ref{thm_H+H-} (X), (D1) and (H+) or (H-)& \multirow{2}{*}{Th. \ref{thm_H+H-_quadra} (X), (D2) and (Q+) or (Q-)} \\
& Th. \ref{AKmodifie} (X), (D1) and ($\widetilde{H+}$) or ($\widetilde{H-}$) &\\
\mathfrak{H}line \multirow{2}{*}{Density existence for $Z$} & \multirow{2}{*}{Th. \ref{thm_density_z_lip} (X), (D2) and (Z+)} & \multirow{2}{*}{Th. \ref{thm_density_z_quadra} (X), (D2) and (Z+)}\\
& &\\
\mathfrak{H}line
\end{tabular}
\end{center}
\noindentrmalsize
\textcolor{blue}egin{eqnarray}gin{thebibliography}{10}
\textcolor{blue}ibitem{AbouraBourguin}
O.~Aboura and S.~Bourguin.
\newblock Density {E}stimates for {S}olutions to {O}ne {D}imensional {B}ackward
{SDE}'s.
\newblock {\em Potential Anal.}, 38(2):573--587, 2013.
\textcolor{blue}ibitem{AIDR}
S.~Ankirchner, P.~Imkeller, and G.~Dos~Reis.
\newblock Classical and variational differentiability of {BSDE}s with quadratic
growth.
\newblock {\em Electron. J. Probab.}, 12:no. 53, 1418--1453 (electronic), 2007.
\textcolor{blue}ibitem{AntonelliKohatsu}
F.~Antonelli and A.~Kohatsu-Higa.
\newblock Densities of one-dimensional backward {SDE}s.
\newblock {\em Potential Anal.}, 22(3):263--287, 2005.
\textcolor{blue}ibitem{barrieu_elk}
P. Barrieu and N. El Karoui.
\newblock {\em Pricing, Hedging and Optimally Designing Derivatives via Minimization of Risk Measures,}
\newblock Priceton university press, 2007.
\textcolor{blue}ibitem{BinghamGoldieTeugels}
N.~H. Bingham, C.M. Goldie, and J.~L. Teugels.
\newblock {\em Regular variation}, volume~27.
\newblock Cambridge university press, 1989.
\textcolor{blue}ibitem{ChassagneuxRichou}
J.-F. Chassagneux and A.~Richou.
\newblock Numerical simulation of quadratic {BSDE}s.
\newblock Preprint. http://arxiv.org/abs/1307.5741, 2013.
\textcolor{blue}ibitem{DosReisPhD}
G.~Dos~Reis.
\newblock {\em On some properties of solutions of quadratic growth {BSDE} and
applications in finance and insurance}.
\newblock PhD thesis, Humboldt University in Berlin,
http://www.math.tu-berlin.de/$\sim$dosreis/publications/GdosReis-PhD-Thesis.pdf,
2010.
\textcolor{blue}ibitem{dosdos}
G.~dos Reis and R.~J.~N. dos Reis.
\newblock A note on comonotonicity and positivity of the control components of
decoupled quadratic {FBSDE}.
\newblock {\em Stoch. Dyn.}, 13(4):1350005, 11, 2013.
\textcolor{blue}ibitem{elk_hamadene_matoussi}
N. El~Karoui, S. Hamadene, and A. Matoussi.
\newblock Backward stochastic differential equations and applications.
Chapter 8 in the book "Indifference Pricing: Theory and Applications" edited by René Carmona, Springer-Verlag pp. 267--320 (2008).
\textcolor{blue}ibitem{ElkarouiPengQuenez}
N.~El~Karoui, S.~Peng, and M.~C. Quenez.
\newblock Backward stochastic differential equations in finance.
\newblock {\em Math. Finance}, 7(1):1--71, 1997.
\textcolor{blue}ibitem{FournierPrintems}
N.~Fournier and J.~Printems.
\newblock Absolute continuity for some one-dimensional processes.
\newblock {\em Bernoulli}, 16(2):343--360, 2010.
\textcolor{blue}ibitem{HIM}
Y.~Hu, P.~Imkeller, and M.~M{{\"u}}ller.
\newblock Utility maximization in incomplete markets.
\newblock {\em Ann. Appl. Probab.}, 15(3):1691--1712, 2005.
\textcolor{blue}ibitem{ImkellerDosreis}
P.~Imkeller and G.~Dos~Reis.
\newblock Path regularity and explicit convergence rate for {BSDE} with
truncated quadratic growth.
\newblock {\em Stochastic Process. Appl.}, 120(3):348--379, 2010.
\textcolor{blue}ibitem{IRR}
P.~Imkeller, A.~R{{\'e}}veillac, and A.~Richter.
\newblock Differentiability of quadratic bsde generated by continuous
martingales and hedging in incomplete markets.
\newblock {\em Annals of Applied Probability}, 22(1):285--336, 2012.
\textcolor{blue}ibitem{jacobson}
D.H. Jacobson.
\newblock Optimal stochastic linear system with exponential criteria and their relation to differential games.
\newblock {\em IEEE Trans. Automat. Control}, 18(2):124-131, 1973.
\textcolor{blue}ibitem{KS}
I.~Karatzas and S.~E. Shreve.
\newblock {\em Brownian motion and stochastic calculus}, volume 113 of {\em
Graduate Texts in Mathematics}.
\newblock Springer-Verlag, New York, second edition, 1991.
\textcolor{blue}ibitem{Kazamaki}
N.~Kazamaki.
\newblock {\em Continuous exponential martingales and {BMO}}, volume 1579 of
{\em Lecture Notes in Mathematics}.
\newblock Springer-Verlag, Berlin, 1994.
\textcolor{blue}ibitem{Kobylanski}
M.~Kobylanski.
\newblock Backward stochastic differential equations and partial differential
equations with quadratic growth.
\newblock {\em Annals of Probability}, 28(2):558--602, 2000.
\textcolor{blue}ibitem{Kohatsu_2003b}
A.~Kohatsu-Higa.
\newblock Lower bounds for densities of uniformly elliptic non-homogeneous
diffusions.
\newblock In {\em Stochastic inequalities and applications}, volume~56 of {\em
Progr. Probab.}, pages 323--338. Birkh{\"a}user, Basel, 2003.
\textcolor{blue}ibitem{Kohatsu_03}
A.~Kohatsu-Higa.
\newblock Lower bounds for densities of uniformly elliptic random variables on
{W}iener space.
\newblock {\em Probab. Theory Related Fields}, 126(3):421--457, 2003.
\textcolor{blue}ibitem{Ma_Zhang_PTRF02}
J.~Ma and J.~Zhang.
\newblock Path regularity for solutions of backward stochastic differential
equations.
\newblock {\em Probab. Theory Related Fields}, 122(2):163--190, 2002.
\textcolor{blue}ibitem{MaZhang}
J.~Ma and J.~Zhang.
\newblock Representation theorems for backward stochastic differential
equations.
\newblock {\em Ann. Appl. Probab.}, 12(4):1390--1418, 2002.
\textcolor{blue}ibitem{NourdinViens}
I.~Nourdin and F.~Viens.
\newblock Density formula and concentration inequalities with {M}alliavin
calculus.
\newblock {\em Electron. J. Probab.}, 14:no. 78, 2287--2309, 2009.
\textcolor{blue}ibitem{Nualartbook}
D.~Nualart.
\newblock {\em The {M}alliavin calculus and related topics}.
\newblock Probability and its Applications (New York). Springer-Verlag, Berlin,
second edition, 2006.
\textcolor{blue}ibitem{Nualart_Quer}
E.~Nualart and L.~Quer-Sardanyons.
\newblock Gaussian estimates for the density of the non-linear stochastic heat
equation in any space dimension.
\newblock {\em Stochastic Process. Appl.}, 122(1):418--447, 2012.
\textcolor{blue}ibitem{PardouxPeng}
{E}. Pardoux and S.~Peng.
\newblock Adapted solution of a backward stochastic differential equation.
\newblock {\em Systems Control Lett.}, 14(1):55--61, 1990.
\textcolor{blue}ibitem{PardouxPeng_92}
E.~Pardoux and S.~Peng.
\newblock Backward stochastic differential equations and quasilinear parabolic
partial differential equations.
\newblock In {\em Stochastic partial differential equations and their
applications}, volume 176 of {\em Lecture Notes in Control and Inform. Sci.},
pages 200--217. Springer, 1992.
\textcolor{blue}ibitem{elk_rouge}
R.~Rouge and N.~El~Karoui.
\newblock Pricing via utility maximization and entropy.
\newblock {\em Math. Finance}, 10(2):259--276, 2000.
\newblock INFORMS Applied Probability Conference (Ulm, 1999).
\textcolor{blue}ibitem{Seneta}
E.~Seneta.
\newblock {\em Regularly varying functions}.
\newblock Lecture Notes in Mathematics, Vol. 508. Springer-Verlag, Berlin,
1976.
\end{thebibliography}
\end{document}
| 3,727 | 50,703 |
en
|
train
|
0.52.0
|
\begin{document}
\date{Draft}
\title{The polyharmonic heat flow of closed plane curves}
\author[S. Parkins]{Scott Parkins}
\address{Institute for Mathematics and its Applications, School of Mathematics and Applied Statistics\\
University of Wollongong\\
Northfields Ave, Wollongong, NSW $2500$,\\
Australia}
\email{[email protected]}
\author[G. Wheeler]{Glen Wheeler}
\address{Institute for Mathematics and its Applications, School of Mathematics and Applied Statistics\\
University of Wollongong\\
Northfields Ave, Wollongong, NSW $2500$,\\
Australia}
\email{[email protected]}
\thanks{The research of the first author was supported by an Australian
Postgraduate Award. The research of the second author was supported in part by
Discovery Projects DP120100097 and DP150100375 of the Australian Research Council.}
\subjclass{53C44}
\begin{abstract}
In this paper we consider the polyharmonic heat flow of a closed curve in the
plane.
Our main result is that closed initial data with initially small normalised
oscillation of curvature and isoperimetric defect flows exponentially fast in
the $C^\infty$-topology to a simple circle.
Our results yield a characterisation of the total amount of time during which
the flow is not strictly convex, quantifying in a sense the failure of the
maximum principle.
\end{abstract}
\maketitle
\begin{section}{Introduction}
Let $\gamma_0:\mathbb{S}\rightarrow\mathbb{R}^2$ be a smooth, closed regular immersed plane curve.
Let $p\in\mathbb{N}_0$.
A one-parameter family $\gamma:\mathbb{S}\times\left[0,T\right)\rightarrow\mathbb{R}^2$ satisfying
\begin{equation}
\frac{\partial}{\partial
t}\gamma=\oo{-1}^{p}\kappa_{s^{2p}}\nu\label{FlowDefinition}\tag{$PF_p$}
\end{equation}
is called the $(2p+2)$-th order polyharmonic heat flow of $\gamma_0$, or the
\emph{polyharmonic flow} for short.
Here $s$ is the regular Euclidean arc length
$s\oo{u}=\int_{0}^{u}{\norm{\gamma_{v}}\,du}$ and $\kappa_{s^{2p}}$ is
$2p$ derivatives of the Euclidean curvature $\kappa$ with respect to
arc length:
\[
\kappa_{s^{2p}} := \partial_s^{2p}\kappa := \frac{\partial^{2p}\kappa}{\partial
s^{2p}}.
\]
We take $\nu$ to be a unit normal vector field to $\gamma$ such that $\kappa\nu =
\partial_s^2\gamma$.
If we take $p=0$, then \eqref{FlowDefinition} is the well-studied \emph{curve
shortening flow} made famous by Hamilton, Gage and Grayson \cite{Hamilton2,grayson1989}:
\[
\partial_t\gamma = \kappa\nu\,.
\]
The curve shortening flow is second-order, and, being a nonlinear geometric
heat equation for the immersion $\gamma$, enjoys the maximum principle and its
standard variations (Harnack inequality, comparison/avoidance principles). This
allows for trademark characteristics such as moving immediately from weak
convexity to strong convexity, preservation of convexity, preservation of
embeddedness, and preservation of graphicality.
The curve shortening flow is the $W^{-0,2} = L^2$ gradient flow for length.
Taking the $H^{-1} = W^{-1,2}$ gradient flow for length yields the fourth order
flow termed the \emph{curve diffusion flow}, whose origins lie in material
science \cite{Mullins1}.
Its gradient flow structure was only later discovered by Fife \cite{Fife}.
The qualitative properties mentioned above for the curve shortening flow do not
hold for the curve diffusion flow (and in fact do not hold for any of the flows
\eqref{FlowDefinition} for $p>0$).
We refer the reader to \cite{Blatt2010,EllPaa2001,EscherIto2005,GI1998,GI1999}
for an overview of these interesting phenomena.
We additionally mention numerical examples contributed by Mayer
\cite{privatecommsmayer} of finite-time singularities arising from embedded
initial data (the resolution of this is an open conjecture that to our
knowledge is due to Giga \cite{privatecommsgiga}).
While local well-posedness belongs by now to standard theory (see for example
\cite{Baker,Mantegazza2}), global analysis and qualitative properties of the
flow remain largely unresolved.
Recently, there have been advances in understanding the stability of the curve
diffusion flow about circles, with work of Elliott-Garcke \cite{EG97}
strengthened by the second author in \cite{Wkosc}.
The result of \cite{Wkosc} relies on the blowup criterion discovered by
Dziuk-Kuwert-Sch\"atzle \cite{DKS}.
The core idea of \cite{Wkosc} is to analyse the normalised oscillation of
curvature:
\[
K_{osc}:=L\intcurve{\oo{\kappa-\bar{\kappa}}^{2}}\,.
\]
The key observation for the curve diffusion flow is that $K_{osc}$ is a natural
energy, being both integrable (in time) for any allowable initial data and
whose blowup characterises finite-time blowup in general.
In this article, we prove that $K_{osc}$ remains a natural energy for every
polyharmonic flow, regardless of how large $p$ is.
In the theorem below and for the remainder of the article we assume $p\in\mathbb{N}$.
\begin{theorem}
\label{TM1}
Suppose $\gamma:\mathbb{S}^{1}\times\left[0,T\right)\rightarrow\mathbb{R}^{2}$
solves $\eqref{FlowDefinition}$. Then there exists a constant
$\varepsilon_{0}>0$ depending only on $p$ such that if
\begin{equation}
\label{EQsmallness}
K_{osc}\oo{0}<\varepsilon_{0}\text{ and }I\oo{0}<e^{\frac{\varepsilon_{0}}{8\pi^{2}}}
\end{equation}
then $\gamma\oo{\mathbb{S}^{1}}$ approaches a round circle exponentially fast
with radius $\sqrt{\frac{A\oo{\gamma_{0}}}{\pi}}$.
\end{theorem}
Although there is a plethora of negative results on the curve diffusion flow
violating positivity over time, there are relatively few results guaranteeing
preservation.
Theorem \ref{TM1} implies that after some \emph{waiting time}, the flow is
uniformly convex and remains forever so.
An estimate for the waiting time for the curve diffusion flow was given in
\cite{Wkosc}.
Here we extend this to each of the \eqref{FlowDefinition} flows.
\begin{proposition}
\label{PN1}
Suppose $\gamma:\mathbb{S}^{1}\times\left[0,T\right)\rightarrow\mathbb{R}^{2}$
solves $\eqref{FlowDefinition}$. If $\gamma(\cdot,0)$ satisfies \eqref{EQsmallness}, then
\[
\mathcal{L}\{t\in[0,\infty)\,:\,k(\cdot,t)\not>0\}
\le \frac{2}{p+1}\bigg[
\bigg( \frac{L(\gamma_0)}{2\pi}\bigg)^{2(p+1)}
- \bigg( \frac{A(\gamma_0)}{\pi} \bigg)^{p+1}
\bigg]
\,.
\]
\end{proposition}
Above we have used $k(\cdot,t) \not> 0$ to mean $k(s_0,t) \le 0$ for at least
one $s_0$.
This estimate is optimal in the sense that the right hand side is zero for a
simple circle.
One may wish to compare this with the case for classical PDE of higher-order,
where exciting progress on eventual positivity continues to be made
\cite{DGK15,FGG08,GG08,GG09}.
The remainder of the present paper is devoted to proving Theorem \ref{TM1} and
Proposition \ref{PN1}.
We cover some basic definitions and integral formulae in Section 2, before
moving on to essential evolution equations for length, area, and curvature in
Section 3. We study $K_{osc}$ directly in Section 4, obtaining precise control
over $K_{osc}$ in the case where the initial data is sufficiently close in a
weak isoperimetric sense to a circle and has $K_{osc}$ initially smaller than
an explicit constant.
We continue by adapting Dziuk-Kuwert-Sch\"atzle's blowup criterion argument to
\eqref{FlowDefinition} flows (Lemma \ref{LongTimeLemma1}), yielding in Section
5 global existence.
Further analysis gives exponentially fast convergence to a circle with specific
radius dependent on the initial enclosed area.
We finish Section 5 by giving the proof of Proposition \ref{PN1}.
\end{section}
\begin{section}{Preliminaries}
\begin{lemma}\label{PrelimLemma1}
Suppose $\gamma:\mathbb{S}^{1}\times\left[0,T\right)\rightarrow\mathbb{R}^{2}$
solves $\eqref{FlowDefinition}$, and
$f:\mathbb{S}^{1}\times\left[0,T\right)\rightarrow\mathbb{R}$ is a periodic
function with the same period as $\gamma$. Then
\begin{equation*}
\frac{d}{dt}\intcurve{f}=\intcurve{f_{t}+\oo{-1}^{p+1}f\cdot\kappa\cdot \kappa_{s^{2p}}}.
\end{equation*}
\end{lemma}
\begin{proof}
We first calculate the evolution of arc length. Because $\nu\perp\tau$, it follows from the Frenet-Serret equations that
\begin{align}
\partial_{s}ds&=\partial_{t}\norm{\gamma_{u}}\,du=\partial_{t}\inner{\gamma_{u},\gamma_{u}}^{\frac{1}{2}}\,du\nonumber\\
&=\norm{\gamma_{u}}^{-1}\inner{\partial_{ut}\gamma,\gamma_{u}}\,du=\inner{\partial_{s}\gamma_{t},\tau}\,ds\nonumber\\
&=\inner{\partial_{s}\oo{\oo{-1}^{p}\kappa_{s^{2p}}\cdot\nu},\tau}\,ds=\oo{-1}^{p}\kappa_{s^{2p}}\inner{\partial_{s}\nu,\tau}\,ds\nonumber\\
&=\oo{-1}^{p+1}\kappa\cdot\kappa_{s^{2p}}\,ds.\label{PrelimLemma1,0}
\end{align}
Next, using the fundamental theorem of calculus and $\oo{\ref{PrelimLemma1,0}}$ we have
\begin{align}
\frac{d}{dt}\int_{\gamma}{f\,ds}&=\frac{d}{dt}\int_{0}^{P\oo{t}}{f\oo{u,t}\left|\gamma_{u}\oo{u,t}\right|\,du}\nonumber\\
&=\intcurve{f_{t}}+\intcurve{f\partial_{t}}+P'\oo{t}\cdot\frac{d}{dP\oo{t}}\int_{0}^{P\oo{t}}{f\oo{u,t}\left|\gamma_{u}\oo{u,t}\right|\,du}\nonumber\\
&=\intcurve{f_{t}+\oo{-1}^{p+1}\kappa\cdot\kappa_{s^{2p}}}+P'\oo{t}f\oo{P\oo{t},t}\left|\gamma_{u}\oo{P\oo{t},t}\right|\nonumber\\
&=\intcurve{f_{t}+\oo{-1}^{p+1}\kappa\cdot\kappa_{s^{2p}}}.\label{PrelimLemma1,1}
\end{align}
Here the last line follows from the fact that
\[
P'(t)\norm{\gamma_{u}\oo{P\oo{t},t}}=\Big(\partial_{t}\big(\gamma\oo{P\oo{t},t}-\gamma\oo{0,t}\big)\Big)^\top=0
\]
because $\partial_{t}\gamma$ is purely normal to $\gamma$.
\end{proof}
| 3,098 | 30,672 |
en
|
train
|
0.52.1
|
\end{section}
\begin{section}{Fundamental evolution equations}
\begin{corollary}\label{PrelimCor1}
Suppose $\gamma:\mathbb{S}^{1}\times\left[0,T\right)\rightarrow\mathbb{R}^{2}$
solves $\eqref{FlowDefinition}$ Then
\[
\frac{d}{dt}L=-\int_{\gamma}{\kappa_{s^{p}}^{2}\,ds}\text{ and }\frac{d}{dt}A=0.
\]
In particular, the isoperimetric ratio decreases in absolute value with velocity
\[
\frac{d}{dt}I=-\frac{2I}{L}\intcurve{\kappa_{s^{p}}^{2}}\leq0.
\]
\end{corollary}
\begin{proof}
Applying Lemma $\ref{PrelimLemma1}$ with $f\equiv1$ gives the statement for $L$:
\[
\frac{d}{dt}L=\frac{d}{dt}\intcurve{}=\oo{-1}^{p+1}\intcurve{\kappa\cdot \kappa_{s^{2p}}}=-\int_{\gamma}{\kappa_{s^{p}}^{2}\,ds}\leq0.
\]
Here we have performed integration by parts $p$ times. For the statement regarding area, we first state the Frenet-Serret formulas with no torsion:
\begin{equation}
\tau_{s}=\kappa\nu\text{ and }\nu_{s}=-\kappa\tau.\label{PrelimCor1,1}
\end{equation}
Using the equations in $\oo{\ref{PrelimCor1,1}}$, we wish to derive a formula for the time derivative of the unit normal $\nu$. We first work out the commutator:
\begin{align}
\partial_{ts}&=\partial_{t}\oo{\partial_{s}}=\partial_{t}\oo{\norm{\gamma_{u}}^{-1}\partial_{u}}=\left|\gamma_{u}\right|^{-1}\partial_{t}\partial_{u}-\left|\gamma_{u}\right|^{-2}\oo{\partial_{t}\left|\gamma_{u}\right|}\partial_{u}\nonumber\\
&=\partial_{st}-\norm{\gamma_{u}}^{-3}\inner{\partial_{u}\gamma_{t},\gamma_{u}}\partial_{u}=\partial_{st}-\inner{\partial_{s}\gamma_{t},\tau}\partial_{s}\nonumber\\
&=\partial_{st}-\inner{\partial_{s}\oo{\oo{-1}^{p}\kappa_{s^{2p}}\cdot\nu},\tau}\partial_{s}\nonumber\\
&=\partial_{st}+\oo{-1}^{p}\kappa\cdot\kappa_{s^{2p}}\partial_{s}.\label{PrelimCor1,2}
\end{align}
We then use $\oo{\ref{PrelimCor1,1}},\oo{\ref{PrelimCor1,2}}$ and the identity $\gamma_{s}=\tau$ to calculate:
\begin{align}
\partial_{t}\tau&=\partial_{ts}\gamma=\partial_{st}\gamma+\oo{-1}^{p}\kappa\cdot\kappa_{s^{2p}}\partial_{s}\gamma\nonumber\\
&=\oo{-1}^{p}\cc{\kappa_{s^{2p+1}}\cdot\nu-\kappa\cdot\kappa_{s^{2p}}\cdot\tau}+\oo{-1}^{p}\kappa\cdot\kappa_{s^{2p}}\cdot\tau\nonumber\\
&=\oo{-1}^{p}\kappa_{s^{2p+1}}\cdot\nu.\label{PrelimCor1,3}
\end{align}
Using the fact that $\nu\perp\tau$ and $\norm{\nu}^{2}=1\implies\partial_{t}\nu\perp\nu$, it then follows from $\oo{\ref{PrelimCor1,3}}$ that
\begin{align}
\partial_{t}\nu&=\inner{\partial_{t}\nu,\tau}\tau=-\inner{\nu,\partial_{t}\tau}\tau\nonumber\\
&=-\inner{\nu,\oo{-1}^{p}\kappa_{s^{2p+1}}\cdot\nu}\tau=\oo{-1}^{p+1}\kappa_{s^{2p+1}}\cdot\tau.\label{PrelimCor1,4}
\end{align}
Applying Lemma $\ref{PrelimLemma1}$ with $f=\inner{\gamma,\nu}$ then gives
\begin{align}
\frac{d}{dt}A&=-\frac{1}{2}\frac{d}{dt}\int_{\gamma}{\inner{\gamma,\nu}\,ds}=-\frac{1}{2}\intcurve{\partial_{t}\inner{\gamma,\nu}+\oo{-1}^{p+1}\inner{\gamma,\nu}\cdot\kappa\cdot\kappa_{s^{2p}}}\nonumber\\
&=-\frac{1}{2}\intcurve{\inner{\oo{-1}^{p}\kappa_{s^{2p}}\cdot\nu,\nu}+\inner{\gamma,\oo{-1}^{p+1}\kappa_{s^{2p+1}}\cdot\tau}+\oo{-1}^{p+1}\inner{\gamma,\nu}\cdot\kappa\cdot\kappa_{s^{2p}}}\nonumber\\
&=-\frac{1}{2}\intcurve{\oo{-1}^{p}\kappa_{s^{2p}}+\oo{-1}^{p+1}\kappa_{s^{2p+1}}\inner{\gamma,\tau}+\oo{-1}^{p+1}\inner{\gamma,\tau_{s}}\kappa_{s^{2p}}}\nonumber\\
&-\frac{1}{2}\intcurve{\oo{-1}^{p}\kappa_{s^{2p}}+\oo{-1}^{p+1}\kappa_{s^{2p+1}}\inner{\gamma,\tau}+\oo{-1}^{p}\cc{\inner{\gamma_{s},\tau}\kappa_{s^{2p}}+\inner{\gamma,\tau}\kappa_{s^{2p+1}}}}\nonumber\\
&=\oo{-1}^{p+1}\intcurve{\kappa_{s^{2p}}}=\oo{-1}^{p+1}\kappa_{s^{2p-1}}\Bigg|_{s=0}^{s=L\oo{\gamma}}\nonumber\\
&=0.\nonumber
\end{align}
Here we have used integration by parts in the third last line. The last step follows from the divergence theorem, and using the periodicity of $\gamma$.
To establish the evolution equaiton for the isoperimetric ratio we simply combine the two established results for $L$ and $A$:
\begin{align}
\frac{\partial}{\partial t}I&=\frac{\partial}{\partial t}\oo{\frac{L^{2}}{4\pi A}}=\frac{1}{4\pi A^{2}}\cc{2AL\frac{\partial}{\partial t}L-L^{2}\frac{\partial}{\partial t}A}\nonumber\\
&=-\frac{2L}{4\pi A}\int_{\gamma}{\kappa_{s^{p}}^{2}\,ds}\nonumber\\
&=-\frac{2I}{L}\int_{\gamma}{\kappa_{s^{p}}^{2}\,ds}\leq0.\nonumber
\end{align}
This completes the proof.
\end{proof}
\begin{lemma}\label{CurvatureLemma1}
Suppose that
$\gamma:\mathbb{S}^{1}\times\left[0,T\right)\rightarrow\mathbb{R}^{2}$ solves
$\eqref{FlowDefinition}$ and
\[
\intcurve{\kappa}\Big|_{t=0}=2\omega\pi.
\]
Then
\[
\intcurve{\kappa}=2\omega\pi
\]
for $t\in\left[0,T\right)$. Moreover, the average curvature $\overline{\kappa}=\frac{1}{L}\intcurve{\kappa}$ increases in absolute value with velocity
\[
\frac{d}{dt}\overline{\kappa}=\frac{2\omega\pi}{L^2}\llll{\kappa_{s^{p}}}_{2}^{2}\geq0.
\]
\end{lemma}
\begin{proof}
We first need to calculate the evolution equation for curvature. Using the definition $\kappa=\inner{\nu,\gamma_{ss}}$ along with previous identities, we have
\begin{align}
\frac{\partial}{\partial t}\kappa&=\frac{\partial}{\partial t}\inner{\nu,\gamma_{ss}}=\inner{\nu_{t},\gamma_{ss}}+\inner{\nu,\partial_{t}\gamma_{ss}}=\inner{\nu_{t},\gamma_{ss}}+\inner{\nu,\partial_{ts}\tau}\nonumber\\
&=\inner{\oo{-1}^{p+1}\kappa_{s^{2p+1}}\cdot\tau,\kappa\cdot\nu}+\inner{\nu,\partial_{st}\tau+\oo{-1}^{p}\kappa\cdot\kappa_{s^{2p}}\cdot\tau_{s}}\nonumber\\
&=\inner{\nu,\partial_{s}\oo{\oo{-1}^{p}\kappa_{s^{2p+1}}\cdot\nu}+\oo{-1}^{p}\kappa^{2}\cdot\kappa_{s^{2p}}\cdot\nu}\nonumber\\
&=\oo{-1}^{p}\oo{\kappa_{s^{2p+2}}+\kappa^{2}\cdot\kappa_{s^{2p}}}.\label{CurvatureLemma1,1}
\end{align}
Then, applying Lemma $\ref{PrelimLemma1}$ with $f=\kappa$ gives us
\begin{equation}
\frac{d}{dt}\intcurve{\kappa}=\intcurve{\kappa_{t}+\oo{-1}^{p+1}\kappa^{2}\cdot\kappa_{s^{2p}}}=0.\label{CurvatureLemma1,3}
\end{equation}
It follows from $\oo{\ref{CurvatureLemma1,3}}$ that the integral $\intcurve{\kappa}$ stays constant on $\left[0,T\right)$. This gives the first assertion of the lemma. For the second assertion, we simply use $\oo{\ref{CurvatureLemma1,3}}$ and Corollary $\ref{PrelimCor1}$ and compute:
\begin{align*}
\frac{d}{dt}\overline{\kappa}&=\frac{d}{dt}\oo{\frac{1}{L}\intcurve{\kappa}}=\frac{1}{L^2}\cc{L\cdot\frac{d}{dt}\intcurve{\kappa}-\intcurve{\kappa}\cdot\frac{d}{dt}L}\\
&=-\frac{2\omega\pi}{L^{2}}\cdot-\intcurve{\kappa_{s^{p}}^{2}}=\frac{2\omega\pi}{L^{2}}\llll{\kappa_{s^{p}}}_{2}^{2}\\
&\geq0.
\end{align*}
This completes the proof.
\end{proof}
| 2,491 | 30,672 |
en
|
train
|
0.52.2
|
\begin{lemma}\label{CurvatureLemma1}
Suppose that
$\gamma:\mathbb{S}^{1}\times\left[0,T\right)\rightarrow\mathbb{R}^{2}$ solves
$\eqref{FlowDefinition}$ and
\[
\intcurve{\kappa}\Big|_{t=0}=2\omega\pi.
\]
Then
\[
\intcurve{\kappa}=2\omega\pi
\]
for $t\in\left[0,T\right)$. Moreover, the average curvature $\overline{\kappa}=\frac{1}{L}\intcurve{\kappa}$ increases in absolute value with velocity
\[
\frac{d}{dt}\overline{\kappa}=\frac{2\omega\pi}{L^2}\llll{\kappa_{s^{p}}}_{2}^{2}\geq0.
\]
\end{lemma}
\begin{proof}
We first need to calculate the evolution equation for curvature. Using the definition $\kappa=\inner{\nu,\gamma_{ss}}$ along with previous identities, we have
\begin{align}
\frac{\partial}{\partial t}\kappa&=\frac{\partial}{\partial t}\inner{\nu,\gamma_{ss}}=\inner{\nu_{t},\gamma_{ss}}+\inner{\nu,\partial_{t}\gamma_{ss}}=\inner{\nu_{t},\gamma_{ss}}+\inner{\nu,\partial_{ts}\tau}\nonumber\\
&=\inner{\oo{-1}^{p+1}\kappa_{s^{2p+1}}\cdot\tau,\kappa\cdot\nu}+\inner{\nu,\partial_{st}\tau+\oo{-1}^{p}\kappa\cdot\kappa_{s^{2p}}\cdot\tau_{s}}\nonumber\\
&=\inner{\nu,\partial_{s}\oo{\oo{-1}^{p}\kappa_{s^{2p+1}}\cdot\nu}+\oo{-1}^{p}\kappa^{2}\cdot\kappa_{s^{2p}}\cdot\nu}\nonumber\\
&=\oo{-1}^{p}\oo{\kappa_{s^{2p+2}}+\kappa^{2}\cdot\kappa_{s^{2p}}}.\label{CurvatureLemma1,1}
\end{align}
Then, applying Lemma $\ref{PrelimLemma1}$ with $f=\kappa$ gives us
\begin{equation}
\frac{d}{dt}\intcurve{\kappa}=\intcurve{\kappa_{t}+\oo{-1}^{p+1}\kappa^{2}\cdot\kappa_{s^{2p}}}=0.\label{CurvatureLemma1,3}
\end{equation}
It follows from $\oo{\ref{CurvatureLemma1,3}}$ that the integral $\intcurve{\kappa}$ stays constant on $\left[0,T\right)$. This gives the first assertion of the lemma. For the second assertion, we simply use $\oo{\ref{CurvatureLemma1,3}}$ and Corollary $\ref{PrelimCor1}$ and compute:
\begin{align*}
\frac{d}{dt}\overline{\kappa}&=\frac{d}{dt}\oo{\frac{1}{L}\intcurve{\kappa}}=\frac{1}{L^2}\cc{L\cdot\frac{d}{dt}\intcurve{\kappa}-\intcurve{\kappa}\cdot\frac{d}{dt}L}\\
&=-\frac{2\omega\pi}{L^{2}}\cdot-\intcurve{\kappa_{s^{p}}^{2}}=\frac{2\omega\pi}{L^{2}}\llll{\kappa_{s^{p}}}_{2}^{2}\\
&\geq0.
\end{align*}
This completes the proof.
\end{proof}
\end{section}
\begin{section}{The Normalised Oscillation of Curvature}
We now introduce a scale-invariant quantity
\[
K_{osc}:=L\intcurve{\oo{\kappa-\bar{\kappa}}^{2}}
\]
which we call the \emph{normalised oscillation of curvature}.
One can deduce from our previous calculations that this quantity is a natural
one, being that for a one parameter family of curves $\gamma_{t}$ that solves
$\eqref{FlowDefinition}$, $K_{osc}\oo{t}$ is a bounded quantity in $L^{1}$
(and in fact is bounded by a quantity that depends on the initial data,
$\gamma_{0}$ and so can be controlled a priori). Indeed, The fact that
$\intcurve{\oo{\kappa-\bar{\kappa}}}=0$ means that we can apply Lemma
$\ref{AppendixLemma1}$, giving
\[
K_{osc}=L\intcurve{\oo{\kappa-\bar{\kappa}}^{2}}\leq L\oo{\frac{L}{2\pi}}^{2}\intcurve{\kappa_{s}^{2}}.
\]
Now the periodicity of $\kappa$ implies that for every $i\geq 1$, $\intcurve{\kappa_{s^{i}}}=0$, so we can apply Lemma $\ref{AppendixLemma1}$ to the right hand side of the above inequality $p$ more times, yielding
\begin{equation}
K_{osc}\leq L\oo{\frac{L}{2\pi}}^{2p}\intcurve{\kappa_{s^{p}}^{2}}=-\frac{L^{2p+1}}{\oo{2\pi}^{2p}}\frac{d}{dt}L=-\frac{1}{2\oo{p+1}\oo{2\pi}^{2p}}\frac{d}{dt}\oo{L^{2\oo{p+1}}}.\label{OscillationOfCurvature1}
\end{equation}
Here we have utilised the evolution of the length functional. We conclude that for any $t\in\left[0,T\right)$
\begin{align}
\int_{0}^{t}{K_{osc}\oo{\tau}\,d\tau}&\leq-\frac{1}{2\oo{p+1}\oo{2\pi}^{2p}}\oo{L^{2\oo{p+1}}\oo{\gamma_{t}}-L^{2p+1}\oo{\gamma_{0}}}\nonumber\\
&\leq\frac{1}{2\oo{p+1}\oo{2\pi}^{2p}}L^{2\oo{p+1}}\oo{\gamma_{0}}.\label{OscillationCurvature2}
\end{align}
We deduce from $\oo{\ref{OscillationCurvature2}}$ that the normalised oscillation of curvature is a priori controlled in $L^{1}$ over the time of existence of the flow. Furthermore, by repeatedly using Lemma $\ref{AppendixLemma2}$ in a similar fashion, one can easily obtain an $L^{1}$ bound for $\llll{\kappa-\bar{\kappa}}_{\infty}^{2}$ over the interval $\left[0,T\right)$. Firstly
\begin{align*}
\llll{\kappa-\bar{\kappa}}_{\infty}^{2}&\leq\frac{L}{2\pi}\intcurve{\kappa_{s}^{2}}\leq \frac{L}{2\pi}\oo{\frac{L}{2\pi}}^{2}\intcurve{\kappa_{s^{2}}^{2}}\\
&\vdots\\
&\leq \frac{L}{2\pi}\oo{\frac{L}{2\pi}}^{2\oo{p-1}}\intcurve{\kappa_{s^{p}}^{2}}=-\frac{L^{2p-1}}{\oo{2\pi}^{2p-1}}\frac{d}{dt}L\\
&=-\frac{1}{2p\oo{2\pi}^{2p-1}}\frac{d}{dt}L^{2p}.
\end{align*}
Hence for any $t\in\left[0,T\right)$,
\begin{equation}
\int_{0}^{t}{\llll{\kappa-\bar{\kappa}}_{\infty}^{2}\,d\tau}\leq-\frac{1}{2p\oo{2\pi}^{2p-1}}\oo{L^{2p}\oo{\gamma_{t}}-L^{2p}\oo{\gamma_{0}}}\leq\frac{1}{2p\oo{2\pi}^{2p-1}}L^{2p}\oo{\gamma_{0}}.\label{OscillationCurvature3}
\end{equation}
Next we formulate the evolution equation for $K_{osc}$.
\begin{lemma}\label{CurvatureLemma2}
Suppose $\gamma:\mathbb{S}^{1}\times\left[0,T\right)\rightarrow\mathbb{R}^{2}$
solves $\eqref{FlowDefinition}$. Then
\begin{align*}
&\frac{d}{dt}\oo{K_{osc}+8\omega^{2}\pi^{2}\ln{L}}+\frac{\llll{\kappa_{s^{p}}}_{2}^{2}}{L}K_{osc}+2L\llll{\kappa_{s^{p+1}}}_{2}^{2}\\
&=L\intcurve{\cc{\oo{\kappa-\bar{\kappa}}^{3}+\bar{\kappa}\oo{\kappa-\bar{\kappa}}^{2}}_{s^{p}}\oo{\kappa-\bar{\kappa}}_{s^{p}}}.
\end{align*}
\end{lemma}
\begin{proof}
We have
\begin{align*}
&\frac{d}{dt}K_{osc}=\frac{d}{dt}L\cdot\intcurve{\oo{\kappa-\bar{\kappa}}^{2}}+L\cdot\frac{d}{dt}\intcurve{\oo{\kappa-\bar{\kappa}}^{2}}\\
&=-\llll{\kappa_{s^{p}}}_{2}^{2}\intcurve{\oo{\kappa-\bar{\kappa}}^{2}}+L\Biggl[2\intcurve{\oo{\kappa-\bar{\kappa}}\kappa_{t}}+\oo{-1}^{p+1}\intM{\oo{\kappa-\bar{\kappa}}^{2}\cdot\kappa\cdot\kappa_{s^{2p}}}\Biggr]\\
&=-\frac{\llll{k_{s^{p}}}_{2}^{2}}{L}K_{osc}+2\oo{-1}^{p}L\intcurve{\oo{\kappa-\bar{\kappa}}\oo{\kappa_{s^{2p+2}}+\kappa^{2}\cdot\kappa_{s^{2p}}}}\\
&+\oo{-1}^{p+1}L\intcurve{\oo{\kappa-\bar{\kappa}}^{2}\cdot\kappa\cdot\kappa_{s^{2p}}}\\
&=-\frac{\llll{\kappa_{s^{p}}}_{2}^{2}}{L}K_{osc}-2L\llll{\kappa_{s^{p+1}}}_{2}^{2}+2\oo{-1}^{p}L\intcurve{\oo{\kappa-\bar{\kappa}}\cdot\kappa^{2}\cdot\kappa_{s^{2p}}}\\
&+\oo{-1}^{p+1}L\intcurve{\oo{\kappa-\bar{\kappa}}^{2}\cdot\kappa\cdot \kappa_{s^{2p}}}.
\end{align*}
Hence
\begin{align*}
&\frac{d}{dt}K_{osc}+\frac{\llll{\kappa_{s^{p}}}_{2}^{2}}{L}K_{osc}+2L\llll{\kappa_{s^{p+1}}}_{2}^{2}\\
&=2\oo{-1}^{p}L\intcurve{\oo{\kappa-\bar{\kappa}}\cdot\cc{\oo{\kappa-\bar{\kappa}}^{2}+2\bar{\kappa}\oo{\kappa-\bar{\kappa}}+\bar{\kappa}^{2}}\cdot \kappa_{s^{2p}}}\\
&+\oo{-1}^{p+1}L\intcurve{\oo{\kappa-\bar{\kappa}}^{2}\cdot\cc{\oo{\kappa-\bar{\kappa}}+\bar{\kappa}}\cdot\kappa_{s^{2p}}}\\
&=\oo{-1}^{p}L\intcurve{\cc{\oo{\kappa-\bar{\kappa}}^{3}+\bar{\kappa}\oo{\kappa-\bar{\kappa}}^{2}+2\bar{\kappa}^{2}\oo{\kappa-\bar{\kappa}}}\oo{\kappa-\bar{\kappa}}_{s^{2p}}}\\
&=L\intcurve{\cc{\oo{\kappa-\bar{\kappa}}^{3}+\bar{\kappa}\oo{\kappa-\bar{\kappa}}^{2}}_{s^{p}}\oo{\kappa-\bar{\kappa}}_{s^{p}}}+2\bar{\kappa}^{2}L\llll{\kappa_{s^{p}}}_{2}^{2}\\
&=L\intcurve{\cc{\oo{\kappa-\bar{\kappa}}^{3}+\bar{\kappa}\oo{\kappa-\bar{\kappa}}^{2}}_{s^{p}}\oo{\kappa-\bar{\kappa}}_{s^{p}}}+\frac{8\omega^{2}\pi^{2}}{L}\llll{\kappa_{s^{p}}}_{2}^{2}\\
&=L\intcurve{\cc{\oo{\kappa-\bar{\kappa}}^{3}+\bar{\kappa}\oo{\kappa-\bar{\kappa}}^{2}}_{s^{p}}\oo{\kappa-\bar{\kappa}}_{s^{p}}}-8\omega^{2}\pi^{2}\frac{d}{dt}\ln{L}
\end{align*}
Here we have used Corollary $\ref{PrelimCor1}$ and Lemma $\ref{CurvatureLemma1}$ in the penultimate step. Rearranging then yields the desired result.
\end{proof}
\begin{lemma}\label{CurvatureLemma3}
\begin{equation}
L\intcurve{\cc{\oo{\kappa-\bar{\kappa}}^{3}+\bar{\kappa}\oo{\kappa-\bar{\kappa}}^{2}}_{s^{p}}\oo{\kappa-\bar{\kappa}}_{s^{p}}}\leq L\oo{c_{1}K_{osc}+c_{2}\sqrt{K_{osc}}}
\end{equation}
for some universal constants $c_{1},c_{2}>0$. Here $c_{i}=c_{i}\oo{p}$.
\end{lemma}
\begin{proof}
The proof follows from an application of a number of interpolation inequalities which can be found in \cite{DKS}. It has been included in the Appendix for the convenience of the reader.
\end{proof}
\begin{corollary}\label{CurvatureCorollary1}
Suppose $\gamma:\mathbb{S}^{1}\times\left[0,T\right)\rightarrow\mathbb{R}^{2}$
solves $\eqref{FlowDefinition}$. Then
\[
\frac{d}{dt}\oo{K_{osc}+8\omega^{2}\pi^{2}\ln{L}}+\frac{\llll{\kappa_{s^{p}}}_{2}^{2}}{L}K_{osc}+L\oo{2-c_{1}K_{osc}-c_{2}\sqrt{K_{osc}}}\llll{\kappa_{s^{p+1}}}_{2}^{2}\leq0.
\]
Here $c_{1}\oo{p},c_{2}\oo{p}$ are the universal constants given in Lemma $\ref{CurvatureLemma3}$. Moreover, if there exists a $T^{*}$ such that for $t\in\left[0,T^{*}\right)$
\begin{equation}
K_{osc}\oo{t}\leq\frac{8c_{1}+2c_{2}^{2}-2c_{2}\sqrt{8c_{1}+c_{2}^{2}}}{4c_{1}^{2}}=2K^{*},\label{CurvatureCorollary1,0}
\end{equation}
then during this time the estimate
\begin{equation}
K_{osc}+8\omega^{2}\pi^{2}\ln{L}+\int_{0}^{t}{K_{osc}\frac{\llll{\kappa_{s}^{p}}_{2}^{2}}{L}\,d\tau}\leq K_{osc}\oo{0}+8\omega^{2}\pi^{2}\ln{L\oo{0}}\label{CurvatureCorollary1,1}
\end{equation}
holds.
\end{corollary}
\begin{proof}
Combining Lemma $\ref{CurvatureLemma2}$ and Lemma $\ref{CurvatureLemma3}$ immediately gives the first result. Using the assumed smallness of $K_{osc}$ then gives the second.
\end{proof}
| 3,750 | 30,672 |
en
|
train
|
0.52.3
|
\begin{lemma}\label{CurvatureLemma3}
\begin{equation}
L\intcurve{\cc{\oo{\kappa-\bar{\kappa}}^{3}+\bar{\kappa}\oo{\kappa-\bar{\kappa}}^{2}}_{s^{p}}\oo{\kappa-\bar{\kappa}}_{s^{p}}}\leq L\oo{c_{1}K_{osc}+c_{2}\sqrt{K_{osc}}}
\end{equation}
for some universal constants $c_{1},c_{2}>0$. Here $c_{i}=c_{i}\oo{p}$.
\end{lemma}
\begin{proof}
The proof follows from an application of a number of interpolation inequalities which can be found in \cite{DKS}. It has been included in the Appendix for the convenience of the reader.
\end{proof}
\begin{corollary}\label{CurvatureCorollary1}
Suppose $\gamma:\mathbb{S}^{1}\times\left[0,T\right)\rightarrow\mathbb{R}^{2}$
solves $\eqref{FlowDefinition}$. Then
\[
\frac{d}{dt}\oo{K_{osc}+8\omega^{2}\pi^{2}\ln{L}}+\frac{\llll{\kappa_{s^{p}}}_{2}^{2}}{L}K_{osc}+L\oo{2-c_{1}K_{osc}-c_{2}\sqrt{K_{osc}}}\llll{\kappa_{s^{p+1}}}_{2}^{2}\leq0.
\]
Here $c_{1}\oo{p},c_{2}\oo{p}$ are the universal constants given in Lemma $\ref{CurvatureLemma3}$. Moreover, if there exists a $T^{*}$ such that for $t\in\left[0,T^{*}\right)$
\begin{equation}
K_{osc}\oo{t}\leq\frac{8c_{1}+2c_{2}^{2}-2c_{2}\sqrt{8c_{1}+c_{2}^{2}}}{4c_{1}^{2}}=2K^{*},\label{CurvatureCorollary1,0}
\end{equation}
then during this time the estimate
\begin{equation}
K_{osc}+8\omega^{2}\pi^{2}\ln{L}+\int_{0}^{t}{K_{osc}\frac{\llll{\kappa_{s}^{p}}_{2}^{2}}{L}\,d\tau}\leq K_{osc}\oo{0}+8\omega^{2}\pi^{2}\ln{L\oo{0}}\label{CurvatureCorollary1,1}
\end{equation}
holds.
\end{corollary}
\begin{proof}
Combining Lemma $\ref{CurvatureLemma2}$ and Lemma $\ref{CurvatureLemma3}$ immediately gives the first result. Using the assumed smallness of $K_{osc}$ then gives the second.
\end{proof}
Note that although Corollary $\ref{CurvatureCorollary1}$ implies that the
normalised oscillation of curvature remains bounded if initially sufficiently
small, it does not seem to give tight control of the quantity per se, because
we already know that $\ln{L}$ (on the left hand side of
$\oo{\ref{CurvatureCorollary1,1}}$) is decreasing, and so without further
analysis, one might think that $K_{osc}$ could be static in time (or even
worse, \emph{increasing}).
However, note by the isoperimetric inequality that for any closed curve solving
$\eqref{FlowDefinition}$ we have
\[
\frac{L^{2}\oo{\gamma}}{4\pi A\oo{\gamma}}\geq1,\text{ and so }\frac{1}{L\oo{\gamma}}\leq\frac{1}{\sqrt{4\pi A\oo{\gamma}}}.
\]
It follows that for any $t\in\left[0,T\right)$,
\begin{equation}
\frac{L\oo{\gamma_{0}}}{L\oo{\gamma_{t}}}\leq\frac{L\oo{\gamma_{0}}}{\sqrt{4\pi A\oo{\gamma_{t}}}}=\frac{L\oo{\gamma_{0}}}{\sqrt{4\pi A\oo{\gamma_{0}}}}=\sqrt{I\oo{\gamma_{0}}}.\label{IsoperimetricConsequence}
\end{equation}
Here we have used the fact that by Corollary $\ref{PrelimCor1}$, the enclosed area of our family of immersed curves is static in time.
So, the quantity $\frac{L\oo{\gamma_{0}}}{L\oo{\gamma_{t}}}$ can be controlled over $\left[0,T\right)$ a priori by assuming that $\gamma_{0}$ is ``sufficiently circular''. In particular, since we may choose $\gamma_{0}$ such that $I\oo{\gamma_{0}}$ is as close to $1$ as we wish (and so $\frac{L\oo{\gamma_{0}}}{L\oo{\gamma_{t}}}$ remains close to $1$ as well), equation $\oo{\ref{CurvatureCorollary1,1}}$ becomes much more appealing because it can be rearranged to give
\begin{equation}
K_{osc}+\int_{0}^{t}{K_{osc}\frac{\llll{\kappa_{s^{p}}}_{2}^{2}}{L}\,d\tau}\leq K_{osc}\oo{0}+8\omega^{2}\pi^{2}\ln{\sqrt{I\oo{0}}}=K_{osc}\oo{0}+4\omega^{2}\pi^{2}\ln\oo{{I\oo{0}}}.\label{OscillationOfCurvature4}
\end{equation}
This of course is an improvement upon Corollary $\ref{CurvatureCorollary1}$ because it tells us that $K_{osc}$ can not get larger than the right hand side of the inequality. One problem is that this inequality as it stands is only valid whilst $K_{osc}$ satisfies $\oo{\ref{CurvatureCorollary1,0}}$, and it is not clear from $\oo{\ref{OscillationOfCurvature4}}$ that this smallness condition should hold for the duration of the flow.
A little bit of tweaking will give us tighter control over $K_{osc}$ for the duration of the flow, and we present this result in the following proposition.
\begin{proposition}\label{FlowProp2}
Let $\gamma:\mathbb{S}^{1}\times\left[0,T\right)\rightarrow\mathbb{R}^{2}$
solve $\eqref{FlowDefinition}$. Additionally, suppose that $\gamma_{0}$ is a
simple closed curve with $\omega=1$, satisfying
\[
K_{osc}\oo{0}\leq K^{\star}\,\,\text{and}\,\,I\oo{0}\leq e^{\frac{K^{\star}}{8\pi^{2}}}.
\]
Then
\[
K_{osc}\oo{t}\leq 2K^{\star}\,\,\text{for}\,\,t\in\left[0,T\right).
\]
\end{proposition}
\begin{proof}
Suppose for the sake of contradiction that $K_{osc}$ does \emph{not} remain bounded by $2K^{\star}$ for the duration of the flow. Then we can find a maximal $T^{\star}<T$ such that
\[
K_{osc}\oo{t}\leq 2K^{\star}\text{ for }t\in\left[0,T^{\star}\right).
\]
Then, by $\oo{\ref{OscillationOfCurvature4}}$, the following identity holds for $t\in\left[0,T^{\star}\right)$:
\begin{equation}
K_{osc}\oo{t}\leq K_{osc}\oo{0}+4\pi^{2}\ln\oo{{I\oo{0}}}\leq K^{\star}+4\pi^{2}\ln\oo{e^{\frac{K^{\star}}{8\pi^{2}}}}=\frac{3K^{\star}}{2}\text{ for }t\in\left[0,T^{\star}\right).\label{FlowProp2,1}
\end{equation}
We have also used the fact that Lemma $\ref{CurvatureLemma1}$ ensures that $\omega=1$ for the duration of the flow.
Taking $t\nearrow T$ in inequality $\oo{\ref{FlowProp2,1}}$ gives $K_{osc}\leq\frac{3K^{\star}}{2}<2K^{\star}$, meaning that by continuity, $K_{osc}\leq 2K^{\star}$ on some larger time interval $\left[0,T^{\star}+\delta\right)$. But $\left[0,T^{\star}\right)$ was chosen to be the largest time interval containing $0$ such that $K_{osc}$ remains bounded by $2K^{\star}$ and so we have arrived at a contradiction. Thus our assumption that $T<T^{\star}$ must have been false, and the result of the proposition follows.
\end{proof}
\begin{corollary}\label{CurvatureCorollary2}
Let $\gamma:\mathbb{S}^{1}\times\left[0,T\right)\rightarrow\mathbb{R}^{2}$
solve $\eqref{FlowDefinition}$. Additionally, suppose that $\gamma_{0}$ is a
simple embedded closed curve satisfying
\[
K_{osc}\oo{0}\leq K^{\star}<\epsilon_{0}\,\,\text{and}\,\,I\oo{0}\leq e^{\frac{K^{\star}}{8\pi^{2}}}\leq e^{\frac{\varepsilon_{0}}{8\pi^{2}}},
\]
where $\epsilon_{0}<32-2\pi^{2}$ is a sufficiently small constant. Then $\gamma$ remains embedded on $\left[0,T\right)$.
\end{corollary}
\begin{proof}
Suppose $\gamma:\mathbb{S}^{1}\rightarrow\mathbb{R}^{2}$ is a smooth immersed curve with winding number $\omega=1$. From the Gauss-Bonnet theorem and Lemma $\ref{CurvatureLemma1}$, we know that for $t\in\left[0,T\right)$ the winding number of $\gamma_{t}$ remains the same. Therefore the hypothesis of the corollary implies that $\omega=1$ for the duration of the flow. Define $m\oo{\gamma}$ to be the maximum number of times that $\gamma$ intersects itself in any one point. That is,
\[
m\oo{\gamma}:=\sup_{x\in\mathbb{R}^{2}}\norm{\gamma^{-1}\oo{x}}.
\]
By Theorem $16$ from \cite{Wkosc}, $m$ satisfies the following inequality:
\begin{equation}
K_{osc}\oo{\gamma}\geq16m^{2}-4\omega^{2}\pi^{2}=16m^{2}-4\pi^{2}.\nonumber
\end{equation}
Hence
\begin{equation}
m^{2}\leq\frac{1}{16}\oo{K_{osc}\oo{\gamma}+4\pi^{2}}.\label{CurvatureCorollary2,1}
\end{equation}
Proposition $\ref{FlowProp2}$ then tells us that by the hypothesis of the corollary, $K_{osc}$ remains bounded above by $2K^{\star}$ for the duration of the flow. We can assume without loss of generality that $K^{\star}<32-2\pi^{2}\approx12.26$, and so we have $K_{osc}\oo{\gamma}<64-4\pi^{2}$ on $\left[0,T\right)$. Therefore by $\oo{\ref{CurvatureCorollary2,1}}$ we have
\[
m^{2}<\frac{1}{16}\oo{64-4\pi^{2}+4\pi^{2}}=4\,\,\text{for}\,\,t\in\left[0,T\right),
\]
and embeddedness follows immediately.
\end{proof}
\begin{lemma}\label{LongTimeLemma1}
Suppose $\gamma:\mathbb{S}^{1}\times \left[0,T\right)\rightarrow\mathbb{R}^{2}$
is a maximal solution to $\eqref{FlowDefinition}$. If $T<\infty$ then
\[
\intcurve{\kappa^{2}}\geq c\oo{T-t}^{-1/2\oo{p+1}}
\]
for a universal constant $c>0$.
\end{lemma}
\begin{proof}
Deriving an evolution equation for $\intcurve{\kappa^{2}}$ in the same manner as Lemma $\ref{CurvatureLemma2}$ and using an interpolation inequality in the same vein as \cite{DKS} gives us
\begin{equation}
\frac{d}{dt}\intcurve{\kappa^{2}}+\intcurve{\kappa_{s^{p+1}}^{2}}\leq c\oo{p}\oo{\intcurve{\kappa^{2}}}^{2\oo{m+p}+3},\nonumber
\end{equation}
which implies that
\begin{equation}
-\frac{1}{2\oo{p+1}}\cc{\oo{\intcurve{\kappa^{2}}\Big|_{t=t_{1}}}^{-1/2\oo{p+1}}-\oo{\intcurve{\kappa^{2}}\Big|_{t=t_{0}}}^{-1/2\oo{p+1}}}\leq c\oo{t_{1}-t_{0}}.\label{LongTimeLemma1,1}
\end{equation}
for any times $t_{0}\leq t_{1}$. Note that if $\limsup_{t\rightarrow T}\intcurve{\kappa^{2}}=\infty$, then taking $t_{1}\nearrow T$ in $\oo{\ref{LongTimeLemma1,1}}$ and rearranging will prove the lemma. Assume for the sake of contradiction that $\intcurve{\kappa^{2}}\leq\varrho$ for all $t<T$. By using an argument similar to Theorem $3.1$ of \cite{Wkosc}, we are able to show that the inequality
\[
\llll{\partial_{u}^{m}\gamma}_{\infty}\leq c_{m}\oo{\varrho,\gamma_{0},T}<\infty
\]
holds for every $m\in\mathbb{N}$ up until time $T$. By short time existence we are then able to extend the life of the flow, contradicting the maximality of $\gamma$. Hence our assumption that $\limsup_{t\rightarrow T}\intcurve{\kappa^{2}}<\infty$ must have been incorrect, and so the limit must diverge. We then conclude the desired result of the lemma from $\oo{\ref{LongTimeLemma1,1}}$.
\end{proof}
\end{section}
\begin{section}{Global analysis}
\begin{corollary}\label{LongTimeCorollary1}
Suppose $\gamma:\mathbb{S}^{1}\times\left[0,T\right)\rightarrow\mathbb{R}^{2}$
solves $\eqref{FlowDefinition}$. Additionally, suppose that $\gamma_{0}$ is
a simple closed curve satisfying
\[
K_{osc}\oo{0}\leq K^{\star}\,\,\text{and}\,\,I\oo{0}\leq e^{\frac{K^{\star}}{8\pi^{2}}}.
\]
Then $T=\infty$.
\end{corollary}
\begin{proof}
Suppose for the sake of contradiction that $T<\infty$. Then by Lemma $\ref{LongTimeLemma1}$ we have
\[
\intcurve{\kappa^{2}}\geq c\oo{T-t}^{-1/2\oo{p+3}}
\]
and so in particular,
\begin{equation}
\intcurve{\kappa^{2}}\rightarrow\infty\text{ as }t\rightarrow T.\label{LongTimeCorollary1,1}
\end{equation}
Next note that $\oo{\ref{IsoperimetricConsequence}}$ gives us an absolute lower bound on the length of $\gamma$:
\[
L\oo{\gamma_{t}}\geq\sqrt{4\pi A\oo{\gamma_{0}}}.
\]
Hence we establish the following following bound on $K_{osc}$:
\[
K_{osc}=L\intcurve{\kappa^{2}}-4\pi^{2}\geq\sqrt{4\pi A\oo{\gamma_{0}}}\intcurve{\kappa^{2}}-4\pi^{2}.
\]
Hence it follows from $\oo{\ref{LongTimeCorollary1,1}}$ that
\[
K_{osc}\oo{t}\rightarrow\infty\,\,\text{as}\,\,t\rightarrow T.
\]
But this directly contradicts the results of Proposition $\ref{FlowProp2}$, and so we conclude that our assumption that $T$ was finite must have been incorrect. Thus $T=\infty$.
\end{proof}
| 3,909 | 30,672 |
en
|
train
|
0.52.4
|
\begin{lemma}\label{LongTimeLemma1}
Suppose $\gamma:\mathbb{S}^{1}\times \left[0,T\right)\rightarrow\mathbb{R}^{2}$
is a maximal solution to $\eqref{FlowDefinition}$. If $T<\infty$ then
\[
\intcurve{\kappa^{2}}\geq c\oo{T-t}^{-1/2\oo{p+1}}
\]
for a universal constant $c>0$.
\end{lemma}
\begin{proof}
Deriving an evolution equation for $\intcurve{\kappa^{2}}$ in the same manner as Lemma $\ref{CurvatureLemma2}$ and using an interpolation inequality in the same vein as \cite{DKS} gives us
\begin{equation}
\frac{d}{dt}\intcurve{\kappa^{2}}+\intcurve{\kappa_{s^{p+1}}^{2}}\leq c\oo{p}\oo{\intcurve{\kappa^{2}}}^{2\oo{m+p}+3},\nonumber
\end{equation}
which implies that
\begin{equation}
-\frac{1}{2\oo{p+1}}\cc{\oo{\intcurve{\kappa^{2}}\Big|_{t=t_{1}}}^{-1/2\oo{p+1}}-\oo{\intcurve{\kappa^{2}}\Big|_{t=t_{0}}}^{-1/2\oo{p+1}}}\leq c\oo{t_{1}-t_{0}}.\label{LongTimeLemma1,1}
\end{equation}
for any times $t_{0}\leq t_{1}$. Note that if $\limsup_{t\rightarrow T}\intcurve{\kappa^{2}}=\infty$, then taking $t_{1}\nearrow T$ in $\oo{\ref{LongTimeLemma1,1}}$ and rearranging will prove the lemma. Assume for the sake of contradiction that $\intcurve{\kappa^{2}}\leq\varrho$ for all $t<T$. By using an argument similar to Theorem $3.1$ of \cite{Wkosc}, we are able to show that the inequality
\[
\llll{\partial_{u}^{m}\gamma}_{\infty}\leq c_{m}\oo{\varrho,\gamma_{0},T}<\infty
\]
holds for every $m\in\mathbb{N}$ up until time $T$. By short time existence we are then able to extend the life of the flow, contradicting the maximality of $\gamma$. Hence our assumption that $\limsup_{t\rightarrow T}\intcurve{\kappa^{2}}<\infty$ must have been incorrect, and so the limit must diverge. We then conclude the desired result of the lemma from $\oo{\ref{LongTimeLemma1,1}}$.
\end{proof}
\end{section}
\begin{section}{Global analysis}
\begin{corollary}\label{LongTimeCorollary1}
Suppose $\gamma:\mathbb{S}^{1}\times\left[0,T\right)\rightarrow\mathbb{R}^{2}$
solves $\eqref{FlowDefinition}$. Additionally, suppose that $\gamma_{0}$ is
a simple closed curve satisfying
\[
K_{osc}\oo{0}\leq K^{\star}\,\,\text{and}\,\,I\oo{0}\leq e^{\frac{K^{\star}}{8\pi^{2}}}.
\]
Then $T=\infty$.
\end{corollary}
\begin{proof}
Suppose for the sake of contradiction that $T<\infty$. Then by Lemma $\ref{LongTimeLemma1}$ we have
\[
\intcurve{\kappa^{2}}\geq c\oo{T-t}^{-1/2\oo{p+3}}
\]
and so in particular,
\begin{equation}
\intcurve{\kappa^{2}}\rightarrow\infty\text{ as }t\rightarrow T.\label{LongTimeCorollary1,1}
\end{equation}
Next note that $\oo{\ref{IsoperimetricConsequence}}$ gives us an absolute lower bound on the length of $\gamma$:
\[
L\oo{\gamma_{t}}\geq\sqrt{4\pi A\oo{\gamma_{0}}}.
\]
Hence we establish the following following bound on $K_{osc}$:
\[
K_{osc}=L\intcurve{\kappa^{2}}-4\pi^{2}\geq\sqrt{4\pi A\oo{\gamma_{0}}}\intcurve{\kappa^{2}}-4\pi^{2}.
\]
Hence it follows from $\oo{\ref{LongTimeCorollary1,1}}$ that
\[
K_{osc}\oo{t}\rightarrow\infty\,\,\text{as}\,\,t\rightarrow T.
\]
But this directly contradicts the results of Proposition $\ref{FlowProp2}$, and so we conclude that our assumption that $T$ was finite must have been incorrect. Thus $T=\infty$.
\end{proof}
Recall we know that if $\gamma:\mathbb{S}^{1}\times\left[0,T\right)\rightarrow\mathbb{R}^{2}$ satisfies the hypothesis of Corollary $\ref{LongTimeCorollary1}$ then $T=\infty$, and then identity $\oo{\ref{OscillationCurvature2}}$ tells us that
\begin{equation}
K_{osc}\in L^{1}\oo{\left[0,\infty\right)},\,\,\text{with}\,\,\int_{0}^{\infty}{K_{osc}\oo{\tau}\,d\tau}\leq \frac{1}{2\oo{p+1}\oo{2\pi}^{2p}}L^{2\oo{p+1}}\oo{0}.\label{OscillationOfCurvatureL1}
\end{equation}
So we can conclude that the ``tail'' of the function $K_{osc}\oo{t}$ must get small as $t\nearrow\infty$. However, at the present time we have not ruled out the possibility that $K_{osc}$ gets smaller and smaller as $t$ gets large, whilst vibrating with higher and higher frequency, remaining in $L^{1}\oo{\left[0,\infty\right)}$ whilst never actually fully dissipating to zero in a smooth sense. To rule out this from happening, it is enough to show that $\norm{\frac{d}{dt}K_{osc}}$ remains bounded by a universal constant for all time. To do so we will need to first show that $\llll{\kappa_{s^{p}}}_{2}^{2}$ remains bounded. We will address this issue with the following proposition.
\begin{proposition}\label{LongTimeProp1}
Suppose $\gamma:\mathbb{S}^{1}\times\left[0,T\right)\rightarrow\mathbb{R}^{2}$
solves $\eqref{FlowDefinition}$ and is simple. There exists a
$\varepsilon_{0}>0$ (with $\varepsilon_{0}\leq K^{\star}$) such that if
\[
K_{osc}\oo{0}<\varepsilon_{0}\text{ and }I\oo{0}<e^{\frac{\varepsilon_{0}}{8\pi^{2}}}
\]
then $\llll{k_{s^{p}}}_{2}^{2}$ remains bounded for all time. In particular,
\[
\intcurve{\kappa_{s^{p}}^{2}}\leq\tilde{c}\oo{\gamma_{0}}
\]
for some constant $\tilde{c}\oo{\gamma_{0}}$ depending only upon the initial immersion.
\end{proposition}
\begin{proof}
We first derive the evolution equation for the quantity $\intcurve{\kappa_{s^{p}}^{2}}$. Applying Lemma $\ref{PrelimLemma1}$ along with repeated applications of the formula for the commutator $\cc{\partial_{t},\partial_{s}}$, we have
\begin{align}
\frac{d}{dt}\intcurve{\kappa_{s^{p}}^{2}}&=2\intcurve{\kappa_{s^{p}}\partial_{t}\kappa_{s^{p}}}+\intcurve{\kappa\cdot\kappa_{s^{p}}^{2}\cdot\kappa_{s^{2p}}}\nonumber\\
&=-2\intcurve{\kappa_{s^{2p+1}}^{2}}+2\oo{-1}^{p}\sum_{j=0}^{p}\oo{-1}^{j}\intcurve{\kappa\cdot\kappa_{s^{p-j}}\cdot\kappa_{s^{p+j}}\cdot\kappa_{s^{2p}}}\nonumber\\
&+\oo{-1}^{p+1}\intcurve{\kappa\cdot\kappa_{s^{p}}^{2}\cdot\kappa_{s^{2p}}}\nonumber\\
&=-2\intcurve{\kappa_{s^{2p+1}}^{2}}+2\intcurve{\kappa^{2}\cdot\kappa_{s^{2p}}^{2}}\nonumber\\
&+2\oo{-1}^{p}\sum_{j=0}^{p-1}\oo{-1}^{j}\intcurve{\cc{\oo{\kappa-\bar{\kappa}}+\bar{\kappa}}\oo{\kappa-\bar{\kappa}}_{s^{p-j}}\oo{\kappa-\bar{\kappa}}_{s^{p+j}}\oo{\kappa_{s^{2p}}}}\nonumber\\
&+\oo{-1}^{p+1}\intcurve{\cc{\oo{\kappa-\bar{\kappa}}+\bar{\kappa}}\oo{\kappa-\bar{\kappa}}_{s^{p}}^{2}\oo{\kappa-\bar{\kappa}}_{s^{2p}}}\nonumber\\
&\leq-2\intcurve{\kappa_{s^{2p+1}}^{2}}+2\intcurve{\kappa^{2}\cdot\kappa_{s^{2p}}^{2}}\nonumber\\
&+c\oo{p}\intcurve{\norm{P_{4}^{4p,2p}\oo{\kappa-\bar{\kappa}}}}+c\oo{p}L^{-1}\intcurve{\norm{P_{3}^{4p,2p}\oo{\kappa-\bar{\kappa}}}}.\label{LongTimeProp1,1}
\end{align}
Here $P_{i}^{j,k}\oo{\cdot}$ stands for a polynomial in $\phi$ of the form
\[
P_{i}^{j,k}\oo{\phi}=\sum_{\mu_{1}+\dots+\mu_{i}=j,\mu_{l}\leq k}\partial_{s}^{\mu_{1}}\phi\star\partial_{s}^{\mu_{2}}\phi\star\cdots\star\partial_{s}^{\mu_{i}}\phi.
\]
(See, for example \cite{DKS} for more details).
Using Lemma $\ref{AppendixLemma5}$, it follows that
\[
\intcurve{\norm{P_{4}^{4p,2p}\oo{\kappa-\bar{\kappa}}}}+L^{-1}\intcurve{\norm{P_{3}^{4p,2p}\oo{\kappa-\bar{\kappa}}}}\leq c\oo{p}\oo{K_{osc}+\sqrt{K_{osc}}}\intcurve{\kappa_{s^{2p+1}}^{2}},
\]
Hence inequality $\oo{\ref{LongTimeProp1,1}}$ can be rearranged to read
\begin{equation}
\frac{d}{dt}\intcurve{\kappa_{s^{p}}^{2}}+\oo{2-c\oo{p}\oo{K_{osc}+\sqrt{K_{osc}}}}\intcurve{\kappa_{s^{2p+1}}^{2}}\leq2\intcurve{\kappa^{2}\cdot\kappa_{s^{2p}}^{2}}.\label{LongTimeProp1,2}
\end{equation}
Next we expand the right hand side of $\ref{LongTimeProp1,2}$ and use the Cauchy-Schwarz inequality on the result:
\begin{align}
2\intcurve{\kappa^{2}\cdot\kappa_{s^{2p}}^{2}}&=2\intcurve{\cc{\oo{\kappa-\bar{\kappa}}^{2}+2\bar{\kappa}\oo{\kappa-\bar{\kappa}}+\bar{\kappa}^{2}}\kappa_{s^{2p}}^{2}}\nonumber\\
&\leq4\intcurve{\oo{\kappa-\bar{\kappa}}^{2}\kappa_{s^{2p}}^{2}}+4\bar{\kappa}^{2}\intcurve{\kappa_{s^{2p}}^{2}}.\label{LongTimeProp1,5}
\end{align}
The first term in $\oo{\ref{LongTimeProp1,5}}$ can be estimated easily, using
Lemma \ref{AppendixLemma2} with $f=\kappa_{s^{2p}}$:
\begin{align}
4\intcurve{\oo{\kappa-\bar{\kappa}}^{2}\kappa_{s^{2p}}^{2}}&\leq4\llll{\kappa_{s^{2p}}}_{\infty}^{2}\intcurve{\oo{\kappa-\bar{\kappa}}^{2}}\nonumber\\
&\leq 4\oo{\frac{L}{2\pi}\intcurve{\kappa_{s^{2p+1}}^{2}}}\intcurve{\oo{\kappa-\bar{\kappa}}^{2}}=\frac{2}{\pi}K_{osc}\intcurve{\kappa_{s^{2p+1}}^{2}}.\label{LongTimeProp1,6}
\end{align}
The second term is dealt with by using Lemma $\ref{AppendixLemma0}$
with $m=2p$:
\begin{align}
4\bar{\kappa}^{2}\intcurve{\kappa_{s^{2p}}^{2}}&=16\pi^{2}L^{-2}\intcurve{\kappa_{s^{2p}}^{2}}\nonumber\\
&\leq16\pi^{2}L^{-2}\oo{\varepsilon L^{2}\intcurve{\kappa_{s^{2p+1}}^{2}}+\frac{1}{4\varepsilon^{2p}}L^{-\oo{4p+1}}K_{osc}}\nonumber\\
&=16\pi^{2}\varepsilon\intcurve{\kappa_{s^{2p+1}}^{2}}+\frac{16\pi^{2}}{4\varepsilon^{2p}}L^{-\oo{4p+3}}K_{osc}.\nonumber
\end{align}
Here, of course $\varepsilon>0$ can be made as small as desired. Letting $\varepsilon^{\star}=16\pi^{2}\varepsilon$ yields
\begin{equation}
4\bar{\kappa}^{2}\intcurve{\kappa_{s^{2p}}^{2}}\leq\varepsilon^{\star}\intcurve{\kappa_{s^{2p+1}}^{2}}+4\pi^{2}\oo{\frac{16\pi^{2}}{\varepsilon^{\star}}}^{2p}L^{-\oo{4p+3}}K_{osc}.\label{LongTimeProp1,7}
\end{equation}
Substituting $\oo{\ref{LongTimeProp1,6}}$ and $\oo{\ref{LongTimeProp1,7}}$ into $\oo{\ref{LongTimeProp1,2}}$ gives
\begin{align}
&\frac{d}{dt}\intcurve{\kappa_{s^{p}}^{2}}+\oo{2-\oo{c\oo{p}+\frac{2}{\pi}+\varepsilon^{\star}}K_{osc}-c\oo{p}\sqrt{K_{osc}}}\intcurve{\kappa_{s^{2p+1}}^{2}}\nonumber\\
&\leq4\pi^{2}\oo{\frac{16\pi^{2}}{\varepsilon^{\star}}}^{2p}L^{-\oo{4p+3}}K_{osc}\leq4\pi^{2}\oo{\frac{16\pi^{2}}{\varepsilon^{\star}}}^{2p}\oo{4\pi A\oo{\gamma_{0}}}^{-\oo{4p+3}/2}K_{osc}.\label{LongTimeProp1,8}
\end{align}
Here we have used the inequality $\oo{\ref{IsoperimetricConsequence}}$ in the last step. Hence Proposition $\ref{FlowProp2}$ tells us that choosing choosing $K_{osc}\oo{0}<\varepsilon_{0}$ for $\varepsilon_{0}>0$ sufficiently small yields the following inequality
\begin{equation}
\frac{d}{dt}\intcurve{\kappa_{s^{p}}^{2}}\leq c\oo{\gamma_{0}}K_{osc}\label{LongTimeProp1,9}
\end{equation}
for some constant $c\oo{\gamma_{0}}$ which only depends upon our initial immersion. This inequality is valid over $\left[0,T\right)$. Note that we have chosen $\varepsilon^{\star}$ to be sufficiently small so that the absorption process is valid in the last step.
Integrating $\oo{\ref{LongTimeProp1,9}}$ while using our $L^{1}$ bound for $K_{osc}$ from $\oo{\ref{OscillationOfCurvatureL1}}$ then yields for any $t\in\left[0,T\right)$ the following inequality:
\[
\intcurve{\kappa_{s^{p}}^{2}}\leq\intcurve{\kappa_{s^{p}}^{2}}\Big|_{t=0}+\frac{c\oo{\gamma_{0}}}{2\oo{p+1}\oo{2\pi}^{2p}}L^{2\oo{p+1}}\oo{\gamma_{0}}\leq \tilde{c}\oo{\gamma_{0}}
\]
for some new constant $\tilde{c}\oo{\gamma_{0}}$ that only depends on the initial immersion. This completes the proof.
\end{proof}
| 3,991 | 30,672 |
en
|
train
|
0.52.5
|
\begin{corollary}\label{LongTimeCorollary2}
Suppose $\gamma:\mathbb{S}^{1}\times\left[0,T\right)\rightarrow\mathbb{R}^{2}$
solves $\eqref{FlowDefinition}$. Then there exists a constant
$\varepsilon_{0}>0$ (with $\varepsilon_{0}\leq K^{\star}$) such that if
\[
K_{osc}\oo{0}<\varepsilon_{0}\text{ and }I\oo{0}<e^{\frac{\varepsilon_{0}}{8\pi^{2}}}
\]
then $\gamma\oo{\mathbb{S}^{1}}$ approaches a round circle with radius $\sqrt{\frac{A\oo{\gamma_{0}}}{\pi}}$.
\end{corollary}
\begin{proof}
Recall from a previous discussion that to show $K_{osc}\searrow0$, it will be enough to show that $\norm{K_{osc}'}$ is bounded for all time.
Firstly, by Corollary $\ref{CurvatureCorollary1}$ and Corollary $\ref{LongTimeCorollary1}$ we know that for $\epsilon_{0}>0$ sufficiently small $T=\infty$ and for all time we have the estimate
\[
\norm{\frac{d}{dt}K_{osc}}\leq\oo{\frac{8\pi^{2}-K_{osc}}{L}}\llll{\kappa_{s^{p}}}_{2}^{2}\leq\frac{8\pi^{2}}{\sqrt{4\pi A\oo{\gamma_{0}}}}\llll{\kappa_{s^{p}}}_{2}^{2}\leq\frac{8\pi^{2}}{\sqrt{4\pi A\oo{\gamma_{0}}}}\cdot\tilde{c}\oo{\gamma_{0}}<\infty.
\]
Here we have used also the results of Proposition $\ref{LongTimeProp1}$.
This immediately tells us that $K_{osc}\searrow0$ as $t\nearrow\infty$.
We will denote the limiting immersion by $\gamma_{\infty}$. That is,
\[
\gamma_{\infty}:=\lim_{t\to\infty}\gamma_{t}\oo{\mathbb{S}^{1}}=\lim_{t\to\infty}\gamma\oo{\cdot,t}.
\]
Our earlier equations imply that $K_{osc}\oo{\gamma_{\infty}}\equiv0$. Note that because the isoperimetric inequality forces $L\oo{\gamma_{\infty}}\geq\sqrt{4\pi A\oo{\gamma_{\infty}}}=\sqrt{4\pi A\oo{\gamma_{0}}}>0$, we can not have $L\searrow0$ and so we may conclude that
\begin{equation}
\int_{\gamma_{\infty}}{\oo{\kappa-\bar{\kappa}}^{2}\,ds}=0.\label{LongTimeCorollary2,1}
\end{equation}
It follow from $\oo{\ref{LongTimeCorollary2,1}}$ that $\kappa\oo{\gamma_{\infty}}\equiv C$ for some constant $C>0$ (note that we know $C$ must be positive because it is impossible for a closed curve with constant curvature to possess negative curvature). That is to say, $\gamma_{t}\oo{\mathbb{S}^{1}}$ approaches a round circle as $t\nearrow\infty$. The final statement of the Corollary regarding the radius of $\gamma_{\infty}$ (which we denote $r\oo{\gamma_{\infty}}$) then follows easily because the enclosed area $A\oo{\gamma_{t}}$ is static in time:
\[
r\oo{\gamma_{\infty}}=\sqrt{\frac{A\oo{\gamma_{\infty}}}{\pi}}=\sqrt{\frac{A\oo{\gamma_{0}}}{\pi}}.
\]
\end{proof}
| 839 | 30,672 |
en
|
train
|
0.52.6
|
Since the previous corollary tells us that $\gamma_{t}\oo{\mathbb{S}^{1}}\rightarrow\mathbb{S}_{\sqrt{\frac{A\oo{\gamma_{0}}}{\pi}}}^{1}$, we can conclude that for every $m\in\mathbb{N}$ there exists a sequence of times $\left\{t_{j}\right\}$ such that
\[
\int{\kappa_{s^{m}}^{2}}\Big|_{t=t_{j}}\searrow0.
\]
Unfortunately, this is only subconvergence, and does not allow us to rule out the possibility of short sharp ``spikes'' (oscillations) in time. Indeed, even if we were to show that for every $m\in\mathbb{N}$ we have $\llll{\kappa_{s^{m}}}_{2}^{2}\in L^{1}\oo{\left[0,\infty\right)}$ (which is true), this would not be enough because these aforementioned ``spikes'' could occur on a time interval approaching that of (Lebesgue) measure zero. To overcome this dilemma, we attempt to control $\norm{\frac{d}{dt}\intcurve{\kappa_{s^{m}}^{2}}}$, and show that his quantity can be bounded by a multiple of $K_{osc}\oo{0}$ (which can be fixed to be as small as desired a priori). We will see this allows to strengthen the subconvergences result above to one of classical exponential convergence.
\begin{corollary}[Exponential Convergence]\label{LongTimeCorollary3}
Suppose $\gamma:\mathbb{S}^{1}\times\left[0,T\right)\rightarrow\mathbb{R}^{2}$
solves $\eqref{FlowDefinition}$ and satisfies the assumptions of Corollary
$\ref{LongTimeCorollary2}$. Then for each $m\in\mathbb{N}$ there are constants
$c_{m},c_{m}^{\star}$ such that we have the estimates
\[
\intcurve{\kappa_{s^{m}}^{2}}\leq c_{m}e^{-c_{m}^{\star}t}\text{ and }\llll{\kappa_{s^{m}}}_{\infty}\leq \sqrt{\frac{L\oo{\gamma_{0}}c_{m+1}}{2\pi}}e^{-\frac{c_{m+1}^{\star}}{2}t}.
\]
\end{corollary}
\begin{proof}
We first derive the evolution equation for $\intcurve{\kappa_{s^{m}}^{2}},m\in\mathbb{N}$ in a similar manner to Proposition $\ref{LongTimeProp1}$:
\begin{align}
\frac{d}{dt}\intcurve{\kappa_{s^{m}}^{2}}&=-2\intcurve{\kappa_{s^{m+p+1}}^{2}}+2\oo{-1}^{p}\sum_{j=1}^{m}\oo{-1}^{j}\intcurve{\kappa\cdot\kappa_{s^{m-j}}\cdot\kappa_{s^{m+j}}\cdot\kappa_{s^{2p}}}\nonumber\\
&+\oo{-1}^{p}\intcurve{\kappa\cdot\kappa_{s^{m}}^{2}\cdot\kappa_{s^{2p}}}.\label{LongTimeCorollary3,1}
\end{align}
We need to be careful in dealing with the extraneous terms in $\oo{\ref{LongTimeCorollary3,1}}$. We wish to apply Lemma $\ref{AppendixLemma5}$
but to do so must consider the cases $p\geq m$ and $p\leq m$ separately.
If $p\geq m$, with $p=m+l,l\in\mathbb{N}_{0}$, then we can perform integration by parts on each term in $\oo{\ref{LongTimeCorollary3,1}}$ $l$ times:
\begin{align}
&2\oo{-1}^{p}\sum_{j=1}^{m}\oo{-1}^{j}\intcurve{\kappa\cdot\kappa_{s^{m-j}}\cdot\kappa_{s^{m+j}}\cdot\kappa_{s^{2p}}}+\oo{-1}^{p}\intcurve{\kappa\cdot\kappa_{s^{m}}^{2}\cdot\kappa_{s^{2p}}}\nonumber\\
&=2\oo{-1}^{p}\sum_{j=1}^{m-1}\oo{-1}^{j}\intcurve{\oo{\kappa-\bar{\kappa}+\bar{\kappa}}\oo{\kappa-\bar{\kappa}}_{s^{m-j}}\oo{\kappa-\bar{\kappa}}_{s^{m+j}}\oo{\kappa-\bar{\kappa}}_{s^{m+p+l}}}\nonumber\\
&+2\oo{-1}^{m+p}\intcurve{\cc{\oo{\kappa-\bar{\kappa}}^{2}+2\bar{\kappa}\oo{\kappa-\bar{\kappa}}+\bar{\kappa}^{2}}\oo{\kappa-\bar{\kappa}}_{s^{2m}}\oo{\kappa-\bar{\kappa}}_{s^{m+p+l}}}\nonumber\\
&+\oo{-1}^{p}\intcurve{\oo{\kappa-\bar{\kappa}+\bar{\kappa}}\oo{\kappa-\bar{\kappa}}_{s^{m}}^{2}\oo{\kappa-\bar{\kappa}}_{s^{m+p+l}}}\nonumber\\
&=2\oo{-1}^{p}\sum_{j=1}^{m-1}\oo{-1}^{j+l}\intcurve{\partial_{s}^{l}\cc{\oo{\kappa-\bar{\kappa}+\bar{\kappa}}\oo{\kappa-\bar{\kappa}}_{s^{m-j}}\oo{\kappa-\bar{\kappa}}_{s^{m+j}}}\oo{\kappa-\bar{\kappa}}_{s^{p+p}}}\nonumber\\
&+2\intcurve{\partial_{s}^{l}\cc{\cc{\oo{\kappa-\bar{\kappa}}^{2}+2\bar{\kappa}\oo{\kappa-\bar{\kappa}}+\bar{\kappa}^{2}}\oo{\kappa-\bar{\kappa}}_{s^{2m}}}\oo{\kappa-\bar{\kappa}}_{s^{m+p}}}\nonumber\\
&+\oo{-1}^{p+l}\intcurve{\partial_{s}^{l}\cc{\oo{\kappa-\bar{\kappa}+\bar{\kappa}}\oo{\kappa-\bar{\kappa}}_{s^{m}}^{2}}\oo{\kappa-\bar{\kappa}}_{s^{m+p}}}\nonumber\\
&\leq c\oo{m,p}\intcurve{\norm{P_{4}^{2\oo{m+p},m+p}\oo{\kappa-\bar{\kappa}}}}+c\cdot L^{-1}\intcurve{\norm{P_{3}^{2\oo{m+p},m+p}\oo{\kappa-\bar{\kappa}}}}\nonumber\\
&+2\bar{\kappa}^{2}\intcurve{\kappa_{s^{2m+l}}\cdot\kappa_{s^{m+p}}}\nonumber\\
&\leq4\pi^{2}L^{-2}\intcurve{\kappa_{s^{m+p}}^{2}}+c\oo{m,p}\oo{K_{osc}+\sqrt{K_{osc}}}\intcurve{\kappa_{s^{m+p+1}}^{2}}.\label{LongTimeCorollary3,2}
\end{align}
Here we have used the energy inequality Lemma \ref{AppendixLemma5} on the $P$-style terms, as well as Lemma $\ref{CurvatureLemma1}$ which tells us that $\bar{\kappa}=\frac{2\pi}{L}$.
If $p<m$ (say with $m=p+l,l\in\mathbb{N}$), then we must proceed slightly differently. In this case, identity $\oo{\ref{LongTimeCorollary3,1}}$ becomes
\begin{align}
&2\oo{-1}^{p}\sum_{j=1}^{m}\oo{-1}^{j}\intcurve{\kappa\cdot\kappa_{s^{m-j}}\cdot\kappa_{s^{m+j}}\cdot\kappa_{s^{2p}}}+\oo{-1}^{p}\intcurve{\kappa\cdot\kappa_{s^{m}}^{2}\cdot\kappa_{s^{2p}}}\nonumber\\
&=2\oo{-1}^{p}\sum_{j=1}^{p+l-1}\oo{-1}^{j}\intcurve{\oo{\kappa-\bar{\kappa}+\bar{\kappa}}\oo{\kappa-\bar{\kappa}}_{s^{p+l-j}}\oo{\kappa-\bar{\kappa}}_{s^{p+l+j}}\oo{\kappa-\bar{\kappa}}_{s^{2p}}}\nonumber\\
&+2\oo{-1}^{m+p}\intcurve{\cc{\oo{\kappa-\bar{\kappa}}^{2}+2\bar{\kappa}\oo{\kappa-\bar{\kappa}}+\bar{\kappa^{2}}}\oo{\kappa-\bar{\kappa}}_{s^{m+p+l}}\oo{\kappa-\bar{\kappa}}_{s^{2p}}}\nonumber\\
&+\intcurve{\oo{\kappa-\bar{\kappa}+\bar{\kappa}}\oo{\kappa-\bar{\kappa}}_{s^{p+l}}^{2}\oo{\kappa-\bar{\kappa}}_{s^{2p}}}\nonumber\\
&=2\oo{-1}^{p}\sum_{j=1}^{p}\oo{-1}^{j}\intcurve{\oo{\kappa-\bar{\kappa}+\bar{\kappa}}\oo{\kappa-\bar{\kappa}}_{s^{p+l-j}}\oo{\kappa-\bar{\kappa}}_{s^{p+l+j}}\oo{\kappa-\bar{\kappa}}_{s^{2p}}}\nonumber\\
&+2\oo{-1}^{p}\sum_{j=p+1}^{p+l-1}\oo{-1}^{j}\intcurve{\oo{\kappa-\bar{\kappa}+\bar{\kappa}}\oo{\kappa-\bar{\kappa}}_{s^{p+l-j}}\oo{\kappa-\bar{\kappa}}_{s^{p+l+j}}\oo{\kappa-\bar{\kappa}}_{s^{2p}}}\nonumber\\
&+2\oo{-1}^{m+p}\intcurve{\cc{\oo{\kappa-\bar{\kappa}}^{2}+2\bar{\kappa}\oo{\kappa-\bar{\kappa}}+\bar{\kappa^{2}}}\oo{\kappa-\bar{\kappa}}_{s^{m+p+l}}\oo{\kappa-\bar{\kappa}}_{s^{2p}}}\nonumber\\
&+\intcurve{\oo{\kappa-\bar{\kappa}+\bar{\kappa}}\oo{\kappa-\bar{\kappa}}_{s^{p+l}}^{2}\oo{\kappa-\bar{\kappa}}_{s^{2p}}}\nonumber\\
&\leq 2\oo{-1}^{m+p}\oo{\frac{2\pi}{L}}^{2}\intcurve{\kappa_{s^{m+p+l}}\cdot\kappa_{s^{2p}}}+c\oo{m,p}\intcurve{\norm{P_{4}^{2\oo{m+p},m+p}\oo{\kappa-\bar{\kappa}}}}\nonumber\\
&+c\oo{m,p}L^{-1}\intcurve{\norm{P_{3}^{2\oo{m+p},m+p}\oo{\kappa-\bar{\kappa}}}}+c\oo{m,p}\intcurve{\norm{P_{4}^{2\oo{m+p},M}\oo{\kappa-\bar{\kappa}}}}\nonumber\\
&+c\oo{m,p}L^{-1}\intcurve{\norm{P_{3}^{2\oo{m+p},M}\oo{\kappa-\bar{\kappa}}}}\nonumber\\
&+2\oo{-1}^{p}\sum_{j=p+1}^{p+l-1}\oo{-1}^{j}\intcurve{\oo{\kappa-\bar{\kappa}+\bar{\kappa}}\oo{\kappa-\bar{\kappa}}_{s^{p+l-j}}\oo{\kappa-\bar{\kappa}}_{s^{p+l+j}}\oo{\kappa-\bar{\kappa}}_{s^{2p}}}\label{LongTimeCorollary3,3}
\end{align}
Here $M:=\max\left\{m,2p\right\}<m+p$.
The second and third terms of $\oo{\ref{LongTimeCorollary3,3}}$ are identical to those in our calculation of $\oo{\ref{LongTimeCorollary3,2}}$, in which we established the identity
\begin{align}
&\intcurve{\norm{P_{4}^{2\oo{m+p},m+p}\oo{\kappa-\bar{\kappa}}}}+L^{-1}\intcurve{\norm{P_{3}^{2\oo{m+p},m+p}\oo{\kappa-\bar{\kappa}}}}\nonumber\\
&\leq c\oo{m,p}\oo{K_{osc}+\sqrt{K_{osc}}}\intcurve{\kappa_{s^{m+p+1}}^{2}}.\label{LongTimeCorollary3,4}
\end{align}
The fourth and fifth terms in $\oo{\ref{LongTimeCorollary3,3}}$ are estimated in a similar way. Because $M<m+p$, we are free to utilise Lemma $\ref{AppendixLemma5}$ wth $K=m+p+1$ and then the terms are estimatable in the same way as the $P$-style terms in $\oo{\ref{LongTimeCorollary3,2}}$. We conclude that
\begin{align}
&\intcurve{\norm{P_{4}^{2\oo{m+p},M}\oo{\kappa-\bar{\kappa}}}}+L^{-1}\intcurve{\norm{P_{3}^{2\oo{m+p},M}\oo{\kappa-\bar{\kappa}}}}\nonumber\\
&\leq c\oo{m,p}\oo{K_{osc}+\sqrt{K_{osc}}}\intcurve{\kappa_{s^{m+p+1}}^{2}}.\label{LongTimeCorollary3,5}
\end{align}
Finally, the last part of $\oo{\ref{LongTimeCorollary3,3}}$ involving the summation can be estimated by applying integrating by parts by parts $j-p$ times to each term in $j$ and then estimating in the same way as above:
\begin{align}
&2\oo{-1}^{p}\sum_{j=p+1}^{p+l-1}\oo{-1}^{j}\intcurve{\oo{\kappa-\bar{\kappa}+\bar{\kappa}}\oo{\kappa-\bar{\kappa}}_{s^{p+l-j}}\oo{\kappa-\bar{\kappa}}_{s^{p+l+j}}\oo{\kappa-\bar{\kappa}}_{s^{2p}}}\nonumber\\
&2\oo{-1}^{p}\sum_{j=p+1}^{p+l-1}\oo{-1}^{2j-p}\intcurve{\partial_{s}^{j-p}\cc{\oo{\kappa-\bar{\kappa}+\bar{\kappa}}\oo{\kappa-\bar{\kappa}}_{s^{p+l-j}}\oo{\kappa-\bar{\kappa}}_{s^{2p}}}\oo{\kappa_{s^{2p+l}}}}\nonumber\\
&\leq c\oo{m,p}\intcurve{\norm{P_{4}^{2\oo{m+p},m+p}\oo{\kappa-\bar{\kappa}}}}+c\oo{m,p}L^{-1}\intcurve{\norm{P_{3}^{2\oo{m+p},m+p}}\oo{\kappa-\bar{\kappa}}}\nonumber\\
&\leq c\oo{m,p}\oo{K_{osc}+\sqrt{K_{osc}}}\intcurve{\kappa_{s^{m+p+1}}^{2}}.\label{LongTimeCorollary3,6}
\end{align}
Combining $\oo{\ref{LongTimeCorollary3,4}}$,$\oo{\ref{LongTimeCorollary3,5}}$ and $\oo{\ref{LongTimeCorollary3,6}}$ and substituting into $\oo{\ref{LongTimeCorollary3,3}}$ then gives
\begin{align}
&2\oo{-1}^{p}\sum_{j=1}^{m}\oo{-1}^{j}\intcurve{\kappa\cdot\kappa_{s^{m-j}}\cdot\kappa_{s^{m+j}}\cdot\kappa_{s^{2p}}}+\oo{-1}^{p}\intcurve{\kappa\cdot\kappa_{s^{m}}^{2}\cdot\kappa_{s^{2p}}}\nonumber\\
&\leq 4\pi^{2}L^{-2}\intcurve{\kappa_{s^{m+p}}^{2}}+c\oo{m,p}\oo{K_{osc}+\sqrt{K_{osc}}}\intcurve{\kappa_{s^{m+p+1}}^{2}}.\label{LongTimeCorollary3,7}
\end{align}
We can clearly see from $\oo{\ref{LongTimeCorollary3,2}}$ and $\oo{\ref{LongTimeCorollary3,7}}$ that the estimates of the extraneous terms in $\oo{\ref{LongTimeCorollary3,1}}$ are of the same form, regardless of the sign of $p-m$. We can conclude that
\begin{equation}
\frac{d}{dt}\intcurve{\kappa_{s^{m}}^{2}}+\oo{2-c\oo{m,p}\oo{K_{osc}+\sqrt{K_{osc}}}}\intcurve{\kappa_{s^{m+p+1}}^{2}}\leq 4\pi^{2}L^{-2}\intcurve{\kappa_{s^{m+p}}^{2}}.\label{LongTimeCorollary3,8}
\end{equation}
Let us step back for a moment and forget about our time parameter, assuming without loss of generality that we are looking at a fixed time slice.
| 4,004 | 30,672 |
en
|
train
|
0.52.7
|
Let us step back for a moment and forget about our time parameter, assuming without loss of generality that we are looking at a fixed time slice.
We claim that for any smooth closed curve $\gamma$ and any $l\in\mathbb{N}$, there exists a universal, bounded constant $c_{l}>0$ such that
\begin{equation}
\intcurve{\kappa_{s^{l}}^{2}}\leq c_{l}L^{2}K_{osc}\intcurve{\kappa_{s^{l+1}}^{2}}.\label{LongTimeCorollary3,9}
\end{equation}
Let us assume for the sake of contradiction that we can not find a suitable constant $c_{l}<\infty$ such that inequality $\oo{\ref{LongTimeCorollary3,9}}$ holds. Then, there exists a sequence of immersions $\left\{\gamma_{j}\right\}$ such that
\begin{equation}
R_{j}:=\frac{\llll{\kappa_{s^{l}}}_{2,\gamma_{j}}^{2}}{L^{2}\oo{\gamma_{j}}K_{osc}\oo{\gamma_{j}}\llll{\kappa_{s^{l+1}}}_{2,\gamma_{j}}^{2}}\nearrow\infty\text{ as }j\rightarrow\infty.\label{LongTimeCorollary3,10}
\end{equation}
Now, Theorem $\ref{AppendixTheorem1}$ implies that for any $j\in\mathbb{N}$ we have
\[
R_{j}\leq \frac{\frac{L^{2}\oo{\gamma_{j}}}{4\pi^{2}}\llll{\kappa_{s^{l+1}}}_{2,\gamma_{j}}^{2}}{L^{2}\oo{\gamma_{j}}K_{osc}\oo{\gamma_{j}}\llll{\kappa_{s^{l+1}}}_{2,\gamma_{j}}^{2}}=\frac{1}{4\pi^{2}K_{osc}\oo{\gamma_{j}}},
\]
and so the only way that $\oo{\ref{LongTimeCorollary3,10}}$ can occur is if we have
\begin{equation}
K_{osc}\oo{\gamma_{j}}\searrow0\text{ as }j\rightarrow\infty.\label{LongTimeCorollary3,11}
\end{equation}
Then, as each $\gamma_{j}$ satisfies the criteria of Theorem $\ref{AppendixTheorem1}$, we conclude there is a subsequence of immersions $\left\{\gamma_{j_{k}}\right\}$ and an immersion $\gamma_{\infty}$ such that $\gamma_{j_{k}}\rightarrow\gamma_{\infty}$ in the $C^{1}$-topology. Moreover, by $\oo{\ref{LongTimeCorollary3,11}}$ we have $K_{osc}\oo{\gamma_{\infty}}=0$. But this implies that $\gamma_{\infty}$ must be a circle, in which case both sides of inequality $\oo{\ref{LongTimeCorollary3,9}}$ are zero. Hence the inequality holds trivially with the immersion $\gamma_{\infty}$ for \emph{any} $c_{l}$ we wish, and so we can not in fact have $R_{j}\nearrow\infty$. This contradicts $\oo{\ref{LongTimeCorollary3,10}}$, and so the assumption that we can not find a constant $c_{l}$ such that the inequality $\oo{\ref{LongTimeCorollary3,9}}$ holds, must be false.
Next, combining $\oo{\ref{LongTimeCorollary3,9}}$ with $\oo{\ref{LongTimeCorollary3,8}}$ gives us
\[
\frac{d}{dt}\intcurve{\kappa_{s^{m}}^{2}}+\oo{2-c\oo{m,p}\oo{K_{osc}+\sqrt{K_{osc}}}-4\tilde{c}_{m}\pi^{2}K_{osc}}\intcurve{\kappa_{s^{m+p+1}}^{2}}\leq0.
\]
Here $\tilde{c}_{m}$ is our new interpolative constant, which we take to be the largest of all optimal constants in inequality $\oo{\ref{LongTimeCorollary3,9}}$ for closed simple curves with length bounded by $L\oo{\gamma_{0}}$. Then, because $K_{osc}\rightarrow0$, we know that there exists a time, say $t_{m}$, such that for $t\geq t_{m}$, $c\oo{m,p}\oo{K_{osc}+\sqrt{K_{osc}}}+4\tilde{c}_{m}\pi^{2}K_{osc}\leq1$. Hence for $t\geq t_{m}$ the previous inequality implies that
\begin{equation}
\frac{d}{dt}\intcurve{\kappa_{s^{m}}^{2}}\leq-\intcurve{\kappa_{s^{m+p+1}}^{2}}.\label{LongTimeCorollary3,12}
\end{equation}
Next, applying inequality Lemma $\ref{AppendixLemma1}$ $p+1$ times and using the monotonicity of $L\oo{\gamma_{t}}$ gives
\[
\intcurve{\kappa_{s^{m}}^{2}}\leq\oo{\frac{L^{2}\oo{\gamma_{t}}}{4\pi^{2}}}^{p+1}\intcurve{\kappa_{s^{m+p+1}}^{2}}\leq\oo{\frac{L^{2}\oo{\gamma_{0}}}{4\pi^{2}}}^{p+1}\intcurve{\kappa_{s^{m+p+1}}^{2}}.
\]
Hence if we define $c_{m}^{\star}:=\oo{\frac{4\pi^{2}}{L^{2}\oo{\gamma_{0}}}}^{p+1}$ then we conclude from $\oo{\ref{LongTimeCorollary3,12}}$ that for any $t\geq t_{m}$ we have the estimate
\[
\frac{d\intcurve{\kappa_{s^{m}}^{2}}}{\intcurve{\kappa_{s^{m}}^{2}}}\leq-c_{m}^{\star}\,dt.
\]
Integrating over $\left[t_{m},t\right]$ and exponentiating yields
\[
\intcurve{\kappa_{s^{m}}^{2}}\leq\int_{\gamma_{t_{m}}}{\kappa_{s^{m}}^{2}\,ds}\cdot e^{-c_{m}^{\star}\oo{t-t_{m}}}=\oo{e^{c_{m}^{\star}t_{m}}\int_{\gamma_{t_{m}}}{\kappa_{s^{m}}^{2}\,ds}}\cdot e^{-c_{m}^{\star}t},
\]
which is the first statement of the corollary. For the second statement, we simply combine the first statement and Lemma $\ref{AppendixLemma2}$ with $f=\kappa_{s^{m}}$:
\[
\llll{\kappa_{s^{m}}}_{\infty}^{2}\leq \frac{L\oo{\gamma_{0}}}{2\pi}\intcurve{\kappa_{s^{m+1}}^{2}}\leq \frac{L\oo{\gamma_{0}}c_{m+1}}{2\pi}e^{-c_{m+1}^{\star}t}.
\]
The pointwise exponential convergence result follows immediately from taking the square root of both sides.
\end{proof}
| 1,654 | 30,672 |
en
|
train
|
0.52.8
|
Let us finish by proving Proposition \ref{PN1}.
\begin{proof}
We follow \cite{Wkosc}.
Rearranging $\gamma$ in time if necessary, we may assume that
\begin{align*}
k(\cdot,t) \not> 0 ,\qquad &\text{ for all }t\in[0,t_0)\\
k(\cdot,t) > 0, \qquad &\text{ for all }t\in[t_0,\infty)
\end{align*}
where
$t_0 > \frac{2}{p+1}\bigg[
\bigg( \frac{L(\gamma_0)}{2\pi}\bigg)^{2(p+1)}
- \bigg( \frac{A(\gamma_0)}{\pi} \bigg)^{p+1}
\bigg]$,
otherwise we have nothing to prove. However in this case we have
\begin{align*}
\frac{d}{dt}L
&= -\vn{k_{s^p}}_2^2
\le -\frac{4\pi^2}{L^2}\vn{k_{s^{p-1}}}_2^2
\le \cdots
\le -\bigg(\frac{4\pi^2}{L^2}\bigg)^{p-1}\vn{k_s}_2^2
\\
&\le -\frac{\pi^2}{L^2}\bigg(\frac{4\pi^2}{L^2}\bigg)^{p-1}\vn{k}_2^2
\\
&\le -\frac{4\pi^4}{L^3}\bigg(\frac{4\pi^2}{L^2}\bigg)^{p-1},&\text{for}\ t\in[0,t_0),
\intertext{where we used the fact that $\gamma$ is closed and that the curvature has a zero. This implies}
L^{2p+2}(t) &\le -\frac{p+1}{2}(2\pi)^{2p+2}t + L^{2p+2}(\gamma_0),& \text{for}\ t\in[0,t_0),
\end{align*}
and thus $L^{2p+2}(t_0) < (4\pi A(\gamma_0))^{2p+2}$.
This is in contradiction with the isoperimetric inequality.
\end{proof}
\end{section}
\begin{section}{Appendix}
\begin{lemma}\label{AppendixLemma0}
Let $\gamma:\mathbb{S}^{1}\rightarrow\mathbb{R}^{2}$ be a smooth closed curve with Euclidean curvature $\kappa$ and arc length element $ds$. Then for any $m\in\mathbb{N}$ we have
\[
\intcurve{\kappa_{s^{m}}^{2}}\leq\varepsilon L^{2}\intcurve{\kappa_{s^{m+1}}^{2}}+\frac{1}{4\varepsilon^{m}}L^{-\oo{2m+1}}K_{osc},
\]
where $\varepsilon>0$ can be made as small as desired.
\end{lemma}
\begin{proof}
We will prove the lemma inductively. The case $m=1$ can be checked quite easily, by applying integration by parts and the Cauchy-Schwarz inequality:
\begin{align}
\intcurve{\kappa_{s}^{2}}&=\intcurve{\oo{\kappa-\bar{\kappa}}_{s}^{2}}=-\intcurve{\oo{\kappa-\bar{\kappa}}\oo{\kappa-\bar{\kappa}}_{s^{2}}}\nonumber\\
&\leq\oo{\intcurve{\oo{\kappa-\bar{\kappa}}^{2}}}^{\frac{1}{2}}\oo{\intcurve{\kappa_{s^{2}}^{2}}}^{\frac{1}{2}}\nonumber\\
&\leq\varepsilon L^{2}\intcurve{\kappa_{s^{2}}^{2}}+\frac{1}{4\varepsilon^{1}}L^{-2}\intcurve{\oo{\kappa-\bar{\kappa}}^{2}}.\nonumber
\end{align}
Next assume inductively that the statement is true for $j=m$. That is, assume that
\begin{equation}
\intcurve{\kappa_{s^{j}}^{2}}\leq\varepsilon L^{2}\intcurve{\kappa_{s^{j+1}}^{2}}+\frac{1}{4\varepsilon^{j}}L^{-\oo{2j+1}}K_{osc}\label{AppendixLemma0,1}
\end{equation}
where $\varepsilon>0$ can be made as small as desired.
Again performing integration by parts and the Cauchy-Schwarz inequality, we have for any $\varepsilon>0$:
\begin{align}
\intcurve{\kappa_{s^{j+1}}^{2}}&=-\intcurve{\kappa_{s^{j}}\cdot\kappa_{s^{j+2}}}\leq\oo{\intcurve{\kappa_{s^{j}}^{2}}}^{\frac{1}{2}}\oo{\intcurve{\kappa_{s^j+2}^{2}}}^{\frac{1}{2}}\nonumber\\
&\leq\frac{\varepsilon}{2} L^{2}\intcurve{\kappa_{s^{j+2}}^{2}}+\frac{1}{2\varepsilon}L^{-2}\intcurve{\kappa_{s^{j}}^{2}}.\label{AppendixLemma0,2}
\end{align}
Substituting the inductive assumption $\oo{\ref{AppendixLemma0,1}}$ into $\oo{\ref{AppendixLemma0,2}}$ then gives
\begin{align}
\intcurve{\kappa_{s^{j+1}}^{2}}&\leq\frac{\varepsilon}{2} L^{2}\intcurve{\kappa_{s^{j+2}}^{2}}+\frac{1}{2\varepsilon}L^{-2}\cc{\varepsilon L^{2}\intcurve{\kappa_{s^{j+1}}^{2}}+\frac{1}{4\varepsilon^{j}}\oo{\varepsilon}L^{-\oo{2j+1}}K_{osc}},\nonumber
\end{align}
meaning that
\[
\frac{1}{2}\intcurve{\kappa_{s^{j+1}}^{2}}\leq\frac{\varepsilon}{2} L^{2}\intcurve{\kappa_{s^{j+2}}^{2}}+\frac{1}{2}\cdot\frac{1}{4\varepsilon^{j+1}}L^{-\oo{2\oo{j+1}+1}}K_{osc}.
\]
Multiplying out by $2$ then gives us the inductive step, completing the lemma.
\end{proof}
\begin{lemma}\label{AppendixLemma1}
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be an absolutely continuous and periodic function of period $P$. Then, if $\int_{0}^{P}{f\,dx}=0$ we have
\[
\int_{0}^{P}{f^{2}\,dx}\leq\frac{P^{2}}{4\pi^{2}}\int_{0}^{P}{f_{x}^{2}\,dx},
\]
with equality if and only if
\[
f\oo{x}=A\cos\oo{\frac{2\pi}{P}x}+B\sin\oo{\frac{2\pi}{P}x}
\]
for some constants $A,B$.
\end{lemma}
\begin{proof}
We will use the calculus of variations. Essentially, we wish to find $f$ that maximises the integral $\int_{0}^{P}{f^{2}\,dx}$, given a fixed value of $\int_{0}^{P}{f_{x}^{2}\,dx}$. We will show that combining this with the requirement that $\int_{0}^{P}{f\,dx}=0$ forces the extremal function to satisfy
\[
\frac{\int_{0}^{P}{f^{2}\,dx}}{\int_{0}^{P}{f_{x}^{2}\,dx}}\leq\frac{P^{2}}{4\pi^{2}}.
\]
For the constrained problem, the associated Euler-Lagrange equation is
\[
L=f^{2}+\lambda f_{x}^{2},
\]
with extremal functions satisfying
\[
\frac{\partial L}{\partial f}-\frac{d}{dx}\oo{\frac{\partial F}{\partial f_{x}}}=2f-2\lambda f_{xx}=0.
\]
That is to say,
\begin{equation}
f_{xx}-\frac{1}{\lambda}f=0.\label{AppendixLemma1,1}
\end{equation}
This means that
\[
0\leq\int_{0}^{P}{f_{x}^{2}\,dx}=-\int_{0}^{P}{ff_{xx}\,dx}=-\frac{1}{\lambda}\int_{0}^{P}{f^{2}\,dx},
\]
which forces $\lambda<0$. By standard arguments, we conclude from $\oo{\ref{AppendixLemma1,1}}$ that our extremal function is
\begin{equation}
f\oo{x}=A\cos\oo{\frac{x}{\sqrt{\norm{\lambda}}}}+B\sin\oo{\frac{x}{\sqrt{\norm{\lambda}}}}.\label{AppendixLemma1,2}
\end{equation}
Here $A,B$ are constants. The periodicity of $f$ forces $f\oo{0}=f\oo{P}$, so
\begin{equation}
A=A\cos\oo{\frac{P}{\sqrt{\norm{\lambda}}}}+B\sin\oo{\frac{P}{\sqrt{\norm{\lambda}}}}.\label{AppendixLemma1,3}
\end{equation}
Also, the requirement that $\int_{0}^{P}{f\,dx}=0$ forces
\begin{equation}
A\sin\oo{\frac{P}{\sqrt{\norm{\lambda}}}}-B\cos\oo{\frac{P}{\sqrt{\norm{\lambda}}}}=-B.\label{AppendixLemma1,4}
\end{equation}
Combining $\oo{\ref{AppendixLemma1,3}}$ and $\oo{\ref{AppendixLemma1,4}}$,
\[
A^{2}=A^{2}\cos\oo{\frac{P}{\sqrt{\norm{\lambda}}}}+AB\sin\oo{\frac{P}{\sqrt{\norm{\lambda}}}}\text{ and }B^{2}=B^{2}\cos\oo{\frac{P}{\sqrt{\norm{\lambda}}}}-AB\sin\oo{\frac{P}{\sqrt{\norm{\lambda}}}},
\]
meaning that
\[
A^{2}+B^{2}=\oo{A^{2}+B^{2}}\cos\oo{\frac{P}{\sqrt{\norm{\lambda}}}}.
\]
We conclude
\[
\frac{P}{\sqrt{\norm{\lambda}}}=2n\pi
\]
for some $n\in\mathbb{Z}\backslash\left\{0\right\}$ to be determined.
Hence
\begin{equation}
f\oo{x}=A\cos\oo{\frac{2n\pi x}{P}}+B\sin\oo{\frac{2n\pi x}{P}}.\label{AppendixLemma1,5}
\end{equation}
A quick calculation yields
\[
\int_{0}^{P}{f^{2}\,dx}=\oo{\frac{A^{2}+B^{2}}{2}}P,\,\,\text{and}\,\,\int_{0}^{P}{f_{x}^{2}\,dx}=\oo{\frac{2n\pi}{P}}^{2}\oo{\frac{A^{2}+B^{2}}{2}}P.
\]
Hence for any of our extremal functions $f$,
\[
\frac{\int_{0}^{P}{f^{2}\,dx}}{\int_{0}^{P}{f_{x}^{2}\,dx}}=\oo{\frac{P}{2n\pi}}^{2}\leq\frac{P^{2}}{4\pi^{2}},
\]
with equality if and only if $n=1$. Thus our constrained function $f$ that maximises the ratio $\frac{\int_{0}^{P}{f^{2}\,dx}}{\int_{0}^{P}{f_{x}^{2}\,dx}}$ is given by
\[
f\oo{x}=A\cos\oo{\frac{2\pi}{P}x}+B\sin\oo{\frac{2\pi}{P}x},
\]
with
\[
\int_{0}^{P}{f^{2}\,dx}\leq\frac{P^{2}}{4\pi^{2}}\int_{0}^{P}{f_{x}^{2}\,dx}
\]
amongst all continuous and $P-$periodic functions with $\int_{0}^{P}{f\,dx}=0$.
\end{proof}
\begin{lemma}\label{AppendixLemma2}
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be an absolutely continuous and periodic function of period $P$. Then, if $\int_{0}^{P}{f\,dx}=0$ we have
\[
\llll{f}_{\infty}^{2}\leq\frac{P}{2\pi}\int_{0}^{P}{f_{x}^{2}\,dx}.
\]
\end{lemma}
\begin{proof}
Since $\int_{0}^{P}{f\,dx}=0$ and $f$ is $P-$periodic we conclude that there exists distinct $0\leq p<q<P$ such that
\[
f\oo{p}=f\oo{q}=0.
\]
Next, the fundamental theorem of calculus tells us that for any $x\in\oo{0,P}$,
\[
\frac{1}{2}\cc{f\oo{x}}^{2}=\int_{p}^{x}{ff_{x}\,dx}=\int_{q}^{x}{ff_{x}\,dx}.
\]
Hence
\begin{align*}
&\cc{f\oo{x}}^{2}=\int_{p}^{x}{ff_{x}\,dx}-\int_{qx}^{q}{ff_{x}\,dx}\leq\int_{p}^{q}{\norm{ff_{x}}\,dx}\leq\int_{0}^{P}{\norm{ff_{x}}\,dx}\\
&\leq\oo{\int_{0}^{P}{f^{2}\,dx}\cdot\int_{0}^{P}{f_{x}^{2}\,dx}}^{\frac{1}{2}}\leq\frac{P}{2\pi}\int_{0}^{P}{f_{x}^{2}\,dx},
\end{align*}
where the last step follows from Lemma $\ref{AppendixLemma1}$. We have also utilised H\"{o}lder's inequality with $p=q=2$.
\end{proof}
\begin{lemma}[\cite{DKS}, Lemma $2.4$]\label{AppendixLemma3}
Let $\gamma:\mathbb{S}^{1}\rightarrow\mathbb{R}^{2}$ be a smooth closed curve. Let $\phi:\mathbb{S}^{1}\rightarrow\mathbb{R}$ be a sufficiently smooth function. Then for any $l\geq 2,K\in\mathbb{N}$ and $0\leq i<K$ we have
\begin{equation}
L^{i+1-\frac{1}{l}}\oo{\intcurve{\oo{\phi}_{s^{i}}^{2}}}^{\frac{1}{l}}\leq c\oo{K}L^{\frac{1-\alpha}{2}}\oo{\intcurve{\phi^{2}}}^{\frac{1-\alpha}{2}}\llll{\phi}_{K,2}^{\alpha}.\label{AppendixLemma3,1}
\end{equation}
Here $\alpha=\frac{i+\frac{1}{2}-\frac{1}{l}}{K}$, and
\[
\llll{\phi}_{K,2}:=\sum_{j=0}^{K}L^{j+\frac{1}{2}}\oo{\intcurve{\oo{\phi}_{s^{j}}^{2}}}^{\frac{1}{2}}.
\]
In particular, if $\phi=\kappa-\bar{\phi}$, then
\begin{equation}
L^{i+1-\frac{1}{l}}\oo{\intcurve{\oo{k-\bar{k}}_{s^{i}}^{2}}}^{\frac{1}{l}}\leq c\oo{K}\oo{K_{osc}}^{\frac{1-\alpha}{2}}\llll{k-\bar{k}}_{K,2}^{\alpha}.\label{AppendixLemma3,2}
\end{equation}
\end{lemma}
\begin{proof}
The proof is identical to that of Lemma $2.4$ from \cite{DKS} and is of a
standard interpolative nature. Note that although we use $k-\bar{k}$ in the
identity (as opposed to \cite{DKS} where $k\nu$ is used).
\end{proof}
| 3,891 | 30,672 |
en
|
train
|
0.52.9
|
\begin{lemma}\label{AppendixLemma2}
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be an absolutely continuous and periodic function of period $P$. Then, if $\int_{0}^{P}{f\,dx}=0$ we have
\[
\llll{f}_{\infty}^{2}\leq\frac{P}{2\pi}\int_{0}^{P}{f_{x}^{2}\,dx}.
\]
\end{lemma}
\begin{proof}
Since $\int_{0}^{P}{f\,dx}=0$ and $f$ is $P-$periodic we conclude that there exists distinct $0\leq p<q<P$ such that
\[
f\oo{p}=f\oo{q}=0.
\]
Next, the fundamental theorem of calculus tells us that for any $x\in\oo{0,P}$,
\[
\frac{1}{2}\cc{f\oo{x}}^{2}=\int_{p}^{x}{ff_{x}\,dx}=\int_{q}^{x}{ff_{x}\,dx}.
\]
Hence
\begin{align*}
&\cc{f\oo{x}}^{2}=\int_{p}^{x}{ff_{x}\,dx}-\int_{qx}^{q}{ff_{x}\,dx}\leq\int_{p}^{q}{\norm{ff_{x}}\,dx}\leq\int_{0}^{P}{\norm{ff_{x}}\,dx}\\
&\leq\oo{\int_{0}^{P}{f^{2}\,dx}\cdot\int_{0}^{P}{f_{x}^{2}\,dx}}^{\frac{1}{2}}\leq\frac{P}{2\pi}\int_{0}^{P}{f_{x}^{2}\,dx},
\end{align*}
where the last step follows from Lemma $\ref{AppendixLemma1}$. We have also utilised H\"{o}lder's inequality with $p=q=2$.
\end{proof}
\begin{lemma}[\cite{DKS}, Lemma $2.4$]\label{AppendixLemma3}
Let $\gamma:\mathbb{S}^{1}\rightarrow\mathbb{R}^{2}$ be a smooth closed curve. Let $\phi:\mathbb{S}^{1}\rightarrow\mathbb{R}$ be a sufficiently smooth function. Then for any $l\geq 2,K\in\mathbb{N}$ and $0\leq i<K$ we have
\begin{equation}
L^{i+1-\frac{1}{l}}\oo{\intcurve{\oo{\phi}_{s^{i}}^{2}}}^{\frac{1}{l}}\leq c\oo{K}L^{\frac{1-\alpha}{2}}\oo{\intcurve{\phi^{2}}}^{\frac{1-\alpha}{2}}\llll{\phi}_{K,2}^{\alpha}.\label{AppendixLemma3,1}
\end{equation}
Here $\alpha=\frac{i+\frac{1}{2}-\frac{1}{l}}{K}$, and
\[
\llll{\phi}_{K,2}:=\sum_{j=0}^{K}L^{j+\frac{1}{2}}\oo{\intcurve{\oo{\phi}_{s^{j}}^{2}}}^{\frac{1}{2}}.
\]
In particular, if $\phi=\kappa-\bar{\phi}$, then
\begin{equation}
L^{i+1-\frac{1}{l}}\oo{\intcurve{\oo{k-\bar{k}}_{s^{i}}^{2}}}^{\frac{1}{l}}\leq c\oo{K}\oo{K_{osc}}^{\frac{1-\alpha}{2}}\llll{k-\bar{k}}_{K,2}^{\alpha}.\label{AppendixLemma3,2}
\end{equation}
\end{lemma}
\begin{proof}
The proof is identical to that of Lemma $2.4$ from \cite{DKS} and is of a
standard interpolative nature. Note that although we use $k-\bar{k}$ in the
identity (as opposed to \cite{DKS} where $k\nu$ is used).
\end{proof}
\begin{lemma}[Proposition $2.5$, \cite{DKS}]\label{AppendixLemma4}
Let $\gamma:\mathbb{S}^{1}\rightarrow\mathbb{R}^{2}$ be a smooth closed curve. Let $\phi:\mathbb{S}^{1}\rightarrow\mathbb{R}$ be a sufficiently smooth function. Then for any term $P_{\nu}^{\mu}\oo{\phi}$ (where $P_{\nu}^{\mu}\oo{\cdot}$ denotes the same $P$-style notation used in for example \cite{DKS}) with $\nu\geq2$ which contains only derivatives of $\kappa$ of order at most $K-1$, we have
\begin{equation}
\intcurve{\norm{P_{\nu}^{\mu}\oo{\phi}}}\leq c\oo{K,\mu,\nu}L^{1-\mu-\nu}\oo{L\intcurve{\phi^{2}}}^{\frac{\nu-\eta}{2}}\llll{\phi}_{K,2}^{\eta}.\label{AppendixLemma4,1}
\end{equation}
In particular, for $\phi=\kappa-\bar{\kappa}$ we have the estimate
\begin{equation}
\intcurve{\norm{P_{\nu}^{\mu}\oo{\kappa-\bar{\kappa}}}}\leq c\oo{K,\mu,\nu}L^{1-\mu-\nu}\oo{K_{osc}}^{\frac{\nu-\eta}{2}}\llll{\kappa-\bar{\kappa}}_{K,2}^{\eta}\label{AppendixLemma4,2}
\end{equation}
where $\eta=\frac{\mu+\frac{\nu}{2}-1}{K}$.
\end{lemma}
\begin{proof}
Using H\"{o}lder's inequality and Lemma $\ref{AppendixLemma3}$ with $K=\nu$, if $\sum_{j=1}^{\nu}i_{j}=\mu$ we have
\begin{align}
&\intcurve{\norm{\phi_{s^{i_{1}}}\star\cdots\star\phi_{s^{i_{\nu}}}}}\nonumber\\
&\leq\prod_{j=1}^{\nu}\oo{\intcurve{\phi_{s^{i_{j}}}^{\nu}}}^{\frac{1}{\nu}}=L^{1-\mu-\nu}\prod_{j=1}^{\nu}L^{i_{j}+1-\frac{1}{\nu}}\oo{\intcurve{\phi_{s^{i_{j}}}^{\nu}}}^{\frac{1}{\nu}}\nonumber\\
&\leq c\oo{K,\mu,\nu}L^{1-\mu-\nu}\prod_{j=1}^{\nu}\oo{L\intcurve{\phi^{2}}}^{\frac{1-\alpha_{j}}{2}}\llll{\phi}_{K,2}^{\alpha_{j}}\label{AppendixLemma4,3}
\end{align}
where $\alpha_{j}=\frac{i_{j}+\frac{1}{2}-\frac{1}{\nu}}{K}$. Now
\[
\sum_{j=1}^{\nu}\alpha_{j}=\frac{1}{K}\sum_{j=1}^{\nu}\oo{i_{j}+\frac{1}{2}-\frac{1}{\nu}}=\frac{\mu+\frac{\nu}{2}-1}{K}=\eta,
\]
ans so substituting this into $\oo{\ref{AppendixLemma4,3}}$ gives the first inequality of the lemma. It is then a simple matter of substituting $\phi=\kappa-\bar{\kappa}$ into this result to prove statement $\oo{\ref{AppendixLemma4,2}}$.
\end{proof}
\begin{lemma}[\cite{DKS}]\label{AppendixLemma5}
Let $\gamma:\mathbb{S}^{1}\rightarrow\mathbb{R}^{2}$ be a smooth closed curve and $\phi:\mathbb{S}^{1}\rightarrow\mathbb{R}$ a sufficiently smooth function.
Then for any term $P_{\nu}^{\mu}\oo{\phi}$ with $\nu\geq2$ which contains only derivatives of $\kappa$ of order at most $K-1$, we have for any $\varepsilon>0$
\begin{equation}
\intcurve{\norm{P_{\nu}^{\mu,K-1}\oo{\phi}}}\leq c\oo{K,\mu,\nu}L^{1-\mu-\nu}\oo{L\intcurve{\phi^{2}}}^{\frac{\nu-\eta}{2}}\oo{L^{2K+1}\intcurve{\phi_{s^{K}}^{2}}+L\intcurve{\phi^{2}}}^{\frac{\eta}{2}}.\label{AppendixLemma5,1}
\end{equation}
Moreover if $\mu+\frac{1}{2}\nu<2K+1$ then $\eta<2$ and we have for any $\varepsilon>0$
\begin{equation}
\intcurve{\norm{P_{\nu}^{\mu,K-1}\oo{\phi}}}\leq\varepsilon\intcurve{\phi_{s^{K}}^{2}}+c\cdot\varepsilon^{-\frac{\eta}{2-\eta}}\oo{\intcurve{\phi^{2}}}^{\frac{\nu-\eta}{2-\eta}}+c\oo{\intcurve{\phi^{2}}}^{\mu+\nu-1}.\label{AppendixLemma5,2}
\end{equation}
In particular, for $\phi=\kappa-\bar{\kappa}$, we have the estimate
\[
\intcurve{\norm{P_{\nu}^{\mu}\oo{k-\bar{k}}}}\leq c\oo{K,\mu,\nu}L^{1-\mu-\nu}\oo{K_{osc}}^{\frac{\nu-\eta}{2}}\oo{L^{2K+1}\intcurve{\oo{k-\bar{k}}_{s^{K}}^{2}}}^{\frac{\eta}{2}}.
\]
Here, as before, $\eta=\frac{\mu+\frac{\nu}{2}-1}{K}$.
\end{lemma}
\begin{proof}
Combining the previous lemma with the following standard interpolation inequality from that follows from repeated applications of Lemma $\ref{AppendixLemma0}$ (and is also found in \cite{Aubin1})
\[
\llll{\phi}_{K,2}^{2}\leq c\oo{K}\oo{L^{2K+1}\intcurve{\phi_{s^{K}}^{2}}+L\intcurve{\phi^{2}}}
\]
yields the identity $\oo{\ref{AppendixLemma5,1}}$ immediately. To prove $\oo{\ref{AppendixLemma5,2}}$ we simply combine $\oo{\ref{AppendixLemma5,1}}$ with the Cauchy-Schwarz identity. The final identity of the Lemma follow by letting $\phi=\kappa-\bar{\kappa}$ in $\oo{\ref{AppendixLemma5,1}}$ and combining this with the identity
\begin{equation}
K_{osc}\leq L\oo{\frac{L^{2}}{4\pi^{2}}}^{K}\intcurve{\oo{\kappa-\bar{\kappa}}_{s^{K}}^{2}}= c\oo{K}L^{2K+1}\intcurve{\oo{\kappa-\bar{\kappa}}_{s^{K}}^{2}},\label{AppendixLemma5,3}
\end{equation}
which is a direct consequence of applying Lemma $\ref{AppendixLemma1}$ $\oo{p+1}$ times repeatedly.
\end{proof}
\begin{theorem}[\cite{Breuning1}, Theorem $1.1$]\label{AppendixTheorem1}
Let $q\in\mathbb{R}^{n}$, $m,p\in\mathbb{N}$ with $p>m$. Additionally, let $\mathcal{A},\mathcal{V}>0$ be some fixed constants. Let $\mathfrak{T}$ be the set of all mappings $f:\mathbb{S}igma:\rightarrow\mathbb{R}^{n}$ with the following properties:
\begin{itemize}
\item $\mathbb{S}igma$ is an $m$-dimensional, compact manifold (without boundary)
\item $f$ is an immersion in $W^{2,p}\oo{\mathbb{S}igma,\mathbb{R}^{n}}$ satisfying
\begin{align*}
\llll{A\oo{f}}_{p}&\leq\mathcal{A},\\
\text{vol}\oo{\mathbb{S}igma}&\leq\mathcal{V},\text{ and }\\
q&\in f\oo{\mathbb{S}igma}.
\end{align*}
\end{itemize}
Then for every sequence $f^{i}:\mathbb{S}igma^{i}\rightarrow\mathbb{R}^{n}$ in $\mathfrak{T}$ there is a subsequence $f^{j}$, a mapping $f:\mathbb{S}igma\rightarrow\mathbb{R}^{n}$ in $\mathfrak{T}$ and a sequence of diffeomorphisms $\phi^{j}:\mathbb{S}igma\rightarrow\mathbb{S}igma^{j}$ such that $f^{j}\circ\phi^{j}$ converges in the $C^{1}$-topology to $f$.
\end{theorem}
\end{section}
\end{document}
| 3,045 | 30,672 |
en
|
train
|
0.53.0
|
\begin{document}
\title{Error rates and resource overheads of encoded three-qubit gates}
\author{Ryuji Takagi, Theodore J. Yoder and Isaac L. Chuang}
\affiliation{Department of Physics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA}
\date{\today}
\begin{abstract}
A non-Clifford gate is required for universal quantum computation, and, typically, this is the most error-prone and resource intensive logical operation on an error-correcting code. Small, single-qubit rotations are popular choices for this non-Clifford gate, but certain three-qubit gates, such as Toffoli or controlled-controlled-$Z$ (CCZ), are equivalent options that are also more suited for implementing some quantum algorithms, for instance, those with coherent classical subroutines.
Here, we calculate error rates and resource overheads for implementing logical CCZ with pieceable fault-tolerance, a non-transversal method for implementing logical gates.
We provide a comparison with a non-local magic-state scheme on a concatenated code and a local magic-state scheme on the surface code. We find the pieceable fault-tolerance scheme particularly advantaged over magic states on concatenated codes and in certain regimes over magic states on the surface code.
Our results suggest that pieceable fault-tolerance is a promising candidate for fault-tolerance in a near-future quantum computer.
\end{abstract}
\mathcal{A}ketitle
\section{Introduction}
Quantum error-correcting codes are the most promising route to scalable quantum computation. However, some of their limitations are well-known. For instance, a major problem is that a single code cannot support a full set of universal, transversal operations \cite{Zeng2007,Chen2008,Eastin2009a}. Often, and always for 2D designs \cite{Bravyi2013a}, the missing gate is not in the normalizer of the Pauli group; that is, it is non-Clifford.
The techniques of gate-teleportation \cite{Gottesman1999b} and magic-states \cite{Bravyi2005a} can overcome the lack of a non-Clifford gate. Different magic-states can be created to implement small $Z$-rotations such as the $T$-gate or 3-qubit operations, like Toffoli or controlled-controlled-$Z$ (CCZ).
However, the process to create a magic-state occurs post-selectively and recursively and leads to large resource overheads. Although improving consistently \cite{Jones2013a,Campbell2017a} approaching believed fundamental limits \cite{Bravyi2012a}, large resource demands remain a serious obstacle for near-future architectures.
Certain other approaches exist in the literature for implementing a universal gate-set while circumventing the use of magic-states. A popular approach is gauge-fixing \cite{Paetznick2013a,Anderson2014a,Bombin2015a}, in which a subsystem code can implement complementary sets of transversal logical gates depending on the settings of the gauge qubits. Another approach \cite{Jochym-OConnor2014a,Nikahd2016a,Nikahd2016b} concatenates different codes with complementary transversal gate sets to achieve the same effect in one larger code. Recently, this approach was shown to lead to asymptotic thresholds around $\sim10^{-3}$ albeit using more physical qubits than, for example, surface code magic-state distillation \cite{Chamberland2016d,Chamberland2017}.
Any fault-tolerant, universal computing scheme operating without magic states is expected to be a promising candidate for near-future architectures where fairly accurate physical components are supplied but space-time resources, like qubit count and circuit depth, are limited.
The primary goal in this near-future regime is to achieve some desired target error rate after a finite-sized computation with small resource overheads.
Such constraints imply that the logical error rates of encoded gates and the first-level pseudothreshold \cite{Svore2005b} (called just pseudothreshold hereafter) are more important measures than asymptotic threshold, which only becomes meaningful with access to huge amounts of resources.
To evaluate near-future fault-tolerant computation, we focus on another magic-less alternative that allows for a logical implementation of three-qubit gates, the pieceable fault-tolerance scheme \cite{Yoder2016c}. In this approach, a logical gate is done non-transversally through the ``round-robin" construction, and made fault-tolerant via partial error-correction performed throughout the circuit.
This construction has recently been used in \cite{Chao2017a} to perform fault-tolerant, universal computing on seven logical qubits requiring only four ancillary qubits and 15 code qubits. The circuit volume metric, a space-time resource measure that counts all gates weighted by the number of qubits involved, was used in \cite{Yoder2016c} to argue that pieceable fault-tolerance reduces logical gate overhead by nearly a factor of two over magic-state creation and injection. However, little was said about error rates of pieceable gates.
In this paper, we calculate these error rates and compare to magic-state schemes for implementing three-qubit non-Clifford gates. Our contenders are (1) a non-local magic-state scheme: magic-states created postselectively on Steane's 7-qubit code (also known as the smallest color code), (2) a local magic-state scheme: surface code magic-state distillation, and (3) pieceable fault-tolerance on the (a) 5-qubit \cite{Yoder2016c}, (b) 7-qubit \cite{Yoder2016c}, (c) $3\times3$ Bacon-Shor \cite{Yodera}, and (d) $3\times9$ Bacon-Shor \cite{Yodera} codes. Our metrics are (I) error rate of the logical gate and (II) circuit volume. Among concatenated schemes (1) and (3), we can definitively declare pieceable $3\times3$ Bacon-Shor the winner in both metrics (I) and (II). When comparing to (2), the picture is more complicated and interesting.
The pieceable $3\times3$ Bacon-Shor beats the surface code in error rate at low distance and in circuit volume when the physical error rate is sufficiently low compared with the desired target logical error.
On the other hand, asymptotically in code distance, the surface code outperforms pieceable $3\times 3$ Bacon-Shor due to better scaling of logical error rate and volume with distance.
\section{Methods}
We first describe our method to evaluate the logical error rates. Evaluating the surface code scheme (2) draws on the extensive literature on the topic \cite{Fowler2012a}.
Our calculations of the logical error rates of schemes (1) and (3) at code distance $d=3$ are done by exact enumeration of all combinations of up to two faults in the circuit extended-rectangle (exREC) \cite{Aliferis2005a} under the standard depolarizing noise model (which serves as a model of average-case noise).
In \cite{Aliferis2005a}, a rigorous upper bound on the logical error rate under depolarizing noise is given.
In contrast, we provide formulas giving a rigorous lower bound as well as a tighter rigorous upper bound.
The lower and upper bounds on logical error rate also determine lower and upper bounds on the pseudothreshold. Having both bounds allows us to definitively prove a separation between two different schemes when it exists.
Our method also confers some advantages over a Monte Carlo simulation. First, we can rigorously verify our circuits are fault-tolerant under the chosen noise model by checking that all single faults are correctable. Second, once the counting is complete, we can independently vary noise for each type of gate.
Our standard noise breakdown assigns single-qubit gates, two-qubit gates, and three-qubit gates each their own failure probabilities $p_1,p_2,$ and $p_3$, respectively.
In the circuit depolarization noise model, an $r$-qubit gate fails with one of the $4^r-1$ $r$-qubit Pauli errors with probability $p_r/(4^r-1)$. In principle, preparation and measurement could be treated separately as well, though we will assign them failure probabilities also equal to $p_1$. Bounds on the error rate can always be written as polynomials in $p_1,p_2,p_3$ as we discuss below.
Our ultimate goal in error-rate estimation is to find the probability the exREC is incorrect given that all ancillas pass verification. Denote this $P_{\text{fail}|\text{acc}}=\text{Pr}\left[\text{fail}|\text{acc}\right]$. Our counting gives the exact values of
\begin{align}
P^{(2)}_{\text{fail},\text{acc}}&=\text{Pr}\left[\text{fail},\text{acc},\le2\text{ faults}\right],\label{eq:fail_acc}\\
P^{(2)}_{\text{succ},\text{acc}}&=\text{Pr}\left[\neg\text{fail},\text{acc},\le2\text{ faults}\right],\label{eq:succ_acc}\\
P^{(2)}_{\text{rej}}&=\text{Pr}\left[\neg\text{acc},\le2\text{ faults}\right],\label{eq:rej}
\end{align}
as polynomials in $p_1,p_2,p_3$ with degree equal to the number of potentially faulty components in the entire exREC. These exactly calculated quantities are enough to bound $P_{\text{fail},\text{acc}}=\text{Pr}\left[\text{fail},\text{acc}\right]$, $P_{\text{succ},\text{acc}}=\text{Pr}\left[\neg\text{fail},\text{acc}\right]$, and $P_{\text{acc}}=\text{Pr}\left[\text{acc}\right]$ as
\begin{comment}
$P^{(2)}_{\text{fail},\text{acc}}\le P_{\text{fail},\text{acc}},
P^{(2)}_{\text{succ},\text{acc}}\le P_{\text{succ},\text{acc}},
P_{\text{acc}} \le 1-P^{(2)}_{\text{rej}}$.
\end{comment}
\begin{align}
P^{(2)}_{\text{fail},\text{acc}}&\le P_{\text{fail},\text{acc}},\\
P^{(2)}_{\text{succ},\text{acc}}&\le P_{\text{succ},\text{acc}},\\
P_{\text{acc}} &\le 1-P^{(2)}_{\text{rej}}.
\end{align}
Thus,
\begin{equation}
\frac{P^{(2)}_{\text{fail},\text{acc}}}{1-P^{(2)}_{\text{rej}}}\le P_{\text{fail}|\text{acc}}=1-P_{\text{succ}|\text{acc}}\le1-\frac{P^{(2)}_{\text{succ},\text{acc}}}{1-P^{(2)}_{\text{rej}}}.
\label{eq:bound}
\end{equation}
More details on the simulation including the description on how to obtain these polynomials can be found in Appendix~\ref{app:sim_details}.
Next, we consider evaluating the resource overhead.
There exist various resource measures such as qubit count, circuit volume, gate counts and so on. The number of reusable physical qubits is often taken as a physical resource measure in the literature.
However, it is not the best, especially when we would like to compare resource overheads between different codes, because there is ambiguity that comes with the level of parallelization we assume.
In this paper, we mainly focus on circuit volume, a space-time resource measure that counts all gates weighted by the number of qubits involved.
Unlike physical qubit count, circuit volume takes into account the trade-off between space and time resources. The circuit volume is a space-time metric in the same vein as the ``quantum volume'' \cite{qvolume}, except for evaluating specific circuits rather than a universal quantum computer.
The circuit volume at a high concatenation level is easy to compute using the volume of the logical construction at the first level of encoding.
Let $V^{(k)}_G$ be the volume for implementing circuit component $G$ at the $k^{\text{th}}$ level of concatenation.
Then, there is a recursion relation between two concatenation levels, $V^{(k+1)}_G=\sum_{G'} N_G^{G'} V^{(k)}_{G'}$ where $N_G^{G'}$ is the number of the circuit component $G'$ in the logical construction of component $G$.
We can understand this as evolution of a vector of circuit volumes of each component via a transformation matrix determined by the logical gate constructions. Namely, we get
\begin{equation}
{\bf V^{(k)}}=A^k {\bf V^{(0)}},
\label{eq:higher_concate}
\end{equation}
where $A$ is the matrix $A_{ij}=N_{G_i}^{G_j}$, ${\bf V^{(k)}}_i = V^{(k)}_{G_i}$, and $V^{(0)}_G$ is the volume of an unencoded component.
We set ${\bf V^{(k)}}=(V^{(k)}_3,V^{(k)}_2,V^{(k)}_1,V^{(k)}_{prep},V^{(k)}_{meas})^T$, where the components refer to the circuit volume of three qubit gates, two qubit gates, single qubit gates, $\ket{0}$ or $\ket{+}$ preparation, and measurement respectively.
Note that $(V^{(0)}_3,V^{(0)}_2,V^{(0)}_1,V^{(0)}_{prep},V^{(0)}_{meas})=(3,2,1,1,1)$.
| 3,367 | 21,149 |
en
|
train
|
0.53.1
|
\section{Logical constructions}\label{circuit_details}
Here, we describe the logical constructions used in the simulation. Explicit descriptions of the circuits at the gate level can be found in \cite{circuit}. All of our constructions begin with a round of syndrome measurement and recovery (the leading error correction) and end with the same (the trailing error correction), in accordance with the exREC formalism \cite{Aliferis2005a}. The rest of the circuit may also include rounds of error correction, called intermediate, in accordance with pieceable fault-tolerance \cite{Yoder2016c}.
For the 5-qubit code, we implement a logical CCZ gate by the round-robin construction \cite{Yoder2016c} with three intermediate error corrections.
The leading error correction and trailing error correction are done by Steane's error correction \cite{Steane1997}.
Since the 5-qubit code is non-CSS, a 10-qubit ancilla is needed to extract the entire syndrome simultaneously.
We actually find that the circuit in \cite{Steane1997} needs some modification for non-CSS codes, which we discuss in the Appendix~\ref{app:generalized_steane} in detail.
For intermediate error corrections, we use Shor-type error correction with CAT states \cite{Shor2011a}.
The size of the CAT states is always four for measuring constant stabilizers (those that commute with the preceeding circuitry), but it varies for measuring non-constant stabilizers because their weight changes as they go through the CCZ gates.
For our circuit, we need to use 9-CAT, 13-CAT, 9-CAT at maximum for the first, second and third intermediate error correction respectively.
For the 7-qubit code, we consider the construction that requires only one intermediate error correction \cite{Yoder2016c}.
All of the error corrections are done by Steane's error correction.
Since the 7-qubit code is a CSS code, correction of $Z$ type errors can be done separately from that of $X$ type errors, and only the encoded states $\ket{\bar{0}}$ and $\ket{\bar{+}}$ are needed. The state
$\ket{\bar{0}}$($\ket{\bar{+}}$) is verified by applying CNOT gates transversally to another noisy $\ket{\bar{0}}$($\ket{\bar{+}}$) and measuring it transversally (a Steane ancilla factory \cite{Steane1998a}).
If some error is detected, we discard the state and start again.
For estimating the circuit volume, we consider a more resource-efficient state preparation method proposed by Goto \cite{Goto2016}.
Although we did not estimate the logical error rate using the Goto's method, we suspect that the change in the logical error rate between different verification methods would be small as indicated in \cite{Goto2016}.
Since intermediate $Z$-type error correction is not needed, we just apply the $X$-type error correction in the middle and notify the trailing error correction about possible locations of $Z$-type errors as described in \cite{Yoder2016c}.
Logical CCZ on the Bacon-Shor code is implemented as proposed in \cite{Yodera}. On the $3\times 3$ Bacon-Shor we need no intermediate correction although we do use a non-Pauli recovery at the end. Furthermore, since the ancilla for the error correction is a tensor product of 3-CAT states, there is no need for verification since, modulo its stabilizers, an error on a 3-CAT is equivalent to a weight one error.
In contrast, the $3\times 9$ Bacon-Shor implements logical CCZ transversally, but it comes with a substantially larger overhead \cite{Yodera}.
For the non-local magic-state scheme, we use magic state injection on the 7-qubit code to implement a logical CCZ gate.
The CCZ magic state is defined by the stabilizers $\left< X_1\text{CZ}(2,3), X_2\text{CZ}(1,3), X_3\text{CZ}(1,2)\right>$.
The protocol consists of two parts, a state preparation circuit and a teleportation circuit.
The state preparation starts with the +1 eigenstate of the second and the third stabilizer, $\ket{\bar{0}}\ket{\bar{+}}\ket{\bar{+}}$, and measures the first stabilizer \cite{Zhou2000}.
Our circuit is a variant of the circuit in \cite{Monroe2014} which we modify to create the CCZ state instead. Two measurements of $X_1\text{CZ}(2,3)$ are done with complete error-correction in between. This makes the circuit fault-tolerant (to one fault).
If the two measurement results do not match, we discard the created state and start over again. If they match and they both show the result -1, we apply $\bar{Z}$ on the first code block to put it back to the desired magic state. If both show +1, we do not need to apply a correction. Like the pieceable 7-qubit case, all the error corrections are done using Steane's method \cite{Steane1997}.
\section{Comparison of concatenated schemes}
We compute the logical error rates and resource overheads of pieceably fault-tolerant CCZ gates on the 5-qubit code, 7-qubit code \cite{Yoder2016c}, $3\times 3$ Bacon-Shor code and $3\times 9$ Bacon-Shor code \cite{Yodera}, and compare them to a magic-state scheme on the 7-qubit code.
Fig.~\fig{logical_error} shows the obtained logical error rates for these cases using two different settings of physical error rate, $p_1=p_2=p_3=p$ and $10p_1=p_2=0.1p_3=p$.
Lower and upper bounds on pseudothresholds are the crossing points of ``break-even" line and the upper and lower bounds for logical error rates. For both settings of physical error rate, the $3\times 3$ Bacon-Shor code has lower logical error rate than the magic-state scheme below pseudothreshold.
For the 7-qubit code, whether the pieceable scheme has a lower rate than the magic-state scheme depends on the physical error rate setting.
The 5-qubit code has a large logical error rate due to a large number of pieces in the round-robin construction.
Similarly, the $3\times 9$ Bacon-Shor code has a higher logical error rate than the $3\times 3$ Bacon-Shor code because the size of the logical code block is obviously much bigger. Moreover, the $3\times 9$ Bacon-Shor needs to implement verification for 9-qubit CAT states.
We now compare the resource overheads. Table~\tbl{resource} shows the resource overheads to implement a logical CCZ gate with these constructions.
We assume that the ancillas are not reusable.
Due to a finite ancilla verification rejection rate, the effective resource count is slightly higher than the values in the table.
However, the rejection rate of the verification is $\mathcal{O}(p)$, and the effective resource count is obtained by multiplying $(1-n_{rej}p)^{-1}$ where $n_{rej}$ is the number of error locations that lead to rejection.
Since we are interested in the region $p<10^{-4}$, and the largest module involving verification is the magic state preparation circuit, which has $n_{rej}\sim 100$, increase in the resource due to verification is within 1\%.
Thus, it is safe to ignore the effects of verification.
Besides using more 3-qubit gates, pieceable constructions on the 7-qubit and $3\times 3$ Bacon-Shor code have smaller resource overheads compared to the magic-state scheme.
In particular, they have a significant reduction in circuit volume.
Fig.~\fig{volume} shows circuit volumes for the pieceable 7-qubit code, $3\times 3$ Bacon-Shor code, and magic-state scheme.
Transformation matrices $A$ (see Eq.~\eqref{eq:higher_concate}) for these codes are given in Appendix~\ref{app:volume_calc}.
Combining the results for the logical error rates and circuit volume, we conclude that the pieceable construction on the $3\times 3$ Bacon-Shor code beats the magic-state scheme on the 7-qubit code in both the criteria. The pieceable construction on the 7-qubit code also beats magic state injection in circuit volume, and in logical error rate when $p_1=p_2=p_3$.
\begin{figure}
\caption{(Color online.) Logical error rates of 3-qubit gate on (a,b) pieceable 7-qubit code (green, dot-dashed), pieceable $3\times 3$ Bacon-Shor code (blue, dashed), (c,d) pieceable 5-qubit code (green, dot-dashed), pieceable $3\times 9$ Bacon-Shor code (blue, dashed), and magic state injection on 7-qubit code (orange, solid) with (a,c) $p_1=p_2=p_3=p$ and (b,d) $10p_1=p_2=0.1p_3=p$ where $p_i$ refers to physical error rate of $i$-qubit gate. Initialization of single-qubit states $\ket{0}
\label{fig:logical_error}
\end{figure}
\begin{table}[htbp]
\centerline{\begin{tabular}{|c|c|c|c|c|}
\hline
& Volume & Qubits & 2-qubit gates & 3-qubit gates\\
\hline\hline
Pieceable 5-qubit & 3841 & 364 & 445 & 46\\
\hline
Pieceable 7-qubit & 771 & 93 & 162 & 21\\
\hline
$3\times 3$ Bacon-Shor & 414 & 81 & 90 & 27\\
\hline
$3\times 9$ Bacon-Shor & 1350 & 252 & 306 & 27\\
\hline
Magic state & 1352 & 154 & 267 & 14\\
\hline\hline
$3\times 3$ BS/Magic & 0.31 & 0.59 & 0.34 & 1.9\\
\hline
\end{tabular}}
\caption{Resource overheads to implement logical CCZ. Volume refers to the circuit volume, which counts all gates weighted by the number of qubits involved. Qubits are the number of physical qubits including data qubits and ancilla qubits where ancilla qubits are assumed to be not reusable. Numbers for the 5-qubit code include all the resources for the adaptive measurements.}
\label{tbl:resource}
\end{table}
\section{Comparison to surface codes}
\text{Next,} we compare logical error rate and resource overheads to a local magic-state scheme on surface codes.
We find that the pieceable construction can have a significant advantage in circuit volume in a certain region in terms of physical error rate and target logical error rate.
\subsection{Logical error rates}
Surface codes are known to have high asymptotic threshold, which is 0.1\%-1\% depending on assumptions and error model \cite{Fowler2012a,Fowler2009,Wang2009b,Wang2011,Wootton2012}, and thus they have attracted attention as a candidate for a scalable quantum computer.
However, having a high asymptotic threshold does not automatically imply that logical error rate is always low for reasonably sized codes.
Firstly, as can be seen in \cite{Fowler2012a}, in the low distance regime the pseudothreshold of the surface code is much smaller than the asymptotic threshold.
Thus, if the physical error rate is lower than the asymptotic threshold but not below the relevant pseudothreshold, encoding at low distance does not help to reduce the error rate.
Secondly, the logical error rate of a logical gate can be large even if the error rate for one surface code {\it cycle} is small, because a logical gate is made up of many cycles. Each cycle consists of measuring the complete error syndrome once via measurement qubits, one per stabilizer generator, as in \cite{Fowler2012a}.
Let $\bar{p}_{cycle}$ be the logical error rate for surface code per surface code cycle.
Let $C_G$ be the number of surface code cycles it takes to implement a logical version of gate $G$.
Then, logical error rate of gate $G$ is $\bar{p}_G\approx C_G \bar{p}_{cycle}$.
Since $C_G \propto d$ and $\bar{p}_{cycle}\propto p^{(d+1)/2}$ where $d$ is the surface code distance, $\bar{p}_{cycle}$ dominates for large distance.
However, when $d$ is small, the contribution to $\bar p_G$ from $C_G$ is not negligible.
In Appendix~\ref{app:surface_calc}, we find a specific form of $C_G$ for the logical Toffoli gate for two different implementations.
Fig.~\fig{error_concate_surface} shows logical error rates of a 3-qubit gate on the surface code using a Toffoli state, and upper bounds of logical error rate of pieceable $3\times 3$ Bacon-Shor code and pieceable 7-qubit code in terms of code distance with three different physical error rates.
Upper bounds are obtained by concatenating the function upper bounding the actual rate in Eq.~\eq{bound}.
Since the 3-qubit gate is the largest component among the components that appear in the logical construction of 3-qubit gate, concatenating the upper bounding error function for the 3-qubit gate upper bounds its error rates at higher concatenation level.
However, because logical 3-qubit gates have an order of magnitude higher error rate than 2-qubit gates and the logical constructions of 3-qubit gates mostly consist of single gates and 2-qubit gates, this upper bound is highly pessimistic.
A careful analysis taking into account error functions for other types of components and possibly even using better decoding algorithm \cite{Poulin2006,Fern2008a} at a higher levels may greatly reduce estimates of logical error rates.
Nevertheless, in Fig.~\fig{error_concate_surface}, we can see that surface codes have better scaling with distance than pieceable concatenated codes, which should be attributed to the high threshold. However, for small $d$, $C_{\mbox{Toffoli}}$ has a significant contribution, and when $d=3$ the logical error rate of the pieceable constructions is two orders of magnitude lower than that of the surface code.
\begin{figure}
\caption{Logical error rates for 3-qubit gate on the surface codes and (a) pieceable $3\times 3$ Bacon-Shor code (b) pieceable 7-qubit code in terms of code distance. Shown rates for pieceable codes are upper bounds obtained by concatenating the upper bounding function from Eq.~\eq{bound}
\label{fig:error_concate_surface}
\end{figure}
| 3,742 | 21,149 |
en
|
train
|
0.53.2
|
\subsection{Logical error rates}
Surface codes are known to have high asymptotic threshold, which is 0.1\%-1\% depending on assumptions and error model \cite{Fowler2012a,Fowler2009,Wang2009b,Wang2011,Wootton2012}, and thus they have attracted attention as a candidate for a scalable quantum computer.
However, having a high asymptotic threshold does not automatically imply that logical error rate is always low for reasonably sized codes.
Firstly, as can be seen in \cite{Fowler2012a}, in the low distance regime the pseudothreshold of the surface code is much smaller than the asymptotic threshold.
Thus, if the physical error rate is lower than the asymptotic threshold but not below the relevant pseudothreshold, encoding at low distance does not help to reduce the error rate.
Secondly, the logical error rate of a logical gate can be large even if the error rate for one surface code {\it cycle} is small, because a logical gate is made up of many cycles. Each cycle consists of measuring the complete error syndrome once via measurement qubits, one per stabilizer generator, as in \cite{Fowler2012a}.
Let $\bar{p}_{cycle}$ be the logical error rate for surface code per surface code cycle.
Let $C_G$ be the number of surface code cycles it takes to implement a logical version of gate $G$.
Then, logical error rate of gate $G$ is $\bar{p}_G\approx C_G \bar{p}_{cycle}$.
Since $C_G \propto d$ and $\bar{p}_{cycle}\propto p^{(d+1)/2}$ where $d$ is the surface code distance, $\bar{p}_{cycle}$ dominates for large distance.
However, when $d$ is small, the contribution to $\bar p_G$ from $C_G$ is not negligible.
In Appendix~\ref{app:surface_calc}, we find a specific form of $C_G$ for the logical Toffoli gate for two different implementations.
Fig.~\fig{error_concate_surface} shows logical error rates of a 3-qubit gate on the surface code using a Toffoli state, and upper bounds of logical error rate of pieceable $3\times 3$ Bacon-Shor code and pieceable 7-qubit code in terms of code distance with three different physical error rates.
Upper bounds are obtained by concatenating the function upper bounding the actual rate in Eq.~\eq{bound}.
Since the 3-qubit gate is the largest component among the components that appear in the logical construction of 3-qubit gate, concatenating the upper bounding error function for the 3-qubit gate upper bounds its error rates at higher concatenation level.
However, because logical 3-qubit gates have an order of magnitude higher error rate than 2-qubit gates and the logical constructions of 3-qubit gates mostly consist of single gates and 2-qubit gates, this upper bound is highly pessimistic.
A careful analysis taking into account error functions for other types of components and possibly even using better decoding algorithm \cite{Poulin2006,Fern2008a} at a higher levels may greatly reduce estimates of logical error rates.
Nevertheless, in Fig.~\fig{error_concate_surface}, we can see that surface codes have better scaling with distance than pieceable concatenated codes, which should be attributed to the high threshold. However, for small $d$, $C_{\mbox{Toffoli}}$ has a significant contribution, and when $d=3$ the logical error rate of the pieceable constructions is two orders of magnitude lower than that of the surface code.
\begin{figure}
\caption{Logical error rates for 3-qubit gate on the surface codes and (a) pieceable $3\times 3$ Bacon-Shor code (b) pieceable 7-qubit code in terms of code distance. Shown rates for pieceable codes are upper bounds obtained by concatenating the upper bounding function from Eq.~\eq{bound}
\label{fig:error_concate_surface}
\end{figure}
\subsection{Resource overheads}
We also count the circuit volume for implementing logical Toffoli on surface codes.
This allows us to compare the circuit volume between pieceable codes and the surface codes, shown in Fig.~\fig{volume}.
Although surface codes have better scaling with distance, pieceable constructions have a significant advantage until three concatenations.
This is especially true at distance three, where the difference is three orders of magnitude.
\begin{figure}
\caption{Circuit volume for logical 3-qubit gate on pieceable 7-qubit code(circles), pieceable $3\times 3$ Bacon-Shor code(squares), and magic-state scheme on 7-qubit code(diamonds) in terms of code distance. The dots correspond to every concatenation level in the range. Although it may be hard to see the data for pieceable 7-qubit code because they are close to the data for the magic-state scheme, the pieceable 7-qubit has slightly lower volume than the magic-state scheme for every distance shown.
}
\label{fig:volume}
\end{figure}
Consider now the space consisting of pairs (physical error rate, target logical error rate)$\equiv (p,p_T)$. Combining volume and error rate estimates, the region of this space where concatenated pieceable constructions require less circuit volume for implementing Toffoli than the surface code can be obtained. Fig.~\fig{error_volume} shows this region for the pieceable $3\times 3$ Bacon-Shor code and the 7-qubit code.
It shows that in large range, pieceable $3\times 3$ Bacon-Shor code has advantage in circuit volume over surface code, and the difference can be significant as can be seen in Fig.~\fig{volume}.
This region is actually determined by the upper bound of error rates at the third level concatenation.
It is because surface code with distance five has already larger volume than $3\times 3$ Bacon-Shor code with three concatenations as can be seen in Fig.~\fig{volume}.
For the 7-qubit code, Fig.~\fig{volume} shows that a 3-qubit logical gate at two concatenations of the 7-qubit code has less circuit volume than the surface code of any size. Thus, whenever two concatenations are sufficient to achieve the target logical error rate, the 7-qubit code will be advantaged, as is represented by the region in Fig.~\ref{fig:error_volume}.
Fig.~\fig{volume} also shows that the volume for the 7-qubit code with three concatenations is slightly larger than that for the surface code with distance seven.
Thus, the surface code is advantaged for the region where distance seven is enough for the surface code but three concatenations are needed for the 7-qubit code, which corresponds to the region between the upper purple region and the lower purple region in Fig.~\ref{fig:error_volume}.
The 7-qubit code again starts to have advantage over surface code for the region where the surface code needs distance nine whereas the 7-qubit code only needs to be concatenated three times, which corresponds to the lower purple region in Fig.~\ref{fig:error_volume}.
\begin{figure}
\caption{(color online.) The region where pieceable $3\times 3$ Bacon-Shor code (orange and purple) and pieceable 7-qubit code (purple) use less volume than surface codes to implement 3-qubit gate to achieve fixed target logical error rate, $p_T$, with fixed physical error rate, $p$. Dashed lines labeled by ${\bf ST_j}
\label{fig:error_volume}
\end{figure}
\section{Conclusions}
In this paper, we calculated logical error rates and resource overheads of 3-qubit gates using pieceable fault-tolerant constructions, a non-local magic-state scheme (on the 7-qubit code), and a local magic-state scheme (on the surface code).
In comparison with the non-local magic-state scheme, we found that while pieceable constructions have comparable, or even lower logical error rate to the magic-state scheme, the required circuit volume can be as little as 30\%. This suggests that the pieceable construction is a promising complement to schemes relying on magic states.
We also compared the pieceable construction to the surface codes and found that in quite a large region in terms of physical error rates and target logical error rates, pieceable constructions can have significantly lower circuit volume than surface codes.
Although realizing physical components with a small physical error rate such that pieceable constructions have a great advantage is challenging, one should notice that surface codes also have as hard a challenge as this in terms of resource overheads.
Just as surface codes are good candidates given access to large overheads, the pieceable construction appears to be a good candidate given access to small physical error rates.
Another difference between pieceable constructions and the surface code is locality, i.e.~the constraint that physical gates involved act only between qubits that are neighboring in some chosen low-dimensional layout.
Although the locality property is desirable in many experimental setups, some systems allow non-local interaction too \cite{Monroe2014}.
Our result indicates that such a non-local techniques can lead to significant reduction of resource use for quantum error correcting codes.
| 2,206 | 21,149 |
en
|
train
|
0.53.3
|
\section{Steane's error correction for arbitrary stabilizer codes}\label{app:generalized_steane}
Here, we describe the error correction used for the leading error correction and trailing error correction of the 5-qubit code. Since the 5-qubit code is not CSS, one might think Steane's error-correction is inappropriate. However, in \cite{Steane1997}, Steane proposes a circuit to do just that for the 5-qubit code. Unfortunately, Steane's construction as written is not quite correct. We present the correct method that works for any stabilizer code. We will also see that this method gives a conceptually simple way to prepare the necessary ancilla state in line with Steane's original proposal \cite{Steane1997}.
Consider a $\llbracket n,k\rrbracket$ stabilizer code $\mathcal{A}thcal{C}$ with stabilizer
\begin{equation}
S=\left(\begin{array}{c|c}S_x&S_z\end{array}\right),
\end{equation}
and logical operators
\begin{equation}
N=\left(\begin{array}{c|c}N_x&N_z\end{array}\right),
\end{equation}
written in symplectic matrix form. That is, ${S_x,S_z\in\mathcal{A}thbb{F}_2^{n-k}\times\mathcal{A}thbb{F}_2^n}$ and ${N_x,N_z\in\mathcal{A}thbb{F}_2^{2k}\times\mathcal{A}thbb{F}_2^{n}}$. Also, if we define ${\Lambda=\left(\begin{smallmatrix}0&I\{\mathbb I}&0\end{smallmatrix}\right)\in\mathcal{A}thbb{F}_2^{2n}\times\mathcal{A}thbb{F}_2^{2n}}$ using $k\times k$ block notation, then the canonical commutation relations are expressed by ${S\Lambda S^T=S\Lambda N^T}$ and ${N\Lambda N^T=A}$ for the ${2k\times 2k}$ matrix $A$ with $1$s on only the antidiagonal.
Following Steane, we propose the circuit in Fig.~\ref{fig:Steane_EC} to extract the syndrome of $\mathcal{A}thcal{C}$. The ancilla state used is twice the size of the code $\mathcal{A}thcal{C}$. The stabilizer of the ancilla state $\ket{\overline{a}}$ can be written
\begin{equation}\label{eq:anc_stabilizer}
S_a=\left(\begin{array}{cc|cc}
S_z&S_x&0&S_z\\
0&0&S_x&S_z\\
0&0&N_x&N_z
\end{array}\right).
\end{equation}
We show that this ancilla state and the circuit in Fig.~\ref{fig:Steane_EC} successfully extract the syndrome without giving information about the logical operators by propagating the observables of the code $\mathcal{A}thcal{C}$ and the stabilizer $S_a$ through the circuit. Begin with,
\begin{align}
\left(\begin{array}{ccc|ccc}aS_x&0&0&S_z&0&0\\bN_x&0&0&N_z&0&0\\0&S_z&S_x&0&0&S_z\\0&0&0&0&S_x&S_z\\0&0&0&0&N_x&N_z\end{array}\right),
\end{align}
where the syndrome is $a\in\mathcal{A}thbb{F}_2^{n-k}$ and logical operator values are $b=\mathcal{A}thbb{F}_2^{2k}$. After the controlled-$Z$ gates,
\begin{equation}
\left(\begin{array}{ccc|ccc}aS_x&0&0&S_z&S_x&0\\bN_x&0&0&N_z&N_x&0\\0&S_z&S_x&S_z&0&S_z\\0&0&0&0&S_x&S_z\\0&0&0&0&N_x&N_z\end{array}\right).
\end{equation}
After the controlled-$X$ gates,
\begin{equation}
\left(\begin{array}{ccc|ccc}aS_x&0&0&S_z&S_x&S_z\\bN_x&0&0&N_z&N_x&N_z\\S_x&S_z&S_x&S_z&0&0\\0&0&0&0&S_x&S_z\\0&0&0&0&N_x&N_z\end{array}\right).
\end{equation}
This is equivalent to the stabilizer,
\begin{equation}
\left(\begin{array}{ccc|ccc}aS_x&0&0&S_z&0&0\\bN_x&0&0&N_z&0&0\\a0&S_z&S_x&0&0&0\\0&0&0&0&S_x&S_z\\0&0&0&0&N_x&N_z\end{array}\right),
\end{equation}
and so we see that measuring all ancilla qubits in the $X$-basis results in a bitstring $m\in\mathcal{A}thbb{F}_2^{2n}$ such that $S\Lambda m=a$.
We note that $\ket{\overline{a}}$ is simply related to a Bell pair $\ket{\Phi}=(\ket{00}+\ket{11})/\sqrt{2}$ encoded in $\mathcal{A}thcal{C}$. If $\text{CX}_{tb}$ denotes $n$ CX gates transversally acting from the top $n$ qubits of the ancilla to the bottom $n$ and $H_t$ denotes $n$ $H$ gates applied to the top $n$ qubits, then $\ket{\overline{a}}=H_t\text{CX}_{tb}\ket{\overline{\Phi}}$. Thus, we can think of $\ket{\overline{a}}$ as an encoded Bell pair that has been ``transversally disentangled". Circuit identities can be used to rearrange Fig.~\ref{fig:Steane_EC} to Knill's error-correction \cite{Knill2005a}. Also, if $\mathcal{A}thcal{C}$ is CSS, Fig.~\ref{fig:Steane_EC} reduces to Steane's error-correction for CSS codes \cite{Steane1998a}.
Steane's original proposal for non-CSS error-correction \cite{Steane1997} omitted the $S_z$ on the right side of the first row of Eq.~\eqref{eq:anc_stabilizer}. Doing the same calculation as above shows that this will not succeed in measuring the syndrome. Steane's proposal suggested that the ancilla state would always be CSS for any code. This, unfortunately, is not true. Indeed, the 7-qubit code from \cite{Yoder2017} has an ancilla that is not even local-Clifford (LC) equivalent to a CSS state.
However, there are non-CSS codes for which $S_a$ is LC equivalent to a CSS code. The 5-qubit code with stabilizer
\begin{equation}
S_5=\left(\begin{array}{ccccc|ccccc}
1&0&1&0&0&0&0&0&1&1\\
0&1&0&0&1&0&0&1&1&0\\
1&0&0&1&0&0&1&1&0&0\\
0&0&1&0&1&1&1&0&0&0
\end{array}\right)
\end{equation}
is one of these. Indeed, $S_a$ can be written using only $Y$-type and $Z$-type generators. This allows us to prepare the ancilla using Fig.~\ref{fig:Steane_prep}, and verify the ancilla against single circuit faults using Fig.~\ref{fig:Steane_verification}, which are both standard constructions for CSS states \cite{Steane1998a,Cross2009}.
\begin{figure}
\caption{Circuit for Steane's error correction on a non-CSS code. $\ket{\bar{\psi}
\label{fig:Steane_EC}
\end{figure}
\begin{figure}
\caption{Preparing the error-correction ancilla state for the 5-qubit code for use in Fig.\fig{Steane_EC}
\label{fig:Steane_prep}
\end{figure}
\begin{figure}
\caption{Verification circuit for the ancilla state prepared by the circuit in Fig.\fig{Steane_prep}
\label{fig:Steane_verification}
\end{figure}
| 2,184 | 21,149 |
en
|
train
|
0.53.4
|
\section{Details of the simulation for logical error estimation}\label{app:sim_details}
Here, we describe some techniques used in the estimation of logical error rates.
For reasons of simulation efficiency, only errors originating from at most two faults are considered, but all such errors are counted. For Clifford circuits, propagating the Pauli errors resulting from circuit depolarizing noise can be done simply using the Gottesman-Knill theorem \cite{Gottesman1998a}. However, some of our circuits are built from non-Clifford CCZ gates. In this case, a tracked error is modified to include controlled-$Z$ (CZ) terms. A Pauli error that propagates through $m$ CCZ gates picks up at most $m$ CZ terms (some may cancel). Upon measurement (e.g.~in the error-correction circuits), the CZ terms must be broken down into a sum of Paulis, only some of which flip measurement bits to cause a signal.
We treat each term as different error element with the probability equal to the square of the amplitude of the term.
There is a subtlety in breaking down CZ errors.
As a sum of Pauli terms, a CZ error is written $(II+ZI+IZ-ZZ)/2$.
If there are multiple CZ errors, this Pauli sum has every possible combination of $I$ and $Z$ on the qubits on which CZ errors are applied, each with a plus or minus sign.
Thus, $m$ CZ errors applied on different qubit pairs are decomposed into a Pauli sum with $4^m$ terms.
If we treated each term as different error element at this point, each term would be assigned the probability square of the amplitude.
However, some terms may be equivalent to other terms up to stabilizers.
Such terms should interfere coherently.
In the simulation of pieceable CCZ on $3\times 3$ Bacon-Shor code, all the terms in the Pauli sum are rewritten in an unambiguous way up to stabilizers, and terms interfere before assigning them a probability.
Now that we recognize the subtle issue as the coherent addition of the Pauli terms, we argue that it does not affect the logical error rate except of the $3\times 3$ Bacon-Shor code case.
Firstly, note that the coherent addition can only happen when the number of qubits in one block on which CZ errors are applied is more than or equal to the weight of stabilizers.
This is because if stabilizers have higher weight, multiplying a stabilizer necessarily gives extra Paulis on the qubits that are not affected by CZ errors.
It prevents the term multiplied by a stabilizer being the same as another term in the Pauli sum.
In pieceable CCZ circuit on the 5-qubit code, CZ errors only occur on the three qubits, the support of logical $Z$. Since stabilizers are weight four, the coherent addition will not happen for the above reason.
For the pieceable CCZ circuit on the 7-qubit code, we argue that although the coherent addition may happen, it will not affect logical error rate.
Since stabilizers for the 7-qubit code are weight four, the coherent addition could happen only when two $X$ errors go through in the same block in the first piece.
However, these $X$ errors cannot be corrected because the 7-qubit code is a perfect CSS code.
Thus, all the error elements where the coherent addition could happen end up with logical errors regardless, and it does not matter whether we accurately interfere the terms.
For the pieceable CCZ circuit on the $3\times 9$ Bacon-Shor code, the situation is similar to the 7-qubit code case; the coherent addition could happen, but will not affect the logical error rate.
Since CCZs on the $3\times 9$ Bacon-Shor code are transversal, the number of qubits in the same code block on which CZ errors are applied is at most two.
Since weight-two $Z$-gauge operators are aligned along a row, the coherent addition could only happen when two CZ errors are applied on the two qubits in the same row in some code block.
However, all the terms in the Pauli sum for the CZ errors in that block are $Z$-type errors whose weight is less than or equal to two, and whose support is in the same row.
Since weight-one errors can be corrected by the standard error correction, and weight-two errors in the same row are equivalent to the identities up to stabilizers for the $Z$-gauge Bacon-Shor code, the terms in the Pauli sum are all correctable when the concerned coherent addition could happen.
Thus, it will not affect the logical error rate.
Considering CZ errors as a Pauli sum is inefficient -- $m$ CZ terms lead to $4^m$ Pauli addends.
However, in the simulation, we do not actually break down all the CZ errors.
Under certain cases, we definitely know that the final error correction will succeed to correct the CZ error.
One of such cases is that the CZ error is applied over different code blocks and those code blocks do not have any $Z$ errors.
The other case is that the CZ error is applied in one code block, there are no $Z$ errors in the block, and an intermediate error correction notifies the correct locations that the CZ error is applied over.
Also, we can reduce the number of CZ errors by removing harmless CZ errors before the measurements in the final error correction take place.
A harmless error is one that does not affect encoded states.
When errors are only Paulis, like in the circuits that only consist of Clifford gates, such errors are just stabilizers.
The following theorem generalizes the condition for the harmless errors to non-Pauli case.
\begin{thm}\label{thm:harmless}
Let $E$ be an error operator, $S=\left<g_1,\dots ,g_{n-k}\right>$ be the stabilizers, $\left<g_{n-k+1},\dots,g_{n+k}\right>$ be the logical operators of the code, and $\ket{\bar{\psi}}$ be an encoded state. If $g_i^{\dagger}E^{\dagger}g_iE\in S$ for all $i=1,\dots,n+k$, then $E\ket{\bar{\psi}}=\ket{\bar{\psi}}$ up to global phase.
\end{thm}
\begin{proof}
By the assumption, there exists a stabilizer $s_l$ such that $g_iE=Eg_is_l, \forall i$.
For $i=1,\dots,n-k$, since
\begin{eqnarray}
g_iE\ket{\bar{\psi}}=Eg_is_l\ket{\bar{\psi}}=E\ket{\bar{\psi}},
\end{eqnarray}
$E$ preserves the codeword space. Now for $i=n-k+1\dots n+k$, let $\ket{g_i^{(\pm)}}$ be the eigenstate of the logical operator $g_i$ with eigenvalue $\pm 1$, then
\begin{eqnarray}
g_i E\ket{g_i^{(\pm)}}=E g_i s_l\ket{g_i^{(\pm)}}=\pm E\ket{g_i^{(\pm)}}.
\end{eqnarray}
Thus, $E$ also preserves the logical space.
\end{proof}
\nonumber\\indent This theorem allows us to ignore the CZ errors that satisfy the above condition, which greatly reduces the computational task.
When intermediate error corrections are present, CZ errors need to be broken down according to the Pauli sum in the intermediate error corrections, and need to be propagated until the error correction at the end.
If the number of intermediate corrections is zero or one, it is rather easy to deal with, because the number of error elements due to the CZ errors that need to be propagated until the end is limited.
Actually, except the pieceable 5-qubit code, all the CZ errors that do not satisfy the condition of Theorem~\ref{thm:harmless} were broken down upon measurement and tracked to see if they end with a logical error.
For the 5-qubit code, to reduce the computational demand, we take the rule where we declare an error to be a logical error as soon as some CZ errors are measured in an intermediate error correction.
Although this strategy would cause some overestimation of the logical error rate, we argue that the probability that CZ errors are measured in an intermediate error correction is rather small.
CZ errors are measured in an intermediate error correction in the following two cases.
The first case is that an $X$ or $Y$-type error is caught by a CCZ gate in the adaptive nonconstant-stabilizer measurement.
It is described in \cite{Yoder2016c} that the adaptive nonconstant-stabilizer measurement is only triggered when some constant stabilizer measurements click due to an $X$ or $Y$-type error only for a single code block.
The adaptive measurement may contain CCZ gates connected between the ancilla block and the code blocks whose constant stabilizers did {\it not} click.
Thus, an $X$ or $Y$-type error is caught by a CCZ gate in the adaptive measurement only when an $X$ or $Y$ error triggers the adaptive measurement, the constant measurement in different code blocks fail with $X$ or $Y$ type error, and it goes to a CCZ gate in the adaptive measurement.
The second case is that a CZ error is caught by a CNOT gate in the adaptive measurement.
Note that CZ error only happens when an $X$ or $Y$-type error propagates through the CCZ gates in the code blocks. CZ errors are then present in code blocks other than the one in which the $X$ or $Y$-type error exists. Also, CNOT gates in the adaptive measurement could be only applied to the code block whose constant stabilizers click.
Thus, a CZ error is caught by a CNOT gate in the adaptive measurement only when an $X$ or $Y$-type error generates CZ errors in different code blocks, a later CCZ gate fails to cancel the first $X$ or $Y$-type error and generate another $X$ or $Y$-type error in the other code block that will make the constant measurement click, and the CZ error goes into CNOT gate in the adaptive measurement.
These two cases are realized in very restricted situations, so the contribution to the total logical error rate from these cases would be rather small.
Another situation arises with two or more intermediate corrections.
The pieceable construction on the 5-qubit code has multiple intermediate corrections, and they detect $X$ errors and notify possible error locations to the final error correction so that the final error correction can correct up to weight two located errors.
However, multiple faults can cause two intermediate error corrections to incorrectly notify more than two locations to the final error correction.
We declare those elements to be logical errors.
| 2,562 | 21,149 |
en
|
train
|
0.53.5
|
\section{Details on the error polynomials}\label{app:error_polys}
Here, we describe how to obtain Eq.~\eq{fail_acc}-\eq{rej} from the exact counting.
We first consider Eq.~\eq{fail_acc}, the probability that one or two faults occur and that pattern is accepted by all the verification modules through the propagation, but ends up with a logical error.
Due to the fault-tolerant property, a single fault never causes a logical error. Thus, it suffices to consider the cases when two faults occur.
In the simulation, each combination of two-fault patterns is assigned a probability $\left(\frac{p_{r}}{4^{r}-1}\right)\left(\frac{p_{s}}{4^{s}-1}\right)$ if the faulty components are an $r$-qubit gate and an $s$-qubit gate.
We propagate all the errors until the end and sum up the probabilities of the errors that lead to logical errors.
During the propagation, these errors may encounter verification processes.
If they are accepted by the verification, we keep propagating them. Otherwise, we stop propagating them so that they do not contribute to the logical error rate.
Let $Q_{\rm fail, acc}$ denote the estimated logical error rate. Since each physical error rate is either $p_1, p_2$ or $p_3$, it looks like
\begin{equation}
Q_{\rm fail,acc}=\sum_{r=1}^3\sum_{s\geq r}^3F_{rs}^{(2)} p_r p_s.
\end{equation}
Let $n_r$ be the total number of $r$-qubit gates.
Since we assume that different components fail independently, Eq.~\eq{fail_acc} is obtained as
\begin{equation}
P_{\rm fail,acc}^{(2)}=\left[\Pi_{t=1}^3 (1-p_t)^{n_t}\right]\left(\sum_{r=1}^3\sum_{s\geq r}^3F_{rs}^{(2)} \frac{p_r}{1-p_r} \frac{p_s}{1-p_s} \right).
\label{eq:fail_acc_formula}
\end{equation}
Similarly, $Q_{\rm succ, acc}$, the sum of the assigned probability of the patterns that are accepted by all the verification modules and do not cause a logical error, looks like
\begin{equation}
Q_{\rm succ,acc}=\sum_{r=1}^3 S_{r}^{(1)}p_r+\sum_{r=1}^3\sum_{s\geq r}^3S_{rs}^{(2)} p_r p_s
\end{equation}
and Eq.~\eq{succ_acc} is obtained as
\begin{eqnarray}
&&P_{\rm succ,acc}^{(2)}=\left[\Pi_{t=1}^3 (1-p_t)^{n_t}\right]\cdot \nonumber\\number\\
&&\left(1+\sum_{r=1}^3 \frac{S_{r}^{(1)} p_r}{1-p_r}+\sum_{r=1}^3\sum_{s\geq r}^3 \frac{S_{rs}^{(2)} p_r p_s}{(1-p_r)(1-p_s)} \right)
\label{eq:succ_acc_formula}
\end{eqnarray}
The patterns that are not counted in either $Q_{\rm fail,acc}$ or $Q_{\rm succ,acc}$ are rejected in some verification module. Thus, we obtain Eq.~\eq{rej} as
\begin{eqnarray}
&&P_{\rm rej}^{(2)}=\left[\Pi_{t=1}^3 (1-p_t)^{n_t}\right]\cdot \nonumber\\number\\
&&\left(\sum_{r=1}^3 \frac{A_r^{(1)} p_r}{1-p_r}+\sum_{r=1}^3\sum_{s> r}^3 \frac{A^{(2)}_{rs} p_r p_s}{(1-p_r)(1-p_s)}\right)
\end{eqnarray}
where
\begin{eqnarray}
A^{(1)}_r&=&n_r-S_{r}^{(1)}\\
A^{(2)}_{rs}&=&
\begin{cases}
n_r n_s - F_{rs}^{(2)}-S_{rs}^{(2)} & (r\neq s)\\
\binom{n_r}{2} - F_{rr}^{(2)}-S_{rr}^{(2)} &(r=s)
\end{cases}
\end{eqnarray}
Special care is required for 5-qubit code because $n_r$ cannot be definitely determined because of the adaptive measurements.
Note that at most two adaptive measurements are triggered when one or two faults occur.
Thus, taking $n_r$ that includes two largest adaptive measurements, which are the ones with 13-CAT and 9-CAT, the lower bound in Eq.~\eq{bound} still holds.
Instead of Eq.~\eq{succ_acc_formula}, we take
\begin{eqnarray}
&&P_{\rm succ,acc}^{(2)}=\left[\Pi_{t=1}^3 (1-p_t)^{n_t'}\right]\cdot \left(1+\sum_{r=1}^3 \frac{S_{r}^{(1)} p_r}{1-p_r}\right)\nonumber\\number \\
&&+\left[\Pi_{t=1}^3 (1-p_t)^{n_t}\right]\cdot\left(\sum_{r=1}^3\sum_{s\geq r}^3 \frac{S_{rs}^{(2)} p_r p_s}{(1-p_r)(1-p_s)} \right)
\end{eqnarray}
where $n_r'$ is the number of fault locations not including adaptive measurements.
For the 5-qubit code we also use
\begin{equation}
P_{\rm acc}=\Pi_j P_{{\rm acc},j}=\Pi_j (1-P_{{\rm rej},j})<\Pi_{j'} (1-P_{{\rm rej},j'}^{(2)})
\end{equation}
where $j$ is taken over all the verification modules and $j'$ is taken over all the verification modules except adaptive measurements.
The following are the obtained values for the parameters for each construction.
\begin{itemize}
\item $3\times 3$ Bacon-Shor
\begin{equation}
n_1=252,n_2=180,n_3=27
\end{equation}
\begin{equation}
S_{r}^{(1)}=(252,180,27)
\end{equation}
\begin{eqnarray}
F_{rs}^{(2)}&=&
\begin{pmatrix}
4216.8& 4271.9 & 783.5 \\
& 1194.5 & 461.5 \\
& & 34.9
\end{pmatrix}\\
S_{rs}^{(2)}&=&
\begin{pmatrix}
27409.2& 41088.1 & 6020.5\\
& 14915.5 & 4398.5 \\
& & 316.1
\end{pmatrix}
\end{eqnarray}
\item Pieceable 7-qubit
\begin{equation}
n_1=648,n_2=480,n_3=21
\end{equation}
\begin{equation}
S_{r}^{(1)}=(383,224,21)
\end{equation}
\begin{eqnarray}
F_{rs}^{(2)}&=&
\begin{pmatrix}
13258.4& 12722.6 & 3581.4 \\
& 3077.3 & 1855.3 \\
& & 176.7
\end{pmatrix}\\
S_{rs}^{(2)}&=&
\begin{pmatrix}
56460.9& 68953.7 & 4461.6\\
& 20748.8 & 2848.7 \\
& & 33.3
\end{pmatrix}
\end{eqnarray}
\item $3\times 9$ Bacon-Shor
\begin{equation}
n_1=2736,n_2=864,n_3=27
\end{equation}
\begin{equation}
S_{r}^{(1)}=(1524,566.4,27)
\end{equation}
\begin{eqnarray}
F_{rs}^{(2)}&=&
\begin{pmatrix}
52074 & 43098.4 & 7049.2 \\
& 8663.0 & 2968.1 \\
& & 183.3
\end{pmatrix}\\
S_{rs}^{(2)}&=&
\begin{pmatrix}
1013640 & 748636 & 34098.8\\
& 138296 & 12324.7 \\
& & 167.7
\end{pmatrix}
\end{eqnarray}
\item Pieceable 5-qubit
\begin{eqnarray}
n_1&=&3365,n_2=1228,n_3=41\\
n_1'&=&2967, n_2'=1152, n_3'=27
\end{eqnarray}
\begin{equation}
S_{r}^{(1)}=(1475,457.6.,27)
\end{equation}
\begin{eqnarray}
F_{rs}^{(2)}&=&
\begin{pmatrix}
113030.0 & 85261.6 & 14679.2 \\
& 16067.4 & 5551.4 \\
& & 332.5
\end{pmatrix}\\
S_{rs}^{(2)}&=&
\begin{pmatrix}
639301.0 & 482043.0 & 20392.4\\
& 90716.0 & 7554.7 \\
& & 59.3
\end{pmatrix}
\end{eqnarray}
\item 7-qubit with magic state
\begin{equation}
n_1=1138,n_2=743,n_3=14
\end{equation}
\begin{equation}
S_{r}^{(1)}=(612,324.3,6.9)
\end{equation}
\begin{eqnarray}
F_{rs}^{(2)}&=&
\begin{pmatrix}
25436.5 & 24565.9 & 1078.5 \\
& 6232.4 & 521.1 \\
& & 26.9
\end{pmatrix}\\
S_{rs}^{(2)}&=&
\begin{pmatrix}
154650 & 166625 & 3921.4\\
& 44308.3 & 2178.5 \\
& & 18.6
\end{pmatrix}
\end{eqnarray}
\end{itemize}
\section{Transformation matrix for volume calculation}\label{app:volume_calc}
As explained in the main text, the circuit volume for concatenated codes at higher concatenation level is described by a transformation matrix $A$ where $A_{ij}=N^{G_j}_{G_i}$.
We show the matrices for pieceable $3\times 3$ Bacon-Shor code, pieceable 7-qubit code, and 7-qubit with magic state, which are denoted by $A_{pBS}$,$A_{p7}$, $A_{m7}$ respectively.
We take the following order for gates; {\bf G}=\{3-qubit gate, 2-qubit gate, single qubit gate, $\ket{0}$ and $\ket{+}$ preparation, $X$ basis and $Z$ basis measurement\}.
For preparation of $\ket{\bar{0}}$ and $\ket{\bar{+}}$ on 7-qubit code, we use the method proposed by Goto \cite{Goto2016}, which requires just one additional ancilla.
\begin{equation*}
A_{pBS}=
\begin{pmatrix}
27 & 90 & 45 & 54 & 54 \\
0 & 69 & 30 & 36 & 36 \\
0 & 30 & 24 & 18 & 18 \\
0 & 6 & 3 & 9 & 0 \\
0 & 0 & 0 & 0 & 9
\end{pmatrix}
\label{eq:pBS_trans}
\end{equation*}
\begin{equation*}
A_{p7}=
\begin{pmatrix}
21 & 162 & 240 & 72 & 72 \\
0 & 79 & 104 & 32 & 32 \\
0 & 36 & 59 & 16 & 16 \\
0 & 11 & 22 & 8 & 1 \\
0 & 0 & 0 & 0 & 7
\end{pmatrix}
\label{eq:p7_trans}
\end{equation*}
\begin{equation*}
A_{m7}=
\begin{pmatrix}
14 & 267 & 504 & 136 & 136 \\
0 & 79 & 104 & 32 & 32 \\
0 & 36 & 59 & 16 & 16 \\
0 & 11 & 22 & 8 & 1 \\
0 & 0 & 0 & 0 & 7
\end{pmatrix}
\label{eq:m7_trans}
\end{equation*}
| 3,449 | 21,149 |
en
|
train
|
0.53.6
|
\section{Detailed resource analysis for surface code}\label{app:surface_calc}
We describe the detailed resource analysis to implement logical Toffoli gate on the surface code.
There are mainly two ways to do it, synthesizing a Toffoli gate using Clifford gates and $T$ gates, and injecting a logical Toffoli state by gate teleportation.
Consider the first method, in the context of the Toffoli implementation proposed by Jones \cite{Jones2013} using four $T$ gates. The
$T$ gates are implemented by $\ket{T}$ state and gate teleportation where $\ket{T}$ state is purified by a distillation protocol.
We use the 15-1 protocol~\cite{Bravyi2005a,Fowler2012a} which reduces error rates of $\ket{T}$ from $\mathcal{O}(p)$ to $\mathcal{O}(p^3)$, because it requires the smallest circuit volume compared to others \cite{Bravyi2012a,Meier2013,Reichardt2004a}.
Since the region of the physical error rate that pieceable construction helps to reduce error rate is $p<10^{-4}$ as can be seen in Fig.\fig{logical_error}, the logical error rate of the magic state distilled once is $<10^{-12}$.
Although the reduction in error rate may not be sufficiently low depending on the goal logical error rate, one distillation already gives large overheads.
Thus, we consider the circuit volume for one distillation as a lower bound and proceed the discussion.
It may come as a surprise that other distillation protocols with better conversion rate between noisy magic state and purified magic state have larger circuit volume.
It comes from that Hadamard gate and phase gate are not transversal on the surface code.
For implementing the Hadamard gate or phase gate fault-tolerantly, some non-trivial techniques, such as state injection, lattice surgery \cite{Horsman2012a}, code deformation \cite{Bombin2009a}, or surface folding \cite{Moussa2016}, are required.
These take many surface code steps, which affect the circuit volume.
Even though conversion rate between noisy $T$ state and purified $T$ state is high, if it requires many costly Clifford gates, the circuit volume will be large.
Especially in the case when only one distillation is required, a poor conversion rate does not hurt circuit volume that much.
Let us analyze the number of surface code cycles and circuit volume for each gate that are necessary to implement the logical Toffoli gate.
Let $C_G$ and $V_G$ be surface code cycles and circuit volume it takes to implement $G$.
We discuss circuit volume in units of [qubit$\cdot$cycle] and then convert it to [qubit$\cdot$step] using the fact that one surface code cycle consists of six steps \cite{Fowler2012a}.
Also, let $d$ be surface code distance, and $n=(2d-1)^2$ be the number of physical qubits on a surface.
Necessary components here are \{$\ket{\bar{0}}$ and $\ket{\bar{+}}$ preparation, CNOT, Hadamard, Phase\}.
For logical state preparation, we initialize a surface with physical $\ket{0}$ for $\ket{\bar{0}}$ preparation, and $\ket{+}$ for $\ket{\bar{+}}$ preparation.
After $d$ rounds of error correction, an appropriate recovery can be determined to prepare the desired logical state fault-tolerantly.
Thus, we find $C_{prep}=d$, $V_{prep}=nd$.
The CNOT gate can be transversally implemented if we allow non-locality or a 3D layered architecture.
However, since one of the striking features of surface codes is local interactions in a 2D architecture, we use lattice surgery to implement the CNOT gate \cite{Horsman2012a}.
First, prepare a surface with $\ket{\bar{+}}$ state between the control surface and the target surfaces.
The control surface and the intermediate surface are merged while obtaining measurement syndromes.
This corresponds to $\bar{Z}\bar{Z}$ measurement.
After that, the surface is split into two original surfaces and the intermediate surface is merged to target surface, which corresponds to $\bar{X}\bar{X}$ measurement.
It ends with splitting it into the two original surfaces.
Since merger and splitting each take $d$ rounds of error correction to stabilize the surface,
\begin{equation}
C_{CNOT}=C_{prep}+4d=5d
\end{equation}
and
\begin{eqnarray}
V_{CNOT}&=&V_{prep}+(3n+2(2d-1))(C_{prep}+4d)\nonumber\\number \\
&=& 6 d - 44 d^2 + 64 d^3.
\end{eqnarray}
The Hadamard gate is also implemented by the lattice surgery.
In the lattice surgery technique, firstly Hadamard gates are applied transversally.
To correct the orientation of the boundary, additional qubits are merged to the boundary and some qubits are split out so that it restores the original boundary orientation. The protocol ends with moving the surface back to the original position.
It takes $d$ cycles to stabilize the original surface after applying transversal $H$, $d$ cycles for lattice merger, $d$ cycles for lattice splitting, and $d$ cycles for SWAP operations to move the lattice back to the original position. Thus, $C_H=4d$.
For circuit volume, we need a bigger surface to carry out merger and split by one more column and row of qubits. Thus, $V_H=(2d)^2 C_H=16d^3$.
For implementing phase gate, we use the circuit in Fig.\fig{S_synthesis}.
A good thing about this circuit is that the ancilla state $\ket{S}=S\ket{+}=(\ket{0}+i\ket{1})/\sqrt{2}$ is preserved.
Thus, once a purified $\ket{S}$ state is prepared at the beginning of the computation, it can be reused whenever a phase gate needs to be applied. After averaging over a whole computation, the volume use for the distillation process at the beginning will be negligible per one logical gate construction. Note that if only local interactions are allowed, it may take additional circuit volume when the qubit to which the phase gate should be applied is far from the stored $\ket{S}$ state.
Thus, our estimation should be considered as a lower bound of the actual circuit volume under the setting in which only local interactions are allowed.
It gives $C_S=2C_{CNOT}+2C_H=18d$ and $V_S=2V_{CNOT}+2V_H+2nC_H=20 d - 120 d^2 + 192 d^3$.
Combining these building blocks, we find the number of cycles and volume required to implement a $T$ gate and a Toffoli gate.
For distilling a $T$ state, $\ket{T}=T\ket{+}$, we use the circuit in \cite{Fowler2012a} which takes 15 $\ket{T}$ states and output 1 $\ket{T}$ with lower error rate.
It takes 7 surface code cycles for CNOTs and 2 steps for transversal $T$ and measurements, which is 1/4 surface code cycle. Ignoring the last 1/4 cycles, we get $C_{\ket{T}}=7C_{CNOT}=35d$.
With some parallelization, we get $V_{\ket{T}}=16V_{prep}+V_{CNOT}+7(5V_{CNOT}+6nC_{CNOT})+\frac{1}{4}\cdot16n d=446 d - 2504 d^2 + 3224 d^3$
For implementing a $T$ gate, we use the usual gate teleportation technique \cite{Nielsen2010}.
The $S$ gate correction is applied with probability $1/2$.
We get $C_T = C_{CNOT}+\frac{1}{2}C_S=14d$ and $V_T =V_{\ket{T}}+V_{CNOT}+\frac{1}{2}V_S=462 d - 2608 d^2 + 3384 d^3$.
Since the surface code is CSS, we can transversally make measurements on all the data qubits, and extract eigenvalue for measurement operator.
Thus, measurement is done with only one time step, which is 1/8 of one surface code cycle, we ignore the volume due to the measurement.
Toffoli gate synthesis in \cite{Jones2013} consists of two steps.
In the first part, one constructs the $\mbox{Toffoli}^*$ gate, which is Toffoli gate followed by controlled-$S^{\dagger}$ gate, where four $T$ gates and two $H$ gates are used. Also, note that one logical ancilla block is used.
The second part takes the $\mbox{Toffoli}^*$ gate to the usual Toffoli gate with help of one additional ancilla block.
By construction of the synthesis circuit, we get
\begin{equation}
C_{\mbox{Toffoli}^*}=2C_H+4C_{CNOT}+C_T=42d
\end{equation}
and
\begin{eqnarray}
V_{\mbox{Toffoli}^*} &=&V_{prep}+6n C_H+2V_H+8V_{CNOT}+4 V_T\nonumber\\number \\
&=&1921 d - 10884 d^2 + 14180 d^3
\end{eqnarray}
Second part of the circuit gives
\begin{equation}
C_{\mbox{Toffoli}}=C_{\mbox{Toffoli}^*}+C_{S}+C_{CNOT}+C_{H}+C_{\ket{T}}=104d
\end{equation}
and
\begin{eqnarray}
V_{\mbox{Toffoli}}&=& V_{\mbox{Toffoli}^*}+V_{prep}+nC_{\mbox{Toffoli}^*}+V_S+V_{CNOT}\nonumber\\number \\
&+&V_H +3n(C_S+C_{CNOT}+C_H)\nonumber\\number \\
&+&n(C_{\mbox{Toffoli}^*}+C_{CNOT}+C_H)\nonumber\\number \\
&=& 2118 d - 11732 d^2 + 15136 d^3
\label{eq:VTof_1st}
\end{eqnarray}
where the unit for the volume is [qubit $\cdot$ cycle].
We included $C_{\ket{T}}$ in $C_{\mbox{Toffoli}}$ because cycles in the distillation circuit also contribute increasing in the final logical error rate.
Note that it includes the ancilla qubits for keeping $\ket{S}$ state that is kept during the whole computation.
Another way to implement logical Toffoli gate on the surface code is to use Toffoli state.
To locally prepare the Toffoli state, we use the protocol that takes eight $\ket{H}$ states and outputs one Toffoli state \cite{Eastin2013}.
In the preparation circuit, there are two $Y(\pi/4)$ gates and four $Y(-\pi/4)$ gates, which are rotations with respect to $Y$ axis.
These gates are implemented using $\ket{H}$ state with $Y$ basis measurement, controlled-$Y$ gate, and $Y(\pm \pi/2)$ gate. To implement these gates on the surface code, we use phase gates and Hadamard gates to rotate them to $X$ basis measurement, CNOT gate, and phase gate.
We then obtain
\begin{eqnarray}
C_{Y(\pi/4)}&=&C_S+C_{CNOT}+C_S+(2C_H+C_S)/2\nonumber\\number \\
&=&54d
\end{eqnarray}
and
\begin{eqnarray}
V_{Y(\pi/4)}&=& V_{prep}+V_S+V_{CNOT}+2V_S+(2V_H+V_S)/2\nonumber\\number \\
&=&77 d - 468 d^2 + 756 d^3
\end{eqnarray}
Using these, we obtain
\begin{eqnarray}
C_{\ket{\mbox{Toffoli}}}&=&7C_{CNOT} + (15/2)C_S + 3C_H\nonumber\\number \\
&=&182d
\end{eqnarray}
and
\begin{eqnarray}
V_{\ket{\mbox{Toffoli}}}&=&4V_{prep} + 3(2V_{CNOT} + 2 V_{Y(\pi/4)} + 2nC_{Y(\pi/4)})\nonumber\\number \\
&&+ V_{CNOT} + 2nC_{CNOT}\nonumber\\number \\
&=&842 d - 4468 d^2 + 6336 d^3
\end{eqnarray}
where $\ket{\mbox{Toffoli}}$ refers to the Toffoli state.
Cycles and volume for the teleportation circuit, which we write $C_{tele}$ and $V_{tele}$ are
\begin{eqnarray}
C_{tele}&=&C_{CNOT}+1/2(3C_{CNOT}+2C_H)\nonumber\\number\\
&=&16.5d
\end{eqnarray}
\begin{eqnarray}
V_{tele}&=&3V_{CNOT}+\{4nC_H+ 3(V_{CNOT}+nC_{CNOT})\}/2 \nonumber\\number \\
&=&42.5 d - 260 d^2 + 350 d^3
\end{eqnarray}
Combining all of them, we get
\begin{eqnarray}
C_{\mbox{Toffoli}}=198.5d
\end{eqnarray}
and
\begin{eqnarray}
V_{\mbox{Toffoli}}=7076 d - 37824 d^2 + 53488 d^3
\label{eq:VTof_2nd}
\end{eqnarray}
where the unit for the volume is [qubit $\cdot$ cycle].
Fig.~\fig{volume_Tofstate_T} shows circuit volume with unit [qubit$\cdot$step] in terms of code distance for both ways of implementation.
We can see that the the scheme with Toffoli state has lower circuit volume.
It is the reason why the scheme with Toffoli state is discussed in the main text.
\begin{figure}
\caption{Circuit identity used for implementing $S$ gate. Here $\ket{S}
\label{fig:S_synthesis}
\end{figure}
\begin{figure}
\caption{Circuit volume for two different implementations of Toffoli gate. Dashed: gate synthesis using $T$ gate. Solid: Toffoli state scheme}
\label{fig:volume_Tofstate_T}
\end{figure}
\end{document}
| 3,639 | 21,149 |
en
|
train
|
0.54.0
|
\begin{document}
\title{ space{-0.7cm}
\begin{abstract}
A typical decomposition question asks whether the edges of some graph $G$ can be partitioned into disjoint copies of another graph $H$.
One of the oldest and best known conjectures in this area, posed by Ringel in 1963, concerns the decomposition of complete graphs into edge-disjoint copies of a tree.
It says that any tree with $n$ edges packs $2n+1$ times into the complete graph $K_{2n+1}$.
In this paper, we prove this conjecture for large $n$.
\epsilonilonnd{abstract}
| 153 | 88,680 |
en
|
train
|
0.54.1
|
\section{Introduction}\label{intro}
The study of decomposition problems for graphs and hypergraphs has a very long history, going back more than two hundred years to the work of Euler on Latin squares. Latin squares are $n \times n$ arrays filled with $n$ symbols such that each symbol appears once in every row and column.
In 1782, Euler asked for which values of $n$ there is a Latin square which can be decomposed into $n$ disjoint transversals, where a transversal is a collection of cells of the Latin square which do not share the same row, column or symbol.
This problem has many equivalent forms. In particular, it is equivalent to a \epsilonilonmph{graph decomposition problem}.
We say that a graph $G$ has a \epsilonilonmph{decomposition into copies of a graph $H$} if the edges of $G$ can be partitioned into edge-disjoint subgraphs isomorphic to $H$. Euler's problem is equivalent to asking for which values of $n$ does the balanced complete $4$-partite graph $K_{n,n,n,n}$ have a decomposition into copies of the complete graph on $4$ vertices, $K_4$.
In 1847, Kirkman studied decompositions of complete graphs $K_n$ and showed that they can be decomposed into copies of a triangle if, and only if, $n\epsilonilonquiv 1 \text{ or } 3 \pmod 6$. Wilson~\cite{wilson1975decompositions} generalized this result by proving necessary and sufficient conditions for a complete graph $K_n$ to be decomposed into copies of \epsilonilonmph{any} graph, for large $n$.
A very old problem in this area, posed in 1853 by Steiner, says that, for every $k$, modulo an obvious divisibility condition every sufficiently large complete $r$-uniform hypergraph can be decomposed into edge-disjoint copies of a complete $r$-uniform hypergraph on $k$ vertices. This problem was the so-called ``existence of designs'' question and has practical relevance to experimental designs. It was resolved only very recently in spectacular
work by Keevash \cite{keevash} (see the subsequent work of \cite{glock-all} for an alternative proof of this result). Over the years graph and hypergraph decomposition problems have been extensively studied and
by now this has become a vast topic with many exciting results and conjectures (see, for example, \cite{gallian2009dynamic,wozniak2004packing,yap1988packing}).
In this paper, we study decompositions of complete graphs into large trees, where a tree is a connected graph with no cycles. By large we mean that the size of the tree is comparable with the size of the complete graph (in contrast with the existence of designs mentioned above, where the decompositions are into small subgraphs).
The earliest such result was obtained more than a century ago by Walecki. In 1882 he proved that a complete graph $K_n$ on an even number of vertices can be partitioned into edge-disjoint Hamilton paths.
A Hamilton path is a path which visits every vertex of the parent graph exactly once. Since paths are a very special kind of tree it is natural to ask which other large trees can be used to decompose a complete graph. This question was raised by Ringel \cite{ringel1963theory}, who in 1963 made the following
appealing conjecture on the decomposition of complete graphs into edge-disjoint copies of a tree with roughly half the size of the complete graph.
\begin{conjecture} \label{ringelconj}
The complete graph $K_{2n+1}$ can be decomposed into copies of any tree with $n$ edges.
\epsilonilonnd{conjecture}
Ringel's conjecture is one of the oldest and best known open conjectures on graph decompositions. It has been established for many very special classes of trees such as caterpillars, trees with $\leq 4$ leaves, firecrackers, diameter $\leq 5$ trees, symmetrical trees, trees with $\leq 35$ vertices, and olive trees (see Chapter 2 of \cite{gallian2009dynamic} and the references therein).
There have also been some partial general results in the direction of Ringel's conjecture. Typically, for these results, an extensive technical method is developed which is capable of almost-packing any appropriately-sized collection of certain sparse graphs, see,
e.g., \cite{bottcher2016approximate, messuti2016packing, ferber2017packing, kim2016blow}. In particular, Joos, Kim, K{\"u}hn and Osthus~\cite{joos2016optimal} have proved Ringel's conjecture for very large bounded-degree trees. Ferber and Samotij~\cite{ferber2016packing} obtained an almost-perfect packing of almost-spanning trees with maximum degree $O(n/\log n)$, thus giving an approximate version of Ringel's conjecture for trees with maximum degree $O(n/\log n)$. A different proof of this was obtained by
Adamaszek, Allen, Grosu, and Hladk{\'y}~\cite{adamaszek2016almost}, using graph labellings. Allen, B\"ottcher, Hladk{\'y} and Piguet~\cite{allen2017packing} almost-perfectly packed arbitrary spanning graphs with maximum degree $O(n/ \log n)$ and constant degeneracy\footnote{A graph is $d$-degenerate if each induced subgraph has a vertex of degree $\leq d$. Trees are exactly the $1$-degenerate, connected graphs.} into large complete graphs.
Recently Allen, B\"ottcher, Clemens, and Taraz~\cite{allen2019perfectly} found perfect packings of complete graphs into specified graphs with maximum degree $o(n/\log n)$, constant degeneracy, and linearly many leaves.
To tackle Ringel's conjecture, the above mentioned papers developed many powerful techniques based on the application of probabilistic methods and
Szemer\'edi's regularity lemma. Yet, despite the variety of these techniques, they all have the same limitation, requiring that the maximum degree of the tree should be much smaller than $n$.
A lot of the work on Ringel's Conjecture has used the \epsilonilonmph{graceful labelling} approach.
This is an elegant approach proposed by R\'osa~\cite{rosa1966certain}. For an $(n+1)$-vertex tree $T$ a bijective labelling of its vertices $f: V(T) \rightarrow \{0, \dots, n\}$ is called graceful
if the values $|f(x)-f(y)|$ are distinct over the edges $(x,y)$ of $T$.
In 1967 R\'osa conjectured that every tree has a graceful labelling. This conjecture has attracted a lot of attention in the last 50 years but has only been proved for some special classes of trees, see e.g., \cite{gallian2009dynamic}.
The most general result for this problem was obtained by Adamaszek, Allen, Grosu, and Hladk{\'y}~\cite{adamaszek2016almost} who proved it asymptotically for trees with maximum degree $O(n/\log n)$.
The main motivation for studying graceful labellings is that one can use them to prove Ringel's conjecture. Indeed,
given a graceful labelling $f: V(T) \rightarrow \{0, \dots, n\}$, think of it as an embedding of $T$ into $\{0, \dots, 2n\}$. Using addition modulo $2n+1$, consider $2n+1$ cyclic shifts $T_0, \ldots, T_{2n}$ of $T$, where the tree $T_i$ is an isomorphic copy of $T$ whose vertices are
$V(T_i)=\{f(v)+i~|~ v \in V(T)\}$ and whose edges are $E(T_i)=\{(f(x)+i,f(y)+i)~|~(x,y)\in E(T)\}$. It is easy to check that the fact that $f$ is graceful implies that the trees $T_i$ are edge disjoint and therefore
decompose $K_{2n+1}$.
R\'osa also introduced a related proof approach to Ringel's conjecture called ``$\rho$-valuations''. We describe it using the language of ``rainbow subgraphs'', since this is the language which we ultimately use in our proofs.
A \epsilonilonmph{rainbow} copy of a graph $H$ in an edge-coloured graph $G$ is a subgraph of $G$ isomorphic to $H$ whose edges have different colours. Rainbow subgraphs are important because many problems in combinatorics can be rephrased as problems asking for rainbow subgraphs (for example the problem of Euler on Latin squares mentioned above).
Ringel's conjecture is implied by the existence of a rainbow copy of every $n$-edge tree $T$ in the following edge-colouring of the complete graph $K_{2n+1}$, which we call the \epsilonilonmph{near distance (ND-)colouring}.
Let $\{0,1,\dots,2n\}$ be the vertex set of $K_{2n+1}$. Colour the edge $ij$ by colour $k$, where $k\in [n]$, if either $i=j+k$ or $j=i+k$ with addition modulo $2n+1$.
Kotzig~\cite{rosa1966certain} noticed that if the ND-coloured $K_{2n+1}$ contains a rainbow copy of a tree $T$, then $K_{2n+1}$ can be decomposed into copies of $T$ by taking $2n+1$ cyclic shifts of the original rainbow copy, as explained above (see also Figure~\ref{FigureIntro}). Motivated by this and Ringel's Conjecture, Kotzig conjectured that the ND-coloured $K_{2n+1}$ contains a rainbow copy of every tree on $n$ edges.
To see the connection with graceful labellings, observe that such a labelling of the tree $T$ is equivalent to a rainbow copy of this tree in the ND-colouring whose vertices are $\{0, \dots, n\}$. Clearly, specifying exactly the vertex set of the tree adds an additional restriction which makes it harder to find such a rainbow copy.
\begin{figure}[b]
\centering
\begin{tikzpicture}
\def0.1cm{0.1cm};
\def1.5{1.5};
\def-6{-6};
\def0.04cm{0.04cm};
\def0.04cm{0.04cm};
\def0.075cm{0.075cm};
\def1{1};
\begin{scope}[shift={(-6,0)}]
\foreach \x in {0,...,8}
{
\draw coordinate (B\x) at (40*\x:1);
}
\draw ($0.9*(B4)+0.9*(B5)$) node {\large $T:$};
\draw [line width=0.04cm] (B0) -- (B5);
\draw [line width=0.04cm] (B4) -- (B5);
\draw [line width=0.04cm] (B3) -- (B5);
\draw [line width=0.04cm] (B0) -- (B6);
\foreach \x in {0,3,4,5,6}
{
\draw [fill=black] (B\x) circle [radius=0.075cm];
}
\epsilonilonnd{scope}
\foreach \x in {0,...,8}
{
\draw coordinate (A\x) at (40*\x:1.5);
}
\draw ($0.8*(A4)+0.8*(A5)$) node {\large $K_9:$};
\foreach \x in {0,...,7}
{
\pgfmathtruncatemacro\y{\x+1};
\draw [line width=0.04cm,darkgreen!50] (A\x) -- (A\y);
}
\foreach \x in {8}
{
\pgfmathtruncatemacro\y{\x-9+1};
\draw [line width=0.04cm,darkgreen!50] (A\x) -- (A\y);
}
\foreach \x in {0,...,6}
{
\pgfmathtruncatemacro\y{\x+2};
\draw [line width=0.04cm,blue!50] (A\x) -- (A\y);
}
\foreach \x in {0,...,5}
{
\pgfmathtruncatemacro\y{\x+3};
\draw [line width=0.04cm,black!50] (A\x) -- (A\y);
}
\foreach \x in {0,...,4}
{
\pgfmathtruncatemacro\y{\x+4};
\draw [line width=0.04cm,red!50] (A\x) -- (A\y);
}
\foreach \x in {7,...,8}
{
\pgfmathtruncatemacro\y{\x-9+2};
\draw [line width=0.04cm,blue!50] (A\x) -- (A\y);
}
\foreach \x in {6,...,8}
{
\pgfmathtruncatemacro\y{\x-9+3};
\draw [line width=0.04cm,black!50] (A\x) -- (A\y);
}
\foreach \x in {5,...,8}
{
\pgfmathtruncatemacro\y{\x-9+4};
\draw [line width=0.04cm,red!50] (A\x) -- (A\y);
}
\draw [line width=0.1cm,red] (A0) -- (A5);
\draw [line width=0.1cm,darkgreen] (A4) -- (A5);
\draw [line width=0.1cm,blue] (A3) -- (A5);
\draw [line width=0.1cm,black] (A0) -- (A6);
\foreach \x in {0,...,8}
{
\draw [fill=black] (A\x) circle [radius=0.1cm];
\draw ($1.2*(A\x)$) node {\x};
}
\epsilonilonnd{tikzpicture}
\caption{The ND-colouring of $K_9$ and a rainbow copy of a tree $T$ with four edges. The colour of each edge corresponds to its Euclidean length. By taking cyclic shifts of this tree around the centre of the picture we obtain $9$ disjoint copies of the tree decomposing $K_9$ (and thus a proof of Ringel's Conjecture for this particular tree). To see that this gives 9 disjoint trees, notice that edges must be shifted to other edges of the same colour (since shifts are isometries).}\label{FigureIntro}
\epsilonilonnd{figure}
In \cite{MPS} we gave a new approach to embedding large trees (with no degree restrictions) into edge-colourings of complete graphs, and used this to prove Conjecture
\ref{ringelconj} asymptotically. Here, we further develop and refine this approach, combining it with several critical new ideas to prove Ringel's conjecture for large complete graphs.
\begin{theorem}\label{main}
For every sufficiently large $n$ the complete graph $K_{2n+1}$ can be decomposed into copies of any tree with $n$ edges.
\epsilonilonnd{theorem}
The proof of Theorem \ref{main} uses the last of the three approaches mentioned above. Instead of working directly with tree decompositions, or studying graceful labellings, we instead prove for large $n$ that every ND-coloured $K_{2n+1}$ contains a rainbow copy of every $n$-edge tree (see Theorem~\ref{Theorem_Ringel_proof}).
Then, we obtain a decomposition of the complete graph by considering cyclic shifts of one copy of a given tree (as in Figure~\ref{FigureIntro}). The existence of such a cyclic decomposition was separately conjectured by Kotzig~\cite{rosa1966certain}. Therefore, this also gives a proof of the conjecture by Kotzig for large $n$.
| 4,046 | 88,680 |
en
|
train
|
0.54.2
|
\def0.1cm{0.1cm};
\def1.5{1.5};
\def-6{-6};
\def0.04cm{0.04cm};
\def0.04cm{0.04cm};
\def0.075cm{0.075cm};
\def1{1};
\begin{scope}[shift={(-6,0)}]
\foreach \x in {0,...,8}
{
\draw coordinate (B\x) at (40*\x:1);
}
\draw ($0.9*(B4)+0.9*(B5)$) node {\large $T:$};
\draw [line width=0.04cm] (B0) -- (B5);
\draw [line width=0.04cm] (B4) -- (B5);
\draw [line width=0.04cm] (B3) -- (B5);
\draw [line width=0.04cm] (B0) -- (B6);
\foreach \x in {0,3,4,5,6}
{
\draw [fill=black] (B\x) circle [radius=0.075cm];
}
\epsilonilonnd{scope}
\foreach \x in {0,...,8}
{
\draw coordinate (A\x) at (40*\x:1.5);
}
\draw ($0.8*(A4)+0.8*(A5)$) node {\large $K_9:$};
\foreach \x in {0,...,7}
{
\pgfmathtruncatemacro\y{\x+1};
\draw [line width=0.04cm,darkgreen!50] (A\x) -- (A\y);
}
\foreach \x in {8}
{
\pgfmathtruncatemacro\y{\x-9+1};
\draw [line width=0.04cm,darkgreen!50] (A\x) -- (A\y);
}
\foreach \x in {0,...,6}
{
\pgfmathtruncatemacro\y{\x+2};
\draw [line width=0.04cm,blue!50] (A\x) -- (A\y);
}
\foreach \x in {0,...,5}
{
\pgfmathtruncatemacro\y{\x+3};
\draw [line width=0.04cm,black!50] (A\x) -- (A\y);
}
\foreach \x in {0,...,4}
{
\pgfmathtruncatemacro\y{\x+4};
\draw [line width=0.04cm,red!50] (A\x) -- (A\y);
}
\foreach \x in {7,...,8}
{
\pgfmathtruncatemacro\y{\x-9+2};
\draw [line width=0.04cm,blue!50] (A\x) -- (A\y);
}
\foreach \x in {6,...,8}
{
\pgfmathtruncatemacro\y{\x-9+3};
\draw [line width=0.04cm,black!50] (A\x) -- (A\y);
}
\foreach \x in {5,...,8}
{
\pgfmathtruncatemacro\y{\x-9+4};
\draw [line width=0.04cm,red!50] (A\x) -- (A\y);
}
\draw [line width=0.1cm,red] (A0) -- (A5);
\draw [line width=0.1cm,darkgreen] (A4) -- (A5);
\draw [line width=0.1cm,blue] (A3) -- (A5);
\draw [line width=0.1cm,black] (A0) -- (A6);
\foreach \x in {0,...,8}
{
\draw [fill=black] (A\x) circle [radius=0.1cm];
\draw ($1.2*(A\x)$) node {\x};
}
\epsilonilonnd{tikzpicture}
\caption{The ND-colouring of $K_9$ and a rainbow copy of a tree $T$ with four edges. The colour of each edge corresponds to its Euclidean length. By taking cyclic shifts of this tree around the centre of the picture we obtain $9$ disjoint copies of the tree decomposing $K_9$ (and thus a proof of Ringel's Conjecture for this particular tree). To see that this gives 9 disjoint trees, notice that edges must be shifted to other edges of the same colour (since shifts are isometries).}\label{FigureIntro}
\epsilonilonnd{figure}
In \cite{MPS} we gave a new approach to embedding large trees (with no degree restrictions) into edge-colourings of complete graphs, and used this to prove Conjecture
\ref{ringelconj} asymptotically. Here, we further develop and refine this approach, combining it with several critical new ideas to prove Ringel's conjecture for large complete graphs.
\begin{theorem}\label{main}
For every sufficiently large $n$ the complete graph $K_{2n+1}$ can be decomposed into copies of any tree with $n$ edges.
\epsilonilonnd{theorem}
The proof of Theorem \ref{main} uses the last of the three approaches mentioned above. Instead of working directly with tree decompositions, or studying graceful labellings, we instead prove for large $n$ that every ND-coloured $K_{2n+1}$ contains a rainbow copy of every $n$-edge tree (see Theorem~\ref{Theorem_Ringel_proof}).
Then, we obtain a decomposition of the complete graph by considering cyclic shifts of one copy of a given tree (as in Figure~\ref{FigureIntro}). The existence of such a cyclic decomposition was separately conjectured by Kotzig~\cite{rosa1966certain}. Therefore, this also gives a proof of the conjecture by Kotzig for large $n$.
Our proof approach builds on ideas from the previous research on both graph decompositions and graceful labellings.
From the work on graph decompositions, our approach is inspired by randomized decompositions and the absorption technique. The rough idea of absorption is as follows. Before the embedding of $T$ we prepare a template which has some useful properties. Next we find a partial embedding of the tree $T$ with some vertices removed such that we did not use the edges of the template.
Finally we use the template to embed the remaining vertices. This idea was introduced as a general method by R\"odl, Ruci\'nski and Szemer\'edi \cite{RRS} and has been used extensively since then. For example, the proof of Ringel's Conjecture for bounded degree trees is based on this technique~\cite{joos2016optimal}.
We are also inspired by graceful labellings. When dealing with trees with very high degree vertices, we use a completely deterministic approach for finding a rainbow copy of the tree. This approach heavily relies on features of the ND-colouring and produces something very close to a graceful labelling of the tree.
Our theorem is the first general result giving a perfect decomposition of a graph into subgraphs with arbitrary degrees. As we mentioned, all previous comparable results placed a bound on the maximum degree of the subgraphs into which they decomposed the complete graph. Therefore, we hope that further development of our techniques can help overcome this ``bounded degree barrier'' in other problems as well.
| 1,852 | 88,680 |
en
|
train
|
0.54.3
|
\section{Proof outline}\label{Section_proof_outline}
From the discussion in the introduction, to prove Theorem~\ref{main} it is sufficient to prove the following result.
\begin{theorem}\label{Theorem_Ringel_proof}
For sufficiently large $n$, every ND-coloured $K_{2n+1}$ has a rainbow copy of every $n$-edge tree.
\epsilonilonnd{theorem}
That is, for large $n$, and each $(n+1)$-vertex tree $T$, we seek a rainbow copy of $T$ in the ND-colouring of the complete graph with $2n+1$ vertices, $K_{2n+1}$. Our approach varies according to which of 3 cases the tree $T$ belongs to. For some small $\delta>0$, we show that, for every large $n$, every $(n+1)$-vertex tree falls in one of the following 3 cases (see Lemma~\ref{Lemma_case_division}), where a bare path is one whose internal vertices have degree 2 in the parent tree.
\begin{enumerate}[label = \Alph{enumi}]
\item $T$ has at least $\delta^6 n$ non-neighbouring leaves.
\item $T$ has at least $\delta n/800$ vertex-disjoint bare paths with length $\delta^{-1}$.
\item Removing leaves next to vertices adjacent to at least $\delta^{-4}$ leaves gives a tree with at most
$n/100$ vertices.
\epsilonilonnd{enumerate}
As defined above, our cases are not mutually disjoint. In practice, we will only use our embeddings for trees in Case A and B which are not in Case C. In~\cite{MPS}, we developed methods to embed any $(1-\epsilonilonps)n$-vertex tree in a rainbow fashion into any 2-factorized $K_{2n+1}$, where $n$ is sufficiently large depending on $\epsilonilonps$. A colouring is a \epsilonilonmph{2-factorization} if every vertex is adjacent to exactly 2 edges of each colour. In this paper, we embed any $(n+1)$-vertex tree $T$ in a rainbow fashion into a specific 2-factorized colouring of $K_{2n+1}$, the ND-colouring, when $n$ is large. To do this, we introduce three key new methods, as follows.
\begin{enumerate}[label = {\bfseries M\arabic{enumi}}]
\item We use our results from \cite{montgomery2018decompositions} to suitably randomize the results of \cite{MPS}. This allows us to randomly embed a $(1-\epsilonilonps)n$-vertex tree into any 2-factorized $K_{2n+1}$, so that the image is rainbow and has certain random properties. These properties allow us to apply a case-appropriate \epsilonilonmph{finishing lemma} with the uncovered colours and vertices.\label{T2}
\item We use a new implementation of absorption to embed a small part of $T$ while using some vertices in a random subset of $V(K_{2n+1})$ and exactly the colours in a random subset of $C(K_{2n+1})$. This uses different \epsilonilonmph{absorption structures} for trees in Case A and in Case B, and in each case gives the finishing lemma for that case.\label{T1}
\item We use an entirely new, deterministic, embedding for trees in Case C.\label{T3}
\epsilonilonnd{enumerate}
For trees in Cases A and B, we start by finding a random rainbow copy of most of the tree using~\ref{T2}, as outlined in Section~\ref{sec:T2}. Then, we embed the rest of the tree using uncovered vertices and exactly the unused colours using~\ref{T1}, which gives a finishing lemma for each case. These finishing lemmas are discussed in Section~\ref{sec:T1}. We use \ref{T3} to embed trees in Case C, which is essentially independent of our embeddings of trees in Cases A and B. This method is outlined in Section \ref{sec:T3}. In Section~\ref{sec:overhead}, we state our main lemmas and theorems, and prove Theorem~\ref{Theorem_Ringel_proof} subject to these results.
The rest of the paper is structured as follows. Following details of our notation, in Section~\ref{sec:prelim} we recall and prove various preliminary results. We then prove the finishing lemma for Case A in Section~\ref{sec:finishA} and the finishing lemma for Case B in Section~\ref{sec:finishB} (together giving \ref{T1}). In Section~\ref{sec:almost}, we give our randomized rainbow embedding of most of the tree (\ref{T2}). In Section~\ref{sec:lastC}, we embed the trees in Case C with a deterministic embedding (\ref{T3}). Finally, in Section~\ref{sec:conc}, we make some concluding remarks.
\subsection{\ref{T2}: Embedding almost all of the tree randomly in Cases A and B}\label{sec:T2}
For a tree $T$ in Case A or B, we carefully choose a large subforest, $T'$ say, of $T$, which contains almost all the edges of $T$. We find a rainbow copy $\hat{T}'$ of $T'$ in the ND-colouring of $K_{2n+1}$ (which exists due to~\cite{MPS}), before applying a finishing lemma to extend $\hat{T}'$ to a rainbow copy of $T$. Extending to a rainbow copy of $T$ is a delicate business --- we must use exactly the $n-e(\hat{T})$ unused colours. Not every rainbow copy of $T'$ will be extendable to a rainbow copy of $T$. However, by combining our methods in \cite{montgomery2018decompositions} and \cite{MPS}, we can take a random rainbow copy $\hat{T}'$ of $T'$ and show that it is likely to be extendable to a rainbow copy of $T$. Therefore, some rainbow copy of $T$ must exist in the ND-colouring of $K_{2n+1}$.
As $\hat{T}'$ is random, the sets $\bar{V}:=V(K_{2n+1})\setminus V(\hat{T})$ and $\bar{C}:=C(K_{2n+1})\setminus C(\hat{T})$ will also be random. The distributions of $\bar{V}$ and $\bar{C}$ will be complicated, but we will not need to know them. It will suffice that there will be large (random) subsets $V\subseteq \bar{V}$ and $C\subseteq \bar{C}$ which each do have a nice, known, distribution.
Here, for example, $V\subseteq V(K_{2n+1})$ has a nice distribution if there is some $q$ so that each element of $V(K_{2n+1})$ appears independently at random in $V$ with probability $q$ --- we say here that $V$ is \epsilonilonmph{$q$-random} if so, and analogously we define a $q$-random subset $C\subseteq C(K_{2n+1})$ (see Section~\ref{sec:prob}). A natural combination of the techniques in~\cite{montgomery2018decompositions,MPS} gives the following.
\begin{theorem}\label{Sketch_Near_Embedding} For each $\epsilonilonpsilon>0$, the following holds for sufficiently large $n$.
Let $K_{2n+1}$ be $2$-factorized and let $T'$ be a forest on $(1-\epsilonilonpsilon)n$ vertices. Then, there is a randomized subgraph $\hat{T}'$ of $K_{2n+1}$ and random subsets $V\subseteq V(K_{2n+1})\setminus V(\hat{T}')$ and $C\subseteq C(K_{2n+1})\setminus C(\hat{T}')$ such that the following hold for some $p:=p(T')$ (defined precisely in Theorem~\ref{nearembedagain}).
\stepcounter{propcounter}
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]
\item $\hat{T}'$ is a rainbow copy of $T'$ with high probability.\label{woo1}
\item $V$ is $(p+\epsilonilonps)/6$-random and $C$ is $(1-\epsilonilonpsilon)\epsilonilonps$-random. ($V$ and $C$ may depend on each other.)\label{woo2}
\epsilonilonnd{enumerate}
\epsilonilonnd{theorem}
We will apply a variant of Theorem~\ref{Sketch_Near_Embedding} (see Theorem~\ref{nearembedagain}) to subforests of trees in Cases A and B, and there we will have that $p\gg \epsilonilonps$. Note that, then, $C$ will likely be much smaller than $V$. This reflects that $\hat{T}'\subseteq K_{2n+1}$ will contain fewer than $n$ out of $2n+1$ vertices, while $C(K_{2n+1})\setminus C(\hat{T}')$ contains exactly $n-e(\hat{T}')$ out of $n$ colours.
As explained in~\cite{MPS}, in general the sets $V$ and $C$ cannot be independent, and this is in fact why we need to treat trees in Case C separately. In order to finish the embedding in Cases A and B, we need, essentially, to find \epsilonilonmph{some} independence between the sets $V$ and $C$ (as discussed below). The embedding is then as follows for some small $\delta$ governing the case division, with $\epsilonilonps=\delta^6$ in Case A and $\bar{\epsilonilonps}\oldgg \delta$ in Case B. Given an $(n+1)$-vertex tree $T$ in Case A or B we delete either $\epsilonilonpsilon n$ non-neighbouring leaves (Case A) or $\bar{\epsilonilonpsilon} n/k$ vertex-disjoint bare paths with length $k=\delta^{-1}$ (Case B) to obtain a forest $T'$. Using (a variant of) Theorem~\ref{Sketch_Near_Embedding}, we find a randomized rainbow copy $\hat{T}'$ of $T'$ along with some random vertex and colour sets and apply a finishing lemma to extend this to a rainbow copy of $T$.
After a quick note on the methods in~\cite{montgomery2018decompositions} and~\cite{MPS}, we will discuss the finishing lemmas, and explain why we need some independence, and how much independence is needed.
\subsubsection*{Randomly embedding nearly-spanning trees}
In~\cite{MPS}, we embedded a $(1-\epsilonilonpsilon)n$-vertex tree $T$ into a 2-factorization of $K_{2n+1}$ by breaking it down mostly into large stars and large matchings. For each of these, we embedded the star or matching using its own random set of vertices and random set of colours (which were not necessarily independent of each other). In doing so, we used almost all of the colours in the random colour set, but only slightly less than one half of the vertices in the random vertex set. (This worked as we had more than twice as many vertices in $K_{2n+1}$ than in $T$.) For trees not in Case C, a substantial portion of a large subtree was broken down into matchings. By embedding these matchings more efficiently, using results from~\cite{montgomery2018decompositions}, we can use a smaller random vertex set. This reduction allows us to have, disjointly from the embedded tree, a large random vertex subset $V$.
More precisely, where $q,\epsilonilonpsilon\oldgg n^{-1}$, using a random set $V$ of $2qn$ vertices and a random set $C$ of $qn$ colours, in~\cite{MPS} we showed that, with high probability, from any set $X\subseteq V(K_{2n+1})\setminus V$ with $|X|\leq (1-\epsilonilonpsilon)qn$, there was a $C$-rainbow matching from $X$ into $V$. Dividing $V$ randomly into two sets $V_1$ and $V_2$, each with $qn$ vertices, and using the results in~\cite{montgomery2018decompositions}, we can use $V_1$ to find the $C$-rainbow matching (see Lemma~\ref{Lemma_MPS_nearly_perfect_matching}).
Thus, we gain the random set $V_2$ of $qn$ vertices which we do not use for the embedding of $T$, and we can instead use it to extend this to an $(n+1)$-vertex tree in $K_{2n+1}$.
Roughly speaking, if in total $pn$ vertices of $T$ are embedded using matchings, then we gain altogether a random set of around $pn$ vertices.
If a tree is not in Case C, then the subtree/subforest we embed using these techniques has plenty of vertices embedded using matchings, so that in this case we will be able to take $p\geq 10^{-3}$ when we apply the full version of Theorem~\ref{Sketch_Near_Embedding} (see Theorem~\ref{nearembedagain}). Therefore, we will have many spare vertices when adding the remaining $\epsilonilonps n$ vertices to the copy of $T$. Our challenges are firstly that we need to use exactly all the colours not used on the copy of $T'$ and secondly that there can be a lot of dependence between the sets $V$ and $C$. We first discuss how we ensure that we use every colour.
| 3,240 | 88,680 |
en
|
train
|
0.54.4
|
\subsection{\ref{T1}: Finishing the embedding in Cases A and B}\label{sec:T1}
To find trees using every colour in an ND-coloured $K_{2n+1}$ we prove two \epsilonilonmph{finishing lemmas} (Lemma~\ref{lem:finishA} and~\ref{lem:finishB}). These lemmas say that, for a given randomized set of vertices $V$ and a given randomized set of colours $C$, we can find a rainbow matching/path-forest which uses exactly the colours in $C$, while using some of the vertices from $V$. These lemmas are used to finish the embedding of the trees in Cases A and B, where the last step is to (respectively) embed a matching or path-forest that we removed from the tree $T$ to get the forest $T'$. Applying (a version of) Theorem~\ref{Sketch_Near_Embedding} we get a random rainbow copy $\hat{T'}$ of $T'$ and random sets $\bar{V}=V(K_n)\setminus V(\hat{T}')$ and $\bar{C}=C(K_n)\setminus C(\hat{T}')$.
In order to apply the case-appropriate finishing lemma, we need some independence between $\bar{V}$ and $\bar{C}$, for reasons we now discuss for Case A, and then Case B. Next, we discuss the independence property we use and how we achieve this independence. (Essentially, this property is that $\bar{V}$ and $\bar{C}$ contain two small random subsets which are independent of each other.) Finally, we discuss the absorption ideas for Case A and Case B.
\subsubsection*{Finishing with matchings (for Case A)}
In Case A, we take the $(n+1)$-vertex tree $T$ and remove a large matching of leaves, $M$ say, to get a tree, $T'$ say, that can be embedded using Theorem~\ref{Sketch_Near_Embedding}. This gives a random copy, $\hat{T}'$ say, of $T'$ along with random sets $V\subseteq V(K_{2n+1})\setminus V(\hat{T}')$ and $C\subseteq C(K_{2n+1})\setminus C(\hat{T}')$
which are $(p+\epsilonilonps)/6$-random and $(1-\epsilonilonps)\epsilonilonps$-random respectively. For trees not in Case C we will have $p\oldgg \epsilonilonps$.
Let $X\subseteq V(\hat{T}')$ be the set of vertices we need to add neighbours to as leaves to make $\hat{T}'$ into a copy of $T$.
We would like to find a perfect matching from $X$ to $\bar{V}:=V(K_{2n+1})\setminus V(\hat{T}')$ with exactly the colours in $\bar{C}:=C(K_{2n+1})\setminus C(\hat{T}')$, using that $V\subseteq \bar{V}$ and $C\subseteq \bar{C}$. (A perfect matching from $X$ to $\bar{V}$ is such a matching covering every vertex in $X$.)
Unfortunately, there may be some $x\in X$ with no edges with colour in $\bar{C}$ leading to $\bar{V}$. (If $C$ and $V$ were independent, then this would not happen with high probability.) If this happens, then the desired matching will not exist.
In Case B, a very similar situation to this may occur, as discussed below, but in Case A there is another potential problem. There may be some colour $c\in \bar{C}$ which does not appear between $X$ and $\bar{V}$, again preventing the desired matching existing. This we will avoid by carefully embedding a small part of $T'$ so that every colour appears between $X$ and $\bar{V}$ on plenty of edges.
\subsubsection*{Finishing with paths (for Case B)}
In Case B, we take the $(n+1)$-vertex tree $T$ and remove a set of vertex-disjoint bare paths to get a forest, $T'$ say, that can be embedded using Theorem~\ref{Sketch_Near_Embedding}. This gives a random copy, $\hat{T}'$ say, of $T'$ along with random sets $V\subseteq V(K_{2n+1})\setminus V(\hat{T}')$ and $C\subseteq C(K_{2n+1})\setminus C(\hat{T}')$
which are $(p+\epsilonilonps)/6$-random and $(1-\epsilonilonps)\epsilonilonps$-random respectively. For trees not in Case C we will have $p\oldgg \epsilonilonps$.
Let $\epsilonilonll$ and $X=\{x_1,\ldots,x_\epsilonilonll,y_1,\ldots,y_\epsilonilonll\}\subseteq V(\hat{T}')$ be such that to get a copy of $T$ from $\hat{T}'$ we need to add vertex-disjointly a suitable path between $x_i$ and $y_i$, for each $i\in [\epsilonilonll]$.
We would like to find these paths with interior vertices in $\bar{V}:= V(K_{2n+1})\setminus V(\hat{T}')$ so that their edges are collectively rainbow with exactly the colours in $\bar{C}:=C(K_{2n+1})\setminus C(\hat{T}')$, using that $V\subseteq \bar{V}$ $C\subseteq \bar{C}$. Unfortunately, there may be some $x\in X$ with no edges with colour in $\bar{C}$ leading to $V$. (If $C$ and $V$ were independent, then, again, this would not happen with high probability.) If this happens, then the desired paths will not exist.
Note that the analogous version of the second problem in Case A does not arise in Case B. Here, it is likely that every colour appears on many edges within $V$, so that we can use any colour by putting an appropriate edge within $V$ in the middle of one of the missing paths.
\subsubsection*{Retaining some independence}
To avoid the problem common to Cases A and B, when proving our version of Theorem~\ref{Sketch_Near_Embedding} (that is, Theorem~\ref{nearembedagain}), we set aside small random sets $V_0$ and $C_0$ early in the embedding, before the dependence between colours and vertices arises. This gives us a version of Theorem~\ref{Sketch_Near_Embedding} with the additional property that, for some $\mu\oldll \epsilonilonps$, there are additional random sets $V_0\subseteq V(K_{2n+1})\setminus (V(\hat{T}')\cup V)$ and $C_0\subseteq C(K_{2n+1})\setminus (C(\hat{T}')\cup C)$ such that the following holds in addition to \ref{woo1} and \ref{woo2}.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]\addtocounter{enumi}{2}
\item $V_0$ is a $\mu$-random subset of $V(K_{2n+1})$, $C_0$ is a $\mu$-random subset of $C(K_{2n+1})$, and they are independent of each other.\label{propq2}
\epsilonilonnd{enumerate}
\noindent
Then, by this independence, with high probability, every vertex in $K_{2n+1}$ will have $\mu^2 n/2$ adjacent edges with colour in $C_0$ going into the set $V_0$ (see Lemma~\ref{Lemma_high_degree_into_random_set}).
To avoid the problem that only arises in Case A, consider the set $U\subseteq V(T')$ of vertices which need leaves added to them to reach $T$ from $T'$. By carefully embedding a small subtree of $T'$ containing plenty of vertices in $U$, we ensure that, with high probability, each colour appears plenty of times between the image of $U$ and $V_0$. That is, we have the following additional property for some $1/n\oldll \xi\oldll \mu$.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]\addtocounter{enumi}{3}
\item With high probability, if $Z$ is the copy of $U$ in $\hat{T}'$, then every colour in $C(K_{2n+1})$, has at least $\xi n$ edges between $Z$ and $V_0$.\label{propq1}
\epsilonilonnd{enumerate}
Of course, \ref{propq2} and \ref{propq1} do not show that our desired matching/path-collection exists, only that (with high probability) there is no single colour or vertex preventing its existence. To move from this to find the actual matching/path-collection we use \epsilonilonmph{distributive absorption}.
\subsubsection*{Distributive absorption}
To prove our finishing lemmas, we use an \epsilonilonmph{absorption strategy}. Absorption has its origins in work by Erd\H{o}s, Gy\'arf\'as and Pyber~\cite{EP} and Krivelevich~\cite{MKtri}, but was codified by R\"odl, Ruci\'nski and Szemer\'edi~\cite{RRS} as a versatile technique
for extending approximate results into exact ones. For both Case A and Case B we use a new implementation of \epsilonilonmph{distributive absorption}, a method introduced by the first author in~\cite{montgomery2018spanning}.
To describe our absorption, let us concentrate on Case A. Our methods in Case B are closely related, and we comment on these afterwards. To recap, we have a random rainbow tree $\hat{T}'$ in the ND-colouring of $K_{2n+1}$ and a set $X\subseteq V(\hat{T})$, so that we need to add a perfect matching from $X$ into $\bar{V}=V(K_{2n+1})\setminus V(\hat{T}')$ to make $\hat{T}'$ into a copy of $T$. We wish to add this matching in a rainbow fashion using (exactly) the colours in $\bar{C}= C(K_{2n+1})\setminus C(\hat{T})$.
To use distributive absorption, we first show that for any set $\hat{C}\subseteq C(K_{2n+1})$ of at most 100 colours, we can find a set $D\subseteq \bar{C}\setminus C$ and sets $X'\subseteq X$ and $V'\subseteq \bar{V}$ with $|D|\leq 10^3$, $|V'|\leq 10^4$ and $|X'|=|D|+1$, so that the following holds.
\stepcounter{propcounter}
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]
\item Given any colour $c\in\hat{C}$, there is a perfect $(D\cup \{c\})$-rainbow matching from $X'$ to $V'$.\label{switchprop}
\epsilonilonnd{enumerate}
We call such a triple $(D,X',V')$ a \epsilonilonmph{switcher} for $\hat{C}$. AS $|\bar{C}|=|X|$, a perfect $(\bar{C}\setminus D)$-rainbow matching from $X\setminus X'$ into $V\setminus V'$ uses all but 1 colour in $\bar{C}\setminus D$. If we can find such a matching whose unused colour, $c$ say, lies in $\hat{C}$, then using \ref{switchprop}, we can find a perfect $(D\cup\{c\})$-rainbow matching from $X'$ to $V'$. Then, the two matchings combined form a perfect $\bar{C}$-rainbow matching from $X$ into $V$, as required.
The switcher outlined above only gives us a tiny local variability property, reducing finding a large perfect matching with exactly the right number of colours to finding a large perfect matching with one spare colour so that the unused colour lies in a small set (the set $\bar{C}$). However, by finding many switchers for carefully chosen sets $\hat{C}$, we can build this into a global variability property. These switchers can be found using different vertices and colours (see Section~\ref{sec:switcherspaths}), so that matchings found using the respective properties \ref{switchprop} can be combined in our embedding.
We choose different sets $\hat{C}$ for which to find a switcher by using an auxillary graph as a template. This template is a \epsilonilonmph{robustly matchable bipartite graph} --- a bipartite graph, $K$ say, with vertex classes $U$ and $Y\cup Z$ (where $Y$ and $Z$ are disjoint), with the following property.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]\addtocounter{enumi}{1}
\item For any set $Z^\ast\subseteq Z$ with size $|U|-|Y|$, there is a perfect matching in $K$ between $U$ and $Y\cup Z^\ast$.
\epsilonilonnd{enumerate}
Such bipartite graphs were shown to exist by the first author~\cite{montgomery2018spanning}, and, furthermore, for large $m$ and $\epsilonilonll\leq m$, we can find such a graph with maximum degree at most 100, $|U|=3m$, $|Y|=2m$ and $|Z|=m+\epsilonilonll$ (see Lemma~\ref{Lemma_H_graph}).
To use the template, we take disjoint sets of colours, $C'=\{c_v:v\in Y\}$ and $C''=\{c_v:v\in Z\}$ in $\bar{C}$. For each $u\in U$, we find a switcher $(D_u,X_u,V_u)$ for the set of colours $\{c_v:v\in N_K(u)\}$. Furthermore, we do this so that the sets $D_u$ are disjoint and in $\bar{C}\setminus (C'\cup C'')$, and the sets $X_u$, and $V_u$, are disjoint and in $X$, and $\bar{V}$, respectively. We can then show we have the following property.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]\addtocounter{enumi}{2}
\item For any set $C^*\subseteq C''$ of $m$ colours, there is a perfect $(C^*\cup C'\cup(\cup_{u\in U}D_u))$-rainbow matching from $\cup_{u\in U}X_u$ into $\cup_{u\in U}V_u$.\label{Kprop}
\epsilonilonnd{enumerate}
Indeed, to see this, take any set of $C^*\subseteq C''$ of $m$ colours, let $Z^*=\{v:c_v\in C^*\}$ and note that $|Z^*|=m$. By \ref{Kprop}, there is a perfect matching in $K$ from $U$ into $Y\cup Z^\ast$, corresponding to the function $f:U\to Y\cup Z^*$ say. For each $u\in U$, using that $(D_u,X_u,V_u)$ is a switcher for $\{c_v:v\in N_K(u)\}$ and $uf(u)\in E(K)$, find a perfect $(D_u\cup \{c_{f(u)}\})$-rainbow matching $M_u$ from $X_u$ to $V_u$.
As the sets $D_u$, $X_u$, $V_u$, $u\in U$, are disjoint, $\cup_{u\in U}M_u$ is a perfect $(C^*\cup C'\cup(\cup_{u\in U}D_u))$-rainbow matching from $\cup_{u\in U}X_u$ into $\cup_{u\in U}V_u$, as required.
Thus, we have a set of colours $C''$ from which we are free to use any $\epsilonilonll$ colours, and then use the remaining colours together with $C'\cup(\cup_{u\in U}D_u)$ to find a perfect rainbow matching from $\cup_{u\in U}X_u$ into $\cup_{u\in U}V_u$. By letting $m$ be as large as allowed by our construction methods, and $C''$ be a random set of colours, we have a useful \epsilonilonmph{reservoir} of colours, so that we can find a structure in $T$ using $\epsilonilonll$ colours in $C''$, and then finish by attaching a matching to $\cup_{u\in U}X_u$.
| 4,003 | 88,680 |
en
|
train
|
0.54.5
|
We call such a triple $(D,X',V')$ a \epsilonilonmph{switcher} for $\hat{C}$. AS $|\bar{C}|=|X|$, a perfect $(\bar{C}\setminus D)$-rainbow matching from $X\setminus X'$ into $V\setminus V'$ uses all but 1 colour in $\bar{C}\setminus D$. If we can find such a matching whose unused colour, $c$ say, lies in $\hat{C}$, then using \ref{switchprop}, we can find a perfect $(D\cup\{c\})$-rainbow matching from $X'$ to $V'$. Then, the two matchings combined form a perfect $\bar{C}$-rainbow matching from $X$ into $V$, as required.
The switcher outlined above only gives us a tiny local variability property, reducing finding a large perfect matching with exactly the right number of colours to finding a large perfect matching with one spare colour so that the unused colour lies in a small set (the set $\bar{C}$). However, by finding many switchers for carefully chosen sets $\hat{C}$, we can build this into a global variability property. These switchers can be found using different vertices and colours (see Section~\ref{sec:switcherspaths}), so that matchings found using the respective properties \ref{switchprop} can be combined in our embedding.
We choose different sets $\hat{C}$ for which to find a switcher by using an auxillary graph as a template. This template is a \epsilonilonmph{robustly matchable bipartite graph} --- a bipartite graph, $K$ say, with vertex classes $U$ and $Y\cup Z$ (where $Y$ and $Z$ are disjoint), with the following property.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]\addtocounter{enumi}{1}
\item For any set $Z^\ast\subseteq Z$ with size $|U|-|Y|$, there is a perfect matching in $K$ between $U$ and $Y\cup Z^\ast$.
\epsilonilonnd{enumerate}
Such bipartite graphs were shown to exist by the first author~\cite{montgomery2018spanning}, and, furthermore, for large $m$ and $\epsilonilonll\leq m$, we can find such a graph with maximum degree at most 100, $|U|=3m$, $|Y|=2m$ and $|Z|=m+\epsilonilonll$ (see Lemma~\ref{Lemma_H_graph}).
To use the template, we take disjoint sets of colours, $C'=\{c_v:v\in Y\}$ and $C''=\{c_v:v\in Z\}$ in $\bar{C}$. For each $u\in U$, we find a switcher $(D_u,X_u,V_u)$ for the set of colours $\{c_v:v\in N_K(u)\}$. Furthermore, we do this so that the sets $D_u$ are disjoint and in $\bar{C}\setminus (C'\cup C'')$, and the sets $X_u$, and $V_u$, are disjoint and in $X$, and $\bar{V}$, respectively. We can then show we have the following property.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]\addtocounter{enumi}{2}
\item For any set $C^*\subseteq C''$ of $m$ colours, there is a perfect $(C^*\cup C'\cup(\cup_{u\in U}D_u))$-rainbow matching from $\cup_{u\in U}X_u$ into $\cup_{u\in U}V_u$.\label{Kprop}
\epsilonilonnd{enumerate}
Indeed, to see this, take any set of $C^*\subseteq C''$ of $m$ colours, let $Z^*=\{v:c_v\in C^*\}$ and note that $|Z^*|=m$. By \ref{Kprop}, there is a perfect matching in $K$ from $U$ into $Y\cup Z^\ast$, corresponding to the function $f:U\to Y\cup Z^*$ say. For each $u\in U$, using that $(D_u,X_u,V_u)$ is a switcher for $\{c_v:v\in N_K(u)\}$ and $uf(u)\in E(K)$, find a perfect $(D_u\cup \{c_{f(u)}\})$-rainbow matching $M_u$ from $X_u$ to $V_u$.
As the sets $D_u$, $X_u$, $V_u$, $u\in U$, are disjoint, $\cup_{u\in U}M_u$ is a perfect $(C^*\cup C'\cup(\cup_{u\in U}D_u))$-rainbow matching from $\cup_{u\in U}X_u$ into $\cup_{u\in U}V_u$, as required.
Thus, we have a set of colours $C''$ from which we are free to use any $\epsilonilonll$ colours, and then use the remaining colours together with $C'\cup(\cup_{u\in U}D_u)$ to find a perfect rainbow matching from $\cup_{u\in U}X_u$ into $\cup_{u\in U}V_u$. By letting $m$ be as large as allowed by our construction methods, and $C''$ be a random set of colours, we have a useful \epsilonilonmph{reservoir} of colours, so that we can find a structure in $T$ using $\epsilonilonll$ colours in $C''$, and then finish by attaching a matching to $\cup_{u\in U}X_u$.
We have two things to consider to fit this final step into our proof structure, which we discuss below. Firstly, we can only absorb colours in $C''$, so after we have covered most of the colours, we need to cover the unused colours outside of $C''$ (essentially achieved by \ref{propp2} below). Secondly, we find the switchers greedily in a random set. There are many more unused colours from this set than we can absorb, and the unused colours no longer have good random properties, so we also need to reduce the unused colours to a number that we can absorb (essentially achieved by \ref{propp1} below).
\subsubsection*{Creating our finishing lemmas using absorption}
To recap, we wish to find a perfect $\bar{C}$-rainbow matching from $X$ into $\bar{V}$. To do this, it is sufficient to find partitions $X=X_1\cup X_2\cup X_3$, $\bar{V}=V_1\cup V_2\cup V_3$ and $\bar{C}=C_1\cup C_2\cup C_3$ with the following properties.
\stepcounter{propcounter}
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]
\item There is a perfect $C_1$-rainbow matching from $X_1$ into $V_1$. \label{propp1}
\item Given any set of colours $C'\subseteq C_1$ with $|C'|\leq |C_1|-|X_1|$, there is a perfect $(C_2\cup C')$-rainbow matching from $X_2$ into $V_2$ which uses each colour in $C'$.\label{propp2}
\item Given any set of colours $C''\subseteq C_2$ with size $|X_3|-|C_3|$, there is a perfect $(C''\cup C_3)$-rainbow matching from $X_3$ into $V_3$.\label{propp3}
\epsilonilonnd{enumerate}
Finding such a partition requires the combination of all our methods for Case A. In brief, however, we develop \ref{propp1} using a result from~\cite{montgomery2018decompositions} (see Lemma~\ref{Lemma_MPS_nearly_perfect_matching}), we develop \ref{propp2} using the condition \ref{propq1}, and we develop \ref{propp3} using the distributive absorption strategy outlined above.
If we can find such a partition, then we can easily show that the matching we want must exist. Indeed, given such a partition, then, using \ref{propp1}, let $M_1$ be a perfect $C_1$-rainbow matching from $X_1$ into $V_1$, and let $C'=C_1\setminus C(M_1)$. Using~\ref{propp2}, let $M_2$ be a perfect $(C_2\cup C')$-rainbow matching from $X_2$ into $V_2$ which uses each colour in $C'$, and let $C''=(C_2\cup C')\setminus C(M)=C_2\setminus C(M)$. Finally, noting that $|C''|+|C_3|=|\bar{C}|-|X_1|-|X_2|=|X_3|$, using~\ref{propp3}, let $M_3$ be a perfect $(C''\cup C_3)$-rainbow matching from $X_3$ into $V_3$. Then, $M_1\cup M_2\cup M_3$ is a $\bar{C}$-rainbow matching from $X$ into $\bar{V}$.
The above outline also lies behind our embedding in Case B, where we finish instead by embedding $\epsilonilonll$ paths vertex-disjointly between certain vertex pairs, for some $\epsilonilonll$. Instead of the partition $X_1\cup X_2\cup X_3$ we have a partition $[\epsilonilonll]=I_1\cup I_2\cup I_3$, and, instead of each matching from $X_i$ to $V_i$, $i\in [3]$, we find a set of vertex-disjoint $x_j,y_j$-paths, $j\in I_i$, with interior vertices in $V_i$ which are collectively $C_i$-rainbow. The main difference is how we construct switchers using paths instead of matchings (see Section~\ref{sec:switchers}).
| 2,418 | 88,680 |
en
|
train
|
0.54.6
|
\subsection{\ref{T3}: The embedding in Case C}\label{sec:T3}
After large clusters of adjacent leaves are removed from a tree in Case C, few vertices remain. We remove these large clusters, from the tree, $T$ say, to get the tree $T'$, and carefully embed $T'$ into the ND-colouring using a deterministic embedding. The image of this deterministic embedding occupies a small interval in the ordering used to create the ND-colouring. Furthermore, the embedded vertices of $T'$ which need leaves added to create a copy of $T$ are well-distributed within this interval. These properties will allow us to embed the missing leaves using the remaining colours. This is given more precisely in Section~\ref{sec:caseC}, but in order to illustrate this in the easiest case, we will give the embedding when there is exactly one vertex with high degree.
Our embedding in this case is rather simple. Removing the leaves incident to a very high degree vertex, we embed the rest of the tree into $[n]$ so that the high degree vertex is embedded to 1. The missing leaves are then embedded into $[2n+1]\setminus [n]$ using the unused colours.
\begin{theorem}[One large vertex]\label{Theorem_one_large_vertex}
Let $n\geq 10^6$. Let $K_{2n+1}$ be $ND$-coloured, and let $T$ be an $(n+1)$-vertex tree containing a vertex $v_1$ which is adjacent to $\geq 2n/3$ leaves.
Then, $K_{2n+1}$ contains a rainbow copy of $T$.
\epsilonilonnd{theorem}
\begin{proof}
See Figure~\ref{Figure_1_vertex} for an illustration of this proof.
Let $T'$ be $T$ with the neighbours of $v_1$ removed and let $m=|T'|$. By assumption, $|T'|\leq n/3+1$. Order the vertices of $T'-v_1$ as $v_2, \dots, v_{m}$ so that $T[v_1, \dots, v_i]$ is a tree for each $i\in [m]$. Embed $v_1$ to $1$ in $K_{2n+1}$, and then greedily embed $v_2, \dots, v_m$ in turn to some vertex in $[n]$ so that the copy
of $T'$ which is formed is rainbow in $K_{2n+1}$. This is possible since at each step at most $|T'|\leq n/3$ of the vertices in $[n]$ are occupied, and at most $e(T')\leq n/3-1$ colours are used. Since the $ND$-colouring has 2 edges of each colour adjacent to each vertex,
this forbids at most $n/3+2(n/3-1)=n-2$ vertices in $[n]$. Thus, we can embed each $v_{i}$, $2\leq i\leq m$ using an unoccupied vertex in $[n]$ so that the edge from $v_i$ to $v_1, \ldots, v_{i-1}$ has a colour that we have not yet used. Let $S'$ be the resulting rainbow copy of $T'$, so that $V(S')\subseteq [n]$.
Let $S$ be $S'$ together with the edges between $1$ and $2n+2-c$ for every $c\in [n]\setminus C(S')$. Note that the neighbours added are all bigger than $n$, and so the resulting graph is a tree. There are exactly $n-e(T')$ edges added, so $S$ is a copy of $T$. Finally, for each $c\in [n]\setminus C(S')$, the edge from $1$ to $2n+2-c$ is colour $c$, so the resulting tree is rainbow.
\epsilonilonnd{proof}
\begin{figure}[h]
\begin{center}
{
\input{caseConevertex}
}
\epsilonilonnd{center}
\caption{Embedding the tree in Case C when there is 1 vertex with many leaves as neighbours.}\label{Figure_1_vertex}
\epsilonilonnd{figure}
The above proof demonstrates the main ideas of our strategy for Case C. Notice that the above proof has two parts --- first we embed the small tree $T'$, and then we find the neighbours of the high degree vertex $v_1$. In order to ensure that the final tree is rainbow we choose the neighbours of $v_1$ in some interval $[n+1, 2n]$ which is disjoint from the copy of $T'$, and to which every colour appears from the image of $v_1$. This way, we were able to use every colour which was not present on the copy of $T'$.
When there are multiple high degree vertices $v_1, \dots, v_{\epsilonilonll}$ the strategy is the same --- first we embed a small rainbow tree $T'$ containing $v_1, \dots, v_{\epsilonilonll}$, then we embed the neighbours of $v_1, \dots, v_{\epsilonilonll}$. This is done in Section~\ref{sec:caseC}.
| 1,211 | 88,680 |
en
|
train
|
0.54.7
|
\subsection{Proof of Theorem~\ref{Theorem_Ringel_proof}}\label{sec:overhead}
Here we will state our main theorems and lemmas, which are proved in later sections, and combine them to prove Theorem~\ref{Theorem_Ringel_proof}. First, we have our randomized embedding of a $(1-\epsilonilonpsilon)n$-vertex tree, which is proved in Section~\ref{sec:almost}. For convenience we use the following definition.
\begin{definition}
Given a vertex set $V\subseteq V(G)$ of a graph $G$, we say $V$ is \epsilonilonmph{$\epsilonilonll$-replete in $G$} if $G[V]$ contains at least $\epsilonilonll$ edges of every colour in $G$. Given, further, $W\subseteq V(G)\setminus V$, we say $(W,V)$ is \epsilonilonmph{$\epsilonilonll$-replete in $G$} if at least $\epsilonilonll$ edges of every colour in $G$ appear in $G$ between $W$ and $V$. When $G=K_{2n+1}$, we simply say that $V$ and $(W,V)$ are \epsilonilonmph{$\epsilonilonll$-replete}.
\epsilonilonnd{definition}
\begin{theorem}[Randomised tree embeddings]\label{nearembedagain} Let $1/n\ll \xi\ll \mu\ll \epsilonilonta\ll \epsilonilonps\ll 1$ and $\xi\ll 1/k\ll \log^{-1}n$.
Let $K_{2n+1}$ be ND-coloured, let $T'$ be a $(1-\epsilonilonpsilon)n$-vertex forest and let $U\subseteq V(T')$ contain $\epsilonilonps n$ vertices. Let $p$ be such that removing leaves around vertices next to $\geq k$ leaves from $T'$ gives a forest with $pn$ vertices.
Then, there is a random subgraph $\hat{T}'\subseteq K_{2n+1}$ and disjoint random subsets $V,V_0\subseteq V(K_{2n+1})\setminus V(\hat{T}')$ and $C,C_0\subseteq C(K_{2n+1})\setminus C(\hat{T}')$ such that the following hold.
\stepcounter{propcounter}
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]
\item With high probability, $\hat{T}'$ is a rainbow copy of $T'$ in which, if $W$ is the copy of $U$, then $(W,V_0)$ is $(\xi n)$-replete,\label{propa1}
\item $V_0$ and $C_0$ are $\mu$-random and independent of each other, and\label{propa2}
\item $V$ is $(p+\epsilonilonpsilon)/6$-random and $C$ is $(1-\epsilonilonta)\epsilonilonpsilon$-random.\label{propa3}
\epsilonilonnd{enumerate}
\epsilonilonnd{theorem}
Next, we have the two finishing lemmas, which are proved in Sections~\ref{sec:finishA} and~\ref{sec:finishB} respectively.
\begin{lemma}[The finishing lemma for Case A]\label{lem:finishA} Let $1/n\ll \xi\ll \mu \ll \epsilonilonta \ll\epsilonilonps\ll p\leq 1$. Let $K_{2n+1}$ be 2-factorized. Suppose that $V,V_0$ are disjoint subsets of $V(K_{2n+1})$ which are $p$- and $\mu$-random respectively. Suppose that $C,C_0$ are disjoint subsets in $C(K_{2n+1})$, so that $C$ is $(1-\epsilonilonta)\epsilonilonps$-random, and
$C_0$ is $\mu$-random and independent of $V_0$. Then, with high probability, the following holds.
Given any disjoint sets $X,Z\subseteq V(K_{2n+1})\setminus (V\cup V_0)$ with $|X|=\epsilonilonps n$, so that $(X,Z)$ is $(\xi n)$-replete, and any set $D\subseteq C(K_{2n+1})$ with $|D|=\epsilonilonps n$ and $C_0\cup C\subseteq D$, there is a perfect $D$-rainbow matching from $X$ into $V\cup V_0\cup Z$.
\epsilonilonnd{lemma}
Note that in the following lemma we implicitly assume that $m$ is an integer. That is, we assume an extra condition on $n$, $k$ and $\epsilonilonps$. We remark on this further in Section~\ref{sec:not}.
\begin{lemma}[The finishing lemma for Case B]\label{lem:finishB} Let $1/n\ll 1/k\ll \mu\ll \epsilonilonta \ll\epsilonilonps\ll p\leq 1$ be such that $k = 7\mod 12$ and $695|k$. Let $K_{2n+1}$ be 2-factorized. Suppose that $V,V_0$ are disjoint subsets of $V(K_{2n+1})$ which are $p$- and $\mu$-random respectively. Suppose that $C,C_0$ are disjoint subsets in $C(K_{2n+1})$, so that $C$ is $(1-\epsilonilonta)\epsilonilonps$-random, and $C_0$ is $\mu$-random and independent of $V_0$. Then, with high probability, the following holds with $m=\epsilonilonps n/k$.
For any set $\{x_1,\ldots,x_m,y_1,\ldots,y_m\}\subseteq V(K_{2n+1})\setminus (V\cup V_0)$, and any set $D\subseteq C(K_{2n+1})$ with $|D|=mk$ and $C\cup C_0\subseteq D$, the following holds. There is a set of vertex-disjoint $x_i,y_i$-paths with length $k$, $i\in [m]$, which have interior vertices in $V\cup V_0$ and which are collectively $D$-rainbow.
\epsilonilonnd{lemma}
Finally, the following theorem, proved in Section~\ref{sec:caseC}, will allow us to embed trees in Case C.
\begin{theorem}[Embedding trees in Case C]\label{Theorem_case_C}
Let $n\geq 10^6$. Let $K_{2n+1}$ be $ND$-coloured, and let $T$ be a tree on $n+1$ vertices with a subtree $T'$ with $\epsilonilonll:=|T'|\leq n/100$ such that $T'$ has vertices $v_1, \dots, v_{\epsilonilonll}$ so that adding $d_i\geq \log^4n$ leaves to each $v_i$ produces $T$.
Then, $K_{2n+1}$ contains a rainbow copy of $T$.
\epsilonilonnd{theorem}
We can now combine these results to prove Theorem~\ref{Theorem_Ringel_proof}. We also use a simple lemma concerning replete random sets, Lemma~\ref{Lemma_inheritence_of_lower_boundedness_random}, which is proved in Section~\ref{sec:prelim}. As used below, it implies that if $V_0$ and $V_1$, with $V_1\subseteq V_0\subseteq V(K_{2n+1})$, are $\mu$- and $\mu/2$-random respectively, then, given any randomised set $X$ such that $(X,V_0)$ is with high probability replete (for some parameter), then $(X,V_1)$ is also with high probability replete (for some suitably reduced parameter).\begin{proof}[Proof of Theorem~\ref{Theorem_Ringel_proof}]
Choose $\xi, \mu,\epsilonilonta,\delta,\bar{\mu},\bar{\epsilonilonta}$ and $\bar{\epsilonilonps}$ such that $1/n \ll\xi \ll\mu\ll \epsilonilonta\ll \delta\ll\bar{\mu}\ll\bar{\epsilonilonta}\ll\bar{\epsilonilonps} \ll\log^{-1} n$ and $k=\delta^{-1}$ is an integer such that $k = 7\mod 12$ and $695|k$. Let $T$ be an $(n+1)$-vertex tree and let $K_{2n+1}$ be ND-coloured.
By Lemma~\ref{Lemma_case_division}, $T$ is in Case A, B or C for this $\delta$. If $T$ is in Case C, then Theorem~\ref{Theorem_case_C} implies that $K_{2n+1}$ has a rainbow copy of $T$. Let us assume then that $T$ is not in Case C. Let $k=\delta^{-4}$, and note that, as $T$ is not in Case $C$, removing from $T$ leaves around any vertex adjacent to at least $k$ leaves gives a tree with at least $n/100$ vertices.
If $T$ is in Case A, then let $\epsilonilonps=\delta^6$, let $L$ be a set of $\epsilonilonps n$ non-neighbouring leaves in $T$, let $U=N_T(L)$ and let $T'=T-L$. Let $p$ be such that removing from $T'$ leaves around any vertex adjacent to at least $k$ leaves gives a tree with $pn$ vertices. Note that each leaf of $T'$ which is not a leaf of $T$ must be adjacent to a vertex in $L$ in $T$. Note further that, if a vertex in $T$ is next to fewer than $k$ leaves in $T$, but at least $k$ leaves in $T'$, then all but at most $k-1$ of those leaves in $T'$ must have a neighbour in $L$ in $T$. Therefore, $p\geq 1/100-(k+1)\epsilonilonps\geq 1/200$.
By Theorem~\ref{nearembedagain}, there is a random subgraph $\hat{T}'\subseteq K_{2n+1}$ and random subsets $V,V_0\subseteq V(K_{2n+1})\setminus V(\hat{T}')$ and $C,C_0\subseteq C(K_{2n+1})\setminus C(\hat{T}')$ such that \ref{propa1}--\ref{propa3} hold.
Using \ref{propa2}, let $V_1,V_2\subseteq V_0$ be disjoint $(\mu/2)$-random subsets of $V(K_{2n+1})$.
By Lemma~\ref{lem:finishA} (applied with $\xi'=\xi/4, \mu'=\mu/2, \epsilonilonta=\epsilonilonta, \epsilonilonpsilon=\epsilonilonpsilon, p'=(p+\epsilonilonpsilon)/6$, $V=V, C=C, V_0'=V_1,$ and $C_0'$ a $(\mu/2)$-random subset of $C_0$)
and \ref{propa2}--\ref{propa3}, and by \ref{propa1} and Lemma~\ref{Lemma_inheritence_of_lower_boundedness_random}, with high probability we have the following properties.
\stepcounter{propcounter}
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]
\item Given any disjoint sets $X,Z\subseteq V(K_{2n+1})\setminus (V\cup V_1)$ so that $|X|=\epsilonilonps n$ and $(X,Z)$ is $(\xi n/4)$-replete, and any set $D\subseteq C(K_{2n+1})$ with $|D|=\epsilonilonps n$ and $C_0\cup C\cup D$, there is a perfect $D$-rainbow matching from $X$ into $V\cup V_1\cup Z$.\label{fine1}
\item $\hat{T}'$ is a rainbow copy of $T'$ in which, letting $W$ be the copy of $U$, $(W,V_2)$ is $(\xi n/4)$-replete.\label{fine2}
\epsilonilonnd{enumerate}
Let $D=C(K_{2n+1})\setminus C(\hat{T}')$, so that $C_0\cup C\subseteq D$, and, as $\hat{T}'$ is rainbow by \ref{fine2}, $|D|=\epsilonilonps n$. Let $W$ be the copy of $U$ in $\hat{T}'$. Then, using \ref{fine1} with $Z=V_2$ and \ref{fine2}, let $M$ be a perfect $D$-rainbow matching from $W$ into $V\cup V_1\cup V_2\subseteq V\cup V_0$. As $V\cup V_0$ is disjoint from $V(\hat{T}')$, $\hat{T}'\cup M$ is a rainbow copy of $T$. Thus, a rainbow copy of $T$ exists with high probability in the ND-colouring of $K_{2n+1}$, and hence certainly some such rainbow copy of $T$ must exist.
If $T$ is in Case B, then recall that $k=\delta^{-1}$ and let $m=\bar{\epsilonilonps} n/k$. Let $P_1,\ldots,P_m$ be vertex-disjoint bare paths with length $k$ in $T$. Let $T'$ be $T$ with the interior vertices of $P_i$, $i\in [m]$, removed.
Let $p$ be such that removing from $T'$ leaves around any vertex adjacent to at least $k$ leaves gives a forest with $pn$ vertices. Note that (reasoning similarly to as in Case A) $p\geq 1/100-2(k+1)m\geq 1/200$.
By Theorem~\ref{nearembedagain}, there is a random subgraph $\hat{T}'\subseteq K_{2n+1}$ and disjoint random subsets $V,V_0\subseteq V(K_{2n+1})\setminus V(\hat{T}')$ and $C,C_0\subseteq C(K_{2n+1})\setminus C(\hat{T}')$ such that \ref{propa1}--\ref{propa3} hold with $\xi=\xi$, $\mu=\bar{\mu}$, $\epsilonilonta=\bar{\epsilonilonta}$ and $\epsilonilonps=\bar{\epsilonilonps}$. By Lemma~\ref{lem:finishB} and \ref{propa1}--\ref{propa3}, and by \ref{propa1}, with high probability we have the following properties.
\stepcounter{propcounter}
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]
\item For any set $\{x_1,\ldots,x_m,y_1,\ldots,y_m\}\subseteq V(K_{2n+1})\setminus (V\cup V_0)$ and any set $D\subseteq C(K_{2n+1})$ with $|D|=mk$ and $C_0\cup C\subseteq D$, there is a set of vertex-disjoint paths $x_i,y_i$-paths with length $k$, $i\in [m]$, which have interior vertices in $V\cup V_0$ and which are collectively $D$-rainbow.\label{fine11}
\item $\hat{T}'$ is a rainbow copy of $T'$.\label{fine12}
\epsilonilonnd{enumerate}
Let $D=C(K_{2n+1})\setminus C(\hat{T}')$, so that $C_0\cup C\subseteq D$, and, as $\hat{T}'$ is rainbow by \ref{fine12}, $|D|=\epsilonilonps n$.
For each path $P_i$, $i\in [m]$, let $x_i$ and $y_i$ be the copy of the endvertices of $P_i$ in $\hat{T}'$. Using \ref{fine11}, let $Q_i$, $i\in [m]$, be a set of vertex-disjoint $x_i,y_i$-paths with length $k$, $i\in [m]$, which have interior vertices in $V\cup V_0$ and which are collectively $D$-rainbow. As $V\cup V_0$ is disjoint from $V(\hat{T}')$,
$\hat{T}'\cup (\cup_{i\in [m]}Q_i)$ is a rainbow copy of $T$. Thus, a rainbow copy of $T$ exists with high probability in the ND-colouring of $K_{2n+1}$, and hence certainly some such rainbow copy of $T$ must exist.
\epsilonilonnd{proof}
| 4,064 | 88,680 |
en
|
train
|
0.54.8
|
\section{Preliminary results and observations}\label{sec:prelim}
\subsection{Notation}\label{sec:not}
For a coloured graph $G$, we denote the set of vertices of $G$ by $V(G)$, the set of edges of $G$ by $E(G)$, and the set of colours of $G$ by $C(G)$.
For a coloured graph $G$, disjoint sets of vertices $A,B\subseteqeq V(G)$ and a set of colours $C\subseteqeq C(G)$ we use $G[A,B,C]$ to denote the subgraph of $G$ consisting of colour $C$ edges from $A$ to $B$, and $G[A,C]$ to be the graph of the colour $C$ edges within $A$.
For a single colour $c$, we denote the set of colour $c$ edges from $A$ to $B$ by $E_c(A,B)$.
A coloured graph is \epsilonilonmph{globally $k$-bounded} if every colour is on at most $k$ edges.
For a set of colours $C$, we say that a graph $H$ is ``$C$-rainbow'' if $H$ is rainbow and $C(H)\subseteqeq C$. We say that a collection of graphs $H_1, \dots, H_k$ is \epsilonilonmph{collectively} rainbow if their union is rainbow.
A star is a tree consisting of a collection of leaves joined to a single vertex (which we call the \epsilonilonmph{centre}).
A \epsilonilonmph{star forest} is a graph consisting of vertex disjoint stars.
For any reals $a,b\in \mathbb{R}$, we say $x=a\pm b$ if $x\in [a-b,a+b]$.
\subsubsectionme{Asymptotic notation}
For any $C\geq 1$ and $x,y\in (0,1]$, we use ``$x\oldll_C y$'' to mean ``$x\leq \frac{y^C}{C}$''. We will write ``$x{\ll} y$'' to mean that there is some absolute constant $C$ for which the proof works with ``$x{\ll}{} y$'' replaced by ``$x\oldll_{C} y$''.
In other words the proof works if $y$ is a small but fixed power of $x$.
This notation compares to the more common notation $x\oldll y$ which means ``there is a fixed positive continuous function $f$ on $(0,1]$ for which the remainder of the proof works with ``$x\oldll y$'' replaced by ``$x\leq f(y)$''. (Equivalently, ``$x\oldll y$'' can be interpreted as ``for all $x\in (0,1]$, there is some $y\in (0,1]$ such that the remainder of the proof works with $x$ and $y$''.) The two notations ``$x{\ll}{} y$'' and ``$x\oldll y$'' are largely interchangeable --- most of our proofs remain correct with all instances of ``${\ll}$'' replaced by ``$\oldll$''. The advantage of using ``${\ll}$'' is that it proves polynomial bounds on the parameters (rather than bounds of the form ``for all $\epsilonilonpsilon>0$ and sufficiently large $n$''). This is important towards the end of this paper, where the proofs need polynomial parameter bounds.
While the constants $C$ will always be implicit in each instance of ``$x{\ll}{} y$'', it is possible to work them out explicitly. To do this one should go through the lemmas in the paper and choose the constants $C$ for a lemma after the constants have been chosen for the lemmas on which it depends. This is because an inequality $x\oldll_C y$ in a lemma may be needed to imply an inequality $x\oldll_{C'} y$ for a lemma it depends on. Within an individual lemma we will often have several inequalities of the form $x{\ll} y$. There the constants $C$ need to be chosen in the reverse order of their occurrence in the text. The reason for this is the same --- as we prove a lemma we may use an inequality $x\oldll_C y$ to imply another inequality $x\oldll_{C'} y$ (and so we should choose $C'$ before choosing $C$).
Throughout the paper, there are four operations we perform with the ``$x{\ll}{} y$'' notation:
\begin{enumerate}[label = (\alph{enumi})]
\item We will use $x_1{\ll} x_2{\ll}\dots{\ll} x_k$ to deduce finitely many inequalities of the form ``$p(x_1, \dots, x_k)\leq q(x_1, \dots, x_k)$'' where $p$ and $q$ are {monomials} with non-negative coefficients and $\min\{i: p(0, \dots, 0, x_{i+1}, \dots, x_k)=0\}< \min\{j: q(0, \dots, 0, x_{j+1}, \dots, x_k)=0\}$ e.g.\ $1000x_1\leq x_2^5x_4^2x_5^3$ is of this form.
\item We will use $x{\ll} y$ to deduce finitely many inequalities of the form ``$x\oldll_C y$'' for a fixed constant $C$.
\item For $x{\ll} y$ and fixed constants $C_1, C_2$, we can choose a variable $z$ with $x\oldll_{C_1} z\oldll_{C_2}y$.
\item For $n^{-1}{\ll} 1$ and any fixed constant $C$, we can deduce $n^{-1}\oldll_C \log^{-1} n\oldll_C 1$.
\epsilonilonnd{enumerate}
See \cite{montgomery2018decompositions} for a detailed explanation of why the above operations are valid.
\subsubsectionme{Rounding}
In several places, we will have, for example, constants $\epsilonilonps$ and integers $n,k$ such that $1/n\ll \epsilonilonps,1/k$ and require that $m=\epsilonilonps n/k$ is an integer, or even divisible by some other small integer. Note that we can arrange this easily with a very small alteration in the value of $\epsilonilonps$. For example, to apply Lemma~\ref{lem:finishB} we assume that $m$ is an integer, and therefore in the proof of Theorem~\ref{Theorem_Ringel_proof}, when we choose $\bar{\epsilonilonps}$ we make sure that when this lemma is applied with $\epsilonilonps=\bar{\epsilonilonps}$ the corresponding value for $m$ is an integer.
\subsection{Probabilistic tools}\label{sec:prob}
For a finite set $V$, a $p$-random subset of $V$ is a set formed by choosing every element of $V$ independently at random with probability $p$. If $V$ is not specified, then we will implicitly assume that $V$ is $V(K_{2n+1})$ or $C(K_{2n+1})$, where this will be clear from context.
If $A,B\subseteqeq V$ with $A$ $p$-random and $B$ $q$-random, we say that $A$ and $B$ are \epsilonilonmph{disjoint} if every $v\in V$ is in $A$ with probability $p$, in $B$ with probability $q$, and outside of $A\cup B$ with probability $1-p-q$ (and this happens independently for each $v\in V$). We say that a $p$-random set $A$ is independent from a $q$-random set $B$ if the choices for $A$ and $B$ are made independently, that is, if $\mathbb{P}(A=A'\land B=B')=\mathbb{P}(A=A')\mathbb{P}(B=B')$ for any outcomes $A'$ and $B'$ of $A$ and $B$.
Often, we will have a $p$-random subset $X$ of $V$ and divide it into two disjoint $(p/2)$-random subsets of $V$. This is possible by choosing which subset each element of $X$ is in independently at random with probability $1/2$ using the following simple lemma.
\begin{lemma}[Random subsets of random sets]\label{Lemma_mixture_of_p_random_sets}
Suppose that $X,Y:\Omega\to 2^V$ where $X$ is a $p$-random subset of $V$ and $Y|X$ is a $q$-random subset of $X$ (i.e. the distribution of $Y$ conditional on the event ``$X=X'$'' is that of a $q$-random subset of $X'$). Then $Y$ is a $pq$-random subset of $V$.
\epsilonilonnd{lemma}
\begin{proof}
First notice that, to show a set $X\subseteqeq V$ is $(pq)$-random, it is sufficient to show that $\mathbb{P}(S\subseteqeq X)=(pq)^{|S|}$ for all $S\subseteqeq V$ (for example, by using inclusion-exclusion).
Now, we prove the lemma.
Since $X$ is $p$-random we have $\mathbb{P}(S\subseteqeq X)=p^{|S|}$. Since $Y|X$ is $q$-random, we have $\mathbb{P}(S\subseteqeq Y|S\subseteqeq X)=q^{|S|}$. This gives $\mathbb{P}(S\subseteqeq Y)= \mathbb{P}(S\subseteqeq Y|S\subseteqeq X)\mathbb{P}(S\subseteqeq X)=(pq)^{|S|}$ for every set $S\subseteqeq V$.
\epsilonilonnd{proof}
We will use the following standard form of Azuma's inequality and a Chernoff Bound. For a probability space $\Omega=\prod_{i=1}^n \Omega_i$, a random variable $X:\Omega\to \mathbb{R}$ is \epsilonilonmph{$k$-Lipschitz} if changing $\omega\in \Omega$ in any one coordinate changes $X(\omega)$ by at most $k$.
\begin{lemma}[Azuma's Inequality]\label{Lemma_Azuma}
Suppose that $X$ is $k$-Lipschitz and influenced by $\leq m$ coordinates in $\{1, \dots, n\}$. Then{, for any $t>0$,}
$$\mathbb{P}\left(|X-\mathbb{E}(X)|>t \right)\leq 2e^{\frac{-t^2}{mk^2}}$$.
\epsilonilonnd{lemma}
Notice that the bound in the above inequality can be rewritten as $\mathbb{P}\left(X\neq \mathbb{E}(X)\pm t \right)\leq 2e^{\frac{-t^2}{mk^2}}$.
\begin{lemma}[Chernoff Bound] \label{Lemma_Chernoff}
Let $X$ be a binomial random variable with parameters $(n,p)$.
Then, for each $\epsilonilonpsilon\in (0,1)$, we have
$$\mathbb P\big(|X-pn|> \epsilonilonpsilon pn\big)\leq
2e^{-\frac{pn\epsilonilonpsilon^2}{3}}.$$
\epsilonilonnd{lemma}
For an event $X$ in a probability space depending on a parameter $n$, we say ``$X$ holds with high probability'' to mean ``$X$ holds with probability $1-o(1)$'' where $o(1)$ is some function $f(n)$ with $f(n)\to 0$ as $n\to \infty$.
We will use this definition for the following operations.
\begin{itemize}
\item \textbf{Chernoff variant:} For $\epsilonilonpsilon \gg n^{-1}$, if $X$ is a $p$-random subset of $[n]$, then, with high probability, $|X|=(1\pm \epsilonilonpsilon)pn$.
\item \textbf{Azuma variant:} For $\epsilonilonpsilon\gg n^{-1}$ and fixed $k$, if $Y$ is a $k$-Lipschitz random variable influenced by at most $n$ coordinates, then, with high probability, $Y=\mathbb{E}(Y)\pm \epsilonilonpsilon n$.
\item \textbf{Union bound variant:} For fixed $k$, if $X_1, \dots, X_{k}$ are events which hold with high probability then they simultaneously occur with high probability.
\epsilonilonnd{itemize}
The first two of these follow directly from Lemmas~\ref{Lemma_Azuma} and~\ref{Lemma_Chernoff}, the latter from the union bound.
| 3,071 | 88,680 |
en
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.